diff --git a/2015/eulers-characteristic-formula/chinese/sentence_translations.json b/2015/eulers-characteristic-formula/chinese/sentence_translations.json index 4c36d5c66..70214858d 100644 --- a/2015/eulers-characteristic-formula/chinese/sentence_translations.json +++ b/2015/eulers-characteristic-formula/chinese/sentence_translations.json @@ -384,7 +384,7 @@ "end": 401.92 }, { - "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge in what will become a spanning tree. ", + "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge that he buys in what will become a spanning tree. ", "translatedText": "或者，在我们的叙述中， 您可以将伦道夫视为从一个顶点开始，并为每个边再获得一个顶点 ，从而形成一棵生成树。", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/eulers-characteristic-formula/japanese/sentence_translations.json b/2015/eulers-characteristic-formula/japanese/sentence_translations.json index 06c8a4faa..3d85a882f 100644 --- a/2015/eulers-characteristic-formula/japanese/sentence_translations.json +++ b/2015/eulers-characteristic-formula/japanese/sentence_translations.json @@ -384,7 +384,7 @@ "end": 401.92 }, { - "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge in what will become a spanning tree.", + "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge that he buys in what will become a spanning tree.", "translatedText": "あるいは、私たちの物語の中で、ランドル フは 1 つの頂点から始めて、スパニング ツリーとなるもののエッジごとにちょうど 1 つずつ 頂点を獲得すると考えることもできます。", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/eulers-characteristic-formula/korean/sentence_translations.json b/2015/eulers-characteristic-formula/korean/sentence_translations.json index d9833ff6d..df5b68499 100644 --- a/2015/eulers-characteristic-formula/korean/sentence_translations.json +++ b/2015/eulers-characteristic-formula/korean/sentence_translations.json @@ -384,7 +384,7 @@ "end": 401.92 }, { - "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge in what will become a spanning tree. ", + "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge that he buys in what will become a spanning tree. ", "translatedText": "또는 우리 이야기 내에서 랜돌프는 하나의 꼭지점으로 시작하여 스패닝 트리가 될 각 모서리에 대해 정확히 하나를 더 얻는 것으로 생각할 수 있습니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/eulers-characteristic-formula/marathi/sentence_translations.json b/2015/eulers-characteristic-formula/marathi/sentence_translations.json index 504eb33a0..1e0113bba 100644 --- a/2015/eulers-characteristic-formula/marathi/sentence_translations.json +++ b/2015/eulers-characteristic-formula/marathi/sentence_translations.json @@ -384,7 +384,7 @@ "end": 401.92 }, { - "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge in what will become a spanning tree.", + "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge that he buys in what will become a spanning tree.", "translatedText": "वैकल्पिकरित्या, आमच्या कथनात, तुम्ही रँडॉल्फचा विचार करू शकता की एका शिरोबिंदूपासून सुरुवात करून आणि प्रत्येक काठासाठी आणखी एक मिळवणे, जे एक पसरलेले झाड होईल.", "model": "google_nmt", "n_reviews": 1, diff --git a/2015/eulers-characteristic-formula/persian/sentence_translations.json b/2015/eulers-characteristic-formula/persian/sentence_translations.json index 9815ece4f..47df4b29d 100644 --- a/2015/eulers-characteristic-formula/persian/sentence_translations.json +++ b/2015/eulers-characteristic-formula/persian/sentence_translations.json @@ -384,7 +384,7 @@ "end": 401.92 }, { - "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge in what will become a spanning tree.", + "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge that he buys in what will become a spanning tree.", "translatedText": "از طرف دیگر، در روایت ما، می‌توانید به راندولف فکر کنید که با یک رأس شروع می‌کند و دقیقاً یک راس دیگر برای هر یال به دست می‌آورد که به یک درخت پوشا تبدیل می‌شود.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/eulers-characteristic-formula/tamil/sentence_translations.json b/2015/eulers-characteristic-formula/tamil/sentence_translations.json index e5d410045..c04cf4d0b 100644 --- a/2015/eulers-characteristic-formula/tamil/sentence_translations.json +++ b/2015/eulers-characteristic-formula/tamil/sentence_translations.json @@ -384,7 +384,7 @@ "end": 401.92 }, { - "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge in what will become a spanning tree.", + "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge that he buys in what will become a spanning tree.", "translatedText": "மாற்றாக, எங்கள் கதையில், ராண்டால்ஃப் ஒரு உச்சியில் தொடங்கி, ஒவ்வொரு விளிம்பிலும் சரியாக ஒன்றைப் பெறுவது, அது பரந்த மரமாக மாறும் என்று நீங்கள் நினைக்கலாம்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/eulers-characteristic-formula/telugu/sentence_translations.json b/2015/eulers-characteristic-formula/telugu/sentence_translations.json index 7f2880f24..92ad40a4f 100644 --- a/2015/eulers-characteristic-formula/telugu/sentence_translations.json +++ b/2015/eulers-characteristic-formula/telugu/sentence_translations.json @@ -384,7 +384,7 @@ "end": 401.92 }, { - "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge in what will become a spanning tree.", + "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge that he buys in what will become a spanning tree.", "translatedText": "ప్రత్యామ్నాయంగా, మా కథనంలో, మీరు రాండోల్ఫ్‌ను ఒక శీర్షంతో ప్రారంభించి, ప్రతి అంచుకు సరిగ్గా మరొకటి పొందడం ద్వారా విస్తరించి ఉన్న చెట్టుగా మారవచ్చు.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/eulers-characteristic-formula/thai/sentence_translations.json b/2015/eulers-characteristic-formula/thai/sentence_translations.json index bf30e4894..7849d7548 100644 --- a/2015/eulers-characteristic-formula/thai/sentence_translations.json +++ b/2015/eulers-characteristic-formula/thai/sentence_translations.json @@ -384,7 +384,7 @@ "end": 401.92 }, { - "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge in what will become a spanning tree. ", + "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge that he buys in what will become a spanning tree. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/eulers-characteristic-formula/turkish/sentence_translations.json b/2015/eulers-characteristic-formula/turkish/sentence_translations.json index 1638d262c..0ff1ffed4 100644 --- a/2015/eulers-characteristic-formula/turkish/sentence_translations.json +++ b/2015/eulers-characteristic-formula/turkish/sentence_translations.json @@ -384,7 +384,7 @@ "end": 401.92 }, { - "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge in what will become a spanning tree.", + "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge that he buys in what will become a spanning tree.", "translatedText": "Alternatif olarak, anlatımızda Randolph'un bir tepe noktasıyla başladığını ve yayılan bir ağaca dönüşecek şekilde her kenar için tam olarak bir tane daha kazandığını düşünebilirsiniz.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/eulers-characteristic-formula/vietnamese/sentence_translations.json b/2015/eulers-characteristic-formula/vietnamese/sentence_translations.json index b37ae71d6..14a27e0ab 100644 --- a/2015/eulers-characteristic-formula/vietnamese/sentence_translations.json +++ b/2015/eulers-characteristic-formula/vietnamese/sentence_translations.json @@ -384,7 +384,7 @@ "end": 401.92 }, { - "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge in what will become a spanning tree.", + "input": "Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge that he buys in what will become a spanning tree.", "translatedText": "Ngoài ra, trong câu chuyện của chúng tôi, bạn có thể nghĩ Randolph bắt đầu với một đỉnh và lấy thêm chính xác một đỉnh nữa cho mỗi cạnh trong cái sẽ trở thành cây bao trùm.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/eulers-formula-old/english/captions.srt b/2015/eulers-formula-old/english/captions.srt index edbe4cc9a..d729d4ad0 100644 --- a/2015/eulers-formula-old/english/captions.srt +++ b/2015/eulers-formula-old/english/captions.srt @@ -1,5 +1,5 @@ 1 -00:00:08,139 --> 00:00:11,963 +00:00:08,140 --> 00:00:11,963 E to the pi i equals negative one is one of the most famous equations in math, 2 diff --git a/2015/inventing-math/arabic/sentence_translations.json b/2015/inventing-math/arabic/sentence_translations.json index 5ee57f9f0..fa515f829 100644 --- a/2015/inventing-math/arabic/sentence_translations.json +++ b/2015/inventing-math/arabic/sentence_translations.json @@ -213,7 +213,7 @@ "end": 209.04 }, { - "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, 1 one-hundredth, 1 one-millionth, or 1 over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny distance of 1.", + "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, one one hundredth, one one millionth, or one over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny tiny distance of 1.", "translatedText": "بعد التفكير في الأمر، تدرك أن ما يجعل الرقم 1 مميزًا هو أن أرقامك يمكن أن تقترب اعتباطيًا من 1، أي بغض النظر عن مدى صغر المسافة التي تريدها، 1 على مائة، أو 1 على مليون، أو 1 على أكبر مسافة. الرقم الذي يمكنك تدوينه، إذا تابعت قائمتك لفترة كافية، فسوف تقع الأرقام في النهاية ضمن تلك المسافة الصغيرة وهي 1.", "model": "google_nmt", "from_community_srt": "بعد التفكير في الأمر ، تدرك أن ما يجعل 1 خاص هو أنه يمكن الحصول على أرقامك بشكل تعسفي بالقرب من 1. وهو ما سيقول، مهما كان صغيرك المسافة المطلوبة ، 1/100 ، 1 / ​​1،000،000 ، أو واحد على أكبر رقم يمكنك كتابته أسفل ، إذا ذهبت إلى أسفل القائمة لفترة كافية ، سوف تقع الأرقام في النهاية ضمن تلك المسافة الصغيرة جدًا من 1.", @@ -276,7 +276,7 @@ "end": 288.76 }, { - "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size ½, ¼, etc., you could have chosen a proportion other than ½.", + "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size one half, one fourth, etc., you could have chosen a proportion other than one half.", "translatedText": "على سبيل المثال، عندما كنت تقوم بتقليص المسافة بين الأشياء، وتقطيع الفاصل الزمني إلى أجزاء بحجم ½، ¼، وما إلى ذلك، كان بإمكانك اختيار نسبة أخرى غير ½.", "model": "google_nmt", "from_community_srt": "على سبيل المثال ، عندما كنت تقلص المسافة بين الأشياء الخاصة بك ، وقطع الفاصل إلى قطع بحجم ½ ، ¼ ، إلخ ، كان بإمكانك اختيار نسبة غير ½.", @@ -294,7 +294,7 @@ "end": 315.82 }, { - "input": "Continuing on and on, you'd see that 9 tenths plus 9 one hundredths plus 9 one thousandths on and on up to infinity equals 1, a fact more popularly written as 0.9 repeating equals 1.", + "input": "Continuing on and on, you'd see that nine tenths plus nine one hundredths plus nine one thousandths on and on up to infinity equals one, a fact more popularly written as point nine repeating equals one.", "translatedText": "وبالاستمرار، سترى أن 9 أعشار زائد 9 أجزاء من مائة زائد 9 أجزاء من الألف وهكذا حتى ما لا نهاية يساوي 1، وهي حقيقة أكثر شيوعًا مكتوبة على أنها 0.9 تكرار يساوي 1.", "model": "google_nmt", "from_community_srt": "9/10 + 9/100 + 9/1000 وفي ما يصل إلى ما لا نهاية يساوي 1 ، حقيقة أكثر مكتوبة بشعبية .9 تكرار = 1.", @@ -312,7 +312,7 @@ "end": 338.58 }, { - "input": "To be general about it, let's say that you cut your interval into pieces of size p and 1-p, where p represents any number between 0 and 1.", + "input": "To be general about it, let's say that you cut your interval into pieces of size p and one minus p, where p represents any number between zero and one.", "translatedText": "لنكون عامًا حول هذا الموضوع، لنفترض أنك قمت بتقطيع الفاصل الزمني إلى أجزاء بالحجم p و1-p، حيث يمثل p أي رقم بين 0 و1.", "model": "google_nmt", "from_community_srt": "يعني نفس الشيء. لكي نكون عامين حول هذا الموضوع ، دعنا نقول أنك أنت قطع الفاصل إلى قطع من حجم ع و (1-p) ، حيث تمثل p أي رقم بين 0 و 1.", @@ -330,7 +330,7 @@ "end": 356.78 }, { - "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that 1-p plus p times 1-p plus p squared times 1-p on and on always adding p to the next power times 1-p equals 1.", + "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that one minus p plus p times one minus p plus p squared times one minus p, on and on always adding p to the next power times one minus p, equals one.", "translatedText": "بالاستمرار بهذه الطريقة، قم دائمًا بتقطيع القطعة الموجودة في أقصى اليمين إلى نفس النسب، ستجد أن 1-p زائد p ضرب 1-p زائد p تربيع مرات 1-p on and on دائمًا إضافة p إلى الأس التالي مرات 1- ع يساوي 1.", "model": "google_nmt", "from_community_srt": "باستمرار في هذا الشكل ، دائماً قطع الجزء أقصى اليمين إلى نفس هذه النسب ، سوف تجد أن (1-p) + p (1-p) + p ^ 2 (1-p) ، على إضافة دائمًا p إلى الطاقة التالية مرات (1-ع) ، يساوي 1.", @@ -365,7 +365,7 @@ "end": 398.4 }, { - "input": "For instance, plugging in negative 1, the equation reads 1 minus 1 plus 1 minus 1 on and on forever alternating between the two, equals one half, which feels both silly and kind of like the only thing it could be.", + "input": "For instance, plugging in negative one, the equation reads one minus one plus one minus one, on and on forever alternating between the two, equals one half, which feels pretty silly and kind of like the only thing it could be.", "translatedText": "على سبيل المثال، عند التعويض بسالب 1، تصبح المعادلة 1 ناقص 1 زائد 1 ناقص 1 مرارًا وتكرارًا بالتناوب بين الاثنين، يساوي نصفًا، وهو ما يبدو سخيفًا ويشبه الشيء الوحيد الذي يمكن أن يكون.", "model": "google_nmt", "from_community_srt": "على سبيل المثال ، يسد في -1 ، المعادلة يقرأ 1-1 + 1-1 على وإلى الأبد بالتناوب بين الاثنين ، يساوي 1/2 التي تبدو سخيفة نوعًا ما مثل الشيء الوحيد الذي يمكن أن يكون.", @@ -374,7 +374,7 @@ "end": 417.86 }, { - "input": "Plugging in 2, the equation reads 1 plus 2 plus 4 plus 8 on and on to infinity equals negative 1, something which doesn't even seem reasonable.", + "input": "Plugging in two, the equation reads one plus two plus four plus eight, on and on to infinity, equals negative one, something which doesn't even seem reasonable.", "translatedText": "بالتعويض بـ 2، تصبح المعادلة 1 زائد 2 زائد 4 زائد 8 وهكذا إلى ما لا نهاية يساوي سالب 1، وهو أمر لا يبدو معقولًا.", "model": "google_nmt", "from_community_srt": "توصيل 2 ، المعادلة يقرأ 1 + 2 + 4 + 8 + ...", @@ -427,7 +427,7 @@ "end": 467.62 }, { - "input": "1, 3, 7, 15, 31, they're all 1 less than a power of 2.", + "input": "One, three, seven, fifteen, thirty-one, they're all one less than a power of two.", "translatedText": "1، 3، 7، 15، 31، كلها أقل بـ 1 من قوة 2.", "model": "google_nmt", "from_community_srt": "هم كل واحد أقل من قوة من 2.", @@ -525,7 +525,7 @@ "end": 562.68 }, { - "input": "You could come up with a completely random notion of distance, where 2 is 7 away from 3, and ½ is 4 fifths away from 100, and all sorts of things, but if you want to actually use a new distance function the way you use the familiar distance function, it should share some of the same properties.", + "input": "You could come up with a completely random notion of distance, where two is seven away from three, and one half is four fifths away from a hundred, and all sorts of things. But if you want to actually use a new distance function the way that you use the familiar distance function, it should share some of the same properties.", "translatedText": "يمكنك التوصل إلى فكرة عشوائية تمامًا للمسافة، حيث 2 هو 7 بعيدًا عن 3، و ½ هو 4 أخماس من 100، وكل أنواع الأشياء، ولكن إذا كنت تريد بالفعل استخدام دالة مسافة جديدة بالطريقة التي تستخدمها دالة المسافة المألوفة، يجب أن تشترك في بعض الخصائص نفسها.", "model": "google_nmt", "from_community_srt": "هل يمكن أن تأتي مع عشوائي تماما مفهوم المسافة ، حيث 2 هي 7 بعيدا عن 3 ، و ½ هو 4 / 5ths بعيدا عن 100 ، وجميع أنواع من الأشياء ، ولكن إذا كنت تريد في الواقع استخدم وظيفة المسافة الجديدة بالطريقة التي تستخدمها استخدام وظيفة المسافة المألوفة ، يجب عليه", @@ -543,7 +543,7 @@ "end": 587.48 }, { - "input": "So 0 and 4 should be the same distance away as 1 and 5, or 2 and 6, even if that same distance is something other than 4 as we're used to.", + "input": "So zero and four should be the same distance away as one and five, or two and six, even if that same distance is something other than four as we're used to.", "translatedText": "لذلك يجب أن تكون 0 و4 على نفس المسافة بين 1 و5، أو 2 و6، حتى لو كانت نفس المسافة غير 4 كما اعتدنا.", "model": "google_nmt", "from_community_srt": "من 0 و 4 يجب أن يكون على مسافة واحدة من 1 و 5 أو 2 و 6 ، حتى لو كانت تلك المسافة نفسها شيء بخلاف أربعة كما اعتدنا.", @@ -569,7 +569,7 @@ "end": 607.24 }, { - "input": "There are other properties that you want your notion of distance to have as well, like the notion of distance could possibly make powers of 2 approach 0, and shift invariant.", + "input": "There are other properties that you want your notion of distance to have as well, like the triangle inequality, but before we start worrying about those, let's start imagining what notion of distance could possibly make powers of two approach zero, and which is shift inva", "translatedText": "هناك خصائص أخرى تريد أن يمتلكها مفهوم المسافة أيضًا، مثل مفهوم المسافة الذي من الممكن أن يجعل قوى 2 تقترب من 0، والتحول ثابتًا.", "model": "google_nmt", "from_community_srt": "دعونا نطلق على هذه الخاصية “shift invariance”. هناك خصائص أخرى تريدها فكرة المسافة لديها كذلك ، مثل مثل عدم المساواة مثلث ، لكن قبل أن نبدأ بالقلق حول هؤلاء ، دعونا نبدأ في تخيل الفكرة من مسافة يمكن أن تجعل من سلطات 2 النهج 0 ، والتي هي ثابتة ثابتة.", @@ -587,7 +587,7 @@ "end": 639.4 }, { - "input": "You think of 0 as being in the same room as all of the powers of 2 greater than 1, as being in the same sub-room as all powers of 2 greater than 2, as being in the same sub-sub-room as powers of 2 greater than 4, and so on, with infinitely many smaller and smaller rooms.", + "input": "You think of zero as being in the same room as all of the powers of two greater than one. As being in the same sub-room as all powers of two greater than two. As being in the same sub-sub-room as powers of two greater than four, and so on, with infinitely many smaller and smaller rooms.", "translatedText": "تعتقد أن 0 موجود في نفس الغرفة حيث توجد جميع قوى 2 أكبر من 1، كما أنه موجود في نفس الغرفة الفرعية مثل جميع قوى 2 الأكبر من 2، كما أنه موجود في نفس الغرفة الفرعية مثل القوى 2 أكبر من 4، وهكذا، مع عدد لا نهائي من الغرف الأصغر فأصغر.", "model": "google_nmt", "from_community_srt": "أنت تفكر في 0 كما لو كانت في نفس الغرفة جميع القوى من 2 أكبر من 1 ، كما يجري في نفس الغرفة الفرعية مثل جميع القوى من 2 أكبر من 2 ، كما يجري في نفس الغرفة الفرعية الفرعية كقوى 2 أكبر من 4 ، وهلم جرا ، مع عدد لا نهائي من الغرف الأصغر والأصغر.", @@ -614,7 +614,7 @@ "end": 677.46 }, { - "input": "For instance, 1 should be as far away from 3 as 2 is from 0.", + "input": "For instance, one should be as far away from three as two is from zero.", "translatedText": "على سبيل المثال، ينبغي أن يكون 1 بعيدًا عن 3 بقدر ما يكون 2 بعيدًا عن 0.", "model": "google_nmt", "from_community_srt": "على سبيل المثال ، يجب أن يكون 1 بعيدًا عن 3 كـ 2 من 0.", @@ -623,7 +623,7 @@ "end": 682.28 }, { - "input": "Likewise, the distance between 0 and 4 should be the same as that between 1 and 5, 2 and 6, and 3 and 7.", + "input": "Likewise, the distance between zero and four should be the same as that between one and five, two and six, and three and seven.", "translatedText": "وبالمثل، يجب أن تكون المسافة بين 0 و4 هي نفس المسافة بين 1 و5، و2 و6، و3 و7.", "model": "google_nmt", "from_community_srt": "وبالمثل يجب أن المسافة بين 0 و 4 يكون نفسه بين 1 و 5 و 2 و 6 و 3 و 7.", @@ -649,7 +649,7 @@ "end": 706.9 }, { - "input": "For example, negative 1 has to be in the same room as 1, in the same sub-room as 3, in the same sub-sub-room as 7, and so on, always in smaller and smaller rooms with numbers 1 less than a power of 2, because 0 is in smaller and smaller rooms with the powers of 2.", + "input": "For example, negative one has to be in the same room as one, in the same sub-room as three, the same sub-sub-room as seven, and so on, always in smaller and smaller rooms with numbers one less than a power of two, because zero is in smaller and smaller rooms with the powers of two.", "translatedText": "على سبيل المثال، سالب 1 يجب أن يكون في نفس الغرفة مثل 1، في نفس الغرفة الفرعية مثل 3، في نفس الغرفة الفرعية مثل 7، وهكذا، دائمًا في غرف أصغر وأصغر بأرقام 1 أقل من قوة 2، لأن 0 موجود في غرف أصغر فأصغر بقوة 2.", "model": "google_nmt", "from_community_srt": "يمكنك أيضا استنتاج الأعداد الصحيحة السلبية يجب أن تقع ، حيث على سبيل المثال -1 يجب أن يكون في نفس الغرفة مثل 1 ، في نفس الغرفة الفرعية مثل 3 ، نفس الغرفة الفرعية الفرعية 7 ، وما إلى ذلك ، دائما في غرف أصغر وأصغر مع أرقام واحد أقل من قوة 2 ، لأن 0 في غرف أصغر وأصغر مع القوى من 2.", @@ -667,7 +667,7 @@ "end": 734.4 }, { - "input": "You can't take this drawing too literally, since it makes 1 look very close to 14 and 0 very far from 13, even though shift invariance should imply that they're the same distance away.", + "input": "You can't take this drawing too literally, since it makes one look very close to fourteen and zero very far from thirteen, even though shift invariance should imply that they're the same distance away.", "translatedText": "لا يمكنك أن تأخذ هذا الرسم بشكل حرفي للغاية، لأنه يجعل 1 يبدو قريبًا جدًا من 14 و0 بعيدًا جدًا عن 13، على الرغم من أن ثبات الإزاحة يجب أن يعني أنهما على نفس المسافة.", "model": "google_nmt", "from_community_srt": "لا يمكنك أخذ هذا الرسم أيضًا حرفيًا ، لأنه يجعل 1 تبدو قريبة جدا من 14 و 0 بعيدا جدا عن 13 ، على الرغم من التحول-الثبات يجب أن تشير إلى أنهم هم نفس المسافة", @@ -721,7 +721,7 @@ "end": 795.78 }, { - "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers, like ⅓ and ½, should fall into.", + "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers like one third and one half should fall into,", "translatedText": "لن نفعل ذلك في هذا الفيديو، لكن لنرى ما إذا كان بإمكانك التفكير في أي الغرف يجب أن تقع الأعداد النسبية الأخرى، مثل ⅓ و½.", "model": "google_nmt", "from_community_srt": "لن نفعل ذلك في هذا الفيديو ، ولكن لنرى ما إذا كان يمكنك التفكير في أي غرف أخرى عقلانية يجب أن تقع أرقام مثل 1/3 و ½ في ، ومعرفة ما إذا كان يمكنك إثبات لماذا هذا المفهوم", diff --git a/2015/inventing-math/english/captions.srt b/2015/inventing-math/english/captions.srt index 0170e5e6c..6f4239270 100644 --- a/2015/inventing-math/english/captions.srt +++ b/2015/inventing-math/english/captions.srt @@ -131,7 +131,7 @@ we should be able to write this thing down as a sum that contains the reciprocal every power of 2. 34 -00:02:09,640 --> 00:02:14,329 +00:02:09,639 --> 00:02:14,329 On the other hand, we can see geometrically that these numbers approach 1, 35 @@ -203,602 +203,618 @@ It's not just that the distance between each number and 1 gets smaller, because for that matter, the distance between each number and 2 also gets smaller. 52 -00:03:29,580 --> 00:03:34,702 -After thinking about it, you realize what makes 1 special is that your numbers can get +00:03:29,580 --> 00:03:33,636 +After thinking about it, you realize what makes 1 special is that your 53 -00:03:34,702 --> 00:03:39,648 -arbitrarily close to 1, which is to say, no matter how small your desired distance, +00:03:33,636 --> 00:03:36,892 +numbers can get arbitrarily close to 1, which is to say, 54 -00:03:39,648 --> 00:03:44,653 -1 one-hundredth, 1 one-millionth, or 1 over the largest number you could write down, +00:03:36,892 --> 00:03:41,519 +no matter how small your desired distance, one one hundredth, one one millionth, 55 -00:03:44,653 --> 00:03:49,835 -if you go down your list long enough, the numbers will eventually fall within that tiny +00:03:41,519 --> 00:03:44,547 +or one over the largest number you could write down, 56 -00:03:49,835 --> 00:03:50,660 -distance of 1. +00:03:44,547 --> 00:03:48,603 +if you go down your list long enough, the numbers will eventually fall 57 +00:03:48,603 --> 00:03:50,660 +within that tiny tiny distance of 1. + +58 00:03:53,280 --> 00:03:56,767 Retrospectively, this might seem like the clear way to solidify what you mean -58 +59 00:03:56,767 --> 00:04:00,120 by approach, but as a first-time endeavor, it's actually incredibly clever. -59 +60 00:04:01,420 --> 00:04:04,934 Now you pull out your pin, and scribble down the definition for -60 +61 00:04:04,934 --> 00:04:08,340 what it means for an infinite sum to equal some number, say x. -61 +62 00:04:09,120 --> 00:04:13,248 It means that when you generate a list of numbers by cutting off your -62 +63 00:04:13,248 --> 00:04:17,436 sum at finite points, the numbers in this list approach x in the sense -63 +64 00:04:17,436 --> 00:04:22,096 that no matter how small the distance you choose, at some point down the list, -64 +65 00:04:22,096 --> 00:04:25,400 all the numbers start falling within that distance of x. -65 +66 00:04:26,860 --> 00:04:30,325 In doing this, you just invented some math, but it never felt like -66 +67 00:04:30,325 --> 00:04:33,686 you were pulling things out of thin air, you were just trying to -67 +68 00:04:33,686 --> 00:04:37,100 justify what it was that the universe gave you in the first place. -68 +69 00:04:39,920 --> 00:04:42,292 You might wonder if you can find other, more general -69 +70 00:04:42,292 --> 00:04:44,800 truths about these infinite sums that you just invented. -70 +71 00:04:45,360 --> 00:04:48,760 To do so, you look for where you made any arbitrary decisions. -71 -00:04:49,340 --> 00:04:53,669 -For instance, when you were shrinking the distance between your objects, - 72 -00:04:53,669 --> 00:04:56,813 -cutting the interval into pieces of size ½, ¼, etc., +00:04:49,340 --> 00:04:53,164 +For instance, when you were shrinking the distance between your objects, 73 -00:04:56,813 --> 00:04:59,660 -you could have chosen a proportion other than ½. +00:04:53,164 --> 00:04:56,464 +cutting the interval into pieces of size one half, one fourth, 74 +00:04:56,464 --> 00:04:59,660 +etc., you could have chosen a proportion other than one half. + +75 00:05:00,340 --> 00:05:04,945 You could have instead cut your interval into pieces of size 9 tenths and 1 tenth, -75 +76 00:05:04,945 --> 00:05:08,329 and then cut that rightmost piece into the same proportions, -76 +77 00:05:08,329 --> 00:05:12,435 giving you smaller pieces of size 9 one hundredths and one one hundredth, -77 +78 00:05:12,435 --> 00:05:15,820 then cut that tiny piece of size one one hundredth similarly. -78 -00:05:16,420 --> 00:05:20,178 -Continuing on and on, you'd see that 9 tenths plus 9 one - 79 -00:05:20,178 --> 00:05:24,727 -hundredths plus 9 one thousandths on and on up to infinity equals 1, +00:05:16,420 --> 00:05:20,162 +Continuing on and on, you'd see that nine tenths plus nine one 80 -00:05:24,727 --> 00:05:28,420 -a fact more popularly written as 0.9 repeating equals 1. +00:05:20,162 --> 00:05:24,558 +hundredths plus nine one thousandths on and on up to infinity equals one, 81 +00:05:24,558 --> 00:05:28,420 +a fact more popularly written as point nine repeating equals one. + +82 00:05:29,040 --> 00:05:33,231 To all of your friends who insist that this doesn't equal 1 and it just approaches it, -82 +83 00:05:33,231 --> 00:05:36,411 you can now just smile, because you know that with infinite sums, -83 +84 00:05:36,411 --> 00:05:38,580 to approach and to equal mean the same thing. -84 -00:05:40,360 --> 00:05:44,645 -To be general about it, let's say that you cut your interval into - 85 -00:05:44,645 --> 00:05:49,320 -pieces of size p and 1-p, where p represents any number between 0 and 1. +00:05:40,360 --> 00:05:44,869 +To be general about it, let's say that you cut your interval into pieces of 86 +00:05:44,869 --> 00:05:49,320 +size p and one minus p, where p represents any number between zero and one. + +87 00:05:49,320 --> 00:05:53,049 Cutting the piece of size p in similar proportions, -87 +88 00:05:53,049 --> 00:05:56,780 we now get pieces of size p times 1-p and p squared. -88 -00:05:59,220 --> 00:06:04,373 -Continuing in this fashion, always cutting up the rightmost piece into - 89 -00:06:04,373 --> 00:06:09,381 -those same proportions, you'll find that 1-p plus p times 1-p plus p +00:05:59,220 --> 00:06:04,337 +Continuing in this fashion, always cutting up the rightmost piece into those same 90 -00:06:09,381 --> 00:06:15,260 -squared times 1-p on and on always adding p to the next power times 1-p equals 1. +00:06:04,337 --> 00:06:09,455 +proportions, you'll find that one minus p plus p times one minus p plus p squared 91 +00:06:09,455 --> 00:06:14,573 +times one minus p, on and on always adding p to the next power times one minus p, + +92 +00:06:14,573 --> 00:06:15,260 +equals one. + +93 00:06:16,200 --> 00:06:19,740 Dividing both sides by 1-p, we get this nice formula. -92 +94 00:06:23,980 --> 00:06:27,520 In this formula, the universe has offered a weird form of nonsense. -93 +95 00:06:28,740 --> 00:06:33,285 Even though the way you discovered it only makes sense for values of p between 0 and 1, -94 +96 00:06:33,285 --> 00:06:37,418 the right hand side still makes sense when you replace p with any other number, -95 +97 00:06:37,418 --> 00:06:38,400 except maybe for 1. -96 -00:06:40,100 --> 00:06:46,103 -For instance, plugging in negative 1, the equation reads 1 minus 1 plus - -97 -00:06:46,103 --> 00:06:52,273 -1 minus 1 on and on forever alternating between the two, equals one half, - 98 -00:06:52,273 --> 00:06:57,860 -which feels both silly and kind of like the only thing it could be. +00:06:40,100 --> 00:06:45,836 +For instance, plugging in negative one, the equation reads one minus one 99 -00:06:59,520 --> 00:07:04,806 -Plugging in 2, the equation reads 1 plus 2 plus 4 plus 8 on and on to +00:06:45,836 --> 00:06:51,101 +plus one minus one, on and on forever alternating between the two, 100 -00:07:04,806 --> 00:07:10,320 -infinity equals negative 1, something which doesn't even seem reasonable. +00:06:51,101 --> 00:06:57,860 +equals one half, which feels pretty silly and kind of like the only thing it could be. 101 +00:06:59,520 --> 00:07:04,312 +Plugging in two, the equation reads one plus two plus four plus eight, + +102 +00:07:04,312 --> 00:07:10,320 +on and on to infinity, equals negative one, something which doesn't even seem reasonable. + +103 00:07:11,200 --> 00:07:14,087 On the one hand, Rigger would dictate that you ignore these, -102 +104 00:07:14,087 --> 00:07:17,260 since the definition of infinite sums doesn't apply in these cases. -103 +105 00:07:17,740 --> 00:07:20,308 The list of numbers that you generate by cutting off -104 +106 00:07:20,308 --> 00:07:22,780 the sum at finite points doesn't approach anything. -105 +107 00:07:30,740 --> 00:07:33,704 But you're a mathematician, not a robot, so you don't -106 +108 00:07:33,704 --> 00:07:36,560 let the fact that something is nonsensical stop you. -107 +109 00:07:37,780 --> 00:07:42,320 I will leave this sum for another day, so that we can jump directly into this monster. -108 +110 00:07:43,360 --> 00:07:47,620 First, to clean things up, notice what you get when you cut off the sum at finite points. -109 +111 00:07:48,220 --> 00:07:54,860 -1, 3, 7, 15, 31, they're all 1 less than a power of 2. +One, three, seven, fifteen, thirty-one, they're all one less than a power of two. -110 +112 00:07:55,680 --> 00:07:59,151 In general, when you add up the first n powers of 2, -111 +113 00:07:59,151 --> 00:08:04,260 you get 2 to the n plus 1 minus 1, which this animation hopefully makes clear. -112 +114 00:08:20,060 --> 00:08:23,652 You decide to humor the universe and pretend that these numbers, -113 +115 00:08:23,652 --> 00:08:27,080 all 1 less than a power of 2, actually do approach negative 1. -114 +116 00:08:27,080 --> 00:08:30,515 It will prove to be cleaner if we add 1 to everything -115 +117 00:08:30,515 --> 00:08:33,059 and say that the powers of 2 approach 0. -116 +118 00:08:35,299 --> 00:08:37,520 Is there any way that this can make sense? -117 +119 00:08:38,539 --> 00:08:42,277 In effect, what you're trying to do is make this formula more general, -118 +120 00:08:42,277 --> 00:08:46,120 by saying that it applies to all numbers, not just those between 0 and 1. -119 +121 00:08:46,800 --> 00:08:49,437 Again, to make things more general, you look for -120 +122 00:08:49,437 --> 00:08:51,860 any place where you made an arbitrary choice. -121 +123 00:08:51,860 --> 00:08:55,923 Here, that place turns out to be very sneaky, so sneaky in fact -122 +124 00:08:55,923 --> 00:08:59,860 that it took mathematicians until the 20th century to find it. -123 +125 00:09:01,440 --> 00:09:05,040 It's the way that we define distance between two rational numbers. -124 +126 00:09:05,780 --> 00:09:08,890 That is to say, organizing them on a line might -125 +127 00:09:08,890 --> 00:09:12,000 not be the only reasonable way to organize them. -126 +128 00:09:15,460 --> 00:09:18,958 The notion of distance is essentially a function that takes in -127 +129 00:09:18,958 --> 00:09:22,680 two numbers and outputs a number indicating how far apart they are. -128 -00:09:24,260 --> 00:09:27,770 -You could come up with a completely random notion of distance, - -129 -00:09:27,770 --> 00:09:32,452 -where 2 is 7 away from 3, and ½ is 4 fifths away from 100, and all sorts of things, - 130 -00:09:32,452 --> 00:09:36,464 -but if you want to actually use a new distance function the way you use +00:09:24,260 --> 00:09:27,437 +You could come up with a completely random notion of distance, 131 -00:09:36,464 --> 00:09:40,700 -the familiar distance function, it should share some of the same properties. +00:09:27,437 --> 00:09:31,723 +where two is seven away from three, and one half is four fifths away from a hundred, 132 -00:09:42,380 --> 00:09:45,100 -For example, the distance between two numbers shouldn't +00:09:31,723 --> 00:09:35,808 +and all sorts of things. But if you want to actually use a new distance function 133 -00:09:45,100 --> 00:09:47,480 -change if you shift them both by the same amount. +00:09:35,808 --> 00:09:38,481 +the way that you use the familiar distance function, 134 -00:09:48,400 --> 00:09:53,091 -So 0 and 4 should be the same distance away as 1 and 5, or 2 and 6, +00:09:38,481 --> 00:09:40,700 +it should share some of the same properties. 135 -00:09:53,091 --> 00:09:57,920 -even if that same distance is something other than 4 as we're used to. +00:09:42,380 --> 00:09:45,100 +For example, the distance between two numbers shouldn't 136 -00:09:59,120 --> 00:10:01,760 -Keeping things general, the distance between two numbers +00:09:45,100 --> 00:09:47,480 +change if you shift them both by the same amount. 137 -00:10:01,760 --> 00:10:04,540 -shouldn't change if you add the same amount to both of them. +00:09:48,400 --> 00:09:53,465 +So zero and four should be the same distance away as one and five, or two and six, 138 -00:10:05,040 --> 00:10:07,240 -Let's call this property shift invariance. +00:09:53,465 --> 00:09:57,920 +even if that same distance is something other than four as we're used to. 139 -00:10:09,460 --> 00:10:16,642 -There are other properties that you want your notion of distance to have as well, +00:09:59,120 --> 00:10:01,760 +Keeping things general, the distance between two numbers 140 -00:10:16,642 --> 00:10:22,948 -like the notion of distance could possibly make powers of 2 approach 0, +00:10:01,760 --> 00:10:04,540 +shouldn't change if you add the same amount to both of them. 141 -00:10:22,948 --> 00:10:24,700 -and shift invariant. +00:10:05,040 --> 00:10:07,240 +Let's call this property shift invariance. 142 -00:10:25,900 --> 00:10:30,496 -At first you might toil for a while to find a frame of mind where this doesn't +00:10:09,460 --> 00:10:14,391 +There are other properties that you want your notion of distance to have as well, 143 -00:10:30,496 --> 00:10:34,337 -feel like utter nonsense, but with enough time and a bit of luck, +00:10:14,391 --> 00:10:18,722 +like the triangle inequality, but before we start worrying about those, 144 -00:10:34,337 --> 00:10:39,400 -you might think to organize your numbers into rooms, subrooms, sub-subrooms, and so on. +00:10:18,722 --> 00:10:22,691 +let's start imagining what notion of distance could possibly make 145 -00:10:40,080 --> 00:10:45,697 -You think of 0 as being in the same room as all of the powers of 2 greater than 1, +00:10:22,691 --> 00:10:26,180 +powers of two approach zero, and which is shift invariant. 146 -00:10:45,697 --> 00:10:50,095 -as being in the same sub-room as all powers of 2 greater than 2, +00:10:26,180 --> 00:10:30,681 +At first you might toil for a while to find a frame of mind where this doesn't 147 -00:10:50,095 --> 00:10:54,494 -as being in the same sub-sub-room as powers of 2 greater than 4, +00:10:30,681 --> 00:10:34,442 +feel like utter nonsense, but with enough time and a bit of luck, 148 -00:10:54,494 --> 00:10:58,420 -and so on, with infinitely many smaller and smaller rooms. +00:10:34,442 --> 00:10:39,400 +you might think to organize your numbers into rooms, subrooms, sub-subrooms, and so on. 149 -00:10:59,860 --> 00:11:04,140 -It's pretty hard to draw infinitely many things, so I'm only going to draw 4 room sizes, +00:10:40,080 --> 00:10:45,831 +You think of zero as being in the same room as all of the powers of two greater than one. 150 -00:11:04,140 --> 00:11:08,180 -but keep in the back of your mind that this process should be able to go on forever. +00:10:45,831 --> 00:10:50,240 +As being in the same sub-room as all powers of two greater than two. 151 -00:11:09,620 --> 00:11:13,793 -If we think of every number as lying in a hierarchy of rooms, not just 0, +00:10:50,240 --> 00:10:54,713 +As being in the same sub-sub-room as powers of two greater than four, 152 -00:11:13,793 --> 00:11:17,460 -shift invariance will tell us where all of the numbers must fall. +00:10:54,713 --> 00:10:58,420 +and so on, with infinitely many smaller and smaller rooms. 153 -00:11:18,220 --> 00:11:22,280 -For instance, 1 should be as far away from 3 as 2 is from 0. +00:10:59,860 --> 00:11:04,140 +It's pretty hard to draw infinitely many things, so I'm only going to draw 4 room sizes, 154 -00:11:24,120 --> 00:11:29,578 -Likewise, the distance between 0 and 4 should be the same as that between 1 and 5, +00:11:04,140 --> 00:11:08,180 +but keep in the back of your mind that this process should be able to go on forever. 155 -00:11:29,578 --> 00:11:30,960 -2 and 6, and 3 and 7. +00:11:09,620 --> 00:11:13,793 +If we think of every number as lying in a hierarchy of rooms, not just 0, 156 -00:11:32,240 --> 00:11:35,815 -Continuing like this, you'll see which rooms, sub-rooms, +00:11:13,793 --> 00:11:17,460 +shift invariance will tell us where all of the numbers must fall. 157 -00:11:35,815 --> 00:11:39,580 -sub-sub-rooms, and so on, successive numbers must fall into. +00:11:18,220 --> 00:11:22,280 +For instance, one should be as far away from three as two is from zero. 158 -00:11:43,540 --> 00:11:46,900 -You can also deduce where negative numbers must fall. +00:11:24,120 --> 00:11:27,566 +Likewise, the distance between zero and four should be the same 159 -00:11:47,320 --> 00:11:53,480 -For example, negative 1 has to be in the same room as 1, in the same sub-room as 3, +00:11:27,566 --> 00:11:30,960 +as that between one and five, two and six, and three and seven. 160 -00:11:53,480 --> 00:11:59,566 -in the same sub-sub-room as 7, and so on, always in smaller and smaller rooms with +00:11:32,240 --> 00:11:35,815 +Continuing like this, you'll see which rooms, sub-rooms, 161 -00:11:59,566 --> 00:12:05,800 -numbers 1 less than a power of 2, because 0 is in smaller and smaller rooms with the +00:11:35,815 --> 00:11:39,580 +sub-sub-rooms, and so on, successive numbers must fall into. 162 -00:12:05,800 --> 00:12:06,680 -powers of 2. +00:11:43,540 --> 00:11:46,900 +You can also deduce where negative numbers must fall. 163 -00:12:07,740 --> 00:12:11,099 -So, how do you turn this general idea of closeness based +00:11:47,320 --> 00:11:51,507 +For example, negative one has to be in the same room as one, 164 -00:12:11,099 --> 00:12:14,400 -on rooms and sub-rooms into an actual distance function? +00:11:51,507 --> 00:11:56,588 +in the same sub-room as three, the same sub-sub-room as seven, and so on, 165 -00:12:15,360 --> 00:12:18,490 -You can't take this drawing too literally, since it makes 1 +00:11:56,588 --> 00:12:02,011 +always in smaller and smaller rooms with numbers one less than a power of two, 166 -00:12:18,490 --> 00:12:21,516 -look very close to 14 and 0 very far from 13, even though +00:12:02,011 --> 00:12:06,680 +because zero is in smaller and smaller rooms with the powers of two. 167 -00:12:21,516 --> 00:12:24,960 -shift invariance should imply that they're the same distance away. +00:12:07,740 --> 00:12:11,099 +So, how do you turn this general idea of closeness based 168 +00:12:11,099 --> 00:12:14,400 +on rooms and sub-rooms into an actual distance function? + +169 +00:12:15,360 --> 00:12:18,560 +You can't take this drawing too literally, since it makes one look + +170 +00:12:18,560 --> 00:12:21,234 +very close to fourteen and zero very far from thirteen, + +171 +00:12:21,234 --> 00:12:24,960 +even though shift invariance should imply that they're the same distance away. + +172 00:12:26,540 --> 00:12:29,825 Again, in the actual process of discovery, you might toil away, -169 +173 00:12:29,825 --> 00:12:33,675 scribbling through many sheets of paper, but if you have the idea that the -170 +174 00:12:33,675 --> 00:12:37,320 only thing which should matter in determining the distance between two -171 +175 00:12:37,320 --> 00:12:41,940 objects is the size of the smallest room they share, you might come up with the following. -172 +176 00:12:43,240 --> 00:12:48,220 Any numbers lying in different large yellow rooms are a distance 1 from each other. -173 +177 00:12:50,540 --> 00:12:54,253 Those which are in the same large room, but not in the -174 +178 00:12:54,253 --> 00:12:57,900 same orange sub-room are a distance ½ from each other. -175 +179 00:12:59,560 --> 00:13:02,928 And those that are in the same orange sub-room, -176 +180 00:13:02,928 --> 00:13:07,560 but not in the same sub-sub-room are a distance ¼ from each other. -177 +181 00:13:09,940 --> 00:13:12,887 And you continue like this, using the reciprocals of -178 +182 00:13:12,887 --> 00:13:15,780 larger and larger powers of 2 to indicate closeness. -179 -00:13:17,620 --> 00:13:21,388 -We won't do it in this video, but see if you can reason about +183 +00:13:17,620 --> 00:13:21,374 +We won't do it in this video, but see if you can reason about which -180 -00:13:21,388 --> 00:13:25,460 -which rooms other rational numbers, like ⅓ and ½, should fall into. +184 +00:13:21,374 --> 00:13:25,460 +rooms other rational numbers like one third and one half should fall into, -181 +185 00:13:26,120 --> 00:13:30,294 And see if you can prove why this notion of distance satisfies many of the nice -182 +186 00:13:30,294 --> 00:13:34,260 properties we expect from a distance function, like the triangle inequality. -183 +187 00:13:35,960 --> 00:13:40,106 Here, I'll just say that this notion of distance is a perfectly legitimate one, -184 +188 00:13:40,106 --> 00:13:44,096 we call it the 2-adic metric, and it falls into a general family of distance -185 +189 00:13:44,096 --> 00:13:47,880 functions called the p-adic metrics, where p stands for any prime number. -186 +190 00:13:48,680 --> 00:13:51,818 These metrics give rise to a completely new type of number, -187 +191 00:13:51,818 --> 00:13:56,160 neither real nor complex, and have become a central notion in modern number theory. -188 +192 00:13:58,540 --> 00:14:02,644 Using the 2-adic metric, the fact that the sum of all powers -189 +193 00:14:02,644 --> 00:14:07,219 of 2 equals negative 1 actually makes sense, because the numbers 1, -190 +194 00:14:07,219 --> 00:14:10,920 3, 7, 15, 31, and so on, genuinely approach negative 1. -191 +195 00:14:12,440 --> 00:14:16,437 This parable does not actually portray the historical trajectory of discoveries, -192 +196 00:14:16,437 --> 00:14:19,497 but nevertheless, I still think it's a good illustration of a -193 +197 00:14:19,497 --> 00:14:21,620 recurring pattern in the discovery of math. -194 -00:14:22,319 --> 00:14:26,500 +198 +00:14:22,320 --> 00:14:26,500 First, nature hands you something that's ill-defined or even nonsensical. -195 +199 00:14:27,480 --> 00:14:31,285 Then you define new concepts that make this fuzzy discovery make sense, -196 +200 00:14:31,285 --> 00:14:35,883 and these new concepts tend to yield genuinely useful math and broaden your mind about -197 +201 00:14:35,883 --> 00:14:36,940 traditional notions. -198 +202 00:14:37,580 --> 00:14:41,787 So, in answer to the age-old question of whether math is invention or discovery, -199 +203 00:14:41,787 --> 00:14:45,319 my personal belief is that discovery of non-rigorous truths is what -200 +204 00:14:45,319 --> 00:14:48,643 leads us to the construction of rigorous terms that are useful, -201 +205 00:14:48,643 --> 00:14:52,020 opening the door for more fuzzy discoveries continuing the cycle. diff --git a/2015/inventing-math/english/sentence_timings.json b/2015/inventing-math/english/sentence_timings.json index a16868368..9b702d428 100644 --- a/2015/inventing-math/english/sentence_timings.json +++ b/2015/inventing-math/english/sentence_timings.json @@ -120,7 +120,7 @@ 209.04 ], [ - "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, 1 one-hundredth, 1 one-millionth, or 1 over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny distance of 1.", + "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, one one hundredth, one one millionth, or one over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny tiny distance of 1.", 209.58, 230.66 ], @@ -155,7 +155,7 @@ 288.76 ], [ - "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size ½, ¼, etc., you could have chosen a proportion other than ½.", + "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size one half, one fourth, etc., you could have chosen a proportion other than one half.", 289.34, 299.66 ], @@ -165,7 +165,7 @@ 315.82 ], [ - "Continuing on and on, you'd see that 9 tenths plus 9 one hundredths plus 9 one thousandths on and on up to infinity equals 1, a fact more popularly written as 0.9 repeating equals 1.", + "Continuing on and on, you'd see that nine tenths plus nine one hundredths plus nine one thousandths on and on up to infinity equals one, a fact more popularly written as point nine repeating equals one.", 316.42, 328.42 ], @@ -175,7 +175,7 @@ 338.58 ], [ - "To be general about it, let's say that you cut your interval into pieces of size p and 1-p, where p represents any number between 0 and 1.", + "To be general about it, let's say that you cut your interval into pieces of size p and one minus p, where p represents any number between zero and one.", 340.36, 349.32 ], @@ -185,7 +185,7 @@ 356.78 ], [ - "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that 1-p plus p times 1-p plus p squared times 1-p on and on always adding p to the next power times 1-p equals 1.", + "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that one minus p plus p times one minus p plus p squared times one minus p, on and on always adding p to the next power times one minus p, equals one.", 359.22, 375.26 ], @@ -205,12 +205,12 @@ 398.4 ], [ - "For instance, plugging in negative 1, the equation reads 1 minus 1 plus 1 minus 1 on and on forever alternating between the two, equals one half, which feels both silly and kind of like the only thing it could be.", + "For instance, plugging in negative one, the equation reads one minus one plus one minus one, on and on forever alternating between the two, equals one half, which feels pretty silly and kind of like the only thing it could be.", 400.1, 417.86 ], [ - "Plugging in 2, the equation reads 1 plus 2 plus 4 plus 8 on and on to infinity equals negative 1, something which doesn't even seem reasonable.", + "Plugging in two, the equation reads one plus two plus four plus eight, on and on to infinity, equals negative one, something which doesn't even seem reasonable.", 419.52, 430.32 ], @@ -240,7 +240,7 @@ 467.62 ], [ - "1, 3, 7, 15, 31, they're all 1 less than a power of 2.", + "One, three, seven, fifteen, thirty-one, they're all one less than a power of two.", 468.22, 474.86 ], @@ -295,7 +295,7 @@ 562.68 ], [ - "You could come up with a completely random notion of distance, where 2 is 7 away from 3, and ½ is 4 fifths away from 100, and all sorts of things, but if you want to actually use a new distance function the way you use the familiar distance function, it should share some of the same properties.", + "You could come up with a completely random notion of distance, where two is seven away from three, and one half is four fifths away from a hundred, and all sorts of things. But if you want to actually use a new distance function the way that you use the familiar distance function, it should share some of the same properties.", 564.26, 580.7 ], @@ -305,7 +305,7 @@ 587.48 ], [ - "So 0 and 4 should be the same distance away as 1 and 5, or 2 and 6, even if that same distance is something other than 4 as we're used to.", + "So zero and four should be the same distance away as one and five, or two and six, even if that same distance is something other than four as we're used to.", 588.4, 597.92 ], @@ -320,17 +320,17 @@ 607.24 ], [ - "There are other properties that you want your notion of distance to have as well, like the notion of distance could possibly make powers of 2 approach 0, and shift invariant.", + "There are other properties that you want your notion of distance to have as well, like the triangle inequality, but before we start worrying about those, let's start imagining what notion of distance could possibly make powers of two approach zero, and which is shift invariant.", 609.46, - 624.7 + 626.18 ], [ "At first you might toil for a while to find a frame of mind where this doesn't feel like utter nonsense, but with enough time and a bit of luck, you might think to organize your numbers into rooms, subrooms, sub-subrooms, and so on.", - 625.9, + 626.18, 639.4 ], [ - "You think of 0 as being in the same room as all of the powers of 2 greater than 1, as being in the same sub-room as all powers of 2 greater than 2, as being in the same sub-sub-room as powers of 2 greater than 4, and so on, with infinitely many smaller and smaller rooms.", + "You think of zero as being in the same room as all of the powers of two greater than one. As being in the same sub-room as all powers of two greater than two. As being in the same sub-sub-room as powers of two greater than four, and so on, with infinitely many smaller and smaller rooms.", 640.08, 658.42 ], @@ -345,12 +345,12 @@ 677.46 ], [ - "For instance, 1 should be as far away from 3 as 2 is from 0.", + "For instance, one should be as far away from three as two is from zero.", 678.22, 682.28 ], [ - "Likewise, the distance between 0 and 4 should be the same as that between 1 and 5, 2 and 6, and 3 and 7.", + "Likewise, the distance between zero and four should be the same as that between one and five, two and six, and three and seven.", 684.12, 690.96 ], @@ -365,7 +365,7 @@ 706.9 ], [ - "For example, negative 1 has to be in the same room as 1, in the same sub-room as 3, in the same sub-sub-room as 7, and so on, always in smaller and smaller rooms with numbers 1 less than a power of 2, because 0 is in smaller and smaller rooms with the powers of 2.", + "For example, negative one has to be in the same room as one, in the same sub-room as three, the same sub-sub-room as seven, and so on, always in smaller and smaller rooms with numbers one less than a power of two, because zero is in smaller and smaller rooms with the powers of two.", 707.32, 726.68 ], @@ -375,7 +375,7 @@ 734.4 ], [ - "You can't take this drawing too literally, since it makes 1 look very close to 14 and 0 very far from 13, even though shift invariance should imply that they're the same distance away.", + "You can't take this drawing too literally, since it makes one look very close to fourteen and zero very far from thirteen, even though shift invariance should imply that they're the same distance away.", 735.36, 744.96 ], @@ -405,7 +405,7 @@ 795.78 ], [ - "We won't do it in this video, but see if you can reason about which rooms other rational numbers, like ⅓ and ½, should fall into.", + "We won't do it in this video, but see if you can reason about which rooms other rational numbers like one third and one half should fall into,", 797.62, 805.46 ], diff --git a/2015/inventing-math/english/transcript.txt b/2015/inventing-math/english/transcript.txt index 197e9eef9..15eb67db4 100644 --- a/2015/inventing-math/english/transcript.txt +++ b/2015/inventing-math/english/transcript.txt @@ -22,31 +22,31 @@ At no point did you actually perform infinitely many operations. You had a list of numbers, a list that could keep going forever if you had the time, and each number came from a perfectly reasonable finite sum. You noticed that the numbers in this list approach 1, but what do you mean by approach? It's not just that the distance between each number and 1 gets smaller, because for that matter, the distance between each number and 2 also gets smaller. -After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, 1 one-hundredth, 1 one-millionth, or 1 over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny distance of 1. +After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, one one hundredth, one one millionth, or one over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny tiny distance of 1. Retrospectively, this might seem like the clear way to solidify what you mean by approach, but as a first-time endeavor, it's actually incredibly clever. Now you pull out your pin, and scribble down the definition for what it means for an infinite sum to equal some number, say x. It means that when you generate a list of numbers by cutting off your sum at finite points, the numbers in this list approach x in the sense that no matter how small the distance you choose, at some point down the list, all the numbers start falling within that distance of x. In doing this, you just invented some math, but it never felt like you were pulling things out of thin air, you were just trying to justify what it was that the universe gave you in the first place. You might wonder if you can find other, more general truths about these infinite sums that you just invented. To do so, you look for where you made any arbitrary decisions. -For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size ½, ¼, etc., you could have chosen a proportion other than ½. +For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size one half, one fourth, etc., you could have chosen a proportion other than one half. You could have instead cut your interval into pieces of size 9 tenths and 1 tenth, and then cut that rightmost piece into the same proportions, giving you smaller pieces of size 9 one hundredths and one one hundredth, then cut that tiny piece of size one one hundredth similarly. -Continuing on and on, you'd see that 9 tenths plus 9 one hundredths plus 9 one thousandths on and on up to infinity equals 1, a fact more popularly written as 0.9 repeating equals 1. +Continuing on and on, you'd see that nine tenths plus nine one hundredths plus nine one thousandths on and on up to infinity equals one, a fact more popularly written as point nine repeating equals one. To all of your friends who insist that this doesn't equal 1 and it just approaches it, you can now just smile, because you know that with infinite sums, to approach and to equal mean the same thing. -To be general about it, let's say that you cut your interval into pieces of size p and 1-p, where p represents any number between 0 and 1. +To be general about it, let's say that you cut your interval into pieces of size p and one minus p, where p represents any number between zero and one. Cutting the piece of size p in similar proportions, we now get pieces of size p times 1-p and p squared. -Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that 1-p plus p times 1-p plus p squared times 1-p on and on always adding p to the next power times 1-p equals 1. +Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that one minus p plus p times one minus p plus p squared times one minus p, on and on always adding p to the next power times one minus p, equals one. Dividing both sides by 1-p, we get this nice formula. In this formula, the universe has offered a weird form of nonsense. Even though the way you discovered it only makes sense for values of p between 0 and 1, the right hand side still makes sense when you replace p with any other number, except maybe for 1. -For instance, plugging in negative 1, the equation reads 1 minus 1 plus 1 minus 1 on and on forever alternating between the two, equals one half, which feels both silly and kind of like the only thing it could be. -Plugging in 2, the equation reads 1 plus 2 plus 4 plus 8 on and on to infinity equals negative 1, something which doesn't even seem reasonable. +For instance, plugging in negative one, the equation reads one minus one plus one minus one, on and on forever alternating between the two, equals one half, which feels pretty silly and kind of like the only thing it could be. +Plugging in two, the equation reads one plus two plus four plus eight, on and on to infinity, equals negative one, something which doesn't even seem reasonable. On the one hand, Rigger would dictate that you ignore these, since the definition of infinite sums doesn't apply in these cases. The list of numbers that you generate by cutting off the sum at finite points doesn't approach anything. But you're a mathematician, not a robot, so you don't let the fact that something is nonsensical stop you. I will leave this sum for another day, so that we can jump directly into this monster. First, to clean things up, notice what you get when you cut off the sum at finite points. -1, 3, 7, 15, 31, they're all 1 less than a power of 2. +One, three, seven, fifteen, thirty-one, they're all one less than a power of two. In general, when you add up the first n powers of 2, you get 2 to the n plus 1 minus 1, which this animation hopefully makes clear. You decide to humor the universe and pretend that these numbers, all 1 less than a power of 2, actually do approach negative 1. It will prove to be cleaner if we add 1 to everything and say that the powers of 2 approach 0. @@ -57,29 +57,29 @@ Here, that place turns out to be very sneaky, so sneaky in fact that it took mat It's the way that we define distance between two rational numbers. That is to say, organizing them on a line might not be the only reasonable way to organize them. The notion of distance is essentially a function that takes in two numbers and outputs a number indicating how far apart they are. -You could come up with a completely random notion of distance, where 2 is 7 away from 3, and ½ is 4 fifths away from 100, and all sorts of things, but if you want to actually use a new distance function the way you use the familiar distance function, it should share some of the same properties. +You could come up with a completely random notion of distance, where two is seven away from three, and one half is four fifths away from a hundred, and all sorts of things. But if you want to actually use a new distance function the way that you use the familiar distance function, it should share some of the same properties. For example, the distance between two numbers shouldn't change if you shift them both by the same amount. -So 0 and 4 should be the same distance away as 1 and 5, or 2 and 6, even if that same distance is something other than 4 as we're used to. +So zero and four should be the same distance away as one and five, or two and six, even if that same distance is something other than four as we're used to. Keeping things general, the distance between two numbers shouldn't change if you add the same amount to both of them. Let's call this property shift invariance. -There are other properties that you want your notion of distance to have as well, like the notion of distance could possibly make powers of 2 approach 0, and shift invariant. +There are other properties that you want your notion of distance to have as well, like the triangle inequality, but before we start worrying about those, let's start imagining what notion of distance could possibly make powers of two approach zero, and which is shift invariant. At first you might toil for a while to find a frame of mind where this doesn't feel like utter nonsense, but with enough time and a bit of luck, you might think to organize your numbers into rooms, subrooms, sub-subrooms, and so on. -You think of 0 as being in the same room as all of the powers of 2 greater than 1, as being in the same sub-room as all powers of 2 greater than 2, as being in the same sub-sub-room as powers of 2 greater than 4, and so on, with infinitely many smaller and smaller rooms. +You think of zero as being in the same room as all of the powers of two greater than one. As being in the same sub-room as all powers of two greater than two. As being in the same sub-sub-room as powers of two greater than four, and so on, with infinitely many smaller and smaller rooms. It's pretty hard to draw infinitely many things, so I'm only going to draw 4 room sizes, but keep in the back of your mind that this process should be able to go on forever. If we think of every number as lying in a hierarchy of rooms, not just 0, shift invariance will tell us where all of the numbers must fall. -For instance, 1 should be as far away from 3 as 2 is from 0. -Likewise, the distance between 0 and 4 should be the same as that between 1 and 5, 2 and 6, and 3 and 7. +For instance, one should be as far away from three as two is from zero. +Likewise, the distance between zero and four should be the same as that between one and five, two and six, and three and seven. Continuing like this, you'll see which rooms, sub-rooms, sub-sub-rooms, and so on, successive numbers must fall into. You can also deduce where negative numbers must fall. -For example, negative 1 has to be in the same room as 1, in the same sub-room as 3, in the same sub-sub-room as 7, and so on, always in smaller and smaller rooms with numbers 1 less than a power of 2, because 0 is in smaller and smaller rooms with the powers of 2. +For example, negative one has to be in the same room as one, in the same sub-room as three, the same sub-sub-room as seven, and so on, always in smaller and smaller rooms with numbers one less than a power of two, because zero is in smaller and smaller rooms with the powers of two. So, how do you turn this general idea of closeness based on rooms and sub-rooms into an actual distance function? -You can't take this drawing too literally, since it makes 1 look very close to 14 and 0 very far from 13, even though shift invariance should imply that they're the same distance away. +You can't take this drawing too literally, since it makes one look very close to fourteen and zero very far from thirteen, even though shift invariance should imply that they're the same distance away. Again, in the actual process of discovery, you might toil away, scribbling through many sheets of paper, but if you have the idea that the only thing which should matter in determining the distance between two objects is the size of the smallest room they share, you might come up with the following. Any numbers lying in different large yellow rooms are a distance 1 from each other. Those which are in the same large room, but not in the same orange sub-room are a distance ½ from each other. And those that are in the same orange sub-room, but not in the same sub-sub-room are a distance ¼ from each other. And you continue like this, using the reciprocals of larger and larger powers of 2 to indicate closeness. -We won't do it in this video, but see if you can reason about which rooms other rational numbers, like ⅓ and ½, should fall into. +We won't do it in this video, but see if you can reason about which rooms other rational numbers like one third and one half should fall into, And see if you can prove why this notion of distance satisfies many of the nice properties we expect from a distance function, like the triangle inequality. Here, I'll just say that this notion of distance is a perfectly legitimate one, we call it the 2-adic metric, and it falls into a general family of distance functions called the p-adic metrics, where p stands for any prime number. These metrics give rise to a completely new type of number, neither real nor complex, and have become a central notion in modern number theory. diff --git a/2015/inventing-math/french/sentence_translations.json b/2015/inventing-math/french/sentence_translations.json index 7b057b868..5c810ecaa 100644 --- a/2015/inventing-math/french/sentence_translations.json +++ b/2015/inventing-math/french/sentence_translations.json @@ -216,7 +216,7 @@ "end": 209.04 }, { - "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, 1 one-hundredth, 1 one-millionth, or 1 over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny distance of 1.", + "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, one one hundredth, one one millionth, or one over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny tiny distance of 1.", "translatedText": "Après y avoir réfléchi, tu réalises que ce qui rend 1 spécial, c'est que tes nombres peuvent s'approcher arbitrairement de 1. En d'autres termes, quelle que soit la distance souhaitée, 1 centième, 1 millionième ou 1 par rapport au plus grand nombre que tu puisses écrire, si tu descends ta liste suffisamment longtemps, les nombres finiront par se situer dans cette minuscule distance de 1.", "model": "DeepL", "from_community_srt": "En y réfléchissant un peu, vous réalisez que ce qui rend \"1\" spécial est que vos nombres peuvent s'approcher autant que vous voulez de 1. C'est-à-dire - peu importe la petitesse de la distance voulue - 1/100ème, 1/1.000.000ème, ou, 1/(le plus grand nombre que vous pouvez écrire): si vous descendez suffisamment bas dans votre liste, les nombres vont finir par se retrouver à l'intérieur de cette ridiculement petite distance de 1.", @@ -279,7 +279,7 @@ "end": 288.76 }, { - "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size ½, ¼, etc., you could have chosen a proportion other than ½.", + "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size one half, one fourth, etc., you could have chosen a proportion other than one half.", "translatedText": "Par exemple, lorsque tu réduisais la distance entre tes objets, en coupant l'intervalle en morceaux de taille ½, ¼, etc, tu aurais pu choisir une proportion autre que ½.", "model": "DeepL", "from_community_srt": "Par exemple, quand vous réduisiez votre distance à votre objets, en coupant les intervalles en morceaux de taille ½, ¼,etc., vous auriez pu choisir une autre proportion que ½.", @@ -297,7 +297,7 @@ "end": 315.82 }, { - "input": "Continuing on and on, you'd see that 9 tenths plus 9 one hundredths plus 9 one thousandths on and on up to infinity equals 1, a fact more popularly written as 0.9 repeating equals 1.", + "input": "Continuing on and on, you'd see that nine tenths plus nine one hundredths plus nine one thousandths on and on up to infinity equals one, a fact more popularly written as point nine repeating equals one.", "translatedText": "En continuant ainsi, tu verras que 9 dixièmes plus 9 centièmes plus 9 millièmes et ainsi de suite jusqu'à l'infini égalent 1, un fait plus populairement écrit comme 0,9 répétant égale 1.", "model": "DeepL", "from_community_srt": "en continuant ainsi sans vous arrêter; vous verriez que 9/10 + 9/100 + 9/1000.. ainsi de suite à l'infini est égal à 1; un fait plus populairement écrit sous la forme 0.9..", @@ -315,7 +315,7 @@ "end": 338.58 }, { - "input": "To be general about it, let's say that you cut your interval into pieces of size p and 1-p, where p represents any number between 0 and 1.", + "input": "To be general about it, let's say that you cut your interval into pieces of size p and one minus p, where p represents any number between zero and one.", "translatedText": "Pour être général à ce sujet, disons que tu découpes ton intervalle en morceaux de taille p et 1-p, où p représente n'importe quel nombre entre 0 et 1.", "model": "DeepL", "from_community_srt": "\"approcher\" et \"être égal\" sont la même chose. Pour généraliser, disons que vous coupez l'intervalle en morceaux de tailles p, et (1-p); où p représente n'importe quel nombre entre 0 et 1.", @@ -333,7 +333,7 @@ "end": 356.78 }, { - "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that 1-p plus p times 1-p plus p squared times 1-p on and on always adding p to the next power times 1-p equals 1.", + "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that one minus p plus p times one minus p plus p squared times one minus p, on and on always adding p to the next power times one minus p, equals one.", "translatedText": "En continuant de cette façon, en découpant toujours le morceau le plus à droite dans les mêmes proportions, tu trouveras que 1-p plus p fois 1-p plus p au carré fois 1-p et ainsi de suite en ajoutant toujours p à la puissance suivante fois 1-p est égal à 1.", "model": "DeepL", "from_community_srt": "nous obtenons alors des morceaux de taille p(1-p) et p². En continuant ces itérations, coupant la partie de droite en mêmes proportions, vous trouverez que (1-p)+p(1-p)+p²(1-p), ainsi de suite en ajoutant toujours p à la puissance suivante, multiplié par (1-p),", @@ -369,7 +369,7 @@ "end": 398.4 }, { - "input": "For instance, plugging in negative 1, the equation reads 1 minus 1 plus 1 minus 1 on and on forever alternating between the two, equals one half, which feels both silly and kind of like the only thing it could be.", + "input": "For instance, plugging in negative one, the equation reads one minus one plus one minus one, on and on forever alternating between the two, equals one half, which feels pretty silly and kind of like the only thing it could be.", "translatedText": "Par exemple, si tu introduis le chiffre négatif 1, l'équation se lit comme suit : 1 moins 1 plus 1 moins 1, et ainsi de suite en alternant les deux, égale la moitié, ce qui semble à la fois stupide et la seule chose possible.", "model": "DeepL", "from_community_srt": "(-1) Par exemple en évaluant la somme en (-1), l'équation est 1-1+1-1... à l'infini, en alternant entre les deux; est égal à 1/2; ce qui semble à la fois un peu stupide... et un peu la seule chose que ça pourrait être.", @@ -378,7 +378,7 @@ "end": 417.86 }, { - "input": "Plugging in 2, the equation reads 1 plus 2 plus 4 plus 8 on and on to infinity equals negative 1, something which doesn't even seem reasonable.", + "input": "Plugging in two, the equation reads one plus two plus four plus eight, on and on to infinity, equals negative one, something which doesn't even seem reasonable.", "translatedText": "En ajoutant 2, l'équation se lit comme suit : 1 plus 2 plus 4 plus 8 et ainsi de suite jusqu'à l'infini est égal à 1 négatif, ce qui ne semble même pas raisonnable.", "model": "DeepL", "from_community_srt": "Pour la somme en 2, l'équation donne 1+2+4+8+...=-1, quelque chose qui ne semble même pas raisonnable.", @@ -432,7 +432,7 @@ "end": 467.62 }, { - "input": "1, 3, 7, 15, 31, they're all 1 less than a power of 2.", + "input": "One, three, seven, fifteen, thirty-one, they're all one less than a power of two.", "translatedText": "1, 3, 7, 15, 31, ils sont tous 1 de moins qu'une puissance de 2.", "model": "DeepL", "from_community_srt": "3, 7, 15, 31... Ils sont tous à 1 de moins qu'une puissance de 2.", @@ -531,7 +531,7 @@ "end": 562.68 }, { - "input": "You could come up with a completely random notion of distance, where 2 is 7 away from 3, and ½ is 4 fifths away from 100, and all sorts of things, but if you want to actually use a new distance function the way you use the familiar distance function, it should share some of the same properties.", + "input": "You could come up with a completely random notion of distance, where two is seven away from three, and one half is four fifths away from a hundred, and all sorts of things. But if you want to actually use a new distance function the way that you use the familiar distance function, it should share some of the same properties.", "translatedText": "Tu pourrais inventer une notion de distance complètement aléatoire, où 2 est à 7 de 3, et ½ est à 4 cinquièmes de 100, et toutes sortes de choses, mais si tu veux réellement utiliser une nouvelle fonction de distance de la même façon que tu utilises la fonction de distance familière, elle devrait partager certaines des mêmes propriétés.", "model": "DeepL", "from_community_srt": "Vous pourriez former une version complètement aléatoire de distance où 2 et à une distance de 7 de 3, et 1/2 est à 4/5èmes de 100. Cependant, si vous voulez réellement utiliser une nouvelle fonction de distance, comme vous utiliseriez la classique, elle doit conserver certaines de ses propriétés.", @@ -549,7 +549,7 @@ "end": 587.48 }, { - "input": "So 0 and 4 should be the same distance away as 1 and 5, or 2 and 6, even if that same distance is something other than 4 as we're used to.", + "input": "So zero and four should be the same distance away as one and five, or two and six, even if that same distance is something other than four as we're used to.", "translatedText": "Ainsi, 0 et 4 devraient être à la même distance que 1 et 5, ou 2 et 6, même si cette même distance est autre chose que 4 comme nous en avons l'habitude.", "model": "DeepL", "from_community_srt": "De 0 à 4 devra être à la même distance que de 1 à 5, ou 2 à 6; même si cette même distance est autre chose que 4,", @@ -575,7 +575,7 @@ "end": 607.24 }, { - "input": "There are other properties that you want your notion of distance to have as well, like the notion of distance could possibly make powers of 2 approach 0, and shift invariant.", + "input": "There are other properties that you want your notion of distance to have as well, like the triangle inequality, but before we start worrying about those, let's start imagining what notion of distance could possibly make powers of two approach zero, and which is shift inva", "translatedText": "Il y a d'autres propriétés que tu veux que ta notion de distance ait aussi, comme la notion de distance pourrait éventuellement faire en sorte que les puissances de 2 s'approchent de 0, et que le décalage soit invariant.", "model": "DeepL", "from_community_srt": "Appelons cette propriété \"invariance par translation\". Il y a d'autres propriétés que vous voulez conserver dans votre nouvelle notion de distance, comme l'inégalité triangulaire; mais avant de s'attaquer à celles-ci, essayons de deviner quelle notion de distance pourrait faire tendre les puissances de 2 vers 0, tout en étant invariante par translation.", @@ -593,7 +593,7 @@ "end": 639.4 }, { - "input": "You think of 0 as being in the same room as all of the powers of 2 greater than 1, as being in the same sub-room as all powers of 2 greater than 2, as being in the same sub-sub-room as powers of 2 greater than 4, and so on, with infinitely many smaller and smaller rooms.", + "input": "You think of zero as being in the same room as all of the powers of two greater than one. As being in the same sub-room as all powers of two greater than two. As being in the same sub-sub-room as powers of two greater than four, and so on, with infinitely many smaller and smaller rooms.", "translatedText": "Tu penses que 0 se trouve dans la même pièce que toutes les puissances de 2 supérieures à 1, qu'il se trouve dans la même sous-pièce que toutes les puissances de 2 supérieures à 2, qu'il se trouve dans la même sous-sous-pièce que les puissances de 2 supérieures à 4, et ainsi de suite, avec une infinité de pièces de plus en plus petites.", "model": "DeepL", "from_community_srt": "sous-sous-salles, etc. Vous concevez 0 comme étant dans la même salle que toutes les puissances de 2 supérieures à 1; comme étant dans la même sous-salle que toutes les puissances de 2 supérieures à 2; comme étant dans la même sous-sous-salle que les puissances de 2 supérieures à 4, ainsi de suite avec des salles indéfiniment de plus en plus petites.", @@ -620,7 +620,7 @@ "end": 677.46 }, { - "input": "For instance, 1 should be as far away from 3 as 2 is from 0.", + "input": "For instance, one should be as far away from three as two is from zero.", "translatedText": "Par exemple, 1 doit être aussi éloigné de 3 que 2 l'est de 0.", "model": "DeepL", "from_community_srt": "Par exemple, 1 devra être à la même distance de 3 que 2 l'est de 0.", @@ -629,7 +629,7 @@ "end": 682.28 }, { - "input": "Likewise, the distance between 0 and 4 should be the same as that between 1 and 5, 2 and 6, and 3 and 7.", + "input": "Likewise, the distance between zero and four should be the same as that between one and five, two and six, and three and seven.", "translatedText": "De même, la distance entre 0 et 4 doit être la même que celle entre 1 et 5, 2 et 6, et 3 et 7.", "model": "DeepL", "from_community_srt": "De la même manière la distance 0 à 4 devra être la même que celle de 1 à 5,", @@ -656,7 +656,7 @@ "end": 706.9 }, { - "input": "For example, negative 1 has to be in the same room as 1, in the same sub-room as 3, in the same sub-sub-room as 7, and so on, always in smaller and smaller rooms with numbers 1 less than a power of 2, because 0 is in smaller and smaller rooms with the powers of 2.", + "input": "For example, negative one has to be in the same room as one, in the same sub-room as three, the same sub-sub-room as seven, and so on, always in smaller and smaller rooms with numbers one less than a power of two, because zero is in smaller and smaller rooms with the powers of two.", "translatedText": "Par exemple, le 1 négatif doit se trouver dans la même pièce que le 1, dans la même sous-pièce que le 3, dans la même sous-sous-pièce que le 7, et ainsi de suite, toujours dans des pièces de plus en plus petites avec des nombres 1 inférieurs à une puissance de 2, parce que le 0 se trouve dans des pièces de plus en plus petites avec les puissances de 2.", "model": "DeepL", "from_community_srt": "Par exemple, -2 doit être dans la même salle que 1, la même sous-salle que 3, même sous-sous-salle que 7, ainsi de suite, toujours dans des salles de plus en plus petites avec des nombres d'un de moins qu'une puissance de 2, car 0 est dans des salles de plus en plus petites avec les puissances de 2.", @@ -674,7 +674,7 @@ "end": 734.4 }, { - "input": "You can't take this drawing too literally, since it makes 1 look very close to 14 and 0 very far from 13, even though shift invariance should imply that they're the same distance away.", + "input": "You can't take this drawing too literally, since it makes one look very close to fourteen and zero very far from thirteen, even though shift invariance should imply that they're the same distance away.", "translatedText": "Tu ne peux pas prendre ce dessin trop au pied de la lettre, car il fait paraître 1 très proche de 14 et 0 très loin de 13, alors que l'invariance des décalages devrait impliquer qu'ils sont à la même distance.", "model": "DeepL", "from_community_srt": "en réelle fonction de distance ? Il ne faut pas prendre cette illustration trop littéralement, car elle fait paraître 1 très proche de 14, et 0 très loin de 13, bien que l'invariance par translation devrait induire qu'ils sont à la même distance.", @@ -728,7 +728,7 @@ "end": 795.78 }, { - "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers, like ⅓ and ½, should fall into.", + "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers like one third and one half should fall into,", "translatedText": "Nous ne le ferons pas dans cette vidéo, mais vois si tu peux raisonner sur les salles dans lesquelles d'autres nombres rationnels, comme ⅓ et ½, devraient tomber.", "model": "DeepL", "from_community_srt": "Nous ne le ferons pas dans cette vidéo, mais voyez si vous pouvez trouver dans quelles salles les autres nombres rationnels, comme 1/3 ou 1/2, devraient se retrouver;", diff --git a/2015/inventing-math/german/sentence_translations.json b/2015/inventing-math/german/sentence_translations.json index 2bcb935bf..8a4e6fd8c 100644 --- a/2015/inventing-math/german/sentence_translations.json +++ b/2015/inventing-math/german/sentence_translations.json @@ -216,7 +216,7 @@ "end": 209.04 }, { - "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, 1 one-hundredth, 1 one-millionth, or 1 over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny distance of 1.", + "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, one one hundredth, one one millionth, or one over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny tiny distance of 1.", "translatedText": "Wenn du darüber nachdenkst, wird dir klar, dass das Besondere an der 1 ist, dass deine Zahlen beliebig nahe an die 1 herankommen können. Das heißt, egal wie klein dein gewünschter Abstand ist - 1 Hundertstel, 1 Millionstel oder 1 über der größten Zahl, die du aufschreiben kannst - wenn du deine Liste lang genug abarbeitest, werden die Zahlen irgendwann innerhalb dieses winzigen Abstands von 1 liegen.", "model": "DeepL", "from_community_srt": "Nach etwas Nachdenken, fällt dir auf was an der 1 so besonders ist, ist das die Zahlen beliebig nahe an die 1 kommen können. Das bedeutet, dass egal wie klein Ihr gewünschter Abstand, 1 / 100, 1 / 1.000.000, oder eins durch die größte Zahl die du schreiben kannst, wenn du die Liste lang genug nach unten gehst, werden die Zahlen irgendwann in diesen winzigen winzigen Abstand zu 1 fallen.", @@ -279,7 +279,7 @@ "end": 288.76 }, { - "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size ½, ¼, etc., you could have chosen a proportion other than ½.", + "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size one half, one fourth, etc., you could have chosen a proportion other than one half.", "translatedText": "Als du zum Beispiel den Abstand zwischen deinen Objekten verkleinert hast, indem du das Intervall in Stücke der Größe ½, ¼ usw. geschnitten hast, hättest du auch ein anderes Verhältnis als ½ wählen können.", "model": "DeepL", "from_community_srt": "wo du willkürliche Entscheidungen getroffen hast Als du zum beispiel den Abstand der beiden Objekte verringert hast, indem du das Intervall in Stücke der Größe ½, ¼, usw. geschnitten hast, hättest du auch irgendeine andere Größe wählen können.", @@ -297,7 +297,7 @@ "end": 315.82 }, { - "input": "Continuing on and on, you'd see that 9 tenths plus 9 one hundredths plus 9 one thousandths on and on up to infinity equals 1, a fact more popularly written as 0.9 repeating equals 1.", + "input": "Continuing on and on, you'd see that nine tenths plus nine one hundredths plus nine one thousandths on and on up to infinity equals one, a fact more popularly written as point nine repeating equals one.", "translatedText": "Wenn du so weitermachst, siehst du, dass 9 Zehntel plus 9 Hundertstel plus 9 Tausendstel und so weiter bis ins Unendliche gleich 1 sind, was im Volksmund als 0,9 Wiederholung gleich 1 geschrieben wird.", "model": "DeepL", "from_community_srt": "Dann würdest du sehen, dass 9/10 + 9/100 + 9/1000 weiter und weiter bis ins Unendliche gleich gleich 1 ist, was man üblicherweise als .9 Periode = 1 schreibt.", @@ -315,7 +315,7 @@ "end": 338.58 }, { - "input": "To be general about it, let's say that you cut your interval into pieces of size p and 1-p, where p represents any number between 0 and 1.", + "input": "To be general about it, let's say that you cut your interval into pieces of size p and one minus p, where p represents any number between zero and one.", "translatedText": "Um es allgemein zu halten, nehmen wir an, dass du dein Intervall in Stücke der Größe p und 1-p unterteilst, wobei p eine beliebige Zahl zwischen 0 und 1 ist.", "model": "DeepL", "from_community_srt": "Um zu verallgemeinern, lass uns sagen, dass Sie ihr Intervall in Stücke der Größe P und (1-P) teilen, wobei P eine beliebige Zahl zwischen 0 und 1 darstellt.", @@ -333,7 +333,7 @@ "end": 356.78 }, { - "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that 1-p plus p times 1-p plus p squared times 1-p on and on always adding p to the next power times 1-p equals 1.", + "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that one minus p plus p times one minus p plus p squared times one minus p, on and on always adding p to the next power times one minus p, equals one.", "translatedText": "Wenn du so weitermachst und das ganz rechte Stück immer in die gleichen Proportionen zerlegst, wirst du feststellen, dass 1-p plus p mal 1-p plus p zum Quadrat mal 1-p und so weiter immer p zur nächsten Potenz mal 1-p gleich 1 ist.", "model": "DeepL", "from_community_srt": "erhalten wir jetzt Stücke der Größe P * (1-P) und P². Wenn wir das so fortsetzen und immer das aller rechte Stück mit dem gleichen Verältiniss teilen Wirst du herausfinden, dass wenn du bei (1-P) + P (1-P) + P ² (1-P) , auf und auf immer P mit der nächsthöheren Potentz mal (1-P) addierst,", @@ -369,7 +369,7 @@ "end": 398.4 }, { - "input": "For instance, plugging in negative 1, the equation reads 1 minus 1 plus 1 minus 1 on and on forever alternating between the two, equals one half, which feels both silly and kind of like the only thing it could be.", + "input": "For instance, plugging in negative one, the equation reads one minus one plus one minus one, on and on forever alternating between the two, equals one half, which feels pretty silly and kind of like the only thing it could be.", "translatedText": "Wenn du zum Beispiel eine negative 1 einsetzt, lautet die Gleichung 1 minus 1 plus 1 minus 1 und so weiter, immer abwechselnd, gleich eine Hälfte, was sich sowohl albern anfühlt als auch so, als wäre es das Einzige, was es sein könnte.", "model": "DeepL", "from_community_srt": "Zum Beispiel wenn man -1 in die Formel Steckt, ergibt die sich Gleichung 1-1+1-1 auf immer und ewig , gleich 1/2 das fühlt sich gleichzeitig ziemlich albern an und auch ein bisschen wie das Einzige,", @@ -378,7 +378,7 @@ "end": 417.86 }, { - "input": "Plugging in 2, the equation reads 1 plus 2 plus 4 plus 8 on and on to infinity equals negative 1, something which doesn't even seem reasonable.", + "input": "Plugging in two, the equation reads one plus two plus four plus eight, on and on to infinity, equals negative one, something which doesn't even seem reasonable.", "translatedText": "Setzt man 2 ein, lautet die Gleichung 1 plus 2 plus 4 plus 8 und so weiter bis ins Unendliche gleich minus 1, was nicht einmal vernünftig erscheint.", "model": "DeepL", "from_community_srt": "was es sein könnte. mit 2, lautet die Gleichung 1 + 2 + 4 + 8 + ... = -1 etwas, das nicht einmal vernünftig scheint.", @@ -432,7 +432,7 @@ "end": 467.62 }, { - "input": "1, 3, 7, 15, 31, they're all 1 less than a power of 2.", + "input": "One, three, seven, fifteen, thirty-one, they're all one less than a power of two.", "translatedText": "1, 3, 7, 15, 31, sie sind alle 1 kleiner als eine Potenz von 2.", "model": "DeepL", "from_community_srt": "3, 7, 15, 31. Sie sind alle eins weniger als eine Zweierpotenz.", @@ -531,7 +531,7 @@ "end": 562.68 }, { - "input": "You could come up with a completely random notion of distance, where 2 is 7 away from 3, and ½ is 4 fifths away from 100, and all sorts of things, but if you want to actually use a new distance function the way you use the familiar distance function, it should share some of the same properties.", + "input": "You could come up with a completely random notion of distance, where two is seven away from three, and one half is four fifths away from a hundred, and all sorts of things. But if you want to actually use a new distance function the way that you use the familiar distance function, it should share some of the same properties.", "translatedText": "Du könntest dir einen völlig willkürlichen Begriff von Abstand ausdenken, bei dem 2 7 von 3 und ½ 4 Fünftel von 100 entfernt ist, und alles Mögliche, aber wenn du eine neue Abstandsfunktion tatsächlich so verwenden willst, wie du die bekannte Abstandsfunktion verwendest, sollte sie einige der gleichen Eigenschaften haben.", "model": "DeepL", "from_community_srt": "wie weit sie voneinander entfernt sind. Du könntest einen völlig zufälligen Begriff des Abstandes erfinden, in dem 2 7 entfernt ist von 3 und 4 ½ ist 4/5 weg von 100, und alle möglichen Dinge, aber wenn du diesen wirklich so verwenden willst wie du die dir bekannte benutzt , sollte sie einige der gleichen Eigenschaften besitzen.", @@ -549,7 +549,7 @@ "end": 587.48 }, { - "input": "So 0 and 4 should be the same distance away as 1 and 5, or 2 and 6, even if that same distance is something other than 4 as we're used to.", + "input": "So zero and four should be the same distance away as one and five, or two and six, even if that same distance is something other than four as we're used to.", "translatedText": "0 und 4 sollten also den gleichen Abstand haben wie 1 und 5 oder 2 und 6, auch wenn dieser Abstand etwas anderes als 4 ist, wie wir es gewohnt sind.", "model": "DeepL", "from_community_srt": "0 und 4 sollten den gleiche Abstand haben, wie 1 und 5 oder 2 und 6 Selbst wenn dieser abstand nicht umbedingt 4 sein muss,", @@ -576,7 +576,7 @@ "end": 607.24 }, { - "input": "There are other properties that you want your notion of distance to have as well, like the notion of distance could possibly make powers of 2 approach 0, and shift invariant.", + "input": "There are other properties that you want your notion of distance to have as well, like the triangle inequality, but before we start worrying about those, let's start imagining what notion of distance could possibly make powers of two approach zero, and which is shift inva", "translatedText": "Es gibt noch andere Eigenschaften, die dein Begriff der Entfernung haben sollte, wie z. B., dass der Begriff der Entfernung Potenzen von 2 gegen 0 gehen kann und dass die Verschiebung invariant ist.", "model": "DeepL", "from_community_srt": "Es gibt noch andere Eigenschaften, die man haben möchte, die der Begriff der Entfernung auch hat, wie die Dreiecksungleichung, aber bevor wir uns darum sorgen überlegen wir was für ein Begriff für die Distanz möglicherweise den Abstand zwischen zweier Potenzen null annähern lässt, und shift invariant ist.", @@ -594,7 +594,7 @@ "end": 639.4 }, { - "input": "You think of 0 as being in the same room as all of the powers of 2 greater than 1, as being in the same sub-room as all powers of 2 greater than 2, as being in the same sub-sub-room as powers of 2 greater than 4, and so on, with infinitely many smaller and smaller rooms.", + "input": "You think of zero as being in the same room as all of the powers of two greater than one. As being in the same sub-room as all powers of two greater than two. As being in the same sub-sub-room as powers of two greater than four, and so on, with infinitely many smaller and smaller rooms.", "translatedText": "Du stellst dir vor, dass 0 im selben Raum ist wie alle Potenzen von 2 größer als 1, dass sie im selben Unterraum ist wie alle Potenzen von 2 größer als 2, dass sie im selben Unter-Unterraum ist wie Potenzen von 2 größer als 4, und so weiter, mit unendlich vielen kleineren und kleineren Räumen.", "model": "DeepL", "from_community_srt": "und so weiter zu unterteilen. Du denkst von 0, als im selben Raum wie alle Potenzen von 2 die größer als 1 sind, in dem gleichen Unterraum wie alle Potenzen von 2 die größer als 2 sind, wie in dem gleichen Unter-Unter-Raum wie Potenzen von 2 größer als 4, und so weiter, mit unendlich vielen kleineren und kleineren Räumen.", @@ -621,7 +621,7 @@ "end": 677.46 }, { - "input": "For instance, 1 should be as far away from 3 as 2 is from 0.", + "input": "For instance, one should be as far away from three as two is from zero.", "translatedText": "Die 1 sollte zum Beispiel so weit von der 3 entfernt sein wie die 2 von der 0.", "model": "DeepL", "from_community_srt": "Zum Beispiel sollte 1 so weit entfernt von 3 wie 2 von 0 sein.", @@ -630,7 +630,7 @@ "end": 682.28 }, { - "input": "Likewise, the distance between 0 and 4 should be the same as that between 1 and 5, 2 and 6, and 3 and 7.", + "input": "Likewise, the distance between zero and four should be the same as that between one and five, two and six, and three and seven.", "translatedText": "Ebenso sollte der Abstand zwischen 0 und 4 derselbe sein wie der zwischen 1 und 5, 2 und 6 und 3 und 7.", "model": "DeepL", "from_community_srt": "Ebenso soll der Abstand zwischen 0 und 4 der gleiche sein wie der zwischen 1 und 5,", @@ -657,7 +657,7 @@ "end": 706.9 }, { - "input": "For example, negative 1 has to be in the same room as 1, in the same sub-room as 3, in the same sub-sub-room as 7, and so on, always in smaller and smaller rooms with numbers 1 less than a power of 2, because 0 is in smaller and smaller rooms with the powers of 2.", + "input": "For example, negative one has to be in the same room as one, in the same sub-room as three, the same sub-sub-room as seven, and so on, always in smaller and smaller rooms with numbers one less than a power of two, because zero is in smaller and smaller rooms with the powers of two.", "translatedText": "Zum Beispiel muss die negative 1 im selben Raum wie die 1 sein, im selben Unterraum wie die 3, im selben Unter-Unterraum wie die 7 und so weiter, immer in immer kleineren Räumen mit Zahlen 1 kleiner als eine Potenz von 2, denn die 0 ist in immer kleineren Räumen mit den Potenzen von 2.", "model": "DeepL", "from_community_srt": "Wo zum beispiel -1 sein muss. im gleichen Raum wie 1, in dem gleichen Unterraum wie 3, der gleiche Unter-Unter-raum wie 7, und so weiter, immer in immer kleineren Räumen mit Zahlen eins weniger als eine Zweierpotenz, weil die 0 in immer kleineren und kleineren Räumen mit den Zweierpotenzen liegt Also,", @@ -675,7 +675,7 @@ "end": 734.4 }, { - "input": "You can't take this drawing too literally, since it makes 1 look very close to 14 and 0 very far from 13, even though shift invariance should imply that they're the same distance away.", + "input": "You can't take this drawing too literally, since it makes one look very close to fourteen and zero very far from thirteen, even though shift invariance should imply that they're the same distance away.", "translatedText": "Du darfst diese Zeichnung nicht zu wörtlich nehmen, denn sie lässt die 1 sehr nah an der 14 und die 0 sehr weit weg von der 13 erscheinen, obwohl die Verschiebungsinvarianz implizieren sollte, dass sie gleich weit weg sind.", "model": "DeepL", "from_community_srt": "Distanzfunktion? Du darfst diese Zeichnung nicht allzu wörtlich nehmen, da sieht es aus als währe die 1 sehr nah an der 14 und die 0 sehr weit von der 13, obwohl die Shift-Invarianz sagt, dass sie den gleichen Abstand haben sollen.", @@ -729,7 +729,7 @@ "end": 795.78 }, { - "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers, like ⅓ and ½, should fall into.", + "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers like one third and one half should fall into,", "translatedText": "Wir werden das in diesem Video nicht tun, aber versuche herauszufinden, in welche Räume andere rationale Zahlen, wie ⅓ und ½, gehören.", "model": "DeepL", "from_community_srt": "Wir werden das in diesem Video nicht tun, aber versuche mal ob du nachvollziehen kannst in welchen Räumen andere rationale Zahlen wie 1/3 und ½ fallen sollten,", diff --git a/2015/inventing-math/polish/sentence_translations.json b/2015/inventing-math/polish/sentence_translations.json index 5973d6ec7..39eeb4b2a 100644 --- a/2015/inventing-math/polish/sentence_translations.json +++ b/2015/inventing-math/polish/sentence_translations.json @@ -192,7 +192,7 @@ "end": 209.04 }, { - "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, 1 one-hundredth, 1 one-millionth, or 1 over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny distance of 1.", + "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, one one hundredth, one one millionth, or one over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny tiny distance of 1.", "translatedText": "", "from_community_srt": "Po przemyśleniu uświadamiasz sobie, że to, co czyni 1 tą wyjątkową liczbą, to fakt, że liczby mogą być dowolnie blisko 1. Chcę powiedzieć, że nieważne, jak mała będzie ta odległość, 1/100, 1/100000, lub nawet jedna z największych liczb, jakie możesz zapisać, jeśli zejdziesz w dół listy odpowiednio daleko, liczby będą tak czy inaczej wpadać do tej maleńkiej odległości od 1.", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 288.76 }, { - "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size ½, ¼, etc., you could have chosen a proportion other than ½.", + "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size one half, one fourth, etc., you could have chosen a proportion other than one half.", "translatedText": "", "from_community_srt": "Na przykład, kiedy zmniejszasz odległość pomiędzy obiektami, dzieląc przedział na kawałki o rozmiarze ½, ¼ itd., to możesz wybrać proporcję inną niż 1/2.", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 315.82 }, { - "input": "Continuing on and on, you'd see that 9 tenths plus 9 one hundredths plus 9 one thousandths on and on up to infinity equals 1, a fact more popularly written as 0.9 repeating equals 1.", + "input": "Continuing on and on, you'd see that nine tenths plus nine one hundredths plus nine one thousandths on and on up to infinity equals one, a fact more popularly written as point nine repeating equals one.", "translatedText": "", "from_community_srt": "i tak dalej, kontynuując, zobaczysz, że 9/10 + 9/100 + 9/1000 i tak dalej aż do nieskończoności, równej 1, czyli faktu bardziej znanego jako 0,99999999...", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 338.58 }, { - "input": "To be general about it, let's say that you cut your interval into pieces of size p and 1-p, where p represents any number between 0 and 1.", + "input": "To be general about it, let's say that you cut your interval into pieces of size p and one minus p, where p represents any number between zero and one.", "translatedText": "", "from_community_srt": "Uogólniając, powiedzmy, że chcesz pociąć przedział na kawałki o rozmiarze p i (1-p), gdzie p to dowolna liczba pomiędzy 0 i 1.", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 356.78 }, { - "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that 1-p plus p times 1-p plus p squared times 1-p on and on always adding p to the next power times 1-p equals 1.", + "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that one minus p plus p times one minus p plus p squared times one minus p, on and on always adding p to the next power times one minus p, equals one.", "translatedText": "", "from_community_srt": "teraz mamy kawałki o rozmiarze p(1-p) i p^2. Kontynuując w ten sposób zawsze będziemy odcinak kawałki o tych samych proporcjach, i odkryjesz, że (1-p)+p(1-p)+p^2(1-p) ciągnie dalej dodawanie p do następnej potęgi razy (1-p), a to się równa 1.", "n_reviews": 0, @@ -327,7 +327,7 @@ "end": 398.4 }, { - "input": "For instance, plugging in negative 1, the equation reads 1 minus 1 plus 1 minus 1 on and on forever alternating between the two, equals one half, which feels both silly and kind of like the only thing it could be.", + "input": "For instance, plugging in negative one, the equation reads one minus one plus one minus one, on and on forever alternating between the two, equals one half, which feels pretty silly and kind of like the only thing it could be.", "translatedText": "", "from_community_srt": "może poza 1. Na przykład podstawiając -1, otrzymasz równanie 1-1+1-1 i tak wiecznie, na zmianę pomiędzy tymi dwoma, co się równa 1/2, i jest jednocześnie zabawne i w sumie jedyne co mogłoby pasować.", "n_reviews": 0, @@ -335,7 +335,7 @@ "end": 417.86 }, { - "input": "Plugging in 2, the equation reads 1 plus 2 plus 4 plus 8 on and on to infinity equals negative 1, something which doesn't even seem reasonable.", + "input": "Plugging in two, the equation reads one plus two plus four plus eight, on and on to infinity, equals negative one, something which doesn't even seem reasonable.", "translatedText": "", "from_community_srt": "Jeśli podstawisz 2, to równanie będzie 1 + 2 + 4 + 8 + ... = -1, czyli coś, co nawet nie wydaje się sensowne.", "n_reviews": 0, @@ -383,7 +383,7 @@ "end": 467.62 }, { - "input": "1, 3, 7, 15, 31, they're all 1 less than a power of 2.", + "input": "One, three, seven, fifteen, thirty-one, they're all one less than a power of two.", "translatedText": "", "from_community_srt": "3, 7, 15, 31. To są liczby o jeden mniejsze od kolejnych potęg 2.", "n_reviews": 0, @@ -471,7 +471,7 @@ "end": 562.68 }, { - "input": "You could come up with a completely random notion of distance, where 2 is 7 away from 3, and ½ is 4 fifths away from 100, and all sorts of things, but if you want to actually use a new distance function the way you use the familiar distance function, it should share some of the same properties.", + "input": "You could come up with a completely random notion of distance, where two is seven away from three, and one half is four fifths away from a hundred, and all sorts of things. But if you want to actually use a new distance function the way that you use the familiar distance function, it should share some of the same properties.", "translatedText": "", "from_community_srt": "Mógłbyś wymyślić zupełnie losowy zapis odległości, gdzie 2 znajduje się w odległości 7 od 3 a 1/2 znajduje się 4/5 od 100, i tak dalej, ale jeśli chcesz używać nowej funkcji odległości w sposób, w który używasz znanej już funkcji odległości, to powinna ona mieć jakieś wspólne własności z tą znaną.", "n_reviews": 0, @@ -487,7 +487,7 @@ "end": 587.48 }, { - "input": "So 0 and 4 should be the same distance away as 1 and 5, or 2 and 6, even if that same distance is something other than 4 as we're used to.", + "input": "So zero and four should be the same distance away as one and five, or two and six, even if that same distance is something other than four as we're used to.", "translatedText": "", "from_community_srt": "Od 0 do 4 powinna być taka sama odległość jak od 1 do 5, lub 2 od 6, nawet jeśli ta sama odległość to coś innego niż cztery,", "n_reviews": 0, @@ -510,7 +510,7 @@ "end": 607.24 }, { - "input": "There are other properties that you want your notion of distance to have as well, like the notion of distance could possibly make powers of 2 approach 0, and shift invariant.", + "input": "There are other properties that you want your notion of distance to have as well, like the triangle inequality, but before we start worrying about those, let's start imagining what notion of distance could possibly make powers of two approach zero, and which is shift inva", "translatedText": "", "from_community_srt": "Nazwijmy tą własność \"niezmiennością przesunięcia\". Chcesz też, żeby twój zapis odległości miał inne własności, jak nierówność boków trójkąta, ale zanim zaczniemy się o to martwić, zacznijmy od wyobrażenia sobie zapisu odległości, który mógłby sprawiać, że potęgi 2 będą dążyć do 0, i który ma niezmienne przesunięcie.", "n_reviews": 0, @@ -526,7 +526,7 @@ "end": 639.4 }, { - "input": "You think of 0 as being in the same room as all of the powers of 2 greater than 1, as being in the same sub-room as all powers of 2 greater than 2, as being in the same sub-sub-room as powers of 2 greater than 4, and so on, with infinitely many smaller and smaller rooms.", + "input": "You think of zero as being in the same room as all of the powers of two greater than one. As being in the same sub-room as all powers of two greater than two. As being in the same sub-sub-room as powers of two greater than four, and so on, with infinitely many smaller and smaller rooms.", "translatedText": "", "from_community_srt": "i tak dalej. Myślisz o tym, że 0 znajduje się w tym samym pokoju, co wszystkie potęgi 2 większe od 1, że znajduje się w tym samym podpokoju co potęgi 2 większe od 2, i w tym samym podpodpokoju, co potęgi 2 większe od 4, i tak dalej, z nieskończoną liczbą coraz mniejszych i mniejszych pokoi.", "n_reviews": 0, @@ -550,7 +550,7 @@ "end": 677.46 }, { - "input": "For instance, 1 should be as far away from 3 as 2 is from 0.", + "input": "For instance, one should be as far away from three as two is from zero.", "translatedText": "", "from_community_srt": "Na przykład jeden powinno być tak daleko od 3,", "n_reviews": 0, @@ -558,7 +558,7 @@ "end": 682.28 }, { - "input": "Likewise, the distance between 0 and 4 should be the same as that between 1 and 5, 2 and 6, and 3 and 7.", + "input": "Likewise, the distance between zero and four should be the same as that between one and five, two and six, and three and seven.", "translatedText": "", "from_community_srt": "jak 2 jest daleko od 0. I dalej, odległość pomiędzy 0 a 4 powinna być taka, jak ta pomiędzy 1 a 5, 2 a 6,", "n_reviews": 0, @@ -582,7 +582,7 @@ "end": 706.9 }, { - "input": "For example, negative 1 has to be in the same room as 1, in the same sub-room as 3, in the same sub-sub-room as 7, and so on, always in smaller and smaller rooms with numbers 1 less than a power of 2, because 0 is in smaller and smaller rooms with the powers of 2.", + "input": "For example, negative one has to be in the same room as one, in the same sub-room as three, the same sub-sub-room as seven, and so on, always in smaller and smaller rooms with numbers one less than a power of two, because zero is in smaller and smaller rooms with the powers of two.", "translatedText": "", "from_community_srt": "gdzie powinny wpaść liczby ujemne, na przykład -1 musi być w tym samym pokoju co 1, w tym samym podpokoju co 3, i w tym samym podpodpokoju co7, i tak dalej, zawsze w mniejszych i mniejszych pokojach z liczbami o jeden mniejszymi niż potęga 2, ponieważ 0 jest w mniejszych i mniejszych pokojach z potęgami 2.", "n_reviews": 0, @@ -598,7 +598,7 @@ "end": 734.4 }, { - "input": "You can't take this drawing too literally, since it makes 1 look very close to 14 and 0 very far from 13, even though shift invariance should imply that they're the same distance away.", + "input": "You can't take this drawing too literally, since it makes one look very close to fourteen and zero very far from thirteen, even though shift invariance should imply that they're the same distance away.", "translatedText": "", "from_community_srt": "Nie możesz brać tego obrazka zbyt dosłownie, ponieważ sprawia on, że 1 wygląda na bardzo bliskie 14, a 0 na bardzo dalekie od 13, nawet jeśli niezmienność przesunięcia powinna powodować, że są od siebie w tej samej odległości.", "n_reviews": 0, @@ -646,7 +646,7 @@ "end": 795.78 }, { - "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers, like ⅓ and ½, should fall into.", + "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers like one third and one half should fall into,", "translatedText": "", "from_community_srt": "Nie zrobimy tego w tym filmie, ale sprawdź, czy możesz wywnioskować coś na temat pokoi, w których znajdą się inne wymierne liczby, jak 1/3 czy ½,", "n_reviews": 0, diff --git a/2015/inventing-math/portuguese/sentence_translations.json b/2015/inventing-math/portuguese/sentence_translations.json index a3ce91ad1..64c5f5ebe 100644 --- a/2015/inventing-math/portuguese/sentence_translations.json +++ b/2015/inventing-math/portuguese/sentence_translations.json @@ -216,7 +216,7 @@ "end": 209.04 }, { - "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, 1 one-hundredth, 1 one-millionth, or 1 over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny distance of 1.", + "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, one one hundredth, one one millionth, or one over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny tiny distance of 1.", "translatedText": "Depois de pensar sobre isso, você percebe que o que torna 1 especial é que seus números podem chegar arbitrariamente perto de 1, ou seja, não importa quão pequena seja a distância desejada, 1 um centésimo, 1 um milionésimo ou 1 sobre o maior número que você poderia anotar, se você percorrer sua lista por tempo suficiente, os números eventualmente cairão naquela pequena distância de 1.", "model": "google_nmt", "from_community_srt": "Depois de pensar sobre isso, você percebe que o que torna 1 especial é que os números ficam arbitrariamente mais próximos de 1. Ou seja, não importa quão pequena seja sua distância, 1/100, 1/1.000.000, ou 1 sobre o maior número que você consegue escrever, se você for mais longe na lista o suficiente, os números eventualmente cairão dentro dessa minúscula distância de 1.", @@ -279,7 +279,7 @@ "end": 288.76 }, { - "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size ½, ¼, etc., you could have chosen a proportion other than ½.", + "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size one half, one fourth, etc., you could have chosen a proportion other than one half.", "translatedText": "Por exemplo, quando você estava diminuindo a distância entre seus objetos, cortando o intervalo em pedaços de tamanho ½, ¼, etc., você poderia ter escolhido uma proporção diferente de ½.", "model": "google_nmt", "from_community_srt": "Por exemplo, quando você estava encolhendo a distância entre os objetos, cortando o intervalo em pedaços de 1/2, 1/4. etc, você poderia ter escolhido outra proporção além de 1/2.", @@ -297,7 +297,7 @@ "end": 315.82 }, { - "input": "Continuing on and on, you'd see that 9 tenths plus 9 one hundredths plus 9 one thousandths on and on up to infinity equals 1, a fact more popularly written as 0.9 repeating equals 1.", + "input": "Continuing on and on, you'd see that nine tenths plus nine one hundredths plus nine one thousandths on and on up to infinity equals one, a fact more popularly written as point nine repeating equals one.", "translatedText": "Continuando, você verá que 9 décimos mais 9 centésimos mais 9 milésimos e assim por diante até o infinito é igual a 1, um fato mais popularmente escrito como 0,9 repetido é igual a 1.", "model": "google_nmt", "from_community_srt": "continuando assim por diante, e você veria que 9/10 + 9/100 + 9/1000 e assim sucessivamente até o infinito é igual a 1, um fato mais popularmente escrito como 0.99999999...", @@ -315,7 +315,7 @@ "end": 338.58 }, { - "input": "To be general about it, let's say that you cut your interval into pieces of size p and 1-p, where p represents any number between 0 and 1.", + "input": "To be general about it, let's say that you cut your interval into pieces of size p and one minus p, where p represents any number between zero and one.", "translatedText": "Para ser geral, digamos que você corte seu intervalo em pedaços de tamanho p e 1-p, onde p representa qualquer número entre 0 e 1.", "model": "google_nmt", "from_community_srt": "Para generalizar, digamos que você divida os intervalos em pedaços de tamanho \"p\" e (1-p), onde \"p\" representa qualquer número entre 0 e 1.", @@ -333,7 +333,7 @@ "end": 356.78 }, { - "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that 1-p plus p times 1-p plus p squared times 1-p on and on always adding p to the next power times 1-p equals 1.", + "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that one minus p plus p times one minus p plus p squared times one minus p, on and on always adding p to the next power times one minus p, equals one.", "translatedText": "Continuando desta forma, sempre cortando a peça mais à direita nas mesmas proporções, você descobrirá que 1-p mais p vezes 1-p mais p ao quadrado vezes 1-p e sempre adicionando p à próxima potência vezes 1- p é igual a 1.", "model": "google_nmt", "from_community_srt": "conseguimos pedaços de tamanho p(1-p) e p². Continuando dessa maneira, sempre cortando os pedaços da direita nas mesmas proporções, você verá que (1-p) + p(1-p) + p²(1-p), e assim sucessivamente, sempre adicionando \"p\" à próxima potência vezes (1-p), é igual a 1.", @@ -369,7 +369,7 @@ "end": 398.4 }, { - "input": "For instance, plugging in negative 1, the equation reads 1 minus 1 plus 1 minus 1 on and on forever alternating between the two, equals one half, which feels both silly and kind of like the only thing it could be.", + "input": "For instance, plugging in negative one, the equation reads one minus one plus one minus one, on and on forever alternating between the two, equals one half, which feels pretty silly and kind of like the only thing it could be.", "translatedText": "Por exemplo, inserindo 1 negativo, a equação mostra 1 menos 1 mais 1 menos 1, alternando indefinidamente entre os dois, igual a metade, o que parece bobo e meio que a única coisa que poderia ser.", "model": "google_nmt", "from_community_srt": "exceto talvez por 1. Por exemplo, substituindo -1, a equação se torna 1-1+1-1... para sempre, alternando entre os dois, e é igual a 1/2, o que soa tanto meio idiota como a única coisa que poderia ser.", @@ -378,7 +378,7 @@ "end": 417.86 }, { - "input": "Plugging in 2, the equation reads 1 plus 2 plus 4 plus 8 on and on to infinity equals negative 1, something which doesn't even seem reasonable.", + "input": "Plugging in two, the equation reads one plus two plus four plus eight, on and on to infinity, equals negative one, something which doesn't even seem reasonable.", "translatedText": "Conectando 2, a equação mostra 1 mais 2 mais 4 mais 8 e assim por diante até o infinito é igual a 1 negativo, algo que nem parece razoável.", "model": "google_nmt", "from_community_srt": "Subsituindo \"2\", a equação se torna 1+2+4+8... = -1, algo que não parece sequer razoável.", @@ -431,7 +431,7 @@ "end": 467.62 }, { - "input": "1, 3, 7, 15, 31, they're all 1 less than a power of 2.", + "input": "One, three, seven, fifteen, thirty-one, they're all one less than a power of two.", "translatedText": "1, 3, 7, 15, 31, são todos 1 a menos que uma potência de 2.", "model": "google_nmt", "from_community_srt": "3, 5, 7, 15, 31. Todos são potências de 2,", @@ -530,7 +530,7 @@ "end": 562.68 }, { - "input": "You could come up with a completely random notion of distance, where 2 is 7 away from 3, and ½ is 4 fifths away from 100, and all sorts of things, but if you want to actually use a new distance function the way you use the familiar distance function, it should share some of the same properties.", + "input": "You could come up with a completely random notion of distance, where two is seven away from three, and one half is four fifths away from a hundred, and all sorts of things. But if you want to actually use a new distance function the way that you use the familiar distance function, it should share some of the same properties.", "translatedText": "Você poderia chegar a uma noção completamente aleatória de distância, onde 2 está a 7 de distância de 3, e ½ está a 4 quintos de 100, e todo tipo de coisa, mas se você quiser realmente usar uma nova função de distância da maneira que você usa a familiar função de distância, ela deve compartilhar algumas das mesmas propriedades.", "model": "google_nmt", "from_community_srt": "Você pode se deparar com uma noção completamente aleatória de distância, onde 2 é 7 distante de 3, e 1/2 é 4/5 distante de 100 e todos tipos de coisas, mas se você quer na verdade usar uma nova função de distância do modo que você usa a familiar função da distância,", @@ -548,7 +548,7 @@ "end": 587.48 }, { - "input": "So 0 and 4 should be the same distance away as 1 and 5, or 2 and 6, even if that same distance is something other than 4 as we're used to.", + "input": "So zero and four should be the same distance away as one and five, or two and six, even if that same distance is something other than four as we're used to.", "translatedText": "Portanto, 0 e 4 devem estar à mesma distância que 1 e 5, ou 2 e 6, mesmo que essa mesma distância seja diferente de 4, como estamos acostumados.", "model": "google_nmt", "from_community_srt": "Entre o 0 e o 4 deve haver a mesma distância entre o 1 e 5 ou o 2 e o 6, mesmo se essa mesma distância é outra coisa além de 4,", @@ -574,7 +574,7 @@ "end": 607.24 }, { - "input": "There are other properties that you want your notion of distance to have as well, like the notion of distance could possibly make powers of 2 approach 0, and shift invariant.", + "input": "There are other properties that you want your notion of distance to have as well, like the triangle inequality, but before we start worrying about those, let's start imagining what notion of distance could possibly make powers of two approach zero, and which is shift inva", "translatedText": "Existem outras propriedades que você deseja que sua noção de distância também tenha, como a noção de distância que poderia fazer com que potências de 2 se aproximassem de 0 e mudassem invariantes.", "model": "google_nmt", "from_community_srt": "Vamos chamar essa propriedade de \"invariância de deslocamento\". Há outras propriedades que você quer que sua noção de distância tenha também, como a desigualdade triangular, mas antes que nos preocupemos com isso, vamos começar a imaginar que noção de distância poderia possivelmente fazer potências de 2 se aproximar de 0, e qual possui \"invariância de deslocamento\".", @@ -592,7 +592,7 @@ "end": 639.4 }, { - "input": "You think of 0 as being in the same room as all of the powers of 2 greater than 1, as being in the same sub-room as all powers of 2 greater than 2, as being in the same sub-sub-room as powers of 2 greater than 4, and so on, with infinitely many smaller and smaller rooms.", + "input": "You think of zero as being in the same room as all of the powers of two greater than one. As being in the same sub-room as all powers of two greater than two. As being in the same sub-sub-room as powers of two greater than four, and so on, with infinitely many smaller and smaller rooms.", "translatedText": "Você pensa em 0 como estando na mesma sala que todas as potências de 2 maiores que 1, como estando na mesma subsala que todas as potências de 2 maiores que 2, como estando na mesma subsala das potências de 2 maior que 4, e assim por diante, com infinitas salas cada vez menores.", "model": "google_nmt", "from_community_srt": "e assim por diante. Você pensa no 0 estando no mesmo quarto de todas as potências de 2 maiores que 1, estando no mesmo subquarto de todas as potências de 2 maiores que 2, e estando no mesmo sub-sub-quarto de todas as potências maiores que 4, e assim por diante,", @@ -619,7 +619,7 @@ "end": 677.46 }, { - "input": "For instance, 1 should be as far away from 3 as 2 is from 0.", + "input": "For instance, one should be as far away from three as two is from zero.", "translatedText": "Por exemplo, 1 deve estar tão longe de 3 quanto 2 está de 0.", "model": "google_nmt", "from_community_srt": "Por exemplo, 1 deve ser tão distante de 3 assim como 2 é de 0.", @@ -628,7 +628,7 @@ "end": 682.28 }, { - "input": "Likewise, the distance between 0 and 4 should be the same as that between 1 and 5, 2 and 6, and 3 and 7.", + "input": "Likewise, the distance between zero and four should be the same as that between one and five, two and six, and three and seven.", "translatedText": "Da mesma forma, a distância entre 0 e 4 deve ser a mesma entre 1 e 5, 2 e 6 e 3 e 7.", "model": "google_nmt", "from_community_srt": "Da mesma maneira, a distância entre 0 e 4 deve ser a mesma entre 1 e 5,", @@ -655,7 +655,7 @@ "end": 706.9 }, { - "input": "For example, negative 1 has to be in the same room as 1, in the same sub-room as 3, in the same sub-sub-room as 7, and so on, always in smaller and smaller rooms with numbers 1 less than a power of 2, because 0 is in smaller and smaller rooms with the powers of 2.", + "input": "For example, negative one has to be in the same room as one, in the same sub-room as three, the same sub-sub-room as seven, and so on, always in smaller and smaller rooms with numbers one less than a power of two, because zero is in smaller and smaller rooms with the powers of two.", "translatedText": "Por exemplo, o negativo 1 tem que estar na mesma sala que 1, na mesma sub-sala que 3, na mesma sub-sub-sala que 7, e assim por diante, sempre em salas cada vez menores com números 1 menores que uma potência de 2, porque 0 está em salas cada vez menores com potências de 2.", "model": "google_nmt", "from_community_srt": "onde, por exemplo, -1 tem que estar no mesmo quarto que 1, no mesmo sub-quarto que 3, no mesmo sub-sub-quarto que 7, e assim por diante, sempre em quartos menores e menores, com números com uma potência de 2 menos 1, pois o 0 está nos quartos menores e menores,", @@ -673,7 +673,7 @@ "end": 734.4 }, { - "input": "You can't take this drawing too literally, since it makes 1 look very close to 14 and 0 very far from 13, even though shift invariance should imply that they're the same distance away.", + "input": "You can't take this drawing too literally, since it makes one look very close to fourteen and zero very far from thirteen, even though shift invariance should imply that they're the same distance away.", "translatedText": "Você não pode interpretar esse desenho muito literalmente, pois faz com que 1 pareça muito próximo de 14 e 0 muito distante de 13, embora a invariância de deslocamento deva implicar que eles estão à mesma distância.", "model": "google_nmt", "from_community_srt": "Você não pode encarar esse desenho tão literalmente, já que parece que o 1 está bem perto do 14, e o 0 bem longe do 13, mesmo que a invariância de deslocamento implique que eles se encontram à mesma distância.", @@ -727,7 +727,7 @@ "end": 795.78 }, { - "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers, like ⅓ and ½, should fall into.", + "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers like one third and one half should fall into,", "translatedText": "Não faremos isso neste vídeo, mas veja se você consegue raciocinar sobre em quais salas outros números racionais, como ⅓ e ½, deveriam cair.", "model": "google_nmt", "from_community_srt": "Nós não vamos fazer isso nesse vídeo, mas veja se você consegue pensar sobre quais quartos outros números racionais como 1/3 e 1/2 devem cair dentro,", diff --git a/2015/inventing-math/russian/sentence_translations.json b/2015/inventing-math/russian/sentence_translations.json index fd24e3cb0..38f704ef5 100644 --- a/2015/inventing-math/russian/sentence_translations.json +++ b/2015/inventing-math/russian/sentence_translations.json @@ -216,7 +216,7 @@ "end": 209.04 }, { - "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, 1 one-hundredth, 1 one-millionth, or 1 over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny distance of 1.", + "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, one one hundredth, one one millionth, or one over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny tiny distance of 1.", "translatedText": "Подумав об этом, ты понимаешь, что особенность 1 заключается в том, что твои числа могут произвольно приближаться к 1, то есть неважно, насколько мало желаемое расстояние, 1 сотая, 1 миллионная или 1 больше самого большого числа, которое ты можешь записать, если ты будешь идти по своему списку достаточно долго, то в конечном итоге числа будут попадать в это крошечное расстояние, равное 1.", "model": "DeepL", "from_community_srt": "После размышления вы осознаете, что 1 - особенное число, потому что ваши числа могут подойти сколь угодно близко к 1. То есть, как бы вы близко не хотели оказаться: на расстоянии одной сотой, одной милионной, или единицы, деленной на самое большое число, которое можно вообразить, если мы пройдем по нашему списку достаточно далеко,", @@ -279,7 +279,7 @@ "end": 288.76 }, { - "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size ½, ¼, etc., you could have chosen a proportion other than ½.", + "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size one half, one fourth, etc., you could have chosen a proportion other than one half.", "translatedText": "Например, когда ты сокращал расстояние между объектами, разрезая интервал на отрезки размером ½, ¼ и т.д., ты мог выбрать пропорцию, отличную от ½.", "model": "DeepL", "from_community_srt": "Например, когда вы сжимали расстояние между объектами, разрезая интервал на кусочки длиной ½, ¼, и так далее, вы могли выбрать не ½,", @@ -297,7 +297,7 @@ "end": 315.82 }, { - "input": "Continuing on and on, you'd see that 9 tenths plus 9 one hundredths plus 9 one thousandths on and on up to infinity equals 1, a fact more popularly written as 0.9 repeating equals 1.", + "input": "Continuing on and on, you'd see that nine tenths plus nine one hundredths plus nine one thousandths on and on up to infinity equals one, a fact more popularly written as point nine repeating equals one.", "translatedText": "Продолжая дальше, ты увидишь, что 9 десятых плюс 9 сотых плюс 9 тысячных и так далее до бесконечности равняется 1. Этот факт более популярно записывать как 0,9 повторения равняется 1.", "model": "DeepL", "from_community_srt": "Продолжая с том же духе, вы увидите что 9/10+9/100+9/1000 и так далее до бесконечности равняется 1. Этот факт чаще записывается как 0,999999...", @@ -315,7 +315,7 @@ "end": 338.58 }, { - "input": "To be general about it, let's say that you cut your interval into pieces of size p and 1-p, where p represents any number between 0 and 1.", + "input": "To be general about it, let's say that you cut your interval into pieces of size p and one minus p, where p represents any number between zero and one.", "translatedText": "Если говорить обобщенно, то допустим, что ты разрезал свой интервал на части размером p и 1-p, где p представляет собой любое число от 0 до 1.", "model": "DeepL", "from_community_srt": "Еще более общими словами, пусть вы разрезали интервал на части размера p и (1-p), где p - любое число между 0 и 1.", @@ -333,7 +333,7 @@ "end": 356.78 }, { - "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that 1-p plus p times 1-p plus p squared times 1-p on and on always adding p to the next power times 1-p equals 1.", + "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that one minus p plus p times one minus p plus p squared times one minus p, on and on always adding p to the next power times one minus p, equals one.", "translatedText": "Продолжая в том же духе, всегда разрезая крайний правый кусок на те же пропорции, ты обнаружишь, что 1-p плюс p, умноженное на 1-p, плюс p в квадрате, умноженное на 1-p, и так далее, всегда добавляя p до следующей степени, умноженное на 1-p, равно 1.", "model": "DeepL", "from_community_srt": "вы получаете части с размерами p(1-p) и p^2. Продолжая в том же духе, каждый раз разрезая крайнюю правую часть в той же пропорции, вы обнаружите, что (1-p) + p(1-p) + p^2(1-p) и так далее, p в следующей степени на (1-p), равняется 1.", @@ -368,7 +368,7 @@ "end": 398.4 }, { - "input": "For instance, plugging in negative 1, the equation reads 1 minus 1 plus 1 minus 1 on and on forever alternating between the two, equals one half, which feels both silly and kind of like the only thing it could be.", + "input": "For instance, plugging in negative one, the equation reads one minus one plus one minus one, on and on forever alternating between the two, equals one half, which feels pretty silly and kind of like the only thing it could be.", "translatedText": "Например, если подставить отрицательную 1, то уравнение будет выглядеть так: 1 минус 1 плюс 1 минус 1 и так далее, вечно чередуя эти два значения, равняется половине, что кажется одновременно глупым и вроде как единственным, чем это может быть.", "model": "DeepL", "from_community_srt": "кроме пожалуй p = 1. Например, для p = -1, получается что 1-1+1-1+1-1+... до бесконечности равняется ½, что с одной стороны как-то глупо, но с другой, и правда, скорее ½,", @@ -377,7 +377,7 @@ "end": 417.86 }, { - "input": "Plugging in 2, the equation reads 1 plus 2 plus 4 plus 8 on and on to infinity equals negative 1, something which doesn't even seem reasonable.", + "input": "Plugging in two, the equation reads one plus two plus four plus eight, on and on to infinity, equals negative one, something which doesn't even seem reasonable.", "translatedText": "Подставляя 2, уравнение читается так: 1 плюс 2 плюс 4 плюс 8 и так далее до бесконечности равняется отрицательной 1, что даже не кажется разумным.", "model": "DeepL", "from_community_srt": "чем что-либо другое. Для p = 2, уравнение говорит, что 1 + 2 + 4 + 8 + ... = -1, это уже вообще не получается как-то оправдать.", @@ -431,7 +431,7 @@ "end": 467.62 }, { - "input": "1, 3, 7, 15, 31, they're all 1 less than a power of 2.", + "input": "One, three, seven, fifteen, thirty-one, they're all one less than a power of two.", "translatedText": "1, 3, 7, 15, 31 - все они на 1 меньше силы 2.", "model": "DeepL", "from_community_srt": "3, 7, 15, 31. Все они на 1 меньше степени двойки.", @@ -530,7 +530,7 @@ "end": 562.68 }, { - "input": "You could come up with a completely random notion of distance, where 2 is 7 away from 3, and ½ is 4 fifths away from 100, and all sorts of things, but if you want to actually use a new distance function the way you use the familiar distance function, it should share some of the same properties.", + "input": "You could come up with a completely random notion of distance, where two is seven away from three, and one half is four fifths away from a hundred, and all sorts of things. But if you want to actually use a new distance function the way that you use the familiar distance function, it should share some of the same properties.", "translatedText": "Ты можешь придумать совершенно произвольное понятие расстояния, где 2 отстоит на 7 от 3, а ½ - на 4 пятых от 100, и все в таком духе, но если ты хочешь использовать новую функцию расстояния так же, как ты используешь привычную функцию расстояния, она должна обладать некоторыми общими свойствами.", "model": "DeepL", "from_community_srt": "Можно например определять расстояние случайным образом, например 2 будет на расстоянии 7 от 3, а 1/2 на расстоянии 4/5 от 100, и вообще что угодно, но если мы хотим как-то использовать это новое расстояние, так же как наше привычное, у них должны быть некоторые общие свойства.", @@ -548,7 +548,7 @@ "end": 587.48 }, { - "input": "So 0 and 4 should be the same distance away as 1 and 5, or 2 and 6, even if that same distance is something other than 4 as we're used to.", + "input": "So zero and four should be the same distance away as one and five, or two and six, even if that same distance is something other than four as we're used to.", "translatedText": "Так что 0 и 4 должны находиться на том же расстоянии, что и 1 и 5, или 2 и 6, даже если это расстояние не 4, как мы привыкли.", "model": "DeepL", "from_community_srt": "Расстояние между 0 и 4 должно быть тем же, что между 1 и 5 или 2 и 6, даже если это расстояние - не наше привычное 4,", @@ -574,7 +574,7 @@ "end": 607.24 }, { - "input": "There are other properties that you want your notion of distance to have as well, like the notion of distance could possibly make powers of 2 approach 0, and shift invariant.", + "input": "There are other properties that you want your notion of distance to have as well, like the triangle inequality, but before we start worrying about those, let's start imagining what notion of distance could possibly make powers of two approach zero, and which is shift inva", "translatedText": "Есть и другие свойства, которыми ты хочешь, чтобы обладало твое понятие расстояния: например, понятие расстояния может приближать силу 2 к 0 и инвариантно к сдвигу.", "model": "DeepL", "from_community_srt": "Назовем это свойство \"неизменность относительно сдвига\". Среди других свойств, которые должно иметь расстояние будет например неравенство треугольника, но прежде чем заботиться от этом, придумаем такое определеие расстояния, в котором степени двойки действительно будут стремиться к 0, и будет сохраняться неизменность относительно сдвига.", @@ -592,7 +592,7 @@ "end": 639.4 }, { - "input": "You think of 0 as being in the same room as all of the powers of 2 greater than 1, as being in the same sub-room as all powers of 2 greater than 2, as being in the same sub-sub-room as powers of 2 greater than 4, and so on, with infinitely many smaller and smaller rooms.", + "input": "You think of zero as being in the same room as all of the powers of two greater than one. As being in the same sub-room as all powers of two greater than two. As being in the same sub-sub-room as powers of two greater than four, and so on, with infinitely many smaller and smaller rooms.", "translatedText": "Ты считаешь, что 0 находится в той же комнате, что и все силы 2 больше 1, что он находится в той же подкомнате, что и все силы 2 больше 2, что он находится в той же под-комнате, что и силы 2 больше 4, и так далее, с бесконечно большим количеством меньших и меньших комнат.", "model": "DeepL", "from_community_srt": "\"под-под-комнатам\" и так далее. Будем думать, что 0 находится в той же комнате, где и все степени двойки, большие чем 1, в той же под-комнате, в которой все степени двойки, большие 2-х, в той же под-под-комнате, где и все степени двойки большие 4-х, и так далее,", @@ -619,7 +619,7 @@ "end": 677.46 }, { - "input": "For instance, 1 should be as far away from 3 as 2 is from 0.", + "input": "For instance, one should be as far away from three as two is from zero.", "translatedText": "Например, 1 должна находиться на таком же расстоянии от 3, как 2 от 0.", "model": "DeepL", "from_community_srt": "1 должен быть на том же расстоянии от 3, как и 2 от 0.", @@ -628,7 +628,7 @@ "end": 682.28 }, { - "input": "Likewise, the distance between 0 and 4 should be the same as that between 1 and 5, 2 and 6, and 3 and 7.", + "input": "Likewise, the distance between zero and four should be the same as that between one and five, two and six, and three and seven.", "translatedText": "Точно так же расстояние между 0 и 4 должно быть таким же, как между 1 и 5, 2 и 6, 3 и 7.", "model": "DeepL", "from_community_srt": "Дистанция между 0 и 4 в свою очередь будет такой же, как между 1 и 5, 2 и 6,", @@ -655,7 +655,7 @@ "end": 706.9 }, { - "input": "For example, negative 1 has to be in the same room as 1, in the same sub-room as 3, in the same sub-sub-room as 7, and so on, always in smaller and smaller rooms with numbers 1 less than a power of 2, because 0 is in smaller and smaller rooms with the powers of 2.", + "input": "For example, negative one has to be in the same room as one, in the same sub-room as three, the same sub-sub-room as seven, and so on, always in smaller and smaller rooms with numbers one less than a power of two, because zero is in smaller and smaller rooms with the powers of two.", "translatedText": "Например, отрицательная 1 должна находиться в той же комнате, что и 1, в той же подкомнате, что и 3, в той же подкомнате, что и 7, и так далее, всегда в меньших и меньших комнатах с числами на 1 меньше силы 2, потому что 0 находится в меньших и меньших комнатах с силами 2.", "model": "DeepL", "from_community_srt": "где должно быть например -1 в той же комнате, что и 1, в под-комнате вместе с 3, в под-под-комнате с 7 и так далее всегда в комнате с числами на 1 меньшими степени двойки, потому что 0 попадает в одну и ту же комнату с самими степенями двойки.", @@ -673,7 +673,7 @@ "end": 734.4 }, { - "input": "You can't take this drawing too literally, since it makes 1 look very close to 14 and 0 very far from 13, even though shift invariance should imply that they're the same distance away.", + "input": "You can't take this drawing too literally, since it makes one look very close to fourteen and zero very far from thirteen, even though shift invariance should imply that they're the same distance away.", "translatedText": "Нельзя воспринимать этот рисунок слишком буквально, так как из-за него 1 выглядит очень близко к 14, а 0 - очень далеко от 13, хотя инвариантность сдвига должна подразумевать, что они находятся на одинаковом расстоянии.", "model": "DeepL", "from_community_srt": "основанной на комнатах в настоящую функцию расстояния? нельзя воспринимать этот рисунок слишком буквально, потому что может показаться, что 1 очень близко к 14, а 0 далеко от 13, хотя это расстояние должно быть одним и тем же. очень близко к 14,", @@ -727,7 +727,7 @@ "end": 795.78 }, { - "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers, like ⅓ and ½, should fall into.", + "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers like one third and one half should fall into,", "translatedText": "В этом видео мы этого делать не будем, но посмотри, сможешь ли ты рассуждать о том, в какие комнаты должны попасть другие рациональные числа, например ⅓ и ½.", "model": "DeepL", "from_community_srt": "В этом видео мы не будем говорить о том, где найти место для рациональных чисел, таких как 1/3 и 1/2, а так же не будем убеждаться в том,", diff --git a/2015/inventing-math/spanish/sentence_translations.json b/2015/inventing-math/spanish/sentence_translations.json index 33ab5060e..dec9e6591 100644 --- a/2015/inventing-math/spanish/sentence_translations.json +++ b/2015/inventing-math/spanish/sentence_translations.json @@ -216,7 +216,7 @@ "end": 209.04 }, { - "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, 1 one-hundredth, 1 one-millionth, or 1 over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny distance of 1.", + "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, one one hundredth, one one millionth, or one over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny tiny distance of 1.", "translatedText": "Después de pensar en ello, te das cuenta de que lo que hace especial al 1 es que tus números pueden acercarse arbitrariamente al 1, es decir, por pequeña que sea la distancia que desees, 1 centésima, 1 millonésima o 1 sobre el número más grande que puedas escribir, si recorres tu lista el tiempo suficiente, los números acabarán cayendo dentro de esa pequeña distancia del 1.", "model": "DeepL", "from_community_srt": "Tras pensar acerca de ello, descubres que lo que hace que el 1 sea especial es que tus números pueden estar arbitrariamente cerca de 1. Lo que quiere decir que, no importa cómo de pequeña sea la distancia deseada, una centésima, una millonésima, o 1 entre el mayor número que puedas escribir, si avanzas lo suficiente en la lista los números se situarán al final dentro de esa distancia de 1.", @@ -279,7 +279,7 @@ "end": 288.76 }, { - "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size ½, ¼, etc., you could have chosen a proportion other than ½.", + "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size one half, one fourth, etc., you could have chosen a proportion other than one half.", "translatedText": "Por ejemplo, al reducir la distancia entre tus objetos, cortando el intervalo en trozos de tamaño ½, ¼, etc., podrías haber elegido una proporción distinta de ½.", "model": "DeepL", "from_community_srt": "Por ejemplo, cuando disminuías la distancia entre objetos, dividiendo el intervalo en trozos de tamaño 1/2, 1/4, etc., podías haber escogido otra proporción distinta a 1/2.", @@ -297,7 +297,7 @@ "end": 315.82 }, { - "input": "Continuing on and on, you'd see that 9 tenths plus 9 one hundredths plus 9 one thousandths on and on up to infinity equals 1, a fact more popularly written as 0.9 repeating equals 1.", + "input": "Continuing on and on, you'd see that nine tenths plus nine one hundredths plus nine one thousandths on and on up to infinity equals one, a fact more popularly written as point nine repeating equals one.", "translatedText": "Siguiendo y siguiendo, verías que 9 décimas más 9 centésimas más 9 milésimas y así sucesivamente hasta el infinito es igual a 1, hecho que se escribe más popularmente como 0,9 que se repite es igual a 1.", "model": "DeepL", "from_community_srt": "siguiendo así, llegarías a que 9/10 + 9/100 + 9/1000 y así hasta infinito es igual a 1, lo que popularmente se escribe como 0.9 periodo =1.", @@ -315,7 +315,7 @@ "end": 338.58 }, { - "input": "To be general about it, let's say that you cut your interval into pieces of size p and 1-p, where p represents any number between 0 and 1.", + "input": "To be general about it, let's say that you cut your interval into pieces of size p and one minus p, where p represents any number between zero and one.", "translatedText": "Para generalizar, digamos que cortas tu intervalo en trozos de tamaño p y 1-p, donde p representa cualquier número entre 0 y 1.", "model": "DeepL", "from_community_srt": "Para ser más generales, digamos que divides el intervalo en trozos de tamaño p y (1-p), donde p representa cualquier número entre 0 y 1.", @@ -333,7 +333,7 @@ "end": 356.78 }, { - "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that 1-p plus p times 1-p plus p squared times 1-p on and on always adding p to the next power times 1-p equals 1.", + "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that one minus p plus p times one minus p plus p squared times one minus p, on and on always adding p to the next power times one minus p, equals one.", "translatedText": "Siguiendo así, cortando siempre el trozo de la derecha en esas mismas proporciones, verás que 1-p más p por 1-p más p al cuadrado por 1-p y así sucesivamente, siempre sumando p a la siguiente potencia por 1-p es igual a 1.", "model": "DeepL", "from_community_srt": "obtenemos ahora trozos de tamaño p(1-p) y p^2. Siguiendo de esta forma, dividiendo siempre el fragmento de la derecha en estas proporciones, verás que (1-p) + p(1-p) + p^2(1-p), y así, añadiendo siempre p a la siguiente potencia multiplicado por (1-p),", @@ -369,7 +369,7 @@ "end": 398.4 }, { - "input": "For instance, plugging in negative 1, the equation reads 1 minus 1 plus 1 minus 1 on and on forever alternating between the two, equals one half, which feels both silly and kind of like the only thing it could be.", + "input": "For instance, plugging in negative one, the equation reads one minus one plus one minus one, on and on forever alternating between the two, equals one half, which feels pretty silly and kind of like the only thing it could be.", "translatedText": "Por ejemplo, al introducir un 1 negativo, la ecuación dice 1 menos 1 más 1 menos 1 y así sucesivamente, alternando siempre entre los dos, es igual a la mitad, lo que parece a la vez una tontería y la única cosa que podría ser.", "model": "DeepL", "from_community_srt": "salvo tal vez por 1. Por ejemplo, poniendo -1, la ecuación dice que 1-1+1-1, así hasta el infinito, alternando entre los dos, es igual a 1/2, lo que parece a la vez una estupidez y el único valor que tendría sentido.", @@ -378,7 +378,7 @@ "end": 417.86 }, { - "input": "Plugging in 2, the equation reads 1 plus 2 plus 4 plus 8 on and on to infinity equals negative 1, something which doesn't even seem reasonable.", + "input": "Plugging in two, the equation reads one plus two plus four plus eight, on and on to infinity, equals negative one, something which doesn't even seem reasonable.", "translatedText": "Introduciendo 2, la ecuación dice que 1 más 2 más 4 más 8 y así hasta el infinito es igual a 1 negativo, algo que ni siquiera parece razonable.", "model": "DeepL", "from_community_srt": "Poniendo 2, la ecuación dice que 1 + 2 + 4 + 8, y así hasta el infinito, es igual a -1, lo que ni siquiera parece razonable.", @@ -431,7 +431,7 @@ "end": 467.62 }, { - "input": "1, 3, 7, 15, 31, they're all 1 less than a power of 2.", + "input": "One, three, seven, fifteen, thirty-one, they're all one less than a power of two.", "translatedText": "1, 3, 7, 15, 31, todos son 1 menos que una potencia de 2.", "model": "DeepL", "from_community_srt": "3, 7, 15, 31. Todos son una potencia de 2 menos 1.", @@ -530,7 +530,7 @@ "end": 562.68 }, { - "input": "You could come up with a completely random notion of distance, where 2 is 7 away from 3, and ½ is 4 fifths away from 100, and all sorts of things, but if you want to actually use a new distance function the way you use the familiar distance function, it should share some of the same properties.", + "input": "You could come up with a completely random notion of distance, where two is seven away from three, and one half is four fifths away from a hundred, and all sorts of things. But if you want to actually use a new distance function the way that you use the familiar distance function, it should share some of the same properties.", "translatedText": "Podrías inventar una noción de distancia completamente aleatoria, en la que 2 esté a 7 de 3, y ½ esté a 4 quintos de 100, y todo tipo de cosas, pero si quieres utilizar realmente una nueva función de distancia del mismo modo que utilizas la función de distancia conocida, debería compartir algunas de las mismas propiedades.", "model": "DeepL", "from_community_srt": "Podrías inventar una noción de distancia completamente aleatoria, donde 2 está a distancia 7 de 3 ,y 1/2 está a distancia 4/5 de 100, y todo tipo de cosas, pero si lo que quieres es usar una nueva función distancia de la forma que usas la función distancia normal, deberían tener algunas propiedades en común.", @@ -548,7 +548,7 @@ "end": 587.48 }, { - "input": "So 0 and 4 should be the same distance away as 1 and 5, or 2 and 6, even if that same distance is something other than 4 as we're used to.", + "input": "So zero and four should be the same distance away as one and five, or two and six, even if that same distance is something other than four as we're used to.", "translatedText": "Por tanto, 0 y 4 deben estar a la misma distancia que 1 y 5, o 2 y 6, aunque esa misma distancia sea algo distinto de 4, como estamos acostumbrados.", "model": "DeepL", "from_community_srt": "De 0 a 4 debería haber la misma distancia que de 1 a 5, o 2 y 6, incluso si esa misma distancia es distinta de 4,", @@ -574,7 +574,7 @@ "end": 607.24 }, { - "input": "There are other properties that you want your notion of distance to have as well, like the notion of distance could possibly make powers of 2 approach 0, and shift invariant.", + "input": "There are other properties that you want your notion of distance to have as well, like the triangle inequality, but before we start worrying about those, let's start imagining what notion of distance could possibly make powers of two approach zero, and which is shift inva", "translatedText": "Hay otras propiedades que también quieres que tenga tu noción de distancia, como que la noción de distancia pueda hacer que las potencias de 2 se aproximen a 0, y que sea invariante de desplazamiento.", "model": "DeepL", "from_community_srt": "Llamemos a esta propiedad \"invariancia por traslación\". Hay otras propiedades que te gustaría que tuviera tu noción de distancia, como la desigualdad triangular, pero antes de empezar a preocuparnos por éstas, empecemos imaginando qué noción de distancia podría hacer que las potencias de 2 se acercaran a 0, y que sea invariante por traslación.", @@ -592,7 +592,7 @@ "end": 639.4 }, { - "input": "You think of 0 as being in the same room as all of the powers of 2 greater than 1, as being in the same sub-room as all powers of 2 greater than 2, as being in the same sub-sub-room as powers of 2 greater than 4, and so on, with infinitely many smaller and smaller rooms.", + "input": "You think of zero as being in the same room as all of the powers of two greater than one. As being in the same sub-room as all powers of two greater than two. As being in the same sub-sub-room as powers of two greater than four, and so on, with infinitely many smaller and smaller rooms.", "translatedText": "Piensa que el 0 está en la misma sala que todas las potencias de 2 mayores que 1, que está en la misma subsala que todas las potencias de 2 mayores que 2, que está en la misma subsala que las potencias de 2 mayores que 4, y así sucesivamente, con infinitas salas cada vez más pequeñas.", "model": "DeepL", "from_community_srt": "y así. Verías al 0 en la misma habitación que todas las potencias de 2 mayores que 1, en la misma sub-habitación que todas las potencias de 2 mayores que 2, en la misma sub-sub-habitación que las potencias de 2 mayores que 4, y así, con un número infinito de habitaciones más y más pequeñas.", @@ -619,7 +619,7 @@ "end": 677.46 }, { - "input": "For instance, 1 should be as far away from 3 as 2 is from 0.", + "input": "For instance, one should be as far away from three as two is from zero.", "translatedText": "Por ejemplo, 1 debe estar tan lejos de 3 como 2 lo está de 0.", "model": "DeepL", "from_community_srt": "Por ejemplo, el 1 debería debería estar tan lejos del 3 como el 2 del 0.", @@ -628,7 +628,7 @@ "end": 682.28 }, { - "input": "Likewise, the distance between 0 and 4 should be the same as that between 1 and 5, 2 and 6, and 3 and 7.", + "input": "Likewise, the distance between zero and four should be the same as that between one and five, two and six, and three and seven.", "translatedText": "Del mismo modo, la distancia entre 0 y 4 debe ser la misma que entre 1 y 5, 2 y 6, y 3 y 7.", "model": "DeepL", "from_community_srt": "De la misma forma la distancia entre el 0 y el 4 debería ser la misma que entre el 1 y el 5, el 2 y el 6, y el 3 y el 7.", @@ -655,7 +655,7 @@ "end": 706.9 }, { - "input": "For example, negative 1 has to be in the same room as 1, in the same sub-room as 3, in the same sub-sub-room as 7, and so on, always in smaller and smaller rooms with numbers 1 less than a power of 2, because 0 is in smaller and smaller rooms with the powers of 2.", + "input": "For example, negative one has to be in the same room as one, in the same sub-room as three, the same sub-sub-room as seven, and so on, always in smaller and smaller rooms with numbers one less than a power of two, because zero is in smaller and smaller rooms with the powers of two.", "translatedText": "Por ejemplo, el 1 negativo tiene que estar en la misma habitación que el 1, en la misma subsala que el 3, en la misma subsala que el 7, y así sucesivamente, siempre en habitaciones cada vez más pequeñas con números 1 menores que una potencia de 2, porque el 0 está en habitaciones cada vez más pequeñas con las potencias de 2.", "model": "DeepL", "from_community_srt": "como por ejemplo el -1, que tiene que estar en la misma habitación que el 1, en la misma sub-habitación que el 3, la misma sub-sub-habitación que el 7, y así, siempre en habitaciones más y más pequeñas junto a potencias de 2 menos 1, puesto que el 0 está en habitaciones más y más pequeñas junto a potencias de 2.", @@ -673,7 +673,7 @@ "end": 734.4 }, { - "input": "You can't take this drawing too literally, since it makes 1 look very close to 14 and 0 very far from 13, even though shift invariance should imply that they're the same distance away.", + "input": "You can't take this drawing too literally, since it makes one look very close to fourteen and zero very far from thirteen, even though shift invariance should imply that they're the same distance away.", "translatedText": "No puedes tomarte este dibujo demasiado al pie de la letra, ya que hace que el 1 parezca estar muy cerca del 14 y el 0 muy lejos del 13, aunque la invarianza de desplazamiento debería implicar que están a la misma distancia.", "model": "DeepL", "from_community_srt": "No puedes considerar este dibujo demasiado literalmente, puesto que hace que el 1 parezca muy cerca del 14, y el 0 muy lejos del 13, a pesar de que la invariancia por traslación debería implicar que están separados en la misma distancia.", @@ -727,7 +727,7 @@ "end": 795.78 }, { - "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers, like ⅓ and ½, should fall into.", + "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers like one third and one half should fall into,", "translatedText": "No lo haremos en este vídeo, pero a ver si eres capaz de razonar sobre en qué habitaciones deberían caer otros números racionales, como ⅓ y ½.", "model": "DeepL", "from_community_srt": "No lo haremos en este vídeo, pero intenta razonar en qué habitaciones se situarían otros números racionales como el 1/3 y 1/2,", diff --git a/2015/inventing-math/turkish/sentence_translations.json b/2015/inventing-math/turkish/sentence_translations.json index 939534cbc..2b80aeff2 100644 --- a/2015/inventing-math/turkish/sentence_translations.json +++ b/2015/inventing-math/turkish/sentence_translations.json @@ -216,7 +216,7 @@ "end": 209.04 }, { - "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, 1 one-hundredth, 1 one-millionth, or 1 over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny distance of 1.", + "input": "After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1, which is to say, no matter how small your desired distance, one one hundredth, one one millionth, or one over the largest number you could write down, if you go down your list long enough, the numbers will eventually fall within that tiny tiny distance of 1.", "translatedText": "Bunu düşündükten sonra, 1'i özel kılan şeyin, sayılarınızın 1'e keyfi olarak yaklaşabilmesi olduğunu fark edersiniz; yani, istediğiniz mesafe ne kadar küçük olursa olsun, yüzde 1, milyonda 1 veya yazabileceğiniz en büyük sayının 1 fazlası, listenizde yeterince uzun süre aşağı inerseniz, sayılar sonunda 1'in o küçük mesafesine düşecektir.", "model": "DeepL", "from_community_srt": "Biraz düşününce, 1'i özel yapan şeyin, sayılarınızın keyfi olarak 1'e yaklaşabilmesidir. Demem o ki, istediğiniz mesafe ne kadar küçük olursa olsun, 1/100'üncü, 1/1,000,000'inci yada alta yazabileceğiniz en büyük sayı, eğer listenin sonuna yeterince yaklaşırsanız, sayılar ile 1'in arasında küçücük mesafe kalacaktır.", @@ -279,7 +279,7 @@ "end": 288.76 }, { - "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size ½, ¼, etc., you could have chosen a proportion other than ½.", + "input": "For instance, when you were shrinking the distance between your objects, cutting the interval into pieces of size one half, one fourth, etc., you could have chosen a proportion other than one half.", "translatedText": "Örneğin, nesneleriniz arasındaki mesafeyi küçültürken, aralığı ½, ¼ vb. boyutlarda parçalara bölerken, ½ dışında bir oran seçebilirdiniz.", "model": "DeepL", "from_community_srt": "Mesela, iki objenin arasındaki mesafeyi kısalttığınızda, her aralığı yarısı kadar, yani ½, ¼ oranında kestiğinizde, ½'den farklı bir oran seçebilirdiniz.", @@ -297,7 +297,7 @@ "end": 315.82 }, { - "input": "Continuing on and on, you'd see that 9 tenths plus 9 one hundredths plus 9 one thousandths on and on up to infinity equals 1, a fact more popularly written as 0.9 repeating equals 1.", + "input": "Continuing on and on, you'd see that nine tenths plus nine one hundredths plus nine one thousandths on and on up to infinity equals one, a fact more popularly written as point nine repeating equals one.", "translatedText": "Devam ederseniz, 9 onda bir artı 9 yüzde bir artı 9 binde birin sonsuza kadar 1'e eşit olduğunu görürsünüz; bu gerçek daha popüler olarak 0,9 tekrar 1'e eşittir şeklinde yazılır.", "model": "DeepL", "from_community_srt": "9/10 + 9/100 + 9/1000... şeklinde devam eden sonsuz toplamın 1'e eşit olduğunu, daha bilinen kullanımıyla 0.99999...=1 olduğunu görürdünüz.", @@ -315,7 +315,7 @@ "end": 338.58 }, { - "input": "To be general about it, let's say that you cut your interval into pieces of size p and 1-p, where p represents any number between 0 and 1.", + "input": "To be general about it, let's say that you cut your interval into pieces of size p and one minus p, where p represents any number between zero and one.", "translatedText": "Bu konuda genel olmak gerekirse, diyelim ki aralığınızı p ve 1-p boyutunda parçalara böldünüz, burada p 0 ile 1 arasındaki herhangi bir sayıyı temsil eder.", "model": "DeepL", "from_community_srt": "Diyelim ki mesafeyi p ve (1-p) uzunluğu kadar ayıracaksınız, p, 0 ile 1 arasındaki herhangi bir sayı olabilir.", @@ -333,7 +333,7 @@ "end": 356.78 }, { - "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that 1-p plus p times 1-p plus p squared times 1-p on and on always adding p to the next power times 1-p equals 1.", + "input": "Continuing in this fashion, always cutting up the rightmost piece into those same proportions, you'll find that one minus p plus p times one minus p plus p squared times one minus p, on and on always adding p to the next power times one minus p, equals one.", "translatedText": "Bu şekilde devam ederek, her zaman en sağdaki parçayı aynı oranlarda keserek, 1-p artı p çarpı 1-p artı p kare çarpı 1-p ve her zaman bir sonraki güce p ekleyerek çarpı 1-p'nin 1'e eşit olduğunu göreceksiniz.", "model": "DeepL", "from_community_srt": "Sürekli olarak en sağdaki parçayı aynı oranlarda kesmeye devam edersek, (1-p)+p(1-p)+p^2(1-p) ve p'nin üstlerinin (1-p) ile çarpılarak devam ettiğini, ve bunun da 1'e eşit olduğunu görürsünüz.", @@ -369,7 +369,7 @@ "end": 398.4 }, { - "input": "For instance, plugging in negative 1, the equation reads 1 minus 1 plus 1 minus 1 on and on forever alternating between the two, equals one half, which feels both silly and kind of like the only thing it could be.", + "input": "For instance, plugging in negative one, the equation reads one minus one plus one minus one, on and on forever alternating between the two, equals one half, which feels pretty silly and kind of like the only thing it could be.", "translatedText": "Örneğin, eksi 1 girildiğinde, denklem 1 eksi 1 artı 1 eksi 1 şeklinde sonsuza dek ikisi arasında gidip gelir ve bir buçuk eder; bu hem aptalca hem de olabilecek tek şeymiş gibi hissettirir.", "model": "DeepL", "from_community_srt": "Örneğin, -1 koyduğunuz zaman, eşitlik 1-1+1-1 şeklinde sonsuza kadar devam eder ve 1/2'ye eşit olur. İkisi de oldukça saçma ama mümkün olan tek şey gibi görünüyor.", @@ -378,7 +378,7 @@ "end": 417.86 }, { - "input": "Plugging in 2, the equation reads 1 plus 2 plus 4 plus 8 on and on to infinity equals negative 1, something which doesn't even seem reasonable.", + "input": "Plugging in two, the equation reads one plus two plus four plus eight, on and on to infinity, equals negative one, something which doesn't even seem reasonable.", "translatedText": "Denkleme 2 eklendiğinde, 1 artı 2 artı 4 artı 8 ve sonsuza kadar eşittir eksi 1 şeklinde bir denklem ortaya çıkar ki, bu hiç de makul görünmemektedir.", "model": "DeepL", "from_community_srt": "2'yi koyarsanız eğer, eşitlik 1 + 2 + 4 + 8 +... = -1 şeklini alır bu da hiç akla yatkın bir şey değil.", @@ -432,7 +432,7 @@ "end": 467.62 }, { - "input": "1, 3, 7, 15, 31, they're all 1 less than a power of 2.", + "input": "One, three, seven, fifteen, thirty-one, they're all one less than a power of two.", "translatedText": "1, 3, 7, 15, 31, hepsi 2'nin kuvvetinden 1 eksiktir.", "model": "DeepL", "from_community_srt": "3, 7, 15, 31 çıktığını farkedersiniz. Ve bunların hepsi 2'nin üslerinin 1 eksiğidir.", @@ -531,7 +531,7 @@ "end": 562.68 }, { - "input": "You could come up with a completely random notion of distance, where 2 is 7 away from 3, and ½ is 4 fifths away from 100, and all sorts of things, but if you want to actually use a new distance function the way you use the familiar distance function, it should share some of the same properties.", + "input": "You could come up with a completely random notion of distance, where two is seven away from three, and one half is four fifths away from a hundred, and all sorts of things. But if you want to actually use a new distance function the way that you use the familiar distance function, it should share some of the same properties.", "translatedText": "2'nin 3'ten 7 uzakta olduğu ve ½'nin 100'den beşte 4 uzakta olduğu gibi tamamen rastgele bir mesafe kavramı ortaya atabilirsiniz, ancak yeni bir mesafe fonksiyonunu bildiğiniz mesafe fonksiyonunu kullandığınız şekilde kullanmak istiyorsanız, aynı özelliklerden bazılarını paylaşmalıdır.", "model": "DeepL", "from_community_srt": "Rasgele bir mesafe belirleyebilirsiniz. 2'nin 3'ten 7 kadar uzak olduğu, ve ya 1/2 'nin 100'den 4/5 kadar uzak olduğu, ve bunun gibi, ama siz diğerlerine benzer yöntemle, tamamen farklı bir mesafe işlemi kullanmak istiyorsanız, bazı özelliklerin aynı olması gerekir.", @@ -549,7 +549,7 @@ "end": 587.48 }, { - "input": "So 0 and 4 should be the same distance away as 1 and 5, or 2 and 6, even if that same distance is something other than 4 as we're used to.", + "input": "So zero and four should be the same distance away as one and five, or two and six, even if that same distance is something other than four as we're used to.", "translatedText": "Yani 0 ve 4, 1 ve 5 ya da 2 ve 6 ile aynı uzaklıkta olmalıdır, bu aynı uzaklık alıştığımız gibi 4'ten farklı bir şey olsa bile.", "model": "DeepL", "from_community_srt": "0 ve 4, 1 ve 5 ile aynı uzaklıkta olmalı, ya da 2 ve 6 ile, kullandığınız mesafe 4'ten faklı olsa bile.", @@ -575,7 +575,7 @@ "end": 607.24 }, { - "input": "There are other properties that you want your notion of distance to have as well, like the notion of distance could possibly make powers of 2 approach 0, and shift invariant.", + "input": "There are other properties that you want your notion of distance to have as well, like the triangle inequality, but before we start worrying about those, let's start imagining what notion of distance could possibly make powers of two approach zero, and which is shift inva", "translatedText": "Mesafe kavramınızın sahip olmasını istediğiniz başka özellikler de vardır, örneğin mesafe kavramı muhtemelen 2'nin kuvvetlerini 0'a yaklaştırabilir ve kayma değişmez.", "model": "DeepL", "from_community_srt": "İsterseniz buna \"sabit değişim\" diyelim. Belirlediğiniz mesafenin başka özellikleri olmasını da isteyebilirsiniz. örneğin üçgen eşitsizliği gibi, ama bunlar için endişelenmeye başlamadan önce, ne kadarlık bir mesafe ve sabit değişim 2'nin üslerini 0'a yaklaştırabilir,", @@ -593,7 +593,7 @@ "end": 639.4 }, { - "input": "You think of 0 as being in the same room as all of the powers of 2 greater than 1, as being in the same sub-room as all powers of 2 greater than 2, as being in the same sub-sub-room as powers of 2 greater than 4, and so on, with infinitely many smaller and smaller rooms.", + "input": "You think of zero as being in the same room as all of the powers of two greater than one. As being in the same sub-room as all powers of two greater than two. As being in the same sub-sub-room as powers of two greater than four, and so on, with infinitely many smaller and smaller rooms.", "translatedText": "0'ı, 2'nin 1'den büyük tüm kuvvetleriyle aynı odada, 2'nin 2'den büyük tüm kuvvetleriyle aynı alt odada, 2'nin 4'ten büyük kuvvetleriyle aynı alt-alt odada ve sonsuz sayıda daha küçük ve daha küçük odalarla aynı odada olarak düşünürsünüz.", "model": "DeepL", "from_community_srt": "ayırmak aklınıza gelebilir. 0'ın, 1'den büyük olan 2'nin üst kuvvetleriyle aynı oda'da olduğunu, ve bunun \"alt-oda\"sında 2'den büyük olan 2'nin üst kuvvetlerinin olduğunu, ve bunun \"alt-alt-oda\"sında 4'ten büyük olanan 2'nin üst kuvvetlerinin olduğunu ve bu şekilde sonsuza kadar küçülen alt-alt-oda'lar olduğunu düşünün.", @@ -620,7 +620,7 @@ "end": 677.46 }, { - "input": "For instance, 1 should be as far away from 3 as 2 is from 0.", + "input": "For instance, one should be as far away from three as two is from zero.", "translatedText": "Örneğin, 1, 3'ten 2'nin 0'dan olduğu kadar uzakta olmalıdır.", "model": "DeepL", "from_community_srt": "Örneğin, 1 3'ten nekadar uzak olması gerekiyorsa,", @@ -629,7 +629,7 @@ "end": 682.28 }, { - "input": "Likewise, the distance between 0 and 4 should be the same as that between 1 and 5, 2 and 6, and 3 and 7.", + "input": "Likewise, the distance between zero and four should be the same as that between one and five, two and six, and three and seven.", "translatedText": "Aynı şekilde, 0 ile 4 arasındaki mesafe 1 ile 5, 2 ile 6 ve 3 ile 7 arasındaki mesafe ile aynı olmalıdır.", "model": "DeepL", "from_community_srt": "2 de 0'dan o kadar uzak olmalı. Aynı şekilde 0 ile 4'ün arasındaki mesafe de, 1 ile 5'in , 2 ile 6'nın,", @@ -656,7 +656,7 @@ "end": 706.9 }, { - "input": "For example, negative 1 has to be in the same room as 1, in the same sub-room as 3, in the same sub-sub-room as 7, and so on, always in smaller and smaller rooms with numbers 1 less than a power of 2, because 0 is in smaller and smaller rooms with the powers of 2.", + "input": "For example, negative one has to be in the same room as one, in the same sub-room as three, the same sub-sub-room as seven, and so on, always in smaller and smaller rooms with numbers one less than a power of two, because zero is in smaller and smaller rooms with the powers of two.", "translatedText": "Örneğin, negatif 1, 1 ile aynı odada, 3 ile aynı alt odada, 7 ile aynı alt-alt odada ve bu şekilde, her zaman 2'nin kuvvetinden 1 eksik sayılarla daha küçük ve daha küçük odalarda olmalıdır, çünkü 0, 2'nin kuvvetleriyle daha küçük ve daha küçük odalarda bulunur.", "model": "DeepL", "from_community_srt": "Örneğin -1 1 ile aynı odada olmalı, aynı alt-odada 3 ile, aynı alt-alt-odada 7 ile olacak şekilde devam eder. 2'nin kuvvetlerinden bir eksik olan sayılarla her zaman küçülen odalarda, çünkü 0 2'nin kuvvetleriyle sürekli küçülen odalardadır.", @@ -674,7 +674,7 @@ "end": 734.4 }, { - "input": "You can't take this drawing too literally, since it makes 1 look very close to 14 and 0 very far from 13, even though shift invariance should imply that they're the same distance away.", + "input": "You can't take this drawing too literally, since it makes one look very close to fourteen and zero very far from thirteen, even though shift invariance should imply that they're the same distance away.", "translatedText": "Bu çizimi tam anlamıyla alamazsınız, çünkü kayma değişmezliği aynı uzaklıkta olduklarını ima etmesine rağmen 1'i 14'e çok yakın ve 0'ı 13'e çok uzak gösteriyor.", "model": "DeepL", "from_community_srt": "1'i 14'e çok yakın ve 0'ı 13'e çok uzak gösterirke hatta sabit değişken ikisinin de aynı uzaklıkta olduğunu ima ederken, bu çizimi çok gerçekçi görmemelisiniz.", @@ -727,7 +727,7 @@ "end": 795.78 }, { - "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers, like ⅓ and ½, should fall into.", + "input": "We won't do it in this video, but see if you can reason about which rooms other rational numbers like one third and one half should fall into,", "translatedText": "Bunu bu videoda yapmayacağız, ancak ⅓ ve ½ gibi diğer rasyonel sayıların hangi odalara girmesi gerektiği konusunda mantık yürütüp yürütemeyeceğinizi görün.", "model": "DeepL", "from_community_srt": "Bu videoda yapmayacağız ama, 1/3 ve 1/2 gibi farklı rasyonel sayıların düştüğü odaları bulmayı ve belirlenen mesafenin,", diff --git a/2015/moser/english/captions.srt b/2015/moser/english/captions.srt index 338ee65b6..5443702e6 100644 --- a/2015/moser/english/captions.srt +++ b/2015/moser/english/captions.srt @@ -1,5 +1,5 @@ 1 -00:00:03,779 --> 00:00:06,420 +00:00:03,780 --> 00:00:06,420 In my last video, I posed the following question. 2 @@ -247,7 +247,7 @@ then subtract the number of edges, then add the number of regions that this graph cuts space into, along with that outer region, we get 2. 63 -00:04:01,079 --> 00:04:06,120 +00:04:01,080 --> 00:04:06,120 If we do the same thing with this other graph, well, we get 2 again. 64 diff --git a/2015/music-and-measure-theory/bengali/sentence_translations.json b/2015/music-and-measure-theory/bengali/sentence_translations.json index 64773be4b..4965ab602 100644 --- a/2015/music-and-measure-theory/bengali/sentence_translations.json +++ b/2015/music-and-measure-theory/bengali/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion. ", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion. ", "translatedText": "এটি করার অনেক উপায় আছে, কিন্তু একটি স্বাভাবিক উপায় যা আমি বেছে নেব তা হল ½ দিয়ে শুরু করা, তারপরে ⅓ এবং ⅔, তারপর ¼ এবং ¾, আমরা ¼ লিখি না যেহেতু এটি ইতিমধ্যে ½ হিসাবে প্রদর্শিত হয়েছে, তারপর সব হর 5 সহ ভগ্নাংশ হ্রাস করা, হর 6 সহ সমস্ত হ্রাসকৃত ভগ্নাংশ, এই পদ্ধতিতে এবং অব্যাহত।", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon. ", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon. ", "translatedText": "শুধু ধনাত্মক পদগুলির সাথে একটি অসীম যোগফল বেছে নিন যা 1-এ রূপান্তরিত হয়, যেমন ½, প্লাস ¼, প্লাস ⅛, প্লাস ⅛, অন এবং অন, তারপর 0-এর মতো 0-এর চেয়ে বড় যে কোনো পছন্দসই মান বেছে নিন।5, এবং যোগফলের সমস্ত পদকে এপসিলন দ্বারা গুণ করুন যাতে আপনার একটি অসীম যোগফল এপসিলনে রূপান্তরিত হয়।", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/chinese/sentence_translations.json b/2015/music-and-measure-theory/chinese/sentence_translations.json index c80d9eaf5..498236a53 100644 --- a/2015/music-and-measure-theory/chinese/sentence_translations.json +++ b/2015/music-and-measure-theory/chinese/sentence_translations.json @@ -393,7 +393,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "有很多方法可以做到这一点，但我选择的一种自然方法是从 1/2 开始，然后是 ⅓ 和 ⅔，然后是 1/4 和 3/4，我 们不写下 1/4，因为它已经显示为 1/2，然后是所有分母为 5 的约化分数，所有分母为 6 的约化分数，以这种方式继续下去。", "model": "google_nmt", "from_community_srt": "有很多方法來這樣做的， 但我將選一種自然的方式就是從½開始， 接著是⅓和⅔， 然後¼和¾， 我們沒有寫下2/4因為它已經以1/2出現了， 然後所有的 以分母為5的簡約分數， 所有以分母我6的簡約分數， 以這種方式一直繼續下去。", @@ -438,7 +438,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "只需选择一个具有收敛于 1 的正项的无限和，例如 1/2、加上 1/4、加上 1/8、加上 1/8，等等，然后选择任何大于 0 的 epsilon 所需值，例如 0。5，将和中的所有项乘以 epsilon，这样就得到 收敛于 epsilon 的无限和。", "model": "google_nmt", "from_community_srt": "只要選擇一個收斂到1的無限的正數項， 像½+¼+⅛+ ...以2的冪， 繼續下去， 然後選擇任何想要的的值ε> 0， 像0.5， 並把所有的項乘以ε 這樣使我們有一個無限的總和收斂到ε。", diff --git a/2015/music-and-measure-theory/german/sentence_translations.json b/2015/music-and-measure-theory/german/sentence_translations.json index 74ea84256..cb571e698 100644 --- a/2015/music-and-measure-theory/german/sentence_translations.json +++ b/2015/music-and-measure-theory/german/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "Es gibt viele Möglichkeiten, dies zu tun, aber eine natürliche Möglichkeit, die ich wählen würde, ist, mit ½ zu beginnen, gefolgt von ⅓ und ⅔, dann ¼ und ¾, wir schreiben ¼ nicht auf, da es bereits als ½ erscheint, und dann alle gekürzten Brüche mit Nenner 5, alle gekürzten Brüche mit Nenner 6 und immer so weiter.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "Wähle einfach eine unendliche Summe mit positiven Termen, die sich 1 annähert, wie ½, plus ¼, plus ⅛, plus ⅛, und so weiter, und wähle dann einen beliebigen Epsilon-Wert größer als 0, wie 0,5 und multipliziere alle Terme in der Summe mit Epsilon, sodass du eine unendliche Summe erhältst, die sich Epsilon annähert.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/hebrew/sentence_translations.json b/2015/music-and-measure-theory/hebrew/sentence_translations.json index 6f155c6e3..3c1d82351 100644 --- a/2015/music-and-measure-theory/hebrew/sentence_translations.json +++ b/2015/music-and-measure-theory/hebrew/sentence_translations.json @@ -308,7 +308,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "ישנן דרכים רבות לעשות זאת, אבל דרך טבעית אחת שאבחר היא להתחיל עם ½, ואחריו ⅓ ו-⅔, ואז ¼ ו-¾, אנחנו לא רושמים ¼ מכיוון שהוא כבר הופיע כ½, ואז הכל שברים מופחתים עם מכנה 5, כל השברים המופחתים עם מכנה 6, ממשיכים הלאה והלאה בצורה זו.", "n_reviews": 0, "start": 433.48, @@ -343,7 +343,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "פשוט בחר סכום אינסופי עם איברים חיוביים שמתכנס ל-1, כמו ½, פלוס ¼, פלוס ⅛, פלוס ⅛, הלאה והלאה, ואז בחר כל ערך רצוי של אפסילון הגדול מ-0, כמו 0.5, והכפילו את כל האיברים בסכום באפסילון כך שיהיה לכם סכום אינסופי שמתכנס לאפסילון.", "n_reviews": 0, "start": 494.24, diff --git a/2015/music-and-measure-theory/hindi/sentence_translations.json b/2015/music-and-measure-theory/hindi/sentence_translations.json index 6eef84cb1..e135ef5ee 100644 --- a/2015/music-and-measure-theory/hindi/sentence_translations.json +++ b/2015/music-and-measure-theory/hindi/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "ऐसा करने के कई तरीके हैं, लेकिन एक प्राकृतिक तरीका जो मैं चुनूंगा वह है ½ से शुरू करना, उसके बाद ⅓ और ⅔, फिर ¼ और ¾, हम ¼ नहीं लिखते हैं क्योंकि यह पहले से ही ½ के रूप में दिखाई देता है, फिर सब कुछ हर 5 के साथ घटी हुई भिन्नें, हर 6 के साथ सभी घटी हुई भिन्नें, इसी तरह से चलती रहती हैं।", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "बस सकारात्मक शब्दों के साथ एक अनंत राशि चुनें जो 1 में परिवर्तित हो, जैसे ½, प्लस ¼, प्लस ⅛, प्लस ⅛, ऑन और ऑन, फिर 0 से अधिक ईपीएसलॉन का कोई भी वांछित मान चुनें, जैसे 0।5, और योग के सभी पदों को ईपीएसलॉन से गुणा करें ताकि आपके पास ईपीएसलॉन में परिवर्तित होने वाला एक अनंत योग हो।", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/hungarian/sentence_translations.json b/2015/music-and-measure-theory/hungarian/sentence_translations.json index e97308455..c86c0c3eb 100644 --- a/2015/music-and-measure-theory/hungarian/sentence_translations.json +++ b/2015/music-and-measure-theory/hungarian/sentence_translations.json @@ -308,7 +308,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "Sokféleképpen lehet ezt megtenni, de az egyik természetes módszert választom, hogy ½-el kezdem, majd ⅓ és ⅔, majd ¼ és ¾, ¼-t nem írunk le, mivel már ½-ként jelent meg, akkor minden kicsinyített törtek 5-ös nevezővel, minden kicsinyített tört 6-os nevezővel, így tovább és tovább.", "n_reviews": 0, "start": 433.48, @@ -343,7 +343,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "Csak válasszon egy végtelen összeget pozitív tagokkal, amelyek konvergálnak 1-hez, például ½, plusz ¼, plusz ⅛, plusz ⅛, majd válasszon egy tetszőleges 0-nál nagyobb epszilon értéket, például 0-t.5, és szorozd meg az összegben szereplő összes tagot epszilonnal, így egy végtelen összeget kapsz, amely epszilonhoz konvergál.", "n_reviews": 0, "start": 494.24, diff --git a/2015/music-and-measure-theory/indonesian/sentence_translations.json b/2015/music-and-measure-theory/indonesian/sentence_translations.json index 40af37cf5..07cd815cc 100644 --- a/2015/music-and-measure-theory/indonesian/sentence_translations.json +++ b/2015/music-and-measure-theory/indonesian/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "Ada banyak cara untuk melakukan ini, tapi satu cara alami yang akan saya pilih adalah memulai dengan ½, diikuti dengan ⅓ dan ⅔, lalu ¼ dan ¾, kita tidak menuliskan ¼ karena sudah muncul sebagai ½, lalu semuanya pecahan tereduksi berpenyebut 5, semua pecahan tereduksi berpenyebut 6, terus menerus seperti ini.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "Pilih saja jumlah tak terhingga dengan suku positif yang konvergen ke 1, seperti ½, ditambah ¼, ditambah ⅛, ditambah ⅛, dan seterusnya, lalu pilih nilai epsilon yang diinginkan lebih besar dari 0, seperti 0.5, dan kalikan semua suku dalam jumlah tersebut dengan epsilon sehingga Anda mendapatkan jumlah tak terhingga yang konvergen ke epsilon.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/japanese/sentence_translations.json b/2015/music-and-measure-theory/japanese/sentence_translations.json index bdd75fe01..ca633e697 100644 --- a/2015/music-and-measure-theory/japanese/sentence_translations.json +++ b/2015/music-and-measure-theory/japanese/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "これを行う方法はたくさんありますが、私が選択する自然な方法の 1 つ は、1/2 から始めて、次に 1/3 と 2/3、次に 1/4 と 3/4 です。 1/4 はすでに 1/2 として表示されているため書き留めません。 その後、すべてを追加します。 分母が 5 の換算分数、分母が 6 のすべての換算分数がこのように続けられます。", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "1 に収束する正の項を含む無限和 (1/2、プラス 1/4、プラス 1/8、プラス 1/8 など) を選択し、その後、0 など、0 より大きい任意のイプシロンの 値を選択するだけです。5 そして、和に含まれるすべての項にイプシロンを掛けて、 イプシロンに収束する無限和を求めます。", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/korean/sentence_translations.json b/2015/music-and-measure-theory/korean/sentence_translations.json index cdb284ef4..f7b7ec703 100644 --- a/2015/music-and-measure-theory/korean/sentence_translations.json +++ b/2015/music-and-measure-theory/korean/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "이를 수행하는 방법에는 여러 가지가 있지만 제가 선택할 자연스러운 방법 중 하나는 ½로 시작하고 그 다음 ⅓과 ⅔, 그 다음 ¼과 ¼로 이어지는 것입니다. ¼은 이미 ½로 표시되었으므로 기록하지 않습니다. 분모가 5인 약분수, 분모가 6인 모든 약분수, 이런 방식으로 계속됩니다.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "½, 더하기 ¼, 더하기 ⅛, 더하기 ⅛ 등 1로 수렴하는 양의 항이 있는 무한합을 선택한 다음 0처럼 0보다 큰 원하는 엡실론 값을 선택하세요.5, 그리고 합의 모든 항에 엡실론을 곱하여 엡실론으로 수렴하는 무한한 합을 얻습니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/marathi/sentence_translations.json b/2015/music-and-measure-theory/marathi/sentence_translations.json index 52d36f411..7bfb1ab3c 100644 --- a/2015/music-and-measure-theory/marathi/sentence_translations.json +++ b/2015/music-and-measure-theory/marathi/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "हे करण्याचे अनेक मार्ग आहेत, परंतु एक नैसर्गिक मार्ग जो मी निवडतो तो म्हणजे ½ ने प्रारंभ करणे, त्यानंतर ⅓ आणि ⅔, नंतर ¼ आणि ¾, आम्ही ¼ लिहून ठेवत नाही कारण ते आधीच ½ असे दिसून आले आहे, नंतर सर्व भाजक 5 सह कमी केलेले अपूर्णांक, भाजक 6 सह सर्व कमी केलेले अपूर्णांक, या पद्धतीने पुढे चालू ठेवा.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "फक्त ½, अधिक ¼, अधिक ⅛, अधिक ⅛, चालू आणि चालू अशा सकारात्मक संज्ञांसह असीम बेरीज निवडा, नंतर 0 पेक्षा जास्त एप्सिलॉनचे कोणतेही इच्छित मूल्य निवडा, जसे 0.5, आणि बेरीजमधील सर्व पदांचा एप्सिलॉनने गुणाकार करा जेणेकरुन तुमच्याकडे एप्सिलॉनमध्ये अपरिमित बेरीज होईल.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/persian/sentence_translations.json b/2015/music-and-measure-theory/persian/sentence_translations.json index f9a62cf1a..3147c78aa 100644 --- a/2015/music-and-measure-theory/persian/sentence_translations.json +++ b/2015/music-and-measure-theory/persian/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion. ", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion. ", "translatedText": "راه‌های زیادی برای انجام این کار وجود دارد، اما یکی از راه‌های طبیعی که من انتخاب می‌کنم این است که با ½ شروع کنم، به دنبال آن ⅓ و ⅔، سپس ¼ و ¾، ما ¼ را یادداشت نمی کنیم زیرا قبلاً به صورت ½ ظاهر شده است، سپس همه کسرهای کاهش یافته با مخرج 5، همه کسرهای کاهش یافته با مخرج 6، به همین شکل ادامه می یابند. ", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon. ", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon. ", "translatedText": "فقط یک مجموع نامتناهی با عبارات مثبت را انتخاب کنید که به 1 همگرا شود، مانند ½، به علاوه ¼، به علاوه ⅛، به علاوه ⅛، و در ادامه، سپس هر مقدار دلخواه اپسیلون بزرگتر از 0، مانند 0 را انتخاب کنید. 5، و همه عبارت های حاصل را در اپسیلون ضرب کنید تا مجموع نامتناهی به اپسیلون همگرا شود. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/russian/sentence_translations.json b/2015/music-and-measure-theory/russian/sentence_translations.json index 5bfbb8eb7..e3639d911 100644 --- a/2015/music-and-measure-theory/russian/sentence_translations.json +++ b/2015/music-and-measure-theory/russian/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "Есть много способов сделать это, но я выберу один естественный способ — начать с ½, затем ⅓ и ⅔, затем ¼ и ¾, мы не записываем ¼, так как она уже появилась как ½, затем все сокращенные дроби со знаменателем 5, все сокращенные дроби со знаменателем 6, и так далее.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "Просто выберите бесконечную сумму с положительными членами, которая сходится к 1, например ½, плюс ¼, плюс ⅛, плюс ⅛, и так далее, а затем выберите любое желаемое значение эпсилона больше 0, например 0.5, и умножьте все члены суммы на эпсилон, чтобы получилась бесконечная сумма, сходящаяся к эпсилону.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/tamil/sentence_translations.json b/2015/music-and-measure-theory/tamil/sentence_translations.json index be69c645b..ad0499ce0 100644 --- a/2015/music-and-measure-theory/tamil/sentence_translations.json +++ b/2015/music-and-measure-theory/tamil/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "இதைச் செய்ய பல வழிகள் உள்ளன, ஆனால் நான் தேர்ந்தெடுக்கும் ஒரு இயற்கையான வழி ½ இல் தொடங்குவது, அதைத் தொடர்ந்து ⅓ மற்றும் ⅔, பின்னர் ¼ மற்றும் ¾, நாங்கள் ¼ என்று எழுத மாட்டோம், ஏனெனில் இது ஏற்கனவே ½ ஆகத் தோன்றியுள்ளது, பின்னர் அனைத்தும் வகுத்தல் 5 உடன் குறைக்கப்பட்ட பின்னங்கள், வகுத்தல் 6 உடன் அனைத்து குறைக்கப்பட்ட பின்னங்களும், இந்த பாணியில் தொடர்கின்றன.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "½, கூட்டல் ¼, கூட்டல் ⅛, கூட்டல் ⅛, ஆன் மற்றும் ஆன் மற்றும் ஆன் மற்றும் ஆன் போன்ற 1 உடன் ஒன்றிணைக்கும் நேர்மறை சொற்களைக் கொண்ட முடிவிலாத் தொகையைத் தேர்வுசெய்து, 0 போன்ற 0 ஐ விட அதிகமான எப்சிலானின் விரும்பிய மதிப்பைத் தேர்வு செய்யவும்.5, மற்றும் கூட்டுத்தொகையில் உள்ள அனைத்து சொற்களையும் எப்சிலானால் பெருக்கவும், இதன் மூலம் எப்சிலானாக மாற்றும் முடிவில்லாத் தொகை கிடைக்கும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/telugu/sentence_translations.json b/2015/music-and-measure-theory/telugu/sentence_translations.json index 1265bc7c2..17e974238 100644 --- a/2015/music-and-measure-theory/telugu/sentence_translations.json +++ b/2015/music-and-measure-theory/telugu/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "దీన్ని చేయడానికి చాలా మార్గాలు ఉన్నాయి, కానీ నేను ఎంచుకునే ఒక సహజ మార్గం ½తో ప్రారంభించడం, తర్వాత ⅓ మరియు ⅔, ఆపై ¼ మరియు ¾, మేము ¼ అని వ్రాయము, ఎందుకంటే ఇది ఇప్పటికే ½గా కనిపించింది, ఆపై అన్నీ హారం 5తో తగ్గిన భిన్నాలు, హారం 6తో అన్ని తగ్గించబడిన భిన్నాలు, ఈ పద్ధతిలో కొనసాగుతాయి.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "½, ప్లస్ ¼, ప్లస్ ⅛, ప్లస్ ⅛, ఆన్ మరియు ఆన్ వంటి 1కి కలిసే సానుకూల పదాలతో అనంతమైన మొత్తాన్ని ఎంచుకోండి, ఆపై 0 వంటి 0 కంటే ఎక్కువ ఎప్సిలాన్ యొక్క ఏదైనా కావలసిన విలువను ఎంచుకోండి.5, మరియు మొత్తంలోని అన్ని నిబంధనలను ఎప్సిలాన్‌తో గుణించండి, తద్వారా మీరు ఎప్సిలాన్‌కి మారే అనంతమైన మొత్తాన్ని కలిగి ఉంటారు.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/thai/sentence_translations.json b/2015/music-and-measure-theory/thai/sentence_translations.json index 9f3779712..4197d2c85 100644 --- a/2015/music-and-measure-theory/thai/sentence_translations.json +++ b/2015/music-and-measure-theory/thai/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion. ", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon. ", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/turkish/sentence_translations.json b/2015/music-and-measure-theory/turkish/sentence_translations.json index 561ce569c..f5aaaaa1d 100644 --- a/2015/music-and-measure-theory/turkish/sentence_translations.json +++ b/2015/music-and-measure-theory/turkish/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "Bunu yapmanın birçok yolu var, ancak seçeceğim doğal yollardan biri ½ ile başlamak, ardından ⅓ ve ⅔, ardından ¼ ve ¾, zaten ½ olarak göründüğü için ¼ yazmıyoruz, sonra hepsi paydası 5 olan azaltılmış kesirler, tümü paydası 6 olan indirgenmiş kesirler, bu şekilde devam edip devam ediyoruz.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "Sadece ½, artı ¼, artı ⅛, artı ⅛ gibi 1'e yakınsayan pozitif terimleri olan sonsuz bir toplam seçin ve ardından 0 gibi 0'dan büyük herhangi bir epsilon değeri seçin.5'i seçin ve toplamdaki tüm terimleri epsilon ile çarpın, böylece epsilon'a yakınsayan sonsuz bir toplam elde edersiniz.", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/ukrainian/sentence_translations.json b/2015/music-and-measure-theory/ukrainian/sentence_translations.json index b3865da4f..a9a74784b 100644 --- a/2015/music-and-measure-theory/ukrainian/sentence_translations.json +++ b/2015/music-and-measure-theory/ukrainian/sentence_translations.json @@ -308,7 +308,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "Є багато способів зробити це, але один природний спосіб, який я оберу, це почати з ½, потім ⅓ і ⅔, потім ¼ і ¾, ми не записуємо ¼, оскільки воно вже з’явилося як ½, а потім усе скорочені дроби зі знаменником 5, усі скорочені дроби зі знаменником 6, продовжуючи так і далі.", "n_reviews": 0, "start": 433.48, @@ -343,7 +343,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "Просто виберіть нескінченну суму з додатними членами, яка збігається до 1, як-от ½, плюс ¼, плюс ⅛, плюс ⅛, далі й далі, а потім виберіть будь-яке бажане значення епсилона, більше за 0, наприклад 0.5, і помножте всі доданки в сумі на епсилон, щоб отримати нескінченну суму, що збігається до епсилон.", "n_reviews": 0, "start": 494.24, diff --git a/2015/music-and-measure-theory/urdu/sentence_translations.json b/2015/music-and-measure-theory/urdu/sentence_translations.json index 09fa0a5c7..0b773181a 100644 --- a/2015/music-and-measure-theory/urdu/sentence_translations.json +++ b/2015/music-and-measure-theory/urdu/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion. ", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion. ", "translatedText": "ایسا کرنے کے بہت سے طریقے ہیں، لیکن ایک فطری طریقہ جس کا میں انتخاب کروں گا وہ ہے ½ سے شروع کرنا، اس کے بعد ⅓ اور ⅔، پھر ¼ اور ¾، ہم ¼ کو نہیں لکھتے کیونکہ یہ پہلے ہی ½ کے طور پر ظاہر ہو چکا ہے، پھر تمام ڈینومینیٹر 5 کے ساتھ کم کیے گئے فریکشنز، ڈینومینیٹر 6 کے ساتھ تمام کم کیے گئے فریکشنز، اس انداز میں جاری و ساری ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon. ", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon. ", "translatedText": "بس مثبت اصطلاحات کے ساتھ ایک لامحدود رقم کا انتخاب کریں جو 1 سے بدل جائے، جیسے ½، جمع ¼، جمع ⅛، جمع ⅛، آن اور آن، پھر 0 سے زیادہ ایپسیلون کی کوئی بھی مطلوبہ قدر منتخب کریں، جیسے 0۔5، اور جمع میں موجود تمام اصطلاحات کو ایپیلون سے ضرب دیں تاکہ آپ کے پاس لامحدود رقم ایپسیلون میں بدل جائے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2015/music-and-measure-theory/vietnamese/sentence_translations.json b/2015/music-and-measure-theory/vietnamese/sentence_translations.json index eb6bc21a3..baa5e88c3 100644 --- a/2015/music-and-measure-theory/vietnamese/sentence_translations.json +++ b/2015/music-and-measure-theory/vietnamese/sentence_translations.json @@ -352,7 +352,7 @@ "end": 432.66 }, { - "input": "There are many ways to do this, but one natural way that I'll choose is to start with ½, followed by ⅓ and ⅔, then ¼ and ¾, we don't write down ¼ since it's already appeared as ½, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", + "input": "There are many ways to do this, but one natural way that I'll choose is to start with 1 half, followed by 1 third and 2 thirds, then 1 fourth and 3 fourths, we don't write down 2 fourths since it's already appeared as 1 half, then all reduced fractions with denominator 5, all reduced fractions with denominator 6, continuing on and on in this fashion.", "translatedText": "Có nhiều cách để làm điều này, nhưng một cách tự nhiên mà tôi sẽ chọn là bắt đầu bằng ½, tiếp theo là ⅓ và ⅔, rồi ¼ và ¾, chúng ta không viết ra ¼ vì nó đã xuất hiện dưới dạng ½, rồi tất cả các phân số rút gọn có mẫu số 5, tất cả các phân số rút gọn có mẫu số 6, cứ tiếp tục như vậy.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 493.22 }, { - "input": "Just choose an infinite sum with positive terms that converges to 1, like ½, plus ¼, plus ⅛, plus ⅛, on and on, then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", + "input": "Just choose an infinite sum with positive terms that converges to 1, like 1 half plus 1 fourth plus 1 eighth, on and on. Then choose any desired value of epsilon greater than 0, like 0.5, and multiply all of the terms in the sum by epsilon so that you have an infinite sum converging to epsilon.", "translatedText": "Chỉ cần chọn một tổng vô hạn với các số hạng dương hội tụ về 1, như ½, cộng ¼, cộng ⅛, cộng ⅛, cứ tiếp tục như vậy, sau đó chọn bất kỳ giá trị mong muốn nào của epsilon lớn hơn 0, chẳng hạn như 0.5 và nhân tất cả các số hạng trong tổng với epsilon để bạn có tổng vô hạn hội tụ về epsilon.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/3d-transformations/bengali/sentence_translations.json b/2016/3d-transformations/bengali/sentence_translations.json index 4887cc933..71312fd2b 100644 --- a/2016/3d-transformations/bengali/sentence_translations.json +++ b/2016/3d-transformations/bengali/sentence_translations.json @@ -144,7 +144,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions. ", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions. ", "translatedText": "স্থানাঙ্কের এই তিনটি সেট একটি ম্যাট্রিক্সের কলামে পরিণত হয় যা সেই ঘূর্ণনকে বর্ণনা করে x, y, z স্থানাঙ্ক সহ একটি ভেক্টর কোথায় অবস্থান করে তা দেখার জন্য, যুক্তিটি দুটি মাত্রার জন্য প্রায় একই রকম। ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/3d-transformations/chinese/sentence_translations.json b/2016/3d-transformations/chinese/sentence_translations.json index 886bc2444..1400f15a4 100644 --- a/2016/3d-transformations/chinese/sentence_translations.json +++ b/2016/3d-transformations/chinese/sentence_translations.json @@ -159,7 +159,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions.", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions.", "translatedText": "这三组坐标成为描述旋转的矩阵的列。 要查看坐标为 x、y、z 的向量落 在哪里，推理几乎与二维的推理相同。", "model": "google_nmt", "from_community_srt": "0) 这三组坐标就成为了描述这一旋转变换的矩阵的三列 要想知道(x, y,", diff --git a/2016/3d-transformations/hindi/sentence_translations.json b/2016/3d-transformations/hindi/sentence_translations.json index c91164a8a..ca3e7a212 100644 --- a/2016/3d-transformations/hindi/sentence_translations.json +++ b/2016/3d-transformations/hindi/sentence_translations.json @@ -126,7 +126,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions.", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions.", "translatedText": "निर्देशांक के वे तीन सेट एक मैट्रिक्स के कॉलम बन जाते हैं जो उस घूर्णन का वर्णन करता है यह देखने के लिए कि निर्देशांक x, y, z वाला एक वेक्टर कहां उतरता है, तर्क लगभग वही है जो दो आयामों के लिए था।", "n_reviews": 0, "start": 160.99, diff --git a/2016/3d-transformations/indonesian/sentence_translations.json b/2016/3d-transformations/indonesian/sentence_translations.json index 43c6e764f..d958351c6 100644 --- a/2016/3d-transformations/indonesian/sentence_translations.json +++ b/2016/3d-transformations/indonesian/sentence_translations.json @@ -144,7 +144,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions.", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions.", "translatedText": "Ketiga himpunan koordinat tersebut menjadi kolom-kolom matriks yang menggambarkan rotasi tersebut. Untuk melihat di mana sebuah vektor dengan koordinat x, y, z mendarat, alasannya hampir sama dengan dua dimensi.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/3d-transformations/japanese/sentence_translations.json b/2016/3d-transformations/japanese/sentence_translations.json index c4c992dbf..23bac2d63 100644 --- a/2016/3d-transformations/japanese/sentence_translations.json +++ b/2016/3d-transformations/japanese/sentence_translations.json @@ -144,7 +144,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions.", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions.", "translatedText": "これら 3 つの座標セットは、その回転を説明する行列の列に なります。 座標 x、y、z を持つベクトルがどこに着地する かを確認するための推論は、2 次元の場合とほぼ同じです。", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/3d-transformations/marathi/sentence_translations.json b/2016/3d-transformations/marathi/sentence_translations.json index a376c7c5a..bbb577e18 100644 --- a/2016/3d-transformations/marathi/sentence_translations.json +++ b/2016/3d-transformations/marathi/sentence_translations.json @@ -144,7 +144,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions.", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions.", "translatedText": "निर्देशांकांचे ते तीन संच मॅट्रिक्सचे स्तंभ बनतात जे त्या रोटेशनचे वर्णन करतात x, y, z सह निर्देशांक असलेला सदिश कोठे उतरतो हे पाहण्यासाठी, तर्क दोन मितींसाठी जवळजवळ समान आहे.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/3d-transformations/persian/sentence_translations.json b/2016/3d-transformations/persian/sentence_translations.json index cf9d5c0d6..f944bd691 100644 --- a/2016/3d-transformations/persian/sentence_translations.json +++ b/2016/3d-transformations/persian/sentence_translations.json @@ -144,7 +144,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions.", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions.", "translatedText": "این سه مجموعه مختصات به ستون‌های یک ماتریس تبدیل می‌شوند که آن چرخش را توصیف می‌کند.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/3d-transformations/tamil/sentence_translations.json b/2016/3d-transformations/tamil/sentence_translations.json index 01e0e0da3..9c3c5dda6 100644 --- a/2016/3d-transformations/tamil/sentence_translations.json +++ b/2016/3d-transformations/tamil/sentence_translations.json @@ -144,7 +144,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions.", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions.", "translatedText": "அந்த மூன்று ஆயத்தொகுப்புகளும் அந்த சுழற்சியை விவரிக்கும் மேட்ரிக்ஸின் நெடுவரிசைகளாக மாறுகின்றன. x, y, z ஆயத்தொலைவுகளைக் கொண்ட ஒரு திசையன் எங்கு இறங்குகிறது என்பதைப் பார்க்க, பகுத்தறிவு இரண்டு பரிமாணங்களில் இருந்ததைப் போலவே இருக்கும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/3d-transformations/telugu/sentence_translations.json b/2016/3d-transformations/telugu/sentence_translations.json index 2e4800818..cc25e6ffa 100644 --- a/2016/3d-transformations/telugu/sentence_translations.json +++ b/2016/3d-transformations/telugu/sentence_translations.json @@ -144,7 +144,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions.", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions.", "translatedText": "ఆ మూడు కోఆర్డినేట్‌లు ఆ భ్రమణాన్ని వివరించే మాతృక యొక్క నిలువు వరుసలుగా మారతాయి, x, y, z కోఆర్డినేట్‌లతో వెక్టార్ ఎక్కడ ల్యాండ్ అవుతుందో చూడటానికి, తార్కికం రెండు కోణాలకు సంబంధించిన దానికి దాదాపు సమానంగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/3d-transformations/thai/sentence_translations.json b/2016/3d-transformations/thai/sentence_translations.json index a78950ee0..de7aa07a0 100644 --- a/2016/3d-transformations/thai/sentence_translations.json +++ b/2016/3d-transformations/thai/sentence_translations.json @@ -144,7 +144,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions. ", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions. ", "translatedText": "พิกัดทั้งสามชุดนั้นกลายเป็นคอลัมน์ของเมทริกซ์ที่อธิบายการหมุนนั้น ในการดูว่าเวกเตอร์ที่มีพิกัด x, y, z ตกลงไปที่ใด การให้เหตุผลเกือบจะเหมือนกันกับที่เป็นในสองมิติ ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/3d-transformations/ukrainian/sentence_translations.json b/2016/3d-transformations/ukrainian/sentence_translations.json index b31f3eaa8..2316de71e 100644 --- a/2016/3d-transformations/ukrainian/sentence_translations.json +++ b/2016/3d-transformations/ukrainian/sentence_translations.json @@ -144,7 +144,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions.", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions.", "translatedText": "Ці три набори координат стають стовпцями матриці, яка описує це обертання. Щоб побачити, де приземляється вектор із координатами x, y, z, міркування майже ідентичне тому, що було для двох вимірів.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/3d-transformations/urdu/sentence_translations.json b/2016/3d-transformations/urdu/sentence_translations.json index 15bf872fb..b344b7240 100644 --- a/2016/3d-transformations/urdu/sentence_translations.json +++ b/2016/3d-transformations/urdu/sentence_translations.json @@ -144,7 +144,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions. ", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions. ", "translatedText": "کوآرڈینیٹ کے وہ تین سیٹ ایک میٹرکس کے کالم بن جاتے ہیں جو اس گردش کو بیان کرتا ہے یہ دیکھنے کے لیے کہ ایکس، y، z کوآرڈینیٹ والا ویکٹر کہاں اترتا ہے، استدلال تقریباً وہی ہے جو دو جہتوں کے لیے تھا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/3d-transformations/vietnamese/sentence_translations.json b/2016/3d-transformations/vietnamese/sentence_translations.json index 5af8f1f94..865a43f8d 100644 --- a/2016/3d-transformations/vietnamese/sentence_translations.json +++ b/2016/3d-transformations/vietnamese/sentence_translations.json @@ -144,7 +144,7 @@ "end": 158.84 }, { - "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation To see where a vector with coordinates x, y, z lands, the reasoning is almost identical to what it was for two dimensions.", + "input": "Those three sets of coordinates become the columns of a matrix that describes that rotation transformation. To see where a vector with coordinates x,y,z lands, the reasoning is almost identical to what it was for two dimensions.", "translatedText": "Ba bộ tọa độ đó trở thành các cột của ma trận mô tả phép quay đó. Để xem vị trí của vectơ có tọa độ x, y, z, lý do gần như giống hệt với trường hợp hai chiều.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/abstract-vector-spaces/arabic/sentence_translations.json b/2016/abstract-vector-spaces/arabic/sentence_translations.json index 67aed2e2e..6f6581b22 100644 --- a/2016/abstract-vector-spaces/arabic/sentence_translations.json +++ b/2016/abstract-vector-spaces/arabic/sentence_translations.json @@ -151,7 +151,7 @@ "end": 153.14 }, { - "input": "The output of this new function at any given input, like negative four, is the sum of the same input, negative four.", + "input": "The output of this new function at any given input, like negative four, is the sum of the outputs of f and g when you evaluate them each at that same input, negative four.", "translatedText": "مخرجات هذه الدالة الجديدة عند أي مدخل، مثل سالب أربعة، هو مجموع المدخل نفسه، سالب أربعة.", "model": "google_nmt", "from_community_srt": "إخراج هذه الوظيفة الجديدة في أي وقت المدخلات ، مثل -4 ، هو مجموع المخرجات من f و g ، عندما تقيم كل منهم عند هذا الإدخال نفسه ، -4.", diff --git a/2016/abstract-vector-spaces/czech/sentence_translations.json b/2016/abstract-vector-spaces/czech/sentence_translations.json index d31b5e2f1..45fbab9ae 100644 --- a/2016/abstract-vector-spaces/czech/sentence_translations.json +++ b/2016/abstract-vector-spaces/czech/sentence_translations.json @@ -153,7 +153,7 @@ "end": 153.14 }, { - "input": "The output of this new function at any given input, like negative four, is the sum of the same input, negative four.", + "input": "The output of this new function at any given input, like negative four, is the sum of the outputs of f and g when you evaluate them each at that same input, negative four.", "translatedText": "Výstupem této nové funkce při jakémkoli zadaném vstupu, například záporné čtyřce, je součet téhož vstupu, záporné čtyřky.", "model": "DeepL", "from_community_srt": "Výstup z nové funkce v nějakém bodě, třeba -4, definujeme jako součet výstupů 'f' a 'g', když je v tomto bodě (např. -4) vyhodnotíme.", diff --git a/2016/abstract-vector-spaces/english/captions.srt b/2016/abstract-vector-spaces/english/captions.srt index a79a9480d..12e346d00 100644 --- a/2016/abstract-vector-spaces/english/captions.srt +++ b/2016/abstract-vector-spaces/english/captions.srt @@ -155,722 +155,726 @@ It's one of those things where you kind of already know what it's going to be, but actually phrasing it is a mouthful. 40 -00:02:33,960 --> 00:02:38,693 -The output of this new function at any given input, +00:02:33,960 --> 00:02:38,406 +The output of this new function at any given input, like negative four, 41 -00:02:38,693 --> 00:02:44,520 -like negative four, is the sum of the same input, negative four. +00:02:38,406 --> 00:02:43,655 +is the sum of the outputs of f and g when you evaluate them each at that same input, 42 +00:02:43,655 --> 00:02:44,520 +negative four. + +43 00:02:45,420 --> 00:02:49,507 Or more generally, the value of the sum function at any -43 +44 00:02:49,507 --> 00:02:53,740 given input x is the sum of the values f of x plus g of x. -44 +45 00:03:00,700 --> 00:03:04,279 This is pretty similar to adding vectors coordinate by coordinate, -45 +46 00:03:04,279 --> 00:03:08,500 it's just that there are, in a sense, infinitely many coordinates to deal with. -46 +47 00:03:11,100 --> 00:03:15,589 Similarly, there's a sensible notion for scaling a function by a real number, -47 +48 00:03:15,589 --> 00:03:18,180 just scale all of the outputs by that number. -48 +49 00:03:20,240 --> 00:03:23,690 And again, this is analogous to scaling a vector coordinate by coordinate, -49 +50 00:03:23,690 --> 00:03:26,220 it just feels like there's infinitely many coordinates. -50 +51 00:03:28,900 --> 00:03:33,546 Now, given that the only thing vectors can really do is get added together or scaled, -51 +52 00:03:33,546 --> 00:03:37,814 it feels like we should be able to take the same useful constructs and problem -52 +53 00:03:37,814 --> 00:03:41,866 solving techniques of linear algebra that were originally thought about in -53 +54 00:03:41,866 --> 00:03:45,540 the context of arrows and space and apply them to functions as well. -54 +55 00:03:46,540 --> 00:03:51,070 For example, there's a perfectly reasonable notion of a linear transformation -55 +56 00:03:51,070 --> 00:03:55,600 for functions, something that takes in one function and turns it into another. -56 +57 00:03:59,820 --> 00:04:02,780 One familiar example comes from calculus, the derivative. -57 +58 00:04:03,420 --> 00:04:07,140 It's something which transforms one function into another function. -58 +59 00:04:08,720 --> 00:04:12,721 Sometimes in this context you'll hear these called operators instead of transformations, -59 +60 00:04:12,721 --> 00:04:13,980 but the meaning is the same. -60 +61 00:04:16,240 --> 00:04:18,864 A natural question you might want to ask is what it -61 +62 00:04:18,864 --> 00:04:21,540 means for a transformation of functions to be linear. -62 +63 00:04:22,440 --> 00:04:26,565 The formal definition of linearity is relatively abstract and symbolically driven -63 +64 00:04:26,565 --> 00:04:30,440 compared to the way that I first talked about it in chapter 3 of this series. -64 +65 00:04:30,440 --> 00:04:33,752 But the reward of abstractness is that we'll get something -65 +66 00:04:33,752 --> 00:04:36,840 general enough to apply to functions as well as arrows. -66 +67 00:04:39,180 --> 00:04:42,683 A transformation is linear if it satisfies two properties, -67 +68 00:04:42,683 --> 00:04:45,000 commonly called additivity and scaling. -68 +69 00:04:46,040 --> 00:04:50,554 Additivity means that if you add two vectors, v and w, -69 +70 00:04:50,554 --> 00:04:57,613 then apply a transformation to their sum, you get the same result as if you added the -70 +71 00:04:57,613 --> 00:05:00,240 transformed versions of v and w. -71 +72 00:05:04,520 --> 00:05:09,606 The scaling property is that when you scale a vector v by some number, -72 +73 00:05:09,606 --> 00:05:14,406 then apply the transformation, you get the same ultimate vector as -73 +74 00:05:14,406 --> 00:05:18,920 if you scaled the transformed version of v by that same amount. -74 +75 00:05:21,700 --> 00:05:25,478 The way you'll often hear this described is that linear transformations -75 +76 00:05:25,478 --> 00:05:29,100 preserve the operations of vector addition and scalar multiplication. -76 +77 00:05:32,200 --> 00:05:36,171 The idea of gridlines remaining parallel and evenly spaced that I've -77 +78 00:05:36,171 --> 00:05:40,028 talked about in past videos is really just an illustration of what -78 +79 00:05:40,028 --> 00:05:44,000 these two properties mean in the specific case of points in 2D space. -79 +80 00:05:44,880 --> 00:05:48,166 One of the most important consequences of these properties, -80 +81 00:05:48,166 --> 00:05:50,960 which makes matrix vector multiplication possible, -81 +82 00:05:50,960 --> 00:05:54,685 is that a linear transformation is completely described by where it -82 +83 00:05:54,685 --> 00:05:56,000 takes the basis vectors. -83 +84 00:05:57,720 --> 00:06:02,595 Since any vector can be expressed by scaling and adding the basis vectors in some way, -84 +85 00:06:02,595 --> 00:06:06,909 finding the transformed version of a vector comes down to scaling and adding -85 +86 00:06:06,909 --> 00:06:10,440 the transformed versions of the basis vectors in that same way. -86 +87 00:06:12,280 --> 00:06:16,780 As you'll see in just a moment, this is as true for functions as it is for arrows. -87 +88 00:06:18,360 --> 00:06:22,437 For example, calculus students are always using the fact that the derivative is -88 +89 00:06:22,437 --> 00:06:26,820 additive and has the scaling property, even if they haven't heard it phrased that way. -89 +90 00:06:28,140 --> 00:06:31,187 If you add two functions, then take the derivative, -90 +91 00:06:31,187 --> 00:06:35,231 it's the same as first taking the derivative of each one separately, -91 +92 00:06:35,231 --> 00:06:36,580 then adding the result. -92 +93 00:06:40,140 --> 00:06:43,305 Similarly, if you scale a function, then take the derivative, -93 +94 00:06:43,305 --> 00:06:46,880 it's the same as first taking the derivative, then scaling the result. -94 +95 00:06:50,280 --> 00:06:53,037 To really drill in the parallel, let's see what it -95 +96 00:06:53,037 --> 00:06:56,120 might look like to describe the derivative with a matrix. -96 +97 00:06:56,980 --> 00:07:00,332 This will be a little tricky, since function spaces have a tendency to be -97 +98 00:07:00,332 --> 00:07:03,820 infinite dimensional, but I think this exercise is actually quite satisfying. -98 +99 00:07:04,840 --> 00:07:09,520 Let's limit ourselves to polynomials, things like x squared plus 3x plus 5, -99 +100 00:07:09,520 --> 00:07:11,860 or 4x to the seventh minus 5x squared. -100 +101 00:07:12,330 --> 00:07:16,439 Each of the polynomials in our space will only have finitely many terms, -101 +102 00:07:16,439 --> 00:07:21,000 but the full space is going to include polynomials with arbitrarily large degree. -102 +103 00:07:22,220 --> 00:07:25,597 The first thing we need to do is give coordinates to this space, -103 +104 00:07:25,597 --> 00:07:27,260 which requires choosing a basis. -104 +105 00:07:28,180 --> 00:07:33,393 Since polynomials are already written down as the sum of scaled powers of the variable x, -105 +106 00:07:33,393 --> 00:07:37,680 it's pretty natural to just choose pure powers of x as the basis function. -106 +107 00:07:38,280 --> 00:07:43,700 In other words, our first basis function will be the constant function, b0 of x equals 1. -107 +108 00:07:44,180 --> 00:07:48,075 The second basis function will be b1 of x equals x, -108 +109 00:07:48,075 --> 00:07:53,320 then b2 of x equals x squared, then b3 of x equals x cubed, and so on. -109 +110 00:07:53,860 --> 00:07:58,229 The role that these basis functions serve will be similar to the roles of i-hat, -110 +111 00:07:58,229 --> 00:08:00,980 j-hat, and k-hat in the world of vectors as arrows. -111 +112 00:08:02,120 --> 00:08:05,269 Since our polynomials can have arbitrarily large degree, -112 +113 00:08:05,269 --> 00:08:07,480 this set of basis functions is infinite. -113 +114 00:08:08,240 --> 00:08:11,823 But that's okay, it just means that when we treat our polynomials as vectors, -114 +115 00:08:11,823 --> 00:08:14,120 they're going to have infinitely many coordinates. -115 +116 00:08:15,600 --> 00:08:19,842 A polynomial like x squared plus 3x plus 5, for example, -116 +117 00:08:19,842 --> 00:08:25,500 would be described with the coordinates 5, 3, 1, then infinitely many zeros. -117 +118 00:08:26,100 --> 00:08:30,121 You'd read this as saying that it's 5 times the first basis function, -118 +119 00:08:30,121 --> 00:08:34,718 plus 3 times that second basis function, plus 1 times the third basis function, -119 +120 00:08:34,718 --> 00:08:39,200 and then none of the other basis functions should be added from that point on. -120 +121 00:08:40,620 --> 00:08:47,176 The polynomial 4x to the seventh minus 5x squared would have the coordinates 0, -121 +122 00:08:47,176 --> 00:08:52,340 0, negative 5, 0, 0, 0, 0, 4, then an infinite string of zeros. -122 +123 00:08:53,260 --> 00:08:57,857 In general, since every individual polynomial has only finitely many terms, -123 +124 00:08:57,857 --> 00:09:03,000 its coordinates will be some finite string of numbers with an infinite tail of zeros. -124 +125 00:09:06,900 --> 00:09:10,427 In this coordinate system, the derivative is described with -125 +126 00:09:10,427 --> 00:09:13,249 an infinite matrix that's mostly full of zeros, -126 +127 00:09:13,249 --> 00:09:17,600 but which has the positive integers counting down on this offset diagonal. -127 +128 00:09:18,400 --> 00:09:21,312 I'll talk about how you could find this matrix in just a moment, -128 +129 00:09:21,312 --> 00:09:24,360 but the best way to get a feel for it is to just watch it in action. -129 +130 00:09:24,970 --> 00:09:31,175 Take the coordinates representing the polynomial x cubed plus 5x squared plus 4x plus 5, -130 +131 00:09:31,175 --> 00:09:34,940 then put those coordinates on the right of the matrix. -131 +132 00:09:40,410 --> 00:09:45,238 The only term that contributes to the first coordinate of the result is 1 times 4, -132 +133 00:09:45,238 --> 00:09:48,380 which means the constant term in the result will be 4. -133 +134 00:09:50,100 --> 00:09:54,380 This corresponds to the fact that the derivative of 4x is the constant 4. -134 +135 00:09:55,640 --> 00:10:00,721 The only term contributing to the second coordinate of the matrix vector product -135 +136 00:10:00,721 --> 00:10:05,740 is 2 times 5, which means the coefficient in front of x in the derivative is 10. -136 +137 00:10:06,500 --> 00:10:09,280 That one corresponds to the derivative of 5x squared. -137 +138 00:10:10,780 --> 00:10:13,430 Similarly, the third coordinate in the matrix -138 +139 00:10:13,430 --> 00:10:16,080 vector product comes down to taking 3 times 1. -139 +140 00:10:17,660 --> 00:10:21,740 This one corresponds to the derivative of x cubed being 3x squared. -140 +141 00:10:23,080 --> 00:10:25,020 And after that, it'll be nothing but zeros. -141 +142 00:10:26,880 --> 00:10:29,800 What makes this possible is that the derivative is linear. -142 +143 00:10:31,640 --> 00:10:34,300 And for those of you who like to pause and ponder, -143 +144 00:10:34,300 --> 00:10:37,691 you could construct this matrix by taking the derivative of each -144 +145 00:10:37,691 --> 00:10:41,500 basis function and putting the coordinates of the results in each column. -145 +146 00:10:59,780 --> 00:11:03,918 So, surprisingly, matrix vector multiplication and taking a derivative, -146 +147 00:11:03,918 --> 00:11:07,080 which at first seem like completely different animals, -147 +148 00:11:07,080 --> 00:11:09,840 are both just really members of the same family. -148 +149 00:11:11,220 --> 00:11:14,849 In fact, most of the concepts I've talked about in this series with -149 +150 00:11:14,849 --> 00:11:19,333 respect to vectors as arrows in space, things like the dot product or eigenvectors, -150 +151 00:11:19,333 --> 00:11:21,842 have direct analogs in the world of functions, -151 +152 00:11:21,842 --> 00:11:26,540 though sometimes they go by different names, things like inner product or eigenfunction. -152 +153 00:11:28,400 --> 00:11:30,880 So back to the question of what is a vector. -153 +154 00:11:31,560 --> 00:11:35,840 The point I want to make here is that there are lots of vectorish things in math. -154 +155 00:11:35,840 --> 00:11:40,536 As long as you're dealing with a set of objects where there's a reasonable notion of -155 +156 00:11:40,536 --> 00:11:45,508 scaling and adding, whether that's a set of arrows in space, lists of numbers, functions, -156 +157 00:11:45,508 --> 00:11:48,382 or whatever other crazy thing you choose to define, -157 +158 00:11:48,382 --> 00:11:51,918 all of the tools developed in linear algebra regarding vectors, -158 +159 00:11:51,918 --> 00:11:55,620 linear transformations and all that stuff, should be able to apply. -159 +160 00:11:57,480 --> 00:11:59,839 Take a moment to imagine yourself right now as a -160 +161 00:11:59,839 --> 00:12:02,440 mathematician developing the theory of linear algebra. -161 +162 00:12:02,440 --> 00:12:06,723 You want all of the definitions and discoveries of your work to apply to -162 +163 00:12:06,723 --> 00:12:11,300 all of the vectorish things in full generality, not just to one specific case. -163 +164 00:12:13,400 --> 00:12:18,186 These sets of vectorish things, like arrows or lists of numbers or functions, -164 +165 00:12:18,186 --> 00:12:19,720 are called vector spaces. -165 +166 00:12:20,580 --> 00:12:23,304 And what you as the mathematician might want to do is say, -166 +167 00:12:23,304 --> 00:12:25,981 hey everyone, I don't want to have to think about all the -167 +168 00:12:25,981 --> 00:12:29,260 different types of crazy vector spaces that you all might come up with. -168 +169 00:12:29,260 --> 00:12:32,387 So what you do is establish a list of rules that -169 +170 00:12:32,387 --> 00:12:35,260 vector addition and scaling have to abide by. -170 +171 00:12:36,400 --> 00:12:40,199 These rules are called axioms, and in the modern theory of linear algebra, -171 +172 00:12:40,199 --> 00:12:43,645 there are eight axioms that any vector space must satisfy if all of -172 +173 00:12:43,645 --> 00:12:47,040 the theory and constructs that we've discovered are going to apply. -173 +174 00:12:47,700 --> 00:12:51,180 I'll leave them on the screen here for anyone who wants to pause and ponder, -174 +175 00:12:51,180 --> 00:12:54,614 but basically it's just a checklist to make sure that the notions of vector -175 +176 00:12:54,614 --> 00:12:58,140 addition and scalar multiplication do the things that you'd expect them to do. -176 +177 00:12:58,720 --> 00:13:02,568 These axioms are not so much fundamental rules of nature as they are an -177 +178 00:13:02,568 --> 00:13:05,936 interface between you, the mathematician, discovering results, -178 +179 00:13:05,936 --> 00:13:10,480 and other people who might want to apply those results to new sorts of vector spaces. -179 +180 00:13:11,420 --> 00:13:14,979 If, for example, someone defines some crazy type of vector space, -180 +181 00:13:14,979 --> 00:13:19,833 like the set of all pi creatures with some definition of adding and scaling pi creatures, -181 +182 00:13:19,833 --> 00:13:23,879 these axioms are like a checklist of things that they need to verify about -182 +183 00:13:23,879 --> 00:13:28,140 their definitions before they can start applying the results of linear algebra. -183 +184 00:13:28,820 --> 00:13:31,580 And you, as the mathematician, never have to think about -184 +185 00:13:31,580 --> 00:13:34,340 all the possible crazy vector spaces people might define. -185 +186 00:13:34,860 --> 00:13:38,356 You just have to prove your results in terms of these axioms so -186 +187 00:13:38,356 --> 00:13:42,617 anyone whose definitions satisfy those axioms can happily apply your results, -187 +188 00:13:42,617 --> 00:13:45,240 even if you never thought about their situation. -188 +189 00:13:46,520 --> 00:13:50,846 As a consequence, you'd tend to phrase all of your results pretty abstractly, -189 +190 00:13:50,846 --> 00:13:53,509 which is to say, only in terms of these axioms, -190 +191 00:13:53,509 --> 00:13:58,280 rather than centering on a specific type of vector, like arrows in space or functions. -191 +192 00:14:01,860 --> 00:14:05,604 For example, this is why just about every textbook you'll find will -192 +193 00:14:05,604 --> 00:14:09,239 define linear transformations in terms of additivity and scaling, -193 +194 00:14:09,239 --> 00:14:13,260 rather than talking about gridlines remaining parallel and evenly spaced. -194 +195 00:14:13,260 --> 00:14:16,956 Even though the latter is more intuitive, and at least in my view, -195 +196 00:14:16,956 --> 00:14:21,260 more helpful for first-time learners, even if it is specific to one situation. -196 +197 00:14:22,620 --> 00:14:26,920 So the mathematician's answer to what are vectors is to just ignore the question. -197 +198 00:14:27,500 --> 00:14:31,260 In the modern theory, the form that vectors take doesn't really matter. -198 +199 00:14:31,860 --> 00:14:36,341 Arrows, lists of numbers, functions, pi creatures, really, it can be anything, -199 +200 00:14:36,341 --> 00:14:41,220 so long as there's some notion of adding and scaling vectors that follows these rules. -200 +201 00:14:41,860 --> 00:14:44,880 It's like asking what the number 3 really is. -201 +202 00:14:45,380 --> 00:14:49,816 Whenever it comes up concretely, it's in the context of some triplet of things, -202 +203 00:14:49,816 --> 00:14:54,308 but in math, it's treated as an abstraction for all possible triplets of things, -203 +204 00:14:54,308 --> 00:14:58,080 and lets you reason about all possible triplets using a single idea. -204 +205 00:14:59,120 --> 00:15:02,283 Same goes with vectors, which have many embodiments, -205 +206 00:15:02,283 --> 00:15:07,000 but math abstracts them all into a single, intangible notion of a vector space. -206 +207 00:15:08,860 --> 00:15:12,354 But, as anyone watching this series knows, I think it's better -207 +208 00:15:12,354 --> 00:15:16,237 to begin reasoning about vectors in a concrete, visualizable setting, -208 +209 00:15:16,237 --> 00:15:18,900 like 2D space, with arrows rooted at the origin. -209 +210 00:15:19,660 --> 00:15:24,435 But as you learn more linear algebra, know that these tools apply much more generally, -210 +211 00:15:24,435 --> 00:15:29,156 and that this is the underlying reason why textbooks and lectures tend to be phrased, -211 +212 00:15:29,156 --> 00:15:30,090 well, abstractly. -212 +213 00:15:31,940 --> 00:15:36,140 So with that, folks, I think I'll call it an in to this essence of linear algebra series. -213 +214 00:15:36,140 --> 00:15:39,837 If you've watched and understood the videos, I really do believe that -214 +215 00:15:39,837 --> 00:15:43,800 you have a solid foundation in the underlying intuitions of linear algebra. -215 +216 00:15:44,640 --> 00:15:47,495 This is not the same thing as learning the full topic, of course, -216 +217 00:15:47,495 --> 00:15:50,697 that's something that can only really come from working through problems, -217 +218 00:15:50,697 --> 00:15:54,548 but the learning you do moving forward could be substantially more efficient if you have -218 +219 00:15:54,548 --> 00:15:56,020 all the right intuitions in place. -219 +220 00:15:56,660 --> 00:16:00,000 So, have fun applying those intuitions, and best of luck with your future learning. diff --git a/2016/abstract-vector-spaces/english/sentence_timings.json b/2016/abstract-vector-spaces/english/sentence_timings.json index 133262403..a3a4e6328 100644 --- a/2016/abstract-vector-spaces/english/sentence_timings.json +++ b/2016/abstract-vector-spaces/english/sentence_timings.json @@ -85,7 +85,7 @@ 153.14 ], [ - "The output of this new function at any given input, like negative four, is the sum of the same input, negative four.", + "The output of this new function at any given input, like negative four, is the sum of the outputs of f and g when you evaluate them each at that same input, negative four.", 153.96, 164.52 ], diff --git a/2016/abstract-vector-spaces/english/transcript.txt b/2016/abstract-vector-spaces/english/transcript.txt index 316cc42a4..0ecce9bb5 100644 --- a/2016/abstract-vector-spaces/english/transcript.txt +++ b/2016/abstract-vector-spaces/english/transcript.txt @@ -15,7 +15,7 @@ To build up to where this is going, I'd actually like to spend the bulk of this You see, there's a sense in which functions are actually just another type of vector. In the same way that you can add two vectors together, there's also a sensible notion for adding two functions, f and g, to get a new function, f plus g. It's one of those things where you kind of already know what it's going to be, but actually phrasing it is a mouthful. -The output of this new function at any given input, like negative four, is the sum of the same input, negative four. +The output of this new function at any given input, like negative four, is the sum of the outputs of f and g when you evaluate them each at that same input, negative four. Or more generally, the value of the sum function at any given input x is the sum of the values f of x plus g of x. This is pretty similar to adding vectors coordinate by coordinate, it's just that there are, in a sense, infinitely many coordinates to deal with. Similarly, there's a sensible notion for scaling a function by a real number, just scale all of the outputs by that number. diff --git a/2016/abstract-vector-spaces/french/sentence_translations.json b/2016/abstract-vector-spaces/french/sentence_translations.json index bcf065314..b330e4ebb 100644 --- a/2016/abstract-vector-spaces/french/sentence_translations.json +++ b/2016/abstract-vector-spaces/french/sentence_translations.json @@ -135,7 +135,7 @@ "end": 153.14 }, { - "input": "The output of this new function at any given input, like negative four, is the sum of the same input, negative four.", + "input": "The output of this new function at any given input, like negative four, is the sum of the outputs of f and g when you evaluate them each at that same input, negative four.", "translatedText": "La sortie de cette nouvelle fonction à n’importe quelle entrée donnée, comme moins quatre, est la somme de la même entrée, moins quatre.", "from_community_srt": "La valeur de cette nouvelle fonction a n'importe quelle valeur, comme -4, est la somme de f et g, quand on les utilise avec le même paramètre,", "n_reviews": 0, diff --git a/2016/abstract-vector-spaces/hungarian/sentence_translations.json b/2016/abstract-vector-spaces/hungarian/sentence_translations.json index 629d58aa0..5843548cc 100644 --- a/2016/abstract-vector-spaces/hungarian/sentence_translations.json +++ b/2016/abstract-vector-spaces/hungarian/sentence_translations.json @@ -136,7 +136,7 @@ "end": 153.14 }, { - "input": "The output of this new function at any given input, like negative four, is the sum of the same input, negative four.", + "input": "The output of this new function at any given input, like negative four, is the sum of the outputs of f and g when you evaluate them each at that same input, negative four.", "translatedText": "Ennek az új függvénynek a kimenete egy adott bemenet, például negatív négy, az ugyanezen bemenet, negatív négy összege.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/abstract-vector-spaces/italian/sentence_translations.json b/2016/abstract-vector-spaces/italian/sentence_translations.json index 8d1422ac5..cc2048996 100644 --- a/2016/abstract-vector-spaces/italian/sentence_translations.json +++ b/2016/abstract-vector-spaces/italian/sentence_translations.json @@ -153,7 +153,7 @@ "end": 153.14 }, { - "input": "The output of this new function at any given input, like negative four, is the sum of the same input, negative four.", + "input": "The output of this new function at any given input, like negative four, is the sum of the outputs of f and g when you evaluate them each at that same input, negative four.", "translatedText": "L'output di questa nuova funzione per ogni dato input, come meno quattro, è la somma dello stesso input, meno quattro.", "model": "google_nmt", "from_community_srt": "ad esempio -4, è la somma degli output di f e di g, quando ne calcoliamo separatamente il valore dato lo stesso input, -4. Più in generale, il valore della funzione somma,", diff --git a/2016/abstract-vector-spaces/korean/sentence_translations.json b/2016/abstract-vector-spaces/korean/sentence_translations.json index 1b662adab..1c8f86328 100644 --- a/2016/abstract-vector-spaces/korean/sentence_translations.json +++ b/2016/abstract-vector-spaces/korean/sentence_translations.json @@ -151,7 +151,7 @@ "end": 153.14 }, { - "input": "The output of this new function at any given input, like negative four, is the sum of the same input, negative four.", + "input": "The output of this new function at any given input, like negative four, is the sum of the outputs of f and g when you evaluate them each at that same input, negative four.", "translatedText": "-4와 같이 주어진 입력에서 이 새로운 함수의 출력은 동일한 입력(-4)의 합입니다.", "model": "google_nmt", "from_community_srt": "-4처럼 어떤 값이 주어지면 새로운 함수의 함수값은 동일하게 주어진 값(-4)에서 계산한 f와 g의 함수값의 합이다 더 일반적으로,", diff --git a/2016/abstract-vector-spaces/polish/sentence_translations.json b/2016/abstract-vector-spaces/polish/sentence_translations.json index 7cc297349..963ce5ecb 100644 --- a/2016/abstract-vector-spaces/polish/sentence_translations.json +++ b/2016/abstract-vector-spaces/polish/sentence_translations.json @@ -151,7 +151,7 @@ "end": 153.14 }, { - "input": "The output of this new function at any given input, like negative four, is the sum of the same input, negative four.", + "input": "The output of this new function at any given input, like negative four, is the sum of the outputs of f and g when you evaluate them each at that same input, negative four.", "translatedText": "Wynik tej nowej funkcji na dowolnym wejściu, na przykład minus cztery, jest sumą tego samego wejścia, minus cztery.", "model": "google_nmt", "from_community_srt": "Wartość tej nowej funkcji dla dowolnego argumentu, np -4, to suma wartości funkcji f oraz g, gdzie obie wartości bierzesz w punkcie -4.", diff --git a/2016/abstract-vector-spaces/portuguese/sentence_translations.json b/2016/abstract-vector-spaces/portuguese/sentence_translations.json index 7d44d7538..1d2d45f12 100644 --- a/2016/abstract-vector-spaces/portuguese/sentence_translations.json +++ b/2016/abstract-vector-spaces/portuguese/sentence_translations.json @@ -153,7 +153,7 @@ "end": 153.14 }, { - "input": "The output of this new function at any given input, like negative four, is the sum of the same input, negative four.", + "input": "The output of this new function at any given input, like negative four, is the sum of the outputs of f and g when you evaluate them each at that same input, negative four.", "translatedText": "A saída desta nova função em qualquer entrada, como menos quatro, é a soma da mesma entrada, menos quatro.", "model": "google_nmt", "from_community_srt": "A saída desta nova função em uma dada entrada, como -4, é a soma das saídas de f e g, quando você as avalia cada uma na mesma entrada,", diff --git a/2016/abstract-vector-spaces/spanish/sentence_translations.json b/2016/abstract-vector-spaces/spanish/sentence_translations.json index c7de8dde4..8f398a882 100644 --- a/2016/abstract-vector-spaces/spanish/sentence_translations.json +++ b/2016/abstract-vector-spaces/spanish/sentence_translations.json @@ -135,7 +135,7 @@ "end": 153.14 }, { - "input": "The output of this new function at any given input, like negative four, is the sum of the same input, negative four.", + "input": "The output of this new function at any given input, like negative four, is the sum of the outputs of f and g when you evaluate them each at that same input, negative four.", "translatedText": "La salida de esta nueva función en cualquier entrada dada, como menos cuatro, es la suma de la misma entrada, menos cuatro.", "from_community_srt": "por ejemplo -4, es la suma de las imágenes de f y g cuando avalúas -4 en cada una de ellas por separado. O, más generalmente,", "n_reviews": 0, diff --git a/2016/brachistochrone/bengali/sentence_translations.json b/2016/brachistochrone/bengali/sentence_translations.json index 602cff31f..600bc166e 100644 --- a/2016/brachistochrone/bengali/sentence_translations.json +++ b/2016/brachistochrone/bengali/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "হ্যাঁ.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "এবং যখন জোহান বার্নোলি প্রথম এটি দেখেছিলেন, আমি ভুল হলে আমাকে সংশোধন করুন, তিনি এটিকে একটি সাইক্লয়েডের ডিফারেনশিয়াল সমীকরণ হিসাবে স্বীকৃতি দিয়েছেন, একটি ঘূর্ণায়মান চাকার রিমের বিন্দু দ্বারা চিহ্নিত আকৃতি।", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/brachistochrone/chinese/sentence_translations.json b/2016/brachistochrone/chinese/sentence_translations.json index 2b3c8686a..278f8e48e 100644 --- a/2016/brachistochrone/chinese/sentence_translations.json +++ b/2016/brachistochrone/chinese/sentence_translations.json @@ -772,7 +772,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "是的。", "model": "google_nmt", "from_community_srt": "嗯， 对。", @@ -835,7 +835,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "当约翰·伯努利第一次看到这个时，如果我错了，请纠正我，他只是将其识别为摆线的微分方程，即滚动轮轮缘上的点所描绘的形状。", "model": "google_nmt", "from_community_srt": "如果我错了， 请纠正：约翰·伯努利最初发觉它时 就认出这就是摆线的微分方程。 摆线是滚轮边缘上的点所描绘的形状。", diff --git a/2016/brachistochrone/english/captions.srt b/2016/brachistochrone/english/captions.srt index 515496bcb..60b6214ef 100644 --- a/2016/brachistochrone/english/captions.srt +++ b/2016/brachistochrone/english/captions.srt @@ -179,12 +179,12 @@ It is. It's a really interesting thing. 46 -00:02:14,420 --> 00:02:17,689 -Most people when they first hear it assume that the shortest path +00:02:14,420 --> 00:02:17,685 +I mean, most people, when they first hear it, assume that the shortest 47 -00:02:17,689 --> 00:02:20,860 -will give the shortest time, that the straight line is the best. +00:02:17,685 --> 00:02:20,860 +path will give the shortest time, that the straight line is the best. 48 00:02:21,620 --> 00:02:26,353 @@ -971,6 +971,14 @@ your objects start at a point a and end at a point b in the xy-space doesn't just look like going from one point to another in the theta-t space. 244 -00:15:33,600 --> 00:15:47,880 -Nevertheless, my challenge to you is this. +00:15:33,600 --> 00:15:38,379 +Nevertheless, my challenge to you is this. Can you find another solution to the + +245 +00:15:38,379 --> 00:15:43,458 +brachistochrone problem by explaining why it must be the case that a time-minimizing + +246 +00:15:43,458 --> 00:15:47,880 +trajectory, when represented in t-theta space, looks like a straight line? diff --git a/2016/brachistochrone/english/sentence_timings.json b/2016/brachistochrone/english/sentence_timings.json index 1a1174be3..dfa3f366c 100644 --- a/2016/brachistochrone/english/sentence_timings.json +++ b/2016/brachistochrone/english/sentence_timings.json @@ -160,7 +160,7 @@ 134.42 ], [ - "Most people when they first hear it assume that the shortest path will give the shortest time, that the straight line is the best.", + "I mean, most people, when they first hear it, assume that the shortest path will give the shortest time, that the straight line is the best.", 134.42, 140.86 ], @@ -685,7 +685,7 @@ 932.84 ], [ - "Nevertheless, my challenge to you is this.", + "Nevertheless, my challenge to you is this. Can you find another solution to the brachistochrone problem by explaining why it must be the case that a time-minimizing trajectory, when represented in t-theta space, looks like a straight line?", 933.6, 947.88 ] diff --git a/2016/brachistochrone/english/transcript.txt b/2016/brachistochrone/english/transcript.txt index 235e686b8..44c61de36 100644 --- a/2016/brachistochrone/english/transcript.txt +++ b/2016/brachistochrone/english/transcript.txt @@ -30,7 +30,7 @@ You want the path to be short, something like a straight line, but you want the But making this quantitative and actually finding the balance with a specific curve, it's not at all obvious and makes for a really interesting problem. It is. It's a really interesting thing. -Most people when they first hear it assume that the shortest path will give the shortest time, that the straight line is the best. +I mean, most people, when they first hear it, assume that the shortest path will give the shortest time, that the straight line is the best. But as you say, it can help to build up some steam by rolling straight down at first, or not necessarily rolling. You could picture it sliding. That doesn't really matter how we phrase it. @@ -135,4 +135,4 @@ You're basically using a differential equation, since what's given is the slope So what's interesting here is that when you look at the solution of the brachistochrone problem not in the xy-plane, but in the t-theta plane, where t is time, theta is the angle of the path, all of the brachistochrone solutions are straight lines, that is to say theta increases at a constant rate with respect to t. When the solution of a curve minimization problem is a straight line, it's highly suggestive that there's some way to view it as a shortest path problem. Here, it's not so straightforward, since the boundary conditions that your objects start at a point a and end at a point b in the xy-space doesn't just look like going from one point to another in the theta-t space. -Nevertheless, my challenge to you is this. \ No newline at end of file +Nevertheless, my challenge to you is this. Can you find another solution to the brachistochrone problem by explaining why it must be the case that a time-minimizing trajectory, when represented in t-theta space, looks like a straight line? \ No newline at end of file diff --git a/2016/brachistochrone/french/sentence_translations.json b/2016/brachistochrone/french/sentence_translations.json index 60d86e640..f98f35137 100644 --- a/2016/brachistochrone/french/sentence_translations.json +++ b/2016/brachistochrone/french/sentence_translations.json @@ -250,7 +250,7 @@ "end": 134.42 }, { - "input": "Most people when they first hear it assume that the shortest path will give the shortest time, that the straight line is the best.", + "input": "I mean, most people, when they first hear it, assume that the shortest path will give the shortest time, that the straight line is the best.", "translatedText": "La plupart des gens, lorsqu’ils l’entendent pour la première fois, supposent que le chemin le plus court donnera le temps le plus court et que la ligne droite est la meilleure.", "from_community_srt": "La plupart des gens pense que que le trajet le plus court donnera le temps le plus court. Que la ligne droite est la meilleure.", "n_reviews": 0, @@ -1087,7 +1087,7 @@ "end": 932.84 }, { - "input": "Nevertheless, my challenge to you is this.", + "input": "Nevertheless, my challenge to you is this. Can you find another solution to the brachistochrone problem by explaining why it must be the case that a time-minimizing trajectory, when represented in t-theta space, looks like a straight line?", "translatedText": "Néanmoins, le défi que je vous lance est le suivant.", "from_community_srt": "Néanmoins, mon défi pour vous est le suivant : Pouvez-vous trouver une autre solution au Brachistochrone en expliquant pourquoi une courbe de temps minimum représentée dans le plan t-θ ressemble à une ligne droite ?", "n_reviews": 0, diff --git a/2016/brachistochrone/german/sentence_translations.json b/2016/brachistochrone/german/sentence_translations.json index e0ef2fdf9..3a4aa9274 100644 --- a/2016/brachistochrone/german/sentence_translations.json +++ b/2016/brachistochrone/german/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "Ja.", "model": "DeepL", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "Und als Johan Bernoulli dies zum ersten Mal sah, erkannte er es einfach als die Differentialgleichung für eine Zykloide, die Form, die durch den Punkt auf der Felge eines rollenden Rades gezeichnet wird, korrigiere mich, wenn ich mich irre.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/brachistochrone/greek/sentence_translations.json b/2016/brachistochrone/greek/sentence_translations.json index f78d655ef..843e2c5a9 100644 --- a/2016/brachistochrone/greek/sentence_translations.json +++ b/2016/brachistochrone/greek/sentence_translations.json @@ -287,7 +287,7 @@ "end": 134.42 }, { - "input": "Most people when they first hear it assume that the shortest path will give the shortest time, that the straight line is the best.", + "input": "I mean, most people, when they first hear it, assume that the shortest path will give the shortest time, that the straight line is the best.", "translatedText": "Οι περισσότεροι άνθρωποι όταν το ακούνε για πρώτη φορά υποθέτουν ότι το συντομότερο μονοπάτι θα δώσει το συντομότερο χρόνο, ότι η ευθεία είναι η καλύτερη.", "model": "google_nmt", "from_community_srt": "Εννοώ οι περισσότεροι άνθρωποι όταν το πρωτακούν, υποθέτουν ότι η συντομότερη διαδρομή θα δώσει το συντομότερο χρονικό διάστημα, ότι η ευθεία γραμμή είναι η καλύτερη.", @@ -1226,7 +1226,7 @@ "end": 932.84 }, { - "input": "Nevertheless, my challenge to you is this.", + "input": "Nevertheless, my challenge to you is this. Can you find another solution to the brachistochrone problem by explaining why it must be the case that a time-minimizing trajectory, when represented in t-theta space, looks like a straight line?", "translatedText": "Παρόλα αυτά, η πρόκληση μου προς εσάς είναι αυτή.", "model": "google_nmt", "from_community_srt": "Παρ 'όλα αυτά, η πρόκλησή μου προς εσάς είναι η εξής: Μπορείτε να βρείτε μια άλλη λύση στο πρόβλημα Brachistochrone εξηγώντας γιατί πρέπει να είναι η περίπτωση που μια φορά την ελαχιστοποίηση τροχιά, όταν παριστάνεται στο διάστημα t-θ, μοιάζει με μια ευθεία γραμμή.", diff --git a/2016/brachistochrone/hebrew/sentence_translations.json b/2016/brachistochrone/hebrew/sentence_translations.json index a60d6a7e8..cb22dc038 100644 --- a/2016/brachistochrone/hebrew/sentence_translations.json +++ b/2016/brachistochrone/hebrew/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "כן.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "וכשיוהאן ברנולי ראה את זה לראשונה, תקן אותי אם אני טועה, הוא פשוט זיהה את זה כמשוואת הדיפרנציאל של ציקלואיד, הצורה המתוארת על ידי הנקודה על שפת גלגל מתגלגל.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/brachistochrone/hindi/sentence_translations.json b/2016/brachistochrone/hindi/sentence_translations.json index 4cbcdf5c4..9d63113d6 100644 --- a/2016/brachistochrone/hindi/sentence_translations.json +++ b/2016/brachistochrone/hindi/sentence_translations.json @@ -609,7 +609,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "हाँ।", "n_reviews": 0, "start": 474.82, @@ -658,7 +658,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "और जब जोहान बर्नौली ने पहली बार इसे देखा, अगर मैं गलत हूं तो मुझे सुधारें, उन्होंने इसे एक चक्रवात के लिए अंतर समीकरण के रूप में पहचाना, एक घूमते हुए पहिये के रिम पर बिंदु द्वारा पता लगाया गया आकार।", "n_reviews": 0, "start": 561.0, diff --git a/2016/brachistochrone/hungarian/sentence_translations.json b/2016/brachistochrone/hungarian/sentence_translations.json index c94ef4a5f..1fadeba27 100644 --- a/2016/brachistochrone/hungarian/sentence_translations.json +++ b/2016/brachistochrone/hungarian/sentence_translations.json @@ -256,7 +256,7 @@ "end": 134.42 }, { - "input": "Most people when they first hear it assume that the shortest path will give the shortest time, that the straight line is the best.", + "input": "I mean, most people, when they first hear it, assume that the shortest path will give the shortest time, that the straight line is the best.", "translatedText": "A legtöbb ember, amikor először hallja, azt feltételezi, hogy a legrövidebb útvonal adja a legrövidebb időt, hogy az egyenes vonal a legjobb.", "model": "DeepL", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 932.84 }, { - "input": "Nevertheless, my challenge to you is this.", + "input": "Nevertheless, my challenge to you is this. Can you find another solution to the brachistochrone problem by explaining why it must be the case that a time-minimizing trajectory, when represented in t-theta space, looks like a straight line?", "translatedText": "Mindazonáltal a kihívásom a következő.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/brachistochrone/indonesian/sentence_translations.json b/2016/brachistochrone/indonesian/sentence_translations.json index e2a245b84..f97d1b756 100644 --- a/2016/brachistochrone/indonesian/sentence_translations.json +++ b/2016/brachistochrone/indonesian/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "Ya.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "Dan ketika Johan Bernoulli pertama kali melihatnya, koreksi saya jika saya salah, dia hanya mengenalinya sebagai persamaan diferensial untuk sikloid, bentuk yang dilacak oleh titik di tepi roda yang berputar.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/brachistochrone/italian/sentence_translations.json b/2016/brachistochrone/italian/sentence_translations.json index d80c3ac45..32f14083d 100644 --- a/2016/brachistochrone/italian/sentence_translations.json +++ b/2016/brachistochrone/italian/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "SÌ.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "E quando Johan Bernoulli lo vide per la prima volta, correggimi se sbaglio, lo riconobbe come l'equazione differenziale di una cicloide, la forma tracciata dal punto sul bordo di una ruota che gira.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/brachistochrone/japanese/sentence_translations.json b/2016/brachistochrone/japanese/sentence_translations.json index 135a2c533..a201ab4f4 100644 --- a/2016/brachistochrone/japanese/sentence_translations.json +++ b/2016/brachistochrone/japanese/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "はい。", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "ヨハン ベルヌーイがこれを初めて見たとき、もし間違っていたら訂正してください。彼はそれがサイクロイドの微分方程式、つまり回転する車輪のリム上の点によってなぞられる形状であると認識しました。", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/brachistochrone/korean/sentence_translations.json b/2016/brachistochrone/korean/sentence_translations.json index 5e103bb18..140791aa2 100644 --- a/2016/brachistochrone/korean/sentence_translations.json +++ b/2016/brachistochrone/korean/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "예.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "그리고 요한 베르누이(Johan Bernoulli)가 이것을 처음 봤을 때, 내가 틀렸다면 정정해 주십시오. 그는 그것을 단지 회전하는 바퀴의 가장자리에 있는 점에 의해 추적되는 모양인 사이클로이드의 미분 방정식으로 인식했습니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/brachistochrone/marathi/sentence_translations.json b/2016/brachistochrone/marathi/sentence_translations.json index 37d38c9b7..888bdffdf 100644 --- a/2016/brachistochrone/marathi/sentence_translations.json +++ b/2016/brachistochrone/marathi/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "होय.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "आणि जेव्हा योहान बर्नौलीने पहिल्यांदा हे पाहिले, मी चुकीचे असल्यास मला दुरुस्त करा, त्याने ते फक्त चक्रीय समीकरण म्हणून ओळखले, रोलिंग व्हीलच्या काठावरील बिंदूद्वारे शोधलेला आकार.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/brachistochrone/polish/sentence_translations.json b/2016/brachistochrone/polish/sentence_translations.json index 1c954cb3a..c70f906ff 100644 --- a/2016/brachistochrone/polish/sentence_translations.json +++ b/2016/brachistochrone/polish/sentence_translations.json @@ -287,7 +287,7 @@ "end": 134.42 }, { - "input": "Most people when they first hear it assume that the shortest path will give the shortest time, that the straight line is the best.", + "input": "I mean, most people, when they first hear it, assume that the shortest path will give the shortest time, that the straight line is the best.", "translatedText": "Większość ludzi, słysząc to po raz pierwszy, zakłada, że najkrótsza ścieżka zapewni najkrótszy czas, a linia prosta jest najlepsza.", "model": "google_nmt", "from_community_srt": "Większość ludzi, gdy to słyszy zakłada, że najkrótszy czas osiągniemy przy najkrótszej ścieżce, czyli przy odcinku.", @@ -1225,7 +1225,7 @@ "end": 932.84 }, { - "input": "Nevertheless, my challenge to you is this.", + "input": "Nevertheless, my challenge to you is this. Can you find another solution to the brachistochrone problem by explaining why it must be the case that a time-minimizing trajectory, when represented in t-theta space, looks like a straight line?", "translatedText": "Niemniej jednak moje wyzwanie dla ciebie jest takie.", "model": "google_nmt", "from_community_srt": "Niemniej jednak, moje wyzwanie dla Ciebie to: Czy możesz znaleźć inne rozwiązanie dla problemu Brachistochrony, wyjaśniając dlaczego musi tak być, że trajektoria minimalizująca czas, reprezentowany w przestrzeni t-θ, wygląda jak linia prosta.", diff --git a/2016/brachistochrone/portuguese/sentence_translations.json b/2016/brachistochrone/portuguese/sentence_translations.json index 4f9073769..a5bcc0c44 100644 --- a/2016/brachistochrone/portuguese/sentence_translations.json +++ b/2016/brachistochrone/portuguese/sentence_translations.json @@ -288,7 +288,7 @@ "end": 134.42 }, { - "input": "Most people when they first hear it assume that the shortest path will give the shortest time, that the straight line is the best.", + "input": "I mean, most people, when they first hear it, assume that the shortest path will give the shortest time, that the straight line is the best.", "translatedText": "A maioria das pessoas, quando ouve isso pela primeira vez, presume que o caminho mais curto proporcionará o tempo mais curto, que a linha reta é a melhor.", "model": "google_nmt", "from_community_srt": "a maioria das pessoas, quando o veem pela primeira vez, assumem que o caminho mais curto gera o tempo mais curto, que a linha reta é o melhor caminho.", @@ -1231,7 +1231,7 @@ "end": 932.84 }, { - "input": "Nevertheless, my challenge to you is this.", + "input": "Nevertheless, my challenge to you is this. Can you find another solution to the brachistochrone problem by explaining why it must be the case that a time-minimizing trajectory, when represented in t-theta space, looks like a straight line?", "translatedText": "No entanto, meu desafio para você é este.", "model": "google_nmt", "from_community_srt": "Ainda assim, meu desafio para você é o seguinte: Você pode encontrar outra solução para o problema da braquistócrona, explicando por que precisa ser o caso em que uma trajetória que minimiza o tempo, quando representada no espaço t-θ, se parece com uma linha reta?", diff --git a/2016/brachistochrone/russian/sentence_translations.json b/2016/brachistochrone/russian/sentence_translations.json index db701e1da..f930355d8 100644 --- a/2016/brachistochrone/russian/sentence_translations.json +++ b/2016/brachistochrone/russian/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "Да.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "И когда Йохан Бернулли впервые увидел это, поправьте меня, если я ошибаюсь, он просто понял, что это дифференциальное уравнение циклоиды, формы, очерченной точкой на ободе катящегося колеса.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/brachistochrone/spanish/sentence_translations.json b/2016/brachistochrone/spanish/sentence_translations.json index 6bbc48dfa..9f0a50db7 100644 --- a/2016/brachistochrone/spanish/sentence_translations.json +++ b/2016/brachistochrone/spanish/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "Sí.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "Y cuando Johan Bernoulli vio esto por primera vez, corríjanme si me equivoco, simplemente lo reconoció como la ecuación diferencial de una cicloide, la forma trazada por el punto en el borde de una rueda en movimiento.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/brachistochrone/tamil/sentence_translations.json b/2016/brachistochrone/tamil/sentence_translations.json index 5c039a315..9c97b6b8b 100644 --- a/2016/brachistochrone/tamil/sentence_translations.json +++ b/2016/brachistochrone/tamil/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "ஆம்.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "ஜோஹன் பெர்னௌலி இதை முதன்முதலில் பார்த்தபோது, நான் தவறாக இருந்தால் என்னைத் திருத்தவும், அவர் அதை ஒரு சைக்ளோயிட்க்கான வேறுபட்ட சமன்பாடு என்று அடையாளம் கண்டுகொண்டார், இது உருளும் சக்கரத்தின் விளிம்பில் உள்ள புள்ளியால் கண்டுபிடிக்கப்பட்டது.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/brachistochrone/telugu/sentence_translations.json b/2016/brachistochrone/telugu/sentence_translations.json index 294158237..0bb652435 100644 --- a/2016/brachistochrone/telugu/sentence_translations.json +++ b/2016/brachistochrone/telugu/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "అవును.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "మరియు జోహన్ బెర్నౌలీ దీన్ని మొదటిసారి చూసినప్పుడు, నేను తప్పుగా ఉంటే నన్ను సరిదిద్దండి, అతను దానిని సైక్లాయిడ్‌కు అవకలన సమీకరణంగా గుర్తించాడు, రోలింగ్ వీల్ యొక్క అంచుపై ఉన్న బిందువు ద్వారా ఆకారాన్ని గుర్తించాడు.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/brachistochrone/turkish/sentence_translations.json b/2016/brachistochrone/turkish/sentence_translations.json index b5b69b874..994ec20b6 100644 --- a/2016/brachistochrone/turkish/sentence_translations.json +++ b/2016/brachistochrone/turkish/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "Evet.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "Ve Johan Bernoulli bunu ilk gördüğünde, yanılıyorsam düzeltin, bunun bir sikloidin diferansiyel denklemi olduğunu, dönen bir tekerleğin kenarındaki nokta tarafından çizilen şeklin olduğunu fark etti.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/brachistochrone/vietnamese/sentence_translations.json b/2016/brachistochrone/vietnamese/sentence_translations.json index 0a094c59f..97989532d 100644 --- a/2016/brachistochrone/vietnamese/sentence_translations.json +++ b/2016/brachistochrone/vietnamese/sentence_translations.json @@ -696,7 +696,7 @@ "end": 474.0 }, { - "input": "Yes.", + "input": "Mm-hmm, yes.", "translatedText": "Đúng.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 560.5 }, { - "input": "And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", + "input": "Mm-hmm. And when Johan Bernoulli first saw this, correct me if I'm wrong, he just recognized it as the differential equation for a cycloid, the shape traced by the point on the rim of a rolling wheel.", "translatedText": "Và khi Johan Bernoulli lần đầu tiên nhìn thấy điều này, hãy sửa cho tôi nếu tôi sai, anh ấy chỉ nhận ra nó là phương trình vi phân của đường cycloid, hình được vẽ bởi điểm trên vành của một bánh xe đang lăn.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/arabic/sentence_translations.json b/2016/change-of-basis/arabic/sentence_translations.json index bb1bb51f5..221b6eb6c 100644 --- a/2016/change-of-basis/arabic/sentence_translations.json +++ b/2016/change-of-basis/arabic/sentence_translations.json @@ -81,7 +81,7 @@ "end": 84.84 }, { - "input": "It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.", + "input": "st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is ti", "translatedText": "يقوم بنقل المتجه الأساسي i-hat إلى الإحداثيات 3 و0 وj-hat إلى 1 و2.", "model": "google_nmt", "from_community_srt": "لدينا نظام الإحداثيات القياسي.", @@ -90,7 +90,7 @@ "end": 91.04 }, { - "input": "So it's represented with a matrix whose columns are 3, 0, and 1, 2.", + "input": "ed up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actual", "translatedText": "لذلك يتم تمثيلها بمصفوفة أعمدتها هي 3، 0، و1، 2.", "model": "google_nmt", "from_community_srt": "ما أود التحدث عنه هنا هي فكرة استخدام مجموعة مختلفة من الأساس ثلاثة أبعاد.", @@ -99,7 +99,7 @@ "end": 95.64 }, { - "input": "Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip.", + "input": "ly scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called th", "translatedText": "ركز على ما يفعله بمتجه معين، وفكر في مدى هذا المتجه، أي الخط الذي يمر عبر نقطة الأصل وطرفه.", "model": "google_nmt", "from_community_srt": "على سبيل المثال ، لنفترض أن لديك صديقًا ، جنيفر الذي يستخدم مجموعة مختلفة من المتجهات الأساسية والتي سأطلق عليها b1 و b2", @@ -108,7 +108,7 @@ "end": 104.16 }, { - "input": "Most vectors are going to get knocked off their span during the transformation.", + "input": "e basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a", "translatedText": "سيتم التخلص من معظم المتجهات خلال عملية التحول.", "model": "google_nmt", "n_reviews": 0, @@ -125,7 +125,7 @@ "end": 115.32 }, { - "input": "But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar.", + "input": "let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up.", "translatedText": "لكن بعض المتجهات الخاصة تظل في امتدادها الخاص، مما يعني أن تأثير المصفوفة على مثل هذا المتجه هو مجرد تمديده أو سحقه، مثل العددية.", "model": "google_nmt", "from_community_srt": "الشخص الذي نود وصفه باستخدامه الإحداثيات [3 ، 2] باستخدام لدينا ناقلات أساس i-hat و j-hat. جنيفر تصف فعلا هذا الناقل مع الإحداثيات [5/3، 1/3] ما يعنيه هذا هو أن طريقة معينة للوصول إلى هذا المتجه", @@ -134,7 +134,7 @@ "end": 127.04 }, { - "input": "For this specific example, the basis vector i-hat is one such special vector.", + "input": "Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vecto", "translatedText": "في هذا المثال المحدد، يعتبر المتجه الأساسي i-hat أحد هذه المتجهات الخاصة.", "model": "google_nmt", "n_reviews": 0, @@ -169,7 +169,7 @@ "end": 164.04 }, { - "input": "It ends up getting stretched by a factor of 2.", + "input": "scale b1 by 5 thirds, scale b2 by 1 third, then add them both togethe", "translatedText": "وينتهي الأمر بالتمدد بعامل 2.", "model": "google_nmt", "n_reviews": 0, @@ -177,7 +177,7 @@ "end": 167.14 }, { - "input": "And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2.", + "input": "r. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she", "translatedText": "ومرة أخرى، الخطية ستعني ضمنًا أن أي متجه آخر على الخط القطري الممتد بواسطة هذا الشخص سوف يتم تمديده بعامل قدره 2.", "model": "google_nmt", "from_community_srt": "لتكون أكثر دقة حول الإعداد هنا لها أول ناقلات أساس b1 هو الشيء الذي نود وصفه مع إحداثيات [2 ، 1] و أساسها الثاني ناقلات b2 هو شيء نود وصفه بـ [-1 ، 1].", @@ -186,7 +186,7 @@ "end": 178.22 }, { - "input": "And for this transformation, those are all the vectors with this special property of staying on their span.", + "input": "thinks of her first coordinate as scali For this specific example, the basis vector i-hat is one such special vector. The span of", "translatedText": "وبالنسبة لهذا التحويل، هذه هي جميع المتجهات التي تتمتع بهذه الخاصية الخاصة وهي البقاء على امتدادها.", "model": "google_nmt", "n_reviews": 0, @@ -194,7 +194,7 @@ "end": 185.18 }, { - "input": "Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2.", + "input": "i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis. What's more, because of the way linear transformations work,", "translatedText": "تلك الموجودة على المحور السيني تتمدد بعامل 3، وتلك الموجودة على هذا الخط القطري تتمدد بعامل 2.", "model": "google_nmt", "from_community_srt": "ولكن من المهم أن ندرك من وجهة نظرها في نظامها تلك المتجهات لها إحداثيات [1، 0] و [0 ، 1] هم ما يعرف معنى الإحداثيات [1 ، 0] و [0 ، 1] في عالمها.", @@ -212,7 +212,7 @@ "end": 198.08 }, { - "input": "As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation.", + "input": "n. A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc t, a way to visualize our coordinate system, and so it depends on our choice of basis", "translatedText": "كما كنت قد خمنت الآن، تسمى هذه المتجهات الخاصة بالمتجهات الذاتية للتحويل، ويرتبط كل ناقل ذاتي به بما يسمى القيمة الذاتية، وهو مجرد العامل الذي يتم من خلاله تمديده أو سحقه أثناء التحويل.", "model": "google_nmt", "from_community_srt": "اسمحوا لي أن أقول كلمة سريعة حول كيف أنا أمثل الأشياء هنا عندما تحرك الفضاء 2D أنا عادة استخدام هذه الشبكة المربعة لكن هذه الشبكة هي مجرد بناء طريقة لتصور نظام الإحداثيات لدينا وذلك يعتمد على اختيارنا من الأساس.", @@ -230,7 +230,7 @@ "end": 225.94 }, { - "input": "In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half.", + "input": "nt as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It", "translatedText": "في مثال آخر، يمكن أن يكون لديك متجه ذاتي قيمته الذاتية سالب 1 نصف، مما يعني أنه يتم قلب المتجه وسحقه بعامل قدره النصف.", "model": "google_nmt", "from_community_srt": "جنيفر قد ترسم شبكتها الخاصة والتي ستكون بناء متساوٍ يعني ليس أكثر من أداة بصرية للمساعدة في متابعة معنى إحداثياتها.", @@ -257,7 +257,7 @@ "end": 249.8 }, { - "input": "If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation.", + "input": "of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewhat during the transformation, k", "translatedText": "إذا تمكنت من العثور على متجه ذاتي لهذا الدوران، أي متجه يظل في امتداده الخاص، فإن ما وجدته هو محور الدوران.", "model": "google_nmt", "from_community_srt": "لذلك ، بعد كل هذا تم إعداده سؤال طبيعي جدا أن نسأل هو كيف نترجم بين أنظمة الإحداثيات؟ إذا ، على سبيل المثال ، تصف جينيفر متجه مع الإحداثيات [-1 ، 2]", @@ -266,7 +266,7 @@ "end": 260.5 }, { - "input": "And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation.", + "input": "nocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", "translatedText": "ومن الأسهل التفكير في دوران ثلاثي الأبعاد من حيث بعض محاور الدوران والزاوية التي يدور بها، بدلاً من التفكير في المصفوفة الكاملة 3x3 المرتبطة بهذا التحويل.", "model": "google_nmt", "from_community_srt": "ماذا سيكون ذلك في نظام الإحداثيات لدينا؟ كيف تترجم من لغتها إلى لنا؟ حسناً ، ما هي إحداثياتنا", @@ -284,7 +284,7 @@ "end": 285.86 }, { - "input": "This pattern shows up a lot in linear algebra.", + "input": "In fact, once you understand matrix vector multiplication as applying", "translatedText": "يظهر هذا النمط كثيرًا في الجبر الخطي.", "model": "google_nmt", "from_community_srt": "ومن وجهة نظرنا يحتوي b1 على إحداثيات [2، 1] و b2 لديه إحداثيات [-1، 1] حتى يمكننا حساب -1 b1 + 2 b2 بالفعل", @@ -293,7 +293,7 @@ "end": 290.02 }, { - "input": "With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors.", + "input": "a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvecto", "translatedText": "مع أي تحويل خطي تصفه المصفوفة، يمكنك فهم ما تفعله من خلال قراءة أعمدة هذه المصفوفة كنقاط هبوط للمتجهات الأساسية.", "model": "google_nmt", "n_reviews": 0, @@ -301,7 +301,7 @@ "end": 299.4 }, { - "input": "But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", + "input": "r with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformati", "translatedText": "لكن في كثير من الأحيان، الطريقة الأفضل للوصول إلى قلب ما يفعله التحويل الخطي فعليًا، بشكل أقل اعتمادًا على نظام الإحداثيات الخاص بك، هي العثور على المتجهات الذاتية والقيم الذاتية.", "model": "google_nmt", "from_community_srt": "كما هي ممثلة في نظام الإحداثيات لدينا ويعمل هذا تحصل على متجه بإحداثيات [-4، 1] إذن ، هكذا سنصف المتجه انها تفكر في [-1 ، 2]", @@ -319,7 +319,7 @@ "end": 326.02 }, { - "input": "Symbolically, here's what the idea of an eigenvector looks like.", + "input": "she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we thi", "translatedText": "رمزيًا، إليك ما تبدو عليه فكرة المتجهات الذاتية.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 339.74 }, { - "input": "What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda.", + "input": "or for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotat", "translatedText": "ما يقوله هذا التعبير هو أن حاصل ضرب المصفوفة والمتجه، A في v، يعطي نفس النتيجة مثل مجرد قياس المتجه الذاتي v ببعض قيمة لامدا.", "model": "google_nmt", "from_community_srt": "مصفوفة تمثل أعمدةها جينيفر ناقلات الأساس يمكن اعتباره بمثابة تحول التي تحرك ناقلات الأساس لدينا ، أنا قبعة وجي هات", @@ -363,7 +363,7 @@ "end": 370.54 }, { - "input": "So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda.", + "input": "never stretch or squish anything, so the length of the vector would remain the same. This pattern shows up a lot in linear algebra. With any linear transformation described by a matrix, you could understand what it's doing by reading of", "translatedText": "لذلك دعونا نبدأ بإعادة كتابة الجانب الأيمن كنوع من ضرب المصفوفة والمتجه، باستخدام مصفوفة لها تأثير في قياس أي متجه بمعامل لامدا.", "model": "google_nmt", "from_community_srt": "قبل التحول الخطي نحن نفكر في هذا الناقل كمجموعة خطية معينة من أساسنا vectors -1 x i-hat + 2 x j-hat.", @@ -372,7 +372,7 @@ "end": 380.62 }, { - "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else.", + "input": "f the columns of this matrix as the landing spots for basis vectors. But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", "translatedText": "ستمثل أعمدة هذه المصفوفة ما يحدث لكل متجه أساسي، ويتم ببساطة ضرب كل متجه أساسي في لامدا، لذلك سيكون لهذه المصفوفة رقم لامدا أسفل القطر، مع وجود أصفار في كل مكان آخر.", "model": "google_nmt", "from_community_srt": "والميزة الرئيسية للتحول الخطي هو أن المتجه الناتج سيكون ذلك نفس التركيبة الخطية ولكن من ناقلات أساس جديد -1 أضعاف المكان الذي تهبط فيه القبعة + مرتين المكان الذي يوجد فيه j-hat.", @@ -381,7 +381,7 @@ "end": 394.32 }, { - "input": "The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal.", + "input": "we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector w", "translatedText": "الطريقة الشائعة لكتابة هذا الرجل هي تحليل لامدا وكتابتها كـ لامدا في i، حيث i هي مصفوفة الهوية مع 1s أسفل القطر.", "model": "google_nmt", "from_community_srt": "فماذا تفعل هذه المصفوفة غيرت مفهومنا الخاطئ لما جينيفر يعني في المتجه الفعلي الذي تشير إليه إلى.", @@ -399,7 +399,7 @@ "end": 411.86 }, { - "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector.", + "input": "that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, with v as the eigenvector", "translatedText": "إذًا ما لدينا الآن هو مصفوفة جديدة، A ناقص لامدا مضروبًا في الهوية، ونحن نبحث عن متجه v بحيث تعطي هذه المصفوفة الجديدة مضروبًا في v المتجه صفرًا.", "model": "google_nmt", "from_community_srt": "لكن من الناحية العددية ، إنها تترجم المتجه موصوفة بلغتها إلى لغتنا. ما الذي جعله ينقر في النهاية بالنسبة لي كان يفكر في كيف يأخذ الفهم الخاطئ لدينا مما تعنيه جينيفر", @@ -425,7 +425,7 @@ "end": 433.64 }, { - "input": "And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", + "input": "and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Je", "translatedText": "وإذا شاهدت الفصلين 5 و6، ستعرف أن الطريقة الوحيدة التي يمكن أن يصبح بها حاصل ضرب مصفوفة ذات متجه غير صفري صفرًا هي أن يؤدي التحويل المرتبط بتلك المصفوفة إلى سحق الفضاء إلى بُعد أقل.", "model": "google_nmt", "from_community_srt": "ماذا عن الذهاب في الاتجاه الآخر؟ في المثال ، استخدمت هذا الفيديو سابقًا عندما يكون لدي متجه مع الإحداثيات [3 ، 2] في نظامنا", @@ -460,7 +460,7 @@ "end": 470.28 }, { - "input": "As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes.", + "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose c", "translatedText": "ومع تغير قيمة لامدا، تتغير المصفوفة نفسها، وبالتالي يتغير محدد المصفوفة.", "model": "google_nmt", "from_community_srt": "في الواقع ، خاصة عندما تعمل في أكثر من بعدين كنت تستخدم جهاز كمبيوتر لحساب المصفوفة هذا يمثل هذا العكس.", @@ -503,7 +503,7 @@ "end": 498.6 }, { - "input": "So this is kind of a lot, but let's unravel what this is saying.", + "input": "And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's importa", "translatedText": "إذن هذا كثير نوعًا ما، لكن دعونا نكشف ما يقوله هذا.", "model": "google_nmt", "n_reviews": 0, @@ -511,7 +511,7 @@ "end": 502.96 }, { - "input": "When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line.", + "input": "nt that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication So let's start by rewriting", "translatedText": "عندما تساوي لامدا 1، فإن المصفوفة A ناقص لامدا مضروبة في الهوية تسحق المساحة على الخط.", "model": "google_nmt", "from_community_srt": "الذي يعمل ليكون [5/3 ، 1/3] لذلك ، باختصار هو كيفية ترجمة وصف الفرد ثلاثة أبعاد ذهابا وإيابا بين أنظمة الإحداثيات.", @@ -520,7 +520,7 @@ "end": 509.56 }, { - "input": "That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector.", + "input": "that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a fact", "translatedText": "هذا يعني أن هناك متجهًا غير صفري v بحيث يكون A ناقص lambda مضروبًا في الهوية مضروبًا في v يساوي المتجه الصفري.", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 518.56 }, { - "input": "And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A.", + "input": "or of lambda. The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else. the columns of our matrix.", "translatedText": "وتذكر أن سبب اهتمامنا بهذا هو أنه يعني A في v يساوي lambda في v، وهو ما يمكنك قراءته كقول إن المتجه v هو متجه ذاتي لـ A، ويظل في امتداده الخاص أثناء التحويل A.", "model": "google_nmt", "from_community_srt": "المصفوفة التي تمثل الأعمدة جينيفر ناقلات الأساس لكن مكتوب في إحداثياتنا يترجم نواقل من لغتها إلى لغتنا. والمصفوفة العكسية تفعل العكس. لكن المتجهات ليست الشيء الوحيد الذي نحن وصف باستخدام الاحداثيات.", @@ -537,7 +537,7 @@ "end": 537.28 }, { - "input": "In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place.", + "input": "But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following", "translatedText": "في هذا المثال، القيمة الذاتية المقابلة هي 1، لذا فإن v ستبقى ثابتة في مكانها.", "model": "google_nmt", "from_community_srt": "لهذا الجزء التالي من المهم أن تكون مرتاحًا تمثل التحولات مع المصفوفات وأنك تعرف كيف الضرب المصفوفة", @@ -555,7 +555,7 @@ "end": 549.5 }, { - "input": "This is the kind of thing I mentioned in the introduction.", + "input": "-hat in the first pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v.", "translatedText": "وهذا هو النوع الذي ذكرته في المقدمة.", "model": "google_nmt", "from_community_srt": "وقفة بالتأكيد وإلقاء نظرة على الفصول 3 و 4 إذا كان أي من ذلك يشعر بعدم الارتياح.", @@ -564,7 +564,7 @@ "end": 555.64 }, { - "input": "If you didn't have a solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to describe those landing spots in her language. Here's a common way to think", "translatedText": "إذا لم يكن لديك فهم قوي للمحددات وسبب ارتباطها بأنظمة المعادلات الخطية التي لها حلول غير صفرية، فإن تعبيرًا مثل هذا سيبدو غريبًا تمامًا.", "model": "google_nmt", "from_community_srt": "خذ بعين الاعتبار بعض التحولات الخطية مثل دوران بمقدار 90 درجة عكس عقارب الساعة. عندما كنت وأنا نمثل هذا مع المصفوفة نتابع حيث المتجهات أساس i-hat و ي-ك كل ذهاب.", @@ -609,7 +609,7 @@ "end": 608.84 }, { - "input": "To figure out what the eigenvectors are that actually have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero.", + "input": "To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry. Now imagine tweaking lambda, turning a knob to change its value. As that value of lambda changes, the matrix itself change", "translatedText": "لمعرفة المتجهات الذاتية التي لها بالفعل إحدى هذه القيم الذاتية، لنفترض أن لامدا تساوي 2، قم بتوصيل قيمة لامدا هذه إلى المصفوفة ثم حدد المتجهات التي ترسلها هذه المصفوفة المعدلة قطريًا إلى الصفر.", "model": "google_nmt", "from_community_srt": "تمثل تلك الأعمدة أين متجهنا الأساسي أنا قبعة وجي هات. لكن المصفوفة التي تريدها جنيفر يجب أن تمثل حيث نواقل أساسها الأرض وتحتاج إلى وصف تلك النقاط الهبوط في لغتها. إليك طريقة شائعة للتفكير في كيفية حدوث ذلك فعله.", @@ -698,7 +698,7 @@ "end": 681.94 }, { - "input": "The only roots of that polynomial are the imaginary numbers, i and negative i.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the oute", "translatedText": "الجذور الوحيدة لذلك كثير الحدود هي الأعداد التخيلية، i والسالب i.", "model": "google_nmt", "from_community_srt": "يأخذ في متجه من لغتها وتبصق النسخة المحولة لذلك متجه بلغتها لهذا المثال بالتحديد عندما تبدو متجهات جينيفر الأساسية مثل [2 ، 1] و [-1 ، 1] بلغتنا", @@ -724,7 +724,7 @@ "end": 699.82 }, { - "input": "This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1.", + "input": "meone else sees it. For those of you wondering why we care about alternate coordinate systems, the next vi", "translatedText": "يؤدي هذا إلى تثبيت i-hat في مكانه وتحريك j-hat 1، بحيث تحتوي المصفوفة على أعمدة 1 و0 و1 و1.", "model": "google_nmt", "n_reviews": 0, @@ -759,7 +759,7 @@ "end": 726.54 }, { - "input": "And the only root of this expression is lambda equals 1.", + "input": "he identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals la", "translatedText": "والجذر الوحيد لهذا التعبير هو لامدا يساوي 1.", "model": "google_nmt", "from_community_srt": "تمثل تلك المصفوفة الوسطى تحولًا من نوع ما ، كما ترونه والمصفوفات الخارجية تمثل التعاطف ، التحول في المنظور", @@ -768,7 +768,7 @@ "end": 732.86 }, { - "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.", + "input": "mbda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corresponding eigenvalue is", "translatedText": "وهذا يتماشى مع ما نراه هندسيًا، وهو أن جميع المتجهات الذاتية لها قيمة ذاتية 1.", "model": "google_nmt", "n_reviews": 0, @@ -827,7 +827,7 @@ "end": 796.38 }, { - "input": "For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2.", + "input": "f equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "على سبيل المثال، ربما يتم تحجيم i-hat بمقدار سالب 1 ويتم تحجيم j-hat بمقدار 2.", "model": "google_nmt", "n_reviews": 0, @@ -851,7 +851,7 @@ "end": 825.42 }, { - "input": "And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.", + "input": "nd compute the determinant. Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda. Since lambda can only be an eigenvalue i", "translatedText": "وطريقة تفسير ذلك هي أن جميع المتجهات الأساسية هي متجهات ذاتية، والمدخلات القطرية لهذه المصفوفة هي قيمها الذاتية.", "model": "google_nmt", "n_reviews": 0, @@ -867,7 +867,7 @@ "end": 841.06 }, { - "input": "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", + "input": "u can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3. To figure out what the eigenvectors are that actu", "translatedText": "أحد أهمها هو أنه من الأسهل حساب ما سيحدث إذا قمت بضرب هذه المصفوفة في نفسها عدة مرات.", "model": "google_nmt", "n_reviews": 0, @@ -875,7 +875,7 @@ "end": 848.34 }, { - "input": "Since all one of these matrices does is scale each basis vector by some eigenvalue, applying that matrix many times, say 100 times, is just going to correspond to scaling each basis vector by the 100th power of the corresponding eigenvalue.", + "input": "ally have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero. If you computed this the way you would any other linear system, you'd see that the solutions are all the vectors on the diagonal line spanned by negative 1, 1.", "translatedText": "نظرًا لأن كل ما تفعله إحدى هذه المصفوفات هو قياس كل متجه أساسي بواسطة بعض القيمة الذاتية، فإن تطبيق هذه المصفوفة عدة مرات، على سبيل المثال 100 مرة، سيتوافق فقط مع قياس كل متجه أساسي بمقدار الأس 100 من القيمة الذاتية المقابلة.", "model": "google_nmt", "n_reviews": 0, @@ -891,7 +891,7 @@ "end": 869.68 }, { - "input": "Really, try it for a moment.", + "input": "x, 3, 0, 1, 2, has the effect of stretching all those vectors by a factor of 2.", "translatedText": "حقا، حاول ذلك للحظة.", "model": "google_nmt", "n_reviews": 0, @@ -923,7 +923,7 @@ "end": 896.54 }, { - "input": "I talked about change of basis last video, but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", + "input": "is, notice what happens. Its matrix has columns 0, 1 and negative 1, 0. Subtract off lambda from the diagonal elements and look for when the determinant is zero. In this case, you get the polynomial lambda squared plus 1. The only roots of that polynomia", "translatedText": "لقد تحدثت عن تغيير الأساس في الفيديو الأخير، ولكنني سأقوم بتذكير سريع جدًا هنا بكيفية التعبير عن التحويل المكتوب حاليًا في نظامنا الإحداثي إلى نظام مختلف.", "model": "google_nmt", "n_reviews": 0, @@ -931,7 +931,7 @@ "end": 907.04 }, { - "input": "Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors, then make those coordinates the columns of a matrix, known as the change of basis matrix.", + "input": "l are the imaginary numbers, i and negative i. The fact that there are no real number solutions indicates that there are no eigenvectors. Another pretty interesting example worth holding in the back of your mind is a shear. This fixes i-hat in place and moves j-hat 1 over, so its mat", "translatedText": "خذ إحداثيات المتجهات التي تريد استخدامها كأساس جديد، وهو ما يعني في هذه الحالة المتجهات الذاتية لدينا، ثم اجعل تلك الإحداثيات أعمدة مصفوفة، تُعرف باسم مصفوفة تغيير الأساس.", "model": "google_nmt", "n_reviews": 0, @@ -955,7 +955,7 @@ "end": 946.68 }, { - "input": "This is because it represents working in a coordinate system where what happens to the basis vectors is that they get scaled during the transformation.", + "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1. Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a lin", "translatedText": "وذلك لأنه يمثل العمل في نظام إحداثي حيث ما يحدث للمتجهات الأساسية هو أنه يتم قياسها أثناء التحويل.", "model": "google_nmt", "n_reviews": 0, @@ -971,7 +971,7 @@ "end": 961.56 }, { - "input": "So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis, compute the 100th power in that system, then convert back to our standard system.", + "input": "A simple example is a matrix that scales everything by 2. The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue. Now is another good time to pause and ponder some of this before I move on to the last topic.", "translatedText": "لذا، على سبيل المثال، إذا كنت بحاجة إلى حساب القوة رقم 100 لهذه المصفوفة، فسيكون من الأسهل كثيرًا التحويل إلى الأساس الذاتي، وحساب القوة رقم 100 في هذا النظام، ثم التحويل مرة أخرى إلى نظامنا القياسي.", "model": "google_nmt", "n_reviews": 0, @@ -979,7 +979,7 @@ "end": 975.68 }, { - "input": "You can't do this with all transformations.", + "input": "I want to finish off here with the idea of an eigenbasis, which relies heavily on ideas from the last video. Take a look at what h", "translatedText": "لا يمكنك القيام بذلك مع كل التحولات.", "model": "google_nmt", "n_reviews": 0, @@ -987,7 +987,7 @@ "end": 978.32 }, { - "input": "A shear, for example, doesn't have enough eigenvectors to span the full space.", + "input": "appens if our basis vectors just so happen to be eigenvectors. For example, maybe i-hat is scale", "translatedText": "القص، على سبيل المثال، لا يحتوي على ما يكفي من المتجهات الذاتية لتغطية المساحة الكاملة.", "model": "google_nmt", "n_reviews": 0, @@ -995,7 +995,7 @@ "end": 982.98 }, { - "input": "But if you can find an eigenbasis, it makes matrix operations really lovely.", + "input": "d by negative 1 and j-hat is scaled by 2. Writing their new coordinates as the columns of a matrix, notice t", "translatedText": "ولكن إذا تمكنت من العثور على الأساس الذاتي، فهذا يجعل عمليات المصفوفة رائعة حقًا.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/bengali/sentence_translations.json b/2016/change-of-basis/bengali/sentence_translations.json index db32af3f2..fc5c4d660 100644 --- a/2016/change-of-basis/bengali/sentence_translations.json +++ b/2016/change-of-basis/bengali/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates. ", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we d ", "translatedText": "যদি আমার কাছে 2D স্পেসে একটি ভেক্টর বসে থাকে, তাহলে আমাদের কাছে স্থানাঙ্কের সাথে এটি বর্ণনা করার একটি আদর্শ উপায় আছে। ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 27.48 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up. ", + "input": "oing this and what does this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin ", "translatedText": "এই ক্ষেত্রে, ভেক্টরের স্থানাঙ্ক রয়েছে 3, 2, যার অর্থ তার লেজ থেকে এর ডগায় যাওয়ার জন্য তিনটি ইউনিট ডানদিকে এবং দুটি ইউনিট উপরে নিয়ে যাওয়া জড়িত। ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.36 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up. ", + "input": "not so much that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of ", "translatedText": "আপনি সেই প্রথম স্থানাঙ্কটিকে স্কেলিং আই-হ্যাট হিসাবে মনে করেন, দৈর্ঘ্য 1 সহ ভেক্টর ডানদিকে নির্দেশ করে, যখন দ্বিতীয় স্থানাঙ্কটি j-হ্যাটকে স্কেল করে, দৈর্ঘ্য 1 সহ ভেক্টরটি সোজা উপরে নির্দেশ করে। ", "model": "google_nmt", "n_reviews": 0, @@ -32,7 +32,7 @@ "end": 57.14 }, { - "input": "The tip-to-tail sum of those two scaled vectors is what the coordinates are meant to describe. ", + "input": "the topics that precede it. Most important here is that you know how to think about matrices as linear transformations, but you also need to be comforta ", "translatedText": "এই দুটি স্কেল করা ভেক্টরের টিপ-টু-টেইল যোগফল হল স্থানাঙ্কগুলিকে বর্ণনা করার জন্য। ", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors. ", + "input": "of that is tied up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors a ", "translatedText": "আমি এখানে যে বিষয়ে কথা বলতে চাই তা হল ভিত্তি ভেক্টরের একটি ভিন্ন সেট ব্যবহার করার ধারণা। ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 133.36 }, { - "input": "In general, whenever Jennifer uses coordinates to describe a vector, she thinks of her first coordinate as scaling b1, the second coordinate as scaling b2, and she adds the results. ", + "input": "vector, b2, points left and up. Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describ ", "translatedText": "সাধারণভাবে, যখনই জেনিফার কোন ভেক্টরকে বর্ণনা করার জন্য স্থানাঙ্ক ব্যবহার করেন, তখন তিনি তার প্রথম স্থানাঙ্কটিকে স্কেলিং b1 হিসাবে, দ্বিতীয় স্থানাঙ্কটিকে স্কেলিং b2 হিসাবে মনে করেন এবং তিনি ফলাফলগুলি যোগ করেন। ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 166.8 }, { - "input": "They are what define the meaning of the coordinates 1,0 and 0,1 in her world. ", + "input": "then add them both together. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. ", "translatedText": "তারাই তার জগতে স্থানাঙ্ক 1,0 এবং 0,1 এর অর্থ সংজ্ঞায়িত করে। ", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 185.86 }, { - "input": "But that grid is just a construct, a way to visualize our coordinate system, and so it depends on our choice of basis. ", + "input": "ample, the basis vector i-hat is one such special vector. The span of i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 ", "translatedText": "কিন্তু সেই গ্রিডটি কেবল একটি নির্মাণ, আমাদের সমন্বয় ব্যবস্থাকে কল্পনা করার একটি উপায় এবং তাই এটি আমাদের পছন্দের ভিত্তিতে নির্ভর করে। ", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid. ", + "input": "times itself, still on that x-axis. What's more, because of the way linear transformations work, ", "translatedText": "স্পেস নিজেই কোন অন্তর্নিহিত গ্রিড নেই. ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 198.08 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. ", + "input": "emains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. ", "translatedText": "যদিও তার উৎপত্তি আসলে আমাদের সাথে সঙ্গতিপূর্ণ হবে, যেহেতু সবাই 0,0 এর অর্থ কী তা নিয়ে একমত। ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 223.72 }, { - "input": "So after all this is set up, a pretty natural question to ask is how we translate between coordinate systems. ", + "input": "self has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct meant as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her ori ", "translatedText": "তাই এই সব সেট আপ করার পরে, একটি সুন্দর স্বাভাবিক প্রশ্ন জিজ্ঞাসা করা হয় কিভাবে আমরা সমন্বয় সিস্টেমের মধ্যে অনুবাদ করি। ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 227.28 }, { - "input": "How do you translate from her language to ours? ", + "input": "hould mean. It's the thing that you get when you scale any vector by zero. ", "translatedText": "আপনি কিভাবে তার ভাষা থেকে আমাদের অনুবাদ করবেন? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 230.76 }, { - "input": "Well, what her coordinates are saying is that this vector is negative 1 times b1 plus 2 times b2. ", + "input": "But the direction of her axes and Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. ", "translatedText": "ওয়েল, তার স্থানাঙ্ক যা বলছে যে এই ভেক্টর নেতিবাচক 1 গুণ b1 প্লাস 2 বার b2। ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 231.76 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1. ", + "input": "Any other vector is going to get rotated somewhat during the transformation, knocked off the line that it spans. ks of as negative 1, 2. ", "translatedText": "এবং আমাদের দৃষ্টিকোণ থেকে, b1 এর স্থানাঙ্ক রয়েছে 2, 1 এবং b2 এর স্থানাঙ্ক নেতিবাচক 1, 1 রয়েছে। ", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 254.28 }, { - "input": "It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vectors in our language. ", + "input": "tand matrix vector multiplication as applying a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or ", "translatedText": "এটি ম্যাট্রিক্স ভেক্টর গুণন, একটি ম্যাট্রিক্স সহ যার কলামগুলি আমাদের ভাষায় জেনিফারের ভিত্তি ভেক্টরকে উপস্থাপন করে। ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 257.08 }, { - "input": "In fact, once you understand matrix vector multiplication as applying a certain linear transformation, say by watching what I view to be the most important video in this series, Chapter 3, there's a pretty intuitive way to think about what's going on here. ", + "input": "the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. ", "translatedText": "প্রকৃতপক্ষে, একবার আপনি ম্যাট্রিক্স ভেক্টর গুণকে একটি নির্দিষ্ট রৈখিক রূপান্তর প্রয়োগ হিসাবে বুঝতে পারলে, আমি এই সিরিজের সবচেয়ে গুরুত্বপূর্ণ ভিডিও, অধ্যায় 3-তে যা দেখছি তা দেখে বলুন, এখানে কী ঘটছে তা ভাবার একটি সুন্দর স্বজ্ঞাত উপায় রয়েছে। ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 384.3 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse. ", + "input": "coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, ", "translatedText": "অনুশীলনে, বিশেষ করে যখন আপনি দুইটির বেশি মাত্রায় কাজ করছেন, আপনি ম্যাট্রিক্স গণনা করার জন্য একটি কম্পিউটার ব্যবহার করবেন যা আসলে এই বিপরীতটি উপস্থাপন করে। ", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 389.6 }, { - "input": "In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. ", + "input": "with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. ", "translatedText": "এই ক্ষেত্রে, ভিত্তি ম্যাট্রিক্সের পরিবর্তনের বিপরীতে যেটির কলামগুলি জেনিফারের ভিত্তি হিসাবে কাজ করে শেষ পর্যন্ত কলামগুলি 1 তৃতীয়াংশ, ঋণাত্মক 1 তৃতীয়াংশ এবং 1 তৃতীয়াংশ, 2 তৃতীয়াংশ রয়েছে৷ ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 443.9 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy. ", + "input": "es, and that you know how matrix multiplication So let's start by rewriting that right-hand ", "translatedText": "অবশ্যই বিরতি দিন এবং অধ্যায় 3 এবং 4 দেখুন যদি এর মধ্যে কেউ অস্বস্তি বোধ করে। ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 446.48 }, { - "input": "Consider some linear transformation, like a 90 degree counterclockwise rotation. ", + "input": "side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda. ", "translatedText": "কিছু রৈখিক রূপান্তর বিবেচনা করুন, যেমন একটি 90 ডিগ্রি ঘড়ির কাঁটার বিপরীতে ঘূর্ণন। ", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 451.06 }, { - "input": "When you and I represent this with a matrix, we follow where the basis vectors i-hat and j-hat each go. ", + "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this ", "translatedText": "যখন আপনি এবং আমি এটিকে একটি ম্যাট্রিক্স দিয়ে উপস্থাপন করি, তখন আমরা অনুসরণ করি যেখানে ভিত্তি ভেক্টর i-hat এবং j-hat প্রতিটি যায়। ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 477.24 }, { - "input": "How would Jennifer describe this same 90 degree rotation of space? ", + "input": "pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v. ", "translatedText": "কিভাবে জেনিফার স্থানের এই একই 90 ডিগ্রী ঘূর্ণন বর্ণনা করবে? ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 484.26 }, { - "input": "But that's not quite right. ", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and ", "translatedText": "কিন্তু এটা পুরোপুরি ঠিক নয়। ", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 515.02 }, { - "input": "Then apply the transformation matrix to what you get by multiplying it on the left. ", + "input": "zero is if the transformation associated with that matrix squishes space into a lower dimension. And that squishification corresponds to a zero determinant for the matr ", "translatedText": "তারপর বাম দিকে গুন করে আপনি যা পাবেন তাতে রূপান্তর ম্যাট্রিক্স প্রয়োগ করুন। ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 515.62 }, { - "input": "This tells us where that vector lands, but still in our language. ", + "input": "ix. To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtract ", "translatedText": "এটি আমাদের বলে যে ভেক্টরটি কোথায় অবতরণ করে, তবে এখনও আমাদের ভাষায়। ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/chinese/sentence_translations.json b/2016/change-of-basis/chinese/sentence_translations.json index 0a1e1510b..d695dca4b 100644 --- a/2016/change-of-basis/chinese/sentence_translations.json +++ b/2016/change-of-basis/chinese/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates. ", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we d ", "translatedText": "如果我在 2D 空间中有一个向量，我们 就有一个标准的方法来用坐标来描述它。", "model": "google_nmt", "from_community_srt": "如果我在二维空间中有一个向量， 我们就有一种用坐标表示它的标准方法 在这种情况下， 这个向量的坐标为(3, 2) 也就意味着从它的起点到它的尖端， 需要向右移动3个单位，", @@ -9,7 +9,7 @@ "end": 27.48 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up. ", + "input": "oing this and what does this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin ", "translatedText": "在本例中，矢量的坐标为 3, 2，这意味着从尾部到尖 端需要向右移动 3 个单位，向上移动 2 个单位。", "model": "google_nmt", "from_community_srt": "并向上移动2个单位 现在以更加线性代数的方法来描述坐标 是将这些数看作拉伸或压缩向量的标量 你将第一个坐标看作缩放i帽的标量 i帽就是指向右方且长度为1的向量 第二个坐标看作缩放j帽的标量", @@ -26,7 +26,7 @@ "end": 42.36 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up. ", + "input": "not so much that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of ", "translatedText": "您可以将第一个坐标视为缩放 i-hat，即长 度为 1 指向右侧的向量，而第二个坐标缩放 j-hat，即长度为 1 指向上方的向量。", "model": "google_nmt", "from_community_srt": "j帽就是指向正上方且长度为1的向量 这两个经过缩放的向量的和就是坐标所要描述的向量 你可以把这两个特殊的向量 看作封装于我们这个坐标系中的隐含假设 第一个数字表示向右的运动 第二个数字表示向上的运动", @@ -35,7 +35,7 @@ "end": 57.14 }, { - "input": "The tip-to-tail sum of those two scaled vectors is what the coordinates are meant to describe. ", + "input": "the topics that precede it. Most important here is that you know how to think about matrices as linear transformations, but you also need to be comforta ", "translatedText": "这两个缩放向量的首尾之和就是坐标所要描述的内容。", "model": "google_nmt", "n_reviews": 0, @@ -68,7 +68,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors. ", + "input": "of that is tied up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors a ", "translatedText": "我想在这里讨论的是使用一组不同的基向量的想法。", "model": "google_nmt", "from_community_srt": "都被称为一个坐标系 而其中两个特殊的向量 - i帽和j帽 被称为我们这个标准坐标系的基向量 我想在此讨论的是使用另一组基向量的思想 比如说你有一个朋友 - 詹妮弗 她使用着一组不同的基向量，", @@ -128,7 +128,7 @@ "end": 133.36 }, { - "input": "In general, whenever Jennifer uses coordinates to describe a vector, she thinks of her first coordinate as scaling b1, the second coordinate as scaling b2, and she adds the results. ", + "input": "vector, b2, points left and up. Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describ ", "translatedText": "一般来说，每当 Jennifer 使用坐标来描述向量时，她都会将第一 个坐标视为缩放 b1，将第二个坐标视为缩放 b2，然后将结果相加。", "model": "google_nmt", "from_community_srt": "b2乘以1/3， 再将两个结果相加 很快我就会向你展示如何计算出这两个数 - 5/3和1/3 总之，", @@ -163,7 +163,7 @@ "end": 166.8 }, { - "input": "They are what define the meaning of the coordinates 1,0 and 0,1 in her world. ", + "input": "then add them both together. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. ", "translatedText": "它们定义了她世界中坐标 1,0 和 0,1 的含义。", "model": "google_nmt", "n_reviews": 0, @@ -206,7 +206,7 @@ "end": 185.86 }, { - "input": "But that grid is just a construct, a way to visualize our coordinate system, and so it depends on our choice of basis. ", + "input": "ample, the basis vector i-hat is one such special vector. The span of i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 ", "translatedText": "但该网格只是一个构造，一种可视化坐标系 的方法，因此它取决于我们对基础的选择。", "model": "google_nmt", "from_community_srt": "1) （因为）它们就是定义坐标(1,", @@ -215,7 +215,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid. ", + "input": "times itself, still on that x-axis. What's more, because of the way linear transformations work, ", "translatedText": "空间本身没有内在的网格。", "model": "google_nmt", "from_community_srt": "0)和(0, 1)含义的向量 所以，", @@ -232,7 +232,7 @@ "end": 198.08 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. ", + "input": "emains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. ", "translatedText": "但她的起源实际上与我们的起源一致，因 为每个人都同意坐标 0,0 的含义。", "model": "google_nmt", "from_community_srt": "我们实际上说着不同的语言 虽然我们都在关注空间中的同一个向量 但是詹妮弗用不同的语言和数字来描述它 我再快速讲讲我是如何表示这些东西的 在制作二维空间的动画时，", @@ -259,7 +259,7 @@ "end": 223.72 }, { - "input": "So after all this is set up, a pretty natural question to ask is how we translate between coordinate systems. ", + "input": "self has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct meant as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her ori ", "translatedText": "因此，在完成所有这些设置之后，一个很自然 的问题是我们如何在坐标系之间进行转换。", "model": "google_nmt", "n_reviews": 0, @@ -275,7 +275,7 @@ "end": 227.28 }, { - "input": "How do you translate from her language to ours? ", + "input": "hould mean. It's the thing that you get when you scale any vector by zero. ", "translatedText": "你如何将她的语言翻译成我们的语言？", "model": "google_nmt", "n_reviews": 0, @@ -283,7 +283,7 @@ "end": 230.76 }, { - "input": "Well, what her coordinates are saying is that this vector is negative 1 times b1 plus 2 times b2. ", + "input": "But the direction of her axes and Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. ", "translatedText": "嗯，她的坐标说明的是这个向量是负的 1 乘以 b1 加 2 乘以 b2。", "model": "google_nmt", "n_reviews": 0, @@ -291,7 +291,7 @@ "end": 231.76 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1. ", + "input": "Any other vector is going to get rotated somewhat during the transformation, knocked off the line that it spans. ks of as negative 1, 2. ", "translatedText": "从我们的角度来看，b1 的坐标为 2, 1，b2 的坐标为负 1, 1。", "model": "google_nmt", "n_reviews": 0, @@ -333,7 +333,7 @@ "end": 254.28 }, { - "input": "It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vectors in our language. ", + "input": "tand matrix vector multiplication as applying a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or ", "translatedText": "它是矩阵向量乘法，矩阵的列代表我们语 言中 Jennifer 的基向量。", "model": "google_nmt", "n_reviews": 0, @@ -341,7 +341,7 @@ "end": 257.08 }, { - "input": "In fact, once you understand matrix vector multiplication as applying a certain linear transformation, say by watching what I view to be the most important video in this series, Chapter 3, there's a pretty intuitive way to think about what's going on here. ", + "input": "the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. ", "translatedText": "事实上，一旦您将矩阵向量乘法理解为应用某种线性变换 ，例如观看我认为是本系列中最重要的视频（第 3 章 ），就会有一种非常直观的方式来思考这里发生的事情。", "model": "google_nmt", "from_community_srt": "比如说， 如果詹妮弗用坐标(-1, 2)描述一个向量 那么这个向量在我们的坐标系中如何描述？ 你如何从她的语言转化到我们的语言？", @@ -456,7 +456,7 @@ "end": 384.3 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse. ", + "input": "coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, ", "translatedText": "在实践中，特别是当您在二维以上工作时，您 将使用计算机来计算实际表示该逆的矩阵。", "model": "google_nmt", "from_community_srt": "不过使用的是新的基向量 -1乘以变换后的i帽，", @@ -465,7 +465,7 @@ "end": 389.6 }, { - "input": "In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. ", + "input": "with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. ", "translatedText": "在这种情况下，以 Jennifer 基为列的基矩 阵变化的逆矩阵最终得出列为 1 三分之一、负 1 三分之一、以及 1 三分之一、2 三分之一。", "model": "google_nmt", "from_community_srt": "加上2乘以变换后的j帽 因此这个矩阵所做的 是将我们对詹妮弗的向量的误解，", @@ -525,7 +525,7 @@ "end": 443.9 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy. ", + "input": "es, and that you know how matrix multiplication So let's start by rewriting that right-hand ", "translatedText": "如果有任何感到不安的地方，一定要停下来看看第三章和第四章。", "model": "google_nmt", "from_community_srt": "2)的向量 我如何计算出它在詹妮弗的坐标系中的坐标为(5/3,", @@ -534,7 +534,7 @@ "end": 446.48 }, { - "input": "Consider some linear transformation, like a 90 degree counterclockwise rotation. ", + "input": "side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda. ", "translatedText": "考虑一些线性变换，例如逆时针旋转 90 度。", "model": "google_nmt", "from_community_srt": "1/3)？", @@ -543,7 +543,7 @@ "end": 451.06 }, { - "input": "When you and I represent this with a matrix, we follow where the basis vectors i-hat and j-hat each go. ", + "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this ", "translatedText": "当你和我用矩阵表示这一点时，我们遵循基向量 i-hat 和 j-hat 各自的走向。", "model": "google_nmt", "from_community_srt": "之前的基变换矩阵从詹妮弗的语言转化到我们的语言 你就此入手，", @@ -577,7 +577,7 @@ "end": 477.24 }, { - "input": "How would Jennifer describe this same 90 degree rotation of space? ", + "input": "pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v. ", "translatedText": "詹妮弗会如何描述同样的 90 度空间旋转？", "model": "google_nmt", "n_reviews": 0, @@ -594,7 +594,7 @@ "end": 484.26 }, { - "input": "But that's not quite right. ", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and ", "translatedText": "但这并不完全正确。", "model": "google_nmt", "from_community_srt": "其逆矩阵的两列为(1/3,", @@ -646,7 +646,7 @@ "end": 515.02 }, { - "input": "Then apply the transformation matrix to what you get by multiplying it on the left. ", + "input": "zero is if the transformation associated with that matrix squishes space into a lower dimension. And that squishification corresponds to a zero determinant for the matr ", "translatedText": "然后将变换矩阵应用到左边相乘得到的结果上。", "model": "google_nmt", "n_reviews": 0, @@ -654,7 +654,7 @@ "end": 515.62 }, { - "input": "This tells us where that vector lands, but still in our language. ", + "input": "ix. To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtract ", "translatedText": "这告诉我们该向量落在哪里，但仍然是我们的语言。", "model": "google_nmt", "from_community_srt": "却是用我们的坐标来描述 对于一个向量， 这个矩阵将她的语言描述转化为我们的语言描述 逆矩阵则与之相反 不过，", diff --git a/2016/change-of-basis/czech/sentence_translations.json b/2016/change-of-basis/czech/sentence_translations.json index 51a23519c..cc49c9a9c 100644 --- a/2016/change-of-basis/czech/sentence_translations.json +++ b/2016/change-of-basis/czech/sentence_translations.json @@ -81,7 +81,7 @@ "end": 84.84 }, { - "input": "It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.", + "input": "st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is ti", "translatedText": "Přesune bázový vektor i-hat na souřadnice 3, 0 a j-hat na 1, 2.", "model": "DeepL", "from_community_srt": "našeho standardního souřadnicového systému.", @@ -90,7 +90,7 @@ "end": 91.04 }, { - "input": "So it's represented with a matrix whose columns are 3, 0, and 1, 2.", + "input": "ed up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actual", "translatedText": "Je tedy reprezentován maticí, jejíž sloupce jsou 3, 0 a 1, 2.", "model": "DeepL", "from_community_srt": "Teď se chci podívat na to, co se stane, když použijeme jinou sadu bázových vektorů.", @@ -99,7 +99,7 @@ "end": 95.64 }, { - "input": "Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip.", + "input": "ly scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called th", "translatedText": "Zaměřte se na to, co to udělá s jedním konkrétním vektorem, a přemýšlejte o rozpětí tohoto vektoru, o přímce procházející jeho počátkem a vrcholem.", "model": "DeepL", "from_community_srt": "Dejme tomu, že máme kamarádku Žanetu, která používá jinou sadu bázových vektorů. Označme je 'b1' a 'b2'.", @@ -108,7 +108,7 @@ "end": 104.16 }, { - "input": "Most vectors are going to get knocked off their span during the transformation.", + "input": "e basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a", "translatedText": "Většina vektorů se během transformace vyřadí ze svého rozpětí.", "model": "DeepL", "from_community_srt": "Její první bázový vektor, 'b1' ukazuje doprava a trochu nahoru, a její druhý bázový vektor,", @@ -126,7 +126,7 @@ "end": 115.32 }, { - "input": "But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar.", + "input": "let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up.", "translatedText": "Některé speciální vektory však zůstávají ve svém vlastním rozpětí, což znamená, že vliv matice na takový vektor je pouze jeho roztažení nebo zmačkání, podobně jako u skaláru.", "model": "DeepL", "from_community_srt": "My jej nazýváme vektorem se souřadnicemi (3, 2) na základě bázových vektorů 'i' a 'j'. Žaneta by ale ten samý vektor popsala jako (5/3, 1/3). To znamená,", @@ -135,7 +135,7 @@ "end": 127.04 }, { - "input": "For this specific example, the basis vector i-hat is one such special vector.", + "input": "Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vecto", "translatedText": "Pro tento konkrétní příklad je jedním z takových speciálních vektorů bázový vektor i-hat.", "model": "DeepL", "from_community_srt": "že tenhle vektor složíme z jejích bázových vektorů tak, že vyškálujeme 'b1' pěti třetinami,", @@ -171,7 +171,7 @@ "end": 164.04 }, { - "input": "It ends up getting stretched by a factor of 2.", + "input": "scale b1 by 5 thirds, scale b2 by 1 third, then add them both togethe", "translatedText": "Nakonec se protáhne na dvojnásobek.", "model": "DeepL", "from_community_srt": "Abychom si naší situaci zpřesnili,", @@ -180,7 +180,7 @@ "end": 167.14 }, { - "input": "And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2.", + "input": "r. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she", "translatedText": "A opět, linearita bude znamenat, že jakýkoli jiný vektor na úhlopříčce, kterou tento chlapík prochází, se prostě protáhne o dvojnásobek.", "model": "DeepL", "from_community_srt": "dejme tomu, že její první bázový vektor 'b1' je něco, co bychom my popsali jako (2, 1) a její druhý bázový vektor 'b2' je něco, co my vidíme jako (-1,", @@ -189,7 +189,7 @@ "end": 178.22 }, { - "input": "And for this transformation, those are all the vectors with this special property of staying on their span.", + "input": "thinks of her first coordinate as scali For this specific example, the basis vector i-hat is one such special vector. The span of", "translatedText": "A pro tuto transformaci jsou to všechny vektory, které mají tuto zvláštní vlastnost, že zůstávají na svém rozpětí.", "model": "DeepL", "from_community_srt": "1). Ale je třeba mít na paměti, že z jejího pohledu mají tyto vektory souřadnice (1,", @@ -198,7 +198,7 @@ "end": 185.18 }, { - "input": "Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2.", + "input": "i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis. What's more, because of the way linear transformations work,", "translatedText": "Ty na ose x se roztáhnou na trojnásobek a ty na této úhlopříčce se roztáhnou na dvojnásobek.", "model": "DeepL", "from_community_srt": "0) a (0, 1). Jsou to ty vektory, které pro ni definují souřadnicím (1, 0) a (0, 1) jejich význam.", @@ -216,7 +216,7 @@ "end": 198.08 }, { - "input": "As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation.", + "input": "n. A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc t, a way to visualize our coordinate system, and so it depends on our choice of basis", "translatedText": "Jak jste již možná uhodli, tyto speciální vektory se nazývají vlastní vektory transformace a ke každému vlastnímu vektoru je přiřazena tzv. vlastní hodnota, což je právě faktor, o který se během transformace roztáhne nebo smrskne.", "model": "DeepL", "from_community_srt": "Oba se díváme na ty samé vektory v rovině, ale Žaneta používá jiná slova na to, aby je popsala. Dovolím si ještě krátkou poznámku o tom, jak si vektory reprezentujeme my. Když animuji rovinu, většinou používám čtvercovou mřížku, ale tahle mřížka je jenom konstrukt, způsob, jak zobrazit náš souřadnicový systém, takže závisí na naší volbě báze.", @@ -234,7 +234,7 @@ "end": 225.94 }, { - "input": "In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half.", + "input": "nt as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It", "translatedText": "V jiném příkladu můžete mít vlastní vektor s vlastní hodnotou zápornou o polovinu, což znamená, že se vektor převrátí a zmenší o polovinu.", "model": "DeepL", "from_community_srt": "tedy nic víc než grafická pomůcka, jak číst souřadnice vektorů z jejího pohledu.", @@ -261,7 +261,7 @@ "end": 249.8 }, { - "input": "If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation.", + "input": "of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewhat during the transformation, k", "translatedText": "Pokud najdete vlastní vektor pro toto natočení, tedy vektor, který zůstává na svém vlastním rozpětí, nalezli jste osu natočení.", "model": "DeepL", "from_community_srt": "V téhle situaci je docela přirozené se ptát: \"Jak se překládá mezi souřadnicovými systémy?\"", @@ -270,7 +270,7 @@ "end": 260.5 }, { - "input": "And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation.", + "input": "nocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", "translatedText": "A je mnohem snazší uvažovat o 3D rotaci v podobě nějaké osy otáčení a úhlu, o který se otáčí, než přemýšlet o celé matici 3x3 spojené s touto transformací.", "model": "DeepL", "from_community_srt": "Když třeba Žaneta popíše vektor souřadnicemi (-1, 2), jak tenhle vektor vyjádříme v našich souřadnicích? Jak to přeložíme z jejího jazyka do našeho? Inu, její souřadnice říkají, že tento vektor je roven -1 b1 + 2 b2,", @@ -288,7 +288,7 @@ "end": 285.86 }, { - "input": "This pattern shows up a lot in linear algebra.", + "input": "In fact, once you understand matrix vector multiplication as applying", "translatedText": "Tento vzorec se často objevuje v lineární algebře.", "model": "DeepL", "from_community_srt": "1) a 'b2' má souřadnice (-1, 1).", @@ -297,7 +297,7 @@ "end": 290.02 }, { - "input": "With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors.", + "input": "a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvecto", "translatedText": "U jakékoli lineární transformace popsané maticí můžete pochopit, co dělá, když sloupce této matice odečtete jako místa, kde přistávají bázové vektory.", "model": "DeepL", "from_community_srt": "Takže můžeme spočítat -1 b1 + 2 b2 v našem souřadnicovém systému.", @@ -306,7 +306,7 @@ "end": 299.4 }, { - "input": "But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", + "input": "r with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformati", "translatedText": "Často je však lepším způsobem, jak se dostat k jádru toho, co lineární transformace skutečně dělá, méně závislým na konkrétním souřadnicovém systému, nalezení vlastních vektorů a vlastních hodnot.", "model": "DeepL", "from_community_srt": "Když to vyčíslíme, vyjde vektor se souřadnicemi (-4,1). Takže my bychom popsali vektor, který Žaneta vidí jako (-1, 2).", @@ -324,7 +324,7 @@ "end": 326.02 }, { - "input": "Symbolically, here's what the idea of an eigenvector looks like.", + "input": "she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we thi", "translatedText": "Symbolicky vypadá představa vlastního vektoru takto.", "model": "DeepL", "from_community_srt": "A protože si násobení matice a vektoru představujeme jako provádění jistého lineárního zobrazení,", @@ -342,7 +342,7 @@ "end": 339.74 }, { - "input": "What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda.", + "input": "or for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotat", "translatedText": "Tento výraz říká, že maticový vektorový součin A krát v dává stejný výsledek jako pouhé škálování vlastního vektoru v nějakou hodnotou lambda.", "model": "DeepL", "from_community_srt": "jejíž sloupečky odpovídají Žanetiným bázovým vektorům se dá chápat jako transformace, která přesune naše bázové vektory 'i', 'j', ty, které my považujeme za (1,", @@ -369,7 +369,7 @@ "end": 370.54 }, { - "input": "So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda.", + "input": "never stretch or squish anything, so the length of the vector would remain the same. This pattern shows up a lot in linear algebra. With any linear transformation described by a matrix, you could understand what it's doing by reading of", "translatedText": "Začněme tedy přepisem této pravé strany jako určitého druhu maticově-vektorového násobení pomocí matice, která má za následek škálování libovolného vektoru koeficientem lambda.", "model": "DeepL", "from_community_srt": "Před transformací jsme se na vektor dívali jako na kombinaci našich bázových vektorů -1i+2j.", @@ -378,7 +378,7 @@ "end": 380.62 }, { - "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else.", + "input": "f the columns of this matrix as the landing spots for basis vectors. But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", "translatedText": "Sloupce takové matice budou představovat to, co se děje s každým bázovým vektorem, a každý bázový vektor se jednoduše vynásobí lambdou, takže tato matice bude mít na diagonále číslo lambda a všude jinde nuly.", "model": "DeepL", "from_community_srt": "Lineární transformace má tu klíčovou vlastnost, že výsledný vektor bude ta samá lineární kombinace, ale nových, bázových vektorů. -1 krát obraz 'i' plus 2 krát obraz 'j'.", @@ -387,7 +387,7 @@ "end": 394.32 }, { - "input": "The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal.", + "input": "we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector w", "translatedText": "Běžný způsob, jak ho zapsat, je vynásobit tuto lambdu a zapsat ji jako lambda krát i, kde i je identická matice s jedničkami na diagonále.", "model": "DeepL", "from_community_srt": "Takže matice přechodu dělá to, že mění naši mylnou představu toho, co má Žaneta na mysli, na skutečný vektor, který popisuje. Pamatuji si,", @@ -405,7 +405,7 @@ "end": 411.86 }, { - "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector.", + "input": "that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, with v as the eigenvector", "translatedText": "Nyní tedy máme novou matici A minus lambda krát identita a hledáme vektor v takový, že tato nová matice krát v dává nulový vektor.", "model": "DeepL", "from_community_srt": "Geometricky matice přechodu přesouvá naši mřížku na Žanetinu mřížku, ale numericky přecházíme z vektorů v jejím jazyce do našeho. Pomohlo mi až, když jsem si to představil,", @@ -431,7 +431,7 @@ "end": 433.64 }, { - "input": "And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", + "input": "and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Je", "translatedText": "A pokud se podíváte na kapitoly 5 a 6, dozvíte se, že jediný způsob, jak je možné, aby se součin matice s nenulovým vektorem stal nulovým, je ten, že transformace spojená s touto maticí zmačká prostor do nižší dimenze.", "model": "DeepL", "from_community_srt": "který měla opravdu na mysli. A co obráceně? Jak jsem třeba ukazoval vektor, který má v našem jazyce souřadnice (3, 2). Jak jsem spočítal, že to je (5/3,", @@ -467,7 +467,7 @@ "end": 470.28 }, { - "input": "As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes.", + "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose c", "translatedText": "Se změnou hodnoty lambda se mění i samotná matice, a tím i její determinant.", "model": "DeepL", "from_community_srt": "obzvlášť, když pracujete ve více než dvou rozměrech, použijete pro výpočet inverzní matice počítač.", @@ -512,7 +512,7 @@ "end": 498.6 }, { - "input": "So this is kind of a lot, but let's unravel what this is saying.", + "input": "And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's importa", "translatedText": "Je toho trochu moc, ale pojďme si říct, co to znamená.", "model": "DeepL", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 502.96 }, { - "input": "When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line.", + "input": "nt that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication So let's start by rewriting", "translatedText": "Když je lambda rovna 1, matice A minus lambda krát identita rozmělní prostor na přímku.", "model": "DeepL", "from_community_srt": "to vyjde (5/3, 1/3).", @@ -529,7 +529,7 @@ "end": 509.56 }, { - "input": "That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector.", + "input": "that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a fact", "translatedText": "To znamená, že existuje nenulový vektor v takový, že A minus lambda krát identita krát v se rovná nulovému vektoru.", "model": "DeepL", "from_community_srt": "Takže takhle se v kostce překládají popisky jednotlivých vektorů mezi jednotlivými souřadnicovými systémy. Matice přechodu, jejíž sloupečky reprezentují Žanetiny bázové vektory, ale zapsané v našich souřadnicích,", @@ -538,7 +538,7 @@ "end": 518.56 }, { - "input": "And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A.", + "input": "or of lambda. The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else. the columns of our matrix.", "translatedText": "A nezapomeňte, že nás to zajímá proto, že to znamená, že A krát v se rovná lambda krát v, což můžete vyčíst jako tvrzení, že vektor v je vlastním vektorem A, který zůstává na svém vlastním rozpětí během transformace A.", "model": "DeepL", "from_community_srt": "překládají vektory z jejího jazyka do našeho. A inverzní matice dělá pravý opak. Ale vektory nejsou to jediné, co popisujeme pomocí souřadnic. V další části bude důležité,", @@ -547,7 +547,7 @@ "end": 537.28 }, { - "input": "In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place.", + "input": "But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following", "translatedText": "V tomto příkladu je odpovídající vlastní číslo 1, takže v by vlastně zůstalo na místě.", "model": "DeepL", "from_community_srt": "abyste si rozuměli s reprezentací transformací pomocí matic a abyste chápali,", @@ -565,7 +565,7 @@ "end": 549.5 }, { - "input": "This is the kind of thing I mentioned in the introduction.", + "input": "-hat in the first pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v.", "translatedText": "O tom jsem se zmínil v úvodu.", "model": "DeepL", "from_community_srt": "rozhodně se vraťte a zopakujte si kapitoly 3 a 4.", @@ -574,7 +574,7 @@ "end": 555.64 }, { - "input": "If you didn't have a solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to describe those landing spots in her language. Here's a common way to think", "translatedText": "Kdybyste neměli solidní znalosti o determinantech a jejich vztahu k lineárním soustavám rovnic s nenulovými řešeními, připadal by vám takový výraz úplně mimo mísu.", "model": "DeepL", "from_community_srt": "Vezměme si nějakou lineární transformaci, jako třeba otočení o 90 stupňů. Když ji reprezentujeme pomocí souřadnic my, díváme se, kde skončí vektory 'i' a 'j'.", @@ -619,7 +619,7 @@ "end": 608.84 }, { - "input": "To figure out what the eigenvectors are that actually have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero.", + "input": "To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry. Now imagine tweaking lambda, turning a knob to change its value. As that value of lambda changes, the matrix itself change", "translatedText": "Chcete-li zjistit, které vlastní vektory mají vlastně jednu z těchto vlastních hodnot, řekněme lambda rovnou 2, dosaďte tuto hodnotu lambda do matice a pak vyřešte, které vektory tato diagonálně změněná matice posílá k nule.", "model": "DeepL", "from_community_srt": "kam se přesunou naše bázové vektory 'i' a 'j'. Ale Žanetu chce matici, ve které jsou transformované verze jejích bázových vektorů, a navíc chce tyto výsledky zaznamenat ve svém jazyce. Běžně se to řeší takto: Začneme s vektorem zapsaným v Žanetině jazyce.", @@ -709,7 +709,7 @@ "end": 681.94 }, { - "input": "The only roots of that polynomial are the imaginary numbers, i and negative i.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the oute", "translatedText": "Jedinými kořeny tohoto polynomu jsou imaginární čísla i a záporné i.", "model": "DeepL", "from_community_srt": "Bere vektor v jejím jazyce a vyplivne transformovaný vektor, opět v jejím jazyce.", @@ -736,7 +736,7 @@ "end": 699.82 }, { - "input": "This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1.", + "input": "meone else sees it. For those of you wondering why we care about alternate coordinate systems, the next vi", "translatedText": "Tím se i-hat zafixuje na místě a j-hat se posune o 1, takže jeho matice má sloupce 1, 0 a 1, 1.", "model": "DeepL", "from_community_srt": "5/3) a (-2/3, -1/3).", @@ -772,7 +772,7 @@ "end": 726.54 }, { - "input": "And the only root of this expression is lambda equals 1.", + "input": "he identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals la", "translatedText": "A jediným kořenem tohoto výrazu je lambda rovná se 1.", "model": "DeepL", "from_community_srt": "Prostřední matice reprezentuje nějaký druh transformace z našeho pohledu, a vnější dvě matice reprezentují empatii -- změnu perspektivy,", @@ -781,7 +781,7 @@ "end": 732.86 }, { - "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.", + "input": "mbda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corresponding eigenvalue is", "translatedText": "To odpovídá tomu, co vidíme geometricky, že všechny vlastní vektory mají vlastní hodnotu 1.", "model": "DeepL", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 796.38 }, { - "input": "For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2.", + "input": "f equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Například i-hat je škálován zápornou hodnotou 1 a j-hat je škálován hodnotou 2.", "model": "DeepL", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 825.42 }, { - "input": "And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.", + "input": "nd compute the determinant. Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda. Since lambda can only be an eigenvalue i", "translatedText": "To lze interpretovat tak, že všechny základní vektory jsou vlastními vektory, přičemž diagonální položky této matice jsou jejich vlastní hodnoty.", "model": "DeepL", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 841.06 }, { - "input": "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", + "input": "u can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3. To figure out what the eigenvectors are that actu", "translatedText": "Jedním z nich je, že je snazší vypočítat, co se stane, když tuto matici vynásobíte celou řadou.", "model": "DeepL", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 848.34 }, { - "input": "Since all one of these matrices does is scale each basis vector by some eigenvalue, applying that matrix many times, say 100 times, is just going to correspond to scaling each basis vector by the 100th power of the corresponding eigenvalue.", + "input": "ally have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero. If you computed this the way you would any other linear system, you'd see that the solutions are all the vectors on the diagonal line spanned by negative 1, 1.", "translatedText": "Protože jediné, co jedna z těchto matic dělá, je škálování každého bázového vektoru určitou vlastní hodnotou, bude použití této matice mnohokrát, řekněme stokrát, odpovídat škálování každého bázového vektoru 100. mocninou příslušné vlastní hodnoty.", "model": "DeepL", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 869.68 }, { - "input": "Really, try it for a moment.", + "input": "x, 3, 0, 1, 2, has the effect of stretching all those vectors by a factor of 2.", "translatedText": "Opravdu, zkuste to na chvíli.", "model": "DeepL", "n_reviews": 0, @@ -936,7 +936,7 @@ "end": 896.54 }, { - "input": "I talked about change of basis last video, but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", + "input": "is, notice what happens. Its matrix has columns 0, 1 and negative 1, 0. Subtract off lambda from the diagonal elements and look for when the determinant is zero. In this case, you get the polynomial lambda squared plus 1. The only roots of that polynomia", "translatedText": "O změně základu jsem mluvil v minulém videu, ale tady si v rychlosti připomenu, jak vyjádřit transformaci zapsanou v našem souřadnicovém systému do jiného systému.", "model": "DeepL", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 907.04 }, { - "input": "Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors, then make those coordinates the columns of a matrix, known as the change of basis matrix.", + "input": "l are the imaginary numbers, i and negative i. The fact that there are no real number solutions indicates that there are no eigenvectors. Another pretty interesting example worth holding in the back of your mind is a shear. This fixes i-hat in place and moves j-hat 1 over, so its mat", "translatedText": "Vezměte souřadnice vektorů, které chcete použít jako novou bázi, což v tomto případě znamená naše dva vlastní vektory, a pak z těchto souřadnic vytvořte sloupce matice, známé jako matice změny báze.", "model": "DeepL", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 946.68 }, { - "input": "This is because it represents working in a coordinate system where what happens to the basis vectors is that they get scaled during the transformation.", + "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1. Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a lin", "translatedText": "Je to proto, že se jedná o práci v souřadnicovém systému, kde se s bázovými vektory děje to, že se při transformaci škálují.", "model": "DeepL", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 961.56 }, { - "input": "So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis, compute the 100th power in that system, then convert back to our standard system.", + "input": "A simple example is a matrix that scales everything by 2. The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue. Now is another good time to pause and ponder some of this before I move on to the last topic.", "translatedText": "Pokud byste tedy například potřebovali vypočítat stou mocninu této matice, bylo by mnohem jednodušší přejít na vlastní základnu, vypočítat stou mocninu v této soustavě a pak ji převést zpět do naší standardní soustavy.", "model": "DeepL", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 975.68 }, { - "input": "You can't do this with all transformations.", + "input": "I want to finish off here with the idea of an eigenbasis, which relies heavily on ideas from the last video. Take a look at what h", "translatedText": "To nelze provést u všech transformací.", "model": "DeepL", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 978.32 }, { - "input": "A shear, for example, doesn't have enough eigenvectors to span the full space.", + "input": "appens if our basis vectors just so happen to be eigenvectors. For example, maybe i-hat is scale", "translatedText": "Například smyk nemá dostatek vlastních vektorů, aby obsáhl celý prostor.", "model": "DeepL", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 982.98 }, { - "input": "But if you can find an eigenbasis, it makes matrix operations really lovely.", + "input": "d by negative 1 and j-hat is scaled by 2. Writing their new coordinates as the columns of a matrix, notice t", "translatedText": "Ale pokud se vám podaří najít vlastní základnu, jsou operace s maticemi opravdu krásné.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/change-of-basis/dutch/sentence_translations.json b/2016/change-of-basis/dutch/sentence_translations.json index c5a2d2f48..82b6a51f6 100644 --- a/2016/change-of-basis/dutch/sentence_translations.json +++ b/2016/change-of-basis/dutch/sentence_translations.json @@ -81,7 +81,7 @@ "end": 84.84 }, { - "input": "It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.", + "input": "st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is ti", "translatedText": "Het verplaatst de basisvector i-hat naar de coördinaten 3, 0, en j-hat naar 1, 2.", "model": "google_nmt", "from_community_srt": "i-hoedje en j-hoedje worden basis vectoren genoemd van ons standaard coördinaten systeem.", @@ -90,7 +90,7 @@ "end": 91.04 }, { - "input": "So it's represented with a matrix whose columns are 3, 0, and 1, 2.", + "input": "ed up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actual", "translatedText": "Het wordt dus weergegeven met een matrix waarvan de kolommen 3, 0 en 1, 2 zijn.", "model": "google_nmt", "from_community_srt": "Waar ik over wil praten is het idee van het gebruik van verschillende basis vectoren.", @@ -99,7 +99,7 @@ "end": 95.64 }, { - "input": "Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip.", + "input": "ly scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called th", "translatedText": "Concentreer u op wat het met een bepaalde vector doet, en denk na over de reikwijdte van die vector, de lijn die door zijn oorsprong en zijn punt loopt.", "model": "google_nmt", "from_community_srt": "Bijvoorbeeld, laten we zeggen dat je een vriendin hebt, Jennifer die andere basis vectoren gebruikt, welke ik b1 en b2 zal noemen.", @@ -108,7 +108,7 @@ "end": 104.16 }, { - "input": "Most vectors are going to get knocked off their span during the transformation.", + "input": "e basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a", "translatedText": "De meeste vectoren zullen tijdens de transformatie uit hun bereik worden gehaald.", "model": "google_nmt", "from_community_srt": "Haar eerste basis vector b1 wijst een klein beetje naar rechts boven en haar tweede vector b2 wijst naar links boven.", @@ -126,7 +126,7 @@ "end": 115.32 }, { - "input": "But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar.", + "input": "let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up.", "translatedText": "Maar sommige speciale vectoren blijven op hun eigen bereik, wat betekent dat het effect dat de matrix op zo'n vector heeft, alleen maar is dat deze wordt uitgerekt of platgedrukt, zoals bij een scalaire vector.", "model": "google_nmt", "from_community_srt": "Degene die je wil beschrijven met het gebruik van de coördinaten [3, 2] nemende onze basis vectoren i-hoedje en j-hoedje. Jennifer zou echter deze vector beschrijven met de coördinaten [5/3, 1/3].", @@ -135,7 +135,7 @@ "end": 127.04 }, { - "input": "For this specific example, the basis vector i-hat is one such special vector.", + "input": "Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vecto", "translatedText": "Voor dit specifieke voorbeeld is de basisvector i-hat zo'n speciale vector.", "model": "google_nmt", "from_community_srt": "Wat betekent dat de manier om die vector te bekomen gebruikende haar 2 basis vectoren is het schalen van b1 met 5/3 en b2 met 1/3", @@ -171,7 +171,7 @@ "end": 164.04 }, { - "input": "It ends up getting stretched by a factor of 2.", + "input": "scale b1 by 5 thirds, scale b2 by 1 third, then add them both togethe", "translatedText": "Het wordt uiteindelijk met een factor 2 uitgerekt.", "model": "google_nmt", "n_reviews": 0, @@ -179,7 +179,7 @@ "end": 167.14 }, { - "input": "And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2.", + "input": "r. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she", "translatedText": "En nogmaals, lineariteit impliceert dat elke andere vector op de diagonale lijn die door deze man wordt overspannen, gewoon met een factor 2 zal worden uitgerekt.", "model": "google_nmt", "from_community_srt": "Om een beetje meer precies over deze opzet hier haar eerste basis vector b1 is iets wat we zouden beschrijven met de coördinaten [2, 1] en haar tweede basis vector b2 is iets wat we zouden beschrijven als [-1,", @@ -188,7 +188,7 @@ "end": 178.22 }, { - "input": "And for this transformation, those are all the vectors with this special property of staying on their span.", + "input": "thinks of her first coordinate as scali For this specific example, the basis vector i-hat is one such special vector. The span of", "translatedText": "En voor deze transformatie zijn dat alle vectoren met de speciale eigenschap om binnen hun bereik te blijven.", "model": "google_nmt", "from_community_srt": "1]. Maar het is belangrijk om te beseffen dat van haar perspectief, in haar systeem deze vectoren coördinaten [1,", @@ -197,7 +197,7 @@ "end": 185.18 }, { - "input": "Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2.", + "input": "i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis. What's more, because of the way linear transformations work,", "translatedText": "Die op de x-as worden met een factor 3 uitgerekt, en die op deze diagonale lijn worden met een factor 2 uitgerekt.", "model": "google_nmt", "from_community_srt": "0] en [0, 1] bezitten Deze zijn wat de betekenis van coördinaten [1, 0] en [0, 1] definieert in haar wereld.", @@ -215,7 +215,7 @@ "end": 198.08 }, { - "input": "As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation.", + "input": "n. A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc t, a way to visualize our coordinate system, and so it depends on our choice of basis", "translatedText": "Zoals je misschien al geraden hebt, worden deze speciale vectoren de eigenvectoren van de transformatie genoemd, en aan elke eigenvector is een zogenaamde eigenwaarde gekoppeld, wat precies de factor is waarmee deze wordt uitgerekt of platgedrukt tijdens de transformatie.", "model": "google_nmt", "from_community_srt": "We kijken naar dezelfde vectoren in de ruimte, maar Jennifer gebruikt verschillende woorden en nummers om ze te beschrijven. Laat me snel iets zeggen over hoe ik de dingen voorstel hier als ik animeer in een 2D ruimte. Meestal gebruik ik een vierkant raster maar het raster is enkel een constructie, een manier om ons coördinaten systeem te visualiseren en zo hangt het af van onze keuze van basis vectoren.", @@ -233,7 +233,7 @@ "end": 225.94 }, { - "input": "In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half.", + "input": "nt as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It", "translatedText": "In een ander voorbeeld zou je een eigenvector kunnen hebben met een eigenwaarde negatief 1 half, wat betekent dat de vector wordt omgedraaid en geplet met een factor 1 half.", "model": "google_nmt", "from_community_srt": "om haar te helpen met de betekenis van haar coördinaten.", @@ -260,7 +260,7 @@ "end": 249.8 }, { - "input": "If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation.", + "input": "of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewhat during the transformation, k", "translatedText": "Als je voor die rotatie een eigenvector kunt vinden, een vector die op zijn eigen bereik blijft, dan heb je de rotatie-as gevonden.", "model": "google_nmt", "from_community_srt": "na al datgene gezegd een natuurlijke vraag om te stellen is Hoe vertalen we tussen de verschillende coördinaten systemen?", @@ -269,7 +269,7 @@ "end": 260.5 }, { - "input": "And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation.", + "input": "nocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", "translatedText": "En het is veel gemakkelijker om aan een 3D-rotatie te denken in termen van een bepaalde rotatie-as en een hoek waarover deze roteert, dan te denken aan de volledige 3x3-matrix die bij die transformatie hoort.", "model": "google_nmt", "from_community_srt": "Als bijvoorbeeld Jennifer een vector beschrijft met coördinaten [-1, 2] wat zou dat dan zijn in ons coördinaten systeem? Hoe vertaal je haar taal in de onze? Wel, wat onze coördinaten zeggen is dat deze vector -1 b1 + 2 b2 is.", @@ -287,7 +287,7 @@ "end": 285.86 }, { - "input": "This pattern shows up a lot in linear algebra.", + "input": "In fact, once you understand matrix vector multiplication as applying", "translatedText": "Dit patroon komt veel voor in de lineaire algebra.", "model": "google_nmt", "from_community_srt": "1] en b2 als coördinaten [-1,", @@ -296,7 +296,7 @@ "end": 290.02 }, { - "input": "With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors.", + "input": "a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvecto", "translatedText": "Bij elke lineaire transformatie die door een matrix wordt beschreven, kun je begrijpen wat deze doet door de kolommen van deze matrix af te lezen als de landingsplaatsen voor basisvectoren.", "model": "google_nmt", "from_community_srt": "1] We kunnen eigenlijk -1 b1 + 2 b2 berekenen als ze zijn gepresenteerd in ons coördinaten systeem", @@ -305,7 +305,7 @@ "end": 299.4 }, { - "input": "But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", + "input": "r with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformati", "translatedText": "Maar vaak is een betere manier om tot de kern te komen van wat de lineaire transformatie feitelijk doet, minder afhankelijk van uw specifieke coördinatensysteem, het vinden van de eigenvectoren en eigenwaarden.", "model": "google_nmt", "from_community_srt": "En dit uitwerkende zal je een vector krijgen met coördinaten [-4, 1] Zo, dat is hoe we de vector zouden beschrijven als zij denkt aan [-1,", @@ -323,7 +323,7 @@ "end": 326.02 }, { - "input": "Symbolically, here's what the idea of an eigenvector looks like.", + "input": "she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we thi", "translatedText": "Symbolisch gezien ziet dit er zo uit hoe het idee van een eigenvector eruit ziet.", "model": "google_nmt", "from_community_srt": "eens je matrix-vector vermenigvuldiging begrijpt als het toepassen van een zekere lineaire transformatie.", @@ -341,7 +341,7 @@ "end": 339.74 }, { - "input": "What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda.", + "input": "or for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotat", "translatedText": "Wat deze uitdrukking zegt is dat het matrix-vectorproduct, A maal v, hetzelfde resultaat geeft als het schalen van de eigenvector v met een bepaalde waarde lambda.", "model": "google_nmt", "from_community_srt": "Een matrix wiens kolommen Jennifers basis vectoren representeren kan worden gezien als een transformatie die onze basis vectoren, i-hoedje en j-hoedje beweegt de dingen waaraan we denken wanneer we [1,", @@ -368,7 +368,7 @@ "end": 370.54 }, { - "input": "So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda.", + "input": "never stretch or squish anything, so the length of the vector would remain the same. This pattern shows up a lot in linear algebra. With any linear transformation described by a matrix, you could understand what it's doing by reading of", "translatedText": "Laten we beginnen met het herschrijven van die rechterkant als een soort matrix-vectorvermenigvuldiging, met behulp van een matrix die het effect heeft dat elke vector met een factor lambda wordt geschaald.", "model": "google_nmt", "from_community_srt": "als een zekere lineaire combinatie van onze basis vectoren -1 x i-hoedje + 2 x j-hoedje en het sleutel kenmerk van een lineaire transformatie", @@ -377,7 +377,7 @@ "end": 380.62 }, { - "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else.", + "input": "f the columns of this matrix as the landing spots for basis vectors. But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", "translatedText": "De kolommen van zo'n matrix vertegenwoordigen wat er met elke basisvector gebeurt, en elke basisvector wordt eenvoudigweg vermenigvuldigd met lambda, dus deze matrix zal het getal lambda onderaan de diagonaal hebben, met overal nullen.", "model": "google_nmt", "from_community_srt": "is dat de uiteindelijke vector diezelfde lineaire combinatie zal zijn maar van de nieuwe basis vectoren -1 keer de plaats waar i-hoedje valt + 2 keer de plaats waar j-hoedje valt Zo wat deze matrix doet", @@ -386,7 +386,7 @@ "end": 394.32 }, { - "input": "The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal.", + "input": "we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector w", "translatedText": "De gebruikelijke manier om deze man te schrijven is door die lambda eruit te halen en het op te schrijven als lambda maal i, waarbij i de identiteitsmatrix is met 1s langs de diagonaal.", "model": "google_nmt", "from_community_srt": "is onze misverstand van wat Jennifer bedoelt veranderen in de echte vector waar ze naar verwijst.", @@ -404,7 +404,7 @@ "end": 411.86 }, { - "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector.", + "input": "that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, with v as the eigenvector", "translatedText": "Dus wat we nu hebben is een nieuwe matrix, A minus lambda maal de identiteit, en we zoeken naar een vector v zodat deze nieuwe matrix maal v de nulvector oplevert.", "model": "google_nmt", "from_community_srt": "vertaalt het een vector beschreven in haar taal naar onze taal wat het me uiteindelijk liet snappen was voor mij", @@ -431,7 +431,7 @@ "end": 433.64 }, { - "input": "And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", + "input": "and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Je", "translatedText": "En als je hoofdstuk 5 en 6 bekijkt, weet je dat de enige manier waarop het product van een matrix met een vector die niet nul is, nul kan worden, is als de transformatie die bij die matrix hoort de ruimte in een lagere dimensie perst.", "model": "google_nmt", "from_community_srt": "Wat over dat omgekeerd gaan? In het voorbeeld dat ik eerder deze video gebruikte waar ik de vector heb met coördinaten [3, 2] in ons systeem Hoe heb ik berekend dat het de coördinaten [5/3,", @@ -467,7 +467,7 @@ "end": 470.28 }, { - "input": "As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes.", + "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose c", "translatedText": "Naarmate de waarde van lambda verandert, verandert de matrix zelf, en dus verandert de determinant van de matrix.", "model": "google_nmt", "from_community_srt": "voornamelijk als je werkt in meer dan 2 dimensies zou je een computer gebruiken om de matrix te berekenen die eigenlijk de inverse voorstelt", @@ -512,7 +512,7 @@ "end": 498.6 }, { - "input": "So this is kind of a lot, but let's unravel what this is saying.", + "input": "And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's importa", "translatedText": "Dit is dus nogal veel, maar laten we ontrafelen wat dit zegt.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 502.96 }, { - "input": "When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line.", + "input": "nt that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication So let's start by rewriting", "translatedText": "Wanneer lambda gelijk is aan 1, drukt de matrix A minus lambda maal de identiteit de ruimte op een lijn.", "model": "google_nmt", "from_community_srt": "2] wat [5/3, 1/3] wordt Zo dat in een notendop is hoe we de beschrijving van individuele vectoren vertalen", @@ -529,7 +529,7 @@ "end": 509.56 }, { - "input": "That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector.", + "input": "that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a fact", "translatedText": "Dat betekent dat er een vector v is die niet nul is, zodat A minus lambda maal de identiteit maal v gelijk is aan de nulvector.", "model": "google_nmt", "from_community_srt": "heen en terug tussen coördinaten systemen De matrix wiens kolommen Jennifers basis vectoren representern", @@ -538,7 +538,7 @@ "end": 518.56 }, { - "input": "And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A.", + "input": "or of lambda. The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else. the columns of our matrix.", "translatedText": "En onthoud, de reden dat dit ons interesseert, is omdat het betekent dat A maal v gelijk is aan lambda maal v, wat je kunt aflezen als te zeggen dat de vector v een eigenvector is van A, die op zijn eigen span blijft tijdens de transformatie A.", "model": "google_nmt", "from_community_srt": "maar geschreven in onze coördinaten vertaalt vectoren van haar taal naar onze taal En de inverse matrix doet het tegen over gestelde. maar vectoren zijn niet het enigste ding dat we gebruiken om coördinaten te beschrijven Voor het volgende deel", @@ -547,7 +547,7 @@ "end": 537.28 }, { - "input": "In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place.", + "input": "But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following", "translatedText": "In dit voorbeeld is de corresponderende eigenwaarde 1, dus v zou eigenlijk gewoon op zijn plaats blijven.", "model": "google_nmt", "from_community_srt": "is het belangrijk dat je comfortabel bent met het representeren van transformaties met matrices en dat je weet hoe matrix vermenigvuldigin overeenkomt met samenstellen van opeenvolgende transformaties", @@ -564,7 +564,7 @@ "end": 549.5 }, { - "input": "This is the kind of thing I mentioned in the introduction.", + "input": "-hat in the first pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v.", "translatedText": "Dit is het soort dingen dat ik in de inleiding noemde.", "model": "google_nmt", "from_community_srt": "Pauzeer en neem een kijkje naar hoofdstuk 3 en 4 als iets van dat ongemakkelijk voelt Bekijk een paar lineaire transformaties", @@ -573,7 +573,7 @@ "end": 555.64 }, { - "input": "If you didn't have a solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to describe those landing spots in her language. Here's a common way to think", "translatedText": "Als je geen goed inzicht had in de determinanten en waarom ze betrekking hebben op lineaire stelsels van vergelijkingen met oplossingen die niet nul zijn, zou een uitdrukking als deze volkomen uit de lucht komen vallen.", "model": "google_nmt", "from_community_srt": "zoals een 90° draaiing tegen de klok in wanneer jij en ik dit representeren met de matrix volgen we waar de basis vectoren i-hoedje en j-hoedje gaan", @@ -618,7 +618,7 @@ "end": 608.84 }, { - "input": "To figure out what the eigenvectors are that actually have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero.", + "input": "To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry. Now imagine tweaking lambda, turning a knob to change its value. As that value of lambda changes, the matrix itself change", "translatedText": "Om erachter te komen wat de eigenvectoren zijn die daadwerkelijk een van deze eigenwaarden hebben, zeg lambda is gelijk aan 2, plug je die waarde van lambda in de matrix in en los je vervolgens op voor welke vectoren deze diagonaal gewijzigde matrix naar nul stuurt.", "model": "google_nmt", "from_community_srt": "Deze kolommen representeren waar onze basis vectoren i-hoedje en j-hoedje gaan. maar de matrix dat Jennifer wil zou moeten voorstellen waar haar basis vectoren landen en het moet beschrijven waar deze landingsplaatsen zijn in haar taal. Here is een veelvoorkomende manier om te denken over hoe het is gebeurt.", @@ -708,7 +708,7 @@ "end": 681.94 }, { - "input": "The only roots of that polynomial are the imaginary numbers, i and negative i.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the oute", "translatedText": "De enige wortels van dat polynoom zijn de denkbeeldige getallen, i en negatieve i.", "model": "google_nmt", "from_community_srt": "en spuugt de getransformeerde versie van die vector uit in onze taal voor dit specifiek voorbeeld", @@ -735,7 +735,7 @@ "end": 699.82 }, { - "input": "This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1.", + "input": "meone else sees it. For those of you wondering why we care about alternate coordinate systems, the next vi", "translatedText": "Hierdoor wordt i-hat op zijn plaats gezet en wordt j-hat 1 verplaatst, zodat de matrix kolommen 1, 0 en 1, 1 heeft.", "model": "google_nmt", "from_community_srt": "heeft kolommen [1/3, 5/3] en [-2/3, -1/3] Zo als Jennifer die matrix vermenigvuldigt met de coördinaten als een vector in haar systeem", @@ -770,7 +770,7 @@ "end": 726.54 }, { - "input": "And the only root of this expression is lambda equals 1.", + "input": "he identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals la", "translatedText": "En de enige wortel van deze uitdrukking is dat lambda gelijk is aan 1.", "model": "google_nmt", "from_community_srt": "De middelste matrix stelt een soort transformatie voor,", @@ -779,7 +779,7 @@ "end": 732.86 }, { - "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.", + "input": "mbda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corresponding eigenvalue is", "translatedText": "Dit komt overeen met wat we geometrisch zien, dat alle eigenvectoren een eigenwaarde 1 hebben.", "model": "google_nmt", "from_community_srt": "zoals je ziet en de twee buitenste matrices stellen de empathie voor, de shift in perspectief en het volledig matrix product stelt dezelfde transformatie voor maar zoals iemand ander het ziet.", @@ -839,7 +839,7 @@ "end": 796.38 }, { - "input": "For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2.", + "input": "f equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Misschien wordt i-hat bijvoorbeeld geschaald met negatief 1 en wordt j-hat geschaald met 2.", "model": "google_nmt", "n_reviews": 0, @@ -863,7 +863,7 @@ "end": 825.42 }, { - "input": "And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.", + "input": "nd compute the determinant. Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda. Since lambda can only be an eigenvalue i", "translatedText": "En de manier om dit te interpreteren is dat alle basisvectoren eigenvectoren zijn, waarbij de diagonale ingangen van deze matrix hun eigenwaarden zijn.", "model": "google_nmt", "n_reviews": 0, @@ -879,7 +879,7 @@ "end": 841.06 }, { - "input": "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", + "input": "u can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3. To figure out what the eigenvectors are that actu", "translatedText": "Een grote daarvan is dat het gemakkelijker is om te berekenen wat er zal gebeuren als je deze matrix een aantal keer met zichzelf vermenigvuldigt.", "model": "google_nmt", "n_reviews": 0, @@ -887,7 +887,7 @@ "end": 848.34 }, { - "input": "Since all one of these matrices does is scale each basis vector by some eigenvalue, applying that matrix many times, say 100 times, is just going to correspond to scaling each basis vector by the 100th power of the corresponding eigenvalue.", + "input": "ally have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero. If you computed this the way you would any other linear system, you'd see that the solutions are all the vectors on the diagonal line spanned by negative 1, 1.", "translatedText": "Omdat al deze matrices elke basisvector met een eigenwaarde schalen, komt het vele malen toepassen van die matrix, bijvoorbeeld 100 keer, overeen met het schalen van elke basisvector met de 100ste macht van de overeenkomstige eigenwaarde.", "model": "google_nmt", "n_reviews": 0, @@ -903,7 +903,7 @@ "end": 869.68 }, { - "input": "Really, try it for a moment.", + "input": "x, 3, 0, 1, 2, has the effect of stretching all those vectors by a factor of 2.", "translatedText": "Probeer het echt even.", "model": "google_nmt", "n_reviews": 0, @@ -935,7 +935,7 @@ "end": 896.54 }, { - "input": "I talked about change of basis last video, but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", + "input": "is, notice what happens. Its matrix has columns 0, 1 and negative 1, 0. Subtract off lambda from the diagonal elements and look for when the determinant is zero. In this case, you get the polynomial lambda squared plus 1. The only roots of that polynomia", "translatedText": "Ik had het in de vorige video over het veranderen van de basis, maar ik zal hier een supersnelle herinnering doornemen over hoe je een transformatie die momenteel in ons coördinatensysteem is geschreven, in een ander systeem kunt uitdrukken.", "model": "google_nmt", "n_reviews": 0, @@ -943,7 +943,7 @@ "end": 907.04 }, { - "input": "Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors, then make those coordinates the columns of a matrix, known as the change of basis matrix.", + "input": "l are the imaginary numbers, i and negative i. The fact that there are no real number solutions indicates that there are no eigenvectors. Another pretty interesting example worth holding in the back of your mind is a shear. This fixes i-hat in place and moves j-hat 1 over, so its mat", "translatedText": "Neem de coördinaten van de vectoren die u als nieuwe basis wilt gebruiken, wat in dit geval onze twee eigenvectoren betekent, en maak van die coördinaten de kolommen van een matrix, ook wel de verandering van de basismatrix genoemd.", "model": "google_nmt", "n_reviews": 0, @@ -967,7 +967,7 @@ "end": 946.68 }, { - "input": "This is because it represents working in a coordinate system where what happens to the basis vectors is that they get scaled during the transformation.", + "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1. Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a lin", "translatedText": "Dit komt omdat het het werken in een coördinatensysteem vertegenwoordigt, waarbij wat er met de basisvectoren gebeurt, is dat ze tijdens de transformatie worden geschaald.", "model": "google_nmt", "n_reviews": 0, @@ -983,7 +983,7 @@ "end": 961.56 }, { - "input": "So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis, compute the 100th power in that system, then convert back to our standard system.", + "input": "A simple example is a matrix that scales everything by 2. The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue. Now is another good time to pause and ponder some of this before I move on to the last topic.", "translatedText": "Dus als je bijvoorbeeld de 100e macht van deze matrix zou moeten berekenen, zou het veel gemakkelijker zijn om naar een eigenbasis te gaan, de 100e macht in dat systeem te berekenen en vervolgens terug te converteren naar ons standaardsysteem.", "model": "google_nmt", "n_reviews": 0, @@ -991,7 +991,7 @@ "end": 975.68 }, { - "input": "You can't do this with all transformations.", + "input": "I want to finish off here with the idea of an eigenbasis, which relies heavily on ideas from the last video. Take a look at what h", "translatedText": "Je kunt dit niet bij alle transformaties doen.", "model": "google_nmt", "n_reviews": 0, @@ -999,7 +999,7 @@ "end": 978.32 }, { - "input": "A shear, for example, doesn't have enough eigenvectors to span the full space.", + "input": "appens if our basis vectors just so happen to be eigenvectors. For example, maybe i-hat is scale", "translatedText": "Een afschuiving heeft bijvoorbeeld niet genoeg eigenvectoren om de volledige ruimte te overspannen.", "model": "google_nmt", "n_reviews": 0, @@ -1007,7 +1007,7 @@ "end": 982.98 }, { - "input": "But if you can find an eigenbasis, it makes matrix operations really lovely.", + "input": "d by negative 1 and j-hat is scaled by 2. Writing their new coordinates as the columns of a matrix, notice t", "translatedText": "Maar als je een eigenbasis kunt vinden, worden matrixbewerkingen heel mooi.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/english/captions.srt b/2016/change-of-basis/english/captions.srt index 25bb7ab42..d065b5dd9 100644 --- a/2016/change-of-basis/english/captions.srt +++ b/2016/change-of-basis/english/captions.srt @@ -71,838 +71,982 @@ one of these topics than it does with eigenvectors and eigenvalues themselves. To start, consider some linear transformation in two dimensions, like the one shown here. 19 -00:01:25,460 --> 00:01:31,040 -It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2. +00:01:25,460 --> 00:01:32,472 +st number indicates rightward motion, that the second one indicates upward motion, 20 -00:01:31,780 --> 00:01:35,640 -So it's represented with a matrix whose columns are 3, 0, and 1, 2. +00:01:32,472 --> 00:01:37,711 +exactly how far a unit of distance is, all of that is tied up 21 -00:01:36,600 --> 00:01:39,353 -Focus in on what it does to one particular vector, +00:01:37,711 --> 00:01:41,260 +in the choice of i-hat and j-hat as the ve 22 -00:01:39,353 --> 00:01:44,160 -and think about the span of that vector, the line passing through its origin and its tip. +00:01:41,260 --> 00:01:45,381 +ctors which are scalar coordinates are meant to actually scale. 23 -00:01:44,920 --> 00:01:48,380 -Most vectors are going to get knocked off their span during the transformation. +00:01:45,381 --> 00:01:51,049 +Any way to translate between vectors and sets of numbers is called a coordinate system, 24 -00:01:48,780 --> 00:01:52,050 -I mean, it would seem pretty coincidental if the place where +00:01:51,049 --> 00:01:51,500 +and the 25 -00:01:52,050 --> 00:01:55,320 -the vector landed also happened to be somewhere on that line. +00:01:51,500 --> 00:01:56,501 +two special vectors i-hat and j-hat are called the basis vectors of our standard 26 -00:01:57,400 --> 00:02:00,653 -But some special vectors do remain on their own span, +00:01:56,501 --> 00:02:01,132 +coordinate system. What I'd like to talk about here is the idea of using a 27 -00:02:00,653 --> 00:02:05,533 -meaning the effect that the matrix has on such a vector is just to stretch it or +00:02:01,132 --> 00:02:06,318 +different set of basis vectors. For example, let's say you have a friend, Jennifer, 28 -00:02:05,533 --> 00:02:07,040 -squish it, like a scalar. +00:02:06,318 --> 00:02:11,320 +who uses a different set of basis vectors, which I'll call b1 and b2. Her first b 29 -00:02:09,460 --> 00:02:14,100 -For this specific example, the basis vector i-hat is one such special vector. +00:02:11,320 --> 00:02:15,349 +asis vector, b1, points up and to the right a little bit, and her second vector, b2, 30 -00:02:14,640 --> 00:02:19,412 -The span of i-hat is the x-axis, and from the first column of the matrix, +00:02:15,349 --> 00:02:19,142 +points left and up. Now take another look at that vector that I showed earlier, 31 -00:02:19,412 --> 00:02:24,120 -we can see that i-hat moves over to 3 times itself, still on that x-axis. +00:02:19,142 --> 00:02:22,223 +the one that you and I would describe using the coordinates 3,2, 32 -00:02:26,320 --> 00:02:30,031 -What's more, because of the way linear transformations work, +00:02:22,223 --> 00:02:24,120 +using our basis vectors i-hat and j-hat. 33 -00:02:30,031 --> 00:02:34,411 -any other vector on the x-axis is also just stretched by a factor of 3, +00:02:26,320 --> 00:02:32,921 +Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third. 34 -00:02:34,411 --> 00:02:36,480 -and hence remains on its own span. +00:02:32,921 --> 00:02:37,797 +What this means is that the particular way to get to that vector 35 -00:02:38,500 --> 00:02:41,325 -A slightly sneakier vector that remains on its own +00:02:37,797 --> 00:02:43,574 +using her two basis vectors is to scale b1 by 5 thirds, scale b2 by 1 third, 36 -00:02:41,325 --> 00:02:44,040 -span during this transformation is negative 1, 1. +00:02:43,574 --> 00:02:46,800 +then add them both together. In a little bi 37 -00:02:44,660 --> 00:02:47,140 -It ends up getting stretched by a factor of 2. +00:02:46,800 --> 00:02:52,073 +t, I'll show you how you could have figured out those two numbers, 38 -00:02:49,000 --> 00:02:53,610 -And again, linearity is going to imply that any other vector on the diagonal +00:02:52,073 --> 00:02:57,504 +5 thirds and 1 third. In general, whenever Jennifer uses coordinates 39 -00:02:53,610 --> 00:02:58,220 -line spanned by this guy is just going to get stretched out by a factor of 2. +00:02:57,504 --> 00:03:02,620 +to describe a vector, she thinks of her first coordinate as scali 40 -00:02:59,820 --> 00:03:02,575 -And for this transformation, those are all the vectors +00:03:02,620 --> 00:03:04,260 +For this specific example, the basis vector i-hat is one such special vector. 41 -00:03:02,575 --> 00:03:05,180 -with this special property of staying on their span. +00:03:04,260 --> 00:03:05,427 +The span of i-hat is the x-axis, and from the first column of the matrix, 42 -00:03:05,620 --> 00:03:08,624 -Those on the x-axis getting stretched out by a factor of 3, +00:03:05,427 --> 00:03:06,580 +we can see that i-hat moves over to 3 times itself, still on that x-axis. 43 -00:03:08,624 --> 00:03:11,980 -and those on this diagonal line getting stretched by a factor of 2. +00:03:06,580 --> 00:03:10,780 +What's more, because of the way linear transformations work, 44 -00:03:12,760 --> 00:03:16,417 -Any other vector is going to get rotated somewhat during the transformation, +00:03:10,780 --> 00:03:15,738 +any other vector on the x-axis is also just stretched by a factor of 3, 45 -00:03:16,417 --> 00:03:18,080 -knocked off the line that it spans. +00:03:15,738 --> 00:03:18,080 +and hence remains on its own span. 46 -00:03:22,520 --> 00:03:27,362 -As you might have guessed by now, these special vectors are called the eigenvectors of +00:03:20,255 --> 00:03:18,080 +A slightly sneakier vector that remains on its own 47 -00:03:27,362 --> 00:03:31,870 -the transformation, and each eigenvector has associated with it what's called an +00:03:22,520 --> 00:03:20,255 +span during this transformation is negative 1, 1. 48 -00:03:31,870 --> 00:03:36,545 -eigenvalue, which is just the factor by which it's stretched or squished during the +00:03:22,520 --> 00:03:29,375 +Let me say a quick word about how I'm representing things here. When I animate 2D space, 49 -00:03:36,545 --> 00:03:37,380 -transformation. +00:03:29,375 --> 00:03:34,460 +I typically use this square grid. But that grid is just a construc 50 -00:03:40,280 --> 00:03:43,399 -Of course, there's nothing special about stretching versus squishing, +00:03:34,460 --> 00:03:38,045 +t, a way to visualize our coordinate system, and so it depends on our choice of basis. 51 -00:03:43,399 --> 00:03:45,940 -or the fact that these eigenvalues happen to be positive. +00:03:38,045 --> 00:03:40,929 +Space itself has no intrinsic grid. Jennifer might draw her own grid, 52 -00:03:46,380 --> 00:03:51,060 -In another example, you could have an eigenvector with eigenvalue negative 1 half, +00:03:40,929 --> 00:03:43,320 +which would be an equally made up construct meant as nothi 53 -00:03:51,060 --> 00:03:55,120 -meaning that the vector gets flipped and squished by a factor of 1 half. +00:03:43,400 --> 00:03:48,952 +ng more than a visual tool to help follow the meaning of her coordinates. 54 -00:03:56,980 --> 00:03:59,737 -But the important part here is that it stays on the +00:03:48,952 --> 00:03:52,854 +Her origin though would actually line up with ours, 55 -00:03:59,737 --> 00:04:02,760 -line that it spans out without getting rotated off of it. +00:03:52,854 --> 00:03:57,656 +since everybody agrees on what the coordinates 0,0 should mean. 56 -00:04:04,460 --> 00:04:07,753 -For a glimpse of why this might be a useful thing to think about, +00:03:57,656 --> 00:04:02,383 +It's the thing that you get when you scale any vector by zero. 57 -00:04:07,753 --> 00:04:09,800 -consider some three-dimensional rotation. +00:04:02,383 --> 00:04:04,860 +But the direction of her axes and 58 -00:04:11,660 --> 00:04:14,983 -If you can find an eigenvector for that rotation, +00:04:04,860 --> 00:04:08,526 +Those on the x-axis getting stretched out by a factor of 3, 59 -00:04:14,983 --> 00:04:20,500 -a vector that remains on its own span, what you have found is the axis of rotation. +00:04:08,526 --> 00:04:12,620 +and those on this diagonal line getting stretched by a factor of 2. 60 -00:04:22,600 --> 00:04:26,587 -And it's much easier to think about a 3D rotation in terms of some +00:04:12,620 --> 00:04:16,414 +Any other vector is going to get rotated somewhat during the transformation, 61 -00:04:26,587 --> 00:04:29,800 -axis of rotation and an angle by which it's rotating, +00:04:16,414 --> 00:04:18,140 +knocked off the line that it spans. 62 -00:04:29,800 --> 00:04:34,740 -rather than thinking about the full 3x3 matrix associated with that transformation. +00:04:18,140 --> 00:04:24,795 +ks of as negative 1, 2. This process here of scaling each of her basis vectors 63 -00:04:37,000 --> 00:04:40,797 -In this case, by the way, the corresponding eigenvalue would have to be 1, +00:04:24,795 --> 00:04:31,198 +by the corresponding coordinates of some vector, then adding them together, 64 -00:04:40,797 --> 00:04:43,328 -since rotations never stretch or squish anything, +00:04:31,198 --> 00:04:36,674 +might feel somewhat familiar. It's matrix vector multiplication, 65 -00:04:43,328 --> 00:04:45,860 -so the length of the vector would remain the same. +00:04:36,674 --> 00:04:43,414 +with a matrix whose columns represent Jennifer's basis vectors in our language. 66 -00:04:48,080 --> 00:04:50,020 -This pattern shows up a lot in linear algebra. +00:04:43,414 --> 00:04:50,155 +In fact, once you understand matrix vector multiplication as applying a certain 67 -00:04:50,440 --> 00:04:53,253 -With any linear transformation described by a matrix, +00:04:50,155 --> 00:04:51,840 +linear transformatio 68 -00:04:53,253 --> 00:04:57,733 -you could understand what it's doing by reading off the columns of this matrix as the +00:04:51,840 --> 00:04:57,142 +Of course, there's nothing special about stretching versus squishing, 69 -00:04:57,733 --> 00:04:59,400 -landing spots for basis vectors. +00:04:57,142 --> 00:05:01,460 +or the fact that these eigenvalues happen to be positive. 70 -00:05:00,020 --> 00:05:03,601 -But often, a better way to get at the heart of what the linear +00:05:01,460 --> 00:05:04,437 +In another example, you could have an eigenvector with eigenvalue negative 1 half, 71 -00:05:03,601 --> 00:05:08,318 -transformation actually does, less dependent on your particular coordinate system, +00:05:04,437 --> 00:05:07,020 +meaning that the vector gets flipped and squished by a factor of 1 half. 72 -00:05:08,318 --> 00:05:10,820 -is to find the eigenvectors and eigenvalues. +00:05:07,020 --> 00:05:11,317 +pretty intuitive way to think about what's going on here. 73 -00:05:15,460 --> 00:05:18,997 -I won't cover the full details on methods for computing eigenvectors +00:05:11,317 --> 00:05:16,726 +A matrix whose columns represent Jennifer's basis vectors can be thought 74 -00:05:18,997 --> 00:05:22,175 -and eigenvalues here, but I'll try to give an overview of the +00:05:16,726 --> 00:05:21,913 +of as a transformation that moves our basis vectors, i-hat and j-hat, 75 -00:05:22,175 --> 00:05:26,020 -computational ideas that are most important for a conceptual understanding. +00:05:21,913 --> 00:05:28,360 +the things we think of when we say 1, 0 and 0, 1, to Jennifer's basis vectors, the thin 76 -00:05:27,180 --> 00:05:30,480 -Symbolically, here's what the idea of an eigenvector looks like. +00:05:28,360 --> 00:05:32,638 +gs she thinks of when she says 1, 0 and 0, 1. To show how this works, 77 -00:05:31,040 --> 00:05:35,929 -A is the matrix representing some transformation, with v as the eigenvector, +00:05:32,638 --> 00:05:36,978 +let's walk through what it would mean to take the vector that we think 78 -00:05:35,929 --> 00:05:39,740 -and lambda is a number, namely the corresponding eigenvalue. +00:05:36,978 --> 00:05:41,380 +of as having coordinates negative 1, 2 and applying that transformation. 79 -00:05:40,680 --> 00:05:45,289 -What this expression is saying is that the matrix-vector product, A times v, +00:05:41,380 --> 00:05:46,432 +If you can find an eigenvector for that rotation, 80 -00:05:45,289 --> 00:05:49,900 -gives the same result as just scaling the eigenvector v by some value lambda. +00:05:46,432 --> 00:05:54,820 +a vector that remains on its own span, what you have found is the axis of rotation. 81 -00:05:51,000 --> 00:05:55,423 -So finding the eigenvectors and their eigenvalues of a matrix A comes +00:05:54,820 --> 00:05:59,982 +And it's much easier to think about a 3D rotation in terms of some 82 -00:05:55,423 --> 00:06:00,100 -down to finding the values of v and lambda that make this expression true. +00:05:59,982 --> 00:06:04,144 +axis of rotation and an angle by which it's rotating, 83 -00:06:01,920 --> 00:06:06,057 -It's a little awkward to work with at first, because that left-hand side represents +00:06:04,144 --> 00:06:10,540 +rather than thinking about the full 3x3 matrix associated with that transformation. 84 -00:06:06,057 --> 00:06:09,801 -matrix-vector multiplication, but the right-hand side here is scalar-vector +00:06:11,120 --> 00:06:15,191 +In this case, by the way, the corresponding eigenvalue would have to be 1, 85 -00:06:09,801 --> 00:06:10,540 -multiplication. +00:06:15,191 --> 00:06:17,905 +since rotations never stretch or squish anything, 86 -00:06:11,120 --> 00:06:15,408 -So let's start by rewriting that right-hand side as some kind of matrix-vector +00:06:17,905 --> 00:06:20,620 +so the length of the vector would remain the same. 87 -00:06:15,408 --> 00:06:20,077 -multiplication, using a matrix which has the effect of scaling any vector by a factor +00:06:21,680 --> 00:06:20,620 +This pattern shows up a lot in linear algebra. 88 -00:06:20,077 --> 00:06:20,620 -of lambda. +00:06:21,680 --> 00:06:23,048 +With any linear transformation described by a matrix, 89 -00:06:21,680 --> 00:06:26,178 -The columns of such a matrix will represent what happens to each basis vector, +00:06:23,048 --> 00:06:25,228 +you could understand what it's doing by reading off the columns of this matrix as the 90 -00:06:26,178 --> 00:06:29,252 -and each basis vector is simply multiplied by lambda, +00:06:25,228 --> 00:06:26,040 +landing spots for basis vectors. 91 -00:06:29,252 --> 00:06:34,320 -so this matrix will have the number lambda down the diagonal, with zeros everywhere else. +00:06:26,040 --> 00:06:30,973 +But often, a better way to get at the heart of what the linear 92 -00:06:36,180 --> 00:06:40,491 -The common way to write this guy is to factor that lambda out and write it +00:06:30,973 --> 00:06:37,474 +transformation actually does, less dependent on your particular coordinate system, 93 -00:06:40,491 --> 00:06:44,860 -as lambda times i, where i is the identity matrix with 1s down the diagonal. +00:06:37,474 --> 00:06:40,920 +is to find the eigenvectors and eigenvalues. 94 -00:06:45,860 --> 00:06:48,785 -With both sides looking like matrix-vector multiplication, +00:06:40,920 --> 00:06:43,611 +we get using the same coordinates but in our system, 95 -00:06:48,785 --> 00:06:51,860 -we can subtract off that right-hand side and factor out the v. +00:06:43,611 --> 00:06:46,709 +then it transforms it into the vector that she really meant. 96 -00:06:54,160 --> 00:06:58,971 -So what we now have is a new matrix, A minus lambda times the identity, +00:06:46,709 --> 00:06:50,823 +What about going the other way around? In the example I used earlier this video, 97 -00:06:58,971 --> 00:07:04,920 -and we're looking for a vector v such that this new matrix times v gives the zero vector. +00:06:50,823 --> 00:06:53,820 +when I had the vector with coordinates 3, 2 in our system, 98 -00:07:06,380 --> 00:07:11,100 -Now, this will always be true if v itself is the zero vector, but that's boring. +00:06:53,820 --> 00:06:57,020 +how did I compute that it would have coordinates 5 thirds and 1 99 -00:07:11,340 --> 00:07:13,640 -What we want is a non-zero eigenvector. +00:06:57,020 --> 00:07:01,280 +Symbolically, here's what the idea of an eigenvector looks like. 100 -00:07:14,420 --> 00:07:18,934 -And if you watch chapter 5 and 6, you'll know that the only way it's possible +00:07:01,280 --> 00:07:08,226 +A is the matrix representing some transformation, with v as the eigenvector, 101 -00:07:18,934 --> 00:07:23,332 -for the product of a matrix with a non-zero vector to become zero is if the +00:07:08,226 --> 00:07:13,640 +and lambda is a number, namely the corresponding eigenvalue. 102 -00:07:23,332 --> 00:07:28,020 -transformation associated with that matrix squishes space into a lower dimension. +00:07:14,420 --> 00:07:19,478 +e. In this case, the inverse of the change of basis matrix that has Jennifer's 103 -00:07:29,300 --> 00:07:34,220 -And that squishification corresponds to a zero determinant for the matrix. +00:07:19,478 --> 00:07:23,703 +basis as its columns ends up working out to have columns 1 third, 104 -00:07:35,480 --> 00:07:39,934 -To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, +00:07:23,703 --> 00:07:29,018 +negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 105 -00:07:39,934 --> 00:07:45,520 -and think about subtracting off a variable amount, lambda, from each diagonal entry. +00:07:29,018 --> 00:07:33,820 +2 looks like in Jennifer's system, we multiply this inverse change of basis 106 -00:07:46,480 --> 00:07:50,280 -Now imagine tweaking lambda, turning a knob to change its value. +00:07:33,820 --> 00:07:39,038 +matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, 107 -00:07:50,940 --> 00:07:54,539 -As that value of lambda changes, the matrix itself changes, +00:07:39,038 --> 00:07:44,325 +in a nutshell, is how to translate the description of individual vectors back 108 -00:07:54,539 --> 00:07:57,240 -and so the determinant of the matrix changes. +00:07:44,325 --> 00:07:49,680 +and forth between coordinate systems. The matrix whose columns represent Jennif 109 -00:07:58,220 --> 00:08:02,964 -The goal here is to find a value of lambda that will make this determinant zero, +00:07:49,680 --> 00:07:53,227 +er's basis vectors, but written in our coordinates, 110 -00:08:02,964 --> 00:08:07,240 -meaning the tweaked transformation squishes space into a lower dimension. +00:07:53,227 --> 00:07:57,047 +translates vectors from her language into our language. 111 -00:08:08,160 --> 00:08:11,160 -In this case, the sweet spot comes when lambda equals 1. +00:07:57,047 --> 00:08:02,231 +And the inverse matrix does the opposite. But vectors aren't the only thing 112 -00:08:12,180 --> 00:08:16,120 -Of course, if we had chosen some other matrix, the eigenvalue might not necessarily be 1. +00:08:02,231 --> 00:08:06,051 +that we describe using coordinates. For this next part, 113 -00:08:16,240 --> 00:08:18,600 -The sweet spot might be hit at some other value of lambda. +00:08:06,051 --> 00:08:11,986 +it's important that you're all comfortable representing transformations with matrices, 114 -00:08:20,080 --> 00:08:22,960 -So this is kind of a lot, but let's unravel what this is saying. +00:08:11,986 --> 00:08:14,920 +and that you know how matrix multiplication 115 -00:08:22,960 --> 00:08:26,330 -When lambda equals 1, the matrix A minus lambda +00:08:14,920 --> 00:08:16,581 +So let's start by rewriting that right-hand side as some kind of matrix-vector 116 -00:08:26,330 --> 00:08:29,560 -times the identity squishes space onto a line. +00:08:16,581 --> 00:08:18,389 +multiplication, using a matrix which has the effect of scaling any vector by a factor 117 -00:08:30,440 --> 00:08:34,500 -That means there's a non-zero vector v such that A minus +00:08:18,389 --> 00:08:18,600 +of lambda. 118 -00:08:34,500 --> 00:08:38,559 -lambda times the identity times v equals the zero vector. +00:08:20,080 --> 00:08:26,656 +The columns of such a matrix will represent what happens to each basis vector, 119 -00:08:40,480 --> 00:08:46,030 -And remember, the reason we care about that is because it means A times v +00:08:26,656 --> 00:08:31,151 +and each basis vector is simply multiplied by lambda, 120 -00:08:46,030 --> 00:08:51,579 -equals lambda times v, which you can read off as saying that the vector v +00:08:31,151 --> 00:08:38,559 +so this matrix will have the number lambda down the diagonal, with zeros everywhere else. 121 -00:08:51,579 --> 00:08:57,280 -is an eigenvector of A, staying on its own span during the transformation A. +00:08:40,480 --> 00:08:45,977 +the columns of our matrix. But this representation is heavily tied up in our choice 122 -00:08:58,320 --> 00:09:01,375 -In this example, the corresponding eigenvalue is 1, +00:08:45,977 --> 00:08:51,540 +of basis vectors, from the fact that we're following i-hat and j-hat in the first pla 123 -00:09:01,375 --> 00:09:04,020 -so v would actually just stay fixed in place. +00:08:51,540 --> 00:08:59,156 +With both sides looking like matrix-vector multiplication, 124 -00:09:06,220 --> 00:09:09,500 -Pause and ponder if you need to make sure that that line of reasoning feels good. +00:08:59,156 --> 00:09:07,160 +we can subtract off that right-hand side and factor out the v. 125 -00:09:13,380 --> 00:09:15,640 -This is the kind of thing I mentioned in the introduction. +00:09:07,160 --> 00:09:10,952 +So what we now have is a new matrix, A minus lambda times the identity, 126 -00:09:16,220 --> 00:09:19,526 -If you didn't have a solid grasp of determinants and why they +00:09:10,952 --> 00:09:15,640 +and we're looking for a vector v such that this new matrix times v gives the zero vector. 127 -00:09:19,526 --> 00:09:22,993 -relate to linear systems of equations having non-zero solutions, +00:09:16,220 --> 00:09:22,149 +s land, and it needs to describe those landing spots in her language. 128 -00:09:22,993 --> 00:09:26,300 -an expression like this would feel completely out of the blue. +00:09:22,149 --> 00:09:26,300 +Here's a common way to think of how this is done. 129 -00:09:28,320 --> 00:09:31,962 -To see this in action, let's revisit the example from the start, +00:09:28,320 --> 00:09:34,540 +What we want is a non-zero eigenvector. 130 -00:09:31,962 --> 00:09:34,540 -with a matrix whose columns are 3, 0 and 1, 2. +00:09:35,350 --> 00:09:43,505 +And if you watch chapter 5 and 6, you'll know that the only way it's possible 131 -00:09:35,350 --> 00:09:39,511 -To find if a value lambda is an eigenvalue, subtract it from +00:09:43,505 --> 00:09:51,451 +for the product of a matrix with a non-zero vector to become zero is if the 132 -00:09:39,511 --> 00:09:43,400 -the diagonals of this matrix and compute the determinant. +00:09:51,451 --> 00:09:59,920 +transformation associated with that matrix squishes space into a lower dimension. 133 -00:09:50,580 --> 00:09:54,441 -Doing this, we get a certain quadratic polynomial in lambda, +00:09:59,920 --> 00:10:02,880 +And that squishification corresponds to a zero determinant for the matrix. 134 -00:09:54,441 --> 00:09:56,720 -3 minus lambda times 2 minus lambda. +00:10:02,880 --> 00:10:05,524 +To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, 135 -00:09:57,800 --> 00:10:02,900 -Since lambda can only be an eigenvalue if this determinant happens to be zero, +00:10:05,524 --> 00:10:08,840 +and think about subtracting off a variable amount, lambda, from each diagonal entry. 136 -00:10:02,900 --> 00:10:08,258 -you can conclude that the only possible eigenvalues are lambda equals 2 and lambda +00:10:09,640 --> 00:10:16,160 +Now imagine tweaking lambda, turning a knob to change its value. 137 -00:10:08,258 --> 00:10:08,840 -equals 3. +00:10:16,160 --> 00:10:17,977 +As that value of lambda changes, the matrix itself changes, 138 -00:10:09,640 --> 00:10:14,979 -To figure out what the eigenvectors are that actually have one of these eigenvalues, +00:10:17,977 --> 00:10:19,340 +and so the determinant of the matrix changes. 139 -00:10:14,979 --> 00:10:19,565 -say lambda equals 2, plug in that value of lambda to the matrix and then +00:10:19,340 --> 00:10:22,666 +ou work through it, has columns one third, five thirds, and negative two thirds, 140 -00:10:19,565 --> 00:10:23,900 -solve for which vectors this diagonally altered matrix sends to zero. +00:10:22,666 --> 00:10:25,336 +negative one third. So if Jennifer multiplies that matrix by the 141 -00:10:24,940 --> 00:10:28,707 -If you computed this the way you would any other linear system, +00:10:25,336 --> 00:10:28,129 +coordinates of a vector in her system, it will return the 90 degree 142 -00:10:28,707 --> 00:10:33,299 -you'd see that the solutions are all the vectors on the diagonal line spanned +00:10:28,129 --> 00:10:30,840 +rotated version of that vector expressed in her coordinate system. 143 -00:10:33,299 --> 00:10:34,300 -by negative 1, 1. +00:10:30,840 --> 00:10:34,300 +In this case, the sweet spot comes when lambda equals 1. 144 -00:10:35,220 --> 00:10:39,277 -This corresponds to the fact that the unaltered matrix, 3, 0, 1, +00:10:35,220 --> 00:10:34,300 +Of course, if we had chosen some other matrix, the eigenvalue might not necessarily be 1. 145 -00:10:39,277 --> 00:10:43,460 -2, has the effect of stretching all those vectors by a factor of 2. +00:10:35,220 --> 00:10:43,308 +A inverse times M times A, it suggests a mathematical sort of empathy. 146 -00:10:46,320 --> 00:10:50,200 -Now, a 2D transformation doesn't have to have eigenvectors. +00:10:43,308 --> 00:10:47,980 +That middle matrix represents a transform 147 -00:10:50,720 --> 00:10:53,400 -For example, consider a rotation by 90 degrees. +00:10:47,980 --> 00:10:49,690 +ation of some kind as you see it, and the outer two matrices represent the empathy, 148 -00:10:53,660 --> 00:10:58,200 -This doesn't have any eigenvectors since it rotates every vector off of its own span. +00:10:49,690 --> 00:10:50,200 +the shift in perspective. 149 -00:11:00,800 --> 00:11:04,513 -If you actually try computing the eigenvalues of a rotation like this, +00:10:50,720 --> 00:10:54,172 +And the full matrix product represents that same transformation, 150 -00:11:04,513 --> 00:11:05,560 -notice what happens. +00:10:54,172 --> 00:10:58,633 +but as someone else sees it. For those of you wondering why we care about alternate 151 -00:11:06,300 --> 00:11:10,140 -Its matrix has columns 0, 1 and negative 1, 0. +00:10:58,633 --> 00:11:03,200 +coordinate systems, the next video on eigenvectors and eigenvalues will give a really 152 -00:11:11,100 --> 00:11:15,800 -Subtract off lambda from the diagonal elements and look for when the determinant is zero. +00:11:03,200 --> 00:11:04,900 +important example of this. See y 153 -00:11:18,140 --> 00:11:21,940 -In this case, you get the polynomial lambda squared plus 1. +00:11:04,900 --> 00:11:11,760 +That means there's a non-zero vector v such that A minus 154 -00:11:22,680 --> 00:11:27,920 -The only roots of that polynomial are the imaginary numbers, i and negative i. +00:11:11,760 --> 00:11:18,620 +lambda times the identity times v equals the zero vector. 155 -00:11:28,840 --> 00:11:33,600 -The fact that there are no real number solutions indicates that there are no eigenvectors. +00:11:18,620 --> 00:11:21,692 +And remember, the reason we care about that is because it means A times v 156 -00:11:35,540 --> 00:11:39,820 -Another pretty interesting example worth holding in the back of your mind is a shear. +00:11:21,692 --> 00:11:24,764 +equals lambda times v, which you can read off as saying that the vector v 157 -00:11:40,560 --> 00:11:47,840 -This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1. +00:11:24,764 --> 00:11:27,920 +is an eigenvector of A, staying on its own span during the transformation A. 158 -00:11:48,740 --> 00:11:51,611 -All of the vectors on the x-axis are eigenvectors +00:11:28,840 --> 00:11:35,744 +In this example, the corresponding eigenvalue is 1, 159 -00:11:51,611 --> 00:11:54,540 -with eigenvalue 1 since they remain fixed in place. +00:11:35,744 --> 00:11:41,720 +so v would actually just stay fixed in place. 160 -00:11:55,680 --> 00:11:57,820 -In fact, these are the only eigenvectors. +00:11:41,720 --> 00:11:47,840 +Pause and ponder if you need to make sure that that line of reasoning feels good. 161 -00:11:58,760 --> 00:12:03,924 -When you subtract off lambda from the diagonals and compute the determinant, +00:11:48,740 --> 00:11:54,540 +This is the kind of thing I mentioned in the introduction. 162 -00:12:03,924 --> 00:12:06,540 -what you get is 1 minus lambda squared. +00:11:55,680 --> 00:12:00,482 +If you didn't have a solid grasp of determinants and why they 163 -00:12:09,320 --> 00:12:12,860 -And the only root of this expression is lambda equals 1. +00:12:00,482 --> 00:12:05,517 +relate to linear systems of equations having non-zero solutions, 164 -00:12:14,560 --> 00:12:17,112 -This lines up with what we see geometrically, +00:12:05,517 --> 00:12:10,320 +an expression like this would feel completely out of the blue. 165 -00:12:17,112 --> 00:12:19,720 -that all of the eigenvectors have eigenvalue 1. +00:12:10,320 --> 00:12:15,824 +To see this in action, let's revisit the example from the start, 166 -00:12:21,080 --> 00:12:25,037 -Keep in mind though, it's also possible to have just one eigenvalue, +00:12:15,824 --> 00:12:19,720 +with a matrix whose columns are 3, 0 and 1, 2. 167 -00:12:25,037 --> 00:12:28,020 -but with more than just a line full of eigenvectors. +00:12:21,080 --> 00:12:22,972 +To find if a value lambda is an eigenvalue, subtract it from 168 -00:12:29,900 --> 00:12:33,180 -A simple example is a matrix that scales everything by 2. +00:12:22,972 --> 00:12:24,740 +the diagonals of this matrix and compute the determinant. 169 -00:12:33,900 --> 00:12:37,200 -The only eigenvalue is 2, but every vector in the +00:12:25,340 --> 00:12:28,723 +Doing this, we get a certain quadratic polynomial in lambda, 170 -00:12:37,200 --> 00:12:40,700 -plane gets to be an eigenvector with that eigenvalue. +00:12:28,723 --> 00:12:30,720 +3 minus lambda times 2 minus lambda. 171 -00:12:42,000 --> 00:12:44,666 -Now is another good time to pause and ponder some +00:12:30,720 --> 00:12:34,323 +Since lambda can only be an eigenvalue if this determinant happens to be zero, 172 -00:12:44,666 --> 00:12:46,960 -of this before I move on to the last topic. +00:12:34,323 --> 00:12:38,109 +you can conclude that the only possible eigenvalues are lambda equals 2 and lambda 173 -00:13:03,540 --> 00:13:06,944 -I want to finish off here with the idea of an eigenbasis, +00:12:38,109 --> 00:12:38,520 +equals 3. 174 -00:13:06,944 --> 00:13:09,880 -which relies heavily on ideas from the last video. +00:12:38,520 --> 00:12:50,262 +To figure out what the eigenvectors are that actually have one of these eigenvalues, 175 -00:13:11,480 --> 00:13:16,380 -Take a look at what happens if our basis vectors just so happen to be eigenvectors. +00:12:50,262 --> 00:13:00,347 +say lambda equals 2, plug in that value of lambda to the matrix and then 176 -00:13:17,120 --> 00:13:22,380 -For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2. +00:13:00,347 --> 00:13:09,880 +solve for which vectors this diagonally altered matrix sends to zero. 177 -00:13:23,420 --> 00:13:27,014 -Writing their new coordinates as the columns of a matrix, +00:13:11,480 --> 00:13:15,207 +If you computed this the way you would any other linear system, 178 -00:13:27,014 --> 00:13:30,361 -notice that those scalar multiples, negative 1 and 2, +00:13:15,207 --> 00:13:19,749 +you'd see that the solutions are all the vectors on the diagonal line spanned 179 -00:13:30,361 --> 00:13:35,382 -which are the eigenvalues of i-hat and j-hat, sit on the diagonal of our matrix, +00:13:19,749 --> 00:13:20,740 +by negative 1, 1. 180 -00:13:35,382 --> 00:13:37,180 -and every other entry is a 0. +00:13:20,740 --> 00:13:23,330 +This corresponds to the fact that the unaltered matrix, 3, 0, 1, 181 -00:13:38,880 --> 00:13:42,551 -Any time a matrix has zeros everywhere other than the diagonal, +00:13:23,330 --> 00:13:26,000 +2, has the effect of stretching all those vectors by a factor of 2. 182 -00:13:42,551 --> 00:13:45,420 -it's called, reasonably enough, a diagonal matrix. +00:13:26,620 --> 00:13:31,140 +Now, a 2D transformation doesn't have to have eigenvectors. 183 -00:13:45,840 --> 00:13:50,509 -And the way to interpret this is that all the basis vectors are eigenvectors, +00:13:31,140 --> 00:13:36,420 +For example, consider a rotation by 90 degrees. 184 -00:13:50,509 --> 00:13:54,400 -with the diagonal entries of this matrix being their eigenvalues. +00:13:36,420 --> 00:13:46,720 +This doesn't have any eigenvectors since it rotates every vector off of its own span. 185 -00:13:57,100 --> 00:14:01,060 -There are a lot of things that make diagonal matrices much nicer to work with. +00:13:46,720 --> 00:13:55,848 +If you actually try computing the eigenvalues of a rotation like this, 186 -00:14:01,780 --> 00:14:05,032 -One big one is that it's easier to compute what will happen +00:13:55,848 --> 00:13:58,420 +notice what happens. 187 -00:14:05,032 --> 00:14:08,340 -if you multiply this matrix by itself a whole bunch of times. +00:13:58,420 --> 00:14:01,060 +Its matrix has columns 0, 1 and negative 1, 0. 188 -00:14:09,420 --> 00:14:14,733 -Since all one of these matrices does is scale each basis vector by some eigenvalue, +00:14:01,780 --> 00:14:03,020 +Subtract off lambda from the diagonal elements and look for when the determinant is zero. 189 -00:14:14,733 --> 00:14:17,769 -applying that matrix many times, say 100 times, +00:14:03,020 --> 00:14:08,340 +In this case, you get the polynomial lambda squared plus 1. 190 -00:14:17,769 --> 00:14:22,765 -is just going to correspond to scaling each basis vector by the 100th power of +00:14:09,420 --> 00:14:09,580 +The only roots of that polynomial are the imaginary numbers, i and negative i. 191 -00:14:22,765 --> 00:14:24,600 -the corresponding eigenvalue. +00:14:09,580 --> 00:14:20,920 +The fact that there are no real number solutions indicates that there are no eigenvectors. 192 -00:14:25,700 --> 00:14:29,680 -In contrast, try computing the 100th power of a non-diagonal matrix. +00:14:20,920 --> 00:14:31,320 +Another pretty interesting example worth holding in the back of your mind is a shear. 193 -00:14:29,680 --> 00:14:31,320 -Really, try it for a moment. +00:14:31,740 --> 00:14:43,620 +This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1. 194 -00:14:31,740 --> 00:14:32,440 -It's a nightmare. +00:14:43,620 --> 00:14:48,313 +All of the vectors on the x-axis are eigenvectors 195 -00:14:36,080 --> 00:14:41,260 -Of course, you'll rarely be so lucky as to have your basis vectors also be eigenvectors. +00:14:48,313 --> 00:14:53,100 +with eigenvalue 1 since they remain fixed in place. 196 -00:14:42,040 --> 00:14:45,110 -But if your transformation has a lot of eigenvectors, +00:14:53,100 --> 00:14:54,860 +In fact, these are the only eigenvectors. 197 -00:14:45,110 --> 00:14:49,887 -like the one from the start of this video, enough so that you can choose a set that +00:14:54,860 --> 00:15:00,475 +When you subtract off lambda from the diagonals and compute the determinant, 198 -00:14:49,887 --> 00:14:54,492 -spans the full space, then you could change your coordinate system so that these +00:15:00,475 --> 00:15:03,320 +what you get is 1 minus lambda squared. 199 -00:14:54,492 --> 00:14:56,540 -eigenvectors are your basis vectors. +00:15:03,320 --> 00:15:11,620 +And the only root of this expression is lambda equals 1. 200 -00:14:57,140 --> 00:15:00,371 -I talked about change of basis last video, but I'll go through +00:15:11,980 --> 00:15:13,028 +This lines up with what we see geometrically, 201 -00:15:00,371 --> 00:15:03,603 -a super quick reminder here of how to express a transformation +00:15:13,028 --> 00:15:14,100 +that all of the eigenvectors have eigenvalue 1. 202 -00:15:03,603 --> 00:15:07,040 -currently written in our coordinate system into a different system. +00:15:14,500 --> 00:15:16,290 +Keep in mind though, it's also possible to have just one eigenvalue, 203 -00:15:08,440 --> 00:15:12,282 -Take the coordinates of the vectors that you want to use as a new basis, +00:15:16,290 --> 00:15:17,640 +but with more than just a line full of eigenvectors. 204 -00:15:12,282 --> 00:15:14,755 -which in this case means our two eigenvectors, +00:15:17,640 --> 00:15:24,540 +A simple example is a matrix that scales everything by 2. 205 -00:15:14,755 --> 00:15:19,440 -then make those coordinates the columns of a matrix, known as the change of basis matrix. +00:15:24,540 --> 00:15:28,044 +The only eigenvalue is 2, but every vector in the 206 -00:15:20,180 --> 00:15:22,834 -When you sandwich the original transformation, +00:15:28,044 --> 00:15:31,760 +plane gets to be an eigenvector with that eigenvalue. 207 -00:15:22,834 --> 00:15:26,843 -putting the change of basis matrix on its right and the inverse of the +00:15:31,760 --> 00:15:34,308 +Now is another good time to pause and ponder some 208 -00:15:26,843 --> 00:15:31,191 -change of basis matrix on its left, the result will be a matrix representing +00:15:34,308 --> 00:15:36,500 +of this before I move on to the last topic. 209 -00:15:31,191 --> 00:15:35,031 -that same transformation, but from the perspective of the new basis +00:15:37,440 --> 00:15:42,402 +I want to finish off here with the idea of an eigenbasis, 210 -00:15:35,031 --> 00:15:36,500 -vectors coordinate system. +00:15:42,402 --> 00:15:46,680 +which relies heavily on ideas from the last video. 211 -00:15:37,440 --> 00:15:41,910 -The whole point of doing this with eigenvectors is that this new matrix is +00:15:46,860 --> 00:15:51,900 +Take a look at what happens if our basis vectors just so happen to be eigenvectors. 212 -00:15:41,910 --> 00:15:46,680 -guaranteed to be diagonal with its corresponding eigenvalues down that diagonal. +00:15:51,900 --> 00:15:53,200 +For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2. 213 -00:15:46,860 --> 00:15:50,417 -This is because it represents working in a coordinate system where what +00:15:53,200 --> 00:15:56,962 +Writing their new coordinates as the columns of a matrix, 214 -00:15:50,417 --> 00:15:54,320 -happens to the basis vectors is that they get scaled during the transformation. +00:15:56,962 --> 00:16:00,464 +notice that those scalar multiples, negative 1 and 2, 215 -00:15:55,800 --> 00:15:59,301 -A set of basis vectors which are also eigenvectors is called, +00:16:00,464 --> 00:16:05,718 +which are the eigenvalues of i-hat and j-hat, sit on the diagonal of our matrix, 216 -00:15:59,301 --> 00:16:01,560 -again, reasonably enough, an eigenbasis. +00:16:05,718 --> 00:16:07,600 +and every other entry is a 0. 217 -00:16:02,340 --> 00:16:07,108 -So if, for example, you needed to compute the 100th power of this matrix, +00:16:07,600 --> 00:16:09,194 +Any time a matrix has zeros everywhere other than the diagonal, 218 -00:16:07,108 --> 00:16:10,460 -it would be much easier to change to an eigenbasis, +00:16:09,194 --> 00:16:10,440 +it's called, reasonably enough, a diagonal matrix. 219 -00:16:10,460 --> 00:16:15,680 -compute the 100th power in that system, then convert back to our standard system. +00:16:10,920 --> 00:16:13,287 +And the way to interpret this is that all the basis vectors are eigenvectors, 220 -00:16:16,620 --> 00:16:18,320 -You can't do this with all transformations. +00:16:13,287 --> 00:16:15,260 +with the diagonal entries of this matrix being their eigenvalues. 221 -00:16:18,320 --> 00:16:22,980 -A shear, for example, doesn't have enough eigenvectors to span the full space. +00:16:15,260 --> 00:16:15,680 +There are a lot of things that make diagonal matrices much nicer to work with. 222 -00:16:23,460 --> 00:16:28,160 -But if you can find an eigenbasis, it makes matrix operations really lovely. +00:16:16,620 --> 00:16:17,066 +One big one is that it's easier to compute what will happen 223 -00:16:29,120 --> 00:16:31,771 -For those of you willing to work through a pretty neat puzzle to +00:16:17,066 --> 00:16:17,520 +if you multiply this matrix by itself a whole bunch of times. 224 -00:16:31,771 --> 00:16:34,586 -see what this looks like in action and how it can be used to produce +00:16:17,520 --> 00:16:20,495 +Since all one of these matrices does is scale each basis vector by some eigenvalue, 225 -00:16:34,586 --> 00:16:37,320 -some surprising results, I'll leave up a prompt here on the screen. +00:16:20,495 --> 00:16:22,194 +applying that matrix many times, say 100 times, 226 -00:16:37,600 --> 00:16:40,280 -It takes a bit of work, but I think you'll enjoy it. +00:16:22,194 --> 00:16:24,992 +is just going to correspond to scaling each basis vector by the 100th power of 227 -00:16:40,840 --> 00:16:46,120 +00:16:24,992 --> 00:16:26,020 +the corresponding eigenvalue. + +228 +00:16:26,020 --> 00:16:27,160 +In contrast, try computing the 100th power of a non-diagonal matrix. + +229 +00:16:27,160 --> 00:16:30,700 +Really, try it for a moment. + +230 +00:16:30,700 --> 00:16:30,860 +It's a nightmare. + +231 +00:16:30,860 --> 00:16:39,060 +Of course, you'll rarely be so lucky as to have your basis vectors also be eigenvectors. + +232 +00:16:39,060 --> 00:16:39,060 +But if your transformation has a lot of eigenvectors, + +233 +00:16:39,060 --> 00:16:39,060 +like the one from the start of this video, enough so that you can choose a set that + +234 +00:16:39,060 --> 00:16:39,060 +spans the full space, then you could change your coordinate system so that these + +235 +00:16:39,060 --> 00:16:39,060 +eigenvectors are your basis vectors. + +236 +00:16:39,060 --> 00:16:41,123 +I talked about change of basis last video, but I'll go through + +237 +00:16:41,123 --> 00:16:43,186 +a super quick reminder here of how to express a transformation + +238 +00:16:43,186 --> 00:16:45,380 +currently written in our coordinate system into a different system. + +239 +00:16:45,426 --> 00:16:45,380 +Take the coordinates of the vectors that you want to use as a new basis, + +240 +00:16:45,549 --> 00:16:45,380 +which in this case means our two eigenvectors, + +241 +00:16:45,601 --> 00:16:45,380 +then make those coordinates the columns of a matrix, known as the change of basis matrix. + +242 +00:16:45,648 --> 00:16:45,380 +When you sandwich the original transformation, + +243 +00:16:45,652 --> 00:16:45,426 +putting the change of basis matrix on its right and the inverse of the + +244 +00:16:45,687 --> 00:16:45,549 +change of basis matrix on its left, the result will be a matrix representing + +245 +00:16:45,718 --> 00:16:45,601 +that same transformation, but from the perspective of the new basis + +246 +00:16:45,815 --> 00:16:45,648 +vectors coordinate system. + +247 +00:16:45,900 --> 00:16:45,652 +The whole point of doing this with eigenvectors is that this new matrix is + +248 +00:16:45,900 --> 00:16:45,687 +guaranteed to be diagonal with its corresponding eigenvalues down that diagonal. + +249 +00:16:45,900 --> 00:16:45,718 +This is because it represents working in a coordinate system where what + +250 +00:16:45,900 --> 00:16:45,815 +happens to the basis vectors is that they get scaled during the transformation. + +251 +00:16:45,900 --> 00:16:46,033 +A set of basis vectors which are also eigenvectors is called, + +252 +00:16:46,033 --> 00:16:46,119 +again, reasonably enough, an eigenbasis. + +253 +00:16:46,119 --> 00:16:46,120 +So if, for example, you needed to compute the 100th power of this matrix, + +254 +00:16:46,120 --> 00:16:46,120 +it would be much easier to change to an eigenbasis, + +255 +00:16:46,120 --> 00:16:46,120 +compute the 100th power in that system, then convert back to our standard system. + +256 +00:16:46,120 --> 00:16:46,120 +You can't do this with all transformations. + +257 +00:16:46,120 --> 00:16:46,120 +A shear, for example, doesn't have enough eigenvectors to span the full space. + +258 +00:16:46,120 --> 00:16:46,120 +But if you can find an eigenbasis, it makes matrix operations really lovely. + +259 +00:16:46,120 --> 00:16:46,120 +For those of you willing to work through a pretty neat puzzle to + +260 +00:16:46,120 --> 00:16:46,120 +see what this looks like in action and how it can be used to produce + +261 +00:16:46,120 --> 00:16:46,120 +some surprising results, I'll leave up a prompt here on the screen. + +262 +00:16:46,120 --> 00:16:46,120 +It takes a bit of work, but I think you'll enjoy it. + +263 +00:16:46,120 --> 00:16:46,120 The next and final video of this series is going to be on abstract vector spaces. diff --git a/2016/change-of-basis/english/sentence_timings.json b/2016/change-of-basis/english/sentence_timings.json index 1378cd3be..8c68b6b58 100644 --- a/2016/change-of-basis/english/sentence_timings.json +++ b/2016/change-of-basis/english/sentence_timings.json @@ -45,553 +45,553 @@ 84.84 ], [ - "It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.", + "st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice of i-hat and j-hat as the ve", 85.46, - 91.04 + 101.26 ], [ - "So it's represented with a matrix whose columns are 3, 0, and 1, 2.", - 91.78, - 95.64 + "ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the", + 101.26, + 111.5 ], [ - "Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip.", - 96.6, - 104.16 + "two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a different set of basis vectors. For example, let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b", + 111.5, + 131.32 ], [ - "Most vectors are going to get knocked off their span during the transformation.", - 104.92, - 108.38 + "asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up. Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat.", + 131.32, + 144.12 ], [ - "I mean, it would seem pretty coincidental if the place where the vector landed also happened to be somewhere on that line.", - 108.78, - 115.32 + "Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third. What this means is that the particular way to get to that vector using her two basis vectors is to scale b1 by 5 thirds, scale b2 by 1 third, then add them both together. In a little bi", + 146.32, + 166.8 ], [ - "But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar.", - 117.4, - 127.04 + "t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she thinks of her first coordinate as scali", + 166.8, + 182.62 ], [ "For this specific example, the basis vector i-hat is one such special vector.", - 129.46, - 134.1 + 182.62, + 184.26 ], [ "The span of i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis.", - 134.64, - 144.12 + 184.26, + 186.58 ], [ "What's more, because of the way linear transformations work, any other vector on the x-axis is also just stretched by a factor of 3, and hence remains on its own span.", - 146.32, - 156.48 + 186.58, + 198.08 ], [ "A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1.", - 158.5, - 164.04 + 202.52, + 198.08 ], [ - "It ends up getting stretched by a factor of 2.", - 164.66, - 167.14 + "Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc", + 202.52, + 214.46 ], [ - "And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2.", - 169.0, - 178.22 + "t, a way to visualize our coordinate system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct meant as nothi", + 214.46, + 223.32 ], [ - "And for this transformation, those are all the vectors with this special property of staying on their span.", - 179.82, - 185.18 + "ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector by zero. But the direction of her axes and", + 223.4, + 244.86 ], [ "Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2.", - 185.62, - 191.98 + 244.86, + 252.62 ], [ "Any other vector is going to get rotated somewhat during the transformation, knocked off the line that it spans.", - 192.76, - 198.08 + 252.62, + 258.14 ], [ - "As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation.", - 202.52, - 217.38 + "ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vectors in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio", + 258.14, + 291.84 ], [ "Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive.", - 220.28, - 225.94 + 291.84, + 301.46 ], [ "In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half.", - 226.38, - 235.12 + 301.46, + 307.02 ], [ - "But the important part here is that it stays on the line that it spans out without getting rotated off of it.", - 236.98, - 242.76 + "pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformation that moves our basis vectors, i-hat and j-hat, the things we think of when we say 1, 0 and 0, 1, to Jennifer's basis vectors, the thin", + 307.02, + 328.36 ], [ - "For a glimpse of why this might be a useful thing to think about, consider some three-dimensional rotation.", - 244.46, - 249.8 + "gs she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation.", + 328.36, + 341.38 ], [ "If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation.", - 251.66, - 260.5 + 341.38, + 354.82 ], [ "And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation.", - 262.6, - 274.74 + 354.82, + 370.54 ], [ "In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length of the vector would remain the same.", - 277.0, - 285.86 + 371.12, + 380.62 ], [ "This pattern shows up a lot in linear algebra.", - 288.08, - 290.02 + 381.68, + 380.62 ], [ "With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors.", - 290.44, - 299.4 + 381.68, + 386.04 ], [ "But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", - 300.02, - 310.82 + 386.04, + 400.92 ], [ - "I won't cover the full details on methods for computing eigenvectors and eigenvalues here, but I'll try to give an overview of the computational ideas that are most important for a conceptual understanding.", - 315.46, - 326.02 + "we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1", + 400.92, + 417.02 ], [ "Symbolically, here's what the idea of an eigenvector looks like.", - 327.18, - 330.48 + 417.02, + 421.28 ], [ "A is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue.", - 331.04, - 339.74 + 421.28, + 433.64 ], [ - "What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda.", - 340.68, - 349.9 + "e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we multiply this inverse change of basis", + 434.42, + 453.82 ], [ - "So finding the eigenvectors and their eigenvalues of a matrix A comes down to finding the values of v and lambda that make this expression true.", - 351.0, - 360.1 + "matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose columns represent Jennif", + 453.82, + 469.68 ], [ - "It's a little awkward to work with at first, because that left-hand side represents matrix-vector multiplication, but the right-hand side here is scalar-vector multiplication.", - 361.92, - 370.54 + "er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's important that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication", + 469.68, + 494.92 ], [ "So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda.", - 371.12, - 380.62 + 494.92, + 498.6 ], [ "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else.", - 381.68, - 394.32 + 500.08, + 518.56 ], [ - "The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal.", - 396.18, - 404.86 + "the columns of our matrix. But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla", + 520.48, + 531.54 ], [ "With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v.", - 405.86, - 411.86 + 531.54, + 547.16 ], [ "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector.", - 414.16, - 424.92 + 547.16, + 555.64 ], [ - "Now, this will always be true if v itself is the zero vector, but that's boring.", - 426.38, - 431.1 + "s land, and it needs to describe those landing spots in her language. Here's a common way to think of how this is done.", + 556.22, + 566.3 ], [ "What we want is a non-zero eigenvector.", - 431.34, - 433.64 + 568.32, + 574.54 ], [ "And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", - 434.42, - 448.02 + 575.35, + 599.92 ], [ "And that squishification corresponds to a zero determinant for the matrix.", - 449.3, - 454.22 + 599.92, + 602.88 ], [ "To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry.", - 455.48, - 465.52 + 602.88, + 608.84 ], [ "Now imagine tweaking lambda, turning a knob to change its value.", - 466.48, - 470.28 + 609.64, + 616.16 ], [ "As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes.", - 470.94, - 477.24 + 616.16, + 619.34 ], [ - "The goal here is to find a value of lambda that will make this determinant zero, meaning the tweaked transformation squishes space into a lower dimension.", - 478.22, - 487.24 + "ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system.", + 619.34, + 630.84 ], [ "In this case, the sweet spot comes when lambda equals 1.", - 488.16, - 491.16 + 630.84, + 634.3 ], [ "Of course, if we had chosen some other matrix, the eigenvalue might not necessarily be 1.", - 492.18, - 496.12 + 635.22, + 634.3 ], [ - "The sweet spot might be hit at some other value of lambda.", - 496.24, - 498.6 + "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform", + 635.22, + 647.98 ], [ - "So this is kind of a lot, but let's unravel what this is saying.", - 500.08, - 502.96 + "ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective.", + 647.98, + 650.2 ], [ - "When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line.", - 502.96, - 509.56 + "And the full matrix product represents that same transformation, but as someone else sees it. For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of this. See y", + 650.72, + 664.9 ], [ "That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector.", - 510.44, - 518.56 + 664.9, + 678.62 ], [ "And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A.", - 520.48, - 537.28 + 678.62, + 687.92 ], [ "In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place.", - 538.32, - 544.02 + 688.84, + 701.72 ], [ "Pause and ponder if you need to make sure that that line of reasoning feels good.", - 546.22, - 549.5 + 701.72, + 707.84 ], [ "This is the kind of thing I mentioned in the introduction.", - 553.38, - 555.64 + 708.74, + 714.54 ], [ "If you didn't have a solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", - 556.22, - 566.3 + 715.68, + 730.32 ], [ "To see this in action, let's revisit the example from the start, with a matrix whose columns are 3, 0 and 1, 2.", - 568.32, - 574.54 + 730.32, + 739.72 ], [ "To find if a value lambda is an eigenvalue, subtract it from the diagonals of this matrix and compute the determinant.", - 575.35, - 583.4 + 741.08, + 744.74 ], [ "Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda.", - 590.58, - 596.72 + 745.34, + 750.72 ], [ "Since lambda can only be an eigenvalue if this determinant happens to be zero, you can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3.", - 597.8, - 608.84 + 750.72, + 758.52 ], [ "To figure out what the eigenvectors are that actually have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero.", - 609.64, - 623.9 + 758.52, + 789.88 ], [ "If you computed this the way you would any other linear system, you'd see that the solutions are all the vectors on the diagonal line spanned by negative 1, 1.", - 624.94, - 634.3 + 791.48, + 800.74 ], [ "This corresponds to the fact that the unaltered matrix, 3, 0, 1, 2, has the effect of stretching all those vectors by a factor of 2.", - 635.22, - 643.46 + 800.74, + 806.0 ], [ "Now, a 2D transformation doesn't have to have eigenvectors.", - 646.32, - 650.2 + 806.62, + 811.14 ], [ "For example, consider a rotation by 90 degrees.", - 650.72, - 653.4 + 811.14, + 816.42 ], [ "This doesn't have any eigenvectors since it rotates every vector off of its own span.", - 653.66, - 658.2 + 816.42, + 826.72 ], [ "If you actually try computing the eigenvalues of a rotation like this, notice what happens.", - 660.8, - 665.56 + 826.72, + 838.42 ], [ "Its matrix has columns 0, 1 and negative 1, 0.", - 666.3, - 670.14 + 838.42, + 841.06 ], [ "Subtract off lambda from the diagonal elements and look for when the determinant is zero.", - 671.1, - 675.8 + 841.78, + 843.02 ], [ "In this case, you get the polynomial lambda squared plus 1.", - 678.14, - 681.94 + 843.02, + 848.34 ], [ "The only roots of that polynomial are the imaginary numbers, i and negative i.", - 682.68, - 687.92 + 849.42, + 849.58 ], [ "The fact that there are no real number solutions indicates that there are no eigenvectors.", - 688.84, - 693.6 + 849.58, + 860.92 ], [ "Another pretty interesting example worth holding in the back of your mind is a shear.", - 695.54, - 699.82 + 860.92, + 871.32 ], [ "This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1.", - 700.56, - 707.84 + 871.74, + 883.62 ], [ "All of the vectors on the x-axis are eigenvectors with eigenvalue 1 since they remain fixed in place.", - 708.74, - 714.54 + 883.62, + 893.1 ], [ "In fact, these are the only eigenvectors.", - 715.68, - 717.82 + 893.1, + 894.86 ], [ "When you subtract off lambda from the diagonals and compute the determinant, what you get is 1 minus lambda squared.", - 718.76, - 726.54 + 894.86, + 903.32 ], [ "And the only root of this expression is lambda equals 1.", - 729.32, - 732.86 + 903.32, + 911.62 ], [ "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.", - 734.56, - 739.72 + 911.98, + 914.1 ], [ "Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a line full of eigenvectors.", - 741.08, - 748.02 + 914.5, + 917.64 ], [ "A simple example is a matrix that scales everything by 2.", - 749.9, - 753.18 + 917.64, + 924.54 ], [ "The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue.", - 753.9, - 760.7 + 924.54, + 931.76 ], [ "Now is another good time to pause and ponder some of this before I move on to the last topic.", - 762.0, - 766.96 + 931.76, + 936.5 ], [ "I want to finish off here with the idea of an eigenbasis, which relies heavily on ideas from the last video.", - 783.54, - 789.88 + 937.44, + 946.68 ], [ "Take a look at what happens if our basis vectors just so happen to be eigenvectors.", - 791.48, - 796.38 + 946.86, + 951.9 ], [ "For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2.", - 797.12, - 802.38 + 951.9, + 953.2 ], [ "Writing their new coordinates as the columns of a matrix, notice that those scalar multiples, negative 1 and 2, which are the eigenvalues of i-hat and j-hat, sit on the diagonal of our matrix, and every other entry is a 0.", - 803.42, - 817.18 + 953.2, + 967.6 ], [ "Any time a matrix has zeros everywhere other than the diagonal, it's called, reasonably enough, a diagonal matrix.", - 818.88, - 825.42 + 967.6, + 970.44 ], [ "And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.", - 825.84, - 834.4 + 970.92, + 975.26 ], [ "There are a lot of things that make diagonal matrices much nicer to work with.", - 837.1, - 841.06 + 975.26, + 975.68 ], [ "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", - 841.78, - 848.34 + 976.62, + 977.52 ], [ "Since all one of these matrices does is scale each basis vector by some eigenvalue, applying that matrix many times, say 100 times, is just going to correspond to scaling each basis vector by the 100th power of the corresponding eigenvalue.", - 849.42, - 864.6 + 977.52, + 986.02 ], [ "In contrast, try computing the 100th power of a non-diagonal matrix.", - 865.7, - 869.68 + 986.02, + 987.16 ], [ "Really, try it for a moment.", - 869.68, - 871.32 + 987.16, + 990.7 ], [ "It's a nightmare.", - 871.74, - 872.44 + 990.7, + 990.86 ], [ "Of course, you'll rarely be so lucky as to have your basis vectors also be eigenvectors.", - 876.08, - 881.26 + 990.86, + 999.06 ], [ "But if your transformation has a lot of eigenvectors, like the one from the start of this video, enough so that you can choose a set that spans the full space, then you could change your coordinate system so that these eigenvectors are your basis vectors.", - 882.04, - 896.54 + 999.06, + 999.06 ], [ "I talked about change of basis last video, but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", - 897.14, - 907.04 + 999.06, + 1005.38 ], [ "Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors, then make those coordinates the columns of a matrix, known as the change of basis matrix.", - 908.44, - 919.44 + 1005.9, + 1005.38 ], [ "When you sandwich the original transformation, putting the change of basis matrix on its right and the inverse of the change of basis matrix on its left, the result will be a matrix representing that same transformation, but from the perspective of the new basis vectors coordinate system.", - 920.18, - 936.5 + 1005.9, + 1005.38 ], [ "The whole point of doing this with eigenvectors is that this new matrix is guaranteed to be diagonal with its corresponding eigenvalues down that diagonal.", - 937.44, - 946.68 + 1005.9, + 1005.38 ], [ "This is because it represents working in a coordinate system where what happens to the basis vectors is that they get scaled during the transformation.", - 946.86, - 954.32 + 1005.9, + 1005.38 ], [ "A set of basis vectors which are also eigenvectors is called, again, reasonably enough, an eigenbasis.", - 955.8, - 961.56 + 1005.9, + 1006.12 ], [ "So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis, compute the 100th power in that system, then convert back to our standard system.", - 962.34, - 975.68 + 1006.12, + 1006.12 ], [ "You can't do this with all transformations.", - 976.62, - 978.32 + 1006.12, + 1006.12 ], [ "A shear, for example, doesn't have enough eigenvectors to span the full space.", - 978.32, - 982.98 + 1006.12, + 1006.12 ], [ "But if you can find an eigenbasis, it makes matrix operations really lovely.", - 983.46, - 988.16 + 1006.12, + 1006.12 ], [ "For those of you willing to work through a pretty neat puzzle to see what this looks like in action and how it can be used to produce some surprising results, I'll leave up a prompt here on the screen.", - 989.12, - 997.32 + 1006.12, + 1006.12 ], [ "It takes a bit of work, but I think you'll enjoy it.", - 997.6, - 1000.28 + 1006.12, + 1006.12 ], [ "The next and final video of this series is going to be on abstract vector spaces.", - 1000.84, + 1006.12, 1006.12 ] ] \ No newline at end of file diff --git a/2016/change-of-basis/english/transcript.txt b/2016/change-of-basis/english/transcript.txt index 11f7346cf..122b8dd7d 100644 --- a/2016/change-of-basis/english/transcript.txt +++ b/2016/change-of-basis/english/transcript.txt @@ -7,56 +7,56 @@ The issue is that it only really makes sense if you have a solid visual understa Most important here is that you know how to think about matrices as linear transformations, but you also need to be comfortable with things like determinants, linear systems of equations, and change of basis. Confusion about eigenstuffs usually has more to do with a shaky foundation in one of these topics than it does with eigenvectors and eigenvalues themselves. To start, consider some linear transformation in two dimensions, like the one shown here. -It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2. -So it's represented with a matrix whose columns are 3, 0, and 1, 2. -Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip. -Most vectors are going to get knocked off their span during the transformation. -I mean, it would seem pretty coincidental if the place where the vector landed also happened to be somewhere on that line. -But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar. +st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice of i-hat and j-hat as the ve +ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the +two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a different set of basis vectors. For example, let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b +asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up. Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. +Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third. What this means is that the particular way to get to that vector using her two basis vectors is to scale b1 by 5 thirds, scale b2 by 1 third, then add them both together. In a little bi +t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she thinks of her first coordinate as scali For this specific example, the basis vector i-hat is one such special vector. The span of i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis. What's more, because of the way linear transformations work, any other vector on the x-axis is also just stretched by a factor of 3, and hence remains on its own span. A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1. -It ends up getting stretched by a factor of 2. -And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2. -And for this transformation, those are all the vectors with this special property of staying on their span. +Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc +t, a way to visualize our coordinate system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct meant as nothi +ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector by zero. But the direction of her axes and Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewhat during the transformation, knocked off the line that it spans. -As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation. +ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vectors in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. -But the important part here is that it stays on the line that it spans out without getting rotated off of it. -For a glimpse of why this might be a useful thing to think about, consider some three-dimensional rotation. +pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformation that moves our basis vectors, i-hat and j-hat, the things we think of when we say 1, 0 and 0, 1, to Jennifer's basis vectors, the thin +gs she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation. If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation. In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length of the vector would remain the same. This pattern shows up a lot in linear algebra. With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors. But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. -I won't cover the full details on methods for computing eigenvectors and eigenvalues here, but I'll try to give an overview of the computational ideas that are most important for a conceptual understanding. +we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. -What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda. -So finding the eigenvectors and their eigenvalues of a matrix A comes down to finding the values of v and lambda that make this expression true. -It's a little awkward to work with at first, because that left-hand side represents matrix-vector multiplication, but the right-hand side here is scalar-vector multiplication. +e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we multiply this inverse change of basis +matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose columns represent Jennif +er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's important that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda. The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else. -The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal. +the columns of our matrix. But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v. So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. -Now, this will always be true if v itself is the zero vector, but that's boring. +s land, and it needs to describe those landing spots in her language. Here's a common way to think of how this is done. What we want is a non-zero eigenvector. And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension. And that squishification corresponds to a zero determinant for the matrix. To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry. Now imagine tweaking lambda, turning a knob to change its value. As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes. -The goal here is to find a value of lambda that will make this determinant zero, meaning the tweaked transformation squishes space into a lower dimension. +ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system. In this case, the sweet spot comes when lambda equals 1. Of course, if we had chosen some other matrix, the eigenvalue might not necessarily be 1. -The sweet spot might be hit at some other value of lambda. -So this is kind of a lot, but let's unravel what this is saying. -When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line. +A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform +ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective. +And the full matrix product represents that same transformation, but as someone else sees it. For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of this. See y That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place. diff --git a/2016/change-of-basis/french/sentence_translations.json b/2016/change-of-basis/french/sentence_translations.json index dc3045884..a9472cbe1 100644 --- a/2016/change-of-basis/french/sentence_translations.json +++ b/2016/change-of-basis/french/sentence_translations.json @@ -72,7 +72,7 @@ "end": 84.84 }, { - "input": "It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.", + "input": "st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is ti", "translatedText": "Il déplace le vecteur de base i-hat vers les coordonnées 3, 0 et j-hat vers 1, 2.", "from_community_srt": "î et ĵ, sont appelés les vecteurs de base de notre système de coordonnées standard.", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 91.04 }, { - "input": "So it's represented with a matrix whose columns are 3, 0, and 1, 2.", + "input": "ed up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actual", "translatedText": "Il est donc représenté par une matrice dont les colonnes sont 3, 0 et 1, 2.", "from_community_srt": "Ce dont j'aimerais parler ici est l'idée d'utiliser un ensemble différent de base vecteurs.", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 95.64 }, { - "input": "Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip.", + "input": "ly scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called th", "translatedText": "Concentrez-vous sur ce qu'il fait à un vecteur particulier et pensez à l'étendue de ce vecteur, à la ligne passant par son origine et sa pointe.", "from_community_srt": "Par exemple, disons que vous avez un ami, Jennifer qui utilise un ensemble différent de vecteurs de base que je vais appeler b1 et b2", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 104.16 }, { - "input": "Most vectors are going to get knocked off their span during the transformation.", + "input": "e basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a", "translatedText": "La plupart des vecteurs vont perdre leur portée pendant la transformation.", "from_community_srt": "Son premier vecteur de base b1 pointe vers le haut,", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 115.32 }, { - "input": "But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar.", + "input": "let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up.", "translatedText": "Mais certains vecteurs spéciaux restent sur leur propre étendue, ce qui signifie que l'effet de la matrice sur un tel vecteur est simplement de l'étirer ou de l'écraser, comme un scalaire.", "from_community_srt": "2] en utilisant nos vecteurs de base î et ĵ. Jennifer décrirait plutôt ce vecteur avec les coordonnées [5/3,", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 127.04 }, { - "input": "For this specific example, the basis vector i-hat is one such special vector.", + "input": "Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vecto", "translatedText": "Pour cet exemple spécifique, le vecteur de base i-hat est l’un de ces vecteurs spéciaux.", "from_community_srt": "1/3] ce qui signifie que la façon particulière pour arriver à ce vecteur en utilisant ses deux vecteurs de base", "n_reviews": 0, @@ -152,14 +152,14 @@ "end": 164.04 }, { - "input": "It ends up getting stretched by a factor of 2.", + "input": "scale b1 by 5 thirds, scale b2 by 1 third, then add them both togethe", "translatedText": "Il finit par être étiré d'un facteur 2.", "n_reviews": 0, "start": 164.66, "end": 167.14 }, { - "input": "And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2.", + "input": "r. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she", "translatedText": "Et encore une fois, la linéarité impliquera que tout autre vecteur sur la diagonale parcourue par ce type sera simplement étiré d'un facteur 2.", "from_community_srt": "Pour être un peu plus précis sur la configuration ici son premier vecteur de base b1 est quelque chose que nous décririons avec le coordonnées [2, 1] et son deuxième vecteur de base b2 est quelque chose que nous décririons comme [-1,", "n_reviews": 0, @@ -167,7 +167,7 @@ "end": 178.22 }, { - "input": "And for this transformation, those are all the vectors with this special property of staying on their span.", + "input": "thinks of her first coordinate as scali For this specific example, the basis vector i-hat is one such special vector. The span of", "translatedText": "Et pour cette transformation, ce sont tous les vecteurs qui ont cette propriété particulière de rester sur leur portée.", "from_community_srt": "1]. Mais il est important de réaliser que de son point de vue, dans son système ces vecteurs ont des coordonnées [1,", "n_reviews": 0, @@ -175,7 +175,7 @@ "end": 185.18 }, { - "input": "Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2.", + "input": "i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis. What's more, because of the way linear transformations work,", "translatedText": "Ceux sur l'axe des x sont étirés d'un facteur 3, et ceux sur cette ligne diagonale sont étirés d'un facteur 2.", "from_community_srt": "0] et [0, 1] Ils sont ce qui définit la signification des coordonnées [1, 0] et [0, 1] dans son monde.", "n_reviews": 0, @@ -191,7 +191,7 @@ "end": 198.08 }, { - "input": "As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation.", + "input": "n. A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc t, a way to visualize our coordinate system, and so it depends on our choice of basis", "translatedText": "Comme vous l'avez peut-être deviné maintenant, ces vecteurs spéciaux sont appelés vecteurs propres de la transformation, et chaque vecteur propre est associé à ce qu'on appelle une valeur propre, qui est simplement le facteur par lequel il est étiré ou écrasé pendant la transformation.", "from_community_srt": "nous parlons des langues différentes Nous voyons tous les mêmes vecteurs dans l'espace mais Jennifer utilise des mots et des nombres différents pour les décrire. Laissez-moi vous dire un mot sur la façon dont je représente les choses ici quand j'anime l'espace 2D J'utilise généralement cette grille carrée Mais cette grille est juste une construction un moyen de visualiser notre système de coordonnées et cela dépend de notre choix de base.", "n_reviews": 0, @@ -207,7 +207,7 @@ "end": 225.94 }, { - "input": "In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half.", + "input": "nt as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It", "translatedText": "Dans un autre exemple, vous pourriez avoir un vecteur propre avec une valeur propre négative de 1 moitié, ce qui signifie que le vecteur est inversé et écrasé d'un facteur de 1 moitié.", "from_community_srt": "qui n'est rien de plus qu'un outil visuel pour aider à suivre la signification de ses coordonnées. Son origine, cependant,", "n_reviews": 0, @@ -231,7 +231,7 @@ "end": 249.8 }, { - "input": "If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation.", + "input": "of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewhat during the transformation, k", "translatedText": "Si vous pouvez trouver un vecteur propre pour cette rotation, un vecteur qui reste sur sa propre étendue, ce que vous avez trouvé est l'axe de rotation.", "from_community_srt": "Donc, après tout cela mis en place une question assez naturelle à poser est Comment nous traduisons entre les systèmes de coordonnées ? Si, par exemple,", "n_reviews": 0, @@ -239,7 +239,7 @@ "end": 260.5 }, { - "input": "And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation.", + "input": "nocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", "translatedText": "Et il est beaucoup plus facile de penser à une rotation 3D en termes d'un axe de rotation et d'un angle de rotation, plutôt que de penser à la matrice 3x3 complète associée à cette transformation.", "from_community_srt": "Jennifer décrit un vecteur avec des coordonnées [-1, 2] que serait-ce dans notre système de coordonnées ? Comment traduiriez-vous de sa langue à la nôtre ? Eh bien,", "n_reviews": 0, @@ -255,7 +255,7 @@ "end": 285.86 }, { - "input": "This pattern shows up a lot in linear algebra.", + "input": "In fact, once you understand matrix vector multiplication as applying", "translatedText": "Ce modèle apparaît souvent en algèbre linéaire.", "from_community_srt": "1] et b2 a les coordonnées [-1,", "n_reviews": 0, @@ -263,7 +263,7 @@ "end": 290.02 }, { - "input": "With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors.", + "input": "a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvecto", "translatedText": "Avec toute transformation linéaire décrite par une matrice, vous pouvez comprendre ce qu'elle fait en lisant les colonnes de cette matrice comme points d'atterrissage pour les vecteurs de base.", "from_community_srt": "1] Nous pouvons donc calculer -1*b1 + 2*b2 comme ils sont représentés dans notre système de coordonnées", "n_reviews": 0, @@ -271,7 +271,7 @@ "end": 299.4 }, { - "input": "But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", + "input": "r with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformati", "translatedText": "Mais souvent, une meilleure façon d'aller au cœur de ce que fait réellement la transformation linéaire, moins dépendante de votre système de coordonnées particulier, est de trouver les vecteurs propres et les valeurs propres.", "from_community_srt": "Et en effectuant vous obtenez un vecteur avec des coordonnées [-4, 1] Donc, voilà comment nous décririons le vecteur qu'elle pense comme [-1,", "n_reviews": 0, @@ -287,7 +287,7 @@ "end": 326.02 }, { - "input": "Symbolically, here's what the idea of an eigenvector looks like.", + "input": "she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we thi", "translatedText": "Symboliquement, voici à quoi ressemble l'idée d'un vecteur propre.", "from_community_srt": "une fois que vous comprenez la multiplication de matrices vectorielles comme une application de certaine transformation linéaire", "n_reviews": 0, @@ -303,7 +303,7 @@ "end": 339.74 }, { - "input": "What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda.", + "input": "or for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotat", "translatedText": "Ce que dit cette expression, c'est que le produit matrice-vecteur, A fois v, donne le même résultat qu'une simple mise à l'échelle du vecteur propre v par une certaine valeur lambda.", "from_community_srt": "Une matrice dont les colonnes représentent les vecteurs de base de Jennifer peut être considéré comme une transformation qui déplace nos vecteurs de base, î et ĵ les choses auquelles nous pensons quand nous disons [1,0] et [0,", "n_reviews": 0, @@ -327,7 +327,7 @@ "end": 370.54 }, { - "input": "So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda.", + "input": "never stretch or squish anything, so the length of the vector would remain the same. This pattern shows up a lot in linear algebra. With any linear transformation described by a matrix, you could understand what it's doing by reading of", "translatedText": "Commençons donc par réécrire ce membre de droite comme une sorte de multiplication matrice-vecteur, en utilisant une matrice qui a pour effet de mettre à l'échelle n'importe quel vecteur par un facteur lambda.", "from_community_srt": "Avant la transformation linéaire nous pensons à ce vecteur comme une certaine combinaison linéaire de notre base vecteurs -1*î + 2*ĵ.", "n_reviews": 0, @@ -335,7 +335,7 @@ "end": 380.62 }, { - "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else.", + "input": "f the columns of this matrix as the landing spots for basis vectors. But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", "translatedText": "Les colonnes d'une telle matrice représenteront ce qui arrive à chaque vecteur de base, et chaque vecteur de base est simplement multiplié par lambda, donc cette matrice aura le nombre lambda sur la diagonale, avec des zéros partout ailleurs.", "from_community_srt": "Et la caractéristique clé d'une transformation linéaire est que le vecteur résultant sera cette même combinaison linéaire mais des les nouveaux vecteurs de base -1 fois l'endroit où î atterrit + 2 fois l'endroit où atterrit ĵ. Alors,", "n_reviews": 0, @@ -343,7 +343,7 @@ "end": 394.32 }, { - "input": "The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal.", + "input": "we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector w", "translatedText": "La façon courante d'écrire ce type est de prendre en compte ce lambda et de l'écrire sous la forme lambda fois i, où i est la matrice d'identité avec des 1 sur la diagonale.", "from_community_srt": "ce que fait cette matrice est transformer notre fausse idée de ce que Jennifer veux dire dans le vrai vecteur auquel elle réfère.", "n_reviews": 0, @@ -359,7 +359,7 @@ "end": 411.86 }, { - "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector.", + "input": "that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, with v as the eigenvector", "translatedText": "Nous avons donc maintenant une nouvelle matrice, A moins lambda fois l'identité, et nous recherchons un vecteur v tel que cette nouvelle matrice multipliée par v donne le vecteur zéro.", "from_community_srt": "Mais numériquement, ça traduit un vecteur décrit dans sa langue à notre langue. Ce qui m'a finalement fait comprendre était en prenant que la transformation prend notre fausse idée de ce que veut dire Jennifer,", "n_reviews": 0, @@ -382,7 +382,7 @@ "end": 433.64 }, { - "input": "And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", + "input": "and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Je", "translatedText": "Et si vous regardez les chapitres 5 et 6, vous saurez que la seule façon pour le produit d'une matrice avec un vecteur non nul de devenir nul est si la transformation associée à cette matrice écrase l'espace dans une dimension inférieure.", "from_community_srt": "Qu'en est-il de l'inverse ? Dans l'exemple que j'ai utilisé plus tôt cette vidéo quand j'ai le vecteur avec les coordonnées [3,2] dans notre système Comment ai-je calculé qu'il aurait des coordonnées [5/3,", "n_reviews": 0, @@ -414,7 +414,7 @@ "end": 470.28 }, { - "input": "As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes.", + "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose c", "translatedText": "À mesure que cette valeur de lambda change, la matrice elle-même change, et donc le déterminant de la matrice change.", "from_community_srt": "En pratique, surtout quand vous travaillez en plus de deux dimensions vous utiliseriez un ordinateur pour calculer la matrice qui représente cette inverse.", "n_reviews": 0, @@ -454,14 +454,14 @@ "end": 498.6 }, { - "input": "So this is kind of a lot, but let's unravel what this is saying.", + "input": "And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's importa", "translatedText": "C'est donc beaucoup, mais voyons ce que cela veut dire.", "n_reviews": 0, "start": 500.08, "end": 502.96 }, { - "input": "When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line.", + "input": "nt that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication So let's start by rewriting", "translatedText": "Lorsque lambda est égal à 1, la matrice A moins lambda multipliée par l'identité écrase l'espace sur une ligne.", "from_community_srt": "2] ce qui fait [5/3, 1/3] Alors,", "n_reviews": 0, @@ -469,7 +469,7 @@ "end": 509.56 }, { - "input": "That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector.", + "input": "that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a fact", "translatedText": "Cela signifie qu'il existe un vecteur v non nul tel que A moins lambda fois l'identité fois v est égal au vecteur zéro.", "from_community_srt": "en un mot c'est comme ça qu'on traduit la description de vecteurs individuels entre les systèmes de coordonnées dans un sens et dans l'autre. La matrice dont les colonnes représentent les vecteurs de base de Jennifer mais écrits dans nos coordonnées", "n_reviews": 0, @@ -477,7 +477,7 @@ "end": 518.56 }, { - "input": "And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A.", + "input": "or of lambda. The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else. the columns of our matrix.", "translatedText": "Et rappelez-vous, la raison pour laquelle nous nous soucions de cela est que cela signifie que A fois v est égal à lambda fois v, ce qui peut être lu comme disant que le vecteur v est un vecteur propre de A, restant sur sa propre étendue pendant la transformation A.", "from_community_srt": "traduit les vecteurs de sa langue en notre langue. Et la matrice inverse fait le contraire. Mais les vecteurs ne sont pas les seules choses que nous décrivons en utilisant des coordonnées.", "n_reviews": 0, @@ -485,7 +485,7 @@ "end": 537.28 }, { - "input": "In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place.", + "input": "But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following", "translatedText": "Dans cet exemple, la valeur propre correspondante est 1, donc v resterait simplement fixe en place.", "from_community_srt": "Pour la prochaine partie il est important que vous soyez tous à l'aise pour représenter des transformations avec des matrices et que vous savez comment la multiplication matricielle correspond à composer des transformations successives.", "n_reviews": 0, @@ -500,7 +500,7 @@ "end": 549.5 }, { - "input": "This is the kind of thing I mentioned in the introduction.", + "input": "-hat in the first pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v.", "translatedText": "C'est le genre de chose que j'ai mentionné dans l'introduction.", "from_community_srt": "Mettez vraiment en pause et jetez un œil aux chapitres 3 et 4 si vous n'êtes pas à l'aise avec un de ces choses.", "n_reviews": 0, @@ -508,7 +508,7 @@ "end": 555.64 }, { - "input": "If you didn't have a solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to describe those landing spots in her language. Here's a common way to think", "translatedText": "Si vous n'aviez pas une solide compréhension des déterminants et de la raison pour laquelle ils se rapportent à des systèmes d'équations linéaires ayant des solutions non nulles, une expression comme celle-ci semblerait complètement inattendue.", "from_community_srt": "Considérez une transformation linéaire comme une rotation de 90 ° dans le sens antihoraire. Quand vous et moi représentons cela avec a matrice nous suivons où les vecteurs de base î et ĵ se retrouvent.", "n_reviews": 0, @@ -548,7 +548,7 @@ "end": 608.84 }, { - "input": "To figure out what the eigenvectors are that actually have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero.", + "input": "To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry. Now imagine tweaking lambda, turning a knob to change its value. As that value of lambda changes, the matrix itself change", "translatedText": "Pour déterminer quels sont les vecteurs propres qui ont réellement l'une de ces valeurs propres, disons que lambda est égal à 2, branchez cette valeur de lambda à la matrice, puis déterminez pour quels vecteurs cette matrice modifiée en diagonale envoie à zéro.", "from_community_srt": "Ces colonnes représentent où nos vecteurs de base î et ĵ vont. Mais la matrice que veut Jennifer devrait représenter où ses vecteurs de base atterrissent et il doit décrire ces points dans sa langue. Voici une façon commune de penser à comment ça marche.", "n_reviews": 0, @@ -627,7 +627,7 @@ "end": 681.94 }, { - "input": "The only roots of that polynomial are the imaginary numbers, i and negative i.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the oute", "translatedText": "Les seules racines de ce polynôme sont les nombres imaginaires i et i négatif.", "from_community_srt": "Elle prend en entrée un vecteur dans la langue de Jennifer et ressort la version transformée de ce vecteur dans sa langue", "n_reviews": 0, @@ -651,7 +651,7 @@ "end": 699.82 }, { - "input": "This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1.", + "input": "meone else sees it. For those of you wondering why we care about alternate coordinate systems, the next vi", "translatedText": "Cela fixe i-hat en place et déplace j-hat 1, de sorte que sa matrice a les colonnes 1, 0 et 1, 1.", "from_community_srt": "5/3] et [-2/3, -1/3] Donc,", "n_reviews": 0, @@ -683,7 +683,7 @@ "end": 726.54 }, { - "input": "And the only root of this expression is lambda equals 1.", + "input": "he identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals la", "translatedText": "Et la seule racine de cette expression est lambda égal à 1.", "from_community_srt": "Cette matrice du milieu représente une certaine transformation, comme vous la voyez et les deux matrices externes représentent l'empathie,", "n_reviews": 0, @@ -691,7 +691,7 @@ "end": 732.86 }, { - "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.", + "input": "mbda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corresponding eigenvalue is", "translatedText": "Cela correspond à ce que nous voyons géométriquement, à savoir que tous les vecteurs propres ont une valeur propre 1.", "from_community_srt": "le changement de perspective et le produit matriciel complet représente cette même transformation", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 796.38 }, { - "input": "For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2.", + "input": "f equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Par exemple, peut-être que i-hat est mis à l'échelle de moins 1 et j-hat est mis à l'échelle de 2.", "n_reviews": 0, "start": 797.12, @@ -765,7 +765,7 @@ "end": 825.42 }, { - "input": "And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.", + "input": "nd compute the determinant. Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda. Since lambda can only be an eigenvalue i", "translatedText": "Et la façon d'interpréter cela est que tous les vecteurs de base sont des vecteurs propres, les entrées diagonales de cette matrice étant leurs valeurs propres.", "n_reviews": 0, "start": 825.84, @@ -779,14 +779,14 @@ "end": 841.06 }, { - "input": "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", + "input": "u can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3. To figure out what the eigenvectors are that actu", "translatedText": "Le plus important est qu'il est plus facile de calculer ce qui se passera si vous multipliez cette matrice par elle-même plusieurs fois.", "n_reviews": 0, "start": 841.78, "end": 848.34 }, { - "input": "Since all one of these matrices does is scale each basis vector by some eigenvalue, applying that matrix many times, say 100 times, is just going to correspond to scaling each basis vector by the 100th power of the corresponding eigenvalue.", + "input": "ally have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero. If you computed this the way you would any other linear system, you'd see that the solutions are all the vectors on the diagonal line spanned by negative 1, 1.", "translatedText": "Puisque toutes ces matrices ne font que mettre à l'échelle chaque vecteur de base par une valeur propre, appliquer cette matrice plusieurs fois, disons 100 fois, va simplement correspondre à la mise à l'échelle de chaque vecteur de base par la puissance 100 de la valeur propre correspondante.", "n_reviews": 0, "start": 849.42, @@ -800,7 +800,7 @@ "end": 869.68 }, { - "input": "Really, try it for a moment.", + "input": "x, 3, 0, 1, 2, has the effect of stretching all those vectors by a factor of 2.", "translatedText": "Vraiment, essayez-le un instant.", "n_reviews": 0, "start": 869.68, @@ -828,14 +828,14 @@ "end": 896.54 }, { - "input": "I talked about change of basis last video, but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", + "input": "is, notice what happens. Its matrix has columns 0, 1 and negative 1, 0. Subtract off lambda from the diagonal elements and look for when the determinant is zero. In this case, you get the polynomial lambda squared plus 1. The only roots of that polynomia", "translatedText": "J'ai parlé du changement de base dans la dernière vidéo, mais je vais faire ici un rappel très rapide de la façon d'exprimer une transformation actuellement écrite dans notre système de coordonnées dans un système différent.", "n_reviews": 0, "start": 897.14, "end": 907.04 }, { - "input": "Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors, then make those coordinates the columns of a matrix, known as the change of basis matrix.", + "input": "l are the imaginary numbers, i and negative i. The fact that there are no real number solutions indicates that there are no eigenvectors. Another pretty interesting example worth holding in the back of your mind is a shear. This fixes i-hat in place and moves j-hat 1 over, so its mat", "translatedText": "Prenez les coordonnées des vecteurs que vous souhaitez utiliser comme nouvelle base, ce qui signifie dans ce cas nos deux vecteurs propres, puis faites de ces coordonnées les colonnes d'une matrice, connue sous le nom de matrice de changement de base.", "n_reviews": 0, "start": 908.44, @@ -856,7 +856,7 @@ "end": 946.68 }, { - "input": "This is because it represents working in a coordinate system where what happens to the basis vectors is that they get scaled during the transformation.", + "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1. Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a lin", "translatedText": "En effet, cela représente un travail dans un système de coordonnées où les vecteurs de base sont mis à l'échelle lors de la transformation.", "n_reviews": 0, "start": 946.86, @@ -870,28 +870,28 @@ "end": 961.56 }, { - "input": "So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis, compute the 100th power in that system, then convert back to our standard system.", + "input": "A simple example is a matrix that scales everything by 2. The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue. Now is another good time to pause and ponder some of this before I move on to the last topic.", "translatedText": "Ainsi, si, par exemple, vous deviez calculer la 100e puissance de cette matrice, il serait beaucoup plus facile de passer à une base propre, de calculer la 100e puissance dans ce système, puis de revenir à notre système standard.", "n_reviews": 0, "start": 962.34, "end": 975.68 }, { - "input": "You can't do this with all transformations.", + "input": "I want to finish off here with the idea of an eigenbasis, which relies heavily on ideas from the last video. Take a look at what h", "translatedText": "Vous ne pouvez pas faire cela avec toutes les transformations.", "n_reviews": 0, "start": 976.62, "end": 978.32 }, { - "input": "A shear, for example, doesn't have enough eigenvectors to span the full space.", + "input": "appens if our basis vectors just so happen to be eigenvectors. For example, maybe i-hat is scale", "translatedText": "Une cisaille, par exemple, n'a pas suffisamment de vecteurs propres pour couvrir tout l'espace.", "n_reviews": 0, "start": 978.32, "end": 982.98 }, { - "input": "But if you can find an eigenbasis, it makes matrix operations really lovely.", + "input": "d by negative 1 and j-hat is scaled by 2. Writing their new coordinates as the columns of a matrix, notice t", "translatedText": "Mais si vous pouvez trouver une base propre, cela rend les opérations matricielles vraiment agréables.", "n_reviews": 0, "start": 983.46, diff --git a/2016/change-of-basis/german/sentence_translations.json b/2016/change-of-basis/german/sentence_translations.json index 6d74a7502..dc1f2fd4c 100644 --- a/2016/change-of-basis/german/sentence_translations.json +++ b/2016/change-of-basis/german/sentence_translations.json @@ -81,7 +81,7 @@ "end": 84.84 }, { - "input": "It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.", + "input": "st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is ti", "translatedText": "Er verschiebt den Basisvektor i-hat auf die Koordinaten 3, 0 und j-hat auf 1, 2.", "model": "DeepL", "from_community_srt": "und die zwei speziellen Vektoren i-Hut und j-Hut sind die sogenannten Basisvektoren unseres Standard Koordinatensystems", @@ -90,7 +90,7 @@ "end": 91.04 }, { - "input": "So it's represented with a matrix whose columns are 3, 0, and 1, 2.", + "input": "ed up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actual", "translatedText": "Sie wird also mit einer Matrix dargestellt, deren Spalten 3, 0 und 1, 2 sind.", "model": "DeepL", "from_community_srt": "Worüber ich hierbei gerne sprechen möchte, ist die Idee unterschiedliche Systeme von Basisvektoren zu nutzen.", @@ -99,7 +99,7 @@ "end": 95.64 }, { - "input": "Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip.", + "input": "ly scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called th", "translatedText": "Konzentriere dich darauf, was es mit einem bestimmten Vektor macht, und denke über die Spannweite dieses Vektors nach, die Linie, die durch seinen Ursprung und seine Spitze verläuft.", "model": "DeepL", "from_community_srt": "Zum Beispiel, lass uns sagen du hast eine Freundin, Jennifer, diese nutzt ein anderes System von Basisvektoren, die ich b1 und b2 nennen werde.", @@ -108,7 +108,7 @@ "end": 104.16 }, { - "input": "Most vectors are going to get knocked off their span during the transformation.", + "input": "e basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a", "translatedText": "Die meisten Vektoren werden bei der Umwandlung aus ihrer Spanne gerissen.", "model": "DeepL", "from_community_srt": "Ihr erster Basisvektor b1 zeigt ein bisschen nach oben rechts und ihr zweiter Basisvektor b2 zeigt nach oben links.", @@ -126,7 +126,7 @@ "end": 115.32 }, { - "input": "But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar.", + "input": "let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up.", "translatedText": "Einige spezielle Vektoren bleiben jedoch in ihrer eigenen Spannweite, d.h. die Wirkung der Matrix auf einen solchen Vektor besteht lediglich darin, ihn zu strecken oder zu stauchen, wie ein Skalar.", "model": "DeepL", "from_community_srt": "den du und ich mit den Koordinaten [3 , 2] beschreiben würden, wenn wir unsere Basisvektoren i-Hut und j-Hut nutzen. Jennifer würde diesen Vektor mit den Koordinaten [5/3, 1/3] beschreiben. Das bedeutet,", @@ -135,7 +135,7 @@ "end": 127.04 }, { - "input": "For this specific example, the basis vector i-hat is one such special vector.", + "input": "Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vecto", "translatedText": "Für dieses spezielle Beispiel ist der Basisvektor i-hat ein solcher spezieller Vektor.", "model": "DeepL", "from_community_srt": "dass der Weg, um zu diesem Vektor zu kommen, mit ihren zwei Basisvektoren ist,", @@ -171,7 +171,7 @@ "end": 164.04 }, { - "input": "It ends up getting stretched by a factor of 2.", + "input": "scale b1 by 5 thirds, scale b2 by 1 third, then add them both togethe", "translatedText": "Am Ende wird er um den Faktor 2 gestreckt.", "model": "DeepL", "n_reviews": 0, @@ -179,7 +179,7 @@ "end": 167.14 }, { - "input": "And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2.", + "input": "r. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she", "translatedText": "Und wieder bedeutet die Linearität, dass jeder andere Vektor auf der Diagonalen, die dieser Typ aufspannt, einfach um den Faktor 2 gestreckt wird.", "model": "DeepL", "from_community_srt": "Um ein bisschen präziser zu sein Ihren erster Basisvektor b1 würden wir mit den Koordinaten [2, 1] beschreiben. Und ihren zweiten Basisvektor b2 würden wir beschreiben als [-1,", @@ -188,7 +188,7 @@ "end": 178.22 }, { - "input": "And for this transformation, those are all the vectors with this special property of staying on their span.", + "input": "thinks of her first coordinate as scali For this specific example, the basis vector i-hat is one such special vector. The span of", "translatedText": "Und für diese Transformation sind das alle Vektoren, die die besondere Eigenschaft haben, auf ihrer Spannweite zu bleiben.", "model": "DeepL", "from_community_srt": "1]. Aber es ist wichtig zu verstehen, dass aus Sicht ihres Systems, diese Vektoren die Koordinaten [1,", @@ -197,7 +197,7 @@ "end": 185.18 }, { - "input": "Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2.", + "input": "i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis. What's more, because of the way linear transformations work,", "translatedText": "Die auf der X-Achse werden um den Faktor 3 gestreckt und die auf dieser diagonalen Linie um den Faktor 2 gestreckt.", "model": "DeepL", "from_community_srt": "0] und [0, 1] haben. SIe beschreiben in ihrer Welt die Bedeutung von [1, 0] und [0,", @@ -215,7 +215,7 @@ "end": 198.08 }, { - "input": "As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation.", + "input": "n. A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc t, a way to visualize our coordinate system, and so it depends on our choice of basis", "translatedText": "Wie du dir vielleicht schon gedacht hast, heißen diese speziellen Vektoren die Eigenvektoren der Transformation, und jedem Eigenvektor ist ein sogenannter Eigenwert zugeordnet, also der Faktor, um den er während der Transformation gestreckt oder gestaucht wird.", "model": "DeepL", "from_community_srt": "aber Jennifer nutzt andere Wörter und Zahlen um diese zu beschreiben Lass mich kurz etwas darüber sagen, wie ich die Dinge hier darstelle, wenn ich den 2D Raum animiere. Ich benutze normalerweise dieses quadratische Raster, aber dieses Raster ist nur ein Konstrukt, ein Weg, um unser Koordinatensystem darzustellen, also ist es abhängig von der Wahl unserer Basisvektoren.", @@ -233,7 +233,7 @@ "end": 225.94 }, { - "input": "In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half.", + "input": "nt as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It", "translatedText": "In einem anderen Beispiel könntest du einen Eigenvektor mit dem Eigenwert negativ 1 halb haben, was bedeutet, dass der Vektor um den Faktor 1 halb gespiegelt und quadriert wird.", "model": "DeepL", "from_community_srt": "um der Bedeutung ihrer Koordinaten zu folgen.", @@ -260,7 +260,7 @@ "end": 249.8 }, { - "input": "If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation.", + "input": "of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewhat during the transformation, k", "translatedText": "Wenn du einen Eigenvektor für diese Drehung finden kannst, also einen Vektor, der auf seiner eigenen Spannweite bleibt, hast du die Drehachse gefunden.", "model": "DeepL", "from_community_srt": "Jetzt wo all dies geklärt ist, ist eine ziemlich natürliche Frage, wie wir zwischen Koordinatensystemen übersetzen. Wenn zum Beispiel Jennifer einen Vektor mit den Koordinaten [-1,", @@ -269,7 +269,7 @@ "end": 260.5 }, { - "input": "And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation.", + "input": "nocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", "translatedText": "Und es ist viel einfacher, über eine 3D-Drehung in Form einer Drehachse und eines Winkels nachzudenken, um den sie sich dreht, als über die gesamte 3x3-Matrix, die mit dieser Transformation verbunden ist.", "model": "DeepL", "from_community_srt": "2] beschreibt, was wäre dieser Vektor in unserem Koordinatensystem? Wie übersetzt du von ihrer Sprache in unsere? Nun, was unserer Koordinaten sagen ist,", @@ -287,7 +287,7 @@ "end": 285.86 }, { - "input": "This pattern shows up a lot in linear algebra.", + "input": "In fact, once you understand matrix vector multiplication as applying", "translatedText": "Dieses Muster taucht häufig in der linearen Algebra auf.", "model": "DeepL", "from_community_srt": "1] und b2 die Koordinaten [-1, 1].", @@ -296,7 +296,7 @@ "end": 290.02 }, { - "input": "With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors.", + "input": "a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvecto", "translatedText": "Bei jeder linearen Transformation, die durch eine Matrix beschrieben wird, kannst du verstehen, was sie tut, wenn du die Spalten dieser Matrix als Landeplätze für Basisvektoren abliest.", "model": "DeepL", "from_community_srt": "Also können wir wirklich -1 b1 + 2 b2 ausführen so wie sie in unserem Koordinatensystem herauskommen.", @@ -305,7 +305,7 @@ "end": 299.4 }, { - "input": "But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", + "input": "r with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformati", "translatedText": "Oft ist es aber besser, die Eigenvektoren und Eigenwerte zu ermitteln, um herauszufinden, was die lineare Transformation tatsächlich bewirkt, und weniger von deinem speziellen Koordinatensystem abhängig zu sein.", "model": "DeepL", "from_community_srt": "Wenn du dies durcharbeitest, kriegst du einen Vektor mit den Koordinaten [-4, 1] . Das ist also wie wir den Vektor, den sie sich als [-1, 2] vorstellt,", @@ -323,7 +323,7 @@ "end": 326.02 }, { - "input": "Symbolically, here's what the idea of an eigenvector looks like.", + "input": "she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we thi", "translatedText": "Symbolisch sieht die Idee eines Eigenvektors folgendermaßen aus.", "model": "DeepL", "from_community_srt": "Wenn du also Matrix-Vektor Multiplikation verstehst, als die Anwendung einer bestimmten Linear-Transformation", @@ -341,7 +341,7 @@ "end": 339.74 }, { - "input": "What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda.", + "input": "or for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotat", "translatedText": "Dieser Ausdruck besagt, dass das Matrix-Vektor-Produkt, A mal v, dasselbe Ergebnis liefert wie die Skalierung des Eigenvektors v mit einem Wert lambda.", "model": "DeepL", "from_community_srt": "Eine Matrix deren Spalten Jennifers Basisvektoren beschreiben kann man sich vorstellen als Transformation die unsere Basisvektoren i-Hut und j-Hut, die Dinger, die wir uns vorstellen,", @@ -368,7 +368,7 @@ "end": 370.54 }, { - "input": "So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda.", + "input": "never stretch or squish anything, so the length of the vector would remain the same. This pattern shows up a lot in linear algebra. With any linear transformation described by a matrix, you could understand what it's doing by reading of", "translatedText": "Beginnen wir also damit, die rechte Seite als eine Art Matrix-Vektor-Multiplikation umzuschreiben, indem wir eine Matrix verwenden, die jeden Vektor um einen Faktor von Lambda skaliert.", "model": "DeepL", "from_community_srt": "Vor der Linear-Transformation stellen wir uns diesen Vektor vor, als eine bestimmte Linearkombination unserer Basisvektoren -1 mal i-Hut + 2 mal j-Hut. Und das Schlüsselkonzept einer Linear-Transformation, ist,", @@ -377,7 +377,7 @@ "end": 380.62 }, { - "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else.", + "input": "f the columns of this matrix as the landing spots for basis vectors. But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", "translatedText": "Die Spalten einer solchen Matrix stellen dar, was mit den einzelnen Basisvektoren passiert. Jeder Basisvektor wird einfach mit Lambda multipliziert, sodass diese Matrix auf der Diagonalen die Zahl Lambda und überall sonst Nullen hat.", "model": "DeepL", "from_community_srt": "dass der resultierende Vektor die selbe Linearkombination sein wird, aber durch die neuen Basisvektoren -1 mal der Ort, an dem i-Hut landet + 2 mal der Ort, an dem j-Hut landet.", @@ -386,7 +386,7 @@ "end": 394.32 }, { - "input": "The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal.", + "input": "we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector w", "translatedText": "Die übliche Art, diesen Typ zu schreiben, ist, das Lambda herauszufaktorisieren und es als Lambda mal i zu schreiben, wobei i die Identitätsmatrix mit 1en auf der Diagonale ist.", "model": "DeepL", "from_community_srt": "Was also diese Matrix tut, ist, sie transformiert unserer falsche Vorstellung davon, was Jennifer meint, in den tatsächlichen Vektor, auf den sie sich bezieht. Ich erinnere mich daran,", @@ -404,7 +404,7 @@ "end": 411.86 }, { - "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector.", + "input": "that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, with v as the eigenvector", "translatedText": "Wir haben jetzt also eine neue Matrix, A minus Lambda mal die Identität, und suchen nach einem Vektor v, bei dem diese neue Matrix mal v den Nullvektor ergibt.", "model": "DeepL", "from_community_srt": "Aber numerisch übersetzt sie einen Vektor aus ihrer Sprache in unsere Sprache. Wodurch es schließlich bei mir Klick gemacht hat, war sich vorzustellen, wie es unsere falsche Vorstellung darüber,", @@ -431,7 +431,7 @@ "end": 433.64 }, { - "input": "And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", + "input": "and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Je", "translatedText": "Und wenn du dir Kapitel 5 und 6 ansiehst, wirst du wissen, dass das Produkt einer Matrix mit einem Nicht-Null-Vektor nur dann Null werden kann, wenn die Transformation, die mit dieser Matrix verbunden ist, den Raum in eine niedrigere Dimension quetscht.", "model": "DeepL", "from_community_srt": "den sie tatsächlich meint. Wie ist es aber andersherum? Im Beispiel, dass ich zu Anfang des Videos benutzte, als ich den Vektor mit den Koordinaten [3, 2] in unserem System habe, wie habe ich berechnet, dass er die Koordinaten [5/3,", @@ -467,7 +467,7 @@ "end": 470.28 }, { - "input": "As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes.", + "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose c", "translatedText": "Wenn sich der Wert von Lambda ändert, ändert sich auch die Matrix selbst und damit auch die Determinante der Matrix.", "model": "DeepL", "from_community_srt": "In der Anwendung, speziell dann, wenn du in mehr als zwei DImensionen arbeitest, würdest du einen Computer nutzen, um die inverse Matrix zu bestimmen.", @@ -512,7 +512,7 @@ "end": 498.6 }, { - "input": "So this is kind of a lot, but let's unravel what this is saying.", + "input": "And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's importa", "translatedText": "Das ist ganz schön viel, aber lass uns enträtseln, was das bedeutet.", "model": "DeepL", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 502.96 }, { - "input": "When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line.", + "input": "nt that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication So let's start by rewriting", "translatedText": "Wenn lambda gleich 1 ist, quetscht die Matrix A minus lambda mal die Identität den Raum auf eine Linie.", "model": "DeepL", "from_community_srt": "2] was herauskommt ist [5/3, 1/3].", @@ -529,7 +529,7 @@ "end": 509.56 }, { - "input": "That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector.", + "input": "that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a fact", "translatedText": "Das bedeutet, dass es einen Nicht-Null-Vektor v gibt, so dass A minus Lambda mal die Identität mal v gleich dem Null-Vektor ist.", "model": "DeepL", "from_community_srt": "Also ist dies in Kürze, wie man die Beschreibung individueller Vektoren vor und zurück zwischen Koordinatensystemen übersetzt. Die Matrix mit den Jennifers Basisvektoren als Spalten, aber beschrieben durch unsere Koordinaten,", @@ -538,7 +538,7 @@ "end": 518.56 }, { - "input": "And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A.", + "input": "or of lambda. The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else. the columns of our matrix.", "translatedText": "Der Grund, warum uns das interessiert, ist, dass es bedeutet, dass A mal v gleich Lambda mal v ist, was bedeutet, dass der Vektor v ein Eigenvektor von A ist und während der Transformation A auf seiner eigenen Spanne bleibt.", "model": "DeepL", "from_community_srt": "übersetzt Vektoren von ihrer Sprache in unsere Sprache. Und die Inverse tut das Gegenteil. Aber Vektoren sind nicht das EInzige, das wir mit Koordinaten beschreiben. Für den nächsten Teil, ist es wichtig that du vertraut damit bist,", @@ -547,7 +547,7 @@ "end": 537.28 }, { - "input": "In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place.", + "input": "But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following", "translatedText": "In diesem Beispiel ist der entsprechende Eigenwert 1, so dass v eigentlich an seinem Platz bleiben würde.", "model": "DeepL", "from_community_srt": "Transformationen mit Matrizen zu beschreiben, dass du weißt wie Matrixmultiplikation", @@ -565,7 +565,7 @@ "end": 549.5 }, { - "input": "This is the kind of thing I mentioned in the introduction.", + "input": "-hat in the first pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v.", "translatedText": "Das ist die Art von Dingen, die ich in der Einleitung erwähnt habe.", "model": "DeepL", "from_community_srt": "Pausiere definitiv und schaue dir Kapitel 3 und 4 an, wenn sich irgendwas davon nicht einfach anfühlt.", @@ -574,7 +574,7 @@ "end": 555.64 }, { - "input": "If you didn't have a solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to describe those landing spots in her language. Here's a common way to think", "translatedText": "Wenn du kein solides Verständnis von Determinanten hättest und wüsstest, warum sie sich auf lineare Gleichungssysteme mit Lösungen ungleich Null beziehen, würde dir ein Ausdruck wie dieser völlig fremd vorkommen.", "model": "DeepL", "from_community_srt": "Nimm eine Lineartransformation wie eine 90° Drehung gegen den Uhrzeigersinn. Wenn du und ich diese mit der Matrix beschreiben, schauen wir, wo unsere Basisvektoren i-Hut und j-Hut landen.", @@ -619,7 +619,7 @@ "end": 608.84 }, { - "input": "To figure out what the eigenvectors are that actually have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero.", + "input": "To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry. Now imagine tweaking lambda, turning a knob to change its value. As that value of lambda changes, the matrix itself change", "translatedText": "Um herauszufinden, welche Eigenvektoren tatsächlich einen dieser Eigenwerte haben, z. B. Lambda gleich 2, fügst du diesen Wert von Lambda in die Matrix ein und löst dann, welche Vektoren diese diagonal veränderte Matrix zu Null macht.", "model": "DeepL", "from_community_srt": "wo unsere Basisvektoren i-Hut und j-Hut landen. Aber die Matrix, die Jennifer will, sollte zeigen, wo ihre Basisvektoren landen und es muss die Landepunkte in ihrer Sprache beschreiben. Hier ist ein normaler Weg, sich vorzustellen, wie das passiert.", @@ -709,7 +709,7 @@ "end": 681.94 }, { - "input": "The only roots of that polynomial are the imaginary numbers, i and negative i.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the oute", "translatedText": "Die einzigen Wurzeln dieses Polynoms sind die imaginären Zahlen, i und negativ i.", "model": "DeepL", "from_community_srt": "Sie nimmt einen Vektor in ihrer Sprache, und gibt uns die transformierte Version des Vektors in ihrer Sprache", @@ -736,7 +736,7 @@ "end": 699.82 }, { - "input": "This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1.", + "input": "meone else sees it. For those of you wondering why we care about alternate coordinate systems, the next vi", "translatedText": "Dadurch wird i-hat an seinem Platz fixiert und j-hat um 1 verschoben, sodass seine Matrix die Spalten 1, 0 und 1, 1 hat.", "model": "DeepL", "from_community_srt": "wenn du dich durcharbeitest, die Matrix mit den Spalten [1/3, 5/3] and [-2/3, -1/3]. Wenn also Jennifer die Matrix multipliziert mit den Koordinaten eines Vektors in ihrem System", @@ -771,7 +771,7 @@ "end": 726.54 }, { - "input": "And the only root of this expression is lambda equals 1.", + "input": "he identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals la", "translatedText": "Und die einzige Wurzel dieses Ausdrucks ist lambda gleich 1.", "model": "DeepL", "from_community_srt": "Die mittlere Matrix repräsentiert in bestimmter Art und Weise eine Transformation,", @@ -780,7 +780,7 @@ "end": 732.86 }, { - "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.", + "input": "mbda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corresponding eigenvalue is", "translatedText": "Dies entspricht dem, was wir geometrisch sehen, nämlich dass alle Eigenvektoren den Eigenwert 1 haben.", "model": "DeepL", "from_community_srt": "wie du sie siehst und die äußeren beiden repräsentieren die Einfühlung, den Sichtwechsel und die die gesamte Matrix repräsentiert die selbe Transformation, aber wie sie jemand anderes sieht.", @@ -840,7 +840,7 @@ "end": 796.38 }, { - "input": "For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2.", + "input": "f equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Zum Beispiel könnte i-hat eine negative Skalierung von 1 und j-hat eine Skalierung von 2 haben.", "model": "DeepL", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 825.42 }, { - "input": "And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.", + "input": "nd compute the determinant. Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda. Since lambda can only be an eigenvalue i", "translatedText": "Das bedeutet, dass alle Basisvektoren Eigenvektoren sind und die Diagonaleinträge dieser Matrix ihre Eigenwerte sind.", "model": "DeepL", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 841.06 }, { - "input": "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", + "input": "u can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3. To figure out what the eigenvectors are that actu", "translatedText": "Eine davon ist, dass es einfacher zu berechnen ist, was passiert, wenn du diese Matrix ein paar Mal mit sich selbst multiplizierst.", "model": "DeepL", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 848.34 }, { - "input": "Since all one of these matrices does is scale each basis vector by some eigenvalue, applying that matrix many times, say 100 times, is just going to correspond to scaling each basis vector by the 100th power of the corresponding eigenvalue.", + "input": "ally have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero. If you computed this the way you would any other linear system, you'd see that the solutions are all the vectors on the diagonal line spanned by negative 1, 1.", "translatedText": "Da eine dieser Matrizen jeden Basisvektor nur um einen Eigenwert skaliert, entspricht die Anwendung dieser Matrix viele Male, z. B. 100 Mal, der Skalierung jedes Basisvektors um die 100-te Potenz des entsprechenden Eigenwertes.", "model": "DeepL", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 869.68 }, { - "input": "Really, try it for a moment.", + "input": "x, 3, 0, 1, 2, has the effect of stretching all those vectors by a factor of 2.", "translatedText": "Probiere es doch mal aus.", "model": "DeepL", "n_reviews": 0, @@ -936,7 +936,7 @@ "end": 896.54 }, { - "input": "I talked about change of basis last video, but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", + "input": "is, notice what happens. Its matrix has columns 0, 1 and negative 1, 0. Subtract off lambda from the diagonal elements and look for when the determinant is zero. In this case, you get the polynomial lambda squared plus 1. The only roots of that polynomia", "translatedText": "Im letzten Video habe ich über den Wechsel der Basis gesprochen, aber ich werde hier noch einmal kurz erklären, wie man eine Transformation, die in unserem Koordinatensystem geschrieben wurde, in einem anderen System ausdrückt.", "model": "DeepL", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 907.04 }, { - "input": "Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors, then make those coordinates the columns of a matrix, known as the change of basis matrix.", + "input": "l are the imaginary numbers, i and negative i. The fact that there are no real number solutions indicates that there are no eigenvectors. Another pretty interesting example worth holding in the back of your mind is a shear. This fixes i-hat in place and moves j-hat 1 over, so its mat", "translatedText": "Nimm die Koordinaten der Vektoren, die du als neue Basis verwenden willst, also in diesem Fall unsere beiden Eigenvektoren, und mache diese Koordinaten zu den Spalten einer Matrix, der sogenannten Basisänderungsmatrix.", "model": "DeepL", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 946.68 }, { - "input": "This is because it represents working in a coordinate system where what happens to the basis vectors is that they get scaled during the transformation.", + "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1. Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a lin", "translatedText": "Das liegt daran, dass es sich um ein Koordinatensystem handelt, in dem die Basisvektoren bei der Transformation skaliert werden.", "model": "DeepL", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 961.56 }, { - "input": "So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis, compute the 100th power in that system, then convert back to our standard system.", + "input": "A simple example is a matrix that scales everything by 2. The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue. Now is another good time to pause and ponder some of this before I move on to the last topic.", "translatedText": "Wenn du also zum Beispiel die 100. Potenz dieser Matrix berechnen müsstest, wäre es viel einfacher, zu einer Eigenbasis zu wechseln, die 100. Potenz in diesem System zu berechnen und dann zu unserem Standardsystem zurückzukehren.", "model": "DeepL", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 975.68 }, { - "input": "You can't do this with all transformations.", + "input": "I want to finish off here with the idea of an eigenbasis, which relies heavily on ideas from the last video. Take a look at what h", "translatedText": "Du kannst das nicht mit allen Transformationen machen.", "model": "DeepL", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 978.32 }, { - "input": "A shear, for example, doesn't have enough eigenvectors to span the full space.", + "input": "appens if our basis vectors just so happen to be eigenvectors. For example, maybe i-hat is scale", "translatedText": "Eine Scherung zum Beispiel hat nicht genug Eigenvektoren, um den gesamten Raum zu erfassen.", "model": "DeepL", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 982.98 }, { - "input": "But if you can find an eigenbasis, it makes matrix operations really lovely.", + "input": "d by negative 1 and j-hat is scaled by 2. Writing their new coordinates as the columns of a matrix, notice t", "translatedText": "Aber wenn du eine Eigenbasis finden kannst, sind Matrixoperationen wirklich schön.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/change-of-basis/hebrew/sentence_translations.json b/2016/change-of-basis/hebrew/sentence_translations.json index 71a5b961a..a4b315074 100644 --- a/2016/change-of-basis/hebrew/sentence_translations.json +++ b/2016/change-of-basis/hebrew/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates.", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we doing this and what d", "translatedText": "אם יש לי וקטור יושב כאן בחלל דו מימדי, יש לנו דרך סטנדרטית לתאר אותו עם קואורדינטות.", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 28.28 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up.", + "input": "oes this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin", "translatedText": "במקרה זה, לוקטור יש קואורדינטות 3, 2, כלומר מעבר מהזנבו לקצהו כרוך בהזזה של שלוש יחידות ימינה ושתי יחידות למעלה.", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.96 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up.", + "input": "ch that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of", "translatedText": "אתה חושב על הקואורדינטה הראשונה כעל קנה מידה של i-hat, הווקטור עם אורך 1 מצביע ימינה, בעוד שהקואורדינטה השנייה מקנה קנה מידה של j-hat, הווקטור עם אורך 1 מצביע ישר כלפי מעלה.", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 60.48 }, { - "input": "You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordinate system.", + "input": "you know how to think about matrices as linear transformations, but you also need to be comfortable with things like determinants, linear systems", "translatedText": "אתה יכול לחשוב על שני הווקטורים המיוחדים האלה כמכילים את כל ההנחות המרומזות של מערכת הקואורדינטות שלנו.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 81.38 }, { - "input": "Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system.", + "input": "some linear transformation in two dimensions, like the one shown here. st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice o", "translatedText": "כל דרך לתרגם בין וקטורים וקבוצות של מספרים נקראת מערכת קואורדינטות, ושני הוקטורים המיוחדים i-hat ו-j-hat נקראים וקטורי הבסיס של מערכת הקואורדינטות הסטנדרטית שלנו.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors.", + "input": "f i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors an", "translatedText": "מה שהייתי רוצה לדבר עליו כאן הוא הרעיון של שימוש בסט שונה של וקטורי בסיס.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.04 }, { - "input": "Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third.", + "input": "nnifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to th", "translatedText": "ג'ניפר למעשה תתאר את הווקטור הזה עם הקואורדינטות 5 שליש ושליש אחד.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 125.16 }, { - "input": "In a little bit, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third.", + "input": "showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describe this vector with the coordinates 5 thirds and", "translatedText": "עוד מעט, אני אראה לך איך יכולת להבין את שני המספרים האלה, 5 שליש ושליש אחד.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 144.12 }, { - "input": "What she gets will typically be completely different from the vector that you and I would think of as having those coordinates.", + "input": "gether. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to", "translatedText": "מה שהיא תקבל יהיה בדרך כלל שונה לחלוטין מהווקטור שאתה ואני היינו חושבים שיש לו את הקואורדינטות האלה.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 167.14 }, { - "input": "So in effect, we're speaking different languages.", + "input": "lumn of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis.", "translatedText": "אז למעשה, אנחנו מדברים בשפות שונות.", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 176.84 }, { - "input": "Let me say a quick word about how I'm representing things here.", + "input": "ector on the x-axis is also just stretched by a factor of 3, and hence remains on its own", "translatedText": "הרשו לי לומר מילה קצרה על איך אני מייצג דברים כאן.", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid.", + "input": "span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here.", "translatedText": "למרחב עצמו אין רשת מהותית.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 197.6 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean.", + "input": "system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct mea", "translatedText": "המקור שלה אמנם היה תואם את שלנו, מכיוון שכולם מסכימים על המשמעות של הקואורדינטות 0,0.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 214.9 }, { - "input": "But the direction of her axes and the spacing of her grid lines will be different, depending on her choice of basis vectors.", + "input": "ollow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector", "translatedText": "אבל כיוון הצירים שלה והמרווח של קווי הרשת שלה יהיו שונים, בהתאם לבחירתה של וקטורי הבסיס.", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 225.94 }, { - "input": "If for example Jennifer describes a vector with coordinates negative 1, 2, what would that be in our coordinate system?", + "input": "on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewh", "translatedText": "אם למשל ג'ניפר מתארת וקטור עם קואורדינטות שליליות 1, 2, מה זה יהיה במערכת הקואורדינטות שלנו?", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 238.98 }, { - "input": "How do you translate from her language to ours?", + "input": "at during the transformation, knocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis", "translatedText": "איך מתרגמים מהשפה שלה לשפה שלנו?", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 260.5 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1.", + "input": "ing them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vector", "translatedText": "ומנקודת המבט שלנו, ל-b1 יש קואורדינטות 2, 1, ול-b2 יש קואורדינטות שליליות 1, 1.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 267.04 }, { - "input": "So we can actually compute negative 1 times b1 plus 2 times b2 as they're represented in our coordinate system.", + "input": "s in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio Of course, there's n", "translatedText": "אז אנחנו יכולים למעשה לחשב שלילי 1 כפול b1 ועוד 2 כפול b2 כפי שהם מיוצגים במערכת הקואורדינטות שלנו.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 270.4 }, { - "input": "And working this out, you get a vector with coordinates negative 4, 1.", + "input": "othing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In", "translatedText": "ולפתור את זה, אתה מקבל וקטור עם קואורדינטות שליליות 4, 1.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 274.74 }, { - "input": "This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", + "input": "alf, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennif", "translatedText": "התהליך הזה כאן של שינוי קנה מידה של כל אחד מהוקטור הבסיס שלה לפי הקואורדינטות המתאימות של וקטור כלשהו, ואז חיבורם יחד, עשוי להרגיש מוכר במקצת.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 330.48 }, { - "input": "To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation.", + "input": "envector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking a", "translatedText": "כדי להראות איך זה עובד, בואו נעבור דרך מה זה אומר לקחת את הווקטור שאנו חושבים עליו כבעל קואורדינטות שליליות 1, 2 וליישם את הטרנספורמציה הזו.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 341.38 }, { - "input": "Before the linear transformation, we're thinking of this vector as a certain linear combination of our basis vectors, negative 1 times i-hat plus 2 times j-hat.", + "input": "bout the full 3x3 matrix associated with that transformation. In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length", "translatedText": "לפני הטרנספורמציה הליניארית, אנו חושבים על הווקטור הזה כצירוף ליניארי מסוים של וקטורי הבסיס שלנו, שלילי 1 כפול i-hat ועוד 2 כפול j-hat.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 375.16 }, { - "input": "Geometrically, this matrix transforms our grid into Jennifer's grid but numerically, it's translating a vector described in her language to our language.", + "input": "t the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. we get using", "translatedText": "מבחינה גיאומטרית, המטריצה הזו הופכת את הרשת שלנו לרשת של ג'ניפר, אבל מבחינה מספרית, היא מתרגמת וקטור המתואר בשפתה לשפה שלנו.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 380.62 }, { - "input": "What made it finally click for me was thinking about how it takes our misconception of what Jennifer means, the vector we get using the same coordinates but in our system, then it transforms it into the vector that she really meant.", + "input": "the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I comp", "translatedText": "מה שגרם לי לבסוף ללחוץ עליו היה לחשוב איך זה לוקח את התפיסה המוטעית שלנו לגבי מה שג'ניפר מתכוונת, הווקטור שאנו מקבלים באמצעות אותן קואורדינטות אבל במערכת שלנו, ואז הוא הופך אותו לווקטור שאליו היא באמת התכוונה.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 398.26 }, { - "input": "What about going the other way around?", + "input": "ute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A", "translatedText": "מה עם ללכת הפוך?", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 404.26 }, { - "input": "In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 third in Jennifer's system?", + "input": "is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis", "translatedText": "בדוגמה שהשתמשתי קודם בסרטון הזה, כשהיה לי את הווקטור עם הקואורדינטות 3, 2 במערכת שלנו, איך חישבתי שיהיו לו קואורדינטות 5 שליש ושליש אחד במערכת של ג'ניפר?", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 409.48 }, { - "input": "You start with that change of basis matrix that translates Jennifer's language into ours, then you take its inverse.", + "input": "as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we", "translatedText": "אתה מתחיל עם השינוי הזה של מטריצת הבסיס שמתרגמת את השפה של ג'ניפר לשפתנו, ואז אתה לוקח את ההיפוך שלה.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 415.48 }, { - "input": "Remember, the inverse of a transformation is a new transformation that corresponds to playing that first one backwards.", + "input": "multiply this inverse change of basis matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and fo", "translatedText": "זכור, ההיפוך של טרנספורמציה הוא טרנספורמציה חדשה המקבילה למשחק הראשון לאחור.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 427.94 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse.", + "input": "rth between coordinate systems. The matrix whose columns represent Jennif er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the invers", "translatedText": "בפועל, במיוחד כשאתה עובד ביותר משני מימדים, תשתמש במחשב כדי לחשב את המטריצה שמייצגת למעשה את היפוך הזה.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 465.52 }, { - "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems.", + "input": "you know how matrix multiplication So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, usi", "translatedText": "אז זה, בקצרה, איך לתרגם את התיאור של וקטורים בודדים הלוך ושוב בין מערכות קואורדינטות.", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 487.24 }, { - "input": "And the inverse matrix does the opposite.", + "input": "The columns of such a matrix will represent what happens to eac", "translatedText": "והמטריצה ההפוכה עושה את ההיפך.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 507.16 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy.", + "input": "heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla With", "translatedText": "בהחלט עצור ותסתכל על פרקים 3 ו-4 אם משהו מזה מרגיש לא נוח.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 529.74 }, { - "input": "i-hat ends up at the spot with coordinates 0, 1, and j-hat ends up at the spot with coordinates negative 1, 0.", + "input": "or out the v. So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to desc", "translatedText": "i-hat מסתיים בנקודה עם קואורדינטות 0, 1, ו-j-hat מסתיים בנקודה עם קואורדינטות שליליות 1, 0.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 566.3 }, { - "input": "But that's not quite right.", + "input": "And that squishification corresponds to a zero determinant for the matrix. To be concrete, let's say your matrix", "translatedText": "אבל זה לא ממש נכון.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.72 }, { - "input": "Here's a common way to think of how this is done.", + "input": "As that value of lambda changes, the matrix itself changes, and so the determina", "translatedText": "הנה דרך נפוצה לחשוב איך זה נעשה.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 603.42 }, { - "input": "Start with any vector written in Jennifer's language.", + "input": "nt of the matrix changes. ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multipl", "translatedText": "התחל עם כל וקטור שנכתב בשפה של ג'ניפר.", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 649.44 }, { - "input": "Since we could do this with any vector written in her language, first applying the change of basis, then the transformation, then the inverse change of basis, that composition of three matrices gives us the transformation matrix in Jennifer's language.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective. And the full matrix product represents that same transformation, but as someone else sees it.", "translatedText": "מכיוון שנוכל לעשות זאת עם כל וקטור שנכתב בשפתה, תחילה ליישם את שינוי הבסיס, אחר כך את השינוי, ואז את השינוי ההפוך של הבסיס, הרכב הזה של שלוש מטריצות נותן לנו את מטריצת הטרנספורמציה בשפתה של ג'ניפר.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 665.56 }, { - "input": "It takes in a vector of her language and spits out the transformed version of that vector in her language.", + "input": "For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of", "translatedText": "הוא קולט וקטור של השפה שלה ויורק את הגרסה שעברה טרנספורמציה של אותו וקטור בשפתה.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 675.8 }, { - "input": "For this specific example, when Jennifer's basis vectors look like 2, 1 and negative in our language, and when the transformation is a 90 degree rotation, the product of these three matrices, if you work through it, has columns one third, five thirds, and negative two thirds, negative one third.", + "input": "this. See y That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corres", "translatedText": "לדוגמא הספציפית הזו, כאשר וקטורי הבסיס של ג'ניפר נראים כמו 2, 1 ושליליים בשפה שלנו, וכאשר הטרנספורמציה היא סיבוב של 90 מעלות, המכפלה של שלוש המטריצות הללו, אם תעבדו דרכה, מכילה עמודות של שליש, חמישה שליש. , ושלילי שני שלישים, שלילי שליש.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 692.2 }, { - "input": "So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system.", + "input": "ponding eigenvalue is 1, so v would actually just stay fixed in place. Pause and ponder if you need to make sure that that line of reasoning feels good. This is the kind of thing I mentioned in the introduction. If you didn't have a", "translatedText": "אז אם ג'ניפר תכפיל את המטריצה הזו בקואורדינטות של וקטור במערכת שלה, היא תחזיר את הגרסה המסובבת ב-90 מעלות של אותו וקטור המתבטא במערכת הקואורדינטות שלה.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 709.82 }, { - "input": "In general, whenever you see an expression like A inverse times M times A, it suggests a mathematical sort of empathy.", + "input": "solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "באופן כללי, בכל פעם שאתה רואה ביטוי כמו A הפוך כפול M כפול A, זה מרמז על סוג מתמטי של אמפתיה.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 714.54 }, { - "input": "That middle matrix represents a transformation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective.", + "input": "To see this in action, let's revisit the example from the start, with a matrix whose columns are 3, 0 and 1, 2. To find if a value lambda is an eigenvalue, subtract it from the diago", "translatedText": "המטריצה האמצעית הזו מייצגת טרנספורמציה מסוג כלשהו כפי שאתה רואה אותה, ושתי המטריצות החיצוניות מייצגות את האמפתיה, את השינוי בפרספקטיבה.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/hindi/sentence_translations.json b/2016/change-of-basis/hindi/sentence_translations.json index 61431e9f1..7de0a0f93 100644 --- a/2016/change-of-basis/hindi/sentence_translations.json +++ b/2016/change-of-basis/hindi/sentence_translations.json @@ -1,13 +1,13 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates.", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we doing this and what d", "translatedText": "यदि मेरे पास 2डी स्पेस में कोई वेक्टर बैठा है, तो हमारे पास निर्देशांक के साथ इसका वर्णन करने का एक मानक तरीका है।", "n_reviews": 0, "start": 19.92, "end": 28.28 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up.", + "input": "oes this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin", "translatedText": "इस मामले में, वेक्टर के निर्देशांक 3, 2 हैं, जिसका अर्थ है कि इसकी पूंछ से इसकी नोक तक जाने में तीन इकाइयों को दाईं ओर और दो इकाइयों को ऊपर ले जाना शामिल है।", "n_reviews": 0, "start": 28.28, @@ -21,7 +21,7 @@ "end": 42.96 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up.", + "input": "ch that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of", "translatedText": "आप उस पहले समन्वय को स्केलिंग आई-हैट के रूप में सोचते हैं, लंबाई 1 वाला वेक्टर दाईं ओर इंगित करता है, जबकि दूसरा समन्वय स्केल जे-हैट, लंबाई 1 वाला वेक्टर सीधे ऊपर की ओर इशारा करता है।", "n_reviews": 0, "start": 42.96, @@ -35,7 +35,7 @@ "end": 60.48 }, { - "input": "You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordinate system.", + "input": "you know how to think about matrices as linear transformations, but you also need to be comfortable with things like determinants, linear systems", "translatedText": "आप इन दो विशेष वैक्टरों के बारे में सोच सकते हैं जो हमारी समन्वय प्रणाली की सभी अंतर्निहित मान्यताओं को समाहित करते हैं।", "n_reviews": 0, "start": 60.48, @@ -49,14 +49,14 @@ "end": 81.38 }, { - "input": "Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system.", + "input": "some linear transformation in two dimensions, like the one shown here. st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice o", "translatedText": "वैक्टर और संख्याओं के सेट के बीच अनुवाद करने के किसी भी तरीके को समन्वय प्रणाली कहा जाता है, और दो विशेष वैक्टर आई-हैट और जे-हैट को हमारे मानक समन्वय प्रणाली के आधार वैक्टर कहा जाता है।", "n_reviews": 0, "start": 81.38, "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors.", + "input": "f i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors an", "translatedText": "मैं यहां जिस बारे में बात करना चाहूंगा वह आधार वैक्टर के एक अलग सेट का उपयोग करने का विचार है।", "n_reviews": 0, "start": 89.5, @@ -84,7 +84,7 @@ "end": 109.04 }, { - "input": "Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third.", + "input": "nnifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to th", "translatedText": "जेनिफ़र वास्तव में इस वेक्टर का वर्णन 5 तिहाई और 1 तिहाई निर्देशांक के साथ करेगी।", "n_reviews": 0, "start": 109.04, @@ -98,7 +98,7 @@ "end": 125.16 }, { - "input": "In a little bit, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third.", + "input": "showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describe this vector with the coordinates 5 thirds and", "translatedText": "थोड़ी देर में, मैं आपको दिखाऊंगा कि आपने उन दो संख्याओं, 5 तिहाई और 1 तिहाई का पता कैसे लगाया होगा।", "n_reviews": 0, "start": 125.16, @@ -112,7 +112,7 @@ "end": 144.12 }, { - "input": "What she gets will typically be completely different from the vector that you and I would think of as having those coordinates.", + "input": "gether. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to", "translatedText": "उसे जो मिलेगा वह आम तौर पर उस वेक्टर से पूरी तरह से अलग होगा जिसे आप और मैं उन निर्देशांक के रूप में सोचेंगे।", "n_reviews": 0, "start": 146.32, @@ -140,7 +140,7 @@ "end": 167.14 }, { - "input": "So in effect, we're speaking different languages.", + "input": "lumn of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis.", "translatedText": "तो वास्तव में, हम अलग-अलग भाषाएँ बोल रहे हैं।", "n_reviews": 0, "start": 169.0, @@ -154,7 +154,7 @@ "end": 176.84 }, { - "input": "Let me say a quick word about how I'm representing things here.", + "input": "ector on the x-axis is also just stretched by a factor of 3, and hence remains on its own", "translatedText": "मैं यहां चीजों का प्रतिनिधित्व कैसे कर रहा हूं इसके बारे में एक संक्षिप्त शब्द कहना चाहता हूं।", "n_reviews": 0, "start": 176.84, @@ -175,7 +175,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid.", + "input": "span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here.", "translatedText": "अंतरिक्ष में स्वयं कोई आंतरिक ग्रिड नहीं है।", "n_reviews": 0, "start": 189.52, @@ -189,7 +189,7 @@ "end": 197.6 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean.", + "input": "system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct mea", "translatedText": "हालाँकि उसका मूल वास्तव में हमारे जैसा ही होगा, क्योंकि हर कोई इस बात पर सहमत है कि निर्देशांक 0,0 का क्या मतलब होना चाहिए।", "n_reviews": 0, "start": 197.6, @@ -203,7 +203,7 @@ "end": 214.9 }, { - "input": "But the direction of her axes and the spacing of her grid lines will be different, depending on her choice of basis vectors.", + "input": "ollow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector", "translatedText": "लेकिन उसकी अक्षों की दिशा और उसकी ग्रिड रेखाओं की दूरी अलग-अलग होगी, जो उसकी पसंद के आधार वैक्टर पर निर्भर करेगी।", "n_reviews": 0, "start": 214.9, @@ -217,14 +217,14 @@ "end": 225.94 }, { - "input": "If for example Jennifer describes a vector with coordinates negative 1, 2, what would that be in our coordinate system?", + "input": "on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewh", "translatedText": "उदाहरण के लिए यदि जेनिफ़र नकारात्मक निर्देशांक 1, 2 वाले एक वेक्टर का वर्णन करता है, तो वह हमारी समन्वय प्रणाली में क्या होगा?", "n_reviews": 0, "start": 226.38, "end": 238.98 }, { - "input": "How do you translate from her language to ours?", + "input": "at during the transformation, knocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis", "translatedText": "आप उसकी भाषा से हमारी भाषा में अनुवाद कैसे करते हैं?", "n_reviews": 0, "start": 238.98, @@ -238,21 +238,21 @@ "end": 260.5 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1.", + "input": "ing them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vector", "translatedText": "और हमारे दृष्टिकोण से, b1 के निर्देशांक 2, 1 हैं, और b2 के निर्देशांक नकारात्मक 1, 1 हैं।", "n_reviews": 0, "start": 262.6, "end": 267.04 }, { - "input": "So we can actually compute negative 1 times b1 plus 2 times b2 as they're represented in our coordinate system.", + "input": "s in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio Of course, there's n", "translatedText": "तो हम वास्तव में नकारात्मक 1 गुना बी1 प्लस 2 गुना बी2 की गणना कर सकते हैं जैसा कि वे हमारी समन्वय प्रणाली में दर्शाए गए हैं।", "n_reviews": 0, "start": 267.04, "end": 270.4 }, { - "input": "And working this out, you get a vector with coordinates negative 4, 1.", + "input": "othing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In", "translatedText": "और इस पर काम करने पर, आपको नकारात्मक निर्देशांक 4, 1 वाला एक वेक्टर मिलता है।", "n_reviews": 0, "start": 270.4, @@ -266,7 +266,7 @@ "end": 274.74 }, { - "input": "This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", + "input": "alf, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennif", "translatedText": "यहाँ उसके प्रत्येक आधार वेक्टर को कुछ वेक्टर के संगत निर्देशांक द्वारा स्केल करने, फिर उन्हें एक साथ जोड़ने की यह प्रक्रिया कुछ हद तक परिचित लग सकती है।", "n_reviews": 0, "start": 277.0, @@ -294,14 +294,14 @@ "end": 330.48 }, { - "input": "To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation.", + "input": "envector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking a", "translatedText": "यह दिखाने के लिए कि यह कैसे काम करता है, आइए देखें कि उस वेक्टर को लेने का क्या मतलब होगा जिसके बारे में हम सोचते हैं कि इसमें नकारात्मक 1, 2 निर्देशांक हैं और उस परिवर्तन को लागू करना है।", "n_reviews": 0, "start": 331.04, "end": 341.38 }, { - "input": "Before the linear transformation, we're thinking of this vector as a certain linear combination of our basis vectors, negative 1 times i-hat plus 2 times j-hat.", + "input": "bout the full 3x3 matrix associated with that transformation. In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length", "translatedText": "रैखिक परिवर्तन से पहले, हम इस वेक्टर को हमारे आधार वैक्टर के एक निश्चित रैखिक संयोजन के रूप में सोच रहे हैं, नकारात्मक 1 गुना आई-हैट प्लस 2 गुना जे-हैट।", "n_reviews": 0, "start": 341.38, @@ -329,49 +329,49 @@ "end": 375.16 }, { - "input": "Geometrically, this matrix transforms our grid into Jennifer's grid but numerically, it's translating a vector described in her language to our language.", + "input": "t the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. we get using", "translatedText": "ज्यामितीय रूप से, यह मैट्रिक्स हमारे ग्रिड को जेनिफर के ग्रिड में बदल देता है लेकिन संख्यात्मक रूप से, यह उसकी भाषा में वर्णित वेक्टर का हमारी भाषा में अनुवाद कर रहा है।", "n_reviews": 0, "start": 375.16, "end": 380.62 }, { - "input": "What made it finally click for me was thinking about how it takes our misconception of what Jennifer means, the vector we get using the same coordinates but in our system, then it transforms it into the vector that she really meant.", + "input": "the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I comp", "translatedText": "आखिरकार मेरे लिए यह सोचने वाली बात थी कि जेनिफ़र के अर्थ के बारे में हमारी ग़लतफ़हमी को कैसे दूर किया जाता है, जो वेक्टर हमें समान निर्देशांक का उपयोग करके मिलता है लेकिन हमारे सिस्टम में, फिर यह उसे उस वेक्टर में बदल देता है जिसका वह वास्तव में मतलब था।", "n_reviews": 0, "start": 381.68, "end": 398.26 }, { - "input": "What about going the other way around?", + "input": "ute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A", "translatedText": "दूसरी ओर जाने के बारे में क्या?", "n_reviews": 0, "start": 398.26, "end": 404.26 }, { - "input": "In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 third in Jennifer's system?", + "input": "is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis", "translatedText": "इस वीडियो के पहले उपयोग किए गए उदाहरण में, जब मेरे पास हमारे सिस्टम में निर्देशांक 3, 2 के साथ वेक्टर था, तो मैंने कैसे गणना की कि जेनिफर के सिस्टम में इसके निर्देशांक 5 तिहाई और 1 तिहाई होंगे?", "n_reviews": 0, "start": 404.26, "end": 409.48 }, { - "input": "You start with that change of basis matrix that translates Jennifer's language into ours, then you take its inverse.", + "input": "as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we", "translatedText": "आप आधार मैट्रिक्स के उस परिवर्तन से शुरू करते हैं जो जेनिफर की भाषा को हमारी भाषा में अनुवादित करता है, फिर आप इसका व्युत्क्रम लेते हैं।", "n_reviews": 0, "start": 409.48, "end": 415.48 }, { - "input": "Remember, the inverse of a transformation is a new transformation that corresponds to playing that first one backwards.", + "input": "multiply this inverse change of basis matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and fo", "translatedText": "याद रखें, परिवर्तन का व्युत्क्रम एक नया परिवर्तन है जो पहले वाले को पीछे की ओर खेलने से मेल खाता है।", "n_reviews": 0, "start": 415.48, "end": 427.94 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse.", + "input": "rth between coordinate systems. The matrix whose columns represent Jennif er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the invers", "translatedText": "व्यवहार में, विशेष रूप से जब आप दो से अधिक आयामों में काम कर रहे हों, तो आप मैट्रिक्स की गणना करने के लिए कंप्यूटर का उपयोग करेंगे जो वास्तव में इस व्युत्क्रम का प्रतिनिधित्व करता है।", "n_reviews": 0, "start": 427.94, @@ -392,7 +392,7 @@ "end": 465.52 }, { - "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems.", + "input": "you know how matrix multiplication So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, usi", "translatedText": "तो, संक्षेप में, यह है कि समन्वय प्रणालियों के बीच अलग-अलग वैक्टरों के विवरण का आगे और पीछे अनुवाद कैसे किया जाए।", "n_reviews": 0, "start": 466.48, @@ -406,7 +406,7 @@ "end": 487.24 }, { - "input": "And the inverse matrix does the opposite.", + "input": "The columns of such a matrix will represent what happens to eac", "translatedText": "और व्युत्क्रम मैट्रिक्स इसके विपरीत कार्य करता है।", "n_reviews": 0, "start": 488.16, @@ -427,7 +427,7 @@ "end": 507.16 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy.", + "input": "heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla With", "translatedText": "यदि इनमें से कोई भी आपको असहज लगता है तो निश्चित रूप से रुकें और अध्याय 3 और 4 पर एक नज़र डालें।", "n_reviews": 0, "start": 507.16, @@ -448,7 +448,7 @@ "end": 529.74 }, { - "input": "i-hat ends up at the spot with coordinates 0, 1, and j-hat ends up at the spot with coordinates negative 1, 0.", + "input": "or out the v. So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to desc", "translatedText": "आई-हैट निर्देशांक 0, 1 के साथ स्थान पर समाप्त होता है और जे-हैट नकारात्मक निर्देशांक 1, 0 के साथ स्थान पर समाप्त होता है।", "n_reviews": 0, "start": 529.74, @@ -483,7 +483,7 @@ "end": 566.3 }, { - "input": "But that's not quite right.", + "input": "And that squishification corresponds to a zero determinant for the matrix. To be concrete, let's say your matrix", "translatedText": "लेकिन यह बिल्कुल सही नहीं है.", "n_reviews": 0, "start": 568.32, @@ -497,14 +497,14 @@ "end": 596.72 }, { - "input": "Here's a common way to think of how this is done.", + "input": "As that value of lambda changes, the matrix itself changes, and so the determina", "translatedText": "यह कैसे किया जाता है, इसके बारे में सोचने का एक सामान्य तरीका यहां दिया गया है।", "n_reviews": 0, "start": 597.8, "end": 603.42 }, { - "input": "Start with any vector written in Jennifer's language.", + "input": "nt of the matrix changes. ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multipl", "translatedText": "जेनिफर की भाषा में लिखे किसी भी वेक्टर से शुरुआत करें।", "n_reviews": 0, "start": 603.42, @@ -546,42 +546,42 @@ "end": 649.44 }, { - "input": "Since we could do this with any vector written in her language, first applying the change of basis, then the transformation, then the inverse change of basis, that composition of three matrices gives us the transformation matrix in Jennifer's language.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective. And the full matrix product represents that same transformation, but as someone else sees it.", "translatedText": "चूँकि हम उसकी भाषा में लिखे किसी भी वेक्टर के साथ ऐसा कर सकते हैं, पहले आधार के परिवर्तन को लागू करना, फिर परिवर्तन, फिर आधार के व्युत्क्रम परिवर्तन को लागू करना, तीन आव्यूहों की रचना हमें जेनिफर की भाषा में परिवर्तन मैट्रिक्स देती है।", "n_reviews": 0, "start": 649.44, "end": 665.56 }, { - "input": "It takes in a vector of her language and spits out the transformed version of that vector in her language.", + "input": "For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of", "translatedText": "यह उसकी भाषा का एक वेक्टर लेता है और उस वेक्टर का रूपांतरित संस्करण उसकी भाषा में उगलता है।", "n_reviews": 0, "start": 666.3, "end": 675.8 }, { - "input": "For this specific example, when Jennifer's basis vectors look like 2, 1 and negative in our language, and when the transformation is a 90 degree rotation, the product of these three matrices, if you work through it, has columns one third, five thirds, and negative two thirds, negative one third.", + "input": "this. See y That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corres", "translatedText": "इस विशिष्ट उदाहरण के लिए, जब जेनिफ़र के आधार वैक्टर हमारी भाषा में 2, 1 और नकारात्मक दिखते हैं, और जब परिवर्तन 90 डिग्री रोटेशन होता है, तो इन तीन मैट्रिक्स का उत्पाद, यदि आप इसके माध्यम से काम करते हैं, तो कॉलम एक तिहाई, पांच तिहाई होते हैं , और नकारात्मक दो तिहाई, नकारात्मक एक तिहाई।", "n_reviews": 0, "start": 678.14, "end": 692.2 }, { - "input": "So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system.", + "input": "ponding eigenvalue is 1, so v would actually just stay fixed in place. Pause and ponder if you need to make sure that that line of reasoning feels good. This is the kind of thing I mentioned in the introduction. If you didn't have a", "translatedText": "इसलिए यदि जेनिफर उस मैट्रिक्स को अपने सिस्टम में एक वेक्टर के निर्देशांक से गुणा करती है, तो यह उसके समन्वय प्रणाली में व्यक्त उस वेक्टर का 90 डिग्री घुमाया गया संस्करण लौटाएगा।", "n_reviews": 0, "start": 692.2, "end": 709.82 }, { - "input": "In general, whenever you see an expression like A inverse times M times A, it suggests a mathematical sort of empathy.", + "input": "solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "सामान्य तौर पर, जब भी आप ए व्युत्क्रम समय एम गुना ए जैसी अभिव्यक्ति देखते हैं, तो यह गणितीय प्रकार की सहानुभूति का सुझाव देता है।", "n_reviews": 0, "start": 709.82, "end": 714.54 }, { - "input": "That middle matrix represents a transformation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective.", + "input": "To see this in action, let's revisit the example from the start, with a matrix whose columns are 3, 0 and 1, 2. To find if a value lambda is an eigenvalue, subtract it from the diago", "translatedText": "जैसा कि आप इसे देखते हैं, वह मध्य मैट्रिक्स किसी प्रकार के परिवर्तन का प्रतिनिधित्व करता है, और बाहरी दो मैट्रिक्स सहानुभूति, परिप्रेक्ष्य में बदलाव का प्रतिनिधित्व करते हैं।", "n_reviews": 0, "start": 715.68, diff --git a/2016/change-of-basis/hungarian/sentence_translations.json b/2016/change-of-basis/hungarian/sentence_translations.json index ea337a9b1..73d218676 100644 --- a/2016/change-of-basis/hungarian/sentence_translations.json +++ b/2016/change-of-basis/hungarian/sentence_translations.json @@ -72,7 +72,7 @@ "end": 84.84 }, { - "input": "It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.", + "input": "st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is ti", "translatedText": "Az i-hat alapvektort a 3, 0 koordinátákra, a j-hatot pedig az 1, 2 koordinátákra helyezi át.", "model": "DeepL", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 91.04 }, { - "input": "So it's represented with a matrix whose columns are 3, 0, and 1, 2.", + "input": "ed up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actual", "translatedText": "Tehát egy olyan mátrixszal ábrázoljuk, amelynek oszlopai a 3, 0 és az 1, 2.", "model": "DeepL", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 95.64 }, { - "input": "Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip.", + "input": "ly scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called th", "translatedText": "Koncentráljon arra, hogy mit tesz egy adott vektorral, és gondoljon a vektor kiterjedésére, az origóján és a csúcsán áthaladó egyenesre.", "model": "DeepL", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 104.16 }, { - "input": "Most vectors are going to get knocked off their span during the transformation.", + "input": "e basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a", "translatedText": "A legtöbb vektor a transzformáció során ki fog esni az átfogásból.", "model": "DeepL", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 115.32 }, { - "input": "But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar.", + "input": "let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up.", "translatedText": "Néhány speciális vektor azonban megmarad a saját tartományában, ami azt jelenti, hogy a mátrix hatása egy ilyen vektorra csak annyi, hogy megnyújtja vagy összenyomja azt, mint egy skalár.", "model": "DeepL", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 127.04 }, { - "input": "For this specific example, the basis vector i-hat is one such special vector.", + "input": "Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vecto", "translatedText": "Ebben a konkrét példában az i-hat alapvektor egy ilyen speciális vektor.", "model": "DeepL", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 164.04 }, { - "input": "It ends up getting stretched by a factor of 2.", + "input": "scale b1 by 5 thirds, scale b2 by 1 third, then add them both togethe", "translatedText": "A végeredmény a 2-szeresére nyúlik.", "model": "DeepL", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 167.14 }, { - "input": "And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2.", + "input": "r. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she", "translatedText": "És ismét, a linearitás azt jelenti, hogy a fickó által felölelt átlós egyenes bármely más vektora 2-szeresére fog megnyúlni.", "model": "DeepL", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 178.22 }, { - "input": "And for this transformation, those are all the vectors with this special property of staying on their span.", + "input": "thinks of her first coordinate as scali For this specific example, the basis vector i-hat is one such special vector. The span of", "translatedText": "És ennél a transzformációnál ezek mind olyan vektorok, amelyeknek megvan az a különleges tulajdonságuk, hogy a tartományukban maradnak.", "model": "DeepL", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 185.18 }, { - "input": "Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2.", + "input": "i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis. What's more, because of the way linear transformations work,", "translatedText": "Az x-tengelyen lévőket 3-szorosára, az átlós vonalon lévőket pedig 2-szeresére nyújtják.", "model": "DeepL", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 198.08 }, { - "input": "As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation.", + "input": "n. A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc t, a way to visualize our coordinate system, and so it depends on our choice of basis", "translatedText": "Ahogyan már kitalálhattad, ezeket a speciális vektorokat a transzformáció sajátvektorainak nevezzük, és minden egyes sajátvektorhoz tartozik egy úgynevezett sajátérték, amely nem más, mint az a tényező, amellyel a transzformáció során megnyújtották vagy összenyomták.", "model": "DeepL", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 225.94 }, { - "input": "In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half.", + "input": "nt as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It", "translatedText": "Egy másik példában egy olyan sajátvektorunk lehet, amelynek sajátértéke negatív 1 fele, ami azt jelenti, hogy a vektort megfordítjuk és 1 fele szorozzuk.", "model": "DeepL", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 249.8 }, { - "input": "If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation.", + "input": "of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewhat during the transformation, k", "translatedText": "Ha találunk egy sajátvektort ehhez a forgáshoz, egy olyan vektort, amely a saját tartományában marad, akkor megtaláltuk a forgástengelyt.", "model": "DeepL", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 260.5 }, { - "input": "And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation.", + "input": "nocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", "translatedText": "És sokkal könnyebb egy 3D-s forgatásról a forgástengely és a forgási szög szempontjából gondolkodni, mint a transzformációhoz tartozó teljes 3x3-as mátrixról.", "model": "DeepL", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 285.86 }, { - "input": "This pattern shows up a lot in linear algebra.", + "input": "In fact, once you understand matrix vector multiplication as applying", "translatedText": "Ez a minta gyakran megjelenik a lineáris algebrában.", "model": "DeepL", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 290.02 }, { - "input": "With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors.", + "input": "a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvecto", "translatedText": "Bármely lineáris transzformáció esetében, amelyet egy mátrix ír le, megérthetjük, hogy mit csinál, ha a mátrix oszlopait a bázisvektorok leszállóhelyeiként olvassuk le.", "model": "DeepL", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 299.4 }, { - "input": "But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", + "input": "r with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformati", "translatedText": "De gyakran jobb módja annak, hogy a lineáris transzformáció tényleges működésének lényegéhez jussunk, és kevésbé függ az adott koordinátarendszertől, ha megkeressük a sajátvektorokat és a sajátértékeket.", "model": "DeepL", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 326.02 }, { - "input": "Symbolically, here's what the idea of an eigenvector looks like.", + "input": "she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we thi", "translatedText": "Szimbolikusan így néz ki egy sajátvektor gondolata.", "model": "DeepL", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 339.74 }, { - "input": "What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda.", + "input": "or for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotat", "translatedText": "Ez a kifejezés azt mondja ki, hogy a mátrix-vektor szorzat, A szorozva v-vel, ugyanazt az eredményt adja, mintha a v sajátvektort csak valamilyen lambda értékkel skáláznánk.", "model": "DeepL", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 370.54 }, { - "input": "So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda.", + "input": "never stretch or squish anything, so the length of the vector would remain the same. This pattern shows up a lot in linear algebra. With any linear transformation described by a matrix, you could understand what it's doing by reading of", "translatedText": "Kezdjük tehát azzal, hogy a jobb oldalt átírjuk valamiféle mátrix-vektor szorzásnak, egy olyan mátrixot használva, amely bármely vektor lambda faktorral való skálázását eredményezi.", "model": "DeepL", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 380.62 }, { - "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else.", + "input": "f the columns of this matrix as the landing spots for basis vectors. But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", "translatedText": "Egy ilyen mátrix oszlopai azt jelölik, hogy mi történik az egyes bázisvektorokkal, és minden bázisvektor egyszerűen megszorozódik lambdával, így ez a mátrix az átló mentén a lambda számot fogja tartalmazni, mindenhol máshol pedig nullákat.", "model": "DeepL", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 394.32 }, { - "input": "The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal.", + "input": "we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector w", "translatedText": "Ezt a fickót általában úgy írjuk le, hogy a lambdát faktorozzuk ki, és úgy írjuk, hogy lambda szorozva i-vel, ahol i az azonossági mátrix az átló mentén lévő 1-esekkel.", "model": "DeepL", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 411.86 }, { - "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector.", + "input": "that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, with v as the eigenvector", "translatedText": "Tehát most van egy új mátrixunk, A mínusz lambda szorozva az azonossággal, és olyan vektort keresünk, hogy ez az új mátrix szorozva v-vel adja a nullvektort.", "model": "DeepL", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 433.64 }, { - "input": "And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", + "input": "and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Je", "translatedText": "És ha megnézed az 5. és 6. fejezetet, tudni fogod, hogy egy mátrix és egy nem nulla vektor szorzata csak akkor válhat nullává, ha a mátrixhoz tartozó transzformáció a teret alacsonyabb dimenzióba szorítja.", "model": "DeepL", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 470.28 }, { - "input": "As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes.", + "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose c", "translatedText": "Ahogy a lambda értéke változik, maga a mátrix is változik, és így a mátrix determinánsa is változik.", "model": "DeepL", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 498.6 }, { - "input": "So this is kind of a lot, but let's unravel what this is saying.", + "input": "And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's importa", "translatedText": "Szóval ez elég sok, de bontsuk ki, mit is jelent ez.", "model": "DeepL", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 502.96 }, { - "input": "When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line.", + "input": "nt that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication So let's start by rewriting", "translatedText": "Ha lambda egyenlő 1-gyel, akkor az A mátrix mínusz lambda szorozva az azonossággal egy vonalra szorítja a teret.", "model": "DeepL", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 509.56 }, { - "input": "That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector.", + "input": "that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a fact", "translatedText": "Ez azt jelenti, hogy van egy olyan nem nulla vektor v, hogy A mínusz lambda szorozva az azonossággal szorozva v-vel egyenlő a nulla vektorral.", "model": "DeepL", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 518.56 }, { - "input": "And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A.", + "input": "or of lambda. The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else. the columns of our matrix.", "translatedText": "És ne feledjük, hogy ez azért érdekel minket, mert ez azt jelenti, hogy A szorozva v-vel egyenlő lambda szorozva v-vel, amit úgy olvashatunk le, hogy a vektor A sajátvektora, és az A transzformáció során a saját tartományában marad.", "model": "DeepL", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 537.28 }, { - "input": "In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place.", + "input": "But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following", "translatedText": "Ebben a példában a megfelelő sajátérték 1, így v valójában csak a helyén marad.", "model": "DeepL", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 549.5 }, { - "input": "This is the kind of thing I mentioned in the introduction.", + "input": "-hat in the first pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v.", "translatedText": "Ez az a fajta dolog, amit a bevezetőben említettem.", "model": "DeepL", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 555.64 }, { - "input": "If you didn't have a solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to describe those landing spots in her language. Here's a common way to think", "translatedText": "Ha nem ismernéd a determinánsokat, és nem tudnád, hogy miért kapcsolódnak a nem nulla megoldású lineáris egyenletrendszerekhez, egy ilyen kifejezés teljesen váratlanul érne téged.", "model": "DeepL", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 608.84 }, { - "input": "To figure out what the eigenvectors are that actually have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero.", + "input": "To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry. Now imagine tweaking lambda, turning a knob to change its value. As that value of lambda changes, the matrix itself change", "translatedText": "Hogy kitaláljuk, melyek azok a sajátvektorok, amelyeknek valóban van egy ilyen sajátértékük, mondjuk lambda egyenlő 2-vel, adjuk meg a lambda értékét a mátrixhoz, majd oldjuk meg, hogy ez az átlósan módosított mátrix mely vektorokat küldi nullára.", "model": "DeepL", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 681.94 }, { - "input": "The only roots of that polynomial are the imaginary numbers, i and negative i.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the oute", "translatedText": "Ennek a polinomnak az egyetlen gyöke a képzeletbeli számok, az i és a negatív i.", "model": "DeepL", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 699.82 }, { - "input": "This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1.", + "input": "meone else sees it. For those of you wondering why we care about alternate coordinate systems, the next vi", "translatedText": "Ez rögzíti az i-kalapot a helyén, és áthelyezi a j-kalapot 1-gyel, így a mátrixának oszlopai 1, 0 és 1, 1.", "model": "DeepL", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 726.54 }, { - "input": "And the only root of this expression is lambda equals 1.", + "input": "he identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals la", "translatedText": "És ennek a kifejezésnek az egyetlen gyökere a lambda egyenlő 1.", "model": "DeepL", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 732.86 }, { - "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.", + "input": "mbda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corresponding eigenvalue is", "translatedText": "Ez összhangban van azzal, amit geometriai szempontból látunk, hogy minden sajátvektornak 1 sajátértéke van.", "model": "DeepL", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 796.38 }, { - "input": "For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2.", + "input": "f equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Például, lehet, hogy az i-hat negatív 1, a j-hat pedig 2 értékkel van skálázva.", "model": "DeepL", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 825.42 }, { - "input": "And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.", + "input": "nd compute the determinant. Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda. Since lambda can only be an eigenvalue i", "translatedText": "Ezt úgy kell értelmezni, hogy az összes bázisvektor sajátvektor, és ennek a mátrixnak az átlós bejegyzései a sajátértékek.", "model": "DeepL", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 841.06 }, { - "input": "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", + "input": "u can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3. To figure out what the eigenvectors are that actu", "translatedText": "Az egyik nagy dolog az, hogy könnyebb kiszámítani, mi fog történni, ha ezt a mátrixot egy csomószor megszorozzuk önmagával.", "model": "DeepL", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 848.34 }, { - "input": "Since all one of these matrices does is scale each basis vector by some eigenvalue, applying that matrix many times, say 100 times, is just going to correspond to scaling each basis vector by the 100th power of the corresponding eigenvalue.", + "input": "ally have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero. If you computed this the way you would any other linear system, you'd see that the solutions are all the vectors on the diagonal line spanned by negative 1, 1.", "translatedText": "Mivel az egyik ilyen mátrix csak annyit tesz, hogy minden egyes bázisvektort valamilyen sajátértékkel skáláz, a mátrix többszöri, mondjuk 100-szoros alkalmazása minden egyes bázisvektornak a megfelelő sajátérték 100. hatványával való skálázásának felel meg.", "model": "DeepL", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 869.68 }, { - "input": "Really, try it for a moment.", + "input": "x, 3, 0, 1, 2, has the effect of stretching all those vectors by a factor of 2.", "translatedText": "Tényleg, próbáld ki egy pillanatra.", "model": "DeepL", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 896.54 }, { - "input": "I talked about change of basis last video, but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", + "input": "is, notice what happens. Its matrix has columns 0, 1 and negative 1, 0. Subtract off lambda from the diagonal elements and look for when the determinant is zero. In this case, you get the polynomial lambda squared plus 1. The only roots of that polynomia", "translatedText": "Az előző videóban már beszéltem az alap megváltoztatásáról, de itt egy szuper gyors emlékeztetővel végigmegyek azon, hogyan fejezhetünk ki egy jelenleg a koordinátarendszerünkben leírt transzformációt egy másik rendszerbe.", "model": "DeepL", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 907.04 }, { - "input": "Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors, then make those coordinates the columns of a matrix, known as the change of basis matrix.", + "input": "l are the imaginary numbers, i and negative i. The fact that there are no real number solutions indicates that there are no eigenvectors. Another pretty interesting example worth holding in the back of your mind is a shear. This fixes i-hat in place and moves j-hat 1 over, so its mat", "translatedText": "Vegyük azoknak a vektoroknak a koordinátáit, amelyeket új bázisként akarunk használni, ami ebben az esetben a két sajátvektorunkat jelenti, majd ezeket a koordinátákat tegyük egy mátrix oszlopává, amelyet bázisváltási mátrixnak nevezünk.", "model": "DeepL", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 946.68 }, { - "input": "This is because it represents working in a coordinate system where what happens to the basis vectors is that they get scaled during the transformation.", + "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1. Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a lin", "translatedText": "Ez azért van így, mert olyan koordinátarendszerben dolgozik, ahol az alapvektorokkal az történik, hogy azok a transzformáció során skálázódnak.", "model": "DeepL", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.56 }, { - "input": "So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis, compute the 100th power in that system, then convert back to our standard system.", + "input": "A simple example is a matrix that scales everything by 2. The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue. Now is another good time to pause and ponder some of this before I move on to the last topic.", "translatedText": "Tehát ha például ennek a mátrixnak a 100. hatványát kellene kiszámítani, sokkal egyszerűbb lenne átváltani egy sajátbázisra, kiszámítani a 100. hatványt abban a rendszerben, majd visszaváltani a mi standard rendszerünkre.", "model": "DeepL", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 975.68 }, { - "input": "You can't do this with all transformations.", + "input": "I want to finish off here with the idea of an eigenbasis, which relies heavily on ideas from the last video. Take a look at what h", "translatedText": "Ezt nem teheti meg minden átalakítással.", "model": "DeepL", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 978.32 }, { - "input": "A shear, for example, doesn't have enough eigenvectors to span the full space.", + "input": "appens if our basis vectors just so happen to be eigenvectors. For example, maybe i-hat is scale", "translatedText": "Egy nyírásnak például nincs elég sajátvektora ahhoz, hogy a teljes teret átfogja.", "model": "DeepL", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 982.98 }, { - "input": "But if you can find an eigenbasis, it makes matrix operations really lovely.", + "input": "d by negative 1 and j-hat is scaled by 2. Writing their new coordinates as the columns of a matrix, notice t", "translatedText": "De ha találsz egy saját bázist, akkor a mátrixműveletek nagyon szépek lesznek.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/change-of-basis/indonesian/sentence_translations.json b/2016/change-of-basis/indonesian/sentence_translations.json index 5e8a70083..99db44740 100644 --- a/2016/change-of-basis/indonesian/sentence_translations.json +++ b/2016/change-of-basis/indonesian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates.", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we doing this and what d", "translatedText": "Jika saya memiliki vektor di sini dalam ruang 2D, kami memiliki cara standar untuk mendeskripsikannya dengan koordinat.", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 28.28 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up.", + "input": "oes this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin", "translatedText": "Dalam hal ini vektor mempunyai koordinat 3, 2 yang berarti perpindahan dari ekor ke ujung melibatkan perpindahan tiga satuan ke kanan dan dua satuan ke atas.", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.96 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up.", + "input": "ch that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of", "translatedText": "Anggap saja koordinat pertama itu berskala i-hat, vektor dengan panjang 1 mengarah ke kanan, sedangkan koordinat kedua berskala j-hat, vektor dengan panjang 1 mengarah lurus ke atas.", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 60.48 }, { - "input": "You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordinate system.", + "input": "you know how to think about matrices as linear transformations, but you also need to be comfortable with things like determinants, linear systems", "translatedText": "Anda dapat menganggap kedua vektor khusus ini merangkum semua asumsi implisit sistem koordinat kita.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 81.38 }, { - "input": "Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system.", + "input": "some linear transformation in two dimensions, like the one shown here. st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice o", "translatedText": "Segala cara untuk menerjemahkan antara vektor dan himpunan bilangan disebut sistem koordinat, dan dua vektor khusus i-hat dan j-hat disebut sebagai vektor basis dari sistem koordinat standar kita.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors.", + "input": "f i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors an", "translatedText": "Apa yang ingin saya bicarakan di sini adalah gagasan untuk menggunakan kumpulan vektor basis yang berbeda.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.04 }, { - "input": "Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third.", + "input": "nnifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to th", "translatedText": "Jennifer sebenarnya menggambarkan vektor ini dengan koordinat 5 pertiga dan 1 pertiga.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 125.16 }, { - "input": "In a little bit, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third.", + "input": "showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describe this vector with the coordinates 5 thirds and", "translatedText": "Sebentar lagi, saya akan menunjukkan kepada Anda bagaimana Anda bisa mengetahui dua angka tersebut, 5 pertiga dan 1 pertiga.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 144.12 }, { - "input": "What she gets will typically be completely different from the vector that you and I would think of as having those coordinates.", + "input": "gether. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to", "translatedText": "Apa yang dia dapatkan biasanya akan sangat berbeda dari vektor yang Anda dan saya anggap memiliki koordinat tersebut.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 167.14 }, { - "input": "So in effect, we're speaking different languages.", + "input": "lumn of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis.", "translatedText": "Jadi sebenarnya kita berbicara dalam bahasa yang berbeda.", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 176.84 }, { - "input": "Let me say a quick word about how I'm representing things here.", + "input": "ector on the x-axis is also just stretched by a factor of 3, and hence remains on its own", "translatedText": "Izinkan saya menyampaikan sepatah kata singkat tentang bagaimana saya mewakili berbagai hal di sini.", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid.", + "input": "span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here.", "translatedText": "Ruang itu sendiri tidak memiliki jaringan intrinsik.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 197.6 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean.", + "input": "system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct mea", "translatedText": "Asal usulnya sebenarnya sejalan dengan asal usul kita, karena semua orang sepakat tentang arti koordinat 0,0.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 214.9 }, { - "input": "But the direction of her axes and the spacing of her grid lines will be different, depending on her choice of basis vectors.", + "input": "ollow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector", "translatedText": "Namun arah sumbunya dan jarak garis gridnya akan berbeda, bergantung pada pilihan vektor basisnya.", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 225.94 }, { - "input": "If for example Jennifer describes a vector with coordinates negative 1, 2, what would that be in our coordinate system?", + "input": "on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewh", "translatedText": "Jika misalnya Jennifer menggambarkan sebuah vektor dengan koordinat negatif 1, 2, apa yang ada dalam sistem koordinat kita?", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 238.98 }, { - "input": "How do you translate from her language to ours?", + "input": "at during the transformation, knocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis", "translatedText": "Bagaimana Anda menerjemahkan dari bahasanya ke bahasa kami?", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 260.5 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1.", + "input": "ing them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vector", "translatedText": "Dan dari sudut pandang kita, b1 memiliki koordinat 2, 1, dan b2 memiliki koordinat negatif 1, 1.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 267.04 }, { - "input": "So we can actually compute negative 1 times b1 plus 2 times b2 as they're represented in our coordinate system.", + "input": "s in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio Of course, there's n", "translatedText": "Jadi kita sebenarnya bisa menghitung negatif 1 kali b1 ditambah 2 kali b2 seperti yang direpresentasikan dalam sistem koordinat kita.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 270.4 }, { - "input": "And working this out, you get a vector with coordinates negative 4, 1.", + "input": "othing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In", "translatedText": "Dan dengan menyelesaikannya, Anda mendapatkan sebuah vektor dengan koordinat negatif 4, 1.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 274.74 }, { - "input": "This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", + "input": "alf, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennif", "translatedText": "Proses penskalaan setiap vektor basisnya dengan koordinat beberapa vektor yang sesuai, lalu menjumlahkannya, mungkin terasa agak familiar.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 330.48 }, { - "input": "To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation.", + "input": "envector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking a", "translatedText": "Untuk menunjukkan cara kerjanya, mari kita bahas apa artinya mengambil vektor yang kita anggap memiliki koordinat negatif 1, 2 dan menerapkan transformasi tersebut.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 341.38 }, { - "input": "Before the linear transformation, we're thinking of this vector as a certain linear combination of our basis vectors, negative 1 times i-hat plus 2 times j-hat.", + "input": "bout the full 3x3 matrix associated with that transformation. In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length", "translatedText": "Sebelum transformasi linier, kita menganggap vektor ini sebagai kombinasi linier tertentu dari vektor basis kita, negatif 1 kali i-hat ditambah 2 kali j-hat.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 375.16 }, { - "input": "Geometrically, this matrix transforms our grid into Jennifer's grid but numerically, it's translating a vector described in her language to our language.", + "input": "t the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. we get using", "translatedText": "Secara geometris, matriks ini mengubah grid kita menjadi grid Jennifer tetapi secara numerik, matriks ini menerjemahkan vektor yang dijelaskan dalam bahasanya ke bahasa kita.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 380.62 }, { - "input": "What made it finally click for me was thinking about how it takes our misconception of what Jennifer means, the vector we get using the same coordinates but in our system, then it transforms it into the vector that she really meant.", + "input": "the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I comp", "translatedText": "Apa yang akhirnya membuat saya cocok adalah memikirkan bagaimana kesalahpahaman kita tentang apa yang dimaksud Jennifer, vektor yang kita peroleh menggunakan koordinat yang sama tetapi dalam sistem kita, kemudian mengubahnya menjadi vektor yang sebenarnya dia maksud.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 398.26 }, { - "input": "What about going the other way around?", + "input": "ute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A", "translatedText": "Bagaimana kalau sebaliknya?", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 404.26 }, { - "input": "In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 third in Jennifer's system?", + "input": "is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis", "translatedText": "Dalam contoh yang saya gunakan sebelumnya di video ini, ketika saya memiliki vektor dengan koordinat 3, 2 di sistem kita, bagaimana saya menghitung bahwa vektor tersebut akan memiliki koordinat 5 pertiga dan 1 pertiga dalam sistem Jennifer?", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 409.48 }, { - "input": "You start with that change of basis matrix that translates Jennifer's language into ours, then you take its inverse.", + "input": "as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we", "translatedText": "Anda mulai dengan perubahan matriks dasar yang menerjemahkan bahasa Jennifer ke dalam bahasa kita, lalu Anda mengambil kebalikannya.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 415.48 }, { - "input": "Remember, the inverse of a transformation is a new transformation that corresponds to playing that first one backwards.", + "input": "multiply this inverse change of basis matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and fo", "translatedText": "Ingat, kebalikan dari transformasi adalah transformasi baru yang berhubungan dengan memutar transformasi pertama secara terbalik.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 427.94 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse.", + "input": "rth between coordinate systems. The matrix whose columns represent Jennif er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the invers", "translatedText": "Dalam praktiknya, terutama saat Anda mengerjakan lebih dari dua dimensi, Anda akan menggunakan komputer untuk menghitung matriks yang benar-benar mewakili invers ini.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 465.52 }, { - "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems.", + "input": "you know how matrix multiplication So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, usi", "translatedText": "Singkatnya, bagaimana menerjemahkan deskripsi vektor individu bolak-balik antar sistem koordinat.", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 487.24 }, { - "input": "And the inverse matrix does the opposite.", + "input": "The columns of such a matrix will represent what happens to eac", "translatedText": "Dan matriks invers melakukan hal sebaliknya.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 507.16 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy.", + "input": "heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla With", "translatedText": "Pastinya berhenti sejenak dan lihat bab 3 dan 4 jika ada yang terasa tidak nyaman.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 529.74 }, { - "input": "i-hat ends up at the spot with coordinates 0, 1, and j-hat ends up at the spot with coordinates negative 1, 0.", + "input": "or out the v. So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to desc", "translatedText": "i-hat berakhir di titik dengan koordinat 0, 1, dan j-hat berakhir di titik dengan koordinat negatif 1, 0.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 566.3 }, { - "input": "But that's not quite right.", + "input": "And that squishification corresponds to a zero determinant for the matrix. To be concrete, let's say your matrix", "translatedText": "Tapi itu kurang tepat.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.72 }, { - "input": "Here's a common way to think of how this is done.", + "input": "As that value of lambda changes, the matrix itself changes, and so the determina", "translatedText": "Inilah cara umum untuk memikirkan bagaimana hal ini dilakukan.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 603.42 }, { - "input": "Start with any vector written in Jennifer's language.", + "input": "nt of the matrix changes. ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multipl", "translatedText": "Mulailah dengan vektor apa pun yang ditulis dalam bahasa Jennifer.", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 649.44 }, { - "input": "Since we could do this with any vector written in her language, first applying the change of basis, then the transformation, then the inverse change of basis, that composition of three matrices gives us the transformation matrix in Jennifer's language.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective. And the full matrix product represents that same transformation, but as someone else sees it.", "translatedText": "Karena kita dapat melakukan ini dengan vektor apa pun yang ditulis dalam bahasanya, pertama-tama terapkan perubahan basis, lalu transformasi, lalu invers perubahan basis, komposisi tiga matriks tersebut menghasilkan matriks transformasi dalam bahasa Jennifer.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 665.56 }, { - "input": "It takes in a vector of her language and spits out the transformed version of that vector in her language.", + "input": "For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of", "translatedText": "Dibutuhkan vektor bahasanya dan mengeluarkan versi transformasi vektor tersebut dalam bahasanya.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 675.8 }, { - "input": "For this specific example, when Jennifer's basis vectors look like 2, 1 and negative in our language, and when the transformation is a 90 degree rotation, the product of these three matrices, if you work through it, has columns one third, five thirds, and negative two thirds, negative one third.", + "input": "this. See y That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corres", "translatedText": "Untuk contoh spesifik ini, ketika vektor basis Jennifer terlihat seperti 2, 1 dan negatif dalam bahasa kita, dan ketika transformasinya adalah rotasi 90 derajat, produk dari ketiga matriks ini, jika Anda mengerjakannya, memiliki kolom sepertiga, lima pertiga , dan negatif dua pertiga, negatif sepertiga.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 692.2 }, { - "input": "So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system.", + "input": "ponding eigenvalue is 1, so v would actually just stay fixed in place. Pause and ponder if you need to make sure that that line of reasoning feels good. This is the kind of thing I mentioned in the introduction. If you didn't have a", "translatedText": "Jadi, jika Jennifer mengalikan matriks tersebut dengan koordinat vektor dalam sistemnya, maka versi vektor tersebut akan dirotasi 90 derajat yang dinyatakan dalam sistem koordinatnya.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 709.82 }, { - "input": "In general, whenever you see an expression like A inverse times M times A, it suggests a mathematical sort of empathy.", + "input": "solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Secara umum, setiap kali Anda melihat ekspresi seperti A terbalik dikali M dikali A, hal ini menunjukkan semacam empati matematis.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 714.54 }, { - "input": "That middle matrix represents a transformation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective.", + "input": "To see this in action, let's revisit the example from the start, with a matrix whose columns are 3, 0 and 1, 2. To find if a value lambda is an eigenvalue, subtract it from the diago", "translatedText": "Matriks tengah tersebut mewakili suatu transformasi seperti yang Anda lihat, dan dua matriks terluar mewakili empati, pergeseran perspektif.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/italian/sentence_translations.json b/2016/change-of-basis/italian/sentence_translations.json index 6ece28677..b3bd3aa21 100644 --- a/2016/change-of-basis/italian/sentence_translations.json +++ b/2016/change-of-basis/italian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates.", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we doing this and what d", "translatedText": "Se ho un vettore qui nello spazio 2D, abbiamo un modo standard per descriverlo con le coordinate.", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 28.28 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up.", + "input": "oes this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin", "translatedText": "In questo caso, il vettore ha coordinate 3, 2, il che significa che andare dalla coda alla punta comporta lo spostamento di tre unità a destra e di due unità in alto.", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.96 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up.", + "input": "ch that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of", "translatedText": "Pensi a quella prima coordinata come se scala i-hat, il vettore con lunghezza 1 che punta verso destra, mentre la seconda coordinata scala j-hat, il vettore con lunghezza 1 che punta verso l'alto.", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 60.48 }, { - "input": "You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordinate system.", + "input": "you know how to think about matrices as linear transformations, but you also need to be comfortable with things like determinants, linear systems", "translatedText": "Puoi pensare a questi due vettori speciali come a incapsulare tutti i presupposti impliciti del nostro sistema di coordinate.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 81.38 }, { - "input": "Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system.", + "input": "some linear transformation in two dimensions, like the one shown here. st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice o", "translatedText": "Qualsiasi modo di tradurre tra vettori e insiemi di numeri è chiamato sistema di coordinate, e i due vettori speciali i-hat e j-hat sono chiamati i vettori base del nostro sistema di coordinate standard.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors.", + "input": "f i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors an", "translatedText": "Ciò di cui vorrei parlare qui è l'idea di utilizzare un insieme diverso di vettori base.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.04 }, { - "input": "Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third.", + "input": "nnifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to th", "translatedText": "Jennifer descriverebbe effettivamente questo vettore con le coordinate 5 terzi e 1 terzo.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 125.16 }, { - "input": "In a little bit, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third.", + "input": "showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describe this vector with the coordinates 5 thirds and", "translatedText": "Tra poco ti mostrerò come avresti potuto capire quei due numeri, 5 terzi e 1 terzo.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 144.12 }, { - "input": "What she gets will typically be completely different from the vector that you and I would think of as having those coordinates.", + "input": "gether. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to", "translatedText": "Ciò che otterrà sarà in genere completamente diverso dal vettore che tu e io considereremmo avere quelle coordinate.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 167.14 }, { - "input": "So in effect, we're speaking different languages.", + "input": "lumn of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis.", "translatedText": "Quindi in effetti parliamo lingue diverse.", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 176.84 }, { - "input": "Let me say a quick word about how I'm representing things here.", + "input": "ector on the x-axis is also just stretched by a factor of 3, and hence remains on its own", "translatedText": "Lasciatemi dire una breve parola su come rappresento le cose qui.", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid.", + "input": "span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here.", "translatedText": "Lo spazio stesso non ha una griglia intrinseca.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 197.6 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean.", + "input": "system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct mea", "translatedText": "La sua origine però sarebbe in realtà in linea con la nostra, poiché tutti sono d'accordo su cosa dovrebbero significare le coordinate 0,0.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 214.9 }, { - "input": "But the direction of her axes and the spacing of her grid lines will be different, depending on her choice of basis vectors.", + "input": "ollow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector", "translatedText": "Ma la direzione dei suoi assi e la spaziatura delle linee della griglia saranno diverse, a seconda della scelta dei vettori base.", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 225.94 }, { - "input": "If for example Jennifer describes a vector with coordinates negative 1, 2, what would that be in our coordinate system?", + "input": "on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewh", "translatedText": "Se ad esempio Jennifer descrivesse un vettore con coordinate negative 1, 2, quale sarebbe nel nostro sistema di coordinate?", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 238.98 }, { - "input": "How do you translate from her language to ours?", + "input": "at during the transformation, knocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis", "translatedText": "Come traduci dalla sua lingua alla nostra?", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 260.5 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1.", + "input": "ing them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vector", "translatedText": "E dal nostro punto di vista, b1 ha coordinate 2, 1 e b2 ha coordinate negative 1, 1.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 267.04 }, { - "input": "So we can actually compute negative 1 times b1 plus 2 times b2 as they're represented in our coordinate system.", + "input": "s in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio Of course, there's n", "translatedText": "Quindi possiamo effettivamente calcolare -1 per b1 più 2 per b2 come sono rappresentati nel nostro sistema di coordinate.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 270.4 }, { - "input": "And working this out, you get a vector with coordinates negative 4, 1.", + "input": "othing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In", "translatedText": "E risolvendo questo problema, ottieni un vettore con coordinate negative 4, 1.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 274.74 }, { - "input": "This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", + "input": "alf, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennif", "translatedText": "Questo processo di ridimensionamento di ciascuno dei suoi vettori di base in base alle coordinate corrispondenti di qualche vettore, quindi sommandoli insieme, potrebbe sembrare in qualche modo familiare.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 330.48 }, { - "input": "To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation.", + "input": "envector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking a", "translatedText": "Per mostrare come funziona, esaminiamo cosa significherebbe prendere il vettore che pensiamo abbia coordinate negative 1, 2 e applicare tale trasformazione.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 341.38 }, { - "input": "Before the linear transformation, we're thinking of this vector as a certain linear combination of our basis vectors, negative 1 times i-hat plus 2 times j-hat.", + "input": "bout the full 3x3 matrix associated with that transformation. In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length", "translatedText": "Prima della trasformazione lineare, pensiamo a questo vettore come a una certa combinazione lineare dei nostri vettori base, meno 1 per i-hat più 2 per j-hat.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 375.16 }, { - "input": "Geometrically, this matrix transforms our grid into Jennifer's grid but numerically, it's translating a vector described in her language to our language.", + "input": "t the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. we get using", "translatedText": "Geometricamente, questa matrice trasforma la nostra griglia nella griglia di Jennifer ma numericamente traduce un vettore descritto nella sua lingua nella nostra lingua.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 380.62 }, { - "input": "What made it finally click for me was thinking about how it takes our misconception of what Jennifer means, the vector we get using the same coordinates but in our system, then it transforms it into the vector that she really meant.", + "input": "the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I comp", "translatedText": "Ciò che alla fine mi ha fatto scattare la scintilla è stato pensare a come il nostro malinteso su cosa significhi Jennifer, il vettore che otteniamo utilizzando le stesse coordinate ma nel nostro sistema, lo trasforma nel vettore che intendeva veramente.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 398.26 }, { - "input": "What about going the other way around?", + "input": "ute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A", "translatedText": "Che ne dici di andare al contrario?", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 404.26 }, { - "input": "In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 third in Jennifer's system?", + "input": "is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis", "translatedText": "Nell'esempio che ho usato in precedenza in questo video, quando avevo il vettore con coordinate 3, 2 nel nostro sistema, come ho fatto a calcolare che avrebbe avuto coordinate 5 terzi e 1 terzo nel sistema di Jennifer?", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 409.48 }, { - "input": "You start with that change of basis matrix that translates Jennifer's language into ours, then you take its inverse.", + "input": "as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we", "translatedText": "Inizi con quella matrice di cambio di base che traduce la lingua di Jennifer nella nostra, poi prendi il suo inverso.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 415.48 }, { - "input": "Remember, the inverse of a transformation is a new transformation that corresponds to playing that first one backwards.", + "input": "multiply this inverse change of basis matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and fo", "translatedText": "Ricorda, l'inverso di una trasformazione è una nuova trasformazione che corrisponde a riprodurre la prima al contrario.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 427.94 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse.", + "input": "rth between coordinate systems. The matrix whose columns represent Jennif er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the invers", "translatedText": "In pratica, soprattutto quando lavori in più di due dimensioni, utilizzeresti un computer per calcolare la matrice che rappresenta effettivamente questo inverso.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 465.52 }, { - "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems.", + "input": "you know how matrix multiplication So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, usi", "translatedText": "Questo, in poche parole, è come tradurre la descrizione dei singoli vettori avanti e indietro tra i sistemi di coordinate.", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 487.24 }, { - "input": "And the inverse matrix does the opposite.", + "input": "The columns of such a matrix will represent what happens to eac", "translatedText": "E la matrice inversa fa il contrario.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 507.16 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy.", + "input": "heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla With", "translatedText": "Sicuramente fermati e dai un'occhiata ai capitoli 3 e 4 se qualcosa ti sembra a disagio.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 529.74 }, { - "input": "i-hat ends up at the spot with coordinates 0, 1, and j-hat ends up at the spot with coordinates negative 1, 0.", + "input": "or out the v. So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to desc", "translatedText": "i-hat finisce nel punto con coordinate 0, 1 e j-hat finisce nel punto con coordinate negative 1, 0.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 566.3 }, { - "input": "But that's not quite right.", + "input": "And that squishification corresponds to a zero determinant for the matrix. To be concrete, let's say your matrix", "translatedText": "Ma non è del tutto corretto.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.72 }, { - "input": "Here's a common way to think of how this is done.", + "input": "As that value of lambda changes, the matrix itself changes, and so the determina", "translatedText": "Ecco un modo comune di pensare a come farlo.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 603.42 }, { - "input": "Start with any vector written in Jennifer's language.", + "input": "nt of the matrix changes. ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multipl", "translatedText": "Inizia con qualsiasi vettore scritto nella lingua di Jennifer.", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 649.44 }, { - "input": "Since we could do this with any vector written in her language, first applying the change of basis, then the transformation, then the inverse change of basis, that composition of three matrices gives us the transformation matrix in Jennifer's language.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective. And the full matrix product represents that same transformation, but as someone else sees it.", "translatedText": "Dato che potremmo farlo con qualsiasi vettore scritto nella sua lingua, applicando prima il cambio di base, poi la trasformazione, poi il cambio di base inverso, quella composizione di tre matrici ci dà la matrice di trasformazione nel linguaggio di Jennifer.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 665.56 }, { - "input": "It takes in a vector of her language and spits out the transformed version of that vector in her language.", + "input": "For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of", "translatedText": "Prende un vettore della sua lingua e sputa fuori la versione trasformata di quel vettore nella sua lingua.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 675.8 }, { - "input": "For this specific example, when Jennifer's basis vectors look like 2, 1 and negative in our language, and when the transformation is a 90 degree rotation, the product of these three matrices, if you work through it, has columns one third, five thirds, and negative two thirds, negative one third.", + "input": "this. See y That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corres", "translatedText": "Per questo esempio specifico, quando i vettori della base di Jennifer appaiono come 2, 1 e negativo nella nostra lingua, e quando la trasformazione è una rotazione di 90 gradi, il prodotto di queste tre matrici, se lo si lavora, ha colonne un terzo, cinque terzi e due terzi negativi, un terzo negativo.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 692.2 }, { - "input": "So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system.", + "input": "ponding eigenvalue is 1, so v would actually just stay fixed in place. Pause and ponder if you need to make sure that that line of reasoning feels good. This is the kind of thing I mentioned in the introduction. If you didn't have a", "translatedText": "Quindi, se Jennifer moltiplica quella matrice per le coordinate di un vettore nel suo sistema, restituirà la versione ruotata di 90 gradi di quel vettore espressa nel suo sistema di coordinate.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 709.82 }, { - "input": "In general, whenever you see an expression like A inverse times M times A, it suggests a mathematical sort of empathy.", + "input": "solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "In generale, ogni volta che vedi un’espressione come A inversa per M per A, suggerisce una sorta di empatia matematica.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 714.54 }, { - "input": "That middle matrix represents a transformation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective.", + "input": "To see this in action, let's revisit the example from the start, with a matrix whose columns are 3, 0 and 1, 2. To find if a value lambda is an eigenvalue, subtract it from the diago", "translatedText": "Quella matrice centrale rappresenta una trasformazione di qualche tipo come la vedete, e le due matrici esterne rappresentano l'empatia, il cambiamento di prospettiva.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/japanese/sentence_translations.json b/2016/change-of-basis/japanese/sentence_translations.json index 37a722074..83fc28942 100644 --- a/2016/change-of-basis/japanese/sentence_translations.json +++ b/2016/change-of-basis/japanese/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates. ", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we d ", "translatedText": "2D 空間にベクトルがある場合、それを 座標で記述する標準的な方法があります。", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 27.48 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up. ", + "input": "oing this and what does this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin ", "translatedText": "この場合、ベクトルの座標は 3、2 です。つまり、尾部から先端まで移 動するには、右に 3 単位、上に 2 単位移動する必要があります。", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.36 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up. ", + "input": "not so much that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of ", "translatedText": "最初の座標は右を指す長さ 1 のベクトルである i-hat をスケ ーリングするものであり、2 番目の座標は真上を指す長さ 1 のベク トルである j-hat をスケーリングするものであると考えます。", "model": "google_nmt", "n_reviews": 0, @@ -32,7 +32,7 @@ "end": 57.14 }, { - "input": "The tip-to-tail sum of those two scaled vectors is what the coordinates are meant to describe. ", + "input": "the topics that precede it. Most important here is that you know how to think about matrices as linear transformations, but you also need to be comforta ", "translatedText": "これら 2 つのスケーリングされたベクトルの先端から末尾までの合計が、座標で表現されることになります。", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors. ", + "input": "of that is tied up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors a ", "translatedText": "ここで話したいのは、別の基底ベクトルのセットを使用するというアイデアです。", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 133.36 }, { - "input": "In general, whenever Jennifer uses coordinates to describe a vector, she thinks of her first coordinate as scaling b1, the second coordinate as scaling b2, and she adds the results. ", + "input": "vector, b2, points left and up. Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describ ", "translatedText": "一般に、ジェニファーがベクトルを記述するために座標を使用するときは常に、最初の座標をスケーリ ング b1 として考え、2 番目の座標をスケーリング b2 として考え、結果を加算します。", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 166.8 }, { - "input": "They are what define the meaning of the coordinates 1,0 and 0,1 in her world. ", + "input": "then add them both together. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. ", "translatedText": "それらは、彼女の世界の座標 1,0 と 0,1 の意味を定義するものです。", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 185.86 }, { - "input": "But that grid is just a construct, a way to visualize our coordinate system, and so it depends on our choice of basis. ", + "input": "ample, the basis vector i-hat is one such special vector. The span of i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 ", "translatedText": "しかし、そのグリッドは単なる構成要素であり、座標系を視 覚化する方法であるため、基準の選択によって決まります。", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid. ", + "input": "times itself, still on that x-axis. What's more, because of the way linear transformations work, ", "translatedText": "空間自体には固有のグリッドがありません。", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 198.08 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. ", + "input": "emains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. ", "translatedText": "しかし、座標 0,0 が何を意味するかについては誰もが同意して いるため、彼女の起源は実際には私たちの起源と一致するでしょう。", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 223.72 }, { - "input": "So after all this is set up, a pretty natural question to ask is how we translate between coordinate systems. ", + "input": "self has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct meant as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her ori ", "translatedText": "したがって、これをすべて設定した後、非常に自然な疑問 は、座標系間でどのように変換するかということです。", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 227.28 }, { - "input": "How do you translate from her language to ours? ", + "input": "hould mean. It's the thing that you get when you scale any vector by zero. ", "translatedText": "彼女の言語を私たちの言語にどのように翻訳しますか? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 230.76 }, { - "input": "Well, what her coordinates are saying is that this vector is negative 1 times b1 plus 2 times b2. ", + "input": "But the direction of her axes and Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. ", "translatedText": "さて、彼女の座標が示しているのは、このベクトルは b1 の 1 倍と b2 の 2 倍の負の値であるということです。", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 231.76 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1. ", + "input": "Any other vector is going to get rotated somewhat during the transformation, knocked off the line that it spans. ks of as negative 1, 2. ", "translatedText": "そして、私たちの視点から見ると、b1 の座標は 2, 1 であり、b2 の座標はマイナス 1, 1 です。", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 254.28 }, { - "input": "It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vectors in our language. ", + "input": "tand matrix vector multiplication as applying a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or ", "translatedText": "これは行列ベクトルの乗算であり、その列が言語のジ ェニファーの基底ベクトルを表す行列を使用します。", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 257.08 }, { - "input": "In fact, once you understand matrix vector multiplication as applying a certain linear transformation, say by watching what I view to be the most important video in this series, Chapter 3, there's a pretty intuitive way to think about what's going on here. ", + "input": "the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. ", "translatedText": "実際、行列ベクトルの乗算を特定の線形変換の適用として理解すると、たとえ ば、このシリーズで最も重要だと私が考えるビデオである第 3 章を視聴す ると、ここで何が起こっているのかを非常に直感的に考えることができます。", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 384.3 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse. ", + "input": "coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, ", "translatedText": "実際には、特に 2 次元以上で作業している場合は、コンピュ ーターを使用して、この逆行列を実際に表す行列を計算します。", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 389.6 }, { - "input": "In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. ", + "input": "with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. ", "translatedText": "この場合、ジェニファーの基底を列として持つ基底 変化行列の逆行列は、最終的に列が 1/3、負の 1/3、および 1/3、2/3 になります。", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 443.9 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy. ", + "input": "es, and that you know how matrix multiplication So let's start by rewriting that right-hand ", "translatedText": "不安を感じたら、ぜひ一時停止して第 3 章と第 4 章を見てください。", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 446.48 }, { - "input": "Consider some linear transformation, like a 90 degree counterclockwise rotation. ", + "input": "side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda. ", "translatedText": "反時計回りに 90 度回転するなど、線形変換を考えてみましょう。", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 451.06 }, { - "input": "When you and I represent this with a matrix, we follow where the basis vectors i-hat and j-hat each go. ", + "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this ", "translatedText": "あなたと私がこれを行列で表すとき、基底ベクトル i-hat と j-hat がそれぞれどこに行くのかを追跡します。", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 477.24 }, { - "input": "How would Jennifer describe this same 90 degree rotation of space? ", + "input": "pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v. ", "translatedText": "ジェニファーは、この同じ空間の 90 度の回転をどのように説明しますか? ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 484.26 }, { - "input": "But that's not quite right. ", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and ", "translatedText": "しかし、それは完全に正しくありません。", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 515.02 }, { - "input": "Then apply the transformation matrix to what you get by multiplying it on the left. ", + "input": "zero is if the transformation associated with that matrix squishes space into a lower dimension. And that squishification corresponds to a zero determinant for the matr ", "translatedText": "次に、左側で乗算して得られたものに変換行列を適用します。", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 515.62 }, { - "input": "This tells us where that vector lands, but still in our language. ", + "input": "ix. To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtract ", "translatedText": "これにより、そのベクトルがどこに着地するかがわかりますが、言語は変わりません。", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/korean/sentence_translations.json b/2016/change-of-basis/korean/sentence_translations.json index 359afb851..acef57ab3 100644 --- a/2016/change-of-basis/korean/sentence_translations.json +++ b/2016/change-of-basis/korean/sentence_translations.json @@ -81,7 +81,7 @@ "end": 84.84 }, { - "input": "It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.", + "input": "st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is ti", "translatedText": "기본 벡터 i-hat을 좌표 3, 0으로 이동하고 j-hat을 1, 2로 이동합니다.", "model": "google_nmt", "from_community_srt": "i hat 과 j hat는 기저 벡터 (basis vector)라고 부른다. 우리의 표준 좌표계에서는 말이다.", @@ -90,7 +90,7 @@ "end": 91.04 }, { - "input": "So it's represented with a matrix whose columns are 3, 0, and 1, 2.", + "input": "ed up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actual", "translatedText": "따라서 열이 3, 0, 1, 2인 행렬로 표현됩니다.", "model": "google_nmt", "from_community_srt": "여기에 대해 이야기하고 싶은 것은 다른 기저 벡터의 집합을 사용하는 아이디어이다.", @@ -99,7 +99,7 @@ "end": 95.64 }, { - "input": "Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip.", + "input": "ly scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called th", "translatedText": "하나의 특정 벡터에 어떤 역할을 하는지에 집중하고 해당 벡터의 범위, 원점과 끝을 통과하는 선에 대해 생각해 보세요.", "model": "google_nmt", "from_community_srt": "예를 들어, 당신이 친구, 제니퍼가 있다고 가정 해 보자 그녀는 b1과 b2라고 불리는 다른 집합의 기저 벡터를 사용한다.", @@ -108,7 +108,7 @@ "end": 104.16 }, { - "input": "Most vectors are going to get knocked off their span during the transformation.", + "input": "e basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a", "translatedText": "대부분의 벡터는 변환 중에 해당 범위를 벗어나게 됩니다.", "model": "google_nmt", "from_community_srt": "그녀의 첫번째 기저 벡터 b1은 약간 오른쪽 위를 가르키고 그녀의 두 번째 기저 벡터 b2는 왼쪽 위를 가르킨다.", @@ -126,7 +126,7 @@ "end": 115.32 }, { - "input": "But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar.", + "input": "let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up.", "translatedText": "그러나 일부 특수 벡터는 자체 범위에 남아 있습니다. 즉, 행렬이 그러한 벡터에 미치는 영향은 스칼라처럼 단순히 늘리거나 찌그러뜨리는 것입니다.", "model": "google_nmt", "from_community_srt": "당신과 내가 묘사하고 싶은 [3, 2]는 우리가 사용했던 기저 벡터 i hat과 j hat을 사용해서 나타냈다. 제니퍼는 이 벡터를 기술 할 것 좌표 [5/3, 1/3]라고 기술할 것이다.", @@ -135,7 +135,7 @@ "end": 127.04 }, { - "input": "For this specific example, the basis vector i-hat is one such special vector.", + "input": "Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vecto", "translatedText": "이 특정 예에서 기본 벡터 i-hat은 그러한 특수 벡터 중 하나입니다.", "model": "google_nmt", "from_community_srt": "이것이 의미하는 것은 벡터를 얻는 특정한 방법, 즉 그녀의 2개의 기저벡터를 사용하면, b1에서 5/3,", @@ -171,7 +171,7 @@ "end": 164.04 }, { - "input": "It ends up getting stretched by a factor of 2.", + "input": "scale b1 by 5 thirds, scale b2 by 1 third, then add them both togethe", "translatedText": "결국 2배로 늘어나게 됩니다.", "model": "google_nmt", "from_community_srt": "여기서,", @@ -180,7 +180,7 @@ "end": 167.14 }, { - "input": "And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2.", + "input": "r. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she", "translatedText": "그리고 다시 선형성은 이 사람이 가로지르는 대각선의 다른 벡터가 2배만큼 늘어나게 된다는 것을 의미합니다.", "model": "google_nmt", "from_community_srt": "설정에 대해 좀 더 정확하게하려면 그녀의 첫 기저 벡터 b1는 우리의 좌표계에서는 [2, 1]s로 표현된다. (역자 주: 표준좌표계는 [x,y]s로 표기) 그리고 그녀의 두번째 기저 벡터 b2는 우리의 좌표계에서 [-1,", @@ -189,7 +189,7 @@ "end": 178.22 }, { - "input": "And for this transformation, those are all the vectors with this special property of staying on their span.", + "input": "thinks of her first coordinate as scali For this specific example, the basis vector i-hat is one such special vector. The span of", "translatedText": "그리고 이 변환의 경우, 그것들은 범위를 유지하는 특별한 속성을 가진 모든 벡터입니다.", "model": "google_nmt", "from_community_srt": "1]s로 표현된다. 하지만 그녀의 시스템에서 그녀의 직관으로부터 이해하는 것은 중요합니다. 이 벡터의 좌표가 [1, 0]j 이고, [0, 1]j 이라는 것 말이죠.", @@ -198,7 +198,7 @@ "end": 185.18 }, { - "input": "Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2.", + "input": "i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis. What's more, because of the way linear transformations work,", "translatedText": "x축에 있는 것들은 3배로 늘어나고, 이 대각선에 있는 것들은 2배로 늘어납니다.", "model": "google_nmt", "from_community_srt": "(역자 주: 제니퍼좌표계는 [x,y]j로 표기) 그 좌표들은 그녀의 세상에서는 [0, 1]j이고 [1, 0]j입니다.", @@ -216,7 +216,7 @@ "end": 198.08 }, { - "input": "As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation.", + "input": "n. A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc t, a way to visualize our coordinate system, and so it depends on our choice of basis", "translatedText": "지금쯤 추측할 수 있듯이 이러한 특수 벡터를 변환의 고유 벡터라고 하며 각 고유 벡터는 고유값이라고 불리는 것과 연관되어 있습니다. 이는 변환 중에 늘어나거나 찌그러지는 요소일 뿐입니다.", "model": "google_nmt", "from_community_srt": "공간안에서 같은 벡터를 보고있지만 제니퍼는 그것을 설명하기 위해 다른 단어와 숫자를 사용한다. 내가 여기서 설명하고 있는 몇가지 단어들을 빠르게 설명하고 지나가도록 하자. 나는 2 차원 공간의 애니메이션을 그릴 때, 나는 일반적으로이 사각형 격자를 사용한다. 하지만, 이 그리드는 그냥 구조물일 뿐이다. 우리의 좌표 시스템을 시각화하는 방법으로써, 그리고 그리드는 우리가 선택하는 기저에 따라 달라집니다.", @@ -234,7 +234,7 @@ "end": 225.94 }, { - "input": "In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half.", + "input": "nt as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It", "translatedText": "또 다른 예로, 고유값이 1/2인 고유벡터가 있을 수 있습니다. 이는 벡터가 1/2만큼 뒤집히고 찌그러진다는 의미입니다.", "model": "google_nmt", "from_community_srt": "이건 그녀의 좌표의 의미를 이해하는데에 도움이 되지만요. 하지만, 그녀의 원점은 실제로 우리와 같을 것이다.", @@ -261,7 +261,7 @@ "end": 249.8 }, { - "input": "If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation.", + "input": "of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewhat during the transformation, k", "translatedText": "해당 회전에 대한 고유벡터(자체 범위에 남아 있는 벡터)를 찾을 수 있다면 회전축을 찾은 것입니다.", "model": "google_nmt", "from_community_srt": "그래서, 이 모든 설정 후 자연스럽게 나오는 질문 중 하나는 우리는 어떻게 다른 좌표계 사이를 해석 해야합니까? 일겁니다. 만일, 예를 들어,", @@ -270,7 +270,7 @@ "end": 260.5 }, { - "input": "And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation.", + "input": "nocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", "translatedText": "그리고 해당 변환과 관련된 전체 3x3 행렬에 대해 생각하는 것보다 일부 회전 축과 회전 각도 측면에서 3D 회전을 생각하는 것이 훨씬 쉽습니다.", "model": "google_nmt", "from_community_srt": "제니퍼는 벡터를 설명하는 경우 좌표 값 [-1, 2]j를 이용하는데 그것은 우리 좌표계에서는 어떻게 될 것인가? 당신은 어떻게 그녀의 언어를 우리의 언어로 해석할 수 있을까? 글쎄, 우리의 좌표가 말하는 것은 이 벡터가 b1에 -1을 곱하고,", @@ -288,7 +288,7 @@ "end": 285.86 }, { - "input": "This pattern shows up a lot in linear algebra.", + "input": "In fact, once you understand matrix vector multiplication as applying", "translatedText": "이 패턴은 선형대수학에서 많이 나타납니다.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 290.02 }, { - "input": "With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors.", + "input": "a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvecto", "translatedText": "행렬로 설명되는 모든 선형 변환을 사용하면 이 행렬의 열을 기저 벡터의 착지 지점으로 읽어서 무엇을 하는지 이해할 수 있습니다.", "model": "google_nmt", "from_community_srt": "1]s의 좌표를 갖는다 그래서 우리는 실제 -1*b1 + 2*b2를 계산할 수 있다. 우리의 좌표 시스템에 표시하고 있는 벡터를 사용해서 말이죠. 그리고 이렇게 하면 당신은 좌표 [-4,", @@ -305,7 +305,7 @@ "end": 299.4 }, { - "input": "But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", + "input": "r with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformati", "translatedText": "그러나 특정 좌표계에 덜 의존하면서 선형 변환이 실제로 수행하는 작업의 핵심을 파악하는 더 좋은 방법은 고유벡터와 고유값을 찾는 것입니다.", "model": "google_nmt", "from_community_srt": "1]s에 있는 벡터를 얻을 수 있다. 그래서, 이것이 그녀가 생각하는 [-1, 2]j를 우리가 표현하는 방법이다. 여기서, 그녀의 각 기저벡터의 스케일[b1,", @@ -323,7 +323,7 @@ "end": 326.02 }, { - "input": "Symbolically, here's what the idea of an eigenvector looks like.", + "input": "she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we thi", "translatedText": "상징적으로 고유벡터의 아이디어는 다음과 같습니다.", "model": "google_nmt", "from_community_srt": "당신이 한번 (행렬 * 벡터) 곱셈을 임의의 선형 변환을 가하는 것을써 이해하고 있다면, 즉,", @@ -341,7 +341,7 @@ "end": 339.74 }, { - "input": "What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda.", + "input": "or for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotat", "translatedText": "이 표현식이 말하는 것은 행렬-벡터 곱 A 곱하기 v가 고유벡터 v를 일부 값 람다로 스케일링하는 것과 동일한 결과를 제공한다는 것입니다.", "model": "google_nmt", "from_community_srt": "제니퍼의 기저 벡터를 나타내는 열을 갖고 있는 행렬은 선형 변환으로 간주 될 수있다 우리의 기저 벡터 i hat 과 j hat을 움직이는 변환으로 말이다. 우리가 말하는 [1,", @@ -368,7 +368,7 @@ "end": 370.54 }, { - "input": "So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda.", + "input": "never stretch or squish anything, so the length of the vector would remain the same. This pattern shows up a lot in linear algebra. With any linear transformation described by a matrix, you could understand what it's doing by reading of", "translatedText": "그럼 우변을 일종의 행렬-벡터 곱셈으로 다시 작성하는 것부터 시작하겠습니다. 행렬을 사용하면 모든 벡터를 람다 배율로 스케일링하는 효과가 있습니다.", "model": "google_nmt", "from_community_srt": "선형 변환 전에 우리는이 벡터를 우리 기준의 특정 선형 조합으로서 벡터 -1*i hat + 2 j hat로 나타낼 수 있다.", @@ -377,7 +377,7 @@ "end": 380.62 }, { - "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else.", + "input": "f the columns of this matrix as the landing spots for basis vectors. But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", "translatedText": "그러한 행렬의 열은 각 기저 벡터에 어떤 일이 일어나는지 나타내며, 각 기저 벡터는 단순히 람다와 곱해집니다. 따라서 이 행렬의 대각선 아래 숫자는 람다이고 다른 곳은 모두 0입니다.", "model": "google_nmt", "from_community_srt": "선형 변환의 주요 기능은 그 결과가 다른 기저를 이용하여도 동일한 선형 결합 벡터가 된다는 점입니다. i hat이 있던 장소에서 -1을 곱하고, j hat이 있던 장소에서 2를 곱함으로써 말이죠.", @@ -386,7 +386,7 @@ "end": 394.32 }, { - "input": "The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal.", + "input": "we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector w", "translatedText": "이 함수를 작성하는 일반적인 방법은 람다를 인수분해하여 람다 곱하기 i로 작성하는 것입니다. 여기서 i는 대각선 아래에 1이 있는 단위 행렬입니다.", "model": "google_nmt", "from_community_srt": "그래서 이 행렬이하는 일은 우리의 오해를 제니퍼가 의미하는 바로 변환시키는 것입니다. 그녀가 생각하는 실제의 벡터가 있는 곳으로 변환시키면서 말이죠.", @@ -404,7 +404,7 @@ "end": 411.86 }, { - "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector.", + "input": "that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, with v as the eigenvector", "translatedText": "이제 우리가 가진 것은 새로운 행렬 A - 람다 곱하기 항등식입니다. 그리고 우리는 이 새로운 행렬 곱하기 v가 0 벡터를 제공하는 벡터 v를 찾고 있습니다.", "model": "google_nmt", "from_community_srt": "그러나 수치학적으로, 그녀의 언어에서 우리의 언어로 번역하는 것이죠. 나를 마침내 두드린 것은 제니퍼가 말하는 것의 우리의 오해를 어떻게 다루는지 생각하는 것이다.", @@ -430,7 +430,7 @@ "end": 433.64 }, { - "input": "And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", + "input": "and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Je", "translatedText": "그리고 5장과 6장을 보면 0이 아닌 벡터를 가진 행렬의 곱이 0이 되는 유일한 방법은 해당 행렬과 관련된 변환이 공간을 더 낮은 차원으로 압축하는 것이라는 것을 알게 될 것입니다.", "model": "google_nmt", "from_community_srt": "다른 방법으로 생각하는 건 어떨까요? 내가 이전에 사용했던 비디오를 예로 들어서 우리의 좌표계에서 [3, 2]s를 가질 때 어떻게 나는 제니퍼의 좌표계에서는 [5/3,", @@ -466,7 +466,7 @@ "end": 470.28 }, { - "input": "As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes.", + "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose c", "translatedText": "람다 값이 변경되면 행렬 자체도 변경되므로 행렬의 행렬식도 변경됩니다.", "model": "google_nmt", "from_community_srt": "실제로, 특히 2차원보다 큰 곳에서 생각할 때 이 역행렬을 구하려면 컴퓨터를 사용하는 것이 좋다.", @@ -511,7 +511,7 @@ "end": 498.6 }, { - "input": "So this is kind of a lot, but let's unravel what this is saying.", + "input": "And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's importa", "translatedText": "내용이 좀 많지만 이것이 무엇을 말하는지 풀어보겠습니다.", "model": "google_nmt", "from_community_srt": "2]s 에 곱해야한다.", @@ -520,7 +520,7 @@ "end": 502.96 }, { - "input": "When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line.", + "input": "nt that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication So let's start by rewriting", "translatedText": "람다가 1이면 행렬 A에서 람다를 곱하고 항등식을 곱하여 공간을 선으로 압축합니다.", "model": "google_nmt", "from_community_srt": "그러면 [5/3, 1/3]j이 나온다. 그래서,", @@ -529,7 +529,7 @@ "end": 509.56 }, { - "input": "That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector.", + "input": "that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a fact", "translatedText": "이는 A 마이너스 람다 곱하기 항등 시간 v가 0 벡터와 같은 0이 아닌 벡터 v가 있다는 것을 의미합니다.", "model": "google_nmt", "from_community_srt": "간단히 말해서 이것이 개개의 벡터의 모습을 변환하는 방법입니다. 좌표계 사이를 왔다 갔다 하면서요. 제니퍼의 기저 벡터를 가지고 있는 행렬은, 우리의 좌표계로 나타내어 있지만,", @@ -538,7 +538,7 @@ "end": 518.56 }, { - "input": "And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A.", + "input": "or of lambda. The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else. the columns of our matrix.", "translatedText": "그리고 우리가 그것에 관심을 갖는 이유는 A 곱하기 v가 람다 곱하기 v와 같다는 것을 의미하기 때문이라는 것을 기억하세요. 이는 벡터 v가 A의 고유 벡터이며 변환 A 동안 자체 범위에 머무르는 것으로 읽을 수 있습니다.", "model": "google_nmt", "from_community_srt": "그녀의 언어에서 우리의 언어로 벡터를 변환시켜 줍니다. 그리고 역행렬은 반대로 작용한다. 그러나 벡터는 좌표계를 이용해서만 묘사되는 것이 아니다. 이 다음부터는 말이죠.", @@ -547,7 +547,7 @@ "end": 537.28 }, { - "input": "In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place.", + "input": "But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following", "translatedText": "이 예에서 해당 고유값은 1이므로 v는 실제로 고정된 상태로 유지됩니다.", "model": "google_nmt", "from_community_srt": "변환을 행렬로 생각하는 방법은 중요합니다. 그리고 행렬 곱이 어떻게 연속적인 변환과 연관되는지 알게 될 것입니다.", @@ -565,7 +565,7 @@ "end": 549.5 }, { - "input": "This is the kind of thing I mentioned in the introduction.", + "input": "-hat in the first pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v.", "translatedText": "서문에서 언급한 내용이 바로 이런 내용입니다.", "model": "google_nmt", "from_community_srt": "만약 불안하다면요. 일부 선형 변환을 고려해보죠.", @@ -574,7 +574,7 @@ "end": 555.64 }, { - "input": "If you didn't have a solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to describe those landing spots in her language. Here's a common way to think", "translatedText": "행렬식을 확실히 이해하지 못하고 왜 행렬식이 0이 아닌 해를 갖는 선형 방정식 시스템과 관련되어 있는지 알지 못한다면 이와 같은 표현은 전혀 예상치 못한 일처럼 느껴질 것입니다.", "model": "google_nmt", "from_community_srt": "반 시계 방향으로 90 ° 회전된것 처럼요. 당신과 나는 이 행렬이 표현하는 것은 기저 벡터 i hat 과 j hat이 각각 어디로 가야하는지를 나타냅니다. i hat은 [0,", @@ -619,7 +619,7 @@ "end": 608.84 }, { - "input": "To figure out what the eigenvectors are that actually have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero.", + "input": "To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry. Now imagine tweaking lambda, turning a knob to change its value. As that value of lambda changes, the matrix itself change", "translatedText": "실제로 이러한 고유값 중 하나(예: 람다가 2)를 갖는 고유벡터가 무엇인지 알아내기 위해 해당 람다 값을 행렬에 연결한 다음 대각선으로 변경된 행렬이 0으로 보내는 벡터를 해결합니다.", "model": "google_nmt", "from_community_srt": "그 행렬의 각각의 열 벡터는 우리의 기저벡터 i hat 과 j hat이 어떻게 가는지를 나타낼 뿐이고, 하지만 제니퍼가 원하는 행렬은 그녀의 기저 벡터가 어디로 가야하는지를 나타내야만 한다. 그리고 그 도착 지점 또한 그녀의 언어로 표시해야만 하죠. 다음은, 어떻게 이것이 실행되는지 가장 일반적인 방법을 설명한다.", @@ -709,7 +709,7 @@ "end": 681.94 }, { - "input": "The only roots of that polynomial are the imaginary numbers, i and negative i.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the oute", "translatedText": "해당 다항식의 유일한 근은 허수 i와 음수 i입니다.", "model": "google_nmt", "from_community_srt": "그녀의 언어로 이루어진 벡터를 뱉습니다. 이 구체적인 예를 들어 우리의 언어에서 [2,", @@ -736,7 +736,7 @@ "end": 699.82 }, { - "input": "This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1.", + "input": "meone else sees it. For those of you wondering why we care about alternate coordinate systems, the next vi", "translatedText": "그러면 i-hat이 제자리에 고정되고 j-hat 1이 위로 이동하므로 해당 행렬에는 열 1, 0과 1, 1이 있습니다.", "model": "google_nmt", "from_community_srt": "5/3] 및 [-2/3, -1/3]가 됩니다.", @@ -772,7 +772,7 @@ "end": 726.54 }, { - "input": "And the only root of this expression is lambda equals 1.", + "input": "he identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals la", "translatedText": "그리고 이 표현식의 유일한 근은 람다가 1과 같다는 것입니다.", "model": "google_nmt", "from_community_srt": "중간의 행렬 M 은 선형 변환을 나타내고, 그리고 바깥쪽 행렬 A와 A^(-1)은 관점의 변환을 나타낸다.", @@ -781,7 +781,7 @@ "end": 732.86 }, { - "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.", + "input": "mbda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corresponding eigenvalue is", "translatedText": "이는 우리가 기하학적으로 보는 것과 일치합니다. 모든 고유벡터는 고유값 1을 갖습니다.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 796.38 }, { - "input": "For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2.", + "input": "f equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "예를 들어, i-hat은 -1로 스케일링되고 j-hat은 2로 스케일링될 수 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 825.42 }, { - "input": "And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.", + "input": "nd compute the determinant. Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda. Since lambda can only be an eigenvalue i", "translatedText": "그리고 이것을 해석하는 방법은 모든 기본 벡터가 고유 벡터이고 이 행렬의 대각선 항목이 고유값이라는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 841.06 }, { - "input": "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", + "input": "u can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3. To figure out what the eigenvectors are that actu", "translatedText": "한 가지 큰 점은 이 행렬 자체를 여러 번 곱하면 어떤 일이 일어날지 계산하는 것이 더 쉽다는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 848.34 }, { - "input": "Since all one of these matrices does is scale each basis vector by some eigenvalue, applying that matrix many times, say 100 times, is just going to correspond to scaling each basis vector by the 100th power of the corresponding eigenvalue.", + "input": "ally have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero. If you computed this the way you would any other linear system, you'd see that the solutions are all the vectors on the diagonal line spanned by negative 1, 1.", "translatedText": "이러한 행렬 중 하나는 각 기본 벡터를 일부 고유값만큼 스케일링하므로 해당 행렬을 여러 번 적용하는 것(가령 100번)은 각 기본 벡터를 해당 고유값의 100승으로 스케일링하는 것과 같습니다.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 869.68 }, { - "input": "Really, try it for a moment.", + "input": "x, 3, 0, 1, 2, has the effect of stretching all those vectors by a factor of 2.", "translatedText": "정말로, 한번 시도해 보세요.", "model": "google_nmt", "n_reviews": 0, @@ -936,7 +936,7 @@ "end": 896.54 }, { - "input": "I talked about change of basis last video, but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", + "input": "is, notice what happens. Its matrix has columns 0, 1 and negative 1, 0. Subtract off lambda from the diagonal elements and look for when the determinant is zero. In this case, you get the polynomial lambda squared plus 1. The only roots of that polynomia", "translatedText": "지난 영상에서 기저 변경에 대해 이야기했지만, 여기서는 현재 좌표계에 쓰여진 변환을 다른 시스템으로 표현하는 방법에 대해 매우 빠르게 설명하겠습니다.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 907.04 }, { - "input": "Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors, then make those coordinates the columns of a matrix, known as the change of basis matrix.", + "input": "l are the imaginary numbers, i and negative i. The fact that there are no real number solutions indicates that there are no eigenvectors. Another pretty interesting example worth holding in the back of your mind is a shear. This fixes i-hat in place and moves j-hat 1 over, so its mat", "translatedText": "새 기저로 사용하려는 벡터의 좌표(이 경우에는 두 개의 고유 벡터를 의미)를 선택한 다음 해당 좌표를 기저 행렬의 변경이라고 알려진 행렬의 열로 만듭니다.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 946.68 }, { - "input": "This is because it represents working in a coordinate system where what happens to the basis vectors is that they get scaled during the transformation.", + "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1. Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a lin", "translatedText": "이는 기본 벡터에 발생하는 일이 변환 중에 크기가 조정되는 좌표계에서의 작업을 나타내기 때문입니다.", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 961.56 }, { - "input": "So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis, compute the 100th power in that system, then convert back to our standard system.", + "input": "A simple example is a matrix that scales everything by 2. The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue. Now is another good time to pause and ponder some of this before I move on to the last topic.", "translatedText": "따라서 예를 들어 이 행렬의 100제곱을 계산해야 하는 경우 고유기저로 변경하고 해당 시스템에서 100제곱을 계산한 다음 표준 시스템으로 다시 변환하는 것이 훨씬 쉬울 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 975.68 }, { - "input": "You can't do this with all transformations.", + "input": "I want to finish off here with the idea of an eigenbasis, which relies heavily on ideas from the last video. Take a look at what h", "translatedText": "모든 변환에 대해 이 작업을 수행할 수는 없습니다.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 978.32 }, { - "input": "A shear, for example, doesn't have enough eigenvectors to span the full space.", + "input": "appens if our basis vectors just so happen to be eigenvectors. For example, maybe i-hat is scale", "translatedText": "예를 들어 전단에는 전체 공간을 포괄할 만큼 고유벡터가 충분하지 않습니다.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 982.98 }, { - "input": "But if you can find an eigenbasis, it makes matrix operations really lovely.", + "input": "d by negative 1 and j-hat is scaled by 2. Writing their new coordinates as the columns of a matrix, notice t", "translatedText": "그러나 고유기저를 찾을 수 있다면 행렬 연산이 정말 멋질 것입니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/marathi/sentence_translations.json b/2016/change-of-basis/marathi/sentence_translations.json index 02201fd6c..727b50806 100644 --- a/2016/change-of-basis/marathi/sentence_translations.json +++ b/2016/change-of-basis/marathi/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates.", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we doing this and what d", "translatedText": "जर माझ्याकडे येथे 2D जागेत वेक्टर बसला असेल, तर आमच्याकडे निर्देशांकांसह त्याचे वर्णन करण्याचा एक मानक मार्ग आहे.", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 28.28 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up.", + "input": "oes this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin", "translatedText": "या प्रकरणात, वेक्टरमध्ये 3, 2 समन्वय असतात, याचा अर्थ त्याच्या शेपटीपासून त्याच्या टोकापर्यंत जाणे म्हणजे तीन युनिट्स उजवीकडे आणि दोन युनिट्स वर हलवणे.", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.96 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up.", + "input": "ch that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of", "translatedText": "तुम्ही त्या पहिल्या समन्वयाचा विचार करा i-hat स्केलिंग करा, लांबी 1 सह वेक्टर उजवीकडे निर्देशित करतो, तर दुसरा समन्वय j-हॅट स्केल करतो, लांबी 1 सह वेक्टर सरळ वर निर्देशित करतो.", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 60.48 }, { - "input": "You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordinate system.", + "input": "you know how to think about matrices as linear transformations, but you also need to be comfortable with things like determinants, linear systems", "translatedText": "आपण या दोन विशेष सदिशांचा विचार करू शकता की ते आमच्या समन्वय प्रणालीच्या सर्व अंतर्निहित गृहितकांना अंतर्भूत करतात.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 81.38 }, { - "input": "Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system.", + "input": "some linear transformation in two dimensions, like the one shown here. st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice o", "translatedText": "सदिश आणि संख्यांच्या संचामध्ये भाषांतर करण्याच्या कोणत्याही मार्गाला समन्वय प्रणाली म्हणतात आणि दोन विशेष वेक्टर i-hat आणि j-hat यांना आमच्या मानक समन्वय प्रणालीचे आधारभूत वेक्टर म्हणतात.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors.", + "input": "f i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors an", "translatedText": "मी येथे ज्याबद्दल बोलू इच्छितो ती म्हणजे भिन्न आधार वेक्टर वापरण्याची कल्पना.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.04 }, { - "input": "Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third.", + "input": "nnifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to th", "translatedText": "जेनिफर या वेक्टरचे वर्णन 5 तृतीयांश आणि 1 तृतीयांश समन्वयांसह करेल.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 125.16 }, { - "input": "In a little bit, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third.", + "input": "showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describe this vector with the coordinates 5 thirds and", "translatedText": "थोड्या वेळाने, मी तुम्हाला ते दोन संख्या, 5 तृतीयांश आणि 1 तृतीयांश कसे काढू शकले असते ते दाखवतो.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 144.12 }, { - "input": "What she gets will typically be completely different from the vector that you and I would think of as having those coordinates.", + "input": "gether. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to", "translatedText": "तिला जे मिळते ते सामान्यत: त्या वेक्टरपेक्षा पूर्णपणे भिन्न असेल ज्याचा तुम्ही आणि मी ते समन्वयक म्हणून विचार करू.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 167.14 }, { - "input": "So in effect, we're speaking different languages.", + "input": "lumn of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis.", "translatedText": "त्यामुळे प्रत्यक्षात, आम्ही वेगवेगळ्या भाषा बोलत आहोत.", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 176.84 }, { - "input": "Let me say a quick word about how I'm representing things here.", + "input": "ector on the x-axis is also just stretched by a factor of 3, and hence remains on its own", "translatedText": "मी येथे गोष्टींचे प्रतिनिधित्व कसे करत आहे याबद्दल मला एक द्रुत शब्द सांगू दे.", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid.", + "input": "span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here.", "translatedText": "स्पेसमध्ये स्वतःला कोणतीही आंतरिक ग्रिड नसते.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 197.6 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean.", + "input": "system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct mea", "translatedText": "तिची उत्पत्ती असली तरी ती प्रत्यक्षात आपल्याशी जुळते, कारण ०.० चा अर्थ काय असावा यावर सर्वजण सहमत आहेत.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 214.9 }, { - "input": "But the direction of her axes and the spacing of her grid lines will be different, depending on her choice of basis vectors.", + "input": "ollow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector", "translatedText": "परंतु तिच्या अक्षांची दिशा आणि तिच्या ग्रीड रेषांचे अंतर भिन्न असेल, तिच्या आधारभूत वेक्टरच्या निवडीनुसार.", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 225.94 }, { - "input": "If for example Jennifer describes a vector with coordinates negative 1, 2, what would that be in our coordinate system?", + "input": "on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewh", "translatedText": "उदाहरणार्थ, जेनिफरने ऋण 1, 2 समन्वयांसह वेक्टरचे वर्णन केले, तर ते आपल्या समन्वय प्रणालीमध्ये काय असेल?", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 238.98 }, { - "input": "How do you translate from her language to ours?", + "input": "at during the transformation, knocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis", "translatedText": "तुम्ही तिच्या भाषेतून आमच्या भाषेत भाषांतर कसे करता?", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 260.5 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1.", + "input": "ing them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vector", "translatedText": "आणि आमच्या दृष्टीकोनातून, b1 चे समन्वय 2, 1 आहेत आणि b2 चे समन्वय ऋण 1, 1 आहेत.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 267.04 }, { - "input": "So we can actually compute negative 1 times b1 plus 2 times b2 as they're represented in our coordinate system.", + "input": "s in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio Of course, there's n", "translatedText": "त्यामुळे आपण 1 गुणिले b1 अधिक 2 वेळा b2 ची गणना करू शकतो कारण ते आपल्या समन्वय प्रणालीमध्ये दर्शवले जातात.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 270.4 }, { - "input": "And working this out, you get a vector with coordinates negative 4, 1.", + "input": "othing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In", "translatedText": "आणि हे समजून घेतल्यास, तुम्हाला ऋण 4, 1 समन्वयांसह एक वेक्टर मिळेल.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 274.74 }, { - "input": "This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", + "input": "alf, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennif", "translatedText": "काही वेक्टरच्या संबंधित निर्देशांकांद्वारे तिच्या प्रत्येक आधारभूत व्हेक्टरचे मोजमाप करण्याची आणि नंतर त्यांना एकत्र जोडण्याची ही प्रक्रिया काहीशी परिचित वाटू शकते.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 330.48 }, { - "input": "To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation.", + "input": "envector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking a", "translatedText": "हे कसे कार्य करते हे दाखवण्यासाठी, ऋण 1, 2 सह समन्वयक असण्याचा आणि ते परिवर्तन लागू करण्याचा आपण विचार करत असलेल्या वेक्टरचा काय अर्थ होतो ते पाहू या.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 341.38 }, { - "input": "Before the linear transformation, we're thinking of this vector as a certain linear combination of our basis vectors, negative 1 times i-hat plus 2 times j-hat.", + "input": "bout the full 3x3 matrix associated with that transformation. In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length", "translatedText": "रेखीय परिवर्तनापूर्वी, आम्ही या सदिशाचा आमच्या आधारभूत व्हेक्टरचे एक विशिष्ट रेखीय संयोजन म्हणून विचार करत आहोत, ऋण 1 पट i-hat अधिक 2 पट j-hat.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 375.16 }, { - "input": "Geometrically, this matrix transforms our grid into Jennifer's grid but numerically, it's translating a vector described in her language to our language.", + "input": "t the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. we get using", "translatedText": "भौमितिकदृष्ट्या, हे मॅट्रिक्स आपल्या ग्रिडचे जेनिफरच्या ग्रिडमध्ये रूपांतर करते परंतु संख्यात्मकदृष्ट्या, ते तिच्या भाषेत वर्णन केलेल्या वेक्टरचे आपल्या भाषेत भाषांतर करत आहे.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 380.62 }, { - "input": "What made it finally click for me was thinking about how it takes our misconception of what Jennifer means, the vector we get using the same coordinates but in our system, then it transforms it into the vector that she really meant.", + "input": "the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I comp", "translatedText": "जेनिफर म्हणजे काय हे आपल्या चुकीच्या समजुतीचे कारण काय आहे याचा विचार करून शेवटी माझ्यासाठी क्लिक केले, व्हेक्टर आपल्याला समान निर्देशांक वापरून मिळतो परंतु आपल्या सिस्टममध्ये, नंतर ते व्हेक्टरमध्ये रूपांतरित होते जे तिला खरोखर म्हणायचे होते.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 398.26 }, { - "input": "What about going the other way around?", + "input": "ute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A", "translatedText": "दुसरीकडे जाण्याबद्दल काय?", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 404.26 }, { - "input": "In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 third in Jennifer's system?", + "input": "is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis", "translatedText": "मी या व्हिडिओच्या आधी वापरलेल्या उदाहरणामध्ये, जेव्हा माझ्याकडे आमच्या सिस्टममध्ये 3, 2 सह निर्देशांक असलेले वेक्टर होते, तेव्हा जेनिफरच्या सिस्टममध्ये 5 तृतीयांश आणि 1 तृतीयांश समन्वय असतील याची मी गणना कशी केली?", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 409.48 }, { - "input": "You start with that change of basis matrix that translates Jennifer's language into ours, then you take its inverse.", + "input": "as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we", "translatedText": "तुम्ही आधार मॅट्रिक्सच्या त्या बदलापासून सुरुवात करा जी जेनिफरची भाषा आमच्या भाषेत अनुवादित करते, नंतर तुम्ही तिचा उलट घ्या.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 415.48 }, { - "input": "Remember, the inverse of a transformation is a new transformation that corresponds to playing that first one backwards.", + "input": "multiply this inverse change of basis matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and fo", "translatedText": "लक्षात ठेवा, परिवर्तनाचा व्युत्क्रम हे एक नवीन परिवर्तन आहे जे त्या पहिल्याला मागे खेळण्याशी संबंधित आहे.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 427.94 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse.", + "input": "rth between coordinate systems. The matrix whose columns represent Jennif er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the invers", "translatedText": "व्यवहारात, विशेषत: जेव्हा तुम्ही दोन पेक्षा जास्त आयामांमध्ये काम करत असाल, तेव्हा तुम्ही मॅट्रिक्सची गणना करण्यासाठी संगणकाचा वापर कराल जे प्रत्यक्षात या व्युत्क्रमाचे प्रतिनिधित्व करतात.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 465.52 }, { - "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems.", + "input": "you know how matrix multiplication So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, usi", "translatedText": "जेणेकरून, थोडक्यात, समन्वय प्रणालींमध्ये वैयक्तिक वेक्टरचे वर्णन कसे भाषांतरित करायचे ते आहे.", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 487.24 }, { - "input": "And the inverse matrix does the opposite.", + "input": "The columns of such a matrix will represent what happens to eac", "translatedText": "आणि व्यस्त मॅट्रिक्स उलट करते.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 507.16 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy.", + "input": "heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla With", "translatedText": "जर काही अस्वस्थ वाटत असेल तर निश्चितपणे थांबा आणि अध्याय 3 आणि 4 पहा.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 529.74 }, { - "input": "i-hat ends up at the spot with coordinates 0, 1, and j-hat ends up at the spot with coordinates negative 1, 0.", + "input": "or out the v. So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to desc", "translatedText": "आय-हॅट स्पॉटवर कोऑर्डिनेट्स 0, 1 सह संपतो आणि j-हॅट स्पॉटवर कोऑर्डिनेट्स 1, 0 ने संपतो.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 566.3 }, { - "input": "But that's not quite right.", + "input": "And that squishification corresponds to a zero determinant for the matrix. To be concrete, let's say your matrix", "translatedText": "पण ते अगदी योग्य नाही.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.72 }, { - "input": "Here's a common way to think of how this is done.", + "input": "As that value of lambda changes, the matrix itself changes, and so the determina", "translatedText": "हे कसे केले जाते याचा विचार करण्याचा एक सामान्य मार्ग येथे आहे.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 603.42 }, { - "input": "Start with any vector written in Jennifer's language.", + "input": "nt of the matrix changes. ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multipl", "translatedText": "जेनिफरच्या भाषेत लिहिलेल्या कोणत्याही वेक्टरसह प्रारंभ करा.", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 649.44 }, { - "input": "Since we could do this with any vector written in her language, first applying the change of basis, then the transformation, then the inverse change of basis, that composition of three matrices gives us the transformation matrix in Jennifer's language.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective. And the full matrix product represents that same transformation, but as someone else sees it.", "translatedText": "आपण हे तिच्या भाषेत लिहिलेल्या कोणत्याही सदिशाने करू शकत असल्याने, प्रथम आधार बदल, नंतर परिवर्तन, नंतर आधाराचा व्यस्त बदल, तीन मॅट्रिक्सची रचना आपल्याला जेनिफरच्या भाषेत परिवर्तन मॅट्रिक्स देते.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 665.56 }, { - "input": "It takes in a vector of her language and spits out the transformed version of that vector in her language.", + "input": "For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of", "translatedText": "ती तिच्या भाषेतील एक वेक्टर घेते आणि त्या वेक्टरची रूपांतरित आवृत्ती तिच्या भाषेत बाहेर टाकते.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 675.8 }, { - "input": "For this specific example, when Jennifer's basis vectors look like 2, 1 and negative in our language, and when the transformation is a 90 degree rotation, the product of these three matrices, if you work through it, has columns one third, five thirds, and negative two thirds, negative one third.", + "input": "this. See y That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corres", "translatedText": "या विशिष्ट उदाहरणासाठी, जेव्हा जेनिफरचे आधारभूत व्हेक्टर आपल्या भाषेत 2, 1 आणि ऋणासारखे दिसतात आणि जेव्हा परिवर्तन हे 90 अंश फिरते तेव्हा, या तीन मॅट्रिक्सच्या गुणाकाराने, जर तुम्ही त्यावर कार्य केले तर, स्तंभ एक तृतीयांश, पाच तृतीयांश असतात. , आणि ऋण दोन तृतीयांश, ऋण एक तृतीयांश.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 692.2 }, { - "input": "So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system.", + "input": "ponding eigenvalue is 1, so v would actually just stay fixed in place. Pause and ponder if you need to make sure that that line of reasoning feels good. This is the kind of thing I mentioned in the introduction. If you didn't have a", "translatedText": "त्यामुळे जेनिफरने त्या मॅट्रिक्सला तिच्या सिस्टीममधील वेक्टरच्या निर्देशांकाने गुणाकार केल्यास, ती तिच्या समन्वय प्रणालीमध्ये व्यक्त केलेल्या वेक्टरची 90 अंश फिरवलेली आवृत्ती परत करेल.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 709.82 }, { - "input": "In general, whenever you see an expression like A inverse times M times A, it suggests a mathematical sort of empathy.", + "input": "solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "सर्वसाधारणपणे, जेव्हा तुम्ही ए व्युत्क्रम गुणा एम गुणिले अ सारखी अभिव्यक्ती पाहता तेव्हा ते एक गणितीय सहानुभूती सूचित करते.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 714.54 }, { - "input": "That middle matrix represents a transformation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective.", + "input": "To see this in action, let's revisit the example from the start, with a matrix whose columns are 3, 0 and 1, 2. To find if a value lambda is an eigenvalue, subtract it from the diago", "translatedText": "ते मधले मॅट्रिक्स हे काही प्रकारचे परिवर्तन दर्शवते जसे तुम्ही ते पाहता, आणि बाह्य दोन मॅट्रिक्स सहानुभूती, दृष्टीकोनातील बदलाचे प्रतिनिधित्व करतात.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/persian/sentence_translations.json b/2016/change-of-basis/persian/sentence_translations.json index 0d9977c86..51a9ca813 100644 --- a/2016/change-of-basis/persian/sentence_translations.json +++ b/2016/change-of-basis/persian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates. ", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we d ", "translatedText": "اگر من یک بردار اینجا در فضای دوبعدی نشسته داشته باشم، یک روش استاندارد برای توصیف آن با مختصات داریم. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 27.48 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up. ", + "input": "oing this and what does this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin ", "translatedText": "در این مورد، بردار دارای مختصات 3، 2 است، یعنی رفتن از دم به نوک آن شامل حرکت سه واحد به سمت راست و دو واحد به بالا است. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.36 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up. ", + "input": "not so much that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of ", "translatedText": "شما مختصات اول را به عنوان مقیاس i-hat در نظر می گیرید، بردار با طول 1 که به سمت راست اشاره می کند، در حالی که مختصات دوم j-hat را مقیاس می کند، بردار با طول 1 که مستقیماً به سمت بالا است. ", "model": "google_nmt", "n_reviews": 0, @@ -32,7 +32,7 @@ "end": 57.14 }, { - "input": "The tip-to-tail sum of those two scaled vectors is what the coordinates are meant to describe. ", + "input": "the topics that precede it. Most important here is that you know how to think about matrices as linear transformations, but you also need to be comforta ", "translatedText": "مجموع نوک به دم آن دو بردار مقیاس شده همان چیزی است که مختصات برای توصیف آن است. ", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors. ", + "input": "of that is tied up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors a ", "translatedText": "چیزی که می خواهم در اینجا درباره آن صحبت کنم، ایده استفاده از مجموعه متفاوتی از بردارهای پایه است. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 133.36 }, { - "input": "In general, whenever Jennifer uses coordinates to describe a vector, she thinks of her first coordinate as scaling b1, the second coordinate as scaling b2, and she adds the results. ", + "input": "vector, b2, points left and up. Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describ ", "translatedText": "به طور کلی، هر زمان که جنیفر از مختصات برای توصیف یک بردار استفاده می کند، اولین مختصات خود را مقیاس بندی b1، مختصات دوم را مقیاس بندی b2 در نظر می گیرد و نتایج را اضافه می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 166.8 }, { - "input": "They are what define the meaning of the coordinates 1,0 and 0,1 in her world. ", + "input": "then add them both together. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. ", "translatedText": "آنها همان چیزی هستند که معنای مختصات 1،0 و 0،1 را در دنیای او تعریف می کنند. ", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 185.86 }, { - "input": "But that grid is just a construct, a way to visualize our coordinate system, and so it depends on our choice of basis. ", + "input": "ample, the basis vector i-hat is one such special vector. The span of i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 ", "translatedText": "اما این شبکه فقط یک ساختار است، راهی برای تجسم سیستم مختصات ما، و بنابراین بستگی به انتخاب مبنای ما دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid. ", + "input": "times itself, still on that x-axis. What's more, because of the way linear transformations work, ", "translatedText": "خود فضا شبکه ذاتی ندارد. ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 198.08 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. ", + "input": "emains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. ", "translatedText": "اگرچه منشأ او در واقع با ما مطابقت دارد، زیرا همه در مورد معنای مختصات 0،0 توافق دارند. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 223.72 }, { - "input": "So after all this is set up, a pretty natural question to ask is how we translate between coordinate systems. ", + "input": "self has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct meant as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her ori ", "translatedText": "بنابراین پس از تنظیم همه اینها، یک سوال کاملاً طبیعی برای پرسیدن این است که چگونه بین سیستم های مختصات ترجمه می کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 227.28 }, { - "input": "How do you translate from her language to ours? ", + "input": "hould mean. It's the thing that you get when you scale any vector by zero. ", "translatedText": "چگونه از زبان او به زبان ما ترجمه می کنید؟ ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 230.76 }, { - "input": "Well, what her coordinates are saying is that this vector is negative 1 times b1 plus 2 times b2. ", + "input": "But the direction of her axes and Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. ", "translatedText": "خوب، آنچه مختصات او می گوید این است که این بردار منفی 1 برابر b1 به اضافه 2 برابر b2 است. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 231.76 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1. ", + "input": "Any other vector is going to get rotated somewhat during the transformation, knocked off the line that it spans. ks of as negative 1, 2. ", "translatedText": "و از دیدگاه ما، b1 دارای مختصات 2، 1، و b2 دارای مختصات منفی 1، 1 است. ", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 254.28 }, { - "input": "It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vectors in our language. ", + "input": "tand matrix vector multiplication as applying a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or ", "translatedText": "این ضرب بردار ماتریسی است، با ماتریسی که ستون‌های آن بردارهای پایه جنیفر را در زبان ما نشان می‌دهند. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 257.08 }, { - "input": "In fact, once you understand matrix vector multiplication as applying a certain linear transformation, say by watching what I view to be the most important video in this series, Chapter 3, there's a pretty intuitive way to think about what's going on here. ", + "input": "the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. ", "translatedText": "در واقع، وقتی ضرب بردار ماتریس را به‌عنوان اعمال یک تبدیل خطی خاص درک کردید، مثلاً با تماشای آنچه به نظر من مهم‌ترین ویدیوی این مجموعه، فصل 3 است، یک راه کاملاً بصری برای فکر کردن در مورد آنچه در اینجا می‌گذرد وجود دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 384.3 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse. ", + "input": "coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, ", "translatedText": "در عمل، به خصوص هنگامی که در بیش از دو بعد کار می کنید، از یک کامپیوتر برای محاسبه ماتریسی استفاده می کنید که در واقع این معکوس را نشان می دهد. ", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 389.6 }, { - "input": "In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. ", + "input": "with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. ", "translatedText": "در این حالت، معکوس ماتریس تغییر مبنا که پایه جنیفر را به عنوان ستون‌های خود دارد، به ستون‌های 1 سوم، منفی 1 سوم و 1 سوم، 2 سوم ختم می‌شود. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 443.9 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy. ", + "input": "es, and that you know how matrix multiplication So let's start by rewriting that right-hand ", "translatedText": "اگر هر یک از اینها احساس ناراحتی کرد، قطعاً مکث کنید و به فصل 3 و 4 نگاهی بیندازید. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 446.48 }, { - "input": "Consider some linear transformation, like a 90 degree counterclockwise rotation. ", + "input": "side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda. ", "translatedText": "مقداری تبدیل خطی را در نظر بگیرید، مانند یک چرخش 90 درجه در خلاف جهت عقربه‌های ساعت. ", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 451.06 }, { - "input": "When you and I represent this with a matrix, we follow where the basis vectors i-hat and j-hat each go. ", + "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this ", "translatedText": "وقتی من و شما این را با یک ماتریس نشان می‌دهیم، بردارهای پایه i-hat و j-hat را دنبال می‌کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 477.24 }, { - "input": "How would Jennifer describe this same 90 degree rotation of space? ", + "input": "pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v. ", "translatedText": "جنیفر همین چرخش 90 درجه ای فضا را چگونه توصیف می کند؟ ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 484.26 }, { - "input": "But that's not quite right. ", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and ", "translatedText": "اما این کاملا درست نیست. ", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 515.02 }, { - "input": "Then apply the transformation matrix to what you get by multiplying it on the left. ", + "input": "zero is if the transformation associated with that matrix squishes space into a lower dimension. And that squishification corresponds to a zero determinant for the matr ", "translatedText": "سپس با ضرب کردن آن در سمت چپ، ماتریس تبدیل را به آنچه به دست می آورید اعمال کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 515.62 }, { - "input": "This tells us where that vector lands, but still in our language. ", + "input": "ix. To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtract ", "translatedText": "این به ما می گوید که آن بردار کجا قرار می گیرد، اما همچنان در زبان ماست. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/polish/sentence_translations.json b/2016/change-of-basis/polish/sentence_translations.json index 46a108985..cd2a4ea8f 100644 --- a/2016/change-of-basis/polish/sentence_translations.json +++ b/2016/change-of-basis/polish/sentence_translations.json @@ -81,7 +81,7 @@ "end": 84.84 }, { - "input": "It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.", + "input": "st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is ti", "translatedText": "Przesuwa wektor bazowy i-hat do współrzędnych 3, 0, a j-hat do 1, 2.", "model": "google_nmt", "from_community_srt": "i-z-daszkiem i j-z-daszkiem nazywamy wektorami bazowymi z bazy standardowej.", @@ -90,7 +90,7 @@ "end": 91.04 }, { - "input": "So it's represented with a matrix whose columns are 3, 0, and 1, 2.", + "input": "ed up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actual", "translatedText": "Jest to więc reprezentowane przez macierz, której kolumny to 3, 0 i 1, 2.", "model": "google_nmt", "from_community_srt": "(w Polsce często nazywane e1 i e2) Chciałbym teraz powiedzieć trochę o koncepcie używania różnych wektorów bazowych.", @@ -99,7 +99,7 @@ "end": 95.64 }, { - "input": "Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip.", + "input": "ly scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called th", "translatedText": "Skoncentruj się na tym, co robi z jednym konkretnym wektorem i pomyśl o rozpiętości tego wektora, linii przechodzącej przez jego początek i wierzchołek.", "model": "google_nmt", "from_community_srt": "Na przykład powiedzmy, że mamy przyjaciółkę Jennifer która używa innych wektorów bazowych, które nazwę b1 i b2.", @@ -108,7 +108,7 @@ "end": 104.16 }, { - "input": "Most vectors are going to get knocked off their span during the transformation.", + "input": "e basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a", "translatedText": "Większość wektorów zostanie wyrzucona ze swojego zakresu podczas transformacji.", "model": "google_nmt", "from_community_srt": "Jej pierwszy wektor z bazy, b1,", @@ -126,7 +126,7 @@ "end": 115.32 }, { - "input": "But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar.", + "input": "let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up.", "translatedText": "Jednak niektóre wektory specjalne pozostają na swoim własnym obszarze, co oznacza, że macierz wywiera wpływ na taki wektor po prostu go rozciągając lub zgniatając, jak skalar.", "model": "google_nmt", "from_community_srt": "który opisaliśmy współrzędnymi [3, 2] używając wektorów bazowych i-z-daszkiem i j-z-daszkiem. Jennifer natomiast opisałaby ten wektor współrzędnymi [5/3, 1/3] Co znaczy,", @@ -135,7 +135,7 @@ "end": 127.04 }, { - "input": "For this specific example, the basis vector i-hat is one such special vector.", + "input": "Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vecto", "translatedText": "W tym konkretnym przykładzie wektor bazowy i-hat jest jednym z takich wektorów specjalnych.", "model": "google_nmt", "from_community_srt": "że aby otrzymać ten wektor za pomocą jej dwóch wektorów bazowych trzeba przeskalować b1 przez 5/3,", @@ -171,7 +171,7 @@ "end": 164.04 }, { - "input": "It ends up getting stretched by a factor of 2.", + "input": "scale b1 by 5 thirds, scale b2 by 1 third, then add them both togethe", "translatedText": "Skończyło się na rozciągnięciu 2-krotnym.", "model": "google_nmt", "n_reviews": 0, @@ -179,7 +179,7 @@ "end": 167.14 }, { - "input": "And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2.", + "input": "r. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she", "translatedText": "I znowu, liniowość będzie oznaczać, że każdy inny wektor na linii ukośnej rozpiętej przez tego gościa zostanie rozciągnięty dwukrotnie.", "model": "google_nmt", "from_community_srt": "Dla nieco większej precyzji, jej pierwszy wektor bazowy, b1, opisalibyśmy u nas jako [2, 1] a drugi wektor b2 opisalibyśmy jako [-1,", @@ -188,7 +188,7 @@ "end": 178.22 }, { - "input": "And for this transformation, those are all the vectors with this special property of staying on their span.", + "input": "thinks of her first coordinate as scali For this specific example, the basis vector i-hat is one such special vector. The span of", "translatedText": "I dla tej transformacji są to wszystkie wektory posiadające tę szczególną właściwość pozostawania na swojej rozpiętości.", "model": "google_nmt", "from_community_srt": "1]. Co ważne, trzeba zauważyć że z jej perspektywy, wektory b1 i b2 mają współrzędne [1,", @@ -197,7 +197,7 @@ "end": 185.18 }, { - "input": "Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2.", + "input": "i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis. What's more, because of the way linear transformations work,", "translatedText": "Te na osi x zostaną rozciągnięte 3-krotnie, a te na tej ukośnej linii zostaną rozciągnięte 2-krotnie.", "model": "google_nmt", "from_community_srt": "0] oraz [0, 1]. To właśnie b1 i b2 definiują, co w jej świecie oznacza [1, 0] i [0,", @@ -215,7 +215,7 @@ "end": 198.08 }, { - "input": "As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation.", + "input": "n. A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc t, a way to visualize our coordinate system, and so it depends on our choice of basis", "translatedText": "Jak można się już domyślić, te specjalne wektory nazywane są wektorami własnymi transformacji, a każdy wektor własny ma przypisaną tak zwaną wartość własną, która jest po prostu czynnikiem, przez który jest on rozciągany lub zgniatany podczas transformacji.", "model": "google_nmt", "from_community_srt": "Wszyscy patrzymy na te same wektory w przestrzeni, lecz Jennifer używa innych słów i liczb na ich opisanie. Teraz szybkie słówko o reprezentacji przestrzeni, której używam w 2D. Zwykle używam tej kwadratowej kraty, ale ta krata to tylko umowa, sposób wizualizacji naszego układu współrzędnych, który zależy od wyboru bazy.", @@ -233,7 +233,7 @@ "end": 225.94 }, { - "input": "In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half.", + "input": "nt as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It", "translatedText": "W innym przykładzie możesz mieć wektor własny z wartością własną ujemną o 1 połowę, co oznacza, że wektor zostanie odwrócony i zgnieciony 1-krotnie.", "model": "google_nmt", "from_community_srt": "która tak samo nie oznacza więcej niż tylko sposób wizualizacji, który pomaga nam używać jej współrzędnych. Początek układu, natomiast,", @@ -260,7 +260,7 @@ "end": 249.8 }, { - "input": "If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation.", + "input": "of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewhat during the transformation, k", "translatedText": "Jeśli możesz znaleźć wektor własny dla tego obrotu, wektor, który pozostaje na swoim własnym rozpiętości, to tym, co znalazłeś, jest oś obrotu.", "model": "google_nmt", "from_community_srt": "Zatem, po całym tym wstępnie, dość naturalnym pytaniem jest: Jak przetłumaczać między różnymi układami współrzędnych? Jeżeli na przykład Jennifer opisze wektor współrzędnymi [-1,", @@ -269,7 +269,7 @@ "end": 260.5 }, { - "input": "And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation.", + "input": "nocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", "translatedText": "O wiele łatwiej jest myśleć o obrocie 3D w kategoriach jakiejś osi obrotu i kąta, o jaki się obraca, niż myśleć o pełnej macierzy 3x3 powiązanej z tą transformacją.", "model": "google_nmt", "from_community_srt": "2] to jak zapisać go w naszym układzie współrzędnych? Jak tłumaczyć z jej języka na nasz? Cóż, jej współrzędne mówią, że ten wektor to -1 b1 + 2 b2.", @@ -287,7 +287,7 @@ "end": 285.86 }, { - "input": "This pattern shows up a lot in linear algebra.", + "input": "In fact, once you understand matrix vector multiplication as applying", "translatedText": "Ten wzór często pojawia się w algebrze liniowej.", "model": "google_nmt", "from_community_srt": "1] i b2 ma współrzędne [-1,", @@ -296,7 +296,7 @@ "end": 290.02 }, { - "input": "With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors.", + "input": "a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvecto", "translatedText": "W przypadku dowolnej transformacji liniowej opisanej przez macierz można zrozumieć, co ona robi, odczytując kolumny tej macierzy jako miejsca lądowania dla wektorów bazowych.", "model": "google_nmt", "from_community_srt": "1] Więc możemy po prostu obliczyć -1 b1 + 2 b2 w naszym układzie współrzędnych.", @@ -305,7 +305,7 @@ "end": 299.4 }, { - "input": "But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", + "input": "r with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformati", "translatedText": "Często jednak lepszym sposobem na dotarcie do sedna tego, co faktycznie robi transformacja liniowa, mniej zależnego od konkretnego układu współrzędnych, jest znalezienie wektorów własnych i wartości własnych.", "model": "google_nmt", "from_community_srt": "Obliczając to otrzymamy wektor [-4, 1] Tak zatem opisalibyśmy wektor o którym ona myśli jako [-1,", @@ -323,7 +323,7 @@ "end": 326.02 }, { - "input": "Symbolically, here's what the idea of an eigenvector looks like.", + "input": "she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we thi", "translatedText": "Symbolicznie, oto jak wygląda idea wektora własnego.", "model": "google_nmt", "from_community_srt": "Tak naprawdę, jeżeli zrozumiesz mnożenie macierzy przez wektor jako przykładanie pewnego przekształcenia liniowego", @@ -341,7 +341,7 @@ "end": 339.74 }, { - "input": "What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda.", + "input": "or for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotat", "translatedText": "To wyrażenie mówi, że iloczyn wektora macierzowego A razy v daje taki sam wynik, jak samo skalowanie wektora własnego v o pewną wartość lambda.", "model": "google_nmt", "from_community_srt": "Macierz której kolumny opisują wektory z bazy Jennifer może być interpretowana jako przekształcenie które wysyła naszą bazę, i-z-daszkiem i j-z-daszkiem czyli wektory o których myślimy,", @@ -368,7 +368,7 @@ "end": 370.54 }, { - "input": "So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda.", + "input": "never stretch or squish anything, so the length of the vector would remain the same. This pattern shows up a lot in linear algebra. With any linear transformation described by a matrix, you could understand what it's doing by reading of", "translatedText": "Zacznijmy więc od przepisania tej prawej strony jako pewnego rodzaju mnożenia macierzy przez wektor, używając macierzy, która powoduje skalowanie dowolnego wektora przez współczynnik lambda.", "model": "google_nmt", "from_community_srt": "Przed przekształceniem myślimy o tym wektorze jako o pewnej kombinacji liniowej naszych wektorów bazowych: -1 razy i-z-daszkiem + 2 razy j-z-daszkiem A główną własnością przekształcenia liniowego jest fakt że wektor wyjściowy będzie tą samą liniową kombinacją", @@ -377,7 +377,7 @@ "end": 380.62 }, { - "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else.", + "input": "f the columns of this matrix as the landing spots for basis vectors. But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", "translatedText": "Kolumny takiej macierzy będą przedstawiać, co dzieje się z każdym wektorem bazowym, a każdy wektor bazowy jest po prostu mnożony przez lambda, zatem w tej macierzy liczba lambda będzie znajdować się wzdłuż przekątnej, z zerami wszędzie indziej.", "model": "google_nmt", "from_community_srt": "ale nowych wektorów bazowych -1 razy obraz i-z-daszkiem + 2 razy obraz j-z-daszkiem. Więc to,", @@ -386,7 +386,7 @@ "end": 394.32 }, { - "input": "The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal.", + "input": "we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector w", "translatedText": "Powszechnym sposobem zapisywania tego faceta jest rozłożenie tej lambdy na czynniki i zapisanie jej jako lambda razy i, gdzie i jest macierzą tożsamości z jedynkami wzdłuż przekątnej.", "model": "google_nmt", "from_community_srt": "co robi macierz to przekształca coś, co jest naszym złym przekonaniem o czym myśli Jennifer w prawdziwy wektor, o którym ona myśli. Pamiętam,", @@ -404,7 +404,7 @@ "end": 411.86 }, { - "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector.", + "input": "that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, with v as the eigenvector", "translatedText": "Mamy więc nową macierz, A minus lambda razy tożsamość i szukamy wektora v takiego, że ta nowa macierz razy v daje wektor zerowy.", "model": "google_nmt", "from_community_srt": "Jednak obliczeniowo, przekształca opis wektora z jej języka na nasz. To, co w końcu otworzyło mi oczy, to pomyślenie o tym jak bierze ten zły,", @@ -431,7 +431,7 @@ "end": 433.64 }, { - "input": "And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", + "input": "and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Je", "translatedText": "A jeśli obejrzysz rozdziały 5 i 6, będziesz wiedział, że jedyny sposób, w jaki iloczyn macierzy z niezerowym wektorem może stać się zerem, polega na tym, że transformacja związana z tą macierzą zgniata przestrzeń do niższego wymiaru.", "model": "google_nmt", "from_community_srt": "A co z przekształceniem z powrotem? W przypadku którego użyłem wcześniej w tym video jeżeli mam wektor o współrzędnych [3, 2] w naszym układzie jak w końcu obliczyłem że będzie miał współrzędne [5/3,", @@ -467,7 +467,7 @@ "end": 470.28 }, { - "input": "As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes.", + "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose c", "translatedText": "Gdy zmienia się wartość lambda, zmienia się sama macierz, a zatem zmienia się wyznacznik macierzy.", "model": "google_nmt", "from_community_srt": "W praktyce, szczególnie gdy działamy w więcej niż 2 wymiarach używalibyśmy komputera do obliczania macierzy odwrotnej.", @@ -511,7 +511,7 @@ "end": 498.6 }, { - "input": "So this is kind of a lot, but let's unravel what this is saying.", + "input": "And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's importa", "translatedText": "To dość dużo, ale spójrzmy, co to mówi.", "model": "google_nmt", "n_reviews": 0, @@ -519,7 +519,7 @@ "end": 502.96 }, { - "input": "When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line.", + "input": "nt that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication So let's start by rewriting", "translatedText": "Gdy lambda jest równa 1, macierz A minus lambda razy tożsamość spację spacji na linii.", "model": "google_nmt", "from_community_srt": "2] co okazuje się być [5/3, 1/3] Więc tak,", @@ -528,7 +528,7 @@ "end": 509.56 }, { - "input": "That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector.", + "input": "that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a fact", "translatedText": "To oznacza, że istnieje niezerowy wektor v taki, że A minus lambda razy identyczność v równa się wektorowi zerowemu.", "model": "google_nmt", "from_community_srt": "pokrótce, przetłumaczamy opis wektorów z jednego układu do drugiego i z powrotem.", @@ -537,7 +537,7 @@ "end": 518.56 }, { - "input": "And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A.", + "input": "or of lambda. The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else. the columns of our matrix.", "translatedText": "I pamiętajcie, przejmujemy się tym, ponieważ oznacza to, że A razy v równa się lambda razy v, co można odczytać w ten sposób, że wektor v jest wektorem własnym A, pozostającym na swoim własnym rozpiętości podczas transformacji A.", "model": "google_nmt", "from_community_srt": "Macierz której kolumny opisują wektory z bazy Jennifer ale opisane w naszych współrzędnych przekształca wektory z jej języka na nasz. A macierz odwrotna robi rzecz odwrotną. Jednak wektory nie są jedyną rzeczą jaką opisujemy współrzędnymi. Przed tą kolejną częścią, ważne jest że umiecie posługiwać się", @@ -546,7 +546,7 @@ "end": 537.28 }, { - "input": "In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place.", + "input": "But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following", "translatedText": "W tym przykładzie odpowiadająca wartość własna wynosi 1, więc v faktycznie pozostanie na miejscu.", "model": "google_nmt", "from_community_srt": "reprezentacją przekształceń macierzami i wiecie, jak mnożenie macierzy odpowiada złożeniu przekształceń.", @@ -563,7 +563,7 @@ "end": 549.5 }, { - "input": "This is the kind of thing I mentioned in the introduction.", + "input": "-hat in the first pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v.", "translatedText": "To jest ten rodzaj rzeczy, o którym wspomniałem we wstępie.", "model": "google_nmt", "from_community_srt": "Z pewnością nie zaszkodzi zapauzować i spojrzeć na rozdziały 3 i 4 jeżeli coś z tego nie wydaje się wam jasne.", @@ -572,7 +572,7 @@ "end": 555.64 }, { - "input": "If you didn't have a solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to describe those landing spots in her language. Here's a common way to think", "translatedText": "Gdybyś nie miał solidnego pojęcia o wyznacznikach i o tym, dlaczego odnoszą się one do liniowych układów równań mających rozwiązania niezerowe, takie wyrażenie wydawałoby się zupełnie niespodziewane.", "model": "google_nmt", "from_community_srt": "Rozpatrzmy pewne przekształcenie liniowe, na przykład obrót o 90° przeciwnie do wskazówek zegara. jeżeli wyrazimy to macierzą, patrzymy gdzie trafiają i-z-daszkiem i j-z-daszkiem.", @@ -617,7 +617,7 @@ "end": 608.84 }, { - "input": "To figure out what the eigenvectors are that actually have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero.", + "input": "To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry. Now imagine tweaking lambda, turning a knob to change its value. As that value of lambda changes, the matrix itself change", "translatedText": "Aby dowiedzieć się, jakie wektory własne faktycznie mają jedną z tych wartości własnych, powiedzmy, że lambda równa się 2, podłącz tę wartość lambda do macierzy, a następnie oblicz, dla jakich wektorów ta zmieniona po przekątnej macierz ma wartość zero.", "model": "google_nmt", "from_community_srt": "gdzie wylądują nasze wektory i oraz j. A Jennifer chce macierz, która reprezentuje gdzie lądują jej wektory bazowe i musi opisać te wektory na których lądują w jej języku. Pokażę popularny sposób myślenia, jak to robić.", @@ -707,7 +707,7 @@ "end": 681.94 }, { - "input": "The only roots of that polynomial are the imaginary numbers, i and negative i.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the oute", "translatedText": "Jedynymi pierwiastkami tego wielomianu są liczby urojone i oraz ujemne i.", "model": "google_nmt", "from_community_srt": "Bierze ono wektor w jej języku i wypluwa jego obrót, też w jej języku.", @@ -734,7 +734,7 @@ "end": 699.82 }, { - "input": "This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1.", + "input": "meone else sees it. For those of you wondering why we care about alternate coordinate systems, the next vi", "translatedText": "To ustawia i-hat na miejscu i przesuwa j-hat 1, więc jego macierz ma kolumny 1, 0 i 1, 1.", "model": "google_nmt", "from_community_srt": "5/3] i [-2/3,", @@ -770,7 +770,7 @@ "end": 726.54 }, { - "input": "And the only root of this expression is lambda equals 1.", + "input": "he identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals la", "translatedText": "Jedynym pierwiastkiem tego wyrażenia jest lambda równa 1.", "model": "google_nmt", "from_community_srt": "macierz środkowa to pewne przekształcenie tak,", @@ -779,7 +779,7 @@ "end": 732.86 }, { - "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.", + "input": "mbda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corresponding eigenvalue is", "translatedText": "Zgadza się to z tym, co widzimy geometrycznie, że wszystkie wektory własne mają wartość własną 1.", "model": "google_nmt", "from_community_srt": "jak my je widzimy, a dwie zewnętrzne mówią o pewnej empatii - zmianie perspektywy,", @@ -839,7 +839,7 @@ "end": 796.38 }, { - "input": "For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2.", + "input": "f equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Na przykład może i-hat jest skalowany przez -1, a j-hat jest skalowany przez 2.", "model": "google_nmt", "n_reviews": 0, @@ -863,7 +863,7 @@ "end": 825.42 }, { - "input": "And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.", + "input": "nd compute the determinant. Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda. Since lambda can only be an eigenvalue i", "translatedText": "Można to zinterpretować w ten sposób, że wszystkie wektory bazowe są wektorami własnymi, a elementy diagonalne tej macierzy są ich wartościami własnymi.", "model": "google_nmt", "n_reviews": 0, @@ -879,7 +879,7 @@ "end": 841.06 }, { - "input": "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", + "input": "u can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3. To figure out what the eigenvectors are that actu", "translatedText": "Najważniejszą z nich jest to, że łatwiej jest obliczyć, co się stanie, jeśli pomnożysz tę macierz przez samą siebie wiele razy.", "model": "google_nmt", "n_reviews": 0, @@ -887,7 +887,7 @@ "end": 848.34 }, { - "input": "Since all one of these matrices does is scale each basis vector by some eigenvalue, applying that matrix many times, say 100 times, is just going to correspond to scaling each basis vector by the 100th power of the corresponding eigenvalue.", + "input": "ally have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero. If you computed this the way you would any other linear system, you'd see that the solutions are all the vectors on the diagonal line spanned by negative 1, 1.", "translatedText": "Ponieważ wszystkie te macierze skalują każdy wektor bazowy o pewną wartość własną, wielokrotne zastosowanie tej macierzy, powiedzmy 100 razy, będzie po prostu odpowiadać skalowaniu każdego wektora bazowego przez setną potęgę odpowiedniej wartości własnej.", "model": "google_nmt", "n_reviews": 0, @@ -903,7 +903,7 @@ "end": 869.68 }, { - "input": "Really, try it for a moment.", + "input": "x, 3, 0, 1, 2, has the effect of stretching all those vectors by a factor of 2.", "translatedText": "Naprawdę, spróbuj przez chwilę.", "model": "google_nmt", "n_reviews": 0, @@ -935,7 +935,7 @@ "end": 896.54 }, { - "input": "I talked about change of basis last video, but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", + "input": "is, notice what happens. Its matrix has columns 0, 1 and negative 1, 0. Subtract off lambda from the diagonal elements and look for when the determinant is zero. In this case, you get the polynomial lambda squared plus 1. The only roots of that polynomia", "translatedText": "Mówiłem o zmianie podstawy w poprzednim filmie, ale tutaj bardzo szybko przypomnę, jak wyrazić transformację aktualnie zapisaną w naszym układzie współrzędnych na inny układ.", "model": "google_nmt", "n_reviews": 0, @@ -943,7 +943,7 @@ "end": 907.04 }, { - "input": "Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors, then make those coordinates the columns of a matrix, known as the change of basis matrix.", + "input": "l are the imaginary numbers, i and negative i. The fact that there are no real number solutions indicates that there are no eigenvectors. Another pretty interesting example worth holding in the back of your mind is a shear. This fixes i-hat in place and moves j-hat 1 over, so its mat", "translatedText": "Weź współrzędne wektorów, których chcesz użyć jako nowej podstawy, co w tym przypadku oznacza nasze dwa wektory własne, a następnie uczyń te współrzędne kolumnami macierzy, znanej jako macierz zmiany podstawy.", "model": "google_nmt", "n_reviews": 0, @@ -967,7 +967,7 @@ "end": 946.68 }, { - "input": "This is because it represents working in a coordinate system where what happens to the basis vectors is that they get scaled during the transformation.", + "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1. Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a lin", "translatedText": "Dzieje się tak, ponieważ reprezentuje pracę w układzie współrzędnych, w którym wektory bazowe ulegają skalowaniu podczas transformacji.", "model": "google_nmt", "n_reviews": 0, @@ -983,7 +983,7 @@ "end": 961.56 }, { - "input": "So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis, compute the 100th power in that system, then convert back to our standard system.", + "input": "A simple example is a matrix that scales everything by 2. The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue. Now is another good time to pause and ponder some of this before I move on to the last topic.", "translatedText": "Jeśli więc na przykład trzeba byłoby obliczyć setną potęgę tej macierzy, znacznie łatwiej byłoby przejść na podstawę własną, obliczyć setną potęgę w tym układzie, a następnie przekonwertować z powrotem do naszego standardowego systemu.", "model": "google_nmt", "n_reviews": 0, @@ -991,7 +991,7 @@ "end": 975.68 }, { - "input": "You can't do this with all transformations.", + "input": "I want to finish off here with the idea of an eigenbasis, which relies heavily on ideas from the last video. Take a look at what h", "translatedText": "Nie można tego zrobić w przypadku wszystkich transformacji.", "model": "google_nmt", "n_reviews": 0, @@ -999,7 +999,7 @@ "end": 978.32 }, { - "input": "A shear, for example, doesn't have enough eigenvectors to span the full space.", + "input": "appens if our basis vectors just so happen to be eigenvectors. For example, maybe i-hat is scale", "translatedText": "Na przykład ścinanie nie ma wystarczającej liczby wektorów własnych, aby objąć całą przestrzeń.", "model": "google_nmt", "n_reviews": 0, @@ -1007,7 +1007,7 @@ "end": 982.98 }, { - "input": "But if you can find an eigenbasis, it makes matrix operations really lovely.", + "input": "d by negative 1 and j-hat is scaled by 2. Writing their new coordinates as the columns of a matrix, notice t", "translatedText": "Ale jeśli potrafisz znaleźć bazę własną, operacje na macierzach stają się naprawdę piękne.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/portuguese/sentence_translations.json b/2016/change-of-basis/portuguese/sentence_translations.json index c96fc9a86..b0da41c03 100644 --- a/2016/change-of-basis/portuguese/sentence_translations.json +++ b/2016/change-of-basis/portuguese/sentence_translations.json @@ -81,7 +81,7 @@ "end": 84.84 }, { - "input": "It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.", + "input": "st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is ti", "translatedText": "Ele move o vetor base i-hat para as coordenadas 3, 0 e j-hat para 1, 2.", "model": "google_nmt", "from_community_srt": "î e ĵ, são chamados os vetores de base do nosso sistema padrão de coordenadas.", @@ -90,7 +90,7 @@ "end": 91.04 }, { - "input": "So it's represented with a matrix whose columns are 3, 0, and 1, 2.", + "input": "ed up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actual", "translatedText": "Portanto, é representado por uma matriz cujas colunas são 3, 0 e 1, 2.", "model": "google_nmt", "from_community_srt": "O que eu gostaria de falar aqui é a ideia de usar um conjunto diferente de vetores de base.", @@ -99,7 +99,7 @@ "end": 95.64 }, { - "input": "Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip.", + "input": "ly scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called th", "translatedText": "Concentre-se no que ele faz com um vetor específico e pense na extensão desse vetor, na reta que passa por sua origem e sua ponta.", "model": "google_nmt", "from_community_srt": "Por exemplo, digamos que você tem uma amiga, Jennifer que usa um conjunto diferente de vetores de base, que chamarei de b1 e b2.", @@ -108,7 +108,7 @@ "end": 104.16 }, { - "input": "Most vectors are going to get knocked off their span during the transformation.", + "input": "e basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a", "translatedText": "A maioria dos vetores será eliminada durante a transformação.", "model": "google_nmt", "from_community_srt": "Seu primeiro vetor de base b1 aponta para a direita um pouco e seu segundo vetor,", @@ -126,7 +126,7 @@ "end": 115.32 }, { - "input": "But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar.", + "input": "let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up.", "translatedText": "Mas alguns vetores especiais permanecem em sua própria extensão, o que significa que o efeito que a matriz tem sobre tal vetor é apenas esticá-lo ou comprimi-lo, como um escalar.", "model": "google_nmt", "from_community_srt": "2], usando nossos vetores de base î e ĵ. Jennifer na verdade descreveria este vetor com as coordenadas [5/3, 1/3].", @@ -135,7 +135,7 @@ "end": 127.04 }, { - "input": "For this specific example, the basis vector i-hat is one such special vector.", + "input": "Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vecto", "translatedText": "Para este exemplo específico, o vetor base i-hat é um desses vetores especiais.", "model": "google_nmt", "from_community_srt": "O que isto significa é que o modo particular para chegar a esse vetor usando os dois vetores de base dela", @@ -171,7 +171,7 @@ "end": 164.04 }, { - "input": "It ends up getting stretched by a factor of 2.", + "input": "scale b1 by 5 thirds, scale b2 by 1 third, then add them both togethe", "translatedText": "Acaba sendo esticado por um fator de 2.", "model": "google_nmt", "n_reviews": 0, @@ -179,7 +179,7 @@ "end": 167.14 }, { - "input": "And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2.", + "input": "r. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she", "translatedText": "E, novamente, a linearidade implicará que qualquer outro vetor na reta diagonal gerada por esse cara será esticado por um fator de 2.", "model": "google_nmt", "from_community_srt": "Vamos ser um pouco mais precisos com relação à configuração aqui: o primeiro vetor de base b1 dela é algo que gostaríamos de descrever com as coordenadas [2, 1] e o segundo vetor de base b2 dela, é algo que nós descreveríamos como [-1,", @@ -188,7 +188,7 @@ "end": 178.22 }, { - "input": "And for this transformation, those are all the vectors with this special property of staying on their span.", + "input": "thinks of her first coordinate as scali For this specific example, the basis vector i-hat is one such special vector. The span of", "translatedText": "E para esta transformação, estes são todos os vetores com esta propriedade especial de permanecer no seu vão.", "model": "google_nmt", "from_community_srt": "1]. Mas é importante perceber que, a partir da perspectiva do sistema dela, esses vetores têm coordenadas [1,", @@ -197,7 +197,7 @@ "end": 185.18 }, { - "input": "Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2.", + "input": "i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis. What's more, because of the way linear transformations work,", "translatedText": "Aqueles no eixo x sendo esticados por um fator de 3, e aqueles nesta linha diagonal sendo esticados por um fator de 2.", "model": "google_nmt", "from_community_srt": "0] e [0, 1]. Eles são o que define o significado das coordenadas [1, 0] e [0, 1] no mundo dela.", @@ -215,7 +215,7 @@ "end": 198.08 }, { - "input": "As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation.", + "input": "n. A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc t, a way to visualize our coordinate system, and so it depends on our choice of basis", "translatedText": "Como você já deve ter adivinhado, esses vetores especiais são chamados de autovetores da transformação, e cada autovetor tem associado a ele o que é chamado de autovalor, que é apenas o fator pelo qual ele é esticado ou comprimido durante a transformação.", "model": "google_nmt", "from_community_srt": "mas Jennifer usa diferentes palavras e números para descrevê-los. Deixe-me dizer uma palavra rápida sobre como eu estou representando coisas aqui: quando eu animo o espaço 2D, Eu normalmente uso esta grade quadrada mas essa grade é apenas uma construção, uma forma de visualizar o nosso sistema de coordenadas, e por isso depende da nossa escolha da base.", @@ -233,7 +233,7 @@ "end": 225.94 }, { - "input": "In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half.", + "input": "nt as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It", "translatedText": "Em outro exemplo, você poderia ter um autovetor com autovalor negativo 1 metade, o que significa que o vetor é invertido e comprimido por um fator de 1 metade.", "model": "google_nmt", "from_community_srt": "significando nada mais que uma ferramenta visual para ajudar a seguir o significado de suas coordenadas. A origem dela, no entanto,", @@ -260,7 +260,7 @@ "end": 249.8 }, { - "input": "If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation.", + "input": "of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewhat during the transformation, k", "translatedText": "Se você puder encontrar um autovetor para essa rotação, um vetor que permaneça em seu próprio vão, o que você encontrará é o eixo de rotação.", "model": "google_nmt", "from_community_srt": "Então, depois que tudo isso é configurado, uma pergunta muito natural a se fazer é: \"Como podemos traduzir entre sistemas de coordenadas?\" Se, por exemplo,", @@ -269,7 +269,7 @@ "end": 260.5 }, { - "input": "And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation.", + "input": "nocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", "translatedText": "E é muito mais fácil pensar em uma rotação 3D em termos de algum eixo de rotação e um ângulo pelo qual ela gira, em vez de pensar na matriz 3x3 completa associada a essa transformação.", "model": "google_nmt", "from_community_srt": "Jennifer descreve um vetor com coordenadas [-1, 2], o que seria isso no nosso sistema de coordenadas? Como você traduzir de sua linguagem para a nossa? Bem, o que nossas coordenadas estão dizendo é que este vetor é -1 b1 + 2 b2.", @@ -287,7 +287,7 @@ "end": 285.86 }, { - "input": "This pattern shows up a lot in linear algebra.", + "input": "In fact, once you understand matrix vector multiplication as applying", "translatedText": "Esse padrão aparece muito na álgebra linear.", "model": "google_nmt", "from_community_srt": "1] e b2 tem coordenadas [-1, 1], então,", @@ -296,7 +296,7 @@ "end": 290.02 }, { - "input": "With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors.", + "input": "a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvecto", "translatedText": "Com qualquer transformação linear descrita por uma matriz, você pode entender o que ela está fazendo lendo as colunas dessa matriz como pontos de aterrissagem para vetores de base.", "model": "google_nmt", "from_community_srt": "podemos calcular realmente -1 b1 + 2 b2, como eles são representados em nosso sistema de coordenadas,", @@ -305,7 +305,7 @@ "end": 299.4 }, { - "input": "But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", + "input": "r with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformati", "translatedText": "Mas muitas vezes, a melhor maneira de chegar ao cerne do que a transformação linear realmente faz, menos dependente do seu sistema de coordenadas específico, é encontrar os autovetores e autovalores.", "model": "google_nmt", "from_community_srt": "E fazendo esta conta, você termina com um vetor com coordenadas [-4, 1]. Então, é assim que nós descreveríamos o vetor que ela entende como [-1, 2].", @@ -323,7 +323,7 @@ "end": 326.02 }, { - "input": "Symbolically, here's what the idea of an eigenvector looks like.", + "input": "she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we thi", "translatedText": "Simbolicamente, esta é a aparência da ideia de um autovetor.", "model": "google_nmt", "from_community_srt": "depois que você entende a multiplicação matriz-vector como a aplicação de uma certa transformação linear,", @@ -341,7 +341,7 @@ "end": 339.74 }, { - "input": "What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda.", + "input": "or for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotat", "translatedText": "O que esta expressão está dizendo é que o produto matriz-vetor, A vezes v, dá o mesmo resultado que apenas dimensionar o autovetor v por algum valor lambda.", "model": "google_nmt", "from_community_srt": "A matriz cujas colunas representam os vetores de base de Jennifer pode ser pensada como uma transformação que move nossos vetores de base, î e ĵ (que são as coisas em que pensamos quando dizemos [1,0] e [0,", @@ -368,7 +368,7 @@ "end": 370.54 }, { - "input": "So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda.", + "input": "never stretch or squish anything, so the length of the vector would remain the same. This pattern shows up a lot in linear algebra. With any linear transformation described by a matrix, you could understand what it's doing by reading of", "translatedText": "Então, vamos começar reescrevendo o lado direito como algum tipo de multiplicação matriz-vetor, usando uma matriz que tem o efeito de escalonar qualquer vetor por um fator lambda.", "model": "google_nmt", "from_community_srt": "como uma certa combinação linear de nossos vetores de base, -1 î + 2 ĵ.", @@ -377,7 +377,7 @@ "end": 380.62 }, { - "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else.", + "input": "f the columns of this matrix as the landing spots for basis vectors. But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", "translatedText": "As colunas dessa matriz representarão o que acontece com cada vetor de base, e cada vetor de base é simplesmente multiplicado por lambda, então essa matriz terá o número lambda na diagonal, com zeros em todos os outros lugares.", "model": "google_nmt", "from_community_srt": "E o elemento-chave de uma transformação linear é que o vetor resultante que será a mesma combinação linear mas dos novos vetores de base -1 vezes o lugar onde î vai parar + 2 vezes o lugar onde ĵ vai parar. Então,", @@ -386,7 +386,7 @@ "end": 394.32 }, { - "input": "The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal.", + "input": "we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector w", "translatedText": "A maneira comum de escrever esse cara é fatorar esse lambda e escrevê-lo como lambda vezes i, onde i é a matriz identidade com 1s na diagonal.", "model": "google_nmt", "from_community_srt": "o que esta matriz faz é transformar nosso equívoco do que Jennifer quer dizer no vetor real a que ela está se referindo.", @@ -404,7 +404,7 @@ "end": 411.86 }, { - "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector.", + "input": "that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, with v as the eigenvector", "translatedText": "Então o que temos agora é uma nova matriz, A menos lambda vezes a identidade, e estamos procurando um vetor v tal que esta nova matriz vezes v dê o vetor zero.", "model": "google_nmt", "from_community_srt": "Mas numericamente, está traduzindo um vetor descrito no idioma dela para um no nosso. O que fez tudo finalmente fazer sentido para mim foi pensar em como ele leva nosso equívoco do que Jennifer quer dizer,", @@ -431,7 +431,7 @@ "end": 433.64 }, { - "input": "And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", + "input": "and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Je", "translatedText": "E se você assistir aos capítulos 5 e 6, saberá que a única maneira de o produto de uma matriz com um vetor diferente de zero se tornar zero é se a transformação associada a essa matriz comprimir o espaço em uma dimensão inferior.", "model": "google_nmt", "from_community_srt": "Que tal ir no outro sentido? No exemplo que eu usei no início deste vídeo, quando tenho o vector com coordenadas [3,2] em nosso sistema, Como é que eu calculei que ele teria coordenadas [5/3,", @@ -467,7 +467,7 @@ "end": 470.28 }, { - "input": "As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes.", + "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose c", "translatedText": "À medida que o valor de lambda muda, a própria matriz muda e, portanto, o determinante da matriz muda.", "model": "google_nmt", "from_community_srt": "especialmente quando você está trabalhando em mais de duas dimensões, você usaria um computador para calcular a matriz que representa esta inversa.", @@ -512,7 +512,7 @@ "end": 498.6 }, { - "input": "So this is kind of a lot, but let's unravel what this is saying.", + "input": "And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's importa", "translatedText": "Então isso é bastante, mas vamos desvendar o que isso quer dizer.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 502.96 }, { - "input": "When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line.", + "input": "nt that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication So let's start by rewriting", "translatedText": "Quando lambda é igual a 1, a matriz A menos lambda vezes a identidade comprime o espaço em uma linha.", "model": "google_nmt", "from_community_srt": "2] que termina como [5/3, 1/3]. Então isso,", @@ -529,7 +529,7 @@ "end": 509.56 }, { - "input": "That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector.", + "input": "that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a fact", "translatedText": "Isso significa que existe um vetor diferente de zero v tal que A menos lambda vezes a identidade vezes v é igual ao vetor zero.", "model": "google_nmt", "from_community_srt": "em poucas palavras, é como traduzir a descrição de vetores individuais entre os sistemas de coordenadas. A matriz cujas colunas representam os vetores de base de Jennifer mas escritos em nossas coordenadas", @@ -538,7 +538,7 @@ "end": 518.56 }, { - "input": "And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A.", + "input": "or of lambda. The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else. the columns of our matrix.", "translatedText": "E lembre-se, a razão pela qual nos preocupamos com isso é porque significa A vezes v é igual a lambda vezes v, o que você pode interpretar como dizendo que o vetor v é um autovetor de A, permanecendo em seu próprio intervalo durante a transformação A.", "model": "google_nmt", "from_community_srt": "traduz vetores da língua dela para a nossa língua. E a matriz inversa faz o oposto. Mas vetores não são a única coisa que nós descrevemos utilizando coordenadas.", @@ -547,7 +547,7 @@ "end": 537.28 }, { - "input": "In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place.", + "input": "But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following", "translatedText": "Neste exemplo, o autovalor correspondente é 1, então v permaneceria fixo no lugar.", "model": "google_nmt", "from_community_srt": "Para esta parte seguinte é importante que vocês estejam todos confortáveis representando transformações com matrizes e que vocês saibam como a multiplicação de matrizes corresponde à composição de transformações sucessivas.", @@ -564,7 +564,7 @@ "end": 549.5 }, { - "input": "This is the kind of thing I mentioned in the introduction.", + "input": "-hat in the first pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v.", "translatedText": "Esse é o tipo de coisa que mencionei na introdução.", "model": "google_nmt", "from_community_srt": "Definitivamente pare e dê uma olhada nos capítulos 3 e 4 se você não estiver confortável com algum desses temas.", @@ -573,7 +573,7 @@ "end": 555.64 }, { - "input": "If you didn't have a solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to describe those landing spots in her language. Here's a common way to think", "translatedText": "Se você não tivesse uma compreensão sólida dos determinantes e por que eles se relacionam com sistemas lineares de equações com soluções diferentes de zero, uma expressão como essa pareceria completamente inesperada.", "model": "google_nmt", "from_community_srt": "Considere alguma transformação linear como uma rotação anti-horária de 90°. Quando você e eu a representamos com a matriz, seguimos onde os vetores de base î e ĵ vão.", @@ -618,7 +618,7 @@ "end": 608.84 }, { - "input": "To figure out what the eigenvectors are that actually have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero.", + "input": "To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry. Now imagine tweaking lambda, turning a knob to change its value. As that value of lambda changes, the matrix itself change", "translatedText": "Para descobrir quais são os autovetores que realmente possuem um desses autovalores, digamos que lambda é igual a 2, insira esse valor de lambda na matriz e, em seguida, resolva quais vetores essa matriz alterada diagonalmente envia para zero.", "model": "google_nmt", "from_community_srt": "Essas colunas representam onde nossos vetores de base, î e ĵ vão. Mas a matriz que Jennifer quer deve representar onde os vetores da base dela vão parar, e ele precisa descrever os pontos de pouso dos vetores na língua dela. Aqui está uma maneira comum de pensar em como isto é feito.", @@ -708,7 +708,7 @@ "end": 681.94 }, { - "input": "The only roots of that polynomial are the imaginary numbers, i and negative i.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the oute", "translatedText": "As únicas raízes desse polinômio são os números imaginários, i e negativo i.", "model": "google_nmt", "from_community_srt": "escrito na linguagem dela à versão transformada do vetor, na linguagem dela.", @@ -735,7 +735,7 @@ "end": 699.82 }, { - "input": "This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1.", + "input": "meone else sees it. For those of you wondering why we care about alternate coordinate systems, the next vi", "translatedText": "Isso fixa o i-hat no lugar e move o j-hat 1, de modo que sua matriz tenha as colunas 1, 0 e 1, 1.", "model": "google_nmt", "from_community_srt": "5/3] e [-2/3, -1/3]. Então,", @@ -771,7 +771,7 @@ "end": 726.54 }, { - "input": "And the only root of this expression is lambda equals 1.", + "input": "he identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals la", "translatedText": "E a única raiz desta expressão é lambda igual a 1.", "model": "google_nmt", "from_community_srt": "Essa matriz do meio representa uma transformação de algum tipo, como você a vê, e as duas matrizes exteriores representam a empatia,", @@ -780,7 +780,7 @@ "end": 732.86 }, { - "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.", + "input": "mbda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corresponding eigenvalue is", "translatedText": "Isso está de acordo com o que vemos geometricamente, que todos os autovetores têm autovalor 1.", "model": "google_nmt", "from_community_srt": "a mudança de perspectiva e o produto matricial completo representa aquela mesma transformação,", @@ -840,7 +840,7 @@ "end": 796.38 }, { - "input": "For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2.", + "input": "f equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Por exemplo, talvez i-hat seja dimensionado em menos 1 e j-hat seja dimensionado em 2.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 825.42 }, { - "input": "And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.", + "input": "nd compute the determinant. Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda. Since lambda can only be an eigenvalue i", "translatedText": "E a maneira de interpretar isto é que todos os vetores de base são autovetores, sendo as entradas diagonais desta matriz os seus autovalores.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 841.06 }, { - "input": "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", + "input": "u can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3. To figure out what the eigenvectors are that actu", "translatedText": "Um grande problema é que é mais fácil calcular o que acontecerá se você multiplicar essa matriz por ela mesma várias vezes.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 848.34 }, { - "input": "Since all one of these matrices does is scale each basis vector by some eigenvalue, applying that matrix many times, say 100 times, is just going to correspond to scaling each basis vector by the 100th power of the corresponding eigenvalue.", + "input": "ally have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero. If you computed this the way you would any other linear system, you'd see that the solutions are all the vectors on the diagonal line spanned by negative 1, 1.", "translatedText": "Como tudo o que uma dessas matrizes faz é dimensionar cada vetor de base por algum autovalor, aplicar essa matriz muitas vezes, digamos 100 vezes, corresponderá apenas a dimensionar cada vetor de base pela 100ª potência do autovalor correspondente.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 869.68 }, { - "input": "Really, try it for a moment.", + "input": "x, 3, 0, 1, 2, has the effect of stretching all those vectors by a factor of 2.", "translatedText": "Sério, experimente por um momento.", "model": "google_nmt", "n_reviews": 0, @@ -936,7 +936,7 @@ "end": 896.54 }, { - "input": "I talked about change of basis last video, but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", + "input": "is, notice what happens. Its matrix has columns 0, 1 and negative 1, 0. Subtract off lambda from the diagonal elements and look for when the determinant is zero. In this case, you get the polynomial lambda squared plus 1. The only roots of that polynomia", "translatedText": "Falei sobre mudança de base no vídeo passado, mas vou fazer um lembrete super rápido aqui de como expressar uma transformação atualmente escrita em nosso sistema de coordenadas em um sistema diferente.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 907.04 }, { - "input": "Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors, then make those coordinates the columns of a matrix, known as the change of basis matrix.", + "input": "l are the imaginary numbers, i and negative i. The fact that there are no real number solutions indicates that there are no eigenvectors. Another pretty interesting example worth holding in the back of your mind is a shear. This fixes i-hat in place and moves j-hat 1 over, so its mat", "translatedText": "Pegue as coordenadas dos vetores que deseja usar como uma nova base, que neste caso significa nossos dois autovetores, e depois transforme essas coordenadas nas colunas de uma matriz, conhecida como matriz de mudança de base.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 946.68 }, { - "input": "This is because it represents working in a coordinate system where what happens to the basis vectors is that they get scaled during the transformation.", + "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1. Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a lin", "translatedText": "Isso ocorre porque representa trabalhar em um sistema de coordenadas onde o que acontece com os vetores de base é que eles são escalonados durante a transformação.", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 961.56 }, { - "input": "So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis, compute the 100th power in that system, then convert back to our standard system.", + "input": "A simple example is a matrix that scales everything by 2. The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue. Now is another good time to pause and ponder some of this before I move on to the last topic.", "translatedText": "Portanto, se, por exemplo, você precisasse calcular a centésima potência desta matriz, seria muito mais fácil mudar para uma base própria, calcular a centésima potência nesse sistema e depois converter novamente para o nosso sistema padrão.", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 975.68 }, { - "input": "You can't do this with all transformations.", + "input": "I want to finish off here with the idea of an eigenbasis, which relies heavily on ideas from the last video. Take a look at what h", "translatedText": "Você não pode fazer isso com todas as transformações.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 978.32 }, { - "input": "A shear, for example, doesn't have enough eigenvectors to span the full space.", + "input": "appens if our basis vectors just so happen to be eigenvectors. For example, maybe i-hat is scale", "translatedText": "Um cisalhamento, por exemplo, não possui vetores próprios suficientes para abranger todo o espaço.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 982.98 }, { - "input": "But if you can find an eigenbasis, it makes matrix operations really lovely.", + "input": "d by negative 1 and j-hat is scaled by 2. Writing their new coordinates as the columns of a matrix, notice t", "translatedText": "Mas se você puder encontrar uma base própria, isso tornará as operações matriciais realmente adoráveis.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/russian/sentence_translations.json b/2016/change-of-basis/russian/sentence_translations.json index feb900f45..75a43e974 100644 --- a/2016/change-of-basis/russian/sentence_translations.json +++ b/2016/change-of-basis/russian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates.", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we doing this and what d", "translatedText": "Если у меня есть вектор, находящийся здесь, в 2D-пространстве, у нас есть стандартный способ описать его координатами.", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 28.28 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up.", + "input": "oes this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin", "translatedText": "В данном случае вектор имеет координаты 3, 2, что означает, что переход от его хвоста к кончику предполагает перемещение на три единицы вправо и на две единицы вверх.", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.96 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up.", + "input": "ch that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of", "translatedText": "Вы представляете эту первую координату как масштабирующую i-шляпу, вектор длиной 1 направлен вправо, а вторая координата масштабирует j-шляпу, вектор длиной 1 направлен прямо вверх.", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 60.48 }, { - "input": "You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordinate system.", + "input": "you know how to think about matrices as linear transformations, but you also need to be comfortable with things like determinants, linear systems", "translatedText": "Вы можете думать об этих двух специальных векторах как об инкапсулирующих все неявные предположения нашей системы координат.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 81.38 }, { - "input": "Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system.", + "input": "some linear transformation in two dimensions, like the one shown here. st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice o", "translatedText": "Любой способ перевода между векторами и наборами чисел называется системой координат, а два специальных вектора i-hat и j-hat называются базисными векторами нашей стандартной системы координат.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors.", + "input": "f i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors an", "translatedText": "Здесь я хотел бы поговорить об идее использования другого набора базисных векторов.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.04 }, { - "input": "Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third.", + "input": "nnifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to th", "translatedText": "Дженнифер на самом деле описала бы этот вектор координатами 5 третей и 1 треть.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 125.16 }, { - "input": "In a little bit, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third.", + "input": "showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describe this vector with the coordinates 5 thirds and", "translatedText": "Чуть позже я покажу вам, как можно было вычислить эти два числа: 5 третей и 1 треть.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 144.12 }, { - "input": "What she gets will typically be completely different from the vector that you and I would think of as having those coordinates.", + "input": "gether. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to", "translatedText": "То, что она получит, обычно будет полностью отличаться от вектора, который мы с вами думаем как имеющий эти координаты.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 167.14 }, { - "input": "So in effect, we're speaking different languages.", + "input": "lumn of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis.", "translatedText": "По сути, мы говорим на разных языках.", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 176.84 }, { - "input": "Let me say a quick word about how I'm representing things here.", + "input": "ector on the x-axis is also just stretched by a factor of 3, and hence remains on its own", "translatedText": "Позвольте мне сказать несколько слов о том, как я здесь представляю ситуацию.", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid.", + "input": "span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here.", "translatedText": "Само пространство не имеет внутренней сетки.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 197.6 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean.", + "input": "system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct mea", "translatedText": "Однако ее происхождение на самом деле совпадало бы с нашим, поскольку все согласны с тем, что должны означать координаты 0,0.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 214.9 }, { - "input": "But the direction of her axes and the spacing of her grid lines will be different, depending on her choice of basis vectors.", + "input": "ollow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector", "translatedText": "Но направление ее осей и расстояние между линиями сетки будут разными, в зависимости от выбора базисных векторов.", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 225.94 }, { - "input": "If for example Jennifer describes a vector with coordinates negative 1, 2, what would that be in our coordinate system?", + "input": "on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewh", "translatedText": "Если, например, Дженнифер описывает вектор с отрицательными координатами 1, 2, что это будет за наша система координат?", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 238.98 }, { - "input": "How do you translate from her language to ours?", + "input": "at during the transformation, knocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis", "translatedText": "Как вы переводите с ее языка на наш?", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 260.5 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1.", + "input": "ing them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vector", "translatedText": "И с нашей точки зрения, b1 имеет координаты 2, 1, а b2 имеет отрицательные координаты 1, 1.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 267.04 }, { - "input": "So we can actually compute negative 1 times b1 plus 2 times b2 as they're represented in our coordinate system.", + "input": "s in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio Of course, there's n", "translatedText": "Таким образом, мы можем фактически вычислить отрицательное значение 1, умноженное на b1, плюс 2, умноженное на b2, как они представлены в нашей системе координат.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 270.4 }, { - "input": "And working this out, you get a vector with coordinates negative 4, 1.", + "input": "othing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In", "translatedText": "Решая это, вы получаете вектор с отрицательными координатами 4, 1.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 274.74 }, { - "input": "This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", + "input": "alf, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennif", "translatedText": "Этот процесс масштабирования каждого из ее базисных векторов по соответствующим координатам некоторого вектора с последующим их сложением может показаться несколько знакомым.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 330.48 }, { - "input": "To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation.", + "input": "envector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking a", "translatedText": "Чтобы показать, как это работает, давайте разберемся, что значит взять вектор, который, как мы думаем, имеет отрицательные координаты 1, 2, и применить это преобразование.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 341.38 }, { - "input": "Before the linear transformation, we're thinking of this vector as a certain linear combination of our basis vectors, negative 1 times i-hat plus 2 times j-hat.", + "input": "bout the full 3x3 matrix associated with that transformation. In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length", "translatedText": "Перед линейным преобразованием мы думаем об этом векторе как об определенной линейной комбинации наших базисных векторов, отрицательных 1 раз i-hat плюс 2 раза j-hat.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 375.16 }, { - "input": "Geometrically, this matrix transforms our grid into Jennifer's grid but numerically, it's translating a vector described in her language to our language.", + "input": "t the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. we get using", "translatedText": "Геометрически эта матрица преобразует нашу сетку в сетку Дженнифер, но численно она переводит вектор, описанный на ее языке, на наш язык.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 380.62 }, { - "input": "What made it finally click for me was thinking about how it takes our misconception of what Jennifer means, the vector we get using the same coordinates but in our system, then it transforms it into the vector that she really meant.", + "input": "the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I comp", "translatedText": "Что, наконец, заставило меня задуматься, так это мысль о том, как наше неправильное представление о том, что имеет в виду Дженнифер, вектор, который мы получаем, используя те же координаты, но в нашей системе, затем преобразует его в вектор, который она на самом деле имела в виду.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 398.26 }, { - "input": "What about going the other way around?", + "input": "ute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A", "translatedText": "А как насчет того, чтобы пойти наоборот?", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 404.26 }, { - "input": "In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 third in Jennifer's system?", + "input": "is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis", "translatedText": "В примере, который я использовал ранее в этом видео, когда у меня был вектор с координатами 3, 2 в нашей системе, как я вычислил, что он будет иметь координаты 5 третей и 1 треть в системе Дженнифер?", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 409.48 }, { - "input": "You start with that change of basis matrix that translates Jennifer's language into ours, then you take its inverse.", + "input": "as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we", "translatedText": "Вы начинаете с изменения базовой матрицы, которая переводит язык Дженнифер на наш, а затем берете ее обратную.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 415.48 }, { - "input": "Remember, the inverse of a transformation is a new transformation that corresponds to playing that first one backwards.", + "input": "multiply this inverse change of basis matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and fo", "translatedText": "Помните, что обратная трансформация — это новая трансформация, которая соответствует воспроизведению первой трансформации задом наперед.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 427.94 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse.", + "input": "rth between coordinate systems. The matrix whose columns represent Jennif er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the invers", "translatedText": "На практике, особенно когда вы работаете более чем в двух измерениях, вам придется использовать компьютер для вычисления матрицы, которая фактически представляет это обратное.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 465.52 }, { - "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems.", + "input": "you know how matrix multiplication So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, usi", "translatedText": "Вкратце, это то, как переводить описание отдельных векторов туда и обратно между системами координат.", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 487.24 }, { - "input": "And the inverse matrix does the opposite.", + "input": "The columns of such a matrix will represent what happens to eac", "translatedText": "А обратная матрица делает обратное.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 507.16 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy.", + "input": "heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla With", "translatedText": "Обязательно сделайте паузу и просмотрите главы 3 и 4, если что-то из этого покажется вам непростым.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 529.74 }, { - "input": "i-hat ends up at the spot with coordinates 0, 1, and j-hat ends up at the spot with coordinates negative 1, 0.", + "input": "or out the v. So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to desc", "translatedText": "i-hat попадает в точку с координатами 0, 1, а j-hat оказывается в точке с отрицательными координатами 1, 0.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 566.3 }, { - "input": "But that's not quite right.", + "input": "And that squishification corresponds to a zero determinant for the matrix. To be concrete, let's say your matrix", "translatedText": "Но это не совсем так.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.72 }, { - "input": "Here's a common way to think of how this is done.", + "input": "As that value of lambda changes, the matrix itself changes, and so the determina", "translatedText": "Вот общий способ думать о том, как это делается.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 603.42 }, { - "input": "Start with any vector written in Jennifer's language.", + "input": "nt of the matrix changes. ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multipl", "translatedText": "Начните с любого вектора, написанного на языке Дженнифер.", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 649.44 }, { - "input": "Since we could do this with any vector written in her language, first applying the change of basis, then the transformation, then the inverse change of basis, that composition of three matrices gives us the transformation matrix in Jennifer's language.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective. And the full matrix product represents that same transformation, but as someone else sees it.", "translatedText": "Поскольку мы могли бы сделать это с любым вектором, написанным на ее языке, сначала применив замену базиса, затем преобразование, а затем обратное изменение базиса, эта композиция трех матриц дает нам матрицу преобразования на языке Дженнифер.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 665.56 }, { - "input": "It takes in a vector of her language and spits out the transformed version of that vector in her language.", + "input": "For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of", "translatedText": "Он принимает вектор ее языка и выдает преобразованную версию этого вектора на ее языке.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 675.8 }, { - "input": "For this specific example, when Jennifer's basis vectors look like 2, 1 and negative in our language, and when the transformation is a 90 degree rotation, the product of these three matrices, if you work through it, has columns one third, five thirds, and negative two thirds, negative one third.", + "input": "this. See y That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corres", "translatedText": "В этом конкретном примере, когда базисные векторы Дженнифер на нашем языке выглядят как 2, 1 и отрицательные, и когда преобразование представляет собой поворот на 90 градусов, произведение этих трех матриц, если вы проработаете его, имеет столбцы на одну треть, пять третей. , и отрицательные две трети, отрицательная одна треть.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 692.2 }, { - "input": "So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system.", + "input": "ponding eigenvalue is 1, so v would actually just stay fixed in place. Pause and ponder if you need to make sure that that line of reasoning feels good. This is the kind of thing I mentioned in the introduction. If you didn't have a", "translatedText": "Таким образом, если Дженнифер умножит эту матрицу на координаты вектора в своей системе, она вернет повернутую на 90 градусов версию этого вектора, выраженную в ее системе координат.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 709.82 }, { - "input": "In general, whenever you see an expression like A inverse times M times A, it suggests a mathematical sort of empathy.", + "input": "solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "В общем, всякий раз, когда вы видите такое выражение, как А, обратное умноженному на М, умноженному на А, это предполагает своего рода математическое сочувствие.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 714.54 }, { - "input": "That middle matrix represents a transformation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective.", + "input": "To see this in action, let's revisit the example from the start, with a matrix whose columns are 3, 0 and 1, 2. To find if a value lambda is an eigenvalue, subtract it from the diago", "translatedText": "Эта средняя матрица представляет собой некую трансформацию, как вы ее видите, а две внешние матрицы представляют собой сочувствие, сдвиг в перспективе.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/spanish/sentence_translations.json b/2016/change-of-basis/spanish/sentence_translations.json index 73a6c3e20..05d696c18 100644 --- a/2016/change-of-basis/spanish/sentence_translations.json +++ b/2016/change-of-basis/spanish/sentence_translations.json @@ -72,7 +72,7 @@ "end": 84.84 }, { - "input": "It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.", + "input": "st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is ti", "translatedText": "Mueve el vector base i-hat a las coordenadas 3, 0 y j-hat a 1, 2.", "from_community_srt": "i y j,son los llamados vectores base de nuestro sistema de coordenadas estándar.", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 91.04 }, { - "input": "So it's represented with a matrix whose columns are 3, 0, and 1, 2.", + "input": "ed up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actual", "translatedText": "Entonces se representa con una matriz cuyas columnas son 3, 0 y 1, 2.", "from_community_srt": "De lo que me gustaría hablar aquí es de la idea de utilizar un conjunto distinto de vectores base.", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 95.64 }, { - "input": "Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip.", + "input": "ly scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called th", "translatedText": "Concéntrese en lo que le hace a un vector en particular y piense en el alcance de ese vector, la línea que pasa por su origen y su punta.", "from_community_srt": "Por ejemplo, supongamos que tienes una amiga, Jennifer que usa un conjunto distinto de vectores base a los que llamaré b1 y b2 Su primer vector base b1 apunta hacia arriba y un poco hacia la derecha", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 104.16 }, { - "input": "Most vectors are going to get knocked off their span during the transformation.", + "input": "e basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a", "translatedText": "La mayoría de los vectores quedarán fuera de su alcance durante la transformación.", "n_reviews": 0, "start": 104.92, @@ -111,7 +111,7 @@ "end": 115.32 }, { - "input": "But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar.", + "input": "let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up.", "translatedText": "Pero algunos vectores especiales permanecen en su propio lapso, lo que significa que el efecto que tiene la matriz sobre dicho vector es simplemente estirarlo o aplastarlo, como un escalar.", "from_community_srt": "al usar nuestros vectores base i y j. Jennifer sin embargo describiria a este vector con las coordenadas [5/3,1/3] Lo que esto significa es que la forma particular con la que obtenemos ese vector", "n_reviews": 0, @@ -119,7 +119,7 @@ "end": 127.04 }, { - "input": "For this specific example, the basis vector i-hat is one such special vector.", + "input": "Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vecto", "translatedText": "Para este ejemplo específico, el vector base i-hat es uno de esos vectores especiales.", "from_community_srt": "usando sus dos vectores base es escalando b1 por 5/3,", "n_reviews": 0, @@ -151,14 +151,14 @@ "end": 164.04 }, { - "input": "It ends up getting stretched by a factor of 2.", + "input": "scale b1 by 5 thirds, scale b2 by 1 third, then add them both togethe", "translatedText": "Termina estirándose por un factor de 2.", "n_reviews": 0, "start": 164.66, "end": 167.14 }, { - "input": "And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2.", + "input": "r. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she", "translatedText": "Y nuevamente, la linealidad implicará que cualquier otro vector en la línea diagonal trazada por este tipo simplemente se estirará en un factor de 2.", "from_community_srt": "Siendo más precisos en este ejemplo su vector base b1 es aquel que nosotros describiríamos con las coordenadas [2,1] y su segundo vector base b2 es aquel que describiríamos como [-1,1].", "n_reviews": 0, @@ -166,7 +166,7 @@ "end": 178.22 }, { - "input": "And for this transformation, those are all the vectors with this special property of staying on their span.", + "input": "thinks of her first coordinate as scali For this specific example, the basis vector i-hat is one such special vector. The span of", "translatedText": "Y para esta transformación, esos son todos los vectores con esta propiedad especial de permanecer en su tramo.", "from_community_srt": "Pero es importante darnos cuenta que desde su perspectiva en su sistema esos vectores tienen las coordenadas [1,0] y [0,1]", "n_reviews": 0, @@ -174,7 +174,7 @@ "end": 185.18 }, { - "input": "Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2.", + "input": "i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis. What's more, because of the way linear transformations work,", "translatedText": "Los que están en el eje x se estiran en un factor de 3 y los que están en esta línea diagonal se estiran en un factor de 2.", "from_community_srt": "Son lo que define para ella el significado de las coordenadas [1,0] y [0,1].", "n_reviews": 0, @@ -190,7 +190,7 @@ "end": 198.08 }, { - "input": "As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation.", + "input": "n. A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc t, a way to visualize our coordinate system, and so it depends on our choice of basis", "translatedText": "Como ya habrás adivinado, estos vectores especiales se llaman vectores propios de la transformación, y cada vector propio tiene asociado lo que se llama un valor propio, que es simplemente el factor por el cual se estira o se aplasta durante la transformación.", "from_community_srt": "Miramos a los mismos vectores pero Jennifer usa palabras y números diferentes para describirlos. Permitidme una aclaración rápida sobre como represento las cosas aquí. Cuando animo el espacio bidimensional (R2) suelo utilizar esta cuadricula. pero esa cuadricula es solo una interpretación, una forma de visualizar nuestro sistema de coordenadas y por tanto depende de nuestra elección de la base.", "n_reviews": 0, @@ -206,7 +206,7 @@ "end": 225.94 }, { - "input": "In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half.", + "input": "nt as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It", "translatedText": "En otro ejemplo, podría tener un vector propio con valor propio negativo 1 mitad, lo que significa que el vector se voltea y se aplasta en un factor de 1 mitad.", "from_community_srt": "nada más que una herramienta visual que ayuda a captar el significado de sus coordenadas.", "n_reviews": 0, @@ -230,7 +230,7 @@ "end": 249.8 }, { - "input": "If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation.", + "input": "of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewhat during the transformation, k", "translatedText": "Si puedes encontrar un vector propio para esa rotación, un vector que permanece en su propio lapso, lo que has encontrado es el eje de rotación.", "from_community_srt": "Así que tras presentar todo esto la pregunta natural que surge es: ¿Cómo traducimos entre sistemas de coordenadas? Si, por ejemplo,", "n_reviews": 0, @@ -238,7 +238,7 @@ "end": 260.5 }, { - "input": "And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation.", + "input": "nocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", "translatedText": "Y es mucho más fácil pensar en una rotación 3D en términos de algún eje de rotación y un ángulo según el cual gira, en lugar de pensar en la matriz completa de 3x3 asociada con esa transformación.", "from_community_srt": "Jennifer describe el vector con coordenadas [-1,2] ¿Cuál sería ese vector en nustro sistema de coordenadas? ¿Cómo traducimos de su lenguaje al nuestro? Bueno,", "n_reviews": 0, @@ -254,14 +254,14 @@ "end": 285.86 }, { - "input": "This pattern shows up a lot in linear algebra.", + "input": "In fact, once you understand matrix vector multiplication as applying", "translatedText": "Este patrón aparece mucho en álgebra lineal.", "n_reviews": 0, "start": 288.08, "end": 290.02 }, { - "input": "With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors.", + "input": "a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvecto", "translatedText": "Con cualquier transformación lineal descrita por una matriz, se puede entender lo que hace leyendo las columnas de esta matriz como puntos de aterrizaje para los vectores base.", "from_community_srt": "Así que podemos calcular -1b1 + 2b2 por como están representados en nuestro sistema de coordenadas.", "n_reviews": 0, @@ -269,7 +269,7 @@ "end": 299.4 }, { - "input": "But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", + "input": "r with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformati", "translatedText": "Pero a menudo, una mejor manera de llegar al núcleo de lo que realmente hace la transformación lineal, menos dependiente de su sistema de coordenadas particular, es encontrar los vectores propios y los valores propios.", "from_community_srt": "Y operando así Obtienes el vector con coordenadas [-4,1]. Así es como nosotros describiríamos el vector en el que ella piensa como [-1,2].", "n_reviews": 0, @@ -285,7 +285,7 @@ "end": 326.02 }, { - "input": "Symbolically, here's what the idea of an eigenvector looks like.", + "input": "she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we thi", "translatedText": "Simbólicamente, así es como se ve la idea de un vector propio.", "from_community_srt": "una vez que entiendes la multiplicación matriz-vector como la aplicación de una aplicación lineal,", "n_reviews": 0, @@ -301,7 +301,7 @@ "end": 339.74 }, { - "input": "What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda.", + "input": "or for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotat", "translatedText": "Lo que esta expresión dice es que el producto matriz-vector, A multiplicado por v, da el mismo resultado que simplemente escalar el vector propio v por algún valor lambda.", "from_community_srt": "Una matriz cuyas columnas representan los vectores de la base de Jennifer puede ser interpretada como una transformación que mueve nuestros vectores base, i y j, las cosas en las que nosotros pensamos al decir [1,0] y [0,1],", "n_reviews": 0, @@ -325,7 +325,7 @@ "end": 370.54 }, { - "input": "So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda.", + "input": "never stretch or squish anything, so the length of the vector would remain the same. This pattern shows up a lot in linear algebra. With any linear transformation described by a matrix, you could understand what it's doing by reading of", "translatedText": "Entonces, comencemos reescribiendo ese lado derecho como una especie de multiplicación matriz-vector, usando una matriz que tiene el efecto de escalar cualquier vector por un factor lambda.", "from_community_srt": "Antes de la aplicación lineal pensamos en este vector como una combinación de nuestros vectores base -1i + 2j", "n_reviews": 0, @@ -333,7 +333,7 @@ "end": 380.62 }, { - "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else.", + "input": "f the columns of this matrix as the landing spots for basis vectors. But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", "translatedText": "Las columnas de dicha matriz representarán lo que sucede con cada vector base, y cada vector base simplemente se multiplica por lambda, por lo que esta matriz tendrá el número lambda en la diagonal, con ceros en el resto.", "from_community_srt": "Y la característica clave de una aplicación lineal es que el vector resultante será esa misma combinación lineal pero de los nuevos vectores base -1 veces el vector en el que termina i + 2 veces el vector en el que termina j", "n_reviews": 0, @@ -341,7 +341,7 @@ "end": 394.32 }, { - "input": "The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal.", + "input": "we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector w", "translatedText": "La forma común de escribir este tipo es factorizar esa lambda y escribirla como lambda multiplicada por i, donde i es la matriz de identidad con 1 en la diagonal.", "from_community_srt": "Así que lo que hace esta matriz es transformar nuestra equivocada interpretación de lo que nos decía Jennifer en el vector al que realmente se refería.", "n_reviews": 0, @@ -357,7 +357,7 @@ "end": 411.86 }, { - "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector.", + "input": "that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, with v as the eigenvector", "translatedText": "Entonces, lo que ahora tenemos es una nueva matriz, A menos lambda multiplicada por la identidad, y estamos buscando un vector v tal que esta nueva matriz multiplicada por v dé el vector cero.", "from_community_srt": "Geometricamente esta matriz cambia nuestra cuadricula en la de Jennifer pero numéricamente traduce un vector de su idioma a el nuestro. Lo que hizo que finalmente tuviera sentido para mi fue esta forma de pensar en como lleva nuestra interpretación incorrecta de lo que Jennifer quiere decir,", "n_reviews": 0, @@ -381,7 +381,7 @@ "end": 433.64 }, { - "input": "And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", + "input": "and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Je", "translatedText": "Y si miras los capítulos 5 y 6, sabrás que la única forma en que es posible que el producto de una matriz con un vector distinto de cero se convierta en cero es si la transformación asociada con esa matriz reduce el espacio a una dimensión inferior.", "from_community_srt": "¿Qué ocurre en el sentido contrario? En el ejemplo que he usado al principio del video cuando tenemos un vector de coordenadas [3,2] en nuestro sistema ¿Cómo he calculado que tendría las coordenadas [5/3,", "n_reviews": 0, @@ -413,7 +413,7 @@ "end": 470.28 }, { - "input": "As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes.", + "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose c", "translatedText": "A medida que cambia ese valor de lambda, la matriz misma cambia y, por lo tanto, cambia el determinante de la matriz.", "from_community_srt": "En practica, especialmente cuando trabajas en más de dos dimensiones usarías un ordenador (Con Mathematica aibalahostia) para calcular la matriz inversa.", "n_reviews": 0, @@ -451,14 +451,14 @@ "end": 498.6 }, { - "input": "So this is kind of a lot, but let's unravel what this is saying.", + "input": "And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's importa", "translatedText": "Esto es mucho, pero analicemos lo que dice.", "n_reviews": 0, "start": 500.08, "end": 502.96 }, { - "input": "When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line.", + "input": "nt that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication So let's start by rewriting", "translatedText": "Cuando lambda es igual a 1, la matriz A menos lambda multiplicada por la identidad aplasta el espacio en una línea.", "from_community_srt": "lo que nos da como resultado [5/3,1/3] Asi es como,", "n_reviews": 0, @@ -466,7 +466,7 @@ "end": 509.56 }, { - "input": "That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector.", + "input": "that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a fact", "translatedText": "Eso significa que hay un vector v distinto de cero tal que A menos lambda multiplicado por la identidad multiplicado por v es igual al vector cero.", "from_community_srt": "en resumidas cuentas 'traducimos' la descripcion de vectores individuales entre sistemas de coordenadas. La matriz cuyas columnas representan los vectores de la base de Jennifer pero escritos en nuestro sistema de coordenadas", "n_reviews": 0, @@ -474,7 +474,7 @@ "end": 518.56 }, { - "input": "And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A.", + "input": "or of lambda. The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else. the columns of our matrix.", "translatedText": "Y recuerde, la razón por la que esto nos importa es porque significa que A multiplicado por v es igual a lambda multiplicado por v, lo que se puede interpretar como que el vector v es un vector propio de A, que permanece en su propio lapso durante la transformación A.", "from_community_srt": "\"traduce\" vectores de su lenguaje al nuestro y la inversa de dicha matriz hace lo contrario. Pero los vectores no son la única cosa que describimos con coordenadas.", "n_reviews": 0, @@ -482,7 +482,7 @@ "end": 537.28 }, { - "input": "In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place.", + "input": "But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following", "translatedText": "En este ejemplo, el valor propio correspondiente es 1, por lo que v en realidad permanecería fijo en su lugar.", "from_community_srt": "Para la parte que viene es importante que esteis cómodos representando transformaciones lineales con matrices y que sepáis como la multiplicación de matrices corresponde a la composición sucesiva de estas transformaciones.", "n_reviews": 0, @@ -497,7 +497,7 @@ "end": 549.5 }, { - "input": "This is the kind of thing I mentioned in the introduction.", + "input": "-hat in the first pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v.", "translatedText": "Este es el tipo de cosas que mencioné en la introducción.", "from_community_srt": "Definitivamente tomate un momento y echa un ojo a los capitulos 3 y 4 si algo de esto parece complicado.", "n_reviews": 0, @@ -505,7 +505,7 @@ "end": 555.64 }, { - "input": "If you didn't have a solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to describe those landing spots in her language. Here's a common way to think", "translatedText": "Si no tuvieras una comprensión sólida de los determinantes y de por qué se relacionan con sistemas lineales de ecuaciones que tienen soluciones distintas de cero, una expresión como esta parecería completamente inesperada.", "from_community_srt": "Considera un transformación lineal como un giro 90 grados en la dirección de las agujas del reloj. Cuando representamos esta transformación con la matriz correspondiente seguimos los vectores i y j y a donde va cada uno tras aplicar la aplicación lineal.", "n_reviews": 0, @@ -545,7 +545,7 @@ "end": 608.84 }, { - "input": "To figure out what the eigenvectors are that actually have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero.", + "input": "To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry. Now imagine tweaking lambda, turning a knob to change its value. As that value of lambda changes, the matrix itself change", "translatedText": "Para descubrir cuáles son los vectores propios que realmente tienen uno de estos valores propios, digamos que lambda es igual a 2, conecte ese valor de lambda a la matriz y luego resuelva qué vectores esta matriz alterada diagonalmente envía a cero.", "from_community_srt": "Estas columnas representan a dónde van nuestros vectores de la base i y j pero la matriz que Jennifer quiere debería representar los vectores de su base y los puntos en los que aterrizan deben ser descritos en su lenguaje también. Esta es una manera común de pensar en como se hace esto.", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 681.94 }, { - "input": "The only roots of that polynomial are the imaginary numbers, i and negative i.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the oute", "translatedText": "Las únicas raíces de ese polinomio son los números imaginarios, i y i negativo.", "from_community_srt": "Toma un vector en su idioma y da por salida la versión transformada de ese vector en su idioma", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 699.82 }, { - "input": "This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1.", + "input": "meone else sees it. For those of you wondering why we care about alternate coordinate systems, the next vi", "translatedText": "Esto fija i-hat en su lugar y mueve j-hat 1, por lo que su matriz tiene las columnas 1, 0 y 1, 1.", "from_community_srt": "Así que si Jennifer multiplica dicha matriz por las coordenadas de cualquier vector en su sistema", "n_reviews": 0, @@ -679,7 +679,7 @@ "end": 726.54 }, { - "input": "And the only root of this expression is lambda equals 1.", + "input": "he identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals la", "translatedText": "Y la única raíz de esta expresión es lambda igual a 1.", "from_community_srt": "La matriz del medio representa una transformación de cierto tipo, de la forma en que tú la ves y las otras dos matrices exteriores representan la conexión,", "n_reviews": 0, @@ -687,7 +687,7 @@ "end": 732.86 }, { - "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.", + "input": "mbda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corresponding eigenvalue is", "translatedText": "Esto se alinea con lo que vemos geométricamente, que todos los vectores propios tienen valor propio 1.", "from_community_srt": "el cambio de perspectiva, y la matriz producto completa representa la misma transfomación", "n_reviews": 0, @@ -740,7 +740,7 @@ "end": 796.38 }, { - "input": "For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2.", + "input": "f equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Por ejemplo, tal vez i-hat esté escalado por 1 negativo y j-hat esté escalado por 2.", "n_reviews": 0, "start": 797.12, @@ -761,7 +761,7 @@ "end": 825.42 }, { - "input": "And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.", + "input": "nd compute the determinant. Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda. Since lambda can only be an eigenvalue i", "translatedText": "Y la forma de interpretar esto es que todos los vectores base son vectores propios, siendo las entradas diagonales de esta matriz sus valores propios.", "n_reviews": 0, "start": 825.84, @@ -775,14 +775,14 @@ "end": 841.06 }, { - "input": "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", + "input": "u can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3. To figure out what the eigenvectors are that actu", "translatedText": "Uno de los más importantes es que es más fácil calcular lo que sucederá si multiplicas esta matriz por sí misma muchas veces.", "n_reviews": 0, "start": 841.78, "end": 848.34 }, { - "input": "Since all one of these matrices does is scale each basis vector by some eigenvalue, applying that matrix many times, say 100 times, is just going to correspond to scaling each basis vector by the 100th power of the corresponding eigenvalue.", + "input": "ally have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero. If you computed this the way you would any other linear system, you'd see that the solutions are all the vectors on the diagonal line spanned by negative 1, 1.", "translatedText": "Dado que lo único que hace una de estas matrices es escalar cada vector base según algún valor propio, aplicar esa matriz muchas veces, digamos 100 veces, corresponderá a escalar cada vector base según la centésima potencia del valor propio correspondiente.", "n_reviews": 0, "start": 849.42, @@ -796,7 +796,7 @@ "end": 869.68 }, { - "input": "Really, try it for a moment.", + "input": "x, 3, 0, 1, 2, has the effect of stretching all those vectors by a factor of 2.", "translatedText": "De verdad, pruébalo por un momento.", "n_reviews": 0, "start": 869.68, @@ -824,14 +824,14 @@ "end": 896.54 }, { - "input": "I talked about change of basis last video, but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", + "input": "is, notice what happens. Its matrix has columns 0, 1 and negative 1, 0. Subtract off lambda from the diagonal elements and look for when the determinant is zero. In this case, you get the polynomial lambda squared plus 1. The only roots of that polynomia", "translatedText": "Hablé sobre el cambio de base en el último video, pero aquí haré un recordatorio súper rápido de cómo expresar una transformación actualmente escrita en nuestro sistema de coordenadas en un sistema diferente.", "n_reviews": 0, "start": 897.14, "end": 907.04 }, { - "input": "Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors, then make those coordinates the columns of a matrix, known as the change of basis matrix.", + "input": "l are the imaginary numbers, i and negative i. The fact that there are no real number solutions indicates that there are no eigenvectors. Another pretty interesting example worth holding in the back of your mind is a shear. This fixes i-hat in place and moves j-hat 1 over, so its mat", "translatedText": "Tome las coordenadas de los vectores que desea usar como una nueva base, que en este caso significa nuestros dos vectores propios, luego convierta esas coordenadas en las columnas de una matriz, conocida como matriz de cambio de base.", "n_reviews": 0, "start": 908.44, @@ -852,7 +852,7 @@ "end": 946.68 }, { - "input": "This is because it represents working in a coordinate system where what happens to the basis vectors is that they get scaled during the transformation.", + "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1. Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a lin", "translatedText": "Esto se debe a que representa trabajar en un sistema de coordenadas donde lo que sucede con los vectores base es que se escalan durante la transformación.", "n_reviews": 0, "start": 946.86, @@ -866,28 +866,28 @@ "end": 961.56 }, { - "input": "So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis, compute the 100th power in that system, then convert back to our standard system.", + "input": "A simple example is a matrix that scales everything by 2. The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue. Now is another good time to pause and ponder some of this before I move on to the last topic.", "translatedText": "Entonces, si, por ejemplo, necesitara calcular la potencia número 100 de esta matriz, sería mucho más fácil cambiar a una base propia, calcular la potencia número 100 en ese sistema y luego volver a convertir a nuestro sistema estándar.", "n_reviews": 0, "start": 962.34, "end": 975.68 }, { - "input": "You can't do this with all transformations.", + "input": "I want to finish off here with the idea of an eigenbasis, which relies heavily on ideas from the last video. Take a look at what h", "translatedText": "No puedes hacer esto con todas las transformaciones.", "n_reviews": 0, "start": 976.62, "end": 978.32 }, { - "input": "A shear, for example, doesn't have enough eigenvectors to span the full space.", + "input": "appens if our basis vectors just so happen to be eigenvectors. For example, maybe i-hat is scale", "translatedText": "Un corte, por ejemplo, no tiene suficientes vectores propios para abarcar todo el espacio.", "n_reviews": 0, "start": 978.32, "end": 982.98 }, { - "input": "But if you can find an eigenbasis, it makes matrix operations really lovely.", + "input": "d by negative 1 and j-hat is scaled by 2. Writing their new coordinates as the columns of a matrix, notice t", "translatedText": "Pero si puedes encontrar una base propia, las operaciones matriciales son realmente hermosas.", "n_reviews": 0, "start": 983.46, diff --git a/2016/change-of-basis/swedish/sentence_translations.json b/2016/change-of-basis/swedish/sentence_translations.json index 81dfaab83..0eddebba7 100644 --- a/2016/change-of-basis/swedish/sentence_translations.json +++ b/2016/change-of-basis/swedish/sentence_translations.json @@ -81,7 +81,7 @@ "end": 84.84 }, { - "input": "It moves the basis vector i-hat to the coordinates 3, 0, and j-hat to 1, 2.", + "input": "st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is ti", "translatedText": "Den flyttar basvektorn i-hat till koordinaterna 3, 0 och j-hat till 1, 2.", "model": "google_nmt", "from_community_srt": "i-hatt och j-hatt, kallas för basvektorer i vårt standardkoordinatsystem.", @@ -90,7 +90,7 @@ "end": 91.04 }, { - "input": "So it's represented with a matrix whose columns are 3, 0, and 1, 2.", + "input": "ed up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actual", "translatedText": "Så det representeras med en matris vars kolumner är 3, 0 och 1, 2.", "model": "google_nmt", "from_community_srt": "Vad jag skulle vilja prata om här är idén att använda en annan uppsättning vektorer som basvektorer.", @@ -99,7 +99,7 @@ "end": 95.64 }, { - "input": "Focus in on what it does to one particular vector, and think about the span of that vector, the line passing through its origin and its tip.", + "input": "ly scale. Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called th", "translatedText": "Fokusera på vad den gör med en viss vektor och tänk på spännvidden för den vektorn, linjen som går genom dess ursprung och dess spets.", "model": "google_nmt", "from_community_srt": "Till exempel, låt oss säga att du har en vän, Jennifer, som använder en annan uppsättning basvektorer, som jag kommer kalla b1 och b2.", @@ -108,7 +108,7 @@ "end": 104.16 }, { - "input": "Most vectors are going to get knocked off their span during the transformation.", + "input": "e basis vectors of our standard coordinate system. What I'd like to talk about here is the idea of using a", "translatedText": "De flesta vektorer kommer att slås ur sitt spann under transformationen.", "model": "google_nmt", "from_community_srt": "Hennes första basvektor b1 pekar uppåt och lite till höger, och hennes andra vektor b2 pekar vänster och uppåt.", @@ -126,7 +126,7 @@ "end": 115.32 }, { - "input": "But some special vectors do remain on their own span, meaning the effect that the matrix has on such a vector is just to stretch it or squish it, like a scalar.", + "input": "let's say you have a friend, Jennifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to the right a little bit, and her second vector, b2, points left and up.", "translatedText": "Men vissa speciella vektorer förblir på sitt eget span, vilket betyder att effekten som matrisen har på en sådan vektor bara är att sträcka den eller klämma ihop den, som en skalär.", "model": "google_nmt", "from_community_srt": "Den som du och jag skulle beskriva med koordinaterna [3,2] i våra basvektorer i-hatt och j-hatt. Jennifer skulle faktiskt beskriva denna vektor med koordinaterna [5/3, 1/3].", @@ -135,7 +135,7 @@ "end": 127.04 }, { - "input": "For this specific example, the basis vector i-hat is one such special vector.", + "input": "Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vecto", "translatedText": "För detta specifika exempel är basvektorn i-hat en sådan speciell vektor.", "model": "google_nmt", "from_community_srt": "Vad detta innebär är att det speciella sättet att få den vektorn med hjälp av hennes två basvektorer", @@ -171,7 +171,7 @@ "end": 164.04 }, { - "input": "It ends up getting stretched by a factor of 2.", + "input": "scale b1 by 5 thirds, scale b2 by 1 third, then add them both togethe", "translatedText": "Det slutar med att det sträcks ut med en faktor 2.", "model": "google_nmt", "n_reviews": 0, @@ -179,7 +179,7 @@ "end": 167.14 }, { - "input": "And again, linearity is going to imply that any other vector on the diagonal line spanned by this guy is just going to get stretched out by a factor of 2.", + "input": "r. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to describe a vector, she", "translatedText": "Och återigen, linearitet kommer att innebära att vilken annan vektor som helst på den diagonala linjen som spänner över av den här killen bara kommer att sträckas ut med en faktor 2.", "model": "google_nmt", "from_community_srt": "För att vara lite mer precis angående upplägget här: hennes första basvektor b1 är något vi skulle beskriva med koordinaterna [2,1] och hennes andra basvektor b2 är något vi skulle beskriva som [-1,1].", @@ -188,7 +188,7 @@ "end": 178.22 }, { - "input": "And for this transformation, those are all the vectors with this special property of staying on their span.", + "input": "thinks of her first coordinate as scali For this specific example, the basis vector i-hat is one such special vector. The span of", "translatedText": "Och för denna transformation är det alla vektorer med den här speciella egenskapen att hålla sig på sin spännvidd.", "model": "google_nmt", "from_community_srt": "Men det är viktigt att inse att från hennes perspektiv i hennes system har de vektorerna koordinaterna [1,0] och [0,1].", @@ -197,7 +197,7 @@ "end": 185.18 }, { - "input": "Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2.", + "input": "i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis. What's more, because of the way linear transformations work,", "translatedText": "De på x-axeln sträcks ut med en faktor 3, och de på den här diagonala linjen sträcks ut med en faktor 2.", "model": "google_nmt", "from_community_srt": "De är vad som definierar meningen av koordinaterna [1,0] och [0,1] i hennes värld.", @@ -215,7 +215,7 @@ "end": 198.08 }, { - "input": "As you might have guessed by now, these special vectors are called the eigenvectors of the transformation, and each eigenvector has associated with it what's called an eigenvalue, which is just the factor by which it's stretched or squished during the transformation.", + "input": "n. A slightly sneakier vector that remains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. But that grid is just a construc t, a way to visualize our coordinate system, and so it depends on our choice of basis", "translatedText": "Som du kanske har gissat vid det här laget kallas dessa speciella vektorer för transformationens egenvektorer, och varje egenvektor har associerat med det vad som kallas ett egenvärde, vilket bara är den faktor med vilken den sträcks ut eller kläms ihop under transformationen.", "model": "google_nmt", "from_community_srt": "Vi tittar alla på samma vektorer i rummet men Jennifer använder andra ord och siffror för att beskriva dem. Låt mig säga några korta ord om hur jag representerar saker här när jag animerar 2D-rummet använder jag vanligtvis det här rutnätet Men det rutnätet är bara en konstruktion ett sätt att visualisera vårt koordinatsystem och därför beror det på vårt val av bas.", @@ -233,7 +233,7 @@ "end": 225.94 }, { - "input": "In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half.", + "input": "nt as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It", "translatedText": "I ett annat exempel kan du ha en egenvektor med egenvärde negativt 1 halv, vilket betyder att vektorn vänds och kläms med en faktor på 1 halv.", "model": "google_nmt", "from_community_srt": "för att hjälpa till att visa vad hennes koordinater betyder.", @@ -260,7 +260,7 @@ "end": 249.8 }, { - "input": "If you can find an eigenvector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation.", + "input": "of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewhat during the transformation, k", "translatedText": "Om du kan hitta en egenvektor för den rotationen, en vektor som stannar kvar på sitt eget span, är det du har hittat rotationsaxeln.", "model": "google_nmt", "from_community_srt": "Så, efter att allt detta är upprättat är det ganska naturligt att fråga: hur översätter vi mellan koordinatsystem? Om, till exempel,", @@ -269,7 +269,7 @@ "end": 260.5 }, { - "input": "And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking about the full 3x3 matrix associated with that transformation.", + "input": "nocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", "translatedText": "Och det är mycket lättare att tänka på en 3D-rotation i termer av någon rotationsaxel och en vinkel med vilken den roterar, snarare än att tänka på hela 3x3-matrisen som är förknippad med den transformationen.", "model": "google_nmt", "from_community_srt": "Jennifer beskriver en vektor med koordinaterna [-1,2] vad skulle det vara i vårt koordinatsystem? Hur översätter vi från hennes språk till vårt? Nå, det våra koordinater säger är att denna vektor är -1 b1 + 2 b2.", @@ -287,7 +287,7 @@ "end": 285.86 }, { - "input": "This pattern shows up a lot in linear algebra.", + "input": "In fact, once you understand matrix vector multiplication as applying", "translatedText": "Detta mönster visar sig mycket i linjär algebra.", "model": "google_nmt", "n_reviews": 0, @@ -295,7 +295,7 @@ "end": 290.02 }, { - "input": "With any linear transformation described by a matrix, you could understand what it's doing by reading off the columns of this matrix as the landing spots for basis vectors.", + "input": "a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvecto", "translatedText": "Med vilken linjär transformation som helst som beskrivs av en matris kan du förstå vad den gör genom att läsa av kolumnerna i denna matris som landningspunkter för basvektorer.", "model": "google_nmt", "from_community_srt": "Så vi kan faktiskt beräkna -1 b1 + 2 b2 som de representeras i vårt i koordinatsystem.", @@ -304,7 +304,7 @@ "end": 299.4 }, { - "input": "But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", + "input": "r with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennifer's basis vectors can be thought of as a transformati", "translatedText": "Men ofta är ett bättre sätt att komma till kärnan i vad den linjära transformationen faktiskt gör, mindre beroende av ditt specifika koordinatsystem, att hitta egenvektorerna och egenvärdena.", "model": "google_nmt", "from_community_srt": "Och om du räknar ut detta får du en vektor med koordinaterna [-4,1]. Det är alltså hur vi skulle beskriva den vektor hon tänker på som [-1,2]. Denna process,", @@ -322,7 +322,7 @@ "end": 326.02 }, { - "input": "Symbolically, here's what the idea of an eigenvector looks like.", + "input": "she thinks of when she says 1, 0 and 0, 1. To show how this works, let's walk through what it would mean to take the vector that we thi", "translatedText": "Symboliskt, så här ser idén med en egenvektor ut.", "model": "google_nmt", "from_community_srt": "Faktum är att när du förstått att matrismultiplikation är att applicera en speciell linjär transformation", @@ -340,7 +340,7 @@ "end": 339.74 }, { - "input": "What this expression is saying is that the matrix-vector product, A times v, gives the same result as just scaling the eigenvector v by some value lambda.", + "input": "or for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotat", "translatedText": "Vad detta uttryck säger är att matris-vektorprodukten, A gånger v, ger samma resultat som att bara skala egenvektorn v med något värde lambda.", "model": "google_nmt", "from_community_srt": "En matris vars kolonner representeras Jennifers basvektorer kan ses som en transformation som flyttar våra basvektorer,", @@ -367,7 +367,7 @@ "end": 370.54 }, { - "input": "So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda.", + "input": "never stretch or squish anything, so the length of the vector would remain the same. This pattern shows up a lot in linear algebra. With any linear transformation described by a matrix, you could understand what it's doing by reading of", "translatedText": "Så låt oss börja med att skriva om den högra sidan som någon slags matris-vektormultiplikation, med hjälp av en matris som har effekten att skala vilken vektor som helst med en faktor lambda.", "model": "google_nmt", "from_community_srt": "Innan den linjära transformationen ser vi denna vektor som en speciell linjärkombination av våra basvektorer -1 x i-hatt + 2x j-hatt.", @@ -376,7 +376,7 @@ "end": 380.62 }, { - "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else.", + "input": "f the columns of this matrix as the landing spots for basis vectors. But often, a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues.", "translatedText": "Kolumnerna i en sådan matris kommer att representera vad som händer med varje basvektor, och varje basvektor multipliceras helt enkelt med lambda, så denna matris kommer att ha talet lambda nedåt diagonalen, med nollor överallt annars.", "model": "google_nmt", "from_community_srt": "Och den viktigaste egenskapen hos en linjärkombination är att den resulterande vektorn kommer vara samma linjärkombination men av de nya basvektorerna -1 gånger stället där i-hat landar plus 2 gånger stället där j-hatt landar.", @@ -385,7 +385,7 @@ "end": 394.32 }, { - "input": "The common way to write this guy is to factor that lambda out and write it as lambda times i, where i is the identity matrix with 1s down the diagonal.", + "input": "we get using the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector w", "translatedText": "Det vanliga sättet att skriva den här killen är att faktorisera den lambdan och skriva den som lambda gånger i, där i är identitetsmatrisen med 1:or nedåt diagonalen.", "model": "google_nmt", "from_community_srt": "Så vad denna matris gör är att transformera vår missuppfattning av vad Jennifer menar till den vektor hon faktiskt refererar till.", @@ -403,7 +403,7 @@ "end": 411.86 }, { - "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector.", + "input": "that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, with v as the eigenvector", "translatedText": "Så vad vi nu har är en ny matris, A minus lambda gånger identiteten, och vi letar efter en vektor v så att denna nya matris gånger v ger nollvektorn.", "model": "google_nmt", "from_community_srt": "Men numeriskt översätter den en vektor beskriven i hennes språk till vårt språk. Vad som fick bitarna att falla på plats för mig var att tänka på hur det tar vår missuppfattning av vad Jennifer menar,", @@ -429,7 +429,7 @@ "end": 433.64 }, { - "input": "And if you watch chapter 5 and 6, you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", + "input": "and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Je", "translatedText": "Och om du tittar på kapitel 5 och 6, kommer du att veta att det enda sättet det är möjligt för produkten av en matris med en vektor som inte är noll att bli noll är om transformationen som är associerad med den matrisen klämmer ihop rymden till en lägre dimension.", "model": "google_nmt", "from_community_srt": "Åt andra hållet då? I exemplet jag använde tidigare i denna videon när jag har vektorn med koordinaterna [3,2] i vårt system., Hur räknade jag ut att det skulle ha koordinaterna i [5/3,1/3] i Jennifers system.", @@ -464,7 +464,7 @@ "end": 470.28 }, { - "input": "As that value of lambda changes, the matrix itself changes, and so the determinant of the matrix changes.", + "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems. The matrix whose c", "translatedText": "När värdet på lambda ändras ändras själva matrisen, och så ändras matrisens determinant.", "model": "google_nmt", "from_community_srt": "I praktiken, speciellt om du arbetar i mer än två dimensioner skulle du använda en dator till att beräkna matrisen som faktiskt representerar denna invers.", @@ -508,7 +508,7 @@ "end": 498.6 }, { - "input": "So this is kind of a lot, but let's unravel what this is saying.", + "input": "And the inverse matrix does the opposite. But vectors aren't the only thing that we describe using coordinates. For this next part, it's importa", "translatedText": "Så det här är ganska mycket, men låt oss reda ut vad det här säger.", "model": "google_nmt", "from_community_srt": "multiplicerar vi inversen av basbytesmatrisen med vektorn [3,2] vilket visar sig bli [5/3,1/3].", @@ -517,7 +517,7 @@ "end": 502.96 }, { - "input": "When lambda equals 1, the matrix A minus lambda times the identity squishes space onto a line.", + "input": "nt that you're all comfortable representing transformations with matrices, and that you know how matrix multiplication So let's start by rewriting", "translatedText": "När lambda är lika med 1, pressar matrisen A minus lambda gånger identiteten utrymme på en linje.", "model": "google_nmt", "n_reviews": 0, @@ -525,7 +525,7 @@ "end": 509.56 }, { - "input": "That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector.", + "input": "that right-hand side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a fact", "translatedText": "Det betyder att det finns en vektor v som inte är noll så att A minus lambda gånger identiteten gånger v är lika med nollvektorn.", "model": "google_nmt", "from_community_srt": "Det är i ett nötskal hur man översätter beskrivningen av individuella vektorer fram och tillbaka mellan koordinatsystem. Matrisen vars kolonner representerar Jennifers basvektorer men är skriven i våra koordinater", @@ -534,7 +534,7 @@ "end": 518.56 }, { - "input": "And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A.", + "input": "or of lambda. The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this matrix will have the number lambda down the diagonal, with zeros everywhere else. the columns of our matrix.", "translatedText": "Och kom ihåg, anledningen till att vi bryr oss om det är för att det betyder att A gånger v är lika med lambda gånger v, vilket du kan läsa som att vektorn v är en egenvektor till A, som stannar på sitt eget span under transformationen A.", "model": "google_nmt", "from_community_srt": "översätter vektorer från hennes språk till vårt språk. Och matrisens inverse gör det motsatta. Men vektorer är inte det enda vi beskriver med hjälp av koordinater.", @@ -543,7 +543,7 @@ "end": 537.28 }, { - "input": "In this example, the corresponding eigenvalue is 1, so v would actually just stay fixed in place.", + "input": "But this representation is heavily tied up in our choice of basis vectors, from the fact that we're following", "translatedText": "I det här exemplet är motsvarande egenvärde 1, så v skulle faktiskt bara stanna kvar på plats.", "model": "google_nmt", "from_community_srt": "För nästa del är det viktigt att du är bekväm med att representera transformationer med matriser och att du vet hur matrismultiplikation korresponderar mot att sammansätta påföljande transformationer.", @@ -560,7 +560,7 @@ "end": 549.5 }, { - "input": "This is the kind of thing I mentioned in the introduction.", + "input": "-hat in the first pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v.", "translatedText": "Det är sånt jag nämnde i inledningen.", "model": "google_nmt", "from_community_srt": "Du borde absolut pausa och ta en titt på kapitel 3 och 4 om något av detta känns svårt.", @@ -569,7 +569,7 @@ "end": 555.64 }, { - "input": "If you didn't have a solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to describe those landing spots in her language. Here's a common way to think", "translatedText": "Om du inte hade ett gediget grepp om determinanter och varför de relaterar till linjära ekvationssystem som har lösningar som inte är noll, skulle ett uttryck som detta kännas helt ur det blå.", "model": "google_nmt", "from_community_srt": "Betrakta en linjär transformation så som en 90-gradig moturs rotation. När du och jag representerar detta med matrisen följer vi var basvektorerna i-hatt och j-hatt hamnar.", @@ -614,7 +614,7 @@ "end": 608.84 }, { - "input": "To figure out what the eigenvectors are that actually have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero.", + "input": "To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtracting off a variable amount, lambda, from each diagonal entry. Now imagine tweaking lambda, turning a knob to change its value. As that value of lambda changes, the matrix itself change", "translatedText": "För att ta reda på vilka egenvektorer det är som faktiskt har ett av dessa egenvärden, säg att lambda är lika med 2, koppla in det värdet på lambda till matrisen och lös sedan för vilka vektorer denna diagonalt förändrade matris skickar till noll.", "model": "google_nmt", "from_community_srt": "De kolonnerna representerar var våra basvektorer i-hatt och j-hatt hamnar. Men matrisen Jennifer vill ha borde representera var hennes basvektorer landar och den behöver beskriva dessa landningspunkter i hennes språk. Här är ett vanligt sätt att tänka på hur detta görs.", @@ -704,7 +704,7 @@ "end": 681.94 }, { - "input": "The only roots of that polynomial are the imaginary numbers, i and negative i.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the oute", "translatedText": "De enda rötterna till det polynomet är de imaginära talen, i och negativa i.", "model": "google_nmt", "from_community_srt": "Den tar in en vektor i hennes språk och spottar ut den transformerade versionen av vektorn i hennes språk.", @@ -730,7 +730,7 @@ "end": 699.82 }, { - "input": "This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns 1, 0 and 1, 1.", + "input": "meone else sees it. For those of you wondering why we care about alternate coordinate systems, the next vi", "translatedText": "Detta fixerar i-hat på plats och flyttar j-hat 1 över, så dess matris har kolumnerna 1, 0 och 1, 1.", "model": "google_nmt", "from_community_srt": "har produkten av dessa tre matrisen om du arbetar genom det kolonnerna [1/3,5/3] och [-2/3,-1/3]-", @@ -766,7 +766,7 @@ "end": 726.54 }, { - "input": "And the only root of this expression is lambda equals 1.", + "input": "he identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals la", "translatedText": "Och den enda roten till detta uttryck är lambda lika med 1.", "model": "google_nmt", "from_community_srt": "Mittenmatrisen representerar någon typ av transformation,", @@ -775,7 +775,7 @@ "end": 732.86 }, { - "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.", + "input": "mbda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corresponding eigenvalue is", "translatedText": "Detta stämmer överens med vad vi ser geometriskt, att alla egenvektorer har egenvärde 1.", "model": "google_nmt", "from_community_srt": "som du ser den och de två yttre matriserna representerar empatin, skiftet av perspektiv och hela matrisen representerar samma transformation men som någon annan ser den.", @@ -835,7 +835,7 @@ "end": 796.38 }, { - "input": "For example, maybe i-hat is scaled by negative 1 and j-hat is scaled by 2.", + "input": "f equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Till exempel kanske i-hat skalas med negativ 1 och j-hat skalas med 2.", "model": "google_nmt", "n_reviews": 0, @@ -859,7 +859,7 @@ "end": 825.42 }, { - "input": "And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.", + "input": "nd compute the determinant. Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda. Since lambda can only be an eigenvalue i", "translatedText": "Och sättet att tolka detta är att alla basvektorer är egenvektorer, där de diagonala ingångarna i denna matris är deras egenvärden.", "model": "google_nmt", "n_reviews": 0, @@ -875,7 +875,7 @@ "end": 841.06 }, { - "input": "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", + "input": "u can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3. To figure out what the eigenvectors are that actu", "translatedText": "En stor sak är att det är lättare att beräkna vad som kommer att hända om du multiplicerar den här matrisen med sig själv en hel massa gånger.", "model": "google_nmt", "n_reviews": 0, @@ -883,7 +883,7 @@ "end": 848.34 }, { - "input": "Since all one of these matrices does is scale each basis vector by some eigenvalue, applying that matrix many times, say 100 times, is just going to correspond to scaling each basis vector by the 100th power of the corresponding eigenvalue.", + "input": "ally have one of these eigenvalues, say lambda equals 2, plug in that value of lambda to the matrix and then solve for which vectors this diagonally altered matrix sends to zero. If you computed this the way you would any other linear system, you'd see that the solutions are all the vectors on the diagonal line spanned by negative 1, 1.", "translatedText": "Eftersom allt en av dessa matriser gör är att skala varje basvektor med något egenvärde, att tillämpa den matrisen många gånger, säg 100 gånger, kommer bara att motsvara att skala varje basvektor med 100:e potensen av motsvarande egenvärde.", "model": "google_nmt", "n_reviews": 0, @@ -899,7 +899,7 @@ "end": 869.68 }, { - "input": "Really, try it for a moment.", + "input": "x, 3, 0, 1, 2, has the effect of stretching all those vectors by a factor of 2.", "translatedText": "Verkligen, prova det ett ögonblick.", "model": "google_nmt", "n_reviews": 0, @@ -931,7 +931,7 @@ "end": 896.54 }, { - "input": "I talked about change of basis last video, but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", + "input": "is, notice what happens. Its matrix has columns 0, 1 and negative 1, 0. Subtract off lambda from the diagonal elements and look for when the determinant is zero. In this case, you get the polynomial lambda squared plus 1. The only roots of that polynomia", "translatedText": "Jag pratade om ändring av grund förra videon, men jag ska gå igenom en supersnabb påminnelse här om hur man uttrycker en transformation som för närvarande är skriven i vårt koordinatsystem till ett annat system.", "model": "google_nmt", "n_reviews": 0, @@ -939,7 +939,7 @@ "end": 907.04 }, { - "input": "Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors, then make those coordinates the columns of a matrix, known as the change of basis matrix.", + "input": "l are the imaginary numbers, i and negative i. The fact that there are no real number solutions indicates that there are no eigenvectors. Another pretty interesting example worth holding in the back of your mind is a shear. This fixes i-hat in place and moves j-hat 1 over, so its mat", "translatedText": "Ta koordinaterna för vektorerna som du vill använda som en ny bas, vilket i det här fallet betyder våra två egenvektorer, gör sedan dessa koordinater till kolumnerna i en matris, känd som förändring av basmatrisen.", "model": "google_nmt", "n_reviews": 0, @@ -963,7 +963,7 @@ "end": 946.68 }, { - "input": "This is because it represents working in a coordinate system where what happens to the basis vectors is that they get scaled during the transformation.", + "input": "This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1. Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a lin", "translatedText": "Detta beror på att det representerar att arbeta i ett koordinatsystem där det som händer med basvektorerna är att de skalas under transformationen.", "model": "google_nmt", "n_reviews": 0, @@ -979,7 +979,7 @@ "end": 961.56 }, { - "input": "So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis, compute the 100th power in that system, then convert back to our standard system.", + "input": "A simple example is a matrix that scales everything by 2. The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue. Now is another good time to pause and ponder some of this before I move on to the last topic.", "translatedText": "Så om du till exempel behövde beräkna den 100:e potensen av denna matris, skulle det vara mycket lättare att ändra till en egenbas, beräkna den 100:e potensen i det systemet och sedan konvertera tillbaka till vårt standardsystem.", "model": "google_nmt", "n_reviews": 0, @@ -987,7 +987,7 @@ "end": 975.68 }, { - "input": "You can't do this with all transformations.", + "input": "I want to finish off here with the idea of an eigenbasis, which relies heavily on ideas from the last video. Take a look at what h", "translatedText": "Du kan inte göra det här med alla transformationer.", "model": "google_nmt", "n_reviews": 0, @@ -995,7 +995,7 @@ "end": 978.32 }, { - "input": "A shear, for example, doesn't have enough eigenvectors to span the full space.", + "input": "appens if our basis vectors just so happen to be eigenvectors. For example, maybe i-hat is scale", "translatedText": "En skjuvning, till exempel, har inte tillräckligt med egenvektorer för att spänna över hela utrymmet.", "model": "google_nmt", "n_reviews": 0, @@ -1003,7 +1003,7 @@ "end": 982.98 }, { - "input": "But if you can find an eigenbasis, it makes matrix operations really lovely.", + "input": "d by negative 1 and j-hat is scaled by 2. Writing their new coordinates as the columns of a matrix, notice t", "translatedText": "Men om du kan hitta en egenbas gör det matrisoperationer riktigt härliga.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/tamil/sentence_translations.json b/2016/change-of-basis/tamil/sentence_translations.json index e2870d146..16760db46 100644 --- a/2016/change-of-basis/tamil/sentence_translations.json +++ b/2016/change-of-basis/tamil/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates.", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we doing this and what d", "translatedText": "இங்கு 2டி இடத்தில் ஒரு வெக்டார் அமர்ந்திருந்தால், அதை ஆயத்தொலைவுகளுடன் விவரிக்க எங்களிடம் ஒரு நிலையான வழி உள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 28.28 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up.", + "input": "oes this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin", "translatedText": "இந்த வழக்கில், திசையன் 3, 2 ஆயத்தொலைவுகளைக் கொண்டுள்ளது, அதாவது அதன் வால் இருந்து அதன் முனைக்குச் செல்வது மூன்று அலகுகளை வலப்புறமாகவும் இரண்டு அலகுகள் மேலேயும் நகர்த்துவதை உள்ளடக்குகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.96 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up.", + "input": "ch that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of", "translatedText": "அந்த முதல் ஆயத்தை அளவிடுதல் i-hat என்று நீங்கள் நினைக்கிறீர்கள், நீளம் 1 கொண்ட திசையன் வலதுபுறம் சுட்டிக்காட்டுகிறது, இரண்டாவது ஒருங்கிணைப்பு அளவுகள் j-hat, நீளம் 1 கொண்ட திசையன் நேராக மேலே சுட்டிக்காட்டுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 60.48 }, { - "input": "You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordinate system.", + "input": "you know how to think about matrices as linear transformations, but you also need to be comfortable with things like determinants, linear systems", "translatedText": "இந்த இரண்டு சிறப்பு திசையன்கள் எங்கள் ஒருங்கிணைப்பு அமைப்பின் அனைத்து மறைமுகமான அனுமானங்களையும் உள்ளடக்கியதாக நீங்கள் நினைக்கலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 81.38 }, { - "input": "Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system.", + "input": "some linear transformation in two dimensions, like the one shown here. st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice o", "translatedText": "திசையன்கள் மற்றும் எண்களின் தொகுப்புகளுக்கு இடையில் மொழிபெயர்ப்பதற்கான எந்த வழியும் ஒரு ஒருங்கிணைப்பு அமைப்பு என்று அழைக்கப்படுகிறது, மேலும் இரண்டு சிறப்பு திசையன்களான i-hat மற்றும் j-hat ஆகியவை எங்கள் நிலையான ஒருங்கிணைப்பு அமைப்பின் அடிப்படை திசையன்கள் என்று அழைக்கப்படுகின்றன.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors.", + "input": "f i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors an", "translatedText": "நான் இங்கு பேச விரும்புவது வேறுபட்ட அடிப்படை வெக்டர்களைப் பயன்படுத்துவதற்கான யோசனை.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.04 }, { - "input": "Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third.", + "input": "nnifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to th", "translatedText": "ஜெனிஃபர் உண்மையில் இந்த வெக்டரை 5 மூன்றில் 5 மற்றும் 1 மூன்றில் ஆயத்தொலைவுகளுடன் விவரிப்பார்.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 125.16 }, { - "input": "In a little bit, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third.", + "input": "showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describe this vector with the coordinates 5 thirds and", "translatedText": "சிறிது நேரத்தில், மூன்றில் 5 மற்றும் மூன்றில் 1 எண்களை நீங்கள் எப்படிக் கண்டுபிடித்திருக்க முடியும் என்பதை நான் உங்களுக்குக் காட்டுகிறேன்.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 144.12 }, { - "input": "What she gets will typically be completely different from the vector that you and I would think of as having those coordinates.", + "input": "gether. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to", "translatedText": "அவள் பெறுவது பொதுவாக நீங்களும் நானும் அந்த ஆயத்தொலைவுகளைக் கொண்டிருப்பதாக நினைக்கும் திசையனிலிருந்து முற்றிலும் வேறுபட்டதாக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 167.14 }, { - "input": "So in effect, we're speaking different languages.", + "input": "lumn of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis.", "translatedText": "எனவே, நாம் வெவ்வேறு மொழிகளைப் பேசுகிறோம்.", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 176.84 }, { - "input": "Let me say a quick word about how I'm representing things here.", + "input": "ector on the x-axis is also just stretched by a factor of 3, and hence remains on its own", "translatedText": "நான் இங்கே விஷயங்களை எவ்வாறு பிரதிநிதித்துவப்படுத்துகிறேன் என்பதைப் பற்றி விரைவாகச் சொல்லுகிறேன்.", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid.", + "input": "span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here.", "translatedText": "விண்வெளியில் உள்ளார்ந்த கட்டம் இல்லை.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 197.6 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean.", + "input": "system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct mea", "translatedText": "0,0 ஆயத்தொகுப்புகள் எதைக் குறிக்க வேண்டும் என்பதை அனைவரும் ஒப்புக்கொள்வதால், அவளுடைய தோற்றம் உண்மையில் நம்முடையதுடன் வரிசையாக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 214.9 }, { - "input": "But the direction of her axes and the spacing of her grid lines will be different, depending on her choice of basis vectors.", + "input": "ollow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector", "translatedText": "ஆனால் அவளது அச்சுகளின் திசையும் அவளது கட்டக் கோடுகளின் இடைவெளியும் அவளது அடிப்படை வெக்டார்களின் தேர்வைப் பொறுத்து மாறுபடும்.", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 225.94 }, { - "input": "If for example Jennifer describes a vector with coordinates negative 1, 2, what would that be in our coordinate system?", + "input": "on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewh", "translatedText": "எடுத்துக்காட்டாக, ஜெனிஃபர் எதிர்மறை 1, 2 ஆயத்தொலைவுகளுடன் ஒரு திசையனை விவரித்தால், அது நமது ஒருங்கிணைப்பு அமைப்பில் என்னவாக இருக்கும்?", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 238.98 }, { - "input": "How do you translate from her language to ours?", + "input": "at during the transformation, knocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis", "translatedText": "அவளுடைய மொழியிலிருந்து எங்களுடைய மொழிக்கு எப்படி மொழிபெயர்ப்பது?", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 260.5 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1.", + "input": "ing them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vector", "translatedText": "எங்கள் கண்ணோட்டத்தில், b1 ஆய 2, 1 மற்றும் b2 ஆனது எதிர்மறை 1, 1 ஆயங்களைக் கொண்டுள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 267.04 }, { - "input": "So we can actually compute negative 1 times b1 plus 2 times b2 as they're represented in our coordinate system.", + "input": "s in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio Of course, there's n", "translatedText": "எனவே நாம் உண்மையில் எதிர்மறை 1 முறை b1 மற்றும் 2 மடங்கு b2 ஐக் கணக்கிடலாம், ஏனெனில் அவை நமது ஒருங்கிணைப்பு அமைப்பில் குறிப்பிடப்படுகின்றன.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 270.4 }, { - "input": "And working this out, you get a vector with coordinates negative 4, 1.", + "input": "othing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In", "translatedText": "இதைச் செயல்படுத்தினால், எதிர்மறை 4, 1 ஆயத்தொலைவுகளுடன் ஒரு திசையன் கிடைக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 274.74 }, { - "input": "This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", + "input": "alf, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennif", "translatedText": "சில வெக்டரின் தொடர்புடைய ஆயத்தொலைவுகளால் அதன் அடிப்படை திசையன்கள் ஒவ்வொன்றையும் அளவிடும் இந்த செயல்முறை, பின்னர் அவற்றை ஒன்றாகச் சேர்ப்பது ஓரளவு பரிச்சயமானதாக உணரலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 330.48 }, { - "input": "To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation.", + "input": "envector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking a", "translatedText": "இது எவ்வாறு செயல்படுகிறது என்பதைக் காட்ட, எதிர்மறை 1, 2 ஆயத்தொலைவுகளைக் கொண்டதாக நாம் நினைக்கும் திசையன் மற்றும் அந்த மாற்றத்தைப் பயன்படுத்துவதன் அர்த்தம் என்ன என்பதைப் பார்ப்போம்.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 341.38 }, { - "input": "Before the linear transformation, we're thinking of this vector as a certain linear combination of our basis vectors, negative 1 times i-hat plus 2 times j-hat.", + "input": "bout the full 3x3 matrix associated with that transformation. In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length", "translatedText": "நேரியல் மாற்றத்திற்கு முன், இந்த வெக்டரை நமது அடிப்படை வெக்டார்களின் ஒரு குறிப்பிட்ட நேரியல் கலவையாக கருதுகிறோம், எதிர்மறை 1 முறை i-hat மற்றும் 2 மடங்கு j-hat.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 375.16 }, { - "input": "Geometrically, this matrix transforms our grid into Jennifer's grid but numerically, it's translating a vector described in her language to our language.", + "input": "t the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. we get using", "translatedText": "வடிவியல் ரீதியாக, இந்த மேட்ரிக்ஸ் எங்கள் கட்டத்தை ஜெனிஃபர் கட்டமாக மாற்றுகிறது, ஆனால் எண்ணியல் ரீதியாக, இது அவரது மொழியில் விவரிக்கப்பட்டுள்ள திசையனை நம் மொழியில் மொழிபெயர்க்கிறது.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 380.62 }, { - "input": "What made it finally click for me was thinking about how it takes our misconception of what Jennifer means, the vector we get using the same coordinates but in our system, then it transforms it into the vector that she really meant.", + "input": "the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I comp", "translatedText": "ஜெனிஃபர் என்றால் என்ன என்பது பற்றிய நமது தவறான எண்ணம், அதே ஆயத்தொகுப்புகளைப் பயன்படுத்தி நாம் பெறும் திசையன், ஆனால் எங்கள் அமைப்பில், அது அவள் உண்மையில் நினைத்த திசையனாக மாற்றுகிறது என்பதைப் பற்றி யோசித்து, இறுதியாக என்னைக் கிளிக் செய்தது.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 398.26 }, { - "input": "What about going the other way around?", + "input": "ute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A", "translatedText": "வேறு வழியில் செல்வது பற்றி என்ன?", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 404.26 }, { - "input": "In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 third in Jennifer's system?", + "input": "is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis", "translatedText": "இந்த வீடியோவை நான் முன்பு பயன்படுத்திய எடுத்துக்காட்டில், எங்கள் கணினியில் 3, 2 ஆயத்தொலைவுகளுடன் வெக்டார் இருந்தபோது, ஜெனிஃபர் அமைப்பில் மூன்றில் 5 மற்றும் மூன்றில் 1 ஆயத்தொலைவுகள் இருக்கும் என்று நான் எப்படிக் கணக்கிட்டேன்?", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 409.48 }, { - "input": "You start with that change of basis matrix that translates Jennifer's language into ours, then you take its inverse.", + "input": "as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we", "translatedText": "ஜெனிஃபரின் மொழியை எங்களுடைய மொழியில் மொழிபெயர்க்கும் அடிப்படை மேட்ரிக்ஸின் மாற்றத்துடன் நீங்கள் தொடங்குகிறீர்கள், பின்னர் அதன் தலைகீழாக எடுத்துக் கொள்ளுங்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 415.48 }, { - "input": "Remember, the inverse of a transformation is a new transformation that corresponds to playing that first one backwards.", + "input": "multiply this inverse change of basis matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and fo", "translatedText": "நினைவில் கொள்ளுங்கள், மாற்றத்தின் தலைகீழ் என்பது ஒரு புதிய மாற்றமாகும், இது முதல் ஒன்றை பின்னோக்கி விளையாடுவதற்கு ஒத்திருக்கிறது.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 427.94 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse.", + "input": "rth between coordinate systems. The matrix whose columns represent Jennif er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the invers", "translatedText": "நடைமுறையில், குறிப்பாக நீங்கள் இரண்டு பரிமாணங்களுக்கு மேல் பணிபுரியும் போது, இந்த தலைகீழ் உண்மையில் பிரதிபலிக்கும் மேட்ரிக்ஸைக் கணக்கிட கணினியைப் பயன்படுத்துவீர்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 465.52 }, { - "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems.", + "input": "you know how matrix multiplication So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, usi", "translatedText": "எனவே, சுருக்கமாக, ஒருங்கிணைப்பு அமைப்புகளுக்கு இடையில் தனிப்பட்ட திசையன்களின் விளக்கத்தை முன்னும் பின்னுமாக மொழிபெயர்ப்பது எப்படி.", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 487.24 }, { - "input": "And the inverse matrix does the opposite.", + "input": "The columns of such a matrix will represent what happens to eac", "translatedText": "மற்றும் தலைகீழ் அணி இதற்கு நேர்மாறாக செயல்படுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 507.16 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy.", + "input": "heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla With", "translatedText": "நிச்சயமாக இடைநிறுத்தப்பட்டு, 3 மற்றும் 4 அத்தியாயங்களில் ஏதேனும் அசௌகரியம் ஏற்பட்டால் பாருங்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 529.74 }, { - "input": "i-hat ends up at the spot with coordinates 0, 1, and j-hat ends up at the spot with coordinates negative 1, 0.", + "input": "or out the v. So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to desc", "translatedText": "i-hat ஆனது 0, 1 ஆயத்தொகுதிகளுடன் முடிவடைகிறது, மேலும் j-hat ஆனது எதிர்மறை 1, 0 ஆயத்தொகுதிகளுடன் முடிவடைகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 566.3 }, { - "input": "But that's not quite right.", + "input": "And that squishification corresponds to a zero determinant for the matrix. To be concrete, let's say your matrix", "translatedText": "ஆனால் அது சரியாக இல்லை.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.72 }, { - "input": "Here's a common way to think of how this is done.", + "input": "As that value of lambda changes, the matrix itself changes, and so the determina", "translatedText": "இது எவ்வாறு செய்யப்படுகிறது என்பதைப் பற்றி சிந்திக்க ஒரு பொதுவான வழி இங்கே உள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 603.42 }, { - "input": "Start with any vector written in Jennifer's language.", + "input": "nt of the matrix changes. ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multipl", "translatedText": "ஜெனிஃபர் மொழியில் எழுதப்பட்ட எந்த வெக்டருடன் தொடங்கவும்.", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 649.44 }, { - "input": "Since we could do this with any vector written in her language, first applying the change of basis, then the transformation, then the inverse change of basis, that composition of three matrices gives us the transformation matrix in Jennifer's language.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective. And the full matrix product represents that same transformation, but as someone else sees it.", "translatedText": "அவரது மொழியில் எழுதப்பட்ட எந்த திசையன் மூலமாகவும் இதைச் செய்ய முடியும் என்பதால், முதலில் அடிப்படை மாற்றம், பின்னர் மாற்றம், பின்னர் அடிப்படையின் தலைகீழ் மாற்றம் ஆகியவற்றைப் பயன்படுத்துவதன் மூலம், மூன்று மெட்ரிக்குகளின் கலவை ஜெனிஃபரின் மொழியில் உருமாற்ற மேட்ரிக்ஸை வழங்குகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 665.56 }, { - "input": "It takes in a vector of her language and spits out the transformed version of that vector in her language.", + "input": "For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of", "translatedText": "அது அவளது மொழியின் திசையனை எடுத்துக் கொண்டு, அந்த திசையன் மாற்றப்பட்ட பதிப்பை அவளது மொழியில் துப்புகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 675.8 }, { - "input": "For this specific example, when Jennifer's basis vectors look like 2, 1 and negative in our language, and when the transformation is a 90 degree rotation, the product of these three matrices, if you work through it, has columns one third, five thirds, and negative two thirds, negative one third.", + "input": "this. See y That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corres", "translatedText": "இந்த குறிப்பிட்ட உதாரணத்திற்கு, ஜெனிஃபரின் அடிப்படை திசையன்கள் நம் மொழியில் 2, 1 மற்றும் எதிர்மறையாக இருக்கும் போது, மற்றும் மாற்றம் 90 டிகிரி சுழற்சியாக இருக்கும்போது, இந்த மூன்று மெட்ரிக்குகளின் பலன், நீங்கள் வேலை செய்தால், மூன்றில் ஒரு பங்கு, மூன்றில் ஐந்தில் நெடுவரிசைகள் இருக்கும். , மற்றும் எதிர்மறை மூன்றில் இரண்டு, எதிர்மறை மூன்றில் ஒரு பங்கு.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 692.2 }, { - "input": "So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system.", + "input": "ponding eigenvalue is 1, so v would actually just stay fixed in place. Pause and ponder if you need to make sure that that line of reasoning feels good. This is the kind of thing I mentioned in the introduction. If you didn't have a", "translatedText": "ஜெனிஃபர் அந்த மேட்ரிக்ஸை அவரது அமைப்பில் உள்ள திசையன்களின் ஆயத்தொலைவுகளால் பெருக்கினால், அது அவரது ஆய அமைப்பில் வெளிப்படுத்தப்பட்ட திசையனின் 90 டிகிரி சுழற்றப்பட்ட பதிப்பை வழங்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 709.82 }, { - "input": "In general, whenever you see an expression like A inverse times M times A, it suggests a mathematical sort of empathy.", + "input": "solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "பொதுவாக, A தலைகீழ் முறை M முறை A போன்ற ஒரு வெளிப்பாட்டைக் காணும் போதெல்லாம், அது ஒரு கணித வகையான பச்சாதாபத்தைக் குறிக்கிறது.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 714.54 }, { - "input": "That middle matrix represents a transformation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective.", + "input": "To see this in action, let's revisit the example from the start, with a matrix whose columns are 3, 0 and 1, 2. To find if a value lambda is an eigenvalue, subtract it from the diago", "translatedText": "அந்த நடுத்தர அணி நீங்கள் பார்க்கும் விதத்தில் ஒருவித மாற்றத்தைக் குறிக்கிறது, மேலும் வெளிப்புற இரண்டு மெட்ரிக்குகள் பச்சாதாபம், முன்னோக்கு மாற்றத்தைக் குறிக்கின்றன.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/telugu/sentence_translations.json b/2016/change-of-basis/telugu/sentence_translations.json index 25e78ab60..ecad319ae 100644 --- a/2016/change-of-basis/telugu/sentence_translations.json +++ b/2016/change-of-basis/telugu/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates.", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we doing this and what d", "translatedText": "నేను ఇక్కడ 2D స్పేస్‌లో కూర్చున్న వెక్టార్‌ని కలిగి ఉంటే, దానిని కోఆర్డినేట్‌లతో వివరించడానికి మాకు ప్రామాణిక మార్గం ఉంది.", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 28.28 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up.", + "input": "oes this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin", "translatedText": "ఈ సందర్భంలో, వెక్టర్ 3, 2 కోఆర్డినేట్‌లను కలిగి ఉంటుంది, అంటే దాని తోక నుండి దాని కొనకు వెళ్లడం అంటే మూడు యూనిట్లను కుడి వైపుకు మరియు రెండు యూనిట్లు పైకి తరలించడం.", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.96 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up.", + "input": "ch that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of", "translatedText": "మీరు ఆ మొదటి కోఆర్డినేట్‌ని స్కేలింగ్ i-hatగా భావిస్తారు, పొడవు 1 ఉన్న వెక్టార్ కుడివైపుకి చూపుతుంది, రెండవ కోఆర్డినేట్ స్కేల్స్ j-hat, పొడవు 1 ఉన్న వెక్టార్ నేరుగా పైకి చూపుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 60.48 }, { - "input": "You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordinate system.", + "input": "you know how to think about matrices as linear transformations, but you also need to be comfortable with things like determinants, linear systems", "translatedText": "మీరు ఈ రెండు ప్రత్యేక వెక్టర్‌లను మా కోఆర్డినేట్ సిస్టమ్ యొక్క అవ్యక్త అంచనాలన్నింటినీ కలుపుతున్నట్లు భావించవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 81.38 }, { - "input": "Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system.", + "input": "some linear transformation in two dimensions, like the one shown here. st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice o", "translatedText": "వెక్టర్స్ మరియు సంఖ్యల సెట్ల మధ్య అనువదించడానికి ఏదైనా మార్గాన్ని కోఆర్డినేట్ సిస్టమ్ అని పిలుస్తారు మరియు రెండు ప్రత్యేక వెక్టర్స్ i-hat మరియు j-hat మా ప్రామాణిక కోఆర్డినేట్ సిస్టమ్ యొక్క ఆధార వెక్టర్స్ అని పిలుస్తారు.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors.", + "input": "f i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors an", "translatedText": "నేను ఇక్కడ మాట్లాడదలుచుకున్నది వేరొక బేస్ వెక్టర్స్‌ని ఉపయోగించాలనే ఆలోచన.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.04 }, { - "input": "Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third.", + "input": "nnifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to th", "translatedText": "జెన్నిఫర్ వాస్తవానికి ఈ వెక్టర్‌ను 5 వంతులు మరియు 1 వంతుల అక్షాంశాలతో వివరిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 125.16 }, { - "input": "In a little bit, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third.", + "input": "showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describe this vector with the coordinates 5 thirds and", "translatedText": "కొద్దిసేపటిలో, మీరు ఆ రెండు సంఖ్యలను, 5 వంతులు మరియు 1 వంతులను ఎలా గుర్తించగలిగారో నేను మీకు చూపిస్తాను.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 144.12 }, { - "input": "What she gets will typically be completely different from the vector that you and I would think of as having those coordinates.", + "input": "gether. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to", "translatedText": "ఆమె పొందేది మీరు మరియు నేను ఆ కోఆర్డినేట్‌లను కలిగి ఉన్నట్లు భావించే వెక్టర్ నుండి పూర్తిగా భిన్నంగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 167.14 }, { - "input": "So in effect, we're speaking different languages.", + "input": "lumn of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis.", "translatedText": "కాబట్టి, మేము వివిధ భాషలను మాట్లాడుతున్నాము.", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 176.84 }, { - "input": "Let me say a quick word about how I'm representing things here.", + "input": "ector on the x-axis is also just stretched by a factor of 3, and hence remains on its own", "translatedText": "నేను ఇక్కడ విషయాలను ఎలా సూచిస్తున్నాను అనే దాని గురించి త్వరగా చెప్పనివ్వండి.", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid.", + "input": "span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here.", "translatedText": "అంతరిక్షంలోనే అంతర్గత గ్రిడ్ లేదు.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 197.6 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean.", + "input": "system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct mea", "translatedText": "ఆమె మూలం వాస్తవానికి మనతో సమానంగా ఉంటుంది, ఎందుకంటే 0,0 కోఆర్డినేట్‌ల అర్థం ఏమిటో అందరూ అంగీకరిస్తారు.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 214.9 }, { - "input": "But the direction of her axes and the spacing of her grid lines will be different, depending on her choice of basis vectors.", + "input": "ollow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector", "translatedText": "కానీ ఆమె గొడ్డలి దిశ మరియు ఆమె గ్రిడ్ లైన్ల అంతరం భిన్నంగా ఉంటుంది, ఇది ఆమె ఆధార వెక్టర్‌ల ఎంపికపై ఆధారపడి ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 225.94 }, { - "input": "If for example Jennifer describes a vector with coordinates negative 1, 2, what would that be in our coordinate system?", + "input": "on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewh", "translatedText": "ఉదాహరణకు జెన్నిఫర్ నెగటివ్ 1, 2 కోఆర్డినేట్‌లతో వెక్టార్‌ని వివరిస్తే, అది మన కోఆర్డినేట్ సిస్టమ్‌లో ఎలా ఉంటుంది?", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 238.98 }, { - "input": "How do you translate from her language to ours?", + "input": "at during the transformation, knocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis", "translatedText": "మీరు ఆమె భాష నుండి మా భాషలోకి ఎలా అనువదిస్తారు?", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 260.5 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1.", + "input": "ing them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vector", "translatedText": "మరియు మా దృక్కోణం నుండి, b1 కోఆర్డినేట్‌లు 2, 1, మరియు b2 అక్షాంశాలు ప్రతికూల 1, 1 ఉన్నాయి.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 267.04 }, { - "input": "So we can actually compute negative 1 times b1 plus 2 times b2 as they're represented in our coordinate system.", + "input": "s in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio Of course, there's n", "translatedText": "కాబట్టి మన కోఆర్డినేట్ సిస్టమ్‌లో అవి ప్రాతినిధ్యం వహిస్తున్నందున మనం వాస్తవానికి ప్రతికూల 1 సార్లు b1 ప్లస్ 2 సార్లు b2ని లెక్కించవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 270.4 }, { - "input": "And working this out, you get a vector with coordinates negative 4, 1.", + "input": "othing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In", "translatedText": "మరియు దీన్ని పని చేస్తే, మీరు అక్షాంశాలు నెగటివ్ 4, 1తో వెక్టార్‌ని పొందుతారు.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 274.74 }, { - "input": "This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", + "input": "alf, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennif", "translatedText": "కొన్ని వెక్టర్ యొక్క సంబంధిత కోఆర్డినేట్‌ల ద్వారా ఆమె ప్రాతిపదిక వెక్టర్‌లలో ప్రతిదానిని స్కేల్ చేయడం, ఆపై వాటిని జోడించడం వంటి ఈ ప్రక్రియ కొంతవరకు సుపరిచితమైనదిగా అనిపించవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 330.48 }, { - "input": "To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation.", + "input": "envector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking a", "translatedText": "ఇది ఎలా పని చేస్తుందో చూపించడానికి, ప్రతికూల 1, 2 కోఆర్డినేట్‌లను కలిగి ఉన్నట్లు మరియు ఆ పరివర్తనను వర్తింపజేయడం అని మనం భావించే వెక్టర్‌ను తీసుకోవడం అంటే ఏమిటో తెలుసుకుందాం.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 341.38 }, { - "input": "Before the linear transformation, we're thinking of this vector as a certain linear combination of our basis vectors, negative 1 times i-hat plus 2 times j-hat.", + "input": "bout the full 3x3 matrix associated with that transformation. In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length", "translatedText": "లీనియర్ ట్రాన్స్‌ఫర్మేషన్‌కు ముందు, మేము ఈ వెక్టార్‌ని మా బేస్ వెక్టర్స్ యొక్క నిర్దిష్ట లీనియర్ కలయికగా ఆలోచిస్తున్నాము, ప్రతికూల 1 సార్లు i-hat ప్లస్ 2 సార్లు j-hat.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 375.16 }, { - "input": "Geometrically, this matrix transforms our grid into Jennifer's grid but numerically, it's translating a vector described in her language to our language.", + "input": "t the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. we get using", "translatedText": "జ్యామితీయంగా, ఈ మాతృక మన గ్రిడ్‌ను జెన్నిఫర్ గ్రిడ్‌గా మారుస్తుంది కానీ సంఖ్యాపరంగా, ఇది ఆమె భాషలో వివరించిన వెక్టర్‌ను మన భాషలోకి అనువదిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 380.62 }, { - "input": "What made it finally click for me was thinking about how it takes our misconception of what Jennifer means, the vector we get using the same coordinates but in our system, then it transforms it into the vector that she really meant.", + "input": "the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I comp", "translatedText": "జెన్నిఫర్ అంటే ఏమిటనేది మన అపోహను ఎలా తీసుకుంటుందనే దాని గురించి ఆలోచించడం నాకు చివరకు క్లిక్ చేసింది, అదే కోఆర్డినేట్‌లను ఉపయోగించి మనం పొందే వెక్టర్, కానీ మన సిస్టమ్‌లో, అది ఆమె నిజంగా ఉద్దేశించిన వెక్టర్‌గా మారుస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 398.26 }, { - "input": "What about going the other way around?", + "input": "ute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A", "translatedText": "మరో వైపు వెళ్లడం గురించి ఏమిటి?", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 404.26 }, { - "input": "In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 third in Jennifer's system?", + "input": "is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis", "translatedText": "నేను ఇంతకు ముందు ఈ వీడియోని ఉపయోగించిన ఉదాహరణలో, మా సిస్టమ్‌లో 3, 2 కోఆర్డినేట్‌లతో వెక్టార్ ఉన్నప్పుడు, జెన్నిఫర్ సిస్టమ్‌లో 5 వంతులు మరియు 1 థర్డ్ కోఆర్డినేట్‌లు ఉన్నాయని నేను ఎలా గణించాను?", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 409.48 }, { - "input": "You start with that change of basis matrix that translates Jennifer's language into ours, then you take its inverse.", + "input": "as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we", "translatedText": "మీరు జెన్నిఫర్ భాషను మా భాషలోకి అనువదించే ఆధార మాతృక మార్పుతో ప్రారంభించండి, ఆపై మీరు దాని విలోమాన్ని తీసుకుంటారు.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 415.48 }, { - "input": "Remember, the inverse of a transformation is a new transformation that corresponds to playing that first one backwards.", + "input": "multiply this inverse change of basis matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and fo", "translatedText": "గుర్తుంచుకోండి, పరివర్తన యొక్క విలోమం అనేది మొదటిదాన్ని వెనుకకు ప్లే చేయడానికి అనుగుణంగా ఉండే కొత్త పరివర్తన.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 427.94 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse.", + "input": "rth between coordinate systems. The matrix whose columns represent Jennif er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the invers", "translatedText": "ఆచరణలో, ప్రత్యేకించి మీరు రెండు కంటే ఎక్కువ కోణాలలో పని చేస్తున్నప్పుడు, ఈ విలోమాన్ని సూచించే మాతృకను గణించడానికి మీరు కంప్యూటర్‌ను ఉపయోగించాలి.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 465.52 }, { - "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems.", + "input": "you know how matrix multiplication So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, usi", "translatedText": "కాబట్టి, క్లుప్తంగా, కోఆర్డినేట్ సిస్టమ్‌ల మధ్య వ్యక్తిగత వెక్టర్‌ల వివరణను ముందుకు వెనుకకు ఎలా అనువదించాలి.", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 487.24 }, { - "input": "And the inverse matrix does the opposite.", + "input": "The columns of such a matrix will represent what happens to eac", "translatedText": "మరియు విలోమ మాతృక దీనికి విరుద్ధంగా చేస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 507.16 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy.", + "input": "heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla With", "translatedText": "ఖచ్చితంగా పాజ్ చేసి, 3 మరియు 4 అధ్యాయాలు ఏవైనా అసౌకర్యంగా అనిపిస్తే వాటిని చూడండి.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 529.74 }, { - "input": "i-hat ends up at the spot with coordinates 0, 1, and j-hat ends up at the spot with coordinates negative 1, 0.", + "input": "or out the v. So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to desc", "translatedText": "i-hat 0, 1 కోఆర్డినేట్‌లతో స్పాట్‌లో ముగుస్తుంది మరియు j-hat అక్షాంశాలు ప్రతికూల 1, 0తో స్పాట్‌లో ముగుస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 566.3 }, { - "input": "But that's not quite right.", + "input": "And that squishification corresponds to a zero determinant for the matrix. To be concrete, let's say your matrix", "translatedText": "కానీ అది సరిగ్గా లేదు.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.72 }, { - "input": "Here's a common way to think of how this is done.", + "input": "As that value of lambda changes, the matrix itself changes, and so the determina", "translatedText": "ఇది ఎలా జరుగుతుందో ఆలోచించడానికి ఇక్కడ ఒక సాధారణ మార్గం ఉంది.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 603.42 }, { - "input": "Start with any vector written in Jennifer's language.", + "input": "nt of the matrix changes. ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multipl", "translatedText": "జెన్నిఫర్ భాషలో వ్రాసిన ఏదైనా వెక్టర్‌తో ప్రారంభించండి.", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 649.44 }, { - "input": "Since we could do this with any vector written in her language, first applying the change of basis, then the transformation, then the inverse change of basis, that composition of three matrices gives us the transformation matrix in Jennifer's language.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective. And the full matrix product represents that same transformation, but as someone else sees it.", "translatedText": "మేము ఆమె భాషలో వ్రాసిన ఏదైనా వెక్టర్‌తో దీన్ని చేయగలము కాబట్టి, మొదట ఆధారం యొక్క మార్పును వర్తింపజేయడం, తరువాత రూపాంతరం, ఆ తర్వాత ఆధారం యొక్క విలోమ మార్పు, మూడు మాత్రికల కూర్పు జెన్నిఫర్ భాషలో పరివర్తన మాతృకను అందిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 665.56 }, { - "input": "It takes in a vector of her language and spits out the transformed version of that vector in her language.", + "input": "For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of", "translatedText": "ఇది ఆమె భాషలోని వెక్టార్‌ని తీసుకుంటుంది మరియు ఆమె భాషలో ఆ వెక్టర్ యొక్క రూపాంతరం చెందిన సంస్కరణను ఉమ్మివేస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 675.8 }, { - "input": "For this specific example, when Jennifer's basis vectors look like 2, 1 and negative in our language, and when the transformation is a 90 degree rotation, the product of these three matrices, if you work through it, has columns one third, five thirds, and negative two thirds, negative one third.", + "input": "this. See y That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corres", "translatedText": "ఈ నిర్దిష్ట ఉదాహరణ కోసం, జెన్నిఫర్ యొక్క ఆధార వెక్టర్స్ మన భాషలో 2, 1 మరియు ప్రతికూలంగా కనిపించినప్పుడు మరియు రూపాంతరం 90 డిగ్రీల భ్రమణం అయినప్పుడు, ఈ మూడు మాత్రికల ఉత్పత్తి, మీరు దాని ద్వారా పని చేస్తే, మూడవ వంతు, ఐదు వంతుల నిలువు వరుసలు ఉంటాయి. , మరియు ప్రతికూల మూడింట రెండు, ప్రతికూల మూడవ వంతు.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 692.2 }, { - "input": "So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system.", + "input": "ponding eigenvalue is 1, so v would actually just stay fixed in place. Pause and ponder if you need to make sure that that line of reasoning feels good. This is the kind of thing I mentioned in the introduction. If you didn't have a", "translatedText": "కనుక జెన్నిఫర్ ఆ మాతృకను తన సిస్టమ్‌లోని వెక్టార్ యొక్క కోఆర్డినేట్‌లతో గుణిస్తే, అది ఆమె కోఆర్డినేట్ సిస్టమ్‌లో వ్యక్తీకరించబడిన వెక్టర్ యొక్క 90 డిగ్రీల రొటేట్ వెర్షన్‌ను అందిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 709.82 }, { - "input": "In general, whenever you see an expression like A inverse times M times A, it suggests a mathematical sort of empathy.", + "input": "solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "సాధారణంగా, మీరు A విలోమ సమయాలు M సార్లు A వంటి వ్యక్తీకరణను చూసినప్పుడల్లా, అది గణిత సంబంధమైన తాదాత్మ్యతను సూచిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 714.54 }, { - "input": "That middle matrix represents a transformation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective.", + "input": "To see this in action, let's revisit the example from the start, with a matrix whose columns are 3, 0 and 1, 2. To find if a value lambda is an eigenvalue, subtract it from the diago", "translatedText": "ఆ మధ్య మాతృక మీరు చూసే విధంగా ఒక రకమైన పరివర్తనను సూచిస్తుంది మరియు బయటి రెండు మాత్రికలు తాదాత్మ్యం, దృక్పథంలో మార్పును సూచిస్తాయి.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/thai/sentence_translations.json b/2016/change-of-basis/thai/sentence_translations.json index 3bda97207..4426a3496 100644 --- a/2016/change-of-basis/thai/sentence_translations.json +++ b/2016/change-of-basis/thai/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates. ", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we d ", "translatedText": "ถ้าฉันมีเวกเตอร์นั่งอยู่ตรงนี้ในปริภูมิ 2 มิติ เราก็มีวิธีมาตรฐานในการอธิบายมันด้วยพิกัด ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 27.48 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up. ", + "input": "oing this and what does this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin ", "translatedText": "ในกรณีนี้ เวกเตอร์มีพิกัด 3, 2 ซึ่งหมายความว่าการย้ายจากหางไปยังปลายของมันเกี่ยวข้องกับการเคลื่อนสามหน่วยไปทางขวาและสองหน่วยขึ้นไป ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.36 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up. ", + "input": "not so much that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of ", "translatedText": "คุณคิดว่าพิกัดแรกนั้นเป็นการปรับขนาด i-hat เวกเตอร์ที่มีความยาว 1 ชี้ไปทางขวา ในขณะที่พิกัดที่สองปรับขนาด j-hat เวกเตอร์ที่มีความยาว 1 ชี้ตรงขึ้นไป ", "model": "google_nmt", "n_reviews": 0, @@ -32,7 +32,7 @@ "end": 57.14 }, { - "input": "The tip-to-tail sum of those two scaled vectors is what the coordinates are meant to describe. ", + "input": "the topics that precede it. Most important here is that you know how to think about matrices as linear transformations, but you also need to be comforta ", "translatedText": "ผลรวมจากปลายจรดท้ายของเวกเตอร์ที่ปรับขนาดทั้งสองนั้นคือสิ่งที่พิกัดมีไว้เพื่ออธิบาย ", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors. ", + "input": "of that is tied up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors a ", "translatedText": "สิ่งที่ผมอยากพูดถึงตรงนี้ คือแนวคิดในการใช้เวกเตอร์ฐานชุดอื่น ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 133.36 }, { - "input": "In general, whenever Jennifer uses coordinates to describe a vector, she thinks of her first coordinate as scaling b1, the second coordinate as scaling b2, and she adds the results. ", + "input": "vector, b2, points left and up. Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describ ", "translatedText": "โดยทั่วไป เมื่อใดก็ตามที่ Jennifer ใช้พิกัดเพื่ออธิบายเวกเตอร์ เธอจะนึกถึงพิกัดแรกของเธอเป็นการปรับขนาด b1 พิกัดที่สองเป็นการปรับขนาด b2 แล้วเธอก็บวกผลลัพธ์เข้าไป ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 166.8 }, { - "input": "They are what define the meaning of the coordinates 1,0 and 0,1 in her world. ", + "input": "then add them both together. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. ", "translatedText": "สิ่งเหล่านี้คือสิ่งที่กำหนดความหมายของพิกัด 1,0 และ 0,1 ในโลกของเธอ ", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 185.86 }, { - "input": "But that grid is just a construct, a way to visualize our coordinate system, and so it depends on our choice of basis. ", + "input": "ample, the basis vector i-hat is one such special vector. The span of i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 ", "translatedText": "แต่ตารางนั้นเป็นเพียงโครงสร้าง เป็นวิธีหนึ่งในการแสดงภาพระบบพิกัดของเรา ดังนั้นมันจึงขึ้นอยู่กับการเลือกพื้นฐานของเรา ", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid. ", + "input": "times itself, still on that x-axis. What's more, because of the way linear transformations work, ", "translatedText": "อวกาศนั้นไม่มีกริดที่แท้จริง ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 198.08 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. ", + "input": "emains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. ", "translatedText": "ต้นกำเนิดของเธอจริงๆ แล้วคงจะสอดคล้องกับของเรา เนื่องจากทุกคนเห็นพ้องกันว่าพิกัด 0,0 ควรหมายถึงอะไร ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 223.72 }, { - "input": "So after all this is set up, a pretty natural question to ask is how we translate between coordinate systems. ", + "input": "self has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct meant as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her ori ", "translatedText": "หลังจากตั้งค่าทั้งหมดนี้แล้ว คำถามที่ค่อนข้างเป็นธรรมชาติที่จะถามคือเราแปลระหว่างระบบพิกัดอย่างไร ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 227.28 }, { - "input": "How do you translate from her language to ours? ", + "input": "hould mean. It's the thing that you get when you scale any vector by zero. ", "translatedText": "คุณแปลจากภาษาของเธอเป็นภาษาของเราอย่างไร? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 230.76 }, { - "input": "Well, what her coordinates are saying is that this vector is negative 1 times b1 plus 2 times b2. ", + "input": "But the direction of her axes and Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. ", "translatedText": "ทีนี้, สิ่งที่พิกัดของเธอบอกคือว่า เวกเตอร์นี้เป็นลบ 1 คูณ b1 บวก 2 คูณ b2 ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 231.76 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1. ", + "input": "Any other vector is going to get rotated somewhat during the transformation, knocked off the line that it spans. ks of as negative 1, 2. ", "translatedText": "จากมุมมองของเรา b1 มีพิกัด 2, 1 และ b2 มีพิกัดลบ 1, 1 ", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 254.28 }, { - "input": "It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vectors in our language. ", + "input": "tand matrix vector multiplication as applying a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or ", "translatedText": "เป็นการคูณเมทริกซ์เวกเตอร์, โดยมีเมทริกซ์ที่มีคอลัมน์แทนเวกเตอร์พื้นฐานของเจนนิเฟอร์ในภาษาของเรา ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 257.08 }, { - "input": "In fact, once you understand matrix vector multiplication as applying a certain linear transformation, say by watching what I view to be the most important video in this series, Chapter 3, there's a pretty intuitive way to think about what's going on here. ", + "input": "the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. ", "translatedText": "ที่จริง, เมื่อคุณเข้าใจการคูณเมทริกซ์เวกเตอร์โดยใช้การแปลงเชิงเส้นแล้ว, พูดโดยการดูสิ่งที่ฉันมองว่าเป็นวิดีโอที่สำคัญที่สุดในชุดนี้, บทที่ 3, มันมีวิธีคิดตามสัญชาตญาณว่าเกิดอะไรขึ้นที่นี่ ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 384.3 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse. ", + "input": "coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, ", "translatedText": "ในทางปฏิบัติ โดยเฉพาะอย่างยิ่งเมื่อคุณทำงานในมากกว่าสองมิติ คุณจะต้องใช้คอมพิวเตอร์เพื่อคำนวณเมทริกซ์ที่แทนค่าผกผันนี้จริงๆ ", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 389.6 }, { - "input": "In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. ", + "input": "with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. ", "translatedText": "ในกรณีนี้ ค่าผกผันของการเปลี่ยนแปลงของเมทริกซ์ฐานที่มีฐานของเจนนิเฟอร์ เมื่อคอลัมน์สุดท้ายกลายเป็นคอลัมน์ที่ 1 ใน 3 ลบ 1 ใน 3 และ 1 ใน 3, 2 ใน 3 ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 443.9 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy. ", + "input": "es, and that you know how matrix multiplication So let's start by rewriting that right-hand ", "translatedText": "หยุดชั่วคราวและดูบทที่ 3 และ 4 อย่างแน่นอนหากรู้สึกไม่สบายใจ ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 446.48 }, { - "input": "Consider some linear transformation, like a 90 degree counterclockwise rotation. ", + "input": "side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda. ", "translatedText": "พิจารณาการแปลงเชิงเส้น เช่น การหมุนทวนเข็มนาฬิกา 90 องศา ", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 451.06 }, { - "input": "When you and I represent this with a matrix, we follow where the basis vectors i-hat and j-hat each go. ", + "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this ", "translatedText": "เมื่อคุณและฉันแทนสิ่งนี้ด้วยเมทริกซ์, เราจะตามตรงที่เวกเตอร์พื้นฐาน i-hat และ j-hat แต่ละตัวไป ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 477.24 }, { - "input": "How would Jennifer describe this same 90 degree rotation of space? ", + "input": "pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v. ", "translatedText": "เจนนิเฟอร์จะอธิบายการหมุนของอวกาศ 90 องศาแบบเดียวกันนี้อย่างไร ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 484.26 }, { - "input": "But that's not quite right. ", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and ", "translatedText": "แต่นั่นไม่ถูกต้องนัก ", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 515.02 }, { - "input": "Then apply the transformation matrix to what you get by multiplying it on the left. ", + "input": "zero is if the transformation associated with that matrix squishes space into a lower dimension. And that squishification corresponds to a zero determinant for the matr ", "translatedText": "จากนั้นใช้เมทริกซ์การแปลงกับสิ่งที่คุณได้โดยการคูณมันทางซ้าย ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 515.62 }, { - "input": "This tells us where that vector lands, but still in our language. ", + "input": "ix. To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtract ", "translatedText": "นี่บอกเราว่าเวกเตอร์นั้นไปถึงจุดไหน แต่ยังเป็นภาษาของเรา ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/turkish/sentence_translations.json b/2016/change-of-basis/turkish/sentence_translations.json index 5b5329f74..b40f03331 100644 --- a/2016/change-of-basis/turkish/sentence_translations.json +++ b/2016/change-of-basis/turkish/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates.", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we doing this and what d", "translatedText": "Burada 2 boyutlu uzayda oturan bir vektörüm varsa, onu koordinatlarla tanımlamanın standart bir yolu vardır.", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 28.28 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up.", + "input": "oes this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin", "translatedText": "Bu durumda vektörün koordinatları 3, 2'dir, yani kuyruğundan ucuna gitmek üç birim sağa ve iki birim yukarı hareket etmeyi gerektirir.", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.96 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up.", + "input": "ch that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of", "translatedText": "İlk koordinatı, uzunluğu 1 olan vektör sağa dönük olan i-hat'ı ölçeklendirmek olarak düşünürsünüz; ikinci koordinat ise uzunluğu 1 olan vektör doğrudan yukarıyı gösteren j-hat'ı ölçeklendirir.", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 60.48 }, { - "input": "You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordinate system.", + "input": "you know how to think about matrices as linear transformations, but you also need to be comfortable with things like determinants, linear systems", "translatedText": "Bu iki özel vektörün koordinat sistemimizin tüm örtülü varsayımlarını kapsadığını düşünebilirsiniz.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 81.38 }, { - "input": "Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system.", + "input": "some linear transformation in two dimensions, like the one shown here. st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice o", "translatedText": "Vektörler ve sayı kümeleri arasında çeviri yapmanın herhangi bir yoluna koordinat sistemi denir ve iki özel vektör i-hat ve j-hat, standart koordinat sistemimizin temel vektörleri olarak adlandırılır.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors.", + "input": "f i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors an", "translatedText": "Burada konuşmak istediğim şey farklı bir temel vektörler kümesi kullanma fikridir.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.04 }, { - "input": "Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third.", + "input": "nnifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to th", "translatedText": "Jennifer aslında bu vektörü 5/3 ve 1/3 koordinatlarıyla tanımlayacaktı.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 125.16 }, { - "input": "In a little bit, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third.", + "input": "showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describe this vector with the coordinates 5 thirds and", "translatedText": "Birazdan size bu iki sayıyı, 5/3 ve 1/3 sayılarını nasıl bulacağınızı göstereceğim.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 144.12 }, { - "input": "What she gets will typically be completely different from the vector that you and I would think of as having those coordinates.", + "input": "gether. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to", "translatedText": "Aldığı şey genellikle sizin ve benim bu koordinatlara sahip olduğunu düşündüğümüz vektörden tamamen farklı olacaktır.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 167.14 }, { - "input": "So in effect, we're speaking different languages.", + "input": "lumn of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis.", "translatedText": "Yani aslında farklı diller konuşuyoruz.", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 176.84 }, { - "input": "Let me say a quick word about how I'm representing things here.", + "input": "ector on the x-axis is also just stretched by a factor of 3, and hence remains on its own", "translatedText": "Burada olayları nasıl temsil ettiğime dair kısa bir söz söylememe izin verin.", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid.", + "input": "span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here.", "translatedText": "Uzayın kendisinin kendine özgü bir ızgarası yoktur.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 197.6 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean.", + "input": "system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct mea", "translatedText": "Herkes 0,0 koordinatlarının ne anlama gelmesi gerektiği konusunda hemfikir olduğundan, onun kökeni aslında bizimkine uyuyor.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 214.9 }, { - "input": "But the direction of her axes and the spacing of her grid lines will be different, depending on her choice of basis vectors.", + "input": "ollow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector", "translatedText": "Ancak eksenlerinin yönü ve ızgara çizgilerinin aralığı, temel vektör seçimine bağlı olarak farklı olacaktır.", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 225.94 }, { - "input": "If for example Jennifer describes a vector with coordinates negative 1, 2, what would that be in our coordinate system?", + "input": "on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewh", "translatedText": "Örneğin Jennifer koordinatları negatif 1, 2 olan bir vektör tanımlarsa, bu bizim koordinat sistemimizde ne olur?", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 238.98 }, { - "input": "How do you translate from her language to ours?", + "input": "at during the transformation, knocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis", "translatedText": "Onun dilinden bizim dilimize nasıl tercüme edersiniz?", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 260.5 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1.", + "input": "ing them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vector", "translatedText": "Ve bizim bakış açımıza göre, b1'in koordinatları 2, 1 ve b2'nin koordinatları negatif 1, 1'dir.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 267.04 }, { - "input": "So we can actually compute negative 1 times b1 plus 2 times b2 as they're represented in our coordinate system.", + "input": "s in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio Of course, there's n", "translatedText": "Yani aslında koordinat sistemimizde gösterildiği gibi negatif 1 çarpı b1 artı 2 çarpı b2'yi hesaplayabiliriz.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 270.4 }, { - "input": "And working this out, you get a vector with coordinates negative 4, 1.", + "input": "othing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In", "translatedText": "Bunu çözerek koordinatları negatif 4, 1 olan bir vektör elde ederiz.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 274.74 }, { - "input": "This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", + "input": "alf, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennif", "translatedText": "Buradaki, temel vektörlerin her birini, bir vektörün karşılık gelen koordinatlarına göre ölçeklendirme ve ardından bunları bir araya getirme süreci, biraz tanıdık gelebilir.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 330.48 }, { - "input": "To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation.", + "input": "envector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking a", "translatedText": "Bunun nasıl çalıştığını göstermek için, koordinatları negatif 1, 2 olduğunu düşündüğümüz vektörü alıp bu dönüşümü uygulamanın ne anlama geldiğini inceleyelim.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 341.38 }, { - "input": "Before the linear transformation, we're thinking of this vector as a certain linear combination of our basis vectors, negative 1 times i-hat plus 2 times j-hat.", + "input": "bout the full 3x3 matrix associated with that transformation. In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length", "translatedText": "Doğrusal dönüşümden önce, bu vektörü temel vektörlerimizin belirli bir doğrusal birleşimi olarak düşünüyoruz, eksi 1 çarpı i-hat artı 2 çarpı j-hat.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 375.16 }, { - "input": "Geometrically, this matrix transforms our grid into Jennifer's grid but numerically, it's translating a vector described in her language to our language.", + "input": "t the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. we get using", "translatedText": "Geometrik olarak bu matris bizim ızgaramızı Jennifer'ın ızgarasına dönüştürüyor ama sayısal olarak onun dilinde tanımlanan bir vektörü bizim dilimize çeviriyor.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 380.62 }, { - "input": "What made it finally click for me was thinking about how it takes our misconception of what Jennifer means, the vector we get using the same coordinates but in our system, then it transforms it into the vector that she really meant.", + "input": "the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I comp", "translatedText": "Sonunda benim için işe yarayan şey, Jennifer'ın ne anlama geldiğine dair yanlış anlamamızı nasıl aldığını, aynı koordinatları kullanarak elde ettiğimiz vektörün sistemimizde onu gerçekten kastettiği vektöre nasıl dönüştürdüğünü düşünmekti.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 398.26 }, { - "input": "What about going the other way around?", + "input": "ute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A", "translatedText": "Peki ya diğer tarafa gitmeye ne dersiniz?", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 404.26 }, { - "input": "In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 third in Jennifer's system?", + "input": "is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis", "translatedText": "Bu videonun başında kullandığım örnekte, sistemimizde koordinatları 3, 2 olan bir vektöre sahip olduğumda, bunun Jennifer'ın sisteminde 5/3 ve 1/3 koordinatlarına sahip olacağını nasıl hesapladım?", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 409.48 }, { - "input": "You start with that change of basis matrix that translates Jennifer's language into ours, then you take its inverse.", + "input": "as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we", "translatedText": "Jennifer'ın dilini bizimkine çeviren temel matris değişikliğiyle başlıyorsunuz, sonra bunun tersini alıyorsunuz.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 415.48 }, { - "input": "Remember, the inverse of a transformation is a new transformation that corresponds to playing that first one backwards.", + "input": "multiply this inverse change of basis matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and fo", "translatedText": "Unutmayın, bir dönüşümün tersi, ilkini geriye doğru oynamaya karşılık gelen yeni bir dönüşümdür.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 427.94 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse.", + "input": "rth between coordinate systems. The matrix whose columns represent Jennif er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the invers", "translatedText": "Uygulamada, özellikle ikiden fazla boyutta çalışırken, bu tersini temsil eden matrisi hesaplamak için bir bilgisayar kullanırsınız.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 465.52 }, { - "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems.", + "input": "you know how matrix multiplication So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, usi", "translatedText": "Kısaca, tek tek vektörlerin tanımının koordinat sistemleri arasında ileri geri nasıl çevrileceği budur.", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 487.24 }, { - "input": "And the inverse matrix does the opposite.", + "input": "The columns of such a matrix will represent what happens to eac", "translatedText": "Ters matris ise tam tersini yapar.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 507.16 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy.", + "input": "heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla With", "translatedText": "Bunlardan herhangi biri sizi rahatsız ediyorsa kesinlikle durun ve 3. ve 4. bölümlere bakın.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 529.74 }, { - "input": "i-hat ends up at the spot with coordinates 0, 1, and j-hat ends up at the spot with coordinates negative 1, 0.", + "input": "or out the v. So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to desc", "translatedText": "i-hat koordinatları 0, 1 olan noktada sona erer ve j-hat koordinatları negatif 1, 0 olan noktada sona erer.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 566.3 }, { - "input": "But that's not quite right.", + "input": "And that squishification corresponds to a zero determinant for the matrix. To be concrete, let's say your matrix", "translatedText": "Ama bu pek doğru değil.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.72 }, { - "input": "Here's a common way to think of how this is done.", + "input": "As that value of lambda changes, the matrix itself changes, and so the determina", "translatedText": "İşte bunun nasıl yapıldığını düşünmenin yaygın bir yolu.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 603.42 }, { - "input": "Start with any vector written in Jennifer's language.", + "input": "nt of the matrix changes. ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multipl", "translatedText": "Jennifer'ın dilinde yazılmış herhangi bir vektörle başlayın.", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 649.44 }, { - "input": "Since we could do this with any vector written in her language, first applying the change of basis, then the transformation, then the inverse change of basis, that composition of three matrices gives us the transformation matrix in Jennifer's language.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective. And the full matrix product represents that same transformation, but as someone else sees it.", "translatedText": "Bunu onun dilinde yazılmış herhangi bir vektörle yapabileceğimiz için önce taban değişimini, sonra dönüşümü, sonra da tabanın ters değişimini uygulayarak üç matrisin bileşimi bize Jennifer'ın dilindeki dönüşüm matrisini verir.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 665.56 }, { - "input": "It takes in a vector of her language and spits out the transformed version of that vector in her language.", + "input": "For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of", "translatedText": "Onun dilinin bir vektörünü alır ve bu vektörün dönüştürülmüş versiyonunu onun dilinde yayınlar.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 675.8 }, { - "input": "For this specific example, when Jennifer's basis vectors look like 2, 1 and negative in our language, and when the transformation is a 90 degree rotation, the product of these three matrices, if you work through it, has columns one third, five thirds, and negative two thirds, negative one third.", + "input": "this. See y That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corres", "translatedText": "Bu özel örnek için, Jennifer'ın temel vektörleri dilimizde 2, 1 ve negatif göründüğünde ve dönüşüm 90 derecelik bir dönüş olduğunda, bu üç matrisin çarpımı, eğer üzerinde çalışırsanız, üçte bir, üçte beşlik sütunlara sahiptir. ve negatif üçte iki, negatif üçte bir.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 692.2 }, { - "input": "So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system.", + "input": "ponding eigenvalue is 1, so v would actually just stay fixed in place. Pause and ponder if you need to make sure that that line of reasoning feels good. This is the kind of thing I mentioned in the introduction. If you didn't have a", "translatedText": "Yani eğer Jennifer bu matrisi sistemindeki bir vektörün koordinatlarıyla çarparsa, bu vektörün kendi koordinat sisteminde ifade edilen 90 derece döndürülmüş versiyonunu verecektir.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 709.82 }, { - "input": "In general, whenever you see an expression like A inverse times M times A, it suggests a mathematical sort of empathy.", + "input": "solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Genel olarak, A ters çarpı M çarpı A gibi bir ifade gördüğünüzde, bu matematiksel bir tür empatiyi akla getirir.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 714.54 }, { - "input": "That middle matrix represents a transformation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective.", + "input": "To see this in action, let's revisit the example from the start, with a matrix whose columns are 3, 0 and 1, 2. To find if a value lambda is an eigenvalue, subtract it from the diago", "translatedText": "Ortadaki matris, gördüğünüz gibi bir tür dönüşümü temsil ediyor ve dıştaki iki matris, empatiyi, perspektifteki değişimi temsil ediyor.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/ukrainian/sentence_translations.json b/2016/change-of-basis/ukrainian/sentence_translations.json index 31e80d86d..f49c7efe8 100644 --- a/2016/change-of-basis/ukrainian/sentence_translations.json +++ b/2016/change-of-basis/ukrainian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates.", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we doing this and what d", "translatedText": "Якщо у мене є вектор, який знаходиться тут у 2D-просторі, ми маємо стандартний спосіб описати його за допомогою координат.", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 28.28 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up.", + "input": "oes this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin", "translatedText": "У цьому випадку вектор має координати 3, 2, що означає, що рух від хвоста до кінчика передбачає переміщення на три одиниці праворуч і дві одиниці вгору.", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.96 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up.", + "input": "ch that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of", "translatedText": "Ви думаєте, що перша координата масштабує i-hat, вектор з довжиною 1 вказує праворуч, тоді як друга координата масштабує j-hat, вектор з довжиною 1 вказує прямо вгору.", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 60.48 }, { - "input": "You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordinate system.", + "input": "you know how to think about matrices as linear transformations, but you also need to be comfortable with things like determinants, linear systems", "translatedText": "Ви можете думати про ці два спеціальні вектори як про інкапсуляцію всіх неявних припущень нашої системи координат.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 81.38 }, { - "input": "Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system.", + "input": "some linear transformation in two dimensions, like the one shown here. st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice o", "translatedText": "Будь-який спосіб перекладу між векторами та наборами чисел називається системою координат, а два спеціальні вектори i-hat і j-hat називаються базисними векторами нашої стандартної системи координат.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors.", + "input": "f i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors an", "translatedText": "Я хотів би поговорити про ідею використання іншого набору базисних векторів.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.04 }, { - "input": "Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third.", + "input": "nnifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to th", "translatedText": "Дженніфер фактично описала б цей вектор з координатами 5 третин і 1 третини.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 125.16 }, { - "input": "In a little bit, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third.", + "input": "showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describe this vector with the coordinates 5 thirds and", "translatedText": "Незабаром я покажу вам, як ви могли обчислити ці два числа, 5 третин і 1 третину.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 144.12 }, { - "input": "What she gets will typically be completely different from the vector that you and I would think of as having those coordinates.", + "input": "gether. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to", "translatedText": "Те, що вона отримує, зазвичай буде повністю відрізнятися від вектора, який ми з вами думали б як такий, що має ці координати.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 167.14 }, { - "input": "So in effect, we're speaking different languages.", + "input": "lumn of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis.", "translatedText": "Тож фактично ми розмовляємо різними мовами.", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 176.84 }, { - "input": "Let me say a quick word about how I'm representing things here.", + "input": "ector on the x-axis is also just stretched by a factor of 3, and hence remains on its own", "translatedText": "Дозвольте мені коротко сказати, як я представляю речі тут.", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid.", + "input": "span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here.", "translatedText": "Сам простір не має внутрішньої сітки.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 197.6 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean.", + "input": "system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct mea", "translatedText": "Хоча її походження фактично збігається з нашим, оскільки всі погоджуються, що мають означати координати 0,0.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 214.9 }, { - "input": "But the direction of her axes and the spacing of her grid lines will be different, depending on her choice of basis vectors.", + "input": "ollow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector", "translatedText": "Але напрям її осей і відстань між її лініями сітки будуть різними залежно від її вибору базисних векторів.", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 225.94 }, { - "input": "If for example Jennifer describes a vector with coordinates negative 1, 2, what would that be in our coordinate system?", + "input": "on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewh", "translatedText": "Якщо, наприклад, Дженніфер описує вектор із координатами мінус 1, 2, що це буде в нашій системі координат?", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 238.98 }, { - "input": "How do you translate from her language to ours?", + "input": "at during the transformation, knocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis", "translatedText": "Як ви перекладаєте з її мови на нашу?", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 260.5 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1.", + "input": "ing them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vector", "translatedText": "І з нашої точки зору b1 має координати 2, 1, а b2 має координати мінус 1, 1.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 267.04 }, { - "input": "So we can actually compute negative 1 times b1 plus 2 times b2 as they're represented in our coordinate system.", + "input": "s in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio Of course, there's n", "translatedText": "Тож ми фактично можемо обчислити мінус 1, помножений на b1, плюс 2, помножений на b2, як вони представлені в нашій системі координат.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 270.4 }, { - "input": "And working this out, you get a vector with coordinates negative 4, 1.", + "input": "othing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In", "translatedText": "І обробивши це, ви отримаєте вектор з координатами мінус 4, 1.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 274.74 }, { - "input": "This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", + "input": "alf, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennif", "translatedText": "Цей процес масштабування кожного з її базисних векторів за допомогою відповідних координат деякого вектора, а потім додавання їх разом, може здатися дещо знайомим.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 330.48 }, { - "input": "To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation.", + "input": "envector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking a", "translatedText": "Щоб показати, як це працює, давайте розберемося, що означало б взяти вектор, який, на нашу думку, має координати мінус 1, 2, і застосувати це перетворення.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 341.38 }, { - "input": "Before the linear transformation, we're thinking of this vector as a certain linear combination of our basis vectors, negative 1 times i-hat plus 2 times j-hat.", + "input": "bout the full 3x3 matrix associated with that transformation. In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length", "translatedText": "Перед лінійним перетворенням ми розглядаємо цей вектор як певну лінійну комбінацію наших базових векторів, мінус 1, помножене на i-hat плюс 2, помножене на j-hat.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 375.16 }, { - "input": "Geometrically, this matrix transforms our grid into Jennifer's grid but numerically, it's translating a vector described in her language to our language.", + "input": "t the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. we get using", "translatedText": "Геометрично ця матриця перетворює нашу сітку на сітку Дженніфер, але чисельно вона перекладає вектор, описаний її мовою, на нашу мову.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 380.62 }, { - "input": "What made it finally click for me was thinking about how it takes our misconception of what Jennifer means, the vector we get using the same coordinates but in our system, then it transforms it into the vector that she really meant.", + "input": "the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I comp", "translatedText": "Що змусило мене нарешті клацнути, так це роздуми про те, як наше неправильне уявлення про те, що має на увазі Дженніфер, вектор, який ми отримуємо за допомогою тих самих координат, але в нашій системі, перетворює його на вектор, який вона насправді мала на увазі.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 398.26 }, { - "input": "What about going the other way around?", + "input": "ute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A", "translatedText": "А як щодо того, щоб піти навпаки?", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 404.26 }, { - "input": "In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 third in Jennifer's system?", + "input": "is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis", "translatedText": "У прикладі, який я використовував раніше в цьому відео, коли я мав вектор із координатами 3, 2 у нашій системі, як я обчислив, що він матиме координати 5 третин і 1 третин у системі Дженніфер?", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 409.48 }, { - "input": "You start with that change of basis matrix that translates Jennifer's language into ours, then you take its inverse.", + "input": "as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we", "translatedText": "Ви починаєте зі зміни базисної матриці, яка перекладає мову Дженніфер на нашу, а потім берете її зворотну.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 415.48 }, { - "input": "Remember, the inverse of a transformation is a new transformation that corresponds to playing that first one backwards.", + "input": "multiply this inverse change of basis matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and fo", "translatedText": "Пам’ятайте, що інверсія трансформації – це нова трансформація, яка відповідає відтворенню першої у зворотному напрямку.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 427.94 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse.", + "input": "rth between coordinate systems. The matrix whose columns represent Jennif er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the invers", "translatedText": "На практиці, особливо коли ви працюєте в більш ніж двох вимірах, ви використовуєте комп’ютер для обчислення матриці, яка насправді представляє цю зворотну величину.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 465.52 }, { - "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems.", + "input": "you know how matrix multiplication So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, usi", "translatedText": "Отже, у двох словах, це те, як перекладати опис окремих векторів вперед і назад між системами координат.", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 487.24 }, { - "input": "And the inverse matrix does the opposite.", + "input": "The columns of such a matrix will represent what happens to eac", "translatedText": "А обернена матриця робить навпаки.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 507.16 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy.", + "input": "heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla With", "translatedText": "Обов’язково зробіть паузу та подивіться на розділи 3 і 4, якщо вам щось з цього незручно.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 529.74 }, { - "input": "i-hat ends up at the spot with coordinates 0, 1, and j-hat ends up at the spot with coordinates negative 1, 0.", + "input": "or out the v. So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to desc", "translatedText": "i-hat опиняється в точці з координатами 0, 1, а j-hat опиняється в точці з від’ємними координатами 1, 0.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 566.3 }, { - "input": "But that's not quite right.", + "input": "And that squishification corresponds to a zero determinant for the matrix. To be concrete, let's say your matrix", "translatedText": "Але це не зовсім правильно.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.72 }, { - "input": "Here's a common way to think of how this is done.", + "input": "As that value of lambda changes, the matrix itself changes, and so the determina", "translatedText": "Ось загальний спосіб уявити, як це робиться.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 603.42 }, { - "input": "Start with any vector written in Jennifer's language.", + "input": "nt of the matrix changes. ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multipl", "translatedText": "Почніть з будь-якого вектора, написаного мовою Дженніфер.", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 649.44 }, { - "input": "Since we could do this with any vector written in her language, first applying the change of basis, then the transformation, then the inverse change of basis, that composition of three matrices gives us the transformation matrix in Jennifer's language.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective. And the full matrix product represents that same transformation, but as someone else sees it.", "translatedText": "Оскільки ми могли б зробити це з будь-яким вектором, написаним її мовою, спочатку застосовуючи зміну базису, потім перетворення, а потім інверсну зміну базису, композиція трьох матриць дає нам матрицю перетворення мовою Дженніфер.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 665.56 }, { - "input": "It takes in a vector of her language and spits out the transformed version of that vector in her language.", + "input": "For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of", "translatedText": "Він приймає вектор її мови та викидає трансформовану версію цього вектора в її мові.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 675.8 }, { - "input": "For this specific example, when Jennifer's basis vectors look like 2, 1 and negative in our language, and when the transformation is a 90 degree rotation, the product of these three matrices, if you work through it, has columns one third, five thirds, and negative two thirds, negative one third.", + "input": "this. See y That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corres", "translatedText": "Для цього конкретного прикладу, коли базові вектори Дженніфер виглядають як 2, 1 і є від’ємними нашою мовою, і коли перетворення є поворотом на 90 градусів, добуток цих трьох матриць, якщо ви попрацюєте над ним, має стовпці одну третину, п’ять третин і мінус дві третини, мінус одна третина.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 692.2 }, { - "input": "So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system.", + "input": "ponding eigenvalue is 1, so v would actually just stay fixed in place. Pause and ponder if you need to make sure that that line of reasoning feels good. This is the kind of thing I mentioned in the introduction. If you didn't have a", "translatedText": "Отже, якщо Дженніфер помножить цю матрицю на координати вектора в її системі, вона поверне повернуту на 90 градусів версію цього вектора, виражену в її системі координат.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 709.82 }, { - "input": "In general, whenever you see an expression like A inverse times M times A, it suggests a mathematical sort of empathy.", + "input": "solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Загалом, щоразу, коли ви бачите вираз на зразок A, обернений на M, помножений на A, це свідчить про математичне співпереживання.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 714.54 }, { - "input": "That middle matrix represents a transformation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective.", + "input": "To see this in action, let's revisit the example from the start, with a matrix whose columns are 3, 0 and 1, 2. To find if a value lambda is an eigenvalue, subtract it from the diago", "translatedText": "Ця середня матриця представляє певну трансформацію, як ви це бачите, а дві зовнішні матриці представляють емпатію, зміну перспективи.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/urdu/sentence_translations.json b/2016/change-of-basis/urdu/sentence_translations.json index a25b3fbc9..7d0b09ecf 100644 --- a/2016/change-of-basis/urdu/sentence_translations.json +++ b/2016/change-of-basis/urdu/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates. ", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we d ", "translatedText": "اگر میرے پاس یہاں 2D جگہ میں کوئی ویکٹر بیٹھا ہے، تو ہمارے پاس اسے کوآرڈینیٹ کے ساتھ بیان کرنے کا ایک معیاری طریقہ ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 27.48 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up. ", + "input": "oing this and what does this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin ", "translatedText": "اس صورت میں، ویکٹر کے پاس 3، 2 کوآرڈینیٹ ہوتے ہیں، جس کا مطلب ہے کہ اس کی دم سے اس کے سرے تک جانے میں تین اکائیوں کو دائیں طرف اور دو اکائیوں کو اوپر جانا شامل ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.36 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up. ", + "input": "not so much that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of ", "translatedText": "آپ اس پہلے کوآرڈینیٹ کو اسکیلنگ i-hat کے طور پر سوچتے ہیں، لمبائی 1 والا ویکٹر دائیں طرف اشارہ کرتا ہے، جب کہ دوسرا کوآرڈینیٹ j-hat کو پیمانہ کرتا ہے، لمبائی 1 والا ویکٹر سیدھا اوپر کی طرف اشارہ کرتا ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -32,7 +32,7 @@ "end": 57.14 }, { - "input": "The tip-to-tail sum of those two scaled vectors is what the coordinates are meant to describe. ", + "input": "the topics that precede it. Most important here is that you know how to think about matrices as linear transformations, but you also need to be comforta ", "translatedText": "ان دو اسکیلڈ ویکٹروں کا ٹپ ٹو ٹیل کا مجموعہ وہی ہے جو کوآرڈینیٹس بیان کرنے کے لیے ہیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors. ", + "input": "of that is tied up in the choice of i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors a ", "translatedText": "میں یہاں جس کے بارے میں بات کرنا چاہوں گا وہ ہے بنیاد ویکٹر کے ایک مختلف سیٹ کو استعمال کرنے کا خیال۔ ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 133.36 }, { - "input": "In general, whenever Jennifer uses coordinates to describe a vector, she thinks of her first coordinate as scaling b1, the second coordinate as scaling b2, and she adds the results. ", + "input": "vector, b2, points left and up. Now take another look at that vector that I showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describ ", "translatedText": "عام طور پر، جب بھی جینیفر کسی ویکٹر کو بیان کرنے کے لیے کوآرڈینیٹ استعمال کرتی ہے، وہ اپنے پہلے کوآرڈینیٹ کو اسکیلنگ b1، دوسرے کوآرڈینیٹ کو اسکیلنگ b2 سمجھتی ہے، اور وہ نتائج کو شامل کرتی ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 166.8 }, { - "input": "They are what define the meaning of the coordinates 1,0 and 0,1 in her world. ", + "input": "then add them both together. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. ", "translatedText": "یہ وہی ہیں جو اس کی دنیا میں نقاط 1,0 اور 0,1 کے معنی کی وضاحت کرتے ہیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 185.86 }, { - "input": "But that grid is just a construct, a way to visualize our coordinate system, and so it depends on our choice of basis. ", + "input": "ample, the basis vector i-hat is one such special vector. The span of i-hat is the x-axis, and from the first column of the matrix, we can see that i-hat moves over to 3 ", "translatedText": "لیکن وہ گرڈ صرف ایک تعمیر ہے، ہمارے کوآرڈینیٹ سسٹم کو دیکھنے کا ایک طریقہ، اور اس لیے یہ ہماری بنیاد کے انتخاب پر منحصر ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid. ", + "input": "times itself, still on that x-axis. What's more, because of the way linear transformations work, ", "translatedText": "خلائی بذات خود کوئی اندرونی گرڈ نہیں ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 198.08 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. ", + "input": "emains on its own span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here. When I animate 2D space, I typically use this square grid. ", "translatedText": "اس کی اصل اگرچہ اصل میں ہمارے ساتھ مطابقت رکھتی ہے، کیونکہ ہر کوئی اس بات پر متفق ہے کہ کوآرڈینیٹ 0,0 کا کیا مطلب ہونا چاہیے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 223.72 }, { - "input": "So after all this is set up, a pretty natural question to ask is how we translate between coordinate systems. ", + "input": "self has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct meant as nothi ng more than a visual tool to help follow the meaning of her coordinates. Her ori ", "translatedText": "لہذا یہ سب کچھ ترتیب دینے کے بعد، یہ پوچھنا ایک قدرتی سوال ہے کہ ہم کوآرڈینیٹ سسٹمز کے درمیان کیسے ترجمہ کرتے ہیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 227.28 }, { - "input": "How do you translate from her language to ours? ", + "input": "hould mean. It's the thing that you get when you scale any vector by zero. ", "translatedText": "آپ اس کی زبان سے ہماری زبان میں کیسے ترجمہ کرتے ہیں؟ ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 230.76 }, { - "input": "Well, what her coordinates are saying is that this vector is negative 1 times b1 plus 2 times b2. ", + "input": "But the direction of her axes and Those on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. ", "translatedText": "ٹھیک ہے، اس کے کوآرڈینیٹ کیا کہہ رہے ہیں کہ یہ ویکٹر منفی 1 گنا b1 جمع 2 گنا b2 ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 231.76 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1. ", + "input": "Any other vector is going to get rotated somewhat during the transformation, knocked off the line that it spans. ks of as negative 1, 2. ", "translatedText": "اور ہمارے نقطہ نظر سے، b1 میں نقاط 2، 1 ہیں، اور b2 میں نقاط منفی 1، 1 ہیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 254.28 }, { - "input": "It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vectors in our language. ", + "input": "tand matrix vector multiplication as applying a certain linear transformatio Of course, there's nothing special about stretching versus squishing, or ", "translatedText": "یہ میٹرکس ویکٹر ضرب ہے، ایک میٹرکس کے ساتھ جس کے کالم ہماری زبان میں جینیفر کے بنیادی ویکٹر کی نمائندگی کرتے ہیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 257.08 }, { - "input": "In fact, once you understand matrix vector multiplication as applying a certain linear transformation, say by watching what I view to be the most important video in this series, Chapter 3, there's a pretty intuitive way to think about what's going on here. ", + "input": "the fact that these eigenvalues happen to be positive. In another example, you could have an eigenvector with eigenvalue negative 1 half, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. ", "translatedText": "درحقیقت، ایک بار جب آپ میٹرکس ویکٹر ضرب کو ایک مخصوص لکیری تبدیلی کو لاگو کرنے کے طور پر سمجھ لیں، تو یہ دیکھ کر کہیے کہ میں اس سیریز کی سب سے اہم ویڈیو، باب 3 میں کیا دیکھ رہا ہوں، یہاں کیا ہو رہا ہے اس کے بارے میں سوچنے کا ایک بہت ہی بدیہی طریقہ ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 384.3 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse. ", + "input": "coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A is the matrix representing some transformation, ", "translatedText": "عملی طور پر، خاص طور پر جب آپ دو سے زیادہ جہتوں میں کام کر رہے ہوں، آپ میٹرکس کی گنتی کے لیے کمپیوٹر کا استعمال کریں گے جو دراصل اس الٹا کی نمائندگی کرتا ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 389.6 }, { - "input": "In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. ", + "input": "with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. ", "translatedText": "اس صورت میں، بنیاد میٹرکس کی تبدیلی کا الٹا جس کی جینیفر کی بنیاد ہے جیسا کہ اس کے کالم کالم 1 تہائی، منفی 1 تہائی، اور 1 تہائی، 2 تہائی پر کام کرتے ہیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 443.9 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy. ", + "input": "es, and that you know how matrix multiplication So let's start by rewriting that right-hand ", "translatedText": "یقینی طور پر توقف کریں اور ابواب 3 اور 4 پر ایک نظر ڈالیں اگر ان میں سے کسی کو بھی پریشانی محسوس ہو۔ ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 446.48 }, { - "input": "Consider some linear transformation, like a 90 degree counterclockwise rotation. ", + "input": "side as some kind of matrix-vector multiplication, using a matrix which has the effect of scaling any vector by a factor of lambda. ", "translatedText": "کچھ لکیری تبدیلی پر غور کریں، جیسے 90 ڈگری مخالف گھڑی کی سمت گردش۔ ", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 451.06 }, { - "input": "When you and I represent this with a matrix, we follow where the basis vectors i-hat and j-hat each go. ", + "input": "The columns of such a matrix will represent what happens to each basis vector, and each basis vector is simply multiplied by lambda, so this ", "translatedText": "جب آپ اور میں میٹرکس کے ساتھ اس کی نمائندگی کرتے ہیں، تو ہم اس بات کی پیروی کرتے ہیں کہ i-hat اور j-hat دونوں کی بنیاد کہاں جاتی ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 477.24 }, { - "input": "How would Jennifer describe this same 90 degree rotation of space? ", + "input": "pla With both sides looking like matrix-vector multiplication, we can subtract off that right-hand side and factor out the v. ", "translatedText": "جینیفر خلا کی اسی 90 ڈگری گردش کو کیسے بیان کرے گی؟ ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 484.26 }, { - "input": "But that's not quite right. ", + "input": "So what we now have is a new matrix, A minus lambda times the identity, and ", "translatedText": "لیکن یہ بالکل درست نہیں ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 515.02 }, { - "input": "Then apply the transformation matrix to what you get by multiplying it on the left. ", + "input": "zero is if the transformation associated with that matrix squishes space into a lower dimension. And that squishification corresponds to a zero determinant for the matr ", "translatedText": "پھر تبدیلی میٹرکس کو اس پر لاگو کریں جو آپ حاصل کرتے ہیں اسے بائیں طرف ضرب کر کے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 515.62 }, { - "input": "This tells us where that vector lands, but still in our language. ", + "input": "ix. To be concrete, let's say your matrix A has columns 2, 1 and 2, 3, and think about subtract ", "translatedText": "یہ ہمیں بتاتا ہے کہ وہ ویکٹر کہاں اترتا ہے، لیکن پھر بھی ہماری زبان میں۔ ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/change-of-basis/vietnamese/sentence_translations.json b/2016/change-of-basis/vietnamese/sentence_translations.json index b7debb22f..d740997e8 100644 --- a/2016/change-of-basis/vietnamese/sentence_translations.json +++ b/2016/change-of-basis/vietnamese/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "If I have a vector sitting here in 2D space, we have a standard way to describe it with coordinates.", + "input": "Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we doing this and what d", "translatedText": "Nếu tôi có một vectơ ở đây trong không gian 2D, chúng ta có một cách tiêu chuẩn để mô tả nó bằng tọa độ.", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 28.28 }, { - "input": "In this case, the vector has coordinates 3, 2, which means going from its tail to its tip involves moving three units to the right and two units up.", + "input": "oes this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward to visualizin", "translatedText": "Trong trường hợp này, vectơ có tọa độ 3, 2, có nghĩa là đi từ đuôi đến đỉnh của nó bao gồm việc di chuyển ba đơn vị sang phải và hai đơn vị lên trên.", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 42.96 }, { - "input": "You think of that first coordinate as scaling i-hat, the vector with length 1 pointing to the right, while the second coordinate scales j-hat, the vector with length 1 pointing straight up.", + "input": "ch that eigenthings are particularly complicated or poorly explained. In fact, it's comparatively straightforward, and I think most books do a fine job explaining it. The issue is that it only really makes sense if you have a solid visual understanding for many of", "translatedText": "Bạn coi tọa độ đầu tiên đó là tỷ lệ i-hat, vectơ có độ dài 1 chỉ về bên phải, trong khi tọa độ thứ hai tỷ lệ j-hat, vectơ có độ dài 1 chỉ thẳng lên trên.", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 60.48 }, { - "input": "You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordinate system.", + "input": "you know how to think about matrices as linear transformations, but you also need to be comfortable with things like determinants, linear systems", "translatedText": "Bạn có thể coi hai vectơ đặc biệt này như gói gọn tất cả các giả định ngầm định của hệ tọa độ của chúng ta.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 81.38 }, { - "input": "Any way to translate between vectors and sets of numbers is called a coordinate system, and the two special vectors i-hat and j-hat are called the basis vectors of our standard coordinate system.", + "input": "some linear transformation in two dimensions, like the one shown here. st number indicates rightward motion, that the second one indicates upward motion, exactly how far a unit of distance is, all of that is tied up in the choice o", "translatedText": "Bất kỳ cách nào để dịch giữa vectơ và tập hợp số đều được gọi là hệ tọa độ và hai vectơ đặc biệt i-hat và j-hat được gọi là vectơ cơ sở của hệ tọa độ tiêu chuẩn của chúng ta.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 89.0 }, { - "input": "What I'd like to talk about here is the idea of using a different set of basis vectors.", + "input": "f i-hat and j-hat as the ve ctors which are scalar coordinates are meant to actually scale. Any way to translate between vectors an", "translatedText": "Điều tôi muốn nói ở đây là ý tưởng sử dụng một tập hợp các vectơ cơ sở khác.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.04 }, { - "input": "Jennifer would actually describe this vector with the coordinates 5 thirds and 1 third.", + "input": "nnifer, who uses a different set of basis vectors, which I'll call b1 and b2. Her first b asis vector, b1, points up and to th", "translatedText": "Jennifer thực sự sẽ mô tả vectơ này với tọa độ 5/3 và 1/3.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 125.16 }, { - "input": "In a little bit, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third.", + "input": "showed earlier, the one that you and I would describe using the coordinates 3,2, using our basis vectors i-hat and j-hat. Jennifer would actually describe this vector with the coordinates 5 thirds and", "translatedText": "Chút nữa, tôi sẽ chỉ cho bạn cách bạn có thể tìm ra hai số đó, 5 phần ba và 1 phần ba.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 144.12 }, { - "input": "What she gets will typically be completely different from the vector that you and I would think of as having those coordinates.", + "input": "gether. In a little bi t, I'll show you how you could have figured out those two numbers, 5 thirds and 1 third. In general, whenever Jennifer uses coordinates to", "translatedText": "Những gì cô ấy nhận được thường sẽ hoàn toàn khác với vectơ mà bạn và tôi nghĩ là có những tọa độ đó.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 167.14 }, { - "input": "So in effect, we're speaking different languages.", + "input": "lumn of the matrix, we can see that i-hat moves over to 3 times itself, still on that x-axis.", "translatedText": "Vì vậy, trên thực tế, chúng ta đang nói những ngôn ngữ khác nhau.", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 176.84 }, { - "input": "Let me say a quick word about how I'm representing things here.", + "input": "ector on the x-axis is also just stretched by a factor of 3, and hence remains on its own", "translatedText": "Hãy để tôi nói nhanh về cách tôi thể hiện mọi thứ ở đây.", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 189.52 }, { - "input": "Space itself has no intrinsic grid.", + "input": "span during this transformation is negative 1, 1. Let me say a quick word about how I'm representing things here.", "translatedText": "Bản thân không gian không có lưới nội tại.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 197.6 }, { - "input": "Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean.", + "input": "system, and so it depends on our choice of basis. Space itself has no intrinsic grid. Jennifer might draw her own grid, which would be an equally made up construct mea", "translatedText": "Tuy nhiên, nguồn gốc của cô ấy thực sự sẽ trùng với nguồn gốc của chúng ta, vì mọi người đều đồng ý về tọa độ 0,0 có nghĩa là gì.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 214.9 }, { - "input": "But the direction of her axes and the spacing of her grid lines will be different, depending on her choice of basis vectors.", + "input": "ollow the meaning of her coordinates. Her origin though would actually line up with ours, since everybody agrees on what the coordinates 0,0 should mean. It's the thing that you get when you scale any vector", "translatedText": "Nhưng hướng của các trục và khoảng cách giữa các đường lưới của cô ấy sẽ khác nhau, tùy thuộc vào sự lựa chọn vectơ cơ sở của cô ấy.", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 225.94 }, { - "input": "If for example Jennifer describes a vector with coordinates negative 1, 2, what would that be in our coordinate system?", + "input": "on the x-axis getting stretched out by a factor of 3, and those on this diagonal line getting stretched by a factor of 2. Any other vector is going to get rotated somewh", "translatedText": "Ví dụ: nếu Jennifer mô tả một vectơ có tọa độ âm 1, 2 thì đó sẽ là gì trong hệ tọa độ của chúng ta?", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 238.98 }, { - "input": "How do you translate from her language to ours?", + "input": "at during the transformation, knocked off the line that it spans. ks of as negative 1, 2. This process here of scaling each of her basis", "translatedText": "Làm thế nào để bạn dịch từ ngôn ngữ của cô ấy sang ngôn ngữ của chúng tôi?", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 260.5 }, { - "input": "And from our perspective, b1 has coordinates 2, 1, and b2 has coordinates negative 1, 1.", + "input": "ing them together, might feel somewhat familiar. It's matrix vector multiplication, with a matrix whose columns represent Jennifer's basis vector", "translatedText": "Và theo quan điểm của chúng tôi, b1 có tọa độ 2, 1 và b2 có tọa độ âm 1, 1.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 267.04 }, { - "input": "So we can actually compute negative 1 times b1 plus 2 times b2 as they're represented in our coordinate system.", + "input": "s in our language. In fact, once you understand matrix vector multiplication as applying a certain linear transformatio Of course, there's n", "translatedText": "Vì vậy, chúng ta thực sự có thể tính âm 1 nhân b1 cộng 2 nhân b2 khi chúng được biểu diễn trong hệ tọa độ của chúng ta.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 270.4 }, { - "input": "And working this out, you get a vector with coordinates negative 4, 1.", + "input": "othing special about stretching versus squishing, or the fact that these eigenvalues happen to be positive. In", "translatedText": "Và tính ra điều này, bạn sẽ có được một vectơ có tọa độ âm 4, 1.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 274.74 }, { - "input": "This process here of scaling each of her basis vectors by the corresponding coordinates of some vector, then adding them together, might feel somewhat familiar.", + "input": "alf, meaning that the vector gets flipped and squished by a factor of 1 half. pretty intuitive way to think about what's going on here. A matrix whose columns represent Jennif", "translatedText": "Quá trình chia tỷ lệ từng vectơ cơ sở của cô ấy theo tọa độ tương ứng của một số vectơ ở đây, sau đó cộng chúng lại với nhau, có thể cảm thấy hơi quen thuộc.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 330.48 }, { - "input": "To show how this works, let's walk through what it would mean to take the vector that we think of as having coordinates negative 1, 2 and applying that transformation.", + "input": "envector for that rotation, a vector that remains on its own span, what you have found is the axis of rotation. And it's much easier to think about a 3D rotation in terms of some axis of rotation and an angle by which it's rotating, rather than thinking a", "translatedText": "Để chỉ ra cách thức hoạt động của nó, chúng ta hãy tìm hiểu ý nghĩa của việc lấy vectơ mà chúng ta nghĩ là có tọa độ âm 1, 2 và áp dụng phép biến đổi đó.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 341.38 }, { - "input": "Before the linear transformation, we're thinking of this vector as a certain linear combination of our basis vectors, negative 1 times i-hat plus 2 times j-hat.", + "input": "bout the full 3x3 matrix associated with that transformation. In this case, by the way, the corresponding eigenvalue would have to be 1, since rotations never stretch or squish anything, so the length", "translatedText": "Trước khi chuyển đổi tuyến tính, chúng ta coi vectơ này là một tổ hợp tuyến tính nhất định của các vectơ cơ sở, âm 1 nhân i-hat cộng 2 nhân j-hat.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 375.16 }, { - "input": "Geometrically, this matrix transforms our grid into Jennifer's grid but numerically, it's translating a vector described in her language to our language.", + "input": "t the heart of what the linear transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. we get using", "translatedText": "Về mặt hình học, ma trận này biến lưới của chúng ta thành lưới của Jennifer nhưng về mặt số học, nó đang dịch một vectơ được mô tả bằng ngôn ngữ của cô ấy sang ngôn ngữ của chúng ta.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 380.62 }, { - "input": "What made it finally click for me was thinking about how it takes our misconception of what Jennifer means, the vector we get using the same coordinates but in our system, then it transforms it into the vector that she really meant.", + "input": "the same coordinates but in our system, then it transforms it into the vector that she really meant. What about going the other way around? In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I comp", "translatedText": "Điều cuối cùng khiến tôi nhấp chuột là suy nghĩ về cách chúng ta hiểu sai ý nghĩa của Jennifer, vectơ mà chúng ta có được khi sử dụng cùng tọa độ nhưng trong hệ thống của chúng ta, sau đó nó biến đổi nó thành vectơ mà cô ấy thực sự muốn nói.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 398.26 }, { - "input": "What about going the other way around?", + "input": "ute that it would have coordinates 5 thirds and 1 Symbolically, here's what the idea of an eigenvector looks like. A", "translatedText": "Còn việc đi ngược lại thì sao?", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 404.26 }, { - "input": "In the example I used earlier this video, when I had the vector with coordinates 3, 2 in our system, how did I compute that it would have coordinates 5 thirds and 1 third in Jennifer's system?", + "input": "is the matrix representing some transformation, with v as the eigenvector, and lambda is a number, namely the corresponding eigenvalue. e. In this case, the inverse of the change of basis matrix that has Jennifer's basis", "translatedText": "Trong ví dụ tôi đã sử dụng trước đó trong video này, khi tôi có vectơ có tọa độ 3, 2 trong hệ thống của mình, làm cách nào để tính toán rằng nó sẽ có tọa độ 5/3 và 1/3 trong hệ thống của Jennifer?", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 409.48 }, { - "input": "You start with that change of basis matrix that translates Jennifer's language into ours, then you take its inverse.", + "input": "as its columns ends up working out to have columns 1 third, negative 1 third, and 1 third, 2 thirds. So for example, to see what the vector 3, 2 looks like in Jennifer's system, we", "translatedText": "Bạn bắt đầu với sự thay đổi ma trận cơ sở để dịch ngôn ngữ của Jennifer sang ngôn ngữ của chúng ta, sau đó bạn thực hiện nghịch đảo của nó.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 415.48 }, { - "input": "Remember, the inverse of a transformation is a new transformation that corresponds to playing that first one backwards.", + "input": "multiply this inverse change of basis matrix by the vector 3, 2, which works out to be 5 thirds, 1 third. So that, in a nutshell, is how to translate the description of individual vectors back and fo", "translatedText": "Hãy nhớ rằng, nghịch đảo của một phép biến đổi là một phép biến đổi mới tương ứng với việc chơi ngược lại phép biến đổi đầu tiên đó.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 427.94 }, { - "input": "In practice, especially when you're working in more than two dimensions, you'd use a computer to compute the matrix that actually represents this inverse.", + "input": "rth between coordinate systems. The matrix whose columns represent Jennif er's basis vectors, but written in our coordinates, translates vectors from her language into our language. And the invers", "translatedText": "Trong thực tế, đặc biệt là khi bạn làm việc trong không gian nhiều hơn hai chiều, bạn sẽ sử dụng máy tính để tính ma trận đại diện cho nghịch đảo này.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 465.52 }, { - "input": "So that, in a nutshell, is how to translate the description of individual vectors back and forth between coordinate systems.", + "input": "you know how matrix multiplication So let's start by rewriting that right-hand side as some kind of matrix-vector multiplication, usi", "translatedText": "Vì vậy, tóm lại, đó là cách dịch mô tả của các vectơ riêng lẻ qua lại giữa các hệ tọa độ.", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 487.24 }, { - "input": "And the inverse matrix does the opposite.", + "input": "The columns of such a matrix will represent what happens to eac", "translatedText": "Và ma trận nghịch đảo làm điều ngược lại.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 507.16 }, { - "input": "Definitely pause and take a look at chapters 3 and 4 if any of that feels uneasy.", + "input": "heavily tied up in our choice of basis vectors, from the fact that we're following i-hat and j-hat in the first pla With", "translatedText": "Chắc chắn hãy tạm dừng và xem lại chương 3 và 4 nếu bạn cảm thấy khó chịu.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 529.74 }, { - "input": "i-hat ends up at the spot with coordinates 0, 1, and j-hat ends up at the spot with coordinates negative 1, 0.", + "input": "or out the v. So what we now have is a new matrix, A minus lambda times the identity, and we're looking for a vector v such that this new matrix times v gives the zero vector. s land, and it needs to desc", "translatedText": "i-hat kết thúc tại vị trí có tọa độ 0, 1 và j-hat kết thúc tại vị trí có tọa độ âm 1, 0.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 566.3 }, { - "input": "But that's not quite right.", + "input": "And that squishification corresponds to a zero determinant for the matrix. To be concrete, let's say your matrix", "translatedText": "Nhưng điều đó không hoàn toàn đúng.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.72 }, { - "input": "Here's a common way to think of how this is done.", + "input": "As that value of lambda changes, the matrix itself changes, and so the determina", "translatedText": "Đây là một cách phổ biến để nghĩ về cách thực hiện điều này.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 603.42 }, { - "input": "Start with any vector written in Jennifer's language.", + "input": "nt of the matrix changes. ou work through it, has columns one third, five thirds, and negative two thirds, negative one third. So if Jennifer multipl", "translatedText": "Bắt đầu với bất kỳ vectơ nào được viết bằng ngôn ngữ của Jennifer.", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 649.44 }, { - "input": "Since we could do this with any vector written in her language, first applying the change of basis, then the transformation, then the inverse change of basis, that composition of three matrices gives us the transformation matrix in Jennifer's language.", + "input": "A inverse times M times A, it suggests a mathematical sort of empathy. That middle matrix represents a transform ation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective. And the full matrix product represents that same transformation, but as someone else sees it.", "translatedText": "Vì chúng ta có thể làm điều này với bất kỳ vectơ nào được viết bằng ngôn ngữ của cô ấy, trước tiên áp dụng phép biến đổi cơ số, sau đó là phép biến đổi, sau đó là phép biến đổi cơ số nghịch đảo, nên sự kết hợp của ba ma trận sẽ cho chúng ta ma trận biến đổi trong ngôn ngữ của Jennifer.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 665.56 }, { - "input": "It takes in a vector of her language and spits out the transformed version of that vector in her language.", + "input": "For those of you wondering why we care about alternate coordinate systems, the next video on eigenvectors and eigenvalues will give a really important example of", "translatedText": "Nó lấy một vectơ ngôn ngữ của cô ấy và tạo ra phiên bản biến đổi của vectơ đó trong ngôn ngữ của cô ấy.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 675.8 }, { - "input": "For this specific example, when Jennifer's basis vectors look like 2, 1 and negative in our language, and when the transformation is a 90 degree rotation, the product of these three matrices, if you work through it, has columns one third, five thirds, and negative two thirds, negative one third.", + "input": "this. See y That means there's a non-zero vector v such that A minus lambda times the identity times v equals the zero vector. And remember, the reason we care about that is because it means A times v equals lambda times v, which you can read off as saying that the vector v is an eigenvector of A, staying on its own span during the transformation A. In this example, the corres", "translatedText": "Đối với ví dụ cụ thể này, khi vectơ cơ sở của Jennifer trông giống như 2, 1 và âm trong ngôn ngữ của chúng ta và khi phép biến đổi là phép quay 90 độ, thì tích của ba ma trận này, nếu bạn tính toán thông qua nó, sẽ có các cột một phần ba, năm phần ba , và âm hai phần ba, âm một phần ba.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 692.2 }, { - "input": "So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the 90 degree rotated version of that vector expressed in her coordinate system.", + "input": "ponding eigenvalue is 1, so v would actually just stay fixed in place. Pause and ponder if you need to make sure that that line of reasoning feels good. This is the kind of thing I mentioned in the introduction. If you didn't have a", "translatedText": "Vì vậy, nếu Jennifer nhân ma trận đó với tọa độ của một vectơ trong hệ tọa độ của cô ấy, nó sẽ trả về phiên bản quay 90 độ của vectơ đó được biểu thị trong hệ tọa độ của cô ấy.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 709.82 }, { - "input": "In general, whenever you see an expression like A inverse times M times A, it suggests a mathematical sort of empathy.", + "input": "solid grasp of determinants and why they relate to linear systems of equations having non-zero solutions, an expression like this would feel completely out of the blue.", "translatedText": "Nói chung, bất cứ khi nào bạn nhìn thấy một biểu thức như A nghịch đảo nhân M nhân A, điều đó gợi ý một dạng đồng cảm về mặt toán học.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 714.54 }, { - "input": "That middle matrix represents a transformation of some kind as you see it, and the outer two matrices represent the empathy, the shift in perspective.", + "input": "To see this in action, let's revisit the example from the start, with a matrix whose columns are 3, 0 and 1, 2. To find if a value lambda is an eigenvalue, subtract it from the diago", "translatedText": "Ma trận ở giữa đó đại diện cho một loại biến đổi nào đó như bạn thấy, và hai ma trận bên ngoài đại diện cho sự đồng cảm, sự thay đổi trong quan điểm.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/arabic/sentence_translations.json b/2016/determinant/arabic/sentence_translations.json index 37d45512c..d61b5eb37 100644 --- a/2016/determinant/arabic/sentence_translations.json +++ b/2016/determinant/arabic/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", "translatedText": "سيكون محدد التحول هو النصف إذا سحق جميع المناطق بعامل النصف. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/bengali/sentence_translations.json b/2016/determinant/bengali/sentence_translations.json index 4d439c475..b5f9c5646 100644 --- a/2016/determinant/bengali/sentence_translations.json +++ b/2016/determinant/bengali/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", "translatedText": "একটি রূপান্তরের নির্ধারক হবে 1 অর্ধেক যদি এটি 1 অর্ধেক একটি ফ্যাক্টর দ্বারা সমস্ত এলাকা squishes নিচে. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/chinese/sentence_translations.json b/2016/determinant/chinese/sentence_translations.json index 6c2061101..43a4e5fe9 100644 --- a/2016/determinant/chinese/sentence_translations.json +++ b/2016/determinant/chinese/sentence_translations.json @@ -179,7 +179,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", "translatedText": "如果变换将所有区域压缩为二分之一 ，则变换的行列式将为二分之一。", "model": "google_nmt", "from_community_srt": "一个线性变换的行列式是3 就是说它将一个区域的面积增加为原来的3倍 一个线性变换的行列式是1/2 就是说它将一个区域的面积缩小一半 而一个二维线性变换的行列式为0 说明它将整个平面压缩到一条线，", diff --git a/2016/determinant/czech/sentence_translations.json b/2016/determinant/czech/sentence_translations.json index 48b306ee5..57c8dea9f 100644 --- a/2016/determinant/czech/sentence_translations.json +++ b/2016/determinant/czech/sentence_translations.json @@ -162,7 +162,7 @@ "end": 157.12 }, { - "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation.", + "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation.", "translatedText": "Později v tomto videu ukážu, jak vypočítat determinant transformace pomocí její matice, což je při výpočtu také důležité.", "model": "DeepL", "from_community_srt": "Později vám ukážu, jak spočítat determinant transformace pomocí její matice, ale věřte mi, rozumět tomu, co to znamená, je mnohem důležitější než to umět spočítat.", @@ -189,7 +189,7 @@ "end": 184.34 }, { - "input": "And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", + "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single", "translatedText": "Determinant 2D transformace je roven 0, pokud se celý prostor zmačká na přímku nebo dokonce na jediný bod.", "model": "DeepL", "from_community_srt": "a když je determinant 2D transformace nulový, tak ta transformace všechno splácne na jednu přímku", @@ -341,7 +341,7 @@ "end": 282.4 }, { - "input": "For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3.", + "input": "For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3.", "translatedText": "Například matice se sloupci 1,1 a 2,-1 kóduje transformaci, která má determinant, řeknu vám jen, záporný 3.", "model": "DeepL", "from_community_srt": "Například matice se sloupci (1,1), (2,-1) popisuje transformaci, jejíž determinant je minus tři.", @@ -706,7 +706,7 @@ "end": 577.88 }, { - "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations.", + "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then!", "translatedText": "Dále se budu věnovat myšlence lineárních transformací, kterou jsme se dosud zabývali, a jedné z oblastí, kde je lineární algebra nejužitečnější - lineárním soustavám rovnic.", "model": "DeepL", "from_community_srt": "jestli by se to nedalo vysvětlit jednou větou. Příště se podíváme na myšlenku, kterou lineární zobrazení už pokryly, na jednu z oblastí, kde je lineární algebra nejužitečnější, soustavy lineárních rovnic.", diff --git a/2016/determinant/english/captions.srt b/2016/determinant/english/captions.srt index fbb099b7c..43f9b5a07 100644 --- a/2016/determinant/english/captions.srt +++ b/2016/determinant/english/captions.srt @@ -135,450 +135,458 @@ This very special scaling factor, the factor by which a linear transformation changes any area, is called the determinant of that transformation. 35 -00:02:39,120 --> 00:02:43,673 -I'll show how to compute the determinant of a transformation using its +00:02:39,120 --> 00:02:42,044 +I'll show how to compute the determinant of a transformation 36 -00:02:43,673 --> 00:02:48,420 -matrix later on in this video, which is also important in the computation. +00:02:42,044 --> 00:02:45,975 +using its matrix later on in this video, but understanding what it represents is, 37 +00:02:45,975 --> 00:02:48,420 +trust me, much more important than the computation. + +38 00:02:49,580 --> 00:02:53,167 For example, the determinant of a transformation would be 3 if -38 +39 00:02:53,167 --> 00:02:57,040 that transformation increases the area of a region by a factor of 3. -39 -00:02:58,180 --> 00:03:01,260 -The determinant of a transformation would be ½ - 40 -00:03:01,260 --> 00:03:04,340 -if it squishes down all areas by a factor of ½. +00:02:57,610 --> 00:02:57,040 +The determinant of a transformation would be ½ 41 -00:03:06,000 --> 00:03:11,705 -And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, +00:02:58,180 --> 00:02:57,610 +if it squishes down all areas by a factor of ½. 42 -00:03:11,705 --> 00:03:13,500 -or even onto a single point. +00:02:58,180 --> 00:03:03,148 +The determinant of a transformation would be 1 half if it squishes down 43 +00:03:03,148 --> 00:03:08,393 +all areas by a factor of 1 half. And the determinant of a 2D transformation + +44 +00:03:08,393 --> 00:03:13,500 +is 0 if it squishes all of space onto a line, or even onto a single point. + +45 00:03:14,000 --> 00:03:16,760 Since then, the area of any region would become zero. -44 +46 00:03:17,620 --> 00:03:19,600 That last example will prove to be pretty important. -45 +47 00:03:20,020 --> 00:03:23,260 It means that checking if the determinant of a given matrix is zero -46 +48 00:03:23,260 --> 00:03:26,261 will give a way of computing whether or not the transformation -47 +49 00:03:26,261 --> 00:03:29,740 associated with that matrix squishes everything into a smaller dimension. -48 +50 00:03:30,520 --> 00:03:34,278 You'll see in the next few videos why this is even a useful thing to think about, -49 +51 00:03:34,278 --> 00:03:37,303 but for now, I just want to lay down all of the visual intuition, -50 +52 00:03:37,303 --> 00:03:40,100 which, in and of itself, is a beautiful thing to think about. -51 +53 00:03:42,120 --> 00:03:45,560 Okay, I need to confess that what I've said so far is not quite right. -52 +54 00:03:45,880 --> 00:03:49,280 The full concept of the determinant allows for negative values. -53 +55 00:03:49,720 --> 00:03:53,480 But what would the idea of scaling an area by a negative amount even mean? -54 +56 00:03:54,940 --> 00:03:56,960 This has to do with the idea of orientation. -55 +57 00:03:57,800 --> 00:04:02,680 For example, notice how this transformation gives the sensation of flipping space over. -56 +58 00:04:03,240 --> 00:04:05,888 If you were thinking of 2D space as a sheet of paper, -57 +59 00:04:05,888 --> 00:04:09,860 a transformation like that one seems to turn over that sheet onto the other side. -58 +60 00:04:10,640 --> 00:04:15,040 Any transformations that do this are said to invert the orientation of space. -59 +61 00:04:15,840 --> 00:04:18,600 Another way to think about it is in terms of i-hat and j-hat. -60 +62 00:04:19,160 --> 00:04:23,060 Notice that in their starting positions, j-hat is to the left of i-hat. -61 +63 00:04:23,620 --> 00:04:27,555 If, after a transformation, j-hat is now on the right of i-hat, -62 +64 00:04:27,555 --> 00:04:30,200 the orientation of space has been inverted. -63 +65 00:04:32,120 --> 00:04:35,151 Whenever this happens, whenever the orientation of space is inverted, -64 +66 00:04:35,151 --> 00:04:36,580 the determinant will be negative. -65 +67 00:04:37,460 --> 00:04:39,650 The absolute value of the determinant, though, -66 +68 00:04:39,650 --> 00:04:42,400 still tells you the factor by which areas have been scaled. -67 -00:04:43,020 --> 00:04:46,610 -For example, the matrix with columns 1,1 and 2,-1 encodes a +69 +00:04:43,020 --> 00:04:45,739 +For example, the matrix with columns 1, 1 and 2, -68 -00:04:46,610 --> 00:04:50,680 -transformation that has determinant, I'll just tell you, negative 3. +70 +00:04:45,739 --> 00:04:50,680 +negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3. -69 +71 00:04:51,460 --> 00:04:56,280 And what this means is that space gets flipped over and areas are scaled by a factor of 3. -70 +72 00:04:57,780 --> 00:05:00,546 So why would this idea of a negative area scaling -71 +73 00:05:00,546 --> 00:05:03,700 factor be a natural way to describe orientation flipping? -72 +74 00:05:04,260 --> 00:05:07,227 Think about the series of transformations you get by -73 +75 00:05:07,227 --> 00:05:10,140 slowly letting i-hat get closer and closer to j-hat. -74 +76 00:05:10,720 --> 00:05:15,149 As i-hat gets closer, all of the areas in space are getting squished more and more, -75 +77 00:05:15,149 --> 00:05:17,100 meaning the determinant approaches 0. -76 +78 00:05:17,820 --> 00:05:21,640 Once i-hat lines up perfectly with j-hat, the determinant is 0. -77 +79 00:05:22,440 --> 00:05:24,827 Then, if i-hat continues the way that it was going, -78 +80 00:05:24,827 --> 00:05:28,315 doesn't it kind of feel natural for the determinant to keep decreasing into -79 +81 00:05:28,315 --> 00:05:29,280 the negative numbers? -80 +82 00:05:30,680 --> 00:05:33,560 So that's the understanding of determinants in two dimensions. -81 +83 00:05:33,560 --> 00:05:35,940 What do you think it should mean for three dimensions? -82 +84 00:05:36,920 --> 00:05:40,162 It also tells you how much a transformation scales things, -83 +85 00:05:40,162 --> 00:05:43,240 but this time, it tells you how much volumes get scaled. -84 +86 00:05:45,340 --> 00:05:49,777 Just as in two dimensions, where this is easiest to think about by focusing -85 +87 00:05:49,777 --> 00:05:54,331 on one particular square with an area 1 and watching only what happens to it, -86 +88 00:05:54,331 --> 00:05:58,827 in three dimensions, it helps to focus your attention on the specific 1 by 1 -87 +89 00:05:58,827 --> 00:06:03,440 by 1 cube whose edges are resting on the basis vectors, i-hat, j-hat and k-hat. -88 +90 00:06:04,320 --> 00:06:09,300 After the transformation, that cube might get warped into some kind of slanty slanty cube. -89 +91 00:06:10,340 --> 00:06:13,252 This shape, by the way, has the best name ever, parallelipiped, -90 +92 00:06:13,252 --> 00:06:16,757 a name that's made even more delightful when your professor has a nice thick -91 +93 00:06:16,757 --> 00:06:17,440 Russian accent. -92 +94 00:06:18,520 --> 00:06:21,074 Since this cube starts out with a volume of 1, -93 +95 00:06:21,074 --> 00:06:24,770 and the determinant gives the factor by which any volume is scaled, -94 +96 00:06:24,770 --> 00:06:28,466 you can think of the determinant simply as being the volume of that -95 +97 00:06:28,466 --> 00:06:30,640 parallelipiped that the cube turns into. -96 +98 00:06:32,380 --> 00:06:37,525 A determinant of 0 would mean that all of space is squished onto something with 0 volume, -97 +99 00:06:37,525 --> 00:06:42,500 meaning either a flat plane, a line, or, in the most extreme case, onto a single point. -98 +100 00:06:43,760 --> 00:06:46,408 Those of you who watched chapter 2 will recognize this as -99 +101 00:06:46,408 --> 00:06:49,240 meaning that the columns of the matrix are linearly dependent. -100 +102 00:06:49,760 --> 00:06:50,420 Can you see why? -101 +103 00:06:54,920 --> 00:06:56,640 What about negative determinants? -102 +104 00:06:56,780 --> 00:06:58,100 What should that mean for three dimensions? -103 +105 00:06:58,780 --> 00:07:02,680 One way to describe orientation in 3D is with the right hand rule. -104 +106 00:07:03,300 --> 00:07:06,485 Point the forefinger of your right hand in the direction of i-hat, -105 +107 00:07:06,485 --> 00:07:09,147 stick out your middle finger in the direction of j-hat, -106 +108 00:07:09,147 --> 00:07:12,760 and notice how when you point your thumb up, it's in the direction of k-hat. -107 +109 00:07:14,880 --> 00:07:17,621 If you can still do that after the transformation, -108 +110 00:07:17,621 --> 00:07:20,900 orientation has not changed, and the determinant is positive. -109 +111 00:07:21,540 --> 00:07:25,383 Otherwise, if after the transformation it only makes sense to do that with -110 +112 00:07:25,383 --> 00:07:29,380 your left hand, orientation has been flipped, and the determinant is negative. -111 +113 00:07:31,900 --> 00:07:35,038 So, if you haven't seen it before, you're probably wondering by now, -112 +114 00:07:35,038 --> 00:07:37,040 how do you actually compute the determinant? -113 +115 00:07:37,560 --> 00:07:44,420 For a 2x2 matrix with entries a, b, c, d, the formula is a times d minus b times c. -114 +116 00:07:45,740 --> 00:07:48,500 Here's part of an intuition for where this formula comes from. -115 +117 00:07:48,880 --> 00:07:51,780 Let's say that the terms b and c both happened to be 0. -116 +118 00:07:51,780 --> 00:07:56,565 Then, the term a tells you how much i-hat is stretched in the x direction, -117 +119 00:07:56,565 --> 00:08:01,160 and the term d tells you how much j-hat is stretched in the y direction. -118 +120 00:08:02,760 --> 00:08:06,202 So, since those other terms are 0, it should make sense that a -119 +121 00:08:06,202 --> 00:08:10,682 times d gives the area of the rectangle that our favorite unit square turns into, -120 +122 00:08:10,682 --> 00:08:13,360 kind of like the 3, 0, 0, 2 example from earlier. -121 +123 00:08:15,360 --> 00:08:18,595 Even if only one of b or c are 0, you'll have -122 +124 00:08:18,595 --> 00:08:21,760 a parallelogram with a base a and a height d. -123 +125 00:08:21,780 --> 00:08:24,500 So, the area should still be a times d. -124 +126 00:08:25,460 --> 00:08:28,422 Loosely speaking, if both b and c are non-zero, -125 +127 00:08:28,422 --> 00:08:33,299 then that b times c term tells you how much this parallelogram is stretched or -126 +128 00:08:33,299 --> 00:08:35,460 squished in the diagonal direction. -127 +129 00:08:36,659 --> 00:08:40,220 For those of you hungry for a more precise description of this b times c term, -128 +130 00:08:40,220 --> 00:08:42,880 here's a helpful diagram if you'd like to pause and ponder. -129 +131 00:08:43,980 --> 00:08:48,235 Now, if you feel like computing determinants by hand is something that you need to know, -130 +132 00:08:48,235 --> 00:08:51,200 the only way to get it down is to just practice it with a few. -131 +133 00:08:51,200 --> 00:08:53,386 There's really not that much I can say or animate -132 +134 00:08:53,386 --> 00:08:55,180 that's going to drill in the computation. -133 +135 00:08:56,120 --> 00:08:58,640 This is all triply true for three-dimensional determinants. -134 +136 00:08:59,040 --> 00:09:02,342 There is a formula, and if you feel like that's something you need to know, -135 +137 00:09:02,342 --> 00:09:04,732 you should practice with a few matrices, or, you know, -136 +138 00:09:04,732 --> 00:09:06,340 go watch Sal Khan work through a few. -137 +139 00:09:07,240 --> 00:09:10,390 Honestly, though, I don't think that those computations fall within -138 +140 00:09:10,390 --> 00:09:13,124 the essence of linear algebra, but I definitely think that -139 +141 00:09:13,124 --> 00:09:16,460 understanding what the determinant represents falls within that essence. -140 +142 00:09:18,060 --> 00:09:20,640 Here's kind of a fun question to think about before the next video. -141 +143 00:09:20,640 --> 00:09:25,451 If you multiply two matrices together, the determinant of the resulting matrix -142 +144 00:09:25,451 --> 00:09:30,080 is the same as the product of the determinants of the original two matrices. -143 +145 00:09:31,100 --> 00:09:34,675 If you tried to justify this with numbers, it would take a really long time, -144 +146 00:09:34,675 --> 00:09:37,880 but see if you can explain why this makes sense in just one sentence. -145 -00:09:42,000 --> 00:09:46,710 -Next up, I'll be relating the idea of linear transformations covered so far to +147 +00:09:42,000 --> 00:09:46,717 +Next up, I'll be relating the idea of linear transformations covered so far to one of -146 -00:09:46,710 --> 00:09:51,600 -one of the areas where linear algebra is most useful, linear systems of equations. +148 +00:09:46,717 --> 00:09:51,600 +the areas where linear algebra is most useful, linear systems of equations. See you then! diff --git a/2016/determinant/english/sentence_timings.json b/2016/determinant/english/sentence_timings.json index b52d63714..92f257386 100644 --- a/2016/determinant/english/sentence_timings.json +++ b/2016/determinant/english/sentence_timings.json @@ -90,7 +90,7 @@ 157.12 ], [ - "I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation.", + "I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation.", 159.12, 168.42 ], @@ -102,11 +102,11 @@ [ "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½.", 178.18, - 184.34 + 177.04 ], [ - "And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", - 186.0, + "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", + 178.18, 193.5 ], [ @@ -190,7 +190,7 @@ 282.4 ], [ - "For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3.", + "For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3.", 283.02, 290.68 ], @@ -395,7 +395,7 @@ 577.88 ], [ - "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations.", + "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then!", 582.0, 591.6 ] diff --git a/2016/determinant/english/transcript.txt b/2016/determinant/english/transcript.txt index 520929660..20b1ad213 100644 --- a/2016/determinant/english/transcript.txt +++ b/2016/determinant/english/transcript.txt @@ -16,10 +16,10 @@ This follows from the fact that grid lines remain parallel and evenly spaced. Then, any shape that's not a grid square can be approximated by grid squares pretty well, with arbitrarily good approximations if you use small enough grid squares. So, since the areas of all those tiny grid squares are being scaled by some single amount, the area of the blob as a whole will also be scaled by that same single amount. This very special scaling factor, the factor by which a linear transformation changes any area, is called the determinant of that transformation. -I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation. +I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation. For example, the determinant of a transformation would be 3 if that transformation increases the area of a region by a factor of 3. The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. -And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point. +The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point. Since then, the area of any region would become zero. That last example will prove to be pretty important. It means that checking if the determinant of a given matrix is zero will give a way of computing whether or not the transformation associated with that matrix squishes everything into a smaller dimension. @@ -36,7 +36,7 @@ Notice that in their starting positions, j-hat is to the left of i-hat. If, after a transformation, j-hat is now on the right of i-hat, the orientation of space has been inverted. Whenever this happens, whenever the orientation of space is inverted, the determinant will be negative. The absolute value of the determinant, though, still tells you the factor by which areas have been scaled. -For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3. +For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3. And what this means is that space gets flipped over and areas are scaled by a factor of 3. So why would this idea of a negative area scaling factor be a natural way to describe orientation flipping? Think about the series of transformations you get by slowly letting i-hat get closer and closer to j-hat. @@ -77,4 +77,4 @@ Honestly, though, I don't think that those computations fall within the essence Here's kind of a fun question to think about before the next video. If you multiply two matrices together, the determinant of the resulting matrix is the same as the product of the determinants of the original two matrices. If you tried to justify this with numbers, it would take a really long time, but see if you can explain why this makes sense in just one sentence. -Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. \ No newline at end of file +Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then! \ No newline at end of file diff --git a/2016/determinant/estonian/sentence_translations.json b/2016/determinant/estonian/sentence_translations.json index 209a1b4c5..5feacd9c9 100644 --- a/2016/determinant/estonian/sentence_translations.json +++ b/2016/determinant/estonian/sentence_translations.json @@ -162,7 +162,7 @@ "end": 157.12 }, { - "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation.", + "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation.", "translatedText": "Selles videos näitan hiljem, kuidas arvutada teisenduse determinant selle maatriksi abil, mis on samuti arvutamisel oluline.", "model": "google_nmt", "from_community_srt": "kuidas kujutuse determinanti arvutatakse selle maatriksi kaudu veidi hiljem selles videos, kuid arusaam, mis see on,", @@ -189,7 +189,7 @@ "end": 184.34 }, { - "input": "And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", + "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single", "translatedText": "Ja 2D teisenduse determinant on 0, kui see surub kogu ruumi joonele või isegi ühele punktile.", "model": "google_nmt", "from_community_srt": "Ning kahemõõtmelise kujutuse determinant on 0, kui see surub kogu tasandi kokku sirgele", @@ -339,7 +339,7 @@ "end": 282.4 }, { - "input": "For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3.", + "input": "For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3.", "translatedText": "Näiteks maatriks veergudega 1,1 ja 2,-1 kodeerib teisenduse, millel on determinant, ma ütlen teile, negatiivne 3.", "model": "google_nmt", "from_community_srt": "mitu korda on pindalad muutunud Näiteks: maatriks veergudega 1;1 ja 2;-1 tähistab kujutust, mille determinant on, ma lihtsalt ütlen ette,", @@ -706,7 +706,7 @@ "end": 577.88 }, { - "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations.", + "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then!", "translatedText": "Järgmisena seon ma seni käsitletud lineaarsete teisenduste idee ühe valdkonnaga, kus lineaaralgebra on kõige kasulikum, lineaarsete võrrandisüsteemidega.", "model": "google_nmt", "from_community_srt": "Järgmises osas ma seostan idee lineaarkujutusest, mida me oleme seni vaadanud, ühe lineaaralgebra kasulikuma osaga,", diff --git a/2016/determinant/french/sentence_translations.json b/2016/determinant/french/sentence_translations.json index f9f44f888..5cc6a27f3 100644 --- a/2016/determinant/french/sentence_translations.json +++ b/2016/determinant/french/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", "translatedText": "Le déterminant d’une transformation serait de 1 moitié si elle réduisait toutes les zones d’un facteur de 1 moitié.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/german/sentence_translations.json b/2016/determinant/german/sentence_translations.json index 0ba1c8252..aac8b5765 100644 --- a/2016/determinant/german/sentence_translations.json +++ b/2016/determinant/german/sentence_translations.json @@ -162,7 +162,7 @@ "end": 157.12 }, { - "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation.", + "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation.", "translatedText": "Später in diesem Video zeige ich dir, wie du die Determinante einer Transformation mit Hilfe ihrer Matrix berechnest, die auch für die Berechnung wichtig ist.", "model": "DeepL", "from_community_srt": "Später im Video zeige ich euch wie ihr die Determinante einer Transformation mit Hilfe ihrer Matrix berechnet. Aber glaubt mir: Zu verstehen was die Determinante ist, ist viel wichtiger als das Verstehen ihrer Berechnung.", @@ -189,7 +189,7 @@ "end": 184.34 }, { - "input": "And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", + "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single", "translatedText": "Und die Determinante einer 2D-Transformation ist 0, wenn sie den gesamten Raum auf eine Linie oder sogar auf einen einzigen Punkt quetscht.", "model": "DeepL", "from_community_srt": "Und die Determinante einer 2D-Transformation ist 0 wenn sie den gesamten Raum auf eine Linie bringt, oder sogar auf einen einzigen Punkt.", @@ -341,7 +341,7 @@ "end": 282.4 }, { - "input": "For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3.", + "input": "For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3.", "translatedText": "Die Matrix mit den Spalten 1,1 und 2,-1 kodiert zum Beispiel eine Transformation, deren Determinante, ich sag's mal so, negativ 3 ist.", "model": "DeepL", "from_community_srt": "Zum Beispiel: die Matrix mit Spalten 1; 1 und 2; –1 beschreibt eine Transformation mit der Determinante —das sag ich euch jetzt einfach— –3.", @@ -706,7 +706,7 @@ "end": 577.88 }, { - "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations.", + "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then!", "translatedText": "Als Nächstes werde ich die bisher behandelte Idee der linearen Transformationen mit einem der nützlichsten Bereiche der linearen Algebra in Verbindung bringen: lineare Gleichungssysteme.", "model": "DeepL", "from_community_srt": "warum das logisch ist. Als nächstes werde ich die Idee der linearen Transformationen, die wir besprochen haben, mit einem der Bereiche verbinden, in denen lineare Algebra am nützlichsten ist.", diff --git a/2016/determinant/hebrew/sentence_translations.json b/2016/determinant/hebrew/sentence_translations.json index 571e64689..b1ccdd596 100644 --- a/2016/determinant/hebrew/sentence_translations.json +++ b/2016/determinant/hebrew/sentence_translations.json @@ -162,7 +162,7 @@ "end": 157.12 }, { - "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation.", + "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation.", "translatedText": "אני אראה כיצד לחשב את הקובע של טרנספורמציה באמצעות המטריצה שלה בהמשך הסרטון הזה, שהוא גם חשוב בחישוב.", "model": "google_nmt", "from_community_srt": "אני אראה לך בהמשך איך לחשב דטרמיננטה של טרנספורמציה ע\"י שימוש במטריצה מאוחר יותר בסירטון הזה. אבל להבין את זה, סמוך עליי, זה הרבה יותר חשוב מאשר להבין את תהליך החישוב עצמו.", @@ -189,7 +189,7 @@ "end": 184.34 }, { - "input": "And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", + "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single", "translatedText": "והקביעה של טרנספורמציה דו-ממדית היא 0 אם היא מוחצת את כל החלל על קו, או אפילו על נקודה אחת.", "model": "google_nmt", "from_community_srt": "ו.., הדטרמיננטה בדו-מימד של הטרנספורמציה הוא 0. אם זה מוחץ את כל המרחב לתוך קו. או, אפילו לתוך נקודה בודדת.", @@ -340,7 +340,7 @@ "end": 282.4 }, { - "input": "For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3.", + "input": "For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3.", "translatedText": "לדוגמה, המטריצה עם העמודות 1,1 ו-2,-1 מקודדת טרנספורמציה שיש לה דטרמיננטה, אני רק אגיד לך, שלילי 3.", "model": "google_nmt", "from_community_srt": "לדוגמא: המטריצה עם העמודות 1,1 ו-1-,2 מצפינה את הטרנספורמציה שיש לה דטרמיננטה שעליה אספר לך שהיא מינוס 3 ומה שזה אומר לנו זה שהמרחב מתהפך", @@ -705,7 +705,7 @@ "end": 577.88 }, { - "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations.", + "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then!", "translatedText": "בשלב הבא, אקשר את הרעיון של טרנספורמציות ליניאריות שכוסו עד כה לאחד התחומים שבהם אלגברה לינארית היא השימושית ביותר, מערכות משוואות ליניאריות.", "model": "google_nmt", "from_community_srt": "אבל תבדוק אם אתה יכול להסביר למה זה הגיוני בעזרת משפט אחד בלבד. הבא בתור: אני אקשר בין הרעיון של הטרנספורמציות הלינאריות שכיסינו עד כה לאחד התחומים בו אלגברה לינארית היא הכי שימושית. מערכות של משוואות לינאריות.", diff --git a/2016/determinant/hindi/sentence_translations.json b/2016/determinant/hindi/sentence_translations.json index 617993adc..c682e3a2c 100644 --- a/2016/determinant/hindi/sentence_translations.json +++ b/2016/determinant/hindi/sentence_translations.json @@ -140,7 +140,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", "translatedText": "किसी परिवर्तन का निर्धारक 1 आधा होगा यदि यह सभी क्षेत्रों को 1 आधे के कारक से कम कर देता है।", "n_reviews": 0, "start": 178.18, diff --git a/2016/determinant/hungarian/sentence_translations.json b/2016/determinant/hungarian/sentence_translations.json index bd7d7850d..cb4c733c9 100644 --- a/2016/determinant/hungarian/sentence_translations.json +++ b/2016/determinant/hungarian/sentence_translations.json @@ -144,7 +144,7 @@ "end": 157.12 }, { - "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation.", + "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation.", "translatedText": "A videó későbbi részében megmutatom, hogyan lehet kiszámítani egy transzformáció determinánsát a mátrixa segítségével, ami szintén fontos a számításban.", "model": "DeepL", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 184.34 }, { - "input": "And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", + "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single", "translatedText": "Egy 2D transzformáció determinánsa pedig 0, ha az egész teret egy vonalra, vagy akár egyetlen pontra szorítja.", "model": "DeepL", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 282.4 }, { - "input": "For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3.", + "input": "For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3.", "translatedText": "Például az 1,1 és 2,-1 oszlopú mátrix olyan transzformációt kódol, amelynek determinánsa, csak annyit mondok, hogy negatív 3.", "model": "DeepL", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 577.88 }, { - "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations.", + "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then!", "translatedText": "A következőkben az eddig tárgyalt lineáris transzformációk gondolatát a lineáris algebra egyik leghasznosabb területével, a lineáris egyenletrendszerekkel fogom összefüggésbe hozni.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/determinant/indonesian/sentence_translations.json b/2016/determinant/indonesian/sentence_translations.json index 1cefbb870..f7affb695 100644 --- a/2016/determinant/indonesian/sentence_translations.json +++ b/2016/determinant/indonesian/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", "translatedText": "Penentu suatu transformasi adalah 1 setengah jika semua luas diperkecil dengan faktor 1 setengah.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/italian/sentence_translations.json b/2016/determinant/italian/sentence_translations.json index 779d0bea5..a27c9f35c 100644 --- a/2016/determinant/italian/sentence_translations.json +++ b/2016/determinant/italian/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", "translatedText": "Il fattore determinante di una trasformazione sarebbe 1 metà se riducesse tutte le aree di un fattore pari a 1 metà.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/japanese/sentence_translations.json b/2016/determinant/japanese/sentence_translations.json index ffdeb3c5c..dd0c45d3d 100644 --- a/2016/determinant/japanese/sentence_translations.json +++ b/2016/determinant/japanese/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", "translatedText": "すべての領域を 1/2 の係数で押しつぶす場 合、変換の決定要因は 1/2 になります。", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/korean/sentence_translations.json b/2016/determinant/korean/sentence_translations.json index b3bcefbed..8efea02de 100644 --- a/2016/determinant/korean/sentence_translations.json +++ b/2016/determinant/korean/sentence_translations.json @@ -162,7 +162,7 @@ "end": 157.12 }, { - "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation.", + "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation.", "translatedText": "나중에 이 비디오에서 행렬을 사용하여 변환의 행렬식을 계산하는 방법을 보여 드리겠습니다. 이는 계산에서도 중요합니다.", "model": "google_nmt", "from_community_srt": "행렬식보다 determinant 자체가 이해하기 쉬운듯;) 뒤에서 이 선형변환의 행렬식(determinant) 를 계산하는 방법을 보여줄건데, 그런데 이게 무엇인지 이해하는 것은 계산법을 이해하는 것보다 훨~씬 중요해. 날 믿어봐.", @@ -189,7 +189,7 @@ "end": 184.34 }, { - "input": "And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", + "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single", "translatedText": "그리고 2D 변환의 행렬식은 모든 공간을 선으로 압축하거나 단일 점으로 압축하는 경우 0입니다.", "model": "google_nmt", "from_community_srt": "2차원 변환의 행렬식이 0 이라면, 모든 공간이 찌부려뜨려져서 선이 될 수도 있어. 아니, 어쩌면 한 점이 될 수도 있지.", @@ -341,7 +341,7 @@ "end": 282.4 }, { - "input": "For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3.", + "input": "For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3.", "translatedText": "예를 들어, 열 1,1과 2,-1이 있는 행렬은 행렬식을 갖는 변환을 인코딩합니다. 그냥 말씀드리자면 마이너스 3입니다.", "model": "google_nmt", "from_community_srt": "예를 들면 열 (1,1), (2,-1) 로 구성된 행렬이 나타내는 변환의 행렬값은 바로 -3 이야.", @@ -707,7 +707,7 @@ "end": 577.88 }, { - "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations.", + "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then!", "translatedText": "다음으로, 지금까지 다룬 선형 변환의 아이디어를 선형 대수가 가장 유용한 영역 중 하나인 선형 방정식 시스템과 연관지을 것입니다.", "model": "google_nmt", "from_community_srt": "다음에는 지금까지 다룬 선형변환 개념을 다른 것과 엮어볼거야. 선형대수가 가장 유용한 분야들 하나야. 선형 방정식계를 사용하는 분야지.", diff --git a/2016/determinant/marathi/sentence_translations.json b/2016/determinant/marathi/sentence_translations.json index 1d77687a6..98d6056ac 100644 --- a/2016/determinant/marathi/sentence_translations.json +++ b/2016/determinant/marathi/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", "translatedText": "परिवर्तनाचा निर्धारक 1 अर्धा असेल जर तो 1 अर्ध्या घटकाने सर्व क्षेत्रे खाली करतो.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/persian/sentence_translations.json b/2016/determinant/persian/sentence_translations.json index 314e73c2f..cb9b32152 100644 --- a/2016/determinant/persian/sentence_translations.json +++ b/2016/determinant/persian/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", "translatedText": "اگر یک تبدیل تمام نواحی را با ضریب 1 نصف کاهش دهد، تعیین کننده 1 نصف خواهد بود. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/polish/sentence_translations.json b/2016/determinant/polish/sentence_translations.json index cb8d9f19e..244a58dea 100644 --- a/2016/determinant/polish/sentence_translations.json +++ b/2016/determinant/polish/sentence_translations.json @@ -161,7 +161,7 @@ "end": 157.12 }, { - "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation.", + "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation.", "translatedText": "W dalszej części tego filmu pokażę, jak obliczyć wyznacznik transformacji, korzystając z jej macierzy, co jest również ważne w obliczeniach.", "model": "google_nmt", "from_community_srt": "Pokażę później jak obliczyć wyznacznik transformacji używając jej macierzy później. Niemniej jednak rozumienie czym jest wyznacznik, uwierzcie, jest bardziej istotne niż zrozumienie obliczeń.", @@ -188,7 +188,7 @@ "end": 184.34 }, { - "input": "And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", + "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single", "translatedText": "Wyznacznikiem transformacji 2D jest 0, jeśli zgniata całą przestrzeń na linię lub nawet na pojedynczy punkt.", "model": "google_nmt", "from_community_srt": "A jeśli wyznacznik transformacji 2D jest równy 0, ściska on cała płaszczyznę w linię. Albo, nawet w pojedynczy punkt.", @@ -338,7 +338,7 @@ "end": 282.4 }, { - "input": "For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3.", + "input": "For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3.", "translatedText": "Na przykład macierz z kolumnami 1,1 i 2,-1 koduje transformację, która ma wyznacznik, powiem tylko, minus 3.", "model": "google_nmt", "from_community_srt": "Dla przykładu, macierz o kolumnach 1,1 i 2,-1 opisuje transformację o wyznaczniku o wartości -3.", @@ -704,7 +704,7 @@ "end": 577.88 }, { - "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations.", + "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then!", "translatedText": "Następnie odniosę omówioną dotychczas koncepcję przekształceń liniowych do jednego z obszarów, w których algebra liniowa jest najbardziej przydatna, czyli liniowych układów równań.", "model": "google_nmt", "from_community_srt": "ale spróbuj wyjaśnić czemu to ma sens jednym zdaniem samemu. W następnym filmie odniosę ideę linowych transformacji o których mówiłem do tej pory do jednego z obszarów gdzie algebra liniowa jest najbardziej przydatna: układy równań liniowych.", diff --git a/2016/determinant/portuguese/sentence_translations.json b/2016/determinant/portuguese/sentence_translations.json index 4aaca48a3..09fd8320f 100644 --- a/2016/determinant/portuguese/sentence_translations.json +++ b/2016/determinant/portuguese/sentence_translations.json @@ -162,7 +162,7 @@ "end": 157.12 }, { - "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation.", + "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation.", "translatedText": "Mostrarei como calcular o determinante de uma transformação usando sua matriz mais adiante neste vídeo, o que também é importante no cálculo.", "model": "google_nmt", "from_community_srt": "Eu vou mostrar como calcular o determinante de uma transformação usando sua matriz mais tarde no vídeo mas entender o que ele é, confie em mim, é muito mais importante do que entender o cálculo.", @@ -189,7 +189,7 @@ "end": 184.34 }, { - "input": "And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", + "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single", "translatedText": "E o determinante de uma transformação 2D é 0 se ela comprimir todo o espaço em uma linha, ou mesmo em um único ponto.", "model": "google_nmt", "from_community_srt": "E, o determinante de uma transformação 2D é 0 se ele esmaga todo o espaço numa linha. Ou, mesmo num único ponto.", @@ -341,7 +341,7 @@ "end": 282.4 }, { - "input": "For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3.", + "input": "For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3.", "translatedText": "Por exemplo, a matriz com colunas 1,1 e 2,-1 codifica uma transformação que tem determinante, vou te dizer, menos 3.", "model": "google_nmt", "from_community_srt": "Por exemplo a matriz com colunas (1,1) e (2,-1) codifica uma transformação que tem determinante, eu vou lhe dizer, -3.", @@ -706,7 +706,7 @@ "end": 577.88 }, { - "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations.", + "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then!", "translatedText": "A seguir, relacionarei a ideia de transformações lineares abordada até agora a uma das áreas onde a álgebra linear é mais útil, os sistemas lineares de equações.", "model": "google_nmt", "from_community_srt": "Adiante eu estarei relacionando a idéia de transformações lineares cobridas até agora a uma das áreas onde a Álgebra Linear é mais útil: sistemas de equações lineares.", diff --git a/2016/determinant/russian/sentence_translations.json b/2016/determinant/russian/sentence_translations.json index 9de76aff6..2cb4603e3 100644 --- a/2016/determinant/russian/sentence_translations.json +++ b/2016/determinant/russian/sentence_translations.json @@ -144,7 +144,7 @@ "end": 157.12 }, { - "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation.", + "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation.", "translatedText": "Позже в этом видео я покажу, как вычислить определитель преобразования, используя его матрицу, что также важно при вычислениях.", "from_community_srt": "Я покажу как вычислять определитель преобразования используя его матрицу позже в этом видео Но понимание, что это на самом деле, поверьте мне,", "n_reviews": 0, @@ -167,7 +167,7 @@ "end": 184.34 }, { - "input": "And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", + "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single", "translatedText": "А определитель двумерного преобразования равен 0, если оно сжимает все пространство в линию или даже в одну точку.", "from_community_srt": "Если оно сжимает все площади в два раза И определитель 2D преобразования будет 0, если оно сжимает все пространство в линию. Или даже в одну точку.", "n_reviews": 0, @@ -301,7 +301,7 @@ "end": 282.4 }, { - "input": "For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3.", + "input": "For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3.", "translatedText": "Например, матрица со столбцами 1,1 и 2,-1 кодирует преобразование, определитель которого, я вам скажу, отрицательный 3.", "from_community_srt": "Скажем матрица 1,1 и 2,-1 кодирует преобразование, чей определитель я вам просто назову.", "n_reviews": 0, @@ -625,7 +625,7 @@ "end": 577.88 }, { - "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations.", + "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then!", "translatedText": "Далее я буду связывать рассмотренную до сих пор идею линейных преобразований с одной из областей, где линейная алгебра наиболее полезна, — с линейными системами уравнений.", "from_community_srt": "Далее я свяжу идею линейного преобразования, разобранную ранее с одной из областей, в которых линейная алгебра наиболее полезна Системы линейных уравнений.", "n_reviews": 0, diff --git a/2016/determinant/spanish/sentence_translations.json b/2016/determinant/spanish/sentence_translations.json index 5a11ed825..d9b45cfeb 100644 --- a/2016/determinant/spanish/sentence_translations.json +++ b/2016/determinant/spanish/sentence_translations.json @@ -144,7 +144,7 @@ "end": 157.12 }, { - "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation.", + "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation.", "translatedText": "Más adelante en este video mostraré cómo calcular el determinante de una transformación usando su matriz, que también es importante en el cálculo.", "from_community_srt": "Les mostraré cómo computar el determinante de una transformación usando su matriz más tarde en este video, pero entender lo que representa es, créanme, mucho más importante que cómo computarlo.", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 184.34 }, { - "input": "And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", + "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single", "translatedText": "Y el determinante de una transformación 2D es 0 si aplasta todo el espacio en una línea, o incluso en un solo punto.", "from_community_srt": "Y el determinante de una transformación 2-D es cero si comprime todo el espacio en una línea o inclusive en un punto,", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 282.4 }, { - "input": "For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3.", + "input": "For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3.", "translatedText": "Por ejemplo, la matriz con las columnas 1,1 y 2,-1 codifica una transformación que tiene determinante, solo te diré, negativo 3.", "from_community_srt": "Por ejemplo: la matriz con columnas [1,1] y [2,-1] define una transformación que tiene determinante, simplemente se los diré,", "n_reviews": 0, @@ -630,7 +630,7 @@ "end": 577.88 }, { - "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations.", + "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then!", "translatedText": "A continuación, relacionaré la idea de transformaciones lineales tratadas hasta ahora con una de las áreas donde el álgebra lineal es más útil: los sistemas lineales de ecuaciones.", "from_community_srt": "En lo próximo que viene, vincularé la idea de transformaciónes lineales cubierta hasta ahora, con una de las áreas donde el álgebra lineal es más útil,", "n_reviews": 0, diff --git a/2016/determinant/tamil/sentence_translations.json b/2016/determinant/tamil/sentence_translations.json index 4166af9e9..4d4f73087 100644 --- a/2016/determinant/tamil/sentence_translations.json +++ b/2016/determinant/tamil/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", "translatedText": "எல்லாப் பகுதிகளையும் 1 பாதி மடங்கு குறைத்தால், உருமாற்றத்தின் நிர்ணயம் 1 பாதியாக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/telugu/sentence_translations.json b/2016/determinant/telugu/sentence_translations.json index d12a44fba..c57de0843 100644 --- a/2016/determinant/telugu/sentence_translations.json +++ b/2016/determinant/telugu/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", "translatedText": "అన్ని ప్రాంతాలను 1 సగం కారకంతో తగ్గించినట్లయితే, పరివర్తన యొక్క డిటర్మినేట్ 1 సగం అవుతుంది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/thai/sentence_translations.json b/2016/determinant/thai/sentence_translations.json index 87f5bd4d3..bee70c46a 100644 --- a/2016/determinant/thai/sentence_translations.json +++ b/2016/determinant/thai/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", "translatedText": "ดีเทอร์มีแนนต์ของการแปลงจะเป็น 1 ครึ่งหนึ่ง ถ้ามันลดพื้นที่ทั้งหมดลง 1 ครึ่งหนึ่ง ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/turkish/sentence_translations.json b/2016/determinant/turkish/sentence_translations.json index d95b3bf6a..90b36fbcb 100644 --- a/2016/determinant/turkish/sentence_translations.json +++ b/2016/determinant/turkish/sentence_translations.json @@ -162,7 +162,7 @@ "end": 157.12 }, { - "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, which is also important in the computation.", + "input": "I'll show how to compute the determinant of a transformation using its matrix later on in this video, but understanding what it represents is, trust me, much more important than the computation.", "translatedText": "Bu videonun ilerleyen kısımlarında bir dönüşümün determinantının matrisini kullanarak nasıl hesaplanacağını göstereceğim; bu da hesaplama açısından önemlidir.", "model": "google_nmt", "from_community_srt": "( Belirleyimci'si .çeviren.) Dönüşümün matrix'i ile determinantı hesaplamayı daha sonra göstereceğim bu video içerisinde. ama hesaptan ziyade ne olduğunu anlamak çok daha önemli,", @@ -189,7 +189,7 @@ "end": 184.34 }, { - "input": "And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single point.", + "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. And the determinant of a 2D transformation is 0 if it squishes all of space onto a line, or even onto a single", "translatedText": "Ve 2 boyutlu bir dönüşümün determinantı, eğer uzayın tamamını bir çizgiye, hatta tek bir noktaya sıkıştırıyorsa 0'dır.", "model": "google_nmt", "from_community_srt": "Ve, 2-Boyutlu bir dönüşümün determinantının 0 oması için de tüm düzlemi bir çizgiye sıkıştırması gerekir. ya da hatta tek bir noktaya sıkıştırması gerekir.", @@ -341,7 +341,7 @@ "end": 282.4 }, { - "input": "For example, the matrix with columns 1,1 and 2,-1 encodes a transformation that has determinant, I'll just tell you, negative 3.", + "input": "For example, the matrix with columns 1, 1 and 2, negative 1 encodes a transformation that has determinant, I'll just tell you, negative 3.", "translatedText": "Örneğin, 1,1 ve 2,-1 sütunlu matris, determinantı eksi 3 olan bir dönüşümü kodlar.", "model": "google_nmt", "from_community_srt": "Örneğin, [1,1] ve [2,-1] sütunlu bir matrix, öyle bir dönüşüm dür ki determinantııı söyleyeceğim,, -3 tür.", @@ -706,7 +706,7 @@ "end": 577.88 }, { - "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations.", + "input": "Next up, I'll be relating the idea of linear transformations covered so far to one of the areas where linear algebra is most useful, linear systems of equations. See you then!", "translatedText": "Şimdi, şu ana kadar ele alınan doğrusal dönüşümler fikrini, doğrusal cebirin en kullanışlı olduğu alanlardan biri olan doğrusal denklem sistemleriyle ilişkilendireceğim.", "model": "google_nmt", "from_community_srt": "Sonraki bölümde... Şu ana kadar ele aldığım doğrusal dönüşümlerle doğrusal cebirin en yararlı olduğu alan olan doğrusal sistem eşitliklerini alakalandıracağım.", diff --git a/2016/determinant/ukrainian/sentence_translations.json b/2016/determinant/ukrainian/sentence_translations.json index e28452d90..77d8ab45b 100644 --- a/2016/determinant/ukrainian/sentence_translations.json +++ b/2016/determinant/ukrainian/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", "translatedText": "Детермінант трансформації дорівнює 1 половині, якщо вона зменшує всі області в 1 раз.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/urdu/sentence_translations.json b/2016/determinant/urdu/sentence_translations.json index 5c740dba4..f2dd37f21 100644 --- a/2016/determinant/urdu/sentence_translations.json +++ b/2016/determinant/urdu/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half. ", "translatedText": "تبدیلی کا تعین کنندہ 1 نصف ہوگا اگر یہ تمام علاقوں کو 1 نصف کے عنصر سے نیچے کر دیتا ہے۔ ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/determinant/vietnamese/sentence_translations.json b/2016/determinant/vietnamese/sentence_translations.json index f256d106f..0d4af06ba 100644 --- a/2016/determinant/vietnamese/sentence_translations.json +++ b/2016/determinant/vietnamese/sentence_translations.json @@ -160,7 +160,7 @@ "end": 177.04 }, { - "input": "The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", + "input": "The determinant of a transformation would be ½ if it squishes down all areas by a factor of ½. The determinant of a transformation would be 1 half if it squishes down all areas by a factor of 1 half.", "translatedText": "Định thức của một phép biến đổi sẽ là 1 nửa nếu nó thu gọn tất cả các diện tích theo hệ số 1 nửa.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/arabic/sentence_translations.json b/2016/dot-products/arabic/sentence_translations.json index 3f4811831..7b1a318ef 100644 --- a/2016/dot-products/arabic/sentence_translations.json +++ b/2016/dot-products/arabic/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "["قصيدة الفرح"، لبيتهوفن، تُعزف حتى نهاية البيانو.] تقليديًا، المنتجات النقطية هي شيء تم تقديمه في وقت مبكر جدًا في دورة الجبر الخطي، عادةً في البداية.", "model": "google_nmt", "from_community_srt": "تقليديا ، والمنتجات نقطة أو شيء من هذا القبيل قدم في وقت مبكر حقا في الجبر الخطي دورة عادة في البداية.", diff --git a/2016/dot-products/bengali/sentence_translations.json b/2016/dot-products/bengali/sentence_translations.json index df03f7d3a..5ce486160 100644 --- a/2016/dot-products/bengali/sentence_translations.json +++ b/2016/dot-products/bengali/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "বিথোভেনের "ওড টু জয়", পিয়ানোর শেষ পর্যন্ত বাজায়। ] ঐতিহ্যগতভাবে, ডট পণ্যগুলি এমন কিছু যা একটি রৈখিক বীজগণিত কোর্সের শুরুতে, সাধারণত ঠিক শুরুতে প্রবর্তিত হয়।", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/chinese/sentence_translations.json b/2016/dot-products/chinese/sentence_translations.json index 373e2aeda..146aeb90e 100644 --- a/2016/dot-products/chinese/sentence_translations.json +++ b/2016/dot-products/chinese/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[贝多芬的《欢乐颂》，钢琴演奏到最后。 ] 传统上，点积是在线性代数课程中很早 就引入的东西，通常是在开始时引入的。", "model": "google_nmt", "from_community_srt": "传统上， 点积是线性代数课程中很靠前的内容 一般就在最开始 我把它放得如此靠后，", diff --git a/2016/dot-products/czech/sentence_translations.json b/2016/dot-products/czech/sentence_translations.json index bb94493a8..b38847262 100644 --- a/2016/dot-products/czech/sentence_translations.json +++ b/2016/dot-products/czech/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "["Óda na radost", od Beethovena, hraje do konce klavíru.] Tradičně jsou tečkové produkty něčím, co je v kurzu lineární algebry představeno opravdu brzy, obvykle hned na začátku.", "model": "google_nmt", "from_community_srt": "Skalární součiny se v lineární algebře běžně zavádí celkem brzo typicky rovnou na začátku, tak může vypadat zvláštně,", diff --git a/2016/dot-products/english/captions.srt b/2016/dot-products/english/captions.srt index c6588e371..9ccb116b7 100644 --- a/2016/dot-products/english/captions.srt +++ b/2016/dot-products/english/captions.srt @@ -1,820 +1,824 @@ 1 -00:00:16,580 --> 00:00:21,476 -Traditionally, dot products are something that's introduced really +00:00:16,580 --> 00:00:20,387 +["Ode to Joy", by Beethoven, plays to the end of the piano.] Traditionally, 2 -00:00:21,476 --> 00:00:26,300 -early on in a linear algebra course, typically right at the start. +00:00:20,387 --> 00:00:24,847 +dot products are something that's introduced really early on in a linear algebra course, 3 +00:00:24,847 --> 00:00:26,300 +typically right at the start. + +4 00:00:26,640 --> 00:00:29,580 So it might seem strange that I've pushed them back this far in the series. -4 +5 00:00:29,580 --> 00:00:32,723 I did this because there's a standard way to introduce the topic, -5 +6 00:00:32,723 --> 00:00:35,914 which requires nothing more than a basic understanding of vectors, -6 +7 00:00:35,914 --> 00:00:40,153 but a fuller understanding of the role that dot products play in math can only really be -7 +8 00:00:40,153 --> 00:00:42,440 found under the light of linear transformations. -8 +9 00:00:43,480 --> 00:00:47,092 Before that, though, let me just briefly cover the standard way that dot products are -9 +10 00:00:47,092 --> 00:00:50,620 introduced, which I'm assuming is at least partially review for a number of viewers. -10 +11 00:00:51,440 --> 00:00:55,034 Numerically, if you have two vectors of the same dimension, -11 +12 00:00:55,034 --> 00:00:59,528 two lists of numbers with the same lengths, taking their dot product means -12 +13 00:00:59,528 --> 00:01:03,661 pairing up all of the coordinates, multiplying those pairs together, -13 +14 00:01:03,661 --> 00:01:04,980 and adding the result. -14 +15 00:01:06,860 --> 00:01:13,180 So the vector 1, 2 dotted with 3, 4 would be 1 times 3 plus 2 times 4. -15 +16 00:01:14,580 --> 00:01:19,108 The vector 6, 2, 8, 3 dotted with 1, 8, 5, 3 would be -16 +17 00:01:19,108 --> 00:01:23,720 6 times 1 plus 2 times 8 plus 8 times 5 plus 3 times 3. -17 +18 00:01:24,740 --> 00:01:28,660 Luckily, this computation has a really nice geometric interpretation. -18 +19 00:01:29,340 --> 00:01:33,000 To think about the dot product between two vectors, v and w, -19 +20 00:01:33,000 --> 00:01:37,980 imagine projecting w onto the line that passes through the origin and the tip of v. -20 +21 00:01:38,780 --> 00:01:42,486 Multiplying the length of this projection by the length of v, -21 +22 00:01:42,486 --> 00:01:44,460 you have the dot product v dot w. -22 +23 00:01:46,420 --> 00:01:50,136 Except when this projection of w is pointing in the opposite direction from v, -23 +24 00:01:50,136 --> 00:01:52,160 that dot product will actually be negative. -24 +25 00:01:53,720 --> 00:01:56,566 So when two vectors are generally pointing in the same direction, -25 +26 00:01:56,566 --> 00:01:57,860 their dot product is positive. -26 +27 00:01:59,240 --> 00:02:02,320 When they're perpendicular, meaning the projection of one -27 +28 00:02:02,320 --> 00:02:05,560 onto the other is the zero vector, their dot product is zero. -28 +29 00:02:05,980 --> 00:02:09,600 And if they point in generally the opposite direction, their dot product is negative. -29 +30 00:02:11,620 --> 00:02:14,560 Now, this interpretation is weirdly asymmetric. -30 +31 00:02:14,800 --> 00:02:16,500 It treats the two vectors very differently. -31 +32 00:02:16,880 --> 00:02:20,000 So when I first learned this, I was surprised that order doesn't matter. -32 +33 00:02:20,960 --> 00:02:24,559 You could instead project v onto w, multiply the length of -33 +34 00:02:24,559 --> 00:02:28,220 the projected v by the length of w, and get the same result. -34 +35 00:02:30,400 --> 00:02:32,840 I mean, doesn't that feel like a really different process? -35 +36 00:02:35,320 --> 00:02:37,760 Here's the intuition for why order doesn't matter. -36 +37 00:02:38,440 --> 00:02:42,180 If v and w happened to have the same length, we could leverage some symmetry. -37 +38 00:02:43,080 --> 00:02:47,344 Since projecting w onto v, then multiplying the length of that projection -38 +39 00:02:47,344 --> 00:02:51,436 by the length of v, is a complete mirror image of projecting v onto w, -39 +40 00:02:51,436 --> 00:02:55,240 then multiplying the length of that projection by the length of w. -40 +41 00:02:57,280 --> 00:03:00,877 Now, if you scale one of them, say v, by some constant like 2, -41 +42 00:03:00,877 --> 00:03:04,360 so that they don't have equal length, the symmetry is broken. -42 +43 00:03:05,020 --> 00:03:09,177 But let's think through how to interpret the dot product between this new vector, -43 +44 00:03:09,177 --> 00:03:10,040 2 times v, and w. -44 +45 00:03:10,880 --> 00:03:14,257 If you think of w as getting projected onto v, -45 +46 00:03:14,257 --> 00:03:19,720 then the dot product 2v dot w will be exactly twice the dot product v dot w. -46 +47 00:03:20,460 --> 00:03:24,706 This is because when you scale v by 2, it doesn't change the length of the -47 +48 00:03:24,706 --> 00:03:29,520 projection of w, but it doubles the length of the vector that you're projecting onto. -48 +49 00:03:30,460 --> 00:03:34,200 But on the other hand, let's say you were thinking about v getting projected onto w. -49 +50 00:03:34,900 --> 00:03:38,903 Well, in that case, the length of the projection is the thing that gets scaled when we -50 +51 00:03:38,903 --> 00:03:43,000 multiply v by 2, but the length of the vector that you're projecting onto stays constant. -51 +52 00:03:43,000 --> 00:03:46,660 So the overall effect is still to just double the dot product. -52 +53 00:03:47,280 --> 00:03:49,673 So even though symmetry is broken in this case, -53 +54 00:03:49,673 --> 00:03:53,513 the effect that this scaling has on the value of the dot product is the same -54 +55 00:03:53,513 --> 00:03:54,860 under both interpretations. -55 +56 00:03:56,640 --> 00:04:00,340 There's also one other big question that confused me when I first learned this stuff. -56 +57 00:04:00,840 --> 00:04:04,411 Why on earth does this numerical process of matching coordinates, -57 +58 00:04:04,411 --> 00:04:08,740 multiplying pairs, and adding them together have anything to do with projection? -58 +59 00:04:10,640 --> 00:04:14,191 Well, to give a satisfactory answer, and also to do full justice to -59 +60 00:04:14,191 --> 00:04:17,743 the significance of the dot product, we need to unearth something a -60 +61 00:04:17,743 --> 00:04:21,399 little bit deeper going on here, which often goes by the name duality. -61 +62 00:04:22,140 --> 00:04:25,804 But before getting into that, I need to spend some time talking about linear -62 +63 00:04:25,804 --> 00:04:30,040 transformations from multiple dimensions to one dimension, which is just the number line. -63 +64 00:04:32,420 --> 00:04:35,927 These are functions that take in a 2D vector and spit out some number, -64 +65 00:04:35,927 --> 00:04:39,237 but linear transformations are of course much more restricted than -65 +66 00:04:39,237 --> 00:04:42,300 your run-of-the-mill function with a 2D input and a 1D output. -66 +67 00:04:43,020 --> 00:04:47,080 As with transformations in higher dimensions, like the ones I talked about in chapter 3, -67 +68 00:04:47,080 --> 00:04:50,138 there are some formal properties that make these functions linear, -68 +69 00:04:50,138 --> 00:04:54,199 but I'm going to purposefully ignore those here so as to not distract from our end goal, -69 +70 00:04:54,199 --> 00:04:58,260 and instead focus on a certain visual property that's equivalent to all the formal stuff. -70 +71 00:04:59,040 --> 00:05:03,508 If you take a line of evenly spaced dots and apply a transformation, -71 +72 00:05:03,508 --> 00:05:07,653 a linear transformation will keep those dots evenly spaced once -72 +73 00:05:07,653 --> 00:05:11,280 they land in the output space, which is the number line. -73 +74 00:05:12,420 --> 00:05:15,403 Otherwise, if there's some line of dots that gets unevenly spaced, -74 +75 00:05:15,403 --> 00:05:17,140 then your transformation is not linear. -75 +76 00:05:19,220 --> 00:05:23,507 As with the cases we've seen before, one of these linear transformations is -76 +77 00:05:23,507 --> 00:05:26,722 completely determined by where it takes i-hat and j-hat, -77 +78 00:05:26,722 --> 00:05:30,671 but this time each one of those basis vectors just lands on a number, -78 +79 00:05:30,671 --> 00:05:34,168 so when we record where they land as the columns of a matrix, -79 +80 00:05:34,168 --> 00:05:36,820 each of those columns just has a single number. -80 +81 00:05:38,460 --> 00:05:39,840 This is a 1x2 matrix. -81 +82 00:05:41,860 --> 00:05:43,701 Let's walk through an example of what it means -82 +83 00:05:43,701 --> 00:05:45,660 to apply one of these transformations to a vector. -83 +84 00:05:46,380 --> 00:05:51,680 Let's say you have a linear transformation that takes i-hat to 1 and j-hat to negative 2. -84 +85 00:05:52,420 --> 00:05:56,490 To follow where a vector with coordinates, say, 4, 3 ends up, -85 +86 00:05:56,490 --> 00:06:01,020 think of breaking up this vector as 4 times i-hat plus 3 times j-hat. -86 +87 00:06:01,840 --> 00:06:05,553 A consequence of linearity is that after the transformation, -87 +88 00:06:05,553 --> 00:06:09,144 the vector will be 4 times the place where i-hat lands, 1, -88 +89 00:06:09,144 --> 00:06:12,431 plus 3 times the place where j-hat lands, negative 2, -89 +90 00:06:12,431 --> 00:06:15,780 which in this case implies that it lands on negative 2. -90 +91 00:06:18,020 --> 00:06:22,360 When you do this calculation purely numerically, it's matrix vector multiplication. -91 +92 00:06:25,700 --> 00:06:29,222 Now, this numerical operation of multiplying a 1x2 matrix by -92 +93 00:06:29,222 --> 00:06:32,860 a vector feels just like taking the dot product of two vectors. -93 +94 00:06:33,460 --> 00:06:36,800 Doesn't that 1x2 matrix just look like a vector that we tipped on its side? -94 +95 00:06:37,960 --> 00:06:42,721 In fact, we could say right now that there's a nice association between 1x2 matrices -95 +96 00:06:42,721 --> 00:06:47,650 and 2D vectors, defined by tilting the numerical representation of a vector on its side -96 +97 00:06:47,650 --> 00:06:52,580 to get the associated matrix, or to tip the matrix back up to get the associated vector. -97 +98 00:06:53,560 --> 00:06:56,509 Since we're just looking at numerical expressions right now, -98 +99 00:06:56,509 --> 00:07:00,860 going back and forth between vectors and 1x2 matrices might feel like a silly thing to do. -99 +100 00:07:01,460 --> 00:07:05,120 But this suggests something that's truly awesome from the geometric view. -100 +101 00:07:05,380 --> 00:07:08,853 There's some kind of connection between linear transformations -101 +102 00:07:08,853 --> 00:07:11,720 that take vectors to numbers and vectors themselves. -102 +103 00:07:14,780 --> 00:07:17,558 Let me show an example that clarifies the significance, -103 +104 00:07:17,558 --> 00:07:21,380 and which just so happens to also answer the dot product puzzle from earlier. -104 +105 00:07:22,140 --> 00:07:24,704 Unlearn what you have learned, and imagine that you don't -105 +106 00:07:24,704 --> 00:07:27,180 already know that the dot product relates to projection. -106 +107 00:07:28,860 --> 00:07:32,409 What I'm going to do here is take a copy of the number line and place -107 +108 00:07:32,409 --> 00:07:36,060 it diagonally in space somehow, with the number 0 sitting at the origin. -108 +109 00:07:36,900 --> 00:07:39,566 Now think of the two-dimensional unit vector whose -109 +110 00:07:39,566 --> 00:07:41,920 tip sits where the number 1 on the number is. -110 +111 00:07:42,400 --> 00:07:44,560 I want to give that guy a name, u-hat. -111 +112 00:07:45,620 --> 00:07:48,324 This little guy plays an important role in what's about to happen, -112 +113 00:07:48,324 --> 00:07:50,020 so just keep him in the back of your mind. -113 +114 00:07:50,740 --> 00:07:54,615 If we project 2d vectors straight onto this diagonal number line, -114 +115 00:07:54,615 --> 00:07:58,960 in effect, we've just defined a function that takes 2d vectors to numbers. -115 +116 00:07:59,660 --> 00:08:02,231 What's more, this function is actually linear, -116 +117 00:08:02,231 --> 00:08:06,771 since it passes our visual test that any line of evenly spaced dots remains evenly -117 +118 00:08:06,771 --> 00:08:08,960 spaced once it lands on the number line. -118 +119 00:08:11,640 --> 00:08:16,202 Just to be clear, even though I've embedded the number line in 2d space like this, -119 +120 00:08:16,202 --> 00:08:19,280 the outputs of the function are numbers, not 2d vectors. -120 +121 00:08:19,960 --> 00:08:21,919 You should think of a function that takes in two -121 +122 00:08:21,919 --> 00:08:23,680 coordinates and outputs a single coordinate. -122 +123 00:08:25,060 --> 00:08:29,020 But that vector u-hat is a two-dimensional vector, living in the input space. -123 +124 00:08:29,440 --> 00:08:33,220 It's just situated in such a way that overlaps with the embedding of the number line. -124 +125 00:08:34,600 --> 00:08:39,518 With this projection, we just defined a linear transformation from 2d vectors to numbers, -125 +126 00:08:39,518 --> 00:08:42,960 so we're going to be able to find some kind of 1x2 matrix that -126 +127 00:08:42,960 --> 00:08:44,600 describes that transformation. -127 +128 00:08:45,540 --> 00:08:49,141 To find that 1x2 matrix, let's zoom in on this diagonal number -128 +129 00:08:49,141 --> 00:08:52,572 line setup and think about where i-hat and j-hat each land, -129 +130 00:08:52,572 --> 00:08:56,460 since those landing spots are going to be the columns of the matrix. -130 +131 00:08:58,480 --> 00:08:59,440 This part's super cool. -131 +132 00:08:59,700 --> 00:09:02,420 We can reason through it with a really elegant piece of symmetry. -132 +133 00:09:03,020 --> 00:09:07,897 Since i-hat and u-hat are both unit vectors, projecting i-hat onto the line -133 +134 00:09:07,897 --> 00:09:13,160 passing through u-hat looks totally symmetric to projecting u-hat onto the x-axis. -134 +135 00:09:13,840 --> 00:09:17,373 So when we ask what number does i-hat land on when it gets projected, -135 +136 00:09:17,373 --> 00:09:21,512 the answer is going to be the same as whatever u-hat lands on when it's projected -136 +137 00:09:21,512 --> 00:09:22,320 onto the x-axis. -137 +138 00:09:22,920 --> 00:09:28,600 But projecting u-hat onto the x-axis just means taking the x-coordinate of u-hat. -138 +139 00:09:29,020 --> 00:09:32,903 So by symmetry, the number where i-hat lands when it's projected onto -139 +140 00:09:32,903 --> 00:09:36,620 that diagonal number line is going to be the x-coordinate of u-hat. -140 +141 00:09:37,160 --> 00:09:37,660 Isn't that cool? -141 +142 00:09:39,200 --> 00:09:41,800 The reasoning is almost identical for the j-hat case. -142 +143 00:09:42,180 --> 00:09:43,260 Think about it for a moment. -143 +144 00:09:49,120 --> 00:09:52,694 For all the same reasons, the y-coordinate of u-hat gives us the -144 +145 00:09:52,694 --> 00:09:56,600 number where j-hat lands when it's projected onto the number line copy. -145 +146 00:09:57,580 --> 00:09:58,720 Pause and ponder that for a moment. -146 +147 00:09:58,780 --> 00:10:00,200 I just think that's really cool. -147 +148 00:10:00,920 --> 00:10:04,172 So the entries of the 1x2 matrix describing the projection -148 +149 00:10:04,172 --> 00:10:07,260 transformation are going to be the coordinates of u-hat. -149 +150 00:10:08,040 --> 00:10:12,255 And computing this projection transformation for arbitrary vectors in space, -150 +151 00:10:12,255 --> 00:10:15,376 which requires multiplying that matrix by those vectors, -151 +152 00:10:15,376 --> 00:10:18,880 is computationally identical to taking a dot product with u-hat. -152 +153 00:10:21,460 --> 00:10:26,025 This is why taking the dot product with a unit vector can be interpreted as -153 +154 00:10:26,025 --> 00:10:30,590 projecting a vector onto the span of that unit vector and taking the length. -154 +155 00:10:34,030 --> 00:10:35,790 So what about non-unit vectors? -155 +156 00:10:36,310 --> 00:10:38,920 For example, let's say we take that unit vector u-hat, -156 +157 00:10:38,920 --> 00:10:40,630 but we scale it up by a factor of 3. -157 +158 00:10:41,350 --> 00:10:44,390 Numerically, each of its components gets multiplied by 3. -158 +159 00:10:44,810 --> 00:10:47,958 So looking at the matrix associated with that vector, -159 +160 00:10:47,958 --> 00:10:52,390 it takes i-hat and j-hat to three times the values where they landed before. -160 +161 00:10:55,230 --> 00:10:59,410 Since this is all linear, it implies more generally that the new matrix can be -161 +162 00:10:59,410 --> 00:11:04,067 interpreted as projecting any vector onto the number line copy and multiplying where it -162 +163 00:11:04,067 --> 00:11:04,650 lands by 3. -163 +164 00:11:05,470 --> 00:11:08,491 This is why the dot product with a non-unit vector can be -164 +165 00:11:08,491 --> 00:11:11,095 interpreted as first projecting onto that vector, -165 +166 00:11:11,095 --> 00:11:14,950 then scaling up the length of that projection by the length of the vector. -166 +167 00:11:17,590 --> 00:11:19,550 Take a moment to think about what happened here. -167 +168 00:11:19,890 --> 00:11:23,081 We had a linear transformation from 2D space to the number line, -168 +169 00:11:23,081 --> 00:11:26,961 which was not defined in terms of numerical vectors or numerical dot products, -169 +170 00:11:26,961 --> 00:11:30,890 it was just defined by projecting space onto a diagonal copy of the number line. -170 +171 00:11:31,670 --> 00:11:36,830 But because the transformation is linear, it was necessarily described by some 1x2 matrix. -171 +172 00:11:37,330 --> 00:11:40,875 And since multiplying a 1x2 matrix by a 2D vector is the same -172 +173 00:11:40,875 --> 00:11:44,364 as turning that matrix on its side and taking a dot product, -173 +174 00:11:44,364 --> 00:11:47,910 this transformation was inescapably related to some 2D vector. -174 +175 00:11:49,410 --> 00:11:53,708 The lesson here is that any time you have one of these linear transformations whose -175 +176 00:11:53,708 --> 00:11:56,933 output space is the number line, no matter how it was defined, -176 +177 00:11:56,933 --> 00:12:00,976 there's going to be some unique vector v corresponding to that transformation, -177 +178 00:12:00,976 --> 00:12:05,070 in the sense that applying the transformation is the same thing as taking a dot -178 +179 00:12:05,070 --> 00:12:06,350 product with that vector. -179 +180 00:12:09,930 --> 00:12:12,030 To me, this is utterly beautiful. -180 +181 00:12:12,730 --> 00:12:15,390 It's an example of something in math called duality. -181 +182 00:12:16,270 --> 00:12:19,781 Duality shows up in many different ways and forms throughout math, -182 +183 00:12:19,781 --> 00:12:21,930 and it's super tricky to actually define. -183 +184 00:12:22,670 --> 00:12:26,367 Loosely speaking, it refers to situations where you have a natural -184 +185 00:12:26,367 --> 00:12:30,230 but surprising correspondence between two types of mathematical thing. -185 +186 00:12:31,010 --> 00:12:34,170 For the linear algebra case that you just learned about, -186 +187 00:12:34,170 --> 00:12:38,717 you'd say that the dual of a vector is the linear transformation that it encodes, -187 +188 00:12:38,717 --> 00:12:43,042 and the dual of a linear transformation from some space to one dimension is a -188 +189 00:12:43,042 --> 00:12:44,650 certain vector in that space. -189 +190 00:12:46,730 --> 00:12:50,028 So to sum up, on the surface, the dot product is a very useful -190 +191 00:12:50,028 --> 00:12:53,221 geometric tool for understanding projections and for testing -191 +192 00:12:53,221 --> 00:12:56,310 whether or not vectors tend to point in the same direction. -192 +193 00:12:56,970 --> 00:13:00,790 And that's probably the most important thing for you to remember about the dot product. -193 +194 00:13:01,270 --> 00:13:04,553 But at a deeper level, dotting two vectors together is a way -194 +195 00:13:04,553 --> 00:13:07,730 to translate one of them into the world of transformations. -195 +196 00:13:08,670 --> 00:13:11,550 Again, numerically, this might feel like a silly point to emphasize. -196 +197 00:13:11,670 --> 00:13:14,490 It's just two computations that happen to look similar. -197 +198 00:13:14,490 --> 00:13:18,047 But the reason I find this so important is that throughout math, -198 +199 00:13:18,047 --> 00:13:22,426 when you're dealing with a vector, once you really get to know its personality, -199 +200 00:13:22,426 --> 00:13:26,915 sometimes you realize that it's easier to understand it not as an arrow in space, -200 +201 00:13:26,915 --> 00:13:30,090 but as the physical embodiment of a linear transformation. -201 +202 00:13:30,730 --> 00:13:35,292 It's as if the vector is really just a conceptual shorthand for a certain transformation, -202 +203 00:13:35,292 --> 00:13:38,739 since it's easier for us to think about arrows in space rather than -203 +204 00:13:38,739 --> 00:13:40,970 moving all of that space to the number line. -204 +205 00:13:42,610 --> 00:13:47,310 In the next video, you'll see another really cool example of this duality in action, -205 +206 00:13:47,310 --> 00:13:49,190 as I talk about the cross product. diff --git a/2016/dot-products/english/sentence_timings.json b/2016/dot-products/english/sentence_timings.json index 3dcd496c5..4d1d65e09 100644 --- a/2016/dot-products/english/sentence_timings.json +++ b/2016/dot-products/english/sentence_timings.json @@ -1,6 +1,6 @@ [ [ - "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", 16.58, 26.3 ], diff --git a/2016/dot-products/english/transcript.txt b/2016/dot-products/english/transcript.txt index 344dea3f1..dce815b75 100644 --- a/2016/dot-products/english/transcript.txt +++ b/2016/dot-products/english/transcript.txt @@ -1,4 +1,4 @@ -Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start. +["Ode to Joy", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start. So it might seem strange that I've pushed them back this far in the series. I did this because there's a standard way to introduce the topic, which requires nothing more than a basic understanding of vectors, but a fuller understanding of the role that dot products play in math can only really be found under the light of linear transformations. Before that, though, let me just briefly cover the standard way that dot products are introduced, which I'm assuming is at least partially review for a number of viewers. diff --git a/2016/dot-products/french/sentence_translations.json b/2016/dot-products/french/sentence_translations.json index 62b20b3e2..a68b3b1d4 100644 --- a/2016/dot-products/french/sentence_translations.json +++ b/2016/dot-products/french/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "["Ode to Joy", de Beethoven, joue jusqu'au bout du piano.] Traditionnellement, les produits scalaires sont quelque chose qui est introduit très tôt dans un cours d'algèbre linéaire, généralement dès le début.", "from_community_srt": "Traditionnellement, les produits dot ou quelque chose qui est introduit très tôt dans une algèbre linéaire cours généralement juste au début.", "n_reviews": 0, diff --git a/2016/dot-products/german/sentence_translations.json b/2016/dot-products/german/sentence_translations.json index ad3091d08..76df536df 100644 --- a/2016/dot-products/german/sentence_translations.json +++ b/2016/dot-products/german/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "Traditionell werden Punktprodukte schon sehr früh in einem Kurs über lineare Algebra eingeführt, normalerweise gleich zu Beginn.", "model": "DeepL", "from_community_srt": "klassischerweise, Skalarprodukt oder irgendwas, dass sehr früher in lineare Algebra eingeführt wurde kurs in der Regel gleich zu Beginn.", diff --git a/2016/dot-products/hebrew/sentence_translations.json b/2016/dot-products/hebrew/sentence_translations.json index 2a54300c1..f057f5728 100644 --- a/2016/dot-products/hebrew/sentence_translations.json +++ b/2016/dot-products/hebrew/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "["אודה לשמחה", מאת בטהובן, מנגן עד קצה הפסנתר.] באופן מסורתי, מוצרי נקודות הם משהו שמוצג ממש מוקדם בקורס אלגברה ליניארי, בדרך כלל ממש בהתחלה.", "model": "google_nmt", "from_community_srt": "-קאלווין: \"אתה יודע, אני לא חושב שמתמטיקה זה מדע, אני חושב שזו דת.\" -הובס: \"דת?\" -קאלווין: \" כן. כל המשוואות הללו הן כמו נס. אתה לוקח שני מספרים וכשאתה מחבר אותם, הם באופן פלאי הופכים למספר אחד חדש! אף אחד לא יכול להגיד איך זה יכול לקרות. זה או שאתה מאמין לזה או שלא\" באופן מסורתי, מכפלה סקלרית היא דבר שמציגים בדרך כלל מאוד מוקדם בקורס של אלגברה לינארית. בדרך כלל,", diff --git a/2016/dot-products/hindi/sentence_translations.json b/2016/dot-products/hindi/sentence_translations.json index 4a6edb004..c1c1a5b92 100644 --- a/2016/dot-products/hindi/sentence_translations.json +++ b/2016/dot-products/hindi/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[संगीत] परंपरागत रूप से, डॉट उत्पाद कुछ ऐसी चीजें हैं जिन्हें रैखिक बीजगणित पाठ्यक्रम में वास्तव में बहुत पहले पेश किया जाता है, आमतौर पर शुरुआत में ही।", "n_reviews": 0, "start": 16.58, diff --git a/2016/dot-products/hungarian/sentence_translations.json b/2016/dot-products/hungarian/sentence_translations.json index c9df69a81..9d3899bdf 100644 --- a/2016/dot-products/hungarian/sentence_translations.json +++ b/2016/dot-products/hungarian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[\"Oda az örömhöz\", Beethoven, a zongora végére.] Hagyományosan a ponttöredékeket a lineáris algebra kurzusokon nagyon korán, jellemzően rögtön az elején bevezetik.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/dot-products/indonesian/sentence_translations.json b/2016/dot-products/indonesian/sentence_translations.json index db9ba8335..7ea839adf 100644 --- a/2016/dot-products/indonesian/sentence_translations.json +++ b/2016/dot-products/indonesian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "["Ode to Joy", oleh Beethoven, dimainkan sampai akhir piano. ] Secara tradisional, perkalian titik adalah sesuatu yang diperkenalkan sejak awal dalam kursus aljabar linier, biasanya tepat di awal.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/italian/sentence_translations.json b/2016/dot-products/italian/sentence_translations.json index 5f8361cea..3bfa6945b 100644 --- a/2016/dot-products/italian/sentence_translations.json +++ b/2016/dot-products/italian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[Musica] Tradizionalmente, i prodotti scalari sono qualcosa che viene introdotto molto presto in un corso di algebra lineare, in genere proprio all'inizio.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/japanese/sentence_translations.json b/2016/dot-products/japanese/sentence_translations.json index 59c506106..bed507c90 100644 --- a/2016/dot-products/japanese/sentence_translations.json +++ b/2016/dot-products/japanese/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[ベートーベンの「歓喜の歌」がピアノの最後まで演奏されます。 ] 伝統的に、ドット積は線形代数コースの非常に早 い段階で、通常は開始直後に導入されるものです。", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/korean/sentence_translations.json b/2016/dot-products/korean/sentence_translations.json index a2d129d4c..0769b25a8 100644 --- a/2016/dot-products/korean/sentence_translations.json +++ b/2016/dot-products/korean/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[베토벤의 "환희의 송가"는 피아노 끝까지 연주됩니다.] 전통적으로 내적은 선형 대수 과정의 초기 단계, 일반적으로 시작 부분에 소개되는 것입니다.", "model": "google_nmt", "from_community_srt": "전통적으로, 내적(dot product) 같은 것은 선형대수 강의에서 상당히 앞쪽에 나와. 보통 시작 부근에 있지.", diff --git a/2016/dot-products/marathi/sentence_translations.json b/2016/dot-products/marathi/sentence_translations.json index 742956713..fd850a796 100644 --- a/2016/dot-products/marathi/sentence_translations.json +++ b/2016/dot-products/marathi/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[बीथोव्हेनचे "ओड टू जॉय", पियानोच्या शेवटी वाजते. ] पारंपारिकपणे, डॉट उत्पादने अशी काही आहे जी रेखीय बीजगणित अभ्यासक्रमात अगदी सुरुवातीस, विशेषत: अगदी सुरुवातीस सादर केली जाते.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/persian/sentence_translations.json b/2016/dot-products/persian/sentence_translations.json index 6240c588f..a91b36e30 100644 --- a/2016/dot-products/persian/sentence_translations.json +++ b/2016/dot-products/persian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[موسیقی] به‌طور سنتی، محصولات نقطه‌ای چیزی هستند که در اوایل دوره جبر خطی، معمولاً در ابتدا معرفی می‌شوند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/polish/sentence_translations.json b/2016/dot-products/polish/sentence_translations.json index a47248203..dc5807a12 100644 --- a/2016/dot-products/polish/sentence_translations.json +++ b/2016/dot-products/polish/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[„Oda do radości” Beethovena gra do końca fortepianu.] Tradycyjnie iloczyny kropkowe są czymś, co wprowadza się bardzo wcześnie w kursie algebry liniowej, zazwyczaj zaraz na początku.", "model": "google_nmt", "from_community_srt": "Calvin: Wiesz co, moim zdaniem matematyka nie jest nauką. Myślę, że to religia. Hobbes: Religia? Calvin: Tak. Wszystkie te równania są jak cuda. Bierzesz dwie liczby i kiedy je dodasz, nagle magicznie stają się jedną, NOWĄ liczbą! Nikt nie wie, jak to się dzieje. Albo w to wierzysz, albo nie. Zazwyczaj iloczyny skalarne wprowadzane są bardzo wcześnie w kursach Algebry Liniowej zwykle na samym początku.", diff --git a/2016/dot-products/portuguese/sentence_translations.json b/2016/dot-products/portuguese/sentence_translations.json index 1b838c0ab..3281908d5 100644 --- a/2016/dot-products/portuguese/sentence_translations.json +++ b/2016/dot-products/portuguese/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "["Ode to Joy", de Beethoven, toca até o final do piano.] Tradicionalmente, produtos escalares são algo introduzido bem no início de um curso de álgebra linear, normalmente logo no início.", "model": "google_nmt", "from_community_srt": "Calvin: \"Sabe, não acho que matemática seja uma ciência. Acho que é uma religião.\" Hobbes: \"Uma religião?\" Calvin: \"É. Todas aquelas equações são como milagres. Você pega dois números, e quando você os soma, eles magicamente viram um número NOVO! Calvin: \"E ninguém sabe dizer como isso acontece. Ou você acredita, ou não acredita.\" Tradicionalmente, o produto escalar é algo introduzido bem cedo num curso de Álgebra Linear, geralmente bem no começo.", diff --git a/2016/dot-products/russian/sentence_translations.json b/2016/dot-products/russian/sentence_translations.json index 936419e73..e992c63c3 100644 --- a/2016/dot-products/russian/sentence_translations.json +++ b/2016/dot-products/russian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[«Ода к радости» Бетховена играет до конца фортепиано.] Традиционно скалярные произведения представляют собой то, что вводится в самом начале курса линейной алгебры, обычно в самом начале.", "from_community_srt": "Обычно,скалярное произведение вводят на ранней стадии изучения курса линейной алгебры, обычно в самом начале.", "n_reviews": 0, diff --git a/2016/dot-products/spanish/sentence_translations.json b/2016/dot-products/spanish/sentence_translations.json index 9be6b912b..8a33116b3 100644 --- a/2016/dot-products/spanish/sentence_translations.json +++ b/2016/dot-products/spanish/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "["Oda a la alegría", de Beethoven, se toca hasta el final del piano.] Tradicionalmente, los productos escalares son algo que se introduce muy temprano en un curso de álgebra lineal, normalmente desde el principio.", "from_community_srt": "Calvin: Sabes, no creo que la matemática sea una ciencia, creo que es una religión. Hobbes: ¿Una religión? Hobbes: ¿Una religión? Calvin: Sí. Todas estas ecuaciones son como milagros. Tomas dos números y luego cuando los sumas ¡mágicamente se vuelven un número NUEVO! Nadie puede decir cómo ocurre. Simplemente los crees o no. Tradicionalmente, el producto punto es algo que es introducido en las primeras partes de un curso de álgebra lineal usualmente en el comienzo.", "n_reviews": 0, diff --git a/2016/dot-products/tamil/sentence_translations.json b/2016/dot-products/tamil/sentence_translations.json index 964d97174..fea23fa25 100644 --- a/2016/dot-products/tamil/sentence_translations.json +++ b/2016/dot-products/tamil/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "பீத்தோவனின் "ஓட் டு ஜாய்", பியானோவின் இறுதிவரை வாசிக்கிறது. ] பாரம்பரியமாக, புள்ளித் தயாரிப்புகள் என்பது ஒரு நேரியல் இயற்கணிதப் பாடத்தின் ஆரம்பத்தில் அறிமுகப்படுத்தப்பட்ட ஒன்று, பொதுவாக தொடக்கத்திலேயே.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/telugu/sentence_translations.json b/2016/dot-products/telugu/sentence_translations.json index 93f5a0438..20a5f5083 100644 --- a/2016/dot-products/telugu/sentence_translations.json +++ b/2016/dot-products/telugu/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[బీథోవెన్ రచించిన "ఓడ్ టు జాయ్", పియానో చివరి వరకు ప్లే చేస్తుంది. ] సాంప్రదాయకంగా, డాట్ ఉత్పత్తులు అనేది ఒక సరళ బీజగణిత కోర్సులో చాలా ప్రారంభంలో పరిచయం చేయబడినవి, సాధారణంగా ప్రారంభంలోనే.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/thai/sentence_translations.json b/2016/dot-products/thai/sentence_translations.json index 61de13e9b..76eddcf8f 100644 --- a/2016/dot-products/thai/sentence_translations.json +++ b/2016/dot-products/thai/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[ดนตรี] ตามเนื้อผ้า ผลิตภัณฑ์ดอทเป็นสิ่งที่ถูกนำเสนอตั้งแต่เนิ่นๆ ในหลักสูตรพีชคณิตเชิงเส้น โดยทั่วไปจะเริ่มต้นตั้งแต่เริ่มต้น ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/turkish/sentence_translations.json b/2016/dot-products/turkish/sentence_translations.json index 8d2bbec1b..a03359e81 100644 --- a/2016/dot-products/turkish/sentence_translations.json +++ b/2016/dot-products/turkish/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[Beethoven'ın "Ode to Joy" adlı eseri piyanonun sonuna kadar çalıyor. ] Geleneksel olarak nokta çarpımlar, doğrusal cebir dersinin çok erken safhalarında, genellikle de en başında tanıtılan bir şeydir.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/ukrainian/sentence_translations.json b/2016/dot-products/ukrainian/sentence_translations.json index defb48295..584fc7cc9 100644 --- a/2016/dot-products/ukrainian/sentence_translations.json +++ b/2016/dot-products/ukrainian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[Музика] Традиційно скалярний добуток вводиться в курс лінійної алгебри дуже рано, як правило, на початку.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/urdu/sentence_translations.json b/2016/dot-products/urdu/sentence_translations.json index 08c78ba2e..b4f843857 100644 --- a/2016/dot-products/urdu/sentence_translations.json +++ b/2016/dot-products/urdu/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "[موسیقی] روایتی طور پر، ڈاٹ پروڈکٹس ایسی چیز ہوتی ہیں جو واقعی ابتدائی طور پر ایک لکیری الجبرا کورس میں متعارف کرائی جاتی ہیں، عام طور پر شروع میں۔ ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/dot-products/vietnamese/sentence_translations.json b/2016/dot-products/vietnamese/sentence_translations.json index 0d7e135b9..68a1e0910 100644 --- a/2016/dot-products/vietnamese/sentence_translations.json +++ b/2016/dot-products/vietnamese/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", + "input": "[\"Ode to Joy\", by Beethoven, plays to the end of the piano.] Traditionally, dot products are something that's introduced really early on in a linear algebra course, typically right at the start.", "translatedText": "["Ode to Joy" của Beethoven chơi đến cuối cây đàn piano. ] Theo truyền thống, tích số chấm là thứ được giới thiệu rất sớm trong khóa học đại số tuyến tính, thường là ngay khi bắt đầu.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/eigenvalues/arabic/sentence_translations.json b/2016/eigenvalues/arabic/sentence_translations.json index 8ae17f8c2..d4c638745 100644 --- a/2016/eigenvalues/arabic/sentence_translations.json +++ b/2016/eigenvalues/arabic/sentence_translations.json @@ -1052,7 +1052,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "الفيديو التالي والأخير من هذه السلسلة سيكون عن المساحات المتجهة المجردة.", "model": "google_nmt", "from_community_srt": "سيكون الفيديو التالي والأخير من هذه السلسلة على مساحات ناقلات مجردة.", diff --git a/2016/eigenvalues/czech/sentence_translations.json b/2016/eigenvalues/czech/sentence_translations.json index e90eff7be..a4291f451 100644 --- a/2016/eigenvalues/czech/sentence_translations.json +++ b/2016/eigenvalues/czech/sentence_translations.json @@ -1061,7 +1061,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "Další a poslední video této série se bude věnovat abstraktním vektorovým prostorům.", "model": "DeepL", "from_community_srt": "Příští a poslední video téhle série bude o \"abstraktních vektorových prostorech\".", diff --git a/2016/eigenvalues/english/captions.srt b/2016/eigenvalues/english/captions.srt index 25bb7ab42..ea21f65b6 100644 --- a/2016/eigenvalues/english/captions.srt +++ b/2016/eigenvalues/english/captions.srt @@ -903,6 +903,10 @@ some surprising results, I'll leave up a prompt here on the screen. It takes a bit of work, but I think you'll enjoy it. 227 -00:16:40,840 --> 00:16:46,120 -The next and final video of this series is going to be on abstract vector spaces. +00:16:40,840 --> 00:16:45,397 +The next and final video of this series is going to be on abstract vector spaces. + +228 +00:16:45,397 --> 00:16:46,120 +See you then! diff --git a/2016/eigenvalues/english/sentence_timings.json b/2016/eigenvalues/english/sentence_timings.json index 1378cd3be..1c5f2fb3e 100644 --- a/2016/eigenvalues/english/sentence_timings.json +++ b/2016/eigenvalues/english/sentence_timings.json @@ -590,7 +590,7 @@ 1000.28 ], [ - "The next and final video of this series is going to be on abstract vector spaces.", + "The next and final video of this series is going to be on abstract vector spaces. See you then!", 1000.84, 1006.12 ] diff --git a/2016/eigenvalues/english/transcript.txt b/2016/eigenvalues/english/transcript.txt index 11f7346cf..cda243b3d 100644 --- a/2016/eigenvalues/english/transcript.txt +++ b/2016/eigenvalues/english/transcript.txt @@ -116,4 +116,4 @@ A shear, for example, doesn't have enough eigenvectors to span the full space. But if you can find an eigenbasis, it makes matrix operations really lovely. For those of you willing to work through a pretty neat puzzle to see what this looks like in action and how it can be used to produce some surprising results, I'll leave up a prompt here on the screen. It takes a bit of work, but I think you'll enjoy it. -The next and final video of this series is going to be on abstract vector spaces. \ No newline at end of file +The next and final video of this series is going to be on abstract vector spaces. See you then! \ No newline at end of file diff --git a/2016/eigenvalues/french/sentence_translations.json b/2016/eigenvalues/french/sentence_translations.json index c0de2d946..3a6006ed7 100644 --- a/2016/eigenvalues/french/sentence_translations.json +++ b/2016/eigenvalues/french/sentence_translations.json @@ -939,7 +939,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "La prochaine et dernière vidéo de cette série portera sur les espaces vectoriels abstraits.", "from_community_srt": "La prochaine, et dernière vidéo de cette série sera sur les espaces vectoriels abstraits.", "n_reviews": 0, diff --git a/2016/eigenvalues/german/sentence_translations.json b/2016/eigenvalues/german/sentence_translations.json index 1688acc7b..08f56e9e7 100644 --- a/2016/eigenvalues/german/sentence_translations.json +++ b/2016/eigenvalues/german/sentence_translations.json @@ -1059,7 +1059,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "Das nächste und letzte Video dieser Reihe wird sich mit abstrakten Vektorräumen beschäftigen.", "model": "DeepL", "from_community_srt": "Das nächste und finale Video dieser Serie wird über abstrakte Vektorräume sein.", diff --git a/2016/eigenvalues/greek/sentence_translations.json b/2016/eigenvalues/greek/sentence_translations.json index 9170c3950..43e40cd15 100644 --- a/2016/eigenvalues/greek/sentence_translations.json +++ b/2016/eigenvalues/greek/sentence_translations.json @@ -1062,7 +1062,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "Το επόμενο και τελευταίο βίντεο αυτής της σειράς θα είναι σε αφηρημένα διανυσματικά κενά.", "model": "google_nmt", "from_community_srt": "Το επόμενο και τελευταίο βίντεο αυτής της σειράς θα είναι σε αφηρημένους διανυσματικούς χώρους.", diff --git a/2016/eigenvalues/hungarian/sentence_translations.json b/2016/eigenvalues/hungarian/sentence_translations.json index e0f4c309d..522ba5e7b 100644 --- a/2016/eigenvalues/hungarian/sentence_translations.json +++ b/2016/eigenvalues/hungarian/sentence_translations.json @@ -944,7 +944,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "A sorozat következő és egyben utolsó videója az absztrakt vektorterekről fog szólni.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/eigenvalues/italian/sentence_translations.json b/2016/eigenvalues/italian/sentence_translations.json index a87e806ea..dfe9c1428 100644 --- a/2016/eigenvalues/italian/sentence_translations.json +++ b/2016/eigenvalues/italian/sentence_translations.json @@ -1061,7 +1061,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "Il prossimo e ultimo video di questa serie riguarderà gli spazi vettoriali astratti.", "model": "google_nmt", "from_community_srt": "Il prossimo e ultimo video di questa serie riguarderà gli spazi vettoriali astratti.", diff --git a/2016/eigenvalues/korean/sentence_translations.json b/2016/eigenvalues/korean/sentence_translations.json index c35c4e56a..c060b958c 100644 --- a/2016/eigenvalues/korean/sentence_translations.json +++ b/2016/eigenvalues/korean/sentence_translations.json @@ -1020,7 +1020,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "이 시리즈의 다음이자 마지막 비디오는 추상적인 벡터 공간에 관한 것입니다.", "model": "google_nmt", "from_community_srt": "\"추상 벡터 공간(abstract vector spaces)\"에 대한 것입니다 다음에 봐요!", diff --git a/2016/eigenvalues/polish/sentence_translations.json b/2016/eigenvalues/polish/sentence_translations.json index 1345a172a..b1eaa4484 100644 --- a/2016/eigenvalues/polish/sentence_translations.json +++ b/2016/eigenvalues/polish/sentence_translations.json @@ -1061,7 +1061,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "Następny i ostatni film z tej serii będzie dotyczył abstrakcyjnych przestrzeni wektorowych.", "model": "google_nmt", "from_community_srt": "Następny, ostatni film z serii będzie o abstrakcyjnych przestrzeniach liniowych.", diff --git a/2016/eigenvalues/portuguese/sentence_translations.json b/2016/eigenvalues/portuguese/sentence_translations.json index 6bf6c2bd7..8d5fb815b 100644 --- a/2016/eigenvalues/portuguese/sentence_translations.json +++ b/2016/eigenvalues/portuguese/sentence_translations.json @@ -1060,7 +1060,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "O próximo e último vídeo desta série será sobre espaços vetoriais abstratos.", "model": "google_nmt", "from_community_srt": "O próximo e último vídeo desta série vai ser em “espaços vetoriais abstratos”.", diff --git a/2016/eigenvalues/russian/sentence_translations.json b/2016/eigenvalues/russian/sentence_translations.json index a7579efa0..d2cdc23cb 100644 --- a/2016/eigenvalues/russian/sentence_translations.json +++ b/2016/eigenvalues/russian/sentence_translations.json @@ -942,7 +942,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "Следующее и последнее видео из этой серии будет посвящено абстрактным векторным пространствам.", "from_community_srt": "Следующее и последнее видео этой серии будет посвящено абстрактным векторым пространствам.", "n_reviews": 0, diff --git a/2016/eigenvalues/spanish/sentence_translations.json b/2016/eigenvalues/spanish/sentence_translations.json index ba8ee21f5..5b45f4560 100644 --- a/2016/eigenvalues/spanish/sentence_translations.json +++ b/2016/eigenvalues/spanish/sentence_translations.json @@ -936,7 +936,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "El siguiente y último vídeo de esta serie tratará sobre espacios vectoriales abstractos.", "from_community_srt": "El próximo y último vídeo de esta serie va a ser sobre \"espacios vectoriales abstractos\" ¡Nos vemos hasta entonces!", "n_reviews": 0, diff --git a/2016/eigenvalues/turkish/sentence_translations.json b/2016/eigenvalues/turkish/sentence_translations.json index 89865c9b4..8e94f69c5 100644 --- a/2016/eigenvalues/turkish/sentence_translations.json +++ b/2016/eigenvalues/turkish/sentence_translations.json @@ -1061,7 +1061,7 @@ "end": 1000.28 }, { - "input": "The next and final video of this series is going to be on abstract vector spaces.", + "input": "The next and final video of this series is going to be on abstract vector spaces. See you then!", "translatedText": "Bu serinin bir sonraki ve son videosu soyut vektör uzayları üzerine olacak.", "model": "google_nmt", "from_community_srt": "Bu serinin bir sonraki ve son videosu soyut vektör uzaylarında olacak.", diff --git a/2016/eola-preview/french/sentence_translations.json b/2016/eola-preview/french/sentence_translations.json index c7a43a6a2..9817291bd 100644 --- a/2016/eola-preview/french/sentence_translations.json +++ b/2016/eola-preview/french/sentence_translations.json @@ -245,4 +245,4 @@ "start": 292.42, "end": 294.54 } -] +] \ No newline at end of file diff --git a/2016/hanoi-and-sierpinski/arabic/sentence_translations.json b/2016/hanoi-and-sierpinski/arabic/sentence_translations.json index dd4a42bb6..2e1291517 100644 --- a/2016/hanoi-and-sierpinski/arabic/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/arabic/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg. ", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg. ", "translatedText": "تعتقد أن هذه الأقراص تحتوي على ثقب في المنتصف بحيث يمكنك وضعها على الوتد. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 109.02 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. ", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associating ", "translatedText": "يبدأ أي وصف للثنائي عادة بالتأمل في طريقتنا المعتادة لتمثيل الأرقام، ما نسميه الأساس 10، لأننا نستخدم 10 أرقام منفصلة، 0، 1، 2، 3، 4، 5، 6، 7، 8، 9 . ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar. ", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits. ", "translatedText": "بهذه الطريقة، يكون إيقاع العد مشابهًا ذاتيًا نوعًا ما. ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable. ", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover. ", "translatedText": "أصبح إيقاع العد الآن أسرع كثيرًا، لكن هذا يجعله أكثر وضوحًا. ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once. ", + "input": "ce. Flip the last, roll over twice. Flip the last, roll over once. ", "translatedText": "الوجه الأخير، يتدحرج مرة واحدة. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 286.76 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again. ", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then ", "translatedText": "على نطاق أكبر، مثل العد حتى 15، وهو 1-1-1-1، تتمثل العملية في السماح لآخر 3 بت بالعد حتى 7، ثم الانتقال إلى خانة 8، ثم السماح للثلاث بتات الأخيرة بالعد مرة أخرى . ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 296.14 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again. ", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till th ", "translatedText": "عند العد حتى 255، أي 8 أرقام متتالية، يبدو هذا وكأنه ترك آخر 7 بتات تحسب حتى تمتلئ، ثم تنتقل إلى خانة 128، ثم تترك آخر 7 بتات تحسب مرة أخرى. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 313.24 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. ", + "input": "ey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. ", "translatedText": "حسنًا، بهذه المقدمة البسيطة، الحقيقة المفاجئة التي أظهرها لي كيث هي أنه يمكننا استخدام هذا الإيقاع لحل أبراج هانوي. ", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 324.72 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right. ", + "input": "lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0. ", "translatedText": "عندما تقوم بقلب الجزء الأخير فقط، من 0 إلى 1، قم بتحريك القرص 0 وتدًا واحدًا إلى اليمين. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 340.8 }, { - "input": "Where do you move it, you might ask? ", + "input": "There's something magical At the small scale, say counting up to 3, which is 11 in binary, this means flip the last bit, rol ", "translatedText": "قد تسأل أين يمكنك نقله؟ ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 345.94 }, { - "input": "You have no choice, you can't put it on top of disk 0, and there's only one other peg, so you move it where you're forced to move it. ", + "input": "l over to the twos, then flip the last bit. At a larger scale, like counting up to 15, which is 1111 in binary, the process is to let the last 3 count up to 7, roll over to the eights place, then let the ", "translatedText": "ليس لديك خيار، ولا يمكنك وضعه أعلى القرص 0، ولا يوجد سوى ربط واحد آخر، لذا يمكنك نقله حيث تضطر إلى نقله. ", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 386.02 }, { - "input": "There's something magical about it, like when I first saw this, I was like, this can't work. ", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to sol ", "translatedText": "هناك شيء سحري في الأمر، عندما رأيت هذا لأول مرة، قلت، هذا لا يمكن أن ينجح. ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 390.82 }, { - "input": "I don't know how this works, I don't know why this works, now I know, but it's just magical when you see it, and I remember putting together animation for this for when I was teaching this, and just like, you know, I know how this works, I know all the things in it, it's still fun to just sit and just like, you know, watch it play out. ", + "input": "ve the towers of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective. Disk 3 is thinking, okay, 2, 1, and 0, like you have to get off of me, like I can't really functi ", "translatedText": "لا أعرف كيف يعمل هذا، لا أعرف لماذا يعمل هذا، الآن أعلم، لكنه أمر سحري عندما تراه، وأتذكر تجميع الرسوم المتحركة لهذا عندما كنت أقوم بتدريس هذا، ومثل، كما تعلمون، أنا أعرف كيف يعمل هذا، أعرف كل الأشياء الموجودة فيه، ولا يزال من الممتع مجرد الجلوس ومشاهدة ما يحدث. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 405.08 }, { - "input": "Oh yeah. ", + "input": "on under this much weight and pressure. And so just from disk 3's perspecti If, in your binary co ", "translatedText": "أوه نعم. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 405.26 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves. ", + "input": "unting, you roll over once to the twos place, meaning you flip the last two bits, you move dis ", "translatedText": "أعني أنه ليس من الواضح في البداية أن هذا سيؤدي دائمًا إلى تحركات قانونية. ", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 406.02 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move? ", + "input": "k number 1. Where do you move it, you might ask? Well, you have no choice. this disk to work, I can turn my bigger problem into something slightly smaller. And then how do I solve that? Well, it's exact ", "translatedText": "على سبيل المثال، كيف تعرف أنه في كل مرة تنتقل فيها إلى مكان الرقم 8، سيتم بالضرورة تحرير هذا القرص 3 للتحرك؟ ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 410.96 }, { - "input": "At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than having to do 2 to the n minus 1 steps? ", + "input": "ly the same thing. If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to where it needs to go. Then disk 1 and 0 can ", "translatedText": "في الوقت نفسه، يثير الحل على الفور هذه الأسئلة مثل، من أين يأتي هذا، ولماذا يعمل، وهل هناك طريقة أفضل للقيام بذلك من الاضطرار إلى القيام بخطوات 2 إلى n ناقص 1؟ ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 446.6 }, { - "input": "That's the only way it can move. ", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. ", "translatedText": "هذه هي الطريقة الوحيدة التي يمكن أن تتحرك بها. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 459.04 }, { - "input": "And then disk 3 says, I'm set. ", + "input": "And if you think about it for a bit, it becomes clear that this has to be the ", "translatedText": "ثم يقول القرص 3، أنا مستعد. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 461.22 }, { - "input": "Everyone else just figure out how to get here. ", + "input": "solution. At every step, you're only doing what's forced upon you. You have to get disk 0 ", "translatedText": "الجميع يعرفون فقط كيفية الوصول إلى هنا. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 470.54 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C. ", + "input": "e you can move disk 3. And you have to move disk 3. Flip the last two, move disk 1. Flip the last, move disk 0. And here, we're going to have to roll over three times t ", "translatedText": "الآن لديك القرص 0 و1 و2 مثبتًا على المغزل B، وعليك نقلهم إلى C. ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 495.54 }, { - "input": "If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. ", + "input": "I don't know how this works, I don't know why this works. Now I know, but it's jus ", "translatedText": "إذا كان القرص 2 سيقول، القرص 1 والقرص 0، فهو ليس أنت، إنه أنا. ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 503.26 }, { - "input": "They need to move somewhere. ", + "input": "remember putting together an animation for this when I was teaching this, and just like, I know how this works. ", "translatedText": "إنهم بحاجة إلى التحرك في مكان ما. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 511.26 }, { - "input": "Then disk 1 and 0 can do this. ", + "input": "ove disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. ", "translatedText": "ثم يمكن للقرص 1 و 0 القيام بذلك. ", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 522.24 }, { - "input": "Then I'm going to move. ", + "input": "he last bit. It's still fun to just sit and just watch it play out. At a slightly larger scale, solving towers of H ", "translatedText": "ثم سأتحرك. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 549.3 }, { - "input": "At every step, you're only doing what's forced upon you. ", + "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counti ", "translatedText": "في كل خطوة، أنت تفعل فقط ما فُرض عليك. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 557.16 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3. ", + "input": "ng up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this wor ", "translatedText": "يجب عليك إيقاف القرص من 0 إلى 2 قبل أن تتمكن من نقل القرص 3. ", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 562.28 }, { - "input": "And you have to move disk 3. ", + "input": "k, and is there a better way of doing this than having to do 2 to the n minus 1 steps? ", "translatedText": "ويجب عليك نقل القرص 3. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 568.24 }, { - "input": "So why does counting in binary capture this algorithm? ", + "input": "t only does this solve Towers of Hanoi, but it does it in the most efficient way possible. Understanding why this works and how it works and what ", "translatedText": "فلماذا يلتقط العد الثنائي هذه الخوارزمية؟ ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 602.8 }, { - "input": "For example, at a pretty small scale, solving towers of Hanoi for two disks, move disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. ", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. T ", "translatedText": "على سبيل المثال، على نطاق صغير جدًا، حل أبراج هانوي لقرصين، حرك القرص 0، حرك القرص 1، حرك القرص 0، ينعكس من خلال العد حتى 3 في النظام الثنائي. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 617.18 }, { - "input": "Flip the last bit, roll over once, flip the last bit. ", + "input": "hat's the only way it can move. access to these videos before I publish the ", "translatedText": "اقلب الجزء الأخير، ثم قم بالتمرير مرة واحدة، ثم اقلب الجزء الأخير. ", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 686.98 }, { - "input": "But it actually gets cooler. ", + "input": "s disk to work, I can turn my bigger problem into something slightly smaller. ", "translatedText": "لكنها في الواقع تصبح أكثر برودة. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 689.96 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle. ", + "input": "and students, you can check out the careers page that I've linked in the description. Personally, I think they'r ", "translatedText": "لم أتوصل حتى إلى كيفية ارتباط هذا بمثلث سيربينسكي. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 691.36 }, { - "input": "And that's exactly what I'm going to do in the follow-on video, part 2. ", + "input": "Well, it's exactly the same thing. Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space. Get off. ", "translatedText": "وهذا بالضبط ما سأفعله في الفيديو التالي، الجزء الثاني. ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 714.02 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring. ", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra ", "translatedText": "يتم أيضًا دعم هذا الفيديو والفيديو التالي جزئيًا بواسطة Desmos، وقبل الفيديو التالي، أريد فقط أن أتوقف لحظة وأشارككم يا رفاق القليل عن هويتهم وحقيقة أنهم يوظفون. ", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 732.08 }, { - "input": "The real meat of their offering is in their classroom activities. ", + "input": "ecomes clear that this has to be the most efficient solution. At every step, you're only doing what's forced upon you. ", "translatedText": "اللحم الحقيقي لعروضهم هو في أنشطة الفصول الدراسية. ", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 740.96 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint. ", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it. ", "translatedText": "من جهتي، أنا معجب جدًا بمدى التفكير الجيد لهذه الأنشطة من وجهة النظر التربوية. ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 750.34 }, { - "input": "And like I said, they're hiring. ", + "input": "So why does counting in binary capture this algorithm? Well, what's going on ", "translatedText": "وكما قلت، إنهم يوظفون. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/bengali/sentence_translations.json b/2016/hanoi-and-sierpinski/bengali/sentence_translations.json index 4ba56de48..822825de2 100644 --- a/2016/hanoi-and-sierpinski/bengali/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/bengali/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg. ", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg. ", "translatedText": "আপনি এই ডিস্কগুলিকে মাঝখানে একটি গর্ত বলে মনে করেন যাতে আপনি সেগুলিকে একটি খুঁটিতে ফিট করতে পারেন। ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 109.02 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. ", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associating ", "translatedText": "বাইনারির যেকোন বর্ণনা সাধারণত সংখ্যাগুলিকে উপস্থাপন করার আমাদের স্বাভাবিক উপায় সম্পর্কে একটি আত্মদর্শন দিয়ে শুরু হয়, যাকে আমরা বেস 10 বলি, যেহেতু আমরা 10টি পৃথক সংখ্যা ব্যবহার করি, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 . ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar. ", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits. ", "translatedText": "এইভাবে, গণনার ছন্দটি স্ব-অনুরূপ। ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable. ", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover. ", "translatedText": "গণনার ছন্দ এখন অনেক দ্রুত, কিন্তু এটি প্রায় আরও লক্ষণীয় করে তোলে। ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once. ", + "input": "ce. Flip the last, roll over twice. Flip the last, roll over once. ", "translatedText": "শেষটি ফ্লিপ করুন, একবার রোল করুন। ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 286.76 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again. ", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then ", "translatedText": "বৃহত্তর স্কেলে, 15 পর্যন্ত গণনা করার মতো, যা 1-1-1-1, প্রক্রিয়াটি হল শেষ 3টি 7 পর্যন্ত গণনা করা, 8 এর জায়গায় রোল ওভার করা, তারপর শেষ 3টি বিটকে আবার গণনা করা . ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 296.14 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again. ", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till th ", "translatedText": "255 পর্যন্ত গণনা করা, যা 8টি পরপর 1 এর, এটি মনে হচ্ছে শেষ 7 বিটগুলি পূর্ণ না হওয়া পর্যন্ত গণনা করতে দেওয়া, 128 এর জায়গায় রোল ওভার করা, তারপর শেষ 7 বিটগুলিকে আবার গণনা করতে দেওয়া। ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 313.24 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. ", + "input": "ey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. ", "translatedText": "ঠিক আছে, তাই সেই মিনি-পরিচয় দিয়ে, কিথ আমাকে যে আশ্চর্যজনক সত্যটি দেখিয়েছেন তা হল যে আমরা হ্যানয়ের টাওয়ারগুলি সমাধান করতে এই ছন্দটি ব্যবহার করতে পারি। ", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 324.72 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right. ", + "input": "lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0. ", "translatedText": "যখনই আপনি শুধুমাত্র সেই শেষ বিটটি ফ্লিপ করছেন, 0 থেকে 1 পর্যন্ত, ডিস্ক 0 এক পেগ ডানদিকে সরান। ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 340.8 }, { - "input": "Where do you move it, you might ask? ", + "input": "There's something magical At the small scale, say counting up to 3, which is 11 in binary, this means flip the last bit, rol ", "translatedText": "আপনি এটি কোথায় সরান, আপনি জিজ্ঞাসা করতে পারেন? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 345.94 }, { - "input": "You have no choice, you can't put it on top of disk 0, and there's only one other peg, so you move it where you're forced to move it. ", + "input": "l over to the twos, then flip the last bit. At a larger scale, like counting up to 15, which is 1111 in binary, the process is to let the last 3 count up to 7, roll over to the eights place, then let the ", "translatedText": "আপনার কোন বিকল্প নেই, আপনি এটি ডিস্ক 0 এর উপরে রাখতে পারবেন না, এবং শুধুমাত্র একটি অন্য পেগ আছে, তাই আপনি যেখানে এটি সরাতে বাধ্য হচ্ছেন সেখানে এটি সরিয়ে ফেলুন। ", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 386.02 }, { - "input": "There's something magical about it, like when I first saw this, I was like, this can't work. ", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to sol ", "translatedText": "এটিতে কিছু জাদু আছে, যেমন আমি যখন এটি প্রথম দেখেছিলাম, তখন আমার মত ছিল, এটি কাজ করতে পারে না। ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 390.82 }, { - "input": "I don't know how this works, I don't know why this works, now I know, but it's just magical when you see it, and I remember putting together animation for this for when I was teaching this, and just like, you know, I know how this works, I know all the things in it, it's still fun to just sit and just like, you know, watch it play out. ", + "input": "ve the towers of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective. Disk 3 is thinking, okay, 2, 1, and 0, like you have to get off of me, like I can't really functi ", "translatedText": "আমি জানি না এটি কীভাবে কাজ করে, আমি জানি না কেন এটি কাজ করে, এখন আমি জানি, কিন্তু আপনি যখন এটি দেখেন তখন এটি জাদুকর, এবং আমি যখন এটি শেখাচ্ছিলাম তখন এটির জন্য অ্যানিমেশন একত্রিত করার কথা মনে আছে, এবং ঠিক যেমন, আপনি জানেন, আমি জানি এটি কীভাবে কাজ করে, আমি এর সমস্ত জিনিস জানি, এটি এখনও বসে থাকা মজাদার এবং ঠিক যেমন, আপনি জানেন, এটি খেলা দেখতে দেখুন। ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 405.08 }, { - "input": "Oh yeah. ", + "input": "on under this much weight and pressure. And so just from disk 3's perspecti If, in your binary co ", "translatedText": "ও আচ্ছা. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 405.26 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves. ", + "input": "unting, you roll over once to the twos place, meaning you flip the last two bits, you move dis ", "translatedText": "আমি বলতে চাচ্ছি, এটি প্রথমে স্পষ্ট নয় যে এটি সর্বদা আইনি পদক্ষেপ নিতে চলেছে। ", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 406.02 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move? ", + "input": "k number 1. Where do you move it, you might ask? Well, you have no choice. this disk to work, I can turn my bigger problem into something slightly smaller. And then how do I solve that? Well, it's exact ", "translatedText": "উদাহরণস্বরূপ, আপনি কিভাবে জানবেন যে প্রতিবার আপনি 8 এর জায়গায় ঘূর্ণায়মান করছেন, সেই ডিস্ক 3 অগত্যা সরানোর জন্য খালি হয়ে যাচ্ছে? ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 410.96 }, { - "input": "At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than having to do 2 to the n minus 1 steps? ", + "input": "ly the same thing. If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to where it needs to go. Then disk 1 and 0 can ", "translatedText": "একই সময়ে, সমাধানটি অবিলম্বে এই প্রশ্নগুলি উত্থাপন করে যেমন, এটি কোথা থেকে আসে, কেন এটি কাজ করে এবং 2 থেকে বিয়োগ 1 ধাপ করার চেয়ে এটি করার একটি ভাল উপায় আছে কি? ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 446.6 }, { - "input": "That's the only way it can move. ", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. ", "translatedText": "এটি সরানো যেতে পারে একমাত্র উপায়. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 459.04 }, { - "input": "And then disk 3 says, I'm set. ", + "input": "And if you think about it for a bit, it becomes clear that this has to be the ", "translatedText": "এবং তারপর ডিস্ক 3 বলে, আমি সেট করছি. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 461.22 }, { - "input": "Everyone else just figure out how to get here. ", + "input": "solution. At every step, you're only doing what's forced upon you. You have to get disk 0 ", "translatedText": "বাকি সবাই শুধু এখানে কিভাবে পেতে চিন্তা. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 470.54 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C. ", + "input": "e you can move disk 3. And you have to move disk 3. Flip the last two, move disk 1. Flip the last, move disk 0. And here, we're going to have to roll over three times t ", "translatedText": "এখন আপনি ডিস্ক 0, 1, এবং 2 টাকু বি-তে বসে আছেন, আপনাকে সেগুলি সি-তে নিয়ে যেতে হবে। ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 495.54 }, { - "input": "If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. ", + "input": "I don't know how this works, I don't know why this works. Now I know, but it's jus ", "translatedText": "যদি ডিস্ক 2 বলতে যাচ্ছে, ডিস্ক 1 এবং ডিস্ক 0, এটি আপনি নন, এটি আমি। ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 503.26 }, { - "input": "They need to move somewhere. ", + "input": "remember putting together an animation for this when I was teaching this, and just like, I know how this works. ", "translatedText": "তাদের কোথাও সরে যেতে হবে। ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 511.26 }, { - "input": "Then disk 1 and 0 can do this. ", + "input": "ove disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. ", "translatedText": "তারপর ডিস্ক 1 এবং 0 এটি করতে পারে। ", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 522.24 }, { - "input": "Then I'm going to move. ", + "input": "he last bit. It's still fun to just sit and just watch it play out. At a slightly larger scale, solving towers of H ", "translatedText": "তারপর আমি সরানো যাচ্ছি. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 549.3 }, { - "input": "At every step, you're only doing what's forced upon you. ", + "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counti ", "translatedText": "প্রতিটি পদক্ষেপে, আপনি কেবল তা করছেন যা আপনার উপর জোর করা হয়েছে। ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 557.16 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3. ", + "input": "ng up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this wor ", "translatedText": "ডিস্ক 3 সরানোর আগে আপনাকে ডিস্ক 0 থেকে 2 পর্যন্ত ছাড় পেতে হবে। ", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 562.28 }, { - "input": "And you have to move disk 3. ", + "input": "k, and is there a better way of doing this than having to do 2 to the n minus 1 steps? ", "translatedText": "এবং আপনাকে ডিস্ক 3 সরাতে হবে। ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 568.24 }, { - "input": "So why does counting in binary capture this algorithm? ", + "input": "t only does this solve Towers of Hanoi, but it does it in the most efficient way possible. Understanding why this works and how it works and what ", "translatedText": "তাহলে কেন বাইনারি গণনা এই অ্যালগরিদম ক্যাপচার? ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 602.8 }, { - "input": "For example, at a pretty small scale, solving towers of Hanoi for two disks, move disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. ", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. T ", "translatedText": "উদাহরণস্বরূপ, একটি সুন্দর ছোট স্কেলে, দুটি ডিস্কের জন্য হ্যানয়ের টাওয়ার সমাধান করা, ডিস্ক 0 সরানো, ডিস্ক 1 সরানো, ডিস্ক 0 সরানো, বাইনারিতে 3 পর্যন্ত গণনা করে প্রতিফলিত হয়। ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 617.18 }, { - "input": "Flip the last bit, roll over once, flip the last bit. ", + "input": "hat's the only way it can move. access to these videos before I publish the ", "translatedText": "শেষ বিটটি ফ্লিপ করুন, একবার রোল করুন, শেষ বিটটি উল্টান। ", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 686.98 }, { - "input": "But it actually gets cooler. ", + "input": "s disk to work, I can turn my bigger problem into something slightly smaller. ", "translatedText": "কিন্তু এটা আসলে ঠান্ডা পায়. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 689.96 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle. ", + "input": "and students, you can check out the careers page that I've linked in the description. Personally, I think they'r ", "translatedText": "সিয়ারপিনস্কির ত্রিভুজের সাথে এটি কীভাবে সম্পর্কিত তাও আমি বুঝতে পারিনি। ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 691.36 }, { - "input": "And that's exactly what I'm going to do in the follow-on video, part 2. ", + "input": "Well, it's exactly the same thing. Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space. Get off. ", "translatedText": "এবং ফলো-অন ভিডিও, পার্ট 2-এ আমি ঠিক সেটাই করতে যাচ্ছি। ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 714.02 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring. ", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra ", "translatedText": "এই ভিডিওটি এবং পরেরটিও Desmos দ্বারা আংশিকভাবে সমর্থিত, এবং পরবর্তী ভিডিওর আগে আমি শুধু কিছুক্ষণ সময় নিয়ে আপনাদের সাথে শেয়ার করতে চাই তারা কারা এবং তারা যে নিয়োগ করছে সে সম্পর্কে একটু শেয়ার করতে চাই৷ ", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 732.08 }, { - "input": "The real meat of their offering is in their classroom activities. ", + "input": "ecomes clear that this has to be the most efficient solution. At every step, you're only doing what's forced upon you. ", "translatedText": "তাদের নৈবেদ্য প্রকৃত মাংস তাদের শ্রেণীকক্ষ কার্যক্রম. ", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 740.96 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint. ", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it. ", "translatedText": "আমার অংশের জন্য, শিক্ষাগত দৃষ্টিকোণ থেকে এই কার্যক্রমগুলি কতটা সুচিন্তিত তা দেখে আমি অত্যন্ত মুগ্ধ। ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 750.34 }, { - "input": "And like I said, they're hiring. ", + "input": "So why does counting in binary capture this algorithm? Well, what's going on ", "translatedText": "এবং আমি যেমন বলেছি, তারা নিয়োগ করছে। ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/chinese/sentence_translations.json b/2016/hanoi-and-sierpinski/chinese/sentence_translations.json index 2911474d8..270f3a790 100644 --- a/2016/hanoi-and-sierpinski/chinese/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/chinese/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "您可以将这些圆盘视为中间有一个孔 ，以便可以将它们安装到钉子上。", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "对二进制的任何描述通常都是从反思我们通常表示数字的方式 开始，我们称之为基数 10，因为我们使用 10 个独 立的数字：0、1、2、3、4、5、6、7、8、9。", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "然后，用完新数字后，用两位数字 1、0 表示下一个数字 10。", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "这样，计数的节奏就有点自相似了。", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "即使你缩小到更大的比例，这个过程看起 来就像做某事，滚动，做同样的事情，滚 动，并在更大的滚动之前重复 9 次。", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "计数的节奏现在快了很多，但这几乎使它更加引人注目。", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "最后翻面，翻滚一次。", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "规模更大的话，比如数到15，也就是1-1-1-1，过程就 是让最后3位数到7，滚到8位，然后让最后3位再数起来。", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "计数到 255，即 8 个连续的 1，这看 起来就像让最后 7 位计数到满，滚动到 1 28 的位置，然后让最后 7 位再次计数。", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "好吧，通过这个简短的介绍，基思向我展示的令人惊讶的 事实是我们可以使用这种节奏来解决河内塔楼的问题。", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "每当您只翻转最后一位（从 0 到 1） 时，请将磁盘 0 向右移动一个钉子。", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "我的意思是，一开始甚至不清楚这是否总是会带来合法的行动。", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "例如，您如何知道每次滚动到 8 的位置 时，磁盘 3 一定会被释放以进行移动？", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "理解它为什么有效、它是如何工作的以及到底发生了什么，可以归结 为对这个谜题的某种视角，计算机科学人士可能称之为递归视角。", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "这是它能够移动的唯一方式。", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "完成后，我们可以将磁盘 3 移到那里。", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "然后磁盘 3 说，我已经准备好了。", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "从某种意义上说，您现在遇到了同一问题的较小版本。", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "现在磁盘 0、1 和 2 位于主轴 B 上，您必须将它们转移到 C 上。", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "他们需要搬到某个地方。", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "好吧，大家都回去吧。", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "如果您稍微思考一下，就会发现 这肯定是最有效的解决方案。", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "您必须先卸下磁盘 0 到 2，然后才能移动磁盘 3。", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "并且您必须移动磁盘 3。", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "从这个角度来看，没有任何低效率的空间。", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "那么为什么二进制计数会捕获这个算法呢？", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "嗯，这里发生的事情是，解决子问题、 移动大磁盘，然后再次解决子问题的这 种模式与二进制计数的模式完全并行。", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "翻转最后一点，翻转一次，翻转最后一点。", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "类似地，在二进制中计数到 111 需要计 数到 3，翻转所有三位，然后再计数 3。", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "在所有规模上，这两个过程都有相同的故障。", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "所以从某种意义上说，这个二进制解决方案起作用的原因 ，或者至少是一个解释，我觉得没有一个解释，但我认为 最自然的一个是你用来生成这些二进制数的模式具有完全 相同的结构就像河内塔所使用的模式一样，这就是为什么 如果你看一下翻转的位，你实际上正在逆转这个过程。", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "你是说，什么过程产生了这些？", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "这很酷，对吧？", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "但它实际上变得更酷。", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "我什至还没有弄清楚这与谢尔宾斯基三角形有何关系。", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "这个视频和下一个视频也得到了 Desmos 的 部分支持，在下一个视频之前，我只想花点时间与 大家分享一下他们是谁以及他们正在招聘的事实。", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "所以 Desmos 真的很酷。", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "他们提供的真正内容是课堂活动。", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "就我而言，这些活动从教学角度来看是经 过深思熟虑的，给我留下了深刻的印象。", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "就像我说的，他们正在招聘。", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "就我个人而言，我认为他们正在做一些真正有意义的事情。", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/english/captions.srt b/2016/hanoi-and-sierpinski/english/captions.srt index 2005fdc13..6bc015ed2 100644 --- a/2016/hanoi-and-sierpinski/english/captions.srt +++ b/2016/hanoi-and-sierpinski/english/captions.srt @@ -55,806 +55,922 @@ The setup pictured here has five disks, which I'll label 0, 1, 2, 3, 4, but in principle, you could have as many disks as you want. 15 -00:01:07,460 --> 00:01:11,788 -So they all start up stacked up from biggest to smallest on one spindle, +00:01:07,460 --> 00:01:10,596 +For example, your first move must involve moving disk 0, 16 -00:01:11,788 --> 00:01:15,880 -and the goal is to move the entire tower from one spindle to another. +00:01:10,596 --> 00:01:14,834 +since any other disk has stuff on top of it that needs to get out of the way 17 -00:01:15,880 --> 00:01:18,741 -The rule is you can only move one disk at a time, +00:01:14,834 --> 00:01:15,880 +before it can move. 18 -00:01:18,741 --> 00:01:22,060 -and you can't move a bigger disk on top of a smaller disk. +00:01:15,880 --> 00:01:18,533 +After that, you can move disk 1, but it has to go on whatever peg doesn't 19 -00:01:23,720 --> 00:01:26,990 -For example, your first move must involve moving disk 0, +00:01:18,533 --> 00:01:21,080 +currently have disk 0, since otherwise you'd be putting a bigger disk o 20 -00:01:26,990 --> 00:01:31,409 -since any other disk has stuff on top of it that needs to get out of the way +00:01:21,080 --> 00:01:24,477 +n a smaller one, which isn't allowed. If you've never seen this before, 21 -00:01:31,409 --> 00:01:32,500 -before it can move. +00:01:24,477 --> 00:01:28,300 +I highly encourage you to pause and pull out some books of varying sizes and try 22 -00:01:33,080 --> 00:01:36,704 -After that, you can move disk 1, but it has to go on whatever +00:01:28,300 --> 00:01:32,500 +it out for yourself. Now Keith showed me something truly surprising about this puzzle, wh 23 -00:01:36,704 --> 00:01:40,212 -peg doesn't currently have disk 0, since otherwise you'd be +00:01:33,080 --> 00:01:36,682 +ich is that you can solve it just by counting up in binary and associating the rhythm 24 -00:01:40,212 --> 00:01:43,720 -putting a bigger disk on a smaller one, which isn't allowed. +00:01:36,682 --> 00:01:39,112 +of that counting with a certain rhythm of disk movements. 25 -00:01:44,580 --> 00:01:48,168 -If you've never seen this before, I highly encourage you to pause +00:01:39,112 --> 00:01:42,672 +For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview 26 -00:01:48,168 --> 00:01:51,920 -and pull out some books of varying sizes and try it out for yourself. +00:01:42,672 --> 00:01:43,720 +here first. Actually, eve 27 +00:01:44,580 --> 00:01:48,250 +n if you're familiar with binary, I want to explain it with a focus on the + +28 +00:01:48,250 --> 00:01:51,920 +rhythm of counting, which you may or may not have thought about before. Any + +29 00:01:52,300 --> 00:01:55,142 Just kind of get a feel for what the puzzle is, if it's hard, -28 +30 00:01:55,142 --> 00:01:57,940 why it's hard, if it's not, why it's not, that kind of stuff. -29 +31 00:02:00,300 --> 00:02:03,784 Now Keith showed me something truly surprising about this puzzle, -30 +32 00:02:03,784 --> 00:02:07,216 which is that you can solve it just by counting up in binary and -31 +33 00:02:07,216 --> 00:02:11,440 associating the rhythm of that counting with a certain rhythm of disk movements. -32 +34 00:02:12,100 --> 00:02:14,459 For anyone unfamiliar with binary, I'm going to -33 +35 00:02:14,459 --> 00:02:16,820 take a moment to do a quick overview here first. -34 -00:02:17,460 --> 00:02:19,769 -Actually, even if you are familiar with binary, - -35 -00:02:19,769 --> 00:02:22,705 -I want to explain it with a focus on the rhythm of counting, - 36 -00:02:22,705 --> 00:02:25,160 -which you may or may not have thought about before. +00:02:17,460 --> 00:02:20,125 +10 that you've already counted up to so far, while freeing the ones place to reset to 0. 37 -00:02:26,400 --> 00:02:30,384 -Any description of binary typically starts off with an introspection +00:02:20,125 --> 00:02:21,832 +The rhythm of counting repeats like this, counting up 9, 38 -00:02:30,384 --> 00:02:34,079 -about our usual way to represent numbers, what we call base 10, +00:02:21,832 --> 00:02:23,420 +rolling over to the tens place, counting up 9 more, r 39 -00:02:34,079 --> 00:02:37,660 -since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. +00:02:23,420 --> 00:02:28,226 +olling over to the tens place, etc. Well, after repeating that process 9 times, 40 +00:02:28,226 --> 00:02:32,973 +you roll over to a hundreds place, a digit that keeps track of how many groups + +41 +00:02:32,973 --> 00:02:37,660 +of 100 you've hit, freeing up the other two digits to reset to 0. In this way, + +42 00:02:38,180 --> 00:02:42,600 The rhythm of counting begins by walking through all 10 of these digits. -41 +43 00:02:45,340 --> 00:02:49,668 Then, having run out of new digits, you express the next number, -42 +44 00:02:49,668 --> 00:02:51,400 10, with two digits, 1, 0. -43 +45 00:02:52,200 --> 00:02:56,709 You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 -44 +46 00:02:56,709 --> 00:03:01,220 that you've already counted up to so far, while freeing the ones place to reset to 0. -45 -00:03:02,079 --> 00:03:05,657 -The rhythm of counting repeats like this, counting up 9, - -46 -00:03:05,657 --> 00:03:11,180 -rolling over to the tens place, counting up 9 more, rolling over to the tens place, etc. - 47 -00:03:12,620 --> 00:03:18,148 -Until, after repeating that process 9 times, you roll over to a hundreds place, +00:03:02,080 --> 00:03:06,686 +t when you're counting, you have to roll over all the time. After counting 0, 1, 48 -00:03:18,148 --> 00:03:22,501 -a digit that keeps track of how many groups of 100 you've hit, +00:03:06,686 --> 00:03:11,180 +you've already run out of bits, so you need to roll over to a two's place, writ 49 -00:03:22,501 --> 00:03:25,680 -freeing up the other two digits to reset to 0. +00:03:12,620 --> 00:03:18,621 +ing 1-0, and resisting every urge in your base-10-trained brain to read this as 10, 50 -00:03:29,519 --> 00:03:33,240 -In this way, the rhythm of counting is kind of self-similar. +00:03:18,621 --> 00:03:24,408 +and instead understand it to mean 1 group of 2 plus 0. Then increment up to 1-1, 51 -00:03:33,820 --> 00:03:38,654 -Even if you zoom out to a larger scale, the process looks like doing something, +00:03:24,408 --> 00:03:28,766 +which represents 3, and already you have to roll over again, 52 -00:03:38,654 --> 00:03:41,736 -rolling over, doing that same thing, rolling over, +00:03:28,766 --> 00:03:34,053 +and since there's a 1 in that two's place, that has to roll over as well, 53 -00:03:41,736 --> 00:03:44,940 -and repeating 9 times before an even larger rollover. +00:03:34,053 --> 00:03:39,340 +giving you 1-0-0, which represents 1 group of 4 plus 0 groups of 2 plus 0. 54 -00:03:49,519 --> 00:03:54,296 -In binary, also known as base-2, you limit yourself to two digits, +00:03:39,340 --> 00:03:45,867 +In the same way that digits in base-10 represent powers of 10, 55 -00:03:54,296 --> 00:03:58,860 -0 and 1, commonly called bits, which is short for binary digits. +00:03:45,867 --> 00:03:55,193 +bits in base-2 represent different powers of 2. So instead of talking about a 10's place, 56 -00:03:59,640 --> 00:04:03,180 -The result is that when you're counting, you have to roll over all the time. +00:03:55,193 --> 00:04:04,518 +a 100's place, a 1000's place, things like that, you talk about a 2's place, a 4's place, 57 -00:04:03,740 --> 00:04:07,044 -After counting 01, you've already run out of bits, +00:04:04,518 --> 00:04:06,280 +and an 8's place. 58 -00:04:07,044 --> 00:04:10,542 -so you need to roll over to a twos place, writing 10, +00:04:06,280 --> 00:04:14,610 +Even if you zoom out to a larger scale, the process looks like doing something, 59 -00:04:10,542 --> 00:04:15,401 -and resisting every urge in your base-10 trained brain to read this as 10, +00:04:14,610 --> 00:04:19,921 +rolling over, doing that same thing, rolling over, 60 -00:04:15,401 --> 00:04:18,899 -and instead understand it to mean 1 group of 2 plus 0. +00:04:19,921 --> 00:04:25,440 +and repeating 9 times before an even larger rollover. 61 -00:04:19,899 --> 00:04:26,214 -Then increment up to 11, which represents 3, and already you have to roll over again, +00:04:25,440 --> 00:04:31,015 +ce. Flip the last, roll over twice. Flip the last, roll over once. Flip the last, 62 -00:04:26,214 --> 00:04:31,573 -and since there's a 1 in that twos place, that has to roll over as well, +00:04:31,015 --> 00:04:36,524 +roll over three times. Again, there's a certain self-similarity to this pattern. 63 -00:04:31,573 --> 00:04:36,860 -giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. +00:04:36,524 --> 00:04:42,644 +At every scale, the process is to do something, roll over, then do that same thing again. 64 -00:04:36,860 --> 00:04:41,490 -In the same way that digits in base-10 represent powers of 10, +00:04:42,644 --> 00:04:47,540 +At the small scale, say counting up to 3, which is 1-1 in binary, this m 65 -00:04:41,490 --> 00:04:47,076 -bits in base-2 represent different powers of 2, so instead of a tens place, +00:04:47,540 --> 00:04:55,920 +eans flip the last bit, roll over to the two's, then flip the last bit. At a larger scale 66 -00:04:47,076 --> 00:04:53,030 -a hundreds place, a thousands place, you talk about a twos place, a fours place, +00:04:55,920 --> 00:04:59,995 +After counting 01, you've already run out of bits, 67 -00:04:53,030 --> 00:04:54,500 -and an eights place. +00:04:59,995 --> 00:05:04,311 +so you need to roll over to a twos place, writing 10, 68 -00:04:55,820 --> 00:05:00,020 -The rhythm of counting is now a lot faster, but that almost makes it more noticeable. +00:05:04,311 --> 00:05:10,304 +and resisting every urge in your base-10 trained brain to read this as 10, 69 -00:05:07,400 --> 00:05:13,240 -Again, there's a certain self-similarity to this pattern. +00:05:10,304 --> 00:05:14,620 +and instead understand it to mean 1 group of 2 plus 0. 70 -00:05:13,920 --> 00:05:19,780 -At every scale, the process is to do something, roll over, then do that same thing again. +00:05:15,060 --> 00:05:19,691 +Then increment up to 11, which represents 3, and already you have to roll over again, 71 -00:05:22,360 --> 00:05:26,638 -At the small scale, say counting up to 3, which is 11 in binary, +00:05:19,691 --> 00:05:23,622 +and since there's a 1 in that twos place, that has to roll over as well, 72 -00:05:26,638 --> 00:05:31,640 -this means flip the last bit, roll over to the twos, then flip the last bit. +00:05:23,622 --> 00:05:27,500 +giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. 73 -00:05:32,960 --> 00:05:37,680 -At a larger scale, like counting up to 15, which is 1111 in binary, +00:05:27,500 --> 00:05:32,547 +tting the last 7 bits count up till they're full, rolling over to the 128's place, 74 -00:05:37,680 --> 00:05:41,011 -the process is to let the last 3 count up to 7, +00:05:32,547 --> 00:05:37,777 +then letting the last 7 bits count up again. Alright, so with that mini-introduction, 75 -00:05:41,011 --> 00:05:45,940 -roll over to the eights place, then let the last 3 bits count up again. +00:05:37,777 --> 00:05:42,946 +the surprising fact that Keith showed me is that we can use this rhythm to solve the 76 -00:05:46,960 --> 00:05:50,078 -Counting up to 255, which is 8 successive ones, +00:05:42,946 --> 00:05:43,920 +towers of Hanoi. 77 -00:05:50,078 --> 00:05:54,496 -this looks like letting the last 7 bits count up till they're full, +00:05:43,920 --> 00:05:51,920 +The rhythm of counting is now a lot faster, but that almost makes it more noticeable. 78 -00:05:54,496 --> 00:05:59,500 -rolling over to the 128th place, then letting the last 7 bits count up again. +00:05:51,920 --> 00:05:59,500 +Again, there's a certain self-similarity to this pattern. 79 -00:06:01,340 --> 00:06:05,264 -Alright, so with that mini-introduction, the surprising fact that Keith +00:06:01,340 --> 00:06:09,136 +ontinues like this. Flip the last, move disk 0. Flip the last two, move disk 1. 80 -00:06:05,264 --> 00:06:09,080 -showed me is that we can use this rhythm to solve the towers of Hanoi. +00:06:09,136 --> 00:06:17,128 +Flip the last, move disk 0. And here we're going to have to roll over three times 81 -00:06:10,380 --> 00:06:11,840 -You start by counting from 0. +00:06:17,128 --> 00:06:25,120 +to the 8's place, and that corresponds to moving disk 3. There's something magical 82 -00:06:12,660 --> 00:06:16,797 -Whenever you're only flipping that last bit, from a 0 to a 1, +00:06:25,120 --> 00:06:29,444 +At the small scale, say counting up to 3, which is 11 in binary, 83 -00:06:16,797 --> 00:06:19,000 -move disk 0 one peg to the right. +00:06:29,444 --> 00:06:34,500 +this means flip the last bit, roll over to the twos, then flip the last bit. 84 -00:06:22,020 --> 00:06:26,020 -If it was already on the right-most peg, you just loop it back to the first peg. +00:06:34,500 --> 00:06:39,169 +At a larger scale, like counting up to 15, which is 1111 in binary, 85 -00:06:28,800 --> 00:06:32,990 -If, in your binary counting, you roll over once to the twos place, +00:06:39,169 --> 00:06:42,464 +the process is to let the last 3 count up to 7, 86 -00:06:32,990 --> 00:06:36,680 -meaning you flip the last two bits, you move disk number 1. +00:06:42,464 --> 00:06:47,340 +roll over to the eights place, then let the last 3 bits count up again. 87 -00:06:37,620 --> 00:06:38,980 -Where do you move it, you might ask? +00:06:47,520 --> 00:06:49,270 +Counting up to 255, which is 8 successive ones, 88 -00:06:39,300 --> 00:06:40,400 -Well, you have no choice. +00:06:49,270 --> 00:06:51,751 +this looks like letting the last 7 bits count up till they're full, 89 -00:06:40,620 --> 00:06:43,821 -You can't put it on top of disk 0, and there's only one other peg, +00:06:51,751 --> 00:06:54,560 +rolling over to the 128th place, then letting the last 7 bits count up again. 90 -00:06:43,821 --> 00:06:46,020 -so you move it where you're forced to move it. +00:06:54,560 --> 00:06:57,490 +Alright, so with that mini-introduction, the surprising fact that Keith 91 -00:06:46,659 --> 00:06:50,676 -So after this, counting up to 1,1, that involves just flipping the last bit, +00:06:57,490 --> 00:07:00,340 +showed me is that we can use this rhythm to solve the towers of Hanoi. 92 -00:06:50,676 --> 00:06:51,980 -so you move disk 0 again. +00:07:00,340 --> 00:07:05,954 +i, but it does it in the most efficient way possible. 93 -00:06:52,640 --> 00:06:57,235 -Then when your binary counting rolls over twice to the fours place, +00:07:05,954 --> 00:07:11,880 +Understanding why this works and how it works and what th 94 -00:06:57,235 --> 00:07:01,020 -move disk number 2, and the pattern continues like this. +00:07:11,880 --> 00:07:16,284 +e heck is going on comes down to a certain perspective on the puzzle, 95 -00:07:01,320 --> 00:07:02,880 -Flip the last, move disk 0. +00:07:16,284 --> 00:07:21,821 +what CS folk might call a recursive perspective. Disk 3 is thinking, okay, 2, 1, and 0, 96 -00:07:03,260 --> 00:07:04,900 -Flip the last two, move disk 1. +00:07:21,821 --> 00:07:23,080 +like you have to get 97 -00:07:05,760 --> 00:07:07,200 -Flip the last, move disk 0. +00:07:23,080 --> 00:07:30,161 +off of me, like I can't really function under this much weight and pressure. 98 -00:07:07,980 --> 00:07:11,660 -And here, we're going to have to roll over three times to the eights place, +00:07:30,161 --> 00:07:33,380 +And so just from disk 3's perspecti 99 -00:07:11,660 --> 00:07:13,840 -and that corresponds to moving disk number 3. +00:07:33,380 --> 00:07:38,070 +If, in your binary counting, you roll over once to the twos place, 100 -00:07:14,800 --> 00:07:16,180 -There's something magical about it. +00:07:38,070 --> 00:07:42,200 +meaning you flip the last two bits, you move disk number 1. 101 -00:07:16,300 --> 00:07:17,920 -When I first saw this, I was like, this can't work. +00:07:42,480 --> 00:07:43,140 +Where do you move it, you might ask? 102 -00:07:18,540 --> 00:07:21,080 -I don't know how this works, I don't know why this works. +00:07:43,140 --> 00:07:45,220 +Well, you have no choice. 103 -00:07:21,200 --> 00:07:24,280 -Now I know, but it's just magical when you see it. +00:07:45,660 --> 00:07:50,645 +this disk to work, I can turn my bigger problem into something slightly smaller. 104 -00:07:24,400 --> 00:07:29,161 -I remember putting together an animation for this when I was teaching this, +00:07:50,645 --> 00:07:54,645 +And then how do I solve that? Well, it's exactly the same thing. 105 -00:07:29,161 --> 00:07:31,480 -and just like, I know how this works. +00:07:54,645 --> 00:07:58,891 +If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. 106 -00:07:31,620 --> 00:07:32,840 -I know all the things in it. +00:07:58,891 --> 00:08:02,400 +I just need some space. Get off. They need to move somewh 107 -00:07:32,900 --> 00:07:36,220 -It's still fun to just sit and just watch it play out. +00:08:02,400 --> 00:08:04,954 +ere. Then disk 2 can move to where it needs to go. Then disk 1 and 0 can do this. 108 -00:07:36,380 --> 00:07:36,640 -Oh yeah. +00:08:04,954 --> 00:08:07,197 +But the interesting point is that every single disk pretty much has the 109 -00:07:37,180 --> 00:07:41,220 -I mean, it's not even clear at first that this is always going to give legal moves. +00:08:07,197 --> 00:08:09,160 +exact same strategy. They all say, everybody above me, get off. 110 -00:07:41,640 --> 00:07:46,109 -For example, how do you know that every time you're rolling over to the eights place, +00:08:09,160 --> 00:08:13,763 +Then I'm going to move. Okay, everyone pile back on. When you know that insight, 111 -00:07:46,109 --> 00:07:49,020 -that disk 3 is necessarily going to be freed up to move? +00:08:13,763 --> 00:08:17,116 +you can code up something that will solve towers of Hanoi, 112 -00:07:49,740 --> 00:07:53,511 -At the same time, the solution just immediately raises these questions like, +00:08:17,116 --> 00:08:21,947 +like five or six lines of code, which probably has the highest ratio of intellectual 113 -00:07:53,511 --> 00:07:55,814 -where does this come from, why does this work, +00:08:21,947 --> 00:08:25,982 +investment to lines of code ever. And if you think about it for a bit, 114 -00:07:55,814 --> 00:07:59,880 -and is there a better way of doing this than having to do 2 to the n minus 1 steps? +00:08:25,982 --> 00:08:27,460 +it becomes clear that this 115 -00:08:00,520 --> 00:08:03,393 -It turns out, not only does this solve Towers of Hanoi, +00:08:27,460 --> 00:08:28,682 +has to be the most efficient solution. At every step, 116 -00:08:03,393 --> 00:08:05,960 -but it does it in the most efficient way possible. +00:08:28,682 --> 00:08:30,109 +you're only doing what's forced upon you. You have to get disk 117 -00:08:06,700 --> 00:08:11,094 -Understanding why this works and how it works and what the heck is going on comes down +00:08:30,109 --> 00:08:31,740 +0 through 2 off before you can move disk 3. And you have to move disk 3. 118 -00:08:11,094 --> 00:08:15,540 -to a certain perspective on the puzzle, what CS folk might call a recursive perspective. +00:08:31,740 --> 00:08:32,179 +Flip the last two, move disk 1. 119 -00:08:16,800 --> 00:08:20,280 -Disk 3 is thinking, okay, 2, 1, and 0, you have to get off of me. +00:08:32,200 --> 00:08:32,179 +Flip the last, move disk 0. 120 -00:08:20,420 --> 00:08:23,620 -I can't really function under this much weight and pressure. +00:08:32,200 --> 00:08:33,305 +And here, we're going to have to roll over three times to the eights place, 121 -00:08:24,900 --> 00:08:29,377 -And so just from disk 3's perspective, if you want to figure out how is disk 3 going +00:08:33,305 --> 00:08:33,960 +and that corresponds to moving disk number 3. 122 -00:08:29,377 --> 00:08:33,960 -to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. +00:08:34,020 --> 00:08:33,960 +There's something magical about it. 123 -00:08:34,020 --> 00:08:35,940 -That's the only way it can move. +00:08:34,020 --> 00:08:38,000 +There's just not any room for inefficiency from this perspective. 124 -00:08:36,080 --> 00:08:38,000 -If any of these disks are on top of 3, it can't move. +00:08:38,000 --> 00:08:41,940 +I don't know how this works, I don't know why this works. 125 -00:08:38,000 --> 00:08:40,600 -If any of them are in spindle C, it can't move there. +00:08:41,940 --> 00:08:46,500 +Now I know, but it's just magical when you see it. 126 -00:08:41,000 --> 00:08:43,059 -So somehow we have to get 2, 1, and 0 off. +00:08:46,500 --> 00:08:54,301 +I remember putting together an animation for this when I was teaching this, 127 -00:08:43,580 --> 00:08:47,600 -Having done that, then we can move disk 3 over there. +00:08:54,301 --> 00:08:58,100 +and just like, I know how this works. 128 -00:08:48,120 --> 00:08:49,380 -And then disk 3 says, I'm set. +00:08:58,100 --> 00:09:01,076 +ove disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. 129 -00:08:49,720 --> 00:08:51,280 -You never need to move me again. +00:09:01,076 --> 00:09:03,000 +Flip the last bit, roll over once, flip the last bit. 130 -00:08:51,740 --> 00:08:53,380 -Everyone else just figure out how to get here. +00:09:03,000 --> 00:09:09,300 +It's still fun to just sit and just watch it play out. 131 -00:08:53,980 --> 00:08:57,660 -And in a sense, you now have a smaller version of the same problem. +00:09:09,300 --> 00:09:10,556 +At a slightly larger scale, solving towers of 132 -00:08:57,740 --> 00:09:01,520 -Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C. +00:09:10,556 --> 00:09:11,840 +Hanoi for three disks looks like doing whatever 133 -00:09:02,080 --> 00:09:04,716 -So the idea is that if I just focus on one disk and I think +00:09:11,940 --> 00:09:22,140 +I mean, it's not even clear at first that this is always going to give legal moves. 134 -00:09:04,716 --> 00:09:07,440 -about what am I going to have to do to get this disk to work, +00:09:22,140 --> 00:09:24,713 +k number 2, then do whatever it takes to solve two disks again. 135 -00:09:07,440 --> 00:09:10,120 -I can turn my bigger problem into something slightly smaller. +00:09:24,713 --> 00:09:27,488 +Analogously, counting up to 111 in binary involves counting up to 3, 136 -00:09:10,340 --> 00:09:11,840 -And then how do I solve that? +00:09:27,488 --> 00:09:29,780 +rolling over all three bits, then counting up three more. 137 -00:09:11,940 --> 00:09:13,260 -Well, it's exactly the same thing. +00:09:29,780 --> 00:09:32,428 +At the same time, the solution just immediately raises these questions like, 138 -00:09:13,400 --> 00:09:17,160 -Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. +00:09:32,428 --> 00:09:34,045 +where does this come from, why does this work, 139 -00:09:17,220 --> 00:09:17,860 -I just need some space. +00:09:34,045 --> 00:09:36,900 +and is there a better way of doing this than having to do 2 to the n minus 1 steps? 140 -00:09:17,960 --> 00:09:18,380 -Get off. +00:09:36,900 --> 00:09:37,618 +It turns out, not only does this solve Towers of Hanoi, 141 -00:09:18,820 --> 00:09:20,120 -They need to move somewhere. +00:09:37,618 --> 00:09:38,260 +but it does it in the most efficient way possible. 142 -00:09:20,340 --> 00:09:22,940 -Then disk 2 can move to where it needs to go. +00:09:38,260 --> 00:09:44,036 +Understanding why this works and how it works and what the heck is going on comes down 143 -00:09:23,020 --> 00:09:24,800 -Then disk 1 and 0 can do this. +00:09:44,036 --> 00:09:49,880 +to a certain perspective on the puzzle, what CS folk might call a recursive perspective. 144 -00:09:25,020 --> 00:09:28,044 -But the interesting point is that every single +00:09:50,420 --> 00:09:52,575 +give me this thing, you're effectively reversing the recursive algorithm for towers of 145 -00:09:28,044 --> 00:09:30,940 -disk pretty much has the exact same strategy. +00:09:52,575 --> 00:09:54,780 +Hanoi, which is why it works out. That's pretty cool, right? But it actually gets cooler. 146 -00:09:31,020 --> 00:09:32,800 -They all say, everybody above me, get off. +00:09:54,780 --> 00:10:03,820 +I can't really function under this much weight and pressure. 147 -00:09:32,800 --> 00:09:35,340 -Then I'm going to move, OK, everyone pile back on. +00:10:04,340 --> 00:10:06,791 +And so just from disk 3's perspective, if you want to figure out how is disk 3 going 148 -00:09:36,320 --> 00:09:42,074 -When you know that insight, you can code up something that will solve Towers of Hanoi, +00:10:06,791 --> 00:10:09,300 +to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. 149 -00:09:42,074 --> 00:09:46,638 -like five or six lines of code, which probably has the highest ratio +00:10:09,880 --> 00:10:11,820 +That's the only way it can move. 150 -00:09:46,638 --> 00:09:49,880 -of intellectual investment to lines of code ever. +00:10:11,820 --> 00:10:13,799 +access to these videos before I publish the full series in a few months. 151 -00:09:50,420 --> 00:09:53,168 -And if you think about it for a bit, it becomes +00:10:13,799 --> 00:10:15,480 +This video and the next one are also supported in part by Desm 152 -00:09:53,168 --> 00:09:56,260 -clear that this has to be the most efficient solution. +00:10:15,480 --> 00:10:21,580 +If any of them are in spindle C, it can't move there. 153 -00:09:56,760 --> 00:09:59,480 -At every step, you're only doing what's forced upon you. +00:10:21,580 --> 00:10:22,880 +So somehow we have to get 2, 1, and 0 off. 154 -00:09:59,920 --> 00:10:03,820 -You have to get disk 0 through 2 off before you can move disk 3. +00:10:23,200 --> 00:10:23,420 +Having done that, then we can move disk 3 over there. 155 -00:10:04,340 --> 00:10:05,900 -And you have to move disk 3. +00:10:23,420 --> 00:10:27,040 +And then disk 3 says, I'm set. 156 -00:10:06,460 --> 00:10:09,300 -And then you have to move disk 0 through 2 back onto it. +00:10:27,040 --> 00:10:29,208 +impressed by just how well-thought-out these activities are from a pedagogical standpoint. 157 -00:10:09,880 --> 00:10:13,640 -There's just not any room for inefficiency from this perspective. +00:10:29,208 --> 00:10:31,280 + The team clearly knows their stuff, and they know where they stand to make a differen 158 -00:10:15,200 --> 00:10:18,400 -So why does counting in binary capture this algorithm? +00:10:31,280 --> 00:10:33,480 +Everyone else just figure out how to get here. 159 -00:10:19,460 --> 00:10:23,401 -Well, what's going on here is that this pattern of solving a subproblem, +00:10:35,140 --> 00:10:41,040 +And in a sense, you now have a smaller version of the same problem. 160 -00:10:23,401 --> 00:10:26,208 -moving a big disk, then solving a subproblem again, +00:10:41,040 --> 00:10:46,880 +Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C. 161 -00:10:26,208 --> 00:10:29,340 -is perfectly paralleled by the pattern of binary counting. +00:10:46,880 --> 00:10:49,207 +So the idea is that if I just focus on one disk and I think 162 -00:10:30,080 --> 00:10:33,480 -Count up some amount, roll over, count up to that same amount again. +00:10:49,207 --> 00:10:51,613 +about what am I going to have to do to get this disk to work, 163 -00:10:35,140 --> 00:10:39,998 -And this Towers of Hanoi algorithm and binary counting are both self-similar processes, +00:10:51,613 --> 00:10:53,980 +I can turn my bigger problem into something slightly smaller. 164 -00:10:39,998 --> 00:10:43,917 -in the sense that if you zoom out and count up to a larger power of 2, +00:10:53,980 --> 00:11:03,378 +and students, you can check out the careers page that I've linked in the description. 165 -00:10:43,917 --> 00:10:48,500 -or solve Towers of Hanoi with more disks, they both still have that same structure. +00:11:03,378 --> 00:11:06,220 +Personally, I think they'r 166 -00:10:49,040 --> 00:10:51,160 -Subproblem, do a thing, subproblem. +00:11:06,220 --> 00:11:06,540 +Well, it's exactly the same thing. 167 -00:10:52,480 --> 00:10:57,593 -For example, at a pretty small scale, solving Towers of Hanoi for two disks, +00:11:07,380 --> 00:11:07,700 +Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. 168 -00:10:57,593 --> 00:11:03,040 -move disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. +00:11:07,700 --> 00:11:13,380 +I just need some space. 169 -00:11:03,680 --> 00:11:06,540 -Flip the last bit, roll over once, flip the last bit. +00:11:13,380 --> 00:11:16,720 +Get off. 170 -00:11:07,380 --> 00:11:10,850 -At a slightly larger scale, solving Towers of Hanoi for three +00:11:16,720 --> 00:11:17,360 +They need to move somewhere. 171 -00:11:10,850 --> 00:11:14,265 -disks looks like doing whatever it takes to solve two disks, +00:11:17,360 --> 00:11:17,360 +Then disk 2 can move to where it needs to go. 172 -00:11:14,265 --> 00:11:18,240 -move disk number 2, then do whatever it takes to solve two disks again. +00:11:17,360 --> 00:11:25,220 +Then disk 1 and 0 can do this. 173 -00:11:19,100 --> 00:11:23,449 -Analogously, counting up to 111 in binary involves counting up to 3, +00:11:25,220 --> 00:11:28,356 +But the interesting point is that every single 174 -00:11:23,449 --> 00:11:26,980 -rolling over all three bits, and counting up three more. +00:11:28,356 --> 00:11:31,360 +disk pretty much has the exact same strategy. 175 -00:11:27,600 --> 00:11:31,360 -At all scales, both processes have this same breakdown. +00:11:31,820 --> 00:11:37,500 +They all say, everybody above me, get off. 176 -00:11:31,820 --> 00:11:36,176 -So in a sense, the reason that this binary solution works, or at least an explanation, +00:11:37,660 --> 00:11:40,500 +Then I'm going to move, OK, everyone pile back on. 177 -00:11:36,176 --> 00:11:40,232 -I feel like there's no one explanation, but I think the most natural one is that +00:11:40,500 --> 00:11:43,411 +When you know that insight, you can code up something that will solve Towers of Hanoi, 178 -00:11:40,232 --> 00:11:44,238 -the pattern you would use to generate these binary numbers has exactly the same +00:11:43,411 --> 00:11:45,720 +like five or six lines of code, which probably has the highest ratio 179 -00:11:44,238 --> 00:11:47,243 -structure as the pattern you would use for Towers of Hanoi, +00:11:45,720 --> 00:11:47,360 +of intellectual investment to lines of code ever. 180 -00:11:47,243 --> 00:11:51,700 -which is why if you look at the bits flipping, you're effectively reversing this process. +00:11:47,360 --> 00:11:52,837 +And if you think about it for a bit, it becomes 181 -00:11:51,820 --> 00:11:54,020 -You're saying, what process generated these? +00:11:52,837 --> 00:11:59,000 +clear that this has to be the most efficient solution. 182 -00:11:54,020 --> 00:11:59,153 -If I were trying to understand how these bits were flipped to give me this thing, +00:11:59,000 --> 00:12:04,400 +At every step, you're only doing what's forced upon you. 183 -00:11:59,153 --> 00:12:04,412 -you're effectively reverse engineering the recursive algorithm for Towers of Hanoi, +00:12:04,400 --> 00:12:04,960 +You have to get disk 0 through 2 off before you can move disk 3. 184 -00:12:04,412 --> 00:12:06,040 -which is why it works out. +00:12:04,960 --> 00:12:11,720 +And you have to move disk 3. 185 -00:12:07,620 --> 00:12:09,000 -That's pretty cool, right? +00:12:11,720 --> 00:12:12,740 +And then you have to move disk 0 through 2 back onto it. 186 -00:12:09,420 --> 00:12:10,740 -But it actually gets cooler. +00:12:12,740 --> 00:12:12,740 +There's just not any room for inefficiency from this perspective. 187 -00:12:10,960 --> 00:12:13,640 -I haven't even gotten to how this relates to Sierpinski's triangle. +00:12:12,740 --> 00:12:14,760 +So why does counting in binary capture this algorithm? 188 -00:12:14,260 --> 00:12:17,780 -And that's exactly what I'm going to do in the follow-on video, part 2. +00:12:14,760 --> 00:12:19,179 +Well, what's going on here is that this pattern of solving a subproblem, 189 -00:12:18,820 --> 00:12:21,860 -Many thanks to everybody who's supporting these videos on Patreon. +00:12:19,179 --> 00:12:22,328 +moving a big disk, then solving a subproblem again, 190 -00:12:21,860 --> 00:12:24,862 -I just finished the first chapter of Essence of Calculus, +00:12:22,328 --> 00:12:25,840 +is perfectly paralleled by the pattern of binary counting. 191 -00:12:24,862 --> 00:12:28,588 -and I'm working on the second one right now, and Patreon supporters are +00:12:25,840 --> 00:12:29,800 +Count up some amount, roll over, count up to that same amount again. 192 -00:12:28,588 --> 00:12:33,040 -getting early access to these videos before I publish the full series in a few months. +00:12:29,800 --> 00:12:34,992 +And this Towers of Hanoi algorithm and binary counting are both self-similar processes, 193 -00:12:34,500 --> 00:12:37,569 -This video and the next one are also supported in part by Desmos, +00:12:34,992 --> 00:12:39,182 +in the sense that if you zoom out and count up to a larger power of 2, 194 -00:12:37,569 --> 00:12:40,824 -and before the next video I just want to take a moment and share with +00:12:39,182 --> 00:12:44,080 +or solve Towers of Hanoi with more disks, they both still have that same structure. 195 -00:12:40,824 --> 00:12:44,080 -you guys a little about who they are and the fact that they're hiring. +00:12:44,740 --> 00:12:50,880 +Subproblem, do a thing, subproblem. 196 -00:12:44,740 --> 00:12:46,480 -So Desmos is actually really cool. +00:12:51,480 --> 00:12:57,233 +For example, at a pretty small scale, solving Towers of Hanoi for two disks, 197 -00:12:46,880 --> 00:12:49,209 -They make a lot of these interactive math activities +00:12:57,233 --> 00:13:03,360 +move disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. 198 -00:12:49,209 --> 00:12:50,880 -for classrooms and tools for teachers. +00:13:03,360 --> 00:13:04,940 +Flip the last bit, roll over once, flip the last bit. 199 -00:12:51,480 --> 00:12:55,180 -The real meat of their offering is in their classroom activities. +00:13:04,940 --> 00:13:09,343 +At a slightly larger scale, solving Towers of Hanoi for three 200 -00:12:56,040 --> 00:12:59,211 -For my part, I'm super impressed by just how well-thought-out +00:13:09,343 --> 00:13:13,676 +disks looks like doing whatever it takes to solve two disks, 201 -00:12:59,211 --> 00:13:01,820 -these activities are from a pedagogical standpoint. +00:13:13,676 --> 00:13:18,720 +move disk number 2, then do whatever it takes to solve two disks again. 202 -00:13:02,500 --> 00:13:04,970 -The team clearly knows their stuff, and they know where they +00:13:19,220 --> 00:13:21,163 +Analogously, counting up to 111 in binary involves counting up to 3, 203 -00:13:04,970 --> 00:13:07,400 -stand to make a difference in students' and teachers' lives. +00:13:21,163 --> 00:13:22,740 +rolling over all three bits, and counting up three more. 204 -00:13:08,080 --> 00:13:09,500 -And like I said, they're hiring. +00:13:22,740 --> 00:13:23,500 +At all scales, both processes have this same breakdown. 205 -00:13:10,060 --> 00:13:14,515 -They are always looking to bring in more good talent, whether that's engineering talent, +00:13:23,500 --> 00:13:26,655 +So in a sense, the reason that this binary solution works, or at least an explanation, 206 -00:13:14,515 --> 00:13:18,720 -designers, teachers, or whatever other skill sets line up with what they want to do. +00:13:26,655 --> 00:13:29,593 +I feel like there's no one explanation, but I think the most natural one is that 207 -00:13:19,220 --> 00:13:21,599 -If any of you out there are interested in joining them, +00:13:29,593 --> 00:13:32,495 +the pattern you would use to generate these binary numbers has exactly the same 208 -00:13:21,599 --> 00:13:24,615 -helping them make some of these great tools for teachers and students, +00:13:32,495 --> 00:13:34,671 +structure as the pattern you would use for Towers of Hanoi, 209 -00:13:24,615 --> 00:13:27,420 -you can check out the careers page I've linked in the description. +00:13:34,671 --> 00:13:37,900 +which is why if you look at the bits flipping, you're effectively reversing this process. 210 -00:13:28,040 --> 00:13:30,400 -Personally, I think they're doing some really meaningful stuff. +00:13:37,900 --> 00:13:39,100 +You're saying, what process generated these? 211 -00:13:30,500 --> 00:13:34,558 -I think their activities are building genuinely good math intuitions for students, +00:13:39,100 --> 00:13:47,171 +If I were trying to understand how these bits were flipped to give me this thing, 212 -00:13:34,558 --> 00:13:37,492 -and the world could use a few more talented people pointing +00:13:47,171 --> 00:13:55,440 +you're effectively reverse engineering the recursive algorithm for Towers of Hanoi, 213 -00:13:37,492 --> 00:13:39,840 -their efforts towards education the way they do. +00:13:55,440 --> 00:13:58,000 +which is why it works out. 214 -00:13:41,260 --> 00:13:49,078 -Alright so with that, I'll see you guys next video, +00:13:58,000 --> 00:13:58,099 +That's pretty cool, right? 215 -00:13:49,078 --> 00:13:58,100 +00:13:58,099 --> 00:13:58,099 +But it actually gets cooler. + +216 +00:13:58,099 --> 00:13:58,099 +I haven't even gotten to how this relates to Sierpinski's triangle. + +217 +00:13:58,099 --> 00:13:58,100 +And that's exactly what I'm going to do in the follow-on video, part 2. + +218 +00:13:58,100 --> 00:13:58,100 +Many thanks to everybody who's supporting these videos on Patreon. + +219 +00:13:58,100 --> 00:13:58,100 +I just finished the first chapter of Essence of Calculus, + +220 +00:13:58,100 --> 00:13:58,100 +and I'm working on the second one right now, and Patreon supporters are + +221 +00:13:58,100 --> 00:13:58,100 +getting early access to these videos before I publish the full series in a few months. + +222 +00:13:58,100 --> 00:13:58,100 +This video and the next one are also supported in part by Desmos, + +223 +00:13:58,100 --> 00:13:58,100 +and before the next video I just want to take a moment and share with + +224 +00:13:58,100 --> 00:13:58,100 +you guys a little about who they are and the fact that they're hiring. + +225 +00:13:58,100 --> 00:13:58,100 +So Desmos is actually really cool. + +226 +00:13:58,100 --> 00:13:58,100 +They make a lot of these interactive math activities + +227 +00:13:58,100 --> 00:13:58,100 +for classrooms and tools for teachers. + +228 +00:13:58,100 --> 00:13:58,100 +The real meat of their offering is in their classroom activities. + +229 +00:13:58,100 --> 00:13:58,100 +For my part, I'm super impressed by just how well-thought-out + +230 +00:13:58,100 --> 00:13:58,100 +these activities are from a pedagogical standpoint. + +231 +00:13:58,100 --> 00:13:58,100 +The team clearly knows their stuff, and they know where they + +232 +00:13:58,100 --> 00:13:58,100 +stand to make a difference in students' and teachers' lives. + +233 +00:13:58,100 --> 00:13:58,100 +And like I said, they're hiring. + +234 +00:13:58,100 --> 00:13:58,100 +They are always looking to bring in more good talent, whether that's engineering talent, + +235 +00:13:58,100 --> 00:13:58,100 +designers, teachers, or whatever other skill sets line up with what they want to do. + +236 +00:13:58,100 --> 00:13:58,100 +If any of you out there are interested in joining them, + +237 +00:13:58,100 --> 00:13:58,100 +helping them make some of these great tools for teachers and students, + +238 +00:13:58,100 --> 00:13:58,100 +you can check out the careers page I've linked in the description. + +239 +00:13:58,100 --> 00:13:58,100 +Personally, I think they're doing some really meaningful stuff. + +240 +00:13:58,100 --> 00:13:58,100 +I think their activities are building genuinely good math intuitions for students, + +241 +00:13:58,100 --> 00:13:58,100 +and the world could use a few more talented people pointing + +242 +00:13:58,100 --> 00:13:58,100 +their efforts towards education the way they do. + +243 +00:13:58,100 --> 00:13:58,100 +Alright so with that, I'll see you guys next video, + +244 +00:13:58,100 --> 00:13:58,100 and I think you're really going to like where this is going. diff --git a/2016/hanoi-and-sierpinski/english/sentence_timings.json b/2016/hanoi-and-sierpinski/english/sentence_timings.json index 12f639d6b..0a546dd39 100644 --- a/2016/hanoi-and-sierpinski/english/sentence_timings.json +++ b/2016/hanoi-and-sierpinski/english/sentence_timings.json @@ -40,27 +40,27 @@ 66.76 ], [ - "So they all start up stacked up from biggest to smallest on one spindle, and the goal is to move the entire tower from one spindle to another.", + "For example, your first move must involve moving disk 0, since any other disk has stuff on top of it that needs to get out of the way before it can move.", 67.46, 75.88 ], [ - "The rule is you can only move one disk at a time, and you can't move a bigger disk on top of a smaller disk.", + "After that, you can move disk 1, but it has to go on whatever peg doesn't currently have disk 0, since otherwise you'd be putting a bigger disk o", 75.88, - 82.06 + 81.08 ], [ - "For example, your first move must involve moving disk 0, since any other disk has stuff on top of it that needs to get out of the way before it can move.", - 83.72, + "n a smaller one, which isn't allowed. If you've never seen this before, I highly encourage you to pause and pull out some books of varying sizes and try it out for yourself. Now Keith showed me something truly surprising about this puzzle, wh", + 81.08, 92.5 ], [ - "After that, you can move disk 1, but it has to go on whatever peg doesn't currently have disk 0, since otherwise you'd be putting a bigger disk on a smaller one, which isn't allowed.", + "ich is that you can solve it just by counting up in binary and associating the rhythm of that counting with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. Actually, eve", 93.08, 103.72 ], [ - "If you've never seen this before, I highly encourage you to pause and pull out some books of varying sizes and try it out for yourself.", + "n if you're familiar with binary, I want to explain it with a focus on the rhythm of counting, which you may or may not have thought about before. Any", 104.58, 111.92 ], @@ -80,13 +80,13 @@ 136.82 ], [ - "Actually, even if you are familiar with binary, I want to explain it with a focus on the rhythm of counting, which you may or may not have thought about before.", + "10 that you've already counted up to so far, while freeing the ones place to reset to 0. The rhythm of counting repeats like this, counting up 9, rolling over to the tens place, counting up 9 more, r", 137.46, - 145.16 + 143.42 ], [ - "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", - 146.4, + "olling over to the tens place, etc. Well, after repeating that process 9 times, you roll over to a hundreds place, a digit that keeps track of how many groups of 100 you've hit, freeing up the other two digits to reset to 0. In this way,", + 143.42, 157.66 ], [ @@ -105,528 +105,528 @@ 181.22 ], [ - "The rhythm of counting repeats like this, counting up 9, rolling over to the tens place, counting up 9 more, rolling over to the tens place, etc.", + "t when you're counting, you have to roll over all the time. After counting 0, 1, you've already run out of bits, so you need to roll over to a two's place, writ", 182.08, 191.18 ], [ - "Until, after repeating that process 9 times, you roll over to a hundreds place, a digit that keeps track of how many groups of 100 you've hit, freeing up the other two digits to reset to 0.", + "ing 1-0, and resisting every urge in your base-10-trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then increment up to 1-1, which represents 3, and already you have to roll over again, and since there's a 1 in that two's place, that has to roll over as well, giving you 1-0-0, which represents 1 group of 4 plus 0 groups of 2 plus 0.", 192.62, - 205.68 + 219.34 ], [ - "In this way, the rhythm of counting is kind of self-similar.", - 209.52, - 213.24 + "In the same way that digits in base-10 represent powers of 10, bits in base-2 represent different powers of 2. So instead of talking about a 10's place, a 100's place, a 1000's place, things like that, you talk about a 2's place, a 4's place, and an 8's place.", + 219.34, + 246.28 ], [ "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", - 213.82, - 224.94 + 246.28, + 265.44 ], [ - "In binary, also known as base-2, you limit yourself to two digits, 0 and 1, commonly called bits, which is short for binary digits.", - 229.52, - 238.86 + "ce. Flip the last, roll over twice. Flip the last, roll over once. Flip the last, roll over three times. Again, there's a certain self-similarity to this pattern. At every scale, the process is to do something, roll over, then do that same thing again. At the small scale, say counting up to 3, which is 1-1 in binary, this m", + 265.44, + 287.54 ], [ - "The result is that when you're counting, you have to roll over all the time.", - 239.64, - 243.18 + "eans flip the last bit, roll over to the two's, then flip the last bit. At a larger scale", + 287.54, + 295.92 ], [ "After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0.", - 243.74, - 258.9 + 295.92, + 314.62 ], [ "Then increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0.", - 259.9, - 276.86 + 315.06, + 327.5 ], [ - "In the same way that digits in base-10 represent powers of 10, bits in base-2 represent different powers of 2, so instead of a tens place, a hundreds place, a thousands place, you talk about a twos place, a fours place, and an eights place.", - 276.86, - 294.5 + "tting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + 327.5, + 343.92 ], [ "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", - 295.82, - 300.02 + 343.92, + 351.92 ], [ "Again, there's a certain self-similarity to this pattern.", - 307.4, - 313.24 + 351.92, + 359.5 ], [ - "At every scale, the process is to do something, roll over, then do that same thing again.", - 313.92, - 319.78 + "ontinues like this. Flip the last, move disk 0. Flip the last two, move disk 1. Flip the last, move disk 0. And here we're going to have to roll over three times to the 8's place, and that corresponds to moving disk 3. There's something magical", + 361.34, + 385.12 ], [ "At the small scale, say counting up to 3, which is 11 in binary, this means flip the last bit, roll over to the twos, then flip the last bit.", - 322.36, - 331.64 + 385.12, + 394.5 ], [ "At a larger scale, like counting up to 15, which is 1111 in binary, the process is to let the last 3 count up to 7, roll over to the eights place, then let the last 3 bits count up again.", - 332.96, - 345.94 + 394.5, + 407.34 ], [ "Counting up to 255, which is 8 successive ones, this looks like letting the last 7 bits count up till they're full, rolling over to the 128th place, then letting the last 7 bits count up again.", - 346.96, - 359.5 + 407.52, + 414.56 ], [ "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", - 361.34, - 369.08 + 414.56, + 420.34 ], [ - "You start by counting from 0.", - 370.38, - 371.84 + "i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th", + 420.34, + 431.88 ], [ - "Whenever you're only flipping that last bit, from a 0 to a 1, move disk 0 one peg to the right.", - 372.66, - 379.0 + "e heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective. Disk 3 is thinking, okay, 2, 1, and 0, like you have to get", + 431.88, + 443.08 ], [ - "If it was already on the right-most peg, you just loop it back to the first peg.", - 382.02, - 386.02 + "off of me, like I can't really function under this much weight and pressure. And so just from disk 3's perspecti", + 443.08, + 453.38 ], [ "If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1.", - 388.8, - 396.68 + 453.38, + 462.2 ], [ "Where do you move it, you might ask?", - 397.62, - 398.98 + 462.48, + 463.14 ], [ "Well, you have no choice.", - 399.3, - 400.4 + 463.14, + 465.22 ], [ - "You can't put it on top of disk 0, and there's only one other peg, so you move it where you're forced to move it.", - 400.62, - 406.02 + "this disk to work, I can turn my bigger problem into something slightly smaller. And then how do I solve that? Well, it's exactly the same thing. If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. I just need some space. Get off. They need to move somewh", + 465.66, + 482.4 ], [ - "So after this, counting up to 1,1, that involves just flipping the last bit, so you move disk 0 again.", - 406.66, - 411.98 + "ere. Then disk 2 can move to where it needs to go. Then disk 1 and 0 can do this. But the interesting point is that every single disk pretty much has the exact same strategy. They all say, everybody above me, get off.", + 482.4, + 489.16 ], [ - "Then when your binary counting rolls over twice to the fours place, move disk number 2, and the pattern continues like this.", - 412.64, - 421.02 + "Then I'm going to move. Okay, everyone pile back on. When you know that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which probably has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear that this", + 489.16, + 507.46 ], [ - "Flip the last, move disk 0.", - 421.32, - 422.88 + "has to be the most efficient solution. At every step, you're only doing what's forced upon you. You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3.", + 507.46, + 511.74 ], [ "Flip the last two, move disk 1.", - 423.26, - 424.9 + 511.74, + 512.18 ], [ "Flip the last, move disk 0.", - 425.76, - 427.2 + 512.2, + 512.18 ], [ "And here, we're going to have to roll over three times to the eights place, and that corresponds to moving disk number 3.", - 427.98, - 433.84 + 512.2, + 513.96 ], [ "There's something magical about it.", - 434.8, - 436.18 + 514.02, + 513.96 ], [ - "When I first saw this, I was like, this can't work.", - 436.3, - 437.92 + "There's just not any room for inefficiency from this perspective.", + 514.02, + 518.0 ], [ "I don't know how this works, I don't know why this works.", - 438.54, - 441.08 + 518.0, + 521.94 ], [ "Now I know, but it's just magical when you see it.", - 441.2, - 444.28 + 521.94, + 526.5 ], [ "I remember putting together an animation for this when I was teaching this, and just like, I know how this works.", - 444.4, - 451.48 + 526.5, + 538.1 ], [ - "I know all the things in it.", - 451.62, - 452.84 + "ove disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. Flip the last bit, roll over once, flip the last bit.", + 538.1, + 543.0 ], [ "It's still fun to just sit and just watch it play out.", - 452.9, - 456.22 + 543.0, + 549.3 ], [ - "Oh yeah.", - 456.38, - 456.64 + "At a slightly larger scale, solving towers of Hanoi for three disks looks like doing whatever", + 549.3, + 551.84 ], [ "I mean, it's not even clear at first that this is always going to give legal moves.", - 457.18, - 461.22 + 551.94, + 562.14 ], [ - "For example, how do you know that every time you're rolling over to the eights place, that disk 3 is necessarily going to be freed up to move?", - 461.64, - 469.02 + "k number 2, then do whatever it takes to solve two disks again. Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + 562.14, + 569.78 ], [ "At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than having to do 2 to the n minus 1 steps?", - 469.74, - 479.88 + 569.78, + 576.9 ], [ "It turns out, not only does this solve Towers of Hanoi, but it does it in the most efficient way possible.", - 480.52, - 485.96 + 576.9, + 578.26 ], [ "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", - 486.7, - 495.54 + 578.26, + 589.88 ], [ - "Disk 3 is thinking, okay, 2, 1, and 0, you have to get off of me.", - 496.8, - 500.28 + "give me this thing, you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out. That's pretty cool, right? But it actually gets cooler.", + 590.42, + 594.78 ], [ "I can't really function under this much weight and pressure.", - 500.42, - 503.62 + 594.78, + 603.82 ], [ "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B.", - 504.9, - 513.96 + 604.34, + 609.3 ], [ "That's the only way it can move.", - 514.02, - 515.94 + 609.88, + 611.82 ], [ - "If any of these disks are on top of 3, it can't move.", - 516.08, - 518.0 + "access to these videos before I publish the full series in a few months. This video and the next one are also supported in part by Desm", + 611.82, + 615.48 ], [ "If any of them are in spindle C, it can't move there.", - 518.0, - 520.6 + 615.48, + 621.58 ], [ "So somehow we have to get 2, 1, and 0 off.", - 521.0, - 523.06 + 621.58, + 622.88 ], [ "Having done that, then we can move disk 3 over there.", - 523.58, - 527.6 + 623.2, + 623.42 ], [ "And then disk 3 says, I'm set.", - 528.12, - 529.38 + 623.42, + 627.04 ], [ - "You never need to move me again.", - 529.72, - 531.28 + "impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen", + 627.04, + 631.28 ], [ "Everyone else just figure out how to get here.", - 531.74, - 533.38 + 631.28, + 633.48 ], [ "And in a sense, you now have a smaller version of the same problem.", - 533.98, - 537.66 + 635.14, + 641.04 ], [ "Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", - 537.74, - 541.52 + 641.04, + 646.88 ], [ "So the idea is that if I just focus on one disk and I think about what am I going to have to do to get this disk to work, I can turn my bigger problem into something slightly smaller.", - 542.08, - 550.12 + 646.88, + 653.98 ], [ - "And then how do I solve that?", - 550.34, - 551.84 + "and students, you can check out the careers page that I've linked in the description. Personally, I think they'r", + 653.98, + 666.22 ], [ "Well, it's exactly the same thing.", - 551.94, - 553.26 + 666.22, + 666.54 ], [ "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me.", - 553.4, - 557.16 + 667.38, + 667.7 ], [ "I just need some space.", - 557.22, - 557.86 + 667.7, + 673.38 ], [ "Get off.", - 557.96, - 558.38 + 673.38, + 676.72 ], [ "They need to move somewhere.", - 558.82, - 560.12 + 676.72, + 677.36 ], [ "Then disk 2 can move to where it needs to go.", - 560.34, - 562.94 + 677.36, + 677.36 ], [ "Then disk 1 and 0 can do this.", - 563.02, - 564.8 + 677.36, + 685.22 ], [ "But the interesting point is that every single disk pretty much has the exact same strategy.", - 565.02, - 570.94 + 685.22, + 691.36 ], [ "They all say, everybody above me, get off.", - 571.02, - 572.8 + 691.82, + 697.5 ], [ "Then I'm going to move, OK, everyone pile back on.", - 572.8, - 575.34 + 697.66, + 700.5 ], [ "When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ratio of intellectual investment to lines of code ever.", - 576.32, - 589.88 + 700.5, + 707.36 ], [ "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", - 590.42, - 596.26 + 707.36, + 719.0 ], [ "At every step, you're only doing what's forced upon you.", - 596.76, - 599.48 + 719.0, + 724.4 ], [ "You have to get disk 0 through 2 off before you can move disk 3.", - 599.92, - 603.82 + 724.4, + 724.96 ], [ "And you have to move disk 3.", - 604.34, - 605.9 + 724.96, + 731.72 ], [ "And then you have to move disk 0 through 2 back onto it.", - 606.46, - 609.3 + 731.72, + 732.74 ], [ "There's just not any room for inefficiency from this perspective.", - 609.88, - 613.64 + 732.74, + 732.74 ], [ "So why does counting in binary capture this algorithm?", - 615.2, - 618.4 + 732.74, + 734.76 ], [ "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", - 619.46, - 629.34 + 734.76, + 745.84 ], [ "Count up some amount, roll over, count up to that same amount again.", - 630.08, - 633.48 + 745.84, + 749.8 ], [ "And this Towers of Hanoi algorithm and binary counting are both self-similar processes, in the sense that if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with more disks, they both still have that same structure.", - 635.14, - 648.5 + 749.8, + 764.08 ], [ "Subproblem, do a thing, subproblem.", - 649.04, - 651.16 + 764.74, + 770.88 ], [ "For example, at a pretty small scale, solving Towers of Hanoi for two disks, move disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary.", - 652.48, - 663.04 + 771.48, + 783.36 ], [ "Flip the last bit, roll over once, flip the last bit.", - 663.68, - 666.54 + 783.36, + 784.94 ], [ "At a slightly larger scale, solving Towers of Hanoi for three disks looks like doing whatever it takes to solve two disks, move disk number 2, then do whatever it takes to solve two disks again.", - 667.38, - 678.24 + 784.94, + 798.72 ], [ "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, and counting up three more.", - 679.1, - 686.98 + 799.22, + 802.74 ], [ "At all scales, both processes have this same breakdown.", - 687.6, - 691.36 + 802.74, + 803.5 ], [ "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for Towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", - 691.82, - 711.7 + 803.5, + 817.9 ], [ "You're saying, what process generated these?", - 711.82, - 714.02 + 817.9, + 819.1 ], [ "If I were trying to understand how these bits were flipped to give me this thing, you're effectively reverse engineering the recursive algorithm for Towers of Hanoi, which is why it works out.", - 714.02, - 726.04 + 819.1, + 838.0 ], [ "That's pretty cool, right?", - 727.62, - 729.0 + 838.0, + 838.1 ], [ "But it actually gets cooler.", - 729.42, - 730.74 + 838.1, + 838.1 ], [ "I haven't even gotten to how this relates to Sierpinski's triangle.", - 730.96, - 733.64 + 838.1, + 838.1 ], [ "And that's exactly what I'm going to do in the follow-on video, part 2.", - 734.26, - 737.78 + 838.1, + 838.1 ], [ "Many thanks to everybody who's supporting these videos on Patreon.", - 738.82, - 741.86 + 838.1, + 838.1 ], [ "I just finished the first chapter of Essence of Calculus, and I'm working on the second one right now, and Patreon supporters are getting early access to these videos before I publish the full series in a few months.", - 741.86, - 753.04 + 838.1, + 838.1 ], [ "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", - 754.5, - 764.08 + 838.1, + 838.1 ], [ "So Desmos is actually really cool.", - 764.74, - 766.48 + 838.1, + 838.1 ], [ "They make a lot of these interactive math activities for classrooms and tools for teachers.", - 766.88, - 770.88 + 838.1, + 838.1 ], [ "The real meat of their offering is in their classroom activities.", - 771.48, - 775.18 + 838.1, + 838.1 ], [ "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", - 776.04, - 781.82 + 838.1, + 838.1 ], [ "The team clearly knows their stuff, and they know where they stand to make a difference in students' and teachers' lives.", - 782.5, - 787.4 + 838.1, + 838.1 ], [ "And like I said, they're hiring.", - 788.08, - 789.5 + 838.1, + 838.1 ], [ "They are always looking to bring in more good talent, whether that's engineering talent, designers, teachers, or whatever other skill sets line up with what they want to do.", - 790.06, - 798.72 + 838.1, + 838.1 ], [ "If any of you out there are interested in joining them, helping them make some of these great tools for teachers and students, you can check out the careers page I've linked in the description.", - 799.22, - 807.42 + 838.1, + 838.1 ], [ "Personally, I think they're doing some really meaningful stuff.", - 808.04, - 810.4 + 838.1, + 838.1 ], [ "I think their activities are building genuinely good math intuitions for students, and the world could use a few more talented people pointing their efforts towards education the way they do.", - 810.5, - 819.84 + 838.1, + 838.1 ], [ "Alright so with that, I'll see you guys next video, and I think you're really going to like where this is going.", - 821.26, + 838.1, 838.1 ] ] \ No newline at end of file diff --git a/2016/hanoi-and-sierpinski/english/transcript.txt b/2016/hanoi-and-sierpinski/english/transcript.txt index 139c4d782..6cd5691e8 100644 --- a/2016/hanoi-and-sierpinski/english/transcript.txt +++ b/2016/hanoi-and-sierpinski/english/transcript.txt @@ -6,76 +6,76 @@ In case you're unfamiliar, let's just lay down what the Towers of Hanoi puzzle a So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg. The setup pictured here has five disks, which I'll label 0, 1, 2, 3, 4, but in principle, you could have as many disks as you want. -So they all start up stacked up from biggest to smallest on one spindle, and the goal is to move the entire tower from one spindle to another. -The rule is you can only move one disk at a time, and you can't move a bigger disk on top of a smaller disk. -For example, your first move must involve moving disk 0, since any other disk has stuff on top of it that needs to get out of the way before it can move. -After that, you can move disk 1, but it has to go on whatever peg doesn't currently have disk 0, since otherwise you'd be putting a bigger disk on a smaller one, which isn't allowed. -If you've never seen this before, I highly encourage you to pause and pull out some books of varying sizes and try it out for yourself. +For example, your first move must involve moving disk 0, since any other disk has stuff on top of it that needs to get out of the way before it can move. +After that, you can move disk 1, but it has to go on whatever peg doesn't currently have disk 0, since otherwise you'd be putting a bigger disk o +n a smaller one, which isn't allowed. If you've never seen this before, I highly encourage you to pause and pull out some books of varying sizes and try it out for yourself. Now Keith showed me something truly surprising about this puzzle, wh +ich is that you can solve it just by counting up in binary and associating the rhythm of that counting with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. Actually, eve +n if you're familiar with binary, I want to explain it with a focus on the rhythm of counting, which you may or may not have thought about before. Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associating the rhythm of that counting with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. -Actually, even if you are familiar with binary, I want to explain it with a focus on the rhythm of counting, which you may or may not have thought about before. -Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. +10 that you've already counted up to so far, while freeing the ones place to reset to 0. The rhythm of counting repeats like this, counting up 9, rolling over to the tens place, counting up 9 more, r +olling over to the tens place, etc. Well, after repeating that process 9 times, you roll over to a hundreds place, a digit that keeps track of how many groups of 100 you've hit, freeing up the other two digits to reset to 0. In this way, The rhythm of counting begins by walking through all 10 of these digits. Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. -The rhythm of counting repeats like this, counting up 9, rolling over to the tens place, counting up 9 more, rolling over to the tens place, etc. -Until, after repeating that process 9 times, you roll over to a hundreds place, a digit that keeps track of how many groups of 100 you've hit, freeing up the other two digits to reset to 0. -In this way, the rhythm of counting is kind of self-similar. +t when you're counting, you have to roll over all the time. After counting 0, 1, you've already run out of bits, so you need to roll over to a two's place, writ +ing 1-0, and resisting every urge in your base-10-trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then increment up to 1-1, which represents 3, and already you have to roll over again, and since there's a 1 in that two's place, that has to roll over as well, giving you 1-0-0, which represents 1 group of 4 plus 0 groups of 2 plus 0. +In the same way that digits in base-10 represent powers of 10, bits in base-2 represent different powers of 2. So instead of talking about a 10's place, a 100's place, a 1000's place, things like that, you talk about a 2's place, a 4's place, and an 8's place. Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover. -In binary, also known as base-2, you limit yourself to two digits, 0 and 1, commonly called bits, which is short for binary digits. -The result is that when you're counting, you have to roll over all the time. +ce. Flip the last, roll over twice. Flip the last, roll over once. Flip the last, roll over three times. Again, there's a certain self-similarity to this pattern. At every scale, the process is to do something, roll over, then do that same thing again. At the small scale, say counting up to 3, which is 1-1 in binary, this m +eans flip the last bit, roll over to the two's, then flip the last bit. At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. -In the same way that digits in base-10 represent powers of 10, bits in base-2 represent different powers of 2, so instead of a tens place, a hundreds place, a thousands place, you talk about a twos place, a fours place, and an eights place. +tting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. The rhythm of counting is now a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. -At every scale, the process is to do something, roll over, then do that same thing again. +ontinues like this. Flip the last, move disk 0. Flip the last two, move disk 1. Flip the last, move disk 0. And here we're going to have to roll over three times to the 8's place, and that corresponds to moving disk 3. There's something magical At the small scale, say counting up to 3, which is 11 in binary, this means flip the last bit, roll over to the twos, then flip the last bit. At a larger scale, like counting up to 15, which is 1111 in binary, the process is to let the last 3 count up to 7, roll over to the eights place, then let the last 3 bits count up again. Counting up to 255, which is 8 successive ones, this looks like letting the last 7 bits count up till they're full, rolling over to the 128th place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. -You start by counting from 0. -Whenever you're only flipping that last bit, from a 0 to a 1, move disk 0 one peg to the right. -If it was already on the right-most peg, you just loop it back to the first peg. +i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th +e heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective. Disk 3 is thinking, okay, 2, 1, and 0, like you have to get +off of me, like I can't really function under this much weight and pressure. And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice. -You can't put it on top of disk 0, and there's only one other peg, so you move it where you're forced to move it. -So after this, counting up to 1,1, that involves just flipping the last bit, so you move disk 0 again. -Then when your binary counting rolls over twice to the fours place, move disk number 2, and the pattern continues like this. -Flip the last, move disk 0. +this disk to work, I can turn my bigger problem into something slightly smaller. And then how do I solve that? Well, it's exactly the same thing. If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. I just need some space. Get off. They need to move somewh +ere. Then disk 2 can move to where it needs to go. Then disk 1 and 0 can do this. But the interesting point is that every single disk pretty much has the exact same strategy. They all say, everybody above me, get off. +Then I'm going to move. Okay, everyone pile back on. When you know that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which probably has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear that this +has to be the most efficient solution. At every step, you're only doing what's forced upon you. You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. Flip the last two, move disk 1. Flip the last, move disk 0. And here, we're going to have to roll over three times to the eights place, and that corresponds to moving disk number 3. There's something magical about it. -When I first saw this, I was like, this can't work. +There's just not any room for inefficiency from this perspective. I don't know how this works, I don't know why this works. Now I know, but it's just magical when you see it. I remember putting together an animation for this when I was teaching this, and just like, I know how this works. -I know all the things in it. +ove disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit and just watch it play out. -Oh yeah. +At a slightly larger scale, solving towers of Hanoi for three disks looks like doing whatever I mean, it's not even clear at first that this is always going to give legal moves. -For example, how do you know that every time you're rolling over to the eights place, that disk 3 is necessarily going to be freed up to move? +k number 2, then do whatever it takes to solve two disks again. Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than having to do 2 to the n minus 1 steps? It turns out, not only does this solve Towers of Hanoi, but it does it in the most efficient way possible. Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective. -Disk 3 is thinking, okay, 2, 1, and 0, you have to get off of me. +give me this thing, you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out. That's pretty cool, right? But it actually gets cooler. I can't really function under this much weight and pressure. And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move. -If any of these disks are on top of 3, it can't move. +access to these videos before I publish the full series in a few months. This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. -You never need to move me again. +impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense, you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C. So the idea is that if I just focus on one disk and I think about what am I going to have to do to get this disk to work, I can turn my bigger problem into something slightly smaller. -And then how do I solve that? +and students, you can check out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing. Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space. diff --git a/2016/hanoi-and-sierpinski/french/sentence_translations.json b/2016/hanoi-and-sierpinski/french/sentence_translations.json index 7bd2baf95..e0f089046 100644 --- a/2016/hanoi-and-sierpinski/french/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/french/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "Vous considérez ces disques comme ayant un trou au milieu afin que vous puissiez les insérer sur une cheville.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "Toute description du binaire commence généralement par une introspection sur notre façon habituelle de représenter les nombres, ce que nous appelons la base 10, puisque nous utilisons 10 chiffres distincts, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "Ensuite, n’ayant plus de nouveaux chiffres, vous exprimez le nombre suivant, 10, avec deux chiffres, 1, 0.", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "De cette façon, le rythme de comptage est en quelque sorte similaire.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "Même si vous effectuez un zoom arrière à plus grande échelle, le processus ressemble à faire quelque chose, à effectuer un survol, à faire la même chose, à effectuer un survol et à répéter 9 fois avant un survol encore plus important.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "Le rythme du comptage est désormais beaucoup plus rapide, mais cela le rend presque plus perceptible.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "Retournez le dernier, retournez-le une fois.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "À une plus grande échelle, comme compter jusqu'à 15, ce qui équivaut à 1-1-1-1, le processus consiste à laisser les 3 derniers compter jusqu'à 7, à passer à la place du 8, puis à laisser les 3 derniers bits compter à nouveau.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "En comptant jusqu'à 255, soit 8 1 successifs, cela revient à laisser les 7 derniers bits compter jusqu'à ce qu'ils soient pleins, à passer à la place des 128, puis à laisser les 7 derniers bits compter à nouveau.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "Très bien, donc avec cette mini-introduction, le fait surprenant que Keith m'a montré est que nous pouvons utiliser ce rythme pour résoudre les tours de Hanoi.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "Chaque fois que vous retournez uniquement ce dernier bit, de 0 à 1, déplacez le disque 0 d'un cran vers la droite.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "Je veux dire, il n'est même pas clair au début que cela donnera toujours lieu à des actions légales.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "Par exemple, comment savez-vous qu'à chaque fois que vous passez à la place du 8, ce disque 3 va nécessairement être libéré pour se déplacer?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "Comprendre pourquoi cela fonctionne, comment cela fonctionne et ce qui se passe revient à une certaine perspective du puzzle, ce que les gens de CS pourraient appeler une perspective récursive.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "C'est la seule façon dont il peut bouger.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "Cela fait, nous pouvons déplacer le disque 3 là-bas.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "Et puis le disque 3 dit, je suis prêt.", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "Et dans un sens, vous disposez désormais d’une version réduite du même problème.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "Maintenant que vous avez les disques 0, 1 et 2 sur la broche B, vous devez les amener sur C.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "Ils doivent déménager quelque part.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "Bon, tout le monde se remet.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "Et si l’on y réfléchit un peu, il apparaît clairement que cela doit être la solution la plus efficace.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "Vous devez retirer les disques 0 à 2 avant de pouvoir déplacer le disque 3.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "Et vous devez déplacer le disque 3.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "De ce point de vue, il n’y a tout simplement pas de place pour l’inefficacité.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "Alors pourquoi le comptage en binaire capture-t-il cet algorithme?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "Eh bien, ce qui se passe ici, c'est que ce modèle de résolution d'un sous-problème, en déplaçant un gros disque, puis en résolvant à nouveau un sous-problème, est parfaitement parallèle au modèle de comptage binaire.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "Retournez le dernier morceau, retournez une fois, retournez le dernier morceau.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "De manière analogue, compter jusqu'à 111 en binaire implique de compter jusqu'à 3, de parcourir les trois bits, puis d'en compter trois de plus.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "À toutes les échelles, les deux processus présentent la même répartition.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "Donc, dans un sens, la raison pour laquelle cette solution binaire fonctionne, ou du moins une explication, j'ai l'impression qu'il n'y a pas une seule explication, mais je pense que la plus naturelle est que le modèle que vous utiliseriez pour générer ces nombres binaires a exactement le même structure comme le modèle que vous utiliseriez pour les tours de Hanoï, c'est pourquoi si vous regardez les bits qui s'inversent, vous inversez effectivement ce processus.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "Vous demandez : quel processus a généré cela?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "C'est plutôt cool, non?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "Mais en fait, ça devient plus frais.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "Je n'ai même pas compris comment cela se rapporte au triangle de Sierpinski.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "Cette vidéo et la suivante sont également soutenues en partie par Desmos, et avant la prochaine vidéo, je veux juste prendre un moment et partager un peu avec vous qui ils sont et le fait qu'ils embauchent.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "Donc Desmos est vraiment vraiment cool.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "La véritable substance de leur offre réside dans leurs activités en classe.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "Pour ma part, je suis très impressionné par la qualité pédagogique de ces activités.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "Et comme je l'ai dit, ils embauchent.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "Personnellement, je pense qu'ils font des choses vraiment significatives.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/german/sentence_translations.json b/2016/hanoi-and-sierpinski/german/sentence_translations.json index 64c60230e..966bc0fbb 100644 --- a/2016/hanoi-and-sierpinski/german/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/german/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "Stellen Sie sich diese Scheiben so vor, als hätten sie in der Mitte ein Loch, damit Sie sie auf einen Stift stecken können.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "Jede Beschreibung von Binärzahlen beginnt typischerweise mit einer Betrachtung unserer üblichen Art, Zahlen darzustellen, was wir Basis 10 nennen, da wir 10 separate Ziffern verwenden, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "Wenn Ihnen dann die neuen Ziffern ausgehen, drücken Sie die nächste Zahl, 10, mit zwei Ziffern aus, 1, 0.", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "Auf diese Weise ist der Rhythmus des Zählens gewissermaßen selbstähnlich.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "Selbst wenn Sie auf einen größeren Maßstab herauszoomen, sieht der Vorgang so aus, als würden Sie etwas tun, sich umdrehen, das Gleiche tun, umdrehen und neunmal wiederholen, bevor es zu einem noch größeren Überschlag kommt.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "Der Zählrhythmus ist jetzt viel schneller, aber das macht es fast deutlicher.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "Den letzten umdrehen, einmal umdrehen.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "In einem größeren Maßstab, etwa beim Zählen bis 15, also 1-1-1-1, besteht der Vorgang darin, die letzten 3 bis 7 zählen zu lassen, an die Stelle der 8 zu wechseln und dann die letzten 3 Bits wieder hochzählen zu lassen.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "Wenn man bis 255 zählt, was 8 aufeinanderfolgenden Einsen entspricht, sieht das so aus, als würde man die letzten 7 Bits hochzählen, bis sie voll sind, dann zur Stelle der 128 übergehen und dann die letzten 7 Bits wieder hochzählen lassen.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "Also gut, die überraschende Tatsache, die Keith mir mit dieser Mini-Einführung gezeigt hat, ist, dass wir diesen Rhythmus verwenden können, um die Türme von Hanoi zu lösen.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "Wenn Sie nur das letzte Bit von 0 auf 1 umdrehen, verschieben Sie die Scheibe 0 um einen Stift nach rechts.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "Ich meine, es ist zunächst nicht einmal klar, dass dies immer zu rechtlichen Schritten führen wird.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "Woher wissen Sie beispielsweise, dass jedes Mal, wenn Sie zur Stelle der 8 wechseln, zwangsläufig die Scheibe 3 zum Bewegen freigegeben wird?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "Um zu verstehen, warum das funktioniert und wie es funktioniert und was zum Teufel vor sich geht, kommt es auf eine bestimmte Perspektive auf das Rätsel an, die CS-Leute eine rekursive Perspektive nennen könnten.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "Nur so kann es sich bewegen.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "Nachdem wir das getan haben, können wir Diskette 3 dorthin verschieben.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "Und dann sagt Diskette 3: „Ich bin fertig.", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "Und in gewisser Weise haben Sie jetzt eine kleinere Version des gleichen Problems.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "Jetzt haben Sie die Festplatten 0, 1 und 2 auf Spindel B und müssen sie nach C bringen.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "Sie müssen irgendwohin ziehen.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "Okay, macht alle weiter.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "Und wenn man kurz darüber nachdenkt, wird klar, dass dies die effizienteste Lösung sein muss.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "Sie müssen die Datenträger 0 bis 2 entfernen, bevor Sie Datenträger 3 verschieben können.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "Und Sie müssen Diskette 3 verschieben.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "Aus dieser Perspektive gibt es einfach keinen Raum für Ineffizienz.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "Warum erfasst die binäre Zählung diesen Algorithmus?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "Nun, hier geht es darum, dass dieses Muster, ein Teilproblem zu lösen, eine große Scheibe zu bewegen und dann ein Teilproblem erneut zu lösen, perfekt mit dem Muster der binären Zählung übereinstimmt.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "Drehen Sie das letzte Stück um, drehen Sie es einmal um, drehen Sie das letzte Stück um.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "Analog dazu umfasst das Binärzählen bis 111 das Zählen bis 3, das Durchlaufen aller drei Bits und das anschließende Hochzählen um drei weitere.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "Auf allen Skalen weisen beide Prozesse die gleiche Aufschlüsselung auf.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "In gewisser Weise ist der Grund, warum diese binäre Lösung funktioniert, oder zumindest eine Erklärung, meiner Meinung nach keine eindeutige Erklärung, aber ich denke, die natürlichste ist, dass das Muster, das Sie zum Generieren dieser Binärzahlen verwenden würden, genau das gleiche hat Struktur wie das Muster, das Sie für die Türme von Hanoi verwenden würden. Wenn Sie sich also die umgedrehten Teile ansehen, kehren Sie diesen Prozess praktisch um.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "Sie fragen sich: Welcher Prozess hat diese erzeugt?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "Das ist ziemlich cool, oder?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "Aber es wird tatsächlich cooler.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "Ich habe noch nicht einmal verstanden, wie das mit dem Sierpinski-Dreieck zusammenhängt.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "Dieses und das nächste Video werden teilweise auch von Desmos unterstützt, und vor dem nächsten Video möchte ich mir kurz einen Moment Zeit nehmen und euch ein wenig darüber erzählen, wer sie sind und welche Mitarbeiter sie einstellen.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "Desmos ist also wirklich cool.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "Der wahre Kern ihres Angebots liegt in ihren Unterrichtsaktivitäten.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "Ich für meinen Teil bin sehr beeindruckt davon, wie gut diese Aktivitäten aus pädagogischer Sicht durchdacht sind.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "Und wie gesagt, sie stellen ein.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "Persönlich denke ich, dass sie einige wirklich sinnvolle Dinge tun.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/hindi/sentence_translations.json b/2016/hanoi-and-sierpinski/hindi/sentence_translations.json index 7e9be69f0..208821276 100644 --- a/2016/hanoi-and-sierpinski/hindi/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/hindi/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "आप इन डिस्क को बीच में एक छेद के रूप में सोचते हैं ताकि आप उन्हें एक खूंटी पर फिट कर सकें।", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "बाइनरी का कोई भी विवरण आम तौर पर संख्याओं को दर्शाने के हमारे सामान्य तरीके के बारे में आत्मनिरीक्षण से शुरू होता है, जिसे हम आधार 10 कहते हैं, क्योंकि हम 10 अलग-अलग अंकों, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 का उपयोग करते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "फिर, नए अंक समाप्त होने पर, आप अगली संख्या, 10 को दो अंकों, 1, 0 के साथ व्यक्त करते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "इस प्रकार, गिनती की लय एक तरह से स्व-समान होती है।", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "यहां तक कि अगर आप बड़े पैमाने पर ज़ूम आउट करते हैं, तो प्रक्रिया कुछ करने, पलटने, वही काम करने, पलटने और इससे भी बड़े रोलओवर से पहले 9 बार दोहराने जैसी दिखती है।", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "गिनती की लय अब बहुत तेज़ हो गई है, लेकिन इससे यह लगभग अधिक ध्यान देने योग्य हो गई है।", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "आखिरी को पलटें, एक बार पलटें।", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "बड़े पैमाने पर, जैसे 15 तक गिनती, जो 1-1-1-1 है, प्रक्रिया यह है कि अंतिम 3 को 7 तक गिनने दें, 8 के स्थान पर रोल करें, फिर अंतिम 3 बिट्स को फिर से गिनने दें.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "255 तक गिनती, जो कि लगातार 8 1 है, ऐसा लगता है कि अंतिम 7 बिट्स को तब तक गिनने दें जब तक कि वे पूरे न हो जाएं, 128 के स्थान पर आ जाएं, फिर अंतिम 7 बिट्स को फिर से गिनने दें।", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "ठीक है, तो उस लघु परिचय के साथ, कीथ ने मुझे जो आश्चर्यजनक तथ्य दिखाया वह यह है कि हम हनोई के टावरों को हल करने के लिए इस लय का उपयोग कर सकते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "जब भी आप केवल अंतिम बिट को 0 से 1 तक फ़्लिप कर रहे हों, तो डिस्क 0 को एक खूंटी से दाईं ओर ले जाएँ।", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "मेरा मतलब है, पहले तो यह भी स्पष्ट नहीं है कि यह हमेशा कानूनी कदम उठाने वाला है।", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "उदाहरण के लिए, आप कैसे जानते हैं कि हर बार जब आप 8 के स्थान पर घूम रहे हैं, तो डिस्क 3 को स्थानांतरित करने के लिए आवश्यक रूप से मुक्त किया जाएगा?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "यह समझना कि यह क्यों काम करता है और यह कैसे काम करता है और क्या हो रहा है, पहेली पर एक निश्चित परिप्रेक्ष्य में आता है, जिसे सीएस लोग पुनरावर्ती परिप्रेक्ष्य कह सकते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "यही एकमात्र तरीका है जिससे वह आगे बढ़ सकता है।", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "ऐसा करने के बाद, हम डिस्क 3 को वहां ले जा सकते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "और फिर डिस्क 3 कहती है, मैं तैयार हूं।", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "और एक तरह से, अब आपके पास उसी समस्या का एक छोटा संस्करण है।", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "अब आपके पास स्पिंडल बी पर डिस्क 0, 1, और 2 हैं, आपको उन्हें सी पर लाना होगा।", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "उन्हें कहीं और जाने की जरूरत है.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "ठीक है, हर कोई वापस ढेर हो गया।", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "और यदि आप इसके बारे में थोड़ा सोचें, तो यह स्पष्ट हो जाता है कि यह सबसे कुशल समाधान होना चाहिए।", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "डिस्क 3 को स्थानांतरित करने से पहले आपको डिस्क 0 से 2 को बंद करना होगा।", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "और आपको डिस्क 3 को स्थानांतरित करना होगा।", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "इस दृष्टिकोण से अक्षमता के लिए कोई जगह नहीं है।", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "तो बाइनरी में गिनती इस एल्गोरिदम को क्यों पकड़ती है?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "खैर, यहाँ जो हो रहा है वह यह है कि एक उप-समस्या को हल करने, एक बड़ी डिस्क को स्थानांतरित करने, फिर एक उप-समस्या को फिर से हल करने का यह पैटर्न, बाइनरी काउंटिंग के पैटर्न से पूरी तरह से समानांतर है।", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "आखिरी टुकड़े को पलटें, एक बार पलटें, आखिरी टुकड़े को पलटें।", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "समान रूप से, बाइनरी में 111 तक की गिनती में 3 तक की गिनती, सभी तीन बिट्स को रोल करना, फिर तीन और को गिनना शामिल है।", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "सभी पैमानों पर, दोनों प्रक्रियाओं में समान टूट-फूट होती है।", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "तो एक अर्थ में, यही कारण है कि यह बाइनरी समाधान काम करता है, या कम से कम एक स्पष्टीकरण, मुझे ऐसा लगता है कि कोई एक स्पष्टीकरण नहीं है, लेकिन मुझे लगता है कि सबसे स्वाभाविक बात यह है कि इन बाइनरी संख्याओं को उत्पन्न करने के लिए आप जिस पैटर्न का उपयोग करेंगे वह बिल्कुल वैसा ही है संरचना उस पैटर्न के समान है जिसका उपयोग आप हनोई के टावरों के लिए करेंगे, यही कारण है कि यदि आप बिट्स को पलटते हुए देखते हैं, तो आप प्रभावी रूप से इस प्रक्रिया को उलट रहे हैं।", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "आप कह रहे हैं, किस प्रक्रिया ने इन्हें उत्पन्न किया?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "यह बहुत अच्छा है, है ना?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "लेकिन यह वास्तव में ठंडा हो जाता है।", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "मैं यह भी नहीं समझ पाया कि इसका सिएरपिंस्की के त्रिकोण से क्या संबंध है।", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "यह वीडियो और अगला वीडियो भी आंशिक रूप से डेस्मोस द्वारा समर्थित है, और अगले वीडियो से पहले मैं बस एक क्षण लेना चाहता हूं और आप लोगों के साथ थोड़ा सा साझा करना चाहता हूं कि वे कौन हैं और तथ्य यह है कि वे काम पर रख रहे हैं।", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "तो डेस्मोस वास्तव में बहुत अच्छा है।", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "उनकी पेशकश का असली सार उनकी कक्षा की गतिविधियाँ हैं।", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "अपनी ओर से, मैं इस बात से बहुत प्रभावित हूँ कि शैक्षणिक दृष्टिकोण से ये गतिविधियाँ कितनी अच्छी तरह से सोची-समझी गई हैं।", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "और जैसा मैंने कहा, वे नियुक्ति कर रहे हैं।", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "व्यक्तिगत रूप से, मुझे लगता है कि वे वास्तव में कुछ सार्थक काम कर रहे हैं।", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/indonesian/sentence_translations.json b/2016/hanoi-and-sierpinski/indonesian/sentence_translations.json index 54a74a401..9ebcbf82b 100644 --- a/2016/hanoi-and-sierpinski/indonesian/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/indonesian/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "Anda menganggap disk ini memiliki lubang di tengahnya sehingga Anda dapat memasangnya pada pasak.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "Setiap deskripsi biner biasanya dimulai dengan introspeksi tentang cara kita biasa merepresentasikan angka, yang kita sebut basis 10, karena kita menggunakan 10 digit terpisah, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "Kemudian, setelah kehabisan angka baru, nyatakan angka berikutnya, 10, dengan dua angka, 1, 0.", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "Dengan cara ini, ritme penghitungan menjadi serupa.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "Bahkan jika Anda memperkecil ke skala yang lebih besar, prosesnya terlihat seperti melakukan sesuatu, berguling, melakukan hal yang sama, berguling, dan mengulanginya sebanyak 9 kali sebelum melakukan rollover yang lebih besar.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "Ritme berhitung sekarang jauh lebih cepat, tetapi hal itu hampir membuatnya lebih terlihat.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "Balik yang terakhir, gulung sekali.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "Pada skala yang lebih besar, seperti menghitung sampai 15, yaitu 1-1-1-1, prosesnya membiarkan 3 bit terakhir dihitung hingga 7, digulingkan ke tempat 8, lalu biarkan 3 bit terakhir dihitung lagi.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "Menghitung hingga 255, yaitu 8 angka 1 berturut-turut, ini seperti membiarkan 7 bit terakhir dihitung hingga penuh, berpindah ke tempat 128, lalu membiarkan 7 bit terakhir dihitung lagi.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "Baiklah, jadi dengan perkenalan singkat itu, fakta mengejutkan yang ditunjukkan Keith kepada saya adalah kita bisa menggunakan ritme ini untuk memecahkan menara Hanoi.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "Setiap kali Anda hanya membalik bit terakhir, dari 0 ke 1, pindahkan disk 0 satu pasak ke kanan.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "Maksud saya, pada awalnya tidak jelas apakah hal ini akan selalu menghasilkan tindakan hukum.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "Misalnya, bagaimana Anda tahu bahwa setiap kali Anda berguling ke tempat 8, disk 3 itu akan dibebaskan untuk bergerak?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "Memahami mengapa ini berhasil dan bagaimana cara kerjanya serta apa yang sedang terjadi bermuara pada perspektif tertentu pada teka-teki tersebut, yang mungkin disebut oleh orang-orang CS sebagai perspektif rekursif.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "Itulah satu-satunya cara agar ia bisa bergerak.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "Setelah melakukan itu, barulah kita bisa memindahkan disk 3 ke sana.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "Dan kemudian disk 3 berkata, saya siap.", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "Dan dalam arti tertentu, Anda sekarang memiliki versi yang lebih kecil dari masalah yang sama.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "Sekarang Anda memiliki disk 0, 1, dan 2 pada spindel B, Anda harus memindahkannya ke C.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "Mereka perlu pindah ke suatu tempat.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "Oke, semuanya kembali berkumpul.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "Dan jika Anda memikirkannya sebentar, menjadi jelas bahwa ini adalah solusi yang paling efisien.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "Anda harus melepas disk 0 hingga 2 sebelum Anda dapat memindahkan disk 3.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "Dan Anda harus memindahkan disk 3.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "Tidak ada ruang untuk inefisiensi dari sudut pandang ini.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "Jadi mengapa penghitungan dalam biner menangkap algoritma ini?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "Apa yang terjadi di sini adalah pola penyelesaian submasalah, memindahkan disk besar, lalu menyelesaikan submasalah lagi, sangat paralel dengan pola penghitungan biner.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "Balik bagian terakhir, putar sekali, balik bagian terakhir.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "Secara analogi, menghitung hingga 111 dalam biner melibatkan penghitungan hingga 3, menggulirkan ketiga bit, lalu menghitung tiga bit lagi.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "Pada semua skala, kedua proses memiliki rincian yang sama.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "Jadi dalam arti tertentu, alasan mengapa solusi biner ini berhasil, atau setidaknya penjelasannya, saya rasa tidak ada satu penjelasan pun, tapi menurut saya yang paling alami adalah pola yang akan Anda gunakan untuk menghasilkan bilangan biner ini memiliki pola yang persis sama. struktur sebagai pola yang akan Anda gunakan untuk menara Hanoi, itulah sebabnya jika Anda melihat bagian-bagiannya yang terbalik, Anda secara efektif membalikkan proses ini.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "Maksud Anda, proses apa yang menghasilkan hal ini?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "Itu cukup keren, bukan?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "Tapi justru menjadi lebih keren.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "Saya bahkan belum mengetahui hubungannya dengan segitiga Sierpinski.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "Video ini dan video berikutnya juga didukung sebagian oleh Desmos, dan sebelum video berikutnya saya hanya ingin meluangkan waktu sejenak dan berbagi sedikit dengan kalian tentang siapa mereka dan fakta bahwa mereka sedang merekrut.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "Jadi Desmos sebenarnya sangat keren.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "Inti sebenarnya dari persembahan mereka adalah dalam kegiatan kelas mereka.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "Bagi saya, saya sangat terkesan dengan betapa matangnya kegiatan ini dari sudut pandang pedagogi.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "Dan seperti yang saya katakan, mereka sedang merekrut.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "Secara pribadi, menurut saya mereka melakukan beberapa hal yang sangat berarti.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/japanese/sentence_translations.json b/2016/hanoi-and-sierpinski/japanese/sentence_translations.json index 2724cb28a..9a13de920 100644 --- a/2016/hanoi-and-sierpinski/japanese/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/japanese/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "これらのディスクの中央には穴があり、ペグ に取り付けることができると考えられます。", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "バイナリの説明は通常、10 個の個別の数字 0、1、2、3、 4、5、6、7、8、9 を使用するため、私たちが基数 10 と呼ぶ通常の数値の表現方法についての内省から始まります。", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "次に、新しい桁がなくなったら、次の数値 10 を 2 桁の 1、0 で表します。", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "このように、数を数えるリズムは一種の自己相似です。", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "より大きなスケールにズームアウトした場合でも、このプロセス は、何かをして、寝返りし、同じことをして、寝返りを 9 回 繰り返した後、さらに大きな寝返りが行われるように見えます。", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "数を数えるリズムはかなり速くなりましたが、その分、より顕著に感じられるようになりました。", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "最後を裏返し、一度裏返します。", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "より大きなスケールでは、15 までカウントする (1-1-1-1) ようなプロセスでは、最後の 3 を 7 までカウントし、8 の位にロールオーバーして、最後の 3 ビットを再びカウントアップします。", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "8 個の連続する 1 である 255 までカウントすると、最後の 7 ビットがいっぱいになるまでカウントアップし、128 の位にロールオーバ ーしてから、最後の 7 ビットを再びカウントアップするように見えます。", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "さて、そのミニ紹介はこのくらいにして、キースが私に見せてくれた驚くべき 事実は、このリズムを使ってハノイの塔を解くことができるということです。", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "最後のビットだけを 0 から 1 に反転するとき は、ディスク 0 を 1 ペグ右に移動します。", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "つまり、これが常に合法的な動きをもたらすかどうかは、最初は明らかではありません。", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "たとえば、8 の位にロールオーバーするたびに、ディスク 3 が必然的 に解放されて移動できることをどうやって知ることができるでしょうか。", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "これがなぜ機能するのか、どのように機能するのか、そして一体何が起こっているのかを理解す るには、パズルに対する特定の視点、CS の人々が再帰的視点と呼ぶものにかかっています。", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "それがそれが動くことができる唯一の方法です。", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "それが完了したら、ディスク 3 をそこに移動できます。", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "そしてディスク 3 では、「準備完了」と表示されます。", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "ある意味、同じ問題の小規模バージョンが手に入ったことになります。", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "ディスク 0、1、2 がスピンドル B に配置されているので、それらを C に移動する必要があります。", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "彼らはどこかに移動する必要があります。", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "さて、みんなまた積み上げてください。", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "少し考えてみれば、これが最も効率的な 解決策であることが明らかになります。", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "ディスク 3 を移動する前に、ディスク 0 ～ 2 を外す必要があります。", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "そしてディスク 3 を移動する必要があります。", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "この観点から見ると、非効率が許される余地はありません。", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "では、なぜバイナリでのカウントがこのアルゴリズムを捉えるのでしょうか?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "さて、ここで何が起こっているのかというと、部分問題を解決し、大き な円盤を移動し、その後、再度部分問題を解決するというこのパターン が、二進数カウントのパターンと完全に類似しているということです。", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "最後のビットを反転し、一度ロールオーバーして、最後のビットを反転します。", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "同様に、2 進数で 111 までカウントするには、3 までカウントし、3 ビッ トすべてをロールオーバーして、さらに 3 つカウントアップする必要があります。", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "どの規模においても、両方のプロセスでこれと同じ内訳が発生します。", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "ある意味、このバイナリ ソリューションが機能する理由、または少なくと も説明はありませんが、最も自然な説明は、これらの 2 進数を生成する ために使用するパターンがまったく同じであるということだと思います。 こ の構造は、ハノイの塔に使用されるパターンと同じです。 そのため、ビット の反転を見ると、事実上、このプロセスを逆にしていることになります。", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "どのようなプロセスでこれらが生成されたのか、ということですね。", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "それはとてもクールですよね？", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "でも実際はもっと涼しくなるんです。", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "これがシェルピンスキーの三角形とどのように関係するのかさえ分かりません。", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "このビデオと次のビデオも Desmos によって部分的にサポートされて います。 次のビデオの前に、少し時間をとって、彼らが誰であるか、そして彼 らが雇用しているという事実について少し皆さんと共有したいと思います。", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "デスモスは本当にクールです。", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "彼らの真骨頂は教室活動にあります。", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "私としては、これらの活動が教育的な観点から非常によ く考えられていることに非常に感銘を受けています。", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "そして、先ほども言ったように、彼らは雇用を行っています。", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "個人的には、彼らは本当に意味のあることをしていると思います。", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/korean/sentence_translations.json b/2016/hanoi-and-sierpinski/korean/sentence_translations.json index 223b5e9f4..16b0a2750 100644 --- a/2016/hanoi-and-sierpinski/korean/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/korean/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "이 디스크는 중앙에 구멍이 있어서 못에 끼울 수 있다고 생각합니다.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "이진법에 대한 모든 설명은 일반적으로 숫자를 표현하는 일반적인 방법, 즉 10진법이라고 부르는 방식에 대한 성찰로 시작됩니다. 왜냐하면 우리는 0, 1, 2, 3, 4, 5, 6, 7, 8, 9라는 10개의 개별 숫자를 사용하기 때문입니다.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "그런 다음 새 숫자가 부족해지면 다음 숫자인 10을 두 자리 숫자 1, 0으로 표현합니다.", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "이런 식으로 계산의 리듬은 일종의 자기 유사성입니다.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "더 큰 규모로 축소하더라도 프로세스는 더 큰 롤오버 전에 무언가를 하고, 롤오버하고, 같은 일을 하고, 롤오버하고, 9번 반복하는 것처럼 보입니다.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "이제 계산의 리듬이 훨씬 빨라졌지만 거의 눈에 띄게 되었습니다.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "마지막을 뒤집고 한 번 뒤집습니다.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "1-1-1-1인 15까지 세는 것과 같이 더 큰 규모에서 프로세스는 마지막 3을 7까지 세고 8의 자리로 롤오버한 다음 마지막 3비트를 다시 세는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "8개의 연속된 1인 255까지 계산하면 마지막 7비트가 가득 찰 때까지 계산하고 128의 자리로 롤오버한 다음 마지막 7비트를 다시 계산하는 것처럼 보입니다.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "자, 그 짧은 소개를 통해 Keith가 나에게 보여준 놀라운 사실은 우리가 이 리듬을 사용하여 하노이의 탑을 해결할 수 있다는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "마지막 비트만 0에서 1로 뒤집을 때마다 디스크 0을 오른쪽으로 한 페그 이동합니다.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "내 말은, 이것이 항상 법적 조치를 취할 것인지가 처음에는 명확하지 않다는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "예를 들어, 8의 자리로 롤오버할 때마다 해당 디스크 3이 반드시 이동을 위해 비워질 것이라는 것을 어떻게 알 수 있습니까?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "이것이 왜 작동하는지, 어떻게 작동하는지, 도대체 무슨 일이 일어나고 있는지 이해하는 것은 CS 담당자가 재귀적 관점이라고 부를 수 있는 퍼즐에 대한 특정 관점으로 귀결됩니다.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "그래야만 움직일 수 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "그런 다음 디스크 3을 저기로 이동할 수 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "그리고 디스크 3에서는 '설정되었습니다'라고 말합니다.", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "그리고 어떤 의미에서는 이제 동일한 문제의 더 작은 버전이 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "이제 디스크 0, 1, 2가 스핀들 B에 있으므로 이를 C로 가져와야 합니다.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "그들은 어딘가로 이동해야 합니다.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "좋아, 모두 다시 모여라.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "그리고 조금만 생각해 보면 이것이 가장 효율적인 솔루션임이 분명해집니다.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "디스크 3을 이동하려면 먼저 디스크 0부터 2까지를 꺼야 합니다.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "그리고 디스크 3을 옮겨야 합니다.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "이러한 관점에서는 비효율적일 여지가 전혀 없습니다.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "그렇다면 이진수 계산이 이 알고리즘을 포착하는 이유는 무엇입니까?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "음, 여기서 일어나는 일은 하위 문제를 해결하고 큰 디스크를 이동한 다음 다시 하위 문제를 해결하는 패턴이 이진수 계산 패턴과 완벽하게 유사하다는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "마지막 비트를 뒤집고 한 번 뒤집은 다음 마지막 비트를 뒤집습니다.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "마찬가지로, 이진수로 최대 111을 세는 것은 최대 3을 세고 세 비트를 모두 롤오버한 다음 세 개를 더 세는 것을 포함합니다.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "모든 규모에서 두 프로세스 모두 이와 동일한 분석을 갖습니다.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "따라서 어떤 의미에서 이 이진법이 작동하는 이유 또는 적어도 설명은 설명이 없는 것 같지만 가장 자연스러운 것은 이러한 이진수를 생성하는 데 사용하는 패턴이 정확히 동일하다는 것입니다. 구조는 하노이 타워에 사용할 패턴이므로 비트 뒤집기를 보면 이 프로세스가 효과적으로 역전되는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "어떤 프로세스에서 이러한 현상이 발생했다는 말씀이신가요?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "정말 멋지죠?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "하지만 실제로는 더 시원해집니다.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "나는 이것이 시에르핀스키의 삼각형과 어떤 관련이 있는지조차 알지 못했습니다.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "이 영상과 다음 영상도 부분적으로 Desmos의 지원을 받고 있습니다. 다음 영상을 보기 전에 잠시 시간을 내어 Desmos가 누구인지, 채용 중인지에 대해 조금 공유하고 싶습니다.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "그래서 Desmos는 실제로 정말 멋집니다.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "그들이 제공하는 것의 진정한 핵심은 교실 활동에 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "저는 이러한 활동이 교육학적 관점에서 얼마나 세심하게 계획되어 있는지에 깊은 인상을 받았습니다.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "그리고 내가 말했듯이, 그들은 채용 중입니다.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "개인적으로는 정말 의미있는 일을 하고 있다고 생각합니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/marathi/sentence_translations.json b/2016/hanoi-and-sierpinski/marathi/sentence_translations.json index 1d43e15c0..b16813816 100644 --- a/2016/hanoi-and-sierpinski/marathi/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/marathi/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "तुम्ही या डिस्क्सना मध्यभागी एक छिद्र असल्यासारखे समजता जेणेकरून तुम्ही त्यांना खुंटीवर बसवू शकाल.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "बायनरीचे कोणतेही वर्णन सामान्यत: अंकांचे प्रतिनिधित्व करण्याच्या आपल्या नेहमीच्या पद्धतीच्या आत्मनिरीक्षणाने सुरू होते, ज्याला आपण बेस 10 म्हणतो, कारण आपण 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 हे वेगळे अंक वापरतो.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "नंतर, नवीन अंक संपल्यानंतर, तुम्ही पुढील संख्या, 10, दोन अंकांसह, 1, 0 व्यक्त करता.", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "अशाप्रकारे, मोजणीची लय एक प्रकारची स्वयं-समान आहे.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "जरी तुम्ही मोठ्या प्रमाणात झूम आउट केले तरीही, प्रक्रिया काहीतरी करणे, रोल ओव्हर करणे, तीच गोष्ट करणे, रोल ओव्हर करणे आणि आणखी मोठ्या रोलओव्हरपूर्वी 9 वेळा पुनरावृत्ती केल्यासारखे दिसते.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "मोजणीची लय आता खूप वेगवान आहे, परंतु ती जवळजवळ अधिक लक्षणीय बनवते.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "शेवटचा फ्लिप करा, एकदा रोल करा.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "मोठ्या प्रमाणावर, जसे की 15 पर्यंत मोजणे, जे 1-1-1-1 आहे, प्रक्रिया म्हणजे शेवटच्या 3 ला 7 पर्यंत मोजू द्या, 8 च्या जागी फिरू द्या, नंतर शेवटचे 3 बिट्स पुन्हा मोजू द्या.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "255 पर्यंत मोजणे, जे 8 लागोपाठ 1 आहे, असे दिसते की शेवटचे 7 बिट्स पूर्ण होईपर्यंत मोजू द्या, 128 च्या जागी फिरू द्या, नंतर शेवटचे 7 बिट्स पुन्हा मोजू द्या.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "ठीक आहे, म्हणून त्या लघु-परिचयातून, किथने मला दाखवलेली आश्चर्यकारक वस्तुस्थिती अशी आहे की आपण हनोईच्या टॉवर्सचे निराकरण करण्यासाठी ही लय वापरू शकतो.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "जेव्हा तुम्ही फक्त शेवटचा बिट फ्लिप करत असाल, 0 ते 1 पर्यंत, डिस्क 0 एक पेग उजवीकडे हलवा.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "मला असे म्हणायचे आहे की हे नेहमीच कायदेशीर हालचाली देत असते हे प्रथम स्पष्ट नाही.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "उदाहरणार्थ, प्रत्येक वेळी तुम्ही 8 च्या जागी फिरत असताना, ती डिस्क 3 हलवण्याकरता मोकळी केली जाईल हे तुम्हाला कसे कळेल?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "हे का कार्य करते आणि ते कसे कार्य करते आणि हे काय चालले आहे हे समजून घेणे कोडेवर एका विशिष्ट दृष्टीकोनातून खाली येते, ज्याला CS लोक रिकर्सिव्ह परिप्रेक्ष्य म्हणू शकतात.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "तो हलवू शकतो एकमेव मार्ग आहे.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "ते केल्यावर, आपण डिस्क 3 तिथे हलवू शकतो.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "आणि मग डिस्क 3 म्हणते, मी सेट आहे.", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "आणि एका अर्थाने, आपल्याकडे आता त्याच समस्येची एक छोटी आवृत्ती आहे.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "आता तुम्हाला डिस्क 0, 1, आणि 2 स्पिंडल B वर बसली आहे, तुम्हाला ती C वर आणायची आहेत.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "त्यांना कुठेतरी हलवावे लागेल.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "ठीक आहे, प्रत्येकजण परत जा.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "आणि जर आपण थोडासा विचार केला तर हे स्पष्ट होते की हा सर्वात कार्यक्षम उपाय आहे.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "तुम्ही डिस्क 3 हलवण्यापूर्वी तुम्हाला डिस्क 0 ते 2 ची सूट मिळावी लागेल.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "आणि तुम्हाला डिस्क 3 हलवावी लागेल.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "या दृष्टीकोनातून अकार्यक्षमतेसाठी कोणतीही जागा नाही.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "मग बायनरीमध्ये मोजणी हे अल्गोरिदम का कॅप्चर करते?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "बरं, इथे काय चाललं आहे की सबप्रॉब्लेम सोडवण्याचा हा पॅटर्न, मोठी डिस्क हलवायची, नंतर पुन्हा सबप्रॉब्लेम सोडवायची, बायनरी मोजणीच्या पॅटर्नशी अगदी समांतर आहे.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "शेवटचा बिट फ्लिप करा, एकदा रोल करा, शेवटचा बिट फ्लिप करा.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "समान रीतीने, बायनरीमध्ये 111 पर्यंत मोजणे 3 पर्यंत मोजणे, सर्व तीन बिट्सवर फिरणे, नंतर आणखी तीन मोजणे समाविष्ट आहे.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "सर्व स्केलवर, दोन्ही प्रक्रियांमध्ये समान ब्रेकडाउन आहे.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "त्यामुळे एका अर्थाने, हे बायनरी सोल्यूशन कार्य करते याचे कारण किंवा किमान एक स्पष्टीकरण, मला असे वाटते की याचे कोणतेही स्पष्टीकरण नाही, परंतु मला वाटते की सर्वात नैसर्गिक आहे की तुम्ही या बायनरी संख्या तयार करण्यासाठी वापरत असलेल्या पॅटर्नमध्ये अगदी समान आहे. हनोईच्या टॉवर्ससाठी तुम्ही वापरत असलेल्या पॅटर्नप्रमाणे रचना करा, म्हणूनच जर तुम्ही बिट्स फ्लिपिंग पाहिल्यास, तुम्ही ही प्रक्रिया प्रभावीपणे उलट करत आहात.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "तुम्ही म्हणता, हे कोणत्या प्रक्रियेतून निर्माण झाले?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "ते खूपच छान आहे, बरोबर?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "पण प्रत्यक्षात ते थंड होते.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "हे सिएरपिन्स्कीच्या त्रिकोणाशी कसे संबंधित आहे हे मला समजले नाही.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "हा व्हिडिओ आणि पुढचा व्हिडिओ देखील Desmos द्वारे काही प्रमाणात समर्थित आहे, आणि पुढील व्हिडिओच्या आधी मला फक्त थोडा वेळ घ्यायचा आहे आणि ते कोण आहेत आणि ते नेमले आहेत याबद्दल थोडेसे सामायिक करू इच्छितो.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "त्यामुळे Desmos खरोखर छान आहे.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "त्यांच्या ऑफरचे खरे मांस त्यांच्या वर्गातील क्रियाकलापांमध्ये आहे.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "माझ्या भागासाठी, अध्यापनशास्त्रीय दृष्टिकोनातून या क्रियाकलाप किती सुविचारित आहेत हे पाहून मी खूप प्रभावित झालो आहे.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "आणि मी म्हटल्याप्रमाणे, ते कामावर घेत आहेत.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "वैयक्तिकरित्या, मला वाटते की ते काही खरोखर अर्थपूर्ण गोष्टी करत आहेत.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/persian/sentence_translations.json b/2016/hanoi-and-sierpinski/persian/sentence_translations.json index ba603180f..bfc4b9108 100644 --- a/2016/hanoi-and-sierpinski/persian/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/persian/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg. ", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg. ", "translatedText": "فکر می کنید این دیسک ها دارای سوراخی در وسط هستند تا بتوانید آنها را روی یک میخ قرار دهید. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 109.02 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. ", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associating ", "translatedText": "هر توصیفی از باینری معمولاً با یک درون نگری در مورد روش معمول ما برای نشان دادن اعداد شروع می شود، چیزی که ما آن را پایه 10 می نامیم، زیرا از 10 رقم جداگانه استفاده می کنیم، 0، 1، 2، 3، 4، 5، 6، 7، 8، 9. . ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar. ", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits. ", "translatedText": "به این ترتیب، ریتم شمارش به نوعی خود مشابه است. ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable. ", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover. ", "translatedText": "ریتم شمارش در حال حاضر بسیار سریعتر است، اما این تقریبا آن را بیشتر قابل توجه می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once. ", + "input": "ce. Flip the last, roll over twice. Flip the last, roll over once. ", "translatedText": "آخری را ورق بزنید، یک بار بغلتانید. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 286.76 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again. ", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then ", "translatedText": "در مقیاس بزرگ‌تر، مانند شمارش تا 15، که 1-1-1-1 است، فرآیند به این صورت است که اجازه می‌دهیم 3 بیت آخر تا 7 بشمارند، به جای 8 بچرخند، سپس اجازه دهیم 3 بیت آخر دوباره شمارش شوند. . ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 296.14 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again. ", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till th ", "translatedText": "با شمارش تا 255، که 8 عدد 1 متوالی است، به نظر می رسد که اجازه دهید تا 7 بیت آخر شمارش شود تا زمانی که پر شوند، به جای 128 بچرخید، سپس اجازه دهید 7 بیت آخر دوباره شمارش شود. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 313.24 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. ", + "input": "ey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. ", "translatedText": "خوب، پس با آن مقدمه کوچک، واقعیت شگفت انگیزی که کیث به من نشان داد این است که ما می توانیم از این ریتم برای حل برج های هانوی استفاده کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 324.72 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right. ", + "input": "lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0. ", "translatedText": "هر زمان که فقط آخرین بیت را برگردانید، از 0 به 1، دیسک 0 را یک میخ به سمت راست حرکت دهید. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 340.8 }, { - "input": "Where do you move it, you might ask? ", + "input": "There's something magical At the small scale, say counting up to 3, which is 11 in binary, this means flip the last bit, rol ", "translatedText": "ممکن است بپرسید آن را به کجا منتقل می کنید؟ ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 345.94 }, { - "input": "You have no choice, you can't put it on top of disk 0, and there's only one other peg, so you move it where you're forced to move it. ", + "input": "l over to the twos, then flip the last bit. At a larger scale, like counting up to 15, which is 1111 in binary, the process is to let the last 3 count up to 7, roll over to the eights place, then let the ", "translatedText": "شما چاره ای ندارید، نمی توانید آن را روی دیسک 0 قرار دهید، و فقط یک میخ دیگر وجود دارد، بنابراین آن را به جایی که مجبور به حرکت آن هستید حرکت می دهید. ", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 386.02 }, { - "input": "There's something magical about it, like when I first saw this, I was like, this can't work. ", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to sol ", "translatedText": "چیزی جادویی در آن وجود دارد، مثلاً وقتی برای اولین بار این را دیدم، فکر کردم این کار نمی‌تواند کار کند. ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 390.82 }, { - "input": "I don't know how this works, I don't know why this works, now I know, but it's just magical when you see it, and I remember putting together animation for this for when I was teaching this, and just like, you know, I know how this works, I know all the things in it, it's still fun to just sit and just like, you know, watch it play out. ", + "input": "ve the towers of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective. Disk 3 is thinking, okay, 2, 1, and 0, like you have to get off of me, like I can't really functi ", "translatedText": "من نمی‌دانم این چگونه کار می‌کند، نمی‌دانم چرا این کار می‌کند، حالا می‌دانم، اما وقتی آن را می‌بینید فقط جادویی است، و یادم می‌آید برای زمانی که این کار را آموزش می‌دادم، انیمیشنی را برای این کار جمع کردم، و درست مثل می‌دانی، من می‌دانم این چگونه کار می‌کند، من همه چیزهای موجود در آن را می‌دانم، هنوز هم لذت‌بخش است که فقط بنشینی و دوست داشته باشی، می‌دانی، بازی آن را تماشا کنی. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 405.08 }, { - "input": "Oh yeah. ", + "input": "on under this much weight and pressure. And so just from disk 3's perspecti If, in your binary co ", "translatedText": "اوه بله. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 405.26 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves. ", + "input": "unting, you roll over once to the twos place, meaning you flip the last two bits, you move dis ", "translatedText": "منظورم این است که حتی در ابتدا مشخص نیست که این همیشه حرکت های قانونی را انجام می دهد. ", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 406.02 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move? ", + "input": "k number 1. Where do you move it, you might ask? Well, you have no choice. this disk to work, I can turn my bigger problem into something slightly smaller. And then how do I solve that? Well, it's exact ", "translatedText": "برای مثال، از کجا می‌دانید که هر بار که به مکان 8 می‌چرخید، دیسک 3 لزوماً برای حرکت آزاد می‌شود؟ ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 410.96 }, { - "input": "At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than having to do 2 to the n minus 1 steps? ", + "input": "ly the same thing. If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to where it needs to go. Then disk 1 and 0 can ", "translatedText": "در همان زمان، راه حل بلافاصله این سؤالات را ایجاد می کند مانند، این از کجا می آید، چرا کار می کند، و آیا راهی بهتر از انجام 2 تا n منهای 1 برای انجام این کار وجود دارد؟ ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 446.6 }, { - "input": "That's the only way it can move. ", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. ", "translatedText": "این تنها راهی است که می تواند حرکت کند. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 459.04 }, { - "input": "And then disk 3 says, I'm set. ", + "input": "And if you think about it for a bit, it becomes clear that this has to be the ", "translatedText": "و سپس دیسک 3 می گوید، من تنظیم شده ام. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 461.22 }, { - "input": "Everyone else just figure out how to get here. ", + "input": "solution. At every step, you're only doing what's forced upon you. You have to get disk 0 ", "translatedText": "بقیه فقط بفهمند که چگونه به اینجا برسند. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 470.54 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C. ", + "input": "e you can move disk 3. And you have to move disk 3. Flip the last two, move disk 1. Flip the last, move disk 0. And here, we're going to have to roll over three times t ", "translatedText": "اکنون دیسک 0، 1 و 2 را دارید که روی اسپیندل B نشسته اند، باید آنها را به C برسانید. ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 495.54 }, { - "input": "If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. ", + "input": "I don't know how this works, I don't know why this works. Now I know, but it's jus ", "translatedText": "اگر قرار است دیسک 2 بگوید، دیسک 1 و دیسک 0، شما نیستید، من هستم. ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 503.26 }, { - "input": "They need to move somewhere. ", + "input": "remember putting together an animation for this when I was teaching this, and just like, I know how this works. ", "translatedText": "آنها باید به جایی حرکت کنند. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 511.26 }, { - "input": "Then disk 1 and 0 can do this. ", + "input": "ove disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. ", "translatedText": "سپس دیسک 1 و 0 می توانند این کار را انجام دهند. ", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 522.24 }, { - "input": "Then I'm going to move. ", + "input": "he last bit. It's still fun to just sit and just watch it play out. At a slightly larger scale, solving towers of H ", "translatedText": "سپس من قصد دارم حرکت کنم. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 549.3 }, { - "input": "At every step, you're only doing what's forced upon you. ", + "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counti ", "translatedText": "در هر مرحله، شما فقط کاری را انجام می دهید که به شما تحمیل شده است. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 557.16 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3. ", + "input": "ng up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this wor ", "translatedText": "قبل از اینکه بتوانید دیسک 3 را جابجا کنید، باید دیسک 0 تا 2 را خاموش کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 562.28 }, { - "input": "And you have to move disk 3. ", + "input": "k, and is there a better way of doing this than having to do 2 to the n minus 1 steps? ", "translatedText": "و باید دیسک 3 را جابجا کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 568.24 }, { - "input": "So why does counting in binary capture this algorithm? ", + "input": "t only does this solve Towers of Hanoi, but it does it in the most efficient way possible. Understanding why this works and how it works and what ", "translatedText": "پس چرا شمارش در باینری این الگوریتم را به تصویر می‌کشد؟ ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 602.8 }, { - "input": "For example, at a pretty small scale, solving towers of Hanoi for two disks, move disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. ", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. T ", "translatedText": "به عنوان مثال، در یک مقیاس بسیار کوچک، حل برج های هانوی برای دو دیسک، حرکت دیسک 0، حرکت دیسک 1، حرکت دیسک 0، با شمارش تا 3 در باینری منعکس می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 617.18 }, { - "input": "Flip the last bit, roll over once, flip the last bit. ", + "input": "hat's the only way it can move. access to these videos before I publish the ", "translatedText": "بیت آخر را ورق بزنید، یک بار بغلتانید، بیت آخر را ورق بزنید. ", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 686.98 }, { - "input": "But it actually gets cooler. ", + "input": "s disk to work, I can turn my bigger problem into something slightly smaller. ", "translatedText": "اما در واقع خنک تر می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 689.96 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle. ", + "input": "and students, you can check out the careers page that I've linked in the description. Personally, I think they'r ", "translatedText": "من حتی به ارتباط این مثلث سیرپینسکی هم نرسیدم. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 691.36 }, { - "input": "And that's exactly what I'm going to do in the follow-on video, part 2. ", + "input": "Well, it's exactly the same thing. Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space. Get off. ", "translatedText": "و این دقیقاً همان کاری است که من در ویدیوی بعدی، قسمت 2 انجام خواهم داد. ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 714.02 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring. ", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra ", "translatedText": "این ویدیو و ویدیوی بعدی نیز تا حدی توسط دسموس پشتیبانی می‌شوند، و قبل از ویدیوی بعدی، من فقط می‌خواهم کمی وقت بگذارم و کمی در مورد اینکه آنها چه کسانی هستند و اینکه در حال استخدام هستند به اشتراک بگذارم. ", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 732.08 }, { - "input": "The real meat of their offering is in their classroom activities. ", + "input": "ecomes clear that this has to be the most efficient solution. At every step, you're only doing what's forced upon you. ", "translatedText": "گوشت واقعی ارائه آنها در فعالیت های کلاسی آنهاست. ", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 740.96 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint. ", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it. ", "translatedText": "به نوبه خود، از این که این فعالیت ها از نقطه نظر آموزشی چقدر خوب فکر شده اند، بسیار تحت تأثیر قرار گرفته ام. ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 750.34 }, { - "input": "And like I said, they're hiring. ", + "input": "So why does counting in binary capture this algorithm? Well, what's going on ", "translatedText": "و همانطور که گفتم، آنها در حال استخدام هستند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/portuguese/sentence_translations.json b/2016/hanoi-and-sierpinski/portuguese/sentence_translations.json index cad1f0749..10662ce62 100644 --- a/2016/hanoi-and-sierpinski/portuguese/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/portuguese/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "Você pensa nesses discos como tendo um buraco no meio para que você possa encaixá-los em um pino.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "Qualquer descrição de binário normalmente começa com uma introspecção sobre nossa maneira usual de representar números, o que chamamos de base 10, já que usamos 10 dígitos separados, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "Então, quando os novos dígitos acabarem, você expressa o próximo número, 10, com dois dígitos, 1, 0.", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "Desta forma, o ritmo da contagem é auto-semelhante.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "Mesmo se você diminuir o zoom para uma escala maior, o processo parece fazer algo, rolar, fazer a mesma coisa, rolar e repetir 9 vezes antes de um rollover ainda maior.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "O ritmo da contagem é agora muito mais rápido, mas isso quase o torna mais perceptível.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "Vire o último e role uma vez.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "Em uma escala maior, como contar até 15, que é 1-1-1-1, o processo consiste em deixar os últimos 3 contarem até 7, passar para a casa do 8 e depois deixar os últimos 3 bits contarem novamente.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "Contar até 255, que são 8 1s sucessivos, é como deixar os últimos 7 bits contarem até ficarem cheios, passar para o lugar dos 128 e depois deixar os últimos 7 bits contarem novamente.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "Tudo bem, então com essa mini-introdução, o fato surpreendente que Keith me mostrou é que podemos usar esse ritmo para resolver as torres de Hanói.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "Sempre que você estiver invertendo apenas o último bit, de 0 para 1, mova o disco 0 um pino para a direita.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "Quero dizer, a princípio nem está claro se isso sempre dará movimentos legais.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "Por exemplo, como você sabe que toda vez que você passa para a posição 8, o disco 3 será necessariamente liberado para se mover?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "Entender por que isso funciona, como funciona e o que diabos está acontecendo se resume a uma certa perspectiva do quebra-cabeça, o que o pessoal do CS pode chamar de perspectiva recursiva.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "Essa é a única maneira pela qual ele pode se mover.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "Feito isso, podemos mover o disco 3 para lá.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "E então o disco 3 diz, estou pronto.", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "E, de certa forma, agora você tem uma versão menor do mesmo problema.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "Agora que você tem os discos 0, 1 e 2 no fuso B, você precisa levá-los para C.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "Eles precisam se mudar para algum lugar.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "Ok, todos voltem.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "E se você pensar um pouco, fica claro que essa deve ser a solução mais eficiente.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "Você precisa desligar o disco 0 a 2 antes de mover o disco 3.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "E você tem que mover o disco 3.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "Simplesmente não há espaço para ineficiência nesta perspectiva.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "Então, por que a contagem em binário captura esse algoritmo?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "Bem, o que está acontecendo aqui é que esse padrão de resolver um subproblema, mover um disco grande e depois resolver um subproblema novamente, é perfeitamente paralelo ao padrão de contagem binária.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "Vire a última parte, role uma vez, vire a última parte.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "Analogamente, contar até 111 em binário envolve contar até 3, rolar todos os três bits e depois contar mais três.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "Em todas as escalas, ambos os processos apresentam o mesmo colapso.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "Então, de certa forma, a razão pela qual esta solução binária funciona, ou pelo menos uma explicação, sinto que não há uma explicação, mas acho que a mais natural é que o padrão que você usaria para gerar esses números binários tenha exatamente o mesmo estrutura como o padrão que você usaria para as torres de Hanói, e é por isso que se você observar a inversão dos bits, estará efetivamente revertendo esse processo.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "Você está dizendo: que processo gerou isso?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "Isso é muito legal, certo?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "Mas na verdade fica mais legal.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "Ainda nem entendi como isso se relaciona com o triângulo de Sierpinski.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "Este vídeo e o próximo também são apoiados em parte pela Desmos, e antes do próximo vídeo eu só quero parar um momento e compartilhar com vocês um pouco sobre quem eles são e o fato de que estão contratando.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "Então Desmos é realmente muito legal.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "A verdadeira essência de sua oferta está nas atividades em sala de aula.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "De minha parte, estou super impressionado com o quão bem pensadas essas atividades são do ponto de vista pedagógico.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "E como eu disse, eles estão contratando.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "Pessoalmente, acho que eles estão fazendo coisas realmente significativas.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/russian/sentence_translations.json b/2016/hanoi-and-sierpinski/russian/sentence_translations.json index 76e5279f7..1ca27e0a4 100644 --- a/2016/hanoi-and-sierpinski/russian/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/russian/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "Вы думаете, что эти диски имеют отверстие посередине, чтобы их можно было надеть на крючок.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "Любое описание двоичного кода обычно начинается с самоанализа нашего обычного способа представления чисел, который мы называем десятичной системой, поскольку мы используем 10 отдельных цифр: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "Затем, когда новые цифры закончатся, вы выражаете следующее число 10 двумя цифрами 1 и 0.", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "Таким образом, ритм счета является своего рода самоподобным.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "Даже если вы увеличите масштаб, процесс будет выглядеть как что-то сделать, перевернуться, сделать то же самое, перевернуться и повторить 9 раз, прежде чем перевернуться еще больше.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "Ритм счета теперь намного быстрее, но это делает его чуть ли не более заметным.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "Переверните последний, переверните один раз.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "В более крупном масштабе, например, при счете до 15, то есть 1-1-1-1, процесс состоит в том, чтобы позволить последним 3 битам считать до 7, перевернуться на 8-е место, а затем позволить последним 3 битам подсчитать снова.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "Считая до 255, то есть 8 последовательных единиц, это похоже на то, что последние 7 бит подсчитываются до тех пор, пока они не заполнятся, с переходом на место 128, а затем снова подсчитываются последним 7 битам.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "Итак, после этого мини-представления, Кит показал мне удивительный факт: мы можем использовать этот ритм для решения башен Ханоя.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "Всякий раз, когда вы меняете только этот последний бит с 0 на 1, переместите диск 0 на один колышек вправо.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "Я имею в виду, что поначалу даже неясно, всегда ли это приведет к законным действиям.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "Например, откуда вы знаете, что каждый раз, когда вы перекатываетесь на цифру 8, диск 3 обязательно освобождается для перемещения?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "Понимание того, почему это работает, как это работает и что, черт возьми, происходит, сводится к определенному взгляду на головоломку, который специалисты по компьютерной технике могли бы назвать рекурсивной перспективой.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "Только так он может двигаться.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "Сделав это, мы сможем переместить туда диск 3.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "И тут диск 3 говорит: я готов.", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "И в каком-то смысле теперь у вас есть уменьшенная версия той же проблемы.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "Теперь у вас есть диски 0, 1 и 2, находящиеся на шпинделе B, вам нужно перенести их в C.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "Им нужно куда-то переехать.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "Хорошо, все возвращаются.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "И если немного подумать, становится ясно, что это должно быть наиболее эффективное решение.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "Вам необходимо отключить диски с 0 по 2, прежде чем вы сможете переместить диск 3.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "И вам нужно переместить диск 3.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "С этой точки зрения просто нет места неэффективности.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "Так почему же двоичный подсчет фиксирует этот алгоритм?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "Что здесь происходит, так это то, что этот шаблон решения подзадачи, перемещения большого диска и последующего решения подзадачи снова прекрасно параллелен шаблону двоичного счета.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "Переверните последний бит, переверните один раз, переверните последний бит.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "Аналогично, двоичный счет до 111 включает в себя счет до 3, прокрутку всех трех битов и затем досчет еще трех.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "На всех уровнях оба процесса имеют одинаковую структуру.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "Итак, в каком-то смысле причина, по которой это двоичное решение работает, или, по крайней мере, объяснение, я чувствую, что не существует единого объяснения, но я думаю, что наиболее естественным является то, что шаблон, который вы бы использовали для генерации этих двоичных чисел, имеет точно такой же структура похожа на шаблон, который вы бы использовали для ханойских башен, поэтому, если вы посмотрите на переворачивание битов, вы фактически обращаете этот процесс вспять.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "Вы говорите, какой процесс их породил?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "Это довольно круто, правда?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "Но на самом деле становится прохладнее.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "Я даже не понял, как это связано с треугольником Серпинского.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "Это и следующее видео также частично поддерживаются Desmos, и перед следующим видео я просто хочу воспользоваться моментом и рассказать вам, ребята, немного о том, кто они и о том, что они нанимают.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "Так что Десмос на самом деле очень крутой.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "Настоящая суть их предложений – это занятия в классе.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "Я, со своей стороны, очень впечатлен тем, насколько хорошо продуманы эти занятия с педагогической точки зрения.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "И, как я уже сказал, они нанимают сотрудников.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "Лично я думаю, что они делают действительно значимые вещи.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/spanish/sentence_translations.json b/2016/hanoi-and-sierpinski/spanish/sentence_translations.json index eb5bac99d..14af80eea 100644 --- a/2016/hanoi-and-sierpinski/spanish/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/spanish/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "Piensas que estos discos tienen un agujero en el medio para poder colocarlos en una clavija.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "Cualquier descripción del binario generalmente comienza con una introspección sobre nuestra forma habitual de representar números, lo que llamamos base 10, ya que usamos 10 dígitos separados, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "Luego, al quedarse sin nuevos dígitos, expresas el siguiente número, 10, con dos dígitos, 1, 0.", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "De esta manera, el ritmo de conteo es algo similar.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "Incluso si se aleja a una escala mayor, el proceso parece hacer algo, girar, hacer lo mismo, girar y repetir 9 veces antes de un desplazamiento aún mayor.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "El ritmo de conteo es ahora mucho más rápido, pero eso casi lo hace más notorio.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "Voltee el último, déle la vuelta una vez.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "A una escala mayor, como contar hasta 15, que es 1-1-1-1, el proceso consiste en dejar que los últimos 3 cuenten hasta 7, pasar al lugar de los 8 y luego dejar que los últimos 3 bits cuenten nuevamente.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "Contar hasta 255, que son 8 unos sucesivos, parece dejar que los últimos 7 bits cuenten hasta que estén llenos, pasar al lugar de 128 y luego dejar que los últimos 7 bits cuenten nuevamente.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "Muy bien, con esa mini introducción, el hecho sorprendente que Keith me mostró es que podemos usar este ritmo para resolver las torres de Hanoi.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "Siempre que solo estés volteando el último bit, de 0 a 1, mueve el disco 0 una clavija hacia la derecha.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "Quiero decir, al principio ni siquiera está claro que esto siempre vaya a dar lugar a movimientos legales.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "Por ejemplo, ¿cómo sabes que cada vez que te desplazas hacia el lugar del 8, el disco 3 necesariamente quedará libre para moverse?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "Comprender por qué funciona esto, cómo funciona y qué diablos está pasando se reduce a una cierta perspectiva del rompecabezas, lo que la gente de CS podría llamar una perspectiva recursiva.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "Esa es la única forma en que puede moverse.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "Una vez hecho esto, podemos mover el disco 3 allí.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "Y luego el disco 3 dice: Estoy listo.", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "Y en cierto sentido, ahora tienes una versión más pequeña del mismo problema.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "Ahora que tiene los discos 0, 1 y 2 en el eje B, debe llevarlos a C.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "Necesitan mudarse a alguna parte.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "Bien, todos vuelvan a ponerse manos a la obra.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "Y si lo piensas un poco, queda claro que esta tiene que ser la solución más eficiente.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "Debe quitar los discos 0 a 2 antes de poder mover el disco 3.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "Y hay que mover el disco 3.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "Desde esta perspectiva, simplemente no hay lugar para la ineficiencia.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "Entonces, ¿por qué el conteo binario captura este algoritmo?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "Bueno, lo que está sucediendo aquí es que este patrón de resolver un subproblema, mover un disco grande y luego resolver un subproblema nuevamente, es perfectamente paralelo al patrón de conteo binario.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "Voltee el último trozo, déle la vuelta una vez, voltee el último trozo.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "De manera análoga, contar hasta 111 en binario implica contar hasta 3, pasar los tres bits y luego contar tres más.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "En todas las escalas, ambos procesos tienen el mismo desglose.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "Entonces, en cierto sentido, la razón por la que esta solución binaria funciona, o al menos una explicación, siento que no hay una explicación única, pero creo que la más natural es que el patrón que usarías para generar estos números binarios tiene exactamente el mismo. estructura como el patrón que usarías para las torres de Hanoi, razón por la cual si observas los bits volteándose, efectivamente estás invirtiendo este proceso.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "Estás diciendo, ¿qué proceso generó estos?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "Eso es genial, ¿verdad?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "Pero en realidad hace más frío.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "Ni siquiera he llegado a explicar cómo se relaciona esto con el triángulo de Sierpinski.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "Este video y el siguiente también cuentan con el apoyo parcial de Desmos, y antes del siguiente video solo quiero tomarme un momento y compartir con ustedes un poco sobre quiénes son y el hecho de que están contratando.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "Entonces Desmos es realmente genial.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "La verdadera esencia de su oferta está en sus actividades en el aula.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "Por mi parte, estoy muy impresionado por lo bien pensadas que están estas actividades desde el punto de vista pedagógico.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "Y como dije, están contratando.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "Personalmente, creo que están haciendo cosas realmente significativas.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/tamil/sentence_translations.json b/2016/hanoi-and-sierpinski/tamil/sentence_translations.json index 9982237dd..02c1f41c8 100644 --- a/2016/hanoi-and-sierpinski/tamil/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/tamil/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "இந்த வட்டுகளுக்கு நடுவில் ஒரு துளை இருப்பதாக நீங்கள் நினைக்கிறீர்கள், அதனால் அவற்றை ஒரு பெக்கில் பொருத்தலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "0, 1, 2, 3, 4, 5, 6, 7, 8, 9 என்ற தனித்தனி இலக்கங்களைப் பயன்படுத்துவதால், பைனரியின் எந்த விளக்கமும், எண்களைக் குறிக்கும் வழக்கமான வழியைப் பற்றிய சுயபரிசோதனையுடன் தொடங்குகிறது, அடிப்படை 10 என்று அழைக்கிறோம்.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "பின்னர், புதிய இலக்கங்கள் தீர்ந்துவிட்டதால், அடுத்த எண்ணான 10 ஐ இரண்டு இலக்கங்களுடன், 1, 0 ஐ வெளிப்படுத்துங்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "இந்த வழியில், எண்ணும் தாளம் சுயமாக ஒத்திருக்கிறது.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "நீங்கள் பெரிய அளவில் பெரிதாக்கினாலும், செயல்முறையானது ஏதாவது செய்வது, உருட்டுவது, அதையே செய்வது, உருட்டுவது மற்றும் இன்னும் பெரிய மாற்றத்திற்கு முன் 9 முறை மீண்டும் செய்வது போன்ற தோற்றமளிக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "எண்ணும் தாளம் இப்போது மிக வேகமாக உள்ளது, ஆனால் அது கிட்டத்தட்ட அதை மேலும் கவனிக்க வைக்கிறது.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "கடைசியாக புரட்டவும், ஒரு முறை உருட்டவும்.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "பெரிய அளவில், 15 வரை எண்ணுவது போல, அதாவது 1-1-1-1, செயல்முறையானது கடைசி 3 ஐ 7 வரை எண்ணி, 8 இன் இடத்திற்கு உருட்டவும், கடைசி 3 பிட்களை மீண்டும் கணக்கிடவும்.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "255 வரை எண்ணினால், அதாவது 8 அடுத்தடுத்த 1கள், கடைசி 7 பிட்கள் நிரம்பும் வரை எண்ண விடாமல், 128 இன் இடத்திற்குச் சென்று, கடைசி 7 பிட்களை மீண்டும் கணக்கிட அனுமதிப்பது போல் தெரிகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "சரி, அந்த சிறிய அறிமுகத்துடன், கீத் எனக்குக் காட்டிய ஆச்சரியமான உண்மை என்னவென்றால், ஹனோயின் கோபுரங்களைத் தீர்க்க இந்த ரிதத்தைப் பயன்படுத்தலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "நீங்கள் கடைசி பிட்டை மட்டும் புரட்டினால், 0 முதல் 1 வரை, வட்டு 0 ஒரு பெக் வலதுபுறமாக நகர்த்தவும்.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "அதாவது, இது எப்பொழுதும் சட்டப்பூர்வ நகர்வுகளைக் கொடுக்கப் போகிறது என்பது கூட முதலில் தெளிவாகத் தெரியவில்லை.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "எடுத்துக்காட்டாக, ஒவ்வொரு முறையும் நீங்கள் 8 இன் இடத்திற்குச் செல்லும்போது, அந்த வட்டு 3 நகர்த்துவதற்கு அவசியமாக விடுவிக்கப்படும் என்பதை நீங்கள் எப்படி அறிவீர்கள்?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "இது ஏன் வேலை செய்கிறது மற்றும் எப்படி வேலை செய்கிறது மற்றும் என்ன நடக்கிறது என்பதைப் புரிந்துகொள்வது புதிரின் ஒரு குறிப்பிட்ட கண்ணோட்டத்திற்கு வருகிறது, சிஎஸ் மக்கள் எதைச் சுழல்நிலை முன்னோக்கு என்று அழைக்கலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "அதன் மூலம் தான் நகர முடியும்.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "அதைச் செய்த பிறகு, வட்டு 3 ஐ அங்கு நகர்த்தலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "பின்னர் வட்டு 3 கூறுகிறது, நான் அமைக்கப்பட்டுவிட்டேன்.", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "ஒரு வகையில், இப்போது அதே சிக்கலின் சிறிய பதிப்பு உங்களிடம் உள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "இப்போது நீங்கள் வட்டு 0, 1 மற்றும் 2 சுழல் B இல் அமர்ந்திருக்கிறீர்கள், நீங்கள் அவற்றை C க்கு கொண்டு வர வேண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "அவர்கள் எங்காவது செல்ல வேண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "சரி, எல்லோரும் மீண்டும் குவியுங்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "நீங்கள் கொஞ்சம் யோசித்துப் பார்த்தால், இது மிகவும் திறமையான தீர்வாக இருக்க வேண்டும் என்பது தெளிவாகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "வட்டு 3 ஐ நகர்த்துவதற்கு முன் நீங்கள் வட்டு 0 முதல் 2 வரை தள்ளுபடி பெற வேண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "நீங்கள் வட்டு 3 ஐ நகர்த்த வேண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "இந்த கண்ணோட்டத்தில் திறமையின்மைக்கு எந்த இடமும் இல்லை.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "பைனரியில் எண்ணுவது ஏன் இந்த அல்காரிதத்தைப் பிடிக்கிறது?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "சரி, இங்கே என்ன நடக்கிறது என்றால், ஒரு துணைப் பிரச்சனையைத் தீர்ப்பது, ஒரு பெரிய வட்டை நகர்த்துவது, பின்னர் ஒரு துணைப் பிரச்சனையைத் தீர்ப்பது போன்ற இந்த முறை பைனரி எண்ணும் முறைக்கு இணையாக உள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "கடைசி பிட்டை புரட்டவும், ஒரு முறை உருட்டவும், கடைசி பிட்டை புரட்டவும்.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "ஒத்ததாக, பைனரியில் 111 வரை எண்ணுவது, 3 வரை எண்ணுவது, மூன்று பிட்கள் மீதும் உருட்டுவது, பிறகு மேலும் மூன்றைக் கணக்கிடுவது ஆகியவை அடங்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "அனைத்து அளவீடுகளிலும், இரண்டு செயல்முறைகளும் ஒரே முறிவைக் கொண்டுள்ளன.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "ஒரு வகையில், இந்த பைனரி தீர்வு செயல்படுவதற்கான காரணம், அல்லது குறைந்தபட்சம் ஒரு விளக்கமாவது, எந்த விளக்கமும் இல்லை என்று நான் உணர்கிறேன், ஆனால் இந்த பைனரி எண்களை உருவாக்க நீங்கள் பயன்படுத்தும் பேட்டர்ன் சரியாக இருப்பதுதான் மிகவும் இயல்பானது என்று நினைக்கிறேன். ஹனோய் கோபுரங்களுக்கு நீங்கள் பயன்படுத்தும் மாதிரியின் அமைப்பு, அதனால்தான் பிட்கள் புரட்டுவதைப் பார்த்தால், இந்த செயல்முறையை நீங்கள் திறம்பட மாற்றுகிறீர்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "நீங்கள் சொல்கிறீர்கள், எந்த செயல்முறை இதை உருவாக்கியது?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "அது மிகவும் அருமையாக இருக்கிறது, இல்லையா?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "ஆனால் அது உண்மையில் குளிர்ச்சியாகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "இது சியர்பின்ஸ்கியின் முக்கோணத்துடன் எவ்வாறு தொடர்புடையது என்று கூட எனக்குப் புரியவில்லை.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "இந்த வீடியோவும் அடுத்த வீடியோவும் டெஸ்மோஸால் ஓரளவு ஆதரிக்கப்படுகிறது, அடுத்த வீடியோவிற்கு முன் நான் சிறிது நேரம் ஒதுக்கி, அவர்கள் யார் மற்றும் அவர்கள் பணியமர்த்துகிறார்கள் என்பதைப் பற்றி கொஞ்சம் உங்களுடன் பகிர்ந்து கொள்ள விரும்புகிறேன்.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "எனவே டெஸ்மோஸ் உண்மையில் மிகவும் அருமையாக உள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "அவர்களின் பிரசாதத்தின் உண்மையான இறைச்சி அவர்களின் வகுப்பறை நடவடிக்கைகளில் உள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "என் பங்கிற்கு, கல்வியியல் நிலைப்பாட்டில் இருந்து இந்தச் செயல்பாடுகள் எவ்வளவு நன்றாகச் சிந்திக்கப்படுகின்றன என்பதில் நான் மிகவும் ஈர்க்கப்பட்டேன்.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "நான் சொன்னது போல், அவர்கள் பணியமர்த்துகிறார்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "தனிப்பட்ட முறையில், அவர்கள் சில அர்த்தமுள்ள விஷயங்களைச் செய்கிறார்கள் என்று நான் நினைக்கிறேன்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/telugu/sentence_translations.json b/2016/hanoi-and-sierpinski/telugu/sentence_translations.json index ec8cd4e65..beb2b3879 100644 --- a/2016/hanoi-and-sierpinski/telugu/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/telugu/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "మీరు ఈ డిస్క్‌లను ఒక పెగ్‌లో అమర్చడానికి మధ్యలో రంధ్రం ఉన్నట్లు భావిస్తారు.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "బైనరీ యొక్క ఏదైనా వివరణ సాధారణంగా సంఖ్యలను సూచించడానికి మన సాధారణ మార్గం గురించి ఆత్మపరిశీలనతో ప్రారంభమవుతుంది, మనం బేస్ 10 అని పిలుస్తాము, ఎందుకంటే మేము 10 వేర్వేరు అంకెలను ఉపయోగిస్తాము, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "అప్పుడు, కొత్త అంకెలు అయిపోయిన తర్వాత, మీరు తదుపరి సంఖ్య, 10, రెండు అంకెలతో, 1, 0ని వ్యక్తపరుస్తారు.", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "ఈ విధంగా, లెక్కింపు యొక్క లయ ఒక రకమైన స్వీయ-సారూప్యంగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "మీరు పెద్ద స్థాయికి జూమ్ అవుట్ చేసినప్పటికీ, ప్రక్రియ ఏదైనా చేయడం, రోలింగ్ ఓవర్, అదే పని చేయడం, రోలింగ్ ఓవర్ చేయడం మరియు మరింత పెద్ద రోల్‌ఓవర్‌కు ముందు 9 సార్లు పునరావృతం చేసినట్లు కనిపిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "లెక్కింపు యొక్క లయ ఇప్పుడు చాలా వేగంగా ఉంది, కానీ అది దాదాపుగా గుర్తించదగినదిగా చేస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "చివరిగా తిప్పండి, ఒకసారి తిరగండి.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "పెద్ద స్కేల్‌లో, 15 వరకు లెక్కించడం వంటిది, అంటే 1-1-1-1, చివరి 3ని 7 వరకు గణించడం, 8 యొక్క స్థానానికి వెళ్లడం, ఆపై చివరి 3 బిట్‌లను మళ్లీ లెక్కించడం వంటి ప్రక్రియ.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "255 వరకు గణించడం, అంటే 8 వరుస 1లు, చివరి 7 బిట్‌లు పూర్తి అయ్యే వరకు వాటిని లెక్కించేలా, 128 స్థానానికి రోలింగ్ చేసి, చివరి 7 బిట్‌లను మళ్లీ లెక్కించేలా చూస్తారు.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "సరే, ఆ చిన్న పరిచయంతో, కీత్ నాకు చూపించిన ఆశ్చర్యకరమైన వాస్తవం ఏమిటంటే, హనోయి టవర్‌లను పరిష్కరించడానికి మనం ఈ రిథమ్‌ని ఉపయోగించవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "మీరు ఆ చివరి బిట్‌ను 0 నుండి 1కి మాత్రమే తిప్పివేసినప్పుడల్లా, డిస్క్ 0ని ఒక పెగ్‌ని కుడివైపుకి తరలించండి.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "నా ఉద్దేశ్యం, ఇది ఎల్లప్పుడూ చట్టపరమైన కదలికలను ఇవ్వబోతోందని మొదట్లో కూడా స్పష్టంగా తెలియదు.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "ఉదాహరణకు, మీరు 8 స్థానంలోకి వెళ్లే ప్రతిసారీ, ఆ డిస్క్ 3 తప్పనిసరిగా తరలించబడుతుందని మీకు ఎలా తెలుసు?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "ఇది ఎందుకు పని చేస్తుంది మరియు ఇది ఎలా పని చేస్తుంది మరియు ఏమి జరుగుతుందో అర్థం చేసుకోవడం పజిల్‌పై ఒక నిర్దిష్ట దృక్పథానికి వస్తుంది, CS ఫోక్ రికర్సివ్ దృక్పథం అని పిలుస్తారు.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "అది కదలగల ఏకైక మార్గం.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "అలా చేసిన తర్వాత, మనం డిస్క్ 3ని అక్కడికి తరలించవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "ఆపై డిస్క్ 3 చెప్పింది, నేను సెట్ అయ్యాను.", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "మరియు ఒక కోణంలో, మీరు ఇప్పుడు అదే సమస్య యొక్క చిన్న సంస్కరణను కలిగి ఉన్నారు.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "ఇప్పుడు మీకు డిస్క్ 0, 1, మరియు 2 స్పిండిల్ Bపై కూర్చున్నాయి, మీరు వాటిని Cకి పొందాలి.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "వారు ఎక్కడికో తరలించాలి.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "సరే, అందరూ తిరిగి పోతారు.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "మరియు మీరు దాని గురించి కొంచెం ఆలోచిస్తే, ఇది అత్యంత సమర్థవంతమైన పరిష్కారం అని స్పష్టమవుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "మీరు డిస్క్ 3ని తరలించడానికి ముందు మీరు డిస్క్ 0 నుండి 2 వరకు ఆఫ్ పొందాలి.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "మరియు మీరు డిస్క్ 3ని తరలించాలి.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "ఈ దృక్కోణం నుండి అసమర్థతకు ఎటువంటి స్థలం లేదు.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "కాబట్టి బైనరీలో లెక్కింపు ఈ అల్గారిథమ్‌ను ఎందుకు సంగ్రహిస్తుంది?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "సరే, ఇక్కడ ఏమి జరుగుతోందంటే, ఉపసమస్యను పరిష్కరించడం, పెద్ద డిస్క్‌ను తరలించడం, ఆపై ఉపసమస్యను మళ్లీ పరిష్కరించడం వంటి ఈ నమూనా బైనరీ లెక్కింపు నమూనాతో ఖచ్చితంగా సమాంతరంగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "చివరి బిట్‌ను తిప్పండి, ఒకసారి తిప్పండి, చివరి బిట్‌ను తిప్పండి.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "సారూప్యంగా, బైనరీలో 111 వరకు లెక్కించడం అనేది 3 వరకు లెక్కించడం, మూడు బిట్‌లను చుట్టడం, ఆపై మరో మూడు వరకు లెక్కించడం.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "అన్ని ప్రమాణాల వద్ద, రెండు ప్రక్రియలు ఒకే విచ్ఛిన్నతను కలిగి ఉంటాయి.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "కాబట్టి ఒక కోణంలో, ఈ బైనరీ సొల్యూషన్ పని చేయడానికి కారణం, లేదా కనీసం ఒక వివరణ, ఎవరికీ వివరణ లేదని నేను భావిస్తున్నాను, కానీ ఈ బైనరీ సంఖ్యలను రూపొందించడానికి మీరు ఉపయోగించే నమూనా సరిగ్గా అదే విధంగా ఉండటం చాలా సహజమైనది అని నేను భావిస్తున్నాను. మీరు హనోయి టవర్‌ల కోసం ఉపయోగించే నమూనాగా నిర్మాణం, అందుకే మీరు బిట్‌లను తిప్పడం చూస్తే, మీరు ఈ ప్రక్రియను సమర్థవంతంగా తిప్పికొడుతున్నారు.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "మీరు చెప్తున్నారు, వీటిని ఏ ప్రక్రియ రూపొందించింది?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "అది చాలా బాగుంది, సరియైనదా?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "కానీ వాస్తవానికి ఇది చల్లగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "ఇది సియర్పిన్స్కి త్రిభుజానికి ఎలా సంబంధం కలిగి ఉందో కూడా నాకు అర్థం కాలేదు.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "ఈ వీడియో మరియు తదుపరి వీడియో కూడా డెస్మోస్ ద్వారా కొంతవరకు మద్దతిస్తోంది మరియు తదుపరి వీడియోకి ముందు నేను కొంత సమయం కేటాయించి, వారు ఎవరో మరియు వారు నియామకం చేస్తున్నారనే దాని గురించి కొంచెం మీతో పంచుకోవాలనుకుంటున్నాను.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "కాబట్టి డెస్మోస్ నిజంగా బాగుంది.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "వారి సమర్పణ యొక్క నిజమైన మాంసం వారి తరగతి గది కార్యకలాపాలలో ఉంది.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "నా వంతుగా, ఈ కార్యకలాపాలు బోధనా దృక్కోణం నుండి ఎంత బాగా ఆలోచించబడ్డాయో నేను చాలా ఆకట్టుకున్నాను.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "మరియు నేను చెప్పినట్లుగా, వారు నియామకం చేస్తున్నారు.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "వ్యక్తిగతంగా, వారు కొన్ని నిజంగా అర్ధవంతమైన అంశాలను చేస్తున్నారని నేను భావిస్తున్నాను.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/thai/sentence_translations.json b/2016/hanoi-and-sierpinski/thai/sentence_translations.json index c808f8bcb..d589245d9 100644 --- a/2016/hanoi-and-sierpinski/thai/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/thai/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg. ", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg. ", "translatedText": "คุณคิดว่าดิสก์เหล่านี้มีรูตรงกลางเพื่อให้คุณใส่เข้ากับหมุดได้ ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 109.02 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. ", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associating ", "translatedText": "โดยทั่วไปคำอธิบายใดๆ ของเลขฐานสองจะเริ่มต้นด้วยการวิปัสสนาเกี่ยวกับวิธีการปกติของเราในการแสดงตัวเลข สิ่งที่เราเรียกว่าฐาน 10 เนื่องจากเราใช้ตัวเลข 10 หลักแยกกัน ได้แก่ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 . ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar. ", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits. ", "translatedText": "ด้วยวิธีนี้ จังหวะการนับจึงคล้ายกับตัวเอง ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable. ", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover. ", "translatedText": "ตอนนี้จังหวะการนับเร็วขึ้นมาก แต่ก็เกือบจะทำให้สังเกตเห็นได้ชัดเจนยิ่งขึ้น ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once. ", + "input": "ce. Flip the last, roll over twice. Flip the last, roll over once. ", "translatedText": "พลิกอันสุดท้าย เกลือกกลิ้งหนึ่งครั้ง ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 286.76 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again. ", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then ", "translatedText": "ในสเกลที่ใหญ่กว่า เช่น การนับถึง 15 ซึ่งก็คือ 1-1-1-1 กระบวนการคือให้ 3 บิตสุดท้ายนับถึง 7 เกลือกกลิ้งไปที่ตำแหน่ง 8 จากนั้นให้ 3 บิตสุดท้ายนับอีกครั้ง . ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 296.14 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again. ", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till th ", "translatedText": "เมื่อนับได้ถึง 255 ซึ่งเป็น 8 1 ติดต่อกัน ดูเหมือนว่าปล่อยให้ 7 บิตสุดท้ายนับจนเต็ม กลิ้งไปที่ตำแหน่ง 128 จากนั้นปล่อยให้ 7 บิตสุดท้ายนับอีกครั้ง ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 313.24 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. ", + "input": "ey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. ", "translatedText": "เอาล่ะ ด้วยการแนะนำสั้นๆ ข้อเท็จจริงที่น่าประหลาดใจที่คีธแสดงให้ผมเห็นก็คือ เราสามารถใช้จังหวะนี้เพื่อไขปริศนาหอคอยแห่งฮานอยได้ ", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 324.72 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right. ", + "input": "lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0. ", "translatedText": "เมื่อใดก็ตามที่คุณพลิกเฉพาะบิตสุดท้ายนั้น จาก 0 ถึง 1 ให้ย้ายดิสก์ 0 ไปหนึ่งหมุดไปทางขวา ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 340.8 }, { - "input": "Where do you move it, you might ask? ", + "input": "There's something magical At the small scale, say counting up to 3, which is 11 in binary, this means flip the last bit, rol ", "translatedText": "ย้ายไปไหน สอบถามได้ครับ ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 345.94 }, { - "input": "You have no choice, you can't put it on top of disk 0, and there's only one other peg, so you move it where you're forced to move it. ", + "input": "l over to the twos, then flip the last bit. At a larger scale, like counting up to 15, which is 1111 in binary, the process is to let the last 3 count up to 7, roll over to the eights place, then let the ", "translatedText": "คุณไม่มีทางเลือก คุณไม่สามารถวางมันไว้บนดิสก์ 0 ได้ และยังมีหมุดอีกอันเดียว ดังนั้นคุณจึงย้ายมันไปยังตำแหน่งที่คุณถูกบังคับให้ย้าย ", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 386.02 }, { - "input": "There's something magical about it, like when I first saw this, I was like, this can't work. ", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to sol ", "translatedText": "มีบางอย่างมหัศจรรย์เกี่ยวกับมัน เหมือนตอนที่ฉันเห็นสิ่งนี้ครั้งแรก ฉันก็แบบว่า นี่มันใช้ไม่ได้ ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 390.82 }, { - "input": "I don't know how this works, I don't know why this works, now I know, but it's just magical when you see it, and I remember putting together animation for this for when I was teaching this, and just like, you know, I know how this works, I know all the things in it, it's still fun to just sit and just like, you know, watch it play out. ", + "input": "ve the towers of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective. Disk 3 is thinking, okay, 2, 1, and 0, like you have to get off of me, like I can't really functi ", "translatedText": "ฉันไม่รู้ว่ามันทำงานยังไง ฉันไม่รู้ว่าทำไมมันถึงได้ผล ตอนนี้ฉันรู้แล้ว แต่มันก็วิเศษมากเมื่อคุณเห็นมัน และฉันจำได้ว่าเอาแอนิเมชั่นสำหรับสิ่งนี้ ตอนที่ฉันสอนเรื่องนี้ และเหมือนกับว่า คุณรู้ไหม ฉันรู้ว่ามันทำงานอย่างไร ฉันรู้ทุกสิ่งในนั้น แค่นั่งดูก็สนุกแล้ว ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 405.08 }, { - "input": "Oh yeah. ", + "input": "on under this much weight and pressure. And so just from disk 3's perspecti If, in your binary co ", "translatedText": "โอ้ใช่. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 405.26 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves. ", + "input": "unting, you roll over once to the twos place, meaning you flip the last two bits, you move dis ", "translatedText": "ฉันหมายความว่า ในตอนแรกยังไม่ชัดเจนด้วยซ้ำว่าการดำเนินการทางกฎหมายจะเกิดขึ้นเสมอ ", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 406.02 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move? ", + "input": "k number 1. Where do you move it, you might ask? Well, you have no choice. this disk to work, I can turn my bigger problem into something slightly smaller. And then how do I solve that? Well, it's exact ", "translatedText": "ตัวอย่างเช่น คุณจะรู้ได้อย่างไรว่าทุกครั้งที่คุณกลิ้งไปที่ตำแหน่ง 8 ดิสก์นั้น 3 จะต้องถูกปล่อยว่างเพื่อย้าย ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 410.96 }, { - "input": "At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than having to do 2 to the n minus 1 steps? ", + "input": "ly the same thing. If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to where it needs to go. Then disk 1 and 0 can ", "translatedText": "ในขณะเดียวกัน, ผลเฉลยก็ทำให้เกิดคำถามพวกนี้ขึ้นมาทันที เช่น, มันมาจากไหน, ทำไมมันถึงได้ผล และมีวิธีใดที่ดีกว่าการทำ 2 กำลัง n ลบ 1 ขั้นไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 446.6 }, { - "input": "That's the only way it can move. ", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. ", "translatedText": "นั่นเป็นวิธีเดียวที่มันสามารถเคลื่อนที่ได้ ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 459.04 }, { - "input": "And then disk 3 says, I'm set. ", + "input": "And if you think about it for a bit, it becomes clear that this has to be the ", "translatedText": "แล้วดิสก์ 3 ก็บอกว่า "พร้อมแล้ว" ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 461.22 }, { - "input": "Everyone else just figure out how to get here. ", + "input": "solution. At every step, you're only doing what's forced upon you. You have to get disk 0 ", "translatedText": "คนอื่นๆ แค่คิดออกว่าจะมาที่นี่ได้อย่างไร ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 470.54 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C. ", + "input": "e you can move disk 3. And you have to move disk 3. Flip the last two, move disk 1. Flip the last, move disk 0. And here, we're going to have to roll over three times t ", "translatedText": "ตอนนี้คุณมีดิสก์ 0, 1 และ 2 อยู่บนสปินเดิล B คุณต้องไปที่ C ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 495.54 }, { - "input": "If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. ", + "input": "I don't know how this works, I don't know why this works. Now I know, but it's jus ", "translatedText": "ถ้าดิสก์ 2 จะบอกว่าดิสก์ 1 และดิสก์ 0 ไม่ใช่คุณ แต่เป็นฉันเอง ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 503.26 }, { - "input": "They need to move somewhere. ", + "input": "remember putting together an animation for this when I was teaching this, and just like, I know how this works. ", "translatedText": "พวกเขาจำเป็นต้องย้ายไปที่ไหนสักแห่ง ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 511.26 }, { - "input": "Then disk 1 and 0 can do this. ", + "input": "ove disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. ", "translatedText": "จากนั้นดิสก์ 1 และ 0 ก็สามารถทำได้ ", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 522.24 }, { - "input": "Then I'm going to move. ", + "input": "he last bit. It's still fun to just sit and just watch it play out. At a slightly larger scale, solving towers of H ", "translatedText": "แล้วฉันจะย้าย. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 549.3 }, { - "input": "At every step, you're only doing what's forced upon you. ", + "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counti ", "translatedText": "ในทุกขั้นตอน คุณจะทำเฉพาะสิ่งที่ถูกบังคับเท่านั้น ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 557.16 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3. ", + "input": "ng up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this wor ", "translatedText": "คุณต้องปิดดิสก์ 0 ถึง 2 ก่อนจึงจะสามารถย้ายดิสก์ 3 ได้ ", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 562.28 }, { - "input": "And you have to move disk 3. ", + "input": "k, and is there a better way of doing this than having to do 2 to the n minus 1 steps? ", "translatedText": "และคุณต้องย้ายดิสก์ 3 ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 568.24 }, { - "input": "So why does counting in binary capture this algorithm? ", + "input": "t only does this solve Towers of Hanoi, but it does it in the most efficient way possible. Understanding why this works and how it works and what ", "translatedText": "เหตุใดการนับไบนารีจึงจับอัลกอริธึมนี้ ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 602.8 }, { - "input": "For example, at a pretty small scale, solving towers of Hanoi for two disks, move disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. ", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. T ", "translatedText": "ตัวอย่างเช่น ในระดับที่ค่อนข้างเล็ก การแก้หอคอยแห่งฮานอยด้วยดิสก์สองแผ่น ย้ายดิสก์ 0 ย้ายดิสก์ 1 ย้ายดิสก์ 0 จะสะท้อนให้เห็นโดยการนับถึง 3 ในรูปแบบไบนารี ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 617.18 }, { - "input": "Flip the last bit, roll over once, flip the last bit. ", + "input": "hat's the only way it can move. access to these videos before I publish the ", "translatedText": "พลิกบิตสุดท้าย เกลือกกลิ้งหนึ่งครั้ง พลิกบิตสุดท้าย ", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 686.98 }, { - "input": "But it actually gets cooler. ", + "input": "s disk to work, I can turn my bigger problem into something slightly smaller. ", "translatedText": "แต่จริงๆ แล้วอากาศจะเย็นลง ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 689.96 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle. ", + "input": "and students, you can check out the careers page that I've linked in the description. Personally, I think they'r ", "translatedText": "ฉันไม่เข้าใจด้วยซ้ำว่าสิ่งนี้เกี่ยวข้องกับสามเหลี่ยมของเซียร์ปินสกี้อย่างไร ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 691.36 }, { - "input": "And that's exactly what I'm going to do in the follow-on video, part 2. ", + "input": "Well, it's exactly the same thing. Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space. Get off. ", "translatedText": "และนั่นคือสิ่งที่ฉันจะทำในวิดีโอต่อจากตอนที่ 2 ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 714.02 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring. ", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra ", "translatedText": "วิดีโอนี้และวิดีโอถัดไปได้รับการสนับสนุนบางส่วนจาก Desmos และก่อนวิดีโอถัดไป ฉันแค่อยากใช้เวลาสักครู่และแบ่งปันกับพวกคุณเล็กน้อยว่าพวกเขาเป็นใครและความจริงที่ว่าพวกเขากำลังจ้างงานอยู่ ", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 732.08 }, { - "input": "The real meat of their offering is in their classroom activities. ", + "input": "ecomes clear that this has to be the most efficient solution. At every step, you're only doing what's forced upon you. ", "translatedText": "เนื้อแท้ของการถวายของพวกเขาอยู่ที่กิจกรรมในชั้นเรียนของพวกเขา ", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 740.96 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint. ", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it. ", "translatedText": "ในส่วนของฉัน ฉันรู้สึกประทับใจอย่างยิ่งกับความคิดสร้างสรรค์ของกิจกรรมเหล่านี้จากมุมมองด้านการสอน ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 750.34 }, { - "input": "And like I said, they're hiring. ", + "input": "So why does counting in binary capture this algorithm? Well, what's going on ", "translatedText": "และอย่างที่ฉันบอกไป พวกเขากำลังจ้างงาน ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/turkish/sentence_translations.json b/2016/hanoi-and-sierpinski/turkish/sentence_translations.json index e8c276ef1..c28a80f7e 100644 --- a/2016/hanoi-and-sierpinski/turkish/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/turkish/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "Bu disklerin ortasında bir delik olduğunu düşünüyorsunuz, böylece onları bir çiviye takabilirsiniz.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "İkili sayının herhangi bir açıklaması tipik olarak, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 olmak üzere 10 ayrı rakam kullandığımız için 10 tabanı olarak adlandırdığımız, sayıları temsil etmenin olağan yolu hakkında bir iç gözlemle başlar.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "Daha sonra, yeni rakamlar tükendiğinde, bir sonraki sayı olan 10'u iki rakam olan 1, 0 ile ifade edersiniz.", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "Bu şekilde saymanın ritmi bir nevi kendine benzer.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "Daha büyük bir ölçeğe uzaklaştırsanız bile, süreç bir şey yapmak, yuvarlanmak, aynı şeyi yapmak, yuvarlanmak ve daha da büyük bir yuvarlanmadan önce 9 kez tekrarlamak gibi görünür.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "Saymanın ritmi artık çok daha hızlı ama bu onu neredeyse daha fark edilir kılıyor.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "Sonuncuyu çevirin, bir kez çevirin.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "Daha büyük bir ölçekte, 1-1-1-1 olan 15'e kadar saymak gibi, süreç son 3'ün 7'ye kadar sayılmasına izin vermek, 8'inci basamağa yuvarlamak ve ardından son 3 bitin tekrar sayılmasına izin vermektir.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "Art arda 8 1 olan 255'e kadar saymak, son 7 bitin dolana kadar sayılmasına, 128'inci basamağa yuvarlanmasına ve ardından son 7 bitin tekrar sayılmasına izin vermeye benziyor.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "Pekala, bu mini girişte Keith'in bana gösterdiği şaşırtıcı gerçek şu ki bu ritmi Hanoi'nin kulelerini çözmek için kullanabiliriz.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "Yalnızca son biti 0'dan 1'e çevirdiğinizde, disk 0'ı bir çivi sağa hareket ettirin.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "Yani bunun her zaman yasal hamlelere yol açacağı ilk başta net değil.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "Örneğin, 8'inci basamağa her döndüğünüzde, 3 numaralı diskin hareket etmek üzere mutlaka serbest bırakılacağını nereden biliyorsunuz?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "Bunun neden işe yaradığını, nasıl çalıştığını ve neler olup bittiğini anlamak, bulmaca üzerinde belirli bir bakış açısına, CS halkının özyinelemeli bir bakış açısı olarak adlandırabileceği bir bakış açısına gelir.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "Hareket edebilmesinin tek yolu budur.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "Bunu yaptıktan sonra disk 3'ü oraya taşıyabiliriz.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "Ve sonra disk 3 diyor ki, hazırım.", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "Ve bir bakıma artık aynı problemin daha küçük bir versiyonuna sahipsiniz.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "Artık B milinin üzerinde 0, 1 ve 2 numaralı diskler var, onları C'ye götürmelisiniz.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "Bir yere taşınmaları gerekiyor.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "Tamam, herkes geri çekilsin.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "Ve eğer biraz düşünürseniz bunun en etkili çözüm olması gerektiği açıkça ortaya çıkıyor.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "Disk 3'ü taşımadan önce 0'dan 2'ye kadar olan diskleri çıkarmanız gerekir.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "Ve 3. diski taşımanız gerekiyor.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "Bu açıdan verimsizliğe yer yok.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "Peki neden ikili sayımda sayma bu algoritmayı yakalıyor?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "Burada olup biten şu ki, bir alt problemi çözme, büyük bir diski hareket ettirme ve sonra bir alt problemi tekrar çözme modeli, ikili sayma modeliyle mükemmel bir şekilde paraleldir.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "Son parçayı çevirin, bir kez çevirin, son parçayı çevirin.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "Benzer şekilde, ikili sistemde 111'e kadar saymak, 3'e kadar saymayı, üç bitin tamamını yuvarlamayı ve ardından üç tane daha saymayı içerir.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "Tüm ölçeklerde her iki süreç de aynı arızaya sahiptir.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "Yani bir anlamda, bu ikili çözümün işe yaramasının nedeni ya da en azından bir açıklaması, tek bir açıklaması yok gibi hissediyorum, ama bence en doğal olanı bu ikili sayıları oluşturmak için kullanacağınız modelin tamamen aynı olmasıdır. Yapı, Hanoi kuleleri için kullanacağınız desen gibidir; bu nedenle, parçaların ters döndüğüne baktığınızda, bu süreci etkili bir şekilde tersine çeviriyorsunuz.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "Bunları hangi süreç yarattı diyorsunuz?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "Bu oldukça hoş, değil mi?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "Ama aslında daha da serinliyor.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "Bunun Sierpinski üçgeniyle nasıl bir ilişkisi olduğunu bile anlamadım.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "Bu video ve bir sonraki video da kısmen Desmos tarafından destekleniyor ve bir sonraki videodan önce biraz durup sizinle onların kim oldukları ve işe aldıkları gerçeği hakkında biraz bilgi paylaşmak istiyorum.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "Yani Desmos aslında gerçekten harika.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "Tekliflerinin gerçek anlamı sınıf aktivitelerindedir.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "Kendi adıma, bu etkinliklerin pedagojik açıdan ne kadar iyi düşünülmüş olmasından çok etkilendim.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "Ve dediğim gibi işe alıyorlar.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "Şahsen ben gerçekten anlamlı şeyler yaptıklarını düşünüyorum.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/urdu/sentence_translations.json b/2016/hanoi-and-sierpinski/urdu/sentence_translations.json index 24e966774..bb73ad9ed 100644 --- a/2016/hanoi-and-sierpinski/urdu/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/urdu/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg. ", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg. ", "translatedText": "آپ ان ڈسکوں کے بارے میں سوچتے ہیں کہ درمیان میں ایک سوراخ ہے تاکہ آپ انہیں کھونٹی پر فٹ کر سکیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 109.02 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. ", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associating ", "translatedText": "بائنری کی کوئی بھی تفصیل عام طور پر اعداد کی نمائندگی کرنے کے ہمارے معمول کے طریقے کے بارے میں ایک خود شناسی کے ساتھ شروع ہوتی ہے، جسے ہم بیس 10 کہتے ہیں، کیونکہ ہم 10 علیحدہ ہندسے استعمال کرتے ہیں، 0، 1، 2، 3، 4، 5، 6، 7، 8، 9 . ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar. ", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits. ", "translatedText": "اس طرح، گنتی کی تال خود سے ملتی جلتی ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable. ", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover. ", "translatedText": "گنتی کی تال اب بہت تیز ہے، لیکن یہ تقریباً اسے زیادہ نمایاں کر دیتا ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once. ", + "input": "ce. Flip the last, roll over twice. Flip the last, roll over once. ", "translatedText": "آخری پلٹائیں، ایک بار رول کریں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 286.76 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again. ", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then ", "translatedText": "بڑے پیمانے پر، جیسے کہ 15 تک کی گنتی، جو کہ 1-1-1-1 ہے، یہ عمل یہ ہے کہ آخری 3 کو 7 تک گننے دیں، 8 کی جگہ پر جائیں، پھر آخری 3 بٹس کو دوبارہ گننے دیں۔ . ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 296.14 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again. ", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till th ", "translatedText": "255 تک کی گنتی، جو کہ 8 لگاتار 1 ہے، ایسا لگتا ہے کہ آخری 7 بٹس کو اس وقت تک گننے دیا جائے جب تک کہ وہ مکمل نہ ہو جائیں، 128 کی جگہ پر جائیں، پھر آخری 7 بٹس کو دوبارہ گننے دیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 313.24 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. ", + "input": "ey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi. ", "translatedText": "ٹھیک ہے، تو اس چھوٹے تعارف کے ساتھ، حیرت انگیز حقیقت جو کیتھ نے مجھے دکھائی وہ یہ ہے کہ ہم اس تال کو ہنوئی کے ٹاورز کو حل کرنے کے لیے استعمال کر سکتے ہیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 324.72 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right. ", + "input": "lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0. ", "translatedText": "جب بھی آپ صرف اس آخری بٹ کو 0 سے 1 تک پلٹ رہے ہیں، ڈسک 0 ایک پیگ کو دائیں طرف لے جائیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 340.8 }, { - "input": "Where do you move it, you might ask? ", + "input": "There's something magical At the small scale, say counting up to 3, which is 11 in binary, this means flip the last bit, rol ", "translatedText": "آپ اسے کہاں منتقل کرتے ہیں، آپ پوچھ سکتے ہیں؟ ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 345.94 }, { - "input": "You have no choice, you can't put it on top of disk 0, and there's only one other peg, so you move it where you're forced to move it. ", + "input": "l over to the twos, then flip the last bit. At a larger scale, like counting up to 15, which is 1111 in binary, the process is to let the last 3 count up to 7, roll over to the eights place, then let the ", "translatedText": "آپ کے پاس کوئی چارہ نہیں ہے، آپ اسے ڈسک 0 کے اوپر نہیں رکھ سکتے، اور صرف ایک اور پیگ ہے، لہذا آپ اسے وہیں منتقل کریں جہاں آپ اسے منتقل کرنے پر مجبور ہوں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 386.02 }, { - "input": "There's something magical about it, like when I first saw this, I was like, this can't work. ", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to sol ", "translatedText": "اس کے بارے میں کچھ جادوئی بات ہے، جیسے جب میں نے پہلی بار اسے دیکھا تھا، میں ایسا ہی تھا، یہ کام نہیں کر سکتا۔ ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 390.82 }, { - "input": "I don't know how this works, I don't know why this works, now I know, but it's just magical when you see it, and I remember putting together animation for this for when I was teaching this, and just like, you know, I know how this works, I know all the things in it, it's still fun to just sit and just like, you know, watch it play out. ", + "input": "ve the towers of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective. Disk 3 is thinking, okay, 2, 1, and 0, like you have to get off of me, like I can't really functi ", "translatedText": "میں نہیں جانتا کہ یہ کیسے کام کرتا ہے، مجھے نہیں معلوم کہ یہ کیوں کام کرتا ہے، اب میں جانتا ہوں، لیکن جب آپ اسے دیکھتے ہیں تو یہ صرف جادوئی ہوتا ہے، اور مجھے یاد ہے کہ جب میں یہ پڑھا رہا تھا تو اس کے لیے اینیمیشن ایک ساتھ ڈالنا تھا، اور بالکل اسی طرح، آپ جانتے ہیں، میں جانتا ہوں کہ یہ کیسے کام کرتا ہے، میں اس میں موجود تمام چیزیں جانتا ہوں، بس بیٹھنا اور بالکل اسی طرح مزہ آتا ہے، آپ جانتے ہیں، اسے کھیلتے ہوئے دیکھیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 405.08 }, { - "input": "Oh yeah. ", + "input": "on under this much weight and pressure. And so just from disk 3's perspecti If, in your binary co ", "translatedText": "ارے ہان. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 405.26 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves. ", + "input": "unting, you roll over once to the twos place, meaning you flip the last two bits, you move dis ", "translatedText": "میرا مطلب ہے، پہلے تو یہ بھی واضح نہیں ہے کہ یہ ہمیشہ قانونی اقدام کرتا ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 406.02 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move? ", + "input": "k number 1. Where do you move it, you might ask? Well, you have no choice. this disk to work, I can turn my bigger problem into something slightly smaller. And then how do I solve that? Well, it's exact ", "translatedText": "مثال کے طور پر، آپ کیسے جانتے ہیں کہ جب بھی آپ 8 کی جگہ پر گھوم رہے ہیں، وہ ڈسک 3 لازمی طور پر منتقل ہونے کے لیے آزاد ہو جائے گی؟ ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 410.96 }, { - "input": "At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than having to do 2 to the n minus 1 steps? ", + "input": "ly the same thing. If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to where it needs to go. Then disk 1 and 0 can ", "translatedText": "ایک ہی وقت میں، حل فوری طور پر یہ سوالات اٹھاتا ہے جیسے، یہ کہاں سے آتا ہے، یہ کیوں کام کرتا ہے، اور کیا ایسا کرنے کا کوئی بہتر طریقہ ہے جو کہ 2 سے مائنس 1 قدم تک کرنا پڑے؟ ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 446.6 }, { - "input": "That's the only way it can move. ", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. ", "translatedText": "یہ واحد راستہ ہے جس سے یہ حرکت کرسکتا ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 459.04 }, { - "input": "And then disk 3 says, I'm set. ", + "input": "And if you think about it for a bit, it becomes clear that this has to be the ", "translatedText": "اور پھر ڈسک 3 کہتی ہے، میں تیار ہوں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 461.22 }, { - "input": "Everyone else just figure out how to get here. ", + "input": "solution. At every step, you're only doing what's forced upon you. You have to get disk 0 ", "translatedText": "باقی سب صرف یہ جان لیں کہ یہاں کیسے جانا ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 470.54 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C. ", + "input": "e you can move disk 3. And you have to move disk 3. Flip the last two, move disk 1. Flip the last, move disk 0. And here, we're going to have to roll over three times t ", "translatedText": "اب آپ کے پاس ڈسک 0، 1، اور 2 سپنڈل B پر بیٹھے ہیں، آپ کو انہیں C تک پہنچانا ہوگا۔ ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 495.54 }, { - "input": "If disk 2 is going to say, disk 1 and disk 0, it's not you, it's me. ", + "input": "I don't know how this works, I don't know why this works. Now I know, but it's jus ", "translatedText": "اگر ڈسک 2 کہنے جا رہی ہے، ڈسک 1 اور ڈسک 0، یہ آپ نہیں، میں ہوں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 503.26 }, { - "input": "They need to move somewhere. ", + "input": "remember putting together an animation for this when I was teaching this, and just like, I know how this works. ", "translatedText": "انہیں کہیں منتقل ہونے کی ضرورت ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 511.26 }, { - "input": "Then disk 1 and 0 can do this. ", + "input": "ove disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. ", "translatedText": "پھر ڈسک 1 اور 0 یہ کر سکتے ہیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 522.24 }, { - "input": "Then I'm going to move. ", + "input": "he last bit. It's still fun to just sit and just watch it play out. At a slightly larger scale, solving towers of H ", "translatedText": "پھر میں منتقل ہونے جا رہا ہوں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 549.3 }, { - "input": "At every step, you're only doing what's forced upon you. ", + "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counti ", "translatedText": "ہر قدم پر، آپ صرف وہی کر رہے ہیں جو آپ پر مجبور ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 557.16 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3. ", + "input": "ng up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this wor ", "translatedText": "اس سے پہلے کہ آپ ڈسک 3 کو منتقل کر سکیں آپ کو ڈسک 0 سے 2 تک کی چھٹی حاصل کرنی ہوگی۔ ", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 562.28 }, { - "input": "And you have to move disk 3. ", + "input": "k, and is there a better way of doing this than having to do 2 to the n minus 1 steps? ", "translatedText": "اور آپ کو ڈسک 3 کو منتقل کرنا ہوگا۔ ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 568.24 }, { - "input": "So why does counting in binary capture this algorithm? ", + "input": "t only does this solve Towers of Hanoi, but it does it in the most efficient way possible. Understanding why this works and how it works and what ", "translatedText": "تو بائنری میں گنتی اس الگورتھم کو کیوں پکڑتی ہے؟ ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 602.8 }, { - "input": "For example, at a pretty small scale, solving towers of Hanoi for two disks, move disk 0, move disk 1, move disk 0, is reflected by counting up to 3 in binary. ", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. T ", "translatedText": "مثال کے طور پر، بہت چھوٹے پیمانے پر، دو ڈسکوں کے لیے ہنوئی کے ٹاورز کو حل کرنا، ڈسک 0 منتقل کرنا، ڈسک 1 منتقل کرنا، ڈسک 0 منتقل کرنا، بائنری میں 3 تک گننے سے ظاہر ہوتا ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 617.18 }, { - "input": "Flip the last bit, roll over once, flip the last bit. ", + "input": "hat's the only way it can move. access to these videos before I publish the ", "translatedText": "آخری بٹ پلٹائیں، ایک بار رول کریں، آخری بٹ پلٹائیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 686.98 }, { - "input": "But it actually gets cooler. ", + "input": "s disk to work, I can turn my bigger problem into something slightly smaller. ", "translatedText": "لیکن یہ اصل میں ٹھنڈا ہو جاتا ہے. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 689.96 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle. ", + "input": "and students, you can check out the careers page that I've linked in the description. Personally, I think they'r ", "translatedText": "میں یہ بھی نہیں جان سکا کہ اس کا سیرپینسکی کے مثلث سے کیا تعلق ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 691.36 }, { - "input": "And that's exactly what I'm going to do in the follow-on video, part 2. ", + "input": "Well, it's exactly the same thing. Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space. Get off. ", "translatedText": "اور بالکل وہی ہے جو میں فالو آن ویڈیو، حصہ 2 میں کرنے جا رہا ہوں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 714.02 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring. ", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra ", "translatedText": "یہ ویڈیو اور اگلی ویڈیو بھی جزوی طور پر Desmos کی طرف سے سپورٹ کی گئی ہے، اور اگلی ویڈیو سے پہلے میں صرف ایک لمحہ نکالنا چاہتا ہوں اور آپ لوگوں کے ساتھ تھوڑا سا شیئر کرنا چاہتا ہوں کہ وہ کون ہیں اور اس حقیقت کے بارے میں کہ وہ ملازمت کر رہے ہیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 732.08 }, { - "input": "The real meat of their offering is in their classroom activities. ", + "input": "ecomes clear that this has to be the most efficient solution. At every step, you're only doing what's forced upon you. ", "translatedText": "ان کی پیشکش کا اصل گوشت ان کی کلاس روم کی سرگرمیوں میں ہے۔ ", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 740.96 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint. ", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it. ", "translatedText": "میری طرف سے، میں اس بات سے بہت متاثر ہوا ہوں کہ یہ سرگرمیاں تدریسی نقطہ نظر سے کتنی اچھی طرح سے سوچی گئی ہیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 750.34 }, { - "input": "And like I said, they're hiring. ", + "input": "So why does counting in binary capture this algorithm? Well, what's going on ", "translatedText": "اور جیسا کہ میں نے کہا، وہ بھرتی کر رہے ہیں۔ ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/hanoi-and-sierpinski/vietnamese/sentence_translations.json b/2016/hanoi-and-sierpinski/vietnamese/sentence_translations.json index bd5956742..cb5c870d1 100644 --- a/2016/hanoi-and-sierpinski/vietnamese/sentence_translations.json +++ b/2016/hanoi-and-sierpinski/vietnamese/sentence_translations.json @@ -40,7 +40,7 @@ "end": 47.64 }, { - "input": "You think of these disks as having a hole in the middle so that you can fit them onto a peg.", + "input": "So you have a collection of three pegs, and you have these disks of descending size. You think of these disks as having a hole in the middle so that you can fit them onto a peg.", "translatedText": "Bạn nghĩ những chiếc đĩa này có một lỗ ở giữa để bạn có thể lắp chúng vào một cái chốt.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 111.5 }, { - "input": "Any description of binary typically starts off with an introspection about our usual way to represent numbers, what we call base 10, since we use 10 separate digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", + "input": "Any Just kind of get a feel for what the puzzle is, if it's hard, why it's hard, if it's not, why it's not, that kind of stuff. Now Keith showed me something truly surprising about this puzzle, which is that you can solve it just by counting up in binary and associatin", "translatedText": "Bất kỳ mô tả nào về hệ nhị phân thường bắt đầu bằng việc xem xét nội tâm về cách biểu diễn số thông thường của chúng ta, cái mà chúng ta gọi là cơ số 10, vì chúng ta sử dụng 10 chữ số riêng biệt, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 129.14 }, { - "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0.", + "input": "with a certain rhythm of disk movements. For anyone unfamiliar with binary, I'm going to take a moment to do a quick overview here first. 10 that you've already counted up to so far, while", "translatedText": "Sau đó, khi hết chữ số mới, bạn biểu thị số tiếp theo, 10, có hai chữ số 1, 0.", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 171.16 }, { - "input": "In this way, the rhythm of counting is kind of self-similar.", + "input": "In this way, The rhythm of counting begins by walking through all 10 of these digits.", "translatedText": "Bằng cách này, nhịp đếm gần như giống nhau.", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 175.74 }, { - "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover.", + "input": "Then, having run out of new digits, you express the next number, 10, with two digits, 1, 0. You say that 1 is in the tens place, since it's meant to encapsulate the group of 10 that you've already counted up to so far, while freeing the ones place to reset to 0. t when you're co", "translatedText": "Ngay cả khi bạn thu nhỏ ở quy mô lớn hơn, quá trình này trông giống như thực hiện một điều gì đó, lăn qua, làm điều tương tự, lăn qua và lặp lại 9 lần trước khi cuộn qua thậm chí còn lớn hơn.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 268.02 }, { - "input": "The rhythm of counting is now a lot faster, but that almost makes it more noticeable.", + "input": "Even if you zoom out to a larger scale, the process looks like doing something, rolling over, doing that same thing, rolling over, and repeating 9 times before an even larger rollover", "translatedText": "Nhịp đếm bây giờ nhanh hơn rất nhiều, nhưng điều đó gần như khiến nó dễ nhận thấy hơn.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 268.02 }, { - "input": "Flip the last, roll over once.", + "input": ". ce. Flip the last, roll over twice. Flip the last, roll over once.", "translatedText": "Lật cái cuối cùng, lăn lại một lần.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 291.6 }, { - "input": "At a larger scale, like counting up to 15, which is 1-1-1-1, the process is to let the last 3 count up to 7, roll over to the 8's place, then let the last 3 bits count up again.", + "input": "At a larger scale After counting 01, you've already run out of bits, so you need to roll over to a twos place, writing 10, and resisting every urge in your base-10 trained brain to read this as 10, and instead understand it to mean 1 group of 2 plus 0. Then", "translatedText": "Ở quy mô lớn hơn, chẳng hạn như đếm đến 15, tức là 1-1-1-1, quy trình là để 3 số cuối cùng đếm lên đến 7, chuyển sang vị trí số 8, sau đó để 3 bit cuối cùng đếm ngược lại.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 313.24 }, { - "input": "Counting up to 255, which is 8 successive 1's, this looks like letting the last 7 bits count up till they're full, rolling over to the 128's place, then letting the last 7 bits count up again.", + "input": "increment up to 11, which represents 3, and already you have to roll over again, and since there's a 1 in that twos place, that has to roll over as well, giving you 100, which represents 1 group of 4 plus 0 groups of 2 plus 0. tting the last 7 bits count up till t", "translatedText": "Đếm tới 255, tức là 8 số 1 liên tiếp, điều này giống như đếm 7 bit cuối cùng cho đến khi đầy, chuyển sang vị trí 128, sau đó để 7 bit cuối cùng đếm lại.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 335.1 }, { - "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", + "input": "hey're full, rolling over to the 128's place, then letting the last 7 bits count up again. Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers of Hanoi.", "translatedText": "Được rồi, với phần giới thiệu nhỏ đó, một sự thật đáng ngạc nhiên mà Keith đã chỉ cho tôi là chúng ta có thể sử dụng nhịp điệu này để giải các tòa tháp của Hà Nội.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 341.46 }, { - "input": "Whenever you're only flipping that last bit, from 0 to 1, move disk 0 one peg to the right.", + "input": "a lot faster, but that almost makes it more noticeable. Again, there's a certain self-similarity to this pattern. ontinues like this. Flip the last, move disk 0.", "translatedText": "Bất cứ khi nào bạn chỉ lật bit cuối cùng đó, từ 0 lên 1, hãy di chuyển đĩa 0 một chốt sang phải.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 409.9 }, { - "input": "I mean, it's not even clear at first that this is always going to give legal moves.", + "input": "Alright, so with that mini-introduction, the surprising fact that Keith showed me is that we can use this rhythm to solve the towers", "translatedText": "Ý tôi là, ban đầu thậm chí còn không rõ ràng rằng điều này sẽ luôn đưa ra những động thái hợp pháp.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 415.8 }, { - "input": "For example, how do you know that every time you're rolling over to the 8's place, that disk 3 is necessarily going to be freed up to move?", + "input": "of Hanoi. i, but it does it in the most efficient way possible. Understanding why this works and how it works and what th e heck is going on comes down to a c", "translatedText": "Ví dụ, làm sao bạn biết mỗi lần bạn lăn tới vị trí số 8 thì đĩa 3 đó nhất thiết sẽ được giải phóng để di chuyển?", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 447.9 }, { - "input": "Understanding why this works and how it works and what the heck is going on comes down to a certain perspective on the puzzle, what CS folk might call a recursive perspective.", + "input": "And so just from disk 3's perspecti If, in your binary counting, you roll over once to the twos place, meaning you flip the last two bits, you move disk number 1. Where do you move it, you might ask? Well, you have no choice.", "translatedText": "Hiểu lý do tại sao điều này hoạt động và cách thức hoạt động cũng như chuyện quái gì đang diễn ra đều phụ thuộc vào một góc nhìn nhất định về câu đố, điều mà dân gian CS có thể gọi là góc nhìn đệ quy.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 484.48 }, { - "input": "That's the only way it can move.", + "input": "I just need some space. Get off. They need to move somewh ere. Then disk 2 can move to", "translatedText": "Đó là cách duy nhất nó có thể di chuyển.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 501.1 }, { - "input": "Having done that, then we can move disk 3 over there.", + "input": "do this. But the interesting point is that every single disk pretty much", "translatedText": "Làm xong việc đó thì chúng ta có thể chuyển đĩa 3 sang đó.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 508.92 }, { - "input": "And then disk 3 says, I'm set.", + "input": "has the exact same strategy. They all say, everybody above me, get off.", "translatedText": "Và sau đó đĩa 3 nói, tôi đã sẵn sàng.", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 515.94 }, { - "input": "And in a sense, you now have a smaller version of the same problem.", + "input": "w that insight, you can code up something that will solve towers of Hanoi, like five or six lines of code, which proba", "translatedText": "Và theo một nghĩa nào đó, bây giờ bạn có một phiên bản nhỏ hơn của cùng một vấn đề.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 517.04 }, { - "input": "Now you've got disk 0, 1, and 2 sitting on spindle B, you got to get them to C.", + "input": "bly has the highest ratio of intellectual investment to lines of code ever. And if you think about it for a bit, it", "translatedText": "Bây giờ bạn đã có đĩa 0, 1 và 2 trên trục quay B, bạn phải chuyển chúng sang C.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 545.54 }, { - "input": "They need to move somewhere.", + "input": "the last, move disk 0. And here, we're going to have to roll ove", "translatedText": "Họ cần phải di chuyển đi đâu đó.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 559.54 }, { - "input": "Okay, everyone pile back on.", + "input": "There's just not any room for inefficiency from this perspective.", "translatedText": "Được rồi, mọi người tập trung trở lại.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 577.82 }, { - "input": "And if you think about it for a bit, it becomes clear that this has to be the most efficient solution.", + "input": "putting together an animation for this when I was teaching this, and just like, I know how this works. ove disk 0, move disk", "translatedText": "Và nếu bạn nghĩ về nó một chút, bạn sẽ thấy rõ rằng đây phải là giải pháp hiệu quả nhất.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 580.36 }, { - "input": "You have to get disk 0 through 2 off before you can move disk 3.", + "input": "nting up to 3 in binary. Flip the last bit, roll over once, flip the last bit. It's still fun to just sit", "translatedText": "Bạn phải tắt đĩa 0 đến đĩa 2 trước khi có thể di chuyển đĩa 3.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 590.74 }, { - "input": "And you have to move disk 3.", + "input": "and just watch it play out. At a slightly larger scale, solving towers of Hanoi for three disks loo", "translatedText": "Và bạn phải di chuyển đĩa 3.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 592.84 }, { - "input": "There's just not any room for inefficiency from this perspective.", + "input": "clear at first that this is always going to give legal moves. k number 2, then do whatever it takes to solve two disks again. A", "translatedText": "Không có chỗ cho sự thiếu hiệu quả từ góc độ này.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 594.6 }, { - "input": "So why does counting in binary capture this algorithm?", + "input": "nalogously, counting up to 111 in binary involves counting up to 3, rolling over all three", "translatedText": "Vậy tại sao việc đếm nhị phân lại nắm bắt được thuật toán này?", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 598.02 }, { - "input": "Well, what's going on here is that this pattern of solving a subproblem, moving a big disk, then solving a subproblem again, is perfectly paralleled by the pattern of binary counting.", + "input": "bits, then counting up three more. At the same time, the solution just immediately raises these questions like, where does this come from, why does this work, and is there a better way of doing this than", "translatedText": "Vâng, những gì đang diễn ra ở đây là mô hình giải một bài toán con, di chuyển một đĩa lớn, sau đó giải lại một bài toán con, hoàn toàn song song với mô hình đếm nhị phân.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 647.36 }, { - "input": "Flip the last bit, roll over once, flip the last bit.", + "input": "you're effectively reversing the recursive algorithm for towers of Hanoi, which is why it works out.", "translatedText": "Lật miếng cuối cùng, lăn qua một lần, lật miếng cuối cùng.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 660.22 }, { - "input": "Analogously, counting up to 111 in binary involves counting up to 3, rolling over all three bits, then counting up three more.", + "input": "And so just from disk 3's perspective, if you want to figure out how is disk 3 going to get over here, somehow, I don't care how, disk 2, 1, and 0 have to get to spindle B. That's the only way it can move.", "translatedText": "Tương tự, đếm tới 111 trong hệ nhị phân bao gồm việc đếm đến 3, cuộn qua cả ba bit, sau đó đếm thêm ba bit nữa.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 662.62 }, { - "input": "At all scales, both processes have this same breakdown.", + "input": "access to these videos before I publish the full series in a few months.", "translatedText": "Ở mọi quy mô, cả hai quá trình đều có sự phân tích giống nhau.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 664.78 }, { - "input": "So in a sense, the reason that this binary solution works, or at least an explanation, I feel like there's no one explanation, but I think the most natural one is that the pattern you would use to generate these binary numbers has exactly the same structure as the pattern you would use for towers of Hanoi, which is why if you look at the bits flipping, you're effectively reversing this process.", + "input": "This video and the next one are also supported in part by Desm If any of them are in spindle C, it can't move there. So somehow we have to get 2, 1, and 0 off. Having done that, then we can move disk 3 over there. And then disk 3 says, I'm set. impressed by just how well-thought-out these activities are from a pedagogical standpoint. The team clearly knows their stuff, and they know where they stand to make a differen Everyone else just figure out how to get here. And in a sense,", "translatedText": "Vì vậy, theo một nghĩa nào đó, lý do mà giải pháp nhị phân này hoạt động, hoặc ít nhất là một lời giải thích, tôi cảm thấy như không có lời giải thích nào, nhưng tôi nghĩ điều tự nhiên nhất là mẫu bạn sẽ sử dụng để tạo ra các số nhị phân này hoàn toàn giống nhau cấu trúc giống như mô hình bạn sẽ sử dụng cho các tòa tháp của Hà Nội, đó là lý do tại sao nếu bạn nhìn vào sự thay đổi từng phần, bạn đang đảo ngược quá trình này một cách hiệu quả.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 686.98 }, { - "input": "You're saying, what process generated these?", + "input": "you now have a smaller version of the same problem. Now you've got disk 0, 1, and 2 sitting on spindle B, you've got to get them to C.", "translatedText": "Bạn đang nói, quá trình nào đã tạo ra những thứ này?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 703.96 }, { - "input": "That's pretty cool, right?", + "input": "this disk to work, I can turn my bigger problem into something slightly smaller. and students, you can c", "translatedText": "Điều đó khá tuyệt phải không?", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 707.52 }, { - "input": "But it actually gets cooler.", + "input": "heck out the careers page that I've linked in the description. Personally, I think they'r Well, it's exactly the same thing.", "translatedText": "Nhưng nó thực sự trở nên mát hơn.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 711.7 }, { - "input": "I haven't even gotten to how this relates to Sierpinski's triangle.", + "input": "Disk 2 is going to say, disk 1, disk 0, it's not you, it's me. I just need some space.", "translatedText": "Tôi thậm chí còn chưa hiểu điều này liên quan thế nào đến tam giác Sierpinski.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 735.62 }, { - "input": "This video and the next one are also supported in part by Desmos, and before the next video I just want to take a moment and share with you guys a little about who they are and the fact that they're hiring.", + "input": "They all say, everybody above me, get off. Then I'm going to move, OK, everyone pile back on. When you know that insight, you can code up something that will solve Towers of Hanoi, like five or six lines of code, which probably has the highest ra", "translatedText": "Video này và video tiếp theo cũng được Desmos hỗ trợ một phần, và trước video tiếp theo tôi chỉ muốn dành chút thời gian chia sẻ với các bạn một chút về họ là ai và việc họ đang tuyển dụng.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 743.88 }, { - "input": "So Desmos is actually really cool.", + "input": "tio of intellectual investment to lines of code ever. And if you think about it for a bit, it becomes clear", "translatedText": "Vì vậy, Desmos thực sự rất tuyệt.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 751.32 }, { - "input": "The real meat of their offering is in their classroom activities.", + "input": "t this has to be the most efficient solution. At every step, you're only doing what's forced upon you.", "translatedText": "Phần thực sự của sản phẩm của họ nằm trong các hoạt động trong lớp học của họ.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 756.34 }, { - "input": "For my part, I'm super impressed by just how well-thought-out these activities are from a pedagogical standpoint.", + "input": "You have to get disk 0 through 2 off before you can move disk 3. And you have to move disk 3. And then you have to move disk 0 through 2 back onto it.", "translatedText": "Về phần mình, tôi vô cùng ấn tượng trước sự chu đáo của những hoạt động này xét từ quan điểm sư phạm.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 772.14 }, { - "input": "And like I said, they're hiring.", + "input": "So why does counting in binary capture this algorithm? Well, what's going on here is that th", "translatedText": "Và như tôi đã nói, họ đang tuyển dụng.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 796.92 }, { - "input": "Personally, I think they're doing some really meaningful stuff.", + "input": "if you zoom out and count up to a larger power of 2, or solve Towers of Hanoi with mor", "translatedText": "Cá nhân tôi nghĩ họ đang làm một số việc thực sự có ý nghĩa.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inscribed-rectangle-problem/bengali/sentence_translations.json b/2016/inscribed-rectangle-problem/bengali/sentence_translations.json index fcc985c91..b91420b3c 100644 --- a/2016/inscribed-rectangle-problem/bengali/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/bengali/sentence_translations.json @@ -608,7 +608,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart. ", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart. ", "translatedText": "সর্বোপরি, আমরা এটি করার পুরো কারণটি দেখানোর জন্য যে লুপের দুটি পৃথক জোড়া বিন্দু একটি মধ্যবিন্দু ভাগ করে এবং একই দূরত্ব আলাদা।", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. ", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you. ", "translatedText": "তারা একটি লুপে পয়েন্ট জোড়া বোঝার একটি প্রাকৃতিক উপায় হিসাবে এসেছে, এবং এটি এমন কিছু যা আমাদের একটি কংক্রিট সমস্যা সমাধানের জন্য প্রয়োজন।", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inscribed-rectangle-problem/chinese/sentence_translations.json b/2016/inscribed-rectangle-problem/chinese/sentence_translations.json index d5bb637cb..6ceeaddff 100644 --- a/2016/inscribed-rectangle-problem/chinese/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/chinese/sentence_translations.json @@ -681,7 +681,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart. ", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart. ", "translatedText": "毕竟，我们这样做的全部原因是为了表明环上的两 对不同的点共享一个中点并且相距相同的距离。", "model": "google_nmt", "from_community_srt": "总之， 我们之所以寻找这个模型 是想证明闭环上存在两个点对 它们有相同的中点和距离。", @@ -1099,7 +1099,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. ", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you. ", "translatedText": "它们是理解循环上的点对的一种自然方 式，这是我们解决具体问题所需要的。", "model": "google_nmt", "from_community_srt": "它们本身就是我们理解闭环点对的数学模型。 它们正是我们解决具体问题所需要的工具。 好吧， 我最后还有一个动画给你看 但我先讲讲有关是什么 让我能够完成这个视频以及未来更多的视频。 首先， 我衷心感谢所有在Patreon上支持的人。", diff --git a/2016/inscribed-rectangle-problem/german/sentence_translations.json b/2016/inscribed-rectangle-problem/german/sentence_translations.json index 7e437ecd0..813aa4e22 100644 --- a/2016/inscribed-rectangle-problem/german/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/german/sentence_translations.json @@ -608,7 +608,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart.", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart.", "translatedText": "Schließlich machen wir das nur, um zu zeigen, dass zwei unterschiedliche Punktpaare auf der Schleife einen gemeinsamen Mittelpunkt haben und den gleichen Abstand voneinander haben.", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem.", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you.", "translatedText": "Sie stellten eine natürliche Möglichkeit dar, Punktpaare auf einer Schleife zu verstehen, und das war etwas, was wir brauchten, um ein konkretes Problem zu lösen.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inscribed-rectangle-problem/indonesian/sentence_translations.json b/2016/inscribed-rectangle-problem/indonesian/sentence_translations.json index ba9829bce..70e11e07c 100644 --- a/2016/inscribed-rectangle-problem/indonesian/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/indonesian/sentence_translations.json @@ -608,7 +608,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart. ", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart. ", "translatedText": "Lagi pula, alasan utama kami melakukan hal ini adalah untuk menunjukkan bahwa dua pasang titik berbeda pada lingkaran berbagi titik tengah dan memiliki jarak yang sama. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. ", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you. ", "translatedText": "Mereka muncul sebagai cara alami untuk memahami pasangan titik dalam satu lingkaran, dan itu adalah sesuatu yang kami perlukan untuk memecahkan masalah konkret. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inscribed-rectangle-problem/japanese/sentence_translations.json b/2016/inscribed-rectangle-problem/japanese/sentence_translations.json index 2fe642e19..daa2da4c9 100644 --- a/2016/inscribed-rectangle-problem/japanese/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/japanese/sentence_translations.json @@ -608,7 +608,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart. ", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart. ", "translatedText": "結局のところ、これを行う理由は、ループ上の 2 つの異なる点 のペアが中点を共有し、同じ距離離れていることを示すことです。", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. ", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you. ", "translatedText": "これらは、ループ上の点のペアを理解するための自然な方法として 考え出され、具体的な問題を解決するために必要なものでした。", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inscribed-rectangle-problem/marathi/sentence_translations.json b/2016/inscribed-rectangle-problem/marathi/sentence_translations.json index 14f87e5ce..6ef58043a 100644 --- a/2016/inscribed-rectangle-problem/marathi/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/marathi/sentence_translations.json @@ -608,7 +608,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart. ", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart. ", "translatedText": "शेवटी, आम्ही असे करत आहोत याचे संपूर्ण कारण म्हणजे लूपवरील बिंदूंच्या दोन भिन्न जोड्या मध्यबिंदू सामायिक करतात आणि समान अंतरावर आहेत. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. ", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you. ", "translatedText": "लूपवरील बिंदूंच्या जोड्या समजून घेण्याचा एक नैसर्गिक मार्ग म्हणून ते समोर आले आणि आम्हाला ठोस समस्येचे निराकरण करण्यासाठी हे आवश्यक आहे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inscribed-rectangle-problem/persian/sentence_translations.json b/2016/inscribed-rectangle-problem/persian/sentence_translations.json index 2d4757720..c628efb5b 100644 --- a/2016/inscribed-rectangle-problem/persian/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/persian/sentence_translations.json @@ -608,7 +608,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart. ", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart. ", "translatedText": "به هر حال، دلیل اصلی انجام این کار این است که نشان دهیم دو جفت نقطه متمایز در حلقه یک نقطه میانی را به اشتراک می گذارند و از هم فاصله دارند. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. ", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you. ", "translatedText": "آنها به عنوان یک روش طبیعی برای درک جفت نقاط در یک حلقه مطرح شدند، و این چیزی است که ما برای حل یک مشکل مشخص به آن نیاز داشتیم. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inscribed-rectangle-problem/tamil/sentence_translations.json b/2016/inscribed-rectangle-problem/tamil/sentence_translations.json index d41eba932..3056e6904 100644 --- a/2016/inscribed-rectangle-problem/tamil/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/tamil/sentence_translations.json @@ -608,7 +608,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart. ", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart. ", "translatedText": "எல்லாவற்றிற்கும் மேலாக, நாங்கள் இதைச் செய்வதற்கான முழுக் காரணம், லூப்பில் உள்ள இரண்டு தனித்தனி ஜோடி புள்ளிகள் ஒரு நடுப்புள்ளியைப் பகிர்ந்து கொள்கின்றன மற்றும் ஒரே தூரத்தில் இருப்பதைக் காட்டுவதாகும். ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. ", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you. ", "translatedText": "ஒரு சுழற்சியில் ஜோடி புள்ளிகளைப் புரிந்துகொள்வதற்கான இயற்கையான வழியாக அவை வந்தன, மேலும் இது ஒரு உறுதியான சிக்கலைத் தீர்க்க நமக்குத் தேவையான ஒன்று. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inscribed-rectangle-problem/telugu/sentence_translations.json b/2016/inscribed-rectangle-problem/telugu/sentence_translations.json index c4cbc220e..e9d434ff0 100644 --- a/2016/inscribed-rectangle-problem/telugu/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/telugu/sentence_translations.json @@ -608,7 +608,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart. ", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart. ", "translatedText": "అన్నింటికంటే, మేము దీన్ని చేయడానికి పూర్తి కారణం ఏమిటంటే, లూప్‌లోని రెండు విభిన్న జతల పాయింట్లు మధ్య బిందువును పంచుకుంటాయి మరియు ఒకే దూరం వేరుగా ఉన్నాయని చూపడం. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. ", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you. ", "translatedText": "లూప్‌లోని జతల పాయింట్‌లను అర్థం చేసుకోవడానికి అవి సహజమైన మార్గంగా వచ్చాయి మరియు ఇది ఒక నిర్దిష్ట సమస్యను పరిష్కరించడానికి మాకు అవసరమైనది. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inscribed-rectangle-problem/thai/sentence_translations.json b/2016/inscribed-rectangle-problem/thai/sentence_translations.json index 7553d6abc..77aa20e45 100644 --- a/2016/inscribed-rectangle-problem/thai/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/thai/sentence_translations.json @@ -608,7 +608,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart. ", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart. ", "translatedText": "ท้ายที่สุดแล้ว เหตุผลทั้งหมดที่เราทำก็เพื่อแสดงให้เห็นว่าจุดสองคู่ที่แตกต่างกันบนลูปมีจุดกึ่งกลางร่วมกันและอยู่ห่างกันเท่ากัน ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. ", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you. ", "translatedText": "มันเกิดขึ้นเป็นวิธีธรรมชาติในการทำความเข้าใจจุดคู่บนลูป และนั่นคือสิ่งที่เราต้องการเพื่อแก้ปัญหาที่เป็นรูปธรรม ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inscribed-rectangle-problem/turkish/sentence_translations.json b/2016/inscribed-rectangle-problem/turkish/sentence_translations.json index 6f73e6a05..04b4f8a51 100644 --- a/2016/inscribed-rectangle-problem/turkish/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/turkish/sentence_translations.json @@ -608,7 +608,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart. ", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart. ", "translatedText": "Sonuçta, bunu yapmamızın tek nedeni, döngü üzerindeki iki farklı nokta çiftinin bir orta noktayı paylaştığını ve birbirlerinden aynı uzaklıkta olduklarını göstermektir. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. ", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you. ", "translatedText": "Bir döngüdeki nokta çiftlerini anlamanın doğal bir yolu olarak ortaya çıktılar ve bu, somut bir sorunu çözmek için ihtiyacımız olan bir şeydi. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inscribed-rectangle-problem/ukrainian/sentence_translations.json b/2016/inscribed-rectangle-problem/ukrainian/sentence_translations.json index 66b90e42d..6a5977b02 100644 --- a/2016/inscribed-rectangle-problem/ukrainian/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/ukrainian/sentence_translations.json @@ -532,7 +532,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart.", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart.", "translatedText": "Зрештою, вся причина, чому ми це робимо, полягає в тому, щоб показати, що дві різні пари точок на петлі мають спільну середину та однакову відстань одна від одної.", "n_reviews": 0, "start": 612.2, @@ -861,7 +861,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem.", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you.", "translatedText": "Вони придумали як природний спосіб зрозуміти пари точок у циклі, і це те, що нам потрібно для вирішення конкретної проблеми.", "n_reviews": 0, "start": 967.26, diff --git a/2016/inscribed-rectangle-problem/urdu/sentence_translations.json b/2016/inscribed-rectangle-problem/urdu/sentence_translations.json index 1a34044e7..013225f3e 100644 --- a/2016/inscribed-rectangle-problem/urdu/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/urdu/sentence_translations.json @@ -608,7 +608,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart. ", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart. ", "translatedText": "بہر حال، ہم ایسا کرنے کی پوری وجہ یہ ظاہر کرنا ہے کہ لوپ پر پوائنٹس کے دو الگ الگ جوڑے ایک وسط پوائنٹ کا اشتراک کرتے ہیں اور ایک ہی فاصلے پر ہیں۔ ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. ", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you. ", "translatedText": "وہ لوپ پر پوائنٹس کے جوڑے کو سمجھنے کے قدرتی طریقے کے طور پر سامنے آئے، اور یہ وہ چیز ہے جس کی ہمیں ایک ٹھوس مسئلے کو حل کرنے کی ضرورت ہے۔ ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inscribed-rectangle-problem/vietnamese/sentence_translations.json b/2016/inscribed-rectangle-problem/vietnamese/sentence_translations.json index bb3c48e92..e837d2350 100644 --- a/2016/inscribed-rectangle-problem/vietnamese/sentence_translations.json +++ b/2016/inscribed-rectangle-problem/vietnamese/sentence_translations.json @@ -608,7 +608,7 @@ "end": 611.44 }, { - "input": "After all, the whole reason we're doing this is to show that two distinct pairs of points on the loop share a midpoint and are the same distance apart. ", + "input": "After all, the whole reason we're doing this is to show that two distinct pairs of pairs of points on the loop share a midpoint and are the same distance apart. ", "translatedText": "Xét cho cùng, toàn bộ lý do chúng ta làm điều này là để chứng tỏ rằng hai cặp điểm phân biệt trên vòng lặp có chung một điểm giữa và cách nhau một khoảng cách như nhau. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 966.34 }, { - "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. ", + "input": "They came up as a natural way to understand pairs of points on a loop, and that's something that we needed to solve a concrete problem. Thank you. ", "translatedText": "Chúng đưa ra một cách tự nhiên để hiểu các cặp điểm trên một vòng lặp và đó là điều chúng tôi cần để giải quyết một vấn đề cụ thể. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/inverse-matrices/arabic/sentence_translations.json b/2016/inverse-matrices/arabic/sentence_translations.json index 5e9424d61..61552f2f5 100644 --- a/2016/inverse-matrices/arabic/sentence_translations.json +++ b/2016/inverse-matrices/arabic/sentence_translations.json @@ -787,7 +787,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "وبعد ذلك، سأعطيكم وجهة نظري حول المنتجات النقطية، وشيء رائع جدًا يحدث عندما تشاهدها تحت ضوء التحولات الخطية.", "model": "google_nmt", "from_community_srt": "ثم ، بعد ذلك ، سأقدم لك تأخذ على منتجات نقطة ، وشيء رائع يحدث عند عرضها تحت ضوء التحولات الخطية.", diff --git a/2016/inverse-matrices/czech/sentence_translations.json b/2016/inverse-matrices/czech/sentence_translations.json index 094176bbd..900430d0f 100644 --- a/2016/inverse-matrices/czech/sentence_translations.json +++ b/2016/inverse-matrices/czech/sentence_translations.json @@ -791,7 +791,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "Poté vám řeknu svůj pohled na bodové součin a něco docela zajímavého, co se stane, když se na ně podíváte ve světle lineárních transformací.", "model": "DeepL", "from_community_srt": "a nějakými prima triky, když se na něj díváme z pohledu lineárních transformací.", diff --git a/2016/inverse-matrices/danish/sentence_translations.json b/2016/inverse-matrices/danish/sentence_translations.json index e86cc6b9f..301770020 100644 --- a/2016/inverse-matrices/danish/sentence_translations.json +++ b/2016/inverse-matrices/danish/sentence_translations.json @@ -790,7 +790,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "Derefter vil jeg give dig mit bud på dot-produkter, og noget ret cool, der sker, når du ser dem i lyset af lineære transformationer.", "model": "google_nmt", "from_community_srt": "når du se dem i lyset af lineære transformationer. Vi ses!", diff --git a/2016/inverse-matrices/english/captions.srt b/2016/inverse-matrices/english/captions.srt index 272310f46..0d32296a0 100644 --- a/2016/inverse-matrices/english/captions.srt +++ b/2016/inverse-matrices/english/captions.srt @@ -691,14 +691,14 @@ learning that you do more fruitful. Next video, by popular request, will be a brief footnote about non-square matrices. 174 -00:11:51,880 --> 00:11:54,916 +00:11:51,880 --> 00:11:54,677 Then after that, I'm going to give you my take on dot products, 175 -00:11:54,916 --> 00:11:58,900 +00:11:54,677 --> 00:11:58,348 and something pretty cool that happens when you view them under the light of linear 176 -00:11:58,900 --> 00:11:59,660 -transformations. +00:11:58,348 --> 00:11:59,660 +transformations. See you then! diff --git a/2016/inverse-matrices/english/sentence_timings.json b/2016/inverse-matrices/english/sentence_timings.json index 319d6a55b..fd419bc01 100644 --- a/2016/inverse-matrices/english/sentence_timings.json +++ b/2016/inverse-matrices/english/sentence_timings.json @@ -440,7 +440,7 @@ 711.88 ], [ - "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", 711.88, 719.66 ] diff --git a/2016/inverse-matrices/english/transcript.txt b/2016/inverse-matrices/english/transcript.txt index cb2fe840b..e2f61e51a 100644 --- a/2016/inverse-matrices/english/transcript.txt +++ b/2016/inverse-matrices/english/transcript.txt @@ -86,4 +86,4 @@ Again, there's a lot that I haven't covered here, most notably how to compute th I also had to limit my scope to examples where the number of equations equals the number of unknowns. But the goal here is not to try to teach everything, it's that you come away with a strong intuition for inverse matrices, column space, and null space, and that those intuitions make any future learning that you do more fruitful. Next video, by popular request, will be a brief footnote about non-square matrices. -Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. \ No newline at end of file +Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then! \ No newline at end of file diff --git a/2016/inverse-matrices/french/sentence_translations.json b/2016/inverse-matrices/french/sentence_translations.json index 40d5cdb46..3ddd05d16 100644 --- a/2016/inverse-matrices/french/sentence_translations.json +++ b/2016/inverse-matrices/french/sentence_translations.json @@ -703,7 +703,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "Ensuite, je vais vous donner mon point de vue sur les produits scalaires, et quelque chose d'assez cool qui se produit lorsque vous les visualisez à la lumière de transformations linéaires.", "from_community_srt": "Ensuite, après ca, je vais vous donner mon interprétation sur les produits scalaires, et quelque chose d'assez cool que l'on observe lorsqu'on les regarde sous la lumière des transformations linéaires.", "n_reviews": 0, diff --git a/2016/inverse-matrices/german/sentence_translations.json b/2016/inverse-matrices/german/sentence_translations.json index 5e7c1918f..b359c6384 100644 --- a/2016/inverse-matrices/german/sentence_translations.json +++ b/2016/inverse-matrices/german/sentence_translations.json @@ -791,7 +791,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "Danach werde ich dir meine Sicht auf Punktprodukte erläutern und dir etwas ziemlich Cooles zeigen, das passiert, wenn du sie unter dem Licht von linearen Transformationen betrachtest.", "model": "DeepL", "from_community_srt": "Dann, danach, werde ich meinen Versuch mit Kreuzprodukten liefern. und etwas ziemlich cooles, das passiert, wenn man sie aus dem Blickwinkel linearer Abbildungen sieht.", diff --git a/2016/inverse-matrices/hebrew/sentence_translations.json b/2016/inverse-matrices/hebrew/sentence_translations.json index ba3f183af..ca7681534 100644 --- a/2016/inverse-matrices/hebrew/sentence_translations.json +++ b/2016/inverse-matrices/hebrew/sentence_translations.json @@ -792,7 +792,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "ואז אחרי זה, אני אתן לך את ההשקפה שלי לגבי מוצרי נקודות, ומשהו די מגניב שקורה כשאתה צופה בהם באור של טרנספורמציות ליניאריות.", "model": "google_nmt", "from_community_srt": "ואז, לאחר מכן, אני הולך לדבר על מכפלה סקלרית משהו מאוד מגניב כשאתה מראה אותם תחת האור של טרנספורמציות לינאריות", diff --git a/2016/inverse-matrices/hungarian/sentence_translations.json b/2016/inverse-matrices/hungarian/sentence_translations.json index 960f3b57d..2ffa5e0e3 100644 --- a/2016/inverse-matrices/hungarian/sentence_translations.json +++ b/2016/inverse-matrices/hungarian/sentence_translations.json @@ -704,7 +704,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "Ezután elmondom a véleményemet a ponttermékről, és valami nagyon klassz dolog történik, ha a lineáris transzformációk fényében nézzük őket.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/inverse-matrices/korean/sentence_translations.json b/2016/inverse-matrices/korean/sentence_translations.json index eeec6757e..9cb7cafae 100644 --- a/2016/inverse-matrices/korean/sentence_translations.json +++ b/2016/inverse-matrices/korean/sentence_translations.json @@ -791,7 +791,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "그런 다음 내적에 대한 내 생각과 이를 선형 변환의 관점에서 볼 때 발생하는 매우 멋진 현상에 대해 설명하겠습니다.", "model": "google_nmt", "from_community_srt": "그런 다음, 그 후, 나는 dot product 에 대해 다룰거야. 나중에 너가 보면 알겠지만, 정말 멋질 거야. 선형변환이라는 빛 아래서 살펴보게되면 말야.", diff --git a/2016/inverse-matrices/polish/sentence_translations.json b/2016/inverse-matrices/polish/sentence_translations.json index 0e5c1e479..73024f913 100644 --- a/2016/inverse-matrices/polish/sentence_translations.json +++ b/2016/inverse-matrices/polish/sentence_translations.json @@ -792,7 +792,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "Potem przedstawię wam moje podejście do produktów skalarnych i coś całkiem fajnego, co się dzieje, gdy spojrzymy na nie w świetle transformacji liniowych.", "model": "google_nmt", "from_community_srt": "Później po tym opowiem o iloczynie skalarnym, i tym jak świetny jest kiedy patrzysz na niego poprzez perspektywę transformacji liniowych.", diff --git a/2016/inverse-matrices/portuguese/sentence_translations.json b/2016/inverse-matrices/portuguese/sentence_translations.json index f2e2136e3..020d5d2be 100644 --- a/2016/inverse-matrices/portuguese/sentence_translations.json +++ b/2016/inverse-matrices/portuguese/sentence_translations.json @@ -791,7 +791,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "Depois disso, vou dar minha opinião sobre produtos escalares e algo muito legal que acontece quando você os vê sob a luz de transformações lineares.", "model": "google_nmt", "from_community_srt": "E depois dele, vou passar minha visão sobre produto escalar (interno) e algo muito legal que acontece quando você o enxerga sob a luz das transformações lineares.", diff --git a/2016/inverse-matrices/russian/sentence_translations.json b/2016/inverse-matrices/russian/sentence_translations.json index 06648af8f..fcdf75de8 100644 --- a/2016/inverse-matrices/russian/sentence_translations.json +++ b/2016/inverse-matrices/russian/sentence_translations.json @@ -704,7 +704,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "Затем я собираюсь поделиться с вами своим взглядом на скалярные произведения и кое-что интересное, что происходит, когда вы рассматриваете их в свете линейных преобразований.", "from_community_srt": "Затем, после этого, я попробую вам объяснить поэлементное умножение матриц и то крутое, что с ним происходит, когда вы это рассматриваете под углом линейных преобразований.", "n_reviews": 0, diff --git a/2016/inverse-matrices/spanish/sentence_translations.json b/2016/inverse-matrices/spanish/sentence_translations.json index e07c5e0a4..5594579b8 100644 --- a/2016/inverse-matrices/spanish/sentence_translations.json +++ b/2016/inverse-matrices/spanish/sentence_translations.json @@ -703,7 +703,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "Luego, después de eso, les daré mi opinión sobre los productos punto y algo muy interesante que sucede cuando los ve bajo la luz de transformaciones lineales.", "from_community_srt": "Luego, después de eso, les daré mi perspectiva del producto punto, y una consecuencia bastante interesante de verlo bajo la perspectiva de las transformaciones lineales.", "n_reviews": 0, diff --git a/2016/inverse-matrices/turkish/sentence_translations.json b/2016/inverse-matrices/turkish/sentence_translations.json index 604b9f9f9..3b0b62570 100644 --- a/2016/inverse-matrices/turkish/sentence_translations.json +++ b/2016/inverse-matrices/turkish/sentence_translations.json @@ -791,7 +791,7 @@ "end": 711.88 }, { - "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations.", + "input": "Then after that, I'm going to give you my take on dot products, and something pretty cool that happens when you view them under the light of linear transformations. See you then!", "translatedText": "Daha sonra, size nokta çarpımları hakkındaki görüşlerimi anlatacağım ve bunlara doğrusal dönüşümlerin ışığı altında baktığınızda oldukça hoş bir şey oluyor.", "model": "google_nmt", "from_community_srt": "Sonrasında, Nokta Ürünü hakkında notlarımı aktaracağım. İşlemleri doğrusal dönüşümler ışığında görünce aydınlanma gibi bir duygu yaşayacaksınız.", diff --git a/2016/linear-transformations/czech/sentence_translations.json b/2016/linear-transformations/czech/sentence_translations.json index 0e6d03402..c3c8c57b2 100644 --- a/2016/linear-transformations/czech/sentence_translations.json +++ b/2016/linear-transformations/czech/sentence_translations.json @@ -756,7 +756,7 @@ "end": 640.56 }, { - "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", "translatedText": "V příštím videu budu mluvit o násobení dvou matic dohromady.", "model": "DeepL", "from_community_srt": "V bezprostředně následujícím videu budu mluvit o násobení matic mezi sebou.", diff --git a/2016/linear-transformations/english/captions.srt b/2016/linear-transformations/english/captions.srt index 9f3c191d6..9c8bfe4cd 100644 --- a/2016/linear-transformations/english/captions.srt +++ b/2016/linear-transformations/english/captions.srt @@ -623,10 +623,10 @@ change of basis, eigenvalues, all of these will become easier to understand once you start thinking about matrices as transformations of space. 157 -00:10:41,300 --> 00:10:44,160 -Most immediately, in the next video, I'll be talking +00:10:41,300 --> 00:10:43,560 +Most immediately, in the next video, I'll be talking about 158 -00:10:44,160 --> 00:10:46,320 -about multiplying two matrices together. +00:10:43,560 --> 00:10:46,320 +multiplying two matrices together. See you then! Thank you for watching! diff --git a/2016/linear-transformations/english/sentence_timings.json b/2016/linear-transformations/english/sentence_timings.json index f91d79d18..621efe8f5 100644 --- a/2016/linear-transformations/english/sentence_timings.json +++ b/2016/linear-transformations/english/sentence_timings.json @@ -420,7 +420,7 @@ 640.56 ], [ - "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", 641.3, 646.32 ] diff --git a/2016/linear-transformations/english/transcript.txt b/2016/linear-transformations/english/transcript.txt index 920210241..fba2237de 100644 --- a/2016/linear-transformations/english/transcript.txt +++ b/2016/linear-transformations/english/transcript.txt @@ -82,4 +82,4 @@ Matrices give us a language to describe these transformations, where the columns The important takeaway here is that every time you see a matrix, you can interpret it as a certain transformation of space. Once you really digest this idea, you're in a great position to understand linear algebra deeply. Almost all of the topics coming up, from matrix multiplication to determinants, change of basis, eigenvalues, all of these will become easier to understand once you start thinking about matrices as transformations of space. -Most immediately, in the next video, I'll be talking about multiplying two matrices together. \ No newline at end of file +Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching! \ No newline at end of file diff --git a/2016/linear-transformations/french/sentence_translations.json b/2016/linear-transformations/french/sentence_translations.json index 87c490756..556a6fe23 100644 --- a/2016/linear-transformations/french/sentence_translations.json +++ b/2016/linear-transformations/french/sentence_translations.json @@ -670,7 +670,7 @@ "end": 640.56 }, { - "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", "translatedText": "Dans l'immédiat, dans la prochaine vidéo, je parlerai de la multiplication de deux matrices ensemble.", "from_community_srt": "D'ailleurs, dans la prochaine vidéo Je vous parlerai de multiplications entre deux matrices ensemble.", "n_reviews": 0, diff --git a/2016/linear-transformations/german/sentence_translations.json b/2016/linear-transformations/german/sentence_translations.json index 9e61d7930..d06eddb17 100644 --- a/2016/linear-transformations/german/sentence_translations.json +++ b/2016/linear-transformations/german/sentence_translations.json @@ -755,7 +755,7 @@ "end": 640.56 }, { - "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", "translatedText": "Im nächsten Video werde ich über die Multiplikation zweier Matrizen sprechen.", "model": "DeepL", "from_community_srt": "Im nächsten Video, bespreche ich wie man zwei Matrizen miteinander multipliziert.", diff --git a/2016/linear-transformations/greek/sentence_translations.json b/2016/linear-transformations/greek/sentence_translations.json index c3ff0f39a..c747d23cb 100644 --- a/2016/linear-transformations/greek/sentence_translations.json +++ b/2016/linear-transformations/greek/sentence_translations.json @@ -755,7 +755,7 @@ "end": 640.56 }, { - "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", "translatedText": "Αμέσως, στο επόμενο βίντεο, θα μιλήσω για τον πολλαπλασιασμό δύο πινάκων μαζί.", "model": "google_nmt", "from_community_srt": "Αμέσως μετά, στο επόμενο βίντεο μιλάω για το πώς πολλαπλασιάζουμε δύο πίνακες μεταξύ τους.", diff --git a/2016/linear-transformations/hebrew/sentence_translations.json b/2016/linear-transformations/hebrew/sentence_translations.json index 802317ead..eae9cf5ff 100644 --- a/2016/linear-transformations/hebrew/sentence_translations.json +++ b/2016/linear-transformations/hebrew/sentence_translations.json @@ -755,7 +755,7 @@ "end": 640.56 }, { - "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", "translatedText": "הכי מיד, בסרטון הבא, אדבר על הכפלת שתי מטריצות יחד.", "model": "google_nmt", "from_community_srt": "באופן הכי מיידי, בסירטון הבא אני אדבר על הכפלה 2 מטריצות ביחד.", diff --git a/2016/linear-transformations/hungarian/sentence_translations.json b/2016/linear-transformations/hungarian/sentence_translations.json index 6b236f0ad..bfc1f8399 100644 --- a/2016/linear-transformations/hungarian/sentence_translations.json +++ b/2016/linear-transformations/hungarian/sentence_translations.json @@ -672,7 +672,7 @@ "end": 640.56 }, { - "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", "translatedText": "A következő videóban két mátrix összeszorzásáról fogok beszélni.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/linear-transformations/korean/sentence_translations.json b/2016/linear-transformations/korean/sentence_translations.json index 640b3027f..38175cb24 100644 --- a/2016/linear-transformations/korean/sentence_translations.json +++ b/2016/linear-transformations/korean/sentence_translations.json @@ -754,7 +754,7 @@ "end": 640.56 }, { - "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", "translatedText": "가장 즉시 다음 비디오에서는 두 행렬을 곱하는 것에 대해 이야기하겠습니다.", "model": "google_nmt", "from_community_srt": "다음 동영상에 나오는, 두 행렬의 곱셈에 대한 것입니다.", diff --git a/2016/linear-transformations/polish/sentence_translations.json b/2016/linear-transformations/polish/sentence_translations.json index 2bcf84403..6e5f05014 100644 --- a/2016/linear-transformations/polish/sentence_translations.json +++ b/2016/linear-transformations/polish/sentence_translations.json @@ -754,7 +754,7 @@ "end": 640.56 }, { - "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", "translatedText": "W następnym filmie od razu opowiem o mnożeniu dwóch macierzy przez siebie.", "model": "google_nmt", "from_community_srt": "W następnym filmie będę mówić o mnożeniu dwóch macierzy przez siebie.", diff --git a/2016/linear-transformations/portuguese/sentence_translations.json b/2016/linear-transformations/portuguese/sentence_translations.json index 2ab54e756..d8c3eba53 100644 --- a/2016/linear-transformations/portuguese/sentence_translations.json +++ b/2016/linear-transformations/portuguese/sentence_translations.json @@ -754,7 +754,7 @@ "end": 640.56 }, { - "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", "translatedText": "Mais imediatamente, no próximo vídeo, falarei sobre a multiplicação de duas matrizes.", "model": "google_nmt", "from_community_srt": "Quase imediatamente, no próximo video, Eu falarei sobre multiplicação de duas matrizes.", diff --git a/2016/linear-transformations/russian/sentence_translations.json b/2016/linear-transformations/russian/sentence_translations.json index 64c4a4e41..4955cf38b 100644 --- a/2016/linear-transformations/russian/sentence_translations.json +++ b/2016/linear-transformations/russian/sentence_translations.json @@ -671,7 +671,7 @@ "end": 640.56 }, { - "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", "translatedText": "Сразу же, в следующем видео, я расскажу об умножении двух матриц вместе.", "from_community_srt": "Уже в следующем видео, я буду говорить о перемножении двух матриц. Увидимся! Технически,", "n_reviews": 0, diff --git a/2016/linear-transformations/spanish/sentence_translations.json b/2016/linear-transformations/spanish/sentence_translations.json index cad9092b4..04c2e9379 100644 --- a/2016/linear-transformations/spanish/sentence_translations.json +++ b/2016/linear-transformations/spanish/sentence_translations.json @@ -588,7 +588,7 @@ "end": 640.56 }, { - "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", "translatedText": "De manera más inmediata, en el siguiente video, hablaré sobre cómo multiplicar dos matrices.", "n_reviews": 1, "start": 641.3, diff --git a/2016/linear-transformations/turkish/sentence_translations.json b/2016/linear-transformations/turkish/sentence_translations.json index 47b0464a1..bf81f1fa7 100644 --- a/2016/linear-transformations/turkish/sentence_translations.json +++ b/2016/linear-transformations/turkish/sentence_translations.json @@ -754,7 +754,7 @@ "end": 640.56 }, { - "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together.", + "input": "Most immediately, in the next video, I'll be talking about multiplying two matrices together. See you then! Thank you for watching!", "translatedText": "Hemen bir sonraki videoda iki matrisin birbiriyle çarpılmasından bahsedeceğim.", "model": "google_nmt", "from_community_srt": "En kısa sürede, sonraki video'da iki matrisi çarpmak hakkında konuşacağım.", diff --git a/2016/matrix-multiplication/arabic/sentence_translations.json b/2016/matrix-multiplication/arabic/sentence_translations.json index 94f644967..bc6f8286d 100644 --- a/2016/matrix-multiplication/arabic/sentence_translations.json +++ b/2016/matrix-multiplication/arabic/sentence_translations.json @@ -692,7 +692,7 @@ "end": 586.44 }, { - "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", "translatedText": "في الفيديو التالي، سأبدأ بالحديث عن توسيع هذه الأفكار إلى ما هو أبعد من بعدين فقط.", "model": "google_nmt", "from_community_srt": "ثق بي ، هذا هو نوع وقت اللعب يجعل حقا الفكرة تغرق في الفيديو التالي سأبدأ بالحديث عنه توسيع هذه الأفكار أبعد من بعدين فقط.", diff --git a/2016/matrix-multiplication/bengali/sentence_translations.json b/2016/matrix-multiplication/bengali/sentence_translations.json index 1c728289c..52be56383 100644 --- a/2016/matrix-multiplication/bengali/sentence_translations.json +++ b/2016/matrix-multiplication/bengali/sentence_translations.json @@ -496,7 +496,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart. ", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart. ", "translatedText": "আপনি যদি প্রথমে ঘোরান, তারপর শিয়ার করুন, i-hat 1,1-এ শেষ হয়, এবং j-hat বন্ধ হয় একটি ভিন্ন দিকে ঋণাত্মক 1,0 এ, এবং তারা আরও দূরে নির্দেশ করছে। ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/matrix-multiplication/chinese/sentence_translations.json b/2016/matrix-multiplication/chinese/sentence_translations.json index e3f94015a..76c670fb7 100644 --- a/2016/matrix-multiplication/chinese/sentence_translations.json +++ b/2016/matrix-multiplication/chinese/sentence_translations.json @@ -547,7 +547,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart. ", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart. ", "translatedText": "如果你先旋转，然后进行剪切，i-hat 最终会在 1,1 处结束，而 j-hat 在负 1,0 处朝不同方向偏离，并且它们指向的距离更远。", "model": "google_nmt", "from_community_srt": "1) 它们彼此靠得很近 如果你首先旋转， 然后剪切 i帽落在(1, 1)， 而j帽落在一个不同的方向(-1,", diff --git a/2016/matrix-multiplication/czech/sentence_translations.json b/2016/matrix-multiplication/czech/sentence_translations.json index e8063a921..9e8d55d3a 100644 --- a/2016/matrix-multiplication/czech/sentence_translations.json +++ b/2016/matrix-multiplication/czech/sentence_translations.json @@ -700,7 +700,7 @@ "end": 586.44 }, { - "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", "translatedText": "V příštím videu začnu mluvit o rozšíření těchto myšlenek nad rámec dvou rozměrů.", "model": "DeepL", "from_community_srt": "V následujícím videu se pustíme do rozšíření této představy do více rozměrů.", diff --git a/2016/matrix-multiplication/english/captions.srt b/2016/matrix-multiplication/english/captions.srt index a844b56fa..4d04b190b 100644 --- a/2016/matrix-multiplication/english/captions.srt +++ b/2016/matrix-multiplication/english/captions.srt @@ -579,10 +579,10 @@ you apply one after the other, and then working out the matrix product numerical Trust me, this is the kind of playtime that really makes the idea sink in. 146 -00:09:47,200 --> 00:09:50,091 +00:09:47,200 --> 00:09:49,713 In the next video, I'll start talking about extending 147 -00:09:50,091 --> 00:09:52,180 -these ideas beyond just two dimensions. +00:09:49,713 --> 00:09:52,180 +these ideas beyond just two dimensions. See you then! diff --git a/2016/matrix-multiplication/english/sentence_timings.json b/2016/matrix-multiplication/english/sentence_timings.json index c1de3ea37..ab531873d 100644 --- a/2016/matrix-multiplication/english/sentence_timings.json +++ b/2016/matrix-multiplication/english/sentence_timings.json @@ -390,7 +390,7 @@ 586.44 ], [ - "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", 587.2, 592.18 ] diff --git a/2016/matrix-multiplication/english/transcript.txt b/2016/matrix-multiplication/english/transcript.txt index ffdb8bb85..64ea66270 100644 --- a/2016/matrix-multiplication/english/transcript.txt +++ b/2016/matrix-multiplication/english/transcript.txt @@ -76,4 +76,4 @@ This might feel like cheating, but it's not. This is an honest-to-goodness proof that matrix multiplication is associative, and even better than that, it's a good explanation for why that property should be true. I really do encourage you to play around more with this idea, imagining two different transformations, thinking about what happens when you apply one after the other, and then working out the matrix product numerically. Trust me, this is the kind of playtime that really makes the idea sink in. -In the next video, I'll start talking about extending these ideas beyond just two dimensions. \ No newline at end of file +In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then! \ No newline at end of file diff --git a/2016/matrix-multiplication/french/sentence_translations.json b/2016/matrix-multiplication/french/sentence_translations.json index 1a5500f05..0fdb38e8c 100644 --- a/2016/matrix-multiplication/french/sentence_translations.json +++ b/2016/matrix-multiplication/french/sentence_translations.json @@ -623,7 +623,7 @@ "end": 586.44 }, { - "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", "translatedText": "Dans la prochaine vidéo, je commencerai à parler de l'extension de ces idées au-delà de deux dimensions.", "from_community_srt": "Dans la vidéo suivante, je vais commencer à parler l'extension de ces idées au-delà de deux dimensions seulement.", "n_reviews": 0, diff --git a/2016/matrix-multiplication/german/sentence_translations.json b/2016/matrix-multiplication/german/sentence_translations.json index dfd833903..8a8fe9de0 100644 --- a/2016/matrix-multiplication/german/sentence_translations.json +++ b/2016/matrix-multiplication/german/sentence_translations.json @@ -700,7 +700,7 @@ "end": 586.44 }, { - "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", "translatedText": "Im nächsten Video werde ich darüber sprechen, wie man diese Ideen auf mehr als nur zwei Dimensionen ausweiten kann.", "model": "DeepL", "from_community_srt": "Im nächsten Video werde ich beginnen,", diff --git a/2016/matrix-multiplication/hebrew/sentence_translations.json b/2016/matrix-multiplication/hebrew/sentence_translations.json index cb78af02a..326483fcb 100644 --- a/2016/matrix-multiplication/hebrew/sentence_translations.json +++ b/2016/matrix-multiplication/hebrew/sentence_translations.json @@ -698,7 +698,7 @@ "end": 586.44 }, { - "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", "translatedText": "בסרטון הבא, אתחיל לדבר על הרחבת הרעיונות הללו מעבר לשני ממדים בלבד.", "model": "google_nmt", "from_community_srt": "בסירטון הבא, אני אתחיל לדבר על הרחבת הרעיונות הללו מעבר לעולם הדו-מימדי.", diff --git a/2016/matrix-multiplication/hindi/sentence_translations.json b/2016/matrix-multiplication/hindi/sentence_translations.json index 00aa1f620..7409a1d7e 100644 --- a/2016/matrix-multiplication/hindi/sentence_translations.json +++ b/2016/matrix-multiplication/hindi/sentence_translations.json @@ -434,7 +434,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart.", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart.", "translatedText": "यदि आप पहले घुमाते हैं, फिर कतरनी करते हैं, तो आई-हैट 1,1 पर समाप्त होता है, और जे-हैट नकारात्मक 1,0 पर एक अलग दिशा में बंद होता है, और वे दूर की ओर इशारा कर रहे हैं।", "n_reviews": 0, "start": 473.86, diff --git a/2016/matrix-multiplication/hungarian/sentence_translations.json b/2016/matrix-multiplication/hungarian/sentence_translations.json index 4acd90d05..205d8f1b3 100644 --- a/2016/matrix-multiplication/hungarian/sentence_translations.json +++ b/2016/matrix-multiplication/hungarian/sentence_translations.json @@ -624,7 +624,7 @@ "end": 586.44 }, { - "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", "translatedText": "A következő videóban arról fogok beszélni, hogyan lehet ezeket az ötleteket két dimenzión túl is kiterjeszteni.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/matrix-multiplication/indonesian/sentence_translations.json b/2016/matrix-multiplication/indonesian/sentence_translations.json index 977ed7f84..938f128b0 100644 --- a/2016/matrix-multiplication/indonesian/sentence_translations.json +++ b/2016/matrix-multiplication/indonesian/sentence_translations.json @@ -496,7 +496,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart.", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart.", "translatedText": "Jika Anda pertama kali memutar, kemudian melakukan geser, i-hat berakhir pada 1,1, dan j-hat menyimpang ke arah yang berbeda pada negatif 1,0, dan keduanya menunjuk lebih jauh satu sama lain.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/matrix-multiplication/italian/sentence_translations.json b/2016/matrix-multiplication/italian/sentence_translations.json index bc5452174..3c541db1a 100644 --- a/2016/matrix-multiplication/italian/sentence_translations.json +++ b/2016/matrix-multiplication/italian/sentence_translations.json @@ -496,7 +496,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart.", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart.", "translatedText": "Se prima ruoti, poi esegui il taglio, i-hat finisce a 1,1 e j-hat si sposta in una direzione diversa a meno 1,0 e puntano più distanti.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/matrix-multiplication/japanese/sentence_translations.json b/2016/matrix-multiplication/japanese/sentence_translations.json index 1ee340358..d801fd96e 100644 --- a/2016/matrix-multiplication/japanese/sentence_translations.json +++ b/2016/matrix-multiplication/japanese/sentence_translations.json @@ -496,7 +496,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart. ", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart. ", "translatedText": "最初に回転してからせん断を行うと、i-hat は 1,1 で終了し、j- hat は負の 1,0 で別の方向にずれて、さらに離れた方向を指します。", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/matrix-multiplication/korean/sentence_translations.json b/2016/matrix-multiplication/korean/sentence_translations.json index f5e5ca2cb..8603d76e7 100644 --- a/2016/matrix-multiplication/korean/sentence_translations.json +++ b/2016/matrix-multiplication/korean/sentence_translations.json @@ -701,7 +701,7 @@ "end": 586.44 }, { - "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", "translatedText": "다음 비디오에서는 이러한 아이디어를 2차원 이상으로 확장하는 방법에 대해 이야기하겠습니다.", "model": "google_nmt", "from_community_srt": "다음 동영상에서는 이 아이디어를 2차원 이상으로 확장해보겠습니다.", diff --git a/2016/matrix-multiplication/marathi/sentence_translations.json b/2016/matrix-multiplication/marathi/sentence_translations.json index c0e439efe..aaed604cc 100644 --- a/2016/matrix-multiplication/marathi/sentence_translations.json +++ b/2016/matrix-multiplication/marathi/sentence_translations.json @@ -496,7 +496,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart.", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart.", "translatedText": "तुम्ही प्रथम फिरवल्यास, नंतर कातरणे करा, i-hat 1,1 वर संपेल आणि j-हॅट ऋण 1,0 वर वेगळ्या दिशेने बंद आहे, आणि ते दूरवर निर्देशित करत आहेत.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/matrix-multiplication/persian/sentence_translations.json b/2016/matrix-multiplication/persian/sentence_translations.json index d2657ca1a..752b19e6b 100644 --- a/2016/matrix-multiplication/persian/sentence_translations.json +++ b/2016/matrix-multiplication/persian/sentence_translations.json @@ -496,7 +496,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart. ", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart. ", "translatedText": "اگر ابتدا بچرخانید، سپس برش را انجام دهید، i-hat در 1،1 به پایان می رسد، و j-hat در جهت دیگری در منفی 1.0 خاموش است، و آنها دورتر از هم را نشان می دهند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/matrix-multiplication/polish/sentence_translations.json b/2016/matrix-multiplication/polish/sentence_translations.json index 5db1d2c32..a8bd49f8e 100644 --- a/2016/matrix-multiplication/polish/sentence_translations.json +++ b/2016/matrix-multiplication/polish/sentence_translations.json @@ -699,7 +699,7 @@ "end": 586.44 }, { - "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", "translatedText": "W następnym filmie zacznę mówić o rozszerzeniu tych pomysłów poza dwa wymiary.", "model": "google_nmt", "from_community_srt": "W następnym filmie zacznę omawiać co się stanie przy rozszerzeniu tych idei ponad dwa wymiary. Do zobaczenia zatem!", diff --git a/2016/matrix-multiplication/portuguese/sentence_translations.json b/2016/matrix-multiplication/portuguese/sentence_translations.json index 5fe883878..295080a9e 100644 --- a/2016/matrix-multiplication/portuguese/sentence_translations.json +++ b/2016/matrix-multiplication/portuguese/sentence_translations.json @@ -699,7 +699,7 @@ "end": 586.44 }, { - "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", "translatedText": "No próximo vídeo, começarei falando sobre como estender essas ideias além de apenas duas dimensões.", "model": "google_nmt", "from_community_srt": "No próximo vídeo, eu começarei a falar sobre estender essas idéias além de somente duas dimensões.", diff --git a/2016/matrix-multiplication/russian/sentence_translations.json b/2016/matrix-multiplication/russian/sentence_translations.json index 9df71297f..f57c87059 100644 --- a/2016/matrix-multiplication/russian/sentence_translations.json +++ b/2016/matrix-multiplication/russian/sentence_translations.json @@ -624,7 +624,7 @@ "end": 586.44 }, { - "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", "translatedText": "В следующем видео я начну говорить о расширении этих идей за пределы двух измерений.", "from_community_srt": "В следующем видео мы начнём рассматривать эти идеи более чем в двух измерениях.", "n_reviews": 0, diff --git a/2016/matrix-multiplication/spanish/sentence_translations.json b/2016/matrix-multiplication/spanish/sentence_translations.json index 6af839162..123bdd093 100644 --- a/2016/matrix-multiplication/spanish/sentence_translations.json +++ b/2016/matrix-multiplication/spanish/sentence_translations.json @@ -546,7 +546,7 @@ "end": 586.44 }, { - "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", "translatedText": "En el siguiente vídeo, empezaré a hablar sobre cómo ampliar estas ideas más allá de sólo dos dimensiones.", "n_reviews": 1, "start": 587.2, diff --git a/2016/matrix-multiplication/tamil/sentence_translations.json b/2016/matrix-multiplication/tamil/sentence_translations.json index ff026347d..878af68a2 100644 --- a/2016/matrix-multiplication/tamil/sentence_translations.json +++ b/2016/matrix-multiplication/tamil/sentence_translations.json @@ -496,7 +496,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart.", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart.", "translatedText": "நீங்கள் முதலில் சுழற்றினால், கத்தரிக்கவும், i-hat முடிவடையும் 1,1, மற்றும் j-hat வேறு திசையில் எதிர்மறை 1,0 இல் உள்ளது, மேலும் அவை தொலைவில் சுட்டிக்காட்டுகின்றன.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/matrix-multiplication/telugu/sentence_translations.json b/2016/matrix-multiplication/telugu/sentence_translations.json index 3b5186123..39a000780 100644 --- a/2016/matrix-multiplication/telugu/sentence_translations.json +++ b/2016/matrix-multiplication/telugu/sentence_translations.json @@ -496,7 +496,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart.", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart.", "translatedText": "మీరు మొదట రొటేట్ చేస్తే, షీర్ చేయండి, i-hat ముగుస్తుంది 1,1, మరియు j-hat నెగెటివ్ 1,0 వద్ద వేరే దిశలో ఆఫ్‌లో ఉంటుంది మరియు అవి దూరంగా ఉంటాయి.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/matrix-multiplication/thai/sentence_translations.json b/2016/matrix-multiplication/thai/sentence_translations.json index a2a7a0ba1..41edd2314 100644 --- a/2016/matrix-multiplication/thai/sentence_translations.json +++ b/2016/matrix-multiplication/thai/sentence_translations.json @@ -496,7 +496,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart. ", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart. ", "translatedText": "หากคุณหมุนครั้งแรก, แล้วทำแรงเฉือน, i-hat จบลงที่ 1,1, และ j-hat ออกไปอีกทางหนึ่งที่ลบ 1,0 และพวกมันชี้ออกจากกันมากขึ้น ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/matrix-multiplication/turkish/sentence_translations.json b/2016/matrix-multiplication/turkish/sentence_translations.json index 54762929f..1ae2c5872 100644 --- a/2016/matrix-multiplication/turkish/sentence_translations.json +++ b/2016/matrix-multiplication/turkish/sentence_translations.json @@ -700,7 +700,7 @@ "end": 586.44 }, { - "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions.", + "input": "In the next video, I'll start talking about extending these ideas beyond just two dimensions. See you then!", "translatedText": "Bir sonraki videoda bu fikirleri iki boyutun ötesine taşımaktan bahsetmeye başlayacağım.", "model": "google_nmt", "from_community_srt": "Gelecek videoda, bu fikirleri iki boyutun ötesine taşımak hakkında konuşmaya başlayacağım.", diff --git a/2016/matrix-multiplication/ukrainian/sentence_translations.json b/2016/matrix-multiplication/ukrainian/sentence_translations.json index f68d8dbd2..6e3d4ecf8 100644 --- a/2016/matrix-multiplication/ukrainian/sentence_translations.json +++ b/2016/matrix-multiplication/ukrainian/sentence_translations.json @@ -496,7 +496,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart.", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart.", "translatedText": "Якщо ви спочатку обертаєте, а потім виконуєте зсув, i-hat закінчується на 1,1, а j-hat змінюється в іншому напрямку на мінус 1,0, і вони спрямовані далі один від одного.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/matrix-multiplication/urdu/sentence_translations.json b/2016/matrix-multiplication/urdu/sentence_translations.json index af254ac2f..0c43f209c 100644 --- a/2016/matrix-multiplication/urdu/sentence_translations.json +++ b/2016/matrix-multiplication/urdu/sentence_translations.json @@ -496,7 +496,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart. ", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart. ", "translatedText": "اگر آپ پہلے گھماتے ہیں، تو قینچی کریں، i-ہیٹ 1,1 پر ختم ہوتا ہے، اور j-hat منفی 1,0 پر مختلف سمت میں بند ہوتا ہے، اور وہ دور کی طرف اشارہ کر رہے ہوتے ہیں۔ ", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/matrix-multiplication/vietnamese/sentence_translations.json b/2016/matrix-multiplication/vietnamese/sentence_translations.json index 84b9b0955..a0255874a 100644 --- a/2016/matrix-multiplication/vietnamese/sentence_translations.json +++ b/2016/matrix-multiplication/vietnamese/sentence_translations.json @@ -496,7 +496,7 @@ "end": 473.06 }, { - "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing farther apart.", + "input": "If you first rotate, then do the shear, i-hat ends up over at 1,1, and j-hat is off in a different direction at negative 1,0, and they're pointing, you know, farther apart.", "translatedText": "Nếu bạn xoay lần đầu tiên, sau đó thực hiện cắt, i-hat sẽ kết thúc ở mức 1,1 và j-hat lệch theo một hướng khác ở âm 1,0 và chúng đang hướng ra xa nhau hơn.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/nonsquare-matrices/arabic/sentence_translations.json b/2016/nonsquare-matrices/arabic/sentence_translations.json index d3cdccac0..54bafafa7 100644 --- a/2016/nonsquare-matrices/arabic/sentence_translations.json +++ b/2016/nonsquare-matrices/arabic/sentence_translations.json @@ -216,7 +216,7 @@ "end": 245.7 }, { - "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions.", + "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun!", "translatedText": "وحتى ذلك الحين، أشجعك على تجربة هذه الفكرة بنفسك، والتفكير في معاني أشياء مثل ضرب المصفوفات وأنظمة المعادلات الخطية في سياق التحولات بين الأبعاد المختلفة.", "model": "google_nmt", "from_community_srt": "حتى ذلك الحين ، أشجعك على اللعب بهذه الفكرة بنفسك ، تفكر في معاني أشياء مثل الضرب المصفري والأنظمة الخطية معادلات في سياق التحولات بين أبعاد مختلفة. إستمتع!", diff --git a/2016/nonsquare-matrices/czech/sentence_translations.json b/2016/nonsquare-matrices/czech/sentence_translations.json index 303f5201b..8606ebbce 100644 --- a/2016/nonsquare-matrices/czech/sentence_translations.json +++ b/2016/nonsquare-matrices/czech/sentence_translations.json @@ -216,7 +216,7 @@ "end": 245.7 }, { - "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions.", + "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun!", "translatedText": "Do té doby vám doporučuji, abyste si s touto myšlenkou pohráli sami a zamysleli se nad významem věcí, jako je násobení matic a lineární soustavy rovnic v kontextu transformací mezi různými dimenzemi.", "model": "DeepL", "from_community_srt": "Do té doby vás chci povzbudit, abyste si hráli s touhle myšlenkou sami, a zamysleli se nad třeba násobením matic nebo soustavami rovnic. ve světle zobrazení mezi různými dimenzemi. Příjemnou zábavu!", diff --git a/2016/nonsquare-matrices/english/captions.srt b/2016/nonsquare-matrices/english/captions.srt index efc37b8d3..61e7cc89b 100644 --- a/2016/nonsquare-matrices/english/captions.srt +++ b/2016/nonsquare-matrices/english/captions.srt @@ -243,14 +243,14 @@ This is actually a surprisingly meaningful type of transformation with close ties to the dot product, and I'll be talking about that next video. 62 -00:04:06,400 --> 00:04:10,079 +00:04:06,400 --> 00:04:09,926 Until then, I encourage you to play around with this idea on your own, 63 -00:04:10,079 --> 00:04:13,966 -contemplating the meanings of things like matrix multiplication and linear +00:04:09,926 --> 00:04:14,048 +contemplating the meanings of things like matrix multiplication and linear systems 64 -00:04:13,966 --> 00:04:18,320 -systems of equations in the context of transformations between different dimensions. +00:04:14,048 --> 00:04:18,320 +of equations in the context of transformations between different dimensions. Have fun! diff --git a/2016/nonsquare-matrices/english/sentence_timings.json b/2016/nonsquare-matrices/english/sentence_timings.json index 703282ed7..66dd14267 100644 --- a/2016/nonsquare-matrices/english/sentence_timings.json +++ b/2016/nonsquare-matrices/english/sentence_timings.json @@ -120,7 +120,7 @@ 245.7 ], [ - "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions.", + "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun!", 246.4, 258.32 ] diff --git a/2016/nonsquare-matrices/english/transcript.txt b/2016/nonsquare-matrices/english/transcript.txt index 00543f569..2b7eef15f 100644 --- a/2016/nonsquare-matrices/english/transcript.txt +++ b/2016/nonsquare-matrices/english/transcript.txt @@ -22,4 +22,4 @@ Thinking about grid lines remaining parallel and evenly spaced is a little bit m One of these transformations is encoded with a 1x2 matrix, each of whose two columns has just a single entry. The two columns represent where the basis vectors land, and each one of those columns requires just one number, the number that that basis vector landed on. This is actually a surprisingly meaningful type of transformation with close ties to the dot product, and I'll be talking about that next video. -Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. \ No newline at end of file +Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun! \ No newline at end of file diff --git a/2016/nonsquare-matrices/german/sentence_translations.json b/2016/nonsquare-matrices/german/sentence_translations.json index f4bbae9ba..2d915ad25 100644 --- a/2016/nonsquare-matrices/german/sentence_translations.json +++ b/2016/nonsquare-matrices/german/sentence_translations.json @@ -216,7 +216,7 @@ "end": 245.7 }, { - "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions.", + "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun!", "translatedText": "Bis dahin möchte ich dich ermutigen, selbst mit dieser Idee zu spielen und über die Bedeutung von Dingen wie Matrixmultiplikation und linearen Gleichungssystemen im Kontext von Transformationen zwischen verschiedenen Dimensionen nachzudenken.", "model": "DeepL", "from_community_srt": "Bis dahin ermutige ich Sie, herumzuspielen mit dieser Idee auf eigene Faust, Nachdenken über die Bedeutung von Dingen wie Matrixmultiplikation und lineare Systeme von Gleichungen im Kontext von Transformationen zwischen verschiedene Dimensionen. Habe Spaß!", diff --git a/2016/nonsquare-matrices/hebrew/sentence_translations.json b/2016/nonsquare-matrices/hebrew/sentence_translations.json index c99d7b1bd..fb1badd69 100644 --- a/2016/nonsquare-matrices/hebrew/sentence_translations.json +++ b/2016/nonsquare-matrices/hebrew/sentence_translations.json @@ -216,7 +216,7 @@ "end": 245.7 }, { - "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions.", + "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun!", "translatedText": "עד אז, אני ממליץ לך לשחק עם הרעיון הזה בעצמך, להרהר במשמעויות של דברים כמו כפל מטריצה ומערכות לינאריות של משוואות בהקשר של טרנספורמציות בין ממדים שונים.", "model": "google_nmt", "from_community_srt": "עד אז, אני מעודד אותך לשחק קצת עם הרעיון עצמו בעצמך, להשלים את המשמעויות של דברים כמו כפל מטריצות ומערכות לינאריות של משוואות במובן של טרנספורמציה בין מימדים שונים. תהנה!", diff --git a/2016/nonsquare-matrices/hungarian/sentence_translations.json b/2016/nonsquare-matrices/hungarian/sentence_translations.json index 5e6446847..cd59e07e1 100644 --- a/2016/nonsquare-matrices/hungarian/sentence_translations.json +++ b/2016/nonsquare-matrices/hungarian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 245.7 }, { - "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions.", + "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun!", "translatedText": "Addig is arra bátorítom, hogy játszadozzon ezzel az ötlettel, és gondolkodjon el az olyan dolgok jelentésén, mint a mátrixszorzás és a lineáris egyenletrendszerek a különböző dimenziók közötti transzformációk kontextusában.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/nonsquare-matrices/italian/sentence_translations.json b/2016/nonsquare-matrices/italian/sentence_translations.json index aa632defc..afb8a980f 100644 --- a/2016/nonsquare-matrices/italian/sentence_translations.json +++ b/2016/nonsquare-matrices/italian/sentence_translations.json @@ -216,7 +216,7 @@ "end": 245.7 }, { - "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions.", + "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun!", "translatedText": "Fino ad allora, ti incoraggio a giocare con questa idea da solo, contemplando il significato di cose come la moltiplicazione di matrici e i sistemi lineari di equazioni nel contesto delle trasformazioni tra dimensioni diverse.", "model": "google_nmt", "from_community_srt": "Prima di allora, ti incoraggio a giocare con quest'idea per conto tuo, contemplando il significato di cose come la moltiplicazione di matrici e i sistemi lineari di equazioni nel contesto delle trasformazioni tra dimensioni diverse. Divertiti!", diff --git a/2016/nonsquare-matrices/korean/sentence_translations.json b/2016/nonsquare-matrices/korean/sentence_translations.json index ae39effd8..7e526fd26 100644 --- a/2016/nonsquare-matrices/korean/sentence_translations.json +++ b/2016/nonsquare-matrices/korean/sentence_translations.json @@ -216,7 +216,7 @@ "end": 245.7 }, { - "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions.", + "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun!", "translatedText": "그때까지는 서로 다른 차원 간의 변환이라는 맥락에서 행렬 곱셈이나 선형 방정식 시스템과 같은 것의 의미를 숙고하면서 이 아이디어를 스스로 시험해 보시기 바랍니다.", "model": "google_nmt", "from_community_srt": "그때까지, 너가 이 개념을 자신의 것으로 만드는데 시간을 갖길 바래. 행렬 곱셈과 선형방정식계에 대해서 고민해봐. 다른 차원들 사이에서의 변화이라는 문맥에서 고민해봐. (역주: 높은차원으로 가는 것은 평면이 3차원 공간에서 비틀리는 것과 유사하고, 낮은 차원으로 가는 것은 투사하는 것과 유사한듯) 재미있게 보내!", diff --git a/2016/nonsquare-matrices/polish/sentence_translations.json b/2016/nonsquare-matrices/polish/sentence_translations.json index 150bedd84..37fb1ee93 100644 --- a/2016/nonsquare-matrices/polish/sentence_translations.json +++ b/2016/nonsquare-matrices/polish/sentence_translations.json @@ -216,7 +216,7 @@ "end": 245.7 }, { - "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions.", + "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun!", "translatedText": "Do tego czasu zachęcam do samodzielnej zabawy tym pomysłem, kontemplując znaczenie takich rzeczy, jak mnożenie macierzy i liniowe układy równań w kontekście transformacji pomiędzy różnymi wymiarami.", "model": "google_nmt", "from_community_srt": "Zanim to nastąpi, zachęcam was do pobawienia się tym pomysłem samodzielnie, oraz kontemplacji nad znaczeniem takich rzeczy jak mnożenie macierzy i układy równań liniowych, w kontekście przekształceń pomiędzy różnymi wymiarami.", diff --git a/2016/nonsquare-matrices/portuguese/sentence_translations.json b/2016/nonsquare-matrices/portuguese/sentence_translations.json index 71425e65c..af8588815 100644 --- a/2016/nonsquare-matrices/portuguese/sentence_translations.json +++ b/2016/nonsquare-matrices/portuguese/sentence_translations.json @@ -216,7 +216,7 @@ "end": 245.7 }, { - "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions.", + "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun!", "translatedText": "Até então, encorajo você a brincar com essa ideia por conta própria, contemplando os significados de coisas como multiplicação de matrizes e sistemas lineares de equações no contexto de transformações entre diferentes dimensões.", "model": "google_nmt", "from_community_srt": "Até lá, eu o encorajo a brincar um pouco com essa ideia, contemplando as significações de coisas como multiplicação matricial e sistemas lineares, no contexto de transformações entre dimensões diferentes. Divirta-se!", diff --git a/2016/nonsquare-matrices/spanish/sentence_translations.json b/2016/nonsquare-matrices/spanish/sentence_translations.json index d5131e2ab..dbd2e9fcf 100644 --- a/2016/nonsquare-matrices/spanish/sentence_translations.json +++ b/2016/nonsquare-matrices/spanish/sentence_translations.json @@ -192,7 +192,7 @@ "end": 245.7 }, { - "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions.", + "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun!", "translatedText": "Hasta entonces, te animo a que juegues con esta idea por tu cuenta, contemplando los significados de cosas como la multiplicación de matrices y los sistemas lineales de ecuaciones en el contexto de las transformaciones entre diferentes dimensiones.", "from_community_srt": "Hasta entonces los invito a que reflexionen esta idea por su cuenta contemplando el significado de cosas como el producto matricial y sistemas de ecuaciones linales en el contexto de transformaciones entre dimensiones ditintas.", "n_reviews": 0, diff --git a/2016/nonsquare-matrices/turkish/sentence_translations.json b/2016/nonsquare-matrices/turkish/sentence_translations.json index b3b929177..fac110aea 100644 --- a/2016/nonsquare-matrices/turkish/sentence_translations.json +++ b/2016/nonsquare-matrices/turkish/sentence_translations.json @@ -216,7 +216,7 @@ "end": 245.7 }, { - "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions.", + "input": "Until then, I encourage you to play around with this idea on your own, contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of transformations between different dimensions. Have fun!", "translatedText": "O zamana kadar, farklı boyutlar arasındaki dönüşümler bağlamında matris çarpımı ve doğrusal denklem sistemleri gibi şeylerin anlamları üzerinde düşünerek bu fikir üzerinde kendi başınıza oynamanızı tavsiye ediyorum.", "model": "google_nmt", "from_community_srt": "O zamana kadar bu fikri kendi başına oynamanı tavsiye ederim. boyutlar arası dönüşümün konularına kafa yorun mesela matris çarpımı, lineer doğru denklemleri gibi İyi eğlenceler!", diff --git a/2016/span/arabic/sentence_translations.json b/2016/span/arabic/sentence_translations.json index 28723ff71..a3c2cbaf9 100644 --- a/2016/span/arabic/sentence_translations.json +++ b/2016/span/arabic/sentence_translations.json @@ -630,7 +630,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "في الفيديو التالي، سأتحدث عن المصفوفات في تحويل الفضاء.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/span/czech/sentence_translations.json b/2016/span/czech/sentence_translations.json index 8b052561e..f6e4a3ed0 100644 --- a/2016/span/czech/sentence_translations.json +++ b/2016/span/czech/sentence_translations.json @@ -633,7 +633,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "V příštím videu se budu věnovat maticím v transformačním prostoru.", "model": "DeepL", "from_community_srt": "proč by tahle definice měla dávat smysl. V příštím videu se podíváme na matice a transformace prostoru.", diff --git a/2016/span/english/captions.srt b/2016/span/english/captions.srt index 983bdfa62..22f378a31 100644 --- a/2016/span/english/captions.srt +++ b/2016/span/english/captions.srt @@ -576,5 +576,5 @@ words span and linearly independent, think about why this definition would make 145 00:09:33,880 --> 00:09:37,880 -In the next video, I'll get into matrices in transforming space. +In the next video, I'll get into matrices in transforming space. See you then! diff --git a/2016/span/english/sentence_timings.json b/2016/span/english/sentence_timings.json index 4192ac016..58370c0fc 100644 --- a/2016/span/english/sentence_timings.json +++ b/2016/span/english/sentence_timings.json @@ -355,7 +355,7 @@ 571.7 ], [ - "In the next video, I'll get into matrices in transforming space.", + "In the next video, I'll get into matrices in transforming space. See you then!", 573.88, 577.88 ] diff --git a/2016/span/english/transcript.txt b/2016/span/english/transcript.txt index 797802cab..51a742c31 100644 --- a/2016/span/english/transcript.txt +++ b/2016/span/english/transcript.txt @@ -69,4 +69,4 @@ On the other hand, if each vector really does add another dimension to the span, So with all of that terminology, and hopefully with some good mental images to go with it, let me leave you with a puzzle before we go. The technical definition of a basis of a space is a set of linearly independent vectors that span that space. Now, given how I described a basis earlier, and given your current understanding of the words span and linearly independent, think about why this definition would make sense. -In the next video, I'll get into matrices in transforming space. \ No newline at end of file +In the next video, I'll get into matrices in transforming space. See you then! \ No newline at end of file diff --git a/2016/span/french/sentence_translations.json b/2016/span/french/sentence_translations.json index 4cbcd2619..bad63856a 100644 --- a/2016/span/french/sentence_translations.json +++ b/2016/span/french/sentence_translations.json @@ -561,7 +561,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "Dans la prochaine vidéo, j'aborderai les matrices dans la transformation de l'espace.", "from_community_srt": "Dans la prochaine vidéo, je toucherai à la notion de matrice et à la transformation de l'espace.", "n_reviews": 0, diff --git a/2016/span/german/sentence_translations.json b/2016/span/german/sentence_translations.json index 21e17ba43..16e5ab4d7 100644 --- a/2016/span/german/sentence_translations.json +++ b/2016/span/german/sentence_translations.json @@ -633,7 +633,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "Im nächsten Video werde ich mich mit Matrizen im Transformationsraum beschäftigen.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/span/hebrew/sentence_translations.json b/2016/span/hebrew/sentence_translations.json index d903aa778..3aa136ac6 100644 --- a/2016/span/hebrew/sentence_translations.json +++ b/2016/span/hebrew/sentence_translations.json @@ -635,7 +635,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "בסרטון הבא, אכנס למטריצות בהפיכת החלל.", "model": "google_nmt", "from_community_srt": "בסירטון הבא, אני אכנס למטריצות ולטרנספורמציות במרחב.", diff --git a/2016/span/hungarian/sentence_translations.json b/2016/span/hungarian/sentence_translations.json index 13968c3e1..e0edb961e 100644 --- a/2016/span/hungarian/sentence_translations.json +++ b/2016/span/hungarian/sentence_translations.json @@ -568,7 +568,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "A következő videóban a mátrixok transzformációs térben történő átalakításával foglalkozom.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/span/italian/sentence_translations.json b/2016/span/italian/sentence_translations.json index fa7419104..cbb50af12 100644 --- a/2016/span/italian/sentence_translations.json +++ b/2016/span/italian/sentence_translations.json @@ -635,7 +635,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "Nel prossimo video parlerò delle matrici nella trasformazione dello spazio.", "model": "google_nmt", "from_community_srt": "Nel prossimo video,", diff --git a/2016/span/korean/sentence_translations.json b/2016/span/korean/sentence_translations.json index 8412caa86..0f046db2a 100644 --- a/2016/span/korean/sentence_translations.json +++ b/2016/span/korean/sentence_translations.json @@ -635,7 +635,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "다음 영상에서는 공간을 변형하는 행렬에 대해 알아볼게요.", "model": "google_nmt", "from_community_srt": "다음 동영상에서는 행렬과 공간변형에 대해 다룰 것입니다.", diff --git a/2016/span/polish/sentence_translations.json b/2016/span/polish/sentence_translations.json index c6bb189c6..1174627b4 100644 --- a/2016/span/polish/sentence_translations.json +++ b/2016/span/polish/sentence_translations.json @@ -635,7 +635,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "W następnym filmie zajmę się macierzami przekształcającymi przestrzeń.", "model": "google_nmt", "from_community_srt": "W następnym filmie, zabierzemy się za macierze i transformaty przestrzenne.", diff --git a/2016/span/portuguese/sentence_translations.json b/2016/span/portuguese/sentence_translations.json index 0f5a5b795..966e49fb1 100644 --- a/2016/span/portuguese/sentence_translations.json +++ b/2016/span/portuguese/sentence_translations.json @@ -635,7 +635,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "No próximo vídeo, abordarei matrizes na transformação do espaço.", "model": "google_nmt", "from_community_srt": "pense por que essa definição faria sentido. No próximo vídeo, irei abordar matrizes e transformações lineares.", diff --git a/2016/span/russian/sentence_translations.json b/2016/span/russian/sentence_translations.json index 2d7b06f53..7806515c4 100644 --- a/2016/span/russian/sentence_translations.json +++ b/2016/span/russian/sentence_translations.json @@ -566,7 +566,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "В следующем видео я займусь матрицами в преобразовании пространства.", "from_community_srt": "почему это определение имеет смысл. В следующем видео, я расскажу о матрицах и преобразованиях пространства.", "n_reviews": 0, diff --git a/2016/span/spanish/sentence_translations.json b/2016/span/spanish/sentence_translations.json index b8b7febf2..54d08ddad 100644 --- a/2016/span/spanish/sentence_translations.json +++ b/2016/span/spanish/sentence_translations.json @@ -497,7 +497,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "En el siguiente vídeo, abordaré las matrices en la transformación del espacio.", "n_reviews": 1, "start": 573.88, diff --git a/2016/span/turkish/sentence_translations.json b/2016/span/turkish/sentence_translations.json index d165b0a90..c682a93dc 100644 --- a/2016/span/turkish/sentence_translations.json +++ b/2016/span/turkish/sentence_translations.json @@ -635,7 +635,7 @@ "end": 571.7 }, { - "input": "In the next video, I'll get into matrices in transforming space.", + "input": "In the next video, I'll get into matrices in transforming space. See you then!", "translatedText": "Bir sonraki videoda uzayı dönüştürmede matrislere gireceğim.", "model": "google_nmt", "from_community_srt": "Bir sonraki videoda, matrislere ve uzay dönüşümü konularına gireceğim.", diff --git a/2016/triangle-of-power/english/captions.srt b/2016/triangle-of-power/english/captions.srt index feb2bc0b2..8b559027e 100644 --- a/2016/triangle-of-power/english/captions.srt +++ b/2016/triangle-of-power/english/captions.srt @@ -1,5 +1,5 @@ 1 -00:00:03,899 --> 00:00:06,820 +00:00:03,900 --> 00:00:06,820 Usually, I don't think notation and math matters that much. 2 @@ -263,7 +263,7 @@ The corresponding fact for logarithms is that log of x times y equals log of x plus log of y. 67 -00:04:09,239 --> 00:04:11,310 +00:04:09,240 --> 00:04:11,310 When you write this with the triangle of power, 68 diff --git a/2016/vectors/arabic/sentence_translations.json b/2016/vectors/arabic/sentence_translations.json index 12c785ed9..a45738576 100644 --- a/2016/vectors/arabic/sentence_translations.json +++ b/2016/vectors/arabic/sentence_translations.json @@ -206,7 +206,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "عندما أريد أن أنقل فكرة الفضاء ثنائي الأبعاد ككل، والتي سترون أنها تظهر قليلاً في الطريق، ولكن الآن سوف يعيقون قليلاً في الطريق.", "model": "google_nmt", "from_community_srt": "إذا أردت تمثيل فكرة الفراغ ثنائي الأبعاد بشكل صحيح والتي كثيراً ما سترونها في هذه الفيديوهات سأقوم بتمدد الفواصل لإنشاء شبكة، ولكن سأخفيهم الآن لبعض الوقت سأقوم بتمدد الفواصل لإنشاء شبكة، ولكن سأخفيهم الآن لبعض الوقت", @@ -596,7 +596,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "وعلى الجانب الآخر، فهو يمنح الأشخاص مثل الفيزيائيين ومبرمجي رسومات الكمبيوتر لغة لوصف الفضاء والكمبيوتر.", "model": "google_nmt", "from_community_srt": "من جهة أخرى فإنه يعطي الأشخاص كالفيزيائيين ومبرمجي الجرافيك لغة لوصف الفراغ وتعديله من جهة أخرى فإنه يعطي الأشخاص كالفيزيائيين ومبرمجي الجرافيك لغة لوصف الفراغ وتعديله باستخدام الأرقام التي يمكن صياغتها من خلال الكمبيوتر باستخدام الأرقام التي يمكن صياغتها من خلال الكمبيوتر", diff --git a/2016/vectors/czech/sentence_translations.json b/2016/vectors/czech/sentence_translations.json index 61117303e..35ac7f1b5 100644 --- a/2016/vectors/czech/sentence_translations.json +++ b/2016/vectors/czech/sentence_translations.json @@ -207,7 +207,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "Až budu chtít zprostředkovat myšlenku 2D prostoru jako celku, což uvidíte, přijde mi trochu do cesty, ale právě teď mi budou trochu překážet.", "model": "DeepL", "from_community_srt": "Když chci znázornit dvou-rozměrnou rovinu jako celek, jak to uvidíte v mnoha následujících videích, rozšířím rysky na celou mřížku. Ale zrovna teď by to spíš překáželo.", @@ -601,7 +601,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "Na druhou stranu dává lidem, jako jsou fyzici a programátoři počítačové grafiky, jazyk pro popis prostoru a počítače.", "model": "DeepL", "from_community_srt": "co jisté operace provedou, a na druhé straně mince dává lineární algebra fyzikům a programátorům počítačové grafiky jazyk, jak popsat prostor a manipulace s ním pomocí čísel, které mohou být zchroustána a prohnána počítačem.", @@ -645,4 +645,4 @@ "start": 575.72, "end": 591.82 } -] +] \ No newline at end of file diff --git a/2016/vectors/english/captions.srt b/2016/vectors/english/captions.srt index 5fa47661b..accade434 100644 --- a/2016/vectors/english/captions.srt +++ b/2016/vectors/english/captions.srt @@ -207,406 +207,414 @@ After choosing an arbitrary length to represent one, you make tick marks on each axis to represent this distance. 53 -00:03:12,320 --> 00:03:15,656 +00:03:12,320 --> 00:03:14,654 When I want to convey the idea of 2D space as a whole, 54 -00:03:15,656 --> 00:03:20,025 -which you'll see comes up a bit in the way, but right now they'll get a +00:03:14,654 --> 00:03:16,733 +which you'll see comes up a lot in these videos, 55 -00:03:20,025 --> 00:03:21,360 -little bit in the way. +00:03:16,733 --> 00:03:19,450 +I'll extend these tick marks to make grid lines, but right now, 56 +00:03:19,450 --> 00:03:21,360 +they'll actually get a little bit in the way. + +57 00:03:22,000 --> 00:03:25,757 The coordinates of a vector is a pair of numbers that basically gives -57 +58 00:03:25,757 --> 00:03:30,160 instructions for how to get from the tail of that vector at the origin to its tip. -58 +59 00:03:30,880 --> 00:03:34,004 The first number tells you how far to walk along the x-axis, -59 +60 00:03:34,004 --> 00:03:38,256 positive numbers indicating rightward motion, negative numbers indicating leftward -60 +61 00:03:38,256 --> 00:03:42,303 motion, and the second number tells you how far to walk parallel to the y-axis -61 +62 00:03:42,303 --> 00:03:45,121 after that, positive numbers indicating upward motion, -62 +63 00:03:45,121 --> 00:03:47,580 and negative numbers indicating downward motion. -63 +64 00:03:48,140 --> 00:03:51,191 To distinguish vectors from points, the convention is to write -64 +65 00:03:51,191 --> 00:03:54,340 this pair of numbers vertically with square brackets around them. -65 +66 00:03:56,340 --> 00:03:59,660 Every pair of numbers gives you one and only one vector, -66 +67 00:03:59,660 --> 00:04:03,680 and every vector is associated with one and only one pair of numbers. -67 +68 00:04:04,640 --> 00:04:05,500 What about in three dimensions? -68 +69 00:04:06,200 --> 00:04:08,970 Well, you add a third axis, called the z-axis, -69 +70 00:04:08,970 --> 00:04:12,920 which is perpendicular to both the x and y-axes, and in this case, -70 +71 00:04:12,920 --> 00:04:16,339 each vector is associated with ordered triplet of numbers. -71 +72 00:04:16,860 --> 00:04:19,903 The first tells you how far to move along the x-axis, -72 +73 00:04:19,903 --> 00:04:23,340 the second tells you how far to move parallel to the y-axis, -73 +74 00:04:23,340 --> 00:04:27,680 and the third one tells you how far to then move parallel to this new z-axis. -74 +75 00:04:28,400 --> 00:04:31,869 Every triplet of numbers gives you one unique vector in space, -75 +76 00:04:31,869 --> 00:04:35,560 and every vector in space gives you exactly one triplet of numbers. -76 +77 00:04:36,900 --> 00:04:40,100 All right, so back to vector addition and multiplication by numbers. -77 +78 00:04:40,460 --> 00:04:44,780 After all, every topic in linear algebra is going to center around these two operations. -78 +79 00:04:45,440 --> 00:04:47,640 Luckily, each one's pretty straightforward to define. -79 +80 00:04:48,480 --> 00:04:51,561 Let's say we have two vectors, one pointing up and a little to the right, -80 +81 00:04:51,561 --> 00:04:53,560 and the other one pointing right and down a bit. -81 +82 00:04:53,960 --> 00:04:56,879 To add these two vectors, move the second one so -82 +83 00:04:56,879 --> 00:04:59,680 that its tail sits at the tip of the first one. -83 +84 00:05:00,300 --> 00:05:04,456 Then, if you draw a new vector from the tail of the first one to -84 +85 00:05:04,456 --> 00:05:08,740 where the tip of the second one sits, that new vector is their sum. -85 +86 00:05:12,080 --> 00:05:15,569 This definition of addition, by the way, is pretty much the only time -86 +87 00:05:15,569 --> 00:05:18,860 in linear algebra where we let vectors stray away from the origin. -87 +88 00:05:19,720 --> 00:05:21,480 Now, why is this a reasonable thing to do? -88 +89 00:05:21,740 --> 00:05:24,020 Why this definition of addition and not some other one? -89 +90 00:05:25,520 --> 00:05:29,995 Well, the way I like to think about it is that each vector represents a certain movement, -90 +91 00:05:29,995 --> 00:05:32,680 a step with a certain distance and direction in space. -91 +92 00:05:33,980 --> 00:05:37,709 If you take a step along the first vector, then take a step in the direction -92 +93 00:05:37,709 --> 00:05:41,196 and distance described by the second vector, the overall effect is just -93 +94 00:05:41,196 --> 00:05:44,780 the same as if you moved along the sum of those two vectors to start with. -94 +95 00:05:45,260 --> 00:05:47,251 You could think about this as an extension of -95 +96 00:05:47,251 --> 00:05:49,460 how we think about adding numbers on a number line. -96 +97 00:05:50,180 --> 00:05:53,521 One way that we teach kids to think about this, say with 2 plus 5, -97 +98 00:05:53,521 --> 00:05:57,960 is to think of moving two steps to the right followed by another five steps to the right. -98 +99 00:05:57,960 --> 00:06:01,720 The overall effect is the same as if you just took seven steps to the right. -99 +100 00:06:02,660 --> 00:06:05,480 In fact, let's see how vector addition looks numerically. -100 +101 00:06:06,020 --> 00:06:11,698 The first vector here has coordinates 1, 2, and the second one has coordinates 3, -101 +102 00:06:11,698 --> 00:06:12,460 negative 1. -102 +103 00:06:14,360 --> 00:06:17,343 When you take the vector sum using this tip-to-tail method, -103 +104 00:06:17,343 --> 00:06:21,420 you can think of a four-step path from the origin to the tip of the second vector. -104 +105 00:06:21,840 --> 00:06:25,620 Walk 1 to the right, then 2 up, then 3 to the right, then 1 down. -105 +106 00:06:26,920 --> 00:06:31,341 Reorganizing these steps so that you first do all of the rightward motion, -106 +107 00:06:31,341 --> 00:06:35,173 then do all the vertical motion, you can read it as saying first -107 +108 00:06:35,173 --> 00:06:38,180 move 1 plus 3 to the right, then move 2 minus 1 up. -108 +109 00:06:40,080 --> 00:06:44,920 So the new vector has coordinates 1 plus 3 and 2 plus negative 1. -109 +110 00:06:45,600 --> 00:06:49,122 In general, vector addition in this list of numbers conception -110 +111 00:06:49,122 --> 00:06:52,700 looks like matching up their terms and adding each one together. -111 +112 00:06:54,640 --> 00:06:58,360 The other fundamental vector operation is multiplication by a number. -112 +113 00:06:58,860 --> 00:07:01,380 Now this is best understood just by looking at a few examples. -113 +114 00:07:01,840 --> 00:07:04,994 If you take the number 2 and multiply it by a given vector, -114 +115 00:07:04,994 --> 00:07:09,620 it means you stretch out that vector so that it's two times as long as when you started. -115 +116 00:07:10,500 --> 00:07:13,065 If you multiply that vector by, say, one-third, -116 +117 00:07:13,065 --> 00:07:16,860 it means you squish it down so that it's one-third the original length. -117 +118 00:07:17,640 --> 00:07:21,317 When you multiply it by a negative number, like negative 1.8, -118 +119 00:07:21,317 --> 00:07:26,300 then the vector first gets flipped around, then stretched out by that factor of 1.8. -119 +120 00:07:27,360 --> 00:07:31,991 This process of stretching or squishing or sometimes reversing the direction of -120 +121 00:07:31,991 --> 00:07:36,739 a vector is called scaling, and whenever you catch a number like two or one-third -121 +122 00:07:36,739 --> 00:07:41,140 or negative 1.8 acting like this, scaling some vector, you call it a scalar. -122 +123 00:07:41,940 --> 00:07:46,252 In fact, throughout linear algebra, one of the main things that numbers do is scale -123 +124 00:07:46,252 --> 00:07:50,820 vectors, so it's common to use the word scalar pretty much interchangeably with the word -124 +125 00:07:50,820 --> 00:07:51,180 number. -125 +126 00:07:52,020 --> 00:07:55,540 Numerically, stretching out a vector by a factor of, say, 2, -126 +127 00:07:55,540 --> 00:07:59,580 corresponds with multiplying each of its components by that factor, 2. -127 +128 00:08:00,300 --> 00:08:03,115 So in the conception of vectors as lists of numbers, -128 +129 00:08:03,115 --> 00:08:07,098 multiplying a given vector by a scalar means multiplying each one of those -129 +130 00:08:07,098 --> 00:08:08,480 components by that scalar. -130 +131 00:08:10,220 --> 00:08:14,644 You'll see in the following videos what I mean when I say linear algebra topics tend to -131 +132 00:08:14,644 --> 00:08:17,108 revolve around these two fundamental operations, -132 +133 00:08:17,108 --> 00:08:19,220 vector addition and scalar multiplication. -133 +134 00:08:19,980 --> 00:08:22,879 And I'll talk more in the last video about how and why the -134 +135 00:08:22,879 --> 00:08:25,336 mathematician thinks only about these operations, -135 +136 00:08:25,336 --> 00:08:29,120 independent and abstracted away from however you choose to represent vectors. -136 +137 00:08:29,800 --> 00:08:33,866 In truth, it doesn't matter whether you think about vectors as fundamentally being arrows -137 +138 00:08:33,866 --> 00:08:37,255 in space, like I'm suggesting you do, that happen to have a nice numerical -138 +139 00:08:37,255 --> 00:08:41,322 representation, or fundamentally as lists of numbers that happen to have a nice geometric -139 +140 00:08:41,322 --> 00:08:42,000 interpretation. -140 +141 00:08:42,520 --> 00:08:46,024 The usefulness of linear algebra has less to do with either one of these -141 +142 00:08:46,024 --> 00:08:49,720 views than it does with the ability to translate back and forth between them. -142 +143 00:08:50,140 --> 00:08:53,590 It gives the data analyst a nice way to conceptualize many lists -143 +144 00:08:53,590 --> 00:08:57,041 of numbers in a visual way, which can seriously clarify patterns -144 +145 00:08:57,041 --> 00:09:00,280 in data and give a global view of what certain operations do. -145 -00:09:00,820 --> 00:09:06,100 -And on the flip side, it gives people like physicists and computer - 146 -00:09:06,100 --> 00:09:11,380 -graphics programmers a language to describe space and the computer. +00:09:00,820 --> 00:09:04,205 +And on the flip side, it gives people like physicists and computer 147 +00:09:04,205 --> 00:09:07,792 +graphics programmers a language to describe space and the manipulation + +148 +00:09:07,792 --> 00:09:11,380 +of space using numbers that can be crunched and run through a computer. + +149 00:09:12,300 --> 00:09:15,918 When I do math-y animations, for example, I start by thinking about what's -148 +150 00:09:15,918 --> 00:09:20,116 actually going on in space, and then get the computer to represent things numerically, -149 +151 00:09:20,116 --> 00:09:23,060 thereby figuring out where to place the pixels on the screen. -150 +152 00:09:23,480 --> 00:09:26,580 And doing that usually relies on a lot of linear algebra understanding. -151 -00:09:27,840 --> 00:09:31,574 -So there are your vector basics, and in the next video I'll start getting into some +153 +00:09:27,840 --> 00:09:35,702 +So there are your vector basics, and in the next video I'll -152 -00:09:31,574 --> 00:09:35,220 -pretty neat concepts surrounding vectors, like span, bases, and linear dependence. +154 +00:09:35,702 --> 00:09:45,661 +start getting into some pretty neat concepts surrounding vectors like span, -153 -00:09:35,720 --> 00:09:51,820 -See you then! +155 +00:09:45,661 --> 00:09:51,820 +bases, and linear dependence. See you then! you diff --git a/2016/vectors/english/sentence_timings.json b/2016/vectors/english/sentence_timings.json index 49600ad3f..4f99e98c3 100644 --- a/2016/vectors/english/sentence_timings.json +++ b/2016/vectors/english/sentence_timings.json @@ -115,7 +115,7 @@ 191.36 ], [ - "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", 192.32, 201.36 ], @@ -335,7 +335,7 @@ 540.28 ], [ - "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", 540.82, 551.38 ], @@ -350,13 +350,8 @@ 566.58 ], [ - "So there are your vector basics, and in the next video I'll start getting into some pretty neat concepts surrounding vectors, like span, bases, and linear dependence.", + "So there are your vector basics, and in the next video I'll start getting into some pretty neat concepts surrounding vectors like span, bases, and linear dependence. See you then! you", 567.84, - 575.22 - ], - [ - "See you then!", - 575.72, 591.82 ] ] \ No newline at end of file diff --git a/2016/vectors/english/transcript.txt b/2016/vectors/english/transcript.txt index 5831321c3..8ff29eb5a 100644 --- a/2016/vectors/english/transcript.txt +++ b/2016/vectors/english/transcript.txt @@ -21,7 +21,7 @@ Now, while I'm sure that many of you are already familiar with this coordinate s Focusing our attention on two dimensions for the moment, you have a horizontal line, called the x-axis, and a vertical line, called the y-axis. The place where they intersect is called the origin, which you should think of as the center of space and the root of all vectors. After choosing an arbitrary length to represent one, you make tick marks on each axis to represent this distance. -When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way. +When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way. The coordinates of a vector is a pair of numbers that basically gives instructions for how to get from the tail of that vector at the origin to its tip. The first number tells you how far to walk along the x-axis, positive numbers indicating rightward motion, negative numbers indicating leftward motion, and the second number tells you how far to walk parallel to the y-axis after that, positive numbers indicating upward motion, and negative numbers indicating downward motion. To distinguish vectors from points, the convention is to write this pair of numbers vertically with square brackets around them. @@ -65,7 +65,7 @@ And I'll talk more in the last video about how and why the mathematician thinks In truth, it doesn't matter whether you think about vectors as fundamentally being arrows in space, like I'm suggesting you do, that happen to have a nice numerical representation, or fundamentally as lists of numbers that happen to have a nice geometric interpretation. The usefulness of linear algebra has less to do with either one of these views than it does with the ability to translate back and forth between them. It gives the data analyst a nice way to conceptualize many lists of numbers in a visual way, which can seriously clarify patterns in data and give a global view of what certain operations do. -And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer. +And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer. When I do math-y animations, for example, I start by thinking about what's actually going on in space, and then get the computer to represent things numerically, thereby figuring out where to place the pixels on the screen. And doing that usually relies on a lot of linear algebra understanding. -So there are your vector basics, and in the next video I'll start getting into some pretty neat concepts surrounding vectors, like span, bases, and linear dependence. \ No newline at end of file +So there are your vector basics, and in the next video I'll start getting into some pretty neat concepts surrounding vectors like span, bases, and linear dependence. See you then! you \ No newline at end of file diff --git a/2016/vectors/french/sentence_translations.json b/2016/vectors/french/sentence_translations.json index b717a958d..33459f567 100644 --- a/2016/vectors/french/sentence_translations.json +++ b/2016/vectors/french/sentence_translations.json @@ -184,7 +184,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "Lorsque je veux transmettre l'idée de l'espace 2D dans son ensemble, ce qui, vous le verrez, gênera un peu, mais pour le moment, cela gênera un peu.", "from_community_srt": "Quand je veux transmettre l'idée d'un espace 2D dans son ensemble, ce qui arrivera beaucoup dans ces vidéos, j'étendrai ces marques pour faire une grille, mais pour le moment elles sont superflues.", "n_reviews": 0, @@ -535,7 +535,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "Et d’un autre côté, cela donne à des personnes comme les physiciens et les programmeurs infographistes un langage pour décrire l’espace et l’ordinateur.", "from_community_srt": "d'un autre côté, cela donne aux physiciens ou aux infographistes un langage pour décrire de voir l'espace et une façon de le manipuler en utilisant des nombres qui peuvent être traités par l'ordinateur.", "n_reviews": 0, diff --git a/2016/vectors/german/sentence_translations.json b/2016/vectors/german/sentence_translations.json index 961635b62..0568d9bc2 100644 --- a/2016/vectors/german/sentence_translations.json +++ b/2016/vectors/german/sentence_translations.json @@ -207,7 +207,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "Wenn ich die Idee des 2D-Raums als Ganzes vermitteln will, was du sehen wirst, kommen sie ein bisschen in die Quere, aber im Moment sind sie ein bisschen im Weg.", "model": "DeepL", "from_community_srt": "Wenn ich das Konzept des gesamten zweidimensionalen Raumes verdeutlichen will - was (wie ihr sehen werdet) häufiger passieren wird - , erweitere ich das Koordinatensystem mit einem Raster Aber im Moment wären sie ein bisschen im Weg", @@ -598,7 +598,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "Und auf der anderen Seite gibt es Leuten wie Physikern und Computergrafikprogrammierern eine Sprache, um den Raum und den Computer zu beschreiben.", "model": "DeepL", "from_community_srt": "was bestimmte Operationen tun Und auf der anderen Seite gibt es z.B. Physikern und Computergrafikern eine Sprache zur Beschreibung des Raumes und der Veränderung des Raumes mit Zahlen, die durch einen Computer gejagt werden können.", diff --git a/2016/vectors/greek/sentence_translations.json b/2016/vectors/greek/sentence_translations.json index 22f04f299..534865c02 100644 --- a/2016/vectors/greek/sentence_translations.json +++ b/2016/vectors/greek/sentence_translations.json @@ -207,7 +207,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "Όταν θέλω να μεταφέρω την ιδέα του δισδιάστατου χώρου στο σύνολό της, που θα δείτε να εμφανίζεται λίγο με τον τρόπο, αλλά αυτή τη στιγμή θα είναι λίγο εμπόδιο.", "model": "google_nmt", "from_community_srt": "Όταν θέλω να μεταφέρω την ιδέα του διδιάστατου χώρου ως σύνολο, κάτι που θα δείτε ότι κάνω συχνά σε αυτά τα βίντεο, τότε θα επεκτείνω αυτά τα σημάδια για να σχηματίσουν τις γραμμές πλέγματος, αλλά για τώρα μπορούμε να τις παραλείξουμε για λόγους απλότητας.", @@ -602,7 +602,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "Και από την άλλη πλευρά, δίνει σε ανθρώπους όπως φυσικούς και προγραμματιστές γραφικών υπολογιστών μια γλώσσα για να περιγράψουν το διάστημα και τον υπολογιστή.", "model": "google_nmt", "from_community_srt": "Και από την άλλη πλευρά: δίνει σε διάφορες ομάδες ανθρώπων όπως οι φυσικοί και οι προγραμματιστές γραφικών μια γλώσσα για να περιγράψουν το χώρο και πως να χειραγωγήσουν το χώρο, χρησιμοποιώντας αριθμούς τους οποίους μπορούν να επεξεργαστούν με ευκολία τα πρόγραμματα υπολογιστών.", diff --git a/2016/vectors/hebrew/sentence_translations.json b/2016/vectors/hebrew/sentence_translations.json index 7160642b0..f6d7ceffc 100644 --- a/2016/vectors/hebrew/sentence_translations.json +++ b/2016/vectors/hebrew/sentence_translations.json @@ -207,7 +207,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "כשאני רוצה להעביר את הרעיון של חלל דו-ממדי בכללותו, שתראו קצת מפריע, אבל כרגע הם קצת יפריעו.", "model": "google_nmt", "from_community_srt": "כשאני רוצה להעביר את הרעיון של מרחב דו-מימדי באופן מקיף, מה שאתה תראה שיצוץ הרבה בסירטונים האלה, אני ארחיב את הסימונים הללו על מנת ליצור קווים של רשת, אבל כרגע הם יפריעו לנו קצת בדרך.", @@ -603,7 +603,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "ובצד השני, זה נותן לאנשים כמו פיזיקאים ומתכנתי גרפיקה ממוחשבת שפה לתאר את החלל והמחשב.", "model": "google_nmt", "from_community_srt": "זה נותן לאנשים כמו פיזיקאים ומתכנתים של גראפיקה שפה לתאר מרחב והשינויים במרחב בעזרת שימוש במספרים שיכולים לנתח מידע מספרים ולרוץ במחשב.", diff --git a/2016/vectors/hungarian/sentence_translations.json b/2016/vectors/hungarian/sentence_translations.json index 893ea48ee..36a5f79e6 100644 --- a/2016/vectors/hungarian/sentence_translations.json +++ b/2016/vectors/hungarian/sentence_translations.json @@ -184,7 +184,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "Amikor a 2D-s tér egészének gondolatát akarom közvetíteni, ami, mint látni fogod, egy kicsit útban lesz, de most egy kicsit útban lesznek.", "model": "DeepL", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "A másik oldalról pedig olyan embereknek, mint a fizikusok és a számítógépes grafikusok, egy nyelvet ad, amellyel leírhatják a teret és a számítógépet.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/vectors/italian/sentence_translations.json b/2016/vectors/italian/sentence_translations.json index 43faac9ca..cbdbce477 100644 --- a/2016/vectors/italian/sentence_translations.json +++ b/2016/vectors/italian/sentence_translations.json @@ -207,7 +207,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "Quando voglio trasmettere l'idea dello spazio 2D nel suo insieme, vedrete che è un po' d'intralcio, ma in questo momento sarà un po' d'intralcio.", "model": "google_nmt", "from_community_srt": "Quando vorrò trasmettere l'idea di spazio dimensionale nella sua totalità, che come vedremo verrà fuori spesso in questi video, estenderò queste tacche a formare una griglia, ma per adesso mi risulterebbe scomodo.", @@ -603,7 +603,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "D’altro canto, offre a persone come fisici e programmatori di computer grafica un linguaggio per descrivere lo spazio e il computer.", "model": "google_nmt", "from_community_srt": "e dare una visione globale di certe operazioni, e dall'altra parte dà a fisici e programmatori grafici un linguaggio con cui descrivere lo spazio e la manipolazione dello spazio mediante numeri che possono essere dati in pasto a un computer. Quando faccio animazioni matematiche, per esempio,", diff --git a/2016/vectors/korean/sentence_translations.json b/2016/vectors/korean/sentence_translations.json index debc51bca..b263d48ad 100644 --- a/2016/vectors/korean/sentence_translations.json +++ b/2016/vectors/korean/sentence_translations.json @@ -207,7 +207,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "2D 공간에 대한 아이디어를 전체적으로 전달하고 싶을 때 여러분이 보게 될 것이 약간 방해가 되지만 지금은 약간 방해가 될 것입니다.", "model": "google_nmt", "from_community_srt": "제가 2 차원 공간의 개념을 전달하고자 할 때, 앞으로 비디오들에서 보게시겠지만, 저는 눈금을 확장해 격자선을 만들 것입니다. 하지만 당장은 좀 방해가 되서 빼겠습니다.", @@ -601,7 +601,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "그리고 다른 한편으로는 물리학자나 컴퓨터 그래픽 프로그래머 같은 사람들에게 공간과 컴퓨터를 설명하는 언어를 제공합니다.", "model": "google_nmt", "from_community_srt": "물리학 및 컴퓨터 그래픽 분야의 사람들에게는 공간과 조작을 숫자로 표현하여, 컴퓨터를 통해 동작시킬 수 있게 해줍니다. 예를들어, 수학-Y 애니메이션같은 작업을 할 때, 저는 공간에서 실제로 무엇이 일어나는지 생각합니다.", diff --git a/2016/vectors/persian/sentence_translations.json b/2016/vectors/persian/sentence_translations.json index 614540cfc..712596cf3 100644 --- a/2016/vectors/persian/sentence_translations.json +++ b/2016/vectors/persian/sentence_translations.json @@ -206,7 +206,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "وقتی می‌خواهم ایده فضای دوبعدی را به‌عنوان یک کل منتقل کنم، که می‌بینید تا حدودی در این راه پیش می‌آید، اما در حال حاضر آنها کمی مانع خواهند شد.", "model": "google_nmt", "from_community_srt": "همانطور که در این سری ویدئو زیاد خواهید دید، برای رساندن مفهوم فضای دو بعدی به صورت یکپارچه این علامت ها را ادامه میدهم تا شبکه ای از خطوط تشکیل دهند", @@ -597,7 +597,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "و از طرف دیگر، به افرادی مانند فیزیکدانان و برنامه نویسان گرافیک کامپیوتری زبانی برای توصیف فضا و کامپیوتر می دهد.", "model": "google_nmt", "from_community_srt": "ارائه کند همینطور در آن طرف سکه زبانی در اختیار افرادی نظیر فیزیکدان ها و برنامه نویسان گرافیک کامپیوتری برای توصیف و دستکاری فضا توسط اعداد، که میتواند خرد شده و روی یک کامپیوتر اجرا شود میگذارد.", diff --git a/2016/vectors/polish/sentence_translations.json b/2016/vectors/polish/sentence_translations.json index c8f1e1161..7d8e65549 100644 --- a/2016/vectors/polish/sentence_translations.json +++ b/2016/vectors/polish/sentence_translations.json @@ -207,7 +207,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "Kiedy chcę przekazać ideę przestrzeni 2D jako całości, co, jak zobaczycie, będzie trochę przeszkadzać, ale teraz będzie trochę przeszkadzać.", "model": "google_nmt", "from_community_srt": "Kiedy będe chciał rozszerzyć je na cała przestrzeń, co będzie się zdarzało często w tych filmach, poszerzę te podziałki za pomocą linii przerywanych.", @@ -602,7 +602,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "Z drugiej strony daje ludziom takim jak fizycy i programiści grafiki komputerowej język do opisu przestrzeni i komputera.", "model": "google_nmt", "from_community_srt": "i dać całościowe spojrzenie co robią konkretne operacje, a z drugiej strony - daje ludziom takim jak fizycy czy programiści grafiki komputerowej język w którym mogą opisywać przestrzeń i przekształcanie przestrzeni za pomocą liczb które mogą być przetwarzana na komputerze.", diff --git a/2016/vectors/portuguese/sentence_translations.json b/2016/vectors/portuguese/sentence_translations.json index e7db169d3..c71461a00 100644 --- a/2016/vectors/portuguese/sentence_translations.json +++ b/2016/vectors/portuguese/sentence_translations.json @@ -206,7 +206,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "Quando eu quero transmitir a ideia do espaço 2D como um todo, o que vocês verão atrapalha um pouco, mas agora eles vão atrapalhar um pouco.", "model": "google_nmt", "from_community_srt": "Quando eu quiser transmitir a ideia do espaço em duas dimensões como um todo — que, como você verá, aparece frequentemente nesses vídeos — eu estenderei essas marcações para fazer uma grade, mas por enquanto ela pode atrapalhar um pouco. ela pode atrapalhar um pouco.", @@ -601,7 +601,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "E, por outro lado, dá a pessoas como físicos e programadores de computação gráfica uma linguagem para descrever o espaço e o computador.", "model": "google_nmt", "from_community_srt": "e dar uma visão global do que certas operações fazem; e, por outro lado, ela dá a físicos e programadores gráficos uma linguagem e, por outro lado, ela dá a físicos e programadores gráficos uma linguagem para descrever o espaço e manipulá-lo usando números que podem ser calculados", diff --git a/2016/vectors/russian/sentence_translations.json b/2016/vectors/russian/sentence_translations.json index 739be9d06..79f72c99d 100644 --- a/2016/vectors/russian/sentence_translations.json +++ b/2016/vectors/russian/sentence_translations.json @@ -184,7 +184,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "Когда я хочу передать идею 2D-пространства в целом, это, как вы увидите, немного мешает, но сейчас они немного мешают.", "from_community_srt": "В попытке передать идею \"всего двумерного пространства как целого\" , которая, как вы увидите, будет часто появлятся в этих видео, я буду строить координатную сетку на этих отметках, но пока что они будут нам только мешать.", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "С другой стороны, это дает физикам и программистам компьютерной графики язык для описания космоса и компьютера.", "from_community_srt": "и с другой стороны, для таких людей, как физики и программисты компьютерной графики, это язык описания пространства и манипуляций над ним, легко переводимый в машинные вычисления.", "n_reviews": 0, diff --git a/2016/vectors/spanish/sentence_translations.json b/2016/vectors/spanish/sentence_translations.json index a47e471e8..4c72cf96e 100644 --- a/2016/vectors/spanish/sentence_translations.json +++ b/2016/vectors/spanish/sentence_translations.json @@ -161,7 +161,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "Cuando quiero transmitir la idea del espacio 2D como un todo, verán que interfiere un poco, pero ahora se interpondrán un poco.", "n_reviews": 1, "start": 192.32, @@ -469,7 +469,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "Y, por otro lado, brinda a personas como físicos y programadores de gráficos por computadora un lenguaje para describir el espacio y la computadora.", "n_reviews": 1, "start": 540.82, diff --git a/2016/vectors/turkish/sentence_translations.json b/2016/vectors/turkish/sentence_translations.json index 9b5dfb367..0c61293fe 100644 --- a/2016/vectors/turkish/sentence_translations.json +++ b/2016/vectors/turkish/sentence_translations.json @@ -206,7 +206,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "2B uzay fikrini bir bütün olarak aktarmak istediğimde, bunun biraz engel teşkil ettiğini göreceksiniz, ancak şu anda bunlar yolumuza biraz engel olacak.", "model": "google_nmt", "from_community_srt": "2-B uzayını tamamen göstermek istediğimde, ki sık sık olacak bu çentikleri ızgara çizgileri şeklinde uzatacağım ama şimdilik kafa karıştırıcı oluyorlar.", @@ -600,7 +600,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "Diğer taraftan, fizikçiler ve bilgisayar grafiği programcıları gibi insanlara uzayı ve bilgisayarı tanımlayacak bir dil sağlıyor.", "model": "google_nmt", "from_community_srt": "öte taraftan bilgisayar programcısı ya da fizikçi gibi kimselere bir dil sağlar. bu dil ile bu kimseler uzayı betimlemek için ve onu şekillendirmek için işleyebilir bilgisayarda çalıştırabilirler.", diff --git a/2016/vectors/vietnamese/sentence_translations.json b/2016/vectors/vietnamese/sentence_translations.json index 4e9685885..bb1f198d5 100644 --- a/2016/vectors/vietnamese/sentence_translations.json +++ b/2016/vectors/vietnamese/sentence_translations.json @@ -207,7 +207,7 @@ "end": 191.36 }, { - "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a bit in the way, but right now they'll get a little bit in the way.", + "input": "When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines, but right now, they'll actually get a little bit in the way.", "translatedText": "Khi tôi muốn truyền tải ý tưởng về không gian 2D một cách tổng thể, bạn sẽ thấy điều đó có một chút cản trở, nhưng ngay bây giờ chúng sẽ có một chút cản trở.", "model": "google_nmt", "from_community_srt": "Khi tôi muốn truyền tải ý tưởng về không gian 2-D một cách tổng quát- -điều bạn sẽ thấy rất nhiều ở những video như thế này, tôi sẽ mở rộng những dấu tick này để tạo nên những mạng lưới, nhưng hiện tại, chúng hơi thừa.", @@ -601,7 +601,7 @@ "end": 540.28 }, { - "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the computer.", + "input": "And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space using numbers that can be crunched and run through a computer.", "translatedText": "Và mặt khác, nó mang lại cho những người như nhà vật lý và lập trình viên đồ họa máy tính một ngôn ngữ để mô tả không gian và máy tính.", "model": "google_nmt", "from_community_srt": "Ngược lại, nó cung cấp cho những người như nhà vật lý học và lập trình viên đồ họa máy tính một thứ ngôn ngữ để mô tả không gian và sự thao tác lên không gian dùng những con số có thể được \"nén ép\" và chạy bằng máy tính.", diff --git a/2016/zeta/arabic/sentence_translations.json b/2016/zeta/arabic/sentence_translations.json index e9c2b1dbb..b761e6b4c 100644 --- a/2016/zeta/arabic/sentence_translations.json +++ b/2016/zeta/arabic/sentence_translations.json @@ -425,7 +425,7 @@ "end": 372.06 }, { - "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", + "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", "translatedText": "لذا، إذا كنت تريد توصيل 2 زائد i إلى دالة زيتا، فإحدى الطرق للتفكير فيما ستفعله هي البدء بجميع الحدود مرفوعة للأس 2، والتي يمكنك التفكير فيها على أنها تجميع الأسطر التي الأطوال هي مقلوب مربعات الأرقام، والتي، كما قلت من قبل، تتقارب إلى باي تربيع على 6.", "model": "google_nmt", "from_community_srt": "حتى لو كنت لتوصيل 2 + ط ل وزيتا وظيفة طريقة واحدة للتفكير ما تقوم به هو أن تبدأ مع كل من شروط مرفوع إلى قوة من 2 التي يمكن ان يخطر لك هو التفكيك معا خطوط طوله من مقلوب من الساحات من الأرقام التي كما قلت قبل CONVERGES إلى pi² أكثر من ستة", @@ -1279,7 +1279,7 @@ "end": 1196.62 }, { - "input": "That is, solving the jigsaw puzzle that began on the right half of the plane.", + "input": "That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane.", "translatedText": "أي حل اللغز الذي بدأ في النصف الأيمن من الطائرة.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/zeta/english/captions.srt b/2016/zeta/english/captions.srt index 20c01ea1e..77da04130 100644 --- a/2016/zeta/english/captions.srt +++ b/2016/zeta/english/captions.srt @@ -375,850 +375,858 @@ So when you multiply it, it doesn't change the size of the number, it just takes that 1 fourth and rotates it somewhat. 95 -00:06:15,100 --> 00:06:18,421 +00:06:15,100 --> 00:06:17,863 So, if you were to plug in 2 plus i to the zeta function, 96 -00:06:18,421 --> 00:06:23,060 -one way to think about what it does is to start off with all of the terms raised +00:06:17,863 --> 00:06:21,484 +one way to think about what it does is to take the 1 half to the i part and 97 -00:06:23,060 --> 00:06:27,756 -to the power of 2, which you can think of as piecing together lines whose lengths +00:06:21,484 --> 00:06:25,725 +think about what it does is to start off with all of the terms raised to the power of 2, 98 -00:06:27,756 --> 00:06:31,764 -are the reciprocals of squares of numbers, which, like I said before, +00:06:25,725 --> 00:06:29,108 +which you can think of as piecing together lines whose lengths are the 99 -00:06:31,764 --> 00:06:33,540 -converges to pi squared over 6. +00:06:29,108 --> 00:06:32,062 +reciprocals of squares of numbers, which, like I said before, 100 +00:06:32,062 --> 00:06:33,540 +converges to pi squared over 6. + +101 00:06:34,300 --> 00:06:37,525 Then when you change that input from 2 up to 2 plus i, -101 +102 00:06:37,525 --> 00:06:40,340 each of these lines gets rotated by some amount. -102 +103 00:06:40,340 --> 00:06:45,369 But importantly, the lengths of those lines won't change, so the sum still converges, -103 +104 00:06:45,369 --> 00:06:49,580 it just does so in a spiral to some specific point on the complex plane. -104 +105 00:06:50,880 --> 00:06:54,059 Here, let me show what it looks like when I vary the input s, -105 +106 00:06:54,059 --> 00:06:56,879 represented with this yellow dot on the complex plane, -106 +107 00:06:56,879 --> 00:07:01,340 where this spiral sum is always going to be showing the converging value for zeta of s. -107 +108 00:07:12,820 --> 00:07:17,012 What this means is that zeta of s, defined as this infinite sum, -108 +109 00:07:17,012 --> 00:07:22,236 is a perfectly reasonable complex function as long as the real part of the input -109 +110 00:07:22,236 --> 00:07:27,397 is greater than 1, meaning the input s sits somewhere on this right half of the -110 +111 00:07:27,397 --> 00:07:28,300 complex plane. -111 +112 00:07:29,140 --> 00:07:33,708 Again, this is because it's the real part of s that determines the size of each number, -112 +113 00:07:33,708 --> 00:07:36,460 while the imaginary part just dictates some rotation. -113 +114 00:07:39,160 --> 00:07:42,360 So now what I want to do is visualize this function. -114 +115 00:07:42,540 --> 00:07:45,780 It takes in inputs on the right half of the complex plane -115 +116 00:07:45,780 --> 00:07:49,020 and spits out outputs somewhere else in the complex plane. -116 +117 00:07:49,760 --> 00:07:54,748 A super nice way to understand complex functions is to visualize them as transformations, -117 +118 00:07:54,748 --> 00:07:58,405 meaning you look at every possible input to the function and just -118 +119 00:07:58,405 --> 00:08:00,900 let it move over to the corresponding output. -119 +120 00:08:01,940 --> 00:08:04,079 For example, let's take a moment and try to visualize -120 +121 00:08:04,079 --> 00:08:06,180 something a little bit easier than the zeta function. -121 +122 00:08:06,180 --> 00:08:08,820 Say f of s is equal to s squared. -122 +123 00:08:09,400 --> 00:08:12,647 When you plug in s equals 2, you get 4, so we'll -123 +124 00:08:12,647 --> 00:08:16,160 end up moving that point at 2 over to the point at 4. -124 +125 00:08:16,880 --> 00:08:20,490 When you plug in negative 1, you get 1, so the point over here -125 +126 00:08:20,490 --> 00:08:24,100 at negative 1 is going to end up moving over to the point at 1. -126 +127 00:08:24,980 --> 00:08:28,602 When you plug in i, by definition its square is negative 1, -127 +128 00:08:28,602 --> 00:08:31,380 so it's going to move over here to negative 1. -128 +129 00:08:32,179 --> 00:08:34,831 Now I'm going to add on a more colorful grid, and this is just -129 +130 00:08:34,831 --> 00:08:37,524 because things are about to start moving, and it's kind of nice -130 +131 00:08:37,524 --> 00:08:40,260 to have something to distinguish grid lines during that movement. -131 +132 00:08:40,860 --> 00:08:45,132 From here, I'll tell the computer to move every single point on this grid -132 +133 00:08:45,132 --> 00:08:49,520 over to its corresponding output under the function f of s equals s squared. -133 +134 00:08:50,140 --> 00:08:51,340 Here's what it looks like. -134 +135 00:08:55,420 --> 00:08:58,260 That can be a lot to take in, so I'll go ahead and play it again. -135 +136 00:08:58,260 --> 00:09:01,018 And this time, focus on one of the marked points, -136 +137 00:09:01,018 --> 00:09:04,880 and notice how it moves over to the point corresponding to its square. -137 +138 00:09:07,240 --> 00:09:10,951 It can be a little complicated to see all of the points moving all at once, -138 +139 00:09:10,951 --> 00:09:14,370 but the reward is that this gives us a very rich picture for what the -139 +140 00:09:14,370 --> 00:09:18,180 complex function is actually doing, and it all happens in just two dimensions. -140 +141 00:09:20,340 --> 00:09:21,800 So, back to the zeta function. -141 +142 00:09:22,120 --> 00:09:26,131 We have this infinite sum, which is a function of some complex number s, -142 +143 00:09:26,131 --> 00:09:30,308 and we feel good and happy about plugging in values of s whose real part is -143 +144 00:09:30,308 --> 00:09:34,760 greater than 1, and getting some meaningful output via the converging spiral sum. -144 +145 00:09:35,600 --> 00:09:39,806 So to visualize this function, I'm going to take the portion of the grid sitting on the -145 +146 00:09:39,806 --> 00:09:44,013 right side of the complex plane here, where the real part of numbers is greater than 1, -146 +147 00:09:44,013 --> 00:09:48,125 and I'm going to tell the computer to move each point of this grid to the appropriate -147 +148 00:09:48,125 --> 00:09:48,460 output. -148 +149 00:09:49,220 --> 00:09:52,375 It actually helps if I add a few more grid lines around the number 1, -149 +150 00:09:52,375 --> 00:09:54,720 since that region gets stretched out by quite a bit. -150 +151 00:09:59,520 --> 00:10:03,580 Alright, so first of all, let's all just appreciate how beautiful that is. -151 +152 00:10:04,000 --> 00:10:08,076 I mean, damn, if that doesn't make you want to learn more about complex functions, -152 +153 00:10:08,076 --> 00:10:08,960 you have no heart. -153 +154 00:10:10,880 --> 00:10:15,700 But also, this transformed grid is just begging to be extended a little bit. -154 +155 00:10:16,880 --> 00:10:19,737 For example, let's highlight these lines here, -155 +156 00:10:19,737 --> 00:10:24,600 which represent all of the complex numbers with imaginary part i, or negative i. -156 +157 00:10:26,940 --> 00:10:29,756 After the transformation, these lines make such -157 +158 00:10:29,756 --> 00:10:32,280 lovely arcs before they just abruptly stop. -158 +159 00:10:33,000 --> 00:10:35,760 Don't you want to just, you know, continue those arcs? -159 +160 00:10:36,800 --> 00:10:40,208 In fact, you can imagine how some altered version of the function, -160 +161 00:10:40,208 --> 00:10:43,515 with a definition that extends into this left half of the plane, -161 +162 00:10:43,515 --> 00:10:47,280 might be able to complete this picture with something that's quite pretty. -162 +163 00:10:48,260 --> 00:10:52,360 Well, this is exactly what mathematicians working with complex functions do. -163 +164 00:10:52,360 --> 00:10:57,280 They continue the function beyond the original domain where it was defined. -164 +165 00:10:58,000 --> 00:11:02,222 Now, as soon as we branch over into inputs where the real part is less than 1, -165 +166 00:11:02,222 --> 00:11:06,712 this infinite sum that we originally used to define the function doesn't make sense -166 +167 00:11:06,712 --> 00:11:07,140 anymore. -167 +168 00:11:07,420 --> 00:11:11,560 You'll get nonsense, like adding 1 plus 2 plus 3 plus 4 on and on up to infinity. -168 +169 00:11:12,260 --> 00:11:16,142 But just looking at this transformed version of the right half of the plane, -169 +170 00:11:16,142 --> 00:11:19,268 where the sum does make sense, it's just begging us to extend -170 +171 00:11:19,268 --> 00:11:21,840 the set of points that we're considering as inputs. -171 +172 00:11:22,360 --> 00:11:25,190 Even if that means defining the extended function -172 +173 00:11:25,190 --> 00:11:28,020 in some way that doesn't necessarily use that sum. -173 +174 00:11:29,220 --> 00:11:31,114 Of course, that leaves us with the question, how -174 +175 00:11:31,114 --> 00:11:33,280 would you define that function on the rest of the plane? -175 +176 00:11:34,840 --> 00:11:37,680 You might think that you could extend it any number of ways. -176 +177 00:11:38,260 --> 00:11:41,591 Maybe you define an extension that makes it so the point at, -177 +178 00:11:41,591 --> 00:11:44,760 say, s equals negative 1 moves over to negative 1 twelfth. -178 +179 00:11:47,620 --> 00:11:51,280 But maybe you squiggle on some extension that makes it land on any other value. -179 +180 00:11:51,280 --> 00:11:56,570 I mean, as soon as you open yourself up to the idea of defining the function differently -180 +181 00:11:56,570 --> 00:12:01,801 for values outside that domain of convergence, that is, not based on this infinite sum, -181 +182 00:12:01,801 --> 00:12:06,260 the world is your oyster, and you can have any number of extensions, right? -182 +183 00:12:07,320 --> 00:12:08,940 Well, not exactly. -183 +184 00:12:09,420 --> 00:12:14,162 I mean, yes, you can give any child a marker and have them extend these lines any -184 +185 00:12:14,162 --> 00:12:19,020 which way, but if you add on the restriction that this new extended function has to -185 +186 00:12:19,020 --> 00:12:23,820 have a derivative everywhere, it locks us into one and only one possible extension. -186 +187 00:12:25,340 --> 00:12:29,175 I know, I know, I said that you wouldn't need to know about derivatives for this video, -187 +188 00:12:29,175 --> 00:12:32,051 and even if you do know calculus, maybe you have yet to learn how -188 +189 00:12:32,051 --> 00:12:34,100 to interpret derivatives for complex functions. -189 +190 00:12:34,880 --> 00:12:38,433 But luckily for us, there is a very nice geometric intuition that you -190 +191 00:12:38,433 --> 00:12:42,240 can keep in mind for when I say a phrase like, has a derivative everywhere. -191 +192 00:12:43,260 --> 00:12:47,220 Here, to show you what I mean, let's look back at that f of s equals s squared example. -192 +193 00:12:47,860 --> 00:12:50,951 Again, we think of this function as a transformation, -193 +194 00:12:50,951 --> 00:12:54,960 moving every point s of the complex plane over to the point s squared. -194 +195 00:12:56,080 --> 00:12:59,822 For those of you who know calculus, you know that you can take the derivative -195 +196 00:12:59,822 --> 00:13:03,709 of this function at any given input, but there's an interesting property of that -196 +197 00:13:03,709 --> 00:13:07,500 transformation that turns out to be related and almost equivalent to that fact. -197 +198 00:13:08,760 --> 00:13:13,092 If you look at any two lines in the input space that intersect at some angle, -198 +199 00:13:13,092 --> 00:13:16,369 and consider what they turn into after the transformation, -199 +200 00:13:16,369 --> 00:13:19,480 they will still intersect each other at that same angle. -200 +201 00:13:21,020 --> 00:13:24,464 The lines might get curved, and that's okay, but the important -201 +202 00:13:24,464 --> 00:13:28,072 part is that the angle at which they intersect remains unchanged, -202 +203 00:13:28,072 --> 00:13:31,080 and this is true for any pair of lines that you choose. -203 +204 00:13:34,780 --> 00:13:37,812 So when I say a function has a derivative everywhere, -204 +205 00:13:37,812 --> 00:13:41,068 I want you to think about this angle-preserving property, -205 +206 00:13:41,068 --> 00:13:44,774 that any time two lines intersect, the angle between them remains -206 +207 00:13:44,774 --> 00:13:46,740 unchanged after the transformation. -207 +208 00:13:47,860 --> 00:13:52,028 At a glance, this is easiest to appreciate by noticing how all of the curves -208 +209 00:13:52,028 --> 00:13:55,980 that the grid lines turn into still intersect each other at right angles. -209 +210 00:13:58,580 --> 00:14:02,355 Complex functions that have a derivative everywhere are called analytic, -210 +211 00:14:02,355 --> 00:14:05,820 so you can think of this term analytic as meaning angle-preserving. -211 +212 00:14:06,680 --> 00:14:09,980 Admittedly, I'm lying to you a little here, but only a little bit. -212 +213 00:14:10,400 --> 00:14:13,312 A slight caveat for those of you who want the full details is -213 +214 00:14:13,312 --> 00:14:16,084 that at inputs where the derivative of a function is zero, -214 +215 00:14:16,084 --> 00:14:19,420 instead of angles being preserved, they get multiplied by some integer. -215 +216 00:14:20,600 --> 00:14:23,690 But those points are by far the minority, and for almost -216 +217 00:14:23,690 --> 00:14:26,780 all inputs to an analytic function, angles are preserved. -217 +218 00:14:29,520 --> 00:14:32,368 So if when I say analytic, you think angle-preserving, -218 +219 00:14:32,368 --> 00:14:34,440 I think that's a fine intuition to have. -219 +220 00:14:39,000 --> 00:14:42,304 Now, if you think about it for a moment, and this is a point that -220 +221 00:14:42,304 --> 00:14:45,760 I really want you to appreciate, this is a very restrictive property. -221 +222 00:14:46,420 --> 00:14:50,680 The angle between any pair of intersecting lines has to remain unchanged. -222 +223 00:14:51,560 --> 00:14:55,780 And yet, pretty much any function out there that has a name turns out to be analytic. -223 +224 00:14:58,420 --> 00:15:02,970 The field of complex analysis, which Riemann helped to establish in its modern form, -224 +225 00:15:02,970 --> 00:15:07,039 is almost entirely about leveraging the properties of analytic functions to -225 +226 00:15:07,039 --> 00:15:10,680 understand results and patterns in other fields of math and science. -226 +227 00:15:12,900 --> 00:15:17,238 The zeta function, defined by this infinite sum on the right half of the plane, -227 +228 00:15:17,238 --> 00:15:18,540 is an analytic function. -228 +229 00:15:19,340 --> 00:15:21,929 Notice how all of these curves that the grid lines -229 +230 00:15:21,929 --> 00:15:24,620 turn into still intersect each other at right angles. -230 +231 00:15:28,100 --> 00:15:32,892 So the surprising fact about complex functions is that if you want to extend an -231 +232 00:15:32,892 --> 00:15:37,804 analytic function beyond the domain where it was originally defined, for example, -232 +233 00:15:37,804 --> 00:15:41,518 extending this zeta function into the left half of the plane, -233 +234 00:15:41,518 --> 00:15:46,250 then if you require that the new extended function still be analytic, that is, -234 +235 00:15:46,250 --> 00:15:51,043 that it still preserves angles everywhere, it forces you into only one possible -235 +236 00:15:51,043 --> 00:15:52,960 extension, if one exists at all. -236 +237 00:15:53,500 --> 00:15:56,720 It's kind of like an infinite continuous jigsaw puzzle, -237 +238 00:15:56,720 --> 00:16:01,492 where this requirement of preserving angles locks you into one and only one choice -238 +239 00:16:01,492 --> 00:16:02,700 for how to extend it. -239 +240 00:16:04,400 --> 00:16:08,641 This process of extending an analytic function in the only way possible that's -240 +241 00:16:08,641 --> 00:16:12,560 still analytic is called, as you may have guessed, analytic continuation. -241 +242 00:16:14,920 --> 00:16:17,720 So that's how the full Riemann zeta function is defined. -242 +243 00:16:18,240 --> 00:16:22,428 For values of s on the right half of the plane, where the real part is greater than 1, -243 +244 00:16:22,428 --> 00:16:25,220 we can plug them into this sum and see where it converges. -244 +245 00:16:25,680 --> 00:16:28,465 And that convergence might look like some kind of spiral, -245 +246 00:16:28,465 --> 00:16:32,740 since raising each of these terms to a complex power has the effect of rotating each one. -246 +247 00:16:33,520 --> 00:16:37,159 Then for the rest of the plane, we know that there exists one and only -247 +248 00:16:37,159 --> 00:16:41,208 one way to extend this definition so that the function will still be analytic, -248 +249 00:16:41,208 --> 00:16:44,540 that is, so that it still preserves angles at every single point. -249 +250 00:16:45,300 --> 00:16:48,609 So we just say that by definition, the zeta function on the -250 +251 00:16:48,609 --> 00:16:52,140 left half of the plane is whatever that extension happens to be. -251 +252 00:16:52,960 --> 00:16:57,260 And that's a valid definition because there's only one possible analytic continuation. -252 +253 00:16:58,600 --> 00:17:00,900 Notice, that's a very implicit definition. -253 +254 00:17:01,420 --> 00:17:04,525 It just says, use the solution of this jigsaw puzzle, -254 +255 00:17:04,525 --> 00:17:07,917 which through more abstract derivation we know must exist, -255 +256 00:17:07,917 --> 00:17:10,619 but it doesn't specify exactly how to solve it. -256 +257 00:17:11,160 --> 00:17:14,611 Mathematicians have a pretty good grasp on what this extension looks like, -257 +258 00:17:14,611 --> 00:17:16,819 but some important parts of it remain a mystery. -258 +259 00:17:17,339 --> 00:17:18,940 A million dollar mystery, in fact. -259 +260 00:17:19,640 --> 00:17:22,453 Let's actually take a moment and talk about the Riemann hypothesis, -260 +261 00:17:22,453 --> 00:17:23,859 which is a million dollar problem. -261 +262 00:17:24,980 --> 00:17:29,157 The places where this function equals zero turn out to be quite important, -262 +263 00:17:29,157 --> 00:17:33,280 that is, which points get mapped onto the origin after the transformation. -263 +264 00:17:34,480 --> 00:17:36,921 One thing we know about this extension is that -264 +265 00:17:36,921 --> 00:17:39,260 the negative even numbers get mapped to zero. -265 +266 00:17:41,160 --> 00:17:43,560 These are commonly called the trivial zeros. -266 +267 00:17:44,300 --> 00:17:47,671 The naming here stems from a long-standing tradition of mathematicians -267 +268 00:17:47,671 --> 00:17:50,473 to call things trivial when they understand it quite well, -268 +269 00:17:50,473 --> 00:17:53,560 even when it's a fact that is not at all obvious from the outset. -269 +270 00:17:54,560 --> 00:17:59,086 We also know that the rest of the points that get mapped to zero sit somewhere -270 +271 00:17:59,086 --> 00:18:02,008 in this vertical strip, called the critical strip, -271 +272 00:18:02,008 --> 00:18:06,306 and the specific placement of those non-trivial zeros encodes a surprising -272 +273 00:18:06,306 --> 00:18:08,140 information about prime numbers. -273 +274 00:18:09,120 --> 00:18:12,536 It's actually pretty interesting why this function carries so much information -274 +275 00:18:12,536 --> 00:18:15,822 about primes, and I definitely think I'll make a video about that later on, -275 +276 00:18:15,822 --> 00:18:18,720 but right now things are long enough, so I'll leave it unexplained. -276 +277 00:18:19,780 --> 00:18:24,180 Riemann hypothesized that all of these non-trivial zeros sit right in the -277 +278 00:18:24,180 --> 00:18:28,640 middle of the strip, on the line of numbers s, whose real part is one half. -278 +279 00:18:29,460 --> 00:18:30,880 This is called the critical line. -279 +280 00:18:31,780 --> 00:18:36,335 If that's true, it gives us a remarkably tight grasp on the pattern of prime numbers, -280 +281 00:18:36,335 --> 00:18:39,460 as well as many other patterns in math that stem from this. -281 +282 00:18:40,340 --> 00:18:43,669 Now, so far, when I've shown what the zeta function looks like, -282 +283 00:18:43,669 --> 00:18:47,363 I've only shown what it does to the portion of the grid on the screen, -283 +284 00:18:47,363 --> 00:18:49,600 and that kind of undersells its complexity. -284 +285 00:18:50,320 --> 00:18:54,270 So if I were to highlight this critical line and apply the transformation, -285 +286 00:18:54,270 --> 00:18:56,640 it might not seem to cross the origin at all. -286 +287 00:18:57,200 --> 00:19:01,960 However, here's what the transformed version of more and more of that line looks like. -287 +288 00:19:06,440 --> 00:19:09,820 Notice how it's passing through the number zero many, many times. -288 +289 00:19:10,500 --> 00:19:14,836 If you can prove that all of the non-trivial zeros sit somewhere on this line, -289 +290 00:19:14,836 --> 00:19:17,800 the Clay Math Institute gives you one million dollars. -290 +291 00:19:18,240 --> 00:19:20,855 And you'd also be proving hundreds, if not thousands, -291 +292 00:19:20,855 --> 00:19:24,294 of modern math results that have already been shown contingent on this -292 +293 00:19:24,294 --> 00:19:25,360 hypothesis being true. -293 +294 00:19:26,520 --> 00:19:29,296 Another thing we know about this extended function is that -294 +295 00:19:29,296 --> 00:19:32,120 it maps the point negative one over to negative one twelfth. -295 +296 00:19:34,160 --> 00:19:38,200 And if you plug this into the original sum, it looks like we're saying one plus -296 +297 00:19:38,200 --> 00:19:42,240 two plus three plus four, on and on up to infinity, equals negative one twelfth. -297 +298 00:19:42,240 --> 00:19:45,305 Now, it might seem disingenuous to still call this a sum, -298 +299 00:19:45,305 --> 00:19:49,904 since the definition of the zeta function on the left half of the plane is not defined -299 +300 00:19:49,904 --> 00:19:51,120 directly from this sum. -300 +301 00:19:51,740 --> 00:19:54,233 Instead, it comes from analytically continuing -301 +302 00:19:54,233 --> 00:19:56,620 the sum beyond the domain where it converges. -302 -00:19:57,120 --> 00:20:01,060 -That is, solving the jigsaw puzzle that began on the right half of the plane. - 303 +00:19:57,120 --> 00:19:59,212 +That is, solving the jigsaw puzzle that began on the first line of the line, + +304 +00:19:59,212 --> 00:20:01,060 +solving the jigsaw puzzle that began on the right half of the plane. + +305 00:20:01,880 --> 00:20:06,146 That said, you have to admit that the uniqueness of this analytic continuation, -304 +306 00:20:06,146 --> 00:20:09,080 the fact that the jigsaw puzzle has only one solution, -305 +307 00:20:09,080 --> 00:20:13,240 is very suggestive of some intrinsic connection between these extended values -306 +308 00:20:13,240 --> 00:20:14,360 and the original sum. diff --git a/2016/zeta/english/sentence_timings.json b/2016/zeta/english/sentence_timings.json index 9775a7bc9..6ef611a01 100644 --- a/2016/zeta/english/sentence_timings.json +++ b/2016/zeta/english/sentence_timings.json @@ -240,7 +240,7 @@ 372.06 ], [ - "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", + "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", 375.1, 393.54 ], @@ -725,7 +725,7 @@ 1196.62 ], [ - "That is, solving the jigsaw puzzle that began on the right half of the plane.", + "That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane.", 1197.12, 1201.06 ], diff --git a/2016/zeta/english/transcript.txt b/2016/zeta/english/transcript.txt index 00fe202de..3de8ae5fd 100644 --- a/2016/zeta/english/transcript.txt +++ b/2016/zeta/english/transcript.txt @@ -46,7 +46,7 @@ If you've never seen this and you're wondering why on earth this happens, I've l For here, I'm just going to move forward with the what without the why. The main takeaway is that when you raise something like 1 half to the power of 2 plus i, which is 1 half squared times 1 half to the i, that 1 half to the i part is going to be on the unit circle, meaning it has an absolute value of 1. So when you multiply it, it doesn't change the size of the number, it just takes that 1 fourth and rotates it somewhat. -So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6. +So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6. Then when you change that input from 2 up to 2 plus i, each of these lines gets rotated by some amount. But importantly, the lengths of those lines won't change, so the sum still converges, it just does so in a spiral to some specific point on the complex plane. Here, let me show what it looks like when I vary the input s, represented with this yellow dot on the complex plane, where this spiral sum is always going to be showing the converging value for zeta of s. @@ -143,5 +143,5 @@ Another thing we know about this extended function is that it maps the point neg And if you plug this into the original sum, it looks like we're saying one plus two plus three plus four, on and on up to infinity, equals negative one twelfth. Now, it might seem disingenuous to still call this a sum, since the definition of the zeta function on the left half of the plane is not defined directly from this sum. Instead, it comes from analytically continuing the sum beyond the domain where it converges. -That is, solving the jigsaw puzzle that began on the right half of the plane. +That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane. That said, you have to admit that the uniqueness of this analytic continuation, the fact that the jigsaw puzzle has only one solution, is very suggestive of some intrinsic connection between these extended values and the original sum. \ No newline at end of file diff --git a/2016/zeta/french/sentence_translations.json b/2016/zeta/french/sentence_translations.json index b273d633f..2da522755 100644 --- a/2016/zeta/french/sentence_translations.json +++ b/2016/zeta/french/sentence_translations.json @@ -383,7 +383,7 @@ "end": 372.06 }, { - "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", + "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", "translatedText": "Donc, si vous deviez brancher 2 plus i à la fonction zêta, une façon de réfléchir à ce qu'elle fait est de commencer avec tous les termes élevés à la puissance 2, que vous pouvez considérer comme un assemblage de lignes dont les longueurs sont les réciproques des carrés des nombres qui, comme je l'ai déjà dit, convergent vers pi au carré sur 6.", "from_community_srt": "Donc, si vous deviez évaluer en 2 + i la la fonction zeta, une façon de se représenter ce qui se passe est de commencer avec les termes élevés à la puissance de deux, que vous pouvez visualiser comme des segments qui s'articulent et dont les longueurs sont les inverses des carrés d'entiers, ce qui comme je l'ai dit converge vers pi au carré sur six.", "n_reviews": 0, @@ -1157,7 +1157,7 @@ "end": 1196.62 }, { - "input": "That is, solving the jigsaw puzzle that began on the right half of the plane.", + "input": "That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane.", "translatedText": "C’est-à-dire résoudre le puzzle qui a commencé sur la moitié droite de l’avion.", "from_community_srt": "à savoir résoudre le puzzle qui a commencé sur la moitié droite du plan.", "n_reviews": 0, diff --git a/2016/zeta/german/sentence_translations.json b/2016/zeta/german/sentence_translations.json index 47221e294..ff74da776 100644 --- a/2016/zeta/german/sentence_translations.json +++ b/2016/zeta/german/sentence_translations.json @@ -431,7 +431,7 @@ "end": 372.06 }, { - "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", + "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", "translatedText": "Wenn Sie also 2 plus i in die Zeta-Funktion einfügen, können Sie sich überlegen, was sie bewirkt, indem Sie mit allen Termen hoch 2 beginnen, was Sie sich als Zusammenfügen von Zeilen vorstellen können, deren Längen sind die Kehrwerte der Quadrate von Zahlen, die, wie ich bereits sagte, gegen das Quadrat von Pi über 6 konvergieren.", "model": "google_nmt", "from_community_srt": "Also wenn du 2 + i anschliessen würdest die Zeta-Funktion eine Möglichkeit, darüber nachzudenken Was es macht, ist mit allem anzufangen die Begriffe auf die Macht von 2 erhöht, die Sie können sich vorstellen, zusammen zu finden Linien, deren Länge der Kehrwerte von Quadrate von Zahlen, die wie ich sagte bevor konvergiert zu pi² über sechs dann,", @@ -1301,7 +1301,7 @@ "end": 1196.62 }, { - "input": "That is, solving the jigsaw puzzle that began on the right half of the plane.", + "input": "That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane.", "translatedText": "Das heißt, das Puzzle zu lösen, das in der rechten Hälfte des Flugzeugs begann.", "model": "google_nmt", "from_community_srt": "das löst die Puzzle, das auf der rechten Seite begann die Hälfte des Flugzeugs,", diff --git a/2016/zeta/greek/sentence_translations.json b/2016/zeta/greek/sentence_translations.json index 34b146d88..c14d10575 100644 --- a/2016/zeta/greek/sentence_translations.json +++ b/2016/zeta/greek/sentence_translations.json @@ -430,7 +430,7 @@ "end": 372.06 }, { - "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", + "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", "translatedText": "Έτσι, εάν επρόκειτο να συνδέσετε το 2 συν i στη συνάρτηση zeta, ένας τρόπος για να σκεφτείτε τι κάνει είναι να ξεκινήσετε με όλους τους όρους που ανεβαίνουν στην ισχύ του 2, τους οποίους μπορείτε να σκεφτείτε ότι συνδυάζουν γραμμές Τα μήκη είναι τα αντίστροφα των τετραγώνων των αριθμών, τα οποία, όπως είπα προηγουμένως, συγκλίνουν στο pi στο τετράγωνο του 6.", "model": "google_nmt", "from_community_srt": "ένα τέταρτο και περιστρέφεται κάπου. Έτσι, αν συνδεθείτε 2 + i στο το zeta λειτουργεί ως ένας τρόπος σκέψης αυτό που κάνει είναι να ξεκινήσει με όλα οι όροι που τέθηκαν για τη δύναμη του 2 το οποίο που μπορείτε να σκεφτείτε είναι συνδέοντας μαζί γραμμές των οποίων το μήκος των οπισθίων γραμμών του τετράγωνα αριθμών που όπως είπα πριν συγκλίνει σε p² πάνω από έξι", @@ -1297,7 +1297,7 @@ "end": 1196.62 }, { - "input": "That is, solving the jigsaw puzzle that began on the right half of the plane.", + "input": "That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane.", "translatedText": "Λύνοντας δηλαδή το παζλ που ξεκίνησε στο δεξί μισό του αεροπλάνου.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/zeta/hungarian/sentence_translations.json b/2016/zeta/hungarian/sentence_translations.json index 1ba2aef49..eb2918b8d 100644 --- a/2016/zeta/hungarian/sentence_translations.json +++ b/2016/zeta/hungarian/sentence_translations.json @@ -384,7 +384,7 @@ "end": 372.06 }, { - "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", + "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", "translatedText": "Ha tehát a zéta-függvénybe 2 plusz i-t teszünk, akkor a zéta-függvényt úgy is el lehet képzelni, hogy az összes kifejezést a 2 hatványára emeljük, amit úgy képzelhetünk el, hogy olyan vonalakat rakunk össze, amelyek hossza a számok négyzetének reciproka, ami, mint már mondtam, 6 felett a pí négyzetéhez konvergál.", "model": "DeepL", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1196.62 }, { - "input": "That is, solving the jigsaw puzzle that began on the right half of the plane.", + "input": "That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane.", "translatedText": "Vagyis a kirakós játék megoldása, amely a gép jobb felén kezdődött.", "model": "DeepL", "n_reviews": 0, diff --git a/2016/zeta/italian/sentence_translations.json b/2016/zeta/italian/sentence_translations.json index 294c3eac4..752cdfffe 100644 --- a/2016/zeta/italian/sentence_translations.json +++ b/2016/zeta/italian/sentence_translations.json @@ -431,7 +431,7 @@ "end": 372.06 }, { - "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", + "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", "translatedText": "Quindi, se dovessi collegare 2 più i alla funzione zeta, un modo di pensare a cosa fa è iniziare con tutti i termini elevati alla potenza di 2, che puoi pensare come se mettessero insieme le linee il cui le lunghezze sono i reciproci dei quadrati dei numeri, che, come ho detto prima, converge a pi quadrato su 6.", "model": "google_nmt", "from_community_srt": "Quindi se dovessi collegare 2 + a la funzione zeta un modo per pensare quello che fa è iniziare con tutti i termini sollevati alla potenza di 2 che puoi pensare di mettere insieme linee la cui lunghezza dei reciproci di quadrati di numeri che come ho detto prima converge a pi² su sei allora quando cambi questo input da due", @@ -1297,7 +1297,7 @@ "end": 1196.62 }, { - "input": "That is, solving the jigsaw puzzle that began on the right half of the plane.", + "input": "That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane.", "translatedText": "Cioè, risolvere il puzzle iniziato nella metà destra dell'aereo.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/zeta/korean/sentence_translations.json b/2016/zeta/korean/sentence_translations.json index 8b2b946f8..eaecf8de6 100644 --- a/2016/zeta/korean/sentence_translations.json +++ b/2016/zeta/korean/sentence_translations.json @@ -429,7 +429,7 @@ "end": 372.06 }, { - "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", + "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", "translatedText": "따라서 2 더하기 i를 제타 함수에 연결하면 그것이 무엇을 하는지 생각하는 한 가지 방법은 모든 항을 2의 거듭제곱으로 시작하는 것입니다. 길이는 숫자의 제곱의 역수입니다. 앞서 말했듯이 6 분의 파이 제곱으로 수렴됩니다.", "model": "google_nmt", "from_community_srt": "오직 방향만이 변하게 된다는 것입니다. 이제 제타 함수에 2+i를 대입하고, 차근차근 생각해 보죠. 일단 각 항목에 제곱을 취한 후 각 항의 길이를 시각화 하여 x축 위에 올려 놓는 것을 상상 할 수 있습니다. 그리고 그 길이는 전에 말했듯이 π^2/6으로 수렴합니다.", @@ -1302,7 +1302,7 @@ "end": 1196.62 }, { - "input": "That is, solving the jigsaw puzzle that began on the right half of the plane.", + "input": "That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane.", "translatedText": "즉, 비행기의 오른쪽 절반에서 시작된 직소 퍼즐을 푸는 것입니다.", "model": "google_nmt", "from_community_srt": "그것은 오른쪽에서 시작해서 직쏘퍼즐을 푸는것과 같습니다. 그것은 오른쪽에서 시작해서 직쏘퍼즐을 푸는것과 같습니다.", diff --git a/2016/zeta/polish/sentence_translations.json b/2016/zeta/polish/sentence_translations.json index 8f2aa0183..186ba1135 100644 --- a/2016/zeta/polish/sentence_translations.json +++ b/2016/zeta/polish/sentence_translations.json @@ -432,7 +432,7 @@ "end": 372.06 }, { - "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", + "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", "translatedText": "Tak więc, jeśli podłączysz 2 plus i do funkcji zeta, jednym ze sposobów myślenia o tym, co ona robi, jest rozpoczęcie od wszystkich wyrazów podniesionych do potęgi 2, co można uznać za składanie razem linii, których długości to odwrotność kwadratów liczb, które, jak powiedziałem wcześniej, zbiegają się do pi do kwadratu przez 6.", "model": "google_nmt", "from_community_srt": "Zatem, jeśli mielibyście wstawić 2+i do funkcji dzeta, to jednym ze sposobów myślenia o tym, co się stanie, jest podniesienie na początku wszystkich wyrazów do potęgi 2. Możecie to rozumieć jako składanie ze sobą linii o długościach równych odwrotnościom kwadratów liczb naturalnych. Jak już mówiłem, to zbiega do pi^2/6.", @@ -1304,7 +1304,7 @@ "end": 1196.62 }, { - "input": "That is, solving the jigsaw puzzle that began on the right half of the plane.", + "input": "That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane.", "translatedText": "Czyli ułożenie układanki, która zaczęła się w prawej połowie samolotu.", "model": "google_nmt", "from_community_srt": "Czyli od ułożenia układanki, która zaczęła się na prawej połowie płaszczyzny.", diff --git a/2016/zeta/portuguese/sentence_translations.json b/2016/zeta/portuguese/sentence_translations.json index b8e4aea0a..20f72c2ad 100644 --- a/2016/zeta/portuguese/sentence_translations.json +++ b/2016/zeta/portuguese/sentence_translations.json @@ -431,7 +431,7 @@ "end": 372.06 }, { - "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", + "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", "translatedText": "Então, se você inserir 2 mais i na função zeta, uma maneira de pensar sobre o que ela faz é começar com todos os termos elevados à potência de 2, o que você pode imaginar como juntar as linhas cujas comprimentos são os inversos dos quadrados dos números, que, como eu disse antes, convergem para pi ao quadrado sobre 6.", "model": "google_nmt", "from_community_srt": "Então, se você fosse conectar 2 + i para a função zeta é uma forma de pensar sobre O que faz é começar com todos os os termos aumentados para o poder de 2 que você pode pensar que está juntando linhas cujo comprimento dos reciprocais de quadrados de números como eu disse antes converge para pi² sobre seis então,", @@ -1298,7 +1298,7 @@ "end": 1196.62 }, { - "input": "That is, solving the jigsaw puzzle that began on the right half of the plane.", + "input": "That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane.", "translatedText": "Ou seja, resolvendo o quebra-cabeça que começava na metade direita do avião.", "model": "google_nmt", "n_reviews": 0, diff --git a/2016/zeta/russian/sentence_translations.json b/2016/zeta/russian/sentence_translations.json index 483e24040..b6c573344 100644 --- a/2016/zeta/russian/sentence_translations.json +++ b/2016/zeta/russian/sentence_translations.json @@ -380,7 +380,7 @@ "end": 372.06 }, { - "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", + "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", "translatedText": "Итак, если бы вы подставили 2 плюс i к дзета-функции, один из способов понять, что она делает, — это начать со всех членов, возведенных в степень 2, что можно рассматривать как соединение воедино строк, чьи длины — это обратные квадраты чисел, которые, как я уже говорил, сходятся к числу пи, возведенному в квадрат более чем 6.", "from_community_srt": "Для того чтобы понять, как работает дзета-функция в точке, скажем, 2 + i нужно сперва взять все дроби, и возвести их в квадраты, которые графически можно представить, как серию отрезков, длины которых равны обратным квадратам наших целых чисел и, как уже было сказано ранее, в сумме все это сходится к π/6.", "n_reviews": 0, @@ -1155,7 +1155,7 @@ "end": 1196.62 }, { - "input": "That is, solving the jigsaw puzzle that began on the right half of the plane.", + "input": "That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane.", "translatedText": "То есть решение головоломки, которая началась в правой половине самолета.", "from_community_srt": "начавшегося в правой полуплоскости.", "n_reviews": 0, diff --git a/2016/zeta/spanish/sentence_translations.json b/2016/zeta/spanish/sentence_translations.json index 4864aa344..0c367abbf 100644 --- a/2016/zeta/spanish/sentence_translations.json +++ b/2016/zeta/spanish/sentence_translations.json @@ -380,7 +380,7 @@ "end": 372.06 }, { - "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", + "input": "So, if you were to plug in 2 plus i to the zeta function, one way to think about what it does is to take the 1 half to the i part and think about what it does is to start off with all of the terms raised to the power of 2, which you can think of as piecing together lines whose lengths are the reciprocals of squares of numbers, which, like I said before, converges to pi squared over 6.", "translatedText": "Entonces, si conectaras 2 más i a la función zeta, una forma de pensar en lo que hace es comenzar con todos los términos elevados a la potencia de 2, que puedes considerar como unir líneas cuyos las longitudes son los recíprocos de los cuadrados de los números, que, como dije antes, convergen en pi al cuadrado sobre 6.", "from_community_srt": "sólo toma a ese 1/4 y lo gira de alguna manera Así que al alimentar la función Z con 2+i Una forma de ver lo que hace Es comenzar con todos los términos elevados a la potencia de 2 lo que puedes pensar que equivale a conectar líneas cuyas longitudes son los recíprocos de los números al cuadrado Que, como dije anteriormente,", "n_reviews": 0, @@ -1146,7 +1146,7 @@ "end": 1196.62 }, { - "input": "That is, solving the jigsaw puzzle that began on the right half of the plane.", + "input": "That is, solving the jigsaw puzzle that began on the first line of the line, solving the jigsaw puzzle that began on the right half of the plane.", "translatedText": "Es decir, resolver el rompecabezas que comenzaba en la mitad derecha del avión.", "from_community_srt": "procede de la continuación analítica de la suma más allá del dominio en el que converge es decir de la solución del rompecabezas que comienza en el lado derecho del plano", "n_reviews": 0, diff --git a/2017/256-bit-security/arabic/sentence_translations.json b/2017/256-bit-security/arabic/sentence_translations.json index 1cd0fb609..aaaae979b 100644 --- a/2017/256-bit-security/arabic/sentence_translations.json +++ b/2017/256-bit-security/arabic/sentence_translations.json @@ -192,7 +192,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", "translatedText": "بعد ذلك، تخيل 4 مليارات نسخة من مجرة درب التبانة، وسمي هذا الكمبيوتر العملاق المجري الذي يعمل بحوالي 2 إلى 160 تخمينًا في كل ثانية. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/256-bit-security/bengali/sentence_translations.json b/2017/256-bit-security/bengali/sentence_translations.json index e92fabf9e..6f789414d 100644 --- a/2017/256-bit-security/bengali/sentence_translations.json +++ b/2017/256-bit-security/bengali/sentence_translations.json @@ -192,7 +192,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", "translatedText": "এরপরে, আকাশগঙ্গার 4 বিলিয়ন কপি কল্পনা করুন এবং এটিকে আপনার গিগা-গ্যালাকটিক সুপার কম্পিউটার বলুন, প্রতি সেকেন্ডে 2 থেকে 160টি অনুমান চালাচ্ছে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/256-bit-security/chinese/sentence_translations.json b/2017/256-bit-security/chinese/sentence_translations.json index 5c4ac4691..2d0e30b09 100644 --- a/2017/256-bit-security/chinese/sentence_translations.json +++ b/2017/256-bit-security/chinese/sentence_translations.json @@ -200,7 +200,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", "translatedText": "接下来，想象一下银河系的 40 亿个副本，并将其称为您的千 兆银河超级计算机，每秒运行大约 2 到 160 次猜测。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/256-bit-security/croatian/sentence_translations.json b/2017/256-bit-security/croatian/sentence_translations.json index e2e6a2e7f..f45100eb4 100644 --- a/2017/256-bit-security/croatian/sentence_translations.json +++ b/2017/256-bit-security/croatian/sentence_translations.json @@ -88,7 +88,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "", "from_community_srt": "Dakle prvih 4 milijarde predstavljat će broj hasheva u sekundi po računalu.", "n_reviews": 0, @@ -279,7 +279,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "", "from_community_srt": "Ostavio sam vezu u opisu videozapisa na \"Reddit\" diskusiju, gdje možete postavljati pitanja i doznati odgovore I vjerojatno ću u sljedećem videu na twitteru ili nečem sličnom, najaviti oblik u kojem želim napisati odgovore. Doviđenja do tada.", "n_reviews": 0, diff --git a/2017/256-bit-security/czech/sentence_translations.json b/2017/256-bit-security/czech/sentence_translations.json index 6ccd8e738..ada51bf8c 100644 --- a/2017/256-bit-security/czech/sentence_translations.json +++ b/2017/256-bit-security/czech/sentence_translations.json @@ -88,7 +88,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "", "from_community_srt": "Takže první čtyři miliardy zde představují počet hashů za sekundu na jeden počítač.", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "", "from_community_srt": "Nechal jsem odkaz na vlákno v popisu. Tam můžete posílat otázky, na které chcete slyšet odpovědi a pravděpodobně v příštím videu nebo na Twitteru, vám řeknu v jakém formátu, vám budu odpovídat. Naviděnou.", "n_reviews": 0, diff --git a/2017/256-bit-security/dutch/sentence_translations.json b/2017/256-bit-security/dutch/sentence_translations.json index 341270a4f..3ddd6757f 100644 --- a/2017/256-bit-security/dutch/sentence_translations.json +++ b/2017/256-bit-security/dutch/sentence_translations.json @@ -88,7 +88,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "De eerste 4 miljard staat hier voor het aantal hashes per seconde per computer.", "model": "DeepL", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "Ik heb in de beschrijving een link achtergelaten naar een Reddit thread waar je vragen kunt plaatsen en de vragen waarop je antwoorden wilt horen kunt upvoten, en waarschijnlijk zal ik in de volgende video of op Twitter het format aankondigen waarin ik antwoorden wil geven.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/256-bit-security/english/captions.srt b/2017/256-bit-security/english/captions.srt index c1c79d346..eb3ade34d 100644 --- a/2017/256-bit-security/english/captions.srt +++ b/2017/256-bit-security/english/captions.srt @@ -83,178 +83,182 @@ Let's say you just take a bunch of those and cram your computer full of extra GPUs so that your computer can run 4 billion hashes per second. 22 -00:01:35,420 --> 00:01:40,320 -The first 4 billion here represents the number of hashes per second per computer. +00:01:35,420 --> 00:01:37,998 +So the first 4 billion here is going to represent 23 +00:01:37,998 --> 00:01:40,320 +the number of hashes per second per computer. + +24 00:01:41,160 --> 00:01:45,360 Now, picture 4 billion of these GPU-packed computers. -24 +25 00:01:46,240 --> 00:01:50,803 For comparison, even though Google does not at all make their number of servers public, -25 +26 00:01:50,803 --> 00:01:53,760 estimates have it somewhere in the single-digit millions. -26 +27 00:01:54,600 --> 00:01:57,422 In reality, most of those servers are going to be much -27 +28 00:01:57,422 --> 00:02:00,040 less powerful than our imagined GPU-packed machine. -28 +29 00:02:00,580 --> 00:02:05,428 But let's say that Google replaced all of its millions of servers with a machine like -29 +30 00:02:05,428 --> 00:02:10,220 this, then 4 billion machines would mean about 1,000 copies of this souped-up Google. -30 +31 00:02:10,800 --> 00:02:13,360 Let's call that 1 kilo-Google worth of computing power. -31 +32 00:02:14,940 --> 00:02:17,500 There's about 7.3 billion people on Earth. -32 +33 00:02:18,060 --> 00:02:21,169 So next, imagine giving a little over half of every -33 +34 00:02:21,169 --> 00:02:24,220 individual on Earth their own personal kilo-Google. -34 +35 00:02:25,460 --> 00:02:28,820 Now, imagine 4 billion copies of this Earth. -35 +36 00:02:29,900 --> 00:02:34,820 For comparison, the Milky Way has somewhere between 100 and 400 billion stars. -36 +37 00:02:35,280 --> 00:02:37,540 We don't really know, but the estimates tend to be in that range. -37 +38 00:02:38,400 --> 00:02:43,126 This would be akin to a full 1% of every star in the galaxy having a copy -38 +39 00:02:43,126 --> 00:02:47,980 of Earth where half the people on Earth have their own personal kilo-Google. -39 +40 00:02:49,140 --> 00:02:53,680 Next, try to imagine 4 billion copies of the Milky Way. -40 +41 00:02:53,680 --> 00:02:57,868 And we're going to call this your giga-galactic supercomputer, -41 +42 00:02:57,868 --> 00:03:01,060 running about 2 to the 160 guesses every second. -42 +43 00:03:03,200 --> 00:03:07,140 Now, 4 billion seconds, that's about 126.8 years. -43 +44 00:03:07,800 --> 00:03:11,040 Four billion of those, well that's 507 billion years, -44 +45 00:03:11,040 --> 00:03:13,920 which is about 37 times the age of the universe. -45 +46 00:03:14,960 --> 00:03:20,134 So even if you were to have your GPU-packed kilo-Google-per-person multiplanetary -46 +47 00:03:20,134 --> 00:03:25,057 giga-galactic computer guessing numbers for 37 times the age of the universe, -47 +48 00:03:25,057 --> 00:03:29,980 it would still only have a 1 in 4 billion chance of finding the correct guess. -48 +49 00:03:32,280 --> 00:03:37,179 By the way, the state of Bitcoin hashing these days is that all of the miners put -49 +50 00:03:37,179 --> 00:03:41,960 together guess and check at a rate of about 5 billion billion hashes per second. -50 +51 00:03:42,600 --> 00:03:45,960 That corresponds to one third of what I just described as a kilo-Google. -51 +52 00:03:46,520 --> 00:03:50,782 This is not because there are billions of GPU-packed machines out there, -52 +53 00:03:50,782 --> 00:03:55,745 but because miners actually use something that's about 1000 times better than a GPU, -53 +54 00:03:55,745 --> 00:03:58,140 application-specific integrated circuits. -54 +55 00:03:58,920 --> 00:04:03,280 These are pieces of hardware specifically designed for Bitcoin mining, -55 +56 00:04:03,280 --> 00:04:06,720 for running a bunch of SHA-256 hashes, and nothing else. -56 +57 00:04:07,500 --> 00:04:11,704 Turns out, there's a lot of efficiency gains to be had when you throw out the need -57 +58 00:04:11,704 --> 00:04:16,060 for general computation and design your integrated circuits for one and only one task. -58 +59 00:04:17,940 --> 00:04:21,996 Also, on the topic of large powers of two that I personally find it hard to -59 +60 00:04:21,996 --> 00:04:26,160 get my mind around, this channel recently surpassed 2 to the 18th subscribers. -60 +61 00:04:26,940 --> 00:04:30,897 And to engage a little more with some sub-portion of those 2 to the 18 people, -61 +62 00:04:30,897 --> 00:04:32,400 I'm going to do a Q&A session. -62 -00:04:33,000 --> 00:04:36,690 -I've left a link in the description to a Reddit thread where you can post - 63 -00:04:36,690 --> 00:04:39,633 -questions and upvote the ones you want to hear answers to, +00:04:33,000 --> 00:04:35,976 +I've left a link in the description to a Reddit thread where you can 64 -00:04:39,633 --> 00:04:43,573 -and probably in the next video or on Twitter I'll announce the format in which +00:04:35,976 --> 00:04:38,737 +post questions and upvote the ones you want to hear answers to. 65 -00:04:43,573 --> 00:04:44,820 -I'd like to give answers. +00:04:38,737 --> 00:04:41,670 +And probably in the next video or on Twitter or something like that + +66 +00:04:41,670 --> 00:04:44,820 +I'll announce the format in which I'd like to give answers. See you then! diff --git a/2017/256-bit-security/english/sentence_timings.json b/2017/256-bit-security/english/sentence_timings.json index 890fead6b..bba9ae91d 100644 --- a/2017/256-bit-security/english/sentence_timings.json +++ b/2017/256-bit-security/english/sentence_timings.json @@ -55,7 +55,7 @@ 94.74 ], [ - "The first 4 billion here represents the number of hashes per second per computer.", + "So the first 4 billion here is going to represent the number of hashes per second per computer.", 95.42, 100.32 ], @@ -175,7 +175,7 @@ 272.4 ], [ - "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", 273.0, 284.82 ] diff --git a/2017/256-bit-security/english/transcript.txt b/2017/256-bit-security/english/transcript.txt index 4e380d6a5..16e538799 100644 --- a/2017/256-bit-security/english/transcript.txt +++ b/2017/256-bit-security/english/transcript.txt @@ -9,7 +9,7 @@ All we need to do is appreciate what multiplying 4 billion times itself 8 succes As many of you know, the GPU on your computer can let you run a bunch of computations in parallel incredibly quickly. If you were to specially program a GPU to run a cryptographic hash function over and over, a really good one might be able to do a little less than a billion hashes per second. Let's say you just take a bunch of those and cram your computer full of extra GPUs so that your computer can run 4 billion hashes per second. -The first 4 billion here represents the number of hashes per second per computer. +So the first 4 billion here is going to represent the number of hashes per second per computer. Now, picture 4 billion of these GPU-packed computers. For comparison, even though Google does not at all make their number of servers public, estimates have it somewhere in the single-digit millions. In reality, most of those servers are going to be much less powerful than our imagined GPU-packed machine. @@ -33,4 +33,4 @@ These are pieces of hardware specifically designed for Bitcoin mining, for runni Turns out, there's a lot of efficiency gains to be had when you throw out the need for general computation and design your integrated circuits for one and only one task. Also, on the topic of large powers of two that I personally find it hard to get my mind around, this channel recently surpassed 2 to the 18th subscribers. And to engage a little more with some sub-portion of those 2 to the 18 people, I'm going to do a Q&A session. -I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers. \ No newline at end of file +I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then! \ No newline at end of file diff --git a/2017/256-bit-security/french/sentence_translations.json b/2017/256-bit-security/french/sentence_translations.json index b846e9c93..a591a0d35 100644 --- a/2017/256-bit-security/french/sentence_translations.json +++ b/2017/256-bit-security/french/sentence_translations.json @@ -99,7 +99,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "Les 4 premiers milliards représentent le nombre de hachages par seconde et par ordinateur.", "model": "DeepL", "from_community_srt": "Donc les premiers 4 milliards ici vont représenter le nombre de hashs par seconde de l'ordinateur.", @@ -315,7 +315,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "J'ai laissé un lien dans la description vers un fil Reddit où tu peux poster des questions et upvoter celles dont tu veux entendre les réponses, et probablement dans la prochaine vidéo ou sur Twitter, j'annoncerai le format dans lequel j'aimerais donner des réponses.", "model": "DeepL", "from_community_srt": "J'ai mis dans la description un lien vers une discussion Reddit où vous pouvez poser des questions et voter pour celles pour lesquelles vous souhaiteriez entendre des réponses, et sans doute dans la prochaine vidéo ou sur Twitter, ou quelque chose comme ça, j'annoncerai le format sous lequel j'aimerais donner les réponses. A bientôt donc.", diff --git a/2017/256-bit-security/german/sentence_translations.json b/2017/256-bit-security/german/sentence_translations.json index aaaff435c..f413a1d98 100644 --- a/2017/256-bit-security/german/sentence_translations.json +++ b/2017/256-bit-security/german/sentence_translations.json @@ -99,7 +99,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "Die ersten 4 Milliarden stehen hier für die Anzahl der Hashes pro Sekunde und Computer.", "model": "DeepL", "from_community_srt": "Also werden die ersten 4 Milliarden durch die Anzahl an Hashes pro Sekunde pro Computer repräsentiert.", @@ -315,7 +315,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "Ich habe in der Beschreibung einen Link zu einem Reddit-Thread hinterlassen, in dem du Fragen posten und die hoch bewerten kannst, auf die du Antworten hören möchtest. Wahrscheinlich werde ich im nächsten Video oder auf Twitter das Format ankündigen, in dem ich Antworten geben möchte.", "model": "DeepL", "from_community_srt": "Ich habe einen Link zu einem Reddit-Thread in der Beschreibung hinterlassen, auf dem ihr Fragen posten und upvoten könnt, zu denen ihr die Antwort wissen wollt. Und im nächsten Video oder auf Twitter werde ich das Format ankündigen, in dem ich die Fragen beantworten werde.", diff --git a/2017/256-bit-security/greek/sentence_translations.json b/2017/256-bit-security/greek/sentence_translations.json index 054db6902..0b0677f76 100644 --- a/2017/256-bit-security/greek/sentence_translations.json +++ b/2017/256-bit-security/greek/sentence_translations.json @@ -88,7 +88,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "", "from_community_srt": "Άρα τα πρώτα 4 δισεκατομμύρια εδώ θα αντιπροσωπεύουν τον αριθμό των τιμών ανά δευτερόλεπτο ανά υπολογιστή.", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "", "from_community_srt": "Έχω αφήσει ένα σύνδεσμο στην περιγραφή σε ένα Reddit thread, όπου μπορείτε να δημοσιεύετε ερωτήσεις και να ανεβάζετε αυτές στις οποίες θέλετε να ακούσετε απαντήσεις, και πιθανώς στο επόμενο βίντεο ή στο Twitter ή σε κάτι τέτοιο, θα ανακοινώσω τη μορφή με την οποία θα ήθελα να δίνω απαντήσεις. Τα λέμε τότε.", "n_reviews": 0, diff --git a/2017/256-bit-security/hebrew/sentence_translations.json b/2017/256-bit-security/hebrew/sentence_translations.json index 326037d04..87b454858 100644 --- a/2017/256-bit-security/hebrew/sentence_translations.json +++ b/2017/256-bit-security/hebrew/sentence_translations.json @@ -168,7 +168,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", "translatedText": "לאחר מכן, דמיינו לעצמכם 4 מיליארד עותקים של שביל החלב, וקראו לזה מחשב העל הג'יגה-גלקטי שלכם, שרץ בערך 2 עד 160 ניחושים בכל שנייה.", "n_reviews": 0, "start": 169.14, diff --git a/2017/256-bit-security/hindi/sentence_translations.json b/2017/256-bit-security/hindi/sentence_translations.json index 732b96010..2f9436ab0 100644 --- a/2017/256-bit-security/hindi/sentence_translations.json +++ b/2017/256-bit-security/hindi/sentence_translations.json @@ -192,7 +192,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", "translatedText": "इसके बाद, आकाशगंगा की 4 अरब प्रतियों की कल्पना करें, और इसे अपना गीगा-गैलेक्टिक सुपरकंप्यूटर कहें, जो हर सेकंड लगभग 2 से 160 अनुमानों तक चलता है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/256-bit-security/hungarian/sentence_translations.json b/2017/256-bit-security/hungarian/sentence_translations.json index efa9a3512..07831144b 100644 --- a/2017/256-bit-security/hungarian/sentence_translations.json +++ b/2017/256-bit-security/hungarian/sentence_translations.json @@ -88,7 +88,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "Az első 4 milliárd itt a másodpercenkénti hash-ek számát jelenti számítógépenként.", "model": "DeepL", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "A leírásban hagytam egy linket egy Reddit-témára, ahol kérdéseket tehetsz fel, és feljebb szavazhatod azokat, amelyekre szeretnéd hallani a válaszokat, és valószínűleg a következő videóban vagy a Twitteren bejelentem, hogy milyen formában szeretnék válaszokat adni.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/256-bit-security/indonesian/sentence_translations.json b/2017/256-bit-security/indonesian/sentence_translations.json index d8b048267..33a0e74dc 100644 --- a/2017/256-bit-security/indonesian/sentence_translations.json +++ b/2017/256-bit-security/indonesian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", "translatedText": "Selanjutnya, bayangkan 4 miliar salinan Bima Sakti, dan sebut saja ini superkomputer giga-galaksi Anda, yang menjalankan sekitar 2 hingga 160 tebakan setiap detik.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/256-bit-security/italian/sentence_translations.json b/2017/256-bit-security/italian/sentence_translations.json index a80dc3962..435fe761c 100644 --- a/2017/256-bit-security/italian/sentence_translations.json +++ b/2017/256-bit-security/italian/sentence_translations.json @@ -99,7 +99,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "I primi 4 miliardi rappresentano il numero di hash al secondo per computer.", "model": "DeepL", "from_community_srt": "Il primo “4 miliardi” allora rappresenterà il numero di hash al secondo per computer.", @@ -315,7 +315,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "Nella descrizione ho lasciato un link a un thread di Reddit in cui puoi postare le domande e votare quelle a cui vuoi ricevere le risposte, e probabilmente nel prossimo video o su Twitter annuncerò il formato in cui vorrei dare le risposte.", "model": "DeepL", "from_community_srt": "Ho lasciato nella descrizione un link ad un thread di Reddit in cui potrete postare domande e caricare quelle alle quali volete sentire una risposta, e probabilmente nel prossimo video su twitter o qualcosa del genere annuncerò il formato in cui vorrei dare le risposte. Ci vediamo Lì.", diff --git a/2017/256-bit-security/japanese/sentence_translations.json b/2017/256-bit-security/japanese/sentence_translations.json index 824dd0da0..d434f8cc5 100644 --- a/2017/256-bit-security/japanese/sentence_translations.json +++ b/2017/256-bit-security/japanese/sentence_translations.json @@ -214,7 +214,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", "translatedText": "次に、天の川銀河の 40 億個のコピーを想像して、これをギガ銀河スーパ ーコンピューターと呼び、毎秒 160 の推測の約 2 を実行します。", "model": "google_nmt", "from_community_srt": "続けて考えるのは、40億個分の天の川銀河である。 これをギガギャラクティック・スーパーコンピュータと名付けよう。 1秒間に2の160乗分の計算が出来る。", diff --git a/2017/256-bit-security/korean/sentence_translations.json b/2017/256-bit-security/korean/sentence_translations.json index e61238d2a..657cfffa9 100644 --- a/2017/256-bit-security/korean/sentence_translations.json +++ b/2017/256-bit-security/korean/sentence_translations.json @@ -97,7 +97,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "여기서 처음 40억 개는 컴퓨터당 초당 해시 수를 나타냅니다.", "model": "DeepL", "from_community_srt": "40억^8 개 중에서 40억개는 이에 해당하겠군요 (1개의 컴퓨터가 초당 계산할 수 있는 양) 이제 40억대의 컴퓨터 (방금 GPU 넣었던)를 생각해 봅시다.", @@ -309,7 +309,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "설명에 질문을 게시하고 답변을 듣고 싶은 질문에 업보팅할 수 있는 Reddit 스레드 링크를 남겨두었으며, 다음 동영상이나 트위터에서 제가 어떤 형식으로 답변을 드릴지 발표할 예정입니다.", "model": "DeepL", "from_community_srt": "설명란에 답변을 듣고 싶은 질문들을 올릴 수 있는 레딧 스레드를 달아놓았습니다. 다음 동영상이나 트위터 같은 곳에 제가 답을 드리고 싶은 질문들을 올릴께요", diff --git a/2017/256-bit-security/marathi/sentence_translations.json b/2017/256-bit-security/marathi/sentence_translations.json index 352930cd0..7637c3611 100644 --- a/2017/256-bit-security/marathi/sentence_translations.json +++ b/2017/256-bit-security/marathi/sentence_translations.json @@ -192,7 +192,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", "translatedText": "पुढे, आकाशगंगेच्या 4 अब्ज प्रतींची कल्पना करा आणि याला तुमचा गिगा-गॅलेक्टिक सुपर कॉम्प्युटर म्हणा, प्रत्येक सेकंदाला सुमारे 2 ते 160 अंदाज चालवतात.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/256-bit-security/persian/sentence_translations.json b/2017/256-bit-security/persian/sentence_translations.json index 16d0418ee..78dcc9449 100644 --- a/2017/256-bit-security/persian/sentence_translations.json +++ b/2017/256-bit-security/persian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", "translatedText": "در مرحله بعد، 4 میلیارد نسخه از کهکشان راه شیری را تصور کنید و آن را ابررایانه گیگا-کهکشانی خود بنامید، که در هر ثانیه حدود 2 تا 160 حدس زده می شود. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/256-bit-security/polish/sentence_translations.json b/2017/256-bit-security/polish/sentence_translations.json index f8216d5ec..efce91e73 100644 --- a/2017/256-bit-security/polish/sentence_translations.json +++ b/2017/256-bit-security/polish/sentence_translations.json @@ -88,7 +88,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "", "from_community_srt": "Więc pierwsze 4 mld będzie tutaj reprezentować liczbę mieszań na sekundę, na komputer.", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "", "from_community_srt": "Zostawiłem link w opisie do wątku Reddit, gdzie możecie dodawać pytania i przesłać te, na które chcecie usłyszeć odpowiedzi i prawdopodobnie w następnym filmie lub na Twitterze ogłoszę format, w którym chciałbym udzielić odpowiedzi. Do zobaczenia.", "n_reviews": 0, diff --git a/2017/256-bit-security/portuguese/sentence_translations.json b/2017/256-bit-security/portuguese/sentence_translations.json index ebd42098c..e0aedf26c 100644 --- a/2017/256-bit-security/portuguese/sentence_translations.json +++ b/2017/256-bit-security/portuguese/sentence_translations.json @@ -99,7 +99,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "Os primeiros 4 bilhões aqui representam o número de hashes por segundo por computador.", "model": "google_nmt", "from_community_srt": "Assim, as primeiras 4 bilhões aqui vão representar o número de hashes por segundo por computador.", @@ -315,7 +315,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "Deixei um link na descrição para um tópico do Reddit onde você pode postar perguntas e votar naquelas para as quais deseja ouvir respostas, e provavelmente no próximo vídeo ou no Twitter anunciarei o formato em que gostaria para dar respostas.", "model": "google_nmt", "from_community_srt": "Eu deixei um link na descrição de um post no Reddit, onde você pode enviar perguntas e votar nas que você quer ouvir respostas e, provavelmente, no próximo vídeo ou no Twitter ou algo parecido, Vou anunciar o formato em que eu gostaria de dar respostas. Vejo você lá!", diff --git a/2017/256-bit-security/russian/sentence_translations.json b/2017/256-bit-security/russian/sentence_translations.json index 5315921ca..34263e8d0 100644 --- a/2017/256-bit-security/russian/sentence_translations.json +++ b/2017/256-bit-security/russian/sentence_translations.json @@ -99,7 +99,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "Первые 4 миллиарда здесь представляют собой количество хэшей в секунду на один компьютер.", "model": "DeepL", "from_community_srt": "Так что первые 4 миллиарда представляют количество хэшей в секунду на компьютер.", @@ -313,7 +313,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "В описании я оставил ссылку на тему Reddit, где ты можешь публиковать вопросы и поднимать голоса за те, на которые хочешь услышать ответы, и, возможно, в следующем видео или в Twitter я объявлю формат, в котором я хотел бы давать ответы.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/256-bit-security/spanish/sentence_translations.json b/2017/256-bit-security/spanish/sentence_translations.json index 24545c897..8266e2773 100644 --- a/2017/256-bit-security/spanish/sentence_translations.json +++ b/2017/256-bit-security/spanish/sentence_translations.json @@ -99,7 +99,7 @@ "end": 94.74 }, { - "input": "The first 4 billion here represents the number of hashes per second per computer.", + "input": "So the first 4 billion here is going to represent the number of hashes per second per computer.", "translatedText": "Los primeros 4.000 millones representan el número de hashes por segundo por ordenador.", "model": "DeepL", "from_community_srt": "4 mil millones de hashes por segundo Este primer \"4 mil millones\" va a representar el numero de Hashes por segundo.", @@ -313,7 +313,7 @@ "end": 272.4 }, { - "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to, and probably in the next video or on Twitter I'll announce the format in which I'd like to give answers.", + "input": "I've left a link in the description to a Reddit thread where you can post questions and upvote the ones you want to hear answers to. And probably in the next video or on Twitter or something like that I'll announce the format in which I'd like to give answers. See you then!", "translatedText": "He dejado un enlace en la descripción a un hilo de Reddit en el que puedes publicar preguntas y votar las que quieras que se respondan, y probablemente en el próximo vídeo o en Twitter anunciaré el formato en el que me gustaría dar las respuestas.", "model": "DeepL", "from_community_srt": "He dejado el link en la descripción a un foro de Reddit donde pueden poner preguntas y subir de las que deseen escuchar respuestas, y probablemente en el siguiente video, o en Twitter o algo por el estilo, anunciaré la forma en la que responderé a las preguntas. Nos vemos entonces.", diff --git a/2017/256-bit-security/tamil/sentence_translations.json b/2017/256-bit-security/tamil/sentence_translations.json index 4f488a850..d4b69fa11 100644 --- a/2017/256-bit-security/tamil/sentence_translations.json +++ b/2017/256-bit-security/tamil/sentence_translations.json @@ -192,7 +192,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", "translatedText": "அடுத்து, பால்வீதியின் 4 பில்லியன் பிரதிகளை கற்பனை செய்து பாருங்கள், இதை உங்கள் கிகா-கேலக்டிக் சூப்பர் கம்ப்யூட்டர் என்று அழைக்கவும், ஒவ்வொரு நொடியும் சுமார் 2 முதல் 160 யூகங்கள் வரை இயங்கும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/256-bit-security/telugu/sentence_translations.json b/2017/256-bit-security/telugu/sentence_translations.json index 5f2ab8d68..8f23c504e 100644 --- a/2017/256-bit-security/telugu/sentence_translations.json +++ b/2017/256-bit-security/telugu/sentence_translations.json @@ -192,7 +192,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", "translatedText": "తర్వాత, పాలపుంత యొక్క 4 బిలియన్ కాపీలను ఊహించుకోండి మరియు దీన్ని మీ గిగా-గెలాక్సీ సూపర్ కంప్యూటర్ అని పిలవండి, ప్రతి సెకనుకు 2 నుండి 160 అంచనాలు నడుస్తాయి.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/256-bit-security/thai/sentence_translations.json b/2017/256-bit-security/thai/sentence_translations.json index fe7b8f1ce..af88fcbc5 100644 --- a/2017/256-bit-security/thai/sentence_translations.json +++ b/2017/256-bit-security/thai/sentence_translations.json @@ -192,7 +192,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/256-bit-security/turkish/sentence_translations.json b/2017/256-bit-security/turkish/sentence_translations.json index 065e46478..87fa07676 100644 --- a/2017/256-bit-security/turkish/sentence_translations.json +++ b/2017/256-bit-security/turkish/sentence_translations.json @@ -192,7 +192,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", "translatedText": "Sonra, Samanyolu'nun 4 milyar kopyasını hayal edin ve buna saniyede 2 ila 160 tahmin çalıştıran devasa galaktik süper bilgisayarınız adını verin.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/256-bit-security/ukrainian/sentence_translations.json b/2017/256-bit-security/ukrainian/sentence_translations.json index 49578c8fc..4ff38afaf 100644 --- a/2017/256-bit-security/ukrainian/sentence_translations.json +++ b/2017/256-bit-security/ukrainian/sentence_translations.json @@ -168,7 +168,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", "translatedText": "Далі уявіть собі 4 мільярди копій Чумацького Шляху і назвіть це своїм гіга-галактичним суперкомп’ютером, що запускає приблизно 2 до 160 припущень щосекунди.", "n_reviews": 0, "start": 169.14, diff --git a/2017/256-bit-security/urdu/sentence_translations.json b/2017/256-bit-security/urdu/sentence_translations.json index 5048010d4..47a22a507 100644 --- a/2017/256-bit-security/urdu/sentence_translations.json +++ b/2017/256-bit-security/urdu/sentence_translations.json @@ -192,7 +192,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second. ", "translatedText": "اس کے بعد، آکاشگنگا کی 4 بلین کاپیوں کا تصور کریں، اور اسے اپنا گیگا-گیلیکٹک سپر کمپیوٹر کہیں، جو ہر سیکنڈ میں تقریباً 2 سے 160 اندازے چلاتا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/256-bit-security/vietnamese/sentence_translations.json b/2017/256-bit-security/vietnamese/sentence_translations.json index 7ec8af67d..110c42dc6 100644 --- a/2017/256-bit-security/vietnamese/sentence_translations.json +++ b/2017/256-bit-security/vietnamese/sentence_translations.json @@ -192,7 +192,7 @@ "end": 167.98 }, { - "input": "Next, imagine 4 billion copies of the Milky Way, and call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", + "input": "Next, try to imagine 4 billion copies of the Milky Way. And we're going to call this your giga-galactic supercomputer, running about 2 to the 160 guesses every second.", "translatedText": "Tiếp theo, hãy tưởng tượng 4 tỷ bản sao của Dải Ngân hà và gọi đây là siêu máy tính khổng lồ của thiên hà, chạy khoảng 2 đến 160 lần đoán mỗi giây.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/arabic/sentence_translations.json b/2017/area-and-slope/arabic/sentence_translations.json index 9aec966e8..fa43495dc 100644 --- a/2017/area-and-slope/arabic/sentence_translations.json +++ b/2017/area-and-slope/arabic/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are? ", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are? ", "translatedText": "والآن إذا أخبرتك أن المسافة بين هذه النقاط هي 0. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints. ", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints. ", "translatedText": "حل مشكلة المتوسط هو التغير في ارتفاع هذا الرسم البياني الجديد مقسومًا على التغير في قيمة x بين a وb، وبعبارة أخرى ميل الرسم البياني للمشتق العكسي بين نقطتي النهاية. ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible. ", + "input": "My thanks, as always, go to those making these videos possible. Thank you. ", "translatedText": "كما أتوجه بالشكر، كما هو الحال دائمًا، إلى أولئك الذين جعلوا مقاطع الفيديو هذه ممكنة. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/bengali/sentence_translations.json b/2017/area-and-slope/bengali/sentence_translations.json index f379eeed4..c45bb884e 100644 --- a/2017/area-and-slope/bengali/sentence_translations.json +++ b/2017/area-and-slope/bengali/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are? ", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are? ", "translatedText": "এবং এখন যদি আমি আপনাকে বলি যে এই বিন্দুগুলির মধ্যে ব্যবধান 0।1, এবং আপনি জানেন যে সেগুলি 0 থেকে পাই পর্যন্ত, আপনি আমাকে বলতে পারেন কতগুলি আছে? ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints. ", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints. ", "translatedText": "গড় সমস্যার সমাধান হল এই নতুন গ্রাফের উচ্চতার পরিবর্তনকে a এবং b এর মধ্যে x মানের পরিবর্তন দ্বারা ভাগ করা, অন্য কথায় দুটি শেষ বিন্দুর মধ্যে অ্যান্টিডেরিভেটিভ গ্রাফের ঢাল।", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible. ", + "input": "My thanks, as always, go to those making these videos possible. Thank you. ", "translatedText": "আমার ধন্যবাদ, বরাবরের মতো, যারা এই ভিডিওগুলিকে সম্ভব করছে তাদের কাছে যান।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/bulgarian/sentence_translations.json b/2017/area-and-slope/bulgarian/sentence_translations.json index 07153f619..71c5981cc 100644 --- a/2017/area-and-slope/bulgarian/sentence_translations.json +++ b/2017/area-and-slope/bulgarian/sentence_translations.json @@ -480,7 +480,7 @@ "end": 507.52 }, { - "input": "For any function f of x, if you want to find its average value on some interval, say between a and b, you take the integral of f on that interval divided by the width of that interval, b minus a.", + "input": "For any function f of x, if you want to find its average value on some interval, say between a and b, what you do is take the integral of f on that interval divided by the width of that interval, b minus a.", "translatedText": "За всяка функция f на x, ако искате да намерите нейната средна стойност на някакъв интервал, например между a и b, вземете интеграла на f на този интервал, разделен на ширината на този интервал, b минус a.", "model": "DeepL", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 656.5 }, { - "input": "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints rather than having to tally up all the points in between.", + "input": "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints, rather than having to actually tally up all the points in between.", "translatedText": "Любимата ми интуиция все още е тази, която показах в последния видеоклип, но втората гледна точка е, че когато преформулирате въпроса за намиране на средна стойност на непрекъсната стойност като намиране на средния наклон на няколко допирателни линии, това ви позволява да видите отговора само чрез сравняване на крайните точки, вместо да се налага да събирате всички точки между тях.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/area-and-slope/chinese/sentence_translations.json b/2017/area-and-slope/chinese/sentence_translations.json index a97c389c6..9db0426be 100644 --- a/2017/area-and-slope/chinese/sentence_translations.json +++ b/2017/area-and-slope/chinese/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are? ", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are? ", "translatedText": "现在如果我告诉你这些点之间的间距是 0。1，你知道它们 的范围是从 0 到 pi，你能告诉我有多少个吗？", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints. ", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints. ", "translatedText": "平均问题的解决方案是这个新图的高度变化除 以 a 和 b 之间 x 值的变化，换 句话说就是两个端点之间的反导图的斜率。", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible. ", + "input": "My thanks, as always, go to those making these videos possible. Thank you. ", "translatedText": "我一如既往地感谢那些使这些视频成为可能的人。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/english/captions.srt b/2017/area-and-slope/english/captions.srt index 5c90b74d0..f4f79a29c 100644 --- a/2017/area-and-slope/english/captions.srt +++ b/2017/area-and-slope/english/captions.srt @@ -447,16 +447,16 @@ range should equal the total slope between the start and end points? To digest this idea, it helps to think about what it looks like for a general function. 113 -00:08:28,320 --> 00:08:34,010 +00:08:28,320 --> 00:08:33,706 For any function f of x, if you want to find its average value on some interval, 114 -00:08:34,010 --> 00:08:38,577 -say between a and b, you take the integral of f on that interval +00:08:33,706 --> 00:08:38,162 +say between a and b, what you do is take the integral of f on that 115 -00:08:38,577 --> 00:08:42,020 -divided by the width of that interval, b minus a. +00:08:38,162 --> 00:08:42,020 +interval divided by the width of that interval, b minus a. 116 00:08:43,080 --> 00:08:47,321 @@ -571,24 +571,24 @@ After all, it is by definition the derivative of capital F. So why are antiderivatives the key to solving integrals? 144 -00:10:57,600 --> 00:11:01,373 +00:10:57,600 --> 00:11:01,267 My favorite intuition is still the one I showed last video, 145 -00:11:01,373 --> 00:11:06,720 -but a second perspective is that when you reframe the question of finding an average +00:11:01,267 --> 00:11:06,646 +but a second perspective is that when you reframe the question of finding an average of 146 -00:11:06,720 --> 00:11:12,255 -of a continuous value as instead finding the average slope of a bunch of tangent lines, +00:11:06,646 --> 00:11:11,842 +a continuous value as instead finding the average slope of a bunch of tangent lines, 147 -00:11:12,255 --> 00:11:17,664 -it lets you see the answer just by comparing endpoints rather than having to tally up +00:11:11,842 --> 00:11:15,265 +it lets you see the answer just by comparing endpoints, 148 -00:11:17,664 --> 00:11:19,300 -all the points in between. +00:11:15,265 --> 00:11:19,300 +rather than having to actually tally up all the points in between. 149 00:11:23,120 --> 00:11:27,658 diff --git a/2017/area-and-slope/english/sentence_timings.json b/2017/area-and-slope/english/sentence_timings.json index 8832cffc4..f02781348 100644 --- a/2017/area-and-slope/english/sentence_timings.json +++ b/2017/area-and-slope/english/sentence_timings.json @@ -300,7 +300,7 @@ 507.52 ], [ - "For any function f of x, if you want to find its average value on some interval, say between a and b, you take the integral of f on that interval divided by the width of that interval, b minus a.", + "For any function f of x, if you want to find its average value on some interval, say between a and b, what you do is take the integral of f on that interval divided by the width of that interval, b minus a.", 508.32, 522.02 ], @@ -370,7 +370,7 @@ 656.5 ], [ - "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints rather than having to tally up all the points in between.", + "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints, rather than having to actually tally up all the points in between.", 657.6, 679.3 ], diff --git a/2017/area-and-slope/english/transcript.txt b/2017/area-and-slope/english/transcript.txt index f39476c5b..9ec84ecd9 100644 --- a/2017/area-and-slope/english/transcript.txt +++ b/2017/area-and-slope/english/transcript.txt @@ -58,7 +58,7 @@ By definition, sine of x is the derivative of this antiderivative graph, giving Another way to think about the average value of sine of x is as the average slope over all tangent lines between 0 and pi. And when you view things like that, doesn't it make a lot of sense that the average slope of a graph over all its points in a certain range should equal the total slope between the start and end points? To digest this idea, it helps to think about what it looks like for a general function. -For any function f of x, if you want to find its average value on some interval, say between a and b, you take the integral of f on that interval divided by the width of that interval, b minus a. +For any function f of x, if you want to find its average value on some interval, say between a and b, what you do is take the integral of f on that interval divided by the width of that interval, b minus a. You can think of this as the area under the graph divided by its width, or more accurately, it is the signed area of that graph, since any area below the x-axis is counted as negative. And it's worth taking a moment to remember what this area has to do with the usual notion of a finite average, where you add up many numbers and divide by how many there are. When you take some sample of points spaced out by dx, the number of samples is about equal to the length of the interval divided by dx. @@ -72,7 +72,7 @@ In other words, it is the slope of the antiderivative graph between the two endp And again, when you stop to think about it, that should make a lot of sense, because little gives us the slope of the tangent line to this graph at each point. After all, it is by definition the derivative of capital F. So why are antiderivatives the key to solving integrals? -My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints rather than having to tally up all the points in between. +My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints, rather than having to actually tally up all the points in between. In the last video I described a sensation that should bring integrals to your mind, namely if you feel like the problem you're solving could be approximated by breaking it up somehow and adding up a large number of small things. Here I want you to come away recognizing a second sensation that should also bring integrals to your mind. If ever there's some idea that you understand in a finite context, and which involves adding up multiple values, like taking the average of a bunch of numbers, and if you want to generalize that idea to apply to an infinite continuous range of values, try seeing if you can phrase things in terms of an integral. diff --git a/2017/area-and-slope/french/sentence_translations.json b/2017/area-and-slope/french/sentence_translations.json index 2c7ab1d86..96ddb54b6 100644 --- a/2017/area-and-slope/french/sentence_translations.json +++ b/2017/area-and-slope/french/sentence_translations.json @@ -479,7 +479,7 @@ "end": 507.52 }, { - "input": "For any function f of x, if you want to find its average value on some interval, say between a and b, you take the integral of f on that interval divided by the width of that interval, b minus a.", + "input": "For any function f of x, if you want to find its average value on some interval, say between a and b, what you do is take the integral of f on that interval divided by the width of that interval, b minus a.", "translatedText": "Pour toute fonction f de x, si vous voulez trouver sa valeur moyenne sur un intervalle, disons entre a et b, vous prenez l'intégrale de f sur cet intervalle divisée par la largeur de cet intervalle, b moins a.", "from_community_srt": "Pour n'importe quelle fonction f(x), si vous voulez trouver sa valeur moyenne sur un certain intervalle, disons entre a et b, ce que vous faites est de prendre l'intégrale de f sur cet intervalle, divisé par l'épaisseur de cet intervalle.", "n_reviews": 0, @@ -591,7 +591,7 @@ "end": 656.5 }, { - "input": "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints rather than having to tally up all the points in between.", + "input": "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints, rather than having to actually tally up all the points in between.", "translatedText": "Mon intuition préférée est toujours celle que j'ai montrée dans la dernière vidéo, mais une deuxième perspective est que lorsque vous reformulez la question de trouver la moyenne d'une valeur continue en trouvant plutôt la pente moyenne d'un ensemble de lignes tangentes, cela vous permet de voir la réponse. simplement en comparant les points finaux plutôt que d'avoir à compter tous les points intermédiaires.", "from_community_srt": "mon intuition préférée reste celle que j'ai montrée dans la dernière vidéo, mais une seconde perspective est que lorsque l'on redemande la question de trouver la moyenne d'une valeur continue comme trouver la pente moyenne d'un groupe de tangentes, ceci vous permet de voir la réponse simplement en comparant les extrêmes,", "n_reviews": 0, diff --git a/2017/area-and-slope/german/sentence_translations.json b/2017/area-and-slope/german/sentence_translations.json index 8e92cad52..ca6254061 100644 --- a/2017/area-and-slope/german/sentence_translations.json +++ b/2017/area-and-slope/german/sentence_translations.json @@ -538,7 +538,7 @@ "end": 507.52 }, { - "input": "For any function f of x, if you want to find its average value on some interval, say between a and b, you take the integral of f on that interval divided by the width of that interval, b minus a.", + "input": "For any function f of x, if you want to find its average value on some interval, say between a and b, what you do is take the integral of f on that interval divided by the width of that interval, b minus a.", "translatedText": "Wenn du für eine beliebige Funktion f von x ihren Durchschnittswert auf einem Intervall, z. B. zwischen a und b, finden willst, nimmst du das Integral von f auf diesem Intervall geteilt durch die Breite des Intervalls, b minus a.", "model": "DeepL", "from_community_srt": "Für jede Funktion f (x), wenn Sie suchen möchten sein Durchschnittswert in einem Intervall, sagen wir zwischen a und b, was Sie tun, ist das Integral zu nehmen von f in diesem Intervall, geteilt durch die Breite des Intervalls.", @@ -664,7 +664,7 @@ "end": 656.5 }, { - "input": "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints rather than having to tally up all the points in between.", + "input": "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints, rather than having to actually tally up all the points in between.", "translatedText": "Meine Lieblingsintuition ist immer noch die, die ich im letzten Video gezeigt habe, aber eine zweite Perspektive ist, dass, wenn du die Frage nach dem Durchschnitt eines kontinuierlichen Wertes neu formulierst und stattdessen die durchschnittliche Steigung einer Reihe von Tangenten findest, du die Antwort einfach durch den Vergleich der Endpunkte siehst, anstatt alle Punkte dazwischen zu zählen.", "model": "DeepL", "from_community_srt": "Nun, meine Lieblingsintuition ist immer noch die Ich habe das letzte Video gezeigt, aber eine zweite Perspektive ist das, wenn Sie die Frage des Findens neu formulieren der Durchschnitt eines kontinuierlichen Wertes als Befund die durchschnittliche Steigung eines Bündels von Tangentenlinien, Sie können die Antwort nur durch Vergleichen sehen Endpunkte,", diff --git a/2017/area-and-slope/hebrew/sentence_translations.json b/2017/area-and-slope/hebrew/sentence_translations.json index 3c3f80ed6..bd3ac9cd7 100644 --- a/2017/area-and-slope/hebrew/sentence_translations.json +++ b/2017/area-and-slope/hebrew/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are? ", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are? ", "translatedText": "ועכשיו אם אני אומר לך שהרווח בין הנקודות האלה הוא 0.1, ואתה יודע שהם נעים בין 0 ל-pi, אתה יכול להגיד לי כמה יש? ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints. ", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints. ", "translatedText": "הפתרון לבעיה הממוצעת הוא השינוי בגובה הגרף החדש הזה חלקי השינוי בערך x בין a ל-b, במילים אחרות השיפוע של הגרף האנטי-נגזרת בין שתי נקודות הקצה. ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible. ", + "input": "My thanks, as always, go to those making these videos possible. Thank you. ", "translatedText": "תודתי, כמו תמיד, מגיעה לאלו שמאפשרים את הסרטונים האלה. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/hindi/sentence_translations.json b/2017/area-and-slope/hindi/sentence_translations.json index cbda7f409..71549048c 100644 --- a/2017/area-and-slope/hindi/sentence_translations.json +++ b/2017/area-and-slope/hindi/sentence_translations.json @@ -168,7 +168,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are?", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are?", "translatedText": "और अब अगर मैं आपसे कहूं कि इन बिंदुओं के बीच का अंतर 0 है।1, और आप जानते हैं कि उनकी सीमा 0 से पाई तक है, क्या आप मुझे बता सकते हैं कि उनकी संख्या कितनी है?", "n_reviews": 0, "start": 239.04, @@ -469,7 +469,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints.", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints.", "translatedText": "औसत समस्या का समाधान इस नए ग्राफ़ की ऊंचाई में परिवर्तन को ए और बी के बीच x मान में परिवर्तन से विभाजित करना है, दूसरे शब्दों में दो समापन बिंदुओं के बीच एंटीडेरिवेटिव ग्राफ़ का ढलान।", "n_reviews": 0, "start": 621.32, @@ -532,7 +532,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible.", + "input": "My thanks, as always, go to those making these videos possible. Thank you.", "translatedText": "हमेशा की तरह, इन वीडियो को संभव बनाने वालों को मेरा धन्यवाद।", "n_reviews": 0, "start": 729.04, diff --git a/2017/area-and-slope/hungarian/sentence_translations.json b/2017/area-and-slope/hungarian/sentence_translations.json index 171062a21..1a4aef8a8 100644 --- a/2017/area-and-slope/hungarian/sentence_translations.json +++ b/2017/area-and-slope/hungarian/sentence_translations.json @@ -480,7 +480,7 @@ "end": 507.52 }, { - "input": "For any function f of x, if you want to find its average value on some interval, say between a and b, you take the integral of f on that interval divided by the width of that interval, b minus a.", + "input": "For any function f of x, if you want to find its average value on some interval, say between a and b, what you do is take the integral of f on that interval divided by the width of that interval, b minus a.", "translatedText": "Ha az x bármely f függvényének átlagértékét szeretnénk megtalálni egy intervallumon, mondjuk a és b között, akkor az f integrálját vesszük az intervallumon, osztva az intervallum szélességével, b mínusz a-val.", "model": "DeepL", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 656.5 }, { - "input": "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints rather than having to tally up all the points in between.", + "input": "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints, rather than having to actually tally up all the points in between.", "translatedText": "A kedvenc intuícióm még mindig az, amit a múltkori videóban mutattam, de egy másik nézőpont az, hogy ha egy folytonos érték átlagának megtalálására vonatkozó kérdést úgy fogalmazzuk meg, hogy egy csomó érintővonal átlagos meredekségét keressük, akkor a végpontok összehasonlításával láthatjuk a választ, ahelyett, hogy a köztes pontokat kellene összeszámolnunk.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/area-and-slope/indonesian/sentence_translations.json b/2017/area-and-slope/indonesian/sentence_translations.json index 5ed339cf2..cd3f153c7 100644 --- a/2017/area-and-slope/indonesian/sentence_translations.json +++ b/2017/area-and-slope/indonesian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are?", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are?", "translatedText": "Dan sekarang jika saya beri tahu Anda bahwa jarak antara titik-titik ini adalah 0.1, dan Anda tahu rentangnya dari 0 hingga pi, dapatkah Anda memberi tahu saya berapa jumlahnya?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints.", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints.", "translatedText": "Penyelesaian permasalahan rata-rata adalah perubahan tinggi grafik baru ini dibagi dengan perubahan nilai x antara a dan b, dengan kata lain kemiringan grafik antiturunan antara kedua titik ujung.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible.", + "input": "My thanks, as always, go to those making these videos possible. Thank you.", "translatedText": "Terima kasih saya, seperti biasa, ditujukan kepada mereka yang membuat video ini menjadi kenyataan.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/italian/sentence_translations.json b/2017/area-and-slope/italian/sentence_translations.json index 90eb3f376..ffe956410 100644 --- a/2017/area-and-slope/italian/sentence_translations.json +++ b/2017/area-and-slope/italian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are?", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are?", "translatedText": "E ora se ti dico che la spaziatura tra questi punti è 0.1, e sai che vanno da 0 a pi greco, puoi dirmi quanti sono?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints.", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints.", "translatedText": "La soluzione al problema della media è la variazione dell'altezza di questo nuovo grafico divisa per la variazione del valore x tra a e b, in altre parole la pendenza del grafico antiderivativa tra i due punti finali.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible.", + "input": "My thanks, as always, go to those making these videos possible. Thank you.", "translatedText": "I miei ringraziamenti, come sempre, vanno a coloro che hanno reso possibile questi video.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/japanese/sentence_translations.json b/2017/area-and-slope/japanese/sentence_translations.json index d58466f73..ec6697f10 100644 --- a/2017/area-and-slope/japanese/sentence_translations.json +++ b/2017/area-and-slope/japanese/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are? ", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are? ", "translatedText": "ここで、これらの点間の間隔が 0 であるとします。1 で、0 から pi までの範囲があることは知っていますが、いくつあるか教えていただけますか? ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints. ", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints. ", "translatedText": "平均問題の解は、この新しいグラフの高さの変化を a と b の間の x 値の変化、つまり 2 つ の端点間の逆微分グラフの傾きで割ったものです。", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible. ", + "input": "My thanks, as always, go to those making these videos possible. Thank you. ", "translatedText": "いつものように、これらのビデオを可能にしてくれた方々に感謝します。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/korean/sentence_translations.json b/2017/area-and-slope/korean/sentence_translations.json index 172ab027a..1630867d6 100644 --- a/2017/area-and-slope/korean/sentence_translations.json +++ b/2017/area-and-slope/korean/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are? ", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are? ", "translatedText": "그리고 이제 이 점들 사이의 간격이 0이라고 말하면. 1이고 그 범위는 0부터 pi까지입니다. 얼마나 많은지 알려주실 수 있나요? ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints. ", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints. ", "translatedText": "평균 문제에 대한 해결책은 이 새로운 그래프의 높이 변화를 a와 b 사이의 x 값 변화, 즉 두 끝점 사이의 역도함수 그래프의 기울기로 나눈 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible. ", + "input": "My thanks, as always, go to those making these videos possible. Thank you. ", "translatedText": "언제나 그렇듯 이 영상을 제작해주신 분들께 감사드립니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/marathi/sentence_translations.json b/2017/area-and-slope/marathi/sentence_translations.json index 4871dd125..fd180a4d3 100644 --- a/2017/area-and-slope/marathi/sentence_translations.json +++ b/2017/area-and-slope/marathi/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are? ", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are? ", "translatedText": "आणि आता जर मी तुम्हाला सांगितले की या बिंदूंमधील अंतर 0 आहे. 1, आणि तुम्हाला माहिती आहे की ते 0 ते pi पर्यंत आहेत, तुम्ही मला सांगू शकता की किती आहेत? ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints. ", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints. ", "translatedText": "सरासरी समस्येचे निराकरण म्हणजे या नवीन आलेखाच्या उंचीतील बदल a आणि b मधील x मूल्यातील बदलाने भागणे, दुसऱ्या शब्दांत दोन अंतबिंदूंमधील अँटीडेरिव्हेटिव्ह आलेखाचा उतार. ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible. ", + "input": "My thanks, as always, go to those making these videos possible. Thank you. ", "translatedText": "माझे आभार, नेहमीप्रमाणे, ज्यांना हे व्हिडिओ शक्य आहेत त्यांच्याकडे जा. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/persian/sentence_translations.json b/2017/area-and-slope/persian/sentence_translations.json index 5980ccfa2..a6cecaa3e 100644 --- a/2017/area-and-slope/persian/sentence_translations.json +++ b/2017/area-and-slope/persian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are? ", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are? ", "translatedText": "و حالا اگر به شما بگویم که فاصله بین این نقاط 0 است. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints. ", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints. ", "translatedText": "راه حل مشکل میانگین، تغییر ارتفاع این نمودار جدید تقسیم بر تغییر مقدار x بین a و b، به عبارت دیگر شیب نمودار ضد مشتق بین دو نقطه پایانی است. ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible. ", + "input": "My thanks, as always, go to those making these videos possible. Thank you. ", "translatedText": "مثل همیشه از کسانی که این ویدیوها را امکان پذیر می کنند تشکر می کنم. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/portuguese/sentence_translations.json b/2017/area-and-slope/portuguese/sentence_translations.json index ddbc1be4e..6cd5b20ea 100644 --- a/2017/area-and-slope/portuguese/sentence_translations.json +++ b/2017/area-and-slope/portuguese/sentence_translations.json @@ -539,7 +539,7 @@ "end": 507.52 }, { - "input": "For any function f of x, if you want to find its average value on some interval, say between a and b, you take the integral of f on that interval divided by the width of that interval, b minus a.", + "input": "For any function f of x, if you want to find its average value on some interval, say between a and b, what you do is take the integral of f on that interval divided by the width of that interval, b minus a.", "translatedText": "Para qualquer função f de x, se você quiser encontrar seu valor médio em algum intervalo, digamos entre a e b, você calcula a integral de f nesse intervalo dividida pela largura desse intervalo, b menos a.", "model": "google_nmt", "from_community_srt": "Para qualquer função f (x), se você quiser encontrar seu valor médio em algum intervalo, digamos entre a e b, o que você faz é tomar a integral de f nesse intervalo, dividido pela largura do intervalo.", @@ -665,7 +665,7 @@ "end": 656.5 }, { - "input": "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints rather than having to tally up all the points in between.", + "input": "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints, rather than having to actually tally up all the points in between.", "translatedText": "Minha intuição favorita ainda é aquela que mostrei no último vídeo, mas uma segunda perspectiva é que quando você reformula a questão de encontrar uma média de um valor contínuo, em vez de encontrar a inclinação média de um monte de linhas tangentes, isso permite que você veja a resposta apenas comparando os pontos finais, em vez de ter que somar todos os pontos intermediários.", "model": "google_nmt", "from_community_srt": "minha intuição favorita ainda é a que mostrei no último vídeo, mas uma segunda perspectiva é que quando você reformula a questão de encontrar a média de um valor contínuo como encontrar a inclinação média do feixe de linhas tangentes, permite que você veja a resposta apenas comparando pontos finais,", diff --git a/2017/area-and-slope/russian/sentence_translations.json b/2017/area-and-slope/russian/sentence_translations.json index 19f3dee98..9bfd352f9 100644 --- a/2017/area-and-slope/russian/sentence_translations.json +++ b/2017/area-and-slope/russian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are?", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are?", "translatedText": "А теперь если я скажу вам, что расстояние между этими точками равно 0.1, и вы знаете, что они варьируются от 0 до пи, можете ли вы сказать мне, сколько их?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints.", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints.", "translatedText": "Решением средней задачи является изменение высоты этого нового графика, деленное на изменение значения x между a и b, другими словами, наклон графика первообразных между двумя конечными точками.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible.", + "input": "My thanks, as always, go to those making these videos possible. Thank you.", "translatedText": "Я, как всегда, благодарен тем, кто сделал эти видео возможными.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/spanish/sentence_translations.json b/2017/area-and-slope/spanish/sentence_translations.json index d585608b9..3b95f4b0c 100644 --- a/2017/area-and-slope/spanish/sentence_translations.json +++ b/2017/area-and-slope/spanish/sentence_translations.json @@ -479,7 +479,7 @@ "end": 507.52 }, { - "input": "For any function f of x, if you want to find its average value on some interval, say between a and b, you take the integral of f on that interval divided by the width of that interval, b minus a.", + "input": "For any function f of x, if you want to find its average value on some interval, say between a and b, what you do is take the integral of f on that interval divided by the width of that interval, b minus a.", "translatedText": "Para cualquier función f de x, si desea encontrar su valor promedio en algún intervalo, digamos entre a y b, toma la integral de f en ese intervalo dividida por el ancho de ese intervalo, b menos a.", "from_community_srt": "Para cualquier función f (x), si desea encontrar su valor medio en algún intervalo, digamos entre a y b, lo que haces es tomar la integral de f en ese intervalo, dividido por el ancho Del intervalo.", "n_reviews": 0, @@ -591,7 +591,7 @@ "end": 656.5 }, { - "input": "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints rather than having to tally up all the points in between.", + "input": "My favorite intuition is still the one I showed last video, but a second perspective is that when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines, it lets you see the answer just by comparing endpoints, rather than having to actually tally up all the points in between.", "translatedText": "Mi intuición favorita sigue siendo la que mostré en el último video, pero una segunda perspectiva es que cuando replanteas la pregunta de encontrar un promedio de un valor continuo en lugar de encontrar la pendiente promedio de un conjunto de líneas tangentes, te permite ver la respuesta. simplemente comparando los puntos finales en lugar de tener que sumar todos los puntos intermedios.", "from_community_srt": "mi intuición favorita sigue siendo la que mostré el último video, pero una segunda perspectiva Es que cuando se replantea la cuestión de encontrar la media de un valor continuo como hallazgo La pendiente media de un montón de líneas tangentes, que le permite ver la respuesta sólo por la comparación Endpoints,", "n_reviews": 0, diff --git a/2017/area-and-slope/tamil/sentence_translations.json b/2017/area-and-slope/tamil/sentence_translations.json index 82972cb39..cd2dc66e1 100644 --- a/2017/area-and-slope/tamil/sentence_translations.json +++ b/2017/area-and-slope/tamil/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are?", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are?", "translatedText": "இப்போது இந்த புள்ளிகளுக்கு இடையிலான இடைவெளி 0 என்று நான் சொன்னால்.1, மற்றும் அவை 0 முதல் பை வரை இருக்கும் என்பது உங்களுக்குத் தெரியும், எத்தனை உள்ளன என்று சொல்ல முடியுமா?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints.", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints.", "translatedText": "சராசரி சிக்கலுக்கான தீர்வு, இந்த புதிய வரைபடத்தின் உயரத்தை a மற்றும் b இடையே உள்ள x மதிப்பின் மாற்றத்தால் வகுக்கப்படும், வேறுவிதமாகக் கூறினால், இரண்டு முனைப்புள்ளிகளுக்கு இடையே உள்ள எதிர்வழி வரைபடத்தின் சாய்வு.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible.", + "input": "My thanks, as always, go to those making these videos possible. Thank you.", "translatedText": "எப்பொழுதும் போல் இந்த வீடியோக்களை சாத்தியமாக்கியவர்களுக்கு எனது நன்றிகள்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/telugu/sentence_translations.json b/2017/area-and-slope/telugu/sentence_translations.json index ee6c1a65f..1f4f761b5 100644 --- a/2017/area-and-slope/telugu/sentence_translations.json +++ b/2017/area-and-slope/telugu/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are?", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are?", "translatedText": "మరియు ఇప్పుడు ఈ పాయింట్ల మధ్య అంతరం 0 అని నేను మీకు చెబితే.1, మరియు అవి 0 నుండి pi వరకు ఉన్నాయని మీకు తెలుసు, ఎన్ని ఉన్నాయో నాకు చెప్పగలరా?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints.", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints.", "translatedText": "సగటు సమస్యకు పరిష్కారం ఈ కొత్త గ్రాఫ్ యొక్క ఎత్తులో మార్పు, a మరియు b మధ్య x విలువకు మార్పుతో భాగించబడుతుంది, మరో మాటలో చెప్పాలంటే రెండు ముగింపు బిందువుల మధ్య యాంటీడెరివేటివ్ గ్రాఫ్ యొక్క వాలు.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible.", + "input": "My thanks, as always, go to those making these videos possible. Thank you.", "translatedText": "ఎప్పటిలాగే, ఈ వీడియోలను సాధ్యం చేస్తున్న వారికి నా ధన్యవాదాలు.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/thai/sentence_translations.json b/2017/area-and-slope/thai/sentence_translations.json index c59927906..98ff612bb 100644 --- a/2017/area-and-slope/thai/sentence_translations.json +++ b/2017/area-and-slope/thai/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are? ", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints. ", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible. ", + "input": "My thanks, as always, go to those making these videos possible. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/turkish/sentence_translations.json b/2017/area-and-slope/turkish/sentence_translations.json index 5748660d4..a2b0eefbc 100644 --- a/2017/area-and-slope/turkish/sentence_translations.json +++ b/2017/area-and-slope/turkish/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are?", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are?", "translatedText": "Şimdi size bu noktalar arasındaki aralığın 0 olduğunu söylersem.1 ve bunların 0 ile pi arasında değiştiğini biliyorsunuz, kaç tane olduğunu bana söyleyebilir misiniz?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints.", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints.", "translatedText": "Ortalama problemin çözümü, bu yeni grafiğin yüksekliğindeki değişimin, a ile b arasındaki x değerindeki değişime, yani iki uç nokta arasındaki antiderivatif grafiğin eğimine bölünmesidir.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible.", + "input": "My thanks, as always, go to those making these videos possible. Thank you.", "translatedText": "Her zaman olduğu gibi bu videoları mümkün kılanlara teşekkürlerimi sunuyorum.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/ukrainian/sentence_translations.json b/2017/area-and-slope/ukrainian/sentence_translations.json index 442d4dd65..3d9492eb3 100644 --- a/2017/area-and-slope/ukrainian/sentence_translations.json +++ b/2017/area-and-slope/ukrainian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are?", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are?", "translatedText": "А тепер, якщо я скажу вам, що відстань між цими точками дорівнює 0.1, і ви знаєте, що вони варіюються від 0 до пі, можете сказати мені, скільки їх?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints.", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints.", "translatedText": "Розв’язанням середньої проблеми є зміна висоти цього нового графіка, поділена на зміну значення x між a і b, іншими словами, нахил графіка першої похідної між двома кінцевими точками.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible.", + "input": "My thanks, as always, go to those making these videos possible. Thank you.", "translatedText": "Я, як завжди, дякую тим, хто зробив ці відео можливими.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/urdu/sentence_translations.json b/2017/area-and-slope/urdu/sentence_translations.json index f33102725..0f13d5ae2 100644 --- a/2017/area-and-slope/urdu/sentence_translations.json +++ b/2017/area-and-slope/urdu/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are? ", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are? ", "translatedText": "اور اب اگر میں آپ کو بتاؤں کہ ان پوائنٹس کے درمیان فاصلہ 0 ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints. ", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints. ", "translatedText": "اوسط مسئلے کا حل اس نئے گراف کی اونچائی میں تبدیلی ہے جسے a اور b کے درمیان x قدر میں تبدیلی سے تقسیم کیا جاتا ہے، دوسرے لفظوں میں دو اختتامی نقطوں کے درمیان antiderivative گراف کی ڈھلوان۔", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible. ", + "input": "My thanks, as always, go to those making these videos possible. Thank you. ", "translatedText": "میرا شکریہ، ہمیشہ کی طرح، ان ویڈیوز کو ممکن بنانے والوں کے پاس جائیں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/area-and-slope/vietnamese/sentence_translations.json b/2017/area-and-slope/vietnamese/sentence_translations.json index f92d6126e..45caccf04 100644 --- a/2017/area-and-slope/vietnamese/sentence_translations.json +++ b/2017/area-and-slope/vietnamese/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.44 }, { - "input": "And now if I tell you that the spacing between these points is 0.1, and you know they range from 0 to pi, can you tell me how many there are?", + "input": "And now, if I tell you that the spacing between these points is, say, 0.1, and you know that they range from 0 to pi, can you tell me how many there are?", "translatedText": "Và bây giờ nếu tôi nói với bạn rằng khoảng cách giữa các điểm này là 0.1, và bạn biết chúng nằm trong khoảng từ 0 đến pi, bạn có thể cho tôi biết có bao nhiêu không?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 620.0 }, { - "input": "The solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b, in other words the slope of the antiderivative graph between the two endpoints.", + "input": "So the solution to the average problem is the change in the height of this new graph divided by the change to the x value between a and b. In other words, it is the slope of the antiderivative graph between the two endpoints.", "translatedText": "Lời giải cho bài toán trung bình là sự thay đổi độ cao của đồ thị mới này chia cho sự thay đổi giá trị x giữa a và b, nói cách khác là độ dốc của đồ thị nguyên hàm giữa hai điểm cuối.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 727.78 }, { - "input": "My thanks, as always, go to those making these videos possible.", + "input": "My thanks, as always, go to those making these videos possible. Thank you.", "translatedText": "Xin gửi lời cảm ơn của tôi, như mọi khi, đến những người đã biến những video này thành hiện thực.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/arabic/sentence_translations.json b/2017/backpropagation-calculus/arabic/sentence_translations.json index fbad7762d..bfa4a94b2 100644 --- a/2017/backpropagation-calculus/arabic/sentence_translations.json +++ b/2017/backpropagation-calculus/arabic/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons. ", + "input": "And we're just going to focus on the connection between the last two neurons. ", "translatedText": "سنركز فقط على الاتصال بين آخر خليتين عصبيتين. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL. ", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL. ", "translatedText": "هذه هي قاعدة السلسلة، حيث أن ضرب هذه النسب الثلاث يعطينا حساسية c للتغيرات الصغيرة في wL. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases. ", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases. ", "translatedText": "بالطبع، هذا مجرد عنصر واحد من متجه التدرج، والذي تم إنشاؤه من المشتقات الجزئية لدالة التكلفة فيما يتعلق بكل تلك الأوزان والتحيزات. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/bengali/sentence_translations.json b/2017/backpropagation-calculus/bengali/sentence_translations.json index 76ea7aab1..7be6d97bc 100644 --- a/2017/backpropagation-calculus/bengali/sentence_translations.json +++ b/2017/backpropagation-calculus/bengali/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons. ", + "input": "And we're just going to focus on the connection between the last two neurons. ", "translatedText": "আমরা শুধু শেষ দুটি নিউরনের মধ্যে সংযোগের উপর ফোকাস করব।", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL. ", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL. ", "translatedText": "এটি এখানে চেইন নিয়ম, যেখানে এই তিনটি অনুপাতকে গুণ করলে wL-তে ছোট পরিবর্তনের প্রতি c-এর সংবেদনশীলতা পাওয়া যায়।", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases. ", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases. ", "translatedText": "অবশ্যই, এটি গ্রেডিয়েন্ট ভেক্টরের শুধুমাত্র একটি উপাদান, যা সেই সমস্ত ওজন এবং পক্ষপাতের সাথে সাপেক্ষে খরচ ফাংশনের আংশিক ডেরিভেটিভ থেকে তৈরি করা হয়েছে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/chinese/sentence_translations.json b/2017/backpropagation-calculus/chinese/sentence_translations.json index b58508162..b92a91150 100644 --- a/2017/backpropagation-calculus/chinese/sentence_translations.json +++ b/2017/backpropagation-calculus/chinese/sentence_translations.json @@ -67,7 +67,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons.", + "input": "And we're just going to focus on the connection between the last two neurons.", "translatedText": "我们将只关注最后两个神经元之间的连接。", "model": "google_nmt", "n_reviews": 0, @@ -243,7 +243,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL.", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL.", "translatedText": "这就是链式法则，将这三个比率相乘即可得 出 c 对 wL 微小变化的敏感度。", "model": "google_nmt", "n_reviews": 0, @@ -335,7 +335,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases.", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases.", "translatedText": "当然，这只是梯度向量的一个组成部分，梯度向量是根据 成本函数相对于所有这些权重和偏差的偏导数构建的。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/german/sentence_translations.json b/2017/backpropagation-calculus/german/sentence_translations.json index a5d57fd16..d30641d92 100644 --- a/2017/backpropagation-calculus/german/sentence_translations.json +++ b/2017/backpropagation-calculus/german/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons.", + "input": "And we're just going to focus on the connection between the last two neurons.", "translatedText": "Wir konzentrieren uns nur auf die Verbindung zwischen den letzten beiden Neuronen.", "model": "DeepL", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL.", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL.", "translatedText": "Das hier ist die Kettenregel, bei der die Multiplikation dieser drei Verhältnisse die Empfindlichkeit von c auf kleine Änderungen von wL ergibt.", "model": "DeepL", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases.", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases.", "translatedText": "Das ist natürlich nur eine Komponente des Gradientenvektors, der sich aus den partiellen Ableitungen der Kostenfunktion in Bezug auf all diese Gewichte und Verzerrungen zusammensetzt.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/hebrew/sentence_translations.json b/2017/backpropagation-calculus/hebrew/sentence_translations.json index 32bdb77d7..875ea40cb 100644 --- a/2017/backpropagation-calculus/hebrew/sentence_translations.json +++ b/2017/backpropagation-calculus/hebrew/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons.", + "input": "And we're just going to focus on the connection between the last two neurons.", "translatedText": "אנחנו רק נתמקד בקשר בין שני הנוירונים האחרונים.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL.", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL.", "translatedText": "זה ממש כאן הוא כלל השרשרת, שבו הכפלת שלושת היחסים הללו נותנת לנו את הרגישות של c לשינויים קטנים ב-wL.", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases.", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases.", "translatedText": "כמובן, זה רק מרכיב אחד של וקטור הגרדיאנט, אשר בנוי מהנגזרות החלקיות של פונקציית העלות ביחס לכל אותם משקלים והטיות.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/hindi/sentence_translations.json b/2017/backpropagation-calculus/hindi/sentence_translations.json index c8e1ffc7f..97f5c3731 100644 --- a/2017/backpropagation-calculus/hindi/sentence_translations.json +++ b/2017/backpropagation-calculus/hindi/sentence_translations.json @@ -56,7 +56,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons.", + "input": "And we're just going to focus on the connection between the last two neurons.", "translatedText": "हम केवल अंतिम दो न्यूरॉन्स के बीच संबंध पर ध्यान केंद्रित करेंगे।", "n_reviews": 0, "start": 61.96, @@ -203,7 +203,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL.", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL.", "translatedText": "यहीं श्रृंखला नियम है, जहां इन तीन अनुपातों को गुणा करने से हमें डब्ल्यूएल में छोटे बदलावों के प्रति सी की संवेदनशीलता मिलती है।", "n_reviews": 0, "start": 225.74, @@ -280,7 +280,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases.", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases.", "translatedText": "बेशक, यह ग्रेडिएंट वेक्टर का सिर्फ एक घटक है, जो उन सभी भारों और पूर्वाग्रहों के संबंध में लागत फ़ंक्शन के आंशिक डेरिवेटिव से बनाया गया है।", "n_reviews": 0, "start": 328.38, diff --git a/2017/backpropagation-calculus/indonesian/sentence_translations.json b/2017/backpropagation-calculus/indonesian/sentence_translations.json index 920e124b8..7f176f15d 100644 --- a/2017/backpropagation-calculus/indonesian/sentence_translations.json +++ b/2017/backpropagation-calculus/indonesian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons.", + "input": "And we're just going to focus on the connection between the last two neurons.", "translatedText": "Kami hanya akan fokus pada hubungan antara dua neuron terakhir.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL.", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL.", "translatedText": "Ini adalah aturan rantai, di mana mengalikan ketiga rasio ini memberi kita sensitivitas c terhadap perubahan kecil pada wL.", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases.", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases.", "translatedText": "Tentu saja, itu hanyalah salah satu komponen vektor gradien, yang dibangun dari turunan parsial fungsi biaya terhadap semua bobot dan bias tersebut.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/italian/sentence_translations.json b/2017/backpropagation-calculus/italian/sentence_translations.json index 5b8dacb79..b37766469 100644 --- a/2017/backpropagation-calculus/italian/sentence_translations.json +++ b/2017/backpropagation-calculus/italian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons.", + "input": "And we're just going to focus on the connection between the last two neurons.", "translatedText": "Ci concentreremo solo sulla connessione tra gli ultimi due neuroni.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL.", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL.", "translatedText": "Questa qui è la regola della catena, dove moltiplicando questi tre rapporti ci dà la sensibilità di c a piccoli cambiamenti in wL.", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases.", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases.", "translatedText": "Naturalmente, questa è solo una componente del vettore del gradiente, che è costruito dalle derivate parziali della funzione di costo rispetto a tutti questi pesi e distorsioni.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/japanese/sentence_translations.json b/2017/backpropagation-calculus/japanese/sentence_translations.json index c839016fa..225ed2c01 100644 --- a/2017/backpropagation-calculus/japanese/sentence_translations.json +++ b/2017/backpropagation-calculus/japanese/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons.", + "input": "And we're just going to focus on the connection between the last two neurons.", "translatedText": "最後の 2 つのニューロン間の接続にのみ焦点を当てます。", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL.", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL.", "translatedText": "これは連鎖則であり、これら 3 つの比率を乗算すると、 wL の小さな変化に対する c の感度が得られます。", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases.", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases.", "translatedText": "もちろん、これは勾配ベクトルの 1 つのコンポーネントにすぎず、す べての重みとバイアスに関するコスト関数の偏導関数から構築されます。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/marathi/sentence_translations.json b/2017/backpropagation-calculus/marathi/sentence_translations.json index 2e1969bf8..59c59a4e3 100644 --- a/2017/backpropagation-calculus/marathi/sentence_translations.json +++ b/2017/backpropagation-calculus/marathi/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons.", + "input": "And we're just going to focus on the connection between the last two neurons.", "translatedText": "आम्ही फक्त शेवटच्या दोन न्यूरॉन्समधील कनेक्शनवर लक्ष केंद्रित करू.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL.", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL.", "translatedText": "येथे हा साखळी नियम आहे, जेथे या तीन गुणोत्तरांचा गुणाकार केल्याने wL मधील लहान बदलांना c ची संवेदनशीलता मिळते.", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases.", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases.", "translatedText": "अर्थात, ग्रेडियंट व्हेक्टरचा हा फक्त एक घटक आहे, जो त्या सर्व वजन आणि पूर्वाग्रहांच्या संदर्भात खर्च फंक्शनच्या आंशिक डेरिव्हेटिव्हमधून तयार केला जातो.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/persian/sentence_translations.json b/2017/backpropagation-calculus/persian/sentence_translations.json index 150e6d501..530010926 100644 --- a/2017/backpropagation-calculus/persian/sentence_translations.json +++ b/2017/backpropagation-calculus/persian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons. ", + "input": "And we're just going to focus on the connection between the last two neurons. ", "translatedText": "ما فقط بر ارتباط بین دو نورون آخر تمرکز خواهیم کرد. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL. ", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL. ", "translatedText": "این دقیقاً در اینجا قانون زنجیره است، که در آن ضرب این سه نسبت به ما حساسیت c را به تغییرات کوچک در wL می دهد. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases. ", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases. ", "translatedText": "البته، این فقط یک جزء از بردار گرادیان است که از مشتقات جزئی تابع هزینه با توجه به تمام آن وزن ها و سوگیری ها ساخته شده است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/tamil/sentence_translations.json b/2017/backpropagation-calculus/tamil/sentence_translations.json index ab6fe1052..fb5e9fee3 100644 --- a/2017/backpropagation-calculus/tamil/sentence_translations.json +++ b/2017/backpropagation-calculus/tamil/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons.", + "input": "And we're just going to focus on the connection between the last two neurons.", "translatedText": "கடைசி இரண்டு நியூரான்களுக்கு இடையேயான தொடர்பைப் பற்றி மட்டும் கவனம் செலுத்துவோம்.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL.", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL.", "translatedText": "இது இங்கே சங்கிலி விதி, இந்த மூன்று விகிதங்களைப் பெருக்குவது wL இல் சிறிய மாற்றங்களுக்கு c இன் உணர்திறனை அளிக்கிறது.", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases.", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases.", "translatedText": "நிச்சயமாக, இது சாய்வு திசையனின் ஒரு கூறு மட்டுமே, இது அனைத்து எடைகள் மற்றும் சார்புகளைப் பொறுத்து செலவு செயல்பாட்டின் பகுதி வழித்தோன்றல்களிலிருந்து கட்டமைக்கப்பட்டுள்ளது.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/telugu/sentence_translations.json b/2017/backpropagation-calculus/telugu/sentence_translations.json index 7b852e0c3..ff8978d14 100644 --- a/2017/backpropagation-calculus/telugu/sentence_translations.json +++ b/2017/backpropagation-calculus/telugu/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons.", + "input": "And we're just going to focus on the connection between the last two neurons.", "translatedText": "మేము చివరి రెండు న్యూరాన్ల మధ్య కనెక్షన్‌పై దృష్టి పెడతాము.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL.", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL.", "translatedText": "ఇది ఇక్కడే గొలుసు నియమం, ఇక్కడ ఈ మూడు నిష్పత్తులను గుణించడం ద్వారా wLలో చిన్న మార్పులకు c యొక్క సున్నితత్వాన్ని ఇస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases.", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases.", "translatedText": "వాస్తవానికి, ఇది గ్రేడియంట్ వెక్టర్‌లోని ఒక భాగం మాత్రమే, ఇది అన్ని బరువులు మరియు పక్షపాతాలకు సంబంధించి ఖర్చు ఫంక్షన్ యొక్క పాక్షిక ఉత్పన్నాల నుండి రూపొందించబడింది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/thai/sentence_translations.json b/2017/backpropagation-calculus/thai/sentence_translations.json index 44d5fdd26..b3df5c80f 100644 --- a/2017/backpropagation-calculus/thai/sentence_translations.json +++ b/2017/backpropagation-calculus/thai/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons. ", + "input": "And we're just going to focus on the connection between the last two neurons. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL. ", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases. ", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/ukrainian/sentence_translations.json b/2017/backpropagation-calculus/ukrainian/sentence_translations.json index ae5403d87..dee611645 100644 --- a/2017/backpropagation-calculus/ukrainian/sentence_translations.json +++ b/2017/backpropagation-calculus/ukrainian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons.", + "input": "And we're just going to focus on the connection between the last two neurons.", "translatedText": "Ми просто зосередимося на зв’язку між двома останніми нейронами.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL.", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL.", "translatedText": "Це ланцюгове правило, де множення цих трьох співвідношень дає нам чутливість c до невеликих змін wL.", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases.", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases.", "translatedText": "Звичайно, це лише один компонент вектора градієнта, який складається з часткових похідних функції вартості щодо всіх цих ваг і зміщень.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/urdu/sentence_translations.json b/2017/backpropagation-calculus/urdu/sentence_translations.json index 678364a2a..e9d18de87 100644 --- a/2017/backpropagation-calculus/urdu/sentence_translations.json +++ b/2017/backpropagation-calculus/urdu/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons. ", + "input": "And we're just going to focus on the connection between the last two neurons. ", "translatedText": "ہم صرف آخری دو نیوران کے درمیان تعلق پر توجہ مرکوز کریں گے۔", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL. ", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL. ", "translatedText": "یہاں یہ سلسلہ اصول ہے، جہاں ان تینوں تناسب کو ضرب کرنے سے ہمیں wL میں چھوٹی تبدیلیوں کے لیے c کی حساسیت ملتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases. ", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases. ", "translatedText": "بلاشبہ، یہ گریڈینٹ ویکٹر کا صرف ایک جزو ہے، جو ان تمام وزنوں اور تعصبات کے حوالے سے لاگت کے فنکشن کے جزوی مشتقات سے بنایا گیا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation-calculus/vietnamese/sentence_translations.json b/2017/backpropagation-calculus/vietnamese/sentence_translations.json index 1674b1431..9e4b165bc 100644 --- a/2017/backpropagation-calculus/vietnamese/sentence_translations.json +++ b/2017/backpropagation-calculus/vietnamese/sentence_translations.json @@ -64,7 +64,7 @@ "end": 60.82 }, { - "input": "We'll just focus on the connection between the last two neurons.", + "input": "And we're just going to focus on the connection between the last two neurons.", "translatedText": "Chúng ta sẽ chỉ tập trung vào kết nối giữa hai nơ-ron cuối cùng.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 224.66 }, { - "input": "This right here is the chain rule, where multiplying these three ratios gives us the sensitivity of c to small changes in wL.", + "input": "This right here is the chain rule, where multiplying together these three ratios gives us the sensitivity of c to small changes in WL.", "translatedText": "Đây chính là quy tắc dây chuyền, trong đó việc nhân ba tỷ lệ này cho chúng ta độ nhạy của c với những thay đổi nhỏ trong wL.", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 327.46 }, { - "input": "Of course, that's just one component of the gradient vector, which is built up from the partial derivatives of the cost function with respect to all those weights and biases.", + "input": "And of course, that is just one component of the gradient vector, which itself is built up from the partial derivatives of the cost function with respect to all those weights and biases.", "translatedText": "Tất nhiên, đó chỉ là một thành phần của vectơ gradient, được xây dựng từ đạo hàm riêng của hàm chi phí đối với tất cả các trọng số và độ lệch đó.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/arabic/sentence_translations.json b/2017/backpropagation/arabic/sentence_translations.json index d513b1efb..a15c6d0a0 100644 --- a/2017/backpropagation/arabic/sentence_translations.json +++ b/2017/backpropagation/arabic/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "بمعنى ما، فإن الخلايا العصبية التي تنشط أثناء رؤية الرقم 2 تصبح أكثر ارتباطًا بتلك الخلايا العصبية التي تنشط عند التفكير في الأمر. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "إنه ليس التدرج الفعلي لدالة التكلفة، والذي يعتمد على جميع بيانات التدريب، وليس هذه المجموعة الفرعية الصغيرة، لذا فهي ليست الخطوة الأكثر كفاءة إلى أسفل، ولكن كل دفعة صغيرة تمنحك تقريبًا جيدًا، والأهم من ذلك أنها يمنحك تسريعًا حسابيًا كبيرًا. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/bengali/sentence_translations.json b/2017/backpropagation/bengali/sentence_translations.json index b67647843..2b18733df 100644 --- a/2017/backpropagation/bengali/sentence_translations.json +++ b/2017/backpropagation/bengali/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "এটি খরচ ফাংশনের প্রকৃত গ্রেডিয়েন্ট নয়, যা প্রশিক্ষণের সমস্ত ডেটার উপর নির্ভর করে, এই ক্ষুদ্র উপসেট নয়, তাই এটি সবচেয়ে কার্যকর পদক্ষেপ নয়, তবে প্রতিটি মিনি-ব্যাচ আপনাকে একটি সুন্দর অনুমান দেয় এবং আরও গুরুত্বপূর্ণভাবে এটি আপনাকে একটি উল্লেখযোগ্য গণনাগত গতি দেয়।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/chinese/sentence_translations.json b/2017/backpropagation/chinese/sentence_translations.json index c36d17fbd..c8437c54e 100644 --- a/2017/backpropagation/chinese/sentence_translations.json +++ b/2017/backpropagation/chinese/sentence_translations.json @@ -305,7 +305,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "从某种意义上说，看到 2 时放电的神经元与思 考 2 时放电的神经元之间的联系更加紧密。", "model": "google_nmt", "n_reviews": 0, @@ -492,7 +492,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "它不是成本函数的实际梯度，它取决于所有训练数 据，而不是这个微小的子集，所以它不是最有效的 下坡步骤，但每个小批量确实给你一个非常好的近 似值，更重要的是它为您带来显着的计算加速。", "model": "google_nmt", "from_community_srt": "毕竟计算真实梯度得用上所有的样本 而非这个子集 所以这也不是下山最高效的一步 然而 每个minibatch都会给你一个不错的近似 而且更重要的是 你的计算量会减轻不少 你如果想把网络沿代价函数的表面下山的路径画出来的话", diff --git a/2017/backpropagation/french/sentence_translations.json b/2017/backpropagation/french/sentence_translations.json index c04e8a10c..b984fe946 100644 --- a/2017/backpropagation/french/sentence_translations.json +++ b/2017/backpropagation/french/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "Dans un sens, les neurones qui s’activent en voyant un 2 sont plus fortement liés à ceux qui s’activent lorsqu’on y pense. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "Ce n'est pas le gradient réel de la fonction de coût, qui dépend de toutes les données d'entraînement, ni de ce petit sous-ensemble, ce n'est donc pas l'étape de descente la plus efficace, mais chaque mini-lot vous donne une assez bonne approximation, et plus important encore. vous donne une accélération de calcul significative. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/hebrew/sentence_translations.json b/2017/backpropagation/hebrew/sentence_translations.json index 194a496b1..04ab21440 100644 --- a/2017/backpropagation/hebrew/sentence_translations.json +++ b/2017/backpropagation/hebrew/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "במובן מסוים, הנוירונים שיורים בזמן שהם רואים 2 מקבלים קשר חזק יותר לאלו היורים כשחושבים על זה. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "זה לא השיפוע האמיתי של פונקציית העלות, שתלוי בכל נתוני האימון, לא תת-הקבוצה הקטנה הזו, אז זה לא הצעד היעיל ביותר בירידה, אבל כל מיני-אצט נותן לך קירוב די טוב, וחשוב מכך. נותן לך זירוז חישוב משמעותי. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/hindi/sentence_translations.json b/2017/backpropagation/hindi/sentence_translations.json index 44003a6a9..e10d392f3 100644 --- a/2017/backpropagation/hindi/sentence_translations.json +++ b/2017/backpropagation/hindi/sentence_translations.json @@ -252,7 +252,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it.", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2.", "translatedText": "एक अर्थ में, 2 को देखते समय जो न्यूरॉन्स सक्रिय होते हैं, वे इसके बारे में सोचते समय सक्रिय होने वाले न्यूरॉन्स से अधिक मजबूती से जुड़ जाते हैं।", "n_reviews": 0, "start": 377.94, @@ -406,7 +406,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup.", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup.", "translatedText": "यह लागत फ़ंक्शन का वास्तविक ग्रेडिएंट नहीं है, जो सभी प्रशिक्षण डेटा पर निर्भर करता है, न कि इस छोटे उपसमुच्चय पर, इसलिए यह डाउनहिल का सबसे कुशल कदम नहीं है, लेकिन प्रत्येक मिनी-बैच आपको एक बहुत अच्छा अनुमान देता है, और इससे भी महत्वपूर्ण बात यह है आपको एक महत्वपूर्ण कम्प्यूटेशनल स्पीडअप देता है।", "n_reviews": 0, "start": 596.96, diff --git a/2017/backpropagation/indonesian/sentence_translations.json b/2017/backpropagation/indonesian/sentence_translations.json index e908d3603..f5e04e7f0 100644 --- a/2017/backpropagation/indonesian/sentence_translations.json +++ b/2017/backpropagation/indonesian/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "Dalam arti tertentu, neuron yang terpicu saat melihat angka 2 menjadi lebih terkait erat dengan neuron yang terpicu saat memikirkannya. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "Ini bukan gradien sebenarnya dari fungsi biaya, yang bergantung pada semua data pelatihan, bukan subset kecil ini, jadi ini bukan langkah menurun yang paling efisien, tetapi setiap mini-batch memberi Anda perkiraan yang cukup bagus, dan yang lebih penting itu memberi Anda kecepatan komputasi yang signifikan. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/italian/sentence_translations.json b/2017/backpropagation/italian/sentence_translations.json index b3fa3707d..d8b36ea03 100644 --- a/2017/backpropagation/italian/sentence_translations.json +++ b/2017/backpropagation/italian/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "In un certo senso, i neuroni che si attivano mentre vedono un 2 si collegano più fortemente a quelli che si attivano quando ci si pensa. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "Non è il gradiente effettivo della funzione di costo, che dipende da tutti i dati di addestramento, non da questo piccolo sottoinsieme, quindi non è il passo più efficiente in discesa, ma ogni mini-batch fornisce un'approssimazione abbastanza buona e, cosa più importante, ti dà una notevole accelerazione computazionale. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/japanese/sentence_translations.json b/2017/backpropagation/japanese/sentence_translations.json index b3d80b2f0..b6fa33d41 100644 --- a/2017/backpropagation/japanese/sentence_translations.json +++ b/2017/backpropagation/japanese/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "ある意味、「2」を見ているときに発火しているニューロンは、それについ て考えているときに発火しているニューロンとより強く結びついています。", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "これはコスト関数の実際の勾配ではなく、この小さなサブセットでは なくすべてのトレーニング データに依存するため、最も効率的な下 り坂のステップではありませんが、各ミニバッチからかなり良好な近 似が得られます。さらに重要なのは、計算速度が大幅に向上します。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/marathi/sentence_translations.json b/2017/backpropagation/marathi/sentence_translations.json index 13eb32559..468e327f9 100644 --- a/2017/backpropagation/marathi/sentence_translations.json +++ b/2017/backpropagation/marathi/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "एका अर्थाने, 2 पाहताना फायरिंग होणारे न्यूरॉन्स त्याबद्दल विचार करताना गोळीबार करणाऱ्यांशी अधिक दृढपणे जोडले जातात. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "हा खर्च फंक्शनचा वास्तविक ग्रेडियंट नाही, जो सर्व प्रशिक्षण डेटावर अवलंबून असतो, या लहान उपसंचावर नाही, म्हणून ही सर्वात कार्यक्षम पायरी उतरणीवर नाही, परंतु प्रत्येक मिनी-बॅच तुम्हाला एक चांगला अंदाज देते आणि अधिक महत्त्वाचे म्हणजे ते तुम्हाला महत्त्वपूर्ण संगणकीय गती देते. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/persian/sentence_translations.json b/2017/backpropagation/persian/sentence_translations.json index b06735a55..f5f56c9e6 100644 --- a/2017/backpropagation/persian/sentence_translations.json +++ b/2017/backpropagation/persian/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "به یک معنا، نورون هایی که هنگام دیدن یک 2 شلیک می کنند، هنگام فکر کردن به آن، ارتباط قوی تری با نورون هایی دارند که شلیک می کنند. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "این شیب واقعی تابع هزینه نیست، که به تمام داده های آموزشی بستگی دارد، نه این زیر مجموعه کوچک، بنابراین کارآمدترین گام در سراشیبی نیست، اما هر دسته کوچک تقریب بسیار خوبی به شما می دهد، و مهمتر از آن سرعت محاسباتی قابل توجهی به شما می دهد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/tamil/sentence_translations.json b/2017/backpropagation/tamil/sentence_translations.json index 5d9aa769c..122b55e91 100644 --- a/2017/backpropagation/tamil/sentence_translations.json +++ b/2017/backpropagation/tamil/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "ஒரு வகையில், 2 ஐப் பார்க்கும்போது சுடும் நியூரான்கள் அதைப் பற்றி சிந்திக்கும்போது சுடுபவர்களுடன் மிகவும் வலுவாக இணைக்கப்படுகின்றன. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "இது செலவு செயல்பாட்டின் உண்மையான சாய்வு அல்ல, இது அனைத்து பயிற்சி தரவையும் சார்ந்துள்ளது, இந்த சிறிய துணைக்குழு அல்ல, எனவே இது மிகவும் திறமையான கீழ்நோக்கிய படி அல்ல, ஆனால் ஒவ்வொரு சிறு தொகுதியும் உங்களுக்கு ஒரு நல்ல தோராயத்தை தருகிறது, மேலும் முக்கியமாக இது குறிப்பிடத்தக்க கணக்கீட்டு வேகத்தை உங்களுக்கு வழங்குகிறது. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/telugu/sentence_translations.json b/2017/backpropagation/telugu/sentence_translations.json index b9ae8ccca..ea995750a 100644 --- a/2017/backpropagation/telugu/sentence_translations.json +++ b/2017/backpropagation/telugu/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "ఒక రకంగా చెప్పాలంటే, 2ని చూసినప్పుడు కాల్చే న్యూరాన్‌లు దాని గురించి ఆలోచిస్తున్నప్పుడు కాల్చే వాటితో మరింత బలంగా ముడిపడి ఉంటాయి. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "ఇది ఖర్చు ఫంక్షన్ యొక్క అసలు గ్రేడియంట్ కాదు, ఇది మొత్తం శిక్షణ డేటాపై ఆధారపడి ఉంటుంది, ఈ చిన్న ఉపసమితి కాదు, కాబట్టి ఇది లోతువైపు అత్యంత ప్రభావవంతమైన దశ కాదు, కానీ ప్రతి చిన్న-బ్యాచ్ మీకు చాలా మంచి ఉజ్జాయింపుని ఇస్తుంది మరియు మరింత ముఖ్యంగా ఇది మీకు ముఖ్యమైన గణన వేగాన్ని అందిస్తుంది. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/thai/sentence_translations.json b/2017/backpropagation/thai/sentence_translations.json index a66f97585..44e1a52f1 100644 --- a/2017/backpropagation/thai/sentence_translations.json +++ b/2017/backpropagation/thai/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/ukrainian/sentence_translations.json b/2017/backpropagation/ukrainian/sentence_translations.json index 058f6c697..133fc7169 100644 --- a/2017/backpropagation/ukrainian/sentence_translations.json +++ b/2017/backpropagation/ukrainian/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "У певному сенсі нейрони, які спрацьовують, коли бачать 2, стають сильніше пов’язаними з тими, хто спрацьовує, коли думає про це. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "Це не фактичний градієнт функції витрат, який залежить від усіх навчальних даних, а не ця крихітна підмножина, тому це не найефективніший крок униз, але кожна міні-серія дає вам досить гарне наближення, і, що важливіше, це дає вам значне прискорення обчислень. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/urdu/sentence_translations.json b/2017/backpropagation/urdu/sentence_translations.json index 33c36b7c5..6f73cd213 100644 --- a/2017/backpropagation/urdu/sentence_translations.json +++ b/2017/backpropagation/urdu/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "ایک لحاظ سے، نیوران جو 2 کو دیکھتے ہوئے فائرنگ کر رہے ہیں، اس کے بارے میں سوچتے ہوئے فائرنگ کرنے والوں سے زیادہ مضبوطی سے جڑ جاتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "یہ لاگت کے فنکشن کا اصل میلان نہیں ہے، جس کا انحصار تمام تربیتی اعداد و شمار پر ہے، نہ کہ اس چھوٹے سب سیٹ پر، اس لیے یہ نیچے کی طرف سب سے زیادہ موثر قدم نہیں ہے، لیکن ہر منی بیچ آپ کو بہت اچھا تخمینہ فراہم کرتا ہے، اور اس سے بھی اہم بات یہ ہے۔آپ کو ایک اہم کمپیوٹیشنل اسپیڈ اپ دیتا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/backpropagation/vietnamese/sentence_translations.json b/2017/backpropagation/vietnamese/sentence_translations.json index 29c2b429e..a6ebdb4d7 100644 --- a/2017/backpropagation/vietnamese/sentence_translations.json +++ b/2017/backpropagation/vietnamese/sentence_translations.json @@ -288,7 +288,7 @@ "end": 377.28 }, { - "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those firing when thinking about it. ", + "input": "In a sense, the neurons that are firing while seeing a 2 get more strongly linked to those are the ones firing when thinking about a 2. ", "translatedText": "Theo một nghĩa nào đó, các nơ-ron kích hoạt khi nhìn thấy số 2 sẽ có mối liên kết chặt chẽ hơn với những nơ-ron kích hoạt khi nghĩ về nó. ", "model": "google_nmt", "n_reviews": 1, @@ -464,7 +464,7 @@ "end": 596.2 }, { - "input": "It's not the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly it gives you a significant computational speedup. ", + "input": "It's not going to be the actual gradient of the cost function, which depends on all of the training data, not this tiny subset, so it's not the most efficient step downhill, but each mini-batch does give you a pretty good approximation, and more importantly, it gives you a significant computational speedup. ", "translatedText": "Đó không phải là gradient thực tế của hàm chi phí, nó phụ thuộc vào tất cả dữ liệu huấn luyện, không phải tập hợp con nhỏ này, vì vậy đây không phải là bước xuống dốc hiệu quả nhất, nhưng mỗi lô nhỏ sẽ đưa ra cho bạn một ước lượng khá tốt và quan trọng hơn là nó cho bạn một tốc độ tính toán đáng kể. ", "model": "google_nmt", "n_reviews": 1, diff --git a/2017/bitcoin/arabic/sentence_translations.json b/2017/bitcoin/arabic/sentence_translations.json index fd8fa993d..c3d9ac078 100644 --- a/2017/bitcoin/arabic/sentence_translations.json +++ b/2017/bitcoin/arabic/sentence_translations.json @@ -729,7 +729,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "ثم عندما تريد إجراء معاملة، مثلما تدفع أليس لبوب 100 دولار، فإنك تبث ذلك إلى العالم ليسمعه الناس ويسجلوه في دفاتر حساباتهم الخاصة.", "model": "google_nmt", "from_community_srt": "ثم عندما تريد إنشاء صفقة، مثل (أليس) ستدفع لـ(بوب) 100 LD. عليك أن تبث هذه العملية للجميع حتى يتمكن الآخرون من سماعها وتسجيلها في سجلاتهم الخاصة.", @@ -755,7 +755,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "عندما يتلقى بوب معاملة، مثلما تدفع أليس لبوب 10 دولارات، كيف يمكنه التأكد من أن الجميع قد استلموا نفس المعاملة ويصدقونها؟", "model": "google_nmt", "from_community_srt": "كيف ستجعل الجميع يتفقون على إحدى هذه النسخ أنها هي الصحيحة؟ عندما يستلم (بوب) صفقة، مثلًا (أليس) ستدفع لـ(بوب) 10 LD،", @@ -764,7 +764,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "هل سيتمكن لاحقًا من الذهاب إلى تشارلي واستخدام نفس هذه الدولارات العشرة لإجراء معاملة؟", "model": "google_nmt", "from_community_srt": "كيف سيتأكد من أن الجميع حصل على المعلومات المتعلقة بهذه الصفقة كما تمت كتابتها لكي يستطيع بعد ذلك أن يذهب إلى (تشارلي) ويستخدم هذه الـ10 LD لإبرام صفقة أخرى؟", @@ -886,7 +886,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "يمكن أن تكون المدخلات الخاصة بإحدى هذه الوظائف أي نوع من الرسائل أو الملفات، فهي تبدو في الحقيقة بحجم 256 بت.", "model": "google_nmt", "from_community_srt": "أولًا، ما هي وظيفة دالّة التّهشير؟ مدخلات إحدى هذه الدوال ممكن أن تكون أي نوع من الرسائل أو الملفات، ذلك لا يهم، ومخرجات هذه الدالة ستكون سلسلة من الأرقام الثنائية بطول محدد ما، مثل 256 بت.", @@ -949,7 +949,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "إذا عرضت عليك سلسلة من 1 و0 وطلبت منك العثور على مدخلات لتجزئة SHA256، فلن يكون لديك طريقة أفضل من مجرد التخمين والتحقق.", "model": "google_nmt", "from_community_srt": "إذا أعطيك سلسلة من الآحاد والأصفار، و طلبت من إيجاد المدخلات التي إن أدخلتها إلى الدالة SHA256 ستعطيك نفس السلسلة من الآحاد والأصفار، ليس لديك طريقة أفضل من التخمين والتحقق.", @@ -958,7 +958,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "ومرة أخرى، إذا كنت تريد معرفة مقدار العمليات الحسابية المطلوبة لإجراء 256 تخمينًا، فما عليك سوى إلقاء نظرة على الفيديو الإضافي.", "model": "google_nmt", "from_community_srt": "ومجددًا، إذا كنت تريد أن تدرك كمية الحوسبة التي تحتاجها لتخمين 2 أس 256 قيمة؛ فقط قم بالاطلاع على الفيديو الملحق.", @@ -1171,7 +1171,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "في الوقت الحالي، لنفترض أن الأمر يجب أن يبدأ بـ 60 صفرًا، ولكننا سنعود لاحقًا إلى طريقة أكثر منهجية قد ترغب في تغييرها.", "model": "google_nmt", "from_community_srt": "الآن، دعنا نقول أنه يجب أن يبدأ بـ ... لنقل 60 صفرًا ولكننا سنعود لاحقًا لشرح طريقة أكثر منهجيةً لاختيار هذا العدد.", @@ -1180,7 +1180,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "تعتبر الكتلة صالحة فقط إذا كان لديها دليل على العمل.", "model": "google_nmt", "from_community_srt": "بالطريقة نفسها التي تعتبر فيها العملية صالحة فقط عندما تُوقع من المُرسِل، تعتبر الكتلة صالحة فقط إذا كان لديها رمز إثبات العمل.", @@ -1493,7 +1493,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "حسنًا، الطريقة التي يعمل بها بروتوكول البيتكوين الفعلي هي تغيير هذا العدد من الأصفار بشكل دوري بحيث يستغرق الأمر 10 دقائق للعثور على كتلة جديدة.", "model": "google_nmt", "from_community_srt": "حسنًا، الطريقة التي تعمل بها البيتكوين الحقيقية هي أن عدد الأصفار هذا يتغير من وقت لآخر بحيث تستغرق بالمتوسط 10 دقائق لإيجاد كتلة جديدة.", @@ -1538,7 +1538,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "هناك موقع ويب رائع يسمى Block Explorer يجعل من السهل البحث في blockchain الخاص بالبيتكوين.", "model": "google_nmt", "from_community_srt": "في الواقع، يوجد موقع ويب رائع يمكنك زيارته يسمى \"Block Explorer\" والذي يجعل الاطلاع على سلسلة كتل البيتكوين سهل.", diff --git a/2017/bitcoin/bengali/sentence_translations.json b/2017/bitcoin/bengali/sentence_translations.json index f6dcf905c..651ed144e 100644 --- a/2017/bitcoin/bengali/sentence_translations.json +++ b/2017/bitcoin/bengali/sentence_translations.json @@ -224,7 +224,7 @@ "end": 228.54 }, { - "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves she has seen it and approved of it, and it should be infeasible for anyone else to forge that signature. ", + "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves that she has seen it and that she's approved of it, and it should be infeasible for anyone else to forge that signature. ", "translatedText": "হাতে লেখা স্বাক্ষরের মতো, এখানে ধারণাটি হল যে অ্যালিস সেই লেনদেনের পাশে এমন কিছু যোগ করতে সক্ষম হওয়া উচিত যা প্রমাণ করে যে তিনি এটি দেখেছেন এবং এটি অনুমোদন করেছেন এবং অন্য কারও পক্ষে সেই স্বাক্ষর জাল করা অসম্ভব।", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 554.26 }, { - "input": "You are free to exchange ledger dollars for real US dollars. ", + "input": "You are of course free to exchange ledger dollars for real US dollars. ", "translatedText": "আপনি প্রকৃত মার্কিন ডলারের জন্য লেজার ডলার বিনিময় করতে পারবেন।", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 896.7 }, { - "input": "For a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 230, which is about 1 in a billion. ", + "input": "Well, for a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 2 to the 30, which is about 1 in a billion. ", "translatedText": "একটি এলোমেলো বার্তার জন্য, হ্যাশের 30টি পরপর শূন্য দিয়ে শুরু হওয়ার সম্ভাবনা 230-এর মধ্যে 1, যা এক বিলিয়নে প্রায় 1।", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 949.64 }, { - "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the list starts with 30 zeros. ", + "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the altered list together with this new number starts with 30 zeros. ", "translatedText": "সুতরাং আপনাকে কাজের একটি নতুন প্রমাণ খুঁজে পেতে আরও বিলিয়ন অনুমানের মধ্য দিয়ে যেতে হবে, একটি নতুন সংখ্যা যা এটি তৈরি করে যাতে তালিকার হ্যাশ 30টি শূন্য দিয়ে শুরু হয়।", "model": "google_nmt", "n_reviews": 0, @@ -1232,7 +1232,7 @@ "end": 1182.64 }, { - "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's really helpful to walk through exactly what it would take to fool someone using this system. ", + "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's actually really helpful to walk through exactly what it would take to fool someone using this system. ", "translatedText": "কেন এটি একটি বিশ্বস্ত সিস্টেমের জন্য তৈরি করে তা দেখতে, এবং কোন পর্যায়ে আপনার বিশ্বাস করা উচিত যে একটি অর্থপ্রদান বৈধ তা বোঝার জন্য, এই সিস্টেমটি ব্যবহার করে কাউকে বোকা বানানোর জন্য ঠিক কী লাগবে তা বোঝার জন্য এটি সত্যিই সহায়ক।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/bitcoin/chinese/sentence_translations.json b/2017/bitcoin/chinese/sentence_translations.json index 62eda1dcc..1b9c473c4 100644 --- a/2017/bitcoin/chinese/sentence_translations.json +++ b/2017/bitcoin/chinese/sentence_translations.json @@ -243,7 +243,7 @@ "end": 228.54 }, { - "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves she has seen it and approved of it, and it should be infeasible for anyone else to forge that signature.", + "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves that she has seen it and that she's approved of it, and it should be infeasible for anyone else to forge that signature.", "translatedText": "就像手写签 名一样，这里的想法是，爱丽丝应该能够在该交易旁边添加 一些东西，证明她已经看到并批准了它，并且其他任何人 都无法伪造该签名。", "model": "google_nmt", "from_community_srt": "就像手写签名一样， Alice需能在交易信息边上留下记录 以证明她了解并且允许这笔交易发生 而且这个签名不能被他人获取并伪造 乍一思索电子签名似乎不太可能实现 无论电子签名是如何存储的，", @@ -594,7 +594,7 @@ "end": 554.26 }, { - "input": "You are free to exchange ledger dollars for real US dollars.", + "input": "You are of course free to exchange ledger dollars for real US dollars.", "translatedText": "您可以 自由地将账本美元兑换成真实的美元。", "model": "google_nmt", "from_community_srt": "并简称为LD 你也可以将账元自由地兑换成真的美元 举个例子，", @@ -1041,7 +1041,7 @@ "end": 896.7 }, { - "input": "For a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 230, which is about 1 in a billion.", + "input": "Well, for a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 2 to the 30, which is about 1 in a billion.", "translatedText": "对于随机消息，散列恰好以 3 0 个连续零开头的概率为 230 分之一，约为十亿分之 一。", "model": "google_nmt", "from_community_srt": "对于一个随机的信息，", @@ -1111,7 +1111,7 @@ "end": 949.64 }, { - "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the list starts with 30 zeros.", + "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the altered list together with this new number starts with 30 zeros.", "translatedText": "因此，你必须进行另外十亿次猜测才能找到 新的工作证明，一个新的数字，使列表的哈希值以 30 个 零开头。", "model": "google_nmt", "from_community_srt": "即便是轻微的改动 也会完全改变最终的散列值 所以就又需要经过十亿次尝试才能找到新的工作量证明 即找到那个特别数字 和它对应的交易记录的散列值会以30个0开头 现在回过头来考虑我们的分布式账本的情形：", @@ -1330,7 +1330,7 @@ "end": 1182.64 }, { - "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's really helpful to walk through exactly what it would take to fool someone using this system.", + "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's actually really helpful to walk through exactly what it would take to fool someone using this system.", "translatedText": "要了解为什么这会成为一个值得信赖的系统，并了解在什 么时候您应该相信付款是合法的，了解如何欺骗使用该系 统的人确实很有帮助。", "model": "google_nmt", "from_community_srt": "并理解这个系统里的交易到底有多可信，", diff --git a/2017/bitcoin/czech/sentence_translations.json b/2017/bitcoin/czech/sentence_translations.json index c9f492e2b..0d383e681 100644 --- a/2017/bitcoin/czech/sentence_translations.json +++ b/2017/bitcoin/czech/sentence_translations.json @@ -654,7 +654,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "", "from_community_srt": "Když budete chtít provést transakci, např. \"Alice dluží Bobovi 100 LD\", tak tu transakci vyšlete do světa, ostatní ji uslyší, a zapíšou ji do jejich kopií účetní knihy.", "n_reviews": 0, @@ -678,7 +678,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "", "from_community_srt": "která z knih je ta správná? Když Bob obdrží zprávu o transakci \"Alice dluží Bobovi 10 LD\", jak si může být jistý, že všichni ostatní ji také obdrželi a věří jí? Jak ví,", "n_reviews": 0, @@ -686,7 +686,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "", "from_community_srt": "že později bude moct jít za Charliem a použít těchto 10 LD pro svou vlastní transakci? Jen si to představte sami.", "n_reviews": 0, @@ -796,7 +796,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "", "from_community_srt": "[též hashovací funkce] Do takové funkce vstupuje jakákoliv zpráva, soubor, na tom nesejde. Výstup je řetězec bitů s pevně danou délkou, jako třeba 256 bitů.", "n_reviews": 0, @@ -852,7 +852,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "", "from_community_srt": "Pokud vám dám řetězec jedniček a nul, a budu po vás chtít najít vstup tak, aby SHA256 otisk vašeho vstupu vrátil původní řetězec jedniček a nul, nezbyde vám nic lepšího než jen hádat a kontrolovat.", "n_reviews": 0, @@ -860,7 +860,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "", "from_community_srt": "Ještě jednou, pokud chcete vědět, jakou práci by dalo projít 2^256 pokusů, podívejte se na pomocné video.", "n_reviews": 0, @@ -1051,7 +1051,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "", "from_community_srt": "Pro teď řekněme, že musí začínat 60 nulami, ale k tomu se později vrátíme a vymyslíme systematičtější způsob, jakým požadovaný počet nul zvolit.", "n_reviews": 0, @@ -1059,7 +1059,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "", "from_community_srt": "Stejně jako je transakce platná pouze pokud je podepsaná odesílatelem, blok je validní pouze pokud obsahuje důkaz práce.", "n_reviews": 0, @@ -1339,7 +1339,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "", "from_community_srt": "Způsob, jakým skutečný Bitcoin funguje spočívá v tom, že pravidelně mění počet nul tak, aby nalezení nového bloku trvalo cca 10 minut.", "n_reviews": 0, @@ -1379,7 +1379,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "", "from_community_srt": "Existuje webová stránka \"Block Explorer\", kde se můžete podívat na celý Bitcoin řetěz.", "n_reviews": 0, diff --git a/2017/bitcoin/dutch/sentence_translations.json b/2017/bitcoin/dutch/sentence_translations.json index 07ecbdb27..c4586733f 100644 --- a/2017/bitcoin/dutch/sentence_translations.json +++ b/2017/bitcoin/dutch/sentence_translations.json @@ -656,7 +656,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "Als je dan een transactie wilt doen, bijvoorbeeld Alice betaalt Bob $100, dan zend je dat de wereld in zodat mensen het kunnen horen en op hun eigen privé grootboek kunnen zetten.", "model": "DeepL", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "Als Bob een transactie ontvangt, zoals Alice die Bob $10 betaalt, hoe kan hij er dan zeker van zijn dat iedereen diezelfde transactie heeft ontvangen en gelooft?", "model": "DeepL", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "Dat hij later naar Charlie kan gaan en diezelfde $10 kan gebruiken om een transactie te doen?", "model": "DeepL", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "De invoer voor een van deze functies kan elk soort bericht of bestand zijn, het lijkt echt op 256 bits.", "model": "DeepL", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "Als ik je een reeks van 1's en 0's laat zien en je vraag om een ingang te vinden voor de SHA256 hash, dan heb je geen betere methode dan gewoon raden en controleren.", "model": "DeepL", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "En nogmaals, als je wilt weten hoeveel rekenwerk er nodig is om 256 gissingen te doen, kijk dan eens naar de aanvullende video.", "model": "DeepL", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "Laten we voorlopig zeggen dat het met 60 nullen moet beginnen, maar later komen we terug op een meer systematische manier die je misschien wilt veranderen.", "model": "DeepL", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "Een blok wordt alleen als geldig beschouwd als het een bewijs van werk heeft.", "model": "DeepL", "n_reviews": 0, @@ -1344,7 +1344,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "Nou, de manier waarop het Bitcoin protocol werkt is om periodiek het aantal nullen te veranderen zodat het 10 minuten duurt om een nieuw blok te vinden.", "model": "DeepL", "n_reviews": 0, @@ -1384,7 +1384,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "Er is een geweldige website genaamd Block Explorer waarmee je eenvoudig door de Bitcoin blockchain kunt kijken.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/bitcoin/english/captions.srt b/2017/bitcoin/english/captions.srt index 95fdbe1ba..a3fae1617 100644 --- a/2017/bitcoin/english/captions.srt +++ b/2017/bitcoin/english/captions.srt @@ -647,16 +647,16 @@ who hosts the website, who controls the rules of adding new lines. To remove that bit of trust, we'll have everybody keep their own copy of the ledger. 163 -00:10:32,660 --> 00:10:37,015 -Then when you want to make a transaction, like Alice pays Bob $100, +00:10:32,660 --> 00:10:37,507 +Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, 164 -00:10:37,015 --> 00:10:42,395 -you broadcast that out into the world for people to hear and to record on their own +00:10:37,507 --> 00:10:41,055 +you do broadcast that out into the world for people to hear 165 -00:10:42,395 --> 00:10:43,420 -private ledgers. +00:10:41,055 --> 00:10:43,420 +and record on their own private ledgers. 166 00:10:44,840 --> 00:10:49,260 @@ -667,834 +667,854 @@ But unless you do something more, this system is absurdly bad. How could you get everyone to agree on what the right ledger is? 168 -00:10:53,440 --> 00:10:56,853 -When Bob receives a transaction, like Alice pays Bob $10, +00:10:53,440 --> 00:10:57,292 +When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, 169 -00:10:56,853 --> 00:11:01,680 +00:10:57,292 --> 00:11:01,680 how can he be sure that everyone else received and believes that same transaction? 170 -00:11:02,340 --> 00:11:07,200 -That he'll be able to later on go to Charlie and use those same $10 to make a transaction? +00:11:02,340 --> 00:11:04,816 +That he'll be able to later on go to Charlie and use 171 +00:11:04,816 --> 00:11:07,200 +those same 10 Ledger Dollars to make a transaction? + +172 00:11:08,240 --> 00:11:12,060 Really, imagine yourself just listening to transactions being broadcast. -172 +173 00:11:12,760 --> 00:11:15,717 How can you be sure that everyone else is recording -173 +174 00:11:15,717 --> 00:11:18,220 the same transactions and in the same order? -174 +175 00:11:19,420 --> 00:11:21,360 This is really the heart of the issue. -175 +176 00:11:21,600 --> 00:11:22,740 This is an interesting puzzle. -176 +177 00:11:23,420 --> 00:11:27,798 Can you come up with a protocol for how to accept or reject transactions, -177 +178 00:11:27,798 --> 00:11:32,531 and in what order, so that you can feel confident that anyone else in the world -178 +179 00:11:32,531 --> 00:11:37,620 who's following that same protocol has a personal ledger that looks the same as yours? -179 +180 00:11:38,300 --> 00:11:41,580 This is the problem addressed in the original Bitcoin paper. -180 +181 00:11:44,060 --> 00:11:48,142 At a high level, the solution that Bitcoin offers is to trust -181 +182 00:11:48,142 --> 00:11:52,160 whichever ledger has the most computational work put into it. -182 +183 00:11:52,540 --> 00:11:54,860 I'll take a moment to explain exactly what that means. -183 +184 00:11:55,320 --> 00:11:58,120 It involves a cryptographic hash function. -184 +185 00:11:58,460 --> 00:12:03,027 The general idea that we'll build to is that if you use computational work as -185 +186 00:12:03,027 --> 00:12:07,653 a basis for what to trust, you can make it so that fraudulent transactions and -186 +187 00:12:07,653 --> 00:12:12,280 conflicting ledgers require an infeasible amount of computation to bring about. -187 +188 00:12:13,040 --> 00:12:16,140 Again, I'll remind you that this is getting well into the weeds -188 +189 00:12:16,140 --> 00:12:19,580 beyond what anyone would need to know just to use a currency like this. -189 +190 00:12:20,120 --> 00:12:22,935 But it's a really cool idea, and if you understand it, -190 +191 00:12:22,935 --> 00:12:26,160 you understand the heart of Bitcoin and other cryptocurrencies. -191 +192 00:12:28,100 --> 00:12:29,940 So first things first, what's a hash function? -192 -00:12:30,800 --> 00:12:37,787 -The inputs for one of these functions can be any kind of message or file, - 193 -00:12:37,787 --> 00:12:40,620 -it really looks like 256 bits. +00:12:30,800 --> 00:12:34,814 +The inputs for one of these functions can be any kind of message or file, 194 +00:12:34,814 --> 00:12:38,124 +it really doesn't matter. And the output is a string of bits + +195 +00:12:38,124 --> 00:12:40,620 +with some kind of fixed length, like 256 bits. + +196 00:12:41,180 --> 00:12:45,027 This output is called the hash or digest of the message, -195 +197 00:12:45,027 --> 00:12:47,660 and the intent is that it looks random. -196 +198 00:12:48,000 --> 00:12:51,660 It's not random, it always gives the same output for a given input. -197 +199 00:12:52,200 --> 00:12:55,418 But the idea is that if you slightly change the input, -198 +200 00:12:55,418 --> 00:13:00,100 maybe editing just one of the characters, the resulting hash changes completely. -199 +201 00:13:00,820 --> 00:13:05,321 In fact, for the hash function I'm showing here, called SHA256, -200 +202 00:13:05,321 --> 00:13:11,440 the way the output changes as you slightly change that input is entirely unpredictable. -201 +203 00:13:12,440 --> 00:13:17,060 You see, this is not just any hash function, it's a cryptographic hash function. -202 +204 00:13:17,340 --> 00:13:20,660 That means it's infeasible to compute in the reverse direction. -203 -00:13:21,260 --> 00:13:29,324 -If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, - -204 -00:13:29,324 --> 00:13:34,640 -you'll have no better method than to just guess and check. - 205 -00:13:35,700 --> 00:13:39,917 -And again, if you want to feel for how much computation would be needed +00:13:21,260 --> 00:13:25,474 +If I show you some string of 1s and 0s, and ask you to find an 206 -00:13:39,917 --> 00:13:43,900 -to go through 256 guesses, just take a look at the supplement video. +00:13:25,474 --> 00:13:30,626 +input so that the SHA256 hash of that input gives this exact string of bits, 207 +00:13:30,626 --> 00:13:34,640 +you will have no better method than to just guess and check. + +208 +00:13:35,700 --> 00:13:39,772 +And again, if you want to feel for how much computation would be needed to + +209 +00:13:39,772 --> 00:13:43,900 +go through two to the 256 guesses, just take a look at the supplement video. + +210 00:13:44,380 --> 00:13:46,660 I actually had way too much fun writing that thing. -208 +211 00:13:48,560 --> 00:13:53,040 You might think that if you just really dig into the details of how exactly this function -209 +212 00:13:53,040 --> 00:13:57,520 works, you could reverse engineer the appropriate input without having to guess and check. -210 +213 00:13:58,240 --> 00:14:00,840 But no one has ever figured out a way to do that. -211 +214 00:14:01,600 --> 00:14:04,280 Interestingly, there's no cold hard rigorous proof -212 +215 00:14:04,280 --> 00:14:06,960 that it's hard to compute in the reverse direction. -213 +216 00:14:07,620 --> 00:14:11,175 And yet, a huge amount of modern security depends on cryptographic -214 +217 00:14:11,175 --> 00:14:14,200 hash functions and the idea that they have this property. -215 +218 00:14:14,940 --> 00:14:18,643 If you were to look at what algorithms underlie the secure connection -216 +219 00:14:18,643 --> 00:14:21,395 that your browser is making with YouTube right now, -217 +220 00:14:21,395 --> 00:14:25,840 or that it makes with your bank, you'll likely see the name SHA256 show up in there. -218 +221 00:14:27,340 --> 00:14:32,318 For right now, our focus will be on how such a function can prove that a particular -219 +222 00:14:32,318 --> 00:14:37,000 list of transactions is associated with a large amount of computational effort. -220 +223 00:14:38,040 --> 00:14:42,152 Imagine someone shows you a list of transactions, and they say, hey, -221 +224 00:14:42,152 --> 00:14:47,219 I found a special number so that when you put that number at the end of this list of -222 +225 00:14:47,219 --> 00:14:50,318 transactions, and apply SHA256 to the entire thing, -223 +226 00:14:50,318 --> 00:14:53,120 the first 30 bits of that output are all zeros. -224 +227 00:14:54,100 --> 00:14:56,700 How hard do you think it was for them to find that number? -225 +228 00:14:58,060 --> 00:15:02,528 Well, for a random message, the probability that a hash happens to start -226 +229 00:15:02,528 --> 00:15:07,180 with 30 successive zeros is 1 in 2 to the 30, which is about 1 in a billion. -227 +230 00:15:08,200 --> 00:15:11,290 And because SHA256 is a cryptographic hash function, -228 +231 00:15:11,290 --> 00:15:15,840 the only way to find a special number like that is just guessing and checking. -229 +232 00:15:16,660 --> 00:15:19,520 So this person almost certainly had to go through about a -230 +233 00:15:19,520 --> 00:15:22,380 billion different numbers before finding this special one. -231 +234 00:15:23,380 --> 00:15:26,253 And once you know that number, it's really quick to verify, -232 +235 00:15:26,253 --> 00:15:28,840 you just run the hash and see that there are 30 zeros. -233 +236 00:15:29,800 --> 00:15:33,618 So in other words, you can verify that they went through a large amount of work, -234 +237 00:15:33,618 --> 00:15:36,400 but without having to go through that same effort yourself. -235 +238 00:15:37,200 --> 00:15:38,800 This is called a proof of work. -236 +239 00:15:39,460 --> 00:15:44,220 And importantly, all of this work is intrinsically tied to the list of transactions. -237 +240 00:15:44,900 --> 00:15:47,785 If you change one of those transactions, even slightly, -238 +241 00:15:47,785 --> 00:15:49,640 it would completely change the hash. -239 +242 00:15:50,080 --> 00:15:54,427 So you'd have to go through another billion guesses to find a new proof of work, -240 +243 00:15:54,427 --> 00:15:57,862 a new number that makes it so that the hash of the altered list -241 +244 00:15:57,862 --> 00:16:00,600 together with this new number starts with 30 zeros. -242 +245 00:16:01,500 --> 00:16:04,100 So now think back to our distributed ledger situation. -243 +246 00:16:04,680 --> 00:16:07,816 Everyone is there broadcasting transactions and we want -244 +247 00:16:07,816 --> 00:16:10,840 a way for them to agree on what the correct ledger is. -245 +248 00:16:12,100 --> 00:16:15,330 As I mentioned, the idea behind the original Bitcoin paper is to -246 +249 00:16:15,330 --> 00:16:18,660 have everyone trust whichever ledger has the most work put into it. -247 +250 00:16:19,280 --> 00:16:22,906 The way this works is to first organize a given ledger into blocks, -248 +251 00:16:22,906 --> 00:16:27,280 where each block consists of a list of transactions together with a proof of work. -249 +252 00:16:27,720 --> 00:16:30,035 That is, a special number so that the hash of -250 +253 00:16:30,035 --> 00:16:32,300 the whole block starts with a bunch of zeros. -251 -00:16:33,140 --> 00:16:38,358 +254 +00:16:33,140 --> 00:16:36,183 For the moment, let's say it has to start with 60 zeros, -252 -00:16:38,358 --> 00:16:45,500 -but later we'll return back to a more systematic way you might want to change. +255 +00:16:36,183 --> 00:16:40,988 +but later we'll return back to a more systematic way you might want to choose that number. -253 +256 +00:16:40,988 --> 00:16:45,152 + In the same way that a transaction is only considered valid when it's signed + +257 +00:16:45,152 --> 00:16:45,900 +by the sender, + +258 00:16:45,900 --> 00:16:50,040 A block is only considered valid if it has a proof of work. -254 +259 00:16:50,960 --> 00:16:54,393 Also, to make sure there's a standard order to these blocks, -255 +260 00:16:54,393 --> 00:16:59,460 we'll make it so that a block has to contain the hash of the previous block at its header. -256 +261 00:17:00,060 --> 00:17:03,857 That way, if you were to go back and change any one of the blocks, -257 +262 00:17:03,857 --> 00:17:08,562 or to swap the order of two blocks, it would change the block that comes after it, -258 +263 00:17:08,562 --> 00:17:13,380 which changes the block's hash, which changes the one that comes after it, and so on. -259 +264 00:17:13,980 --> 00:17:17,804 That would require redoing all of the work, finding a new special number -260 +265 00:17:17,804 --> 00:17:21,420 for each of these blocks that makes their hashes start with 60 zeros. -261 +266 00:17:22,440 --> 00:17:24,843 Because blocks are chained together like this, -262 +267 00:17:24,843 --> 00:17:28,319 instead of calling it a ledger, it's common to call it a blockchain. -263 +268 00:17:30,080 --> 00:17:32,416 As part of our updated protocol, we'll now allow -264 +269 00:17:32,416 --> 00:17:34,420 anyone in the world to be a block creator. -265 +270 00:17:35,240 --> 00:17:39,219 What that means is that they're going to listen for transactions being broadcast, -266 +271 00:17:39,219 --> 00:17:42,811 collect them into some block, and then do a whole bunch of work to find a -267 +272 00:17:42,811 --> 00:17:46,160 special number that makes the hash of that block start with 60 zeros. -268 +273 00:17:46,960 --> 00:17:49,900 Once they find it, they broadcast out the block they found. -269 +274 00:17:50,860 --> 00:17:55,204 To reward a block creator for all this work, when she puts together a block, -270 +275 00:17:55,204 --> 00:17:59,267 we'll allow her to include a very special transaction at the top of it, -271 +276 00:17:59,267 --> 00:18:02,540 in which she gets, say, 10 ledger dollars out of thin air. -272 +277 00:18:03,080 --> 00:18:06,186 This is called the block reward, and it's an exception to -273 +278 00:18:06,186 --> 00:18:09,400 our usual rules about whether or not to accept transactions. -274 +279 00:18:10,040 --> 00:18:12,920 It doesn't come from anyone, so it doesn't have to be signed. -275 +280 00:18:13,660 --> 00:18:16,429 It also means that the total number of ledger -276 +281 00:18:16,429 --> 00:18:19,620 dollars in our economy increases with each new block. -277 +282 00:18:20,900 --> 00:18:25,160 Creating blocks is often called mining, since it requires doing a lot of work, -278 +283 00:18:25,160 --> 00:18:28,180 and it introduces new bits of currency into the economy. -279 +284 00:18:29,020 --> 00:18:33,088 But when you hear or read about miners, keep in mind that what they're -280 +285 00:18:33,088 --> 00:18:36,584 really doing is listening for transactions, creating blocks, -281 +286 00:18:36,584 --> 00:18:40,940 broadcasting those blocks, and getting rewarded with new money for doing so. -282 +287 00:18:41,780 --> 00:18:45,706 From the miners' perspective, each block is like a miniature lottery, -283 +288 00:18:45,706 --> 00:18:48,847 where everyone is guessing numbers as fast as they can, -284 +289 00:18:48,847 --> 00:18:53,503 until one lucky individual finds a special number that makes the hash of the block -285 +290 00:18:53,503 --> 00:18:56,140 start with many zeros, and they get the reward. -286 +291 00:18:57,620 --> 00:19:01,131 For anyone else who just wants to use this system to make payments, -287 +292 00:19:01,131 --> 00:19:05,262 instead of listening for transactions, they all start listening just for blocks -288 +293 00:19:05,262 --> 00:19:09,600 being broadcast by miners, and updating their own personal copies of the blockchain. -289 +294 00:19:10,560 --> 00:19:14,309 Now the key addition to our protocol is that if you hear two -290 +295 00:19:14,309 --> 00:19:18,058 distinct blockchains with conflicting transaction histories, -291 +296 00:19:18,058 --> 00:19:22,300 you defer to the longest one, the one with the most work put into it. -292 +297 00:19:22,860 --> 00:19:25,263 If there's a tie, just wait until you hear an -293 +298 00:19:25,263 --> 00:19:27,720 additional block that makes one of them longer. -294 +299 00:19:28,720 --> 00:19:33,323 So even though there's no central authority, and everyone is maintaining their own -295 +300 00:19:33,323 --> 00:19:38,092 copy of the blockchain, if everyone agrees to give preference to whichever blockchain -296 +301 00:19:38,092 --> 00:19:42,640 has the most work put into it, we have a way to arrive at decentralized consensus. -297 +302 00:19:43,560 --> 00:19:45,901 To see why this makes for a trustworthy system, -298 +303 00:19:45,901 --> 00:19:49,510 and to understand at what point you should trust that a payment is legit, -299 +304 00:19:49,510 --> 00:19:53,168 it's actually really helpful to walk through exactly what it would take to -300 +305 00:19:53,168 --> 00:19:54,680 fool someone using this system. -301 +306 00:19:55,600 --> 00:19:58,967 Maybe Alice is trying to fool Bob with a fraudulent block, -302 +307 00:19:58,967 --> 00:20:03,648 namely she tries to send him one that includes her paying him 100 Ledger dollars, -303 +308 00:20:03,648 --> 00:20:07,301 but without broadcasting that block to the rest of the network, -304 +309 00:20:07,301 --> 00:20:11,240 that way everyone else still thinks she has those 100 Ledger dollars. -305 +310 00:20:11,960 --> 00:20:16,857 To do this, she would have to find a valid proof of work before all the other miners, -306 +311 00:20:16,857 --> 00:20:18,680 each working on their own block. -307 +312 00:20:19,500 --> 00:20:21,966 And that could definitely happen, maybe Alice just -308 +313 00:20:21,966 --> 00:20:24,820 happens to win this miniature lottery before everyone else. -309 +314 00:20:25,680 --> 00:20:29,736 But Bob is still going to be hearing the broadcasts made by other miners, -310 +315 00:20:29,736 --> 00:20:32,367 so to keep him believing this fraudulent block, -311 +316 00:20:32,367 --> 00:20:36,423 Alice would have to do all the work herself to keep adding blocks on this -312 +317 00:20:36,423 --> 00:20:40,425 special fork in Bob's blockchain that's different from what he's hearing -313 +318 00:20:40,425 --> 00:20:41,960 from the rest of the miners. -314 +319 00:20:42,740 --> 00:20:48,260 Remember, as per the protocol, Bob always trusts the longest chain he knows about. -315 +320 00:20:49,260 --> 00:20:53,291 Alice might be able to keep this up for a few blocks if just by chance she -316 +321 00:20:53,291 --> 00:20:57,700 finds blocks more quickly than the rest of the miners on the network all combined. -317 +322 00:20:58,480 --> 00:21:03,480 But unless she has close to 50% of the computing resources among all of the miners, -318 +323 00:21:03,480 --> 00:21:08,422 the probability becomes overwhelming that the blockchain that all the other miners -319 +324 00:21:08,422 --> 00:21:13,780 are working on grows faster than the single fraudulent blockchain Alice is feeding to Bob. -320 +325 00:21:15,000 --> 00:21:19,009 So after enough time, Bob will just reject what he's hearing from -321 +326 00:21:19,009 --> 00:21:23,140 Alice in favor of the longer chain that everyone else is working on. -322 +327 00:21:23,960 --> 00:21:28,920 Notice, that means you shouldn't necessarily trust a new block you hear immediately. -323 +328 00:21:29,500 --> 00:21:33,400 Instead, you should wait for several new blocks to be added on top of it. -324 +329 00:21:33,820 --> 00:21:36,449 If you still haven't heard of any longer blockchains, -325 +330 00:21:36,449 --> 00:21:40,540 you can trust that this block is part of the same chain that everyone else is using. -326 +331 00:21:42,120 --> 00:21:45,220 And with that, we've hit all the main ideas. -327 +332 00:21:45,780 --> 00:21:49,702 This distributed ledger system based on a proof of work is more or less -328 +333 00:21:49,702 --> 00:21:53,680 how the Bitcoin protocol works, and how many other cryptocurrencies work. -329 +334 00:21:54,300 --> 00:21:56,160 There's just a few details to clear up. -330 +335 00:21:56,300 --> 00:21:59,210 Earlier I said that the proof of work might be to find a -331 +336 00:21:59,210 --> 00:22:02,580 special number so that the hash of the block starts with 60 zeros. -332 -00:22:03,220 --> 00:22:07,530 -Well, the way the actual Bitcoin protocol works is to periodically change +337 +00:22:03,220 --> 00:22:07,452 +Well, the way the actual Bitcoin protocol works is to periodically change that -333 -00:22:07,530 --> 00:22:11,900 -that number of zeros so that it should take 10 minutes to find a new block. +338 +00:22:07,452 --> 00:22:11,900 +number of zeros so that it should take, on average, 10 minutes to find a new block. -334 +339 00:22:12,780 --> 00:22:16,098 So as there are more and more miners added to the network, -335 +340 00:22:16,098 --> 00:22:19,529 the challenge gets harder and harder in such a way that this -336 +341 00:22:19,529 --> 00:22:22,960 miniature lottery only has about one winner every 10 minutes. -337 +342 00:22:23,920 --> 00:22:27,880 Many newer cryptocurrencies have much shorter block times than that. -338 +343 00:22:28,580 --> 00:22:32,460 And all of the money in Bitcoin ultimately comes from some block reward. -339 +344 00:22:32,920 --> 00:22:35,740 In the beginning, these rewards were 50 Bitcoin per block. -340 -00:22:36,140 --> 00:22:38,729 -There's a great website called Block Explorer that +345 +00:22:36,140 --> 00:22:38,634 +There's actually a great website you can go to called Block -341 -00:22:38,729 --> 00:22:41,420 -makes it easy to look through the Bitcoin blockchain. +346 +00:22:38,634 --> 00:22:41,420 +Explorer that makes it easy to look through the Bitcoin blockchain. -342 +347 00:22:41,960 --> 00:22:45,141 And if you look at the very first few blocks on the chain, -343 +348 00:22:45,141 --> 00:22:49,240 they contain no transactions other than that 50 Bitcoin reward to the miner. -344 +349 00:22:49,860 --> 00:22:56,340 But every 210,000 blocks, which is about every 4 years, that reward gets cut in half. -345 +350 00:22:56,860 --> 00:23:00,140 So right now, the reward is 12.5 Bitcoin per block. -346 +351 00:23:00,720 --> 00:23:04,623 And because this reward decreases geometrically over time, -347 +352 00:23:04,623 --> 00:23:09,320 it means there will never be more than 21 million Bitcoin in existence. -348 +353 00:23:10,280 --> 00:23:13,280 However, this doesn't mean miners will stop earning money. -349 +354 00:23:13,820 --> 00:23:17,940 In addition to the block reward, miners can also pick up transaction fees. -350 +355 00:23:18,520 --> 00:23:21,494 The way this works is that whenever you make a payment, -351 +356 00:23:21,494 --> 00:23:24,681 you can purely optionally include a transaction fee with it -352 +357 00:23:24,681 --> 00:23:28,240 that will go to the miner of whichever block includes that payment. -353 +358 00:23:29,020 --> 00:23:32,692 The reason you might do that is to incentivize miners to actually -354 +359 00:23:32,692 --> 00:23:35,920 include the transaction you broadcast into the next block. -355 +360 00:23:36,440 --> 00:23:41,313 You see, in Bitcoin, each block is limited to about 2400 transactions, -356 +361 00:23:41,313 --> 00:23:45,020 which many critics argue is unnecessarily restrictive. -357 +362 00:23:45,860 --> 00:23:51,294 For comparison, Visa processes an average of about 1700 transactions per second, -358 +363 00:23:51,294 --> 00:23:55,320 and they're capable of handling more than 24,000 per second. -359 +364 00:23:56,020 --> 00:24:00,899 This comparatively slow processing on Bitcoin makes for higher transaction fees, -360 +365 00:24:00,899 --> 00:24:06,200 since that's what determines which transactions miners choose to include in a new block. -361 +366 00:24:07,820 --> 00:24:11,500 All of this is far from a comprehensive coverage of cryptocurrencies. -362 +367 00:24:12,160 --> 00:24:16,180 There are still many nuances and alternate design choices that I haven't even touched. -363 +368 00:24:16,640 --> 00:24:20,402 But my hope is that this can provide a stable WaitButWhy-style tree-trunk of -364 +369 00:24:20,402 --> 00:24:24,360 understanding for anyone looking to add a few more branches with further reading. -365 +370 00:24:25,180 --> 00:24:28,968 Like I said at the start, one of the motives behind this is that a lot of money has -366 +371 00:24:28,968 --> 00:24:32,712 started flowing towards cryptocurrencies, and even though I don't want to make any -367 +372 00:24:32,712 --> 00:24:35,147 claims about whether that's a good or bad investment, -368 +373 00:24:35,147 --> 00:24:38,981 I really do think it's healthy for people getting into the game to at least know the -369 +374 00:24:38,981 --> 00:24:40,380 fundamentals of the technology. -370 +375 00:24:41,340 --> 00:24:45,420 As always, my sincerest thanks to those of you making this channel possible on Patreon. -371 +376 00:24:46,080 --> 00:24:49,036 I understand that not everyone is in a position to contribute, -372 +377 00:24:49,036 --> 00:24:51,242 but if you're still interested in helping out, -373 +378 00:24:51,242 --> 00:24:54,668 one of the best ways to do that is simply to share videos that you think -374 +379 00:24:54,668 --> 00:24:56,640 might be interesting or helpful to others. -375 +380 00:24:57,280 --> 00:24:59,320 I know you know that, but it really does help. diff --git a/2017/bitcoin/english/sentence_timings.json b/2017/bitcoin/english/sentence_timings.json index 1351dc26a..34460e5cd 100644 --- a/2017/bitcoin/english/sentence_timings.json +++ b/2017/bitcoin/english/sentence_timings.json @@ -410,7 +410,7 @@ 631.96 ], [ - "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", 632.66, 643.42 ], @@ -425,12 +425,12 @@ 652.74 ], [ - "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", 653.44, 661.68 ], [ - "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", 662.34, 667.2 ], @@ -500,7 +500,7 @@ 749.94 ], [ - "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", 750.8, 760.62 ], @@ -535,12 +535,12 @@ 800.66 ], [ - "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", 801.26, 814.64 ], [ - "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", 815.7, 823.9 ], @@ -660,9 +660,9 @@ 992.3 ], [ - "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a transaction is only considered valid when it's signed by the sender,", 993.14, - 1005.5 + 1005.9 ], [ "A block is only considered valid if it has a proof of work.", @@ -840,7 +840,7 @@ 1322.58 ], [ - "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", 1323.22, 1331.9 ], @@ -865,7 +865,7 @@ 1355.74 ], [ - "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", 1356.14, 1361.42 ], diff --git a/2017/bitcoin/english/transcript.txt b/2017/bitcoin/english/transcript.txt index 0642cd7a3..b5e8fbdb0 100644 --- a/2017/bitcoin/english/transcript.txt +++ b/2017/bitcoin/english/transcript.txt @@ -80,11 +80,11 @@ But before that, there's actually an even more significant difference between ou So far, I've said that this ledger is in some public place, like a website where anyone can add new lines. But that would require trusting a central location, namely, who hosts the website, who controls the rules of adding new lines. To remove that bit of trust, we'll have everybody keep their own copy of the ledger. -Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers. +Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers. But unless you do something more, this system is absurdly bad. How could you get everyone to agree on what the right ledger is? -When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction? -That he'll be able to later on go to Charlie and use those same $10 to make a transaction? +When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction? +That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction? Really, imagine yourself just listening to transactions being broadcast. How can you be sure that everyone else is recording the same transactions and in the same order? This is really the heart of the issue. @@ -98,15 +98,15 @@ The general idea that we'll build to is that if you use computational work as a Again, I'll remind you that this is getting well into the weeds beyond what anyone would need to know just to use a currency like this. But it's a really cool idea, and if you understand it, you understand the heart of Bitcoin and other cryptocurrencies. So first things first, what's a hash function? -The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits. +The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits. This output is called the hash or digest of the message, and the intent is that it looks random. It's not random, it always gives the same output for a given input. But the idea is that if you slightly change the input, maybe editing just one of the characters, the resulting hash changes completely. In fact, for the hash function I'm showing here, called SHA256, the way the output changes as you slightly change that input is entirely unpredictable. You see, this is not just any hash function, it's a cryptographic hash function. That means it's infeasible to compute in the reverse direction. -If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check. -And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video. +If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check. +And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video. I actually had way too much fun writing that thing. You might think that if you just really dig into the details of how exactly this function works, you could reverse engineer the appropriate input without having to guess and check. But no one has ever figured out a way to do that. @@ -130,7 +130,7 @@ Everyone is there broadcasting transactions and we want a way for them to agree As I mentioned, the idea behind the original Bitcoin paper is to have everyone trust whichever ledger has the most work put into it. The way this works is to first organize a given ledger into blocks, where each block consists of a list of transactions together with a proof of work. That is, a special number so that the hash of the whole block starts with a bunch of zeros. -For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change. +For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a transaction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work. Also, to make sure there's a standard order to these blocks, we'll make it so that a block has to contain the hash of the previous block at its header. That way, if you were to go back and change any one of the blocks, or to swap the order of two blocks, it would change the block that comes after it, which changes the block's hash, which changes the one that comes after it, and so on. @@ -166,12 +166,12 @@ And with that, we've hit all the main ideas. This distributed ledger system based on a proof of work is more or less how the Bitcoin protocol works, and how many other cryptocurrencies work. There's just a few details to clear up. Earlier I said that the proof of work might be to find a special number so that the hash of the block starts with 60 zeros. -Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block. +Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block. So as there are more and more miners added to the network, the challenge gets harder and harder in such a way that this miniature lottery only has about one winner every 10 minutes. Many newer cryptocurrencies have much shorter block times than that. And all of the money in Bitcoin ultimately comes from some block reward. In the beginning, these rewards were 50 Bitcoin per block. -There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain. +There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain. And if you look at the very first few blocks on the chain, they contain no transactions other than that 50 Bitcoin reward to the miner. But every 210,000 blocks, which is about every 4 years, that reward gets cut in half. So right now, the reward is 12.5 Bitcoin per block. diff --git a/2017/bitcoin/french/sentence_translations.json b/2017/bitcoin/french/sentence_translations.json index e112ccbb4..2d73c6a48 100644 --- a/2017/bitcoin/french/sentence_translations.json +++ b/2017/bitcoin/french/sentence_translations.json @@ -734,7 +734,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "Ensuite, lorsque tu veux effectuer une transaction, par exemple Alice paie 100 dollars à Bob, tu la diffuses dans le monde entier pour que les gens l'entendent et l'enregistrent dans leurs propres registres privés.", "model": "DeepL", "from_community_srt": "Alors lorsque vous voulez faire une transaction comme Alice paie Bob 100 LD. Ce que vous faites, c'est diffuser cela dans le monde entier pour que les gens entendent et enregistrent sur leur propre livre de compte privé.", @@ -760,7 +760,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "Lorsque Bob reçoit une transaction, par exemple Alice verse 10 $ à Bob, comment peut-il être sûr que tous les autres ont reçu et croient cette même transaction ?", "model": "DeepL", "from_community_srt": "Comment pouvez vous obtenir le consensus sur ce que le bon livre de compte est ? Lorsque Bob reçoit une transaction comme Alice paie Bob 10 LD. Comment pouvez-vous être sûr que personne d'autre reçoive et croit cette même transaction ?", @@ -769,7 +769,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "Qu'il pourra plus tard se rendre à Charlie et utiliser ces mêmes 10 $ pour effectuer une transaction ?", "model": "DeepL", "from_community_srt": "Qu'il sera capable d'aller voir Charlie plus tard et d'utiliser ces mêmes 10 LD pour faire une transaction.", @@ -894,7 +894,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "Les entrées de l'une de ces fonctions peuvent être n'importe quel type de message ou de fichier, cela ressemble vraiment à 256 bits.", "model": "DeepL", "from_community_srt": "Qu'est-ce qu'une fonction de hash ? Les entrées pour ce type de fonctions peuvent être n'importe quel sorte de message ou fichier. Cela importe peu. Et la sortie est une chaîne de bits avec une sorte de longueur fixe comme 256 bits.", @@ -957,7 +957,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "Si je te montre une chaîne de 1 et de 0 et que je te demande de trouver une entrée pour le hachage SHA256, tu n'auras pas de meilleure méthode que de deviner et de vérifier.", "model": "DeepL", "from_community_srt": "Si je vous montre une chaîne de 1 et de 0 et que je vous demande de trouver une entrée de sorte que le hash SHA256 de cette entrée donne exactement cette chaîne de bits vous n'auriez pas de meilleure méthode que de juste deviner et vérifier.", @@ -966,7 +966,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "Et encore une fois, si tu veux avoir une idée de la quantité de calcul nécessaire pour faire 256 suppositions, jette un coup d'œil à la vidéo du supplément.", "model": "DeepL", "from_community_srt": "Et de nouveau, si vous voulez sentir quelle puissance de calcul serait nécessaire pour aller à travers les 2 puissances 256 devinettes juste jetez un oeil à la vidéo supplémentaire.", @@ -1180,7 +1180,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "Pour l'instant, disons qu'il doit commencer par 60 zéros, mais nous reviendrons plus tard sur une façon plus systématique que tu pourrais vouloir modifier.", "model": "DeepL", "from_community_srt": "Pour le moment, disons qu'il doit commencer avec 60 zéros. Mais plus tard, nous reviendrons de manière plus systématique, vous voudrez peut-être choisir ce nombre. De la même manière qu'une transaction n'est considérée comme valide que lorsqu'elle est signée par l'expéditeur,", @@ -1189,7 +1189,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "Un bloc n'est considéré comme valide que s'il possède une preuve de travail.", "model": "DeepL", "from_community_srt": "un bloc est seulement considéré comme valide s'il a une preuve de travail.", @@ -1504,7 +1504,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "Eh bien, la façon dont le protocole Bitcoin proprement dit fonctionne est de changer périodiquement ce nombre de zéros de façon à ce qu'il faille 10 minutes pour trouver un nouveau bloc.", "model": "DeepL", "from_community_srt": "et bien, la façon dont le vrai protocole Bitcoin fonctionne est que ce nombre de zéros change périodiquement de sorte que cela devrait prendre en moyenne 10 minutes pour trouver un nouveau bloc.", @@ -1549,7 +1549,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "Il existe un excellent site Web appelé Block Explorer qui permet de consulter facilement la chaîne de blocs Bitcoin.", "model": "DeepL", "from_community_srt": "ces récompenses étaient de 50 bitcoins par bloc. Il y a en fait un super site web sur lequel vous pouvez aller appelé \"Block Explorer\" qui rend les choses faciles pour parcourir la blockchain Bitcoin.", diff --git a/2017/bitcoin/german/sentence_translations.json b/2017/bitcoin/german/sentence_translations.json index 85dfca709..2a103aecf 100644 --- a/2017/bitcoin/german/sentence_translations.json +++ b/2017/bitcoin/german/sentence_translations.json @@ -735,7 +735,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "Wenn du dann eine Transaktion durchführst, z.B. Alice zahlt Bob 100 Dollar, sendest du das in die Welt hinaus, damit die Leute es hören und in ihren eigenen privaten Ledgern aufzeichnen können.", "model": "DeepL", "from_community_srt": "Wenn du dann eine Transaktion durchführen möchtest, beispielsweise \"Alice zahlt Bob 100 LD.\", dann sendest du diese Nachricht in die Welt hinaus, damit die Leute sie hören und auf ihren eigenen, privaten Bestandsbüchern vermerken.", @@ -762,7 +762,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "Wenn Bob eine Transaktion erhält, z. B. wenn Alice ihm 10 Dollar zahlt, wie kann er dann sicher sein, dass alle anderen dieselbe Transaktion erhalten haben und daran glauben?", "model": "DeepL", "from_community_srt": "welches das richtige Bestandsbuch ist? Wenn Bob eine Zahlung empfängt, \"Alice zahlt Bob 10 LD.\", wie kann er dann sicher sein, dass alle anderen diese Transaktion empfangen haben und sie für glaubwürdig halten? Woher weiß er sicher,", @@ -771,7 +771,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "Dass er später zu Charlie gehen und die gleichen 10 Dollar für eine Transaktion verwenden kann?", "model": "DeepL", "from_community_srt": "dass er später zu Charlie gehen kann,", @@ -895,7 +895,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "Die Eingaben für eine dieser Funktionen können jede Art von Nachricht oder Datei sein, es sieht wirklich nach 256 Bit aus.", "model": "DeepL", "from_community_srt": "Also eins nach dem anderen – was ist eine Hashfunktion? Die Eingabe für eine dieser Funktionen kann irgendeine Art Nachricht oder Datei sein, das spielt wirklich keine große Rolle. Die Ausgabe ist eine Bitfolge mit einer bestimmten Länge, zum Beispiel 256 Bits.", @@ -958,7 +958,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "Wenn ich dir eine Reihe von 1en und 0en zeige und dich bitte, eine Eingabe für den SHA256-Hash zu finden, wirst du keine bessere Methode haben, als einfach zu raten und zu prüfen.", "model": "DeepL", "from_community_srt": "Wenn ich dir eine Folge Nullen und Einsen zeige, und dich auffordere, eine Eingabe zu finden, mit der SHA256 genau diese Folge Nullen und Einsen ausgibt. Dann hast du keine bessere Möglichkeit als einfach zu raten und zu überprüfen.", @@ -967,7 +967,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "Und wenn du ein Gefühl dafür bekommen willst, wie viel Rechenarbeit nötig wäre, um 256 Vermutungen durchzugehen, dann schau dir das Zusatzvideo an.", "model": "DeepL", "from_community_srt": "Und wie gesagt, wenn du ein Gefühl dafür entwickeln möchtest, wieviel rechnerische Arbeit vonnöten ist um 2 hoch 256 Vermutungen zu überprüfen, dann schau dir einfach das Zusatzvideo an.", @@ -1181,7 +1181,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "Für den Moment nehmen wir an, dass sie mit 60 Nullen beginnen muss, aber später werden wir auf eine systematischere Art und Weise zurückkommen, die du vielleicht ändern möchtest.", "model": "DeepL", "from_community_srt": "Lass uns einfach mal festlegen, dass der Hashwert mit…60 Nullen beginnen muss. Wir kommen später nochmal zu einem systematischeren Weg die Anzahl der Nullen auszuwählen. Genau so wie eine Transaktion nur gültig ist,", @@ -1190,7 +1190,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "Ein Block wird nur dann als gültig angesehen, wenn er einen Arbeitsnachweis hat.", "model": "DeepL", "from_community_srt": "wenn sie vom Sender unterzeichnet wurde, so ist auch ein Block nur gültig, wenn er einen Ausführungsnachweis hat.", @@ -1505,7 +1505,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "Das Bitcoin-Protokoll funktioniert so, dass die Anzahl der Nullen regelmäßig geändert wird, so dass es 10 Minuten dauert, einen neuen Block zu finden.", "model": "DeepL", "from_community_srt": "Eigentlich funktioniert es so, dass die Anzahl dieser Nullen regelmäßig geändert wird, so dass im Durchschnitt alle 10 Minuten ein neuer Block gefunden wird.", @@ -1550,7 +1550,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "Es gibt eine großartige Website namens Block Explorer, die es einfach macht, die Bitcoin-Blockchain zu durchsuchen.", "model": "DeepL", "from_community_srt": "Es gibt eine gute Website auf die du gehen kannst, der \"Block Explorer\", auf der du dir die Bitcoin-Blockkette ansehen kannst.", diff --git a/2017/bitcoin/greek/sentence_translations.json b/2017/bitcoin/greek/sentence_translations.json index c0e0f7f6f..f6922487f 100644 --- a/2017/bitcoin/greek/sentence_translations.json +++ b/2017/bitcoin/greek/sentence_translations.json @@ -654,7 +654,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "", "from_community_srt": "Στη συνέχεια, όταν θέλετε να κάνετε μια συναλλαγή, όπως η Alice πληρώνει τον Bob 100 LD, αυτό που κάνουμε είναι να το εκπέψουμε σε όλους τους άλλους τους ανθρώπους ώστε να το \"ακούσουν\" και να το καταγράψουν στο δικό τους ιδιωτικό βιβλίο Αλλά,", "n_reviews": 0, @@ -678,7 +678,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "", "from_community_srt": "όπως η Alice πληρώνει τον Bob 10 LD, πώς μπορεί να είναι σίγουρος ότι όλοι οι άλλοι έλαβαν και πίστεψαν ότι η ίδια συναλλαγή", "n_reviews": 0, @@ -686,7 +686,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "", "from_community_srt": "ότι θα είναι σε θέση να πάει αργότερα στον Charlie και να χρησιμοποιήσει τα ίδια αυτά 10 LD για να κάνει μια συναλλαγή;", "n_reviews": 0, @@ -797,7 +797,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "", "from_community_srt": "τι είναι μία συνάρτηση κατακερματισμού; Οι είσοδοι για μία από αυτές τις συναρτήσεις μπορεί να είναι οποιοδήποτε είδος μηνύματος ή αρχείου, δεν έχει σημασία. Και η έξοδος είναι μια συμβολοσειρά δυαδικών ψηφίων ένα είδος σταθερού μήκους, όπως 256 bits.", "n_reviews": 0, @@ -853,7 +853,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "", "from_community_srt": "Αν σας δείξω κάποια σειρά από 1 και 0, και σας ζητήσω να βρείτε μία είσοδο έτσι ώστε η \"σύνοψη\" SHA256 της εν λόγω εισόδου να δίνει αυτή την συγκεκριμένη σειρά από bits, δεν θα έχετε καμία καλύτερη μέθοδο από το να μαντεύετε και να ελέγχετε.", "n_reviews": 0, @@ -861,7 +861,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "", "from_community_srt": "Και πάλι, αν θέλετε να καταλάβετε το πόσοι υπολογισμοί θα χρειαστούν για όλες τις 2^256 μαντεψιές, απλά ρίξτε μια ματιά στο συμπληρωματικό βίντεο.", "n_reviews": 0, @@ -1052,7 +1052,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "", "from_community_srt": "Προς το παρόν, ας πούμε ότι πρέπει να ξεκινήσει με ... 60 μηδενικά, αλλά αργότερα θα επιστρέψουμε σε ένα πιο συστηματικό τρόπο για την επιλογή του αριθμού αυτού. Με τον ίδιο τρόπο που μια συναλλαγή είναι μόνο θεωρείται έγκυρη όταν είναι υπογεγραμμένη από τον αποστολέα,", "n_reviews": 0, @@ -1060,7 +1060,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "", "from_community_srt": "ένα μπλοκ θεωρείται έγκυρο μόνο αν έχει μια απόδειξη δουλειάς.", "n_reviews": 0, @@ -1340,7 +1340,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "", "from_community_srt": "Λοιπόν, ο τρόπος που το πραγματικό πρωτόκολλο Bitcoin δουλεύει είναι να αλλάξει περιοδικά ο αριθμός των μηδενικών έτσι ώστε να πρέπει να παίρνει κατά μέσο όρο 10 λεπτά για να βρεθεί ένα νέο μπλοκ.", "n_reviews": 0, @@ -1380,7 +1380,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "", "from_community_srt": "Υπάρχει πραγματικά μια πολύ καλή ιστοσελίδα μπορείτε να πάτε που ονομάζεται «Block Explorer» που καθιστά εύκολο να κοιτάξετε μέσα στο blockchain του Bitcoin.", "n_reviews": 0, diff --git a/2017/bitcoin/hebrew/sentence_translations.json b/2017/bitcoin/hebrew/sentence_translations.json index 6e7f45bcd..b7e1a0e5f 100644 --- a/2017/bitcoin/hebrew/sentence_translations.json +++ b/2017/bitcoin/hebrew/sentence_translations.json @@ -735,7 +735,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "ואז כשאתה רוצה לבצע עסקה, כמו שאליס משלמת לבוב 100 דולר, אתה משדר את זה לעולם כדי שאנשים ישמעו ויקליטו בפנקסים הפרטיים שלהם.", "model": "google_nmt", "from_community_srt": "ואז, כאשר אתם רוצים לבצע עסקה, כמו אליס משלמת לבוב מאה LD, מה שאתם צריכים לעשות זה לשדר את העסקה הזו לעולם כדי שאנשים אחרים יוכלו לשמוע ולתעד אותה בספרי החשבונות הפרטיים שלהם.", @@ -762,7 +762,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "כשבוב מקבל עסקה, כמו שאליס משלמת לבוב 10 דולר, איך הוא יכול להיות בטוח שכולם קיבלו ומאמינים באותה עסקה?", "model": "google_nmt", "from_community_srt": "כמו אליס משלמת עשרה LD לבוב, איך הוא יכול להיות בטוח שכולם קיבלו את המידע אודות אותה העסקה כפי שהיא נכתבה", @@ -771,7 +771,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "שהוא יוכל ללכת מאוחר יותר לצ'רלי ולהשתמש באותם 10 דולר כדי לבצע עסקה?", "model": "google_nmt", "from_community_srt": "כדי שהוא יוכל, בהמשך,", @@ -896,7 +896,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "הכניסות לאחת מהפונקציות הללו יכולות להיות כל סוג של הודעה או קובץ, זה באמת נראה כמו 256 סיביות.", "model": "google_nmt", "from_community_srt": "מהי פונקציית hash? הקלט לפונקציה כזו יכול להיות כל הודעה או קובץ, זה באמת לא משנה. והפלט הוא מחרוזת של ביטים עם איזה אורך קבוע, נניח 256 סיביות.", @@ -959,7 +959,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "אם אני אראה לך מחרוזת של 1 ו-0 ואבקש ממך למצוא קלט ל-hash SHA256, לא תהיה לך שיטה טובה יותר מאשר פשוט לנחש ולבדוק.", "model": "google_nmt", "from_community_srt": "אם אני מראה לכם איזו מחרוזת של אחדות ואפסים, ואבקש מכם למצוא קלט כך שתוצאת ה-SHA256 של הקלט תהיה המחרוזת המדויקת הזו של ביטים, לא תהיה לכם שום שיטה טובה יותר למצוא קלט כזה מאשר סתם לנחש ולבדוק.", @@ -968,7 +968,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "ושוב, אם אתה רוצה להרגיש כמה חישוב יידרש כדי לעבור 256 ניחושים, פשוט תסתכל בסרטון התוסף.", "model": "google_nmt", "from_community_srt": "ושוב, אם אתם רוצים לקבל תחושה של כמה חישובים דרושים כדי לעבור על שתיים בחזקת מאתיים חמישים ושישה ניחושים, פשוט תסתכלו על הוידאו הנוסף.", @@ -1183,7 +1183,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "כרגע, נניח שצריך להתחיל ב-60 אפסים, אבל בהמשך נחזור לדרך שיטתית יותר שאולי תרצה לשנות.", "model": "google_nmt", "from_community_srt": "לעת עתה, נניח שיש להתחיל עם ... שישים אפסים, אך מאוחר יותר נחזור באופן שיטתי יותר לאיך כדאי לבחור את המספר הזה.", @@ -1192,7 +1192,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "בלוק נחשב תקף רק אם יש לו הוכחת עבודה.", "model": "google_nmt", "from_community_srt": "באותו אופן שבו עסקה נחשבת תקפה רק אם מצורפת לה חתימה של השולח, בלוק נחשב תקף רק אם יש לו הוכחה של עבודה.", @@ -1507,7 +1507,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "ובכן, הדרך בה פועל פרוטוקול הביטקוין בפועל היא לשנות מעת לעת את מספר האפסים כך שייקח 10 דקות למצוא בלוק חדש.", "model": "google_nmt", "from_community_srt": "ובכן, פרוטוקול הביטקוין בפועל משנה מעת לעת את המספר הזה של אפסים כך שיקח עשר דקות בממוצע למצוא בלוק חדש.", @@ -1551,7 +1551,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "יש אתר נהדר בשם Block Explorer שמקל על העיון ב-Blockchain של ביטקוין.", "model": "google_nmt", "from_community_srt": "יש למעשה אתר אינטרנט גדול שאתם יכולים ללכת אליו שנקרא \"Block Explorer\" שמקל על החיפוש של בלוקצ'יינס של ביטקוין", diff --git a/2017/bitcoin/hindi/sentence_translations.json b/2017/bitcoin/hindi/sentence_translations.json index 2f89a4a70..2b6b2b634 100644 --- a/2017/bitcoin/hindi/sentence_translations.json +++ b/2017/bitcoin/hindi/sentence_translations.json @@ -224,7 +224,7 @@ "end": 228.54 }, { - "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves she has seen it and approved of it, and it should be infeasible for anyone else to forge that signature.", + "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves that she has seen it and that she's approved of it, and it should be infeasible for anyone else to forge that signature.", "translatedText": "हस्तलिखित हस्ताक्षरों की तरह, यहां विचार यह है कि ऐलिस को उस लेनदेन के आगे कुछ जोड़ने में सक्षम होना चाहिए जो साबित करता है कि उसने इसे देखा है और इसे मंजूरी दे दी है, और किसी और के लिए उस हस्ताक्षर को बनाना असंभव होना चाहिए।", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 554.26 }, { - "input": "You are free to exchange ledger dollars for real US dollars.", + "input": "You are of course free to exchange ledger dollars for real US dollars.", "translatedText": "आप वास्तविक अमेरिकी डॉलर के लिए लेजर डॉलर का आदान-प्रदान करने के लिए स्वतंत्र हैं।", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 896.7 }, { - "input": "For a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 230, which is about 1 in a billion.", + "input": "Well, for a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 2 to the 30, which is about 1 in a billion.", "translatedText": "एक यादृच्छिक संदेश के लिए, संभावना है कि एक हैश 30 लगातार शून्य से शुरू होता है 230 में 1 है, जो एक अरब में लगभग 1 है।", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 949.64 }, { - "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the list starts with 30 zeros.", + "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the altered list together with this new number starts with 30 zeros.", "translatedText": "तो आपको काम का एक नया प्रमाण, एक नई संख्या खोजने के लिए अन्य अरब अनुमानों से गुजरना होगा जो इसे बनाता है ताकि सूची का हैश 30 शून्य से शुरू हो।", "model": "google_nmt", "n_reviews": 0, @@ -1232,7 +1232,7 @@ "end": 1182.64 }, { - "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's really helpful to walk through exactly what it would take to fool someone using this system.", + "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's actually really helpful to walk through exactly what it would take to fool someone using this system.", "translatedText": "यह देखने के लिए कि यह एक भरोसेमंद प्रणाली क्यों बनती है, और यह समझने के लिए कि आपको किस बिंदु पर भरोसा करना चाहिए कि भुगतान वैध है, इस प्रणाली का उपयोग करने वाले किसी व्यक्ति को बेवकूफ बनाने के लिए वास्तव में क्या करना होगा, यह जानना वास्तव में मददगार है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/bitcoin/hungarian/sentence_translations.json b/2017/bitcoin/hungarian/sentence_translations.json index b2139b336..d5ffa99b5 100644 --- a/2017/bitcoin/hungarian/sentence_translations.json +++ b/2017/bitcoin/hungarian/sentence_translations.json @@ -735,7 +735,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "Aztán amikor tranzakciót akarsz végrehajtani, például Alice 100 dollárt fizet Bobnak, akkor ezt a világba küldöd, hogy az emberek hallhassák és rögzíthessék a saját magánkönyvükben.", "model": "DeepL", "from_community_srt": "Azután amikor egy tranzakciót szeretnél csinálni, például \"Alice fizet Bobnak 100LD-t\", akkor azt közvetíted mindenkinek a világban, hogy az emberek az alapján elmenthessék a saját privát főkönyvjükben.", @@ -762,7 +762,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "Amikor Bob kap egy tranzakciót, például Alice 10 dollárt fizet Bobnak, hogyan lehet biztos abban, hogy mindenki más is megkapta és elhiszi ugyanezt a tranzakciót?", "model": "DeepL", "from_community_srt": "hogy melyik a helyes főkönyv? Amikor Bobnak küldenek egy tranzakciót, például \"Alice fizet Bobnak 10LD-t\", hogyan győződhet meg arról, hogy mindenki más is megkapta és elhiszi ugyanazt a tranzakciót, ha majd később odamegy Charlie-hoz,", @@ -771,7 +771,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "Hogy később képes lesz elmenni Charlie-hoz, és ugyanazzal a 10 dollárral tranzakciót kötni?", "model": "DeepL", "from_community_srt": "hogy felasználja ugyanazt a 10LD-t egy tranzakcióhoz? De tényleg,", @@ -896,7 +896,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "Az egyik ilyen függvény bemenete bármilyen üzenet vagy fájl lehet, valóban úgy néz ki, mintha 256 bit lenne.", "model": "DeepL", "from_community_srt": "mi az a hasító függvény (*hash function*)? Ezeknek a függvényeknek a bemente egy tetszőleges üzenet vagy fájl, ez valójában mindegy. A kimenete pedig bitek egy meghatározott hosszúságú láncolata, pl. 256 bit.", @@ -959,7 +959,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "Ha mutatok neked egy 1-ekből és 0-kból álló sorozatot, és megkérlek, hogy találd meg az SHA256 hash bemenetét, nem lesz jobb módszered, minthogy csak találgatsz és ellenőrzöl.", "model": "DeepL", "from_community_srt": "Hogy ha mutatok neked néhány egyes és nullás láncolatát, és megkérlek arra hogy keresd meg azt a bemenetet, amely bemenetre a SHA256 hash-e pontosan ugyanazt a bitek láncolatát adja, akkor nem lenne jobb módszered annál, mint hogy találgass és leellenőrizd.", @@ -968,7 +968,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "És még egyszer, ha érezni akarod, hogy mennyi számításra lenne szükség a 256 találgatáshoz, csak nézd meg a kiegészítő videót.", "model": "DeepL", "from_community_srt": "Ha szeretnéd ismét átérezni, hogy mennyi számításra lenne szükséged ahhoz, hogy végigmenj 2^256-ik találgatáson, akkor csak nézd meg az erről szóló videónkat.", @@ -1183,7 +1183,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "Egyelőre mondjuk, hogy 60 nullával kell kezdődnie, de később visszatérünk egy szisztematikusabb módra, amit esetleg módosítani szeretne.", "model": "DeepL", "from_community_srt": "Egy pillanatra mondjuk azt, hogy 60 darab nullással kell kezdődnie, de később majd vissza fogunk térni ennek a számnak a kiválasztásának egy sokkal szisztematikusabb módjára. Ugyanazon a módon ahogy egy tranzakciót csak akkor tekintük érvényesnek,", @@ -1192,7 +1192,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "Egy blokk csak akkor tekinthető érvényesnek, ha van munkabizonylata.", "model": "DeepL", "from_community_srt": "hogy ha az alá van írva a küldő fél által, egy blokk csak akkor tekinthető érvényesnek, hogy ha a belefektett munkának van bizonyítéka, azaz, hogy ha érvényes \"proof of work\"-je.", @@ -1507,7 +1507,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "Nos, a Bitcoin protokoll úgy működik, hogy a nullák számát rendszeresen megváltoztatja, így egy új blokk megtalálása 10 percet vesz igénybe.", "model": "DeepL", "from_community_srt": "Nos, a Bitcoin protokoll valójában úgy működik, hogy időközönként megváltoztatják a nullák számát, hogy így átlagosan 10 perc szükségeltessen ahhoz, hogy megtaláljanak egy új blokkot.", @@ -1552,7 +1552,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "Van egy nagyszerű weboldal, a Block Explorer, amely megkönnyíti a Bitcoin blokkláncának áttekintését.", "model": "DeepL", "from_community_srt": "Igazából van egy nagyszerű weboldal amit \"Block Explorer\"-nek hívnak, ami segít átlátni a Bitcoin blokkláncot.", diff --git a/2017/bitcoin/indonesian/sentence_translations.json b/2017/bitcoin/indonesian/sentence_translations.json index ecc870450..3b611cc22 100644 --- a/2017/bitcoin/indonesian/sentence_translations.json +++ b/2017/bitcoin/indonesian/sentence_translations.json @@ -735,7 +735,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "Kemudian ketika Anda ingin melakukan transaksi, misalnya Alice membayar Bob $100, Anda menyiarkannya ke seluruh dunia untuk didengar dan dicatat oleh orang-orang di buku besar pribadi mereka.", "model": "DeepL", "from_community_srt": "Kemudian ketika Anda ingin melakukan transaksi, seperti Alice membayar Bob 100 LD, apa yang Anda lakukan adalah menyiarkannya keluar ke dunia agar orang bisa mendengar dan mencatat di buku besar pribadi mereka sendiri.", @@ -762,7 +762,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "Ketika Bob menerima transaksi, misalnya Alice membayar Bob $10, bagaimana dia bisa yakin bahwa semua orang lain menerima dan mempercayai transaksi yang sama?", "model": "DeepL", "from_community_srt": "seperti Alice membayar Bob 10 LD, bagaimana dia bisa memastikan bahwa semua orang menerima dan percaya bahwa transaksi yang sama", @@ -771,7 +771,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "Bahwa dia nantinya bisa pergi ke Charlie dan menggunakan $10 yang sama untuk melakukan transaksi?", "model": "DeepL", "from_community_srt": "bahwa dia akan bisa kemudian pergi ke Charlie dan menggunakan 10 LD yang sama untuk melakukan transaksi?", @@ -896,7 +896,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "Input untuk salah satu fungsi ini dapat berupa pesan atau file apa pun, benar-benar terlihat seperti 256 bit.", "model": "DeepL", "from_community_srt": "apa itu fungsi hash? data masukan ke salah satu fungsi ini bisa berupa pesan atau file apa pun, itu tidak masalah Dan hasilnya adalah untaian beberapa bit dengan panjang tetap, seperti 256 bit.", @@ -959,7 +959,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "Jika saya menunjukkan kepada Anda beberapa string 1 dan 0 dan meminta Anda untuk menemukan input ke hash SHA256, Anda tidak akan memiliki metode yang lebih baik daripada hanya menebak dan memeriksa.", "model": "DeepL", "from_community_srt": "Jika saya menunjukkan beberapa string angka dan angka nol, dan meminta Anda untuk mencari tahu data masukan sehingga hash SHA256 dari masukan tersebut memberikan rangkaian bit yang tepat ini, Anda tidak akan memiliki metode yang lebih baik daripada hanya menebak dan memeriksa.", @@ -968,7 +968,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "Dan sekali lagi, jika Anda ingin merasakan berapa banyak komputasi yang diperlukan untuk melakukan 256 tebakan, lihat saja video suplemennya.", "model": "DeepL", "from_community_srt": "Dan lagi, jika Anda ingin merasakan berapa banyak perhitungan yang dibutuhkan untuk melewati 2^256 terkaan, coba lihat saja video pelengkapnya.", @@ -1183,7 +1183,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "Untuk saat ini, katakanlah harus dimulai dengan 60 angka nol, tetapi nanti kita akan kembali ke cara yang lebih sistematis yang mungkin ingin Anda ubah.", "model": "DeepL", "from_community_srt": "Untuk saat ini, katakanlah bahwa itu harus dimulai dengan ... 60 angka nol, Tapi kemudian kita akan kembali ke cara yang lebih sistematis sehingga Anda mungkin ingin memilih nomor itu. Dengan cara yang sama bahwa transaksi hanya dianggap valid saat ditandatangani oleh pengirim,", @@ -1192,7 +1192,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "Sebuah blok hanya dianggap sah jika memiliki bukti kerja.", "model": "DeepL", "from_community_srt": "blok hanya dianggap sah jika memiliki bukti kerja.", @@ -1507,7 +1507,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "Cara kerja protokol Bitcoin yang sebenarnya adalah mengubah jumlah angka nol tersebut secara berkala sehingga dibutuhkan waktu 10 menit untuk menemukan blok baru.", "model": "DeepL", "from_community_srt": "Nah, cara kerja protokol Bitcoin sebenarnya adalah mengubah jumlah angka nol secara berkala sehingga harus memakan waktu rata-rata 10 menit untuk menemukan blok baru.", @@ -1552,7 +1552,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "Ada sebuah situs web hebat bernama Block Explorer yang memudahkan Anda untuk melihat-lihat blockchain Bitcoin.", "model": "DeepL", "from_community_srt": "Sebenarnya ada situs web hebat yang bisa Anda kunjungi \"Block Explorer\" yang membuatnya mudah untuk melihat melalui blokir Bitcoin.", diff --git a/2017/bitcoin/italian/sentence_translations.json b/2017/bitcoin/italian/sentence_translations.json index 8a838d5ba..48d1d47b8 100644 --- a/2017/bitcoin/italian/sentence_translations.json +++ b/2017/bitcoin/italian/sentence_translations.json @@ -734,7 +734,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "Poi, quando si vuole effettuare una transazione, ad esempio Alice paga 100 dollari a Bob, la si trasmette al mondo perché le persone la sentano e la registrino nei loro libri mastri privati.", "model": "DeepL", "from_community_srt": "Quindi quando vuoi fare una transazione come \"Alice paga a Bob 100 LD\", quello che fai è trasmetterlo al mondo così che le altre persone possano sentirlo e aggiungerlo al proprio libro mastro.", @@ -761,7 +761,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "Quando Bob riceve una transazione, ad esempio Alice paga a Bob 10 dollari, come può essere sicuro che tutti gli altri abbiano ricevuto e credano alla stessa transazione?", "model": "DeepL", "from_community_srt": "come \"Alice paga a Bob 10 LD\", come fa ad essere sicuro che tutti gli altri abbiano ricevuto e credano alla stessa transazione", @@ -770,7 +770,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "Che poi potrà andare da Charlie e usare quegli stessi 10 dollari per fare una transazione?", "model": "DeepL", "from_community_srt": "così che possa andare più tardi da Charlie e usare questi stessi 10 LD per fare una nuova transazione?", @@ -895,7 +895,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "Gli input per una di queste funzioni possono essere qualsiasi tipo di messaggio o file, in realtà sembra che si tratti di 256 bit.", "model": "DeepL", "from_community_srt": "cos'è una funzione crittografica di hash? Gli argomenti per una di queste funzioni possono essere ogni sorta di messaggi o file, è davvero irrilevante. E restituisce una stringa di bit di lunghezza fissa, ad esempio 256 bit.", @@ -958,7 +958,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "Se ti mostro una stringa di 1 e 0 e ti chiedo di trovare un input per l'hash SHA256, non avrai un metodo migliore che tirare a indovinare e controllare.", "model": "DeepL", "from_community_srt": "Se ti mostrassi una stringa di 1 e 0, e ti chiedessi di trovare un argomento per cui la funzione SHA256 restituisse esattamente quella stringa di bit, non avresti miglior sistema che tirare ad indovinare.", @@ -967,7 +967,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "E ancora, se vuoi farti un'idea della quantità di calcoli necessari per eseguire 256 ipotesi, dai un'occhiata al video del supplemento.", "model": "DeepL", "from_community_srt": "E di nuovo, se vuoi avere una sensazione di quanta computazione sarebbe necessaria per provare 2^256 possibili combinazioni, dai un'occhiata al video di supplemento.", @@ -1182,7 +1182,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "Per il momento diciamo che deve iniziare con 60 zeri, ma più avanti torneremo su un modo più sistematico in cui potresti voler cambiare.", "model": "DeepL", "from_community_srt": "Per il momento, diciamo che debba iniziare con... 60 zeri, ma più tardi torneremo su un modo più sistematico per scegliere questo numero.", @@ -1191,7 +1191,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "Un blocco è considerato valido solo se ha una prova di lavoro.", "model": "DeepL", "from_community_srt": "Allo stesso modo in cui una transazione è considerata valida solo se firmata dal mittente, un blocco è considerato valido solo se ha una \"proof of work\".", @@ -1506,7 +1506,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "Il protocollo Bitcoin funziona in modo da cambiare periodicamente il numero di zeri in modo da impiegare 10 minuti per trovare un nuovo blocco.", "model": "DeepL", "from_community_srt": "Ebbene, la maniera in cui il protocollo Bitcoin funziona è cambiando periodicamente il numero di zeri così che in media trovare un nuovo blocco richieda 10 minuti.", @@ -1551,7 +1551,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "Esiste un ottimo sito web chiamato Block Explorer che permette di consultare facilmente la blockchain di Bitcoin.", "model": "DeepL", "from_community_srt": "C'è un bel sito web che puoi visitare chiamato \"Block Explorer\" che rende semplice scorrere la \"blockchain\" del Bitcoin.", diff --git a/2017/bitcoin/japanese/sentence_translations.json b/2017/bitcoin/japanese/sentence_translations.json index e7460826e..47c351ade 100644 --- a/2017/bitcoin/japanese/sentence_translations.json +++ b/2017/bitcoin/japanese/sentence_translations.json @@ -250,7 +250,7 @@ "end": 228.54 }, { - "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves she has seen it and approved of it, and it should be infeasible for anyone else to forge that signature.", + "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves that she has seen it and that she's approved of it, and it should be infeasible for anyone else to forge that signature.", "translatedText": "手書きの署名と同 様に、ここでの考え方は、アリスがトランザクションを見て承認したことを証明する何かを そのトランザクションの隣に追加できるべきであり、その署名は他人が偽造できないようにす る必要があるということです。", "model": "google_nmt", "from_community_srt": "手で書く署名と同じく、 アリスは取引記録の横に何か付随するものを加える。 これによりアリスの意図を持って行われたことを証明する。 当然、誰もその署名を真似出来ないようにしなければらない。", @@ -617,7 +617,7 @@ "end": 554.26 }, { - "input": "You are free to exchange ledger dollars for real US dollars.", + "input": "You are of course free to exchange ledger dollars for real US dollars.", "translatedText": "元帳ド ルを実際の米ドルに自由に交換できます。", "model": "google_nmt", "from_community_srt": "もちろん元帳ドルと米ドルの両替は自由である。", @@ -1080,7 +1080,7 @@ "end": 896.7 }, { - "input": "For a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 230, which is about 1 in a billion.", + "input": "Well, for a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 2 to the 30, which is about 1 in a billion.", "translatedText": "ランダムなメッセージの場合、ハッシュが 3 0 個の連続するゼロで始まる確率は 230 分の 1、つまり約 10 億分の 1 です。", "model": "google_nmt", "from_community_srt": "記録はランダムだから、 30のゼロの列が始めに来る確率は、2の30乗分の1であり、 10億分の1に近似できる。", @@ -1152,7 +1152,7 @@ "end": 949.64 }, { - "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the list starts with 30 zeros.", + "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the altered list together with this new number starts with 30 zeros.", "translatedText": "したがって、新しいプルーフ オブ ワーク、つまりリストのハッシュが 30 個のゼロで始まるようにする新しい番号を見つけるには、さらに 10 億回の推測を行う必 要があります。", "model": "google_nmt", "from_community_srt": "こうなってしまえば、新たなプルーフ・オブ・ワークを見つけるために、 再び億単位の作業を行って、 ハッシュの始めが30のゼロになるようにしなければならない。", @@ -1375,7 +1375,7 @@ "end": 1182.64 }, { - "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's really helpful to walk through exactly what it would take to fool someone using this system.", + "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's actually really helpful to walk through exactly what it would take to fool someone using this system.", "translatedText": "これがなぜ信頼できるシステムになるのかを確認し、どの時点で支払いが正当であると信頼す べきかを理解するには、このシステムを使用して誰かをだますために何が必要になるかを正確 に説明することが非常に役立ちます。", "model": "google_nmt", "from_community_srt": "このシステム体系が信頼に値するものである事と、 支払いが正当と見なせるのはどのタイミングなのかを理解するために、 逆にこのシステムを使って悪事を働くことが出来るか試してみよう。", diff --git a/2017/bitcoin/korean/sentence_translations.json b/2017/bitcoin/korean/sentence_translations.json index 06bda0b48..564404689 100644 --- a/2017/bitcoin/korean/sentence_translations.json +++ b/2017/bitcoin/korean/sentence_translations.json @@ -735,7 +735,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "그런 다음 앨리스가 밥에게 100달러를 지불하는 것처럼 거래를 하고 싶을 때 사람들이 듣고 자신의 개인 장부에 기록할 수 있도록 전 세계로 방송합니다.", "model": "DeepL", "from_community_srt": "그리고 \"앨리스가 밥에게 100 장부달러 지불\"이라는 거래를 기록하려면 장부를 가지고 있는 모든 사람들에게 이 거래를 알려야 하지. 그래야 사람들이 그 거래 정보를 듣고 자기들이 각자 보유한 장부에 기록할 수 있을테니까.", @@ -761,7 +761,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "앨리스가 밥에게 10달러를 지불한 것처럼 밥이 거래를 수신할 때, 다른 모든 사람이 동일한 거래를 수신하고 믿었는지 어떻게 확신할 수 있을까요?", "model": "DeepL", "from_community_srt": "수많은 장부 중에서 어떤 장부가 올바른 장부인지 어떻게 알 수 있겠어? 밥이 \"앨리스가 밥에게 10 장부달러 지불\"이라는 거래 정보를 수신받았다면,", @@ -770,7 +770,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "나중에 찰리에게 가서 같은 10달러를 사용하여 거래를 할 수 있을까요?", "model": "DeepL", "from_community_srt": "다른 모든 사람들도 똑같은 거래 정보를 수신 받고 그 거래 정보를 신뢰해서, 밥이 앨리스에게 받은 그 10 장부달러를 찰리에게 지불할 수 있다는 것을 밥은 어떻게 확신할 수 있을까?", @@ -894,7 +894,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "이러한 함수 중 하나에 대한 입력은 모든 종류의 메시지 또는 파일일 수 있으며 실제로 256비트처럼 보입니다.", "model": "DeepL", "from_community_srt": "해쉬 함수가 뭘까? 어떤 종류의 메시지나 파일도 해쉬 함수의 입력값이 될 수 있어. 입력값의 종류는 중요하지 않아. 해쉬 함수의 결과값은 256비트처럼, 고정된 길이의 연속된 비트값이야.", @@ -957,7 +957,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "1과 0으로 이루어진 문자열을 보여주고 SHA256 해시에 대한 입력을 찾으라고 하면, 그냥 추측해서 확인하는 것 외에 더 좋은 방법은 없습니다.", "model": "DeepL", "from_community_srt": "내가 0과 1로된 숫자를 여러분에게 보여주고, \"SHA256함수로 계산해서 이런 해쉬값이 나오려면 입력값은 무엇일까요?\" 하고 묻는다면, 답을 찾는 방법은 그저 찍어서 맞추는 수 밖에 없어.", @@ -966,7 +966,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "256번의 추측을 거치는 데 얼마나 많은 계산이 필요한지 느껴보고 싶다면 보충 동영상을 살펴보세요.", "model": "DeepL", "from_community_srt": "그리고 그 찍을 수 있는 가지수가 2의 256승이나 돼. 2의 256승 개 중에서 여러분이 찍어서 맞출 수 있겠어? 2의 256이 얼마나 큰 수인지는(https://youtu.be/S9JGmA5_unY)를 보면 알 수 있을거야.", @@ -1180,7 +1180,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "지금은 0부터 60개로 시작해야 한다고 가정해 보지만 나중에 좀 더 체계적으로 변경할 수 있는 방법으로 다시 돌아가겠습니다.", "model": "DeepL", "from_community_srt": "먼저 블록의 해쉬값이 60자리의 0으로 시작해야 한다고 해보자. 나중에는 이 60이라는 값을 결정하는데도 어떤 체계가 있다는 것을 알게될 거야.", @@ -1189,7 +1189,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "블록은 작업 증명이 있는 경우에만 유효한 것으로 간주됩니다.", "model": "DeepL", "from_community_srt": "거래는 지불하는 송금자의 디지털 서명이 있어야만 유효한 거래로 인식되는 것처럼 블록도 마찬가지로 작업 증명이 있어야만 유효한 블록으로 인식될 수 있어.", @@ -1504,7 +1504,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "실제 비트코인 프로토콜이 작동하는 방식은 주기적으로 0의 수를 변경하여 새로운 블록을 찾는 데 10분이 걸리도록 하는 것입니다.", "model": "DeepL", "from_community_srt": "정확하게는 비트코인은 그 0의 갯수를 주기적으로 조절하는 규약을 가지고 있어. 그래서 새 블록을 생성하는데 드는 시간이 평균적으로 10분이 되도록 주기적으로 0의 갯수를 조절해.", @@ -1549,7 +1549,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "비트코인 블록체인을 쉽게 살펴볼 수 있는 블록 익스플로러라는 훌륭한 웹사이트가 있습니다.", "model": "DeepL", "from_community_srt": "\"Block Explorer\"라는 웹사이트에 가면 비트코인 블록체인 정보를 쉽게 살펴볼 수 있어.", diff --git a/2017/bitcoin/lithuanian/sentence_translations.json b/2017/bitcoin/lithuanian/sentence_translations.json index b005d5f75..6627b6c42 100644 --- a/2017/bitcoin/lithuanian/sentence_translations.json +++ b/2017/bitcoin/lithuanian/sentence_translations.json @@ -654,7 +654,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "", "from_community_srt": "Tada, kai tu nori padaryti transakciją, kaip Alisa sumoka Bobui 100 LD, ką Tu darai, tai transliuoji šitai į pasaulį, kad žmonės išgirstų ir įsirašytų į savo asmenines apskaitas.", "n_reviews": 0, @@ -678,7 +678,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "", "from_community_srt": "kuri apskaita yra teisinga? Kai Bobas gauna transakciją, kaip Alisa moka Bobui 10 LD, kaip jis gali būti tikras,", "n_reviews": 0, @@ -686,7 +686,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "", "from_community_srt": "kad visi kiti gavo ir pasitiki ta pačia transakcija, kad jis galės vėliau nueiti pas Čarlį ir panaudoti tuos pačius 10 LD,", "n_reviews": 0, @@ -797,7 +797,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "", "from_community_srt": "kas yra hešo funkcija? Įvestys vienai iš šių funkcijų gali būti bet kokio tipo žinutė arba failas, specifika tikrai nėra svarbi. O išvestis yra bitų eilutė su fiksuotu ilgiu, pavyzdžiui 256 bitai.", "n_reviews": 0, @@ -853,7 +853,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "", "from_community_srt": "Jei parodysiu tau kažkokią vienetų ir nulių eilutę ir paprašysiu tavęs surasti įvestį taip, kad tos įvesties SHA256 hešas duotų būtent šią bitų eilutę, tu neturėsi geresnio metodo nei tiesiog spėti ir patikrinti.", "n_reviews": 0, @@ -861,7 +861,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "", "from_community_srt": "Ir vėl, jei nori pajausti kiek skaičiavimo reikėtų, kad praeitum per 2^256 spėjimų, tiesiog pažiūrėk į papildinį video.", "n_reviews": 0, @@ -1051,7 +1051,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "", "from_community_srt": "Šiam kartui, sakykim, kad jis turi prasidėti su... 60 nulių, bet vėliau sugrįšim prie labiau sistematiško būdo, kaip galėtum pasirinkti tą skaičių. Taip pat, kaip transakcija yra laikoma validžia,", "n_reviews": 0, @@ -1059,7 +1059,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "", "from_community_srt": "tik kai ji yra pasirašyta siuntėjo, blokas yra laikomas validžiu, tik kai jis turi darbo įrodymą.", "n_reviews": 0, @@ -1339,7 +1339,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "", "from_community_srt": "Na, tai, kaip tikrasis Bitkoino protokolas veikia, yra periodiškai pakeisti nulių skaičių taip, kad vidutiniškai užtruktų dešimt minučių surasti naują bloką.", "n_reviews": 0, @@ -1379,7 +1379,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "", "from_community_srt": "Iš tiesų yra geras internetinis puslapis, pavadinimu \"Blokų tyrinėtojas\", kuris lengvai padeda peržiūrėti Bitkoino blokų grandinę.", "n_reviews": 0, diff --git a/2017/bitcoin/marathi/sentence_translations.json b/2017/bitcoin/marathi/sentence_translations.json index 5e479bac0..f47a9fd4a 100644 --- a/2017/bitcoin/marathi/sentence_translations.json +++ b/2017/bitcoin/marathi/sentence_translations.json @@ -224,7 +224,7 @@ "end": 228.54 }, { - "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves she has seen it and approved of it, and it should be infeasible for anyone else to forge that signature.", + "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves that she has seen it and that she's approved of it, and it should be infeasible for anyone else to forge that signature.", "translatedText": "हस्तलिखित स्वाक्षरींप्रमाणे, येथे कल्पना अशी आहे की अॅलिस त्या व्यवहाराच्या पुढे काहीतरी जोडण्यास सक्षम असावी ज्यामुळे तिने ते पाहिले आहे आणि त्यास मान्यता दिली आहे आणि ती स्वाक्षरी खोटी करणे इतर कोणासाठीही अशक्य आहे.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 554.26 }, { - "input": "You are free to exchange ledger dollars for real US dollars.", + "input": "You are of course free to exchange ledger dollars for real US dollars.", "translatedText": "खऱ्या यूएस डॉलर्ससाठी लेजर डॉलर्सची देवाणघेवाण करण्यासाठी तुम्ही मोकळे आहात.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 896.7 }, { - "input": "For a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 230, which is about 1 in a billion.", + "input": "Well, for a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 2 to the 30, which is about 1 in a billion.", "translatedText": "यादृच्छिक संदेशासाठी, हॅश 30 सलग शून्यांसह सुरू होण्याची संभाव्यता 230 मधील 1 आहे, जी एक अब्जापैकी 1 आहे.", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 949.64 }, { - "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the list starts with 30 zeros.", + "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the altered list together with this new number starts with 30 zeros.", "translatedText": "त्यामुळे तुम्हाला कामाचा नवीन पुरावा शोधण्यासाठी आणखी एक अब्ज अंदाजे जावे लागतील, एक नवीन क्रमांक जो तो बनवतो जेणेकरून सूचीचा हॅश ३० शून्यांनी सुरू होईल.", "model": "google_nmt", "n_reviews": 0, @@ -1232,7 +1232,7 @@ "end": 1182.64 }, { - "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's really helpful to walk through exactly what it would take to fool someone using this system.", + "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's actually really helpful to walk through exactly what it would take to fool someone using this system.", "translatedText": "ही एक विश्वासार्ह प्रणाली का बनवते हे पाहण्यासाठी आणि पेमेंट कायदेशीर आहे यावर तुम्‍हाला कोणत्‍या वेळी विश्‍वास ठेवावा हे समजून घेण्‍यासाठी, ही सिस्‍टम वापरणार्‍या एखाद्याला मूर्ख बनवण्‍यासाठी नेमके काय करावे लागेल हे जाणून घेणे खरोखर उपयुक्त आहे.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/bitcoin/persian/sentence_translations.json b/2017/bitcoin/persian/sentence_translations.json index 22f2e0860..28807d924 100644 --- a/2017/bitcoin/persian/sentence_translations.json +++ b/2017/bitcoin/persian/sentence_translations.json @@ -224,7 +224,7 @@ "end": 228.54 }, { - "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves she has seen it and approved of it, and it should be infeasible for anyone else to forge that signature. ", + "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves that she has seen it and that she's approved of it, and it should be infeasible for anyone else to forge that signature. ", "translatedText": "مانند امضاهای دست نویس، ایده در اینجا این است که آلیس باید بتواند چیزی را در کنار آن تراکنش اضافه کند که ثابت کند آن را دیده و آن را تأیید کرده است، و جعل آن امضا برای دیگران غیرممکن است. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 554.26 }, { - "input": "You are free to exchange ledger dollars for real US dollars. ", + "input": "You are of course free to exchange ledger dollars for real US dollars. ", "translatedText": "شما آزاد هستید که دلارهای دفتر کل را با دلارهای واقعی آمریکا مبادله کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 896.7 }, { - "input": "For a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 230, which is about 1 in a billion. ", + "input": "Well, for a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 2 to the 30, which is about 1 in a billion. ", "translatedText": "همه صفر هستند به نظر شما یافتن این عدد برای آنها چقدر سخت بود؟ برای یک پیام تصادفی، احتمال اینکه یک هش با 30 صفر متوالی شروع شود، 1 در 230 است، که حدود 1 در یک میلیارد است. ", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 949.64 }, { - "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the list starts with 30 zeros. ", + "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the altered list together with this new number starts with 30 zeros. ", "translatedText": "بنابراین، برای یافتن یک اثبات کار جدید، باید یک میلیارد حدس دیگر را مرور کنید، یک عدد جدید که باعث می‌شود هش لیست با 30 صفر شروع شود. ", "model": "google_nmt", "n_reviews": 0, @@ -1232,7 +1232,7 @@ "end": 1182.64 }, { - "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's really helpful to walk through exactly what it would take to fool someone using this system. ", + "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's actually really helpful to walk through exactly what it would take to fool someone using this system. ", "translatedText": "برای اینکه بفهمید چرا این یک سیستم قابل اعتماد را ایجاد می کند، و برای درک اینکه در چه مرحله ای باید به قانونی بودن پرداخت اعتماد کنید، واقعاً مفید است که دقیقاً آنچه را که برای فریب دادن شخصی که از این سیستم استفاده می کند لازم است بررسی کنید. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/bitcoin/polish/sentence_translations.json b/2017/bitcoin/polish/sentence_translations.json index ec2840fd5..19e44300b 100644 --- a/2017/bitcoin/polish/sentence_translations.json +++ b/2017/bitcoin/polish/sentence_translations.json @@ -654,7 +654,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "", "from_community_srt": "Wtedy, gdy chcemy dokonać transakcji, jak \"Alicja płaci Bobowi 100 LD\", trzeba to wyemitować w świat aby ludzie to usłyszeli i zarejestrowali we własnych prywatnych księgach.", "n_reviews": 0, @@ -678,7 +678,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "", "from_community_srt": "jaka księga jest właściwa? Kiedy Bob odbiera transakcję, jak \"Alicja płaci Bobowi 10 LD\", jak on może mieć pewność,", "n_reviews": 0, @@ -686,7 +686,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "", "from_community_srt": "że wszyscy otrzymali i wierzą w prawdziwość tej samej transakcji, że on będzie mógł później udać się do Charliego i używać tych samych 10 LD,", "n_reviews": 0, @@ -797,7 +797,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "", "from_community_srt": "co to jest funkcja skrótu? Danymi wejściowymi jednej z takich funkcji może być dowolny rodzaj wiadomości lub pliku, to naprawdę nie ma znaczenia. A danymi wyjściowymi jest ciąg bitów pewnej stałej długości, na przykład 256 bitów.", "n_reviews": 0, @@ -853,7 +853,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "", "from_community_srt": "Jeśli pokażę ci ciąg zer i jedynek, i poproszę, aby znaleźć dane wejściowe takie, że funkcja SHA256 dla tych danych daje dokładnie ten ciąg bitów, nie będzie lepszego sposobu niż zgadywać i sprawdzać.", "n_reviews": 0, @@ -861,7 +861,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "", "from_community_srt": "Przypominam, jeśli chcesz lepiej poczuć, ile obliczeń byłoby potrzebnych, aby przeprowadzić 2^256 prób, spójrz na dodatkowe wideo.", "n_reviews": 0, @@ -1052,7 +1052,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "", "from_community_srt": "Na razie powiedzmy, że ma się zacząć ... 60 zerami, ale później wrócimy do bardziej systematycznego sposobu, w którym możesz wybrać ten numer. W ten sam sposób jak transakcja jest uważana za ważną tylko wtedy,", "n_reviews": 0, @@ -1060,7 +1060,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "", "from_community_srt": "gdy jest podpisana przez nadawcę, blok jest uważany za ważny tylko, jeśli ma dowód pracy.", "n_reviews": 0, @@ -1340,7 +1340,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "", "from_community_srt": "Cóż, rzeczywisty protokół Bitcoin działa w ten sposób, że liczba zer jest okresowo zmieniana, tak, że powinno zająć średnio 10 minut, aby znaleźć nowy blok.", "n_reviews": 0, @@ -1380,7 +1380,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "", "from_community_srt": "Jest świetna strona internetowa , którą możesz odwiedzić, „Block Explorer”, na której można łatwo przejrzeć Block Chain Bitcoin.", "n_reviews": 0, diff --git a/2017/bitcoin/portuguese/sentence_translations.json b/2017/bitcoin/portuguese/sentence_translations.json index 7c77b332c..b887887ba 100644 --- a/2017/bitcoin/portuguese/sentence_translations.json +++ b/2017/bitcoin/portuguese/sentence_translations.json @@ -731,7 +731,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "Então, quando você quiser fazer uma transação, como Alice paga a Bob US$ 100, você transmite isso para o mundo para que as pessoas ouçam e registrem em seus próprios registros privados.", "model": "google_nmt", "from_community_srt": "Então quando quiser fazer uma transação, como Alice paga 100 LD para Bob o que você faz é transmitir isso para o resto do mundo para as pessoas ouvirem e gravarem no seu próprio registro privado", @@ -757,7 +757,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "Quando Bob recebe uma transação, como Alice paga a Bob US$ 10, como ele pode ter certeza de que todos os outros receberam e acreditam na mesma transação?", "model": "google_nmt", "from_community_srt": "Quando Bob recebe uma transação como, Alice paga 10 LD para Bob Como ele pode ter certeza de que todos receberam e acreditam nessa transação?", @@ -766,7 +766,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "Que mais tarde ele poderá ir até Charlie e usar os mesmos US$ 10 para fazer uma transação?", "model": "google_nmt", "from_community_srt": "que ele vai poder, depois, ir até o Charlie e usar os mesmo 10 LD para fazer uma transação? Sério,", @@ -889,7 +889,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "As entradas para uma dessas funções podem ser qualquer tipo de mensagem ou arquivo, na verdade parecem 256 bits.", "model": "google_nmt", "from_community_srt": "o que é uma função Hash? O valor de entrada dessas funções pode ser qualquer tipo de mensagem ou arquivo, realmente não importa e o resultado é uma sequência de bits com algum tipo de comprimento fixo", @@ -951,7 +951,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "Se eu mostrar uma sequência de 1s e 0s e pedir que você encontre uma entrada para o hash SHA256, você não terá método melhor do que apenas adivinhar e verificar.", "model": "google_nmt", "from_community_srt": "se eu te mostrar uma sequência de 1 e 0 e pedir para achar o valor de entrada, para que o Hash SHA256 dessa entrada resulte nesta exata sequência de bits você não terá método melhor do que adivinhar e testar E novamente,", @@ -960,7 +960,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "E, novamente, se você quiser saber quanto cálculo seria necessário para realizar 256 suposições, basta dar uma olhada no vídeo complementar.", "model": "google_nmt", "from_community_srt": "se você quer imaginar quanto trabalho computacional seria necessário para verificar 2 elevado a 256 tentativas", @@ -1175,7 +1175,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "No momento, digamos que tenha que começar com 60 zeros, mas depois voltaremos a uma forma mais sistemática que você pode querer alterar.", "model": "google_nmt", "from_community_srt": "vamos dizer que deve começar com 60 zeros mas depois vamos retornar a uma forma mais sistemática que você possa querer escolher esse número Da mesma forma que uma transação só é considerada válida", @@ -1184,7 +1184,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "Um bloco só é considerado válido se possuir uma prova de trabalho.", "model": "google_nmt", "from_community_srt": "quando ela está assinada pelo expedidor um bloco só é considerado válido se possuir uma Prova de Trabalho", @@ -1497,7 +1497,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "Bem, a forma como o protocolo Bitcoin real funciona é alterar periodicamente esse número de zeros para que demore 10 minutos para encontrar um novo bloco.", "model": "google_nmt", "from_community_srt": "Bem, a forma que o verdadeiro protocolo Bitcoin funciona é periodicamente mudar esse número de zeros para que demore, em média, 10 minutos para achar um novo bloco.", @@ -1541,7 +1541,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "Existe um ótimo site chamado Block Explorer que facilita a visualização do blockchain Bitcoin.", "model": "google_nmt", "from_community_srt": "essa recompensa eram 50 Bitcoin por bloco há um ótimo website chamado Block Explorer que deixa fácil olhar a cadeia de blocos do Bitcoin", diff --git a/2017/bitcoin/russian/sentence_translations.json b/2017/bitcoin/russian/sentence_translations.json index f9cf9c6d3..7f9dd62b6 100644 --- a/2017/bitcoin/russian/sentence_translations.json +++ b/2017/bitcoin/russian/sentence_translations.json @@ -735,7 +735,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "Затем, когда ты хочешь совершить транзакцию, например, Алиса платит Бобу 100 долларов, ты транслируешь это в мир, чтобы люди услышали и записали в свои личные бухгалтерские книги.", "model": "DeepL", "from_community_srt": "Когда нужно совершить операцию, например, Алиса захочет заплатить Бобу 100 LD, это просто нужно сообщить всему миру, чтобы каждый записал это в своем личном реестре.", @@ -762,7 +762,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "Когда Боб получает транзакцию, например, Алиса платит Бобу 10 долларов, как он может быть уверен, что все остальные получили и считают эту же транзакцию?", "model": "DeepL", "from_community_srt": "чей реестр верный? Когда Боб получает платеж, например, Алиса платит Бобу 10 LD, как он может быть уверен, что все знают и верят,", @@ -771,7 +771,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "Что позже он сможет пойти к Чарли и использовать те же 10 долларов для совершения сделки?", "model": "DeepL", "from_community_srt": "что операция произошла, и что потом он сможет прийти к Чарли и потратить те самые 10 LD на другую операцию?", @@ -897,7 +897,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "Входом для одной из этих функций может быть любое сообщение или файл, на самом деле это выглядит как 256 бит.", "model": "DeepL", "from_community_srt": "что же такое хеш-функция? Аргументом такой функции может быть сообщение или файл, и вообще что угодно. А значение - цепочка битов определенной длины, например, 256 бит.", @@ -960,7 +960,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "Если я покажу тебе какую-нибудь строку из 1 и 0 и попрошу найти входной хэш SHA256, то у тебя не будет лучшего метода, чем просто угадывать и проверять.", "model": "DeepL", "from_community_srt": "Допустим, я покажу вам цепочку из нулей и единиц и попрошу вас найти такой аргумент, для которого значение SHA256-хеш-функции точно совпадет с этой цепочкой. У вас не останется лучшего метода кроме подбора и проверки.", @@ -969,7 +969,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "И опять же, если ты хочешь почувствовать, сколько вычислений потребуется, чтобы перебрать 256 догадок, просто посмотри видео с дополнением.", "model": "DeepL", "from_community_srt": "Опять же, если вы хотите осознать, какого объема вычислений потребует подбор из 2^256 вариантов, просто посмотрите моё дополнительное видео.", @@ -1184,7 +1184,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "На данный момент скажем, что он должен начинаться с 60 нулей, но позже мы вернемся к более систематизированному способу, который ты, возможно, захочешь изменить.", "model": "DeepL", "from_community_srt": "Сейчас допустим, что он должен начинаться с 60 нулей, но позже мы вернемся к более последовательному способу выбора этого числа. По аналогу с операцией, которая подтверждается,", @@ -1193,7 +1193,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "Блок считается действительным только в том случае, если у него есть подтверждение работы.", "model": "DeepL", "from_community_srt": "только если её подписывает отправитель, блок действителен, только если в нём есть доказательство выполнения работы.", @@ -1508,7 +1508,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "Ну, на самом деле протокол Bitcoin работает так: периодически изменяется количество нулей, так что на поиск нового блока должно уходить 10 минут.", "model": "DeepL", "from_community_srt": "Но протоколу Биткоина требуемое число нулей меняется периодично, таким образом, чтобы в среднем потребовалось 10 минут для создания нового блока То есть,", @@ -1552,7 +1552,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "Есть отличный сайт под названием Block Explorer, который позволяет легко просматривать блокчейн Биткойна.", "model": "DeepL", "from_community_srt": "который называется BlockExplorer Он позволяет легко путешествовать по истории транзакций в блокчейне", diff --git a/2017/bitcoin/spanish/sentence_translations.json b/2017/bitcoin/spanish/sentence_translations.json index d9cd5757b..956ae6c69 100644 --- a/2017/bitcoin/spanish/sentence_translations.json +++ b/2017/bitcoin/spanish/sentence_translations.json @@ -734,7 +734,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "Luego, cuando quieras hacer una transacción, como que Alicia pague 100 $ a Bob, lo difundes por el mundo para que la gente lo oiga y lo registre en sus propios libros de contabilidad privados.", "model": "DeepL", "from_community_srt": "Entonces cuando quieres hacer una transacción, como Alice paga a Bob 100 LD. lo que haces es emitirla al resto de participantes para que la reciban y almacenen en su copia privada del libro de cuentas.", @@ -761,7 +761,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "Cuando Bob recibe una transacción, como que Alice le paga 10 $, ¿cómo puede estar seguro de que todos los demás recibieron y creen esa misma transacción?", "model": "DeepL", "from_community_srt": "Cuando Bob recibe una transacción, como Alice paga a Bob 10 LD, cómo puede estar seguro de que todos los demás han recibido y se creen esa transacción", @@ -770,7 +770,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "¿Que más tarde podrá ir a Charlie y utilizar esos mismos 10$ para hacer una transacción?", "model": "DeepL", "from_community_srt": "para que más tarde pueda ir a Charlie y usar esos mismos 10 LD para hacer una transacción? De verdad,", @@ -894,7 +894,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "Las entradas para una de estas funciones pueden ser cualquier tipo de mensaje o archivo, en realidad parecen 256 bits.", "model": "DeepL", "from_community_srt": "qué es una función hash? La entrada para una de estas funciones puede ser cualquier tipo de mensaje o archivo no importa en realidad. Y la salida es una cadena de bits de longitud fija, como 256 bits.", @@ -957,7 +957,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "Si te muestro una cadena de 1s y 0s y te pido que encuentres una entrada para el hash SHA256, no tendrás mejor método que simplemente adivinar y comprobar.", "model": "DeepL", "from_community_srt": "Si yo te enseño una cadena de unos y ceros y te pido que encuentres una entrada tal que el hash SHA256 de esa entrada devuelva exactamente esta cadena de bits, no tendrás mejor método que adivinar y comprobar.", @@ -966,7 +966,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "Y de nuevo, si quieres hacerte una idea de cuántos cálculos serían necesarios para realizar 256 conjeturas, sólo tienes que echar un vistazo al vídeo del suplemento.", "model": "DeepL", "from_community_srt": "Y de nuevo, si quieres hacerte una idea de cuanta computación seria necesaria para recorrer 2^256 intentos, echale un vistazo al video extra.", @@ -1180,7 +1180,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "De momento, digamos que tiene que empezar por 60 ceros, pero más adelante volveremos sobre una forma más sistemática que tal vez quieras cambiar.", "model": "DeepL", "from_community_srt": "Por el momento, digamos que tiene que empezar con... 60 ceros, pero más tarde volveremos con una forma más sistemática que podrías querer utilizar para elegir ese numero. De la misma forma que una transaccion solo es considerada válida cuando está firmada por el emisor,", @@ -1189,7 +1189,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "Un bloque sólo se considera válido si tiene una prueba de trabajo.", "model": "DeepL", "from_community_srt": "un bloque solo es considerado válido si contiene una prueba de trabajo.", @@ -1504,7 +1504,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "Pues bien, la forma en que funciona el protocolo real de Bitcoin es cambiar periódicamente ese número de ceros para que se tarde 10 minutos en encontrar un nuevo bloque.", "model": "DeepL", "from_community_srt": "Bueno, la manera en que funciona con Bitcoin es cambiando periodicamente el numero de ceros de tal forma que debería llevar de media 10 minutos encontrar un bloque nuevo.", @@ -1549,7 +1549,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "Hay un gran sitio web llamado Block Explorer que facilita la consulta de la cadena de bloques de Bitcoin.", "model": "DeepL", "from_community_srt": "De hecho hay una muy buena página a la que puedes ir llamada \"Block Explorer\" que facilita visualizar la cadena de Bitcoin.", diff --git a/2017/bitcoin/tamil/sentence_translations.json b/2017/bitcoin/tamil/sentence_translations.json index 8939891fe..5137ca363 100644 --- a/2017/bitcoin/tamil/sentence_translations.json +++ b/2017/bitcoin/tamil/sentence_translations.json @@ -224,7 +224,7 @@ "end": 228.54 }, { - "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves she has seen it and approved of it, and it should be infeasible for anyone else to forge that signature.", + "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves that she has seen it and that she's approved of it, and it should be infeasible for anyone else to forge that signature.", "translatedText": "கையால் எழுதப்பட்ட கையொப்பங்களைப் போலவே, இங்குள்ள யோசனை என்னவென்றால், ஆலிஸ் அந்த பரிவர்த்தனைக்கு அடுத்ததாக ஏதாவது ஒன்றைச் சேர்க்க வேண்டும், அது தான் அதைப் பார்த்து ஒப்புதல் அளித்ததை நிரூபிக்கிறது, மேலும் அந்த கையொப்பத்தை வேறு யாரும் போலியாக உருவாக்குவது சாத்தியமில்லை.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 554.26 }, { - "input": "You are free to exchange ledger dollars for real US dollars.", + "input": "You are of course free to exchange ledger dollars for real US dollars.", "translatedText": "உண்மையான அமெரிக்க டாலர்களுக்கு லெட்ஜர் டாலர்களை மாற்றிக்கொள்ள நீங்கள் சுதந்திரமாக இருக்கிறீர்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 896.7 }, { - "input": "For a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 230, which is about 1 in a billion.", + "input": "Well, for a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 2 to the 30, which is about 1 in a billion.", "translatedText": "ஒரு சீரற்ற செய்திக்கு, ஹாஷ் 30 தொடர்ச்சியான பூஜ்ஜியங்களுடன் தொடங்குவதற்கான நிகழ்தகவு 230 இல் 1 ஆகும், இது ஒரு பில்லியனில் 1 ஆகும்.", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 949.64 }, { - "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the list starts with 30 zeros.", + "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the altered list together with this new number starts with 30 zeros.", "translatedText": "எனவே, வேலைக்கான புதிய சான்றைக் கண்டுபிடிக்க நீங்கள் மற்றொரு பில்லியன் யூகங்களைச் செய்ய வேண்டும், புதிய எண்ணை உருவாக்குகிறது, இதனால் பட்டியலின் ஹாஷ் 30 பூஜ்ஜியங்களுடன் தொடங்குகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -1232,7 +1232,7 @@ "end": 1182.64 }, { - "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's really helpful to walk through exactly what it would take to fool someone using this system.", + "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's actually really helpful to walk through exactly what it would take to fool someone using this system.", "translatedText": "இது ஏன் நம்பகமான அமைப்பை உருவாக்குகிறது என்பதைப் பார்க்கவும், எந்த நேரத்தில் பணம் செலுத்துவது முறையானது என்று நீங்கள் நம்ப வேண்டும் என்பதைப் புரிந்து கொள்ளவும், இந்த முறையைப் பயன்படுத்தும் ஒருவரை முட்டாளாக்குவதற்கு என்ன செய்ய வேண்டும் என்பதைச் சரியாகப் பார்ப்பது மிகவும் பயனுள்ளதாக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/bitcoin/telugu/sentence_translations.json b/2017/bitcoin/telugu/sentence_translations.json index fc37093bc..a5dd107d9 100644 --- a/2017/bitcoin/telugu/sentence_translations.json +++ b/2017/bitcoin/telugu/sentence_translations.json @@ -224,7 +224,7 @@ "end": 228.54 }, { - "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves she has seen it and approved of it, and it should be infeasible for anyone else to forge that signature.", + "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves that she has seen it and that she's approved of it, and it should be infeasible for anyone else to forge that signature.", "translatedText": "చేతితో వ్రాసిన సంతకాల వలె, ఇక్కడ ఆలోచన ఏమిటంటే, ఆలిస్ ఆ లావాదేవీని చూసినట్లు మరియు దానిని ఆమోదించినట్లు రుజువు చేసే దాని పక్కన ఏదైనా జోడించగలగాలి మరియు ఆ సంతకాన్ని మరెవరికైనా నకిలీ చేయడం సాధ్యం కాదు.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 554.26 }, { - "input": "You are free to exchange ledger dollars for real US dollars.", + "input": "You are of course free to exchange ledger dollars for real US dollars.", "translatedText": "మీరు నిజమైన US డాలర్లకు లెడ్జర్ డాలర్లను మార్పిడి చేసుకోవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 896.7 }, { - "input": "For a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 230, which is about 1 in a billion.", + "input": "Well, for a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 2 to the 30, which is about 1 in a billion.", "translatedText": "యాదృచ్ఛిక సందేశం కోసం, హాష్ 30 వరుస సున్నాలతో ప్రారంభమయ్యే సంభావ్యత 230లో 1, ఇది బిలియన్‌లో 1.", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 949.64 }, { - "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the list starts with 30 zeros.", + "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the altered list together with this new number starts with 30 zeros.", "translatedText": "కాబట్టి మీరు పని యొక్క కొత్త రుజువును కనుగొనడానికి మరొక బిలియన్ అంచనాల ద్వారా వెళ్ళవలసి ఉంటుంది, ఇది కొత్త సంఖ్యను చేస్తుంది, తద్వారా జాబితా యొక్క హాష్ 30 సున్నాలతో ప్రారంభమవుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1232,7 +1232,7 @@ "end": 1182.64 }, { - "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's really helpful to walk through exactly what it would take to fool someone using this system.", + "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's actually really helpful to walk through exactly what it would take to fool someone using this system.", "translatedText": "ఇది నమ్మదగిన సిస్టమ్‌ను ఎందుకు తయారు చేస్తుందో చూడడానికి మరియు చెల్లింపు చట్టబద్ధమైనదని మీరు ఏ సమయంలో విశ్వసించాలో అర్థం చేసుకోవడానికి, ఈ సిస్టమ్‌ని ఉపయోగించి ఎవరైనా మోసం చేయడానికి ఖచ్చితంగా ఏమి తీసుకుంటారో తెలుసుకోవడం నిజంగా సహాయకరంగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/bitcoin/thai/sentence_translations.json b/2017/bitcoin/thai/sentence_translations.json index e4572ef36..e1dfdba29 100644 --- a/2017/bitcoin/thai/sentence_translations.json +++ b/2017/bitcoin/thai/sentence_translations.json @@ -224,7 +224,7 @@ "end": 228.54 }, { - "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves she has seen it and approved of it, and it should be infeasible for anyone else to forge that signature. ", + "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves that she has seen it and that she's approved of it, and it should be infeasible for anyone else to forge that signature. ", "translatedText": "นี่คือที่มาของการเข้ารหัสบิตแรก ซึ่งก็คือลายเซ็นดิจิทัล เช่นเดียวกับลายเซ็นที่เขียนด้วยลายมือ แนวคิดก็คืออลิซควรสามารถเพิ่มบางสิ่งถัดจากธุรกรรมนั้นเพื่อพิสูจน์ว่าเธอได้เห็นและอนุมัติแล้ว และบุคคลอื่นจะปลอมแปลงลายเซ็นนั้นไม่ได้ ในตอนแรก อาจดูเหมือนว่าลายเซ็นดิจิทัลไม่ควรเป็นไปได้ด้วยซ้ำ ฉันหมายถึงว่าข้อมูลใดก็ตามที่ประกอบขึ้นเป็นลายเซ็นนั้นก็สามารถอ่านและคัดลอกได้โดยคอมพิวเตอร์เท่านั้น แล้วจะป้องกันการปลอมแปลงได้อย่างไร? ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 554.26 }, { - "input": "You are free to exchange ledger dollars for real US dollars. ", + "input": "You are of course free to exchange ledger dollars for real US dollars. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 896.7 }, { - "input": "For a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 230, which is about 1 in a billion. ", + "input": "Well, for a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 2 to the 30, which is about 1 in a billion. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 949.64 }, { - "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the list starts with 30 zeros. ", + "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the altered list together with this new number starts with 30 zeros. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1232,7 +1232,7 @@ "end": 1182.64 }, { - "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's really helpful to walk through exactly what it would take to fool someone using this system. ", + "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's actually really helpful to walk through exactly what it would take to fool someone using this system. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/bitcoin/turkish/sentence_translations.json b/2017/bitcoin/turkish/sentence_translations.json index 1cc3f720b..c9197bfdc 100644 --- a/2017/bitcoin/turkish/sentence_translations.json +++ b/2017/bitcoin/turkish/sentence_translations.json @@ -733,7 +733,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "Daha sonra bir işlem yapmak istediğinizde, örneğin Alice Bob'a 100 dolar ödediğinde, bunu insanların duyması ve kendi özel defterlerine kaydetmesi için dünyaya yayınlarsınız.", "model": "DeepL", "from_community_srt": "Sonra da \" Alice Bob'a 100 LD öder\" gibi parasal işlem yapmak istediğinde, yaptığın şey; dünyaya yayın yapmak.. ..insanların duymasını ve kendi özel defterlerine kaydetmesi için.", @@ -759,7 +759,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "Bob bir işlem aldığında, örneğin Alice Bob'a 10 dolar ödediğinde, diğer herkesin aynı işlemi aldığından ve buna inandığından nasıl emin olabilir?", "model": "DeepL", "from_community_srt": "Herkes doğru defterin ne olduğunu kabul etmesini nasıl sağlardın? \"Alice Bob'a 10 LD öder\" gibi Bob bir parasal işlem alırsa,", @@ -768,7 +768,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "Daha sonra Charlie'ye gidip işlem yapmak için aynı 10 doları kullanabilecek mi?", "model": "DeepL", "from_community_srt": "diğer herkesin aynı parasal işlemini aldığından ve inandığından nasıl emin olur ki o daha sonra aynı parasal işlemi yapmak için 10 LD yi kullanıp daha sonra Charlie'e gidebilir", @@ -893,7 +893,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "Bu fonksiyonlardan birinin girdileri herhangi bir tür mesaj veya dosya olabilir, gerçekten 256 bit gibi görünür.", "model": "DeepL", "from_community_srt": "bir hash fonksiyonu nedir? Bu fonksiyona girişler herhangi bir mesaj veya dosya olabilir, bunun gerçekten bir önemi yok. Ve çıkış ise 256 bit gibi değişmez uzunlukta bitlerdir Bu çıkışa \"hash\" veya mesajın \"özeti\" denir.", @@ -956,7 +956,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "Size 1 ve 0'lardan oluşan bir dizi gösterip SHA256 hash'ine bir girdi bulmanızı istersem, tahmin edip kontrol etmekten daha iyi bir yönteminiz olmayacaktır.", "model": "DeepL", "from_community_srt": "Eğer ben sana bir ve sıfırlardan oluşan bir dizi gösterseydim ve senden bir girişi bulmanı istesem böylece SHA256 hash fonksiyonun girişi bu bit dizisini verir tahmin ve kontrol etmekten daha iyi bir yöntemin yok.", @@ -965,7 +965,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "Ve yine, 256 tahminden geçmek için ne kadar hesaplama yapılması gerektiğini hissetmek istiyorsanız, ek videoya bir göz atın.", "model": "DeepL", "from_community_srt": "Ve tekrardan, 2^256 için ne kadar hesaplama gerektiğini sezmek istersen ilave video ya bir göz at.", @@ -1179,7 +1179,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "Şimdilik 60 sıfırla başlaması gerektiğini söyleyelim, ancak daha sonra değiştirmek isteyebileceğiniz daha sistematik bir yola geri döneceğiz.", "model": "DeepL", "from_community_srt": "Şimdilik, diyelim ki 60 tane sıfır ile başlamak zorunda olsun, ama sonra da sayıyı seçmek isteyeceğiniz için daha sistematik bir şekilde geriye döneceğiz.", @@ -1188,7 +1188,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "Bir blok yalnızca bir çalışma kanıtına sahipse geçerli kabul edilir.", "model": "DeepL", "from_community_srt": "Aynı şekilde sadece gönderici tarafından imzalandığında para işlemi geçerli sayılır, eğer ki bir çalışma prensibi varsa bir blok geçerli sayılır.", @@ -1471,7 +1471,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "Gerçek Bitcoin protokolünün çalışma şekli, yeni bir blok bulmanın 10 dakika sürmesi için bu sıfır sayısını periyodik olarak değiştirmektir.", "model": "DeepL", "n_reviews": 0, @@ -1511,7 +1511,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "Bitcoin blok zincirine bakmayı kolaylaştıran Block Explorer adlı harika bir web sitesi var.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/bitcoin/ukrainian/sentence_translations.json b/2017/bitcoin/ukrainian/sentence_translations.json index 0bc071e3d..db2df8182 100644 --- a/2017/bitcoin/ukrainian/sentence_translations.json +++ b/2017/bitcoin/ukrainian/sentence_translations.json @@ -196,7 +196,7 @@ "end": 228.54 }, { - "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves she has seen it and approved of it, and it should be infeasible for anyone else to forge that signature.", + "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves that she has seen it and that she's approved of it, and it should be infeasible for anyone else to forge that signature.", "translatedText": "Подібно до рукописних підписів, ідея полягає в тому, що Аліса повинна мати можливість додати щось поруч із цією транзакцією, що доводить, що вона бачила це та схвалила це, і будь-кому іншому має бути неможливо підробити цей підпис.", "n_reviews": 0, "start": 229.48, @@ -483,7 +483,7 @@ "end": 554.26 }, { - "input": "You are free to exchange ledger dollars for real US dollars.", + "input": "You are of course free to exchange ledger dollars for real US dollars.", "translatedText": "Ви можете вільно обмінювати долари бухгалтерської книги на справжні долари США.", "n_reviews": 0, "start": 554.82, @@ -847,7 +847,7 @@ "end": 896.7 }, { - "input": "For a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 230, which is about 1 in a billion.", + "input": "Well, for a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 2 to the 30, which is about 1 in a billion.", "translatedText": "Для випадкового повідомлення ймовірність того, що хеш починається з 30 послідовних нулів, становить 1 до 230, тобто приблизно 1 до мільярда.", "n_reviews": 0, "start": 898.06, @@ -903,7 +903,7 @@ "end": 949.64 }, { - "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the list starts with 30 zeros.", + "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the altered list together with this new number starts with 30 zeros.", "translatedText": "Тож вам доведеться пройти через ще один мільярд здогадок, щоб знайти нове підтвердження роботи, нове число, яке робить так, щоб хеш списку починався з 30 нулів.", "n_reviews": 0, "start": 950.08, @@ -1078,7 +1078,7 @@ "end": 1182.64 }, { - "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's really helpful to walk through exactly what it would take to fool someone using this system.", + "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's actually really helpful to walk through exactly what it would take to fool someone using this system.", "translatedText": "Щоб зрозуміти, чому ця система є надійною, і зрозуміти, на якому етапі ви повинні вірити, що платіж є законним, було б дуже корисно пройтися через те, що потрібно, щоб обдурити когось, хто використовує цю систему.", "n_reviews": 0, "start": 1183.56, diff --git a/2017/bitcoin/urdu/sentence_translations.json b/2017/bitcoin/urdu/sentence_translations.json index a4f07aa16..909e34669 100644 --- a/2017/bitcoin/urdu/sentence_translations.json +++ b/2017/bitcoin/urdu/sentence_translations.json @@ -224,7 +224,7 @@ "end": 228.54 }, { - "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves she has seen it and approved of it, and it should be infeasible for anyone else to forge that signature. ", + "input": "Like handwritten signatures, the idea here is that Alice should be able to add something next to that transaction that proves that she has seen it and that she's approved of it, and it should be infeasible for anyone else to forge that signature. ", "translatedText": "ہاتھ سے لکھے ہوئے دستخطوں کی طرح، یہاں خیال یہ ہے کہ ایلس کو اس لین دین کے آگے کچھ شامل کرنے کے قابل ہونا چاہیے جس سے یہ ثابت ہو کہ اس نے اسے دیکھا ہے اور اس کی منظوری دی ہے، اور کسی اور کے لیے اس دستخط کو جعل سازی کرنا ناقابل عمل ہونا چاہیے۔", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 554.26 }, { - "input": "You are free to exchange ledger dollars for real US dollars. ", + "input": "You are of course free to exchange ledger dollars for real US dollars. ", "translatedText": "آپ لیجر ڈالر کو حقیقی امریکی ڈالر کے بدلے کرنے کے لیے آزاد ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 896.7 }, { - "input": "For a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 230, which is about 1 in a billion. ", + "input": "Well, for a random message, the probability that a hash happens to start with 30 successive zeros is 1 in 2 to the 30, which is about 1 in a billion. ", "translatedText": "آپ کے خیال میں ان کے لیے اس نمبر کو تلاش کرنا کتنا مشکل تھا؟ بے ترتیب پیغام کے لیے، ہیش کے 30 لگاتار زیرو سے شروع ہونے کا امکان 230 میں 1 ہے، جو کہ ایک ارب میں تقریباً 1 ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 949.64 }, { - "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the list starts with 30 zeros. ", + "input": "So you'd have to go through another billion guesses to find a new proof of work, a new number that makes it so that the hash of the altered list together with this new number starts with 30 zeros. ", "translatedText": "لہذا آپ کو کام کا ایک نیا ثبوت تلاش کرنے کے لیے ایک اور ارب اندازے سے گزرنا پڑے گا، ایک نیا نمبر جو اسے بناتا ہے تاکہ فہرست کا ہیش 30 صفر سے شروع ہو۔", "model": "google_nmt", "n_reviews": 0, @@ -1232,7 +1232,7 @@ "end": 1182.64 }, { - "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's really helpful to walk through exactly what it would take to fool someone using this system. ", + "input": "To see why this makes for a trustworthy system, and to understand at what point you should trust that a payment is legit, it's actually really helpful to walk through exactly what it would take to fool someone using this system. ", "translatedText": "یہ دیکھنے کے لیے کہ یہ ایک قابل اعتماد نظام کیوں بناتا ہے، اور یہ سمجھنے کے لیے کہ آپ کو کس مقام پر بھروسہ کرنا چاہیے کہ ادائیگی جائز ہے، یہ واقعی مددگار ہے کہ اس سسٹم کو استعمال کرنے والے کسی کو بے وقوف بنانے میں کیا کرنا پڑے گا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/bitcoin/vietnamese/sentence_translations.json b/2017/bitcoin/vietnamese/sentence_translations.json index e8cdff116..c1c42d5cf 100644 --- a/2017/bitcoin/vietnamese/sentence_translations.json +++ b/2017/bitcoin/vietnamese/sentence_translations.json @@ -733,7 +733,7 @@ "end": 631.96 }, { - "input": "Then when you want to make a transaction, like Alice pays Bob $100, you broadcast that out into the world for people to hear and to record on their own private ledgers.", + "input": "Then when you want to make a transaction, like Alice pays Bob 100 Ledger Dollars, you do broadcast that out into the world for people to hear and record on their own private ledgers.", "translatedText": "Sau đó, khi bạn muốn thực hiện một giao dịch, chẳng hạn như Alice trả cho Bob 100 đô la, bạn sẽ công bố thông tin đó ra thế giới để mọi người nghe và ghi lại vào sổ cái riêng của họ.", "model": "google_nmt", "from_community_srt": "Sau đó, khi bạn muốn thực hiện giao dịch, như \"Alice trả cho Bob 100 LD\", bạn phát thông tin ra cho thế giới để mọi người nghe và ghi lại vào sổ cái riêng của họ.", @@ -760,7 +760,7 @@ "end": 652.74 }, { - "input": "When Bob receives a transaction, like Alice pays Bob $10, how can he be sure that everyone else received and believes that same transaction?", + "input": "When Bob receives a transaction, like Alice pays Bob 10 Ledger Dollars, how can he be sure that everyone else received and believes that same transaction?", "translatedText": "Khi Bob nhận được một giao dịch, chẳng hạn như Alice trả cho Bob 10 đô la, làm sao anh ấy có thể chắc chắn rằng những người khác đã nhận và tin vào giao dịch đó?", "model": "google_nmt", "from_community_srt": "như \"Alice trả cho Bob 10 LD\", làm thế nào anh ta có thể chắc rằng mọi người khác đã biết về và tin tưởng giao dịch đó", @@ -769,7 +769,7 @@ "end": 661.68 }, { - "input": "That he'll be able to later on go to Charlie and use those same $10 to make a transaction?", + "input": "That he'll be able to later on go to Charlie and use those same 10 Ledger Dollars to make a transaction?", "translatedText": "Rằng sau này anh ta có thể đến gặp Charlie và sử dụng chính 10 đô la đó để thực hiện giao dịch?", "model": "google_nmt", "from_community_srt": "rằng sau đó, anh ta sẽ có thể sử dụng cùng 10 LD đó để thực hiện giao dịch với Charlie? Thật vậy,", @@ -893,7 +893,7 @@ "end": 749.94 }, { - "input": "The inputs for one of these functions can be any kind of message or file, it really looks like 256 bits.", + "input": "The inputs for one of these functions can be any kind of message or file, it really doesn't matter. And the output is a string of bits with some kind of fixed length, like 256 bits.", "translatedText": "Đầu vào cho một trong các chức năng này có thể là bất kỳ loại tin nhắn hoặc tệp nào, nó thực sự trông giống như 256 bit.", "model": "google_nmt", "from_community_srt": "một hàm băm (hash function) là gì? Các đầu vào cho những hàm này có thể là bất kỳ loại thông điệp hoặc tập tin, nó thực sự không quan trọng. Và đầu ra là một chuỗi các bit với một chiều dài nhất định, như 256 bit.", @@ -956,7 +956,7 @@ "end": 800.66 }, { - "input": "If I show you some string of 1s and 0s and ask you to find an input to the SHA256 hash, you'll have no better method than to just guess and check.", + "input": "If I show you some string of 1s and 0s, and ask you to find an input so that the SHA256 hash of that input gives this exact string of bits, you will have no better method than to just guess and check.", "translatedText": "Nếu tôi chỉ cho bạn một số chuỗi 1 và 0 và yêu cầu bạn tìm đầu vào cho hàm băm SHA256, bạn sẽ không có phương pháp nào tốt hơn là chỉ đoán và kiểm tra.", "model": "google_nmt", "from_community_srt": "Nếu tôi chỉ cho bạn một chuỗi những số 1 và 0, và yêu cầu bạn tìm một đầu vào để băm SHA256 cho ra chính cái chuỗi các bit đó, bạn sẽ không có phương pháp tốt hơn ngoài chỉ đoán và kiểm tra.", @@ -965,7 +965,7 @@ "end": 814.64 }, { - "input": "And again, if you want to feel for how much computation would be needed to go through 256 guesses, just take a look at the supplement video.", + "input": "And again, if you want to feel for how much computation would be needed to go through two to the 256 guesses, just take a look at the supplement video.", "translatedText": "Và một lần nữa, nếu bạn muốn biết cần bao nhiêu tính toán để thực hiện 256 lần đoán, chỉ cần xem video bổ sung.", "model": "google_nmt", "from_community_srt": "Và một lần nữa, nếu bạn muốn nắm được cần bao nhiêu công tính toán để đoán 2 ^ 256 lần, chỉ cần đi xem video bổ sung.", @@ -1180,7 +1180,7 @@ "end": 992.3 }, { - "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to change.", + "input": "For the moment, let's say it has to start with 60 zeros, but later we'll return back to a more systematic way you might want to choose that number. In the same way that a tran", "translatedText": "Hiện tại, giả sử nó phải bắt đầu bằng 60 số 0, nhưng sau đó chúng ta sẽ quay lại theo cách có hệ thống hơn mà bạn có thể muốn thay đổi.", "model": "google_nmt", "from_community_srt": "Bây giờ, giả sử rằng nó phải bắt đầu với ... 60 số 0, nhưng sau đó chúng ta sẽ trở lại và nói về một cách chọn số số 0 có hệ thống hơn.", @@ -1189,7 +1189,7 @@ "end": 1005.5 }, { - "input": "A block is only considered valid if it has a proof of work.", + "input": "saction is only considered valid when it's signed by the sender, A block is only considered valid if it has a proof of work.", "translatedText": "Một khối chỉ được coi là hợp lệ nếu nó có bằng chứng hoạt động.", "model": "google_nmt", "from_community_srt": "Tương tự như cách một giao dịch chỉ được coi là hợp lệ khi nó được ký bởi người gửi, một khối chỉ được coi là có giá trị chỉ khi nó có proof of work.", @@ -1504,7 +1504,7 @@ "end": 1322.58 }, { - "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take 10 minutes to find a new block.", + "input": "Well, the way the actual Bitcoin protocol works is to periodically change that number of zeros so that it should take, on average, 10 minutes to find a new block.", "translatedText": "Chà, cách thức hoạt động của giao thức Bitcoin thực tế là thay đổi định kỳ số số 0 đó sao cho phải mất 10 phút để tìm một khối mới.", "model": "google_nmt", "from_community_srt": "Vâng, cách mà giao thức Bitcoin thực tế hoạt động là định kỳ thay đổi số lượng số 0 để việc tìm một khối mới phải mất trung bình 10 phút.", @@ -1549,7 +1549,7 @@ "end": 1355.74 }, { - "input": "There's a great website called Block Explorer that makes it easy to look through the Bitcoin blockchain.", + "input": "There's actually a great website you can go to called Block Explorer that makes it easy to look through the Bitcoin blockchain.", "translatedText": "Có một trang web tuyệt vời tên là Block Explorer giúp bạn dễ dàng xem qua chuỗi khối Bitcoin.", "model": "google_nmt", "from_community_srt": "Có một trang web thú vị mà bạn có thể ghé thăm, tên là \"Block Explorer\", nó giúp ta xem xét các blockchain của Bitcoin một cách dễ dàng.", diff --git a/2017/chain-rule-and-product-rule/arabic/sentence_translations.json b/2017/chain-rule-and-product-rule/arabic/sentence_translations.json index 12eab9e39..24e8a8786 100644 --- a/2017/chain-rule-and-product-rule/arabic/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/arabic/sentence_translations.json @@ -603,7 +603,7 @@ "end": 592.58 }, { - "input": "So for the derivative, let's again start by nudging that x value by dx.", + "input": "So, for the derivative, let's again start by nudging that x value by some little dx.", "translatedText": "لذا بالنسبة للمشتقة، فلنبدأ مرة أخرى بدفع قيمة x بواسطة dx.", "model": "google_nmt", "from_community_srt": "حيثما يجب أن تكون (sin(9 إذاً لأجل المشتقة ..", @@ -666,7 +666,7 @@ "end": 653.68 }, { - "input": "The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", + "input": "By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", "translatedText": "حقيقة أنه يتحرك إلى اليسار بينما يتجه نتوء dh إلى اليمين يعني فقط أن هذا التغيير، d sin of h، سيكون عددًا سالبًا نوعًا ما.", "model": "google_nmt", "from_community_srt": "حقيقة كونها تتحرك إلى اليسار، بينما تتحرك نتأة dh إلى اليمين .. هذا يعني فحسب أن هذا التغير: [(d[sin(h ..", @@ -962,7 +962,7 @@ "end": 910.4 }, { - "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice.", + "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice.", "translatedText": "أتمنى حقًا أن أعرض عليك القيام بذلك، ولكن أخشى أن تكون الكرة في ملعبك للبحث عن هذه الممارسة.", "model": "google_nmt", "from_community_srt": "أتمنى حقاً لو أنني أستطيع أن أوفر لك ذلك .. لكن أخشى أن الكرة في ملعبك يا صديقي، لتسعى إلى التدريب ما أستطيعه أن أوفره ..", diff --git a/2017/chain-rule-and-product-rule/bengali/sentence_translations.json b/2017/chain-rule-and-product-rule/bengali/sentence_translations.json index 879908ed4..9b3de2fda 100644 --- a/2017/chain-rule-and-product-rule/bengali/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/bengali/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking. ", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking. ", "translatedText": "কিন্তু দুটি ফাংশনের সমষ্টির ডেরিভেটিভ নেওয়ার অর্থ কী তা নিয়ে সত্যিই চিন্তা করে এই উদাহরণের সাথে উষ্ণতা লাভ করা মূল্যবান, যেহেতু পণ্য এবং ফাংশন কম্পোজিশনের ডেরিভেটিভ প্যাটার্নগুলি এত সোজা হবে না এবং তাদের এই ধরণের প্রয়োজন হবে।গভীর চিন্তা।", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar. ", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar. ", "translatedText": "উদাহরণস্বরূপ, ধরা যাক x সমান 0।5, সাইন গ্রাফের উচ্চতা এই উল্লম্ব বার দ্বারা দেওয়া হয়, এবং x বর্গক্ষেত্র প্যারাবোলার উচ্চতা এই ছোট উল্লম্ব বার দ্বারা দেওয়া হয়।", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x. ", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x. ", "translatedText": "এটি আমাদের নতুন ক্ষেত্রফলের তিনটি ছোট স্নিপেট দেয়, নীচে একটি পাতলা আয়তক্ষেত্র যার ক্ষেত্রফল তার প্রস্থ, x এর সাইন, তার পাতলা উচ্চতার গুণ, dx বর্গক্ষেত্র এবং ডানদিকে এই পাতলা আয়তক্ষেত্র যার ক্ষেত্রফল তার উচ্চতা, x বর্গক্ষেত্র, বার এর পাতলা প্রস্থ, x এর d সাইন।", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0. ", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean, ", "translatedText": "এর ক্ষেত্রফল শেষ পর্যন্ত dx বর্গক্ষেত্রের সমানুপাতিক, এবং আমরা আগে দেখেছি, dx 0-তে গেলে এটি নগণ্য হয়ে যায়।", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small. ", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small. ", "translatedText": "এবং ঠিক তখনই, মনে রাখবেন যে আমি জিনিসগুলি আঁকতে এখানে কিছুটা বিফী পরিবর্তনগুলি ব্যবহার করছি যাতে আমরা আসলে সেগুলি দেখতে পারি, তবে নীতিগতভাবে dx হল খুব ছোট কিছু, এবং এর মানে হল x এর dx বর্গ এবং d সাইনও খুব খুব ছোট. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx. ", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx. ", "translatedText": "তাই সাইন এবং x বর্গক্ষেত্রের ডেরিভেটিভ সম্পর্কে আমরা যা জানি তা প্রয়োগ করলে, dx বর্গক্ষেত্রের ক্ষুদ্র পরিবর্তন হবে প্রায় 2x গুণ dx, এবং x এর সেই ক্ষুদ্র পরিবর্তন d সাইনটি হবে x গুণ dx এর কোসাইন সম্পর্কে।", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", "translatedText": "আমি তিনটি ভিন্ন সংখ্যা রেখা রাখব, উপরেরটি x এর মান ধরবে, দ্বিতীয়টি x বর্গের মান ধরবে এবং তৃতীয় লাইনটি x বর্গক্ষেত্রের সাইনের মান ধরবে, অর্থাৎ ফাংশনটি x বর্গক্ষেত্র আপনাকে লাইন 1 থেকে লাইন 2 এ নিয়ে যায় এবং সাইন ফাংশনটি আপনাকে লাইন 2 থেকে লাইন 3 এ নিয়ে যায়।", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", "translatedText": "আমরা এটিকে 2x গুণ dx হিসাবে প্রসারিত করতে পারি, যা আমাদের নির্দিষ্ট ইনপুটের জন্য 2 গুণ 1 হবে।5 বার dx, কিন্তু এটি অন্তত আপাতত dx বর্গ হিসাবে লেখা জিনিস রাখতে সাহায্য করে।", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h. ", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h. ", "translatedText": "যদি আপনার কোন দুটি ফাংশন থাকে, x এর g এবং x এর h, তাদের রচনার ডেরিভেটিভ, x এর h এর g, h এর ডেরিভেটিভ দ্বারা গুণিত, h এর উপর মূল্যায়ন করা g এর ডেরিভেটিভ।", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule. ", + "input": "This pattern right here is what we usually call the chain rule. ", "translatedText": "এই প্যাটার্নটিকে আমরা সাধারণত চেইন নিয়ম বলি।", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h. ", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h. ", "translatedText": "কিন্তু আমরা ডেরিভেটিভকে সেই মধ্যবর্তী চলক, h এর সাপেক্ষে নিতে পারি।", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", "translatedText": "লক্ষ্য করুন, সেই ডিএইচগুলি বাতিল করে দেয় এবং আমাদেরকে সেই চূড়ান্ত আউটপুট এবং ইনপুটের পরিবর্তনের মধ্যে একটি অনুপাত দেয় যা একটি নির্দিষ্ট ঘটনা শৃঙ্খলের মাধ্যমে এটি নিয়ে আসে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/bulgarian/sentence_translations.json b/2017/chain-rule-and-product-rule/bulgarian/sentence_translations.json index b32cc5c83..7bfd90702 100644 --- a/2017/chain-rule-and-product-rule/bulgarian/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/bulgarian/sentence_translations.json @@ -544,7 +544,7 @@ "end": 592.58 }, { - "input": "So for the derivative, let's again start by nudging that x value by dx.", + "input": "So, for the derivative, let's again start by nudging that x value by some little dx.", "translatedText": "И така, за производната нека отново започнем с побутване на стойността x с dx.", "model": "DeepL", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 653.68 }, { - "input": "The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", + "input": "By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", "translatedText": "Фактът, че тя се движи наляво, докато буцата dh се движи надясно, означава, че тази промяна, синусът d на h, ще бъде някакво отрицателно число.", "model": "DeepL", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 910.4 }, { - "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice.", + "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice.", "translatedText": "Много ми се иска да мога да ви предложа да го направя, но се опасявам, че топката е във вашето поле, за да потърсите практиката.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/chinese/sentence_translations.json b/2017/chain-rule-and-product-rule/chinese/sentence_translations.json index ef3f90c3b..f6404eebe 100644 --- a/2017/chain-rule-and-product-rule/chinese/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/chinese/sentence_translations.json @@ -106,7 +106,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking.", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking.", "translatedText": "但是，值得通过真正思 考对两个函数之和求导的含义来热身这个例 子，因为乘积和函数组合的导数模式不会那 么简单，并且它们需要这种更深入的思考。", "model": "google_nmt", "from_community_srt": "但是这很值得通过这个例子去认真思考 对两个函数的和求导 是什么意思， 因为函数乘积 和组装函数的求导模式不会那么简单直接 ，", @@ -132,7 +132,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar.", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar.", "translatedText": "例如，假设 x 等于 0。如图 5 所示，正弦 图的高度由该竖条给出，x 平方抛物线的高度由该较小的 竖条给出。", "model": "google_nmt", "from_community_srt": "例如， 在x = 0.5时， 正弦图像的高度由这个bar表示 的x2抛物线的高度则由这个bar表示，", @@ -337,7 +337,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x.", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x.", "translatedText": "这给了我们新区域的三个小片段，底部 的一个薄矩形，其面积是其宽度，x的正弦，乘以其薄的高度，dx 平方，右侧的这个薄矩形，其面积是其高度，x的平方，乘以它的薄宽 度，x 的 d 正弦值。", "model": "google_nmt", "from_community_srt": "这给了我们三个新的小片段 面积：底部的一个薄的长方形 面积是它的宽度， sin（x）， 乘以其薄 身高， d（x2）;有一个薄的矩形 在右侧， 其面积是其高度x2， 乘以其细宽d（sin（x））。", @@ -355,7 +355,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0.", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean,", "translatedText": "它的面积最终与 dx 的平方成正比，正如我们 之前所看到的，当 dx 变为 0 时，面积可以忽略不计。", "model": "google_nmt", "from_community_srt": "但是我们可以忽略它， 因为它的区域会 最终与dx2成比例 由于DX变为0， 可以忽略不计。", @@ -373,7 +373,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small.", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small.", "translatedText": "就像那时一样，请记住，我在这里使用了一些强大的更改来绘制东西 ，以便我们实际上可以看到它们，但原则上 dx 是非常非常小的 东西，这意味着 x 的 dx 平方和 d 正弦也非常小很小。", "model": "google_nmt", "from_community_srt": "与x2图。 就像那样， 请记住， 我正在使用 如此一些有趣的改变来画东西 我们可以看到他们， 但原则上可以想到 dx非常小， 意思是d（x2）和d（sin（x）） 也非常非常小。", @@ -382,7 +382,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx.", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx.", "translatedText": "因此，应用我们对正弦和 x 平方的导数的了解，dx 平方的微小变 化将约为 2x 乘以 dx，而 x 的微小变化 d 正弦将约为 x 乘以 dx 的余弦。", "model": "google_nmt", "from_community_srt": "应用我们所了解的衍生物 正弦和x2， 微小的变化d（x2）是 2x * dx， 那么微小的变化d（sin（x））就是cos（x）dx。", @@ -534,7 +534,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3.", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3.", "translatedText": "我将放置三个不同的数轴，第一 行将保存 x 的值，第二行将保存 x 平方的值，第三行将保存 x 平方的正弦值，即函数x squared 使您从第 1 行到第 2 行，函数 sine 使您从第 2 行到第 3 行。", "model": "google_nmt", "from_community_srt": "我会提出三个号码。 最高的一个将保持x的值 第二个将代表x2的值， 那第三行将保持价值 SIN（X2）。 也就是说， 函数x2从行中获取 1到第2行， 函数正弦得到你 从第2行到第3行。", @@ -579,7 +579,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now.", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now.", "translatedText": "我们可以将其扩展为 2x 乘以 dx，对于我们的特定输入来说，这将是 2 乘以 1。5 倍 dx，但它有助于将内容写成 dx 平方，至少目前如此。", "model": "google_nmt", "from_community_srt": "你可以把它扩展为2x * dx， 这对于我们的 具体的输入长度将是2 *（1.5）* dx， 但它有助于将其保存为d（x2） 现在。", @@ -728,7 +728,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h.", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h.", "translatedText": "如果有任何两个函数，x 的 g 和 x 的 h，它们的组合的导数，x 的 h 的 g，就是 g 在 h 上计算的导数乘以 h 的导数。", "model": "google_nmt", "from_community_srt": "和x2。 如果你有两个函数g（x）和h（x）， 那么 其组成函数g（h（x））的导数 是在h（x）处评估的g的导数，", @@ -737,7 +737,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule.", + "input": "This pattern right here is what we usually call the chain rule.", "translatedText": "这种模式就是我们通常所说的链式法则。", "model": "google_nmt", "from_community_srt": "乘以h的导数。 这就是我们所说的“连锁规则”。", @@ -791,7 +791,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h.", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h.", "translatedText": "但我们可以对中间变量 h 求导数。", "model": "google_nmt", "from_community_srt": "这就是我们所有的东西 试图弄清楚， 但我们可以采取的 相对于中间产品而言 变量h。", @@ -836,7 +836,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about.", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about.", "translatedText": "请注意，这些 dh 相互抵消，并 为我们提供了最终输出的变化与通过特 定事件链带来的输入变化之间的比率。", "model": "google_nmt", "from_community_srt": "这个dh的取消给了之间的比例 最终产出的微小变化 微小的变化， 通过一定的输入 事件链，", diff --git a/2017/chain-rule-and-product-rule/dutch/sentence_translations.json b/2017/chain-rule-and-product-rule/dutch/sentence_translations.json index 8b14ebdbb..b72bf5e38 100644 --- a/2017/chain-rule-and-product-rule/dutch/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/dutch/sentence_translations.json @@ -544,7 +544,7 @@ "end": 592.58 }, { - "input": "So for the derivative, let's again start by nudging that x value by dx.", + "input": "So, for the derivative, let's again start by nudging that x value by some little dx.", "translatedText": "Dus laten we voor de afgeleide opnieuw beginnen met die x-waarde te verhogen met dx.", "model": "DeepL", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 653.68 }, { - "input": "The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", + "input": "By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", "translatedText": "Het feit dat het naar links beweegt terwijl de dh hobbel naar rechts gaat, betekent alleen dat deze verandering, d sinus van h, een negatief getal wordt.", "model": "DeepL", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 910.4 }, { - "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice.", + "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice.", "translatedText": "Ik zou echt willen dat ik kon aanbieden om dat voor je te doen, maar ik ben bang dat de bal bij jou ligt om de praktijk op te zoeken.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/english/captions.srt b/2017/chain-rule-and-product-rule/english/captions.srt index 498536a83..a68a8d98f 100644 --- a/2017/chain-rule-and-product-rule/english/captions.srt +++ b/2017/chain-rule-and-product-rule/english/captions.srt @@ -544,7 +544,7 @@ going to go to whatever sine of 9 happens to be. 137 00:09:54,900 --> 00:10:00,400 -So for the derivative, let's again start by nudging that x value by dx. +So, for the derivative, let's again start by nudging that x value by some little dx. 138 00:10:01,540 --> 00:10:04,800 @@ -591,12 +591,12 @@ This makes it easier to think about that third value, which is now pegged at sin Its change is d sine of h, the tiny change caused by the nudge dh. 149 -00:10:55,000 --> 00:11:00,081 -The fact that it's moving to the left while the dh bump is going to the right just +00:10:55,000 --> 00:11:00,134 +By the way, the fact that it's moving to the left while the dh bump is going to the right 150 -00:11:00,081 --> 00:11:05,040 -means that this change, d sine of h, is going to be some kind of negative number. +00:11:00,134 --> 00:11:05,040 +just means that this change, d sine of h, is going to be some kind of negative number. 151 00:11:06,140 --> 00:11:09,640 @@ -831,12 +831,12 @@ never going to substitute for practicing those mechanics yourself, and building up the muscles to do these computations yourself. 209 -00:15:11,240 --> 00:15:13,850 +00:15:11,240 --> 00:15:13,601 I really wish I could offer to do that for you, 210 -00:15:13,850 --> 00:15:17,440 -but I'm afraid the ball is in your court to seek out the practice. +00:15:13,601 --> 00:15:17,440 +but I'm afraid the ball is in your court, my friend, to seek out the practice. 211 00:15:18,040 --> 00:15:20,941 diff --git a/2017/chain-rule-and-product-rule/english/sentence_timings.json b/2017/chain-rule-and-product-rule/english/sentence_timings.json index 98da6a402..7ef904e7e 100644 --- a/2017/chain-rule-and-product-rule/english/sentence_timings.json +++ b/2017/chain-rule-and-product-rule/english/sentence_timings.json @@ -340,7 +340,7 @@ 592.58 ], [ - "So for the derivative, let's again start by nudging that x value by dx.", + "So, for the derivative, let's again start by nudging that x value by some little dx.", 594.9, 600.4 ], @@ -375,7 +375,7 @@ 653.68 ], [ - "The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", + "By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", 655.0, 665.04 ], @@ -540,7 +540,7 @@ 910.4 ], [ - "I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice.", + "I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice.", 911.24, 917.44 ], diff --git a/2017/chain-rule-and-product-rule/english/transcript.txt b/2017/chain-rule-and-product-rule/english/transcript.txt index 8ab5bf957..dac92aab0 100644 --- a/2017/chain-rule-and-product-rule/english/transcript.txt +++ b/2017/chain-rule-and-product-rule/english/transcript.txt @@ -66,14 +66,14 @@ I'll put up three different number lines, the top one is going to hold the value That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. As I shift around this value of x, maybe moving it up to the value 3, that second value stays pegged to whatever x squared is, in this case moving up to 9. That bottom value, being sine of x squared, is going to go to whatever sine of 9 happens to be. -So for the derivative, let's again start by nudging that x value by dx. +So, for the derivative, let's again start by nudging that x value by some little dx. I always think that it's helpful to think of x as starting at some actual concrete number, maybe 1.5 in this case. The resulting nudge to that second value, the change in x squared caused by such a dx, is dx squared. We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. In fact, I'm going to go one step further, give a new name to this x squared, maybe h, so instead of writing dx squared for this nudge, we write dh. This makes it easier to think about that third value, which is now pegged at sine of h. Its change is d sine of h, the tiny change caused by the nudge dh. -The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number. +By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number. Once again, we can use our knowledge of the derivative of the sine. This d sine of h is going to be about cosine of h times dh. That's what it means for the derivative of sine to be cosine. @@ -106,6 +106,6 @@ So those are the three basic tools to have in your belt to handle derivatives of You've got the sum rule, the product rule, and the chain rule. And I'll be honest with you, there is a big difference between knowing what the chain rule is and what the product rule is, and actually being fluent with applying them in even the most hairy of situations. Watching videos, any videos, about the mechanics of calculus is never going to substitute for practicing those mechanics yourself, and building up the muscles to do these computations yourself. -I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice. +I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice. What I can offer, and what I hope I have offered, is to show you where these rules actually come from. To show that they're not just something to be memorized and hammered away, but they're natural patterns, things that you too could have discovered just by patiently thinking through what a derivative actually means. \ No newline at end of file diff --git a/2017/chain-rule-and-product-rule/french/sentence_translations.json b/2017/chain-rule-and-product-rule/french/sentence_translations.json index 7ebc39d44..1a0086cca 100644 --- a/2017/chain-rule-and-product-rule/french/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/french/sentence_translations.json @@ -543,7 +543,7 @@ "end": 592.58 }, { - "input": "So for the derivative, let's again start by nudging that x value by dx.", + "input": "So, for the derivative, let's again start by nudging that x value by some little dx.", "translatedText": "Donc, pour la dérivée, commençons à nouveau par augmenter cette valeur x de dx.", "from_community_srt": "ira vers quel que soit le péché (9) est. Donc, pour le dérivé,", "n_reviews": 0, @@ -599,7 +599,7 @@ "end": 653.68 }, { - "input": "The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", + "input": "By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", "translatedText": "Le fait qu'il se déplace vers la gauche tandis que la bosse dh va vers la droite signifie simplement que ce changement, d sinus de h, va être une sorte de nombre négatif.", "from_community_srt": "D'ailleurs, le fait qu'il se déplace à gauche tandis que la bosse dh est à droite seulement des moyens que ce changement d (sin (h)) est un peu négatif nombre.", "n_reviews": 0, @@ -862,7 +862,7 @@ "end": 910.4 }, { - "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice.", + "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice.", "translatedText": "J'aimerais vraiment pouvoir proposer de le faire pour vous, mais j'ai bien peur que la balle soit dans votre camp pour rechercher l'entraînement.", "from_community_srt": "Je voudrais pouvoir offrir de le faire pour vous, mais Je crains que la balle est dans votre cour, mon ami, de rechercher la pratique.", "n_reviews": 0, @@ -885,4 +885,4 @@ "start": 924.14, "end": 934.56 } -] +] \ No newline at end of file diff --git a/2017/chain-rule-and-product-rule/german/sentence_translations.json b/2017/chain-rule-and-product-rule/german/sentence_translations.json index e986ff53d..df5699d5c 100644 --- a/2017/chain-rule-and-product-rule/german/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/german/sentence_translations.json @@ -610,7 +610,7 @@ "end": 592.58 }, { - "input": "So for the derivative, let's again start by nudging that x value by dx.", + "input": "So, for the derivative, let's again start by nudging that x value by some little dx.", "translatedText": "Für die Ableitung fangen wir also wieder damit an, dass wir den x-Wert um dx verschieben.", "model": "DeepL", "from_community_srt": "was auch immer die Sünde ist (9). Denken wir also noch einmal über die Ableitung nach diesen x-Wert um ein wenig dx zu stupsen,", @@ -673,7 +673,7 @@ "end": 653.68 }, { - "input": "The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", + "input": "By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", "translatedText": "Die Tatsache, dass er sich nach links bewegt, während der dh-Buckel nach rechts geht, bedeutet nur, dass diese Veränderung, d Sinus von h, eine Art negative Zahl sein wird.", "model": "DeepL", "from_community_srt": "Übrigens die Tatsache, dass es sich nach links bewegt während die dh Beule rechts ist nur gemeint dass diese Änderung d (sin (h)) etwas negativ ist Nummer.", @@ -970,7 +970,7 @@ "end": 910.4 }, { - "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice.", + "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice.", "translatedText": "Ich wünschte wirklich, ich könnte das für dich tun, aber ich fürchte, es liegt an dir, die Praxis aufzusuchen.", "model": "DeepL", "from_community_srt": "Ich wünschte, ich könnte anbieten, das für Sie zu tun, aber Ich fürchte, der Ball ist in deinem Spielfeld, mein Gott Freund,", diff --git a/2017/chain-rule-and-product-rule/hebrew/sentence_translations.json b/2017/chain-rule-and-product-rule/hebrew/sentence_translations.json index 1c5eddb3a..279454a18 100644 --- a/2017/chain-rule-and-product-rule/hebrew/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/hebrew/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking. ", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking. ", "translatedText": "אבל כדאי להתחמם עם הדוגמה הזו על ידי מחשבה אמיתית מה המשמעות של לקחת נגזרת של סכום של שתי פונקציות, שכן דפוסי הנגזרת של מוצרים והרכב פונקציות לא יהיו כל כך פשוטים, והם ידרשו סוג זה של חשיבה עמוקה יותר. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar. ", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar. ", "translatedText": "לדוגמה, נניח ש-x שווה ל-0.5, גובה גרף הסינוס ניתן על ידי פס אנכי זה, וגובה הפרבולה בריבוע x ניתן על ידי פס אנכי קטן יותר. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x. ", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x. ", "translatedText": "זה נותן לנו שלושה קטעים קטנים של שטח חדש, מלבן דק בתחתית ששטחו הוא הרוחב שלו, סינוס של x, כפול גובהו הדק, dx בריבוע, והמלבן הדק הזה מימין ששטחו הוא גובהו, x בריבוע, כפול הרוחב הדק שלו, סינוס d של x. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0. ", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean, ", "translatedText": "השטח שלו הוא בסופו של דבר פרופורציונלי ל-dx בריבוע, וכפי שראינו בעבר, זה הופך זניח ככל ש-dx הולך ל-0. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small. ", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small. ", "translatedText": "ובדיוק כמו אז, זכור שאני משתמש כאן בשינויים קצת בשרניים כדי לצייר דברים כדי שנוכל לראות אותם, אבל באופן עקרוני dx הוא משהו מאוד מאוד קטן, וזה אומר ש-Dx בריבוע ו-d סינוס של x הם גם מאוד קטן מאוד. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx. ", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx. ", "translatedText": "אז אם מיישמים את מה שאנחנו יודעים על הנגזרת של סינוס ו-x בריבוע, השינוי הזעיר של dx בריבוע יהיה בערך פי 2 כפול dx, והשינוי הזעיר הזה d סינוס של x יהיה בערך קוסינוס של x כפול dx. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", "translatedText": "אני אשים שלושה קווי מספר שונים, העליון יחזיק את הערך של x, השני יחזיק את הערך של x בריבוע, והשורה השלישית תכיל את הערך של סינוס של x בריבוע, כלומר, הפונקציה x בריבוע מעביר אותך משורה 1 לקו 2, והפונקציה סינוס מביאה אותך משורה 2 לקו 3. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", "translatedText": "נוכל להרחיב את זה לפי 2x dx, אשר עבור הקלט הספציפי שלנו יהיה 2 כפול 1.5 פעמים dx, אבל זה עוזר לשמור דברים כתובים כ-dx בריבוע, לפחות לעת עתה. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h. ", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h. ", "translatedText": "אם יש לך שתי פונקציות כלשהן, g של x ו-h של x, הנגזרת של הרכבן, g של h של x, היא הנגזרת של g המוערכת ב-h, כפולה בנגזרת של h. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule. ", + "input": "This pattern right here is what we usually call the chain rule. ", "translatedText": "דפוס זה הוא מה שאנו מכנים בדרך כלל כלל השרשרת. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h. ", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h. ", "translatedText": "אבל נוכל לקחת את הנגזרת ביחס למשתנה הביניים הזה, h. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", "translatedText": "שימו לב, ה-dh's האלה מבטלים ונותנים לנו יחס בין השינוי בתפוקה הסופית הזו לבין השינוי לקלט שדרך שרשרת אירועים מסוימת הביא אותו. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/hindi/sentence_translations.json b/2017/chain-rule-and-product-rule/hindi/sentence_translations.json index dc7be6604..ecf2ed572 100644 --- a/2017/chain-rule-and-product-rule/hindi/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/hindi/sentence_translations.json @@ -84,7 +84,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking.", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking.", "translatedText": "लेकिन दो कार्यों के योग का व्युत्पन्न लेने का क्या मतलब है, इस पर वास्तव में विचार करके इस उदाहरण को समझना उचित है, क्योंकि उत्पादों और फ़ंक्शन संरचना के लिए व्युत्पन्न पैटर्न इतने सीधे नहीं होंगे, और उन्हें इस तरह की आवश्यकता होगी गहरी सोच.", "n_reviews": 0, "start": 119.8, @@ -105,7 +105,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar.", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar.", "translatedText": "उदाहरण के लिए, मान लीजिए कि x 0 के बराबर है।5, साइन ग्राफ़ की ऊँचाई इस ऊर्ध्वाधर पट्टी द्वारा दी गई है, और x वर्ग परवलय की ऊँचाई इस छोटी ऊर्ध्वाधर पट्टी द्वारा दी गई है।", "n_reviews": 0, "start": 149.76, @@ -266,7 +266,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x.", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x.", "translatedText": "इससे हमें नए क्षेत्र के तीन छोटे टुकड़े मिलते हैं, तल पर एक पतला आयत जिसका क्षेत्रफल इसकी चौड़ाई, x की ज्या, इसकी पतली ऊंचाई का गुना, dx वर्ग है, और दाईं ओर यह पतला आयत है जिसका क्षेत्रफल इसकी ऊंचाई, x वर्ग है, इसकी पतली चौड़ाई का गुना, x की d साइन।", "n_reviews": 0, "start": 350.18, @@ -280,7 +280,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0.", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean,", "translatedText": "इसका क्षेत्रफल अंततः dx वर्ग के समानुपाती होता है, और जैसा कि हमने पहले देखा है, dx के 0 पर जाने पर यह नगण्य हो जाता है।", "n_reviews": 0, "start": 374.44, @@ -294,14 +294,14 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small.", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small.", "translatedText": "और ठीक उसी तरह, ध्यान रखें कि मैं चीजों को चित्रित करने के लिए यहां कुछ बड़े बदलावों का उपयोग कर रहा हूं ताकि हम वास्तव में उन्हें देख सकें, लेकिन सिद्धांत रूप में dx कुछ बहुत छोटा है, और इसका मतलब है कि dx का वर्ग और x का d साइन भी बहुत है बहुत छोटे से।", "n_reviews": 0, "start": 389.46, "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx.", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx.", "translatedText": "तो साइन और x वर्ग के व्युत्पन्न के बारे में हम जो जानते हैं उसे लागू करते हुए, वह छोटा परिवर्तन dx वर्ग लगभग 2x गुना dx होने वाला है, और x का वह छोटा परिवर्तन d साइन x गुना dx के कोसाइन के बारे में होने वाला है।", "n_reviews": 0, "start": 405.98, @@ -420,7 +420,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3.", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3.", "translatedText": "मैं तीन अलग-अलग संख्या रेखाएँ रखूँगा, शीर्ष रेखा में x का मान होगा, दूसरी में x वर्ग का मान होगा, और तीसरी पंक्ति में x वर्ग की ज्या का मान होगा, अर्थात फ़ंक्शन x स्क्वेर्ड आपको लाइन 1 से लाइन 2 तक ले जाता है, और फ़ंक्शन साइन आपको लाइन 2 से लाइन 3 तक ले जाता है।", "n_reviews": 0, "start": 553.32, @@ -455,7 +455,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now.", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now.", "translatedText": "हम इसे 2x गुना dx के रूप में विस्तारित कर सकते हैं, जो हमारे विशिष्ट इनपुट के लिए 2 गुना 1 होगा।5 गुना dx, लेकिन यह कम से कम अभी के लिए, dx वर्ग के रूप में लिखी गई चीजों को रखने में मदद करता है।", "n_reviews": 0, "start": 616.96, @@ -574,14 +574,14 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h.", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h.", "translatedText": "यदि आपके पास कोई दो फ़ंक्शन हैं, x का g और x का h, तो उनकी संरचना का व्युत्पन्न, x के h का g, h पर मूल्यांकन किए गए g का व्युत्पन्न है, जिसे h के व्युत्पन्न से गुणा किया जाता है।", "n_reviews": 0, "start": 749.74, "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule.", + "input": "This pattern right here is what we usually call the chain rule.", "translatedText": "इस पैटर्न को हम आमतौर पर श्रृंखला नियम कहते हैं।", "n_reviews": 0, "start": 767.14, @@ -623,7 +623,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h.", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h.", "translatedText": "लेकिन हम उस मध्यवर्ती चर, एच के संबंध में व्युत्पन्न ले सकते हैं।", "n_reviews": 0, "start": 810.42, @@ -658,7 +658,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about.", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about.", "translatedText": "ध्यान दें, वे dh रद्द हो जाते हैं और हमें उस अंतिम आउटपुट में परिवर्तन और इनपुट में परिवर्तन के बीच एक अनुपात देते हैं, जो घटनाओं की एक निश्चित श्रृंखला के माध्यम से इसे लाता है।", "n_reviews": 0, "start": 852.3, diff --git a/2017/chain-rule-and-product-rule/hungarian/sentence_translations.json b/2017/chain-rule-and-product-rule/hungarian/sentence_translations.json index ffe9a6a94..6ced327e7 100644 --- a/2017/chain-rule-and-product-rule/hungarian/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/hungarian/sentence_translations.json @@ -544,7 +544,7 @@ "end": 592.58 }, { - "input": "So for the derivative, let's again start by nudging that x value by dx.", + "input": "So, for the derivative, let's again start by nudging that x value by some little dx.", "translatedText": "Tehát a derivált esetében kezdjük megint azzal, hogy az x értéket dx-sel eltoljuk.", "model": "DeepL", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 653.68 }, { - "input": "The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", + "input": "By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", "translatedText": "Az a tény, hogy balra mozog, míg a dh dudor jobbra megy, csak azt jelenti, hogy ez a változás, a h d szinusza, valamilyen negatív szám lesz.", "model": "DeepL", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 910.4 }, { - "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice.", + "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice.", "translatedText": "Nagyon szeretném, ha felajánlhatnám, hogy megteszem ezt önnek, de attól tartok, hogy a labda az önök térfelén van, hogy felkeressék a gyakorlatot.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/indonesian/sentence_translations.json b/2017/chain-rule-and-product-rule/indonesian/sentence_translations.json index f6354cb7a..78f3f807b 100644 --- a/2017/chain-rule-and-product-rule/indonesian/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/indonesian/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking. ", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking. ", "translatedText": "Namun ada baiknya kita mulai dengan contoh ini dengan benar-benar memikirkan apa yang dimaksud dengan mengambil turunan dari penjumlahan dua fungsi, karena pola turunan untuk komposisi perkalian dan fungsi tidak akan begitu mudah, dan memerlukan hal-hal seperti ini. pemikiran yang lebih mendalam. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar. ", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar. ", "translatedText": "Misalnya, di x sama dengan 0.5, tinggi grafik sinus ditentukan oleh batang vertikal ini, dan tinggi parabola x kuadrat ditentukan oleh batang vertikal yang lebih kecil. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x. ", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x. ", "translatedText": "Hasilnya adalah tiga potongan kecil luas baru, sebuah persegi panjang tipis di bagian bawah yang luasnya adalah lebarnya, sinus x, dikalikan tinggi tipisnya, dx kuadrat, dan persegi panjang tipis di sebelah kanan yang luasnya adalah tingginya, x kuadrat, kali lebar tipisnya, d sinus x. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0. ", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean, ", "translatedText": "Luasnya pada akhirnya sebanding dengan dx kuadrat, dan seperti yang telah kita lihat sebelumnya, luasnya menjadi dapat diabaikan jika dx bernilai 0. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small. ", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small. ", "translatedText": "Dan seperti itu, perlu diingat bahwa saya menggunakan perubahan yang agak besar di sini untuk menggambar sesuatu sehingga kita benar-benar dapat melihatnya, tetapi pada prinsipnya dx adalah sesuatu yang sangat sangat kecil, dan itu berarti dx kuadrat dan d sinus dari x juga sangat kecil. sangat kecil. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx. ", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx. ", "translatedText": "Jadi dengan menerapkan apa yang kita ketahui tentang turunan sinus dan x kuadrat, perubahan kecil dx kuadrat akan menjadi sekitar 2x kali dx, dan perubahan kecil d sinus dari x akan menjadi sekitar cosinus x kali dx. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", "translatedText": "Saya akan membuat tiga garis bilangan yang berbeda, baris paling atas berisi nilai x, baris kedua berisi nilai x kuadrat, dan baris ketiga berisi nilai sinus x kuadrat, yaitu fungsi x kuadrat membawa Anda dari baris 1 ke baris 2, dan fungsi sinus membawa Anda dari baris 2 ke baris 3. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", "translatedText": "Kita dapat memperluasnya menjadi 2x kali dx, yang untuk masukan spesifik kita adalah 2 kali 1.5 kali dx, tetapi akan membantu jika semuanya tetap ditulis sebagai dx kuadrat, setidaknya untuk saat ini. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h. ", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h. ", "translatedText": "Jika Anda mempunyai dua fungsi, g dari x dan h dari x, turunan komposisinya, g dari h dari x, adalah turunan dari g yang dievaluasi pada h, dikalikan dengan turunan dari h. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule. ", + "input": "This pattern right here is what we usually call the chain rule. ", "translatedText": "Pola inilah yang biasa kita sebut dengan aturan rantai. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h. ", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h. ", "translatedText": "Namun kita dapat mengambil turunannya terhadap variabel perantara tersebut, h. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", "translatedText": "Perhatikan, dh tersebut saling meniadakan dan memberi kita rasio antara perubahan keluaran akhir dan perubahan masukan yang, melalui rangkaian peristiwa tertentu, menghasilkan hal tersebut. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/italian/sentence_translations.json b/2017/chain-rule-and-product-rule/italian/sentence_translations.json index c205eb90b..ebe26a32c 100644 --- a/2017/chain-rule-and-product-rule/italian/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/italian/sentence_translations.json @@ -611,7 +611,7 @@ "end": 592.58 }, { - "input": "So for the derivative, let's again start by nudging that x value by dx.", + "input": "So, for the derivative, let's again start by nudging that x value by some little dx.", "translatedText": "Quindi, per quanto riguarda la derivata, iniziamo ancora una volta a modificare il valore di x con dx.", "model": "DeepL", "from_community_srt": "passerà a qualunque valore sia il sin(9). Quindi, per la derivata, di nuovo,", @@ -674,7 +674,7 @@ "end": 653.68 }, { - "input": "The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", + "input": "By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", "translatedText": "Il fatto che si muova verso sinistra mentre l'urto dh va verso destra significa che questo cambiamento, il seno d di h, sarà un numero negativo.", "model": "DeepL", "from_community_srt": "A proposito, il fatto che si muova a sinistra mentre dh bump (?) è a destra significa solo che questo cambiamento d (sin(h)) è un numero negativo.", @@ -964,7 +964,7 @@ "end": 910.4 }, { - "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice.", + "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice.", "translatedText": "Vorrei davvero potermi offrire di farlo per te, ma temo che sia tu a dover cercare la pratica.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/japanese/sentence_translations.json b/2017/chain-rule-and-product-rule/japanese/sentence_translations.json index 8987f9290..6cd2d77a7 100644 --- a/2017/chain-rule-and-product-rule/japanese/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/japanese/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking.", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking.", "translatedText": "ただし、積と関数合成の導関数パ ターンはそれほど単純ではなく、この種の関数が必要になるため、2 つの関数の和の導関数を取ることが何を意味するのかをよく考え て、この例でウォームアップする価値があります。 より深い思考。", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar.", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar.", "translatedText": "たとえば、x が 0 に等しいとします。図５において、正弦グラフの高 さはこの縦棒によって与えられ、ｘ二乗放物線の高さはこの小さな縦棒によっ て与えられる。", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x.", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x.", "translatedText": "これにより、新しい領域の 3 つの小さなスニペッ トが得られます。 下部の薄い長方形の面積は、幅と x の正弦に、薄い高さの dx の 2 乗を掛けたものです。 右側の薄い長方形の面積は、高さの x の 2 乗です。 その細い 幅に x の d サインを掛けます。", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0.", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean,", "translatedText": "その面積は最終的には dx の 2 乗に比例し、これ までに見たように、dx が 0 に近づくと無視できるほどになります。", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small.", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small.", "translatedText": "そして、その時と同じように、ここでも実際に見えるように描画するために多少大胆な変更を使用し ていることに注意してください。 しかし、原則として、dx は非常に非常に小さいものであり、 つまり、x の dx 2 乗と d サインも非常に小さいことを意味します。 非常に少ない。", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx.", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx.", "translatedText": "したがって、サインと x の 2 乗の微分について私たちが知っていることを適用すると、その小 さな変化 dx の 2 乗は dx の約 2x 倍になり、x の小さな変化 d サインは dx の x 倍のコサインになります。", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3.", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3.", "translatedText": "3 つの異なる数直線を用意します。 一番上の 直線は x の値を保持し、2 番目の直線は x の 2 乗の値を保持し、3 番目の直線は x の 2 乗の正弦の値、つまり関数を保持します。 x 2 乗を使用すると 1 行目 から 2 行目まで移動でき、関数 sine を使用すると 2 行目から 3 行目まで移動 できます。", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now.", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now.", "translatedText": "これを 2x x dx として拡張できます。 これは、特定の入力では 2 x 1 になります。d x の 5 倍ですが、少なくとも現時点では、物事を dx の 2 乗として記述するのに役立ちます。", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h.", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h.", "translatedText": "x の g と x の h と いう 2 つの関数がある場合、それらの合成の導関数、g of h of x は、h で評価された g の導関数に h の導関数を乗算したものになります。", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule.", + "input": "This pattern right here is what we usually call the chain rule.", "translatedText": "このパターンは、通常、連鎖規則と呼ばれるものです。", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h.", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h.", "translatedText": "しかし、中間変数 h に関して導関数を求めることはできます。", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about.", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about.", "translatedText": "これらの dh が相殺され、最終的な出力の変化と 、特定の一連のイベントによってそれがもたらされた入力 の変化との間の比率が得られることに注意してください。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/korean/sentence_translations.json b/2017/chain-rule-and-product-rule/korean/sentence_translations.json index a4067f269..68be4bd07 100644 --- a/2017/chain-rule-and-product-rule/korean/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/korean/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking.", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking.", "translatedText": "하지만 두 함수의 합에 대한 도함수를 구하는 것이 무엇을 의미하는지 곰곰이 생각해 보면 이 예를 통해 워밍업할 가치가 있습니다. 곱셈과 함수 구성에 대한 도함수 패턴은 그렇게 간단하지 않고 이런 유형의 함수가 필요하기 때문입니다. 더 깊은 생각.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar.", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar.", "translatedText": "예를 들어 x가 0이라고 가정해 보겠습니다.도 5에서, 사인 그래프의 높이는 이 수직 막대에 의해 주어지고, x 제곱 포물선의 높이는 이 작은 수직 막대에 의해 주어진다.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x.", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x.", "translatedText": "그러면 새 영역에 대한 세 개의 작은 조각이 제공됩니다. 아래쪽의 얇은 직사각형은 너비(사인 x)에 얇은 높이(dx 제곱)를 곱한 면적이고, 오른쪽의 이 얇은 직사각형(넓이는 높이 x 제곱)입니다. 얇은 너비를 곱한 x의 사인입니다.", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0.", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean,", "translatedText": "그 면적은 궁극적으로 dx 제곱에 비례하며, 이전에 본 것처럼 dx가 0으로 갈수록 무시할 수 있게 됩니다.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small.", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small.", "translatedText": "그리고 그 때와 마찬가지로 여기서는 우리가 실제로 볼 수 있도록 그리기 위해 다소 강력한 변화를 사용하고 있다는 점을 명심하십시오. 그러나 원칙적으로 dx는 매우 작은 값이며 이는 dx 제곱과 x의 d 사인도 매우 작다는 것을 의미합니다. 매우 작은.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx.", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx.", "translatedText": "따라서 우리가 사인과 x 제곱의 도함수에 대해 알고 있는 것을 적용하면 dx 제곱의 작은 변화는 약 2x 곱하기 dx가 될 것이고 x의 작은 변화 d 사인은 대략 코사인 x 곱하기 dx가 될 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3.", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3.", "translatedText": "세 개의 서로 다른 수직선을 표시하겠습니다. 맨 위의 줄은 x 값, 두 번째 줄은 x 제곱 값, 세 번째 줄은 x 제곱의 사인 값, 즉 함수를 담을 것입니다. x squared는 1행에서 2행으로 이동하고, sine 함수는 2행에서 3행으로 이동합니다.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now.", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now.", "translatedText": "이를 2x dx로 확장할 수 있으며, 특정 입력의 경우 2x 1이 됩니다.5 곱하기 dx. 하지만 적어도 지금은 dx 제곱으로 기록하는 것이 도움이 됩니다.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h.", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h.", "translatedText": "두 개의 함수(x의 g와 x의 h)가 있는 경우 해당 구성의 도함수인 x의 h의 g는 h에서 평가된 g의 도함수에 h의 도함수를 곱한 값입니다.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule.", + "input": "This pattern right here is what we usually call the chain rule.", "translatedText": "이 패턴은 우리가 일반적으로 체인 규칙이라고 부르는 패턴입니다.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h.", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h.", "translatedText": "그러나 우리는 중간 변수 h에 대해 미분을 취할 수 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about.", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about.", "translatedText": "dh는 상쇄되어 최종 출력의 변화와 특정 일련의 사건을 통해 발생하는 입력의 변화 사이의 비율을 제공합니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/marathi/sentence_translations.json b/2017/chain-rule-and-product-rule/marathi/sentence_translations.json index 2ae4d2c85..331479ad5 100644 --- a/2017/chain-rule-and-product-rule/marathi/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/marathi/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking.", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking.", "translatedText": "परंतु दोन फंक्शन्सच्या बेरीजचे डेरिव्हेटिव्ह घेणे म्हणजे काय याचा विचार करून या उदाहरणासह वार्म अप करणे फायदेशीर आहे, कारण उत्पादने आणि फंक्शन कंपोझिशनसाठी व्युत्पन्न नमुने इतके सरळ नसतील आणि त्यांना या प्रकारची आवश्यकता असेल. सखोल विचार.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar.", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar.", "translatedText": "उदाहरणार्थ, x बरोबर 0 असे म्हणू.5, साइन आलेखाची उंची या उभ्या पट्टीने दिली आहे आणि x चौरस पॅराबोलाची उंची या लहान उभ्या पट्टीने दिली आहे.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x.", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x.", "translatedText": "हे आपल्याला नवीन क्षेत्रफळाचे तीन छोटे तुकडे देते, तळाशी एक पातळ आयत ज्याचे क्षेत्रफळ त्याची रुंदी आहे, x चा साइन, त्याच्या पातळ उंचीच्या पट, dx चौरस, आणि उजवीकडे असलेला हा पातळ आयत ज्याचे क्षेत्रफळ त्याची उंची आहे, x वर्ग, त्याच्या पातळ रुंदीच्या पट, x च्या d साइन.", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0.", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean,", "translatedText": "त्याचे क्षेत्रफळ शेवटी dx वर्गाच्या प्रमाणात आहे, आणि जसे आपण आधी पाहिले आहे, dx 0 वर गेल्यावर ते नगण्य होते.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small.", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small.", "translatedText": "आणि त्याचप्रमाणे, हे लक्षात ठेवा की मी येथे काही गोमांस बदल गोष्टी काढण्यासाठी वापरत आहे जेणेकरुन आपण ते प्रत्यक्षात पाहू शकू, परंतु तत्वतः dx ही खूप लहान गोष्ट आहे आणि याचा अर्थ x चा dx वर्ग आणि d साइन देखील खूप आहेत. खूप लहान.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx.", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx.", "translatedText": "म्हणून आपल्याला साइन आणि x स्क्वेअरच्या व्युत्पन्नाबद्दल जे माहिती आहे ते लागू केल्यास, तो लहान बदल dx वर्ग dx च्या 2x पट असेल आणि x चा d साइन हा x गुणा dx च्या कोसाइन बद्दल असेल.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3.", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3.", "translatedText": "मी तीन वेगवेगळ्या क्रमांकाच्या रेषा ठेवीन, वरच्या ओळीत x चे मूल्य असेल, दुसऱ्यामध्ये x वर्गाचे मूल्य असेल आणि तिसऱ्या ओळीत x वर्गाच्या sine चे मूल्य असेल, म्हणजेच फंक्शन. x वर्ग तुम्हाला ओळ 1 पासून ओळ 2 पर्यंत आणतो आणि फंक्शन sine तुम्हाला ओळ 2 पासून ओळ 3 पर्यंत मिळवतो.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now.", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now.", "translatedText": "आम्ही हे 2x पट dx म्हणून वाढवू शकतो, जे आमच्या विशिष्ट इनपुटसाठी 2 पट 1 असेल.5 वेळा dx, परंतु हे किमान आत्तासाठी dx स्क्वेअर म्हणून लिहिलेल्या गोष्टी ठेवण्यास मदत करते.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h.", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h.", "translatedText": "तुमच्याकडे x चे g आणि x चे h ही दोन कार्ये असल्यास, त्यांच्या रचनेचे व्युत्पन्न, x च्या h चा g, h वर मूल्यमापन केलेले g चे व्युत्पन्न आहे, h च्या व्युत्पन्नाने गुणाकार केला आहे.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule.", + "input": "This pattern right here is what we usually call the chain rule.", "translatedText": "या पॅटर्नला आपण सहसा साखळी नियम म्हणतो.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h.", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h.", "translatedText": "पण त्या इंटरमीडिएट व्हेरिएबलच्या संदर्भात आपण व्युत्पन्न घेऊ शकतो, h.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about.", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about.", "translatedText": "लक्ष द्या, त्या dh च्या रद्द करा आणि आम्हाला त्या अंतिम आउटपुटमधील बदल आणि इनपुटमधील बदल यांच्यातील एक गुणोत्तर द्या, ज्याने विशिष्ट घटनांच्या साखळीद्वारे ते घडवून आणले.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/persian/sentence_translations.json b/2017/chain-rule-and-product-rule/persian/sentence_translations.json index 4c1822cd1..6a17982f2 100644 --- a/2017/chain-rule-and-product-rule/persian/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/persian/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking. ", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking. ", "translatedText": "اما ارزش آن را دارد که با این مثال به معنای واقعی گرفتن مشتق از مجموع دو تابع فکر کنید، زیرا الگوهای مشتق برای محصولات و ترکیب تابع چندان ساده نیستند و به این نوع نیاز دارند. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar. ", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar. ", "translatedText": "به عنوان مثال، فرض کنید در x برابر با 0 است. در شکل 5، ارتفاع نمودار سینوسی با این میله عمودی، و ارتفاع سهمی مربع x توسط این میله عمودی کوچکتر داده می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x. ", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x. ", "translatedText": "این به ما سه قطعه کوچک از ناحیه جدید می دهد، یک مستطیل نازک در پایین که مساحت آن عرض، سینوس x، ضربدر ارتفاع نازک آن، dx مربع است، و این مستطیل نازک در سمت راست که مساحت آن ارتفاع آن است، x مربع، برابر عرض نازک آن، d سینوس x. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0. ", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean, ", "translatedText": "مساحت آن در نهایت متناسب با dx مربع است و همانطور که قبلاً دیدیم، با رفتن dx به 0 این مقدار ناچیز می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small. ", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small. ", "translatedText": "و دقیقاً مانند آن زمان، به خاطر داشته باشید که من از تغییرات کمی در اینجا برای ترسیم چیزها استفاده می کنم تا بتوانیم آنها را واقعاً ببینیم، اما در اصل dx چیزی بسیار بسیار کوچک است، و این بدان معناست که dx مربع و d سینوس x نیز بسیار هستند. خیلی کوچک. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx. ", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx. ", "translatedText": "بنابراین، با اعمال آنچه در مورد مشتق سینوس و x مربع می دانیم، آن تغییر کوچک dx مربع حدود 2x برابر dx خواهد بود، و آن تغییر کوچک d sine از x حدود کسینوس x برابر dx خواهد بود. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", "translatedText": "من سه خط اعداد مختلف را قرار می‌دهم، خط بالایی مقدار x را نگه می‌دارد، دومی مقدار x را در مجذور نگه می‌دارد و خط سوم مقدار سینوس x را در مجذور نگه می‌دارد، یعنی تابع x مربع شما را از خط 1 به خط 2 می برد و تابع سینوس شما را از خط 2 به خط 3 می رساند. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", "translatedText": "ما می توانیم این را به صورت 2x ضربدر dx گسترش دهیم، که برای ورودی خاص ما 2 برابر 1 خواهد بود. 5 برابر dx، اما کمک می کند تا چیزهایی که به صورت dx مربع نوشته می شوند، حداقل در حال حاضر حفظ شوند. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h. ", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h. ", "translatedText": "اگر هر دو تابع g از x و h از x دارید، مشتق ترکیب آنها، g از h از x، مشتق g ارزیابی شده در h است که در مشتق h ضرب می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule. ", + "input": "This pattern right here is what we usually call the chain rule. ", "translatedText": "این الگو همان چیزی است که ما معمولاً به آن قانون زنجیره ای می گوییم. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h. ", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h. ", "translatedText": "اما می‌توانیم مشتق را با توجه به آن متغیر میانی، h، بگیریم. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", "translatedText": "توجه کنید، آن dh ها لغو می شوند و نسبتی بین تغییر در آن خروجی نهایی و تغییر به ورودی که از طریق زنجیره خاصی از رویدادها باعث شده است، به ما می دهند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/polish/sentence_translations.json b/2017/chain-rule-and-product-rule/polish/sentence_translations.json index 3c2f9174b..b6842a051 100644 --- a/2017/chain-rule-and-product-rule/polish/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/polish/sentence_translations.json @@ -541,7 +541,7 @@ "end": 592.58 }, { - "input": "So for the derivative, let's again start by nudging that x value by dx.", + "input": "So, for the derivative, let's again start by nudging that x value by some little dx.", "translatedText": "", "from_community_srt": "Aby obliczyć pochodną, zmieńmy x o małe dx.", "n_reviews": 0, @@ -597,7 +597,7 @@ "end": 653.68 }, { - "input": "The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", + "input": "By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", "translatedText": "", "from_community_srt": "Przy okazji, wartość sin(h) przesuwa się w lewo, gdy h przesuwamy w prawo. Oznacza to, że d(sin(h)) jest liczbą ujemną.", "n_reviews": 0, @@ -860,7 +860,7 @@ "end": 910.4 }, { - "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice.", + "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice.", "translatedText": "", "from_community_srt": "Bardzo chciałbym ci w tym pomóc, ale w tym przypadku musisz się sam wykazać.", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/portuguese/sentence_translations.json b/2017/chain-rule-and-product-rule/portuguese/sentence_translations.json index 1754e3247..55df0b3a6 100644 --- a/2017/chain-rule-and-product-rule/portuguese/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/portuguese/sentence_translations.json @@ -611,7 +611,7 @@ "end": 592.58 }, { - "input": "So for the derivative, let's again start by nudging that x value by dx.", + "input": "So, for the derivative, let's again start by nudging that x value by some little dx.", "translatedText": "Então, para a derivada, vamos novamente começar ajustando o valor de x por dx.", "model": "google_nmt", "from_community_srt": "Logo, para a derivada, vamos voltar a pensar em peteleco no valor de x por um infinitesimal dx.", @@ -674,7 +674,7 @@ "end": 653.68 }, { - "input": "The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", + "input": "By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", "translatedText": "O fato de ele estar se movendo para a esquerda enquanto a colisão dh está indo para a direita significa apenas que essa mudança, d seno de h, será algum tipo de número negativo.", "model": "google_nmt", "from_community_srt": "E a propósito, o fato que o movimento para a esquerda enquanto o dh empurra para a direita significa que essa mundança d(sin(h)) é um número negativo.", @@ -971,7 +971,7 @@ "end": 910.4 }, { - "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice.", + "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice.", "translatedText": "Eu realmente gostaria de poder me oferecer para fazer isso por você, mas temo que a bola esteja do seu lado para buscar a prática.", "model": "google_nmt", "from_community_srt": "Eu realmente gostaria de poder oferecer isso para você, mas temo que a bola esteja em sua quadra, meu amigo, para procurar a prática.", diff --git a/2017/chain-rule-and-product-rule/russian/sentence_translations.json b/2017/chain-rule-and-product-rule/russian/sentence_translations.json index 36a54c58a..6ddf0f2f7 100644 --- a/2017/chain-rule-and-product-rule/russian/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/russian/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking. ", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking. ", "translatedText": "Но стоит потренироваться на этом примере, хорошенько поразмыслив над тем, что значит взять производную от суммы двух функций, поскольку шаблоны производных для произведений и композиции функций не будут такими простыми и потребуют такого рода более глубокое мышление. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar. ", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar. ", "translatedText": "Например, скажем, что x равен 0.5 высота синусоидального графика определяется этой вертикальной полосой, а высота параболы в квадрате x определяется этой меньшей вертикальной полосой. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x. ", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x. ", "translatedText": "Это дает нам три маленьких фрагмента новой области: тонкий прямоугольник внизу, площадь которого равна его ширине, синусу x, умноженному на его тонкую высоту, dx в квадрате, и этот тонкий прямоугольник справа, площадь которого равна его высоте, x в квадрате, умноженное на его тонкую ширину, d синус x. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0. ", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean, ", "translatedText": "Его площадь в конечном счете пропорциональна квадрату dx, и, как мы видели ранее, она становится незначительной, когда dx стремится к 0. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small. ", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small. ", "translatedText": "И так же, как и тогда, имейте в виду, что я использую здесь несколько существенные изменения, чтобы рисовать вещи, чтобы мы могли их действительно видеть, но в принципе dx - это что-то очень-очень маленькое, а это означает, что dx в квадрате и d синус x также очень малы. очень маленький. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx. ", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx. ", "translatedText": "Итак, применяя то, что мы знаем о производной синуса и x в квадрате, это крошечное изменение dx в квадрате будет примерно в 2 раза dx, и это крошечное изменение d синуса x будет примерно в 2 раза больше косинуса x, умноженного на dx. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", "translatedText": "Я помещу три разные числовые строки: верхняя будет содержать значение x, вторая будет содержать значение x в квадрате, а третья строка будет содержать значение синуса x в квадрате, то есть функцию x в квадрате переводит вас из строки 1 в строку 2, а функция синус переводит из строки 2 в строку 3. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", "translatedText": "Мы могли бы расширить это значение как 2x, умноженное на dx, что для наших конкретных входных данных будет равно 2, умноженному на 1.5 умножить на dx, но это помогает сохранить запись в квадрате dx, по крайней мере, на данный момент. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h. ", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h. ", "translatedText": "Если у вас есть какие-либо две функции, g от x и h от x, производная их композиции g от h от x — это производная от g, вычисленная по h, умноженная на производную от h. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule. ", + "input": "This pattern right here is what we usually call the chain rule. ", "translatedText": "Эту закономерность мы обычно называем правилом цепочки. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h. ", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h. ", "translatedText": "Но мы могли бы взять производную по этой промежуточной переменной h. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", "translatedText": "Обратите внимание: эти dh компенсируются и дают нам соотношение между изменением конечного результата и изменением входных данных, которое в результате определенной цепочки событий привело к этому. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/spanish/sentence_translations.json b/2017/chain-rule-and-product-rule/spanish/sentence_translations.json index 0c9dc375a..243f964c1 100644 --- a/2017/chain-rule-and-product-rule/spanish/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/spanish/sentence_translations.json @@ -541,7 +541,7 @@ "end": 592.58 }, { - "input": "So for the derivative, let's again start by nudging that x value by dx.", + "input": "So, for the derivative, let's again start by nudging that x value by some little dx.", "translatedText": "Entonces, para la derivada, comencemos nuevamente empujando ese valor de x por dx.", "from_community_srt": "Así que para la derivada, vamos a pensar de nuevo,", "n_reviews": 0, @@ -597,7 +597,7 @@ "end": 653.68 }, { - "input": "The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", + "input": "By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", "translatedText": "El hecho de que se mueva hacia la izquierda mientras el relieve dh va hacia la derecha simplemente significa que este cambio, d seno de h, será una especie de número negativo.", "from_community_srt": "Por cierto, el hecho de que se está moviendo hacia la izquierda mientras que la protuberancia dh va hacia a la derecha siginifica", "n_reviews": 0, @@ -861,7 +861,7 @@ "end": 910.4 }, { - "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice.", + "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice.", "translatedText": "Realmente desearía poder ofrecerme a hacer eso por usted, pero me temo que la pelota está en su tejado para buscar la práctica.", "from_community_srt": "Me gustaría poder ofrecerme a hacer eso por ti, pero Me temo que la pelota está en tu tejado, mi amigo, ve a buscar a la práctica.", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/tagalog/sentence_translations.json b/2017/chain-rule-and-product-rule/tagalog/sentence_translations.json index 3cebf90d0..cf0edf10a 100644 --- a/2017/chain-rule-and-product-rule/tagalog/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/tagalog/sentence_translations.json @@ -544,7 +544,7 @@ "end": 592.58 }, { - "input": "So for the derivative, let's again start by nudging that x value by dx.", + "input": "So, for the derivative, let's again start by nudging that x value by some little dx.", "translatedText": "Kaya para sa derivative, muli nating simulan sa pamamagitan ng pag-nudging na x value ng dx.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 653.68 }, { - "input": "The fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", + "input": "By the way, the fact that it's moving to the left while the dh bump is going to the right just means that this change, d sine of h, is going to be some kind of negative number.", "translatedText": "Ang katotohanan na ito ay gumagalaw sa kaliwa habang ang dh bump ay papunta sa kanan ay nangangahulugan lamang na ang pagbabagong ito, d sine ng h, ay magiging isang uri ng negatibong numero.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 910.4 }, { - "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court to seek out the practice.", + "input": "I really wish I could offer to do that for you, but I'm afraid the ball is in your court, my friend, to seek out the practice.", "translatedText": "Gusto ko talagang mag-alok na gawin iyon para sa iyo, ngunit natatakot ako na ang bola ay nasa iyong korte upang hanapin ang pagsasanay.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/tamil/sentence_translations.json b/2017/chain-rule-and-product-rule/tamil/sentence_translations.json index cc349e20f..ad1f50c88 100644 --- a/2017/chain-rule-and-product-rule/tamil/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/tamil/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking.", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking.", "translatedText": "ஆனால் இரண்டு செயல்பாடுகளின் ஒரு தொகையின் வழித்தோன்றலை எடுப்பது என்றால் என்ன என்று சிந்தித்து இந்த உதாரணத்துடன் வெப்பமடைவது மதிப்புக்குரியது, ஏனெனில் தயாரிப்புகள் மற்றும் செயல்பாட்டு கலவைக்கான வழித்தோன்றல் வடிவங்கள் அவ்வளவு நேரடியானவை அல்ல, மேலும் அவை இந்த வகையானவை தேவைப்படும். ஆழ்ந்த சிந்தனை.", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar.", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar.", "translatedText": "எடுத்துக்காட்டாக, x க்கு சமம் 0 என்று வைத்துக் கொள்வோம்.5, சைன் வரைபடத்தின் உயரம் இந்த செங்குத்து பட்டையால் வழங்கப்படுகிறது, மேலும் x ஸ்கொயர் பாரபோலாவின் உயரம் இந்த சிறிய செங்குத்து பட்டியால் வழங்கப்படுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x.", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x.", "translatedText": "இது புதிய பகுதியின் மூன்று சிறிய துணுக்குகளை நமக்கு வழங்குகிறது, கீழே ஒரு மெல்லிய செவ்வகம் அதன் அகலம், x இன் சைன், அதன் மெல்லிய உயரத்தின் மடங்கு, dx சதுரம் மற்றும் வலதுபுறத்தில் இந்த மெல்லிய செவ்வகம் அதன் உயரம், x சதுரம், அதன் மெல்லிய அகலத்தின் மடங்கு, x இன் d சைன்.", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0.", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean,", "translatedText": "அதன் பரப்பளவு இறுதியில் dx ஸ்கொயர்க்கு விகிதாசாரமாகும், மேலும் நாம் முன்பு பார்த்தது போல், dx 0 க்கு செல்லும் போது அது மிகக் குறைவாக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small.", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small.", "translatedText": "அதைப் போலவே, விஷயங்களை வரைவதற்கு நான் இங்கு சற்றே மாட்டிறைச்சி மாற்றங்களைப் பயன்படுத்துகிறேன் என்பதை நினைவில் கொள்ளுங்கள், அதனால் அவற்றை நாம் உண்மையில் பார்க்க முடியும், ஆனால் கொள்கையளவில் dx என்பது மிகச் சிறிய ஒன்று, அதாவது dx ஸ்கொயர் மற்றும் x இன் d சைன் ஆகியவையும் மிக அதிகம். மிகவும் சிறியது.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx.", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx.", "translatedText": "எனவே சைன் மற்றும் x ஸ்கொயர்டின் வழித்தோன்றலைப் பற்றி நமக்குத் தெரிந்ததைப் பயன்படுத்தினால், அந்த சிறிய மாற்றம் dx ஸ்கொயர் 2x மடங்கு dx ஆக இருக்கும், மேலும் x இன் சிறிய மாற்றம் d sine x மடங்கு dx இன் கொசைனைப் பற்றியதாக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3.", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3.", "translatedText": "நான் மூன்று வெவ்வேறு எண் கோடுகளை வைப்பேன், மேலே உள்ள ஒன்று x இன் மதிப்பைக் கொண்டிருக்கும், இரண்டாவது x ஸ்கொயர் மதிப்பைக் கொண்டிருக்கும், மூன்றாவது வரி x ஸ்கொயர்டின் சைனின் மதிப்பைக் கொண்டிருக்கும், அதாவது செயல்பாடு x ஸ்கொயர் உங்களை வரி 1 முதல் வரி 2 வரையும், சைன் செயல்பாடு வரி 2 முதல் வரி 3 வரையும் அழைத்துச் செல்லும்.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now.", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now.", "translatedText": "இதை 2x மடங்கு dx ஆக விரிவாக்கலாம், இது நமது குறிப்பிட்ட உள்ளீட்டிற்கு 2 முறை 1 ஆக இருக்கும்.5 மடங்கு dx, ஆனால் குறைந்தபட்சம் இப்போதைக்கு dx ஸ்கொயர் என்று எழுதப்பட்ட விஷயங்களை வைத்திருக்க உதவுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h.", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h.", "translatedText": "உங்களிடம் ஏதேனும் இரண்டு செயல்பாடுகள் இருந்தால், x இன் g மற்றும் x இன் h, அவற்றின் கலவையின் வழித்தோன்றல், g இன் h இன் x, h இல் மதிப்பிடப்பட்ட g இன் வழித்தோன்றலாகும், இது h இன் வழித்தோன்றலால் பெருக்கப்படுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule.", + "input": "This pattern right here is what we usually call the chain rule.", "translatedText": "இந்த மாதிரியை நாம் வழக்கமாக சங்கிலி விதி என்று அழைக்கிறோம்.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h.", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h.", "translatedText": "ஆனால் அந்த இடைநிலை மாறி, h ஐப் பொறுத்து நாம் வழித்தோன்றலை எடுக்கலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about.", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about.", "translatedText": "கவனிக்கவும், அந்த டிஹெச்கள் ரத்து செய்யப்பட்டு, அந்த இறுதி வெளியீட்டின் மாற்றத்திற்கும் உள்ளீட்டிற்கான மாற்றத்திற்கும் இடையே ஒரு விகிதத்தை எங்களுக்கு வழங்குகின்றன, இது ஒரு குறிப்பிட்ட நிகழ்வுகளின் மூலம், அதை ஏற்படுத்தியது.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/telugu/sentence_translations.json b/2017/chain-rule-and-product-rule/telugu/sentence_translations.json index 80a1f24e4..0d75d3f47 100644 --- a/2017/chain-rule-and-product-rule/telugu/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/telugu/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking. ", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking. ", "translatedText": "ఉత్పత్తులు మరియు ఫంక్షన్ కూర్పు కోసం ఉత్పన్న నమూనాలు అంత సూటిగా ఉండవు మరియు వాటికి ఈ రకమైనవి అవసరం కాబట్టి, రెండు ఫంక్షన్‌ల మొత్తాన్ని ఉత్పన్నం చేయడం అంటే ఏమిటో ఆలోచించడం ద్వారా ఈ ఉదాహరణతో వేడెక్కడం విలువైనదే లోతైన ఆలోచన. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar. ", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar. ", "translatedText": "ఉదాహరణకు, x వద్ద 0కి సమానం అనుకుందాం. 5, సైన్ గ్రాఫ్ యొక్క ఎత్తు ఈ నిలువు పట్టీ ద్వారా ఇవ్వబడుతుంది మరియు x స్క్వేర్డ్ పారాబొలా యొక్క ఎత్తు ఈ చిన్న నిలువు పట్టీ ద్వారా ఇవ్వబడుతుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x. ", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x. ", "translatedText": "ఇది మాకు మూడు చిన్న చిన్న స్నిప్పెట్‌లను కొత్త వైశాల్యం, దిగువన ఒక సన్నని దీర్ఘచతురస్రాన్ని అందిస్తుంది, దీని వైశాల్యం దాని వెడల్పు, x యొక్క సైన్, దాని సన్నని ఎత్తుకు రెట్లు, dx స్క్వేర్డ్ మరియు కుడి వైపున ఉన్న ఈ సన్నని దీర్ఘచతురస్రం దాని ఎత్తు, x స్క్వేర్డ్, దాని సన్నని వెడల్పు రెట్లు, x యొక్క d సైన్. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0. ", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean, ", "translatedText": "దీని వైశాల్యం అంతిమంగా dx స్క్వేర్‌కు అనులోమానుపాతంలో ఉంటుంది మరియు మనం ఇంతకు ముందు చూసినట్లుగా, dx 0కి వెళ్లినప్పుడు అది చాలా తక్కువ అవుతుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small. ", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small. ", "translatedText": "మరియు అప్పటిలాగే, వస్తువులను గీయడానికి నేను ఇక్కడ కొంత గొడ్డు మార్పులను ఉపయోగిస్తున్నానని గుర్తుంచుకోండి, కాబట్టి మనం వాటిని నిజంగా చూడగలం, కానీ సూత్రప్రాయంగా dx చాలా చిన్నది, మరియు dx స్క్వేర్డ్ మరియు x యొక్క d సైన్ కూడా చాలా ఎక్కువ. చాల చిన్నది. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx. ", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx. ", "translatedText": "కాబట్టి సైన్ మరియు x స్క్వేర్డ్ యొక్క ఉత్పన్నం గురించి మనకు తెలిసిన వాటిని వర్తింపజేస్తే, ఆ చిన్న మార్పు dx స్క్వేర్డ్ దాదాపు 2x రెట్లు dx అవుతుంది మరియు x యొక్క చిన్న మార్పు d సైన్ x సార్లు dx యొక్క కొసైన్ గురించి ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", "translatedText": "నేను మూడు వేర్వేరు సంఖ్యల పంక్తులను ఉంచుతాను, పైభాగంలో x విలువ ఉంటుంది, రెండవది x స్క్వేర్డ్ విలువను కలిగి ఉంటుంది మరియు మూడవ పంక్తి x స్క్వేర్డ్ యొక్క సైన్ విలువను కలిగి ఉంటుంది, అంటే ఫంక్షన్ x స్క్వేర్డ్ మిమ్మల్ని లైన్ 1 నుండి లైన్ 2 వరకు తీసుకువస్తుంది మరియు సైన్ ఫంక్షన్ మిమ్మల్ని లైన్ 2 నుండి లైన్ 3కి తీసుకువస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", "translatedText": "మేము దీన్ని 2x రెట్లు dxగా విస్తరించవచ్చు, ఇది మా నిర్దిష్ట ఇన్‌పుట్ కోసం 2 సార్లు 1 అవుతుంది. 5 రెట్లు dx, కానీ కనీసం ఇప్పటికైనా విషయాలను dx స్క్వేర్డ్‌గా వ్రాయడానికి ఇది సహాయపడుతుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h. ", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h. ", "translatedText": "మీరు ఏవైనా రెండు ఫంక్షన్‌లను కలిగి ఉంటే, x యొక్క g మరియు x యొక్క h, వాటి కూర్పు యొక్క ఉత్పన్నం, g యొక్క h యొక్క x, h పై మూల్యాంకనం చేయబడిన g యొక్క ఉత్పన్నం, h యొక్క ఉత్పన్నంతో గుణించబడుతుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule. ", + "input": "This pattern right here is what we usually call the chain rule. ", "translatedText": "ఈ నమూనానే మనం సాధారణంగా చైన్ రూల్ అని పిలుస్తాము. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h. ", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h. ", "translatedText": "కానీ మనం ఆ ఇంటర్మీడియట్ వేరియబుల్‌కి సంబంధించి ఉత్పన్నం తీసుకోవచ్చు, h. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", "translatedText": "గమనించండి, ఆ dhలు రద్దు చేయబడి, ఆ తుది అవుట్‌పుట్‌లో మార్పు మరియు ఇన్‌పుట్‌కి మార్పు మధ్య నిష్పత్తిని అందించండి, ఇది నిర్దిష్ట సంఘటనల గొలుసు ద్వారా దానిని తీసుకువచ్చింది. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/thai/sentence_translations.json b/2017/chain-rule-and-product-rule/thai/sentence_translations.json index 0f00adaff..5144e437d 100644 --- a/2017/chain-rule-and-product-rule/thai/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/thai/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking. ", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar. ", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x. ", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0. ", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small. ", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx. ", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h. ", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule. ", + "input": "This pattern right here is what we usually call the chain rule. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h. ", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/turkish/sentence_translations.json b/2017/chain-rule-and-product-rule/turkish/sentence_translations.json index 24978d7d7..dfdf9280f 100644 --- a/2017/chain-rule-and-product-rule/turkish/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/turkish/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking. ", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking. ", "translatedText": "Ancak iki fonksiyonun toplamının türevini almanın ne anlama geldiğini gerçekten düşünerek bu örneğe ısınmaya değer, çünkü çarpımlar ve fonksiyon bileşimi için türev kalıpları o kadar basit olmayacak ve bu tür bir işlem gerektirecekler. daha derin düşünme. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar. ", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar. ", "translatedText": "Örneğin x'in 0'a eşit olduğunu varsayalım. Şekil 5'te sinüs grafiğinin yüksekliği bu dikey çubukla, x kare parabolün yüksekliği ise bu daha küçük dikey çubukla verilmektedir. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x. ", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x. ", "translatedText": "Bu bize üç küçük yeni alan parçacığı verir; altta alanı genişliği sinüs x çarpı ince yüksekliği dx kare olan ince bir dikdörtgen ve sağdaki alanı yüksekliği x kare olan bu ince dikdörtgen, çarpı ince genişliği, d sinüs x. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0. ", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean, ", "translatedText": "Alanı sonuçta dx kareyle orantılıdır ve daha önce gördüğümüz gibi, dx 0'a giderken bu ihmal edilebilir hale gelir. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small. ", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small. ", "translatedText": "Ve tıpkı o zamanki gibi, burada bir şeyleri gerçekten görebilelim diye çizmek için biraz büyük değişiklikler kullandığımı unutmayın, ama prensipte dx çok çok küçük bir şeydir ve bu da dx kare ve d sinüs x'in de çok olduğu anlamına gelir. çok küçük. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx. ", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx. ", "translatedText": "Sinüs ve x karenin türevi hakkında bildiklerimizi uygularsak, bu küçük dx kare değişimi yaklaşık 2x çarpı dx olacak ve bu küçük d sinüs x değişimi de kosinüs x çarpı dx civarında olacak. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", "translatedText": "Üç farklı sayı doğrusu koyacağım, üstteki x'in değerini, ikincisi x karenin değerini ve üçüncü satır sinüs x karenin değerini, yani fonksiyonu tutacak x kare sizi 1. satırdan 2. satıra götürür ve sinüs fonksiyonu sizi 2. satırdan 3. satıra götürür. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", "translatedText": "Bunu 2x çarpı dx olarak genişletebiliriz, bu bizim spesifik girdimiz için 2 çarpı 1 olacaktır. 5 çarpı dx, ancak en azından şimdilik her şeyin dx kare olarak yazılmasına yardımcı olur. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h. ", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h. ", "translatedText": "Herhangi iki fonksiyonunuz varsa, g x ve h x, bunların bileşimlerinin türevi, g h x, h'ye göre değerlendirilen g'nin türevi ile h'nin türevinin çarpımıdır. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule. ", + "input": "This pattern right here is what we usually call the chain rule. ", "translatedText": "Bu modele genellikle zincir kuralı dediğimiz şeydir. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h. ", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h. ", "translatedText": "Ama ara değişken h'ye göre türev alabiliriz. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", "translatedText": "Dikkat edin, bu dh'ler birbirini götürür ve bize, son çıktıdaki değişiklik ile belirli bir olaylar zinciri aracılığıyla girdideki değişiklik arasında bir oran verir. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/ukrainian/sentence_translations.json b/2017/chain-rule-and-product-rule/ukrainian/sentence_translations.json index 471602f8e..da2d9affb 100644 --- a/2017/chain-rule-and-product-rule/ukrainian/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/ukrainian/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking. ", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking. ", "translatedText": "Але варто розігрітися з цим прикладом, подумавши про те, що означає взяти похідну від суми двох функцій, оскільки шаблони похідних для продуктів і композиції функцій не будуть такими простими, і вони вимагатимуть такого роду глибше мислення. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar. ", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar. ", "translatedText": "Наприклад, припустимо, що x дорівнює 0.5, висота графіка синуса задана цією вертикальною смугою, а висота параболи x у квадраті задається цією меншою вертикальною смугою. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x. ", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x. ", "translatedText": "Це дає нам три маленькі фрагменти нової площі: тонкий прямокутник внизу, площа якого дорівнює його ширині, синус х, помноженій на його тонку висоту, dx у квадраті, і цей тонкий прямокутник праворуч, площа якого дорівнює його висоті, х у квадраті, помножити на його тонку ширину, d синус x. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0. ", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean, ", "translatedText": "Його площа остаточно пропорційна квадрату dx, і, як ми бачили раніше, це стає незначним, коли dx дорівнює 0. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small. ", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small. ", "translatedText": "І так само, як і тоді, майте на увазі, що я використовую дещо потужні зміни тут, щоб намалювати речі, щоб ми могли їх побачити, але в принципі dx є чимось дуже маленьким, а це означає, що dx у квадраті та d синус від x також дуже дуже мало. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx. ", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx. ", "translatedText": "Отже, застосовуючи те, що ми знаємо про похідну від синуса та х у квадраті, ця невелика зміна dx у квадраті буде приблизно 2x помножена на dx, і ця незначна зміна d на синус від x буде приблизно косинус x на dx. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", "translatedText": "Я розмістю три різні числові рядки, верхній буде містити значення х, другий міститиме значення х у квадраті, а третій рядок міститиме значення синуса х у квадраті, тобто функцію x у квадраті переведе вас із рядка 1 у рядок 2, а функція синус переведе вас із рядка 2 у рядок 3. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", "translatedText": "Ми могли б розширити це як 2x, помножене на dx, що для наших конкретних вхідних даних буде 2x 1.5 разів на dx, але це допомагає зберегти речі, записані як dx у квадраті, принаймні на даний момент. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h. ", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h. ", "translatedText": "Якщо у вас є будь-які дві функції, g від x і h від x, похідна їх складу, g від h від x, є похідною від g, обчисленою за h, помноженою на похідну від h. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule. ", + "input": "This pattern right here is what we usually call the chain rule. ", "translatedText": "Цю закономірність ми зазвичай називаємо ланцюговим правилом. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h. ", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h. ", "translatedText": "Але ми можемо взяти похідну відносно цієї проміжної змінної, h. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", "translatedText": "Зауважте, ці dh компенсуються і дають нам співвідношення між зміною в цьому кінцевому виході та зміною на вході, яка через певний ланцюжок подій спричинила це. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/urdu/sentence_translations.json b/2017/chain-rule-and-product-rule/urdu/sentence_translations.json index b0761fe4b..e2c11a924 100644 --- a/2017/chain-rule-and-product-rule/urdu/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/urdu/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking. ", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking. ", "translatedText": "لیکن اس مثال کے ساتھ واقعی یہ سوچنے کے قابل ہے کہ دو افعال کے مجموعے سے مشتق ہونے کا کیا مطلب ہے، کیونکہ مصنوعات اور فنکشن کمپوزیشن کے مشتق نمونے اتنے سیدھے نہیں ہوں گے، اور انہیں اس قسم کی ضرورت ہوگی۔گہری سوچ. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar. ", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar. ", "translatedText": "مثال کے طور پر، ہم کہتے ہیں کہ x برابر ہے 0۔5، سائن گراف کی اونچائی اس عمودی بار سے دی گئی ہے، اور x مربع پیرابولا کی اونچائی اس چھوٹی عمودی بار سے دی گئی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x. ", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x. ", "translatedText": "اس سے ہمیں نئے رقبہ کے تین چھوٹے چھوٹے ٹکڑے ملتے ہیں، نیچے ایک پتلا مستطیل جس کا رقبہ اس کی چوڑائی ہے، x کا سائن، اس کی پتلی اونچائی کا گنا، dx مربع، اور دائیں جانب یہ پتلا مستطیل جس کا رقبہ اس کی اونچائی ہے، x مربع، بار اس کی پتلی چوڑائی، x کی d سائن۔", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0. ", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean, ", "translatedText": "اس کا رقبہ بالآخر dx مربع کے متناسب ہے، اور جیسا کہ ہم پہلے دیکھ چکے ہیں، یہ نہ ہونے کے برابر ہو جاتا ہے کیونکہ dx 0 تک جاتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small. ", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small. ", "translatedText": "اور بالکل اسی طرح، یہ بات ذہن میں رکھیں کہ میں یہاں چیزوں کو کھینچنے کے لیے کچھ خوبصورت تبدیلیاں استعمال کر رہا ہوں تاکہ ہم انہیں حقیقت میں دیکھ سکیں، لیکن اصولی طور پر dx بہت چھوٹی چیز ہے، اور اس کا مطلب ہے کہ dx مربع اور x کا d سائن بھی بہت ہیں۔بہت چھوٹا. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx. ", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx. ", "translatedText": "تو جو کچھ ہم سائن اور ایکس اسکوائر کے مشتق کے بارے میں جانتے ہیں اس کو لاگو کرتے ہوئے، وہ چھوٹی تبدیلی dx مربع تقریباً 2x بار dx ہونے والی ہے، اور x کی وہ چھوٹی تبدیلی d سائن x اوقات dx کے cosine کے بارے میں ہوگی۔", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", "translatedText": "میں تین مختلف نمبر لائنیں ڈالوں گا، سب سے اوپر والی x کی قدر رکھے گی، دوسری میں x مربع کی قدر ہوگی، اور تیسری لائن میں x مربع کی سائن کی قدر ہوگی، یعنی فنکشن x مربع آپ کو لائن 1 سے لائن 2 تک لے جاتا ہے، اور فنکشن سائن آپ کو لائن 2 سے لائن 3 تک لے جاتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", "translatedText": "ہم اسے 2x بار dx کے طور پر بڑھا سکتے ہیں، جو ہمارے مخصوص ان پٹ کے لیے 2 گنا 1 ہوگا۔5 بار dx، لیکن یہ چیزوں کو dx مربع کے طور پر لکھے رکھنے میں مدد کرتا ہے، کم از کم ابھی کے لیے۔", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h. ", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h. ", "translatedText": "اگر آپ کے پاس کوئی دو فنکشنز ہیں، x کا g اور x کا h، ان کی ساخت کا مشتق، x کا h کا g، h کے مشتق سے ضرب کردہ، h پر تشخیص شدہ g کا مشتق ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule. ", + "input": "This pattern right here is what we usually call the chain rule. ", "translatedText": "اس پیٹرن کو ہم عام طور پر سلسلہ اصول کہتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h. ", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h. ", "translatedText": "لیکن ہم اس انٹرمیڈیٹ متغیر کے حوالے سے مشتق لے سکتے ہیں، h۔", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", "translatedText": "غور کریں، وہ ڈی ایچ کینسل آؤٹ اور ہمیں اس حتمی آؤٹ پٹ میں تبدیلی اور ان پٹ میں تبدیلی کے درمیان ایک تناسب دیتے ہیں جو کہ واقعات کی ایک خاص زنجیر کے ذریعے، اسے سامنے لایا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/chain-rule-and-product-rule/vietnamese/sentence_translations.json b/2017/chain-rule-and-product-rule/vietnamese/sentence_translations.json index c5f2ef527..670f05643 100644 --- a/2017/chain-rule-and-product-rule/vietnamese/sentence_translations.json +++ b/2017/chain-rule-and-product-rule/vietnamese/sentence_translations.json @@ -96,7 +96,7 @@ "end": 118.6 }, { - "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they'll require this kind of deeper thinking. ", + "input": "But it's worth warming up with this example by really thinking through what it means to take a derivative of a sum of two functions, since the derivative patterns for products and function composition won't be so straightforward, and they're going to require this kind of deeper thinking. ", "translatedText": "Nhưng cũng đáng để khởi động với ví dụ này bằng cách thực sự suy nghĩ xem việc lấy đạo hàm của tổng hai hàm có ý nghĩa gì, vì các mẫu đạo hàm của tích và hàm hợp sẽ không đơn giản như vậy, và chúng sẽ yêu cầu loại phép tính này. suy nghĩ sâu sắc hơn. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 147.96 }, { - "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this smaller vertical bar. ", + "input": "For example, let's say at x equals 0.5, the height of the sine graph is given by this vertical bar, and the height of the x squared parabola is given by this slightly smaller vertical bar. ", "translatedText": "Ví dụ: giả sử tại x bằng 0.5, chiều cao của đồ thị hình sin được cho bởi thanh dọc này và chiều cao của parabol bình phương x được cho bởi thanh dọc nhỏ hơn này. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 347.92 }, { - "input": "This gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared, and this thin rectangle on the right whose area is its height, x squared, times its thin width, d sine of x. ", + "input": "And this gives us three little snippets of new area, a thin rectangle on the bottom whose area is its width, sine of x, times its thin height, dx squared. And there's this thin rectangle on the right, whose area is its height, x squared, times its thin width, d sine of x. ", "translatedText": "Điều này mang lại cho chúng ta ba đoạn nhỏ về diện tích mới, một hình chữ nhật mỏng ở phía dưới có diện tích là chiều rộng, sin x, nhân chiều cao mỏng của nó, dx bình phương, và hình chữ nhật mỏng ở bên phải có diện tích là chiều cao, x bình phương, nhân với chiều rộng mỏng của nó, d sin x. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 374.14 }, { - "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to 0. ", + "input": "Its area is ultimately proportional to dx squared, and as we've seen before, that becomes negligible as dx goes to zero. I mean, ", "translatedText": "Diện tích của nó cuối cùng tỷ lệ thuận với dx bình phương, và như chúng ta đã thấy trước đây, diện tích đó trở nên không đáng kể khi dx tiến về 0. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 388.7 }, { - "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things so we can actually see them, but in principle dx is something very very small, and that means dx squared and d sine of x are also very very small. ", + "input": "And just like then, keep in mind that I'm using somewhat beefy changes here to draw things, just so we can actually see them. But in principle, dx is something very very small, and that means that dx squared and d sine of x are also very very small. ", "translatedText": "Và cũng giống như vậy, hãy nhớ rằng tôi đang sử dụng những thay đổi mạnh mẽ ở đây để vẽ mọi thứ sao cho chúng ta có thể thực sự nhìn thấy chúng, nhưng về nguyên tắc dx là một cái gì đó rất rất nhỏ, và điều đó có nghĩa là dx bình phương và d sin x cũng rất rất nhỏ. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 404.7 }, { - "input": "So applying what we know about the derivative of sine and x squared, that tiny change dx squared is going to be about 2x times dx, and that tiny change d sine of x is going to be about cosine of x times dx. ", + "input": "So, applying what we know about the derivative of sine and of x squared, that tiny change, dx squared, is going to be about 2x times dx. And that tiny change, d sine of x, well that's going to be about cosine of x times dx. ", "translatedText": "Vì vậy, áp dụng những gì chúng ta biết về đạo hàm của sin và x bình phương, sự thay đổi nhỏ dx bình phương đó sẽ bằng khoảng 2x nhân dx, và sự thay đổi nhỏ d sin của x sẽ bằng cosin của x nhân dx. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 552.54 }, { - "input": "I'll put up three different number lines, the top one will hold the value of x, the second one will hold the value of x squared, and the third line will hold the value of sine of x squared, that is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", + "input": "I'll put up three different number lines, the top one is going to hold the value of x, the second one is going to hold the x squared, and the third line is going to hold the value of sine of x squared. That is, the function x squared gets you from line 1 to line 2, and the function sine gets you from line 2 to line 3. ", "translatedText": "Tôi sẽ đặt ba dòng số khác nhau, dòng trên cùng sẽ chứa giá trị của x, dòng thứ hai sẽ chứa giá trị của x bình phương, và dòng thứ ba sẽ chứa giá trị sin của x bình phương, tức là hàm số x bình phương sẽ đưa bạn từ dòng 1 đến dòng 2, và hàm sin sẽ đưa bạn từ dòng 2 đến dòng 3. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 615.7 }, { - "input": "We could expand this as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", + "input": "We could expand this like we have before, as 2x times dx, which for our specific input would be 2 times 1.5 times dx, but it helps to keep things written as dx squared, at least for now. ", "translatedText": "Chúng ta có thể mở rộng giá trị này thành 2x nhân dx, đối với đầu vào cụ thể của chúng ta sẽ là 2 nhân 1.5 nhân dx, nhưng nó giúp giữ cho mọi thứ được viết dưới dạng dx bình phương, ít nhất là vào lúc này. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 749.22 }, { - "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is the derivative of g evaluated on h, multiplied by the derivative of h. ", + "input": "If you have any two functions, g of x and h of x, the derivative of their composition, g of h of x, is going to be the derivative of g evaluated on h, multiplied by the derivative of h. ", "translatedText": "Nếu bạn có bất kỳ hai hàm số nào, g của x và h của x, thì đạo hàm của thành phần của chúng, g của h của x, là đạo hàm của g tính theo h, nhân với đạo hàm của h. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 763.66 }, { - "input": "This pattern is what we usually call the chain rule. ", + "input": "This pattern right here is what we usually call the chain rule. ", "translatedText": "Mô hình này là những gì chúng ta thường gọi là quy tắc dây chuyền. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 809.84 }, { - "input": "But we could take the derivative with respect to that intermediate variable, h. ", + "input": "That's kind of the whole thing we were trying to figure out. But we could take the derivative with respect to that intermediate variable, h. ", "translatedText": "Nhưng chúng ta có thể lấy đạo hàm theo biến trung gian h. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 851.2 }, { - "input": "Notice, those dh's cancel out and give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", + "input": "So notice, those dh's cancel out, and they give us a ratio between the change in that final output and the change to the input that, through a certain chain of events, brought it about. ", "translatedText": "Lưu ý, những dh đó bị loại bỏ và cho chúng ta một tỷ lệ giữa sự thay đổi ở đầu ra cuối cùng đó và sự thay đổi ở đầu vào mà nó xảy ra thông qua một chuỗi sự kiện nhất định. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivative-formulas-geometrically/arabic/sentence_translations.json b/2017/derivative-formulas-geometrically/arabic/sentence_translations.json index 56f72e54f..3ebb7f19e 100644 --- a/2017/derivative-formulas-geometrically/arabic/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/arabic/sentence_translations.json @@ -202,7 +202,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "بالنسبة لتلك المساحة الجديدة من هذين المستطيلين الرفيعين ستكون 2 ضرب 3 ضرب 0.01، أي 0.06، حوالي 6 أضعاف حجم dx.", "model": "google_nmt", "from_community_srt": "لنقل أن x=3, dx=0.01 إذاً المساحة الجديدة من هذين المستطيلين النحيفين ستكون 2*3*0.01 = 0.06 تقريباً 6 أضعاف قياس dx", diff --git a/2017/derivative-formulas-geometrically/bulgarian/sentence_translations.json b/2017/derivative-formulas-geometrically/bulgarian/sentence_translations.json index 58fa33478..2b5d08700 100644 --- a/2017/derivative-formulas-geometrically/bulgarian/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/bulgarian/sentence_translations.json @@ -184,7 +184,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "За тази нова площ от тези два тънки правоъгълника ще бъде 2 пъти 3 пъти 0,01, което е 0,06, около 6 пъти размера на dx.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/derivative-formulas-geometrically/dutch/sentence_translations.json b/2017/derivative-formulas-geometrically/dutch/sentence_translations.json index 8d217ac5a..00d1bceef 100644 --- a/2017/derivative-formulas-geometrically/dutch/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/dutch/sentence_translations.json @@ -184,7 +184,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "Want dat nieuwe gebied van deze twee dunne rechthoeken zou 2 keer 3 keer 0,01 zijn, wat 0,06 is, ongeveer 6 keer de grootte van dx.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/derivative-formulas-geometrically/english/captions.srt b/2017/derivative-formulas-geometrically/english/captions.srt index d2cbab346..7b2b11151 100644 --- a/2017/derivative-formulas-geometrically/english/captions.srt +++ b/2017/derivative-formulas-geometrically/english/captions.srt @@ -195,806 +195,810 @@ The two thin rectangles each have side lengths of x and dx, so they account for 2 times x times dx units of new area. 50 -00:03:18,240 --> 00:03:24,730 -For that new area from these two thin rectangles would be 2 times 3 times 0.01, +00:03:18,240 --> 00:03:21,031 +For example, let's say x was 3 and dx was 0.01, 51 -00:03:24,730 --> 00:03:28,300 -which is 0.06, about 6 times the size of dx. +00:03:21,031 --> 00:03:25,741 +then that new area from these two thin rectangles would be 2 times 3 times 0.01, 52 +00:03:25,741 --> 00:03:28,300 +which is 0.06, about 6 times the size of dx. + +53 00:03:29,700 --> 00:03:32,872 That little square there has an area of dx squared, -53 +54 00:03:32,872 --> 00:03:36,960 but you should think of that as being really tiny, negligibly tiny. -54 +55 00:03:37,700 --> 00:03:41,340 For example, if dx was 0.01, that would be only 0.0001, -55 +56 00:03:41,340 --> 00:03:46,086 and keep in mind I'm drawing dx with a fair bit of width here just so we -56 +57 00:03:46,086 --> 00:03:49,662 can actually see it, but always remember in principle, -57 +58 00:03:49,662 --> 00:03:54,993 dx should be thought of as a truly tiny amount, and for those truly tiny amounts, -58 +59 00:03:54,993 --> 00:03:59,674 a good rule of thumb is that you can ignore anything that includes a dx -59 +60 00:03:59,674 --> 00:04:01,820 raised to a power greater than 1. -60 +61 00:04:02,400 --> 00:04:05,880 That is, a tiny change squared is a negligible change. -61 +62 00:04:07,500 --> 00:04:13,055 What this leaves us with is that dF is just some multiple of dx, and that multiple 2x, -62 +63 00:04:13,055 --> 00:04:18,100 which you could also write as dF divided by dx, is the derivative of x squared. -63 +64 00:04:19,040 --> 00:04:24,297 For example, if you were starting at x equals 3, then as you slightly increase x, -64 +65 00:04:24,297 --> 00:04:29,683 the rate of change in the area per unit change in length added, dx squared over dx, -65 +66 00:04:29,683 --> 00:04:34,427 would be 2 times 3, or 6, and if instead you were starting at x equals 5, -66 +67 00:04:34,427 --> 00:04:38,980 then the rate of change would be 10 units of area per unit change in x. -67 +68 00:04:41,220 --> 00:04:45,420 Let's go ahead and try a different simple function, f of x equals x cubed. -68 +69 00:04:45,940 --> 00:04:48,039 This is going to be the geometric view of the stuff -69 +70 00:04:48,039 --> 00:04:50,140 that I went through algebraically in the last video. -70 +71 00:04:51,020 --> 00:04:55,641 What's nice here is that we can think of x cubed as the volume of an actual -71 +72 00:04:55,641 --> 00:05:00,020 cube whose side lengths are x, and when you increase x by a tiny nudge, -72 +73 00:05:00,020 --> 00:05:04,520 a tiny dx, the resulting increase in volume is what I have here in yellow. -73 +74 00:05:04,860 --> 00:05:08,748 That represents all the volume in a cube with side lengths x plus dx -74 +75 00:05:08,748 --> 00:05:12,580 that's not already in the original cube, the one with side length x. -75 +76 00:05:13,580 --> 00:05:18,602 It's nice to think of this new volume as broken up into multiple components, -76 +77 00:05:18,602 --> 00:05:22,386 but almost all of it comes from these three square faces, -77 +78 00:05:22,386 --> 00:05:25,843 or said a little more precisely, as dx approaches 0, -78 +79 00:05:25,843 --> 00:05:30,279 those three squares comprise a portion closer and closer to 100% of -79 +80 00:05:30,279 --> 00:05:31,780 that new yellow volume. -80 +81 00:05:33,840 --> 00:05:38,058 Each of those thin squares has a volume of x squared times dx, -81 +82 00:05:38,058 --> 00:05:41,540 the area of the face times that little thickness dx. -82 +83 00:05:42,220 --> 00:05:46,260 So in total this gives us 3x squared dx of volume change. -83 +84 00:05:47,300 --> 00:05:51,062 And to be sure there are other slivers of volume here along the edges -84 +85 00:05:51,062 --> 00:05:54,877 and that tiny one in the corner, but all of that volume is going to be -85 +86 00:05:54,877 --> 00:05:58,640 proportional to dx squared, or dx cubed, so we can safely ignore them. -86 +87 00:05:59,460 --> 00:06:03,466 Again this is ultimately because they're going to be divided by dx, -87 +88 00:06:03,466 --> 00:06:07,118 and if there's still any dx remaining then those terms aren't -88 +89 00:06:07,118 --> 00:06:10,300 going to survive the process of letting dx approach 0. -89 +90 00:06:11,280 --> 00:06:14,435 What this means is that the derivative of x cubed, -90 +91 00:06:14,435 --> 00:06:19,200 the rate at which x cubed changes per unit change of x, is 3 times x squared. -91 +92 00:06:20,640 --> 00:06:25,185 What that means in terms of graphical intuition is that the slope of -92 +93 00:06:25,185 --> 00:06:29,600 the graph of x cubed at every single point x is exactly 3x squared. -93 +94 00:06:34,080 --> 00:06:38,786 And reasoning about that slope, it should make sense that this derivative is high on the -94 +95 00:06:38,786 --> 00:06:42,805 left and then 0 at the origin and then high again as you move to the right, -95 +96 00:06:42,805 --> 00:06:47,142 but just thinking in terms of the graph would never have landed us on the precise -96 +97 00:06:47,142 --> 00:06:48,200 quantity 3x squared. -97 +98 00:06:48,880 --> 00:06:53,060 For that we had to take a much more direct look at what x cubed actually means. -98 +99 00:06:54,260 --> 00:06:57,561 Now in practice you wouldn't necessarily think of the square every -99 +100 00:06:57,561 --> 00:06:59,927 time you're taking the derivative of x squared, -100 +101 00:06:59,927 --> 00:07:03,278 nor would you necessarily think of this cube whenever you're taking -101 +102 00:07:03,278 --> 00:07:04,560 the derivative of x cubed. -102 +103 00:07:04,880 --> 00:07:08,400 Both of them fall under a pretty recognizable pattern for polynomial terms. -103 +104 00:07:09,200 --> 00:07:13,341 The derivative of x to the fourth turns out to be 4x cubed, -104 +105 00:07:13,341 --> 00:07:17,760 the derivative of x to the fifth is 5x to the fourth, and so on. -105 +106 00:07:18,880 --> 00:07:22,792 Abstractly you'd write this as the derivative of x to -106 +107 00:07:22,792 --> 00:07:26,560 the n for any power n is n times x to the n minus 1. -107 +108 00:07:27,300 --> 00:07:30,560 This right here is what's known in the business as the power rule. -108 +109 00:07:31,740 --> 00:07:35,913 In practice we all quickly just get jaded and think about this symbolically as -109 +110 00:07:35,913 --> 00:07:39,769 the exponent hopping down in front, leaving behind one less than itself, -110 +111 00:07:39,769 --> 00:07:44,260 rarely pausing to think about the geometric delights that underlie these derivatives. -111 +112 00:07:45,240 --> 00:07:47,296 That's the kind of thing that happens when these tend -112 +113 00:07:47,296 --> 00:07:49,200 to fall in the middle of much longer computations. -113 +114 00:07:50,640 --> 00:07:53,328 But rather than tracking it all off to symbolic patterns, -114 +115 00:07:53,328 --> 00:07:57,360 let's just take a moment and think about why this works for powers beyond just 2 and 3. -115 +116 00:07:58,440 --> 00:08:02,773 When you nudge that input x, increasing it slightly to x plus dx, -116 +117 00:08:02,773 --> 00:08:06,974 working out the exact value of that nudged output would involve -117 +118 00:08:06,974 --> 00:08:10,520 multiplying together these n separate x plus dx terms. -118 +119 00:08:11,340 --> 00:08:13,890 The full expansion would be really complicated, -119 +120 00:08:13,890 --> 00:08:18,460 but part of the point of derivatives is that most of that complication can be ignored. -120 +121 00:08:19,280 --> 00:08:22,020 The first term in your expansion is x to the n. -121 +122 00:08:22,680 --> 00:08:25,584 This is analogous to the area of the original square, -122 +123 00:08:25,584 --> 00:08:28,920 or the volume of the original cube from our previous examples. -123 +124 00:08:30,820 --> 00:08:36,039 For the next terms in the expansion you can choose mostly x's with a single dx. -124 +125 00:08:41,720 --> 00:08:46,804 Since there are n different parentheticals from which you could have chosen -125 +126 00:08:46,804 --> 00:08:50,016 that single dx, this gives us n separate terms, -126 +127 00:08:50,016 --> 00:08:53,160 all of which include n minus 1 x's times a dx, -127 +128 00:08:53,160 --> 00:08:56,640 giving a value of x to the power n minus 1 times dx. -128 +129 00:08:57,580 --> 00:09:02,820 This is analogous to how the majority of the new area in the square came from those -129 +130 00:09:02,820 --> 00:09:07,997 two bars, each with area x times dx, or how the bulk of the new volume in the cube -130 +131 00:09:07,997 --> 00:09:13,300 came from those three thin squares, each of which had a volume of x squared times dx. -131 +132 00:09:14,540 --> 00:09:17,432 There will be many other terms of this expansion, -132 +133 00:09:17,432 --> 00:09:21,251 but all of them are just going to be some multiple of dx squared, -133 +134 00:09:21,251 --> 00:09:25,185 so we can safely ignore them, and what that means is that all but a -134 +135 00:09:25,185 --> 00:09:29,350 negligible portion of the increase in the output comes from n copies of -135 +136 00:09:29,350 --> 00:09:31,260 this x to the n minus 1 times dx. -136 +137 00:09:31,940 --> 00:09:37,520 That's what it means for the derivative of x to the n to be n times x to the n minus 1. -137 +138 00:09:38,960 --> 00:09:43,220 And even though, like I said in practice, you'll find yourself performing this -138 +139 00:09:43,220 --> 00:09:47,911 derivative quickly and symbolically, imagining the exponent hopping down to the front, -139 +140 00:09:47,911 --> 00:09:52,280 every now and then it's nice to just step back and remember why these rules work. -140 +141 00:09:52,820 --> 00:09:56,880 Not just because it's pretty, and not just because it helps remind us that math -141 +142 00:09:56,880 --> 00:10:00,332 actually makes sense and isn't just a pile of formulas to memorize, -142 +143 00:10:00,332 --> 00:10:04,494 but because it flexes that very important muscle of thinking about derivatives in -143 +144 00:10:04,494 --> 00:10:05,560 terms of tiny nudges. -144 +145 00:10:07,500 --> 00:10:11,640 As another example, think of the function f of x equals 1 divided by x. -145 +146 00:10:12,700 --> 00:10:16,738 Now on the hand you could just blindly try applying the power rule, -146 +147 00:10:16,738 --> 00:10:20,540 since 1 divided by x is the same as writing x to the negative 1. -147 +148 00:10:21,100 --> 00:10:24,433 That would involve letting the negative 1 hop down in front, -148 +149 00:10:24,433 --> 00:10:27,440 leaving behind 1 less than itself, which is negative 2. -149 +150 00:10:28,240 --> 00:10:31,444 But let's have some fun and see if we can reason about this geometrically, -150 +151 00:10:31,444 --> 00:10:33,580 rather than just plugging it through some formula. -151 +152 00:10:34,860 --> 00:10:40,180 The value 1 over x is asking what number multiplied by x equals 1. -152 +153 00:10:40,960 --> 00:10:42,820 So here's how I'd like to visualize it. -153 +154 00:10:42,820 --> 00:10:48,120 Imagine a little rectangular puddle of water sitting in two dimensions whose area is 1. -154 +155 00:10:48,960 --> 00:10:53,766 And let's say that its width is x, which means that the height has to be 1 over x, -155 +156 00:10:53,766 --> 00:10:55,620 since the total area of it is 1. -156 +157 00:10:56,360 --> 00:11:01,040 So if x was stretched out to 2, then that height is forced down to 1 half. -157 +158 00:11:01,780 --> 00:11:05,920 And if you increased x up to 3, then the other side has to be squished down to 1 third. -158 +159 00:11:07,040 --> 00:11:10,680 This is a nice way to think about the graph of 1 over x, by the way. -159 +160 00:11:11,280 --> 00:11:15,360 If you think of this width x of the puddle as being in the xy-plane, -160 +161 00:11:15,360 --> 00:11:20,623 then that corresponding output 1 divided by x, the height of the graph above that point, -161 +162 00:11:20,623 --> 00:11:24,940 is whatever the height of your puddle has to be to maintain an area of 1. -162 +163 00:11:26,360 --> 00:11:29,358 So with this visual in mind, for the derivative, -163 +164 00:11:29,358 --> 00:11:33,580 imagine nudging up that value of x by some tiny amount, some tiny dx. -164 +165 00:11:34,580 --> 00:11:37,401 How must the height of this rectangle change so -165 +166 00:11:37,401 --> 00:11:40,340 that the area of the puddle remains constant at 1? -166 +167 00:11:41,340 --> 00:11:46,020 That is, increasing the width by dx adds some new area to the right here. -167 +168 00:11:46,260 --> 00:11:50,355 So the puddle has to decrease in height by some d 1 over x, -168 +169 00:11:50,355 --> 00:11:54,860 so that the area lost off of that top cancels out the area gained. -169 +170 00:11:56,100 --> 00:11:59,259 You should think of that d 1 over x as being a negative amount, -170 +171 00:11:59,259 --> 00:12:02,320 by the way, since it's decreasing the height of the rectangle. -171 +172 00:12:03,540 --> 00:12:04,400 And you know what? -172 +173 00:12:04,840 --> 00:12:07,027 I'm going to leave the last few steps here for you, -173 +174 00:12:07,027 --> 00:12:09,720 for you to pause and ponder and work out an ultimate expression. -174 +175 00:12:10,560 --> 00:12:14,121 And once you reason out what d of 1 over x divided by dx should be, -175 +176 00:12:14,121 --> 00:12:17,839 I want you to compare it to what you would have gotten if you had just -176 +177 00:12:17,839 --> 00:12:21,820 blindly applied the power rule, purely symbolically, to x to the negative 1. -177 +178 00:12:23,980 --> 00:12:26,143 And while I'm encouraging you to pause and ponder, -178 +179 00:12:26,143 --> 00:12:28,520 here's another fun challenge if you're feeling up to it. -179 +180 00:12:29,060 --> 00:12:33,420 See if you can reason through what the derivative of the square root of x should be. -180 +181 00:12:36,400 --> 00:12:40,053 To finish things off, I want to tackle one more type of function, -181 +182 00:12:40,053 --> 00:12:44,260 trigonometric functions, and in particular let's focus on the sine function. -182 +183 00:12:45,320 --> 00:12:48,230 So for this section I'm going to assume that you're already -183 +184 00:12:48,230 --> 00:12:51,674 familiar with how to think about trig functions using the unit circle, -184 +185 00:12:51,674 --> 00:12:54,100 the circle with a radius 1 centered at the origin. -185 +186 00:12:55,240 --> 00:12:59,152 For a given value of theta, like say 0.8, you imagine yourself -186 +187 00:12:59,152 --> 00:13:02,878 walking around the circle starting from the rightmost point -187 +188 00:13:02,878 --> 00:13:06,480 until you've traversed that distance of 0.8 in arc length. -188 +189 00:13:06,760 --> 00:13:11,718 This is the same thing as saying that the angle right here is exactly theta radians, -189 +190 00:13:11,718 --> 00:13:13,760 since the circle has a radius of 1. -190 +191 00:13:14,760 --> 00:13:20,013 Then what sine of theta means is the height of that point above the x-axis, -191 +192 00:13:20,013 --> 00:13:24,507 and as your theta value increases and you walk around the circle -192 +193 00:13:24,507 --> 00:13:28,240 your height bobs up and down between negative 1 and 1. -193 +194 00:13:29,020 --> 00:13:33,616 So when you graph sine of theta versus theta you get this wave pattern, -194 +195 00:13:33,616 --> 00:13:35,660 the quintessential wave pattern. -195 +196 00:13:37,600 --> 00:13:40,311 And just from looking at this graph we can start to -196 +197 00:13:40,311 --> 00:13:43,180 get a feel for the shape of the derivative of the sine. -197 +198 00:13:44,020 --> 00:13:48,828 The slope at 0 is something positive since sine of theta is increasing there, -198 +199 00:13:48,828 --> 00:13:54,191 and as we move to the right and sine of theta approaches its peak that slope goes down -199 +200 00:13:54,191 --> 00:13:54,500 to 0. -200 +201 00:13:55,720 --> 00:13:58,340 Then the slope is negative for a little while, -201 +202 00:13:58,340 --> 00:14:03,080 while the sine is decreasing before coming back up to 0 as the sine graph levels out. -202 +203 00:14:04,460 --> 00:14:07,433 And as you continue thinking this through and drawing it out, -203 +204 00:14:07,433 --> 00:14:11,174 if you're familiar with the graph of trig functions you might guess that this -204 +205 00:14:11,174 --> 00:14:13,668 derivative graph should be exactly cosine of theta, -205 +206 00:14:13,668 --> 00:14:17,265 since all the peaks and valleys line up perfectly with where the peaks and -206 +207 00:14:17,265 --> 00:14:19,280 valleys for the cosine function should be. -207 +208 00:14:20,340 --> 00:14:23,910 And spoiler alert, the derivative is in fact the cosine of theta, -208 +209 00:14:23,910 --> 00:14:27,860 but aren't you a little curious about why it's precisely cosine of theta? -209 +210 00:14:28,240 --> 00:14:32,158 I mean you could have all sorts of functions with peaks and valleys at the same points -210 +211 00:14:32,158 --> 00:14:34,365 that have roughly the same shape, but who knows, -211 +212 00:14:34,365 --> 00:14:38,103 maybe the derivative of sine could have turned out to be some entirely new type of -212 +213 00:14:38,103 --> 00:14:40,400 function that just happens to have a similar shape. -213 +214 00:14:41,600 --> 00:14:44,832 Well just like the previous examples, a more exact understanding -214 +215 00:14:44,832 --> 00:14:48,662 of the derivative requires looking at what the function actually represents, -215 +216 00:14:48,662 --> 00:14:51,100 rather than looking at the graph of the function. -216 +217 00:14:52,400 --> 00:14:54,996 So think back to that walk around the unit circle, -217 +218 00:14:54,996 --> 00:14:58,967 having traversed an arc with length theta and thinking about sine of theta as -218 +219 00:14:58,967 --> 00:15:00,240 the height of that point. -219 +220 00:15:01,700 --> 00:15:06,247 Now zoom into that point on the circle and consider a slight nudge of d theta -220 +221 00:15:06,247 --> 00:15:10,620 along their circumference, a tiny step in your walk around the unit circle. -221 +222 00:15:11,480 --> 00:15:14,640 How much does that tiny step change the sine of theta? -222 +223 00:15:15,440 --> 00:15:20,420 How much does this increase d theta of arc length increase the height above the x-axis? -223 +224 00:15:21,640 --> 00:15:26,181 Well zoomed in close enough, the circle basically looks like a straight line in this -224 +225 00:15:26,181 --> 00:15:30,777 neighborhood, so let's go ahead and think of this right triangle where the hypotenuse -225 +226 00:15:30,777 --> 00:15:34,891 of that right triangle represents the nudge d theta along the circumference, -226 +227 00:15:34,891 --> 00:15:39,540 and that left side here represents the change in height, the resulting d sine of theta. -227 +228 00:15:40,140 --> 00:15:44,185 Now this tiny triangle is actually similar to this larger triangle here, -228 +229 00:15:44,185 --> 00:15:48,841 with the defining angle theta and whose hypotenuse is the radius of the circle with -229 +230 00:15:48,841 --> 00:15:49,340 length 1. -230 +231 00:15:50,960 --> 00:15:55,940 Specifically this little angle right here is precisely equal to theta radians. -231 +232 00:15:57,420 --> 00:16:00,520 Now think about what the derivative of sine is supposed to mean. -232 +233 00:16:01,220 --> 00:16:05,585 It's the ratio between that d sine of theta, the tiny change to the height, -233 +234 00:16:05,585 --> 00:16:09,320 divided by d theta, the tiny change to the input of the function. -234 +235 00:16:10,520 --> 00:16:14,052 And from the picture we can see that that's the ratio between the -235 +236 00:16:14,052 --> 00:16:17,960 length of the side adjacent to the angle theta divided by the hypotenuse. -236 +237 00:16:18,800 --> 00:16:21,518 Well let's see, adjacent divided by hypotenuse, -237 +238 00:16:21,518 --> 00:16:26,220 that's exactly what the cosine of theta means, that's the definition of the cosine. -238 +239 00:16:27,540 --> 00:16:30,223 So this gives us two different really nice ways of -239 +240 00:16:30,223 --> 00:16:32,960 thinking about how the derivative of sine is cosine. -240 +241 00:16:33,140 --> 00:16:36,642 One of them is looking at the graph and getting a loose feel for the shape of -241 +242 00:16:36,642 --> 00:16:40,280 things based on thinking about the slope of the sine graph at every single point. -242 +243 00:16:41,100 --> 00:16:45,400 And the other is a more precise line of reasoning looking at the unit circle itself. -243 +244 00:16:47,080 --> 00:16:49,276 For those of you that like to pause and ponder, -244 +245 00:16:49,276 --> 00:16:52,846 see if you can try a similar line of reasoning to find what the derivative of -245 +246 00:16:52,846 --> 00:16:54,220 the cosine of theta should be. -246 +247 00:16:56,320 --> 00:16:59,496 In the next video I'll talk about how you can take derivatives -247 +248 00:16:59,496 --> 00:17:02,470 of functions who combine simple functions like these ones, -248 +249 00:17:02,470 --> 00:17:06,000 either as sums or products or function compositions, things like that. -249 +250 00:17:06,560 --> 00:17:09,643 And similar to this video the goal is going to be to understand each one -250 +251 00:17:09,643 --> 00:17:13,359 geometrically in a way that makes it intuitively reasonable and somewhat more memorable. diff --git a/2017/derivative-formulas-geometrically/english/sentence_timings.json b/2017/derivative-formulas-geometrically/english/sentence_timings.json index 2a51e528e..3cb05bfb5 100644 --- a/2017/derivative-formulas-geometrically/english/sentence_timings.json +++ b/2017/derivative-formulas-geometrically/english/sentence_timings.json @@ -115,7 +115,7 @@ 193.78 ], [ - "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", 198.24, 208.3 ], diff --git a/2017/derivative-formulas-geometrically/english/transcript.txt b/2017/derivative-formulas-geometrically/english/transcript.txt index 197929d05..af0b098b0 100644 --- a/2017/derivative-formulas-geometrically/english/transcript.txt +++ b/2017/derivative-formulas-geometrically/english/transcript.txt @@ -21,7 +21,7 @@ That slight change in area is what dF means in this context. It's the tiny increase to the value of f of x equals x squared, caused by increasing x by that tiny nudge dx. Now you can see that there's three new bits of area in this diagram, two thin rectangles and a minuscule square. The two thin rectangles each have side lengths of x and dx, so they account for 2 times x times dx units of new area. -For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx. +For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx. That little square there has an area of dx squared, but you should think of that as being really tiny, negligibly tiny. For example, if dx was 0.01, that would be only 0.0001, and keep in mind I'm drawing dx with a fair bit of width here just so we can actually see it, but always remember in principle, dx should be thought of as a truly tiny amount, and for those truly tiny amounts, a good rule of thumb is that you can ignore anything that includes a dx raised to a power greater than 1. That is, a tiny change squared is a negligible change. diff --git a/2017/derivative-formulas-geometrically/french/sentence_translations.json b/2017/derivative-formulas-geometrically/french/sentence_translations.json index ba9a65fc7..91ecc1cd4 100644 --- a/2017/derivative-formulas-geometrically/french/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/french/sentence_translations.json @@ -184,7 +184,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "Car cette nouvelle zone de ces deux rectangles minces serait de 2 fois 3 fois 0,01, soit 0,06, soit environ 6 fois la taille de dx.", "from_community_srt": "disons que x valait 3 et que dx valait 0,01. Cette nouvelle aire apparue via ces deux fins rectangles serait alors de 2 × 3 × 0,01 = 0,06 environ 6 fois la taille de dx.", "n_reviews": 0, diff --git a/2017/derivative-formulas-geometrically/german/sentence_translations.json b/2017/derivative-formulas-geometrically/german/sentence_translations.json index 9618db025..30c8c437c 100644 --- a/2017/derivative-formulas-geometrically/german/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/german/sentence_translations.json @@ -204,7 +204,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "Denn die neue Fläche aus diesen beiden dünnen Rechtecken wäre 2 mal 3 mal 0,01, also 0,06, etwa 6 mal so groß wie dx.", "model": "DeepL", "from_community_srt": "Wenn zum Beispiel x gleich 3 war und dx gleich 0,01 Dann wäre die neue Fläche von diesen beiden dünnen Rechtecken 2 * 3 * 0,01, was 0,06 ergibt, etwa 6 mal die Größe von dx.", diff --git a/2017/derivative-formulas-geometrically/hungarian/sentence_translations.json b/2017/derivative-formulas-geometrically/hungarian/sentence_translations.json index 6d9ace53f..66a607da9 100644 --- a/2017/derivative-formulas-geometrically/hungarian/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/hungarian/sentence_translations.json @@ -184,7 +184,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "A két vékony téglalapból származó új terület 2-szer 3szor 0,01, ami 0,06, azaz körülbelül 6-szor akkora, mint a dx.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/derivative-formulas-geometrically/italian/sentence_translations.json b/2017/derivative-formulas-geometrically/italian/sentence_translations.json index 73da75e8f..8b2dfd84b 100644 --- a/2017/derivative-formulas-geometrically/italian/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/italian/sentence_translations.json @@ -203,7 +203,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "La nuova area di questi due rettangoli sottili sarebbe 2 volte 3 volte 0,01, ovvero 0,06, circa 6 volte la dimensione di dx.", "model": "DeepL", "from_community_srt": "Per esempio, sia x= 3 e dx=0.01 Quindi la nuova area da questi 2 rettangoli sarebbe 2*3*0.01 che è 0.06,", diff --git a/2017/derivative-formulas-geometrically/polish/sentence_translations.json b/2017/derivative-formulas-geometrically/polish/sentence_translations.json index a1a3d9bab..183527b2e 100644 --- a/2017/derivative-formulas-geometrically/polish/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/polish/sentence_translations.json @@ -181,7 +181,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "", "from_community_srt": "Weźmy np. x = 3 i dx = 0.01. Wtedy pole prostokątów jest równe 2 * 3 * 0.01 = 0.06, czyli 6 * dx.", "n_reviews": 0, diff --git a/2017/derivative-formulas-geometrically/portuguese/sentence_translations.json b/2017/derivative-formulas-geometrically/portuguese/sentence_translations.json index c86899a4a..7d408898b 100644 --- a/2017/derivative-formulas-geometrically/portuguese/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/portuguese/sentence_translations.json @@ -203,7 +203,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "Pois essa nova área desses dois retângulos finos seria 2 vezes 3 vezes 0,01, que é 0,06, cerca de 6 vezes o tamanho de dx.", "model": "google_nmt", "from_community_srt": "Por exemplo, digamos que x foi 3 e dx foi 0,01. Então essa nova área desses dois retângulos finos seria 2 * 3 * 0,01, que é 0,06, cerca de 6 vezes o tamanho de dx. Aquela pequena praça lá tem uma área de dx ^ 2,", diff --git a/2017/derivative-formulas-geometrically/spanish/sentence_translations.json b/2017/derivative-formulas-geometrically/spanish/sentence_translations.json index d1308f444..cf71dcb32 100644 --- a/2017/derivative-formulas-geometrically/spanish/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/spanish/sentence_translations.json @@ -181,7 +181,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "Porque esa nueva área de estos dos rectángulos delgados sería 2 por 3 por 0,01, que es 0,06, aproximadamente 6 veces el tamaño de dx.", "from_community_srt": "Por ejemplo, digamos que x=3 y dx=0.01, entonces el área añadida de estos dos finos rectángulos será 2 por 3 por 0.01 que es 0.06, como unas 6 veces el tamaño de dx.", "n_reviews": 0, diff --git a/2017/derivative-formulas-geometrically/swedish/sentence_translations.json b/2017/derivative-formulas-geometrically/swedish/sentence_translations.json index 7845609c0..c32160ebe 100644 --- a/2017/derivative-formulas-geometrically/swedish/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/swedish/sentence_translations.json @@ -181,7 +181,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "", "from_community_srt": "Till exempel säg att x är 3 och dx 0.01. Då blir den nya arean från rektanglarna 2*3*0.01 vilket är 0.06. 6 gånger dx's storlek.", "n_reviews": 0, diff --git a/2017/derivative-formulas-geometrically/tagalog/sentence_translations.json b/2017/derivative-formulas-geometrically/tagalog/sentence_translations.json index 07013ee24..8e9d852f4 100644 --- a/2017/derivative-formulas-geometrically/tagalog/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/tagalog/sentence_translations.json @@ -184,7 +184,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "Para sa bagong lugar mula sa dalawang manipis na parihaba na ito ay magiging 2 beses 3 beses 0.01, na 0.06, mga 6 na beses ang laki ng dx.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivative-formulas-geometrically/turkish/sentence_translations.json b/2017/derivative-formulas-geometrically/turkish/sentence_translations.json index 9e42f43a3..1239ac446 100644 --- a/2017/derivative-formulas-geometrically/turkish/sentence_translations.json +++ b/2017/derivative-formulas-geometrically/turkish/sentence_translations.json @@ -204,7 +204,7 @@ "end": 193.78 }, { - "input": "For that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", + "input": "For example, let's say x was 3 and dx was 0.01, then that new area from these two thin rectangles would be 2 times 3 times 0.01, which is 0.06, about 6 times the size of dx.", "translatedText": "Bu iki ince dikdörtgenin oluşturduğu yeni alan 2 kere 3 kere 0.01, yani 0.06, yani dx'in yaklaşık 6 katı büyüklüğünde olacaktır.", "model": "DeepL", "from_community_srt": "Örneğin, x=3 ve dx=0.01 olsun. Böylece bu iki ince dikdörtgenden gelen yeni alan 2 * 3 * 0.01, yani 0.06 olur, yani dx'in büyüklüğünün 6 katı kadar.", diff --git a/2017/derivatives/arabic/sentence_translations.json b/2017/derivatives/arabic/sentence_translations.json index eedfce7b2..461ebb2be 100644 --- a/2017/derivatives/arabic/sentence_translations.json +++ b/2017/derivatives/arabic/sentence_translations.json @@ -549,7 +549,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "وعلى الرغم من أن عداد سرعة السيارة سينظر فعليًا إلى تغير في الوقت، مثل 0.01 ثانية، وعلى الرغم من أن برنامج الرسم هنا ينظر إلى تغير فعلي في الوقت، إلا أن المشتق في الرياضيات البحتة ليس هذه النسبة ds على dt لنسبة محددة اختيار dt، بدلاً من ذلك هو ما تقترب منه هذه النسبة حيث يقترب اختيارك لـ dt من 0.", "model": "google_nmt", "from_community_srt": "هذه تـقـرريـبـاً هي المشتقة ورغم أن عداد السرعة للسيارة سينظر فعلاً إلى تغير واقعي في الزمن، مثل 0.01 ورغم أن برنامج الرسم هنا ينظر إلى تغير واقعي وحقيقي في الزمن في الرياضيات البحتة المشتقة ليست هي النسبة ds على dt، باختيار محدد لـ dt عوضاً عن ذلك.. هي ما تؤول إليه تلك النسبة عندما يقترب اختيارك لـ dt من الصفر :لحسن الحظ..", @@ -689,7 +689,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "على سبيل المثال، يتم كتابة مشتق الرياضيات البحتة الصادق إلى الخير كـ ds مقسومًا على dt، على الرغم من أنه من الناحية الفنية ليس كسرًا في حد ذاته، ولكن مهما كان هذا الكسر يقترب من دفعات أصغر في t.", "model": "google_nmt", "from_community_srt": "كمثال: المشتقة الخالصة الرياضية البحتة تكتب كـ ds مقسومة على dt رغم أنها تقنياً ليست كسراً بحد ذاته، لكن ما يؤول إليه ذلك الكسر لدفعات أصغر فأصغر لـ t", diff --git a/2017/derivatives/bengali/sentence_translations.json b/2017/derivatives/bengali/sentence_translations.json index 28d7b6a26..ea4fbe9a0 100644 --- a/2017/derivatives/bengali/sentence_translations.json +++ b/2017/derivatives/bengali/sentence_translations.json @@ -592,7 +592,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph. ", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph. ", "translatedText": "এটি স্পর্শ করার প্রয়োজন ছাড়াই তাত্ক্ষণিক পরিবর্তনের প্যারাডক্সের সাথে ফ্লার্ট করার ধরণের, এবং এটি গ্রাফের একটি একক বিন্দুতে একটি স্পর্শক রেখার ঢাল হিসাবে একটি চমৎকার ভিজ্যুয়াল অন্তর্দৃষ্টির সাথে আসে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivatives/bulgarian/sentence_translations.json b/2017/derivatives/bulgarian/sentence_translations.json index 1c427ee35..c065e0b01 100644 --- a/2017/derivatives/bulgarian/sentence_translations.json +++ b/2017/derivatives/bulgarian/sentence_translations.json @@ -504,7 +504,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "И въпреки че скоростомерът на автомобила всъщност отчита промяна във времето, например 0,01 секунди, и въпреки че програмата за чертане тук отчита действителна промяна във времето, в чистата математика производната не е това съотношение ds спрямо dt за конкретен избор на dt, а е каквото и да е това съотношение, когато вашият избор за dt се приближава до 0.", "model": "DeepL", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "Например чисто математическата производна се записва като ds, разделено на dt, въпреки че технически не е дроб сама по себе си, а каквото и да е, което се приближава до тази дроб за по-малките промени в t.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/derivatives/chinese/sentence_translations.json b/2017/derivatives/chinese/sentence_translations.json index d69eeadc4..4478bfd6a 100644 --- a/2017/derivatives/chinese/sentence_translations.json +++ b/2017/derivatives/chinese/sentence_translations.json @@ -665,7 +665,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", "translatedText": "它有点与瞬间变化的悖论调情，而无需触 摸它，并且它也具有很好的视觉直觉， 就像图表上单个点的切线的斜率一样。", "model": "google_nmt", "from_community_srt": "它在和這個關於「瞬間」的悖論周旋， 甚至無需接觸 「瞬間」本身。 而且它在視覺上有個很好的直覺， 如斜率是圖上在某個點上 的切線。", diff --git a/2017/derivatives/dutch/sentence_translations.json b/2017/derivatives/dutch/sentence_translations.json index 314b1c43c..f3629d8d6 100644 --- a/2017/derivatives/dutch/sentence_translations.json +++ b/2017/derivatives/dutch/sentence_translations.json @@ -504,7 +504,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "En ook al kijkt de snelheidsmeter van een auto feitelijk naar een verandering in tijd, zoals 0,01 seconde, en ook al kijkt het tekenprogramma hier naar een feitelijke verandering in tijd, in pure wiskunde is de afgeleide niet deze verhouding ds over dt voor een specifieke keuze van dt, in plaats daarvan is het wat die verhouding benadert als je keuze voor dt de 0 nadert.", "model": "DeepL", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "Zo wordt de pure wiskunde-afgeleide in alle eerlijkheid geschreven als ds gedeeld door dt, ook al is het technisch gezien niet per se een breuk, maar wat die breuk ook benadert voor kleinere stappen in t.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/derivatives/english/captions.srt b/2017/derivatives/english/captions.srt index 631e6edbd..ca6bc580b 100644 --- a/2017/derivatives/english/captions.srt +++ b/2017/derivatives/english/captions.srt @@ -455,23 +455,23 @@ This idea of ds over dt, a tiny change in the value of the function s divided by the tiny change in the input that caused it, that's almost what a derivative is. 115 -00:07:38,700 --> 00:07:43,616 -And even though a car's speedometer will actually look at a change in time, +00:07:38,700 --> 00:07:43,908 +And even though a car's speedometer will actually look at a concrete change in time, 116 -00:07:43,616 --> 00:07:49,050 +00:07:43,908 --> 00:07:49,055 like 0.01 seconds, and even though the drawing program here is looking at an actual 117 -00:07:49,050 --> 00:07:54,743 -change in time, in pure math the derivative is not this ratio ds over dt for a specific +00:07:49,055 --> 00:07:54,448 +concrete change in time, in pure math the derivative is not this ratio ds over dt for a 118 -00:07:54,743 --> 00:07:59,919 -choice of dt, instead it's whatever that ratio approaches as your choice for dt +00:07:54,448 --> 00:07:59,963 +specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt 119 -00:07:59,919 --> 00:08:00,760 +00:07:59,963 --> 00:08:00,760 approaches 0. 120 @@ -615,16 +615,16 @@ you're kind of announcing your intention that eventually you're going to see what happens as dt approaches 0. 155 -00:10:11,920 --> 00:10:17,119 +00:10:11,920 --> 00:10:16,829 For example, the honest-to-goodness pure math derivative is written as ds divided by dt, 156 -00:10:17,119 --> 00:10:20,157 +00:10:16,829 --> 00:10:19,697 even though it's technically not a fraction per se, 157 -00:10:20,157 --> 00:10:23,780 -but whatever that fraction approaches for smaller nudges in t. +00:10:19,697 --> 00:10:23,780 +but whatever that fraction approaches for smaller and smaller nudges in t. 158 00:10:25,780 --> 00:10:27,680 diff --git a/2017/derivatives/english/sentence_timings.json b/2017/derivatives/english/sentence_timings.json index 1b2ab765a..ab562bcd7 100644 --- a/2017/derivatives/english/sentence_timings.json +++ b/2017/derivatives/english/sentence_timings.json @@ -315,7 +315,7 @@ 458.0 ], [ - "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", 458.7, 480.76 ], @@ -395,7 +395,7 @@ 611.1 ], [ - "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", 611.92, 623.78 ], diff --git a/2017/derivatives/english/transcript.txt b/2017/derivatives/english/transcript.txt index 70bc0253b..67bc7d6ba 100644 --- a/2017/derivatives/english/transcript.txt +++ b/2017/derivatives/english/transcript.txt @@ -61,7 +61,7 @@ Then you can just divide that difference by the change in time, dt, and that giv So with a formula like this, you could give the computer any curve representing any distance function s of t, and it could figure out the curve representing velocity. Now would be a good time to pause, reflect, and make sure this idea of relating distance to velocity by looking at tiny changes makes sense, because we're going to tackle the paradox of the derivative head on. This idea of ds over dt, a tiny change in the value of the function s divided by the tiny change in the input that caused it, that's almost what a derivative is. -And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0. +And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0. Luckily there is a really nice visual understanding for what it means to ask what this ratio approaches, Remember, for any specific choice of dt, this ratio ds over dt is the slope of a line passing through two separate points on the graph, right? Well as dt approaches 0, and as those two points approach each other, the slope of the line approaches the slope of a line that's tangent to the graph at whatever point t we're looking at. So the true honest-to-goodness pure math derivative is not the rise over run slope between two nearby points on the graph, it's equal to the slope of a line tangent to the graph at a single point. @@ -77,7 +77,7 @@ And because change in an instant still makes no sense, I think it's healthiest f By the way, it's worth saying a couple words on notation here. Throughout this video I've been using dt to refer to a tiny change in t with some actual size, and ds to refer to the resulting change in s, which again has an actual size, and this is because that's how I want you to think about them. But the convention in calculus is that whenever you're using the letter d like this, you're kind of announcing your intention that eventually you're going to see what happens as dt approaches 0. -For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t. +For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t. I think a specific example should help here. You might think that asking about what this ratio approaches for smaller and smaller values would make it much more difficult to compute, but weirdly it kind of makes things easier. Let's say you have a given distance vs time function that happens to be exactly t cubed. diff --git a/2017/derivatives/french/sentence_translations.json b/2017/derivatives/french/sentence_translations.json index f6bc9cf7d..f67a41f4b 100644 --- a/2017/derivatives/french/sentence_translations.json +++ b/2017/derivatives/french/sentence_translations.json @@ -502,7 +502,7 @@ "end": 458 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "Et même si le compteur de vitesse d'une voiture indique en réalité un changement de temps, par exemple 0,01 seconde, et même si le programme de dessin ici examine un changement de temps réel, en mathématiques pures, la dérivée n'est pas ce rapport ds sur dt pour un temps spécifique. choix de dt, c'est plutôt ce que ce rapport approche lorsque votre choix pour dt s'approche de 0.", "from_community_srt": "Même si le compteur de vitesse dans notre voiture regardera un changement réel dans le temps comme 0.01 secondes, et même si mon logiciel de graphismes trouve une fonction de vitesse donnée ayant une fonction de position regarde pour une valeur concrète de dt, en mathématiques pur, le dérivé n'est pas ce rapport ds/dt pour tout choix spécifique de dt. Il est plutôt n'importe quel valeur qu'approche le rapport comme le choix pour dt approchant 0", "n_reviews": 0, @@ -630,7 +630,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "Par exemple, la dérivée mathématique pure et honnête s'écrit sous la forme ds divisé par dt, même si ce n'est techniquement pas une fraction en soi, mais quelle que soit l'approche de cette fraction pour des coups de pouce plus petits dans t.", "from_community_srt": "le dérivé authentique de la fonction s(t) est écrit sous la forme ds/dt, même si le dérivé n'est pas une fraction en soi, mais la valeur qu'approche la fraction pour de plus en plus petits valeurs de t.", "n_reviews": 0, @@ -1068,4 +1068,4 @@ "start": 999.18, "end": 1008.4 } -] +] \ No newline at end of file diff --git a/2017/derivatives/german/sentence_translations.json b/2017/derivatives/german/sentence_translations.json index 53b80beff..1cb11b200 100644 --- a/2017/derivatives/german/sentence_translations.json +++ b/2017/derivatives/german/sentence_translations.json @@ -562,7 +562,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "Und obwohl der Tachometer eines Autos eine Zeitänderung wie 0,01 Sekunden anzeigt und obwohl das Zeichenprogramm hier eine tatsächliche Zeitänderung anzeigt, ist die Ableitung in der reinen Mathematik nicht dieses Verhältnis ds über dt für eine bestimmte Wahl von dt, sondern das, was sich diesem Verhältnis annähert, wenn deine Wahl für dt sich 0 nähert.", "model": "DeepL", "from_community_srt": "Auch wenn der Tacho des Autos eine tatsächliche Änderung der Zeit um z.B. 0,01 Sekunden betrachtet um die Geschwindigkeit zu berechnen, und obwohl mein Programm zum Finden einer Geschwindigkeitsfunktion einen konkreten Wert dt verwendet, in reiner Mathematik, ist die Ableitung nicht dieses Verhältnis ds / dt für eine bestimmte Wahl von dt. Es ist, welchem Wert auch immer sich dieses Verhältnis nähert,", @@ -705,7 +705,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "So wird zum Beispiel die ehrliche, rein mathematische Ableitung als ds geteilt durch dt geschrieben, auch wenn es technisch gesehen kein Bruch an sich ist, sondern das, was sich diesem Bruch bei kleineren Verschiebungen von t nähert.", "model": "DeepL", "from_community_srt": "Zum Beispiel, die Ableitung der Funktion s(t) wird als ds / dt geschrieben, obwohl die Ableitung kein Bruch an sich ist, aber was auch immer dieser Bruch sich nähert für kleinere und kleinere Stücke t.", diff --git a/2017/derivatives/hebrew/sentence_translations.json b/2017/derivatives/hebrew/sentence_translations.json index 1c53b3223..0a7c39107 100644 --- a/2017/derivatives/hebrew/sentence_translations.json +++ b/2017/derivatives/hebrew/sentence_translations.json @@ -556,7 +556,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "ולמרות שמד המהירות של מכונית יסתכל למעשה על שינוי בזמן, כמו 0.01 שניות, ולמרות שתוכנית השרטוט כאן מסתכלת על שינוי בפועל בזמן, במתמטיקה טהורה הנגזרת היא לא היחס הזה ds על dt עבור ספציפי הבחירה של dt, במקום זאת זה לא משנה מה היחס הזה שיתקרב כאשר הבחירה שלך עבור dt מתקרבת ל-0.", "model": "google_nmt", "from_community_srt": "כדי לחשב מהירות, ולמרות שהתוכנה שלי למציאת פונקציית מהירות משתמשת בערך קונקרטי של dt, במתמטיקה טהורה, הנגזרת אינה היחס ds/dt עבור בחירה ספציפית של dt הנגזרת היא הערך שהיחס שואף אליו כאשר dt שואף ל-0", @@ -697,7 +697,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "לדוגמה, הנגזרת המתמטית הטהורה של כנה לטובה כתובה כ-ds חלקי dt, למרות שבאופן טכני זה לא שבר כשלעצמו, אלא כל מה שהשבר הזה מתקרב לתנודות קטנות יותר ב-t.", "model": "google_nmt", "from_community_srt": "לדוגמה, הנגזרת ה\"כמעט נכונה\" שרשומה בתור ds/dt למרות שהנגזרת אינה שבר כשלעצמה,", diff --git a/2017/derivatives/hindi/sentence_translations.json b/2017/derivatives/hindi/sentence_translations.json index 215ca19db..98f51ca12 100644 --- a/2017/derivatives/hindi/sentence_translations.json +++ b/2017/derivatives/hindi/sentence_translations.json @@ -518,7 +518,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", "translatedText": "यह बिना किसी स्पर्श की आवश्यकता के एक पल में परिवर्तन के विरोधाभास के साथ खिलवाड़ करने जैसा है, और यह एक अच्छे दृश्य अंतर्ज्ञान के साथ भी आता है, जैसे कि ग्राफ़ पर एक बिंदु पर स्पर्शरेखा रेखा का ढलान।", "n_reviews": 0, "start": 558.06, diff --git a/2017/derivatives/hungarian/sentence_translations.json b/2017/derivatives/hungarian/sentence_translations.json index 6da07e52c..88c7be627 100644 --- a/2017/derivatives/hungarian/sentence_translations.json +++ b/2017/derivatives/hungarian/sentence_translations.json @@ -504,7 +504,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "És bár az autó sebességmérője valójában egy időbeli változást, például 0,01 másodpercet mutat, és bár a rajzolóprogram itt egy tényleges időbeli változást vizsgál, a tiszta matematikában a derivált nem a ds és a dt közötti arány egy adott dt értéknél, hanem az, amit ez az arány megközelít, ahogy a dt értéke a 0-hoz közelít.", "model": "DeepL", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "Például az őszinte tiszta matematikai deriváltat úgy írjuk, hogy ds osztva dt-vel, bár ez technikailag nem egy tört önmagában, hanem bármi, amit ez a tört megközelít a t kisebb lökéseinél.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/derivatives/indonesian/sentence_translations.json b/2017/derivatives/indonesian/sentence_translations.json index b320c1873..c37d29890 100644 --- a/2017/derivatives/indonesian/sentence_translations.json +++ b/2017/derivatives/indonesian/sentence_translations.json @@ -592,7 +592,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", "translatedText": "Ini semacam menggoda dengan paradoks perubahan dalam sekejap tanpa perlu menyentuhnya, dan ia hadir dengan intuisi visual yang bagus juga, seperti kemiringan garis singgung ke satu titik pada grafik.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivatives/italian/sentence_translations.json b/2017/derivatives/italian/sentence_translations.json index b7d91f95f..4a77c46f5 100644 --- a/2017/derivatives/italian/sentence_translations.json +++ b/2017/derivatives/italian/sentence_translations.json @@ -563,7 +563,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "E anche se il tachimetro di un'auto guarda effettivamente una variazione di tempo, come 0,01 secondi, e anche se il programma di disegno qui sta guardando un'effettiva variazione di tempo, in matematica pura la derivata non è questo rapporto ds su dt per una specifica scelta di dt, ma è qualsiasi rapporto si avvicini quando la tua scelta di dt si avvicina a 0.", "model": "DeepL", "from_community_srt": "Anche se il tachimetro della vettura utilizza un reale intervallo di tempo, ad esempio 0,01 secondi e anche il mio programma di disegno calcola un valore concreto di dt nella matematica pura, la derivata non è questo rapporto ds / dt per una qualisasi scelta specifica di dt. Invece, Si tratta di un qualsiasi valore a cui il rapporto tende quando la scelta di dt tende a zero", @@ -705,7 +705,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "Ad esempio, la derivata puramente matematica si scrive come ds diviso dt, anche se tecnicamente non si tratta di una frazione in sé, ma di qualsiasi cosa si avvicini a tale frazione per piccole variazioni in t.", "model": "DeepL", "from_community_srt": "la buona vecchia derivata della funzione s (t) è scritta come ds / dt, anche se la derivato non è una frazione, di per sé,", diff --git a/2017/derivatives/japanese/sentence_translations.json b/2017/derivatives/japanese/sentence_translations.json index e3334103b..d7f272728 100644 --- a/2017/derivatives/japanese/sentence_translations.json +++ b/2017/derivatives/japanese/sentence_translations.json @@ -592,7 +592,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", "translatedText": "これは、変化のパラドックスに触れる必要もなく瞬時 に理解できるもので、グラフ上の 1 点に対する接 線の傾きなど、優れた視覚的直観も備えています。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivatives/korean/sentence_translations.json b/2017/derivatives/korean/sentence_translations.json index d3d9f2aa1..34f9638b6 100644 --- a/2017/derivatives/korean/sentence_translations.json +++ b/2017/derivatives/korean/sentence_translations.json @@ -560,7 +560,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "자동차의 속도계는 실제로 0.01초와 같은 시간의 변화를 보고, 여기서 그리기 프로그램은 실제 시간의 변화를 보고 있지만, 순수 수학에서 미분은 특정 선택의 dt에 대한 ds의 비율이 아니라, 선택의 dt가 0에 가까워질수록 그 비율이 어떻게 다가오는지를 나타냅니다.", "model": "DeepL", "from_community_srt": "우리가 보고 있는 속도계의 숫자가 실제로 이렇게 짧은 구간의 거리 차이를 이용하여 속력을 구할지라도, 그리고 이 방법으로 어떤 곡선 s(t)에 대해 속력 곡선을 그릴 수 있을지라도 어떤 정해진 수 dt를 정해서 이 계산을 하는 것은 순수 수학에서 정의하는 미분은 아닙니다. 미분은 이 계산 방법에서, dt가 0으로 접근하는 값입니다.", @@ -701,7 +701,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "예를 들어, 정직하게 순수한 수학 도함수는 기술적으로는 분수 자체가 아니지만, 그 분수가 t의 작은 넛지에 접근하는 것이 무엇이든 간에 ds를 dt로 나눈 값으로 표기됩니다.", "model": "DeepL", "from_community_srt": "예를 들어서 s(t)의 도함수는 실제로 ds/dt입니다. 비록 도함수가 분수는 아니지만 어찌되었건분수에서 t가 점점 작은 값으로 향한다는 것입니다.", diff --git a/2017/derivatives/marathi/sentence_translations.json b/2017/derivatives/marathi/sentence_translations.json index be28f664a..268e2b953 100644 --- a/2017/derivatives/marathi/sentence_translations.json +++ b/2017/derivatives/marathi/sentence_translations.json @@ -592,7 +592,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", "translatedText": "एका क्षणात बदल करण्याच्या विरोधाभासाने त्याला स्पर्श करण्याची गरज न पडता फ्लर्टिंग करण्याचा हा प्रकार आहे, आणि ते आलेखावरील एका बिंदूपर्यंत स्पर्शरेषेच्या उताराप्रमाणे एक छान दृश्य अंतर्ज्ञान देखील देते.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivatives/persian/sentence_translations.json b/2017/derivatives/persian/sentence_translations.json index de77dbcbd..b7df64cff 100644 --- a/2017/derivatives/persian/sentence_translations.json +++ b/2017/derivatives/persian/sentence_translations.json @@ -592,7 +592,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph. ", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph. ", "translatedText": "این تمیز نیست؟ این نوعی معاشقه با پارادوکس تغییر در یک لحظه بدون نیاز به لمس آن است، و همچنین با شهود بصری خوبی همراه است، به عنوان شیب یک خط مماس به یک نقطه در نمودار. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivatives/polish/sentence_translations.json b/2017/derivatives/polish/sentence_translations.json index b30677f60..4d58663a7 100644 --- a/2017/derivatives/polish/sentence_translations.json +++ b/2017/derivatives/polish/sentence_translations.json @@ -500,7 +500,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "", "from_community_srt": "Nawet, jeśli prędkościomierz patrzy na zmianę w konkretnym czasie, np. 0.01s, tak samo jak program do rysowania wykresów patrzy na konkretną wartość dt, w matematyce pochodna nie jest ilorazem ds/dt dla konkretnego dt. To wartość, do której zbliża się ten ułamek, gdy dt zbliża się do 0.", "n_reviews": 0, @@ -627,7 +627,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "", "from_community_srt": "Przykładowo, w matematyce pochodną funkcji s(t) oznaczamy ds/dt nawet, jeśli pochodna nie jest ułamkiem, ale tym, do czego dąży.", "n_reviews": 0, diff --git a/2017/derivatives/portuguese/sentence_translations.json b/2017/derivatives/portuguese/sentence_translations.json index 5aa96cb19..e203101fe 100644 --- a/2017/derivatives/portuguese/sentence_translations.json +++ b/2017/derivatives/portuguese/sentence_translations.json @@ -564,7 +564,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "E mesmo que o velocímetro de um carro realmente observe uma mudança no tempo, como 0,01 segundo, e mesmo que o programa de desenho aqui esteja olhando para uma mudança real no tempo, em matemática pura a derivada não é esta razão ds sobre dt para um determinado escolha de dt, em vez disso, é qualquer proporção que se aproxime à medida que sua escolha para dt se aproxima de 0.", "model": "google_nmt", "from_community_srt": "Mesmo que o velocímetro do nosso carro se baseie em uma mudança no tempo - como 0,01s - e mesmo que o programa de computador analise uma passagem de tempo, na Matemática, a derivada não é a razão ds/dt para um valor específico de \"dt\". Ao invés disso, ela é o valor para o qual essa razão converge quando \"dt\" se aproxima de zero.", @@ -708,7 +708,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "Por exemplo, a derivada matemática pura, honesta, é escrita como ds dividido por dt, embora tecnicamente não seja uma fração em si, mas o que quer que essa fração se aproxime para pequenos deslocamentos em t.", "model": "google_nmt", "from_community_srt": "Por exemplo, a forma correta da derivada da função \"s(t)\"' é escrita como \"ds/dt\", mesmo que ela não seja tecnicamente uma fração, e sim uma aproximação de tal razão para partes cade vez menores em \"t\".", diff --git a/2017/derivatives/russian/sentence_translations.json b/2017/derivatives/russian/sentence_translations.json index 5469169d4..5752bae72 100644 --- a/2017/derivatives/russian/sentence_translations.json +++ b/2017/derivatives/russian/sentence_translations.json @@ -592,7 +592,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph. ", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph. ", "translatedText": "Это своего рода заигрывание с парадоксом мгновенных изменений, даже не прикасаясь к нему, и это также сопровождается приятной визуальной интуицией, например, наклоном касательной линии к одной точке на графике. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivatives/spanish/sentence_translations.json b/2017/derivatives/spanish/sentence_translations.json index 758287563..bd067a39f 100644 --- a/2017/derivatives/spanish/sentence_translations.json +++ b/2017/derivatives/spanish/sentence_translations.json @@ -497,7 +497,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "Y aunque el velocímetro de un automóvil en realidad observará un cambio en el tiempo, como 0,01 segundos, y aunque el programa de dibujo aquí observa un cambio real en el tiempo, en matemáticas puras la derivada no es esta relación ds sobre dt para un tiempo específico. elección de dt, en lugar de eso, es lo que sea que se acerque esa relación a medida que su elección de dt se acerca a 0.", "from_community_srt": "Aunque el auto compare dos puntos a distancia fija para calcular la velocidad y aunque yo también elegí un dt fijo para poder graficar mi función de velocidad, en matemáticas, la derivada no es en verdad \"ds/dt\" para un dt cualquiera, no importa cual elijamos. Es en verdad esa división para cuando dt tiende a 0.", "n_reviews": 0, @@ -622,7 +622,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "Por ejemplo, la derivada matemática pura, sinceramente, se escribe como ds dividido por dt, aunque técnicamente no es una fracción per se, sino lo que sea que esa fracción se acerque para empujones más pequeños en t.", "from_community_srt": "Por ejemplo, la verdadera derivada de s(t) se escribe como ds/dt, aunque la derivada no es una división en sí sino el valor al cual esa división se acerca para dt's cada vez más chiquitos.", "n_reviews": 0, diff --git a/2017/derivatives/swedish/sentence_translations.json b/2017/derivatives/swedish/sentence_translations.json index 03347d462..411bc8fdb 100644 --- a/2017/derivatives/swedish/sentence_translations.json +++ b/2017/derivatives/swedish/sentence_translations.json @@ -498,7 +498,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "", "from_community_srt": "Även om vår bils hastighetsmätare kommer undersöka en faktisk förändring i tid som 0.01 sekunder för att beräknahastighet, och även om mitt datorprogram här använder ett visst litet dt för att finna en hastighetsfunktion I den rena matematiken är derivatan inte den här kvoten för något specifikt val av dt. Istället är det vilket värde den kvoten närmar sig då valet för dt går mot 0.", "n_reviews": 0, @@ -625,7 +625,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "", "from_community_srt": "Till exempel, den ärliga och sanna derivatan av funktionen s(t) är skriven som ds/dt, även om derivatan inte är ett bråk, men vad det bråket närmar sig för mindre och mindre hopp i t.", "n_reviews": 0, diff --git a/2017/derivatives/tagalog/sentence_translations.json b/2017/derivatives/tagalog/sentence_translations.json index f778c8ac0..a22634961 100644 --- a/2017/derivatives/tagalog/sentence_translations.json +++ b/2017/derivatives/tagalog/sentence_translations.json @@ -504,7 +504,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "At kahit na ang speedometer ng kotse ay aktwal na titingnan ang pagbabago sa oras, tulad ng 0.01 segundo, at kahit na ang drawing program dito ay tumitingin sa isang aktwal na pagbabago sa oras, sa purong matematika ang derivative ay hindi ang ratio na ito ds sa dt para sa isang partikular na pagpili ng dt, sa halip ito ay anuman ang lumalapit na ratio habang ang iyong pinili para sa dt ay lumalapit sa 0.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "Halimbawa, ang honest-to-goodness pure math derivative ay isinulat bilang ds na hinati sa dt, kahit na ito ay teknikal na hindi isang fraction per se, ngunit anuman ang fraction na iyon na lumalapit para sa mas maliliit na nudges sa t.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivatives/tamil/sentence_translations.json b/2017/derivatives/tamil/sentence_translations.json index 5e2c5a105..485b8d8e7 100644 --- a/2017/derivatives/tamil/sentence_translations.json +++ b/2017/derivatives/tamil/sentence_translations.json @@ -592,7 +592,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", "translatedText": "அதைத் தொடத் தேவையில்லாமல் ஒரு நொடியில் மாற்றத்தின் முரண்பாட்டுடன் ஊர்சுற்றுவது போன்றது, மேலும் இது வரைபடத்தில் ஒரு புள்ளியில் ஒரு தொடுகோட்டின் சாய்வாக ஒரு நல்ல காட்சி உள்ளுணர்வுடன் வருகிறது.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivatives/telugu/sentence_translations.json b/2017/derivatives/telugu/sentence_translations.json index e16f88889..8239a2ebb 100644 --- a/2017/derivatives/telugu/sentence_translations.json +++ b/2017/derivatives/telugu/sentence_translations.json @@ -592,7 +592,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", "translatedText": "ఇది ఎప్పుడూ తాకాల్సిన అవసరం లేకుండా తక్షణమే మార్పు యొక్క వైరుధ్యంతో సరసాలాడుట మరియు ఇది గ్రాఫ్‌లోని ఒక బిందువుకు టాంజెంట్ లైన్ యొక్క వాలు వలె చక్కటి దృశ్యమాన అంతర్ దృష్టితో కూడా వస్తుంది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivatives/thai/sentence_translations.json b/2017/derivatives/thai/sentence_translations.json index fce8109f0..e2410a401 100644 --- a/2017/derivatives/thai/sentence_translations.json +++ b/2017/derivatives/thai/sentence_translations.json @@ -592,7 +592,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph. ", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivatives/turkish/sentence_translations.json b/2017/derivatives/turkish/sentence_translations.json index 771fabc14..6cdd5a372 100644 --- a/2017/derivatives/turkish/sentence_translations.json +++ b/2017/derivatives/turkish/sentence_translations.json @@ -561,7 +561,7 @@ "end": 458.0 }, { - "input": "And even though a car's speedometer will actually look at a change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt, instead it's whatever that ratio approaches as your choice for dt approaches 0.", + "input": "And even though a car's speedometer will actually look at a concrete change in time, like 0.01 seconds, and even though the drawing program here is looking at an actual concrete change in time, in pure math the derivative is not this ratio ds over dt for a specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt approaches 0.", "translatedText": "Her ne kadar bir arabanın hız göstergesi 0,01 saniye gibi gerçek bir zaman değişimine bakıyor olsa da ve buradaki çizim programı gerçek bir zaman değişimine bakıyor olsa da, saf matematikte türev, belirli bir dt seçimi için ds'nin dt'ye oranı değildir, bunun yerine dt seçiminiz 0'a yaklaştıkça bu oranın yaklaştığı şeydir.", "model": "DeepL", "from_community_srt": "her ne kadar aracımızın hız ölçeri, hızı hesaplamak için gerçek bir 0.01 zaman değeri kullansa bile, ve burada yazdığımız hız bulma fonksiyonumuz da somut bir dt değeri kullanıyor olsa bile, saf matematikte türev, spesifik bir dt değeri için ds/dt değildir. Aslında türev, ds/dt'nin, dt 0'a yaklaşırken yakınsadığı orandır.", @@ -703,7 +703,7 @@ "end": 611.1 }, { - "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller nudges in t.", + "input": "For example, the honest-to-goodness pure math derivative is written as ds divided by dt, even though it's technically not a fraction per se, but whatever that fraction approaches for smaller and smaller nudges in t.", "translatedText": "Örneğin, saf matematik türevi teknik olarak bir kesir olmasa da ds bölü dt olarak yazılır, ancak bu kesir t'deki daha küçük dürtmeler için neye yaklaşıyorsa odur.", "model": "DeepL", "from_community_srt": "Örnek olarak, bir s(t) fonksiyonunun hakiki türevi ds/dt olarak yazılmıştır. türev, her bir \"s\" için bir kesit olmasa bile küçük bir t değeri değişikliği için kesitin yakınsadığı şeydir.", diff --git a/2017/derivatives/ukrainian/sentence_translations.json b/2017/derivatives/ukrainian/sentence_translations.json index 269d65040..ecde5162f 100644 --- a/2017/derivatives/ukrainian/sentence_translations.json +++ b/2017/derivatives/ukrainian/sentence_translations.json @@ -592,7 +592,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph.", "translatedText": "Він ніби заграє з парадоксом зміни миттєво, навіть не торкаючись до нього, а також має гарну візуальну інтуїцію, як нахил дотичної лінії до однієї точки на графіку.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivatives/urdu/sentence_translations.json b/2017/derivatives/urdu/sentence_translations.json index 7099d1c89..fdb2c94d7 100644 --- a/2017/derivatives/urdu/sentence_translations.json +++ b/2017/derivatives/urdu/sentence_translations.json @@ -592,7 +592,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph. ", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph. ", "translatedText": "کیا یہ صاف نہیں ہے؟ یہ ایک لمحے میں تبدیلی کے تضاد کے ساتھ چھیڑ چھاڑ کرنے کی طرح ہے اسے چھونے کی ضرورت کے بغیر، اور یہ ایک اچھی بصری وجدان کے ساتھ بھی آتا ہے، جیسا کہ گراف پر ایک نقطہ تک ٹینجنٹ لائن کی ڈھلوان۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/derivatives/vietnamese/sentence_translations.json b/2017/derivatives/vietnamese/sentence_translations.json index 137585789..c4fde66e1 100644 --- a/2017/derivatives/vietnamese/sentence_translations.json +++ b/2017/derivatives/vietnamese/sentence_translations.json @@ -592,7 +592,7 @@ "end": 557.52 }, { - "input": "It's kind of flirting with the paradox of change in an instant without ever needing to touch it, and it comes with a nice visual intuition too, as the slope of a tangent line to a single point on the graph. ", + "input": "It's kind of flirting with the paradox of change in an instant without ever needing to actually touch it. And it comes with such a nice visual intuition too, as the slope of a tangent line to a single point on the graph. ", "translatedText": "Nó giống như đang đùa giỡn với nghịch lý của sự thay đổi tức thì mà không cần phải chạm vào nó, và nó cũng đi kèm với một trực giác trực quan tuyệt vời, giống như độ dốc của một đường tiếp tuyến với một điểm duy nhất trên biểu đồ. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/essence-of-calculus/bengali/sentence_translations.json b/2017/essence-of-calculus/bengali/sentence_translations.json index 822ed2e5d..f82c392f5 100644 --- a/2017/essence-of-calculus/bengali/sentence_translations.json +++ b/2017/essence-of-calculus/bengali/sentence_translations.json @@ -248,7 +248,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1. ", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1. ", "translatedText": "সুতরাং আমরা যেখানে আছি তা যোগ করার জন্য, আপনি বৃত্তের ক্ষেত্রফলকে এই সমস্ত রিংগুলিতে বিভক্ত করেছেন, এবং আপনি তাদের প্রতিটির ক্ষেত্রফলকে আনুমানিক 2 পাই এর ব্যাসার্ধের গুন ডর হিসাবে অনুমান করছেন, যেখানে নির্দিষ্ট মান সেই অভ্যন্তরীণ ব্যাসার্ধের জন্য ক্ষুদ্রতম রিংয়ের জন্য 0 থেকে, সবচেয়ে বড় রিংয়ের জন্য মাত্র 3-এর নিচে, আপনি dr-এর জন্য যে বেধটি বেছে নিন তার মধ্যে ব্যবধান 0 এর মতো।1. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen. ", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen. ", "translatedText": "আমি বলতে চাচ্ছি, 2 গুণ পাই গুণ 3 19 এর কাছাকাছি, তাই আসুন একটি y-অক্ষ ছুঁড়ে ফেলি যা একটু ভিন্নভাবে স্কেল করা হয়েছে যাতে আমরা এই সমস্ত আয়তক্ষেত্রগুলিকে স্ক্রিনে ফিট করতে পারি।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/essence-of-calculus/chinese/sentence_translations.json b/2017/essence-of-calculus/chinese/sentence_translations.json index b2552077f..05c4037a8 100644 --- a/2017/essence-of-calculus/chinese/sentence_translations.json +++ b/2017/essence-of-calculus/chinese/sentence_translations.json @@ -271,7 +271,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1. ", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1. ", "translatedText": "总结一下我们现在的情况，您已将圆的面积分解为所 有这些环，并且将每个环的面积近似为 2 pi 乘以其半径乘以 dr，其中具体值因为内半径范 围从最小环的 0 到最大环的略低于 3，间隔 由您为 dr 选择的任何厚度（例如 0）。1. ", "model": "google_nmt", "from_community_srt": "我們會把圓切成越來越小的環 總結目前的進度 你把整個圓分割成這些環 而且你用2πR乘上dr來近似每個環的面積 而內部半徑的值(r)則依序從 0(最小的環)一直到3(最大的環) 中間間隔的值是你所選的dr，", @@ -324,7 +324,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen. ", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen. ", "translatedText": "我的意思是，2 乘以 pi 乘以 3 大约为 19，所以我们只需要设置一 个缩放比例稍有不同的 y 轴，这样我们就可以将所有这些矩形放入屏幕上。", "model": "google_nmt", "from_community_srt": "2πr對於螢幕來說太高了 2*π*3大約是19 所以讓我們把y軸延伸， 稍微縮放一下 好讓螢幕可以容下所有的矩形 為了更好理解它，", diff --git a/2017/essence-of-calculus/french/sentence_translations.json b/2017/essence-of-calculus/french/sentence_translations.json index b0a5cf023..b91f329b9 100644 --- a/2017/essence-of-calculus/french/sentence_translations.json +++ b/2017/essence-of-calculus/french/sentence_translations.json @@ -908,4 +908,4 @@ "start": 997.02, "end": 1003.42 } -] +] \ No newline at end of file diff --git a/2017/essence-of-calculus/hindi/sentence_translations.json b/2017/essence-of-calculus/hindi/sentence_translations.json index 33205a009..5fe24b425 100644 --- a/2017/essence-of-calculus/hindi/sentence_translations.json +++ b/2017/essence-of-calculus/hindi/sentence_translations.json @@ -217,7 +217,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1.", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1.", "translatedText": "तो संक्षेप में कहें तो हम कहां हैं, आपने वृत्त के क्षेत्रफल को इन सभी छल्लों में तोड़ दिया है, और आप उनमें से प्रत्येक के क्षेत्रफल का अनुमान उसकी त्रिज्या के 2 pi गुणा dr के रूप में लगा रहे हैं, जहां विशिष्ट मान उस आंतरिक त्रिज्या के लिए सबसे छोटी अंगूठी के लिए 0 से लेकर, सबसे बड़ी अंगूठी के लिए सिर्फ 3 से कम तक, जो भी मोटाई आप डॉ के लिए चुनते हैं, उसके बीच 0 जैसा कुछ होता है।1.", "n_reviews": 0, "start": 247.7, @@ -259,7 +259,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen.", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen.", "translatedText": "मेरा मतलब है, 2 गुना पाई गुना 3 19 के आसपास है, तो चलिए बस एक y-अक्ष फेंकते हैं जिसे थोड़ा अलग तरीके से स्केल किया गया है ताकि हम इन सभी आयतों को स्क्रीन पर फिट कर सकें।", "n_reviews": 0, "start": 312.8, diff --git a/2017/essence-of-calculus/indonesian/sentence_translations.json b/2017/essence-of-calculus/indonesian/sentence_translations.json index 309df0682..f256b8550 100644 --- a/2017/essence-of-calculus/indonesian/sentence_translations.json +++ b/2017/essence-of-calculus/indonesian/sentence_translations.json @@ -248,7 +248,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1.", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1.", "translatedText": "Jadi untuk meringkas di mana kita berada, Anda telah membagi luas lingkaran menjadi semua cincin ini, dan Anda memperkirakan luas setiap cincin adalah 2 pi dikali jari-jarinya dikali dr, yang mana nilai spesifiknya untuk itu jari-jari dalam berkisar dari 0 untuk cincin terkecil, hingga sedikit di bawah 3 untuk cincin terbesar, diberi jarak sesuai ketebalan apa pun yang Anda pilih untuk dr, kira-kira 0.1.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen.", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen.", "translatedText": "Maksud saya, 2 kali pi dikali 3 adalah sekitar 19, jadi mari kita buat sumbu y yang skalanya sedikit berbeda sehingga kita bisa memuat semua persegi panjang ini di layar.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/essence-of-calculus/italian/sentence_translations.json b/2017/essence-of-calculus/italian/sentence_translations.json index 1f3d44044..eafe19698 100644 --- a/2017/essence-of-calculus/italian/sentence_translations.json +++ b/2017/essence-of-calculus/italian/sentence_translations.json @@ -1019,4 +1019,4 @@ "start": 997.02, "end": 1003.42 } -] +] \ No newline at end of file diff --git a/2017/essence-of-calculus/japanese/sentence_translations.json b/2017/essence-of-calculus/japanese/sentence_translations.json index 9b5b0efe7..f616093fe 100644 --- a/2017/essence-of-calculus/japanese/sentence_translations.json +++ b/2017/essence-of-calculus/japanese/sentence_translations.json @@ -248,7 +248,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1. ", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1. ", "translatedText": "ここでの状況を要約すると、円の面積をこれらすべてのリングに分割し 、それぞれのリングの面積を 2 pi とその半径の掛け合わせ dr で近似します。ここで、特定の値はその内側の半径の範囲は、最 小リングの 0 から、最大リングの 3 未満までの範囲で、dr に選択した厚さ (0 など) によって間隔があけられます。1. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen. ", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen. ", "translatedText": "つまり、円周率の 2 倍と 3 の積は約 19 です。そこで、これらの長方形をすべ て画面に収めることができるように、少し異なるスケールの Y 軸を作成しましょう。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/essence-of-calculus/korean/sentence_translations.json b/2017/essence-of-calculus/korean/sentence_translations.json index 94e829af5..251d29137 100644 --- a/2017/essence-of-calculus/korean/sentence_translations.json +++ b/2017/essence-of-calculus/korean/sentence_translations.json @@ -1021,4 +1021,4 @@ "start": 997.02, "end": 1003.42 } -] +] \ No newline at end of file diff --git a/2017/essence-of-calculus/marathi/sentence_translations.json b/2017/essence-of-calculus/marathi/sentence_translations.json index a9b925ef5..406ad2bed 100644 --- a/2017/essence-of-calculus/marathi/sentence_translations.json +++ b/2017/essence-of-calculus/marathi/sentence_translations.json @@ -248,7 +248,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1. ", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1. ", "translatedText": "तर आपण कुठे आहोत याचा सारांश देण्यासाठी, आपण वर्तुळाचे क्षेत्रफळ या सर्व वलयांमध्ये मोडले आहे, आणि आपण त्या प्रत्येकाचे क्षेत्रफळ अंदाजे 2 pi पट त्याच्या त्रिज्या गुणा dr आहे, जेथे विशिष्ट मूल्य त्या अंतर्गत त्रिज्या सर्वात लहान रिंगसाठी 0 पासून, सर्वात मोठ्या रिंगसाठी फक्त 3 च्या खाली, dr साठी तुम्ही निवडलेल्या कोणत्याही जाडीने अंतर ठेवा, 0 सारखे काहीतरी. १. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen. ", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen. ", "translatedText": "म्हणजे, 2 गुणिले pi गुणिले 3 म्हणजे 19 च्या आसपास, तर चला y-अक्ष टाकू या ज्याचा आकार थोडा वेगळा आहे जेणेकरून आपण हे सर्व आयत स्क्रीनवर बसवू शकू. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/essence-of-calculus/persian/sentence_translations.json b/2017/essence-of-calculus/persian/sentence_translations.json index 586444fd1..0ef2f3131 100644 --- a/2017/essence-of-calculus/persian/sentence_translations.json +++ b/2017/essence-of-calculus/persian/sentence_translations.json @@ -248,7 +248,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1. ", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1. ", "translatedText": "بنابراین، برای جمع‌بندی اینکه کجا هستیم، شما مساحت دایره را به همه این حلقه‌ها تقسیم کرده‌اید، و مساحت هر یک از این حلقه‌ها را 2 پی برابر شعاع آن ضربدر dr، که مقدار خاص آن است، تقریب می‌زنید. برای آن شعاع داخلی از 0 برای کوچکترین حلقه، تا کمی کمتر از 3 برای بزرگترین حلقه، با هر ضخامتی که برای dr انتخاب می کنید، فاصله دارد، چیزی شبیه 0.1. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen. ", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen. ", "translatedText": "منظورم این است که 2 ضربدر پی ضربدر 3 حدود 19 است، بنابراین اجازه دهید یک محور y را که کمی متفاوت مقیاس شده است بالا بیاوریم تا بتوانیم همه این مستطیل ها را روی صفحه قرار دهیم. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/essence-of-calculus/portuguese/sentence_translations.json b/2017/essence-of-calculus/portuguese/sentence_translations.json index 9f6b0d3c2..3c90114ea 100644 --- a/2017/essence-of-calculus/portuguese/sentence_translations.json +++ b/2017/essence-of-calculus/portuguese/sentence_translations.json @@ -1014,4 +1014,4 @@ "start": 997.02, "end": 1003.42 } -] +] \ No newline at end of file diff --git a/2017/essence-of-calculus/tamil/sentence_translations.json b/2017/essence-of-calculus/tamil/sentence_translations.json index 91099d87e..07f7523eb 100644 --- a/2017/essence-of-calculus/tamil/sentence_translations.json +++ b/2017/essence-of-calculus/tamil/sentence_translations.json @@ -248,7 +248,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1. ", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1. ", "translatedText": "நாம் இருக்கும் இடத்தைச் சுருக்கமாகச் சொல்ல வேண்டுமானால், வட்டத்தின் பரப்பளவை இந்த அனைத்து வளையங்களாகப் பிரித்துவிட்டீர்கள், மேலும் அவை ஒவ்வொன்றின் பரப்பளவையும் தோராயமாக மதிப்பிடுகிறீர்கள், அதன் ஆரம் மடங்கு dr ஐ விட 2 மடங்கு அதிகமாக இருக்கும். அந்த உள் ஆரம் சிறிய வளையத்திற்கு 0 முதல் பெரிய வளையத்திற்கு 3 வரை இருக்கும், dr க்கு நீங்கள் தேர்ந்தெடுக்கும் தடிமன், 0 போன்றது. 1. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen. ", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen. ", "translatedText": "அதாவது, 2 பெருக்கல் pi பெருக்கல் 3 என்பது 19 ஆகும், எனவே கொஞ்சம் வித்தியாசமாக அளவிடப்பட்ட ஒரு y-அச்சியை மேலே தூக்கி எறிவோம், இந்த செவ்வகங்கள் அனைத்தையும் திரையில் பொருத்த முடியும். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/essence-of-calculus/telugu/sentence_translations.json b/2017/essence-of-calculus/telugu/sentence_translations.json index 1b553ba2b..2490dde40 100644 --- a/2017/essence-of-calculus/telugu/sentence_translations.json +++ b/2017/essence-of-calculus/telugu/sentence_translations.json @@ -248,7 +248,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1. ", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1. ", "translatedText": "కాబట్టి మనం ఎక్కడ ఉన్నామో సంక్షిప్తీకరించడానికి, మీరు సర్కిల్ యొక్క వైశాల్యాన్ని ఈ అన్ని రింగ్‌లుగా విభజించారు మరియు మీరు వాటిలో ప్రతి ఒక్కటి యొక్క వైశాల్యాన్ని దాని వ్యాసార్థం సార్లు dr కంటే 2 pi రెట్లు అంచనా వేస్తున్నారు, ఇక్కడ నిర్దిష్ట విలువ ఆ లోపలి వ్యాసార్థం చిన్న రింగ్‌కు 0 నుండి పెద్ద రింగ్‌కు 3 కంటే తక్కువ వరకు ఉంటుంది, dr కోసం మీరు ఎంచుకున్న మందంతో 0 వంటిది ఉంటుంది. 1. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen. ", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen. ", "translatedText": "నా ఉద్దేశ్యం, 2 సార్లు pi సార్లు 3 అనేది దాదాపు 19, కాబట్టి కొంచెం విభిన్నంగా స్కేల్ చేయబడిన y-యాక్సిస్‌ని త్రోసివేద్దాం, తద్వారా మనం స్క్రీన్‌పై ఈ దీర్ఘచతురస్రాలన్నీ అమర్చవచ్చు. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/essence-of-calculus/thai/sentence_translations.json b/2017/essence-of-calculus/thai/sentence_translations.json index 336477472..96349cda5 100644 --- a/2017/essence-of-calculus/thai/sentence_translations.json +++ b/2017/essence-of-calculus/thai/sentence_translations.json @@ -248,7 +248,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1. ", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1. ", "translatedText": "ที่เล็กลงเรื่อยๆ นั่นคือถ้าเราแบ่งวงกลมออกเป็นวงแหวนที่บางลงและบางลง เพื่อสรุปว่าเราอยู่ที่ไหน คุณแบ่งพื้นที่ของวงกลมออกเป็นวงแหวนเหล่านี้ทั้งหมด แล้วคุณก็ประมาณพื้นที่ของแต่ละวงแหวนเป็น 2 ไพ คูณรัศมีคูณ dr โดยที่ค่าเฉพาะ สำหรับรัศมีภายในนั้นมีตั้งแต่ 0 สำหรับวงแหวนที่เล็กที่สุด ไปจนถึงต่ำกว่า 3 สำหรับวงแหวนที่ใหญ่ที่สุด โดยเว้นระยะห่างตามความหนาใดก็ตามที่คุณเลือกสำหรับ dr ประมาณ 0 1. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen. ", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/essence-of-calculus/ukrainian/sentence_translations.json b/2017/essence-of-calculus/ukrainian/sentence_translations.json index e21b7373d..eda57e206 100644 --- a/2017/essence-of-calculus/ukrainian/sentence_translations.json +++ b/2017/essence-of-calculus/ukrainian/sentence_translations.json @@ -248,7 +248,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1. ", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1. ", "translatedText": "Отже, щоб підбити підсумок, ви розбили площу кола на всі ці кільця, і ви наблизили площу кожного з них як 2 пі, помножене на його радіус, помножене на dr, де конкретне значення оскільки цей внутрішній радіус коливається від 0 для найменшого кільця до трохи менше 3 для найбільшого кільця, розділених на будь-яку товщину, яку ви виберете для dr, приблизно 0.1. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen. ", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen. ", "translatedText": "Я маю на увазі, що 2, помножене на пі, помножене на 3, дорівнює приблизно 19, тож давайте просто відобразимо вісь ординат, яка має трохи інший масштаб, щоб ми могли розмістити всі ці прямокутники на екрані. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/essence-of-calculus/urdu/sentence_translations.json b/2017/essence-of-calculus/urdu/sentence_translations.json index 3c18f817c..bdb413d74 100644 --- a/2017/essence-of-calculus/urdu/sentence_translations.json +++ b/2017/essence-of-calculus/urdu/sentence_translations.json @@ -248,7 +248,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1. ", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1. ", "translatedText": "تو صرف خلاصہ کرنے کے لئے کہ ہم کہاں ہیں، آپ نے دائرے کے رقبے کو ان تمام حلقوں میں توڑ دیا ہے، اور آپ ان میں سے ہر ایک کے رقبے کا تخمینہ 2 pi گنا اس کے رداس گنا dr کے طور پر کر رہے ہیں، جہاں مخصوص قدر اس اندرونی رداس کے لیے سب سے چھوٹی انگوٹھی کے لیے 0 سے لے کر، سب سے بڑی انگوٹھی کے لیے صرف 3 سے کم تک، آپ dr کے لیے جو بھی موٹائی منتخب کرتے ہیں، اس سے فاصلہ 0 جیسا ہے۔1۔", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen. ", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen. ", "translatedText": "میرا مطلب ہے، 2 گنا pi گنا 3 19 کے لگ بھگ ہے، تو آئیے صرف ایک y-axis کو پھینکتے ہیں جو تھوڑا سا مختلف انداز میں بنایا گیا ہے تاکہ ہم ان تمام مستطیلوں کو اسکرین پر فٹ کر سکیں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/essence-of-calculus/vietnamese/sentence_translations.json b/2017/essence-of-calculus/vietnamese/sentence_translations.json index 663b83cd0..2c2d55978 100644 --- a/2017/essence-of-calculus/vietnamese/sentence_translations.json +++ b/2017/essence-of-calculus/vietnamese/sentence_translations.json @@ -248,7 +248,7 @@ "end": 246.4 }, { - "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring, up to just under 3 for the biggest ring, spaced out by whatever thickness you choose for dr, something like 0.1. ", + "input": "So just to sum up where we are, you've broken up the area of the circle into all of these rings, and you're approximating the area of each one of those as 2 pi times its radius times dr, where the specific value for that inner radius ranges from 0 for the smallest ring up to just under 3 for the biggest ring, spaced out by whatever the thickness is that you choose for dr, something like 0.1. ", "translatedText": "Vì vậy, để tóm tắt vị trí của chúng ta, bạn đã chia diện tích hình tròn thành tất cả các vòng này, và bạn đang tính gần đúng diện tích của mỗi vòng đó là 2 pi nhân bán kính của nó nhân dr, trong đó giá trị cụ thể đối với bán kính bên trong đó dao động từ 0 đối với vòng nhỏ nhất, lên đến chỉ dưới 3 đối với vòng lớn nhất, cách nhau bất kỳ độ dày nào bạn chọn cho dr, chẳng hạn như 0.1. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 312.32 }, { - "input": "I mean, 2 times pi times 3 is around 19, so let's just throw up a y-axis that's scaled a little differently so we can fit all of these rectangles on the screen. ", + "input": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's scaled a little differently so that we can actually fit all of these rectangles on the screen. ", "translatedText": "Ý tôi là, 2 nhân pi nhân 3 là khoảng 19, vậy hãy dựng trục y có tỷ lệ khác đi một chút để chúng ta có thể vừa tất cả các hình chữ nhật này trên màn hình. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/arabic/sentence_translations.json b/2017/eulers-formula-via-group-theory/arabic/sentence_translations.json index cbe23d442..942657bd0 100644 --- a/2017/eulers-formula-via-group-theory/arabic/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/arabic/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group. ", + "input": "And there are actually two separate ways to think about numbers as a group. ", "translatedText": "هناك طريقتان منفصلتان للتفكير في الأرقام كمجموعة. ", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing. ", + "input": "The number zero itself, well, that's associated with the action of just doing nothing. ", "translatedText": "الرقم 0 نفسه مرتبط بفعل عدم القيام بأي شيء. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with. ", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with. ", "translatedText": "وبشكل عام، فإن تطبيق أحد هذه الإجراءات متبوعًا بآخر يتوافق مع مضاعفة الأرقام المرتبطة بها. ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group. ", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group. ", "translatedText": "ستكون هذه المقارنة لكيفية تقسيم الإجراءات في كل مجموعة أمرًا مهمًا، لذا تذكر في كل إجراء، يمكنك تقسيم أي إجراء كإجراء عددي حقيقي بحت، متبوعًا بشيء محدد للأعداد المركبة، سواء كان ذلك عبارة عن شرائح رأسية للجمع المجموعة، أو التناوب النقي للمجموعة المضاعفة. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication. ", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and ", "translatedText": "أول مقدمة لنا عن الأسس هي أن ننظر إليها من حيث الضرب المتكرر. ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication. ", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication. ", "translatedText": "لكن التعبيرات مثل 2 إلى ½، أو 2 إلى -1، وأقل بكثير من 2 إلى i لا معنى لها حقًا عندما تفكر في الأسس على أنها ضرب متكرر. ", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time? ", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time? ", "translatedText": "ماذا يعني ضرب 2 في نفسه نصف مرة، أو -1 من المرة؟ ", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined. ", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined. ", "translatedText": "الإجابة الكاملة تكمن في حساب التفاضل والتكامل، وهو مسقط رأس e، وحيث يتم تعريفه. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/bengali/sentence_translations.json b/2017/eulers-formula-via-group-theory/bengali/sentence_translations.json index 2318eaa83..a8cd65517 100644 --- a/2017/eulers-formula-via-group-theory/bengali/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/bengali/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group. ", + "input": "And there are actually two separate ways to think about numbers as a group. ", "translatedText": "একটি গ্রুপ হিসাবে সংখ্যা সম্পর্কে চিন্তা করার দুটি পৃথক উপায় আছে।", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing. ", + "input": "The number zero itself, well, that's associated with the action of just doing nothing. ", "translatedText": "0 নম্বরটি নিজেই কিছুই না করার ক্রিয়াটির সাথে যুক্ত।", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with. ", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with. ", "translatedText": "সাধারণভাবে, এই ক্রিয়াগুলির মধ্যে একটি প্রয়োগ করার পরে অন্যটি প্রয়োগ করা সংখ্যাগুলিকে গুণিত করার সাথে সম্পর্কিত।", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group. ", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group. ", "translatedText": "প্রতিটি গোষ্ঠীর ক্রিয়াগুলি কীভাবে ভেঙে যায় তার তুলনাটি গুরুত্বপূর্ণ হতে চলেছে, তাই মনে রাখবেন প্রতিটিতে, আপনি যে কোনও ক্রিয়াকে কিছু বিশুদ্ধভাবে বাস্তব সংখ্যা ক্রিয়া হিসাবে ভেঙে ফেলতে পারেন, তারপরে জটিল সংখ্যাগুলির জন্য নির্দিষ্ট কিছু, এটি সংযোজনের জন্য উল্লম্ব স্লাইড হোক না কেন।গোষ্ঠী বা গুণক গোষ্ঠীর জন্য বিশুদ্ধ ঘূর্ণন।", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication. ", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and ", "translatedText": "সূচকগুলির সাথে আমাদের প্রথম ভূমিকা হল বারবার গুণের পরিপ্রেক্ষিতে তাদের চিন্তা করা।", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication. ", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication. ", "translatedText": "কিন্তু 2 থেকে ½, অথবা 2 থেকে -1, এবং i-এর থেকে অনেক কম 2-এর মতো রাশিগুলি যখন আপনি সূচকগুলিকে বারবার গুণন হিসাবে ভাবেন তখন সত্যিই কোন অর্থ হয় না।", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time? ", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time? ", "translatedText": "একটি সময়ের অর্ধেক বা –1কে নিজের দ্বারা গুন করার অর্থ কী? ", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined. ", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined. ", "translatedText": "সম্পূর্ণ উত্তরটি ক্যালকুলাসে থাকে, এটি ই এর জন্মস্থান এবং যেখানে এটি এমনকি সংজ্ঞায়িত করা হয়েছে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/chinese/sentence_translations.json b/2017/eulers-formula-via-group-theory/chinese/sentence_translations.json index 1378e90ff..172810d37 100644 --- a/2017/eulers-formula-via-group-theory/chinese/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/chinese/sentence_translations.json @@ -368,7 +368,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "有两种不同的方法可以将数字视为一个整体。", "model": "google_nmt", "n_reviews": 0, @@ -401,7 +401,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "想一想可以沿着数轴向左或向右滑动的所有方法。", "model": "google_nmt", "n_reviews": 0, @@ -451,7 +451,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "数字 0 本身与无所事事的行为相关。", "model": "google_nmt", "n_reviews": 0, @@ -694,7 +694,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "一般来说，应用这些操作之一，然后再应用 另一个操作相当于将它们关联的数字相乘。", "model": "google_nmt", "n_reviews": 0, @@ -820,7 +820,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "比较每组中的动作如何分解非常重要，因此请记 住，在每一组中，您都可以将任何动作分解为 一些纯实数动作，然后是特定于复数的动作，无 论是加法的垂直滑动群，或乘法群的纯旋转。", "model": "google_nmt", "n_reviews": 0, @@ -879,7 +879,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "我们对指数的第一个介绍是从重复乘法的角度来思考它们。", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "但是，当您将指数视为重复乘法时，诸如 2 到 ½、或 2 到 –1 以及更不用说 2 到 i 之类的表达式实际上没有意义。", "model": "google_nmt", "from_community_srt": "但是对于2^(1/2)或2^(-1) 更不必说2^i 当你把指数看作是多次相乘时 这些表达式并没有意义 2和自己相乘\"半次\"或者\"-1次\"是什么意思 所以我们做一件在数学上很常见的事", @@ -921,7 +921,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "2 乘以自身一半时间或 –1 时间意味着什么？", "model": "google_nmt", "n_reviews": 0, @@ -1014,7 +1014,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "想想这对于将复平面中的加法群与复 平面中的乘法群相关联意味着什么。", "model": "google_nmt", "n_reviews": 0, @@ -1090,7 +1090,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "两个单位的垂直滑动将映射为两个弧度的旋转。", "model": "google_nmt", "from_community_srt": "向上滑动1个单位的数i 恰恰会映射为1弧度的旋转 也就是沿单位圆走过的距离正好是1个单位 所以向上滑动2个单位会映射为2弧度的旋转 向上滑动3个单位对应3弧度的旋转 而向上滑动π个单位 对应于输入值πi", @@ -1132,7 +1132,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "完整的答案在于微积分，这是 e 的诞生地，甚至是 e 的定义地。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/english/captions.srt b/2017/eulers-formula-via-group-theory/english/captions.srt index 779ab47a8..ca2e91e0d 100644 --- a/2017/eulers-formula-via-group-theory/english/captions.srt +++ b/2017/eulers-formula-via-group-theory/english/captions.srt @@ -55,20 +55,20 @@ So here, two years later, lets you and me go through an introduction to the basi of group theory, building up to how Euler's formula comes to life under this light. 15 -00:00:59,660 --> 00:01:02,796 +00:00:59,660 --> 00:01:02,604 If all you want is a quick explanation of Euler's formula, 16 -00:01:02,796 --> 00:01:05,348 +00:01:02,604 --> 00:01:04,999 and if you're comfortable with vector calculus, 17 -00:01:05,348 --> 00:01:08,751 -I'll put up a particularly short explanation on the screen that +00:01:04,999 --> 00:01:08,593 +I'll go ahead and put up a particularly short explanation on the screen 18 -00:01:08,751 --> 00:01:10,240 -you can pause and ponder on. +00:01:08,593 --> 00:01:10,240 +that you can pause and ponder on. 19 00:01:10,640 --> 00:01:14,040 diff --git a/2017/eulers-formula-via-group-theory/english/sentence_timings.json b/2017/eulers-formula-via-group-theory/english/sentence_timings.json index a20f1a753..02f136c07 100644 --- a/2017/eulers-formula-via-group-theory/english/sentence_timings.json +++ b/2017/eulers-formula-via-group-theory/english/sentence_timings.json @@ -35,7 +35,7 @@ 58.88 ], [ - "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll put up a particularly short explanation on the screen that you can pause and ponder on.", + "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll go ahead and put up a particularly short explanation on the screen that you can pause and ponder on.", 59.66, 70.24 ], diff --git a/2017/eulers-formula-via-group-theory/english/transcript.txt b/2017/eulers-formula-via-group-theory/english/transcript.txt index 1a72a0a09..399e1d450 100644 --- a/2017/eulers-formula-via-group-theory/english/transcript.txt +++ b/2017/eulers-formula-via-group-theory/english/transcript.txt @@ -5,7 +5,7 @@ You see, the idea underlying that video was to take certain concepts from a fiel And two years ago, I thought it might be fun to use those ideas without referencing group theory itself, or any of the technical terms within it. But I've come to see that you all actually quite like getting into the math itself, even if it takes some time. So here, two years later, lets you and me go through an introduction to the basics of group theory, building up to how Euler's formula comes to life under this light. -If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll put up a particularly short explanation on the screen that you can pause and ponder on. +If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll go ahead and put up a particularly short explanation on the screen that you can pause and ponder on. If it doesn't make sense, don't worry about it, it's not needed for where we're going. The reason I want to put out this group theory view, though, is not because I think it's a better explanation. Heck, it's not even a complete proof, it's just an intuition really. diff --git a/2017/eulers-formula-via-group-theory/french/sentence_translations.json b/2017/eulers-formula-via-group-theory/french/sentence_translations.json index 80d48a0fd..caef58fa4 100644 --- a/2017/eulers-formula-via-group-theory/french/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/french/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "Il existe deux manières distinctes de considérer les nombres en tant que groupe.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "Pensez à toutes les façons dont vous pouvez faire glisser une droite numérique vers la gauche ou la droite le long d’elle-même.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "Le chiffre 0 lui-même est associé à l’action de ne rien faire.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "En général, appliquer une de ces actions suivie d'une autre correspond à multiplier les nombres auxquels elles sont associées.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "Cette comparaison de la façon dont les actions de chaque groupe se décomposent sera importante, alors rappelez-vous que dans chacun d'eux, vous pouvez décomposer n'importe quelle action en une action numérique purement réelle, suivie de quelque chose de spécifique aux nombres complexes, qu'il s'agisse de diapositives verticales pour l'additif. groupe, ou rotations pures pour le groupe multiplicatif.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "Notre première introduction aux exposants consiste à les considérer en termes de multiplication répétée.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "Mais des expressions comme 2 pour ½, ou 2 pour –1, et encore moins 2 pour i n'ont pas vraiment de sens quand on considère les exposants comme des multiplications répétées.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "Que signifie multiplier 2 par lui-même la moitié d'une fois, ou –1 d'une fois?", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "Pensez à ce que tout cela signifie pour associer le groupe additif dans le plan complexe au groupe multiplicatif dans le plan complexe.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "Un glissement vertical de deux unités correspondrait à une rotation de deux radians.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "La réponse complète réside dans le calcul, c'est le lieu de naissance de e, et où il est même défini.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/german/sentence_translations.json b/2017/eulers-formula-via-group-theory/german/sentence_translations.json index 62522767d..e8101cce8 100644 --- a/2017/eulers-formula-via-group-theory/german/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/german/sentence_translations.json @@ -63,7 +63,7 @@ "end": 58.88 }, { - "input": "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll put up a particularly short explanation on the screen that you can pause and ponder on.", + "input": "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll go ahead and put up a particularly short explanation on the screen that you can pause and ponder on.", "translatedText": "Wenn du nur eine kurze Erklärung der Eulerschen Formel brauchst und dich mit der Vektorrechnung auskennst, zeige ich dir eine besonders kurze Erklärung auf dem Bildschirm an, über die du in Ruhe nachdenken kannst.", "model": "DeepL", "from_community_srt": "Falls du nur eine schnelle Erklärung der Euelerformel benötigst und du mit Vektor-Analysis vertraut bist, werde ich hier eine besonders kurze Erklärung einblenden, so dass du pausieren und darüber nachgrübeln kannst.", diff --git a/2017/eulers-formula-via-group-theory/hebrew/sentence_translations.json b/2017/eulers-formula-via-group-theory/hebrew/sentence_translations.json index cf080684d..fd8426283 100644 --- a/2017/eulers-formula-via-group-theory/hebrew/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/hebrew/sentence_translations.json @@ -308,7 +308,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "ישנן שתי דרכים נפרדות לחשוב על מספרים כקבוצה.", "n_reviews": 0, "start": 365.1, @@ -336,7 +336,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "תחשוב על כל הדרכים שבהן אתה יכול להחליק קו מספר ימינה או שמאלה לאורך עצמו.", "n_reviews": 0, "start": 384.74, @@ -378,7 +378,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "המספר 0 עצמו קשור לפעולה של לא לעשות כלום.", "n_reviews": 0, "start": 435.22, @@ -581,7 +581,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "באופן כללי, יישום של אחת מהפעולות הללו ואחריה אחר תואם להכפלת המספרים שאליהם הם משויכים.", "n_reviews": 0, "start": 695.76, @@ -686,7 +686,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "ההשוואה הזו של האופן שבו פעולות בכל קבוצה מתפרקות תהיה חשובה, אז זכור שבכל פעולה, אתה יכול לפרק כל פעולה כפעולה של מספרים ממשיים בלבד, ואחריה משהו ספציפי למספרים מרוכבים, בין אם זה שקופיות אנכיות עבור התוספת קבוצה, או סיבובים טהורים עבור קבוצת הכפל.", "n_reviews": 0, "start": 852.6, @@ -735,7 +735,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "ההקדמה הראשונה שלנו למעריכים היא לחשוב עליהם במונחים של כפל חוזר.", "n_reviews": 0, "start": 926.74, @@ -763,14 +763,14 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "אבל ביטויים כמו 2 ל-½, או 2 ל-1, והרבה פחות 2 ל-i לא ממש הגיוניים כשחושבים על מעריכי כפל חוזר.", "n_reviews": 0, "start": 963.2, "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "מה זה אומר להכפיל 2 בעצמו חצי זמן, או -1 של זמן?", "n_reviews": 0, "start": 973.8, @@ -847,7 +847,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "חשבו מה כל זה אומר על שיוך הקבוצה הנוספת במישור המורכב עם קבוצת הכפל במישור המורכב.", "n_reviews": 0, "start": 1103.62, @@ -910,7 +910,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "החלקה אנכית של שתי יחידות תמפה לסיבוב של שני רדיאנים.", "n_reviews": 0, "start": 1227.78, @@ -945,7 +945,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "התשובה המלאה נמצאת בחשבון, זה מקום הולדתו של e, והיכן הוא אפילו מוגדר.", "n_reviews": 0, "start": 1260.14, diff --git a/2017/eulers-formula-via-group-theory/hindi/sentence_translations.json b/2017/eulers-formula-via-group-theory/hindi/sentence_translations.json index 92d7ff36a..cf68e3c2f 100644 --- a/2017/eulers-formula-via-group-theory/hindi/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/hindi/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "एक समूह के रूप में संख्याओं के बारे में सोचने के दो अलग-अलग तरीके हैं।", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "उन सभी तरीकों के बारे में सोचें जिनसे आप किसी संख्या रेखा को उसके साथ बाएँ या दाएँ सरका सकते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "संख्या 0 स्वयं कुछ न करने की क्रिया से जुड़ी है।", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "सामान्य तौर पर, इनमें से एक क्रिया के बाद दूसरी क्रिया लागू करना उन संख्याओं को गुणा करने से मेल खाता है जिनसे वे संबद्ध हैं।", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "प्रत्येक समूह में क्रियाएं कैसे विभाजित होती हैं, इसकी तुलना महत्वपूर्ण होगी, इसलिए याद रखें कि प्रत्येक में, आप किसी भी क्रिया को कुछ विशुद्ध रूप से वास्तविक संख्या क्रिया के रूप में तोड़ सकते हैं, उसके बाद जटिल संख्याओं के लिए कुछ विशिष्ट कर सकते हैं, चाहे वह योगात्मक के लिए ऊर्ध्वाधर स्लाइड हो। समूह, या गुणक समूह के लिए शुद्ध घुमाव।", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "घातांकों से हमारा पहला परिचय बार-बार गुणन के संदर्भ में उनके बारे में सोचना है।", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "लेकिन 2 से ½, या 2 से -1, और इससे भी कम 2 से i जैसे भाव वास्तव में तब समझ में नहीं आते जब आप घातांक को बार-बार गुणा के रूप में सोचते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "2 को एक समय के आधे से या एक समय को -1 से गुणा करने का क्या मतलब है?", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "इस बारे में सोचें कि जटिल तल में योगात्मक समूह को जटिल तल में गुणक समूह के साथ जोड़ने के लिए इसका क्या मतलब है।", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "दो इकाइयों की एक ऊर्ध्वाधर स्लाइड दो रेडियन के घूर्णन को मैप करेगी।", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "पूरा उत्तर कैलकुलस में निहित है, यही ई का जन्मस्थान है, और जहां इसे परिभाषित भी किया गया है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/hungarian/sentence_translations.json b/2017/eulers-formula-via-group-theory/hungarian/sentence_translations.json index 00b89be5c..cf9f6a319 100644 --- a/2017/eulers-formula-via-group-theory/hungarian/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/hungarian/sentence_translations.json @@ -56,7 +56,7 @@ "end": 58.88 }, { - "input": "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll put up a particularly short explanation on the screen that you can pause and ponder on.", + "input": "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll go ahead and put up a particularly short explanation on the screen that you can pause and ponder on.", "translatedText": "Ha csak az Euler-képlet gyors magyarázatára van szükséged, és ha a vektorszámítással is tisztában vagy, akkor egy különösen rövid magyarázatot teszek fel a képernyőre, amelyen megállhatsz és elgondolkodhatsz.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/indonesian/sentence_translations.json b/2017/eulers-formula-via-group-theory/indonesian/sentence_translations.json index 309ad06f1..98f1fcbd5 100644 --- a/2017/eulers-formula-via-group-theory/indonesian/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/indonesian/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "Ada dua cara berbeda untuk memikirkan angka sebagai sebuah kelompok.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "Pikirkan semua cara Anda dapat menggeser garis bilangan ke kiri atau ke kanan sepanjang garis itu sendiri.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "Angka 0 sendiri dikaitkan dengan tindakan tidak berbuat apa-apa.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "Secara umum, menerapkan salah satu tindakan berikut diikuti tindakan lainnya sama dengan mengalikan angka-angka yang terkait dengannya.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "Perbandingan mengenai bagaimana tindakan dalam masing-masing kelompok dipecah akan menjadi hal yang penting, jadi ingat. Dalam masing-masing kelompok, Anda dapat memecah tindakan apa pun sebagai tindakan bilangan real murni, diikuti oleh sesuatu yang spesifik untuk bilangan kompleks, baik itu slide vertikal untuk penjumlahannya. grup, atau rotasi murni untuk grup perkalian.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "Pengenalan pertama kita tentang eksponen adalah dengan menganggapnya sebagai perkalian berulang.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "Namun ekspresi seperti 2 hingga ½, atau 2 hingga –1, dan apalagi 2 hingga i tidak masuk akal jika Anda menganggap eksponen sebagai perkalian berulang.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "Apa yang dimaksud dengan mengalikan 2 dengan dirinya sendiri pada separuh waktu, atau –1 kali?", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "Pikirkan tentang apa artinya mengasosiasikan grup aditif di bidang kompleks dengan grup perkalian di bidang kompleks.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "Perosotan vertikal sebanyak dua unit akan dipetakan ke rotasi dua radian.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "Jawaban lengkapnya terletak pada kalkulus, itulah tempat lahirnya e, dan bahkan didefinisikan.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/italian/sentence_translations.json b/2017/eulers-formula-via-group-theory/italian/sentence_translations.json index b0f6035af..1213ebc7e 100644 --- a/2017/eulers-formula-via-group-theory/italian/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/italian/sentence_translations.json @@ -308,7 +308,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "Esistono due modi separati di pensare ai numeri come gruppo.", "n_reviews": 0, "start": 365.1, @@ -336,7 +336,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "Pensa a tutti i modi in cui puoi far scorrere una linea numerica a sinistra o a destra lungo se stessa.", "n_reviews": 0, "start": 384.74, @@ -378,7 +378,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "Il numero 0 stesso è associato all’azione di non fare nulla.", "n_reviews": 0, "start": 435.22, @@ -581,7 +581,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "In generale, applicare una di queste azioni seguita da un'altra corrisponde a moltiplicare i numeri a cui sono associate.", "n_reviews": 0, "start": 695.76, @@ -686,7 +686,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "Il confronto su come le azioni in ciascun gruppo si scompongono sarà importante, quindi ricorda. In ognuno di essi puoi scomporre qualsiasi azione come un'azione puramente numerica reale, seguita da qualcosa di specifico per i numeri complessi, sia che si tratti di diapositive verticali per l'additivo gruppo, o rotazioni pure per il gruppo moltiplicativo.", "n_reviews": 0, "start": 852.6, @@ -735,7 +735,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "La nostra prima introduzione agli esponenti è pensarli in termini di moltiplicazioni ripetute.", "n_reviews": 0, "start": 926.74, @@ -763,14 +763,14 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "Ma espressioni come 2 alla ½, o 2 alla –1, e ancor meno 2 alla i, non hanno davvero senso se si pensa agli esponenti come a una moltiplicazione ripetuta.", "n_reviews": 0, "start": 963.2, "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "Cosa significa moltiplicare 2 per se stesso metà di un tempo, oppure –1 di un tempo?", "n_reviews": 0, "start": 973.8, @@ -847,7 +847,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "Pensate a cosa significa tutto ciò per associare il gruppo additivo nel piano complesso al gruppo moltiplicativo nel piano complesso.", "n_reviews": 0, "start": 1103.62, @@ -910,7 +910,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "Uno scorrimento verticale di due unità corrisponderebbe ad una rotazione di due radianti.", "n_reviews": 0, "start": 1227.78, @@ -945,7 +945,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "La risposta completa risiede nel calcolo infinitesimale, che è il luogo di nascita di e, e dove viene anche definito.", "n_reviews": 0, "start": 1260.14, @@ -1070,4 +1070,4 @@ "start": 1431.16, "end": 1448.28 } -] +] \ No newline at end of file diff --git a/2017/eulers-formula-via-group-theory/japanese/sentence_translations.json b/2017/eulers-formula-via-group-theory/japanese/sentence_translations.json index 710f3f8e8..573c2807d 100644 --- a/2017/eulers-formula-via-group-theory/japanese/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/japanese/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "数値をグループとして考えるには、2 つの異なる方法があります。", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "数直線をそれ自体に沿って左または右にスライドさせるすべての方法を考えてください。", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "数字の0自体は、何もしないという行為に関連付けられています。", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "一般に、これらのアクションの 1 つを適用してから別のアクションを適用 することは、それらに関連付けられている数値を乗算することに相当します。", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "各グループのアクションがどのように分解されるかを比較することが重要になるた め、それぞれのグループで、任意のアクションを純粋な実数のアクションとして分 解し、その後に加算器の垂直スライドなど、複素数に固有の何かを続けることがで きることを覚えておいてください。 グループ、または乗法グループの純粋な回転。", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "指数について最初に紹介するのは、繰り返しの乗算の観点から指数を考えることです。", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "しかし、指数を繰り返しの乗算と考える場合、2 を 1/2 にする、または 2 を –1 にする、ましてや 2 を i にするなどの式は、実際には意味がありません。", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "2 を半分だけ掛ける、あるいは –1 を掛けるというのは何を意味するのでしょうか?", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "複素平面内の加法群を複素平面内の乗法群に関連付け る場合に、これが何を意味するかを考えてください。", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "2 単位の垂直スライドは 2 ラジアンの回転にマッピングされます。", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "完全な答えは微積分にあり、それが e の発 祥の地であり、定義された場所でもあります。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/korean/sentence_translations.json b/2017/eulers-formula-via-group-theory/korean/sentence_translations.json index f642e0302..7dc0e719c 100644 --- a/2017/eulers-formula-via-group-theory/korean/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/korean/sentence_translations.json @@ -62,7 +62,7 @@ "end": 58.88 }, { - "input": "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll put up a particularly short explanation on the screen that you can pause and ponder on.", + "input": "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll go ahead and put up a particularly short explanation on the screen that you can pause and ponder on.", "translatedText": "오일러 공식에 대한 간단한 설명만 원하시는 분, 벡터 미적분에 익숙하신 분을 위해 잠시 멈추고 숙고하실 수 있도록 특별히 짧은 설명을 화면에 띄워드리겠습니다.", "model": "DeepL", "from_community_srt": "벡터 미적분에 익숙하다면 화면에 특별히 짧은 설명을 올려놓겠습니다. 당신이 일시정지하고 생각해 볼 수 있게요 그 설명이 만약 이해가 잘 안 된다면,", diff --git a/2017/eulers-formula-via-group-theory/marathi/sentence_translations.json b/2017/eulers-formula-via-group-theory/marathi/sentence_translations.json index 16ac92892..11edbf714 100644 --- a/2017/eulers-formula-via-group-theory/marathi/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/marathi/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "संख्यांचा समूह म्हणून विचार करण्याचे दोन वेगळे मार्ग आहेत.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "आपण संख्या रेषा डावीकडे किंवा उजवीकडे स्लाइड करू शकता अशा सर्व मार्गांचा विचार करा.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "संख्या 0 स्वतः काहीही न करण्याच्या क्रियेशी संबंधित आहे.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "सर्वसाधारणपणे, यापैकी एक क्रिया लागू करणे आणि त्यानंतर दुसरी क्रिया करणे हे त्यांच्याशी संबंधित असलेल्या संख्यांच्या गुणाकाराशी संबंधित आहे.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "प्रत्येक गटातील क्रिया कशा मोडतात याची तुलना महत्त्वाची असणार आहे, म्हणून लक्षात ठेवा की प्रत्येकामध्ये, तुम्ही कोणतीही क्रिया पूर्णपणे वास्तविक संख्या क्रिया म्हणून खंडित करू शकता, त्यानंतर जटिल संख्यांसाठी विशिष्ट काहीतरी असू शकते, मग ती अॅडिटीव्हसाठी अनुलंब स्लाइड्स असोत. समूह, किंवा गुणाकार गटासाठी शुद्ध आवर्तन.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "घातांकांची आमची पहिली ओळख म्हणजे त्यांचा वारंवार गुणाकार करण्याच्या दृष्टीने विचार करणे.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "परंतु 2 ते ½, किंवा 2 ते -1, आणि i मधील 2 सारख्या अभिव्यक्तींचा अर्थ नाही जेव्हा तुम्ही घातांकांचा पुनरावृत्ती गुणाकार म्हणून विचार करता.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "वेळेच्या अर्ध्या किंवा -1 चा स्वतः 2 गुणाकार करणे म्हणजे काय?", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "कॉम्प्लेक्स प्लेनमधील अॅडिटीव्ह ग्रुपला कॉम्प्लेक्स प्लेनमधील गुणाकार गटाशी जोडण्याचा या सर्वांचा अर्थ काय आहे याचा विचार करा.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "दोन युनिट्सची उभी स्लाइड दोन रेडियनच्या रोटेशनवर मॅप करेल.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "संपूर्ण उत्तर कॅल्क्युलसमध्ये आहे, ते e चे जन्मस्थान आहे आणि जिथे ते अगदी परिभाषित केले आहे.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/persian/sentence_translations.json b/2017/eulers-formula-via-group-theory/persian/sentence_translations.json index 93eacd43d..9722b917f 100644 --- a/2017/eulers-formula-via-group-theory/persian/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/persian/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group. ", + "input": "And there are actually two separate ways to think about numbers as a group. ", "translatedText": "دو راه جداگانه برای فکر کردن به اعداد به صورت گروهی وجود دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing. ", + "input": "The number zero itself, well, that's associated with the action of just doing nothing. ", "translatedText": "خود عدد 0 با عمل انجام هیچ کاری مرتبط است. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with. ", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with. ", "translatedText": "به طور کلی، اعمال یکی از این اقدامات و به دنبال آن یکی دیگر با ضرب اعداد مربوط به آنها مطابقت دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group. ", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group. ", "translatedText": "این مقایسه نحوه تجزیه اقدامات در هر گروه بسیار مهم خواهد بود، بنابراین به یاد داشته باشید که در هر یک، شما می توانید هر عمل را به عنوان یک عمل اعداد کاملا واقعی، و به دنبال آن چیزی خاص برای اعداد مختلط، چه اسلایدهای عمودی برای افزودنی، تجزیه کنید. گروه، یا چرخش های خالص برای گروه ضربی. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication. ", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and ", "translatedText": "اولین مقدمه ما برای توان این است که آنها را بر حسب ضرب مکرر در نظر بگیریم. ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication. ", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication. ", "translatedText": "و وقتی چیزها را گسترش می دهید، این به اندازه کافی معقول به نظر می رسد، درست است؟ اما عباراتی مانند 2 به ½، یا 2 به 1-، و بسیار کمتر از 2 به i واقعاً معنی ندارند وقتی که شما توان ها را به عنوان ضرب مکرر در نظر بگیرید. ", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time? ", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined. ", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined. ", "translatedText": "حالا ممکن است بپرسید چرا e؟ چرا پایگاه دیگری نیست؟ پاسخ کامل در حساب دیفرانسیل و انتگرال قرار دارد، این زادگاه e است و حتی در کجا تعریف شده است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/polish/sentence_translations.json b/2017/eulers-formula-via-group-theory/polish/sentence_translations.json index 49ff1ef6c..26c290f0b 100644 --- a/2017/eulers-formula-via-group-theory/polish/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/polish/sentence_translations.json @@ -56,7 +56,7 @@ "end": 58.88 }, { - "input": "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll put up a particularly short explanation on the screen that you can pause and ponder on.", + "input": "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll go ahead and put up a particularly short explanation on the screen that you can pause and ponder on.", "translatedText": "", "from_community_srt": "Jeśli chcesz, abym tylko szybko wytłumaczył ci wzór Eulera i jeśli znasz analizę wektorową, to zacznę teraz pokazywać na ekranie wyjątkowo krótkie wyjaśnienie. Możesz zatrzymać film i je przemyśleć.", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/portuguese/sentence_translations.json b/2017/eulers-formula-via-group-theory/portuguese/sentence_translations.json index 079b71a4f..6a9c3a1bc 100644 --- a/2017/eulers-formula-via-group-theory/portuguese/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/portuguese/sentence_translations.json @@ -63,7 +63,7 @@ "end": 58.88 }, { - "input": "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll put up a particularly short explanation on the screen that you can pause and ponder on.", + "input": "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll go ahead and put up a particularly short explanation on the screen that you can pause and ponder on.", "translatedText": "Se tudo o que você deseja é uma explicação rápida da fórmula de Euler, e se você se sente confortável com o cálculo vetorial, colocarei uma explicação particularmente curta na tela, na qual você poderá fazer uma pausa e refletir.", "model": "google_nmt", "from_community_srt": "Se tudo o que você quer é uma explicação rápida da Fórmula de Euler e você sabe Cálculo Vetorial, eu colocarei uma explicação curta na tela, que você pode pausar e ponderar, se não fizer sentido,", diff --git a/2017/eulers-formula-via-group-theory/russian/sentence_translations.json b/2017/eulers-formula-via-group-theory/russian/sentence_translations.json index cc4cf524a..72b782bf0 100644 --- a/2017/eulers-formula-via-group-theory/russian/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/russian/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "Есть два разных способа рассматривать числа как группу.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "Подумайте обо всех способах скольжения числовой прямой влево или вправо.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "Само число 0 связано с действием ничегонеделания.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "В общем, применение одного из этих действий, за которым следует другое, соответствует умножению чисел, с которыми они связаны.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "Это сравнение того, как разбиваются действия в каждой группе, будет важным, поэтому помните, что в каждой из них вы можете разбить любое действие на какое-то действие с чисто действительным числом, за которым следует что-то специфическое для комплексных чисел, будь то вертикальные слайды для добавки. группа или чистые вращения для мультипликативной группы.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "Наше первое знакомство с экспонентами состоит в том, чтобы представить их в терминах многократного умножения.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "Но такие выражения, как 2 до ½ или 2 до –1 и тем более 2 до i, на самом деле не имеют смысла, когда вы думаете о показателях степени как о повторяющемся умножении.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "Что значит умножить 2 само на себя половину времени или –1 времени?", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "Подумайте, что все это означает для связи аддитивной группы в комплексной плоскости с мультипликативной группой в комплексной плоскости.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "Вертикальное скольжение на две единицы соответствует вращению на два радиана.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "Полный ответ находится в исчислении, это место рождения e и место его определения.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/spanish/sentence_translations.json b/2017/eulers-formula-via-group-theory/spanish/sentence_translations.json index ddffc1264..ecbf69d65 100644 --- a/2017/eulers-formula-via-group-theory/spanish/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/spanish/sentence_translations.json @@ -63,7 +63,7 @@ "end": 58.88 }, { - "input": "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll put up a particularly short explanation on the screen that you can pause and ponder on.", + "input": "If all you want is a quick explanation of Euler's formula, and if you're comfortable with vector calculus, I'll go ahead and put up a particularly short explanation on the screen that you can pause and ponder on.", "translatedText": "Si lo único que quieres es una explicación rápida de la fórmula de Euler, y si te sientes cómodo con el cálculo vectorial, pondré en la pantalla una explicación especialmente breve en la que podrás detenerte y reflexionar.", "model": "DeepL", "from_community_srt": "si todo lo que quieres es una explicación rápida de la fórmula de Euler y te sientes cómodo con el cálculo vectorial más adelante pondré una explicación particularmente breve en la pantalla, para que pueda detener y reflexionar sobre ella si no le encuentra el sentido,", diff --git a/2017/eulers-formula-via-group-theory/tamil/sentence_translations.json b/2017/eulers-formula-via-group-theory/tamil/sentence_translations.json index 9cf907d30..cb6460739 100644 --- a/2017/eulers-formula-via-group-theory/tamil/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/tamil/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "ஒரு குழுவாக எண்களைப் பற்றி சிந்திக்க இரண்டு தனித்தனி வழிகள் உள்ளன.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "ஒரு எண் கோட்டை இடது அல்லது வலதுபுறமாக ஸ்லைடு செய்வதற்கான அனைத்து வழிகளையும் சிந்தியுங்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "எண் 0 தானே எதுவும் செய்யாத செயலுடன் தொடர்புடையது.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "பொதுவாக, இந்த செயல்களில் ஒன்றைத் தொடர்ந்து மற்றொன்றைப் பயன்படுத்துவது, அவை தொடர்புடைய எண்களைப் பெருக்குவதற்கு ஒத்திருக்கிறது.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "ஒவ்வொரு குழுவிலும் உள்ள செயல்கள் எவ்வாறு உடைகின்றன என்பதற்கான ஒப்பீடு முக்கியமானதாக இருக்கும், எனவே ஒவ்வொன்றிலும் நினைவில் கொள்ளுங்கள், நீங்கள் எந்த செயலையும் முற்றிலும் உண்மையான எண் செயலாக உடைக்கலாம், அதைத் தொடர்ந்து சிக்கலான எண்களுக்கு குறிப்பிட்ட ஏதாவது சேர்க்கலாம், அது செங்குத்து ஸ்லைடுகளாக இருந்தாலும் சரி குழு, அல்லது பெருக்கல் குழுவிற்கான தூய சுழற்சிகள்.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "அடுக்குகளைப் பற்றிய நமது முதல் அறிமுகம், மீண்டும் மீண்டும் பெருக்குவதன் அடிப்படையில் அவற்றைப் பற்றி சிந்திக்க வேண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "ஆனால் 2 முதல் ½ வரை, அல்லது 2 முதல் –1 வரை, மற்றும் மிகக் குறைவான 2 ஐப் போன்ற வெளிப்பாடுகள், அடுக்குகளை மீண்டும் மீண்டும் பெருக்குவதாக நீங்கள் நினைக்கும் போது உண்மையில் அர்த்தமில்லை.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "2 ஐ ஒரு நேரத்தின் பாதி, அல்லது ஒரு நேரத்தின் –1 என்று பெருக்கினால் என்ன அர்த்தம்?", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "சிக்கலான விமானத்தில் உள்ள கூட்டல் குழுவை சிக்கலான விமானத்தில் உள்ள பெருக்கல் குழுவுடன் இணைப்பதன் அர்த்தம் என்ன என்று சிந்தியுங்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "இரண்டு அலகுகளின் செங்குத்து ஸ்லைடு இரண்டு ரேடியன்களின் சுழற்சிக்கு வரைபடமாக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "முழுப் பதிலும் கால்குலஸில் உள்ளது, அதுவே e இன் பிறப்பிடமாகும், மேலும் அது வரையறுக்கப்பட்ட இடமாகும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/telugu/sentence_translations.json b/2017/eulers-formula-via-group-theory/telugu/sentence_translations.json index e967632b2..02b12b5a7 100644 --- a/2017/eulers-formula-via-group-theory/telugu/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/telugu/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "సమూహంగా సంఖ్యల గురించి ఆలోచించడానికి రెండు వేర్వేరు మార్గాలు ఉన్నాయి.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "మీరు నంబర్ లైన్‌ను ఎడమ లేదా కుడివైపు స్లైడ్ చేసే అన్ని మార్గాల గురించి ఆలోచించండి.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "సంఖ్య 0 కూడా ఏమీ చేయని చర్యతో ముడిపడి ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "సాధారణంగా, ఈ చర్యలలో ఒకదానిని వర్తింపజేయడం ద్వారా మరొకటి వాటితో అనుబంధించబడిన సంఖ్యలను గుణించడంతో సమానంగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "ప్రతి సమూహంలోని చర్యలు ఎలా విచ్ఛిన్నమవుతాయి అనే దాని పోలిక ముఖ్యమైనది, కాబట్టి ప్రతిదానిలో గుర్తుంచుకోండి, మీరు సంకలితం కోసం నిలువు స్లయిడ్‌లు అయినా సంక్లిష్ట సంఖ్యలకు నిర్దిష్టమైన దాని తర్వాత ఏదైనా చర్యను పూర్తిగా వాస్తవ సంఖ్య చర్యగా విభజించవచ్చు. సమూహం, లేదా గుణకార సమూహం కోసం స్వచ్ఛమైన భ్రమణాలు.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "ఘాతాంకాలకు మా మొదటి పరిచయం వాటిని పునరావృత గుణకారంలో ఆలోచించడం.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "కానీ మీరు ఘాతాంకాలను పునరావృత గుణకారంగా భావించినప్పుడు 2 నుండి ½ వరకు, లేదా 2 నుండి –1 వరకు మరియు చాలా తక్కువ 2 వంటి వ్యక్తీకరణలు నిజంగా అర్థం కావు.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "2ని ఒక సమయంలో సగం లేదా -1ని గుణించడం అంటే ఏమిటి?", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "కాంప్లెక్స్ ప్లేన్‌లోని సంకలిత సమూహాన్ని కాంప్లెక్స్ ప్లేన్‌లోని గుణకార సమూహంతో అనుబంధించడం అంటే ఏమిటో ఆలోచించండి.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "రెండు యూనిట్ల నిలువు స్లయిడ్ రెండు రేడియన్ల భ్రమణానికి మ్యాప్ చేస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "పూర్తి సమాధానం కాలిక్యులస్‌లో ఉంటుంది, అది ఇ యొక్క జన్మస్థలం మరియు అది ఎక్కడ నిర్వచించబడింది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/thai/sentence_translations.json b/2017/eulers-formula-via-group-theory/thai/sentence_translations.json index 274320bbe..9faf2dac4 100644 --- a/2017/eulers-formula-via-group-theory/thai/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/thai/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group. ", + "input": "And there are actually two separate ways to think about numbers as a group. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing. ", + "input": "The number zero itself, well, that's associated with the action of just doing nothing. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with. ", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group. ", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication. ", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication. ", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time? ", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined. ", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/turkish/sentence_translations.json b/2017/eulers-formula-via-group-theory/turkish/sentence_translations.json index 58bfc1f89..5145a8b34 100644 --- a/2017/eulers-formula-via-group-theory/turkish/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/turkish/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "Sayıları grup olarak düşünmenin iki ayrı yolu vardır.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "Bir sayı doğrusunu kendi boyunca sola veya sağa kaydırmanın tüm yollarını düşünün.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "0 sayısının kendisi hiçbir şey yapmama eylemiyle ilişkilidir.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "Genel olarak, bu eylemlerden birini ve ardından diğerini uygulamak, ilişkili oldukları sayıları çarpmaya karşılık gelir.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "Her gruptaki eylemlerin nasıl bozulduğuna dair karşılaştırma önemli olacak, bu yüzden şunu unutmayın: Her birinde, herhangi bir eylemi tamamen gerçek sayı eylemi olarak, ardından karmaşık sayılara özgü bir şeyle (katkı maddesi için dikey slaytlar) ayırabilirsiniz. grup veya çarpımsal grup için saf rotasyonlar.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "Üslü sayılarla ilk tanışmamız onları tekrarlı çarpma açısından düşünmektir.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "Ancak 2 üzeri ½ veya 2 üzeri –1 ve çok daha az 2 üzeri i gibi ifadeler, üsleri tekrarlanan çarpma olarak düşündüğünüzde pek mantıklı gelmiyor.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "2'yi kendisiyle yarısı kadar veya -1'i ile çarpmak ne anlama gelir?", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "Karmaşık düzlemdeki toplamsal grubu, karmaşık düzlemdeki çarpımsal grupla ilişkilendirmenin tüm bunların ne anlama geldiğini düşünün.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "İki birimlik dikey bir kayma, iki radyanlık bir dönüşle eşleşir.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "Tam cevap matematikte yatıyor; burası e'nin doğum yeri ve hatta tanımlandığı yer.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/ukrainian/sentence_translations.json b/2017/eulers-formula-via-group-theory/ukrainian/sentence_translations.json index 410a9e5d6..3fa126797 100644 --- a/2017/eulers-formula-via-group-theory/ukrainian/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/ukrainian/sentence_translations.json @@ -308,7 +308,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "Є два різні способи розглядати числа як групу.", "n_reviews": 0, "start": 365.1, @@ -336,7 +336,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "Подумайте, якими способами можна пересунути числову пряму вліво або вправо вздовж неї.", "n_reviews": 0, "start": 384.74, @@ -378,7 +378,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "Саме число 0 асоціюється з дією нічого не робити.", "n_reviews": 0, "start": 435.22, @@ -581,7 +581,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "Загалом застосування однієї з цих дій з наступною іншою відповідає множенню чисел, з якими вони пов’язані.", "n_reviews": 0, "start": 695.76, @@ -686,7 +686,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "Це порівняння того, як розбиваються дії в кожній групі, буде важливим, тому пам’ятайте, що в кожній з них ви можете розбити будь-яку дію як певну дію з дійсними числами, за якою слідує щось специфічне для комплексних чисел, незалежно від того, чи це вертикальні слайди для додавання або чисті обертання для мультиплікативної групи.", "n_reviews": 0, "start": 852.6, @@ -735,7 +735,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "Наше перше знайомство з експонентами — це подумати про них у термінах багаторазового множення.", "n_reviews": 0, "start": 926.74, @@ -763,14 +763,14 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "Але такі вирази, як 2 до ½, або 2 до –1, і набагато менше, 2 до i, насправді не мають сенсу, якщо ви думаєте про показник як про повторне множення.", "n_reviews": 0, "start": 963.2, "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "Що означає помножити 2 на себе половину часу або –1 разу?", "n_reviews": 0, "start": 973.8, @@ -847,7 +847,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "Подумайте, що все це означає для асоціації адитивної групи на комплексній площині з мультиплікативною групою на комплексній площині.", "n_reviews": 0, "start": 1103.62, @@ -910,7 +910,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "Вертикальне ковзання на дві одиниці буде відповідати повороту на два радіани.", "n_reviews": 0, "start": 1227.78, @@ -945,7 +945,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "Повна відповідь міститься в численні, де народилася e, і де воно навіть визначено.", "n_reviews": 0, "start": 1260.14, diff --git a/2017/eulers-formula-via-group-theory/urdu/sentence_translations.json b/2017/eulers-formula-via-group-theory/urdu/sentence_translations.json index 63335f93e..ccb2817ec 100644 --- a/2017/eulers-formula-via-group-theory/urdu/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/urdu/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group. ", + "input": "And there are actually two separate ways to think about numbers as a group. ", "translatedText": "ایک گروپ کے طور پر نمبروں کے بارے میں سوچنے کے دو الگ الگ طریقے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing. ", + "input": "The number zero itself, well, that's associated with the action of just doing nothing. ", "translatedText": "نمبر 0 خود کچھ نہ کرنے کے عمل سے وابستہ ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with. ", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with. ", "translatedText": "عام طور پر، ان میں سے کسی ایک کو لاگو کرنا اور اس کے بعد دوسرا عمل کرنا ان نمبروں کو ضرب دینے کے مساوی ہے جن سے وہ وابستہ ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group. ", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group. ", "translatedText": "یہ موازنہ اس بات کا ہے کہ ہر گروپ میں اعمال کیسے ٹوٹتے ہیں، لہذا یاد رکھیں کہ ہر ایک میں، آپ کسی بھی عمل کو مکمل طور پر حقیقی نمبر کی کارروائی کے طور پر توڑ سکتے ہیں، اس کے بعد پیچیدہ نمبروں کے لیے مخصوص کوئی چیز، چاہے وہ اضافی کے لیے عمودی سلائیڈز ہو۔گروپ، یا ضرب گروپ کے لیے خالص گردش۔", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication. ", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and ", "translatedText": "مفہوم کے ساتھ ہمارا پہلا تعارف یہ ہے کہ انہیں بار بار ضرب کے لحاظ سے سوچنا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication. ", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication. ", "translatedText": "اور جب آپ چیزوں کو بڑھاتے ہیں، تو یہ کافی معقول لگتا ہے، ٹھیک ہے؟ لیکن 2 سے ½، یا 2 سے -1، اور i کے لیے بہت کم 2 جیسے تاثرات اس وقت کوئی معنی نہیں رکھتے جب آپ ایکسپوننٹ کو بار بار ضرب کے طور پر سوچتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time? ", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined. ", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined. ", "translatedText": "اب آپ پوچھ سکتے ہیں، کیوں ای؟ کوئی اور بنیاد کیوں نہیں؟ مکمل جواب کیلکولس میں موجود ہے، یہ ای کی جائے پیدائش ہے، اور جہاں اس کی وضاحت بھی کی گئی ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-formula-via-group-theory/vietnamese/sentence_translations.json b/2017/eulers-formula-via-group-theory/vietnamese/sentence_translations.json index f67766ccd..009370471 100644 --- a/2017/eulers-formula-via-group-theory/vietnamese/sentence_translations.json +++ b/2017/eulers-formula-via-group-theory/vietnamese/sentence_translations.json @@ -352,7 +352,7 @@ "end": 364.54 }, { - "input": "There are two separate ways to think about numbers as a group.", + "input": "And there are actually two separate ways to think about numbers as a group.", "translatedText": "Có hai cách riêng biệt để suy nghĩ về các con số như một nhóm.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 383.92 }, { - "input": "Think of all the ways you can slide a number line left or right along itself.", + "input": "Think of all of the ways that you can slide a number line left or right along itself.", "translatedText": "Hãy nghĩ đến tất cả các cách bạn có thể trượt một trục số sang trái hoặc sang phải dọc theo chính nó.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 434.22 }, { - "input": "The number 0 itself is associated with the action of doing nothing.", + "input": "The number zero itself, well, that's associated with the action of just doing nothing.", "translatedText": "Bản thân số 0 gắn liền với hành động không làm gì cả.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 694.5 }, { - "input": "In general, applying one of these actions followed by another corresponds with multiplying the numbers they're associated with.", + "input": "And in general, applying one of these actions followed by another corresponds with multiplying the numbers that they're associated with.", "translatedText": "Nói chung, việc áp dụng một trong những hành động này theo sau một hành động khác tương ứng với việc nhân các số mà chúng liên kết với nhau.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 851.56 }, { - "input": "That comparison of how actions in each group breaks down is going to be important, so remember In each one, you can break down any action as some purely real number action, followed by something specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", + "input": "That comparison of how actions in each group breaks down is going to be important, so remember it. In each one, you can break down any action as some purely real number action, followed by something that's specific to complex numbers, whether that's vertical slides for the additive group, or pure rotations for the multiplicative group.", "translatedText": "Sự so sánh về cách phân tích các hành động trong mỗi nhóm sẽ rất quan trọng, vì vậy hãy nhớ rằng Trong mỗi nhóm, bạn có thể chia nhỏ bất kỳ hành động nào dưới dạng một số hành động thuần túy số thực, theo sau là một số hành động cụ thể cho số phức, cho dù đó là các trang trình bày dọc cho phép cộng nhóm, hoặc phép quay thuần túy cho nhóm nhân.", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 925.18 }, { - "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication.", + "input": "Our first introduction to exponents is to think of them in terms of repeated multiplication, right? I mean, the meaning of something like 2 cubed is to take 2 times 2 times 2, and", "translatedText": "Phần giới thiệu đầu tiên của chúng ta về số mũ là nghĩ về chúng dưới dạng phép nhân lặp lại.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 962.52 }, { - "input": "But expressions like 2 to the ½, or 2 to the –1, and much less 2 to the i don't really make sense when you think of exponents as repeated multiplication.", + "input": "But expressions like 2 to the 1 half, or 2 to the negative 1, and much less 2 to the i, don't really make sense when you think of exponents as repeated multiplication.", "translatedText": "Nhưng các biểu thức như 2 với ½, hoặc 2 với –1, và ít hơn nhiều 2 với i thực sự không có ý nghĩa khi bạn coi số mũ là phép nhân lặp lại.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 973.8 }, { - "input": "What does it mean to multiply 2 by itself half of a time, or –1 of a time?", + "input": "I mean, what does it mean to multiply 2 by itself half of a time, or negative 1 of a time?", "translatedText": "Việc nhân 2 với chính nó một nửa thời gian hoặc -1 thời gian có nghĩa là gì?", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 1101.26 }, { - "input": "Think about what all this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", + "input": "Now, think about what all of this means for associating the additive group in the complex plane with the multiplicative group in the complex plane.", "translatedText": "Hãy nghĩ xem tất cả những điều này có ý nghĩa gì đối với việc liên kết nhóm cộng trong mặt phẳng phức với nhóm nhân trong mặt phẳng phức.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1227.78 }, { - "input": "A vertical slide of two units would map to a rotation of two radians.", + "input": "And so, a vertical slide of two units would map to a rotation of two radians.", "translatedText": "Một slide dọc gồm hai đơn vị sẽ ánh xạ tới một góc quay hai radian.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1259.28 }, { - "input": "The full answer resides in calculus, that's the birthplace of e, and where it's even defined.", + "input": "Well, the full answer resides in calculus. I mean, that's the birthplace of e and where it's even defined.", "translatedText": "Câu trả lời đầy đủ nằm trong phép tính, đó là nơi sinh của e và thậm chí là nơi nó được xác định.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/arabic/sentence_translations.json b/2017/eulers-number/arabic/sentence_translations.json index 068d186ff..f55dd891f 100644 --- a/2017/eulers-number/arabic/sentence_translations.json +++ b/2017/eulers-number/arabic/sentence_translations.json @@ -559,7 +559,7 @@ "end": 558.42 }, { - "input": "Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", + "input": "Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", "translatedText": "أو بدلاً من تطبيق القاعدة بشكل أعمى، يمكنك تخصيص هذه اللحظة للتدرب على الحدس الخاص بقاعدة السلسلة التي تحدثت عنها في الفيديو الأخير، والتفكير في كيفية تغيير دفعة طفيفة إلى t قيمة 3t، وكيف يؤدي هذا التغيير الوسيط إلى دفع القيمة النهائية من ه إلى 3t.", "model": "google_nmt", "from_community_srt": "أو ، بدلاً من مجرد تطبيق هذه القاعدة بطريقة عمياء ، يمكنك أن تأخذ هذه اللحظة لممارسة الحدس لقاعدة السلسلة التي تحدثت خلال الفيديو الأخير ، والتفكير في كيفية دفعة صغيرة إلى t يغير قيمة 3t وكيف يؤدي هذا التغيير المتوسط إلى دفع القيمة النهائية لـ e إلى 3t.", @@ -640,7 +640,7 @@ "end": 646.22 }, { - "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base.", + "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base.", "translatedText": "ثابت التناسب الغامض الذي يظهر عند أخذ المشتقات هو مجرد اللوغاريتم الطبيعي للأساس.", "model": "google_nmt", "from_community_srt": "ثابت التناسب الغامض الذي ينبثق عند أخذ المشتقات هو مجرد اللوغارتم الطبيعي للأساس، الجواب على السؤال ، \"e أس ماذا يساوي تلك الأساس؟\"", diff --git a/2017/eulers-number/bengali/sentence_translations.json b/2017/eulers-number/bengali/sentence_translations.json index 812f8392d..cc49617ea 100644 --- a/2017/eulers-number/bengali/sentence_translations.json +++ b/2017/eulers-number/bengali/sentence_translations.json @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931. ", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931. ", "translatedText": "এবং স্পষ্টতই, অনেক ছোট টাইমস্কেলের উপর এই ফাংশনের পরিবর্তনের হার 0 এর এই অদ্ভুত আনুপাতিকতার ধ্রুবক সহ, নিজের সাথে সমানুপাতিক নয়।6931।", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself. ", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself. ", "translatedText": "আপনি যদি একটি শীতল ঘরে এক কাপ গরম জল রাখেন, তাহলে যে হারে জল ঠান্ডা হয় তা ঘর এবং জলের মধ্যে তাপমাত্রার পার্থক্যের সমানুপাতিক হয়, বা যে হারে সেই পার্থক্যটি পরিবর্তিত হয় তা নিজেই সমানুপাতিক।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/bulgarian/sentence_translations.json b/2017/eulers-number/bulgarian/sentence_translations.json index eb172eb05..2544fea5a 100644 --- a/2017/eulers-number/bulgarian/sentence_translations.json +++ b/2017/eulers-number/bulgarian/sentence_translations.json @@ -504,7 +504,7 @@ "end": 558.42 }, { - "input": "Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", + "input": "Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", "translatedText": "Или вместо да прилагате правило на сляпо, можете да използвате този момент, за да упражните интуицията си за верижното правило, за което говорих в последното видео, като помислите как леко побутване на t променя стойността на 3t и как тази междинна промяна побутва крайната стойност на e към 3t.", "model": "DeepL", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 646.22 }, { - "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base.", + "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base.", "translatedText": "Мистериозната константа на пропорционалност, която се появява при вземането на производни, е просто естественият логаритъм на основата.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/eulers-number/chinese/sentence_translations.json b/2017/eulers-number/chinese/sentence_translations.json index 18fead12f..87e2f7bb3 100644 --- a/2017/eulers-number/chinese/sentence_translations.json +++ b/2017/eulers-number/chinese/sentence_translations.json @@ -358,7 +358,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931. ", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931. ", "translatedText": "显然，这个函数在更小的时间尺度上的变化 率并不完全等于自身，而是与自身成正比 ，这个特殊的比例常数为 0。6931。", "model": "google_nmt", "from_community_srt": "当这个变化发生在一个很小的时间尺度内 变化率就不简单的等于这个函数的值了 而是与这个函数的值成比例关系 这个比例就是一个古怪的常数0.6931 还有，", @@ -729,7 +729,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself. ", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself. ", "translatedText": "如果将一杯热水放在凉爽的房间里，水冷 却的速度与房间和水之间的温差成正比， 或者温差变化的速度与其本身成正比。", "model": "google_nmt", "from_community_srt": "的温度差成比 或者这么说 这杯水的温度的变化速度和这杯水的温度变化成比例 如果你存入一笔钱，", diff --git a/2017/eulers-number/english/captions.srt b/2017/eulers-number/english/captions.srt index efcc0b1de..cf12f6a59 100644 --- a/2017/eulers-number/english/captions.srt +++ b/2017/eulers-number/english/captions.srt @@ -503,19 +503,19 @@ which due to this special nature of e is just itself, and multiply by the derivative of that inner function 3t, which is the constant 3. 127 -00:09:19,460 --> 00:09:23,435 -Or rather than applying a rule blindly, you could take this moment +00:09:19,460 --> 00:09:23,553 +Or rather than just applying a rule blindly, you could take this moment 128 -00:09:23,435 --> 00:09:27,708 -to practice the intuition for the chain rule I talked about last video, +00:09:23,553 --> 00:09:28,044 +to practice the intuition for the chain rule that I talked through last video, 129 -00:09:27,708 --> 00:09:31,506 +00:09:28,044 --> 00:09:31,683 thinking about how a slight nudge to t changes the value of 3t, 130 -00:09:31,506 --> 00:09:35,720 +00:09:31,683 --> 00:09:35,720 and how that intermediate change nudges the final value of e to the 3t. 131 @@ -579,12 +579,12 @@ you'll find that it's 0.6931, the mystery constant we ran into earlier. And the same goes for all the other bases. 146 -00:10:46,760 --> 00:10:50,089 -The mystery proportionality constant that pops up when +00:10:46,760 --> 00:10:49,991 +The mystery proportionality constant that pops up when taking derivatives is just 147 -00:10:50,089 --> 00:10:53,420 -taking derivatives is just the natural log of the base. +00:10:49,991 --> 00:10:53,420 +the natural log of the base. The answer to the question e to the what equals that base. 148 00:10:53,420 --> 00:11:00,787 diff --git a/2017/eulers-number/english/sentence_timings.json b/2017/eulers-number/english/sentence_timings.json index 44fa9e56e..afbbdba1b 100644 --- a/2017/eulers-number/english/sentence_timings.json +++ b/2017/eulers-number/english/sentence_timings.json @@ -315,7 +315,7 @@ 558.42 ], [ - "Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", + "Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", 559.46, 575.72 ], @@ -360,7 +360,7 @@ 646.22 ], [ - "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base.", + "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base.", 646.76, 653.42 ], diff --git a/2017/eulers-number/english/transcript.txt b/2017/eulers-number/english/transcript.txt index d6ef7c386..23c43af88 100644 --- a/2017/eulers-number/english/transcript.txt +++ b/2017/eulers-number/english/transcript.txt @@ -61,7 +61,7 @@ The existence of a function like this answers the question of the mystery consta The key is to use the chain rule. For example, what is the derivative of e to the 3t? Well, you take the derivative of the outermost function, which due to this special nature of e is just itself, and multiply by the derivative of that inner function 3t, which is the constant 3. -Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t. +Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t. Either way, the point is e to the power of some constant times t is equal to that same constant times itself. And from here, the question of those mystery constants really just comes down to a certain algebraic manipulation. The number 2 can also be written as e to the natural log of 2. @@ -70,7 +70,7 @@ So the function 2 to the t is the same as the function e to the power of the nat And from what we just saw, combining the fact that e to the t is its own derivative with the chain rule, the derivative of this function is proportional to itself, with a proportionality constant equal to the natural log of 2. And indeed, if you go plug in the natural log of 2 to a calculator, you'll find that it's 0.6931, the mystery constant we ran into earlier. And the same goes for all the other bases. -The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. +The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base. In fact, throughout applications of calculus, you rarely see exponentials written as some base to a power t. Instead, you almost always write the exponential as e to the power of some constant times t. It's all equivalent, I mean any function like 2 to the t or 3 to the t can also be written as e to some constant times t. diff --git a/2017/eulers-number/french/sentence_translations.json b/2017/eulers-number/french/sentence_translations.json index 9134919bb..132775667 100644 --- a/2017/eulers-number/french/sentence_translations.json +++ b/2017/eulers-number/french/sentence_translations.json @@ -503,7 +503,7 @@ "end": 558.42 }, { - "input": "Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", + "input": "Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", "translatedText": "Ou plutôt que d'appliquer une règle aveuglément, vous pouvez profiter de ce moment pour mettre en pratique l'intuition de la règle de chaîne dont j'ai parlé dans la dernière vidéo, en réfléchissant à la façon dont un léger coup de pouce vers t modifie la valeur de 3t, et comment ce changement intermédiaire modifie la valeur finale. de e au 3t.", "from_community_srt": "Ou, plutôt que d'appliquer une règle aveuglément, vous pouvez prendre ce moment pour pratiquer votre intuition de la règle de la chaîne dont je parlais à travers la dernière vidéo, en pensant à la façon dont une légère poussée de t modifie la valeur de 3t et comment ce changement intermédiaire pousse la valeur finale de e à la puissance 3t.", "n_reviews": 0, @@ -575,7 +575,7 @@ "end": 646.22 }, { - "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base.", + "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base.", "translatedText": "La constante mystérieuse de proportionnalité qui apparaît lors de la prise de dérivés n’est que le logarithme naturel de la base.", "from_community_srt": "La constante de proportionnalité mystère qui apparaît lorsqu'on calcule la dérivée est juste le logarithme naturel de la base,", "n_reviews": 0, diff --git a/2017/eulers-number/german/sentence_translations.json b/2017/eulers-number/german/sentence_translations.json index fb9c4a17e..df1fd1c87 100644 --- a/2017/eulers-number/german/sentence_translations.json +++ b/2017/eulers-number/german/sentence_translations.json @@ -565,7 +565,7 @@ "end": 558.42 }, { - "input": "Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", + "input": "Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", "translatedText": "Anstatt eine Regel blind anzuwenden, könntest du in diesem Moment auch die Intuition für die Kettenregel üben, über die ich im letzten Video gesprochen habe, indem du darüber nachdenkst, wie eine kleine Änderung von t den Wert von 3t verändert und wie diese zwischenzeitliche Änderung den endgültigen Wert von e auf die 3t bringt.", "model": "DeepL", "from_community_srt": "3. Anstatt eine Regel nur blind anzuwenden, können Sie diesen Moment nutzen, um die Intuition für die Kettenregel zu üben dass ich das letzte Video durchgesprochen habe und darüber nachgedacht habe, wie ein leichter Anstoß zu t den Wert von 3t verändert und wie diese Zwischenänderung den Endwert von e auf 3t bringt.", @@ -646,7 +646,7 @@ "end": 646.22 }, { - "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base.", + "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base.", "translatedText": "Die mysteriöse Proportionalitätskonstante, die bei Ableitungen auftaucht, ist einfach der natürliche Logarithmus der Basis.", "model": "DeepL", "from_community_srt": "Die mysteriöse Proportionalitätskonstante, die beim Einnehmen von Derivaten auftritt ist nur das natürliche Protokoll der Basis, die Antwort auf die Frage:", diff --git a/2017/eulers-number/hebrew/sentence_translations.json b/2017/eulers-number/hebrew/sentence_translations.json index b27c8d60a..11b0414ba 100644 --- a/2017/eulers-number/hebrew/sentence_translations.json +++ b/2017/eulers-number/hebrew/sentence_translations.json @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931. ", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931. ", "translatedText": "ולראיה, קצב השינוי של פונקציה זו על פני טווחי זמן קטנים בהרבה אינו שווה לגמרי לעצמה, אלא פרופורציונלי לעצמה, עם קבוע המידתיות המוזר הזה של 0.6931. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself. ", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself. ", "translatedText": "אם שמים כוס מים חמים בחדר קריר, הקצב שבו המים מתקררים הוא פרופורציונלי להפרש הטמפרטורה בין החדר למים, או הקצב שבו משתנה ההבדל הוא פרופורציונלי לעצמו. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/hindi/sentence_translations.json b/2017/eulers-number/hindi/sentence_translations.json index 93b17ed56..8ddb22c90 100644 --- a/2017/eulers-number/hindi/sentence_translations.json +++ b/2017/eulers-number/hindi/sentence_translations.json @@ -210,7 +210,7 @@ "end": 277.68 }, { - "input": "Just look at what happens here.", + "input": "I mean, just look at what happens here.", "translatedText": "जरा देखो यहां क्या होता है.", "n_reviews": 0, "start": 278.42, @@ -287,7 +287,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931.", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931.", "translatedText": "और जाहिर है, इस फ़ंक्शन के लिए बहुत छोटे समय के पैमाने पर परिवर्तन की दर स्वयं के बराबर नहीं है, बल्कि स्वयं के लिए आनुपातिक है, इस विशिष्ट आनुपातिकता स्थिरांक 0 के साथ।6931.", "n_reviews": 0, "start": 369.03, @@ -588,7 +588,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself.", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself.", "translatedText": "यदि आप एक ठंडे कमरे में एक कप गर्म पानी डालते हैं, तो पानी के ठंडा होने की दर कमरे और पानी के बीच के तापमान के अंतर के समानुपाती होती है, या जिस दर पर अंतर बदलता है वह उसी के समानुपाती होता है।", "n_reviews": 0, "start": 734.1, diff --git a/2017/eulers-number/hungarian/sentence_translations.json b/2017/eulers-number/hungarian/sentence_translations.json index 2a4efa561..22f6be863 100644 --- a/2017/eulers-number/hungarian/sentence_translations.json +++ b/2017/eulers-number/hungarian/sentence_translations.json @@ -504,7 +504,7 @@ "end": 558.42 }, { - "input": "Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", + "input": "Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", "translatedText": "Vagy ahelyett, hogy vakon alkalmaznál egy szabályt, megragadhatnád ezt a pillanatot, hogy gyakorold a láncszabály intuícióját, amiről a múltkori videóban beszéltem, gondolkodva azon, hogy a t enyhe lökése hogyan változtatja meg a 3t értékét, és hogy ez a köztes változás hogyan löki az e végső értékét a 3t-hez.", "model": "DeepL", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 646.22 }, { - "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base.", + "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base.", "translatedText": "A rejtélyes arányossági állandó, amely a deriváltak felvételekor felbukkan, nem más, mint a bázis természetes logaritmusa.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/eulers-number/indonesian/sentence_translations.json b/2017/eulers-number/indonesian/sentence_translations.json index a59fe192e..7dcc2d243 100644 --- a/2017/eulers-number/indonesian/sentence_translations.json +++ b/2017/eulers-number/indonesian/sentence_translations.json @@ -240,7 +240,7 @@ "end": 277.68 }, { - "input": "Just look at what happens here.", + "input": "I mean, just look at what happens here.", "translatedText": "Lihat saja apa yang terjadi di sini.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931.", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931.", "translatedText": "Dan terbukti, laju perubahan fungsi ini dalam rentang waktu yang jauh lebih kecil tidak sama dengan fungsi itu sendiri, namun sebanding dengan fungsi itu sendiri, dengan konstanta proporsionalitas khusus sebesar 0.6931.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself.", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself.", "translatedText": "Jika Anda menaruh secangkir air panas di ruangan yang sejuk, laju pendinginan air sebanding dengan perbedaan suhu antara ruangan dan air, atau laju perubahan perbedaan tersebut sebanding dengan suhu itu sendiri.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/italian/sentence_translations.json b/2017/eulers-number/italian/sentence_translations.json index 85ea470f4..775cbd84f 100644 --- a/2017/eulers-number/italian/sentence_translations.json +++ b/2017/eulers-number/italian/sentence_translations.json @@ -240,7 +240,7 @@ "end": 277.68 }, { - "input": "Just look at what happens here.", + "input": "I mean, just look at what happens here.", "translatedText": "Guarda cosa succede qui.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931.", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931.", "translatedText": "Ed evidentemente, il tasso di cambiamento di questa funzione su scale temporali molto più piccole non è del tutto uguale a se stesso, ma proporzionale a se stesso, con questa peculiare costante di proporzionalità pari a 0.6931.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself.", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself.", "translatedText": "Se metti una tazza di acqua calda in una stanza fresca, la velocità con cui l'acqua si raffredda è proporzionale alla differenza di temperatura tra la stanza e l'acqua, oppure la velocità con cui tale differenza cambia è proporzionale a se stessa.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/japanese/sentence_translations.json b/2017/eulers-number/japanese/sentence_translations.json index 7da78da39..2826831dc 100644 --- a/2017/eulers-number/japanese/sentence_translations.json +++ b/2017/eulers-number/japanese/sentence_translations.json @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931. ", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931. ", "translatedText": "そして明らかに、はるかに小さいタイムスケールにわたるこの関数の 変化率は、それ自体と完全に等しいわけではありませんが、この独特 の比例定数 0 により、それ自体に比例します。6931。", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself. ", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself. ", "translatedText": "涼しい部屋に熱湯の入ったカップを入れると、水 が冷える速度は部屋と水の温度差に比例するか、 その差が変化する速度はそれ自体に比例します。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/korean/sentence_translations.json b/2017/eulers-number/korean/sentence_translations.json index 93bcba3cb..e54861d89 100644 --- a/2017/eulers-number/korean/sentence_translations.json +++ b/2017/eulers-number/korean/sentence_translations.json @@ -566,7 +566,7 @@ "end": 558.42 }, { - "input": "Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", + "input": "Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", "translatedText": "또는 무턱대고 규칙을 적용하기보다는 지난 동영상에서 말씀드린 연쇄 규칙에 대한 직관을 연습하는 시간을 가지면서, t를 살짝 넛지하면 3t의 값이 어떻게 바뀌고, 그 중간 변화가 어떻게 최종 값인 e를 3t로 넛지하는지 생각해 볼 수도 있습니다.", "model": "DeepL", "from_community_srt": "또는 맹목적으로 법칙을 적용하는 것보다는, 연쇄법칙에 대한 직관을 연습하는 데 시간을 쓸 수 있습니다. 제가 가장 최근의 비디오를 통해 이야기 한 t의 미소 변화량이 3t 미소 변화량을 어떻게 바꾸는지, 그리고 그 중간의 변화가 어떻게 e의 3t제곱을 바꾸는지 생각해보는 것입니다.", @@ -647,7 +647,7 @@ "end": 646.22 }, { - "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base.", + "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base.", "translatedText": "파생상품을 취할 때 나타나는 미스테리 비례 상수는 기본의 자연 로그일 뿐입니다.", "model": "DeepL", "from_community_srt": "미분을 할 때 나타나는 수수께끼의 비례 상수는 'e의 몇제곱이 밑과 같아지는가?'라는 질문의 대답인 밑의 자연로그일 뿐입니다.", diff --git a/2017/eulers-number/marathi/sentence_translations.json b/2017/eulers-number/marathi/sentence_translations.json index 0efa0e8ad..884396be7 100644 --- a/2017/eulers-number/marathi/sentence_translations.json +++ b/2017/eulers-number/marathi/sentence_translations.json @@ -240,7 +240,7 @@ "end": 277.68 }, { - "input": "Just look at what happens here.", + "input": "I mean, just look at what happens here.", "translatedText": "फक्त इथे काय होते ते पहा.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931.", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931.", "translatedText": "आणि स्पष्टपणे, या फंक्शनच्या बदलाचा दर खूपच लहान वेळापत्रकांवरील बदलाचा दर 0 च्या या विचित्र आनुपातिक स्थिरतेसह, स्वतःच्या समान नाही, परंतु स्वतःच्या प्रमाणात आहे.६९३१.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself.", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself.", "translatedText": "जर तुम्ही थंड खोलीत एक कप गरम पाणी ठेवले तर, पाणी ज्या दराने थंड होते ते खोली आणि पाणी यांच्यातील तापमानातील फरकाच्या प्रमाणात असते किंवा ज्या दराने तो फरक बदलतो तो दर स्वतःच्या प्रमाणात असतो.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/persian/sentence_translations.json b/2017/eulers-number/persian/sentence_translations.json index d5c64f891..e29da4b59 100644 --- a/2017/eulers-number/persian/sentence_translations.json +++ b/2017/eulers-number/persian/sentence_translations.json @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931. ", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931. ", "translatedText": "و بدیهی است که نرخ تغییر این تابع در بازه‌های زمانی بسیار کوچک‌تر کاملاً برابر نیست، بلکه متناسب با خودش است، با این ثابت تناسب عجیب و غریب 0.6931. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself. ", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself. ", "translatedText": "اگر یک فنجان آب داغ را در یک اتاق خنک قرار دهید، سرعت سرد شدن آب متناسب با اختلاف دمای اتاق و آب است، یا سرعت تغییر آن تفاوت با خودش متناسب است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/polish/sentence_translations.json b/2017/eulers-number/polish/sentence_translations.json index a4327c98f..4955225e0 100644 --- a/2017/eulers-number/polish/sentence_translations.json +++ b/2017/eulers-number/polish/sentence_translations.json @@ -500,7 +500,7 @@ "end": 558.42 }, { - "input": "Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", + "input": "Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", "translatedText": "", "from_community_srt": "Mógłbyś też, zamiast stosować tu poznaną regułę, przećwiczyć to, co robiliśmy w poprzednim filmie i zastanowić się, jak mała zmiana t wpływa na wartość 3t oraz jak zmiana funkcji pośredniej wpływa na e^(3t).", "n_reviews": 0, @@ -572,7 +572,7 @@ "end": 646.22 }, { - "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base.", + "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base.", "translatedText": "", "from_community_srt": "który pojawia się przy pochodnej, jest równy logarytmowi podstawy.", "n_reviews": 0, diff --git a/2017/eulers-number/portuguese/sentence_translations.json b/2017/eulers-number/portuguese/sentence_translations.json index 121689d06..e03688fbf 100644 --- a/2017/eulers-number/portuguese/sentence_translations.json +++ b/2017/eulers-number/portuguese/sentence_translations.json @@ -565,7 +565,7 @@ "end": 558.42 }, { - "input": "Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", + "input": "Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", "translatedText": "Ou, em vez de aplicar uma regra cegamente, você pode aproveitar este momento para praticar a intuição da regra da cadeia de que falei no vídeo passado, pensando em como um leve empurrão em t altera o valor de 3t e como essa mudança intermediária altera o valor final. de e elevado a 3t.", "model": "google_nmt", "from_community_srt": "3. Ou, ao invés de apenas aplicar uma regra cegamente, você poderia aproveitar este momento para praticar a intuição para a regra da cadeia que mostrei no último vídeo, pensando em como uma 'cutucada' em 't' muda o valor de '3t' e como essa mudança intermediária leva o valor final de 'e' elevado a '3t'.", @@ -646,7 +646,7 @@ "end": 646.22 }, { - "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base.", + "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base.", "translatedText": "A misteriosa constante de proporcionalidade que surge ao derivar é apenas o logaritmo natural da base.", "model": "google_nmt", "from_community_srt": "A constante de proporcionalidade misteriosa que aparece quando se toma derivadas é apenas o logaritmo natural da base,", diff --git a/2017/eulers-number/russian/sentence_translations.json b/2017/eulers-number/russian/sentence_translations.json index 4c55fe051..50d6a37c5 100644 --- a/2017/eulers-number/russian/sentence_translations.json +++ b/2017/eulers-number/russian/sentence_translations.json @@ -240,7 +240,7 @@ "end": 277.68 }, { - "input": "Just look at what happens here.", + "input": "I mean, just look at what happens here.", "translatedText": "Просто посмотрите, что здесь происходит.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931.", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931.", "translatedText": "И очевидно, что скорость изменения этой функции в гораздо меньших масштабах времени не совсем равна самой себе, а пропорциональна самой себе, с этой своеобразной константой пропорциональности, равной 0.6931.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself.", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself.", "translatedText": "Если вы поместите чашку с горячей водой в прохладную комнату, скорость, с которой вода остывает, пропорциональна разнице температур между комнатой и водой, или скорость, с которой эта разница изменяется, пропорциональна самой себе.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/spanish/sentence_translations.json b/2017/eulers-number/spanish/sentence_translations.json index eb1952eac..9bb75fab6 100644 --- a/2017/eulers-number/spanish/sentence_translations.json +++ b/2017/eulers-number/spanish/sentence_translations.json @@ -500,7 +500,7 @@ "end": 558.42 }, { - "input": "Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", + "input": "Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", "translatedText": "O en lugar de aplicar una regla a ciegas, podrías aprovechar este momento para practicar la intuición de la regla de la cadena de la que hablé en el último vídeo, pensando en cómo un ligero empujón a t cambia el valor de 3t, y cómo ese cambio intermedio empuja el valor final. de e al 3t.", "from_community_srt": "3. O, en lugar de aplicar una regla ciegamente, se podría aprovechar este momento para practicar la intuición para la regla de la cadena de la que hablé en el último vídeo, pensando cómo un ligero empujón para t cambia el valor de 3t y cómo ese cambio intermedio empuja el valor final de e elevado a 3T.", "n_reviews": 0, @@ -572,7 +572,7 @@ "end": 646.22 }, { - "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base.", + "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base.", "translatedText": "La misteriosa constante de proporcionalidad que aparece al tomar derivadas es simplemente el logaritmo natural de la base.", "from_community_srt": "La constante de proporcionalidad misteriosa que aparece cuando se toman derivadas es el logaritmo natural de la base,", "n_reviews": 0, diff --git a/2017/eulers-number/swedish/sentence_translations.json b/2017/eulers-number/swedish/sentence_translations.json index b57bae76d..c51def52c 100644 --- a/2017/eulers-number/swedish/sentence_translations.json +++ b/2017/eulers-number/swedish/sentence_translations.json @@ -500,7 +500,7 @@ "end": 558.42 }, { - "input": "Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", + "input": "Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", "translatedText": "", "from_community_srt": "3. Eller, snarare än att bara tillämpa en regel blint, du kan ta detta tillfälle att öva intuition för kedjeregeln att jag talade genom sista video, tänka på hur en liten knuff till t ändrar värdet av 3t och hur det mellan förändring knuffar det slutliga värdet av e till 3t.", "n_reviews": 0, @@ -572,7 +572,7 @@ "end": 646.22 }, { - "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base.", + "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base.", "translatedText": "", "from_community_srt": "Mysteriet proportionalitetskonstanten som dyker upp när man tar derivatan är bara den naturliga logaritmen av basen,", "n_reviews": 0, diff --git a/2017/eulers-number/tagalog/sentence_translations.json b/2017/eulers-number/tagalog/sentence_translations.json index 0154353e7..f24229c61 100644 --- a/2017/eulers-number/tagalog/sentence_translations.json +++ b/2017/eulers-number/tagalog/sentence_translations.json @@ -504,7 +504,7 @@ "end": 558.42 }, { - "input": "Or rather than applying a rule blindly, you could take this moment to practice the intuition for the chain rule I talked about last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", + "input": "Or rather than just applying a rule blindly, you could take this moment to practice the intuition for the chain rule that I talked through last video, thinking about how a slight nudge to t changes the value of 3t, and how that intermediate change nudges the final value of e to the 3t.", "translatedText": "O sa halip na ilapat ang isang panuntunan nang walang taros, maaari mong gawin ang sandaling ito para sanayin ang intuwisyon para sa chain rule na binanggit ko tungkol sa huling video, iniisip kung paano binabago ng bahagyang pag-udyok sa t ang halaga ng 3t, at kung paano itinutulak ng intermediate na pagbabago ang huling halaga. ng e hanggang sa 3t.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 646.22 }, { - "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base.", + "input": "The mystery proportionality constant that pops up when taking derivatives is just the natural log of the base. The answer to the question e to the what equals that base.", "translatedText": "Ang mystery proportionality constant na lumalabas kapag kumukuha ng derivatives ay ang natural na log ng base.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/tamil/sentence_translations.json b/2017/eulers-number/tamil/sentence_translations.json index 81409ec85..d17b55911 100644 --- a/2017/eulers-number/tamil/sentence_translations.json +++ b/2017/eulers-number/tamil/sentence_translations.json @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931. ", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931. ", "translatedText": "மேலும் வெளிப்படையாக, இந்தச் செயல்பாட்டிற்கான மாற்றத்தின் வீதம் மிகவும் சிறிய நேர அளவீடுகளுக்குச் சமமாக இல்லை, ஆனால் அதற்கு விகிதாசாரமாக, இந்த வினோதமான விகிதாசார மாறிலி 0 உடன் உள்ளது. 6931. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself. ", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself. ", "translatedText": "நீங்கள் குளிர்ந்த அறையில் ஒரு கப் சூடான நீரை வைத்தால், நீர் குளிர்ச்சியடையும் வீதம் அறைக்கும் தண்ணீருக்கும் இடையிலான வெப்பநிலை வேறுபாட்டிற்கு விகிதாசாரமாக இருக்கும், அல்லது அந்த வித்தியாசம் மாறும் விகிதம் தனக்கு விகிதாசாரமாக இருக்கும். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/telugu/sentence_translations.json b/2017/eulers-number/telugu/sentence_translations.json index f587c93ce..f4ee85cf0 100644 --- a/2017/eulers-number/telugu/sentence_translations.json +++ b/2017/eulers-number/telugu/sentence_translations.json @@ -240,7 +240,7 @@ "end": 277.68 }, { - "input": "Just look at what happens here.", + "input": "I mean, just look at what happens here.", "translatedText": "ఇక్కడ ఏమి జరుగుతుందో చూడండి.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931.", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931.", "translatedText": "మరియు స్పష్టంగా, చాలా చిన్న సమయ ప్రమాణాలలో ఈ ఫంక్షన్ యొక్క మార్పు రేటు దానికదే సమానంగా ఉండదు, కానీ దానికి అనులోమానుపాతంలో ఉంటుంది, ఈ విచిత్రమైన అనుపాత స్థిరాంకం 0.6931.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself.", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself.", "translatedText": "మీరు చల్లని గదిలో ఒక కప్పు వేడి నీటిని ఉంచినట్లయితే, నీరు చల్లబడే రేటు గది మరియు నీటి మధ్య ఉష్ణోగ్రతలో ఉన్న వ్యత్యాసానికి అనులోమానుపాతంలో ఉంటుంది లేదా ఆ వ్యత్యాసం మారే రేటు దానికి అనులోమానుపాతంలో ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/thai/sentence_translations.json b/2017/eulers-number/thai/sentence_translations.json index 1e44ec0ed..4b9bc63c2 100644 --- a/2017/eulers-number/thai/sentence_translations.json +++ b/2017/eulers-number/thai/sentence_translations.json @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931. ", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself. ", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/turkish/sentence_translations.json b/2017/eulers-number/turkish/sentence_translations.json index fd156bfa6..9f63f02ca 100644 --- a/2017/eulers-number/turkish/sentence_translations.json +++ b/2017/eulers-number/turkish/sentence_translations.json @@ -240,7 +240,7 @@ "end": 277.68 }, { - "input": "Just look at what happens here.", + "input": "I mean, just look at what happens here.", "translatedText": "Burada neler olduğuna bir bakın.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931.", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931.", "translatedText": "Ve açıkça görülüyor ki, çok daha küçük zaman ölçeklerinde bu fonksiyonun değişim oranı kendisine tam olarak eşit değil, kendisine orantılıdır ve bu tuhaf orantı sabiti 0'dır.6931.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself.", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself.", "translatedText": "Soğuk bir odaya bir bardak sıcak su koyarsanız, suyun soğuma hızı oda ile su arasındaki sıcaklık farkıyla ya da bu farkın değişme hızı kendisiyle orantılıdır.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/ukrainian/sentence_translations.json b/2017/eulers-number/ukrainian/sentence_translations.json index ebf940f64..cc5b3c2fd 100644 --- a/2017/eulers-number/ukrainian/sentence_translations.json +++ b/2017/eulers-number/ukrainian/sentence_translations.json @@ -240,7 +240,7 @@ "end": 277.68 }, { - "input": "Just look at what happens here.", + "input": "I mean, just look at what happens here.", "translatedText": "Тільки подивіться, що тут відбувається.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931.", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931.", "translatedText": "І, очевидно, швидкість зміни для цієї функції протягом значно менших часових масштабів не зовсім дорівнює самій собі, а пропорційна сама собі, з цією незвичайною константою пропорційності 0.6931.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself.", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself.", "translatedText": "Якщо ви поставите чашку гарячої води в прохолодну кімнату, швидкість, з якою вода охолоджується, буде пропорційна різниці температур між кімнатою та водою, або швидкість, з якою ця різниця змінюється, пропорційна самій собі.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/urdu/sentence_translations.json b/2017/eulers-number/urdu/sentence_translations.json index 984289539..c72c65840 100644 --- a/2017/eulers-number/urdu/sentence_translations.json +++ b/2017/eulers-number/urdu/sentence_translations.json @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931. ", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931. ", "translatedText": "اور واضح طور پر، اس فنکشن کی تبدیلی کی شرح بہت چھوٹے اوقات میں اپنے آپ سے بالکل مساوی نہیں ہے، بلکہ خود کے لیے متناسب ہے، اس مخصوص تناسب کے مستقل 0 کے ساتھ۔6931. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself. ", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself. ", "translatedText": "اگر آپ ٹھنڈے کمرے میں ایک کپ گرم پانی ڈالتے ہیں، تو جس شرح سے پانی ٹھنڈا ہوتا ہے وہ کمرے اور پانی کے درجہ حرارت کے فرق کے متناسب ہوتا ہے، یا جس شرح پر یہ فرق بدلتا ہے وہ خود متناسب ہوتا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/eulers-number/vietnamese/sentence_translations.json b/2017/eulers-number/vietnamese/sentence_translations.json index 7804c029b..f06cb2f29 100644 --- a/2017/eulers-number/vietnamese/sentence_translations.json +++ b/2017/eulers-number/vietnamese/sentence_translations.json @@ -240,7 +240,7 @@ "end": 277.68 }, { - "input": "Just look at what happens here.", + "input": "I mean, just look at what happens here.", "translatedText": "Chỉ cần nhìn vào những gì xảy ra ở đây.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 368.44 }, { - "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but proportional to itself, with this peculiar proportionality constant of 0.6931.", + "input": "And evidently, the rate of change for this function over much smaller timescales is not quite equal to itself, but it's proportional to itself, with this very peculiar proportionality constant of 0.6931.", "translatedText": "Và rõ ràng, tốc độ thay đổi của hàm này trong những khoảng thời gian nhỏ hơn nhiều không hoàn toàn bằng chính nó mà tỷ lệ thuận với chính nó, với hằng số tỷ lệ đặc biệt này là 0.6931.", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 733.48 }, { - "input": "If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or the rate at which that difference changes is proportional to itself.", + "input": "And if you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water, or said a little differently, the rate at which that difference changes is proportional to itself.", "translatedText": "Nếu bạn đặt một cốc nước nóng trong phòng mát, tốc độ nước nguội đi tỷ lệ thuận với chênh lệch nhiệt độ giữa phòng và nước, hoặc tốc độ thay đổi chênh lệch đó tỷ lệ với chính nó.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/arabic/sentence_translations.json b/2017/fractal-dimension/arabic/sentence_translations.json index 4c1d70ca9..be4ddac2e 100644 --- a/2017/fractal-dimension/arabic/sentence_translations.json +++ b/2017/fractal-dimension/arabic/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "الخط ذو بعد واحد، والمستوى ذو بعدين، والفضاء الذي نعيش فيه ثلاثي الأبعاد، وهكذا. ", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "إليك طريقة واحدة للتفكير في هذا الأمر. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "إذا كنت تريد رسم عامل القياس مقارنة بعدد المربعات التي يلمسها القرص الذي تم تغيير حجمه، فيجب أن تتوافق بياناتك بشكل وثيق مع القطع المكافئ المثالي، نظرًا لأن عدد المربعات التي تم لمسها يتناسب تقريبًا مع مربع عامل القياس. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "بالنسبة لقيم القياس الأكبر والأكبر، والتي تعادل في الواقع مجرد النظر إلى شبكة أكثر دقة، فإن هذه البيانات ستكون أكثر ملاءمة لهذا القطع المكافئ. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "الآن نعود إلى الفركتلات، لنلعب هذه اللعبة مع مثلث سيربينسكي، ونحسب عدد المربعات التي تتلامس مع نقاط في هذا الشكل. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "حسنًا، يجب أن تكون نسبة الصناديق التي لمسها الصندوق الكبير إلى عدد الصناديق التي لمسها الصندوق الصغير حوالي ثلاثة. ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "بعد كل شيء، هذه النسخة الأكبر مكونة من ثلاث نسخ من النسخة الأصغر. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "يمكنك أيضًا التفكير في هذا باعتباره اثنين مرفوعًا إلى بُعد الفراكتل، والذي رأيناه للتو يساوي حوالي 1.585. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "لذا، إذا أردت رسم عامل القياس في هذه الحالة مقابل عدد المربعات التي تم لمسها بمثلث سيربينسكي، فإن البيانات ستتناسب بشكل وثيق مع منحنى بشكل y يساوي x للأس 1.585، مضروبًا في ثابت التناسب. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "كما هو الحال في، تخيل أنني أعطيك بعض الشكل، وأنت مبرمج ماهر، كيف يمكنك العثور على هذا الرقم؟ ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "لذا فإن الخدعة الشائعة هي أخذ لوغاريتم كلا الطرفين. ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "الأمر الرائع في ذلك هو أنها طريقة كمية للقول بأن هذه الأشكال خشنة، وأنها تظل خشنة حتى عند تكبيرها. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "في الوضع ثلاثي الأبعاد، بالمناسبة، عندما تقوم بعد الصناديق، يكون لديك شبكة ثلاثية الأبعاد مليئة بالمكعبات الصغيرة بدلاً من المربعات الصغيرة، ولكنها تعمل بنفس الطريقة. ", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "في هذا المقياس، حيث يكون سمك الشكل أصغر من حجم الصناديق، فإنه يبدو أحادي البعد، أي أن عدد المربعات التي يلمسها يتناسب مع طوله. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "ولكن عندما تقوم بتوسيع نطاقه، فإنه يبدأ بالتصرف مثل الأنبوب، حيث يلامس الصناديق الموجودة على سطح ذلك الأنبوب، وبالتالي سيبدو ثنائي الأبعاد، حيث يتناسب عدد الصناديق التي تم لمسها مع مربع الشكل عامل التحجيم. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "لذا فإن تعيين رقم لشكل لأبعاده يمكن أن يكون أمرًا صعبًا، ويترك مجالًا لتعريفات مختلفة وأعراف مختلفة. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "لكن في بيئة أكثر تطبيقًا، مثل النظر إلى ساحل بريطانيا، ليس من المنطقي حقًا التحدث عن الحد كلما قمت بالتكبير أكثر فأكثر، أعني أنه في مرحلة ما ستصطدم بالذرات فقط. ", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "وبدلاً من ذلك، فإن ما تفعله هو النظر إلى نطاق واسع بما فيه الكفاية من المقاييس، بدءًا من المقاييس المُصغرة جدًا وحتى المُكبرة جدًا، وحساب البعد في كل منها. ", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "تلعب الأشكال المتشابهة تمامًا دورًا مهمًا في الهندسة الكسورية. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/bengali/sentence_translations.json b/2017/fractal-dimension/bengali/sentence_translations.json index 4148968e4..34390b89d 100644 --- a/2017/fractal-dimension/bengali/sentence_translations.json +++ b/2017/fractal-dimension/bengali/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "একটি রেখা এক মাত্রিক, একটি সমতল দ্বিমাত্রিক, আমরা যে স্থানটিতে বাস করি তা ত্রিমাত্রিক, ইত্যাদি।", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "এই সম্পর্কে চিন্তা করার একটি উপায় এখানে. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "স্কেল করা ডিস্ক স্পর্শ করা বাক্সের সংখ্যার তুলনায় যদি আপনি স্কেলিং ফ্যাক্টরটি প্লট করতে চান, আপনার ডেটা একটি নিখুঁত প্যারাবোলার সাথে খুব ঘনিষ্ঠভাবে ফিট করা উচিত, যেহেতু স্পর্শ করা বাক্সের সংখ্যাটি স্কেলিং ফ্যাক্টরের বর্গক্ষেত্রের সমানুপাতিক।", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "বৃহত্তর এবং বৃহত্তর স্কেলিং মানগুলির জন্য, যা আসলে একটি সূক্ষ্ম গ্রিডের দিকে তাকানোর সমতুল্য, সেই ডেটা সেই প্যারাবোলার সাথে আরও পুরোপুরি ফিট হতে চলেছে।", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "এখন ফ্র্যাক্টালগুলিতে ফিরে আসা যাক, আসুন এই গেমটি সিয়েরপিনস্কি ত্রিভুজ দিয়ে খেলি, কতগুলি বাক্স সেই আকারে স্পর্শকারী পয়েন্টগুলি গণনা করি।", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "বড় দ্বারা স্পর্শ করা বাক্সের অনুপাত এবং ছোটটি দ্বারা স্পর্শ করা বাক্সের সংখ্যা প্রায় তিনটি হওয়া উচিত।", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "সর্বোপরি, সেই বড় সংস্করণটি ছোট সংস্করণের তিনটি অনুলিপি দ্বারা নির্মিত।", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "আপনি এটিকে ফ্র্যাক্টালের মাত্রায় দুটি উত্থাপিত হিসাবেও ভাবতে পারেন, যা আমরা এইমাত্র দেখেছি প্রায় 1।585।", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "এবং তাই যদি আপনি যান এবং এই ক্ষেত্রে স্কেলিং ফ্যাক্টরটি সিয়েরপিনস্কি ত্রিভুজ দ্বারা স্পর্শ করা বাক্সের সংখ্যার বিপরীতে প্লট করেন, তথ্যটি একটি বক্ররেখার সাথে ঘনিষ্ঠভাবে মানানসই হবে যার আকার y সমান x শক্তি 1।585, কিছু সমানুপাতিক ধ্রুবক দ্বারা গুণিত।", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "যেমন, কল্পনা করুন আমি আপনাকে কিছু আকার দিই, এবং আপনি একজন বুদ্ধিমান প্রোগ্রামার, আপনি কীভাবে এই নম্বরটি খুঁজে পাবেন? ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "সুতরাং একটি সাধারণ কৌশল হল উভয় পক্ষের লগারিদম নেওয়া।", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "3D তে, যাইহোক, আপনি যখন একটি বাক্স-গণনা করেন, তখন আপনার কাছে ছোট স্কোয়ারের পরিবর্তে ছোট ছোট কিউব দিয়ে পূর্ণ একটি 3D গ্রিড থাকে, কিন্তু এটি একইভাবে কাজ করে।", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "এই স্কেলে, যেখানে আকৃতির পুরুত্ব বাক্সগুলির আকারের চেয়ে ছোট, সেখানে এটি এক-মাত্রিক দেখায়, যার অর্থ এটি যে বাক্সগুলি স্পর্শ করে তার দৈর্ঘ্যের সমানুপাতিক।", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "কিন্তু যখন আপনি এটিকে স্কেল করেন, তখন এটি একটি টিউবের মতো আচরণ করতে শুরু করে, সেই টিউবের পৃষ্ঠের বাক্সগুলিকে স্পর্শ করে, এবং তাই এটি দ্বি-মাত্রিক দেখাবে, স্পর্শ করা বাক্সের সংখ্যাটি বর্গের বর্গক্ষেত্রের সমানুপাতিক।", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "সুতরাং প্রকৃতপক্ষে এর মাত্রার জন্য একটি আকৃতিতে একটি সংখ্যা নির্ধারণ করা কঠিন হতে পারে এবং এটি বিভিন্ন সংজ্ঞা এবং ভিন্ন প্রথার জন্য জায়গা ছেড়ে দেয়।", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "তবে আরও প্রয়োগযোগ্য সেটিংয়ে, যেমন ব্রিটেনের উপকূলরেখার দিকে তাকানো, আপনি যত বেশি বেশি করে জুম করবেন তত সীমা সম্পর্কে কথা বলার অর্থ নেই, আমি বলতে চাচ্ছি যে কোনও সময়ে আপনি কেবল পরমাণুকে আঘাত করবেন।", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "পরিবর্তে আপনি যা করেন তা হল পর্যাপ্ত পরিমাণে বিস্তৃত স্কেলগুলি দেখুন, খুব জুম আউট থেকে খুব জুম ইন পর্যন্ত, এবং প্রতিটিতে মাত্রা গণনা করুন।", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "সম্পূর্ণরূপে স্ব-সদৃশ আকারগুলি ফ্র্যাক্টাল জ্যামিতিতে একটি গুরুত্বপূর্ণ ভূমিকা পালন করে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/chinese/sentence_translations.json b/2017/fractal-dimension/chinese/sentence_translations.json index f9fc97e87..6187a81d4 100644 --- a/2017/fractal-dimension/chinese/sentence_translations.json +++ b/2017/fractal-dimension/chinese/sentence_translations.json @@ -151,7 +151,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "一条线是一 维的，一个平面是二维的，我们居住的空间是三维的，等等 。", "model": "google_nmt", "n_reviews": 0, @@ -766,7 +766,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "这是思考这个问题的一种方法。", "model": "google_nmt", "from_community_srt": "我一点也没预料到这个值会如此缓慢地逼近4 有一种方法可以用来考虑这个 如果你关于放大倍数 作圆盘碰到的格子数量的函数图像的话 你的数据应该能很好地拟合成一条抛物线 因为被碰到的格子数 大约和放大倍数的平方成正比", @@ -775,7 +775,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "如果您要绘制缩放因子与缩 放圆盘接触的盒子数量的比较，则您的数 据应该非常接近完美的抛物线，因为接触的 盒子数量大致与缩放因子的平方成正比。", "model": "google_nmt", "from_community_srt": "当放大倍数越来越大 也就是网格越来越密的时候 数据点就能更好地拟合成抛物线 回到分形上 我们在谢尔宾斯基三角形上也做同样的事 数出有多少个格子碰上了它 那么你觉得在把三角形放大两倍之后", @@ -784,7 +784,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "对于越来越大的缩放值（实际上相当于查看更精细 的网格），该数据将更加完美地拟合该抛物线。", "model": "google_nmt", "from_community_srt": "被碰到格子数量会如何变化呢 嗯 大三角碰到的格子数除以小三角碰到的格子数 应该约为3 毕竟 大三角只不过是把3个小三角拼起来罢了 你也可以把它想做是2的分形维度次方 也就是我们刚刚看到的1.585次方", @@ -793,7 +793,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "现在回到分形，让我们用谢尔宾斯基三角形玩这 个游戏，计算有多少个盒子与该形状的接触点。", "model": "google_nmt", "n_reviews": 0, @@ -809,7 +809,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "那么，大的盒子接触到的盒子与小盒子接触的盒子数量的 比例应该是大约 3。", "model": "google_nmt", "from_community_srt": "那么 假如你关于这个情况中的放大倍数 作谢尔宾斯基三角形碰到的格子数的函数图像的话 你的数据就应该能很好地拟合成曲线y=x1.585 再乘以某个比例常数 但重要的是 我之所以讲这个 是因为对于非自相似 但有粗糙度的图形 我们也可以如法炮制", @@ -818,7 +818,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "毕竟，那个较大的版本 只是由较小版本的三个副本组成的。", "model": "google_nmt", "n_reviews": 0, @@ -826,7 +826,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "您也可以将其视为分形维数的 2，我 们刚刚看到分形维数约为 1。585. ", "model": "google_nmt", "n_reviews": 0, @@ -834,7 +834,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "因此，如果您要根据谢尔宾斯基三角形所接触的框的数量绘制本 例中的缩放因子，则数据将紧密拟合一条形状为 y 等于 x 1 次方的曲线。585，只是乘以一些比例常数。", "model": "google_nmt", "from_community_srt": "英国的海岸线就是一个经典的例子 假如你把海岸线画到平面上 数数碰上了多少个格子 再把它放大 接着重新数出碰到的格子个数 你会发现 海岸线碰到的格子数 大约正比于放大倍数的1.21次方", @@ -877,7 +877,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "假设我给你一些形状，而你是一位精明 的程序员，你会如何找到这个数字？", "model": "google_nmt", "n_reviews": 0, @@ -902,7 +902,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "所以一个常见的技巧就是两边取对数。", "model": "google_nmt", "n_reviews": 0, @@ -961,7 +961,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "很酷的一点是，这是一种定量的方式来表示它们是 粗糙的形状，并且即使放大，它们仍然保持粗糙。", "model": "google_nmt", "n_reviews": 0, @@ -994,7 +994,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "顺便说一句，在 3D 中，当您进行盒子计数时，您会得到一个充 满小立方体而不是小正方形的 3D 网格，但其工作方式相同。", "model": "google_nmt", "from_community_srt": "你就要用小立方体网格 而不是小正方形的了 不过方法还是一样的 当形状的粗细小于方块的尺寸时 它看似是一维的 换言之 碰到的方块数和长度成正比 但把它放大之后 它看上去就会更像一根管子", @@ -1003,7 +1003,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "在这个比例下，形状的厚度小于盒子的 大小，它看起来是一维的，这意味着它 接触的盒子的数量与其长度成正比。", "model": "google_nmt", "n_reviews": 0, @@ -1011,7 +1011,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "但是当你放大它时，它开始表现得更像一个管子， 接触管子表面上的盒子，所以它看起来是二维的， 接触的盒子数量与管子的平方成正比。比例因子。", "model": "google_nmt", "from_community_srt": "管子的表面会碰上方块 因此它看起来就是二维的 也就是说它碰上的方块数和放大倍数的平方成正比 但它并不是一根管子 而是由细丝绕成的一个致密的螺线圈 所以当你进一步放大 直到方块能捕捉到这些线条的细节时", @@ -1029,7 +1029,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "因此，实际上为形状的尺寸分配一个数字可能很棘手 ，并且它为不同的定义和不同的约定留下了空间。", "model": "google_nmt", "from_community_srt": "但它们都侧重于当图形被放得越来越大时 图形维度的极限 或者用之前的图像来说 你要看的就是当图像越来越靠右时 斜率的极限 因此 一个纯几何图形要是能称得上是分形的话 那么它即使被放大到无穷大 看上去仍然会同样粗糙", @@ -1062,7 +1062,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "但在更实际的环境中，比如看英国的海岸线，当 你放大得越来越大时，谈论极限并没有多大意 义，我的意思是在某些时候你只会撞击原子。", "model": "google_nmt", "from_community_srt": "不过在现实中 比如在考虑英国海岸线的时候 考虑放大的极限其实并没有什么道理 因为放大到一定程度之后 就变成挨个数原子了 这种情况下 你就需要考虑一个足够大的尺度范围 从缩得很小一直到放得很大", @@ -1071,7 +1071,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "相反，您要做的是查看足够广泛的尺度（从非常 缩小到非常放大），并计算每个尺度的尺寸。", "model": "google_nmt", "from_community_srt": "然后计算不同尺度下的维度即可 在这个更实际的情形中 要说一个形状是分形 那么测出来的维度就需要在许多不同的尺度下都大致恒定 例如 英国海岸线并不只有远看时才是1.21维 就算被放大了一千倍 它的粗糙程度也大约是1.21", @@ -1105,7 +1105,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "完全自相似的形状确实在分形几何中发挥着重要作用。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/french/sentence_translations.json b/2017/fractal-dimension/french/sentence_translations.json index 6a2add69a..5c828609c 100644 --- a/2017/fractal-dimension/french/sentence_translations.json +++ b/2017/fractal-dimension/french/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "Une ligne est unidimensionnelle, un plan est bidimensionnel, l’espace dans lequel nous vivons est tridimensionnel, etc. ", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "Voici une façon d’y penser. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "Si vous deviez tracer le facteur d'échelle par rapport au nombre de cases touchées par le disque mis à l'échelle, vos données devraient correspondre très étroitement à une parabole parfaite, car le nombre de cases touchées est à peu près proportionnel au carré du facteur d'échelle. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "Pour des valeurs d'échelle de plus en plus grandes, ce qui équivaut en fait à simplement regarder une grille plus fine, ces données s'adapteront plus parfaitement à cette parabole. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "Revenons maintenant aux fractales, jouons à ce jeu avec le triangle de Sierpinski, en comptant combien de cases touchent des points dans cette forme. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "Eh bien, la proportion de cases touchées par la grande par rapport au nombre de cases touchées par la petite devrait être d'environ trois. ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "Après tout, cette version plus grande est simplement constituée de trois copies de la version plus petite. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "Vous pourriez également considérer cela comme deux élevés à la dimension de la fractale, dont nous venons de voir qu'elle est d'environ 1.585. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "Et donc, si vous deviez tracer le facteur d'échelle dans ce cas en fonction du nombre de cases touchées par le triangle de Sierpinski, les données correspondraient étroitement à une courbe dont la forme est y égale x à la puissance 1.585, juste multiplié par une constante de proportionnalité. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "Comme dans, imaginez que je vous donne une forme et que vous êtes un programmeur averti, comment trouveriez-vous ce numéro ? ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "Une astuce courante consiste donc à prendre le logarithme des deux côtés. ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "Ce qui est cool, c'est que c'est une façon quantitative de dire que ce sont des formes qui sont grossières et qu'elles restent grossières même lorsque vous zoomez. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "En 3D, d'ailleurs, lorsque vous comptez des boîtes, vous avez une grille 3D pleine de petits cubes au lieu de petits carrés, mais cela fonctionne de la même manière. ", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "À cette échelle, où l'épaisseur de la forme est inférieure à la taille des cases, elle semble unidimensionnelle, ce qui signifie que le nombre de cases qu'elle touche est proportionnel à sa longueur. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "Mais lorsque vous l'agrandissez, il commence à se comporter beaucoup plus comme un tube, touchant les cases à la surface de ce tube, et il aura donc un aspect bidimensionnel, le nombre de cases touchées étant proportionnel au carré du tube. facteur d'échelle. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "Ainsi, attribuer un numéro à une forme pour sa dimension peut être délicat, et cela laisse place à des définitions et des conventions différentes. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "Mais dans un contexte plus appliqué, comme regarder le littoral britannique, cela n'a pas vraiment de sens de parler de limite lorsque vous zoomez de plus en plus, je veux dire qu'à un moment donné, vous ne feriez que toucher des atomes. ", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "Au lieu de cela, vous regardez une gamme suffisamment large d'échelles, de très zoomée à très zoomée, et calculez la dimension de chacune d'entre elles. ", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "Les formes parfaitement similaires jouent un rôle important dans la géométrie fractale. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/german/sentence_translations.json b/2017/fractal-dimension/german/sentence_translations.json index 3ad3c62f5..af49bbf26 100644 --- a/2017/fractal-dimension/german/sentence_translations.json +++ b/2017/fractal-dimension/german/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "Eine Linie ist eindimensional, eine Ebene ist zweidimensional, der Raum, in dem wir leben, ist dreidimensional und so weiter. ", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "Hier ist eine Möglichkeit, darüber nachzudenken. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "Wenn Sie den Skalierungsfaktor im Vergleich zur Anzahl der Kästchen, die die skalierte Scheibe berührt, grafisch darstellen, sollten Ihre Daten einer perfekten Parabel sehr genau entsprechen, da die Anzahl der berührten Kästchen ungefähr proportional zum Quadrat des Skalierungsfaktors ist. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "Bei immer größeren Skalierungswerten, was eigentlich der bloßen Betrachtung eines feineren Gitters entspricht, passen diese Daten perfekter zu dieser Parabel. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "Kommen wir nun zurück zu den Fraktalen. Spielen wir dieses Spiel mit dem Sierpinski-Dreieck und zählen wir, wie viele Kästchen sich berührende Punkte in dieser Form haben. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "Nun, das Verhältnis der vom Großen berührten Kisten zur Anzahl der vom Kleinen berührten Kisten sollte etwa drei betragen. ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "Schließlich besteht diese größere Version nur aus drei Kopien der kleineren Version. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "Man könnte sich das auch als zwei auf die Dimension des Fraktals angehoben vorstellen, die, wie wir gerade gesehen haben, etwa 1 beträgt. 585. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "Wenn Sie also den Skalierungsfaktor in diesem Fall gegen die Anzahl der vom Sierpinski-Dreieck berührten Kästchen grafisch darstellen würden, würden die Daten einer Kurve mit der Form y gleich x hoch 1 genau entsprechen. 585, einfach mit einer Proportionalitätskonstante multipliziert. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "Stellen Sie sich vor, ich gebe Ihnen eine Form, und Sie sind ein versierter Programmierer. Wie würden Sie diese Zahl finden? ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "Ein üblicher Trick besteht also darin, den Logarithmus beider Seiten zu bilden. ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "Das Coole daran ist, dass es eine quantitative Möglichkeit ist, auszudrücken, dass es sich um Formen handelt, die rau sind und dass sie rau bleiben, auch wenn man hineinzoomt. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "In 3D haben Sie übrigens beim Kästchenzählen ein 3D-Gitter voller kleiner Würfel statt kleiner Quadrate, aber es funktioniert genauso. ", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "In diesem Maßstab, in dem die Dicke der Form kleiner als die Größe der Kästchen ist, sieht sie eindimensional aus, was bedeutet, dass die Anzahl der Kästchen, die sie berührt, proportional zu ihrer Länge ist. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "Aber wenn man es vergrößert, verhält es sich viel mehr wie eine Röhre und berührt die Kästchen auf der Oberfläche dieser Röhre, so dass es zweidimensional aussieht, wobei die Anzahl der berührten Kästchen proportional zum Quadrat der Röhre ist Vergößerungsfaktor, Verkleinerungsfaktor. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "Daher kann es schwierig sein, einer Form tatsächlich eine Zahl für ihre Dimension zuzuordnen, und es lässt Raum für unterschiedliche Definitionen und unterschiedliche Konventionen. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "Aber in einer eher angewandten Umgebung, etwa wenn man sich die Küstenlinie Großbritanniens anschaut, macht es keinen Sinn, über die Grenze zu sprechen, wenn man immer weiter hineinzoomt, ich meine, irgendwann würde man nur noch auf Atome treffen. ", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "Stattdessen betrachten Sie einen ausreichend breiten Bereich von Maßstäben, von stark verkleinert bis sehr vergrößert, und berechnen die Dimension für jeden einzelnen. ", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "Perfekt selbstähnliche Formen spielen in der fraktalen Geometrie eine wichtige Rolle. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/hebrew/sentence_translations.json b/2017/fractal-dimension/hebrew/sentence_translations.json index 4ee1f7e45..19f9703dd 100644 --- a/2017/fractal-dimension/hebrew/sentence_translations.json +++ b/2017/fractal-dimension/hebrew/sentence_translations.json @@ -126,7 +126,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on.", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on.", "translatedText": "קו הוא חד מימדי, מישור הוא דו מימדי, המרחב בו אנו חיים הוא תלת מימדי וכן הלאה.", "n_reviews": 0, "start": 154.08, @@ -644,21 +644,21 @@ "end": 690.32 }, { - "input": "Here's one way to think about this.", + "input": "Here's one example. For larger and larger scaling values, which is actually", "translatedText": "הנה דרך אחת לחשוב על זה.", "n_reviews": 0, "start": 696.48, "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor.", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the", "translatedText": "אם היית מתווה את גורם קנה המידה בהשוואה למספר הקופסאות שהדיסק המוקטן נוגע בהן, הנתונים שלך צריכים להתאים מאוד לפרבולה מושלמת, מכיוון שמספר התיבות שנגעת בהן הוא פרופורציונלי בערך לריבוע של גורם קנה המידה.", "n_reviews": 0, "start": 713.64, "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola.", + "input": "triangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve", "translatedText": "עבור ערכי קנה מידה גדולים יותר ויותר, שהם למעשה מקבילים רק להסתכלות על רשת עדינה יותר, הנתונים האלה הולכים להתאים בצורה מושלמת יותר לפרבולה הזו.", "n_reviews": 0, "start": 733.28, @@ -672,7 +672,7 @@ "end": 750.54 }, { - "input": "How would you imagine that number compares to scaling up the triangle by a factor of two and counting the new number of boxes touched?", + "input": "version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. And so if you were to go and plot the scaling factor in this case against the number of boxes touched", "translatedText": "איך הייתם מתארים לעצמכם את המספר הזה בהשוואה להגדלת המשולש בפקטור של שניים וספירת המספר החדש של תיבות שנגעו בהן?", "n_reviews": 0, "start": 750.54, @@ -721,7 +721,7 @@ "end": 801.04 }, { - "input": "If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21.", + "input": "pproximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that number empirically. As in, imagine I give you some shape, and you're a savvy programmer. How would you find this number? So what I'm saying here is that if you scale this shape by some factor, which I'll call S, the number of boxe", "translatedText": "אם תזרקו את קו החוף הזה לתוך המטוס ותספור כמה קופסאות נוגעות בו, ואז מדרגים אותו בכמות מסוימת, ותספור כמה קופסאות נוגעות באותה גרסה מוקטנת קנה מידה חדשה, מה שתגלה הוא שמספר הקופסאות שנוגעות ב- קו החוף גדל בערך ביחס לגורם קנה המידה המועלה בחזקת 1.21.", "n_reviews": 0, "start": 801.04, @@ -742,7 +742,7 @@ "end": 832.22 }, { - "input": "So what I'm saying here is that if you scale this shape by some factor, which I'll call S, the number of boxes touching that shape should equal some constant multiplied by that scaling factor raised to whatever the dimension is, the value we're looking for.", + "input": "ied by that scaling factor raised to whatever the dimension is, the value that we're looking for. Now, if you have some data plot that closely fits a curve that looks like the input raised to some power, it can be hard to see exactly what that power should be. So a common trick is to take the logarithm of both sides. That way, the dimension is going to d", "translatedText": "אז מה שאני אומר כאן הוא שאם אתה משנה את הצורה הזו לפי גורם כלשהו, שאקרא לו S, מספר התיבות שנוגעות בצורה הזו אמור להיות שווה לאיזשהו קבוע כפול גורם קנה המידה הזה שהועלה לכל הממד, הערך אנחנו מחפשים.", "n_reviews": 0, "start": 832.22, @@ -756,7 +756,7 @@ "end": 859.82 }, { - "input": "So a common trick is to take the logarithm of both sides.", + "input": "lot the log of the scaling factor against the log of the number of boxes touching the coastline, the relationship should look like a l", "translatedText": "אז טריק נפוץ הוא לקחת את הלוגריתם של שני הצדדים.", "n_reviews": 0, "start": 859.82, @@ -770,14 +770,14 @@ "end": 871.36 }, { - "input": "What this suggests is that if you were to plot the log of the scaling factor against the log of the number of boxes touching the coastline, the relationship should look like a line, and that line should have a slope equal to the dimension.", + "input": "So what that means is that if you tried out a whole bunch of scaling factors, counted the number of boxes touching the coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regression to find the best fit line to your data set, and when you look at the slope of th", "translatedText": "מה שזה מרמז הוא שאם היית מתווה את היומן של גורם קנה המידה מול היומן של מספר הקופסאות הנוגעות בקו החוף, הקשר צריך להיראות כמו קו, ולקו הזה צריך להיות שיפוע שווה לממד.", "n_reviews": 0, "start": 873.44, "end": 889.52 }, { - "input": "So what that means is that if you tried out a whole bunch of scaling factors, counted the number of boxes touching the coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regression to find the best fit line to your dataset, and when you look at the slope of that line, that tells you the empirical measurement for the dimension of what you're examining.", + "input": "at line, that tells you the empirical measurement for the dimension of what you're examining. I just think that makes this idea of fractal dimension so much more real and visceral compared to abstract, artificially perfect shapes. And once you're comfortable thinking about dimension like this, you, my friend, have become ready to hear the definition of a fractal. Essentially, fractals are shapes whose dimension is not an integer, but instead some fractional amount. What's cool", "translatedText": "אז מה שזה אומר הוא שאם תנסה חבורה שלמה של גורמי קנה מידה, ספרת את מספר הקופסאות שנוגעות בחוף בכל רגע, ואז משרטטת את הנקודות על חלקת היומן, תוכל לבצע איזושהי רגרסיה ליניארית כדי למצוא את הקו המתאים ביותר למערך הנתונים שלך, וכשאתה מסתכל על השיפוע של הקו הזה, זה אומר לך את המדידה האמפירית עבור הממד של מה שאתה בוחן.", "n_reviews": 0, "start": 889.52, @@ -805,14 +805,14 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in.", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance th", "translatedText": "מה שמגניב בזה הוא שזו דרך כמותית לומר שהן צורות מחוספסות, ושהן נשארות מחוספסות גם כשאתה מתקרב.", "n_reviews": 0, "start": 928.94, "end": 939.46 }, { - "input": "Technically there's a slightly more accurate definition, and I've included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for.", + "input": "ough that I haven't brought up yet, but it's worth pointing out, which is that this dimension, at least as I've described it so far using the box counting method, can sometimes change based on how far zoomed in you are. For example, here's a shape sitting in thr", "translatedText": "מבחינה טכנית יש הגדרה קצת יותר מדויקת, וכללתי אותה בתיאור הסרטון, אבל הרעיון הזה כאן של מימד לא שלם לוכד כמעט לחלוטין את רעיון החספוס שאליו אנחנו הולכים.", "n_reviews": 0, "start": 939.46, diff --git a/2017/fractal-dimension/hindi/sentence_translations.json b/2017/fractal-dimension/hindi/sentence_translations.json index 8e81ff512..705e27af6 100644 --- a/2017/fractal-dimension/hindi/sentence_translations.json +++ b/2017/fractal-dimension/hindi/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "एक रेखा एक आयामी है, एक विमान दो आयामी है, जिस स्थान पर हम रहते हैं वह तीन आयामी है, इत्यादि।", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "इस बारे में सोचने का एक तरीका यहां दिया गया है।", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "यदि आपको स्केलिंग डिस्क को छूने वाले बक्सों की संख्या की तुलना में स्केलिंग कारक को प्लॉट करना था, तो आपका डेटा एक पूर्ण परवलय में बहुत बारीकी से फिट होना चाहिए, क्योंकि छुए गए बक्सों की संख्या स्केलिंग कारक के वर्ग के लगभग आनुपातिक है।", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "बड़े और बड़े स्केलिंग मानों के लिए, जो वास्तव में एक बेहतर ग्रिड को देखने के बराबर है, वह डेटा उस परवलय में अधिक सटीक रूप से फिट होगा।", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "अब फ्रैक्टल्स पर वापस आते हैं, आइए इस गेम को सिएरपिंस्की त्रिकोण के साथ खेलें, गिनती करें कि उस आकार में कितने बक्से स्पर्श बिंदु हैं।", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "वैसे बड़े व्यक्ति द्वारा छुए गए बक्सों की संख्या और छोटे वाले द्वारा छुए गए बक्सों की संख्या का अनुपात लगभग तीन होना चाहिए।", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "आख़िरकार, वह बड़ा संस्करण छोटे संस्करण की तीन प्रतियों से बना है।", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "आप इसे फ्रैक्टल के आयाम तक उठाए गए दो के रूप में भी सोच सकते हैं, जिसे हमने अभी लगभग 1 देखा है।585. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "और इसलिए यदि आपको इस मामले में सिएरपिंस्की त्रिकोण द्वारा छूए गए बक्सों की संख्या के विरुद्ध स्केलिंग कारक को प्लॉट करना था, तो डेटा y के आकार के साथ एक वक्र में बारीकी से फिट होगा, जो कि x की शक्ति 1 के बराबर है।585, बस कुछ आनुपातिकता स्थिरांक से गुणा किया गया।", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "जैसे, कल्पना कीजिए कि मैं आपको कुछ आकार देता हूं, और आप एक समझदार प्रोग्रामर हैं, तो आप यह संख्या कैसे खोजेंगे? ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "इसलिए एक सामान्य युक्ति दोनों पक्षों का लघुगणक लेना है।", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "इसमें अच्छी बात यह है कि यह यह कहने का एक मात्रात्मक तरीका है कि वे ऐसी आकृतियाँ हैं जो खुरदरी हैं, और जब आप ज़ूम इन करते हैं तब भी वे खुरदरी रहती हैं।", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "3डी में, वैसे, जब आप बॉक्स-गिनती करते हैं, तो आपके पास छोटे वर्गों के बजाय छोटे क्यूब्स से भरा 3डी ग्रिड होता है, लेकिन यह उसी तरह काम करता है।", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "इस पैमाने पर, जहां आकृति की मोटाई बक्सों के आकार से छोटी होती है, यह एक-आयामी दिखती है, जिसका अर्थ है कि यह जितने बक्सों को छूता है वह इसकी लंबाई के समानुपाती होती है।", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "लेकिन जब आप इसे बढ़ाते हैं, तो यह एक ट्यूब की तरह व्यवहार करना शुरू कर देता है, उस ट्यूब की सतह पर बक्सों को छूता है, और इसलिए यह द्वि-आयामी दिखाई देगा, जिसमें छुए गए बक्सों की संख्या इसके वर्ग के समानुपाती होती है।मापन कारक।", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "तो वास्तव में किसी आकृति को उसके आयाम के लिए एक संख्या निर्दिष्ट करना मुश्किल हो सकता है, और यह अलग-अलग परिभाषाओं और अलग-अलग परंपराओं के लिए जगह छोड़ देता है।", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "लेकिन अधिक व्यावहारिक सेटिंग में, जैसे कि ब्रिटेन के समुद्र तट को देखना, वास्तव में सीमा के बारे में बात करने का कोई मतलब नहीं है क्योंकि आप अधिक से अधिक ज़ूम करते हैं, मेरा मतलब है कि किसी बिंदु पर आप बस परमाणुओं से टकरा रहे होंगे।", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "इसके बजाय आप जो करते हैं वह स्केल की पर्याप्त विस्तृत श्रृंखला को देखते हैं, बहुत ज़ूम आउट से लेकर बहुत ज़ूम इन तक, और प्रत्येक पर आयाम की गणना करते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "बिल्कुल स्व-समान आकृतियाँ फ्रैक्टल ज्यामिति में एक महत्वपूर्ण भूमिका निभाती हैं।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/indonesian/sentence_translations.json b/2017/fractal-dimension/indonesian/sentence_translations.json index 45614a6fe..5f1650ebd 100644 --- a/2017/fractal-dimension/indonesian/sentence_translations.json +++ b/2017/fractal-dimension/indonesian/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on.", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on.", "translatedText": "Garis adalah satu dimensi, bidang adalah dua dimensi, ruang yang kita tinggali adalah tiga dimensi, dan seterusnya.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this.", + "input": "Here's one example. For larger and larger scaling values, which is actually", "translatedText": "Inilah salah satu cara untuk memikirkan hal ini.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor.", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the", "translatedText": "Jika Anda memplot faktor penskalaan dibandingkan dengan jumlah kotak yang disentuh oleh disk berskala, data Anda harus sangat sesuai dengan parabola sempurna, karena jumlah kotak yang disentuh kira-kira sebanding dengan kuadrat faktor penskalaan.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola.", + "input": "triangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve", "translatedText": "Untuk nilai skala yang semakin besar, yang sebenarnya setara dengan hanya melihat grid yang lebih halus, data tersebut akan lebih sesuai dengan parabola tersebut.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 750.54 }, { - "input": "How would you imagine that number compares to scaling up the triangle by a factor of two and counting the new number of boxes touched?", + "input": "version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. And so if you were to go and plot the scaling factor in this case against the number of boxes touched", "translatedText": "Bagaimana Anda membayangkan angka tersebut dibandingkan dengan memperbesar segitiga sebanyak dua kali lipat dan menghitung jumlah kotak baru yang disentuh?", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 801.04 }, { - "input": "If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21.", + "input": "pproximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that number empirically. As in, imagine I give you some shape, and you're a savvy programmer. How would you find this number? So what I'm saying here is that if you scale this shape by some factor, which I'll call S, the number of boxe", "translatedText": "Jika Anda memasukkan garis pantai tersebut ke dalam bidang dan menghitung berapa banyak kotak yang menyentuhnya, lalu menskalakannya dengan jumlah tertentu, dan menghitung berapa banyak kotak yang menyentuh versi skala baru tersebut, yang akan Anda temukan adalah jumlah kotak yang menyentuh garis pantai tersebut. garis pantai meningkat kira-kira sebanding dengan faktor skala yang dipangkatkan 1.21.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 832.22 }, { - "input": "So what I'm saying here is that if you scale this shape by some factor, which I'll call S, the number of boxes touching that shape should equal some constant multiplied by that scaling factor raised to whatever the dimension is, the value we're looking for.", + "input": "ied by that scaling factor raised to whatever the dimension is, the value that we're looking for. Now, if you have some data plot that closely fits a curve that looks like the input raised to some power, it can be hard to see exactly what that power should be. So a common trick is to take the logarithm of both sides. That way, the dimension is going to d", "translatedText": "Jadi maksud saya di sini adalah jika Anda menskalakan bentuk ini dengan beberapa faktor, yang saya sebut S, jumlah kotak yang menyentuh bentuk itu harus sama dengan konstanta dikalikan dengan faktor penskalaan yang dipangkatkan ke dimensi berapa pun, nilainya kami sedang mencari.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 859.82 }, { - "input": "So a common trick is to take the logarithm of both sides.", + "input": "lot the log of the scaling factor against the log of the number of boxes touching the coastline, the relationship should look like a l", "translatedText": "Jadi trik yang umum adalah dengan mengambil logaritma kedua sisi.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 871.36 }, { - "input": "What this suggests is that if you were to plot the log of the scaling factor against the log of the number of boxes touching the coastline, the relationship should look like a line, and that line should have a slope equal to the dimension.", + "input": "So what that means is that if you tried out a whole bunch of scaling factors, counted the number of boxes touching the coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regression to find the best fit line to your data set, and when you look at the slope of th", "translatedText": "Hal ini menunjukkan bahwa jika Anda memplot log faktor penskalaan terhadap log jumlah kotak yang menyentuh garis pantai, hubungannya akan terlihat seperti sebuah garis, dan garis tersebut harus memiliki kemiringan yang sama dengan dimensinya.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 889.52 }, { - "input": "So what that means is that if you tried out a whole bunch of scaling factors, counted the number of boxes touching the coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regression to find the best fit line to your dataset, and when you look at the slope of that line, that tells you the empirical measurement for the dimension of what you're examining.", + "input": "at line, that tells you the empirical measurement for the dimension of what you're examining. I just think that makes this idea of fractal dimension so much more real and visceral compared to abstract, artificially perfect shapes. And once you're comfortable thinking about dimension like this, you, my friend, have become ready to hear the definition of a fractal. Essentially, fractals are shapes whose dimension is not an integer, but instead some fractional amount. What's cool", "translatedText": "Artinya, jika Anda mencoba sejumlah faktor penskalaan, menghitung jumlah kotak yang menyentuh pantai setiap saat, lalu memplot titik-titik tersebut pada plot log-log, Anda kemudian dapat melakukan semacam regresi linier untuk menemukan garis yang paling sesuai dengan kumpulan data Anda, dan saat Anda melihat kemiringan garis tersebut, hal itu memberi tahu Anda pengukuran empiris untuk dimensi yang Anda periksa.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in.", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance th", "translatedText": "Yang menarik dari hal ini adalah bahwa ini adalah cara kuantitatif untuk mengatakan bahwa bentuk tersebut kasar, dan tetap kasar meskipun Anda memperbesarnya.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 939.46 }, { - "input": "Technically there's a slightly more accurate definition, and I've included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for.", + "input": "ough that I haven't brought up yet, but it's worth pointing out, which is that this dimension, at least as I've described it so far using the box counting method, can sometimes change based on how far zoomed in you are. For example, here's a shape sitting in thr", "translatedText": "Secara teknis ada definisi yang sedikit lebih akurat, dan saya telah memasukkannya ke dalam deskripsi video, tetapi gagasan tentang dimensi non-integer ini hampir seluruhnya menangkap gagasan tentang kekasaran yang ingin kita capai.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/japanese/sentence_translations.json b/2017/fractal-dimension/japanese/sentence_translations.json index e163aea05..6666ba86d 100644 --- a/2017/fractal-dimension/japanese/sentence_translations.json +++ b/2017/fractal-dimension/japanese/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "線は 1 次 元、平面は 2 次元、私たちが住んでいる空間は 3 次元などです 。", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "これについては 1 つの考え方があります。", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "スケーリングされたディスクが接触する ボックスの数と比較してスケーリング係数をプロットすると、 接触したボックスの数はスケーリング係数の 2 乗にほぼ比例 するため、データは完全な放物線に非常に近くなるはずです。", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "スケーリング値がどんどん大きくなるにつれて、実際にはより細かいグリッド を見ることと同じになり、そのデータはその放物線により完全に適合します。", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "フラクタルの話に戻って、シェルピンスキー三角形を使ってこのゲームをプ レイして、その形状の点に接触しているボックスの数を数えてみましょう。", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "そうですね、小さいボックスがタッチしたボックスの数に対する、大きいボックスがタッチしたボッ クスの割合は約 3 になるはずです。", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "結局のところ、その大きなバージョンは、 小さなバージョンの 3 つのコピーから構築されているだけです。", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "これを 2 をフラクタルの次元に引き上げたものと考える こともできます。先ほど見たのは約 1 です。585. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "したがって、シェルピンスキー三角形が接触するボックスの数に対してこの場合のスケー リング係数をプロットすると、データは、y の形状が x の 1 乗に等しい曲線に 厳密に近似することになります。585、何らかの比例定数を掛けただけです。", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "たとえば、私があなたに何らかの形を与えたと想像してください。あなた は精通したプログラマーで、この数値をどうやって見つけるでしょうか? ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "したがって、一般的なトリックは、両辺の対数を取ることです。", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "これのすばらしい点は、形状が粗いこと、およびズームインして も粗いままであることを定量的に示すことができることです。", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "ちなみに、3D では、箱を数えるとき、小さな正方形ではなく小さな立方 体でいっぱいの 3D グリッドが表示されますが、仕組みは同じです。", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "このスケールでは、シェイプの厚さがボックスのサイズより 小さいため、シェイプは 1 次元に見えます。つまり、シ ェイプが接触するボックスの数はその長さに比例します。", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "しかし、スケールアップすると、よりチューブのように動作し始め、その チューブの表面上のボックスに触れるので、触れたボックスの数が 2 乗に比例して 2 次元に見えるようになります。スケーリング係数。", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "そのため、実際に形状に寸法の数値を割り当てるのは難しい場合 があり、異なる定義や異なる規則が存在する余地が残ります。", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "しかし、イギリスの海岸線を見るような、より応用的な設定では、ズ ームインを進めていくと限界について話すのはあまり意味がありま せん。つまり、ある時点では原子にぶつかることになるだけです。", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "代わりに、非常にズームアウトしたものから非常にズームインしたものまで、 十分に広い範囲のスケールを調べ、それぞれのスケールで寸法を計算します。", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "完全に自己相似な形状は、フラクタル幾何学において重要な役割を果たします。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/marathi/sentence_translations.json b/2017/fractal-dimension/marathi/sentence_translations.json index 2964e0508..6821c8318 100644 --- a/2017/fractal-dimension/marathi/sentence_translations.json +++ b/2017/fractal-dimension/marathi/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "एक रेषा एक मितीय आहे, विमान द्विमितीय आहे, आपण ज्या जागेत राहतो ते त्रिमितीय आहे, इत्यादी. ", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "यावर विचार करण्याचा एक मार्ग येथे आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "स्केल केलेल्या डिस्कने स्पर्श केलेल्या बॉक्सच्या संख्येच्या तुलनेत तुम्ही स्केलिंग फॅक्टर प्लॉट कराल, तर तुमचा डेटा परिपूर्ण पॅराबोलाशी अगदी जवळून बसला पाहिजे, कारण स्पर्श केलेल्या बॉक्सची संख्या स्केलिंग घटकाच्या चौरसाच्या प्रमाणात असते. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "मोठ्या आणि मोठ्या स्केलिंग मूल्यांसाठी, जे प्रत्यक्षात फक्त एक बारीक ग्रिड पाहण्यासारखे आहे, तो डेटा त्या पॅराबोलामध्ये अधिक अचूकपणे फिट होणार आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "आता फ्रॅक्टल्सकडे परत जाताना, सिएरपिन्स्की त्रिकोणासह हा गेम खेळू या, त्या आकारात किती बॉक्स टचिंग पॉइंट आहेत ते मोजू. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "बरं, मोठ्याने स्पर्श केलेल्या बॉक्सचे प्रमाण आणि लहान बॉक्सने स्पर्श केलेल्या बॉक्सचे प्रमाण सुमारे तीन असावे. ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "शेवटी, ती मोठी आवृत्ती फक्त लहान आवृत्तीच्या तीन प्रतींनी बनलेली आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "तुम्ही याचा विचार करू शकता की फ्रॅक्टलच्या परिमाणापर्यंत दोन वाढवले आहेत, जे आम्ही आत्ताच पाहिले आहे सुमारे 1.५८५. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "आणि म्हणून जर तुम्ही या केसमध्ये सिएरपिन्स्की त्रिकोणाने स्पर्श केलेल्या बॉक्सच्या संख्येच्या विरूद्ध स्केलिंग फॅक्टर प्लॉट कराल, तर डेटा 1 च्या पॉवर 1 च्या x बरोबर y च्या आकारासह वक्र फिट होईल. 585, फक्त काही आनुपातिक स्थिरांकाने गुणाकार केला. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "जसे की, कल्पना करा मी तुम्हाला काही आकार देतो, आणि तुम्ही जाणकार प्रोग्रामर आहात, तुम्हाला हा नंबर कसा मिळेल? ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "त्यामुळे दोन्ही बाजूंचे लॉगरिदम घेणे ही एक सामान्य युक्ती आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "त्याबद्दल काय छान आहे ते असे म्हणण्याचा एक परिमाणात्मक मार्ग आहे की ते आकार खडबडीत आहेत आणि तुम्ही झूम वाढवले तरीही ते खडबडीत राहतात. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "3D मध्ये, तसे, जेव्हा तुम्ही बॉक्स-काउंटिंग करता, तेव्हा तुमच्याकडे लहान चौरसांऐवजी लहान चौकोनी तुकड्यांनी भरलेला 3D ग्रिड असतो, परंतु ते त्याच प्रकारे कार्य करते. ", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "या स्केलवर, जिथे आकाराची जाडी बॉक्सच्या आकारापेक्षा लहान असते, तिथे ती एक-आयामी दिसते, म्हणजे तो स्पर्श केलेल्या बॉक्सची संख्या त्याच्या लांबीच्या प्रमाणात असते. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "परंतु जेव्हा तुम्ही त्याचे प्रमाण वाढवता तेव्हा ते त्या नळीच्या पृष्ठभागावर असलेल्या बॉक्सला स्पर्श करून, ट्यूबसारखे वागू लागते आणि त्यामुळे ते द्विमितीय दिसेल, स्पर्श केलेल्या बॉक्सच्या संख्येच्या चौरसाच्या प्रमाणात. स्केलिंग घटक. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "त्यामुळे प्रत्यक्षात आकारासाठी संख्या निश्चित करणे अवघड असू शकते आणि ते भिन्न व्याख्या आणि भिन्न परंपरांसाठी जागा सोडते. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "परंतु अधिक लागू केलेल्या सेटिंगमध्ये, जसे की ब्रिटनच्या किनारपट्टीकडे पाहणे, आपण अधिकाधिक झूम वाढवत असताना मर्यादेबद्दल बोलण्यात खरोखर अर्थ नाही, मला असे म्हणायचे आहे की आपण कधीतरी अणूंना मारत असाल. ", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "त्याऐवजी तुम्ही काय करता ते स्केलची पुरेशी विस्तृत श्रेणी पहा, अगदी झूम आउटपासून ते खूप झूम इन पर्यंत, आणि प्रत्येकाच्या परिमाणाची गणना करा. ", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "फ्रॅक्टल भूमितीमध्ये पूर्णतः स्व-समान आकार महत्त्वाची भूमिका बजावतात. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/persian/sentence_translations.json b/2017/fractal-dimension/persian/sentence_translations.json index d3128be9e..735654299 100644 --- a/2017/fractal-dimension/persian/sentence_translations.json +++ b/2017/fractal-dimension/persian/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "بعد چیزی است که معمولاً فقط برای اعداد طبیعی معنا دارد، درست است؟ یک خط یک بعدی است، یک هواپیما دو بعدی است، فضایی که ما در آن زندگی می کنیم سه بعدی است و غیره. ", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "در اینجا یک راه برای فکر کردن در مورد این وجود دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "اگر بخواهید ضریب مقیاس را در مقایسه با تعداد جعبه هایی که دیسک مقیاس شده لمس می کند ترسیم کنید، داده های شما باید خیلی نزدیک با یک سهمی کامل مطابقت داشته باشد، زیرا تعداد جعبه های لمس شده تقریباً با مربع ضریب مقیاس بندی متناسب است. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "برای مقادیر مقیاس‌بندی بزرگ‌تر و بزرگ‌تر، که در واقع معادل نگاه کردن به یک شبکه ظریف‌تر است، این داده‌ها کاملاً با آن سهمی مطابقت دارند. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "اکنون که به فراکتال ها برمی گردیم، بیایید این بازی را با مثلث Sierpinski انجام دهیم و شمارش کنیم که چند جعبه در آن شکل نقاط تماس دارند. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "از این گذشته، آن نسخه بزرگتر فقط از سه نسخه از نسخه کوچکتر ساخته شده است. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "شما همچنین می توانید این را به عنوان دو افزایش یافته تا بعد فراکتال در نظر بگیرید، که اکنون دیدیم که حدود 1 است. 585. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "بنابراین، اگر بخواهید ضریب مقیاس را در این مورد بر اساس تعداد جعبه های لمس شده توسط مثلث Sierpinski رسم کنید، داده ها به منحنی با شکل y برابر با x به توان 1 نزدیک می شوند. 585، فقط در مقداری ثابت تناسب ضرب شده است. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "بنابراین یک ترفند رایج این است که لگاریتم هر دو طرف را بگیرید. ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "نکته جالب در مورد آن این است که روشی کمی برای گفتن این است که آنها اشکالی هستند که خشن هستند، و حتی وقتی بزرگنمایی می کنید، خشن باقی می مانند. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "اتفاقاً در سه بعدی، وقتی جعبه شماری می کنید، به جای مربع های کوچک، یک شبکه سه بعدی پر از مکعب های کوچک دارید، اما به همین صورت عمل می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "در این مقیاس، جایی که ضخامت شکل کوچکتر از اندازه جعبه ها است، یک بعدی به نظر می رسد، به این معنی که تعداد جعبه هایی که لمس می کند با طول آن متناسب است. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "اما هنگامی که آن را بزرگتر می کنید، بسیار شبیه به یک لوله رفتار می کند، جعبه های روی سطح آن لوله را لمس می کند، و بنابراین دو بعدی به نظر می رسد، به طوری که تعداد جعبه های لمس شده متناسب با مربع است. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "بنابراین در واقع تخصیص یک عدد به یک شکل برای ابعاد آن می تواند مشکل باشد، و جایی برای تعاریف متفاوت و قراردادهای متفاوت باقی می گذارد. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "اما در یک محیط کاربردی تر، مانند نگاه کردن به خط ساحلی بریتانیا، واقعاً منطقی نیست که با بزرگنمایی بیشتر و بیشتر در مورد محدودیت صحبت کنیم، منظورم این است که در مقطعی فقط به اتم ها برخورد می کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "در عوض کاری که شما انجام می دهید این است که به طیف وسیعی از مقیاس ها، از بزرگنمایی بسیار کوچک تا بزرگنمایی بسیار، نگاه کنید و بعد را در هر یک محاسبه کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "اشکال کاملاً خود مشابه نقش مهمی در هندسه فراکتال ایفا می کنند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/portuguese/sentence_translations.json b/2017/fractal-dimension/portuguese/sentence_translations.json index 52e4c81e6..be1a7b38b 100644 --- a/2017/fractal-dimension/portuguese/sentence_translations.json +++ b/2017/fractal-dimension/portuguese/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "Uma linha é unidimensional, um plano é bidimensional, o espaço em que vivemos é tridimensional e assim por diante. ", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "Aqui está uma maneira de pensar sobre isso. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "Se você plotasse o fator de escala em comparação com o número de caixas que o disco dimensionado toca, seus dados deveriam se ajustar perfeitamente a uma parábola perfeita, já que o número de caixas tocadas é aproximadamente proporcional ao quadrado do fator de escala. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "Para valores de escala cada vez maiores, o que na verdade equivale a apenas olhar para uma grade mais fina, esses dados se ajustarão mais perfeitamente a essa parábola. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "Agora voltando aos fractais, vamos jogar este jogo com o triângulo de Sierpinski, contando quantas caixas estão tocando pontos nessa forma. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "Bem, a proporção de caixas tocadas pela grande em relação ao número de caixas tocadas pela pequena deve ser cerca de três. ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "Afinal, essa versão maior é composta apenas de três cópias da versão menor. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "Você também pode pensar nisso como dois elevados à dimensão do fractal, que acabamos de ver é cerca de 1.585. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "E então, se você representasse graficamente o fator de escala neste caso em relação ao número de caixas tocadas pelo triângulo de Sierpinski, os dados se ajustariam perfeitamente a uma curva com a forma de y igual a x elevado à potência 1.585, apenas multiplicado por alguma constante de proporcionalidade. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "Tipo, imagine que eu lhe dê uma forma e você seja um programador experiente, como encontraria esse número? ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "Portanto, um truque comum é calcular o logaritmo de ambos os lados. ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "O que é legal nisso é que é uma maneira quantitativa de dizer que são formas ásperas e que permanecem ásperas mesmo quando você aumenta o zoom. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "A propósito, em 3D, quando você faz uma contagem de caixas, você tem uma grade 3D cheia de cubinhos em vez de quadradinhos, mas funciona da mesma maneira. ", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "Nessa escala, onde a espessura da forma é menor que o tamanho das caixas, ela parece unidimensional, ou seja, o número de caixas que toca é proporcional ao seu comprimento. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "Mas quando você aumenta a escala, ele começa a se comportar muito mais como um tubo, tocando as caixas na superfície desse tubo, e assim parecerá bidimensional, com o número de caixas tocadas sendo proporcional ao quadrado do fator de escala. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "Portanto, atribuir um número a uma forma para sua dimensão pode ser complicado e deixa espaço para definições e convenções diferentes. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "Mas num cenário mais aplicado, como olhar para a costa da Grã-Bretanha, não faz realmente sentido falar sobre o limite à medida que aumentamos cada vez mais o zoom, quero dizer, em algum momento estaríamos apenas a atingir átomos. ", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "Em vez disso, o que você faz é observar uma gama suficientemente ampla de escalas, desde muito reduzidas até muito ampliadas, e calcular a dimensão de cada uma delas. ", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "Formas perfeitamente semelhantes desempenham um papel importante na geometria fractal. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/tamil/sentence_translations.json b/2017/fractal-dimension/tamil/sentence_translations.json index 6e4641de4..5e4d22c03 100644 --- a/2017/fractal-dimension/tamil/sentence_translations.json +++ b/2017/fractal-dimension/tamil/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "ஒரு கோடு ஒரு பரிமாணம், ஒரு விமானம் இரு பரிமாணம், நாம் வாழும் இடம் முப்பரிமாணம், மற்றும் பல. ", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "இதைப் பற்றி சிந்திக்க ஒரு வழி இங்கே. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "அளவிடப்பட்ட வட்டு தொடும் பெட்டிகளின் எண்ணிக்கையுடன் ஒப்பிடும்போது அளவிடுதல் காரணியை நீங்கள் திட்டமிடினால், உங்கள் தரவு ஒரு சரியான பரவளையத்துடன் மிகவும் நெருக்கமாக பொருந்த வேண்டும், ஏனெனில் தொடப்பட்ட பெட்டிகளின் எண்ணிக்கை அளவிடுதல் காரணியின் வர்க்கத்திற்கு தோராயமாக விகிதாசாரமாக இருக்கும். ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "பெரிய மற்றும் பெரிய அளவிடுதல் மதிப்புகளுக்கு, இது உண்மையில் ஒரு நுண்ணிய கட்டத்தைப் பார்ப்பதற்குச் சமமானதாக இருக்கும், அந்த தரவு பரவளையத்தை மிகவும் சரியாகப் பொருத்தும். ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "இப்போது ஃப்ராக்டல்களுக்கு வருகிறேன், சியர்பின்ஸ்கி முக்கோணத்துடன் இந்த விளையாட்டை விளையாடுவோம், அந்த வடிவத்தில் எத்தனை பெட்டிகள் தொட்டுப் புள்ளிகள் உள்ளன என்பதைக் கணக்கிடுவோம். ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "பெரியவர் தொட்ட பெட்டிகளின் விகிதம் சிறியது தொட்ட பெட்டிகளின் எண்ணிக்கை சுமார் மூன்றாக இருக்க வேண்டும். ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "எல்லாவற்றிற்கும் மேலாக, அந்த பெரிய பதிப்பு சிறிய பதிப்பின் மூன்று நகல்களால் கட்டமைக்கப்பட்டுள்ளது. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "இதைப் பிரிவின் பரிமாணத்திற்கு உயர்த்தப்பட்ட இரண்டாகவும் நீங்கள் நினைக்கலாம், நாங்கள் இப்போது பார்த்தது சுமார் 1.585. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "எனவே, சியர்பின்ஸ்கி முக்கோணத்தால் தொடப்பட்ட பெட்டிகளின் எண்ணிக்கைக்கு எதிராக இந்த வழக்கில் அளவிடும் காரணியை நீங்கள் திட்டமிட வேண்டும் என்றால், தரவு y யின் வடிவத்துடன் x க்கு சமம் சக்தி 1 உடன் நெருக்கமாகப் பொருந்தும். 585, சில விகிதாசார மாறிலியால் பெருக்கப்படுகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "நான் உங்களுக்கு சில வடிவங்களை தருகிறேன் என்று கற்பனை செய்து பாருங்கள், நீங்கள் ஒரு ஆர்வமுள்ள புரோகிராமர், இந்த எண்ணை எப்படி கண்டுபிடிப்பீர்கள்? ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "எனவே இரு பக்கங்களின் மடக்கையை எடுப்பது ஒரு பொதுவான தந்திரம். ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "இதில் சிறப்பான விஷயம் என்னவென்றால், அவை கரடுமுரடான வடிவங்கள் என்றும், நீங்கள் பெரிதாக்கும்போது கூட அவை கரடுமுரடானவை என்றும் கூறுவதற்கான ஒரு அளவு வழி. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "3D இல், நீங்கள் ஒரு பெட்டியை எண்ணும் போது, சிறிய சதுரங்களுக்கு பதிலாக சிறிய கனசதுரங்கள் நிறைந்த 3D கட்டம் உள்ளது, ஆனால் அது அதே வழியில் வேலை செய்கிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "இந்த அளவில், வடிவத்தின் தடிமன் பெட்டிகளின் அளவை விட சிறியதாக இருந்தால், அது ஒரு பரிமாணமாகத் தெரிகிறது, அதாவது அது தொடும் பெட்டிகளின் எண்ணிக்கை அதன் நீளத்திற்கு விகிதாசாரமாகும். ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "ஆனால் நீங்கள் அதை அளவிடும்போது, அது ஒரு குழாயைப் போலவே செயல்படத் தொடங்குகிறது, அந்த குழாயின் மேற்பரப்பில் உள்ள பெட்டிகளைத் தொடுகிறது, எனவே அது இரு பரிமாணமாகத் தோன்றும், தொடப்பட்ட பெட்டிகளின் எண்ணிக்கை சதுரத்திற்கு விகிதாசாரமாக இருக்கும். அளவிடுதல் காரணி. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "எனவே உண்மையில் ஒரு எண்ணை அதன் பரிமாணத்திற்கு ஒரு வடிவத்திற்கு ஒதுக்குவது தந்திரமானதாக இருக்கலாம், மேலும் இது மாறுபட்ட வரையறைகள் மற்றும் மாறுபட்ட மரபுகளுக்கு இடமளிக்கிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "ஆனால் பிரித்தானியாவின் கடற்கரைப் பகுதியைப் பார்ப்பது போன்ற ஒரு நடைமுறை அமைப்பில், நீங்கள் மேலும் மேலும் பெரிதாக்கும்போது வரம்பைப் பற்றி பேசுவதில் அர்த்தமில்லை, அதாவது ஒரு கட்டத்தில் நீங்கள் அணுக்களைத் தாக்கியிருப்பீர்கள். ", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "அதற்குப் பதிலாக, நீங்கள் செய்வது, மிகவும் பெரிதாக்கப்பட்டது முதல் பெரிதாக்கப்பட்டது வரை, போதுமான அளவு பரந்த அளவிலான அளவைப் பார்த்து, ஒவ்வொன்றின் பரிமாணத்தையும் கணக்கிடுங்கள். ", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "முற்றிலும் சுய-ஒத்த வடிவங்கள் பின்ன வடிவவியலில் முக்கிய பங்கு வகிக்கின்றன. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/telugu/sentence_translations.json b/2017/fractal-dimension/telugu/sentence_translations.json index f7a51c4fc..1baa4ecf4 100644 --- a/2017/fractal-dimension/telugu/sentence_translations.json +++ b/2017/fractal-dimension/telugu/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "ఒక రేఖ ఒక డైమెన్షనల్, ఒక విమానం రెండు డైమెన్షనల్, మనం నివసించే స్థలం త్రిమితీయ, మొదలైనవి. ", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "దీని గురించి ఆలోచించడానికి ఇక్కడ ఒక మార్గం ఉంది. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "మీరు స్కేల్ చేయబడిన డిస్క్ తాకిన పెట్టెల సంఖ్యతో పోల్చితే స్కేలింగ్ కారకాన్ని ప్లాట్ చేస్తే, మీ డేటా చాలా దగ్గరగా ఒక ఖచ్చితమైన పారాబొలాకు సరిపోతుంది, ఎందుకంటే తాకిన పెట్టెల సంఖ్య స్కేలింగ్ కారకం యొక్క వర్గానికి దాదాపు అనులోమానుపాతంలో ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "పెద్ద మరియు పెద్ద స్కేలింగ్ విలువల కోసం, ఇది కేవలం సూక్ష్మమైన గ్రిడ్‌ను చూడడానికి సమానం, ఆ డేటా ఆ పారాబొలాకు మరింత ఖచ్చితంగా సరిపోతుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "ఇప్పుడు ఫ్రాక్టల్స్‌కి తిరిగి వస్తున్నప్పుడు, సియర్‌పిన్స్‌కి ట్రయాంగిల్‌తో ఈ గేమ్‌ను ఆడుదాం, ఆ ఆకారంలో ఎన్ని పెట్టెలు తాకుతున్నాయో లెక్కించండి. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "పెద్దది తాకిన పెట్టెల నిష్పత్తికి చిన్నది తాకిన పెట్టెల సంఖ్యకు దాదాపు మూడు ఉండాలి. ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "అన్నింటికంటే, ఆ పెద్ద వెర్షన్ కేవలం చిన్న వెర్షన్ యొక్క మూడు కాపీలతో రూపొందించబడింది. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "మీరు దీనిని ఫ్రాక్టల్ యొక్క పరిమాణంలో పెంచినట్లు కూడా భావించవచ్చు, ఇది మేము ఇప్పుడే చూసింది 1.585. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "కాబట్టి మీరు వెళ్లి, సియర్‌పిన్స్‌కి త్రిభుజం తాకిన పెట్టెల సంఖ్యకు వ్యతిరేకంగా ఈ సందర్భంలో స్కేలింగ్ కారకాన్ని ప్లాట్ చేస్తే, డేటా శక్తి 1కి xకి సమానమైన y ఆకారంతో వక్రరేఖకు దగ్గరగా సరిపోతుంది. 585, కేవలం కొంత అనుపాత స్థిరాంకంతో గుణించబడింది. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "అలాగే, నేను మీకు కొంత ఆకారాన్ని ఇస్తానని ఊహించుకోండి మరియు మీరు అవగాహన ఉన్న ప్రోగ్రామర్, మీరు ఈ సంఖ్యను ఎలా కనుగొంటారు? ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "కాబట్టి రెండు వైపుల సంవర్గమానాన్ని తీసుకోవడం ఒక సాధారణ ఉపాయం. ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "దాని గురించి గొప్ప విషయం ఏమిటంటే, అవి కఠినమైన ఆకారాలు అని మరియు మీరు జూమ్ చేసినప్పటికీ అవి కఠినమైనవిగా ఉన్నాయని చెప్పడానికి ఇది పరిమాణాత్మక మార్గం. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "3Dలో, మీరు బాక్స్-కౌంటింగ్ చేసినప్పుడు, మీరు చిన్న చతురస్రాలకు బదులుగా చిన్న క్యూబ్‌లతో నిండిన 3D గ్రిడ్‌ను కలిగి ఉంటారు, కానీ ఇది అదే విధంగా పని చేస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "ఈ స్కేల్‌లో, ఆకారపు మందం పెట్టెల పరిమాణం కంటే తక్కువగా ఉంటే, అది ఒక డైమెన్షనల్‌గా కనిపిస్తుంది, అంటే అది తాకిన పెట్టెల సంఖ్య దాని పొడవుకు అనులోమానుపాతంలో ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "కానీ మీరు దానిని స్కేల్ చేసినప్పుడు, అది ఒక ట్యూబ్ లాగా ప్రవర్తించడం ప్రారంభిస్తుంది, ఆ ట్యూబ్ ఉపరితలంపై ఉన్న పెట్టెలను తాకడం ప్రారంభమవుతుంది, తద్వారా అది రెండు డైమెన్షనల్‌గా కనిపిస్తుంది, తాకిన పెట్టెల సంఖ్య చతురస్రానికి అనులోమానుపాతంలో ఉంటుంది. స్కేలింగ్ కారకం. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "కాబట్టి వాస్తవానికి ఒక సంఖ్యను దాని పరిమాణం కోసం ఆకారానికి కేటాయించడం గమ్మత్తైనది, మరియు ఇది విభిన్న నిర్వచనాలు మరియు విభిన్న సంప్రదాయాలకు స్థలాన్ని వదిలివేస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "కానీ బ్రిటన్ తీరప్రాంతాన్ని చూడటం వంటి మరింత అనువర్తిత సెట్టింగ్‌లో, మీరు మరింత ఎక్కువ జూమ్ చేస్తున్నప్పుడు పరిమితి గురించి మాట్లాడటం నిజంగా అర్ధవంతం కాదు, నా ఉద్దేశ్యం ఏదో ఒక సమయంలో మీరు అణువులను తాకినట్లు. ", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "బదులుగా మీరు చేసేది చాలా జూమ్ అవుట్ నుండి చాలా జూమ్ ఇన్ వరకు తగినంత విస్తృత శ్రేణి స్కేల్‌లను చూడండి మరియు ప్రతి దానిలో పరిమాణాన్ని గణించడం. ", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "ఫ్రాక్టల్ జ్యామితిలో సంపూర్ణ స్వీయ-సారూప్య ఆకారాలు ముఖ్యమైన పాత్ర పోషిస్తాయి. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/thai/sentence_translations.json b/2017/fractal-dimension/thai/sentence_translations.json index 1d157aac7..0e1d6e783 100644 --- a/2017/fractal-dimension/thai/sentence_translations.json +++ b/2017/fractal-dimension/thai/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/turkish/sentence_translations.json b/2017/fractal-dimension/turkish/sentence_translations.json index a28548b7e..f4d495515 100644 --- a/2017/fractal-dimension/turkish/sentence_translations.json +++ b/2017/fractal-dimension/turkish/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "Çizgi tek boyutludur, düzlem iki boyutludur, yaşadığımız uzay üç boyutludur vb. ", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "İşte bunu düşünmenin bir yolu. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "Ölçeklendirme faktörünü, ölçeklendirilmiş diskin dokunduğu kutu sayısına göre çizecekseniz, dokunulan kutuların sayısı ölçeklendirme faktörünün karesiyle kabaca orantılı olduğundan, verileriniz mükemmel bir parabole çok yakın bir şekilde uymalıdır. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "Daha büyük ve daha büyük ölçeklendirme değerleri için, ki bu aslında sadece daha ince bir ızgaraya bakmaya eşdeğerdir, bu veriler o parabole daha mükemmel bir şekilde uyacaktır. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "Şimdi fraktallara geri dönersek, Sierpinski üçgeni ile bu oyunu oynayalım ve bu şekildeki noktalara kaç kutunun temas ettiğini sayalım. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "Büyük olanın dokunduğu kutuların küçük olanın dokunduğu kutuların sayısına oranı yaklaşık üç olmalıdır. ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "Sonuçta, bu daha büyük versiyon, daha küçük versiyonun üç kopyasından oluşuyor. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "Bunu aynı zamanda fraktalın boyutuna yükseltilmiş iki olarak da düşünebilirsiniz, bunun da az önce yaklaşık 1 olduğunu gördük. 585. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "Ve eğer bu durumda ölçeklendirme faktörünü Sierpinski üçgeninin dokunduğu kutu sayısına göre çizerseniz, veriler y eşittir x üzeri 1 şeklindeki bir eğriye çok yakın bir şekilde uyacaktır. 585, bir orantı sabitiyle çarpıldı. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "Mesela, size bir şekil verdiğimi ve bilgili bir programcı olduğunuzu hayal edin, bu sayıyı nasıl bulursunuz? ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "Yani ortak bir numara her iki tarafın logaritmasını almaktır. ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "Bunun güzel yanı, bunların kaba şekiller olduğunu ve siz yakınlaştırdığınızda bile kaba kaldıklarını söylemenin niceliksel bir yolu olmasıdır. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "Bu arada 3B'de kutu sayımı yaptığınızda küçük kareler yerine küçük küplerle dolu bir 3B ızgaranız olur, ancak aynı şekilde çalışır. ", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "Şeklin kalınlığının kutuların boyutundan küçük olduğu bu ölçekte tek boyutlu görünüyor, yani dokunduğu kutu sayısı uzunluğuyla orantılı. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "Ancak ölçeği büyüttüğünüzde, daha çok bir tüp gibi davranmaya başlar, bu tüpün yüzeyindeki kutulara dokunur ve böylece iki boyutlu görünür, dokunulan kutuların sayısı kareyle orantılıdır. ölçekleme faktörü. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "Yani aslında bir şekle boyutu için bir sayı atamak zor olabilir ve farklı tanımlara ve farklı geleneklere yer bırakır. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "Ancak Britanya'nın kıyı şeridine bakmak gibi daha uygulamalı bir ortamda, yakınlaştırdıkça sınır hakkında konuşmak gerçekten mantıklı değil, yani bir noktada sadece atomlara çarpıyor olacaksınız. ", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "Bunun yerine, çok uzaklaştırılmıştan çok yakınlaştırılmışa kadar yeterince geniş bir ölçek aralığına bakmak ve her birinin boyutunu hesaplamaktır. ", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "Kendine tamamen benzeyen şekiller fraktal geometride önemli bir rol oynar. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/ukrainian/sentence_translations.json b/2017/fractal-dimension/ukrainian/sentence_translations.json index 75a8e7c71..06443209d 100644 --- a/2017/fractal-dimension/ukrainian/sentence_translations.json +++ b/2017/fractal-dimension/ukrainian/sentence_translations.json @@ -126,7 +126,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on.", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on.", "translatedText": "Лінія одновимірна, площина двовимірна, простір, у якому ми живемо, тривимірний і так далі.", "n_reviews": 0, "start": 154.08, @@ -644,21 +644,21 @@ "end": 690.32 }, { - "input": "Here's one way to think about this.", + "input": "Here's one example. For larger and larger scaling values, which is actually", "translatedText": "Ось один спосіб подумати про це.", "n_reviews": 0, "start": 696.48, "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor.", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the", "translatedText": "Якщо ви побудуєте коефіцієнт масштабування порівняно з кількістю квадратів, яких торкається масштабований диск, ваші дані мають дуже точно відповідати ідеальній параболі, оскільки кількість квадратів, до яких торкається, приблизно пропорційна квадрату коефіцієнта масштабування.", "n_reviews": 0, "start": 713.64, "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola.", + "input": "triangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve", "translatedText": "Для все більших і більших значень масштабування, що фактично еквівалентно простому перегляду дрібнішої сітки, ці дані точніше відповідатимуть цій параболі.", "n_reviews": 0, "start": 733.28, @@ -672,7 +672,7 @@ "end": 750.54 }, { - "input": "How would you imagine that number compares to scaling up the triangle by a factor of two and counting the new number of boxes touched?", + "input": "version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. And so if you were to go and plot the scaling factor in this case against the number of boxes touched", "translatedText": "Як ви можете собі уявити це число порівняно зі збільшенням трикутника у два рази та підрахунком нової кількості торканих ящиків?", "n_reviews": 0, "start": 750.54, @@ -721,7 +721,7 @@ "end": 801.04 }, { - "input": "If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21.", + "input": "pproximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that number empirically. As in, imagine I give you some shape, and you're a savvy programmer. How would you find this number? So what I'm saying here is that if you scale this shape by some factor, which I'll call S, the number of boxe", "translatedText": "Якщо ви покладете цю берегову лінію на площину та порахуєте, скільки коробок торкається її, а потім збільшите її на деяку величину та порахуєте, скільки коробок торкається цієї нової масштабованої версії, ви побачите, що кількість коробок торкається берегова лінія збільшується приблизно пропорційно коефіцієнту масштабування, зведеному до степеня 1.21.", "n_reviews": 0, "start": 801.04, @@ -742,7 +742,7 @@ "end": 832.22 }, { - "input": "So what I'm saying here is that if you scale this shape by some factor, which I'll call S, the number of boxes touching that shape should equal some constant multiplied by that scaling factor raised to whatever the dimension is, the value we're looking for.", + "input": "ied by that scaling factor raised to whatever the dimension is, the value that we're looking for. Now, if you have some data plot that closely fits a curve that looks like the input raised to some power, it can be hard to see exactly what that power should be. So a common trick is to take the logarithm of both sides. That way, the dimension is going to d", "translatedText": "Тож я хочу сказати, що якщо ви масштабуєте цю фігуру за деяким коефіцієнтом, який я назву S, кількість коробок, які торкаються цієї фігури, має дорівнювати деякій константі, помноженій на коефіцієнт масштабування, зведений до будь-якого розміру, значення ми шукаємо.", "n_reviews": 0, "start": 832.22, @@ -756,7 +756,7 @@ "end": 859.82 }, { - "input": "So a common trick is to take the logarithm of both sides.", + "input": "lot the log of the scaling factor against the log of the number of boxes touching the coastline, the relationship should look like a l", "translatedText": "Тому поширений трюк полягає в тому, щоб взяти логарифм обох сторін.", "n_reviews": 0, "start": 859.82, @@ -770,14 +770,14 @@ "end": 871.36 }, { - "input": "What this suggests is that if you were to plot the log of the scaling factor against the log of the number of boxes touching the coastline, the relationship should look like a line, and that line should have a slope equal to the dimension.", + "input": "So what that means is that if you tried out a whole bunch of scaling factors, counted the number of boxes touching the coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regression to find the best fit line to your data set, and when you look at the slope of th", "translatedText": "Це свідчить про те, що якщо ви побудуєте логарифм коефіцієнта масштабування проти логарифму кількості ящиків, які торкаються берегової лінії, зв’язок має виглядати як лінія, і ця лінія має мати нахил, рівний розміру.", "n_reviews": 0, "start": 873.44, "end": 889.52 }, { - "input": "So what that means is that if you tried out a whole bunch of scaling factors, counted the number of boxes touching the coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regression to find the best fit line to your dataset, and when you look at the slope of that line, that tells you the empirical measurement for the dimension of what you're examining.", + "input": "at line, that tells you the empirical measurement for the dimension of what you're examining. I just think that makes this idea of fractal dimension so much more real and visceral compared to abstract, artificially perfect shapes. And once you're comfortable thinking about dimension like this, you, my friend, have become ready to hear the definition of a fractal. Essentially, fractals are shapes whose dimension is not an integer, but instead some fractional amount. What's cool", "translatedText": "Отже, це означає, що якщо ви випробували цілу групу коефіцієнтів масштабування, підрахували кількість ящиків, які торкаються узбережжя за кожну мить, а потім нанесли точки на логарифмічний графік, ви могли б виконати якусь лінійну регресію щоб знайти лінію, яка найкраще підходить для вашого набору даних, і коли ви дивитеся на нахил цієї лінії, це говорить вам про емпіричне вимірювання для розмірності того, що ви досліджуєте.", "n_reviews": 0, "start": 889.52, @@ -805,14 +805,14 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in.", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance th", "translatedText": "Що круто в цьому, так це те, що це кількісний спосіб сказати, що це грубі форми, які залишаються грубими, навіть коли ви збільшуєте масштаб.", "n_reviews": 0, "start": 928.94, "end": 939.46 }, { - "input": "Technically there's a slightly more accurate definition, and I've included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for.", + "input": "ough that I haven't brought up yet, but it's worth pointing out, which is that this dimension, at least as I've described it so far using the box counting method, can sometimes change based on how far zoomed in you are. For example, here's a shape sitting in thr", "translatedText": "Технічно є дещо точніше визначення, і я включив його в опис відео, але ця ідея тут про нецілочисельний вимір майже повністю відображає ідею шорсткості, до якої ми прагнемо.", "n_reviews": 0, "start": 939.46, diff --git a/2017/fractal-dimension/urdu/sentence_translations.json b/2017/fractal-dimension/urdu/sentence_translations.json index 608c15d6b..82cc3750c 100644 --- a/2017/fractal-dimension/urdu/sentence_translations.json +++ b/2017/fractal-dimension/urdu/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "طول و عرض ایک ایسی چیز ہے جو عام طور پر صرف قدرتی نمبروں کے لئے معنی رکھتی ہے، ٹھیک ہے؟ ایک لائن ایک جہتی ہے، ایک طیارہ دو جہتی ہے، ہم جس جگہ میں رہتے ہیں وہ تین جہتی ہے، وغیرہ۔", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "اس کے بارے میں سوچنے کا ایک طریقہ یہ ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "اگر آپ اسکیل ڈسک کے چھونے والے خانوں کی تعداد کے مقابلے میں اسکیلنگ فیکٹر کو پلاٹ کرنا چاہتے ہیں، تو آپ کا ڈیٹا ایک بہترین پیرابولا کے ساتھ بہت قریب سے فٹ ہونا چاہیے، کیونکہ چھوئے جانے والے خانوں کی تعداد تقریباً اسکیلنگ فیکٹر کے مربع کے متناسب ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "بڑے اور بڑے پیمانے کی قدروں کے لیے، جو حقیقت میں صرف ایک باریک گرڈ کو دیکھنے کے مترادف ہے، وہ ڈیٹا اس پیرابولا میں بالکل فٹ ہونے والا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "اب فریکٹلز پر واپس آتے ہیں، آئیے اس گیم کو سیرپینسکی مثلث کے ساتھ کھیلیں، گنتے ہیں کہ اس شکل میں کتنے خانے چھونے والے پوائنٹس ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "سب کے بعد، وہ بڑا ورژن صرف چھوٹے ورژن کی تین کاپیاں پر مشتمل ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "آپ اس کے بارے میں یہ بھی سوچ سکتے ہیں کہ فریکٹل کے طول و عرض میں دو بڑھے ہوئے ہیں، جو ہم نے ابھی دیکھا ہے تقریباً 1۔585. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "اور اس لیے اگر آپ اس معاملے میں سیرپینسکی مثلث کے ذریعے چھوئے گئے خانوں کی تعداد کے خلاف اسکیلنگ فیکٹر کو پلاٹ کرتے ہیں، تو ڈیٹا ایک منحنی خطوط کو قریب سے فٹ کرے گا جس کی شکل y کے برابر x کی طاقت 1 ہے۔585، صرف کچھ تناسب مستقل سے ضرب۔", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "لہذا ایک عام چال یہ ہے کہ دونوں اطراف کا لوگارتھم لیا جائے۔", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "اس میں اچھی بات یہ ہے کہ یہ کہنے کا ایک مقداری طریقہ ہے کہ وہ ایسی شکلیں ہیں جو کھردری ہیں، اور جب آپ زوم ان کرتے ہیں تو بھی وہ کھردری رہتی ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "3D میں، ویسے، جب آپ باکس گنتی کرتے ہیں، تو آپ کے پاس چھوٹے مربعوں کی بجائے چھوٹے کیوبز سے بھرا 3D گرڈ ہوتا ہے، لیکن یہ اسی طرح کام کرتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "اس پیمانے پر، جہاں شکل کی موٹائی خانوں کے سائز سے چھوٹی ہے، یہ ایک جہتی نظر آتی ہے، یعنی اس کے چھونے والے خانوں کی تعداد اس کی لمبائی کے متناسب ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "لیکن جب آپ اسے پیمانہ بناتے ہیں، تو یہ ٹیوب کی سطح پر موجود خانوں کو چھوتے ہوئے، ٹیوب کی طرح بہت زیادہ برتاؤ کرنے لگتا ہے، اور اس طرح یہ دو جہتی نظر آئے گا، چھوئے جانے والے خانوں کی تعداد کے مربع کے متناسب ہونے کے ساتھ۔", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "اس لیے اصل میں کسی نمبر کو اس کے طول و عرض کے لیے شکل دینا مشکل ہو سکتا ہے، اور یہ مختلف تعریفوں اور مختلف کنونشنز کے لیے گنجائش چھوڑ دیتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "لیکن زیادہ لاگو ترتیب میں، جیسے برطانیہ کے ساحل کو دیکھنا، حد کے بارے میں بات کرنا واقعی کوئی معنی نہیں رکھتا کیونکہ آپ زیادہ سے زیادہ زوم کرتے ہیں، میرا مطلب ہے کہ کسی وقت آپ صرف ایٹموں کو مار رہے ہوں گے۔", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "اس کے بجائے آپ جو کچھ کرتے ہیں وہ یہ ہے کہ بہت زیادہ زوم آؤٹ سے لے کر بہت زوم ان تک کے پیمانے کی کافی وسیع رینج کو دیکھیں، اور ہر ایک پر طول و عرض کی گنتی کریں۔", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "مکمل طور پر خود سے ملتی جلتی شکلیں فریکٹل جیومیٹری میں اہم کردار ادا کرتی ہیں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/fractal-dimension/vietnamese/sentence_translations.json b/2017/fractal-dimension/vietnamese/sentence_translations.json index a52ab6ba0..c1255e0e3 100644 --- a/2017/fractal-dimension/vietnamese/sentence_translations.json +++ b/2017/fractal-dimension/vietnamese/sentence_translations.json @@ -144,7 +144,7 @@ "end": 153.78 }, { - "input": "A line is one dimensional, a plane is two dimensional, the space we live in is three dimensional, and so on. ", + "input": "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on. ", "translatedText": "Đường thẳng là một chiều, mặt phẳng là hai chiều, không gian chúng ta đang sống là ba chiều, v.v. ", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 690.32 }, { - "input": "Here's one way to think about this. ", + "input": "Here's one example. For larger and larger scaling values, which is actually ", "translatedText": "Đây là một cách để suy nghĩ về điều này. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 713.64 }, { - "input": "If you were to plot the scaling factor compared to the number of boxes that the scaled disc touches, your data should very closely fit a perfect parabola, since the number of boxes touched is roughly proportional to the square of the scaling factor. ", + "input": "equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. How would you imagine that number compares to scaling up the tr ", "translatedText": "Nếu bạn định vẽ hệ số tỷ lệ so với số hộp mà đĩa đã chia tỷ lệ chạm vào, thì dữ liệu của bạn phải rất khớp với một hình parabol hoàn hảo, vì số hộp được chạm gần tỷ lệ thuận với bình phương của hệ số tỷ lệ. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 733.28 }, { - "input": "For larger and larger scaling values, which is actually equivalent to just looking at a finer grid, that data is going to more perfectly fit that parabola. ", + "input": "iangle by a factor of two and counting the new number of boxes touched? Well, the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. After all, that bigger ve ", "translatedText": "Đối với các giá trị tỷ lệ lớn hơn và lớn hơn, thực sự tương đương với việc chỉ nhìn vào một lưới mịn hơn, dữ liệu đó sẽ khớp hoàn toàn hơn với parabol đó. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 747.72 }, { - "input": "Now getting back to fractals, let's play this game with the Sierpinski triangle, counting how many boxes are touching points in that shape. ", + "input": "rsion is just built up of three copies of the smaller version. You could also think about this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", "translatedText": "Bây giờ quay trở lại với fractal, chúng ta hãy chơi trò chơi này với tam giác Sierpinski, đếm xem có bao nhiêu hộp là điểm tiếp xúc trong hình đó. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 765.28 }, { - "input": "Well the proportion of boxes touched by the big one to the number of boxes touched by the small one should be about three. ", + "input": "by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. But importantly, the wh ", "translatedText": "Chà, tỷ lệ hộp mà hộp lớn chạm vào so với số hộp mà hộp nhỏ chạm vào phải là khoảng ba. ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 778.04 }, { - "input": "After all, that bigger version is just built up of three copies of the smaller version. ", + "input": "ole reason that I'm talking about this is that we can play the same game with non-self-similar shapes that still have some k ", "translatedText": "Suy cho cùng, phiên bản lớn hơn đó chỉ được tạo thành từ ba bản sao của phiên bản nhỏ hơn. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 783.6 }, { - "input": "You could also think of this as two raised to the dimension of the fractal, which we just saw is about 1.585. ", + "input": "ind of roughness. And the classic example here is the coastline of Britain. If you plop that coastline into the plane and count how many boxes are touching it, and then scale it by some amount, ", "translatedText": "Bạn cũng có thể coi đây là hai số được nâng lên theo chiều của fractal, mà chúng ta vừa thấy là khoảng 1.585. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 793.68 }, { - "input": "And so if you were to go and plot the scaling factor in this case against the number of boxes touched by the Sierpinski triangle, the data would closely fit a curve with the shape of y equals x to the power 1.585, just multiplied by some proportionality constant. ", + "input": "and count how many boxes are touching that new scaled version, what you'd find is that the number of boxes touching the coastline increases approximately in proportion to the scaling factor raised to the power of 1.21. Here, it's kind of fun to think about how you would actually compute that ", "translatedText": "Và vì vậy nếu bạn định vẽ hệ số tỷ lệ trong trường hợp này theo số hộp được tam giác Sierpinski chạm vào, dữ liệu sẽ khớp chặt với một đường cong có dạng y bằng x lũy thừa 1.585, chỉ cần nhân với một hằng số tỷ lệ nào đó. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 848.64 }, { - "input": "As in, imagine I give you some shape, and you're a savvy programmer, how would you find this number? ", + "input": "rick is to take the logarithm of both sides. That way, the dimension is going to drop down from the exponent, and we'll have a nice clean linear relationship. What this suggests ", "translatedText": "Như trong, hãy tưởng tượng tôi cho bạn một số hình dạng, và bạn là một lập trình viên hiểu biết, bạn sẽ tìm thấy con số này như thế nào? ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 878.64 }, { - "input": "So a common trick is to take the logarithm of both sides. ", + "input": "he coast in each instant, and then plotted the points on the log-log plot, you could then do some kind of linear regres ", "translatedText": "Vì vậy, một thủ thuật phổ biến là lấy logarit của cả hai vế. ", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 928.94 }, { - "input": "What's cool about that is that it's a quantitative way to say that they're shapes that are rough, and that they stay rough even as you zoom in. ", + "input": "ve included it in the video description, but this idea here of a non-integer dimension almost entirely captures the idea of roughness that we're going for. There is one nuance though that I haven't brought up yet, but it's worth pointing ", "translatedText": "Điều thú vị ở đây là đó là một cách định lượng để nói rằng chúng có hình dạng thô và chúng vẫn thô ngay cả khi bạn phóng to. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 965.48 }, { - "input": "In 3D, by the way, when you do a box-counting, you have a 3D grid full of little cubes instead of little squares, but it works the same way. ", + "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", "translatedText": "Nhân tiện, trong 3D, khi bạn đếm hộp, bạn có một lưới 3D chứa đầy các hình khối nhỏ thay vì các hình vuông nhỏ, nhưng nó hoạt động theo cách tương tự. ", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 976.64 }, { - "input": "At this scale, where the shape's thickness is smaller than the size of the boxes, it looks one-dimensional, meaning the number of boxes it touches is proportional to its length. ", + "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", "translatedText": "Ở tỷ lệ này, khi độ dày của hình dạng nhỏ hơn kích thước của các hộp, nó trông có vẻ một chiều, nghĩa là số lượng hộp mà nó chạm vào tỷ lệ thuận với chiều dài của nó. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 990.76 }, { - "input": "But when you scale it up, it starts behaving a lot more like a tube, touching the boxes on the surface of that tube, and so it'll look two-dimensional, with the number of boxes touched being proportional to the square of the scaling factor. ", + "input": "But it's not really a tube, it's made of these rapidly winding little curves, so once you scale it up even more, to the point where the boxes can pick up on the details of those curves, it looks one-dimensional again, with the number of boxes touched scaling directly in proportion to the scaling co ", "translatedText": "Nhưng khi bạn phóng to nó lên, nó bắt đầu hoạt động giống một cái ống hơn, chạm vào các hộp trên bề mặt của ống đó, và do đó nó sẽ trông có dạng hai chiều, với số lượng hộp chạm vào tỉ lệ với bình phương của hình ống đó. yếu tố nhân rộng. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1025.0 }, { - "input": "So actually assigning a number to a shape for its dimension can be tricky, and it leaves room for differing definitions and differing conventions. ", + "input": "sion, but all of them focus on what the limit of this dimension is at closer and closer zoom levels. You can think of that in terms of the plot as the limit of this slope as you move farther and farther to the right. ", "translatedText": "Vì vậy, thực sự việc gán một số cho một hình dạng theo kích thước của nó có thể phức tạp và nó có chỗ cho các định nghĩa khác nhau và các quy ước khác nhau. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 1057.68 }, { - "input": "But in a more applied setting, like looking at the coastline of Britain, it doesn't really make sense to talk about the limit as you zoom in more and more, I mean at some point you'd just be hitting atoms. ", + "input": "I mean, at some point you'd just be hitting atoms. Instead what you do is you look at a sufficiently wide range of scales from very zoomed out up to very zoomed in, and compute the dimension at each one. And in this more applied setting, a sh ", "translatedText": "Nhưng trong một bối cảnh ứng dụng hơn, chẳng hạn như nhìn vào bờ biển nước Anh, sẽ không thực sự có ý nghĩa khi nói về giới hạn khi bạn phóng to ngày càng nhiều, ý tôi là đến một lúc nào đó bạn sẽ chạm vào các nguyên tử. ", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 1071.92 }, { - "input": "Instead what you do is look at a sufficiently wide range of scales, from very zoomed out up to very zoomed in, and compute the dimension at each one. ", + "input": "ape is typically considered to be a fractal only when the measured dimension stays approximately constant even across multiple different scales. For example, the coastline of Britain doesn't just look 1.21 dimensional at a dista ", "translatedText": "Thay vào đó, những gì bạn làm là xem xét một phạm vi tỷ lệ đủ rộng, từ rất thu nhỏ đến rất phóng to và tính toán kích thước ở mỗi tỷ lệ. ", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1111.24 }, { - "input": "Perfectly self-similar shapes do play an important role in fractal geometry. ", + "input": "different scales and at arbitrarily close scales. And that's important, it gives us the primitive tools for modeling these fractal phenomena. ", "translatedText": "Những hình dạng hoàn toàn giống nhau đóng một vai trò quan trọng trong hình học fractal. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/gradient-descent/arabic/sentence_translations.json b/2017/gradient-descent/arabic/sentence_translations.json index f68282c2a..f76585680 100644 --- a/2017/gradient-descent/arabic/sentence_translations.json +++ b/2017/gradient-descent/arabic/sentence_translations.json @@ -384,7 +384,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "أريدك أيضًا أن تلاحظ كيف أنه إذا جعلت أحجام خطواتك متناسبة مع المنحدر، فعندما يصبح المنحدر مسطحًا نحو الحد الأدنى، تصبح خطواتك أصغر فأصغر، وهذا يساعدك على عدم التجاوز.", "model": "google_nmt", "from_community_srt": "هذا سوف ينتقل إلى حالة الشبكة العصبية أيضًا ، وأريد أيضًا أن تلاحظ ذلك كيف إذا كنت تجعل أحجام الخطوة الخاصة بك متناسبة مع المنحدر ثم عندما يميل المنحدر نحو الحد الأدنى ، تصبح خطواتك أصغر وأصغر وهذا النوع يساعدك على تجاوز الحدود", @@ -1025,7 +1025,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "من الواضح أن دقة الاختبار هنا لم تكن أفضل من العشوائية، نظرًا لأن كل شيء تم تصنيفه بشكل عشوائي، لكنه كان لا يزال قادرًا على تحقيق نفس دقة التدريب كما تفعل في مجموعة بيانات مصنفة بشكل صحيح.", "model": "google_nmt", "from_community_srt": "من الواضح أن دقة الاختبار هنا لن تكون أفضل من العشوائية حيث أن كل شيء مكتوب بشكل عشوائي ولكنها كانت لا تزال قادرة على تحقيق نفس دقة التدريب كما تفعل في مجموعة بيانات مصنفة بشكل صحيح", @@ -1043,7 +1043,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "إذا نظرت إلى منحنى الدقة هذا، إذا كنت تتدرب فقط على مجموعة بيانات عشوائية، فقد انخفض هذا المنحنى ببطء شديد بطريقة خطية تقريبًا، لذا فأنت تكافح حقًا للعثور على الحد الأدنى المحلي الممكن، كما تعلم ، الأوزان المناسبة التي من شأنها أن تمنحك تلك الدقة.", "model": "google_nmt", "from_community_srt": "أم هل أنت فقط تعرف؟ احفظ كامل مجموعة البيانات الخاصة بالتصنيف الصحيح ، ومن ثم نعرف بعضًا منكم بعد مرور نصف عام على ICML هذا العام لم يكن هناك بالضبط ورقة ورقة الطعن التي عالجت بعض طلب مثل يا في الواقع ، تقوم هذه الشبكات بعمل شيء أكثر ذكاءً من ذلك إذا نظرت إلى منحنى الدقة هذا إذا كنت مجرد تدريب على مجموعة بيانات عشوائية منحنى نوع من انحدار جدا جدا تعرف ببطء شديد تقريبا في شكل خطي لذلك أنت حقا تكافح من أجل العثور على الحد الأدنى المحلي ممكن أنت تعرف الأوزان الصحيحة التي من شأنها أن تحصل على هذه الدقة في حين إذا كنت في الواقع التدريب على مجموعة البيانات المنظمة التي لديها", @@ -1087,7 +1087,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "أريد أيضًا أن أتقدم بشكر خاص لشركة VC Amplify Partners، لدعمها لمقاطع الفيديو الأولية هذه في السلسلة.", "model": "google_nmt", "from_community_srt": "تريد أيضا أن تعطي خاص. شكرًا لشركاء VC firmifi في دعمهم لمقاطع الفيديو الأولية هذه في السلسلة", diff --git a/2017/gradient-descent/bengali/sentence_translations.json b/2017/gradient-descent/bengali/sentence_translations.json index d72d6d8ab..5b5758594 100644 --- a/2017/gradient-descent/bengali/sentence_translations.json +++ b/2017/gradient-descent/bengali/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "আমি এও চাই যে আপনি লক্ষ্য করুন কিভাবে আপনি যদি আপনার ধাপের আকার ঢালের সমানুপাতিক করেন, তাহলে ঢাল যখন ন্যূনতম দিকে সমতল হয়, তখন আপনার পদক্ষেপগুলি ছোট থেকে ছোট হয় এবং এটি আপনাকে ওভারশুটিং থেকে সাহায্য করে।", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "স্পষ্টতই এখানে পরীক্ষার নির্ভুলতা র্যান্ডম থেকে ভাল ছিল না, যেহেতু সবকিছুই কেবল এলোমেলোভাবে লেবেলযুক্ত, তবে এটি এখনও সঠিকভাবে লেবেলযুক্ত ডেটাসেটের মতো একই প্রশিক্ষণ নির্ভুলতা অর্জন করতে সক্ষম হয়েছিল।", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "আপনি যদি সেই নির্ভুলতার বক্ররেখাটি দেখেন, যদি আপনি শুধুমাত্র একটি এলোমেলো ডেটাসেটের প্রশিক্ষণ নিচ্ছেন, সেই বক্ররেখাটি প্রায় এক ধরনের রৈখিক ফ্যাশনে খুব ধীরে ধীরে নিচে নেমে গেছে, তাই আপনি সত্যিই সম্ভাব্য স্থানীয় মিনিমাম খুঁজে পেতে সংগ্রাম করছেন, আপনি জানেন , সঠিক ওজন যা আপনাকে সেই নির্ভুলতা পাবে।", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "সিরিজের এই প্রাথমিক ভিডিওগুলির সমর্থনে আমি VC সংস্থা Amplify Partners-কেও বিশেষ ধন্যবাদ জানাতে চাই৷", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/chinese/sentence_translations.json b/2017/gradient-descent/chinese/sentence_translations.json index 764985b7b..90f49100e 100644 --- a/2017/gradient-descent/chinese/sentence_translations.json +++ b/2017/gradient-descent/chinese/sentence_translations.json @@ -333,7 +333,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "我还希望您注意，如果您使步长与斜率成正比，那么当斜率趋于最小值时，您的步数会变得越来越小，这有助于您避免过度调整。", "from_community_srt": "也是一樣的情況 另外需要注意的是， 如果你的步長和斜率成比例 那麽當越接近最小值時， 你的步長就越小，", "n_reviews": 0, @@ -902,7 +902,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "显然，这里的测试准确性并不比随机测试更好，因为所有内容都是随机标记的，但它仍然能够达到与在正确标记的数据集上相同的训练准确性。", "from_community_srt": "測試準確度不會比隨機結果好到哪去， 因為標籤本身就是混亂的 但是一但你使用了正確標記的數據集， 依然可以達到相同的識別精度 基本上，", "n_reviews": 0, @@ -918,7 +918,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "如果你看一下准确率曲线，如果你只是在随机数据集上进行训练，那么该曲线几乎以线性方式缓慢下降，所以你真的很难找到可能的局部最小值，你知道，正确的权重可以让您获得准确度。", "from_community_srt": "記憶整個正確分類的數據集 今年在ICML 沒有反駁的論文， 只有一些簡單提及的論文 事實上這些神經網路做的更聰明一些 如果你看這個準確度曲線 如果你只是訓練一個隨機數據集 這個曲線會非常非常慢的下降， 近似於線性 所以你可能真的很難找到局部最小值 只要你用正確標記過的結構化的數據集 正確的權重會讓你得到一定的準確度 一開始你可能會反覆折騰，", "n_reviews": 0, @@ -957,7 +957,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "我还要特别感谢风险投资公司 Amplify Partners 对本系列初始视频的支持。", "from_community_srt": "但是如果沒有你是不可能的做出這些影片的 同時也要特別感謝VC公司的合作夥伴對這些系列影片的支持", "n_reviews": 0, diff --git a/2017/gradient-descent/english/captions.srt b/2017/gradient-descent/english/captions.srt index bffb6c29c..f623b4021 100644 --- a/2017/gradient-descent/english/captions.srt +++ b/2017/gradient-descent/english/captions.srt @@ -403,16 +403,16 @@ value of the cost function. That will carry over to our neural network case as well. 102 -00:06:43,180 --> 00:06:47,653 -I also want you to notice how if you make your step sizes proportional to the slope, +00:06:43,180 --> 00:06:47,618 +And I also want you to notice how if you make your step sizes proportional to the slope, 103 -00:06:47,653 --> 00:06:50,758 +00:06:47,618 --> 00:06:50,560 then when the slope is flattening out towards the minimum, 104 -00:06:50,758 --> 00:06:54,600 -your steps get smaller and smaller, and that helps you from overshooting. +00:06:50,560 --> 00:06:54,600 +your steps get smaller and smaller, and that kind of helps you from overshooting. 105 00:06:55,940 --> 00:06:58,653 @@ -1099,16 +1099,16 @@ at image recognition, and instead of training it on a properly labeled dataset, shuffled all the labels around before training. 276 -00:18:09,480 --> 00:18:12,916 -Obviously the testing accuracy here was no better than random, +00:18:09,480 --> 00:18:13,366 +Obviously the testing accuracy here was going to be no better than random, 277 -00:18:12,916 --> 00:18:16,625 -since everything is just randomly labeled, but it was still able to +00:18:13,366 --> 00:18:17,252 +since everything's just randomly labeled. But it was still able to achieve 278 -00:18:16,625 --> 00:18:20,880 -achieve the same training accuracy as you would on a properly labeled dataset. +00:18:17,252 --> 00:18:20,880 +the same training accuracy as you would on a properly labeled dataset. 279 00:18:21,600 --> 00:18:25,313 @@ -1127,78 +1127,82 @@ which raises the question for whether minimizing this cost function actually corresponds to any sort of structure in the image, or is it just memorization? 283 -00:18:51,440 --> 00:18:57,405 -If you look at that accuracy curve, if you were just training on a random dataset, +00:18:51,440 --> 00:18:55,937 +...to memorize the entire dataset of what the correct classification is. 284 -00:18:57,405 --> 00:19:02,940 -that curve sort of went down very slowly in almost kind of a linear fashion, +00:18:55,937 --> 00:19:00,065 +And so a couple of, you know, half a year later at ICML this year, 285 -00:19:02,940 --> 00:19:07,755 -so you're really struggling to find that local minima of possible, +00:19:00,065 --> 00:19:05,301 +there was not exactly rebuttal paper, but paper that addressed some aspects of like, 286 -00:19:07,755 --> 00:19:12,140 -you know, the right weights that would get you that accuracy. +00:19:05,301 --> 00:19:10,291 +hey, actually these networks are doing something a little bit smarter than that. 287 +00:19:10,291 --> 00:19:12,140 +If you look at that accuracy c + +288 00:19:12,240 --> 00:19:15,823 Whereas if you're actually training on a structured dataset, -288 +289 00:19:15,823 --> 00:19:20,523 one that has the right labels, you fiddle around a little bit in the beginning, -289 +290 00:19:20,523 --> 00:19:24,636 but then you kind of dropped very fast to get to that accuracy level, -290 +291 00:19:24,636 --> 00:19:28,220 and so in some sense it was easier to find that local maxima. -291 +292 00:19:28,540 --> 00:19:33,614 And so what was also interesting about that is it brings into light another paper from -292 +293 00:19:33,614 --> 00:19:38,688 actually a couple of years ago, which has a lot more simplifications about the network -293 +294 00:19:38,688 --> 00:19:43,879 layers, but one of the results was saying how if you look at the optimization landscape, -294 +295 00:19:43,879 --> 00:19:48,662 the local minima that these networks tend to learn are actually of equal quality, -295 +296 00:19:48,662 --> 00:19:51,462 so in some sense if your dataset is structured, -296 +297 00:19:51,462 --> 00:19:54,320 you should be able to find that much more easily. -297 +298 00:19:58,160 --> 00:20:01,180 My thanks, as always, to those of you supporting on Patreon. -298 +299 00:20:01,520 --> 00:20:04,065 I've said before just what a game changer Patreon is, -299 +300 00:20:04,065 --> 00:20:06,800 but these videos really would not be possible without you. -300 -00:20:07,460 --> 00:20:10,439 -I also want to give a special thanks to the VC firm Amplify Partners, - 301 -00:20:10,439 --> 00:20:12,780 -in their support of these initial videos in the series. +00:20:07,460 --> 00:20:10,159 +I also want to give a special thanks to the VC firm Amplify Partners + +302 +00:20:10,159 --> 00:20:12,780 +and their support of these initial videos in the series. Thank you. diff --git a/2017/gradient-descent/english/sentence_timings.json b/2017/gradient-descent/english/sentence_timings.json index 367b16ff7..4362586a5 100644 --- a/2017/gradient-descent/english/sentence_timings.json +++ b/2017/gradient-descent/english/sentence_timings.json @@ -220,7 +220,7 @@ 402.62 ], [ - "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", 403.18, 414.6 ], @@ -585,7 +585,7 @@ 1088.74 ], [ - "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", 1089.48, 1100.88 ], @@ -595,7 +595,7 @@ 1116.4 ], [ - "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", 1131.44, 1152.14 ], @@ -620,7 +620,7 @@ 1206.8 ], [ - "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", 1207.46, 1212.78 ] diff --git a/2017/gradient-descent/english/transcript.txt b/2017/gradient-descent/english/transcript.txt index b5e5e015b..d3e958874 100644 --- a/2017/gradient-descent/english/transcript.txt +++ b/2017/gradient-descent/english/transcript.txt @@ -42,7 +42,7 @@ If you do this repeatedly, at each point checking the new slope and taking the a The image you might have in mind here is a ball rolling down a hill. Notice, even for this really simplified single input function, there are many possible valleys that you might land in, depending on which random input you start at, and there's no guarantee that the local minimum you land in is going to be the smallest possible value of the cost function. That will carry over to our neural network case as well. -I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting. +And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting. Bumping up the complexity a bit, imagine instead a function with two inputs and one output. You might think of the input space as the xy-plane, and the cost function as being graphed as a surface above it. Instead of asking about the slope of the function, you have to ask which direction you should step in this input space so as to decrease the output of the function most quickly. @@ -115,11 +115,11 @@ To close things off here for the last few minutes, I want to jump back into a sn You might remember her from the last video, she did her PhD work in deep learning. In this little snippet she talks about two recent papers that really dig into how some of the more modern image recognition networks are actually learning. Just to set up where we were in the conversation, the first paper took one of these particularly deep neural networks that's really good at image recognition, and instead of training it on a properly labeled dataset, shuffled all the labels around before training. -Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset. +Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset. Basically, the millions of weights for this particular network were enough for it to just memorize the random data, which raises the question for whether minimizing this cost function actually corresponds to any sort of structure in the image, or is it just memorization? -If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy. +...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c Whereas if you're actually training on a structured dataset, one that has the right labels, you fiddle around a little bit in the beginning, but then you kind of dropped very fast to get to that accuracy level, and so in some sense it was easier to find that local maxima. And so what was also interesting about that is it brings into light another paper from actually a couple of years ago, which has a lot more simplifications about the network layers, but one of the results was saying how if you look at the optimization landscape, the local minima that these networks tend to learn are actually of equal quality, so in some sense if your dataset is structured, you should be able to find that much more easily. My thanks, as always, to those of you supporting on Patreon. I've said before just what a game changer Patreon is, but these videos really would not be possible without you. -I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series. \ No newline at end of file +I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you. \ No newline at end of file diff --git a/2017/gradient-descent/french/sentence_translations.json b/2017/gradient-descent/french/sentence_translations.json index 3490660d9..0501b0208 100644 --- a/2017/gradient-descent/french/sentence_translations.json +++ b/2017/gradient-descent/french/sentence_translations.json @@ -349,7 +349,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "Je veux également que vous remarquiez que si vous rendez la taille de vos pas proportionnelle à la pente, lorsque la pente s'aplatit vers le minimum, vos pas deviennent de plus en plus petits, ce qui vous aide à ne pas dépasser.", "from_community_srt": "et je veux aussi que vous remarquiez Comment faire en sorte que vos pas soient proportionnels à la pente Alors, lorsque la pente s’aplatit vers le minimum, vos pas deviennent de plus en plus petits et cela vous aide à ne pas dépasser En remontant un peu la complexité,", "n_reviews": 0, @@ -923,7 +923,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "De toute évidence, la précision des tests ici n'était pas meilleure que celle du hasard, puisque tout est étiqueté de manière aléatoire, mais il était toujours possible d'obtenir la même précision de formation que celle que vous obtiendriez sur un ensemble de données correctement étiqueté.", "from_community_srt": "la précision des tests ne serait pas meilleure que aléatoire puisque tout est étiqueté au hasard Mais il était toujours capable d'obtenir la même précision de formation que sur un ensemble de données correctement étiqueté", "n_reviews": 0, @@ -939,7 +939,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Si vous regardez cette courbe de précision, si vous vous entraîniez simplement sur un ensemble de données aléatoires, cette courbe descendait très lentement, de manière presque linéaire, donc vous avez vraiment du mal à trouver ce minimum local de possible, vous savez. , les bons poids qui vous apporteraient cette précision.", "from_community_srt": "Ensemble de données de ce qu'est la classification correcte et deux d'entre vous connaissent une demi-année plus tard à ICML cette année Il n'y avait pas exactement papier de réfutation qui a adressé à certains demandé comme hey En fait, ces réseaux font quelque chose d'un peu plus intelligent que si vous regardez cette courbe de précision si vous vous entraîniez sur un Un ensemble de données aléatoires dont la courbe est en quelque sorte tombée très lentement, presque de façon linéaire Donc, vous avez vraiment du mal à trouver les minima locaux possibles vous connaissez les bons poids qui vous apporteraient cette précision alors que si vous vous entraînez réellement sur un ensemble de données structuré", "n_reviews": 0, @@ -979,7 +979,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "Je tiens également à remercier tout particulièrement la société de capital-risque Amplify Partners, pour son soutien à ces premières vidéos de la série.", "from_community_srt": "je Aussi envie de donner une spéciale. Merci à la firme de capital-risque partenaires amplifi dans leur soutien de ces premières vidéos de la série", "n_reviews": 0, diff --git a/2017/gradient-descent/german/sentence_translations.json b/2017/gradient-descent/german/sentence_translations.json index ee298119a..c339fd2b2 100644 --- a/2017/gradient-descent/german/sentence_translations.json +++ b/2017/gradient-descent/german/sentence_translations.json @@ -352,7 +352,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "Außerdem solltest du beachten, dass die Schrittgröße proportional zur Steigung ist. Wenn die Steigung zum Minimum hin abflacht, werden deine Schritte immer kleiner und das hilft dir, nicht zu weit zu gehen.", "model": "DeepL", "n_reviews": 0, @@ -936,7 +936,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Natürlich war die Testgenauigkeit hier nicht besser als beim Zufallsprinzip, da alles nur zufällig beschriftet ist, aber es konnte trotzdem die gleiche Trainingsgenauigkeit wie bei einem richtig beschrifteten Datensatz erreicht werden.", "model": "DeepL", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Wenn du dir die Genauigkeitskurve ansiehst und mit einem zufälligen Datensatz trainierst, geht die Kurve sehr langsam und fast linear nach unten, sodass du wirklich darum kämpfst, ein lokales Minimum möglicher Gewichte zu finden, die dir diese Genauigkeit bringen.", "model": "DeepL", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "Ein besonderer Dank gilt auch der VC-Firma Amplify Partners, die diese ersten Videos der Reihe unterstützt hat.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/gradient-descent/greek/sentence_translations.json b/2017/gradient-descent/greek/sentence_translations.json index c74eb5b82..7b2d65963 100644 --- a/2017/gradient-descent/greek/sentence_translations.json +++ b/2017/gradient-descent/greek/sentence_translations.json @@ -352,7 +352,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "Θέλω επίσης να παρατηρήσετε ότι αν κάνετε τα μεγέθη των βημάτων σας ανάλογα με την κλίση, τότε όταν η κλίση εξομαλύνεται προς το ελάχιστο, τα βήματά σας γίνονται όλο και μικρότερα, και αυτό σας βοηθάει να μην υπερβείτε τα όρια.", "model": "DeepL", "n_reviews": 0, @@ -936,7 +936,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Προφανώς, η ακρίβεια δοκιμής εδώ δεν ήταν καλύτερη από την τυχαία, αφού όλα είναι τυχαία επισημασμένα, αλλά ήταν ακόμα σε θέση να επιτύχει την ίδια ακρίβεια εκπαίδευσης με αυτή που θα είχατε σε ένα σωστά επισημασμένο σύνολο δεδομένων.", "model": "DeepL", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Αν κοιτάξετε αυτή την καμπύλη ακρίβειας, αν απλά εκπαιδεύατε σε ένα τυχαίο σύνολο δεδομένων, αυτή η καμπύλη θα κατέβαινε πολύ αργά με σχεδόν γραμμικό τρόπο, οπότε πραγματικά αγωνίζεστε να βρείτε τα τοπικά ελάχιστα των πιθανών, ξέρετε, των σωστών βαρών που θα σας έδιναν αυτή την ακρίβεια.", "model": "DeepL", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "Θέλω επίσης να ευχαριστήσω ιδιαίτερα την εταιρεία VC Amplify Partners για την υποστήριξή της σε αυτά τα πρώτα βίντεο της σειράς.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/gradient-descent/hebrew/sentence_translations.json b/2017/gradient-descent/hebrew/sentence_translations.json index e035ee094..e3b209a87 100644 --- a/2017/gradient-descent/hebrew/sentence_translations.json +++ b/2017/gradient-descent/hebrew/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "אני גם רוצה שתשים לב איך אם אתה הופך את גדלי הצעדים שלך לפרופורציונליים למדרון, אז כשהשיפוע משתטח לכיוון המינימום, הצעדים שלך הולכים וקטנים, וזה עוזר לך לחרוג.", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "ברור שדיוק הבדיקה כאן לא היה טוב יותר מאקראי, מכיוון שהכל פשוט מסומן באקראי, אבל הוא עדיין הצליח להשיג את אותו דיוק אימון כפי שאתה משיג במערך נתונים מסומן כהלכה.", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "אם אתה מסתכל על עקומת הדיוק הזו, אם רק היית מתאמן על מערך נתונים אקראי, העקומה הזו ירדה לאט מאוד בצורה כמעט ליניארית, אז אתה באמת מתקשה למצוא את המינימום המקומי האפשרי הזה, אתה יודע , המשקולות הנכונות שיביאו לך את הדיוק הזה.", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "אני גם רוצה להודות במיוחד לחברת ה-VC Amplify Partners, בתמיכתם בסרטונים הראשונים בסדרה.", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/hindi/sentence_translations.json b/2017/gradient-descent/hindi/sentence_translations.json index 27b0a4ba6..3d95f6a74 100644 --- a/2017/gradient-descent/hindi/sentence_translations.json +++ b/2017/gradient-descent/hindi/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "मैं यह भी देखना चाहता हूं कि यदि आप अपने कदमों का आकार ढलान के समानुपाती बनाते हैं, तो जब ढलान न्यूनतम की ओर समतल हो जाता है, तो आपके कदम छोटे और छोटे होते जाते हैं, और इससे आपको ओवरशूटिंग से बचने में मदद मिलती है।", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "स्पष्ट रूप से यहां परीक्षण सटीकता यादृच्छिक से बेहतर नहीं थी, क्योंकि सब कुछ बस यादृच्छिक रूप से लेबल किया गया है, लेकिन यह अभी भी उसी प्रशिक्षण सटीकता को प्राप्त करने में सक्षम था जैसा कि आप उचित रूप से लेबल किए गए डेटासेट पर करेंगे।", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "यदि आप उस सटीकता वक्र को देखते हैं, यदि आप बस एक यादृच्छिक डेटासेट पर प्रशिक्षण ले रहे थे, तो वह वक्र लगभग एक रैखिक फैशन में बहुत धीरे-धीरे नीचे चला गया, इसलिए आप वास्तव में संभव के उस स्थानीय न्यूनतम को खोजने के लिए संघर्ष कर रहे हैं, आप जानते हैं , सही वज़न जो आपको वह सटीकता दिलाएगा।", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "मैं श्रृंखला के इन शुरुआती वीडियो के समर्थन के लिए वीसी फर्म एम्प्लीफाई पार्टनर्स को भी विशेष धन्यवाद देना चाहता हूं।", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/hungarian/sentence_translations.json b/2017/gradient-descent/hungarian/sentence_translations.json index e677ad307..82736b773 100644 --- a/2017/gradient-descent/hungarian/sentence_translations.json +++ b/2017/gradient-descent/hungarian/sentence_translations.json @@ -394,7 +394,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "Azt is szeretném, ha észrevennéd, hogy ha a lépések méretét a lejtővel arányossá teszed, akkor amikor a lejtő a minimum felé ellaposodik, a lépéseid egyre kisebbek lesznek, és ez segít a túllövéstől.", "model": "DeepL", "from_community_srt": "és azt is szeretném, ha észreveszed Hogyan csinálhatja lépcsőméreteit a lejtővel arányosnak? Akkor, amikor a lejtés a minimálisra süllyed, lépései kisebbek és kisebbek lesznek,", @@ -1049,7 +1049,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Nyilvánvaló, hogy a tesztelési pontosság itt sem volt jobb, mint a véletlenszerű, mivel minden csak véletlenszerűen van címkézve, de még mindig képes volt ugyanazt a képzési pontosságot elérni, mint egy megfelelően címkézett adathalmazon.", "model": "DeepL", "from_community_srt": "hogy a tesztelési pontosság itt nem jobb, mint véletlenszerű, mivel mindent véletlenszerűen címkézett De még mindig képes volt ugyanazt a képzési pontosságot elérni,", @@ -1067,7 +1067,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Ha megnézzük a pontossági görbét, ha csak egy véletlenszerű adathalmazon edzenénk, akkor ez a görbe nagyon lassan, szinte lineárisan csökkenne, tehát tényleg küzdünk, hogy megtaláljuk a lehetséges helyi minimumokat, a megfelelő súlyokat, amelyekkel elérhetjük a pontosságot.", "model": "DeepL", "from_community_srt": "memorizálni az egészet Adja meg, hogy mi a helyes besorolás, tehát néhányan fél évvel később az ICML-ben ismerkedtek meg idén Nem volt pontosan a fellebbező papíralap, amely a megkérdezetteknek szólt, mint a hé Valójában ezek a hálózatok valami okosabbat csinálnak, mint ha megnézzük azt a pontossági görbét ha csak egy a Véletlen adatkészlet, hogy a görbe fajta lement nagyon tudod nagyon lassan szinte egyfajta lineáris módon Tehát tényleg igyekszel megtalálni a lehetséges helyi minimumokat tudod a megfelelő súlyokat, amelyek megkapnák a pontosságot,", @@ -1112,7 +1112,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "Külön köszönetet szeretnék mondani az Amplify Partners nevű kockázatitőke-cégnek is, amely támogatta a sorozat első videóit.", "model": "DeepL", "from_community_srt": "hogy egy különleges. A VC cégnek köszönhetően erősíti a partnereit a sorozat kezdeti videóinak támogatásában", diff --git a/2017/gradient-descent/indonesian/sentence_translations.json b/2017/gradient-descent/indonesian/sentence_translations.json index b32a546f7..1f3c8c518 100644 --- a/2017/gradient-descent/indonesian/sentence_translations.json +++ b/2017/gradient-descent/indonesian/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "Saya juga ingin Anda memperhatikan bagaimana jika Anda membuat ukuran langkah Anda proporsional dengan kemiringan, maka ketika kemiringannya mendatar ke arah minimum, langkah Anda akan semakin kecil, dan itu membantu Anda menghindari overshooting.", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Tentu saja akurasi pengujian di sini tidak lebih baik daripada pengujian acak, karena semuanya hanya diberi label secara acak, namun akurasi pelatihan masih dapat dicapai seperti yang Anda lakukan pada kumpulan data yang diberi label dengan benar.", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Jika Anda melihat kurva akurasi tersebut, jika Anda hanya berlatih pada kumpulan data acak, kurva tersebut turun sangat lambat hampir seperti gaya linier, jadi Anda benar-benar kesulitan menemukan kemungkinan minimum lokal tersebut, Anda tahu , bobot yang tepat yang akan memberi Anda akurasi tersebut.", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "Saya juga ingin mengucapkan terima kasih khusus kepada perusahaan VC Amplify Partners, atas dukungan mereka terhadap video awal dalam seri ini.", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/italian/sentence_translations.json b/2017/gradient-descent/italian/sentence_translations.json index e28f92a30..a29cbfe25 100644 --- a/2017/gradient-descent/italian/sentence_translations.json +++ b/2017/gradient-descent/italian/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "Voglio anche che tu noti come se rendi le dimensioni dei tuoi passi proporzionali alla pendenza, quando la pendenza si appiattisce verso il minimo, i tuoi passi diventano sempre più piccoli e questo ti aiuta a non superare il limite.", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Ovviamente la precisione del test qui non era migliore di quella casuale, poiché tutto è etichettato in modo casuale, ma è stato comunque in grado di ottenere la stessa precisione di addestramento che si otterrebbe su un set di dati correttamente etichettato.", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Se guardi quella curva di precisione, se ti stessi allenando su un set di dati casuale, quella curva scendeva molto lentamente in modo quasi lineare, quindi fai davvero fatica a trovare quel minimo locale di possibile, sai , i pesi giusti che ti darebbero quella precisione.", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "Voglio anche ringraziare in modo speciale la società di VC Amplify Partners, per il suo supporto a questi video iniziali della serie.", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/japanese/sentence_translations.json b/2017/gradient-descent/japanese/sentence_translations.json index b2518b937..795a96c1c 100644 --- a/2017/gradient-descent/japanese/sentence_translations.json +++ b/2017/gradient-descent/japanese/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "また、ステップ サイズを勾配に比例させた場合、勾配が最小に向かって平坦になるとステップがどんどん小さくなり、オーバーシュートが防止されることにも注目してください。", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "すべてがランダムにラベル付けされているだけであるため、ここでのテスト精度は明らかにランダムよりも優れていませんでしたが、それでも適切にラベル付けされたデータセットで行うのと同じトレーニング精度を達成することができました。", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "その精度曲線を見ると、ランダムなデータセットでトレーニングしているだけだとすると、その曲線はほぼ直線的に非常にゆっくりと下がっていくので、可能な極小値を見つけるのに本当に苦労しています。 、その精度を実現する適切な重み。", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "また、シリーズの最初のビデオをサポートしてくれたベンチャーキャピタル企業 Amplify Partners にも特別な感謝を表したいと思います。", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/korean/sentence_translations.json b/2017/gradient-descent/korean/sentence_translations.json index c4f41b645..3a3a56709 100644 --- a/2017/gradient-descent/korean/sentence_translations.json +++ b/2017/gradient-descent/korean/sentence_translations.json @@ -388,7 +388,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "또한 스텝 크기를 경사에 비례하게 만들면 경사가 최소가 될수록 스텝이 점점 작아져 오버슈팅을 방지하는 데 도움이 된다는 점도 알아두세요.", "model": "DeepL", "from_community_srt": "또 여러분에게 알려드릴 게 있습니다. 만약 당신이 한번에 이동할 거리를 기울기에 비례해서 결정한다면 기울기가 줄어들수록 한번에 이동하는 거리가 점점 작아지고 이는 오버 슈팅을 방지하는 데 도움이됩니다.", @@ -1040,7 +1040,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "모든 것이 무작위로 레이블이 지정되었기 때문에 테스트 정확도는 무작위보다 떨어지지만, 그래도 제대로 레이블이 지정된 데이터 세트에서와 동일한 학습 정확도를 달성할 수 있었습니다.", "model": "DeepL", "from_community_srt": "테스트 정확도는 그냥 무작위로 나올 것처럼 보입니다. 모두 무작위로 라벨링 돼있으니까요. 하지만 제대로 라벨링된 데이터셋으로 학습한 것과 동일한 테스트 정확도를 달성했습니다.", @@ -1058,7 +1058,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "정확도 곡선을 보면, 무작위 데이터 세트로 훈련하는 경우 이 곡선은 거의 선형적인 방식으로 매우 느리게 내려가므로 정확도를 얻을 수 있는 적절한 가중치를 찾기 위해 정말 고군분투하고 있습니다.", "model": "DeepL", "from_community_srt": "올바른 분류가 무엇인지에 대한 데이터셋을 기억하는 거예요. 올해 ICML엔 반 년 동안 이를 반박하는 논문이 없었어요. 이렇게요. \"야! 신경망은 그것보다 더 똑똑해.\" 이 정확도 곡선을 보면 만약 당신이 무작위 데이터셋으로 훈련을 시작했다면 곡선이 거의 직선에 가깝게 아주 천천히 하강합니다. 당신이 구조화된 데이터셋,", @@ -1103,7 +1103,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "또한 이 시리즈의 첫 번째 동영상을 지원해 준 VC 회사인 Amplify Partners에 특별히 감사의 말씀을 전하고 싶습니다.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/gradient-descent/marathi/sentence_translations.json b/2017/gradient-descent/marathi/sentence_translations.json index a7c7a0087..8e551410a 100644 --- a/2017/gradient-descent/marathi/sentence_translations.json +++ b/2017/gradient-descent/marathi/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "मी तुम्हाला हे देखील लक्षात घ्यायचे आहे की जर तुम्ही तुमच्या पायऱ्यांचा आकार उताराच्या प्रमाणात कसा बनवलात, तर जेव्हा उतार किमान दिशेने सपाट होत असेल, तेव्हा तुमच्या पायऱ्या लहान होत जातात आणि त्यामुळे तुम्हाला ओव्हरशूटिंगपासून मदत होते.", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "साहजिकच येथे चाचणी अचूकता यादृच्छिक पेक्षा चांगली नव्हती, कारण प्रत्येक गोष्ट यादृच्छिकपणे लेबल केलेली आहे, परंतु तरीही ती योग्यरित्या लेबल केलेल्या डेटासेटवर तुम्ही समान प्रशिक्षण अचूकता प्राप्त करण्यास सक्षम होती.", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "तुम्ही अचूकता वक्र पाहिल्यास, जर तुम्ही फक्त एका यादृच्छिक डेटासेटवर प्रशिक्षण घेत असाल, तर वक्र क्रमवारी जवळजवळ एक रेखीय फॅशनमध्ये अतिशय हळू हळू खाली गेली आहे, त्यामुळे तुम्हाला शक्य तितके स्थानिक किमान शोधण्यासाठी खरोखरच संघर्ष करावा लागत आहे, तुम्हाला माहिती आहे. , योग्य वजन ज्यामुळे तुम्हाला अचूकता मिळेल.", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "VC फर्म Amplify Partners चे देखील मी विशेष आभार मानू इच्छितो, त्यांनी मालिकेतील या सुरुवातीच्या व्हिडिओंच्या समर्थनार्थ.", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/persian/sentence_translations.json b/2017/gradient-descent/persian/sentence_translations.json index 49c727884..a7d29750a 100644 --- a/2017/gradient-descent/persian/sentence_translations.json +++ b/2017/gradient-descent/persian/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "همچنین می‌خواهم توجه داشته باشید که اگر اندازه‌های گام‌هایتان را با شیب متناسب کنید، وقتی شیب به سمت حداقل می‌رود، قدم‌هایتان کوچک‌تر و کوچک‌تر می‌شوند و این به شما کمک می‌کند که بیش از حد شیب نکنید.", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "بدیهی است که دقت آزمایش در اینجا بهتر از تصادفی نبود، زیرا همه چیز فقط به صورت تصادفی برچسب‌گذاری شده است، اما همچنان می‌توانست به همان دقت آموزشی که در مجموعه داده‌های دارای برچسب مناسب می‌رسید، دست یابد.", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "اگر به آن منحنی دقت نگاه کنید، اگر فقط روی یک مجموعه داده تصادفی تمرین می‌کردید، آن منحنی به آرامی به شکلی تقریباً خطی پایین می‌آید، بنابراین شما واقعاً در تلاش برای یافتن آن حداقل ممکن محلی هستید، می‌دانید ، وزنه های مناسبی که به شما این دقت را می دهد.", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "همچنین می‌خواهم تشکر ویژه‌ای از شرکت VC Amplify Partners داشته باشم که از این ویدیوهای اولیه در این سری حمایت می‌کند.", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/portuguese/sentence_translations.json b/2017/gradient-descent/portuguese/sentence_translations.json index 8d3be90b8..ae1cade49 100644 --- a/2017/gradient-descent/portuguese/sentence_translations.json +++ b/2017/gradient-descent/portuguese/sentence_translations.json @@ -396,7 +396,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "Também quero que você observe como se você fizer o tamanho dos seus passos proporcionais à inclinação, quando a inclinação estiver se achatando em direção ao mínimo, seus passos ficarão cada vez menores, e isso o ajudará a não ultrapassar os limites.", "model": "google_nmt", "from_community_srt": "E também quero que observe que, se você tornar os tamanhos das etapas proporcionais à inclinação, quando a inclinação estiver se achatando em direção ao mínimo, as etapas vão ficar cada vez menores, e isso ajuda a evitar passar do ponto.", @@ -1053,7 +1053,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Obviamente, a precisão do teste aqui não foi melhor do que aleatória, já que tudo é rotulado aleatoriamente, mas ainda foi capaz de atingir a mesma precisão de treinamento que você alcançaria em um conjunto de dados devidamente rotulado.", "model": "google_nmt", "from_community_srt": "Óbvio que a precisão do teste não seria melhor do que o acaso, já que tudo estava rotulado aleatoriamente. Mas ela ainda conseguiu a mesma precisão de treinamento que conseguiria num conjunto de dados rotulado adequadamente.", @@ -1071,7 +1071,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Se você olhar para aquela curva de precisão, se você estivesse apenas treinando em um conjunto de dados aleatório, essa curva desceu muito lentamente, quase de forma linear, então você está realmente lutando para encontrar os mínimos locais possíveis, você sabe , os pesos certos que proporcionariam essa precisão.", "model": "google_nmt", "from_community_srt": "memorizou todo o conjunto de dados sobre qual é a classificação correta. E, então, seis meses depois, no ICML este ano, houve não exatamente um artigo-réplica, um artigo que aborda a ideia de que, na verdade, essas redes estão fazendo algo um pouco mais inteligente do que isso. Se você olhar aquela curva de precisão, se você estivesse só treinando num conjunto de dados aleatório, essa curva desceria muito devagar, quase linearmente. Então, você tem muita dificuldade para encontrar o mínimo local, os pesos corretos que lhe dariam essa precisão.", @@ -1116,7 +1116,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "Também quero agradecer especialmente à empresa de capital de risco Amplify Partners, pelo apoio a esses vídeos iniciais da série.", "model": "google_nmt", "from_community_srt": "Também quero agradecer especialmente à empresa de capital de risco Amplify Partners, e ao seu apoio a estes vídeos iniciais da série.", diff --git a/2017/gradient-descent/romanian/sentence_translations.json b/2017/gradient-descent/romanian/sentence_translations.json index f2bc1fe0e..49434d689 100644 --- a/2017/gradient-descent/romanian/sentence_translations.json +++ b/2017/gradient-descent/romanian/sentence_translations.json @@ -352,7 +352,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "De asemenea, vreau să observați cum, dacă mărim pașii proporțional cu panta, atunci când panta se aplatizează spre minim, pașii devin din ce în ce mai mici, ceea ce vă ajută să nu depășiți limita.", "model": "DeepL", "n_reviews": 0, @@ -936,7 +936,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Evident, acuratețea testelor nu a fost mai bună decât cea aleatorie, deoarece totul este etichetat la întâmplare, dar a reușit să obțină aceeași acuratețe de instruire ca și în cazul unui set de date etichetat corespunzător.", "model": "DeepL", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Dacă vă uitați la curba de acuratețe, dacă v-ați antrena pe un set de date aleatoriu, curba a coborât foarte încet, aproape liniar, astfel încât vă străduiți să găsiți minimul local al posibilelor ponderi corecte care să vă asigure această acuratețe.", "model": "DeepL", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "De asemenea, doresc să mulțumesc în mod special firmei de capital de risc Amplify Partners, care a sprijinit aceste videoclipuri inițiale din serie.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/gradient-descent/russian/sentence_translations.json b/2017/gradient-descent/russian/sentence_translations.json index 3e22b0813..62e18502b 100644 --- a/2017/gradient-descent/russian/sentence_translations.json +++ b/2017/gradient-descent/russian/sentence_translations.json @@ -351,7 +351,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "Я также хочу, чтобы вы заметили, что если вы сделаете размер шага пропорциональным наклону, то, когда уклон выравнивается к минимуму, ваши шаги будут становиться все меньше и меньше, и это поможет вам не промахнуться.", "from_community_srt": "и я также хочу, чтобы вы заметили Как сделать размеры шага пропорциональными наклону Затем, когда наклон сглаживается к минимуму, ваши шаги становятся все меньше и меньше, и этот вид помогает вам избежать перерегулирования Вместо того,", "n_reviews": 0, @@ -935,7 +935,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Очевидно, что точность тестирования здесь была не лучше, чем случайная, поскольку все помечено случайным образом, но все равно удалось достичь той же точности обучения, что и на правильно маркированном наборе данных.", "from_community_srt": "что точность тестирования здесь не лучше, чем случайная, поскольку все просто случайно помечено Но он все еще мог достичь такой же точности обучения,", "n_reviews": 0, @@ -951,7 +951,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Если вы посмотрите на эту кривую точности, если бы вы просто тренировались на случайном наборе данных, эта кривая как бы снижалась очень медленно, почти линейно, так что вы действительно изо всех сил пытаетесь найти этот локальный минимум возможных, вы знаете , правильные веса, которые обеспечат вам такую точность.", "from_community_srt": "Набор данных о том, какова правильная классификация, и поэтому пара из вас знает полгода спустя в ICML в этом году Не было точно отформатированной бумажной бумаги, которая обращалась к некоторым из них: Фактически, эти сети делают что-то немного умнее, если вы посмотрите на эту кривую точности если вы просто тренировались на Случайные данные показывают, что тип кривой спустился очень точно, вы знаете очень медленно почти линейным способом Таким образом, вы действительно пытаетесь найти, что местные минимумы возможных вы знаете правильные веса, которые доставят вам такую ​​точность,", "n_reviews": 0, @@ -991,7 +991,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "Я также хочу выразить особую благодарность венчурной фирме Amplify Partners за поддержку этих первых видеороликов из этой серии.", "from_community_srt": "Также хочу дать специальный. Благодаря партнерам VC фирмы amplifi в поддержке этих первых видеороликов в серии", "n_reviews": 0, diff --git a/2017/gradient-descent/spanish/sentence_translations.json b/2017/gradient-descent/spanish/sentence_translations.json index 8c8d7adac..fae06c161 100644 --- a/2017/gradient-descent/spanish/sentence_translations.json +++ b/2017/gradient-descent/spanish/sentence_translations.json @@ -348,7 +348,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "También quiero que notes cómo si haces que el tamaño de tus pasos sea proporcional a la pendiente, cuando la pendiente se aplana hacia el mínimo, tus pasos se vuelven cada vez más pequeños, y eso te ayuda a no sobrepasarte.", "from_community_srt": "y también quiero que se den cuenta Cómo si haces el tamaño proporcional a la pendiente de su paso Luego, cuando la pendiente se aplana hacia el mínimo de sus pasos se hacen más pequeños y más pequeños y ese tipo de ayuda que caigan demasiado", "n_reviews": 0, @@ -922,7 +922,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Obviamente, la precisión de las pruebas aquí no fue mejor que la aleatoria, ya que todo está etiquetado simplemente al azar, pero aun así fue capaz de lograr la misma precisión de entrenamiento que lo haría en un conjunto de datos correctamente etiquetado.", "from_community_srt": "la exactitud de las pruebas que aquí iba a ser mejor que el azar ya que todo acaba de etiquetado al azar Pero todavía era capaz de conseguir la misma precisión formación como lo haría en un conjunto de datos debidamente etiquetado", "n_reviews": 0, @@ -938,7 +938,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Si observas esa curva de precisión, si solo estuvieras entrenando en un conjunto de datos aleatorio, esa curva descendió muy lentamente de manera casi lineal, por lo que realmente estás luchando por encontrar esos mínimos locales posibles, ya sabes. , los pesos correctos que le brindarían esa precisión.", "from_community_srt": "conjunto de datos de lo que es la correcta clasificación y así un par de ustedes saben la mitad de un año después en ICML este año No hubo réplicas por exactamente el papel de papel que se refirió a algunas pidió gusta Hey En realidad, estas redes están haciendo algo un poco más inteligente que eso si nos fijamos en que la curva de precisión si se acaba de entrenamiento en una datos aleatorios establecidos que la curva de tipo de fuimos muy sabes muy lentamente en casi una especie de forma lineal Por lo que está realmente luchando para encontrar que los mínimos locales de la posible conoce las ponderaciones adecuadas que le conseguiría que la precisión mientras que si en realidad estás entrenando en un conjunto estructurado de datos que tiene el", "n_reviews": 0, @@ -978,7 +978,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "También quiero agradecer especialmente a la firma de capital riesgo Amplify Partners, por su apoyo a estos videos iniciales de la serie.", "from_community_srt": "Gracias a los socios amplifi firmes en su apoyo VC de estos videos iniciales de la serie", "n_reviews": 0, diff --git a/2017/gradient-descent/tagalog/sentence_translations.json b/2017/gradient-descent/tagalog/sentence_translations.json index 663365155..2ea6e662d 100644 --- a/2017/gradient-descent/tagalog/sentence_translations.json +++ b/2017/gradient-descent/tagalog/sentence_translations.json @@ -352,7 +352,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "Gusto ko ring mapansin mo kung paano kung gagawin mong proporsyonal ang mga laki ng iyong hakbang sa slope, at kapag ang slope ay unti-unting lumalapad, unti-unting lumiliit ang iyong mga hakbang, at nakakatulong iyon sa iyong mag-overshoot.", "model": "google_nmt", "n_reviews": 0, @@ -936,7 +936,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Malinaw na ang katumpakan ng pagsubok dito ay hindi mas mahusay kaysa sa random, dahil ang lahat ay random na may label, ngunit nagawa pa rin nitong makamit ang parehong katumpakan ng pagsasanay tulad ng gagawin mo sa isang maayos na naka-label na dataset.", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Kung titingnan mo ang curve ng katumpakan na iyon, kung nagsasanay ka lang sa isang random na dataset, ang uri ng curve na iyon ay bumaba nang napakabagal sa halos uri ng isang linear na paraan, kaya talagang nahihirapan kang hanapin ang lokal na minimum na posible, alam mo , ang mga tamang timbang na magbibigay sa iyo ng ganoong katumpakan.", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "Gusto ko ring magbigay ng espesyal na pasasalamat sa VC firm na Amplify Partners, sa kanilang suporta sa mga unang video na ito sa serye.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/gradient-descent/tamil/sentence_translations.json b/2017/gradient-descent/tamil/sentence_translations.json index 7b41d707c..67e850ee7 100644 --- a/2017/gradient-descent/tamil/sentence_translations.json +++ b/2017/gradient-descent/tamil/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "உங்கள் படி அளவுகளை சாய்வுக்கு விகிதாசாரமாக மாற்றினால், சாய்வு குறைந்தபட்சத்தை நோக்கி தட்டையாக இருக்கும்போது, உங்கள் படிகள் சிறியதாகவும் சிறியதாகவும் இருக்கும், மேலும் அது எப்படி அதிகமாக படமெடுப்பதில் இருந்து உங்களுக்கு உதவுகிறது என்பதையும் நீங்கள் கவனிக்க வேண்டும்.", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "இங்கே சோதனை துல்லியம் சீரற்றதை விட சிறப்பாக இல்லை, ஏனென்றால் எல்லாமே தோராயமாக லேபிளிடப்பட்டுள்ளன, ஆனால் சரியாக லேபிளிடப்பட்ட தரவுத்தொகுப்பில் நீங்கள் அடையும் அதே பயிற்சி துல்லியத்தை இன்னும் அடைய முடிந்தது.", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "நீங்கள் அந்த துல்லிய வளைவைப் பார்த்தால், நீங்கள் ஒரு சீரற்ற தரவுத்தொகுப்பில் பயிற்சி பெற்றிருந்தால், அந்த வளைவு கிட்டத்தட்ட ஒரு நேரியல் பாணியில் மிக மெதுவாக கீழே சென்றது, எனவே சாத்தியமான உள்ளூர் மினிமாவைக் கண்டுபிடிக்க நீங்கள் மிகவும் சிரமப்படுகிறீர்கள் என்பது உங்களுக்குத் தெரியும். , சரியான எடைகள் உங்களுக்கு அந்த துல்லியத்தைப் பெறும்.", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "இந்தத் தொடரில் இந்த ஆரம்ப வீடியோக்களுக்கு ஆதரவளித்த VC நிறுவனமான ஆம்ப்லிஃபை பார்ட்னர்களுக்கும் நான் சிறப்பு நன்றியைத் தெரிவிக்க விரும்புகிறேன்.", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/telugu/sentence_translations.json b/2017/gradient-descent/telugu/sentence_translations.json index 90fbab1a8..212522116 100644 --- a/2017/gradient-descent/telugu/sentence_translations.json +++ b/2017/gradient-descent/telugu/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "మీరు మీ దశల పరిమాణాలను వాలుకు అనులోమానుపాతంలో చేస్తే, వాలు కనిష్ట స్థాయికి చదునుగా ఉన్నప్పుడు, మీ అడుగులు చిన్నవిగా మరియు చిన్నవిగా ఉంటాయి మరియు అది ఓవర్‌షూట్ నుండి మీకు ఎలా సహాయపడుతుందో కూడా మీరు గమనించాలని నేను కోరుకుంటున్నాను.", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "సహజంగానే ఇక్కడ టెస్టింగ్ ఖచ్చితత్వం యాదృచ్ఛికం కంటే మెరుగైనది కాదు, ఎందుకంటే ప్రతిదీ యాదృచ్ఛికంగా లేబుల్ చేయబడింది, కానీ సరిగ్గా లేబుల్ చేయబడిన డేటాసెట్‌లో మీరు సాధించిన అదే శిక్షణ ఖచ్చితత్వాన్ని ఇది ఇప్పటికీ సాధించగలిగింది.", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "మీరు ఆ ఖచ్చితత్వ వక్రతను పరిశీలిస్తే, మీరు కేవలం యాదృచ్ఛిక డేటాసెట్‌లో శిక్షణ పొందుతున్నట్లయితే, ఆ వక్రరేఖ దాదాపు సరళ పద్ధతిలో చాలా నెమ్మదిగా తగ్గుతుంది, కాబట్టి మీరు సాధ్యమయ్యే స్థానిక కనిష్టాన్ని కనుగొనడంలో నిజంగా కష్టపడుతున్నారని మీకు తెలుసు. , మీరు ఆ ఖచ్చితత్వాన్ని పొందే సరైన బరువులు.", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "సిరీస్‌లోని ఈ ప్రారంభ వీడియోలకు మద్దతుగా VC సంస్థ యాంప్లిఫై పార్ట్‌నర్‌లకు కూడా నేను ప్రత్యేక ధన్యవాదాలు చెప్పాలనుకుంటున్నాను.", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/thai/sentence_translations.json b/2017/gradient-descent/thai/sentence_translations.json index fdc688ecb..39dcd85d3 100644 --- a/2017/gradient-descent/thai/sentence_translations.json +++ b/2017/gradient-descent/thai/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "ฉันยังอยากให้คุณสังเกตด้วยว่าหากคุณทำให้ขนาดขั้นบันไดเป็นสัดส่วนกับความชัน แล้วเมื่อความชันแบนราบไปทางขั้นต่ำ ขั้นตอนของคุณจะเล็กลงเรื่อยๆ และนั่นจะช่วยคุณจากการถ่ายภาพเกินขนาด", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "แน่นอนว่าความแม่นยำในการทดสอบที่นี่ไม่ได้ดีไปกว่าการสุ่ม เนื่องจากทุกอย่างเป็นเพียงป้ายกำกับแบบสุ่ม แต่ก็ยังสามารถบรรลุความแม่นยำในการฝึกอบรมแบบเดียวกับที่คุณทำในชุดข้อมูลที่มีป้ายกำกับอย่างถูกต้อง", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "หากคุณดูกราฟความแม่นยำนั้น หากคุณแค่ฝึกกับชุดข้อมูลสุ่ม เส้นโค้งนั้นจะลดลงอย่างช้าๆ ในลักษณะเชิงเส้นตรง ดังนั้นคุณจึงลำบากมากที่จะหาค่าต่ำสุดเฉพาะจุดที่เป็นไปได้ ตุ้มน้ำหนักที่เหมาะสมซึ่งจะทำให้คุณได้รับความแม่นยำนั้น", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "ฉันยังอยากจะแสดงความขอบคุณเป็นพิเศษต่อบริษัท VC Amplify Partners ในการสนับสนุนวิดีโอเริ่มต้นเหล่านี้ในซีรีส์นี้", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/turkish/sentence_translations.json b/2017/gradient-descent/turkish/sentence_translations.json index 46c8f713c..c5c4bd25e 100644 --- a/2017/gradient-descent/turkish/sentence_translations.json +++ b/2017/gradient-descent/turkish/sentence_translations.json @@ -393,7 +393,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "Ayrıca, adım boyutlarınızı eğimle orantılı hale getirirseniz, eğim minimuma doğru düzleşirken adımlarınızın nasıl küçüldüğünü ve bunun aşırıya kaçmanıza nasıl yardımcı olduğunu fark etmenizi istiyorum.", "model": "DeepL", "from_community_srt": "ve ben de seni fark etmeni istiyorum. Adım boyutlarınızı eğim ile orantılı olarak nasıl yaparsınız? Daha sonra, eğim minimum düzeye doğru düzleştiğinde, adımlarınız küçülür ve daha küçüktür ve bu tipler aşırı çekimden size yardımcı olur.", @@ -1041,7 +1041,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Açıkçası buradaki test doğruluğu rastgele olandan daha iyi değildi, çünkü her şey rastgele etiketlenmişti, ancak yine de uygun şekilde etiketlenmiş bir veri kümesinde elde edeceğiniz aynı eğitim doğruluğunu elde edebildi.", "model": "DeepL", "from_community_srt": "her şey rastgele etiketlendiğinden, rasgele olmaktan daha iyi olmayacaktı. Fakat yine de düzgün etiketlenmiş veri kümesinde yaptığınız gibi aynı eğitim doğruluğunu elde edebildi.", @@ -1059,7 +1059,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Bu doğruluk eğrisine bakarsanız, rastgele bir veri kümesi üzerinde eğitim yapıyor olsaydınız, bu eğri neredeyse doğrusal bir şekilde çok yavaş bir şekilde aşağı inerdi, bu nedenle size bu doğruluğu sağlayacak olası doğru ağırlıkların yerel minimumunu bulmak için gerçekten mücadele ediyorsunuz.", "model": "DeepL", "from_community_srt": "tüm ezberlemek Doğru sınıflandırmanın ne olduğuna dair veri seti ve bu yüzden birkaçınız bu yıl ICML'de yarım yıl sonra biliyorsunuz Tamamen çığlık atan bir kağıt kağıdı yoktu. Aslında bu ağlar, bu doğruluk eğrisine bakarsanız biraz daha akıllı bir şey yapıyorlar. eğer sadece bir antrenman yapıyor olsaydın Rasgele veri seti, eğri türünün çok azaldığını biliyorsunuz. Yani gerçekten mümkün olan yerel azamı bulmakta zorlanıyorsunuz Eğer gerçek bir yapıya sahip bir veri seti üzerinde eğitim alıyorsanız,", @@ -1104,7 +1104,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "Ayrıca, serinin bu ilk videolarına destek veren VC firması Amplify Partners'a da özel olarak teşekkür etmek istiyorum.", "model": "DeepL", "from_community_srt": "Ayrıca özel bir vermek istiyorum. Serideki bu ilk videoları desteklemelerinde VC firması amplifi ortakları sayesinde", diff --git a/2017/gradient-descent/ukrainian/sentence_translations.json b/2017/gradient-descent/ukrainian/sentence_translations.json index dcfb42a44..bc8593b8a 100644 --- a/2017/gradient-descent/ukrainian/sentence_translations.json +++ b/2017/gradient-descent/ukrainian/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "Я також хочу, щоб ви помітили, що якщо ви робите розмір кроку пропорційним схилу, тоді, коли схил вирівнюється до мінімуму, ваші кроки стають все меншими, і це допомагає вам уникнути перевищення.", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Очевидно, що точність тестування тут була не кращою, ніж випадкова, оскільки все просто випадково позначено, але все одно вдалося досягти такої ж точності навчання, як і на правильно позначеному наборі даних.", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Якщо ви подивіться на цю криву точності, якби ви просто тренувалися на випадковому наборі даних, ця крива начебто опускалася дуже повільно майже лінійним способом, тож вам справді важко знайти той локальний мінімум можливого, ви знаєте правильні ваги, які забезпечать вам таку точність.", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "Я також хочу особливо подякувати фірмі венчурного капіталу Amplify Partners за підтримку цих перших відео в серії.", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/urdu/sentence_translations.json b/2017/gradient-descent/urdu/sentence_translations.json index 2ad4cf5e8..721c489dc 100644 --- a/2017/gradient-descent/urdu/sentence_translations.json +++ b/2017/gradient-descent/urdu/sentence_translations.json @@ -308,7 +308,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "میں آپ کو یہ بھی دیکھنا چاہتا ہوں کہ اگر آپ اپنے قدموں کے سائز کو ڈھلوان کے متناسب بناتے ہیں، تو جب ڈھلوان کم سے کم کی طرف چپٹا ہوتا ہے، تو آپ کے قدم چھوٹے سے چھوٹے ہوتے جاتے ہیں، اور یہ آپ کو اوور شوٹنگ سے بچانے میں مدد کرتا ہے۔", "n_reviews": 0, "start": 403.18, @@ -819,7 +819,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "ظاہر ہے کہ یہاں جانچ کی درستگی بے ترتیب سے بہتر نہیں تھی، کیونکہ ہر چیز پر صرف تصادفی طور پر لیبل لگا ہوا ہے، لیکن یہ پھر بھی وہی تربیت کی درستگی حاصل کرنے میں کامیاب رہا جیسا کہ آپ مناسب طریقے سے لیبل والے ڈیٹاسیٹ پر کرتے ہیں۔", "n_reviews": 0, "start": 1089.48, @@ -833,7 +833,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "اگر آپ اس درستگی کے منحنی خطوط پر نظر ڈالتے ہیں، اگر آپ صرف ایک بے ترتیب ڈیٹاسیٹ پر تربیت لے رہے تھے، تو وہ وکر کی قسم تقریباً ایک لکیری انداز میں بہت آہستہ آہستہ نیچے جاتی ہے، لہذا آپ واقعی اس مقامی کم سے کم ممکنہ کو تلاش کرنے کے لیے جدوجہد کر رہے ہیں، آپ جانتے ہیں ، صحیح وزن جو آپ کو درستگی حاصل کرے گا۔", "n_reviews": 0, "start": 1131.44, @@ -868,7 +868,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "میں سیریز میں ان ابتدائی ویڈیوز کی حمایت میں VC فرم Amplify Partners کا بھی خصوصی شکریہ ادا کرنا چاہتا ہوں۔", "n_reviews": 0, "start": 1207.46, diff --git a/2017/gradient-descent/vietnamese/sentence_translations.json b/2017/gradient-descent/vietnamese/sentence_translations.json index 905d6e4ad..979a731e2 100644 --- a/2017/gradient-descent/vietnamese/sentence_translations.json +++ b/2017/gradient-descent/vietnamese/sentence_translations.json @@ -390,7 +390,7 @@ "end": 402.62 }, { - "input": "I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that helps you from overshooting.", + "input": "And I also want you to notice how if you make your step sizes proportional to the slope, then when the slope is flattening out towards the minimum, your steps get smaller and smaller, and that kind of helps you from overshooting.", "translatedText": "Tôi cũng muốn bạn lưu ý rằng nếu bạn làm cho kích thước bước của mình tỷ lệ thuận với độ dốc, thì khi độ dốc giảm dần về mức tối thiểu, các bước của bạn sẽ ngày càng nhỏ hơn và điều đó giúp bạn không bị vượt quá.", "model": "google_nmt", "from_community_srt": "và tôi cũng muốn bạn chú ý Làm thế nào nếu bạn thực hiện kích thước bước của bạn tỷ lệ thuận với độ dốc Sau đó, khi độ dốc được làm phẳng về phía mức tối thiểu,", @@ -1042,7 +1042,7 @@ "end": 1088.74 }, { - "input": "Obviously the testing accuracy here was no better than random, since everything is just randomly labeled, but it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", + "input": "Obviously the testing accuracy here was going to be no better than random, since everything's just randomly labeled. But it was still able to achieve the same training accuracy as you would on a properly labeled dataset.", "translatedText": "Rõ ràng độ chính xác của thử nghiệm ở đây không tốt hơn ngẫu nhiên, vì mọi thứ chỉ được gắn nhãn ngẫu nhiên, nhưng nó vẫn có thể đạt được độ chính xác huấn luyện giống như bạn làm trên một tập dữ liệu được gắn nhãn chính xác.", "model": "google_nmt", "from_community_srt": "Đặt nó xáo trộn tất cả các nhãn xung quanh trước khi đào tạo Rõ ràng độ chính xác thử nghiệm ở đây sẽ không tốt hơn ngẫu nhiên vì mọi thứ chỉ được dán nhãn ngẫu nhiên Nhưng nó vẫn có thể đạt được độ chính xác đào tạo giống như bạn có trên một tập dữ liệu được dán nhãn đúng cách", @@ -1060,7 +1060,7 @@ "end": 1116.4 }, { - "input": "If you look at that accuracy curve, if you were just training on a random dataset, that curve sort of went down very slowly in almost kind of a linear fashion, so you're really struggling to find that local minima of possible, you know, the right weights that would get you that accuracy.", + "input": "...to memorize the entire dataset of what the correct classification is. And so a couple of, you know, half a year later at ICML this year, there was not exactly rebuttal paper, but paper that addressed some aspects of like, hey, actually these networks are doing something a little bit smarter than that. If you look at that accuracy c", "translatedText": "Nếu bạn nhìn vào đường cong chính xác đó, nếu bạn chỉ đang đào tạo trên một tập dữ liệu ngẫu nhiên, thì đường cong đó sẽ đi xuống rất chậm theo kiểu gần như tuyến tính, vì vậy bạn thực sự đang gặp khó khăn để tìm ra mức tối thiểu cục bộ có thể có, bạn biết đấy , trọng lượng phù hợp sẽ giúp bạn có được độ chính xác đó.", "model": "google_nmt", "from_community_srt": "Tập dữ liệu về phân loại chính xác là gì và vì vậy một vài bạn biết nửa năm sau tại ICML năm nay Không có giấy giấy chính xác nào đề cập đến một số người được hỏi như này Trên thực tế, các mạng này đang hoạt động thông minh hơn một chút nếu bạn nhìn vào đường cong chính xác đó nếu bạn chỉ tập luyện Bộ dữ liệu ngẫu nhiên mà đường cong sắp xếp đi xuống rất bạn biết rất chậm trong hầu hết các loại thời trang tuyến tính Vì vậy, bạn đang thực sự đấu tranh để tìm thấy rằng minima địa phương có thể bạn biết đúng trọng lượng sẽ giúp bạn có được độ chính xác đó trong khi nếu bạn đang thực sự đào tạo trên bộ dữ liệu có cấu trúc có", @@ -1104,7 +1104,7 @@ "end": 1206.8 }, { - "input": "I also want to give a special thanks to the VC firm Amplify Partners, in their support of these initial videos in the series.", + "input": "I also want to give a special thanks to the VC firm Amplify Partners and their support of these initial videos in the series. Thank you.", "translatedText": "Tôi cũng muốn gửi lời cảm ơn đặc biệt đến công ty Amplify Partners của VC vì đã hỗ trợ những video đầu tiên trong chuỗi này.", "model": "google_nmt", "from_community_srt": "Cũng muốn đưa ra một đặc biệt. Cảm ơn các đối tác amplifi của VC trong việc hỗ trợ những video ban đầu này trong bộ phim", diff --git a/2017/hardest-problem/arabic/sentence_translations.json b/2017/hardest-problem/arabic/sentence_translations.json index eb6f3e5ec..267b4028b 100644 --- a/2017/hardest-problem/arabic/sentence_translations.json +++ b/2017/hardest-problem/arabic/sentence_translations.json @@ -392,7 +392,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got. ", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got. ", "translatedText": "حسنًا، هناك شيء واحد يمكنك القيام به وهو العودة إلى الحالة ثنائية الأبعاد والتفكير فيما إذا كانت هناك طريقة مختلفة للتفكير في نفس الإجابة التي حصلنا عليها. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/hardest-problem/bengali/sentence_translations.json b/2017/hardest-problem/bengali/sentence_translations.json index ca0c2c4af..68987341d 100644 --- a/2017/hardest-problem/bengali/sentence_translations.json +++ b/2017/hardest-problem/bengali/sentence_translations.json @@ -392,7 +392,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got. ", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got. ", "translatedText": "ঠিক আছে, আপনি একটি জিনিস করতে পারেন তা হল 2D ক্ষেত্রে ব্যাক আপ করা এবং আমরা যে উত্তর পেয়েছি সেই একই উত্তর সম্পর্কে চিন্তা করার একটি ভিন্ন উপায় আছে কিনা তা বিবেচনা করুন।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/hardest-problem/chinese/sentence_translations.json b/2017/hardest-problem/chinese/sentence_translations.json index c3f91b69a..87ddf0ed7 100644 --- a/2017/hardest-problem/chinese/sentence_translations.json +++ b/2017/hardest-problem/chinese/sentence_translations.json @@ -414,7 +414,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got. ", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got. ", "translatedText": "好吧，你可以做的一件事就 是回到 2D 案例，并思考是否有不同的方式来思考我们得到的相 同答案。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/hardest-problem/croatian/sentence_translations.json b/2017/hardest-problem/croatian/sentence_translations.json index 928d67ebb..df7a3b6cd 100644 --- a/2017/hardest-problem/croatian/sentence_translations.json +++ b/2017/hardest-problem/croatian/sentence_translations.json @@ -450,7 +450,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "", "from_community_srt": "U ovom slučaju, umjesto da nastojimo izabrati tri točke nasumično, počnimo ovako:", "n_reviews": 0, @@ -490,7 +490,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "", "from_community_srt": "Jer vidite, jednom kad su dvije linije i treća točka determinirane, Postoje samo četiri mogućnosti za to gdje će se P1 i P2 nalaziti. Svaka od mogućnosti je jednako vjerojatna.", "n_reviews": 0, diff --git a/2017/hardest-problem/english/captions.srt b/2017/hardest-problem/english/captions.srt index a790fddd0..16b5def41 100644 --- a/2017/hardest-problem/english/captions.srt +++ b/2017/hardest-problem/english/captions.srt @@ -419,12 +419,12 @@ setup that makes it conceptually easier, see if you can reframe the entire question in terms of those things you just added. 106 -00:06:18,820 --> 00:06:23,615 -In this case, rather than choosing three points randomly, start by saying, +00:06:18,820 --> 00:06:23,027 +In this case, rather than thinking about choosing three points randomly, 107 -00:06:23,615 --> 00:06:27,580 -choose two random lines that pass through the circle's center. +00:06:23,027 --> 00:06:27,580 +start by saying, choose two random lines that pass through the circle's center. 108 00:06:28,460 --> 00:06:31,662 @@ -463,15 +463,15 @@ We'll still think about that third point, p3, as just being a random point on th but imagine that it was chosen before you do the two coin flips. 117 -00:07:02,560 --> 00:07:05,885 -Once the two lines and the third point are set in stone, +00:07:02,560 --> 00:07:06,504 +Because you see, once the two lines and the third point are set in stone, 118 -00:07:05,885 --> 00:07:09,735 -there's only four possibilities for where p1 and p2 might end up, +00:07:06,504 --> 00:07:10,021 +there's only four possibilities for where P1 and P2 might end up, 119 -00:07:09,735 --> 00:07:13,060 +00:07:10,021 --> 00:07:13,060 based on those coin flips, each one being equally likely. 120 diff --git a/2017/hardest-problem/english/sentence_timings.json b/2017/hardest-problem/english/sentence_timings.json index 9c8b70aa9..62b843028 100644 --- a/2017/hardest-problem/english/sentence_timings.json +++ b/2017/hardest-problem/english/sentence_timings.json @@ -285,7 +285,7 @@ 378.14 ], [ - "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", 378.82, 387.58 ], @@ -310,7 +310,7 @@ 421.72 ], [ - "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", 422.56, 433.06 ], diff --git a/2017/hardest-problem/english/transcript.txt b/2017/hardest-problem/english/transcript.txt index 4216577cb..cbe24d4ba 100644 --- a/2017/hardest-problem/english/transcript.txt +++ b/2017/hardest-problem/english/transcript.txt @@ -55,12 +55,12 @@ One of the main reasons I wanted to make a video about this particular problem i Think about those two lines we drew for p1 and p2 through the origin. They made the problem a lot easier to think about. And in general, whenever you've added something to the problem setup that makes it conceptually easier, see if you can reframe the entire question in terms of those things you just added. -In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center. +In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center. For each line, there's two possible points it could correspond to, so just flip a coin for each one to choose which of the endpoints is going to be p1, and likewise for the other, which endpoint is going to be p2. Choosing a random line and flipping a coin like this is the same thing as choosing a random point on the circle, it just feels a little bit convoluted at first. But the reason for thinking about the random process this way is that things are actually about to become easier. We'll still think about that third point, p3, as just being a random point on the circle, but imagine that it was chosen before you do the two coin flips. -Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely. +Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely. But one and only one of those four outcomes leaves p1 and p2 on the opposite side of the circle as p3, with the triangle they form containing the center. So no matter where those two lines end up, and where that p3 ends up, it's always a 1 fourth chance that the coin flips leave us with a triangle containing the center. Now that's very subtle. diff --git a/2017/hardest-problem/french/sentence_translations.json b/2017/hardest-problem/french/sentence_translations.json index eefd98a44..096d58ccc 100644 --- a/2017/hardest-problem/french/sentence_translations.json +++ b/2017/hardest-problem/french/sentence_translations.json @@ -451,7 +451,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "Dans ce cas, plutôt que de choisir trois points au hasard, commencez par dire : choisissez deux lignes au hasard qui passent par le centre du cercle.", "from_community_srt": "Dans ce cas, plutôt que de penser à choisir trois points au hasard, commencez en disant :", "n_reviews": 0, @@ -491,7 +491,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "Une fois que les deux lignes et le troisième point sont gravés dans le marbre, il n'y a que quatre possibilités pour savoir où p1 et p2 pourraient aboutir, sur la base de ces lancers de pièces, chacune étant également probable.", "from_community_srt": "Parce que vous voyez que, une fois les deux diamètres et le troisième point choisis, il y a quatre possibilités pour P1 et P2, basées sur les lancers, chacun étant également probable.", "n_reviews": 0, diff --git a/2017/hardest-problem/german/sentence_translations.json b/2017/hardest-problem/german/sentence_translations.json index ba54cd312..28f13e278 100644 --- a/2017/hardest-problem/german/sentence_translations.json +++ b/2017/hardest-problem/german/sentence_translations.json @@ -511,7 +511,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "In diesem Fall solltest du nicht drei Punkte zufällig auswählen, sondern zwei zufällige Linien wählen, die durch den Mittelpunkt des Kreises gehen.", "model": "DeepL", "from_community_srt": "In diesem Fall, anstatt darüber nachzudenken, drei Punkte zufällig zu wählen, wählen Sie zwei zufällige Linien,", @@ -556,7 +556,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "Sobald die beiden Linien und der dritte Punkt feststehen, gibt es nur noch vier Möglichkeiten, wo p1 und p2 landen könnten, basierend auf den Münzwürfen, wobei jede gleich wahrscheinlich ist.", "model": "DeepL", "from_community_srt": "Denn ihr seht, sobald die beiden Lienien und der dritte Punkt festgelegt sind, gibt es nur vier Möglichkeiten, wo P1 und P2, basierend auf dem Münzwurf, liegen können, jeder gleich wahrscheinlich.", diff --git a/2017/hardest-problem/greek/sentence_translations.json b/2017/hardest-problem/greek/sentence_translations.json index f8086ea46..c0d0f6921 100644 --- a/2017/hardest-problem/greek/sentence_translations.json +++ b/2017/hardest-problem/greek/sentence_translations.json @@ -450,7 +450,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "", "from_community_srt": "σε αυτή την περίπτωση, αντί να σκεφτείς να επιλέξεις 3 σημεία τυχαία, ξεκίνα λέγοντας: επέλεξε δυο τυχαίες γραμμές που περνάνε από το κέντρο του κύκλου; για κάθε γραμμή,", "n_reviews": 0, @@ -490,7 +490,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "", "from_community_srt": "Γιατί βλέπεις, όταν αυτές οι δυο γραμμές και το τρίτο σημείο έχουν τεθεί σε πέτρα, υπάρχουν 4 πιθανότητες για το που το P1 και το P2 μπορεί να είναι βασισμένη στο στρίψιμο εκείνων των κερμάτων, κάθε μια ίση πιθανοτήτων.", "n_reviews": 0, diff --git a/2017/hardest-problem/hebrew/sentence_translations.json b/2017/hardest-problem/hebrew/sentence_translations.json index 38ea6e143..e64f0c303 100644 --- a/2017/hardest-problem/hebrew/sentence_translations.json +++ b/2017/hardest-problem/hebrew/sentence_translations.json @@ -392,7 +392,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got. ", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got. ", "translatedText": "ובכן, דבר אחד שאתה יכול לעשות הוא לגבות למקרה הדו-ממדי ולחשוב אם יש דרך אחרת לחשוב על אותה תשובה שקיבלנו. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/hardest-problem/hindi/sentence_translations.json b/2017/hardest-problem/hindi/sentence_translations.json index f03cf5b5e..3f08b6ca8 100644 --- a/2017/hardest-problem/hindi/sentence_translations.json +++ b/2017/hardest-problem/hindi/sentence_translations.json @@ -343,7 +343,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got.", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got.", "translatedText": "ठीक है, एक चीज़ जो आप कर सकते हैं वह है 2डी केस का बैकअप लेना और इस पर विचार करना कि क्या हमें जो उत्तर मिला है, उसके बारे में सोचने का कोई अलग तरीका है या नहीं।", "n_reviews": 0, "start": 339.06, @@ -441,7 +441,7 @@ "end": 456.46 }, { - "input": "Just by reframing how we think about the random process for choosing points, the answer ¼ popped out in a very different way from how it did before.", + "input": "Just by reframing how we think about the random process for choosing points, the answer 1 quarter popped out in a very different way from how it did before.", "translatedText": "अंक चुनने की यादृच्छिक प्रक्रिया के बारे में हम कैसे सोचते हैं, इसे पुनः निर्धारित करने से, उत्तर ¼ पहले की तुलना में बहुत अलग तरीके से सामने आया।", "n_reviews": 0, "start": 457.04, diff --git a/2017/hardest-problem/hungarian/sentence_translations.json b/2017/hardest-problem/hungarian/sentence_translations.json index c234a3ca2..b2377a09d 100644 --- a/2017/hardest-problem/hungarian/sentence_translations.json +++ b/2017/hardest-problem/hungarian/sentence_translations.json @@ -456,7 +456,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "Ebben az esetben ahelyett, hogy véletlenszerűen választanánk ki három pontot, kezdjük azzal, hogy válasszunk két véletlenszerű egyenest, amelyek áthaladnak a kör középpontján.", "model": "DeepL", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "Miután a két vonal és a harmadik pont kőbe van vésve, csak négy lehetőség van arra, hogy p1 és p2 hová kerülhet, az érmefeldobások alapján, és mindegyik egyformán valószínű.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/hardest-problem/indonesian/sentence_translations.json b/2017/hardest-problem/indonesian/sentence_translations.json index 412d703a3..248096b07 100644 --- a/2017/hardest-problem/indonesian/sentence_translations.json +++ b/2017/hardest-problem/indonesian/sentence_translations.json @@ -392,7 +392,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got. ", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got. ", "translatedText": "Nah, satu hal yang dapat Anda lakukan adalah kembali ke kasus 2D dan merenungkan apakah ada cara lain untuk memikirkan jawaban yang sama yang kita dapatkan. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/hardest-problem/italian/sentence_translations.json b/2017/hardest-problem/italian/sentence_translations.json index ec2b8defa..a9997a303 100644 --- a/2017/hardest-problem/italian/sentence_translations.json +++ b/2017/hardest-problem/italian/sentence_translations.json @@ -509,7 +509,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "In questo caso, invece di scegliere tre punti a caso, inizia a dire: scegli due linee a caso che passino per il centro del cerchio.", "model": "DeepL", "from_community_srt": "In questo caso, piuttosto che pensare di scegliere tre punti in modo casuale, iniziate dicendo: scegliamo due linee casuali che passano attraverso il centro del cerchio; per ogni linea,", @@ -554,7 +554,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "Una volta fissate le due linee e il terzo punto, ci sono solo quattro possibilità per la posizione di p1 e p2, basate sul lancio della moneta, ognuna delle quali è ugualmente probabile.", "model": "DeepL", "from_community_srt": "Perché si vede, una volta che le due linee e che il terzo punto sono fissati, ci sono solo quattro possibilità su cui P1 e P2 possono finire basate sui lanci della moneta, ognuno con uguale probabilità.", diff --git a/2017/hardest-problem/japanese/sentence_translations.json b/2017/hardest-problem/japanese/sentence_translations.json index 531bceda2..720cf7034 100644 --- a/2017/hardest-problem/japanese/sentence_translations.json +++ b/2017/hardest-problem/japanese/sentence_translations.json @@ -392,7 +392,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got. ", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got. ", "translatedText": "そうですね、できることの 1 つは、2D のケースに戻って、得られた同じ答えについて別の方法で考えることができるかどうかを熟考することです。 ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/hardest-problem/korean/sentence_translations.json b/2017/hardest-problem/korean/sentence_translations.json index 458b0b8ed..0c44acfcf 100644 --- a/2017/hardest-problem/korean/sentence_translations.json +++ b/2017/hardest-problem/korean/sentence_translations.json @@ -499,7 +499,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "이 경우 세 점을 무작위로 선택하는 대신 원의 중심을 통과하는 두 개의 임의의 선을 선택하는 것부터 시작하세요.", "model": "DeepL", "from_community_srt": "이 경우에는, 무작위로 3개의 점을 선택하는 것 보단, 이렇게 시작해봅시다: 원의 중심을 지나는 무작위의 선분을 2개 고르는것이죠 각 선분에는,", @@ -543,7 +543,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "두 줄과 세 번째 지점이 정해지면, 동전 던지기를 기준으로 p1과 p2가 어디로 나올지는 네 가지 가능성만 남게 되며, 각 가능성은 똑같습니다.", "model": "DeepL", "from_community_srt": "왜냐하면, 보시다시피, 두 선과 세번째 지점을 고정시키고 동전 던지기로 결정할 점P1,", diff --git a/2017/hardest-problem/marathi/sentence_translations.json b/2017/hardest-problem/marathi/sentence_translations.json index e0fb4300d..c795f1e95 100644 --- a/2017/hardest-problem/marathi/sentence_translations.json +++ b/2017/hardest-problem/marathi/sentence_translations.json @@ -392,7 +392,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got. ", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got. ", "translatedText": "बरं, तुम्ही एक गोष्ट करू शकता ती म्हणजे 2D केसचा बॅकअप घ्या आणि आम्हाला मिळालेल्या त्याच उत्तराबद्दल विचार करण्याचा वेगळा मार्ग आहे का याचा विचार करा. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/hardest-problem/persian/sentence_translations.json b/2017/hardest-problem/persian/sentence_translations.json index 2a5d052c0..062313d8c 100644 --- a/2017/hardest-problem/persian/sentence_translations.json +++ b/2017/hardest-problem/persian/sentence_translations.json @@ -392,7 +392,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got. ", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/hardest-problem/polish/sentence_translations.json b/2017/hardest-problem/polish/sentence_translations.json index eb39820bb..16a98e19c 100644 --- a/2017/hardest-problem/polish/sentence_translations.json +++ b/2017/hardest-problem/polish/sentence_translations.json @@ -450,7 +450,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "", "from_community_srt": "W tym przypadku zamiast myśleć o wybieraniu losowo trzech punktów, zacznij od stwierdzenia: wybierz dwie losowe linie,", "n_reviews": 0, @@ -490,7 +490,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "", "from_community_srt": "Ponieważ widzisz, kiedy dwie linie i ten trzeci punkt są ustawione w jednym punkcie istnieją tylko cztery możliwości, w których P1 i P2 mogą się skończyć w oparciu o te rzuty monetą, każde z nich jest jednakowo prawdopodobne.", "n_reviews": 0, diff --git a/2017/hardest-problem/portuguese/sentence_translations.json b/2017/hardest-problem/portuguese/sentence_translations.json index 0b39ce012..469f42d60 100644 --- a/2017/hardest-problem/portuguese/sentence_translations.json +++ b/2017/hardest-problem/portuguese/sentence_translations.json @@ -510,7 +510,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "Neste caso, em vez de escolher três pontos aleatoriamente, comece dizendo, escolha duas linhas aleatórias que passem pelo centro do círculo.", "model": "google_nmt", "from_community_srt": "Nesse caso, ao invés de pensar em escolher três pontos aleatórios, comece dizendo:", @@ -555,7 +555,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "Uma vez que as duas linhas e o terceiro ponto estejam gravados em pedra, há apenas quatro possibilidades de onde p1 e p2 podem terminar, com base nesses lançamentos de moeda, sendo cada uma igualmente provável.", "model": "google_nmt", "from_community_srt": "Como você pode vê, ambas as linhas e o terceiro ponto estão fixos, só existem 4 possibilidades para onde P1 e P2 irão estar (baseado naquela troca de posições de P1 e P2,", diff --git a/2017/hardest-problem/russian/sentence_translations.json b/2017/hardest-problem/russian/sentence_translations.json index af8169fd7..8383c5325 100644 --- a/2017/hardest-problem/russian/sentence_translations.json +++ b/2017/hardest-problem/russian/sentence_translations.json @@ -455,7 +455,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "В этом случае вместо того, чтобы выбирать три точки случайным образом, начните с выбора двух случайных линий, проходящих через центр круга.", "from_community_srt": "В этом случае, вместо того, чтобы думать о том, чтобы выбрать три точки случайным образом, начните, сказав: выберите две случайные линии,", "n_reviews": 0, @@ -495,7 +495,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "Как только две линии и третья точка высечены в камне, существует только четыре возможности того, где могут оказаться p1 и p2, исходя из этих подбрасываний монеты, причем каждая из них одинаково вероятны.", "from_community_srt": "Поскольку вы видите, как только две линии и эта третья точка установлены в камне, есть только четыре возможности для того, где P1 и P2 могут закончиться на основе этих переводов монет, каждый из которых в равной степени вероятен.", "n_reviews": 0, diff --git a/2017/hardest-problem/spanish/sentence_translations.json b/2017/hardest-problem/spanish/sentence_translations.json index 449998e00..7da6ccee5 100644 --- a/2017/hardest-problem/spanish/sentence_translations.json +++ b/2017/hardest-problem/spanish/sentence_translations.json @@ -452,7 +452,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "En este caso, en lugar de elegir tres puntos al azar, comience diciendo, elija dos líneas aleatorias que pasen por el centro del círculo.", "from_community_srt": "En este caso, en lugar de pensar acerca de elegir tres puntos al azar, comencemos por decir: elegimos dos líneas al azar que pasan a través del centro del círculo; para cada línea,", "n_reviews": 0, @@ -492,7 +492,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "Una vez que las dos líneas y el tercer punto están escritos en piedra, solo hay cuatro posibilidades de dónde podrían terminar p1 y p2, según esos lanzamientos de moneda, siendo cada una de ellas igualmente probable.", "from_community_srt": "Porque, una vez que las dos líneas y el tercer punto están fijados, sólo hay cuatro posibilidades para dónde P1 y P2 pueden estar, basado en los lanzamientos de la moneda, cada una con igual probabilidad.", "n_reviews": 0, diff --git a/2017/hardest-problem/tamil/sentence_translations.json b/2017/hardest-problem/tamil/sentence_translations.json index 2892220a8..dec3f789a 100644 --- a/2017/hardest-problem/tamil/sentence_translations.json +++ b/2017/hardest-problem/tamil/sentence_translations.json @@ -392,7 +392,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got. ", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got. ", "translatedText": "சரி, நீங்கள் செய்யக்கூடிய ஒன்று, 2D கேஸை மீண்டும் எடுத்து, நாங்கள் பெற்ற அதே பதிலைப் பற்றி சிந்திக்க வேறு வழி இருக்கிறதா என்று சிந்தியுங்கள். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/hardest-problem/telugu/sentence_translations.json b/2017/hardest-problem/telugu/sentence_translations.json index c3870b053..8c84ad8ad 100644 --- a/2017/hardest-problem/telugu/sentence_translations.json +++ b/2017/hardest-problem/telugu/sentence_translations.json @@ -392,7 +392,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got. ", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got. ", "translatedText": "సరే, మీరు చేయగలిగినది 2D కేస్‌కు బ్యాకప్ చేసి, మాకు లభించిన అదే సమాధానం గురించి ఆలోచించడానికి వేరే మార్గం ఉందా అని ఆలోచించండి. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/hardest-problem/thai/sentence_translations.json b/2017/hardest-problem/thai/sentence_translations.json index c0f016e67..bdc3304be 100644 --- a/2017/hardest-problem/thai/sentence_translations.json +++ b/2017/hardest-problem/thai/sentence_translations.json @@ -392,7 +392,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got. ", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/hardest-problem/turkish/sentence_translations.json b/2017/hardest-problem/turkish/sentence_translations.json index ee2f23f8d..e009a95bc 100644 --- a/2017/hardest-problem/turkish/sentence_translations.json +++ b/2017/hardest-problem/turkish/sentence_translations.json @@ -509,7 +509,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "Bu durumda, rastgele üç nokta seçmek yerine, dairenin merkezinden geçen rastgele iki çizgi seçin diyerek başlayın.", "model": "DeepL", "from_community_srt": "İlk olarak, rastgele 3 nokta seçerek başlamaktansa, merkezden geçen iki çap düşünün.", @@ -554,7 +554,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "İki çizgi ve üçüncü nokta belirlendikten sonra, bu yazı tura atışlarına dayalı olarak p1 ve p2'nin nerede sonlanabileceğine dair yalnızca dört olasılık vardır ve her biri eşit derecede olasıdır.", "model": "DeepL", "from_community_srt": "Çünkü iki çizgimizi ve P3'ü seçtikten sonra, P1 ve P2'nin olabileceği sadece dört yer kalmış oluyor. Her biri eşit olasılıklı.", diff --git a/2017/hardest-problem/ukrainian/sentence_translations.json b/2017/hardest-problem/ukrainian/sentence_translations.json index 455e5cdd4..f6450f98b 100644 --- a/2017/hardest-problem/ukrainian/sentence_translations.json +++ b/2017/hardest-problem/ukrainian/sentence_translations.json @@ -510,7 +510,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "У цьому випадку, замість того, щоб вибирати три випадкові точки, почніть з того, що виберіть дві випадкові лінії, які проходять через центр кола.", "model": "DeepL", "from_community_srt": "У цьому випадку, замість роздумів про вибір трьох точок випадково, почніть зі слів: виберіть дві випадкові лінії,", @@ -555,7 +555,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "Після того, як дві лінії та третя точка визначені, є лише чотири можливості того, де можуть опинитися p1 та p2, на основі цих підкидань монети, причому кожна з них однаково вірогідна.", "model": "DeepL", "from_community_srt": "Тому що ви бачите, як тільки дві лінії і ця третя точка стоять на своїх місцях лишається лише чотири ймовірності, у яких Р1 і Р2 можуть закінчитись, базуючись на підкиданні монетки, кожна з яких однаковго вірогідна.", diff --git a/2017/hardest-problem/urdu/sentence_translations.json b/2017/hardest-problem/urdu/sentence_translations.json index 589c9748b..27ea41619 100644 --- a/2017/hardest-problem/urdu/sentence_translations.json +++ b/2017/hardest-problem/urdu/sentence_translations.json @@ -392,7 +392,7 @@ "end": 338.52 }, { - "input": "Well, one thing you can do is back up to the 2D case and contemplate if there is a different way to think about the same answer we got. ", + "input": "Well, one thing you can do is back up to the two-dimensional case and contemplate if there is a different way to think about the same answer we got. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/hardest-problem/vietnamese/sentence_translations.json b/2017/hardest-problem/vietnamese/sentence_translations.json index 1c4052e8c..6f3cb0a24 100644 --- a/2017/hardest-problem/vietnamese/sentence_translations.json +++ b/2017/hardest-problem/vietnamese/sentence_translations.json @@ -508,7 +508,7 @@ "end": 378.14 }, { - "input": "In this case, rather than choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", + "input": "In this case, rather than thinking about choosing three points randomly, start by saying, choose two random lines that pass through the circle's center.", "translatedText": "Trong trường hợp này, thay vì chọn ngẫu nhiên ba điểm, hãy bắt đầu bằng cách nói, hãy chọn hai đường thẳng ngẫu nhiên đi qua tâm vòng tròn.", "model": "google_nmt", "from_community_srt": "Trong trường hợp này, thay vì nhìn nó theo kiểu chọn 3 điểm ngẫu nhiên, hãy bắt đầu bằng việc nói rằng:", @@ -553,7 +553,7 @@ "end": 421.72 }, { - "input": "Once the two lines and the third point are set in stone, there's only four possibilities for where p1 and p2 might end up, based on those coin flips, each one being equally likely.", + "input": "Because you see, once the two lines and the third point are set in stone, there's only four possibilities for where P1 and P2 might end up, based on those coin flips, each one being equally likely.", "translatedText": "Khi hai đường thẳng và điểm thứ ba đã được xác định rõ ràng, chỉ có bốn khả năng về vị trí mà p1 và p2 có thể kết thúc, dựa trên những lần tung đồng xu đó, mỗi khả năng đều có khả năng như nhau.", "model": "google_nmt", "from_community_srt": "Bởi vì, như bạn thấy, một khi 2 đường và điểm thứ ba được đặt, Chỉ có bốn khả năng cho vị trí của P1 và P2, dựa trên việc tung đồng tiền, mỗi khả năng đều như nhau.", diff --git a/2017/higher-dimensions/arabic/sentence_translations.json b/2017/higher-dimensions/arabic/sentence_translations.json index d36d6cfc4..8f81e5dfa 100644 --- a/2017/higher-dimensions/arabic/sentence_translations.json +++ b/2017/higher-dimensions/arabic/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "ارسم مكعبًا مقاس 2×2×2 تحتوي زواياه على رؤوس 1،1،1،1،1،1،1،1، ثم سنأخذ ثمانية مجالات مختلفة لكل منها نصف قطر 1 ونركزها على هذه الرؤوس بحيث كل واحد مماس لثلاثة من جيرانه. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/bengali/sentence_translations.json b/2017/higher-dimensions/bengali/sentence_translations.json index 96a03ec54..59a4d5f7d 100644 --- a/2017/higher-dimensions/bengali/sentence_translations.json +++ b/2017/higher-dimensions/bengali/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "একটি 2x2x2 ঘনক আঁকুন যার কোণে শীর্ষবিন্দু রয়েছে 1,1,1,1,1,1,1,1, এবং তারপরে আমরা আটটি ভিন্ন গোলক নেব যার প্রতিটির ব্যাসার্ধ 1 এবং এই শীর্ষবিন্দুগুলিতে কেন্দ্র করে যাতে প্রতিটি তার তিনটি প্রতিবেশীর স্পর্শক।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/chinese/sentence_translations.json b/2017/higher-dimensions/chinese/sentence_translations.json index f7366ec43..21dbb20af 100644 --- a/2017/higher-dimensions/chinese/sentence_translations.json +++ b/2017/higher-dimensions/chinese/sentence_translations.json @@ -558,7 +558,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "画一个 2x2x 2 的立方体，其角的顶点为 1,1,1,1,1,1,1,1， 然后我们将取八个不同的球体，每个球体的半径为 1，并将它们以 这些顶点为中心，这样每一个都与其三个邻居相切。", "model": "google_nmt", "from_community_srt": "1, 1) (1, 1,", diff --git a/2017/higher-dimensions/english/captions.srt b/2017/higher-dimensions/english/captions.srt index 2e73ad6e4..10fba8cf2 100644 --- a/2017/higher-dimensions/english/captions.srt +++ b/2017/higher-dimensions/english/captions.srt @@ -543,842 +543,846 @@ Here in two dimensions we can use the Pythagorean theorem to see that the distan from the origin to the corner of the box is the square root of 2, which is around 1.414. 137 -00:09:58,320 --> 00:10:06,235 -Then you can subtract off this portion here, the radius of the corner circle, +00:09:58,320 --> 00:10:02,317 +Then you can subtract off this portion here the radius of the 138 -00:10:06,235 --> 00:10:10,700 -which by definition is 1, and that is 1,414. +00:10:02,317 --> 00:10:06,573 +corner circle which by definition is 1, and that means the radius 139 +00:10:06,573 --> 00:10:10,700 +of the inner circle is square root of 2 minus 1, or about 0.414. + +140 00:10:11,540 --> 00:10:13,820 No surprises here, that seems pretty reasonable. -140 +141 00:10:15,060 --> 00:10:17,080 Now do something analogous in three dimensions. -141 +142 00:10:17,740 --> 00:10:23,455 Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, -142 +143 00:10:23,455 --> 00:10:29,611 and then we're going to take eight different spheres, each of which has a radius 1, -143 +144 00:10:29,611 --> 00:10:36,060 and center them on these vertices so that each one is tangent to three of its neighbors. -144 +145 00:10:37,220 --> 00:10:40,783 Now again, think about the sphere centered at the origin which is -145 +146 00:10:40,783 --> 00:10:44,400 just large enough to be barely touching those eight corner spheres. -146 +147 00:10:45,160 --> 00:10:48,690 As before, we can start by thinking about the distance from -147 +148 00:10:48,690 --> 00:10:52,280 the origin to the corner of the box, say the corner at 1,1,1. -148 +149 00:10:53,280 --> 00:10:56,923 By the way, I guess I still haven't yet explicitly said that the -149 +150 00:10:56,923 --> 00:11:00,511 way distances work in higher dimensions is always to add up the -150 +151 00:11:00,511 --> 00:11:04,380 squares of the components in each direction and take the square root. -151 +152 00:11:05,040 --> 00:11:08,911 If you've never seen why this follows from the Pythagorean theorem just in the -152 +153 00:11:08,911 --> 00:11:12,439 two-dimensional case, it's actually a really fun puzzle to think about, -153 +154 00:11:12,439 --> 00:11:16,506 and I've left the relevant image up on the screen for any of you who want to pause -154 +155 00:11:16,506 --> 00:11:17,340 and ponder on it. -155 +156 00:11:18,080 --> 00:11:23,358 Anyway, in our case the distance between the origin and the corner 1,1,1 is the square -156 +157 00:11:23,358 --> 00:11:28,820 root of 1 squared plus 1 squared plus 1 squared, or square root of 3, which is about 1.73. -157 +158 00:11:29,880 --> 00:11:33,730 So the radius of that inner sphere is going to be this quantity -158 +159 00:11:33,730 --> 00:11:37,460 minus the radius of a corner sphere, which by definition is 1. -159 +160 00:11:38,160 --> 00:11:43,060 And again, 0.73 seems like a reasonable radius for that inner sphere. -160 +161 00:11:43,820 --> 00:11:47,040 But what happens to that inner radius as you increase dimensions? -161 +162 00:11:48,100 --> 00:11:51,842 Obviously the reason I bring this up is that something surprising will happen, -162 +163 00:11:51,842 --> 00:11:54,069 and some of you might see where this is going, -163 +164 00:11:54,069 --> 00:11:56,580 but I actually don't want it to feel like a surprise. -164 +165 00:11:57,180 --> 00:12:01,041 As fun as it is to wow people with counterintuitive facts and math, -165 +166 00:12:01,041 --> 00:12:03,880 the goal here is genuine understanding, not shock. -166 +167 00:12:04,500 --> 00:12:08,703 For higher dimensions we'll be using sliders to get a gut feel for what's going on, -167 +168 00:12:08,703 --> 00:12:13,156 but since it's kind of a different way of viewing things it helps to get a running start -168 +169 00:12:13,156 --> 00:12:17,559 by looking back at how to analyze the two and three-dimensional cases in the context of -169 +170 00:12:17,559 --> 00:12:17,960 sliders. -170 +171 00:12:19,100 --> 00:12:24,160 First things first, how do you think about a circle centered at a corner like 1,-1? -171 +172 00:12:25,180 --> 00:12:28,309 Well previously, for a circle centered at the origin, -172 +173 00:12:28,309 --> 00:12:32,655 the amount of real estate belonging to both x and y was dependent on their -173 +174 00:12:32,655 --> 00:12:34,220 distance from the number 0. -174 +175 00:12:35,140 --> 00:12:38,041 And it's the same basic idea here as you move around the center, -175 +176 00:12:38,041 --> 00:12:42,058 it's just that the real estate might be dependent on the distance between each coordinate -176 +177 00:12:42,058 --> 00:12:43,040 and some other number. -177 +178 00:12:43,720 --> 00:12:47,562 So for this circle, centered at 1,-1, the amount of real -178 +179 00:12:47,562 --> 00:12:51,540 estate belonging to x is the square of its distance from 1. -179 +180 00:12:52,280 --> 00:12:57,420 Likewise, the real estate belonging to y is the square of its distance from negative 1. -180 +181 00:12:58,100 --> 00:13:00,545 Other than that, the look and feel with this piston -181 +182 00:13:00,545 --> 00:13:02,380 dance trade-off is completely the same. -182 +183 00:13:03,480 --> 00:13:07,820 For simplicity, we'll only focus on one of these circles, the one centered at 1,-1. -183 +184 00:13:08,780 --> 00:13:13,002 Now ask yourself, what does it mean to find a circle centered at the origin large -184 +185 00:13:13,002 --> 00:13:17,020 enough to be tangent to this guy when we're thinking just in terms of sliders? -185 +186 00:13:20,120 --> 00:13:22,805 Well notice how this point of tangency happens -186 +187 00:13:22,805 --> 00:13:25,490 when the x and y coordinates are both the same. -187 +188 00:13:26,310 --> 00:13:31,492 Or phrased differently, at the point of this corner circle closest to the origin, -188 +189 00:13:31,492 --> 00:13:34,590 the real estate is shared evenly between x and y. -189 +190 00:13:35,410 --> 00:13:39,090 This will be important for later, so let's really dig in and think about why it's true. -190 +191 00:13:40,070 --> 00:13:44,302 Imagine perturbing that point slightly, maybe moving x a little closer to 0, -191 +192 00:13:44,302 --> 00:13:47,270 which means y would have to move a little away from 0. -192 +193 00:13:47,910 --> 00:13:51,897 The change in x would have to be a little smaller than the change in y, -193 +194 00:13:51,897 --> 00:13:55,719 since the real estate it gains by moving farther away from 1 is more -194 +195 00:13:55,719 --> 00:13:59,430 expensive than the real estate that y loses by getting closer to 1. -195 +196 00:14:00,310 --> 00:14:04,610 But from the perspective of the origin point 0,0 that trade-off is reversed. -196 +197 00:14:05,330 --> 00:14:11,324 The resulting change to x squared is smaller than the resulting change to y squared, -197 +198 00:14:11,324 --> 00:14:15,273 since when real estate is measured with respect to 0,0, -198 +199 00:14:15,273 --> 00:14:18,870 that move of y towards 1 is the more expensive one. -199 +200 00:14:20,150 --> 00:14:24,523 What this means is that any slight perturbation away from this point where -200 +201 00:14:24,523 --> 00:14:29,130 real estate is shared evenly results in an increasing distance from the origin. -201 +202 00:14:30,470 --> 00:14:34,036 The reason we care is that this point is tangent to the inner circle, -202 +203 00:14:34,036 --> 00:14:37,450 so we can also think about it as being a point of the inner circle. -203 +204 00:14:38,130 --> 00:14:40,070 And this will be very useful for higher dimensions. -204 +205 00:14:40,530 --> 00:14:44,470 It gives us a reference point to understanding the radius of that inner circle. -205 +206 00:14:45,290 --> 00:14:49,888 Specifically, you can ask how much real estate is shared between x and y at -206 +207 00:14:49,888 --> 00:14:54,790 this point when real estate measurements are done with respect to the origin 0,0. -207 +208 00:14:55,890 --> 00:15:00,894 For example, down here in two dimensions both x and y dip below -208 +209 00:15:00,894 --> 00:15:05,821 0.5 in this configuration, so the total value x squared plus y -209 +210 00:15:05,821 --> 00:15:10,670 squared is going to be less than 0.5 squared plus 0.5 squared. -210 +211 00:15:11,670 --> 00:15:14,442 Comparing to this halfway point is really going to come in handy -211 +212 00:15:14,442 --> 00:15:17,130 for wrapping our mind around what happens in higher dimensions. -212 +213 00:15:18,010 --> 00:15:20,750 Taking things one step at a time, let's bump it up to three dimensions. -213 +214 00:15:22,690 --> 00:15:26,270 Consider the corner sphere with radius 1 centered at 1,1,1. -214 +215 00:15:26,970 --> 00:15:32,393 The point on that sphere that's closest to the origin corresponds to the configuration -215 +216 00:15:32,393 --> 00:15:37,630 of sliders where x, y, and z are all reaching down toward 0 and equal to each other. -216 +217 00:15:38,450 --> 00:15:44,017 Again, they all have to go a little beyond that halfway point because the -217 +218 00:15:44,017 --> 00:15:49,510 position 0.5 only accounts for 0.5 squared, or 0.25 units of real estate. -218 +219 00:15:50,630 --> 00:15:54,460 So with all three coordinates getting a third of a unit of real estate, -219 +220 00:15:54,460 --> 00:15:55,950 they need to be farther out. -220 +221 00:15:56,750 --> 00:16:01,024 And again, since this is a point where the corner sphere is tangent to the inner sphere, -221 +222 00:16:01,024 --> 00:16:02,850 it's also a point of the inner sphere. -222 +223 00:16:03,610 --> 00:16:08,782 So with reference to the origin 0,0,0, think about the amount of real estate -223 +224 00:16:08,782 --> 00:16:14,090 shared between x, y, and z in this position corresponding to the tangent point. -224 +225 00:16:14,830 --> 00:16:21,430 It's definitely less than 0.75 since all three of these are smaller than 0.5, -225 +226 00:16:21,430 --> 00:16:25,830 so each one has less than 0.25 units of real estate. -226 +227 00:16:26,890 --> 00:16:29,950 And again, we sit back and feel comfortable with this result, right? -227 +228 00:16:30,070 --> 00:16:32,590 The inner sphere is smaller than the corner spheres. -228 +229 00:16:33,550 --> 00:16:36,270 But things get interesting when we move up into four dimensions. -229 -00:16:37,950 --> 00:16:44,294 -Our 2x2x2x2 box is going to have 16 vertices at 1111, 111-1, - 230 -00:16:44,294 --> 00:16:51,470 -and so on, with all possible binary combinations of 1 and negative 1. +00:16:37,950 --> 00:16:44,757 +Our 2x2x2x2 box is going to have 16 vertices at 1 1 1 1 1 1 1 negative 231 +00:16:44,757 --> 00:16:51,470 +1 and so on with all possible binary combinations of 1 and negative 1. + +232 00:16:52,310 --> 00:16:56,541 What this means is that there are 16 unit spheres centered at these corners, -232 +233 00:16:56,541 --> 00:16:58,850 each one tangent to four of its neighbors. -233 +234 00:17:00,150 --> 00:17:04,550 As before, we'll just be focusing on one of them, the one centered at 1111. -234 +235 00:17:04,990 --> 00:17:10,035 The point of the sphere closest to the origin corresponds to the configuration -235 +236 00:17:10,035 --> 00:17:14,890 of sliders where all four coordinates reach exactly halfway between 1 and 0. -236 +237 00:17:15,770 --> 00:17:20,574 And that's because when one of the coordinates is 0.5 units away from 1, -237 +238 00:17:20,574 --> 00:17:24,589 it has 0.25 units of real estate with respect to the point 1. -238 +239 00:17:25,369 --> 00:17:28,379 We do the same trick as before, thinking of this now as a point -239 +240 00:17:28,379 --> 00:17:31,623 of the inner sphere and measuring things with respect to the origin, -240 +241 00:17:31,623 --> 00:17:34,350 but you can already see what's cool about four dimensions. -241 +242 00:17:34,930 --> 00:17:39,030 As you switch to thinking of real estate with respect to 0000, -242 +243 00:17:39,030 --> 00:17:44,694 it's still the case that each of these four coordinates has 0.25 units of real estate, -243 +244 00:17:44,694 --> 00:17:48,730 making for a total of one shared between the four coordinates. -244 +245 00:17:50,110 --> 00:17:54,710 In other words, that inner sphere is precisely the same size as the corner spheres. -245 +246 00:17:55,670 --> 00:17:58,812 This matches with what you see numerically, by the way, -246 +247 00:17:58,812 --> 00:18:03,076 where you can compute the distance between the origin and the corner, 1111, -247 +248 00:18:03,076 --> 00:18:07,622 is the square root of 4, and then when you subtract off the radius of one of the -248 +249 00:18:07,622 --> 00:18:09,530 corner spheres, what you get is 1. -249 +250 00:18:10,430 --> 00:18:14,790 But there's something much more satisfying about seeing it, rather than just computing it. -250 +251 00:18:15,590 --> 00:18:19,570 In particular, here's a cool aspect of the fact that that inner sphere has radius 1. -251 +252 00:18:20,190 --> 00:18:24,518 Move things around so that all of the real estate goes to the coordinate x, -252 +253 00:18:24,518 --> 00:18:26,910 and you'll end up at the point 1, 0, 0, 0. -253 +254 00:18:27,490 --> 00:18:30,730 This point is actually touching the 2x2x2x2 box, -254 +255 00:18:30,730 --> 00:18:35,358 and when you're stuck thinking in the two or three dimensional cases, -255 +256 00:18:35,358 --> 00:18:40,847 this fact that the inner sphere has radius 1, the same size as the corner spheres, -256 +257 00:18:40,847 --> 00:18:44,550 and that it touches the box, well it just seems too big. -257 +258 00:18:45,290 --> 00:18:49,887 But it's important to realize this is fundamentally a four-dimensional phenomenon, -258 +259 00:18:49,887 --> 00:18:52,990 and you just can't cram it down into smaller dimensions. -259 +260 00:18:54,170 --> 00:18:55,070 But things get weirder. -260 +261 00:18:55,350 --> 00:18:56,750 Let's knock it up to five dimensions. -261 +262 00:18:57,370 --> 00:19:00,873 In this case we have quite a few corner spheres, 32 in total, -262 +263 00:19:00,873 --> 00:19:05,450 but again for simplicity we'll only be thinking about the ones centered at 11111. -263 +264 00:19:06,190 --> 00:19:09,226 Think about the point of the sphere closest to the origin, -264 +265 00:19:09,226 --> 00:19:13,550 where all five coordinates are equally splitting the one unit of shared real estate. -265 +266 00:19:14,430 --> 00:19:18,110 This time each coordinate is a little higher than 0.5. -266 +267 00:19:18,530 --> 00:19:23,914 If they reach down to 0.5, each one would have 0.25 units of real estate, -267 +268 00:19:23,914 --> 00:19:26,970 giving a total of 1.25, which is too much. -268 +269 00:19:27,750 --> 00:19:32,123 But the tables are turned when you view this as a point on the inner sphere, -269 +270 00:19:32,123 --> 00:19:37,008 because with respect to the origin, this configuration has much more than one unit of -270 +271 00:19:37,008 --> 00:19:37,690 real estate. -271 +272 00:19:40,130 --> 00:19:43,903 Not only is every coordinate more than 0.5 units away from 0, -272 +273 00:19:43,903 --> 00:19:48,954 but the larger number of dimensions means that there's more total real estate when -273 +274 00:19:48,954 --> 00:19:50,050 you add it all up. -274 +275 00:19:50,650 --> 00:19:55,690 Specifically you can compute that the radius of that inner sphere is about 1.24. -275 +276 00:19:56,530 --> 00:20:00,575 The intuitive feel for what that means is that the sliders can roam over -276 +277 00:20:00,575 --> 00:20:04,510 more territory than what just a single unit of real estate would allow. -277 +278 00:20:05,610 --> 00:20:08,310 One fun way to see what this means is to adjust everything -278 +279 00:20:08,310 --> 00:20:11,010 so that all of the real estate goes to just one coordinate. -279 +280 00:20:12,010 --> 00:20:16,347 Because this coordinate can reach beyond one, what you are seeing is -280 +281 00:20:16,347 --> 00:20:20,810 that this five dimensional inner sphere actually pokes outside the box. -281 +282 00:20:22,670 --> 00:20:25,500 But to really get a feel for how strange things become, -282 +283 00:20:25,500 --> 00:20:28,330 as a last example I want to jump up into ten dimensions. -283 +284 00:20:29,130 --> 00:20:32,110 Remember, all this means is that points have ten coordinates. -284 +285 00:20:32,770 --> 00:20:35,498 For a sphere with radius 1, a single unit of real -285 +286 00:20:35,498 --> 00:20:38,610 estate must be shared among all ten of those coordinates. -286 +287 00:20:39,630 --> 00:20:43,152 As always, the point of this corner sphere closest to the origin -287 +288 00:20:43,152 --> 00:20:46,730 is the one where all ten coordinates split the real estate evenly. -288 +289 00:20:47,450 --> 00:20:51,310 And here you can really see just how far away this feels from the origin. -289 +290 00:20:52,230 --> 00:20:55,233 Or phrased differently, that inner sphere is allowed -290 +291 00:20:55,233 --> 00:20:57,670 to have a very large amount of real estate. -291 +292 00:20:58,690 --> 00:21:03,690 In fact, you can compute that the radius of the inner sphere is about 2.16. -292 +293 00:21:04,750 --> 00:21:09,011 And viewed from this perspective, where you have ten full dimensions to share -293 +294 00:21:09,011 --> 00:21:13,273 that real estate, doesn't it actually feel somewhat reasonable that the inner -294 +295 00:21:13,273 --> 00:21:17,590 sphere should have a radius more than twice as big as all those corner spheres? -295 +296 00:21:18,830 --> 00:21:22,172 To get a sense for just how big this inner sphere is, -296 +297 00:21:22,172 --> 00:21:26,877 look back in two dimensions and imagine a 4x4 box bounding all four circles -297 +298 00:21:26,877 --> 00:21:27,930 from the outside. -298 +299 00:21:28,750 --> 00:21:32,153 Or go to three dimensions and imagine a 4x4x4 box -299 +300 00:21:32,153 --> 00:21:35,830 bounding all of those corner spheres from the outside. -300 +301 00:21:36,350 --> 00:21:42,057 Way up here in ten dimensions, that quote-unquote inner sphere is actually large enough -301 +302 00:21:42,057 --> 00:21:47,570 to poke outside of that outer bounding box, since it has a diameter bigger than four. -302 +303 00:21:50,070 --> 00:21:54,953 I know that seems crazy, but you have to realize that the face of the box is -303 +304 00:21:54,953 --> 00:21:59,710 always two units away from the origin, no matter how high the dimension is. -304 +305 00:22:00,170 --> 00:22:04,250 And fundamentally it's because it only involves moving along a single axis. -305 +306 00:22:05,030 --> 00:22:09,394 But the point 1111111111, which determines the inner sphere's radius, -306 +307 00:22:09,394 --> 00:22:14,570 is actually really far away from the center, all the way up here in ten dimensions. -307 +308 00:22:15,250 --> 00:22:18,211 And it's because all ten of those dimensions add -308 +309 00:22:18,211 --> 00:22:20,750 a full unit of real estate for that point. -309 +310 00:22:22,390 --> 00:22:24,869 And of course as you keep upping the dimensions, -310 +311 00:22:24,869 --> 00:22:27,450 that inner sphere just keeps growing without bound. -311 +312 00:22:27,950 --> 00:22:33,472 Not only is it poking outside of these boxes, but the proportion of the inner sphere -312 +313 00:22:33,472 --> 00:22:38,735 lying inside the box decreases exponentially towards zero as the dimension keeps -313 +314 00:22:38,735 --> 00:22:39,450 increasing. -314 +315 00:22:41,610 --> 00:22:45,384 So taking a step back, one of the things I like about using this slider method -315 +316 00:22:45,384 --> 00:22:48,156 for teaching is that when I shared it with a few friends, -316 +317 00:22:48,156 --> 00:22:51,691 the way they started to talk about higher dimensions became a little less -317 +318 00:22:51,691 --> 00:22:55,466 metaphysical and started to sound more like how you would hear a mathematician -318 +319 00:22:55,466 --> 00:22:56,470 talk about the topic. -319 +320 00:22:57,010 --> 00:23:01,805 Rather than skeptically asking whether or not ten-dimensional space is a real thing, -320 +321 00:23:01,805 --> 00:23:04,852 recognizing that it's exactly as real as numbers are, -321 +322 00:23:04,852 --> 00:23:08,802 people would actually probe at what other properties high-dimensional -322 +323 00:23:08,802 --> 00:23:12,470 spheres have and what other shapes feel like in terms of sliders. -323 +324 00:23:13,450 --> 00:23:17,179 This box situation is just one in a number of things that feel very -324 +325 00:23:17,179 --> 00:23:20,635 crazy about higher dimensional spheres, and it's really fun to -325 +326 00:23:20,635 --> 00:23:24,310 think about these others in the context of sliders and real estate. -326 +327 00:23:25,230 --> 00:23:28,809 It's obviously limited, I mean you're a bug on the surface of these objects, -327 +328 00:23:28,809 --> 00:23:32,250 only getting a feel for one point at a time and for the rules of movement. -328 +329 00:23:33,230 --> 00:23:36,509 Also, geometry can be quite nice when it's coordinate-free, -329 +330 00:23:36,509 --> 00:23:39,898 and this is the opposite of that, but it does give a foothold -330 +331 00:23:39,898 --> 00:23:43,670 into thinking about high-dimensional shapes a little more concretely. -331 +332 00:23:46,030 --> 00:23:48,980 Now you could say that viewing things with sliders is no -332 +333 00:23:48,980 --> 00:23:51,930 different from thinking about things purely analytically. -333 +334 00:23:52,490 --> 00:23:56,217 I mean, it's honestly little more than representing each coordinate literally, -334 +335 00:23:56,217 --> 00:23:58,530 it's kind of the most obvious thing you might do. -335 +336 00:23:59,250 --> 00:24:03,468 But this small move makes it much more possible to play with the thought of a -336 +337 00:24:03,468 --> 00:24:07,741 high-dimensional point, and even little things like thinking about the squares -337 +338 00:24:07,741 --> 00:24:12,014 of coordinates as real estate can shed light on some seemingly strange aspects -338 +339 00:24:12,014 --> 00:24:16,450 of high dimensions, like just how far away the corner of a box is from its center. -339 +340 00:24:17,290 --> 00:24:20,978 If anything, the fact that it's such a direct representation of a -340 +341 00:24:20,978 --> 00:24:24,834 purely analytic description is exactly what makes it such a faithful -341 +342 00:24:24,834 --> 00:24:28,690 reflection of what genuinely doing math in higher dimensions entails. -342 +343 00:24:29,390 --> 00:24:33,503 We're still flying in the clouds, trusting the instruments of analytic reasoning, -343 +344 00:24:33,503 --> 00:24:37,566 but this is a redesign of those instruments, one which better takes advantage of -344 +345 00:24:37,566 --> 00:24:41,530 the fact that such a large portion of our brains goes towards image processing. -345 +346 00:24:42,410 --> 00:24:44,705 I mean, just because you can't visualize something -346 +347 00:24:44,705 --> 00:24:47,090 doesn't mean you can't still think about it visually. diff --git a/2017/higher-dimensions/english/sentence_timings.json b/2017/higher-dimensions/english/sentence_timings.json index a69fe9c09..2b763d028 100644 --- a/2017/higher-dimensions/english/sentence_timings.json +++ b/2017/higher-dimensions/english/sentence_timings.json @@ -280,7 +280,7 @@ 597.22 ], [ - "Then you can subtract off this portion here, the radius of the corner circle, which by definition is 1, and that is 1,414.", + "Then you can subtract off this portion here the radius of the corner circle which by definition is 1, and that means the radius of the inner circle is square root of 2 minus 1, or about 0.414.", 598.32, 610.7 ], @@ -520,7 +520,7 @@ 996.27 ], [ - "Our 2x2x2x2 box is going to have 16 vertices at 1111, 111-1, and so on, with all possible binary combinations of 1 and negative 1.", + "Our 2x2x2x2 box is going to have 16 vertices at 1 1 1 1 1 1 1 negative 1 and so on with all possible binary combinations of 1 and negative 1.", 997.95, 1011.47 ], diff --git a/2017/higher-dimensions/english/transcript.txt b/2017/higher-dimensions/english/transcript.txt index ae11e6db6..10cd71cb1 100644 --- a/2017/higher-dimensions/english/transcript.txt +++ b/2017/higher-dimensions/english/transcript.txt @@ -54,7 +54,7 @@ Draw four circles, each with radius 1, centered at these four vertices, so each Now I want you to think of the circle centered at the origin which is just large enough to be touching those corner circles, tangent to each one of them. What we want to do for this setup and for its analogies in higher dimensions is find the radius of that inner circle. Here in two dimensions we can use the Pythagorean theorem to see that the distance from the origin to the corner of the box is the square root of 2, which is around 1.414. -Then you can subtract off this portion here, the radius of the corner circle, which by definition is 1, and that is 1,414. +Then you can subtract off this portion here the radius of the corner circle which by definition is 1, and that means the radius of the inner circle is square root of 2 minus 1, or about 0.414. No surprises here, that seems pretty reasonable. Now do something analogous in three dimensions. Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. @@ -102,7 +102,7 @@ It's definitely less than 0.75 since all three of these are smaller than 0.5, so And again, we sit back and feel comfortable with this result, right? The inner sphere is smaller than the corner spheres. But things get interesting when we move up into four dimensions. -Our 2x2x2x2 box is going to have 16 vertices at 1111, 111-1, and so on, with all possible binary combinations of 1 and negative 1. +Our 2x2x2x2 box is going to have 16 vertices at 1 1 1 1 1 1 1 negative 1 and so on with all possible binary combinations of 1 and negative 1. What this means is that there are 16 unit spheres centered at these corners, each one tangent to four of its neighbors. As before, we'll just be focusing on one of them, the one centered at 1111. The point of the sphere closest to the origin corresponds to the configuration of sliders where all four coordinates reach exactly halfway between 1 and 0. diff --git a/2017/higher-dimensions/french/sentence_translations.json b/2017/higher-dimensions/french/sentence_translations.json index aa0cf244f..5927b7315 100644 --- a/2017/higher-dimensions/french/sentence_translations.json +++ b/2017/higher-dimensions/french/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "Dessinez un cube 2x2x2 dont les coins ont des sommets 1,1,1,1,1,1,1,1, puis nous allons prendre huit sphères différentes dont chacune a un rayon 1 et les centrer sur ces sommets pour que chacun est tangent à trois de ses voisins. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/german/sentence_translations.json b/2017/higher-dimensions/german/sentence_translations.json index 001469d8e..070379934 100644 --- a/2017/higher-dimensions/german/sentence_translations.json +++ b/2017/higher-dimensions/german/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "Zeichnen Sie einen 2x2x2-Würfel, dessen Ecken die Eckpunkte 1,1,1,1,1,1,1,1 haben, und dann nehmen wir acht verschiedene Kugeln, von denen jede einen Radius 1 hat, und zentrieren sie auf diesen Eckpunkten, so dass jeder tangiert drei seiner Nachbarn. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/hebrew/sentence_translations.json b/2017/higher-dimensions/hebrew/sentence_translations.json index 8a7285f5d..3d4575375 100644 --- a/2017/higher-dimensions/hebrew/sentence_translations.json +++ b/2017/higher-dimensions/hebrew/sentence_translations.json @@ -462,7 +462,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors.", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors.", "translatedText": "צייר קובייה בגודל 2x2x2 שלפינותיה יש קודקודים 1,1,1,1,1,1,1,1, ואז ניקח שמונה כדורים שונים שלכל אחד מהם יש רדיוס 1 ונרכז אותם על הקודקודים האלה. כל אחד משיק לשלושה מהשכנים שלו.", "n_reviews": 0, "start": 617.74, diff --git a/2017/higher-dimensions/hindi/sentence_translations.json b/2017/higher-dimensions/hindi/sentence_translations.json index 79871182e..f1e71a6bb 100644 --- a/2017/higher-dimensions/hindi/sentence_translations.json +++ b/2017/higher-dimensions/hindi/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "एक 2x2x2 घन बनाएं जिसके कोनों पर शीर्ष 1,1,1,1,1,1,1,1 हों, और फिर हम आठ अलग-अलग गोले लेंगे जिनमें से प्रत्येक की त्रिज्या 1 होगी और उन्हें इन शीर्षों पर केन्द्रित करेंगे ताकि प्रत्येक अपने तीन पड़ोसियों से स्पर्शरेखा है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/hungarian/sentence_translations.json b/2017/higher-dimensions/hungarian/sentence_translations.json index 6f24b48b8..fb5e3766c 100644 --- a/2017/higher-dimensions/hungarian/sentence_translations.json +++ b/2017/higher-dimensions/hungarian/sentence_translations.json @@ -448,7 +448,7 @@ "end": 597.22 }, { - "input": "Then you can subtract off this portion here, the radius of the corner circle, which by definition is 1, and that is 1,414.", + "input": "Then you can subtract off this portion here the radius of the corner circle which by definition is 1, and that means the radius of the inner circle is square root of 2 minus 1, or about 0.414.", "translatedText": "Ezután kivonhatjuk ezt a részt itt, a sarokkör sugarát, ami a definíció szerint 1, és ez 1,414.", "model": "DeepL", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 996.27 }, { - "input": "Our 2x2x2x2 box is going to have 16 vertices at 1111, 111-1, and so on, with all possible binary combinations of 1 and negative 1.", + "input": "Our 2x2x2x2 box is going to have 16 vertices at 1 1 1 1 1 1 1 negative 1 and so on with all possible binary combinations of 1 and negative 1.", "translatedText": "A 2x2x2x2x2 dobozunknak 16 csúcsa lesz 1111, 111-1, és így tovább, az 1 és a negatív 1 minden lehetséges bináris kombinációjával.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/higher-dimensions/indonesian/sentence_translations.json b/2017/higher-dimensions/indonesian/sentence_translations.json index b5c17d1bb..eed2967dd 100644 --- a/2017/higher-dimensions/indonesian/sentence_translations.json +++ b/2017/higher-dimensions/indonesian/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "Gambarlah sebuah kubus berukuran 2x2x2 yang sudut-sudutnya memiliki simpul 1,1,1,1,1,1,1,1, lalu kita akan mengambil delapan bola berbeda yang masing-masing berjari-jari 1 dan memusatkannya pada simpul-simpul tersebut sehingga masing-masing bersinggungan dengan tiga tetangganya. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/italian/sentence_translations.json b/2017/higher-dimensions/italian/sentence_translations.json index 9c09c1ffd..8601c62a9 100644 --- a/2017/higher-dimensions/italian/sentence_translations.json +++ b/2017/higher-dimensions/italian/sentence_translations.json @@ -462,7 +462,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors.", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors.", "translatedText": "Disegna un cubo 2x2x2 i cui angoli hanno vertici 1,1,1,1,1,1,1,1, quindi prenderemo otto sfere diverse, ciascuna delle quali ha un raggio 1 e le centreremo su questi vertici in modo che ognuno è tangente a tre dei suoi vicini.", "n_reviews": 0, "start": 617.74, diff --git a/2017/higher-dimensions/japanese/sentence_translations.json b/2017/higher-dimensions/japanese/sentence_translations.json index 7733e5008..40cb3518e 100644 --- a/2017/higher-dimensions/japanese/sentence_translations.json +++ b/2017/higher-dimensions/japanese/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "角が頂点 1,1, 1,1,1,1,1,1 を持つ 2x2x2 の立方体を描画します。次に、 それぞれ半径 1 の 8 つの異なる球を取り、これらの頂点を中心に配置し ます。それぞれは隣接する 3 つのノードに接しています。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/korean/sentence_translations.json b/2017/higher-dimensions/korean/sentence_translations.json index 939046ee4..1b0a15c49 100644 --- a/2017/higher-dimensions/korean/sentence_translations.json +++ b/2017/higher-dimensions/korean/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "모서리의 정점이 1,1,1,1,1,1,1,1인 2x2x2 큐브를 그린 다음 각각의 반경이 1인 8개의 서로 다른 구를 가져와 이 정점의 중심에 두겠습니다. 각각은 이웃 세 개에 접합니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/marathi/sentence_translations.json b/2017/higher-dimensions/marathi/sentence_translations.json index a8509d2bc..d4d9db4f7 100644 --- a/2017/higher-dimensions/marathi/sentence_translations.json +++ b/2017/higher-dimensions/marathi/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "2x2x2 घन काढा ज्याच्या कोपऱ्यात 1,1,1,1,1,1,1,1 शिरोबिंदु आहेत आणि मग आपण आठ वेगवेगळे गोल घेणार आहोत ज्यातील प्रत्येकाची त्रिज्या 1 आहे आणि त्यांना या शिरोबिंदूंवर केंद्रस्थानी ठेवू. प्रत्येक त्याच्या तीन शेजाऱ्यांना स्पर्शिका आहे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/persian/sentence_translations.json b/2017/higher-dimensions/persian/sentence_translations.json index d12fffa71..4761af777 100644 --- a/2017/higher-dimensions/persian/sentence_translations.json +++ b/2017/higher-dimensions/persian/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "یک مکعب 2x2x2 رسم کنید که گوشه های آن دارای رئوس 1،1،1،1،1،1،1،1، و سپس هشت کره مختلف که هر کدام شعاع 1 دارند را می گیریم و آنها را روی این رئوس متمرکز می کنیم تا هر کدام مماس بر سه همسایه خود است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/portuguese/sentence_translations.json b/2017/higher-dimensions/portuguese/sentence_translations.json index cacf0f9ba..2873f787c 100644 --- a/2017/higher-dimensions/portuguese/sentence_translations.json +++ b/2017/higher-dimensions/portuguese/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "Desenhe um cubo 2x2x2 cujos cantos tenham vértices 1,1,1,1,1,1,1,1, e então pegaremos oito esferas diferentes, cada uma com raio 1 e centralizá-las nesses vértices para que cada um é tangente a três de seus vizinhos. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/russian/sentence_translations.json b/2017/higher-dimensions/russian/sentence_translations.json index f4c8d404f..a645ec987 100644 --- a/2017/higher-dimensions/russian/sentence_translations.json +++ b/2017/higher-dimensions/russian/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "Нарисуйте куб 2х2х2, углы которого имеют вершины 1,1,1,1,1,1,1,1, а затем мы возьмем восемь разных сфер, каждая из которых имеет радиус 1, и отцентрируем их по этим вершинам так, чтобы каждый из них касается трех своих соседей. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/spanish/sentence_translations.json b/2017/higher-dimensions/spanish/sentence_translations.json index bafaa98da..403822bdf 100644 --- a/2017/higher-dimensions/spanish/sentence_translations.json +++ b/2017/higher-dimensions/spanish/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "Dibuja un cubo de 2x2x2 cuyas esquinas tengan vértices 1,1,1,1,1,1,1,1, y luego tomaremos ocho esferas diferentes, cada una de las cuales tiene un radio 1 y las centraremos en estos vértices para que cada uno es tangente a tres de sus vecinos. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/tamil/sentence_translations.json b/2017/higher-dimensions/tamil/sentence_translations.json index 84f1845ce..a85e02e5a 100644 --- a/2017/higher-dimensions/tamil/sentence_translations.json +++ b/2017/higher-dimensions/tamil/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "2x2x2 கனசதுரத்தை வரையவும், அதன் மூலைகளில் 1,1,1,1,1,1,1,1,1 செங்குத்துகள் உள்ளன, பின்னர் நாம் எட்டு வெவ்வேறு கோளங்களை எடுக்கப் போகிறோம், அவை ஒவ்வொன்றும் ஒரு ஆரம் 1 மற்றும் அவற்றை இந்த செங்குத்துகளில் மையப்படுத்துவோம். ஒவ்வொன்றும் அதன் மூன்று அண்டை நாடுகளுக்கு தொடுவானது. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/telugu/sentence_translations.json b/2017/higher-dimensions/telugu/sentence_translations.json index c817a9840..2b3e6bbdb 100644 --- a/2017/higher-dimensions/telugu/sentence_translations.json +++ b/2017/higher-dimensions/telugu/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "మూలల్లో 1,1,1,1,1,1,1,1,1 శీర్షాలు ఉన్న 2x2x2 క్యూబ్‌ను గీయండి, ఆపై మేము ఎనిమిది వేర్వేరు గోళాలను తీయబోతున్నాము, వాటిలో ప్రతి ఒక్కటి వ్యాసార్థం 1ని కలిగి ఉంటుంది మరియు వాటిని ఈ శీర్షాలపై మధ్యలో ఉంచుతాము. ప్రతి ఒక్కటి దాని పొరుగువారిలో ముగ్గురికి టాంజెంట్‌గా ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/thai/sentence_translations.json b/2017/higher-dimensions/thai/sentence_translations.json index 68f25aacb..0307ee900 100644 --- a/2017/higher-dimensions/thai/sentence_translations.json +++ b/2017/higher-dimensions/thai/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/turkish/sentence_translations.json b/2017/higher-dimensions/turkish/sentence_translations.json index 9e4a0a08c..5cac7d4da 100644 --- a/2017/higher-dimensions/turkish/sentence_translations.json +++ b/2017/higher-dimensions/turkish/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "Köşeleri 1,1,1,1,1,1,1,1 olan 2x2x2'lik bir küp çizin ve sonra her birinin yarıçapı 1 olan sekiz farklı küre alıp bunları bu köşelere ortalayacağız, böylece her biri komşularının üçüne teğettir. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/ukrainian/sentence_translations.json b/2017/higher-dimensions/ukrainian/sentence_translations.json index 97d7860ef..24d97af9a 100644 --- a/2017/higher-dimensions/ukrainian/sentence_translations.json +++ b/2017/higher-dimensions/ukrainian/sentence_translations.json @@ -462,7 +462,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors.", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors.", "translatedText": "Намалюйте куб 2x2x2, кути якого мають вершини 1,1,1,1,1,1,1,1, а потім візьмемо вісім різних сфер, кожна з яких має радіус 1, і відцентруємо їх у цих вершинах так, щоб кожен з них дотичний до трьох своїх сусідів.", "n_reviews": 0, "start": 617.74, diff --git a/2017/higher-dimensions/urdu/sentence_translations.json b/2017/higher-dimensions/urdu/sentence_translations.json index 9cbaf6ce5..373aa1200 100644 --- a/2017/higher-dimensions/urdu/sentence_translations.json +++ b/2017/higher-dimensions/urdu/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "ایک 2x2x2 مکعب کھینچیں جس کے کونوں میں عمودی 1,1,1,1,1,1,1,1, اور پھر ہم آٹھ مختلف دائرے لیں گے جن میں سے ہر ایک کا رداس 1 ہے اور ان کو ان عمودی عمودی پر مرکز کریں گے تاکہ ہر ایک اپنے تین پڑوسیوں کا مماس ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-dimensions/vietnamese/sentence_translations.json b/2017/higher-dimensions/vietnamese/sentence_translations.json index cc6ba359c..ff82e7e4b 100644 --- a/2017/higher-dimensions/vietnamese/sentence_translations.json +++ b/2017/higher-dimensions/vietnamese/sentence_translations.json @@ -528,7 +528,7 @@ "end": 617.08 }, { - "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,1,1,1, and then we're going to take eight different spheres each of which has a radius 1 and center them on these vertices so that each one is tangent to three of its neighbors. ", + "input": "Draw a 2x2x2 cube whose corners have vertices 1,1,1,1,1,-1, on and on and on, and then we're going to take eight different spheres, each of which has a radius 1, and center them on these vertices so that each one is tangent to three of its neighbors. ", "translatedText": "Vẽ một khối lập phương 2x2x2 có các góc có các đỉnh 1,1,1,1,1,1,1,1, sau đó chúng ta sẽ lấy tám hình cầu khác nhau, mỗi hình cầu có bán kính 1 và căn giữa chúng trên các đỉnh này sao cho mỗi cái tiếp tuyến với ba lân cận của nó. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/arabic/sentence_translations.json b/2017/higher-order-derivatives/arabic/sentence_translations.json index 900b6ef08..ddcb62431 100644 --- a/2017/higher-order-derivatives/arabic/sentence_translations.json +++ b/2017/higher-order-derivatives/arabic/sentence_translations.json @@ -184,7 +184,7 @@ "end": 221.64 }, { - "input": "Even though it's not like this letter d is a variable being multiplied by f, for the sake of more compact notation you'd write it as d2f divided by dx2, and you don't typically bother with any parentheses on the bottom. ", + "input": "eleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for ex ", "translatedText": "على الرغم من أن هذا الحرف d ليس متغيرًا يتم ضربه في f، إلا أنه من أجل تدوين أكثر إحكامًا، يمكنك كتابته بالشكل d2f مقسومًا على dx2، ولا تهتم عادةً بأي قوسين في الأسفل. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. ", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over t ", "translatedText": "ثم يخبرك مشتقها بالسرعة عند كل نقطة زمنية، على سبيل المثال، قد يبدو الرسم البياني مثل هذا النتوء، حيث يزيد إلى الحد الأقصى، ثم يتناقص مرة أخرى إلى الصفر. ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward. ", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d ", "translatedText": "في هذا المثال، يكون المشتق الثاني موجبًا للنصف الأول من الرحلة، وهو ما يشير إلى السرعة، وهو الإحساس بأنك تُدفع للخلف إلى مقعد السيارة، أو بالأحرى، أن مقعد السيارة يدفعك للأمام. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/bengali/sentence_translations.json b/2017/higher-order-derivatives/bengali/sentence_translations.json index add485bb2..021434a12 100644 --- a/2017/higher-order-derivatives/bengali/sentence_translations.json +++ b/2017/higher-order-derivatives/bengali/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "সম্ভবত দ্বিতীয় ডেরিভেটিভের সবচেয়ে ভিসারাল বোঝাপড়া হল যে এটি ত্বরণকে প্রতিনিধিত্ব করে।", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "তারপর এর ডেরিভেটিভ আপনাকে সময়ের প্রতিটি বিন্দুতে বেগ বলে, উদাহরণস্বরূপ গ্রাফটি এই বাম্পের মতো দেখাতে পারে, কিছু সর্বোচ্চ পর্যন্ত বৃদ্ধি পাচ্ছে এবং শূন্যে ফিরে আসছে।", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "এই উদাহরণে, দ্বিতীয় ডেরিভেটিভটি যাত্রার প্রথমার্ধের জন্য ইতিবাচক, যা গতি বাড়ার ইঙ্গিত দেয়, এটি আপনার গাড়ির সিটে পিছনে ঠেলে দেওয়ার সংবেদন, বা বরং, গাড়ির আসনটি আপনাকে সামনের দিকে ঠেলে দেয়।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/bulgarian/sentence_translations.json b/2017/higher-order-derivatives/bulgarian/sentence_translations.json index df557f590..469ad7f22 100644 --- a/2017/higher-order-derivatives/bulgarian/sentence_translations.json +++ b/2017/higher-order-derivatives/bulgarian/sentence_translations.json @@ -120,7 +120,7 @@ "end": 110.9 }, { - "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where, as always, the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx approaches 0.", + "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where as always the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx, both dx's in this case, approach 0.", "translatedText": "Що се отнася до записването, можете да опитате да го напишете по този начин, като посочите някаква малка промяна в производната функция, разделена на някаква малка промяна в x, където, както винаги, използването на тази буква d предполага, че това, което наистина искате да разгледате, е какво е това съотношение, когато dx се приближава до 0.", "model": "DeepL", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 134.44 }, { - "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d2f divided by dx2.", + "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d squared f divided by dx squared.", "translatedText": "Това е доста неудобно и тромаво, така че стандартът е да се съкращава като d2f, разделено на dx2.", "model": "DeepL", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "Може би най-ясното разбиране за втората производна е, че тя представлява ускорение.", "model": "DeepL", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "След това неговата производна ви показва скоростта във всеки един момент от време, например графиката може да изглежда като този удар, нарастващ до някакъв максимум и намаляващ обратно до нула.", "model": "DeepL", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "В този пример втората производна е положителна за първата половина от пътуването, което показва ускоряване, това е усещането, че сте избутани обратно в седалката на колата или по-скоро, че седалката на колата ви избутва напред.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/chinese/sentence_translations.json b/2017/higher-order-derivatives/chinese/sentence_translations.json index ab5a2e54e..0a54d09e3 100644 --- a/2017/higher-order-derivatives/chinese/sentence_translations.json +++ b/2017/higher-order-derivatives/chinese/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "也许对二阶导数最本能的理解是它代表加速度。", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "然后它的导数告诉你每个时间点的速度，例如图表可能看 起来像这个凹凸，增加到某个最大值，然后减少到零。", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "在这个例子中，前半段旅程的二阶导数为正， 这表示加速，即被推回汽车座椅的感觉，或者 更确切地说，是汽车座椅推动您前进的感觉。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/english/captions.srt b/2017/higher-order-derivatives/english/captions.srt index 390acdad1..3173725ab 100644 --- a/2017/higher-order-derivatives/english/captions.srt +++ b/2017/higher-order-derivatives/english/captions.srt @@ -111,32 +111,32 @@ but it's smaller, the slope only increases slowly. At points where there's not really any curvature, the second derivative is just 0. 29 -00:01:53,380 --> 00:01:57,779 +00:01:53,380 --> 00:01:57,484 As far as notation goes, you could try writing it like this, 30 -00:01:57,779 --> 00:02:01,890 +00:01:57,484 --> 00:02:01,319 indicating some small change to the derivative function, 31 -00:02:01,890 --> 00:02:05,713 -divided by some small change to x, where, as always, +00:02:01,319 --> 00:02:06,433 +divided by some small change to x, where as always the use of this letter d 32 -00:02:05,713 --> 00:02:10,905 -the use of this letter d suggests that what you really want to consider +00:02:06,433 --> 00:02:12,085 +suggests that what you really want to consider is what this ratio approaches as dx, 33 -00:02:10,905 --> 00:02:14,440 -is what this ratio approaches as dx approaches 0. +00:02:12,085 --> 00:02:14,440 +both dx's in this case, approach 0. 34 -00:02:15,540 --> 00:02:19,603 -That's pretty awkward and clunky, so the standard +00:02:15,540 --> 00:02:19,254 +That's pretty awkward and clunky, so the standard is 35 -00:02:19,603 --> 00:02:23,180 -is to abbreviate this as d2f divided by dx2. +00:02:19,254 --> 00:02:23,180 +to abbreviate this as d squared f divided by dx squared. 36 00:02:24,360 --> 00:02:28,504 @@ -207,90 +207,110 @@ divided by the size of dx2, or more precisely, whatever that ratio approaches as dx approaches 0. 53 -00:03:43,000 --> 00:03:48,445 -Even though it's not like this letter d is a variable being multiplied by f, +00:03:43,000 --> 00:03:47,248 +eleration. Given some movement along a line, suppose you have some function that 54 -00:03:48,445 --> 00:03:53,819 -for the sake of more compact notation you'd write it as d2f divided by dx2, +00:03:47,248 --> 00:03:51,233 +records the distance traveled versus time, maybe its graph looks like this, 55 -00:03:53,819 --> 00:03:57,780 -and you don't bother with any parentheses on the bottom. +00:03:51,233 --> 00:03:55,324 +steadily increasing over time. Then its derivative tells you velocity at each 56 -00:03:59,040 --> 00:04:01,799 -Maybe the most visceral understanding of the second +00:03:55,324 --> 00:03:58,680 +point in time, for example the graph might look like this bump, 57 -00:04:01,799 --> 00:04:04,240 -derivative is that it represents acceleration. +00:03:58,680 --> 00:04:01,827 +increasing up to some maximum, and decreasing back to zero. 58 +00:04:01,827 --> 00:04:04,240 +So the second derivative tells you the rate of + +59 +00:04:04,681 --> 00:04:04,240 +Maybe the most visceral understanding of the second + +60 +00:04:05,180 --> 00:04:04,681 +derivative is that it represents acceleration. + +61 00:04:05,180 --> 00:04:08,898 Given some movement along a line, suppose you have some function -59 +62 00:04:08,898 --> 00:04:11,644 that records the distance traveled versus time, -60 +63 00:04:11,644 --> 00:04:15,820 maybe its graph looks something like this, steadily increasing over time. -61 +64 00:04:16,740 --> 00:04:20,226 Then its derivative tells you velocity at each point in time, -62 +65 00:04:20,226 --> 00:04:24,725 for example the graph might look like this bump, increasing up to some maximum, -63 +66 00:04:24,725 --> 00:04:26,300 and decreasing back to zero. -64 -00:04:27,200 --> 00:04:31,219 -So the second derivative tells you the rate of change for the velocity, +67 +00:04:27,200 --> 00:04:30,922 +The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, -65 -00:04:31,219 --> 00:04:33,900 -which is the acceleration at each point in time. +68 +00:04:30,922 --> 00:04:33,693 +it means that the strength of the acceleration itself is changing. -66 -00:04:34,920 --> 00:04:39,468 +69 +00:04:33,693 --> 00:04:37,167 +One of the most useful things about higher order derivatives is how they help us in + +70 +00:04:37,167 --> 00:04:38,160 +approximating functions, + +71 +00:04:38,160 --> 00:04:41,470 In this example, the second derivative is positive for the first half of the journey, -67 -00:04:39,468 --> 00:04:43,488 +72 +00:04:41,470 --> 00:04:44,395 which indicates speeding up, that's the sensation of being pushed back into -68 -00:04:43,488 --> 00:04:46,820 +73 +00:04:44,395 --> 00:04:46,820 your car seat, or rather, having the car seat push you forward. -69 +74 00:04:47,540 --> 00:04:52,520 A negative second derivative indicates slowing down, negative acceleration. -70 +75 00:04:54,000 --> 00:04:57,080 The third derivative, and this is not a joke, is called jerk. -71 +76 00:04:57,840 --> 00:05:03,920 So if the jerk is not zero, it means the strength of the acceleration itself is changing. -72 +77 00:05:06,280 --> 00:05:09,656 One of the most useful things about higher order derivatives is -73 +78 00:05:09,656 --> 00:05:13,138 how they help us in approximating functions, which is exactly the -74 +79 00:05:13,138 --> 00:05:16,620 topic of the next chapter on Taylor series, so I'll see you there. diff --git a/2017/higher-order-derivatives/english/sentence_timings.json b/2017/higher-order-derivatives/english/sentence_timings.json index 3660d8d75..45089e85d 100644 --- a/2017/higher-order-derivatives/english/sentence_timings.json +++ b/2017/higher-order-derivatives/english/sentence_timings.json @@ -75,12 +75,12 @@ 110.9 ], [ - "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where, as always, the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx approaches 0.", + "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where as always the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx, both dx's in this case, approach 0.", 113.38, 134.44 ], [ - "That's pretty awkward and clunky, so the standard is to abbreviate this as d2f divided by dx2.", + "That's pretty awkward and clunky, so the standard is to abbreviate this as d squared f divided by dx squared.", 135.54, 143.18 ], @@ -125,13 +125,13 @@ 221.64 ], [ - "Even though it's not like this letter d is a variable being multiplied by f, for the sake of more compact notation you'd write it as d2f divided by dx2, and you don't bother with any parentheses on the bottom.", + "eleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. So the second derivative tells you the rate of", 223.0, - 237.78 + 244.24 ], [ "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", - 239.04, + 245.18, 244.24 ], [ @@ -145,13 +145,13 @@ 266.3 ], [ - "So the second derivative tells you the rate of change for the velocity, which is the acceleration at each point in time.", + "The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order derivatives is how they help us in approximating functions,", 267.2, - 273.9 + 278.16 ], [ "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", - 274.92, + 278.16, 286.82 ], [ diff --git a/2017/higher-order-derivatives/english/transcript.txt b/2017/higher-order-derivatives/english/transcript.txt index 88e22e66a..b7bb78258 100644 --- a/2017/higher-order-derivatives/english/transcript.txt +++ b/2017/higher-order-derivatives/english/transcript.txt @@ -13,8 +13,8 @@ At points where it curves upwards, the slope is increasing, and that means the s At points where it's curving downwards, the slope is decreasing, so the second derivative is negative. For example, a graph like this one has a very positive second derivative at the point 4, since the slope is rapidly increasing around that point, whereas a graph like this one still has a positive second derivative at the same point, but it's smaller, the slope only increases slowly. At points where there's not really any curvature, the second derivative is just 0. -As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where, as always, the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx approaches 0. -That's pretty awkward and clunky, so the standard is to abbreviate this as d2f divided by dx2. +As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where as always the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx, both dx's in this case, approach 0. +That's pretty awkward and clunky, so the standard is to abbreviate this as d squared f divided by dx squared. And even though it's not terribly important for getting an intuition for the second derivative, I think it might be worth showing you how you can read this notation. To start off, think of some input to your function, and then take two small steps to the right, each one with a size of dx. I'm choosing rather big steps here so we'll be able to see what's going on, but in principle keep in the back of your mind that dx should be rather tiny. @@ -23,11 +23,11 @@ The difference between these changes, the change in how the function changes, is You should think of this as really small, typically proportional to the size of dx2. So if, for example, you substituted in 0.01 for dx, you would expect this ddf to be about proportional to 0.0001. The second derivative is the size of this change to the change, divided by the size of dx2, or more precisely, whatever that ratio approaches as dx approaches 0. -Even though it's not like this letter d is a variable being multiplied by f, for the sake of more compact notation you'd write it as d2f divided by dx2, and you don't bother with any parentheses on the bottom. +eleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. So the second derivative tells you the rate of Maybe the most visceral understanding of the second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. -So the second derivative tells you the rate of change for the velocity, which is the acceleration at each point in time. +The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order derivatives is how they help us in approximating functions, In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward. A negative second derivative indicates slowing down, negative acceleration. The third derivative, and this is not a joke, is called jerk. diff --git a/2017/higher-order-derivatives/french/sentence_translations.json b/2017/higher-order-derivatives/french/sentence_translations.json index 10e3b24d0..da4e0aef2 100644 --- a/2017/higher-order-derivatives/french/sentence_translations.json +++ b/2017/higher-order-derivatives/french/sentence_translations.json @@ -119,7 +119,7 @@ "end": 110.9 }, { - "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where, as always, the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx approaches 0.", + "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where as always the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx, both dx's in this case, approach 0.", "translatedText": "En ce qui concerne la notation, vous pouvez essayer de l'écrire comme ceci, indiquant un petit changement dans la fonction dérivée, divisé par un petit changement dans x, où, comme toujours, l'utilisation de cette lettre d suggère que ce que vous voulez vraiment considérer C'est ce à quoi ce rapport se rapproche lorsque dx se rapproche de 0.", "from_community_srt": "la dérivée seconde est nulle. En ce qui concerne la notation, vous pouvez essayer de l'écrire comme ceci, ce qui indique un petit changement à la fonction dérivée divisée par un certain petit changement de x, comme toujours l'utilisation de cette lettre d suggère que vous ne considérez uniquement ce vers quoi tend la valeur du rapport lorsque dx, ici les deux dx dans ce cas,", "n_reviews": 0, @@ -127,7 +127,7 @@ "end": 134.44 }, { - "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d2f divided by dx2.", + "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d squared f divided by dx squared.", "translatedText": "C'est assez gênant et maladroit, donc la norme est d'abréger cela en d2f divisé par dx2.", "from_community_srt": "approchent 0. C'est assez étrange et maladroit, de sorte que la norme est d'abréger comme d2f/dx2.", "n_reviews": 0, @@ -207,7 +207,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "La compréhension la plus viscérale de la dérivée seconde est peut-être qu’elle représente une accélération.", "from_community_srt": "Peut-être que la compréhension la plus viscérale de la dérivée seconde est qu'elle représente l'accélération.", "n_reviews": 0, @@ -223,7 +223,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "Ensuite, sa dérivée vous indique la vitesse à chaque instant, par exemple le graphique pourrait ressembler à cette bosse, augmentant jusqu'à un maximum et diminuant jusqu'à zéro.", "from_community_srt": "Ainsi, sa dérivée nous indique la vitesse à chaque point, OK ? Par exemple, le graphique pourrait ressembler à cette bosse, ce qui augmente jusqu'à un certain maximum, puis diminuant jusqu'à 0.", "n_reviews": 0, @@ -239,7 +239,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "Dans cet exemple, la dérivée seconde est positive pour la première moitié du trajet, ce qui indique une accélération, c'est la sensation d'être repoussé dans son siège auto, ou plutôt, d'être poussé vers l'avant par le siège auto.", "from_community_srt": "Dans cet exemple, la dérivée seconde est positive pour la première moitié du voyage, ce qui indique indique une vitesse qui augmente. C'est la sensation d'être tiré en arrière dans votre siège. Ou plutôt, d'avoir le siège de la voiture qui vous pousse vers l'avant.", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/german/sentence_translations.json b/2017/higher-order-derivatives/german/sentence_translations.json index 80a1845e6..0926685b5 100644 --- a/2017/higher-order-derivatives/german/sentence_translations.json +++ b/2017/higher-order-derivatives/german/sentence_translations.json @@ -134,7 +134,7 @@ "end": 110.9 }, { - "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where, as always, the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx approaches 0.", + "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where as always the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx, both dx's in this case, approach 0.", "translatedText": "Was die Schreibweise angeht, könntest du versuchen, es so zu schreiben: eine kleine Änderung der Ableitungsfunktion, geteilt durch eine kleine Änderung von x. Wie immer deutet die Verwendung des Buchstabens d darauf hin, dass du in Wirklichkeit betrachten willst, wie sich dieses Verhältnis nähert, wenn dx gegen 0 geht.", "model": "DeepL", "from_community_srt": "ist die zweite Ableitung null. So weit wie Definitionen gehen, könntest du versuchen, sie so zu schreiben, also eine kleine Änderung in der Ableitungsfunktion geteilt durch eine kleine Änderung von x, wo der Buchstabe d zeigt, dass du wirklich beachtest, welchem Wert sich dieser Quotient bzw. dx annähern - hier werden beide dx null.", @@ -143,7 +143,7 @@ "end": 134.44 }, { - "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d2f divided by dx2.", + "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d squared f divided by dx squared.", "translatedText": "Das ist ziemlich umständlich und klobig, deshalb ist es üblich, dies als d2f geteilt durch dx2 abzukürzen.", "model": "DeepL", "from_community_srt": "Das ist ziemlich komisch und schwerfällig, deshalb kürzt man es normalerweise mit d²f/dx² ab.", @@ -233,7 +233,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "Das intuitivste Verständnis der zweiten Ableitung ist vielleicht, dass sie für die Beschleunigung steht.", "model": "DeepL", "from_community_srt": "Die instinktivste Erklärung der zweiten Ableitung ist vielleicht, dass sie die Beschleunigung darstellt.", @@ -251,7 +251,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "Die Ableitung zeigt dir dann die Geschwindigkeit zu jedem Zeitpunkt an. Der Graph könnte zum Beispiel so aussehen: Er steigt bis zu einem bestimmten Maximum und fällt dann wieder auf Null.", "model": "DeepL", "from_community_srt": "Dann ist dessen Ableitung die Geschwindigkeit an jedem Zeitpunkt, richtig? Der Graph könnte zum Beispiel so aussehen, bis zu einem bestimmten Maximum zu- und dann bis 0 wieder abnehmend.", @@ -269,7 +269,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "In diesem Beispiel ist die zweite Ableitung für die erste Hälfte der Fahrt positiv, was auf eine Beschleunigung hindeutet. Das ist das Gefühl, in den Autositz zurückgeschoben zu werden, oder besser gesagt, dass der Autositz dich nach vorne schiebt.", "model": "DeepL", "from_community_srt": "In diesem Beispiel ist die zweite Ableitung in der ersten Hälfte positiv, was so viel heißt wie schneller zu werden. Das ist das Gefühl, mit einer konstanten Kraft in seinen Sitz gedrückt zu werden. Beziehungsweise eher, dass der Sitz dich mit einer konstanten Kraft drückt.", diff --git a/2017/higher-order-derivatives/hebrew/sentence_translations.json b/2017/higher-order-derivatives/hebrew/sentence_translations.json index f7012c273..03d1474a1 100644 --- a/2017/higher-order-derivatives/hebrew/sentence_translations.json +++ b/2017/higher-order-derivatives/hebrew/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "אולי ההבנה הקרבית ביותר של הנגזרת השנייה היא שהיא מייצגת תאוצה.", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "ואז הנגזרת שלו אומרת לך את המהירות בכל נקודת זמן, למשל הגרף עשוי להיראות כמו הבליטה הזו, גדל עד למקסימום מסוים, ויורד בחזרה לאפס.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "בדוגמה זו, הנגזרת השנייה חיובית עבור המחצית הראשונה של הנסיעה, מה שמצביע על האצה, זו התחושה של דחיפה לאחור למושב המכונית שלך, או ליתר דיוק, כאשר מושב המכונית דוחף אותך קדימה.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/hindi/sentence_translations.json b/2017/higher-order-derivatives/hindi/sentence_translations.json index 38a225387..7c0c1b368 100644 --- a/2017/higher-order-derivatives/hindi/sentence_translations.json +++ b/2017/higher-order-derivatives/hindi/sentence_translations.json @@ -168,7 +168,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "शायद दूसरे व्युत्पन्न की सबसे गहरी समझ यह है कि यह त्वरण का प्रतिनिधित्व करता है।", "n_reviews": 0, "start": 239.04, @@ -182,7 +182,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "फिर इसका व्युत्पन्न आपको समय के प्रत्येक बिंदु पर वेग बताता है, उदाहरण के लिए ग्राफ़ इस उछाल की तरह दिख सकता है, जो कुछ अधिकतम तक बढ़ रहा है, और वापस शून्य तक घट रहा है।", "n_reviews": 0, "start": 256.74, @@ -196,7 +196,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "इस उदाहरण में, दूसरा व्युत्पन्न यात्रा के पहले भाग के लिए सकारात्मक है, जो गति को इंगित करता है, यह आपकी कार की सीट पर पीछे धकेले जाने की अनुभूति है, या यूं कहें कि कार की सीट आपको आगे की ओर धकेलती है।", "n_reviews": 0, "start": 274.92, diff --git a/2017/higher-order-derivatives/hungarian/sentence_translations.json b/2017/higher-order-derivatives/hungarian/sentence_translations.json index 84fa74e40..684313b74 100644 --- a/2017/higher-order-derivatives/hungarian/sentence_translations.json +++ b/2017/higher-order-derivatives/hungarian/sentence_translations.json @@ -120,7 +120,7 @@ "end": 110.9 }, { - "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where, as always, the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx approaches 0.", + "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where as always the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx, both dx's in this case, approach 0.", "translatedText": "Ami a jelölést illeti, megpróbálhatnád így írni, jelezve a derivált függvény egy kis változását, osztva az x egy kis változásával, ahol, mint mindig, a d betű használata azt sugallja, hogy amit valójában figyelembe akarsz venni, az az, hogy ez az arány mit közelít, ahogy dx közelít a 0-hoz.", "model": "DeepL", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 134.44 }, { - "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d2f divided by dx2.", + "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d squared f divided by dx squared.", "translatedText": "Ez elég kényelmetlen és nehézkes, ezért a szabvány szerint d2f osztva dx2-vel rövidítik.", "model": "DeepL", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "Talán a második derivált legnyilvánvalóbb megértése az, hogy a gyorsulást jelképezi.", "model": "DeepL", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "Ezután a deriváltja megmondja a sebességet minden egyes időpontban, például a grafikon úgy nézhet ki, mint ez a dudor, amely növekszik egy bizonyos maximumig, majd csökken vissza nullára.", "model": "DeepL", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "Ebben a példában a második derivált pozitív az út első felében, ami a gyorsulást jelzi, vagyis azt az érzést, amikor visszatolnak az autósülésbe, vagy inkább azt, amikor az autósülés előre tolja az embert.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/indonesian/sentence_translations.json b/2017/higher-order-derivatives/indonesian/sentence_translations.json index 4de0674c5..ac56afee6 100644 --- a/2017/higher-order-derivatives/indonesian/sentence_translations.json +++ b/2017/higher-order-derivatives/indonesian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "Mungkin pemahaman paling mendalam tentang turunan kedua adalah bahwa turunan tersebut melambangkan percepatan.", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "Kemudian turunannya memberi tahu Anda kecepatan pada setiap titik waktu, misalnya grafiknya mungkin terlihat seperti benjolan ini, meningkat hingga maksimum, dan menurun kembali ke nol.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "Dalam contoh ini, turunan kedua bernilai positif untuk paruh pertama perjalanan, yang menunjukkan percepatan, yaitu sensasi didorong kembali ke kursi mobil Anda, atau lebih tepatnya, kursi mobil mendorong Anda ke depan.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/italian/sentence_translations.json b/2017/higher-order-derivatives/italian/sentence_translations.json index 540f7337c..36088a3e2 100644 --- a/2017/higher-order-derivatives/italian/sentence_translations.json +++ b/2017/higher-order-derivatives/italian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "Forse la comprensione più viscerale della derivata seconda è che rappresenta l'accelerazione.", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "Quindi la sua derivata ti dice la velocità in ogni momento nel tempo, ad esempio il grafico potrebbe assomigliare a questo dosso, aumentando fino a un massimo e diminuendo fino a zero.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "In questo esempio la derivata seconda è positiva per la prima metà del viaggio, che indica un'accelerazione, cioè la sensazione di essere spinti all'indietro sul sedile dell'auto, o meglio, di avere il sedile dell'auto che ti spinge in avanti.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/japanese/sentence_translations.json b/2017/higher-order-derivatives/japanese/sentence_translations.json index bacb923c2..36e0ed365 100644 --- a/2017/higher-order-derivatives/japanese/sentence_translations.json +++ b/2017/higher-order-derivatives/japanese/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "おそらく、二次導関数について最も直感的に理解できるのは、二次導関数が加速度を表すということです。", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "次に、その微分値から各時点での速度がわかります。 たとえば、グラフ はこのバンプのようになり、ある最大値まで増加し、ゼロに戻ります。", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "この例では、2 次導関数は移動の前半で正であり、速度が上 がっていることを示しています。 これは、車のシートに押し戻 される感覚、または車のシートに前に押し戻される感覚です。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/korean/sentence_translations.json b/2017/higher-order-derivatives/korean/sentence_translations.json index 1ccc7e92b..d2f1992c0 100644 --- a/2017/higher-order-derivatives/korean/sentence_translations.json +++ b/2017/higher-order-derivatives/korean/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "아마도 2차 미분에 대한 가장 본능적인 이해는 그것이 가속도를 나타낸다는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "그런 다음 그 파생물은 각 시점의 속도를 알려줍니다. 예를 들어 그래프는 이 범프처럼 보일 수 있으며 최대치까지 증가했다가 다시 0으로 감소합니다.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "이 예에서 2차 도함수는 여행의 전반부에 대해 양수입니다. 이는 속도 증가, 즉 자동차 좌석에 뒤로 밀려나는 느낌, 또는 오히려 자동차 좌석이 당신을 앞으로 밀어주는 느낌을 나타냅니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/marathi/sentence_translations.json b/2017/higher-order-derivatives/marathi/sentence_translations.json index 7da4e0358..2d23f14f6 100644 --- a/2017/higher-order-derivatives/marathi/sentence_translations.json +++ b/2017/higher-order-derivatives/marathi/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "कदाचित दुसर्‍या व्युत्पन्नाची सर्वात आंतरीक समज अशी आहे की ती प्रवेग दर्शवते.", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "मग त्याचे व्युत्पन्न तुम्हाला वेळेच्या प्रत्येक बिंदूवर वेग सांगते, उदाहरणार्थ आलेख या धक्क्यासारखा दिसू शकतो, काही कमाल पर्यंत वाढतो आणि परत शून्यावर कमी होतो.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "या उदाहरणात, दुसरा व्युत्पन्न प्रवासाच्या पहिल्या सहामाहीसाठी सकारात्मक आहे, जो वेग वाढवण्याचा संकेत देतो, म्हणजे तुमच्या कारच्या सीटवर मागे ढकलले गेल्याची संवेदना, किंवा त्याऐवजी, कार सीटने तुम्हाला पुढे ढकलले आहे.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/persian/sentence_translations.json b/2017/higher-order-derivatives/persian/sentence_translations.json index 0e2d9ee8c..703db4925 100644 --- a/2017/higher-order-derivatives/persian/sentence_translations.json +++ b/2017/higher-order-derivatives/persian/sentence_translations.json @@ -184,7 +184,7 @@ "end": 221.64 }, { - "input": "Even though it's not like this letter d is a variable being multiplied by f, for the sake of more compact notation you'd write it as d2f divided by dx2, and you don't typically bother with any parentheses on the bottom. ", + "input": "eleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for ex ", "translatedText": "حتی اگر این حرف d متغیری نیست که در f ضرب شود، به خاطر نماد فشرده تر، آن را به صورت d2f تقسیم بر dx2 بنویسید، و معمولاً با هیچ پرانتزی در پایین آن را خسته نمی کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. ", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over t ", "translatedText": "سپس مشتق آن سرعت را در هر نقطه از زمان به شما می گوید، برای مثال نمودار ممکن است شبیه این برآمدگی باشد، تا مقداری حداکثر افزایش یابد، و دوباره به صفر کاهش یابد. ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward. ", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d ", "translatedText": "در این مثال، مشتق دوم برای نیمه اول سفر مثبت است، که نشان دهنده افزایش سرعت است، این احساس به عقب رانده شدن به صندلی ماشین شما است، یا بهتر است بگوییم، اینکه صندلی ماشین شما را به جلو می راند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/polish/sentence_translations.json b/2017/higher-order-derivatives/polish/sentence_translations.json index e35e589d3..84afcf54a 100644 --- a/2017/higher-order-derivatives/polish/sentence_translations.json +++ b/2017/higher-order-derivatives/polish/sentence_translations.json @@ -119,7 +119,7 @@ "end": 110.9 }, { - "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where, as always, the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx approaches 0.", + "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where as always the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx, both dx's in this case, approach 0.", "translatedText": "", "from_community_srt": "Jeśli chodzi o notację, to mógłbyś pisać to tak, myśląc o stosunku małej zmiany pochodnej do małej zmiany x. Jak zwykle, używamy litery d, bo będziemy chcieli wiedzieć, do czego dąży ten stosunek, gdy oba dx zbliżają się do 0.", "n_reviews": 0, @@ -127,7 +127,7 @@ "end": 134.44 }, { - "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d2f divided by dx2.", + "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d squared f divided by dx squared.", "translatedText": "", "from_community_srt": "Jest to dość niezgrabna notacja, więc skraca się ją do d^2f/dx^2.", "n_reviews": 0, @@ -207,7 +207,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "", "from_community_srt": "Najbardziej oklepanym zastosowaniem drugiej pochodnej jest opis przyspieszenia.", "n_reviews": 0, @@ -223,7 +223,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "", "from_community_srt": "Wtedy pochodna tej funkcji to szybkość. Jej wykres może wyglądać np. tak. Prędkość rośnie do pewnego maksimum, a potem maleje z powrotem do 0.", "n_reviews": 0, @@ -239,7 +239,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "", "from_community_srt": "W tym przykładzie przyspieszenie jest dodatnie w pierwszej połowie trasy, co oznacza, że samochód przyspiesza. Wtedy czujesz się wgnieciony w fotel. Lub inaczej, to samochód pcha cię do przodu.", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/portuguese/sentence_translations.json b/2017/higher-order-derivatives/portuguese/sentence_translations.json index 8c55fe328..d9d457451 100644 --- a/2017/higher-order-derivatives/portuguese/sentence_translations.json +++ b/2017/higher-order-derivatives/portuguese/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "Talvez a compreensão mais visceral da segunda derivada seja que ela representa aceleração.", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "Então, sua derivada informa a velocidade em cada ponto no tempo, por exemplo, o gráfico pode se parecer com esta colisão, aumentando até um máximo e diminuindo de volta a zero.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "Neste exemplo, a segunda derivada é positiva para a primeira metade da viagem, o que indica aceleração, é a sensação de ser empurrado para trás na cadeirinha, ou melhor, ter a cadeirinha empurrando você para frente.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/russian/sentence_translations.json b/2017/higher-order-derivatives/russian/sentence_translations.json index 3ac676d55..591ed3b63 100644 --- a/2017/higher-order-derivatives/russian/sentence_translations.json +++ b/2017/higher-order-derivatives/russian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "Возможно, самое интуитивное понимание второй производной заключается в том, что она представляет собой ускорение.", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "Затем ее производная сообщает вам скорость в каждый момент времени, например, график может выглядеть как этот выступ, увеличивающийся до некоторого максимума и уменьшающийся обратно до нуля.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "В этом примере вторая производная положительна для первой половины пути, что указывает на ускорение, то есть ощущение, будто вас толкают обратно в автокресло или, скорее, когда автокресло толкает вас вперед.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/slovak/sentence_translations.json b/2017/higher-order-derivatives/slovak/sentence_translations.json index c0ec06f68..548f79441 100644 --- a/2017/higher-order-derivatives/slovak/sentence_translations.json +++ b/2017/higher-order-derivatives/slovak/sentence_translations.json @@ -119,7 +119,7 @@ "end": 110.9 }, { - "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where, as always, the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx approaches 0.", + "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where as always the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx, both dx's in this case, approach 0.", "translatedText": "", "from_community_srt": "je druhá derivácia rovná nule. Čo sa týka zápisu, mohli by sme to skúsiť zapísať nejako takto, naznačiť nejakú malú zmenu v derivácii funkcie vydelenú nejakou malou hodnotou x, kde, ako vždy, použitie písmena \"d\" naznačuje, že by sme mali rozmýšľať, čo sa stane s týmto pomerom, keď dx, obe dx v tomto prípade, sa priblížia 0.", "n_reviews": 0, @@ -127,7 +127,7 @@ "end": 134.44 }, { - "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d2f divided by dx2.", + "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d squared f divided by dx squared.", "translatedText": "", "from_community_srt": "Je to celkom divné a zložité, takže sa to štandardne zjednoduší na (d^2)f/dx^2 a aj keď to nie je veľmi dôležité,", "n_reviews": 0, @@ -207,7 +207,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "", "from_community_srt": "Najlepšie sa to asi vysvetlí na grafe zrýchľovania.", "n_reviews": 0, @@ -223,7 +223,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "", "from_community_srt": "Potom jeho derivácia vám povie rýchlosť v určitom bode v čase, však? Ku príkladu, graf môže vyzerať ako kopec, stúpajúci k svojmu maximu a potom klesajúc naspäť na 0.", "n_reviews": 0, @@ -239,7 +239,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "", "from_community_srt": "Ku príkladu, druhotná derivácia je pozitívna pre prvú polovicu jazdy, čo naznačuje zrýchľovanie. To je ten pocit, keď ste tlačení do sedadla určitou silou, respektíve, že vás auto tlačí dopredu určitou silou.", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/spanish/sentence_translations.json b/2017/higher-order-derivatives/spanish/sentence_translations.json index 199c962f8..bb6690d66 100644 --- a/2017/higher-order-derivatives/spanish/sentence_translations.json +++ b/2017/higher-order-derivatives/spanish/sentence_translations.json @@ -119,7 +119,7 @@ "end": 110.9 }, { - "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where, as always, the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx approaches 0.", + "input": "As far as notation goes, you could try writing it like this, indicating some small change to the derivative function, divided by some small change to x, where as always the use of this letter d suggests that what you really want to consider is what this ratio approaches as dx, both dx's in this case, approach 0.", "translatedText": "En lo que respecta a la notación, podrías intentar escribirla así, indicando algún pequeño cambio en la función derivada, dividida por algún pequeño cambio en x, donde, como siempre, el uso de esta letra d sugiere que lo que realmente quieres considerar es a lo que se acerca esta relación cuando dx se acerca a 0.", "from_community_srt": "la segunda derivada es cero. En cuanto a la notación , tú podrías intentar escribirla como esta, lo que indica un pequeño cambio a la función derivada dividido por alguna pequeño cambio en x, donde como siempre el uso de esa letra d que realmente siguiere que deseas considerar lo esta razón aproxima como dx dx, ambos dx en este caso,", "n_reviews": 0, @@ -127,7 +127,7 @@ "end": 134.44 }, { - "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d2f divided by dx2.", + "input": "That's pretty awkward and clunky, so the standard is to abbreviate this as d squared f divided by dx squared.", "translatedText": "Esto es bastante incómodo y complicado, por lo que el estándar es abreviarlo como d2f dividido por dx2.", "from_community_srt": "aproxima a 0. Eso es bastante incómodo y torpe, por lo que el estándar es abreviarlo como d**2f / dx**2.", "n_reviews": 0, @@ -207,7 +207,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "Quizás la comprensión más visceral de la segunda derivada es que representa la aceleración.", "from_community_srt": "Tal vez la comprensión más visceral de la segunda derivada es que representa la aceleración.", "n_reviews": 0, @@ -223,7 +223,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "Luego, su derivada le indica la velocidad en cada momento; por ejemplo, el gráfico podría verse como este aumento, aumentando hasta un máximo y disminuyendo de nuevo a cero.", "from_community_srt": "Entonces su derivada te dice velocidad en cada punto en el tiempo, ¿verdad? Para el ejemplo, el gráfico puede ser como este bache, aumentando hasta cierto máximo, luego disminuyendo de nuevo a 0.", "n_reviews": 0, @@ -239,7 +239,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "En este ejemplo, la segunda derivada es positiva para la primera mitad del viaje, lo que indica aceleración, es decir, la sensación de ser empujado hacia atrás en el asiento del automóvil, o mejor dicho, de que el asiento del automóvil lo empuja hacia adelante.", "from_community_srt": "En el ejemplo, la segunda derivada es positiva para la primera mitad del viaje, lo que indica indica la aceleración. Esa es la sensación de ser empujado de nuevo en su asiento de seguridad con una fuerza constante. O más bien, teniendo el asiento de coche que empuja con una fuerza constante.", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/tamil/sentence_translations.json b/2017/higher-order-derivatives/tamil/sentence_translations.json index d3a483011..97da13a30 100644 --- a/2017/higher-order-derivatives/tamil/sentence_translations.json +++ b/2017/higher-order-derivatives/tamil/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "இரண்டாவது வழித்தோன்றலின் மிகவும் உள்ளுறுப்பு புரிதல் அது முடுக்கத்தைக் குறிக்கிறது.", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "அதன் வழித்தோன்றல் ஒவ்வொரு நேரத்திலும் வேகத்தை உங்களுக்குக் கூறுகிறது, எடுத்துக்காட்டாக, வரைபடம் இந்த பம்ப் போல் தோன்றலாம், அதிகபட்சம் வரை அதிகரித்து, மீண்டும் பூஜ்ஜியமாகக் குறையும்.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "இந்த எடுத்துக்காட்டில், பயணத்தின் முதல் பாதியில் இரண்டாவது வழித்தோன்றல் நேர்மறையானது, இது வேகத்தை குறிக்கிறது, அது உங்கள் கார் இருக்கைக்கு பின்னால் தள்ளப்படும் உணர்வு, அல்லது கார் இருக்கை உங்களை முன்னோக்கி தள்ளும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/telugu/sentence_translations.json b/2017/higher-order-derivatives/telugu/sentence_translations.json index 988e4daa3..cca64f9fd 100644 --- a/2017/higher-order-derivatives/telugu/sentence_translations.json +++ b/2017/higher-order-derivatives/telugu/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "రెండవ ఉత్పన్నం యొక్క అత్యంత విసెరల్ అవగాహన ఏమిటంటే అది త్వరణాన్ని సూచిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "అప్పుడు దాని ఉత్పన్నం మీకు ప్రతి సమయంలో వేగాన్ని తెలియజేస్తుంది, ఉదాహరణకు గ్రాఫ్ ఈ బంప్ లాగా ఉండవచ్చు, కొంత గరిష్టంగా పెరుగుతుంది మరియు తిరిగి సున్నాకి తగ్గుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "ఈ ఉదాహరణలో, రెండవ డెరివేటివ్ ప్రయాణంలో మొదటి సగానికి సానుకూలంగా ఉంటుంది, ఇది వేగాన్ని సూచిస్తుంది, అది మీ కారు సీటులోకి వెనుకకు నెట్టబడిన అనుభూతి, లేదా కారు సీటు మిమ్మల్ని ముందుకు నెట్టడం.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/thai/sentence_translations.json b/2017/higher-order-derivatives/thai/sentence_translations.json index a834f294a..84b701a0c 100644 --- a/2017/higher-order-derivatives/thai/sentence_translations.json +++ b/2017/higher-order-derivatives/thai/sentence_translations.json @@ -184,7 +184,7 @@ "end": 221.64 }, { - "input": "Even though it's not like this letter d is a variable being multiplied by f, for the sake of more compact notation you'd write it as d2f divided by dx2, and you don't typically bother with any parentheses on the bottom. ", + "input": "eleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for ex ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. ", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over t ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward. ", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/turkish/sentence_translations.json b/2017/higher-order-derivatives/turkish/sentence_translations.json index 8060f76eb..942b61282 100644 --- a/2017/higher-order-derivatives/turkish/sentence_translations.json +++ b/2017/higher-order-derivatives/turkish/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "Belki de ikinci türevin en içten anlaşılması, onun ivmeyi temsil etmesidir.", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "Daha sonra türevi size zamanın her noktasındaki hızı söyler; örneğin grafik bu tümsek gibi görünebilir, bir maksimuma kadar artabilir ve tekrar sıfıra düşebilir.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "Bu örnekte, ikinci türev yolculuğun ilk yarısı için pozitiftir, bu da hızlanmayı, yani araba koltuğunuza geri itilme hissini, daha doğrusu araba koltuğunun sizi ileri doğru itmesini gösterir.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/ukrainian/sentence_translations.json b/2017/higher-order-derivatives/ukrainian/sentence_translations.json index eaea10b2a..c019b076d 100644 --- a/2017/higher-order-derivatives/ukrainian/sentence_translations.json +++ b/2017/higher-order-derivatives/ukrainian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "Можливо, найбільш логічне розуміння другої похідної полягає в тому, що вона представляє прискорення.", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "Тоді його похідна повідомляє вам швидкість у кожен момент часу, наприклад, графік може виглядати як ця горба, зростаючи до деякого максимуму та зменшуючись назад до нуля.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "У цьому прикладі друга похідна додатна для першої половини подорожі, що вказує на прискорення, тобто відчуття, коли вас штовхають назад у сидіння автомобіля, точніше, коли сидіння автомобіля штовхає вас вперед.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/urdu/sentence_translations.json b/2017/higher-order-derivatives/urdu/sentence_translations.json index aac65d7cb..e9bfc9d3b 100644 --- a/2017/higher-order-derivatives/urdu/sentence_translations.json +++ b/2017/higher-order-derivatives/urdu/sentence_translations.json @@ -184,7 +184,7 @@ "end": 221.64 }, { - "input": "Even though it's not like this letter d is a variable being multiplied by f, for the sake of more compact notation you'd write it as d2f divided by dx2, and you don't typically bother with any parentheses on the bottom. ", + "input": "eleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for ex ", "translatedText": "اگرچہ یہ ایسا نہیں ہے کہ یہ حرف d ایک متغیر ہے جس کو f سے ضرب کیا جاتا ہے، زیادہ کمپیکٹ اشارے کی خاطر آپ اسے d2f کو dx2 سے تقسیم کرکے لکھیں گے، اور آپ عام طور پر نیچے والے قوسین سے پریشان نہیں ہوتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. ", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over t ", "translatedText": "پھر اس کا مشتق آپ کو وقت کے ہر نقطہ پر رفتار بتاتا ہے، مثال کے طور پر گراف اس ٹکرانے کی طرح نظر آتا ہے، کچھ زیادہ سے زیادہ تک بڑھتا ہے، اور واپس صفر تک کم ہوتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward. ", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d ", "translatedText": "اس مثال میں، دوسرا مشتق سفر کے پہلے نصف کے لیے مثبت ہے، جو تیز رفتاری کی طرف اشارہ کرتا ہے، یہ آپ کی کار سیٹ پر پیچھے دھکیلنے کا احساس ہے، یا اس کے بجائے، گاڑی کی سیٹ آپ کو آگے دھکیلتی ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/higher-order-derivatives/vietnamese/sentence_translations.json b/2017/higher-order-derivatives/vietnamese/sentence_translations.json index e787f1565..4edf9ee48 100644 --- a/2017/higher-order-derivatives/vietnamese/sentence_translations.json +++ b/2017/higher-order-derivatives/vietnamese/sentence_translations.json @@ -192,7 +192,7 @@ "end": 237.78 }, { - "input": "Maybe the most visceral understanding of the second derivative is that it represents acceleration.", + "input": "aybe its graph looks like this, steadily increasing over time. Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing", "translatedText": "Có lẽ sự hiểu biết trực quan nhất về đạo hàm bậc hai là nó đại diện cho gia tốc.", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 255.82 }, { - "input": "Then its derivative tells you velocity at each point in time, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero.", + "input": "e second derivative is that it represents acceleration. Given some movement along a line, suppose you have some function that records the distance traveled versus time, maybe its graph looks something like this, steadily increasing over time.", "translatedText": "Sau đó, đạo hàm của nó cho bạn biết vận tốc tại mỗi thời điểm, ví dụ: đồ thị có thể trông giống như vết lồi này, tăng lên đến mức tối đa nào đó và giảm về 0.", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 273.9 }, { - "input": "In this example, the second derivative is positive for the first half of the journey, which indicates speeding up, that's the sensation of being pushed back into your car seat, or rather, having the car seat push you forward.", + "input": "ime, for example the graph might look like this bump, increasing up to some maximum, and decreasing back to zero. The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero, it means that the strength of the acceleration itself is changing. One of the most useful things about higher order d", "translatedText": "Trong ví dụ này, đạo hàm thứ hai dương trong nửa đầu của hành trình, biểu thị việc tăng tốc, đó là cảm giác bị đẩy lùi vào ghế ô tô của bạn, hay nói đúng hơn là bị ghế ô tô đẩy bạn về phía trước.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/arabic/sentence_translations.json b/2017/implicit-differentiation/arabic/sentence_translations.json index 55382f095..4aa7215a9 100644 --- a/2017/implicit-differentiation/arabic/sentence_translations.json +++ b/2017/implicit-differentiation/arabic/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "بعد ذلك، سأتحدث عن النهايات وكيفية استخدامها لإضفاء الطابع الرسمي على فكرة المشتقة. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/bengali/sentence_translations.json b/2017/implicit-differentiation/bengali/sentence_translations.json index e615f078c..213d55553 100644 --- a/2017/implicit-differentiation/bengali/sentence_translations.json +++ b/2017/implicit-differentiation/bengali/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "পরবর্তীতে আমি সীমা সম্পর্কে কথা বলতে যাচ্ছি এবং কীভাবে সেগুলি একটি ডেরিভেটিভের ধারণাকে আনুষ্ঠানিক করতে ব্যবহৃত হয়।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/bulgarian/sentence_translations.json b/2017/implicit-differentiation/bulgarian/sentence_translations.json index 4f3cd272f..282c21dec 100644 --- a/2017/implicit-differentiation/bulgarian/sentence_translations.json +++ b/2017/implicit-differentiation/bulgarian/sentence_translations.json @@ -424,7 +424,7 @@ "end": 495.66 }, { - "input": "For points on this circle, that number is 25.", + "input": "For points on the circle, that number happens to be 25.", "translatedText": "За точките от тази окръжност това число е 25.", "model": "DeepL", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 504.4 }, { - "input": "For other points xy closer to the derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", + "input": "For other points xy closer to the origin, that value would be smaller. Now what it means to take a derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", "translatedText": "За други точки xy по-близо до производната на този израз, производна на s, е да се разгледа малка промяна на двете променливи, някаква малка промяна dx на x и някаква малка промяна dy на y, и не е задължително да е такава, че да ви държи на кръга, между другото, това е просто всяка малка стъпка в която и да е посока на равнината xy.", "model": "DeepL", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 560.18 }, { - "input": "Then the decrease in s, the amount that x2 plus y2 changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", + "input": "Then the decrease in s, the amount that x squared plus y squared changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", "translatedText": "Тогава намалението на s, количеството, което x2 плюс y2 се променя за тази стъпка, ще бъде около 2 пъти 3 пъти отрицателна стойност 0,02 плюс 2 пъти 4 пъти отрицателна стойност 0,01.", "model": "DeepL", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 580.8 }, { - "input": "It's a recipe for telling you how much the value x2 plus y2 changes as determined by the point xy where you start and the tiny step dx dy you take.", + "input": "It's a recipe for telling you how much the value x squared plus y squared changes as determined by the point xy where you start and the tiny step dx dy that you take. And", "translatedText": "Това е рецепта, която ви казва с колко се променя стойността x2 плюс y2 в зависимост от точката xy, от която започвате, и малката стъпка dx dy, която правите.", "model": "DeepL", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 643.9 }, { - "input": "Of course, there's nothing special about the expression x2 plus y2 equals 5 squared.", + "input": "Of course, there's nothing special about the expression x squared plus y squared equals 5 squared.", "translatedText": "Разбира се, в израза x2 плюс y2 е равно на 5 на квадрат няма нищо особено.", "model": "DeepL", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 661.64 }, { - "input": "Those curves represent all the points xy where the value of sin of x times y2 equals x.", + "input": "And those curves, remember, represent all of the points xy where the value of sine of x times y squared happens to equal the value of x.", "translatedText": "Тези криви представляват всички точки xy, в които стойността на sin от x, умножена по y2, е равна на x.", "model": "DeepL", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 691.44 }, { - "input": "On the left side, the product rule tells us that this should be left d right plus right d left.", + "input": "On the left side, the product rule that we talked through last video tells us that this should be left d right plus right d left.", "translatedText": "От лявата страна правилото за произведение ни казва, че това трябва да бъде ляво d дясно плюс дясно d ляво.", "model": "DeepL", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 698.84 }, { - "input": "That is, sin of x times the change to y2, which is 2y dy, plus y2 times the change to sin of x, which is cos x times dx.", + "input": "That is sine of x times the change to y squared, which is 2y times dy, plus y squared times the change to sine of x, which is cosine of x times dx.", "translatedText": "Това означава, че sin на x се умножава по промяната на y2, което е 2y dy, плюс y2 се умножава по промяната на sin на x, което е cos x пъти dx.", "model": "DeepL", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 710.98 }, { - "input": "The right side is simply x, so the size of a change is exactly dx.", + "input": "The right side is simply x, so the size of a change to that value is exactly dx, right? Now", "translatedText": "Дясната страна е просто x, така че размерът на промяната е точно dx.", "model": "DeepL", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 717.62 }, { - "input": "Setting these two sides equal to each other is a way of saying, whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left and right side must change by the same amount.", + "input": "setting these two sides equal to each other is a way of saying whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left hand side and the right hand side must change by the same amount.", "translatedText": "Задаването на равни стойности на тези две страни е начин да се каже, че каквато и да е вашата малка стъпка с координати dx и dy, ако тя ще ни държи на кривата, стойностите на лявата и дясната страна трябва да се променят с една и съща стойност.", "model": "DeepL", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 761.71 }, { - "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x, can be thought of as an implicit curve.", + "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x? Well the graph of the natural log of x can be thought of as an implicit curve.", "translatedText": "Споменах, че производната на e спрямо x е самата тя, но какво да кажем за производната на обратната ѝ функция, естествения лог на x, може да се разглежда като имплицитна крива.", "model": "DeepL", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 795.41 }, { - "input": "Well, to e to the y equals x.", + "input": "Well to find that, first rearrange this equation y equals ln of x to be e to the y equals x.", "translatedText": "Е, за e към y е равно на x.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/implicit-differentiation/chinese/sentence_translations.json b/2017/implicit-differentiation/chinese/sentence_translations.json index c38cc9bb1..1fea76f90 100644 --- a/2017/implicit-differentiation/chinese/sentence_translations.json +++ b/2017/implicit-differentiation/chinese/sentence_translations.json @@ -874,7 +874,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "接下来我将讨论极限以及如 何使用它们来形式化导数的概念。", "model": "google_nmt", "from_community_srt": "和他们中是如何精确地影响彼此的 接下来， 我将谈谈极限究竟是什么， 它是如何 使导数的想法正式化的。", diff --git a/2017/implicit-differentiation/english/captions.srt b/2017/implicit-differentiation/english/captions.srt index 4b3716a3e..01929cb12 100644 --- a/2017/implicit-differentiation/english/captions.srt +++ b/2017/implicit-differentiation/english/captions.srt @@ -452,333 +452,353 @@ It takes every point xy on the plane and associates it with a number. 114 00:08:16,620 --> 00:08:19,660 -For points on this circle, that number is 25. +For points on the circle, that number happens to be 25. 115 00:08:20,560 --> 00:08:24,400 If you stepped off the circle away from the center, that value would be bigger. 116 -00:08:25,060 --> 00:08:31,743 -For other points xy closer to the derivative of this expression, a derivative of s, +00:08:25,060 --> 00:08:29,755 +For other points xy closer to the origin, that value would be smaller. 117 -00:08:31,743 --> 00:08:38,346 -is to consider a tiny change to both of these variables, some tiny change dx to x, +00:08:29,755 --> 00:08:34,913 +Now what it means to take a derivative of this expression, a derivative of s, 118 -00:08:38,346 --> 00:08:45,030 -and some tiny change dy to y, and not necessarily one that keeps you on the circle, +00:08:34,913 --> 00:08:40,402 +is to consider a tiny change to both of these variables, some tiny change dx to x, 119 -00:08:45,030 --> 00:08:50,520 -by the way, it's just any tiny step in any direction of the xy plane. +00:08:40,402 --> 00:08:45,957 +and some tiny change dy to y, and not necessarily one that keeps you on the circle, 120 +00:08:45,957 --> 00:08:50,520 +by the way, it's just any tiny step in any direction of the xy plane. + +121 00:08:51,520 --> 00:08:55,020 From there you ask how much does the value of s change? -121 +122 00:08:56,000 --> 00:09:01,696 That difference, the difference in the value of s before the nudge and after the nudge, -122 +123 00:09:01,696 --> 00:09:03,380 is what I'm writing as ds. -123 +124 00:09:04,480 --> 00:09:09,460 For example, in this picture we're starting off at a point -124 +125 00:09:09,460 --> 00:09:14,777 where x equals 3 and where y equals 4, and let's just say that -125 +126 00:09:14,777 --> 00:09:20,180 the step I drew has dx at negative 0.02 and dy at negative 0.01. -126 -00:09:21,120 --> 00:09:27,729 -Then the decrease in s, the amount that x2 plus y2 changes over that step, - 127 -00:09:27,729 --> 00:09:34,780 -would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01. +00:09:21,120 --> 00:09:28,313 +Then the decrease in s, the amount that x squared plus y squared changes over that step, 128 -00:09:35,600 --> 00:09:40,800 -That's what this derivative expression, 2x dx plus 2y dy, actually means. +00:09:28,313 --> 00:09:34,780 +would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01. 129 -00:09:41,380 --> 00:09:46,538 -It's a recipe for telling you how much the value x2 plus y2 changes as +00:09:35,600 --> 00:09:40,800 +That's what this derivative expression, 2x dx plus 2y dy, actually means. 130 -00:09:46,538 --> 00:09:52,060 -determined by the point xy where you start and the tiny step dx dy you take. +00:09:41,380 --> 00:09:46,720 +It's a recipe for telling you how much the value x squared plus y squared changes as 131 +00:09:46,720 --> 00:09:52,060 +determined by the point xy where you start and the tiny step dx dy that you take. And + +132 00:09:53,080 --> 00:09:56,689 As with all things derivative, this is only an approximation, -132 +133 00:09:56,689 --> 00:10:01,580 but it's one that gets truer and truer for smaller and smaller choices of dx and dy. -133 +134 00:10:02,500 --> 00:10:07,110 The key point here is that when you restrict yourself to steps along the circle, -134 +135 00:10:07,110 --> 00:10:11,720 you're essentially saying you want to ensure that this value of s doesn't change. -135 +136 00:10:12,240 --> 00:10:16,520 It starts at a value of 25 and you want to keep it at a value of 25. -136 +137 00:10:17,180 --> 00:10:19,100 That is, ds should be 0. -137 +138 00:10:20,200 --> 00:10:25,159 So setting the expression 2x dx plus 2y dy equal to 0 is the condition -138 +139 00:10:25,159 --> 00:10:29,700 under which one of these tiny steps actually stays on the circle. -139 +140 00:10:30,620 --> 00:10:32,460 Again, this is only an approximation. -140 +141 00:10:33,040 --> 00:10:36,489 Speaking more precisely, that condition is what keeps you -141 +142 00:10:36,489 --> 00:10:39,880 on the tangent line of the circle, not the circle itself. -142 +143 00:10:40,580 --> 00:10:43,900 But for tiny enough steps, those are essentially the same thing. -143 -00:10:45,180 --> 00:10:49,780 -Of course, there's nothing special about the expression x2 plus y2 equals 5 squared. - 144 +00:10:45,180 --> 00:10:47,808 +Of course, there's nothing special about the expression + +145 +00:10:47,808 --> 00:10:49,780 +x squared plus y squared equals 5 squared. + +146 00:10:50,440 --> 00:10:53,584 It's always nice to think through more examples, -145 +147 00:10:53,584 --> 00:10:57,500 so let's consider this expression sin of x times y2 equals x. -146 +148 00:10:58,160 --> 00:11:01,640 This corresponds to a whole bunch of u-shaped curves on the plane. -147 -00:11:02,420 --> 00:11:11,340 -Those curves represent all the points xy where the value of sin of x times y2 equals x. +149 +00:11:02,420 --> 00:11:06,945 +And those curves, remember, represent all of the points xy where the -148 +150 +00:11:06,945 --> 00:11:11,340 +value of sine of x times y squared happens to equal the value of x. + +151 00:11:16,000 --> 00:11:19,616 Now imagine taking some tiny step with components dx and dy, -149 +152 00:11:19,616 --> 00:11:22,700 and not necessarily one that keeps you on the curve. -150 +153 00:11:23,820 --> 00:11:27,724 Taking the derivative of each side of this equation will tell -151 +154 00:11:27,724 --> 00:11:31,440 us how much the value of that side changes during the step. -152 -00:11:32,460 --> 00:11:35,750 -On the left side, the product rule tells us that - -153 -00:11:35,750 --> 00:11:38,840 -this should be left d right plus right d left. - -154 -00:11:39,480 --> 00:11:45,038 -That is, sin of x times the change to y2, which is 2y dy, - 155 -00:11:45,038 --> 00:11:50,980 -plus y2 times the change to sin of x, which is cos x times dx. +00:11:32,460 --> 00:11:35,575 +On the left side, the product rule that we talked through last 156 -00:11:52,020 --> 00:11:57,620 -The right side is simply x, so the size of a change is exactly dx. +00:11:35,575 --> 00:11:38,840 +video tells us that this should be left d right plus right d left. 157 -00:11:59,160 --> 00:12:03,551 -Setting these two sides equal to each other is a way of saying, +00:11:39,480 --> 00:11:45,034 +That is sine of x times the change to y squared, which is 2y times dy, 158 -00:12:03,551 --> 00:12:07,325 -whatever your tiny step with coordinates dx and dy is, +00:11:45,034 --> 00:11:50,980 +plus y squared times the change to sine of x, which is cosine of x times dx. 159 -00:12:07,325 --> 00:12:12,609 -if it's going to keep us on the curve, the values of both the left and right +00:11:52,020 --> 00:11:57,004 +The right side is simply x, so the size of a change to that value is exactly dx, 160 -00:12:12,609 --> 00:12:15,080 -side must change by the same amount. +00:11:57,004 --> 00:11:57,620 +right? Now 161 +00:11:59,160 --> 00:12:04,381 +setting these two sides equal to each other is a way of saying whatever your tiny + +162 +00:12:04,381 --> 00:12:09,157 +step with coordinates dx and dy is, if it's going to keep us on the curve, + +163 +00:12:09,157 --> 00:12:14,315 +the values of both the left hand side and the right hand side must change by the + +164 +00:12:14,315 --> 00:12:15,080 +same amount. + +165 00:12:15,640 --> 00:12:18,860 That's the only way this top equation can remain true. -162 +166 00:12:20,220 --> 00:12:23,792 From there, depending on what problem you're trying to solve, -163 +167 00:12:23,792 --> 00:12:26,500 you have something to work with algebraically, -164 +168 00:12:26,500 --> 00:12:31,110 and maybe the most common goal is to try to figure out what dy divided by dx is. -165 +169 00:12:33,210 --> 00:12:37,128 As a final example here, I want to show you how you can use this -166 +170 00:12:37,128 --> 00:12:41,710 technique of implicit differentiation to figure out new derivative formulas. -167 -00:12:42,630 --> 00:12:46,939 +171 +00:12:42,630 --> 00:12:46,157 I've mentioned that the derivative of e to the x is itself, -168 -00:12:46,939 --> 00:12:52,469 -but what about the derivative of its inverse function, the natural log of x, +172 +00:12:46,157 --> 00:12:50,684 +but what about the derivative of its inverse function, the natural log of x? -169 -00:12:52,469 --> 00:12:55,270 -can be thought of as an implicit curve. +173 +00:12:50,684 --> 00:12:55,270 +Well the graph of the natural log of x can be thought of as an implicit curve. -170 +174 00:12:56,050 --> 00:13:00,830 It's all of the points xy on the plane where y happens to equal ln of x. -171 +175 00:13:01,550 --> 00:13:04,658 It just happens to be the case that the x's and y's of this -172 +176 00:13:04,658 --> 00:13:08,130 equation aren't as intermingled as they were in our other examples. -173 +177 00:13:09,350 --> 00:13:15,410 The slope of this graph, dy divided by dx, should be the derivative of ln of x, right? -174 -00:13:16,650 --> 00:13:24,030 -Well, to e to the y equals x. +178 +00:13:16,650 --> 00:13:20,580 +Well to find that, first rearrange this equation -175 +179 +00:13:20,580 --> 00:13:24,030 +y equals ln of x to be e to the y equals x. + +180 00:13:24,650 --> 00:13:30,850 This is exactly what the natural log of x means, it's saying e to the what equals x. -176 +181 00:13:31,870 --> 00:13:35,378 Since we know the derivative of e to the y, we can take the -177 +182 00:13:35,378 --> 00:13:39,238 derivative of both sides here, effectively asking how a tiny step -178 +183 00:13:39,238 --> 00:13:43,390 with components dx and dy changes the value of each one of these sides. -179 +184 00:13:44,530 --> 00:13:50,138 To ensure that a step stays on the curve, the change to the left side of the equation, -180 +185 00:13:50,138 --> 00:13:54,715 which is e to the y times dy, must equal the change to the right side, -181 +186 00:13:54,715 --> 00:13:56,650 which in this case is just dx. -182 +187 00:13:57,870 --> 00:14:03,531 Rearranging, that means dy divided by dx, the slope of our graph, -183 +188 00:14:03,531 --> 00:14:06,190 equals 1 divided by e to the y. -184 +189 00:14:06,910 --> 00:14:11,822 When we're on the curve, e to the y is by definition the same thing as x, -185 +190 00:14:11,822 --> 00:14:14,610 so evidently this slope is 1 divided by x. -186 +191 00:14:15,830 --> 00:14:19,987 And of course, an expression for the slope of a graph of a function -187 +192 00:14:19,987 --> 00:14:24,145 written in terms of x like this is the derivative of that function, -188 +193 00:14:24,145 --> 00:14:27,630 so evidently the derivative of ln of x is 1 divided by x. -189 +194 00:14:32,610 --> 00:14:37,194 By the way, all of this is a little sneak peek into multivariable calculus, -190 +195 00:14:37,194 --> 00:14:40,994 where you consider functions that have multiple inputs and how -191 +196 00:14:40,994 --> 00:14:43,830 they change as you tweak those multiple inputs. -192 +197 00:14:44,870 --> 00:14:48,939 The key, as always, is to have a clear image in your head of what -193 +198 00:14:48,939 --> 00:14:53,070 tiny nudges are at play, and how exactly they depend on each other. -194 +199 00:14:54,530 --> 00:14:56,933 Next up, I'm going to be talking about limits, -195 +200 00:14:56,933 --> 00:14:59,950 and how they're used to formalize the idea of a derivative. -196 +201 00:15:17,490 --> 00:15:22,730 Thank you. diff --git a/2017/implicit-differentiation/english/sentence_timings.json b/2017/implicit-differentiation/english/sentence_timings.json index b19bb12dc..c27149073 100644 --- a/2017/implicit-differentiation/english/sentence_timings.json +++ b/2017/implicit-differentiation/english/sentence_timings.json @@ -265,7 +265,7 @@ 495.66 ], [ - "For points on this circle, that number is 25.", + "For points on the circle, that number happens to be 25.", 496.62, 499.66 ], @@ -275,7 +275,7 @@ 504.4 ], [ - "For other points xy closer to the derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", + "For other points xy closer to the origin, that value would be smaller. Now what it means to take a derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", 505.06, 530.52 ], @@ -295,7 +295,7 @@ 560.18 ], [ - "Then the decrease in s, the amount that x2 plus y2 changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", + "Then the decrease in s, the amount that x squared plus y squared changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", 561.12, 574.78 ], @@ -305,7 +305,7 @@ 580.8 ], [ - "It's a recipe for telling you how much the value x2 plus y2 changes as determined by the point xy where you start and the tiny step dx dy you take.", + "It's a recipe for telling you how much the value x squared plus y squared changes as determined by the point xy where you start and the tiny step dx dy that you take. And", 581.38, 592.06 ], @@ -350,7 +350,7 @@ 643.9 ], [ - "Of course, there's nothing special about the expression x2 plus y2 equals 5 squared.", + "Of course, there's nothing special about the expression x squared plus y squared equals 5 squared.", 645.18, 649.78 ], @@ -365,7 +365,7 @@ 661.64 ], [ - "Those curves represent all the points xy where the value of sin of x times y2 equals x.", + "And those curves, remember, represent all of the points xy where the value of sine of x times y squared happens to equal the value of x.", 662.42, 671.34 ], @@ -380,22 +380,22 @@ 691.44 ], [ - "On the left side, the product rule tells us that this should be left d right plus right d left.", + "On the left side, the product rule that we talked through last video tells us that this should be left d right plus right d left.", 692.46, 698.84 ], [ - "That is, sin of x times the change to y2, which is 2y dy, plus y2 times the change to sin of x, which is cos x times dx.", + "That is sine of x times the change to y squared, which is 2y times dy, plus y squared times the change to sine of x, which is cosine of x times dx.", 699.48, 710.98 ], [ - "The right side is simply x, so the size of a change is exactly dx.", + "The right side is simply x, so the size of a change to that value is exactly dx, right? Now", 712.02, 717.62 ], [ - "Setting these two sides equal to each other is a way of saying, whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left and right side must change by the same amount.", + "setting these two sides equal to each other is a way of saying whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left hand side and the right hand side must change by the same amount.", 719.16, 735.08 ], @@ -415,7 +415,7 @@ 761.71 ], [ - "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x, can be thought of as an implicit curve.", + "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x? Well the graph of the natural log of x can be thought of as an implicit curve.", 762.63, 775.27 ], @@ -435,7 +435,7 @@ 795.41 ], [ - "Well, to e to the y equals x.", + "Well to find that, first rearrange this equation y equals ln of x to be e to the y equals x.", 796.65, 804.03 ], diff --git a/2017/implicit-differentiation/english/transcript.txt b/2017/implicit-differentiation/english/transcript.txt index 260d71c68..39b7a43fa 100644 --- a/2017/implicit-differentiation/english/transcript.txt +++ b/2017/implicit-differentiation/english/transcript.txt @@ -51,15 +51,15 @@ Let me show you a nice way to think about this. Let's give this expression x squared plus y squared a name, maybe s. s is essentially a function of two variables. It takes every point xy on the plane and associates it with a number. -For points on this circle, that number is 25. +For points on the circle, that number happens to be 25. If you stepped off the circle away from the center, that value would be bigger. -For other points xy closer to the derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane. +For other points xy closer to the origin, that value would be smaller. Now what it means to take a derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane. From there you ask how much does the value of s change? That difference, the difference in the value of s before the nudge and after the nudge, is what I'm writing as ds. For example, in this picture we're starting off at a point where x equals 3 and where y equals 4, and let's just say that the step I drew has dx at negative 0.02 and dy at negative 0.01. -Then the decrease in s, the amount that x2 plus y2 changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01. +Then the decrease in s, the amount that x squared plus y squared changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01. That's what this derivative expression, 2x dx plus 2y dy, actually means. -It's a recipe for telling you how much the value x2 plus y2 changes as determined by the point xy where you start and the tiny step dx dy you take. +It's a recipe for telling you how much the value x squared plus y squared changes as determined by the point xy where you start and the tiny step dx dy that you take. And As with all things derivative, this is only an approximation, but it's one that gets truer and truer for smaller and smaller choices of dx and dy. The key point here is that when you restrict yourself to steps along the circle, you're essentially saying you want to ensure that this value of s doesn't change. It starts at a value of 25 and you want to keep it at a value of 25. @@ -68,24 +68,24 @@ So setting the expression 2x dx plus 2y dy equal to 0 is the condition under whi Again, this is only an approximation. Speaking more precisely, that condition is what keeps you on the tangent line of the circle, not the circle itself. But for tiny enough steps, those are essentially the same thing. -Of course, there's nothing special about the expression x2 plus y2 equals 5 squared. +Of course, there's nothing special about the expression x squared plus y squared equals 5 squared. It's always nice to think through more examples, so let's consider this expression sin of x times y2 equals x. This corresponds to a whole bunch of u-shaped curves on the plane. -Those curves represent all the points xy where the value of sin of x times y2 equals x. +And those curves, remember, represent all of the points xy where the value of sine of x times y squared happens to equal the value of x. Now imagine taking some tiny step with components dx and dy, and not necessarily one that keeps you on the curve. Taking the derivative of each side of this equation will tell us how much the value of that side changes during the step. -On the left side, the product rule tells us that this should be left d right plus right d left. -That is, sin of x times the change to y2, which is 2y dy, plus y2 times the change to sin of x, which is cos x times dx. -The right side is simply x, so the size of a change is exactly dx. -Setting these two sides equal to each other is a way of saying, whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left and right side must change by the same amount. +On the left side, the product rule that we talked through last video tells us that this should be left d right plus right d left. +That is sine of x times the change to y squared, which is 2y times dy, plus y squared times the change to sine of x, which is cosine of x times dx. +The right side is simply x, so the size of a change to that value is exactly dx, right? Now +setting these two sides equal to each other is a way of saying whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left hand side and the right hand side must change by the same amount. That's the only way this top equation can remain true. From there, depending on what problem you're trying to solve, you have something to work with algebraically, and maybe the most common goal is to try to figure out what dy divided by dx is. As a final example here, I want to show you how you can use this technique of implicit differentiation to figure out new derivative formulas. -I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x, can be thought of as an implicit curve. +I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x? Well the graph of the natural log of x can be thought of as an implicit curve. It's all of the points xy on the plane where y happens to equal ln of x. It just happens to be the case that the x's and y's of this equation aren't as intermingled as they were in our other examples. The slope of this graph, dy divided by dx, should be the derivative of ln of x, right? -Well, to e to the y equals x. +Well to find that, first rearrange this equation y equals ln of x to be e to the y equals x. This is exactly what the natural log of x means, it's saying e to the what equals x. Since we know the derivative of e to the y, we can take the derivative of both sides here, effectively asking how a tiny step with components dx and dy changes the value of each one of these sides. To ensure that a step stays on the curve, the change to the left side of the equation, which is e to the y times dy, must equal the change to the right side, which in this case is just dx. diff --git a/2017/implicit-differentiation/french/sentence_translations.json b/2017/implicit-differentiation/french/sentence_translations.json index a14b1f033..9a40d4781 100644 --- a/2017/implicit-differentiation/french/sentence_translations.json +++ b/2017/implicit-differentiation/french/sentence_translations.json @@ -422,7 +422,7 @@ "end": 495.66 }, { - "input": "For points on this circle, that number is 25.", + "input": "For points on the circle, that number happens to be 25.", "translatedText": "Pour les points de ce cercle, ce nombre est 25.", "from_community_srt": "Pour les points du cercle,", "n_reviews": 0, @@ -438,7 +438,7 @@ "end": 504.4 }, { - "input": "For other points xy closer to the derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", + "input": "For other points xy closer to the origin, that value would be smaller. Now what it means to take a derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", "translatedText": "Pour d'autres points xy plus proches de la dérivée de cette expression, une dérivée de s, il faut considérer un petit changement de ces deux variables, un petit changement dx en x, et un petit changement dy en y, et pas nécessairement celui qui conserve au fait, vous êtes sur le cercle, c'est juste n'importe quel petit pas dans n'importe quelle direction du plan xy.", "from_community_srt": "Pour d'autres points (x, y) plus proche de l'origine, cette valeur serait plus petite. Ce que cela signifie de prendre une dérivée de cette expression, une dérivée de S, c'est de considérer un petit changement pour les deux variables, une déviation dx pour x et dy pour y. Pas nécessairement une qui vous laisse sur le cercle d'ailleurs. C'est juste uns petite déviation dans n'importe quelle direction du plan x-y.", "n_reviews": 0, @@ -470,7 +470,7 @@ "end": 560.18 }, { - "input": "Then the decrease in s, the amount that x2 plus y2 changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", + "input": "Then the decrease in s, the amount that x squared plus y squared changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", "translatedText": "Ensuite, la diminution de s, la quantité que x2 plus y2 change au cours de cette étape, serait d'environ 2 fois 3 fois moins 0,02 plus 2 fois 4 fois moins 0,01.", "from_community_srt": "Ensuite, la diminution de S, le changement dans la quantité x^2 + y^2 au cours de ce processus, est d'environ 2*3*(-0,02) + 2*4*(- 0,01).", "n_reviews": 0, @@ -485,7 +485,7 @@ "end": 580.8 }, { - "input": "It's a recipe for telling you how much the value x2 plus y2 changes as determined by the point xy where you start and the tiny step dx dy you take.", + "input": "It's a recipe for telling you how much the value x squared plus y squared changes as determined by the point xy where you start and the tiny step dx dy that you take. And", "translatedText": "C'est une recette pour vous dire à quel point la valeur x2 plus y2 change en fonction du point xy où vous commencez et du petit pas dx dy que vous faites.", "from_community_srt": "C'est ce que cette expression dérivée 2x*dx + 2y*dy signifie, C'est une recette qui vous indique de combien la valeur x^2 + y^2 change, tel que déterminé par le point (x, y) d'où vous avez commencé, et la petite déviation (dx,", "n_reviews": 0, @@ -556,7 +556,7 @@ "end": 643.9 }, { - "input": "Of course, there's nothing special about the expression x2 plus y2 equals 5 squared.", + "input": "Of course, there's nothing special about the expression x squared plus y squared equals 5 squared.", "translatedText": "Bien sûr, l’expression x2 plus y2 égale 5 au carré n’a rien de spécial.", "from_community_srt": "Bien sûr, il n'y a rien de spécial avec l'expression x^2 + y^2 = 5^2.", "n_reviews": 0, @@ -580,7 +580,7 @@ "end": 661.64 }, { - "input": "Those curves represent all the points xy where the value of sin of x times y2 equals x.", + "input": "And those curves, remember, represent all of the points xy where the value of sine of x times y squared happens to equal the value of x.", "translatedText": "Ces courbes représentent tous les points xy où la valeur de sin de x fois y2 est égale à x.", "from_community_srt": "Ces courbes représentent tous les points (x, y) du plan où la valeur de sin(x)*y^2 est égale à la valeur de x.", "n_reviews": 0, @@ -604,7 +604,7 @@ "end": 691.44 }, { - "input": "On the left side, the product rule tells us that this should be left d right plus right d left.", + "input": "On the left side, the product rule that we talked through last video tells us that this should be left d right plus right d left.", "translatedText": "Sur le côté gauche, la règle du produit nous dit que cela doit être gauche d droite plus droite d gauche.", "from_community_srt": "Sur le membre de gauche, la règle du produit que nous avons découvert dans la vidéo précédent nous dit que cela devrait être", "n_reviews": 0, @@ -612,7 +612,7 @@ "end": 698.84 }, { - "input": "That is, sin of x times the change to y2, which is 2y dy, plus y2 times the change to sin of x, which is cos x times dx.", + "input": "That is sine of x times the change to y squared, which is 2y times dy, plus y squared times the change to sine of x, which is cosine of x times dx.", "translatedText": "Autrement dit, sin de x fois la modification en y2, qui est 2y dy, plus y2 fois la modification en sin de x, qui est cos x fois dx.", "from_community_srt": "« gauche d droite plus droite d-gauche » sin(x)*(la déviation de y^2), qui est 2y*dy, plus y^2*(la déviation de sin(x)),", "n_reviews": 0, @@ -620,7 +620,7 @@ "end": 710.98 }, { - "input": "The right side is simply x, so the size of a change is exactly dx.", + "input": "The right side is simply x, so the size of a change to that value is exactly dx, right? Now", "translatedText": "Le côté droit est simplement x, donc la taille d’un changement est exactement dx.", "from_community_srt": "qui est cos(x)*dx. Le membre de droite est tout simplement x, de sorte que la modification de la valeur est exactement dx,", "n_reviews": 0, @@ -628,7 +628,7 @@ "end": 717.62 }, { - "input": "Setting these two sides equal to each other is a way of saying, whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left and right side must change by the same amount.", + "input": "setting these two sides equal to each other is a way of saying whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left hand side and the right hand side must change by the same amount.", "translatedText": "Mettre ces deux côtés égaux est une façon de dire, quel que soit votre petit pas avec les coordonnées dx et dy, si cela veut nous maintenir sur la courbe, les valeurs des côtés gauche et droit doivent changer du même montant. .", "from_community_srt": "non? Régler ces deux côtés pour qu'ils soient égaux est une manière de dire : « quelque-soit les déviations de coordonnées (dx, dy), si elle nous laisse sur cette courbe, les valeurs des deux membres, gauche et droite, doivent changer de la même quantité.", "n_reviews": 0, @@ -660,7 +660,7 @@ "end": 761.71 }, { - "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x, can be thought of as an implicit curve.", + "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x? Well the graph of the natural log of x can be thought of as an implicit curve.", "translatedText": "J'ai mentionné que la dérivée de e par rapport à x est elle-même, mais qu'en est-il de la dérivée de sa fonction inverse, le logarithme naturel de x, peut être considérée comme une courbe implicite.", "from_community_srt": "J'ai déjà mentionné dans une vidéo que la dérivée de la fonction e^x est elle-même. Mais qu'en est-il de la dérivée de sa fonction inverse, le logarithme naturel de x ? Le graphique de ln(x) peut être vu comme une courbe implicite ;", "n_reviews": 0, @@ -691,7 +691,7 @@ "end": 795.41 }, { - "input": "Well, to e to the y equals x.", + "input": "Well to find that, first rearrange this equation y equals ln of x to be e to the y equals x.", "translatedText": "Eh bien, e au y est égal à x.", "from_community_srt": "pas vrai ? Eh bien, pour découvrir cela, réorganisons d'abord cette équation : y = ln(x), donc e^y = x.", "n_reviews": 0, diff --git a/2017/implicit-differentiation/german/sentence_translations.json b/2017/implicit-differentiation/german/sentence_translations.json index 181278456..2305ba95b 100644 --- a/2017/implicit-differentiation/german/sentence_translations.json +++ b/2017/implicit-differentiation/german/sentence_translations.json @@ -477,7 +477,7 @@ "end": 495.66 }, { - "input": "For points on this circle, that number is 25.", + "input": "For points on the circle, that number happens to be 25.", "translatedText": "Für Punkte auf diesem Kreis ist diese Zahl 25.", "model": "DeepL", "from_community_srt": "Für Punkte auf diesem Kreis ist diese Zahl 25.", @@ -495,7 +495,7 @@ "end": 504.4 }, { - "input": "For other points xy closer to the derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", + "input": "For other points xy closer to the origin, that value would be smaller. Now what it means to take a derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", "translatedText": "Für andere Punkte xy, die näher an der Ableitung dieses Ausdrucks liegen, ist eine Ableitung von s eine winzige Änderung dieser beiden Variablen, eine winzige Änderung dx zu x und eine winzige Änderung dy zu y, und zwar nicht unbedingt eine, die dich auf dem Kreis hält, sondern einfach ein winziger Schritt in eine beliebige Richtung der xy-Ebene.", "model": "DeepL", "from_community_srt": "Zum andere Punkte (x, y) näher am Ursprung, Dieser Wert ist kleiner. Was es bedeutet, eine Ableitung davon zu nehmen Ausdruck, eine Ableitung von S, ist zu berücksichtigen eine winzige Änderung an diesen beiden Variablen, einige winzige Änderung dx zu x und einige winzige Änderung dy to y - und nicht unbedingt eine, die hält Sie in diesem Kreis, übrigens, es ist nur ein winziger Schritt in jede Richtung auf der xy-Ebene - und fragen Sie,", @@ -531,7 +531,7 @@ "end": 560.18 }, { - "input": "Then the decrease in s, the amount that x2 plus y2 changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", + "input": "Then the decrease in s, the amount that x squared plus y squared changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", "translatedText": "Dann würde die Abnahme von s, also der Betrag, um den sich x2 plus y2 in diesem Schritt ändert, etwa 2 mal 3 mal negative 0,02 plus 2 mal 4 mal negative 0,01 betragen.", "model": "DeepL", "from_community_srt": "Dann die Abnahme auf S, die Menge Das x2 + y2 ändert sich in diesem Schritt um 2 (3) (- 0,02) + 2 (4) (- 0,01).", @@ -549,7 +549,7 @@ "end": 580.8 }, { - "input": "It's a recipe for telling you how much the value x2 plus y2 changes as determined by the point xy where you start and the tiny step dx dy you take.", + "input": "It's a recipe for telling you how much the value x squared plus y squared changes as determined by the point xy where you start and the tiny step dx dy that you take. And", "translatedText": "Es ist ein Rezept, das dir sagt, wie sehr sich der Wert x2 plus y2 durch den Punkt xy, an dem du beginnst, und den kleinen Schritt dx dy, den du machst, verändert.", "model": "DeepL", "from_community_srt": "es sagt Ihnen, wie viel der Wert x2 + y2 Änderungen, bestimmt durch den Punkt (x, y) wo du angefangen hast und der winzige Schritt (dx,", @@ -630,7 +630,7 @@ "end": 643.9 }, { - "input": "Of course, there's nothing special about the expression x2 plus y2 equals 5 squared.", + "input": "Of course, there's nothing special about the expression x squared plus y squared equals 5 squared.", "translatedText": "Natürlich ist der Ausdruck x2 plus y2 gleich 5 zum Quadrat nichts Besonderes.", "model": "DeepL", "from_community_srt": "Natürlich gibt es nichts Besonderes der Ausdruck x2 + y2 = 52 hier. Du könntest habe einen anderen Ausdruck mit x und ys, die eine andere Kurve darstellen, und nehmen die Ableitung beider Seiten wie Dies würde Ihnen eine Möglichkeit geben,", @@ -657,7 +657,7 @@ "end": 661.64 }, { - "input": "Those curves represent all the points xy where the value of sin of x times y2 equals x.", + "input": "And those curves, remember, represent all of the points xy where the value of sine of x times y squared happens to equal the value of x.", "translatedText": "Diese Kurven stellen alle Punkte xy dar, bei denen der Wert von sin von x mal y2 gleich x ist.", "model": "DeepL", "from_community_srt": "Diese Kurven repräsentieren alle Punkte (x, y) der Ebene, auf der der Wert liegt von sin (x) * y2 entspricht dem Wert von x.", @@ -684,7 +684,7 @@ "end": 691.44 }, { - "input": "On the left side, the product rule tells us that this should be left d right plus right d left.", + "input": "On the left side, the product rule that we talked through last video tells us that this should be left d right plus right d left.", "translatedText": "Auf der linken Seite sagt uns die Produktregel, dass dies links d rechts plus rechts d links sein sollte.", "model": "DeepL", "from_community_srt": "Auf der linken Seite regelt das Produkt, dass wir gefunden im letzten Video sagt uns,", @@ -693,7 +693,7 @@ "end": 698.84 }, { - "input": "That is, sin of x times the change to y2, which is 2y dy, plus y2 times the change to sin of x, which is cos x times dx.", + "input": "That is sine of x times the change to y squared, which is 2y times dy, plus y squared times the change to sine of x, which is cosine of x times dx.", "translatedText": "Das heißt, sin of x mal die Änderung zu y2, also 2y dy, plus y2 mal die Änderung zu sin of x, also cos x mal dx.", "model": "DeepL", "from_community_srt": "dass dies sollte \"links d-rechts plus rechts d-links\" sein: sin (x) * (die Änderung zu y2), die 2y * dy ist, plus y2 * (die Änderung zu sin (x)),", @@ -702,7 +702,7 @@ "end": 710.98 }, { - "input": "The right side is simply x, so the size of a change is exactly dx.", + "input": "The right side is simply x, so the size of a change to that value is exactly dx, right? Now", "translatedText": "Die rechte Seite ist einfach x, also ist die Größe einer Veränderung genau dx.", "model": "DeepL", "from_community_srt": "was cos (x) * dx ist. Die rechte Seite ist einfach x, also die Größe von Eine Änderung des Wertes ist genau dx,", @@ -711,7 +711,7 @@ "end": 717.62 }, { - "input": "Setting these two sides equal to each other is a way of saying, whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left and right side must change by the same amount.", + "input": "setting these two sides equal to each other is a way of saying whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left hand side and the right hand side must change by the same amount.", "translatedText": "Wenn du diese beiden Seiten gleich setzt, bedeutet das, dass sich die Werte der linken und der rechten Seite um denselben Betrag ändern müssen, damit wir auf der Kurve bleiben, egal wie klein dein Schritt mit den Koordinaten dx und dy ist.", "model": "DeepL", "from_community_srt": "oder? Stellen Sie diese beiden Seiten gleich ein ist eine Art zu sagen: „Was auch immer dein kleiner Schritt ist mit Koordinaten (dx, dy) ist, wenn es geht um uns auf dieser Kurve zu halten, die Werte von beiden die linke Seite und die rechte Seite muss sich um den gleichen Betrag ändern.", @@ -747,7 +747,7 @@ "end": 761.71 }, { - "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x, can be thought of as an implicit curve.", + "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x? Well the graph of the natural log of x can be thought of as an implicit curve.", "translatedText": "Ich habe erwähnt, dass die Ableitung von e nach x selbst ist, aber was ist mit der Ableitung seiner Umkehrfunktion, dem natürlichen Logarithmus von x, die man sich als implizite Kurve vorstellen kann.", "model": "DeepL", "from_community_srt": "Ich habe das in einem Fußnotenvideo erwähnt Die Ableitung von ex ist selbst, aber was ist mit die Ableitung seiner Umkehrfunktion die natürliches Protokoll von x? Der Graph von ln (x) kann als betrachtet werden implizite Kurve;", @@ -783,7 +783,7 @@ "end": 795.41 }, { - "input": "Well, to e to the y equals x.", + "input": "Well to find that, first rearrange this equation y equals ln of x to be e to the y equals x.", "translatedText": "Nun, zu e zum y ist gleich x.", "model": "DeepL", "from_community_srt": "richtig? Um das zu finden, ordnen Sie diese Gleichung zuerst neu y = ln (x) soll ey = x sein.", diff --git a/2017/implicit-differentiation/hebrew/sentence_translations.json b/2017/implicit-differentiation/hebrew/sentence_translations.json index 91982fc62..23fd7615c 100644 --- a/2017/implicit-differentiation/hebrew/sentence_translations.json +++ b/2017/implicit-differentiation/hebrew/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "בשלב הבא אני הולך לדבר על גבולות וכיצד הם משמשים לפורמליזציה של הרעיון של נגזרת. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/hindi/sentence_translations.json b/2017/implicit-differentiation/hindi/sentence_translations.json index 487138afd..6d8bf1b93 100644 --- a/2017/implicit-differentiation/hindi/sentence_translations.json +++ b/2017/implicit-differentiation/hindi/sentence_translations.json @@ -686,7 +686,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative.", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you.", "translatedText": "आगे मैं सीमाओं के बारे में बात करने जा रहा हूं और व्युत्पन्न के विचार को औपचारिक बनाने के लिए उनका उपयोग कैसे किया जाता है।", "n_reviews": 0, "start": 894.53, diff --git a/2017/implicit-differentiation/hungarian/sentence_translations.json b/2017/implicit-differentiation/hungarian/sentence_translations.json index 0340f506a..9d1392ee5 100644 --- a/2017/implicit-differentiation/hungarian/sentence_translations.json +++ b/2017/implicit-differentiation/hungarian/sentence_translations.json @@ -424,7 +424,7 @@ "end": 495.66 }, { - "input": "For points on this circle, that number is 25.", + "input": "For points on the circle, that number happens to be 25.", "translatedText": "A körön lévő pontok esetében ez a szám 25.", "model": "DeepL", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 504.4 }, { - "input": "For other points xy closer to the derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", + "input": "For other points xy closer to the origin, that value would be smaller. Now what it means to take a derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", "translatedText": "Más, xy pontokhoz közelebb eső pontok esetén ennek a kifejezésnek a deriváltja, s deriváltja, az, hogy mindkét változó egy apró változását, az x-hez képest valamilyen apró dx változást, az y-hoz képest pedig valamilyen apró dy változást kell figyelembe venni, és nem feltétlenül olyat, ami a körön tart, egyébként ez csak egy bármilyen apró lépés az xy sík bármely irányában.", "model": "DeepL", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 560.18 }, { - "input": "Then the decrease in s, the amount that x2 plus y2 changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", + "input": "Then the decrease in s, the amount that x squared plus y squared changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", "translatedText": "Ekkor az s csökkenése, azaz az x2 plusz y2 változása a lépés során körülbelül 2-szer 3-szor negatív 0,02 plus 2-szer 4-szer negatív 0,01.", "model": "DeepL", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 580.8 }, { - "input": "It's a recipe for telling you how much the value x2 plus y2 changes as determined by the point xy where you start and the tiny step dx dy you take.", + "input": "It's a recipe for telling you how much the value x squared plus y squared changes as determined by the point xy where you start and the tiny step dx dy that you take. And", "translatedText": "Ez egy olyan recept, amely megmondja, hogy az x2 plusz y2 érték mennyit változik az xy ponttól, ahonnan indulunk, és a megtett apró lépés dx dy-től függően.", "model": "DeepL", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 643.9 }, { - "input": "Of course, there's nothing special about the expression x2 plus y2 equals 5 squared.", + "input": "Of course, there's nothing special about the expression x squared plus y squared equals 5 squared.", "translatedText": "Természetesen semmi különös nincs az x2 plusz y2 egyenlő 5 négyzete kifejezésben.", "model": "DeepL", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 661.64 }, { - "input": "Those curves represent all the points xy where the value of sin of x times y2 equals x.", + "input": "And those curves, remember, represent all of the points xy where the value of sine of x times y squared happens to equal the value of x.", "translatedText": "Ezek a görbék az összes olyan xy pontot ábrázolják, ahol az x x-szer y2 sin értéke egyenlő x-szel.", "model": "DeepL", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 691.44 }, { - "input": "On the left side, the product rule tells us that this should be left d right plus right d left.", + "input": "On the left side, the product rule that we talked through last video tells us that this should be left d right plus right d left.", "translatedText": "A bal oldalon a szorzási szabály azt mondja, hogy ennek bal d jobb plusz jobb d balnak kell lennie.", "model": "DeepL", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 698.84 }, { - "input": "That is, sin of x times the change to y2, which is 2y dy, plus y2 times the change to sin of x, which is cos x times dx.", + "input": "That is sine of x times the change to y squared, which is 2y times dy, plus y squared times the change to sine of x, which is cosine of x times dx.", "translatedText": "Vagyis az x sin-je szorozva az y2 változásával, ami 2y dy, plusz y2 szorozva az x sin-jének változásával, ami cos x szorozva dx.", "model": "DeepL", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 710.98 }, { - "input": "The right side is simply x, so the size of a change is exactly dx.", + "input": "The right side is simply x, so the size of a change to that value is exactly dx, right? Now", "translatedText": "A jobb oldal egyszerűen x, tehát a változás nagysága pontosan dx.", "model": "DeepL", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 717.62 }, { - "input": "Setting these two sides equal to each other is a way of saying, whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left and right side must change by the same amount.", + "input": "setting these two sides equal to each other is a way of saying whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left hand side and the right hand side must change by the same amount.", "translatedText": "A két oldal egyenlővé tétele azt jelenti, hogy bármi is legyen a dx és dy koordinátákkal végrehajtott apró lépésünk, ha ez a görbén tart minket, akkor a bal és a jobb oldal értékeinek ugyanannyira kell változniuk.", "model": "DeepL", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 761.71 }, { - "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x, can be thought of as an implicit curve.", + "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x? Well the graph of the natural log of x can be thought of as an implicit curve.", "translatedText": "Említettem, hogy az e deriváltja az x-hez maga, de mi a helyzet az inverz függvényének, az x természetes logaritmusának deriváltjával, amit egy implicit görbének lehet elképzelni.", "model": "DeepL", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 795.41 }, { - "input": "Well, to e to the y equals x.", + "input": "Well to find that, first rearrange this equation y equals ln of x to be e to the y equals x.", "translatedText": "Nos, hogy e az y egyenlő x-szel.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/implicit-differentiation/indonesian/sentence_translations.json b/2017/implicit-differentiation/indonesian/sentence_translations.json index f3f00504d..0eaec1913 100644 --- a/2017/implicit-differentiation/indonesian/sentence_translations.json +++ b/2017/implicit-differentiation/indonesian/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "Selanjutnya saya akan membahas tentang limit dan cara penggunaannya untuk memformalkan gagasan turunan. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/italian/sentence_translations.json b/2017/implicit-differentiation/italian/sentence_translations.json index ca79688fe..46d3240ad 100644 --- a/2017/implicit-differentiation/italian/sentence_translations.json +++ b/2017/implicit-differentiation/italian/sentence_translations.json @@ -476,7 +476,7 @@ "end": 495.66 }, { - "input": "For points on this circle, that number is 25.", + "input": "For points on the circle, that number happens to be 25.", "translatedText": "Per i punti di questo cerchio, quel numero è 25.", "model": "DeepL", "from_community_srt": "Per i punti sul cerchio, quel numero è 25.", @@ -494,7 +494,7 @@ "end": 504.4 }, { - "input": "For other points xy closer to the derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", + "input": "For other points xy closer to the origin, that value would be smaller. Now what it means to take a derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", "translatedText": "Per altri punti xy più vicini alla derivata di questa espressione, una derivata di s, è considerare una piccola variazione di entrambe le variabili, una piccola variazione dx su x e una piccola variazione dy su y, e non necessariamente una variazione che ti mantiene sulla circonferenza, tra l'altro, è solo un piccolo passo in qualsiasi direzione del piano xy.", "model": "DeepL", "from_community_srt": "Per altri punti (x, y) più vicini all'origine, sarà più piccolo. Il significato di prendere la derivata di questa espressione, una derivata di S, è di considerare una piccola variazione di entrambe queste variabili. Consideriamo cioè una piccola variazione dx della coordinata x, e una piccola variazione dy di y. E non sto parlando necessariamente di una variazione che ti mantenga sul cerchio, ma un qualsiasi step in una qualsiasi direzione nel piano.", @@ -530,7 +530,7 @@ "end": 560.18 }, { - "input": "Then the decrease in s, the amount that x2 plus y2 changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", + "input": "Then the decrease in s, the amount that x squared plus y squared changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", "translatedText": "Allora la diminuzione di s, ovvero la quantità di variazioni di x2 più y2 in quel passaggio, sarebbe di circa 2 volte 3 volte 0,02 negativo più 2 volte 4 volte 0,01 negativo.", "model": "DeepL", "from_community_srt": "quanto cambia la quantità x^2+y^2 in questo step, sarà data da 2*3*(-0.02) + 2*4*(-0.01).", @@ -548,7 +548,7 @@ "end": 580.8 }, { - "input": "It's a recipe for telling you how much the value x2 plus y2 changes as determined by the point xy where you start and the tiny step dx dy you take.", + "input": "It's a recipe for telling you how much the value x squared plus y squared changes as determined by the point xy where you start and the tiny step dx dy that you take. And", "translatedText": "È una ricetta che ti dice di quanto cambia il valore x2 più y2 in base al punto xy da cui parti e al piccolo passo dx dy che fai.", "model": "DeepL", "from_community_srt": "y) in cui inizi e il piccolo step (dx, dy) di cui ti sposti.", @@ -629,7 +629,7 @@ "end": 643.9 }, { - "input": "Of course, there's nothing special about the expression x2 plus y2 equals 5 squared.", + "input": "Of course, there's nothing special about the expression x squared plus y squared equals 5 squared.", "translatedText": "Naturalmente, non c'è nulla di speciale nell'espressione x2 più y2 uguale a 5 al quadrato.", "model": "DeepL", "from_community_srt": "Ovviamente, non c'è niente di speciale nell'espressione x^2 + y^2 = 5^2. Potresti avere qualsiasi altra espressione in termini di x e y,", @@ -656,7 +656,7 @@ "end": 661.64 }, { - "input": "Those curves represent all the points xy where the value of sin of x times y2 equals x.", + "input": "And those curves, remember, represent all of the points xy where the value of sine of x times y squared happens to equal the value of x.", "translatedText": "Queste curve rappresentano tutti i punti xy in cui il valore di sin di x per y2 è uguale a x.", "model": "DeepL", "from_community_srt": "Queste curve rappresentano tutti i punti (x, y) del piano in cui il valore di sin(x)*y^2 è pari al valore di x.", @@ -683,7 +683,7 @@ "end": 691.44 }, { - "input": "On the left side, the product rule tells us that this should be left d right plus right d left.", + "input": "On the left side, the product rule that we talked through last video tells us that this should be left d right plus right d left.", "translatedText": "Sul lato sinistro, la regola del prodotto ci dice che dovrebbe essere sinistra d destra più destra d sinistra.", "model": "DeepL", "from_community_srt": "la regola del prodotto, spiegata anch'essa nel video precedente, ci dice che questa variazione sarà sin(x)*(la variazione di y^2),", @@ -692,7 +692,7 @@ "end": 698.84 }, { - "input": "That is, sin of x times the change to y2, which is 2y dy, plus y2 times the change to sin of x, which is cos x times dx.", + "input": "That is sine of x times the change to y squared, which is 2y times dy, plus y squared times the change to sine of x, which is cosine of x times dx.", "translatedText": "Vale a dire, il seno di x moltiplicato per la variazione di y2, che è 2y dy, più y2 moltiplicato per la variazione di x, che è cos x per dx.", "model": "DeepL", "from_community_srt": "che è 2y*dy, più y^2*(la variazione di sin(x)), che è cos(x)*dx.", @@ -701,7 +701,7 @@ "end": 710.98 }, { - "input": "The right side is simply x, so the size of a change is exactly dx.", + "input": "The right side is simply x, so the size of a change to that value is exactly dx, right? Now", "translatedText": "Il lato destro è semplicemente x, quindi la dimensione di una variazione è esattamente dx.", "model": "DeepL", "from_community_srt": "A destra abbiamo solo x, dunque la sua variazione è semplicemente dx,", @@ -710,7 +710,7 @@ "end": 717.62 }, { - "input": "Setting these two sides equal to each other is a way of saying, whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left and right side must change by the same amount.", + "input": "setting these two sides equal to each other is a way of saying whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left hand side and the right hand side must change by the same amount.", "translatedText": "Fissare questi due lati uguali tra loro è un modo per dire che, qualunque sia il tuo piccolo passo con le coordinate dx e dy, se ci manterrà sulla curva, i valori di entrambi i lati sinistro e destro devono cambiare della stessa quantità.", "model": "DeepL", "from_community_srt": "non vi pare? Eguagliando queste due espressioni ha lo stesso significato della frase \"qualunque sia il tuo piccolo step con coordinate (dx, dy), se ci dovrà mantenere sulla curva, il valore di entrambi i membri dell'equazione devono variare della stessa quantità\".", @@ -746,7 +746,7 @@ "end": 761.71 }, { - "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x, can be thought of as an implicit curve.", + "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x? Well the graph of the natural log of x can be thought of as an implicit curve.", "translatedText": "Ho detto che la derivata di e rispetto alla x è se stessa, ma la derivata della sua funzione inversa, il log naturale di x, può essere pensata come una curva implicita.", "model": "DeepL", "from_community_srt": "Ho menzionato in un video che la derivata dell'esponenziale e^x è sé stesso, ma cosa possiamo dire sulla derivata della sua funzione inversa, il logaritmo naturale di x? Il grafico di ln(x) può essere pensato come curva implicita: tutti i punti sul piano xy dove y=ln(x).", @@ -781,7 +781,7 @@ "end": 795.41 }, { - "input": "Well, to e to the y equals x.", + "input": "Well to find that, first rearrange this equation y equals ln of x to be e to the y equals x.", "translatedText": "Ebbene, alla e alla y corrisponde la x.", "model": "DeepL", "from_community_srt": "giusto? Per scoprirlo, prima riscriviamo l'equazione y = ln(x) come e^y = x.", diff --git a/2017/implicit-differentiation/japanese/sentence_translations.json b/2017/implicit-differentiation/japanese/sentence_translations.json index 9f9ad4a2c..54711d14a 100644 --- a/2017/implicit-differentiation/japanese/sentence_translations.json +++ b/2017/implicit-differentiation/japanese/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative.", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you.", "translatedText": "次に、限界と、導関数のアイデアを形式化する ために限界がどのように使用されるかについて説明します。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/korean/sentence_translations.json b/2017/implicit-differentiation/korean/sentence_translations.json index 7c0eadd48..f63969307 100644 --- a/2017/implicit-differentiation/korean/sentence_translations.json +++ b/2017/implicit-differentiation/korean/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "다음에는 극한과 극한이 도함수의 개념을 공식화하는 데 어떻게 사용되는지에 대해 이야기하겠습니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/marathi/sentence_translations.json b/2017/implicit-differentiation/marathi/sentence_translations.json index c28bf1d40..6ea855575 100644 --- a/2017/implicit-differentiation/marathi/sentence_translations.json +++ b/2017/implicit-differentiation/marathi/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "पुढे मी मर्यादांबद्दल आणि डेरिव्हेटिव्हची कल्पना औपचारिक करण्यासाठी त्यांचा वापर कसा केला जातो याबद्दल बोलणार आहे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/persian/sentence_translations.json b/2017/implicit-differentiation/persian/sentence_translations.json index 8c19119bc..f1e8c2f61 100644 --- a/2017/implicit-differentiation/persian/sentence_translations.json +++ b/2017/implicit-differentiation/persian/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "در ادامه در مورد محدودیت ها و نحوه استفاده از آنها برای رسمی کردن ایده مشتق صحبت خواهم کرد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/polish/sentence_translations.json b/2017/implicit-differentiation/polish/sentence_translations.json index 841365d63..2074c2220 100644 --- a/2017/implicit-differentiation/polish/sentence_translations.json +++ b/2017/implicit-differentiation/polish/sentence_translations.json @@ -421,7 +421,7 @@ "end": 495.66 }, { - "input": "For points on this circle, that number is 25.", + "input": "For points on the circle, that number happens to be 25.", "translatedText": "", "from_community_srt": "Dla punktów na okręgu ta liczba to 25.", "n_reviews": 0, @@ -437,7 +437,7 @@ "end": 504.4 }, { - "input": "For other points xy closer to the derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", + "input": "For other points xy closer to the origin, that value would be smaller. Now what it means to take a derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", "translatedText": "", "from_community_srt": "Dla punktów wewnątrz okręgu jest mniejsze. Branie pochodnej funkcji S polega na rozważeniu małej zmiany obu argumentów: zmiany x o dx i y o dy. Niekoniecznie takiej, która utrzyma cię na okręgu.", "n_reviews": 0, @@ -469,7 +469,7 @@ "end": 560.18 }, { - "input": "Then the decrease in s, the amount that x2 plus y2 changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", + "input": "Then the decrease in s, the amount that x squared plus y squared changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", "translatedText": "", "from_community_srt": "Wtedy mała zmiana dS wartości funkcji S między tymi punktami jest równa 2 * 3 * (-0.02) + 2 * 4 * (-0.01).", "n_reviews": 0, @@ -485,7 +485,7 @@ "end": 580.8 }, { - "input": "It's a recipe for telling you how much the value x2 plus y2 changes as determined by the point xy where you start and the tiny step dx dy you take.", + "input": "It's a recipe for telling you how much the value x squared plus y squared changes as determined by the point xy where you start and the tiny step dx dy that you take. And", "translatedText": "", "from_community_srt": "Mówi ona, jak zmienia się wartość x^2 + y^2 w zależności od punktu początkowego (x, y) i wielkości zmiany (dx, dy).", "n_reviews": 0, @@ -557,7 +557,7 @@ "end": 643.9 }, { - "input": "Of course, there's nothing special about the expression x2 plus y2 equals 5 squared.", + "input": "Of course, there's nothing special about the expression x squared plus y squared equals 5 squared.", "translatedText": "", "from_community_srt": "Oczywiście, równanie x^2 + y^2 = 25 nie jest wyjątkowe.", "n_reviews": 0, @@ -581,7 +581,7 @@ "end": 661.64 }, { - "input": "Those curves represent all the points xy where the value of sin of x times y2 equals x.", + "input": "And those curves, remember, represent all of the points xy where the value of sine of x times y squared happens to equal the value of x.", "translatedText": "", "from_community_srt": "Te krzywe zawierają wszystkie punkty (x, y), które spełniają zależność sin(x) * y^2 = x.", "n_reviews": 0, @@ -605,7 +605,7 @@ "end": 691.44 }, { - "input": "On the left side, the product rule tells us that this should be left d right plus right d left.", + "input": "On the left side, the product rule that we talked through last video tells us that this should be left d right plus right d left.", "translatedText": "", "from_community_srt": "Wiemy, jak obliczyć pochodną iloczynu,", "n_reviews": 0, @@ -613,7 +613,7 @@ "end": 698.84 }, { - "input": "That is, sin of x times the change to y2, which is 2y dy, plus y2 times the change to sin of x, which is cos x times dx.", + "input": "That is sine of x times the change to y squared, which is 2y times dy, plus y squared times the change to sine of x, which is cosine of x times dx.", "translatedText": "", "from_community_srt": "więc lewa strona po zróżniczkowaniu jest równa sin(x) * 2y * dy + cos(x) * dx * y^2.", "n_reviews": 0, @@ -621,7 +621,7 @@ "end": 710.98 }, { - "input": "The right side is simply x, so the size of a change is exactly dx.", + "input": "The right side is simply x, so the size of a change to that value is exactly dx, right? Now", "translatedText": "", "from_community_srt": "Prawa strona równania jest równa x, więc zmiana jej wartości jest równa dokładnie dx.", "n_reviews": 0, @@ -629,7 +629,7 @@ "end": 717.62 }, { - "input": "Setting these two sides equal to each other is a way of saying, whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left and right side must change by the same amount.", + "input": "setting these two sides equal to each other is a way of saying whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left hand side and the right hand side must change by the same amount.", "translatedText": "", "from_community_srt": "Jeśli każemy obu stronom równania po zróżniczkowaniu być równe, to to znaczy tyle, co \"po przejściu tego małego kroku (dx, dy) punkt pozostanie na krzywej, bo lewa i prawa strona zmienią się o tyle samo\".", "n_reviews": 0, @@ -661,7 +661,7 @@ "end": 761.71 }, { - "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x, can be thought of as an implicit curve.", + "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x? Well the graph of the natural log of x can be thought of as an implicit curve.", "translatedText": "", "from_community_srt": "Wiemy już, że pochodną funkcji e^x jest e^x. Ale jaka jest pochodna funkcji odwrotnej - logarytmu? Możesz pomyśleć o wykresie funkcji ln(x) jak o wykresie funkcji uwikłanej.", "n_reviews": 0, @@ -693,7 +693,7 @@ "end": 795.41 }, { - "input": "Well, to e to the y equals x.", + "input": "Well to find that, first rearrange this equation y equals ln of x to be e to the y equals x.", "translatedText": "", "from_community_srt": "Aby obliczyć ten stosunek, przekształćmy równanie: zamiast y = ln(x) mamy e^y = x.", "n_reviews": 0, diff --git a/2017/implicit-differentiation/portuguese/sentence_translations.json b/2017/implicit-differentiation/portuguese/sentence_translations.json index 68afbb083..ae6f989b1 100644 --- a/2017/implicit-differentiation/portuguese/sentence_translations.json +++ b/2017/implicit-differentiation/portuguese/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "A seguir falarei sobre limites e como eles são usados para formalizar a ideia de derivada. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/russian/sentence_translations.json b/2017/implicit-differentiation/russian/sentence_translations.json index 13a1d4e29..92c07b69c 100644 --- a/2017/implicit-differentiation/russian/sentence_translations.json +++ b/2017/implicit-differentiation/russian/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "Далее я собираюсь поговорить об ограничениях и о том, как они используются для формализации идеи производной. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/spanish/sentence_translations.json b/2017/implicit-differentiation/spanish/sentence_translations.json index 688845112..b558761b6 100644 --- a/2017/implicit-differentiation/spanish/sentence_translations.json +++ b/2017/implicit-differentiation/spanish/sentence_translations.json @@ -423,7 +423,7 @@ "end": 495.66 }, { - "input": "For points on this circle, that number is 25.", + "input": "For points on the circle, that number happens to be 25.", "translatedText": "Para los puntos de este círculo, ese número es 25.", "from_community_srt": "Para los puntos en este círculo, ese número es 25.", "n_reviews": 0, @@ -439,7 +439,7 @@ "end": 504.4 }, { - "input": "For other points xy closer to the derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", + "input": "For other points xy closer to the origin, that value would be smaller. Now what it means to take a derivative of this expression, a derivative of s, is to consider a tiny change to both of these variables, some tiny change dx to x, and some tiny change dy to y, and not necessarily one that keeps you on the circle, by the way, it's just any tiny step in any direction of the xy plane.", "translatedText": "Para otros puntos xy más cercanos a la derivada de esta expresión, una derivada de s, es considerar un pequeño cambio en ambas variables, algún pequeño cambio de dx a x y algún pequeño cambio de dy a y, y no necesariamente uno que mantenga Por cierto, estás en el círculo, es solo un pequeño paso en cualquier dirección del plano xy.", "from_community_srt": "por en otros puntos (x, y) más cercanos al origen, ese valor sería menor. Lo que significa tomar una derivada de esta expresión, una derivada de S, es considerar un pequeño cambio en estas dos variables, algunas pequeño cambio dx para x, y algun cambio pequeño dy para y , y no necesariamente uno que se mantiene en este círculo, por cierto, es sólo alguno pequeño paso en cualquier dirección en el Plano X;Y y de ahí,", "n_reviews": 0, @@ -471,7 +471,7 @@ "end": 560.18 }, { - "input": "Then the decrease in s, the amount that x2 plus y2 changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", + "input": "Then the decrease in s, the amount that x squared plus y squared changes over that step, would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01.", "translatedText": "Entonces, la disminución en s, la cantidad que x2 más y2 cambia en ese paso, sería aproximadamente 2 veces 3 veces menos 0,02 más 2 veces 4 veces menos 0,01.", "from_community_srt": "la disminución de S, la cantidad a la que x^2 + y^2 cambia, en ese paso sería: alrededor de 2 (3) (- 0,02) + 2 (4) (- 0,01).", "n_reviews": 0, @@ -487,7 +487,7 @@ "end": 580.8 }, { - "input": "It's a recipe for telling you how much the value x2 plus y2 changes as determined by the point xy where you start and the tiny step dx dy you take.", + "input": "It's a recipe for telling you how much the value x squared plus y squared changes as determined by the point xy where you start and the tiny step dx dy that you take. And", "translatedText": "Es una receta para decirte cuánto cambia el valor x2 más y2 según lo determinado por el punto xy donde comienzas y el pequeño paso dx dy que das.", "from_community_srt": "Es una receta para decirte lo mucho que el valor de x2 + y2 cambia, tal como es determinado por el punto (x, y) donde empezaste , y el pequeño paso (dx,", "n_reviews": 0, @@ -559,7 +559,7 @@ "end": 643.9 }, { - "input": "Of course, there's nothing special about the expression x2 plus y2 equals 5 squared.", + "input": "Of course, there's nothing special about the expression x squared plus y squared equals 5 squared.", "translatedText": "Por supuesto, no hay nada especial en la expresión x2 más y2 es igual a 5 al cuadrado.", "from_community_srt": "Por supuesto, no hay nada especial aquí con la expresión x^2 + y^2 = 5^2.", "n_reviews": 0, @@ -583,7 +583,7 @@ "end": 661.64 }, { - "input": "Those curves represent all the points xy where the value of sin of x times y2 equals x.", + "input": "And those curves, remember, represent all of the points xy where the value of sine of x times y squared happens to equal the value of x.", "translatedText": "Esas curvas representan todos los puntos xy donde el valor de sen de x multiplicado por y2 es igual a x.", "from_community_srt": "Y recuerda que esas curvas representan todos los puntos (x, y) del plano en el que el valor de sin (x) * y^2 es igual al valor de x.", "n_reviews": 0, @@ -607,7 +607,7 @@ "end": 691.44 }, { - "input": "On the left side, the product rule tells us that this should be left d right plus right d left.", + "input": "On the left side, the product rule that we talked through last video tells us that this should be left d right plus right d left.", "translatedText": "En el lado izquierdo, la regla del producto nos dice que debe ser izquierda d derecha más derecha d izquierda.", "from_community_srt": "sin (x) * (el cambio para y^2), que es 2y * dy, mas y^2 * (el cambio de sin (x)),", "n_reviews": 0, @@ -615,7 +615,7 @@ "end": 698.84 }, { - "input": "That is, sin of x times the change to y2, which is 2y dy, plus y2 times the change to sin of x, which is cos x times dx.", + "input": "That is sine of x times the change to y squared, which is 2y times dy, plus y squared times the change to sine of x, which is cosine of x times dx.", "translatedText": "Es decir, sen de x multiplicado por el cambio a y2, que es 2y dy, más y2 multiplicado por el cambio a sen de x, que es cos x multiplicado por dx.", "from_community_srt": "que es cos (x) * dx. sin (x) * (el cambio para y^2), que es 2y * dy, mas y^2 * (el cambio de sin (x)),", "n_reviews": 0, @@ -623,7 +623,7 @@ "end": 710.98 }, { - "input": "The right side is simply x, so the size of a change is exactly dx.", + "input": "The right side is simply x, so the size of a change to that value is exactly dx, right? Now", "translatedText": "El lado derecho es simplemente x, por lo que el tamaño de un cambio es exactamente dx.", "from_community_srt": "que es cos (x) * dx. Igualando estos dos lados uno al otro es una forma de decir :“sea cual sea tu pequeño paso", "n_reviews": 0, @@ -631,7 +631,7 @@ "end": 717.62 }, { - "input": "Setting these two sides equal to each other is a way of saying, whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left and right side must change by the same amount.", + "input": "setting these two sides equal to each other is a way of saying whatever your tiny step with coordinates dx and dy is, if it's going to keep us on the curve, the values of both the left hand side and the right hand side must change by the same amount.", "translatedText": "Igualar estos dos lados entre sí es una forma de decir, cualquiera que sea su pequeño paso con coordenadas dx y dy, si nos mantendrá en la curva, los valores de los lados izquierdo y derecho deben cambiar en la misma cantidad. .", "from_community_srt": "con coordenadas (dx, dy) , si va a mantenernos en esta curva, los valores de ambos, lado izquierdo y lado derecho, debe cambiar por la misma cantidad.”Eso es la única forma en que la ecuación superior puede mantenerse verdadera.", "n_reviews": 0, @@ -663,7 +663,7 @@ "end": 761.71 }, { - "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x, can be thought of as an implicit curve.", + "input": "I've mentioned that the derivative of e to the x is itself, but what about the derivative of its inverse function, the natural log of x? Well the graph of the natural log of x can be thought of as an implicit curve.", "translatedText": "He mencionado que la derivada de e respecto de x es ella misma, pero ¿qué pasa con la derivada de su función inversa, el logaritmo natural de x, que puede considerarse como una curva implícita?", "from_community_srt": "pero ¿qué pasa con derivada de su función inversa , logaritmo natural de x? La gráfica de ln (x) puede ser pensada como una curva implícita; todos los puntos en el plano xy donde y = ln (x),", "n_reviews": 0, @@ -695,7 +695,7 @@ "end": 795.41 }, { - "input": "Well, to e to the y equals x.", + "input": "Well to find that, first rearrange this equation y equals ln of x to be e to the y equals x.", "translatedText": "Bueno, e a y es igual a x.", "from_community_srt": "Esto es exactamente lo el logaritmo natural des x, realmente significa e elevado a lo que sea y es igual a x.", "n_reviews": 0, diff --git a/2017/implicit-differentiation/tamil/sentence_translations.json b/2017/implicit-differentiation/tamil/sentence_translations.json index 75a90bcd0..602e5cc1a 100644 --- a/2017/implicit-differentiation/tamil/sentence_translations.json +++ b/2017/implicit-differentiation/tamil/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "அடுத்து நான் வரம்புகள் மற்றும் வழித்தோன்றல் யோசனையை முறைப்படுத்த அவை எவ்வாறு பயன்படுத்தப்படுகின்றன என்பதைப் பற்றி பேசப் போகிறேன். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/telugu/sentence_translations.json b/2017/implicit-differentiation/telugu/sentence_translations.json index 6e43f818a..7960c19a0 100644 --- a/2017/implicit-differentiation/telugu/sentence_translations.json +++ b/2017/implicit-differentiation/telugu/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative.", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you.", "translatedText": "తదుపరి నేను పరిమితుల గురించి మాట్లాడబోతున్నాను మరియు అవి ఉత్పన్నం యొక్క ఆలోచనను అధికారికం చేయడానికి ఎలా ఉపయోగించబడుతున్నాయి.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/thai/sentence_translations.json b/2017/implicit-differentiation/thai/sentence_translations.json index 41074ab64..03489e114 100644 --- a/2017/implicit-differentiation/thai/sentence_translations.json +++ b/2017/implicit-differentiation/thai/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/turkish/sentence_translations.json b/2017/implicit-differentiation/turkish/sentence_translations.json index ab8313bb1..45ebecd39 100644 --- a/2017/implicit-differentiation/turkish/sentence_translations.json +++ b/2017/implicit-differentiation/turkish/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "Şimdi limitlerden ve bunların türev fikrini resmileştirmek için nasıl kullanıldıklarından bahsedeceğim. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/ukrainian/sentence_translations.json b/2017/implicit-differentiation/ukrainian/sentence_translations.json index bed84201a..897098569 100644 --- a/2017/implicit-differentiation/ukrainian/sentence_translations.json +++ b/2017/implicit-differentiation/ukrainian/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "Далі я буду говорити про обмеження та те, як вони використовуються для формалізації ідеї похідної. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/urdu/sentence_translations.json b/2017/implicit-differentiation/urdu/sentence_translations.json index d5c6e4750..d7032d6f3 100644 --- a/2017/implicit-differentiation/urdu/sentence_translations.json +++ b/2017/implicit-differentiation/urdu/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "اس کے بعد میں حدود کے بارے میں بات کرنے جا رہا ہوں اور یہ کہ وہ کس طرح مشتق کے خیال کو باضابطہ بنانے کے لیے استعمال ہوتے ہیں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/implicit-differentiation/vietnamese/sentence_translations.json b/2017/implicit-differentiation/vietnamese/sentence_translations.json index ec49c532a..11d43c1c3 100644 --- a/2017/implicit-differentiation/vietnamese/sentence_translations.json +++ b/2017/implicit-differentiation/vietnamese/sentence_translations.json @@ -784,7 +784,7 @@ "end": 893.07 }, { - "input": "Next up I'm going to be talking about limits and how they're used to formalize the idea of a derivative. ", + "input": "Next up, I'm going to be talking about limits, and how they're used to formalize the idea of a derivative. Thank you. ", "translatedText": "Tiếp theo tôi sẽ nói về các giới hạn và cách chúng được sử dụng để chính thức hóa ý tưởng về đạo hàm. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/arabic/sentence_translations.json b/2017/integration/arabic/sentence_translations.json index 2486ca63e..6640ec132 100644 --- a/2017/integration/arabic/sentence_translations.json +++ b/2017/integration/arabic/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "ثم في الفيديو التالي، سنرى كيف يمكن تعميم هذه الفكرة نفسها، ولكن على سياقين آخرين. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "تلعب هذه القيمة dt دورين منفصلين. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "إذا قدرت أن تكون السرعة ثابتة على فترات متعددة، فيمكنك معرفة المسافة التي تقطعها السيارة في كل فترة زمنية مع الضرب، ثم جمع كل هذه المسافة. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/bengali/sentence_translations.json b/2017/integration/bengali/sentence_translations.json index b157ff027..6c498c1b6 100644 --- a/2017/integration/bengali/sentence_translations.json +++ b/2017/integration/bengali/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "তারপরে পরবর্তী ভিডিওতে আমরা দেখব যে এই একই ধারণাটি কীভাবে সাধারণীকরণ করে, তবে কয়েকটি অন্যান্য প্রসঙ্গে।", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "সেই মান dt দুটি পৃথক ভূমিকা পালন করে।", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "আপনি যদি আনুমানিক গতিবেগ একাধিক ব্যবধানে ধ্রুবক হতে চান, তাহলে আপনি গুণের সাথে প্রতিটি ব্যবধানে গাড়িটি কতদূর যায় তা বের করতে পারেন এবং তারপরে সেগুলিকে যোগ করুন।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/chinese/sentence_translations.json b/2017/integration/chinese/sentence_translations.json index 1c19ac0fb..c20021635 100644 --- a/2017/integration/chinese/sentence_translations.json +++ b/2017/integration/chinese/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "然后在下一个视频中，我们将看到相同的 想法如何推广，但适用于其他几个环境。", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "该值 dt 扮演两个不同的角色。", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "如果您将多个时间间隔内的速度近似为恒定 ，则可以通过乘法计算出汽车在每个时间间 隔内行驶的距离，然后将所有这些相加。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/french/sentence_translations.json b/2017/integration/french/sentence_translations.json index a718f5638..3f40769de 100644 --- a/2017/integration/french/sentence_translations.json +++ b/2017/integration/french/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts.", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts.", "translatedText": "Ensuite, dans la vidéo suivante, nous verrons comment cette même idée se généralise, mais à quelques autres contextes.", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 184.16 }, { - "input": "And if visualizing distance as area seems weird, I'm right there with you.", + "input": "And if visualizing distance as area seems kind of weird, I'm right there with you.", "translatedText": "Et si visualiser la distance sous forme de surface vous semble bizarre, je suis à vos côtés.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles.", + "input": "First of all, that value dt plays two separate roles.", "translatedText": "Cette valeur dt joue deux rôles distincts.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 884.68 }, { - "input": "Notice that it doesn't matter which antiderivative we chose here, if for some reason it had a constant added to it, like 5, that constant would cancel out.", + "input": "Notice, by the way, it doesn't matter which antiderivative we chose here. If for some reason it had a constant added to it, like 5, that constant would cancel out.", "translatedText": "Notez que peu importe la primitive que nous avons choisie ici, si, pour une raison quelconque, une constante y était ajoutée, comme 5, cette constante s'annulerait.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up.", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up.", "translatedText": "Si vous estimez que la vitesse est constante sur plusieurs intervalles, vous pouvez déterminer la distance parcourue par la voiture sur chaque intervalle avec la multiplication, puis additionner tout cela.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1134.18 }, { - "input": "Next up, I'll bring up more context where this idea of an integral and area under curves comes up, along with some other intuitions for this fundamental theorem of calculus.", + "input": "Next up, I'm going to bring up more context where this idea of an integral and area under curves comes up, along with some other intuitions for this fundamental theorem of calculus.", "translatedText": "Ensuite, j'évoquerai plus de contexte où apparaît cette idée d'intégrale et d'aire sous les courbes, ainsi que d'autres intuitions pour ce théorème fondamental du calcul.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/hebrew/sentence_translations.json b/2017/integration/hebrew/sentence_translations.json index 029882231..e9a08f4ae 100644 --- a/2017/integration/hebrew/sentence_translations.json +++ b/2017/integration/hebrew/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "ואז בסרטון הבא נראה איך אותו רעיון מתכלל, אבל לכמה הקשרים אחרים. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "הערך הזה dt ממלא שני תפקידים נפרדים. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "אם אתה מעריך את המהירות קבועה במרווחים מרובים, אתה יכול להבין כמה רחוק המכונית מגיעה בכל מרווח עם כפל, ואז לחבר את כל אלה. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/hindi/sentence_translations.json b/2017/integration/hindi/sentence_translations.json index 3f45fc5d5..99e4e3d95 100644 --- a/2017/integration/hindi/sentence_translations.json +++ b/2017/integration/hindi/sentence_translations.json @@ -28,7 +28,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts.", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts.", "translatedText": "फिर अगले वीडियो में हम देखेंगे कि इसी विचार को कैसे सामान्यीकृत किया जाता है, लेकिन कुछ अन्य संदर्भों में।", "n_reviews": 0, "start": 49.18, @@ -154,7 +154,7 @@ "end": 184.16 }, { - "input": "And if visualizing distance as area seems weird, I'm right there with you.", + "input": "And if visualizing distance as area seems kind of weird, I'm right there with you.", "translatedText": "और अगर दूरी को क्षेत्र के रूप में देखना अजीब लगता है, तो मैं आपके साथ हूं।", "n_reviews": 0, "start": 185.0, @@ -336,7 +336,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles.", + "input": "First of all, that value dt plays two separate roles.", "translatedText": "वह मान dt दो अलग-अलग भूमिकाएँ निभाता है।", "n_reviews": 0, "start": 418.18, @@ -777,7 +777,7 @@ "end": 884.68 }, { - "input": "Notice that it doesn't matter which antiderivative we chose here, if for some reason it had a constant added to it, like 5, that constant would cancel out.", + "input": "Notice, by the way, it doesn't matter which antiderivative we chose here. If for some reason it had a constant added to it, like 5, that constant would cancel out.", "translatedText": "ध्यान दें कि इससे कोई फर्क नहीं पड़ता कि हमने यहां कौन सा प्रतिअवकलन चुना है, अगर किसी कारण से इसमें कोई स्थिरांक जोड़ा गया है, जैसे 5, तो वह स्थिरांक रद्द हो जाएगा।", "n_reviews": 0, "start": 885.9, @@ -882,7 +882,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up.", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up.", "translatedText": "यदि आप कई अंतरालों पर वेग को स्थिर मानते हैं, तो आप गुणन के साथ यह पता लगा सकते हैं कि कार प्रत्येक अंतराल पर कितनी दूर तक जाती है, और फिर उन सभी को जोड़ सकते हैं।", "n_reviews": 0, "start": 995.08, @@ -987,7 +987,7 @@ "end": 1134.18 }, { - "input": "Next up, I'll bring up more context where this idea of an integral and area under curves comes up, along with some other intuitions for this fundamental theorem of calculus.", + "input": "Next up, I'm going to bring up more context where this idea of an integral and area under curves comes up, along with some other intuitions for this fundamental theorem of calculus.", "translatedText": "आगे, मैं और अधिक संदर्भ लाऊंगा जहां एक अभिन्न अंग और वक्रों के नीचे के क्षेत्र का विचार सामने आता है, साथ ही कैलकुलस के इस मौलिक प्रमेय के लिए कुछ अन्य अंतर्ज्ञान भी सामने आते हैं।", "n_reviews": 0, "start": 1135.68, diff --git a/2017/integration/indonesian/sentence_translations.json b/2017/integration/indonesian/sentence_translations.json index 9256d8ec3..7c4c39691 100644 --- a/2017/integration/indonesian/sentence_translations.json +++ b/2017/integration/indonesian/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts.", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts.", "translatedText": "Kemudian di video berikutnya kita akan melihat bagaimana gagasan yang sama ini digeneralisasikan, tetapi ke beberapa konteks lain.", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 184.16 }, { - "input": "And if visualizing distance as area seems weird, I'm right there with you.", + "input": "And if visualizing distance as area seems kind of weird, I'm right there with you.", "translatedText": "Dan jika memvisualisasikan jarak sebagai luas terasa aneh, saya siap membantu Anda.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles.", + "input": "First of all, that value dt plays two separate roles.", "translatedText": "Nilai tersebut dt memainkan dua peran terpisah.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 884.68 }, { - "input": "Notice that it doesn't matter which antiderivative we chose here, if for some reason it had a constant added to it, like 5, that constant would cancel out.", + "input": "Notice, by the way, it doesn't matter which antiderivative we chose here. If for some reason it had a constant added to it, like 5, that constant would cancel out.", "translatedText": "Perhatikan bahwa tidak masalah antiturunan mana yang kita pilih di sini, jika karena alasan tertentu ada konstanta yang ditambahkan ke dalamnya, seperti 5, konstanta tersebut akan hilang.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up.", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up.", "translatedText": "Jika Anda memperkirakan kecepatan konstan pada beberapa interval, Anda dapat mengetahui seberapa jauh mobil melaju pada setiap interval dengan mengalikan, lalu menjumlahkan semuanya.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1134.18 }, { - "input": "Next up, I'll bring up more context where this idea of an integral and area under curves comes up, along with some other intuitions for this fundamental theorem of calculus.", + "input": "Next up, I'm going to bring up more context where this idea of an integral and area under curves comes up, along with some other intuitions for this fundamental theorem of calculus.", "translatedText": "Selanjutnya, saya akan mengemukakan lebih banyak konteks di mana gagasan tentang integral dan luas di bawah kurva muncul, bersama dengan beberapa intuisi lain untuk teorema dasar kalkulus ini.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/italian/sentence_translations.json b/2017/integration/italian/sentence_translations.json index c8b159ec7..160f0e75d 100644 --- a/2017/integration/italian/sentence_translations.json +++ b/2017/integration/italian/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "Poi nel prossimo video vedremo come questa stessa idea si generalizza, ma in un paio di altri contesti. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "Quel valore dt svolge due ruoli separati. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "Se calcoli che la velocità sia costante su più intervalli, potresti calcolare la distanza percorsa dall'auto in ciascun intervallo con la moltiplicazione e quindi sommarli tutti. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/japanese/sentence_translations.json b/2017/integration/japanese/sentence_translations.json index de69edcdf..30a71b882 100644 --- a/2017/integration/japanese/sentence_translations.json +++ b/2017/integration/japanese/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "次のビデオでは、これと同じアイデアが他のいくつか の文脈でどのように一般化されるかを見ていきます。", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "その値 dt は 2 つの別々の役割を果たします。", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "複数の区間で速度が一定であると近似する場合、乗算 によって各区間で車がどれくらいの距離を進むかを 計算し、それらをすべて加算することができます。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/korean/sentence_translations.json b/2017/integration/korean/sentence_translations.json index 4e0ffe59e..483313192 100644 --- a/2017/integration/korean/sentence_translations.json +++ b/2017/integration/korean/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "그런 다음 다음 비디오에서는 동일한 아이디어가 몇 가지 다른 맥락으로 일반화되는 방법을 살펴보겠습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "해당 값 dt는 두 가지 별도의 역할을 합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "여러 간격에서 속도가 일정하도록 대략적으로 계산하면 곱셈을 통해 자동차가 각 간격에서 얼마나 멀리 가는지 알아낸 다음 이를 모두 더할 수 있습니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/marathi/sentence_translations.json b/2017/integration/marathi/sentence_translations.json index 08f46a8ae..51abe0f66 100644 --- a/2017/integration/marathi/sentence_translations.json +++ b/2017/integration/marathi/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "मग पुढील व्हिडिओमध्ये आपण हीच कल्पना सामान्यीकरण कशी होते ते पाहू, परंतु काही इतर संदर्भांसाठी. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "ते मूल्य dt दोन स्वतंत्र भूमिका बजावते. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "जर तुम्ही अंदाजे वेग एकाहून अधिक मध्यांतरांवर स्थिर असेल, तर तुम्ही गुणाकाराने प्रत्येक मध्यांतरावर गाडी किती अंतरावर जाते हे शोधून काढू शकता आणि नंतर ते सर्व जोडा. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/persian/sentence_translations.json b/2017/integration/persian/sentence_translations.json index 914e43436..e491ca498 100644 --- a/2017/integration/persian/sentence_translations.json +++ b/2017/integration/persian/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "سپس در ویدیوی بعدی خواهیم دید که چگونه این همان ایده تعمیم می یابد، اما به چند زمینه دیگر. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "این مقدار dt دو نقش جداگانه ایفا می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "اگر سرعت را به طور تقریبی ثابت کنید در فواصل متعدد، می توانید بفهمید که ماشین در هر بازه با ضرب چقدر مسافت را طی می کند و سپس همه آنها را جمع کنید. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/russian/sentence_translations.json b/2017/integration/russian/sentence_translations.json index dcd0cfb06..9b77fe5e5 100644 --- a/2017/integration/russian/sentence_translations.json +++ b/2017/integration/russian/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "Затем в следующем видео мы увидим, как эта же идея обобщается, но на пару других контекстов. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "Это значение dt играет две отдельные роли. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "Если вы приближаете скорость к постоянной на нескольких интервалах, вы можете выяснить, как далеко машина проедет на каждом интервале, с помощью умножения, а затем сложить все эти значения. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/tamil/sentence_translations.json b/2017/integration/tamil/sentence_translations.json index 2ef0fd981..c4b760367 100644 --- a/2017/integration/tamil/sentence_translations.json +++ b/2017/integration/tamil/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "அடுத்த காணொளியில் இதே கருத்தை எவ்வாறு பொதுமைப்படுத்துகிறது என்பதைப் பார்ப்போம், ஆனால் வேறு சில சூழல்களுக்கு. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "அந்த மதிப்பு dt இரண்டு தனித்தனி பாத்திரங்களை வகிக்கிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "நீங்கள் தோராயமான வேகம் பல இடைவெளிகளில் நிலையானதாக இருந்தால், பெருக்கல் மூலம் ஒவ்வொரு இடைவெளியிலும் கார் எவ்வளவு தூரம் செல்கிறது என்பதை நீங்கள் கண்டுபிடிக்கலாம், பின்னர் அவை அனைத்தையும் சேர்க்கலாம். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/telugu/sentence_translations.json b/2017/integration/telugu/sentence_translations.json index 8bcbfd5a8..8c018f307 100644 --- a/2017/integration/telugu/sentence_translations.json +++ b/2017/integration/telugu/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "తరువాతి వీడియోలో ఇదే ఆలోచన ఎలా సాధారణీకరించబడుతుందో చూద్దాం, కానీ రెండు ఇతర సందర్భాలలో. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "ఆ విలువ dt రెండు వేర్వేరు పాత్రలను పోషిస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "మీరు వేగాన్ని బహుళ విరామాలలో స్థిరంగా ఉండేలా అంచనా వేస్తే, గుణకారంతో ప్రతి విరామంలో కారు ఎంత దూరం వెళుతుందో మీరు గుర్తించవచ్చు, ఆపై వాటన్నింటినీ జోడించవచ్చు. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/thai/sentence_translations.json b/2017/integration/thai/sentence_translations.json index 330391466..5af16b5a8 100644 --- a/2017/integration/thai/sentence_translations.json +++ b/2017/integration/thai/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "โกรเธนดิเอค ผู้ชายคนนี้ค่อนข้างจะเป็นไอดอลทางคณิตศาสตร์สำหรับฉัน และฉันก็ชอบคำพูดนี้ใช่ไหม บ่อยครั้งในวิชาคณิตศาสตร์ เราเจาะลึกในการแสดงให้เห็นว่าข้อเท็จจริงบางอย่างเป็นจริงด้วยชุดสูตรยาวๆ ก่อนที่จะถอยกลับไปและทำให้แน่ใจว่ารู้สึกว่าสมเหตุสมผล และควรชัดเจน อย่างน้อยก็ในระดับที่เป็นธรรมชาติ ในวิดีโอนี้ ผมอยากพูดถึงอินทิกรัล และสิ่งที่ผมอยากทำให้ชัดเจนคือ พวกมันเป็นอินเวอร์สของอนุพันธ์ ต่อไปนี้เราจะเน้นไปที่ตัวอย่างหนึ่ง ซึ่งเป็นลักษณะเดียวกับตัวอย่างของรถที่กำลังเคลื่อนที่ ที่ผมพูดถึงในบทที่ 2 ของซีรีส์นี้ โดยแนะนำอนุพันธ์ จากนั้นในวิดีโอหน้า เราจะดูว่าแนวคิดเดียวกันนี้สรุปได้อย่างไร แต่ในบริบทอื่นๆ สองสามอย่าง ลองจินตนาการว่าคุณกำลังนั่งอยู่ในรถ และคุณไม่สามารถมองเห็นออกไปนอกหน้าต่างได้ สิ่งที่คุณเห็นก็คือมาตรวัดความเร็ว เมื่อถึงจุดหนึ่ง รถก็เริ่มเคลื่อนที่ เร่งความเร็ว แล้วลดความเร็วลงจนหยุด ตลอดระยะเวลา 8 วินาที คำถามคือ มีวิธีที่ดีในการพิจารณาว่าคุณเดินทางมาไกลแค่ไหนในช่วงเวลานั้นโดยดูจากมาตรวัดความเร็วเท่านั้น หรือดีกว่านั้น คุณสามารถหาฟังก์ชันระยะทาง s ของ t ที่บอกคุณว่าคุณเดินทางได้ไกลแค่ไหนหลังจากระยะเวลาที่กำหนด t ที่ไหนสักแห่งระหว่าง 0 ถึง 8 วินาที? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/turkish/sentence_translations.json b/2017/integration/turkish/sentence_translations.json index 20ba5bcb1..3dc942ce6 100644 --- a/2017/integration/turkish/sentence_translations.json +++ b/2017/integration/turkish/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "Bir sonraki videoda aynı fikrin birkaç başka bağlam için de nasıl genelleştirildiğini göreceğiz. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "Bu dt değeri iki ayrı rol oynar. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "Hızın birden fazla aralıkta sabit olacağını yaklaşık olarak hesaplarsanız, arabanın her aralıkta ne kadar uzağa gittiğini çarpma işlemiyle bulabilir ve sonra hepsini toplayabilirsiniz. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/ukrainian/sentence_translations.json b/2017/integration/ukrainian/sentence_translations.json index 510f15e59..2d0dd2c48 100644 --- a/2017/integration/ukrainian/sentence_translations.json +++ b/2017/integration/ukrainian/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "Потім у наступному відео ми побачимо, як ця сама ідея узагальнюється, але в кількох інших контекстах. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "Це значення dt відіграє дві окремі ролі. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "Якщо ви наближено вважаєте, що швидкість буде постійною на кількох інтервалах, ви можете визначити, яку відстань проїжджає автомобіль на кожному інтервалі, множивши, а потім додати все це. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/urdu/sentence_translations.json b/2017/integration/urdu/sentence_translations.json index 212da2303..fcd2ba7de 100644 --- a/2017/integration/urdu/sentence_translations.json +++ b/2017/integration/urdu/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "پھر اگلی ویڈیو میں ہم دیکھیں گے کہ یہ ایک ہی خیال کیسے عام ہوتا ہے، لیکن کچھ دوسرے سیاق و سباق میں۔", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "وہ قدر dt دو الگ الگ کردار ادا کرتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "اگر آپ ایک سے زیادہ وقفوں پر رفتار کے مستقل ہونے کا تخمینہ لگاتے ہیں، تو آپ اندازہ لگا سکتے ہیں کہ ضرب کے ساتھ ہر وقفہ پر کار کتنی دور جاتی ہے، اور پھر ان سب کو شامل کریں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/integration/vietnamese/sentence_translations.json b/2017/integration/vietnamese/sentence_translations.json index 18afdfa73..c56ec085b 100644 --- a/2017/integration/vietnamese/sentence_translations.json +++ b/2017/integration/vietnamese/sentence_translations.json @@ -32,7 +32,7 @@ "end": 48.14 }, { - "input": "Then in the next video we'll see how this same idea generalizes, but to a couple other contexts. ", + "input": "Then in the next video we're going to see how this same idea generalizes, but to a couple other contexts. ", "translatedText": "Sau đó, trong video tiếp theo, chúng ta sẽ thấy ý tưởng này được khái quát hóa như thế nào nhưng trong một số bối cảnh khác. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 417.46 }, { - "input": "That value dt plays two separate roles. ", + "input": "First of all, that value dt plays two separate roles. ", "translatedText": "Giá trị đó dt đóng hai vai trò riêng biệt. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 994.22 }, { - "input": "If you approximate velocity to be constant on multiple intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", + "input": "If you approximate velocity to be constant on multiple different intervals, you could figure out how far the car goes on each interval with multiplication, and then add all of those up. ", "translatedText": "Nếu bạn ước chừng vận tốc không đổi trong nhiều khoảng thời gian, bạn có thể tính ra quãng đường ô tô đi được trong mỗi khoảng thời gian bằng phép nhân, rồi cộng tất cả những khoảng đó lại. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/arabic/sentence_translations.json b/2017/leibniz-formula/arabic/sentence_translations.json index e29f4e11a..1b76f98f2 100644 --- a/2017/leibniz-formula/arabic/sentence_translations.json +++ b/2017/leibniz-formula/arabic/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points. ", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points. ", "translatedText": "إذا نظرت إلى نصف القطر 1، فهذا يصل إلى 4 نقاط شبكية. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points. ", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice poi ", "translatedText": "الجذر التربيعي لنصف القطر 5 يصل في الواقع إلى 8 نقاط شبكية. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", "translatedText": "لذا بدلًا من هذه النقطة الشبكية هنا كزوج من الإحداثيات الصحيحة، 3,4، بدلًا من ذلك فكر فيها كرقم مركب واحد، 3 زائد 4i. ", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes. ", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes. ", "translatedText": "الأعداد الأولية التي تكون أعلى من مضاعف العدد 4، مثل 5 أو 13 أو 17، يمكن دائمًا تحليلها إلى اثنين من الأعداد الأولية الغوسية المتميزة. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers. ", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers. ", "translatedText": "من ناحية أخرى، فإن الأعداد الأولية التي تكون 3 أعلى من مضاعفات 4، مثل 3 أو 7 أو 11، لا يمكن تحليلها بشكل أكبر داخل الأعداد الصحيحة الغوسية. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25. ", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25. ", "translatedText": "نظرًا لأن كل شيء على اليمين هو مرافق مع كل شيء على اليسار، فإن الناتج هو زوج مترافق معقد مضروبًا في ٢٥. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i? ", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i? ", "translatedText": "هل تتذكر كيف ذكرت أن التحليل إلى أعداد أولية غوسية يمكن أن يبدو مختلفًا إذا ضربت بعضها في i أو -1 أو -i؟ ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i. ", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i. ", "translatedText": "في هذه الحالة، يمكنك كتابة تحليل 25 بشكل مختلف، وربما تقسيم أحد تلك الـ 5 إلى -1 زائد 2i في -1 ناقص 2i. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i. ", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i. ", "translatedText": "التأثير الوحيد الذي سيحدثه هذا هو ضرب إجمالي الناتج في i أو -1 أو -i. ", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees. ", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees. ", "translatedText": "خذ هذا المنتج من العمود الأيسر، واختر ضربه في 1، أو i، أو –1، أو –i، بما يتوافق مع عمليات الدوران التي تكون بعض مضاعفات 90 درجة. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", "translatedText": "هذه الاختيارات الأربعة، مضروبة في الاختيارات الأربعة الأخيرة لضرب الناتج من العمود الأيسر في 1 أو i أو -1 أو -i، تشير إلى أن هناك إجمالي 16 نقطة شبكية على مسافة جذر تربيعي قدره 125 بعيدًا عن النقطة أصل. ", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", "translatedText": "إذا كانت قوة زوجية، مثل 4 في هذه الحالة، فإن المجموع يصبح 1، وهو ما يلخص حقيقة أن هناك خيارًا واحدًا فقط لما يجب فعله بتلك الأرقام الثلاثة غير القابلة للتجزئة. ", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point. ", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point. ", "translatedText": "نحن نقترب من الذروة الآن، وبدأت الأمور تبدو منظمة، لذا توقف وتأمل، وتأكد من أن كل شيء على ما يرام حتى هذه اللحظة. ", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", "translatedText": "دعونا نتجاهل نقطة الأصل التي يبلغ نصف قطرها 0، فهي لا تتبع نمط الباقي، ونقطة صغيرة واحدة لن تحدث فرقًا عندما نترك r ينمو نحو اللانهاية. ", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4. ", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4. ", "translatedText": "من كل هذا الأعداد الصحيحة الغوسية والتحليل إلى عوامل ودالة تشي التي كنا نقوم بها، تبدو إجابة كل n وكأنها جمع قيمة تشي على كل مقسوم على n، وضربها في 4. ", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3. ", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2. ", "translatedText": "حوالي ثلث هذه الصفوف تحتوي على كاي 3، لذلك يمكننا وضع r2 مقسومًا على 3 ضرب كاي 3. ", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better. ", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will g ", "translatedText": "ضع في اعتبارك أننا تقريبيون، نظرًا لأن r2 قد لا يقسم 2 أو 3 بشكل مثالي، ولكن مع نمو r نحو ما لا نهاية، فإن هذا التقريب سوف يتحسن. ", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", + "input": "et better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", "translatedText": "وعندما تستمر بهذه الطريقة، تحصل على تعبير منظم جدًا لإجمالي عدد نقاط الشبكة. ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum. ", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum. ", "translatedText": "وإذا قمت بتحليل r2 وأعدت الـ 4 التي يجب ضربها، فإن ما يعنيه ذلك هو أن إجمالي عدد نقاط الشبكة داخل هذه الدائرة الكبيرة يبلغ تقريبًا 4 أضعاف r2 مضروبًا في هذا المجموع. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/bengali/sentence_translations.json b/2017/leibniz-formula/bengali/sentence_translations.json index 56133f47b..5d0843087 100644 --- a/2017/leibniz-formula/bengali/sentence_translations.json +++ b/2017/leibniz-formula/bengali/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points. ", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points. ", "translatedText": "আপনি যদি ব্যাসার্ধ 1 দেখেন, এটি 4টি জালি বিন্দুতে আঘাত করে।", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points. ", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice poi ", "translatedText": "5 এর একটি ব্যাসার্ধ বর্গমূল আসলে 8টি জালি বিন্দুতে আঘাত করে।", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", "translatedText": "সুতরাং এখানে এই জালি বিন্দুর পরিবর্তে পূর্ণসংখ্যা স্থানাঙ্কের জোড়া হিসাবে, 3,4, পরিবর্তে এটিকে একক জটিল সংখ্যা, 3 যোগ 4i হিসাবে ভাবুন।", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes. ", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes. ", "translatedText": "প্রাইম সংখ্যা যেগুলি 4-এর গুণিতকের উপরে একটি, যেমন 5, বা 13, বা 17, সর্বদা ঠিক দুটি স্বতন্ত্র গাউসিয়ান মৌলিক সংখ্যায় ফ্যাক্টর করা যেতে পারে।", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers. ", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers. ", "translatedText": "অন্যদিকে, মৌলিক সংখ্যাগুলি যেগুলি 4 এর গুণিতকের উপরে 3, যেমন 3, বা 7, বা 11, গাউসিয়ান পূর্ণসংখ্যার ভিতরে আরও গুণিত হতে পারে না।", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25. ", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25. ", "translatedText": "কারণ ডানদিকের সবকিছু বাম দিকের সবকিছুর সাথে একটি সংযোজক, যা বেরিয়ে আসে তা হল একটি জটিল সংযোজক জোড়া যা 25-এ গুণ করে।", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i? ", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i? ", "translatedText": "মনে রাখবেন কিভাবে আমি উল্লেখ করেছি যে গাউসিয়ান প্রাইমগুলিতে একটি ফ্যাক্টরাইজেশন ভিন্ন দেখাতে পারে যদি আপনি তাদের কয়েকটিকে i, বা –1, বা –i দ্বারা গুণ করেন? ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i. ", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i. ", "translatedText": "এই ক্ষেত্রে, আপনি 25 এর ফ্যাক্টরাইজেশন ভিন্নভাবে লিখতে পারেন, হয়ত সেই 5s-এর একটিকে –1 প্লাস 2i বার –1 বিয়োগ 2i হিসাবে বিভক্ত করতে পারেন।", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i. ", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i. ", "translatedText": "এর একমাত্র প্রভাব হল মোট আউটপুটকে i, বা –1, বা –i দ্বারা গুণ করা।", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees. ", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees. ", "translatedText": "বাম কলাম থেকে সেই পণ্যটি নিন, এবং এটিকে 1, i, –1, বা –i দ্বারা গুণ করতে বেছে নিন, যা 90 ডিগ্রির কিছু একাধিক ঘূর্ণনের সাথে সম্পর্কিত।", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", "translatedText": "এই চারটি পছন্দ, বাম কলাম থেকে গুণফলকে 1, i, –1, বা –i দ্বারা গুণ করার চূড়ান্ত চারটি পছন্দ দ্বারা গুণ করা হলে, প্রস্তাবিত হবে যে মোট 16টি জালি বিন্দু রয়েছে একটি দূরত্ব বর্গমূল থেকে 125 দূরে।", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", "translatedText": "যদি এটি একটি জোড় শক্তি হয়, এই ক্ষেত্রে 4 এর মত, যোগফলটি 1 হবে, যা এই সত্যকে ধারণ করে যে এই অবিভক্ত 3 এর সাথে কী করতে হবে তার জন্য শুধুমাত্র একটি বিকল্প রয়েছে।", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point. ", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point. ", "translatedText": "আমরা এখন চূড়ান্তের কাছাকাছি চলে এসেছি, জিনিসগুলি সংগঠিত দেখাতে শুরু করেছে, তাই বিরতি দিন এবং চিন্তা করুন, নিশ্চিত করুন যে এই বিন্দু পর্যন্ত সবকিছু ভাল লাগছে।", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", "translatedText": "চলুন ব্যাসার্ধ 0 সহ সেই উৎপত্তি বিন্দুটিকে উপেক্ষা করা যাক, এটি বাকিগুলির প্যাটার্ন অনুসরণ করে না, এবং একটি ছোট বিন্দু একটি পার্থক্য তৈরি করতে যাচ্ছে না কারণ আমরা r অসীমের দিকে বাড়তে দিই।", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4. ", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4. ", "translatedText": "এই সমস্ত গাউসিয়ান পূর্ণসংখ্যা এবং ফ্যাক্টরিং এবং চি ফাংশন স্টাফ থেকে আমরা করছি, প্রতিটি n-এর উত্তরটি n-এর প্রতিটি ভাজকের উপর chi-এর মান যোগ করা এবং 4 দ্বারা গুণ করার মত দেখাচ্ছে।", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3. ", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2. ", "translatedText": "এই সারির প্রায় এক-তৃতীয়াংশে 3-এর chi আছে, তাই আমরা 3-এর 3 গুণ chi দিয়ে ভাগ করলে r2 বসাতে পারি।", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better. ", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will g ", "translatedText": "মনে রাখবেন আমরা আনুমানিক হচ্ছে, যেহেতু r2 পুরোপুরি 2 বা 3কে ভাগ করতে পারে না, কিন্তু r অসীমের দিকে বাড়ার সাথে সাথে এই আনুমানিকতা আরও ভাল হবে।", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", + "input": "et better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", "translatedText": "এবং যখন আপনি এভাবে চলতে থাকুন, আপনি মোট জালি পয়েন্টের জন্য একটি সুন্দর সংগঠিত অভিব্যক্তি পাবেন।", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum. ", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum. ", "translatedText": "এবং আপনি যদি সেই r2 গুণনীয়ক বের করেন এবং 4টি ফিরিয়ে আনেন যেটিকে গুণ করতে হবে, তাহলে এর অর্থ হল এই বড় বৃত্তের ভিতরে মোট জাল বিন্দুর সংখ্যা এই যোগফলের প্রায় 4 গুণ r2 গুণ।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/chinese/sentence_translations.json b/2017/leibniz-formula/chinese/sentence_translations.json index ba2279d62..3bda3529d 100644 --- a/2017/leibniz-formula/chinese/sentence_translations.json +++ b/2017/leibniz-formula/chinese/sentence_translations.json @@ -234,7 +234,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points.", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points.", "translatedText": "如果观察半径 1，就会碰到 4 个晶格点。", "model": "google_nmt", "n_reviews": 0, @@ -268,7 +268,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "5 的半径平方根实际上击中了 8 个格点。", "model": "google_nmt", "from_community_srt": "而根号5会经过8个格点。", @@ -403,7 +403,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", "translatedText": "因此，不要将此格点视为一对整 数坐标 3,4，而是将其视为单个复数 3 加 4i。", "model": "google_nmt", "from_community_srt": "所以与其将这些格点看做形如 (3,4)的整数对， 不如将格点视作一个单一复数3+4i 换言之，", @@ -439,7 +439,7 @@ "end": 397.06 }, { - "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers will come into play.", + "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers are going to come into play.", "translatedText": "它将我们的问题变成了因式分解问题，这就是素数之间的模式最 终发挥作用的原因。", "model": "google_nmt", "from_community_srt": "这就将原问题转化成了分解质因数的问题。 这也是为什么质数会在其中出现的原因。", @@ -636,7 +636,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes.", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes.", "translatedText": "大于 4 的倍数的素数（例如 5、13 或 17）始终可以因式分解 为两个不同的高斯素数。", "model": "google_nmt", "from_community_srt": "形如4n+1的质数 如5， 13， 17总能将其分解为两个不同的高斯质数。", @@ -663,7 +663,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers.", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers.", "translatedText": "另一方面，大于 4 的倍数 3 的素数（例如 3、7 或 11）不能在高斯整数内进一步分解。", "model": "google_nmt", "from_community_srt": "形如4n+3的质数 比如3， 7， 11等等不能被进一步分解为高斯质数。", @@ -788,7 +788,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25.", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25.", "translatedText": "因为右边的所有内容都与左边的所 有内容共轭，所以结果是一个乘以 25 的复共轭 对。", "model": "google_nmt", "from_community_srt": "因为左右两边互为共轭， 乘积得到的也是 共轭复数对， 他们相乘就是25.", @@ -859,7 +859,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i?", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i?", "translatedText": "还记得我曾提到过，如果将其中一些素数乘以 i、–1 或 –i，则分解为高斯素数可能看起来会有所不同吗？", "model": "google_nmt", "from_community_srt": "之前提过， 将这些因子乘以i， -1或-i就能得到 看起来不同的分解方法。", @@ -868,7 +868,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i.", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i.", "translatedText": "在这种情况下，您可以以不同的方式编写 25 的因式分解，也许 将其中一个 5 分解为 –1 加 2i 乘以 –1 减 2i。", "model": "google_nmt", "from_community_srt": "稍微改写一下， 将左侧其中一个5写成(-1+2i)(-1-2i) 这会影响最后的结果，", @@ -885,7 +885,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i.", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i.", "translatedText": "唯一的效果是将总输出乘以 i、或 –1、或 –i。", "model": "google_nmt", "from_community_srt": "但仅仅只是将这些结果 乘以i， -1或-i。", @@ -903,7 +903,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees.", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees.", "translatedText": "从左列中取出该乘积，然后选择将其乘以 1、i、– 1 或 –i，对应于 90 度的某个倍数的旋转。", "model": "google_nmt", "from_community_srt": "i， -1或-i。 这三个格点 乘以1， i，", @@ -957,7 +957,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", "translatedText": "这四个选项，乘以左列的乘积乘以 1 、i、–1 或 –i 的最后四个选项，表明 距 125 的距离平方根总共有 16 个格 点。 起源。", "model": "google_nmt", "from_community_srt": "这四个不同的分配方式， 左侧最后的乘积 再乘以1， i， -1或-i， 就会得到 总共16个符合要求的高斯整点，", @@ -1391,7 +1391,7 @@ "end": 1374.96 }, { - "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, 1 minus 1 plus 1 minus 1 plus 1.", + "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, it goes 1 minus 1 plus 1 minus 1 plus 1.", "translatedText": "但在这种情况下，由于 3 的 chi 为负 1，因此该和会振荡，即 1 减 1 加 1 减 1 加 1。", "model": "google_nmt", "from_community_srt": "因为χ(3)=-1， χ值会在1， -1之前来回变动 本例中，", @@ -1400,7 +1400,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", "translatedText": "如果它 是偶次幂，例如本例中的 4，则总和为 1，这概括了这样一个 事实：对于如何处理这些不可分割的 3，只有一种选择。", "model": "google_nmt", "from_community_srt": "因为3的幂是偶数， 所以最后的和会是1 也即对于这些3， 只有一种方法将其平等分开。", @@ -1445,7 +1445,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point.", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point.", "translatedText": "我们现在 已经接近高潮了，事情开始看起来有条理，所以停 下来思考一下，确保到目前为止一切都感觉良好。", "model": "google_nmt", "from_community_srt": "所以前面有一个系数4 我们马上就要揭晓谜底了。 截至目前， 分析逐渐有了调理 暂停思考一会儿，", @@ -1571,7 +1571,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", "translatedText": "让我们忽略半径为 0 的原点，它不 遵循其余点的模式，并且当我们让 r 向 无穷大增长时，一个小点不会产生影响。", "model": "google_nmt", "from_community_srt": "因为它可以看成半径为0的圆 不过它不像其他整数有这么好的形式， 但一个点 对最终结论并不会有太的影响，", @@ -1580,7 +1580,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4.", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4.", "translatedText": "从我们一直在做的所有高斯整数、因式分解和 chi 函数的内容来看，每 个 n 的答案看起来就像将 n 的每个除数上的 chi 值相加，然 后乘以 4。", "model": "google_nmt", "from_community_srt": "因为我们会让R趋于无穷 有了前面讲的高斯整数， 质因数分解和χ函数这些内容 对一个给定的半径N的圆， 其经过的格点数实际上就是将N的每个因数的χ函数值加起来最后乘以4 先把4放在一边，", @@ -1675,7 +1675,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3.", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2.", "translatedText": "这些行中大约有三分之一的 chi 为 3，因此我们可以 将 r2 除以 3 的 chi 乘以 3。", "model": "google_nmt", "from_community_srt": "所以加上R^2/2*χ(2)这一项 有大约1/3的行有χ(3)， 所以加上R^2/3*χ(3) 记住，", @@ -1684,7 +1684,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better.", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will", "translatedText": "请记住，我们是近似值，因为 r2 可 能无法完美地整除 2 或 3，但随着 r 向无穷大增长，这种近似值会 变得更好。", "model": "google_nmt", "from_community_srt": "这只是一个估计。 R^2并不总能被2、3...整除， 但是当R趋于无穷时， 这种近似效果就越好。", @@ -1693,7 +1693,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", + "input": "get better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", "translatedText": "当你继续这样做时，你会得到一个非常有组 织的格点总数表达式。", "model": "google_nmt", "from_community_srt": "以此类推，", @@ -1702,7 +1702,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum.", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum.", "translatedText": "如果你将 r2 分解出来并带 回需要相乘的 4，那么这意味着这个大圆内的格点总数 大约是 r2 乘以这个总和的 4 倍。", "model": "google_nmt", "from_community_srt": "圆内总格点数会是一个非常有规律的表达式 提取公因数， 再把4乘回来 所以足够大的圆内总的格点数 将会近似于4*R^2再乘以这个和。", diff --git a/2017/leibniz-formula/english/captions.srt b/2017/leibniz-formula/english/captions.srt index bbb20e5f5..32a0abc42 100644 --- a/2017/leibniz-formula/english/captions.srt +++ b/2017/leibniz-formula/english/captions.srt @@ -243,1438 +243,1450 @@ A radius of 0 just gives you that single origin point. If you look at the radius 1, that hits 4 different lattice points. 62 -00:04:12,440 --> 00:04:15,980 -Radius square root of 2, well that also hits 4 lattice points. +00:04:12,440 --> 00:04:13,808 +Radius square root of 2, well that also hits 4 lattice points. 63 +00:04:13,808 --> 00:04:15,002 +Radius square root of 3 doesn't actually hit anything. + +64 +00:04:15,002 --> 00:04:15,980 +Square root of 4 again hits 4 lattice points. + +65 00:04:16,920 --> 00:04:19,180 A radius square root of 3 doesn't actually hit anything. -64 +66 00:04:19,899 --> 00:04:22,180 Square root of 4 again hits 4 lattice points. -65 +67 00:04:22,840 --> 00:04:26,140 A radius square root of 5 actually hits 8 lattice points. -66 +68 00:04:27,260 --> 00:04:32,019 And what we want is a systematic way to count how many lattice points are on a -67 +69 00:04:32,019 --> 00:04:36,960 given one of these rings, a given distance from the origin, and tally them all up. -68 +70 00:04:37,720 --> 00:04:41,338 And if you pause and try this for a moment, what you'll find is that -69 +71 00:04:41,338 --> 00:04:45,220 the pattern seems really chaotic, just very hard to find order under here. -70 +72 00:04:45,680 --> 00:04:49,260 And that's a good sign that some very interesting math is about to come into play. -71 +73 00:04:50,140 --> 00:04:54,980 In fact, as you'll see, this pattern is rooted in the distribution of primes. -72 +74 00:04:56,460 --> 00:04:59,880 As an example, let's look at the ring with radius square root of 25. -73 +75 00:05:00,700 --> 00:05:05,180 It hits the point 5,0, since 5 squared plus 0 squared is 25. -74 +76 00:05:06,100 --> 00:05:10,740 It also hits 4,3, since 4 squared plus 3 squared gives 25. -75 +77 00:05:12,780 --> 00:05:17,580 And likewise it hits 3,4, and also 0,5. -76 +78 00:05:18,660 --> 00:05:23,725 And what's really happening here is that you're counting how many pairs of integers, -77 +79 00:05:23,725 --> 00:05:27,480 a,b, have the property that a squared plus b squared equals 25. -78 +80 00:05:28,120 --> 00:05:32,000 And looking at this circle, it looks like there's a total of 12 of them. -79 +81 00:05:32,700 --> 00:05:35,980 As another example, take a look at the ring with radius square root of 11. -80 +82 00:05:36,600 --> 00:05:38,200 It doesn't hit any lattice points. -81 +83 00:05:38,700 --> 00:05:41,754 And that corresponds to the fact that you cannot -82 +84 00:05:41,754 --> 00:05:44,560 find two integers whose squares add up to 11. -83 +85 00:05:45,140 --> 00:05:45,820 Try it. -84 +86 00:05:48,240 --> 00:05:52,648 Now, many times in math, when you see a question that has to do with the 2D plane, -85 +87 00:05:52,648 --> 00:05:56,259 it can be surprisingly fruitful to just ask what it looks like when -86 +88 00:05:56,259 --> 00:05:59,340 you think of this plane as the set of all complex numbers. -87 +89 00:06:00,400 --> 00:06:05,514 So instead of thinking of this lattice point here as the pair of integer coordinates, -88 +90 00:06:05,514 --> 00:06:09,380 3,4, instead think of it as the single complex number, 3 plus 4i. -89 +91 00:06:10,620 --> 00:06:15,724 That way, another way to think about the sum of the squares of its coordinates, -90 +92 00:06:15,724 --> 00:06:20,000 3 squared plus 4 squared, is to multiply this number by 3 minus 4i. -91 +93 00:06:20,760 --> 00:06:22,480 This is called its complex conjugate. -92 +94 00:06:22,480 --> 00:06:27,300 It's what you get by reflecting over the real axis, replacing i with negative i. -93 +95 00:06:28,340 --> 00:06:30,365 And this might seem like a strange step if you -94 +96 00:06:30,365 --> 00:06:32,520 don't have much of a history with complex numbers. -95 +97 00:06:33,220 --> 00:06:37,060 But describing this distance as a product can be unexpectedly useful. -96 +98 00:06:37,780 --> 00:06:40,034 It turns our question into a factoring problem, -97 +99 00:06:40,034 --> 00:06:43,840 which is ultimately why patterns among prime numbers are going to come into play. -98 +100 00:06:45,060 --> 00:06:48,280 Algebraically, this relation is straightforward enough to verify. -99 +101 00:06:48,560 --> 00:06:55,180 You get a 3 squared, and then the 3 times minus 4i cancels out with the 4i times 3. -100 +102 00:06:55,920 --> 00:07:01,279 And then you have negative 4i squared, which, because i squared is negative 1, -101 +103 00:07:01,279 --> 00:07:02,840 becomes plus 4 squared. -102 +104 00:07:04,160 --> 00:07:06,080 This is also quite nice to see geometrically. -103 +105 00:07:06,580 --> 00:07:09,900 And if you're a little rusty with how complex multiplication works, -104 +106 00:07:09,900 --> 00:07:14,294 I do have another video that goes more into detail about why complex multiplication looks -105 +107 00:07:14,294 --> 00:07:15,320 the way that it does. -106 +108 00:07:15,780 --> 00:07:19,631 The way you might think about a case like this is that the number -107 +109 00:07:19,631 --> 00:07:23,600 3 plus 4i has a magnitude of 5 and some angle off of the horizontal. -108 +110 00:07:24,580 --> 00:07:29,726 And what it means to multiply it by 3 minus 4i is to rotate by that same angle in the -109 +111 00:07:29,726 --> 00:07:33,197 opposite direction, putting it on the positive real axis, -110 +112 00:07:33,197 --> 00:07:38,524 and then to stretch out by a factor of 5, which in this case lands you on the output 25, -111 +113 00:07:38,524 --> 00:07:40,200 the square of the magnitude. -112 +114 00:07:43,100 --> 00:07:46,125 The collection of all of these lattice points, -113 +115 00:07:46,125 --> 00:07:49,860 a plus bi, where a and b are integers, has a special name. -114 +116 00:07:50,240 --> 00:07:53,720 They're called the Gaussian integers, named after Martin Sheen. -115 +117 00:07:54,500 --> 00:07:56,820 Geometrically, you'll still be asking the same question. -116 +118 00:07:57,420 --> 00:08:00,341 How many of these lattice points, Gaussian integers, -117 +119 00:08:00,341 --> 00:08:03,980 are a given distance away from the origin, like square root of 25? -118 +120 00:08:04,880 --> 00:08:07,760 But we'll be phrasing it in a slightly more algebraic way. -119 +121 00:08:07,760 --> 00:08:11,210 How many Gaussian integers have the property that -120 +122 00:08:11,210 --> 00:08:14,800 multiplying by their complex conjugate gives you 25? -121 +123 00:08:16,540 --> 00:08:21,053 This might seem needlessly complex, but it's the key to understanding the seemingly -122 +124 00:08:21,053 --> 00:08:25,620 random pattern for how many lattice points are a given distance away from the origin. -123 +125 00:08:26,580 --> 00:08:32,220 To see why, we first need to understand how numbers factor inside the Gaussian integers. -124 +126 00:08:33,220 --> 00:08:36,784 As a refresher, among ordinary integers, every number -125 +127 00:08:36,784 --> 00:08:40,679 can be factored as some unique collection of prime numbers. -126 +128 00:08:41,559 --> 00:08:48,320 For example, 2250 can be factored as 2 times 3 squared times 5 cubed. -127 +129 00:08:48,580 --> 00:08:54,600 And there is no other collection of prime numbers that also multiplies to make 2250. -128 +130 00:08:55,760 --> 00:08:58,312 Unless you let negative numbers into the picture, -129 +131 00:08:58,312 --> 00:09:02,600 in which case you could just make some of the primes in this factorization negative. -130 +132 00:09:03,640 --> 00:09:08,820 So really, within the integers, factorization is not perfectly unique. -131 +133 00:09:09,060 --> 00:09:12,301 It's almost unique, with the exception that you can get a different -132 +134 00:09:12,301 --> 00:09:15,400 looking product by multiplying some of the factors by negative 1. -133 +135 00:09:17,960 --> 00:09:20,579 The reason I bring that up is that factoring works -134 +136 00:09:20,579 --> 00:09:22,840 very similarly inside the Gaussian integers. -135 +137 00:09:23,540 --> 00:09:28,529 Some numbers, like 5, can be factored into smaller Gaussian integers, -136 +138 00:09:28,529 --> 00:09:31,880 which in this case is 2 plus i times 2 minus i. -137 +139 00:09:32,880 --> 00:09:38,025 This Gaussian integer here, 2 plus i, cannot be factored into anything smaller, -138 +140 00:09:38,025 --> 00:09:40,020 so we call it a Gaussian prime. -139 +141 00:09:41,080 --> 00:09:45,946 Again, this factorization is almost unique, but this time not only -140 +142 00:09:45,946 --> 00:09:50,739 can you multiply each one of those factors by negative 1 to get a -141 +143 00:09:50,739 --> 00:09:55,750 factorization that looks different, you can also be extra sneaky and -142 +144 00:09:55,750 --> 00:10:00,980 multiply one of these factors by i and then the other one by negative i. -143 +145 00:10:02,180 --> 00:10:07,440 This will give you a different way to factor 5 into two distinct Gaussian primes. -144 +146 00:10:08,420 --> 00:10:13,341 But other than the things that you can get by multiplying some of these factors by -145 +147 00:10:13,341 --> 00:10:18,440 negative 1, or i, or negative i, factorization within the Gaussian integers is unique. -146 +148 00:10:20,120 --> 00:10:25,067 And if you can figure out how ordinary prime numbers factor inside the Gaussian integers, -147 +149 00:10:25,067 --> 00:10:28,256 that'll be enough to tell us how any other natural number -148 +150 00:10:28,256 --> 00:10:30,400 factors inside these Gaussian integers. -149 +151 00:10:31,240 --> 00:10:35,040 And so here, we pull in a crucial and pretty surprising fact. -150 +152 00:10:35,960 --> 00:10:41,521 Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, -151 +153 00:10:41,521 --> 00:10:47,560 these guys can always be factored into exactly two distinct Gaussian primes. -152 +154 00:10:48,860 --> 00:10:52,310 This corresponds with the fact that rings with a radius equal to the -153 +155 00:10:52,310 --> 00:10:55,960 square root of one of these prime numbers always hit some lattice points. -154 +156 00:10:56,620 --> 00:11:00,440 In fact, they always hit exactly 8 lattice points, as you'll see in just a moment. -155 +157 00:11:03,440 --> 00:11:08,394 On the other hand, prime numbers that are 3 above a multiple of 4, like 3, -156 +158 00:11:08,394 --> 00:11:13,680 or 7, or 11, these guys cannot be factored further inside the Gaussian integers. -157 +159 00:11:14,600 --> 00:11:17,180 Not only are they primes in the normal numbers, -158 +160 00:11:17,180 --> 00:11:21,320 but they're also Gaussian primes, unsplittable even when i is in the picture. -159 +161 00:11:22,200 --> 00:11:25,540 And this corresponds with the fact that a ring whose radius is the -160 +162 00:11:25,540 --> 00:11:28,980 square root of one of those primes will never hit any lattice points. -161 +163 00:11:33,180 --> 00:11:36,030 And this pattern right here is the regularity within -162 +164 00:11:36,030 --> 00:11:38,880 prime numbers that we're going to ultimately exploit. -163 +165 00:11:39,660 --> 00:11:43,273 And in a later video, I might explain why on earth this is true, -164 +166 00:11:43,273 --> 00:11:47,054 why a prime number's remainder when divided by 4 has anything to do -165 +167 00:11:47,054 --> 00:11:50,667 with whether or not it factors inside the Gaussian integers, or, -166 +168 00:11:50,667 --> 00:11:55,060 said differently, whether or not it can be expressed as the sum of two squares. -167 +169 00:11:55,980 --> 00:11:58,580 But here, and now, we'll just have to take it as a given. -168 +170 00:11:59,680 --> 00:12:04,542 The prime number 2, by the way, is a little special, because it does factor, -169 +171 00:12:04,542 --> 00:12:09,910 you can write it as 1 plus i times 1 minus i, but these two Gaussian primes are a 90 -170 +172 00:12:09,910 --> 00:12:15,088 degree rotation away from each other, so you can multiply one of them by i to get -171 +173 00:12:15,088 --> 00:12:15,720 the other. -172 +174 00:12:16,560 --> 00:12:20,194 And that fact is going to make us want to treat the prime number 2 a little bit -173 +175 00:12:20,194 --> 00:12:22,466 differently for where all of this stuff is going, -174 +176 00:12:22,466 --> 00:12:24,420 so just keep that in the back of your mind. -175 +177 00:12:26,860 --> 00:12:30,476 Remember, our goal here is to count how many lattice points are a -176 +178 00:12:30,476 --> 00:12:34,148 given distance away from the origin, and doing this systematically -177 +179 00:12:34,148 --> 00:12:37,820 for all distances square root of n can lead us to a formula for pi. -178 +180 00:12:38,920 --> 00:12:42,914 And counting the number of lattice points with a given magnitude, -179 +181 00:12:42,914 --> 00:12:47,333 like square root of 25, is the same as asking how many Gaussian integers -180 +182 00:12:47,333 --> 00:12:52,660 have the special property that multiplying them by their complex conjugate gives you 25. -181 +183 00:12:54,000 --> 00:12:58,000 So here's the recipe for finding all Gaussian integers that have this property. -182 +184 00:12:58,000 --> 00:13:04,030 Step 1, factor 25, which inside the ordinary integers looks like 5 squared, -183 +185 00:13:04,030 --> 00:13:09,029 but since 5 factors even further, as 2 plus i times 2 minus i, -184 +186 00:13:09,029 --> 00:13:12,600 25 breaks down as these four Gaussian primes. -185 +187 00:13:13,500 --> 00:13:16,414 Step 2, organize these into two different columns, -186 +188 00:13:16,414 --> 00:13:19,500 with conjugate pairs sitting right next to each other. -187 +189 00:13:20,260 --> 00:13:22,929 And once you do that, multiply what's in each column, -188 +190 00:13:22,929 --> 00:13:26,440 and you'll come out with two different Gaussian integers on the bottom. -189 +191 00:13:26,440 --> 00:13:31,553 And because everything on the right is a conjugate with everything on the left, -190 +192 00:13:31,553 --> 00:13:36,540 what comes out is going to be a complex conjugate pair which multiplies to 25. -191 +193 00:13:37,980 --> 00:13:40,368 Picking an arbitrary standard, let's say that the -192 +194 00:13:40,368 --> 00:13:43,140 product from that left column is the output of our recipe. -193 +195 00:13:44,680 --> 00:13:47,415 Now notice, there are three choices for how you -194 +196 00:13:47,415 --> 00:13:50,380 can divvy up the primes that can affect this output. -195 +197 00:13:51,300 --> 00:13:55,224 Pictured right here, both copies of 2 plus i are in the left column, -196 +198 00:13:55,224 --> 00:13:57,500 and that gives us the product 3 plus 4i. -197 +199 00:13:58,460 --> 00:14:02,925 You could also have chosen to have only one copy of 2 plus i in this left column, -198 +200 00:14:02,925 --> 00:14:04,940 in which case the product would be 5. -199 +201 00:14:05,720 --> 00:14:09,422 Or you could have both copies of 2 plus i in that right column, -200 +202 00:14:09,422 --> 00:14:13,240 in which case the output of our recipe would have been 3 minus 4i. -201 +203 00:14:15,920 --> 00:14:18,923 And those three possible outputs are all different -202 +204 00:14:18,923 --> 00:14:22,280 lattice points on a circle with radius square root of 25. -203 +205 00:14:24,340 --> 00:14:29,120 But why does this recipe not yet capture all 12 of the lattice points? -204 +206 00:14:30,180 --> 00:14:34,012 Remember how I mentioned that a factorization into Gaussian primes can -205 +207 00:14:34,012 --> 00:14:38,060 look different if you multiply some of them by i or negative 1, negative i? -206 +208 00:14:38,880 --> 00:14:42,978 In this case, you could write the factorization of 25 differently, -207 +209 00:14:42,978 --> 00:14:48,240 maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i. -208 +210 00:14:48,240 --> 00:14:52,500 And if you do that, running through the same recipe, it can affect the result. -209 +211 00:14:52,760 --> 00:14:54,980 You'll get a different product out of that left column. -210 +212 00:14:56,000 --> 00:15:01,369 But the only effect that this is going to have is to multiply that total output by i, -211 +213 00:15:01,369 --> 00:15:03,180 or negative 1, or negative i. -212 +214 00:15:03,900 --> 00:15:08,140 So as a final step for our recipe, let's say that you have to make one of four choices. -213 +215 00:15:08,800 --> 00:15:14,304 Take that product from the left column and choose to multiply it by 1, i, negative 1, -214 +216 00:15:14,304 --> 00:15:19,360 or negative i, corresponding to rotations that are some multiple of 90 degrees. -215 +217 00:15:21,540 --> 00:15:25,071 That will account for all 12 different ways of constructing -216 +218 00:15:25,071 --> 00:15:28,720 a Gaussian integer whose product with its own conjugate is 25. -217 +219 00:15:30,520 --> 00:15:33,240 This process is a little complicated, so I think the best way -218 +220 00:15:33,240 --> 00:15:35,960 to get a feel for it is to just try it out with more examples. -219 +221 00:15:36,760 --> 00:15:40,760 Let's say instead we were looking at 125, which is 5 cubed. -220 +222 00:15:40,760 --> 00:15:44,250 In that case, we would have four different choices for how -221 +223 00:15:44,250 --> 00:15:47,860 to divvy up the prime conjugate pairs into these two columns. -222 +224 00:15:48,520 --> 00:15:53,110 You can either have zero copies of 2 plus i in the left column, -223 +225 00:15:53,110 --> 00:15:58,920 one copy in there, two copies in there, or all three of them in that left column. -224 +226 00:15:59,660 --> 00:16:05,560 Those four choices multiplied by the final four choices of multiplying the product from -225 +227 00:16:05,560 --> 00:16:09,651 the left column by 1, or by i, or negative 1, or negative i, -226 +228 00:16:09,651 --> 00:16:15,283 would suggest that there are a total of 16 lattice points a distance square root of -227 +229 00:16:15,283 --> 00:16:16,960 125 away from the origin. -228 +230 00:16:19,000 --> 00:16:21,874 And indeed, if you draw that circle out and count, -229 +231 00:16:21,874 --> 00:16:25,200 what you'll find is that it hits exactly 16 lattice points. -230 +232 00:16:26,680 --> 00:16:30,041 But what if you introduce a factor like 3, which doesn't -231 +233 00:16:30,041 --> 00:16:33,520 break down as the product of two conjugate Gaussian primes? -232 +234 00:16:34,420 --> 00:16:36,440 Well that really mucks up the whole system. -233 +235 00:16:36,940 --> 00:16:39,928 When you're divvying up the primes between the two columns, -234 +236 00:16:39,928 --> 00:16:42,120 there's no way that you can split up this 3. -235 +237 00:16:42,520 --> 00:16:45,620 No matter where you put it, it leaves the columns imbalanced. -236 +238 00:16:46,260 --> 00:16:49,558 And what that means is that when you take the product of all of the -237 +239 00:16:49,558 --> 00:16:53,100 numbers in each column, you're not going to end up with a conjugate pair. -238 +240 00:16:53,660 --> 00:16:57,752 So for a number like this, 3 times 5 cubed, which is 375, -239 +241 00:16:57,752 --> 00:17:01,280 there's actually no lattice point that you'll hit. -240 +242 00:17:01,880 --> 00:17:06,680 No Gaussian integer whose product with its own conjugate gives you 375. -241 +243 00:17:08,119 --> 00:17:12,420 However, if you introduce a second factor of 3, then you have an option. -242 +244 00:17:12,920 --> 00:17:17,200 You can throw one 3 in the left column, and the other 3 in the right column. -243 +245 00:17:17,200 --> 00:17:21,388 Since 3 is its own complex conjugate, this leaves things balanced, -244 +246 00:17:21,388 --> 00:17:26,702 in the sense that the product of the left and right columns will indeed be a complex -245 +247 00:17:26,702 --> 00:17:27,640 conjugate pair. -246 +248 00:17:29,380 --> 00:17:31,520 But it doesn't add any new options. -247 +249 00:17:31,940 --> 00:17:36,752 There's still going to be a total of 4 choices for how to divvy up those factors of 5, -248 +250 00:17:36,752 --> 00:17:41,400 multiplied by the final 4 choices of multiplying by 1, i, negative 1, or negative i. -249 +251 00:17:42,000 --> 00:17:45,894 So just like the square root of 125 circle, this guy is -250 +252 00:17:45,894 --> 00:17:49,720 also going to end up hitting exactly 16 lattice points. -251 +253 00:17:51,240 --> 00:17:52,620 Let's just sum up where we are. -252 +254 00:17:53,020 --> 00:17:56,382 When you're counting up how many lattice points lie on a circle -253 +255 00:17:56,382 --> 00:17:59,640 with a radius square root of n, the first step is to factor n. -254 +256 00:18:01,000 --> 00:18:05,435 And for prime numbers like 5, or 13, or 17, which factor further into a -255 +257 00:18:05,435 --> 00:18:10,056 complex conjugate pair of Gaussian primes, the number of choices they give -256 +258 00:18:10,056 --> 00:18:14,800 you will always be one more than the exponent that shows up with that factor. -257 +259 00:18:17,200 --> 00:18:20,462 On the other hand, for prime factors like 3, or 7, or 11, -258 +260 00:18:20,462 --> 00:18:23,556 which are already Gaussian primes and cannot be split, -259 +261 00:18:23,556 --> 00:18:27,437 if they show up with an even power, you have one and only one choice -260 +262 00:18:27,437 --> 00:18:28,900 with what to do with them. -261 +263 00:18:29,440 --> 00:18:33,240 But if it's an odd exponent, you're screwed, and you just have zero choices. -262 +264 00:18:34,200 --> 00:18:37,640 And always, no matter what, you have those final 4 choices at the end. -263 +265 00:18:39,860 --> 00:18:42,280 By the way, I do think that this process right -264 +266 00:18:42,280 --> 00:18:44,700 here is the most complicated part of the video. -265 +267 00:18:45,380 --> 00:18:47,996 It took me a couple times to think through that, yes, -266 +268 00:18:47,996 --> 00:18:50,953 this is a valid way to count lattice points, so don't be shy -267 +269 00:18:50,953 --> 00:18:54,200 if you want to pause and scribble things down to get a feel for it. -268 +270 00:18:54,920 --> 00:19:00,080 The one last thing to mention about this recipe is how factors of 2 affect the count. -269 +271 00:19:01,020 --> 00:19:07,680 If your number is even, then that factor of 2 breaks down as 1 plus i times 1 minus i. -270 +272 00:19:07,680 --> 00:19:11,820 So you can divvy up that complex conjugate pair between the two columns. -271 +273 00:19:12,780 --> 00:19:16,052 And at first, it might look like this doubles your options, -272 +274 00:19:16,052 --> 00:19:20,580 depending on how you choose to place those two Gaussian primes between the columns. -273 +275 00:19:21,460 --> 00:19:26,156 However, since multiplying one of these guys by i gives you the other one, -274 +276 00:19:26,156 --> 00:19:30,665 when you swap them between the columns, the effect that that has on the -275 +277 00:19:30,665 --> 00:19:35,300 output from the left column is to just multiply it by i, or by negative i. -276 +278 00:19:35,300 --> 00:19:38,464 So that's actually redundant with the final step, -277 +279 00:19:38,464 --> 00:19:43,844 where we take the product of this left column and choose to multiply it by either 1, -278 +280 00:19:43,844 --> 00:19:45,680 i, negative 1, or negative i. -279 +281 00:19:46,640 --> 00:19:50,506 What this means is that a factor of 2, or any power of 2, -280 +282 00:19:50,506 --> 00:19:53,240 doesn't actually change the count at all. -281 +283 00:19:53,720 --> 00:19:55,620 It doesn't hurt, but it doesn't help. -282 +284 00:19:56,420 --> 00:20:00,860 For example, a circle with radius square root of 5 hits 8 lattice points. -283 +285 00:20:00,860 --> 00:20:05,620 And if you grow that radius to square root of 10, then you also hit 8 lattice points. -284 +286 00:20:06,220 --> 00:20:10,160 And square root of 20 also hits 8 lattice points, as does square root of 40. -285 +287 00:20:11,020 --> 00:20:13,080 Factors of 2 just don't make a difference. -286 +288 00:20:15,580 --> 00:20:18,320 Now what's about to happen is number theory at its best. -287 +289 00:20:18,980 --> 00:20:23,513 We have this complicated recipe telling us how many lattice points sit on a circle -288 +290 00:20:23,513 --> 00:20:27,720 with radius square root of n, and it depends on the prime factorization of n. -289 +291 00:20:27,720 --> 00:20:32,013 To turn this into something simpler, something we can actually deal with, -290 +292 00:20:32,013 --> 00:20:36,016 we're going to exploit the regularity of primes that those which are -291 +293 00:20:36,016 --> 00:20:39,961 1 above a multiple of 4 split into distinct Gaussian prime factors, -292 +294 00:20:39,961 --> 00:20:43,500 while those that are 3 above a multiple of 4 cannot be split. -293 +295 00:20:44,300 --> 00:20:46,530 To do this, let's introduce a simple function, -294 +296 00:20:46,530 --> 00:20:48,760 one which I'll label with the Greek letter chi. -295 +297 00:20:49,680 --> 00:20:54,820 For inputs that are 1 above a multiple of 4, the output of chi is just 1. -296 +298 00:20:55,380 --> 00:21:00,900 If it takes in an input 3 above a multiple of 4, then the output of chi is negative 1. -297 +299 00:21:01,880 --> 00:21:05,240 And then on all even numbers, it gives 0. -298 +300 00:21:09,680 --> 00:21:15,827 So if you evaluate chi on the natural numbers, it gives this very nice cyclic pattern, -299 +301 00:21:15,827 --> 00:21:19,360 1, 0, negative 1, 0, and then repeat indefinitely. -300 +302 00:21:20,860 --> 00:21:24,160 And this cyclic function chi has a very special property. -301 +303 00:21:24,160 --> 00:21:26,660 It's what's called a multiplicative function. -302 +304 00:21:27,620 --> 00:21:31,940 If you evaluate it on two different numbers and multiply the results, -303 +305 00:21:31,940 --> 00:21:35,829 like chi of 3 times chi of 5, it's the same as if you evaluate -304 +306 00:21:35,829 --> 00:21:39,780 chi on the product of those two numbers, in this case chi of 15. -305 +307 00:21:40,880 --> 00:21:44,713 Likewise, chi of 5 times chi of 5 is equal to chi of 25, -306 +308 00:21:44,713 --> 00:21:50,160 and no matter what two natural numbers you put in there, this property will hold. -307 +309 00:21:50,700 --> 00:21:51,880 Go ahead, try it if you want. -308 +310 00:21:52,820 --> 00:21:56,837 So for our central question of counting lattice points in this way that -309 +311 00:21:56,837 --> 00:22:01,022 involves factoring a number, what I'm going to do is write down the number -310 +312 00:22:01,022 --> 00:22:06,044 of choices we have but using chi in what at first seems like a much more complicated way, -311 +313 00:22:06,044 --> 00:22:09,560 but this has the benefit of treating all prime factors equally. -312 +314 00:22:10,660 --> 00:22:15,029 For each prime power, like 5 cubed, what you write down is chi -313 +315 00:22:15,029 --> 00:22:19,260 of 1 plus chi of 5 plus chi of 5 squared plus chi of 5 cubed. -314 +316 00:22:19,260 --> 00:22:22,586 You add up the value of chi on all the powers of this -315 +317 00:22:22,586 --> 00:22:26,220 prime up to the one that shows up inside the factorization. -316 +318 00:22:27,340 --> 00:22:32,327 In this case, since 5 is 1 above a multiple of 4, all of these are just 1, -317 +319 00:22:32,327 --> 00:22:37,648 so this sum comes out to be 4, which reflects the fact that a factor of 5 cubed -318 +320 00:22:37,648 --> 00:22:43,167 gives you 4 options for how to divvy up the two Gaussian prime factors between the -319 +321 00:22:43,167 --> 00:22:43,700 columns. -320 +322 00:22:46,340 --> 00:22:51,164 For a factor like 3 to the 4th, what you write down looks totally similar, -321 +323 00:22:51,164 --> 00:22:54,960 chi of 1 plus chi of 3 on and on up to chi of 3 to the 4th. -322 +324 00:22:55,040 --> 00:22:58,862 But in this case, since chi of 3 is negative 1, -323 +325 00:22:58,862 --> 00:23:03,720 this sum oscillates, it goes 1 minus 1 plus 1 minus 1 plus 1. -324 +326 00:23:04,420 --> 00:23:07,400 And if it's an even power, like 4 in this case, -325 +327 00:23:07,400 --> 00:23:11,187 the total sum comes out to be 1, which encapsulates the fact -326 +328 00:23:11,187 --> 00:23:15,720 that there is only one choice for what to do with those unsplittable 3's. -327 +329 00:23:16,200 --> 00:23:19,037 But if it's an odd power, that sum comes out to 0, -328 +330 00:23:19,037 --> 00:23:22,820 indicating that you're screwed, you can't place that unsplittable 3. -329 +331 00:23:24,580 --> 00:23:28,945 When you do this for a power of 2, what it looks like is 1 plus -330 +332 00:23:28,945 --> 00:23:33,380 0 plus 0 plus 0 on and on, since chi is always 0 on even numbers. -331 +333 00:23:33,920 --> 00:23:38,151 And this reflects the fact that a factor of 2 doesn't help and it doesn't hurt, -332 +334 00:23:38,151 --> 00:23:41,060 you always have just one option for what to do with it. -333 +335 00:23:41,940 --> 00:23:46,410 And as always, we keep a 4 in front to indicate that final choice of multiplying by 1, -334 +336 00:23:46,410 --> 00:23:47,900 i, negative 1, or negative i. -335 +337 00:23:49,080 --> 00:23:50,700 We're getting close to the culmination now. -336 +338 00:23:51,040 --> 00:23:53,522 Things are starting to look organized, so take a moment, -337 +339 00:23:53,522 --> 00:23:56,440 pause and ponder, make sure everything feels good up to this point. -338 +340 00:23:57,140 --> 00:23:59,460 Take the number 45 as an example. -339 +341 00:24:00,140 --> 00:24:04,285 This guy factors as 3 squared times 5, so the expression for -340 +342 00:24:04,285 --> 00:24:08,362 the total number of lattice points is 4 times chi of 1 plus -341 +343 00:24:08,362 --> 00:24:12,440 chi of 3 plus chi of 3 squared times chi of 1 plus chi of 5. -342 +344 00:24:13,160 --> 00:24:17,237 You can think about this as 4 times the one choice for what to do with the -343 +345 00:24:17,237 --> 00:24:21,260 3's times two choices for how to divvy up the Gaussian prime factors of 5. -344 +346 00:24:22,020 --> 00:24:25,822 It might seem like expanding out this sum is really complicated, -345 +347 00:24:25,822 --> 00:24:29,860 because it involves all possible combinations of these prime factors. -346 +348 00:24:30,640 --> 00:24:31,380 And it kind of is. -347 +349 00:24:32,020 --> 00:24:35,547 However, because chi is multiplicative, each one of -348 +350 00:24:35,547 --> 00:24:38,940 those combinations corresponds to a divisor of 45. -349 +351 00:24:38,940 --> 00:24:44,868 In this case, we get 4 times chi of 1 plus chi of 3 plus -350 +352 00:24:44,868 --> 00:24:50,380 chi of 5 plus chi of 9 plus chi of 15 plus chi of 45. -351 +353 00:24:51,360 --> 00:24:56,868 What you'll notice is that this covers every number that divides evenly into 45, -352 +354 00:24:56,868 --> 00:24:58,160 once and only once. -353 +355 00:24:58,940 --> 00:25:02,480 And it works like this for any number, there's nothing special about 45. -354 +356 00:25:03,220 --> 00:25:06,760 And that to me is pretty interesting, and I think wholly unexpected. -355 +357 00:25:07,380 --> 00:25:11,117 This question of counting the number of lattice points a distance -356 +358 00:25:11,117 --> 00:25:14,628 square root of n away from the origin, involves adding up the -357 +359 00:25:14,628 --> 00:25:18,480 value of this relatively simple function over all the divisors of n. -358 +360 00:25:20,100 --> 00:25:22,720 To bring it all together, remember why we're doing this. -359 +361 00:25:23,100 --> 00:25:26,241 The total number of lattice points inside a big circle -360 +362 00:25:26,241 --> 00:25:29,040 with radius r should be about pi times r squared. -361 +363 00:25:29,040 --> 00:25:32,910 But on the other hand, we can count those same lattice points by looking -362 +364 00:25:32,910 --> 00:25:35,773 through all of the numbers n between 0 and r squared, -363 +365 00:25:35,773 --> 00:25:40,280 and counting how many lattice points are a distance square root of n from the origin. -364 +366 00:25:41,280 --> 00:25:44,342 Let's go ahead and just ignore that origin dot with radius 0, -365 +367 00:25:44,342 --> 00:25:46,812 it doesn't really follow the pattern of the rest, -366 +368 00:25:46,812 --> 00:25:51,060 and one little dot isn't going to make a difference as we let r grow towards infinity. -367 +369 00:25:52,200 --> 00:25:56,846 Now from all of this Gaussian integer and factoring and chi function -368 +370 00:25:56,846 --> 00:26:01,493 stuff that we've been doing, the answer for each n looks like adding -369 +371 00:26:01,493 --> 00:26:06,140 up the value of chi on every divisor of n, and then multiplying by 4. -370 +372 00:26:07,220 --> 00:26:09,974 And for now let's just take that 4 and put it in the corner, -371 +373 00:26:09,974 --> 00:26:11,600 and remember to bring it back later. -372 +374 00:26:12,720 --> 00:26:18,340 At first, adding up the values for each one of these rows seems super random, right? -373 +375 00:26:18,840 --> 00:26:22,507 I mean, numbers with a lot of factors have a lot of divisors, -374 +376 00:26:22,507 --> 00:26:25,880 whereas prime numbers will always only have two divisors. -375 +377 00:26:25,880 --> 00:26:29,591 So it initially seems like you would have to have perfect knowledge -376 +378 00:26:29,591 --> 00:26:33,140 of the distribution of primes to get anything useful out of this. -377 +379 00:26:34,180 --> 00:26:39,040 But if instead you organize these into columns, the puzzle starts to fit together. -378 -00:26:40,100 --> 00:26:43,920 -How many numbers between 1 and r squared have 1 as a divisor? +380 +00:26:40,100 --> 00:26:41,831 +How many numbers between 1 and r2 have 1 as a divisor? All of them. -379 +381 +00:26:41,831 --> 00:26:43,920 +So our sum should include r2 times chi of 1. How many of them have 2 as a divisor? + +382 00:26:44,560 --> 00:26:45,400 Well, all of them. -380 +383 00:26:45,900 --> 00:26:49,200 So our sum should include r squared times chi of 1. -381 +384 00:26:50,060 --> 00:26:52,320 How many of them have 2 as a divisor? -382 +385 00:26:52,820 --> 00:26:54,320 Well, about half of them. -383 +386 00:26:54,840 --> 00:26:58,520 So that would account for about r squared over 2 times chi of 2. -384 +387 00:26:59,120 --> 00:27:02,120 About a third of these rows have chi of 3, so we -385 +388 00:27:02,120 --> 00:27:05,120 can put in r squared divided by 3 times chi of 3. -386 +389 00:27:06,020 --> 00:27:10,120 And keep in mind we're being approximate since r squared might not perfectly -387 +390 00:27:10,120 --> 00:27:14,540 divide 2 or 3, but as r grows towards infinity, this approximation will get better. -388 +391 00:27:15,360 --> 00:27:17,607 And when you keep going like this, you get a pretty -389 +392 00:27:17,607 --> 00:27:20,200 organized expression for the total number of lattice points. -390 +393 00:27:22,980 --> 00:27:27,919 And if you factor out that r squared and then bring back the 4 that needs -391 +394 00:27:27,919 --> 00:27:33,126 to be multiplied in, what it means is that the total number of lattice points -392 +395 00:27:33,126 --> 00:27:38,000 inside this big circle is approximately 4 times r squared times this sum. -393 +396 00:27:38,660 --> 00:27:45,708 And because chi is 0 on every even number, and it oscillates between 1 and negative 1 for -394 +397 00:27:45,708 --> 00:27:52,600 odd numbers, this sum looks like 1 minus 1 third plus a fifth minus 1 seventh and so on. -395 +398 00:27:53,020 --> 00:27:54,360 And this is exactly what we wanted. -396 +399 00:27:54,760 --> 00:27:58,883 What we have here is an alternate expression for the total number of lattice -397 +400 00:27:58,883 --> 00:28:03,060 points inside a big circle, which we know should be around pi times r squared. -398 +401 00:28:04,000 --> 00:28:07,826 And the bigger r is, the more accurate both of these estimates are, -399 +402 00:28:07,826 --> 00:28:12,327 so the percent error between the left-hand side and the right-hand side can get -400 +403 00:28:12,327 --> 00:28:13,340 arbitrarily small. -401 +404 00:28:13,900 --> 00:28:17,086 So divide out by that r squared, and this gives -402 +405 00:28:17,086 --> 00:28:20,140 us an infinite sum that should converge to pi. -403 +406 00:28:21,300 --> 00:28:23,160 And keep in mind, I just think this is really cool. -404 +407 00:28:23,420 --> 00:28:26,424 The reason that this sum came out to be so simple, -405 +408 00:28:26,424 --> 00:28:29,369 requiring relatively low information to describe, -406 +409 00:28:29,369 --> 00:28:33,139 ultimately stems from the regular pattern and how prime numbers -407 +410 00:28:33,139 --> 00:28:35,260 factor inside the Gaussian integers. -408 +411 00:28:36,540 --> 00:28:40,496 If you're curious, there are two main branches of number theory, -409 +412 00:28:40,496 --> 00:28:43,600 algebraic number theory and analytic number theory. -410 +413 00:28:44,420 --> 00:28:47,825 Very loosely speaking, the former deals with new number systems, -411 +414 00:28:47,825 --> 00:28:51,860 things like these Gaussian integers that you and I looked at, and a lot more. -412 +415 00:28:52,280 --> 00:28:55,728 And the latter deals with things like the Riemann zeta function, -413 +416 00:28:55,728 --> 00:28:59,176 or its cousins, called L-functions, which involve multiplicative -414 +417 00:28:59,176 --> 00:29:02,200 functions like this central character chi from our story. -415 +418 00:29:02,780 --> 00:29:07,300 And the path that we just walked is a little glimpse at where those two fields intersect. -416 +419 00:29:07,920 --> 00:29:10,046 And both of these are pretty heavy-duty fields -417 +420 00:29:10,046 --> 00:29:12,400 with a lot of active research and unsolved problems. -418 +421 00:29:13,000 --> 00:29:17,058 So if all this feels like something that takes time to mentally digest, -419 +422 00:29:17,058 --> 00:29:21,510 like there's more patterns to be uncovered and understood, it's because it is, -420 +423 00:29:21,510 --> 00:29:22,300 and there are. diff --git a/2017/leibniz-formula/english/sentence_timings.json b/2017/leibniz-formula/english/sentence_timings.json index 70f575e02..966ac86ee 100644 --- a/2017/leibniz-formula/english/sentence_timings.json +++ b/2017/leibniz-formula/english/sentence_timings.json @@ -130,7 +130,7 @@ 251.64 ], [ - "Radius square root of 2, well that also hits 4 lattice points.", + "Radius square root of 2, well that also hits 4 lattice points. Radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points.", 252.44, 255.98 ], @@ -1000,7 +1000,7 @@ 1599.04 ], [ - "How many numbers between 1 and r squared have 1 as a divisor?", + "How many numbers between 1 and r2 have 1 as a divisor? All of them. So our sum should include r2 times chi of 1. How many of them have 2 as a divisor?", 1600.1, 1603.92 ], diff --git a/2017/leibniz-formula/english/transcript.txt b/2017/leibniz-formula/english/transcript.txt index f46d2a79f..53776810d 100644 --- a/2017/leibniz-formula/english/transcript.txt +++ b/2017/leibniz-formula/english/transcript.txt @@ -24,7 +24,7 @@ If you think about it, for each one of these lattice points AB, its distance fro And since a and b are both integers, a squared plus b squared is also some integer, so you only have to consider rings whose radii are the square roots of some whole number. A radius of 0 just gives you that single origin point. If you look at the radius 1, that hits 4 different lattice points. -Radius square root of 2, well that also hits 4 lattice points. +Radius square root of 2, well that also hits 4 lattice points. Radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice points. @@ -198,7 +198,7 @@ At first, adding up the values for each one of these rows seems super random, ri I mean, numbers with a lot of factors have a lot of divisors, whereas prime numbers will always only have two divisors. So it initially seems like you would have to have perfect knowledge of the distribution of primes to get anything useful out of this. But if instead you organize these into columns, the puzzle starts to fit together. -How many numbers between 1 and r squared have 1 as a divisor? +How many numbers between 1 and r2 have 1 as a divisor? All of them. So our sum should include r2 times chi of 1. How many of them have 2 as a divisor? Well, all of them. So our sum should include r squared times chi of 1. How many of them have 2 as a divisor? diff --git a/2017/leibniz-formula/french/sentence_translations.json b/2017/leibniz-formula/french/sentence_translations.json index ce310e9c2..14a660763 100644 --- a/2017/leibniz-formula/french/sentence_translations.json +++ b/2017/leibniz-formula/french/sentence_translations.json @@ -261,7 +261,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "Une racine carrée de rayon de 5 touche en fait 8 points de treillis.", "model": "DeepL", "from_community_srt": "Un rayon de √5 en touche en fait 8.", @@ -1796,7 +1796,7 @@ "end": 1599.04 }, { - "input": "How many numbers between 1 and r squared have 1 as a divisor?", + "input": "How many numbers between 1 and r2 have 1 as a divisor? All of them. So our sum should include r2 times chi of 1. How many of them have 2 as a divisor?", "translatedText": "Combien de nombres compris entre 1 et r au carré ont 1 comme diviseur ?", "model": "DeepL", "from_community_srt": "Combien de numéros entre 1 et R2 ont 1 un diviseur;", diff --git a/2017/leibniz-formula/german/sentence_translations.json b/2017/leibniz-formula/german/sentence_translations.json index 99c93a13c..db4244c82 100644 --- a/2017/leibniz-formula/german/sentence_translations.json +++ b/2017/leibniz-formula/german/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points.", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points.", "translatedText": "Wenn man sich den Radius 1 anschaut, trifft das auf 4 Gitterpunkte.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "Eine Radius-Quadratwurzel von 5 trifft tatsächlich auf 8 Gitterpunkte.", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", "translatedText": "Stellen Sie sich diesen Gitterpunkt hier also nicht als das ganzzahlige Koordinatenpaar 3,4 vor, sondern als die einzelne komplexe Zahl 3 plus 4i.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 397.06 }, { - "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers will come into play.", + "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers are going to come into play.", "translatedText": "Dadurch wird unsere Frage zu einem Faktorisierungsproblem, das letztendlich der Grund dafür ist, dass Muster zwischen Primzahlen ins Spiel kommen.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes.", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes.", "translatedText": "Primzahlen, die eins über einem Vielfachen von 4 liegen, wie 5, 13 oder 17, können immer in genau zwei verschiedene Gaußsche Primzahlen zerlegt werden.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers.", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers.", "translatedText": "Andererseits können Primzahlen, die 3 über einem Vielfachen von 4 liegen, wie 3, 7 oder 11, nicht weiter in die Gaußschen ganzen Zahlen zerlegt werden.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25.", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25.", "translatedText": "Da alles auf der rechten Seite mit allem auf der linken Seite konjugiert ist, entsteht ein komplexes konjugiertes Paar, das sich mit 25 multipliziert.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i?", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i?", "translatedText": "Erinnern Sie sich, wie ich erwähnte, dass eine Faktorisierung in Gaußsche Primzahlen anders aussehen kann, wenn Sie einige davon mit i, oder –1, oder –i multiplizieren?", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i.", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i.", "translatedText": "In diesem Fall könnten Sie die Faktorisierung von 25 anders schreiben und vielleicht eine dieser 5 als –1 plus 2i mal –1 minus 2i aufteilen.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i.", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i.", "translatedText": "Der einzige Effekt, den dies hat, ist die Multiplikation der Gesamtleistung mit i, oder –1, oder –i.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees.", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees.", "translatedText": "Nehmen Sie das Produkt aus der linken Spalte und multiplizieren Sie es mit 1, i, –1 oder –i, was Drehungen entspricht, die ein Vielfaches von 90 Grad sind.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", "translatedText": "Diese vier Möglichkeiten, multipliziert mit den letzten vier Möglichkeiten, das Produkt aus der linken Spalte mit 1, i, –1 oder –i zu multiplizieren, würden darauf hindeuten, dass es insgesamt 16 Gitterpunkte mit einem Abstand der Quadratwurzel von 125 von gibt Herkunft.", "model": "google_nmt", "n_reviews": 0, @@ -1248,7 +1248,7 @@ "end": 1374.96 }, { - "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, 1 minus 1 plus 1 minus 1 plus 1.", + "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, it goes 1 minus 1 plus 1 minus 1 plus 1.", "translatedText": "Aber in diesem Fall schwankt diese Summe, da Chi von 3 minus 1 ist, 1 minus 1 plus 1 minus 1 plus 1.", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", "translatedText": "Wenn es eine gerade Potenz ist, wie in diesem Fall 4, ergibt sich als Summe 1, was die Tatsache zum Ausdruck bringt, dass es nur eine Wahl gibt, was mit diesen unteilbaren Dreien geschehen soll.", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point.", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point.", "translatedText": "Wir nähern uns jetzt dem Höhepunkt, die Dinge scheinen organisiert zu sein, also halten Sie inne und denken Sie nach, stellen Sie sicher, dass sich bis zu diesem Punkt alles gut anfühlt.", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", "translatedText": "Ignorieren wir diesen Ursprungspunkt mit dem Radius 0, er folgt nicht dem Muster der anderen, und ein kleiner Punkt wird keinen Unterschied machen, wenn wir r in Richtung Unendlich wachsen lassen.", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4.", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4.", "translatedText": "Nach all dem Gaußschen Ganzzahl- und Faktorisierungs- und Chi-Funktions-Zeug, das wir gemacht haben, sieht die Antwort für jedes n so aus, als würde man den Wert von chi auf jedem Teiler von n addieren und mit 4 multiplizieren.", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3.", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2.", "translatedText": "Ungefähr ein Drittel dieser Reihen hat Chi von 3, also können wir r2 geteilt durch 3 mal Chi von 3 einsetzen.", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better.", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will", "translatedText": "Bedenken Sie, dass es sich um Näherungswerte handelt, da r2 2 oder 3 möglicherweise nicht perfekt teilt, aber wenn r in Richtung Unendlich wächst, wird diese Näherung besser.", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", + "input": "get better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", "translatedText": "Und wenn Sie so weitermachen, erhalten Sie einen ziemlich organisierten Ausdruck für die Gesamtzahl der Gitterpunkte.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum.", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum.", "translatedText": "Und wenn man r2 herausrechnet und die 4 zurückbringt, die multipliziert werden muss, bedeutet das, dass die Gesamtzahl der Gitterpunkte innerhalb dieses großen Kreises ungefähr 4 mal r2 mal diese Summe beträgt.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/hebrew/sentence_translations.json b/2017/leibniz-formula/hebrew/sentence_translations.json index 19a68285e..97d8b5614 100644 --- a/2017/leibniz-formula/hebrew/sentence_translations.json +++ b/2017/leibniz-formula/hebrew/sentence_translations.json @@ -182,7 +182,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points.", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points.", "translatedText": "אם אתה מסתכל על רדיוס 1, זה פוגע ב-4 נקודות סריג.", "n_reviews": 0, "start": 248.22, @@ -210,7 +210,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "שורש רדיוס ריבועי של 5 פוגע למעשה ב-8 נקודות סריג.", "n_reviews": 0, "start": 262.84, @@ -315,7 +315,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", "translatedText": "אז במקום נקודת הסריג הזו כאן בתור צמד הקואורדינטות של מספרים שלמים, 3,4, במקום זאת חשבו עליה כעל המספר המרוכב היחיד, 3 ועוד 4i.", "n_reviews": 0, "start": 360.4, @@ -343,7 +343,7 @@ "end": 397.06 }, { - "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers will come into play.", + "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers are going to come into play.", "translatedText": "זה הופך את השאלה שלנו לבעיית פקטורינג, וזו בסופו של דבר הסיבה שדפוסים בין מספרים ראשוניים יכנסו לתמונה.", "n_reviews": 0, "start": 397.78, @@ -497,7 +497,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes.", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes.", "translatedText": "מספרים ראשוניים שהם אחד מעל מכפילה של 4, כמו 5, או 13, או 17, תמיד יכולים להיות מוערכים לשני ראשוני גאוס נפרדים בדיוק.", "n_reviews": 0, "start": 635.96, @@ -518,7 +518,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers.", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers.", "translatedText": "מצד שני, מספרים ראשוניים שנמצאים 3 מעל מכפילה של 4, כמו 3, או 7 או 11, לא ניתנים לגורם נוסף בתוך המספרים השלמים של גאוס.", "n_reviews": 0, "start": 663.44, @@ -616,7 +616,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25.", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25.", "translatedText": "מכיוון שכל דבר מימין הוא צמוד עם כל דבר משמאל, מה שיוצא זה זוג מצומד מורכב שמכפיל ל-25.", "n_reviews": 0, "start": 806.44, @@ -672,14 +672,14 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i?", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i?", "translatedText": "זוכר איך ציינתי שפירוק לגורמים ראשוניים גאוסים יכול להיראות אחרת אם מכפילים חלק מהם ב-i, או –1, או –i?", "n_reviews": 0, "start": 870.18, "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i.", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i.", "translatedText": "במקרה זה, אתה יכול לכתוב את הפירוק לגורמים של 25 אחרת, אולי לפצל את אחת מה-5s האלה כ-1 פלוס 2i כפול -1 פחות 2i.", "n_reviews": 0, "start": 878.88, @@ -693,7 +693,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i.", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i.", "translatedText": "ההשפעה היחידה שתהיה לכך היא הכפלת הפלט הכולל ב-i, או -1, או -i.", "n_reviews": 0, "start": 896.0, @@ -707,7 +707,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees.", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees.", "translatedText": "קח את המוצר הזה מהעמודה השמאלית ובחר להכפיל אותו ב-1, i, -1 או -i, בהתאמה לסיבובים שהם כפולה כלשהי של 90 מעלות.", "n_reviews": 0, "start": 908.8, @@ -749,7 +749,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", "translatedText": "ארבע האפשרויות הללו, מוכפלות בארבע האפשרויות האחרונות של הכפלת המכפלה מהעמודה השמאלית ב-1, i, –1, או –i, רומזות שיש סה"כ 16 נקודות סריג במרחק שורש ריבועי של 125 מה- מָקוֹר.", "n_reviews": 0, "start": 959.66, @@ -1092,14 +1092,14 @@ "end": 1374.96 }, { - "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, 1 minus 1 plus 1 minus 1 plus 1.", + "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, it goes 1 minus 1 plus 1 minus 1 plus 1.", "translatedText": "אבל במקרה זה, מכיוון שצ'י של 3 הוא 1 שלילי, הסכום הזה מתנודד, 1 מינוס 1 ועוד 1 מינוס 1 ועוד 1.", "n_reviews": 0, "start": 1375.04, "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", "translatedText": "אם מדובר בחזק זוגי, כמו 4 במקרה הזה, הסכום יוצא כ-1, מה שמכיל את העובדה שיש רק בחירה אחת מה לעשות עם ה-3 הבלתי ניתנים לחלוקה.", "n_reviews": 0, "start": 1384.42, @@ -1134,7 +1134,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point.", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point.", "translatedText": "אנחנו מתקרבים לשיא עכשיו, הדברים מתחילים להיראות מאורגנים, אז עצרו ותחשבו, וודאו שהכל מרגיש טוב עד לנקודה זו.", "n_reviews": 0, "start": 1429.08, @@ -1232,14 +1232,14 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", "translatedText": "בוא נתעלם מנקודת המוצא הזו עם רדיוס 0, היא לא עוקבת אחר הדפוס של השאר, ונקודה קטנה אחת לא תעשה הבדל כשאנחנו נותנים ל-r לצמוח לעבר אינסוף.", "n_reviews": 0, "start": 1541.28, "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4.", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4.", "translatedText": "מכל הדברים האלה של מספר שלם גאוס ופירוק ופונקציות צ'י שעשינו, התשובה עבור כל n נראית כמו חיבור הערך של צ'י על כל מחלק של n, והכפלה ב-4.", "n_reviews": 0, "start": 1552.2, @@ -1316,28 +1316,28 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3.", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2.", "translatedText": "בערך שליש מהשורות האלה יש צ'י של 3, אז אנחנו יכולים להכניס את r2 חלקי פי 3 צ'י של 3.", "n_reviews": 0, "start": 1619.12, "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better.", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will", "translatedText": "זכור שאנו משוערים, מכיוון ש-r2 אולי לא מחלק בצורה מושלמת 2 או 3, אבל ככל ש-r יגדל לקראת אינסוף, הקירוב הזה ישתפר.", "n_reviews": 0, "start": 1626.02, "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", + "input": "get better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", "translatedText": "וכשאתה ממשיך כך, אתה מקבל ביטוי די מאורגן עבור המספר הכולל של נקודות הסריג.", "n_reviews": 0, "start": 1635.36, "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum.", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum.", "translatedText": "ואם תפרק את ה-r2 הזה ותחזיר את ה-4 שצריך להכפיל, המשמעות היא שהמספר הכולל של נקודות הסריג בתוך המעגל הגדול הזה הוא בערך פי 4 כפול r2 כפול הסכום הזה.", "n_reviews": 0, "start": 1642.98, diff --git a/2017/leibniz-formula/hindi/sentence_translations.json b/2017/leibniz-formula/hindi/sentence_translations.json index 43d737a9b..7d7868ca3 100644 --- a/2017/leibniz-formula/hindi/sentence_translations.json +++ b/2017/leibniz-formula/hindi/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points.", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points.", "translatedText": "यदि आप त्रिज्या 1 को देखें, तो वह 4 जाली बिंदुओं से टकराता है।", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "5 का त्रिज्या वर्गमूल वास्तव में 8 जालक बिंदुओं से टकराता है।", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", "translatedText": "तो इस जाली बिंदु के बजाय पूर्णांक निर्देशांक की जोड़ी के रूप में, 3,4, इसके बजाय इसे एकल जटिल संख्या, 3 प्लस 4i के रूप में सोचें।", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 397.06 }, { - "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers will come into play.", + "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers are going to come into play.", "translatedText": "यह हमारे प्रश्न को फैक्टरिंग समस्या में बदल देता है, जिसके कारण अंततः अभाज्य संख्याओं के बीच पैटर्न चलन में आएगा।", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes.", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes.", "translatedText": "अभाज्य संख्याएँ जो 4 के गुणज से ऊपर एक हैं, जैसे 5, या 13, या 17, को हमेशा दो अलग-अलग गॉसियन अभाज्य संख्याओं में विभाजित किया जा सकता है।", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers.", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers.", "translatedText": "दूसरी ओर, अभाज्य संख्याएँ जो 4 के गुणज से 3 ऊपर हैं, जैसे 3, या 7, या 11, को गाऊसी पूर्णांक के अंदर आगे गुणनखंडित नहीं किया जा सकता है।", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25.", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25.", "translatedText": "चूँकि दाहिनी ओर की हर चीज़ बायीं ओर की हर चीज़ के साथ संयुग्मित है, जो सामने आता है वह एक जटिल संयुग्मी युग्म है जो 25 से गुणा हो जाता है।", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i?", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i?", "translatedText": "याद रखें कि मैंने कैसे उल्लेख किया था कि यदि आप उनमें से कुछ को i, या -1, या -i से गुणा करते हैं तो गाऊसी अभाज्य संख्याओं का गुणनखंड अलग दिख सकता है?", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i.", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i.", "translatedText": "इस मामले में, आप 25 के गुणनखंडन को अलग-अलग तरीके से लिख सकते हैं, हो सकता है कि उन 5 में से किसी एक को -1 प्लस 2i गुना -1 माइनस 2i के रूप में विभाजित किया जाए।", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i.", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i.", "translatedText": "इसका एकमात्र प्रभाव कुल आउटपुट को i, या -1, या -i से गुणा करना होगा।", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees.", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees.", "translatedText": "उस उत्पाद को बाएं कॉलम से लें, और इसे 1, i, -1, या -i से गुणा करना चुनें, जो कि 90 डिग्री के कुछ गुणज घुमावों के अनुरूप है।", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", "translatedText": "उन चार विकल्पों को, बाएं कॉलम से उत्पाद को 1, i, -1, या -i से गुणा करने के अंतिम चार विकल्पों से गुणा करने पर पता चलता है कि कुल 16 जाली बिंदु हैं जो 125 की दूरी के वर्गमूल से दूर हैं। मूल।", "model": "google_nmt", "n_reviews": 0, @@ -1248,7 +1248,7 @@ "end": 1374.96 }, { - "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, 1 minus 1 plus 1 minus 1 plus 1.", + "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, it goes 1 minus 1 plus 1 minus 1 plus 1.", "translatedText": "लेकिन इस मामले में, चूँकि 3 की ची ऋणात्मक 1 है, यह योग दोलन करता है, 1 घटा 1 जमा 1 घटा 1 जमा 1।", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", "translatedText": "यदि यह एक सम घात है, जैसे इस मामले में 4, तो योग 1 आता है, जो इस तथ्य को दर्शाता है कि उन अविभाजित 3 के साथ क्या करना है इसके लिए केवल एक ही विकल्प है।", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point.", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point.", "translatedText": "हम अब चरमोत्कर्ष के करीब पहुंच रहे हैं, चीजें व्यवस्थित दिखने लगी हैं, इसलिए रुकें और विचार करें, सुनिश्चित करें कि इस बिंदु तक सब कुछ अच्छा लगे।", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", "translatedText": "आइए त्रिज्या 0 वाले मूल बिंदु को नजरअंदाज करें, यह बाकी के पैटर्न का पालन नहीं करता है, और एक छोटे बिंदु से कोई फर्क नहीं पड़ने वाला है क्योंकि हम r को अनंत की ओर बढ़ने देते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4.", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4.", "translatedText": "इस सभी गॉसियन पूर्णांक और फैक्टरिंग और ची फ़ंक्शन सामग्री से जो हम कर रहे हैं, प्रत्येक एन का उत्तर एन के प्रत्येक विभाजक पर ची का मान जोड़ने और 4 से गुणा करने जैसा दिखता है।", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3.", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2.", "translatedText": "इनमें से लगभग एक तिहाई पंक्तियों में 3 की ची है, इसलिए हम 3 की 3 गुना ची से विभाजित आर2 डाल सकते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better.", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will", "translatedText": "ध्यान रखें कि हम अनुमानित हैं, क्योंकि r2 2 या 3 को पूरी तरह से विभाजित नहीं कर सकता है, लेकिन जैसे-जैसे r अनंत की ओर बढ़ता है, यह अनुमान बेहतर होता जाएगा।", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", + "input": "get better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", "translatedText": "और जब आप इस तरह से चलते रहते हैं, तो आपको जाली बिंदुओं की कुल संख्या के लिए एक बहुत ही व्यवस्थित अभिव्यक्ति मिलती है।", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum.", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum.", "translatedText": "और यदि आप उस r2 का गुणनखंड करते हैं और उस 4 को वापस लाते हैं जिसे गुणा करने की आवश्यकता है, तो इसका मतलब यह है कि इस बड़े वृत्त के अंदर जाली बिंदुओं की कुल संख्या इस योग का लगभग 4 गुना r2 गुना है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/hungarian/sentence_translations.json b/2017/leibniz-formula/hungarian/sentence_translations.json index 0699c8869..7392dcec4 100644 --- a/2017/leibniz-formula/hungarian/sentence_translations.json +++ b/2017/leibniz-formula/hungarian/sentence_translations.json @@ -232,7 +232,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "Az 5 sugarú négyzetgyök valójában 8 rácspontot érint.", "model": "DeepL", "n_reviews": 0, @@ -1600,7 +1600,7 @@ "end": 1599.04 }, { - "input": "How many numbers between 1 and r squared have 1 as a divisor?", + "input": "How many numbers between 1 and r2 have 1 as a divisor? All of them. So our sum should include r2 times chi of 1. How many of them have 2 as a divisor?", "translatedText": "Hány 1 és r négyzete közötti szám osztója az 1?", "model": "DeepL", "n_reviews": 0, diff --git a/2017/leibniz-formula/indonesian/sentence_translations.json b/2017/leibniz-formula/indonesian/sentence_translations.json index c030bdbda..767800306 100644 --- a/2017/leibniz-formula/indonesian/sentence_translations.json +++ b/2017/leibniz-formula/indonesian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points.", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points.", "translatedText": "Jika dilihat radiusnya 1, maka mengenai 4 titik kisi.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "Akar kuadrat radius 5 sebenarnya mencapai 8 titik kisi.", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", "translatedText": "Jadi, alih-alih titik kisi di sini sebagai pasangan koordinat bilangan bulat, 3,4, anggap saja sebagai bilangan kompleks tunggal, 3 ditambah 4i.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 397.06 }, { - "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers will come into play.", + "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers are going to come into play.", "translatedText": "Hal ini mengubah pertanyaan kita menjadi masalah pemfaktoran, yang pada akhirnya menjadi alasan mengapa pola di antara bilangan prima akan ikut berperan.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes.", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes.", "translatedText": "Bilangan prima yang satu di atas kelipatan 4, seperti 5, atau 13, atau 17, selalu dapat difaktorkan menjadi dua bilangan prima Gaussian yang berbeda.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers.", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers.", "translatedText": "Sebaliknya, bilangan prima yang merupakan 3 di atas kelipatan 4, seperti 3, atau 7, atau 11, tidak dapat difaktorkan lebih jauh ke dalam bilangan bulat Gaussian.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25.", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25.", "translatedText": "Karena semua yang di sebelah kanan adalah konjugasi dengan semua yang di sebelah kiri, maka yang dihasilkan adalah pasangan konjugasi kompleks yang dikalikan dengan 25.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i?", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i?", "translatedText": "Ingat bagaimana saya menyebutkan bahwa faktorisasi menjadi bilangan prima Gaussian bisa terlihat berbeda jika Anda mengalikan beberapa di antaranya dengan i, atau –1, atau –i?", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i.", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i.", "translatedText": "Dalam hal ini, Anda dapat menulis faktorisasi 25 secara berbeda, mungkin membagi salah satu dari 5 tersebut menjadi –1 ditambah 2i dikalikan –1 dikurangi 2i.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i.", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i.", "translatedText": "Satu-satunya dampak yang ditimbulkan adalah mengalikan total output dengan i, atau –1, atau –i.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees.", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees.", "translatedText": "Ambil hasil kali tersebut dari kolom kiri, dan pilih untuk mengalikannya dengan 1, i, –1, atau –i, sesuai dengan rotasi yang merupakan kelipatan 90 derajat.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", "translatedText": "Keempat pilihan tersebut, dikalikan dengan empat pilihan terakhir yaitu mengalikan hasil kali dari kolom kiri dengan 1, i, –1, atau –i, akan menunjukkan bahwa terdapat total 16 titik kisi dengan jarak akar kuadrat 125 dari titik kisi. asal.", "model": "google_nmt", "n_reviews": 0, @@ -1248,7 +1248,7 @@ "end": 1374.96 }, { - "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, 1 minus 1 plus 1 minus 1 plus 1.", + "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, it goes 1 minus 1 plus 1 minus 1 plus 1.", "translatedText": "Namun dalam kasus ini, karena chi dari 3 adalah negatif 1, jumlah ini berosilasi, 1 dikurangi 1 ditambah 1 dikurangi 1 ditambah 1.", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", "translatedText": "Jika pangkatnya genap, seperti 4 dalam kasus ini, hasilnya adalah 1, yang merangkum fakta bahwa hanya ada satu pilihan tentang apa yang harus dilakukan dengan angka 3 yang tidak dapat dipisahkan tersebut.", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point.", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point.", "translatedText": "Kita sudah mendekati puncaknya sekarang, segalanya mulai terlihat terorganisir, jadi berhentilah sejenak dan renungkan, pastikan semuanya terasa baik hingga saat ini.", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", "translatedText": "Mari kita abaikan titik asal dengan radius 0, titik tersebut tidak mengikuti pola titik lainnya, dan satu titik kecil tidak akan membuat perbedaan saat kita membiarkan r tumbuh menuju tak terhingga.", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4.", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4.", "translatedText": "Dari semua bilangan bulat Gaussian dan pemfaktoran serta fungsi chi yang telah kita lakukan, jawaban untuk setiap n terlihat seperti menjumlahkan nilai chi pada setiap pembagi n, dan mengalikannya dengan 4.", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3.", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2.", "translatedText": "Sekitar sepertiga baris ini mempunyai chi sebesar 3, sehingga kita dapat memasukkan r2 dibagi 3 kali chi dari 3.", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better.", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will", "translatedText": "Ingatlah bahwa kita sedang melakukan perkiraan, karena r2 mungkin tidak dapat membagi 2 atau 3 dengan sempurna, namun seiring dengan bertambahnya r menuju tak terhingga, perkiraan ini akan menjadi lebih baik.", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", + "input": "get better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", "translatedText": "Dan jika Anda melanjutkan seperti ini, Anda mendapatkan ekspresi yang cukup terorganisir untuk jumlah total titik kisi.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum.", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum.", "translatedText": "Dan jika Anda memfaktorkan r2 tersebut dan mengembalikan angka 4 yang perlu dikalikan, artinya jumlah titik kisi di dalam lingkaran besar ini kira-kira 4 kali r2 dikalikan jumlah tersebut.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/italian/sentence_translations.json b/2017/leibniz-formula/italian/sentence_translations.json index 5ed21c712..5f5f75b9c 100644 --- a/2017/leibniz-formula/italian/sentence_translations.json +++ b/2017/leibniz-formula/italian/sentence_translations.json @@ -182,7 +182,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points.", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points.", "translatedText": "Se guardi il raggio 1, colpisce 4 punti del reticolo.", "n_reviews": 0, "start": 248.22, @@ -210,7 +210,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "Una radice quadrata con raggio di 5 colpisce effettivamente 8 punti del reticolo.", "n_reviews": 0, "start": 262.84, @@ -315,7 +315,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", "translatedText": "Quindi invece di questo punto del reticolo qui come la coppia di coordinate intere, 3,4, pensalo invece come il singolo numero complesso, 3 più 4i.", "n_reviews": 0, "start": 360.4, @@ -343,7 +343,7 @@ "end": 397.06 }, { - "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers will come into play.", + "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers are going to come into play.", "translatedText": "Trasforma la nostra domanda in un problema di fattorizzazione, che è in definitiva il motivo per cui entreranno in gioco i modelli tra i numeri primi.", "n_reviews": 0, "start": 397.78, @@ -497,7 +497,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes.", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes.", "translatedText": "I numeri primi che sono uno sopra un multiplo di 4, come 5, o 13, o 17, possono sempre essere scomposti esattamente in due numeri primi gaussiani distinti.", "n_reviews": 0, "start": 635.96, @@ -518,7 +518,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers.", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers.", "translatedText": "D'altra parte, i numeri primi che sono 3 sopra un multiplo di 4, come 3, o 7, o 11, non possono essere ulteriormente scomposti all'interno degli interi gaussiani.", "n_reviews": 0, "start": 663.44, @@ -616,7 +616,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25.", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25.", "translatedText": "Poiché tutto a destra è un coniugato con tutto a sinistra, ciò che ne risulta è una coppia coniugata complessa che si moltiplica fino a 25.", "n_reviews": 0, "start": 806.44, @@ -672,14 +672,14 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i?", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i?", "translatedText": "Ricordi come ho detto che una fattorizzazione in numeri primi gaussiani può apparire diversa se ne moltiplichi alcuni per i, o –1, o –i?", "n_reviews": 0, "start": 870.18, "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i.", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i.", "translatedText": "In questo caso, potresti scrivere la fattorizzazione di 25 in modo diverso, magari dividendo uno di questi 5 come –1 più 2i volte –1 meno 2i.", "n_reviews": 0, "start": 878.88, @@ -693,7 +693,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i.", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i.", "translatedText": "L’unico effetto che ciò avrà sarà quello di moltiplicare la produzione totale per i, o –1, o –i.", "n_reviews": 0, "start": 896.0, @@ -707,7 +707,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees.", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees.", "translatedText": "Prendi il prodotto dalla colonna di sinistra e scegli di moltiplicarlo per 1, i, –1 o –i, corrispondenti a rotazioni che sono multipli di 90 gradi.", "n_reviews": 0, "start": 908.8, @@ -749,7 +749,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", "translatedText": "Queste quattro scelte, moltiplicate per le ultime quattro scelte di moltiplicare il prodotto della colonna di sinistra per 1, i, –1 o –i, suggerirebbero che ci sono un totale di 16 punti del reticolo a una distanza radice quadrata di 125 dal punto origine.", "n_reviews": 0, "start": 959.66, @@ -1092,14 +1092,14 @@ "end": 1374.96 }, { - "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, 1 minus 1 plus 1 minus 1 plus 1.", + "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, it goes 1 minus 1 plus 1 minus 1 plus 1.", "translatedText": "Ma in questo caso, poiché chi di 3 è negativo 1, questa somma oscilla, 1 meno 1 più 1 meno 1 più 1.", "n_reviews": 0, "start": 1375.04, "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", "translatedText": "Se è una potenza pari, come 4 in questo caso, la somma risulta essere 1, il che incapsula il fatto che c'è solo una scelta su cosa fare con quei 3 indivisibili.", "n_reviews": 0, "start": 1384.42, @@ -1134,7 +1134,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point.", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point.", "translatedText": "Ci stiamo avvicinando al culmine ora, le cose iniziano a sembrare organizzate, quindi fermati e riflette, assicurati che tutto vada bene fino a questo punto.", "n_reviews": 0, "start": 1429.08, @@ -1232,14 +1232,14 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", "translatedText": "Ignoriamo quel punto di origine con raggio 0, non segue lo schema del resto, e un piccolo punto non farà la differenza mentre lasciamo che r cresca verso l'infinito.", "n_reviews": 0, "start": 1541.28, "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4.", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4.", "translatedText": "Da tutto questo intero gaussiano, fattorizzazione e funzione chi che abbiamo fatto, la risposta per ogni n sembra sommare il valore di chi su ogni divisore di n e moltiplicarlo per 4.", "n_reviews": 0, "start": 1552.2, @@ -1316,28 +1316,28 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3.", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2.", "translatedText": "Circa un terzo di queste righe hanno chi pari a 3, quindi possiamo inserire r2 diviso per 3 volte chi pari a 3.", "n_reviews": 0, "start": 1619.12, "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better.", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will", "translatedText": "Tieni presente che siamo approssimativi, poiché r2 potrebbe non dividere perfettamente 2 o 3, ma man mano che r cresce verso l'infinito, questa approssimazione migliorerà.", "n_reviews": 0, "start": 1626.02, "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", + "input": "get better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", "translatedText": "E continuando così, ottieni un'espressione abbastanza organizzata per il numero totale di punti del reticolo.", "n_reviews": 0, "start": 1635.36, "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum.", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum.", "translatedText": "E se estraiamo r2 e riportiamo il 4 che deve essere moltiplicato, significa che il numero totale di punti del reticolo all'interno di questo grande cerchio è circa 4 volte r2 per questa somma.", "n_reviews": 0, "start": 1642.98, diff --git a/2017/leibniz-formula/japanese/sentence_translations.json b/2017/leibniz-formula/japanese/sentence_translations.json index 9880581b0..1fd0f1106 100644 --- a/2017/leibniz-formula/japanese/sentence_translations.json +++ b/2017/leibniz-formula/japanese/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points. ", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points. ", "translatedText": "半径 1 を見ると、4 つの格子点に当たります。", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points. ", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice poi ", "translatedText": "半径の平方根 5 は、実際には 8 つの格 子点に当たります。", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", "translatedText": "したがって、この格子点を整数座標のペア 3, 4 として表すのではなく、単一の複素数 3 プラス 4i として考えて ください。", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes. ", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes. ", "translatedText": "5、13、1 7 など、4 の倍数の 1 つ上の素数は、常に 2 つの異なるガ ウス素数に因数分解できます。", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers. ", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers. ", "translatedText": "一方、3、7、11 など、4 の倍数より 3 つ大きい素数 は、ガウス整数内でさらに因数分解できません。", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25. ", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25. ", "translatedText": "右側のすべては左側のすべてと共役であるため 、結果として得られるのは、乗算すると 25 になる複素共役ペアになり ます。", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i? ", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i? ", "translatedText": "ガウス素数への因数分解は、一部の素数に i、-1、または -i を掛けると異なって見える可能性があると述べたことを覚えていますか? ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i. ", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i. ", "translatedText": "この場合、25 の因数分解を別の方法で記述し、これらの 5 の 1 つを –1 プラス 2i 倍 –1 マイナス 2i として分割することもできます。 ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i. ", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i. ", "translatedText": "これによる唯一の影響は、合計出力に i、または –1、または –i を乗算することです。", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees. ", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees. ", "translatedText": "左の列からその積を取得し、90 度の倍数の回転に対応する 1、i、-1、または -i を乗算することを選択します。", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", "translatedText": "これら 4 つの選択肢に、左列の積に 1、i、-1、 または -i を掛ける最後の 4 つの選択肢を掛け合わせると、距 離平方根 125 の距離に合計 16 個の格子点があることがわか ります。起源。", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", "translatedText": "この場合の 4 のように、偶数乗の場合、合計は 1 になります。これは、分割できない 3 をどうするかについて選択肢が 1 つしかないという事実を要約しています。", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point. ", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point. ", "translatedText": "私たちは今、最 高潮に近づいています。物事が整理され始めているように見えます。そこで、立ち止 まって熟考し、この時点までのすべてが順調に進んでいることを確認してください。", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", "translatedText": "半径 0 の原点のドットは無視しましょう。これは 残りのパターンに従いません。また、r を無限に向かって 成長させると、1 つの小さなドットは違いを生みません。", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4. ", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4. ", "translatedText": "私たちがこれまで行ってきたガウス整数、因数分解、カイ関数の処理から、 各 n の答えは、n の約数ごとにカイの値を加算し、4 を掛けるよう に見えます。", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3. ", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2. ", "translatedText": "これらの行の約 3 分の 1 にはカイ 3 が含まれているため、r 2 を 3 のカイの 3 倍で割った値を入力できます。", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better. ", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will g ", "translatedText": "r2 が 2 または 3 を完全に除算できない可能性が あるため、近似値であることに留意してください。ただし、r が無限大に向かって大きくなるにつれて、この近似値 は改善されます。", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", + "input": "et better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", "translatedText": "このように続けると、格子点の総数についてかなり整 理された式が得られます。", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum. ", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum. ", "translatedText": "そして、その r2 を因数分解して、乗算 する必要がある 4 を戻すと、それが意味するのは、この大きな円内の格子点 の総数は、r2 の合計の約 4 倍であるということです。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/korean/sentence_translations.json b/2017/leibniz-formula/korean/sentence_translations.json index ab04e71de..75c9a6b69 100644 --- a/2017/leibniz-formula/korean/sentence_translations.json +++ b/2017/leibniz-formula/korean/sentence_translations.json @@ -259,7 +259,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "반지름 제곱근 5는 실제로 8개의 격자점에 닿습니다.", "model": "DeepL", "n_reviews": 0, @@ -1787,7 +1787,7 @@ "end": 1599.04 }, { - "input": "How many numbers between 1 and r squared have 1 as a divisor?", + "input": "How many numbers between 1 and r2 have 1 as a divisor? All of them. So our sum should include r2 times chi of 1. How many of them have 2 as a divisor?", "translatedText": "1과 제곱의 제곱 사이에 1을 제수로 갖는 숫자는 몇 개인가요?", "model": "DeepL", "from_community_srt": "1을 약수로 갖는 숫자가 1과 R² 사이에 얼마나 있을까요?", diff --git a/2017/leibniz-formula/marathi/sentence_translations.json b/2017/leibniz-formula/marathi/sentence_translations.json index 85d969f5d..a1883da08 100644 --- a/2017/leibniz-formula/marathi/sentence_translations.json +++ b/2017/leibniz-formula/marathi/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points. ", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points. ", "translatedText": "तुम्ही त्रिज्या 1 पाहिल्यास, ते 4 जाळीच्या बिंदूंना मारते. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points. ", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice poi ", "translatedText": "5 ची त्रिज्या वर्गमूळ प्रत्यक्षात 8 जाळी बिंदूंना मारते. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", "translatedText": "तर येथे पूर्णांक समन्वयांची जोडी म्हणून या जाळीच्या बिंदूऐवजी, 3,4, त्याऐवजी एकल कॉम्प्लेक्स संख्या म्हणून विचार करा, 3 अधिक 4i. ", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes. ", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes. ", "translatedText": "5, किंवा 13, किंवा 17 सारख्या 4 च्या गुणाकाराच्या वर असलेल्या अविभाज्य संख्या नेहमी दोन वेगळ्या गॉसियन प्राइममध्ये घटक बनवल्या जाऊ शकतात. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers. ", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers. ", "translatedText": "दुसरीकडे, 3, किंवा 7, किंवा 11 सारख्या, 4 च्या गुणाकाराच्या वर 3 असलेल्या मूळ संख्यांना गॉसियन पूर्णांकांमध्ये अधिक गुणांकन करता येत नाही. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25. ", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25. ", "translatedText": "कारण उजवीकडील प्रत्येक गोष्ट डावीकडील सर्व गोष्टींसह एक संयुग्मित आहे, जे बाहेर येते ते एक जटिल संयुग्मित जोडी आहे जी 25 ने गुणाकार करते. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i? ", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i? ", "translatedText": "लक्षात ठेवा की गॉसियन प्राइममध्ये फॅक्टरायझेशन वेगळे दिसू शकते जर तुम्ही त्यापैकी काहींना i, किंवा –1, किंवा –i ने गुणाकार केला तर? ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i. ", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i. ", "translatedText": "या प्रकरणात, तुम्ही 25 चे फॅक्टरायझेशन वेगळ्या पद्धतीने लिहू शकता, कदाचित त्या 5s पैकी एकाला -1 अधिक 2i गुणा -1 वजा 2i असे विभाजित करू शकता. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i. ", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i. ", "translatedText": "एकूण आउटपुटला i, किंवा –1, किंवा –i ने गुणाकार करणे हा एकमात्र परिणाम होईल. ", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees. ", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees. ", "translatedText": "डाव्या स्तंभातून ते उत्पादन घ्या आणि 90 अंशांच्या काही गुणाकार असलेल्या रोटेशनशी संबंधित 1, i, –1, किंवा –i ने गुणाकार करणे निवडा. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", "translatedText": "त्या चार निवडी, डाव्या स्तंभातील उत्पादनास 1, i, –1, किंवा –i ने गुणाकार करण्याच्या अंतिम चार निवडींनी गुणाकार केल्यास, एकूण 16 जाळीचे बिंदू असे सूचित करतात की 125 च्या अंतराचे वर्गमूळ आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", "translatedText": "जर ती सम पॉवर असेल तर, या प्रकरणात 4 प्रमाणे, बेरीज 1 होईल, जे या अविभाज्य 3 चे काय करावे यासाठी एकच पर्याय आहे हे तथ्य समाविष्ट करते. ", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point. ", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point. ", "translatedText": "आम्ही आता पराकाष्ठा जवळ येत आहोत, गोष्टी व्यवस्थित दिसू लागल्या आहेत, म्हणून थांबा आणि विचार करा, या क्षणापर्यंत सर्वकाही चांगले वाटेल याची खात्री करा. ", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", "translatedText": "त्रिज्या 0 सह मूळ बिंदूकडे दुर्लक्ष करूया, ते बाकीच्या पॅटर्नचे अनुसरण करत नाही आणि एक लहान बिंदू काही फरक करणार नाही कारण आपण r अनंताकडे वाढू देतो. ", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4. ", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4. ", "translatedText": "आम्ही करत असलेल्या या सर्व गॉसियन पूर्णांक आणि फॅक्टरिंग आणि ची फंक्शन सामग्रीवरून, प्रत्येक n चे उत्तर n च्या प्रत्येक विभाजकावर ची चे मूल्य जोडून, 4 ने गुणाकार केल्यासारखे दिसते. ", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3. ", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2. ", "translatedText": "या पंक्तींपैकी सुमारे एक तृतीयांश ची ची 3 आहे, म्हणून आपण r2 ला भागिले 3 ची ची 3 गुणिले घालू शकतो. ", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better. ", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will g ", "translatedText": "लक्षात ठेवा की आम्ही अंदाजे आहोत, कारण r2 कदाचित 2 किंवा 3 पूर्णतः विभाजित करू शकत नाही, परंतु जसजसा r अनंताकडे वाढेल, तसतसे हे अंदाजे अधिक चांगले होईल. ", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", + "input": "et better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", "translatedText": "आणि जेव्हा तुम्ही असेच पुढे जात राहता, तेव्हा तुम्हाला एकूण जाळीच्या बिंदूंसाठी एक सुंदर संघटित अभिव्यक्ती मिळते. ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum. ", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum. ", "translatedText": "आणि जर तुम्ही ते r2 काढले आणि 4 परत आणले ज्याचा गुणाकार करणे आवश्यक आहे, तर याचा अर्थ काय आहे की या मोठ्या वर्तुळातील एकूण जाळी बिंदूंची संख्या या बेरजेच्या r2 च्या अंदाजे 4 पट आहे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/persian/sentence_translations.json b/2017/leibniz-formula/persian/sentence_translations.json index 4fe2e643b..7d90bab78 100644 --- a/2017/leibniz-formula/persian/sentence_translations.json +++ b/2017/leibniz-formula/persian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points. ", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points. ", "translatedText": "اگر به شعاع 1 نگاه کنید، به 4 نقطه شبکه برخورد می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points. ", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice poi ", "translatedText": "شعاع جذر 5 در واقع به 8 نقطه شبکه برخورد می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", "translatedText": "بنابراین به جای این نقطه شبکه در اینجا به عنوان جفت مختصات اعداد صحیح، 3،4، به جای آن به عنوان یک عدد مختلط واحد، 3 به علاوه 4i فکر کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes. ", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes. ", "translatedText": "اعداد اول که یک بالای مضرب 4 هستند، مانند 5، یا 13، یا 17، همیشه می توانند دقیقاً در دو عدد اول گاوسی مجزا در نظر گرفته شوند. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers. ", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers. ", "translatedText": "از طرف دیگر، اعداد اولی که 3 بالای مضرب 4 هستند، مانند 3، یا 7، یا 11، نمی توانند بیشتر در داخل اعداد صحیح گاوسی فاکتور شوند. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25. ", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25. ", "translatedText": "از آنجایی که هر چیزی در سمت راست مزدوج است با هر چیزی که در سمت چپ است، آنچه بیرون می آید یک جفت مزدوج پیچیده است که تا 25 ضرب می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i? ", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i? ", "translatedText": "اما چرا این دستور العمل هنوز همه 12 نقطه شبکه را نمی گیرد؟ به یاد داشته باشید که چگونه ذکر کردم که فاکتورسازی به اعداد اول گاوسی می تواند متفاوت به نظر برسد اگر تعدادی از آنها را در i، یا –1، یا –i ضرب کنید؟ در این مورد، می توانید فاکتورسازی 25 را متفاوت بنویسید، شاید یکی از آن 5 ها را به صورت -1 به علاوه 2i ضربدر -1 منهای 2i تقسیم کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i. ", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i. ", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i. ", "translatedText": "تنها اثری که این کار خواهد داشت این است که کل خروجی را در i یا -1 یا -i ضرب کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees. ", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees. ", "translatedText": "آن حاصل ضرب را از ستون سمت چپ بگیرید، و ضرب آن را در 1، i، -1، یا -i انتخاب کنید، که مربوط به چرخش هایی است که مضرب 90 درجه هستند. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", "translatedText": "این چهار گزینه، ضرب در چهار گزینه نهایی ضرب ضرب حاصل از ستون سمت چپ در 1، i، -1، یا -i، نشان می دهد که در مجموع 16 نقطه شبکه با فاصله 125 ریشه مربع از ستون وجود دارد. اصل و نسب. ", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", "translatedText": "اگر یک توان زوج باشد، مانند 4 در این مورد، مجموع به 1 می رسد، که این واقعیت را در بر می گیرد که تنها یک انتخاب وجود دارد که با آن 3 های غیرقابل تقسیم چه باید کرد. ", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point. ", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point. ", "translatedText": "اکنون به نقطه اوج نزدیک می شویم، همه چیز شروع به منظم شدن می کند، پس مکث و تامل کنید، مطمئن شوید که همه چیز تا این لحظه خوب است. ", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", "translatedText": "بیایید آن نقطه مبدا با شعاع 0 را نادیده بگیریم، از الگوی بقیه پیروی نمی کند، و یک نقطه کوچک نیز تفاوتی ایجاد نمی کند زیرا اجازه می دهیم r به سمت بی نهایت رشد کند. ", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4. ", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4. ", "translatedText": "از تمام این اعداد صحیح گاوسی و فاکتورسازی و تابع چی که انجام داده‌ایم، پاسخ هر n به نظر می‌رسد که مقدار chi را در هر مقسوم‌گیرنده n جمع کرده و در 4 ضرب می‌کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3. ", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2. ", "translatedText": "حدود یک سوم از این ردیف ها دارای chi 3 هستند، بنابراین می توانیم r2 را تقسیم بر 3 ضرب در چی از 3 قرار دهیم. ", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better. ", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will g ", "translatedText": "به خاطر داشته باشید که ما تقریبی هستیم، زیرا r2 ممکن است 2 یا 3 را به طور کامل تقسیم نکند، اما با رشد r به سمت بی نهایت، این تقریب بهتر می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", + "input": "et better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", "translatedText": "و هنگامی که به این شکل ادامه می دهید، یک عبارت بسیار سازمان یافته برای تعداد کل نقاط شبکه دریافت می کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum. ", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum. ", "translatedText": "و اگر آن r2 را در نظر بگیرید و 4 را که باید در آن ضرب شود برگردانید، به این معنی است که تعداد کل نقاط شبکه داخل این دایره بزرگ تقریباً 4 برابر r2 برابر این مجموع است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/portuguese/sentence_translations.json b/2017/leibniz-formula/portuguese/sentence_translations.json index ac0d653f8..b18fec8e0 100644 --- a/2017/leibniz-formula/portuguese/sentence_translations.json +++ b/2017/leibniz-formula/portuguese/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points. ", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points. ", "translatedText": "Se você olhar para o raio 1, isso atinge 4 pontos da rede. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points. ", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice poi ", "translatedText": "Uma raiz quadrada de raio de 5 atinge, na verdade, 8 pontos da rede. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", "translatedText": "Então, em vez deste ponto da rede aqui como o par de coordenadas inteiras, 3,4, pense nele como o único número complexo, 3 mais 4i. ", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes. ", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes. ", "translatedText": "Os números primos que estão um acima de um múltiplo de 4, como 5, ou 13, ou 17, sempre podem ser fatorados em exatamente dois primos gaussianos distintos. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers. ", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers. ", "translatedText": "Por outro lado, os números primos 3 acima de um múltiplo de 4, como 3, ou 7, ou 11, não podem ser fatorados posteriormente dentro dos inteiros gaussianos. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25. ", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25. ", "translatedText": "Como tudo à direita é conjugado com tudo à esquerda, o que resulta é um par conjugado complexo que se multiplica por 25. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i? ", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i? ", "translatedText": "Lembra como mencionei que uma fatoração em primos gaussianos pode parecer diferente se você multiplicar alguns deles por i, ou –1, ou –i? ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i. ", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i. ", "translatedText": "Neste caso, você poderia escrever a fatoração de 25 de forma diferente, talvez dividindo um desses 5 como –1 mais 2i vezes –1 menos 2i. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i. ", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i. ", "translatedText": "O único efeito que isso terá é multiplicar a produção total por i, ou –1, ou –i. ", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees. ", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees. ", "translatedText": "Pegue esse produto da coluna da esquerda e escolha multiplicá-lo por 1, i, –1 ou –i, correspondendo a rotações que são múltiplos de 90 graus. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", "translatedText": "Essas quatro opções, multiplicadas pelas quatro opções finais de multiplicar o produto da coluna da esquerda por 1, i, –1 ou –i, sugeririam que há um total de 16 pontos na rede a uma distância da raiz quadrada de 125 do origem. ", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", "translatedText": "Se for uma potência par, como 4 neste caso, a soma será 1, o que resume o fato de que há apenas uma escolha sobre o que fazer com esses 3 indivisíveis. ", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point. ", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point. ", "translatedText": "Estamos chegando perto do ponto culminante agora, as coisas estão começando a parecer organizadas, então faça uma pausa e pondere, certifique-se de que tudo está bem até este ponto. ", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", "translatedText": "Vamos ignorar aquele ponto de origem com raio 0, ele não segue o padrão dos demais, e um pequeno ponto não fará diferença à medida que deixamos r crescer em direção ao infinito. ", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4. ", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4. ", "translatedText": "De todo esse trabalho de inteiro gaussiano, fatoração e função chi que temos feito, a resposta para cada n parece somar o valor de chi em cada divisor de n e multiplicar por 4. ", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3. ", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2. ", "translatedText": "Cerca de um terço dessas linhas tem chi de 3, então podemos colocar r2 dividido por 3 vezes chi de 3. ", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better. ", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will g ", "translatedText": "Lembre-se de que estamos sendo aproximados, pois r2 pode não dividir perfeitamente 2 ou 3, mas à medida que r cresce em direção ao infinito, essa aproximação ficará melhor. ", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", + "input": "et better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", "translatedText": "E quando continuamos assim, obtemos uma expressão bastante organizada para o número total de pontos da rede. ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum. ", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum. ", "translatedText": "E se você fatorar esse r2 e trazer de volta o 4 que precisa ser multiplicado, o que isso significa é que o número total de pontos da rede dentro deste grande círculo é aproximadamente 4 vezes r2 vezes esta soma. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/russian/sentence_translations.json b/2017/leibniz-formula/russian/sentence_translations.json index b0865dbcd..d8d794356 100644 --- a/2017/leibniz-formula/russian/sentence_translations.json +++ b/2017/leibniz-formula/russian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points. ", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points. ", "translatedText": "Если вы посмотрите на радиус 1, он соответствует 4 точкам решетки. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points. ", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice poi ", "translatedText": "Квадратный корень радиуса из 5 фактически попадает в 8 точек решетки. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", "translatedText": "Поэтому вместо этой точки решетки здесь как пары целочисленных координат, 3,4, представьте себе ее как одно комплексное число, 3 плюс 4i. ", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes. ", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes. ", "translatedText": "Простые числа, которые на единицу больше кратного 4, например 5, 13 или 17, всегда можно разложить ровно на два различных гауссовских простых числа. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers. ", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers. ", "translatedText": "С другой стороны, простые числа, число которых больше 3, кратного 4, например 3, 7 или 11, не могут быть дополнительно факторизованы внутри гауссовских целых чисел. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25. ", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25. ", "translatedText": "Поскольку все, что справа, сопряжено со всем, что слева, получается комплексно-сопряженная пара, которая умножается на 25. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i? ", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i? ", "translatedText": "Помните, как я упоминал, что факторизация в простые гауссовы числа может выглядеть иначе, если вы умножите некоторые из них на i, или –1, или –i? ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i. ", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i. ", "translatedText": "В этом случае вы могли бы записать факторизацию 25 по-другому, возможно, разделив одну из этих пятерок как –1 плюс 2i, умноженную на –1 минус 2i. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i. ", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i. ", "translatedText": "Единственный эффект, который это будет иметь, — это умножение общего объема производства на i, или –1, или –i. ", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees. ", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees. ", "translatedText": "Возьмите это произведение из левого столбца и выберите его умножение на 1, i, –1 или –i, что соответствует повороту, кратному 90 градусам. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", "translatedText": "Эти четыре варианта, умноженные на последние четыре варианта умножения произведения из левого столбца на 1, i, –1 или –i, предполагают, что всего существует 16 точек решетки, находящихся на расстоянии квадратного корня из 125 от источник. ", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", "translatedText": "Если это четная степень, например 4 в данном случае, сумма оказывается равной 1, что отражает тот факт, что есть только один выбор, что делать с этими неделимыми тройками. ", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point. ", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point. ", "translatedText": "Сейчас мы приближаемся к кульминации, все начинает выглядеть организованным, так что сделайте паузу и подумайте, убедитесь, что до этого момента все идет хорошо. ", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", "translatedText": "Давайте проигнорируем исходную точку с радиусом 0, она не повторяет шаблон остальных, и одна маленькая точка не будет иметь значения, поскольку мы позволяем r расти до бесконечности. ", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4. ", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4. ", "translatedText": "Судя по всем этим гауссовым целым числам, факторингу и функциям хи, которые мы проделали, ответ для каждого n выглядит как сложение значения chi для каждого делителя n и умножение на 4. ", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3. ", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2. ", "translatedText": "Около трети этих строк имеют хи, равный 3, поэтому мы можем положить r2, разделенный на 3, умноженный на хи, равный 3. ", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better. ", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will g ", "translatedText": "Имейте в виду, что мы приблизительны, поскольку r2 может не полностью делить 2 или 3, но по мере того, как r приближается к бесконечности, это приближение будет улучшаться. ", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", + "input": "et better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", "translatedText": "И если вы продолжите в том же духе, вы получите довольно организованное выражение для общего количества точек решетки. ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum. ", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum. ", "translatedText": "И если вы вычтете это r2 и вернете 4, которые нужно умножить, это будет означать, что общее количество точек решетки внутри этого большого круга примерно в 4 раза умножит r2 на эту сумму. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/spanish/sentence_translations.json b/2017/leibniz-formula/spanish/sentence_translations.json index cbbedae9d..3258fdd2c 100644 --- a/2017/leibniz-formula/spanish/sentence_translations.json +++ b/2017/leibniz-formula/spanish/sentence_translations.json @@ -261,7 +261,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "Una raíz cuadrada de radio 5 alcanza en realidad 8 puntos de entramado.", "model": "DeepL", "from_community_srt": "Uno de radio raíz cuadrada de 5 nos da 8 puntos de corte.", @@ -1799,7 +1799,7 @@ "end": 1599.04 }, { - "input": "How many numbers between 1 and r squared have 1 as a divisor?", + "input": "How many numbers between 1 and r2 have 1 as a divisor? All of them. So our sum should include r2 times chi of 1. How many of them have 2 as a divisor?", "translatedText": "¿Cuántos números entre 1 y r al cuadrado tienen 1 como divisor?", "model": "DeepL", "from_community_srt": "Cuántos números entre el 1 y R^2 tienen 1 como divisor; bueno,", diff --git a/2017/leibniz-formula/tamil/sentence_translations.json b/2017/leibniz-formula/tamil/sentence_translations.json index 1db120475..ad0d91274 100644 --- a/2017/leibniz-formula/tamil/sentence_translations.json +++ b/2017/leibniz-formula/tamil/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points.", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points.", "translatedText": "நீங்கள் ஆரம் 1 ஐப் பார்த்தால், அது 4 லட்டு புள்ளிகளைத் தாக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "5 இன் ஆரம் வர்க்கமூலம் உண்மையில் 8 லட்டுப் புள்ளிகளைத் தாக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", "translatedText": "எனவே இங்கே இந்த லட்டு புள்ளிக்கு பதிலாக முழு எண் ஒருங்கிணைப்புகளின் ஜோடி, 3,4, அதற்கு பதிலாக ஒற்றை கலப்பு எண், 3 கூட்டல் 4i என நினைத்துப் பாருங்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 397.06 }, { - "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers will come into play.", + "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers are going to come into play.", "translatedText": "இது எங்கள் கேள்வியை ஒரு காரணியாக்கல் சிக்கலாக மாற்றுகிறது, அதனால்தான் பகா எண்களுக்கு இடையே உள்ள வடிவங்கள் செயல்படும்.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes.", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes.", "translatedText": "5, அல்லது 13, அல்லது 17 போன்ற 4 இன் பெருக்கல் ஒன்றின் மேல் இருக்கும் முதன்மை எண்களை எப்போதும் இரண்டு தனித்துவமான காஸியன் பகா எண்களாகக் கணக்கிடலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers.", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers.", "translatedText": "மறுபுறம், 3, அல்லது 7, அல்லது 11 போன்ற 4 இன் பெருக்கத்திற்கு மேல் 3 இருக்கும் பகா எண்களை காஸியன் முழு எண்களுக்குள் மேலும் காரணியாக்க முடியாது.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25.", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25.", "translatedText": "வலதுபுறத்தில் உள்ள அனைத்தும் இடதுபுறத்தில் உள்ள எல்லாவற்றோடும் இணைந்திருப்பதால், வெளிவருவது சிக்கலான கூட்டு ஜோடியாகும், இது 25 ஆக பெருகும்.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i?", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i?", "translatedText": "காஸியன் ப்ரைம்களில் சிலவற்றை நீங்கள் i, அல்லது –1, அல்லது –i ஆல் பெருக்கினால், ஒரு காரணியாக்கம் வித்தியாசமாக இருக்கும் என்று நான் குறிப்பிட்டதை நினைவிருக்கிறதா?", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i.", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i.", "translatedText": "இந்த வழக்கில், நீங்கள் 25 இன் காரணியாக்கத்தை வித்தியாசமாக எழுதலாம், ஒருவேளை அந்த 5களில் ஒன்றை –1 கூட்டல் 2i முறை –1 கழித்தல் 2i என பிரிக்கலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i.", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i.", "translatedText": "மொத்த வெளியீட்டை i, அல்லது –1, அல்லது –i ஆல் பெருக்குவது மட்டுமே இதன் விளைவு.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees.", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees.", "translatedText": "இடது நெடுவரிசையிலிருந்து அந்தத் தயாரிப்பை எடுத்து, 90 டிகிரியில் சில மடங்குகளாக இருக்கும் சுழற்சிகளுடன் தொடர்புடைய 1, i, –1, அல்லது –i ஆல் பெருக்க தேர்வு செய்யவும்.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", "translatedText": "அந்த நான்கு தேர்வுகள், இடது நெடுவரிசையிலிருந்து 1, i, –1, அல்லது –i ஆல் பெருக்குவதற்கான இறுதி நான்கு தேர்வுகளால் பெருக்கப்படும், மொத்தம் 16 லட்டுப் புள்ளிகள் 125 தொலைவில் உள்ள வர்க்க மூலத்தில் உள்ளன என்று பரிந்துரைக்கும். தோற்றம்.", "model": "google_nmt", "n_reviews": 0, @@ -1248,7 +1248,7 @@ "end": 1374.96 }, { - "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, 1 minus 1 plus 1 minus 1 plus 1.", + "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, it goes 1 minus 1 plus 1 minus 1 plus 1.", "translatedText": "ஆனால் இந்த வழக்கில், 3 இன் chi எதிர்மறை 1 என்பதால், இந்த கூட்டுத்தொகை ஊசலாடுகிறது, 1 கழித்தல் 1 கூட்டல் 1 கழித்தல் 1 கூட்டல் 1.", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", "translatedText": "இந்த வழக்கில் 4 போன்ற சம சக்தியாக இருந்தால், கூட்டுத்தொகை 1 ஆக இருக்கும், இது பிரிக்க முடியாத 3 ஐ என்ன செய்வது என்பதற்கு ஒரே ஒரு தேர்வு மட்டுமே உள்ளது என்ற உண்மையை உள்ளடக்கியது.", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point.", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point.", "translatedText": "நாங்கள் இப்போது உச்சக்கட்டத்தை நெருங்கிவிட்டோம், விஷயங்கள் ஒழுங்கமைக்கத் தொடங்கிவிட்டன, எனவே இடைநிறுத்தி யோசித்துப் பாருங்கள், இது வரை எல்லாம் நன்றாக இருக்கிறது என்பதை உறுதிப்படுத்திக் கொள்ளுங்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", "translatedText": "0 ஆரம் கொண்ட அந்த மூலப் புள்ளியைப் புறக்கணிப்போம், அது மற்றவற்றின் வடிவத்தைப் பின்பற்றாது, மேலும் ஒரு சிறிய புள்ளியானது r ஐ முடிவிலியை நோக்கி வளர விடுவதால் எந்த மாற்றத்தையும் ஏற்படுத்தப் போவதில்லை.", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4.", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4.", "translatedText": "இந்த அனைத்து காஸியன் முழு எண் மற்றும் காரணியாக்கம் மற்றும் chi செயல்பாடு ஸ்டஃப் இருந்து, ஒவ்வொரு n க்கான பதில், n இன் ஒவ்வொரு வகுப்பியில் உள்ள chi இன் மதிப்பைக் கூட்டி, 4 ஆல் பெருக்குவது போல் தெரிகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3.", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2.", "translatedText": "இந்த வரிசைகளில் மூன்றில் ஒரு பங்கு 3 இன் chi ஐக் கொண்டுள்ளது, எனவே r2 ஐ 3 இன் 3 மடங்கு chi ஆல் வகுக்கலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better.", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will", "translatedText": "r2 ஆனது 2 அல்லது 3ஐ முழுமையாகப் பிரிக்காது என்பதால், நாம் தோராயமாக இருக்கிறோம் என்பதை நினைவில் கொள்ளுங்கள், ஆனால் r முடிவிலியை நோக்கி வளரும்போது, இந்த தோராயமானது சிறப்பாக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", + "input": "get better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", "translatedText": "நீங்கள் தொடர்ந்து இப்படிச் செல்லும்போது, மொத்த லட்டுப் புள்ளிகளுக்கு அழகான ஒழுங்கமைக்கப்பட்ட வெளிப்பாடு கிடைக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum.", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum.", "translatedText": "நீங்கள் அந்த r2 ஐக் கணக்கிட்டு, பெருக்க வேண்டிய 4 ஐ மீண்டும் கொண்டு வந்தால், இதன் பொருள் என்னவென்றால், இந்த பெரிய வட்டத்திற்குள் உள்ள மொத்த லட்டு புள்ளிகளின் எண்ணிக்கை இந்த தொகையின் 4 மடங்கு r2 மடங்கு ஆகும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/telugu/sentence_translations.json b/2017/leibniz-formula/telugu/sentence_translations.json index 29c6a3ef4..c8ad2e592 100644 --- a/2017/leibniz-formula/telugu/sentence_translations.json +++ b/2017/leibniz-formula/telugu/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points.", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points.", "translatedText": "మీరు వ్యాసార్థం 1ని చూస్తే, అది 4 లాటిస్ పాయింట్లను తాకుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "5 యొక్క వ్యాసార్థం వర్గమూలం వాస్తవానికి 8 లాటిస్ పాయింట్లను తాకుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", "translatedText": "కాబట్టి ఇక్కడ ఈ లాటిస్ పాయింట్‌కి బదులుగా పూర్ణాంకాల కోఆర్డినేట్‌ల జత, 3,4, బదులుగా దీనిని సింగిల్ కాంప్లెక్స్ సంఖ్య, 3 ప్లస్ 4iగా భావించండి.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 397.06 }, { - "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers will come into play.", + "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers are going to come into play.", "translatedText": "ఇది మా ప్రశ్నను ఫ్యాక్టరింగ్ సమస్యగా మారుస్తుంది, చివరికి ప్రధాన సంఖ్యల మధ్య నమూనాలు ఎందుకు అమలులోకి వస్తాయి.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes.", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes.", "translatedText": "5, లేదా 13, లేదా 17 వంటి 4 యొక్క గుణకం పైన ఒకటిగా ఉండే ప్రధాన సంఖ్యలను ఎల్లప్పుడూ ఖచ్చితంగా రెండు విభిన్న గాస్సియన్ ప్రైమ్‌లుగా మార్చవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers.", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers.", "translatedText": "మరోవైపు, 3, లేదా 7, లేదా 11 వంటి 4 యొక్క గుణకం కంటే 3 పైన ఉన్న ప్రధాన సంఖ్యలు గాస్సియన్ పూర్ణాంకాల లోపల మరింతగా కారకం చేయబడవు.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25.", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25.", "translatedText": "కుడి వైపున ఉన్న ప్రతిదీ ఎడమ వైపు ఉన్న ప్రతిదానితో సంయోగం అయినందున, బయటకు వచ్చేది 25కి గుణించే సంక్లిష్ట సంయోగ జత.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i?", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i?", "translatedText": "మీరు వాటిలో కొన్నింటిని i, లేదా –1, లేదా –iతో గుణిస్తే గాస్సియన్ ప్రైమ్‌లలోకి కారకం భిన్నంగా కనిపిస్తుందని నేను ఎలా పేర్కొన్నానో గుర్తుందా?", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i.", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i.", "translatedText": "ఈ సందర్భంలో, మీరు 25 యొక్క కారకాన్ని విభిన్నంగా వ్రాయవచ్చు, బహుశా ఆ 5లలో ఒకదానిని –1 ప్లస్ 2i సార్లు –1 మైనస్ 2iగా విభజించవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i.", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i.", "translatedText": "మొత్తం అవుట్‌పుట్‌ను i, లేదా –1, లేదా –i ద్వారా గుణించడం మాత్రమే దీని ప్రభావం.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees.", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees.", "translatedText": "ఎడమ నిలువు వరుస నుండి ఆ ఉత్పత్తిని తీసుకోండి మరియు 90 డిగ్రీల గుణకారంగా ఉండే భ్రమణాలకు అనుగుణంగా 1, i, –1, లేదా –iతో గుణించడాన్ని ఎంచుకోండి.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", "translatedText": "ఆ నాలుగు ఎంపికలు, ఎడమ కాలమ్ నుండి ఉత్పత్తిని 1, i, –1, లేదా –i ద్వారా గుణించే చివరి నాలుగు ఎంపికలతో గుణిస్తే, 125 దూరంలో ఉన్న వర్గమూలం నుండి మొత్తం 16 లాటిస్ పాయింట్లు ఉన్నాయని సూచిస్తాయి మూలం.", "model": "google_nmt", "n_reviews": 0, @@ -1248,7 +1248,7 @@ "end": 1374.96 }, { - "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, 1 minus 1 plus 1 minus 1 plus 1.", + "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, it goes 1 minus 1 plus 1 minus 1 plus 1.", "translatedText": "కానీ ఈ సందర్భంలో, 3 యొక్క చి ప్రతికూల 1 అయినందున, ఈ మొత్తం డోలనం అవుతుంది, 1 మైనస్ 1 ప్లస్ 1 మైనస్ 1 ప్లస్ 1.", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", "translatedText": "ఈ సందర్భంలో 4 లాగా ఇది సమానమైన శక్తి అయితే, మొత్తం 1 అవుతుంది, ఇది విభజించబడని 3లతో ఏమి చేయాలనేది ఒకే ఒక ఎంపిక అనే వాస్తవాన్ని నిక్షిప్తం చేస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point.", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point.", "translatedText": "మేము ఇప్పుడు పరాకాష్టకు చేరుకుంటున్నాము, విషయాలు క్రమబద్ధంగా కనిపించడం ప్రారంభించాయి, కాబట్టి పాజ్ చేసి, ఆలోచించండి, ఈ సమయం వరకు ప్రతిదీ బాగానే ఉందని నిర్ధారించుకోండి.", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", "translatedText": "వ్యాసార్థం 0తో ఉన్న ఆ మూల బిందువును విస్మరించండి, అది మిగిలిన వాటి నమూనాను అనుసరించదు మరియు మేము rని అనంతం వైపుగా ఎదగనివ్వడం వలన ఒక చిన్న చుక్క తేడాను కలిగించదు.", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4.", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4.", "translatedText": "మేము చేస్తున్న ఈ గాస్సియన్ పూర్ణాంకం మరియు కారకం మరియు చి ఫంక్షన్ అంశాల నుండి, ప్రతి nకి సమాధానం n యొక్క ప్రతి భాగహారంపై చి విలువను జోడించి, 4తో గుణించినట్లుగా కనిపిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3.", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2.", "translatedText": "ఈ అడ్డు వరుసలలో దాదాపు మూడింట ఒక వంతు ఛి 3ని కలిగి ఉంటుంది, కాబట్టి మనం r2ని 3కి 3 సార్లు చితో భాగించవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better.", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will", "translatedText": "మేము సుమారుగా ఉన్నామని గుర్తుంచుకోండి, ఎందుకంటే r2 సంపూర్ణంగా 2 లేదా 3ని విభజించకపోవచ్చు, కానీ r అనంతం వైపు పెరిగే కొద్దీ, ఈ ఉజ్జాయింపు మెరుగవుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", + "input": "get better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", "translatedText": "మరియు మీరు ఇలాగే కొనసాగించినప్పుడు, మొత్తం లాటిస్ పాయింట్ల సంఖ్యకు మీరు అందంగా వ్యవస్థీకృత వ్యక్తీకరణను పొందుతారు.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum.", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum.", "translatedText": "మరియు మీరు ఆ r2ని కారకం చేసి, గుణించాల్సిన 4ని తిరిగి తీసుకువస్తే, దీని అర్థం ఏమిటంటే, ఈ పెద్ద సర్కిల్‌లోని మొత్తం లాటిస్ పాయింట్ల సంఖ్య ఈ మొత్తానికి దాదాపు 4 రెట్లు r2 రెట్లు ఎక్కువ.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/thai/sentence_translations.json b/2017/leibniz-formula/thai/sentence_translations.json index 9ed76e396..d0ce713de 100644 --- a/2017/leibniz-formula/thai/sentence_translations.json +++ b/2017/leibniz-formula/thai/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points. ", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points. ", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice poi ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes. ", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers. ", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25. ", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i? ", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i. ", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i. ", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees. ", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees. ", "translatedText": "จำได้ไหมที่ฉันบอกว่าการแยกตัวประกอบของจำนวนเฉพาะแบบเกาส์เซียนอาจดูแตกต่างออกไป หากคุณคูณบางตัวด้วย i หรือ –1 หรือ –i ในกรณีนี้, คุณอาจเขียนการแยกตัวประกอบของ 25 ต่างกันออกไป, อาจแบ่ง 5 อันใดอันหนึ่งเป็น –1 บวก 2i คูณ –1 ลบ 2i และถ้าคุณทำเช่นนั้น โดยใช้สูตรเดียวกัน อาจส่งผลต่อผลลัพธ์ คุณจะได้ผลิตภัณฑ์อื่นจากคอลัมน์ด้านซ้าย ผลกระทบเดียวที่จะเกิดขึ้นคือการคูณผลลัพธ์ทั้งหมดด้วย i หรือ –1 หรือ –i ขั้นตอนสุดท้ายสำหรับสูตรของเรา สมมติว่าคุณต้องเลือกหนึ่งในสี่ตัวเลือก นำผลคูณนั้นจากคอลัมน์ด้านซ้าย แล้วเลือกคูณด้วย 1, i, –1 หรือ –i ซึ่งสอดคล้องกับการหมุนที่เป็นผลคูณของ 90 องศา นั่นจะอธิบายถึงวิธีการสร้างจำนวนเต็มเกาส์เซียนที่แตกต่างกันทั้งหมด 12 วิธีซึ่งมีผลคูณกับคอนจูเกตของตัวเองคือ 25 กระบวนการนี้ซับซ้อนเล็กน้อย ดังนั้นฉันคิดว่าวิธีที่ดีที่สุดในการทำความเข้าใจคือลองใช้ตัวอย่างเพิ่มเติม สมมุติว่าเราดูที่ 125 ซึ่งก็คือ 5 ลูกบาศก์แทน ในกรณีนั้น เราจะมีสี่ตัวเลือกที่แตกต่างกันสำหรับวิธีแบ่งคู่คอนจูเกตเฉพาะออกเป็นสองคอลัมน์นี้ คุณสามารถมีสำเนา 2 บวก i เป็นศูนย์ในคอลัมน์ด้านซ้าย, สำเนาหนึ่งชุดในนั้น, สองสำเนาในนั้น, หรือทั้งสามชุดในคอลัมน์ด้านซ้าย ตัวเลือกสี่ตัวนั้นคูณด้วยตัวเลือกสี่ตัวสุดท้ายในการคูณผลคูณจากคอลัมน์ด้านซ้ายด้วย 1, i, –1 หรือ –i จะแนะนำว่ามีจุดขัดแตะทั้งหมด 16 จุด โดยมีรากที่สองอยู่ห่างจากจุด 125 ต้นทาง. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", "translatedText": "ปัจจัยของ 2 ไม่ได้สร้างความแตกต่าง ตอนนี้สิ่งที่กำลังจะเกิดขึ้นคือทฤษฎีจำนวนที่ดีที่สุด เรามีสูตรที่ซับซ้อนนี้บอกเราว่ามีจุดขัดแตะจำนวนกี่จุดบนวงกลมที่มีรัศมีรากที่สองของ n และมันขึ้นอยู่กับการแยกตัวประกอบเฉพาะของ n เพื่อเปลี่ยนสิ่งนี้ให้เป็นสิ่งที่ง่ายกว่า ซึ่งเป็นสิ่งที่เราจัดการได้จริงๆ เราจะใช้ประโยชน์จากความสม่ำเสมอของจำนวนเฉพาะที่ค่าที่เป็น 1 ส่วนเหนือของ 4 แบ่งออกเป็นตัวประกอบเฉพาะแบบเกาส์เซียนที่แตกต่างกัน ในขณะที่ตัวที่มีค่าเป็น 3 มากกว่าตัวคูณของ 4 ไม่สามารถแบ่งได้ เพื่อที่จะทำสิ่งนี้ เรามาแนะนำฟังก์ชันง่ายๆ กัน ซึ่งฉันจะเขียนกำกับด้วยอักษรกรีก chi สำหรับอินพุตที่เป็น 1 เหนือผลคูณของ 4 ผลลัพธ์ของไคคือ 1 หากรับอินพุต 3 เหนือผลคูณของ 4 ผลลัพธ์ของไคจะเป็นลบ 1 แล้วเลขคู่ทั้งหมดจะให้ 0 ดังนั้น หากคุณประเมินค่าไคจากจำนวนธรรมชาติ มันจะให้รูปแบบวงจรที่สวยงามมาก 1, 0, ลบ 1, 0 แล้วทำซ้ำไปเรื่อยๆ และฟังก์ชันไซคลิกไคนี้มีคุณสมบัติพิเศษมาก มันเรียกว่าฟังก์ชันคูณ หากคุณประเมินค่าไคจากตัวเลขสองตัวที่ต่างกันและคูณผลลัพธ์ เช่น ไคของ 3 คูณไคของ 5 ก็จะเหมือนกับที่คุณประเมินค่าไคจากผลคูณของตัวเลขสองตัวนั้น ในกรณีนี้ ไคของ 15 ในทำนองเดียวกัน ไคของ 5 คูณไคของ 5 เท่ากับไคของ 25 และไม่ว่าคุณจะใส่จำนวนธรรมชาติสองตัวลงไป คุณสมบัตินี้จะคงอยู่ เอาเลยลองดูถ้าคุณต้องการ สำหรับคำถามหลักของเราเกี่ยวกับการนับจุดขัดแตะในลักษณะนี้ ที่เกี่ยวข้องกับการแยกตัวประกอบของตัวเลข สิ่งที่ผมจะทำคือเขียนจำนวนตัวเลือกที่เรามี แต่ใช้ไคในสิ่งที่ตอนแรกดูเหมือนจะซับซ้อนกว่ามาก แต่ สิ่งนี้มีประโยชน์ในการปฏิบัติต่อปัจจัยสำคัญทั้งหมดอย่างเท่าเทียมกัน สำหรับกำลังไพรม์แต่ละตัว เช่น 5 ลูกบาศก์ สิ่งที่คุณเขียนลงไปคือ ไคของ 1 บวก ไคของ 5 บวก ไคของ 5 กำลังสอง บวก ไคของ 5 ลูกบาศก์ คุณบวกค่าไคจากกำลังทั้งหมดของไพรม์นี้ กับค่าที่ปรากฎภายในการแยกตัวประกอบ ในกรณีนี้ เนื่องจาก 5 คือ 1 เหนือผลคูณของ 4 ทั้งหมดจึงเป็นเพียง 1 ดังนั้นผลรวมนี้จึงได้เป็น 4 ซึ่งสะท้อนความจริงที่ว่าตัวประกอบของ 5 กำลังสามให้ตัวเลือก 4 ทางแก่คุณในการหารค่า ตัวประกอบเฉพาะแบบเกาส์เซียนสองตัวระหว่างคอลัมน์ สำหรับปัจจัยอย่าง 3 ยกกำลัง 4 สิ่งที่คุณเขียนลงไปดูคล้ายกันมาก ไคของ 1 บวกไคของ 3 ขึ้นไปถึงไคของ 3 ยกที่ 4 แต่ในกรณีนี้ เนื่องจากไคของ 3 เป็นลบ 1 ผลรวมนี้จึงแกว่ง 1 ลบ 1 บวก 1 ลบ 1 บวก 1 หากเป็นกำลังคู่ เช่น 4 ในกรณีนี้ ผลรวมจะเป็น 1 ซึ่งสรุปข้อเท็จจริงที่ว่ามีทางเลือกเดียวเท่านั้นว่าจะทำอย่างไรกับ 3 ที่แยกไม่ได้ แต่ถ้าเป็นเลขยกกำลังคี่ ผลรวมนั้นออกมาเป็น 0 แสดงว่า คุณมันแย่ คุณไม่สามารถวาง 3 ที่แยกไม่ได้นั้นได้ เมื่อคุณทำสิ่งนี้เพื่อยกกำลัง 2 มันจะดูเหมือน 1 บวก 0 บวก 0 บวก 0 ไปเรื่อยๆ เนื่องจากไคจะเป็น 0 ในเลขคู่เสมอ และนี่สะท้อนความจริงที่ว่าปัจจัยของ 2 ไม่ได้ช่วยอะไรและไม่เจ็บ คุณมีตัวเลือกเดียวเสมอว่าจะทำอย่างไรกับมัน และเช่นเคย เราคงเลข 4 ไว้ข้างหน้าเพื่อระบุตัวเลือกสุดท้ายที่จะคูณด้วย 1, i, ลบ 1 หรือลบ i ตอนนี้เราใกล้ถึงจุดสุดยอดแล้ว สิ่งต่างๆ เริ่มดูเป็นระเบียบ ดังนั้นให้หยุดและไตร่ตรอง ตรวจสอบให้แน่ใจว่าทุกอย่างรู้สึกดีจนถึงจุดนี้ ใช้หมายเลข 45 เป็นตัวอย่าง เจ้านี่แยกตัวประกอบเป็น 3 กำลังสองคูณ 5 ดังนั้นพจน์ของจำนวนจุดขัดแตะทั้งหมดคือ 4 คูณไคของ 1 บวกไคของ 3 บวกไคของ 3 กำลังสองคูณไคของ 1 บวกไคของ 5 คุณคิดว่านี่เป็น 4 คูณตัวเลือกเดียวว่าจะทำอย่างไรกับ 3 คูณ 2 ตัวเลือกในการหารตัวประกอบเฉพาะแบบเกาส์เซียนของ 5 มันอาจดูเหมือนการขยายผลรวมออกไปนั้นซับซ้อนมาก เพราะมันเกี่ยวข้องกับการรวมตัวประกอบเฉพาะที่เป็นไปได้ทั้งหมด และมันก็เป็นเช่นนั้น อย่างไรก็ตาม เนื่องจากไคเป็นแบบคูณ ดังนั้นแต่ละค่าผสมจึงมีค่าตัวหาร 45 ในกรณีนี้สิ่งที่เราได้คือ 4 คูณไคของ 1 บวกไคของ 3 บวกไคของ 5 บวกไคของ 9 บวกไคของ 15 บวกไคของ 45 สิ่งที่คุณจะสังเกตเห็นก็คือ ค่านี้ครอบคลุมทุกตัวเลขที่หาร 45 เท่าๆ กัน เพียงครั้งเดียวเท่านั้น และมันใช้ได้แบบนี้กับจำนวนใดๆ ก็ได้ ไม่มีอะไรพิเศษเกี่ยวกับ 45 และนั่นสำหรับฉันมันค่อนข้างน่าสนใจ และฉันคิดว่าเป็นเรื่องที่คาดไม่ถึงเลย คำถามเกี่ยวกับการนับจำนวนจุดขัดแตะซึ่งระยะห่างจากรากที่สองของ n ที่อยู่ห่างจากจุดกำเนิดเกี่ยวข้องกับการบวกค่าของฟังก์ชันที่ค่อนข้างง่ายนี้กับตัวหารทั้งหมดของ n เพื่อรวบรวมทุกอย่างเข้าด้วยกัน จำไว้ว่าทำไมเราถึงทำเช่นนี้ จำนวนจุดขัดแตะทั้งหมดภายในวงกลมใหญ่ที่มีรัศมี r ควรอยู่ที่ประมาณ pi คูณ r กำลังสอง แต่ในทางกลับกัน เราสามารถนับจุดขัดแตะเดียวกันเหล่านั้นได้โดยการดูตัวเลขทั้งหมด n ระหว่าง 0 ถึง r กำลังสอง แล้วนับจำนวนจุดขัดแตะที่เป็นระยะห่างของรากที่สองของ n จากจุดกำเนิด ลองมองข้ามจุดกำเนิดที่มีรัศมี 0 มันไม่เป็นไปตามรูปแบบของจุดที่เหลือ และจุดเล็กๆ หนึ่งจุดจะไม่สร้างความแตกต่างเมื่อเราปล่อยให้ r เติบโตไปสู่อนันต์ จากจำนวนเต็มเกาส์เซียน การแยกตัวประกอบ และฟังก์ชันไคที่เราเคยทำมา คำตอบของแต่ละ n ดูเหมือนจะบวกค่าไคบนตัวหารทุกตัวของ n แล้วคูณด้วย 4 ตอนนี้เราเอา 4 อันนั้นไปวางไว้ตรงมุม แล้วอย่าลืมนำกลับมาทีหลัง ในตอนแรกการเพิ่มค่าสำหรับแต่ละแถวเหล่านี้ดูเหมือนจะสุ่มสุด ๆ ใช่ไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point. ", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4. ", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3. ", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better. ", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will g ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", + "input": "et better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", "translatedText": "ประมาณครึ่งหนึ่งของพวกเขา นั่นจะคิดเป็น r2 ส่วน 2 คูณไคของ 2 ประมาณหนึ่งในสามของแถวนี้มีไคเป็น 3 เราก็ใส่ r2 หารด้วย 3 คูณไคของ 3 ได้ จำไว้ว่าเราเป็นแบบประมาณ เนื่องจาก r2 อาจหาร 2 หรือ 3 ได้ไม่ลงตัว แต่เมื่อ r ขยายไปสู่อนันต์ การประมาณนี้ก็จะดีขึ้น และเมื่อคุณทำต่อไปแบบนี้ คุณจะได้นิพจน์ที่ค่อนข้างเป็นระเบียบ สำหรับจำนวนจุดขัดแตะทั้งหมด และถ้าคุณแยก r2 นั้นออกมา แล้วนำ 4 กลับมาที่ต้องคูณ หมายความว่าจำนวนจุดขัดแตะทั้งหมดภายในวงกลมใหญ่นี้ จะอยู่ที่ประมาณ 4 คูณ r2 คูณผลรวมนี้ และเนื่องจากไคเป็น 0 ในทุกเลขคู่ และมันแกว่งระหว่าง 1 ถึงลบ 1 สำหรับเลขคี่ ผลรวมนี้จึงดูเหมือน 1 ลบ 1 ใน 3 บวก 1 ใน 5 ลบ 1 ใน 7 และต่อๆ ไป และนี่คือสิ่งที่เราต้องการจริงๆ! ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum. ", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/turkish/sentence_translations.json b/2017/leibniz-formula/turkish/sentence_translations.json index 40352e5e8..80cc1104b 100644 --- a/2017/leibniz-formula/turkish/sentence_translations.json +++ b/2017/leibniz-formula/turkish/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points.", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points.", "translatedText": "Eğer yarıçap 1'e bakarsanız, bu 4 kafes noktasına ulaşır.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "5'lik bir yarıçap karekökü aslında 8 kafes noktasına çarpıyor.", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", "translatedText": "Yani buradaki kafes noktasının tamsayı koordinat çifti olan 3,4 yerine, onu tek bir karmaşık sayı olan 3 artı 4i olarak düşünün.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 397.06 }, { - "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers will come into play.", + "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers are going to come into play.", "translatedText": "Sorumuzu bir çarpanlara ayırma problemine dönüştürüyor, bu da sonuçta asal sayılar arasındaki kalıpların devreye girmesine neden oluyor.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes.", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes.", "translatedText": "5, 13 veya 17 gibi 4'ün bir katının üzerinde olan asal sayılar her zaman tam olarak iki farklı Gauss asalına ayrılabilir.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers.", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers.", "translatedText": "Öte yandan, 3, 7 veya 11 gibi 4'ün katının üzerinde 3 olan asal sayılar Gauss tamsayılarının içinde daha fazla çarpanlara ayrılamaz.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25.", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25.", "translatedText": "Sağdaki her şey soldaki her şeyle eşlenik olduğundan, ortaya çıkan şey 25 ile çarpan karmaşık bir eşlenik çiftidir.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i?", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i?", "translatedText": "Gauss asal sayılarını çarpanlara ayırmanın, bunlardan bazılarını i, –1 veya –i ile çarptığınızda farklı görünebileceğini söylediğimi hatırlıyor musunuz?", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i.", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i.", "translatedText": "Bu durumda, 25'in çarpanlara ayrılmasını farklı şekilde yazabilirsiniz, belki bu 5'lerden birini –1 artı 2i çarpı –1 eksi 2i olarak bölebilirsiniz.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i.", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i.", "translatedText": "Bunun yaratacağı tek etki, toplam çıktıyı i, –1 veya –i ile çarpmak olacaktır.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees.", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees.", "translatedText": "Sol sütundan bu çarpımı alın ve bunu 90 derecenin katları olan dönüşlere karşılık gelen 1, i, –1 veya –i ile çarpmayı seçin.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", "translatedText": "Bu dört seçenek, sol sütundaki çarpımın 1, i, –1 veya –i ile çarpılması şeklindeki son dört seçenekle çarpıldığında, karekök uzaklığı 125 olan toplam 16 kafes noktası olduğu ortaya çıkar. Menşei.", "model": "google_nmt", "n_reviews": 0, @@ -1248,7 +1248,7 @@ "end": 1374.96 }, { - "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, 1 minus 1 plus 1 minus 1 plus 1.", + "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, it goes 1 minus 1 plus 1 minus 1 plus 1.", "translatedText": "Ancak bu durumda chi 3 eksi 1 olduğundan bu toplam 1 eksi 1 artı 1 eksi 1 artı 1 şeklinde salınır.", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", "translatedText": "Bu durumda 4 gibi çift bir kuvvetse, toplam 1 olur, bu da bölünemeyen 3'lerle ne yapılacağına dair tek bir seçeneğin olduğu gerçeğini özetler.", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point.", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point.", "translatedText": "Artık doruğa yaklaşıyoruz, işler düzenli görünmeye başlıyor, o yüzden durun ve düşünün, bu noktaya kadar her şeyin yolunda olduğundan emin olun.", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", "translatedText": "Yarıçapı 0 olan başlangıç noktasını göz ardı edelim, geri kalanın modelini takip etmiyor ve r'nin sonsuza doğru büyümesine izin verdiğimizde küçük bir nokta fark yaratmayacaktır.", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4.", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4.", "translatedText": "Yaptığımız tüm Gauss tamsayı, çarpanlara ayırma ve chi fonksiyonu işlemlerinden, her n'nin cevabı, n'nin her bölenindeki chi değerini toplamak ve 4 ile çarpmak gibi görünüyor.", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3.", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2.", "translatedText": "Bu satırların yaklaşık üçte biri 3'lük chi'ye sahiptir, dolayısıyla r2'yi 3 çarpı chi 3'e bölebiliriz.", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better.", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will", "translatedText": "Yaklaşık olduğumuzu unutmayın, çünkü r2 2 veya 3'ü tam olarak bölmeyebilir, ancak r sonsuza doğru büyüdükçe bu yaklaşım daha iyi hale gelecektir.", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", + "input": "get better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", "translatedText": "Ve böyle devam ettiğinizde, toplam kafes noktalarının sayısı için oldukça düzenli bir ifade elde edersiniz.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum.", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum.", "translatedText": "Ve eğer r2'yi dışarıda bırakırsanız ve çarpılması gereken 4'ü geri getirirseniz, bu, bu büyük daire içindeki kafes noktalarının toplam sayısının yaklaşık olarak 4 çarpı r2 çarpı bu toplam olduğu anlamına gelir.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/ukrainian/sentence_translations.json b/2017/leibniz-formula/ukrainian/sentence_translations.json index 3b8ba6d96..45dc64643 100644 --- a/2017/leibniz-formula/ukrainian/sentence_translations.json +++ b/2017/leibniz-formula/ukrainian/sentence_translations.json @@ -182,7 +182,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points.", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points.", "translatedText": "Якщо ви подивіться на радіус 1, це досягне 4 точок решітки.", "n_reviews": 0, "start": 248.22, @@ -210,7 +210,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points.", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice po", "translatedText": "Квадратний корінь радіуса з 5 насправді потрапляє на 8 точок решітки.", "n_reviews": 0, "start": 262.84, @@ -315,7 +315,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i.", "translatedText": "Отже, замість цієї точки решітки як пари цілих координат, 3,4, уявіть її як одне комплексне число, 3 плюс 4i.", "n_reviews": 0, "start": 360.4, @@ -343,7 +343,7 @@ "end": 397.06 }, { - "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers will come into play.", + "input": "It turns our question into a factoring problem, which is ultimately why patterns among prime numbers are going to come into play.", "translatedText": "Це перетворює наше запитання на проблему розкладу на множники, тому, зрештою, в гру вступають закономірності серед простих чисел.", "n_reviews": 0, "start": 397.78, @@ -497,7 +497,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes.", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes.", "translatedText": "Прості числа, які на одиницю перевищують число, кратне 4, наприклад 5, 13 або 17, завжди можна розкласти на два різних простих числа Гауса.", "n_reviews": 0, "start": 635.96, @@ -518,7 +518,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers.", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers.", "translatedText": "З іншого боку, прості числа, які на 3 перевищують число, кратне 4, наприклад 3, або 7, або 11, не можуть бути розкладені далі всередині цілих чисел Гауса.", "n_reviews": 0, "start": 663.44, @@ -616,7 +616,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25.", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25.", "translatedText": "Оскільки все праворуч є спряженим з усім ліворуч, виходить комплексно спряжена пара, яка множиться на 25.", "n_reviews": 0, "start": 806.44, @@ -672,14 +672,14 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i?", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i?", "translatedText": "Пам’ятаєте, як я згадував, що розкладання на прості числа Гауса може виглядати інакше, якщо ви помножите деякі з них на i, або –1, або –i?", "n_reviews": 0, "start": 870.18, "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i.", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i.", "translatedText": "У цьому випадку ви можете записати розклад 25 на множники по-іншому, можливо, розділивши одну з цих 5 на –1 плюс 2i помножити на –1 мінус 2i.", "n_reviews": 0, "start": 878.88, @@ -693,7 +693,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i.", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i.", "translatedText": "Єдиний ефект, який це матиме, це множення загального виходу на i, або –1, або –i.", "n_reviews": 0, "start": 896.0, @@ -707,7 +707,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees.", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees.", "translatedText": "Візьміть цей добуток із лівого стовпця та помножте його на 1, i, –1 або –i, що відповідає обертам, кратним 90 градусам.", "n_reviews": 0, "start": 908.8, @@ -749,7 +749,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin.", "translatedText": "Ці чотири варіанти, помножені на останні чотири варіанти множення добутку з лівого стовпця на 1, i, –1 або –i, означають, що існує загалом 16 точок решітки на відстані квадратного кореня 125 від походження.", "n_reviews": 0, "start": 959.66, @@ -1092,14 +1092,14 @@ "end": 1374.96 }, { - "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, 1 minus 1 plus 1 minus 1 plus 1.", + "input": "But in this case, since chi of 3 is negative 1, this sum oscillates, it goes 1 minus 1 plus 1 minus 1 plus 1.", "translatedText": "Але в цьому випадку, оскільки хі 3 дорівнює мінус 1, ця сума коливається: 1 мінус 1 плюс 1 мінус 1 плюс 1.", "n_reviews": 0, "start": 1375.04, "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's.", "translatedText": "Якщо це парний ступінь, наприклад 4 у цьому випадку, сума виходить рівною 1, що втілює той факт, що є лише один вибір, що робити з цими нерозділеними трійками.", "n_reviews": 0, "start": 1384.42, @@ -1134,7 +1134,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point.", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point.", "translatedText": "Зараз ми наближаємося до кульмінації, все починає виглядати організованим, тому зупиніться та подумайте, переконайтеся, що до цього моменту все добре.", "n_reviews": 0, "start": 1429.08, @@ -1232,14 +1232,14 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity.", "translatedText": "Давайте проігноруємо цю вихідну крапку з радіусом 0, вона не слідує шаблону решти, і одна маленька крапка не матиме значення, оскільки ми дозволимо r зростати до нескінченності.", "n_reviews": 0, "start": 1541.28, "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4.", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4.", "translatedText": "З усіх цих цілих чисел Гауса, розкладання на множники та функції хі, які ми робили, відповідь для кожного n виглядає як додавання значення хі до кожного дільника n і множення на 4.", "n_reviews": 0, "start": 1552.2, @@ -1316,28 +1316,28 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3.", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2.", "translatedText": "Приблизно третина цих рядків має хі, що дорівнює 3, тому ми можемо поділити r2 на 3, помножене на хі, 3.", "n_reviews": 0, "start": 1619.12, "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better.", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will", "translatedText": "Майте на увазі, що ми наближені, оскільки r2 може не ідеально ділити 2 або 3, але коли r зростає до нескінченності, це наближення ставатиме кращим.", "n_reviews": 0, "start": 1626.02, "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", + "input": "get better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points.", "translatedText": "І якщо ви продовжуєте так, ви отримуєте досить організований вираз для загальної кількості точок решітки.", "n_reviews": 0, "start": 1635.36, "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum.", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum.", "translatedText": "І якщо ви вилучите це r2 і повернете 4, на які потрібно помножити, це означатиме, що загальна кількість точок решітки всередині цього великого кола приблизно в 4 рази помножена на r2, помножену на цю суму.", "n_reviews": 0, "start": 1642.98, diff --git a/2017/leibniz-formula/urdu/sentence_translations.json b/2017/leibniz-formula/urdu/sentence_translations.json index 148b017ef..88d84e3ff 100644 --- a/2017/leibniz-formula/urdu/sentence_translations.json +++ b/2017/leibniz-formula/urdu/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points. ", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points. ", "translatedText": "اگر آپ رداس 1 کو دیکھیں تو یہ 4 جالی پوائنٹس سے ٹکراتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points. ", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice poi ", "translatedText": "5 کا ایک رداس مربع جڑ دراصل 8 جالی پوائنٹس سے ٹکراتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", "translatedText": "لہٰذا یہاں اس جالی نقطہ کے بجائے انٹیجر کوآرڈینیٹ کے جوڑے کے طور پر، 3,4، اس کے بجائے اسے واحد کمپلیکس نمبر، 3 جمع 4i سمجھیں۔", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes. ", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes. ", "translatedText": "پرائم نمبرز جو 4 کے ضرب سے اوپر ایک ہیں، جیسے 5، یا 13، یا 17، ہمیشہ بالکل دو الگ الگ گاوسی پرائمز میں فیکٹر کیے جا سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers. ", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers. ", "translatedText": "دوسری طرف، بنیادی اعداد جو کہ 4 کے ضرب سے اوپر 3 ہیں، جیسے 3، یا 7، یا 11، کو گاوسی انٹیجرز کے اندر مزید فیکٹر نہیں کیا جا سکتا۔", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25. ", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25. ", "translatedText": "چونکہ دائیں طرف کی ہر چیز بائیں طرف کی ہر چیز کے ساتھ ایک کنجوجٹ ہے، اس لیے جو نکلتا ہے وہ ایک پیچیدہ کنجوگیٹ جوڑا ہے جو 25 تک ضرب کرتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i? ", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i? ", "translatedText": "لیکن یہ نسخہ ابھی تک تمام 12 جعلی نکات کو کیوں نہیں پکڑتا ہے؟ یاد رکھیں کہ میں نے کس طرح ذکر کیا تھا کہ اگر آپ ان میں سے کچھ کو i، یا -1، یا -i سے ضرب دیتے ہیں تو گاوسی پرائمز میں فیکٹرائزیشن مختلف نظر آسکتی ہے؟ اس صورت میں، آپ 25 کی فیکٹرائزیشن کو مختلف طریقے سے لکھ سکتے ہیں، ہو سکتا ہے کہ ان 5s میں سے ایک کو -1 جمع 2i بار -1 مائنس 2i کے طور پر تقسیم کریں۔", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i. ", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i. ", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i. ", "translatedText": "اس کا واحد اثر یہ ہوگا کہ کل آؤٹ پٹ کو i، یا -1، یا -i سے ضرب کیا جائے۔", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees. ", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees. ", "translatedText": "اس پروڈکٹ کو بائیں کالم سے لیں، اور اسے 1، i، -1، یا -i سے ضرب کرنے کا انتخاب کریں، ان گردشوں کے مطابق جو 90 ڈگری کے کچھ کثیر ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", "translatedText": "وہ چار انتخاب، جن کو بائیں کالم سے مصنوع کو 1، i، -1، یا -i سے ضرب کرنے کے آخری چار انتخاب سے ضرب دیا گیا ہے، یہ تجویز کریں گے کہ کل 16 جالی پوائنٹس ہیں، ایک فاصلہ مربع جڑ سے 125 کے فاصلے پر۔اصل. ", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", "translatedText": "اگر یہ یکساں طاقت ہے، جیسا کہ اس معاملے میں 4، رقم نکلتی ہے 1، جو اس حقیقت کو سمیٹتی ہے کہ ان غیر تقسیم شدہ 3 کے ساتھ کیا کرنا ہے اس کے لیے صرف ایک ہی انتخاب ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point. ", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point. ", "translatedText": "ہم اب انتہا کے قریب پہنچ رہے ہیں، چیزیں منظم نظر آنے لگی ہیں، اس لیے رکیں اور غور کریں، اس بات کو یقینی بنائیں کہ اس مقام تک سب کچھ اچھا لگتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", "translatedText": "آئیے رداس 0 کے ساتھ اس اصلی نقطے کو نظر انداز کرتے ہیں، یہ باقی کے پیٹرن کی پیروی نہیں کرتا ہے، اور ایک چھوٹا سا نقطہ فرق نہیں کرے گا کیونکہ ہم اسے لامحدودیت کی طرف بڑھنے دیتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4. ", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4. ", "translatedText": "اس تمام گاوسی انٹیجر اور فیکٹرنگ اور چی فنکشن سے جو ہم کر رہے ہیں، ہر n کا جواب n کے ہر تقسیم پر chi کی قدر کو شامل کرنے اور 4 سے ضرب کرنے جیسا لگتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3. ", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2. ", "translatedText": "ان قطاروں میں سے تقریباً ایک تہائی میں 3 کی chi ہوتی ہے، لہذا ہم r2 کو 3 کے chi سے تقسیم کر کے رکھ سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better. ", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will g ", "translatedText": "ذہن میں رکھیں کہ ہم تخمینہ لگا رہے ہیں، کیونکہ r2 شاید 2 یا 3 کو مکمل طور پر تقسیم نہ کرے، لیکن جیسے جیسے r لامحدودیت کی طرف بڑھتا جائے گا، یہ تخمینہ بہتر ہوتا جائے گا۔", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", + "input": "et better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", "translatedText": "اور جب آپ اس طرح چلتے رہتے ہیں، تو آپ کو جالی پوائنٹس کی کل تعداد کے لیے ایک خوبصورت منظم اظہار ملتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum. ", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum. ", "translatedText": "اور اگر آپ اس r2 کو فیکٹر کرتے ہیں اور 4 کو واپس لاتے ہیں جس میں ضرب لگانے کی ضرورت ہے، تو اس کا مطلب یہ ہے کہ اس بڑے دائرے کے اندر جالی پوائنٹس کی کل تعداد اس رقم سے تقریباً 4 گنا r2 گنا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/leibniz-formula/vietnamese/sentence_translations.json b/2017/leibniz-formula/vietnamese/sentence_translations.json index 7b65fd885..7e896ff69 100644 --- a/2017/leibniz-formula/vietnamese/sentence_translations.json +++ b/2017/leibniz-formula/vietnamese/sentence_translations.json @@ -208,7 +208,7 @@ "end": 247.36 }, { - "input": "If you look at the radius 1, that hits 4 lattice points. ", + "input": "If you look at the radius 1, that hits 4 different lattice points. Radius square root of 2, well that also hits 4 lattice points. ", "translatedText": "Nếu bạn nhìn vào bán kính 1, nó chạm tới 4 điểm mạng. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 262.18 }, { - "input": "A radius square root of 5 actually hits 8 lattice points. ", + "input": "A radius square root of 3 doesn't actually hit anything. Square root of 4 again hits 4 lattice points. A radius square root of 5 actually hits 8 lattice poi ", "translatedText": "Căn bậc hai bán kính của 5 thực sự chạm tới 8 điểm mạng. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.34 }, { - "input": "So instead of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", + "input": "So instead of thinking of this lattice point here as the pair of integer coordinates, 3,4, instead think of it as the single complex number, 3 plus 4i. ", "translatedText": "Vì vậy, thay vì điểm mạng này ở đây là cặp tọa độ nguyên, 3,4, hãy nghĩ nó như một số phức đơn, 3 cộng 4i. ", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 635.04 }, { - "input": "Prime numbers that are one above a multiple of 4, like 5, or 13, or 17, can always be factored into exactly two distinct Gaussian primes. ", + "input": "Prime numbers that are 1 above a multiple of 4, like 5, or 13, or 17, these guys can always be factored into exactly two distinct Gaussian primes. ", "translatedText": "Các số nguyên tố lớn hơn bội số của 4 một đơn vị, chẳng hạn như 5, 13 hoặc 17, luôn có thể được phân tích thành chính xác hai số nguyên tố Gaussian riêng biệt. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 660.44 }, { - "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, cannot be factored further inside the Gaussian integers. ", + "input": "On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11, these guys cannot be factored further inside the Gaussian integers. ", "translatedText": "Mặt khác, các số nguyên tố lớn hơn 3 trên bội số của 4, như 3, 7 hoặc 11, không thể được phân tích thêm bên trong số nguyên Gaussian. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 806.44 }, { - "input": "Because everything on the right is a conjugate with everything on the left, what comes out is a complex conjugate pair which multiplies to 25. ", + "input": "And because everything on the right is a conjugate with everything on the left, what comes out is going to be a complex conjugate pair which multiplies to 25. ", "translatedText": "Bởi vì mọi thứ ở bên phải đều là liên hợp với mọi thứ ở bên trái, kết quả là một cặp liên hợp phức tạp nhân lên 25. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 869.12 }, { - "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i, or –1, or –i? ", + "input": "Remember how I mentioned that a factorization into Gaussian primes can look different if you multiply some of them by i or negative 1, negative i? ", "translatedText": "Hãy nhớ tôi đã đề cập rằng việc phân tích thành thừa số nguyên tố Gaussian có thể trông khác nếu bạn nhân một số trong số chúng với i, hoặc –1 hoặc –i? ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 878.06 }, { - "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those 5s as –1 plus 2i times –1 minus 2i. ", + "input": "In this case, you could write the factorization of 25 differently, maybe splitting up one of those fives as negative 1 plus 2i times negative 1 minus 2i. ", "translatedText": "Trong trường hợp này, bạn có thể viết hệ số của 25 theo cách khác, có thể chia một trong 5 số đó thành –1 cộng 2i nhân –1 trừ 2i. ", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 894.98 }, { - "input": "The only effect this will have is to multiply the total output by i, or –1, or –i. ", + "input": "But the only effect that this is going to have is to multiply that total output by i, or negative 1, or negative i. ", "translatedText": "Tác dụng duy nhất mà điều này mang lại là nhân tổng sản lượng với i, hoặc –1 hoặc –i. ", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 908.14 }, { - "input": "Take that product from the left column, and choose to multiply it by 1, i, –1, or –i, corresponding to rotations that are some multiple of 90 degrees. ", + "input": "Take that product from the left column and choose to multiply it by 1, i, negative 1, or negative i, corresponding to rotations that are some multiple of 90 degrees. ", "translatedText": "Lấy tích đó từ cột bên trái và chọn nhân nó với 1, i, –1 hoặc –i, tương ứng với các phép quay là bội số của 90 độ. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 958.92 }, { - "input": "Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1, i, –1, or –i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", + "input": "Those four choices multiplied by the final four choices of multiplying the product from the left column by 1, or by i, or negative 1, or negative i, would suggest that there are a total of 16 lattice points a distance square root of 125 away from the origin. ", "translatedText": "Bốn lựa chọn đó, nhân với bốn lựa chọn cuối cùng là nhân kết quả ở cột bên trái với 1, i, –1 hoặc –i, sẽ gợi ý rằng có tổng cộng 16 điểm mạng, cách căn bậc hai khoảng cách là 125. nguồn gốc. ", "model": "google_nmt", "n_reviews": 0, @@ -1256,7 +1256,7 @@ "end": 1383.72 }, { - "input": "If it's an even power, like 4 in this case, the sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", + "input": "And if it's an even power, like 4 in this case, the total sum comes out to be 1, which encapsulates the fact that there is only one choice for what to do with those unsplittable 3's. ", "translatedText": "Nếu đó là lũy thừa chẵn, chẳng hạn như 4 trong trường hợp này, thì tổng sẽ là 1, điều này gói gọn thực tế là chỉ có một lựa chọn về việc phải làm gì với những số 3 không thể chia được đó. ", "model": "google_nmt", "n_reviews": 0, @@ -1296,7 +1296,7 @@ "end": 1427.9 }, { - "input": "We're getting close to the culmination now, things are starting to look organized, so pause and ponder, make sure everything feels good up to this point. ", + "input": "We're getting close to the culmination now. Things are starting to look organized, so take a moment, pause and ponder, make sure everything feels good up to this point. ", "translatedText": "Bây giờ chúng ta đang tiến gần đến đỉnh điểm, mọi thứ đang bắt đầu có vẻ ngăn nắp, vì vậy hãy tạm dừng và suy ngẫm, đảm bảo rằng mọi thứ đều ổn cho đến thời điểm này. ", "model": "google_nmt", "n_reviews": 0, @@ -1408,7 +1408,7 @@ "end": 1540.28 }, { - "input": "Let's ignore that origin dot with radius 0, it doesn't follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", + "input": "Let's go ahead and just ignore that origin dot with radius 0, it doesn't really follow the pattern of the rest, and one little dot isn't going to make a difference as we let r grow towards infinity. ", "translatedText": "Hãy bỏ qua dấu chấm gốc có bán kính 0, nó không tuân theo quy luật của những dấu chấm còn lại và một dấu chấm nhỏ sẽ không tạo ra sự khác biệt khi chúng ta để r phát triển về phía vô cùng. ", "model": "google_nmt", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1551.06 }, { - "input": "From all this Gaussian integer and factoring and chi function stuff we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and multiplying by 4. ", + "input": "Now from all of this Gaussian integer and factoring and chi function stuff that we've been doing, the answer for each n looks like adding up the value of chi on every divisor of n, and then multiplying by 4. ", "translatedText": "Từ tất cả những thứ về số nguyên Gaussian, phân tích nhân tử và hàm chi mà chúng ta đang làm, câu trả lời cho mỗi n trông giống như việc cộng giá trị của chi trên mọi ước số của n và nhân với 4. ", "model": "google_nmt", "n_reviews": 0, @@ -1504,7 +1504,7 @@ "end": 1618.52 }, { - "input": "About a third of these rows have chi of 3, so we can put in r2 divided by 3 times chi of 3. ", + "input": "How many of them have 2 as a divisor? Well, about half of them. So that would account for about r squared over 2 times chi of 2. ", "translatedText": "Khoảng một phần ba số hàng này có chi bằng 3, vì vậy chúng ta có thể đặt r2 chia cho 3 nhân chi bằng 3. ", "model": "google_nmt", "n_reviews": 0, @@ -1512,7 +1512,7 @@ "end": 1625.12 }, { - "input": "Keep in mind we're being approximate, since r2 might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will get better. ", + "input": "About a third of these rows have chi of 3, so we can put in r squared divided by 3 times chi of 3. And keep in mind we're being approximate since r squared might not perfectly divide 2 or 3, but as r grows towards infinity, this approximation will g ", "translatedText": "Hãy nhớ rằng chúng ta đang tính gần đúng, vì r2 có thể không chia hoàn toàn cho 2 hoặc 3, nhưng khi r tiến dần đến vô cùng, phép tính gần đúng này sẽ tốt hơn. ", "model": "google_nmt", "n_reviews": 0, @@ -1520,7 +1520,7 @@ "end": 1634.54 }, { - "input": "And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", + "input": "et better. And when you keep going like this, you get a pretty organized expression for the total number of lattice points. ", "translatedText": "Và khi bạn tiếp tục như thế này, bạn sẽ có được một biểu thức có tổ chức khá tốt cho tổng số điểm mạng. ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1640.2 }, { - "input": "And if you factor out that r2 and bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r2 times this sum. ", + "input": "And if you factor out that r squared and then bring back the 4 that needs to be multiplied in, what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum. ", "translatedText": "Và nếu bạn tính r2 ra nhân tử và mang về 4 cần nhân, điều đó có nghĩa là tổng số điểm mạng bên trong vòng tròn lớn này xấp xỉ 4 nhân r2 nhân tổng này. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/bengali/sentence_translations.json b/2017/light-quantum-mechanics/bengali/sentence_translations.json index 44cde1bec..bbb3e219d 100644 --- a/2017/light-quantum-mechanics/bengali/sentence_translations.json +++ b/2017/light-quantum-mechanics/bengali/sentence_translations.json @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number. ", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number. ", "translatedText": "ফোটনকে এখনও দুটি দোদুল্যমান উপাদানের একটি সুপারপজিশন হিসাবে বর্ণনা করা হয়েছে, প্রতিটিতে কিছু ফেজ এবং প্রশস্ততা রয়েছে, প্রায়শই একটি একক জটিল সংখ্যা ব্যবহার করে এনকোড করা হয়।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/chinese/sentence_translations.json b/2017/light-quantum-mechanics/chinese/sentence_translations.json index f32e8552a..6b95259a0 100644 --- a/2017/light-quantum-mechanics/chinese/sentence_translations.json +++ b/2017/light-quantum-mechanics/chinese/sentence_translations.json @@ -1049,7 +1049,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number. ", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number. ", "translatedText": "光子仍然被描述为两个振荡分量的叠加，每个振荡分量 都有一定的相位和幅度，通常使用单个复数进行编码。", "model": "google_nmt", "from_community_srt": "我想强调一点：波动方程并没有改变 光子还是用两个振荡分量的叠加描述 每个分量都有相位和振幅 这两点信息通常用一个复数表示 区别之处在于 在经典的观点中 每个分量的振幅的平方 指的是波在这个方向上的能量大小", diff --git a/2017/light-quantum-mechanics/english/captions.srt b/2017/light-quantum-mechanics/english/captions.srt index 6556c0be1..69b5415ae 100644 --- a/2017/light-quantum-mechanics/english/captions.srt +++ b/2017/light-quantum-mechanics/english/captions.srt @@ -1143,11 +1143,11 @@ Classically, you might think of its horizontal component as having energy proportional to 0.38 squared, which is around 0.15. 287 -00:17:49,820 --> 00:17:54,836 +00:17:49,820 --> 00:17:53,780 Likewise, you might think of the vertical component as having an energy proportional to 0. 288 -00:17:54,836 --> 00:17:57,400 +00:17:53,780 --> 00:17:57,400 92 squared, which comes out to be around 0.85. 289 diff --git a/2017/light-quantum-mechanics/english/sentence_timings.json b/2017/light-quantum-mechanics/english/sentence_timings.json index 595287a0f..0948a20dc 100644 --- a/2017/light-quantum-mechanics/english/sentence_timings.json +++ b/2017/light-quantum-mechanics/english/sentence_timings.json @@ -600,8 +600,13 @@ 1069.2 ], [ - "Likewise, you might think of the vertical component as having an energy proportional to 0.92 squared, which comes out to be around 0.85.", + "Likewise, you might think of the vertical component as having an energy proportional to 0.", 1069.82, + 1073.78 + ], + [ + "92 squared, which comes out to be around 0.85.", + 1073.78, 1077.4 ], [ diff --git a/2017/light-quantum-mechanics/german/sentence_translations.json b/2017/light-quantum-mechanics/german/sentence_translations.json index 73929c69d..f627545ec 100644 --- a/2017/light-quantum-mechanics/german/sentence_translations.json +++ b/2017/light-quantum-mechanics/german/sentence_translations.json @@ -376,7 +376,7 @@ "end": 453.38 }, { - "input": "Right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", + "input": "Now right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, and in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", "translatedText": "Im Moment ist die resultierende Überlagerung eine Welle, die in diagonaler Richtung wackelt, aber wenn die horizontalen und vertikalen Komponenten zueinander phasenverschoben wären, was passieren könnte, wenn man die Phasenverschiebung in einer von ihnen erhöht, könnte ihre Summe stattdessen verschwinden Eine Art Ellipse. Wenn die Phasen genau um 90 Grad voneinander abweichen und die Amplituden beide gleich sind, nennen wir dies zirkular polarisiertes Licht.", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number.", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number.", "translatedText": "Das Photon wird immer noch als Überlagerung zweier oszillierender Komponenten mit jeweils einer bestimmten Phase und Amplitude beschrieben, die oft mit einer einzigen komplexen Zahl codiert werden.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/hebrew/sentence_translations.json b/2017/light-quantum-mechanics/hebrew/sentence_translations.json index 4e0d68912..355ced442 100644 --- a/2017/light-quantum-mechanics/hebrew/sentence_translations.json +++ b/2017/light-quantum-mechanics/hebrew/sentence_translations.json @@ -329,7 +329,7 @@ "end": 453.38 }, { - "input": "Right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", + "input": "Now right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, and in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", "translatedText": "כרגע, הסופרפוזיציה המתקבלת היא גל שמתנועע בכיוון האלכסוני, אבל אם הרכיבים האופקיים והאנכיים היו מחוץ לפאזה זה עם זה, מה שעלול לקרות אם תגדיל את הסטת הפאזה באחד מהם, הסכום שלהם עשוי להתחקות איזושהי אליפסה, במקרה שבו השלבים אינם מסונכרנים זה עם זה בדיוק ב-90 מעלות, והמשרעות שתיהן שוות, זה מה שאנו מכנים אור מקוטב מעגלי.", "n_reviews": 0, "start": 454.3, @@ -868,7 +868,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number.", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number.", "translatedText": "הפוטון עדיין מתואר כסופרפוזיציה של שני רכיבים מתנודדים, שלכל אחד מהם פאזה ומשרעת, לרוב מקודדים באמצעות מספר מרוכב יחיד.", "n_reviews": 0, "start": 1109.04, diff --git a/2017/light-quantum-mechanics/hindi/sentence_translations.json b/2017/light-quantum-mechanics/hindi/sentence_translations.json index a0970e6b6..c5b003971 100644 --- a/2017/light-quantum-mechanics/hindi/sentence_translations.json +++ b/2017/light-quantum-mechanics/hindi/sentence_translations.json @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number. ", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number. ", "translatedText": "फोटॉन को अभी भी दो दोलनशील घटकों के सुपरपोजिशन के रूप में वर्णित किया गया है, जिनमें से प्रत्येक में कुछ चरण और आयाम हैं, जिन्हें अक्सर एकल जटिल संख्या का उपयोग करके एन्कोड किया जाता है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/indonesian/sentence_translations.json b/2017/light-quantum-mechanics/indonesian/sentence_translations.json index 9a7944317..f8bf60a0d 100644 --- a/2017/light-quantum-mechanics/indonesian/sentence_translations.json +++ b/2017/light-quantum-mechanics/indonesian/sentence_translations.json @@ -376,7 +376,7 @@ "end": 453.38 }, { - "input": "Right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", + "input": "Now right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, and in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", "translatedText": "Saat ini, superposisi yang dihasilkan adalah gelombang yang bergerak ke arah diagonal, tetapi jika komponen horizontal dan vertikal tidak sefase satu sama lain, yang mungkin terjadi jika Anda meningkatkan pergeseran fasa pada salah satu komponen tersebut, jumlah keduanya mungkin akan terlacak. semacam elips, dalam kasus di mana fase-fasanya tepat 90 derajat tidak sinkron satu sama lain, dan amplitudo keduanya sama, inilah yang kita sebut cahaya terpolarisasi sirkuler.", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number.", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number.", "translatedText": "Foton masih digambarkan sebagai superposisi dua komponen yang berosilasi, masing-masing dengan fase dan amplitudo tertentu, sering kali dikodekan menggunakan bilangan kompleks tunggal.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/japanese/sentence_translations.json b/2017/light-quantum-mechanics/japanese/sentence_translations.json index f7f88d471..8e8769ce0 100644 --- a/2017/light-quantum-mechanics/japanese/sentence_translations.json +++ b/2017/light-quantum-mechanics/japanese/sentence_translations.json @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number. ", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number. ", "translatedText": "光子は依然として 2 つの振動成分の重ね合わせとして説明され、それぞれが何ら かの位相と振幅を持ち、多くの場合単一の複素数を使用してエンコードされます。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/marathi/sentence_translations.json b/2017/light-quantum-mechanics/marathi/sentence_translations.json index f06e8129c..37ba79135 100644 --- a/2017/light-quantum-mechanics/marathi/sentence_translations.json +++ b/2017/light-quantum-mechanics/marathi/sentence_translations.json @@ -376,7 +376,7 @@ "end": 453.38 }, { - "input": "Right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", + "input": "Now right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, and in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", "translatedText": "आत्ता, परिणामी सुपरपोझिशन ही कर्ण दिशेने फिरणारी लहर आहे, परंतु जर क्षैतिज आणि अनुलंब घटक एकमेकांच्या फेजच्या बाहेर असतील, जे तुम्ही त्यापैकी एकामध्ये फेज शिफ्ट वाढवल्यास, त्यांची बेरीज कदाचित ट्रेस आउट होऊ शकते. काही प्रकारचे लंबवर्तुळ, जेव्हा टप्प्यांचे एकमेकांशी बरोबर 90 अंश समक्रमित नसतात आणि मोठेपणा दोन्ही समान असतात, यालाच आपण वर्तुळाकार ध्रुवीकृत प्रकाश म्हणतो.", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number.", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number.", "translatedText": "फोटॉनचे वर्णन अजूनही दोन दोलायमान घटकांचे सुपरपोझिशन म्हणून केले जाते, प्रत्येक काही फेज आणि मोठेपणासह, अनेकदा एकच जटिल संख्या वापरून एन्कोड केले जाते.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/persian/sentence_translations.json b/2017/light-quantum-mechanics/persian/sentence_translations.json index 466ac2c95..f909ae0d2 100644 --- a/2017/light-quantum-mechanics/persian/sentence_translations.json +++ b/2017/light-quantum-mechanics/persian/sentence_translations.json @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number. ", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number. ", "translatedText": "فوتون هنوز به عنوان برهم نهی از دو جزء نوسانی توصیف می شود که هر کدام دارای فاز و دامنه ای هستند که اغلب با استفاده از یک عدد مختلط رمزگذاری می شوند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/spanish/sentence_translations.json b/2017/light-quantum-mechanics/spanish/sentence_translations.json index f3375ebcd..451333bf2 100644 --- a/2017/light-quantum-mechanics/spanish/sentence_translations.json +++ b/2017/light-quantum-mechanics/spanish/sentence_translations.json @@ -376,7 +376,7 @@ "end": 453.38 }, { - "input": "Right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", + "input": "Now right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, and in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", "translatedText": "En este momento, la superposición resultante es una onda que se mueve en dirección diagonal, pero si los componentes horizontal y vertical estuvieran desfasados entre sí, lo que podría suceder si aumenta el cambio de fase en uno de ellos, su suma podría trazarse una especie de elipse, en el caso en que las fases están exactamente desincronizadas 90 grados entre sí y las amplitudes son ambas iguales, esto es lo que llamamos luz polarizada circularmente.", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number.", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number.", "translatedText": "El fotón todavía se describe como una superposición de dos componentes oscilantes, cada uno con cierta fase y amplitud, a menudo codificados mediante un único número complejo.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/tamil/sentence_translations.json b/2017/light-quantum-mechanics/tamil/sentence_translations.json index 3ed81489e..06a21f08b 100644 --- a/2017/light-quantum-mechanics/tamil/sentence_translations.json +++ b/2017/light-quantum-mechanics/tamil/sentence_translations.json @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number. ", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number. ", "translatedText": "ஃபோட்டான் இன்னும் இரண்டு ஊசலாடும் கூறுகளின் சூப்பர்போசிஷனாக விவரிக்கப்படுகிறது, ஒவ்வொன்றும் சில கட்டம் மற்றும் வீச்சு, பெரும்பாலும் ஒற்றை கலப்பு எண்ணைப் பயன்படுத்தி குறியாக்கம் செய்யப்படுகிறது. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/telugu/sentence_translations.json b/2017/light-quantum-mechanics/telugu/sentence_translations.json index 56c22e821..c46d390b8 100644 --- a/2017/light-quantum-mechanics/telugu/sentence_translations.json +++ b/2017/light-quantum-mechanics/telugu/sentence_translations.json @@ -376,7 +376,7 @@ "end": 453.38 }, { - "input": "Right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", + "input": "Now right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, and in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", "translatedText": "ప్రస్తుతం, ఫలితంగా ఏర్పడిన సూపర్‌పొజిషన్ అనేది వికర్ణ దిశలో కదలాడుతోంది, అయితే క్షితిజ సమాంతర మరియు నిలువు భాగాలు ఒకదానికొకటి దశకు దూరంగా ఉంటే, మీరు వాటిలో ఒకదానిలో దశ మార్పును పెంచినట్లయితే, వాటి మొత్తం కనుగొనబడవచ్చు. ఒకరకమైన దీర్ఘవృత్తాకారంలో, దశలు ఒకదానితో ఒకటి సరిగ్గా 90 డిగ్రీలు సమకాలీకరించబడనప్పుడు మరియు ఆంప్లిట్యూడ్‌లు రెండూ సమానంగా ఉంటే, దీనిని మనం వృత్తాకార ధ్రువణ కాంతి అని పిలుస్తాము.", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number.", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number.", "translatedText": "ఫోటాన్ ఇప్పటికీ రెండు డోలనం భాగాల యొక్క సూపర్‌పొజిషన్‌గా వర్ణించబడింది, ప్రతి ఒక్కటి కొంత దశ మరియు వ్యాప్తితో, తరచుగా ఒకే సంక్లిష్ట సంఖ్యను ఉపయోగించి ఎన్‌కోడ్ చేయబడుతుంది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/thai/sentence_translations.json b/2017/light-quantum-mechanics/thai/sentence_translations.json index 1482e0317..632726583 100644 --- a/2017/light-quantum-mechanics/thai/sentence_translations.json +++ b/2017/light-quantum-mechanics/thai/sentence_translations.json @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number. ", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/turkish/sentence_translations.json b/2017/light-quantum-mechanics/turkish/sentence_translations.json index a9c1a91f7..fb29945f0 100644 --- a/2017/light-quantum-mechanics/turkish/sentence_translations.json +++ b/2017/light-quantum-mechanics/turkish/sentence_translations.json @@ -376,7 +376,7 @@ "end": 453.38 }, { - "input": "Right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", + "input": "Now right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, and in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", "translatedText": "Şu anda ortaya çıkan süperpozisyon çapraz yönde hareket eden bir dalgadır, ancak yatay ve dikey bileşenler birbiriyle faz dışıysa, ki bu içlerinden birindeki faz kaymasını arttırırsanız meydana gelebilir, bunların toplamı bunun yerine izlenebilir. bir tür elips, fazların birbiriyle tam olarak 90 derece uyumsuz olduğu ve genliklerin her ikisinin de eşit olduğu durumda, buna dairesel polarize ışık diyoruz.", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number.", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number.", "translatedText": "Foton hâlâ, her biri belirli bir faza ve genliğe sahip olan ve sıklıkla tek bir karmaşık sayı kullanılarak kodlanan iki salınımlı bileşenin süperpozisyonu olarak tanımlanmaktadır.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/ukrainian/sentence_translations.json b/2017/light-quantum-mechanics/ukrainian/sentence_translations.json index b88bffd3a..044345b5e 100644 --- a/2017/light-quantum-mechanics/ukrainian/sentence_translations.json +++ b/2017/light-quantum-mechanics/ukrainian/sentence_translations.json @@ -329,7 +329,7 @@ "end": 453.38 }, { - "input": "Right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", + "input": "Now right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, and in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", "translatedText": "На даний момент отримана суперпозиція — це хвиля, що коливається в діагональному напрямку, але якщо горизонтальна і вертикальна складові не мають фази одна з одною, що може статися, якщо ви збільшите фазовий зсув в одній із них, їхня сума може натомість вийти свого роду еліпс, у випадку, коли фази точно не синхронізовані одна з одною на 90 градусів, а амплітуди обидві рівні, це те, що ми називаємо циркулярно поляризованим світлом.", "n_reviews": 0, "start": 454.3, @@ -868,7 +868,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number.", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number.", "translatedText": "Фотон все ще описується як суперпозиція двох осцилюючих компонентів, кожна з яких має певну фазу та амплітуду, часто кодовану за допомогою одного комплексного числа.", "n_reviews": 0, "start": 1109.04, diff --git a/2017/light-quantum-mechanics/urdu/sentence_translations.json b/2017/light-quantum-mechanics/urdu/sentence_translations.json index 005e4565f..1088c491a 100644 --- a/2017/light-quantum-mechanics/urdu/sentence_translations.json +++ b/2017/light-quantum-mechanics/urdu/sentence_translations.json @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number. ", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number. ", "translatedText": "فوٹون کو اب بھی دو دوغلی اجزاء کی ایک سپرپوزیشن کے طور پر بیان کیا جاتا ہے، ہر ایک کچھ فیز اور طول و عرض کے ساتھ، اکثر ایک ہی پیچیدہ نمبر کا استعمال کرتے ہوئے انکوڈ کیا جاتا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/light-quantum-mechanics/vietnamese/sentence_translations.json b/2017/light-quantum-mechanics/vietnamese/sentence_translations.json index ad3e417e9..b9ddc3c27 100644 --- a/2017/light-quantum-mechanics/vietnamese/sentence_translations.json +++ b/2017/light-quantum-mechanics/vietnamese/sentence_translations.json @@ -376,7 +376,7 @@ "end": 453.38 }, { - "input": "Right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", + "input": "Now right now, the resulting superposition is a wave wiggling in the diagonal direction, but if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them, their sum might instead trace out some sort of ellipse, and in the case where the phases are exactly 90 degrees out of sync with each other, and the amplitudes are both equal, this is what we call circularly polarized light.", "translatedText": "Hiện tại, sự chồng chất thu được là một sóng lắc lư theo hướng chéo, nhưng nếu các thành phần ngang và dọc lệch pha nhau, điều này có thể xảy ra nếu bạn tăng độ lệch pha ở một trong số chúng, thay vào đó, tổng của chúng có thể bị lệch pha một loại hình elip nào đó, trong trường hợp các pha không đồng bộ với nhau chính xác 90 độ và biên độ đều bằng nhau, đây là cái mà chúng ta gọi là ánh sáng phân cực tròn.", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1108.76 }, { - "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, often encoded using a single complex number.", + "input": "The photon is still described as a superposition of two oscillating components, each with some phase and amplitude, and these are often encoded using a single complex number.", "translatedText": "Photon vẫn được mô tả là sự chồng chất của hai thành phần dao động, mỗi thành phần có pha và biên độ nào đó, thường được mã hóa bằng một số phức duy nhất.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/arabic/sentence_translations.json b/2017/limits/arabic/sentence_translations.json index 3bbd3ff29..aa179d6ac 100644 --- a/2017/limits/arabic/sentence_translations.json +++ b/2017/limits/arabic/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", "translatedText": "هناك آخرون يرغبون في تفسير dx على أنه تغيير صغير للغاية، أو مجرد القول بأن dx وdf ليسا أكثر من مجرد رموز لا ينبغي لنا أن نأخذها على محمل الجد. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach? ", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach? ", "translatedText": "لكن تخيل أنك عالم رياضيات تخترع حساب التفاضل والتكامل، ويسألك أحدهم، حسنًا، ماذا تقصد بالضبط بالمقاربة؟ ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 379.86 }, { - "input": "We all know what it means for one value to get closer to another. ", + "input": "I mean, come on, we all know what it means for one value to get closer to another. ", "translatedText": "نعلم جميعًا ما يعنيه أن تقترب قيمة ما من قيمة أخرى. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus. ", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus. ", "translatedText": "يمكنك القول بأن هذا واجب ثقيل بلا داعٍ كمقدمة لحساب التفاضل والتكامل. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", "translatedText": "إذا كنت تعرف ما تعنيه كلمة نهج، فأنت تعرف بالفعل ما يعنيه الحد، فلا يوجد شيء جديد على المستوى المفاهيمي هنا. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/bengali/sentence_translations.json b/2017/limits/bengali/sentence_translations.json index a63b46f36..b3eaa5eb6 100644 --- a/2017/limits/bengali/sentence_translations.json +++ b/2017/limits/bengali/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", "translatedText": "অন্য কেউ আছেন যারা এই dxটিকে একটি অসীম ছোট পরিবর্তন হিসাবে ব্যাখ্যা করতে চান, বা শুধু বলতে চান যে dx এবং df প্রতীক ছাড়া আর কিছুই নয় যা আমাদের খুব বেশি গুরুত্ব সহকারে নেওয়া উচিত নয়।", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach? ", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach? ", "translatedText": "কিন্তু কল্পনা করুন আপনি একজন গণিতবিদ ক্যালকুলাস উদ্ভাবন করছেন, এবং কেউ আপনাকে জিজ্ঞাসা করবে, আচ্ছা, পদ্ধতি বলতে আপনি ঠিক কী বোঝেন? ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 379.86 }, { - "input": "We all know what it means for one value to get closer to another. ", + "input": "I mean, come on, we all know what it means for one value to get closer to another. ", "translatedText": "আমরা সকলেই জানি যে একটি মান অন্যটির কাছাকাছি যাওয়ার অর্থ কী।", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus. ", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus. ", "translatedText": "আপনি যুক্তি দিতে পারেন যে এটি ক্যালকুলাসের ভূমিকার জন্য অপ্রয়োজনীয়ভাবে ভারী দায়িত্ব।", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", "translatedText": "অ্যাপ্রোচ শব্দের অর্থ কী তা যদি আপনি জানেন, আপনি ইতিমধ্যেই জানেন একটি সীমা মানে কী, এখানে ধারণাগত স্তরে নতুন কিছু নেই।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/bulgarian/sentence_translations.json b/2017/limits/bulgarian/sentence_translations.json index a8a67cfe5..1825fb562 100644 --- a/2017/limits/bulgarian/sentence_translations.json +++ b/2017/limits/bulgarian/sentence_translations.json @@ -648,7 +648,7 @@ "end": 683.6 }, { - "input": "Some systematic process to take an expression like this one, that looks like 0 divided by and ask, what is its limit as x approaches that input?", + "input": "Some systematic process to take an expression like this one, that looks like 0 divided by 0 at some input, and ask what is its limit as x approaches that input?", "translatedText": "Някакъв систематичен процес за вземане на израз като този, който изглежда като 0, разделено на, и задаване на въпроса каква е неговата граница, когато x се приближи до този вход?", "model": "DeepL", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1073.96 }, { - "input": "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it kinda depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there.", + "input": "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there. Thanks for watching!", "translatedText": "Все още обсъждам дали да направя предварителна партида от плюшени същества за пай, това зависи от това колко зрители се интересуват от магазина като цяло, но нека ми кажете в коментарите какви други неща бихте искали да видите там.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/limits/chinese/sentence_translations.json b/2017/limits/chinese/sentence_translations.json index 6fb91d5ad..31f356556 100644 --- a/2017/limits/chinese/sentence_translations.json +++ b/2017/limits/chinese/sentence_translations.json @@ -233,7 +233,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", "translatedText": "还有一些人喜欢将这个 dx 解释为无限小的变化，或者只 是说 dx 和 df 只不过是我们不应该太认真的符号。", "model": "google_nmt", "from_community_srt": "你看， 还有其他人喜欢解释 DX是一个“无限小的变化”， 无论如何 这将意味着， 或只是说， DX和 DF只不过是不应该的符号而已 被认真对待。", @@ -386,7 +386,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach? ", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach? ", "translatedText": "但是想象一下，你是一位发明微积分的数学家 ，有人问你，你所说的方法到底是什么意思？", "model": "google_nmt", "from_community_srt": "但是， 想象一下， 你是一个数学家发明 微积分， 有人怀疑地问“好” 你的意思是什么意思？", @@ -404,7 +404,7 @@ "end": 379.86 }, { - "input": "We all know what it means for one value to get closer to another. ", + "input": "I mean, come on, we all know what it means for one value to get closer to another. ", "translatedText": "我们都知道一种价值观接近另一种价值观意味着什么。", "model": "google_nmt", "from_community_srt": "我的意思是， 来吧， 我们都知道这对一个人意味着什么 价值接近另一个。", @@ -502,7 +502,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus. ", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus. ", "translatedText": "你可能会说，对于微积分入门来说，这是不必要的繁重任务。", "model": "google_nmt", "from_community_srt": "你可以争辩这不必要的重负 对微积分的介绍。", @@ -511,7 +511,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", "translatedText": "如果您知道“方法”一词的含义，那么您就已经知道“ 限制”的含义，这里在概念层面上没有什么新内容。", "model": "google_nmt", "from_community_srt": "就像我说的， 如果你知道“方法”这个词的意思， 你知道什么是限制， 所以有 这里的概念层面没有新东西。", diff --git a/2017/limits/english/captions.srt b/2017/limits/english/captions.srt index 88bdb5228..82055c0f6 100644 --- a/2017/limits/english/captions.srt +++ b/2017/limits/english/captions.srt @@ -647,382 +647,390 @@ Doing that, you'd find that this should be a number around negative 1.57. But is there a way to know precisely what it is? 163 -00:11:23,960 --> 00:11:28,001 +00:11:23,960 --> 00:11:27,597 Some systematic process to take an expression like this one, 164 -00:11:28,001 --> 00:11:33,500 -that looks like 0 divided by and ask, what is its limit as x approaches that input? +00:11:27,597 --> 00:11:32,188 +that looks like 0 divided by 0 at some input, and ask what is its limit as x 165 +00:11:32,188 --> 00:11:33,500 +approaches that input? + +166 00:11:36,440 --> 00:11:40,157 After limits, so helpfully let us write the definition for derivatives, -166 +167 00:11:40,157 --> 00:11:44,700 derivatives can actually come back here and return the favor to help us evaluate limits. -167 +168 00:11:45,200 --> 00:11:46,020 Let me show you what I mean. -168 +169 00:11:47,020 --> 00:11:50,368 Here's what the graph of sin of pi times x looks like, -169 +170 00:11:50,368 --> 00:11:53,900 and here's what the graph of x squared minus 1 looks like. -170 +171 00:11:53,900 --> 00:11:56,780 That's a lot to have up on the screen, but just -171 +172 00:11:56,780 --> 00:11:59,420 focus on what's happening around x equals 1. -172 +173 00:12:00,180 --> 00:12:06,302 The point here is that sin of pi times x and x squared minus 1 are both 0 at that point, -173 +174 00:12:06,302 --> 00:12:08,160 they both cross the x axis. -174 +175 00:12:09,000 --> 00:12:14,277 In the same spirit as plugging in a specific value near 1, like 1.00001, -175 +176 00:12:14,277 --> 00:12:20,640 let's zoom in on that point and consider what happens just a tiny nudge dx away from it. -176 +177 00:12:21,300 --> 00:12:26,381 The value sin of pi times x is bumped down, and the value of that nudge, -177 +178 00:12:26,381 --> 00:12:32,160 which was caused by the nudge dx to the input, is what we might call d sin of pi x. -178 +179 00:12:33,040 --> 00:12:37,260 And from our knowledge of derivatives, using the chain rule, -179 +180 00:12:37,260 --> 00:12:41,480 that should be around cosine of pi times x times pi times dx. -180 +181 00:12:42,700 --> 00:12:47,700 Since the starting value was x equals 1, we plug in x equals 1 to that expression. -181 +182 00:12:51,260 --> 00:12:56,707 In other words, the amount that this sin of pi times x graph changes is roughly -182 +183 00:12:56,707 --> 00:13:02,360 proportional to dx, with a proportionality constant equal to cosine of pi times pi. -183 +184 00:13:03,360 --> 00:13:06,646 And cosine of pi, if we think back to our trig knowledge, -184 +185 00:13:06,646 --> 00:13:11,180 is exactly negative 1, so we can write this whole thing as negative pi times dx. -185 +186 00:13:12,220 --> 00:13:18,445 Similarly, the value of the x squared minus 1 graph changes by some dx squared minus 1, -186 +187 00:13:18,445 --> 00:13:23,540 and taking the derivative, the size of that nudge should be 2x times dx. -187 +188 00:13:24,480 --> 00:13:29,407 Again, we were starting at x equals 1, so we plug in x equals 1 to that expression, -188 +189 00:13:29,407 --> 00:13:33,280 meaning the size of that output nudge is about 2 times 1 times dx. -189 +190 00:13:34,920 --> 00:13:41,278 What this means is that for values of x which are just a tiny nudge dx away from 1, -190 +191 00:13:41,278 --> 00:13:46,425 the ratio sin of pi x divided by x squared minus 1 is approximately -191 +192 00:13:46,425 --> 00:13:49,680 negative pi times dx divided by 2 times dx. -192 +193 00:13:50,900 --> 00:13:54,740 The dx's cancel out, so what's left is negative pi over 2. -193 +194 00:13:55,720 --> 00:13:58,591 And importantly, those approximations get more and more -194 +195 00:13:58,591 --> 00:14:01,360 accurate for smaller and smaller choices of dx, right? -195 +196 00:14:02,310 --> 00:14:05,619 This ratio, negative pi over 2, actually tells -196 +197 00:14:05,619 --> 00:14:09,000 us the precise limiting value as x approaches 1. -197 +198 00:14:09,540 --> 00:14:13,107 Remember, what that means is that the limiting height on -198 +199 00:14:13,107 --> 00:14:16,800 our original graph is evidently exactly negative pi over 2. -199 +200 00:14:18,220 --> 00:14:21,602 What happened there is a little subtle, so I want to go through it again, -200 +201 00:14:21,602 --> 00:14:23,340 but this time a little more generally. -201 +202 00:14:24,120 --> 00:14:29,388 Instead of these two specific functions, which are both equal to 0 at x equals 1, -202 +203 00:14:29,388 --> 00:14:34,913 think of any two functions, f of x and g of x, which are both 0 at some common value, -203 +204 00:14:34,913 --> 00:14:35,620 x equals a. -204 +205 00:14:36,280 --> 00:14:39,563 The only constraint is that these have to be functions where you're -205 +206 00:14:39,563 --> 00:14:41,929 able to take a derivative of them at x equals a, -206 +207 00:14:41,929 --> 00:14:45,406 which means they each basically look like a line when you zoom in close -207 +208 00:14:45,406 --> 00:14:46,420 enough to that value. -208 +209 00:14:47,800 --> 00:14:52,393 Even though you can't compute f divided by g at this trouble point, -209 +210 00:14:52,393 --> 00:14:56,514 since both of them equal 0, you can ask about this ratio for -210 +211 00:14:56,514 --> 00:15:00,500 values of x really close to a, the limit as x approaches a. -211 +212 00:15:01,220 --> 00:15:06,200 It's helpful to think of those nearby inputs as just a tiny nudge, dx, away from a. -212 +213 00:15:06,760 --> 00:15:12,161 The value of f at that nudged point is approximately its derivative, -213 +214 00:15:12,161 --> 00:15:14,980 df over dx, evaluated at a times dx. -214 +215 00:15:15,980 --> 00:15:22,124 Likewise, the value of g at that nudged point is approximately the derivative of g, -215 +216 00:15:22,124 --> 00:15:23,880 evaluated at a times dx. -216 +217 00:15:25,060 --> 00:15:31,060 Near that trouble point, the ratio between the outputs of f and g is actually about the -217 +218 00:15:31,060 --> 00:15:37,060 same as the derivative of f at a times dx, divided by the derivative of g at a times dx. -218 +219 00:15:37,880 --> 00:15:41,119 Those dx's cancel out, so the ratio of f and g near a -219 +220 00:15:41,119 --> 00:15:44,540 is about the same as the ratio between their derivatives. -220 +221 00:15:45,860 --> 00:15:50,307 Because each of those approximations gets more and more accurate for smaller and -221 +222 00:15:50,307 --> 00:15:54,700 smaller nudges, this ratio of derivatives gives the precise value for the limit. -222 +223 00:15:55,540 --> 00:15:58,500 This is a really handy trick for computing a lot of limits. -223 +224 00:15:58,920 --> 00:16:02,938 Whenever you come across some expression that seems to equal 0 divided by -224 +225 00:16:02,938 --> 00:16:06,901 0 when you plug in some particular input, just try taking the derivative -225 +226 00:16:06,901 --> 00:16:10,920 of the top and bottom expressions and plugging in that same trouble input. -226 +227 00:16:13,980 --> 00:16:16,300 This clever trick is called L'Hopital's Rule. -227 +228 00:16:17,240 --> 00:16:20,182 Interestingly, it was actually discovered by Johann Bernoulli, -228 +229 00:16:20,182 --> 00:16:22,844 but L'Hopital was this wealthy dude who essentially paid -229 +230 00:16:22,844 --> 00:16:25,880 Bernoulli for the rights to some of his mathematical discoveries. -230 +231 00:16:26,740 --> 00:16:30,076 Academia is weird back then, but in a very literal way, -231 +232 00:16:30,076 --> 00:16:32,460 it pays to understand these tiny nudges. -232 +233 00:16:34,960 --> 00:16:38,717 Right now, you might be remembering that the definition of a derivative -233 +234 00:16:38,717 --> 00:16:42,265 for a given function comes down to computing the limit of a certain -234 +235 00:16:42,265 --> 00:16:45,657 fraction that looks like 0 divided by 0, so you might think that -235 +236 00:16:45,657 --> 00:16:49,780 L'Hopital's Rule could give us a handy way to discover new derivative formulas. -236 +237 00:16:50,680 --> 00:16:53,474 But that would actually be cheating, since presumably -237 +238 00:16:53,474 --> 00:16:56,320 you don't know what the derivative of the numerator is. -238 +239 00:16:57,020 --> 00:16:59,594 When it comes to discovering derivative formulas, -239 +240 00:16:59,594 --> 00:17:02,374 something we've been doing a fair amount this series, -240 +241 00:17:02,374 --> 00:17:04,640 there is no systematic plug-and-chug method. -241 +242 00:17:05,119 --> 00:17:05,960 But that's a good thing! -242 +243 00:17:06,400 --> 00:17:09,373 Whenever creativity is needed to solve problems like these, -243 +244 00:17:09,373 --> 00:17:11,901 it's a good sign that you're doing something real, -244 +245 00:17:11,901 --> 00:17:15,420 something that might give you a powerful tool to solve future problems. -245 +246 00:17:18,260 --> 00:17:22,908 And speaking of powerful tools, up next I'm going to be talking about what an integral -246 +247 00:17:22,908 --> 00:17:25,687 is, as well as the fundamental theorem of calculus, -247 +248 00:17:25,687 --> 00:17:30,443 another example of where limits can be used to give a clear meaning to a pretty delicate -248 +249 00:17:30,443 --> 00:17:32,100 idea that flirts with infinity. -249 +250 00:17:33,580 --> 00:17:36,819 As you know, most support for this channel comes through Patreon, -250 +251 00:17:36,819 --> 00:17:40,795 and the primary perk for patrons is early access to future series like this one, -251 +252 00:17:40,795 --> 00:17:43,200 where the next one is going to be on probability. -252 +253 00:17:44,260 --> 00:17:47,755 But for those of you who want a more tangible way to flag that -253 +254 00:17:47,755 --> 00:17:51,640 you're part of the community, there is also a small 3blue1brown store. -254 +255 00:17:52,300 --> 00:17:53,960 Links on the screen and in the description. -255 -00:17:54,680 --> 00:18:05,700 -I'm still debating whether or not to make a preliminary batch of plushie pie creatures, - 256 -00:18:05,700 --> 00:18:15,970 -it kinda depends on how many viewers seem interested in the store more generally, +00:17:54,680 --> 00:18:05,081 +I'm still debating whether or not to make a preliminary batch of plushie pie creatures, 257 -00:18:15,970 --> 00:18:26,240 -but let me know in comments what other kinds of things you'd like to see in there. +00:18:05,081 --> 00:18:14,065 +it depends on how many viewers seem interested in the store more generally, + +258 +00:18:14,065 --> 00:18:23,875 +but let me know in comments what other kinds of things you'd like to see in there. + +259 +00:18:23,875 --> 00:18:26,240 +Thanks for watching! diff --git a/2017/limits/english/sentence_timings.json b/2017/limits/english/sentence_timings.json index d90cfd267..b6b2585ac 100644 --- a/2017/limits/english/sentence_timings.json +++ b/2017/limits/english/sentence_timings.json @@ -405,7 +405,7 @@ 683.6 ], [ - "Some systematic process to take an expression like this one, that looks like 0 divided by and ask, what is its limit as x approaches that input?", + "Some systematic process to take an expression like this one, that looks like 0 divided by 0 at some input, and ask what is its limit as x approaches that input?", 683.96, 693.5 ], @@ -620,7 +620,7 @@ 1073.96 ], [ - "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it kinda depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there.", + "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there. Thanks for watching!", 1074.68, 1106.24 ] diff --git a/2017/limits/english/transcript.txt b/2017/limits/english/transcript.txt index da8f91ea1..9cc684c35 100644 --- a/2017/limits/english/transcript.txt +++ b/2017/limits/english/transcript.txt @@ -79,7 +79,7 @@ So you might ask, how exactly do you find what output this approaches as x appro Well, one way to approximate it would be to plug in a number that's just really close to 1, like 1.00001. Doing that, you'd find that this should be a number around negative 1.57. But is there a way to know precisely what it is? -Some systematic process to take an expression like this one, that looks like 0 divided by and ask, what is its limit as x approaches that input? +Some systematic process to take an expression like this one, that looks like 0 divided by 0 at some input, and ask what is its limit as x approaches that input? After limits, so helpfully let us write the definition for derivatives, derivatives can actually come back here and return the favor to help us evaluate limits. Let me show you what I mean. Here's what the graph of sin of pi times x looks like, and here's what the graph of x squared minus 1 looks like. @@ -122,4 +122,4 @@ And speaking of powerful tools, up next I'm going to be talking about what an in As you know, most support for this channel comes through Patreon, and the primary perk for patrons is early access to future series like this one, where the next one is going to be on probability. But for those of you who want a more tangible way to flag that you're part of the community, there is also a small 3blue1brown store. Links on the screen and in the description. -I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it kinda depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there. \ No newline at end of file +I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there. Thanks for watching! \ No newline at end of file diff --git a/2017/limits/french/sentence_translations.json b/2017/limits/french/sentence_translations.json index 9c854c110..d7eb5d2ea 100644 --- a/2017/limits/french/sentence_translations.json +++ b/2017/limits/french/sentence_translations.json @@ -647,7 +647,7 @@ "end": 683.6 }, { - "input": "Some systematic process to take an expression like this one, that looks like 0 divided by and ask, what is its limit as x approaches that input?", + "input": "Some systematic process to take an expression like this one, that looks like 0 divided by 0 at some input, and ask what is its limit as x approaches that input?", "translatedText": "Un processus systématique pour prendre une expression comme celle-ci, qui ressemble à 0 divisé par et demander quelle est sa limite lorsque x s'approche de cette entrée ?", "from_community_srt": "qui prenne une expression comme celle-ci, qui ressemble à 0/0 pour une entrée,", "n_reviews": 0, @@ -991,7 +991,7 @@ "end": 1073.96 }, { - "input": "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it kinda depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there.", + "input": "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there. Thanks for watching!", "translatedText": "Je me demande encore s'il faut ou non créer un lot préliminaire de créatures en peluche, cela dépend un peu du nombre de téléspectateurs qui semblent intéressés par le magasin de manière plus générale, mais faites-moi savoir dans les commentaires quels autres types de choses vous aimeriez voir. là-dedans.", "from_community_srt": "Je me demande encore si je dois commander un lot préliminaire de doudou pi, cela dépend du nombre de spectateurs qui semblent intéressés par le magasin en général, mais laissez faites le moi savoir dans les commentaires quel type de produit vous voudriez y voir.", "n_reviews": 0, diff --git a/2017/limits/german/sentence_translations.json b/2017/limits/german/sentence_translations.json index 852ac3f4a..e77a85b00 100644 --- a/2017/limits/german/sentence_translations.json +++ b/2017/limits/german/sentence_translations.json @@ -729,7 +729,7 @@ "end": 683.6 }, { - "input": "Some systematic process to take an expression like this one, that looks like 0 divided by and ask, what is its limit as x approaches that input?", + "input": "Some systematic process to take an expression like this one, that looks like 0 divided by 0 at some input, and ask what is its limit as x approaches that input?", "translatedText": "Ein systematischer Prozess, um einen Ausdruck wie diesen zu nehmen, der wie 0 geteilt durch aussieht, und zu fragen, was sein Grenzwert ist, wenn x sich diesem Eingang nähert?", "model": "DeepL", "from_community_srt": "Ein systematischer Prozess, um einen Ausdruck zu finden wie dieser, der bei manchen wie 0/0 aussieht Geben Sie ein und fragen Sie, wo seine Grenze liegt,", @@ -1116,7 +1116,7 @@ "end": 1073.96 }, { - "input": "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it kinda depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there.", + "input": "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there. Thanks for watching!", "translatedText": "Ich bin noch am Überlegen, ob ich eine erste Ladung Plüschtier-Kreaturen machen soll. Das hängt davon ab, wie viele Zuschauer sich für den Laden interessieren.", "model": "DeepL", "from_community_srt": "Ich überlege immer noch, ob ich eine machen soll oder nicht vorläufige Charge von Plüschtierkreaturen, es hängt irgendwie davon ab, wie viele Zuschauer scheinen interessiert an dem Laden im Allgemeinen, aber lassen Ich weiß in Kommentaren, was für andere Dinge Sie möchten dort sehen.", diff --git a/2017/limits/hebrew/sentence_translations.json b/2017/limits/hebrew/sentence_translations.json index bbd287bcc..9c71492f5 100644 --- a/2017/limits/hebrew/sentence_translations.json +++ b/2017/limits/hebrew/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", "translatedText": "יש אחרים שאוהבים לפרש את ה-dx הזה כשינוי קטן לאין שיעור, או פשוט לומר ש-dx ו-df הם לא יותר מסמלים שאסור לנו לקחת יותר מדי ברצינות. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach? ", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach? ", "translatedText": "אבל תאר לעצמך שאתה מתמטיקאי שממציא חשבון, ומישהו שואל אותך, ובכן, למה בדיוק אתה מתכוון בגישה? ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 379.86 }, { - "input": "We all know what it means for one value to get closer to another. ", + "input": "I mean, come on, we all know what it means for one value to get closer to another. ", "translatedText": "כולנו יודעים מה המשמעות של ערך אחד להתקרב לאחר. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus. ", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus. ", "translatedText": "אתה יכול לטעון שזו חובה כבדה מיותר למבוא לחשבון. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", "translatedText": "אם אתה יודע מה פירוש המילה גישה, אתה כבר יודע מה המשמעות של גבול, אין כאן שום דבר חדש ברמה הרעיונית. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/hindi/sentence_translations.json b/2017/limits/hindi/sentence_translations.json index dd7f302d9..a2562eb25 100644 --- a/2017/limits/hindi/sentence_translations.json +++ b/2017/limits/hindi/sentence_translations.json @@ -182,7 +182,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", "translatedText": "ऐसे अन्य लोग भी हैं जो इस dx को एक असीम रूप से छोटे परिवर्तन के रूप में व्याख्या करना पसंद करते हैं, या बस यह कहें कि dx और df प्रतीकों से ज्यादा कुछ नहीं हैं जिन्हें हमें बहुत गंभीरता से नहीं लेना चाहिए।", "n_reviews": 0, "start": 234.38, @@ -301,14 +301,14 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach?", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach?", "translatedText": "लेकिन कल्पना कीजिए कि आप एक गणितज्ञ हैं और कैलकुलस का आविष्कार कर रहे हैं, और कोई आपसे पूछता है, ठीक है, दृष्टिकोण से आपका वास्तव में क्या मतलब है?", "n_reviews": 0, "start": 369.36, "end": 377.48 }, { - "input": "That would be kind of an annoying question.", + "input": "That would be kind of an annoying question, I mean, come on,", "translatedText": "यह एक प्रकार का कष्टप्रद प्रश्न होगा।", "n_reviews": 0, "start": 378.44, @@ -392,14 +392,14 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus.", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus.", "translatedText": "आप यह तर्क दे सकते हैं कि कैलकुलस के परिचय के लिए यह अनावश्यक रूप से कठिन काम है।", "n_reviews": 0, "start": 481.22, "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", "translatedText": "यदि आप जानते हैं कि दृष्टिकोण शब्द का क्या अर्थ है, आप पहले से ही जानते हैं कि सीमा का क्या अर्थ है, तो यहां वैचारिक स्तर पर कुछ भी नया नहीं है।", "n_reviews": 0, "start": 486.06, diff --git a/2017/limits/hungarian/sentence_translations.json b/2017/limits/hungarian/sentence_translations.json index 8377a9aaa..7e7fb37f0 100644 --- a/2017/limits/hungarian/sentence_translations.json +++ b/2017/limits/hungarian/sentence_translations.json @@ -648,7 +648,7 @@ "end": 683.6 }, { - "input": "Some systematic process to take an expression like this one, that looks like 0 divided by and ask, what is its limit as x approaches that input?", + "input": "Some systematic process to take an expression like this one, that looks like 0 divided by 0 at some input, and ask what is its limit as x approaches that input?", "translatedText": "Valamilyen szisztematikus folyamat, hogy egy ilyen kifejezést, mint ez, ami úgy néz ki, mint 0 osztva és kérdezzük meg, mi a határértéke, ahogy x megközelíti ezt a bemeneti értéket?", "model": "DeepL", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 1073.96 }, { - "input": "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it kinda depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there.", + "input": "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there. Thanks for watching!", "translatedText": "Még gondolkodom azon, hogy készítsek-e egy előzetes adag plüss pite lényt, ez attól függ, hogy a nézők mennyien érdeklődnek a bolt iránt, de írjátok meg kommentben, hogy milyen másfajta dolgokat szeretnétek látni benne.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/limits/indonesian/sentence_translations.json b/2017/limits/indonesian/sentence_translations.json index db3750f85..63fe421ec 100644 --- a/2017/limits/indonesian/sentence_translations.json +++ b/2017/limits/indonesian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", "translatedText": "Ada orang lain yang suka menafsirkan dx ini sebagai perubahan yang sangat kecil, atau sekadar mengatakan bahwa dx dan df tidak lebih dari simbol yang tidak boleh kita anggap terlalu serius.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach?", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach?", "translatedText": "Tapi bayangkan Anda seorang ahli matematika yang menemukan kalkulus, dan seseorang bertanya kepada Anda, apa sebenarnya yang Anda maksud dengan pendekatan?", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 377.48 }, { - "input": "That would be kind of an annoying question.", + "input": "That would be kind of an annoying question, I mean, come on,", "translatedText": "Itu akan menjadi pertanyaan yang menjengkelkan.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus.", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus.", "translatedText": "Anda dapat berargumen bahwa ini adalah tugas berat yang tidak perlu untuk pengenalan kalkulus.", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", "translatedText": "Jika Anda tahu apa arti kata pendekatan, Anda sudah tahu apa arti batas, tidak ada yang baru dalam tataran konseptual di sini.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/italian/sentence_translations.json b/2017/limits/italian/sentence_translations.json index 73c37c579..6f005a78c 100644 --- a/2017/limits/italian/sentence_translations.json +++ b/2017/limits/italian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", "translatedText": "Ci sono altri a cui piace interpretare questo dx come un cambiamento infinitamente piccolo, o semplicemente dire che dx e df non sono altro che simboli che non dovremmo prendere troppo sul serio.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach?", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach?", "translatedText": "Ma immagina di essere un matematico che inventa il calcolo infinitesimale e qualcuno ti chiede, cosa intendi esattamente per approccio?", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 377.48 }, { - "input": "That would be kind of an annoying question.", + "input": "That would be kind of an annoying question, I mean, come on,", "translatedText": "Sarebbe una domanda un po' fastidiosa.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus.", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus.", "translatedText": "Si potrebbe sostenere che questo sia inutilmente pesante per un'introduzione al calcolo infinitesimale.", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", "translatedText": "Se sai cosa significa la parola approccio, sai già cosa significa limite, qui non c'è niente di nuovo a livello concettuale.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/japanese/sentence_translations.json b/2017/limits/japanese/sentence_translations.json index 9dc5ec173..fffe13f60 100644 --- a/2017/limits/japanese/sentence_translations.json +++ b/2017/limits/japanese/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", "translatedText": "この dx を限りなく小さな変化として解釈したり、dx と df はあま り真剣に受け止めるべきではない単なる記号にすぎないと言いたい人もいます。", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach? ", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach? ", "translatedText": "しかし、あなたが微積分を発明した数学者で、誰かがあなたに「アプロ ーチとは正確には何を意味しますか?」と尋ねたと想像してください。", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 379.86 }, { - "input": "We all know what it means for one value to get closer to another. ", + "input": "I mean, come on, we all know what it means for one value to get closer to another. ", "translatedText": "ある値が別の値に近づくことが何を意味するかは誰もが知っています。", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus. ", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus. ", "translatedText": "これは微積分の入門としては不必要に負担が大きいと言えるでしょう。", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", "translatedText": "アプローチという言葉の意味を知っていれば、制限が何を意味するかすでに 知っているでしょう。ここでは概念レベルで新しいことは何もありません。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/korean/sentence_translations.json b/2017/limits/korean/sentence_translations.json index 3f278b0cb..fc82a7d00 100644 --- a/2017/limits/korean/sentence_translations.json +++ b/2017/limits/korean/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", "translatedText": "이 dx를 무한히 작은 변화로 해석하거나 dx와 df는 너무 심각하게 받아들이지 말아야 할 기호에 불과하다고 말하는 사람들도 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach? ", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach? ", "translatedText": "하지만 당신이 미적분학을 발명한 수학자라고 상상해 보세요. 누군가가 당신에게 접근 방식이 정확히 무엇을 의미하는지 묻습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 379.86 }, { - "input": "We all know what it means for one value to get closer to another. ", + "input": "I mean, come on, we all know what it means for one value to get closer to another. ", "translatedText": "우리 모두는 한 가치가 다른 가치에 가까워진다는 것이 무엇을 의미하는지 알고 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus. ", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus. ", "translatedText": "미적분학을 소개하는 데 있어서 이것이 불필요하게 무거운 의무라고 주장할 수 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", "translatedText": "접근이라는 단어가 무엇을 의미하는지 알고 있다면 한계가 무엇을 의미하는지 이미 알고 있으므로 개념적 수준에서는 새로운 것이 없습니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/marathi/sentence_translations.json b/2017/limits/marathi/sentence_translations.json index ede256b7b..e0ad5e653 100644 --- a/2017/limits/marathi/sentence_translations.json +++ b/2017/limits/marathi/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", "translatedText": "असे काही लोक आहेत ज्यांना या dx चा अनंत लहान बदल म्हणून अर्थ लावणे आवडते किंवा फक्त असे म्हणायचे आहे की dx आणि df हे चिन्हांशिवाय दुसरे काहीही नाही ज्यांना आपण फारसे गांभीर्याने घेऊ नये. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach? ", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach? ", "translatedText": "पण कल्पना करा की तुम्ही कॅल्क्युलसचा शोध लावणारे गणितज्ञ आहात आणि कोणीतरी तुम्हाला विचारेल, बरं, तुम्हाला दृष्टिकोनाचा नेमका अर्थ काय आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 379.86 }, { - "input": "We all know what it means for one value to get closer to another. ", + "input": "I mean, come on, we all know what it means for one value to get closer to another. ", "translatedText": "आपल्या सर्वांना माहित आहे की एका मूल्यासाठी दुसर्‍याच्या जवळ जाण्याचा अर्थ काय आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus. ", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus. ", "translatedText": "तुम्ही असा तर्क करू शकता की कॅल्क्युलसच्या परिचयासाठी हे अनावश्यकपणे भारी कर्तव्य आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", "translatedText": "दृष्टीकोन या शब्दाचा अर्थ काय हे तुम्हाला माहीत असेल, तर तुम्हाला मर्यादा म्हणजे काय हे आधीच माहीत असेल, येथे वैचारिक पातळीवर काहीही नवीन नाही. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/persian/sentence_translations.json b/2017/limits/persian/sentence_translations.json index 718e737dc..15052a519 100644 --- a/2017/limits/persian/sentence_translations.json +++ b/2017/limits/persian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", "translatedText": "دیگرانی هستند که دوست دارند این dx را به عنوان یک تغییر بی نهایت کوچک تعبیر کنند، یا فقط بگویند dx و df چیزی بیش از نمادهایی نیستند که ما نباید زیاد آنها را جدی بگیریم. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach? ", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach? ", "translatedText": "اما تصور کنید که شما یک ریاضیدان هستید که حساب دیفرانسیل و انتگرال را اختراع می کند، و یکی از شما می پرسد، خوب، منظورتان از رویکرد دقیقا چیست؟ این یک نوع سوال آزاردهنده خواهد بود. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 379.86 }, { - "input": "We all know what it means for one value to get closer to another. ", + "input": "I mean, come on, we all know what it means for one value to get closer to another. ", "translatedText": "همه ما می دانیم که نزدیک شدن یک ارزش به ارزش دیگر به چه معناست. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus. ", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus. ", "translatedText": "می توانید استدلال کنید که این کار بیهوده سنگینی برای مقدمه ای بر حساب دیفرانسیل و انتگرال است. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", "translatedText": "اگر معنی کلمه رویکرد را می‌دانید، از قبل می‌دانید محدودیت به چه معناست، هیچ چیز جدیدی در سطح مفهومی در اینجا وجود ندارد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/polish/sentence_translations.json b/2017/limits/polish/sentence_translations.json index 3328c6e3b..88cf1c92f 100644 --- a/2017/limits/polish/sentence_translations.json +++ b/2017/limits/polish/sentence_translations.json @@ -647,7 +647,7 @@ "end": 683.6 }, { - "input": "Some systematic process to take an expression like this one, that looks like 0 divided by and ask, what is its limit as x approaches that input?", + "input": "Some systematic process to take an expression like this one, that looks like 0 divided by 0 at some input, and ask what is its limit as x approaches that input?", "translatedText": "", "from_community_srt": "za pomocą którego można przetworzyć funkcję, która dla pewnej wartości daje 0/0, żeby dowiedzieć się,", "n_reviews": 0, @@ -989,7 +989,7 @@ "end": 1073.96 }, { - "input": "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it kinda depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there.", + "input": "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there. Thanks for watching!", "translatedText": "", "from_community_srt": "Jeszcze nie wiem, czy powstaną maskotki pi-stworzeń. To zależy od tego, ile osób będzie zainteresowanych. Jeśli chciałbyś zobaczyć w sklepie jakiś inny gadżet, daj mi znać w komentarzach.", "n_reviews": 0, diff --git a/2017/limits/portuguese/sentence_translations.json b/2017/limits/portuguese/sentence_translations.json index 7ae530b54..aa1be7233 100644 --- a/2017/limits/portuguese/sentence_translations.json +++ b/2017/limits/portuguese/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", "translatedText": "Há outros que gostam de interpretar este dx como uma mudança infinitamente pequena, ou apenas dizer que dx e df nada mais são do que símbolos que não devemos levar muito a sério.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach?", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach?", "translatedText": "Mas imagine que você é um matemático inventando o cálculo e alguém lhe pergunta: bem, o que exatamente você quer dizer com aproximar?", "model": "google_nmt", "n_reviews": 1, @@ -352,7 +352,7 @@ "end": 377.48 }, { - "input": "That would be kind of an annoying question.", + "input": "That would be kind of an annoying question, I mean, come on,", "translatedText": "Essa seria uma pergunta meio chatinha.", "model": "google_nmt", "n_reviews": 1, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus.", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus.", "translatedText": "Você poderia argumentar que isso é uma tarefa desnecessariamente robusta para uma introdução ao cálculo.", "model": "google_nmt", "n_reviews": 1, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", "translatedText": "Se você sabe o que o verbo aproximar significa, você já sabe o que limite significa, não há nada de novo no nível conceitual aqui.", "model": "google_nmt", "n_reviews": 1, diff --git a/2017/limits/russian/sentence_translations.json b/2017/limits/russian/sentence_translations.json index 905bff7af..e11ae57fd 100644 --- a/2017/limits/russian/sentence_translations.json +++ b/2017/limits/russian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", "translatedText": "Есть и другие, которым нравится интерпретировать dx как бесконечно маленькое изменение или просто говорить, что dx и df — не более чем символы, к которым не следует относиться слишком серьезно.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach?", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach?", "translatedText": "Но представьте, что вы математик, изобретающий исчисление, и кто-то спрашивает вас: что именно вы подразумеваете под подходом?", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 377.48 }, { - "input": "That would be kind of an annoying question.", + "input": "That would be kind of an annoying question, I mean, come on,", "translatedText": "Это был бы довольно неприятный вопрос.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus.", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus.", "translatedText": "Вы можете возразить, что это излишне тяжелая работа для введения в исчисление.", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", "translatedText": "Если вы знаете, что означает слово «подход», вы уже знаете, что означает предел, на концептуальном уровне здесь нет ничего нового.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/spanish/sentence_translations.json b/2017/limits/spanish/sentence_translations.json index fd9d60e3e..c9766d3b4 100644 --- a/2017/limits/spanish/sentence_translations.json +++ b/2017/limits/spanish/sentence_translations.json @@ -647,7 +647,7 @@ "end": 683.6 }, { - "input": "Some systematic process to take an expression like this one, that looks like 0 divided by and ask, what is its limit as x approaches that input?", + "input": "Some systematic process to take an expression like this one, that looks like 0 divided by 0 at some input, and ask what is its limit as x approaches that input?", "translatedText": "Algún proceso sistemático para tomar una expresión como esta, que parece 0 dividido por y preguntar, ¿cuál es su límite cuando x se acerca a esa entrada?", "from_community_srt": "Algunos proceso sistemático para tomar una expresión como éste, que se parece a 0/0 en algún de entrada, y preguntar lo que su límite es cuando se acerca x esa entrada? Bueno,", "n_reviews": 0, @@ -991,7 +991,7 @@ "end": 1073.96 }, { - "input": "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it kinda depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there.", + "input": "I'm still debating whether or not to make a preliminary batch of plushie pie creatures, it depends on how many viewers seem interested in the store more generally, but let me know in comments what other kinds of things you'd like to see in there. Thanks for watching!", "translatedText": "Todavía estoy debatiendo si hacer o no un lote preliminar de criaturas de peluche, depende un poco de cuántos espectadores parezcan interesados en la tienda en general, pero déjame saber en los comentarios qué otro tipo de cosas te gustaría ver. ahí.", "from_community_srt": "Todavía estoy debatiendo si o para hacer una lotes preliminar de criaturas peluche pi, En cierto modo depende del número de espectadores parecen interesados ​​en la tienda en general, pero vamos Quiero saber en los comentarios qué tipo de otras cosas desea ver allí.", "n_reviews": 0, diff --git a/2017/limits/tamil/sentence_translations.json b/2017/limits/tamil/sentence_translations.json index 22f05c577..5e03daec4 100644 --- a/2017/limits/tamil/sentence_translations.json +++ b/2017/limits/tamil/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", "translatedText": "இந்த dx ஐ எண்ணற்ற சிறிய மாற்றமாக விளக்க விரும்புபவர்கள் அல்லது dx மற்றும் df என்பது நாம் பெரிதாக எடுத்துக் கொள்ளக் கூடாத சின்னங்களைத் தவிர வேறில்லை. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach? ", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach? ", "translatedText": "ஆனால் நீங்கள் கால்குலஸைக் கண்டுபிடிக்கும் ஒரு கணிதவியலாளர் என்று கற்பனை செய்து பாருங்கள், யாரோ உங்களிடம் கேட்கிறார்கள், நீங்கள் சரியாக அணுகுவதன் மூலம் என்ன சொல்கிறீர்கள்? ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 379.86 }, { - "input": "We all know what it means for one value to get closer to another. ", + "input": "I mean, come on, we all know what it means for one value to get closer to another. ", "translatedText": "ஒரு மதிப்பு மற்றொன்றை நெருங்குவது என்றால் என்ன என்பதை நாம் அனைவரும் அறிவோம். ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus. ", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus. ", "translatedText": "கால்குலஸ் அறிமுகத்திற்கு இது தேவையில்லாமல் கடுமையான கடமை என்று நீங்கள் வாதிடலாம். ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", "translatedText": "அணுகுமுறை என்ற வார்த்தையின் அர்த்தம் என்னவென்று உங்களுக்குத் தெரிந்தால், வரம்பு என்றால் என்ன என்பதை நீங்கள் ஏற்கனவே அறிந்திருக்கிறீர்கள், கருத்தியல் மட்டத்தில் புதிதாக எதுவும் இல்லை. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/telugu/sentence_translations.json b/2017/limits/telugu/sentence_translations.json index cb3027bd2..2df295f0a 100644 --- a/2017/limits/telugu/sentence_translations.json +++ b/2017/limits/telugu/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", "translatedText": "ఈ dxని అనంతమైన చిన్న మార్పుగా అర్థం చేసుకోవడానికి లేదా dx మరియు df అనేవి మనం చాలా సీరియస్‌గా తీసుకోకూడని చిహ్నాలు తప్ప మరేమీ కాదని చెప్పడానికి ఇష్టపడే ఇతరులు కూడా ఉన్నారు. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach? ", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach? ", "translatedText": "అయితే మీరు కాలిక్యులస్‌ని కనిపెట్టే గణిత శాస్త్రజ్ఞుడని ఊహించుకోండి, మరియు ఎవరైనా మిమ్మల్ని అడిగారు, అలాగే, సరిగ్గా మీరు విధానం అంటే ఏమిటి? ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 379.86 }, { - "input": "We all know what it means for one value to get closer to another. ", + "input": "I mean, come on, we all know what it means for one value to get closer to another. ", "translatedText": "ఒక విలువ మరొకదానికి చేరువ కావడం అంటే ఏమిటో మనందరికీ తెలుసు. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus. ", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus. ", "translatedText": "కాలిక్యులస్ పరిచయం కోసం ఇది అనవసరంగా హెవీ డ్యూటీ అని మీరు వాదించవచ్చు. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", "translatedText": "విధానం అనే పదానికి అర్థం ఏమిటో మీకు తెలిస్తే, పరిమితి అంటే ఏమిటో మీకు ఇప్పటికే తెలుసు, ఇక్కడ సంభావిత స్థాయిలో కొత్తది ఏమీ లేదు. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/thai/sentence_translations.json b/2017/limits/thai/sentence_translations.json index 02fed5ab4..0773ab69e 100644 --- a/2017/limits/thai/sentence_translations.json +++ b/2017/limits/thai/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach? ", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 379.86 }, { - "input": "We all know what it means for one value to get closer to another. ", + "input": "I mean, come on, we all know what it means for one value to get closer to another. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus. ", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/turkish/sentence_translations.json b/2017/limits/turkish/sentence_translations.json index 9419811fd..ab917e583 100644 --- a/2017/limits/turkish/sentence_translations.json +++ b/2017/limits/turkish/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", "translatedText": "Bu dx'i sonsuz küçük bir değişiklik olarak yorumlamaktan hoşlananlar ya da sadece dx ve df'nin fazla ciddiye almamamız gereken sembollerden başka bir şey olmadığını söylemekten hoşlananlar da var.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach?", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach?", "translatedText": "Ancak, hesabı icat eden bir matematikçi olduğunuzu ve birisinin size yaklaşımla tam olarak ne demek istediğinizi sorduğunu hayal edin.", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 377.48 }, { - "input": "That would be kind of an annoying question.", + "input": "That would be kind of an annoying question, I mean, come on,", "translatedText": "Bu biraz sinir bozucu bir soru olurdu.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus.", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus.", "translatedText": "Bunun matematiğe giriş için gereksiz derecede ağır bir görev olduğunu iddia edebilirsiniz.", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", "translatedText": "Yaklaşım kelimesinin ne anlama geldiğini biliyorsanız, limitin ne anlama geldiğini zaten biliyorsunuzdur, burada kavramsal düzeyde yeni bir şey yoktur.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/ukrainian/sentence_translations.json b/2017/limits/ukrainian/sentence_translations.json index 755af5e0a..e4e946ada 100644 --- a/2017/limits/ukrainian/sentence_translations.json +++ b/2017/limits/ukrainian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", "translatedText": "Є інші, які люблять інтерпретувати dx як нескінченно малу зміну, або просто кажуть, що dx і df — це не що інше, як символи, які ми не повинні сприймати надто серйозно.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach?", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach?", "translatedText": "Але уявіть, що ви математик, який винаходить обчислення, і хтось запитує вас, що саме ви маєте на увазі під підходом?", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 377.48 }, { - "input": "That would be kind of an annoying question.", + "input": "That would be kind of an annoying question, I mean, come on,", "translatedText": "Це було б неприємне питання.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus.", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus.", "translatedText": "Ви можете заперечити, що це непотрібно важкий обов’язок для вступу в обчислення.", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", "translatedText": "Якщо ви знаєте, що означає слово підхід, ви вже знаєте, що означає межа, тут немає нічого нового на концептуальному рівні.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/urdu/sentence_translations.json b/2017/limits/urdu/sentence_translations.json index 58ed47f8c..e3ab9c5c5 100644 --- a/2017/limits/urdu/sentence_translations.json +++ b/2017/limits/urdu/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously. ", "translatedText": "کچھ اور بھی ہیں جو اس dx کو ایک لامحدود چھوٹی تبدیلی کے طور پر تشریح کرنا چاہتے ہیں، یا صرف یہ کہنا چاہتے ہیں کہ dx اور df علامتوں کے علاوہ کچھ نہیں ہیں جنہیں ہمیں زیادہ سنجیدگی سے نہیں لینا چاہیے۔", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach? ", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach? ", "translatedText": "لیکن تصور کریں کہ آپ ایک ریاضی دان ہیں جو کیلکولس ایجاد کر رہے ہیں، اور کوئی آپ سے پوچھے، ٹھیک ہے، نقطہ نظر سے آپ کا کیا مطلب ہے؟ یہ ایک پریشان کن سوال ہوگا۔", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 379.86 }, { - "input": "We all know what it means for one value to get closer to another. ", + "input": "I mean, come on, we all know what it means for one value to get closer to another. ", "translatedText": "ہم سب جانتے ہیں کہ ایک قدر کا دوسری قدر کے قریب جانے کا کیا مطلب ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus. ", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus. ", "translatedText": "آپ یہ بحث کر سکتے ہیں کہ حساب کتاب کے تعارف کے لیے یہ بلا ضرورت بھاری ذمہ داری ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here. ", "translatedText": "اگر آپ جانتے ہیں کہ لفظ نقطہ نظر کا کیا مطلب ہے، تو آپ پہلے ہی جانتے ہیں کہ حد کا کیا مطلب ہے، یہاں تصوراتی سطح پر کوئی نئی بات نہیں ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/limits/vietnamese/sentence_translations.json b/2017/limits/vietnamese/sentence_translations.json index 0774869cd..c64ced95b 100644 --- a/2017/limits/vietnamese/sentence_translations.json +++ b/2017/limits/vietnamese/sentence_translations.json @@ -208,7 +208,7 @@ "end": 233.4 }, { - "input": "There are others who like to interpret this dx as an infinitely small change, or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", + "input": "There are others who like to interpret this dx as an infinitely small change, whatever Or to just say that dx and df are nothing more than symbols that we shouldn't take too seriously.", "translatedText": "Có những người khác thích giải thích dx này như một sự thay đổi vô cùng nhỏ, hoặc chỉ nói rằng dx và df không gì khác hơn là những biểu tượng mà chúng ta không nên quá coi trọng.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 368.2 }, { - "input": "But imagine you're a mathematician inventing calculus, and someone asks you, well, what exactly do you mean by approach?", + "input": "But imagine you're a mathematician inventing calculus, and someone skeptically asks you, well, what exactly do you mean by approach?", "translatedText": "Nhưng hãy tưởng tượng bạn là một nhà toán học phát minh ra phép tính, và ai đó hỏi bạn, chính xác thì bạn muốn nói gì khi nói cách tiếp cận?", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 377.48 }, { - "input": "That would be kind of an annoying question.", + "input": "That would be kind of an annoying question, I mean, come on,", "translatedText": "Đó sẽ là một câu hỏi khó chịu.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 480.28 }, { - "input": "You could argue that this is needlessly heavy duty for an introduction to calculus.", + "input": "Now I should tell you, you could argue that this is needlessly heavy duty for an introduction to calculus.", "translatedText": "Bạn có thể lập luận rằng đây là một nhiệm vụ nặng nề không cần thiết đối với phần giới thiệu về giải tích.", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 485.5 }, { - "input": "If you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", + "input": "Like I said, if you know what the word approach means, you already know what a limit means, there's nothing new on the conceptual level here.", "translatedText": "Nếu bạn biết từ tiếp cận nghĩa là gì thì bạn đã biết giới hạn nghĩa là gì rồi, ở đây không có gì mới ở cấp độ khái niệm.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/neural-networks/arabic/sentence_translations.json b/2017/neural-networks/arabic/sentence_translations.json index 5c97fa99b..1d7be5697 100644 --- a/2017/neural-networks/arabic/sentence_translations.json +++ b/2017/neural-networks/arabic/sentence_translations.json @@ -452,7 +452,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "لكن بالعودة إلى كيفية عمل أي من هذا فعليًا، تصور نفسك الآن وأنت تصمم كيف يمكن لعمليات التنشيط في طبقة واحدة أن تحدد الطبقة التالية.", "from_community_srt": "تصور نفسك الآن تصمم كيف بالضبط أن تفعيل بعض الخلايا العصبية فى طبقة واحدة يؤثر على تفعيل الخلايا العصبية فى الطبقة التي تليها", "n_reviews": 0, diff --git a/2017/neural-networks/bengali/sentence_translations.json b/2017/neural-networks/bengali/sentence_translations.json index 002362123..1b4c85c49 100644 --- a/2017/neural-networks/bengali/sentence_translations.json +++ b/2017/neural-networks/bengali/sentence_translations.json @@ -472,7 +472,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "কিন্তু এর যেকোনটি আসলে কীভাবে কাজ করে সে সম্পর্কে ফিরে যান, এখনই নিজেকে ডিজাইন করুন যে একটি স্তরের অ্যাক্টিভেশনগুলি ঠিক কীভাবে পরবর্তী অ্যাক্টিভেশনগুলি নির্ধারণ করতে পারে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/neural-networks/chinese/sentence_translations.json b/2017/neural-networks/chinese/sentence_translations.json index 3bec58636..ec82bff98 100644 --- a/2017/neural-networks/chinese/sentence_translations.json +++ b/2017/neural-networks/chinese/sentence_translations.json @@ -496,7 +496,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "我们的目标是建立一种机制，可以将像素组合成边缘，或将边缘组合成图案，或将图案组合成数字。", "model": "DeepL", "n_reviews": 0, diff --git a/2017/neural-networks/czech/sentence_translations.json b/2017/neural-networks/czech/sentence_translations.json index c8a99ab51..7dba57890 100644 --- a/2017/neural-networks/czech/sentence_translations.json +++ b/2017/neural-networks/czech/sentence_translations.json @@ -462,7 +462,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "", "from_community_srt": "A jak cokoliv z toho vlastně funguje? Představte si, že vy jste ten, kdo má navrhnout jak přesně mají aktivace jedné vrstvy ovlivnit aktivace v další vrstvě Cílem je mít nějaký mechanismus,", "n_reviews": 0, diff --git a/2017/neural-networks/english/captions.srt b/2017/neural-networks/english/captions.srt index d522a9b72..c429baea3 100644 --- a/2017/neural-networks/english/captions.srt +++ b/2017/neural-networks/english/captions.srt @@ -519,16 +519,16 @@ which combine to make certain syllables, which combine to form words, which combine to make up phrases and more abstract thoughts, etc. 131 -00:08:21,100 --> 00:08:24,058 +00:08:21,100 --> 00:08:23,735 But getting back to how any of this actually works, 132 -00:08:24,058 --> 00:08:28,497 -picture yourself right now designing how exactly the activations in one layer +00:08:23,735 --> 00:08:27,993 +picture yourself right now designing how exactly the activations in one layer might 133 -00:08:28,497 --> 00:08:29,920 -might determine the next. +00:08:27,993 --> 00:08:29,920 +determine the activations in the next. 134 00:08:30,860 --> 00:08:36,049 diff --git a/2017/neural-networks/english/sentence_timings.json b/2017/neural-networks/english/sentence_timings.json index af03879c8..c307ac43c 100644 --- a/2017/neural-networks/english/sentence_timings.json +++ b/2017/neural-networks/english/sentence_timings.json @@ -290,7 +290,7 @@ 500.06 ], [ - "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", 501.1, 509.92 ], diff --git a/2017/neural-networks/english/transcript.txt b/2017/neural-networks/english/transcript.txt index 260c74309..6a54f1a7b 100644 --- a/2017/neural-networks/english/transcript.txt +++ b/2017/neural-networks/english/transcript.txt @@ -56,7 +56,7 @@ Whether or not this is what our final network actually does is another question, Moreover, you can imagine how being able to detect edges and patterns like this would be really useful for other image recognition tasks. And even beyond image recognition, there are all sorts of intelligent things you might want to do that break down into layers of abstraction. Parsing speech, for example, involves taking raw audio and picking out distinct sounds, which combine to make certain syllables, which combine to form words, which combine to make up phrases and more abstract thoughts, etc. -But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next. +But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next. The goal is to have some mechanism that could conceivably combine pixels into edges, or edges into patterns, or patterns into digits. And to zoom in on one very specific example, let's say the hope is for one particular neuron in the second layer to pick up on whether or not the image has an edge in this region here. The question at hand is what parameters should the network have? diff --git a/2017/neural-networks/french/sentence_translations.json b/2017/neural-networks/french/sentence_translations.json index 254eeb566..e786e9ca9 100644 --- a/2017/neural-networks/french/sentence_translations.json +++ b/2017/neural-networks/french/sentence_translations.json @@ -464,7 +464,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Mais pour en revenir à la façon dont tout cela fonctionne réellement, imagine-toi en train de concevoir comment les activations d'une couche peuvent déterminer la suivante.", "from_community_srt": "Mais pour en revenir à comment tout ceci marche, imaginez-vous décider comment les activations d'une couche pourraient déterminer les activations de la suivante.", "n_reviews": 0, diff --git a/2017/neural-networks/german/sentence_translations.json b/2017/neural-networks/german/sentence_translations.json index a21c0ec7a..1b5cd8328 100644 --- a/2017/neural-networks/german/sentence_translations.json +++ b/2017/neural-networks/german/sentence_translations.json @@ -522,7 +522,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Aber um darauf zurückzukommen, wie das eigentlich funktioniert, stell dir vor, wie genau die Aktivierungen in einer Schicht die nächste Schicht bestimmen könnten.", "model": "DeepL", "from_community_srt": "usw. Zurück dazu, wie das alles tatsächlich funktioniert. Stellen Sie sich vor den Prozess zu gestalten. Wie genau könnte die Aktivierungen in einer Schicht die Aktivierungen in den nächsten bestimmen?", diff --git a/2017/neural-networks/greek/sentence_translations.json b/2017/neural-networks/greek/sentence_translations.json index f7b7049ae..a748e3037 100644 --- a/2017/neural-networks/greek/sentence_translations.json +++ b/2017/neural-networks/greek/sentence_translations.json @@ -522,7 +522,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Αλλά για να επιστρέψουμε στο πώς λειτουργούν όλα αυτά στην πραγματικότητα, φανταστείτε τον εαυτό σας τώρα να σχεδιάζει πώς ακριβώς οι ενεργοποιήσεις σε ένα επίπεδο μπορεί να καθορίζουν το επόμενο.", "model": "DeepL", "from_community_srt": "Ας γυρίσουμε όμως στο πώς δουλεύει καθετί από αυτά. Φανταστείτε τον εαυτό σας αυτήν τη στιγμή να σχεδιάζει το πώς ακριβώς οι ενεργοποιήσεις σε ένα στρώμα μπορεί να καθορίζουν τις ενεργοποιήσεις του επόμενου.", diff --git a/2017/neural-networks/hebrew/sentence_translations.json b/2017/neural-networks/hebrew/sentence_translations.json index 251d7be5d..74279cb4a 100644 --- a/2017/neural-networks/hebrew/sentence_translations.json +++ b/2017/neural-networks/hebrew/sentence_translations.json @@ -472,7 +472,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "אבל אם נחזור לאופן שבו כל זה עובד בפועל, דמיינו את עצמכם כעת מעצבים כיצד בדיוק ההפעלה בשכבה אחת עשויות לקבוע את ההפעלה בשכבה הבאה. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/neural-networks/hindi/sentence_translations.json b/2017/neural-networks/hindi/sentence_translations.json index 12af13f6c..9c4ca214b 100644 --- a/2017/neural-networks/hindi/sentence_translations.json +++ b/2017/neural-networks/hindi/sentence_translations.json @@ -413,7 +413,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "लेकिन इनमें से कोई भी वास्तव में कैसे काम करता है, इस पर वापस लौटते हुए, अभी स्वयं कल्पना करें कि एक परत में सक्रियता वास्तव में अगली परत में सक्रियता कैसे निर्धारित कर सकती है।", "n_reviews": 0, "start": 501.1, diff --git a/2017/neural-networks/hungarian/sentence_translations.json b/2017/neural-networks/hungarian/sentence_translations.json index 0daeeadcb..6c82575f7 100644 --- a/2017/neural-networks/hungarian/sentence_translations.json +++ b/2017/neural-networks/hungarian/sentence_translations.json @@ -462,7 +462,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "De visszatérve arra, hogy mindez hogyan is működik valójában, képzelje el magát most, amint megtervezi, hogy pontosan hogyan határozhatják meg az egyik réteg aktivációi a következőt.", "from_community_srt": "De térjünk vissza arra, hogy mindez hogy is működik. Képzeld el, hogy azt tervezed éppen meg, hogyan határozza meg egy réteg aktivitása a következő rétegét.", "n_reviews": 0, @@ -1042,4 +1042,4 @@ "start": 1105.1, "end": 1105.64 } -] +] \ No newline at end of file diff --git a/2017/neural-networks/indonesian/sentence_translations.json b/2017/neural-networks/indonesian/sentence_translations.json index adb8d4892..b21432c5a 100644 --- a/2017/neural-networks/indonesian/sentence_translations.json +++ b/2017/neural-networks/indonesian/sentence_translations.json @@ -460,7 +460,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Tetapi kembali ke cara kerja semua ini, bayangkan diri Anda sekarang sedang merancang bagaimana tepatnya aktivasi di satu lapisan dapat menentukan lapisan berikutnya.", "from_community_srt": "Tetapi kembali ke bagaimana semua ini benar-benar bekerja menggambarkan diri Anda saat ini merancang Bagaimana sebenarnya aktivasi dalam satu lapisan dapat menentukan aktivasi di lapisan berikutnya?", "n_reviews": 0, diff --git a/2017/neural-networks/italian/sentence_translations.json b/2017/neural-networks/italian/sentence_translations.json index 65d8a2af0..65db05ee6 100644 --- a/2017/neural-networks/italian/sentence_translations.json +++ b/2017/neural-networks/italian/sentence_translations.json @@ -462,7 +462,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Ma tornando al funzionamento effettivo di tutto questo, immagina di progettare in questo momento come le attivazioni di uno strato possano determinare il successivo.", "from_community_srt": "ecc Ma torniamo a vedere come funzionano raffigurandole visivamente In che modo esattamente le attivazioni in un livello potrebbero determinare le attivazioni nel prossimo?", "n_reviews": 0, @@ -1028,4 +1028,4 @@ "start": 1105.1, "end": 1105.64 } -] +] \ No newline at end of file diff --git a/2017/neural-networks/japanese/sentence_translations.json b/2017/neural-networks/japanese/sentence_translations.json index 3b4dc12cc..8c9425c8e 100644 --- a/2017/neural-networks/japanese/sentence_translations.json +++ b/2017/neural-networks/japanese/sentence_translations.json @@ -514,7 +514,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "ピクセルをエッジに、エッジをパターンに、パターンを数字に結合するようなメカニズムが考えられる。", "model": "DeepL", "from_community_srt": "1つの層の活性化が次の層の活性化をどの程度正確に決定するか？", diff --git a/2017/neural-networks/korean/sentence_translations.json b/2017/neural-networks/korean/sentence_translations.json index 9813e0ccd..23e2226fe 100644 --- a/2017/neural-networks/korean/sentence_translations.json +++ b/2017/neural-networks/korean/sentence_translations.json @@ -462,7 +462,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "하지만 이 모든 것이 실제로 어떻게 작동하는지 다시 돌아가서, 한 레이어의 활성화가 다음 레이어를 어떻게 결정할지 설계하는 모습을 상상해 보세요.", "from_community_srt": "다시 돌아와서 지금 당신이 이것이 어떻게 작동할 것인지 설계한다고 상상해봅시다. 한 층의 활성이 어떻게 다음 층에서의 정확한 활성을 이끌어 내는 걸까요? 목표는 픽셀을 테두리로 결합시키거나,", "n_reviews": 0, diff --git a/2017/neural-networks/marathi/sentence_translations.json b/2017/neural-networks/marathi/sentence_translations.json index ea94014d7..8788e8d63 100644 --- a/2017/neural-networks/marathi/sentence_translations.json +++ b/2017/neural-networks/marathi/sentence_translations.json @@ -472,7 +472,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "परंतु यापैकी कोणतेही प्रत्यक्षात कसे कार्य करते याकडे परत जाण्यासाठी, एका लेयरमधील अ‍ॅक्टिव्हेशन्स पुढील अ‍ॅक्टिव्हेशन्स कशी निश्चित करू शकतात हे आत्ताच डिझाईन करत आहे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/neural-networks/persian/sentence_translations.json b/2017/neural-networks/persian/sentence_translations.json index 07feb80c1..50d825ecd 100644 --- a/2017/neural-networks/persian/sentence_translations.json +++ b/2017/neural-networks/persian/sentence_translations.json @@ -457,7 +457,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "اما برای بازگشت به نحوه عملکرد هر یک از اینها، خود را در حال طراحی تصور کنید که دقیقاً چگونه فعال سازی در یک لایه ممکن است لایه بعدی را تعیین کند.", "from_community_srt": "ما به عقب بر گردیم به این که چگونه هر یک از اینها در حال حاضر طراحی تصویر خود را انجام می دهد", "n_reviews": 0, diff --git a/2017/neural-networks/polish/sentence_translations.json b/2017/neural-networks/polish/sentence_translations.json index 48d35da20..d341c3358 100644 --- a/2017/neural-networks/polish/sentence_translations.json +++ b/2017/neural-networks/polish/sentence_translations.json @@ -460,7 +460,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "", "from_community_srt": "jak to wszystko działa, wyobraź sobie teraz projektowanie Jak dokładnie aktywacje w jednej warstwie mogą decydować o aktywacjach w następnej?", "n_reviews": 0, diff --git a/2017/neural-networks/portuguese/sentence_translations.json b/2017/neural-networks/portuguese/sentence_translations.json index 60742063a..aa6ea4cfb 100644 --- a/2017/neural-networks/portuguese/sentence_translations.json +++ b/2017/neural-networks/portuguese/sentence_translations.json @@ -463,7 +463,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Mas voltando ao modo como tudo isso realmente funciona, imagine-se agora mesmo projetando como exatamente as ativações em uma camada podem determinar a próxima.", "from_community_srt": "etc. Mas voltando para como isso realmente funciona, imagine-se projetando o modo como as ativações numa camada podem determinar as ativações da seguinte.", "n_reviews": 0, diff --git a/2017/neural-networks/romanian/sentence_translations.json b/2017/neural-networks/romanian/sentence_translations.json index 3151f599a..4d3b678c2 100644 --- a/2017/neural-networks/romanian/sentence_translations.json +++ b/2017/neural-networks/romanian/sentence_translations.json @@ -464,7 +464,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Dar, revenind la modul în care funcționează de fapt toate acestea, imaginați-vă chiar acum cum anume activările dintr-un strat ar putea determina următorul.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/neural-networks/russian/sentence_translations.json b/2017/neural-networks/russian/sentence_translations.json index 366c3e5ce..3d4e5c6a1 100644 --- a/2017/neural-networks/russian/sentence_translations.json +++ b/2017/neural-networks/russian/sentence_translations.json @@ -464,7 +464,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Но, возвращаясь к тому, как все это работает на самом деле, представь, что ты прямо сейчас проектируешь, как именно активации в одном слое могут определять следующий.", "from_community_srt": "и т.д. Но возвращаясь к тому как собственно что-либо из этого работает. Представьте себе сейчас идею того как активации в одном слое могут определять активации в следующем.", "n_reviews": 0, diff --git a/2017/neural-networks/spanish/sentence_translations.json b/2017/neural-networks/spanish/sentence_translations.json index 965d926c7..930c930f9 100644 --- a/2017/neural-networks/spanish/sentence_translations.json +++ b/2017/neural-networks/spanish/sentence_translations.json @@ -460,7 +460,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Pero volviendo a cómo funciona realmente todo esto, imagínate ahora mismo diseñando cómo exactamente las activaciones de una capa podrían determinar la siguiente.", "from_community_srt": "etc. pero, recordando, cómo esto funciona , , ahora vizualizate a ti diseñando cómo la activación en una capa debería determinar la activación en la siguiente,", "n_reviews": 0, diff --git a/2017/neural-networks/tagalog/sentence_translations.json b/2017/neural-networks/tagalog/sentence_translations.json index 426485575..c01d4cd62 100644 --- a/2017/neural-networks/tagalog/sentence_translations.json +++ b/2017/neural-networks/tagalog/sentence_translations.json @@ -464,7 +464,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Ngunit sa pagbabalik sa kung paano aktwal na gumagana ang alinman sa mga ito, isipin ang iyong sarili ngayon na nagdidisenyo kung paano eksaktong matukoy ng mga pag-activate sa isang layer ang susunod.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/neural-networks/tamil/sentence_translations.json b/2017/neural-networks/tamil/sentence_translations.json index ddcd76a48..8a9f3c75a 100644 --- a/2017/neural-networks/tamil/sentence_translations.json +++ b/2017/neural-networks/tamil/sentence_translations.json @@ -472,7 +472,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "ஆனால், இதில் எது உண்மையில் வேலை செய்கிறது என்பதைத் தெரிந்துகொள்வதன் மூலம், ஒரு லேயரில் உள்ள ஆக்டிவேஷன்கள் அடுத்த ஆக்டிவேஷனை எப்படித் துல்லியமாகத் தீர்மானிக்கலாம் என்பதை இப்போது நீங்களே கற்பனை செய்து பாருங்கள். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/neural-networks/telugu/sentence_translations.json b/2017/neural-networks/telugu/sentence_translations.json index 2c800f13a..3058d1669 100644 --- a/2017/neural-networks/telugu/sentence_translations.json +++ b/2017/neural-networks/telugu/sentence_translations.json @@ -472,7 +472,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "అయితే వీటిలో ఏదైనా వాస్తవానికి ఎలా పనిచేస్తుందో తిరిగి పొందడం, ఒక లేయర్‌లోని యాక్టివేషన్‌లు తదుపరి యాక్టివేషన్‌లను ఎలా నిర్ధారిస్తాయో ఇప్పుడే డిజైన్ చేసుకోండి. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/neural-networks/thai/sentence_translations.json b/2017/neural-networks/thai/sentence_translations.json index 1217e29df..f88dd800d 100644 --- a/2017/neural-networks/thai/sentence_translations.json +++ b/2017/neural-networks/thai/sentence_translations.json @@ -450,7 +450,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "แต่เมื่อกลับมาดูว่าสิ่งเหล่านี้ใช้งานได้จริงอย่างไร ลองนึกภาพตัวคุณเองในตอนนี้ว่ากำลังออกแบบว่าการเปิดใช้งานในเลเยอร์หนึ่งสามารถกำหนดเลเยอร์ถัดไปได้อย่างไร", "from_community_srt": "ซึ่งรวมกันเป็นคำที่รวมกันเพื่อทำเป็นวลีและความคิดที่เป็นนามธรรมมากขึ้นเป็นต้น แต่กลับไปที่วิธีการใด ๆ นี้จริงการทำงานภาพตัวเองในขณะนี้การออกแบบ", "n_reviews": 0, diff --git a/2017/neural-networks/turkish/sentence_translations.json b/2017/neural-networks/turkish/sentence_translations.json index 743145758..8ec93fa44 100644 --- a/2017/neural-networks/turkish/sentence_translations.json +++ b/2017/neural-networks/turkish/sentence_translations.json @@ -459,7 +459,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Ancak bunların gerçekte nasıl çalıştığına geri dönersek, şu anda kendinizi bir katmandaki aktivasyonların bir sonrakini tam olarak nasıl belirleyebileceğini tasarlarken hayal edin.", "from_community_srt": "Ancak bunlardan herhangi birinin resmin kendisinin şu anda tasarımında nasıl çalıştığını öğrenmek. Bir katmandaki aktivasyonların sonraki etkinlikleri nasıl belirleyeceği? Hedef,", "n_reviews": 0, diff --git a/2017/neural-networks/ukrainian/sentence_translations.json b/2017/neural-networks/ukrainian/sentence_translations.json index 3a62bfe49..9e1fdde4c 100644 --- a/2017/neural-networks/ukrainian/sentence_translations.json +++ b/2017/neural-networks/ukrainian/sentence_translations.json @@ -472,7 +472,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Мета полягає в тому, щоб мати якийсь механізм, який міг би об'єднувати пікселі в ребра, або ребра в візерунки, або візерунки в цифри.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/neural-networks/urdu/sentence_translations.json b/2017/neural-networks/urdu/sentence_translations.json index d58308821..55935b021 100644 --- a/2017/neural-networks/urdu/sentence_translations.json +++ b/2017/neural-networks/urdu/sentence_translations.json @@ -472,7 +472,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "لیکن اس بات پر واپس جانا کہ اس میں سے کوئی بھی حقیقت میں کیسے کام کرتا ہے، اپنے آپ کو ابھی یہ ڈیزائن کرتے ہوئے تصویر بنائیں کہ ایک پرت میں موجود ایکٹیویشن اگلے میں ایکٹیویشن کا تعین کیسے کر سکتی ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/neural-networks/vietnamese/sentence_translations.json b/2017/neural-networks/vietnamese/sentence_translations.json index af4c5980c..2f1fcdf2c 100644 --- a/2017/neural-networks/vietnamese/sentence_translations.json +++ b/2017/neural-networks/vietnamese/sentence_translations.json @@ -460,7 +460,7 @@ "end": 500.06 }, { - "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the next.", + "input": "But getting back to how any of this actually works, picture yourself right now designing how exactly the activations in one layer might determine the activations in the next.", "translatedText": "Nhưng quay lại cách thức hoạt động của bất kỳ thứ nào trong số này thực sự hoạt động, hãy hình dung ngay bây giờ chính bạn đang thiết kế cách chính xác các kích hoạt trong một lớp có thể xác định lớp tiếp theo.", "from_community_srt": "Nhưng nhận được trở lại như thế nào những điều này thực sự hoạt động hình ảnh chính mình ngay bây giờ thiết kế", "n_reviews": 0, diff --git a/2017/pythagorean-triples/arabic/sentence_translations.json b/2017/pythagorean-triples/arabic/sentence_translations.json index f22ac2c98..fb0d7c9d4 100644 --- a/2017/pythagorean-triples/arabic/sentence_translations.json +++ b/2017/pythagorean-triples/arabic/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "في الواقع، لأسباب سأشرحها قريبًا، كل ثلاثية فيثاغورس محتملة نغفلها هي مجرد مضاعفات لثلاثية مختلفة ضربناها.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "عند القيام بذلك لجميع النقاط الممكنة، فسوف تقوم بحساب كل ثلاثية فيثاغورس محتملة.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "وهذا يعني أن طريقتنا يجب أن تصل إلى كل ثلاثية فيثاغورس محتملة.", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/bengali/sentence_translations.json b/2017/pythagorean-triples/bengali/sentence_translations.json index 42e2e7421..679ba4e77 100644 --- a/2017/pythagorean-triples/bengali/sentence_translations.json +++ b/2017/pythagorean-triples/bengali/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "প্রকৃতপক্ষে, যে কারণে আমি শীঘ্রই ব্যাখ্যা করব, প্রতিটি সম্ভাব্য পিথাগোরিয়ান ট্রিপল যা আমরা মিস করি তা হল আমাদের আঘাত করা একটি ভিন্ন ট্রিপলের কয়েকটি গুণ।", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "সমস্ত সম্ভাব্য পয়েন্টের জন্য এটি করা, আপনি প্রতিটি সম্ভাব্য পিথাগোরিয়ান ট্রিপলের জন্য হিসাব করবেন।", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "এবং এর অর্থ হল আমাদের পদ্ধতি অবশ্যই প্রতিটি সম্ভাব্য পিথাগোরিয়ান ট্রিপলকে আঘাত করতে হবে।", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/chinese/sentence_translations.json b/2017/pythagorean-triples/chinese/sentence_translations.json index f123022d2..c4e9291f2 100644 --- a/2017/pythagorean-triples/chinese/sentence_translations.json +++ b/2017/pythagorean-triples/chinese/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "事实上，出于我将很快解释的原因，我们错过的每个可能的毕达哥拉斯三元组只是我们击中的不同三元组的一些倍数。", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "对所有可能的点执行此操作，您将解释每个可能的毕达哥拉斯三元组。", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "这意味着我们的方法必须命中所有可能的毕达哥拉斯三元组。", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/english/captions.srt b/2017/pythagorean-triples/english/captions.srt index 27d9a5e44..03a7ba438 100644 --- a/2017/pythagorean-triples/english/captions.srt +++ b/2017/pythagorean-triples/english/captions.srt @@ -515,16 +515,16 @@ since you can get each one of them by scaling up the familiar triple 3 4 5, which is accounted for in our method. 130 -00:09:02,180 --> 00:09:05,121 +00:09:02,180 --> 00:09:04,926 In fact, for reasons that I'll explain shortly, 131 -00:09:05,121 --> 00:09:09,349 -every possible pythagorean triple we miss is just some multiple of a +00:09:04,926 --> 00:09:09,160 +every possible pythagorean triple that we miss is just some multiple of a 132 -00:09:09,349 --> 00:09:10,820 -different triple we hit. +00:09:09,160 --> 00:09:10,820 +different triple that we hit. 133 00:09:11,560 --> 00:09:14,880 @@ -571,254 +571,270 @@ Marking all of the lattice points that this line hits will account for any multiples of these points that we might have missed. 144 -00:10:03,700 --> 00:10:09,220 -Doing this for all possible points, you'll account for every possible pythagorean triple. +00:10:03,700 --> 00:10:07,674 +Doing this for all possible points, you'll account for every possible point that you miss. 145 +00:10:07,674 --> 00:10:09,220 + Every possible Pythagorean triple, + +146 00:10:10,000 --> 00:10:14,148 Every right triangle that you ever have seen or ever will see that has -146 +147 00:10:14,148 --> 00:10:18,180 whole number side lengths is accounted for somewhere in this diagram. -147 +148 00:10:22,760 --> 00:10:27,375 To see why, we'll now shift to a different view of the pythagorean triple problem, -148 +149 00:10:27,375 --> 00:10:31,880 one that involves finding points on a unit circle that have rational coordinates. -149 +150 00:10:33,080 --> 00:10:38,763 If you take the expression a squared plus b squared equals c squared and divide out -150 +151 00:10:38,763 --> 00:10:44,380 by that c squared, what you get is a over c squared plus b over c squared equals 1. -151 +152 00:10:45,200 --> 00:10:48,600 This gives us some point on the unit circle x squared plus y -152 +153 00:10:48,600 --> 00:10:52,000 squared equals 1 whose coordinates are each rational numbers. -153 +154 00:10:52,400 --> 00:10:55,680 This is what we call a rational point of the unit circle. -154 +155 00:10:56,220 --> 00:10:59,692 And going the other way around, if you find some rational point on -155 +156 00:10:59,692 --> 00:11:03,112 the unit circle when you multiply out by a common denominator for -156 +157 00:11:03,112 --> 00:11:06,584 each of those coordinates, what you'll land on is a point that has -157 +158 00:11:06,584 --> 00:11:10,420 integer coordinates and whose distance from the origin is also an integer. -158 +159 00:11:11,700 --> 00:11:14,873 With that in mind, consider our diagram, where we squared every -159 +160 00:11:14,873 --> 00:11:18,046 possible lattice point and then drew these radial lines through -160 +161 00:11:18,046 --> 00:11:21,220 each one to account for any multiples that we might have missed. -161 +162 00:11:22,040 --> 00:11:25,176 If you project all of these points onto the unit circle, -162 +163 00:11:25,176 --> 00:11:28,092 each one moving along its corresponding radial line, -163 +164 00:11:28,092 --> 00:11:32,220 what you'll end up with is a whole bunch of rational points on that circle. -164 +165 00:11:33,440 --> 00:11:38,012 And keep in mind, by the way, I'm drawing only finitely many of these dots and lines, -165 +166 00:11:38,012 --> 00:11:42,265 but if I drew all infinitely many lines corresponding to every possible squared -166 +167 00:11:42,265 --> 00:11:46,040 lattice point, it would actually fill every single pixel of the screen. -167 +168 00:11:47,660 --> 00:11:51,522 Now if our method was incomplete, if we were missing a Pythagorean triple -168 +169 00:11:51,522 --> 00:11:55,229 out there somewhere, it would mean that there's some rational point on -169 +170 00:11:55,229 --> 00:11:59,040 this circle that we never hit once we project everything onto the circle. -170 +171 00:11:59,900 --> 00:12:02,100 And let me show you why that cannot happen. -171 +172 00:12:03,120 --> 00:12:05,950 Take any one of those rational points and draw -172 +173 00:12:05,950 --> 00:12:08,720 a line between it and the point at negative 1. -173 +174 00:12:09,340 --> 00:12:12,935 When you compute the rise over run slope of this line, -174 +175 00:12:12,935 --> 00:12:17,772 the rise between the two points is rational and the run is also rational, -175 +176 00:12:17,772 --> 00:12:21,760 so the slope itself is just going to be some rational number. -176 +177 00:12:22,520 --> 00:12:26,682 So if we can show that our method of squaring complex numbers accounts -177 +178 00:12:26,682 --> 00:12:30,726 for every possible rational slope here, it's going to guarantee that -178 +179 00:12:30,726 --> 00:12:34,420 we hit every possible rational point of the unit circle, right? -179 +180 00:12:36,720 --> 00:12:38,580 Well, let's think through our method. -180 +181 00:12:39,340 --> 00:12:43,266 We start off with some point u plus vi that has integer coordinates, -181 +182 00:12:43,266 --> 00:12:48,160 and this number makes some angle off of the horizontal, which I'm going to call theta. -182 +183 00:12:48,900 --> 00:12:54,220 Squaring this number, the resulting angle off the horizontal, is 2 times theta. -183 +184 00:12:56,160 --> 00:12:59,586 And of course, when you project that onto the unit circle, -184 +185 00:12:59,586 --> 00:13:03,245 it's along the same radial line, so the corresponding rational -185 +186 00:13:03,245 --> 00:13:07,020 point of the unit circle also has that same angle, 2 times theta. -186 +187 00:13:08,140 --> 00:13:11,358 And here, I'll bring in a nice little bit of circle geometry, -187 +188 00:13:11,358 --> 00:13:14,836 which is that any time you have an angle between two points on the -188 +189 00:13:14,836 --> 00:13:18,522 circumference of a circle and its center, that turns out to be exactly -189 +190 00:13:18,522 --> 00:13:22,312 two times the angle made by those same points and any other point on the -190 +191 00:13:22,312 --> 00:13:26,101 circle's circumference, provided that that other point isn't between the -191 +192 00:13:26,101 --> 00:13:27,140 original two points. -192 +193 00:13:28,400 --> 00:13:32,762 What this means for our situation is that the line between negative 1 and -193 +194 00:13:32,762 --> 00:13:37,360 the rational point on the circle must make an angle theta with the horizontal. -194 +195 00:13:38,740 --> 00:13:42,324 In other words, that line has the same slope as the line -195 +196 00:13:42,324 --> 00:13:46,160 between the origin and our initial complex number, u plus vi. -196 -00:13:46,780 --> 00:13:51,978 -But look at the rise over run slope of the line defined by our choice of integers, - 197 -00:13:51,978 --> 00:13:52,480 -u and v. +00:13:46,780 --> 00:13:49,800 +But look at the rise over run slope, which is the same as the line 198 -00:13:53,280 --> 00:13:55,220 -The slope is v divided by u. +00:13:49,800 --> 00:13:52,911 +between the original and the original. So the rise over run slope is 199 -00:13:56,060 --> 00:14:00,390 -And of course, we can choose v and u to be whatever integers we want, +00:13:52,911 --> 00:13:55,617 +the same as the line between the original and the original, 200 -00:14:00,390 --> 00:14:04,660 -and therefore we do indeed account for every possible rational slope. +00:13:55,617 --> 00:13:59,359 +but look at the rise over run slope of the line defined by our choice of integers, 201 +00:13:59,359 --> 00:13:59,720 +u and v. + +202 +00:14:00,420 --> 00:13:59,720 +The slope is v divided by u. + +203 +00:14:00,420 --> 00:14:02,555 +And of course, we can choose v and u to be whatever integers we want, + +204 +00:14:02,555 --> 00:14:04,660 +and therefore we do indeed account for every possible rational slope. + +205 00:14:05,820 --> 00:14:06,460 So there you go! -202 +206 00:14:07,080 --> 00:14:12,336 The radial lines from our method, determined by all possible choices of u and v, -203 +207 00:14:12,336 --> 00:14:15,840 must pass through every rational point on this circle. -204 +208 00:14:16,160 --> 00:14:20,400 And that means our method must hit every possible Pythagorean triple. -205 -00:14:27,540 --> 00:14:42,409 +209 +00:14:27,540 --> 00:14:40,136 If you haven't already watched the video about pi hiding in prime regularities, -206 -00:14:42,409 --> 00:14:52,260 -the topics there are highly related to the ones here. +210 +00:14:40,136 --> 00:14:52,260 +the topics there are highly related to the ones here. Thank you for watching! diff --git a/2017/pythagorean-triples/english/sentence_timings.json b/2017/pythagorean-triples/english/sentence_timings.json index 9cffea355..f6e0be90e 100644 --- a/2017/pythagorean-triples/english/sentence_timings.json +++ b/2017/pythagorean-triples/english/sentence_timings.json @@ -320,7 +320,7 @@ 541.1 ], [ - "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", 542.18, 550.82 ], @@ -365,7 +365,7 @@ 599.88 ], [ - "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", 603.7, 609.22 ], @@ -475,18 +475,18 @@ 826.16 ], [ - "But look at the rise over run slope of the line defined by our choice of integers, u and v.", + "But look at the rise over run slope, which is the same as the line between the original and the original. So the rise over run slope is the same as the line between the original and the original, but look at the rise over run slope of the line defined by our choice of integers, u and v.", 826.78, - 832.48 + 839.72 ], [ "The slope is v divided by u.", - 833.28, - 835.22 + 840.42, + 839.72 ], [ "And of course, we can choose v and u to be whatever integers we want, and therefore we do indeed account for every possible rational slope.", - 836.06, + 840.42, 844.66 ], [ @@ -505,7 +505,7 @@ 860.4 ], [ - "If you haven't already watched the video about pi hiding in prime regularities, the topics there are highly related to the ones here.", + "If you haven't already watched the video about pi hiding in prime regularities, the topics there are highly related to the ones here. Thank you for watching!", 867.54, 892.26 ] diff --git a/2017/pythagorean-triples/english/transcript.txt b/2017/pythagorean-triples/english/transcript.txt index 42a6cfd6c..4f94cee21 100644 --- a/2017/pythagorean-triples/english/transcript.txt +++ b/2017/pythagorean-triples/english/transcript.txt @@ -62,7 +62,7 @@ For example, you will never get the point 6 plus 8i using this method, even thou There are simply no integers u and v where u plus vi squared is 6 plus 8i. Likewise, you will never hit 9 plus 12i. But these don't really feel like anything new, do they, since you can get each one of them by scaling up the familiar triple 3 4 5, which is accounted for in our method. -In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit. +In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit. To give another example, we miss the point 4 plus 3i. There are no integers u and v, so that u plus vi squared is 4 plus 3i. In fact, you'll never hit any points whose imaginary component is odd. @@ -71,7 +71,7 @@ So even though we miss 4 plus 3i, it's just one half times the point we do hit. And by the way, you'll never have to scale down by anything smaller than one half. A nice way to think about these multiples that we miss is to take each point that we get using this squaring method and draw a line from the origin through that point out to infinity. Marking all of the lattice points that this line hits will account for any multiples of these points that we might have missed. -Doing this for all possible points, you'll account for every possible pythagorean triple. +Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple, Every right triangle that you ever have seen or ever will see that has whole number side lengths is accounted for somewhere in this diagram. To see why, we'll now shift to a different view of the pythagorean triple problem, one that involves finding points on a unit circle that have rational coordinates. If you take the expression a squared plus b squared equals c squared and divide out by that c squared, what you get is a over c squared plus b over c squared equals 1. @@ -93,10 +93,10 @@ And of course, when you project that onto the unit circle, it's along the same r And here, I'll bring in a nice little bit of circle geometry, which is that any time you have an angle between two points on the circumference of a circle and its center, that turns out to be exactly two times the angle made by those same points and any other point on the circle's circumference, provided that that other point isn't between the original two points. What this means for our situation is that the line between negative 1 and the rational point on the circle must make an angle theta with the horizontal. In other words, that line has the same slope as the line between the origin and our initial complex number, u plus vi. -But look at the rise over run slope of the line defined by our choice of integers, u and v. +But look at the rise over run slope, which is the same as the line between the original and the original. So the rise over run slope is the same as the line between the original and the original, but look at the rise over run slope of the line defined by our choice of integers, u and v. The slope is v divided by u. And of course, we can choose v and u to be whatever integers we want, and therefore we do indeed account for every possible rational slope. So there you go! The radial lines from our method, determined by all possible choices of u and v, must pass through every rational point on this circle. And that means our method must hit every possible Pythagorean triple. -If you haven't already watched the video about pi hiding in prime regularities, the topics there are highly related to the ones here. \ No newline at end of file +If you haven't already watched the video about pi hiding in prime regularities, the topics there are highly related to the ones here. Thank you for watching! \ No newline at end of file diff --git a/2017/pythagorean-triples/french/sentence_translations.json b/2017/pythagorean-triples/french/sentence_translations.json index 5c357b08b..379fd0268 100644 --- a/2017/pythagorean-triples/french/sentence_translations.json +++ b/2017/pythagorean-triples/french/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "En fait, pour des raisons que j'expliquerai sous peu, chaque triplet pythagoricien possible que nous manquons n'est qu'un multiple d'un triplet différent que nous avons atteint.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "En faisant cela pour tous les points possibles, vous prendrez en compte tous les triples pythagoriciens possibles.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "Et cela signifie que notre méthode doit atteindre tous les triplets pythagoriciens possibles.", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/german/sentence_translations.json b/2017/pythagorean-triples/german/sentence_translations.json index 8ea31eff5..00116343c 100644 --- a/2017/pythagorean-triples/german/sentence_translations.json +++ b/2017/pythagorean-triples/german/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "Aus Gründen, die ich gleich erläutern werde, ist tatsächlich jedes mögliche pythagoreische Tripel, das wir verpassen, nur ein Vielfaches eines anderen Tripels, das wir treffen.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "Wenn Sie dies für alle möglichen Punkte tun, berücksichtigen Sie jedes mögliche pythagoräische Tripel.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "Und das bedeutet, dass unsere Methode jedes mögliche pythagoräische Tripel treffen muss.", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/hebrew/sentence_translations.json b/2017/pythagorean-triples/hebrew/sentence_translations.json index 87bba097c..a46502014 100644 --- a/2017/pythagorean-triples/hebrew/sentence_translations.json +++ b/2017/pythagorean-triples/hebrew/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "למעשה, מסיבות שאסביר בקרוב, כל משולשת פיתגורית אפשרית שאנו מפספסים היא רק כפולה של משולש אחר שאנו מכים.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "אם תעשה זאת עבור כל הנקודות האפשריות, תחשבו על כל טריפל אפשרי של פיתגורס.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "וזה אומר שהשיטה שלנו חייבת לפגוע בכל משולש פיתגורי אפשרי.", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/hindi/sentence_translations.json b/2017/pythagorean-triples/hindi/sentence_translations.json index f13167117..82aed8b05 100644 --- a/2017/pythagorean-triples/hindi/sentence_translations.json +++ b/2017/pythagorean-triples/hindi/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "वास्तव में, जिन कारणों के बारे में मैं जल्द ही बताऊंगा, उनमें से प्रत्येक संभावित पायथागॉरियन ट्रिपल जो हम चूक जाते हैं, वह हमारे द्वारा प्राप्त किए गए अलग-अलग ट्रिपल का कुछ गुणज है।", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "सभी संभावित बिंदुओं के लिए ऐसा करने पर, आप प्रत्येक संभावित पायथागॉरियन ट्रिपल का हिसाब लगा लेंगे।", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "और इसका मतलब है कि हमारी पद्धति को हर संभव पायथागॉरियन ट्रिपल को हिट करना होगा।", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/hungarian/sentence_translations.json b/2017/pythagorean-triples/hungarian/sentence_translations.json index 3aae03249..76bf5e589 100644 --- a/2017/pythagorean-triples/hungarian/sentence_translations.json +++ b/2017/pythagorean-triples/hungarian/sentence_translations.json @@ -512,7 +512,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "Valójában, olyan okokból, amelyeket rövidesen elmagyarázok, minden lehetséges pitagorai hármas, amit kihagyunk, csak egy másik hármas többszöröse, amit eltalálunk.", "model": "DeepL", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "Ha ezt az összes lehetséges pontra elvégezzük, akkor minden lehetséges pitagoraszi hármas számot figyelembe veszünk.", "model": "DeepL", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "Ez azt jelenti, hogy a módszerünknek minden lehetséges Pitagorasz-hármast el kell találnia.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/pythagorean-triples/indonesian/sentence_translations.json b/2017/pythagorean-triples/indonesian/sentence_translations.json index e81767d9a..cb17c53cc 100644 --- a/2017/pythagorean-triples/indonesian/sentence_translations.json +++ b/2017/pythagorean-triples/indonesian/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "Faktanya, untuk alasan yang akan saya jelaskan segera, setiap kemungkinan tripel pythagoras yang kita lewatkan hanyalah kelipatan dari tripel berbeda yang kita peroleh.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "Dengan melakukan hal ini untuk semua kemungkinan poin, Anda akan memperhitungkan setiap kemungkinan tripel Pythagoras.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "Artinya, metode kita harus mencapai semua kemungkinan tripel Pythagoras.", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/italian/sentence_translations.json b/2017/pythagorean-triples/italian/sentence_translations.json index e1ff8527a..e6e835882 100644 --- a/2017/pythagorean-triples/italian/sentence_translations.json +++ b/2017/pythagorean-triples/italian/sentence_translations.json @@ -575,7 +575,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "Infatti, per ragioni che spiegherò a breve, ogni possibile tripla pitagorica mancata è solo un multiplo di una tripla diversa che abbiamo colpito.", "model": "DeepL", "from_community_srt": "In effetti, per ragioni che spiegherò a breve, ogni possibile tripla pitagorica che \"manchiamo\" è solo un qualche multiplo di un'altra tripla che otteniamo.", @@ -656,7 +656,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "Facendo questo per tutti i punti possibili, si terrà conto di tutte le possibili triple pitagoriche.", "model": "DeepL", "from_community_srt": "Facendolo per tutti i possibili punti, otterremo ogni possibile tripla pitagorica.", @@ -899,7 +899,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "Questo significa che il nostro metodo deve colpire ogni possibile triplo pitagorico.", "model": "DeepL", "from_community_srt": "e questo significa che il nostro metodo deve colpire ogni possibile tripla pitagorica.", diff --git a/2017/pythagorean-triples/japanese/sentence_translations.json b/2017/pythagorean-triples/japanese/sentence_translations.json index 32da6a6bc..8a7a50678 100644 --- a/2017/pythagorean-triples/japanese/sentence_translations.json +++ b/2017/pythagorean-triples/japanese/sentence_translations.json @@ -510,7 +510,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "実際、すぐに説明する理由により、私たちがミスする可能性のあるすべてのピタゴラス トリプルは、ヒットした別のトリプルの倍数にすぎません。", "from_community_srt": "つまり、簡単に言えば、探し当てることができないピタゴラス数は、 探し当てられるピタゴラス数の何倍かしたものである。", "n_reviews": 0, @@ -580,7 +580,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "これをすべての可能な点に対して行うと、すべての可能なピタゴラス トリプルが計算されます。", "from_community_srt": "他の点でも同様のことをすると、さらにピタゴラス数が見つかる。", "n_reviews": 0, @@ -794,7 +794,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "それは、私たちの方法がすべての可能なピタゴラス トリプルに一致する必要があることを意味します。", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/korean/sentence_translations.json b/2017/pythagorean-triples/korean/sentence_translations.json index 4ec5be2f9..2f8d30a43 100644 --- a/2017/pythagorean-triples/korean/sentence_translations.json +++ b/2017/pythagorean-triples/korean/sentence_translations.json @@ -547,7 +547,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "사실, 곧 설명할 이유 때문에 우리가 놓칠 수 있는 모든 피타고라스 삼각형은 다른 삼각형의 일부 배수일 뿐입니다.", "model": "DeepL", "n_reviews": 0, @@ -619,7 +619,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "가능한 모든 포인트에 대해 이 작업을 수행하면 가능한 모든 피타고라스의 세 배를 설명할 수 있습니다.", "model": "DeepL", "n_reviews": 0, @@ -835,7 +835,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "즉, 우리의 방법은 가능한 모든 피타고라스 삼중 항을 맞혀야 합니다.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/pythagorean-triples/marathi/sentence_translations.json b/2017/pythagorean-triples/marathi/sentence_translations.json index d9c045bb0..9de50cf80 100644 --- a/2017/pythagorean-triples/marathi/sentence_translations.json +++ b/2017/pythagorean-triples/marathi/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "खरं तर, ज्या कारणांसाठी मी लवकरच समजावून सांगेन, आपण गमावलेला प्रत्येक संभाव्य पायथागोरियन ट्रिपल हा आपण मारलेल्या वेगळ्या तिहेरीचा काही गुणक असतो.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "सर्व संभाव्य बिंदूंसाठी असे केल्याने, आपण प्रत्येक संभाव्य पायथागोरियन ट्रिपलचा हिशोब घ्याल.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "आणि याचा अर्थ असा आहे की आमच्या पद्धतीने प्रत्येक संभाव्य पायथागोरियन ट्रिपलला मारले पाहिजे.", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/persian/sentence_translations.json b/2017/pythagorean-triples/persian/sentence_translations.json index cfe69ac95..c5e4a1a25 100644 --- a/2017/pythagorean-triples/persian/sentence_translations.json +++ b/2017/pythagorean-triples/persian/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "در واقع، به دلایلی که به‌زودی توضیح خواهم داد، هر سه‌گانه فیثاغورثی ممکنی که از دست می‌دهیم، فقط چند مضرب از یک سه‌گانه متفاوت است که به آن برخورد می‌کنیم.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "با انجام این کار برای تمام نقاط ممکن، هر سه گانه فیثاغورثی ممکن را محاسبه خواهید کرد.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "و این بدان معناست که روش ما باید به هر سه گانه فیثاغورثی ممکن ضربه بزند.", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/polish/sentence_translations.json b/2017/pythagorean-triples/polish/sentence_translations.json index 3b6b80aba..6cbf316b9 100644 --- a/2017/pythagorean-triples/polish/sentence_translations.json +++ b/2017/pythagorean-triples/polish/sentence_translations.json @@ -512,7 +512,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "", "from_community_srt": "Właściwie, z powodów, które wytłumaczę wkrótce, wszelkie możliwe trójki pitagorejskie, które omijamy, są tylko wielokrotnościami innych trójek, na które trafiliśmy.", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "", "from_community_srt": "Robiąc tak dla wszystkich możliwych punktów wskażesz wszystkie możliwe trójki pitagorejskie.", "n_reviews": 0, @@ -799,7 +799,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "", "from_community_srt": "a to oznacza, że ​​nasza metoda musi znaleźć każdą możliwą trójkę pitagorejską.", "n_reviews": 0, diff --git a/2017/pythagorean-triples/portuguese/sentence_translations.json b/2017/pythagorean-triples/portuguese/sentence_translations.json index bcaecce06..9fb171f2c 100644 --- a/2017/pythagorean-triples/portuguese/sentence_translations.json +++ b/2017/pythagorean-triples/portuguese/sentence_translations.json @@ -568,7 +568,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "Na verdade, por razões que explicarei em breve, cada triplo pitagórico possível que perdemos é apenas um múltiplo de um triplo diferente que atingimos.", "model": "google_nmt", "from_community_srt": "por razões as quais explicarei em breve Cada triangulo pitagórico possível que perdemos é apenas um múltiplo de um outro terno que obtemos", @@ -649,7 +649,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "Fazendo isso para todos os pontos possíveis, você contabilizará todos os triplos pitagóricos possíveis.", "model": "google_nmt", "from_community_srt": "fazendo isso para todos os pontos possíveis você irá contabilizar cada triangulo pitagórico possível", @@ -889,7 +889,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "E isso significa que o nosso método deve atingir todos os triplos pitagóricos possíveis.", "model": "google_nmt", "from_community_srt": "devem passar por todos os pontos racionais neste círculo e isto significa que nosso método deve obter cada terno pitagórico possível", diff --git a/2017/pythagorean-triples/russian/sentence_translations.json b/2017/pythagorean-triples/russian/sentence_translations.json index 6d5c3b5cf..e9d3f4b2c 100644 --- a/2017/pythagorean-triples/russian/sentence_translations.json +++ b/2017/pythagorean-triples/russian/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "Фактически, по причинам, которые я вскоре объясню, каждая возможная тройка Пифагора, которую мы упускаем, является всего лишь кратным другой тройки, в которую мы попадаем.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "Проделав это для всех возможных точек, вы учтете каждую возможную пифагорову тройку.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "А это значит, что наш метод должен поразить каждую возможную пифагорову тройку.", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/spanish/sentence_translations.json b/2017/pythagorean-triples/spanish/sentence_translations.json index 2b1b3f8bd..ebdc0930f 100644 --- a/2017/pythagorean-triples/spanish/sentence_translations.json +++ b/2017/pythagorean-triples/spanish/sentence_translations.json @@ -509,7 +509,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "De hecho, por razones que explicaré en breve, cada triplete pitagórico posible que omitimos es sólo un múltiplo de un triplete diferente que acertamos.", "from_community_srt": "En realidad, por razones que explicaré en breve, cada terna pitagórica que se nos escapa es tan sólo un múltiplo de otra terna que sí generamos.", "n_reviews": 0, @@ -581,7 +581,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "Al hacer esto para todos los puntos posibles, tendrás en cuenta cada terna pitagórica posible.", "from_community_srt": "Si hacemos esto con todos los puntos posibles, incluiremos todas las ternas pitagóricas.", "n_reviews": 0, @@ -795,7 +795,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "Y eso significa que nuestro método debe alcanzar todas las ternas pitagóricas posibles.", "from_community_srt": "lo cual significa que nuestro método encontrará todas las ternas pitagóricas posibles.", "n_reviews": 0, diff --git a/2017/pythagorean-triples/tamil/sentence_translations.json b/2017/pythagorean-triples/tamil/sentence_translations.json index af0abdb66..5511d3fce 100644 --- a/2017/pythagorean-triples/tamil/sentence_translations.json +++ b/2017/pythagorean-triples/tamil/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "உண்மையில், நான் விரைவில் விளக்கக்கூடிய காரணங்களுக்காக, நாம் தவறவிடக்கூடிய ஒவ்வொரு பித்தகோரியன் ட்ரிப்பிளும் நாம் அடித்த வெவ்வேறு ட்ரிப்பிளின் சில மடங்குகளாகும்.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "சாத்தியமான எல்லா புள்ளிகளுக்கும் இதைச் செய்வதன் மூலம், சாத்தியமான ஒவ்வொரு பித்தகோரியன் மும்மடங்கையும் நீங்கள் கணக்கிடுவீர்கள்.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "எங்கள் முறை சாத்தியமான ஒவ்வொரு பித்தகோரியன் மும்மடங்கைத் தாக்க வேண்டும் என்பதாகும்.", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/telugu/sentence_translations.json b/2017/pythagorean-triples/telugu/sentence_translations.json index 57d73973d..d028d4504 100644 --- a/2017/pythagorean-triples/telugu/sentence_translations.json +++ b/2017/pythagorean-triples/telugu/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "వాస్తవానికి, నేను త్వరలో వివరించే కారణాల వల్ల, మనం మిస్ అయ్యే ప్రతి పైథాగరియన్ ట్రిపుల్ మనం కొట్టే వేరే ట్రిపుల్‌లో కొన్ని మల్టిపుల్ మాత్రమే.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "సాధ్యమయ్యే అన్ని పాయింట్ల కోసం ఇలా చేయడం ద్వారా, మీరు సాధ్యమయ్యే ప్రతి పైథాగరియన్ ట్రిపుల్‌కు ఖాతా పొందుతారు.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "మరియు మా పద్ధతి సాధ్యమయ్యే ప్రతి పైథాగరియన్ ట్రిపుల్‌ను కొట్టాలి.", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/thai/sentence_translations.json b/2017/pythagorean-triples/thai/sentence_translations.json index 88d09f2ae..3d53c4815 100644 --- a/2017/pythagorean-triples/thai/sentence_translations.json +++ b/2017/pythagorean-triples/thai/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "ที่จริง ด้วยเหตุผลที่ผมจะอธิบายเร็วๆ นี้ ทุกทริปเปิ้ลพีทาโกรัสที่เป็นไปได้ที่เราพลาดไป เป็นเพียงผลคูณของทริปเปิ้ลต่างๆ ที่เราทำได้", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "การทำเช่นนี้กับคะแนนที่เป็นไปได้ทั้งหมด คุณจะคิดเป็นสามเท่าของพีทาโกรัสที่เป็นไปได้ด้วย", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "และนั่นหมายความว่าวิธีการของเราต้องตีทุกสามเท่าของพีทาโกรัสที่เป็นไปได้", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/turkish/sentence_translations.json b/2017/pythagorean-triples/turkish/sentence_translations.json index 6fc1ed574..70192248d 100644 --- a/2017/pythagorean-triples/turkish/sentence_translations.json +++ b/2017/pythagorean-triples/turkish/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "Aslında, kısaca açıklayacağım nedenlerden ötürü, kaçırdığımız her olası Pisagor üçlüsü, bulduğumuz farklı bir üçlünün sadece birkaç katıdır.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "Bunu mümkün olan tüm noktalar için yaparsanız, mümkün olan her pisagor üçlüsünü hesaba katmış olursunuz.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "Bu da yöntemimizin mümkün olan her Pisagor üçlüsünü tutturması gerektiği anlamına geliyor.", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/ukrainian/sentence_translations.json b/2017/pythagorean-triples/ukrainian/sentence_translations.json index b3e92fd4c..f5466bd78 100644 --- a/2017/pythagorean-triples/ukrainian/sentence_translations.json +++ b/2017/pythagorean-triples/ukrainian/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "Фактично, з причин, які я коротко поясню, кожна можлива трійка Піфагора, яку ми пропускаємо, є лише деяким кратним іншій трійці, яку ми влучили.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "Зробивши це для всіх можливих точок, ви врахуєте кожну можливу трійку Піфагора.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "А це означає, що наш метод має вражати кожну можливу трійку Піфагора.", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/urdu/sentence_translations.json b/2017/pythagorean-triples/urdu/sentence_translations.json index e8a1908a2..ff97bb75a 100644 --- a/2017/pythagorean-triples/urdu/sentence_translations.json +++ b/2017/pythagorean-triples/urdu/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "درحقیقت، ان وجوہات کی بناء پر جن کی میں جلد ہی وضاحت کروں گا، ہر ممکنہ پائتھاگورین ٹرپل جو ہم نے کھو دیا ہے وہ ایک مختلف ٹرپل کا کچھ ملٹیپل ہے جسے ہم مارتے ہیں۔", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "تمام ممکنہ پوائنٹس کے لیے ایسا کرنے سے، آپ کو ہر ممکنہ پائیتھاگورین ٹرپل کا حساب ملے گا۔", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "اور اس کا مطلب ہے کہ ہمارے طریقہ کار کو ہر ممکن Pythagorean Triple کو مارنا چاہیے۔", "n_reviews": 0, "start": 856.16, diff --git a/2017/pythagorean-triples/vietnamese/sentence_translations.json b/2017/pythagorean-triples/vietnamese/sentence_translations.json index c547c992a..2cbd31263 100644 --- a/2017/pythagorean-triples/vietnamese/sentence_translations.json +++ b/2017/pythagorean-triples/vietnamese/sentence_translations.json @@ -448,7 +448,7 @@ "end": 541.1 }, { - "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple we miss is just some multiple of a different triple we hit.", + "input": "In fact, for reasons that I'll explain shortly, every possible pythagorean triple that we miss is just some multiple of a different triple that we hit.", "translatedText": "Trên thực tế, vì những lý do mà tôi sẽ giải thích ngắn gọn, mọi bộ ba số pythagore mà chúng ta bỏ lỡ chỉ là bội số của một bộ ba khác mà chúng ta gặp phải.", "n_reviews": 0, "start": 542.18, @@ -511,7 +511,7 @@ "end": 599.88 }, { - "input": "Doing this for all possible points, you'll account for every possible pythagorean triple.", + "input": "Doing this for all possible points, you'll account for every possible point that you miss. Every possible Pythagorean triple,", "translatedText": "Làm điều này cho tất cả các điểm có thể, bạn sẽ tính được mọi bộ ba số pythagore có thể.", "n_reviews": 0, "start": 603.7, @@ -700,7 +700,7 @@ "end": 855.84 }, { - "input": "And that means our method must hit every possible Pythagorean triple.", + "input": "l and the original, but look at the rise over run slope of the line defined by", "translatedText": "Và điều đó có nghĩa là phương pháp của chúng ta phải đạt tới mọi bộ ba Pythagore có thể.", "n_reviews": 0, "start": 856.16, diff --git a/2017/tattoos-on-math/arabic/sentence_translations.json b/2017/tattoos-on-math/arabic/sentence_translations.json index 08638b8c5..98db24d9a 100644 --- a/2017/tattoos-on-math/arabic/sentence_translations.json +++ b/2017/tattoos-on-math/arabic/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse. ", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse. ", "translatedText": "في كل مثلث من هذه المثلثات، فكر في نسبة طول الضلع المقابل لثيتا إلى طول الوتر. ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant. ", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant. ", "translatedText": "المثلث الأكبر، الضلع المقابل لثيتا، هو الخط الشعاعي الذي يبلغ طوله 1، والوتر الآن هو هذا الطول على المحور الصادي، الذي أدعيه هو قاطع التمام. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/bengali/sentence_translations.json b/2017/tattoos-on-math/bengali/sentence_translations.json index 5f469b5af..ea3fe5057 100644 --- a/2017/tattoos-on-math/bengali/sentence_translations.json +++ b/2017/tattoos-on-math/bengali/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "এই ত্রিভুজগুলির প্রতিটির জন্য, থিটার বিপরীত বাহুর দৈর্ঘ্যের সাথে কর্ণের দৈর্ঘ্যের অনুপাত সম্পর্কে চিন্তা করুন।", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "বৃহত্তর ত্রিভুজ, থিটার বিপরীত দিকটি হল দৈর্ঘ্য 1 এর রেডিয়াল রেখা, এবং কর্ণটি এখন y-অক্ষের এই দৈর্ঘ্য, আমি যাকে কোসেক্যান্ট দাবি করছি।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/chinese/sentence_translations.json b/2017/tattoos-on-math/chinese/sentence_translations.json index 4399bbaee..014c56481 100644 --- a/2017/tattoos-on-math/chinese/sentence_translations.json +++ b/2017/tattoos-on-math/chinese/sentence_translations.json @@ -210,7 +210,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "对于每个三角形，请考虑 θ 相 对边的长度与斜边长度的比率。", "model": "google_nmt", "from_community_srt": "上面这个角是θ 也就是最初在圆里的角 这个证明留给你作为习题 现在 对于每一个三角形 我希望你思考一下θ对应边的长度与斜边长度的比值 对小三角形而言 对边的长度是θ的正弦 斜边则是我们定义的长度为1的半径", @@ -227,7 +227,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "较大的三角形，与 theta 相对的一侧，是长度为 1 的 径向线，斜边现在是 y 轴上的这个长度，我所说的就是余割。", "model": "google_nmt", "from_community_srt": "所以比值是θ的正弦除以1 当我们把视线转向大三角形时 θ的对边是长度为1的半径 斜边是y轴上的这段距离 这就是我之前声称的余割 如果对两边同时取倒数 你会发现它和“θ的余割是1除以正弦”这个事实相符", diff --git a/2017/tattoos-on-math/english/captions.srt b/2017/tattoos-on-math/english/captions.srt index 0861d08e3..6570a093f 100644 --- a/2017/tattoos-on-math/english/captions.srt +++ b/2017/tattoos-on-math/english/captions.srt @@ -1,5 +1,5 @@ 1 -00:00:03,939 --> 00:00:07,480 +00:00:03,940 --> 00:00:07,480 Hey folks, just a short, out of the ordinary video for you today. 2 @@ -127,302 +127,306 @@ And these links give a really wonderful understanding for what cosine and sine are all about. 33 -00:02:07,520 --> 00:02:12,990 +00:02:07,520 --> 00:02:11,529 People might learn that the tangent of an angle is sine divided by cosine, 34 -00:02:12,990 --> 00:02:17,730 -and that relatively few learn that there's also a nice geometric +00:02:11,529 --> 00:02:15,379 +and that the cotangent is the other way around, cosine divided by sine, 35 -00:02:17,730 --> 00:02:20,940 -interpretation for each of those quantities. +00:02:15,379 --> 00:02:19,870 +but relatively few learn that there's also a nice geometric interpretation for each 36 +00:02:19,870 --> 00:02:20,940 +of those quantities. + +37 00:02:21,780 --> 00:02:25,630 If you draw a line tangent to the circle at this point, -37 +38 00:02:25,630 --> 00:02:31,820 the distance from that point to the x-axis along that tangent is the tangent of the angle. -38 +39 00:02:32,540 --> 00:02:35,258 And the distance along that line to the point where -39 +40 00:02:35,258 --> 00:02:37,820 it hits the y-axis is the cotangent of the angle. -40 +41 00:02:38,660 --> 00:02:41,920 Again, this gives a really intuitive feel for what those quantities mean. -41 +42 00:02:41,920 --> 00:02:45,856 You can imagine tweaking that theta and seeing when cotangent gets smaller and when -42 +43 00:02:45,856 --> 00:02:49,700 tangent gets larger, and it's a good gut check for any students working with them. -43 +44 00:02:50,740 --> 00:02:55,180 Likewise, secant, which is defined as 1 divided by the cosine, and cosecant, -44 +45 00:02:55,180 --> 00:02:58,180 which is defined as 1 divided by the sine of theta, -45 +46 00:02:58,180 --> 00:03:00,660 each have their own places on this diagram. -46 +47 00:03:01,720 --> 00:03:06,073 If you look at that point where this tangent line crosses the x-axis, -47 +48 00:03:06,073 --> 00:03:10,489 the distance from that point to the origin is the secant of the angle, -48 +49 00:03:10,489 --> 00:03:12,480 that is 1 divided by the cosine. -49 +50 00:03:13,120 --> 00:03:17,450 Likewise, the distance between where this tangent line crosses the y-axis -50 +51 00:03:17,450 --> 00:03:21,840 and the origin is the cosecant of the angle, that is 1 divided by the sine. -51 +52 00:03:22,560 --> 00:03:26,387 If you're wondering why on earth that's true, notice that we have -52 +53 00:03:26,387 --> 00:03:30,274 two similar right triangles here, one small one inside the circle, -53 +54 00:03:30,274 --> 00:03:34,160 and this larger triangle whose hypotenuse is resting on the y-axis. -54 +55 00:03:34,160 --> 00:03:38,343 I'll leave it to you to check that the interior angle up at the tip there is theta, -55 +56 00:03:38,343 --> 00:03:41,580 the angle that we originally started with over inside the circle. -56 +57 00:03:42,800 --> 00:03:46,521 For each one of those triangles, I want you to think about the ratio of -57 +58 00:03:46,521 --> 00:03:50,140 the length of the side opposite theta to the length of the hypotenuse. -58 +59 00:03:50,880 --> 00:03:55,205 For the small triangle, the length of the opposite side is sine of theta, -59 +60 00:03:55,205 --> 00:03:59,414 and the hypotenuse is that radius, the one we defined to have length 1, -60 +61 00:03:59,414 --> 00:04:02,220 so the ratio is just sine of theta divided by 1. -61 +62 00:04:02,780 --> 00:04:06,392 Now, when we look at the larger triangle, the side opposite -62 +63 00:04:06,392 --> 00:04:10,065 theta is that radial line of length 1, and the hypotenuse is -63 +64 00:04:10,065 --> 00:04:14,160 now this length on the y-axis, the one I'm claiming is the cosecant. -64 +65 00:04:16,040 --> 00:04:19,347 If you take the reciprocal of each side here, you see that this -65 +66 00:04:19,347 --> 00:04:23,120 matches up with the fact that the cosecant of theta is 1 divided by sine. -66 +67 00:04:23,980 --> 00:04:24,700 Kinda cool, right? -67 +68 00:04:25,520 --> 00:04:30,331 It's also kind of nice that sine, tangent, and secant all correspond to lengths of -68 +69 00:04:30,331 --> 00:04:35,142 lines that somehow go to the x-axis, and then the corresponding cosine, cotangent, -69 +70 00:04:35,142 --> 00:04:40,360 and cosecant are all then lengths of lines going to the corresponding spots on the y-axis. -70 +71 00:04:41,180 --> 00:04:43,466 And on a diagram like this, it might be pleasing -71 +72 00:04:43,466 --> 00:04:45,940 that all six of these are separately named functions. -72 +73 00:04:46,480 --> 00:04:50,290 But in any practical use of trigonometry, you can get by just using sine, -73 +74 00:04:50,290 --> 00:04:51,320 cosine, and tangent. -74 +75 00:04:52,000 --> 00:04:55,880 In fact, if you really wanted, you could define all six of these in terms of sine alone. -75 +76 00:04:56,720 --> 00:04:59,800 But the sort of things that cosine and tangent correspond to come up -76 +77 00:04:59,800 --> 00:05:03,060 frequently enough that it's more convenient to give them their own names. -77 +78 00:05:03,740 --> 00:05:07,880 But cosecant, secant, and cotangent never really come up in problem solving in a -78 +79 00:05:07,880 --> 00:05:12,020 way that's not just as convenient to write in terms of sine, cosine, and tangent. -79 +80 00:05:12,560 --> 00:05:15,769 At that point, it's really just adding more words for students to learn, -80 +81 00:05:15,769 --> 00:05:17,220 with not that much added utility. -81 +82 00:05:17,980 --> 00:05:21,432 And if anything, if you only introduce secant as 1 over cosine, -82 +83 00:05:21,432 --> 00:05:25,748 and cosecant as 1 over sine, the mismatch of this co-prefix is probably just an -83 +84 00:05:25,748 --> 00:05:29,956 added point of confusion in a class that's prone enough to confusion for many -84 +85 00:05:29,956 --> 00:05:30,820 of its students. -85 +86 00:05:31,980 --> 00:05:35,807 The reason that all six of these functions have separate names, by the way, -86 +87 00:05:35,807 --> 00:05:39,534 is that before computers and calculators, if you were doing trigonometry, -87 +88 00:05:39,534 --> 00:05:43,311 maybe because you're a sailor, or an astronomer, or some kind of engineer, -88 +89 00:05:43,311 --> 00:05:47,340 you'd find the values for these functions using large charts that just recorded -89 +90 00:05:47,340 --> 00:05:48,600 known input-output pairs. -90 +91 00:05:48,600 --> 00:05:52,198 And when you can't easily plug in something like 1 divided by the -91 +92 00:05:52,198 --> 00:05:55,850 sine of 30 degrees into a calculator, it might actually make sense -92 +93 00:05:55,850 --> 00:05:59,340 to have a dedicated column to this value, with a dedicated name. -93 +94 00:06:00,280 --> 00:06:04,296 And if you have a diagram like this one in mind when you're taking measurements, -94 +95 00:06:04,296 --> 00:06:08,511 with sine, tangent, and secant having nicely mirrored meanings to cosine, cotangent, -95 +96 00:06:08,511 --> 00:06:12,726 and cosecant, calling this cosecant instead of 1 divided by sine might actually make -96 +97 00:06:12,726 --> 00:06:17,140 some sense, and it might actually make it easier to remember what it means geometrically. -97 +98 00:06:17,940 --> 00:06:21,062 But times have changed, and most use cases for trig just -98 +99 00:06:21,062 --> 00:06:24,020 don't involve charts of values and diagrams like this. -99 +100 00:06:24,600 --> 00:06:27,933 Hence, the cosecant and its brothers are tattoos on math, -100 +101 00:06:27,933 --> 00:06:31,381 ideas whose permanence in our conventions is our own doing, -101 +102 00:06:31,381 --> 00:06:33,220 not the result of nature itself. -102 +103 00:06:34,140 --> 00:06:36,599 And in general, I actually think this is a good lesson for -103 +104 00:06:36,599 --> 00:06:39,100 any student learning a new piece of math, at whatever level. -104 +105 00:06:39,640 --> 00:06:42,977 You just gotta take a moment and ask yourself whether what you're -105 +106 00:06:42,977 --> 00:06:46,417 learning is core to the flesh of math itself, and to nature itself, -106 +107 00:06:46,417 --> 00:06:50,007 or if what you're looking at is actually just inked on to the subject, -107 +108 00:06:50,007 --> 00:06:53,700 and could just as easily have been inked on in some completely other way. diff --git a/2017/tattoos-on-math/english/sentence_timings.json b/2017/tattoos-on-math/english/sentence_timings.json index dd5bd77fe..3859731bb 100644 --- a/2017/tattoos-on-math/english/sentence_timings.json +++ b/2017/tattoos-on-math/english/sentence_timings.json @@ -80,7 +80,7 @@ 126.52 ], [ - "People might learn that the tangent of an angle is sine divided by cosine, and that relatively few learn that there's also a nice geometric interpretation for each of those quantities.", + "People might learn that the tangent of an angle is sine divided by cosine, and that the cotangent is the other way around, cosine divided by sine, but relatively few learn that there's also a nice geometric interpretation for each of those quantities.", 127.52, 140.94 ], diff --git a/2017/tattoos-on-math/english/transcript.txt b/2017/tattoos-on-math/english/transcript.txt index 25cf13278..a9405fa34 100644 --- a/2017/tattoos-on-math/english/transcript.txt +++ b/2017/tattoos-on-math/english/transcript.txt @@ -14,7 +14,7 @@ Here, let me show you a picture of the tattoo he chose, because not a lot of peo Whenever you have an angle, typically represented with the Greek letter theta, it's common in trigonometry to relate it to a corresponding point on the unit circle, the circle with the radius 1 centered at the origin in the xy plane. Most trigonometry students learn that the distance between this point here on the circle and the x-axis is the sine of the angle, and the distance between that point and the y-axis is the cosine of the angle. And these links give a really wonderful understanding for what cosine and sine are all about. -People might learn that the tangent of an angle is sine divided by cosine, and that relatively few learn that there's also a nice geometric interpretation for each of those quantities. +People might learn that the tangent of an angle is sine divided by cosine, and that the cotangent is the other way around, cosine divided by sine, but relatively few learn that there's also a nice geometric interpretation for each of those quantities. If you draw a line tangent to the circle at this point, the distance from that point to the x-axis along that tangent is the tangent of the angle. And the distance along that line to the point where it hits the y-axis is the cotangent of the angle. Again, this gives a really intuitive feel for what those quantities mean. diff --git a/2017/tattoos-on-math/french/sentence_translations.json b/2017/tattoos-on-math/french/sentence_translations.json index 9d2066df1..ec9664923 100644 --- a/2017/tattoos-on-math/french/sentence_translations.json +++ b/2017/tattoos-on-math/french/sentence_translations.json @@ -144,7 +144,7 @@ "end": 126.52 }, { - "input": "People might learn that the tangent of an angle is sine divided by cosine, and that relatively few learn that there's also a nice geometric interpretation for each of those quantities.", + "input": "People might learn that the tangent of an angle is sine divided by cosine, and that the cotangent is the other way around, cosine divided by sine, but relatively few learn that there's also a nice geometric interpretation for each of those quantities.", "translatedText": "Les gens peuvent apprendre que la tangente d'un angle est le sinus divisé par le cosinus, et que relativement peu apprennent qu'il y a aussi une belle interprétation géométrique pour chacune de ces quantités.", "model": "DeepL", "from_community_srt": "Les gens peuvent aussi apprendre que la tangente d'un angle est le sinus divisé par le cosinus, et que la cotangente est l'inverse, le cosinus divisé par le sinus, mais peu de gens apprennent qu'il existe aussi une jolie représentation géométrique pour l'interprétation de ce que sont ces quantités.", diff --git a/2017/tattoos-on-math/german/sentence_translations.json b/2017/tattoos-on-math/german/sentence_translations.json index 265443f93..304cd116a 100644 --- a/2017/tattoos-on-math/german/sentence_translations.json +++ b/2017/tattoos-on-math/german/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "Denken Sie für jedes dieser Dreiecke über das Verhältnis der Länge der gegenüberliegenden Theta-Seite zur Länge der Hypotenuse nach.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "Das größere Dreieck, die Theta gegenüberliegende Seite, ist die radiale Linie der Länge 1, und die Hypotenuse hat jetzt diese Länge auf der y-Achse, diejenige, die ich behaupte, ist der Kosekans.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/hindi/sentence_translations.json b/2017/tattoos-on-math/hindi/sentence_translations.json index 8a025f7e5..0d0d4a430 100644 --- a/2017/tattoos-on-math/hindi/sentence_translations.json +++ b/2017/tattoos-on-math/hindi/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "उन त्रिभुजों में से प्रत्येक के लिए, थीटा के विपरीत भुजा की लंबाई और कर्ण की लंबाई के अनुपात के बारे में सोचें।", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "बड़ा त्रिभुज, थीटा के विपरीत भुजा, लंबाई 1 की रेडियल रेखा है, और कर्ण अब y-अक्ष पर यह लंबाई है, जिसके बारे में मैं दावा कर रहा हूं वह सहसंयोजक है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/indonesian/sentence_translations.json b/2017/tattoos-on-math/indonesian/sentence_translations.json index dd90d5db5..2bcf8e38e 100644 --- a/2017/tattoos-on-math/indonesian/sentence_translations.json +++ b/2017/tattoos-on-math/indonesian/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "Untuk masing-masing segitiga tersebut, pikirkan perbandingan panjang sisi yang berhadapan dengan panjang sisi miringnya.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "Segitiga yang lebih besar, sisi yang berhadapan dengan theta, adalah garis radial dengan panjang 1, dan sisi miringnya sekarang panjangnya pada sumbu y, yang saya klaim adalah kosekan.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/japanese/sentence_translations.json b/2017/tattoos-on-math/japanese/sentence_translations.json index 8a94d294e..d19184bd6 100644 --- a/2017/tattoos-on-math/japanese/sentence_translations.json +++ b/2017/tattoos-on-math/japanese/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "これらの三角形のそれぞれについて、シータの反対 側の辺の長さと斜辺の長さの比を考えてください。", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "シータの反対側の大きい三角形は長さ 1 の放射状の線であり、斜辺は y 軸上のこの長さであり、私が主張しているのはコセカントです。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/korean/sentence_translations.json b/2017/tattoos-on-math/korean/sentence_translations.json index 4e4868b10..0f423361e 100644 --- a/2017/tattoos-on-math/korean/sentence_translations.json +++ b/2017/tattoos-on-math/korean/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "각 삼각형에 대해 빗변의 길이에 대한 세타 반대쪽 변의 길이의 비율을 생각해 보세요.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "세타 반대편에 있는 더 큰 삼각형은 길이가 1인 방사형 선이고 빗변은 이제 y축의 이 길이입니다. 제가 주장하는 것은 코시컨트입니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/marathi/sentence_translations.json b/2017/tattoos-on-math/marathi/sentence_translations.json index eaecc782c..2dced2693 100644 --- a/2017/tattoos-on-math/marathi/sentence_translations.json +++ b/2017/tattoos-on-math/marathi/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "त्या प्रत्येक त्रिकोणासाठी, थीटाच्या विरुद्ध बाजूच्या लांबीच्या कर्णाच्या लांबीच्या गुणोत्तराचा विचार करा.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "मोठा त्रिकोण, थीटा विरुद्ध बाजू, लांबी 1 ची रेडियल रेषा आहे, आणि कर्ण ही आता y-अक्षावर ही लांबी आहे, ज्याचा मी दावा करत आहे तो कोसेकंट आहे.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/persian/sentence_translations.json b/2017/tattoos-on-math/persian/sentence_translations.json index 099cc2a7d..af6236782 100644 --- a/2017/tattoos-on-math/persian/sentence_translations.json +++ b/2017/tattoos-on-math/persian/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse. ", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse. ", "translatedText": "برای هر یک از آن مثلث ها، به نسبت طول ضلع مقابل تتا به طول هیپوتانوس فکر کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant. ", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant. ", "translatedText": "مثلث بزرگتر، ضلع مقابل تتا، خط شعاعی به طول 1 است، و هیپوتونوس اکنون به این طول در محور y است، چیزی که من ادعا می کنم همسانت است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/portuguese/sentence_translations.json b/2017/tattoos-on-math/portuguese/sentence_translations.json index ca5f3498e..78f8165b7 100644 --- a/2017/tattoos-on-math/portuguese/sentence_translations.json +++ b/2017/tattoos-on-math/portuguese/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "Para cada um desses triângulos, pense na razão entre o comprimento do lado oposto a teta e o comprimento da hipotenusa.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "O triângulo maior, o lado oposto a teta, é a linha radial de comprimento 1, e a hipotenusa agora tem esse comprimento no eixo y, aquela que estou afirmando ser a cossecante.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/russian/sentence_translations.json b/2017/tattoos-on-math/russian/sentence_translations.json index ff86d68c5..9e8c14ee9 100644 --- a/2017/tattoos-on-math/russian/sentence_translations.json +++ b/2017/tattoos-on-math/russian/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "Для каждого из этих треугольников подумайте о соотношении длины стороны, противоположной тэте, к длине гипотенузы.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "Больший треугольник, сторона, противоположная тете, представляет собой радиальную линию длиной 1, а гипотенуза теперь имеет эту длину на оси Y, та, которую я утверждаю, является косекансом.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/spanish/sentence_translations.json b/2017/tattoos-on-math/spanish/sentence_translations.json index 3215dc62d..6e0ace17f 100644 --- a/2017/tattoos-on-math/spanish/sentence_translations.json +++ b/2017/tattoos-on-math/spanish/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "Para cada uno de esos triángulos, piensa en la razón entre la longitud del lado opuesto a theta y la longitud de la hipotenusa.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "El triángulo más grande, el lado opuesto a theta, es la línea radial de longitud 1, y la hipotenusa ahora tiene esta longitud en el eje y, la que digo es la cosecante.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/tamil/sentence_translations.json b/2017/tattoos-on-math/tamil/sentence_translations.json index 5ba5a677f..c11af204a 100644 --- a/2017/tattoos-on-math/tamil/sentence_translations.json +++ b/2017/tattoos-on-math/tamil/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "அந்த முக்கோணங்களில் ஒவ்வொன்றிற்கும், தீட்டாவுக்கு எதிரே உள்ள பக்கத்தின் நீளம் மற்றும் ஹைப்போடென்யூஸின் நீளத்தின் விகிதத்தைப் பற்றி சிந்தியுங்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "பெரிய முக்கோணம், தீட்டாவுக்கு எதிரே உள்ள பக்கம், நீளம் 1 இன் ஆரக் கோடு, மற்றும் ஹைப்போடென்யூஸ் இப்போது y-அச்சில் இந்த நீளம் உள்ளது, நான் கூறுவது கோசெகண்ட்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/telugu/sentence_translations.json b/2017/tattoos-on-math/telugu/sentence_translations.json index 1a6251f34..e9d7c64ef 100644 --- a/2017/tattoos-on-math/telugu/sentence_translations.json +++ b/2017/tattoos-on-math/telugu/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "ఆ త్రిభుజాలలో ప్రతి ఒక్కదానికి, తీటాకు ఎదురుగా ఉన్న భుజం యొక్క పొడవు మరియు హైపోటెన్యూస్ పొడవు యొక్క నిష్పత్తి గురించి ఆలోచించండి.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "పెద్ద త్రిభుజం, తీటాకు ఎదురుగా, పొడవు 1 యొక్క రేడియల్ లైన్, మరియు హైపోటెన్యూస్ ఇప్పుడు y-యాక్సిస్‌పై ఈ పొడవు, నేను క్లెయిమ్ చేస్తున్నది కోసెకెంట్.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/thai/sentence_translations.json b/2017/tattoos-on-math/thai/sentence_translations.json index 39b70bff3..6a4ed3221 100644 --- a/2017/tattoos-on-math/thai/sentence_translations.json +++ b/2017/tattoos-on-math/thai/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse. ", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant. ", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/turkish/sentence_translations.json b/2017/tattoos-on-math/turkish/sentence_translations.json index cb42595b7..dd82e7eeb 100644 --- a/2017/tattoos-on-math/turkish/sentence_translations.json +++ b/2017/tattoos-on-math/turkish/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "Bu üçgenlerin her biri için tetanın karşısındaki kenarın uzunluğunun hipotenüs uzunluğuna oranını düşünün.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "Tetanın karşısındaki daha büyük üçgen, uzunluğu 1 olan radyal çizgidir ve hipotenüs şimdi y eksenindeki bu uzunluktur, benim iddia ettiğim kosekanttır.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/urdu/sentence_translations.json b/2017/tattoos-on-math/urdu/sentence_translations.json index 78e532d3a..a09af9480 100644 --- a/2017/tattoos-on-math/urdu/sentence_translations.json +++ b/2017/tattoos-on-math/urdu/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse. ", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse. ", "translatedText": "ان مثلثوں میں سے ہر ایک کے لیے، تھیٹا کے مخالف پہلو کی لمبائی کے تناسب کے بارے میں سوچیں اور hypotenuse کی لمبائی۔", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant. ", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant. ", "translatedText": "بڑا مثلث، تھیٹا کا مخالف سمت، لمبائی 1 کی شعاعی لکیر ہے، اور فرضی اب یہ لمبائی y-axis پر ہے، جس کا میں دعویٰ کر رہا ہوں کہ cosecant ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/tattoos-on-math/vietnamese/sentence_translations.json b/2017/tattoos-on-math/vietnamese/sentence_translations.json index e29155dae..76ef36461 100644 --- a/2017/tattoos-on-math/vietnamese/sentence_translations.json +++ b/2017/tattoos-on-math/vietnamese/sentence_translations.json @@ -200,7 +200,7 @@ "end": 221.58 }, { - "input": "For each one of those triangles, think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", + "input": "For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to the length of the hypotenuse.", "translatedText": "Đối với mỗi tam giác đó, hãy nghĩ về tỉ số giữa độ dài cạnh đối diện theta với chiều dài cạnh huyền.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 242.22 }, { - "input": "The larger triangle, the side opposite theta, is the radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", + "input": "Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one I'm claiming is the cosecant.", "translatedText": "Tam giác lớn hơn, cạnh đối diện với theta, là đường xuyên tâm có độ dài 1, và cạnh huyền bây giờ có độ dài này trên trục y, cái mà tôi đang khẳng định là cosecant.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/arabic/sentence_translations.json b/2017/taylor-series/arabic/sentence_translations.json index 789324509..73351e925 100644 --- a/2017/taylor-series/arabic/sentence_translations.json +++ b/2017/taylor-series/arabic/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "ستكون السلسلة التالية مثلها على أساس الاحتمالية، وإذا كنت تريد الوصول المبكر أثناء إنشاء مقاطع الفيديو هذه، فأنت تعرف إلى أين تذهب. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/bengali/sentence_translations.json b/2017/taylor-series/bengali/sentence_translations.json index f24ca8c89..27c0640dc 100644 --- a/2017/taylor-series/bengali/sentence_translations.json +++ b/2017/taylor-series/bengali/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "এর মতো পরবর্তী সিরিজগুলি সম্ভাব্যতার উপর থাকবে, এবং আপনি যদি সেই ভিডিওগুলি তৈরি হওয়ার সাথে সাথে প্রাথমিক অ্যাক্সেস চান তবে আপনি কোথায় যেতে হবে তা জানেন।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/bulgarian/sentence_translations.json b/2017/taylor-series/bulgarian/sentence_translations.json index c579f2713..4ee464e4f 100644 --- a/2017/taylor-series/bulgarian/sentence_translations.json +++ b/2017/taylor-series/bulgarian/sentence_translations.json @@ -96,7 +96,7 @@ "end": 152.66 }, { - "input": "First of all, at the input 0, the value of cosine of x is 1, so if our approximation is any good at all, it should also equal 1 at the input x equals 0.", + "input": "Well, first of all, at the input 0, the value of cosine of x is 1, so if our approximation is going to be any good at all, it should also equal 1 at the input x equals 0.", "translatedText": "На първо място, при вход 0 стойността на косинуса на x е 1, така че ако нашето приближение е добро, то също трябва да е равно на 1 при вход x, равен на 0.", "model": "DeepL", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 547.78 }, { - "input": "The second thing to notice is that adding on new terms, like this c4 times x to the old terms should be, and that's really important.", + "input": "The second thing to notice is that adding on new terms, like this c4 times x to the fourth, doesn't mess up what the old terms should be, and that's really important.", "translatedText": "Второто нещо, което трябва да забележите, е, че добавянето на нови термини, като този c4 пъти x към старите термини, трябва да бъде и това е наистина важно.", "model": "DeepL", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 713.16 }, { - "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and evaluate each one of them at x equals 0.", + "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and you would evaluate each one of them at x equals 0.", "translatedText": "По-общо и следователно по-абстрактно, ако се занимавахме с някаква друга функция, различна от косинус, щяхте да изчислите нейната производна, втората ѝ производна и т.н., като получите толкова членове, колкото искате, и да оцените всеки от тях при x, равно на 0.", "model": "DeepL", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 838.24 }, { - "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x2 plus 1 over 3 factorial times x3, and so on, depending on how many terms you want.", + "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed, and so on, depending on how many terms you want.", "translatedText": "Това означава, че нашата полиномна апроксимация трябва да изглежда като 1 плюс 1 пъти x плюс 1 над 2 пъти x2 плюс 1 над 3 пъти x3 и т.н., в зависимост от това колко члена искате.", "model": "DeepL", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 1110.16 }, { - "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or that it equals the number e.", + "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or what's saying the same thing, that it equals the number e.", "translatedText": "С добавянето на все повече и повече членове на полинома общата сума се приближава все повече до стойността e, така че се казва, че тази безкрайна редица се схожда с числото e или че е равна на числото e.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/taylor-series/chinese/sentence_translations.json b/2017/taylor-series/chinese/sentence_translations.json index dccfe03c7..f295d1d0f 100644 --- a/2017/taylor-series/chinese/sentence_translations.json +++ b/2017/taylor-series/chinese/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "下一个类似的系列将是关于概率的，如果您想在 这些视频制作时尽早访问，您知道该去哪里。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/english/captions.srt b/2017/taylor-series/english/captions.srt index 2d1483465..dc6d1f7c5 100644 --- a/2017/taylor-series/english/captions.srt +++ b/2017/taylor-series/english/captions.srt @@ -135,16 +135,16 @@ c1, and c2, find the one that most resembles cosine of x near x equals 0, whose graph kind of spoons with the graph of cosine x at that point. 35 -00:02:33,860 --> 00:02:38,298 -First of all, at the input 0, the value of cosine of x is 1, +00:02:33,860 --> 00:02:38,218 +Well, first of all, at the input 0, the value of cosine of x is 1, 36 -00:02:38,298 --> 00:02:43,683 -so if our approximation is any good at all, it should also equal 1 at the +00:02:38,218 --> 00:02:41,862 +so if our approximation is going to be any good at all, 37 -00:02:43,683 --> 00:02:44,920 -input x equals 0. +00:02:41,862 --> 00:02:44,920 +it should also equal 1 at the input x equals 0. 38 00:02:45,820 --> 00:02:50,940 @@ -475,678 +475,686 @@ For example, that x to the fourth coefficient was the fourth derivative of cosin 1, but divided by 4 factorial, 24. 120 -00:09:09,400 --> 00:09:13,568 +00:09:09,400 --> 00:09:12,739 The second thing to notice is that adding on new terms, 121 -00:09:13,568 --> 00:09:19,300 -like this c4 times x to the old terms should be, and that's really important. +00:09:12,739 --> 00:09:17,630 +like this c4 times x to the fourth, doesn't mess up what the old terms should be, 122 +00:09:17,630 --> 00:09:19,300 +and that's really important. + +123 00:09:20,100 --> 00:09:25,213 For example, the second derivative of this polynomial at x equals 0 is still equal -123 +124 00:09:25,213 --> 00:09:30,080 to 2 times the second coefficient, even after you introduce higher order terms. -124 +125 00:09:30,960 --> 00:09:33,879 And it's because we're plugging in x equals 0, -125 +126 00:09:33,879 --> 00:09:38,537 so the second derivative of any higher order term, which all include an x, -126 +127 00:09:38,537 --> 00:09:39,780 will just wash away. -127 +128 00:09:40,740 --> 00:09:45,479 And the same goes for any other derivative, which is why each derivative of a -128 +129 00:09:45,479 --> 00:09:50,280 polynomial at x equals 0 is controlled by one and only one of the coefficients. -129 +130 00:09:52,640 --> 00:09:57,353 If instead you were approximating near an input other than 0, like x equals pi, -130 +131 00:09:57,353 --> 00:10:01,772 in order to get the same effect you would have to write your polynomial in -131 +132 00:10:01,772 --> 00:10:05,720 terms of powers of x minus pi, or whatever input you're looking at. -132 +133 00:10:06,320 --> 00:10:09,208 This makes it look noticeably more complicated, -133 +134 00:10:09,208 --> 00:10:13,961 but all we're doing is making sure that the point pi looks and behaves like 0, -134 +135 00:10:13,961 --> 00:10:18,715 so that plugging in x equals pi will result in a lot of nice cancellation that -135 +136 00:10:18,715 --> 00:10:20,220 leaves only one constant. -136 +137 00:10:22,380 --> 00:10:27,731 And finally, on a more philosophical level, notice how what we're doing here is basically -137 +138 00:10:27,731 --> 00:10:32,666 taking information about higher order derivatives of a function at a single point, -138 +139 00:10:32,666 --> 00:10:37,780 and translating that into information about the value of the function near that point. -139 +140 00:10:40,960 --> 00:10:44,120 You can take as many derivatives of cosine as you want. -140 +141 00:10:44,600 --> 00:10:47,544 It follows this nice cyclic pattern, cosine of x, -141 +142 00:10:47,544 --> 00:10:51,020 negative sine of x, negative cosine, sine, and then repeat. -142 +143 00:10:52,320 --> 00:10:55,660 And the value of each one of these is easy to compute at x equals 0. -143 +144 00:10:56,100 --> 00:11:01,100 It gives this cyclic pattern 1, 0, negative 1, 0, and then repeat. -144 +145 00:11:02,000 --> 00:11:07,149 And knowing the values of all those higher order derivatives is a lot of information -145 +146 00:11:07,149 --> 00:11:12,480 about cosine of x, even though it only involves plugging in a single number, x equals 0. -146 +147 00:11:14,260 --> 00:11:19,603 So what we're doing is leveraging that information to get an approximation around this -147 +148 00:11:19,603 --> 00:11:25,131 input, and you do it by creating a polynomial whose higher order derivatives are designed -148 +149 00:11:25,131 --> 00:11:30,660 to match up with those of cosine, following this same 1, 0, negative 1, 0, cyclic pattern. -149 +150 00:11:31,420 --> 00:11:35,482 And to do that, you just make each coefficient of the polynomial follow that -150 +151 00:11:35,482 --> 00:11:39,440 same pattern, but you have to divide each one by the appropriate factorial. -151 +152 00:11:40,120 --> 00:11:42,615 Like I mentioned before, this is what cancels out -152 +153 00:11:42,615 --> 00:11:45,260 the cascading effect of many power rule applications. -153 +154 00:11:47,280 --> 00:11:50,111 The polynomials you get by stopping this process at -154 +155 00:11:50,111 --> 00:11:53,160 any point are called Taylor polynomials for cosine of x. -155 -00:11:53,900 --> 00:11:58,847 -More generally, and hence more abstractly, if we were dealing with some other function - 156 -00:11:58,847 --> 00:12:03,794 -other than cosine, you would compute its derivative, its second derivative, and so on, +00:11:53,900 --> 00:11:58,660 +More generally, and hence more abstractly, if we were dealing with some other function 157 -00:12:03,794 --> 00:12:08,400 -getting as many terms as you'd like, and evaluate each one of them at x equals 0. +00:11:58,660 --> 00:12:03,420 +other than cosine, you would compute its derivative, its second derivative, and so on, 158 +00:12:03,420 --> 00:12:08,290 +getting as many terms as you'd like, and you would evaluate each one of them at x equals + +159 +00:12:08,290 --> 00:12:08,400 +0. + +160 00:12:09,580 --> 00:12:15,938 Then for the polynomial approximation, the coefficient of each x to the n term should be -159 +161 00:12:15,938 --> 00:12:20,511 the value of the nth derivative of the function evaluated at 0, -160 +162 00:12:20,511 --> 00:12:22,440 but divided by n factorial. -161 +163 00:12:23,480 --> 00:12:27,371 This whole rather abstract formula is something you'll likely -162 +164 00:12:27,371 --> 00:12:31,200 see in any text or course that touches on Taylor polynomials. -163 +165 00:12:31,780 --> 00:12:36,139 And when you see it, think to yourself that the constant term ensures that -164 +166 00:12:36,139 --> 00:12:39,452 the value of the polynomial matches with the value of f, -165 +167 00:12:39,452 --> 00:12:43,696 the next term ensures that the slope of the polynomial matches the slope -166 +168 00:12:43,696 --> 00:12:48,114 of the function at x equals 0, the next term ensures that the rate at which -167 +169 00:12:48,114 --> 00:12:51,369 the slope changes is the same at that point, and so on, -168 +170 00:12:51,369 --> 00:12:53,520 depending on how many terms you want. -169 +171 00:12:54,620 --> 00:12:57,451 And the more terms you choose, the closer the approximation, -170 +172 00:12:57,451 --> 00:13:00,980 but the tradeoff is that the polynomial you'd get would be more complicated. -171 +173 00:13:02,640 --> 00:13:07,727 And to make things even more general, if you wanted to approximate near some input -172 +174 00:13:07,727 --> 00:13:12,937 other than 0, which we'll call a, you would write this polynomial in terms of powers -173 +175 00:13:12,937 --> 00:13:17,780 of x minus a, and you would evaluate all the derivatives of f at that input, a. -174 +176 00:13:18,680 --> 00:13:23,120 This is what Taylor polynomials look like in their fullest generality. -175 +177 00:13:24,000 --> 00:13:28,534 Changing the value of a changes where this approximation is hugging the original -176 +178 00:13:28,534 --> 00:13:33,236 function, where its higher order derivatives will be equal to those of the original -177 +179 00:13:33,236 --> 00:13:33,740 function. -178 +180 00:13:35,880 --> 00:13:38,860 One of the simplest meaningful examples of this is -179 +181 00:13:38,860 --> 00:13:41,900 the function e to the x around the input x equals 0. -180 +182 00:13:42,760 --> 00:13:46,406 Computing the derivatives is super nice, as nice as it gets, -181 +183 00:13:46,406 --> 00:13:49,275 because the derivative of e to the x is itself, -182 +184 00:13:49,275 --> 00:13:53,580 so the second derivative is also e to the x, as is its third, and so on. -183 +185 00:13:54,340 --> 00:13:58,240 So at the point x equals 0, all of these are equal to 1. -184 -00:13:59,120 --> 00:14:05,627 -And what that means is our polynomial approximation should look +186 +00:13:59,120 --> 00:14:05,720 +And what that means is our polynomial approximation should look like -185 -00:14:05,627 --> 00:14:13,659 -like 1 plus 1 times x plus 1 over 2 times x2 plus 1 over 3 factorial times x3, +187 +00:14:05,720 --> 00:14:13,948 +1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed, -186 -00:14:13,659 --> 00:14:18,540 +188 +00:14:13,948 --> 00:14:18,540 and so on, depending on how many terms you want. -187 +189 00:14:19,400 --> 00:14:22,700 These are the Taylor polynomials for e to the x. -188 +190 00:14:26,380 --> 00:14:31,039 Ok, so with that as a foundation, in the spirit of showing you just how connected all -189 +191 00:14:31,039 --> 00:14:34,614 the topics of calculus are, let me turn to something kind of fun, -190 +192 00:14:34,614 --> 00:14:38,840 a completely different way to understand this second order term of the Taylor -191 +193 00:14:38,840 --> 00:14:40,520 polynomials, but geometrically. -192 +194 00:14:41,400 --> 00:14:43,904 It's related to the fundamental theorem of calculus, -193 +195 00:14:43,904 --> 00:14:47,260 which I talked about in chapters 1 and 8 if you need a quick refresher. -194 +196 00:14:47,980 --> 00:14:52,001 Like we did in those videos, consider a function that gives the area -195 +197 00:14:52,001 --> 00:14:56,140 under some graph between a fixed left point and a variable right point. -196 +198 00:14:56,980 --> 00:15:00,915 What we're going to do here is think about how to approximate this area function, -197 +199 00:15:00,915 --> 00:15:04,180 not the function for the graph itself, like we've been doing before. -198 +200 00:15:04,900 --> 00:15:09,440 Focusing on that area is what's going to make the second order term pop out. -199 +201 00:15:10,440 --> 00:15:16,575 Remember, the fundamental theorem of calculus is that this graph itself represents the -200 +202 00:15:16,575 --> 00:15:22,711 derivative of the area function, and it's because a slight nudge dx to the right bound -201 +203 00:15:22,711 --> 00:15:28,988 of the area gives a new bit of area approximately equal to the height of the graph times -202 +204 00:15:28,988 --> 00:15:29,200 dx. -203 +205 00:15:30,040 --> 00:15:34,480 And that approximation is increasingly accurate for smaller and smaller choices of dx. -204 +206 00:15:35,980 --> 00:15:39,601 But if you wanted to be more accurate about this change in area, -205 +207 00:15:39,601 --> 00:15:42,666 given some change in x that isn't meant to approach 0, -206 +208 00:15:42,666 --> 00:15:46,065 you would have to take into account this portion right here, -207 +209 00:15:46,065 --> 00:15:47,960 which is approximately a triangle. -208 +210 00:15:49,600 --> 00:15:57,460 Let's name the starting input a, and the nudged input above it x, so that change is x-a. -209 +211 00:15:58,100 --> 00:16:02,985 The base of that little triangle is that change, x-a, -210 +212 00:16:02,985 --> 00:16:07,600 and its height is the slope of the graph times x-a. -211 +213 00:16:08,420 --> 00:16:11,987 Since this graph is the derivative of the area function, -212 +214 00:16:11,987 --> 00:16:17,120 its slope is the second derivative of the area function, evaluated at the input a. -213 +215 00:16:18,440 --> 00:16:22,662 So the area of this triangle, 1 half base times height, -214 +216 00:16:22,662 --> 00:16:28,467 is 1 half times the second derivative of this area function, evaluated at a, -215 +217 00:16:28,467 --> 00:16:29,900 multiplied by x-a2. -216 +218 00:16:30,960 --> 00:16:34,380 And this is exactly what you would see with a Taylor polynomial. -217 +219 00:16:34,880 --> 00:16:40,478 If you knew the various derivative information about this area function at the point a, -218 +220 00:16:40,478 --> 00:16:43,660 how would you approximate the area at the point x? -219 +221 00:16:45,360 --> 00:16:49,316 Well you have to include all that area up to a, f of a, -220 +222 00:16:49,316 --> 00:16:54,968 plus the area of this rectangle here, which is the first derivative, times x-a, -221 +223 00:16:54,968 --> 00:17:00,902 plus the area of that little triangle, which is 1 half times the second derivative, -222 +224 00:17:00,902 --> 00:17:01,680 times x-a2. -223 +225 00:17:02,560 --> 00:17:06,539 I really like this, because even though it looks a bit messy all written out, -224 +226 00:17:06,539 --> 00:17:11,079 each one of the terms has a very clear meaning that you can just point to on the diagram. -225 +227 00:17:13,400 --> 00:17:16,877 If you wanted, we could call it an end here, and you would have a -226 +228 00:17:16,877 --> 00:17:20,460 phenomenally useful tool for approximating these Taylor polynomials. -227 +229 00:17:21,400 --> 00:17:25,812 But if you're thinking like a mathematician, one question you might ask is -228 +230 00:17:25,812 --> 00:17:30,460 whether or not it makes sense to never stop and just add infinitely many terms. -229 +231 00:17:31,380 --> 00:17:35,883 In math, an infinite sum is called a series, so even though one of these -230 +232 00:17:35,883 --> 00:17:40,263 approximations with finitely many terms is called a Taylor polynomial, -231 +233 00:17:40,263 --> 00:17:44,520 adding all infinitely many terms gives what's called a Taylor series. -232 +234 00:17:45,260 --> 00:17:48,848 You have to be really careful with the idea of an infinite series, -233 +235 00:17:48,848 --> 00:17:52,598 because it doesn't actually make sense to add infinitely many things, -234 +236 00:17:52,598 --> 00:17:56,080 you can only hit the plus button on the calculator so many times. -235 +237 00:17:57,440 --> 00:18:01,380 But if you have a series where adding more and more of the terms, -236 +238 00:18:01,380 --> 00:18:06,396 which makes sense at each step, gets you increasingly close to some specific value, -237 +239 00:18:06,396 --> 00:18:09,740 what you say is that the series converges to that value. -238 +240 00:18:10,320 --> 00:18:14,293 Or, if you're comfortable extending the definition of equality to -239 +241 00:18:14,293 --> 00:18:19,049 include this kind of series convergence, you'd say that the series as a whole, -240 +242 00:18:19,049 --> 00:18:22,360 this infinite sum, equals the value it's converging to. -241 +243 00:18:23,460 --> 00:18:27,452 For example, look at the Taylor polynomial for e to the x, -242 +244 00:18:27,452 --> 00:18:30,160 and plug in some input, like x equals 1. -243 -00:18:31,140 --> 00:18:36,190 -As you add more and more polynomial terms, the total sum gets - -244 -00:18:36,190 --> 00:18:41,404 -closer and closer to the value e, so you say that this infinite - 245 -00:18:41,404 --> 00:18:46,700 -series converges to the number e, or that it equals the number e. +00:18:31,140 --> 00:18:36,279 +As you add more and more polynomial terms, the total sum gets closer and 246 +00:18:36,279 --> 00:18:42,405 +closer to the value e, so you say that this infinite series converges to the number e, + +247 +00:18:42,405 --> 00:18:46,700 +or what's saying the same thing, that it equals the number e. + +248 00:18:47,840 --> 00:18:53,639 In fact, it turns out that if you plug in any other value of x, like x equals 2, -247 +249 00:18:53,639 --> 00:18:59,867 and look at the value of the higher and higher order Taylor polynomials at this value, -248 +250 00:18:59,867 --> 00:19:04,020 they will converge towards e to the x, which is e squared. -249 +251 00:19:04,680 --> 00:19:08,951 This is true for any input, no matter how far away from 0 it is, -250 +252 00:19:08,951 --> 00:19:14,602 even though these Taylor polynomials are constructed only from derivative information -251 +253 00:19:14,602 --> 00:19:16,180 gathered at the input 0. -252 +254 00:19:18,270 --> 00:19:24,276 In a case like this, we say that e to the x equals its own Taylor series at all inputs x, -253 +255 00:19:24,276 --> 00:19:27,480 which is kind of a magical thing to have happen. -254 +256 00:19:28,380 --> 00:19:32,400 Even though this is also true for a couple other important functions, -255 +257 00:19:32,400 --> 00:19:36,306 like sine and cosine, sometimes these series only converge within a -256 +258 00:19:36,306 --> 00:19:40,500 certain range around the input whose derivative information you're using. -257 +259 00:19:41,580 --> 00:19:47,273 If you work out the Taylor series for the natural log of x around the input x equals 1, -258 +260 00:19:47,273 --> 00:19:51,996 which is built by evaluating the higher order derivatives of the natural -259 +261 00:19:51,996 --> 00:19:55,620 log of x at x equals 1, this is what it would look like. -260 +262 00:19:56,080 --> 00:20:00,800 When you plug in an input between 0 and 2, adding more and more terms of this -261 +263 00:20:00,800 --> 00:20:05,520 series will indeed get you closer and closer to the natural log of that input. -262 +264 00:20:06,400 --> 00:20:09,510 But outside of that range, even by just a little bit, -263 +265 00:20:09,510 --> 00:20:11,700 the series fails to approach anything. -264 +266 00:20:12,480 --> 00:20:17,440 As you add on more and more terms, the sum bounces back and forth wildly. -265 +267 00:20:18,100 --> 00:20:22,695 It does not, as you might expect, approach the natural log of that value, -266 +268 00:20:22,695 --> 00:20:27,540 even though the natural log of x is perfectly well defined for inputs above 2. -267 +269 00:20:28,460 --> 00:20:31,839 In some sense, the derivative information of ln -268 +270 00:20:31,839 --> 00:20:35,360 of x at x equals 1 doesn't propagate out that far. -269 +271 00:20:36,580 --> 00:20:41,277 In a case like this, where adding more terms of the series doesn't approach anything, -270 +272 00:20:41,277 --> 00:20:43,080 you say that the series diverges. -271 +273 00:20:44,180 --> 00:20:47,953 And that maximum distance between the input you're approximating -272 +274 00:20:47,953 --> 00:20:51,669 near and points where the outputs of these polynomials actually -273 +275 00:20:51,669 --> 00:20:55,560 converge is called the radius of convergence for the Taylor series. -274 +276 00:20:56,840 --> 00:20:59,160 There remains more to learn about Taylor series. -275 +277 00:20:59,500 --> 00:21:03,242 There are many use cases, tactics for placing bounds on the error of -276 +278 00:21:03,242 --> 00:21:07,636 these approximations, tests for understanding when series do and don't converge, -277 +279 00:21:07,636 --> 00:21:11,759 and for that matter, there remains more to learn about calculus as a whole, -278 +280 00:21:11,759 --> 00:21:14,580 and the countless topics not touched by this series. -279 +281 00:21:15,320 --> 00:21:19,241 The goal with these videos is to give you the fundamental intuitions -280 +282 00:21:19,241 --> 00:21:23,389 that make you feel confident and efficient in learning more on your own, -281 +283 00:21:23,389 --> 00:21:27,140 and potentially even rediscovering more of the topic for yourself. -282 +284 00:21:28,060 --> 00:21:32,327 In the case of Taylor series, the fundamental intuition to keep in mind -283 +285 00:21:32,327 --> 00:21:36,595 as you explore more of what there is, is that they translate derivative -284 +286 00:21:36,595 --> 00:21:41,160 information at a single point to approximation information around that point. -285 +287 00:21:43,920 --> 00:21:46,600 Thank you once again to everybody who supported this series. -286 +288 00:21:47,300 --> 00:21:49,529 The next series like it will be on probability, -287 +289 00:21:49,529 --> 00:21:53,060 and if you want early access as those videos are made, you know where to go. -288 +290 00:22:11,160 --> 00:22:19,060 Thank you. diff --git a/2017/taylor-series/english/sentence_timings.json b/2017/taylor-series/english/sentence_timings.json index 6344efe0d..8cc8c1bc5 100644 --- a/2017/taylor-series/english/sentence_timings.json +++ b/2017/taylor-series/english/sentence_timings.json @@ -60,7 +60,7 @@ 152.66 ], [ - "First of all, at the input 0, the value of cosine of x is 1, so if our approximation is any good at all, it should also equal 1 at the input x equals 0.", + "Well, first of all, at the input 0, the value of cosine of x is 1, so if our approximation is going to be any good at all, it should also equal 1 at the input x equals 0.", 153.86, 164.92 ], @@ -275,7 +275,7 @@ 547.78 ], [ - "The second thing to notice is that adding on new terms, like this c4 times x to the old terms should be, and that's really important.", + "The second thing to notice is that adding on new terms, like this c4 times x to the fourth, doesn't mess up what the old terms should be, and that's really important.", 549.4, 559.3 ], @@ -355,7 +355,7 @@ 713.16 ], [ - "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and evaluate each one of them at x equals 0.", + "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and you would evaluate each one of them at x equals 0.", 713.9, 728.4 ], @@ -410,7 +410,7 @@ 838.24 ], [ - "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x2 plus 1 over 3 factorial times x3, and so on, depending on how many terms you want.", + "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed, and so on, depending on how many terms you want.", 839.12, 858.54 ], @@ -535,7 +535,7 @@ 1110.16 ], [ - "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or that it equals the number e.", + "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or what's saying the same thing, that it equals the number e.", 1111.14, 1126.7 ], diff --git a/2017/taylor-series/english/transcript.txt b/2017/taylor-series/english/transcript.txt index 1bf04ee5e..005005417 100644 --- a/2017/taylor-series/english/transcript.txt +++ b/2017/taylor-series/english/transcript.txt @@ -10,7 +10,7 @@ The study of Taylor series is largely about taking non-polynomial functions and The motive here is that polynomials tend to be much easier to deal with than other functions, they're easier to compute, easier to take derivatives, easier to integrate, just all around more friendly. So let's take a look at that function, cosine of x, and really take a moment to think about how you might construct a quadratic approximation near x equals 0. That is, among all of the possible polynomials that look like c0 plus c1 times x plus c2 times x squared, for some choice of these constants, c0, c1, and c2, find the one that most resembles cosine of x near x equals 0, whose graph kind of spoons with the graph of cosine x at that point. -First of all, at the input 0, the value of cosine of x is 1, so if our approximation is any good at all, it should also equal 1 at the input x equals 0. +Well, first of all, at the input 0, the value of cosine of x is 1, so if our approximation is going to be any good at all, it should also equal 1 at the input x equals 0. Plugging in 0 just results in whatever c0 is, so we can set that equal to 1. This leaves us free to choose constants c1 and c2 to make this approximation as good as we can, but nothing we do with them is going to change the fact that the polynomial equals 1 at x equals 0. It would also be good if our approximation had the same tangent slope as cosine x at this point of interest. @@ -53,7 +53,7 @@ First of all, factorial terms come up very naturally in this process. When you take n successive derivatives of the function x to the n, letting the power rule keep cascading on down, what you'll be left with is 1 times 2 times 3 on and on up to whatever n is. So you don't simply set the coefficients of the polynomial equal to whatever derivative you want, you have to divide by the appropriate factorial to cancel out this effect. For example, that x to the fourth coefficient was the fourth derivative of cosine, 1, but divided by 4 factorial, 24. -The second thing to notice is that adding on new terms, like this c4 times x to the old terms should be, and that's really important. +The second thing to notice is that adding on new terms, like this c4 times x to the fourth, doesn't mess up what the old terms should be, and that's really important. For example, the second derivative of this polynomial at x equals 0 is still equal to 2 times the second coefficient, even after you introduce higher order terms. And it's because we're plugging in x equals 0, so the second derivative of any higher order term, which all include an x, will just wash away. And the same goes for any other derivative, which is why each derivative of a polynomial at x equals 0 is controlled by one and only one of the coefficients. @@ -69,7 +69,7 @@ So what we're doing is leveraging that information to get an approximation aroun And to do that, you just make each coefficient of the polynomial follow that same pattern, but you have to divide each one by the appropriate factorial. Like I mentioned before, this is what cancels out the cascading effect of many power rule applications. The polynomials you get by stopping this process at any point are called Taylor polynomials for cosine of x. -More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and evaluate each one of them at x equals 0. +More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and you would evaluate each one of them at x equals 0. Then for the polynomial approximation, the coefficient of each x to the n term should be the value of the nth derivative of the function evaluated at 0, but divided by n factorial. This whole rather abstract formula is something you'll likely see in any text or course that touches on Taylor polynomials. And when you see it, think to yourself that the constant term ensures that the value of the polynomial matches with the value of f, the next term ensures that the slope of the polynomial matches the slope of the function at x equals 0, the next term ensures that the rate at which the slope changes is the same at that point, and so on, depending on how many terms you want. @@ -80,7 +80,7 @@ Changing the value of a changes where this approximation is hugging the original One of the simplest meaningful examples of this is the function e to the x around the input x equals 0. Computing the derivatives is super nice, as nice as it gets, because the derivative of e to the x is itself, so the second derivative is also e to the x, as is its third, and so on. So at the point x equals 0, all of these are equal to 1. -And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x2 plus 1 over 3 factorial times x3, and so on, depending on how many terms you want. +And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed, and so on, depending on how many terms you want. These are the Taylor polynomials for e to the x. Ok, so with that as a foundation, in the spirit of showing you just how connected all the topics of calculus are, let me turn to something kind of fun, a completely different way to understand this second order term of the Taylor polynomials, but geometrically. It's related to the fundamental theorem of calculus, which I talked about in chapters 1 and 8 if you need a quick refresher. @@ -105,7 +105,7 @@ You have to be really careful with the idea of an infinite series, because it do But if you have a series where adding more and more of the terms, which makes sense at each step, gets you increasingly close to some specific value, what you say is that the series converges to that value. Or, if you're comfortable extending the definition of equality to include this kind of series convergence, you'd say that the series as a whole, this infinite sum, equals the value it's converging to. For example, look at the Taylor polynomial for e to the x, and plug in some input, like x equals 1. -As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or that it equals the number e. +As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or what's saying the same thing, that it equals the number e. In fact, it turns out that if you plug in any other value of x, like x equals 2, and look at the value of the higher and higher order Taylor polynomials at this value, they will converge towards e to the x, which is e squared. This is true for any input, no matter how far away from 0 it is, even though these Taylor polynomials are constructed only from derivative information gathered at the input 0. In a case like this, we say that e to the x equals its own Taylor series at all inputs x, which is kind of a magical thing to have happen. diff --git a/2017/taylor-series/french/sentence_translations.json b/2017/taylor-series/french/sentence_translations.json index e0cd646cc..63d3c59da 100644 --- a/2017/taylor-series/french/sentence_translations.json +++ b/2017/taylor-series/french/sentence_translations.json @@ -96,7 +96,7 @@ "end": 152.66 }, { - "input": "First of all, at the input 0, the value of cosine of x is 1, so if our approximation is any good at all, it should also equal 1 at the input x equals 0.", + "input": "Well, first of all, at the input 0, the value of cosine of x is 1, so if our approximation is going to be any good at all, it should also equal 1 at the input x equals 0.", "translatedText": "Tout d'abord, à l'entrée 0, la valeur du cosinus de x est 1, donc si notre approximation est bonne, elle devrait également être égale à 1 à l'entrée x est égal à 0.", "from_community_srt": "Eh bien, tout d'abord, pour la valeur 0, cos(x) est égal à 1, donc si l'on souhaite que notre approximation ait une quelconque valeur, elle devrait également valoir 1 quand on y met 0.", "n_reviews": 0, @@ -439,7 +439,7 @@ "end": 547.78 }, { - "input": "The second thing to notice is that adding on new terms, like this c4 times x to the old terms should be, and that's really important.", + "input": "The second thing to notice is that adding on new terms, like this c4 times x to the fourth, doesn't mess up what the old terms should be, and that's really important.", "translatedText": "La deuxième chose à remarquer est que l'ajout de nouveaux termes, comme celui-ci c4 fois x aux anciens termes, devrait l'être, et c'est vraiment important.", "from_community_srt": "24. La deuxième chose à noter est que l'ajout de nouveaux termes, comme ce c4*x^4, ne perturbe pas la valeur des anciens termes, et c'est très important.", "n_reviews": 0, @@ -567,7 +567,7 @@ "end": 713.16 }, { - "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and evaluate each one of them at x equals 0.", + "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and you would evaluate each one of them at x equals 0.", "translatedText": "Plus généralement, et donc de manière plus abstraite, si nous avions affaire à une autre fonction que le cosinus, vous calculeriez sa dérivée, sa dérivée seconde, etc., en obtenant autant de termes que vous le souhaitez, et évalueriez chacun d'eux. à x est égal à 0.", "from_community_srt": "De manière plus générale, et donc plus abstraire, si nous avions affaire à une autre fonction que cosinus, vous devriez calculer sa dérivée, sa dérivée seconde, et ainsi de suite, obtenant ainsi autant de termes que vous le souhaiteriez, et vous auriez évalueriez chacune en x=0.", "n_reviews": 0, @@ -655,7 +655,7 @@ "end": 838.24 }, { - "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x2 plus 1 over 3 factorial times x3, and so on, depending on how many terms you want.", + "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed, and so on, depending on how many terms you want.", "translatedText": "Et cela signifie que notre approximation polynomiale devrait ressembler à 1 plus 1 fois x plus 1 sur 2 fois x2 plus 1 sur 3 fois factorielle x3, et ainsi de suite, selon le nombre de termes souhaités.", "from_community_srt": "elles valent toutes 1. Cela signifie que notre approximation polynomiale ressemble à 1 + x + ½ x^2 + 1/(3!) x^3 + 1/(4!) x^4, et ainsi de suite, selon le nombre de termes que vous voulez.", "n_reviews": 0, @@ -854,7 +854,7 @@ "end": 1110.16 }, { - "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or that it equals the number e.", + "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or what's saying the same thing, that it equals the number e.", "translatedText": "À mesure que vous ajoutez de plus en plus de termes polynomiaux, la somme totale se rapproche de plus en plus de la valeur e, on dit donc que cette série infinie converge vers le nombre e, ou qu'elle est égale au nombre e.", "from_community_srt": "Plus vous ajoutez de termes dans votre polynôme, plus la somme totale se rapproche de la valeur e, donc nous disons que la série infinie converge vers le nombre e. Ou, ce qui est la même chose,", "n_reviews": 0, diff --git a/2017/taylor-series/german/sentence_translations.json b/2017/taylor-series/german/sentence_translations.json index 85886a345..c67598fc4 100644 --- a/2017/taylor-series/german/sentence_translations.json +++ b/2017/taylor-series/german/sentence_translations.json @@ -108,7 +108,7 @@ "end": 152.66 }, { - "input": "First of all, at the input 0, the value of cosine of x is 1, so if our approximation is any good at all, it should also equal 1 at the input x equals 0.", + "input": "Well, first of all, at the input 0, the value of cosine of x is 1, so if our approximation is going to be any good at all, it should also equal 1 at the input x equals 0.", "translatedText": "Zunächst einmal ist der Wert des Kosinus von x bei Eingang 0 gleich 1. Wenn unsere Annäherung also überhaupt etwas taugt, sollte sie auch bei Eingang x gleich 0 gleich 1 sein.", "model": "DeepL", "from_community_srt": "Nun, zuallererst am Eingang 0 den Wert von cos (x) ist 1, wenn also unsere Näherung ist wird überhaupt gut sein, sollte es auch gleich 1, wenn Sie 0 einstecken.", @@ -494,7 +494,7 @@ "end": 547.78 }, { - "input": "The second thing to notice is that adding on new terms, like this c4 times x to the old terms should be, and that's really important.", + "input": "The second thing to notice is that adding on new terms, like this c4 times x to the fourth, doesn't mess up what the old terms should be, and that's really important.", "translatedText": "Die zweite Sache, die du beachten solltest, ist, dass das Hinzufügen neuer Begriffe, wie dieser c4 mal x zu den alten Begriffen sein sollte, und das ist wirklich wichtig.", "model": "DeepL", "from_community_srt": "24. Das zweite, was zu bemerken ist, ist das Hinzufügen Neue Begriffe wie dieser c4x4 bringen nichts durcheinander up was alte Begriffe sein sollten,", @@ -638,7 +638,7 @@ "end": 713.16 }, { - "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and evaluate each one of them at x equals 0.", + "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and you would evaluate each one of them at x equals 0.", "translatedText": "Allgemeiner und damit abstrakter ausgedrückt: Wenn wir es mit einer anderen Funktion als dem Kosinus zu tun hätten, würdest du ihre Ableitung, ihre zweite Ableitung und so weiter berechnen, so viele Terme wie du willst, und jeden einzelnen davon bei x gleich 0 auswerten.", "model": "DeepL", "from_community_srt": "Allgemeiner und damit abstrakter wenn wir es mit einer anderen Funktion zu tun hätten als Kosinus würden Sie seine Ableitung berechnen, zweite Ableitung und so weiter, so viele bekommen Begriffe, wie Sie möchten, und Sie würden bewerten jeweils bei x = 0.", @@ -737,7 +737,7 @@ "end": 838.24 }, { - "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x2 plus 1 over 3 factorial times x3, and so on, depending on how many terms you want.", + "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed, and so on, depending on how many terms you want.", "translatedText": "Das bedeutet, dass unsere Polynomnäherung wie 1 plus 1 mal x plus 1 über 2 mal x2 plus 1 über 3 mal x3 aussehen sollte, je nachdem, wie viele Terme du haben willst.", "model": "DeepL", "from_community_srt": "Dies bedeutet, dass unsere Polynomnäherung so aussieht 1 + x + ½ x2 + 1 / (3!) X3 + 1 / (4!) X4 und usw., je nachdem,", @@ -962,7 +962,7 @@ "end": 1110.16 }, { - "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or that it equals the number e.", + "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or what's saying the same thing, that it equals the number e.", "translatedText": "Wenn du immer mehr Polynomterme hinzufügst, nähert sich die Gesamtsumme immer mehr dem Wert e. Man sagt also, dass diese unendliche Reihe gegen die Zahl e konvergiert oder dass sie gleich der Zahl e ist.", "model": "DeepL", "from_community_srt": "Wenn Sie mehr und mehr Polynomterme hinzufügen, Die Gesamtsumme kommt dem immer näher Wert e, also sagen wir, dass die unendliche Reihe konvergiert gegen die Zahl e. Oder was sagt das? das gleiche, dass es gleich der Zahl ist e.", diff --git a/2017/taylor-series/hebrew/sentence_translations.json b/2017/taylor-series/hebrew/sentence_translations.json index cc2f5e186..01005fa75 100644 --- a/2017/taylor-series/hebrew/sentence_translations.json +++ b/2017/taylor-series/hebrew/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "הסדרה הבאה כמוה תהיה בהסתברות, ואם אתה רוצה גישה מוקדמת עם יצירת הסרטונים האלה, אתה יודע לאן ללכת. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/hindi/sentence_translations.json b/2017/taylor-series/hindi/sentence_translations.json index ef6633a68..2200f8f0a 100644 --- a/2017/taylor-series/hindi/sentence_translations.json +++ b/2017/taylor-series/hindi/sentence_translations.json @@ -871,7 +871,7 @@ "end": 1076.08 }, { - "input": "But if you have a series where adding more and more of the terms, which makes sense at each step, gets you increasingly close to some specific value, you say that the series converges to that value.", + "input": "But if you have a series where adding more and more of the terms, which makes sense at each step, gets you increasingly close to some specific value, what you say is that the series converges to that value.", "translatedText": "लेकिन यदि आपके पास एक श्रृंखला है जहां अधिक से अधिक शब्दों को जोड़ने पर, जो प्रत्येक चरण में समझ में आता है, तो आप तेजी से कुछ विशिष्ट मूल्य के करीब पहुंच जाते हैं, तो आप कहते हैं कि श्रृंखला उस मूल्य पर परिवर्तित हो जाती है।", "from_community_srt": "लेकिन अगर आपके पास एक श्रृंखला है जहां अधिक जोड़ना है और अधिक शर्तें आपको तेजी से बंद कर देती हैं कुछ विशिष्ट मूल्य के लिए, आप श्रृंखला कहते हैं उस मूल्य में अभिसरण।", "n_reviews": 0, @@ -1039,7 +1039,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go.", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you.", "translatedText": "इस तरह की अगली श्रृंखला संभाव्यता पर होगी, और यदि आप वीडियो बनाते समय शीघ्र पहुंच चाहते हैं, तो आप जानते हैं कि कहां जाना है।", "from_community_srt": "इस तरह की अगली श्रृंखला संभावना पर होगी, और यदि आप उन वीडियो के रूप में प्रारंभिक पहुंच चाहते हैं बने होते हैं, आप जानते हैं कि कहां जाना है।", "n_reviews": 0, diff --git a/2017/taylor-series/hungarian/sentence_translations.json b/2017/taylor-series/hungarian/sentence_translations.json index 5e870b65a..2dd13d949 100644 --- a/2017/taylor-series/hungarian/sentence_translations.json +++ b/2017/taylor-series/hungarian/sentence_translations.json @@ -96,7 +96,7 @@ "end": 152.66 }, { - "input": "First of all, at the input 0, the value of cosine of x is 1, so if our approximation is any good at all, it should also equal 1 at the input x equals 0.", + "input": "Well, first of all, at the input 0, the value of cosine of x is 1, so if our approximation is going to be any good at all, it should also equal 1 at the input x equals 0.", "translatedText": "Először is, a 0 bemenetnél az x koszinuszának értéke 1, tehát ha a közelítésünk egyáltalán jó, akkor az x 0-nak megfelelő bemenetnél is 1-nek kell lennie.", "model": "DeepL", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 547.78 }, { - "input": "The second thing to notice is that adding on new terms, like this c4 times x to the old terms should be, and that's really important.", + "input": "The second thing to notice is that adding on new terms, like this c4 times x to the fourth, doesn't mess up what the old terms should be, and that's really important.", "translatedText": "A második dolog, amit észre kell venni, hogy az új kifejezések hozzáadása, mint ez a c4-szer x a régi kifejezésekhez, és ez nagyon fontos.", "model": "DeepL", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 713.16 }, { - "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and evaluate each one of them at x equals 0.", + "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and you would evaluate each one of them at x equals 0.", "translatedText": "Általánosabban, és így absztraktabb módon, ha a koszinuszon kívül más függvénnyel foglalkoznánk, akkor kiszámítanánk a deriváltját, a második deriváltját, és így tovább, annyi tagot kapnánk, amennyit csak szeretnénk, és mindegyiküket kiértékelnénk, ha x egyenlő 0-val.", "model": "DeepL", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 838.24 }, { - "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x2 plus 1 over 3 factorial times x3, and so on, depending on how many terms you want.", + "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed, and so on, depending on how many terms you want.", "translatedText": "Ez azt jelenti, hogy a polinomiális közelítésünknek úgy kell kinéznie, hogy 1 plusz 1 x-szor x plusz 1 x 2 x x2 plus 1 x 3 faktoriális x3, és így tovább, attól függően, hogy hány tagot szeretnénk.", "model": "DeepL", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 1110.16 }, { - "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or that it equals the number e.", + "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or what's saying the same thing, that it equals the number e.", "translatedText": "Ahogy egyre több és több polinomiális tagot adunk hozzá, a végösszeg egyre közelebb kerül az e értékhez, ezért azt mondjuk, hogy ez a végtelen sorozat konvergál az e számhoz, vagy hogy megegyezik az e számmal.", "model": "DeepL", "n_reviews": 0, diff --git a/2017/taylor-series/indonesian/sentence_translations.json b/2017/taylor-series/indonesian/sentence_translations.json index fc9767a50..9522244fa 100644 --- a/2017/taylor-series/indonesian/sentence_translations.json +++ b/2017/taylor-series/indonesian/sentence_translations.json @@ -872,7 +872,7 @@ "end": 1076.08 }, { - "input": "But if you have a series where adding more and more of the terms, which makes sense at each step, gets you increasingly close to some specific value, you say that the series converges to that value.", + "input": "But if you have a series where adding more and more of the terms, which makes sense at each step, gets you increasingly close to some specific value, what you say is that the series converges to that value.", "translatedText": "Namun jika Anda memiliki rangkaian yang menambahkan lebih banyak suku, yang masuk akal pada setiap langkah, membuat Anda semakin mendekati nilai tertentu, Anda mengatakan bahwa rangkaian tersebut menyatu dengan nilai tersebut.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go.", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you.", "translatedText": "Seri berikutnya yang serupa kemungkinan besar akan ada, dan jika Anda menginginkan akses awal saat video tersebut dibuat, Anda tahu ke mana harus pergi.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/italian/sentence_translations.json b/2017/taylor-series/italian/sentence_translations.json index 61c34fc10..f4ddf33b5 100644 --- a/2017/taylor-series/italian/sentence_translations.json +++ b/2017/taylor-series/italian/sentence_translations.json @@ -108,7 +108,7 @@ "end": 152.66 }, { - "input": "First of all, at the input 0, the value of cosine of x is 1, so if our approximation is any good at all, it should also equal 1 at the input x equals 0.", + "input": "Well, first of all, at the input 0, the value of cosine of x is 1, so if our approximation is going to be any good at all, it should also equal 1 at the input x equals 0.", "translatedText": "Innanzitutto, in corrispondenza dell'ingresso 0, il valore del coseno di x è 1, quindi se la nostra approssimazione è buona, dovrebbe essere uguale a 1 anche in corrispondenza dell'ingresso x uguale a 0.", "model": "DeepL", "from_community_srt": "Bene, prima di tutto, all'ingresso 0 il valore di cos (x) è 1, quindi se la nostra approssimazione è andrà bene, dovrebbe anche uguale a 1 quando si inserisce 0.", @@ -494,7 +494,7 @@ "end": 547.78 }, { - "input": "The second thing to notice is that adding on new terms, like this c4 times x to the old terms should be, and that's really important.", + "input": "The second thing to notice is that adding on new terms, like this c4 times x to the fourth, doesn't mess up what the old terms should be, and that's really important.", "translatedText": "La seconda cosa da notare è che l'aggiunta di nuovi termini, come questo c4 volte x, ai vecchi termini dovrebbe essere, e questo è davvero importante.", "model": "DeepL", "from_community_srt": "24. La seconda cosa da notare è l'aggiunta nuovi termini, come questo c4x4, non pasticciano su quali vecchi termini dovrebbero essere,", @@ -638,7 +638,7 @@ "end": 713.16 }, { - "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and evaluate each one of them at x equals 0.", + "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and you would evaluate each one of them at x equals 0.", "translatedText": "Più in generale, e quindi in modo più astratto, se avessimo a che fare con qualche altra funzione diversa dal coseno, calcoleresti la sua derivata, la sua derivata seconda e così via, ottenendo tutti i termini che desideri, e valutando ognuno di essi a x uguale a 0.", "model": "DeepL", "from_community_srt": "Più in generale, e quindi in modo più astratto, se avessimo a che fare con qualche altra funzione di coseno, calcoleresti la sua derivata, seconda derivata, e così via, ottenendo altrettanti termini che vorresti e valuterai ognuno a x = 0.", @@ -737,7 +737,7 @@ "end": 838.24 }, { - "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x2 plus 1 over 3 factorial times x3, and so on, depending on how many terms you want.", + "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed, and so on, depending on how many terms you want.", "translatedText": "Questo significa che l'approssimazione del polinomio dovrebbe assomigliare a 1 più 1 volte x, più 1 su 2 volte x2, più 1 su 3 fattoriale x3 e così via, a seconda del numero di termini desiderati.", "model": "DeepL", "from_community_srt": "Questo significa che la nostra approssimazione polinomiale è simile 1 + x + ½ x2 + 1 / (3!) X3 + 1 / (4!) X4, e così via,", @@ -962,7 +962,7 @@ "end": 1110.16 }, { - "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or that it equals the number e.", + "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or what's saying the same thing, that it equals the number e.", "translatedText": "Aggiungendo sempre più termini polinomiali, la somma totale si avvicina sempre di più al valore e, per cui si dice che questa serie infinita converge al numero e, o che è uguale al numero e.", "model": "DeepL", "from_community_srt": "Mentre aggiungi sempre più termini polinomiali, la somma totale si avvicina sempre di più al valore e, quindi diciamo che la serie infinita converge al numero e. O, cosa sta dicendo la stessa cosa, che è uguale al numero e.", diff --git a/2017/taylor-series/japanese/sentence_translations.json b/2017/taylor-series/japanese/sentence_translations.json index 3b70b3df4..a0d59d327 100644 --- a/2017/taylor-series/japanese/sentence_translations.json +++ b/2017/taylor-series/japanese/sentence_translations.json @@ -872,7 +872,7 @@ "end": 1076.08 }, { - "input": "But if you have a series where adding more and more of the terms, which makes sense at each step, gets you increasingly close to some specific value, you say that the series converges to that value.", + "input": "But if you have a series where adding more and more of the terms, which makes sense at each step, gets you increasingly close to some specific value, what you say is that the series converges to that value.", "translatedText": "しかし、各ステップで意味のある項をどんどん追加し ていくことで特定の値にどんどん近づいていく系列が ある場合、その系列はその値に収束すると言います。", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go.", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you.", "translatedText": "次のシリーズは確率に基づいたものになるでしょう。 ビデオが作成され たときに早期アクセスしたい場合は、どこに行くべきか知っています。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/korean/sentence_translations.json b/2017/taylor-series/korean/sentence_translations.json index 8dfea8316..44cee5ba4 100644 --- a/2017/taylor-series/korean/sentence_translations.json +++ b/2017/taylor-series/korean/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "이와 같은 다음 시리즈는 확률적으로 나올 것이며 해당 비디오가 제작될 때 조기 액세스를 원한다면 어디로 가야할지 알 수 있습니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/marathi/sentence_translations.json b/2017/taylor-series/marathi/sentence_translations.json index aedec4e43..27494f0de 100644 --- a/2017/taylor-series/marathi/sentence_translations.json +++ b/2017/taylor-series/marathi/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "यासारखी पुढील मालिका संभाव्यतेवर असेल आणि तुम्हाला ते व्हिडिओ बनवताना लवकर प्रवेश हवा असल्यास, तुम्हाला कुठे जायचे हे माहित आहे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/persian/sentence_translations.json b/2017/taylor-series/persian/sentence_translations.json index 01f14f844..c5421fe3c 100644 --- a/2017/taylor-series/persian/sentence_translations.json +++ b/2017/taylor-series/persian/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "سری بعدی مانند آن به احتمال زیاد خواهد بود، و اگر می‌خواهید با ساخت آن ویدیوها دسترسی اولیه داشته باشید، می‌دانید کجا باید بروید. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/polish/sentence_translations.json b/2017/taylor-series/polish/sentence_translations.json index 18218da98..e221eebf0 100644 --- a/2017/taylor-series/polish/sentence_translations.json +++ b/2017/taylor-series/polish/sentence_translations.json @@ -96,7 +96,7 @@ "end": 152.66 }, { - "input": "First of all, at the input 0, the value of cosine of x is 1, so if our approximation is any good at all, it should also equal 1 at the input x equals 0.", + "input": "Well, first of all, at the input 0, the value of cosine of x is 1, so if our approximation is going to be any good at all, it should also equal 1 at the input x equals 0.", "translatedText": "", "from_community_srt": "Zacznijmy od tego, że dla x=0, cos(x)=1, więc aby nasze przybliżenie było jak najlepsze, powinno także dawać 1 dla x=0.", "n_reviews": 0, @@ -439,7 +439,7 @@ "end": 547.78 }, { - "input": "The second thing to notice is that adding on new terms, like this c4 times x to the old terms should be, and that's really important.", + "input": "The second thing to notice is that adding on new terms, like this c4 times x to the fourth, doesn't mess up what the old terms should be, and that's really important.", "translatedText": "", "from_community_srt": "czyli 24 Drugą sprawą jest to, że dodawanie nowych współczynników nie psuje efektów działania starych, co jest ważne.", "n_reviews": 0, @@ -567,7 +567,7 @@ "end": 713.16 }, { - "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and evaluate each one of them at x equals 0.", + "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and you would evaluate each one of them at x equals 0.", "translatedText": "", "from_community_srt": "Bardziej ogólnie i tym samym bardziej abstrakcyjnie, jeśli mamy do czynienia z jakąś funkcją inną niż cosinus, liczyłbyś jej pochodną, drugą pochodną i tak dalej, dostając tyle wyrazów, ile byś chciał, i wyliczałbyś każdy w x=0.", "n_reviews": 0, @@ -655,7 +655,7 @@ "end": 838.24 }, { - "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x2 plus 1 over 3 factorial times x3, and so on, depending on how many terms you want.", + "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed, and so on, depending on how many terms you want.", "translatedText": "", "from_community_srt": "wszystkie są równe 1. Oznacza to, że nasz wielomian wygląda jak 1 + x + ½ x$2 + 1/(3!) x$3 + 1/(4!) x^4, i tak dalej,", "n_reviews": 0, @@ -855,7 +855,7 @@ "end": 1110.16 }, { - "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or that it equals the number e.", + "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or what's saying the same thing, that it equals the number e.", "translatedText": "", "from_community_srt": "jak x=1. Dodając coraz więcej wyrazów wielomianowych, łączna suma zbliża się do wartości e, więc mówimy, że nieskończony szereg zbiega do liczby e. Lub, równoważnie , że równa się liczbie e.", "n_reviews": 0, diff --git a/2017/taylor-series/portuguese/sentence_translations.json b/2017/taylor-series/portuguese/sentence_translations.json index 03891d2bb..07861b5d3 100644 --- a/2017/taylor-series/portuguese/sentence_translations.json +++ b/2017/taylor-series/portuguese/sentence_translations.json @@ -872,7 +872,7 @@ "end": 1076.08 }, { - "input": "But if you have a series where adding more and more of the terms, which makes sense at each step, gets you increasingly close to some specific value, you say that the series converges to that value.", + "input": "But if you have a series where adding more and more of the terms, which makes sense at each step, gets you increasingly close to some specific value, what you say is that the series converges to that value.", "translatedText": "Mas se tivermos uma série em que adicionar cada vez mais termos, o que faz sentido a cada passo, nos aproxima cada vez mais de algum valor específico, dizemos que a série converge para esse valor.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go.", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you.", "translatedText": "A próxima série como essa será sobre probabilidade, e se você quiser ter acesso antecipado à medida que esses vídeos são feitos, você sabe aonde ir.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/russian/sentence_translations.json b/2017/taylor-series/russian/sentence_translations.json index 1f65ffec8..0bfa6dc76 100644 --- a/2017/taylor-series/russian/sentence_translations.json +++ b/2017/taylor-series/russian/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "Следующая серия, подобная этой, будет посвящена вероятности, и если вам нужен ранний доступ по мере создания этих видео, вы знаете, куда идти. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/spanish/sentence_translations.json b/2017/taylor-series/spanish/sentence_translations.json index 434789d34..f69e16fc4 100644 --- a/2017/taylor-series/spanish/sentence_translations.json +++ b/2017/taylor-series/spanish/sentence_translations.json @@ -96,7 +96,7 @@ "end": 152.66 }, { - "input": "First of all, at the input 0, the value of cosine of x is 1, so if our approximation is any good at all, it should also equal 1 at the input x equals 0.", + "input": "Well, first of all, at the input 0, the value of cosine of x is 1, so if our approximation is going to be any good at all, it should also equal 1 at the input x equals 0.", "translatedText": "En primer lugar, en la entrada 0, el valor del coseno de x es 1, por lo que si nuestra aproximación es buena, también debería ser igual a 1 en la entrada x es igual a 0.", "from_community_srt": "Bien, antes que nada, en el valor de entrada 0 el valor de cos(x) es 1, así que si nuestra aproximación será algo precisa, debería ser 1 cuando el valor de entrada es 0.", "n_reviews": 0, @@ -439,7 +439,7 @@ "end": 547.78 }, { - "input": "The second thing to notice is that adding on new terms, like this c4 times x to the old terms should be, and that's really important.", + "input": "The second thing to notice is that adding on new terms, like this c4 times x to the fourth, doesn't mess up what the old terms should be, and that's really important.", "translatedText": "La segunda cosa a tener en cuenta es que debería ser agregar nuevos términos, como este c4 multiplicado por x a los términos antiguos, y eso es realmente importante.", "from_community_srt": "24. La segunda cosa a notar es que añadiendo nuevos terminos, como este c4*x^4", "n_reviews": 0, @@ -567,7 +567,7 @@ "end": 713.16 }, { - "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and evaluate each one of them at x equals 0.", + "input": "More generally, and hence more abstractly, if we were dealing with some other function other than cosine, you would compute its derivative, its second derivative, and so on, getting as many terms as you'd like, and you would evaluate each one of them at x equals 0.", "translatedText": "De manera más general, y por lo tanto de manera más abstracta, si estuviéramos tratando con alguna otra función distinta del coseno, calcularíamos su derivada, su segunda derivada, etc., obtendríamos tantos términos como quisieramos y evaluaríamos cada uno de ellos. en x es igual a 0.", "from_community_srt": "De manera más general, y por lo tanto de manera más abstracta, si estuviéramos tratando con alguna otra función distinta que el coseno, calcularías su derivada, segunda derivada, y así sucesivamente, obtenieniendo tantos términos como desees,", "n_reviews": 0, @@ -655,7 +655,7 @@ "end": 838.24 }, { - "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x2 plus 1 over 3 factorial times x3, and so on, depending on how many terms you want.", + "input": "And what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed, and so on, depending on how many terms you want.", "translatedText": "Y lo que eso significa es que nuestra aproximación polinómica debería verse como 1 más 1 por x más 1 sobre 2 por x2 más 1 sobre 3 factorial por x3, y así sucesivamente, dependiendo de cuántos términos quieras.", "from_community_srt": "Esto significa que nuestra aproximación polinómica es 1 + x + ½ x² + 1 / (3!) x³ + 1 / (4!) x⁴ etc. dependiendo de cuántos términos se deseen.", "n_reviews": 0, @@ -855,7 +855,7 @@ "end": 1110.16 }, { - "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or that it equals the number e.", + "input": "As you add more and more polynomial terms, the total sum gets closer and closer to the value e, so you say that this infinite series converges to the number e, or what's saying the same thing, that it equals the number e.", "translatedText": "A medida que agregas más y más términos polinomiales, la suma total se acerca cada vez más al valor e, por lo que dices que esta serie infinita converge al número e, o que es igual al número e.", "from_community_srt": "A medida que se agregan más y más términos del polinomio, la suma total se acerca cada vez más al valor de e, entonces decimos que la serie infinita converge al número e. O lo que es lo mismo, que es igual al número e.", "n_reviews": 0, diff --git a/2017/taylor-series/tamil/sentence_translations.json b/2017/taylor-series/tamil/sentence_translations.json index ed381fef9..5739e74e7 100644 --- a/2017/taylor-series/tamil/sentence_translations.json +++ b/2017/taylor-series/tamil/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "இது போன்ற அடுத்த தொடர் நிகழ்தகவில் இருக்கும், மேலும் அந்த வீடியோக்கள் உருவாக்கப்பட்டதால் நீங்கள் முன்கூட்டியே அணுக விரும்பினால், எங்கு செல்ல வேண்டும் என்பது உங்களுக்குத் தெரியும். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/telugu/sentence_translations.json b/2017/taylor-series/telugu/sentence_translations.json index 2b07e314d..2f74fbc9c 100644 --- a/2017/taylor-series/telugu/sentence_translations.json +++ b/2017/taylor-series/telugu/sentence_translations.json @@ -872,7 +872,7 @@ "end": 1076.08 }, { - "input": "But if you have a series where adding more and more of the terms, which makes sense at each step, gets you increasingly close to some specific value, you say that the series converges to that value.", + "input": "But if you have a series where adding more and more of the terms, which makes sense at each step, gets you increasingly close to some specific value, what you say is that the series converges to that value.", "translatedText": "కానీ మీరు ఒక సిరీస్‌ని కలిగి ఉన్నట్లయితే, ప్రతి దశలోనూ అర్థవంతంగా ఉండే మరిన్ని నిబంధనలను జోడించడం వలన, మీరు కొంత నిర్దిష్ట విలువకు దగ్గరగా ఉండేలా చేస్తుంది, సిరీస్ ఆ విలువకు కలుస్తుందని మీరు అంటున్నారు.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go.", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you.", "translatedText": "ఇది వంటి తదుపరి సిరీస్ సంభావ్యతపై ఉంటుంది మరియు ఆ వీడియోలు రూపొందించబడినందున మీకు ముందస్తు యాక్సెస్ కావాలంటే, ఎక్కడికి వెళ్లాలో మీకు తెలుసు.", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/thai/sentence_translations.json b/2017/taylor-series/thai/sentence_translations.json index a8117fe60..1dcbe8940 100644 --- a/2017/taylor-series/thai/sentence_translations.json +++ b/2017/taylor-series/thai/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/turkish/sentence_translations.json b/2017/taylor-series/turkish/sentence_translations.json index 148708dcc..23e82094a 100644 --- a/2017/taylor-series/turkish/sentence_translations.json +++ b/2017/taylor-series/turkish/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "Bunun gibi bir sonraki seri olasılık üzerine olacak ve eğer bu videolar hazırlanırken erken erişim istiyorsanız nereye gideceğinizi biliyorsunuz. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/ukrainian/sentence_translations.json b/2017/taylor-series/ukrainian/sentence_translations.json index 442eeb495..2814267c1 100644 --- a/2017/taylor-series/ukrainian/sentence_translations.json +++ b/2017/taylor-series/ukrainian/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "Подібна наступна серія, ймовірно, буде доступною, і якщо ви хочете мати ранній доступ, коли ці відео будуть створені, ви знаєте, куди йти. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/urdu/sentence_translations.json b/2017/taylor-series/urdu/sentence_translations.json index 626e208c0..dbd5b8eae 100644 --- a/2017/taylor-series/urdu/sentence_translations.json +++ b/2017/taylor-series/urdu/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "اس کی طرح اگلی سیریز امکان پر ہوگی، اور اگر آپ ان ویڈیوز کے بنتے ہی جلد رسائی چاہتے ہیں، تو آپ جانتے ہیں کہ کہاں جانا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/taylor-series/vietnamese/sentence_translations.json b/2017/taylor-series/vietnamese/sentence_translations.json index b8b10564b..2aa93a9e8 100644 --- a/2017/taylor-series/vietnamese/sentence_translations.json +++ b/2017/taylor-series/vietnamese/sentence_translations.json @@ -1040,7 +1040,7 @@ "end": 1306.6 }, { - "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. ", + "input": "The next series like it will be on probability, and if you want early access as those videos are made, you know where to go. Thank you. ", "translatedText": "Loạt phim tiếp theo giống như vậy sẽ có khả năng xảy ra và nếu bạn muốn truy cập sớm khi những video đó được thực hiện, bạn sẽ biết phải đi đâu. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/arabic/sentence_translations.json b/2017/three-utilities/arabic/sentence_translations.json index 6a4336fa1..d35fb56f0 100644 --- a/2017/three-utilities/arabic/sentence_translations.json +++ b/2017/three-utilities/arabic/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "حسنًا، سأضعه في المنزل الثاني. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "إذا مررت بكل منطقة من هذه المناطق وقمت بجمع عدد الحواف التي تحتوي عليها، حسنًا، سينتهي بك الأمر بـ 5 ضرب 4، أو 20. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/bengali/sentence_translations.json b/2017/three-utilities/bengali/sentence_translations.json index 4b3e02ead..995591609 100644 --- a/2017/three-utilities/bengali/sentence_translations.json +++ b/2017/three-utilities/bengali/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "ঠিক আছে, আমি এটাকে দ্বিতীয় বাড়িতে রাখব।", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "আপনি যদি এই অঞ্চলগুলির প্রতিটির মধ্য দিয়ে যান এবং এটির প্রান্তের সংখ্যা যোগ করেন, তাহলে আপনি 5 গুণ 4 বা 20 দিয়ে শেষ করবেন।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/chinese/sentence_translations.json b/2017/three-utilities/chinese/sentence_translations.json index d1e879158..3c59e4698 100644 --- a/2017/three-utilities/chinese/sentence_translations.json +++ b/2017/three-utilities/chinese/sentence_translations.json @@ -451,7 +451,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "好的，我会把它放到第二个房子里。", "model": "google_nmt", "from_community_srt": "这么画挺好 这么画……等一下……这么画 差不离是这样 这倒挺简单 然后我们只要从这里连到那里 这是1，", @@ -1409,7 +1409,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "如果您遍历这些区域中的每一个并将其具有的边数相 加，那么您最终会得到 5 乘以 4，即 20。", "model": "google_nmt", "from_community_srt": "并把其 各自边的数量相加 那么你就会得到五乘以四 也就是二十。", diff --git a/2017/three-utilities/english/captions.srt b/2017/three-utilities/english/captions.srt index 1676d2d58..8d316cfb8 100644 --- a/2017/three-utilities/english/captions.srt +++ b/2017/three-utilities/english/captions.srt @@ -224,15 +224,15 @@ I should have put it over there, but oh well. 57 00:02:15,480 --> 00:02:18,340 -Oh, that should go to the second house. +Oh, that should go to the, that should go to the second house. 58 -00:02:18,620 --> 00:02:20,000 +00:02:18,620 --> 00:02:18,340 Okay, I'll put it to the second house. 59 -00:02:21,920 --> 00:02:23,640 -Presumably this is easy enough. +00:02:18,620 --> 00:02:23,640 +Okay, I'll put it to the second house. This is easy enough. 60 00:02:25,060 --> 00:02:26,800 @@ -659,818 +659,826 @@ light up the new one. So at first, each new edge lights up one more vertex. 166 -00:07:58,080 --> 00:08:03,780 -But if you connect to an already lit up vertex, notice how this closes off a new region. +00:07:58,080 --> 00:08:00,995 +But if you connect to an already lit up vertex, notice how this closes off a new region. 167 +00:08:00,995 --> 00:08:03,780 +And this is a very simple way to do it. You can see how this closes off a new region. + +168 00:08:04,640 --> 00:08:06,320 And this gives us a super useful fact. -168 +169 00:08:06,920 --> 00:08:11,957 Each new edge either increases the number of lit up nodes by one, -169 +170 00:08:11,957 --> 00:08:16,080 or it increases the number of enclosed regions by one. -170 +171 00:08:20,020 --> 00:08:23,286 This fact is something that we can use to figure out the number of -171 +172 00:08:23,286 --> 00:08:26,700 regions that a hypothetical solution to this would cut the plane into. -172 +173 00:08:27,320 --> 00:08:27,880 Can you see how? -173 +174 00:08:31,400 --> 00:08:35,600 When you start off, there's one node lit up and one region, all of 2D space. -174 +175 00:08:36,419 --> 00:08:39,024 By the end, we're going to need to draw nine lines, -175 +176 00:08:39,024 --> 00:08:42,880 since each of the three utilities gets connected to each of the three houses. -176 +177 00:08:43,900 --> 00:08:47,520 Five of those lines are going to light up the initially dim vertices. -177 +178 00:08:59,260 --> 00:09:04,080 So the other four lines each must introduce a new region. -178 -00:09:13,180 --> 00:09:18,020 -So a hypothetical solution would cut the plane into five separate regions. - 179 +00:09:13,180 --> 00:09:15,599 +So a hypothetical solution that has turned up thehew times one node through a difficult + +180 +00:09:15,599 --> 00:09:18,020 +Bye� ham die So, a hypothetical solution would cut the plane into five separate regions. + +181 00:09:18,780 --> 00:09:23,000 And you might say, okay, that's a cute fact, but why should that make things impossible? -180 +182 00:09:23,000 --> 00:09:24,520 What's wrong with having five regions? -181 +183 00:09:25,340 --> 00:09:27,580 Well again, take a look at this partially complete graph. -182 +184 00:09:27,960 --> 00:09:30,920 Notice that each region is bounded by four edges. -183 +185 00:09:32,220 --> 00:09:36,660 And in fact, for this graph, you could never have a cycle with fewer than four edges. -184 +186 00:09:37,220 --> 00:09:40,970 Say you start at a house, then the next line has to be to some utility, -185 +187 00:09:40,970 --> 00:09:44,147 and then a line out of that is going to go to another house, -186 +188 00:09:44,147 --> 00:09:47,220 and you can't cycle back to where you started immediately, -187 +189 00:09:47,220 --> 00:09:51,700 because you have to go to another utility before you can get back to that first house. -188 +190 00:09:52,340 --> 00:09:54,940 So all cycles have at least four edges. -189 +191 00:09:55,560 --> 00:09:58,860 And this right here gives us enough to prove the impossibility of our puzzle. -190 +192 00:09:59,460 --> 00:10:03,061 Having five regions, each with a boundary of at least four edges, -191 +193 00:10:03,061 --> 00:10:05,680 would require more edges than we have available. -192 +194 00:10:06,520 --> 00:10:10,693 Here, let me draw a planar graph that's completely different from our utilities puzzle, -193 +195 00:10:10,693 --> 00:10:14,440 but useful for illustrating what five regions with four edges each would imply. -194 +196 00:10:14,880 --> 00:10:22,053 If you went through each of these regions and add up the number of edges that it has, -195 +197 00:10:22,053 --> 00:10:26,140 well, you end up with five times four, or twenty. -196 +198 00:10:27,100 --> 00:10:30,403 And of course, this way over counts the total number of edges in the graph, -197 +199 00:10:30,403 --> 00:10:32,360 since each edge is touching multiple regions. -198 +200 00:10:33,280 --> 00:10:36,774 But in fact, each edge is touching exactly two regions, -199 +201 00:10:36,774 --> 00:10:40,580 so this number twenty is precisely double counting the edges. -200 +202 00:10:41,380 --> 00:10:44,403 So any graph that cuts the plane into five regions, -201 +203 00:10:44,403 --> 00:10:48,880 where each region is touching four edges, would have to have ten total edges. -202 +204 00:10:50,500 --> 00:10:53,880 But our utilities puzzle has only nine edges available. -203 +205 00:10:55,100 --> 00:10:58,861 So even though we concluded that it would have to cut the plane into five regions, -204 +206 00:10:58,861 --> 00:11:00,720 it would be impossible for it to do that. -205 +207 00:11:01,280 --> 00:11:03,958 So there you go, bada boom bada bang, it is impossible to -206 +208 00:11:03,958 --> 00:11:06,960 solve this puzzle on a piece of paper without intersecting lines. -207 +209 00:11:07,540 --> 00:11:08,700 Tell me that's not a slick proof. -208 +210 00:11:10,200 --> 00:11:12,496 And before getting back to our friends and the mug, -209 +211 00:11:12,496 --> 00:11:15,940 it's worth taking a moment to pull out a general truth sitting inside of this. -210 +212 00:11:15,940 --> 00:11:20,939 Think back to the key rule where each new edge was introducing either a new -211 +213 00:11:20,939 --> 00:11:26,400 vertex by being drawn to an untouched spot, or it introduced a new enclosed region. -212 +214 00:11:27,340 --> 00:11:30,092 That same logic applies to any planar graph, not -213 +215 00:11:30,092 --> 00:11:32,620 just our specific utilities puzzle situation. -214 +216 00:11:35,620 --> 00:11:39,839 In other words, the number of vertices minus the number of edges plus -215 +217 00:11:39,839 --> 00:11:44,120 the number of regions remains unchanged, no matter what graph you draw. -216 +218 00:11:44,680 --> 00:11:47,540 Namely, it started at two, so it always stays at two. -217 +219 00:11:48,320 --> 00:11:53,000 And this relation, true for any planar graph, is called Euler's characteristic formula. -218 +220 00:11:54,100 --> 00:11:58,204 Historically, by the way, the formula came up in the context of convex polyhedra, -219 +221 00:11:58,204 --> 00:12:01,757 like a cube for example, where the number of vertices minus the number -220 +222 00:12:01,757 --> 00:12:04,360 of edges plus the number of faces always equals two. -221 +223 00:12:04,960 --> 00:12:07,066 So when you see it written down, you often see it -222 +224 00:12:07,066 --> 00:12:09,300 with an F for faces instead of talking about regions. -223 +225 00:12:10,660 --> 00:12:13,747 Now before you go thinking of me as some kind of grinch that sends friends -224 +226 00:12:13,747 --> 00:12:16,918 an impossible puzzle and then makes them film themselves trying to solve it, -225 +227 00:12:16,918 --> 00:12:19,800 keep in mind, I didn't give this puzzle to people on a piece of paper. -226 +228 00:12:21,340 --> 00:12:23,500 And I'm betting the handle has something to do with this. -227 +229 00:12:23,580 --> 00:12:23,900 Okay. -228 +230 00:12:24,060 --> 00:12:28,400 Otherwise, why would you have brought a mug over here and not a piece of paper? -229 +231 00:12:29,720 --> 00:12:32,060 This is a valid observation. -230 +232 00:12:32,760 --> 00:12:35,460 Ooh, I have one cool idea, maybe. -231 +233 00:12:36,080 --> 00:12:38,680 Use the mug handle to be... -232 +234 00:12:38,680 --> 00:12:39,740 Oh yeah, I think I see it. -233 +235 00:12:39,960 --> 00:12:39,960 Okay. -234 +236 00:12:39,960 --> 00:12:42,588 I feel like it has to do something with the handle and -235 +237 00:12:42,588 --> 00:12:45,360 because that's our ability to hop one line over the other. -236 +238 00:12:45,860 --> 00:12:50,089 I'm gonna start by, I think, taking advantage of the -237 +239 00:12:50,089 --> 00:12:54,240 handle because I think that that is the key to this. -238 +240 00:12:54,480 --> 00:12:54,880 You know what? -239 +241 00:12:55,180 --> 00:12:58,920 I think actually a sphere is the wrong thing to be thinking about. -240 +242 00:12:59,900 --> 00:13:05,200 I mean like famously, a mug is topologically the same as a donut. -241 +243 00:13:05,200 --> 00:13:10,780 So to solve this thing, you're gonna have to use the torusiness of the mug. -242 +244 00:13:10,900 --> 00:13:12,960 You're gonna have to use the handle somehow. -243 +245 00:13:13,120 --> 00:13:14,540 That's the thing that makes this a torus. -244 +246 00:13:15,700 --> 00:13:22,380 So let's take the green and go over the handle here. -245 +247 00:13:22,780 --> 00:13:22,880 Okay. -246 +248 00:13:25,360 --> 00:13:27,280 And then the red can kind of come under. -247 +249 00:13:27,900 --> 00:13:28,160 Nice. -248 +250 00:13:29,140 --> 00:13:29,920 And there we go. -249 +251 00:13:30,160 --> 00:13:30,580 There you go. -250 +252 00:13:30,780 --> 00:13:31,420 I think I did it. -251 +253 00:13:31,480 --> 00:13:31,840 All right. -252 +254 00:13:32,340 --> 00:13:32,680 Wow. -253 +255 00:13:32,680 --> 00:13:38,553 My approach is to get as far as you can with, as far as you -254 +256 00:13:38,553 --> 00:13:44,720 can as if you were on a plane and then see where you get stuck. -255 +257 00:13:45,020 --> 00:13:50,120 So look, I'm gonna draw this to here like that. -256 +258 00:13:50,680 --> 00:13:55,660 And now I've come across a problem because electricity can't be joined to this house. -257 +259 00:13:55,780 --> 00:13:57,560 This is where you have to use the handle. -258 +260 00:13:57,760 --> 00:14:00,880 So whatever you did, do it again, but go around the handle. -259 +261 00:14:00,880 --> 00:14:02,220 So I'm gonna go down here. -260 +262 00:14:03,480 --> 00:14:09,500 I'm gonna loop underneath, come back around and back to where I started. -261 +263 00:14:09,980 --> 00:14:14,200 And now I'm free to get my electricity through. -262 +264 00:14:14,360 --> 00:14:16,640 I'm gonna go over the handle like that. -263 +265 00:14:17,000 --> 00:14:17,280 There we go. -264 +266 00:14:17,420 --> 00:14:18,320 Bit messy, but there you go. -265 +267 00:14:18,920 --> 00:14:24,480 And then I'm gonna go on the inside of the handle. -266 +268 00:14:24,620 --> 00:14:28,020 Go all the way around the inside of the handle. -267 +269 00:14:28,020 --> 00:14:34,300 And finally connect to the gas company. -268 +270 00:14:34,720 --> 00:14:38,520 To solve this puzzle, just join the M and there's three more connections to go. -269 +271 00:14:39,180 --> 00:14:44,760 Let's just make them one, two, and now we have to connect those two guys, right? -270 +272 00:14:45,020 --> 00:14:45,560 Just watch it. -271 +273 00:14:46,900 --> 00:14:48,960 In through the front door, out through the back door. -272 +274 00:14:49,540 --> 00:14:49,720 Done. -273 -00:14:50,519 --> 00:14:51,400 +275 +00:14:50,520 --> 00:14:51,400 No intersections. -274 +276 00:14:52,340 --> 00:14:53,720 Maybe you think that's cheating. -275 +277 00:14:53,720 --> 00:14:56,548 Well, so it's a topological puzzle, so it means -276 +278 00:14:56,548 --> 00:14:59,260 the relative positions of things don't matter. -277 +279 00:14:59,460 --> 00:15:04,320 What that means is we can take this handle and move it here, creating another connection. -278 +280 00:15:04,880 --> 00:15:05,520 Ho, ho, ho. -279 +281 00:15:06,220 --> 00:15:07,180 Oh my god, am I done? -280 +282 00:15:08,220 --> 00:15:08,840 Is this over? -281 +283 00:15:10,240 --> 00:15:11,400 I think I might have gotten it. -282 +284 00:15:12,420 --> 00:15:13,280 24 minutes. -283 +285 00:15:14,060 --> 00:15:15,540 Grant, you said this would take 15 minutes. -284 +286 00:15:16,600 --> 00:15:17,440 There you go. -285 +287 00:15:17,700 --> 00:15:19,120 I think I've solved it. -286 +288 00:15:19,720 --> 00:15:20,620 You're in the success category. -287 +289 00:15:20,900 --> 00:15:21,800 Hard but not impossible. -288 +290 00:15:21,920 --> 00:15:22,860 Hard but not impossible. -289 +291 00:15:22,860 --> 00:15:28,360 This is maybe perhaps not the most elegant solution to this problem. -290 +292 00:15:29,060 --> 00:15:32,880 If I drew this line here, you'll think, oh no, he's blocked that house. -291 +293 00:15:32,900 --> 00:15:35,380 There's no way to get the gas in, but this is why it's on a mug, right? -292 +294 00:15:35,380 --> 00:15:39,942 Because if you take the gas line all the way up here to the top, -293 +295 00:15:39,942 --> 00:15:42,680 you then take it over and into the mug. -294 +296 00:15:42,800 --> 00:15:46,240 If you draw the line under the coffee, it wets the pen. -295 +297 00:15:46,520 --> 00:15:50,040 So when the line comes back out again, the pen's not working anymore. -296 +298 00:15:50,040 --> 00:15:52,789 You can go straight across there in and join it up, -297 +299 00:15:52,789 --> 00:15:56,280 and because it wasn't drawing, you haven't had to cross the lines. -298 +300 00:15:57,040 --> 00:15:57,260 Easy. -299 +301 00:15:57,840 --> 00:15:58,820 By the way, funny story. -300 +302 00:15:59,160 --> 00:16:03,460 So I was originally given this mug as a gift, and I didn't really know where it came from. -301 +303 00:16:03,640 --> 00:16:06,974 And it was only after I had invited people to be a part of this -302 +304 00:16:06,974 --> 00:16:09,579 that I realized the origin of the mug, MathsGear, -303 +305 00:16:09,579 --> 00:16:14,060 is a website run by three of the YouTubers I had just invited, Matt, James, and Steve. -304 +306 00:16:14,620 --> 00:16:15,100 Small world. -305 +307 00:16:15,600 --> 00:16:19,062 Given just how helpful these three guys were in the logistics of a lot of this, -306 +308 00:16:19,062 --> 00:16:22,351 really the least I could do to thank them is give a small plug for how gift -307 +309 00:16:22,351 --> 00:16:25,640 cards from MathsGear could make a pretty good last minute Christmas present. -308 +310 00:16:26,380 --> 00:16:29,424 Back to the puzzle though, this is one of those things where once you see it, -309 +311 00:16:29,424 --> 00:16:30,400 it kind of feels obvious. -310 +312 00:16:30,720 --> 00:16:32,806 The handle of the mug can basically be used as -311 +313 00:16:32,806 --> 00:16:34,760 a bridge to prevent two lines from crossing. -312 +314 00:16:35,460 --> 00:16:38,180 But this raises a really interesting mathematical question. -313 +315 00:16:38,660 --> 00:16:42,460 We just proved that this task is impossible for graphs on a plane. -314 +316 00:16:42,840 --> 00:16:46,560 So where exactly does that proof break down on the surface of a mug? -315 +317 00:16:47,380 --> 00:16:49,620 And I'm actually not going to tell you the answer here. -316 +318 00:16:49,940 --> 00:16:51,780 I want you to think about this on your own. -317 +319 00:16:52,240 --> 00:16:54,219 And I don't just mean saying, oh it's because -318 +320 00:16:54,219 --> 00:16:56,500 Euler's formula is different on surfaces with a hole. -319 +321 00:16:56,920 --> 00:16:58,200 Really, think about this. -320 +322 00:16:58,720 --> 00:17:01,190 Where specifically does the line of reasoning that -321 +323 00:17:01,190 --> 00:17:03,660 I laid out break down when you're working on a mug? -322 +324 00:17:04,300 --> 00:17:08,099 I promise you, thinking this through will give you a deeper understanding of math. -323 +325 00:17:09,240 --> 00:17:11,704 Like anyone tackling a tricky problem, you will -324 +326 00:17:11,704 --> 00:17:14,220 likely run into walls and moments of frustration. -325 +327 00:17:14,780 --> 00:17:17,963 But the smartest people I know actively seek out new challenges, -326 +328 00:17:17,963 --> 00:17:19,579 even if they're just toy puzzles. -327 +329 00:17:20,140 --> 00:17:23,401 They ask new questions, they aren't afraid to start over many times, -328 +330 00:17:23,401 --> 00:17:25,339 and they embrace every moment of failure. -329 +331 00:17:26,740 --> 00:17:30,840 So give this and other puzzles an earnest try, and never stop asking questions. -330 +332 00:17:34,060 --> 00:17:37,840 But Grant, I hear you complaining, how am I supposed to practice my problem solving -331 +333 00:17:37,840 --> 00:17:41,440 if I don't have someone shipping me puzzles on topologically interesting shapes? -332 +334 00:17:41,440 --> 00:17:44,471 Well, let's close things off by going through a couple puzzles -333 +335 00:17:44,471 --> 00:17:47,840 created by this week's mathematically oriented sponsor, brilliant.org. -334 +336 00:17:48,480 --> 00:17:50,760 So here I'm in their intro to problem solving course and -335 +337 00:17:50,760 --> 00:17:53,080 going through a particular sequence called flipping pairs. -336 +338 00:17:53,620 --> 00:17:56,988 And the rules here seem to be that we can flip adjacent pairs of coins, -337 +339 00:17:56,988 --> 00:17:58,720 but we can't flip them one at a time. -338 +340 00:17:59,080 --> 00:18:03,760 And we are asked, is it possible to get it so that all three coins are gold side up? -339 +341 00:18:04,320 --> 00:18:06,440 Well, clearly I just did it, so yes. -340 +342 00:18:07,400 --> 00:18:10,149 And the next question, we start with a different configuration, -341 +343 00:18:10,149 --> 00:18:12,556 have the same rules, and we're asked the same question, -342 +344 00:18:12,556 --> 00:18:15,220 can we get it so that all three of the coins are gold side up? -343 +345 00:18:15,920 --> 00:18:19,193 And, you know, there's not really that many degrees of freedom we have here, -344 +346 00:18:19,193 --> 00:18:22,934 just two different spots to click, so you might quickly come to the conclusion that no, -345 +347 00:18:22,934 --> 00:18:23,360 you can't. -346 +348 00:18:23,360 --> 00:18:26,420 Even if you don't necessarily know the theoretical reason yet, that's totally fine. -347 +349 00:18:26,820 --> 00:18:28,600 So no, and we kind of move along. -348 +350 00:18:29,460 --> 00:18:34,182 So next, it's kind of showing us every possible starting configuration that there is, -349 +351 00:18:34,182 --> 00:18:39,015 and asking for how many of them can we get it to a point where all three gold coins are -350 +352 00:18:39,015 --> 00:18:39,180 up? -351 +353 00:18:40,040 --> 00:18:41,842 Obviously, I'm kind of giving away the answer, -352 +354 00:18:41,842 --> 00:18:44,720 it's sitting here four on the right, because I've gone through this before. -353 +355 00:18:45,200 --> 00:18:47,741 But if you want to go through it yourself, this particular -354 +356 00:18:47,741 --> 00:18:50,240 quiz has a really nice resolution, and a lot of others in -355 +357 00:18:50,240 --> 00:18:53,040 this course do build up genuinely good problem solving instincts. -356 +358 00:18:53,440 --> 00:18:58,141 So you can go to brilliant.org slash 3b1b to let them know that you came from here, -357 +359 00:18:58,141 --> 00:19:01,500 or even slash 3b1b flipping to jump straight into this quiz. -358 +360 00:19:02,060 --> 00:19:05,101 And you can make an account for free, a lot of what they offer is free, -359 +361 00:19:05,101 --> 00:19:07,214 but they also have a annual subscription service, -360 +362 00:19:07,214 --> 00:19:09,960 if you want to get the full suite of experiences that they offer. -361 +363 00:19:10,400 --> 00:19:11,520 And I just think they're really good. -362 +364 00:19:11,660 --> 00:19:13,957 I know a couple of the people there, and they're incredibly -363 +365 00:19:13,957 --> 00:19:16,140 thoughtful about how they put together math explanations. -364 +366 00:19:17,260 --> 00:19:20,800 Water goes to one, and then wraps around to the other. -365 -00:19:22,219 --> 00:19:23,879 +367 +00:19:22,220 --> 00:19:23,880 And naively at this point... -366 +368 00:19:24,740 --> 00:19:26,040 Oh, wait, I've already messed up. -367 +369 00:19:27,360 --> 00:19:31,340 Then from there, water can make its way to cottage number three. -368 +370 00:19:31,740 --> 00:19:32,740 Ah, I'm trapped! -369 +371 00:19:33,740 --> 00:19:34,580 I've done this wrong again. diff --git a/2017/three-utilities/english/sentence_timings.json b/2017/three-utilities/english/sentence_timings.json index 41d452d25..dc9680dca 100644 --- a/2017/three-utilities/english/sentence_timings.json +++ b/2017/three-utilities/english/sentence_timings.json @@ -245,18 +245,18 @@ 135.22 ], [ - "Oh, that should go to the second house.", + "Oh, that should go to the, that should go to the second house.", 135.48, 138.34 ], [ "Okay, I'll put it to the second house.", 138.62, - 140.0 + 138.34 ], [ - "Presumably this is easy enough.", - 141.92, + "Okay, I'll put it to the second house. This is easy enough.", + 138.62, 143.64 ], [ @@ -685,7 +685,7 @@ 473.42 ], [ - "But if you connect to an already lit up vertex, notice how this closes off a new region.", + "But if you connect to an already lit up vertex, notice how this closes off a new region. And this is a very simple way to do it. You can see how this closes off a new region.", 478.08, 483.78 ], @@ -730,7 +730,7 @@ 544.08 ], [ - "So a hypothetical solution would cut the plane into five separate regions.", + "So a hypothetical solution that has turned up thehew times one node through a difficult Bye� ham die So, a hypothetical solution would cut the plane into five separate regions.", 553.18, 558.02 ], diff --git a/2017/three-utilities/english/transcript.txt b/2017/three-utilities/english/transcript.txt index 477f34fb6..da1d8f7a6 100644 --- a/2017/three-utilities/english/transcript.txt +++ b/2017/three-utilities/english/transcript.txt @@ -47,9 +47,9 @@ I can do another one and now we're up to five. We have four to go. I'm looking at my display over here. I should have put it over there, but oh well. -Oh, that should go to the second house. +Oh, that should go to the, that should go to the second house. Okay, I'll put it to the second house. -Presumably this is easy enough. +Okay, I'll put it to the second house. This is easy enough. So we just need to get from here to there. I have one, two, three, four, five, six, seven lines, two to go. So I have that one connected to that one. @@ -135,7 +135,7 @@ Here, think of it like this. Start by imagining one of your nodes as lit up while the other five are dim. And then every time you draw an edge from a lit up vertex to a dim vertex, light up the new one. So at first, each new edge lights up one more vertex. -But if you connect to an already lit up vertex, notice how this closes off a new region. +But if you connect to an already lit up vertex, notice how this closes off a new region. And this is a very simple way to do it. You can see how this closes off a new region. And this gives us a super useful fact. Each new edge either increases the number of lit up nodes by one, or it increases the number of enclosed regions by one. This fact is something that we can use to figure out the number of regions that a hypothetical solution to this would cut the plane into. @@ -144,7 +144,7 @@ When you start off, there's one node lit up and one region, all of 2D space. By the end, we're going to need to draw nine lines, since each of the three utilities gets connected to each of the three houses. Five of those lines are going to light up the initially dim vertices. So the other four lines each must introduce a new region. -So a hypothetical solution would cut the plane into five separate regions. +So a hypothetical solution that has turned up thehew times one node through a difficult Bye� ham die So, a hypothetical solution would cut the plane into five separate regions. And you might say, okay, that's a cute fact, but why should that make things impossible? What's wrong with having five regions? Well again, take a look at this partially complete graph. diff --git a/2017/three-utilities/french/sentence_translations.json b/2017/three-utilities/french/sentence_translations.json index 68793668d..04f845bd0 100644 --- a/2017/three-utilities/french/sentence_translations.json +++ b/2017/three-utilities/french/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "D'accord, je vais le mettre dans la deuxième maison. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "Si vous parcourez chacune de ces régions et additionnez le nombre d’arêtes qu’elle possède, vous obtenez 5 fois 4, soit 20. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/german/sentence_translations.json b/2017/three-utilities/german/sentence_translations.json index d1404f164..c1b207add 100644 --- a/2017/three-utilities/german/sentence_translations.json +++ b/2017/three-utilities/german/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "Okay, ich stecke es in das zweite Haus. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "Wenn Sie jeden dieser Bereiche durchgehen und die Anzahl seiner Kanten addieren, erhalten Sie am Ende 5 mal 4, also 20. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/hebrew/sentence_translations.json b/2017/three-utilities/hebrew/sentence_translations.json index 30accad49..b402257ef 100644 --- a/2017/three-utilities/hebrew/sentence_translations.json +++ b/2017/three-utilities/hebrew/sentence_translations.json @@ -385,7 +385,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house.", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house.", "translatedText": "בסדר, אני אשים את זה לבית השני.", "n_reviews": 0, "start": 138.62, @@ -1162,7 +1162,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20.", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty.", "translatedText": "אם עברת על כל אחד מהאזורים האלה וסכמת את מספר הקצוות שיש לו, ובכן, אתה מסיים עם 5 כפול 4, או 20.", "n_reviews": 0, "start": 614.88, @@ -1470,7 +1470,7 @@ "end": 817.54 }, { - "input": ".", + "input": "as far as you can as if you were on a plane and then see where you get stuck.", "translatedText": ".", "n_reviews": 0, "start": 817.64, diff --git a/2017/three-utilities/hindi/sentence_translations.json b/2017/three-utilities/hindi/sentence_translations.json index b7cb1c9f9..a2b3a3c45 100644 --- a/2017/three-utilities/hindi/sentence_translations.json +++ b/2017/three-utilities/hindi/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "ठीक है, मैं इसे दूसरे घर में रख दूँगा।", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "यदि आप इनमें से प्रत्येक क्षेत्र से गुजरे और उसके किनारों की संख्या जोड़ दें, तो अंत में आपको 5 गुना 4, या 20 प्राप्त होता है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/hungarian/sentence_translations.json b/2017/three-utilities/hungarian/sentence_translations.json index 98581d6a5..1af29127d 100644 --- a/2017/three-utilities/hungarian/sentence_translations.json +++ b/2017/three-utilities/hungarian/sentence_translations.json @@ -385,7 +385,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house.", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house.", "translatedText": "Oké, felteszem a második házba.", "n_reviews": 0, "start": 138.62, @@ -1162,7 +1162,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20.", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty.", "translatedText": "Ha végigment ezeken a területeken, és összeadja az élek számát, akkor 5-ször 4-et vagy 20-at kap.", "n_reviews": 0, "start": 614.88, @@ -1470,7 +1470,7 @@ "end": 817.54 }, { - "input": ".", + "input": "as far as you can as if you were on a plane and then see where you get stuck.", "translatedText": ".", "n_reviews": 0, "start": 817.64, diff --git a/2017/three-utilities/indonesian/sentence_translations.json b/2017/three-utilities/indonesian/sentence_translations.json index 81a3edfdc..2494e73be 100644 --- a/2017/three-utilities/indonesian/sentence_translations.json +++ b/2017/three-utilities/indonesian/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "Oke, saya akan taruh di rumah kedua. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "Jika Anda menelusuri masing-masing wilayah ini dan menjumlahkan jumlah sisi yang dimilikinya, Anda akan mendapatkan 5 kali 4, atau 20. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/italian/sentence_translations.json b/2017/three-utilities/italian/sentence_translations.json index 2487a0d96..43bab64f9 100644 --- a/2017/three-utilities/italian/sentence_translations.json +++ b/2017/three-utilities/italian/sentence_translations.json @@ -385,7 +385,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house.", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house.", "translatedText": "Ok, lo metto nella seconda casa.", "n_reviews": 0, "start": 138.62, @@ -1162,7 +1162,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20.", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty.", "translatedText": "Se attraversi ciascuna di queste regioni e sommi il numero di bordi che ha, beh, ti ritroverai con 5 per 4, ovvero 20.", "n_reviews": 0, "start": 614.88, @@ -1470,7 +1470,7 @@ "end": 817.54 }, { - "input": ".", + "input": "as far as you can as if you were on a plane and then see where you get stuck.", "translatedText": ".", "n_reviews": 0, "start": 817.64, diff --git a/2017/three-utilities/japanese/sentence_translations.json b/2017/three-utilities/japanese/sentence_translations.json index 5c7dabaa1..ca1439865 100644 --- a/2017/three-utilities/japanese/sentence_translations.json +++ b/2017/three-utilities/japanese/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "わかりました、2軒目に入れます。", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "これらの各領域を調べて、その領域にあるエッジの数を合計する と、最終的には 5 掛ける 4、つまり 20 になります。", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/korean/sentence_translations.json b/2017/three-utilities/korean/sentence_translations.json index c108f263f..560ac55f2 100644 --- a/2017/three-utilities/korean/sentence_translations.json +++ b/2017/three-utilities/korean/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "알았어, 두 번째 집에 놓을게. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "각 영역을 살펴보고 모서리 수를 더하면 5 곱하기 4 또는 20이 됩니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/marathi/sentence_translations.json b/2017/three-utilities/marathi/sentence_translations.json index 54336de70..e8d5e254e 100644 --- a/2017/three-utilities/marathi/sentence_translations.json +++ b/2017/three-utilities/marathi/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "ठीक आहे, मी ते दुसऱ्या घरात ठेवतो. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "जर तुम्ही यापैकी प्रत्येक प्रदेशातून गेलात आणि त्यात असलेल्या कडांची संख्या जोडली, तर तुम्ही 5 गुणिले 4 किंवा 20 असाल. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/persian/sentence_translations.json b/2017/three-utilities/persian/sentence_translations.json index 69f9768cf..464e597b0 100644 --- a/2017/three-utilities/persian/sentence_translations.json +++ b/2017/three-utilities/persian/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "باشه، میذارمش خونه دوم. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "اگر از هر یک از این مناطق عبور کنید و تعداد یال های آن را جمع کنید، در نهایت به 5 ضربدر 4 یا 20 خواهید رسید. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/portuguese/sentence_translations.json b/2017/three-utilities/portuguese/sentence_translations.json index bb227cdf9..af57c145f 100644 --- a/2017/three-utilities/portuguese/sentence_translations.json +++ b/2017/three-utilities/portuguese/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "Ok, vou colocar na segunda casa. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "Se você passar por cada uma dessas regiões e somar o número de arestas que ela possui, você acabará com 5 vezes 4, ou 20. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/russian/sentence_translations.json b/2017/three-utilities/russian/sentence_translations.json index 70f08318d..019c3e9a1 100644 --- a/2017/three-utilities/russian/sentence_translations.json +++ b/2017/three-utilities/russian/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "Хорошо, я отнесу это ко второму дому. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "Если вы прошли через каждую из этих областей и сложили количество ребер, которые она имеет, в итоге вы получите 5 умножить на 4 или 20. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/spanish/sentence_translations.json b/2017/three-utilities/spanish/sentence_translations.json index 1906d588d..c05c2284d 100644 --- a/2017/three-utilities/spanish/sentence_translations.json +++ b/2017/three-utilities/spanish/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "Bien, lo pondré en la segunda casa. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "Si pasaste por cada una de estas regiones y sumas el número de aristas que tiene, terminas con 5 por 4, o 20. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/tamil/sentence_translations.json b/2017/three-utilities/tamil/sentence_translations.json index 48e5c6830..dc3dfd39c 100644 --- a/2017/three-utilities/tamil/sentence_translations.json +++ b/2017/three-utilities/tamil/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "சரி, நான் அதை இரண்டாவது வீட்டிற்கு வைக்கிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "நீங்கள் இந்தப் பகுதிகள் ஒவ்வொன்றிலும் சென்று அதன் விளிம்புகளின் எண்ணிக்கையைக் கூட்டினால், நீங்கள் 5 பெருக்கல் 4 அல்லது 20 இல் முடிவடையும். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/telugu/sentence_translations.json b/2017/three-utilities/telugu/sentence_translations.json index 8b3bab7aa..bc82a41ad 100644 --- a/2017/three-utilities/telugu/sentence_translations.json +++ b/2017/three-utilities/telugu/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "సరే, రెండో ఇంటికి పెడతాను. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "మీరు ఈ ప్రాంతాలలో ప్రతి దాని గుండా వెళ్లి, దాని అంచుల సంఖ్యను జోడిస్తే, మీరు 5 సార్లు 4 లేదా 20తో ముగుస్తుంది. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/thai/sentence_translations.json b/2017/three-utilities/thai/sentence_translations.json index 4ad150d50..8f60da234 100644 --- a/2017/three-utilities/thai/sentence_translations.json +++ b/2017/three-utilities/thai/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/turkish/sentence_translations.json b/2017/three-utilities/turkish/sentence_translations.json index 65b0c8307..618d01d9c 100644 --- a/2017/three-utilities/turkish/sentence_translations.json +++ b/2017/three-utilities/turkish/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "Tamam, onu ikinci eve koyacağım. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "Bu bölgelerin her birinden geçip sahip olduğu kenar sayısını toplarsanız, 5 çarpı 4 veya 20 elde edersiniz. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/ukrainian/sentence_translations.json b/2017/three-utilities/ukrainian/sentence_translations.json index 99e629301..0118ae11d 100644 --- a/2017/three-utilities/ukrainian/sentence_translations.json +++ b/2017/three-utilities/ukrainian/sentence_translations.json @@ -385,7 +385,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house.", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house.", "translatedText": "Гаразд, поставлю до другого будинку.", "n_reviews": 0, "start": 138.62, @@ -1162,7 +1162,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20.", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty.", "translatedText": "Якщо ви пройшли через кожну з цих областей і склали кількість ребер, які вона має, то ви отримаєте 5 помножити на 4 або 20.", "n_reviews": 0, "start": 614.88, @@ -1470,7 +1470,7 @@ "end": 817.54 }, { - "input": ".", + "input": "as far as you can as if you were on a plane and then see where you get stuck.", "translatedText": ".", "n_reviews": 0, "start": 817.64, diff --git a/2017/three-utilities/urdu/sentence_translations.json b/2017/three-utilities/urdu/sentence_translations.json index 5099ccc41..aa2b7c3ba 100644 --- a/2017/three-utilities/urdu/sentence_translations.json +++ b/2017/three-utilities/urdu/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "ٹھیک ہے، میں اسے دوسرے گھر میں رکھ دوں گا۔", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "اگر آپ ان علاقوں میں سے ہر ایک سے گزرتے ہیں اور اس کے کناروں کی تعداد کو جوڑ دیتے ہیں، تو آپ کا اختتام 5 گنا 4، یا 20 ہوتا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2017/three-utilities/vietnamese/sentence_translations.json b/2017/three-utilities/vietnamese/sentence_translations.json index 2b3d8c974..1c41836ed 100644 --- a/2017/three-utilities/vietnamese/sentence_translations.json +++ b/2017/three-utilities/vietnamese/sentence_translations.json @@ -424,7 +424,7 @@ "end": 138.34 }, { - "input": "Okay, I'll put it to the second house. ", + "input": "Okay, I'll put it to the second house. Okay, I'll put it to the second house. ", "translatedText": "Được rồi, tôi sẽ đặt nó ở ngôi nhà thứ hai. ", "model": "google_nmt", "n_reviews": 0, @@ -1312,7 +1312,7 @@ "end": 614.44 }, { - "input": "If you went through each of these regions and add up the number of edges that it has, well you end up with 5 times 4, or 20. ", + "input": "If you went through each of these regions and add up the number of edges that it has, well, you end up with five times four, or twenty. ", "translatedText": "Nếu bạn đi qua từng vùng này và cộng số cạnh mà nó có, bạn sẽ có 5 nhân 4 hoặc 20. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/arabic/sentence_translations.json b/2018/basel-problem/arabic/sentence_translations.json index 1f0d28a1a..cb6e0612d 100644 --- a/2018/basel-problem/arabic/sentence_translations.json +++ b/2018/basel-problem/arabic/sentence_translations.json @@ -69,7 +69,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "في الهندسة الكروية، تتحدث أحيانًا عن الزاوية الصلبة لشكل ما، وهي نسبة الكرة التي يغطيها عند النظر إليها من نقطة معينة، ترى أول مكانين ستكون هذه القصة التي نفكر فيها بشأن الشاشات مفيدة فهم قانون التربيع العكسي وهو ظاهرة ثلاثية الأبعاد بشكل واضح، فكر في كل أشعة الضوء التي تضرب الشاشة على بعد وحدة واحدة من المصدر حيث تقوم بمضاعفة المسافة التي ستغطيها هذه الأشعة الآن مساحة ذات ضعف العرض وضعف الارتفاع لذلك سيستغرق الأمر أربع نسخ من تلك الشاشة الأصلية لتلقي نفس الأشعة على تلك المسافة، وبالتالي يتلقى كل فرد ربع كمية الضوء، وهذا هو المعنى الذي أعني به أن الضوء سيظهر بمقدار ربع سطوع ضعف المسافة البعيدة. وبالمثل، عندما تكون على بعد ثلاث مرات، ستحتاج إلى تسع نسخ من تلك الشاشة الأصلية لتلقي نفس الأشعة، بحيث تتلقى كل شاشة فردية فقط 1/9 من كمية الضوء ويستمر هذا النمط لأن المساحة التي يضربها الضوء تزيد بمقدار مربع المسافة، يتناقص سطوع ذلك الضوء بمقدار المربع العكسي لتلك المسافة، وأنا متأكد من أن الكثير منكم يعرف أن قانون التربيع العكسي هذا ليس خاصًا بالضوء على الإطلاق، فهو يظهر عندما يكون لديك نوع من الكمية التي تنتشر بالتساوي من مصدر نقطي سواء كان صوتًا أو حرارة أو إشارات راديوية وأشياء كهذه ومجموعة لا نهائية من المنارات المتباعدة بشكل متساوٍ تنفذ فعليًا مشكلة بازل ولكن مرة أخرى ما نحتاجه إذا أردنا تحقيق أي تقدم هنا هو فهم كيف يمكننا التعامل مع الإعدادات مع مصادر الضوء مثل هذه دون تغيير السطوع الإجمالي للمراقب ووحدة البناء الرئيسية هي طريقة لطيفة بشكل خاص لتحويل منارة واحدة إلى اثنتين فكر في مراقب في أصل المستوى XY ومنارة واحدة تجلس في مكان ما على ذلك المستوى الآن ارسم خطًا من تلك المنارة إلى الراصد ثم خطًا آخر متعامدًا مع الخط الموجود في المنارة. الآن ضع منارتين حيث يتقاطع هذا الخط الجديد مع محاور الإحداثيات التي سأمضي قدمًا وأسميها المنارة هنا على اليسار و المنارة B على الجانب العلوي اتضح وسترى سبب صحة ذلك في دقيقة واحدة فقط، السطوع الذي يختبره الراصد من تلك المنارة الأولى يساوي السطوع المشترك الذي شهدته المنارتان A وB معًا ويجب أن أقول الطريقة التي يتم بها الافتراض خلال هذا الفيديو هي أن جميع المنارات متكافئة، فهم يستخدمون نفس المصباح الكهربائي الذي يصدر نفس القوة، كل ذلك بعبارة أخرى، تعيين متغيرات للأشياء هنا إذا أطلقنا على المسافة من الراصد إلى المنارة القليل أ والمسافة من المراقب إلى المنارة ب القليل ب والمسافة إلى المنارة الأولى ح لدينا العلاقة 1 على a تربيع زائد 1 على b تربيع يساوي 1 على h تربيع هذه هي نظرية فيثاغورس العكسية الأقل شهرة والتي قد يتعرف عليها البعض منكم من أحدث فيديو لعلماء الرياضيات وسأقول إنه الفيديو الأكثر ممتازة عن العديد من أبناء عمومة نظرية فيثاغورس، ألا تعتقدون أنها علاقة رائعة جدًا، وإذا كنت عالم رياضيات في القلب، فقد تتساءل الآن كيف تثبت ذلك؟ هناك بعض الطرق المباشرة التي يمكنك من خلالها التعبير عن مساحة المثلثات بطريقتين منفصلتين وتطبيق نظرية فيثاغورس المعتادة ولكن هناك طريقة أخرى جميلة جدًا أود أن أوضحها بإيجاز هنا والتي تقع بشكل أفضل في قصتنا لأنه يستخدم حدس الضوء والشاشات مرة أخرى، تخيل تصغير المثلث القائم بالكامل إلى نسخة أصغر وفكر في هذا الوتر المصغر كشاشة تتلقى الضوء من المنارة الأولى إذا قمت بإعادة تشكيل تلك الشاشة لتكون مزيجًا من ساقي المنارة مثلث مصغر مثل هذا حسنًا، لا يزال يتلقى نفس كمية الضوء، أليس كذلك؟", "model": "google_nmt", "from_community_srt": "ما هي الزاوية التي يغطيها الضوء في كلا الاتجاهين المتعامدين مع المصدر في شكل كروي الهندسة ، تتحدث في بعض الأحيان ، حول الزاوية الصلبة للشكل وهي نسبة الكرة التي تغطيها عند مشاهدتها من نقطة معينة شاهد أول مكانين في هذه القصة ، نفكر في أن الشاشات ستكون مفيدة في فهم قانون المربع المعكوس وهي ظاهرة ثلاثية الأبعاد تفكر بوضوح في جميع أشعة الضوء التي تضرب شاشة وحدة واحدة ، بعيدا من المصدر وأنت تضاعف المسافة ستغطي هذه الأشعة الآن منطقة بها ضعف العرض ومرتين الارتفاع لذلك سيستغرق الأمر 4 نسخ من تلك الشاشة الأصلية لتلقي نفس الزيادة على تلك المسافة وهكذا كل منها الفرد يحصل على ربع كمية الضوء هذا ، هو المعنى الذي أعني فيه الضوء ، سيظهر 1/4 كمشرق مرتين على مسافة بعيدة بالمثل عندما تكون أبعد ثلاث مرات عنك؟ أنت ، ستحتاج إلى تسعة نسخ من تلك الشاشة الأصلية لتلقي نفس الأشعة بحيث تتلقى كل شاشة فردية 1/9 فقط قدر الإضاءة والضوء يستمر هذا النمط نظرًا لأن المنطقة ضربت ، بواسطة زيادات خفيفة من خلال مربع المسافة ينقص سطوع ذلك الضوء، بواسطة المربع العكسي لتلك المسافة و كما أنا متأكد من أن العديد منكم يعرف ، هذا قانون مربع معكوس ليس على الإطلاق خاصة على ضوء ذلك للملوثات العضوية الثابتة كلما كان لديك نوع من الكمية التي تنتشر بالتساوي من مصدر نقطة سواء كان ذلك الصوت أو الحرارة أو إشارات الراديو مثل هذه الأشياء. وتذكر أنه بسبب قانون المربع المعكوس هذا ، مجموعة لا حصر لها من المنارات المتساوية التباعد تنفذ ماديًا مشكلة بازل. ولكن مرة أخرى ، ما نحتاجه ، إذا كنا سنحرز أي تقدم هنا ، هو أن نفهم كيف يمكننا التلاعب بالأجهزة مصادر الضوء مثل هذا دون تغيير السطوع الكلي للمراقب. والبناء الأساسي هو طريقة لطيفة خاصة لتحويل منارة واحدة إلى فكر في مراقب من أصل الطائرة xy ومنارة واحدة تجلس في مكان ما على تلك الطائرة الآن رسم خط من تلك المنارة إلى المراقب ثم خط آخر ، عمودي على ذلك في المنارة الآن وضع منارتين حيث يتقاطع هذا الخط الجديد مع محاور الإحداثيات التي سأذهب إلى الأمام ومنارة من فوق هنا على اليسار والمنارة ب على الجانب العلوي اتضح وسترى لماذا هذا صحيح في دقيقة واحدة فقط السطوع الذي يختبره المراقب من تلك المنارة الأولى يساوي السطوع مجتمعة من ذوي الخبرة من المنارات أ و ب معا ويجب أن أقول بالمناسبة أن الافتراض القائم في كل مكان هذا الفيديو ، هو أن جميع المنارات تعادل هم باستخدام نفس المصباح الكهربائي ينبثق نفس القوة كل ذلك وبعبارة أخرى تعيين المتغيرات للأشياء هنا إذا كنا نسمي المسافة من المراقب إلى المنارة أ قليل و المسافة من المراقب إلى المنارة ب الصغير و المسافة إلى المنارة الأولى لدينا علاقة 1 على مربع تربيعي زائد 1 فوق b يساوي مربعاً 1 على h مربعة هذا ، هو أقل من ذلك بكثير ، معروف جيدا معكوس نظرية فيثاغورس ، التي البعض منكم؟ ربما ، التعرف على الأساطير من أحدث وأغلى الفيديوهات على العديد من أبناء عمومة نظرية فيثاغورس رائع جدا ، لا تظن وعلاقة إذا كنت ، عالم رياضيات في القلب قد تسأل الآن. ماذا أنت ، تثبت ذلك ، وهناك بعض الطرق المباشرة ، حيث يمكنك التعبير عن منطقة المثلثات بطريقتين منفصلتين؟ وتطبيق نظرية فيثاغورس المعتادة ولكن هناك طريقة أخرى جميلة للغاية أود أن أصفها بإيجاز هنا والتي تقع بشكل أكثر جاذبية في قصة لدينا مرة أخرى يستخدم الحدس من الضوء والشاشات تخيل تدرج المثلث الأيمن بأكمله في نسخة أصغر وفكر في هذه الوتر المصغرة كشاشة تلقي الضوء من المنارة الأولى إذا أنت ، إعادة تشكيل تلك الشاشة ، لتكون مزيجا من ساقي المثلث المصغر ، مثل هذا حسناً ، إنه لا يزال يتلقى نفس المقدار من الضوء ، أعني أشعة الضوء التي تضرب واحدة من تلك الساقين", @@ -132,7 +132,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "هذا يعني أن السطوع الظاهري سينخفض بمعامل أربعة وهذا أيضًا جبر مباشر نسبيًا ينتقل من مجموع كل الأعداد الصحيحة إلى مجموع الأعداد الصحيحة الزوجية ويتضمن الضرب في 1 على 4 وما يعنيه ذلك هو أن الانتقال من كل الأعداد الصحيحة سيتم ضرب الأعداد الصحيحة إلى الأعداد الفردية في 3 على 4 نظرًا لأن الأعداد الزوجية بالإضافة إلى الاحتمالات يجب أن تعطينا الأمر برمته، لذلك إذا قلبنا ذلك فهذا يعني أن الانتقال من المجموع على الأعداد الفردية إلى المجموع على كل الأعداد الصحيحة الموجبة يتطلب الضرب بمقدار 4 أثلاث إذن، بأخذ باي تربيع على 8 مضروبًا في 4 أثلاث بادا بوم بادا بنج، حصلنا على حل لمشكلة الريحان. الآن هذا الفيديو الذي شاهدته للتو تمت كتابته وتحريكه بشكل أساسي بواسطة واحد من الثلاثة الجدد باللون الأزرق والبني أعضاء الفريق بن هامبريشت إضافة ممكنة.", "model": "google_nmt", "from_community_srt": "لأن ذلك ينطوي على مضاعفة المسافة لكل منارة ، يعني أنه سيتم تقليل السطوع الظاهر بعامل 4 و؟ هذا أيضا الجبر بسيط نسبيا الذهاب من المجموع على كل الأعداد الصحيحة إلى المجموع على الأعداد الصحيحة الزوجية ينطوي على ضرب بواسطة 1/4. وماذا يعني ذلك؟ هل هذا الانتقال من كل الأعداد الصحيحة إلى الأعداد الفردية؟ سوف يتضاعف 3/4 منذ أن يتقدم بالإضافة إلى الصعاب أن يعطينا كل شيء لذلك إذا كنا نقلب ذلك حول ذلك يعني الذهاب ، من المجموع على الأرقام الفردية إلى المجموع على جميع الأعداد الصحيحة الموجبة يتطلب ضرب 4/3 حتى أخذ ذلك pi squared فوق 8. Multiplying ، بواسطة 4/3 لكن a ، boom ، bada ، bing لدينا أنفسنا حلاً لمشكلة basel هذا الفيديو الذي شاهدته للتو تم كتابته وتحفيزه بشكل أساسي من خلال واحدة من الجديد", diff --git a/2018/basel-problem/bangla/sentence_translations.json b/2018/basel-problem/bangla/sentence_translations.json index 3800a7b69..0b8f6644d 100644 --- a/2018/basel-problem/bangla/sentence_translations.json +++ b/2018/basel-problem/bangla/sentence_translations.json @@ -63,7 +63,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "", "from_community_srt": "ত্রিমাত্রিক জ্যামিতিতে আমরা অনেকসময় কোনও আকৃতির \"ঘনকোণ\"(solid angle)-এর কথা বলি। মানে, উৎসের চারদিকে একটা গোলকের তলের কতটা অংশ এটা ঢাকছে। আমাদের গোটা গল্পের মধ্যে স্ক্রিন দুভাবে কাজে লাগবে-- তার মধ্যে একটা হল ব্যস্তানুপাতিক বর্গ সূত্রকে (inverse square law) বোঝা। যা একান্তই ত্রিমাত্রিক একটা ঘটনা। যদি উৎস থেকে এক মিটার দূরে একটা স্ক্রিনে পৌঁছোনো রশ্মিগুলোকে ধরি, সেই রশ্মিগুলো দুই মিটার দূরত্বে ঐ স্ক্রিনের দ্বিগুণ চওড়া আর দ্বিগুণ উচ্চতার একটা ক্ষেত্রফল অধিকার করবে। মানে, আগেরটার মতো চারটে স্ক্রিন লাগবে ঐ পরিমাণ আলো পেতে, আর প্রতিটা স্ক্রিন নিজে ঐ আলোর এক-চতুর্থাংশ পাবে। এইভাবে ভাবলে বলা যায়, দ্বিগুণ দূরত্বে একটা আলোর ঔজ্জ্বল্য চারভাগের একভাগ হয়ে যাবে। একইভাবে, তিনগুণ দূরত্বে...? তখন চারটের জায়গায় ন'টা স্ক্রিন লাগবে ঐ পরিমাণ আলো পেতে, মানে প্রতিটা স্ক্রিন মোট আলোর 1/9 অংশ পাবে। এই প্যাটার্নটা চলবে, অর্থাৎ আলোর অধিকৃত ক্ষেত্রফলটা দূরত্বের বর্গের সমানুপাতিক হারে বাড়বে, আর ফলে আলোর ঔজ্জ্বল্যটা সেই হারে, মানে দূরত্বের বর্গের ব্যস্তানুপাতে কমবে। তবে হ্যাঁ, আশা করি অনেকেই জানো, এই ব্যস্তানুপাতিক বর্গ সূত্র... আলোকবিজ্ঞানের একচেটিয়া নয়। যখনই কোনো বিন্দু উৎস থেকে কোনো কিছু, যেমন শব্দ, তাপ বা রেডিও তরঙ্গের মত জিনিস, সুষমভাবে সবদিকে ছড়িয়ে পড়ে, তখনই এই সূত্রের আবির্ভাব হয়। আর উল্লেখ্য এই ব্যস্তানুপাতিক বর্গ সূত্রের জন্যেই, বাতিঘরের এই অসীম সজ্জাটা ব্যাসেল প্রবলেমের প্রাকৃতিক একটা রূপায়ণ। কিন্তু তাও, এর সমাধানের দিকে এগোতে চাইলে এটা বুঝতে হবে, যে এই ধরনের আলোকসজ্জাকে ... কীভাবে পাল্টানো যায়, যাতে আবার সেই দর্শকের চোখে মোট ঔজ্জ্বল্যটা অপরিবর্তিত থাকে৷ আর সেইটার গোড়ার কথা হল, একটা বাতিঘরকে কীভাবে দুটোতে ভাঙা যায়। ধরো, একজন দর্শক x-y দ্বিমাত্রিক তলের মূল বিন্দুতে দাঁড়িয়ে, আর একটাই বাতিঘর ঐ তলে কোথাও রয়েছে। এইবার বাতিঘর আর দর্শকের মধ্যে একটা রেখাংশ টানো, আর বাতিঘরে ঐ রেখাংশের ওপর লম্ব আঁকো। এখন ঐ লম্বটা, x আর y অক্ষকে যে দুই বিন্দুতে ছেদ করছে সেগুলোতে দুটো বাতিঘর বসাও। ডানদিকের-টাকে বলছি A,বাঁদিকের-টাকে B. যেটা দেখা যায়, (কেন হয় এক্ষুনি বুঝতে পারবে), দর্শক প্রথম বাতিঘরটার যতটা ঔজ্জ্বল্য দেখবে সেটা, A আর B বাতিঘরদুটোর যে আপাত ঔজ্জ্বল্য সে দেখবে তাদের সমষ্টির সমান। ও হ্যাঁ, গোটা ভিডিওটায় সবকটা বাতিঘরকে আমরা সমতুল্য ধরে নিচ্ছি। যেন ওরা সবাই একই বাল্ব ব্যবহার করছে, একই পরিমাণ শক্তি বিকিরণ করছে, ইত্যাদি। অন্যভাবে বললে, যদি বিভিন্ন দূরত্বের নাম দিই এরকম যে, দর্শকের থেকে... বাতিঘর A-র দূরত্ব a, বাতিঘর B-এর দূরত্ব b, আর h দূরত্বে অবস্থিত প্রথম বাতিঘরটা; তখন এই সম্পর্কটা পাবো: (1/a^2+ 1/b^2= 1/h^2). এটা তুলনায় বেশ অখ্যাত, পিথাগোরাসের বিপরীত সূত্র, যেটা, তোমাদের কেউ কেউ হয়তো... ম্যাথোলজার-এর সাম্প্রতিক অসাধারণ একটা ভিডিওতে দেখে থাকবে, যেটা পিথাগোরাসের উপপাদ্যের বেশকিছু রূপান্তর সম্পর্কে। বেশ মজার সম্পর্ক, না? আর যদি তুমি মনেপ্রাণে গণিতবিদ হয়ে থাকো, তবে তুমি নির্ঘাত ভাবছো... এটাকে প্রমাণ করে কীভাবে? হ্যাঁ, একটা সহজ-সরল পদ্ধতি আছে বটে, ত্রিভুজের ক্ষেত্রফলটাকে দুটো আলাদা আকারে প্রকাশ করে... সাধারণ পিথাগোরাসের উপপাদ্য প্রয়োগ করলেই চলে। তবে আরেকটা সুন্দর পদ্ধতিও আছে, যেটা আমি ছোট্ট করে বলব, কারণ এটাও আলো-স্ক্রিনের গল্প থেকে অনুপ্রাণিত, আর আমাদের গল্পে সুন্দর খাপ খেয়ে যায়। ধরো, সমকোণী ত্রিভুজটাকে আমরা আকারে অনেক ছোটো করে দিলাম, আর এই ছোট্ট অতিভুজটাকে একটা স্ক্রিন ভাবলাম, যেটা শুধু প্রথম বাতিঘরটা থেকে আলো পাচ্ছে। এবার স্ক্রিনটাকে 'ভেঙে' দি, যাতে ওটা ছোটো ত্রিভুজের অন্য দুটো বাহুর মতো হয়, মানে এমনি। এখন, নতুন স্ক্রিনটা পুরোনোটার সমপরিমাণ আলোই পাচ্ছে।", "n_reviews": 0, @@ -119,7 +119,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "", "from_community_srt": "ফলে আপাত ঔজ্জ্বল্য চার ভাগের এক ভাগ হয়ে যাবে। অবশ্য এটা আরও সহজ বীজগণিত থেকেই চট করে পেয়ে যাবে যে শুধু জোড় সংখ্যার উপর করা যোগফলটা, সমস্ত পূর্ণসংখ্যার উপর করা যোগফলের চার ভাগের এক ভাগ। তাহলে এর থেকে আমরা সমস্ত বিজোড় সংখ্যার উপর করা যোগফল থেকে কি পাই? যে সেটি সমস্ত পূর্ণসংখ্যার উপর করা যোগফলটির চার ভাগের তিন (4/3) ভাগ; কারণ বিজোড় আর জোড় সংখ্যাদের উপর করা যোগফলের সমষ্টিতে সমগ্র পূর্ণসংখ্যার উপর করা যোগফলটা আমরা ফেরত পাই। তো এটাকে ঘুরিয়ে বললে বিজোড় সংখ্যাদের উপর করা যোগফলকে কে 4/3 দিয়ে গুণ করলে সমস্ত ধনাত্মক পূর্ণসংখ্যার যোগফলটা ফেরত পাই। তো π²/8, আর তার সাথে 4/3 গুন করলেই কেল্লা ফতে! আমরা বেসেল প্রবলেমের সমাধান পেয়ে গেছি! এই ভিডিওটি, যেটা তোমরা এক্ষুনি দেখলে, তার মূল লেখক ও animator আমাদের", "n_reviews": 0, diff --git a/2018/basel-problem/bengali/sentence_translations.json b/2018/basel-problem/bengali/sentence_translations.json index d8d91a0bc..388fa0554 100644 --- a/2018/basel-problem/bengali/sentence_translations.json +++ b/2018/basel-problem/bengali/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right? ", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right? ", "translatedText": "আমি বলতে চাচ্ছি যে আলোর রশ্মিগুলি এই দুটি পায়ের মধ্যে একটিকে আঘাত করে সেই রশ্মিগুলি অবিকল সেই রশ্মির মতো যা কর্ণকে আঘাত করে তারপর মূল বিষয় হল প্রথম বাতিঘর থেকে যে পরিমাণ আলো এই বাম দিকে আঘাত করে সেই রশ্মির সীমিত কোণে আঘাত করে এই স্ক্রীনটি এখানে বাতিঘর থেকে আসা আলোর পরিমাণ ঠিক একই যেটি যে দিকে আঘাত করে সেটি রশ্মির একই কোণ হবে এবং প্রতিসাম্যভাবে আমাদের পর্দার নীচের অংশে আঘাতকারী প্রথম ঘর থেকে আলোর পরিমাণ একই।বাতিঘর বি থেকে সেই অংশে আলোর পরিমাণ আঘাত কেন আপনি ভালভাবে জিজ্ঞাসা করতে পারেন, এটি অনুরূপ ত্রিভুজগুলির একটি বিষয় এই অ্যানিমেশনটি ইতিমধ্যেই আপনাকে এটি কীভাবে কাজ করে তার জন্য একটি শক্তিশালী ইঙ্গিত দেয় এবং আমরা একটি সাধারণ জিওজেব্রার বিবরণে একটি লিঙ্কও রেখেছি অ্যাপলেট তাদের জন্য যারা একটু বেশি ইন্টারেক্টিভ পরিবেশে এবং এটির সাথে খেলতে খেলতে এখানে একটি গুরুত্বপূর্ণ তথ্য যা আপনি দেখতে সক্ষম হবেন যে অনুরূপ ত্রিভুজগুলি শুধুমাত্র একটি খুব ছোট পর্দার জন্য সীমাবদ্ধ ক্ষেত্রে প্রযোজ্য সব ঠিক আছে এখন বক আপ কারণ এখানে যেখানে জিনিস ভাল হয় আমরা এই বিপরীত পিথাগোরিয়ান উপপাদ্য পেয়েছি, তাই না? ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right. ", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right. ", "translatedText": "ঠিক আছে সেই বাতিঘরগুলি থেকে কেন্দ্র পর্যন্ত রেখাগুলি একে অপরের সাথে 90 ডিগ্রি কোণে রয়েছে তাই যেহেতু জিনিসগুলি বাম থেকে ডানদিকে প্রতিসাম্য যার মানে পরিধি বরাবর দূরত্বগুলি 1 2 2 2 এবং 1 ঠিক আছে, আপনি দেখতে পারেন এটি কোথায় যাচ্ছে, কিন্তু আমি আর মাত্র এক ধাপের জন্য এর মধ্য দিয়ে যেতে চাই আপনি এখন 16 এর পরিধির দ্বিগুণ বড় একটি বৃত্ত আঁকুন এবং প্রতিটি বাতিঘরের জন্য আপনি সেই বাতিঘর থেকে ছোট বৃত্তের শীর্ষ দিয়ে একটি রেখা আঁকুন যা বড় বৃত্তের কেন্দ্র।এবং তারপরে দুটি নতুন বাতিঘর তৈরি করুন যেখানে সেই রেখাটি বৃহত্তর বৃত্তের সাথে ছেদ করে ঠিক আগের মতো কারণ দীর্ঘ রেখাটি বড় বৃত্তের একটি ব্যাস এই দুটি নতুন বাতিঘর পর্যবেক্ষকের ডানদিকে একটি সমকোণ তৈরি করে এবং ঠিক যেমনটি পর্যবেক্ষকের থেকে লাইনের আগে মূল বাতিঘরটি দীর্ঘ রেখার লম্ব এবং এই দুটি তথ্য যা আমাদের বিপরীত পিথাগোরিয়ান উপপাদ্য ব্যবহার করার ক্ষেত্রে ন্যায্যতা দেয় তবে যেটি স্পষ্ট নাও হতে পারে তা হল যে আপনি যখন সমস্ত বাতিঘরগুলির জন্য এটি করেন তখন আটটি বড় থিওরেম পাওয়ার জন্য লেক সেই আটটি নতুন বাতিঘর সমানভাবে ব্যবধানে চলে যাচ্ছে এটি চূড়ান্ত থ্রাস্টের আগে জ্যামিতি প্রমাণের চূড়ান্ত বিট এটি দেখতে মনে রাখবেন যে আপনি যদি ছোট লেকের উপর দুটি সংলগ্ন বাতিঘর থেকে কেন্দ্রে লাইন আঁকেন তবে তারা একটি 90 ডিগ্রি কোণ তৈরি করে পরিবর্তে আপনি বৃত্তের পরিধির কোথাও একটি বিন্দুতে রেখা আঁকুন যা তাদের মধ্যে নয় জ্যামিতি থেকে খুব দরকারী খোদাই করা কোণ উপপাদ্যটি আমাদের বলে যে এটি এই ক্ষেত্রে কেন্দ্রের সাথে যে কোণ তৈরি করে তার ঠিক অর্ধেক হবে 45 ডিগ্রি কিন্তু যখন আমরা হ্রদের শীর্ষে সেই নতুন বিন্দুটিকে অবস্থান করি এই দুটি লাইন যা বৃহত্তর হ্রদে নতুন বাতিঘরের অবস্থান নির্ধারণ করে তাহলে এর অর্থ হল যে আপনি যখন সেই আটটি নতুন বাতিঘর থেকে কেন্দ্রে রেখা আঁকবেন তখন তারা বৃত্তকে বিভক্ত করবে সমানভাবে 45 ডিগ্রি কোণে টুকরো টুকরো করে এবং এর অর্থ হল আটটি বাতিঘর পরিধির চারপাশে সমানভাবে ব্যবধানে তাদের প্রত্যেকটির মধ্যে দুটির দূরত্ব রয়েছে এবং এখন কল্পনা করুন এই জিনিসটি প্রতিটি ধাপে বাজছে প্রতিটি বৃত্তের আকার দ্বিগুণ করে এবং প্রতিটি বাতিঘরকে রূপান্তরিত করছে বৃহত্তর বৃত্তের কেন্দ্রের মধ্য দিয়ে আঁকা একটি রেখা বরাবর দুটি নতুন প্রতিটি ধাপে পর্যবেক্ষকের কাছে আপাত উজ্জ্বলতা একই pi বর্গক্ষেত্রে 4 থেকে যায় এবং প্রতিটি ধাপে বাতিঘরগুলি সমানভাবে ব্যবধানে থাকে যার প্রতিটির মধ্যে 2 দূরত্ব থাকে পরিধি এবং সীমার মধ্যে আমরা এখানে যা পাচ্ছি তা হল একটি সমতল অনুভূমিক রেখা যেখানে অসীম সংখ্যক বাতিঘর রয়েছে উভয় দিকে সমানভাবে ব্যবধানে এবং কারণ আপাত উজ্জ্বলতা pi 4 এর উপর বর্গক্ষেত্র ছিল পুরো পথ যা এই সীমাবদ্ধ ক্ষেত্রেও সত্য হবে এবং এটি আমাদের একটি চমত্কার অসীম সিরিজ দেয় বিপরীত বর্গগুলির যোগফল 1 ওভার n বর্গ যেখানে n সমস্ত বিজোড় পূর্ণসংখ্যা 1 3 5 এবং আরও অনেকগুলিকে কভার করে তবে বাম দিকের দিকে ঋণাত্মক 1 ঋণাত্মক 3 ঋণাত্মক 5 বন্ধ করে সেই সমস্তগুলিকে যোগ করে আমাদেরকে 4 এর উপরে পাই বর্গ দিতে যাচ্ছে এটা আশ্চর্যজনক এবং আমি আপনাকে যা দেখাতে চাই তার মূল বিষয় এবং শুধু একধাপ পিছিয়ে যান এবং ভাবুন এটি কতটা অবাস্তব বলে মনে হচ্ছে সাধারণ ভগ্নাংশের সমষ্টি যা প্রথম দর্শনে জ্যামিতির সাথে কোন সম্পর্ক নেই বৃত্তের সাথে কিছুই করার নেই দৃশ্যত আমাদের এই ফলাফলটি দেয় যা পাই এর সাথে সম্পর্কিত যা এখন আপনি আসলে দেখতে পাচ্ছেন জ্যামিতির সাথে এটির কী সম্পর্ক রয়েছে সংখ্যা রেখাটি একধরনের ক্রমবর্ধমান চেনাশোনাগুলির একটি সীমার মতো এবং আপনি যে সংখ্যাটি জুড়ে যোগ করেন লাইনের উভয় দিকে অসীম পর্যন্ত সমস্ত পথের যোগফল নিশ্চিত করা এটা অনেকটা এমন যে আপনি একটি অসীম বৃহৎ বৃত্তের সীমানা বরাবর যোগ করছেন এবং একটি খুব আলগা কিন্তু কথা বলার খুব মজার উপায় কিন্তু অপেক্ষা করুন আপনি বলতে পারেন এটি যোগফল নয় যে আপনি ভিডিওর শুরুতে আমাদের প্রতিশ্রুতি দিয়েছিলেন এবং আপনি ঠিকই বলেছেন।", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. ", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/chinese/sentence_translations.json b/2018/basel-problem/chinese/sentence_translations.json index 40b0343b9..f6bd94cd8 100644 --- a/2018/basel-problem/chinese/sentence_translations.json +++ b/2018/basel-problem/chinese/sentence_translations.json @@ -80,7 +80,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", "translatedText": "我的意思是，照射到这两条腿中的一条腿的光线与照射 到斜边的光线完全相同那么关键是来自第一个灯塔的 光线照射到左侧的光线量最终照射到的光线的有限角 度那个屏幕与这里从灯塔 a 射到那一侧的光量完 全相同，光线的角度相同，并且对称地，从第一座房 子射到屏幕底部的光量是相同的作为从灯塔 B 照 射到该部分的光量 为什么你可能会问，这是一个相 似三角形的问题 这个动画已经给了你一个关于它如何 工作的强烈提示 我们还在描述中留下了一个简单 GeoGebra 的链接对于那些想要在一个稍微 更具交互性的环境中思考这一点的人来说，并且在玩 弄一个重要事实时，您将能够看到，相似的三角形仅 适用于非常小的屏幕的限制情况，这是一个小程序好 吧，现在系好安全带，因为这就是事情变得更好的地 方。 我们已经得到了这个逆毕达哥拉斯定理，对吗？", "model": "google_nmt", "from_community_srt": "像这样 把它变成小三角形的两条直角边 它接收到的光线还是一样多的 因为照在两条直角边上的光线 正好就是照在斜边上的光线 关键就在于 从原始的灯塔出发 最终到达左边屏幕的 这一角度内部的光线能量 正好等于灯塔A照在左边屏幕的光线能量 光线的角度大小是相同的 与之对称 从原始的灯塔出发 最终到达下边屏幕的光线数量 正好等于灯塔B照在下边屏幕的光线数量 你可能会问：“为什么？ ” 这用到了相似三角形原理 这个动画已经给出了不少提示 而且我在简介中留了一个简单的GeoGebra小程序 愿意亲自动手操作来想通这一点的朋友 不妨试试看 在把玩的过程中 你会注意到很重要的一点 相似三角形只适用于极限情况 屏幕要非常小才行 现在坐稳咯 精彩的部分要来了 我们学会了倒数勾股定理 就可以利用它来把一座灯塔转化成另外两座灯塔 而且不改变观察者看到的总亮度 掌握了这个方法", @@ -116,7 +116,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right.", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right.", "translatedText": "好吧，从这些灯塔到中心的线彼此成 90 度角所 以因为东西是左右对称的，这意味着沿圆周的距离是 1 2 2 2 和 1 好吧，你可能会看到这是 怎么回事，但我想再走一步你画一个两倍大的圆，现在 周长为 16，对于每个灯塔，你从那个灯塔穿过小 圆的顶部画一条线，这是大圆的中心然后创建两个新的 灯塔，该线与大圆相交就像以前一样，因为长线是大 圆的直径，这两个新灯塔与观察者右侧成直角，就像之 前从观察者到原始灯塔垂直于长线，这两个事实证明 我们使用逆毕达哥拉斯定理是正确的，但可能不太清楚 的是，当您对所有灯塔执行此操作时，会在大灯塔上 获得八个新灯塔湖中那八座新灯塔将均匀分布 这是最 后的推力之前几何验证的最后一点 要看到这一点 请记住，如果您从小湖上的两个相邻灯塔到中心画线 它们会形成 90 度角 如果相反，您可以在圆的圆 周上的任何位置画线，该点不在它们之间，几何学中 非常有用的内切角定理告诉我们，这将恰好是它们与圆 心形成的角度的一半，在这种情况下为 45 度但 是当我们将这个新点定位在湖的顶部 这是两条线，它 们定义了新灯塔在较大湖上的位置 这意味着，当您 从这八个新灯塔到中心画线时，它们将圆分开均匀地分 成 45 度角的部分，这意味着八个灯塔在圆周上 均匀分布，每个灯塔之间的距离为 2，现在想象一下 这个东西在每一步都将每个圆圈的大小加倍，并将每 个灯塔变成沿着穿过大圆中心绘制的一条线的两个新灯 塔，每一步，观察者的表观亮度都保持在 4 的 pi 平方相同，并且每一步，灯塔都保持均匀分布， 每个灯塔之间的距离为 2周长并且在极限情况下， 我们在这里得到的是一条平坦的水平线，在两个方向上 均匀分布着无限数量的灯塔，并且因为表观亮度是 pi 的平方超过 4，所以在这种极限情况下也是如 此并且这给了我们一个非常棒的无限级数，即反平方 1 与 n 平方之和，其中 n 涵盖所有奇数 1 3 5 等，但也包括向左方向的负 1 负 3 负 5 将所有这些相加将为我们提供 pi 的 平方超过 4 这太神奇了，这是我想向您展示的核 心内容，退后一步，想想这看起来多么不真实乍一看与 几何无关的简单分数的总和显然与圆无关 给我们这 个与 pi 有关的结果 除了现在你可以真正看到它 与几何有什么关系之外，数轴有点像不断增长的圆的 极限 当你对这个数字求和时确保两边的总和一直到无 穷大 这有点像你沿着一个无限大的圆的边界加起来 ，并且是一种非常松散但非常有趣的说话方式 但是等 等，你可能会说这不是总和你在视频开始时向我们承诺 过，你是对的。", "model": "google_nmt", "from_community_srt": "因为这些灯塔和大圆圆心的连线 构成了许多直角 而且图形左右对称 因此 圆周上的各段距离分别是 1 2 2 2 1 你大概知道是怎么回事了 但是我还想再讲一步 画一个两倍大的圆 现在周长是16了 对每一座灯塔来说 你都经过它和小圆顶端画一条线 （小圆顶端还是大圆圆心） 然后把两座灯塔放在这条线和大圆的交点上 和之前一样 因为这条长线是大圆的直径 所以两座新灯塔和观察者之间的连线形成了直角 还是和之前一样 观察者到原始灯塔的连线 垂直于那条长线 这两条结论确保我们可以使用倒数勾股定理 但是还有一点并不太清楚 当你对灯塔操作完之后 大湖的岸边就有八座新灯塔 这八座灯塔还是在岸边均匀分布的 这是高潮来临前的最后一点几何证明 回想一下 如果你把小湖边的相邻灯塔和小湖中心相连的话 它们会形成90°角 但是 如果它和小圆上的任一点相连 并且这个点不在两座灯塔之间 几何学中非常有用的圆周角定理告诉我们 这个角度刚好是圆心角的一半 在这个情况下 角度是45° 当我们把这个点放在小湖顶端时 这两条线正好确定了大湖上灯塔的位置 意思就是说 当你把八座新灯塔和湖中心连起来时 它们会将圆均分成几个45°角的区域 因此 这八座灯塔确实均匀分布在圆周上 相邻灯塔之间的距离是2 现在想象这个动画一直播放下去 在每一步过程中 让圆的大小加倍 对于每座灯塔 经过过它和大圆圆心画一条直线 把它转化成两座新灯塔 在每一步过程中 观察者看到的表观亮度一直保持在π^2/4 在每一步过程中 灯塔依旧均匀分布 而且相邻灯塔在圆周上的距离是2 在极限情况下 我们会得到一条水平线 在两个方向上均匀分布着无限多座灯塔 因为表观亮度一直都是π^2/4 所以在极限情况下 这个值还是π^2/4 这就给出了一个相当漂亮的无穷级数 它是一系列平方数倒数1/n^2的和 其中n是所有的奇数 包括1 3 5等正数 也包括-1 -3 -5等负数 把它们都加起来就是π^2/4 太了不起了 这也是我想向你展示的核心内容 退一步看 想想这个结论有多么奇幻 这些简单分数的和 初看起来 跟几何没什么关系 “显然”也跟圆一点都不搭边 但它的结果和π有关 不过现在 你应该明白它和几何的关系是什么了 这条数轴像是一个不断增长的圆的极限 当你在数轴上不断求和 保证能一直加到两边无穷远处 就好像在一个无限大的圆的边界上求和一样 这种说法很粗略 但是非常有趣 你可能会说 “等一下！ ” “你一开始答应计算的和不是这个！", @@ -125,7 +125,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible.", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "我们确实还有一点思考首先，首先让我们将 总和限制为正奇数，这样我们就可以得到 pi 的平方除以 8 现在，这与我们正在寻找的总和之间的唯一区别就过去了所有 正整数奇数和偶数是它缺少偶数的倒数之和，我在这里用红色着 色现在你可以将那个缺少的系列视为我们想要的总系列的缩放副 本每个灯塔在哪里移动到距原点的两倍远，一个移动到两个，两 个移动到四个，三个移动到六，依此类推，因为这涉及到每个灯 塔的距离加倍，这意味着表观亮度将减少一个因子四的和这也是相 对简单的代数，从所有整数的总和到偶数整数的总和涉及乘以 1 4 次，这意味着从所有整数到奇数将乘以 3 4 次因 为偶数加上奇数必须给我们整个结果所以如果我们只是翻转它意 味着从奇数的总和到所有正整数的总和需要乘以三分之四所以将 pi 的平方除以 8 乘以4 三分之二 baddaoom badda bing 我们已经找到了解决罗勒问题的方法 现在，您刚刚观看的这段视频主要是由新的三名蓝一棕团队成 员之一 Ben Hambricht 编写和制作动画的，另 外还有一个补充使之成为可能。", "model": "google_nmt", "from_community_srt": "” 你说得没错 我们确实要再动点脑筋 首先 我们把加和范围限制为正奇数 这样就等于π^2/8 这样 唯一的区别就是 我们准备计算的和是在正整数范围上的 有奇数 也有偶数 而这个式子缺少所有偶数平方的倒数和 就是上面这些红色的项 你可以把缺失的级数 看成待求的完整级数的放大版 每座灯塔到原点的距离全都加倍 位于1的灯塔挪到2 位于2的灯塔挪到4 位于3的灯塔挪到6 依此类推 这样做的话 每座灯塔到原点的距离加倍 所以表观亮度就会减小为原来的1/4 代数上的关系也很直接 从在正整数范围内求和 到在偶数范围内求和 只需要乘上1/4就行 这也就意味着 从所有正整数 到奇数部分 级数和需要乘上3/4 因为偶数部分和奇数部分合起来 才能得到整体 反过来看的话 从在奇数范围内求和 到在正整数范围内求和 只需要乘上4/3 计算π^2/8乘以4/3的话 瞧瞧 我们就算出了巴塞尔问题的答案！ 这期视频的脚本撰写和动画制作 主要是由3Blue1Brown团队的新成员Ben Hambrecht完成的", diff --git a/2018/basel-problem/czech/sentence_translations.json b/2018/basel-problem/czech/sentence_translations.json index 9e5dc4072..53a493072 100644 --- a/2018/basel-problem/czech/sentence_translations.json +++ b/2018/basel-problem/czech/sentence_translations.json @@ -63,7 +63,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "", "from_community_srt": "někdy mluvíte, o pevném úhlu tvaru Který je podíl koule, kterou pokrývá při pohledu z daného místa Podívejte se na první ze dvou míst, kde tento příběh, o kterém přemýšlíme o obrazovkách, bude užitečný, je pochopení inverzního zákona o čtvercích Což je zřetelně trojrozměrný jev, pomyslete na všechny paprsky světla dopadající na obrazovku o jednu jednotku dál Ze zdroje, když zdvojnásobíte vzdálenost, tyto paprsky budou nyní pokrývat oblast s dvojnásobnou šířkou a dvakrát vyšší Takže to by trvalo 4 kopie té původní obrazovky, aby bylo přijato stejné zvýšení v této vzdálenosti, a tak každou Jednotlivci obdrží čtvrtinu tolik světla To je smysl, ve kterém mám na mysli světlo, by se objevilo 1/4 jako jasné dvojnásobek vzdálenosti Podobně, když jste třikrát dál, pryč? Vy byste potřebovali devět kopií té původní obrazovky, abyste dostali stejné paprsky, takže každá jednotlivá obrazovka obdrží pouze 1/9 tolik světla a Tento vzorec pokračuje, protože se oblast zasáhne světlem o druhou mocninu vzdálenosti klesá jas tohoto světla, o inverzní čtverec této vzdálenosti a Jak jsem si jistý, mnozí z vás vědí, tento inverzní zákon o čtvercích Není vůbec zvláštní na to, že se rozsvítí, kdykoli máte nějaké množství, které se rovnoměrně šíří z bodového zdroje, ať už je to zvuk, teplo nebo rádiový signál. A pamatujte si, že díky tomuto inverznímu čtvercovému zákonu je to nekonečná řada rovnoměrně rozmístěných majáků fyzicky implementuje Basilejský problém. Ale znovu, co potřebujeme, pokud zde uděláme nějaký pokrok, je pochopit, jak můžeme manipulovat s nastavením světelné zdroje, jako je tento, bez změny celkové jasnosti pro pozorovatele. A klíčový stavební blok je obzvláště pěkný způsob, jak přeměnit jediný maják na Pomyslete na pozorovatele na počátku xy letounu a na jediný maják, který sedí někde v tomto letadle Nyní nakreslete čáru od tohoto majáku k pozorovateli a pak další čáru, kolmou k té, která je v majáku nyní umístěte dva majáky, kde tato nová linie protíná souřadné osy Který půjdu dopředu a zavolám maják sem vlevo a maják b na horní stranu Ukázalo se, že to uvidíte je pravda, za pouhou minutu jas, který pozorovatel zažívá od prvního majáku, je Stejné jako v kombinaci jasu, který zažíváme společně s majáky aab, a měl bych říct Mimochodem, že stálý předpoklad celé Toto video je, že všechny majáky jsou stejné Použití stejné žárovky vyzařující stejnou energii Jinými slovy, přiřazení proměnných k věcem zde, pokud nazýváme vzdálenost od pozorovatele k majáku malý a vzdálenost od pozorovatele k majáku b malý ba vzdálenost k prvnímu, maják h Máme vztah 1 na druhou a 1 na b na druhou se rovná 1 na h na druhou Toto je mnohem méně známé Inverzní pythagorova věta, který z vás? Květen, rozeznávejte z mytologií nejnovější a také nejlepší video o mnoha bratrancích pythagorovy věty docela v pohodě, vztah nemyslíš a Pokud jste, v jádru matematik, můžete se zeptat právě teď. Jak Dokážete to a existují některé přímé způsoby, jak vyjádřit oblast trojúhelníků dvěma různými způsoby? a aplikovat obvyklou pythagorovu větu Ale je tu další docela pěkná metoda, kterou bych zde chtěl stručně nastínit a která mnohem hezky spadá do našeho příběhu, protože Opět používá intuice světla a obrazovek Představte si, jak se zmenšuje pravý trojúhelník do jemnější verze a uvažujte o této miniaturní přepěně jako obrazovce přijímající světlo z prvního majáku, pokud Vy, přetvořte tuto obrazovku, abyste byli kombinací obou nohou miniaturního trojúhelníku No to stále dostává stejné množství světla,", "n_reviews": 0, @@ -119,7 +119,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "", "from_community_srt": "znamená to, že by se zjevná jasnost snížila o faktor 4 a? To je také relativně přímá algebra Od součtu přes všechna celá čísla k součtu přes sudá celá čísla se vynásobí 1/4. A co to znamená? Je to přechod ze všech celých čísel na lichá? Vynásobilo by se 3/4, protože nám zlo a šance musí dát celou věc Takže pokud to prostě převrátíme znamená jít, od součtu přes lichá čísla k součtu přes všechna kladná celá čísla vynásobí 4/3 Takže to, že pi na druhou mocninu přes 8, vynásobením 4/3, ale, boom, bada, bing, máme sami řešení problému Basilej Toto video, které jste právě sledovali, bylo primárně napsáno a animováno jedním z nových", "n_reviews": 0, diff --git a/2018/basel-problem/english/captions.srt b/2018/basel-problem/english/captions.srt index 7da3c1259..205597b82 100644 --- a/2018/basel-problem/english/captions.srt +++ b/2018/basel-problem/english/captions.srt @@ -179,614 +179,622 @@ It might be more accurate to ask What is the angle the light covers in both directions perpendicular to the source? 46 -00:03:33,260 --> 00:03:38,606 +00:03:33,260 --> 00:03:38,525 In spherical geometry you sometimes talk about the solid angle of a shape Which is the 47 -00:03:38,606 --> 00:03:43,953 +00:03:38,525 --> 00:03:43,791 proportion of a sphere it covers as viewed from a given point You see the first of two 48 -00:03:43,953 --> 00:03:49,484 -places this story we're thinking of screens is going to be useful is in understanding the +00:03:43,791 --> 00:03:48,996 +places this story we're thinking of screens is going to be useful is in understanding 49 -00:03:49,484 --> 00:03:55,015 -inverse square law Which is a distinctly three-dimensional phenomenon think of all of the +00:03:48,996 --> 00:03:54,262 +the inverse square law Which is a distinctly three-dimensional phenomenon think of all 50 -00:03:55,015 --> 00:04:00,423 -rays of light hitting a screen one unit away from the source as You double the distance +00:03:54,262 --> 00:03:59,467 +of the rays of light hitting a screen one unit away from the source as You double the 51 -00:04:00,423 --> 00:04:05,832 -those rays will now cover an area with twice the width and twice the height So it would +00:03:59,467 --> 00:04:04,794 +distance those rays will now cover an area with twice the width and twice the height So 52 -00:04:05,832 --> 00:04:11,363 -take four copies of that original screen to receive the same rays at that distance And so +00:04:04,794 --> 00:04:09,817 +it would take four copies of that original screen to receive the same rays at that 53 -00:04:11,363 --> 00:04:16,771 -each individual one receives 1 fourth as much light This is the sense in which I mean a +00:04:09,817 --> 00:04:15,204 +distance And so each individual one receives 1 fourth as much light This is the sense in 54 -00:04:16,771 --> 00:04:22,118 -light would appear 1 fourth as bright two times the distance away Likewise when you're +00:04:15,204 --> 00:04:20,651 +which I mean a light would appear 1 fourth as bright two times the distance away Likewise 55 -00:04:22,118 --> 00:04:27,464 -three times farther away You would need nine copies of that original screen to receive +00:04:20,651 --> 00:04:25,978 +when you're three times farther away You would need nine copies of that original screen 56 -00:04:27,464 --> 00:04:32,565 -the same rays so each individual screen only receives 1 9th as much light and This +00:04:25,978 --> 00:04:31,365 +to receive the same rays so each individual screen only receives 1 9th as much light and 57 -00:04:32,565 --> 00:04:38,096 -pattern continues because the area hit by a light increases by the square of the distance +00:04:31,365 --> 00:04:36,570 +This pattern continues because the area hit by a light increases by the square of the 58 -00:04:38,096 --> 00:04:43,566 -the brightness of that light decreases by the inverse square of that distance and As I'm +00:04:36,570 --> 00:04:41,836 +distance the brightness of that light decreases by the inverse square of that distance 59 -00:04:43,566 --> 00:04:48,974 -sure many of you know this inverse square law is not at all special to light It pops up +00:04:41,836 --> 00:04:47,162 +and As I'm sure many of you know this inverse square law is not at all special to light 60 -00:04:48,974 --> 00:04:54,137 -whenever you have some kind of quantity that spreads out evenly from a point source +00:04:47,162 --> 00:04:52,488 +It pops up whenever you have some kind of quantity that spreads out evenly from a point 61 -00:04:54,137 --> 00:04:59,299 -whether that's sound or heat or radio signal things like that and Infinite array of +00:04:52,488 --> 00:04:57,875 +source whether that's sound or heat or a radio signal things like that and Remember it's 62 -00:04:59,299 --> 00:05:04,769 -evenly spaced lighthouses physically implements the basel problem But again what we need +00:04:57,875 --> 00:05:03,141 +because of this inverse square law that an infinite array of evenly spaced lighthouses 63 -00:05:04,769 --> 00:05:10,116 -if we're going to make any progress here is to understand how we can manipulate setups +00:05:03,141 --> 00:05:08,346 +physically implements the Basel problem But again what we need if we're going to make 64 -00:05:10,116 --> 00:05:15,524 -with light sources like this without changing the total brightness for the observer and +00:05:08,346 --> 00:05:13,672 +any progress here is to understand how we can manipulate setups with light sources like 65 -00:05:15,524 --> 00:05:20,870 -The key building block is an especially nice way to transform a single lighthouse into +00:05:13,672 --> 00:05:18,938 +this without changing the total brightness for the observer and The key building block 66 -00:05:20,870 --> 00:05:26,217 -two Think Of an observer at the origin of the XY plane and a single lighthouse sitting +00:05:18,938 --> 00:05:24,325 +is an especially nice way to transform a single lighthouse into two Think of an observer 67 -00:05:26,217 --> 00:05:31,748 -out somewhere on that plane Now draw a line from that lighthouse to the observer and then +00:05:24,325 --> 00:05:29,772 +at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane 68 -00:05:31,748 --> 00:05:37,218 -another line perpendicular to that one at the lighthouse Now place two lighthouses where +00:05:29,772 --> 00:05:35,159 +Now draw a line from that lighthouse to the observer and then another line perpendicular 69 -00:05:37,218 --> 00:05:42,565 -this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a +00:05:35,159 --> 00:05:40,425 +to that one at the lighthouse Now place two lighthouses where this new line intersects 70 -00:05:42,565 --> 00:05:48,034 -over here on the left and lighthouse B on the upper side It turns out and you'll see why +00:05:40,425 --> 00:05:45,751 +the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and 71 -00:05:48,034 --> 00:05:53,258 -this is true in just a minute the brightness that the observer Experiences from that +00:05:45,751 --> 00:05:50,957 +lighthouse B on the upper side It turns out and you'll see why this is true in just a 72 -00:05:53,258 --> 00:05:58,666 -first lighthouse is equal to the combined brightness experienced from lighthouses A and +00:05:50,957 --> 00:05:56,283 +minute the brightness that the observer Experiences from that first lighthouse is equal 73 -00:05:58,666 --> 00:06:04,198 -B together And I should say by the way that the standing assumption throughout this video +00:05:56,283 --> 00:06:01,488 +to the combined brightness experienced from lighthouses A and B together And I should 74 -00:06:04,198 --> 00:06:09,544 -is that all lighthouses are equivalent They're using the same light bulb emanating the +00:06:01,488 --> 00:06:06,935 +say by the way that the standing assumption throughout this video is that all lighthouses 75 -00:06:09,544 --> 00:06:14,891 -same power all of that So in other words assigning variables to things here if we call +00:06:06,935 --> 00:06:12,322 +are equivalent They're using the same light bulb emanating the same power all of that So 76 -00:06:14,891 --> 00:06:19,931 -the distance from the observer to lighthouse a little a And the distance from the +00:06:12,322 --> 00:06:17,346 +in other words assigning variables to things here if we call the distance from the 77 -00:06:19,931 --> 00:06:25,400 -observer to lighthouse B little B and the distance to the first lighthouse H We have the +00:06:17,346 --> 00:06:22,491 +observer to lighthouse a little a and The distance from the observer to lighthouse B 78 -00:06:25,400 --> 00:06:30,870 -relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much +00:06:22,491 --> 00:06:27,938 +little B and the distance to the first lighthouse H We have the relation 1 over a squared 79 -00:06:30,870 --> 00:06:36,155 -less well-known Inverse Pythagorean theorem which some of you may recognize from math +00:06:27,938 --> 00:06:33,204 +plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse 80 -00:06:36,155 --> 00:06:41,195 -ologer's most recent and I'll say most excellent video on the many cousins of the +00:06:33,204 --> 00:06:38,470 +Pythagorean theorem which some of you may recognize from math ologer's most recent and 81 -00:06:41,195 --> 00:06:46,726 -Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at +00:06:38,470 --> 00:06:43,856 +I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool 82 -00:06:46,726 --> 00:06:52,134 -heart you might be asking right now how you prove it and There are some straightforward +00:06:43,856 --> 00:06:49,304 +relation don't you think and if you're a mathematician at heart you might be asking right 83 -00:06:52,134 --> 00:06:57,235 -ways where you express the triangles area in two separate ways and apply the usual +00:06:49,304 --> 00:06:54,327 +now how you prove it and There are some straightforward ways where you express the 84 -00:06:57,235 --> 00:07:02,520 -Pythagorean theorem But there is another quite pretty method that I'd like to briefly +00:06:54,327 --> 00:06:59,714 +triangles area in two separate ways and apply the usual Pythagorean theorem But there is 85 -00:07:02,520 --> 00:07:07,560 -outline here that falls much more nicely into our storyline because again It uses +00:06:59,714 --> 00:07:04,980 +another quite pretty method that I'd like to briefly outline here that falls much more 86 -00:07:07,560 --> 00:07:12,783 -intuitions of light and screens Imagine scaling down the whole right triangle into a +00:07:04,980 --> 00:07:10,306 +nicely into our storyline because again It uses intuitions of light and screens Imagine 87 -00:07:12,783 --> 00:07:18,130 -tinier version and think of this miniature Hypotenuse as a screen receiving light from +00:07:10,306 --> 00:07:15,633 +scaling down the whole right triangle into a tinier version and think of this miniature 88 -00:07:18,130 --> 00:07:23,600 -the first lighthouse If you reshape that screen to be the combination of the two legs of +00:07:15,633 --> 00:07:20,777 +Hypotenuse as a screen receiving light from the first lighthouse If you reshape that 89 -00:07:23,600 --> 00:07:29,070 -the miniature triangle like this Well, it still receives the same amount of light, right? +00:07:20,777 --> 00:07:26,043 +screen to be the combination of the two legs of the miniature triangle like this Well, 90 +00:07:26,043 --> 00:07:29,070 +it still receives the same amount of light, right? + +91 00:07:29,270 --> 00:07:34,890 I mean the rays of light hitting one of those two legs are precisely the same as the rays -91 +92 00:07:34,890 --> 00:07:39,886 that hit the hypotenuse Then the key is that the amount of light from the first -92 +93 00:07:39,886 --> 00:07:45,319 lighthouse that hits this left side the limited angle of rays that end up hitting that -93 +94 00:07:45,319 --> 00:07:50,628 screen is Exactly the same as the amount of light over here coming from lighthouse a -94 +95 00:07:50,628 --> 00:07:55,936 which hits that side it'll be the same angle of rays and Symmetrically the amount of -95 +96 00:07:55,936 --> 00:08:01,369 light from the first house hitting the bottom portion of our screen is The same as the -96 +97 00:08:01,369 --> 00:08:06,303 amount of light hitting that portion from lighthouse B Why you might ask well, -97 +98 00:08:06,303 --> 00:08:11,923 it's a matter of similar triangles This animation already gives you a strong hint for how -98 +99 00:08:11,923 --> 00:08:17,356 it works And we've also left a link in the description to a simple GeoGebra applet for -99 +100 00:08:17,356 --> 00:08:22,789 those of you who want to think this through in a slightly more interactive environment -100 +101 00:08:22,789 --> 00:08:28,285 and in playing with that One important fact here that you'll be able to see is that the -101 +102 00:08:28,285 --> 00:08:33,593 similar triangles only apply in the limiting case for a very tiny screen The inverse -102 +103 00:08:33,593 --> 00:08:39,151 Pythagorean theorem Alright buckle up now because here's where things get good We've got -103 +104 00:08:39,151 --> 00:08:41,650 this inverse Pythagorean theorem, right? -104 +105 00:08:41,929 --> 00:08:46,960 And that's going to let us transform a single lighthouse into two others without -105 +106 00:08:46,960 --> 00:08:52,116 changing the brightness experienced by the observer With that in hand and no small -106 +107 00:08:52,116 --> 00:08:57,146 amount of cleverness we can use this to build up the infinite array that we need -107 +108 00:08:57,146 --> 00:09:02,426 Picture yourself at the edge of a circular lake directly opposite a lighthouse We're -108 +109 00:09:02,426 --> 00:09:07,829 going to want it to be the case that the distance between you and the lighthouse Along -109 +110 00:09:07,829 --> 00:09:12,985 the border of the lake is one so we'll say the lake has a circumference of two now -110 +111 00:09:12,985 --> 00:09:18,202 the apparent brightness is one divided by the diameter squared and In this case the -111 +112 00:09:18,202 --> 00:09:23,419 diameter is that circumference 2 divided by pi so the apparent brightness works out -112 +113 00:09:23,419 --> 00:09:28,823 to be pi squared divided by 4 Now for our first transformation draw a new circle twice -113 +114 00:09:28,823 --> 00:09:34,164 as big so circumference 4 and Draw a tangent line to the top of the small circle then -114 +115 00:09:34,164 --> 00:09:39,443 replace the original lighthouse with two new ones where this tangent line intersects -115 +116 00:09:39,443 --> 00:09:44,661 the larger circle an Important fact from geometry that we'll be using over and over -116 +117 00:09:44,661 --> 00:09:49,940 here Is that if you take the diameter of a circle and form a triangle with any point -117 +118 00:09:49,940 --> 00:09:50,810 on the circle? -118 +119 00:09:51,330 --> 00:09:56,695 The angle at that new point will always be 90 degrees the significance of that in our -119 +120 00:09:56,695 --> 00:10:02,247 diagram here is that it means the inverse Pythagorean theorem applies and the brightness -120 +121 00:10:02,247 --> 00:10:07,862 from those two new lighthouses equals the brightness from the first one namely pi squared -121 +122 00:10:07,862 --> 00:10:12,853 divided by 4 as The next step draw a new circle twice as big as the last with a -122 +123 00:10:12,853 --> 00:10:18,156 circumference 8 Now for each lighthouse take a line from that lighthouse through the -123 +124 00:10:18,156 --> 00:10:23,647 top of the smaller circle Which is the center of the larger circle and consider the two -124 +125 00:10:23,647 --> 00:10:27,328 points where that intersects with the larger circle Again, -125 +126 00:10:27,328 --> 00:10:32,631 since this line is a diameter of that large circle Then the lines from those two new -126 +127 00:10:32,631 --> 00:10:38,183 points to the observer are going to form a right angle Likewise by looking at this right -127 +128 00:10:38,183 --> 00:10:43,549 triangle here whose hypotenuse is the diameter of the smaller circle You can see that -128 +129 00:10:43,549 --> 00:10:48,914 the line from the observer to that original lighthouse is at a right angle With a new -129 +130 00:10:48,914 --> 00:10:51,410 long line that we drew Good news, right? -130 +131 00:10:51,670 --> 00:10:56,249 because that means we can apply the inverse Pythagorean theorem and that means -131 +132 00:10:56,249 --> 00:11:00,713 that the apparent brightness from the original lighthouse is the same as the -132 +133 00:11:00,713 --> 00:11:04,133 combined brightness from the two newer ones and Of course, -133 +134 00:11:04,133 --> 00:11:08,597 you can do that same thing over on the other side drawing a line through the -134 +135 00:11:08,597 --> 00:11:13,176 top of the smaller circle and getting two new lighthouses on the larger circle -135 +136 00:11:13,176 --> 00:11:17,756 and Even nicer these four lighthouses are all going to be evenly spaced around -136 +137 00:11:17,756 --> 00:11:18,510 the lake Why? -137 +138 00:11:19,270 --> 00:11:24,843 Well, the lines from those lighthouses to the center are at 90 degree angles with each -138 +139 00:11:24,843 --> 00:11:30,352 other So since things are symmetric left to right that means that the distances along -139 +140 00:11:30,352 --> 00:11:35,732 the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, -140 +141 00:11:35,732 --> 00:11:41,434 but I want to walk through this for just one more step You draw a circle twice as big so -141 +142 00:11:41,434 --> 00:11:46,878 circumference of 16 now and for each lighthouse You draw a line from that lighthouse -142 +143 00:11:46,878 --> 00:11:52,516 through the top of the smaller circle Which is the center of the bigger circle and then -143 +144 00:11:52,516 --> 00:11:57,960 create two new lighthouses where that line intersects with the larger circle Just as -144 +145 00:11:57,960 --> 00:12:03,534 before because the long line is a diameter of the big circle those two new lighthouses -145 +146 00:12:03,534 --> 00:12:09,299 make a right angle with the observer, right and Just as before the line from the observer -146 +147 00:12:09,299 --> 00:12:15,000 to the original lighthouse is Perpendicular to the long line and those are the two facts -147 +148 00:12:15,000 --> 00:12:20,637 that justify us in using the inverse Pythagorean theorem But what might not be as clear -148 +149 00:12:20,637 --> 00:12:26,402 is that when you do this for all of the lighthouses to get eight new ones on the Big lake -149 +150 00:12:26,402 --> 00:12:31,719 those eight new lighthouses are going to be evenly spaced This is the final bit of -150 +151 00:12:31,719 --> 00:12:37,356 geometry proofiness before the final thrust To see this remember that if you draw lines -151 +152 00:12:37,356 --> 00:12:43,121 from two adjacent lighthouses on the small lake to the center They make a 90 degree angle -152 +153 00:12:43,121 --> 00:12:48,758 If instead you draw lines to a point anywhere on the circumference of the circle that's -153 +154 00:12:48,758 --> 00:12:54,524 not between them the very useful inscribed angle theorem from geometry tells us that this -154 +155 00:12:54,524 --> 00:13:00,225 will be Exactly half of the angle that they make with the center in this case 45 degrees -155 +156 00:13:00,225 --> 00:13:05,926 But when we position that new point at the top of the lake These are the two lines which -156 +157 00:13:05,926 --> 00:13:11,435 define the position of the new lighthouses on the larger lake What that means then is -157 +158 00:13:11,435 --> 00:13:16,944 that when you draw lines from those eight new lighthouses into the center They divide -158 +159 00:13:16,944 --> 00:13:22,517 the circle evenly into 45 degree angle pieces and that means the eight lighthouses are -159 +160 00:13:22,517 --> 00:13:28,218 evenly spaced around the circumference with the distance of two between each one of them -160 +161 00:13:28,218 --> 00:13:33,983 and Now just imagine this thing playing on at every step doubling the size of each circle -161 +162 00:13:33,983 --> 00:13:39,684 and Transforming each lighthouse into two new ones along a line drawn through the center -162 +163 00:13:39,684 --> 00:13:45,257 of the larger circle at every step the apparent brightness to the observer remains the -163 +164 00:13:45,257 --> 00:13:50,895 same pi squared over 4 and at every step the lighthouse has remained evenly spaced with -164 +165 00:13:50,895 --> 00:13:56,468 a distance 2 between each one of them on the circumference and In the limit what we're -165 +166 00:13:56,468 --> 00:14:01,912 getting here is a flat horizontal line with an infinite number of lighthouses evenly -166 +167 00:14:01,912 --> 00:14:07,550 spaced in both directions and Because the apparent brightness was pi squared over 4 the -167 +168 00:14:07,550 --> 00:14:12,866 entire way that will also be true in this limiting case And This gives us a pretty -168 +169 00:14:12,866 --> 00:14:18,439 awesome infinite series the sum of the inverse squares 1 over n squared Where n covers -169 +170 00:14:18,439 --> 00:14:24,141 all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in -170 +171 00:14:24,141 --> 00:14:29,521 the leftward direction Adding all of those up is going to give us pi squared over 4 -171 +172 00:14:29,521 --> 00:14:35,030 That's amazing and it's the core of what I want to show you and Just take a step back -172 +173 00:14:35,030 --> 00:14:40,539 and think about how unreal this seems The sum of simple fractions that at first sight -173 +174 00:14:40,539 --> 00:14:46,112 have nothing to do with geometry nothing to do with circles at all Apparently gives us -174 +175 00:14:46,112 --> 00:14:51,750 this result that's related to pi Except now you can actually see what it has to do with -175 +176 00:14:51,750 --> 00:14:57,387 geometry the number line is kind of like a limit of ever-growing circles and As you sum -176 +177 00:14:57,387 --> 00:15:02,960 across that number line making sure to sum all the way to infinity on either side It's -177 +178 00:15:02,960 --> 00:15:08,725 sort of like you're adding up along the boundary of an infinitely large circle and a very -178 +179 00:15:08,725 --> 00:15:14,362 loose But very fun way of speaking But wait, you might say this is not the sum that you -179 +180 00:15:14,362 --> 00:15:18,270 promised us at the start of the video And well, you're right. -180 +181 00:15:18,570 --> 00:15:22,146 We do have a little bit of thinking left First things first, -181 +182 00:15:22,146 --> 00:15:27,072 let's just restrict the sum to only being the positive odd numbers which gets us pi -182 +183 00:15:27,072 --> 00:15:32,291 squared divided by 8 Now the only difference between this and the sum that we're looking -183 +184 00:15:32,291 --> 00:15:37,392 for that goes over all the positive integers odd and even is That it's missing the sum -184 +185 00:15:37,392 --> 00:15:42,611 of the reciprocals of even numbers what I'm coloring in red up here Now you can think of -185 +186 00:15:42,611 --> 00:15:47,361 that missing series as a scaled copy of the total series that we want Where each -186 +187 00:15:47,361 --> 00:15:52,521 lighthouse moves to being twice as far away from the origin one gets shifted to two two -187 +188 00:15:52,521 --> 00:15:57,388 gets shifted to four three gets shifted to six and so on and Because that involves -188 +189 00:15:57,388 --> 00:15:59,910 doubling the distance for every lighthouse. -189 -00:15:59,930 --> 00:16:04,907 -It means that the apparent brightness would be decreased by a factor of four and - 190 -00:16:04,907 --> 00:16:10,314 -That's also relatively straightforward algebra going from the sum over all the integers +00:15:59,930 --> 00:16:05,086 +it means that the apparent brightness would be decreased by a factor of four and That's 191 -00:16:10,314 --> 00:16:15,475 -to the sum over the even integers Involves multiplying by 1 4th And what that means +00:16:05,086 --> 00:16:10,243 +also relatively straightforward algebra going from the sum over all the integers to the 192 -00:16:15,475 --> 00:16:20,575 -is that going from all the integers to the odd ones would be multiplying by 3 4ths +00:16:10,243 --> 00:16:15,224 +sum over the even integers Involves multiplying by 1 4th and what that means is that 193 -00:16:20,575 --> 00:16:25,859 -Since the evens plus the odds have to give us the whole thing So if we just flip that +00:16:15,224 --> 00:16:20,205 +going from all the integers to the odd ones Would be multiplying by 3 4ths since the 194 -00:16:25,859 --> 00:16:31,205 -around that means going from the sum over the odd numbers to the sum over all positive +00:16:20,205 --> 00:16:25,361 +evens plus the odds have to give us the whole thing So if we just flip that around that 195 -00:16:31,205 --> 00:16:36,551 -integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying +00:16:25,361 --> 00:16:30,284 +means going from the sum over the odd numbers to the sum over all positive integers 196 -00:16:36,551 --> 00:16:41,712 -by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem +00:16:30,284 --> 00:16:35,558 +requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds 197 -00:16:41,712 --> 00:16:46,751 -Now this video that you just watched was primarily written and animated by one of +00:16:35,558 --> 00:16:40,773 +badda boom badda bing We've got ourselves a solution to the basil problem Now this video 198 -00:16:46,751 --> 00:16:51,790 -the new three blue one brown team members Ben Hambricht An addition made possible. +00:16:40,773 --> 00:16:45,812 +that you just watched was primarily written and animated by one of the new three blue + +199 +00:16:45,812 --> 00:16:49,563 +one brown team members Ben Hambricht an addition made possible. + +200 +00:16:49,563 --> 00:16:51,790 +Thanks to you guys through patreon You diff --git a/2018/basel-problem/english/sentence_timings.json b/2018/basel-problem/english/sentence_timings.json index f6fe03c1f..c68809728 100644 --- a/2018/basel-problem/english/sentence_timings.json +++ b/2018/basel-problem/english/sentence_timings.json @@ -40,7 +40,7 @@ 213.26 ], [ - "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", 213.26, 449.07 ], @@ -75,7 +75,7 @@ 959.91 ], [ - "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", 959.93, 1011.79 ] diff --git a/2018/basel-problem/english/transcript.txt b/2018/basel-problem/english/transcript.txt index fc849a264..5f25741f3 100644 --- a/2018/basel-problem/english/transcript.txt +++ b/2018/basel-problem/english/transcript.txt @@ -6,11 +6,11 @@ And why is it squared? We don't usually see it squared in honor of Euler whose hometown was basil This infinite sum is often referred to as the basil problem But the proof that I'd like to show you is very different from the one that Euler had I've said in a previous video that whenever you see pi show up There will be some connection to circles and there are those who like to say that pi is not fundamentally about circles and Insisting on connecting equations like these ones with a geometric intuition stems from a stubborn insistence on only understanding pi in the context where we first discovered it and That's all well and good But whatever your own perspective holds as fundamental the fact is pi is very much tied to circles So if you see it show up there will be a path somewhere in the massive interconnected web of mathematics Leading you back to circles and geometry The question is just how long and convoluted that path might be and in the case of the basil problem It's a lot shorter than you might first think and it all starts with light Here's the basic idea Imagine standing at the origin of a positive number line and putting a little lighthouse on all of the positive integers one two three four and so on that first lighthouse has some Apparent brightness from your point of view some amount of energy that your eye is receiving from the light per unit time and Let's just call that a brightness of one For reasons I'll explain shortly the apparent brightness of the second lighthouse is 1 fourth as much as the first and the apparent brightness of the third is 1 9th as much as the first and then 1 16th and so on and you can probably see why this is useful for the basil problem It gives us a physical representation of what's being asked Since the brightness received from the whole infinite line of lighthouses is going to be 1 plus 1 4th plus 1 9th Plus the 16th and so on So the result that we are aiming to show is that this total brightness is equal to pi squared divided by 6 times the brightness of that first lighthouse And at first that might seem useless I mean, we're just re-asking the same original question But the progress comes from a new question that this framing raises are there ways that we can rearrange these lighthouses That don't change the total brightness for the observer And if so, can you show this to be equivalent to a setup that's somehow easier to compute To start let's be clear about what we mean when we reference apparent brightness to an observer Imagine a little screen which maybe represents the retina of your eye or a digital camera sensor or something like that You could ask what proportion of the rays coming out of the source hit that screen or phrase differently What is the angle between the ray hitting the bottom of that screen and the ray hitting the top? Or rather since we should be thinking of these lights as being in three dimensions. It might be more accurate to ask What is the angle the light covers in both directions perpendicular to the source? -In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right? +In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right? I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right? And that's going to let us transform a single lighthouse into two others without changing the brightness experienced by the observer With that in hand and no small amount of cleverness we can use this to build up the infinite array that we need Picture yourself at the edge of a circular lake directly opposite a lighthouse We're going to want it to be the case that the distance between you and the lighthouse Along the border of the lake is one so we'll say the lake has a circumference of two now the apparent brightness is one divided by the diameter squared and In this case the diameter is that circumference 2 divided by pi so the apparent brightness works out to be pi squared divided by 4 Now for our first transformation draw a new circle twice as big so circumference 4 and Draw a tangent line to the top of the small circle then replace the original lighthouse with two new ones where this tangent line intersects the larger circle an Important fact from geometry that we'll be using over and over here Is that if you take the diameter of a circle and form a triangle with any point on the circle? The angle at that new point will always be 90 degrees the significance of that in our diagram here is that it means the inverse Pythagorean theorem applies and the brightness from those two new lighthouses equals the brightness from the first one namely pi squared divided by 4 as The next step draw a new circle twice as big as the last with a circumference 8 Now for each lighthouse take a line from that lighthouse through the top of the smaller circle Which is the center of the larger circle and consider the two points where that intersects with the larger circle Again, since this line is a diameter of that large circle Then the lines from those two new points to the observer are going to form a right angle Likewise by looking at this right triangle here whose hypotenuse is the diameter of the smaller circle You can see that the line from the observer to that original lighthouse is at a right angle With a new long line that we drew Good news, right? because that means we can apply the inverse Pythagorean theorem and that means that the apparent brightness from the original lighthouse is the same as the combined brightness from the two newer ones and Of course, you can do that same thing over on the other side drawing a line through the top of the smaller circle and getting two new lighthouses on the larger circle and Even nicer these four lighthouses are all going to be evenly spaced around the lake Why? Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right. We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. -It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible. \ No newline at end of file +it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You \ No newline at end of file diff --git a/2018/basel-problem/french/sentence_translations.json b/2018/basel-problem/french/sentence_translations.json index 77d196611..ac93e637a 100644 --- a/2018/basel-problem/french/sentence_translations.json +++ b/2018/basel-problem/french/sentence_translations.json @@ -63,7 +63,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "En géométrie sphérique, vous parlez parfois de l'angle solide d'une forme, qui est la proportion d'une sphère qu'elle couvre vue d'un point donné. Vous voyez le premier des deux endroits où cette histoire à laquelle nous pensons est utile. comprendre la loi du carré inverse qui est un phénomène distinctement tridimensionnel, pensez à tous les rayons de lumière frappant un écran à une unité de la source alors que vous doublez la distance, ces rayons couvriront désormais une zone de deux fois la largeur et deux fois la hauteur Il faudrait donc quatre copies de cet écran original pour recevoir les mêmes rayons à cette distance. Ainsi, chaque individu reçoit 1 quart de lumière. C'est dans le sens dans lequel je veux dire qu'une lumière apparaîtrait 1 quart plus brillante deux fois la distance. De même, lorsque vous êtes trois fois plus loin, vous auriez besoin de neuf copies de cet écran d'origine pour recevoir les mêmes rayons, de sorte que chaque écran individuel ne reçoive que 1 9ème de lumière et ce modèle continue car la zone touchée par une lumière augmente du carré de à la distance, la luminosité de cette lumière diminue du carré inverse de cette distance et comme je suis sûr que beaucoup d'entre vous le savent, cette loi du carré inverse n'est pas du tout spéciale pour la lumière. Elle apparaît chaque fois que vous avez une sorte de quantité qui s'étale uniformément à partir d'une source ponctuelle, qu'il s'agisse de son, de chaleur ou de signal radio, un réseau infini de phares uniformément espacés met physiquement en œuvre le problème de Bâle. Mais encore une fois, ce dont nous avons besoin si nous voulons faire des progrès ici, c'est de comprendre comment nous pouvons manipuler les configurations. avec des sources de lumière comme celle-ci sans modifier la luminosité totale de l'observateur et l'élément de base clé est un moyen particulièrement intéressant de transformer un seul phare en deux. Pensez à un observateur à l'origine du plan XY et à un seul phare assis quelque part sur celui-ci. plan Maintenant, tracez une ligne de ce phare à l'observateur, puis une autre ligne perpendiculaire à celle du phare. Maintenant, placez deux phares à l'endroit où cette nouvelle ligne coupe les axes de coordonnées. Je vais continuer et appeler le phare a ici à gauche et phare B sur la face supérieure Il s'avère et vous verrez pourquoi cela est vrai dans une minute seulement, la luminosité que l'observateur ressent de ce premier phare est égale à la luminosité combinée ressentie par les phares A et B ensemble Et je devrais dire par la façon dont l'hypothèse dominante tout au long de cette vidéo est que tous les phares sont équivalents. Ils utilisent la même ampoule émettant la même puissance, tout cela. Donc, en d'autres termes, attribuer des variables aux choses ici si nous appelons la distance entre l'observateur et le phare une petit a Et la distance de l'observateur au phare B petit B et la distance au premier phare H Nous avons la relation 1 sur a au carré plus 1 sur b au carré est égal à 1 sur h au carré C'est le théorème de Pythagore inverse beaucoup moins connu que certains d'entre vous reconnaîtront peut-être grâce à la vidéo la plus récente et, je dirais, la plus excellente, d'un mathématicien sur les nombreux cousins du théorème de Pythagore. Une relation assez cool, ne pensez-vous pas et si vous êtes un mathématicien dans l'âme, vous vous demandez peut-être en ce moment comment le prouver et il existe des moyens simples d'exprimer l'aire des triangles de deux manières distinctes et d'appliquer le théorème de Pythagore habituel. Mais il existe une autre méthode assez jolie que j'aimerais décrire brièvement ici et qui s'inscrit beaucoup plus bien dans notre scénario. parce qu'encore une fois, il utilise des intuitions de lumière et des écrans. Imaginez réduire tout le triangle rectangle en une version plus petite et pensez à cette hypoténuse miniature comme un écran recevant la lumière du premier phare. Si vous remodelez cet écran pour qu'il soit la combinaison des deux jambes du premier phare. triangle miniature comme celui-ci. Eh bien, il reçoit toujours la même quantité de lumière, n'est-ce pas ?", "from_community_srt": "En géométrie sphérique on parle parfois de \"l'angle solide\" d'une figure qui est la proportion de la sphère que la figure recouvre, vue d'un certain point Voyez, une première utilité de cette histoire d'écrans est de comprendre la loi des carrés inverse qui est un phénomène typique de la 3D. Pensez à tous les rayons atteignant un écran distant d'une unité de longueur quand on double la distance, ces rayons couvrent une aire deux fois plus large et deux fois plus haute donc il nous faudrait 4 copies de l'écran original pour recevoir la même chose à cette distance, donc chaque écran individuel reçoit 1/4 de la lumière. C'est dans ce sens que je dis qu'une lumière semble 1/4 aussi lumineuse quand elle est 2 fois plus loin Et de même, quand elle est 3 fois plus loin, Il vous faudrait 9 copies de l'écran d'origine pour recevoir les mêmes rayons, donc chacun reçoit 1/9 de la lumière et ce motif se répète, puisque l'aire atteindte par une lumière augmente comme le carré de la distance, donc la luminosité perçue diminue comme l'inverse du carré de cette distance et, je suis sûr que beaucoup parmi vous le savent, cette loi du carré inverse n'est pas spécifique à la lumière, elle apparaît dès que vous avez une quantité de quelque chose qui se propage uniformément à partir d'un point source, que ce soit du son, de la chaleur ou un signal radio Et rappelez-vous, c'est à cause de cette loi qu'une ligne infinie de phares espacés régulièrement implémente physiquement le problème de Bâle. mais, encore une fois, ce qu'il nous faut pour progresser, c'est comprendre comment manipuler les arrangements de sources lumineuses sans changer la luminosité totale perçue par l'observateur et la composante clé est une façon particulièrement agréable de transformer un phare en deux phares. Imaginez un observateur à l'origine du plan x-y, et un unique phare posé quelque part sur ce plan maintenant, tracez une droite de ce phare à l'observateur, et une autre, perpendiculaire à la première au niveau du phare et placez deux phares là où cette droite intersecte les axes que l'on va appeler Phare A là bas à gauche et Phare B là-haut. Il se trouve, vous allez voir pourquoi dans une minute, que la luminosité perçue par l'observateur du premier phare est égale à la luminosité combinée venant des phares A et B, et au fait, je devrais ajouter qu'on fait l'hypothèse, tout au long de la vidéo, que les phares sont tous équivalents, utilisent la même ampoule qui émet la même puissance, etc. Donc, en d'autres termes, en nommant les variables, si on appelle la distance de l'observateur au phare A petit a, et la distance de l'observateur au phare B petit b, et la distance de l'observateur au premier phare, h on a la relation 1 sur a au carré plus 1 sur b au carré égale 1 sur h au carré ce qui est le beaucoup moins célèbre Théorème de Pythagore inverse, que certains d'entre vous reconnaîtront peut-être de la dernière (et excellente) vidéo de Mathologer sur les cousins du théorème de Pythagore Plutôt cool, cette relation, vous ne trouvez pas ? Si vous êtes un mathématicien dans l'âme, vous vous demandez peut-être comment on le prouve, et il y a des méthodes directes où l'on exprime l'aire des triangles de deux façons différentes et on utilise le théorème de Pythagore habituel Mais il y a une autre méthode, très jolie, que j'aimerais mentionner brièvement, qui va beaucoup mieux avec notre histoire puisqu'elle utilise aussi des intuitions de lumière et d'écrans. Imaginez rétrécir tout le triangle rectangle en une version minuscule et pensez à l'hypothénuse miniature comme un écran recevant de la lumière du premier phare Si vous changez la forme de cet écran, pour qu'il couvre les deux autres côtés du triangle miniature, comme ceci Eh bien, il reçoit toujours la même quantité de lumière,", "n_reviews": 0, @@ -119,7 +119,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "Cela signifie que la luminosité apparente serait diminuée d'un facteur quatre et c'est aussi une algèbre relativement simple allant de la somme sur tous les entiers à la somme sur les entiers pairs Implique une multiplication par 1 4ème Et ce que cela signifie, c'est que passer de tous les entiers pairs les entiers aux impairs se multiplieraient par 3 quarts Puisque les pairs plus les impairs doivent nous donner le tout Donc si nous retournons simplement cela, cela signifie que passer de la somme sur les nombres impairs à la somme sur tous les entiers positifs nécessite de multiplier par 4 tiers Donc, en prenant ce pi au carré sur 8 en multipliant par 4 tiers bada boom bada bing Nous avons nous-mêmes une solution au problème du basilic Maintenant, cette vidéo que vous venez de regarder a été principalement écrite et animée par l'un des trois nouveaux bleu un marron membres de l'équipe Ben Hambricht Un ajout rendu possible.", "from_community_srt": "la luminosité totale est divisée par 4 C'est aussi direct avec un peu de calcul, passer de la somme sur tous les entiers à la somme sur les pairs implique de multiplier par 1/4. Ce que cela signifie, c'est que passer de tous les entiers à seulement les impairs revient à multiplier par 3/4, puisque les pairs plus les impairs doit nous redonner la somme totale Donc, si on retourne l'argument, passer de la somme sur les impairs à la somme sur tous les entiers positifs requiert de multiplier par 4/3 Donc, en partant de pi au carré sur 8, et en multipliant par 4/3, boum badabing, on a une solution du problème de Bâle Cette vidéo a été principalement écrite et animée par un nouveau membre de l'équipe 3blue1brown,", "n_reviews": 0, diff --git a/2018/basel-problem/german/sentence_translations.json b/2018/basel-problem/german/sentence_translations.json index ff1738d11..b4cb26960 100644 --- a/2018/basel-problem/german/sentence_translations.json +++ b/2018/basel-problem/german/sentence_translations.json @@ -72,7 +72,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "In der sphärischen Geometrie spricht man manchmal vom Raumwinkel einer Form. Das ist der Anteil einer Kugel, den sie von einem bestimmten Punkt aus gesehen abdeckt. Der erste von zwei Punkten, an denen diese Geschichte, in der wir an Bildschirme denken, nützlich ist, ist das Verständnis des inversen Quadratgesetzes.Wenn du die Entfernung verdoppelst, decken diese Strahlen nun einen Bereich ab, der doppelt so breit und hoch ist. Man bräuchte also vier Kopien des ursprünglichen Bildschirms, um dieselben Strahlen in dieser Entfernung zu empfangen, und jede einzelne Das bedeutet, dass das Licht in der doppelten Entfernung ein Viertel so hell erscheint In der dreifachen Entfernung bräuchtest du neun Kopien des ursprünglichen Bildschirms, um die gleichen Strahlen zu empfangen, so dass jeder einzelne Bildschirm nur noch ein Neuntel der Lichtmenge empfängt Und dieses Muster setzt sich fort, weil die Fläche, die von einem Licht getroffen wird, mit dem Quadrat der Entfernung zunimmt, während die Helligkeit des Lichts mit dem umgekehrten Quadrat der Entfernung abnimmt, und wie viele von euch sicher wissen, gilt dieses Gesetz des umgekehrten Quadrats nicht nur für Licht. Es taucht immer dann auf, wenn sich eine Größe gleichmäßig von einer Punktquelle aus ausbreitet, sei es Schall oder Wärme oder ein Funksignal und eine unendliche Reihe gleichmäßig verteilter Leuchttürme setzt das Basel-Problem physikalisch um. Aber auch hier müssen wir, wenn wir Fortschritte machen wollen, verstehen, wie wir Aufstellungen mit Lichtquellen wie dieser manipulieren können ohne die Gesamthelligkeit für den Beobachter und den Leuchtturm zu verändern Der wichtigste Baustein ist eine besonders schöne Art, einen einzelnen Leuchtturm in zwei zu verwandeln Denke an einen Beobachter im Ursprung der XY-Ebene und einen einzelnen Leuchtturm, der irgendwo auf dieser Ebene steht Zeichne nun eine Linie von diesem Leuchtturm zum Beobachter und dann Nun platziere zwei Leuchttürme dort, wo diese neue Linie die Koordinatenachsen schneidet. Ich nenne sie Leuchtturm A hier auf der linken Seite und Leuchtturm B auf der oberen Seite Es stellt sich heraus, dass die Helligkeit, die der Beobachter Die Helligkeit, die der Beobachter von diesem ersten Leuchtturm erfährt, ist gleich der kombinierten Helligkeit von Leuchtturm A und B. Ich sollte übrigens sagen, dass in diesem Video davon ausgegangen wird, dass alle Leuchttürme gleichwertig sind. Sie verwenden dieselbe Glühbirne, die dieselbe Energie ausstrahlt. Wenn wir also die Entfernung vom Beobachter zum Leuchtturm a als kleines a und die Entfernung vom Beobachter zum Leuchtturm B als kleines B und die Entfernung zum ersten Leuchtturm H bezeichnen, haben wir die Beziehung 1 über a zum Quadrat plus 1 über b zum Quadrat gleich 1 über h zum Quadrat.Dies ist der weniger bekannte umgekehrte Satz des Pythagoras, den einige von euch vielleicht aus dem neuesten und ich würde sagen, ausgezeichneten Video von math ologer über die vielen Verwandten des Satzes des Pythagoras kennen Ziemlich coole Beziehung, findet ihr nicht auch und wenn ihr im Herzen Mathematiker seid, fragt ihr euch vielleicht gerade, wie ihr das beweisen könnt und Es gibt einige einfache Möglichkeiten, bei denen ihr die Fläche des Dreiecks auf zwei verschiedene Arten ausdrückt und den üblichen Satz des Pythagoras anwendet. Aber es gibt noch eine andere, sehr hübsche Methode, die ich hier kurz vorstellen möchte und die viel besser zu unserer Geschichte passt, denn sie nutzt die Intuition des Lichts und des Bildschirms Stell dir vor, du verkleinerst das ganze rechtwinklige Dreieck und stellst dir diese Miniatur-Hypotenuse als einen Bildschirm vor, der das Licht vom ersten Leuchtturm empfängt, empfängt er immer noch die gleiche Menge Licht, oder?", "model": "DeepL", "from_community_srt": "den Raumwinkel, der Anteil einer Kugel der vom Leuchtturm im Zentrum der Kugel angestrahlt wird. Dies ist von Bedeutung, um das Auftauchen der Reziproken der Quadrate zu erklären. Betrachte alle Strahlen die den Schirm bei einer Distanz von 1 treffen. Pro Einheit (1) wird die Höhe und die Breite nun um 1 erhöht da der \"Anstieg\" der Strahlen gleich bleibt. Ist die Distanz 2 werden jene Strahlen eine Fläche von 4 mal dem Schirm bestrahlen könne. Daher kann aber jeder einzelne jener Schirme nur 1/4 des Lichtes abfangen. Daher erscheint der 2 Einheiten entfernte Leuchtturm 1/4 so hell wie der Erste. Ist man 3 Einheiten entfernt benötigt man folglich 9 Schirme wobei jeder 1/9 des Lichts abfängt. Diese Muster setzt sich fort. Da der Flächeninhalt dem Quadrat der Distanz gleicht wird eine ... Einheit von dem Reziproken des Quadrates bestrahlt, ist eben so Hell. Und wie viele von euch Wissen tritt dies immer auf wenn etwas kreisförmig von einem Punkt ausgestrahlt wird Mag dies Wärme, Schall, Radiowellen oder elektrische Feldstärke sein. Und wegen dieses Verhaltens ist die Helligkeit der unendlich vielen Leuchttürme dem Basel Problem gleich Aber was wir wirklich benötigen um voranzukommen ist ein Weg die Anordnung der Türme zu verändern ... ohne dabei jedoch die Helligkeit an 0 zu verändern. Die Magie wirkt durch eine speziellen Methode einen Leuchtturm durch Zwei zu ersetzen. Du sitzt im Koordinaten Ursprung eines x-y Diagramms Irgendwo steht ein Leuchtturm Zeichne die Linie Turm-Ich und dann eine weiter welche zu der ersten rechtwinklig steht Stelle nun Türme an den Durchstoßpunkten auf Nennen wir die neuen Türme A und B und den ursprünglichen H Dann gilt für die Helligkeiten dass, H = A + B Und ja alle Türme sind identisch Wenn a die Strecke 0-A ist und b := 0-B und h := 0-h Dann gilt : dies ist die weniger bekannte Umkehrung des Pythagoras welche die die besonders schnuckligen unter euch von Mathologers epochalem Meisterwerk kennen. Ziemlich süße Formel, findest du nicht ? Und bist du ein stahlhartgesottener Mathematiker fragst du nun natürlich wie man dies Beweist Und es gibt sehr direkte Wege wenn man den Flächeninhalt des Dreiecks unterschiedlich ausdrückt... ... und dann mit standart Pythagoras verfährt blablabla ABER es gibt noch eine andere sahnige Methode welche ich kurz zeigen will da sie gut in jene Geschichte passt welche wir hier versuchen zu schreiben. Eine Geschichte über Licht. Skaliere das Dreieck zu einer handlicheren Version und betrachte diese winz Hypothenuse als Schirm der Licht des Turmes abfängt wenn du nun jenen Schirm durch die Katheten ausdrückst ... ...", @@ -135,7 +135,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "Das bedeutet, dass die scheinbare Helligkeit um den Faktor vier abnimmt und Das ist auch eine relativ einfache Algebra Wenn man von der Summe über alle ganzen Zahlen zur Summe über die geraden ganzen Zahlen geht, muss man mit 1 Viertel multiplizieren Und das bedeutet, dass man von allen ganzen Zahlen zu den ungeraden mit 3 Vierteln multiplizieren muss, da die geraden plus die ungeraden Zahlen das Ganze ergeben müssen Wenn wir das also einfach umdrehen Wenn wir das also umdrehen, bedeutet das, dass wir von der Summe über die ungeraden Zahlen zur Summe über alle positiven ganzen Zahlen kommen, indem wir mit 4 Terzen multiplizieren. Also nehmen wir Pi Quadrat über 8 und multiplizieren mit 4 Terzen bada boom bada bing Wir haben eine Lösung für das Basilikum-Problem Dieses Video, das du gerade gesehen hast, wurde hauptsächlich von einem der neuen drei blauen und einem braunen Teammitglieder geschrieben und animiert: Ben Hambricht Eine Addition wird möglich.", "model": "DeepL", "from_community_srt": "Und daher musst du die Ursprüngliche Reihe durch 2^2 Teilen um die Summe der Geraden Quadrate zu erhalten Und dies ist simple Algebra. Die Summe aller Zahlen mal 2 ist die Summe aller geraden Zahlen und darum folgt dann das Vorgehen bei den Reziproken Daher muss mann aber wenn man von allen Reziproken zu denen der ungeraden Quadrate will mal 3/4 rechnen Da die geraden und die ungeraden uns das ganze geben müssen A = 0.25A + xA ; x = 0.75 Wer dann von den Reziproken der ungeraden Quadrate zu allen gehen will muss dann mit mal 4/3 rechnen Daher wird π^2 / 8 zu π^2 / 6 Daher wird π^2 / 8 zu π^2 / 6 HMMMMMMMMMMMMMMMMM", diff --git a/2018/basel-problem/greek/sentence_translations.json b/2018/basel-problem/greek/sentence_translations.json index 3cfc1afbb..ed7bfd4e5 100644 --- a/2018/basel-problem/greek/sentence_translations.json +++ b/2018/basel-problem/greek/sentence_translations.json @@ -63,7 +63,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "", "from_community_srt": "Στη γεωμετρία των σφαιρών , αναφερόμαστε στη στερεά γωνία ενός σχήματος... ... η οποία είναι το ποσοστό μιας σφαίρας που \"καλύπτεται\" από τις ακτίνες που εκκινούν από την κορυφή της γωνίας αυτής. Βλέπετε , το πρώτο από τα δύο μέρη αυτής της ιστορίας στα οποία σκέφτομαι ότι οι \"οθόνες\" θα μας είναι χρήσιμες είναι η κατανόηση του νόμου του \"αντιστρόφου τετραγώνου\" ... ...το οποίο είναι ένα χαρακτηριστικά τρισδιάστατο φαινόμενο. Σκεφτείτε όλες τις ακτίνες του φωτός η οποίες καταλήγουν σε μια οθόνη , μία μονάδα μήκους μακριά από την πηγή. Καθώς διπλασιάζετε την απόσταση, αυτές οι ακτίνες θα καλύπτουν τώρα μια περιοχή με διπλάσιο πλάτος... ...και διπλάσιο ύψος. Έτσι θα χρειαζόταν 4 αντίγραφα αυτής της αρχικής οθόνης για να λάβουμε την ίδια ένταση σε αυτή την απόσταση οπότε το κάθε... ...κομμάτι λαμβάνει το ένα τέταρτο της έντασης φωτός της πρώτης οθόνης. Αυτό εννοώ όταν λέω πως ένα φως, στη διπλάσια απόσταση , θα φαινόταν φωτεινό όσο το 1/4 του πρώτου. Ομοίως όταν είμαστε τρεις φορές μακρύτερα; Θα χρειαζόμασταν εννέα αντίγραφα της αρχικής οθόνης για να λάβουμε τις ίδιες ακτίνες , οπότε κάθε μεμονωμένη οθόνη λαμβάνει μόνο το 1/9 της πρώτης. Αυτό το μοτίβο συνεχίζεται επειδή η περιοχή που καλύπτεται από τις ακτίνες φωτός αυξάνεται βάση του τετραγώνου της απόστασης , και η φωτεινότητα του κάθε \"κομματιού\" μειώνεται, κατά το αντίστροφο τετράγωνο αυτής της απόστασης και... ...είμαι σίγουρος ότι πολλοί από εσάς γνωρίζετε ότι αυτός ο νόμος του αντίστροφου τετραγώνου... ...δεν ισχύει αποκλειστικά για το φως , αλλά εμφανίζεται κάθε φορά που έχετε... ...κάποιο είδος ποσότητας που εξαπλώνεται ομοιόμορφα από μια πηγή εκπομπής.... ...είτε πρόκειται για ήχο , είτε για θερμότητα , ραδιόφωνο , ή κάτι παρόμοιο. Σημειώστε ότι εξαιτίας του νόμου του αντιστρόφου τετραγώνου, μια άπειρη σειρά... ...από ομοιόμορφα τοποθετημένους φάρους , υλοποιεί \"έμπρακτα\" το πρόβλημα της Βασιλείας. Αλλά και πάλι, αυτό που χρειαζόμαστε, αν θέλουμε να σημειώσουμε κάποια πρόοδο , είναι να κατανοήσουμε πώς μπορούμε να χειριστούμε τη διαρύθμιση... ...των πηγών φωτός όπως οι φάροι , χωρίς να αλλάξει η συνολική φωτεινότητα για τον παρατηρητή. Και το βασικό δομικό στοιχείο είναι ένας ιδιαίτερα καλός τρόπος να μετατραπεί ένας μόνο φάρος σε ... δύο! Σκεφτείτε έναν παρατηρητή στην αρχή των αξόνων του επιπέδου xy και έναν φάρο που βρίσκεται κάπου σε αυτό το επίπεδο. Τραβήξτε μια γραμμή από τον φάρο στον παρατηρητή και στη συνέχεια μια άλλη γραμμή, κάθετη προς εκείνη στο σημείο που βρίσκεται ο φάρος. Τοποθετήστε τώρα δύο φάρους στα σημεία που η νέα αυτή γραμμή τέμνει τους άξονες του επιπέδου... ...τους οποίους θα ονομάσω \"Α\" τον φάρο εδώ στα αριστερά και \"Β\" τον φάρο στην επάνω πλευρά... Αποδεικνύεται , και θα δούμε αυτό αληθεύει σε λίγο , ότι.... ...η φωτεινότητα που αντιλαμβάνεται ο παρατηρητής από τον πρώτο φάρο... ...είναι ίση με τη συνδυασμένη φωτεινότητα των φάρων Α και Β μαζί , και πρέπει να πούμε πως... υποθέτουμε καθ' όλη τη διάρκεια αυτού του βίντεο.... ...ότι όλοι οι φάροι είναι ισοδύναμοι , ... ...χρησιμοποιούν τον ίδιο λαμπτήρα , εκπέμπουν την ίδια ισχύ κτλ.... Έτσι, με άλλα λόγια, αναθέτοντας μεταβλητές στα πράγματα εδώ , αν ονομάσουμε την απόσταση από τον παρατηρητή στο φάρο Α \"a\" και την απόσταση από τον παρατηρητή στο φάρο Β \"b\" και τέλος ,την απόσταση από τον πρώτο φάρο \"h\"... ...θα έχουμε τη σχέση 1 προς a^2 συν 1 προς b^2 ισούται με 1 προς h^2 . Αυτό, είναι το πολύ λιγότερο γνωστό... ... \"αντίστροφο θεώρημα του Πυθαγόρα\", το οποίο μερικοί από σας... ...μπορεί να γνωρίσατε από το πρόσφατο εξαιρετικό βίντεο του \"Mathologist\" σχετικά με τα πολλά \"ξαδέλφια\" του θεωρήματος... ...πολύ ενδιαφέρον, δεν συμφωνείτε; Και εάν έχετε \"μαθηματική καρδιά\" ίσως να αναρωτιέστε τώρα \"Πως αποδεικνύεται αυτό\"; Υπάρχουν λοιπόν μερικοί απλοί τρόποι υπολογισμού της επιφάνειας των τριγώνων με δύο διαφορετικούς τρόπους... ...και εφαρμογή στη συνέχεια του συνηθισμένου θεωρήματος του Πυθαγόρα. Αλλά υπάρχει μια άλλη πολύ όμορφη μέθοδος που θα ήθελα να περιγράψω εν συντομία εδώ που \"δένει\" πολύ καλύτερα με την ιστορία μας επειδή... ...χρησιμοποιεί και αυτή διαισθητικά \"φως\" και \"οθόνες\". Φανταστείτε την σμίκρυνση ολόκληρου του ορθογωνίου τριγώνου σε μια μικρότερη έκδοση... ....και \"δείτε\" αυτήν την μικροσκοπική υποτείνουσα μεταφορικά ως οθόνη που λαμβάνει φως από τον πρώτο φάρο. Εάν αναδιαμορφώσουμε την οθόνη, έτσι ώστε να αποτελείται από τον συνδυασμό των δύο πλευρών... ...του μικροσκοπικού τριγώνου, θα εξακολουθεί να λαμβάνει την ίδια ποσότητα φωτός...", "n_reviews": 0, @@ -119,7 +119,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "", "from_community_srt": "σημαίνει ότι η φαινομενική φωτεινότητα θα μειωθεί κατά ένα συντελεστή 4 και; Αυτό είναι επίσης σχετικά απλή άλγεβρα πηγαίνει Από το άθροισμα πάνω από όλους τους ακέραιους έως το άθροισμα πάνω από τους ζυγούς ακεραίους συμπεριλαμβάνεται πολλαπλασιασμός κατά 1/4. Και τι σημαίνει αυτό; Είναι αυτό που πηγαίνει από όλους τους ακεραίους στους περίεργους; Θα πολλαπλασιάζονταν κατά 3/4 από τα ζυγώματα και τις πιθανότητες πρέπει να μας δώσουν το όλο θέμα Έτσι αν το απλά γυρίσουμε γύρω από αυτό σημαίνει να πηγαίνεις, από το άθροισμα πάνω από τους μονούς αριθμούς μέχρι το άθροισμα πάνω από όλους τους θετικούς ακέραιους, απαιτεί πολλαπλασιασμό κατά 4/3 Λαμβάνοντας λοιπόν αυτό το π εις το τετράγωνο δια 8 πολλαπλασιάζοντας, κατά 4/3 αλλά ένα, έχουμε λύσει το πρόβλημα της βασιλείας Αυτό, το βίντεο που παρακολουθήσατε ήταν κυρίως γραμμένο και κινούμενο από ένα από τα νέα", "n_reviews": 0, diff --git a/2018/basel-problem/hebrew/sentence_translations.json b/2018/basel-problem/hebrew/sentence_translations.json index 3b5bbba6a..a4ccd1539 100644 --- a/2018/basel-problem/hebrew/sentence_translations.json +++ b/2018/basel-problem/hebrew/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", "translatedText": "אני מתכוון שקרני האור הפוגעות באחת משתי הרגליים האלה זהות בדיוק לקרניים שפוגעות בתחתית האדמה אז המפתח הוא שכמות האור מהמגדלור הראשון שהוא פוגע בצד שמאל זה הזווית המוגבלת של הקרניים שבסופו של דבר פוגעות. המסך הזה זהה בדיוק לכמות האור כאן שמגיע ממגדלור a שפוגע בצד זה תהיה אותה זווית של קרניים ובאופן סימטרי כמות האור מהבית הראשון שפוגע בחלק התחתון של המסך שלנו זהה כמו כמות האור שפוגעת בחלק הזה ממגדלור B למה אתה יכול לשאול טוב, זה עניין של משולשים דומים האנימציה הזו כבר נותנת לך רמז חזק לאיך זה עובד וגם השארנו קישור בתיאור לגיאוגברה פשוטה יישומון לאלו מכם שרוצים לחשוב על זה בסביבה קצת יותר אינטראקטיבית ובמשחק עם העובדה החשובה הזו כאן שתוכלו לראות היא שהמשולשים הדומים חלים רק במקרה המגביל עבור מסך זעיר מאוד בסדר, תתכופף עכשיו כי כאן הדברים משתפרים יש לנו את משפט פיתגורס ההפוך הזה, נכון?", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right.", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right.", "translatedText": "ובכן, הקווים מהמגדלורים האלה למרכז נמצאים בזוויות של 90 מעלות זה עם זה אז מכיוון שהדברים הם סימטריים משמאל לימין זה אומר שהמרחקים לאורך ההיקף הם 1 2 2 2 ו-1 בסדר, אולי תראה לאן זה הולך, אבל אני רוצה לעבור דרך זה רק עוד שלב אחד אתה מצייר עיגול גדול פי שניים אז היקף של 16 עכשיו ולכל מגדלור אתה מצייר קו מהמגדלור הזה דרך החלק העליון של המעגל הקטן יותר שהוא מרכז המעגל הגדול יותר ולאחר מכן ליצור שני מגדלורים חדשים שבהם הקו הזה נחתך עם המעגל הגדול יותר בדיוק כמו קודם מכיוון שהקו הארוך הוא קוטר של המעגל הגדול שני המגדלורים החדשים האלה יוצרים זווית ישרה עם הצופה ימינה וממש כמו לפני הקו מהצופה אל המגדלור המקורי הוא בניצב לקו הארוך ואלה שתי העובדות שמצדיקות אותנו בשימוש במשפט פיתגורס ההפוך אבל מה שאולי לא כל כך ברור הוא שכשאתה עושה זאת כדי שכל המגדלורים יקבלו שמונה חדשים ב-The Big אגם שמונת המגדלורים החדשים האלה יהיו מרווחים באופן שווה. זו החלק האחרון של הוכחת גיאומטריה לפני הדחף הסופי כדי לראות את זה, זכור שאם אתה מצייר קווים משני מגדלורים סמוכים על האגם הקטן למרכז הם יוצרים זווית של 90 מעלות. במקום זאת אתה מצייר קווים לנקודה בכל מקום על היקף המעגל שאינה ביניהם, משפט הזווית הכתובה מאוד שימושית מהגיאומטריה אומר לנו שזה יהיה בדיוק חצי מהזווית שהם יוצרים עם המרכז במקרה זה 45 מעלות אבל כאשר אנו ממקמים את הנקודה החדשה בראש האגם אלו שני הקווים המגדירים את מיקומם של המגדלורים החדשים באגם הגדול יותר. המשמעות היא שכאשר אתה מצייר קווים משמונת המגדלורים החדשים אל המרכז, הם מחלקים את המעגל באופן שווה לחתיכות זווית של 45 מעלות וזה אומר ששמונת המגדלורים מרווחים באופן שווה סביב ההיקף עם מרחק של שניים בין כל אחד מהם ועכשיו רק דמיינו את הדבר הזה משחק בכל שלב מכפיל את גודלו של כל עיגול והופך כל מגדלור ל שניים חדשים לאורך קו הנמשך דרך מרכז המעגל הגדול יותר בכל צעד הבהירות הנראית לעין למתבונן נשארת זהה פי בריבוע על פני 4 ובכל צעד המגדלורים נשארים מרווחים באופן שווה עם המרחק 2 בין כל אחד מהם על היקף ובמגבלה מה שאנחנו מקבלים כאן הוא קו אופקי שטוח עם מספר אינסופי של מגדלורים מרווחים באופן שווה בשני הכיוונים ומכיוון שהבהירות הנראית לעין הייתה פי בריבוע על פני 4 לאורך כל הדרך, זה יהיה נכון גם במקרה המגביל הזה. זה נותן לנו סדרה אינסופית די מדהימה סכום הריבועים ההפוכים 1 על n בריבוע כאשר n מכסה את כל המספרים השלמים האי-זוגיים 1 3 5 וכן הלאה, אך גם שלילי 1 שלילי 3 שלילי 5 כבוי בכיוון שמאלה חיבור כל אלה למעלה הולך לתת לנו פאי בריבוע על 4 זה מדהים וזה הליבה של מה שאני רוצה להראות לך ופשוט קח צעד אחורה ותחשוב על כמה זה נראה לא אמיתי סכום השברים הפשוטים שבמבט ראשון אין להם שום קשר לגיאומטריה אין שום קשר למעגלים בכלל, כנראה נותן לנו את התוצאה הזו שקשורה ל-pi אלא שעכשיו אתה יכול לראות מה זה קשור לגיאומטריה, קו המספרים הוא סוג של גבול של עיגולים שגדלים כל הזמן, וכאשר אתה מסכם על פני המספר הזה קו מקפיד לסכם כל הדרך עד האינסוף משני הצדדים זה בערך כמו שאתה מצטבר לאורך הגבול של עיגול גדול לאין שיעור ודרך דיבור מאוד רופפת אבל מאוד כיפית אבל רגע אולי תגיד שזה לא הסכום שהבטחת לנו בתחילת הסרטון ובכן אתה צודק.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible.", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "נותרה לנו קצת חשיבה ראשית דבר ראשון, בואו רק נגביל את הסכום להיות המספרים האי-זוגיים החיוביים בלבד, מה שמביא אותנו בריבוע ה-Pi חלקי 8 כעת ההבדל היחיד בין זה לסכום שאנו מחפשים עובר כל המספרים השלמים החיוביים אי-זוגיים וזוגיים זה שחסר את סכום ההדדיות של המספרים הזוגיים מה שאני צובע באדום כאן למעלה עכשיו אתה יכול לחשוב על הסדרה החסרה כעותק מוקטן של כל הסדרה שאנחנו רוצים איפה כל מגדלור עובר למרחק פי שניים מהמקור אחד מוזז לשניים שניים מוזז לארבע שלוש מוזז לשש וכן הלאה ומכיוון שזה כרוך בהכפלת המרחק עבור כל מגדלור זה אומר שהבהירות הנראית לעין תפחת בפקטור של ארבעה וזו גם אלגברה פשוטה יחסית מעבר מהסכום על כל המספרים השלמים לסכום על המספרים הזוגיים כולל הכפלה ב-1 4 ומה זה אומר שמעבר מכל המספרים השלמים לא-אי-זוגיים תהיה הכפלה ב-3 4יות שכן הזוגות פלוס הסיכויים צריכים לתת לנו את כל העניין אז אם רק נהפוך את זה לאחור, זה אומר לעבור מהסכום על המספרים האי-זוגיים לסכום על כל המספרים השלמים החיוביים דורש הכפלה ב-4 שליש אז לקחת את ה-pi בריבוע על 8 וכפל ב- 4 שלישים badda בום badda bing יש לנו לעצמנו פתרון לבעיית הבזיליקום עכשיו הסרטון הזה שצפיתם זה עתה נכתב והונפש בעיקר על ידי אחד משלושת חברי הצוות החדשים כחול אחד חום בן המבריכט, תוספת שאפשרה.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/hindi/sentence_translations.json b/2018/basel-problem/hindi/sentence_translations.json index dbb17fe8f..807ea8be1 100644 --- a/2018/basel-problem/hindi/sentence_translations.json +++ b/2018/basel-problem/hindi/sentence_translations.json @@ -63,7 +63,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", "translatedText": "मेरा मतलब है कि उन दो पैरों में से एक पर पड़ने वाली प्रकाश की किरणें बिल्कुल वैसी ही होती हैं जैसी कि किरणें कर्ण पर पड़ती हैं, फिर मुख्य बात यह है कि पहले प्रकाशस्तंभ से प्रकाश की मात्रा इस बाईं ओर गिरती है, किरणों का सीमित कोण होता है जो अंत में टकराता है वह स्क्रीन बिल्कुल वैसी ही है जैसे यहां लाइटहाउस से आने वाली रोशनी की मात्रा उस तरफ पड़ती है, यह किरणों का कोण समान होगा और सममित रूप से हमारी स्क्रीन के निचले हिस्से से टकराने वाले पहले घर से प्रकाश की मात्रा समान है लाइटहाउस बी से उस हिस्से पर पड़ने वाले प्रकाश की मात्रा के रूप में आप क्यों पूछ सकते हैं, यह समान त्रिकोणों का मामला है यह एनीमेशन आपको पहले से ही एक मजबूत संकेत देता है कि यह कैसे काम करता है और हमने विवरण में एक सरल जियोजेब्रा के लिए एक लिंक भी छोड़ा है एप्लेट आप में से उन लोगों के लिए है जो इसे थोड़ा अधिक इंटरैक्टिव वातावरण में सोचना चाहते हैं और इसके साथ खेलना चाहते हैं, यहां एक महत्वपूर्ण तथ्य यह है कि आप देख पाएंगे कि समान त्रिकोण केवल बहुत छोटी स्क्रीन के लिए सीमित मामले में लागू होते हैं ठीक है अब कमर कस लें क्योंकि यहीं चीजें अच्छी होती हैं हमें यह व्युत्क्रम पायथागॉरियन प्रमेय मिल गया है, है ना?", "n_reviews": 0, "start": 449.27, @@ -91,14 +91,14 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right.", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right.", "translatedText": "खैर, उन प्रकाशस्तंभों से केंद्र तक की रेखाएं एक दूसरे के साथ 90 डिग्री के कोण पर हैं, इसलिए चूंकि चीजें बाएं से दाएं सममित हैं, इसका मतलब है कि परिधि के साथ दूरियां 1 2 2 2 और 1 हैं, ठीक है, आप देख सकते हैं कि यह कहाँ जा रहा है, लेकिन मैं बस एक और कदम के लिए इसके माध्यम से चलना चाहता हूं आप एक वृत्त को दोगुना बड़ा बनाते हैं इसलिए अब 16 की परिधि है और प्रत्येक प्रकाशस्तंभ के लिए आप उस प्रकाशस्तंभ से छोटे वृत्त के शीर्ष के माध्यम से एक रेखा खींचते हैं जो बड़े वृत्त का केंद्र है और फिर दो नए प्रकाशस्तंभ बनाएं जहां वह रेखा बड़े वृत्त के साथ प्रतिच्छेद करती है, ठीक पहले की तरह क्योंकि लंबी रेखा बड़े वृत्त का व्यास है, वे दो नए प्रकाशस्तंभ प्रेक्षक के दाईं ओर एक समकोण बनाते हैं और ठीक वैसे ही जैसे कि प्रेक्षक से रेखा के पहले मूल प्रकाशस्तंभ लंबी रेखा के लंबवत है और ये दो तथ्य हैं जो हमें व्युत्क्रम पाइथागोरस प्रमेय का उपयोग करने के लिए उचित ठहराते हैं लेकिन जो बात स्पष्ट नहीं हो सकती है वह यह है कि जब आप सभी प्रकाशस्तंभों के लिए ऐसा करते हैं तो बड़े पर आठ नए प्रकाशस्तंभ प्राप्त होते हैं झील उन आठ नए प्रकाशस्तंभों को समान दूरी पर रखने जा रही है यह अंतिम जोर से पहले ज्यामिति प्रमाणता का अंतिम बिट है इसे देखने के लिए याद रखें कि यदि आप छोटी झील पर दो आसन्न प्रकाशस्तंभों से केंद्र तक रेखाएँ खींचते हैं तो वे 90 डिग्री का कोण बनाते हैं यदि इसके बजाय आप वृत्त की परिधि पर कहीं भी एक बिंदु पर रेखाएँ खींचते हैं जो उनके बीच नहीं है, ज्यामिति से बहुत उपयोगी उत्कीर्ण कोण प्रमेय हमें बताता है कि यह उस कोण का बिल्कुल आधा होगा जो वे इस मामले में केंद्र के साथ बनाते हैं 45 डिग्री लेकिन जब हम उस नए बिंदु को झील के शीर्ष पर रखते हैं। ये दो रेखाएं हैं जो बड़ी झील पर नए प्रकाशस्तंभों की स्थिति को परिभाषित करती हैं। इसका मतलब यह है कि जब आप उन आठ नए प्रकाशस्तंभों से केंद्र में रेखाएं खींचते हैं तो वे वृत्त को विभाजित करते हैं। समान रूप से 45 डिग्री के कोण वाले टुकड़ों में और इसका मतलब है कि आठ प्रकाशस्तंभ परिधि के चारों ओर समान रूप से दूरी पर हैं और उनमें से प्रत्येक के बीच दो की दूरी है और अब बस कल्पना करें कि यह चीज़ प्रत्येक चरण पर प्रत्येक वृत्त के आकार को दोगुना कर रही है और प्रत्येक प्रकाशस्तंभ को एक में बदल रही है। प्रत्येक चरण पर बड़े वृत्त के केंद्र के माध्यम से खींची गई एक रेखा के साथ दो नए, पर्यवेक्षक के लिए स्पष्ट चमक 4 से अधिक pi वर्ग के समान बनी रहती है और प्रत्येक चरण पर प्रकाशस्तंभ उनमें से प्रत्येक के बीच की दूरी 2 के साथ समान दूरी पर रहते हैं। परिधि और सीमा में हम यहां जो प्राप्त कर रहे हैं वह एक सपाट क्षैतिज रेखा है जिसमें अनंत संख्या में प्रकाशस्तंभ दोनों दिशाओं में समान रूप से दूरी पर हैं और क्योंकि स्पष्ट चमक पूरे रास्ते में पाई वर्ग 4 से अधिक थी जो इस सीमित मामले में भी सच होगी और यह हमें एक बहुत ही शानदार अनंत श्रृंखला देता है, जिसमें n वर्ग के ऊपर व्युत्क्रम वर्गों का योग 1 होता है, जहां n सभी विषम पूर्णांकों 1 3 5 इत्यादि को कवर करता है, लेकिन बाईं ओर नकारात्मक 1 नकारात्मक 3 नकारात्मक 5 बंद भी करता है, उन सभी को जोड़ना हमें पाई को 4 से अधिक वर्ग देने जा रहा है यह आश्चर्यजनक है और यह उसका मूल है जो मैं आपको दिखाना चाहता हूं और बस एक कदम पीछे हटें और सोचें कि यह कितना अवास्तविक लगता है सरल भिन्नों का योग जिसका पहली नजर में ज्यामिति से कोई लेना-देना नहीं है स्पष्ट रूप से वृत्तों से इसका कोई लेना-देना नहीं है, यह हमें यह परिणाम देता है जो पाई से संबंधित है, सिवाय इसके कि अब आप वास्तव में देख सकते हैं कि इसका ज्यामिति से क्या लेना-देना है, संख्या रेखा एक तरह से लगातार बढ़ते हुए वृत्तों की सीमा की तरह है और जैसे ही आप उस संख्या का योग करते हैं रेखा यह सुनिश्चित करती है कि दोनों तरफ अनंत तक सभी का योग हो। यह ऐसा है जैसे आप एक अनंत बड़े वृत्त की सीमा के साथ जोड़ रहे हैं और बोलने का एक बहुत ही ढीला लेकिन बहुत मजेदार तरीका है लेकिन रुकिए, आप कह सकते हैं कि यह योग नहीं है आपने वीडियो की शुरुआत में हमसे वादा किया था और आप सही हैं।", "n_reviews": 0, "start": 679.27, "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible.", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "हमारे पास थोड़ी सी सोच बाकी है, सबसे पहली बात तो यह है कि आइए योग को केवल सकारात्मक विषम संख्याओं तक ही सीमित रखें, जो हमें पाई को 8 से विभाजित करने पर वर्ग प्राप्त कराता है। अब इसके और जिस योग की हम तलाश कर रहे हैं, उसके बीच एकमात्र अंतर खत्म हो गया है। सभी सकारात्मक पूर्णांक विषम और सम हैं, इसमें सम संख्याओं के व्युत्क्रमों का योग गायब है जिसे मैं यहां लाल रंग में रंग रहा हूं अब आप उस लुप्त श्रृंखला को उस कुल श्रृंखला की एक स्केल की गई प्रतिलिपि के रूप में सोच सकते हैं जो हम चाहते हैं जहां प्रत्येक प्रकाशस्तंभ मूल से दुगनी दूरी पर चला जाता है, एक को दो पर स्थानांतरित कर दिया जाता है, दो को चार पर स्थानांतरित कर दिया जाता है, तीन को छह पर स्थानांतरित कर दिया जाता है और इसी तरह और क्योंकि इसमें प्रत्येक प्रकाशस्तंभ के लिए दूरी को दोगुना करना शामिल है, इसका मतलब है कि स्पष्ट चमक एक कारक से कम हो जाएगी चार में से और यह भी अपेक्षाकृत सरल बीजगणित है जो सभी पूर्णांकों के योग से लेकर सम पूर्णांकों के योग तक जाता है जिसमें 1/4वें से गुणा करना शामिल है और इसका मतलब यह है कि सभी पूर्णांकों से विषम पूर्णांकों तक जाने पर 3/4वें से गुणा करना होगा क्योंकि सम और विषम संख्याओं से हमें पूरी चीज़ मिलनी चाहिए, इसलिए यदि हम इसे पलट दें तो इसका मतलब है कि विषम संख्याओं के योग से सभी सकारात्मक पूर्णांकों के योग तक जाने के लिए 4 तिहाई से गुणा करने की आवश्यकता होती है, इसलिए उस पाई को 8 से गुणा करके वर्ग बनाते हैं। 4 तिहाई बड्डा बूम बड्डा बिंग, हमने तुलसी की समस्या का समाधान ढूंढ लिया है। अब यह वीडियो जो आपने अभी देखा है, वह मुख्य रूप से तीन ब्लू वन ब्राउन टीम के नए सदस्यों में से एक बेन हैम्ब्रिच द्वारा लिखा और एनिमेटेड किया गया था, जिसे जोड़ना संभव हो गया।", "n_reviews": 0, "start": 918.57, diff --git a/2018/basel-problem/hungarian/sentence_translations.json b/2018/basel-problem/hungarian/sentence_translations.json index 4611a28ce..73cb19e84 100644 --- a/2018/basel-problem/hungarian/sentence_translations.json +++ b/2018/basel-problem/hungarian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "A gömbi geometriában néha beszélünk egy alakzat térszögéről, ami a gömbnek egy adott pontból nézve azt az arányát jelenti, amit a gömb lefed, Látod, a két hely közül az első, ahol ez a történet, amire gondolunk, a képernyők hasznosak lesznek, az a fordított négyzetes törvény megértése, ami egy kifejezetten három- és egy négyzet alakú törvény.gondoljunk a fénysugarakra, amelyek a forrástól egy egységnyire lévő képernyőre esnek, ha megduplázzuk a távolságot, akkor ezek a sugarak kétszer olyan széles és kétszer olyan magas területet fognak lefedni. Tehát az eredeti képernyő négy példányára lenne szükség ahhoz, hogy ugyanazokat a sugarakat kapjuk a távolságból. egyenként 1 negyedannyi fényt kap Ez az az értelem, ami alatt azt értem, hogy a fény kétszer akkora távolságban 1 negyedannyi fényesnek tűnik Hasonlóképpen, ha háromszor messzebb vagyunk, akkor kilenc példányra lenne szükség az eredeti képernyőből, hogy ugyanazokat a sugarakat kapja, tehát minden egyes képernyő csak 1 9-ed annyi fényt kap. és ez a minta folytatódik, mert a fény által érintett terület a távolság négyzetével nő, a fény fény fényereje pedig a távolság fordított négyzetével csökken, és ahogyan azt bizonyára sokan tudják, ez a fordított négyzet törvény egyáltalán nem csak a fényre vonatkozik. egyenletesen terjed egy pontforrástól, legyen az hang, hő vagy rádiójel, vagy ilyesmi, és az egyenletes távolságban lévő világítótornyok végtelen sokasága fizikailag megvalósítja a bázeli problémát. De ismétlem, amire szükségünk van, ha előre akarunk lépni, az az, hogy megértsük, hogyan manipulálhatjuk az ilyen fényforrásokkal való elrendezéseket. anélkül, hogy a megfigyelő teljes fényerejét megváltoztatnánk és A legfontosabb építőelem egy különösen szép módja annak, hogy egyetlen világítótornyot kettővé alakítsunk Gondoljunk egy megfigyelőre az XY sík origójában és egy egyetlen világítótoronyra, amely valahol ezen a síkon áll Most húzzunk egy vonalat a világítótoronytól a megfigyelőig, majd pedig egy másik, erre merőleges vonalat a világítótoronynál Most helyezzünk el két világítótornyot ott, ahol ez az új vonal metszi a koordináta tengelyeket, amit én most elnevezek A világítótoronynak itt a bal oldalon és B világítótoronynak a felső oldalon Kiderül, és egy perc múlva látni fogjátok, hogy ez miért igaz, hogy a megfigyelő fényerejét Az első világítótorony fényereje megegyezik az A és B világítótorony együttes fényerejével. Egyébként meg kell jegyeznem, hogy a videó során az a feltételezés, hogy az összes világítótorony egyenértékű. Ugyanazt az izzót használják, és ugyanazt a teljesítményt sugározzák. Ha a megfigyelő és az a világítótorony távolságát a kis a-nak nevezzük, a megfigyelő és a B világítótorony távolságát a kis B-nek, és az első világítótorony távolságát H-nak, akkor az 1 a négyzet plusz 1 b négyzet egyenlő 1 h négyzet. Ez a sokkal kevésbé jó...inverz Pitagorasz-tétel, amit néhányan talán felismerhettek a matek ologer legutóbbi és mondhatom, hogy a legkiválóbb videójából a Pitagorasz-tétel számos rokonáról. Elég király kapcsolat, nem gondoljátok, és ha a szívetek mélyén matematikusok vagytok, akkor most azt kérdezhetitek, hogyan bizonyítjátok be, és van néhány egyszerű módszer, ahol a háromszögek területét két különböző módon fejezitek ki, és a szokásos Pitagorasz-tételt alkalmazzátok. van egy másik nagyon szép módszer, amit szeretnék röviden felvázolni, ami sokkal jobban beleillik a történetünkbe, mert ismét a fény és a képernyő intuícióit használja fel Képzeljük el, hogy az egész derékszögű háromszöget kicsinyítjük egy kisebb változatra, és ezt a miniatűr hipotenúzát úgy képzeljük el, mint egy képernyőt, ami fényt kap az első világítótoronyból Ha ezt a képernyőt úgy alakítjuk át, hogy az a miniatűr háromszög két lábának kombinációja legyen, így Nos, még mindig ugyanannyi fényt kap, igaz?", "model": "DeepL", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "Ez azt jelenti, hogy a látszólagos fényerő négyszeresére csökkenne, és Ez is viszonylag egyszerű algebrai feladat, ha az összes egész szám összegéből a páros egész számok összegébe megyünk, akkor 1 4-gyel kell szorozni És ez azt jelenti, hogy ha az összes egész számból a páratlanokba megyünk, akkor 3 4-gyel kell szorozni, mivel a párosak és a páratlanok adják az egészet Tehát ha ezt megfordítjuk, akkor ez azt jelenti, hogy a páratlan számok összegétől az összes pozitív egész szám összegéig eljutva 4 harmaddal kell szoroznunk Tehát ha a pi négyzetét 8-ra vesszük és 4 harmaddal szorozzuk bada bumm bada bing Megvan a megoldás a bazilika problémára Ezt a videót, amit most láttatok, elsősorban a három kék egy barna csapat egyik új tagja, Ben Hambricht írta és animálta Az összeadás lehetővé vált.", "model": "DeepL", "n_reviews": 0, diff --git a/2018/basel-problem/indonesian/sentence_translations.json b/2018/basel-problem/indonesian/sentence_translations.json index 901a0baad..b6b2773da 100644 --- a/2018/basel-problem/indonesian/sentence_translations.json +++ b/2018/basel-problem/indonesian/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", "translatedText": "Maksud saya sinar cahaya yang mengenai salah satu dari kedua kaki itu sama persis dengan sinar yang mengenai sisi miring. Maka kuncinya adalah banyaknya cahaya dari mercusuar pertama yang mengenai sisi kiri ini terbatasnya sudut sinar yang akhirnya mengenai layar itu persis sama dengan jumlah cahaya di sini yang datang dari mercusuar a yang mengenai sisi itu, sudut sinarnya akan sama dan secara simetris jumlah cahaya dari rumah pertama yang mengenai bagian bawah layar kita adalah sama sebagai jumlah cahaya yang mengenai bagian itu dari mercusuar B Mengapa Anda mungkin bertanya, ini soal segitiga sebangun Animasi ini sudah memberi Anda petunjuk kuat tentang cara kerjanya Dan kami juga meninggalkan tautan dalam deskripsi ke GeoGebra sederhana applet untuk Anda yang ingin memikirkan hal ini dalam lingkungan yang sedikit lebih interaktif dan bermain dengan satu fakta penting di sini yang dapat Anda lihat adalah bahwa segitiga serupa hanya berlaku dalam kasus pembatas untuk layar yang sangat kecil Baiklah kencangkan sabuk pengamanmu sekarang karena di sinilah segalanya menjadi lebih baik. Kita punya teorema Pythagoras terbalik, kan?", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right.", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right.", "translatedText": "Nah, garis-garis dari mercusuar ke pusat itu membentuk sudut 90 derajat satu sama lain. Jadi karena semuanya simetris dari kiri ke kanan, berarti jarak sepanjang kelilingnya adalah 1 2 2 2 dan 1 Baiklah, Anda mungkin bisa melihat ke mana arahnya, tapi aku ingin melewati ini satu langkah lagi Kamu menggambar sebuah lingkaran dua kali lebih besar jadi kelilingnya sekarang menjadi 16 dan untuk setiap mercusuar Kamu menarik garis dari mercusuar itu melalui puncak lingkaran yang lebih kecil Yang merupakan pusat dari lingkaran yang lebih besar lalu buat dua mercusuar baru yang garisnya berpotongan dengan lingkaran yang lebih besar. Sama seperti sebelumnya, karena garis panjang adalah diameter lingkaran besar, kedua mercusuar baru itu membentuk sudut siku-siku dengan pengamat di kanan dan persis seperti sebelum garis dari pengamat ke mercusuar aslinya Tegak Lurus terhadap garis panjang dan itulah dua fakta yang membenarkan kita dalam menggunakan teorema terbalik Pythagoras. Tapi yang mungkin kurang jelas adalah ketika Anda melakukan ini untuk semua mercusuar untuk mendapatkan delapan mercusuar baru di The besar danau kedelapan mercusuar baru itu akan diberi jarak yang sama Ini adalah bagian terakhir dari bukti geometri sebelum dorongan terakhir Untuk melihatnya, ingatlah bahwa jika Anda menggambar garis dari dua mercusuar yang berdekatan di danau kecil ke tengahnya, Mereka membentuk sudut 90 derajat Jika alih-alih Anda menggambar garis ke suatu titik di mana saja pada keliling lingkaran yang bukan di antara keduanya, teorema sudut tertulis yang sangat berguna dari geometri memberi tahu kita bahwa ini akan menjadi Persis setengah sudut yang mereka buat dengan pusat dalam hal ini 45 derajat Tapi ketika kita posisikan titik baru itu di puncak danau. Ini adalah dua garis yang menentukan posisi mercusuar baru di danau yang lebih besar. Artinya adalah ketika Anda menarik garis dari delapan mercusuar baru itu ke tengah, mereka membagi lingkaran secara merata menjadi potongan-potongan bersudut 45 derajat dan itu berarti delapan mercusuar ditempatkan secara merata di sekeliling keliling dengan jarak dua di antara masing-masing mercusuar dan Sekarang bayangkan benda ini bermain di setiap langkah menggandakan ukuran setiap lingkaran dan Mengubah setiap mercusuar menjadi dua mercusuar baru di sepanjang garis yang ditarik melalui pusat lingkaran yang lebih besar pada setiap langkah, kecerahan yang tampak bagi pengamat tetap sama pi kuadrat pada 4 dan pada setiap langkah mercusuar tetap berjarak sama dengan jarak 2 antara masing-masing mercusuar di keliling dan Dalam batas yang kita dapatkan di sini adalah garis horizontal datar dengan jumlah mercusuar yang tak terhingga yang berjarak sama di kedua arah dan Karena kecerahan semu adalah pi kuadrat pada 4 keseluruhannya, hal itu juga berlaku dalam kasus pembatas ini Dan Ini memberi kita deret tak terhingga yang mengagumkan, jumlah dari kuadrat terbalik 1 pada n kuadrat Dimana n mencakup semua bilangan bulat ganjil 1 3 5 dan seterusnya, tetapi juga negatif 1 negatif 3 negatif 5 ke arah kiri Menjumlahkan semuanya akan memberi kita pi kuadrat di atas 4 Itu luar biasa dan itulah inti dari apa yang ingin saya tunjukkan kepada Anda dan Coba mundur selangkah dan pikirkan betapa tidak nyatanya hal ini. Jumlah pecahan sederhana yang pada pandangan pertama tidak ada hubungannya dengan geometri tampaknya tidak ada hubungannya dengan lingkaran. Memberi kita hasil yang berhubungan dengan pi. Kecuali sekarang Anda benar-benar dapat melihat apa hubungannya dengan geometri, garis bilangan itu seperti batas lingkaran yang terus bertambah dan Saat Anda menjumlahkan bilangan itu garis memastikan untuk menjumlahkan hingga tak terhingga di kedua sisi Ini seperti Anda menjumlahkan sepanjang batas lingkaran yang sangat besar dan sangat longgar Cara berbicara yang sangat menyenangkan Tapi tunggu, Anda mungkin mengatakan ini bukan jumlah yang Anda janjikan kepada kami di awal video Dan Anda benar.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible.", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "Kita masih punya sedikit pemikiran lagi Hal pertama yang pertama, mari kita batasi jumlahnya menjadi hanya bilangan ganjil positif yang membuat kita pi kuadrat dibagi 8 Sekarang satu-satunya perbedaan antara ini dan jumlah yang kita cari sudah habis semua bilangan bulat positif ganjil dan genap adalah Jumlah kebalikan dari bilangan genap yang hilang Apa yang saya warnai dengan warna merah di sini Sekarang Anda dapat menganggap deret yang hilang itu sebagai salinan skala dari total deret yang kita inginkan Dimana setiap mercusuar berpindah menjadi dua kali lebih jauh dari titik asal, satu digeser ke dua, dua digeser ke empat, tiga digeser ke enam, dan seterusnya, dan karena hal ini melibatkan penggandaan jarak untuk setiap mercusuar, maka kecerahan semu akan berkurang sebesar satu faktor dari empat dan Itu juga aljabar yang relatif mudah mulai dari jumlah semua bilangan bulat ke jumlah bilangan bulat genap Melibatkan perkalian dengan 1 4 dan artinya adalah mengubah semua bilangan bulat ke bilangan ganjil Akan menjadi perkalian dengan 3 4 sejak angka genap ditambah peluang harus memberi kita segalanya. Jadi jika kita membalikkannya, itu berarti beralih dari jumlah bilangan ganjil ke jumlah semua bilangan bulat positif memerlukan perkalian dengan 4 pertiga. Jadi, ambil pi kuadrat dari 8 dikalikan dengan 4 pertiga badda boom badda bing Kami punya solusi untuk masalah basil Sekarang video yang baru saja Anda tonton ini sebagian besar ditulis dan dianimasikan oleh salah satu dari tiga anggota tim biru satu coklat Ben Hambricht dan penambahan dimungkinkan.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/italian/sentence_translations.json b/2018/basel-problem/italian/sentence_translations.json index aea5428f7..3c974cf9b 100644 --- a/2018/basel-problem/italian/sentence_translations.json +++ b/2018/basel-problem/italian/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", "translatedText": "Voglio dire, i raggi di luce che colpiscono una di queste due gambe sono esattamente gli stessi raggi che colpiscono l'ipotenusa Quindi la chiave è che la quantità di luce proveniente dal primo faro colpisce questo lato sinistro l'angolo limitato dei raggi che finiscono per colpire quello schermo è esattamente uguale alla quantità di luce qui proveniente dal faro che colpisce quel lato avrà lo stesso angolo di raggi e simmetricamente la quantità di luce proveniente dalla prima casa che colpisce la parte inferiore del nostro schermo è la stessa come la quantità di luce che colpisce quella porzione del faro B Perché potresti chiedere bene, è una questione di triangoli simili Questa animazione ti dà già un forte suggerimento su come funziona E abbiamo anche lasciato un collegamento nella descrizione a un semplice GeoGebra applet per quelli di voi che vogliono riflettere su questo in un ambiente leggermente più interattivo e giocare con un fatto importante che potrete vedere è che i triangoli simili si applicano solo nel caso limite di uno schermo molto piccolo Va bene, allacciatevi le cinture adesso perché è qui che le cose si mettono bene. Abbiamo questo teorema di Pitagora inverso, giusto?", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right.", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right.", "translatedText": "Bene, le linee da quei fari al centro sono ad angoli di 90 gradi tra loro Quindi, poiché le cose sono simmetriche da sinistra a destra, ciò significa che le distanze lungo la circonferenza sono 1 2 2 2 e 1 Va bene, potresti vedere dove stiamo andando, ma voglio proseguire ancora per un altro passo. Disegna un cerchio due volte più grande, quindi una circonferenza di 16 ora e per ogni faro Traccia una linea da quel faro attraverso la parte superiore del cerchio più piccolo Che è il centro del cerchio più grande e poi crea due nuovi fari dove quella linea si interseca con il cerchio più grande Proprio come prima perché la linea lunga è un diametro del cerchio grande quei due nuovi fari formano un angolo retto con l'osservatore a destra e Proprio come prima la linea dall'osservatore a il faro originale è perpendicolare alla lunga linea e questi sono i due fatti che ci giustificano nell'usare il teorema di Pitagora inverso. Ma ciò che potrebbe non essere così chiaro è che quando lo fai per tutti i fari ne otterrai otto nuovi sul grande lago quegli otto nuovi fari saranno distanziati uniformemente Questo è l'ultimo esempio di verifica geometrica prima della spinta finale Per vederlo ricorda che se disegni linee da due fari adiacenti sul laghetto al centro Formano un angolo di 90 gradi Se invece disegna linee fino a un punto qualsiasi della circonferenza del cerchio che non sia compreso tra di loro l'utilissimo teorema dell'angolo inscritto dalla geometria ci dice che questo sarà esattamente la metà dell'angolo che formano con il centro in questo caso 45 gradi Ma quando posizioniamo quel nuovo punto in cima al lago Queste sono le due linee che definiscono la posizione dei nuovi fari sul lago più grande Ciò significa quindi che quando disegni le linee da quegli otto nuovi fari al centro Dividono il cerchio uniformemente in pezzi angolari di 45 gradi e ciò significa che gli otto fari sono equamente distanziati attorno alla circonferenza con la distanza di due tra ciascuno di essi e ora immagina questa cosa che continua ad ogni passo raddoppiando la dimensione di ogni cerchio e trasformando ogni faro in due nuovi lungo una linea tracciata attraverso il centro del cerchio più grande ad ogni passo la luminosità apparente all'osservatore rimane la stessa pi quadrato su 4 e ad ogni passo i fari rimangono equidistanti con la distanza 2 tra ciascuno di essi sul circonferenza e al limite quello che otteniamo qui è una linea orizzontale piatta con un numero infinito di fari equamente distanziati in entrambe le direzioni e poiché la luminosità apparente è stata pi quadrato su 4 per tutto il percorso sarà vero anche in questo caso limite E Questo ci dà una serie infinita davvero fantastica la somma dei quadrati inversi 1 su n al quadrato Dove n copre tutti gli interi dispari 1 3 5 e così via ma anche meno 1 negativo 3 negativo 5 nella direzione verso sinistra Sommandoli tutti su ci darà pi quadrato su 4 È fantastico ed è il nocciolo di ciò che voglio mostrarti e fai un passo indietro e pensa a quanto sembra irreale La somma di frazioni semplici che a prima vista non hanno nulla a che fare con la geometria apparentemente non ha nulla a che fare con i cerchi Ci dà questo risultato che è correlato a pi greco Tranne che ora puoi effettivamente vedere cosa ha a che fare con la geometria la linea numericaèun po' come un limite di cerchi in continua crescita e man mano che sommi quel numero linea assicurandoti di sommare fino all'infinito su entrambi i lati È un po' come se stessi sommando lungo il confine di un cerchio infinitamente grande e un modo di parlare molto vago ma molto divertente Ma aspetta, potresti dire che questa non è la somma che ci avevi promesso all'inizio del video E beh hai ragione.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible.", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "Ci resta ancora un po' di riflessione Per prima cosa limitiamo la somma ai soli numeri dispari positivi che ci danno pi quadrato diviso per 8 Ora l'unica differenza tra questa e la somma che stiamo cercando va oltre tutti gli interi positivi pari e dispari è Che manca la somma dei reciproci dei numeri pari quello che sto colorando di rosso qui sopra Ora puoi pensare a quella serie mancante come ad una copia in scala della serie totale che vogliamo Dove ogni faro si sposta ad essere due volte più lontano dall'origine uno viene spostato a due due viene spostato a quattro tre viene spostato a sei e così via e poiché ciò comporta il raddoppio della distanza per ogni faro significa che la luminosità apparente verrebbe ridotta di un fattore di quattro e Questa è anche un'algebra relativamente semplice che va dalla somma di tutti gli interi alla somma degli interi pari Implica la moltiplicazione per 1 4 e ciò significa che passare da tutti gli interi a quelli dispari si moltiplicherebbe per 3 4 poiché i pari più le probabilità devono darci il tutto Quindi se lo invertiamo significa che passare dalla somma dei numeri dispari alla somma di tutti gli interi positivi richiede la moltiplicazione per 4 terzi Quindi prendendo quel pi quadrato su 8 moltiplicando per 4 terzi badda boom badda bing Abbiamo trovato una soluzione al problema del basilico Ora, questo video che hai appena visto è stato scritto e animato principalmente da uno dei nuovi tre membri del team blu e marrone, Ben Hambricht, un'aggiunta resa possibile.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/japanese/sentence_translations.json b/2018/basel-problem/japanese/sentence_translations.json index fdcf88ea0..d770091e9 100644 --- a/2018/basel-problem/japanese/sentence_translations.json +++ b/2018/basel-problem/japanese/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", "translatedText": "つまり、これらの 2 本の脚のうちの 1 つに当たる光線は、斜 辺に当たる光線と正確に同じです。 そして重要なのは、最初の灯台か らの光の量がこの左側に当たり、最終的に当たる光線の角度は限られ ているということですその画面は、こちら側に当たる灯台から来る 光の量とまったく同じです。 同じ光線の角度になります。 対称的に、 最初の家から画面の下部に当たる光の量も同じです。 灯台 B から その部分に当たる光の量として なぜよく尋ねられるかもしれません が、それは相似の三角形の問題です このアニメーションはすでに 、それがどのように機能するかについての強力なヒントを提供してい ます そして、説明には簡単な GeoGebra へのリンクも残 していますこのアプレットは、もう少しインタラクティブな環境でこ れについて考えてみたい人向けです。 ここで重要な事実を試してみ ると、同様の三角形は非常に小さな画面の限定的な場合にのみ適用さ れることがわかります。 よし、今すぐシートベルトを締めろ、ここか らが物事がうまくいくから 逆ピタゴラスの定理があるんだよね？", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right.", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right.", "translatedText": "さて、これらの灯台から中心までの線は互いに 90 度の角度をしていま す つまり、物事は左右対称なので、円周に沿った距離は 1 2 2 2 と 1 であることを意味します。 でももう一歩だけこれを見て歩きたい のです あなたは 2 倍の大きさの円を描きます、つまり円周が 16 になるように、そして各灯台について あなたはその灯台から小さな円の上 部を通る線を引きます それが大きな円の中心です次に、その線が大きな円と 交差する場所に 2 つの新しい灯台を作成します。 前と同じように、長い 線は大きな円の直径であるため、これらの 2 つの新しい灯台は観測者か ら右に直角を作ります。 また、前と同様に、観測者から右への線が作成され ます。 元の灯台は長い線に対して垂直であり、これらが逆ピタゴラスの定理を 使用することを正当化する 2 つの事実です。 しかし、それほど明確で はないのは、すべての灯台に対してこれを行うと、大きな灯台に 8 つの 新しい灯台が得られるということです。 湖にあるこれら 8 つの新しい灯 台は等間隔に配置されます これは、最後の推進に先立って、ジオメトリの 証明を行う最後の部分です これを確認するには、小さな湖にある 2 つの 隣接する灯台から中心に向かって線を引くと、それらは 90 度の角度を 作ることを思い出してください。 代わりに、円の円周上の、それらの間では ない任意の点に線を引きます。 幾何学からの非常に便利な内接角定理により 、これは、この場合、中心とのなす角度のちょうど半分である 45 度にな ることがわかります。 その新しい点を湖の頂上に配置します これらは、よ り大きな湖上の新しい灯台の位置を定義する 2 本の線です つまり、意 味するのは、これら 8 つの新しい灯台から中心に向かって線を引くと、 それらが円を分割するということですこれは、8 つの灯台が円周上に等間 隔に配置され、各灯台間の距離が 2 であることを意味します。 各円のサイ ズが 2 倍になり、各灯台が次のように変化することを想像してください。 大きな円の中心を通って引かれた線に沿った 2 つの新しい灯台は、各 ステップで観察者にとっての見かけの明るさは同じ円周率 4 の 2 乗 のままであり、各ステップで灯台は等間隔に配置され、灯台間の距離は 2 になります。 円周と制限の中で、ここで得られるのは、両方向に等間隔に配 置された無数の灯台を持つ平らな水平線です そして、見かけの明るさは全 体的に 4 の pi の 2 乗であるため、この制限された場合にも当 てはまりますこれにより、非常に素晴らしい無限級数が得られます。 n の 2 乗に対する 1 の逆二乗の合計です。 ここで、n は奇数のすべて の整数 1 3 5 などをカバーしますが、左方向に負の 1 負の 3 負の 5 もカバーします。 これは 4 の円周率の 2 乗を示します これは驚くべきことであり、これが私がお見せしたいものの核心です 一 歩下がって、これがいかに非現実的であるかを考えてください 一見すると 幾何学とは何の関係もない単純な分数の合計明らかに円とはまったく関係があ りません 円周率に関連するこの結果が得られます ただし、実際に幾何学 と何の関係があるのかがわかります 数直線は成長し続ける円の限界のよう なものであり、その数値を合計するとどちらかの辺の合計が無限大になるよ うに線を引いていきます 無限に大きな円の境界線に沿って足していくよう なもので、非常に緩やかですが、とても楽しい話し方です でも待ってくださ い、これは合計ではないと言うかもしれませんビデオの冒頭で約束してくれ たのは、その通りです。", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible.", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "まだ少し考えが残っています まず最初に、合計を正の奇数の みに制限して、円周率の 2 乗を 8 で割った値を求めます。 これで、これと探してい る合計との唯一の違いは次のとおりです。 すべての正の整数は奇数と偶数です それは偶数 の逆数の合計が欠落しているということです ここで私が赤で色付けしているのは これで 、その欠落している系列を、必要な全系列のスケールコピーとして考えることができます 各灯台はどこにありますか原点から 2 倍の距離に移動します 1 つは 2 に移動し ます 2 は 4 に移動します 3 は 6 に移動します。 というように、すべての灯 台の距離が 2 倍になるため、見かけの明るさが 1 分の 1 減少することを意味し ます。 これは、すべての整数の合計から偶数の整数の合計に至る比較的単純な代数でもあり ます。 1 4 分の 1 を乗算する必要があります。 これが意味するのは、すべての整数 から奇数の整数に行くには、4 分の 3 を掛けることになります。 偶数とオッズで全体が 得られる必要があるため、これをひっくり返すと、奇数の合計からすべての正の整数の合計 に 3 分の 4 を掛ける必要があるということになります。 つまり、円周率の 2 乗 を 8 で乗算することになります。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/korean/sentence_translations.json b/2018/basel-problem/korean/sentence_translations.json index 98cd36596..798af85a1 100644 --- a/2018/basel-problem/korean/sentence_translations.json +++ b/2018/basel-problem/korean/sentence_translations.json @@ -71,7 +71,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "구형 기하학에서는 때때로 도형의 입체각에 대해 이야기합니다. 이는 주어진 지점에서 볼 때 도형이 차지하는 구의 비율입니다. 스크린에 대한 이 이야기가 유용할 두 곳 중 첫 번째는 역제곱 법칙을 이해하는 것입니다. 이는 뚜렷한 3차원 현상입니다.광원에서 한 단위 떨어진 스크린에 닿는 모든 광선을 생각하면 그 광선이 이제 두 배의 너비와 두 배의 높이로 영역을 덮을 것입니다. 따라서 그 거리에서 동일한 광선을 받으려면 원래 화면의 사본이 네 개가 필요하므로 각 개인은 하나는 4분의 1의 빛을 받습니다 이것은 빛이 2 배 떨어진 거리에서 4 분의 1만큼 밝게 보인다는 의미입니다 마찬가지로 3 배 더 멀리 떨어져있을 때 동일한 광선을 받으려면 원본 화면의 9 개 사본이 필요하므로 각 개별 화면은 9 분의 1의 빛만받습니다. 그리고 빛이 닿는 면적은 거리의 제곱만큼 증가하고 그 빛의 밝기는 거리의 역제곱만큼 감소하기 때문에이 패턴은 계속됩니다. 많은 분들이 아시다시피이 역 제곱 법칙은 빛에 전혀 특별한 것이 아닙니다. 어떤 종류의 양이있을 때마다 나타납니다. 소리나 열 또는 무선 신호와 같은 점 광원에서 균등하게 퍼지고 균등 간격의 무한 배열 등대는 물리적으로 바젤 문제를 구현하지만 여기서 진전을 이루기 위해 필요한 것은 이와 같은 광원으로 설정을 조작 할 수있는 방법을 이해하는 것입니다. 관찰자의 총 밝기를 변경하지 않고 핵심 빌딩 블록은 하나의 등대를 두 개로 변환하는 특히 좋은 방법입니다. XY 평면의 원점에 있는 관찰자와 그 평면 어딘가에 있는 하나의 등대를 생각해보세요. 이제 그 등대에서 관찰자까지 선을 그린 다음 등대에서 그 선에 수직 인 또 다른 선 이제이 새 선이 좌표축과 교차하는 곳에 두 개의 등대를 배치합니다. 여기 왼쪽에있는 등대 A와 위쪽에있는 등대 B라고하겠습니다. 이것이 왜 이것이 사실인지 잠시 후에 관찰자가 밝기를 알 수 있습니다. 첫 번째 등대에서 경험하는 밝기는 등대 A와 B에서 함께 경험하는 밝기를 합친 것과 같습니다 그리고이 비디오 전체에서 모든 등대가 동일하다는 가정은 모든 등대가 동일한 전구를 사용하여 동일한 전력을 발산하고 있다는 것입니다 그래서 다른 말로하면 즉, 관찰자에서 등대까지의 거리를 조금 a 그리고 관찰자에서 등대까지의 거리를 조금 B 그리고 첫 번째 등대까지의 거리를 H라고 부르면 여기에 변수를 할당합니다. 우리는 관계 1 제곱에 1 제곱에 1 제곱에 B 제곱을 더하면 1 제곱에 H 제곱이됩니다 이것은 훨씬 덜 잘 알려져 있습니다.여러분 중 일부는 수학 올로거의 가장 최근에서 알아볼 수있는 알려진 역 피타고라스 정리이며 피타고라스 정리의 많은 사촌들에 대한 가장 훌륭한 비디오를 말할 것입니다. 꽤 멋진 관계라고 생각하지 않습니까? 그리고 당신이 수학자라면 지금 당장 그것을 증명하는 방법을 묻고있을 것입니다. 삼각형 영역을 두 가지 방법으로 표현하고 일반적인 피타고라스 정리를 적용하는 몇 가지 간단한 방법이 있지만 여기서 간략하게 설명하고자 하는 또 다른 아주 예쁜 방법이 있는데, 이 방법 역시 빛과 스크린의 직관을 이용하기 때문에 스토리 라인에 훨씬 더 잘 들어맞습니다. 전체 직각삼각형을 더 작은 버전으로 축소하고 이 미니어처 히포테뉴스를 첫 등대에서 빛을 받는 스크린이라고 생각하면 그 스크린을 이렇게 미니어처 삼각형의 두 다리의 조합으로 재구성할 수 있습니다, 여전히 같은 양의 빛을 받겠죠?", "model": "DeepL", "from_community_srt": "하는 것입니다. 구면기하학에서 도형의 입체각이라고 부르는 이것은, 어떤 도형을 특정한 점에서 볼 때, 구형의 면적중에 그것이 차지하는 면적의 분율을 의미합니다. 등대문제에서 처음 두 등대의 위치와 스크린을 생각해보면 , 역제곱의 법칙을 이해하는데 도움이 되는데 이는 분명 삼차원적인 현상이죠. 광원에서 떨어져 있는 스크린의 단위 면적에 닿는 모든 광선을 생각해 보세요 거리가 두 배가 되면 광선이 비추는 면적의 너비는 두 배가 되고 높이도 두 배가 됩니다. 거리가 두 배가 되면 광선이 비추는 면적의 너비는 두 배가 되고 높이도 두 배가 됩니다. 따라서 거리를 늘린 후 기존에 받던 빛과 같은 양의 빛을 받으려면 기존 스크린 4개가 필요하며 각 스크린 한 개가 받는 빛의 양은 원래 받던 빛의 양의 1/4 입니다. 즉, 거리가 2배로 늘어나면 밝기는 4배로 감소한다는 의미입니다. 마찬가지로 거리가 3배로 늘어나면 9개의 스크린이 있어야 같은 양의 빛을 받을 수 있으며, 한 개의 스크린은 원래 빛의 양의 1/9을 받게 됩니다. 이러한 패턴은 반복되는데 빛이 비추는 면적은 거리의 제곱에 비례하여 증가하고, 빛의 밝기는 거리의 제곱에 비례하여 감소하기 때문이죠. 많은 사람들이 알고 있겠지만 , 역제곱 법칙은 법칙은 빛에만 국한된 것은 아닙니다. 어떤 물리량이 한 점에서 균등하게 확산되는 한 이 법칙은 적용됩니다. 소리나 열, 라디오파와 같은 것들 말이죠. 이 역제곱 법칙 때문에 등대를 같은 간격으로 무한히 배열하는 것이 물리학적으로 바젤문제를 시행하는 것이라는 것을 기억하세요. 하지만 다시, 여기서 더 진행하기 위해서는 어떻게 해야 등대의 재배치 하면서도, 관찰자가 받는 빛의 총량은 보존시킬 수 있는지 이해하는 것이 필요합니다. 그리고 이것의 핵심은 하나의 등대를 둘로 변환시키는 것인데 이것은 매우 멋진 방법입니다. 관찰자가 x-y평면의 원점에 있고 한 개의 등대가 이 평면의 어딘가에 있다고 합시다. 그리고 등대에서 관찰자를 잇는 선을 하나 그리고 등대에서 이 선에 수직인 선을 하나 더 그립니다. 그리고 이 선이 좌표축에 교차하는 점에 두개의 등대를 놓습니다. 이제부터 오른쪽에 있는 이 등대를 A, 위쪽에 있는 등대를 B라고 부르죠. 그 이유를 곧 알게 되겠지만 결론은 다음과 같습니다. 처음 등대로부터 관찰자가 받은 빛의 밝기는 A와 B 두 등대에서 받은 빛의 밝기의 합과 같습니다. 그런데 이 영상을 통틀어 먼저 가정할 것은 모든 등대는 동일하다는 것과 모든 등대는 동일한 광량을 발산하는 같은 전구를 사용한다는 것입니다. 이제 변수를 할당해보도록 하죠. 등대 A와 관찰자와의 거리를 소문자 a 등대 B와 관찰자와의 거리를 소문자 b 원래 등대와 관찰자의 거리를 h로 놓습니다. 이 때 1/a^2 + 1/b^2 = 1/h^2 이 성립합니다. 이것은 잘 알려져 있지 않은 역 피타고라스 정리 라고 합니다. 아마 여러분들 중에 알고 있는 분들도 있겠지만 이것은 가장 최근이면서 가장 훌륭한 \"피타고라스 정리의 사촌들\"에 대한 Methologer의 영상에서도 소개된 적이 있습니다. 꽤 멋진 관계이지 않나요? 그리고 당신이 마음에서부터 수학자라면 지금 당장 이것을 어떻게 증명해야 될지 고민하겠죠. 여기 삼각형의 넓이를 표현하는 간단한 두 가지 방법이 있습니다. 그리고 거기에 피타고라스 정리를 적용해 보면 됩니다. 하지만 여기서는 위와 다르면서도 꽤 멋진 방법을 간단히 언급하려 합니다. 이 방법은 원래 우리의 이야기와 좀 더 밀접한데, 빛과 스크린에 대한 직관을 이용하기 때문입니다. 여기 이 삼각형을 작게 축소한다고 생각해보세요. 이 작은 삼각형의 빗변을 첫 번째 등대에서 빛을 받는 스크린이라고 가정합니다. 당신은 이 스크린을 변형시킬 수 있는데 이 작은 삼각형을 이와 같이 두 개의 다리를 가진 도형으로 나누더라도 여전히 같은 양의 빛을 받게 됩니다.", @@ -134,7 +134,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "그것은 겉보기 밝기가 4 배 감소한다는 것을 의미합니다 그리고 그것은 또한 모든 정수의 합에서 짝수 정수의 합으로가는 비교적 간단한 대수입니다 1 4를 곱하는 것이 포함됩니다 그리고 그것이 의미하는 것은 모든 정수에서 홀수로가는 것은 3 4를 곱하는 것입니다 짝수와 확률을 더하면 우리에게 전체를 제공해야하기 때문에 우리가 그것을 뒤집 으면 홀수의 합에서 모든 양의 정수의 합으로 가려면 4분의 3을 곱해야 하므로 8에 파이를 제곱한 값에 4분의 3을 곱하면 바다 붐 바다 빙 우리는 바질 문제에 대한 해결책을 얻었습니다 이제 방금 보신 이 동영상은 새로운 세 명의 블루 원 브라운 팀원 중 한 명인 벤 함브리히트가 직접 제작하고 애니메이션을 추가하여 만들었습니다.", "model": "DeepL", "from_community_srt": "모든 등대는 거리가 2배가 되었으니 겉보기 밝기는 4배 만큼 감소합니다. 그리고 이제 상대적으로 간단한 대수문제만이 남았습니다. 모든 자연수에 대한 합에서 모든 짝수에 대한 합을 구하려면 1/4을 곱하면 됩니다. 이것은 즉, 모든 자연수에서 홀수로 변환하려면 3/4를 곱하면 된다는 의미인데, 홀수와 짝수의 합이 모든 자연수이기 때문입니다. 이제 역으로 생각해보면, 다시말해, 모든 홀수에 대한 합을 자연수에 대한 합으로 바꿀때는 3/4를 곱해야 합니다. 따라서 π^2/8에 4/3을 곱하면, 빠라바라밤! 마침내 바젤 문제의 답을 구해냈습니다! 여러분이 지금 보고 있는 영상은3blue1brown팀의 새로운 멤버인 Ben Hambrecht가 제작했습니다.", diff --git a/2018/basel-problem/marathi/sentence_translations.json b/2018/basel-problem/marathi/sentence_translations.json index 580b074a6..ca2cf1119 100644 --- a/2018/basel-problem/marathi/sentence_translations.json +++ b/2018/basel-problem/marathi/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", "translatedText": "मला असे म्हणायचे आहे की त्या दोन पायांपैकी एकाला आदळणारे प्रकाश किरण कर्णावर आदळणार्‍या किरणांसारखेच असतात मग मुख्य गोष्ट अशी आहे की पहिल्या दीपगृहातील प्रकाशाचे प्रमाण ते या डाव्या बाजूला किरणांच्या मर्यादित कोनावर आदळते. ती स्क्रीन लाइटहाऊसमधून येणा-या प्रकाशाच्या प्रमाणाइतकीच आहे आणि त्या बाजूने आदळणाऱ्या किरणांचा कोन समान असेल आणि आपल्या स्क्रीनच्या खालच्या भागाला आदळणाऱ्या पहिल्या घराच्या प्रकाशाचे प्रमाण समान असेल. लाइटहाऊस बी वरून त्या भागाला किती प्रकाश पडतो म्हणून तुम्ही चांगले का विचारू शकता, ही समान त्रिकोणांची बाब आहे हे अॅनिमेशन तुम्हाला ते कसे कार्य करते याबद्दल आधीच एक मजबूत इशारा देते आणि आम्ही एका साध्या जिओजेब्राच्या वर्णनात एक लिंक देखील सोडली आहे. तुमच्यापैकी ज्यांना थोड्या अधिक परस्परसंवादी वातावरणात आणि त्यासोबत खेळताना याचा विचार करायचा आहे त्यांच्यासाठी ऍपलेट येथे एक महत्त्वाची वस्तुस्थिती आहे जी तुम्ही पाहण्यास सक्षम असाल की समान त्रिकोण केवळ एका अत्यंत लहान स्क्रीनसाठी मर्यादित प्रकरणात लागू होतात. ठीक आहे आता बकल करा कारण इथेच गोष्टी चांगल्या होतात आम्हाला हे इनव्हर्स पायथागोरियन प्रमेय मिळाले आहे, बरोबर?", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right.", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right.", "translatedText": "त्या दीपगृहांपासून केंद्रापर्यंतच्या रेषा एकमेकांशी ९० अंश कोनात आहेत त्यामुळे गोष्टी डावीकडून उजवीकडे सममितीय असल्याने याचा अर्थ परिघाच्या बाजूचे अंतर 1 2 2 2 आणि 1 ठीक आहे, हे कुठे चालले आहे ते तुम्ही पाहू शकता. पण मला अजून एक पाऊल यातून चालायचे आहे तुम्ही आता 16 चा घेर म्हणून दुप्पट मोठे वर्तुळ काढा आणि प्रत्येक दीपगृहासाठी तुम्ही त्या दीपगृहातून लहान वर्तुळाच्या वरच्या बाजूने एक रेषा काढता जे मोठ्या वर्तुळाचे केंद्र आहे. आणि नंतर दोन नवीन दीपगृहे तयार करा जिथे ती रेषा मोठ्या वर्तुळाला छेदते अगदी पूर्वीप्रमाणेच कारण लांब रेषा मोठ्या वर्तुळाचा व्यास आहे ती दोन नवीन दीपगृहे निरीक्षकाच्या उजवीकडे काटकोन करतात आणि निरीक्षकाच्या रेषेच्या आधी मूळ दीपगृह लांब रेषेला लंब आहे आणि ती दोन तथ्ये आहेत जी आपल्याला व्यस्त पायथागोरियन प्रमेय वापरण्यात न्याय्य ठरवतात परंतु जे स्पष्ट होणार नाही ते हे आहे की जेव्हा तुम्ही हे सर्व दीपगृहांसाठी आठ नवीन प्रमेय मिळवण्यासाठी कराल. लेक ते आठ नवीन दीपगृह समान रीतीने अंतरावर असणार आहेत अंतिम जोराच्या आधी भूमिती प्रूफनेसचा हा अंतिम भाग आहे हे पाहण्यासाठी लक्षात ठेवा की जर तुम्ही लहान तलावावरील दोन शेजारील दीपगृहांपासून मध्यभागी रेषा काढल्या तर ते 90 अंश कोन बनवतात. त्याऐवजी तुम्ही वर्तुळाच्या परिघावर कोठेही बिंदूवर रेषा काढा जी त्यांच्या दरम्यान नसली तर भूमितीतील अतिशय उपयुक्त कोरलेले कोन प्रमेय आम्हाला सांगते की या प्रकरणात ते मध्यभागी बनवलेल्या कोनाच्या अगदी अर्धा असेल 45 अंश परंतु जेव्हा आम्ही तलावाच्या शीर्षस्थानी तो नवीन बिंदू ठेवतो या दोन रेषा आहेत ज्या मोठ्या तलावावरील नवीन दीपगृहांची स्थिती परिभाषित करतात याचा अर्थ असा आहे की जेव्हा तुम्ही त्या आठ नवीन दीपगृहांमधून मध्यभागी रेषा काढता तेव्हा ते वर्तुळ विभाजित करतात समान रीतीने 45 अंश कोनाच्या तुकड्यांमध्ये आणि याचा अर्थ असा की आठ दीपगृह परिघाभोवती समान रीतीने अंतर ठेवलेले आहेत आणि त्यातील प्रत्येकामध्ये दोनचे अंतर आहे आणि आता कल्पना करा ही गोष्ट प्रत्येक टप्प्यावर खेळत आहे प्रत्येक वर्तुळाचा आकार दुप्पट करते आणि प्रत्येक दीपगृहाचे रूपांतर प्रत्येक पायरीवर मोठ्या वर्तुळाच्या मध्यभागी काढलेल्या एका रेषेसह दोन नवीन, निरीक्षकांना स्पष्ट चमक 4 वर समान pi चौरस राहते आणि प्रत्येक पायरीवर दीपगृह समान रीतीने अंतरावर राहतात आणि प्रत्येक पायरीवर प्रत्येकाच्या दरम्यान 2 अंतर असते. परिघ आणि मर्यादेत आपण येथे जे मिळवत आहोत ती एक सपाट क्षैतिज रेषा आहे ज्यामध्ये असीम संख्येने दीपगृह दोन्ही दिशांना समान रीतीने अंतरावर आहेत आणि कारण स्पष्ट ब्राइटनेस pi 4 वर वर्ग केला होता संपूर्ण मार्ग जो या मर्यादित प्रकरणात देखील सत्य असेल आणि हे आपल्याला एक सुंदर अनंत मालिका देते व्यस्त वर्गांची बेरीज 1 पेक्षा n स्क्वेअर जेथे n सर्व विषम पूर्णांक 1 3 5 आणि असेच समाविष्ट करते परंतु डावीकडील दिशेने नकारात्मक 1 ऋण 3 ऋण 5 ऑफ देखील समाविष्ट करते त्या सर्व जोडून आम्हाला pi चा वर्ग 4 वर देणार आहे हे आश्चर्यकारक आहे आणि मी तुम्हाला जे दाखवू इच्छितो त्याचा मुख्य भाग आहे आणि फक्त एक पाऊल मागे घ्या आणि हे किती अवास्तव वाटते याचा विचार करा वर्तुळांशी अजिबात संबंध नाही हे वरवर पाहता आम्हाला हा परिणाम देते जो pi शी संबंधित आहे, आता तुम्ही प्रत्यक्षात पाहू शकता की भूमितीशी त्याचा काय संबंध आहे, संख्यारेषा ही एक प्रकारची सतत वाढणाऱ्या वर्तुळांच्या मर्यादेसारखी असते आणि तुम्ही त्या संख्येची बेरीज करता. दोन्ही बाजूंनी अनंतापर्यंत सर्व मार्गांची बेरीज करण्याची खात्री करणे हे असे आहे की आपण एका अमर्याद मोठ्या वर्तुळाच्या सीमेवर जोडत आहात आणि खूप सैल परंतु बोलण्याचा खूप मजेदार मार्ग आहे परंतु थांबा तुम्ही म्हणाल ही बेरीज नाही व्हिडिओच्या सुरुवातीला तुम्ही आम्हाला वचन दिले होते आणि तुम्ही बरोबर आहात.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible.", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "आपल्याकडे थोडा विचार बाकी आहे प्रथम गोष्टी प्रथम आपण फक्त बेरीज फक्त सकारात्मक विषम संख्यांपुरती मर्यादित करू या ज्यामुळे आपल्याला pi चा वर्ग भागाकार 8 मिळतो. सर्व सकारात्मक पूर्णांक विषम आणि सम म्हणजे सम संख्यांच्या परस्परसंख्येची बेरीज मी इथे लाल रंगात रंगवत आहे ती गहाळ आहे उत्पत्तीपासून दुप्पट दूर जाण्याकडे सरकते, एक दोनकडे स्थलांतरित होते, दोन चारवर स्थलांतरित होते, तीन सहाकडे स्थलांतरित होते आणि असेच पुढे आणि कारण त्यात प्रत्येक दीपगृहाचे अंतर दुप्पट होते याचा अर्थ असा होतो की स्पष्ट चमक एका घटकाने कमी होईल. चार पैकी आणि ते देखील तुलनेने सरळ बीजगणित आहे जे सर्व पूर्णांकांच्या बेरजेपासून सम पूर्णांकांच्या बेरजेकडे जाण्यासाठी 1 4थ्याने गुणाकार करणे समाविष्ट आहे आणि याचा अर्थ काय आहे की सर्व पूर्णांकांपासून विषमकडे जाणे 3 4थ्याने गुणाकार केले जाईल. सम आणि विषमता ही आपल्याला संपूर्ण गोष्ट द्यावी लागेल म्हणून जर आपण फक्त त्याभोवती फिरलो तर याचा अर्थ विषम संख्यांच्या बेरजेपासून सर्व धन पूर्णांकांच्या बेरजेपर्यंत जाण्यासाठी 4 तृतीयांश ने गुणाकार करणे आवश्यक आहे तर त्या pi चा वर्ग 8 ने गुणाकार केला पाहिजे 4 थर्ड्स बड्डा बूम बड्डा बिंग आम्ही स्वतःच तुळशीच्या समस्येवर उपाय शोधला आहे आता तुम्ही नुकताच पाहिला हा व्हिडिओ प्रामुख्याने नवीन तीन ब्लू वन ब्राउन टीम सदस्यांपैकी एक बेन हॅम्ब्रिक्ट यांनी लिहिलेला आणि अॅनिमेटेड आहे, ज्यामुळे हे शक्य झाले आहे.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/persian/sentence_translations.json b/2018/basel-problem/persian/sentence_translations.json index 3e2492a81..9762f19dc 100644 --- a/2018/basel-problem/persian/sentence_translations.json +++ b/2018/basel-problem/persian/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right? ", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right? ", "translatedText": "صفحه نمایش ترکیبی از دو پایه مثلث مینیاتوری مانند این باشد خوب، هنوز همان مقدار نور را دریافت می کند، درست است؟ منظورم این است که پرتوهای نوری که به یکی از آن دو پا برخورد می کنند دقیقاً مشابه پرتوهایی هستند که به هیپوتنوس برخورد می کنند، سپس نکته کلیدی این است که مقدار نوری که از اولین فانوس دریایی به سمت چپ می خورد، زاویه محدود پرتوهایی است که در نهایت به آن برخورد می کند. آن صفحه دقیقاً همان مقدار نوری است که در اینجا از فانوس دریایی می‌آید و به آن طرف برخورد می‌کند و به طور متقارن، میزان نور اولین خانه‌ای که به قسمت پایین صفحه ما برخورد می‌کند یکسان است. به عنوان مقدار نوری که از فانوس دریایی B به آن قسمت برخورد می کند چرا ممکن است خوب بپرسید، این موضوع مربوط به مثلث های مشابه است. اپلت برای کسانی از شما که می‌خواهید در یک محیط تعاملی‌تر به این موضوع فکر کنید و در بازی با آن واقعیت مهمی که در اینجا خواهید دید این است که مثلث‌های مشابه فقط در حالت محدود برای یک صفحه نمایش بسیار کوچک کاربرد دارند. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right. ", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right. ", "translatedText": "از بالای دایره کوچکتر و گرفتن دو فانوس دریایی جدید روی دایره بزرگتر و حتی زیباتر این چهار فانوس دریایی همه به طور مساوی در اطراف دریاچه قرار می گیرند چرا؟ خوب خطوط از آن فانوس‌های دریایی تا مرکز در زاویه 90 درجه با یکدیگر قرار دارند، بنابراین از آنجایی که چیزها متقارن از چپ به راست هستند، به این معنی است که فواصل در امتداد محیط 1 2 2 2 و 1 خوب است، ممکن است ببینید این به کجا می‌رود. اما من می‌خواهم فقط برای یک قدم دیگر از آن عبور کنم، شما دایره‌ای دوبرابر بزرگ‌تر از محیط 16 بکشید و برای هر فانوس دریایی یک خط از آن فانوس دریایی از بالای دایره کوچک‌تر که مرکز دایره بزرگ‌تر است می‌کشید. و سپس دو فانوس دریایی جدید در جایی که آن خط با دایره بزرگتر تلاقی می کند درست مانند قبل ایجاد کنید زیرا خط طولانی قطر دایره بزرگ است، آن دو فانوس دریایی جدید با ناظر زاویه ای قائم ایجاد می کنند و درست مانند قبل از خط از ناظر به سمت راست. فانوس اصلی عمود بر خط طولانی است و این دو واقعیت است که ما را در استفاده از قضیه فیثاغورث معکوس توجیه می کند، اما چیزی که ممکن است چندان واضح نباشد این است که وقتی این کار را انجام می دهید برای همه فانوس های دریایی هشت فانوس جدید در The big دریافت کنید. این هشت فانوس دریایی جدید قرار است به طور مساوی با هم فاصله داشته باشند این آخرین بیت اثبات هندسی قبل از رانش نهایی است. در عوض شما خطوطی را به نقطه‌ای از محیط دایره رسم می‌کنید که بین آنها نیست، قضیه زاویه محاطی بسیار مفید از هندسه به ما می‌گوید که این دقیقاً نیمی از زاویه‌ای است که آنها با مرکز در این مورد 45 درجه می‌سازند، اما وقتی ما آن نقطه جدید را در بالای دریاچه قرار می دهیم این دو خط هستند که موقعیت فانوس های دریایی جدید را روی دریاچه بزرگتر مشخص می کنند. معنی آن این است که وقتی خطوطی را از آن هشت فانوس دریایی جدید به مرکز می کشید، دایره را تقسیم می کنند. به طور مساوی به قطعات با زاویه 45 درجه و این بدان معناست که هشت فانوس دریایی به طور مساوی در اطراف محیط قرار گرفته اند و فاصله بین هر یک از آنها دو است و اکنون فقط تصور کنید که این چیز در هر مرحله بازی می کند و اندازه هر دایره دو برابر می شود و هر فانوس دریایی را به شکل تبدیل می کند. دو فانوس جدید در امتداد خطی که از مرکز دایره بزرگتر در هر مرحله کشیده شده است، درخشندگی ظاهری برای ناظر همان عدد پی مربع بر روی 4 باقی می ماند و در هر مرحله فانوس های دریایی به طور مساوی با فاصله 2 بین هر یک از آنها روی محیط و در حد چیزی که در اینجا به دست می آوریم یک خط افقی مسطح با تعداد بی نهایت فانوس دریایی است که در هر دو جهت به طور مساوی فاصله دارند و از آنجا که روشنایی ظاهری مربع پی بر روی 4 بوده است که در این حالت محدود نیز صادق خواهد بود. این یک سری بی نهایت بسیار عالی به ما می دهد مجموع مربع های معکوس 1 بر n مربع که در آن n همه اعداد صحیح فرد 1 3 5 و غیره را پوشش می دهد، اما همچنین منفی 1 منفی 3 منفی 5 خاموش در جهت چپ با جمع کردن همه آنها به بالا به ما پی مربع روی 4 می دهد که شگفت انگیز است و این هسته چیزی است که می خواهم به شما نشان دهم و فقط یک قدم به عقب بردارید و به این فکر کنید که چقدر غیر واقعی به نظر می رسد مجموع کسری های ساده ای که در نگاه اول هیچ ربطی به هندسه ندارند. اصلاً ربطی به دایره ها ندارد ظاهراً این نتیجه را به ما می دهد که مربوط به pi است، به جز اینکه اکنون می توانید ببینید که چه ربطی به هندسه دارد، خط اعداد به نوعی مانند حدی از دایره های همیشه در حال رشد است و همانطور که در آن عدد جمع می کنید خط مطمئن شوید که تا بی نهایت در هر دو طرف جمع می شود این به نوعی مانند این است که در امتداد مرز یک دایره بی نهایت بزرگ و یک روش صحبت بسیار شل اما بسیار سرگرم کننده جمع می کنید اما صبر کنید ممکن است بگویید این مجموع نیست که در ابتدای ویدیو به ما قول دادید و خب حق با شماست. ", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. ", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You ", "translatedText": "کمی تفکر باقی مانده است. اولاً اجازه دهید مجموع را فقط به اعداد فرد مثبت محدود کنیم که مجذور پی تقسیم بر 8 را بدست می‌آورد. همه اعداد صحیح مثبت فرد و زوج این است که مجموع متقابل اعداد زوج را از دست داده است، چیزی که من در اینجا با رنگ قرمز رنگ آمیزی می کنم اکنون می توانید آن سری گمشده را به عنوان یک کپی مقیاس شده از کل مجموعه ای که ما می خواهیم هر فانوس دریایی را کجا در نظر بگیرید. به سمت دو برابر دورتر شدن از مبدأ حرکت می کند، یکی به دو منتقل می شود، دو به چهار منتقل می شود، سه به شش و غیره منتقل می شود و از آنجا که این شامل دو برابر شدن فاصله برای هر فانوس دریایی است، به این معنی است که روشنایی ظاهری یک عامل کاهش می یابد. از چهار و این نیز جبر نسبتاً ساده ای است که از مجموع همه اعداد صحیح به مجموع روی اعداد زوج صحیح می رود شامل ضرب در 1 4 است و معنی آن این است که رفتن از همه اعداد صحیح به افراد فرد در 3 و 4 ضرب می شود. زوج ها به علاوه شانس باید همه چیز را به ما بدهند بنابراین اگر فقط آن را برگردانیم به این معنی است که از مجموع اعداد فرد به مجموع همه اعداد صحیح مثبت باید ضرب در 4 سوم شود، بنابراین با در نظر گرفتن مربع آن پی روی 8 ضرب در 4 سوم badda boom badda bing ما راه حلی برای مشکل ریحان داریم حالا این ویدیویی که شما همین الان تماشا کردید در اصل توسط یکی از سه عضو جدید تیم آبی یک قهوه ای بن هامبریخت نوشته و متحرک شده است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/polish/sentence_translations.json b/2018/basel-problem/polish/sentence_translations.json index 06b950748..04093e9cb 100644 --- a/2018/basel-problem/polish/sentence_translations.json +++ b/2018/basel-problem/polish/sentence_translations.json @@ -63,7 +63,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "", "from_community_srt": "W geometrii sferycznej mówi się czasem o kącie bryłowym figury który jest proporcją sfery którą pokrywa patrząc z danego punktu. Pierwszym przypadkiem, gdzie myślenie o tych ekranach się przyda, jest zrozumienie prawa odwrotnych kwadratów, które jest wyłącznie trójwymiarowym zjawiskiem. Pomyślmy o wszystkich promieniach światła uderzających w ekran oddalony o jedną jednostkę Jeżeli podwoimy ten dystans, to te promienie pokryją powierzchnię o podwójnej szerokości i wysokości więc trzeba 4 kopii naszego ekranu aby otrzymać tyle samo światła w tej odległości, zatem jeden ekran otrzymuje tylko jedną czwartą światła. To w tym sensie mówimy, że światło dwa razy dalej będzie tylko 1/4 razy jasne. A gdy jesteśmy trzy razy dalej? Potrzebowalibyśmy 9 kopii ekranu, zatem jeden ekran odbiera tylko 1/9 światła i ta prawidłowość jest prawdziwa dalej, ponieważ powierzchnia odbierająca światło zwiększa się jak kwadrat odległości. Jak pewnie wielu z was wie, prawo odwrotnych kwadratów nie jest charakterystyczna dla światła, pojawia się wszędzie, gdy mamy jakąś ilość, która rozprowadza się równomiernie od źródła, na przykład dźwięk lub ciepło czy fale radiowe. I pamiętajmy, że to dzięki temu prawo nieskończony zbiór latarni może być użyty do implementacji fizycznej naszego problemu. Jednak znów, jeśli chcemy pójść dalej, musimy zrozumieć jak przesuwać te latarnie bez zmieniania jasności odbieranej przez obserwującego. A głównym pomysłem jest bardzo elegancki sposób przekształcenia jednej latarni w dwie. Pomyślmy o obserwatorze w punkcie (0,0) na płaszczyźnie, i jednej latarni gdzieś w oddali. Narysujmy teraz linię z tej latarni do obserwującego, a potem kolejną, prostopadłą, i postaw dwie latarnie na przecięciach tej prostej z osiami. Nazwiemy je latarnia A i latarnia B Okazuje się, i zobaczycie dlaczego za chwilę, że jasność naszej pierwszej latarni jest taka sama jak suma jasności latarń A oraz B. Przy okazji zakładamy w całym filmie, że wszystkie latarnie są identyczne; mają te same żarówki o tej samej mocy itd. Zatem, w innych słowach, dodając kilka zmiennych, jeśli nazwiemy odległość od latarni A jako a, oraz alogicznie b, a odległość od naszej pierwszej latarni jako h, to okaże się, że 1/a^a + 1/b^2 równa się 1/h^2 Jest to o wiele mniej znane odwrotne twierdzenie pitagorasa (nie mylić z tw. odwrotnym do twierdzenia pitagorasa) które niektórzy mogą kojarzyć z filmu Mathologera, jest to wspaniały film na temat kuzynów tw. pitagorasa Całkiem fajna zależność, co? Jeżeli jesteś matematykiem w sercu, to możesz teraz pytać, jak tego dowieść. Są metody wprost, gdzie liczysz pole trójkąta na dwa sposoby, i używasz tw. pitagorasa, ale jest inna, całkiem ładna metoda, którą chcę tu streścić, bo o wiele bardziej pasuje do naszej historii. Znów używa ona intuicji związanych z światłem i ekranami Wyobraź sobie zmniejszenie tego trójkąta i myślenie o tej malutkiej przekątnej jako o ekranie odbierającym światło od pierwszej latarni. Jeżeli zmienimy kształt tego ekranu, żeby był sumą dwóch boków tego trójkącika, otrzymuje on nadal dokładnie tyle samo światła,", "n_reviews": 0, @@ -119,7 +119,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "", "from_community_srt": "to jasność maleje 4 razy co jest dość proste obliczeniowo; żeby przejść z sumy po wszystkich, do sumy po parzystych, musimy przemnożyć przez 1/4. A co to znaczy? Znaczy to, że jeśli przejść z sumy po wszystkich do sumy po nieparzystych, to przemnożyć przez 3/4 ponieważ te dwie rzeczy muszą dać nam jedynkę, czyli cały szereg. Zatem wywracamy ułamek i mówimy że przejście z sumy po nieparzystych do sumy wszystkich to przemnożenie przez 4/3 Więc mamy pi kwadrat przez 8, pomnożone przez 4/3, i czary mary - mamy rozwiązanie problemu bazylejskiego. To wideo, które właśnie widzieliście, było napisane i animowane głównie przez jednego z naszych nowych członków,", "n_reviews": 0, diff --git a/2018/basel-problem/portuguese/sentence_translations.json b/2018/basel-problem/portuguese/sentence_translations.json index 92fa374b5..88b623e5e 100644 --- a/2018/basel-problem/portuguese/sentence_translations.json +++ b/2018/basel-problem/portuguese/sentence_translations.json @@ -71,7 +71,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "Na geometria esférica você às vezes fala sobre o ângulo sólido de uma forma Que é a proporção de uma esfera que ela cobre quando vista de um determinado ponto Você vê o primeiro dos dois lugares em que esta história que estamos pensando nas telas será útil é entendendo a lei do inverso do quadrado Que é um fenômeno distintamente tridimensional pense em todos os raios de luz atingindo uma tela a uma unidade de distância da fonte enquanto você dobra a distância que esses raios agora cobrirão uma área com o dobro da largura e o dobro da altura Então seriam necessárias quatro cópias daquela tela original para receber os mesmos raios a essa distância E assim cada indivíduo recebe 1 quarto da quantidade de luz Este é o sentido no qual quero dizer que uma luz apareceria 1 quarto mais brilhante duas vezes a distância Da mesma forma, quando você está três vezes mais longe, você precisaria de nove cópias da tela original para receber os mesmos raios, de modo que cada tela individual receba apenas 19 a mais de luz e esse padrão continua porque a área atingida por uma luz aumenta pelo quadrado de a distância, o brilho dessa luz diminui pelo inverso do quadrado dessa distância e, como tenho certeza que muitos de vocês sabem, essa lei do inverso do quadrado não é nada especial para a luz. Ela aparece sempre que você tem algum tipo de quantidade que se espalha uniformemente de uma fonte pontual, seja som, calor ou sinal de rádio, coisas assim e um conjunto infinito de faróis uniformemente espaçados implementa fisicamente o problema de Basileia Mas, novamente, o que precisamos se quisermos fazer algum progresso aqui é entender como podemos manipular as configurações com fontes de luz como esta sem alterar o brilho total para o observador e O alicerce principal é uma maneira especialmente agradável de transformar um único farol em dois Pense em um observador na origem do plano XY e um único farol situado em algum lugar naquele plano Agora desenhe uma linha daquele farol até o observador e depois outra linha perpendicular àquela no farol Agora coloque dois faróis onde esta nova linha cruza os eixos de coordenadas Que irei em frente e chamarei de farol a aqui à esquerda e farol B no lado superior Acontece e você verá por que isso é verdade em apenas um minuto o brilho que o observador experimenta daquele primeiro farol é igual ao brilho combinado experimentado pelos faróis A e B juntos E devo dizer por a suposição constante ao longo deste vídeo é que todos os faróis são equivalentes Eles estão usando a mesma lâmpada emanando a mesma energia tudo isso Então, em outras palavras, atribuindo variáveis às coisas aqui se chamarmos a distância do observador ao farol de pequeno a E a distância do observador ao farol B pequeno B e a distância até o primeiro farol H Temos a relação 1 sobre a ao quadrado mais 1 sobre b ao quadrado é igual a 1 sobre h ao quadrado Este é o muito menos conhecido Teorema de Pitágoras Inverso que alguns de vocês podem reconhecer do vídeo mais recente e direi o mais excelente do matemático sobre os muitos primos do teorema de Pitágoras. Relação muito legal, você não acha? E se você é um matemático de coração, pode estar perguntando agora como você prova isso e Existem algumas maneiras simples de expressar a área dos triângulos de duas maneiras distintas e aplicar o teorema de Pitágoras usual Mas há outro método bastante bonito que eu gostaria de descrever brevemente aqui e que se enquadra muito melhor em nosso enredo porque, novamente, ele usa intuições de luz e telas Imagine reduzir todo o triângulo retângulo em uma versão menor e pense nesta hipotenusa em miniatura como uma tela recebendo luz do primeiro farol Se você remodelar essa tela para ser a combinação das duas pernas do triângulo em miniatura como este Bom, ele ainda recebe a mesma quantidade de luz, certo?", "model": "google_nmt", "from_community_srt": "Em Geometria Esférica às vezes perguntamos sobre o \"ângulo sólido\" de uma forma, que é a proporção de uma esfera que ela cobre, quando vista de um ponto dado. Veja, a primeira entre as duas vezes em que essa estória de telas será útil é em entender a lei do inverso do quadrado que é um fenômeno distintamente tridimensional. Pense em todos os raios de luz que atingem a tela uma unidade adiante da fonte. Quando você dobra essa distância esses raios agora vão cobrir uma área com o dobro da largura e o dobro da altura. Então seriam necessárias 4 cópias da tela original para receber a mesma quantidade a essa distância, de modo que cada cópia individual recebe um quarto da luz. Isto é o que quero dizer quando digo que a luz apareceria com 1/4 do brilho a uma distância dobrada da fonte. Da mesma forma, quando você está três vezes mais distante você precisaria de nove cópias daquela tela original para receber a mesma quantidade de luz, então cada tela individual recebe apenas 1/9 dessa quantidade. E esse padrão continua porque como a área atingida pela luz aumenta com o quadrado da distância, o brilho da luz diminui segundo o inverso do quadrado dessa distância. E, como sei que muitos de vocês sabem, essa lei do inverso do quadrado não é de modo algum especial para a luz. Ela aparece sempre que existe algum tipo de quantidade que se espalha homogeneamente desde uma fonte pontual, quer seja som ou calor ou sinais de rádio, coisas assim. E lembre-se: é por causa dessa lei do inverso do quadrado que uma cadeia infinita de faróis de luz igualmente espaçados efetua fisicamente o problema de Basileia. Mas novamente, o que precisamos para fazer qualquer progresso aqui é entender como podemos manipular configurações com fontes de luz como essa sem mudar o brilho total para o observador. E a peça fundamental é uma maneira especialmente elegante de transformar um único farol em dois. Pense num observador na origem do plano xy e um único farol situado em algum lugar do plano. Agora desenhe uma linha desde esse farol até o observador e então outra linha perpendicular a esta linha, passando pelo farol. Agora posicione dois faróis onde essa linha cruza os eixos coordenados. que agora eu vou chamar de farol A aqui na direita e farol B aqui em cima. Acontece - e você vai ver porque isso é verdade num minuto - que o brilho que o observador experimenta devido àquele primeiro farol é igual aos brilhos combinados dos faróis A e B juntos. E aliás eu devo dizer que a suposição básica durante todo esse vídeo é que todos os faróis são equivalentes. Eles usam o mesmo bulbo de luz, emanam a mesma potência e tudo o mais. Então, em outras palavras, designando variáveis para as coisas aqui, se chamarmos a distância do observador até o farol A de 'a' minúsculo e a distância do observador até o farol B de 'b' minúsculo, e a distância daquele primeiro farol de 'h' nós temos a relação: 1 sobre 'a' ao quadrado mais 1 sobre 'b' ao quadrado é igual a 1 sobre 'h' ao quadrado. Essa é o pouco conhecido teorema de Pitágoras inverso, que talvez alguns de vocês reconheçam do mais recente e, vou dizer, excelente vídeo do Mathologer sobre os vários primos do Teorema de Pitágoras Muito legal a relação, você não acha? E se você é matemático de coração deve estar se perguntando agora mesmo como você a prova, e há algumas provas diretas onde você expressa a área dos triângulos de duas formas separadas e aplica o Teorema de Pitágoras comum. Mas existe outro método bem elegante que eu gostaria de destacar brevemente aqui e que combina bem melhor com a nossa estória porque novamente ele usa intuições de luzes e telas. Imagine redimensionar o triângulo retângulo inteiro em uma versão minúscula e pensar nessa hipotenusa em miniatura como a tela que recebe luz do primeiro farol. Se você remodelar essa tela para ser uma combinação dos dois braços do triângulo em miniatura, dessa forma, bem, ela ainda receve a mesma quantidade de luz, certo?", @@ -134,7 +134,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "Isso significa que o brilho aparente seria diminuído por um fator de quatro. Isso também é uma álgebra relativamente simples, indo da soma de todos os números inteiros até a soma dos números inteiros pares. Envolve a multiplicação por 1 4. E o que isso significa é que indo de todos os números inteiros números inteiros para os ímpares seriam multiplicados por 3 4 Já que os pares mais as probabilidades têm que nos dar a coisa toda Então se invertermos isso significa passar da soma dos números ímpares para a soma de todos os números inteiros positivos requer multiplicação por 4 terços Então, pegando aquele pi ao quadrado sobre 8 multiplicando por 4 terços bada boom bada bing Nós temos uma solução para o problema do manjericão Agora, este vídeo que você acabou de assistir foi escrito e animado principalmente por um dos novos três azuis e um marrom membros da equipe Ben Hambricht Uma adição tornada possível.", "model": "google_nmt", "from_community_srt": "isso quer dizer que o brilho aparente seria reduzido por um fator de 4 e isso também é álgebra relativamente direta, indo desde a soma sobre todos os inteiros à soma sobre todos os inteiros pares envolve multiplicar por 1/4. O que isso significa é que indo de todos os inteiros para os ímpares seria multiplicar por 3/4, já que os pares mais os ímpares tem que nos dar a coisa inteira. Então, se nós apenas virarmos o sentido isso significa que indo da soma sobre os ímpares para a soma sobre todos os positivos requer uma multiplicação por 4/3. Então pegando aquele pi ao quadrado sobre 8, multiplicando por 4/3 - bada bum bada bing! - conseguimos uma solução para o problema de Basileia. O vídeo que você acabou de assistir foi primariamente escrito e animado por um dos novos", diff --git a/2018/basel-problem/russian/sentence_translations.json b/2018/basel-problem/russian/sentence_translations.json index 2e56b66a0..8877e67f7 100644 --- a/2018/basel-problem/russian/sentence_translations.json +++ b/2018/basel-problem/russian/sentence_translations.json @@ -63,7 +63,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "В сферической геометрии иногда говорят о телесном угле формы. Это пропорция сферы, которую она покрывает, если смотреть из данной точки. Вы видите первое из двух мест, в которых, как мы думаем, будет полезна эта история об экранах. понимание закона обратных квадратов, который представляет собой отчетливо трехмерное явление, представьте себе, что все лучи света падают на экран на расстоянии одной единицы от источника, когда вы удваиваете расстояние, которое эти лучи теперь будут покрывать область с удвоенной шириной и удвоенной высотой. Таким образом, потребовалось бы четыре копии исходного экрана, чтобы получить одни и те же лучи на этом расстоянии. И поэтому каждый отдельный экземпляр получает на 1 четверть больше света. В этом смысле я имею в виду, что свет будет казаться на 1 четверть ярче на расстоянии в два раза больше. Аналогично, когда вы находитесь в три раза дальше, вам понадобится девять копий этого исходного экрана, чтобы получить одни и те же лучи, поэтому каждый отдельный экран получает только в 19 раз больше света. Эта закономерность продолжается, потому что площадь, на которую попадает свет, увеличивается на квадрат с расстоянием яркость этого света уменьшается пропорционально квадрату этого расстояния, и, как я уверен, многие из вас знают, что этот закон обратных квадратов не является чем-то особенным для света. Он всплывает всякий раз, когда у вас есть какая-то величина, которая распределяется равномерно. из точечного источника, будь то звук, тепло или радиосигнал и тому подобное, и Бесконечный массив равномерно расположенных маяков физически реализует базельскую задачу. Но опять же, если мы хотим добиться какого-либо прогресса, нам нужно понять, как мы можем манипулировать установками с такими источниками света без изменения общей яркости для наблюдателя, а ключевой строительный блок — особенно хороший способ превратить один маяк в два. Подумайте о наблюдателе в начале плоскости XY и одиночном маяке, расположенном где-то на ней. плоскость. Теперь нарисуйте линию от маяка до наблюдателя, а затем еще одну линию, перпендикулярную этой линии на маяке. Теперь поместите два маяка там, где эта новая линия пересекает оси координат. Я назову это маяком а здесь слева и маяк B на верхней стороне Оказывается, и вы поймете, почему это правда, всего через минуту яркость, которую наблюдатель ощущает от этого первого маяка, равна совокупной яркости, которую испытывает маяк A и B вместе взятые. И я должен сказать, В этом видео постоянное предположение заключается в том, что все маяки эквивалентны. Они используют одну и ту же лампочку, излучающую одинаковую мощность. Другими словами, присвоение переменных вещам здесь, если мы называем расстояние от наблюдателя до маяка немного a И расстояние от наблюдателя до маяка B немного B и расстояние до первого маяка H. У нас есть соотношение 1 в квадрате плюс 1 в квадрате b равно 1 в квадрате h. Это гораздо менее известная обратная теорема Пифагора. который некоторые из вас могут узнать из последнего и, я бы сказал, самого превосходного видео о многих родственниках теоремы Пифагора. Довольно крутое соотношение, не правда ли, и если вы математик в душе, вы можете спросить прямо сейчас как вы это доказываете. Есть несколько простых способов выразить площадь треугольников двумя разными способами и применить обычную теорему Пифагора. Но есть еще один довольно красивый метод, который я хотел бы кратко обрисовать здесь, который гораздо лучше вписывается в нашу сюжетную линию. потому что опять-таки он использует интуицию света и экранов. Представьте себе, что вы уменьшаете весь прямоугольный треугольник до более маленькой версии и думаете об этой миниатюрной Гипотенузе как об экране, получающем свет от первого маяка. Если вы измените форму этого экрана, чтобы он представлял собой комбинацию двух частей маяка. Миниатюрный треугольник вроде этого. Ну, он все равно получает такое же количество света, верно?", "from_community_srt": "наверное, доводилось говорить о телесном угле, который определяется как часть охватываемой им сферы с центром в вершине телесного угла. Это первый момент (из двух), когда воображаемый экран, поможет понять закон обратных квадратов. Однозначно, это связано с пространственным явлением, заключающимся в том, что если все лучи падают на экран единичной площади, и вы решите удвоить расстояние от источника света, то лучи покроют площадку вдвое большей ширины и вдвое большей высоты. То есть мы получим 4 копии исходного экрана получающих на этом расстоянии то же количество лучей. На каждый такой экран приходится четверть света. Вот, что я имел в виду, говоря, что яркость составляет 1/4 при увеличении расстояния в два раза. А что, если увеличить расстояние в три раза? Вам понадобятся девять копий экрана, получающих то же количество лучей, то есть на каждый отдельный экран будет приходиться 1/9 излучаемого света. Дальше по тому же принципу, потому что площадь, освещаемая лучами, возрастает в квадратичной зависимости от расстояния, и яркость убывает пропорционально квадрату расстояния. И, я в этом уверен, многие из вас знают, что закон обратных квадратов присущ не только лучам. Он всплывает всюду, где некоторое количество чего бы то ни было равномерно распространяется от точечного источника. Например, звук, или тепло, или радиоволна и тому подобное. Так вот, благодаря этому самому закону обратных квадратов, бесконечный ряд равномерно расположенных маячков физически воплощает Базельскую задачу. Итак, всё что нам нужно, чтобы получить какое-то продвижение, это понять, каким образом мы можем осуществить расстановку источников света, не меняя общей яркости для наблюдателя. И краеугольным камнем является весьма изящный способ преобразования одного маячка в два. Представьте, что наблюдатель находится в начале координат, а одиночный маячок расположен где-то на координатной плоскости. Проведем прямую линию от маячка к наблюдателю, а также прямую перпендикулярную ей и проходящую через маячок. Теперь поместим два маячка в точках пересечения этой прямой координатных осей. В дальнейшем я буду называть маячок, расположенный справа - A, а тот маячок, что сверху - B. Получается, и вы поймёте почему это так уже через минуту, что по оценке наблюдателя яркость исходного маячка равна суммарной яркости двух маячков A и B. Кстати, на протяжении всего этого видео будем полагать, что все маячки эквивалентны, и в них используется одинаковые лампочки, одинаковой мощности. Условимся, что мы обозначим расстояние от наблюдателя до маячка A маленькой буквой a, расстояние от наблюдателя до маячка B маленькой буквой b, и расстояние до исходного маячка маленькой буквой h, мы получим равенство: 1 разделить на a в квадрате, плюс 1 на b в квадрате, равно 1 разделить на h в квадрате. Это довольно таки известная обратная теорема Пифагора. Знали её? Может, узнали недавнее видео на канале Mathologer о множестве сестёр теоремы Пифагора? Довольно прикольное соотношение. Вам так не кажется? Если вы являетесь математиками в душе, то прямо сейчас вы наверное задаёте вопрос, каким образом это можно доказать? Это можно доказать напрямую, выразив площадь треугольника двумя различными способами, и применив обычную теорему Пифагора. Однако есть другой довольно привлекательный метод, который я опишу вкратце, и он вписывается как нельзя лучше в наше повествование, потому что в нём опять-таки используется наглядная демонстрация со светом и экранами. Мысленно уменьшим правый треугольник до его крошечной копии, и будем рассматривать эту уменьшенную гипотенузу как экран, на который падает свет от исходного маячка. Если вы преобразуете этот экран в пару двух катетов уменьшенного треугольника, то на них по-прежнему будет падать то же самое количество света. То есть, я имею в виду,", "n_reviews": 0, @@ -119,7 +119,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "Это означает, что видимая яркость уменьшится в четыре раза. Это также относительно простая алгебра, переход от суммы по всем целым числам к сумме по четным целым числам. Включает умножение на 1 4. И это означает, что переход от всех целые числа на нечетные будут умножаться на 3 четверти. Поскольку четы плюс шансы должны дать нам все. Итак, если мы просто перевернем это, это означает, что переход от суммы по нечетным числам к сумме по всем положительным целым числам требует умножения на 4 трети. Итак, возведя это число в квадрат к 8, умножив на 4 трети bada boom bada bing У нас есть решение проблемы с базиликом. Теперь это видео, которое вы только что посмотрели, было в основном написано и анимировано одним из трех новых синих и одного коричневого. члены команды Бен Хамбрихт Добавление стало возможным.", "from_community_srt": "их яркость, воспринимаемая наблюдателем, уменьшается в 4 раза. И, как это напрямую вытекает из алгебры, для перехода от суммы обратных квадратов всех чисел к сумме обратных квадратов чётных чисел нужно применить множитель 1/4. Что из этого следует? Что для перехода от суммы полного ряда к сумме обратных квадратов нечётных чисел нужно применить множитель 3/4. В этом случае, сложив суммы обратных кваратов чётных и нечётных чисел, мы получим сумму полного ряда. И, соответственно, наоборот, чтобы перейти от суммы обратных квадратов нечётных чисел, которая нам уже известна, к сумме полного ряда нужно умножить известную сумму на 4/3.", "n_reviews": 0, diff --git a/2018/basel-problem/spanish/sentence_translations.json b/2018/basel-problem/spanish/sentence_translations.json index 91758736c..ccb8c83e0 100644 --- a/2018/basel-problem/spanish/sentence_translations.json +++ b/2018/basel-problem/spanish/sentence_translations.json @@ -63,7 +63,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "En geometría esférica a veces se habla del ángulo sólido de una forma, que es la proporción de una esfera que cubre vista desde un punto determinado. Ves el primero de los dos lugares en los que esta historia en la que estamos pensando sobre las pantallas será útil. Comprender la ley del cuadrado inverso, que es un fenómeno claramente tridimensional. Piense en todos los rayos de luz que inciden en una pantalla a una unidad de distancia de la fuente. Al duplicar la distancia, esos rayos ahora cubrirán un área con el doble de ancho y el doble de alto. Entonces, se necesitarían cuatro copias de esa pantalla original para recibir los mismos rayos a esa distancia. Y entonces cada individuo recibe 1 cuarta parte de la luz. Este es el sentido en el que quiero decir que una luz parecería 1 cuarta parte más brillante dos veces a la distancia. Del mismo modo, cuando estás tres veces más lejos, necesitarías nueve copias de esa pantalla original para recibir los mismos rayos, por lo que cada pantalla individual solo recibe 19 veces más luz y este patrón continúa porque el área alcanzada por una luz aumenta en el cuadrado de a distancia, el brillo de esa luz disminuye en el cuadrado inverso de esa distancia y, como estoy seguro de que muchos de ustedes saben, esta ley del cuadrado inverso no es en absoluto especial para la luz. Aparece cada vez que tienes algún tipo de cantidad que se extiende uniformemente. desde una fuente puntual, ya sea sonido, calor o señales de radio, cosas así y una variedad infinita de faros espaciados uniformemente implementa físicamente el problema de Basilea. Pero nuevamente, lo que necesitamos si queremos hacer algún progreso aquí es comprender cómo podemos manipular las configuraciones. con fuentes de luz como esta sin cambiar el brillo total para el observador. El bloque de construcción clave es una forma especialmente agradable de transformar un solo faro en dos. Piense en un observador en el origen del plano XY y un solo faro en algún lugar de ese. plano Ahora dibuja una línea desde ese faro hasta el observador y luego otra línea perpendicular a la del faro. Ahora coloca dos faros donde esta nueva línea cruza los ejes de coordenadas. Lo llamaré faro a aquí a la izquierda y faro B en la parte superior Resulta y verás por qué esto es cierto en solo un minuto, el brillo que el observador experimenta desde ese primer faro es igual al brillo combinado experimentado por los faros A y B juntos. Y debería decir por la forma en que la suposición vigente a lo largo de este video es que todos los faros son equivalentes. Están usando la misma bombilla que emana la misma potencia. En otras palabras, asignamos variables a las cosas aquí si llamamos a la distancia desde el observador al faro una pequeño a Y la distancia del observador al faro B pequeño B y la distancia al primer faro H Tenemos la relación 1 sobre a al cuadrado más 1 sobre b al cuadrado es igual a 1 sobre h al cuadrado Este es el mucho menos conocido teorema de Pitágoras Inverso que algunos de ustedes pueden reconocer por el video más reciente y excelente de Math Ologer sobre los muchos primos del teorema de Pitágoras. Una relación muy buena, ¿no lo creen? Y si son matemáticos de corazón, podrían estar preguntando ahora mismo. cómo lo demuestras y Hay algunas formas sencillas en las que expresas el área de los triángulos de dos maneras separadas y aplicas el teorema de Pitágoras habitual. Pero hay otro método bastante bonito que me gustaría resumir brevemente aquí y que encaja mucho mejor en nuestra trama. porque, de nuevo, utiliza intuiciones de luz y pantallas. Imagínese reducir todo el triángulo rectángulo a una versión más pequeña y pensar en esta hipotenusa en miniatura como una pantalla que recibe luz del primer faro. Si remodela esa pantalla para que sea la combinación de los dos catetos del triángulo en miniatura como este Bueno, todavía recibe la misma cantidad de luz, ¿verdad?", "from_community_srt": "Geometría, a veces hablas, sobre el ángulo sólido de una forma ¿Cuál es la proporción de una esfera que cubre como se ve desde un punto determinado que Ver el primero de dos lugares esta historia que pensamos que las pantallas serán útiles es para entender la ley del cuadrado inverso Lo cual es un fenómeno claramente tridimensional, piense en todos los rayos de luz que golpean una pantalla, una unidad, lejos Desde la fuente a medida que duplicas la distancia, esos rayos cubrirán ahora un área con el doble del ancho y dos veces la altura Así que tomaría 4 copias de esa pantalla original para recibir el mismo aumento a esa distancia y cada El individuo recibe una cuarta parte de la luz Este, es el sentido en el que me refiero a una luz, aparecería 1/4 como brillante a dos veces la distancia Del mismo modo, cuando estás tres veces más lejos, de distancia? Usted, necesitaría nueve copias de esa pantalla original para recibir los mismos rayos, de modo que cada pantalla individual solo reciba 1/9 de luz y Este patrón continúa porque el área golpeada, por una luz aumenta por el cuadrado de la distancia, el brillo de esa luz disminuye, por el cuadrado inverso de esa distancia y Como estoy seguro que muchos de ustedes saben, esta ley de cuadrado inverso No es nada especial a la luz, aparece cada vez que tienes algún tipo de cantidad que se distribuye uniformemente desde una fuente puntual, ya sea sonido, calor o señales de radio, cosas así. Y recuerde que es debido a esta ley del cuadrado inverso, que una serie infinita de faros espaciados uniformemente implementa físicamente el problema de Basilea. Pero nuevamente, lo que necesitamos, si vamos a hacer algún progreso aquí, es entender cómo, podemos manipular las configuraciones con fuentes de luz como esta sin, cambiando el brillo total para el observador. Y el elemento clave es una forma especialmente agradable de transformar un solo faro en Piense en un observador en el origen del avión x-y y un solo faro sentado en algún lugar de ese plano Ahora traza una línea desde ese faro hasta el observador y luego otra línea, perpendicular a esa en el faro ahora coloque dos faros donde esta nueva línea interseca los ejes de coordenadas Lo que voy a seguir y llamar al faro a aquí a la izquierda y al faro b en la parte superior Resulta que verás por qué esto es cierto en solo un minuto el brillo que el observador experimenta desde ese primer faro es Igual al brillo combinado experimentado en los faros a y b juntos y debo decir Por cierto que la suposición de pie a lo largo Este video, es que todos los faros son equivalentes son Usando la misma bombilla que emana la misma potencia, todo eso Entonces, en otras palabras, asignando variables a las cosas aquí si llamamos la distancia del observador al faro a poco a y la distancia del observador al faro b poco b y la distancia al primero, faro h Tenemos la relación 1 sobre un cuadrado más 1 sobre b al cuadrado es igual a 1 sobre h al cuadrado Este es el mucho menos conocido Teorema de Pitágoras inverso, ¿cuál de ustedes? Mayo, reconozca de las mitologías el video más reciente y también el más excelente sobre los muchos primos del teorema de Pitágoras muy bien, relación, ¿no crees y Si es así, un matemático de corazón podría estar preguntándose ahora mismo. Cómo Tú, demuéstralo y hay algunas formas sencillas, donde expresas el área de triángulos de dos maneras distintas. y aplicar el teorema de Pitágoras habitual Pero hay otro método bastante bonito que me gustaría resumir aquí que encaja mucho mejor en nuestra historia porque Nuevamente usa intuiciones de luz y pantallas Imagine reducir el triángulo rectángulo completo a una versión más pequeña y piense en esta hipotenusa en miniatura como una pantalla que recibe luz del primer faro si Tú, reconfigura esa pantalla, para ser la combinación de las dos patas del triángulo en miniatura, como esta Bueno, todavía recibe la misma cantidad de luz,", "n_reviews": 0, @@ -119,7 +119,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "Significa que el brillo aparente se reduciría en un factor de cuatro y eso también es un álgebra relativamente sencilla: pasar de la suma de todos los números enteros a la suma de los enteros pares. Implica multiplicar por 1 4. Y lo que eso significa es que pasar de todos los números enteros a impares sería multiplicar por 3 cuartos Dado que los pares más las probabilidades tienen que darnos el todo. Entonces, si simplemente le damos la vuelta a eso, eso significa que pasar de la suma de los números impares a la suma de todos los números enteros positivos requiere multiplicar. por 4 tercios Entonces, tomando pi al cuadrado sobre 8 multiplicado por 4 tercios bada boom bada bing Tenemos una solución al problema de la albahaca. Ahora bien, este video que acabas de ver fue escrito y animado principalmente por uno de los nuevos tres azules y uno marrón. miembros del equipo Ben Hambricht Una incorporación hecha posible.", "from_community_srt": "significa que el brillo aparente se reduciría en un factor de 4 y? Eso también es álgebra relativamente directa De la suma de todos los enteros a la suma sobre los enteros pares implica multiplicar por 1/4. ¿Y qué significa eso? ¿Eso va de todos los enteros a los impares? Se estaría multiplicando por 3/4 ya que los pares más las probabilidades tienen que darnos todo Si volteamos eso significa ir, de la suma sobre los números impares a la suma sobre todos los enteros positivos requiere multiplicar por 4/3 Tomando ese pi al cuadrado sobre 8. Multiplicando, por 4/3 pero a, boom, bada, bing, tenemos una solución al problema de basilea Este video que acabas de ver fue escrito y animado principalmente por uno de los nuevos", "n_reviews": 0, diff --git a/2018/basel-problem/tamil/sentence_translations.json b/2018/basel-problem/tamil/sentence_translations.json index d5a65ece8..34a680085 100644 --- a/2018/basel-problem/tamil/sentence_translations.json +++ b/2018/basel-problem/tamil/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", "translatedText": "அதாவது, அந்த இரண்டு கால்களில் ஒன்றைத் தாக்கும் ஒளிக் கதிர்களும், ஹைப்போடென்யூஸைத் தாக்கும் கதிர்களும் துல்லியமாக ஒரே மாதிரியாக இருக்கும். அப்போது முக்கியமானது என்னவென்றால், முதல் கலங்கரை விளக்கத்தில் இருந்து வரும் ஒளியின் அளவு இந்த இடது பக்கத்தைத் தாக்கும் கதிர்களின் வரையறுக்கப்பட்ட கோணத்தில் தாக்குகிறது. அந்தத் திரையானது கலங்கரை விளக்கத்தில் இருந்து வரும் ஒளியின் அளவைப் போலவே இருக்கிறது, அது அந்தப் பக்கத்தைத் தாக்கும் அதே கோணத்தில் கதிர்கள் இருக்கும் மற்றும் சமச்சீராக நமது திரையின் கீழ்ப் பகுதியைத் தாக்கும் முதல் வீட்டிலிருந்து வரும் ஒளியின் அளவு ஒன்றுதான். கலங்கரை விளக்கத்திலிருந்து அந்த பகுதியைத் தாக்கும் ஒளியின் அளவு B ஏன் நீங்கள் நன்றாகக் கேட்கலாம், இது ஒரே மாதிரியான முக்கோணங்களின் விஷயம் தான் இந்த அனிமேஷன் ஏற்கனவே உங்களுக்கு இது எப்படி வேலை செய்கிறது என்பதற்கான வலுவான குறிப்பைத் தருகிறது மேலும் நாங்கள் ஒரு எளிய ஜியோஜிப்ராவுக்கான இணைப்பையும் விளக்கியுள்ளோம். உங்களில் சற்று அதிக ஊடாடும் சூழலில் இதைப் பற்றி சிந்திக்க விரும்புவோருக்கான ஆப்லெட் மற்றும் அந்த ஒரு முக்கியமான உண்மையை இங்கே நீங்கள் பார்க்க முடியும், இது போன்ற முக்கோணங்கள் மிக சிறிய திரையில் மட்டுமே பொருந்தும் சரி, இப்போதே கொக்கி போடுங்கள், ஏனென்றால் இங்குதான் விஷயங்கள் நன்றாக இருக்கும், இந்த தலைகீழ் பித்தகோரியன் தேற்றம் நமக்குக் கிடைத்துள்ளது, இல்லையா?", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right.", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right.", "translatedText": "அந்த கலங்கரை விளக்கங்களிலிருந்து மையத்திற்கு உள்ள கோடுகள் ஒன்றோடொன்று 90 டிகிரி கோணத்தில் உள்ளன, எனவே விஷயங்கள் இடமிருந்து வலமாக சமச்சீராக இருப்பதால், சுற்றளவுடன் உள்ள தூரங்கள் 1 2 2 2 மற்றும் 1 சரி, இது எங்கே போகிறது என்பதை நீங்கள் பார்க்கலாம், ஆனால் நான் இன்னும் ஒரு படிக்கு இதை கடந்து செல்ல விரும்புகிறேன், நீங்கள் இப்போது 16 சுற்றளவை விட இரண்டு மடங்கு பெரிய வட்டத்தை வரைகிறீர்கள், மேலும் ஒவ்வொரு கலங்கரை விளக்கத்திற்கும் அந்த பெரிய வட்டத்தின் மையமாக இருக்கும் சிறிய வட்டத்தின் மேல் ஒரு கோடு வரைகிறீர்கள். நீண்ட கோடு பெரிய வட்டத்தின் விட்டம் என்பதால் முன்பு போலவே இரண்டு புதிய கலங்கரை விளக்கங்களை உருவாக்குங்கள். அசல் கலங்கரை விளக்கம் நீண்ட கோட்டிற்கு செங்குத்தாக உள்ளது மற்றும் தலைகீழ் பித்தகோரியன் தேற்றத்தைப் பயன்படுத்துவதில் நம்மை நியாயப்படுத்தும் இரண்டு உண்மைகள் இவைதான். ஏரி அந்த எட்டு புதிய கலங்கரை விளக்கங்கள் சம இடைவெளியில் இருக்கப் போகிறது, இது இறுதி உந்துதலுக்கு முன் வடிவியல் ஆதாரத்தின் இறுதிப் பிட் ஆகும், இதைப் பார்க்க, சிறிய ஏரியின் இரண்டு அருகிலுள்ள கலங்கரை விளக்கங்களிலிருந்து மையத்திற்கு கோடுகளை வரைந்தால் அவை 90 டிகிரி கோணத்தை உருவாக்குகின்றன என்பதை நினைவில் கொள்ளுங்கள் அதற்குப் பதிலாக வட்டத்தின் சுற்றளவில் எங்கும் ஒரு புள்ளியில் கோடுகளை வரைகிறீர்கள், அவற்றுக்கிடையே இல்லாத மிகவும் பயனுள்ள பொறிக்கப்பட்ட கோணத் தேற்றம் வடிவவியலின் பொறிக்கப்பட்ட கோணத் தேற்றம், இது இந்த விஷயத்தில் 45 டிகிரி மையத்துடன் அவர்கள் செய்யும் கோணத்தின் பாதியாக இருக்கும் என்று நமக்குச் சொல்கிறது. ஏரியின் உச்சியில் புதிய புள்ளியை நிலைநிறுத்துகிறோம், பெரிய ஏரியில் உள்ள புதிய கலங்கரை விளக்கங்களின் நிலையை வரையறுக்கும் இரண்டு கோடுகள் இவையே, அதன் அர்த்தம் என்னவென்றால், அந்த எட்டு புதிய கலங்கரை விளக்கங்களிலிருந்து மையத்தில் கோடுகளை வரையும்போது அவை வட்டத்தைப் பிரிக்கின்றன. 45 டிகிரி கோணத் துண்டுகளாக சமமாக, அதாவது எட்டு கலங்கரை விளக்கங்கள் சுற்றளவைச் சுற்றி சம இடைவெளியில் ஒவ்வொன்றிற்கும் இடையே இரண்டு தூரம் இருக்கும். இப்போது ஒவ்வொரு வட்டத்தின் அளவையும் இரட்டிப்பாக்கி ஒவ்வொரு கலங்கரை விளக்கமாக மாற்றும் ஒவ்வொரு அடியிலும் இந்த விஷயம் விளையாடுவதை கற்பனை செய்து பாருங்கள். ஒவ்வொரு அடியிலும் பெரிய வட்டத்தின் மையத்தில் வரையப்பட்ட ஒரு கோட்டுடன் இரண்டு புதியவை, பார்வையாளருக்கு வெளிப்படையான பிரகாசம் 4 க்கு மேல் ஒரே பையாக இருக்கும், மேலும் ஒவ்வொரு அடியிலும் கலங்கரை விளக்கங்கள் ஒவ்வொன்றிற்கும் இடையே 2 தூரத்துடன் சமமாக இருக்கும் சுற்றளவு மற்றும் வரம்பில் நாம் இங்கு பெறுவது ஒரு தட்டையான கிடைமட்ட கோடு ஆகும் இது நமக்கு அழகான அற்புதமான முடிவிலித் தொடரின் கூட்டுத்தொகையை 1 மேல் n ஸ்கொயர்களின் கூட்டுத்தொகையை வழங்குகிறது, இதில் n என்பது ஒற்றைப்படை முழு எண்கள் 1 3 5 மற்றும் பலவற்றை உள்ளடக்கியது ஆனால் எதிர்மறை 1 எதிர்மறை 3 எதிர்மறை 5 ஆஃப் இடதுபுறத்தில் உள்ள அனைத்தையும் சேர்த்தல் 4 க்கு மேல் பை ஸ்கொயர் கொடுக்கப் போகிறது அது ஆச்சரியமாக இருக்கிறது, நான் உங்களுக்குக் காட்ட விரும்புவதின் முக்கிய அம்சம் இது. ஒரு படி பின்வாங்கி, இது எவ்வளவு உண்மையற்றது என்று யோசித்துப் பாருங்கள். முதல் பார்வையில் வடிவவியலுக்கும் எந்த சம்பந்தமும் இல்லாத எளிய பின்னங்களின் கூட்டுத்தொகை வட்டங்களுடன் எந்த தொடர்பும் இல்லை என்பது வெளிப்படையாக பை தொடர்பான இந்த முடிவை எங்களுக்குத் தருகிறது தவிர, வடிவவியலுக்கும் அதற்கும் என்ன சம்பந்தம் என்பதை இப்போது நீங்கள் உண்மையில் பார்க்க முடியும், எண் கோடு எப்போதும் வளர்ந்து வரும் வட்டங்களின் வரம்பைப் போன்றது மற்றும் நீங்கள் அந்த எண்ணைக் கூட்டினால் கோடு இருபுறமும் முடிவிலிக்கு முழுவதுமாகத் தொகுக்கப்படுவதை உறுதிசெய்கிறது, இது எல்லையற்ற பெரிய வட்டத்தின் எல்லையில் நீங்கள் சேர்ப்பது போன்றது மற்றும் மிகவும் தளர்வான ஆனால் மிகவும் வேடிக்கையாக பேசும் விதம் ஆனால் இது தொகை அல்ல என்று நீங்கள் கூறலாம். வீடியோவின் தொடக்கத்தில் நீங்கள் எங்களுக்கு உறுதியளித்தீர்கள், நீங்கள் சொல்வது சரிதான்.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible.", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "நம்மிடம் கொஞ்சம் யோசிக்க வேண்டியிருக்கிறது முதலில் முதலில் கூட்டுத்தொகையை நேர்மறை ஒற்றைப்படை எண்களாக மட்டும் கட்டுப்படுத்துவோம், அது நம்மை 8 ஆல் வகுத்தால் பை ஸ்கொயர் ஆகும். அனைத்து நேர்மறை முழு எண்கள் ஒற்றைப்படை மற்றும் சம எண்களின் கூட்டுத்தொகையைக் காணவில்லை, நான் இங்கே சிவப்பு நிறத்தில் வண்ணம் தீட்டுகிறேன், இப்போது அந்தத் தொடர் காணாமல் போன தொடரை நாம் விரும்பும் மொத்தத் தொடரின் அளவிடப்பட்ட நகலாக நீங்கள் நினைக்கலாம். தோற்றத்தில் இருந்து இருமடங்கு தொலைவில் இருப்பது ஒன்று இரண்டாக மாற்றப்படுகிறது இரண்டு நான்குக்கு மாற்றப்படுகிறது மூன்று ஆறு ஆக மாறுகிறது மற்றும் பல மற்றும் ஒவ்வொரு கலங்கரை விளக்கத்திற்கும் உள்ள தூரத்தை இரட்டிப்பாக்குவதால், வெளிப்படையான பிரகாசம் ஒரு காரணியால் குறைக்கப்படும் என்று அர்த்தம். நான்கு மற்றும் அதுவும் ஒப்பீட்டளவில் நேரடியான இயற்கணிதம் அனைத்து முழு எண்களின் கூட்டுத்தொகையிலிருந்து கூட்டு முழு எண்களின் கூட்டுத்தொகைக்கு 1 4 ஆல் பெருக்குவதை உள்ளடக்குகிறது மற்றும் இதன் பொருள் என்னவென்றால், அனைத்து முழு எண்களிலிருந்தும் ஒற்றைப்படைக்கு செல்வது 3 4-ல் பெருகும். சமன்பாடுகள் மற்றும் முரண்பாடுகள் நமக்கு முழு விஷயத்தையும் கொடுக்க வேண்டும், எனவே நாம் அதை சுழற்றினால், ஒற்றைப்படை எண்களின் கூட்டுத்தொகையில் இருந்து அனைத்து நேர்மறை முழு எண்களின் கூட்டுத்தொகைக்கு 4 மூன்றில் பெருக்க வேண்டும், எனவே அந்த pi வர்க்கத்தை 8 பெருக்கினால் பெருக்க வேண்டும். மூன்றில் 4 பாடா பூம் பட்டா பிங் துளசி பிரச்சனைக்கு நாமே தீர்வு பெற்றுள்ளோம், இப்போது நீங்கள் பார்த்த இந்த வீடியோவை முதன்மையாக புதிய மூன்று நீல நிற ஒன் பிரவுன் குழு உறுப்பினர்களில் ஒருவரான பென் ஹாம்ப்ரிச்ட் எழுதி அனிமேஷன் செய்துள்ளார்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/telugu/sentence_translations.json b/2018/basel-problem/telugu/sentence_translations.json index fad5c98e0..a7ba4f2b4 100644 --- a/2018/basel-problem/telugu/sentence_translations.json +++ b/2018/basel-problem/telugu/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", "translatedText": "నా ఉద్దేశ్యం ఏమిటంటే, ఆ రెండు కాళ్ళలో ఒకదానిని తాకిన కాంతి కిరణాలు ఖచ్చితంగా హైపోటెన్యూస్‌ను తాకిన కిరణాల మాదిరిగానే ఉంటాయి, అప్పుడు కీలకం ఏమిటంటే, మొదటి లైట్‌హౌస్ నుండి కాంతి మొత్తం ఈ ఎడమ వైపుకి తాకిన కిరణాల పరిమిత కోణాన్ని తాకుతుంది. ఆ స్క్రీన్ లైట్‌హౌస్ నుండి వచ్చే కాంతికి సరిగ్గా సమానంగా ఉంటుంది, అది ఆ వైపుకు తాకితే అది కిరణాల కోణంలో ఉంటుంది మరియు సౌష్టవంగా మన స్క్రీన్ దిగువ భాగాన్ని తాకిన మొదటి ఇంటి నుండి వచ్చే కాంతి మొత్తం అదే విధంగా ఉంటుంది. లైట్‌హౌస్ B నుండి ఆ భాగాన్ని తాకిన కాంతి మొత్తంగా మీరు ఎందుకు బాగా అడగవచ్చు, ఇది సారూప్య త్రిభుజాల విషయం ఈ యానిమేషన్ ఇప్పటికే మీకు ఇది ఎలా పని చేస్తుందనే దాని గురించి బలమైన సూచనను అందిస్తుంది మరియు మేము వివరణలో ఒక సాధారణ జియోజీబ్రాకు లింక్‌ను కూడా ఉంచాము కొంచెం ఎక్కువ ఇంటరాక్టివ్ వాతావరణంలో దీని గురించి ఆలోచించాలనుకునే మీ కోసం ఆప్లెట్ మరియు ఇక్కడ మీరు చూడగలిగే ఒక ముఖ్యమైన వాస్తవం ఏమిటంటే, ఇలాంటి త్రిభుజాలు చాలా చిన్న స్క్రీన్‌కు పరిమితం చేసే సందర్భంలో మాత్రమే వర్తిస్తాయి. సరే, ఇప్పుడే కట్టుకోండి ఎందుకంటే ఇక్కడే విషయాలు బాగా ఉన్నాయి, మనకు ఈ విలోమ పైథాగరియన్ సిద్ధాంతం ఉంది, సరియైనదా?", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right.", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right.", "translatedText": "ఆ లైట్‌హౌస్‌ల నుండి మధ్యలో ఉన్న పంక్తులు ఒకదానికొకటి 90 డిగ్రీల కోణంలో ఉంటాయి కాబట్టి విషయాలు ఎడమ నుండి కుడికి సుష్టంగా ఉంటాయి కాబట్టి చుట్టుకొలతలో ఉన్న దూరాలు 1 2 2 2 మరియు 1 సరే, ఇది ఎక్కడికి వెళుతుందో మీరు చూడవచ్చు, కానీ నేను దీని ద్వారా మరో ఒక్క అడుగు మాత్రమే నడవాలనుకుంటున్నాను, మీరు ఇప్పుడు 16 చుట్టుకొలత కంటే రెండు రెట్లు పెద్ద వృత్తాన్ని గీస్తారు మరియు ప్రతి లైట్‌హౌస్‌కి మీరు ఆ లైట్‌హౌస్ నుండి చిన్న వృత్తం పైభాగంలో ఒక గీతను గీస్తారు, ఇది పెద్ద వృత్తానికి మధ్యలో ఉంటుంది. ఆపై రెండు కొత్త లైట్‌హౌస్‌లను సృష్టించండి, ఆ రేఖ పెద్ద వృత్తంతో కలుస్తుంది, ఎందుకంటే పొడవైన రేఖ పెద్ద వృత్తం యొక్క వ్యాసం కాబట్టి ఆ రెండు కొత్త లైట్‌హౌస్‌లు పరిశీలకుడితో కుడివైపు మరియు పరిశీలకుడి నుండి రేఖకు ముందు ఉన్నట్లే లంబ కోణాన్ని చేస్తాయి. అసలైన లైట్‌హౌస్ దీర్ఘ రేఖకు లంబంగా ఉంటుంది మరియు అవి విలోమ పైథాగరియన్ సిద్ధాంతాన్ని ఉపయోగించడంలో మనల్ని సమర్థించే రెండు వాస్తవాలు కానీ మీరు అన్ని లైట్‌హౌస్‌ల కోసం దీన్ని చేసినప్పుడు పెద్ద మొత్తంలో ఎనిమిది కొత్త వాటిని పొందడానికి మీరు దీన్ని చేసినప్పుడు అంత స్పష్టంగా ఉండకపోవచ్చు. సరస్సులో ఆ ఎనిమిది కొత్త లైట్‌హౌస్‌లు సమానంగా ఉంటాయి బదులుగా మీరు వృత్తం యొక్క చుట్టుకొలతపై ఎక్కడైనా ఒక బిందువుకు గీతలు గీయండి, వాటి మధ్య లేని చాలా ఉపయోగకరమైన లిఖిత కోణ సిద్ధాంతం జ్యామితి నుండి మనకు చెబుతుంది, ఇది ఈ సందర్భంలో 45 డిగ్రీల మధ్యలో వారు చేసే కోణంలో సరిగ్గా సగం ఉంటుంది. మేము సరస్సు ఎగువన ఆ కొత్త బిందువును ఉంచుతాము, ఇవి పెద్ద సరస్సుపై కొత్త లైట్‌హౌస్‌ల స్థానాన్ని నిర్వచించే రెండు పంక్తులు, అంటే మీరు ఆ ఎనిమిది కొత్త లైట్‌హౌస్‌ల నుండి మధ్యలోకి గీతలు గీసినప్పుడు అవి వృత్తాన్ని విభజిస్తాయి సమానంగా 45 డిగ్రీల కోణ ముక్కలుగా మరియు అంటే ఎనిమిది లైట్‌హౌస్‌లు చుట్టుకొలత చుట్టూ సమానంగా ఉంటాయి, వాటిలో ప్రతి దాని మధ్య రెండు దూరం ఉంటుంది మరియు ఇప్పుడు ప్రతి వృత్తం యొక్క పరిమాణాన్ని రెట్టింపు చేయడం మరియు ప్రతి లైట్‌హౌస్‌గా మార్చడం ద్వారా ప్రతి అడుగులో ఈ విషయం ప్లే అవుతుందని ఊహించండి. ప్రతి అడుగులో పెద్ద వృత్తం మధ్యలో గీసిన రేఖ వెంట రెండు కొత్తవి ఉంటాయి, పరిశీలకుడికి కనిపించే ప్రకాశం 4 కంటే ఎక్కువ స్క్వేర్డ్‌గా ఉంటుంది మరియు ప్రతి దశలోనూ లైట్‌హౌస్‌లు వాటి మధ్య 2 దూరంతో సమానంగా ఉంటాయి. చుట్టుకొలత మరియు పరిమితిలో మనం ఇక్కడ పొందుతున్నది ఒక ఫ్లాట్ క్షితిజ సమాంతర రేఖను రెండు దిశలలో సమానంగా ఖాళీగా ఉన్న అనంతమైన లైట్‌హౌస్‌లతో మరియు స్పష్టమైన ప్రకాశం 4 కంటే ఎక్కువ స్క్వేర్ చేయబడింది మరియు ఈ పరిమితి విషయంలో కూడా ఇది నిజం. ఇది మాకు చాలా అద్భుతమైన అనంతమైన శ్రేణిని అందిస్తుంది 1 ఓవర్ n స్క్వేర్డ్ విలోమ చతురస్రాల మొత్తాన్ని ఇక్కడ n అన్ని బేసి పూర్ణాంకాల 1 3 5 మరియు మొదలైనవాటిని కవర్ చేస్తుంది కానీ ఎడమ వైపున ఉన్న వాటన్నింటినీ జోడించడం ద్వారా ప్రతికూల 1 ప్రతికూల 3 ప్రతికూల 5 ఆఫ్ మాకు 4 కంటే ఎక్కువ pi స్క్వేర్‌ని ఇవ్వబోతున్నాం అది అద్భుతంగా ఉంది మరియు ఇది నేను మీకు చూపించాలనుకుంటున్న దాని యొక్క ప్రధాన అంశం మరియు ఒక్క అడుగు వెనక్కి వేసి, ఇది ఎంత అవాస్తవంగా అనిపిస్తుందో ఆలోచించండి మొదటి చూపులో జ్యామితితో సంబంధం లేని సాధారణ భిన్నాల మొత్తం సర్కిల్‌లతో ఏమీ చేయనవసరం లేదు, పైకి సంబంధించిన ఈ ఫలితాన్ని మాకు అందిస్తుంది తప్ప జ్యామితితో దీనికి సంబంధం ఏమిటో ఇప్పుడు మీరు నిజంగా చూడగలరు, సంఖ్యా రేఖ నిరంతరం పెరుగుతున్న సర్కిల్‌ల పరిమితి లాంటిది మరియు మీరు ఆ సంఖ్య అంతటా కలిపితే పంక్తి ఇరువైపులా అనంతం వరకు సంక్షిప్తంగా ఉండేలా చూసుకోవాలి ఇది మీరు ఒక అనంతమైన పెద్ద వృత్తం మరియు చాలా వదులుగా ఉన్న సరిహద్దులో జోడించడం వంటిది, కానీ ఇది మొత్తం కాదు అని మీరు చెప్పవచ్చు వీడియో ప్రారంభంలో మీరు మాకు వాగ్దానం చేసారు మరియు మీరు చెప్పింది నిజమే.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible.", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "మనకు కొంచెం ఆలోచించడం మిగిలి ఉంది మొదటి విషయాలు ముందుగా మొత్తంని కేవలం సానుకూల బేసి సంఖ్యలుగా మాత్రమే పరిమితం చేద్దాం, ఇది pi స్క్వేర్డ్‌ను 8 ద్వారా భాగించబడుతుంది. ఇప్పుడు దీనికి మరియు మనం వెతుకుతున్న మొత్తానికి మధ్య ఉన్న తేడా ఒక్కటే దాటిపోతుంది. అన్ని ధన పూర్ణాంకాల బేసి మరియు సరి సంఖ్యల రెసిప్రోకల్‌ల మొత్తాన్ని నేను ఇక్కడ ఎరుపు రంగులో వేస్తున్నాను, ఇప్పుడు మీరు ఆ తప్పిపోయిన శ్రేణిని ప్రతి లైట్‌హౌస్‌లో మనకు కావలసిన మొత్తం శ్రేణి యొక్క స్కేల్డ్ కాపీగా భావించవచ్చు. మూలం నుండి రెండు రెట్లు దూరంగా ఉండటం వలన ఒకటి రెండుకి మార్చబడుతుంది రెండు నాలుగుకి మార్చబడుతుంది మూడు ఆరుకి మార్చబడుతుంది మరియు మొదలైనవి మరియు ప్రతి లైట్‌హౌస్‌కు దూరాన్ని రెట్టింపు చేయడం వల్ల స్పష్టమైన ప్రకాశం ఒక కారకం ద్వారా తగ్గుతుందని అర్థం. నాలుగు మరియు అది కూడా సాపేక్షంగా సూటిగా ఉండే బీజగణితం అన్ని పూర్ణాంకాలపై మొత్తం నుండి మొత్తం పూర్ణాంకాలపై మొత్తానికి 1 4వ గుణించడాన్ని కలిగి ఉంటుంది మరియు దీని అర్థం ఏమిటంటే అన్ని పూర్ణాంకాల నుండి బేసి వాటికి వెళ్లడం అనేది 3 4వ వంతున గుణించబడుతుంది సరిలు మరియు అసమానతలు మనకు మొత్తం విషయాన్ని అందించాలి కాబట్టి మనం దానిని తిప్పితే, బేసి సంఖ్యల మొత్తం నుండి మొత్తం ధనాత్మక పూర్ణాంకాల మొత్తానికి 4 వంతులు గుణించడం అవసరం కాబట్టి ఆ పైని 8 కంటే గుణించడం ద్వారా గుణించాలి 4 వంతుల బద్దా బూమ్ బద్దా బింగ్ తులసి సమస్యకు మేమే పరిష్కారం పొందాము ఇప్పుడు మీరు చూసిన ఈ వీడియో ప్రధానంగా కొత్త ముగ్గురు బ్లూ వన్ బ్రౌన్ టీమ్ సభ్యులలో ఒకరు బెన్ హాంబ్రిచ్ట్ ద్వారా వ్రాయబడింది మరియు యానిమేట్ చేయబడింది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/thai/sentence_translations.json b/2018/basel-problem/thai/sentence_translations.json index e6b47c401..253a8db9c 100644 --- a/2018/basel-problem/thai/sentence_translations.json +++ b/2018/basel-problem/thai/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right? ", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right? ", "translatedText": "ฉันหมายถึงรังสีที่ตกกระทบกับขาข้างใดข้างหนึ่งนั้นเหมือนกับรังสีที่กระทบด้านตรงข้ามมุมฉากพอดี สิ่งสำคัญคือปริมาณแสงจากประภาคารแรกที่กระทบทางด้านซ้ายนี้จะเป็นมุมที่จำกัดของรังสีที่สุดท้ายจะกระทบกัน หน้าจอนั้นก็เหมือนกับปริมาณแสงตรงนี้ที่มาจากประภาคาร ซึ่งกระทบกับด้านนั้นก็จะเป็นมุมของรังสีเท่ากัน และปริมาณแสงจากบ้านหลังแรกที่กระทบกับส่วนล่างของหน้าจอของเราจะเท่ากันทุกประการ เนื่องจากปริมาณแสงที่ตกกระทบส่วนนั้นจากประภาคาร B ทำไมคุณถึงถามได้ดี มันเป็นเรื่องของสามเหลี่ยมที่คล้ายกัน แอนิเมชั่นนี้ให้คำแนะนำที่ชัดเจนเกี่ยวกับวิธีการทำงานแล้ว และเรายังได้ทิ้งลิงก์ไว้ในคำอธิบายไปยัง GeoGebra แบบง่ายๆ แอปเพล็ตสำหรับผู้ที่ต้องการคิดเรื่องนี้ในสภาพแวดล้อมที่มีการโต้ตอบมากขึ้นเล็กน้อย และในการเล่นกับข้อเท็จจริงที่สำคัญประการหนึ่งที่นี่ คุณจะเห็นได้ว่าสามเหลี่ยมที่คล้ายกันจะใช้เฉพาะในกรณีที่จำกัดสำหรับหน้าจอขนาดเล็กมาก เอาล่ะ รัดเข็มขัดไว้เลย เพราะนี่คือจุดที่อะไรๆ ดีขึ้น เรามีทฤษฎีบทพีทาโกรัสอินเวอร์ส ใช่ไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right. ", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. ", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/turkish/sentence_translations.json b/2018/basel-problem/turkish/sentence_translations.json index 5ce9e29be..67044abe1 100644 --- a/2018/basel-problem/turkish/sentence_translations.json +++ b/2018/basel-problem/turkish/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", "translatedText": "Demek istediğim, bu iki bacaktan birine çarpan ışık ışınları, hipotenüse çarpan ışınlarla tamamen aynıdır. O zaman önemli olan, ilk deniz fenerinden gelen ışık miktarının bu sol tarafa çarptığı ve sonunda vuran ışınların sınırlı açısıdır. bu ekran tam olarak buradaki a deniz fenerinden gelen ve o tarafa çarpan ışık miktarıyla aynı, ışınların açısı aynı olacak ve simetrik olarak ilk evden ekranımızın alt kısmına çarpan ışık miktarı aynı olacak B deniz fenerinden bu kısma çarpan ışık miktarı olarak Neden iyi sorabilirsiniz, benzer üçgenler meselesi Bu animasyon zaten size nasıl çalıştığına dair güçlü bir ipucu veriyor Ve ayrıca açıklamaya basit bir GeoGebra bağlantısı da bıraktık Bunu biraz daha etkileşimli bir ortamda düşünmek isteyenler için bir uygulama ve bununla oynarken burada görebileceğiniz önemli bir gerçek, benzer üçgenlerin yalnızca çok küçük bir ekran için sınırlama durumunda geçerli olduğudur. Pekala, şimdi kemerlerinizi bağlayın çünkü işler burada iyiye gidiyor. Ters Pisagor teoremimiz var, değil mi?", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right.", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right.", "translatedText": "Bu deniz fenerlerinden merkeze doğru çizgiler birbiriyle 90 derecelik açılardadır. Yani her şey soldan sağa simetrik olduğundan bu, çevre boyunca mesafelerin 1 2 2 2 ve 1 olduğu anlamına gelir. Tamam, bunun nereye gittiğini görebilirsiniz, ama ben bunun üzerinden bir adım daha yürümek istiyorum Şimdi iki katı büyüklüğünde bir daire çiziyorsunuz, yani çevresi 16'dır ve her deniz feneri için o deniz fenerinden daha küçük olan dairenin tepesine doğru bir çizgi çiziyorsunuz ki bu büyük dairenin merkezidir ve sonra bu çizginin daha büyük daireyle kesiştiği yerde iki yeni deniz feneri oluşturun. Tıpkı daha önce olduğu gibi, uzun çizgi büyük dairenin çapı olduğundan bu iki yeni deniz feneri gözlemcinin sağına ve gözlemciden gelen çizginin önünde olduğu gibi orijinal deniz feneri uzun çizgiye diktir ve bunlar ters Pisagor teoremini kullanmamızı haklı çıkaran iki gerçektir. göldeki bu sekiz yeni deniz feneri eşit aralıklarla yerleştirilecek Bu, son itişten önceki son geometri kanıtıdır Bunu görmek için şunu unutmayın, küçük göldeki iki bitişik deniz fenerinden merkeze doğru çizgiler çizerseniz 90 derecelik bir açı yaparlar. bunun yerine dairenin çevresi üzerinde aralarında olmayan herhangi bir noktaya çizgiler çizersiniz, geometrideki çok kullanışlı yazılı açı teoremi bize bunun merkezle yaptıkları açının tam olarak yarısı olacağını söyler, bu durumda 45 derece. yeni noktayı gölün tepesine konumlandırıyoruz Bunlar, daha büyük göldeki yeni deniz fenerlerinin konumunu tanımlayan iki çizgidir. Bu şu anlama gelir: Bu sekiz yeni deniz fenerinden merkeze doğru çizgiler çizdiğinizde Çemberi bölerler. 45 derecelik açılı parçalara eşit olarak dağıtın ve bu, sekiz deniz fenerinin her biri arasında iki mesafe olacak şekilde çevre etrafında eşit aralıklarla yerleştirildiği anlamına gelir. Şimdi bu şeyin her adımda oynadığını, her dairenin boyutunu iki katına çıkardığını ve her bir deniz fenerini bir şeye dönüştürdüğünü hayal edin. büyük dairenin merkezinden geçen bir çizgi boyunca iki yeni fener, her adımda gözlemciye görünen parlaklık aynı pi kare bölü 4 olarak kalır ve her adımda fenerler, her biri arasında 2 mesafe olacak şekilde eşit aralıklı kalır. Çevre ve Limitte, burada her iki yönde de eşit aralıklarla yerleştirilmiş sonsuz sayıda deniz fenerinin bulunduğu düz bir yatay çizgi elde ediyoruz. Görünen parlaklık tüm yol boyunca pi kare bölü 4 olduğundan, bu da bu sınırlama durumunda da doğru olacaktır. Bu bize oldukça harika bir sonsuz seri verir, ters karelerin toplamı 1 bölü n kare Burada n, 1 3 5 tek tam sayıların tümünü kapsar ve bu şekilde devam eder, aynı zamanda negatif 1 negatif 3 negatif 5 kapalı sol yönde Bunların hepsini toplarsak yukarıya doğru bize pi kare bölü 4'ü verecek. Bu harika ve size göstermek istediğim şeyin özü bu. Sadece bir adım geri atın ve bunun ne kadar gerçek dışı göründüğünü düşünün. İlk bakışta geometriyle hiçbir ilgisi olmayan basit kesirlerin toplamı Görünüşe göre dairelerle hiçbir ilgisi yok Bize pi ile ilgili bu sonucu veriyor Ancak şimdi aslında bunun geometriyle ne ilgisi olduğunu görebiliyorsunuz, sayı doğrusu bir nevi sürekli büyüyen dairelerin limiti gibidir ve bu sayıyı topladığınızda Her iki tarafın da sonsuza kadar toplamını sağlayan çizgi Bu sanki sonsuz büyüklükte bir dairenin sınırı boyunca toplama yapıyormuşsunuz gibi ve çok gevşek Ama çok eğlenceli bir konuşma tarzı Ama durun durun bunun toplam olmadığını söyleyebilirsiniz videonun başında bize söz vermiştin. Ve haklısın.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible.", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "Biraz düşünmemiz kaldı İlk olarak, toplamı yalnızca pi kare bölü 8'e getiren pozitif tek sayılar olarak sınırlayalım. Şimdi bununla aradığımız toplam arasındaki tek fark şu şekildedir: tüm pozitif tam sayılar tek ve çifttir Burada kırmızıyla renklendirdiğim çift sayıların tersinin toplamı eksiktir Şimdi bu eksik seriyi istediğimiz toplam serinin ölçeklendirilmiş bir kopyası olarak düşünebilirsiniz. başlangıç noktasından iki kat daha uzağa doğru hareket eder bir ikiye kaydırılır iki dörde kaydırılır üç altıya kaydırılır ve bu böyle devam eder ve bu her deniz feneri için mesafenin iki katına çıkarılmasını gerektirdiğinden görünür parlaklığın bir faktör kadar azalacağı anlamına gelir Bu aynı zamanda tüm tam sayıların toplamından çift tam sayıların toplamına giden nispeten basit bir cebirdir. 1 4 ile çarpmayı içerir ve bu, tüm tam sayılardan tek olanlara gitmenin 3 4 ile çarpılacağı anlamına gelir. çift sayılar artı olasılıklar bize her şeyi vermek zorunda. Yani eğer bunu tersine çevirirsek, bu, tek sayıların toplamından tüm pozitif tam sayıların toplamına gitmek için 4/3 ile çarpmayı gerektirir. Yani pi kare bölü 8 ile çarpmayı gerektirir. 4/3 badda boom badda bing Fesleğen sorununa kendimiz için bir çözüm bulduk Şimdi az önce izlediğiniz bu video öncelikle mavi bir kahverengi ekibin yeni üç üyesinden biri olan Ben Hambricht tarafından yazılmış ve canlandırılmıştır ve bir ekleme mümkün olmuştur.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/ukrainian/sentence_translations.json b/2018/basel-problem/ukrainian/sentence_translations.json index 01e267552..afd8ef05d 100644 --- a/2018/basel-problem/ukrainian/sentence_translations.json +++ b/2018/basel-problem/ukrainian/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right?", "translatedText": "Я маю на увазі, що промені світла, що потрапляють на одну з цих двох катетів, точно такі ж, як і промені, які потрапляють на гіпотенузу. Тоді ключовим є те, що кількість світла від першого маяка, яке воно потрапляє на цю ліву сторону, обмежений кут променів, які в кінцевому підсумку потрапляють цей екран точно такий самий, як і кількість світла, що йде від маяка, яке потрапляє на той бік, це буде той самий кут променів. Симетрично кількість світла від першого будинку, що потрапляє на нижню частину нашого екрана, є однаковою як кількість світла, що потрапляє на цю частину від маяка B Чому ви можете запитати добре, це питання подібних трикутників. Ця анімація вже дає вам чітку підказку про те, як це працює. Ми також залишили посилання в описі на простий GeoGebra аплет для тих із вас, хто хоче подумати про це в дещо більш інтерактивному середовищі та граючи з одним важливим фактом, який ви зможете побачити, а саме те, що подібні трикутники застосовуються лише в граничному випадку для дуже маленького екрана Гаразд, пристебніться, тому що тут все налагодиться. У нас є обернена теорема Піфагора, чи не так?", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right.", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right.", "translatedText": "Ну, лінії від цих маяків до центру розташовані під кутом 90 градусів одна до одної. Отже, оскільки речі симетричні зліва направо, це означає, що відстані вздовж кола дорівнюють 1 2 2 2 і 1 Добре, ви можете побачити, куди це йде, але я хочу пройти через це лише один крок. Ви малюєте коло вдвічі більше, отже окружність 16 тепер і для кожного маяка Ви проводите лінію від цього маяка через вершину меншого кола, яке є центром більшого кола. а потім створіть два нових маяки, де ця лінія перетинатиметься з великим колом. Так само, як і раніше, оскільки довга лінія є діаметром великого кола, ці два нові маяки утворюють прямий кут із спостерігачем праворуч і, як і раніше, лінія від спостерігача до оригінальний маяк перпендикулярний до довгої лінії, і це два факти, які виправдовують нас у використанні зворотної теореми Піфагора. Але те, що може бути не таким ясним, так це те, що коли ви робите це для всіх маяків, щоб отримати вісім нових на великому озеро, ці вісім нових маяків будуть розташовані рівномірно. Це остання частина перевірки геометрії перед остаточним поштовхом. Щоб побачити це, пам’ятайте, що якщо ви проводите лінії від двох суміжних маяків на маленькому озері до центру, вони утворюють кут 90 градусів, якщо натомість ви проводите лінії до будь-якої точки на окружності кола, яка не знаходиться між ними. Дуже корисна теорема про вписаний кут із геометрії говорить нам, що це буде рівно половина кута, який вони утворюють із центром у цьому випадку 45 градусів. Але коли ми розташовуємо цю нову точку на вершині озера. Це дві лінії, які визначають положення нових маяків на великому озері. Це означає, що коли ви проводите лінії від цих восьми нових маяків до центру, вони ділять коло рівномірно на частини під кутом 45 градусів, а це означає, що вісім маяків рівномірно розподілені по колу з відстанню в два між кожними з них, а тепер уявіть, що ця річ продовжується на кожному кроці, подвоюючи розмір кожного кола та перетворюючи кожен маяк на два нових вздовж лінії, проведеної через центр більшого кола, на кожному кроці видима яскравість для спостерігача залишається однаковою пі в квадраті на 4, і на кожному кроці маяки залишаються рівномірними з відстанню 2 між кожним з них на окружність і в межі, що ми отримуємо тут, це плоска горизонтальна лінія з нескінченною кількістю маяків, рівномірно розташованих в обох напрямках і Оскільки видима яскравість була квадратом пі на 4 на всьому шляху, це також буде вірно в цьому граничному випадку І Це дає нам досить приголомшливий нескінченний ряд суми обернених квадратів 1 на n у квадраті, де n охоплює всі непарні цілі числа 1 3 5 і так далі, але також мінус 1 мінус 3 мінус 5 у напрямку ліворуч Додаючи всі ці числа дасть нам пі в квадраті на 4. Це дивовижно, і це суть того, що я хочу вам показати. Просто зробіть крок назад і подумайте, наскільки це нереально. Сума простих дробів, які на перший погляд не мають нічого спільного з геометрією Мабуть, не має нічого спільного з колами. Дає нам цей результат, який пов’язаний з пі. За винятком того, що тепер ви можете побачити, що це має спільного з геометрією, числова лінія схожа на межу кіл, що постійно зростають, і коли ви підсумовуєте це число лінія обов’язково підсумовує весь шлях до нескінченності з будь-якої сторони. Це схоже на те, що ви додаєте вздовж межі нескінченно великого кола, і дуже вільний, але дуже веселий спосіб говорити. Але зачекайте, ви можете сказати, що це не сума що ви обіцяли нам на початку відео, і ви маєте рацію.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible.", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "Нам залишилося трохи подумати. Перш за все, давайте обмежимо суму лише додатними непарними числами, що дає нам пі в квадраті, поділене на 8. Тепер єдина різниця між цією сумою та сумою, яку ми шукаємо, це усі додатні цілі непарні та парні є. Відсутня сума зворотних величин парних чисел, які я тут зафарбую червоним. Тепер ви можете розглядати цей відсутній ряд як масштабовану копію загального ряду, який ми хочемо. Де кожен маяк переміщується вдвічі далі від початку координат один зміщується на два два зсуваються на чотири три зсуваються на шість і так далі, і оскільки це передбачає подвоєння відстані для кожного маяка, це означає, що видима яскравість зменшиться на коефіцієнт з чотирьох, і це також відносно проста алгебра, яка переходить від суми всіх цілих чисел до суми парних цілих. Включає множення на 1 4, а це означає, що перехід від усіх цілих до непарних буде множенням на 3 4, оскільки парні плюс шанси повинні дати нам все. Отже, якщо ми просто перевернемо це навколо, це означає, що перехід від суми непарних чисел до суми всіх додатних цілих потребує множення на 4 третини. Отже, беручи це число пі в квадраті на 8, помноживши на 4 терці badda boom badda bing У нас є рішення проблеми з базиліком Тепер це відео, яке ви щойно переглянули, було в основному написано та анімовано одним із трьох нових членів команди Blue One Brown Беном Хамбріхтом, що стало можливим.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/urdu/sentence_translations.json b/2018/basel-problem/urdu/sentence_translations.json index 89bf67dcb..cf4d62320 100644 --- a/2018/basel-problem/urdu/sentence_translations.json +++ b/2018/basel-problem/urdu/sentence_translations.json @@ -72,7 +72,7 @@ "end": 449.07 }, { - "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse it hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that one important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen All Right buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right? ", + "input": "I mean the rays of light hitting one of those two legs are precisely the same as the rays that hit the hypotenuse Then the key is that the amount of light from the first lighthouse that hits this left side the limited angle of rays that end up hitting that screen is Exactly the same as the amount of light over here coming from lighthouse a which hits that side it'll be the same angle of rays and Symmetrically the amount of light from the first house hitting the bottom portion of our screen is The same as the amount of light hitting that portion from lighthouse B Why you might ask well, it's a matter of similar triangles This animation already gives you a strong hint for how it works And we've also left a link in the description to a simple GeoGebra applet for those of you who want to think this through in a slightly more interactive environment and in playing with that One important fact here that you'll be able to see is that the similar triangles only apply in the limiting case for a very tiny screen The inverse Pythagorean theorem Alright buckle up now because here's where things get good We've got this inverse Pythagorean theorem, right? ", "translatedText": "سکرین اس طرح چھوٹے مثلث کی دو ٹانگوں کا مجموعہ ہے ٹھیک ہے، یہ اب بھی روشنی کی ایک ہی مقدار حاصل کرتا ہے، ٹھیک ہے؟ میرا مطلب ہے کہ ان دونوں ٹانگوں میں سے کسی ایک ٹانگ سے ٹکرانے والی روشنی کی شعاعیں بالکل وہی ہوتی ہیں جو شعاعوں سے ٹکراتی ہیں پھر اہم بات یہ ہے کہ پہلے لائٹ ہاؤس سے روشنی کی جتنی مقدار یہ بائیں جانب ٹکراتی ہے وہ شعاعوں کے محدود زاویے سے ٹکراتی ہے۔وہ اسکرین بالکل یکساں ہے جتنی روشنی یہاں لائٹ ہاؤس سے آتی ہے جو اس طرف سے ٹکراتی ہے یہ شعاعوں کا ایک ہی زاویہ ہوگا اور ہم آہنگی کے لحاظ سے ہماری اسکرین کے نیچے والے حصے سے ٹکرانے والے پہلے گھر سے روشنی کی مقدار ایک جیسی ہے۔لائٹ ہاؤس B سے اس حصے کو مارنے والی روشنی کی مقدار کیوں کہ آپ اچھی طرح سے پوچھ سکتے ہیں، یہ اسی طرح کے مثلثوں کا معاملہ ہے یہ اینیمیشن آپ کو پہلے ہی ایک مضبوط اشارہ دیتا ہے کہ یہ کیسے کام کرتا ہے اور ہم نے تفصیل میں ایک سادہ جیو جیبرا کا لنک بھی چھوڑ دیا ہے۔آپ میں سے ان لوگوں کے لیے ایپلٹ جو قدرے زیادہ انٹرایکٹو ماحول میں اور اس کے ساتھ کھیلتے ہوئے یہ سوچنا چاہتے ہیں کہ یہاں ایک اہم حقیقت جو آپ دیکھ سکیں گے وہ یہ ہے کہ ملتے جلتے مثلث صرف ایک بہت چھوٹی اسکرین کے لیے محدود صورت میں لاگو ہوتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 678.51 }, { - "input": "Well the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1 2 2 2 and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on The big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouses remain evenly spaced with the distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all apparently Gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait you might say this is not the sum that you promised us at the start of the video And well you're right. ", + "input": "Well, the lines from those lighthouses to the center are at 90 degree angles with each other So since things are symmetric left to right that means that the distances along the circumference are 1, 2, 2, 2, and 1 Alright, you might see where this is going, but I want to walk through this for just one more step You draw a circle twice as big so circumference of 16 now and for each lighthouse You draw a line from that lighthouse through the top of the smaller circle Which is the center of the bigger circle and then create two new lighthouses where that line intersects with the larger circle Just as before because the long line is a diameter of the big circle those two new lighthouses make a right angle with the observer, right and Just as before the line from the observer to the original lighthouse is Perpendicular to the long line and those are the two facts that justify us in using the inverse Pythagorean theorem But what might not be as clear is that when you do this for all of the lighthouses to get eight new ones on the Big lake those eight new lighthouses are going to be evenly spaced This is the final bit of geometry proofiness before the final thrust To see this remember that if you draw lines from two adjacent lighthouses on the small lake to the center They make a 90 degree angle If instead you draw lines to a point anywhere on the circumference of the circle that's not between them the very useful inscribed angle theorem from geometry tells us that this will be Exactly half of the angle that they make with the center in this case 45 degrees But when we position that new point at the top of the lake These are the two lines which define the position of the new lighthouses on the larger lake What that means then is that when you draw lines from those eight new lighthouses into the center They divide the circle evenly into 45 degree angle pieces and that means the eight lighthouses are evenly spaced around the circumference with the distance of two between each one of them and Now just imagine this thing playing on at every step doubling the size of each circle and Transforming each lighthouse into two new ones along a line drawn through the center of the larger circle at every step the apparent brightness to the observer remains the same pi squared over 4 and at every step the lighthouse has remained evenly spaced with a distance 2 between each one of them on the circumference and In the limit what we're getting here is a flat horizontal line with an infinite number of lighthouses evenly spaced in both directions and Because the apparent brightness was pi squared over 4 the entire way that will also be true in this limiting case And This gives us a pretty awesome infinite series the sum of the inverse squares 1 over n squared Where n covers all of the odd integers 1 3 5 and so on but also negative 1 negative 3 negative 5 off in the leftward direction Adding all of those up is going to give us pi squared over 4 That's amazing and it's the core of what I want to show you and Just take a step back and think about how unreal this seems The sum of simple fractions that at first sight have nothing to do with geometry nothing to do with circles at all Apparently gives us this result that's related to pi Except now you can actually see what it has to do with geometry the number line is kind of like a limit of ever-growing circles and As you sum across that number line making sure to sum all the way to infinity on either side It's sort of like you're adding up along the boundary of an infinitely large circle and a very loose But very fun way of speaking But wait, you might say this is not the sum that you promised us at the start of the video And well, you're right. ", "translatedText": "چھوٹے دائرے کے اوپر سے گزرنا اور بڑے دائرے پر دو نئے لائٹ ہاؤسز حاصل کرنا اور اس سے بھی اچھے یہ چار لائٹ ہاؤسز جھیل کے گرد یکساں فاصلے پر کیوں ہیں؟ ٹھیک ہے ان لائٹ ہاؤسز سے مرکز تک لائنیں ایک دوسرے کے ساتھ 90 ڈگری زاویہ پر ہیں لہذا چونکہ چیزیں بائیں سے دائیں سمت میں ہیں اس کا مطلب ہے کہ فریم کے ساتھ فاصلے 1 2 2 2 اور 1 ٹھیک ہیں، آپ دیکھ سکتے ہیں کہ یہ کہاں جا رہا ہے، لیکن میں صرف ایک قدم کے لیے اس سے گزرنا چاہتا ہوں، آپ ایک دائرہ دو گنا بڑا بناتے ہیں اس لیے اب فریم 16 ہے اور ہر لائٹ ہاؤس کے لیے آپ اس لائٹ ہاؤس سے چھوٹے دائرے کے اوپر سے ایک لکیر کھینچتے ہیں جو بڑے دائرے کا مرکز ہے۔اور پھر دو نئے لائٹ ہاؤسز بنائیں جہاں وہ لکیر بڑے دائرے کے ساتھ ملتی ہے بالکل پہلے کی طرح کیونکہ لمبی لائن بڑے دائرے کا قطر ہے وہ دو نئے لائٹ ہاؤسز مبصر کے دائیں طرف ایک صحیح زاویہ بناتے ہیں اور بالکل اسی طرح جیسے مبصر سے لائن سے پہلے اصل لائٹ ہاؤس لمبی لکیر کے لیے کھڑا ہے اور یہ وہ دو حقائق ہیں جو ہمیں الٹا پائتھاگورین تھیوریم استعمال کرنے کا جواز فراہم کرتے ہیں لیکن جو بات اتنی واضح نہیں ہو سکتی ہے وہ یہ ہے کہ جب آپ تمام لائٹ ہاؤسز کے لیے یہ کرتے ہیں تو بڑے پر آٹھ نئے جھیل کے وہ آٹھ نئے لائٹ ہاؤسز یکساں فاصلہ پر ہونے والے ہیں حتمی زور سے پہلے یہ جیومیٹری پروفنیس کا آخری حصہ ہے اسے دیکھنے کے لیے یاد رکھیں کہ اگر آپ چھوٹی جھیل پر دو ملحقہ لائٹ ہاؤسز سے مرکز کی طرف لکیریں کھینچتے ہیں تو وہ 90 ڈگری کا زاویہ بناتے ہیں۔اس کے بجائے آپ دائرے کے طواف پر کہیں بھی ایک نقطہ کی طرف لکیریں کھینچتے ہیں جو ان کے درمیان نہیں ہے جیومیٹری سے بہت مفید لکھا ہوا زاویہ نظریہ ہمیں بتاتا ہے کہ یہ اس معاملے میں مرکز کے ساتھ جو زاویہ بناتے ہیں اس کا بالکل نصف ہوگا لیکن جب ہم جھیل کے اوپری حصے میں اس نئے پوائنٹ کو پوزیشن دیتے ہیں یہ وہ دو لکیریں ہیں جو بڑی جھیل پر نئے لائٹ ہاؤسز کی پوزیشن کا تعین کرتی ہیں اس کا مطلب یہ ہے کہ جب آپ ان آٹھ نئے لائٹ ہاؤسز سے لائنیں کھینچتے ہیں تو وہ دائرے کو تقسیم کر دیتے ہیں۔یکساں طور پر 45 ڈگری زاویہ کے ٹکڑوں میں اور اس کا مطلب ہے کہ آٹھ لائٹ ہاؤسز فریم کے گرد یکساں طور پر فاصلہ رکھتے ہیں اور ان میں سے ہر ایک کے درمیان دو کا فاصلہ ہے اور اب ذرا تصور کریں کہ اس چیز کو ہر قدم پر چل رہا ہے ہر دائرے کے سائز کو دوگنا کرتا ہے اور ہر لائٹ ہاؤس میں تبدیل ہوتا ہے۔ہر قدم پر بڑے دائرے کے بیچ میں کھینچی گئی لکیر کے ساتھ دو نئے، مبصر کے لیے ظاہری چمک ایک ہی pi مربع پر رہتی ہے اور ہر قدم پر لائٹ ہاؤسز یکساں طور پر فاصلہ رکھتے ہیں اور ان میں سے ہر ایک کے درمیان فاصلہ 2 ہوتا ہے۔فریم اور اس حد میں جو ہم یہاں حاصل کر رہے ہیں وہ ایک فلیٹ افقی لکیر ہے جس میں لائٹ ہاؤسز کی لامحدود تعداد دونوں سمتوں میں یکساں طور پر فاصلہ رکھتی ہے اور کیونکہ ظاہری چمک پوری طرح سے 4 پر مربع تھی جو اس محدود معاملے میں بھی درست ہو گی اور یہ ہمیں ایک بہت ہی زبردست لامحدود سیریز فراہم کرتا ہے الٹا مربعوں کا مجموعہ 1 سے زیادہ n مربع جہاں n تمام طاق عدد 1 3 5 اور اسی طرح کا احاطہ کرتا ہے لیکن بائیں طرف کی سمت میں منفی 1 منفی 3 منفی 5 آف بھی شامل کرتا ہے۔ہمیں pi مربع 4 پر دینے جا رہا ہے یہ حیرت انگیز ہے اور یہ اس کا بنیادی حصہ ہے جو میں آپ کو دکھانا چاہتا ہوں اور ذرا ایک قدم پیچھے ہٹیں اور سوچیں کہ یہ کتنا غیر حقیقی لگتا ہے سادہ فریکشنز کا مجموعہ جن کا پہلی نظر میں جیومیٹری سے کوئی تعلق نہیں ہے۔حلقوں سے کوئی تعلق نہیں بظاہر ہمیں یہ نتیجہ دیتا ہے جو pi سے متعلق ہے سوائے اس کے کہ اب آپ حقیقت میں دیکھ سکتے ہیں کہ اس کا جیومیٹری سے کیا تعلق ہے نمبر لائن ایک طرح سے بڑھتے ہوئے دائروں کی حد کی طرح ہے اور جیسا کہ آپ اس نمبر کو پورا کرتے ہیں لکیر اس بات کو یقینی بنا رہی ہے کہ کسی بھی طرف لامحدودیت کے تمام راستے جمع کریں یہ اس طرح ہے جیسے آپ ایک لامحدود بڑے دائرے کی حد کے ساتھ جوڑ رہے ہیں اور ایک بہت ہی ڈھیلا لیکن بات کرنے کا بہت ہی مزے کا طریقہ لیکن انتظار کریں آپ کہہ سکتے ہیں کہ یہ رقم نہیں ہے۔کہ آپ نے ویڈیو کے آغاز میں ہم سے وعدہ کیا تھا اور آپ ٹھیک کہہ رہے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 918.27 }, { - "input": "We do have a little bit of thinking left First things first let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and because that involves doubling the distance for every lighthouse it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. ", + "input": "We do have a little bit of thinking left First things first, let's just restrict the sum to only being the positive odd numbers which gets us pi squared divided by 8 Now the only difference between this and the sum that we're looking for that goes over all the positive integers odd and even is That it's missing the sum of the reciprocals of even numbers what I'm coloring in red up here Now you can think of that missing series as a scaled copy of the total series that we want Where each lighthouse moves to being twice as far away from the origin one gets shifted to two two gets shifted to four three gets shifted to six and so on and Because that involves doubling the distance for every lighthouse. it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You ", "translatedText": "ہمارے پاس تھوڑا سا سوچنا باقی ہے پہلی چیزیں پہلے آئیے صرف جمع کو صرف مثبت طاق نمبروں تک محدود رکھیں جس سے ہمیں pi مربع کو 8 سے تقسیم کیا جاتا ہے اب اس اور اس رقم کے درمیان فرق صرف اتنا ہے کہ ہم اسے تلاش کر رہے ہیں تمام مثبت انٹیجرز عجیب اور برابر ہیں کہ اس میں جفت ہندسوں کے متواتر کا مجموعہ غائب ہے جسے میں یہاں سرخ رنگ میں رنگ رہا ہوں اب آپ اس گمشدہ سیریز کے بارے میں سوچ سکتے ہیں کہ ہم اس کل سیریز کی ایک سکیلڈ کاپی ہیں جو ہم چاہتے ہیں کہ ہر لائٹ ہاؤس اصل سے دوگنا دور ہونے کی طرف بڑھتا ہے ایک دو میں شفٹ ہو جاتا ہے دو چار میں منتقل ہو جاتا ہے تین چھ پر منتقل ہو جاتا ہے اور اسی طرح اور کیونکہ اس میں ہر لائٹ ہاؤس کے لیے فاصلے کو دوگنا کرنا شامل ہے اس کا مطلب ہے کہ ظاہری چمک ایک عنصر سے کم ہو جائے گی۔چار کا اور یہ نسبتاً سیدھا الجبرا بھی ہے جس میں تمام عدد کے جمع سے جوڑ کر مجموعے پر 1 4ویں سے ضرب کرنا شامل ہے اور اس کا کیا مطلب ہے کہ تمام عدد سے طاق تک جانے سے 3 4ویں سے ضرب ہو جائے گی۔مساوی جمع مشکلات کو ہمیں پوری چیز دینا ہوتی ہے لہذا اگر ہم صرف اس کے ارد گرد پلٹائیں تو اس کا مطلب ہے کہ طاق ہندسوں کی جمع سے تمام مثبت عددوں پر جمع ہونے کے لیے 4 تہائی سے ضرب کی ضرورت ہوتی ہے تو اس pi مربع کو 8 سے ضرب کرنے سے 4 Thirds badda boom badda bing ہم نے خود کو تلسی کے مسئلے کا حل تلاش کر لیا ہے اب یہ ویڈیو جو آپ نے ابھی دیکھی ہے بنیادی طور پر تین بلیو ون براؤن ٹیم کے ممبر بین ہیمبرچٹ میں سے ایک نے لکھا اور اینیمیٹ کیا تھا جس سے ایک اضافہ ممکن ہوا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/basel-problem/vietnamese/sentence_translations.json b/2018/basel-problem/vietnamese/sentence_translations.json index 64aedbedf..2a61b2e66 100644 --- a/2018/basel-problem/vietnamese/sentence_translations.json +++ b/2018/basel-problem/vietnamese/sentence_translations.json @@ -70,7 +70,7 @@ "end": 213.26 }, { - "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or radio signal things like that and Infinite array of evenly spaced lighthouses physically implements the basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think Of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a And the distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over b squared equals 1 over h squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", + "input": "In spherical geometry you sometimes talk about the solid angle of a shape Which is the proportion of a sphere it covers as viewed from a given point You see the first of two places this story we're thinking of screens is going to be useful is in understanding the inverse square law Which is a distinctly three-dimensional phenomenon think of all of the rays of light hitting a screen one unit away from the source as You double the distance those rays will now cover an area with twice the width and twice the height So it would take four copies of that original screen to receive the same rays at that distance And so each individual one receives 1 fourth as much light This is the sense in which I mean a light would appear 1 fourth as bright two times the distance away Likewise when you're three times farther away You would need nine copies of that original screen to receive the same rays so each individual screen only receives 1 9th as much light and This pattern continues because the area hit by a light increases by the square of the distance the brightness of that light decreases by the inverse square of that distance and As I'm sure many of you know this inverse square law is not at all special to light It pops up whenever you have some kind of quantity that spreads out evenly from a point source whether that's sound or heat or a radio signal things like that and Remember it's because of this inverse square law that an infinite array of evenly spaced lighthouses physically implements the Basel problem But again what we need if we're going to make any progress here is to understand how we can manipulate setups with light sources like this without changing the total brightness for the observer and The key building block is an especially nice way to transform a single lighthouse into two Think of an observer at the origin of the XY plane and a single lighthouse sitting out somewhere on that plane Now draw a line from that lighthouse to the observer and then another line perpendicular to that one at the lighthouse Now place two lighthouses where this new line intersects the coordinate axes Which I'll go ahead and call lighthouse a over here on the left and lighthouse B on the upper side It turns out and you'll see why this is true in just a minute the brightness that the observer Experiences from that first lighthouse is equal to the combined brightness experienced from lighthouses A and B together And I should say by the way that the standing assumption throughout this video is that all lighthouses are equivalent They're using the same light bulb emanating the same power all of that So in other words assigning variables to things here if we call the distance from the observer to lighthouse a little a and The distance from the observer to lighthouse B little B and the distance to the first lighthouse H We have the relation 1 over a squared plus 1 over B squared equals 1 over H squared This is the much less well-known Inverse Pythagorean theorem which some of you may recognize from math ologer's most recent and I'll say most excellent video on the many cousins of the Pythagorean theorem Pretty cool relation don't you think and if you're a mathematician at heart you might be asking right now how you prove it and There are some straightforward ways where you express the triangles area in two separate ways and apply the usual Pythagorean theorem But there is another quite pretty method that I'd like to briefly outline here that falls much more nicely into our storyline because again It uses intuitions of light and screens Imagine scaling down the whole right triangle into a tinier version and think of this miniature Hypotenuse as a screen receiving light from the first lighthouse If you reshape that screen to be the combination of the two legs of the miniature triangle like this Well, it still receives the same amount of light, right?", "translatedText": "Trong hình học hình cầu, đôi khi bạn nói về góc đặc của một hình Đó là tỷ lệ của hình cầu mà nó bao phủ khi nhìn từ một điểm nhất định Bạn thấy vị trí đầu tiên trong hai vị trí mà câu chuyện mà chúng ta đang nghĩ về màn hình sẽ hữu ích này hiểu định luật nghịch đảo bình phương Đó là một hiện tượng ba chiều rõ rệt, hãy nghĩ đến tất cả các tia sáng chạm vào màn hình cách nguồn một đơn vị khi Bạn tăng gấp đôi khoảng cách những tia đó bây giờ sẽ bao phủ một khu vực có chiều rộng gấp đôi và chiều cao gấp đôi Vì vậy, sẽ cần bốn bản sao của màn hình gốc đó để nhận được các tia giống nhau ở khoảng cách đó Và do đó, mỗi cá nhân sẽ nhận được lượng ánh sáng bằng 1 phần tư Đây là ý nghĩa mà tôi muốn nói rằng ánh sáng sẽ xuất hiện 1 phần tư sáng gấp đôi khoảng cách Tương tự như vậy, khi bạn ở xa hơn ba lần Bạn sẽ cần chín bản sao của màn hình gốc đó để nhận được các tia giống nhau nên mỗi màn hình riêng lẻ chỉ nhận được lượng ánh sáng bằng 1 phần 9 và Mô hình này tiếp tục vì diện tích bị ánh sáng chiếu vào tăng theo bình phương của khoảng cách độ sáng của ánh sáng đó giảm theo nghịch đảo bình phương của khoảng cách đó và tôi chắc rằng nhiều bạn biết định luật nghịch đảo bình phương này không hề đặc biệt đối với ánh sáng. Nó xuất hiện bất cứ khi nào bạn có một loại đại lượng nào đó trải đều từ một nguồn điểm cho dù đó là âm thanh hay nhiệt hay tín hiệu vô tuyến, những thứ tương tự và vô số ngọn hải đăng cách đều nhau thực hiện bài toán basel Nhưng một lần nữa, điều chúng ta cần nếu muốn đạt được bất kỳ tiến bộ nào ở đây là hiểu cách chúng ta có thể thao tác các thiết lập với các nguồn sáng như thế này mà không làm thay đổi tổng độ sáng cho người quan sát và Khối xây dựng chính là một cách đặc biệt hay để biến một ngọn hải đăng thành hai Hãy nghĩ về một người quan sát ở điểm gốc của mặt phẳng XY và một ngọn hải đăng duy nhất nằm ở đâu đó trên đó mặt phẳng Bây giờ vẽ một đường thẳng từ ngọn hải đăng đó đến người quan sát và sau đó một đường thẳng khác vuông góc với đường đó ở ngọn hải đăng Bây giờ đặt hai ngọn hải đăng nơi đường thẳng mới này giao với các trục tọa độ. Tôi sẽ tiếp tục gọi ngọn hải đăng là ở đây bên trái và ngọn hải đăng B ở phía trên Hóa ra và bạn sẽ hiểu tại sao điều này đúng chỉ sau một phút nữa độ sáng mà người quan sát Trải nghiệm từ ngọn hải đăng đầu tiên đó bằng với độ sáng tổng hợp trải qua từ ngọn hải đăng A và B cùng nhau Và tôi nên nói rằng giả định thường trực xuyên suốt video này là tất cả các ngọn hải đăng đều tương đương nhau. Họ sử dụng cùng một bóng đèn phát ra cùng một công suất. Nói cách khác, việc gán các biến cho mọi thứ ở đây nếu chúng ta gọi khoảng cách từ người quan sát đến ngọn hải đăng là a nhỏ a Và khoảng cách từ người quan sát đến ngọn hải đăng B nhỏ B và khoảng cách đến ngọn hải đăng đầu tiên H Chúng ta có hệ thức 1 trên a bình cộng 1 trên b bình phương bằng 1 trên h bình Đây là định lý Pythagore nghịch đảo ít được biết đến hơn nhiều mà một số bạn có thể nhận ra từ video gần đây nhất của nhà toán học và tôi sẽ nói là video xuất sắc nhất về nhiều họ hàng của định lý Pythagore. Bạn nghĩ mối quan hệ khá thú vị phải không và nếu thực tâm bạn là một nhà toán học thì bạn có thể đang hỏi ngay bây giờ bạn chứng minh điều đó như thế nào và Có một số cách đơn giản để bạn thể hiện diện tích hình tam giác theo hai cách riêng biệt và áp dụng định lý Pythagore thông thường Nhưng có một phương pháp khá hay khác mà tôi muốn phác thảo ngắn gọn ở đây phù hợp hơn nhiều với cốt truyện của chúng ta bởi vì một lần nữa Nó sử dụng trực giác của ánh sáng và màn hình Hãy tưởng tượng thu nhỏ toàn bộ tam giác vuông thành một phiên bản nhỏ hơn và coi Cạnh huyền thu nhỏ này như một màn hình nhận ánh sáng từ ngọn hải đăng đầu tiên Nếu bạn định hình lại màn hình đó thành sự kết hợp của hai chân của ngọn hải đăng đầu tiên. hình tam giác thu nhỏ như thế này Chà, nó vẫn nhận được lượng ánh sáng như nhau phải không?", "model": "google_nmt", "from_community_srt": "Trong hình học, đôi khi ta nói tới \"góc khối\" của một hình Mà nó là một phần bao trùm của một hình cầu được ta quan sát. (*tra wiki để dễ hiểu hơn) Điều thứ nhất trong hai điều mà chúng ta đáng nghĩ tới là cái màn hình sẽ giúp ta hiểu về định luật \"inverse square\" Đó là một hiện tượng ba chiều đặc biệt. Hãy tưởng tượng tất cả các đường sáng chạm vào màn hình bằng 1 đơn vị tính từ nguồn sáng, khi bạn nhân đôi khoảng cách đó lên thì đường sáng sẽ bao trùm một vùng lớn gấp đôi cả về chiều cao lẫn chiều rộng. tính từ nguồn sáng, khi bạn nhân đôi khoảng cách đó lên thì đường sáng sẽ bao trùm một vùng lớn gấp đôi cả về chiều cao lẫn chiều rộng. Vậy sẽ phải lấy 4 copy của cái màn hình đầu tiên, để nhận được đủ ánh sáng từ khoảng cách đó. Mỗi màn hình copy ấy sẽ nhận được 1/4 lượng ánh sáng. Điều này chính là ý của tôi khi nói độ sáng sẽ bằng 1/4 khi khoảng cách là 2. Tương tự như vậy khi khoảng cách tăng lên 3? Bạn sẽ cần 9 copy của màn hình đầu tiên để nhận đủ ánh sáng, và mỗi màn hình sẽ chỉ nhận 1/9 ánh sáng Mô hình này cứ tiếp tục vì diện tích nhận ánh sáng tăng lên bằng đúng bình phương khoảng cách. Lượng sáng nhận được thì bằng nghịch đảo bình phương khoảng cách. Tôi chắc chắn nhiều bạn biết, định lý inverse square này không chỉ riêng cho ánh sáng mà nó còn hiện hữu ở bất cứ trường hợp nào có sự lan tỏa đều đặn từ một điểm như là âm thanh, nhiệt độ, tín hiệu radio... Và hãy nhớ vì luật inverse square này mà hàng vô tận cột hải đăng cách đều nhau chính là một kiểu đại lượng vật lý của vấn đề Basel. Nhắc lại là việc chúng ta đang cố làm ở đây là hiểu là làm thế nào để biến đổi nguồn sáng thành như thế này mà không làm thay đổi tổng độ sáng người quan sát nhận được. Và chìa khóa là biến một cột hải đăng thành hai cột hải đăng. Hãy tưởng tượng người quan sát đứng ở gốc tọa độ mặt phẳng x-y và ngọn hải đăng nằm đâu đó ở mặt phẳng này Bây giờ, vẽ một đường thẳng từ ngọn hải đăng đó đến người quan sát và một đường thẳng khác vuông góc với đường thẳng còn lại Bây giờ đặt hai ngọn hải đăng nơi ở điểm giao nhau của đường thẳng với hai trục tọa độ. Mà tôi sẽ gọi NHĐ ở đây là A và cái ở trên kia là B. Và tý nữa bạn sẽ nhận ra tại sao cái này đúng. Độ sáng mà người quan sát thaays từ ngọn hải đăng đầu tiên bằng với tổng độ sáng từ các ngọn hải đăng A và B và tôi nói trước là chúng ta đang giả dụ trong video này là tất cả các ngọn hải đăng đều giống nhau đều dụng cùng một loại bóng đèn có cùng độ sáng. Và tiếp tục, ta hãy gắn biến cho những thứ ở đây. Khoảng cách từ người quan sát đến cột A gọi là a (nhỏ) và khoảng cách từ người quan sát đến cột B là b (nhỏ) và khoảng cách đến ngọn hải đăng đầu tiên là h Chúng ta có sự liên hệ 1/a^2 + 1/b^2 = 1/h^2. Cái này được biết tới là Định lý Inverse Pythagore, mà một số bạn có thể đã nhận ra từ video của Mathologer về định lý pythagorean. Mối liên hệ khá là hay phải không? Và nếu bạn có tình yêu toán học, bạn sẽ thắc mắc là chứng minh nó thế nào? Có một vào phương pháp trực tiếp như là chia tam giác thành 2 phần và áp dụng định lý pythagore thông thường. Nhưng có một cách khá hay mà tôi muốn trình bày ngắn gọn ở đây, nó khá liên quan với câu hỏi của chúng ta. Vì nó lại sử dụng trực giác về ánh sáng và màn hình. Hãy tưởng tượng thu nhỏ toàn bộ tam giác thành một phiên bản nhỏ hơn và nghĩ về cạnh huyền nhỏ này như một màn hình nhận ánh sáng từ ngọn hải đăng đầu tiên và nếu bạn biến hình cái màn hình đó thành hai cạnh còn lại của tam giác, như này. Chà, nó vẫn nhận được cùng một lượng ánh sáng,", @@ -133,7 +133,7 @@ "end": 959.91 }, { - "input": "It means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th And what that means is that going from all the integers to the odd ones would be multiplying by 3 4ths Since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds bada boom bada bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht An addition made possible.", + "input": "it means that the apparent brightness would be decreased by a factor of four and That's also relatively straightforward algebra going from the sum over all the integers to the sum over the even integers Involves multiplying by 1 4th and what that means is that going from all the integers to the odd ones Would be multiplying by 3 4ths since the evens plus the odds have to give us the whole thing So if we just flip that around that means going from the sum over the odd numbers to the sum over all positive integers requires multiplying by 4 thirds So taking that pi squared over 8 multiplying by 4 thirds badda boom badda bing We've got ourselves a solution to the basil problem Now this video that you just watched was primarily written and animated by one of the new three blue one brown team members Ben Hambricht an addition made possible. Thanks to you guys through patreon You", "translatedText": "Điều đó có nghĩa là độ sáng biểu kiến sẽ giảm đi bốn lần và Đó cũng là đại số tương đối đơn giản đi từ tổng trên tất cả các số nguyên đến tổng trên các số nguyên chẵn Bao gồm việc nhân với 1 4 Và điều đó có nghĩa là đi từ tất cả các số nguyên số nguyên đến số lẻ sẽ được nhân với 3 phần 4 Vì số chẵn cộng với số lẻ sẽ cho chúng ta toàn bộ kết quả Vì vậy, nếu chúng ta chỉ lật ngược lại điều đó có nghĩa là đi từ tổng trên các số lẻ đến tổng trên tất cả các số nguyên dương cần phải nhân với 4 phần ba Vì vậy, lấy số pi bình phương trên 8 nhân với 4 phần ba bada boom bada bing Chúng ta đã có cho mình một giải pháp cho vấn đề húng quế Bây giờ, video mà bạn vừa xem này chủ yếu được viết và hoạt hình bởi một trong ba màu xanh một màu nâu mới các thành viên trong nhóm Ben Hambricht Một sự bổ sung có thể thực hiện được.", "model": "google_nmt", "from_community_srt": "Vì khoảng cách bị nhân đôi cho mỗi NHĐ nên độ sáng sẽ giảm đi 4 lần. Về mặt đại số thì nó khá rõ ràng... (*chuỗi chẵn này lớn gấp 2 chuỗi số nguyên kia nhưng độ sáng bị giảm đi 4 lần) Từ tổng các số nguyên biến thành tổng các số nguyên chẵn sẽ phải nhân nó với 1/4. Và đều nãy nghĩa ra gì? Nó có nghĩa là nếu từ tổng các số nguyên biến thành tổng các số nguyên lẻ thì sẽ phải nhân với 3/4 vì chẵn+lẻ=tổng số nguyên. Vậy nếu ta đảo ngược lại có nghĩa là từ tổng các số lẻ biến thành tổng các số nguyên dương sẽ phải nhân với 4/3 Vậy, lấy số (pi^2)/8 nhân 4/3.. boom bada bing ta có đáp số cho câu hỏi basel. sub by MinHíu...", diff --git a/2018/borsuk-ulam/arabic/sentence_translations.json b/2018/borsuk-ulam/arabic/sentence_translations.json index becb87035..2c911a3aa 100644 --- a/2018/borsuk-ulam/arabic/sentence_translations.json +++ b/2018/borsuk-ulam/arabic/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure. ", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure. ", "translatedText": "وذلك لأن ربط كل نقطة على سطح الأرض بزوج من الأرقام، هو نفس رسم خريطة سطح الأرض على مستوى إحداثي ثنائي الأبعاد، حيث يمثل الإحداثي الأول درجة الحرارة والثاني يمثل الضغط. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/bengali/sentence_translations.json b/2018/borsuk-ulam/bengali/sentence_translations.json index a5b381379..cc00b1d5e 100644 --- a/2018/borsuk-ulam/bengali/sentence_translations.json +++ b/2018/borsuk-ulam/bengali/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure. ", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure. ", "translatedText": "এর কারণ হল পৃথিবীর পৃষ্ঠের প্রতিটি বিন্দুকে এক জোড়া সংখ্যার সাথে সংযুক্ত করা, পৃথিবীর পৃষ্ঠকে একটি 2D স্থানাঙ্ক সমতলে ম্যাপ করার মতো একই জিনিস, যেখানে প্রথম স্থানাঙ্ক তাপমাত্রার প্রতিনিধিত্ব করে এবং দ্বিতীয়টি চাপের প্রতিনিধিত্ব করে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/chinese/sentence_translations.json b/2018/borsuk-ulam/chinese/sentence_translations.json index 043d56107..c915dbeb3 100644 --- a/2018/borsuk-ulam/chinese/sentence_translations.json +++ b/2018/borsuk-ulam/chinese/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "这是因为将地球表面上的每个点与一对数字相关联 ，与将地球表面映射到二维坐标平面上是一样的， 其中第一个坐标代表温度，第二个坐标代表压力。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/english/captions.srt b/2018/borsuk-ulam/english/captions.srt index 857746e01..576c8eee9 100644 --- a/2018/borsuk-ulam/english/captions.srt +++ b/2018/borsuk-ulam/english/captions.srt @@ -1123,15 +1123,15 @@ there must be some antipodal collision, an input x1, x2, x3, x4, where flipping all of the signs wouldn't change the output. 282 -00:18:33,940 --> 00:18:37,273 -I'll leave it to you to pause and ponder and think about how this +00:18:33,940 --> 00:18:37,336 +I'll leave it to you to pause and ponder and think about how this could 283 -00:18:37,273 --> 00:18:41,364 -could apply to the And about what the general statement of Borsuk-Ulam might be, +00:18:37,336 --> 00:18:41,534 +apply to the 3-jewel case, and about what the general statement of Borsuk-Ulam might be, 284 -00:18:41,364 --> 00:18:43,940 +00:18:41,534 --> 00:18:43,940 and how it applies to the general necklace problem. 285 diff --git a/2018/borsuk-ulam/english/sentence_timings.json b/2018/borsuk-ulam/english/sentence_timings.json index 0afbdcd55..ec49ec392 100644 --- a/2018/borsuk-ulam/english/sentence_timings.json +++ b/2018/borsuk-ulam/english/sentence_timings.json @@ -665,7 +665,7 @@ 1113.12 ], [ - "I'll leave it to you to pause and ponder and think about how this could apply to the And about what the general statement of Borsuk-Ulam might be, and how it applies to the general necklace problem.", + "I'll leave it to you to pause and ponder and think about how this could apply to the 3-jewel case, and about what the general statement of Borsuk-Ulam might be, and how it applies to the general necklace problem.", 1113.94, 1123.94 ], diff --git a/2018/borsuk-ulam/english/transcript.txt b/2018/borsuk-ulam/english/transcript.txt index 0f5c14bf7..9571adbdf 100644 --- a/2018/borsuk-ulam/english/transcript.txt +++ b/2018/borsuk-ulam/english/transcript.txt @@ -131,6 +131,6 @@ As an example, Borsuk-Ulam applies to mapping hyperspheres in 4D space into thre And what I mean by a hypersphere is the set of all possible lists of four coordinates where the sum of their squares equals one. Those are the points in 4D space a distance one from the origin. Borsuk-Ulam says that if you try to map that set, all those special quadruplets of numbers, into three-dimensional space, continuously associating each one with some triplet of numbers, there must be some antipodal collision, an input x1, x2, x3, x4, where flipping all of the signs wouldn't change the output. -I'll leave it to you to pause and ponder and think about how this could apply to the And about what the general statement of Borsuk-Ulam might be, and how it applies to the general necklace problem. +I'll leave it to you to pause and ponder and think about how this could apply to the 3-jewel case, and about what the general statement of Borsuk-Ulam might be, and how it applies to the general necklace problem. And maybe, just maybe, this gives you an inkling of why mathematicians care about things like higher dimensional spheres, regardless of whether or not they exist in physical reality. It's not always about the sphere per se, it's about what other problems in math they can be used to encode. \ No newline at end of file diff --git a/2018/borsuk-ulam/french/sentence_translations.json b/2018/borsuk-ulam/french/sentence_translations.json index 1ddacc0df..5046e68df 100644 --- a/2018/borsuk-ulam/french/sentence_translations.json +++ b/2018/borsuk-ulam/french/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "En effet, associer chaque point de la surface de la Terre à une paire de nombres revient à cartographier la surface de la Terre sur un plan de coordonnées 2D, où la première coordonnée représente la température et la seconde la pression.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/german/sentence_translations.json b/2018/borsuk-ulam/german/sentence_translations.json index ec1e9ed25..0a3aa6841 100644 --- a/2018/borsuk-ulam/german/sentence_translations.json +++ b/2018/borsuk-ulam/german/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "Dies liegt daran, dass die Zuordnung jedes Punktes auf der Erdoberfläche zu einem Zahlenpaar dasselbe ist, als würde man die Erdoberfläche auf eine zweidimensionale Koordinatenebene abbilden, wobei die erste Koordinate die Temperatur und die zweite den Druck darstellt.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/hindi/sentence_translations.json b/2018/borsuk-ulam/hindi/sentence_translations.json index e8252055d..32f801235 100644 --- a/2018/borsuk-ulam/hindi/sentence_translations.json +++ b/2018/borsuk-ulam/hindi/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "ऐसा इसलिए है क्योंकि पृथ्वी की सतह पर प्रत्येक बिंदु को संख्याओं की एक जोड़ी के साथ जोड़ना, पृथ्वी की सतह को 2डी समन्वय विमान पर मैप करने के समान है, जहां पहला समन्वय तापमान का प्रतिनिधित्व करता है और दूसरा दबाव का प्रतिनिधित्व करता है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/hungarian/sentence_translations.json b/2018/borsuk-ulam/hungarian/sentence_translations.json index 1dd7b68f8..8552cb259 100644 --- a/2018/borsuk-ulam/hungarian/sentence_translations.json +++ b/2018/borsuk-ulam/hungarian/sentence_translations.json @@ -1064,7 +1064,7 @@ "end": 1113.12 }, { - "input": "I'll leave it to you to pause and ponder and think about how this could apply to the And about what the general statement of Borsuk-Ulam might be, and how it applies to the general necklace problem.", + "input": "I'll leave it to you to pause and ponder and think about how this could apply to the 3-jewel case, and about what the general statement of Borsuk-Ulam might be, and how it applies to the general necklace problem.", "translatedText": "Rád bízom, hogy állj meg, gondolkodj el, és gondolkodj el azon, hogy ez hogyan vonatkozhat az És arról, hogy mi lehet a Borsuk-Ulam általános kijelentése, és hogyan vonatkozik az általános nyakláncproblémára.", "model": "DeepL", "n_reviews": 0, diff --git a/2018/borsuk-ulam/indonesian/sentence_translations.json b/2018/borsuk-ulam/indonesian/sentence_translations.json index b9bad8b8c..a685abecc 100644 --- a/2018/borsuk-ulam/indonesian/sentence_translations.json +++ b/2018/borsuk-ulam/indonesian/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "Sebab, mengaitkan setiap titik di permukaan bumi dengan sepasang angka, sama saja dengan memetakan permukaan bumi ke dalam bidang koordinat 2D, dimana koordinat pertama mewakili suhu dan koordinat kedua mewakili tekanan.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/japanese/sentence_translations.json b/2018/borsuk-ulam/japanese/sentence_translations.json index 63310ad4f..e6f80a0a4 100644 --- a/2018/borsuk-ulam/japanese/sentence_translations.json +++ b/2018/borsuk-ulam/japanese/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "これは、地球の表面上の各点を 1 組の数値に関連付けることは、地 球の表面を 2D 座標平面にマッピングすることと同じであるため です。 最初の座標は温度を表し、2 番目の座標は圧力を表します。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/korean/sentence_translations.json b/2018/borsuk-ulam/korean/sentence_translations.json index 953c2f886..227959626 100644 --- a/2018/borsuk-ulam/korean/sentence_translations.json +++ b/2018/borsuk-ulam/korean/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "이는 지구 표면의 각 지점을 한 쌍의 숫자와 연결하는 것은 지구 표면을 2D 좌표 평면에 매핑하는 것과 동일하기 때문입니다. 여기서 첫 번째 좌표는 온도를 나타내고 두 번째 좌표는 압력을 나타냅니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/marathi/sentence_translations.json b/2018/borsuk-ulam/marathi/sentence_translations.json index 4e676d461..8989475f7 100644 --- a/2018/borsuk-ulam/marathi/sentence_translations.json +++ b/2018/borsuk-ulam/marathi/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "याचे कारण असे की पृथ्वीच्या पृष्ठभागावरील प्रत्येक बिंदूला संख्यांच्या जोडीने जोडणे, पृथ्वीच्या पृष्ठभागाचे 2D समन्वय समतलावर मॅपिंग करण्यासारखेच आहे, जेथे पहिला समन्वय तापमानाचे प्रतिनिधित्व करतो आणि दुसरा दाब दर्शवतो.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/persian/sentence_translations.json b/2018/borsuk-ulam/persian/sentence_translations.json index 0e6199da5..dbc10f6be 100644 --- a/2018/borsuk-ulam/persian/sentence_translations.json +++ b/2018/borsuk-ulam/persian/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure. ", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure. ", "translatedText": "این به این دلیل است که مرتبط کردن هر نقطه از سطح زمین با یک جفت اعداد، همان چیزی است که سطح زمین را بر روی یک صفحه مختصات دوبعدی ترسیم می‌کند، جایی که مختصات اول دما و مختصات دوم نشان دهنده فشار است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/polish/sentence_translations.json b/2018/borsuk-ulam/polish/sentence_translations.json index dacb26c32..4c568227f 100644 --- a/2018/borsuk-ulam/polish/sentence_translations.json +++ b/2018/borsuk-ulam/polish/sentence_translations.json @@ -1064,7 +1064,7 @@ "end": 1113.12 }, { - "input": "I'll leave it to you to pause and ponder and think about how this could apply to the And about what the general statement of Borsuk-Ulam might be, and how it applies to the general necklace problem.", + "input": "I'll leave it to you to pause and ponder and think about how this could apply to the 3-jewel case, and about what the general statement of Borsuk-Ulam might be, and how it applies to the general necklace problem.", "translatedText": "", "from_community_srt": "Zostawię to wam, żeby zastanowić się jak można to zastosować do przypadku w 3 rodzajami klejnotów oraz jak w ogólności brzmi twierdzenie Borsuka-Ulama i jak można je zastosować do ogólniejszego", "n_reviews": 0, diff --git a/2018/borsuk-ulam/portuguese/sentence_translations.json b/2018/borsuk-ulam/portuguese/sentence_translations.json index 851e29629..ac41aa3d0 100644 --- a/2018/borsuk-ulam/portuguese/sentence_translations.json +++ b/2018/borsuk-ulam/portuguese/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "Isso ocorre porque associar cada ponto da superfície da Terra a um par de números é a mesma coisa que mapear a superfície da Terra em um plano de coordenadas 2D, onde a primeira coordenada representa a temperatura e a segunda representa a pressão.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/russian/sentence_translations.json b/2018/borsuk-ulam/russian/sentence_translations.json index 72aa74a89..043cdd55f 100644 --- a/2018/borsuk-ulam/russian/sentence_translations.json +++ b/2018/borsuk-ulam/russian/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "Это связано с тем, что связывание каждой точки на поверхности Земли с парой чисел — это то же самое, что отображение поверхности Земли на двумерной координатной плоскости, где первая координата представляет температуру, а вторая — давление.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/spanish/sentence_translations.json b/2018/borsuk-ulam/spanish/sentence_translations.json index 3dda9a510..077524d8c 100644 --- a/2018/borsuk-ulam/spanish/sentence_translations.json +++ b/2018/borsuk-ulam/spanish/sentence_translations.json @@ -1194,7 +1194,7 @@ "end": 1113.12 }, { - "input": "I'll leave it to you to pause and ponder and think about how this could apply to the And about what the general statement of Borsuk-Ulam might be, and how it applies to the general necklace problem.", + "input": "I'll leave it to you to pause and ponder and think about how this could apply to the 3-jewel case, and about what the general statement of Borsuk-Ulam might be, and how it applies to the general necklace problem.", "translatedText": "Te dejo que te detengas a reflexionar y pienses cómo podría aplicarse esto a la Y sobre cuál podría ser la declaración general de Borsuk-Ulam, y cómo se aplica al problema general del collar.", "model": "DeepL", "from_community_srt": "Te dejo parar y reflexionar acerca de cómo esto se puede aplicar al caso con 3 gemas, y acerca de cómo podría ser el enunciado general del teorema de Borsuk-Ulam, y cómo se aplica al problema general del collar.", diff --git a/2018/borsuk-ulam/tamil/sentence_translations.json b/2018/borsuk-ulam/tamil/sentence_translations.json index 30fa74f74..27cd5648a 100644 --- a/2018/borsuk-ulam/tamil/sentence_translations.json +++ b/2018/borsuk-ulam/tamil/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "ஏனென்றால், பூமியின் மேற்பரப்பில் உள்ள ஒவ்வொரு புள்ளியையும் ஒரு ஜோடி எண்களுடன் தொடர்புபடுத்துவது, பூமியின் மேற்பரப்பை 2D ஒருங்கிணைப்பு விமானத்தில் வரைபடமாக்குவது போன்றது, இதில் முதல் ஒருங்கிணைப்பு வெப்பநிலையையும் இரண்டாவது அழுத்தத்தையும் குறிக்கிறது.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/telugu/sentence_translations.json b/2018/borsuk-ulam/telugu/sentence_translations.json index ba72bebb6..ca9a5401b 100644 --- a/2018/borsuk-ulam/telugu/sentence_translations.json +++ b/2018/borsuk-ulam/telugu/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "ఎందుకంటే భూమి యొక్క ఉపరితలంపై ఉన్న ప్రతి బిందువును ఒక జత సంఖ్యలతో అనుబంధించడం, భూమి యొక్క ఉపరితలాన్ని 2D కోఆర్డినేట్ ప్లేన్‌లో మ్యాప్ చేయడం లాంటిదే, ఇక్కడ మొదటి కోఆర్డినేట్ ఉష్ణోగ్రతను సూచిస్తుంది మరియు రెండవది ఒత్తిడిని సూచిస్తుంది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/thai/sentence_translations.json b/2018/borsuk-ulam/thai/sentence_translations.json index 6b2eb524d..6511bd8b9 100644 --- a/2018/borsuk-ulam/thai/sentence_translations.json +++ b/2018/borsuk-ulam/thai/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure. ", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/turkish/sentence_translations.json b/2018/borsuk-ulam/turkish/sentence_translations.json index ba8691573..b9de854ef 100644 --- a/2018/borsuk-ulam/turkish/sentence_translations.json +++ b/2018/borsuk-ulam/turkish/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "Bunun nedeni, Dünya yüzeyindeki her noktayı bir çift sayıyla ilişkilendirmenin, Dünya yüzeyini 2 boyutlu bir koordinat düzlemine haritalamakla aynı şey olmasıdır; burada ilk koordinat sıcaklığı, ikincisi ise basıncı temsil eder.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/urdu/sentence_translations.json b/2018/borsuk-ulam/urdu/sentence_translations.json index 2d7a69268..1de2f935a 100644 --- a/2018/borsuk-ulam/urdu/sentence_translations.json +++ b/2018/borsuk-ulam/urdu/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure. ", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure. ", "translatedText": "اس کی وجہ یہ ہے کہ زمین کی سطح پر ہر ایک نقطہ کو اعداد کے جوڑے کے ساتھ جوڑنا، زمین کی سطح کو 2D کوآرڈینیٹ طیارے پر نقشہ بنانے کے مترادف ہے، جہاں پہلا کوآرڈینیٹ درجہ حرارت کی نمائندگی کرتا ہے اور دوسرا دباؤ کو ظاہر کرتا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/borsuk-ulam/vietnamese/sentence_translations.json b/2018/borsuk-ulam/vietnamese/sentence_translations.json index 28a81f29a..5e1b11ba1 100644 --- a/2018/borsuk-ulam/vietnamese/sentence_translations.json +++ b/2018/borsuk-ulam/vietnamese/sentence_translations.json @@ -296,7 +296,7 @@ "end": 301.6 }, { - "input": "This is because associating each point on the surface of the Earth with a pair of numbers, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature and the second represents pressure.", + "input": "This is because associating each point on the surface of the Earth with a pair of numbers, temperature and pressure, is the same thing as mapping the surface of the Earth onto a 2D coordinate plane, where the first coordinate represents temperature, and the second represents pressure.", "translatedText": "Điều này là do việc liên kết mỗi điểm trên bề mặt Trái đất với một cặp số, cũng giống như ánh xạ bề mặt Trái đất lên mặt phẳng tọa độ 2D, trong đó tọa độ đầu tiên biểu thị nhiệt độ và tọa độ thứ hai biểu thị áp suất.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/arabic/sentence_translations.json b/2018/derivatives-and-transforms/arabic/sentence_translations.json index 0e9dffb70..921d11006 100644 --- a/2018/derivatives-and-transforms/arabic/sentence_translations.json +++ b/2018/derivatives-and-transforms/arabic/sentence_translations.json @@ -191,7 +191,7 @@ "end": 209.02 }, { - "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a This is what it means for the derivative of x2 at the input x equals 1 to be 2.", + "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a factor of 2. This is what it means for the derivative of x2 at the input x equals 1 to be 2.", "translatedText": "كلما اقتربت من التكبير، كلما بدا هذا السلوك المحلي وكأنه ضرب بـ a. هذا ما يعنيه أن مشتق x2 عند الإدخال x يساوي 1 يساوي 2.", "model": "google_nmt", "from_community_srt": "في الواقع ، يبدو أنها تقريبًا تشبه عامل 2 ، وكلما اقتربت من تكبير هذا السلوك الموضعي يبدو الأمر وكأنه يتضاعف بمعامل 2 هذا ما يعنيه بالنسبة لمشتق x التربيع عند إدخال x يساوي 1 ليكون 2 هذا ما تبدو عليه هذه الحقيقة في سياق التحولات", @@ -259,7 +259,7 @@ "end": 261.68 }, { - "input": "As you zoom in closer and closer, by 100x or by 1000x, it looks more and more like a And this is what it looks like for the derivative to be 0.", + "input": "As you zoom in closer and closer, by 100x, or by 1000x, it looks more and more like a small neighborhood of points around 0 just gets collapsed into 0 itself. This is what it looks like for the derivative to be 0.", "translatedText": "كلما قمت بالتكبير أكثر فأكثر، بمقدار 100x أو 1000x، يبدو أكثر فأكثر مثل a وهذا ما يبدو عليه المشتق ليكون 0.", "model": "google_nmt", "from_community_srt": "وأنت تقترب أكثر وأقرب من 100x أو 1000 X يبدو أكثر وأكثر مثل مقدار تقريبي صغير من النقاط حول الصفر فقط يحصل تساقط في الصفر نفسه", diff --git a/2018/derivatives-and-transforms/bengali/sentence_translations.json b/2018/derivatives-and-transforms/bengali/sentence_translations.json index f352ba844..9b4a44c30 100644 --- a/2018/derivatives-and-transforms/bengali/sentence_translations.json +++ b/2018/derivatives-and-transforms/bengali/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out. ", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out. ", "translatedText": "আপনি যদি ইনপুট 1 এর চারপাশে বিন্দুগুলির একটি ছোট ক্লাস্টারে জুম করেন এবং দেখেন যে তারা প্রাসঙ্গিক আউটপুটের চারপাশে কোথায় অবতরণ করে, আপনি লক্ষ্য করবেন যে তারা প্রসারিত হতে থাকে।", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", "translatedText": "ইনপুট 1 চতুর্থের আশেপাশে, একটি ছোট অঞ্চল 1 অর্ধেক একটি ফ্যাক্টর দ্বারা সংকুচিত হতে থাকে, এবং এটি 1 এর থেকে ছোট একটি ডেরিভেটিভের মত দেখায়।", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem. ", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem. ", "translatedText": "আমি মনে করি আপনি পয়েন্টটি পেয়েছেন, এটি সবই ভাল এবং ভাল, তবে আসুন দেখি এটি একটি সমস্যা সমাধানে কীভাবে কার্যকর।", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", "translatedText": "সুতরাং যে আউটপুট ফাংশন মধ্যে ফিরে প্লাগ, আপনি প্রথম অনুভূমিকভাবে সরানো হতে পারে যতক্ষণ না আপনি লাইন y আঘাত x সমান, এবং যে আপনি একটি অবস্থান দিতে যাচ্ছে যেখানে x মান আপনার পূর্ববর্তী y মান অনুরূপ, ডান? ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you? ", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you? ", "translatedText": "ব্যক্তিগতভাবে, আমি মনে করি এটি একটি ফাংশন বারবার প্রয়োগ করার বিষয়ে চিন্তা করার একটি বিশ্রী উপায়, তাই না? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", "translatedText": "তাই বিভিন্ন নমুনাযুক্ত ইনপুট পয়েন্টগুলি কোথায় যাবে তা নির্দেশ করতে আমি এখানে একগুচ্ছ তীর আঁকার মাধ্যমে শুরু করতে যাচ্ছি।", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", "translatedText": "সেখানে, ডেরিভেটিভের মাত্রা 1 এর চেয়ে বড়, তাই স্থির বিন্দুর কাছাকাছি বিন্দুগুলি এটি থেকে দূরে সরিয়ে দেওয়া হয়।", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you. ", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you. ", "translatedText": "আপনি ফি-এর ছোট ভাইকে অসীম ভগ্নাংশের একটি বৈধ মান বিবেচনা করতে চান কিনা তা আপনার উপর নির্ভর করে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/chinese/sentence_translations.json b/2018/derivatives-and-transforms/chinese/sentence_translations.json index f4e16fcb6..4626e6dd8 100644 --- a/2018/derivatives-and-transforms/chinese/sentence_translations.json +++ b/2018/derivatives-and-transforms/chinese/sentence_translations.json @@ -177,7 +177,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out. ", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out. ", "translatedText": "如果放大输入 1 周围的一小群点，并查看它们落在 相关输出周围的位置，您会注意到它们往往会被拉伸。", "model": "google_nmt", "from_community_srt": "然后看看他们在相关输出周围的位置， 这个函数也恰好是1 你会注意到它们往往会被拉长。", @@ -239,7 +239,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", "translatedText": "在输入 1/4 附近，一个小区域往往会收缩 1/2，这就是导数小于 1 时的情况。", "model": "google_nmt", "from_community_srt": "在输入端1/4周围， 一个小区域实际上往往会收缩 特别是1/2的因子， 这就是导数小于1的情况。", @@ -364,7 +364,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem. ", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem. ", "translatedText": "我想你明白了，这一切都很好，但让 我们看看这在解决问题时有何用处。", "model": "google_nmt", "from_community_srt": "这一切都很好，", @@ -579,7 +579,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", "translatedText": "因此，要将输出插回到函数中，您可能首先水平移动， 直到到达 y 等于 x 的线，这将为您提供一个 位置，其中 x 值对应于之前的 y 值，对吧？", "model": "google_nmt", "from_community_srt": "所以想想把这个输出插回到函数中， 你可能首先水平移动， 直到你碰到y线等于x的线， 这会给你一个位置， 在这里x值 对应于您之前的y值， 对吗？", @@ -597,7 +597,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you? ", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you? ", "translatedText": "就我个人而言，我认为这是重复应用 函数的一种尴尬的方式，不是吗？", "model": "google_nmt", "from_community_srt": "现在我个人认为这是一种屡次想应用某种功能的尴尬方法， 不是吗？", @@ -650,7 +650,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", "translatedText": "因此，我将从这里开始绘制一堆箭 头来指示各种采样输入点的去向。", "model": "google_nmt", "from_community_srt": "所以我要继续前进， 并从这里开始绘制一大堆箭头， 以指示输入点的各个样本的位置，", @@ -765,7 +765,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", "translatedText": "在那里，导数的大小大于 1， 因此固定点附近的点会被排斥。", "model": "google_nmt", "from_community_srt": "在那边， 导数实际上有一个大于1的数值， 所以靠近固定点的点被排斥离开 当你解决这个问题时，", @@ -828,7 +828,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you. ", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you. ", "translatedText": "您是否想将 phi 的弟弟视 为无限分数的有效值取决于您。", "model": "google_nmt", "from_community_srt": "至于你是否想考虑phi的小弟弟这个无限分数的有效值 那么，", diff --git a/2018/derivatives-and-transforms/english/captions.srt b/2018/derivatives-and-transforms/english/captions.srt index 1c29aeb97..ca87a4e3b 100644 --- a/2018/derivatives-and-transforms/english/captions.srt +++ b/2018/derivatives-and-transforms/english/captions.srt @@ -211,622 +211,630 @@ to get stretched out. In fact, it roughly looks like stretching out by a factor of 2. 54 -00:03:29,660 --> 00:03:35,629 -The closer you zoom in, the more this local behavior looks just like multiplying +00:03:29,660 --> 00:03:35,535 +The closer you zoom in, the more this local behavior looks just like multiplying by a 55 -00:03:35,629 --> 00:03:41,820 -by a This is what it means for the derivative of x2 at the input x equals 1 to be 2. +00:03:35,535 --> 00:03:41,683 +factor of 2. This is what it means for the derivative of x2 at the input x equals 1 to be 56 +00:03:41,683 --> 00:03:41,820 +2. + +57 00:03:42,340 --> 00:03:45,400 It's what that fact looks like in the context of transformations. -57 +58 00:03:46,460 --> 00:03:49,732 If you looked at a neighborhood of points around the input 3, -58 +59 00:03:49,732 --> 00:03:52,160 they would get stretched out by a factor of 6. -59 +60 00:03:52,740 --> 00:03:57,440 This is what it means for the derivative of this function at the input 3 to equal 6. -60 +61 00:03:58,980 --> 00:04:03,614 Around the input 1 fourth, a small region tends to get contracted specifically by a -61 +62 00:04:03,614 --> 00:04:08,360 factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. -62 +63 00:04:10,720 --> 00:04:12,600 The input 0 is interesting. -63 +64 00:04:13,120 --> 00:04:15,618 Zooming in by a factor of 10, it doesn't really -64 +65 00:04:15,618 --> 00:04:17,959 look like a constant stretching or squishing. -65 +66 00:04:18,380 --> 00:04:21,680 For one thing, all of the outputs end up on the right positive side of things. -66 -00:04:23,320 --> 00:04:29,720 -As you zoom in closer and closer, by 100x or by 1000x, - 67 -00:04:29,720 --> 00:04:39,960 -it looks more and more like a And this is what it looks like for the derivative to be 0. +00:04:23,320 --> 00:04:27,694 +As you zoom in closer and closer, by 100x, or by 1000x, 68 +00:04:27,694 --> 00:04:33,397 +it looks more and more like a small neighborhood of points around 0 just + +69 +00:04:33,397 --> 00:04:39,960 +gets collapsed into 0 itself. This is what it looks like for the derivative to be 0. + +70 00:04:40,500 --> 00:04:45,020 The local behavior looks more and more like multiplying the whole number line by 0. -69 +71 00:04:45,680 --> 00:04:49,783 It doesn't have to completely collapse everything to a point at a particular zoom level, -70 +72 00:04:49,783 --> 00:04:53,840 instead it's a matter of what the limiting behavior is as you zoom in closer and closer. -71 +73 00:04:55,280 --> 00:04:58,960 It's also instructive to take a look at the negative inputs here. -72 +74 00:05:00,700 --> 00:05:04,536 Things start to feel a little cramped since they collide with where all the positive -73 +75 00:05:04,536 --> 00:05:08,057 input values go, and this is one of the downsides of thinking of functions as -74 +76 00:05:08,057 --> 00:05:08,780 transformations. -75 +77 00:05:09,400 --> 00:05:13,094 But for derivatives, we only really care about the local behavior anyway, -76 +78 00:05:13,094 --> 00:05:15,640 what happens in a small range around a given input. -77 +79 00:05:16,500 --> 00:05:20,189 Here, notice that the inputs in a little neighborhood around, say, -78 +80 00:05:20,189 --> 00:05:24,100 negative 2, don't just get stretched out, they also get flipped around. -79 +81 00:05:24,680 --> 00:05:28,132 Specifically, the action on such a neighborhood looks more -80 +82 00:05:28,132 --> 00:05:31,820 and more like multiplying by negative 4 the closer you zoom in. -81 +83 00:05:32,320 --> 00:05:35,600 This is what it looks like for the derivative of a function to be negative. -82 +84 00:05:38,460 --> 00:05:40,952 And I think you get the point, this is all well and good, -83 +85 00:05:40,952 --> 00:05:43,660 but let's see how this is actually useful in solving a problem. -84 +86 00:05:44,260 --> 00:05:48,305 A friend of mine recently asked me a pretty fun question about the infinite -85 +87 00:05:48,305 --> 00:05:52,137 fraction 1 plus 1 divided by 1 plus 1 divided by 1 plus 1 divided by 1, -86 +88 00:05:52,137 --> 00:05:56,182 and clearly you watch math videos online, so maybe you've seen this before, -87 +89 00:05:56,182 --> 00:05:59,961 but my friend's question actually cuts to something you might not have -88 +90 00:05:59,961 --> 00:06:04,220 thought about before, relevant to the view of derivatives we're looking at here. -89 +91 00:06:05,020 --> 00:06:09,661 The typical way you might evaluate an expression like this is to set it equal to x, -90 +92 00:06:09,661 --> 00:06:13,640 and then notice that there is a copy of the full fraction inside itself. -91 +93 00:06:14,700 --> 00:06:18,780 So you can replace that copy with another x, and then just solve for x. -92 +94 00:06:19,440 --> 00:06:24,580 That is, what you want is to find a fixed point of the function 1 plus 1 divided by x. -93 +95 00:06:27,160 --> 00:06:30,970 But here's the thing, there are actually two solutions for x, -94 +96 00:06:30,970 --> 00:06:36,380 two special numbers where 1 plus 1 divided by that number gives you back the same thing. -95 +97 00:06:36,940 --> 00:06:42,949 One is the golden ratio, phi, around 1.618, and the other is negative 0.618, -96 +98 00:06:42,949 --> 00:06:46,540 which happens to be negative 1 divided by phi. -97 +99 00:06:46,960 --> 00:06:49,682 I like to call this other number phi's little brother, -98 +100 00:06:49,682 --> 00:06:52,900 since just about any property that phi has, this number also has. -99 +101 00:06:53,560 --> 00:06:58,413 And this raises the question, would it be valid to say that the infinite -100 +102 00:06:58,413 --> 00:07:03,600 fraction we saw is somehow also equal to phi's little brother, negative 0.618? -101 +103 00:07:04,520 --> 00:07:08,813 Maybe you initially say, obviously not, everything on the left hand side is positive, -102 +104 00:07:08,813 --> 00:07:11,260 so how could it possibly equal a negative number? -103 +105 00:07:12,500 --> 00:07:17,100 Well, first we should be clear about what we actually mean by an expression like this. -104 +106 00:07:17,780 --> 00:07:21,316 One way you could think about it, and it's not the only way, -105 +107 00:07:21,316 --> 00:07:26,186 there's freedom for choice here, is to imagine starting with some constant, like 1, -106 +108 00:07:26,186 --> 00:07:30,940 and then repeatedly applying the function 1 plus 1 divided by x, and then asking, -107 +109 00:07:30,940 --> 00:07:33,260 what is this approach as you keep going? -108 +110 00:07:36,040 --> 00:07:38,552 I mean, certainly symbolically what you get looks more and more -109 +111 00:07:38,552 --> 00:07:41,300 like our infinite fraction, so maybe if you wanted to equal a number, -110 +112 00:07:41,300 --> 00:07:43,420 you should ask what this series of numbers approaches. -111 +113 00:07:45,120 --> 00:07:48,510 And if that's your view of things, maybe you start off with a negative number, -112 +114 00:07:48,510 --> 00:07:51,300 so it's not so crazy for the whole expression to end up negative. -113 +115 00:07:52,740 --> 00:07:55,837 After all, if you start with negative 1 divided by phi, -114 +116 00:07:55,837 --> 00:07:59,986 then applying this function 1 plus 1 over x, you get back the same number, -115 +117 00:07:59,986 --> 00:08:03,803 negative 1 divided by phi, so no matter how many times you apply it, -116 +118 00:08:03,803 --> 00:08:05,740 you're staying fixed at this value. -117 +119 00:08:07,820 --> 00:08:10,620 But even then, there is one reason you should -118 +120 00:08:10,620 --> 00:08:13,420 view phi as the favorite brother in this pair. -119 +121 00:08:14,020 --> 00:08:19,331 Here, try this, pull up a calculator of some kind, then start with any random number, -120 +122 00:08:19,331 --> 00:08:22,728 and plug it into this function, 1 plus 1 divided by x, -121 +123 00:08:22,728 --> 00:08:28,040 and plug that number into 1 plus 1 over x, and again, and again, and again, and again. -122 +124 00:08:28,480 --> 00:08:33,159 No matter what constant you start with, you eventually end up at 1.618. -123 +125 00:08:33,799 --> 00:08:38,482 Even if you start with a negative number, even one that's really close to phi's -124 +126 00:08:38,482 --> 00:08:43,400 little brother, eventually it shies away from that value and jumps back over to phi. -125 +127 00:08:50,820 --> 00:08:52,460 So, what's going on here? -126 +128 00:08:52,800 --> 00:08:55,920 Why is one of these fixed points favored above the other one? -127 +129 00:08:56,720 --> 00:09:00,159 Maybe you can already see how the transformational understanding of derivatives -128 +130 00:09:00,159 --> 00:09:03,984 is helpful for understanding this setup, but for the sake of having a point of contrast, -129 +131 00:09:03,984 --> 00:09:07,080 I want to show you how a problem like this is often taught using graphs. -130 +132 00:09:07,920 --> 00:09:11,115 If you were to plug in some random input to this function, -131 +133 00:09:11,115 --> 00:09:14,040 the y value tells you the corresponding output, right? -132 +134 00:09:14,040 --> 00:09:17,863 So to think about plugging that output back into the function, -133 +135 00:09:17,863 --> 00:09:22,050 you might first move horizontally until you hit the line y equals x, -134 +136 00:09:22,050 --> 00:09:26,783 and that's going to give you a position where the x value corresponds to your -135 +137 00:09:26,783 --> 00:09:28,240 previous y value, right? -136 +138 00:09:28,920 --> 00:09:34,554 So then from there, you can move vertically to see what output this new x value has, -137 +139 00:09:34,554 --> 00:09:35,880 and then you repeat. -138 +140 00:09:36,340 --> 00:09:40,598 You move horizontally to the line y equals x to find a point whose x value is the same -139 +141 00:09:40,598 --> 00:09:44,760 as the output you just got, and then you move vertically to apply the function again. -140 +142 00:09:45,880 --> 00:09:48,286 Now personally, I think this is kind of an awkward way -141 +143 00:09:48,286 --> 00:09:50,780 to think about repeatedly applying a function, don't you? -142 +144 00:09:51,300 --> 00:09:53,804 I mean, it makes sense, but you kind of have to pause -143 +145 00:09:53,804 --> 00:09:56,540 and think about it to remember which way to draw the lines. -144 +146 00:09:57,120 --> 00:10:01,426 And you can, if you want, think through what conditions make this spiderweb -145 +147 00:10:01,426 --> 00:10:05,280 process narrow in on a fixed point, versus propagating away from it. -146 +148 00:10:05,860 --> 00:10:08,900 In fact, go ahead, pause right now, and try to think it through as an exercise. -147 +149 00:10:09,240 --> 00:10:10,460 It has to do with slopes. -148 +150 00:10:12,020 --> 00:10:15,819 Or if you want to skip the exercise for something that I think gives a much more -149 +151 00:10:15,819 --> 00:10:19,620 satisfying understanding, think about how this function acts as a transformation. -150 +152 00:10:22,280 --> 00:10:24,924 So I'm going to go ahead and start here by drawing a bunch of -151 +153 00:10:24,924 --> 00:10:27,740 arrows to indicate where the various sampled input points will go. -152 +154 00:10:28,320 --> 00:10:31,440 And side note, don't you think this gives a neat emergent pattern? -153 +155 00:10:31,820 --> 00:10:35,020 I wasn't expecting this, but it was cool to see it pop up when animating. -154 +156 00:10:35,020 --> 00:10:38,797 I guess the action of 1 divided by x gives this nice emergent circle, -155 +157 00:10:38,797 --> 00:10:41,280 and then we're just shifting things over by 1. -156 +158 00:10:42,040 --> 00:10:46,622 Anyway, I want you to think about what it means to repeatedly apply some function, -157 +159 00:10:46,622 --> 00:10:48,720 like 1 plus 1 over x, in this context. -158 +160 00:10:50,240 --> 00:10:53,590 Well after letting it map all of the inputs to the outputs, -159 +161 00:10:53,590 --> 00:10:58,504 you could consider those as the new inputs, and then just apply the same process again, -160 +162 00:10:58,504 --> 00:11:01,520 and then again, and do it however many times you want. -161 -00:11:02,580 --> 00:11:06,729 +163 +00:11:02,580 --> 00:11:06,523 Notice, in animating this with a few dots representing the sample points, -162 -00:11:06,729 --> 00:11:11,775 +164 +00:11:06,523 --> 00:11:11,320 it doesn't take many iterations at all before all of those dots kind of clump in around 1. -163 -00:11:11,775 --> 00:11:12,000 +165 +00:11:11,320 --> 00:11:12,000 618. -164 +166 00:11:14,620 --> 00:11:18,355 Now remember, we know that 1.618 and its little brother, -165 +167 00:11:18,355 --> 00:11:23,860 negative 0.618 on and on, stay fixed in place during each iteration of this process. -166 +168 00:11:24,860 --> 00:11:27,480 But zoom in on a neighborhood around phi. -167 +169 00:11:27,480 --> 00:11:32,788 During the map, points in that region get contracted around phi, -168 +170 00:11:32,788 --> 00:11:39,404 meaning that the function 1 plus 1 over x has a derivative with a magnitude less -169 +171 00:11:39,404 --> 00:11:41,120 than 1 at this input. -170 +172 00:11:41,880 --> 00:11:45,200 In fact, this derivative works out to be around negative 0.38. -171 +173 00:11:46,120 --> 00:11:50,312 So what that means is that each repeated application scrunches the neighborhood -172 +174 00:11:50,312 --> 00:11:54,400 around this number smaller and smaller, like a gravitational pull towards phi. -173 +175 00:11:54,960 --> 00:11:58,620 So now tell me what you think happens in the neighborhood of phi's little brother. -174 +176 00:12:01,320 --> 00:12:05,426 Over there, the derivative actually has a magnitude larger than 1, -175 +177 00:12:05,426 --> 00:12:08,920 so points near the fixed point are repelled away from it. -176 +178 00:12:09,520 --> 00:12:11,599 And when you work it out, you can see that they get -177 +179 00:12:11,599 --> 00:12:13,800 stretched by more than a factor of 2 in each iteration. -178 +180 00:12:14,420 --> 00:12:17,675 They also get flipped around, because the derivative is negative here, -179 +181 00:12:17,675 --> 00:12:20,840 but the salient fact for the sake of stability is just the magnitude. -180 +182 00:12:23,440 --> 00:12:26,970 Mathematicians would call this right value a stable fixed point, -181 +183 00:12:26,970 --> 00:12:29,360 and the left one is an unstable fixed point. -182 +184 00:12:30,000 --> 00:12:33,409 Something is considered stable if when you perturb it just a little bit, -183 +185 00:12:33,409 --> 00:12:37,100 it tends to come back towards where it started, rather than going away from it. -184 +186 00:12:38,180 --> 00:12:40,778 So what we're seeing is a very useful little fact, -185 +187 00:12:40,778 --> 00:12:45,312 that the stability of a fixed point is determined by whether or not the magnitude of its -186 +188 00:12:45,312 --> 00:12:47,300 derivative is bigger or smaller than 1. -187 +189 00:12:47,300 --> 00:12:50,480 This explains why phi always shows up in the numerical play, -188 +190 00:12:50,480 --> 00:12:53,922 where you're just hitting enter on your calculator over and over, -189 +191 00:12:53,922 --> 00:12:55,800 but phi's little brother never does. -190 +192 00:12:56,460 --> 00:12:59,621 As to whether or not you want to consider phi's little brother a -191 +193 00:12:59,621 --> 00:13:02,880 valid value of the infinite fraction, well that's really up to you. -192 +194 00:13:03,260 --> 00:13:06,970 Everything we just showed suggests that if you think of this expression -193 +195 00:13:06,970 --> 00:13:10,525 as representing a limiting process, then because every possible seed -194 +196 00:13:10,525 --> 00:13:14,442 value other than phi's little brother gives you a series converging to phi, -195 +197 00:13:14,442 --> 00:13:17,740 it does feel silly to put them on equal footing with each other. -196 +198 00:13:18,260 --> 00:13:21,773 But maybe you don't think of it as a limit, maybe the kind of math -197 +199 00:13:21,773 --> 00:13:25,601 you're doing lends itself to treating this as a purely algebraic object, -198 +200 00:13:25,601 --> 00:13:29,220 like the solutions of a polynomial, which simply has multiple values. -199 +201 00:13:30,340 --> 00:13:34,485 Anyway, that's beside the point, and my point here is not that viewing derivatives -200 +202 00:13:34,485 --> 00:13:38,780 as this change in density is somehow better than the graphical intuition on the whole. -201 +203 00:13:39,600 --> 00:13:42,204 In fact, picturing an entire function this way can be -202 +204 00:13:42,204 --> 00:13:44,760 kind of clunky and impractical as compared to graphs. -203 +205 00:13:45,340 --> 00:13:48,222 My point is that it deserves more of a mention in most of the -204 +206 00:13:48,222 --> 00:13:50,918 introductory calculus courses, because it can help make a -205 +207 00:13:50,918 --> 00:13:53,940 student's understanding of the derivative a little more flexible. -206 +208 00:13:54,900 --> 00:13:58,358 Like I mentioned, the real reason I'd recommend you carry this perspective -207 +209 00:13:58,358 --> 00:14:01,817 with you as you learn new topics is not so much for what it does with your -208 +210 00:14:01,817 --> 00:14:05,000 understanding of single variable calculus, it's for what comes after. diff --git a/2018/derivatives-and-transforms/english/sentence_timings.json b/2018/derivatives-and-transforms/english/sentence_timings.json index d080dc105..71ae877c7 100644 --- a/2018/derivatives-and-transforms/english/sentence_timings.json +++ b/2018/derivatives-and-transforms/english/sentence_timings.json @@ -110,7 +110,7 @@ 209.02 ], [ - "The closer you zoom in, the more this local behavior looks just like multiplying by a This is what it means for the derivative of x2 at the input x equals 1 to be 2.", + "The closer you zoom in, the more this local behavior looks just like multiplying by a factor of 2. This is what it means for the derivative of x2 at the input x equals 1 to be 2.", 209.66, 221.82 ], @@ -150,7 +150,7 @@ 261.68 ], [ - "As you zoom in closer and closer, by 100x or by 1000x, it looks more and more like a And this is what it looks like for the derivative to be 0.", + "As you zoom in closer and closer, by 100x, or by 1000x, it looks more and more like a small neighborhood of points around 0 just gets collapsed into 0 itself. This is what it looks like for the derivative to be 0.", 263.32, 279.96 ], @@ -385,8 +385,13 @@ 661.52 ], [ - "Notice, in animating this with a few dots representing the sample points, it doesn't take many iterations at all before all of those dots kind of clump in around 1.618.", + "Notice, in animating this with a few dots representing the sample points, it doesn't take many iterations at all before all of those dots kind of clump in around 1.", 662.58, + 671.32 + ], + [ + "618.", + 671.32, 672.0 ], [ diff --git a/2018/derivatives-and-transforms/english/transcript.txt b/2018/derivatives-and-transforms/english/transcript.txt index dae548f3b..ba689b148 100644 --- a/2018/derivatives-and-transforms/english/transcript.txt +++ b/2018/derivatives-and-transforms/english/transcript.txt @@ -20,7 +20,7 @@ Take the function x2, it maps 1 to 1, 2 to 4, 3 to 9, and so on. You can also see how it acts on all of the points in between. If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out. In fact, it roughly looks like stretching out by a factor of 2. -The closer you zoom in, the more this local behavior looks just like multiplying by a This is what it means for the derivative of x2 at the input x equals 1 to be 2. +The closer you zoom in, the more this local behavior looks just like multiplying by a factor of 2. This is what it means for the derivative of x2 at the input x equals 1 to be 2. It's what that fact looks like in the context of transformations. If you looked at a neighborhood of points around the input 3, they would get stretched out by a factor of 6. This is what it means for the derivative of this function at the input 3 to equal 6. @@ -28,7 +28,7 @@ Around the input 1 fourth, a small region tends to get contracted specifically b The input 0 is interesting. Zooming in by a factor of 10, it doesn't really look like a constant stretching or squishing. For one thing, all of the outputs end up on the right positive side of things. -As you zoom in closer and closer, by 100x or by 1000x, it looks more and more like a And this is what it looks like for the derivative to be 0. +As you zoom in closer and closer, by 100x, or by 1000x, it looks more and more like a small neighborhood of points around 0 just gets collapsed into 0 itself. This is what it looks like for the derivative to be 0. The local behavior looks more and more like multiplying the whole number line by 0. It doesn't have to completely collapse everything to a point at a particular zoom level, instead it's a matter of what the limiting behavior is as you zoom in closer and closer. It's also instructive to take a look at the negative inputs here. diff --git a/2018/derivatives-and-transforms/french/sentence_translations.json b/2018/derivatives-and-transforms/french/sentence_translations.json index c7922734f..748ea9783 100644 --- a/2018/derivatives-and-transforms/french/sentence_translations.json +++ b/2018/derivatives-and-transforms/french/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out.", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out.", "translatedText": "Si vous zoomez sur un petit groupe de points autour de l'entrée 1 et voyez où ils atterrissent autour de la sortie correspondante, vous remarquerez qu'ils ont tendance à s'étirer.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", "translatedText": "Autour de l'entrée 1 quart, une petite région a tendance à se contracter d'un facteur de moitié, et c'est à cela que ressemble une dérivée inférieure à 1.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem.", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem.", "translatedText": "Je pense que vous comprenez, c'est bien beau, mais voyons comment cela est utile pour résoudre un problème.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", "translatedText": "Donc, pour rebrancher cette sortie dans la fonction, vous pouvez d'abord vous déplacer horizontalement jusqu'à ce que vous atteigniez la ligne y est égale à x, et cela vous donnera une position où la valeur x correspond à votre valeur y précédente, n'est-ce pas?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you?", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you?", "translatedText": "Personnellement, je pense que c'est une façon délicate d'envisager l'application répétée d'une fonction, n'est-ce pas?", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 590.78 }, { - "input": "I mean, it makes sense, but you have to pause and think about it to remember which way to draw the lines.", + "input": "I mean, it makes sense, but you kind of have to pause and think about it to remember which way to draw the lines.", "translatedText": "Je veux dire, c’est logique, mais il faut faire une pause et y réfléchir pour se rappeler dans quelle direction tracer les lignes.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", "translatedText": "Je vais donc commencer ici en dessinant un tas de flèches pour indiquer où iront les différents points d'entrée échantillonnés.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it.", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it.", "translatedText": "Là-bas, la dérivée a une magnitude supérieure à 1, donc les points proches du point fixe en sont repoussés.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you.", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you.", "translatedText": "Que vous souhaitiez ou non considérer le petit frère de phi comme une valeur valide de la fraction infinie dépend de vous.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/german/sentence_translations.json b/2018/derivatives-and-transforms/german/sentence_translations.json index c3e7f59f8..aa90b09d5 100644 --- a/2018/derivatives-and-transforms/german/sentence_translations.json +++ b/2018/derivatives-and-transforms/german/sentence_translations.json @@ -198,7 +198,7 @@ "end": 209.02 }, { - "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a This is what it means for the derivative of x2 at the input x equals 1 to be 2.", + "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a factor of 2. This is what it means for the derivative of x2 at the input x equals 1 to be 2.", "translatedText": "Je näher du heranzoomst, desto mehr sieht dieses lokale Verhalten aus wie die Multiplikation mit einem Dies bedeutet, dass die Ableitung von x2 an der Eingabe x gleich 1 gleich 2 ist.", "model": "DeepL", "from_community_srt": "und je näher man zoomt, desto stärker ist dieses lokale Verhalten Sieht so aus, als würde man mit dem Faktor 2 multiplizieren. Dies bedeutet, dass die Ableitung von x im Quadrat am Eingang x gleich 1 gleich 2 ist.", @@ -269,7 +269,7 @@ "end": 261.68 }, { - "input": "As you zoom in closer and closer, by 100x or by 1000x, it looks more and more like a And this is what it looks like for the derivative to be 0.", + "input": "As you zoom in closer and closer, by 100x, or by 1000x, it looks more and more like a small neighborhood of points around 0 just gets collapsed into 0 itself. This is what it looks like for the derivative to be 0.", "translatedText": "Wenn du näher heranzoomst, 100- oder 1000-fach, sieht es immer mehr wie eine Und so sieht es aus, wenn die Ableitung 0 ist.", "model": "DeepL", "from_community_srt": "zum einen landen alle Ausgänge rechts positive Seite der Dinge und wenn Sie näher und näher um das 100-fache oder um das 1000-fache zoomen Es sieht immer mehr so ​​aus, als würde eine kleine Nachbarschaft von Punkten um Null einfach zu Null zusammenfallen. Und so sieht es aus, wenn die Ableitung Null ist,", diff --git a/2018/derivatives-and-transforms/greek/sentence_translations.json b/2018/derivatives-and-transforms/greek/sentence_translations.json index c72a47267..adf5cb255 100644 --- a/2018/derivatives-and-transforms/greek/sentence_translations.json +++ b/2018/derivatives-and-transforms/greek/sentence_translations.json @@ -176,7 +176,7 @@ "end": 209.02 }, { - "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a This is what it means for the derivative of x2 at the input x equals 1 to be 2.", + "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a factor of 2. This is what it means for the derivative of x2 at the input x equals 1 to be 2.", "translatedText": "", "from_community_srt": "και όσο πιο κοντά μεγεθύνετε, τόσο περισσότερο αυτή η τοπική συμπεριφορά μοιάζει όπως με τον πολλαπλασιασμό με συντελεστή 2. Αυτό είναι το τί σημαίνει η παράγωγος του x τετράγωνο στο δεδομένο σημείο x που ισούται με 1, να είναι 2.", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 261.68 }, { - "input": "As you zoom in closer and closer, by 100x or by 1000x, it looks more and more like a And this is what it looks like for the derivative to be 0.", + "input": "As you zoom in closer and closer, by 100x, or by 1000x, it looks more and more like a small neighborhood of points around 0 just gets collapsed into 0 itself. This is what it looks like for the derivative to be 0.", "translatedText": "", "from_community_srt": "Και καθώς μεγεθύνετε όλο και πιο κοντά κατά 100 x ή κατά 1000 x Μοιάζει όλο και περισσότερο σαν μια μικρή γειτονιά σημείων γύρω από το μηδέν να καταρρέει στο ίδιο το μηδέν. Και αυτό είναι το πώς φαίνεται για τη παράγωγο να είναι μηδέν.", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/hebrew/sentence_translations.json b/2018/derivatives-and-transforms/hebrew/sentence_translations.json index e457b504f..16bfc195f 100644 --- a/2018/derivatives-and-transforms/hebrew/sentence_translations.json +++ b/2018/derivatives-and-transforms/hebrew/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out. ", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out. ", "translatedText": "אם תתקרב למקבץ קטן של נקודות מסביב לקלט 1, ותראה היכן הן נוחתות סביב הפלט הרלוונטי, תשים לב שהן נוטות להימתח. ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", "translatedText": "בסביבות קלט 1 הרביעי, אזור קטן נוטה להתכווץ בפקטור של חצי, וכך זה נראה עבור נגזרת קטנה מ-1. ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem. ", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem. ", "translatedText": "אני חושב שהבנתם את הנקודה, הכל טוב ויפה, אבל בואו נראה איך זה מועיל בפתרון בעיה. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", "translatedText": "אז כדי לחבר את הפלט הזה בחזרה לפונקציה, ייתכן שתזוז תחילה אופקית עד שתפגע בקו y שווה ל-x, וזה ייתן לך מיקום שבו ערך ה-x מתאים לערך ה-y הקודם שלך, נכון? ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you? ", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you? ", "translatedText": "באופן אישי, אני חושב שזו דרך מביכה לחשוב על יישום חוזר של פונקציה, לא? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", "translatedText": "אז אני אתחיל כאן על ידי ציור של חבורה של חצים כדי לציין לאן יעברו נקודות הקלט השונות שנדגמו. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", "translatedText": "שם, לנגזרת יש גודל גדול מ-1, כך שנקודות ליד הנקודה הקבועה נדחות ממנה. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you. ", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you. ", "translatedText": "האם אתה רוצה להתייחס לאחיו הקטן של פי כערך חוקי של השבר האינסופי תלוי בך. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/hindi/sentence_translations.json b/2018/derivatives-and-transforms/hindi/sentence_translations.json index 7544c2c8a..effbcc5a0 100644 --- a/2018/derivatives-and-transforms/hindi/sentence_translations.json +++ b/2018/derivatives-and-transforms/hindi/sentence_translations.json @@ -140,7 +140,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out.", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out.", "translatedText": "यदि आप इनपुट 1 के आसपास बिंदुओं के एक छोटे समूह पर ज़ूम इन करते हैं, और देखते हैं कि वे प्रासंगिक आउटपुट के आसपास कहाँ पहुँचते हैं, तो आप देखेंगे कि वे खिंच जाते हैं।", "n_reviews": 0, "start": 192.72, @@ -189,7 +189,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", "translatedText": "इनपुट 1 चौथाई के आसपास, एक छोटा क्षेत्र 1 आधे के कारक से संकुचित हो जाता है, और 1 से छोटे होने वाले व्युत्पन्न के लिए ऐसा ही दिखता है।", "n_reviews": 0, "start": 238.98, @@ -287,7 +287,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem.", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem.", "translatedText": "मुझे लगता है कि आप बात समझ गए हैं, यह सब ठीक है और अच्छा है, लेकिन आइए देखें कि यह किसी समस्या को हल करने में कैसे उपयोगी है।", "n_reviews": 0, "start": 338.46, @@ -455,7 +455,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", "translatedText": "तो उस आउटपुट को फ़ंक्शन में वापस प्लग करने के लिए, आप पहले क्षैतिज रूप से आगे बढ़ सकते हैं जब तक कि आप लाइन y बराबर x तक नहीं पहुंच जाते, और यह आपको एक स्थिति देगा जहां x मान आपके पिछले y मान से मेल खाता है, है ना?", "n_reviews": 0, "start": 554.04, @@ -469,14 +469,14 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you?", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you?", "translatedText": "व्यक्तिगत रूप से, मुझे लगता है कि किसी फ़ंक्शन को बार-बार लागू करने के बारे में सोचने का यह एक अजीब तरीका है, है ना?", "n_reviews": 0, "start": 585.88, "end": 590.78 }, { - "input": "I mean, it makes sense, but you have to pause and think about it to remember which way to draw the lines.", + "input": "I mean, it makes sense, but you kind of have to pause and think about it to remember which way to draw the lines.", "translatedText": "मेरा मतलब है, यह समझ में आता है, लेकिन आपको यह याद रखने के लिए रुकना होगा और सोचना होगा कि रेखाएं किस तरह खींचनी हैं।", "n_reviews": 0, "start": 591.3, @@ -511,7 +511,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", "translatedText": "इसलिए मैं यह इंगित करने के लिए तीरों का एक समूह बनाकर यहां शुरुआत करने जा रहा हूं कि विभिन्न नमूना इनपुट बिंदु कहां जाएंगे।", "n_reviews": 0, "start": 622.28, @@ -602,7 +602,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it.", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it.", "translatedText": "वहाँ पर, व्युत्पन्न का परिमाण 1 से बड़ा होता है, इसलिए निश्चित बिंदु के निकट के बिंदु इससे दूर हो जाते हैं।", "n_reviews": 0, "start": 721.32, @@ -651,7 +651,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you.", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you.", "translatedText": "आप फी के छोटे भाई को अनंत अंश का वैध मान मानना चाहते हैं या नहीं, यह आप पर निर्भर है।", "n_reviews": 0, "start": 776.46, diff --git a/2018/derivatives-and-transforms/hungarian/sentence_translations.json b/2018/derivatives-and-transforms/hungarian/sentence_translations.json index 9015824b2..dfff09464 100644 --- a/2018/derivatives-and-transforms/hungarian/sentence_translations.json +++ b/2018/derivatives-and-transforms/hungarian/sentence_translations.json @@ -176,7 +176,7 @@ "end": 209.02 }, { - "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a This is what it means for the derivative of x2 at the input x equals 1 to be 2.", + "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a factor of 2. This is what it means for the derivative of x2 at the input x equals 1 to be 2.", "translatedText": "Minél közelebb nagyítunk, annál inkább úgy néz ki ez a helyi viselkedés, mintha szoroznánk egy Ez azt jelenti, hogy az x2 deriváltja az x egyenlő 1 bemenetnél 2 lesz.", "model": "DeepL", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 261.68 }, { - "input": "As you zoom in closer and closer, by 100x or by 1000x, it looks more and more like a And this is what it looks like for the derivative to be 0.", + "input": "As you zoom in closer and closer, by 100x, or by 1000x, it looks more and more like a small neighborhood of points around 0 just gets collapsed into 0 itself. This is what it looks like for the derivative to be 0.", "translatedText": "Ahogy egyre közelebb és közelebb zoomolsz, 100x vagy 1000x, egyre inkább úgy néz ki, mint egy És így néz ki, ha a derivált 0.", "model": "DeepL", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/indonesian/sentence_translations.json b/2018/derivatives-and-transforms/indonesian/sentence_translations.json index 0f836b166..e3dab5a6b 100644 --- a/2018/derivatives-and-transforms/indonesian/sentence_translations.json +++ b/2018/derivatives-and-transforms/indonesian/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out.", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out.", "translatedText": "Jika Anda memperbesar sekelompok kecil titik di sekitar masukan 1, dan melihat di mana titik-titik tersebut berada di sekitar keluaran yang relevan, Anda akan melihat bahwa titik-titik tersebut cenderung melebar.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", "translatedText": "Di sekitar input 1 keempat, wilayah kecil cenderung berkontraksi setengah kali lipat, dan seperti itulah turunannya lebih kecil dari 1.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem.", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem.", "translatedText": "Saya pikir Anda mengerti maksudnya, ini semua baik dan bagus, tapi mari kita lihat bagaimana ini berguna dalam memecahkan suatu masalah.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", "translatedText": "Jadi untuk memasukkan kembali output tersebut ke dalam fungsi, Anda mungkin terlebih dahulu bergerak secara horizontal hingga Anda mencapai garis y sama dengan x, dan itu akan memberi Anda posisi di mana nilai x sesuai dengan nilai y sebelumnya, bukan?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you?", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you?", "translatedText": "Secara pribadi, menurut saya ini adalah cara yang canggung untuk berpikir tentang penerapan suatu fungsi berulang kali, bukan?", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 590.78 }, { - "input": "I mean, it makes sense, but you have to pause and think about it to remember which way to draw the lines.", + "input": "I mean, it makes sense, but you kind of have to pause and think about it to remember which way to draw the lines.", "translatedText": "Maksud saya, ini masuk akal, tetapi Anda harus berhenti sejenak dan memikirkannya untuk mengingat cara menggambar garis.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", "translatedText": "Jadi saya akan mulai di sini dengan menggambar sekumpulan panah untuk menunjukkan ke mana arah berbagai titik masukan sampel.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it.", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it.", "translatedText": "Di sana, turunannya mempunyai besaran lebih besar dari 1, sehingga titik-titik yang dekat dengan titik tetap tersebut ditolak menjauhinya.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you.", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you.", "translatedText": "Apakah Anda ingin menganggap adik laki-laki phi sebagai nilai valid pecahan tak terbatas, itu terserah Anda.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/italian/sentence_translations.json b/2018/derivatives-and-transforms/italian/sentence_translations.json index 25d6d81ce..c39c58009 100644 --- a/2018/derivatives-and-transforms/italian/sentence_translations.json +++ b/2018/derivatives-and-transforms/italian/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out.", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out.", "translatedText": "Se ingrandisci un piccolo gruppo di punti attorno all'input 1 e vedi dove si fermano attorno all'output rilevante, noterai che tendono ad allungarsi.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", "translatedText": "Intorno all'input 1 quarto, una piccola regione tende a contrarsi di un fattore pari a 1 metà, ed è così che sembra che una derivata sia inferiore a 1.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem.", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem.", "translatedText": "Penso che tu abbia capito il punto, va tutto bene, ma vediamo come è utile per risolvere un problema.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", "translatedText": "Quindi per ricollegare l'output alla funzione, potresti prima spostarti orizzontalmente finché non raggiungi la linea y uguale a x, e questo ti darà una posizione in cui il valore x corrisponde al tuo precedente valore y, giusto?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you?", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you?", "translatedText": "Personalmente, penso che questo sia un modo scomodo di pensare all'applicazione ripetuta di una funzione, non è vero?", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 590.78 }, { - "input": "I mean, it makes sense, but you have to pause and think about it to remember which way to draw the lines.", + "input": "I mean, it makes sense, but you kind of have to pause and think about it to remember which way to draw the lines.", "translatedText": "Voglio dire, ha senso, ma devi fermarti e pensarci per ricordare in che modo tracciare le linee.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", "translatedText": "Quindi inizierò da qui disegnando un gruppo di frecce per indicare dove andranno i vari punti di input campionati.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it.", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it.", "translatedText": "Laggiù la derivata ha un modulo maggiore di 1, quindi i punti vicini al punto fisso ne vengono respinti.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you.", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you.", "translatedText": "Sta a te decidere se considerare o meno il fratellino di phi un valore valido della frazione infinita.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/japanese/sentence_translations.json b/2018/derivatives-and-transforms/japanese/sentence_translations.json index a1c77237b..eb27d0dd9 100644 --- a/2018/derivatives-and-transforms/japanese/sentence_translations.json +++ b/2018/derivatives-and-transforms/japanese/sentence_translations.json @@ -174,7 +174,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out. ", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out. ", "translatedText": "入力 1 の周りの小さな点のクラスターを拡大して、それらが関連する出力の周りでど こに到達するかを確認すると、それらが引き伸ばされる傾向があることがわかります。", "model": "google_nmt", "from_community_srt": "1に1をマッピングし、2から4を3から9にマッピングするなど もちろん間にあるすべてのポイントも同様です 入力1の周りの点の群れを拡大する場合は、 それらが入力1の周りのどこに行くのかを見ることができます。 この関数だと1はまた１に行くわけですが、 周りの点は引き伸ばされる傾向があることに気付くでしょう。", @@ -235,7 +235,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", "translatedText": "入力の 1/4 付近では、小さな領域が 1/2 に縮小される傾 向があり、導関数が 1 より小さくなるのはこのようになります。", "model": "google_nmt", "from_community_srt": "入力1/4付近では、小さな領域が収縮する傾向があります 具体的には1/2になり、1よりも小さい時の微分係数はこのような形になりそうです。", @@ -356,7 +356,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem. ", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem. ", "translatedText": "これで要点は理解できたと思いますが、これが問 題解決にどのように役立つかを見てみましょう。", "model": "google_nmt", "from_community_srt": "さらにこれが問題の解法として実際にどのように役立つのかを見てみましょう。", @@ -570,7 +570,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", "translatedText": "その出力を関数に戻すには、まず y が x に等し い行に達するまで水平に移動します。そうすると、x の値が前の y の値に対応する位置が得られます。", "model": "google_nmt", "from_community_srt": "その出力を関数に代入し戻すということを考えると、 最初にy=xに行くまで水平方向に移動するでしょう、そしてここは以前のyと等しい値の xの位置ということになりますね？", @@ -588,7 +588,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you? ", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you? ", "translatedText": "個人的には、関数を繰り返し適用する という考え方は厄介だと思います。", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", "translatedText": "そこで、ここでは、さまざまなサンプリングされた入力ポイントが どこに行くのかを示すために、矢印の束を描くことから始めます。", "model": "google_nmt", "n_reviews": 0, @@ -751,7 +751,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", "translatedText": "そこでは導関数の大きさが 1 より大きいため 、固定点に近い点は固定点から遠ざけられます。", "model": "google_nmt", "n_reviews": 0, @@ -811,7 +811,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you. ", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you. ", "translatedText": "ファイの弟を無限分数の有効な値と みなすかどうかはあなた次第です。", "model": "google_nmt", "from_community_srt": "全く姿をあらわさないのかを説明していることになるのです 今、あなたがφの弟を連分数の有効な値と見なしたいかどうかについては・・・ まあ、それは本当にあなた次第です 私たちはただ次のようなことを言ったのです;", diff --git a/2018/derivatives-and-transforms/korean/sentence_translations.json b/2018/derivatives-and-transforms/korean/sentence_translations.json index 625f038a0..4f84e5030 100644 --- a/2018/derivatives-and-transforms/korean/sentence_translations.json +++ b/2018/derivatives-and-transforms/korean/sentence_translations.json @@ -197,7 +197,7 @@ "end": 209.02 }, { - "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a This is what it means for the derivative of x2 at the input x equals 1 to be 2.", + "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a factor of 2. This is what it means for the derivative of x2 at the input x equals 1 to be 2.", "translatedText": "확대하면 확대할수록 이 국부적인 동작이 마치 a를 곱하는 것처럼 보입니다. 입력 x가 1일 때 x2의 도함수가 2가 되는 것이 바로 이런 의미입니다.", "model": "DeepL", "from_community_srt": "더 많이 확대한다면 2배 확대하는 것과 다를 바가 없어집니다. 이것은 x²을 미분했을 때 x에 1을 넣은 값이 2라는 것을 의미합니다.", @@ -269,7 +269,7 @@ "end": 261.68 }, { - "input": "As you zoom in closer and closer, by 100x or by 1000x, it looks more and more like a And this is what it looks like for the derivative to be 0.", + "input": "As you zoom in closer and closer, by 100x, or by 1000x, it looks more and more like a small neighborhood of points around 0 just gets collapsed into 0 itself. This is what it looks like for the derivative to be 0.", "translatedText": "점점 더 가까이, 100배 또는 1000배로 확대하면 점점 더 미분값이 0이 되는 것처럼 보입니다.", "model": "DeepL", "from_community_srt": "100배, 1000배 더 확대해 보면 더 많이 확대할수록 0 주위의 점들이 0쪽으로 모여드는 것처럼 보입니다. 그리고 이건 미분값이 0이 될 때를 나타내죠.", diff --git a/2018/derivatives-and-transforms/marathi/sentence_translations.json b/2018/derivatives-and-transforms/marathi/sentence_translations.json index 65d9ac7e6..7bb4c7098 100644 --- a/2018/derivatives-and-transforms/marathi/sentence_translations.json +++ b/2018/derivatives-and-transforms/marathi/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out. ", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out. ", "translatedText": "जर तुम्ही इनपुट 1 च्या आजूबाजूच्या बिंदूंच्या थोड्या क्लस्टरवर झूम इन केले आणि ते संबंधित आउटपुटच्या आसपास कुठे उतरतात ते पाहिल्यास, तुमच्या लक्षात येईल की ते पसरलेले आहेत. ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", "translatedText": "इनपुट 1 चौथ्या आसपास, एक लहान प्रदेश 1 अर्ध्या घटकाने आकुंचन पावतो, आणि व्युत्पन्न 1 पेक्षा लहान असण्यासारखे दिसते. ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem. ", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem. ", "translatedText": "मला वाटते की तुम्हाला मुद्दा समजला आहे, हे सर्व चांगले आणि चांगले आहे, परंतु समस्या सोडवण्यासाठी हे कसे उपयुक्त आहे ते पाहूया. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", "translatedText": "तर ते आउटपुट पुन्हा फंक्शनमध्ये जोडण्यासाठी, तुम्ही आधी क्षैतिज हलवू शकता जोपर्यंत तुम्ही y बरोबर x ही रेषा मारत नाही, आणि ते तुम्हाला अशी स्थिती देईल जिथे x मूल्य तुमच्या मागील y मूल्याशी संबंधित असेल, बरोबर? ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you? ", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you? ", "translatedText": "वैयक्तिकरित्या, मला असे वाटते की फंक्शन वारंवार लागू करण्याचा विचार करण्याचा हा एक विचित्र मार्ग आहे, नाही का? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", "translatedText": "म्हणून मी येथे बाणांचा एक गुच्छ रेखाटून प्रारंभ करणार आहे जे विविध नमुना इनपुट पॉइंट्स कुठे जातील हे सूचित करण्यासाठी. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", "translatedText": "तेथे, व्युत्पन्नाचे परिमाण 1 पेक्षा मोठे आहे, म्हणून निश्चित बिंदूजवळील बिंदू त्यापासून दूर दूर केले जातात. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you. ", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you. ", "translatedText": "तुम्हाला फिच्या लहान भावाला अनंत अपूर्णांकाचे वैध मूल्य मानायचे की नाही हे तुमच्यावर अवलंबून आहे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/persian/sentence_translations.json b/2018/derivatives-and-transforms/persian/sentence_translations.json index 68a6e6f76..1a1048f1f 100644 --- a/2018/derivatives-and-transforms/persian/sentence_translations.json +++ b/2018/derivatives-and-transforms/persian/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out. ", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out. ", "translatedText": "اگر روی مجموعه کوچکی از نقاط در اطراف ورودی 1 بزرگنمایی کنید و ببینید که آنها در اطراف خروجی مربوطه کجا قرار می گیرند، متوجه خواهید شد که آنها تمایل به کشیده شدن دارند. ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", "translatedText": "در اطراف ورودی 1 چهارم، یک ناحیه کوچک به ضریب 1 نصف منقبض می شود، و این چیزی است که برای یک مشتق کوچکتر از 1 به نظر می رسد. ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem. ", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem. ", "translatedText": "فکر می‌کنم متوجه موضوع شده‌اید، همه اینها خوب و خوب است، اما بیایید ببینیم که چگونه در حل یک مشکل مفید است. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", "translatedText": "اگر بخواهید یک ورودی تصادفی را به این تابع وصل کنید، مقدار y خروجی مربوطه را به شما می گوید، درست است؟ بنابراین برای وصل کردن آن خروجی به تابع، ممکن است ابتدا به صورت افقی حرکت کنید تا زمانی که خط y برابر با x شود، و این به شما موقعیتی می دهد که در آن مقدار x با مقدار y قبلی شما مطابقت دارد، درست است؟ پس از آنجا، می‌توانید به صورت عمودی حرکت کنید تا ببینید این مقدار x جدید چه خروجی دارد، و سپس تکرار می‌کنید، به صورت افقی به خط y برابر با x حرکت می‌کنید تا نقطه‌ای را پیدا کنید که مقدار x آن همان خروجی است که دریافت کردید. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you? ", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you? ", "translatedText": "من شخصاً فکر می‌کنم این روشی ناخوشایند برای فکر کردن به اعمال مکرر یک تابع است، اینطور نیست؟ منظورم این است که منطقی است، اما باید مکث کرد و به آن فکر کرد تا به یاد بیاوری که خطوط را به کدام سمت بکشی. ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", "translatedText": "بنابراین من می‌خواهم از اینجا با رسم یک دسته فلش شروع کنم تا مشخص کنم نقاط ورودی نمونه‌برداری‌شده مختلف کجا خواهند رفت. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", "translatedText": "در آنجا، مشتق دارای قدر بزرگتر از 1 است، بنابراین نقاط نزدیک به نقطه ثابت از آن دفع می شوند. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you. ", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you. ", "translatedText": "اینکه آیا می خواهید برادر کوچک فی را یک مقدار معتبر از کسر نامتناهی در نظر بگیرید یا نه، به شما بستگی دارد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/polish/sentence_translations.json b/2018/derivatives-and-transforms/polish/sentence_translations.json index 71501dcc3..60c0d74eb 100644 --- a/2018/derivatives-and-transforms/polish/sentence_translations.json +++ b/2018/derivatives-and-transforms/polish/sentence_translations.json @@ -175,7 +175,7 @@ "end": 209.02 }, { - "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a This is what it means for the derivative of x2 at the input x equals 1 to be 2.", + "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a factor of 2. This is what it means for the derivative of x2 at the input x equals 1 to be 2.", "translatedText": "", "from_community_srt": "zbliżając jeszcze bardziej zauważyłbyś, lokalnie to wygląda jak mnożenie długości przedziału przez dwa. To właśnie oznacza fakt, że pochodna funkcji x^2 w 1 wynosi 2.", "n_reviews": 0, @@ -239,7 +239,7 @@ "end": 261.68 }, { - "input": "As you zoom in closer and closer, by 100x or by 1000x, it looks more and more like a And this is what it looks like for the derivative to be 0.", + "input": "As you zoom in closer and closer, by 100x, or by 1000x, it looks more and more like a small neighborhood of points around 0 just gets collapsed into 0 itself. This is what it looks like for the derivative to be 0.", "translatedText": "", "from_community_srt": "np. 100-krotne lub 1000-krotne Wygląda to jakby bardzo małe otoczenia zera, po prosu zapadły się w ten punkt. I to właśnie oznacza mieć pochodna równą zero,", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/portuguese/sentence_translations.json b/2018/derivatives-and-transforms/portuguese/sentence_translations.json index f161d74a8..0a2b447b4 100644 --- a/2018/derivatives-and-transforms/portuguese/sentence_translations.json +++ b/2018/derivatives-and-transforms/portuguese/sentence_translations.json @@ -197,7 +197,7 @@ "end": 209.02 }, { - "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a This is what it means for the derivative of x2 at the input x equals 1 to be 2.", + "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a factor of 2. This is what it means for the derivative of x2 at the input x equals 1 to be 2.", "translatedText": "Quanto mais você aumenta o zoom, mais esse comportamento local se parece com a multiplicação por a. Isso é o que significa que a derivada de x2 na entrada x igual a 1 é 2.", "model": "google_nmt", "from_community_srt": "mais esse comportamento local parece estar multiplicando por um fator de 2. É isso que significa a derivada de x² no dado de entrada x igual a 1 ser 2.", @@ -269,7 +269,7 @@ "end": 261.68 }, { - "input": "As you zoom in closer and closer, by 100x or by 1000x, it looks more and more like a And this is what it looks like for the derivative to be 0.", + "input": "As you zoom in closer and closer, by 100x, or by 1000x, it looks more and more like a small neighborhood of points around 0 just gets collapsed into 0 itself. This is what it looks like for the derivative to be 0.", "translatedText": "À medida que você aumenta o zoom cada vez mais, em 100x ou em 1000x, parece cada vez mais com um E é assim que a derivada é 0.", "model": "google_nmt", "from_community_srt": "E conforme você dá um zoom cada vez mais próximo, 100x ou 1000x, parece cada vez mais que uma pequena vizinhança de pontos ao redor do zero simplesmente colapsam no próprio zero. E é isso que significa uma derivada ser zero.", diff --git a/2018/derivatives-and-transforms/russian/sentence_translations.json b/2018/derivatives-and-transforms/russian/sentence_translations.json index 8c7c21b9b..40b5e152f 100644 --- a/2018/derivatives-and-transforms/russian/sentence_translations.json +++ b/2018/derivatives-and-transforms/russian/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out.", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out.", "translatedText": "Если вы увеличите небольшой кластер точек вокруг входа 1 и увидите, где они располагаются вокруг соответствующего выхода, вы заметите, что они имеют тенденцию растягиваться.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", "translatedText": "Около входной 1-й четверти небольшая область имеет тенденцию сжиматься в половину раза, и именно так выглядит производная, меньшая 1.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem.", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem.", "translatedText": "Я думаю, вы поняли, это все хорошо, но давайте посмотрим, насколько это полезно для решения проблемы.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", "translatedText": "Итак, чтобы подключить этот вывод обратно к функции, вы можете сначала двигаться по горизонтали, пока не дойдете до линии y, равной x, и это даст вам позицию, в которой значение x соответствует вашему предыдущему значению y, верно?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you?", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you?", "translatedText": "Лично я считаю, что это неуклюжий способ многократного применения функции, не так ли?", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 590.78 }, { - "input": "I mean, it makes sense, but you have to pause and think about it to remember which way to draw the lines.", + "input": "I mean, it makes sense, but you kind of have to pause and think about it to remember which way to draw the lines.", "translatedText": "Я имею в виду, что это имеет смысл, но вам нужно остановиться и подумать, чтобы вспомнить, в каком направлении проводить линии.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", "translatedText": "Итак, я собираюсь начать с рисования стрелок, указывающих, куда пойдут различные входные точки выборки.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it.", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it.", "translatedText": "Там производная имеет величину больше 1, поэтому точки вблизи фиксированной точки отталкиваются от нее.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you.", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you.", "translatedText": "Хотите ли вы считать младшего брата Фи действительным значением бесконечной дроби, зависит от вас.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/spanish/sentence_translations.json b/2018/derivatives-and-transforms/spanish/sentence_translations.json index ef23390c6..a517c67c5 100644 --- a/2018/derivatives-and-transforms/spanish/sentence_translations.json +++ b/2018/derivatives-and-transforms/spanish/sentence_translations.json @@ -176,7 +176,7 @@ "end": 209.02 }, { - "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a This is what it means for the derivative of x2 at the input x equals 1 to be 2.", + "input": "The closer you zoom in, the more this local behavior looks just like multiplying by a factor of 2. This is what it means for the derivative of x2 at the input x equals 1 to be 2.", "translatedText": "Cuanto más te acercas, más se parece este comportamiento local a multiplicar por a. Esto es lo que significa que la derivada de x2 en la entrada x es igual a 1 sea 2.", "from_community_srt": "y cuanto más nos acerquemos, más se parece este comportamiento local a multiplicar simplemente por 2. Esto es lo que significa que la derivada de x al cuadrado, en el punto 1, vale 2.", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 261.68 }, { - "input": "As you zoom in closer and closer, by 100x or by 1000x, it looks more and more like a And this is what it looks like for the derivative to be 0.", + "input": "As you zoom in closer and closer, by 100x, or by 1000x, it looks more and more like a small neighborhood of points around 0 just gets collapsed into 0 itself. This is what it looks like for the derivative to be 0.", "translatedText": "A medida que te acercas más y más, 100x o 1000x, se parece cada vez más a Y esto es lo que parece que la derivada sea 0.", "from_community_srt": "Y si hacemos más zoom, aumentando 100 o 1000 veces, se parece cada vez más a un pequeño entorno de puntos alrededor del cero que colapsan hacia el propio cero. y así es como se visualiza que la derivada sea cero.", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/tamil/sentence_translations.json b/2018/derivatives-and-transforms/tamil/sentence_translations.json index 97739e59d..dd198a54d 100644 --- a/2018/derivatives-and-transforms/tamil/sentence_translations.json +++ b/2018/derivatives-and-transforms/tamil/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out.", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out.", "translatedText": "உள்ளீடு 1ஐச் சுற்றியுள்ள புள்ளிகளின் சிறிய தொகுப்பை நீங்கள் பெரிதாக்கினால், அவை தொடர்புடைய வெளியீட்டைச் சுற்றி எங்கு இறங்குகின்றன என்பதைப் பார்த்தால், அவை நீட்டிக்கப்படுவதை நீங்கள் கவனிப்பீர்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", "translatedText": "உள்ளீடு 1 நான்காவது சுற்றி, ஒரு சிறிய பகுதி 1 பாதி மடங்கு மூலம் சுருங்க முனைகிறது, மேலும் ஒரு வழித்தோன்றல் 1 ஐ விட சிறியதாக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem.", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem.", "translatedText": "உங்களுக்குப் புரியும் என்று நினைக்கிறேன், இது எல்லாம் நன்றாக இருக்கிறது, ஆனால் சிக்கலைத் தீர்ப்பதில் இது எவ்வாறு பயனுள்ளதாக இருக்கும் என்பதைப் பார்ப்போம்.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", "translatedText": "எனவே அந்த வெளியீட்டை மீண்டும் செயல்பாட்டில் செருக, நீங்கள் முதலில் கிடைமட்டமாக நகர்த்தலாம், நீங்கள் y x க்கு சமம் என்ற வரியை அடிக்கும் வரை, அது உங்களுக்கு ஒரு நிலையை கொடுக்கப் போகிறது, அங்கு x மதிப்பு உங்கள் முந்தைய y மதிப்புடன் ஒத்துப்போகிறது, இல்லையா?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you?", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you?", "translatedText": "தனிப்பட்ட முறையில், ஒரு செயல்பாட்டை மீண்டும் மீண்டும் பயன்படுத்துவதைப் பற்றி சிந்திக்க இது ஒரு மோசமான வழி என்று நான் நினைக்கிறேன், இல்லையா?", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 590.78 }, { - "input": "I mean, it makes sense, but you have to pause and think about it to remember which way to draw the lines.", + "input": "I mean, it makes sense, but you kind of have to pause and think about it to remember which way to draw the lines.", "translatedText": "அதாவது, இது அர்த்தமுள்ளதாக இருக்கிறது, ஆனால் கோடுகளை எந்த வழியில் வரைய வேண்டும் என்பதை நினைவில் வைத்துக் கொள்ள நீங்கள் இடைநிறுத்தப்பட்டு அதைப் பற்றி சிந்திக்க வேண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", "translatedText": "எனவே பல்வேறு மாதிரி உள்ளீட்டு புள்ளிகள் எங்கு செல்லும் என்பதைக் குறிக்க அம்புகளை வரைவதன் மூலம் நான் இங்கே தொடங்கப் போகிறேன்.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it.", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it.", "translatedText": "அங்கு, வழித்தோன்றல் 1 ஐ விட பெரிய அளவைக் கொண்டுள்ளது, எனவே நிலையான புள்ளிக்கு அருகிலுள்ள புள்ளிகள் அதிலிருந்து விலக்கப்படுகின்றன.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you.", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you.", "translatedText": "ஃபையின் சிறிய சகோதரரை எல்லையற்ற பின்னத்தின் சரியான மதிப்பாக நீங்கள் கருத வேண்டுமா இல்லையா என்பது உங்களுடையது.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/telugu/sentence_translations.json b/2018/derivatives-and-transforms/telugu/sentence_translations.json index 28c0f2325..c575a97e7 100644 --- a/2018/derivatives-and-transforms/telugu/sentence_translations.json +++ b/2018/derivatives-and-transforms/telugu/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out.", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out.", "translatedText": "మీరు ఇన్‌పుట్ 1 చుట్టూ ఉన్న పాయింట్‌ల యొక్క చిన్న క్లస్టర్‌పై జూమ్ చేసి, సంబంధిత అవుట్‌పుట్ చుట్టూ అవి ఎక్కడ ల్యాండ్ అవుతాయో చూస్తే, అవి విస్తరించి ఉన్నట్లు మీరు గమనించవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", "translatedText": "ఇన్‌పుట్ 1 నాల్గవ వంతు చుట్టూ, ఒక చిన్న ప్రాంతం 1 సగం కారకంతో సంకోచించబడుతుంది మరియు ఉత్పన్నం 1 కంటే చిన్నదిగా ఉన్నట్లు కనిపిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem.", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem.", "translatedText": "మీరు పాయింట్‌ని అర్థం చేసుకున్నారని నేను అనుకుంటున్నాను, ఇదంతా బాగానే ఉంది, అయితే సమస్యను పరిష్కరించడంలో ఇది ఎలా ఉపయోగపడుతుందో చూద్దాం.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", "translatedText": "కాబట్టి ఆ అవుట్‌పుట్‌ను తిరిగి ఫంక్షన్‌లోకి ప్లగ్ చేయడానికి, మీరు y xకి సమానం అనే పంక్తిని కొట్టే వరకు మీరు మొదట అడ్డంగా కదలవచ్చు మరియు అది మీకు x విలువ మీ మునుపటి y విలువకు అనుగుణంగా ఉండే స్థితిని ఇస్తుంది, సరియైనదా?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you?", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you?", "translatedText": "వ్యక్తిగతంగా, ఒక ఫంక్షన్‌ని పదే పదే వర్తింపజేయడం గురించి ఆలోచించడానికి ఇది ఇబ్బందికరమైన మార్గం అని నేను భావిస్తున్నాను, కాదా?", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 590.78 }, { - "input": "I mean, it makes sense, but you have to pause and think about it to remember which way to draw the lines.", + "input": "I mean, it makes sense, but you kind of have to pause and think about it to remember which way to draw the lines.", "translatedText": "నా ఉద్దేశ్యం, ఇది అర్ధమే, కానీ పంక్తులు ఏ విధంగా గీస్తామో గుర్తుంచుకోవడానికి మీరు పాజ్ చేసి దాని గురించి ఆలోచించాలి.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", "translatedText": "కాబట్టి నేను వివిధ నమూనా ఇన్‌పుట్ పాయింట్‌లు ఎక్కడికి వెళ్తాయో సూచించడానికి బాణాల సమూహాన్ని గీయడం ద్వారా ఇక్కడ ప్రారంభించబోతున్నాను.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it.", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it.", "translatedText": "అక్కడ, ఉత్పన్నం 1 కంటే పెద్ద పరిమాణంలో ఉంటుంది, కాబట్టి స్థిర బిందువు దగ్గర ఉన్న పాయింట్లు దాని నుండి దూరంగా తిప్పబడతాయి.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you.", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you.", "translatedText": "మీరు ఫై యొక్క చిన్న సోదరుడిని అనంతమైన భిన్నం యొక్క చెల్లుబాటు అయ్యే విలువగా పరిగణించాలనుకుంటున్నారా లేదా అనేది మీ ఇష్టం.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/thai/sentence_translations.json b/2018/derivatives-and-transforms/thai/sentence_translations.json index 75d743457..4bc9b5cf3 100644 --- a/2018/derivatives-and-transforms/thai/sentence_translations.json +++ b/2018/derivatives-and-transforms/thai/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out. ", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem. ", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you? ", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", "translatedText": "จากตรงนั้น คุณสามารถเคลื่อนที่ในแนวตั้งเพื่อดูว่าค่า x ใหม่นี้มีเอาต์พุตเท่าใด แล้วคุณทำซ้ำ คุณเลื่อนในแนวนอนไปที่เส้น y เท่ากับ x เพื่อหาจุดที่มีค่า x เท่ากับผลลัพธ์ที่คุณเพิ่งได้ จากนั้นคุณเลื่อนในแนวตั้งเพื่อใช้ฟังก์ชันอีกครั้ง โดยส่วนตัวแล้ว ฉันคิดว่านี่เป็นวิธีที่น่าอึดอัดใจในการคิดถึงการใช้ฟังก์ชันซ้ำๆ ใช่ไหม ฉันหมายความว่ามันสมเหตุสมผล แต่คุณต้องหยุดและคิดเกี่ยวกับมันเพื่อจำไว้ว่าจะลากเส้นไปทางไหน และถ้าคุณต้องการ คุณสามารถคิดได้ว่าเงื่อนไขใดที่ทำให้กระบวนการใยแมงมุมนี้แคบลงในจุดคงที่ แทนที่จะแพร่กระจายออกไป ที่จริงแล้ว ให้หยุดตอนนี้เลยแล้วลองคิดทบทวนเป็นแบบฝึกหัด มันเกี่ยวข้องกับความลาดชัน หรือถ้าคุณอยากข้ามแบบฝึกหัดไปทำอะไรที่ผมคิดว่าให้ความเข้าใจที่น่าพอใจมากขึ้น ลองคิดว่าฟังก์ชันนี้ทำหน้าที่เป็นการแปลงอย่างไร ฉันจะเริ่มต้นที่นี่ด้วยการวาดลูกศรหลายๆ อันเพื่อระบุว่าจุดอินพุตตัวอย่างต่างๆ จะไปอยู่ที่ใด และหมายเหตุข้างเคียง คุณไม่คิดว่านี่จะให้รูปแบบที่โผล่ออกมาอย่างประณีตใช่ไหม ฉันไม่ได้คาดหวังสิ่งนี้ แต่มันเจ๋งมากที่ได้เห็นมันปรากฏขึ้นตอนสร้างภาพเคลื่อนไหว การกระทำของ 1 หารด้วย x ทำให้เกิดวงกลมสวยงามขึ้นมา แล้วเราก็แค่เลื่อนไปทีละอัน อย่างไรก็ตาม ผมอยากให้คุณลองนึกถึงความหมายของการใช้ฟังก์ชันบางอย่างซ้ำๆ เช่น 1 บวก 1 ส่วน x ในบริบทนี้ หลังจากปล่อยให้มันแมปอินพุตทั้งหมดกับเอาท์พุต คุณสามารถพิจารณาสิ่งเหล่านั้นเป็นอินพุตใหม่ จากนั้นใช้กระบวนการเดิมอีกครั้ง ซ้ำอีกครั้ง และทำกี่ครั้งก็ได้ตามที่คุณต้องการ สังเกตว่าในการสร้างภาพเคลื่อนไหวด้วยจุดสองสามจุดที่เป็นตัวแทนของจุดตัวอย่าง มันไม่ต้องใช้การวนซ้ำมากนัก ก่อนที่จุดเหล่านั้นทั้งหมดจะรวมตัวกันประมาณ 1 618. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you. ", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/turkish/sentence_translations.json b/2018/derivatives-and-transforms/turkish/sentence_translations.json index 67da2a9d8..ad7a685e6 100644 --- a/2018/derivatives-and-transforms/turkish/sentence_translations.json +++ b/2018/derivatives-and-transforms/turkish/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out. ", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out. ", "translatedText": "Giriş 1'in etrafındaki küçük bir nokta kümesini yakınlaştırırsanız ve bunların ilgili çıktının etrafında nereye geldiklerini görürseniz, bunların uzama eğiliminde olduğunu fark edeceksiniz. ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", "translatedText": "Girdinin 1/4'ü civarında, küçük bir bölge 1 buçuk kat daralma eğilimindedir ve bir türevin 1'den küçük olması böyle görünür. ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem. ", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem. ", "translatedText": "Sanırım meseleyi anladınız, her şey yolunda ve güzel, ama bunun bir problemi çözmede ne kadar faydalı olduğunu görelim. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", "translatedText": "Yani bu çıktıyı tekrar fonksiyona bağlamak için, önce y eşittir x çizgisine ulaşıncaya kadar yatay olarak hareket edebilirsiniz ve bu size x değerinin önceki y değerinize karşılık geldiği bir konum verecektir, değil mi? ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you? ", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you? ", "translatedText": "Kişisel olarak bunun bir işlevi tekrar tekrar uygulamayı düşünmenin tuhaf bir yolu olduğunu düşünüyorum, öyle değil mi? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", "translatedText": "Burada çeşitli örneklenen giriş noktalarının nereye gideceğini gösteren bir grup ok çizerek başlayacağım. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", "translatedText": "Orada, türevin büyüklüğü 1'den büyük olduğundan, sabit noktaya yakın noktalar ondan uzağa doğru itilir. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you. ", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you. ", "translatedText": "Phi'nin küçük kardeşini sonsuz kesirin geçerli bir değeri olarak kabul etmek isteyip istemediğiniz size kalmış. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/ukrainian/sentence_translations.json b/2018/derivatives-and-transforms/ukrainian/sentence_translations.json index 41a979fd3..89fe24403 100644 --- a/2018/derivatives-and-transforms/ukrainian/sentence_translations.json +++ b/2018/derivatives-and-transforms/ukrainian/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out. ", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out. ", "translatedText": "Якщо ви збільшите масштаб невеликого скупчення точок навколо входу 1 і побачите, де вони приземляються навколо відповідного виходу, ви помітите, що вони, як правило, розтягуються. ", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", "translatedText": "Навколо вхідного значення 1/4 невелика область має тенденцію скорочуватися на коефіцієнт 1 вдвічі, і саме так виглядає похідна менша за 1. ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem. ", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem. ", "translatedText": "Я думаю, ви зрозуміли суть, це все добре, але давайте подивимося, як це корисно для вирішення проблеми. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", "translatedText": "Отже, щоб підключити цей вихід назад до функції, ви можете спочатку рухатися горизонтально, доки не потрапите на рядок y дорівнює x, і це дасть вам позицію, де значення x відповідає вашому попередньому значенню y, чи не так? ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you? ", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you? ", "translatedText": "Особисто я вважаю, що це незручний спосіб багаторазового застосування функції, чи не так? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", "translatedText": "Тож я збираюся розпочати тут із малювання купи стрілок, щоб вказати, куди підуть різні вибіркові точки введення. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", "translatedText": "Там похідна має величину більшу за 1, тому точки поблизу фіксованої точки відштовхуються від неї. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you. ", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you. ", "translatedText": "Чи хочете ви вважати молодшого брата phi дійсним значенням нескінченного дробу, вирішувати вам. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/urdu/sentence_translations.json b/2018/derivatives-and-transforms/urdu/sentence_translations.json index fbf098593..f13972d92 100644 --- a/2018/derivatives-and-transforms/urdu/sentence_translations.json +++ b/2018/derivatives-and-transforms/urdu/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out. ", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out. ", "translatedText": "اگر آپ ان پٹ 1 کے ارد گرد پوائنٹس کے ایک چھوٹے سے جھرمٹ پر زوم ان کرتے ہیں، اور دیکھتے ہیں کہ وہ متعلقہ آؤٹ پٹ کے ارد گرد کہاں اترتے ہیں، تو آپ دیکھیں گے کہ وہ پھیل جاتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. ", "translatedText": "ان پٹ 1 چوتھے کے ارد گرد، ایک چھوٹا سا خطہ 1 نصف کے فیکٹر سے کنٹریکٹ ہو جاتا ہے، اور یہی ایسا لگتا ہے کہ مشتق کے لیے 1 سے چھوٹا ہونا۔", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem. ", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem. ", "translatedText": "میرے خیال میں آپ کو بات سمجھ میں آئی، یہ سب ٹھیک اور اچھا ہے، لیکن آئیے دیکھتے ہیں کہ یہ کسی مسئلے کو حل کرنے میں کس طرح مفید ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right? ", "translatedText": "اگر آپ اس فنکشن میں کچھ رینڈم ان پٹ لگانا چاہتے ہیں، تو y قدر آپ کو متعلقہ آؤٹ پٹ بتاتی ہے، ٹھیک ہے؟ تو اس آؤٹ پٹ کو فنکشن میں واپس پلگ کرنے کے لیے، آپ پہلے افقی طور پر حرکت کر سکتے ہیں جب تک کہ آپ لائن y کو x کے برابر نہیں مارتے، اور یہ آپ کو ایک ایسی پوزیشن دے گا جہاں x کی قدر آپ کی پچھلی y قدر سے مطابقت رکھتی ہے، ٹھیک ہے؟ تو پھر وہاں سے، آپ یہ دیکھنے کے لیے عمودی طور پر حرکت کر سکتے ہیں کہ اس نئی x ویلیو کا کیا آؤٹ پٹ ہے، اور پھر آپ دہراتے ہیں، آپ افقی طور پر لائن y کے برابر x کی طرف بڑھتے ہیں تاکہ ایک نقطہ تلاش کیا جا سکے جس کی x کی قیمت وہی ہے جو آپ کو ابھی ملی ہے، اور پھر آپ فنکشن کو دوبارہ لاگو کرنے کے لیے عمودی طور پر منتقل ہوتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you? ", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you? ", "translatedText": "ذاتی طور پر، مجھے لگتا ہے کہ کسی فنکشن کو بار بار لگانے کے بارے میں سوچنے کا یہ ایک عجیب طریقہ ہے، کیا آپ نہیں؟ میرا مطلب ہے، یہ سمجھ میں آتا ہے، لیکن آپ کو رک کر اس کے بارے میں سوچنا ہوگا تاکہ یاد رکھیں کہ لکیریں کس طرح کھینچنی ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go. ", "translatedText": "تو میں یہاں تیروں کا ایک گچھا کھینچ کر شروع کرنے جا رہا ہوں تاکہ اس بات کی نشاندہی کی جا سکے کہ مختلف نمونے والے ان پٹ پوائنٹس کہاں جائیں گے۔", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it. ", "translatedText": "تو اب بتائیے آپ کے خیال میں فائی کے چھوٹے بھائی کے پڑوس میں کیا ہوتا ہے؟ وہاں، مشتق کی شدت 1 سے بڑی ہوتی ہے، لہذا مقررہ نقطہ کے قریب پوائنٹس اس سے دور ہو جاتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you. ", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you. ", "translatedText": "آپ phi کے چھوٹے بھائی کو لامحدود کسر کی درست قدر سمجھنا چاہتے ہیں یا نہیں یہ آپ پر منحصر ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/derivatives-and-transforms/vietnamese/sentence_translations.json b/2018/derivatives-and-transforms/vietnamese/sentence_translations.json index 5f3f0dc12..fc4537b24 100644 --- a/2018/derivatives-and-transforms/vietnamese/sentence_translations.json +++ b/2018/derivatives-and-transforms/vietnamese/sentence_translations.json @@ -160,7 +160,7 @@ "end": 189.22 }, { - "input": "If you zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, you'd notice that they tend to get stretched out.", + "input": "If you were to zoom in on a little cluster of points around the input 1, and see where they land around the relevant output, which for this function also happens to be 1, you'd notice that they tend to get stretched out.", "translatedText": "Nếu bạn phóng to một cụm điểm nhỏ xung quanh đầu vào 1 và xem vị trí của chúng xung quanh đầu ra có liên quan, bạn sẽ nhận thấy rằng chúng có xu hướng bị kéo dài ra.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 237.44 }, { - "input": "Around the input 1 fourth, a small region tends to get contracted by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", + "input": "Around the input 1 fourth, a small region tends to get contracted specifically by a factor of 1 half, and that's what it looks like for a derivative to be smaller than 1.", "translatedText": "Xung quanh đầu vào 1 phần tư, một vùng nhỏ có xu hướng bị thu hẹp lại theo hệ số 1 một nửa và đó là hiện tượng đạo hàm nhỏ hơn 1.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 335.6 }, { - "input": "I think you get the point, this is all well and good, but let's see how this is useful in solving a problem.", + "input": "And I think you get the point, this is all well and good, but let's see how this is actually useful in solving a problem.", "translatedText": "Tôi nghĩ bạn hiểu rồi, điều này cũng tốt thôi, nhưng hãy xem điều này hữu ích như thế nào trong việc giải quyết vấn đề.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 554.04 }, { - "input": "So to plug that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", + "input": "So to think about plugging that output back into the function, you might first move horizontally until you hit the line y equals x, and that's going to give you a position where the x value corresponds to your previous y value, right?", "translatedText": "Vì vậy, để cắm kết quả đầu ra đó trở lại hàm, trước tiên bạn có thể di chuyển theo chiều ngang cho đến khi chạm vào dòng y bằng x, và điều đó sẽ cho bạn một vị trí trong đó giá trị x tương ứng với giá trị y trước đó của bạn, phải không?", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 584.76 }, { - "input": "Personally, I think this is an awkward way to think about repeatedly applying a function, don't you?", + "input": "Now personally, I think this is kind of an awkward way to think about repeatedly applying a function, don't you?", "translatedText": "Cá nhân tôi nghĩ rằng đây là một cách khó xử khi nghĩ đến việc áp dụng nhiều lần một hàm, phải không?", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 590.78 }, { - "input": "I mean, it makes sense, but you have to pause and think about it to remember which way to draw the lines.", + "input": "I mean, it makes sense, but you kind of have to pause and think about it to remember which way to draw the lines.", "translatedText": "Ý tôi là, nó có ý nghĩa, nhưng bạn phải tạm dừng và suy nghĩ về nó để nhớ cách vẽ các đường thẳng.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 619.62 }, { - "input": "So I'm going to start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", + "input": "So I'm going to go ahead and start here by drawing a bunch of arrows to indicate where the various sampled input points will go.", "translatedText": "Vì vậy, tôi sẽ bắt đầu ở đây bằng cách vẽ một loạt các mũi tên để chỉ ra các điểm đầu vào được lấy mẫu khác nhau sẽ đi đến đâu.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 718.62 }, { - "input": "Over there, the derivative has a magnitude larger than 1, so points near the fixed point are repelled away from it.", + "input": "Over there, the derivative actually has a magnitude larger than 1, so points near the fixed point are repelled away from it.", "translatedText": "Ở đó, đạo hàm có độ lớn lớn hơn 1 nên các điểm ở gần điểm cố định sẽ bị đẩy ra xa đạo hàm.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 775.8 }, { - "input": "Whether or not you want to consider phi's little brother a valid value of the infinite fraction is up to you.", + "input": "As to whether or not you want to consider phi's little brother a valid value of the infinite fraction, well that's really up to you.", "translatedText": "Việc bạn có muốn coi em trai của Phi là một giá trị hợp lệ của phân số vô hạn hay không là tùy thuộc vào bạn.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/arabic/sentence_translations.json b/2018/divergence-and-curl/arabic/sentence_translations.json index 259f95917..9e1da6d96 100644 --- a/2018/divergence-and-curl/arabic/sentence_translations.json +++ b/2018/divergence-and-curl/arabic/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector. ", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector. ", "translatedText": "لدي الكثير من المحتوى من وقتي في أكاديمية خان يصف هذا بمزيد من التفصيل، ولكن لغرضنا الرئيسي، سأشير فقط إلى الشكل ثنائي الأبعاد للالتفاف، والذي يربط كل نقطة في الفضاء ثنائي الأبعاد برقم واحد، بدلا من ناقلات جديدة. ", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field. ", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field. ", "translatedText": "هذا أيضًا له تفسير مفاده أن أحاديات القطب المغناطيسي، وهو الشيء الذي يعمل تمامًا مثل الطرف الشمالي أو الجنوبي للمغناطيس في عزلة، غير موجودة، ولا يوجد شيء مماثل للشحنات الموجبة والسالبة في المجال الكهربائي. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow. ", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow, ", "translatedText": "هذه فكرة ثلاثية الأبعاد بحتة، وهي خارج نطاق تركيزنا الرئيسي هنا قليلًا، لكن النقطة المهمة هي أن الاختلاف والالتفاف ينشأان في سياقات لا علاقة لها بالتدفق. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves. ", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves. ", "translatedText": "والعكس من هاتين المعادلتين الأخيرتين هو ما يؤدي إلى ظهور موجات ضوئية. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another. ", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another. ", "translatedText": "لنأخذ مثالًا كلاسيكيًا يدرسه غالبًا طلاب المعادلات التفاضلية، لنفترض أنك تريد تتبع أحجام مجموعات نوعين مختلفين، حيث يكون أحدهما مفترسًا للآخر. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery. ", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery. ", "translatedText": "لأكون صادقًا تمامًا معك، بالنسبة لشيء كهذا، غالبًا ما ترغب في جلب أدوات ذات صلة تتجاوز مجرد التباعد والالتفاف، ولكن الإطار الذهني الذي تمارسه مع هاتين الفكرتين يجعلك تمضي قدمًا جيدًا في دراسة إعدادات مثل هذه مع ما شابه ذلك قطع من الآلات الرياضية. ", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar. ", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar. ", "translatedText": "أحب أن أعتقد أنني سأحاول دائمًا تعظيم قيمة التجربة مهما حدث، ولكن في هذا الصدد، أود أيضًا أن أعتقد أنني أستطيع دائمًا الاستيقاظ مبكرًا ومقاومة تناول الكثير من السكر. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/bengali/sentence_translations.json b/2018/divergence-and-curl/bengali/sentence_translations.json index 6a43c8075..d2f1ddf7d 100644 --- a/2018/divergence-and-curl/bengali/sentence_translations.json +++ b/2018/divergence-and-curl/bengali/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector. ", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector. ", "translatedText": "খান একাডেমিতে আমার সময় থেকে আমার কাছে প্রচুর বিষয়বস্তু রয়েছে যা আরও বিস্তারিতভাবে বর্ণনা করে, কিন্তু আমাদের মূল উদ্দেশ্যের জন্য, আমি শুধু কার্ল-এর দ্বি-মাত্রিক বৈকল্পিকের কথা উল্লেখ করব, যা 2D স্থানের প্রতিটি বিন্দুকে একটি একক সংখ্যার সাথে সংযুক্ত করে, একটি নতুন ভেক্টরের পরিবর্তে।", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field. ", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field. ", "translatedText": "এটিরও ব্যাখ্যা রয়েছে যে চৌম্বক মনোপোল, এমন কিছু যা বিচ্ছিন্নভাবে চুম্বকের উত্তর বা দক্ষিণ প্রান্তের মতো কাজ করে, বিদ্যমান নেই, বৈদ্যুতিক ক্ষেত্রে ধনাত্মক এবং ঋণাত্মক চার্জের সাথে সাদৃশ্যপূর্ণ কিছুই নেই।", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow. ", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow, ", "translatedText": "এটি একটি সম্পূর্ণরূপে ত্রিমাত্রিক ধারণা, এবং এখানে আমাদের মূল ফোকাসের একটু বাইরে, কিন্তু বিন্দু হল যে প্রবাহের সাথে সম্পর্কহীন প্রেক্ষাপটে ভিন্নতা এবং কার্ল উদ্ভূত হয়।", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves. ", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves. ", "translatedText": "এবং এই শেষ দুটি সমীকরণের সামনে এবং পিছনে যা আলোক তরঙ্গের জন্ম দেয়।", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another. ", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another. ", "translatedText": "একটি ক্লাসিক উদাহরণ নিতে যে ডিফারেনশিয়াল সমীকরণের ছাত্ররা প্রায়শই অধ্যয়ন করে, ধরা যাক আপনি দুটি ভিন্ন প্রজাতির জনসংখ্যার আকার ট্র্যাক করতে চেয়েছিলেন, যেখানে একটি অন্যটির শিকারী।", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery. ", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery. ", "translatedText": "আপনার সাথে পুরোপুরি সৎ হতে, এইরকম কিছুর জন্য আপনি প্রায়শই কেবল ভিন্নতা এবং কার্ল এর বাইরে সম্পর্কিত সরঞ্জামগুলি আনতে চান, তবে এই দুটি ধারণার সাথে অনুশীলন করার ফলে আপনি একই রকম সেটআপগুলি অধ্যয়ন করতে পারেন গাণিতিক যন্ত্রপাতি টুকরা. ", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar. ", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar. ", "translatedText": "আমি ভাবতে পছন্দ করি যে আমি সবসময় অভিজ্ঞতার মূল্যকে সর্বোচ্চ করার চেষ্টা করব যাই হোক না কেন, তবে সেই বিষয়ে আমি ভাবতেও চাই যে আমি ধারাবাহিকভাবে তাড়াতাড়ি ঘুম থেকে উঠতে পারি এবং খুব বেশি চিনি খাওয়া প্রতিরোধ করতে পারি।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/chinese/sentence_translations.json b/2018/divergence-and-curl/chinese/sentence_translations.json index c7d4a9527..2a912fa1c 100644 --- a/2018/divergence-and-curl/chinese/sentence_translations.json +++ b/2018/divergence-and-curl/chinese/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "我在可汗学院的时间里有很多内容更详细地描述了这一点，但出 于我们的主要目的，我将仅参考curl的二维变体，它将二 维空间中的每个点与单个数字相关联，而不是一个新的向量。", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "这也有这样的解释：磁单极子（其作用就像 孤立的磁体的北端或南端）不存在，没有任 何类似于电场中的正电荷和负电荷的东西。", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "这是一个纯粹的三维想法，有点超出我们的主要焦点， 但要点是发散和旋度出现在与流动无关的上下文中。", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "最后两个方程的反复产生了光波。", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "举一个微分方程学生经常研究的经典例子， 假设您想要跟踪两个不同物种的种群规模 ，其中一个物种是另一个物种的捕食者。", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "老实说，对于这样的事情，您通常会想要引 入除散度和旋度之外的相关工具，但是练习 这两个想法的心态可以让您很好地使用类似 的方法来研究这样的设置数学机器的部件。", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "我喜欢认为无论如何我都会尝试最大化体 验的价值，但就此而言，我也喜欢认为我 可以始终如一地早起并抵制吃太多糖。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/french/sentence_translations.json b/2018/divergence-and-curl/french/sentence_translations.json index 8b358ec21..73634c440 100644 --- a/2018/divergence-and-curl/french/sentence_translations.json +++ b/2018/divergence-and-curl/french/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "J'ai beaucoup de contenu de mon passage à la Khan Academy décrivant cela plus en détail, mais pour notre objectif principal, je ferai simplement référence à la variante bidimensionnelle de curl, qui associe chaque point de l'espace 2D à un seul nombre, plutôt qu'un nouveau vecteur.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "Cela donne également l'interprétation que les monopôles magnétiques, quelque chose qui agit comme une extrémité nord ou sud d'un aimant isolé, n'existent pas, il n'y a rien d'analogue aux charges positives et négatives dans un champ électrique.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "Il s’agit d’une idée purement tridimensionnelle, et un peu en dehors de notre objectif principal ici, mais le fait est que la divergence et la boucle surviennent dans des contextes qui n’ont aucun rapport avec le flux.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "Et le va-et-vient de ces deux dernières équations est ce qui donne naissance aux ondes lumineuses.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "Pour prendre un exemple classique que les étudiants en équations différentielles étudient souvent, disons que vous vouliez suivre la taille des populations de deux espèces différentes, l'une étant le prédateur de l'autre.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "Pour être parfaitement honnête avec vous, pour quelque chose comme celui-ci, vous voudriez souvent utiliser des outils connexes au-delà de la simple divergence et du curl, mais l'état d'esprit que vous amène la pratique de ces deux idées se répercute bien sur l'étude de configurations comme celle-ci avec des configurations similaires. pièces de machinerie mathématique.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "J'aime penser que j'essaierai toujours de maximiser la valeur de l'expérience quoi qu'il arrive, mais d'ailleurs, j'aime aussi penser que je peux toujours me réveiller tôt et résister à manger trop de sucre.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/german/sentence_translations.json b/2018/divergence-and-curl/german/sentence_translations.json index 6dd528dd7..d998cbc2f 100644 --- a/2018/divergence-and-curl/german/sentence_translations.json +++ b/2018/divergence-and-curl/german/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "Aus meiner Zeit bei der Khan Academy habe ich noch viele Inhalte, die dies genauer beschreiben, aber für unser Hauptziel beziehe ich mich nur auf die zweidimensionale Variante von curl, bei der jeder Punkt im 2D-Raum mit einer einzelnen Zahl und nicht mit einem neuen Vektor verknüpft wird.", "model": "DeepL", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "Das bedeutet auch, dass es keine magnetischen Monopole gibt, also etwas, das sich wie ein Nord- oder Südende eines Magneten verhält, es gibt nichts, was positiven und negativen Ladungen in einem elektrischen Feld entspricht.", "model": "DeepL", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "Das ist eine rein dreidimensionale Idee und liegt ein wenig außerhalb unseres Hauptaugenmerks, aber der Punkt ist, dass Divergenz und Krümmung in Zusammenhängen auftreten, die nichts mit dem Fluss zu tun haben.", "model": "DeepL", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "Das Hin und Her zwischen diesen beiden Gleichungen ist die Ursache für die Lichtwellen.", "model": "DeepL", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "Um ein klassisches Beispiel zu nehmen, mit dem sich Studenten von Differentialgleichungen oft beschäftigen: Nehmen wir an, du möchtest die Populationsgrößen von zwei verschiedenen Arten verfolgen, von denen die eine ein Räuber der anderen ist.", "model": "DeepL", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "Um ehrlich zu sein, würdest du für so etwas oft andere Werkzeuge als nur Divergenz und Krümmung verwenden wollen, aber die Denkweise, die du durch die Übung mit diesen beiden Ideen erlangst, überträgt sich gut auf die Untersuchung von Versuchen wie diesem mit ähnlichen mathematischen Instrumenten.", "model": "DeepL", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "Ich denke gerne, dass ich immer versuchen werde, den Wert der Erfahrung zu maximieren, egal was passiert, aber ich denke auch gerne, dass ich immer früh aufstehen und nicht zu viel Zucker essen kann.", "model": "DeepL", "n_reviews": 0, diff --git a/2018/divergence-and-curl/hebrew/sentence_translations.json b/2018/divergence-and-curl/hebrew/sentence_translations.json index 4a8f750c0..d5d70ca38 100644 --- a/2018/divergence-and-curl/hebrew/sentence_translations.json +++ b/2018/divergence-and-curl/hebrew/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "יש לי המון תוכן מהתקופה שלי באקדמיית חאן שמתאר את זה ביתר פירוט, אבל למטרה העיקרית שלנו, אני רק אתייחס לגרסה הדו-ממדית של תלתל, שמשייכת כל נקודה במרחב הדו-ממדי למספר בודד, במקום וקטור חדש.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "יש לזה גם את הפרשנות שמונופולים מגנטיים, משהו שמתנהג בדיוק כמו קצה צפוני או דרומי של מגנט בבידוד, לא קיימים, אין שום דבר מקביל למטענים חיוביים ושליליים בשדה חשמלי.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "זהו רעיון תלת מימדי גרידא, וקצת מחוץ להתמקדות העיקרית שלנו כאן, אבל הנקודה היא שדיברגנציה ותלתלים נוצרים בהקשרים שאינם קשורים לזרימה.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "וההלוך ושוב משתי המשוואות האחרונות הללו הוא מה שמוליד גלי אור.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "כדי לקחת דוגמה קלאסית שתלמידי משוואות דיפרנציאליות לומדים לעתים קרובות, נניח שרצית לעקוב אחר גודל האוכלוסייה של שני מינים שונים, כאשר אחד הוא טורף של אחר.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "למען האמת איתך, בשביל משהו כזה, לעתים קרובות היית רוצה להביא כלים קשורים מעבר לסתירות ותלתלים בלבד, אבל המסגרת התודעה שתרגול עם שני הרעיונות האלה מביא אותך עובר היטב ללימוד מערכים כמו זה עם הגדרות דומות חלקים של מכונות מתמטיות.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "אני אוהב לחשוב שתמיד אנסה למקסם את הערך של החוויה לא משנה מה, אבל לצורך העניין אני גם אוהב לחשוב שאני יכול להתעורר באופן עקבי מוקדם ולהתנגד לאכול יותר מדי סוכר.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/hindi/sentence_translations.json b/2018/divergence-and-curl/hindi/sentence_translations.json index 8e87e5d8a..2ddd522bc 100644 --- a/2018/divergence-and-curl/hindi/sentence_translations.json +++ b/2018/divergence-and-curl/hindi/sentence_translations.json @@ -280,7 +280,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "मेरे पास खान अकादमी में मेरे समय की बहुत सारी सामग्री है जो इसका अधिक विस्तार से वर्णन करती है, लेकिन हमारे मुख्य उद्देश्य के लिए, मैं केवल कर्ल के द्वि-आयामी संस्करण का उल्लेख कर रहा हूं, जो 2डी स्पेस में प्रत्येक बिंदु को एक ही संख्या के साथ जोड़ता है, एक नए वेक्टर के बजाय।", "n_reviews": 0, "start": 331.52, @@ -350,7 +350,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "इसकी यह भी व्याख्या है कि चुंबकीय मोनोपोल, कुछ ऐसा जो अलगाव में चुंबक के उत्तरी या दक्षिणी छोर की तरह कार्य करता है, अस्तित्व में नहीं है, विद्युत क्षेत्र में सकारात्मक और नकारात्मक चार्ज के अनुरूप कुछ भी नहीं है।", "n_reviews": 0, "start": 418.92, @@ -364,14 +364,14 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "यह पूरी तरह से त्रि-आयामी विचार है, और यहां हमारे मुख्य फोकस से थोड़ा बाहर है, लेकिन मुद्दा यह है कि विचलन और कर्ल उन संदर्भों में उत्पन्न होते हैं जो प्रवाह से असंबंधित हैं।", "n_reviews": 0, "start": 441.32, "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "और इन अंतिम दो समीकरणों का आगे-पीछे होना ही प्रकाश तरंगों को जन्म देता है।", "n_reviews": 0, "start": 451.58, @@ -385,7 +385,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "एक उत्कृष्ट उदाहरण लेने के लिए जिसका अध्ययन विभेदक समीकरणों के छात्र अक्सर करते हैं, मान लीजिए कि आप दो अलग-अलग प्रजातियों की जनसंख्या के आकार को ट्रैक करना चाहते हैं, जहां एक दूसरे का शिकारी है।", "n_reviews": 0, "start": 464.0, @@ -490,7 +490,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "आपके साथ पूरी तरह से ईमानदार होने के लिए, इस तरह की किसी चीज़ के लिए आप अक्सर केवल विचलन और कर्ल से परे संबंधित टूल लाना चाहेंगे, लेकिन इन दो विचारों के साथ अभ्यास करने वाला मन का ढांचा आपको इस तरह के समान सेटअपों का अध्ययन करने में अच्छी तरह से सक्षम बनाता है। गणितीय मशीनरी के टुकड़े.", "n_reviews": 0, "start": 601.84, @@ -728,7 +728,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "मुझे यह सोचना पसंद है कि मैं हमेशा अनुभव के मूल्य को अधिकतम करने की कोशिश करूंगा, चाहे कुछ भी हो, लेकिन उस मामले में मुझे यह भी सोचना पसंद है कि मैं लगातार जल्दी उठ सकता हूं और बहुत अधिक चीनी खाने से बच सकता हूं।", "n_reviews": 0, "start": 906.44, diff --git a/2018/divergence-and-curl/indonesian/sentence_translations.json b/2018/divergence-and-curl/indonesian/sentence_translations.json index 1ad8811bd..4a4a906ca 100644 --- a/2018/divergence-and-curl/indonesian/sentence_translations.json +++ b/2018/divergence-and-curl/indonesian/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "Saya punya banyak konten dari waktu saya di Khan Academy yang menjelaskan hal ini secara lebih rinci, tapi untuk tujuan utama kita, saya hanya akan mengacu pada varian curl dua dimensi, yang mengaitkan setiap titik dalam ruang 2D dengan satu angka, daripada vektor baru.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "Hal ini juga memiliki penafsiran bahwa monopole magnet, sesuatu yang bertindak seperti ujung utara atau selatan magnet yang terisolasi, tidak ada, tidak ada analogi muatan positif dan negatif dalam medan listrik.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "Ini murni gagasan tiga dimensi, dan sedikit di luar fokus utama kita di sini, namun intinya adalah divergensi dan curl muncul dalam konteks yang tidak berhubungan dengan aliran.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "Dan bolak-balik dari dua persamaan terakhir inilah yang menimbulkan gelombang cahaya.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "Untuk mengambil contoh klasik yang sering dipelajari oleh para pelajar persamaan diferensial, katakanlah Anda ingin melacak ukuran populasi dua spesies berbeda, yang mana salah satu spesies merupakan pemangsa spesies lainnya.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "Sejujurnya, untuk hal seperti ini Anda sering kali ingin membawa alat terkait lebih dari sekadar divergensi dan curl, namun kerangka berpikir yang dibawa oleh latihan dengan dua ide ini membawa Anda dengan baik untuk mempelajari pengaturan seperti ini dengan yang serupa potongan mesin matematika.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "Saya pikir saya akan selalu berusaha memaksimalkan nilai pengalaman, apa pun yang terjadi, tetapi dalam hal ini saya juga berpikir saya bisa bangun pagi secara konsisten dan menolak makan terlalu banyak gula.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/italian/sentence_translations.json b/2018/divergence-and-curl/italian/sentence_translations.json index 9cc3ce61e..975f36332 100644 --- a/2018/divergence-and-curl/italian/sentence_translations.json +++ b/2018/divergence-and-curl/italian/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "Ho molti contenuti del mio periodo alla Khan Academy che descrivono questo in modo più dettagliato, ma per il nostro scopo principale mi riferirò solo alla variante bidimensionale del ricciolo, che associa ogni punto nello spazio 2D a un singolo numero, piuttosto che un nuovo vettore.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "Questo ha anche l'interpretazione che i monopoli magnetici, qualcosa che agisce proprio come l'estremità nord o sud di un magnete isolato, non esistono, non c'è niente di analogo alle cariche positive e negative in un campo elettrico.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "Questa è un’idea puramente tridimensionale, e un po’ al di fuori del nostro obiettivo principale qui, ma il punto è che la divergenza e l’arricciatura sorgono in contesti che non sono correlati al flusso.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "E l'andata e ritorno di queste ultime due equazioni è ciò che dà origine alle onde luminose.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "Per fare un classico esempio che studiano spesso gli studenti di equazioni differenziali, supponiamo che tu voglia monitorare le dimensioni della popolazione di due specie diverse, di cui una è predatrice di un'altra.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "Per essere del tutto onesto con te, per qualcosa del genere vorresti spesso utilizzare strumenti correlati oltre la semplice divergenza e arricciatura, ma lo stato d'animo che ti porta a praticare con queste due idee si ripercuote bene sullo studio di configurazioni come questa con simili pezzi di macchine matematiche.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "Mi piace pensare che cercherò sempre di massimizzare il valore dell'esperienza, qualunque cosa accada, ma del resto mi piace anche pensare di potermi svegliare costantemente presto e resistere a mangiare troppo zucchero.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/japanese/sentence_translations.json b/2018/divergence-and-curl/japanese/sentence_translations.json index 4a0ad31f5..e4eefda98 100644 --- a/2018/divergence-and-curl/japanese/sentence_translations.json +++ b/2018/divergence-and-curl/japanese/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "これについて詳しく説明しているカーン アカデミー時代のコンテンツがたくさんありま すが、ここでは主な目的として、2D 空間の各点を 1 つの数値に関連付ける、c url の 2 次元バリアントについてのみ説明します。 新しいベクトルではなく。", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "これには、磁気単極子、つまり磁石の北端または南端のよう に単独で機能するものは存在せず、電界における正および負 の電荷に類似したものは存在しないという解釈もあります。", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "これは純粋に 3 次元の考え方であり、ここでの主な焦点からは少し外れますが 、重要なのは、発散とカールは流れとは無関係な文脈で発生するということです。", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "そして、これら最後の 2 つの方程式を行ったり来たりすることで、光の波が生じます。", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "微分方程式の学生がよく研究する古典的な例と して、一方が他方を捕食する 2 つの異な る種の個体数サイズを追跡したいとします。", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "正直に言うと、このようなことを行う場合、発散とカール以外の 関連ツールを導入したくなることがよくありますが、これら 2 つのアイデアを練習するという心構えは、同様のセットアッ プを研究する際にもよく引き継がれます。 数学的機械の一部。", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "私は、何があっても常にその経験の価値を最大化しようと考え たいと思っていますが、それと同時に、常に早起きして砂糖の 過剰摂取を防ぐことができると考えたいとも思っています。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/korean/sentence_translations.json b/2018/divergence-and-curl/korean/sentence_translations.json index e1a81f4f3..e3c78840a 100644 --- a/2018/divergence-and-curl/korean/sentence_translations.json +++ b/2018/divergence-and-curl/korean/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "칸아카데미에서 이에 대해 자세히 설명하는 내용이 많이 있지만, 여기서는 2D 공간의 각 점을 단일 숫자와 연결하는 컬의 2차원 변형에 대해서만 언급하겠습니다.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "이것은 또한 고립된 자석의 북쪽 또는 남쪽 끝처럼 작용하는 자기 단극이 존재하지 않으며 전기장에서 양전하와 음전하와 유사한 것이 없다는 해석도 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "이것은 순전히 3차원적인 아이디어이고 여기에서 우리의 주요 초점에서 약간 벗어났습니다. 그러나 요점은 흐름과 관련 없는 맥락에서 발산과 컬이 발생한다는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "그리고 이 마지막 두 방정식의 앞뒤가 광파를 발생시키는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "미분방정식을 공부하는 학생들이 자주 연구하는 전형적인 예를 들기 위해, 하나가 다른 종의 포식자인 서로 다른 두 종의 개체군 크기를 추적하고 싶다고 가정해 보겠습니다.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "솔직히 말해서, 이와 같은 작업의 경우 발산 및 컬 외에도 관련 도구를 가져오고 싶을 때가 많지만, 이 두 가지 아이디어를 연습하는 마음의 틀은 유사한 설정을 연구하는 데에도 도움이 됩니다. 수학 기계 조각.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "나는 무슨 일이 있어도 항상 경험의 가치를 극대화하려고 노력할 것이라고 생각하고 싶지만, 그 문제에 관해서는 지속적으로 일찍 일어나 설탕을 너무 많이 먹는 것을 거부할 수 있다고 생각하고 싶습니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/marathi/sentence_translations.json b/2018/divergence-and-curl/marathi/sentence_translations.json index 44d3006c4..4ef5fddea 100644 --- a/2018/divergence-and-curl/marathi/sentence_translations.json +++ b/2018/divergence-and-curl/marathi/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "माझ्याकडे खान अकादमीमधील माझ्या काळातील भरपूर सामग्री आहे ज्याचे अधिक तपशीलवार वर्णन केले आहे, परंतु आमच्या मुख्य हेतूसाठी, मी फक्त कर्लच्या द्विमितीय प्रकाराचा संदर्भ देत आहे, जे 2D स्पेसमधील प्रत्येक बिंदूला एकाच संख्येसह संबद्ध करते, नवीन वेक्टर ऐवजी.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "याचा असाही अर्थ आहे की चुंबकीय मोनोपोल, जे चुंबकाच्या उत्तर किंवा दक्षिण टोकासारखे कार्य करतात, ते अस्तित्वात नसतात, विद्युत क्षेत्रामध्ये सकारात्मक आणि नकारात्मक शुल्कासारखे काहीही नसते.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "ही पूर्णपणे त्रिमितीय कल्पना आहे आणि येथे आपल्या मुख्य फोकसच्या थोडे बाहेर आहे, परंतु मुद्दा असा आहे की प्रवाहाशी संबंधित नसलेल्या संदर्भांमध्ये विचलन आणि कर्ल उद्भवतात.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "आणि या शेवटच्या दोन समीकरणांमधुन पुढे आणि मागे हेच प्रकाश लहरींना जन्म देते.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "विभेदक समीकरणांचे विद्यार्थी सहसा अभ्यास करतात असे उत्कृष्ट उदाहरण घेण्यासाठी, समजा तुम्हाला दोन भिन्न प्रजातींच्या लोकसंख्येचा मागोवा घ्यायचा होता, जिथे एक दुसर्‍याचा शिकारी आहे.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "तुमच्याशी अगदी प्रामाणिकपणे सांगायचे तर, यासारख्या गोष्टीसाठी तुम्हाला अनेकदा फक्त विचलन आणि कर्ल यापलीकडे संबंधित साधने आणायची आहेत, परंतु या दोन कल्पनांसह सराव करणारी मनाची चौकट तुम्हाला यासारख्या सेटअप्सचा अभ्यास करण्यासाठी चांगल्या प्रकारे पार पाडते. गणिती यंत्रांचे तुकडे.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "मला असे वाटायला आवडते की मी अनुभवाचे मूल्य काहीही असले तरी जास्तीत जास्त वाढवण्याचा प्रयत्न करेन, परंतु त्या बाबतीत मला हे विचार करायलाही आवडते की मी सातत्याने लवकर उठू शकतो आणि जास्त साखर खाण्याचा प्रतिकार करू शकतो.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/persian/sentence_translations.json b/2018/divergence-and-curl/persian/sentence_translations.json index 954e7e137..59427a0e5 100644 --- a/2018/divergence-and-curl/persian/sentence_translations.json +++ b/2018/divergence-and-curl/persian/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector. ", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector. ", "translatedText": "من مطالب زیادی از زمان حضورم در آکادمی خان دارم که این را با جزئیات بیشتر توصیف می کند، اما برای هدف اصلی ما، فقط به نوع دوبعدی کرل اشاره می کنم که هر نقطه در فضای دوبعدی را با یک عدد مرتبط می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field. ", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field. ", "translatedText": "این همچنین این تفسیر را دارد که تک قطبی های مغناطیسی، چیزی که دقیقاً مانند انتهای شمالی یا جنوبی یک آهنربا به صورت مجزا عمل می کند، وجود ندارد، هیچ چیزی مشابه بارهای مثبت و منفی در یک میدان الکتریکی وجود ندارد. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow. ", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow, ", "translatedText": "این یک ایده کاملاً سه بعدی است و کمی خارج از تمرکز اصلی ما در اینجا است، اما نکته اینجاست که واگرایی و پیچ خوردگی در زمینه‌هایی به وجود می‌آید که به جریان ارتباطی ندارند. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves. ", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves. ", "translatedText": "و عقب و جلوی این دو معادله آخر همان چیزی است که امواج نور را به وجود می آورد. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another. ", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another. ", "translatedText": "برای مثال کلاسیکی که دانش‌آموزان معادلات دیفرانسیل اغلب مطالعه می‌کنند، فرض کنید می‌خواستید اندازه جمعیت دو گونه مختلف را ردیابی کنید، جایی که یکی شکارچی دیگری است. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery. ", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery. ", "translatedText": "آیا اندازه‌های جمعیت به سمت یک جفت اعداد خاص همگرا می‌شوند، یا مقادیری وجود دارند که از آنها فاصله دارند؟ آیا الگوهای چرخه ای وجود دارد و آیا آن چرخه ها پایدار هستند یا ناپایدار؟ برای اینکه کاملاً با شما صادق باشم، برای چنین چیزی اغلب می خواهید ابزارهای مرتبط را فراتر از صرفاً واگرایی و پیچیدن وارد کنید، اما چارچوب ذهنی که تمرین با این دو ایده شما را به خوبی به مطالعه تنظیمات مانند این با موارد مشابه می رساند. ", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar. ", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar. ", "translatedText": "من دوست دارم فکر کنم که همیشه سعی می کنم ارزش تجربه را بدون توجه به هر اتفاقی به حداکثر برسانم، اما برای این موضوع همچنین دوست دارم فکر کنم که می توانم دائماً زود از خواب بیدار شوم و در مقابل خوردن قند زیاد مقاومت کنم. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/russian/sentence_translations.json b/2018/divergence-and-curl/russian/sentence_translations.json index 7a3c9a0e7..19e0579fb 100644 --- a/2018/divergence-and-curl/russian/sentence_translations.json +++ b/2018/divergence-and-curl/russian/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "За время работы в Академии Хана у меня есть множество материалов, описывающих это более подробно, но для нашей основной цели я буду просто сослаться на двумерный вариант завитка, который связывает каждую точку в двухмерном пространстве с одним числом. а не новый вектор.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "Это также интерпретирует, что магнитных монополей, которые действуют точно так же, как северный или южный конец магнита по отдельности, не существует, нет ничего аналогичного положительным и отрицательным зарядам в электрическом поле.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "Это чисто трехмерная идея, которая немного выходит за рамки нашего основного внимания, но суть в том, что дивергенция и закручивание возникают в контекстах, не связанных с потоком.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "И обратное и обратное из этих двух последних уравнений – это то, что порождает световые волны.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "Возьмем классический пример, который часто изучают изучающие дифференциальные уравнения. Предположим, вы хотите отследить размеры популяций двух разных видов, где один является хищником другого.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "Честно говоря, для чего-то подобного вам часто хочется использовать связанные инструменты, помимо дивергенции и скручивания, но образ мыслей, который приводит вас к практике с этими двумя идеями, хорошо переносится и на изучение подобных установок с похожими установками. части математической машины.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "Мне нравится думать, что я всегда буду стараться максимизировать ценность полученного опыта, несмотря ни на что, но в этом отношении мне также нравится думать, что я могу постоянно просыпаться рано и не есть слишком много сахара.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/tamil/sentence_translations.json b/2018/divergence-and-curl/tamil/sentence_translations.json index 7a21df90d..fd1ac77de 100644 --- a/2018/divergence-and-curl/tamil/sentence_translations.json +++ b/2018/divergence-and-curl/tamil/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "கான் அகாடமியில் நான் இருந்த காலத்திலிருந்து இதை இன்னும் விரிவாக விவரிக்கும் உள்ளடக்கம் என்னிடம் நிறைய உள்ளது, ஆனால் எங்கள் முக்கிய நோக்கத்திற்காக, 2D இடத்தில் உள்ள ஒவ்வொரு புள்ளியையும் ஒரே எண்ணுடன் இணைக்கும் இரு பரிமாண மாறுபாட்டைக் குறிப்பிடுகிறேன். புதிய திசையன் அல்ல.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "காந்த மோனோபோல்கள், ஒரு காந்தத்தின் வடக்கு அல்லது தெற்கு முனையில் தனிமையில் செயல்படும் ஒன்று, இல்லை, மின்புலத்தில் நேர்மறை மற்றும் எதிர்மறை கட்டணங்களுக்கு நிகரான எதுவும் இல்லை என்ற விளக்கமும் இதில் உள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "இது முற்றிலும் முப்பரிமாண யோசனையாகும், மேலும் இங்கு நாம் முக்கிய கவனம் செலுத்துவதற்கு சற்று அப்பாற்பட்டது, ஆனால் பாயிண்ட்டுக்கு தொடர்பில்லாத சூழல்களில் வேறுபாடு மற்றும் சுருட்டை எழுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "இந்த கடைசி இரண்டு சமன்பாடுகளிலிருந்து முன்னும் பின்னுமாக ஒளி அலைகளை உருவாக்குகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "வேறுபட்ட சமன்பாடுகளின் மாணவர்கள் அடிக்கடி படிக்கும் ஒரு சிறந்த உதாரணத்தை எடுக்க, நீங்கள் இரண்டு வெவ்வேறு இனங்களின் மக்கள்தொகை அளவைக் கண்காணிக்க விரும்புகிறீர்கள் என்று வைத்துக்கொள்வோம், அங்கு ஒன்று மற்றொன்றை வேட்டையாடும்.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "உங்களுடன் முற்றிலும் நேர்மையாக இருக்க, இது போன்றவற்றுக்கு நீங்கள் அடிக்கடி வேறுபட்ட மற்றும் சுருட்டைத் தாண்டி தொடர்புடைய கருவிகளைக் கொண்டு வர விரும்புகிறீர்கள், ஆனால் இந்த இரண்டு யோசனைகளையும் பயிற்சி செய்யும் மனப்போக்கு உங்களை இது போன்ற அமைப்புகளைப் படிப்பதில் சிறப்பாகச் செல்கிறது. கணித இயந்திரங்களின் துண்டுகள்.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "நான் எப்போதும் அனுபவத்தின் மதிப்பை அதிகரிக்க முயற்சிப்பேன் என்று நினைக்க விரும்புகிறேன், ஆனால் அந்த விஷயத்தில் நான் தொடர்ந்து சீக்கிரம் எழுந்து அதிக சர்க்கரை சாப்பிடுவதை எதிர்க்க முடியும் என்று நினைக்க விரும்புகிறேன்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/telugu/sentence_translations.json b/2018/divergence-and-curl/telugu/sentence_translations.json index 58b0b7c81..3664bf2cd 100644 --- a/2018/divergence-and-curl/telugu/sentence_translations.json +++ b/2018/divergence-and-curl/telugu/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "నేను ఖాన్ అకాడమీలో దీన్ని మరింత వివరంగా వివరించే సమయం నుండి నాకు పుష్కలంగా కంటెంట్ ఉంది, కానీ మా ముఖ్య ఉద్దేశ్యం కోసం, నేను 2D స్పేస్‌లోని ప్రతి పాయింట్‌ను ఒకే సంఖ్యతో అనుబంధించే కర్ల్ యొక్క రెండు-డైమెన్షనల్ వేరియంట్‌ని సూచిస్తున్నాను, కొత్త వెక్టర్ కాకుండా.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "ఇది మాగ్నెటిక్ మోనోపోల్స్, అయస్కాంతం యొక్క ఉత్తరం లేదా దక్షిణం చివరగా ఏకాంతంగా పని చేస్తుంది, ఉనికిలో లేదు, విద్యుత్ క్షేత్రంలో సానుకూల మరియు ప్రతికూల చార్జీలకు సారూప్యంగా ఏమీ ఉండదు.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "ఇది పూర్తిగా త్రిమితీయ ఆలోచన, మరియు ఇక్కడ మా ప్రధాన దృష్టికి కొద్దిగా వెలుపల ఉంది, అయితే విషయం ఏమిటంటే, ప్రవాహానికి సంబంధం లేని సందర్భాలలో విభేదాలు మరియు కర్ల్ తలెత్తుతాయి.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "మరియు ఈ చివరి రెండు సమీకరణాల నుండి ముందుకు వెనుకకు కాంతి తరంగాలు ఏర్పడతాయి.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "అవకలన సమీకరణాల విద్యార్థులు తరచుగా అధ్యయనం చేసే క్లాసిక్ ఉదాహరణను తీసుకోవడానికి, మీరు రెండు వేర్వేరు జాతుల జనాభా పరిమాణాలను ట్రాక్ చేయాలనుకుంటున్నారని అనుకుందాం, ఇక్కడ ఒకటి మరొకటి వేటాడేది.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "మీతో సంపూర్ణంగా నిజాయితీగా ఉండాలంటే, ఇలాంటి వాటి కోసం మీరు తరచుగా వైవిధ్యం మరియు వంకరగా కాకుండా సంబంధిత సాధనాలను తీసుకురావాలని కోరుకుంటారు, అయితే ఈ రెండు ఆలోచనలతో సాధన చేసే మనస్సు మీకు ఇలాంటి సెటప్‌లను బాగా అధ్యయనం చేస్తుంది. గణిత యంత్రాల ముక్కలు.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "నేను ఎల్లప్పుడూ అనుభవం యొక్క విలువను పెంచుకోవడానికి ప్రయత్నిస్తానని అనుకుంటున్నాను, కానీ దాని కోసం నేను స్థిరంగా త్వరగా మేల్కొంటాను మరియు ఎక్కువ చక్కెరను తినకుండా నిరోధించగలనని అనుకుంటున్నాను.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/thai/sentence_translations.json b/2018/divergence-and-curl/thai/sentence_translations.json index ee6663888..edfcd5cf9 100644 --- a/2018/divergence-and-curl/thai/sentence_translations.json +++ b/2018/divergence-and-curl/thai/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector. ", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field. ", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field. ", "translatedText": "บริเวณที่การหมุนตามเข็มนาฬิกาจะบอกว่ามีการโค้งงอเป็นบวก และบริเวณที่หมุนทวนเข็มนาฬิกาจะมีการโค้งงอเป็นลบ และไม่จำเป็นว่าเวกเตอร์ทั้งหมดรอบๆ อินพุทจะชี้ทวนเข็มนาฬิกา หรือเวกเตอร์ทั้งหมดชี้ตามเข็มนาฬิกา ตัวอย่างเช่น จุดภายในขอบเขตเช่นนี้ ก็จะมีการโค้งงอที่ไม่เป็นศูนย์ เนื่องจากการไหลจะช้าที่ด้านล่าง แต่ขึ้นไปด้านบนอย่างรวดเร็ว ส่งผลให้เกิดอิทธิพลสุทธิตามเข็มนาฬิกา และจริงๆ แล้ว การโค้งงอที่เหมาะสมที่แท้จริงนั้นเป็นแนวคิดสามมิติ โดยที่คุณเชื่อมโยงแต่ละจุดในอวกาศ 3 มิติกับเวกเตอร์ใหม่ เพื่อแสดงลักษณะการหมุนรอบจุดนั้น ตามกฎมือขวาที่แน่นอน ฉันมีเนื้อหามากมายจากสมัยที่ Khan Academy ที่อธิบายรายละเอียดมากกว่านี้ แต่สำหรับจุดประสงค์หลักของเรา ฉันจะอ้างถึงรูปแบบสองมิติของ curl ซึ่งเชื่อมโยงแต่ละจุดในปริภูมิ 2 มิติด้วยตัวเลขตัวเดียว แทนที่จะเป็นเวกเตอร์ใหม่ อย่างที่ผมบอกไป แม้ว่าสัญชาตญาณเหล่านี้จะถูกกำหนดไว้ในบริบทของการไหลของของไหล แต่แนวคิดทั้งสองนี้มีความสำคัญสำหรับสนามเวกเตอร์ประเภทอื่น ตัวอย่างที่สำคัญมากประการหนึ่งคือการอธิบายไฟฟ้าและแม่เหล็กด้วยสมการพิเศษสี่สมการ พวกนี้เรียกว่าสมการของแมกซ์เวลล์ และเขียนด้วยภาษาของไดเวอร์เจนซ์และโค้งงอ ตัวอย่างเช่น สิ่งที่สำคัญที่สุดนี้คือกฎของเกาส์ ที่ระบุว่าการเบี่ยงเบนของสนามไฟฟ้า ณ จุดที่กำหนดจะเป็นสัดส่วนกับความหนาแน่นประจุที่จุดนั้น เมื่อแกะสัญชาตญาณออกมา คุณอาจจินตนาการว่าบริเวณที่มีประจุบวกทำหน้าที่เหมือนแหล่งกำเนิดของของเหลวที่จินตนาการไว้ และบริเวณที่มีประจุลบเป็นแหล่งกักเก็บของเหลวนั้น และทั่วทั้งพื้นที่ที่ไม่มีประจุ ของเหลวจะไหลอย่างอัดแน่นไม่ได้ เช่นเดียวกับน้ำ แน่นอนว่าไม่มีของไหลไฟฟ้าตามตัวอักษร แต่มันมีประโยชน์มากและเป็นวิธีอ่านสมการแบบนี้ได้ค่อนข้างดี ในทำนองเดียวกัน สมการที่สำคัญอีกสมการหนึ่งคือความต่างของสนามแม่เหล็กเป็นศูนย์ทุกแห่ง คุณสามารถเข้าใจได้ว่าหากสนามแม่เหล็กแทนการไหลของของไหล ของไหลนั้นจะไม่สามารถอัดตัวได้ โดยไม่มีแหล่งกำเนิดและไม่มีแหล่งกักเก็บ นอกจากนี้ยังตีความได้ว่าไม่มีขั้วแม่เหล็กขั้วเดียว ซึ่งทำหน้าที่เหมือนปลายด้านเหนือหรือทิศใต้ของแม่เหล็กที่แยกออกจากกัน ไม่มีอยู่จริง ไม่มีอะไรเทียบได้กับประจุบวกและประจุลบในสนามไฟฟ้า ในทำนองเดียวกัน สมการสองสมการสุดท้ายบอกเราว่าการเปลี่ยนแปลงของช่องใดช่องหนึ่งนั้นขึ้นอยู่กับความโค้งของช่องอื่น นี่เป็นแนวคิดสามมิติล้วนๆ และอยู่นอกเหนือจุดสนใจหลักของเราเล็กน้อย แต่ประเด็นก็คือความแตกต่างและความโค้งงอเกิดขึ้นในบริบทที่ไม่เกี่ยวข้องกับโฟลว์ และการกลับไปกลับมาของสมการสองสมการสุดท้ายนี้คือสิ่งที่ทำให้เกิดคลื่นแสง และบ่อยครั้งที่แนวคิดเหล่านี้มีประโยชน์ในบริบทที่ดูเหมือนจะไม่มีลักษณะเชิงพื้นที่ด้วยซ้ำในตอนแรก ยกตัวอย่างคลาสสิกที่นักเรียนสมการเชิงอนุพันธ์มักศึกษา สมมติว่าคุณต้องการติดตามขนาดประชากรของสองสายพันธุ์ที่แตกต่างกัน โดยที่สายพันธุ์หนึ่งเป็นนักล่าจากอีกสายพันธุ์หนึ่ง สถานะของระบบนี้ในเวลาที่กำหนด ซึ่งหมายถึงขนาดประชากรทั้งสองนั้น อาจมองว่าเป็นจุดในปริภูมิสองมิติ ซึ่งเป็นสิ่งที่คุณเรียกว่าสเปซเฟสของระบบนี้ สำหรับขนาดประชากรคู่หนึ่ง ประชากรเหล่านี้อาจมีแนวโน้มที่จะเปลี่ยนแปลงขึ้นอยู่กับสิ่งต่างๆ เช่น การสืบพันธุ์ของสัตว์ทั้งสองชนิดนี้ หรือเพียงแค่ว่าสัตว์ชนิดใดชนิดหนึ่งชอบกินอาหารอีกชนิดมากเพียงใด โดยทั่วไปอัตราการเปลี่ยนแปลงเหล่านี้จะถูกเขียนเชิงวิเคราะห์เป็นชุดของสมการเชิงอนุพันธ์ ไม่เป็นไรถ้าคุณไม่เข้าใจสมการเหล่านี้ ฉันแค่ทิ้งมันไว้สำหรับคนที่อยากรู้ และเนื่องจากการแทนที่ตัวแปรด้วยรูปภาพทำให้ฉันหัวเราะนิดหน่อย แต่ความเกี่ยวข้องตรงนี้ก็คือ วิธีที่ดีในการเห็นภาพว่าชุดสมการนั้นบอกอะไรจริงๆ ก็คือการเชื่อมโยงแต่ละจุดบนระนาบ ขนาดประชากรแต่ละคู่ ด้วยเวกเตอร์ที่ระบุอัตราการเปลี่ยนแปลงของตัวแปรทั้งสอง ตัวอย่างเช่น เมื่อมีสุนัขจิ้งจอกจำนวนมาก แต่มีกระต่ายค่อนข้างน้อย จำนวนสุนัขจิ้งจอกอาจมีแนวโน้มลดลงเนื่องจากปริมาณอาหารที่จำกัด และจำนวนกระต่ายก็อาจมีแนวโน้มลดลงเนื่องจากพวกมันถูกกินหมด ของสุนัขจิ้งจอก ซึ่งอาจเร็วเกินกว่าที่จะสืบพันธุ์ได้ เวกเตอร์ที่ให้มาตรงนี้กำลังบอกคุณว่า ขนาดประชากรคู่หนึ่งๆ มีแนวโน้มที่จะเปลี่ยนแปลงอย่างไร และเร็วแค่ไหน โปรดสังเกตว่า นี่เป็นกรณีที่สนามเวกเตอร์ไม่เกี่ยวกับพื้นที่ทางกายภาพ แต่เป็นการเป็นตัวแทนของระบบไดนามิกบางอย่างที่มีตัวแปรสองตัว และระบบนั้นพัฒนาไปตามกาลเวลาอย่างไร นี่อาจทำให้เข้าใจได้ว่าเหตุใดนักคณิตศาสตร์จึงสนใจที่จะศึกษาเรขาคณิตในมิติที่สูงกว่า จะเกิดอะไรขึ้นหากระบบของเราติดตามตัวเลขมากกว่าสองหรือสามหมายเลข? ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow. ", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves. ", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another. ", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery. ", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar. ", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/turkish/sentence_translations.json b/2018/divergence-and-curl/turkish/sentence_translations.json index 74b1b3832..8c5e5ce3d 100644 --- a/2018/divergence-and-curl/turkish/sentence_translations.json +++ b/2018/divergence-and-curl/turkish/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "Khan Academy'de geçirdiğim zamandan bu yana bunu daha ayrıntılı olarak anlatan birçok içeriğim var, ancak asıl amacımız için, sadece 2 boyutlu uzaydaki her noktayı tek bir sayıyla ilişkilendiren iki boyutlu rotasyonel varyantından bahsedeceğim. yeni bir vektör yerine.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "Bu aynı zamanda, bir mıknatısın kuzey veya güney ucu gibi davranan manyetik monopollerin var olmadığı, bir elektrik alanında pozitif ve negatif yüklere benzer hiçbir şeyin bulunmadığı yorumuna da sahiptir.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "Bu tamamen üç boyutlu bir fikir ve buradaki ana odak noktamızın biraz dışında, ancak önemli olan nokta, ıraksama ve dönmenin akışla ilgisi olmayan bağlamlarda ortaya çıkmasıdır.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "Ve bu son iki denklemin ileri geri gidişi ışık dalgalarının oluşmasına neden olur.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "Diferansiyel denklem öğrencilerinin sıklıkla çalıştığı klasik bir örneği ele alırsak, diyelim ki, birinin diğerinin yırtıcısı olduğu iki farklı türün popülasyon büyüklüğünü izlemek istiyorsunuz.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "Dürüst olmak gerekirse, bunun gibi bir şey için genellikle ıraksaklık ve rotasyonelin ötesinde ilgili araçları da dahil etmek istersiniz, ancak bu iki fikirle pratik yapmanın getirdiği zihin yapısı, bunun gibi benzer kurulumları çalışmaya iyi bir şekilde aktarır. matematik makinelerinin parçaları.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "Ne olursa olsun her zaman deneyimin değerini en üst düzeye çıkarmaya çalışacağımı düşünmek hoşuma gidiyor, ancak aynı zamanda sürekli olarak erken kalkabileceğimi ve çok fazla şeker yemeye direnebileceğimi de düşünüyorum.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/ukrainian/sentence_translations.json b/2018/divergence-and-curl/ukrainian/sentence_translations.json index e7285028e..5b5beefff 100644 --- a/2018/divergence-and-curl/ukrainian/sentence_translations.json +++ b/2018/divergence-and-curl/ukrainian/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "У мене є багато інформації про час, проведений в Академії Хана, який описує це більш детально, але для нашої основної мети я буду лише посилатися на двовимірний варіант curl, який пов’язує кожну точку в 2D просторі з одним числом, а не новий вектор.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "Це також тлумачить, що магнітних монополів, щось, що діє так само, як північний або південний кінець магніту в ізоляції, не існує, немає нічого аналогічного позитивним і негативним зарядам в електричному полі.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "Це суто тривимірна ідея, яка трохи виходить за межі нашої головної уваги, але справа в тому, що розбіжність і завиток виникають у контекстах, які не пов’язані з потоком.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "І назад і вперед від цих двох останніх рівнянь є те, що породжує світлові хвилі.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "Щоб взяти класичний приклад, який часто вивчають студенти диференціальних рівнянь, скажімо, ви хочете відстежити розмір популяції двох різних видів, де один є хижаком іншого.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "Якщо бути абсолютно чесним з вами, для чогось подібного вам часто захочеться використовувати пов’язані інструменти, окрім просто розбіжності та викривлення, але настрій розуму, що практика з цими двома ідеями приведе вас, добре переноситься на вивчення подібних установок із подібними частини математичної техніки.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "Мені подобається думати, що я завжди намагатимуся максимізувати цінність досвіду, незважаючи ні на що, але в цьому відношенні мені також подобається думати, що я можу постійно прокидатися рано і не їсти занадто багато цукру.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/urdu/sentence_translations.json b/2018/divergence-and-curl/urdu/sentence_translations.json index 5dc0cef4c..f96923673 100644 --- a/2018/divergence-and-curl/urdu/sentence_translations.json +++ b/2018/divergence-and-curl/urdu/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector. ", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector. ", "translatedText": "میرے پاس خان اکیڈمی میں اپنے وقت کا کافی مواد ہے جو اس کو مزید تفصیل سے بیان کرتا ہے، لیکن ہمارے بنیادی مقصد کے لیے، میں صرف curl کے دو جہتی تغیرات کا حوالہ دوں گا، جو 2D اسپیس میں ہر ایک پوائنٹ کو ایک نمبر کے ساتھ جوڑتا ہے، ایک نئے ویکٹر کے بجائے۔", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field. ", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field. ", "translatedText": "اس کی یہ تشریح بھی ہے کہ مقناطیسی اجارہ داری، کوئی ایسی چیز جو تنہائی میں مقناطیس کے شمالی یا جنوبی سرے کی طرح کام کرتی ہے، موجود نہیں ہے، برقی میدان میں مثبت اور منفی چارجز کے مترادف کوئی چیز نہیں ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow. ", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow, ", "translatedText": "یہ ایک خالصتاً سہ جہتی خیال ہے، اور یہاں ہماری مرکزی توجہ سے تھوڑا باہر ہے، لیکن بات یہ ہے کہ انحراف اور کرل ان سیاق و سباق میں پیدا ہوتے ہیں جن کا بہاؤ سے کوئی تعلق نہیں ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves. ", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves. ", "translatedText": "اور ان آخری دو مساواتوں سے آگے پیچھے روشنی کی لہروں کو جنم دیتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another. ", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another. ", "translatedText": "ایک کلاسک مثال لینے کے لیے جس کا اکثر تفریق مساوات کے طالب علم پڑھتے ہیں، فرض کریں کہ آپ دو مختلف پرجاتیوں کی آبادی کے سائز کو ٹریک کرنا چاہتے ہیں، جہاں ایک دوسرے کا شکاری ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery. ", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery. ", "translatedText": "کیا آبادی کے سائز کا رجحان کسی خاص جوڑے کی تعداد کی طرف ہوتا ہے، یا کیا ایسی کوئی قدریں ہیں جن سے وہ ہٹ جاتے ہیں؟ کیا وہاں چکراتی نمونے ہیں، اور کیا وہ چکر مستحکم ہیں یا غیر مستحکم؟ آپ کے ساتھ مکمل طور پر ایماندار ہونے کے لیے، اس طرح کی کسی چیز کے لیے آپ اکثر متعلقہ ٹولز کو صرف ڈائیورجن اور کرل سے آگے لانا چاہتے ہیں، لیکن ذہن کا فریم جو ان دونوں آئیڈیاز کے ساتھ مشق کرتا ہے آپ کو اس طرح کے سیٹ اپس کا مطالعہ کرنے میں مدد دیتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar. ", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar. ", "translatedText": "میں یہ سوچنا پسند کرتا ہوں کہ میں ہمیشہ تجربے کی قدر کو زیادہ سے زیادہ کرنے کی کوشش کروں گا چاہے کچھ بھی ہو، لیکن اس معاملے کے لیے میں یہ سوچنا بھی پسند کرتا ہوں کہ میں مستقل طور پر جلدی جاگ سکتا ہوں اور بہت زیادہ چینی کھانے کے خلاف مزاحمت کر سکتا ہوں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/divergence-and-curl/vietnamese/sentence_translations.json b/2018/divergence-and-curl/vietnamese/sentence_translations.json index bdcdacbbf..02da527c9 100644 --- a/2018/divergence-and-curl/vietnamese/sentence_translations.json +++ b/2018/divergence-and-curl/vietnamese/sentence_translations.json @@ -320,7 +320,7 @@ "end": 330.94 }, { - "input": "I have plenty of content from my time at Khan Academy describing this in more detail, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2D space with a single number, rather than a new vector.", + "input": "and I have plenty of content from my time at Khan Academy describing this in more detail if you want, but for our main purpose, I'll just be referring to the two-dimensional variant of curl, which associates each point in 2d space with a single number rather than a new vector.", "translatedText": "Tôi có rất nhiều nội dung trong thời gian làm việc tại Học viện Khan mô tả điều này chi tiết hơn, nhưng với mục đích chính của chúng ta, tôi sẽ chỉ đề cập đến biến thể hai chiều của đường cong, liên kết từng điểm trong không gian 2D với một số duy nhất, chứ không phải là một vectơ mới.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 418.68 }, { - "input": "This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", + "input": "it acts just like water. This also has the interpretation that magnetic monopoles, something that acts just like a north or south end of a magnet in isolation, don't exist, there's nothing analogous to positive and negative charges in an electric field.", "translatedText": "Điều này cũng có cách giải thích rằng các đơn cực từ, thứ hoạt động giống như đầu cực bắc hoặc cực nam của nam châm, không tồn tại, không có gì tương tự như điện tích dương và điện tích âm trong điện trường.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 440.54 }, { - "input": "This is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow.", + "input": "And really, this is a purely three-dimensional idea, and a little outside of our main focus here, but the point is that divergence and curl arise in contexts that are unrelated to flow,", "translatedText": "Đây hoàn toàn là một ý tưởng ba chiều và hơi nằm ngoài trọng tâm chính của chúng tôi ở đây, nhưng vấn đề là sự phân kỳ và đường cong phát sinh trong các bối cảnh không liên quan đến dòng chảy.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 450.88 }, { - "input": "And the back and forth from these last two equations is what gives rise to light waves.", + "input": "and side note, the back and forth from these last two equations is what gives rise to light waves.", "translatedText": "Và sự qua lại của hai phương trình cuối cùng này là nguyên nhân tạo ra sóng ánh sáng.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 463.18 }, { - "input": "To take a classic example that students of differential equations often study, let's say you wanted to track the population sizes of two different species, where one is a predator of another.", + "input": "To take a classic example that students of differential equations often study, let's say that you wanted to track the population sizes of two different species, where maybe one of them is a predator of another.", "translatedText": "Lấy một ví dụ kinh điển mà những người nghiên cứu phương trình vi phân thường nghiên cứu, giả sử bạn muốn theo dõi quy mô quần thể của hai loài khác nhau, trong đó một loài là kẻ săn mồi của loài khác.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 600.64 }, { - "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", + "input": "To be perfectly honest with you, for something like this you'd often want to bring in related tools beyond just divergence and curl, those would give you the full story, but the frame of mind that practice with these two ideas brings you carries over well to studying setups like this with similar pieces of mathematical machinery.", "translatedText": "Thành thật mà nói với bạn, đối với những thứ như thế này, bạn thường muốn sử dụng các công cụ liên quan ngoài phân kỳ và cuộn tròn, nhưng việc thực hành với hai ý tưởng này sẽ giúp bạn tiếp tục nghiên cứu các thiết lập như thế này với các thiết lập tương tự. các mảnh của máy toán học.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 905.84 }, { - "input": "I like to think I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think I can consistently wake up early and resist eating too much sugar.", + "input": "I like to think that I'll always try to maximize the value of the experience no matter what, but for that matter I also like to think that I can consistently wake up early and resist eating too much sugar.", "translatedText": "Tôi muốn nghĩ rằng mình sẽ luôn cố gắng tối đa hóa giá trị của trải nghiệm dù thế nào đi nữa, nhưng về vấn đề đó, tôi cũng muốn nghĩ rằng mình có thể thức dậy sớm một cách nhất quán và không ăn quá nhiều đường.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/arabic/sentence_translations.json b/2018/feynmans-lost-lecture/arabic/sentence_translations.json index 6769ffa99..e8f65b6ce 100644 --- a/2018/feynmans-lost-lecture/arabic/sentence_translations.json +++ b/2018/feynmans-lost-lecture/arabic/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", "translatedText": "والسبب المهم، كما سترون لاحقًا، هو أن اتجاه التماس هذا سيتوافق مع سرعة الجسم الذي يدور حوله. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line. ", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line. ", "translatedText": "من حيث المبدأ، سننظر فقط في شرائح صغيرة جدًا، لذلك لن يكون هناك أي غموض فيما أعنيه بنصف القطر، وسيكون طول القوس خطًا مستقيمًا. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play. ", + "input": "Alright, now think about how the inverse square law comes into play. ", "translatedText": "فكر في كيفية تفعيل قانون التربيع العكسي. ", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again? ", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again? ", "translatedText": "وربما تكون هذه هي اللحظة التي يتعين عليك فيها عقد حاجبك والتفكير مرة أخرى، حسنًا، انتظر لحظة، ما الذي كان يحدث في هذا الدليل مرة أخرى؟ ", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be. ", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be. ", "translatedText": "ما يعنيه ذلك هو أن نقطة القطع الناقص لدينا، والتي تقع على بعد درجات ثيتا من المستوى الأفقي بالنسبة إلى مركز الدائرة، لها ميل مماس عمودي على الخط اللامركزي، وبسبب دوران 90 درجة بالكامل، فهذا يعني أنه بالتوازي مع ناقل السرعة الذي نحتاجه. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/bengali/sentence_translations.json b/2018/feynmans-lost-lecture/bengali/sentence_translations.json index e33d454fe..dde5556bc 100644 --- a/2018/feynmans-lost-lecture/bengali/sentence_translations.json +++ b/2018/feynmans-lost-lecture/bengali/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", "translatedText": "গুরুত্বপূর্ণ কারণটি, আপনি পরে দেখতে পাবেন, এই স্পর্শক দিকটি একটি প্রদক্ষিণকারী বস্তুর বেগের সাথে সামঞ্জস্যপূর্ণ হতে চলেছে।", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line. ", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line. ", "translatedText": "নীতিগতভাবে, আমরা শুধুমাত্র খুব ছোট স্লাইস বিবেচনা করতে যাচ্ছি, তাই ব্যাসার্ধ বলতে আমি যা বুঝি তাতে কোনো অস্পষ্টতা থাকবে না এবং চাপের দৈর্ঘ্য হবে একটি সরল রেখা।", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play. ", + "input": "Alright, now think about how the inverse square law comes into play. ", "translatedText": "বিপরীত বর্গ আইন কার্যকর হয় কিভাবে সম্পর্কে চিন্তা করুন. ", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again? ", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again? ", "translatedText": "এবং এটি সম্ভবত সেই মুহূর্ত যেখানে আপনাকে আপনার ভ্রু কুঁচকে ফিরে ভাবতে হবে, ঠিক আছে এক মিনিট অপেক্ষা করুন, সেই প্রমাণে আবার কী ঘটছিল? ", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be. ", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be. ", "translatedText": "এর অর্থ হল আমাদের ছোট উপবৃত্তের বিন্দু, যা বৃত্তের কেন্দ্রের সাপেক্ষে অনুভূমিক থেকে থিটা ডিগ্রি দূরে, একটি স্পর্শক ঢাল রয়েছে উদ্ভট রেখার সাথে লম্ব, এবং পুরো 90 ডিগ্রি ঘূর্ণন জিনিসের কারণে, এর মানে হল এটি বেগ ভেক্টরের সমান্তরাল আমাদের এটি হওয়া দরকার।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/chinese/sentence_translations.json b/2018/feynmans-lost-lecture/chinese/sentence_translations.json index 9b887267f..ccde626b3 100644 --- a/2018/feynmans-lost-lecture/chinese/sentence_translations.json +++ b/2018/feynmans-lost-lecture/chinese/sentence_translations.json @@ -563,7 +563,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", "translatedText": "正如您稍后将看到的，这一点很重要的原因 是该切线方向将对应于轨道物体的速度。", "model": "google_nmt", "from_community_srt": "正如你所说， 这将是重要的原因 后来说， 就是这个相切的方向 将对应于轨道的速度 宾语。", @@ -948,7 +948,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line. ", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line. ", "translatedText": "原则上，我们只会考虑非常小的切片，因此我所说 的半径不会有任何歧义，并且弧长将是一条直线。", "model": "google_nmt", "from_community_srt": "原则上， 我们最终会考虑 非常小的切片， 所以不会有歧义 我的意思是来自行星的半径 给定切片上的太阳， 以及相关的 弧长的位将是有效的直线。", @@ -957,7 +957,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play. ", + "input": "Alright, now think about how the inverse square law comes into play. ", "translatedText": "想想平方反比定律是如何发挥作用的。", "model": "google_nmt", "from_community_srt": "好吧， 现在想想逆方块如何 法律发挥作用。", @@ -1322,7 +1322,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again? ", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again? ", "translatedText": "这可能是你必须皱起眉头回想一下的时刻 ，好吧等一下，那个证明又发生了什么？", "model": "google_nmt", "from_community_srt": "而这正是你有这种感觉的时刻 皱起眉头， 回想一下“等等， 那证据再次发生了什么？", @@ -1349,7 +1349,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be. ", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be. ", "translatedText": "这意味着我们的小椭圆的点（相对于圆心与 水平面成 θ 度角）具有垂直于偏心线的 切线斜率，并且由于整个 90 度旋转， 这意味着它是与我们需要的速度矢量平行。", "model": "google_nmt", "from_community_srt": "这意味着我们的一点点 椭圆是水平的θ度， 关于圆圈的中心， 有 垂直于该偏心轮的切线斜面 线。 而且由于整个90度旋转， 这意味着它与速度平行 矢量我们需要它。", diff --git a/2018/feynmans-lost-lecture/english/captions.srt b/2018/feynmans-lost-lecture/english/captions.srt index 958457ca4..0b533e65b 100644 --- a/2018/feynmans-lost-lecture/english/captions.srt +++ b/2018/feynmans-lost-lecture/english/captions.srt @@ -1,5 +1,5 @@ 1 -00:00:03,019 --> 00:00:05,745 +00:00:03,020 --> 00:00:05,745 You may be aware that I'm a huge fan of the YouTube 2 @@ -59,7 +59,7 @@ rotate it 90 degrees about its midpoint, and when you do that for all of the lin an ellipse emerges in the middle. 16 -00:00:52,889 --> 00:00:55,605 +00:00:52,890 --> 00:00:55,605 Out of context, this is a mildly pleasing curiosity, 17 @@ -487,7 +487,7 @@ Now take a moment to think about the sum of the distances from our proposed focus points to any point Q along this perpendicular bisector. 123 -00:07:22,719 --> 00:07:27,942 +00:07:22,720 --> 00:07:27,942 The key insight here is that you can find two congruent triangles and use them to 124 @@ -539,7 +539,7 @@ specifically an ellipse whose focal sum is equal to the radius of the circle. Isn't that neat? 136 -00:08:13,719 --> 00:08:18,477 +00:08:13,720 --> 00:08:18,477 Second, because the sum of these two lengths at every other point on that 137 @@ -595,15 +595,15 @@ orbiting object. Okay, geometry proofing is done, on to some actual physics and orbital mechanics. 150 -00:09:07,319 --> 00:09:10,979 +00:09:07,320 --> 00:09:10,980 The first fact is to use Kepler's very beautiful second law, 151 -00:09:10,979 --> 00:09:14,099 +00:09:10,980 --> 00:09:14,100 which says that as an object orbits around the sun, 152 -00:09:14,099 --> 00:09:18,180 +00:09:14,100 --> 00:09:18,180 the area it sweeps out during a given amount of time, like one day, 153 @@ -783,300 +783,300 @@ we're going to show the shape traced out by the velocity vectors of that object. Here, let me show you what I mean by that. 197 -00:12:14,360 --> 00:12:17,529 -As the object orbits, its velocity will be changing, +00:12:14,360 --> 00:12:18,418 +As the object orbits, its velocity will be changing, right? It's rotating, 198 -00:12:17,529 --> 00:12:20,580 -rotating always tangent to the curve of the orbit, +00:12:18,418 --> 00:12:22,639 +always tangent to the curve of the orbit, and it's longer at points where the 199 -00:12:20,580 --> 00:12:23,749 -longer at points where the object is moving quickly, +00:12:22,639 --> 00:12:26,860 +object is moving quickly, and shorter at points where it's moving more slowly. 200 -00:12:23,749 --> 00:12:26,860 -and shorter at points where it's moving more slowly. - -201 00:12:27,620 --> 00:12:32,969 What we'll show is that if you take all these velocity vectors and collect them together -202 +201 00:12:32,969 --> 00:12:38,380 so that their tails sit at a single point, their tips actually trace out a perfect circle. -203 +202 00:12:39,040 --> 00:12:41,500 This is an awesome fact, if you ask me. -204 +203 00:12:41,920 --> 00:12:45,798 The velocity spins around, getting faster and slower at various angles, -205 +204 00:12:45,798 --> 00:12:50,431 but evidently the laws of physics cook things up just right so that these trace out a -206 +205 00:12:50,431 --> 00:12:51,240 perfect circle. -207 +206 00:12:52,080 --> 00:12:55,383 The astute among you might have a little internal lightbulb starting -208 +207 00:12:55,383 --> 00:12:58,400 to turn on at the site of this circle with an off-center point. -209 +208 00:12:59,620 --> 00:13:02,480 But again we have to ask, why on earth should this be true? -210 +209 00:13:03,480 --> 00:13:06,800 Feynman describes being unable to easily follow Newton at this point, -211 +210 00:13:06,800 --> 00:13:10,786 so instead he comes up with his own elegant line of reasoning to explain where this -212 +211 00:13:10,786 --> 00:13:11,640 circle comes from. +212 +00:13:11,940 --> 00:13:15,265 +He starts by looking at the orbit, whose shape we don't know, + 213 -00:13:11,940 --> 00:13:15,990 -He starts by looking at the orbit, and slicing it into little +00:13:15,265 --> 00:13:20,040 +and slicing it into little pieces which all cover the same angle with respect to the sun. 214 -00:13:15,990 --> 00:13:20,040 -pieces which all cover the same angle with respect to the sun. - -215 00:13:21,340 --> 00:13:25,609 Think about the amount of time it takes for the orbiting object to traverse one -216 +215 00:13:25,609 --> 00:13:30,040 of these equal angle slices, and how that time changes as you go to a bigger slice. -217 +216 00:13:30,860 --> 00:13:35,840 By Kepler's second law, that time is proportional to the area of the slice, right? -218 +217 00:13:36,220 --> 00:13:41,343 And because these slices have the same angle, as you get farther away from the sun, -219 +218 00:13:41,343 --> 00:13:45,430 not only does the radius increase, but the component of arc length -220 +219 00:13:45,430 --> 00:13:49,700 perpendicular to that radial line goes up in proportion to the radius. -221 +220 00:13:50,580 --> 00:13:54,979 So the area of one of these slices, and hence the time it takes the object -222 +221 00:13:54,979 --> 00:13:59,320 to traverse it, is proportional to the distance away from the sun squared. -223 +222 00:14:00,220 --> 00:14:04,512 In principle, by the way, we're only going to be considering very small slices, -224 +223 00:14:04,512 --> 00:14:07,732 so there will be no ambiguity in what I mean by the radius, -225 +224 00:14:07,732 --> 00:14:10,040 and the arc length will be a straight line. -226 +225 00:14:10,840 --> 00:14:14,480 Alright, now think about how the inverse square law comes into play. -227 +226 00:14:14,800 --> 00:14:18,290 At any given point, the force that the sun imparts on the -228 +227 00:14:18,290 --> 00:14:21,780 object is proportional to 1 divided by the radius squared. -229 +228 00:14:22,400 --> 00:14:23,700 But what does that really mean? -230 +229 00:14:24,380 --> 00:14:28,709 What force is, is the mass of an object times its acceleration, -231 +230 00:14:28,709 --> 00:14:32,160 the amount that its velocity changes per unit time. -232 +231 00:14:32,740 --> 00:14:37,907 This is enough to give us a super useful bit of information about how the velocity of the -233 +232 00:14:37,907 --> 00:14:42,960 orbiting object changes as it goes from the start of one slice to the start of the next. -234 +233 00:14:43,500 --> 00:14:48,160 That change in velocity is the acceleration times the change in time, right? -235 +234 00:14:48,960 --> 00:14:51,994 What that means is that this change to the velocity is -236 +235 00:14:51,994 --> 00:14:55,580 proportional to the change in time divided by the radius squared. -237 +236 00:14:56,480 --> 00:14:59,791 But since the time that it takes to traverse one slice is -238 +237 00:14:59,791 --> 00:15:03,160 proportional to the radius squared, these terms cancel out. -239 +238 00:15:03,700 --> 00:15:07,085 So the change in velocity as it traverses a given slice is -240 +239 00:15:07,085 --> 00:15:10,700 actually some constant that doesn't depend on the slice at all. -241 +240 00:15:11,300 --> 00:15:16,109 Here, unpacking what I mean by that, if you look at the velocity at the start of a slice, -242 +241 00:15:16,109 --> 00:15:19,315 and then you look at the velocity at the end of that slice, -243 +242 00:15:19,315 --> 00:15:22,682 and directly compare those two vectors by joining their tails, -244 +243 00:15:22,682 --> 00:15:27,117 and you look at the difference between them, the little vector joining their tips, -245 +244 00:15:27,117 --> 00:15:31,125 this difference has the same length no matter which slice of the orbit you -246 +245 00:15:31,125 --> 00:15:31,980 were looking at. -247 +246 00:15:32,700 --> 00:15:36,640 So as you compare these velocity vectors at the start of each slice, -248 +247 00:15:36,640 --> 00:15:40,980 they'll be forming some kind of polygon whose side lengths are all the same. -249 +248 00:15:41,880 --> 00:15:45,663 Also, since the force vector is always pointing towards the sun, -250 +249 00:15:45,663 --> 00:15:49,738 as you go from the start of one slice to the next, that force vector, -251 +250 00:15:49,738 --> 00:15:53,580 and hence the acceleration vector, is turning by a constant angle. -252 +251 00:15:54,160 --> 00:15:57,993 In geometry lingo, what this implies is that all the external -253 +252 00:15:57,993 --> 00:16:01,580 angles of our polygon are going to be equal to each other. -254 +253 00:16:02,620 --> 00:16:06,120 I know this is a little tricky, but hang in there, remember, -255 +254 00:16:06,120 --> 00:16:09,220 all you need to follow along is infinite intelligence. -256 +255 00:16:10,000 --> 00:16:11,993 It's worth reiterating just to make sure it's -257 +256 00:16:11,993 --> 00:16:14,160 clear what's happening with this velocity diagram. -258 +257 00:16:14,700 --> 00:16:18,937 The change from one vector to the next, this little difference vector joining the tip -259 +258 00:16:18,937 --> 00:16:21,844 of one to the tip of the next, always has the same length, -260 +259 00:16:21,844 --> 00:16:26,180 that was the consequence of the perfect cancellation between mixing Kepler's second law -261 +260 00:16:26,180 --> 00:16:27,560 with the inverse square law. -262 +261 00:16:27,560 --> 00:16:33,706 And because those constant length change vectors rotate by a constant angle each time, -263 +262 00:16:33,706 --> 00:16:36,320 it means they form a regular polygon. -264 +263 00:16:37,220 --> 00:16:40,848 And as we consider finer and finer slices of the original orbit, -265 +264 00:16:40,848 --> 00:16:43,863 based on smaller and smaller angles for those slices, -266 +265 00:16:43,863 --> 00:16:48,330 the relevant regular polygon defining the tips of these vectors in the velocity -267 +266 00:16:48,330 --> 00:16:50,340 diagram approaches a perfect circle. -268 +267 00:16:51,020 --> 00:16:51,620 Isn't that neat? +268 +00:16:53,200 --> 00:16:55,670 +Hopefully at this point you're looking at the circle, + 269 -00:16:53,200 --> 00:16:57,184 -Hopefully, at this point, you're looking at the circle, the eccentric point, +00:16:55,670 --> 00:16:58,552 +you're looking at the eccentric point, and you're just itching 270 -00:16:57,184 --> 00:17:01,480 -and you're itching to see how this gives rise to an ellipse the way we saw earlier. +00:16:58,552 --> 00:17:01,480 +to see how this gives rise to an ellipse the way we saw earlier. 271 00:17:01,880 --> 00:17:04,099 @@ -1367,6 +1367,6 @@ Watching Feynman do physics, even elementary physics, is like watching Bobby Fischer play chess. 343 -00:21:23,689 --> 00:21:27,090 +00:21:23,690 --> 00:21:27,090 Thanks again to Grant, and you should definitely go check out his videos on 3blue1brown. diff --git a/2018/feynmans-lost-lecture/english/sentence_timings.json b/2018/feynmans-lost-lecture/english/sentence_timings.json index 68b3b6f3d..af3ec3544 100644 --- a/2018/feynmans-lost-lecture/english/sentence_timings.json +++ b/2018/feynmans-lost-lecture/english/sentence_timings.json @@ -465,7 +465,7 @@ 733.9 ], [ - "As the object orbits, its velocity will be changing, rotating always tangent to the curve of the orbit, longer at points where the object is moving quickly, and shorter at points where it's moving more slowly.", + "As the object orbits, its velocity will be changing, right? It's rotating, always tangent to the curve of the orbit, and it's longer at points where the object is moving quickly, and shorter at points where it's moving more slowly.", 734.36, 746.86 ], @@ -500,7 +500,7 @@ 791.64 ], [ - "He starts by looking at the orbit, and slicing it into little pieces which all cover the same angle with respect to the sun.", + "He starts by looking at the orbit, whose shape we don't know, and slicing it into little pieces which all cover the same angle with respect to the sun.", 791.94, 800.04 ], @@ -625,7 +625,7 @@ 1011.62 ], [ - "Hopefully, at this point, you're looking at the circle, the eccentric point, and you're itching to see how this gives rise to an ellipse the way we saw earlier.", + "Hopefully at this point you're looking at the circle, you're looking at the eccentric point, and you're just itching to see how this gives rise to an ellipse the way we saw earlier.", 1013.2, 1021.48 ], diff --git a/2018/feynmans-lost-lecture/english/transcript.txt b/2018/feynmans-lost-lecture/english/transcript.txt index a8ea6a9c4..c861b692e 100644 --- a/2018/feynmans-lost-lecture/english/transcript.txt +++ b/2018/feynmans-lost-lecture/english/transcript.txt @@ -91,14 +91,14 @@ For all we know, it's some wonky non-elliptical egg shape. The inverse square law is going to help us pin down that shape precisely, but the strategy is a little indirect. Before showing the shape of the path traced out by the orbiting object, we're going to show the shape traced out by the velocity vectors of that object. Here, let me show you what I mean by that. -As the object orbits, its velocity will be changing, rotating always tangent to the curve of the orbit, longer at points where the object is moving quickly, and shorter at points where it's moving more slowly. +As the object orbits, its velocity will be changing, right? It's rotating, always tangent to the curve of the orbit, and it's longer at points where the object is moving quickly, and shorter at points where it's moving more slowly. What we'll show is that if you take all these velocity vectors and collect them together so that their tails sit at a single point, their tips actually trace out a perfect circle. This is an awesome fact, if you ask me. The velocity spins around, getting faster and slower at various angles, but evidently the laws of physics cook things up just right so that these trace out a perfect circle. The astute among you might have a little internal lightbulb starting to turn on at the site of this circle with an off-center point. But again we have to ask, why on earth should this be true? Feynman describes being unable to easily follow Newton at this point, so instead he comes up with his own elegant line of reasoning to explain where this circle comes from. -He starts by looking at the orbit, and slicing it into little pieces which all cover the same angle with respect to the sun. +He starts by looking at the orbit, whose shape we don't know, and slicing it into little pieces which all cover the same angle with respect to the sun. Think about the amount of time it takes for the orbiting object to traverse one of these equal angle slices, and how that time changes as you go to a bigger slice. By Kepler's second law, that time is proportional to the area of the slice, right? And because these slices have the same angle, as you get farther away from the sun, not only does the radius increase, but the component of arc length perpendicular to that radial line goes up in proportion to the radius. @@ -123,7 +123,7 @@ The change from one vector to the next, this little difference vector joining th And because those constant length change vectors rotate by a constant angle each time, it means they form a regular polygon. And as we consider finer and finer slices of the original orbit, based on smaller and smaller angles for those slices, the relevant regular polygon defining the tips of these vectors in the velocity diagram approaches a perfect circle. Isn't that neat? -Hopefully, at this point, you're looking at the circle, the eccentric point, and you're itching to see how this gives rise to an ellipse the way we saw earlier. +Hopefully at this point you're looking at the circle, you're looking at the eccentric point, and you're just itching to see how this gives rise to an ellipse the way we saw earlier. But it's a little weird, right? We're looking at this diagram in velocity space, so how do we use that to make conclusions about the actual orbit? What follows is tricky, but clever. diff --git a/2018/feynmans-lost-lecture/french/sentence_translations.json b/2018/feynmans-lost-lecture/french/sentence_translations.json index c47af0215..f9e14e190 100644 --- a/2018/feynmans-lost-lecture/french/sentence_translations.json +++ b/2018/feynmans-lost-lecture/french/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", "translatedText": "La raison qui est importante, comme vous le verrez plus tard, est que cette direction de tangence va correspondre à la vitesse d'un objet en orbite.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line.", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line.", "translatedText": "En principe, nous ne considérerons que de très petites tranches, il n'y aura donc aucune ambiguïté sur ce que j'entends par rayon, et la longueur de l'arc sera une ligne droite.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play.", + "input": "Alright, now think about how the inverse square law comes into play.", "translatedText": "Pensez à la façon dont la loi du carré inverse entre en jeu.", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again?", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again?", "translatedText": "Et c'est probablement le moment où vous devez froncer les sourcils et réfléchir, d'accord, attendez une minute, que se passait-il encore dans cette preuve?", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be.", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be.", "translatedText": "Cela signifie que le point de notre petite ellipse, qui est à thêta degrés de l'horizontale par rapport au centre du cercle, a une pente tangente perpendiculaire à la ligne excentrique, et à cause de toute la rotation de 90 degrés, cela signifie que c'est parallèle au vecteur vitesse dont nous avons besoin.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/german/sentence_translations.json b/2018/feynmans-lost-lecture/german/sentence_translations.json index 22dee1e72..3d27936d2 100644 --- a/2018/feynmans-lost-lecture/german/sentence_translations.json +++ b/2018/feynmans-lost-lecture/german/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", "translatedText": "Wie Sie später sehen werden, ist der Grund dafür wichtig, dass diese Tangentenrichtung der Geschwindigkeit eines umlaufenden Objekts entspricht.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line.", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line.", "translatedText": "Im Prinzip werden wir nur sehr kleine Scheiben in Betracht ziehen, es wird also keine Unklarheit darüber geben, was ich mit dem Radius meine, und die Bogenlänge wird eine gerade Linie sein.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play.", + "input": "Alright, now think about how the inverse square law comes into play.", "translatedText": "Überlegen Sie, wie das Umkehrquadratgesetz ins Spiel kommt.", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again?", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again?", "translatedText": "Und das ist wahrscheinlich der Moment, in dem Sie die Stirn runzeln und zurückdenken müssen: Okay, Moment mal, was war nochmal in diesem Beweis los?", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be.", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be.", "translatedText": "Das bedeutet, dass der Punkt unserer kleinen Ellipse, der Theta-Grad von der Horizontalen in Bezug auf den Mittelpunkt des Kreises entfernt ist, eine tangentiale Steigung senkrecht zur Exzenterlinie aufweist, und aufgrund der ganzen 90-Grad-Rotation bedeutet dies, dass dies der Fall ist parallel zum Geschwindigkeitsvektor muss es sein.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/hebrew/sentence_translations.json b/2018/feynmans-lost-lecture/hebrew/sentence_translations.json index bf4073596..c4006e946 100644 --- a/2018/feynmans-lost-lecture/hebrew/sentence_translations.json +++ b/2018/feynmans-lost-lecture/hebrew/sentence_translations.json @@ -441,7 +441,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", "translatedText": "הסיבה החשובה, כפי שתראו מאוחר יותר, היא שכיוון הכוח הזה הולך להתאים למהירותו של עצם הסובב.", "n_reviews": 0, "start": 533.14, @@ -742,14 +742,14 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line.", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line.", "translatedText": "באופן עקרוני, אנחנו נשקול רק פרוסות קטנות מאוד, כך שלא תהיה אי בהירות למה אני מתכוון ברדיוס, ואורך הקשת יהיה קו ישר.", "n_reviews": 0, "start": 840.22, "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play.", + "input": "Alright, now think about how the inverse square law comes into play.", "translatedText": "חשבו כיצד חוק הריבוע ההפוך נכנס לתמונה.", "n_reviews": 0, "start": 850.84, @@ -1036,7 +1036,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again?", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again?", "translatedText": "וזה כנראה הרגע שבו אתה צריך לקמט את המצח ולחשוב אחורה, אוקיי חכה רגע, מה קרה שוב בהוכחה הזו?", "n_reviews": 0, "start": 1169.97, @@ -1057,7 +1057,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be.", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be.", "translatedText": "מה שזה אומר הוא שלנקודת האליפסה הקטנה שלנו, שהיא מעלות תטא מהאופקי ביחס למרכז המעגל, יש שיפוע משיק מאונך לקו האקסצנטרי, ובגלל כל עניין הסיבוב של 90 מעלות, זה אומר שזה מקביל לווקטור המהירות שאנחנו צריכים שהוא יהיה.", "n_reviews": 0, "start": 1203.25, diff --git a/2018/feynmans-lost-lecture/hindi/sentence_translations.json b/2018/feynmans-lost-lecture/hindi/sentence_translations.json index bf444a282..48c6ecc1c 100644 --- a/2018/feynmans-lost-lecture/hindi/sentence_translations.json +++ b/2018/feynmans-lost-lecture/hindi/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", "translatedText": "इसका महत्वपूर्ण कारण, जैसा कि आप बाद में देखेंगे, यह है कि यह स्पर्शरेखा दिशा एक परिक्रमा करने वाली वस्तु के वेग के अनुरूप होगी।", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line. ", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line. ", "translatedText": "सिद्धांत रूप में, हम केवल बहुत छोटे स्लाइस पर विचार करने जा रहे हैं, इसलिए त्रिज्या से मेरा जो मतलब है उसमें कोई अस्पष्टता नहीं होगी, और चाप की लंबाई एक सीधी रेखा होगी।", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play. ", + "input": "Alright, now think about how the inverse square law comes into play. ", "translatedText": "इस बारे में सोचें कि व्युत्क्रम वर्ग नियम कैसे लागू होता है।", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again? ", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again? ", "translatedText": "और शायद यही वह क्षण है जब आपको अपनी भौहें सिकोड़नी होंगी और पीछे सोचना होगा, ठीक है एक मिनट रुकिए, उस सबूत में फिर क्या चल रहा था? ", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be. ", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be. ", "translatedText": "इसका मतलब यह है कि हमारे छोटे दीर्घवृत्त का बिंदु, जो वृत्त के केंद्र के संबंध में क्षैतिज से थीटा डिग्री दूर है, में विलक्षण रेखा के लंबवत एक स्पर्शरेखा ढलान है, और पूरे 90 डिग्री घूर्णन की वजह से, इसका मतलब है कि यह है वेग वेक्टर के समानांतर हमें इसकी आवश्यकता है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/hungarian/sentence_translations.json b/2018/feynmans-lost-lecture/hungarian/sentence_translations.json index 28b633d38..6299abf09 100644 --- a/2018/feynmans-lost-lecture/hungarian/sentence_translations.json +++ b/2018/feynmans-lost-lecture/hungarian/sentence_translations.json @@ -441,7 +441,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", "translatedText": "Amint később látni fogjuk, az az ok, ami fontos, hogy ez az érintési irány meg fog felelni egy keringő objektum sebességének.", "n_reviews": 0, "start": 533.14, @@ -742,14 +742,14 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line.", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line.", "translatedText": "Elvileg csak nagyon kis szeleteket fogunk figyelembe venni, így nem lesz kétértelmű, hogy mit értek a sugár alatt, és az ív hossza egyenes lesz.", "n_reviews": 0, "start": 840.22, "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play.", + "input": "Alright, now think about how the inverse square law comes into play.", "translatedText": "Gondold át, hogyan lép életbe a fordított négyzettörvény.", "n_reviews": 0, "start": 850.84, @@ -1036,7 +1036,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again?", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again?", "translatedText": "És valószínűleg ez az a pillanat, amikor össze kell ráncolnia a szemöldökét, és vissza kell gondolnia, oké, várjon egy percet, mi történt megint abban a bizonyítékban?", "n_reviews": 0, "start": 1169.97, @@ -1057,7 +1057,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be.", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be.", "translatedText": "Ez azt jelenti, hogy a mi kis ellipszisünk pontja, amely a kör középpontjához képest théta fokban van eltérve a vízszintestől, van egy érintő meredeksége az excentrikus vonalra, és az egész 90 fokos elforgatás miatt ez azt jelenti, hogy párhuzamosnak kell lennie a sebességvektorral.", "n_reviews": 0, "start": 1203.25, diff --git a/2018/feynmans-lost-lecture/indonesian/sentence_translations.json b/2018/feynmans-lost-lecture/indonesian/sentence_translations.json index 97e868d95..2968ed11d 100644 --- a/2018/feynmans-lost-lecture/indonesian/sentence_translations.json +++ b/2018/feynmans-lost-lecture/indonesian/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", "translatedText": "Alasan pentingnya, seperti yang akan Anda lihat nanti, adalah bahwa arah singgung ini akan berhubungan dengan kecepatan benda yang mengorbit.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line.", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line.", "translatedText": "Pada prinsipnya, kita hanya akan mempertimbangkan irisan yang sangat kecil, sehingga tidak akan ada ambiguitas dalam arti jari-jari, dan panjang busur akan menjadi garis lurus.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play.", + "input": "Alright, now think about how the inverse square law comes into play.", "translatedText": "Pikirkan tentang bagaimana hukum kuadrat terbalik berperan.", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again?", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again?", "translatedText": "Dan ini mungkin saat di mana Anda harus mengerutkan alis dan berpikir kembali, oke tunggu sebentar, apa lagi yang terjadi pada bukti itu?", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be.", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be.", "translatedText": "Artinya adalah titik elips kecil kita, yang berjarak satu derajat dari horizontal terhadap pusat lingkaran, mempunyai kemiringan singgung yang tegak lurus terhadap garis eksentrik, dan karena seluruh rotasinya sebesar 90 derajat, ini berarti elips tersebut sejajar dengan vektor kecepatan yang kita inginkan.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/italian/sentence_translations.json b/2018/feynmans-lost-lecture/italian/sentence_translations.json index 32b3a5344..cf1d6fbec 100644 --- a/2018/feynmans-lost-lecture/italian/sentence_translations.json +++ b/2018/feynmans-lost-lecture/italian/sentence_translations.json @@ -834,7 +834,7 @@ "end": 733.9 }, { - "input": "As the object orbits, its velocity will be changing, rotating always tangent to the curve of the orbit, longer at points where the object is moving quickly, and shorter at points where it's moving more slowly.", + "input": "As the object orbits, its velocity will be changing, right? It's rotating, always tangent to the curve of the orbit, and it's longer at points where the object is moving quickly, and shorter at points where it's moving more slowly.", "translatedText": "Mentre l'oggetto orbita, la sua velocità cambierà, ruotando sempre tangente alla curva dell'orbita, più lunga nei punti in cui l'oggetto si muove velocemente e più corta nei punti in cui si muove più lentamente.", "model": "DeepL", "from_community_srt": "Mentre l'oggetto orbita, la sua velocità sarà mutevole, sempre tangente alla curva del orbita, più lunga nei punti in cui si muove l'oggetto rapidamente e più corto nei punti in cui si muove più lentamente.", @@ -897,7 +897,7 @@ "end": 791.64 }, { - "input": "He starts by looking at the orbit, and slicing it into little pieces which all cover the same angle with respect to the sun.", + "input": "He starts by looking at the orbit, whose shape we don't know, and slicing it into little pieces which all cover the same angle with respect to the sun.", "translatedText": "Inizia osservando l'orbita e tagliandola in piccoli pezzi che coprono tutti lo stesso angolo rispetto al sole.", "model": "DeepL", "from_community_srt": "Inizia guardando l'orbita e affettando in piccoli pezzi che coprono tutti il stesso angolo rispetto al sole.", @@ -1121,7 +1121,7 @@ "end": 1011.62 }, { - "input": "Hopefully, at this point, you're looking at the circle, the eccentric point, and you're itching to see how this gives rise to an ellipse the way we saw earlier.", + "input": "Hopefully at this point you're looking at the circle, you're looking at the eccentric point, and you're just itching to see how this gives rise to an ellipse the way we saw earlier.", "translatedText": "Si spera che a questo punto tu stia guardando il cerchio, il punto eccentrico e che ti venga voglia di vedere come questo dia origine a un'ellisse come abbiamo visto prima.", "model": "DeepL", "from_community_srt": "Non è davvero carino? Si spera che a questo punto tu stia cercando a questo cerchio con un punto eccentrico speciale, e hai solo voglia di vederlo dare origine in un'ellisse come abbiamo visto prima.", diff --git a/2018/feynmans-lost-lecture/japanese/sentence_translations.json b/2018/feynmans-lost-lecture/japanese/sentence_translations.json index bb09af308..1040c3e53 100644 --- a/2018/feynmans-lost-lecture/japanese/sentence_translations.json +++ b/2018/feynmans-lost-lecture/japanese/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", "translatedText": "これが重要な理由は、後ほど説明しますが、この接 線方向が周回する物体の速度に対応するためです。", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line. ", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line. ", "translatedText": "原則として、非常に小さなスライスのみを考慮するので、半 径の意味に曖昧さはなく、円弧の長さは直線になります。", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play. ", + "input": "Alright, now think about how the inverse square law comes into play. ", "translatedText": "逆二乗の法則がどのように作用するかを考えてみましょう。", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again? ", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again? ", "translatedText": "そしておそらくこれが、眉間にしわを寄せて、「ちょっと待って、もう一度あ の証明で何が起こっていたのか？」と思い返さなければならない瞬間です。", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be. ", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be. ", "translatedText": "これが意味するのは、円の中心に対して水平からシータ度ず れた小さな楕円の点は、偏心線に対して垂直な接線の傾き を持ち、全体が 90 度回転するため、これは次のことを 意味します。速度ベクトルと平行であることが必要です。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/korean/sentence_translations.json b/2018/feynmans-lost-lecture/korean/sentence_translations.json index 269006f8d..b1e4977a7 100644 --- a/2018/feynmans-lost-lecture/korean/sentence_translations.json +++ b/2018/feynmans-lost-lecture/korean/sentence_translations.json @@ -832,7 +832,7 @@ "end": 733.9 }, { - "input": "As the object orbits, its velocity will be changing, rotating always tangent to the curve of the orbit, longer at points where the object is moving quickly, and shorter at points where it's moving more slowly.", + "input": "As the object orbits, its velocity will be changing, right? It's rotating, always tangent to the curve of the orbit, and it's longer at points where the object is moving quickly, and shorter at points where it's moving more slowly.", "translatedText": "물체가 궤도를 돌면서 속도는 변화하며 항상 궤도 곡선에 접하는 방향으로 회전하고, 물체가 빠르게 움직이는 지점에서는 더 길게, 더 느리게 움직이는 지점에서는 더 짧게 회전합니다.", "model": "DeepL", "from_community_srt": "물체가 궤도를 돌 때 물체의 속도는 변화하며, 항상 궤도 곡선에 접하며, 물체가 빠르게 움직이는 지점에서는 더 길고, 더 느리게 움직이는 지점에서는 더 짧아집니다.", @@ -895,7 +895,7 @@ "end": 791.64 }, { - "input": "He starts by looking at the orbit, and slicing it into little pieces which all cover the same angle with respect to the sun.", + "input": "He starts by looking at the orbit, whose shape we don't know, and slicing it into little pieces which all cover the same angle with respect to the sun.", "translatedText": "그는 궤도를 보고 태양에 대해 모두 같은 각도를 이루는 작은 조각으로 궤도를 잘라내는 것으로 시작합니다.", "model": "DeepL", "from_community_srt": "그는 궤도를 보고, 태양에 대해 같은 각도로 모든 것을 덮는 작은 조각으로 자르는 것으로 시작합니다.", @@ -1118,7 +1118,7 @@ "end": 1011.62 }, { - "input": "Hopefully, at this point, you're looking at the circle, the eccentric point, and you're itching to see how this gives rise to an ellipse the way we saw earlier.", + "input": "Hopefully at this point you're looking at the circle, you're looking at the eccentric point, and you're just itching to see how this gives rise to an ellipse the way we saw earlier.", "translatedText": "이 시점에서 여러분은 원, 즉 편심점을 보고 있으며, 이것이 어떻게 앞서 본 타원을 만드는지 궁금해하고 계실 것입니다.", "model": "DeepL", "from_community_srt": "정말 멋지지 않나요? 바라건대, 이 시점에서 여러분은 이 원을 특별하고 기괴한 점으로 보고있었으면 좋겠습니다. 그리고 그것을 보고 싶어하는 욕구가 이전에 본 것처럼 타원을 생기게 하기를 바랍니다 그런데 좀 이상하지 않나요?", diff --git a/2018/feynmans-lost-lecture/marathi/sentence_translations.json b/2018/feynmans-lost-lecture/marathi/sentence_translations.json index 122e642f2..a1c6bab0e 100644 --- a/2018/feynmans-lost-lecture/marathi/sentence_translations.json +++ b/2018/feynmans-lost-lecture/marathi/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", "translatedText": "हे महत्त्वाचे कारण आहे, जसे आपण नंतर पाहू शकाल, की ही स्पर्शरेषा दिशा परिभ्रमण करणाऱ्या वस्तूच्या वेगाशी संबंधित आहे.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line.", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line.", "translatedText": "तत्वतः, आम्ही फक्त अगदी लहान तुकड्यांचा विचार करणार आहोत, त्यामुळे मला त्रिज्या काय म्हणायचे आहे यात कोणतीही संदिग्धता राहणार नाही आणि कमानीची लांबी सरळ रेषा असेल.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play.", + "input": "Alright, now think about how the inverse square law comes into play.", "translatedText": "व्युत्क्रम वर्ग कायदा कसा लागू होतो याचा विचार करा.", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again?", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again?", "translatedText": "आणि कदाचित हाच तो क्षण आहे जिथे तुम्हाला तुमची कपाळमोक्ष करून परत विचार करावा लागेल, ठीक आहे एक मिनिट थांबा, त्या पुराव्यात पुन्हा काय होते?", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be.", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be.", "translatedText": "याचा अर्थ असा आहे की आपल्या लहान लंबवर्तुळाचा बिंदू, जो वर्तुळाच्या केंद्राच्या संदर्भात क्षैतिज बिंदूपासून दूर आहे, त्याला विक्षिप्त रेषेला स्पर्शिक उतार लंब आहे आणि संपूर्ण 90 अंश रोटेशन गोष्टीमुळे, याचा अर्थ असा आहे की ते आहे. वेग वेक्टरच्या समांतर असणे आवश्यक आहे.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/persian/sentence_translations.json b/2018/feynmans-lost-lecture/persian/sentence_translations.json index c83c92a7a..db0646dfc 100644 --- a/2018/feynmans-lost-lecture/persian/sentence_translations.json +++ b/2018/feynmans-lost-lecture/persian/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", "translatedText": "همانطور که بعداً خواهید دید، دلیل مهم این است که این جهت مماس با سرعت یک جسم در حال گردش مطابقت دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line. ", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line. ", "translatedText": "در اصل، ما فقط برش های بسیار کوچک را در نظر می گیریم، بنابراین هیچ ابهامی در منظور من از شعاع وجود نخواهد داشت و طول قوس یک خط مستقیم خواهد بود. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play. ", + "input": "Alright, now think about how the inverse square law comes into play. ", "translatedText": "به این فکر کنید که چگونه قانون مربع معکوس وارد عمل می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again? ", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be. ", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be. ", "translatedText": "معنی آن این است که نقطه بیضی کوچک ما، که نسبت به مرکز دایره از افقی فاصله دارد، دارای شیب مماس عمود بر خط خارج از مرکز است، و به دلیل کل چرخش 90 درجه، به این معنی است که موازی با بردار سرعت که باید باشد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/portuguese/sentence_translations.json b/2018/feynmans-lost-lecture/portuguese/sentence_translations.json index 476ebab4b..edbfab08b 100644 --- a/2018/feynmans-lost-lecture/portuguese/sentence_translations.json +++ b/2018/feynmans-lost-lecture/portuguese/sentence_translations.json @@ -834,7 +834,7 @@ "end": 733.9 }, { - "input": "As the object orbits, its velocity will be changing, rotating always tangent to the curve of the orbit, longer at points where the object is moving quickly, and shorter at points where it's moving more slowly.", + "input": "As the object orbits, its velocity will be changing, right? It's rotating, always tangent to the curve of the orbit, and it's longer at points where the object is moving quickly, and shorter at points where it's moving more slowly.", "translatedText": "À medida que o objeto orbita, sua velocidade muda, girando sempre tangente à curva da órbita, mais longa nos pontos onde o objeto se move rapidamente e mais curta nos pontos onde se move mais lentamente.", "model": "google_nmt", "from_community_srt": "À medida que o objeto orbita, sua velocidade muda, sempre tangente à curva da órbita, maior em ponto onde o objeto se move rapidamente, e menor em pontos onde se move mais lentamente.", @@ -896,7 +896,7 @@ "end": 791.64 }, { - "input": "He starts by looking at the orbit, and slicing it into little pieces which all cover the same angle with respect to the sun.", + "input": "He starts by looking at the orbit, whose shape we don't know, and slicing it into little pieces which all cover the same angle with respect to the sun.", "translatedText": "Ele começa olhando para a órbita e cortando-a em pequenos pedaços que cobrem o mesmo ângulo em relação ao sol.", "model": "google_nmt", "from_community_srt": "Ele começa olhando para a órbita e divide-a em pedacinhos que tem o mesmo ângulo em relação ao sol.", @@ -1120,7 +1120,7 @@ "end": 1011.62 }, { - "input": "Hopefully, at this point, you're looking at the circle, the eccentric point, and you're itching to see how this gives rise to an ellipse the way we saw earlier.", + "input": "Hopefully at this point you're looking at the circle, you're looking at the eccentric point, and you're just itching to see how this gives rise to an ellipse the way we saw earlier.", "translatedText": "Esperançosamente, neste ponto, você está olhando para o círculo, o ponto excêntrico, e está ansioso para ver como isso dá origem a uma elipse como vimos anteriormente.", "model": "google_nmt", "from_community_srt": "Agora você vê o círculo, vê o ponto de excentricidade e mal pode esperar para vê-lo se transformar numa elipse,", diff --git a/2018/feynmans-lost-lecture/russian/sentence_translations.json b/2018/feynmans-lost-lecture/russian/sentence_translations.json index dad6de2dd..6800b0159 100644 --- a/2018/feynmans-lost-lecture/russian/sentence_translations.json +++ b/2018/feynmans-lost-lecture/russian/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", "translatedText": "Причина, по которой это важно, как вы увидите позже, заключается в том, что это направление касания будет соответствовать скорости вращающегося объекта.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line.", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line.", "translatedText": "В принципе, мы будем рассматривать только очень маленькие срезы, поэтому не будет никакой двусмысленности в том, что я подразумеваю под радиусом, а длина дуги будет прямой линией.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play.", + "input": "Alright, now think about how the inverse square law comes into play.", "translatedText": "Подумайте о том, как действует закон обратных квадратов.", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again?", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again?", "translatedText": "И, вероятно, это тот момент, когда вам придется нахмурить брови и подумать: ладно, подождите минутку, что еще раз происходило в этом доказательстве?", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be.", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be.", "translatedText": "Это означает, что точка нашего маленького эллипса, которая находится на тета-градусах от горизонтали относительно центра круга, имеет наклон касательной, перпендикулярный эксцентричной линии, и из-за всего поворота на 90 градусов это означает, что нам нужно, чтобы оно было параллельно вектору скорости.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/spanish/sentence_translations.json b/2018/feynmans-lost-lecture/spanish/sentence_translations.json index 03813286e..48f205766 100644 --- a/2018/feynmans-lost-lecture/spanish/sentence_translations.json +++ b/2018/feynmans-lost-lecture/spanish/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", "translatedText": "La razón por la que esto es importante, como verá más adelante, es que esta dirección de tangencia corresponderá a la velocidad de un objeto en órbita.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line.", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line.", "translatedText": "En principio, solo consideraremos cortes muy pequeños, por lo que no habrá ninguna ambigüedad en lo que quiero decir con radio y la longitud del arco será una línea recta.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play.", + "input": "Alright, now think about how the inverse square law comes into play.", "translatedText": "Piense en cómo entra en juego la ley del cuadrado inverso.", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again?", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again?", "translatedText": "Y este es probablemente el momento en el que tienes que fruncir el ceño y pensar en retrospectiva, está bien, espera un minuto, ¿qué estaba pasando en esa prueba otra vez?", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be.", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be.", "translatedText": "Lo que eso significa es que el punto de nuestra pequeña elipse, que está a theta grados fuera de la horizontal con respecto al centro del círculo, tiene una pendiente tangente perpendicular a la línea excéntrica, y debido a toda la rotación de 90 grados, esto significa que es paralelo al vector de velocidad que necesitamos que sea.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/tamil/sentence_translations.json b/2018/feynmans-lost-lecture/tamil/sentence_translations.json index 86b541bd1..20f7d00aa 100644 --- a/2018/feynmans-lost-lecture/tamil/sentence_translations.json +++ b/2018/feynmans-lost-lecture/tamil/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", "translatedText": "முக்கியமான காரணம், நீங்கள் பின்னர் பார்ப்பது போல, இந்த தொடுநிலை திசையானது சுற்றும் பொருளின் திசைவேகத்துடன் ஒத்துப்போகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line. ", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line. ", "translatedText": "கொள்கையளவில், நாங்கள் மிகச் சிறிய துண்டுகளை மட்டுமே பரிசீலிக்கப் போகிறோம், எனவே நான் ஆரம் என்ன சொல்கிறேன் என்பதில் எந்த தெளிவின்மையும் இருக்காது, மேலும் வில் நீளம் ஒரு நேர் கோட்டாக இருக்கும். ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play. ", + "input": "Alright, now think about how the inverse square law comes into play. ", "translatedText": "தலைகீழ் சதுர சட்டம் எவ்வாறு செயல்படுகிறது என்பதைப் பற்றி சிந்தியுங்கள். ", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again? ", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again? ", "translatedText": "ஒருவேளை நீங்கள் உங்கள் புருவத்தை சுருக்கி மீண்டும் சிந்திக்க வேண்டிய தருணம் இதுவாக இருக்கலாம், சரி ஒரு நிமிடம் காத்திருங்கள், மீண்டும் அந்த ஆதாரத்தில் என்ன நடக்கிறது? ", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be. ", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be. ", "translatedText": "இதன் பொருள் என்னவென்றால், வட்டத்தின் மையத்தைப் பொறுத்து கிடைமட்டத்தில் இருந்து தீட்டா டிகிரியில் இருக்கும் சிறிய நீள்வட்டத்தின் புள்ளியானது, விசித்திரக் கோட்டிற்கு செங்குத்தாக ஒரு தொடு சாய்வைக் கொண்டுள்ளது, மேலும் முழு 90 டிகிரி சுழற்சியின் காரணமாக, இது திசைவேக வெக்டருக்கு இணையாக அது இருக்க வேண்டும். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/telugu/sentence_translations.json b/2018/feynmans-lost-lecture/telugu/sentence_translations.json index 13c30c5d7..48a8942f6 100644 --- a/2018/feynmans-lost-lecture/telugu/sentence_translations.json +++ b/2018/feynmans-lost-lecture/telugu/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", "translatedText": "ముఖ్యమైన కారణం ఏమిటంటే, మీరు తర్వాత చూస్తారు, ఈ టాంజెన్సీ దిశ కక్ష్యలో ఉన్న వస్తువు యొక్క వేగానికి అనుగుణంగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line.", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line.", "translatedText": "సూత్రప్రాయంగా, మేము చాలా చిన్న ముక్కలను మాత్రమే పరిగణలోకి తీసుకుంటాము, కాబట్టి వ్యాసార్థం ద్వారా నా ఉద్దేశ్యంలో ఎటువంటి అస్పష్టత ఉండదు మరియు ఆర్క్ పొడవు సరళ రేఖగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play.", + "input": "Alright, now think about how the inverse square law comes into play.", "translatedText": "విలోమ చతురస్ర చట్టం ఎలా అమలులోకి వస్తుందో ఆలోచించండి.", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again?", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again?", "translatedText": "మరియు ఇది బహుశా మీరు మీ నుదురు ముడుచుకుని తిరిగి ఆలోచించాల్సిన క్షణం కావచ్చు, సరే ఒక్క నిమిషం ఆగండి, ఆ రుజువులో మళ్లీ ఏమి జరుగుతోంది?", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be.", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be.", "translatedText": "దీని అర్థం ఏమిటంటే, వృత్తం యొక్క కేంద్రానికి సంబంధించి క్షితిజ సమాంతరంగా ఉన్న తీటా డిగ్రీల దూరంలో ఉన్న మన చిన్న దీర్ఘవృత్తాకార బిందువు అసాధారణ రేఖకు లంబంగా టాంజెంట్ వాలును కలిగి ఉంటుంది మరియు మొత్తం 90 డిగ్రీల భ్రమణ విషయం కారణంగా, ఇది మనకు అవసరమైన వేగం వెక్టార్‌కు సమాంతరంగా ఉండాలి.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/thai/sentence_translations.json b/2018/feynmans-lost-lecture/thai/sentence_translations.json index 89a90d72f..6a01fbc61 100644 --- a/2018/feynmans-lost-lecture/thai/sentence_translations.json +++ b/2018/feynmans-lost-lecture/thai/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line. ", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line. ", "translatedText": "เนื่องจากชิ้นเหล่านี้มีมุมเท่ากัน เมื่อคุณอยู่ห่างจากดวงอาทิตย์มากขึ้น รัศมีไม่เพียงเพิ่มขึ้นเท่านั้น แต่องค์ประกอบของความยาวส่วนโค้งที่ตั้งฉากกับเส้นรัศมีนั้นจะเพิ่มขึ้นตามสัดส่วนของรัศมีด้วย พื้นที่ของชิ้นใดชิ้นหนึ่งเหล่านี้ และด้วยเหตุนี้เวลาที่วัตถุใช้ในการเคลื่อนที่ จึงเป็นสัดส่วนกับระยะห่างจากดวงอาทิตย์ยกกำลังสอง โดยหลักการแล้ว เราจะพิจารณาเฉพาะชิ้นที่เล็กมากเท่านั้น ดังนั้นรัศมีที่ฉันหมายถึงจึงไม่มีความคลุมเครือ และความยาวส่วนโค้งจะเป็นเส้นตรง ลองคิดดูว่ากฎกำลังสองผกผันเข้ามามีบทบาทอย่างไร ณ จุดใดก็ตาม แรงที่ดวงอาทิตย์ส่งใส่วัตถุจะเป็นสัดส่วนกับ 1 หารด้วยรัศมียกกำลังสอง แต่นั่นหมายถึงอะไรจริงๆ? ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play. ", + "input": "Alright, now think about how the inverse square law comes into play. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again? ", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be. ", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/turkish/sentence_translations.json b/2018/feynmans-lost-lecture/turkish/sentence_translations.json index a03f1fdff..a7d6982c7 100644 --- a/2018/feynmans-lost-lecture/turkish/sentence_translations.json +++ b/2018/feynmans-lost-lecture/turkish/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", "translatedText": "Bunun önemli olmasının nedeni, daha sonra göreceğiniz gibi, bu teğetlik yönünün yörüngedeki bir nesnenin hızına karşılık gelmesidir.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line.", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line.", "translatedText": "Prensip olarak, yalnızca çok küçük dilimleri dikkate alacağız, dolayısıyla yarıçapla ne kastettiğimde herhangi bir belirsizlik olmayacak ve yay uzunluğu düz bir çizgi olacaktır.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play.", + "input": "Alright, now think about how the inverse square law comes into play.", "translatedText": "Ters kare yasasının nasıl devreye girdiğini düşünün.", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again?", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again?", "translatedText": "Ve bu muhtemelen kaşlarınızı çatmanız ve geriye dönüp düşünmeniz gereken an, tamam durun bir dakika, o kanıtta yine ne oluyordu?", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be.", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be.", "translatedText": "Bunun anlamı, küçük elipsin, dairenin merkezine göre yataydan teta derece uzakta olan noktasının, eksantrik çizgiye dik bir teğet eğime sahip olduğu ve 90 derecelik dönme olayı nedeniyle, bunun anlamı şudur: hız vektörüne paralel olmasını istiyoruz.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/ukrainian/sentence_translations.json b/2018/feynmans-lost-lecture/ukrainian/sentence_translations.json index d970a61c2..f40231f9b 100644 --- a/2018/feynmans-lost-lecture/ukrainian/sentence_translations.json +++ b/2018/feynmans-lost-lecture/ukrainian/sentence_translations.json @@ -441,7 +441,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", "translatedText": "Причина, яка є важливою, як ви побачите пізніше, полягає в тому, що цей напрямок дотику відповідатиме швидкості об’єкта, що обертається.", "n_reviews": 0, "start": 533.14, @@ -742,14 +742,14 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line.", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line.", "translatedText": "В принципі, ми будемо розглядати лише дуже маленькі шматочки, тому не буде ніякої двозначності в тому, що я маю на увазі під радіусом, а довжина дуги буде прямою лінією.", "n_reviews": 0, "start": 840.22, "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play.", + "input": "Alright, now think about how the inverse square law comes into play.", "translatedText": "Подумайте, як діє закон обернених квадратів.", "n_reviews": 0, "start": 850.84, @@ -1036,7 +1036,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again?", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again?", "translatedText": "І це, мабуть, той момент, коли вам доведеться насупити брову і подумати, добре, зачекайте хвилинку, що знову відбувалося в цьому доказі?", "n_reviews": 0, "start": 1169.97, @@ -1057,7 +1057,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be.", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be.", "translatedText": "Це означає, що точка нашого маленького еліпса, яка знаходиться на тета градусів від горизонталі відносно центру кола, має дотичний нахил, перпендикулярний до ексцентричної лінії, і через обертання на 90 градусів це означає, що це паралельно вектору швидкості, який нам потрібен.", "n_reviews": 0, "start": 1203.25, diff --git a/2018/feynmans-lost-lecture/urdu/sentence_translations.json b/2018/feynmans-lost-lecture/urdu/sentence_translations.json index 114dbca75..e4d18d993 100644 --- a/2018/feynmans-lost-lecture/urdu/sentence_translations.json +++ b/2018/feynmans-lost-lecture/urdu/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object. ", "translatedText": "اس کی اہم وجہ، جیسا کہ آپ بعد میں دیکھیں گے، یہ ہے کہ یہ ٹینجنسی سمت کسی گردش کرنے والی شے کی رفتار سے مطابقت رکھتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line. ", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line. ", "translatedText": "اصولی طور پر، ہم صرف بہت چھوٹے ٹکڑوں پر غور کرنے جا رہے ہیں، لہذا رداس سے میرا مطلب کیا ہے اس میں کوئی ابہام نہیں ہوگا، اور قوس کی لمبائی ایک سیدھی لکیر ہوگی۔", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play. ", + "input": "Alright, now think about how the inverse square law comes into play. ", "translatedText": "اس بارے میں سوچیں کہ الٹا مربع قانون کیسے کام میں آتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again? ", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be. ", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be. ", "translatedText": "اس کا مطلب یہ ہے کہ ہمارے چھوٹے بیضوی کا نقطہ، جو دائرے کے مرکز کے حوالے سے افقی سے دور تھیٹا ڈگری ہے، سنکی لکیر کے لیے ایک ٹینجنٹ ڈھلوان ہے، اور پوری 90 ڈگری گردش کی وجہ سے، اس کا مطلب ہے کہ یہ ہے رفتار ویکٹر کے متوازی ہمیں اس کی ضرورت ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/feynmans-lost-lecture/vietnamese/sentence_translations.json b/2018/feynmans-lost-lecture/vietnamese/sentence_translations.json index fa8068789..e20faf122 100644 --- a/2018/feynmans-lost-lecture/vietnamese/sentence_translations.json +++ b/2018/feynmans-lost-lecture/vietnamese/sentence_translations.json @@ -504,7 +504,7 @@ "end": 532.76 }, { - "input": "The reason that's important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", + "input": "The reason that's going to be important, as you'll see later, is that this tangency direction is going to correspond to the velocity of an orbiting object.", "translatedText": "Lý do quan trọng, như bạn sẽ thấy sau, là hướng tiếp tuyến này sẽ tương ứng với vận tốc của một vật thể đang quay quanh.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 839.32 }, { - "input": "In principle, we're only going to be considering very small slices, so there won't be any ambiguity in what I mean by the radius, and the arc length will be a straight line.", + "input": "In principle, by the way, we're only going to be considering very small slices, so there will be no ambiguity in what I mean by the radius, and the arc length will be a straight line.", "translatedText": "Về nguyên tắc, chúng ta sẽ chỉ xem xét những lát cắt rất nhỏ, do đó sẽ không có bất kỳ sự mơ hồ nào về ý nghĩa của bán kính và chiều dài cung sẽ là một đường thẳng.", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 850.04 }, { - "input": "Think about how the inverse square law comes into play.", + "input": "Alright, now think about how the inverse square law comes into play.", "translatedText": "Hãy suy nghĩ về cách phát huy tác dụng của luật bình phương nghịch đảo.", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1169.21 }, { - "input": "And this is probably the moment where you have to furrow your brow and think back, okay wait a minute, what was going on in that proof again?", + "input": "And this is probably the moment where you have to kind of furrow your brow and think back, okay, wait a minute, what was going on in that proof again?", "translatedText": "Và đây có lẽ là lúc bạn phải nhíu mày và nghĩ lại, được rồi đợi một chút, điều gì đang xảy ra trong bằng chứng đó vậy?", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1202.35 }, { - "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector we need it to be.", + "input": "What that means is that the point of our little ellipse, which is theta degrees off the horizontal circle with respect to the circle's center, has a tangent slope perpendicular to the eccentric line, and because of the whole 90 degree rotation thing, this means that it's parallel to the velocity vector that we need it to be.", "translatedText": "Điều đó có nghĩa là điểm của hình elip nhỏ của chúng ta, lệch theta độ so với phương ngang so với tâm của đường tròn, có hệ số góc tiếp tuyến vuông góc với đường lệch tâm, và do toàn bộ vật quay 90 độ, điều này có nghĩa là nó song song với vectơ vận tốc mà chúng ta cần.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/fourier-transforms/arabic/sentence_translations.json b/2018/fourier-transforms/arabic/sentence_translations.json index e81502b9f..2e67a993a 100644 --- a/2018/fourier-transforms/arabic/sentence_translations.json +++ b/2018/fourier-transforms/arabic/sentence_translations.json @@ -217,7 +217,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "الفكرة الأساسية هي أخذ هذا الرسم البياني ولفه حول دائرة.", "model": "google_nmt", "from_community_srt": "في هذه الحالة، الجزء بين الصفر وال4.5 ثانية الفكرة الرئيسية هي أن نأخذ هذه الرسمة البيانية ونقوم بلفها حول دائرة", @@ -1041,7 +1041,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "على سبيل المثال، ما ننظر إليه هنا هو كيف أنه عندما يكون لديك تردد نقي قدره نبضتان في الثانية وتدوره حول الرسم البياني بمعدل دورتين في الثانية، فإن مركز الكتلة يبقى في نفس المكان، ولكن كلما زاد طوله وتستمر هذه الإشارة، كلما زادت قيمة تحويل فورييه عند هذا التردد.", "model": "google_nmt", "from_community_srt": "ما ننظر له هنا هو عندما يكون عندك تردد نقي يساوي ذبذبتين لكل ثانية وتلفه حول المخطط بتردد دورتين لكل ثانية فإن مركز الكتلة يبقي عند نفس النقطة، إنه يتتبع نفس الشكل ولكن كلما بقيت هذه الإشارة لوقت أطول كلما كانت قيمة تحويلة فورير عن هذا التردد أكبر", diff --git a/2018/fourier-transforms/bengali/sentence_translations.json b/2018/fourier-transforms/bengali/sentence_translations.json index a874df18c..5d1e23897 100644 --- a/2018/fourier-transforms/bengali/sentence_translations.json +++ b/2018/fourier-transforms/bengali/sentence_translations.json @@ -408,7 +408,7 @@ "end": 362.98 }, { - "input": "As you increase the winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just wobbles around a bit. ", + "input": "And then as you increase that winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit. ", "translatedText": "আপনি যখন ঘুরার ফ্রিকোয়েন্সি বাড়ান, এবং গ্রাফটি বৃত্তের চারপাশে ভারসাম্য বজায় রাখে, তখন ভরের কেন্দ্রের x-কোঅর্ডিনেটটি শূন্যের কাছাকাছি চলে যায় এবং এটি একটু ঘোরাফেরা করে।", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 1081.44 }, { - "input": "Out of context, you can imagine how seeing this formula would seem daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning. ", + "input": "And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning. ", "translatedText": "প্রেক্ষাপটের বাইরে, আপনি কল্পনা করতে পারেন যে এই সূত্রটি দেখলে কতটা দুঃসাধ্য মনে হবে, কিন্তু আপনি যদি বুঝতে পারেন কীভাবে সূচকগুলি ঘূর্ণনের সাথে সামঞ্জস্যপূর্ণ, t এর ফাংশন দ্বারা এটিকে কীভাবে গুণ করা মানে গ্রাফের একটি ক্ষত-বিক্ষত সংস্করণ অঙ্কন করা, এবং কীভাবে একটি অখণ্ড জটিল মূল্যবান ফাংশনটি ভর ধারণার কেন্দ্রের পরিপ্রেক্ষিতে ব্যাখ্যা করা যেতে পারে, আপনি দেখতে পারেন কিভাবে এই পুরো জিনিসটি এটির সাথে একটি খুব সমৃদ্ধ স্বজ্ঞাত অর্থ বহন করে।", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1170.42 }, { - "input": "So let's say you have a closed bounded convex set C sitting in 3D space, and let B be the boundary of that space, the surface of your complex blob. ", + "input": "So let's say that you have a closed bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob. ", "translatedText": "তাহলে ধরা যাক আপনার কাছে একটি বদ্ধ আবদ্ধ উত্তল সেট C 3D স্পেসে বসে আছে, এবং B কে সেই স্থানের সীমানা হতে দিন, আপনার জটিল ব্লবের পৃষ্ঠ।", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1224.24 }, { - "input": "If you want the answer to that puzzler, or to learn more about what they do, or to apply for open positions, go to janestreet.com slash 3b1b. ", + "input": "If you want the answer to that puzzler, or to learn more about what they do, or to apply for open positions, go to janestreet.com slash 3b1b. Thank you. ", "translatedText": "আপনি যদি সেই ধাঁধাঁর উত্তর চান, বা তারা কী করেন সে সম্পর্কে আরও জানতে বা খোলা পদের জন্য আবেদন করতে, janestreet-এ যান।com স্ল্যাশ 3 বি 1 বি।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/fourier-transforms/chinese/sentence_translations.json b/2018/fourier-transforms/chinese/sentence_translations.json index 69eb6c8f6..4b25e1d41 100644 --- a/2018/fourier-transforms/chinese/sentence_translations.json +++ b/2018/fourier-transforms/chinese/sentence_translations.json @@ -454,7 +454,7 @@ "end": 362.98 }, { - "input": "As you increase the winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just wobbles around a bit.", + "input": "And then as you increase that winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit.", "translatedText": "当您增加缠绕频率时，图形会在圆 周上保持平衡，质心的 x 坐标 会接近零，并且只会稍微摆动。", "model": "google_nmt", "from_community_srt": "然后， 当你增加缠绕频率时， 图像就会平均分布在圆上 该质心的x坐标也就趋近于0， 并且在0附近摆动。", @@ -792,7 +792,7 @@ "end": 722.38 }, { - "input": "This thing is in two dimensions, it's got a y-coordinate as well.", + "input": "I mean, this thing is in two dimensions, it's got a y-coordinate as well.", "translatedText": "这个东西是二维的，它也有一个 y 坐标。", "model": "google_nmt", "from_community_srt": "我的意思是， 这个东西是二维的， 它还有y坐标。 而且， 就像数学中的典型情况一样， 每当你处理二维的东西时，", @@ -1123,7 +1123,7 @@ "end": 1081.44 }, { - "input": "Out of context, you can imagine how seeing this formula would seem daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", + "input": "And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", "translatedText": "脱离上下文，您可以想象看到这个公式会显得多么令人畏 惧，但是如果您了解指数如何对应于旋转，将其乘以 t 的函数 g 如何意味着绘制图形的缠绕版本，以 及如何积分 a复值函数可以用质心的概念来解释，你 可以看到这整个事情如何承载着非常丰富的直观意义。", "model": "google_nmt", "from_community_srt": "而且， 你可以看出这个公式复杂的似乎有点令人生畏。 但是， 如果你明白了指数与旋转的关系... 如果把他和函数g(t)相乘 意味着绘制一张缠绕图， 以及质心的思想， 对应了 函数的积分 就不难看出这个公式带有着非常丰富且直观的意义。", @@ -1212,7 +1212,7 @@ "end": 1170.42 }, { - "input": "So let's say you have a closed bounded convex set C sitting in 3D space, and let B be the boundary of that space, the surface of your complex blob.", + "input": "So let's say that you have a closed bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob.", "translatedText": "假设您有一个位于 3D 空间中的闭有界凸集 C ，并令 B 为该空间的边界，即复杂斑点的表面。", "model": "google_nmt", "n_reviews": 0, @@ -1281,7 +1281,7 @@ "end": 1230.54 }, { - "input": "com slash 3b1b.", + "input": "com slash 3b1b. Thank you.", "translatedText": "com 斜杠 3b1b。", "model": "google_nmt", "from_community_srt": "可以访问janestreet.com/3b1b/", diff --git a/2018/fourier-transforms/english/captions.srt b/2018/fourier-transforms/english/captions.srt index 4d8b8c07d..e876384d4 100644 --- a/2018/fourier-transforms/english/captions.srt +++ b/2018/fourier-transforms/english/captions.srt @@ -184,7 +184,7 @@ in this case the portion between 0 seconds and 4.5 seconds. 47 00:02:55,660 --> 00:03:01,080 -The key idea is to take this graph and sort of wrap it up around a circle. +The key idea is going to be to take this graph and sort of wrap it up around a circle. 48 00:03:04,980 --> 00:03:06,640 @@ -983,266 +983,270 @@ Physically, this has the effect that when a certain frequency persists for a lon then the magnitude of the Fourier transform at that frequency is scaled up more and more. 247 -00:16:36,040 --> 00:16:40,822 -For example, what we're looking at right here is how when you have a pure +00:16:36,040 --> 00:16:40,970 +For example, what we're looking at here is how when you have a pure frequency of 2 248 -00:16:40,822 --> 00:16:46,573 -frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, +00:16:40,970 --> 00:16:45,365 +beats per second and you wind it around the graph at 2 cycles per second, 249 -00:16:46,573 --> 00:16:51,743 -the center of mass stays in the same spot, but the longer that signal persists, +00:16:45,365 --> 00:16:49,880 +the center of mass stays in the same spot, just tracing out the same shape. 250 -00:16:51,743 --> 00:16:55,880 -the larger the value of the Fourier transform at that frequency. +00:16:49,880 --> 00:16:54,810 +But the longer that signal persists, the larger the value of the Fourier transform 251 +00:16:54,810 --> 00:16:55,880 +at that frequency. + +252 00:16:56,500 --> 00:16:59,605 For other frequencies, even if you just increase it by a bit, -252 +253 00:16:59,605 --> 00:17:02,911 this is cancelled out by the fact that for longer time intervals, -253 +254 00:17:02,911 --> 00:17:07,220 you're giving the wound-up graph more of a chance to balance itself around the circle. -254 +255 00:17:08,940 --> 00:17:11,716 That is a lot of different moving parts, so let's -255 +256 00:17:11,716 --> 00:17:14,160 step back and summarize what we have so far. -256 +257 00:17:14,599 --> 00:17:17,540 The Fourier transform of an intensity vs. -257 +258 00:17:17,700 --> 00:17:22,750 time function, like g of t, is a new function, which doesn't have time as an input, -258 +259 00:17:22,750 --> 00:17:27,500 but instead takes in a frequency, what I've been calling the winding frequency. -259 +260 00:17:28,680 --> 00:17:31,900 In terms of notation, by the way, the common convention is to -260 +261 00:17:31,900 --> 00:17:35,380 call this new function g-hat with a little circumflex on top of it. -261 +262 00:17:35,840 --> 00:17:38,780 The output of this function is a complex number, -262 +263 00:17:38,780 --> 00:17:43,040 some point in the 2d plane that corresponds to the strength of a given -263 +264 00:17:43,040 --> 00:17:45,020 frequency in the original signal. -264 +265 00:17:46,060 --> 00:17:49,589 The plot I've been graphing for the Fourier transform is just the real -265 +266 00:17:49,589 --> 00:17:53,020 component of that output, the x-coordinate, but you could also graph -266 +267 00:17:53,020 --> 00:17:56,500 the imaginary component separately if you wanted a fuller description. -267 +268 00:17:57,440 --> 00:18:01,440 And all of this is encapsulated inside that formula we built up. -268 +269 00:18:01,920 --> 00:18:07,017 And out of context, you can imagine how seeing this formula would seem sort of daunting, -269 +270 00:18:07,017 --> 00:18:10,625 but if you understand how exponentials correspond to rotation, -270 +271 00:18:10,625 --> 00:18:15,435 how multiplying that by the function g of t means drawing a wound up version of the -271 +272 00:18:15,435 --> 00:18:20,246 graph, and how an integral of a complex valued function can be interpreted in terms -272 +273 00:18:20,246 --> 00:18:25,229 of a center of mass idea, you can see how this whole thing carries with it a very rich -273 +274 00:18:25,229 --> 00:18:26,260 intuitive meaning. -274 +275 00:18:27,540 --> 00:18:30,640 And by the way, one quick small note before we can call this wrapped up. -275 +276 00:18:30,920 --> 00:18:33,777 Even though in practice, with things like sound editing, -276 +277 00:18:33,777 --> 00:18:36,334 you'll be integrating over a finite time interval, -277 +278 00:18:36,334 --> 00:18:40,094 the theory of Fourier transforms is often phrased where the bounds of this -278 +279 00:18:40,094 --> 00:18:42,300 integral are negative infinity and infinity. -279 +280 00:18:43,140 --> 00:18:46,688 Concretely, what that means is that you consider this expression -280 +281 00:18:46,688 --> 00:18:49,854 for all possible finite time intervals, and you just ask, -281 +282 00:18:49,854 --> 00:18:53,020 what is its limit as that time interval grows to infinity? -282 +283 00:18:54,760 --> 00:18:57,040 And man oh man, there is so much more to say. -283 +284 00:18:57,220 --> 00:18:58,800 So much, I don't want to call it done here. -284 +285 00:18:58,980 --> 00:19:01,083 This transform extends to corners of math well -285 +286 00:19:01,083 --> 00:19:03,500 beyond the idea of extracting frequencies from signal. -286 +287 00:19:04,240 --> 00:19:06,940 So the next video I put out is going to go through a couple of these, -287 +288 00:19:06,940 --> 00:19:09,140 and that's really where things start getting interesting. -288 +289 00:19:10,000 --> 00:19:13,104 So stay subscribed for when that comes out, or an alternate option -289 +290 00:19:13,104 --> 00:19:16,070 is to just binge on a couple 3Blue and Brown videos so that the -290 +291 00:19:16,070 --> 00:19:19,500 YouTube recommender is more inclined to show you new things that come out. -291 +292 00:19:19,880 --> 00:19:20,760 Really the choice is yours. -292 +293 00:19:22,640 --> 00:19:25,201 And to close things off, I have something pretty fun, -293 +294 00:19:25,201 --> 00:19:28,190 a mathematical puzzler from this video's sponsor, Jane Street, -294 +295 00:19:28,190 --> 00:19:30,420 who's looking to recruit more technical talent. -295 +296 00:19:31,200 --> 00:19:36,287 So let's say that you have a closed bounded convex set C sitting in 3D space, -296 +297 00:19:36,287 --> 00:19:41,440 and then let B be the boundary of that space, the surface of your complex blob. -297 +298 00:19:42,200 --> 00:19:47,022 Now imagine taking every possible pair of points on that surface and adding them up, -298 +299 00:19:47,022 --> 00:19:48,100 doing a vector sum. -299 +300 00:19:48,960 --> 00:19:51,320 Let's name this set of all possible sums D. -300 +301 00:19:52,020 --> 00:19:55,920 Your task is to prove that D is also a convex set. -301 +302 00:19:57,200 --> 00:19:59,490 So Jane Street is a quantitative trading firm, -302 +303 00:19:59,490 --> 00:20:03,389 and if you're the kind of person who enjoys math and solving puzzles like this, -303 +304 00:20:03,389 --> 00:20:05,972 the team there really values intellectual curiosity, -304 +305 00:20:05,972 --> 00:20:08,020 so they might be interested in hiring you. -305 +306 00:20:08,360 --> 00:20:10,720 And they're looking both for full-time employees and interns. -306 +307 00:20:11,140 --> 00:20:14,480 For my part, I can say the couple of people I've interacted with there just -307 +308 00:20:14,480 --> 00:20:17,162 seem to love math and sharing math, and when they're hiring, -308 +309 00:20:17,162 --> 00:20:20,371 they look less at a background in finance than they do at how you think, -309 +310 00:20:20,371 --> 00:20:24,240 how you learn, and how you solve problems, hence the sponsorship of a 3Blue1Brown video. -310 +311 00:20:25,000 --> 00:20:29,281 If you want the answer to that puzzler, or to learn more about what they do, -311 +312 00:20:29,281 --> 00:20:32,840 or to apply for open positions, go to janestreet.com slash 3b1b. -312 +313 00:20:41,040 --> 00:20:46,800 Thank you. diff --git a/2018/fourier-transforms/english/sentence_timings.json b/2018/fourier-transforms/english/sentence_timings.json index c118318a4..4c18be648 100644 --- a/2018/fourier-transforms/english/sentence_timings.json +++ b/2018/fourier-transforms/english/sentence_timings.json @@ -125,7 +125,7 @@ 174.94 ], [ - "The key idea is to take this graph and sort of wrap it up around a circle.", + "The key idea is going to be to take this graph and sort of wrap it up around a circle.", 175.66, 181.08 ], @@ -595,7 +595,7 @@ 995.16 ], [ - "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", 996.04, 1015.88 ], diff --git a/2018/fourier-transforms/english/transcript.txt b/2018/fourier-transforms/english/transcript.txt index cef99f857..0d5e31ff6 100644 --- a/2018/fourier-transforms/english/transcript.txt +++ b/2018/fourier-transforms/english/transcript.txt @@ -23,7 +23,7 @@ Adding up those signals really mixes them all together, so pulling them back apa The general strategy is going to be to build for ourselves a mathematical machine that treats signals with a given frequency differently from how it treats other signals. To start, consider simply taking a pure signal, say with a lowly 3 beats per second, so we can plot it easily. And let's limit ourselves to looking at a finite portion of this graph, in this case the portion between 0 seconds and 4.5 seconds. -The key idea is to take this graph and sort of wrap it up around a circle. +The key idea is going to be to take this graph and sort of wrap it up around a circle. Concretely, here's what I mean by that. Imagine a little rotating vector where at each point in time its length is equal to the height of our graph for that time. So high points of the graph correspond to a greater distance from the origin, and low points end up closer to the origin. @@ -117,7 +117,7 @@ What that means is that instead of looking at the center of mass, you would scal If the portion of the original graph you were using spanned 3 seconds, you would multiply the center of mass by 3. If it was spanning 6 seconds, you would multiply the center of mass by 6. Physically, this has the effect that when a certain frequency persists for a long time, then the magnitude of the Fourier transform at that frequency is scaled up more and more. -For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency. +For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency. For other frequencies, even if you just increase it by a bit, this is cancelled out by the fact that for longer time intervals, you're giving the wound-up graph more of a chance to balance itself around the circle. That is a lot of different moving parts, so let's step back and summarize what we have so far. The Fourier transform of an intensity vs. diff --git a/2018/fourier-transforms/french/sentence_translations.json b/2018/fourier-transforms/french/sentence_translations.json index 8a646616f..5f8620099 100644 --- a/2018/fourier-transforms/french/sentence_translations.json +++ b/2018/fourier-transforms/french/sentence_translations.json @@ -197,7 +197,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "L’idée clé est de prendre ce graphique et de l’enrouler autour d’un cercle.", "from_community_srt": "L'idée clé va être de prendre ce graphique, et de l'enrouler autour d'un cercle.", "n_reviews": 0, @@ -948,7 +948,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "Par exemple, ce que nous regardons ici, c'est comment, lorsque vous avez une fréquence pure de 2 battements par seconde et que vous l'enroulez autour du graphique à 2 cycles par seconde, le centre de masse reste au même endroit, mais plus longtemps ce signal persiste, plus la valeur de la transformée de Fourier à cette fréquence est grande.", "from_community_srt": "Par exemple, ce que nous regardons ici est la manière dont, lorsqu'on prend une fréquence pure à 2 oscillations par seconde, et qu'on l'enroule autour du graphique à deux cycles par seconde, le centre de masse reste au même endroit, non? le signal retrace juste la même forme Mais plus ce signal persiste, plus la valeur de la transformée de Fourier est importante, à cette fréquence.", "n_reviews": 0, diff --git a/2018/fourier-transforms/german/sentence_translations.json b/2018/fourier-transforms/german/sentence_translations.json index dae26dd48..2e1b23e6a 100644 --- a/2018/fourier-transforms/german/sentence_translations.json +++ b/2018/fourier-transforms/german/sentence_translations.json @@ -224,7 +224,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "Der Kerngedanke ist, dieses Diagramm zu nehmen und es um einen Kreis zu wickeln.", "model": "DeepL", "from_community_srt": "Die Hauptidee ist es, diesen Graphen zu nehmen und ihn sozusagen um einen Kreis zu wickeln.", @@ -1070,7 +1070,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "Wenn du zum Beispiel eine reine Frequenz von 2 Schlägen pro Sekunde hast und sie mit 2 Zyklen pro Sekunde um den Graphen wickelst, bleibt der Schwerpunkt an der gleichen Stelle, aber je länger das Signal anhält, desto größer wird der Wert der Fourier-Transformation bei dieser Frequenz.", "model": "DeepL", "from_community_srt": "Zum Beispiel: Was wir uns hier anschauen ist, dass wenn man eine reine Frequenz von 2 Hertz hat und diese mit 2 Drehungen pro Sekunde auf einen Kreis wickelt der Massenmittelpunt am gleichen Ort bleibt. Aber je länger dieses Signal besteht, desto größer wird der Wert der Fourier-Transformation bei dieser Frequenz.", diff --git a/2018/fourier-transforms/greek/sentence_translations.json b/2018/fourier-transforms/greek/sentence_translations.json index 2e7efc274..156e643db 100644 --- a/2018/fourier-transforms/greek/sentence_translations.json +++ b/2018/fourier-transforms/greek/sentence_translations.json @@ -200,7 +200,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "", "from_community_srt": "Η βασική ιδέα, θα είναι να πάρουμε αυτό το γράφημα, και κάπως να το τυλίξουμε γύρω από έναν κύκλο.", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "", "from_community_srt": "Για παράδειγμα, εδώ βλέπουμε το πως όταν έχεις μια καθαρή συχνότητα 2 χτυπημάτων ανά δευτερόλεπτο, και την τυλίξεις γύρω από το γράφημα στους 2 κύκλους ανά δευτερόλεπτο, το κέντρο μάζας παραμένει στο ίδιο σημείο, σωστα? Καταγράφει το ίδιο σχήμα. Αλλά όσο περισσότερο επιμένει το σήμα, τόσο μεγαλύτερη γίνεται η τιμή του μετασχηματισμού Fourier, σε αυτή τη συχνότητα.", "n_reviews": 0, diff --git a/2018/fourier-transforms/hebrew/sentence_translations.json b/2018/fourier-transforms/hebrew/sentence_translations.json index 84bc06c6f..cebcb1f65 100644 --- a/2018/fourier-transforms/hebrew/sentence_translations.json +++ b/2018/fourier-transforms/hebrew/sentence_translations.json @@ -225,7 +225,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "הרעיון המרכזי הוא לקחת את הגרף הזה ולעטוף אותו סביב מעגל.", "model": "google_nmt", "from_community_srt": "הרעיון המרכזי הולך להיות כזה, בו אנחנו ניקח את הגרף הזה, ונעטוף אותו מסביב למעגל.", @@ -1071,7 +1071,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "לדוגמה, מה שאנו מסתכלים עליו כאן הוא כיצד כאשר יש לך תדר טהור של 2 פעימות בשנייה ואתה מגלגל אותו סביב הגרף ב-2 מחזורים לשנייה, מרכז המסה נשאר באותו נקודה, אך ככל שיותר זמן. האות הזה נמשך, ככל שהערך של התמרת פורייה בתדר זה גדול יותר.", "model": "google_nmt", "from_community_srt": "למשל, מה שאנחנו מסתכלים עליו כאן הוא כיצד, כי כאשר יש לכם תדירות טהורה של 2 פעימות בשניה, ואתם מריצים אותה סביב הגרף שעושה 2 מעגלים בכל שניה, מרכז המסה נשאר באותו מקום, נכון? הוא פשוט עוקב אחר אותה צורה. אבל ככל שהאות הזה יתמיד לאורך זמן רב יותר, גדול יותר יהיה ערכו של טרנספורם פורייה, בתדירות הזאת.", diff --git a/2018/fourier-transforms/hindi/sentence_translations.json b/2018/fourier-transforms/hindi/sentence_translations.json index 99c6fb56c..7ce257405 100644 --- a/2018/fourier-transforms/hindi/sentence_translations.json +++ b/2018/fourier-transforms/hindi/sentence_translations.json @@ -357,7 +357,7 @@ "end": 362.98 }, { - "input": "As you increase the winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just wobbles around a bit.", + "input": "And then as you increase that winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit.", "translatedText": "जैसे-जैसे आप घुमावदार आवृत्ति बढ़ाते हैं, और ग्राफ सर्कल के चारों ओर संतुलित होता है, द्रव्यमान के उस केंद्र का एक्स-निर्देशांक शून्य के करीब चला जाता है, और यह बस थोड़ा सा घूम जाता है।", "n_reviews": 0, "start": 363.74, @@ -623,7 +623,7 @@ "end": 722.38 }, { - "input": "This thing is in two dimensions, it's got a y-coordinate as well.", + "input": "I mean, this thing is in two dimensions, it's got a y-coordinate as well.", "translatedText": "यह चीज़ दो आयामों में है, इसमें y-निर्देशांक भी है।", "n_reviews": 0, "start": 722.52, @@ -882,7 +882,7 @@ "end": 1081.44 }, { - "input": "Out of context, you can imagine how seeing this formula would seem daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", + "input": "And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", "translatedText": "संदर्भ से बाहर, आप कल्पना कर सकते हैं कि इस सूत्र को देखना कितना कठिन प्रतीत होगा, लेकिन यदि आप समझते हैं कि घातांक रोटेशन के अनुरूप कैसे हैं, तो टी के फ़ंक्शन जी से इसे गुणा करने का मतलब ग्राफ़ का एक घुमावदार संस्करण बनाना है, और ए का एक अभिन्न अंग कैसे है जटिल मूल्यवान फ़ंक्शन की व्याख्या जन विचार के केंद्र के संदर्भ में की जा सकती है, आप देख सकते हैं कि यह पूरी चीज़ अपने साथ एक बहुत ही समृद्ध सहज अर्थ रखती है।", "n_reviews": 0, "start": 1081.92, @@ -952,7 +952,7 @@ "end": 1170.42 }, { - "input": "So let's say you have a closed bounded convex set C sitting in 3D space, and let B be the boundary of that space, the surface of your complex blob.", + "input": "So let's say that you have a closed bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob.", "translatedText": "तो मान लीजिए कि आपके पास 3डी स्थान में एक बंद घिरा हुआ उत्तल सेट सी है, और बी को उस स्थान की सीमा, आपके जटिल ब्लॉब की सतह दें।", "n_reviews": 0, "start": 1171.2, @@ -1008,7 +1008,7 @@ "end": 1230.54 }, { - "input": "com slash 3b1b.", + "input": "com slash 3b1b. Thank you.", "translatedText": "कॉम स्लैश 3बी1बी।", "n_reviews": 0, "start": 1230.54, diff --git a/2018/fourier-transforms/hungarian/sentence_translations.json b/2018/fourier-transforms/hungarian/sentence_translations.json index be5f7dd11..a67add60e 100644 --- a/2018/fourier-transforms/hungarian/sentence_translations.json +++ b/2018/fourier-transforms/hungarian/sentence_translations.json @@ -200,7 +200,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "A kulcsötlet az, hogy fogjuk ezt a grafikont, és egy kör kör köré tekerjük.", "model": "DeepL", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "Itt például azt nézzük, hogy ha van egy 2 ütem/másodperc tiszta frekvenciánk, és másodpercenként 2 ciklusonként tekerjük körbe a grafikonon, akkor a tömegközéppont ugyanott marad, de minél tovább áll fenn ez a jel, annál nagyobb lesz a Fourier-transzformáció értéke az adott frekvencián.", "model": "DeepL", "n_reviews": 0, diff --git a/2018/fourier-transforms/indonesian/sentence_translations.json b/2018/fourier-transforms/indonesian/sentence_translations.json index 13c93447f..e0a6531ed 100644 --- a/2018/fourier-transforms/indonesian/sentence_translations.json +++ b/2018/fourier-transforms/indonesian/sentence_translations.json @@ -408,7 +408,7 @@ "end": 362.98 }, { - "input": "As you increase the winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just wobbles around a bit.", + "input": "And then as you increase that winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit.", "translatedText": "Saat Anda meningkatkan frekuensi belitan, dan grafik menjadi seimbang di sekeliling lingkaran, koordinat x dari pusat massa tersebut semakin mendekati nol, dan grafik tersebut hanya bergoyang sedikit.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 722.38 }, { - "input": "This thing is in two dimensions, it's got a y-coordinate as well.", + "input": "I mean, this thing is in two dimensions, it's got a y-coordinate as well.", "translatedText": "Benda ini berbentuk dua dimensi, dan mempunyai koordinat y juga.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 1081.44 }, { - "input": "Out of context, you can imagine how seeing this formula would seem daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", + "input": "And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", "translatedText": "Di luar konteks, Anda dapat membayangkan betapa sulitnya melihat rumus ini, tetapi jika Anda memahami bagaimana eksponensial berhubungan dengan rotasi, bagaimana mengalikannya dengan fungsi g dari t berarti menggambar versi grafik yang diringkas, dan bagaimana integral dari a fungsi bernilai kompleks dapat diinterpretasikan dalam istilah pusat ide massa, Anda dapat melihat bagaimana semua ini membawa serta makna intuitif yang sangat kaya.", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1170.42 }, { - "input": "So let's say you have a closed bounded convex set C sitting in 3D space, and let B be the boundary of that space, the surface of your complex blob.", + "input": "So let's say that you have a closed bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob.", "translatedText": "Jadi, misalkan Anda memiliki himpunan C cembung berbatas tertutup yang berada dalam ruang 3D, dan misalkan B menjadi batas ruang tersebut, yaitu permukaan gumpalan kompleks Anda.", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1230.54 }, { - "input": "com slash 3b1b.", + "input": "com slash 3b1b. Thank you.", "translatedText": "com garis miring 3b1b.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/fourier-transforms/italian/sentence_translations.json b/2018/fourier-transforms/italian/sentence_translations.json index 5b7db1ba6..0907ec9cf 100644 --- a/2018/fourier-transforms/italian/sentence_translations.json +++ b/2018/fourier-transforms/italian/sentence_translations.json @@ -225,7 +225,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "L'idea chiave è quella di prendere questo grafico e avvolgerlo intorno a un cerchio.", "model": "DeepL", "from_community_srt": "L'idea chiave, sarà quella di prendere questo grafico, e in qualche modo avvolgerlo intorno ad un cerchio.", @@ -1071,7 +1071,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "Ad esempio, quello che stiamo osservando qui è che quando abbiamo una frequenza pura di 2 battiti al secondo e la avvolgiamo intorno al grafico a 2 cicli al secondo, il centro di massa rimane nello stesso punto, ma più a lungo il segnale persiste, più grande è il valore della trasformata di Fourier a quella frequenza.", "model": "DeepL", "from_community_srt": "Ad esempio, quello che stiamo guardando qui è come, quando si ha una frequenza di due battiti al secondo, e la avvolgi intorno al grafico a due giri al secondo, il centro di massa rimane nello stesso punto, giusto? Sta solo disegnando la stessa forma. Ma più a lungo persiste quel segnale, maggiore è il valore della trasformata di Fourier, a quella frequenza.", diff --git a/2018/fourier-transforms/japanese/sentence_translations.json b/2018/fourier-transforms/japanese/sentence_translations.json index 769340f3a..c400ad4f2 100644 --- a/2018/fourier-transforms/japanese/sentence_translations.json +++ b/2018/fourier-transforms/japanese/sentence_translations.json @@ -408,7 +408,7 @@ "end": 362.98 }, { - "input": "As you increase the winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just wobbles around a bit. ", + "input": "And then as you increase that winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit. ", "translatedText": "巻きの周波数を増やすと、円の周りでグラフの バランスが取れ、重心の x 座標がゼロに 近づき、少しだけぐらつくようになります。", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 1081.44 }, { - "input": "Out of context, you can imagine how seeing this formula would seem daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning. ", + "input": "And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning. ", "translatedText": "文脈を無視してこの式を見るとどれほど気が遠くなるか想像できるでしょうが、指数 関数がどのように回転に対応するのか、それに t の関数 g を掛けることがど のようにグラフの巻き上げ版を描くことを意味するのか、そして の積分がどのよう に行われるのかを理解していれば、複素数値関数は重心の概念の観点から解釈できま す。この全体がどのように非常に豊かな直感的な意味を持っているかがわかります。", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1170.42 }, { - "input": "So let's say you have a closed bounded convex set C sitting in 3D space, and let B be the boundary of that space, the surface of your complex blob. ", + "input": "So let's say that you have a closed bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob. ", "translatedText": "したがって、3D 空間にある閉じた有界凸集合 C があり、B をその空間の境界、つまり複雑なブロブの表面であるとします。", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1224.24 }, { - "input": "If you want the answer to that puzzler, or to learn more about what they do, or to apply for open positions, go to janestreet.com slash 3b1b. ", + "input": "If you want the answer to that puzzler, or to learn more about what they do, or to apply for open positions, go to janestreet.com slash 3b1b. Thank you. ", "translatedText": "謎解きの答えが知りたい場合、彼らが何をしているのか詳しく知りたい場合、または募集中のポジション に応募したい場合は、janestreet にアクセスしてください。コムスラッシュ3b1b。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/fourier-transforms/korean/sentence_translations.json b/2018/fourier-transforms/korean/sentence_translations.json index e67fdf7c2..e59bf40a2 100644 --- a/2018/fourier-transforms/korean/sentence_translations.json +++ b/2018/fourier-transforms/korean/sentence_translations.json @@ -224,7 +224,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "핵심 아이디어는 이 그래프를 원으로 감싸는 것입니다.", "model": "DeepL", "from_community_srt": "핵심 아이디어는 이 그래프를 하나의 원 주변에 감는다고 생각해 보는 것입니다.", @@ -1069,7 +1069,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "예를 들어, 초당 2비트인 순수한 주파수를 초당 2사이클로 그래프에 감으면 질량 중심은 같은 위치에 유지되지만 신호가 오래 지속될수록 해당 주파수에서 푸리에 변환의 값이 커지는 것을 볼 수 있습니다.", "model": "DeepL", "from_community_srt": "예를 들어서 주파수 2인 순수한 코사인파 신호가 있고 1초에 2바퀴를 도는 감는 주파수를 택하면 무게중심은 변하지 않을 겁니다. 구간이 바뀌어도 모양은 바뀌지 않으니까요. 하지만 신호가 오래 지속될수록 그 주파수에서의 푸리에 변환 값은 커질 겁니다.", diff --git a/2018/fourier-transforms/marathi/sentence_translations.json b/2018/fourier-transforms/marathi/sentence_translations.json index 112253494..a452de4d2 100644 --- a/2018/fourier-transforms/marathi/sentence_translations.json +++ b/2018/fourier-transforms/marathi/sentence_translations.json @@ -408,7 +408,7 @@ "end": 362.98 }, { - "input": "As you increase the winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just wobbles around a bit.", + "input": "And then as you increase that winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit.", "translatedText": "जसजसे तुम्ही वळणाची वारंवारता वाढवता, आणि आलेख वर्तुळाभोवती संतुलित होतो, तेव्हा वस्तुमानाच्या केंद्राचा x-समन्वय शून्याच्या जवळ जातो आणि तो थोडासा वळवळतो.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 722.38 }, { - "input": "This thing is in two dimensions, it's got a y-coordinate as well.", + "input": "I mean, this thing is in two dimensions, it's got a y-coordinate as well.", "translatedText": "ही गोष्ट दोन आयामांमध्ये आहे, तिला y-समन्वय देखील आहे.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 1081.44 }, { - "input": "Out of context, you can imagine how seeing this formula would seem daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", + "input": "And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", "translatedText": "संदर्भाबाहेर, तुम्ही कल्पना करू शकता की हे सूत्र पाहणे किती त्रासदायक वाटेल, परंतु घातांक रोटेशनशी कसे जुळतात हे समजल्यास, t च्या फंक्शनने किती गुणाकार करणे म्हणजे आलेखाची घसरलेली आवृत्ती काढणे आणि a चा अविभाज्य भाग कसा बनतो. कॉम्प्लेक्स व्हॅल्यूड फंक्शनचा वस्तुमान कल्पनेच्या केंद्राच्या संदर्भात अर्थ लावला जाऊ शकतो, आपण पाहू शकता की या संपूर्ण गोष्टीचा त्याच्यासोबत एक अतिशय समृद्ध अंतर्ज्ञानी अर्थ कसा आहे.", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1170.42 }, { - "input": "So let's say you have a closed bounded convex set C sitting in 3D space, and let B be the boundary of that space, the surface of your complex blob.", + "input": "So let's say that you have a closed bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob.", "translatedText": "तर समजा तुमच्याकडे 3D जागेत बसलेला बंद बद्ध बहिर्वक्र संच C आहे, आणि B ही त्या जागेची सीमा, तुमच्या जटिल ब्लॉबची पृष्ठभाग असू द्या.", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1230.54 }, { - "input": "com slash 3b1b.", + "input": "com slash 3b1b. Thank you.", "translatedText": "com स्लॅश 3b1b.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/fourier-transforms/persian/sentence_translations.json b/2018/fourier-transforms/persian/sentence_translations.json index 7156d4924..85c45a811 100644 --- a/2018/fourier-transforms/persian/sentence_translations.json +++ b/2018/fourier-transforms/persian/sentence_translations.json @@ -220,7 +220,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "ایده اصلی این است که این نمودار را بگیرید و به نوعی آن را دور یک دایره بپیچید.", "model": "google_nmt", "from_community_srt": "ایده کلیدی می خواهد این گراف را بگیریم و آن را در اطراف دایره قرار بدهد.", @@ -1060,7 +1060,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "به عنوان مثال، آنچه ما در اینجا به آن نگاه می کنیم این است که چگونه وقتی فرکانس خالص 2 ضربه در ثانیه دارید و آن را با سرعت 2 سیکل در ثانیه به دور نمودار می پیچید، مرکز جرم در همان نقطه باقی می ماند، اما بیشتر آن سیگنال باقی می ماند، مقدار تبدیل فوریه در آن فرکانس بزرگتر است.", "model": "google_nmt", "from_community_srt": "به عنوان مثال، آنچه ما در اینجا به دنبال آن هستیم این است که اگر یک فرکانس خالص متشکل از دو ضربه در ثانیه را داشته باشیم چگونه شما آن را در اطراف گراف با دو دوره در ثانیه بپیچانیم مرکز جرم در همان نقطه باقی می ماند، درست است؟ این فقط شکل مشابهی را نشان می دهد. اما طولانی تر کردن سیگنال پردازشی باعث، بزرگتر تر شدن مقدار تبدیل فوریه در آن فرکانس میشود.", diff --git a/2018/fourier-transforms/polish/sentence_translations.json b/2018/fourier-transforms/polish/sentence_translations.json index 6a78f18ab..705d69d34 100644 --- a/2018/fourier-transforms/polish/sentence_translations.json +++ b/2018/fourier-transforms/polish/sentence_translations.json @@ -199,7 +199,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "", "from_community_srt": "Kluczowym pomysłem będzie wzięcie tego wykresu i nawinięcie go, w pewien sposób,", "n_reviews": 0, @@ -951,7 +951,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "", "from_community_srt": "Na przykład, tutaj patrzymy na sytuację, gdy częstotliwość wynosi dwa uderzenia na sekundę, i nawijasz ją na wykres w tempie dwóch obrotów na sekundę. Środek masy pozostaje w tym samym punkcie, prawda? Po prostu śledzi ten sam kształt. Jednak im dłużej ten sygnał trwa, tym większa wartość transformaty Fouriera dla tej częstotliwości.", "n_reviews": 0, diff --git a/2018/fourier-transforms/portuguese/sentence_translations.json b/2018/fourier-transforms/portuguese/sentence_translations.json index 9cc3c2237..4bcee4f70 100644 --- a/2018/fourier-transforms/portuguese/sentence_translations.json +++ b/2018/fourier-transforms/portuguese/sentence_translations.json @@ -221,7 +221,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "A ideia principal é pegar esse gráfico e envolvê-lo em um círculo.", "model": "google_nmt", "from_community_srt": "a parte entre 0 segundos e 4,5 segundos A ideia-chave vai ser pegar esse gráfico e meio que,", @@ -1061,7 +1061,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "Por exemplo, o que estamos vendo aqui é como quando você tem uma frequência pura de 2 batimentos por segundo e a gira no gráfico a 2 ciclos por segundo, o centro de massa permanece no mesmo lugar, mas quanto mais tempo esse sinal persistir, maior será o valor da transformada de Fourier nessa frequência.", "model": "google_nmt", "from_community_srt": "mais e mais Por exemplo, o que estamos olhando aqui É que quando você tem uma frequência pura de 2Hz e você o enrola no gráfico a 2 ciclos por segundo, o centro de massa fica no mesmo ponto, certo? Só está desenhando a mesma forma Mas quanto mais o sinal persiste, maior é o valor da Transformada de Fourier naquela frequência", diff --git a/2018/fourier-transforms/russian/sentence_translations.json b/2018/fourier-transforms/russian/sentence_translations.json index 13a09522a..7c01e99b4 100644 --- a/2018/fourier-transforms/russian/sentence_translations.json +++ b/2018/fourier-transforms/russian/sentence_translations.json @@ -197,7 +197,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "Основная идея — взять этот график и как бы обернуть его вокруг круга.", "from_community_srt": "Ключевая идея, состоит в том, чтобы взять этот график и обернуть его вокруг некоторой окружности...", "n_reviews": 0, @@ -949,7 +949,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "Например, здесь мы рассматриваем то, как, когда у вас чистая частота 2 удара в секунду и вы вращаете ее по графику со скоростью 2 цикла в секунду, центр массы остается в том же месте, но чем дольше этот сигнал сохраняется, тем больше значение преобразования Фурье на этой частоте.", "from_community_srt": "Например, то, что мы сейчас видим, это то как выглядит одночастотный сигнал два колебания в секунду, и при равной этому значению частоте намотки центр масс остается на одном и том же месте, не так ли? Форма сигнала повторяется. Но чем дольше сохраняется этот сигнал, тем больше значение преобразования Фурье на этой частоте.", "n_reviews": 0, diff --git a/2018/fourier-transforms/spanish/sentence_translations.json b/2018/fourier-transforms/spanish/sentence_translations.json index ef6750d2b..6c3ad4b28 100644 --- a/2018/fourier-transforms/spanish/sentence_translations.json +++ b/2018/fourier-transforms/spanish/sentence_translations.json @@ -199,7 +199,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "La idea clave es tomar este gráfico y envolverlo alrededor de un círculo.", "from_community_srt": "La idea clave es tomar esta grafica y enrollarla en torno a un circulo.", "n_reviews": 0, @@ -945,7 +945,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "Por ejemplo, lo que estamos viendo aquí es cómo cuando tienes una frecuencia pura de 2 latidos por segundo y la enrollas alrededor del gráfico a 2 ciclos por segundo, el centro de masa permanece en el mismo lugar, pero cuanto más tiempo Mientras más persista esa señal, mayor será el valor de la transformada de Fourier en esa frecuencia.", "from_community_srt": "Por ejemplo, lo que estamos viendo ahora es cómo cuando tenemos una frecuencia pura de 2 vibraciones por segundo, y enrollas la gráfica a 2 ciclos por segundo, el centro de masa permanece en la misma dirección, ¿verdad? Ya que la gráfica tiene la misma forma. Pero a medida que la señal persista, mayor será el valor de la Transformada de Fourier en dicha frecuencia.", "n_reviews": 0, diff --git a/2018/fourier-transforms/tamil/sentence_translations.json b/2018/fourier-transforms/tamil/sentence_translations.json index d23f42d55..38384789a 100644 --- a/2018/fourier-transforms/tamil/sentence_translations.json +++ b/2018/fourier-transforms/tamil/sentence_translations.json @@ -408,7 +408,7 @@ "end": 362.98 }, { - "input": "As you increase the winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just wobbles around a bit.", + "input": "And then as you increase that winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit.", "translatedText": "நீங்கள் முறுக்கு அதிர்வெண்ணை அதிகரிக்கும்போது, வரைபடம் வட்டத்தைச் சுற்றி சமநிலைப்படுத்தும்போது, அந்த வெகுஜன மையத்தின் x- ஒருங்கிணைப்பு பூஜ்ஜியத்திற்கு அருகில் செல்கிறது, மேலும் அது சிறிது தள்ளாடுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 722.38 }, { - "input": "This thing is in two dimensions, it's got a y-coordinate as well.", + "input": "I mean, this thing is in two dimensions, it's got a y-coordinate as well.", "translatedText": "இந்த விஷயம் இரண்டு பரிமாணங்களில் உள்ளது, இது ஒரு y-கோர்டினேட்டையும் பெற்றுள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 1081.44 }, { - "input": "Out of context, you can imagine how seeing this formula would seem daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", + "input": "And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", "translatedText": "சூழலுக்கு வெளியே, இந்த சூத்திரத்தைப் பார்ப்பது எப்படி அச்சுறுத்தலாகத் தோன்றும் என்பதை நீங்கள் கற்பனை செய்யலாம், ஆனால் அதிவேகங்கள் சுழற்சியுடன் எவ்வாறு ஒத்துப்போகின்றன என்பதை நீங்கள் புரிந்து கொண்டால், t இன் செயல்பாட்டின் மூலம் அதை எவ்வாறு பெருக்குவது என்பது வரைபடத்தின் மாற்றப்பட்ட பதிப்பை வரைவது மற்றும் எப்படி ஒரு ஒருங்கிணைந்த சிக்கலான மதிப்புள்ள செயல்பாட்டை வெகுஜன யோசனையின் மையத்தின் அடிப்படையில் விளக்கலாம், இந்த முழு விஷயமும் எவ்வாறு மிகவும் பணக்கார உள்ளுணர்வு பொருளைக் கொண்டுள்ளது என்பதை நீங்கள் பார்க்கலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1170.42 }, { - "input": "So let's say you have a closed bounded convex set C sitting in 3D space, and let B be the boundary of that space, the surface of your complex blob.", + "input": "So let's say that you have a closed bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob.", "translatedText": "எனவே நீங்கள் 3D இடத்தில் அமர்ந்து மூடிய எல்லைக்குட்பட்ட குவிந்த செட் C ஐ வைத்திருப்பதாக வைத்துக்கொள்வோம், மேலும் B ஆனது அந்த இடத்தின் எல்லையாக இருக்கட்டும், உங்கள் சிக்கலான குமிழியின் மேற்பரப்பாகும்.", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1230.54 }, { - "input": "com slash 3b1b.", + "input": "com slash 3b1b. Thank you.", "translatedText": "com ஸ்லாஷ் 3b1b.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/fourier-transforms/telugu/sentence_translations.json b/2018/fourier-transforms/telugu/sentence_translations.json index 78ecb410f..67c09c87e 100644 --- a/2018/fourier-transforms/telugu/sentence_translations.json +++ b/2018/fourier-transforms/telugu/sentence_translations.json @@ -408,7 +408,7 @@ "end": 362.98 }, { - "input": "As you increase the winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just wobbles around a bit.", + "input": "And then as you increase that winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit.", "translatedText": "మీరు వైండింగ్ ఫ్రీక్వెన్సీని పెంచినప్పుడు మరియు వృత్తం చుట్టూ గ్రాఫ్ బ్యాలెన్స్ చేస్తున్నప్పుడు, ఆ ద్రవ్యరాశి కేంద్రం యొక్క x-కోఆర్డినేట్ సున్నాకి దగ్గరగా ఉంటుంది మరియు అది కొంచెం చుట్టూ తిరుగుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 722.38 }, { - "input": "This thing is in two dimensions, it's got a y-coordinate as well.", + "input": "I mean, this thing is in two dimensions, it's got a y-coordinate as well.", "translatedText": "ఈ విషయం రెండు కోణాలలో ఉంది, దీనికి y-కోఆర్డినేట్ కూడా ఉంది.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 1081.44 }, { - "input": "Out of context, you can imagine how seeing this formula would seem daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", + "input": "And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", "translatedText": "సందర్భం లేకుండా, ఈ ఫార్ములాను చూడటం ఎలా భయంకరంగా అనిపిస్తుందో మీరు ఊహించవచ్చు, అయితే ఎక్స్‌పోనెన్షియల్‌లు భ్రమణానికి ఎలా అనుగుణంగా ఉంటాయో మీరు అర్థం చేసుకుంటే, t యొక్క ఫంక్షన్ ద్వారా దాన్ని ఎంత గుణించడం అంటే గ్రాఫ్ యొక్క వ్రాతపూర్వక సంస్కరణను గీయడం మరియు ఒక యొక్క సమగ్రత ఎలా కాంప్లెక్స్ వాల్యూడ్ ఫంక్షన్‌ను మాస్ ఐడియా యొక్క సెంటర్ పరంగా అర్థం చేసుకోవచ్చు, ఈ మొత్తం విషయం దానితో చాలా గొప్ప సహజమైన అర్థాన్ని ఎలా తీసుకువెళుతుందో మీరు చూడవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1170.42 }, { - "input": "So let's say you have a closed bounded convex set C sitting in 3D space, and let B be the boundary of that space, the surface of your complex blob.", + "input": "So let's say that you have a closed bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob.", "translatedText": "కాబట్టి మీరు 3D స్పేస్‌లో కూర్చున్న క్లోజ్డ్ బౌండెడ్ కుంభాకార సెట్‌ని కలిగి ఉన్నారని అనుకుందాం మరియు B ఆ స్థలం యొక్క సరిహద్దుగా, మీ కాంప్లెక్స్ బొట్టు యొక్క ఉపరితలంగా ఉండనివ్వండి.", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1230.54 }, { - "input": "com slash 3b1b.", + "input": "com slash 3b1b. Thank you.", "translatedText": "com స్లాష్ 3b1b.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/fourier-transforms/thai/sentence_translations.json b/2018/fourier-transforms/thai/sentence_translations.json index e8f40160a..3f3d76693 100644 --- a/2018/fourier-transforms/thai/sentence_translations.json +++ b/2018/fourier-transforms/thai/sentence_translations.json @@ -408,7 +408,7 @@ "end": 362.98 }, { - "input": "As you increase the winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just wobbles around a bit. ", + "input": "And then as you increase that winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 1081.44 }, { - "input": "Out of context, you can imagine how seeing this formula would seem daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning. ", + "input": "And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1170.42 }, { - "input": "So let's say you have a closed bounded convex set C sitting in 3D space, and let B be the boundary of that space, the surface of your complex blob. ", + "input": "So let's say that you have a closed bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1224.24 }, { - "input": "If you want the answer to that puzzler, or to learn more about what they do, or to apply for open positions, go to janestreet.com slash 3b1b. ", + "input": "If you want the answer to that puzzler, or to learn more about what they do, or to apply for open positions, go to janestreet.com slash 3b1b. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/fourier-transforms/turkish/sentence_translations.json b/2018/fourier-transforms/turkish/sentence_translations.json index 5a055839c..b69f0a178 100644 --- a/2018/fourier-transforms/turkish/sentence_translations.json +++ b/2018/fourier-transforms/turkish/sentence_translations.json @@ -408,7 +408,7 @@ "end": 362.98 }, { - "input": "As you increase the winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just wobbles around a bit.", + "input": "And then as you increase that winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit.", "translatedText": "Sarma frekansını artırdığınızda ve grafik daire etrafında dengelendiğinde, kütle merkezinin x koordinatı sıfıra yaklaşır ve sadece biraz sallanır.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 722.38 }, { - "input": "This thing is in two dimensions, it's got a y-coordinate as well.", + "input": "I mean, this thing is in two dimensions, it's got a y-coordinate as well.", "translatedText": "Bu şey iki boyutlu, aynı zamanda bir y koordinatı da var.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 1081.44 }, { - "input": "Out of context, you can imagine how seeing this formula would seem daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", + "input": "And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.", "translatedText": "Bağlam dışında, bu formülü görmenin ne kadar göz korkutucu görünebileceğini hayal edebilirsiniz, ancak üstel sayıların dönüşe nasıl karşılık geldiğini, bunu g t fonksiyonuyla çarpmanın grafiğin tamamlanmış bir versiyonunu çizmek anlamına geldiğini ve bir a'nın integralinin nasıl olduğunu anlarsanız karmaşık değerli fonksiyon bir kütle merkezi fikri açısından yorumlanabilir, tüm bu şeyin nasıl çok zengin bir sezgisel anlam taşıdığını görebilirsiniz.", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1170.42 }, { - "input": "So let's say you have a closed bounded convex set C sitting in 3D space, and let B be the boundary of that space, the surface of your complex blob.", + "input": "So let's say that you have a closed bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob.", "translatedText": "Diyelim ki 3 boyutlu uzayda duran kapalı sınırlı dışbükey bir C kümeniz var ve B'nin bu uzayın sınırı, yani karmaşık bloğunuzun yüzeyi olmasına izin verin.", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1230.54 }, { - "input": "com slash 3b1b.", + "input": "com slash 3b1b. Thank you.", "translatedText": "com eğik çizgi 3b1b.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/fourier-transforms/ukrainian/sentence_translations.json b/2018/fourier-transforms/ukrainian/sentence_translations.json index fb8dbb808..3a04b4ca7 100644 --- a/2018/fourier-transforms/ukrainian/sentence_translations.json +++ b/2018/fourier-transforms/ukrainian/sentence_translations.json @@ -408,7 +408,7 @@ "end": 362.98 }, { - "input": "As you increase the winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just wobbles around a bit. ", + "input": "And then as you increase that winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit. ", "translatedText": "Коли ви збільшуєте частоту намотування, і графік балансує навколо кола, координата x цього центру мас наближається до нуля, і він просто трохи коливається. ", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 1081.44 }, { - "input": "Out of context, you can imagine how seeing this formula would seem daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning. ", + "input": "And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning. ", "translatedText": "Поза контекстом ви можете собі уявити, як дивитися на цю формулу буде страшно, але якщо ви зрозумієте, як експоненти відповідають обертанню, як множення цього на функцію g від t означає накреслення завершеної версії графіка, і як інтеграл від a комплекснозначну функцію можна інтерпретувати в термінах ідеї центру маси, ви можете побачити, як усе це несе в собі дуже багате інтуїтивне значення. ", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1170.42 }, { - "input": "So let's say you have a closed bounded convex set C sitting in 3D space, and let B be the boundary of that space, the surface of your complex blob. ", + "input": "So let's say that you have a closed bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob. ", "translatedText": "Отже, припустимо, у вас є замкнена обмежена опукла множина C, яка знаходиться в 3D-просторі, і нехай B буде межею цього простору, поверхнею вашої складної краплі. ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1224.24 }, { - "input": "If you want the answer to that puzzler, or to learn more about what they do, or to apply for open positions, go to janestreet.com slash 3b1b. ", + "input": "If you want the answer to that puzzler, or to learn more about what they do, or to apply for open positions, go to janestreet.com slash 3b1b. Thank you. ", "translatedText": "Якщо ви хочете отримати відповідь на цю головоломку або дізнатися більше про те, що вони роблять, або подати заявку на відкриті вакансії, перейдіть на janestreet. com слеш 3b1b. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/fourier-transforms/urdu/sentence_translations.json b/2018/fourier-transforms/urdu/sentence_translations.json index 12dcc33d5..d5df8f512 100644 --- a/2018/fourier-transforms/urdu/sentence_translations.json +++ b/2018/fourier-transforms/urdu/sentence_translations.json @@ -408,7 +408,7 @@ "end": 362.98 }, { - "input": "As you increase the winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just wobbles around a bit. ", + "input": "And then as you increase that winding frequency, and the graph balances out around the circle, the x-coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit. ", "translatedText": "جیسے جیسے آپ سمیٹنے کی فریکوئنسی میں اضافہ کرتے ہیں، اور گراف دائرے کے گرد بیلنس کرتا ہے، اس مرکز کا ایکس کوآرڈینیٹ صفر کے قریب جاتا ہے، اور یہ تھوڑا سا گھومتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 1081.44 }, { - "input": "Out of context, you can imagine how seeing this formula would seem daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning. ", + "input": "And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function g of t means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning. ", "translatedText": "سیاق و سباق سے ہٹ کر، آپ تصور کر سکتے ہیں کہ اس فارمولے کو دیکھنا کتنا مشکل لگتا ہے، لیکن اگر آپ یہ سمجھتے ہیں کہ ایکسپوینینشل کس طرح گردش سے مطابقت رکھتے ہیں، t کے فنکشن سے اس کو کس طرح ضرب دینے کا مطلب ہے کہ گراف کا ایک زخم اپ ورژن بنانا ہے، اور کس طرح a کا انضمام پیچیدہ قابل قدر فنکشن کی تشریح بڑے پیمانے پر خیال کے مرکز کے لحاظ سے کی جا سکتی ہے، آپ دیکھ سکتے ہیں کہ یہ پوری چیز اپنے ساتھ ایک بہت ہی بھرپور بدیہی معنی کیسے رکھتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1170.42 }, { - "input": "So let's say you have a closed bounded convex set C sitting in 3D space, and let B be the boundary of that space, the surface of your complex blob. ", + "input": "So let's say that you have a closed bounded convex set C sitting in 3D space, and then let B be the boundary of that space, the surface of your complex blob. ", "translatedText": "تو آئیے کہتے ہیں کہ آپ کے پاس 3D اسپیس میں ایک بند محدب سیٹ C بیٹھا ہوا ہے، اور B کو اس جگہ کی باؤنڈری بننے دیں، آپ کے پیچیدہ بلاب کی سطح۔", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1224.24 }, { - "input": "If you want the answer to that puzzler, or to learn more about what they do, or to apply for open positions, go to janestreet.com slash 3b1b. ", + "input": "If you want the answer to that puzzler, or to learn more about what they do, or to apply for open positions, go to janestreet.com slash 3b1b. Thank you. ", "translatedText": "اگر آپ اس پزلر کا جواب چاہتے ہیں، یا وہ کیا کرتے ہیں اس کے بارے میں مزید جاننے کے لیے، یا کھلی جگہوں کے لیے درخواست دینا چاہتے ہیں، تو جینسٹریٹ پر جائیں۔com سلیش 3b1b. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/fourier-transforms/vietnamese/sentence_translations.json b/2018/fourier-transforms/vietnamese/sentence_translations.json index b8919b64c..209516116 100644 --- a/2018/fourier-transforms/vietnamese/sentence_translations.json +++ b/2018/fourier-transforms/vietnamese/sentence_translations.json @@ -224,7 +224,7 @@ "end": 174.94 }, { - "input": "The key idea is to take this graph and sort of wrap it up around a circle.", + "input": "The key idea is going to be to take this graph and sort of wrap it up around a circle.", "translatedText": "Ý tưởng chính là lấy biểu đồ này và gói nó lại thành một vòng tròn.", "model": "google_nmt", "from_community_srt": "Ý tưởng chính, sẽ lấy biểu đồ này và sắp xếp nó quanh một vòng tròn.", @@ -1070,7 +1070,7 @@ "end": 995.16 }, { - "input": "For example, what we're looking at right here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, but the longer that signal persists, the larger the value of the Fourier transform at that frequency.", + "input": "For example, what we're looking at here is how when you have a pure frequency of 2 beats per second and you wind it around the graph at 2 cycles per second, the center of mass stays in the same spot, just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform at that frequency.", "translatedText": "Ví dụ: điều chúng ta đang xem xét ở đây là làm thế nào khi bạn có tần số thuần túy là 2 nhịp mỗi giây và bạn quấn nó quanh biểu đồ với tốc độ 2 chu kỳ mỗi giây, thì khối tâm vẫn ở cùng một vị trí, nhưng lâu hơn tín hiệu đó vẫn tồn tại thì giá trị của biến đổi Fourier ở tần số đó càng lớn.", "model": "google_nmt", "from_community_srt": "Ví dụ, những gì chúng tôi đang xem xét ngay tại đây là khi bạn có tần suất thuần túy là hai nhịp mỗi giây, và bạn xoay quanh biểu đồ ở hai chu kỳ mỗi giây, trung tâm của khối lượng vẫn ở cùng một chỗ, đúng không? Nó chỉ là truy tìm cùng một hình dạng. Nhưng tín hiệu càng dài thì giá trị của biến đổi Fourier càng lớn, ở tần số đó càng lớn.", diff --git a/2018/qa3/english/captions.srt b/2018/qa3/english/captions.srt index 6d942f712..8e29ea12c 100644 --- a/2018/qa3/english/captions.srt +++ b/2018/qa3/english/captions.srt @@ -51,7 +51,7 @@ explanation. But when I do cover quantum computing, which I will at some point... 14 -00:00:41,839 --> 00:00:43,720 +00:00:41,840 --> 00:00:43,720 What would you do professionally if it weren't 15 @@ -572,7 +572,7 @@ Do you have any questions, Jabril? 144 00:07:26,080 --> 00:07:30,100 -I'm just reading from some Reddit ones here but we do something live. +I'm just reading from, reading from some Reddit ones here, but we do something live. 145 00:07:30,320 --> 00:07:33,883 @@ -696,7 +696,7 @@ afternoon and here it's taking me like three weeks to do one video. 175 00:09:09,900 --> 00:09:11,920 -Which of these actually carries more of an impact? +You know, which of these actually carries more of an impact. 176 00:09:12,660 --> 00:09:16,720 @@ -743,16 +743,16 @@ he does really good work and just go you know download some of the music and leave him a little tip if you feel like it's something that you enjoy. 187 -00:09:49,980 --> 00:09:50,620 +00:09:49,980 --> 00:09:51,200 What is your favorite Palomino? 188 -00:09:50,620 --> 00:09:51,960 +00:09:51,560 --> 00:09:51,200 Palomino? 189 -00:09:52,180 --> 00:09:52,560 -Palimano? +00:09:51,560 --> 00:09:52,560 +Palomino? Polymono? 190 00:09:53,620 --> 00:09:54,160 @@ -771,6 +771,14 @@ All right folks, thanks for watching, stick around for whenever the next upload It's going to be on Quaternions and I hope you like it. 194 -00:10:05,920 --> 00:10:15,420 -This is your close-up and this is your wide. +00:10:05,920 --> 00:10:09,203 +This is your close-up and this is your wide. All right. + +195 +00:10:09,203 --> 00:10:12,781 +Does that probably mean I should be looking at the wide one? + +196 +00:10:12,781 --> 00:10:15,420 +Or do I do a dramatic like camera number two? diff --git a/2018/qa3/english/sentence_timings.json b/2018/qa3/english/sentence_timings.json index f3356c24d..7767f32ff 100644 --- a/2018/qa3/english/sentence_timings.json +++ b/2018/qa3/english/sentence_timings.json @@ -345,7 +345,7 @@ 444.26 ], [ - "I'm just reading from some Reddit ones here but we do something live.", + "I'm just reading from, reading from some Reddit ones here, but we do something live.", 446.08, 450.1 ], @@ -420,7 +420,7 @@ 549.48 ], [ - "Which of these actually carries more of an impact?", + "You know, which of these actually carries more of an impact.", 549.9, 551.92 ], @@ -457,16 +457,16 @@ [ "What is your favorite Palomino?", 589.98, - 590.62 + 591.2 ], [ "Palomino?", - 590.62, - 591.96 + 591.56, + 591.2 ], [ - "Palimano?", - 592.18, + "Palomino? Polymono?", + 591.56, 592.56 ], [ @@ -490,7 +490,7 @@ 603.52 ], [ - "This is your close-up and this is your wide.", + "This is your close-up and this is your wide. All right. Does that probably mean I should be looking at the wide one? Or do I do a dramatic like camera number two?", 605.92, 615.42 ] diff --git a/2018/qa3/english/transcript.txt b/2018/qa3/english/transcript.txt index cfd553c10..24085cd16 100644 --- a/2018/qa3/english/transcript.txt +++ b/2018/qa3/english/transcript.txt @@ -67,7 +67,7 @@ And there's a lot of good probability material online. I will probably do something to release the material that I have, either just as it is but on some second channel with the acknowledgement, hey this isn't the greatest work I think I've done, or trying to rework them and make them standalones. But as far as you know essence of blank content, I feel much clearer about how I would want to extend the linear algebra series rather than spinning my wheels on certain scripts and animations that I ultimately don't think are going to deliver something to you guys that I would feel proud of. Do you have any questions, Jabril? -I'm just reading from some Reddit ones here but we do something live. +I'm just reading from, reading from some Reddit ones here, but we do something live. How much compromise if any do you have to give with like what you can animate versus like what your script is trying to convey? Usually if I can't animate a thing, and it's a mathematical thing, it's not like a frivolous cartoonish type thing, I change the tool so that it can animate that thing right and then that might take more time. And it's possible that subconsciously that means I resist topics that I know would be more difficult to animate. @@ -82,7 +82,7 @@ How do you compare making your videos to making videos for Khan Academy? So very different processes right, like Khan Academy you imagine sitting next to someone and tutoring them and just explaining it, you're writing everything by hand, for the most part you do it live. On this channel I obviously script things, I put a lot of time into creating the visuals for it. Sometimes in a way that makes me feel you know if at Khan Academy I could sit down and make like three videos in an afternoon and here it's taking me like three weeks to do one video. -Which of these actually carries more of an impact? +You know, which of these actually carries more of an impact. I think there's a proper balance for both of them and I think there's a lot of people out there who do the Khan style stuff to include Khan Academy but also many others. The way I like to think about things is what wouldn't happen if I wasn't doing it? But there is that little part of me that thinks maybe I should start some sort of second channel of the super cheap, just like me and a notebook and a pencil like scrapping through some sort of explanation super quickly. @@ -91,9 +91,9 @@ Vince Rubenetti. Link in the description, link in all of the descriptions actually, he does really good work and just go you know download some of the music and leave him a little tip if you feel like it's something that you enjoy. What is your favorite Palomino? Palomino? -Palimano? +Palomino? Polymono? This one. I'll figure it out later and insert it on the screen. All right folks, thanks for watching, stick around for whenever the next upload is. It's going to be on Quaternions and I hope you like it. -This is your close-up and this is your wide. \ No newline at end of file +This is your close-up and this is your wide. All right. Does that probably mean I should be looking at the wide one? Or do I do a dramatic like camera number two? \ No newline at end of file diff --git a/2018/quaternions-and-3d-rotation/english/captions.srt b/2018/quaternions-and-3d-rotation/english/captions.srt index 514687c9d..73a566eb4 100644 --- a/2018/quaternions-and-3d-rotation/english/captions.srt +++ b/2018/quaternions-and-3d-rotation/english/captions.srt @@ -1,5 +1,5 @@ 1 -00:00:02,959 --> 00:00:05,824 +00:00:02,960 --> 00:00:05,824 In a moment, I'll point you to a separate website hosting 2 diff --git a/2018/quaternions/arabic/sentence_translations.json b/2018/quaternions/arabic/sentence_translations.json index 144120f87..c4fd36b96 100644 --- a/2018/quaternions/arabic/sentence_translations.json +++ b/2018/quaternions/arabic/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "ثم تقول للينوس، لضرب عددين مركبين، ما عليك سوى استخدام خاصية التوزيع، التي يتعلمها الكثير من الناس في المدرسة باسم FOIL، وتطبيق هذه القاعدة، i ضرب i يساوي سالب 1، لتبسيط الأمور بشكل أكبر. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "بنفس الطريقة التي تتضمن فيها الأعداد المركبة الأعداد الحقيقية ذات البعد التخيلي الإضافي الوحيد، الذي تمثله الوحدة i، وأن النظام الذي ليس في الواقع رقمًا والذي كان لدينا في ثلاثة أبعاد يتضمن اتجاهًا وهميًا ثانيًا، j، تتضمن الكواترنيونات الأعداد الحقيقية مع ثلاثة أبعاد خيالية منفصلة، ممثلة بالوحدات i وj وk. ", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "الدائرة المتعامدة مع تلك الدائرة، والتي تمر عبر i وk، يتم تدويرها بمقدار 90 درجة وفقًا لهذه القاعدة، حيث تشير بإبهامك من 1 إلى j، لذا فإن j في i يساوي سالب k وj في k هو i. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/bengali/sentence_translations.json b/2018/quaternions/bengali/sentence_translations.json index fd99c06f8..8192bff6f 100644 --- a/2018/quaternions/bengali/sentence_translations.json +++ b/2018/quaternions/bengali/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "তারপর আপনি লিনাসকে বলবেন, দুটি জটিল সংখ্যাকে গুণ করার জন্য, আপনি শুধু বন্টনমূলক সম্পত্তি ব্যবহার করুন, যা স্কুলে অনেকে FOIL হিসাবে শিখে, এবং এই নিয়মটি প্রয়োগ করুন, i বার i সমান ঋণাত্মক 1, জিনিসগুলি আরও সহজ করার জন্য।", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "একইভাবে জটিল সংখ্যাগুলি একটি একক অতিরিক্ত কাল্পনিক মাত্রা সহ বাস্তব সংখ্যাগুলিকে অন্তর্ভুক্ত করে, একক i দ্বারা উপস্থাপিত হয়, এবং আমাদের তিনটি মাত্রায় যে-না-বাস্তব-এক-সংখ্যা সিস্টেম জিনিসটি ছিল তার মধ্যে একটি দ্বিতীয় কাল্পনিক দিক অন্তর্ভুক্ত ছিল, j, চতুর্ভুজগুলি i, j এবং k একক দ্বারা উপস্থাপিত তিনটি পৃথক কাল্পনিক মাত্রা সহ বাস্তব সংখ্যাগুলিকে অন্তর্ভুক্ত করে।", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "1 এবং j এর মধ্য দিয়ে বৃত্ত, যেটিকে আমরা উৎপত্তির মধ্য দিয়ে একটি রেখা হিসাবে প্রক্ষিপ্ত দেখি, 90 ডিগ্রি ঘোরানো হয়, 1 কে j পর্যন্ত টেনে নিয়ে যায়, তাই j গুণ 1 হল 1 এবং j গুণ j হল ঋণাত্মক 1৷ সেই বৃত্তটির লম্ব, i এবং k এর মধ্য দিয়ে যাওয়া, এই নিয়ম অনুসারে 90 ডিগ্রি ঘোরানো হয়, যেখানে আপনি 1 থেকে j পর্যন্ত আপনার থাম্ব নির্দেশ করেন, তাই j গুণ i ঋণাত্মক k এবং j গুণ k হল i।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/chinese/sentence_translations.json b/2018/quaternions/chinese/sentence_translations.json index 53ac33b4f..9ea5f244f 100644 --- a/2018/quaternions/chinese/sentence_translations.json +++ b/2018/quaternions/chinese/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "然后 你对 Linus 说，要乘两个复数，你只需使用分配律（许多人在 学校里学的 FOIL），并应用这条规则，i 乘以 i 等于负 1，来进一步简化事情。", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "同样，复数包括具有单个额外 虚数维度的实数，由单位 i 表示，并且我们在三个维 度中拥有的实际上不是数字系统的东西包括第二个虚数方 向 j，四元数包括实数和三个独立的虚数维度，由单位 i、j 和 k 表示。", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "根据这一规则，垂直于该圆的圆经过 i 和 k 旋转 90 度，其中您将拇指从 1 指向 j，因 此 j 乘以 i 为负 k，j 乘以 k 为 i。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/english/captions.srt b/2018/quaternions/english/captions.srt index 2deb6c45c..935d40625 100644 --- a/2018/quaternions/english/captions.srt +++ b/2018/quaternions/english/captions.srt @@ -207,20 +207,20 @@ What I'll show you here is a way to visualize quaternions in their full four-dimensional glory. 53 -00:03:12,240 --> 00:03:15,189 +00:03:12,240 --> 00:03:15,079 It would surprise me if this approach was fully original, 54 -00:03:15,189 --> 00:03:19,105 +00:03:15,079 --> 00:03:18,848 but I can say that it's certainly not the standard way to teach quaternions, 55 -00:03:19,105 --> 00:03:23,427 -and that the specific four-dimensional right-hand rule image I'd like to build up to +00:03:18,848 --> 00:03:23,254 +and that the specific four-dimensional right-hand rule image that I'd like to build up to 56 -00:03:23,427 --> 00:03:25,360 -is something I haven't seen elsewhere. +00:03:23,254 --> 00:03:25,360 +is something that I haven't seen elsewhere. 57 00:03:25,880 --> 00:03:29,364 @@ -795,16 +795,16 @@ south pole through that point and see where it intersects the plane. So the point 1 at the north pole ends up at the center of the plane. 200 -00:13:15,780 --> 00:13:20,131 -All of the points of the northern hemisphere get mapped somewhere +00:13:15,780 --> 00:13:19,891 +all of the points of the northern hemisphere get mapped somewhere 201 -00:13:20,131 --> 00:13:24,218 -inside the unit circle of the i-j plane, and that unit circle +00:13:19,891 --> 00:13:25,249 +inside the unit circle of the i-j plane, and that unit circle which passes through i, 202 -00:13:24,218 --> 00:13:28,240 -which passes through i,j,-i,-j actually stays fixed in place. +00:13:25,249 --> 00:13:28,240 +j, negative i, negative j, stays fixed in place. 203 00:13:28,820 --> 00:13:30,940 @@ -1211,626 +1211,638 @@ Analogous to what we did for Linus and Felix, we stereographically project this hypersphere into 3D space. 304 -00:20:27,320 --> 00:20:32,072 +00:20:27,320 --> 00:20:31,766 This label in the upper right is going to show a given unit quaternion, 305 -00:20:32,072 --> 00:20:37,880 -and this little pink dot will show where that quaternion gets projected in our 3D space. +00:20:31,766 --> 00:20:37,077 +and this little pink dot will show where that particular quaternion gets projected in 306 +00:20:37,077 --> 00:20:37,880 +our 3D space. + +307 00:20:38,580 --> 00:20:41,720 Just as before, we're projecting from the number negative 1, -307 +308 00:20:41,720 --> 00:20:46,301 which sits on the real number line that is somehow perpendicular to all of our 3D space, -308 +309 00:20:46,301 --> 00:20:47,640 and beyond our perception. -309 +310 00:20:48,340 --> 00:20:53,040 Just as before, the number 1 ends up projected straight into the center of our space. -310 -00:20:53,740 --> 00:20:58,817 -And in the same way that i and negative i were fixed in place for Linus, - 311 -00:20:58,817 --> 00:21:03,894 -and that the 3D space and 3D space get a whole sphere passing through i, +00:20:53,740 --> 00:20:58,266 +And in the same way that i and negative i were fixed in place for Linus, 312 -00:21:03,894 --> 00:21:09,180 -j, and k on that unit hypersphere which stays in place under the projection. +00:20:58,266 --> 00:21:01,863 +and that the ij unit circle was fixed in place for Felix, 313 +00:21:01,863 --> 00:21:06,575 +we get a whole sphere passing through i, j, and k on that unit hypersphere, + +314 +00:21:06,575 --> 00:21:09,180 +which stays in place under the projection. + +315 00:21:09,660 --> 00:21:14,361 So what we see as a unit sphere in our 3D space represents the only unaltered -314 +316 00:21:14,361 --> 00:21:18,580 part of the hypersphere of quaternions getting projected down onto us. -315 +317 00:21:18,880 --> 00:21:22,026 It's something analogous to the equator of a 3D sphere, -316 +318 00:21:22,026 --> 00:21:26,014 and it represents all of the unit quaternions whose real part is zero, -317 +319 00:21:26,014 --> 00:21:28,880 what Hamilton would have described as unit vectors. -318 +320 00:21:31,760 --> 00:21:35,322 The unit quaternions with positive real parts, between 0 and 1, -319 +321 00:21:35,322 --> 00:21:39,886 end up somewhere inside this unit sphere, closer to the number 1 in our 3D space, -320 +322 00:21:39,886 --> 00:21:44,395 which should feel analogous to how the northern hemisphere got mapped inside the -321 +323 00:21:44,395 --> 00:21:45,620 unit circle for Felix. -322 +324 00:21:47,480 --> 00:21:50,274 On the other hand, all the unit quaternions with negative -323 +325 00:21:50,274 --> 00:21:52,780 real part end up somewhere outside that unit sphere. -324 +326 00:22:00,100 --> 00:22:03,147 The number negative 1 is sitting off at the point at infinity, -325 +327 00:22:03,147 --> 00:22:05,760 which you can easily find by walking in any direction. -326 +328 00:22:06,980 --> 00:22:10,837 Keep in mind, even though we see the projection of some of these quaternions -327 +329 00:22:10,837 --> 00:22:13,843 as being closer or farther from the origin of our 3D space, -328 +330 00:22:13,843 --> 00:22:16,799 everything you're looking at represents a unit quaternion, -329 +331 00:22:16,799 --> 00:22:19,956 so everything you're looking at really has the same magnitude, -330 +332 00:22:19,956 --> 00:22:21,760 the same distance from the number 0. -331 +333 00:22:22,520 --> 00:22:25,740 And that number 0 itself is nowhere to be found in this picture. -332 +334 00:22:26,160 --> 00:22:29,580 Like all other non-unit quaternions, it's invisible to us. -333 +335 00:22:30,820 --> 00:22:35,468 In the same way that for Felix, the circle passing through 1, i, negative 1, -334 +336 00:22:35,468 --> 00:22:39,151 and negative i got projected into a line through the origin, -335 +337 00:22:39,151 --> 00:22:43,679 when we see this line through the origin passing through i and negative i, -336 +338 00:22:43,679 --> 00:22:47,060 we should understand that it really represents a circle. -337 +339 00:22:47,580 --> 00:22:50,716 Likewise, up on the hypersphere, invisible to us, -338 +340 00:22:50,716 --> 00:22:55,232 there is a unit sphere passing through 1, i, j, negative 1, negative i, -339 +341 00:22:55,232 --> 00:23:00,250 and negative j, and that whole sphere gets projected into the plane that we see -340 +342 00:23:00,250 --> 00:23:05,393 passing through 1, i, negative i, j, negative j, and negative 1, off at infinity, -341 +343 00:23:05,393 --> 00:23:07,840 what you and I might call the xy plane. -342 +344 00:23:08,580 --> 00:23:12,626 In general, any plane that you see here really represents the projection of a -343 +345 00:23:12,626 --> 00:23:16,880 sphere somewhere up on the hypersphere which passes through the number negative 1. -344 +346 00:23:19,180 --> 00:23:24,186 The action of taking a unit quaternion and multiplying it by any other quaternion from -345 +347 00:23:24,186 --> 00:23:27,984 the left can be thought of in terms of two separate 2D rotations, -346 +348 00:23:27,984 --> 00:23:33,048 happening perpendicular to and in sync with each other in a way that could only ever be -347 +349 00:23:33,048 --> 00:23:34,660 possible in four dimensions. -348 +350 00:23:35,480 --> 00:23:38,040 As a first example, let's look at multiplication by i. -349 +351 00:23:38,720 --> 00:23:42,732 We already know what this does to the circle that passes through 1 and i, -350 +352 00:23:42,732 --> 00:23:43,980 which we see as a line. -351 +353 00:23:45,840 --> 00:23:50,206 1 goes to i, i goes to negative 1, off at infinity, -352 +354 00:23:50,206 --> 00:23:56,000 negative 1 comes back around to negative i, and negative i goes to 1. -353 +355 00:23:56,700 --> 00:23:59,426 Remember, just like what Linus saw, all of this is -354 +356 00:23:59,426 --> 00:24:02,260 the stereographic projection of a 90 degree rotation. -355 +357 00:24:03,120 --> 00:24:05,760 Now look at the circle passing through j and k, -356 +358 00:24:05,760 --> 00:24:09,720 which is in a sense perpendicular to the circle passing through 1 and i. -357 +359 00:24:10,280 --> 00:24:14,118 It might feel weird to talk about two circles being perpendicular to each other, -358 +360 00:24:14,118 --> 00:24:18,288 especially when they have the same center and radius and don't touch each other at all, -359 +361 00:24:18,288 --> 00:24:20,800 but nothing could be more natural in four dimensions. -360 +362 00:24:21,640 --> 00:24:25,533 You can think of the action of i on this perpendicular circle as obeying -361 +363 00:24:25,533 --> 00:24:29,373 a certain right-hand rule, if you'll excuse the intrusion of my ghostly -362 +364 00:24:29,373 --> 00:24:33,320 green-screen hand into our otherwise pristine platonic mathematical stage. -363 +365 00:24:33,700 --> 00:24:37,447 You let that thumb of your right hand point from the number 1 to i, -364 +366 00:24:37,447 --> 00:24:38,880 and you curl your fingers. -365 +367 00:24:39,400 --> 00:24:43,020 The j-k circle will rotate in the direction of that curl. -366 +368 00:24:43,660 --> 00:24:44,120 How much? -367 +369 00:24:44,600 --> 00:24:50,020 Well, by the same amount as the 1i circle rotates, which is 90 degrees in this case. -368 +370 00:24:50,560 --> 00:24:55,480 This is what I meant by two rotations perpendicular to and in sync with each other. -369 +371 00:24:55,480 --> 00:25:05,184 So j goes to k, k goes to negative j, negative j goes to negative k, -370 +372 00:25:05,184 --> 00:25:08,700 and negative k goes to j. -371 -00:25:10,820 --> 00:25:14,782 +373 +00:25:10,820 --> 00:25:14,591 This gives us a little table for what the number i does to the other quaternions, -372 -00:25:14,782 --> 00:25:17,198 -but I want this not to be something you memorize, +374 +00:25:14,591 --> 00:25:17,120 +but I want this not to be something that you memorize, -373 -00:25:17,198 --> 00:25:20,340 -but something you could close your eyes and you could really see. +375 +00:25:17,120 --> 00:25:20,340 +but something that you could close your eyes and you could really see. -374 +376 00:25:21,340 --> 00:25:26,272 Computationally, if you know what a quaternion does to the numbers 1, i, j, and k, -375 +377 00:25:26,272 --> 00:25:29,302 you know what it does to any arbitrary quaternion, -376 +378 00:25:29,302 --> 00:25:31,680 since multiplication distributes nicely. -377 +379 00:25:32,240 --> 00:25:36,122 In the language of linear algebra, 1, i, j, and k form a basis of -378 +380 00:25:36,122 --> 00:25:40,122 our 4-dimensional space, so knowing what our transformation does to -379 +381 00:25:40,122 --> 00:25:44,240 them gives us the full information about what it does to all of space. -380 +382 00:25:45,220 --> 00:25:48,665 Geometrically, a 4-dimensional creature would be able to look at -381 +383 00:25:48,665 --> 00:25:51,422 those two perpendicular rotations I just described, -382 +384 00:25:51,422 --> 00:25:56,140 and understand that they lock you into one and only one rigid motion for the hypersphere. -383 +385 00:25:56,620 --> 00:25:59,590 We might lack the intuitions of such a hypothetical creature, -384 +386 00:25:59,590 --> 00:26:01,220 but we can maybe try to get close. -385 +387 00:26:01,680 --> 00:26:04,995 Here's what the action of repeatedly multiplying by i -386 +388 00:26:04,995 --> 00:26:08,740 looks like on our stereographic projection of the ijk sphere. -387 +389 00:26:09,560 --> 00:26:14,320 It gets rotated into what we see as a plane, then gets rotated further back to where it -388 +390 00:26:14,320 --> 00:26:17,132 used to be, though the orientation is reversed now, -389 +391 00:26:17,132 --> 00:26:21,730 then gets rotated again into what we see as a plane, and after the fourth iteration, -390 +392 00:26:21,730 --> 00:26:23,840 it ends up right back where it started. -391 +393 00:26:25,040 --> 00:26:30,077 As another example, think of a quaternion like q equals negative square root -392 +394 00:26:30,077 --> 00:26:33,348 of 2 over 2 plus square root of 2 over 2 times i, -393 +395 00:26:33,348 --> 00:26:36,620 which if we pull up a picture of a complex plane, -394 +396 00:26:36,620 --> 00:26:40,480 is a 135 degree rotation away from 1 in the direction of i. -395 +397 00:26:41,110 --> 00:26:43,910 Under our projection, we see this along the line -396 +398 00:26:43,910 --> 00:26:46,540 from 1 to i somewhere outside the unit sphere. -397 +399 00:26:47,260 --> 00:26:50,680 If that sounds weird, just remember how Linus would have seen this same number. -398 +400 00:26:51,660 --> 00:26:56,453 The action of multiplying this q by all other quaternions will look to us -399 +401 00:26:56,453 --> 00:27:01,182 like dragging the point at 1 all the way to this projected version of q, -400 +402 00:27:01,182 --> 00:27:06,300 while the jk circle gets rotated 135 degrees, according to our right-hand rule. -401 +403 00:27:09,980 --> 00:27:13,220 Multiplication by any other quaternion is completely similar. -402 +404 00:27:13,740 --> 00:27:16,194 For example, let's see what it looks like for j to act -403 +405 00:27:16,194 --> 00:27:18,560 on other quaternions by multiplication from the left. -404 +406 00:27:19,360 --> 00:27:25,312 The circle through 1 and j, which we see projected as a line through the origin, -405 +407 00:27:25,312 --> 00:27:31,853 gets rotated 90 degrees, dragging 1 up to j, so j times 1 is 1 and j times j is negative -406 +408 00:27:31,853 --> 00:27:32,000 1. -407 +409 00:27:38,060 --> 00:27:43,347 The circle perpendicular to that one, passing through i and k, -408 +410 00:27:43,347 --> 00:27:48,298 gets rotated 90 degrees according to this right-hand rule, -409 +411 00:27:48,298 --> 00:27:55,600 where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. -410 +412 00:28:01,740 --> 00:28:05,842 In general, for any other unit quaternion you see somewhere in space, -411 +413 00:28:05,842 --> 00:28:10,003 start by drawing the unit circle passing through 1, q, and negative 1, -412 +414 00:28:10,003 --> 00:28:13,520 which we see in our projection as a line through the origin. -413 +415 00:28:14,220 --> 00:28:18,100 Then draw the circle perpendicular to that 1 on what we see as the unit sphere. -414 +416 00:28:18,900 --> 00:28:22,415 You rotate the first circle so that 1 ends up where q was, -415 +417 00:28:22,415 --> 00:28:27,660 and rotate the perpendicular circle by the same amount according to the right-hand rule. -416 +418 00:28:40,120 --> 00:28:43,680 One thing worth noticing here is that order of multiplication matters. -417 +419 00:28:43,920 --> 00:28:46,700 It's not, as mathematicians would say, commutative. -418 -00:28:47,280 --> 00:28:53,696 -For example, i times j is k, which you might think of in terms of i acting on the +420 +00:28:47,280 --> 00:28:51,673 +For example, i times j is k, which you might think of in terms -419 -00:28:53,696 --> 00:29:00,740 -quaternion j, but if you think of j as acting on i, j times i, it rotates i to negative k. +421 +00:28:51,673 --> 00:28:55,439 +of i acting on the quaternion j, rotating it up to k, -420 +422 +00:28:55,439 --> 00:29:00,740 +but if you think of j as acting on i, j times i, it rotates i to negative k. + +423 00:29:01,740 --> 00:29:05,440 In fact, commutativity, this ability to swap the order of multiplication, -421 +424 00:29:05,440 --> 00:29:08,440 is a way more special property than a lot of people realize. -422 +425 00:29:09,080 --> 00:29:12,360 And most groups of actions on some space don't have it. -423 +426 00:29:12,740 --> 00:29:16,465 It's like how in solving a Rubik's cube, order matters a lot, -424 +427 00:29:16,465 --> 00:29:20,792 or how rotating a cube about the z-axis and then about the x-axis gives -425 +428 00:29:20,792 --> 00:29:25,660 a different final state from rotating it about the x-axis, then about the z-axis. -426 -00:29:29,300 --> 00:29:32,054 +429 +00:29:29,300 --> 00:29:31,962 And last, as one final but rather important point, -427 -00:29:32,054 --> 00:29:36,752 +430 +00:29:31,962 --> 00:29:36,503 so far I've shown you how to think about quaternions as acting by left multiplication, -428 -00:29:36,752 --> 00:29:39,451 +431 +00:29:36,503 --> 00:29:39,113 where when you read an expression like i times j, -429 -00:29:39,451 --> 00:29:42,692 +432 +00:29:39,113 --> 00:29:42,245 you think of i as a kind of function morphing all of space, -430 -00:29:42,692 --> 00:29:44,960 -and j as one of the points it's acting on. +433 +00:29:42,245 --> 00:29:44,960 +and j is just one of the points that it's acting on. -431 +434 00:29:44,960 --> 00:29:48,470 But you can also think of them as a different sort of action, -432 +435 00:29:48,470 --> 00:29:53,000 by multiplying from the right, where in this expression, j would be acting on i. -433 +436 00:29:53,680 --> 00:29:56,420 In that case, the rule for multiplication is very similar. -434 +437 00:29:56,940 --> 00:30:00,522 It's still the case that 1 goes to j, and j goes to negative 1, -435 +438 00:30:00,522 --> 00:30:04,105 etc., but instead of applying the right-hand rule to the circle -436 +439 00:30:04,105 --> 00:30:07,520 perpendicular to the 1j circle, you would use your left hand. -437 +440 00:30:08,100 --> 00:30:13,246 So either way, i times j is equal to k, but you can either think about this -438 +441 00:30:13,246 --> 00:30:18,595 with your right hand curling the number j to the number k as your thumb points -439 +442 00:30:18,595 --> 00:30:24,080 from 1 to i, or as your left hand curling i to k as its thumb points from 1 to j. -440 +443 00:30:24,920 --> 00:30:29,354 Understanding this left-hand rule for multiplication from the other side will be -441 +444 00:30:29,354 --> 00:30:33,897 extremely useful for understanding how unit quaternions describe rotation in three -442 +445 00:30:33,897 --> 00:30:34,500 dimensions. -443 +446 00:30:35,640 --> 00:30:40,700 And so far, it's probably not clear how exactly quaternions do describe 3D rotation. -444 +447 00:30:41,100 --> 00:30:45,185 I mean, if you consider one of these actions on the unit sphere, passing through i, -445 +448 00:30:45,185 --> 00:30:48,980 j, and k, it doesn't leave that sphere in place, it morphs it out of position. -446 +449 00:30:49,500 --> 00:30:53,420 So the way that this works is slightly more complicated than a single quaternion product. -447 +450 00:30:53,840 --> 00:30:57,233 It involves a process called conjugation, and I'll make a full follow-on -448 +451 00:30:57,233 --> 00:31:00,580 video all about it so that we have the time to go through some examples. -449 +452 00:31:02,020 --> 00:31:06,137 In the meantime, for more information on the story of quaternions and their relation -450 +453 00:31:06,137 --> 00:31:10,254 to orientation in 3D space, Quanta, a mathematical publication I'm sure a lot of you -451 +454 00:31:10,254 --> 00:31:14,420 are familiar with, just put out a post in a kind of loose conjunction with this video. -452 +455 00:31:14,760 --> 00:31:15,540 Link in the description. -453 +456 00:31:16,860 --> 00:31:19,769 If you enjoyed this, consider sharing it with some friends, -454 +457 00:31:19,769 --> 00:31:24,134 and if you felt like the narrative structure here was actually helpful for understanding, -455 +458 00:31:24,134 --> 00:31:28,450 maybe reassure those friends who would be turned off by a large timestamp that good math -456 +459 00:31:28,450 --> 00:31:29,760 is actually worth the time. -457 +460 00:31:30,420 --> 00:31:32,480 And many thanks to the patrons among you. -458 +461 00:31:32,480 --> 00:31:35,665 I actually spent way longer than I care to admit on this project, -459 +462 00:31:35,665 --> 00:31:39,140 so your patience and support is especially appreciated this time around. diff --git a/2018/quaternions/english/sentence_timings.json b/2018/quaternions/english/sentence_timings.json index 5f2bc4d31..3f7c7b92e 100644 --- a/2018/quaternions/english/sentence_timings.json +++ b/2018/quaternions/english/sentence_timings.json @@ -90,7 +90,7 @@ 192.24 ], [ - "It would surprise me if this approach was fully original, but I can say that it's certainly not the standard way to teach quaternions, and that the specific four-dimensional right-hand rule image I'd like to build up to is something I haven't seen elsewhere.", + "It would surprise me if this approach was fully original, but I can say that it's certainly not the standard way to teach quaternions, and that the specific four-dimensional right-hand rule image that I'd like to build up to is something that I haven't seen elsewhere.", 192.24, 205.36 ], @@ -395,7 +395,7 @@ 795.52 ], [ - "All of the points of the northern hemisphere get mapped somewhere inside the unit circle of the i-j plane, and that unit circle which passes through i,j,-i,-j actually stays fixed in place.", + "all of the points of the northern hemisphere get mapped somewhere inside the unit circle of the i-j plane, and that unit circle which passes through i, j, negative i, negative j, stays fixed in place.", 795.78, 808.24 ], @@ -600,7 +600,7 @@ 1223.46 ], [ - "This label in the upper right is going to show a given unit quaternion, and this little pink dot will show where that quaternion gets projected in our 3D space.", + "This label in the upper right is going to show a given unit quaternion, and this little pink dot will show where that particular quaternion gets projected in our 3D space.", 1227.32, 1237.88 ], @@ -615,7 +615,7 @@ 1253.04 ], [ - "And in the same way that i and negative i were fixed in place for Linus, and that the 3D space and 3D space get a whole sphere passing through i, j, and k on that unit hypersphere which stays in place under the projection.", + "And in the same way that i and negative i were fixed in place for Linus, and that the ij unit circle was fixed in place for Felix, we get a whole sphere passing through i, j, and k on that unit hypersphere, which stays in place under the projection.", 1253.74, 1269.18 ], @@ -745,7 +745,7 @@ 1508.7 ], [ - "This gives us a little table for what the number i does to the other quaternions, but I want this not to be something you memorize, but something you could close your eyes and you could really see.", + "This gives us a little table for what the number i does to the other quaternions, but I want this not to be something that you memorize, but something that you could close your eyes and you could really see.", 1510.82, 1520.34 ], @@ -845,7 +845,7 @@ 1726.7 ], [ - "For example, i times j is k, which you might think of in terms of i acting on the quaternion j, but if you think of j as acting on i, j times i, it rotates i to negative k.", + "For example, i times j is k, which you might think of in terms of i acting on the quaternion j, rotating it up to k, but if you think of j as acting on i, j times i, it rotates i to negative k.", 1727.28, 1740.74 ], @@ -865,7 +865,7 @@ 1765.66 ], [ - "And last, as one final but rather important point, so far I've shown you how to think about quaternions as acting by left multiplication, where when you read an expression like i times j, you think of i as a kind of function morphing all of space, and j as one of the points it's acting on.", + "And last, as one final but rather important point, so far I've shown you how to think about quaternions as acting by left multiplication, where when you read an expression like i times j, you think of i as a kind of function morphing all of space, and j is just one of the points that it's acting on.", 1769.3, 1784.96 ], diff --git a/2018/quaternions/english/transcript.txt b/2018/quaternions/english/transcript.txt index 8cbefbf2d..c214ba868 100644 --- a/2018/quaternions/english/transcript.txt +++ b/2018/quaternions/english/transcript.txt @@ -16,7 +16,7 @@ And this is because they give an elegant way to describe and compute 3D rotation The 20th century also brought quaternions some more love from a completely different direction, quantum mechanics. The special actions that quaternions describe in four dimensions are actually quite relevant to the way that two-state systems like spin of an electron or the polarization of a photon are described mathematically. What I'll show you here is a way to visualize quaternions in their full four-dimensional glory. -It would surprise me if this approach was fully original, but I can say that it's certainly not the standard way to teach quaternions, and that the specific four-dimensional right-hand rule image I'd like to build up to is something I haven't seen elsewhere. +It would surprise me if this approach was fully original, but I can say that it's certainly not the standard way to teach quaternions, and that the specific four-dimensional right-hand rule image that I'd like to build up to is something that I haven't seen elsewhere. Building up an understanding for this visual will take us meaningful time, but once you have it, there is a very natural and satisfying intuition for how to think about quaternion multiplication. It won't be until the next video that I show you how exactly quaternions describe orientation in three dimensions, which is for some people the whole reason we care about it, but once we're able to go at it armed with the image of what they're doing to a 4D hypersphere, there's a pleasing understanding to be had for the otherwise opaque formulas characterizing this relationship. The structure here will be to start by imagining teaching complex numbers to someone who only understands one dimension, then describing 3D rotations to someone who only understands two dimensions, and ultimately to represent what quaternions are doing up in four dimensions within the constraints of our 3D space. @@ -77,7 +77,7 @@ We can't show all of 3D space to Felix, but what we can do is project this 2D su Analogous to what we did before, stereographic projection will associate almost every point on the unit sphere with a unique point on the horizontal plane defined by the i and the j axes. For each point on the sphere, draw a line from negative 1 at the south pole through that point and see where it intersects the plane. So the point 1 at the north pole ends up at the center of the plane. -All of the points of the northern hemisphere get mapped somewhere inside the unit circle of the i-j plane, and that unit circle which passes through i,j,-i,-j actually stays fixed in place. +all of the points of the northern hemisphere get mapped somewhere inside the unit circle of the i-j plane, and that unit circle which passes through i, j, negative i, negative j, stays fixed in place. And that's an important point to make note of. Even though most points and lines and patches that Felix the Flatlander sees are warped projections of the real sphere, this unit circle is the one thing he has which is an honest part of our unit sphere, unaltered by projection. All of the points in the southern hemisphere get projected outside that unit circle, each getting farther and farther away as you approach negative 1 at the south pole. @@ -118,10 +118,10 @@ And just as the magnitude of a complex number, its distance from zero, is the sq And multiplying one quaternion, q1, by another, q2, has the effect of scaling q2 by the magnitude of q1, followed by a very special type of rotation in four dimensions. And those special 4D rotations, the heart of what we need to understand, correspond to the hypersphere of quaternions, a distance 1 from the origin, both in the sense that the quaternions whose multiplying action is a pure rotation live on that hypersphere, and in the sense that we can understand this weird 4D action just by following points on the hypersphere, rather than trying to look at all the points in the inconceivable stretches of four-dimensional space. Analogous to what we did for Linus and Felix, we stereographically project this hypersphere into 3D space. -This label in the upper right is going to show a given unit quaternion, and this little pink dot will show where that quaternion gets projected in our 3D space. +This label in the upper right is going to show a given unit quaternion, and this little pink dot will show where that particular quaternion gets projected in our 3D space. Just as before, we're projecting from the number negative 1, which sits on the real number line that is somehow perpendicular to all of our 3D space, and beyond our perception. Just as before, the number 1 ends up projected straight into the center of our space. -And in the same way that i and negative i were fixed in place for Linus, and that the 3D space and 3D space get a whole sphere passing through i, j, and k on that unit hypersphere which stays in place under the projection. +And in the same way that i and negative i were fixed in place for Linus, and that the ij unit circle was fixed in place for Felix, we get a whole sphere passing through i, j, and k on that unit hypersphere, which stays in place under the projection. So what we see as a unit sphere in our 3D space represents the only unaltered part of the hypersphere of quaternions getting projected down onto us. It's something analogous to the equator of a 3D sphere, and it represents all of the unit quaternions whose real part is zero, what Hamilton would have described as unit vectors. The unit quaternions with positive real parts, between 0 and 1, end up somewhere inside this unit sphere, closer to the number 1 in our 3D space, which should feel analogous to how the northern hemisphere got mapped inside the unit circle for Felix. @@ -147,7 +147,7 @@ How much? Well, by the same amount as the 1i circle rotates, which is 90 degrees in this case. This is what I meant by two rotations perpendicular to and in sync with each other. So j goes to k, k goes to negative j, negative j goes to negative k, and negative k goes to j. -This gives us a little table for what the number i does to the other quaternions, but I want this not to be something you memorize, but something you could close your eyes and you could really see. +This gives us a little table for what the number i does to the other quaternions, but I want this not to be something that you memorize, but something that you could close your eyes and you could really see. Computationally, if you know what a quaternion does to the numbers 1, i, j, and k, you know what it does to any arbitrary quaternion, since multiplication distributes nicely. In the language of linear algebra, 1, i, j, and k form a basis of our 4-dimensional space, so knowing what our transformation does to them gives us the full information about what it does to all of space. Geometrically, a 4-dimensional creature would be able to look at those two perpendicular rotations I just described, and understand that they lock you into one and only one rigid motion for the hypersphere. @@ -167,11 +167,11 @@ Then draw the circle perpendicular to that 1 on what we see as the unit sphere. You rotate the first circle so that 1 ends up where q was, and rotate the perpendicular circle by the same amount according to the right-hand rule. One thing worth noticing here is that order of multiplication matters. It's not, as mathematicians would say, commutative. -For example, i times j is k, which you might think of in terms of i acting on the quaternion j, but if you think of j as acting on i, j times i, it rotates i to negative k. +For example, i times j is k, which you might think of in terms of i acting on the quaternion j, rotating it up to k, but if you think of j as acting on i, j times i, it rotates i to negative k. In fact, commutativity, this ability to swap the order of multiplication, is a way more special property than a lot of people realize. And most groups of actions on some space don't have it. It's like how in solving a Rubik's cube, order matters a lot, or how rotating a cube about the z-axis and then about the x-axis gives a different final state from rotating it about the x-axis, then about the z-axis. -And last, as one final but rather important point, so far I've shown you how to think about quaternions as acting by left multiplication, where when you read an expression like i times j, you think of i as a kind of function morphing all of space, and j as one of the points it's acting on. +And last, as one final but rather important point, so far I've shown you how to think about quaternions as acting by left multiplication, where when you read an expression like i times j, you think of i as a kind of function morphing all of space, and j is just one of the points that it's acting on. But you can also think of them as a different sort of action, by multiplying from the right, where in this expression, j would be acting on i. In that case, the rule for multiplication is very similar. It's still the case that 1 goes to j, and j goes to negative 1, etc., but instead of applying the right-hand rule to the circle perpendicular to the 1j circle, you would use your left hand. diff --git a/2018/quaternions/french/sentence_translations.json b/2018/quaternions/french/sentence_translations.json index b2eeef221..9e46676da 100644 --- a/2018/quaternions/french/sentence_translations.json +++ b/2018/quaternions/french/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further.", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further.", "translatedText": "Ensuite, vous dites à Linus, pour multiplier deux nombres complexes, vous utilisez simplement la propriété distributive, ce que beaucoup de gens apprennent à l'école sous le nom de FOIL, et vous appliquez cette règle, i fois i est égal à moins 1, pour simplifier davantage les choses.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", "translatedText": "De la même manière que les nombres complexes incluent les nombres réels avec une seule dimension imaginaire supplémentaire, représentée par l'unité i, et que le système numérique qui n'est pas réellement un système numérique que nous avions en trois dimensions comprenait une deuxième direction imaginaire, j, les quaternions incluent les nombres réels ainsi que trois dimensions imaginaires distinctes, représentées par les unités i, j et k.", "model": "google_nmt", "n_reviews": 0, @@ -1328,7 +1328,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i.", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i.", "translatedText": "Le cercle perpendiculaire à celui-ci, passant par i et k, pivote de 90 degrés selon cette règle, où vous pointez votre pouce de 1 à j, donc j fois i est négatif k et j fois k est i.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/german/sentence_translations.json b/2018/quaternions/german/sentence_translations.json index e2bdb1b8a..a539aba9d 100644 --- a/2018/quaternions/german/sentence_translations.json +++ b/2018/quaternions/german/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further.", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further.", "translatedText": "Dann sagen Sie zu Linus: Um zwei komplexe Zahlen zu multiplizieren, verwenden Sie einfach die Verteilungseigenschaft, die viele Leute in der Schule als FOIL lernen, und wenden diese Regel an, i mal i ist gleich minus 1, um die Dinge weiter zu vereinfachen.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", "translatedText": "Auf die gleiche Weise, wie die komplexen Zahlen die reellen Zahlen mit einer einzigen zusätzlichen imaginären Dimension umfassen, dargestellt durch die Einheit i, und dass das System, das eigentlich keine Zahl ist, das wir in drei Dimensionen hatten, eine zweite imaginäre Richtung j enthielt, Die Quaternionen umfassen die reellen Zahlen zusammen mit drei separaten imaginären Dimensionen, dargestellt durch die Einheiten i, j und k.", "model": "google_nmt", "n_reviews": 0, @@ -1328,7 +1328,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i.", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i.", "translatedText": "Der Kreis senkrecht dazu, der durch i und k verläuft, wird gemäß dieser Regel um 90 Grad gedreht, wobei Sie mit dem Daumen von 1 nach j zeigen, sodass j mal i negativ k ist und j mal k gleich i ist.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/hebrew/sentence_translations.json b/2018/quaternions/hebrew/sentence_translations.json index 79044b74c..b4ace9bd0 100644 --- a/2018/quaternions/hebrew/sentence_translations.json +++ b/2018/quaternions/hebrew/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "ואז אתה אומר ללינוס, כדי להכפיל שני מספרים מרוכבים, אתה פשוט משתמש במאפיין החלוקתי, מה שאנשים רבים לומדים בבית הספר בתור FOIL, ומיישם את הכלל הזה, כפול i שווה ל-1, כדי לפשט את הדברים עוד יותר. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "באותו האופן שבו המספרים המרוכבים כוללים את המספרים הממשיים עם ממד דמיוני נוסף יחיד, המיוצג על ידי היחידה i, ושהדבר שאינו למעשה מערכת מספרים שהיה לנו בתלת מימד כלל כיוון דמיוני שני, j, הקווטרניונים כוללים את המספרים הממשיים יחד עם שלושה ממדים דמיוניים נפרדים, המיוצגים על ידי היחידות i, j ו-k. ", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "המעגל המאונך לזה, העובר דרך i ו-k, מסתובב 90 מעלות לפי הכלל הזה, שבו אתה מכוון את האגודל מ-1 ל-j, כך ש-j כפול i הוא שלילי k ו-j כפול k הוא i. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/hindi/sentence_translations.json b/2018/quaternions/hindi/sentence_translations.json index 0d13669ba..ba122ea50 100644 --- a/2018/quaternions/hindi/sentence_translations.json +++ b/2018/quaternions/hindi/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "फिर आप लिनुस से कहते हैं, दो सम्मिश्र संख्याओं को गुणा करने के लिए, आप बस वितरण गुण का उपयोग करते हैं, जो कई लोग स्कूल में FOIL के रूप में सीखते हैं, और चीजों को और अधिक सरल बनाने के लिए, इस नियम को लागू करते हैं, i गुणा i ऋणात्मक 1 के बराबर होता है।", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "उसी तरह से जटिल संख्याओं में एक अतिरिक्त काल्पनिक आयाम के साथ वास्तविक संख्याएं शामिल होती हैं, जिसे इकाई i द्वारा दर्शाया जाता है, और यह कि हमारे पास तीन आयामों में वास्तव में एक संख्या प्रणाली नहीं है, जिसमें एक दूसरी काल्पनिक दिशा, जे शामिल है, चतुर्धातुकों में तीन अलग-अलग काल्पनिक आयामों के साथ वास्तविक संख्याएँ शामिल हैं, जो इकाइयों i, j और k द्वारा दर्शायी जाती हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "उस पर लंबवत वृत्त, i और k से गुजरते हुए, इस नियम के अनुसार 90 डिग्री घूमता है, जहां आप अपना अंगूठा 1 से j तक इंगित करते हैं, इसलिए j गुना i नकारात्मक k है और j गुना k i है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/hungarian/sentence_translations.json b/2018/quaternions/hungarian/sentence_translations.json index 2b982cf99..7617f01f2 100644 --- a/2018/quaternions/hungarian/sentence_translations.json +++ b/2018/quaternions/hungarian/sentence_translations.json @@ -144,7 +144,7 @@ "end": 192.24 }, { - "input": "It would surprise me if this approach was fully original, but I can say that it's certainly not the standard way to teach quaternions, and that the specific four-dimensional right-hand rule image I'd like to build up to is something I haven't seen elsewhere.", + "input": "It would surprise me if this approach was fully original, but I can say that it's certainly not the standard way to teach quaternions, and that the specific four-dimensional right-hand rule image that I'd like to build up to is something that I haven't seen elsewhere.", "translatedText": "Meglepne, ha ez a megközelítés teljesen eredeti lenne, de azt mondhatom, hogy biztosan nem ez a standard módja a kvaternionok tanításának, és hogy a konkrét négydimenziós jobbkéz-szabály képet, amelyre szeretnék felépíteni, még nem láttam máshol.", "model": "DeepL", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 795.52 }, { - "input": "All of the points of the northern hemisphere get mapped somewhere inside the unit circle of the i-j plane, and that unit circle which passes through i,j,-i,-j actually stays fixed in place.", + "input": "all of the points of the northern hemisphere get mapped somewhere inside the unit circle of the i-j plane, and that unit circle which passes through i, j, negative i, negative j, stays fixed in place.", "translatedText": "Az északi félteke minden pontja valahol az i-j sík egységkörén belül helyezkedik el, és az i,j,-i,-j síkot átszelő egységkör valójában a helyén marad.", "model": "DeepL", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 1223.46 }, { - "input": "This label in the upper right is going to show a given unit quaternion, and this little pink dot will show where that quaternion gets projected in our 3D space.", + "input": "This label in the upper right is going to show a given unit quaternion, and this little pink dot will show where that particular quaternion gets projected in our 3D space.", "translatedText": "Ez a címke a jobb felső sarokban egy adott egységnyi kvaterniont fog mutatni, és ez a kis rózsaszín pont fogja megmutatni, hogy ez a kvaternion hova vetül a 3D térben.", "model": "DeepL", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1253.04 }, { - "input": "And in the same way that i and negative i were fixed in place for Linus, and that the 3D space and 3D space get a whole sphere passing through i, j, and k on that unit hypersphere which stays in place under the projection.", + "input": "And in the same way that i and negative i were fixed in place for Linus, and that the ij unit circle was fixed in place for Felix, we get a whole sphere passing through i, j, and k on that unit hypersphere, which stays in place under the projection.", "translatedText": "És ugyanúgy, ahogy Linus számára az i és a negatív i a helyén rögzült, és a 3D tér és a 3D tér kap egy egész gömböt, amely áthalad i-n, j-n és k-n azon az egységnyi hipergömbön, amely a vetítés alatt a helyén marad.", "model": "DeepL", "n_reviews": 0, @@ -1192,7 +1192,7 @@ "end": 1508.7 }, { - "input": "This gives us a little table for what the number i does to the other quaternions, but I want this not to be something you memorize, but something you could close your eyes and you could really see.", + "input": "This gives us a little table for what the number i does to the other quaternions, but I want this not to be something that you memorize, but something that you could close your eyes and you could really see.", "translatedText": "Ez egy kis táblázatot ad nekünk arról, hogy mit tesz az i szám a többi kvaternionnal, de azt akarom, hogy ezt ne kelljen megjegyezned, hanem hogy becsukd a szemed, és tényleg láthasd.", "model": "DeepL", "n_reviews": 0, @@ -1352,7 +1352,7 @@ "end": 1726.7 }, { - "input": "For example, i times j is k, which you might think of in terms of i acting on the quaternion j, but if you think of j as acting on i, j times i, it rotates i to negative k.", + "input": "For example, i times j is k, which you might think of in terms of i acting on the quaternion j, rotating it up to k, but if you think of j as acting on i, j times i, it rotates i to negative k.", "translatedText": "Például i szorozva j-vel k, amit úgy gondolhatunk, hogy i hat a j kvaternionra, de ha úgy gondolunk j-re, mintha i-re hatna, akkor j szorozva i-vel, akkor i-t negatív k-ra forgatja.", "model": "DeepL", "n_reviews": 0, @@ -1384,7 +1384,7 @@ "end": 1765.66 }, { - "input": "And last, as one final but rather important point, so far I've shown you how to think about quaternions as acting by left multiplication, where when you read an expression like i times j, you think of i as a kind of function morphing all of space, and j as one of the points it's acting on.", + "input": "And last, as one final but rather important point, so far I've shown you how to think about quaternions as acting by left multiplication, where when you read an expression like i times j, you think of i as a kind of function morphing all of space, and j is just one of the points that it's acting on.", "translatedText": "És végül, egy utolsó, de meglehetősen fontos pontként, eddig megmutattam, hogyan gondolkodhatunk a kvaternionokról úgy, mint baloldali szorzásról, ahol, amikor egy olyan kifejezést olvasunk, mint i szorozva j-vel, úgy gondolunk i-re, mint egyfajta függvényre, amely az egész teret morfizálja, és j-re, mint az egyik pontra, amelyre hat.", "model": "DeepL", "n_reviews": 0, diff --git a/2018/quaternions/indonesian/sentence_translations.json b/2018/quaternions/indonesian/sentence_translations.json index 694794927..c1d053071 100644 --- a/2018/quaternions/indonesian/sentence_translations.json +++ b/2018/quaternions/indonesian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "Lalu Anda berkata kepada Linus, untuk mengalikan dua bilangan kompleks, Anda cukup menggunakan sifat distributif, yang banyak orang pelajari di sekolah sebagai FOIL, dan terapkan aturan ini, i dikali i sama dengan negatif 1, untuk menyederhanakan semuanya. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "Dengan cara yang sama seperti bilangan kompleks mencakup bilangan real dengan satu dimensi imajiner tambahan, yang diwakili oleh satuan i, dan bahwa benda yang bukan sistem bilangan yang kita miliki dalam tiga dimensi mencakup arah imajiner kedua, j, angka empat mencakup bilangan real bersama dengan tiga dimensi imajiner terpisah, yang diwakili oleh satuan i, j, dan k. ", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "Lingkaran yang tegak lurus dengan lingkaran tersebut, melewati i dan k, diputar 90 derajat menurut aturan ini, di mana Anda mengarahkan ibu jari Anda dari 1 ke j, jadi j dikali i adalah negatif k dan j dikali k adalah i. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/italian/sentence_translations.json b/2018/quaternions/italian/sentence_translations.json index c0ad9d734..d5e76f23a 100644 --- a/2018/quaternions/italian/sentence_translations.json +++ b/2018/quaternions/italian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "Poi dici a Linus che per moltiplicare due numeri complessi usi semplicemente la proprietà distributiva, che molte persone imparano a scuola come FOIL, e applichi questa regola, i per i uguale a meno 1, per semplificare ulteriormente le cose. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "Allo stesso modo in cui i numeri complessi includono i numeri reali con un'unica dimensione immaginaria extra, rappresentata dall'unità i, e che il sistema non realmente numerico che avevamo in tre dimensioni includeva una seconda direzione immaginaria, j, i quaternioni includono i numeri reali insieme a tre dimensioni immaginarie separate, rappresentate dalle unità i, j e k. ", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "Il cerchio perpendicolare a quello, che passa per i e k, viene ruotato di 90 gradi secondo questa regola, dove punti il pollice da 1 a j, quindi j per i è negativo k e j per k è i. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/japanese/sentence_translations.json b/2018/quaternions/japanese/sentence_translations.json index 6d9bde5cf..38407c69f 100644 --- a/2018/quaternions/japanese/sentence_translations.json +++ b/2018/quaternions/japanese/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further.", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further.", "translatedText": "次にあ なたはライナスに、「2 つの複素数を掛けるには、多くの人が学校で FOIL として学ぶ分配特性を使用 するだけです。 そして、物事をさらに単純化するために、このルール (i 掛ける i はマイナス 1 に等しい) を適用します」と言います。", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", "translatedText": "複素数には、単位 i で表される単一 の余分な虚数次元を持つ実数が含まれており、3 次元にある実際には数値で はないものには 2 番目の虚数方向 j が含まれているのと同じように 、クォータニオンには、実数と、単位 i、j、および k で表される 3 つの別々の虚数次元が含まれます。", "model": "google_nmt", "n_reviews": 0, @@ -1328,7 +1328,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i.", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i.", "translatedText": "i と k を通るその円に垂直な円は、親指を 1 から j に向けるこの規則に従って 90 度回転します。 したがって、i の j 倍は k の負になり、k の j 倍は i になります。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/korean/sentence_translations.json b/2018/quaternions/korean/sentence_translations.json index f18cb079c..29eb9fe72 100644 --- a/2018/quaternions/korean/sentence_translations.json +++ b/2018/quaternions/korean/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "그런 다음 리누스에게 두 개의 복소수를 곱하려면 많은 사람들이 학교에서 FOIL로 배우는 분배 법칙을 사용하고 이 규칙(i 곱하기 i = -1과 같음)을 적용하여 상황을 더욱 단순화하라고 말합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "복소수에는 단위 i로 표시되는 단일 추가 허수 차원이 있는 실수가 포함되고, 우리가 3차원에서 가지고 있는 실제로 숫자가 아닌 시스템에는 두 번째 허수 방향 j가 포함되는 것과 같은 방식으로, 쿼터니언에는 단위 i, j 및 k로 표시되는 세 개의 개별 허수 차원과 함께 실수가 포함됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "i와 k를 통과하는 원에 수직인 원은 이 규칙에 따라 90도 회전합니다. 여기서 엄지손가락으로 1에서 j를 가리키므로 j 곱하기 i는 음수 k이고 j 곱하기 k는 i입니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/marathi/sentence_translations.json b/2018/quaternions/marathi/sentence_translations.json index 26e3a3594..628a29a51 100644 --- a/2018/quaternions/marathi/sentence_translations.json +++ b/2018/quaternions/marathi/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "मग तुम्ही लिनसला म्हणाल, दोन जटिल संख्यांचा गुणाकार करण्यासाठी, तुम्ही फक्त वितरण गुणधर्म वापरा, जे अनेक लोक शाळेत शिकतात ते FOIL म्हणून, आणि हा नियम लागू करा, i times i बरोबरी ऋण 1, गोष्टी आणखी सोपी करण्यासाठी. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "ज्या प्रकारे जटिल संख्यांमध्ये एक अतिरिक्त काल्पनिक परिमाण असलेल्या वास्तविक संख्यांचा समावेश होतो, एकक i द्वारे प्रस्तुत केले जाते, आणि आमच्याकडे तीन मितींमध्ये नसलेल्या-वास्तविक-संख्या प्रणालीमध्ये दुसरी काल्पनिक दिशा समाविष्ट असते, j, चतुर्थांश मध्ये तीन स्वतंत्र काल्पनिक परिमाणांसह वास्तविक संख्या समाविष्ट असतात, i, j आणि k या एककांनी दर्शविल्या जातात. ", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "i आणि k मधून जाणारे वर्तुळ लंबवत, या नियमानुसार 90 अंश फिरवले जाते, जेथे तुम्ही तुमचा अंगठा 1 ते j कडे निर्देशित करता, म्हणून j गुणा i ऋण k आणि j गुणा k आहे i. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/persian/sentence_translations.json b/2018/quaternions/persian/sentence_translations.json index 6bbadb6b7..d683f3d61 100644 --- a/2018/quaternions/persian/sentence_translations.json +++ b/2018/quaternions/persian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "سپس به لینوس می‌گویید، برای ضرب دو عدد مختلط، فقط از خاصیت توزیعی استفاده می‌کنید، چیزی که بسیاری از افراد در مدرسه به عنوان FOIL یاد می‌گیرند، و این قانون را به کار می‌برید، i ضربدر i برابر با 1 است، تا همه چیز را ساده‌تر کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "به همان ترتیبی که اعداد مختلط شامل اعداد حقیقی با یک بعد خیالی اضافی منفرد می شود که با واحد i نشان داده می شود، و سیستم نه در واقع عددی که در سه بعد داشتیم شامل جهت خیالی دوم، j، می شود. رباعی ها شامل اعداد حقیقی به همراه سه بعد خیالی مجزا هستند که با واحدهای i، j و k نشان داده می شوند. ", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "دایره عمود بر آن دایره که از i و k می گذرد، طبق این قانون 90 درجه می چرخد، جایی که انگشت شست خود را از 1 به j نشان می دهید، بنابراین j ضربدر i k منفی است و j ضربدر k i است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/polish/sentence_translations.json b/2018/quaternions/polish/sentence_translations.json index ba15a6d41..74cd7e274 100644 --- a/2018/quaternions/polish/sentence_translations.json +++ b/2018/quaternions/polish/sentence_translations.json @@ -144,7 +144,7 @@ "end": 192.24 }, { - "input": "It would surprise me if this approach was fully original, but I can say that it's certainly not the standard way to teach quaternions, and that the specific four-dimensional right-hand rule image I'd like to build up to is something I haven't seen elsewhere.", + "input": "It would surprise me if this approach was fully original, but I can say that it's certainly not the standard way to teach quaternions, and that the specific four-dimensional right-hand rule image that I'd like to build up to is something that I haven't seen elsewhere.", "translatedText": "", "from_community_srt": "Byłbym zaskoczony, gdyby to podejście okazało się całkowicie oryginalne. Ale mogę powiedzieć, że to na pewno nie jest standardowy sposób uczenia kwaternionów. A tego konkretnego obrazka z czterowymiarową zasadą prawej ręki, którą chcę wprowadzić, nie widziałem nigdzie indziej.", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 795.52 }, { - "input": "All of the points of the northern hemisphere get mapped somewhere inside the unit circle of the i-j plane, and that unit circle which passes through i,j,-i,-j actually stays fixed in place.", + "input": "all of the points of the northern hemisphere get mapped somewhere inside the unit circle of the i-j plane, and that unit circle which passes through i, j, negative i, negative j, stays fixed in place.", "translatedText": "", "from_community_srt": "Wszystkie punkty z półkuli północnej zostaną posłane gdzieś do wnętrza okręgu jednostkowego płaszczyzny ,,ij”. A ten okrąg jednostkowy, który przechodzi przez ,,i”, ,,j”, ,,-i” oraz ,,-j” pozostanie na swoim miejscu.", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 1223.46 }, { - "input": "This label in the upper right is going to show a given unit quaternion, and this little pink dot will show where that quaternion gets projected in our 3D space.", + "input": "This label in the upper right is going to show a given unit quaternion, and this little pink dot will show where that particular quaternion gets projected in our 3D space.", "translatedText": "", "from_community_srt": "Ten napis w prawym górnym rogu będzie pokazywał dany jednostkowy kwaternion, a ta mała różowa kropka będzie pokazywała, gdzie ten konkretny kwaternion zostanie zrzutowany w naszej przestrzeni trójwymiarowej.", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 1253.04 }, { - "input": "And in the same way that i and negative i were fixed in place for Linus, and that the 3D space and 3D space get a whole sphere passing through i, j, and k on that unit hypersphere which stays in place under the projection.", + "input": "And in the same way that i and negative i were fixed in place for Linus, and that the ij unit circle was fixed in place for Felix, we get a whole sphere passing through i, j, and k on that unit hypersphere, which stays in place under the projection.", "translatedText": "", "from_community_srt": "Tak jak ,,i” oraz ,,-i” z punktu widzenia Linusa pozostawały na swoim miejscu, a koło jednostkowe ,,ij” pozostawało na swoim miejscu z punktu widzenia Feliksa, tak my mamy całą sferę z tej jednostkowej hipersfery, przechodzącą przez ,,i”, ,,j” oraz ,,k”, która pozostaje na swoim miejscu przy rzutowaniu.", "n_reviews": 0, @@ -1192,7 +1192,7 @@ "end": 1508.7 }, { - "input": "This gives us a little table for what the number i does to the other quaternions, but I want this not to be something you memorize, but something you could close your eyes and you could really see.", + "input": "This gives us a little table for what the number i does to the other quaternions, but I want this not to be something that you memorize, but something that you could close your eyes and you could really see.", "translatedText": "", "from_community_srt": "To daje nam małą tabelkę opisującą, jak liczba ,,i” działa na inne kwaterniony. Ale nie chcę, żebyście uczyli się jej na pamięć, tylko żebyście byli w stanie zamknąć oczy i to naprawdę zobaczyć.", "n_reviews": 0, @@ -1352,7 +1352,7 @@ "end": 1726.7 }, { - "input": "For example, i times j is k, which you might think of in terms of i acting on the quaternion j, but if you think of j as acting on i, j times i, it rotates i to negative k.", + "input": "For example, i times j is k, which you might think of in terms of i acting on the quaternion j, rotating it up to k, but if you think of j as acting on i, j times i, it rotates i to negative k.", "translatedText": "", "from_community_srt": "przemienne. Na przykład ,,i” razy ,,j” równa się ,,k”. Możecie myśleć o tym tak, że ,,i” działa na kwaternion ,,j” obracając go do ,,k”. Ale jeśli zastanowicie się nad działaniem kwaternionem ,,j” na ,,i”, czyli ,,j” razy ,,i”, to ono obróci ,,i” do ,,-k”.", "n_reviews": 0, @@ -1384,7 +1384,7 @@ "end": 1765.66 }, { - "input": "And last, as one final but rather important point, so far I've shown you how to think about quaternions as acting by left multiplication, where when you read an expression like i times j, you think of i as a kind of function morphing all of space, and j as one of the points it's acting on.", + "input": "And last, as one final but rather important point, so far I've shown you how to think about quaternions as acting by left multiplication, where when you read an expression like i times j, you think of i as a kind of function morphing all of space, and j is just one of the points that it's acting on.", "translatedText": "", "from_community_srt": "I na koniec jeszcze ostatnia, ale dość ważna sprawa. Jak dotąd pokazałem wam, jak myśleć o kwaternionach jako o działaniach mnożenia z lewej strony. Czyli, gdy czytacie wyrażenie takie jak ,,i” razy ,,j”, to myślicie o ,,i” jako o jakiejś funkcji przekształcającej całą przestrzeń. A ,,j” jest tylko jednym z punktów, na które ona działa.", "n_reviews": 0, diff --git a/2018/quaternions/portuguese/sentence_translations.json b/2018/quaternions/portuguese/sentence_translations.json index d708be83a..66c2379d1 100644 --- a/2018/quaternions/portuguese/sentence_translations.json +++ b/2018/quaternions/portuguese/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further.", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further.", "translatedText": "Aí você diz ao Linus, para multiplicar dois números complexos, basta usar a propriedade distributiva, o que muitas pessoas aprendem na escola como FOIL, e aplicar esta regra, i vezes i igual a menos 1, para simplificar ainda mais as coisas.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", "translatedText": "Da mesma forma que os números complexos incluem os números reais com uma única dimensão imaginária extra, representada pela unidade i, e que o sistema não-realmente-numérico que tínhamos em três dimensões incluía uma segunda direção imaginária, j, os quaterniões incluem os números reais juntamente com três dimensões imaginárias separadas, representadas pelas unidades i, j e k.", "model": "google_nmt", "n_reviews": 0, @@ -1328,7 +1328,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i.", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i.", "translatedText": "O círculo perpendicular a esse, passando por i e k, é girado 90 graus de acordo com esta regra, onde você aponta o polegar de 1 para j, então j vezes i é negativo k e j vezes k é i.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/russian/sentence_translations.json b/2018/quaternions/russian/sentence_translations.json index d02d5a942..ce3aa6fdf 100644 --- a/2018/quaternions/russian/sentence_translations.json +++ b/2018/quaternions/russian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further.", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further.", "translatedText": "Затем вы говорите Линусу, что для умножения двух комплексных чисел вы просто используете свойство распределения, которое многие люди изучают в школе как FOIL, и применяете это правило: i, умноженное на i, равно отрицательному 1, чтобы еще больше упростить ситуацию.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", "translatedText": "Точно так же, как комплексные числа включают в себя действительные числа с одним дополнительным мнимым измерением, представленным единицей i, и что не-числовая система, которую мы имели в трех измерениях, включала второе мнимое направление j, кватернионы включают действительные числа вместе с тремя отдельными мнимыми измерениями, представленными единицами i, j и k.", "model": "google_nmt", "n_reviews": 0, @@ -1328,7 +1328,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i.", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i.", "translatedText": "Круг, перпендикулярный этому, проходящий через i и k, поворачивается на 90 градусов в соответствии с этим правилом, когда вы указываете большим пальцем от 1 до j, так что j, умноженный на i, будет отрицательным k, а j, умноженным на k, будет i.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/spanish/sentence_translations.json b/2018/quaternions/spanish/sentence_translations.json index 53e010274..52a6ed8a9 100644 --- a/2018/quaternions/spanish/sentence_translations.json +++ b/2018/quaternions/spanish/sentence_translations.json @@ -144,7 +144,7 @@ "end": 192.24 }, { - "input": "It would surprise me if this approach was fully original, but I can say that it's certainly not the standard way to teach quaternions, and that the specific four-dimensional right-hand rule image I'd like to build up to is something I haven't seen elsewhere.", + "input": "It would surprise me if this approach was fully original, but I can say that it's certainly not the standard way to teach quaternions, and that the specific four-dimensional right-hand rule image that I'd like to build up to is something that I haven't seen elsewhere.", "translatedText": "Me sorprendería si este enfoque fuera completamente original, pero puedo decir que ciertamente no es la forma estándar de enseñar cuaterniones, y que la imagen específica de la regla de la mano derecha de cuatro dimensiones que me gustaría desarrollar es algo que ya tengo. No se ve en ningún otro lugar.", "from_community_srt": "Me sorprendería si este enfoque fuera completamente original, pero puedo decir que ciertamente no es la forma estándar de enseñar cuaterniones. Y que estas específicas imagenes de la regla de la mano derecha en cuatro dimensiones que me gustaría construir, es algo que no he visto en ningún otro lugar.", "n_reviews": 0, @@ -630,7 +630,7 @@ "end": 795.52 }, { - "input": "All of the points of the northern hemisphere get mapped somewhere inside the unit circle of the i-j plane, and that unit circle which passes through i,j,-i,-j actually stays fixed in place.", + "input": "all of the points of the northern hemisphere get mapped somewhere inside the unit circle of the i-j plane, and that unit circle which passes through i, j, negative i, negative j, stays fixed in place.", "translatedText": "Todos los puntos del hemisferio norte quedan mapeados en algún lugar dentro del círculo unitario del plano ij, y ese círculo unitario que pasa por i,j,-i,-j en realidad permanece fijo en su lugar.", "from_community_srt": "Así que el punto 1 en el polo norte termina en el centro del plano; todos los puntos del hemisferio norte se asignan en algún lugar dentro del círculo unitario del plano ij; y ese círculo unitario que pasa por i, j, -i, y -j en realidad permanece fijo en su lugar.", "n_reviews": 0, @@ -958,7 +958,7 @@ "end": 1223.46 }, { - "input": "This label in the upper right is going to show a given unit quaternion, and this little pink dot will show where that quaternion gets projected in our 3D space.", + "input": "This label in the upper right is going to show a given unit quaternion, and this little pink dot will show where that particular quaternion gets projected in our 3D space.", "translatedText": "Esta etiqueta en la parte superior derecha mostrará un cuaternión unitario determinado, y este pequeño punto rosa mostrará dónde se proyecta ese cuaternión en nuestro espacio 3D.", "from_community_srt": "Esta etiqueta en la parte superior derecha mostrará un cuaternión unitario dado, y este pequeño punto rosa mostrará dónde se proyecta ese cuaternión en particular en nuestro espacio 3d.", "n_reviews": 0, @@ -982,7 +982,7 @@ "end": 1253.04 }, { - "input": "And in the same way that i and negative i were fixed in place for Linus, and that the 3D space and 3D space get a whole sphere passing through i, j, and k on that unit hypersphere which stays in place under the projection.", + "input": "And in the same way that i and negative i were fixed in place for Linus, and that the ij unit circle was fixed in place for Felix, we get a whole sphere passing through i, j, and k on that unit hypersphere, which stays in place under the projection.", "translatedText": "Y de la misma manera que i y i negativo se fijaron en su lugar para Linus, y que el espacio 3D y el espacio 3D obtienen una esfera completa que pasa por i, j y k en esa hiperesfera unitaria que permanece en su lugar bajo la proyección.", "from_community_srt": "el número 1 termina proyectado directamente hacia el centro de nuestro espacio, y de la misma manera que i y -i se fijaron en su lugar para Linus, y que el círculo de la unidad ij se fijó en su lugar para Felix, obtenemos una esfera completa que pasa por i, j y k en esa hiperesfera unitaria que permanece en su lugar debajo de la proyección.", "n_reviews": 0, @@ -1190,7 +1190,7 @@ "end": 1508.7 }, { - "input": "This gives us a little table for what the number i does to the other quaternions, but I want this not to be something you memorize, but something you could close your eyes and you could really see.", + "input": "This gives us a little table for what the number i does to the other quaternions, but I want this not to be something that you memorize, but something that you could close your eyes and you could really see.", "translatedText": "Esto nos da una pequeña tabla de lo que el número i hace con los otros cuaterniones, pero quiero que esto no sea algo que memorices, sino algo que puedas cerrar los ojos y realmente puedas ver.", "from_community_srt": "Esto nos da una pequeña tabla de lo que el número i hace a los otros cuaterniones. Pero quiero que esto no sea algo que memorices, Sino algo que podrías cerrar los ojos y que realmente pudieras ver.", "n_reviews": 0, @@ -1350,7 +1350,7 @@ "end": 1726.7 }, { - "input": "For example, i times j is k, which you might think of in terms of i acting on the quaternion j, but if you think of j as acting on i, j times i, it rotates i to negative k.", + "input": "For example, i times j is k, which you might think of in terms of i acting on the quaternion j, rotating it up to k, but if you think of j as acting on i, j times i, it rotates i to negative k.", "translatedText": "Por ejemplo, i multiplicado por j es k, lo cual podría pensarse en términos de i actuando sobre el cuaternión j, pero si piensa que j actúa sobre i, j multiplicado por i, rota i a menos k.", "from_community_srt": "Por ejemplo, i veces j es k, lo que podría pensar en términos de i que actúa sobre el cuaternión j, girándolo hasta k. Pero si piensas que j actúa sobre i, j veces i, rota i a -k.", "n_reviews": 0, @@ -1382,7 +1382,7 @@ "end": 1765.66 }, { - "input": "And last, as one final but rather important point, so far I've shown you how to think about quaternions as acting by left multiplication, where when you read an expression like i times j, you think of i as a kind of function morphing all of space, and j as one of the points it's acting on.", + "input": "And last, as one final but rather important point, so far I've shown you how to think about quaternions as acting by left multiplication, where when you read an expression like i times j, you think of i as a kind of function morphing all of space, and j is just one of the points that it's acting on.", "translatedText": "Y por último, como punto final pero bastante importante, hasta ahora te he mostrado cómo pensar en los cuaterniones como si actuaran mediante multiplicación por la izquierda, donde cuando lees una expresión como i multiplicada por j, piensas en i como una especie de función de transformación. todo el espacio, y j como uno de los puntos sobre los que actúa.", "from_community_srt": "Y por último, como punto final pero bastante importante, hasta ahora te he mostrado cómo pensar acerca de los cuaterniones que actúan mediante la multiplicación de la izquierda, donde cuando lees una expresión como i veces j, piensas en i como un tipo de función que transforma todo el espacio y j es solo uno de los puntos en los que está actuando.", "n_reviews": 0, diff --git a/2018/quaternions/tamil/sentence_translations.json b/2018/quaternions/tamil/sentence_translations.json index 9ea30353e..fc04e192b 100644 --- a/2018/quaternions/tamil/sentence_translations.json +++ b/2018/quaternions/tamil/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further.", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further.", "translatedText": "இரண்டு கலப்பு எண்களைப் பெருக்க லினஸிடம் சொல்கிறீர்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", "translatedText": "அதே வழியில், சிக்கலான எண்கள் ஒற்றை கூடுதல் கற்பனை பரிமாணத்துடன் உண்மையான எண்களை உள்ளடக்கியது, ஐ யூனிட் பிரதிநிதித்துவப்படுத்துகிறது, மேலும் முப்பரிமாணத்தில் நம்மிடம் இருந்த எண்-எண்-அல்லாத அமைப்பு இரண்டாவது கற்பனை திசையை உள்ளடக்கியது, j, குவாட்டர்னியன்களில் உண்மையான எண்கள் மற்றும் மூன்று தனித்தனி கற்பனை பரிமாணங்கள் உள்ளன, அவை i, j மற்றும் k அலகுகளால் குறிக்கப்படுகின்றன.", "model": "google_nmt", "n_reviews": 0, @@ -1328,7 +1328,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i.", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i.", "translatedText": "i மற்றும் k வழியாகச் செல்லும் வட்டத்திற்குச் செங்குத்தாக உள்ள வட்டமானது, இந்த விதியின்படி 90 டிகிரியில் சுழற்றப்படுகிறது, அங்கு நீங்கள் உங்கள் கட்டைவிரலை 1 முதல் j வரை சுட்டிக்காட்டுகிறீர்கள், எனவே j முறை i எதிர்மறை k மற்றும் j முறை k என்பது i.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/telugu/sentence_translations.json b/2018/quaternions/telugu/sentence_translations.json index 36231f156..115c2a22d 100644 --- a/2018/quaternions/telugu/sentence_translations.json +++ b/2018/quaternions/telugu/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further.", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further.", "translatedText": "అప్పుడు మీరు లైనస్‌తో చెప్పండి, రెండు సమ్మేళన సంఖ్యలను గుణించడానికి, మీరు డిస్ట్రిబ్యూటివ్ ప్రాపర్టీని, చాలా మంది స్కూల్‌లో నేర్చుకునే వాటిని FOIL లాగా ఉపయోగించుకోండి మరియు ఈ నియమాన్ని వర్తింపజేయండి, i సార్లు నేను నెగటివ్ 1కి సమానం, విషయాలను మరింత సరళీకరించడానికి.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k.", "translatedText": "అదే విధంగా సంక్లిష్ట సంఖ్యలు యూనిట్ i ద్వారా సూచించబడే ఒక అదనపు ఊహాజనిత పరిమాణంతో వాస్తవ సంఖ్యలను కలిగి ఉంటాయి మరియు మేము మూడు కోణాలలో కలిగి ఉన్న నిజానికి-ఒక-సంఖ్య వ్యవస్థలో రెండవ ఊహాత్మక దిశ, j, చతుర్భుజాలలో వాస్తవ సంఖ్యలు మరియు మూడు వేర్వేరు ఊహాత్మక కొలతలు ఉంటాయి, ఇవి i, j మరియు k అనే యూనిట్‌లచే సూచించబడతాయి.", "model": "google_nmt", "n_reviews": 0, @@ -1328,7 +1328,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i.", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i.", "translatedText": "దానికి లంబంగా ఉన్న సర్కిల్, i మరియు k గుండా వెళుతుంది, ఈ నియమం ప్రకారం 90 డిగ్రీలు తిప్పబడుతుంది, ఇక్కడ మీరు మీ బొటనవేలును 1 నుండి j వరకు చూపుతారు, కాబట్టి j సార్లు i నెగటివ్ k మరియు j సార్లు k అనేది i.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/thai/sentence_translations.json b/2018/quaternions/thai/sentence_translations.json index d748929f1..5fd0b7366 100644 --- a/2018/quaternions/thai/sentence_translations.json +++ b/2018/quaternions/thai/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "ตัวละครตัวแรกของเราคือ Linus the Linelander ซึ่งจิตใจสามารถเข้าใจได้เพียงเรขาคณิตมิติเดียวของเส้นและพีชคณิตของจำนวนจริงเท่านั้น เราจะพยายามอธิบายจำนวนเชิงซ้อนให้ไลนัสฟัง และเป็นสิ่งสำคัญมากที่คุณจะต้องเห็นอกเห็นใจเขาให้มากที่สุดเท่าที่จะทำได้ เพราะในไม่กี่นาที คุณจะเป็นเหมือนเขา ในด้านหนึ่ง คุณสามารถกำหนดจำนวนเชิงซ้อนโดยใช้พีชคณิตล้วนๆ ได้ คุณบอกว่าแต่ละค่าแสดงเป็นจำนวนจริงบวกจำนวนจริงอื่นๆ คูณ i โดยที่ i คือค่าคงที่ที่เพิ่งประดิษฐ์ขึ้นใหม่ โดยมีคุณสมบัติในการนิยามคือ i คูณ i เท่ากับลบ 1 จากนั้นคุณบอกกับไลนัสว่า ในการคูณจำนวนเชิงซ้อนสองตัว คุณแค่ใช้สมบัติการกระจาย สิ่งที่หลายๆ คนเรียนรู้ในโรงเรียนเป็น FOIL และใช้กฎนี้ i คูณ i เท่ากับลบ 1 เพื่อให้ทุกอย่างง่ายขึ้นอีก ไม่เป็นไร มันได้ผลจริงๆ และวิธีการแนะนำควอเทอร์เนียนในตำรามาตรฐานก็คล้ายคลึงกับอันนี้ โดยแสดงกฎพีชคณิตแล้วเรียกมันว่าเสร็จแล้ว แต่ฉันคิดว่ามีบางอย่างขาดหายไป ถ้าอย่างน้อยเราไม่พยายามแสดงให้ไลนัสเห็นเรขาคณิตของจำนวนเชิงซ้อน และลักษณะของการคูณที่ซับซ้อน เนื่องจากปัญหาทางคณิตศาสตร์และฟิสิกส์ ซึ่งจำนวนเชิงซ้อนมีประโยชน์อย่างน่าตกใจ มักจะใช้ประโยชน์จากสัญชาตญาณเชิงพื้นที่นี้ คุณและฉันที่เข้าใจสองมิติอาจคิดเช่นนี้ เมื่อคุณคูณจำนวนเชิงซ้อนสองตัว z คูณ w คุณจะนึกถึง z ว่าเป็นฟังก์ชันประเภทหนึ่งที่กระทำกับ w โดยหมุนและยืดมันด้วยวิธีใดวิธีหนึ่ง ฉันชอบคิดเรื่องนี้โดยขยายมุมมองให้กว้างขึ้นแล้วถามว่า z ทำอะไรกับทั้งระนาบ? ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/turkish/sentence_translations.json b/2018/quaternions/turkish/sentence_translations.json index be0fa5ceb..35471df3a 100644 --- a/2018/quaternions/turkish/sentence_translations.json +++ b/2018/quaternions/turkish/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "Sonra Linus'a, iki karmaşık sayıyı çarpmak için, çoğu insanın okulda FOIL olarak öğrendiği dağılım özelliğini kullanacağını ve işleri daha da basitleştirmek için i çarpı i eşittir eksi 1 kuralını uygulayacağını söylüyorsun. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "Aynı şekilde karmaşık sayılar, i birimiyle temsil edilen tek bir sanal boyuta sahip gerçek sayıları içerir ve üç boyutta sahip olduğumuz aslında sayı olmayan sistem şeyi ikinci bir sanal yön olan j'yi içerir, kuaterniyonlar, i, j ve k birimleriyle temsil edilen üç ayrı hayali boyutla birlikte gerçek sayıları içerir. ", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "Buna dik olan, i ve k'den geçen daire, bu kurala göre 90 derece döndürülür, burada başparmağınızı 1'den j'ye doğru tutarsınız, yani j çarpı i negatif k ve j çarpı k, i'dir. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/ukrainian/sentence_translations.json b/2018/quaternions/ukrainian/sentence_translations.json index 8d644e944..084f9b119 100644 --- a/2018/quaternions/ukrainian/sentence_translations.json +++ b/2018/quaternions/ukrainian/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "Тоді ви говорите Лінусу, щоб помножити два комплексних числа, ви просто використовуєте властивість розподілу, яку багато людей вивчають у школі як FOIL, і застосовуєте це правило, i помножене на i дорівнює мінус 1, щоб ще більше спростити речі. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "Так само, як комплексні числа включають дійсні числа з одним додатковим уявним виміром, представленим одиницею i, і що несправжня система чисел, яку ми мали в трьох вимірах, включала другий уявний напрямок, j, кватерніони включають дійсні числа разом із трьома окремими уявними вимірами, представленими одиницями i, j і k. ", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "Коло, перпендикулярне до нього, яке проходить через i і k, повертається на 90 градусів відповідно до цього правила, де ви вказуєте великий палець від 1 до j, тому j помножене на i дорівнює від’ємному k, а j помножене на k дорівнює i. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/urdu/sentence_translations.json b/2018/quaternions/urdu/sentence_translations.json index e095e4634..dc9b6ffe7 100644 --- a/2018/quaternions/urdu/sentence_translations.json +++ b/2018/quaternions/urdu/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "پھر آپ لینس سے کہتے ہیں، دو پیچیدہ نمبروں کو ضرب دینے کے لیے، آپ صرف تقسیمی خاصیت کا استعمال کریں، جسے بہت سے لوگ اسکول میں FOIL کے طور پر سیکھتے ہیں، اور اس اصول کو لاگو کریں، i times i برابر منفی 1، چیزوں کو مزید آسان بنانے کے لیے۔", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "اسی طرح کہ پیچیدہ نمبروں میں ایک اضافی خیالی جہت کے ساتھ حقیقی اعداد شامل ہوتے ہیں، جس کی نمائندگی اکائی i کے ذریعے کی جاتی ہے، اور یہ کہ ہمارے پاس تین جہتوں میں موجود غیر حقیقی نمبر کے نظام کی چیز میں ایک دوسری خیالی سمت شامل ہوتی ہے، j، quaternions میں تین الگ الگ خیالی جہتوں کے ساتھ حقیقی اعداد شامل ہوتے ہیں، جس کی نمائندگی اکائی i، j اور k کرتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "اس کے لیے کھڑا دائرہ، i اور k سے گزرتے ہوئے، اس اصول کے مطابق 90 ڈگری گھمایا جاتا ہے، جہاں آپ اپنے انگوٹھے کو 1 سے j تک اشارہ کرتے ہیں، تو j اوقات i منفی k ہے اور j گنا k i ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/quaternions/vietnamese/sentence_translations.json b/2018/quaternions/vietnamese/sentence_translations.json index 8f98dadb9..347f71443 100644 --- a/2018/quaternions/vietnamese/sentence_translations.json +++ b/2018/quaternions/vietnamese/sentence_translations.json @@ -208,7 +208,7 @@ "end": 288.86 }, { - "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and apply this rule, i times i equals negative 1, to simplify things down further. ", + "input": "Then you say to Linus, to multiply two complex numbers, you just use the distributive property, what many people learn in school as FOIL, and you apply this rule that i times i equals negative one to simplify things down further. ", "translatedText": "Sau đó, bạn nói với Linus, để nhân hai số phức, bạn chỉ cần sử dụng thuộc tính phân phối, thứ mà nhiều người học ở trường là FOIL, và áp dụng quy tắc này, i nhân i bằng âm 1, để đơn giản hóa mọi thứ hơn nữa. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 1041.96 }, { - "input": "In the same way that the complex numbers include the real numbers with a single extra imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", + "input": "In the same way that the complex numbers included the real numbers with a single extra quote unquote imaginary dimension, represented by the unit i, and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction, j, the quaternions include the real numbers together with three separate imaginary dimensions, represented by the units i, j, and k. ", "translatedText": "Theo cách tương tự, các số phức bao gồm các số thực có thêm một chiều ảo, được biểu thị bằng đơn vị i, và hệ thống không thực sự là số mà chúng ta có trong ba chiều bao gồm hướng tưởng tượng thứ hai, j, các bậc bốn bao gồm các số thực cùng với ba chiều ảo riêng biệt, được biểu thị bằng các đơn vị i, j và k. ", "model": "google_nmt", "n_reviews": 0, @@ -1320,7 +1320,7 @@ "end": 1652.0 }, { - "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this rule, where you point your thumb from 1 to j, so j times i is negative k and j times k is i. ", + "input": "The circle perpendicular to that one, passing through i and k, gets rotated 90 degrees according to this right-hand rule, where you point your thumb from 1 to j, so j times i is negative k, and j times k is i. ", "translatedText": "Đường tròn vuông góc với đường tròn đó, đi qua i và k, được quay 90 độ theo quy tắc này, trong đó bạn chỉ ngón cái từ 1 đến j, vậy j nhân i âm k và j nhân k bằng i. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/arabic/sentence_translations.json b/2018/sphere-area/arabic/sentence_translations.json index bae46e65a..733e2b7cc 100644 --- a/2018/sphere-area/arabic/sentence_translations.json +++ b/2018/sphere-area/arabic/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same. ", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same. ", "translatedText": "وبما أنه بالنسبة لأي غطاء محدد، فإن مستطيلات الكرة لها نفس مساحة الأسطوانة، مهما كانت القيمة التي تقترب منها كل من هاتين السلسلتين من التقريبات، فيجب أن تكون هي نفسها في الواقع. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up. ", + "input": "All right, structured exercise time. We'll ease in with a warm-up. ", "translatedText": "سوف يصبح وقت التمرين المنظم أسهل مع عملية الإحماء. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta? ", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta? ", "translatedText": "السؤال 1، ما هو محيط هذه الحلقة، مثلاً عند الحافة الداخلية، بدلالة r وtheta؟ ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", "translatedText": "بمجرد حصولك على ذلك، اضرب الإجابة في السُمك r مرات d-theta للحصول على تقدير تقريبي لمساحة الحلقة، وهو تقريب سيصبح أفضل وأفضل كلما قمت بتقطيع الكرة بشكل أكثر دقة. ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta? ", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta? ", "translatedText": "السؤال 2، ما مساحة ظل إحدى هذه الحلقات على المستوى xy، معبرًا عنها مرة أخرى بدلالة r، وtheta، وd-theta؟ ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", "translatedText": "السؤال الثالث، وهذا هو جوهر الموضوع حقًا، كل ظلال من هذه الحلقات لها بالضبط نصف مساحة إحدى الحلقات الموجودة على الكرة. ", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", "translatedText": "السؤال الرابع، قلت في البداية أن هناك تطابقًا بين كل الظلال من نصف الكرة الشمالي، والتي تشكل دائرة نصف قطرها r، وكل حلقة ثانية على الكرة. ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", "translatedText": "أخيرًا، سأكون مقصّرًا إذا لم أذكر بإيجاز حقيقة أن مساحة سطح الكرة هي مثال محدد جدًا لحقيقة أكثر عمومية. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/bengali/sentence_translations.json b/2018/sphere-area/bengali/sentence_translations.json index 694aabac1..71d60a58a 100644 --- a/2018/sphere-area/bengali/sentence_translations.json +++ b/2018/sphere-area/bengali/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same. ", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same. ", "translatedText": "এবং যেহেতু যেকোন নির্দিষ্ট আবরণের জন্য, গোলক আয়তক্ষেত্রগুলির সিলিন্ডারের মতো একই ক্ষেত্রফল রয়েছে যা-ই হোক না কেন এই দুটি সিরিজের অনুমানগুলির প্রতিটিটি আসলে একই হতে হবে।", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up. ", + "input": "All right, structured exercise time. We'll ease in with a warm-up. ", "translatedText": "স্ট্রাকচার্ড ব্যায়াম সময় একটি ওয়ার্ম আপ সঙ্গে সহজ হবে. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta? ", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta? ", "translatedText": "প্রশ্ন 1, এই বলয়ের পরিধি কত, ভিতরের প্রান্তে বলুন, r এবং theta পরিপ্রেক্ষিতে? ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", "translatedText": "একবার আপনার কাছে এটি হয়ে গেলে, রিং এর ক্ষেত্রফলের আনুমানিকতা পেতে উত্তরটিকে r বার d-থেটা দ্বারা গুণ করুন, একটি আনুমানিকতা যা আপনি গোলকটিকে আরও এবং আরও সূক্ষ্মভাবে কাটার সাথে সাথে আরও ভাল হবে।", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta? ", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta? ", "translatedText": "প্রশ্ন 2, xy-প্লেনে এই বলয়ের একটির ছায়ার ক্ষেত্রফলকে আবার r, theta, এবং d-theta দ্বারা প্রকাশ করা হয়? ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", "translatedText": "প্রশ্ন 3, এবং এটি সত্যিই এটির হৃদয়, এই রিংগুলির প্রতিটি ছায়া গোলাকার রিংগুলির একটির অর্ধেক ক্ষেত্রফল রয়েছে।", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", "translatedText": "প্রশ্ন 4, আমি শুরুতেই বলেছিলাম যে উত্তর গোলার্ধের সমস্ত ছায়াগুলির মধ্যে একটি সঙ্গতি রয়েছে, যা r ব্যাসার্ধের সাথে একটি বৃত্ত তৈরি করে এবং গোলকের প্রতিটি দ্বিতীয় বলয়।", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", "translatedText": "পরিশেষে, আমি এই সত্যটির সংক্ষিপ্ত উল্লেখ না করা থেকে বিরত থাকব যে একটি গোলকের পৃষ্ঠের ক্ষেত্রফল অনেক বেশি সাধারণ সত্যের একটি খুব নির্দিষ্ট উদাহরণ।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/chinese/sentence_translations.json b/2018/sphere-area/chinese/sentence_translations.json index 28e7074ed..ce5e2040b 100644 --- a/2018/sphere-area/chinese/sentence_translations.json +++ b/2018/sphere-area/chinese/sentence_translations.json @@ -662,7 +662,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same. ", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same. ", "translatedText": "由于对于任何特定的覆盖物，球体矩形与圆柱 体具有相同的面积，因此无论这两个系列的近 似值中的每一个接近的值实际上都必须相同。", "model": "google_nmt", "from_community_srt": "因為任何特定的覆蓋， 球體 矩形與圓柱體具有相同的面積 矩形， 無論這些都是什麼價值 兩個系列的近似值即將到來 實際上必須是一樣的。", @@ -842,7 +842,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up. ", + "input": "All right, structured exercise time. We'll ease in with a warm-up. ", "translatedText": "结构化的锻炼时间会随着热身而逐渐放松。", "model": "google_nmt", "from_community_srt": "有條理的運動時間。", @@ -851,7 +851,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta? ", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta? ", "translatedText": "问题 1，这个环的周长（即内边缘）的周长 是多少，用 r 和 theta 表示？", "model": "google_nmt", "from_community_srt": "好 通過熱身來輕鬆自如 問題1：什麼是周長 在R的內邊緣處的這個環 和theta？", @@ -860,7 +860,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", "translatedText": "一旦你得到了这个结果，将答案乘以厚度 r 乘以 d-theta 即可得到环面积的近似值，当你将 球体切得越来越细时，这个近似值会变得越来越好。", "model": "google_nmt", "from_community_srt": "來吧， 再加上你的答案 厚度R * d-θ得到近似值 這個戒指的區域;和近似 當你切碎時， 這會越來越好 球體越來越精細。", @@ -878,7 +878,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta? ", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta? ", "translatedText": "问题 2，这些环之一在 xy 平面上的阴影面积是多少 ，同样用 r、theta 和 d-theta 表示？", "model": "google_nmt", "from_community_srt": "所以… 問題2：影子的面積是多少 在xy平面上的這些環之一？ 再次， 以R，", @@ -896,7 +896,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", "translatedText": "问题 3，这确实是问题的核心，每个环的阴 影恰好是球体上其中一个环的面积的一半。", "model": "google_nmt", "from_community_srt": "問題＃3：每個環形陰影都有 恰好是其中一個戒指面積的一半 在球體上。", @@ -932,7 +932,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", "translatedText": "问题4，我一开始就说过，北半球的所 有阴影（组成一个半径为r的圆）与球 体上每隔一个环之间存在对应关系。", "model": "google_nmt", "from_community_srt": "您可能想要參考一些 觸發身份） 問題＃4：我在一開始就說過 所有陰影之間的對應關係 北半球， 構成一個圓圈 半徑為R， 並且每個其他環都在 球。", @@ -976,7 +976,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", "translatedText": "最后，如果我不简要提及一个事实，那就是我的失职：球 体的表面积是一个更普遍的事实的一个非常具体的例子。", "model": "google_nmt", "from_community_srt": "最後， 我不應該做出一個 簡要提到表面的事實 球體的區域是一個特定的實例 一個更普遍的事實：如果你採取任何 凸形，", diff --git a/2018/sphere-area/english/captions.srt b/2018/sphere-area/english/captions.srt index 3ac79a89b..4793efd28 100644 --- a/2018/sphere-area/english/captions.srt +++ b/2018/sphere-area/english/captions.srt @@ -319,12 +319,12 @@ d is the distance from the bottom of the rectangle. To think about projecting this out to the cylinder, we'll picture two similar triangles. 81 -00:05:24,120 --> 00:05:29,124 +00:05:24,120 --> 00:05:28,575 The first one shares its base with the base of the rectangle on the sphere, 82 -00:05:29,124 --> 00:05:32,680 -and has a height but on the z-axis, a distance d away. +00:05:28,575 --> 00:05:32,680 +and has a tip at the same height but on the z-axis, a distance d away. 83 00:05:33,760 --> 00:05:36,880 @@ -935,6 +935,10 @@ averaged over all possible orientations in 3D space, that average will be exactly one-fourth the surface area of your shape. 235 -00:15:38,900 --> 00:15:42,460 -As to why this is true, I'll have to leave those details for another day. +00:15:38,900 --> 00:15:41,702 +As to why this is true, I'll have to leave those details for another day. + +236 +00:15:41,702 --> 00:15:42,460 +Thanks for watching! diff --git a/2018/sphere-area/english/sentence_timings.json b/2018/sphere-area/english/sentence_timings.json index 6dded976b..6f10dabe3 100644 --- a/2018/sphere-area/english/sentence_timings.json +++ b/2018/sphere-area/english/sentence_timings.json @@ -205,7 +205,7 @@ 323.48 ], [ - "The first one shares its base with the base of the rectangle on the sphere, and has a height but on the z-axis, a distance d away.", + "The first one shares its base with the base of the rectangle on the sphere, and has a tip at the same height but on the z-axis, a distance d away.", 324.12, 332.68 ], @@ -560,7 +560,7 @@ 938.06 ], [ - "As to why this is true, I'll have to leave those details for another day.", + "As to why this is true, I'll have to leave those details for another day. Thanks for watching!", 938.9, 942.46 ] diff --git a/2018/sphere-area/english/transcript.txt b/2018/sphere-area/english/transcript.txt index a18027ec8..d97970166 100644 --- a/2018/sphere-area/english/transcript.txt +++ b/2018/sphere-area/english/transcript.txt @@ -39,7 +39,7 @@ You could rightfully complain that the distance d is a little ambiguous, dependi But for tinier and tinier rectangles, that ambiguity will become negligible, and tinier and tinier is when this approximation with rectangles gets closer to the true surface area anyway. To choose an arbitrary standard, let's say that d is the distance from the bottom of the rectangle. To think about projecting this out to the cylinder, we'll picture two similar triangles. -The first one shares its base with the base of the rectangle on the sphere, and has a height but on the z-axis, a distance d away. +The first one shares its base with the base of the rectangle on the sphere, and has a tip at the same height but on the z-axis, a distance d away. The second triangle is a scaled-up version of this, scaled so that it just barely reaches the cylinder, meaning its long side now has a length r. So the ratio of their bases, which is how much our rectangle's width gets stretched out, is r divided by d. What about the height? @@ -110,4 +110,4 @@ Why does this imply that the area of the circle is exactly one-fourth the surfac If you want answers or hints, I'm quite sure that people in comments and on Reddit will have them waiting for you. And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. If you take any convex shape and look at the average area of all of its shadows, averaged over all possible orientations in 3D space, that average will be exactly one-fourth the surface area of your shape. -As to why this is true, I'll have to leave those details for another day. \ No newline at end of file +As to why this is true, I'll have to leave those details for another day. Thanks for watching! \ No newline at end of file diff --git a/2018/sphere-area/french/sentence_translations.json b/2018/sphere-area/french/sentence_translations.json index 7f76fac27..f73a0aeea 100644 --- a/2018/sphere-area/french/sentence_translations.json +++ b/2018/sphere-area/french/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "Et puisque pour tout revêtement spécifique, les rectangles de la sphère ont la même aire que le cylindre, quelle que soit la valeur proche de chacune de ces deux séries d'approximations, elle doit en réalité être la même.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "Le temps d’exercice structuré s’atténuera avec un échauffement.", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "Question 1, quelle est la circonférence de cet anneau, disons au bord intérieur, en termes de r et thêta?", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "Une fois que vous avez cela, multipliez la réponse par l'épaisseur r fois d-thêta pour obtenir une approximation de la surface de l'anneau, une approximation qui s'améliorera à mesure que vous hacherez la sphère de plus en plus finement.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "Question 2, quelle est l'aire de l'ombre de l'un de ces anneaux sur le plan xy, encore une fois exprimée en termes de r, thêta et d-thêta?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "Question 3, et c'est vraiment le cœur du problème, les ombres de chacun de ces anneaux ont précisément la moitié de la surface d'un des anneaux de la sphère.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "Question 4, j'ai dit au début qu'il y a une correspondance entre toutes les ombres de l'hémisphère nord, qui composent un cercle de rayon r, et un anneau sur deux sur la sphère.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "Enfin, je m’en voudrais de ne pas mentionner brièvement le fait que la surface d’une sphère est un exemple très spécifique d’un fait beaucoup plus général.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/german/sentence_translations.json b/2018/sphere-area/german/sentence_translations.json index 915f03f29..9fbf181b9 100644 --- a/2018/sphere-area/german/sentence_translations.json +++ b/2018/sphere-area/german/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "Und da für jede spezifische Abdeckung die Kugelrechtecke die gleiche Fläche wie der Zylinder haben, muss jeder Wert, dem sich jede dieser beiden Reihen von Näherungen annähert, tatsächlich derselbe sein.", "model": "google_nmt", "n_reviews": 1, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "Ein strukturiertes Training beginnt mit leichtem Aufwärmen.", "model": "google_nmt", "n_reviews": 1, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "Frage 1: Wie groß ist der Umfang dieses Rings, beispielsweise am inneren Rand, ausgedrückt in r und Theta?", "model": "google_nmt", "n_reviews": 1, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "Sobald du das hast, multipliziere das Ergebnis mit der Dicke r mal d-Theta, um eine Näherung für die Ringfläche zu erhalten, eine Näherung, die immer besser wird, je feiner du die Kugel zerteilst.", "model": "google_nmt", "n_reviews": 1, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "Frage 2: Wie groß ist die Schattenfläche eines dieser Ringe auf der XY-Ebene, wiederum ausgedrückt in r, Theta und d-Theta?", "model": "google_nmt", "n_reviews": 1, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "Frage 3, und das ist eigentlich der Kern der Sache: Jeder einzelne Schatten dieser Ringe hat genau die halbe Fläche eines der Ringe auf der Kugel.", "model": "google_nmt", "n_reviews": 1, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "Frage 4: Ich habe zu Beginn gesagt, dass es einen Zusammenhang zwischen allen Schatten der nördlichen Hemisphäre, die einen Kreis mit dem Radius r bilden, und jedem zweiten Ring auf der Kugel gibt.", "model": "google_nmt", "n_reviews": 1, @@ -872,7 +872,7 @@ "end": 914 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "Abschließend wäre es eine Versäumnis, nicht kurz die Tatsache zu erwähnen, dass die Oberfläche einer Kugel ein sehr spezifisches Beispiel einer viel allgemeineren Tatsache ist.", "model": "google_nmt", "n_reviews": 1, @@ -903,4 +903,4 @@ "start": 942.46, "end": 942.46 } -] +] \ No newline at end of file diff --git a/2018/sphere-area/hebrew/sentence_translations.json b/2018/sphere-area/hebrew/sentence_translations.json index b9f9348bb..45eb0fa8c 100644 --- a/2018/sphere-area/hebrew/sentence_translations.json +++ b/2018/sphere-area/hebrew/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "ומכיוון שלכל כיסוי ספציפי, למלבני הכדור יש את אותו שטח כמו הגליל, כל ערך שכל אחת משתי סדרות הקירוב הללו מתקרבות חייב להיות למעשה זהה.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "זמן אימון מובנה יקל בחימום.", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "שאלה 1, מה ההיקף של הטבעת הזו, נניח בקצה הפנימי, מבחינת r ותטא?", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "ברגע שיש לך את זה, הכפילו את התשובה בעובי r כפול d-theta כדי לקבל קירוב עבור שטח הטבעת, קירוב שישתפר יותר ויותר ככל שתחתוך את הכדור דק יותר ויותר.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "שאלה 2, מהו שטח הצל של אחת מהטבעות הללו במישור ה-xy, שוב מבוטא במונחים של r, תטא ו-d-theta?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "שאלה 3, וזה באמת לב ליבה, לכל אחת מהצללים של הטבעות האלה יש בדיוק חצי מהשטח של אחת הטבעות על הכדור.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "שאלה 4, אמרתי בהתחלה שיש התאמה בין כל הצללים מחצי הכדור הצפוני, המרכיבים עיגול ברדיוס r, לבין כל טבעת שנייה בכדור.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "לבסוף, לא אזכור בקצרה את העובדה ששטח הפנים של כדור הוא דוגמה מאוד ספציפית של עובדה כללית הרבה יותר.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/hindi/sentence_translations.json b/2018/sphere-area/hindi/sentence_translations.json index 6a1d29f17..8ea808408 100644 --- a/2018/sphere-area/hindi/sentence_translations.json +++ b/2018/sphere-area/hindi/sentence_translations.json @@ -518,7 +518,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "और चूंकि किसी भी विशिष्ट आवरण के लिए, गोले के आयतों का क्षेत्रफल सिलेंडर के समान होता है, इन दो श्रृंखलाओं में से प्रत्येक का जो भी मूल्य आ रहा है वह वास्तव में समान होना चाहिए।", "n_reviews": 0, "start": 575.02, @@ -658,21 +658,21 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "वार्म-अप के साथ संरचित व्यायाम का समय आसान हो जाएगा।", "n_reviews": 0, "start": 788.96, "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "प्रश्न 1, आर और थीटा के संदर्भ में, इस रिंग की परिधि, आंतरिक किनारे पर क्या है?", "n_reviews": 0, "start": 792.84, "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "एक बार जब आपके पास यह हो, तो रिंग के क्षेत्र का अनुमान प्राप्त करने के लिए उत्तर को मोटाई आर गुना डी-थीटा से गुणा करें, एक अनुमान जो कि गोले को अधिक से अधिक बारीकी से काटने पर बेहतर और बेहतर होता जाएगा।", "n_reviews": 0, "start": 802.04, @@ -686,7 +686,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "प्रश्न 2, xy-तल पर इन छल्लों में से एक की छाया का क्षेत्रफल क्या है, जिसे फिर से r, थीटा और d-थीटा के रूप में व्यक्त किया गया है?", "n_reviews": 0, "start": 828.24, @@ -700,7 +700,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "प्रश्न 3, और यही वास्तव में इसका मूल है, इन छल्लों की छाया में से प्रत्येक का क्षेत्रफल गोले पर मौजूद छल्लों में से एक के क्षेत्रफल का ठीक आधा है।", "n_reviews": 0, "start": 849.36, @@ -728,7 +728,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "प्रश्न 4, मैंने शुरू में ही कहा था कि उत्तरी गोलार्ध की सभी छायाओं के बीच एक पत्राचार है, जो त्रिज्या r के साथ एक वृत्त बनाता है, और गोले पर हर दूसरा वलय है।", "n_reviews": 0, "start": 874.42, @@ -763,7 +763,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "अंत में, इस तथ्य का संक्षिप्त उल्लेख न करना मेरी भूल होगी कि किसी गोले का सतह क्षेत्र कहीं अधिक सामान्य तथ्य का एक बहुत ही विशिष्ट उदाहरण है।", "n_reviews": 0, "start": 915.16, diff --git a/2018/sphere-area/hungarian/sentence_translations.json b/2018/sphere-area/hungarian/sentence_translations.json index b841b60b4..601278c70 100644 --- a/2018/sphere-area/hungarian/sentence_translations.json +++ b/2018/sphere-area/hungarian/sentence_translations.json @@ -328,7 +328,7 @@ "end": 323.48 }, { - "input": "The first one shares its base with the base of the rectangle on the sphere, and has a height but on the z-axis, a distance d away.", + "input": "The first one shares its base with the base of the rectangle on the sphere, and has a tip at the same height but on the z-axis, a distance d away.", "translatedText": "Az elsőnek közös az alapja a gömbön lévő téglalap alapjával, és magassága van, de a z tengelyen, d távolságban.", "model": "DeepL", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 938.06 }, { - "input": "As to why this is true, I'll have to leave those details for another day.", + "input": "As to why this is true, I'll have to leave those details for another day. Thanks for watching!", "translatedText": "Hogy ez miért igaz, azt egy másik napra kell hagynom.", "model": "DeepL", "n_reviews": 0, diff --git a/2018/sphere-area/indonesian/sentence_translations.json b/2018/sphere-area/indonesian/sentence_translations.json index 4cd1a681f..0ec27789e 100644 --- a/2018/sphere-area/indonesian/sentence_translations.json +++ b/2018/sphere-area/indonesian/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "Dan karena untuk penutup tertentu, persegi panjang bola mempunyai luas yang sama dengan silinder, berapa pun nilai yang didekati oleh kedua rangkaian pendekatan ini, sebenarnya harus sama.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "Waktu latihan yang terstruktur akan dipermudah dengan pemanasan.", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "Pertanyaan 1, berapa keliling cincin ini, katakanlah di tepi bagian dalam, dalam r dan theta?", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "Setelah Anda memilikinya, kalikan jawabannya dengan ketebalan r dikalikan d-theta untuk mendapatkan perkiraan luas cincin, perkiraan yang akan semakin baik seiring Anda memotong bola semakin halus.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "Pertanyaan 2, berapakah luas bayangan salah satu cincin pada bidang xy, yang dinyatakan lagi dalam r, theta, dan d-theta?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "Pertanyaan 3, dan inilah intinya, masing-masing bayangan cincin ini mempunyai tepat setengah luas salah satu cincin pada bola.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "Pertanyaan 4, Saya katakan di awal bahwa ada korespondensi antara semua bayangan dari belahan bumi utara, yang membentuk lingkaran dengan jari-jari r, dan setiap detik cincin pada bola tersebut.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "Terakhir, saya akan lalai untuk tidak menyebutkan secara singkat fakta bahwa luas permukaan bola adalah contoh yang sangat spesifik dari fakta yang jauh lebih umum.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/italian/sentence_translations.json b/2018/sphere-area/italian/sentence_translations.json index 752299309..60676e299 100644 --- a/2018/sphere-area/italian/sentence_translations.json +++ b/2018/sphere-area/italian/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "E poiché per qualsiasi rivestimento specifico, i rettangoli della sfera hanno la stessa area del cilindro, qualunque valore si avvicini ciascuna di queste due serie di approssimazioni deve effettivamente essere lo stesso.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "Il tempo di esercizio strutturato si allenterà con un riscaldamento.", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "Domanda 1, qual è la circonferenza di questo anello, diciamo sul bordo interno, in termini di r e theta?", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "Una volta ottenuto questo, moltiplica il risultato per lo spessore r per d-theta per ottenere un'approssimazione dell'area dell'anello, un'approssimazione che migliorerà sempre di più man mano che taglierai la sfera sempre più finemente.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "Domanda 2: qual è l'area dell'ombra di uno di questi anelli sul piano xy, espressa ancora in termini di r, theta e d-theta?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "Domanda 3, e questo è davvero il nocciolo della questione, ciascuna delle ombre di questi anelli ha esattamente la metà dell'area di uno degli anelli sulla sfera.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "Domanda 4, ho detto all'inizio che esiste una corrispondenza tra tutte le ombre dell'emisfero settentrionale, che formano un cerchio di raggio r, e ogni secondo anello della sfera.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "Infine, sarei negligente se non menzionassi brevemente il fatto che l’area superficiale di una sfera è un esempio molto specifico di un fatto molto più generale.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/japanese/sentence_translations.json b/2018/sphere-area/japanese/sentence_translations.json index a47f14499..799972dd4 100644 --- a/2018/sphere-area/japanese/sentence_translations.json +++ b/2018/sphere-area/japanese/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same. ", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same. ", "translatedText": "そして、特定の被覆の場合、球の長方形は円柱と同じ 面積を持つため、これら 2 つの近似系列のそれぞ れが近づく値は実際には同じでなければなりません。", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up. ", + "input": "All right, structured exercise time. We'll ease in with a warm-up. ", "translatedText": "体系的な運動時間は、ウォームアップを行うことで簡単に始められます。", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta? ", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta? ", "translatedText": "質問 1、このリングの円周、たとえば内側の端の 円周は、r と theta でいくらですか? ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", "translatedText": "それがわかったら、その答えに厚さ r と d-θ を乗 算して、リングの面積の近似値を求めます。この近似値は 、球を細かく刻むにつれてどんどん良くなっていきます。", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta? ", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta? ", "translatedText": "質問 2、xy 平面上のこれらのリングの 1 つの影の面積 は、これも r、シータ、および d シータで表されますか? ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", "translatedText": "質問 3、そしてこれがまさにその核心です。これらのリングのそれぞれの 影は、球体上の 1 つのリングのちょうど半分の面積を持っています。", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", "translatedText": "質問 4、私は冒頭で、半径 r の円を構成す る北半球のすべての影と、球上の 2 番目の リングとの間に対応関係があると言いました。", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", "translatedText": "最後に、球の表面積は、より一般的な事実の非常に特殊な例で あるという事実について簡単に言及しないのは不謹慎です。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/korean/sentence_translations.json b/2018/sphere-area/korean/sentence_translations.json index 6fdcc8253..c9d3a8f15 100644 --- a/2018/sphere-area/korean/sentence_translations.json +++ b/2018/sphere-area/korean/sentence_translations.json @@ -368,7 +368,7 @@ "end": 323.48 }, { - "input": "The first one shares its base with the base of the rectangle on the sphere, and has a height but on the z-axis, a distance d away.", + "input": "The first one shares its base with the base of the rectangle on the sphere, and has a tip at the same height but on the z-axis, a distance d away.", "translatedText": "첫 번째는 구의 직사각형 밑변과 밑변을 공유하며 높이가 있지만 Z축에서는 거리가 d 떨어져 있습니다.", "model": "DeepL", "from_community_srt": "이 첫 번째 하나는 그 밑변을 사각형의 밑면과 공유한다. 구에 있고, 같은 높이의 팁을 가지고있다. z 축에 거리 d 멀리.", @@ -1005,7 +1005,7 @@ "end": 938.06 }, { - "input": "As to why this is true, I'll have to leave those details for another day.", + "input": "As to why this is true, I'll have to leave those details for another day. Thanks for watching!", "translatedText": "그 이유에 대해서는 다음 기회에 자세히 설명하겠습니다.", "model": "DeepL", "from_community_srt": "평균 그림자 영역. 이것이 사실 인 이유에 관해서는,", diff --git a/2018/sphere-area/marathi/sentence_translations.json b/2018/sphere-area/marathi/sentence_translations.json index 853fb2ae2..6c66a4e06 100644 --- a/2018/sphere-area/marathi/sentence_translations.json +++ b/2018/sphere-area/marathi/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "आणि कोणत्याही विशिष्ट आच्छादनासाठी, गोल आयतांचे क्षेत्रफळ सिलिंडर सारखेच असते, या दोन अंदाजे मालिकेतील प्रत्येक जवळ येत असलेल्‍या प्रत्येक मूल्याचे असले पाहिजे.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "स्ट्रक्चर्ड व्यायामाचा वेळ वॉर्म-अप सह सहज होईल.", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "प्रश्न 1, या रिंगचा घेर किती आहे, r आणि theta च्या संदर्भात, आतील काठावर म्हणा?", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "तुमच्याकडे ते मिळाल्यावर, रिंगच्या क्षेत्रासाठी अंदाजे मिळवण्यासाठी उत्तराला r गुणाकार d-थेटा जाडीने गुणा, असा अंदाज जो तुम्ही गोलाकार अधिकाधिक बारीक चिरताना अधिक चांगला होईल.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "प्रश्न 2, xy-प्लेनवरील यापैकी एका कड्याच्या सावलीचे क्षेत्रफळ किती आहे, ते पुन्हा r, theta, आणि d-theta या शब्दांनी व्यक्त केले जाते?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "प्रश्न 3, आणि हे खरोखरच त्याचे हृदय आहे, या प्रत्येक रिंगच्या सावलीचे क्षेत्रफळ गोलावरील एका कड्याच्या निम्मे आहे.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "प्रश्न 4, मी सुरवातीलाच म्हटले आहे की उत्तर गोलार्धातील सर्व सावल्यांमध्ये एक पत्रव्यवहार आहे, जे त्रिज्या r असलेले वर्तुळ बनवतात आणि गोलावरील प्रत्येक द्वितीय वलय.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "शेवटी, गोलाच्या पृष्ठभागाचे क्षेत्रफळ हे अधिक सामान्य वस्तुस्थितीचे एक अतिशय विशिष्ट उदाहरण आहे या वस्तुस्थितीचा थोडक्यात उल्लेख न करणे मला कमी पडेल.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/persian/sentence_translations.json b/2018/sphere-area/persian/sentence_translations.json index db9931b27..19decab17 100644 --- a/2018/sphere-area/persian/sentence_translations.json +++ b/2018/sphere-area/persian/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same. ", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same. ", "translatedText": "و از آنجایی که برای هر پوشش خاص، مستطیل‌های کروی مساحتی برابر با استوانه دارند، هر مقداری که هر یک از این دو سری تقریب به آن نزدیک شوند، باید در واقع یکسان باشند. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up. ", + "input": "All right, structured exercise time. We'll ease in with a warm-up. ", "translatedText": "زمان تمرین ساختاریافته با گرم کردن کاهش می یابد. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta? ", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", "translatedText": "سوال 1، محیط این حلقه، مثلاً در لبه داخلی، بر حسب r و تتا چقدر است؟ هنگامی که آن را به دست آوردید، پاسخ را در ضخامت r ضرب در d-تتا کنید تا تقریبی برای مساحت حلقه به دست آورید، تقریبی که وقتی کره را بیشتر و بیشتر ریز می کنید بهتر و بهتر می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta? ", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta? ", "translatedText": "سوال 2 مساحت سایه یکی از این حلقه ها در صفحه xy که دوباره بر حسب r و تتا و d-تتا بیان می شود چقدر است؟ برای این مورد، شاید مفید باشد که به آن مثلث قائم الزاویه کوچکی که قبلاً در مورد آن صحبت کردیم فکر کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", "translatedText": "سوال 3، و این واقعاً قلب آن است، هر یک از سایه‌های این حلقه‌ها دقیقاً نصف مساحت یکی از حلقه‌های روی کره را دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", "translatedText": "سوال 4، در ابتدا گفتم که بین تمام سایه‌های نیمکره شمالی که دایره‌ای با شعاع r تشکیل می‌دهند و هر حلقه ثانیه روی کره مطابقت دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", "translatedText": "در نهایت، نادیده می‌گیرم که به این واقعیت اشاره کوتاهی نکنم که سطح یک کره، نمونه‌ای بسیار خاص از یک واقعیت بسیار کلی‌تر است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/polish/sentence_translations.json b/2018/sphere-area/polish/sentence_translations.json index 74dc80c49..1d2d0f237 100644 --- a/2018/sphere-area/polish/sentence_translations.json +++ b/2018/sphere-area/polish/sentence_translations.json @@ -328,7 +328,7 @@ "end": 323.48 }, { - "input": "The first one shares its base with the base of the rectangle on the sphere, and has a height but on the z-axis, a distance d away.", + "input": "The first one shares its base with the base of the rectangle on the sphere, and has a tip at the same height but on the z-axis, a distance d away.", "translatedText": "", "from_community_srt": "Pierwszy niech dzieli podstawę z naszym prostokątem na sferze, a wierzchołek niech leży na tej samej wysokości, na osi Z,", "n_reviews": 0, @@ -895,7 +895,7 @@ "end": 938.06 }, { - "input": "As to why this is true, I'll have to leave those details for another day.", + "input": "As to why this is true, I'll have to leave those details for another day. Thanks for watching!", "translatedText": "", "from_community_srt": "Dlaczego to jest prawda?", "n_reviews": 0, diff --git a/2018/sphere-area/portuguese/sentence_translations.json b/2018/sphere-area/portuguese/sentence_translations.json index 2f493e53b..741b29049 100644 --- a/2018/sphere-area/portuguese/sentence_translations.json +++ b/2018/sphere-area/portuguese/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "E como para qualquer cobertura específica, os retângulos esféricos têm a mesma área que o cilindro, qualquer valor que cada uma dessas duas séries de aproximações se aproxime deve ser na verdade o mesmo.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "O tempo de exercício estruturado será facilitado com um aquecimento.", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "Pergunta 1, qual é a circunferência deste anel, digamos na borda interna, em termos de r e teta?", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "Depois de fazer isso, multiplique a resposta pela espessura r vezes d-teta para obter uma aproximação para a área do anel, uma aproximação que ficará cada vez melhor à medida que você corta a esfera cada vez mais finamente.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "Questão 2, qual é a área da sombra de um desses anéis no plano xy, novamente expressa em termos de r, teta e d-teta?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "Pergunta 3, e este é realmente o cerne da questão, cada uma das sombras desses anéis tem precisamente metade da área de um dos anéis na esfera.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "Questão 4, eu disse no início que existe uma correspondência entre todas as sombras do hemisfério norte, que formam um círculo com raio r, e cada segundo anel da esfera.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "Finalmente, seria negligente se não fizesse uma breve menção ao facto de que a área superficial de uma esfera é um exemplo muito específico de um facto muito mais geral.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/russian/sentence_translations.json b/2018/sphere-area/russian/sentence_translations.json index 569e7ac15..d105bc1a9 100644 --- a/2018/sphere-area/russian/sentence_translations.json +++ b/2018/sphere-area/russian/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "А поскольку для любого конкретного покрытия прямоугольники сферы имеют ту же площадь, что и цилиндр, к какой бы величине ни приближалась каждая из этих двух серий приближений, она должна фактически быть одинаковой.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "Структурированное время упражнений упростится после разминки.", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "Вопрос 1: какова длина окружности этого кольца, скажем, на внутреннем крае, в терминах r и тета?", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "Как только вы это получите, умножьте ответ на толщину r, умноженную на d-тета, чтобы получить приближение площади кольца, приближение, которое будет становиться все лучше и лучше по мере того, как вы будете измельчать сферу все более и более мелко.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "Вопрос 2: какова площадь тени одного из этих колец на плоскости xy, опять-таки выраженная через r, тета и d-тета?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "Вопрос 3, и в этом его суть: каждая тень этих колец имеет ровно половину площади одного из колец на сфере.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "В четвертом вопросе я вначале сказал, что существует соответствие между всеми тенями северного полушария, образующими круг радиуса r, и каждым вторым кольцом на сфере.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "Наконец, было бы упущением не упомянуть вкратце тот факт, что площадь поверхности сферы является весьма конкретным примером гораздо более общего факта.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/spanish/sentence_translations.json b/2018/sphere-area/spanish/sentence_translations.json index 674a127e4..fe4e7d028 100644 --- a/2018/sphere-area/spanish/sentence_translations.json +++ b/2018/sphere-area/spanish/sentence_translations.json @@ -327,7 +327,7 @@ "end": 323.48 }, { - "input": "The first one shares its base with the base of the rectangle on the sphere, and has a height but on the z-axis, a distance d away.", + "input": "The first one shares its base with the base of the rectangle on the sphere, and has a tip at the same height but on the z-axis, a distance d away.", "translatedText": "El primero comparte su base con la base del rectángulo de la esfera, y tiene una altura pero en el eje z, a una distancia d.", "from_community_srt": "El primero comparte su base con la base del rectángulo en la esfera, y tiene una esquina a la misma altura en el eje z, a una distancia d.", "n_reviews": 0, @@ -893,7 +893,7 @@ "end": 938.06 }, { - "input": "As to why this is true, I'll have to leave those details for another day.", + "input": "As to why this is true, I'll have to leave those details for another day. Thanks for watching!", "translatedText": "En cuanto a por qué esto es cierto, tendré que dejar esos detalles para otro día.", "from_community_srt": "En cuanto a por qué esto es cierto,", "n_reviews": 0, diff --git a/2018/sphere-area/tamil/sentence_translations.json b/2018/sphere-area/tamil/sentence_translations.json index 3f45fadfa..c9dd8139b 100644 --- a/2018/sphere-area/tamil/sentence_translations.json +++ b/2018/sphere-area/tamil/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "மேலும், எந்தவொரு குறிப்பிட்ட உறைக்கும், கோள செவ்வகங்கள் உருளையின் அதே பகுதியைக் கொண்டிருப்பதால், இந்த இரண்டு தோராயத் தொடர்களில் ஒவ்வொன்றும் எந்த மதிப்பை நெருங்குகிறதோ, அது உண்மையில் ஒரே மாதிரியாக இருக்க வேண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "கட்டமைக்கப்பட்ட உடற்பயிற்சி நேரம் ஒரு வார்ம்-அப் மூலம் எளிதாக்கப்படும்.", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "கேள்வி 1, இந்த வளையத்தின் சுற்றளவு என்ன, உள் விளிம்பில், r மற்றும் தீட்டாவின் அடிப்படையில் சொல்லுங்கள்?", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "உங்களிடம் அது கிடைத்ததும், மோதிரத்தின் பகுதிக்கான தோராயத்தைப் பெற, தடிமன் r மடங்கு d-தீட்டாவால் பதிலைப் பெருக்கவும், இது நீங்கள் கோளத்தை மேலும் மேலும் நுணுக்கமாக வெட்டும்போது சிறப்பாகவும் சிறப்பாகவும் இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "கேள்வி 2, xy-விமானத்தில் உள்ள இந்த வளையங்களில் ஒன்றின் நிழலின் பரப்பளவு என்ன, மீண்டும் r, theta மற்றும் d-theta ஆகியவற்றின் அடிப்படையில் வெளிப்படுத்தப்படுகிறது?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "கேள்வி 3, இது உண்மையில் அதன் இதயம், இந்த மோதிரங்களின் நிழல்கள் ஒவ்வொன்றும் கோளத்தில் உள்ள மோதிரங்களில் ஒன்றின் பாதி பரப்பளவைக் கொண்டுள்ளன.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "கேள்வி 4, நான் ஆரம்பத்தில் சொன்னேன், வடக்கு அரைக்கோளத்தின் அனைத்து நிழல்களுக்கும் இடையே ஒரு கடித தொடர்பு உள்ளது, இது r ஆரம் கொண்ட ஒரு வட்டத்தையும், கோளத்தின் ஒவ்வொரு இரண்டாவது வளையத்தையும் உருவாக்குகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "இறுதியாக, ஒரு கோளத்தின் பரப்பளவு என்பது மிகவும் பொதுவான உண்மையின் ஒரு குறிப்பிட்ட நிகழ்வாகும் என்ற உண்மையைப் பற்றி சுருக்கமாக குறிப்பிடாமல் இருப்பேன்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/telugu/sentence_translations.json b/2018/sphere-area/telugu/sentence_translations.json index ac6b04c01..adbc92e0a 100644 --- a/2018/sphere-area/telugu/sentence_translations.json +++ b/2018/sphere-area/telugu/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "మరియు ఏదైనా నిర్దిష్ట కవరింగ్ కోసం, గోళాకార దీర్ఘచతురస్రాలు సిలిండర్‌కు సమానమైన వైశాల్యాన్ని కలిగి ఉంటాయి, ఈ రెండు ఉజ్జాయింపుల శ్రేణిలో ప్రతి ఒక్కటి ఏ విలువను సమీపిస్తున్నా వాస్తవానికి అదే విధంగా ఉండాలి.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "నిర్మాణాత్మక వ్యాయామ సమయం సన్నాహకతతో సులభం అవుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "ప్రశ్న 1, ఈ రింగ్ యొక్క చుట్టుకొలత ఎంత, లోపలి అంచు వద్ద, r మరియు తీటా పరంగా చెప్పండి?", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "మీరు దానిని కలిగి ఉన్న తర్వాత, రింగ్ యొక్క వైశాల్యానికి ఉజ్జాయింపుని పొందడానికి, మీరు గోళాన్ని మరింత మెత్తగా కోసేటప్పుడు మరింత మెరుగ్గా మరియు మెరుగ్గా ఉండే ఉజ్జాయింపుని పొందడానికి, సమాధానాన్ని మందం r సార్లు d-తీటాతో గుణించండి.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "ప్రశ్న 2, xy-ప్లేన్‌లోని ఈ రింగ్‌లలో ఒకదాని నీడ యొక్క వైశాల్యం ఏమిటి, మళ్లీ r, తీటా మరియు d-తీటా పరంగా వ్యక్తీకరించబడింది?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "ప్రశ్న 3, మరియు ఇది నిజంగా దాని హృదయం, ఈ రింగ్‌ల నీడల్లో ప్రతి ఒక్కటి గోళంలోని ఒక రింగులో సగం వైశాల్యాన్ని కలిగి ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "ప్రశ్న 4, నేను మొదట్లో చెప్పాను, ఉత్తర అర్ధగోళంలోని నీడలన్నింటికీ, r వ్యాసార్థంతో ఒక వృత్తాన్ని ఏర్పరుస్తుంది మరియు గోళంలోని ప్రతి రెండవ రింగ్‌కు మధ్య ఒక అనురూప్యం ఉంది.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "చివరగా, గోళం యొక్క ఉపరితల వైశాల్యం చాలా సాధారణ వాస్తవం యొక్క చాలా నిర్దిష్ట ఉదాహరణ అనే వాస్తవాన్ని క్లుప్తంగా ప్రస్తావించకూడదని నేను విస్మరించాను.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/thai/sentence_translations.json b/2018/sphere-area/thai/sentence_translations.json index b30d0b402..75275951b 100644 --- a/2018/sphere-area/thai/sentence_translations.json +++ b/2018/sphere-area/thai/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same. ", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up. ", + "input": "All right, structured exercise time. We'll ease in with a warm-up. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta? ", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta? ", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta? ", "translatedText": "เมื่อคุณได้คำตอบแล้ว ให้คูณคำตอบด้วยความหนา r คูณ d-theta เพื่อให้ได้ค่าประมาณสำหรับพื้นที่ของวงแหวน ซึ่งเป็นค่าประมาณที่จะดีขึ้นเรื่อยๆ เมื่อคุณสับทรงกลมให้ละเอียดมากขึ้นเรื่อยๆ ณ จุดนี้ ถ้าคุณรู้แคลคูลัสของตัวเอง ก็สามารถอินทิเกรตได้ แต่เป้าหมายของเราไม่ใช่แค่การหาคำตอบเท่านั้น แต่ยังรู้สึกถึงความเชื่อมโยงระหว่างทรงกลมกับเงาของมันอีกด้วย คำถามที่ 2 พื้นที่เงาของวงแหวนวงใดวงหนึ่งบนระนาบ xy เป็นเท่าใด ซึ่งแสดงในรูปของ r, theta และ d-theta อีกครั้ง สำหรับอันนี้ การคิดย้อนกลับไปถึงสามเหลี่ยมมุมฉากเล็กๆ ที่เราพูดถึงก่อนหน้านี้อาจเป็นประโยชน์ คำถามที่ 3 และนี่คือหัวใจสำคัญของคำถาม เงาของวงแหวนแต่ละวงมีพื้นที่ครึ่งหนึ่งของวงแหวนวงใดวงหนึ่งบนทรงกลมพอดี ไม่ใช่อันที่เป็นมุมทีต้าตรงเหนือ แต่เป็นอีกอันหนึ่ง คำถามคืออันไหน? ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/turkish/sentence_translations.json b/2018/sphere-area/turkish/sentence_translations.json index 330d7ff46..d1d8135a8 100644 --- a/2018/sphere-area/turkish/sentence_translations.json +++ b/2018/sphere-area/turkish/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "Ve herhangi bir özel kaplama için küre dikdörtgenleri silindirle aynı alana sahip olduğundan, bu iki yaklaşım serisinin her birinin yaklaştığı değer aslında aynı olmalıdır.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "Yapılandırılmış egzersiz süresi ısınmayla kolaylaşacaktır.", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "Soru 1, bu halkanın çevresi, örneğin iç kenarda, r ve teta cinsinden nedir?", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "Bunu elde ettikten sonra, halkanın alanı için bir yaklaşık değer elde etmek için cevabı kalınlık r çarpı d-teta ile çarpın; küreyi gittikçe daha ince bir şekilde doğradıkça bu yaklaşım giderek daha iyi hale gelecektir.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "Soru 2: Bu halkalardan birinin xy düzlemindeki gölgesinin alanı yine r, teta ve d-teta cinsinden ifade edilir?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "Soru 3, ve bu aslında konunun kalbidir, bu halkaların gölgelerinin her biri, küre üzerindeki halkalardan birinin alanının tam olarak yarısı kadardır.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "Soru 4, başlangıçta, yarıçapı r olan bir daire oluşturan kuzey yarımküredeki tüm gölgeler ile küre üzerindeki her ikinci halka arasında bir örtüşme olduğunu söylemiştim.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "Son olarak, bir kürenin yüzey alanının çok daha genel bir gerçeğin çok spesifik bir örneği olduğu gerçeğinden kısaca bahsetmemeyi ihmal etmiş olurum.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/ukrainian/sentence_translations.json b/2018/sphere-area/ukrainian/sentence_translations.json index e1284f689..89112ec61 100644 --- a/2018/sphere-area/ukrainian/sentence_translations.json +++ b/2018/sphere-area/ukrainian/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "А оскільки для будь-якого конкретного покриття прямокутники сфери мають таку саму площу, що й циліндр, будь-яке значення, до якого наближається кожна з цих двох серій наближень, має бути однаковим.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "Структурований час вправ полегшить розминку.", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "Запитання 1, яка окружність цього кільця, скажімо, на внутрішньому краю, в термінах r і тета?", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "Отримавши це, помножте відповідь на товщину r, помножену на d-тета, щоб отримати приблизне значення площі кільця, приблизне значення, яке ставатиме все кращим і кращим, коли ви все дрібніше наріжете сферу.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "Запитання 2, яка площа тіні одного з цих кілець на площині xy, знову виражена через r, тета та d-тета?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "Запитання 3, і це справді суть справи, кожна з тіней цих кілець має рівно половину площі одного з кілець на сфері.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "Запитання 4, я сказав на початку, що існує відповідність між усіма тінями з північної півкулі, які складають коло з радіусом r, і кожним другим кільцем на сфері.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "Нарешті, я був би помилкою коротко не згадати той факт, що площа поверхні сфери є дуже конкретним прикладом набагато більш загального факту.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/urdu/sentence_translations.json b/2018/sphere-area/urdu/sentence_translations.json index bb8b87fac..e55fe3a56 100644 --- a/2018/sphere-area/urdu/sentence_translations.json +++ b/2018/sphere-area/urdu/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same. ", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same. ", "translatedText": "اور چونکہ کسی بھی مخصوص کورنگ کے لیے، کرہ مستطیل کا رقبہ وہی ہوتا ہے جو سلنڈر کا ہوتا ہے جو بھی قدر ہوتی ہے ان دونوں سلسلے کے قریب آنے والے ہر ایک کو درحقیقت وہی ہونا چاہیے۔", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up. ", + "input": "All right, structured exercise time. We'll ease in with a warm-up. ", "translatedText": "منظم ورزش کا وقت وارم اپ کے ساتھ آسان ہو جائے گا۔", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta? ", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely. ", "translatedText": "سوال 1، اس انگوٹھی کا طواف کیا ہے، اندرونی کنارے پر، r اور تھیٹا کے لحاظ سے؟ ایک بار جب آپ کے پاس یہ ہو جائے تو، انگوٹھی کے رقبے کا تخمینہ حاصل کرنے کے لیے جواب کو r بار d-theta سے ضرب دیں، ایک ایسا تخمینہ جو آپ کے کرہ کو زیادہ سے زیادہ باریک کاٹتے ہی بہتر سے بہتر ہوتا جائے گا۔", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta? ", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta? ", "translatedText": "سوال 2، xy-plane پر ان حلقوں میں سے کسی ایک کے سائے کا رقبہ کیا ہے، جسے دوبارہ r، theta، اور d-theta کے لحاظ سے ظاہر کیا گیا ہے؟ اس کے لیے، اس چھوٹے سے دائیں مثلث کے بارے میں سوچنا مددگار ہو سکتا ہے جس کے بارے میں ہم پہلے بات کر رہے تھے۔", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere. ", "translatedText": "سوال نمبر 3، اور یہ واقعی اس کا دل ہے، ان حلقوں کے سائے میں سے ہر ایک کا رقبہ کرہ پر موجود حلقوں میں سے ایک کا نصف ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere. ", "translatedText": "سوال 4، میں نے شروع میں کہا تھا کہ شمالی نصف کرہ کے تمام سائے کے درمیان ایک خط و کتابت ہے، جو رداس r کے ساتھ ایک دائرہ بناتے ہیں، اور کرہ پر ہر دوسری انگوٹھی بنتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact. ", "translatedText": "آخر میں، میں اس حقیقت کا مختصر ذکر نہ کرنے سے گریز کروں گا کہ ایک کرہ کی سطح کا رقبہ بہت زیادہ عمومی حقیقت کی ایک خاص مثال ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/sphere-area/vietnamese/sentence_translations.json b/2018/sphere-area/vietnamese/sentence_translations.json index 580b3c1f1..3e597e5bf 100644 --- a/2018/sphere-area/vietnamese/sentence_translations.json +++ b/2018/sphere-area/vietnamese/sentence_translations.json @@ -592,7 +592,7 @@ "end": 574.58 }, { - "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder whatever value each of these two series of approximations are approaching must actually be the same.", + "input": "And since for any specific covering, the sphere rectangles have the same area as the cylinder rectangles, whatever value each of these two series of approximations are approaching must actually be the same.", "translatedText": "Và vì đối với bất kỳ lớp phủ cụ thể nào, các hình chữ nhật hình cầu có cùng diện tích với hình trụ, bất kể giá trị nào trong hai chuỗi xấp xỉ này đều phải thực sự giống nhau.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 786.92 }, { - "input": "Structured exercise time will ease in with a warm-up.", + "input": "All right, structured exercise time. We'll ease in with a warm-up.", "translatedText": "Thời gian tập thể dục có cấu trúc sẽ dễ dàng hơn khi khởi động.", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 792.64 }, { - "input": "Question 1, what is the circumference of this ring, say at the inner edge, in terms of r and theta?", + "input": "Question number one, what is the circumference of this ring, say, at the inner edge, in terms of r and theta?", "translatedText": "Câu hỏi 1, chu vi của vòng này là bao nhiêu, tính theo r và theta?", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 800.1 }, { - "input": "Once you have that, multiply the answer by the thickness r times d-theta to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", + "input": "Once you have that, go ahead and multiply the answer by the thickness, r times d-theta, to get an approximation for the ring's area, an approximation that will get better and better as you chop up the sphere more and more finely.", "translatedText": "Khi bạn đã có kết quả đó, hãy nhân kết quả với độ dày r nhân với d-theta để có được giá trị gần đúng cho diện tích của chiếc nhẫn, giá trị gần đúng sẽ ngày càng tốt hơn khi bạn cắt quả cầu ngày càng mịn hơn.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 827.4 }, { - "input": "Question 2, what is the area of the shadow of one of these rings on the xy-plane, again expressed in terms of r, theta, and d-theta?", + "input": "So question number two, what is the area of the shadow of one of these rings on the x-y plane, again expressed in terms of r, theta, and d-theta?", "translatedText": "Câu hỏi 2, diện tích bóng của một trong các vòng này trên mặt phẳng xy, một lần nữa được biểu thị theo r, theta và d-theta là bao nhiêu?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 844.52 }, { - "input": "Question 3, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", + "input": "Question number three, and this is really the heart of it, each one of these rings' shadows has precisely half the area of one of the rings on the sphere.", "translatedText": "Câu hỏi 3, và đây thực sự là trọng tâm của nó, bóng của mỗi chiếc nhẫn này có diện tích chính xác bằng một nửa diện tích của một trong những chiếc nhẫn trên quả cầu.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 868.76 }, { - "input": "Question 4, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", + "input": "Question number four, I said at the outset that there's a correspondence between all of the shadows from the northern hemisphere, which make up a circle with radius r, and every second ring on the sphere.", "translatedText": "Câu hỏi 4, tôi đã nói ngay từ đầu rằng có sự tương ứng giữa tất cả các bóng từ bán cầu bắc, tạo thành một vòng tròn có bán kính r, và mỗi giây vòng trên quả cầu.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 914.0 }, { - "input": "Finally, I would be remiss not to make a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", + "input": "And finally, I would be remiss not to make at least a brief mention of the fact that the surface area of a sphere is a very specific instance of a much more general fact.", "translatedText": "Cuối cùng, tôi sẽ thật thiếu sót nếu không đề cập ngắn gọn đến thực tế là diện tích bề mặt của một hình cầu là một ví dụ rất cụ thể của một thực tế tổng quát hơn nhiều.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/arabic/sentence_translations.json b/2018/turbulence/arabic/sentence_translations.json index 3e2e6e336..8af6d56a4 100644 --- a/2018/turbulence/arabic/sentence_translations.json +++ b/2018/turbulence/arabic/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes. ", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes. ", "translatedText": "إذا كنت تريد تجربة ذلك في المنزل، فكن حذرًا للغاية مع الليزر، وتأكد من عدم توجيهه بالقرب من عيون أي شخص. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/bengali/sentence_translations.json b/2018/turbulence/bengali/sentence_translations.json index 192611fac..a3e75b119 100644 --- a/2018/turbulence/bengali/sentence_translations.json +++ b/2018/turbulence/bengali/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes. ", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes. ", "translatedText": "এবং একটি অতিরিক্ত বোনাস হিসাবে, সেটআপটি একটি দুর্দান্ত ডেথ ইটার থিমযুক্ত হ্যালোইন সজ্জা হিসাবে দ্বিগুণ হয়৷ আপনি যদি বাড়িতে এটি চেষ্টা করতে চান তবে লেজারের সাথে খুব সতর্কতা অবলম্বন করুন, নিশ্চিত করুন যে এটি কারও চোখের কাছে নির্দেশ না করে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/chinese/sentence_translations.json b/2018/turbulence/chinese/sentence_translations.json index de7aea4e7..b6c3a1a5c 100644 --- a/2018/turbulence/chinese/sentence_translations.json +++ b/2018/turbulence/chinese/sentence_translations.json @@ -117,7 +117,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "如果您确实想在家尝试此操作，请务必小心 激光，确保不要将其指向任何人的眼睛。", "model": "google_nmt", "from_community_srt": "如果你想在家里试试这个， 我应该说： 激光时要特别小心！ 确保不要将它指向任何人的眼睛附近。", diff --git a/2018/turbulence/english/captions.srt b/2018/turbulence/english/captions.srt index 4cea7871a..2c9d0c373 100644 --- a/2018/turbulence/english/captions.srt +++ b/2018/turbulence/english/captions.srt @@ -543,7 +543,7 @@ the existence of a constant like this is that it suggests there's some predictab however slight, to this whole mass. 137 -00:09:17,859 --> 00:09:22,192 +00:09:17,860 --> 00:09:22,192 There is something ironic about talking about this energy cascade while viewing 138 @@ -583,7 +583,7 @@ One of the mechanisms behind this energy cascade, which could only ever happen in three dimensions, is a process known as vortex stretching. 147 -00:09:55,099 --> 00:09:59,239 +00:09:55,100 --> 00:09:59,239 A rotating part of the fluid will tend to stretch out perpendicular 148 diff --git a/2018/turbulence/french/sentence_translations.json b/2018/turbulence/french/sentence_translations.json index b75c57319..cf4c2939b 100644 --- a/2018/turbulence/french/sentence_translations.json +++ b/2018/turbulence/french/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "Si vous souhaitez essayer cela à la maison, soyez très prudent avec le laser, assurez-vous de ne pas le diriger près des yeux de qui que ce soit.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/german/sentence_translations.json b/2018/turbulence/german/sentence_translations.json index d84f643a3..0ef651023 100644 --- a/2018/turbulence/german/sentence_translations.json +++ b/2018/turbulence/german/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "Wenn Sie dies zu Hause ausprobieren möchten, gehen Sie äußerst vorsichtig mit dem Laser um und richten Sie ihn nicht in die Nähe der Augen anderer Personen.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/hindi/sentence_translations.json b/2018/turbulence/hindi/sentence_translations.json index 80b8d84a2..4fe595b16 100644 --- a/2018/turbulence/hindi/sentence_translations.json +++ b/2018/turbulence/hindi/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "यदि आप इसे घर पर आज़माना चाहते हैं, तो लेज़र के साथ अत्यधिक सावधान रहें, सुनिश्चित करें कि यह किसी की आँखों के पास न जाए।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/indonesian/sentence_translations.json b/2018/turbulence/indonesian/sentence_translations.json index fc1fb8186..f06ab65ac 100644 --- a/2018/turbulence/indonesian/sentence_translations.json +++ b/2018/turbulence/indonesian/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "Jika Anda ingin mencobanya di rumah, berhati-hatilah dengan lasernya, pastikan untuk tidak mengarahkannya ke dekat mata siapa pun.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/japanese/sentence_translations.json b/2018/turbulence/japanese/sentence_translations.json index 33256e863..8b9b79605 100644 --- a/2018/turbulence/japanese/sentence_translations.json +++ b/2018/turbulence/japanese/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "自宅でこれを試したい場合は、レーザーの扱いには細心の注意を 払い、レーザーを人の目に近づけないように注意してください。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/korean/sentence_translations.json b/2018/turbulence/korean/sentence_translations.json index 707c50347..da4f83d97 100644 --- a/2018/turbulence/korean/sentence_translations.json +++ b/2018/turbulence/korean/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "집에서 이것을 시도하고 싶다면 레이저를 사용할 때 매우 조심하고 다른 사람의 눈 근처를 가리키지 않도록 하세요.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/marathi/sentence_translations.json b/2018/turbulence/marathi/sentence_translations.json index e640b1c4f..07aea3a84 100644 --- a/2018/turbulence/marathi/sentence_translations.json +++ b/2018/turbulence/marathi/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "जर तुम्हाला हे घरी करून पहायचे असेल, तर लेसरच्या बाबतीत अत्यंत सावधगिरी बाळगा, ते कोणाच्याही डोळ्यांजवळ न येण्याची खात्री करा.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/persian/sentence_translations.json b/2018/turbulence/persian/sentence_translations.json index 8f19f75c2..fded626f6 100644 --- a/2018/turbulence/persian/sentence_translations.json +++ b/2018/turbulence/persian/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes. ", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes. ", "translatedText": "اگر می خواهید این کار را در خانه امتحان کنید، در مورد لیزر بسیار مراقب باشید، مطمئن شوید که آن را نزدیک چشم کسی قرار ندهید. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/portuguese/sentence_translations.json b/2018/turbulence/portuguese/sentence_translations.json index a99adb5d5..bee1e240e 100644 --- a/2018/turbulence/portuguese/sentence_translations.json +++ b/2018/turbulence/portuguese/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "Se você quiser tentar fazer isso em casa, tenha muito cuidado com o laser, certifique-se de não apontá-lo para perto dos olhos de ninguém.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/russian/sentence_translations.json b/2018/turbulence/russian/sentence_translations.json index ec4c95037..15de18f81 100644 --- a/2018/turbulence/russian/sentence_translations.json +++ b/2018/turbulence/russian/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "Если вы хотите попробовать это дома, будьте очень осторожны с лазером и не направляйте его кому-либо в глаза.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/spanish/sentence_translations.json b/2018/turbulence/spanish/sentence_translations.json index 7fe4f374c..1d4952d7b 100644 --- a/2018/turbulence/spanish/sentence_translations.json +++ b/2018/turbulence/spanish/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "Si quieres probar esto en casa, ten mucho cuidado con el láser, asegúrate de no apuntarlo cerca de los ojos de nadie.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/tamil/sentence_translations.json b/2018/turbulence/tamil/sentence_translations.json index a4cdb2f24..da49a8e41 100644 --- a/2018/turbulence/tamil/sentence_translations.json +++ b/2018/turbulence/tamil/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "நீங்கள் இதை வீட்டிலேயே முயற்சி செய்ய விரும்பினால், லேசர் மூலம் மிகவும் கவனமாக இருங்கள், யாருடைய கண்களுக்கும் அருகில் அதைச் சுட்டிக் காட்டாமல் பார்த்துக் கொள்ளுங்கள்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/telugu/sentence_translations.json b/2018/turbulence/telugu/sentence_translations.json index c9b58c364..b3c5b762c 100644 --- a/2018/turbulence/telugu/sentence_translations.json +++ b/2018/turbulence/telugu/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "మీరు దీన్ని ఇంట్లో ప్రయత్నించాలనుకుంటే, లేజర్‌తో చాలా జాగ్రత్తగా ఉండండి, ఎవరి కళ్లకు దగ్గరగా ఉండకుండా చూసుకోండి.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/thai/sentence_translations.json b/2018/turbulence/thai/sentence_translations.json index 9cdd0a82b..87b5bbadc 100644 --- a/2018/turbulence/thai/sentence_translations.json +++ b/2018/turbulence/thai/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes. ", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/turkish/sentence_translations.json b/2018/turbulence/turkish/sentence_translations.json index 0e90ce3fe..ff5adf374 100644 --- a/2018/turbulence/turkish/sentence_translations.json +++ b/2018/turbulence/turkish/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "Bunu evde denemek istiyorsanız lazere çok dikkat edin, kimsenin gözüne doğrultmadığınızdan emin olun.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/urdu/sentence_translations.json b/2018/turbulence/urdu/sentence_translations.json index 1aff9b26f..5310adfe5 100644 --- a/2018/turbulence/urdu/sentence_translations.json +++ b/2018/turbulence/urdu/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes. ", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes. ", "translatedText": "اگر آپ اسے گھر پر آزمانا چاہتے ہیں تو، لیزر کے ساتھ انتہائی محتاط رہیں، اس بات کو یقینی بنائیں کہ اسے کسی کی آنکھوں کے قریب نہ کریں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/turbulence/vietnamese/sentence_translations.json b/2018/turbulence/vietnamese/sentence_translations.json index 9bab4063b..c9c1e5881 100644 --- a/2018/turbulence/vietnamese/sentence_translations.json +++ b/2018/turbulence/vietnamese/sentence_translations.json @@ -104,7 +104,7 @@ "end": 140.76 }, { - "input": "If you do want to try this at home, be super careful with the laser, make sure not to point it near anyone's eyes.", + "input": "If you do want to try this at home, I should say, be super careful with the laser, make sure not to point it near anyone's eyes.", "translatedText": "Nếu bạn muốn thử điều này ở nhà, hãy hết sức cẩn thận với tia laser, đảm bảo không chĩa nó vào gần mắt của bất kỳ ai.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/arabic/sentence_translations.json b/2018/uncertainty-principle/arabic/sentence_translations.json index 6e63c1970..7679e283d 100644 --- a/2018/uncertainty-principle/arabic/sentence_translations.json +++ b/2018/uncertainty-principle/arabic/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went. ", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went. ", "translatedText": "آخر مقطع فيديو قمت بنشره كان حدسًا بصريًا لهذا التحويل، ونعم، سيكون من المفيد إذا كنت قد شاهدته، لكنني سأقدم ملخصًا سريعًا هنا لتذكير أنفسنا كيف سار الأمر. ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region. ", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region. ", "translatedText": "عندما تقيس هذه الجسيمات، لنفترض أن محاولة اكتشاف ما إذا كانت موجودة في منطقة معينة، وسواء وجدتها هناك أم لا يبدو الأمر احتماليًا، حيث يتناسب احتمال العثور عليها مع قوة الموجة في تلك المنطقة. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/bengali/sentence_translations.json b/2018/uncertainty-principle/bengali/sentence_translations.json index f16e1df5d..bcc18dbd5 100644 --- a/2018/uncertainty-principle/bengali/sentence_translations.json +++ b/2018/uncertainty-principle/bengali/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went. ", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went. ", "translatedText": "আমার দেওয়া শেষ ভিডিওটি এই রূপান্তরের জন্য একটি ভিজ্যুয়াল অন্তর্দৃষ্টি ছিল, এবং হ্যাঁ, আপনি যদি এটি দেখে থাকেন তবে এটি সহায়ক হবে, তবে এটি কীভাবে হয়েছে তা মনে করিয়ে দেওয়ার জন্য আমি এখানে একটি দ্রুত সংকলন দেব।", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region. ", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region. ", "translatedText": "যখন আপনি এই কণাগুলি পরিমাপ করেন, বলুন যে এটি একটি প্রদত্ত অঞ্চলে আছে কিনা তা সনাক্ত করার চেষ্টা করছেন, আপনি এটি খুঁজে পান কিনা তা সম্ভাব্য বলে মনে হচ্ছে, যেখানে এটি খুঁজে পাওয়ার সম্ভাবনা সেই অঞ্চলের তরঙ্গের শক্তির সমানুপাতিক।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/chinese/sentence_translations.json b/2018/uncertainty-principle/chinese/sentence_translations.json index da67a5e14..c6838f1e0 100644 --- a/2018/uncertainty-principle/chinese/sentence_translations.json +++ b/2018/uncertainty-principle/chinese/sentence_translations.json @@ -151,7 +151,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "我发布的最后一个视频是对这种转换的视觉直觉，是的，如果您看过它，那将 会很有帮助，但我将在这里快速回顾一下，以提醒我们自己是如何进行的。", "model": "google_nmt", "from_community_srt": "我推出的是这种变换的视觉直觉， 是的， 如果你看过它可能会有所帮助 但是我要继续这里， 快速回顾一下。", @@ -852,7 +852,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "更充分地使用它需要一些狭义相对论的知识，所以我们只需要等待亨利 ·赖克（Henry Reich）关于该主题的系列文章的出版。", "model": "google_nmt", "from_community_srt": "这些事件实际上可能在不同的时间发生 更全面地理解这一点需要一些狭义相对论的知识 因此， 我们都必须等待亨利赖斯关于该主题的系列节目出来 就在这里，", @@ -959,7 +959,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "当您测量这些粒子时，例如尝试检测它是否在给定 区域中，无论您是否找到它，似乎都是概率性的， 其中找到它的概率与该区域中波的强度成正比。", "model": "google_nmt", "from_community_srt": "试图检测它是否在给定区域 无论你是否找到它， 似乎存在找到概率的概率 它与该区域的波浪强度成正比，", diff --git a/2018/uncertainty-principle/english/captions.srt b/2018/uncertainty-principle/english/captions.srt index 3938c482f..bc2ee1a27 100644 --- a/2018/uncertainty-principle/english/captions.srt +++ b/2018/uncertainty-principle/english/captions.srt @@ -459,24 +459,24 @@ and a signal with a concentrated Fourier transform has to be spread out in time. And one other place where this comes up in a really tangible way is Doppler radar. 116 -00:07:08,220 --> 00:07:13,714 -With radar, you send out a radio wave pulse, and the pulse might reflect off of objects, +00:07:08,220 --> 00:07:11,782 +So with radar, the idea is you send out some radio wave pulse, 117 -00:07:13,714 --> 00:07:17,727 -and the time it takes for this echo signal to return to you lets +00:07:11,782 --> 00:07:15,740 +and the pulse might reflect off of objects, and the time it takes for 118 -00:07:17,727 --> 00:07:20,320 -you deduce how far away those objects are. +00:07:15,740 --> 00:07:20,320 +this echo signal to return to you lets you deduce how far away those objects are. 119 -00:07:20,780 --> 00:07:23,819 -You can take this one step further and make deductions about +00:07:20,780 --> 00:07:23,832 +And you can actually take this one step further and make deductions 120 -00:07:23,819 --> 00:07:26,660 -the velocities of those objects using the Doppler effect. +00:07:23,832 --> 00:07:26,660 +about the velocities of those objects using the Doppler effect. 121 00:07:27,360 --> 00:07:29,220 diff --git a/2018/uncertainty-principle/english/sentence_timings.json b/2018/uncertainty-principle/english/sentence_timings.json index 389f0258e..ab1894e08 100644 --- a/2018/uncertainty-principle/english/sentence_timings.json +++ b/2018/uncertainty-principle/english/sentence_timings.json @@ -240,12 +240,12 @@ 428.22 ], [ - "With radar, you send out a radio wave pulse, and the pulse might reflect off of objects, and the time it takes for this echo signal to return to you lets you deduce how far away those objects are.", + "So with radar, the idea is you send out some radio wave pulse, and the pulse might reflect off of objects, and the time it takes for this echo signal to return to you lets you deduce how far away those objects are.", 428.22, 440.32 ], [ - "You can take this one step further and make deductions about the velocities of those objects using the Doppler effect.", + "And you can actually take this one step further and make deductions about the velocities of those objects using the Doppler effect.", 440.78, 446.66 ], diff --git a/2018/uncertainty-principle/english/transcript.txt b/2018/uncertainty-principle/english/transcript.txt index aad12c6c8..313f64d0c 100644 --- a/2018/uncertainty-principle/english/transcript.txt +++ b/2018/uncertainty-principle/english/transcript.txt @@ -46,8 +46,8 @@ Over on the frequency plot, that corresponds to a much broader peak around the 5 And that's the uncertainty principle, just phrased a little bit more mathematically. A signal concentrated in time must have a spread out Fourier transform, meaning it correlates with a wide range of frequencies, and a signal with a concentrated Fourier transform has to be spread out in time. And one other place where this comes up in a really tangible way is Doppler radar. -With radar, you send out a radio wave pulse, and the pulse might reflect off of objects, and the time it takes for this echo signal to return to you lets you deduce how far away those objects are. -You can take this one step further and make deductions about the velocities of those objects using the Doppler effect. +So with radar, the idea is you send out some radio wave pulse, and the pulse might reflect off of objects, and the time it takes for this echo signal to return to you lets you deduce how far away those objects are. +And you can actually take this one step further and make deductions about the velocities of those objects using the Doppler effect. Think about sending out a pulse with some frequency. If this gets reflected off an object moving towards you, then the beats of that wave get kind of smushed together, so the echo you hear back is going to be a slightly higher frequency. Fourier transforms give a neat way to view this. diff --git a/2018/uncertainty-principle/french/sentence_translations.json b/2018/uncertainty-principle/french/sentence_translations.json index 2379541a9..5ca7f27b9 100644 --- a/2018/uncertainty-principle/french/sentence_translations.json +++ b/2018/uncertainty-principle/french/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "La dernière vidéo que j'ai publiée était une intuition visuelle de cette transformation, et oui, il serait utile que vous l'ayez vue, mais je vais vous donner un bref récapitulatif ici pour nous rappeler comment cela s'est passé.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "L'utiliser plus pleinement nécessite une certaine connaissance de la relativité restreinte, nous devrons donc tous simplement attendre la sortie de la série d'Henry Reich sur ce sujet.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "Lorsque vous mesurez ces particules, par exemple essayer de détecter si elles se trouvent dans une région donnée, que vous les trouviez ou non là-bas semble être probabiliste, où la probabilité de les trouver est proportionnelle à la force de l'onde dans cette région.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/german/sentence_translations.json b/2018/uncertainty-principle/german/sentence_translations.json index 5ad306fb1..a4ee5d8ae 100644 --- a/2018/uncertainty-principle/german/sentence_translations.json +++ b/2018/uncertainty-principle/german/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "Das letzte Video, das ich veröffentlicht habe, war eine visuelle Vorstellung dieser Transformation, und ja, es wäre hilfreich, wenn Sie es gesehen hätten, aber ich werde hier eine kurze Zusammenfassung geben, um uns daran zu erinnern, wie es gelaufen ist.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "Um dies umfassender nutzen zu können, sind einige Kenntnisse der speziellen Relativitätstheorie erforderlich. Wir müssen also alle nur darauf warten, dass Henry Reichs Serie zu diesem Thema erscheint.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "Wenn man diese Partikel misst, beispielsweise um herauszufinden, ob sie sich in einer bestimmten Region befinden, scheint es probabilistisch zu sein, ob man sie dort findet oder nicht, wobei die Wahrscheinlichkeit, sie zu finden, proportional zur Stärke der Welle in dieser Region ist.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/hebrew/sentence_translations.json b/2018/uncertainty-principle/hebrew/sentence_translations.json index 65c51e380..84f12284c 100644 --- a/2018/uncertainty-principle/hebrew/sentence_translations.json +++ b/2018/uncertainty-principle/hebrew/sentence_translations.json @@ -119,7 +119,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "הסרטון האחרון שהוצאתי היה אינטואיציה חזותית לטרנספורמציה הזו, וכן, זה יעזור אם ראיתם אותו, אבל אני אתן כאן סיכום קצר כדי להזכיר לעצמנו איך זה הלך.", "n_reviews": 0, "start": 159.44, @@ -679,7 +679,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "שימוש מלא יותר בזה דורש ידע מסוים בתורת היחסות הפרטית, אז כולנו נצטרך רק לחכות שהסדרה של הנרי רייך על הנושא הזה תצא.", "n_reviews": 0, "start": 829.6, @@ -763,7 +763,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "כאשר אתה מודד את החלקיקים האלה, נניח לנסות לזהות אם הם נמצאים באזור נתון, אם אתה מוצא אותו או לא שם נראה הסתברותי, כאשר ההסתברות למצוא אותם היא פרופורציונלית לעוצמת הגל באזור זה.", "n_reviews": 0, "start": 982.56, diff --git a/2018/uncertainty-principle/hindi/sentence_translations.json b/2018/uncertainty-principle/hindi/sentence_translations.json index 48359d059..c9fe806cd 100644 --- a/2018/uncertainty-principle/hindi/sentence_translations.json +++ b/2018/uncertainty-principle/hindi/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "आखिरी वीडियो जो मैंने डाला था वह इस परिवर्तन के लिए एक दृश्य अंतर्ज्ञान था, और हां, यदि आपने इसे देखा है तो यह मददगार होगा, लेकिन मैं खुद को याद दिलाने के लिए यहां एक त्वरित पुनर्कथन दूंगा कि यह कैसे हुआ।", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "इसे और अधिक पूर्ण रूप से उपयोग करने के लिए विशेष सापेक्षता के कुछ ज्ञान की आवश्यकता होती है, इसलिए हम सभी को उस विषय पर हेनरी रीच की श्रृंखला के आने का इंतजार करना होगा।", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "जब आप इन कणों को मापते हैं, तो मान लीजिए कि यह पता लगाने की कोशिश करना कि क्या यह किसी दिए गए क्षेत्र में है, आपको यह वहां मिलता है या नहीं, यह संभाव्य प्रतीत होता है, जहां इसे खोजने की संभावना उस क्षेत्र में तरंग की ताकत के समानुपाती होती है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/hungarian/sentence_translations.json b/2018/uncertainty-principle/hungarian/sentence_translations.json index 7345bb118..67c747245 100644 --- a/2018/uncertainty-principle/hungarian/sentence_translations.json +++ b/2018/uncertainty-principle/hungarian/sentence_translations.json @@ -384,7 +384,7 @@ "end": 428.22 }, { - "input": "With radar, you send out a radio wave pulse, and the pulse might reflect off of objects, and the time it takes for this echo signal to return to you lets you deduce how far away those objects are.", + "input": "So with radar, the idea is you send out some radio wave pulse, and the pulse might reflect off of objects, and the time it takes for this echo signal to return to you lets you deduce how far away those objects are.", "translatedText": "A radarral egy rádióhullám-impulzust küldünk ki, amely visszaverődhet a tárgyakról, és az idő, amely alatt ez a visszhangjel visszatér hozzánk, lehetővé teszi, hogy megállapítsuk, milyen messze vannak az adott tárgyak.", "model": "DeepL", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 440.32 }, { - "input": "You can take this one step further and make deductions about the velocities of those objects using the Doppler effect.", + "input": "And you can actually take this one step further and make deductions about the velocities of those objects using the Doppler effect.", "translatedText": "Ezt egy lépéssel tovább lehet vinni, és a Doppler-effektus segítségével következtetéseket lehet levonni az objektumok sebességére.", "model": "DeepL", "n_reviews": 0, diff --git a/2018/uncertainty-principle/indonesian/sentence_translations.json b/2018/uncertainty-principle/indonesian/sentence_translations.json index c11c91906..ddb2d932c 100644 --- a/2018/uncertainty-principle/indonesian/sentence_translations.json +++ b/2018/uncertainty-principle/indonesian/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "Video terakhir yang saya keluarkan adalah intuisi visual untuk transformasi ini, dan ya, akan sangat membantu jika Anda pernah melihatnya, tapi saya akan memberikan rekap singkat di sini untuk mengingatkan kita bagaimana hasilnya.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "Untuk menggunakan teori ini secara lebih lengkap diperlukan pengetahuan tentang relativitas khusus, jadi kita semua hanya perlu menunggu seri Henry Reich tentang topik tersebut diterbitkan.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "Saat Anda mengukur partikel-partikel ini, katakanlah mencoba mendeteksi apakah ia berada di wilayah tertentu, apakah Anda menemukannya atau tidak, tampaknya bersifat probabilistik, yang mana kemungkinan menemukannya sebanding dengan kekuatan gelombang di wilayah tersebut.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/italian/sentence_translations.json b/2018/uncertainty-principle/italian/sentence_translations.json index c4287519f..9cc5c423f 100644 --- a/2018/uncertainty-principle/italian/sentence_translations.json +++ b/2018/uncertainty-principle/italian/sentence_translations.json @@ -119,7 +119,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "L'ultimo video che ho pubblicato era un'intuizione visiva per questa trasformazione, e sì, sarebbe utile se l'avessi visto, ma farò un breve riepilogo qui per ricordarci come è andata.", "n_reviews": 0, "start": 159.44, @@ -679,7 +679,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "Usarlo in modo più completo richiede una certa conoscenza della relatività speciale, quindi dovremo solo aspettare che venga pubblicata la serie di Henry Reich su quell'argomento.", "n_reviews": 0, "start": 829.6, @@ -763,7 +763,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "Quando misuri queste particelle, ad esempio cercando di rilevare se si trova in una determinata regione, se la trovi o meno sembra essere probabilistico, dove la probabilità di trovarla è proporzionale alla forza dell'onda in quella regione.", "n_reviews": 0, "start": 982.56, diff --git a/2018/uncertainty-principle/japanese/sentence_translations.json b/2018/uncertainty-principle/japanese/sentence_translations.json index 5bb79a776..eb78d5304 100644 --- a/2018/uncertainty-principle/japanese/sentence_translations.json +++ b/2018/uncertainty-principle/japanese/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "私が最後に公開したビデオは、この変換の視覚的な直観でした。 はい、それを見たことがあれば 役立つでしょう。 しかし、どのように行われたかを思い出すために、ここで簡単に要約します。", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "これをより完全に使用するには、特殊相対性理論に関するある程度の知識が必要なので、 このテーマに関するヘンリー・ライヒのシリーズが出版されるのを待つ必要があります。", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "これらの粒子を測定するとき、たとえば粒子が特定の領域にあるかど うかを検出しようとすると、そこで粒子が見つかるかどうかは確率論 的になり、粒子が見つかる確率はその領域の波の強さに比例します。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/korean/sentence_translations.json b/2018/uncertainty-principle/korean/sentence_translations.json index bce639621..1efac8e33 100644 --- a/2018/uncertainty-principle/korean/sentence_translations.json +++ b/2018/uncertainty-principle/korean/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "제가 마지막으로 내놓은 비디오는 이 변환에 대한 시각적 직관이었습니다. 그렇습니다. 본 적이 있다면 도움이 될 것입니다. 하지만 어떻게 진행되었는지 상기시키기 위해 여기서 간단히 요약하겠습니다.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "이것을 더 완벽하게 사용하려면 특수 상대성 이론에 대한 지식이 필요하므로 우리는 해당 주제에 대한 Henry Reich의 시리즈가 나올 때까지 기다려야 합니다.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "이러한 입자를 측정할 때 해당 입자가 특정 지역에 있는지 감지하려고 시도한다고 가정해 보겠습니다. 발견 여부는 확률적으로 나타나며, 입자를 발견할 확률은 해당 지역의 파동 강도에 비례합니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/marathi/sentence_translations.json b/2018/uncertainty-principle/marathi/sentence_translations.json index a57789df9..11fdc990a 100644 --- a/2018/uncertainty-principle/marathi/sentence_translations.json +++ b/2018/uncertainty-principle/marathi/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "मी टाकलेला शेवटचा व्हिडिओ हा या परिवर्तनासाठी एक दृश्य अंतर्ज्ञान होता, आणि हो, तुम्ही तो पाहिला असेल तर ते उपयुक्त ठरेल, परंतु ते कसे घडले याची आठवण करून देण्यासाठी मी येथे एक द्रुत संक्षेप देईन.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "हे अधिक पूर्णपणे वापरण्यासाठी विशेष सापेक्षतेचे काही ज्ञान आवश्यक आहे, म्हणून आपल्या सर्वांना त्या विषयावरील हेन्री रीचची मालिका बाहेर येण्याची प्रतीक्षा करावी लागेल.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "जेव्हा तुम्ही या कणांचे मोजमाप करता, तेव्हा ते दिलेल्या प्रदेशात आहे की नाही हे शोधण्याचा प्रयत्न करा म्हणा, तुम्हाला ते सापडले की नाही हे संभाव्य आहे असे दिसते, जेथे ते शोधण्याची संभाव्यता त्या प्रदेशातील लहरीच्या सामर्थ्याच्या प्रमाणात असते.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/persian/sentence_translations.json b/2018/uncertainty-principle/persian/sentence_translations.json index f36d30539..7ba0ab8d7 100644 --- a/2018/uncertainty-principle/persian/sentence_translations.json +++ b/2018/uncertainty-principle/persian/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went. ", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went. ", "translatedText": "آخرین ویدیویی که من منتشر کردم یک شهود بصری برای این تغییر بود، و بله، اگر آن را دیده باشید مفید خواهد بود، اما من در اینجا خلاصه‌ای سریع ارائه می‌دهم تا به خودمان یادآوری کنیم که چگونه پیش رفت. ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region. ", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region. ", "translatedText": "وقتی این ذرات را اندازه می‌گیرید، بگویید تلاش برای تشخیص اینکه آیا در یک منطقه معین است، چه آن را پیدا کنید یا نه، احتمالی است، جایی که احتمال یافتن آن متناسب با قدرت موج در آن منطقه است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/polish/sentence_translations.json b/2018/uncertainty-principle/polish/sentence_translations.json index 33a3f7b7b..66d0821db 100644 --- a/2018/uncertainty-principle/polish/sentence_translations.json +++ b/2018/uncertainty-principle/polish/sentence_translations.json @@ -384,7 +384,7 @@ "end": 428.22 }, { - "input": "With radar, you send out a radio wave pulse, and the pulse might reflect off of objects, and the time it takes for this echo signal to return to you lets you deduce how far away those objects are.", + "input": "So with radar, the idea is you send out some radio wave pulse, and the pulse might reflect off of objects, and the time it takes for this echo signal to return to you lets you deduce how far away those objects are.", "translatedText": "", "from_community_srt": "jest radar dopplerowski więc z radarem, pomysł jest taki, żeby wysłać impuls fali radiowej, który może się odbijać od obiektów z czasu potrzebny na powrót tego sygnału echa do Ciebie, wywnioskujemy,", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 440.32 }, { - "input": "You can take this one step further and make deductions about the velocities of those objects using the Doppler effect.", + "input": "And you can actually take this one step further and make deductions about the velocities of those objects using the Doppler effect.", "translatedText": "", "from_community_srt": "jak daleko te obiekty są i możemy zrobić krok na przód i wywnioskować prędkość tych obiektów przy użyciu efektu Dopplera.", "n_reviews": 0, diff --git a/2018/uncertainty-principle/portuguese/sentence_translations.json b/2018/uncertainty-principle/portuguese/sentence_translations.json index 1fb98576f..7edaf6602 100644 --- a/2018/uncertainty-principle/portuguese/sentence_translations.json +++ b/2018/uncertainty-principle/portuguese/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "O último vídeo que publiquei foi uma intuição visual dessa transformação e, sim, seria útil se você já tivesse visto, mas vou fazer uma rápida recapitulação aqui para nos lembrarmos de como foi.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "Usar isso de forma mais completa requer algum conhecimento da relatividade especial, então todos teremos que esperar que a série de Henry Reich sobre esse assunto seja lançada.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "Quando você mede essas partículas, digamos, tentando detectar se elas estão em uma determinada região, se você as encontra ou não, parece ser probabilístico, onde a probabilidade de encontrá-las é proporcional à força da onda naquela região.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/russian/sentence_translations.json b/2018/uncertainty-principle/russian/sentence_translations.json index cc3f4a70e..5cd0411b3 100644 --- a/2018/uncertainty-principle/russian/sentence_translations.json +++ b/2018/uncertainty-principle/russian/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "Последнее видео, которое я выложил, представляло собой визуальное представление об этом преобразовании, и да, было бы полезно, если бы вы его видели, но я дам здесь краткий обзор, чтобы напомнить себе, как все прошло.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "Более полное использование этого требует некоторых знаний специальной теории относительности, поэтому нам всем просто придется дождаться выхода серии статей Генри Райха по этой теме.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "Когда вы измеряете эти частицы, скажем, пытаетесь определить, находятся ли они в определенной области, независимо от того, найдете ли вы их там или нет, это кажется вероятностным, где вероятность найти их пропорциональна силе волны в этой области.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/spanish/sentence_translations.json b/2018/uncertainty-principle/spanish/sentence_translations.json index f8a9ea07d..1c17dc7f0 100644 --- a/2018/uncertainty-principle/spanish/sentence_translations.json +++ b/2018/uncertainty-principle/spanish/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "El último vídeo que publiqué fue una intuición visual de esta transformación y sí, sería útil si lo hubieras visto, pero haré un resumen rápido aquí para recordar cómo fue.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "Usar esto más completamente requiere cierto conocimiento de la relatividad especial, por lo que todos tendremos que esperar a que salga la serie de Henry Reich sobre ese tema.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "Cuando se miden estas partículas, digamos que se trata de detectar si están en una región determinada, si se encuentran o no allí parece ser probabilístico, donde la probabilidad de encontrarlas es proporcional a la fuerza de la onda en esa región.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/tamil/sentence_translations.json b/2018/uncertainty-principle/tamil/sentence_translations.json index ac6a7cc8f..e2622b738 100644 --- a/2018/uncertainty-principle/tamil/sentence_translations.json +++ b/2018/uncertainty-principle/tamil/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "நான் கடைசியாக வெளியிட்ட வீடியோ, இந்த மாற்றத்திற்கான ஒரு காட்சி உள்ளுணர்வு ஆகும், ஆம், நீங்கள் அதைப் பார்த்திருந்தால் உதவியாக இருக்கும், ஆனால் அது எப்படிச் சென்றது என்பதை நினைவூட்டுவதற்காக இங்கே ஒரு விரைவான மறுபரிசீலனையைத் தருகிறேன்.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "இதை இன்னும் முழுமையாகப் பயன்படுத்துவதற்கு சிறப்பு சார்பியல் பற்றிய சில அறிவு தேவைப்படுகிறது, எனவே அந்த தலைப்பில் ஹென்றி ரீச்சின் தொடர் வெளிவருவதற்கு நாம் அனைவரும் காத்திருக்க வேண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "இந்தத் துகள்களை நீங்கள் அளவிடும்போது, அது கொடுக்கப்பட்ட பகுதியில் உள்ளதா என்பதைக் கண்டறிய முயல்கிறீர்கள் என்று சொல்லுங்கள், அது நிகழ்தகவு இருப்பதாகத் தோன்றுகிறதா இல்லையா என்பதைக் கண்டறிய முயற்சிக்கவும், அங்கு அதைக் கண்டுபிடிப்பதற்கான நிகழ்தகவு அந்த பிராந்தியத்தில் உள்ள அலையின் வலிமைக்கு விகிதாசாரமாகும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/telugu/sentence_translations.json b/2018/uncertainty-principle/telugu/sentence_translations.json index f1adc39fd..62917c1f6 100644 --- a/2018/uncertainty-principle/telugu/sentence_translations.json +++ b/2018/uncertainty-principle/telugu/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "నేను ఉంచిన చివరి వీడియో ఈ రూపాంతరం కోసం ఒక దృశ్యమాన అంతర్ దృష్టి, మరియు అవును, మీరు దీన్ని చూసినట్లయితే అది ఉపయోగకరంగా ఉంటుంది, కానీ అది ఎలా జరిగిందో మనకు గుర్తు చేసుకోవడానికి నేను ఇక్కడ శీఘ్ర రీక్యాప్ ఇస్తాను.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "దీన్ని పూర్తిగా ఉపయోగించడం కోసం ప్రత్యేక సాపేక్షత గురించి కొంత జ్ఞానం అవసరం, కాబట్టి ఆ అంశంపై హెన్రీ రీచ్ యొక్క సిరీస్ బయటకు వచ్చే వరకు మనమందరం వేచి ఉండాలి.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "మీరు ఈ కణాలను కొలిచినప్పుడు, అది ఒక నిర్దిష్ట ప్రాంతంలో ఉందో లేదో గుర్తించడానికి ప్రయత్నిస్తున్నారని చెప్పండి, మీరు అక్కడ అది సంభావ్యత ఉన్నట్లు కనిపిస్తుందో లేదో, అక్కడ దాన్ని కనుగొనే సంభావ్యత ఆ ప్రాంతంలోని తరంగ బలానికి అనులోమానుపాతంలో ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/thai/sentence_translations.json b/2018/uncertainty-principle/thai/sentence_translations.json index c3811af71..4496f5c69 100644 --- a/2018/uncertainty-principle/thai/sentence_translations.json +++ b/2018/uncertainty-principle/thai/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went. ", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region. ", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/turkish/sentence_translations.json b/2018/uncertainty-principle/turkish/sentence_translations.json index b8d1c4211..92860bff9 100644 --- a/2018/uncertainty-principle/turkish/sentence_translations.json +++ b/2018/uncertainty-principle/turkish/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "Yayınladığım son video bu dönüşüme dair görsel bir sezgiydi ve evet, izlemiş olmanız faydalı olacaktır, ancak nasıl gittiğini kendimize hatırlatmak için burada kısa bir özet vereceğim.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "Bunu daha kapsamlı bir şekilde kullanmak, özel görelilik hakkında biraz bilgi gerektirir, bu yüzden Henry Reich'ın bu konudaki serisinin çıkmasını beklememiz gerekecek.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "Bu parçacıkları ölçtüğünüzde, örneğin belirli bir bölgede olup olmadığını, onu orada bulup bulmadığınızı tespit etmeye çalıştığınızda, olasılıksal görünüyor, burada onu bulma olasılığı o bölgedeki dalganın gücüyle orantılıdır.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/ukrainian/sentence_translations.json b/2018/uncertainty-principle/ukrainian/sentence_translations.json index 80b13a6d7..df2d8b7eb 100644 --- a/2018/uncertainty-principle/ukrainian/sentence_translations.json +++ b/2018/uncertainty-principle/ukrainian/sentence_translations.json @@ -119,7 +119,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "Останнє відео, яке я опублікував, було візуальною інтуїцією для цього перетворення, і так, було б корисно, якщо ви його бачили, але я дам короткий підсумок тут, щоб нагадати нам, як це було.", "n_reviews": 0, "start": 159.44, @@ -679,7 +679,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "Щоб використовувати це більш повно, потрібні певні знання спеціальної теорії відносності, тому нам усім доведеться просто почекати, поки вийде серія Генрі Райха на цю тему.", "n_reviews": 0, "start": 829.6, @@ -763,7 +763,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "Коли ви вимірюєте ці частинки, скажімо, намагаючись виявити, чи є вони в певній області, незалежно від того, чи знайдете ви їх там, видається імовірнісним, де ймовірність знайти їх пропорційна силі хвилі в цій області.", "n_reviews": 0, "start": 982.56, diff --git a/2018/uncertainty-principle/urdu/sentence_translations.json b/2018/uncertainty-principle/urdu/sentence_translations.json index 8187d44cb..2455fd860 100644 --- a/2018/uncertainty-principle/urdu/sentence_translations.json +++ b/2018/uncertainty-principle/urdu/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went. ", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went. ", "translatedText": "آخری ویڈیو جو میں نے پیش کی تھی وہ اس تبدیلی کے لیے ایک بصری ادراک تھی، اور ہاں، اگر آپ نے اسے دیکھا ہے تو یہ مددگار ثابت ہوگا، لیکن میں یہاں ایک مختصر جائزہ پیش کروں گا تاکہ خود کو یاد دلایا جا سکے کہ یہ کیسے ہوا۔", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region. ", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region. ", "translatedText": "جب آپ ان ذرات کی پیمائش کرتے ہیں، تو کہیں کہ یہ معلوم کرنے کی کوشش کر رہے ہیں کہ آیا یہ کسی مخصوص علاقے میں ہے، چاہے آپ اسے وہاں پائے یا نہیں، امکانی معلوم ہوتا ہے، جہاں اسے تلاش کرنے کا امکان اس خطے میں لہر کی طاقت کے متناسب ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/uncertainty-principle/vietnamese/sentence_translations.json b/2018/uncertainty-principle/vietnamese/sentence_translations.json index 47ddbc55d..884b1562d 100644 --- a/2018/uncertainty-principle/vietnamese/sentence_translations.json +++ b/2018/uncertainty-principle/vietnamese/sentence_translations.json @@ -136,7 +136,7 @@ "end": 158.88 }, { - "input": "The last video I put out was a visual intuition for this transform, and yes, it would be helpful if you've seen it, but I'll give a quick recap here to remind ourselves how it went.", + "input": "The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to give a quick recap here to remind ourselves how it went.", "translatedText": "Video cuối cùng mà tôi đưa ra là trực quan trực quan về sự chuyển đổi này và vâng, sẽ rất hữu ích nếu bạn xem nó, nhưng tôi sẽ tóm tắt nhanh ở đây để nhắc nhở bản thân về quá trình diễn ra.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 826.04 }, { - "input": "Using this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", + "input": "Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.", "translatedText": "Để sử dụng điều này một cách đầy đủ hơn đòi hỏi một số kiến thức về thuyết tương đối đặc biệt, vì vậy tất cả chúng ta sẽ phải đợi loạt bài về chủ đề đó của Henry Reich ra mắt.", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 982.0 }, { - "input": "When you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", + "input": "You see, when you measure these particles, say trying to detect if it's in a given region, whether or not you find it there appears to be probabilistic, where the probability of finding it is proportional to the strength of the wave in that region.", "translatedText": "Khi bạn đo những hạt này, chẳng hạn như cố gắng phát hiện xem nó có ở một vùng nhất định hay không, bạn có tìm thấy nó ở đó hay không, dường như có xác suất, trong đó xác suất tìm thấy nó tỷ lệ thuận với cường độ sóng trong vùng đó.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/wallis-product/english/captions.srt b/2018/wallis-product/english/captions.srt index c84c97e82..a9f3a8dee 100644 --- a/2018/wallis-product/english/captions.srt +++ b/2018/wallis-product/english/captions.srt @@ -1,5 +1,5 @@ 1 -00:00:03,979 --> 00:00:05,540 +00:00:03,980 --> 00:00:05,540 Alright, I think you're going to like this. 2 @@ -123,7 +123,7 @@ itself from the public view. So, without further ado, let's dive into the math. 32 -00:01:48,979 --> 00:01:53,585 +00:01:48,980 --> 00:01:53,585 Consider the product 2 over 1 times 4 over 3 times 6 over 5 on and on and on, 33 @@ -267,7 +267,7 @@ since the inverse square law gave a really nice physical interpretation to this it was the total amount of light received by that observer. 68 -00:04:05,579 --> 00:04:07,937 +00:04:05,580 --> 00:04:07,937 But despite that nice physical interpretation, 69 @@ -303,7 +303,7 @@ we'll be multiplying them, giving a quantity I'll be referring to as the distance product for the observer. 77 -00:04:39,259 --> 00:04:43,971 +00:04:39,260 --> 00:04:43,971 And even though this distance product no longer has a nice physical analogy, 78 @@ -579,7 +579,7 @@ It's only the fraction of the way between adjacent lighthouses that describes our observer which will come into play. 146 -00:09:28,219 --> 00:09:34,086 +00:09:28,220 --> 00:09:34,086 If this fraction is f, then observer to the power n lands f of the way around a 147 @@ -707,7 +707,7 @@ And that's our first key fact, so just tuck that away. For the next key fact, imagine putting the observer right on one of the lighthouses. 178 -00:11:52,079 --> 00:11:54,920 +00:11:52,080 --> 00:11:54,920 Well, then of course the distance product is 0. 179 @@ -843,662 +843,666 @@ Put the keeper directly on one of the lighthouses, and put the sailor halfway between that point and the next lighthouse. 212 -00:14:01,480 --> 00:14:06,175 -The idea here will be to look at the distance product for the keeper divided by the +00:14:01,480 --> 00:14:04,453 +The idea here will be to look at the distance product for 213 -00:14:06,175 --> 00:14:10,760 -distance product for the sailor, and then compute this ratio in two separate ways. +00:14:04,453 --> 00:14:07,478 +the keeper divided by the distance product for the sailor, 214 +00:14:07,478 --> 00:14:10,760 +and then we're going to compute this ratio in two separate ways. + +215 00:14:11,580 --> 00:14:16,320 From the first key fact, we know that the total distance product for the sailor is 2. -215 +216 00:14:17,980 --> 00:14:19,240 And the distance product for the keeper? -216 +217 00:14:20,040 --> 00:14:22,820 Well, it's 0, since he's standing right on top of 1. -217 +218 00:14:23,160 --> 00:14:26,673 But if we got rid of that lighthouse, then by our second key fact, -218 +219 00:14:26,673 --> 00:14:29,400 the remaining distance product for that keeper is n. -219 +220 00:14:31,120 --> 00:14:33,472 And of course, by getting rid of that lighthouse, -220 +221 00:14:33,472 --> 00:14:37,048 we've also gotten rid of its contribution to the sailor's distance product, -221 +222 00:14:37,048 --> 00:14:41,000 so that denominator now has to be divided by the distance between the two observers. -222 +223 00:14:42,100 --> 00:14:45,659 And simplifying this just a little bit, it means that the ratio -223 +224 00:14:45,659 --> 00:14:48,996 between the keeper's distance product and the sailor's is n -224 +225 00:14:48,996 --> 00:14:52,500 times the distance between the two observers, all divided by 2. -225 +226 00:14:53,360 --> 00:14:56,723 But we could also compute this ratio in a different way, -226 +227 00:14:56,723 --> 00:14:59,320 by considering each lighthouse individually. -227 +228 00:15:00,040 --> 00:15:04,702 For each lighthouse, think about its contribution to the keeper's distance product, -228 +229 00:15:04,702 --> 00:15:08,420 meaning its distance to the keeper, divided by its contribution to -229 +230 00:15:08,420 --> 00:15:11,640 the sailor's distance product, its distance to the sailor. -230 +231 00:15:12,480 --> 00:15:16,074 And when we multiply all of these factors up over each lighthouse, -231 +232 00:15:16,074 --> 00:15:20,688 we have to get the same ratio in the end, n times the distance between the observers, -232 +233 00:15:20,688 --> 00:15:21,600 all divided by 2. -233 +234 00:15:22,460 --> 00:15:26,561 Now that might seem like a super messy calculation, but as n gets larger, -234 +235 00:15:26,561 --> 00:15:29,720 this actually gets simpler for any particular lighthouse. -235 +236 00:15:30,300 --> 00:15:33,661 For example, think about the first lighthouse after the keeper, -236 +237 00:15:33,661 --> 00:15:35,920 in the sense of counter-clockwise from him. -237 +238 00:15:36,600 --> 00:15:39,842 This is a bit closer to the sailor than it is to the keeper, -238 +239 00:15:39,842 --> 00:15:43,084 specifically the angle from this lighthouse to the keeper is -239 +240 00:15:43,084 --> 00:15:46,220 exactly twice the angle from this lighthouse to the sailor. -240 +241 00:15:47,100 --> 00:15:51,162 And those angles aren't exactly proportional to the straight line distances, -241 +242 00:15:51,162 --> 00:15:55,120 but as n gets larger and larger, the correspondence gets better and better. -242 +243 00:15:55,480 --> 00:15:59,344 And for a very large n, the distance from the lighthouse to the keeper -243 +244 00:15:59,344 --> 00:16:03,100 is very nearly twice the distance from that lighthouse to the sailor. -244 +245 00:16:04,900 --> 00:16:09,231 And in the same way, looking at the second lighthouse after the keeper, -245 +246 00:16:09,231 --> 00:16:14,043 it has an angle to keeper divided by angle to sailor ratio of exactly 4 thirds, -246 +247 00:16:14,043 --> 00:16:19,216 which is very nearly the same as the distance to keeper divided by distance to sailor -247 +248 00:16:19,216 --> 00:16:20,540 ratio as n gets large. -248 +249 00:16:21,140 --> 00:16:24,933 And that third lighthouse, L3, is going to contribute a fraction -249 +250 00:16:24,933 --> 00:16:28,960 that gets closer and closer to 6 fifths as n is approaching infinity. -250 +251 00:16:31,880 --> 00:16:35,095 Now for this proof, we're going to want to consider all the lighthouses -251 +252 00:16:35,095 --> 00:16:37,507 on the bottom of the circle a little bit differently, -252 +253 00:16:37,507 --> 00:16:41,080 which is why I've enumerated them negative 1, negative 2, negative 3, and so on. -253 +254 00:16:41,580 --> 00:16:44,899 If you look at that first lighthouse before the keeper, -254 +255 00:16:44,899 --> 00:16:49,818 it has a distance to keeper over distance to sailor ratio that approaches 2 thirds -255 +256 00:16:49,818 --> 00:16:51,300 as n approaches infinity. -256 +257 00:16:52,100 --> 00:16:55,616 And then the second lighthouse before it, L-2 here, -257 +258 00:16:55,616 --> 00:16:59,740 contributes a ratio that gets closer and closer to 4 fifths, -258 +259 00:16:59,740 --> 00:17:05,623 and the third lighthouse, L-3, contributes a fraction closer and closer to 6 sevenths, -259 +260 00:17:05,623 --> 00:17:06,300 and so on. -260 +261 00:17:07,540 --> 00:17:13,523 Combining this over all of the lighthouses, we get the product 2 over 1 times 2 -261 +262 00:17:13,523 --> 00:17:19,880 over 3 times 4 over 3 times 4 over 5 times 6 over 5 times 6 over 7, on and on and on. -262 +263 00:17:20,260 --> 00:17:23,295 This is the product that we're interested in studying, -263 +264 00:17:23,295 --> 00:17:26,772 and in this context, each one of those terms reflects what the -264 +265 00:17:26,772 --> 00:17:30,580 contribution for a particular lighthouse is as n approaches infinity. -265 +266 00:17:31,880 --> 00:17:35,647 And when I say contribution, I mean the contribution to this ratio of the -266 +267 00:17:35,647 --> 00:17:38,701 keeper's distance product to the sailor's distance product, -267 +268 00:17:38,701 --> 00:17:42,469 which we know at every step has to equal n times the distance between the -268 +269 00:17:42,469 --> 00:17:43,640 observers divided by 2. -269 +270 00:17:44,500 --> 00:17:47,780 So what does that value approach as n approaches infinity? -270 +271 00:17:48,740 --> 00:17:55,152 The distance between the observers is half of 1 over n of a full turn around the circle, -271 +272 00:17:55,152 --> 00:17:59,908 and since this is a unit circle, its total circumference is 2 pi, -272 +273 00:17:59,908 --> 00:18:04,663 so the distance between the observers approaches pi divided by n, -273 +274 00:18:04,663 --> 00:18:10,140 and therefore n times this distance divided by 2 approaches pi divided by 2. -274 +275 00:18:10,660 --> 00:18:12,220 So there you have it! -275 +276 00:18:12,520 --> 00:18:16,893 Our product, 2 over 1 times 2 over 3 times 4 over 3 times 4 over 5, -276 +277 00:18:16,893 --> 00:18:19,980 on and on and on, must approach pi divided by 2. -277 +278 00:18:21,040 --> 00:18:24,839 This is a truly marvelous result, and it's known as the Wallace product, -278 +279 00:18:24,839 --> 00:18:27,597 named after 17th century mathematician John Wallace, -279 +280 00:18:27,597 --> 00:18:30,720 who first discovered this fact in a way more convoluted way. -280 +281 00:18:31,320 --> 00:18:35,425 And also, a little bit of trivia, this is the same guy who discovered, -281 +282 00:18:35,425 --> 00:18:37,680 or well, invented, the infinity symbol. -282 +283 00:18:43,060 --> 00:18:45,440 And, actually, if you look back at this argument, -283 +284 00:18:45,440 --> 00:18:48,772 we've pulled a little bit of sleight of hand in the informality here, -284 +285 00:18:48,772 --> 00:18:52,580 which the particularly mathematically sophisticated among you might have caught. -285 +286 00:18:53,460 --> 00:18:57,741 What we have here is a whole bunch of factors which we knew multiplied together -286 +287 00:18:57,741 --> 00:19:01,167 to get n times the distance between the observers divided by 2, -287 +288 00:19:01,167 --> 00:19:05,610 and then we looked at the limit of each factor individually as n went to infinity, -288 +289 00:19:05,610 --> 00:19:09,624 and concluded that the product of all of those limiting terms had to equal -289 +290 00:19:09,624 --> 00:19:13,960 whatever the limit of n times the distance between the observers divided by 2 is. -290 +291 00:19:14,680 --> 00:19:19,252 But what that assumes is that the product of limits is equal to the limit of products, -291 +292 00:19:19,252 --> 00:19:21,460 even when there's infinitely many factors. -292 +293 00:19:22,340 --> 00:19:28,120 And this kind of commuting of limits in infinitary arithmetic, well, it's not always true. -293 +294 00:19:28,500 --> 00:19:30,780 It often holds, but it sometimes fails. -294 +295 00:19:31,660 --> 00:19:34,110 Here, let me show you a simple example of a case where -295 +296 00:19:34,110 --> 00:19:36,740 this kind of commuting of limits doesn't actually work out. -296 +297 00:19:37,080 --> 00:19:42,240 So we've got a grid here where every row has a single 7 and then a whole bunch of ones. -297 +298 00:19:42,420 --> 00:19:45,190 So if you were to take the infinite product of each row, -298 +299 00:19:45,190 --> 00:19:46,940 you just get 7 for each one of them. -299 +300 00:19:47,420 --> 00:19:52,560 So since every one of these products is 7, the limit of the products is also 7. -300 +301 00:19:53,100 --> 00:19:55,040 But look at what happens if you take the limits first. -301 +302 00:19:55,320 --> 00:19:59,734 If you look at each column, the limit of a given column is going to be 1, -302 +303 00:19:59,734 --> 00:20:02,120 since at some point it's nothing but 1s. -303 +304 00:20:02,120 --> 00:20:05,038 But then, if you're taking the product of those limits, -304 +305 00:20:05,038 --> 00:20:09,311 you're just taking the product of a bunch of ones, so you get a different answer, -305 +306 00:20:09,311 --> 00:20:09,780 namely 1. -306 +307 00:20:13,240 --> 00:20:16,972 Luckily, mathematicians have spent a lot of time thinking about this phenomenon, -307 +308 00:20:16,972 --> 00:20:20,013 and they've developed tools for quickly seeing certain conditions -308 +309 00:20:20,013 --> 00:20:22,640 under which this exchanging of the limits actually works. -309 +310 00:20:23,320 --> 00:20:27,295 In this case, a particular standard result known as dominated convergence -310 +311 00:20:27,295 --> 00:20:31,700 quickly assures us that the argument we just showed will go through in full rigor. -311 +312 00:20:32,260 --> 00:20:35,956 For those of you who are interested, Sridhar has written up a supplemental -312 +313 00:20:35,956 --> 00:20:39,900 blog post to this video which covers those details, along with many more things. -313 +314 00:20:40,740 --> 00:20:42,722 And I should also say, we need to be a little -314 +315 00:20:42,722 --> 00:20:44,920 careful about how to interpret a product like this. -315 +316 00:20:45,400 --> 00:20:49,709 Remember, we have contributions from lighthouses counterclockwise from the keeper, -316 +317 00:20:49,709 --> 00:20:52,305 as well as lighthouses clockwise from the keeper, -317 +318 00:20:52,305 --> 00:20:55,680 and what we did was interleave these in order to get our product. -318 +319 00:20:55,680 --> 00:21:02,070 The lighthouses counterclockwise from the keeper contribute 2 over 1, 4 over 3, 6 over 5, -319 +320 00:21:02,070 --> 00:21:08,460 on and on, and the ones clockwise from the keeper contribute 2 over 3, 4 over 5, 6 over 7. -320 +321 00:21:09,080 --> 00:21:12,808 And like I said before, if you play around with those individual series, -321 +322 00:21:12,808 --> 00:21:16,893 you'll find that the first one gets larger and larger and blows up to infinity, -322 +323 00:21:16,893 --> 00:21:20,060 and the second one gets smaller and smaller, approaching zero. -323 +324 00:21:20,660 --> 00:21:24,751 So it's actually pretty delicate to make sense out of this overall product -324 +325 00:21:24,751 --> 00:21:28,680 in terms of computing the two halves separately and then combining them. -325 +326 00:21:29,240 --> 00:21:32,983 And indeed, we'll find that if you intermix these two halves differently, -326 +327 00:21:32,983 --> 00:21:37,536 for example taking twice as many factors from one of them for each factor from the other, -327 +328 00:21:37,536 --> 00:21:40,420 you could get a different result for the overall product. -328 +329 00:21:40,740 --> 00:21:44,026 It's only when you specifically combine them in this one-for-one -329 +330 00:21:44,026 --> 00:21:46,960 manner that you get a product that converges to pi halves. -330 +331 00:21:47,620 --> 00:21:51,885 This is something that falls out of the way that dominated convergence justifies us in -331 +332 00:21:51,885 --> 00:21:56,200 commuting limits the way we did, and again, for more details, see the supplemental post. -332 +333 00:21:57,140 --> 00:21:58,800 Still, those are just technicalities. -333 +334 00:21:59,140 --> 00:22:02,840 The conceptual gist for what's going on here is exactly what we just showed. -334 +335 00:22:07,660 --> 00:22:11,138 And in fact, after doing all that work, it would be a shame not to take a -335 +336 00:22:11,138 --> 00:22:14,900 quick moment to talk about one more neat result that falls out of this argument. -336 +337 00:22:14,900 --> 00:22:17,680 Arguably, this is the coolest part of the whole proof. -337 +338 00:22:18,240 --> 00:22:20,420 You see, we can generalize this whole discussion. -338 +339 00:22:21,100 --> 00:22:24,002 Think back to when we discovered our first key fact, -339 +340 00:22:24,002 --> 00:22:28,110 where we saw that you could not only consider placing the sailor precisely -340 +341 00:22:28,110 --> 00:22:33,040 halfway between lighthouses, but any fraction, f, of the way between adjacent lighthouses. -341 +342 00:22:33,720 --> 00:22:38,873 In that more general setting, the distance product for the sailor wasn't necessarily 2, -342 +343 00:22:38,873 --> 00:22:43,500 but it was chord of f, where f is that fraction of the way between lighthouses. -343 +344 00:22:44,200 --> 00:22:49,496 If we go through the same reasoning that we just did with the sailor at this location -344 +345 00:22:49,496 --> 00:22:54,608 instead and change nothing else, what we'll find is that the ratio of the keeper's -345 +346 00:22:54,608 --> 00:22:59,905 distance product to the sailor's distance product is now n times the distance between -346 +347 00:22:59,905 --> 00:23:05,448 them divided by chord of f, which approaches f times 2 pi divided by chord of f as n gets -347 +348 00:23:05,448 --> 00:23:05,880 larger. -348 +349 00:23:08,800 --> 00:23:12,006 And in the same way as before, you could alternatively calculate -349 +350 00:23:12,006 --> 00:23:15,460 this by considering the contributions from each individual lighthouse. -350 +351 00:23:16,340 --> 00:23:20,988 If you take the time to work this out, the kth lighthouse after the -351 +352 00:23:20,988 --> 00:23:25,500 keeper will contribute a factor of k divided by k-f to this ratio. -352 +353 00:23:26,240 --> 00:23:29,624 And all the lighthouses before the keeper, they contribute the same thing, -353 +354 00:23:29,624 --> 00:23:31,880 but you're just plugging in negative values for k. -354 +355 00:23:32,720 --> 00:23:36,982 If you combine all those contributions over all non-zero integers k, -355 +356 00:23:36,982 --> 00:23:41,924 where in the same way as before you have to be careful about how you bundle the -356 +357 00:23:41,924 --> 00:23:46,927 positive and negative k terms together, what you'll get is that the product of k -357 +358 00:23:46,927 --> 00:23:52,055 divided by k-f over all non-zero integers k is going to equal f times 2 pi divided -358 +359 00:23:52,055 --> 00:23:52,920 by chord of f. -359 +360 00:23:53,580 --> 00:23:59,522 Put another way, since chord of f is 2 times the sine of f pi, -360 +361 00:23:59,522 --> 00:24:06,501 this product is the same as f times 2 pi divided by 2 times sine of f pi, -361 +362 00:24:06,501 --> 00:24:09,520 which is f pi over sine of f pi. -362 +363 00:24:10,320 --> 00:24:14,800 Now rewriting this a little bit more, what you get is a pretty interesting fact. -363 +364 00:24:15,420 --> 00:24:20,518 Sine of f times pi is equal to f pi times this really big product, -364 +365 00:24:20,518 --> 00:24:25,160 the product of 1 minus f over k over all non-zero integers k. -365 +366 00:24:25,920 --> 00:24:30,877 So what we found is a way to express sine of x as an infinite product, -366 +367 00:24:30,877 --> 00:24:33,880 which is really cool if you think about it. -367 +368 00:24:34,300 --> 00:24:37,325 So not only does this proof give us the Wallace product, -368 +369 00:24:37,325 --> 00:24:41,625 which is incredible in its own right, it also generalizes to give us the product -369 +370 00:24:41,625 --> 00:24:42,740 formula for the sine. -370 +371 00:24:43,260 --> 00:24:46,621 And what's neat about that is that it connects to how Euler originally -371 +372 00:24:46,621 --> 00:24:49,840 solved the Basel problem, the sum that we saw in the previous video. -372 +373 00:24:50,160 --> 00:24:52,880 He was looking at this very infinite product for sine. -373 +374 00:24:53,600 --> 00:24:56,858 I mean, connecting these formulas for pi to circles is one thing, -374 +375 00:24:56,858 --> 00:24:59,820 but connecting them to each other is another thing entirely. -375 -00:25:00,520 --> 00:25:02,956 +376 +00:25:00,520 --> 00:25:02,703 And once again, if you want more details on all of this, -376 -00:25:02,956 --> 00:25:04,580 -check out the supplementary blog post. +377 +00:25:02,703 --> 00:25:04,580 +check out the supplementary blog post. Thank you. diff --git a/2018/wallis-product/english/sentence_timings.json b/2018/wallis-product/english/sentence_timings.json index c3f599bbd..2a6b85b66 100644 --- a/2018/wallis-product/english/sentence_timings.json +++ b/2018/wallis-product/english/sentence_timings.json @@ -450,7 +450,7 @@ 840.82 ], [ - "The idea here will be to look at the distance product for the keeper divided by the distance product for the sailor, and then compute this ratio in two separate ways.", + "The idea here will be to look at the distance product for the keeper divided by the distance product for the sailor, and then we're going to compute this ratio in two separate ways.", 841.48, 850.76 ], @@ -810,7 +810,7 @@ 1499.82 ], [ - "And once again, if you want more details on all of this, check out the supplementary blog post.", + "And once again, if you want more details on all of this, check out the supplementary blog post. Thank you.", 1500.52, 1504.58 ] diff --git a/2018/wallis-product/english/transcript.txt b/2018/wallis-product/english/transcript.txt index 06953c6cd..e5a1db08b 100644 --- a/2018/wallis-product/english/transcript.txt +++ b/2018/wallis-product/english/transcript.txt @@ -88,7 +88,7 @@ And now, so can you! So next, with both these facts in our back pocket, let's see how to use them to understand the product we're interested in, and how it relates to pi. Take this setup, with n lighthouses evenly spaced around a unit circle, and imagine two separate observers, what I'll call the keeper and the sailor. Put the keeper directly on one of the lighthouses, and put the sailor halfway between that point and the next lighthouse. -The idea here will be to look at the distance product for the keeper divided by the distance product for the sailor, and then compute this ratio in two separate ways. +The idea here will be to look at the distance product for the keeper divided by the distance product for the sailor, and then we're going to compute this ratio in two separate ways. From the first key fact, we know that the total distance product for the sailor is 2. And the distance product for the keeper? Well, it's 0, since he's standing right on top of 1. @@ -160,4 +160,4 @@ So not only does this proof give us the Wallace product, which is incredible in And what's neat about that is that it connects to how Euler originally solved the Basel problem, the sum that we saw in the previous video. He was looking at this very infinite product for sine. I mean, connecting these formulas for pi to circles is one thing, but connecting them to each other is another thing entirely. -And once again, if you want more details on all of this, check out the supplementary blog post. \ No newline at end of file +And once again, if you want more details on all of this, check out the supplementary blog post. Thank you. \ No newline at end of file diff --git a/2018/wallis-product/french/sentence_translations.json b/2018/wallis-product/french/sentence_translations.json index 08bb07132..4eb263fb8 100644 --- a/2018/wallis-product/french/sentence_translations.json +++ b/2018/wallis-product/french/sentence_translations.json @@ -600,7 +600,7 @@ "end": 691.22 }, { - "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two results in a distance product of precisely 2.", + "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two of them results in a distance product of precisely 2.", "translatedText": "On voit donc que quel que soit le nombre de phares, répartis également autour du cercle unité, placer un observateur exactement à mi-chemin du cercle entre deux donne un produit de distance précisément de 2.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/wallis-product/german/sentence_translations.json b/2018/wallis-product/german/sentence_translations.json index c1e63c4d9..2102d4c48 100644 --- a/2018/wallis-product/german/sentence_translations.json +++ b/2018/wallis-product/german/sentence_translations.json @@ -600,7 +600,7 @@ "end": 691.22 }, { - "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two results in a distance product of precisely 2.", + "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two of them results in a distance product of precisely 2.", "translatedText": "Wir sehen also, dass unabhängig von der Anzahl der Leuchttürme, die gleichmäßig über den Einheitskreis verteilt sind, ein Distanzprodukt von genau 2 entsteht, wenn man einen Beobachter genau in der Mitte des Kreises zwischen zwei Leuchttürmen platziert.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/wallis-product/indonesian/sentence_translations.json b/2018/wallis-product/indonesian/sentence_translations.json index 869df0a55..ee095b36e 100644 --- a/2018/wallis-product/indonesian/sentence_translations.json +++ b/2018/wallis-product/indonesian/sentence_translations.json @@ -600,7 +600,7 @@ "end": 691.22 }, { - "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two results in a distance product of precisely 2.", + "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two of them results in a distance product of precisely 2.", "translatedText": "Jadi kita melihat bahwa tidak peduli berapa banyak mercusuar yang ada, tersebar secara merata di sekeliling lingkaran satuan, menempatkan seorang pengamat tepat di tengah lingkaran di antara keduanya akan menghasilkan hasil kali jarak yang tepat 2.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/wallis-product/marathi/sentence_translations.json b/2018/wallis-product/marathi/sentence_translations.json index 8251c1ccf..8e49560a7 100644 --- a/2018/wallis-product/marathi/sentence_translations.json +++ b/2018/wallis-product/marathi/sentence_translations.json @@ -600,7 +600,7 @@ "end": 691.22 }, { - "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two results in a distance product of precisely 2.", + "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two of them results in a distance product of precisely 2.", "translatedText": "म्हणून आपण पाहतो की, कितीही दीपगृह असले तरीही, एकक वर्तुळाभोवती तितकेच पसरलेले, निरीक्षकाला दोन वर्तुळाच्या मध्यभागी नेमके अर्धे ठेवल्याने त्याचे परिणाम 2 च्या अंतराचे होते.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/wallis-product/polish/sentence_translations.json b/2018/wallis-product/polish/sentence_translations.json index b2446217c..de8a3e51f 100644 --- a/2018/wallis-product/polish/sentence_translations.json +++ b/2018/wallis-product/polish/sentence_translations.json @@ -719,7 +719,7 @@ "end": 840.82 }, { - "input": "The idea here will be to look at the distance product for the keeper divided by the distance product for the sailor, and then compute this ratio in two separate ways.", + "input": "The idea here will be to look at the distance product for the keeper divided by the distance product for the sailor, and then we're going to compute this ratio in two separate ways.", "translatedText": "", "from_community_srt": "Pomysł jest taki, żeby patrzeć na iloczyn odległości dla latarnika podzielony przez iloczyn odległości dla żeglarza. Potem obliczymy ten stosunek na dwa odrębne sposoby.", "n_reviews": 0, @@ -1295,7 +1295,7 @@ "end": 1499.82 }, { - "input": "And once again, if you want more details on all of this, check out the supplementary blog post.", + "input": "And once again, if you want more details on all of this, check out the supplementary blog post. Thank you.", "translatedText": "", "from_community_srt": "I jeszcze raz przypomnę: jeśli chcecie znać więcej szczegółów dotyczących tego wszystkiego,", "n_reviews": 0, diff --git a/2018/wallis-product/portuguese/sentence_translations.json b/2018/wallis-product/portuguese/sentence_translations.json index f1b30476f..5b4b51fc6 100644 --- a/2018/wallis-product/portuguese/sentence_translations.json +++ b/2018/wallis-product/portuguese/sentence_translations.json @@ -600,7 +600,7 @@ "end": 691.22 }, { - "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two results in a distance product of precisely 2.", + "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two of them results in a distance product of precisely 2.", "translatedText": "Portanto, vemos que não importa quantos faróis existam, distribuídos igualmente em torno do círculo unitário, colocar um observador exatamente a meio caminho do círculo entre dois resulta num produto de distância de precisamente 2.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/wallis-product/russian/sentence_translations.json b/2018/wallis-product/russian/sentence_translations.json index d946642ac..fb5f71439 100644 --- a/2018/wallis-product/russian/sentence_translations.json +++ b/2018/wallis-product/russian/sentence_translations.json @@ -600,7 +600,7 @@ "end": 691.22 }, { - "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two results in a distance product of precisely 2.", + "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two of them results in a distance product of precisely 2.", "translatedText": "Итак, мы видим, что независимо от того, сколько маяков равномерно распределено по единичному кругу, размещение наблюдателя ровно на середине круга между двумя приводит к произведению расстояний, равному точно 2.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/wallis-product/spanish/sentence_translations.json b/2018/wallis-product/spanish/sentence_translations.json index 5393a0378..1742418dd 100644 --- a/2018/wallis-product/spanish/sentence_translations.json +++ b/2018/wallis-product/spanish/sentence_translations.json @@ -798,7 +798,7 @@ "end": 840.82 }, { - "input": "The idea here will be to look at the distance product for the keeper divided by the distance product for the sailor, and then compute this ratio in two separate ways.", + "input": "The idea here will be to look at the distance product for the keeper divided by the distance product for the sailor, and then we're going to compute this ratio in two separate ways.", "translatedText": "La idea aquí será observar el producto de la distancia para el guardián dividido por el producto de la distancia para el navegante, y luego calcular esta relación de dos formas distintas.", "model": "DeepL", "from_community_srt": "La idea aquí será: Mirar el \"producto distancia\" del guardián dividido por el \"producto distancia\" del marinero", @@ -1439,7 +1439,7 @@ "end": 1499.82 }, { - "input": "And once again, if you want more details on all of this, check out the supplementary blog post.", + "input": "And once again, if you want more details on all of this, check out the supplementary blog post. Thank you.", "translatedText": "Y una vez más, si quieres más detalles sobre todo esto, consulta la entrada complementaria del blog.", "model": "DeepL", "from_community_srt": "y de nuevo,", diff --git a/2018/wallis-product/tamil/sentence_translations.json b/2018/wallis-product/tamil/sentence_translations.json index eaa42f9a7..9b8a4b32e 100644 --- a/2018/wallis-product/tamil/sentence_translations.json +++ b/2018/wallis-product/tamil/sentence_translations.json @@ -600,7 +600,7 @@ "end": 691.22 }, { - "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two results in a distance product of precisely 2.", + "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two of them results in a distance product of precisely 2.", "translatedText": "எனவே, எத்தனை கலங்கரை விளக்கங்கள் இருந்தாலும், யூனிட் வட்டத்தைச் சுற்றிலும் சமமாகப் பரவி, ஒரு பார்வையாளரை வட்டத்தின் நடுவில் சரியாகப் பாதியில் வைத்து இரண்டு முடிவுகளுக்கு இடையே துல்லியமாக 2 என்ற தொலைவுப் பெருக்கத்தில் இருப்பதைக் காண்கிறோம்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/wallis-product/telugu/sentence_translations.json b/2018/wallis-product/telugu/sentence_translations.json index 56006a9e2..f56e8872c 100644 --- a/2018/wallis-product/telugu/sentence_translations.json +++ b/2018/wallis-product/telugu/sentence_translations.json @@ -600,7 +600,7 @@ "end": 691.22 }, { - "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two results in a distance product of precisely 2.", + "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two of them results in a distance product of precisely 2.", "translatedText": "కాబట్టి, ఎన్ని లైట్‌హౌస్‌లు ఉన్నప్పటికీ, యూనిట్ సర్కిల్ చుట్టూ సమానంగా విస్తరించి, రెండు ఫలితాల మధ్య వృత్తం పొడవునా సరిగ్గా సగానికి ఒక పరిశీలకుడిని ఉంచడం ద్వారా ఖచ్చితంగా 2 దూర ఉత్పత్తిని పొందడం మనం చూస్తాము.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/wallis-product/turkish/sentence_translations.json b/2018/wallis-product/turkish/sentence_translations.json index 7024aa30f..74bd012e6 100644 --- a/2018/wallis-product/turkish/sentence_translations.json +++ b/2018/wallis-product/turkish/sentence_translations.json @@ -600,7 +600,7 @@ "end": 691.22 }, { - "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two results in a distance product of precisely 2.", + "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two of them results in a distance product of precisely 2.", "translatedText": "Yani, kaç tane deniz feneri olursa olsun, birim çemberin çevresine eşit olarak yayılmış bir gözlemciyi çemberin tam yarısına iki noktanın arasına koyarsak, mesafe çarpımının tam olarak 2 olacağını görüyoruz.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/wallis-product/vietnamese/sentence_translations.json b/2018/wallis-product/vietnamese/sentence_translations.json index 571a34de0..a029dbbbb 100644 --- a/2018/wallis-product/vietnamese/sentence_translations.json +++ b/2018/wallis-product/vietnamese/sentence_translations.json @@ -600,7 +600,7 @@ "end": 691.22 }, { - "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two results in a distance product of precisely 2.", + "input": "So we see that no matter how many lighthouses there are, equally spread around the unit circle, putting an observer exactly halfway along the circle between two of them results in a distance product of precisely 2.", "translatedText": "Vì vậy, chúng ta thấy rằng cho dù có bao nhiêu ngọn hải đăng, trải đều xung quanh vòng tròn đơn vị, việc đặt một người quan sát chính xác ở nửa đường tròn giữa hai kết quả sẽ có tích khoảng cách chính xác là 2.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/arabic/sentence_translations.json b/2018/winding-numbers/arabic/sentence_translations.json index c3405e7a7..2fa1ade19 100644 --- a/2018/winding-numbers/arabic/sentence_translations.json +++ b/2018/winding-numbers/arabic/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs. ", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs. ", "translatedText": "قمنا بتقسيمه إلى قسمين، والنصف الذي ضيّقنا انتباهنا عليه هو النصف الذي كانت لأطرافه الخارجية علامات مختلفة. ", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero. ", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero. ", "translatedText": "تذكر، في أحد الأبعاد، كانت الرؤية الرئيسية هي أنه إذا كانت الدالة المستمرة موجبة عند نقطة ما وسالبة عند نقطة أخرى، فيجب أن تكون في مكان ما بينهما صفرًا. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again. ", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again. ", "translatedText": "المخرجات التي صادفناها تدور حول إجمالي ثلاث دورات كاملة في اتجاه عقارب الساعة، وتأرجحت الألوان عبر قوس قزح، بالترتيب، من الأحمر إلى الأحمر مرة أخرى، ثم مرة أخرى، ومرة أخرى. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths. ", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths. ", "translatedText": "الشيء الرئيسي الذي يجب ملاحظته بشأن هذه الأرقام المتعرجة هو أنها تتراكم بشكل جيد عند دمج المسارات في مسارات أكبر. ", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. ", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching. ", "translatedText": "المؤلف الرئيسي لهذا الفيديو هو أحد أحدث أعضاء فريق 3blue1brown، سريدار راميش. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/bengali/sentence_translations.json b/2018/winding-numbers/bengali/sentence_translations.json index ce2675b3a..535dcf8be 100644 --- a/2018/winding-numbers/bengali/sentence_translations.json +++ b/2018/winding-numbers/bengali/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs. ", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs. ", "translatedText": "আমরা এটিকে দুটি ভাগে বিভক্ত করেছি, এবং অর্ধেকের দিকে আমরা আমাদের মনোযোগ সংকুচিত করেছি যেখানে বাইরের বিন্দুতে বিভিন্ন চিহ্ন রয়েছে।", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero. ", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero. ", "translatedText": "মনে রাখবেন, একটি মাত্রায় মূল অন্তর্দৃষ্টি ছিল যে একটি অবিচ্ছিন্ন ফাংশন যদি এক বিন্দুতে ধনাত্মক এবং অন্যটিতে ঋণাত্মক হয়, তবে এর মধ্যে কোথাও এটি শূন্য হতে হবে।", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again. ", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again. ", "translatedText": "আমরা যে আউটপুটগুলিকে ঘড়ির কাঁটার চারপাশে মোট তিনটি পূর্ণ ঘড়ির চারপাশে বাতাসের সাথে মিলিয়ে দেখি, রংগুলি রংধনুর মধ্য দিয়ে, ক্রমানুসারে, লাল থেকে লালে আবার, এবং তারপরে আবার এবং আবার।", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths. ", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths. ", "translatedText": "এই উইন্ডিং নম্বরগুলি সম্পর্কে লক্ষ্য করার প্রধান বিষয় হল যে আপনি যখন পাথগুলিকে বড় পাথগুলিতে একত্রিত করেন তখন এগুলি সুন্দরভাবে যুক্ত হয়৷ কিন্তু আমরা যা চাই তা হল অঞ্চলগুলির সীমানা বরাবর ঘুরানো সংখ্যাগুলি সুন্দরভাবে যোগ করার জন্য যখন আমরা অঞ্চলগুলিকে একত্রিত করে বড় অঞ্চল তৈরি করি।", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. ", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/chinese/sentence_translations.json b/2018/winding-numbers/chinese/sentence_translations.json index 30d51811c..61b788734 100644 --- a/2018/winding-numbers/chinese/sentence_translations.json +++ b/2018/winding-numbers/chinese/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "我们把它分成两半，我们将注意力集中 到最外面的点有不同符号的那一半。", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "请记住，要创建这样的图表，我们已经让计算机在 平面上的所有像素上计算函数，但我们的目标是 找到一种更有效的算法，只需要在尽可能少的点 上计算函数，可以这么说，对颜色的了解有限。", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "请记住，在一个维度上，主要见解是，如果连续函数在某一点 为正，在另一点为负，则介于两者之间的某个位置必须为零。", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "我们遇到的输出总共绕了三个完整的顺时针方向，颜色按顺 序在彩虹中摆动，从红色再次变为红色，然后再次，再次。", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "关于这些蜿蜒的数字，需要注意的主要一点是，当您 将路径组合成更大的路径时，它们会很好地相加。", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "该视频的主要作者是 3blue1brown 团队 的最新成员之一 Sridhar Ramesh。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/english/captions.srt b/2018/winding-numbers/english/captions.srt index a3043100b..4083c25fc 100644 --- a/2018/winding-numbers/english/captions.srt +++ b/2018/winding-numbers/english/captions.srt @@ -215,7 +215,7 @@ this kind of thinking into two-dimensional equations, equations between functions whose inputs and outputs are both two-dimensional. 55 -00:03:14,739 --> 00:03:17,851 +00:03:14,740 --> 00:03:17,851 For example, complex numbers are 2D, and this tool we're 56 @@ -243,7 +243,7 @@ and that's not going to work so well in our 3D world on our 2D screens, but we still have a couple good options. 62 -00:03:40,859 --> 00:03:45,440 +00:03:40,860 --> 00:03:45,440 One is to just look at both the input space and the output space side by side. 63 @@ -351,7 +351,7 @@ And so on, all we're doing here is assigning a specific color to each direction, all changing continuously. 89 -00:05:25,359 --> 00:05:29,142 +00:05:25,360 --> 00:05:29,142 You might notice the darkness and brightness differences here are pretty subtle, 90 @@ -411,12 +411,12 @@ you can get a sense just by looking at that input space for roughly where the function takes each point. 104 -00:06:19,120 --> 00:06:23,758 -For example, this stripe of pink points on the left tells us that all of those +00:06:19,120 --> 00:06:23,865 +For example, this stripe of pink points on the left tells us that all of those points 105 -00:06:23,758 --> 00:06:28,280 -points get mapped in the pink direction, that lower left of the output space. +00:06:23,865 --> 00:06:28,280 +get mapped somewhere in the pink direction, that lower left of the output space. 106 00:06:29,780 --> 00:06:32,314 diff --git a/2018/winding-numbers/english/sentence_timings.json b/2018/winding-numbers/english/sentence_timings.json index 7f7e85cc6..30ea2e1e5 100644 --- a/2018/winding-numbers/english/sentence_timings.json +++ b/2018/winding-numbers/english/sentence_timings.json @@ -205,7 +205,7 @@ 378.42 ], [ - "For example, this stripe of pink points on the left tells us that all of those points get mapped in the pink direction, that lower left of the output space.", + "For example, this stripe of pink points on the left tells us that all of those points get mapped somewhere in the pink direction, that lower left of the output space.", 379.12, 388.28 ], diff --git a/2018/winding-numbers/english/transcript.txt b/2018/winding-numbers/english/transcript.txt index 7d02a730e..065fbd7d6 100644 --- a/2018/winding-numbers/english/transcript.txt +++ b/2018/winding-numbers/english/transcript.txt @@ -39,7 +39,7 @@ The one important thing about brightness for you to notice is that near the orig So for thinking about functions, now that we've decided on a color for each output, we can visualize 2D functions by coloring each point in the input space based on the color of the point where it lands in the output space. I like to imagine many different points from that input space hopping over to their corresponding outputs in the output space, then getting painted based on the color of the point where they land, and then hopping back to where they came from in the input space. Doing this for every point in the input space, you can get a sense just by looking at that input space for roughly where the function takes each point. -For example, this stripe of pink points on the left tells us that all of those points get mapped in the pink direction, that lower left of the output space. +For example, this stripe of pink points on the left tells us that all of those points get mapped somewhere in the pink direction, that lower left of the output space. Also those three points which are black with lots of colors around them are the ones that go to zero. Alright, so just like the 1D case, solving equations of 2D functions can always be reframed by asking when a certain function is equal to zero. So that's our challenge right now, create an algorithm that finds which input points of a given 2D function go to zero. diff --git a/2018/winding-numbers/french/sentence_translations.json b/2018/winding-numbers/french/sentence_translations.json index 366f9eae8..3a7af9eb4 100644 --- a/2018/winding-numbers/french/sentence_translations.json +++ b/2018/winding-numbers/french/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "Nous l'avons divisé en deux, et la moitié sur laquelle nous avons concentré notre attention était celle où les points les plus extérieurs présentaient des signes différents.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "Gardez à l'esprit que pour créer un diagramme comme celui-ci, nous avons demandé à l'ordinateur de calculer la fonction sur tous les pixels du plan, mais notre objectif est de trouver un algorithme plus efficace qui nécessite uniquement de calculer la fonction sur le moins de points possible. , n'ayant qu'une vision limitée des couleurs, pour ainsi dire.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "Rappelez-vous, dans une dimension, l’idée principale était que si une fonction continue est positive à un moment donné et négative à un autre, quelque part entre les deux, elle doit être nulle.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "Les sorties que nous rencontrons s'enroulent sur un total de trois tours complets dans le sens des aiguilles d'une montre, les couleurs oscillent à travers l'arc-en-ciel, dans l'ordre, du rouge au rouge encore, puis encore et encore.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "La principale chose à remarquer à propos de ces nombres sinueux est qu’ils s’additionnent bien lorsque vous combinez des chemins en chemins plus grands.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "L'auteur principal de cette vidéo est l'un des nouveaux membres de l'équipe 3blue1brown, Sridhar Ramesh.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/german/sentence_translations.json b/2018/winding-numbers/german/sentence_translations.json index 1885378d6..732ce12d9 100644 --- a/2018/winding-numbers/german/sentence_translations.json +++ b/2018/winding-numbers/german/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "Wir haben es in zwei Teile geteilt, und die Hälfte, auf die wir unsere Aufmerksamkeit konzentrierten, war diejenige, bei der die äußersten Punkte unterschiedliche Vorzeichen hatten.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "Bedenken Sie, dass wir zum Erstellen eines Diagramms wie diesem den Computer die Funktion für alle Pixel auf der Ebene berechnen lassen müssen. Unser Ziel besteht jedoch darin, einen effizienteren Algorithmus zu finden, der nur die Berechnung der Funktion an so wenigen Punkten wie möglich erfordert , man hat sozusagen nur einen eingeschränkten Blick auf die Farben.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "Denken Sie daran, dass in einer Dimension die wichtigste Erkenntnis darin bestand, dass eine stetige Funktion, wenn sie an einem Punkt positiv und an einem anderen negativ ist, irgendwo dazwischen Null sein muss.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "Die Ausgänge, auf die wir stoßen, winden sich insgesamt um drei volle Umdrehungen im Uhrzeigersinn, die Farben schwankten durch den Regenbogen, der Reihe nach, von Rot zu Rot, wieder und dann noch einmal und noch einmal.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "Das Wichtigste an diesen gewundenen Zahlen ist, dass sie sich gut addieren, wenn man Pfade zu größeren Pfaden kombiniert.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "Der Hauptautor dieses Videos ist eines der neuesten 3blue1brown-Teammitglieder, Sridhar Ramesh.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/hindi/sentence_translations.json b/2018/winding-numbers/hindi/sentence_translations.json index e1d3430c9..68749171f 100644 --- a/2018/winding-numbers/hindi/sentence_translations.json +++ b/2018/winding-numbers/hindi/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "हमने इसे दो भागों में विभाजित किया, और जिस आधे हिस्से पर हमने अपना ध्यान केंद्रित किया वह वह था जहां सबसे बाहरी बिंदुओं पर अलग-अलग संकेत थे।", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "ध्यान रखें कि इस तरह का आरेख बनाने के लिए, हमने कंप्यूटर को समतल पर सभी पिक्सेल पर फ़ंक्शन की गणना करने के लिए कहा है, लेकिन हमारा लक्ष्य एक अधिक कुशल एल्गोरिदम ढूंढना है जिसके लिए केवल यथासंभव कुछ बिंदुओं पर फ़ंक्शन की गणना करने की आवश्यकता होती है , ऐसा कहने के लिए, केवल रंगों का एक सीमित दृष्टिकोण है।", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "याद रखें, एक आयाम में मुख्य अंतर्दृष्टि यह थी कि यदि एक सतत कार्य एक बिंदु पर सकारात्मक है और दूसरे पर नकारात्मक है, तो बीच में कहीं यह शून्य होना चाहिए।", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "आउटपुट में हमें कुल तीन पूर्ण दक्षिणावर्त घुमावों के आसपास हवा का सामना करना पड़ता है, रंग इंद्रधनुष के माध्यम से घूमते हैं, क्रम में, लाल से फिर से लाल, और फिर से, और फिर से।", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "इन घुमावदार संख्याओं के बारे में ध्यान देने वाली मुख्य बात यह है कि जब आप पथों को बड़े पथों में जोड़ते हैं तो वे अच्छी तरह से जुड़ जाते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "इस वीडियो के प्राथमिक लेखक 3ब्लू1ब्राउन टीम के नवीनतम सदस्यों में से एक, श्रीधर रमेश हैं।", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/indonesian/sentence_translations.json b/2018/winding-numbers/indonesian/sentence_translations.json index e95016075..30c57bca4 100644 --- a/2018/winding-numbers/indonesian/sentence_translations.json +++ b/2018/winding-numbers/indonesian/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "Kami membaginya menjadi dua, dan bagian yang kami persempit perhatiannya adalah bagian yang titik terluarnya memiliki tanda yang berbeda-beda.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "Ingatlah bahwa untuk membuat diagram seperti ini, kita telah meminta komputer menghitung fungsi di semua piksel pada bidang, namun tujuan kita adalah menemukan algoritma yang lebih efisien yang hanya memerlukan komputasi fungsi pada titik sesedikit mungkin. , hanya memiliki pandangan terbatas terhadap warna.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "Ingat, dalam satu dimensi, pemahaman utamanya adalah jika suatu fungsi kontinu bernilai positif di satu titik dan negatif di titik lain, maka di antara keduanya pasti ada nol.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "Keluaran yang kita temui berputar sebanyak tiga kali putaran penuh searah jarum jam, warna-warna berayun melalui pelangi, secara berurutan, dari merah ke merah lagi, lalu lagi, dan lagi.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "Hal utama yang perlu diperhatikan tentang angka-angka berliku ini adalah angka-angka tersebut bertambah dengan baik saat Anda menggabungkan jalur menjadi jalur yang lebih besar.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "Penulis utama video ini adalah salah satu anggota tim 3blue1 brown terbaru, Sridhar Ramesh.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/japanese/sentence_translations.json b/2018/winding-numbers/japanese/sentence_translations.json index 4d09878c8..14bcc4a17 100644 --- a/2018/winding-numbers/japanese/sentence_translations.json +++ b/2018/winding-numbers/japanese/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "私たちはそれを 2 つに分割し、最も外側の点に さまざまな兆候がある半分に注意を絞りました。", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "このような図を作成するために、コンピューターに平面上のすべてのピクセ ルで関数を計算させましたが、私たちの目標は、できるだけ少ない点で関 数を計算するだけで済む、より効率的なアルゴリズムを見つけることである ことに注意してください。 いわば、色の見え方が限られているだけです。", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "ある次元での主要な洞察は、連続関数がある点で正で別の点で負である場合 、その間のどこかでゼロになるはずだということを思い出してください。", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "私たちが目にする出力は時計回りに合計 3 回転し、色は 虹の中を順番に、赤から赤、そしてまた赤へと変化します。", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "これらの曲がりくねった数値について注目すべき主な点は、パスをより 大きなパスに結合すると、それらの数値が適切に加算されることです。", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "このビデオの主な作成者は、3blue1brown チームの最 新メンバーの 1 人、Sridhar Ramesh です。", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/korean/sentence_translations.json b/2018/winding-numbers/korean/sentence_translations.json index 3457c921d..e67c32f8f 100644 --- a/2018/winding-numbers/korean/sentence_translations.json +++ b/2018/winding-numbers/korean/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "우리는 그것을 두 개로 나누고, 우리가 주의를 좁힌 절반은 가장 바깥쪽 지점에 다양한 부호가 있는 부분이었습니다.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "이와 같은 다이어그램을 만들기 위해 컴퓨터가 평면의 모든 픽셀에서 함수를 계산하도록 했지만 우리의 목표는 가능한 한 적은 수의 지점에서만 함수를 계산해야 하는 보다 효율적인 알고리즘을 찾는 것입니다. , 말하자면 색상에 대한 제한된 시각만 가지고 있는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "한 차원에서 주요 통찰력은 연속 함수가 한 지점에서는 양수이고 다른 지점에서는 음수이면 그 사이의 어딘가는 0이어야 한다는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "우리가 접하게 되는 출력은 시계 방향으로 총 3바퀴를 돌며, 색상은 빨간색에서 다시 빨간색으로, 그리고 다시 빨간색으로 순서대로 무지개를 통해 회전합니다.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "이러한 구불구불한 숫자에 대해 주목해야 할 가장 중요한 점은 경로를 더 큰 경로로 결합할 때 그 숫자가 멋지게 합산된다는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "이 비디오의 주요 작성자는 3blue1brown의 최신 팀원 중 한 명인 Sridhar Ramesh입니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/marathi/sentence_translations.json b/2018/winding-numbers/marathi/sentence_translations.json index dbb6c5631..f9922a6bf 100644 --- a/2018/winding-numbers/marathi/sentence_translations.json +++ b/2018/winding-numbers/marathi/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "आम्ही त्याचे दोन भाग केले आणि अर्ध्या भागाकडे आमचे लक्ष वेधले ते एक होते जेथे सर्वात बाहेरील बिंदूंमध्ये भिन्न चिन्हे होती.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "लक्षात ठेवा की यासारखे आकृती तयार करण्यासाठी, आमच्याकडे कॉम्प्युटरने विमानातील सर्व पिक्सेलवर फंक्शनची गणना केली आहे, परंतु आमचे ध्येय अधिक कार्यक्षम अल्गोरिदम शोधणे आहे ज्यासाठी शक्य तितक्या कमी बिंदूंवर फंक्शनची गणना करणे आवश्यक आहे. , फक्त रंगांचे मर्यादित दृश्य आहे, म्हणून बोलायचे आहे.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "लक्षात ठेवा, एका परिमाणात मुख्य अंतर्दृष्टी अशी होती की जर सतत कार्य एका बिंदूवर सकारात्मक असेल आणि दुसर्‍या ठिकाणी नकारात्मक असेल तर, ते कुठेतरी शून्य असले पाहिजे.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "एकूण तीन पूर्ण घड्याळाच्या दिशेने वळणाच्या आसपास वाऱ्यावर येणारे आऊटपुट, रंग इंद्रधनुष्यातून, क्रमाने, लाल ते पुन्हा लाल, आणि नंतर पुन्हा आणि पुन्हा.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "या वळण संख्यांबद्दल लक्षात येण्यासारखी मुख्य गोष्ट म्हणजे जेव्हा तुम्ही मोठ्या पाथांमध्ये पथ एकत्र करता तेव्हा ते छान जोडतात.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "या व्हिडिओचे प्राथमिक लेखक नवीन 3ब्लू1ब्राउन टीम सदस्यांपैकी एक आहेत, श्रीधर रमेश.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/persian/sentence_translations.json b/2018/winding-numbers/persian/sentence_translations.json index 75ab30e43..6b0bce198 100644 --- a/2018/winding-numbers/persian/sentence_translations.json +++ b/2018/winding-numbers/persian/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs. ", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs. ", "translatedText": "ما آن را به دو قسمت تقسیم کردیم، و نیمه‌ای که توجهمان را به آن محدود کردیم، جایی بود که بیرونی‌ترین نقاط آن نشانه‌های متفاوتی داشتند. ", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero. ", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero. ", "translatedText": "به یاد داشته باشید، در یک بعد بینش اصلی این بود که اگر یک تابع پیوسته در یک نقطه مثبت و در نقطه دیگر منفی باشد، جایی در بین آن باید صفر باشد. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again. ", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again. ", "translatedText": "خروجی‌هایی که با آن مواجه می‌شویم، در مجموع سه دور کامل در جهت عقربه‌های ساعت می‌پیچند، رنگ‌ها به ترتیب از قرمز به قرمز و دوباره و دوباره و دوباره در رنگین کمان می‌چرخند. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths. ", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths. ", "translatedText": "نکته اصلی در مورد این اعداد پیچ در پیچ این است که وقتی مسیرها را در مسیرهای بزرگتر ترکیب می کنید، به خوبی جمع می شوند. ", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. ", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching. ", "translatedText": "نویسنده اصلی این ویدیو یکی از جدیدترین اعضای تیم 3blue1brown، سریدار رامش است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/portuguese/sentence_translations.json b/2018/winding-numbers/portuguese/sentence_translations.json index c7b8f490c..66e2729f4 100644 --- a/2018/winding-numbers/portuguese/sentence_translations.json +++ b/2018/winding-numbers/portuguese/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "Nós o dividimos em dois, e a metade para a qual concentramos nossa atenção foi aquela em que os pontos mais externos apresentavam sinais variados.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "Tenha em mente que para criar um diagrama como este, fizemos com que o computador calculasse a função em todos os pixels do plano, mas nosso objetivo é encontrar um algoritmo mais eficiente que exija apenas calcular a função no menor número possível de pontos. , tendo apenas uma visão limitada das cores, por assim dizer.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "Lembre-se, em uma dimensão o principal insight foi que se uma função contínua é positiva em um ponto e negativa em outro, em algum ponto intermediário ela deve ser zero.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "As saídas que encontramos giram em torno de um total de três voltas completas no sentido horário, as cores oscilavam no arco-íris, em ordem, de vermelho para vermelho novamente, e novamente, e novamente.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "A principal coisa a notar sobre esses números sinuosos é que eles somam muito bem quando você combina caminhos em caminhos maiores.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "O autor principal deste vídeo é um dos mais novos membros da equipe 3blue1brown, Sridhar Ramesh.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/russian/sentence_translations.json b/2018/winding-numbers/russian/sentence_translations.json index 28209b859..fb7bbcd2f 100644 --- a/2018/winding-numbers/russian/sentence_translations.json +++ b/2018/winding-numbers/russian/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "Мы разделили его на две части, и та половина, на которой мы сузили свое внимание, была той, где самые крайние точки имели разные знаки.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "Имейте в виду, что для создания такой диаграммы мы заставили компьютер вычислить функцию во всех пикселях плоскости, но наша цель — найти более эффективный алгоритм, который требует вычисления функции только в как можно меньшем количестве точек. , так сказать, имея лишь ограниченное представление о цветах.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "Помните, что в одном измерении основная идея заключалась в том, что если непрерывная функция положительна в одной точке и отрицательна в другой, где-то посередине она должна быть равна нулю.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "Выходные данные, с которыми мы сталкиваемся, совершают в общей сложности три полных оборота по часовой стрелке, цвета менялись по радуге по порядку: снова от красного к красному, а затем снова и снова.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "Главное, что следует отметить в этих извилистых числах, — это то, что они хорошо складываются, когда вы объединяете пути в более крупные пути.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "Основным автором этого видео является один из новых участников команды 3blue1brown Шридхар Рамеш.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/spanish/sentence_translations.json b/2018/winding-numbers/spanish/sentence_translations.json index d164697ad..3f346d469 100644 --- a/2018/winding-numbers/spanish/sentence_translations.json +++ b/2018/winding-numbers/spanish/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "Lo dividimos en dos, y la mitad a la que centramos nuestra atención fue aquella en la que los puntos más externos tenían signos variables.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "Tenga en cuenta que para crear un diagrama como este, hemos hecho que la computadora calcule la función en todos los píxeles del plano, pero nuestro objetivo es encontrar un algoritmo más eficiente que solo requiera calcular la función en la menor cantidad de puntos posible. , teniendo sólo una visión limitada de los colores, por así decirlo.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "Recuerde, en una dimensión la idea principal fue que si una función continua es positiva en un punto y negativa en otro, en algún punto intermedio debe ser cero.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "Las salidas con las que nos encontramos giran alrededor de un total de tres vueltas completas en el sentido de las agujas del reloj, los colores oscilan a través del arco iris, en orden, de rojo a rojo nuevamente, y luego otra vez, y otra vez.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "Lo principal a tener en cuenta sobre estos números sinuosos es que se suman muy bien cuando combinas caminos en caminos más grandes.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "El autor principal de este video es uno de los miembros más nuevos del equipo 3blue1brown, Sridhar Ramesh.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/tamil/sentence_translations.json b/2018/winding-numbers/tamil/sentence_translations.json index acfdedfac..46356526d 100644 --- a/2018/winding-numbers/tamil/sentence_translations.json +++ b/2018/winding-numbers/tamil/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "நாங்கள் அதை இரண்டாகப் பிரித்தோம், எங்கள் கவனத்தைச் சுருக்கிய பாதி வெளிப்புற புள்ளிகள் வெவ்வேறு அறிகுறிகளைக் கொண்டிருந்தது.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "இது போன்ற ஒரு வரைபடத்தை உருவாக்க, விமானத்தில் உள்ள அனைத்து பிக்சல்களிலும் செயல்பாட்டை கணினி கணக்கிட வேண்டும் என்பதை நினைவில் கொள்ளுங்கள், ஆனால் எங்கள் குறிக்கோள் மிகவும் திறமையான அல்காரிதத்தைக் கண்டுபிடிப்பதாகும், இது முடிந்தவரை சில புள்ளிகளில் மட்டுமே செயல்பாட்டைக் கணக்கிட வேண்டும். , பேசுவதற்கு, வண்ணங்களின் வரையறுக்கப்பட்ட பார்வையை மட்டுமே கொண்டுள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "நினைவில் கொள்ளுங்கள், ஒரு பரிமாணத்தில் முக்கிய நுண்ணறிவு என்னவென்றால், தொடர்ச்சியான செயல்பாடு ஒரு கட்டத்தில் நேர்மறையாகவும் மற்றொரு கட்டத்தில் எதிர்மறையாகவும் இருந்தால், இடையில் எங்காவது அது பூஜ்ஜியமாக இருக்க வேண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "மொத்தம் மூன்று முழு கடிகாரத் திசையில் காற்று வீசும் வெளியீடுகள், வண்ணங்கள் வானவில் வழியாகச் சென்றன, வரிசையாக, சிவப்பு நிறத்தில் இருந்து மீண்டும் சிவப்பு, பின்னர் மீண்டும், மீண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "இந்த முறுக்கு எண்களைப் பற்றி கவனிக்க வேண்டிய முக்கிய விஷயம் என்னவென்றால், நீங்கள் பாதைகளை பெரிய பாதைகளாக இணைக்கும்போது அவை நன்றாகச் சேர்கின்றன.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "இந்த வீடியோவின் முதன்மை ஆசிரியர், புதிய 3ப்ளூ1பிரவுன் குழு உறுப்பினர்களில் ஒருவர், ஸ்ரீதர் ரமேஷ்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/telugu/sentence_translations.json b/2018/winding-numbers/telugu/sentence_translations.json index 24c2f4aa5..ec93a3c51 100644 --- a/2018/winding-numbers/telugu/sentence_translations.json +++ b/2018/winding-numbers/telugu/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "మేము దానిని రెండుగా విభజించాము మరియు మేము మా దృష్టిని కుదించిన సగం బయటి పాయింట్లు వివిధ సంకేతాలను కలిగి ఉండేవి.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "ఇలాంటి రేఖాచిత్రాన్ని రూపొందించడానికి, మేము విమానంలోని అన్ని పిక్సెల్‌ల వద్ద ఫంక్షన్‌ను కంప్యూట్ చేసాము, అయితే మా లక్ష్యం మరింత సమర్థవంతమైన అల్గారిథమ్‌ను కనుగొనడం, ఇది ఫంక్షన్‌ను వీలైనంత తక్కువ పాయింట్ల వద్ద మాత్రమే గణించడం అవసరం. , చెప్పాలంటే రంగుల పరిమిత వీక్షణను మాత్రమే కలిగి ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "గుర్తుంచుకోండి, ఒక కోణంలో ప్రధాన అంతర్దృష్టి ఏమిటంటే, నిరంతర ఫంక్షన్ ఒక పాయింట్ వద్ద సానుకూలంగా మరియు మరొక సమయంలో ప్రతికూలంగా ఉంటే, మధ్యలో ఎక్కడో అది సున్నాగా ఉండాలి.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "అవుట్‌పుట్‌లు మొత్తం మూడు సవ్యదిశలో గాలిని చూస్తాము, రంగులు ఇంద్రధనస్సు గుండా తిరుగుతాయి, క్రమంలో, ఎరుపు నుండి ఎరుపుకు మళ్లీ, ఆపై మళ్లీ, మళ్లీ.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "ఈ వైండింగ్ నంబర్‌ల గురించి గమనించవలసిన ప్రధాన విషయం ఏమిటంటే, మీరు పాత్‌లను పెద్ద పాత్‌లుగా కలిపినప్పుడు అవి చక్కగా జోడించబడతాయి.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "ఈ వీడియో యొక్క ప్రాథమిక రచయిత సరికొత్త 3blue1brown టీమ్ సభ్యులలో ఒకరు, శ్రీధర్ రమేష్.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/thai/sentence_translations.json b/2018/winding-numbers/thai/sentence_translations.json index 9bd521655..6eae81c3c 100644 --- a/2018/winding-numbers/thai/sentence_translations.json +++ b/2018/winding-numbers/thai/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs. ", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero. ", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again. ", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths. ", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. ", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/turkish/sentence_translations.json b/2018/winding-numbers/turkish/sentence_translations.json index 2c9da599b..1a0416d95 100644 --- a/2018/winding-numbers/turkish/sentence_translations.json +++ b/2018/winding-numbers/turkish/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "Onu ikiye böldük ve dikkatimizi daralttığımız kısım, en dıştaki noktaların farklı işaretlerin olduğu kısımdı.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "Bunun gibi bir diyagram oluşturmak için, bilgisayarın düzlemdeki tüm piksellerdeki işlevi hesaplamasını sağladığımızı unutmayın; ancak amacımız, işlevi yalnızca mümkün olduğunca az sayıda noktada hesaplamayı gerektiren daha verimli bir algoritma bulmaktır. tabiri caizse yalnızca sınırlı bir renk görüşüne sahip.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "Unutmayın, bir boyuttaki temel anlayış, sürekli bir fonksiyonun bir noktada pozitif, diğer noktada negatif olması durumunda bu ikisinin arasında bir yerde sıfır olması gerektiğiydi.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "Karşılaştığımız çıktılar saat yönünde toplam üç tam dönüş etrafında dönüyor, renkler gökkuşağının içinde sırayla kırmızıdan kırmızıya, sonra tekrar ve tekrar sallanıyordu.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "Bu dolambaçlı sayılarla ilgili dikkat edilmesi gereken en önemli şey, yolları daha büyük yollarla birleştirdiğinizde güzel bir şekilde bir araya gelmeleridir.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "Bu videonun ana yazarı, 3blue1brown ekibinin en yeni üyelerinden biri olan Sridhar Ramesh'tir.", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/urdu/sentence_translations.json b/2018/winding-numbers/urdu/sentence_translations.json index a2144c98e..df82cc933 100644 --- a/2018/winding-numbers/urdu/sentence_translations.json +++ b/2018/winding-numbers/urdu/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs. ", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs. ", "translatedText": "ہم نے اسے دو حصوں میں تقسیم کیا، اور جس آدھے حصے پر ہم نے اپنی توجہ کم کی وہ وہ تھا جہاں سب سے باہری پوائنٹس میں مختلف علامات تھے۔", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero. ", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero. ", "translatedText": "یاد رکھیں، ایک جہت میں بنیادی بصیرت یہ تھی کہ اگر کوئی مسلسل فعل ایک نقطہ پر مثبت اور دوسرے مقام پر منفی ہے، تو اس کے درمیان کہیں صفر ہونا چاہیے۔", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again. ", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again. ", "translatedText": "آؤٹ پٹ جو ہم ہوا میں کل تین مکمل گھڑی کی سمت موڑ کے ارد گرد آتے ہیں، رنگ قوس قزح کے ذریعے، ترتیب سے، سرخ سے دوبارہ سرخ، اور پھر دوبارہ، اور پھر۔", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths. ", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths. ", "translatedText": "ان سمیٹنے والے نمبروں کے بارے میں سب سے اہم بات یہ ہے کہ جب آپ راستوں کو بڑے راستوں میں جوڑتے ہیں تو وہ اچھی طرح سے شامل ہوتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. ", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching. ", "translatedText": "اس ویڈیو کے بنیادی مصنف 3 بلیو 1 براؤن ٹیم کے نئے ارکان میں سے ایک ہیں، سریدھر رمیش۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2018/winding-numbers/vietnamese/sentence_translations.json b/2018/winding-numbers/vietnamese/sentence_translations.json index 5eb28b2a7..979e2100c 100644 --- a/2018/winding-numbers/vietnamese/sentence_translations.json +++ b/2018/winding-numbers/vietnamese/sentence_translations.json @@ -144,7 +144,7 @@ "end": 143.34 }, { - "input": "We split it into two, and the half we narrowed our attention to was the one where the outermost points had varying signs.", + "input": "And then we split it into two, and the half that we narrowed our attention to was the one where the outermost points had varying signs.", "translatedText": "Chúng tôi chia nó thành hai, và nửa mà chúng tôi thu hẹp sự chú ý là phần mà các điểm ngoài cùng có các dấu hiệu khác nhau.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 430.0 }, { - "input": "Keep in mind that to create a diagram like this, we've had the computer compute the function at all the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", + "input": "Well keep in mind that to create a diagram like this, we've had the computer compute the function at all of the pixels on the plane, but our goal is to find a more efficient algorithm that only requires computing the function at as few points as possible, only having a limited view of the colors, so to speak.", "translatedText": "Hãy nhớ rằng để tạo một sơ đồ như thế này, chúng ta đã dùng máy tính tính toán hàm ở tất cả các pixel trên mặt phẳng, nhưng mục tiêu của chúng ta là tìm ra một thuật toán hiệu quả hơn mà chỉ yêu cầu tính toán hàm ở càng ít điểm càng tốt , có thể nói là chỉ có một cái nhìn hạn chế về màu sắc.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 457.64 }, { - "input": "Remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, somewhere in between it must be zero.", + "input": "Now remember, in one dimension the main insight was that if a continuous function is positive at one point and negative at another, then somewhere in between it must be zero.", "translatedText": "Hãy nhớ rằng, trong một chiều, cái nhìn sâu sắc chính là nếu một hàm liên tục dương tại một điểm và âm tại một điểm khác, thì đâu đó ở giữa nó phải bằng 0.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 933.68 }, { - "input": "The outputs we come across wind around a total of three full clockwise turns, the colors swung through the rainbow, in order, from red to red again, and then again, and again.", + "input": "The outputs that we come across wind around a total of three full clockwise turns. The colors swung through the rainbow, ROYGBIV, in order, from red to red again, and then again, and again.", "translatedText": "Các kết quả đầu ra mà chúng ta gặp phải xoay quanh tổng cộng ba vòng theo chiều kim đồng hồ, các màu sắc chuyển động qua cầu vồng, theo thứ tự, từ đỏ sang đỏ, rồi lại, rồi lại.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 1004.04 }, { - "input": "The main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", + "input": "Alright, so the main thing to notice about these winding numbers is that they add up nicely when you combine paths into bigger paths.", "translatedText": "Điều chính cần chú ý về những con số quanh co này là chúng cộng lại rất đẹp khi bạn kết hợp các đường dẫn thành các đường dẫn lớn hơn.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1369.6 }, { - "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh.", + "input": "The primary author of this video is one of the newest 3blue1brown team members, Sridhar Ramesh. Thank you for watching.", "translatedText": "Tác giả chính của video này là một trong những thành viên mới nhất của nhóm 3blue1Brown, Sridhar Ramesh.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/arabic/sentence_translations.json b/2019/bayes-theorem/arabic/sentence_translations.json index 33b463ba7..190f99cd0 100644 --- a/2019/bayes-theorem/arabic/sentence_translations.json +++ b/2019/bayes-theorem/arabic/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence. ", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence. ", "translatedText": "تذكر أول رقم ذي صلة استخدمناه، وهو الاحتمال الذي تحمله الفرضية قبل النظر في أي من تلك الأدلة الجديدة. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", "translatedText": "كيفما تكتبها، أنا في الواقع أشجعك على عدم محاولة حفظ الصيغة، ولكن بدلا من ذلك رسم هذا الرسم البياني حسب الحاجة. ", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", "translatedText": "الآن، إذا كنت تعتقد أن المزارع من المرجح أن يتناسب مع الأدلة كأمين مكتبة، فإن النسبة لن تتغير، وهو ما يجب أن يكون منطقيًا، أليس كذلك؟ ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error. ", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error. ", "translatedText": "وبدلاً من ذلك، إذا قيل للمشاركين أن هناك 100 شخص ينطبق عليهم هذا الوصف، ثم طُلب منهم تقدير عدد صرافين البنوك من بين هؤلاء المائة، وكم عدد صرافين البنوك الناشطين في الحركة النسوية، فلن يرتكب أحد الخطأ. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", "translatedText": "غالبًا ما يفكر الناس في الاحتمالية على أنها دراسة عدم اليقين، وهذا بالطبع هو كيفية تطبيقها في العلوم، ولكن الرياضيات الفعلية للاحتمال، حيث تأتي جميع الصيغ، هي مجرد رياضيات النسب، وفي هذا السياق ننتقل إلى الهندسة مفيدة للغاية. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/bengali/sentence_translations.json b/2019/bayes-theorem/bengali/sentence_translations.json index a4a289de0..230c608e9 100644 --- a/2019/bayes-theorem/bengali/sentence_translations.json +++ b/2019/bayes-theorem/bengali/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence. ", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence. ", "translatedText": "আমরা যে প্রথম প্রাসঙ্গিক সংখ্যাটি ব্যবহার করেছি তা মনে রাখবেন, সেই নতুন প্রমাণের কোনোটি বিবেচনা করার আগে হাইপোথিসিস ধারণ করার সম্ভাবনা।", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", "translatedText": "যাইহোক আপনি এটি লিখুন, আমি আসলে আপনাকে উত্সাহিত করি সূত্রটি মুখস্থ করার চেষ্টা না করে বরং প্রয়োজন অনুসারে এই চিত্রটি আঁকতে।", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", "translatedText": "এখন আপনি যদি মনে করেন যে একজন কৃষক গ্রন্থাগারিকের মতো প্রমাণের সাথে মানানসই হওয়ার সম্ভাবনা রয়েছে, তাহলে অনুপাতটি পরিবর্তিত হয় না, যার অর্থ হওয়া উচিত, তাই না? ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error. ", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error. ", "translatedText": "পরিবর্তে, যদি অংশগ্রহণকারীদের বলা হয় যে এই বর্ণনার সাথে মানানসই 100 জন লোক আছে, এবং তারপর অনুমান করতে বলা হয় যে এই 100 জনের মধ্যে কতজন ব্যাঙ্ক টেলার, এবং কতজন ব্যাঙ্ক টেলার নারীবাদী আন্দোলনে সক্রিয়, কেউ ভুল করে না।", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", "translatedText": "লোকেরা প্রায়শই সম্ভাব্যতাকে অনিশ্চয়তার অধ্যয়ন হিসাবে মনে করে, এবং এটি অবশ্যই বিজ্ঞানে কীভাবে প্রয়োগ করা হয়, তবে সম্ভাব্যতার প্রকৃত গণিত, যেখানে সমস্ত সূত্র আসে, কেবলমাত্র অনুপাতের গণিত, এবং সেই প্রসঙ্গে মোড় নেয় জ্যামিতি অত্যন্ত সহায়ক।", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/chinese/sentence_translations.json b/2019/bayes-theorem/chinese/sentence_translations.json index 67c867011..0331ed597 100644 --- a/2019/bayes-theorem/chinese/sentence_translations.json +++ b/2019/bayes-theorem/chinese/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "记住我们使用的第一个相关数字，即在 考虑任何新证据之前假设成立的概率。", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "不管你怎么写，我实际上鼓励你不要尝试 记住公式，而是根据需要画出这个图表。", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "现在，如果您认为农民与图书管理员一样有可能符合证 据，那么比例就不会改变，这应该是有道理的，对吗？", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "相反，如果参与者被告知有 100 人符合这种描述，然后 要求估计这 100 人中有多少人是银行出纳员，以及有多 少银行出纳员活跃于女权主义运动，那么没有人会犯错误。", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "人们经常认为概率是对不确定性的研究，这 当然就是它在科学中的应用方式，但所有公 式的来源，概率的实际数学只是比例的数学 ，在这种情况下转向几何学非常有帮助。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/french/sentence_translations.json b/2019/bayes-theorem/french/sentence_translations.json index 2e638d613..7d6e1d4fc 100644 --- a/2019/bayes-theorem/french/sentence_translations.json +++ b/2019/bayes-theorem/french/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "Rappelez-vous le premier nombre pertinent que nous avons utilisé, la probabilité que l'hypothèse soit vérifiée, avant de considérer l'une de ces nouvelles preuves.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "Quelle que soit la manière dont vous l'écrivez, je vous encourage en fait à ne pas essayer de mémoriser la formule, mais plutôt à dessiner ce diagramme si nécessaire.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "Maintenant, si vous pensez qu’un agriculteur est tout aussi susceptible de correspondre aux preuves qu’un bibliothécaire, alors la proportion ne change pas, ce qui devrait être logique, n’est-ce pas?", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "Au lieu de cela, si l’on disait aux participants qu’il y a 100 personnes qui correspondent à cette description, et qu’on leur demandait ensuite d’estimer combien de ces 100 sont des caissiers de banque et combien sont des caissiers de banque actifs dans le mouvement féministe, personne ne commettrait d’erreur.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "Les gens pensent souvent aux probabilités comme à l'étude de l'incertitude, et c'est bien sûr ainsi qu'elles sont appliquées en science, mais les mathématiques réelles des probabilités, d'où proviennent toutes les formules, ne sont que des mathématiques de proportions, et dans ce contexte, se tourner vers la géométrie est extrêmement utile.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/gujarati/sentence_translations.json b/2019/bayes-theorem/gujarati/sentence_translations.json index b027e7812..147d5f11d 100644 --- a/2019/bayes-theorem/gujarati/sentence_translations.json +++ b/2019/bayes-theorem/gujarati/sentence_translations.json @@ -273,7 +273,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "આપણે ઉપયોગમાં લીધેલા પ્રથમ સંબંધિત નંબરને યાદ કરો , તે કોઈપણ નવા પુરાવાને ધ્યાનમાં લેતા પહેલા, પૂર્વધારણા ની સંભાવના કેટલી છે.", "n_reviews": 1, "start": 320.22, @@ -455,7 +455,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "ભલે તમે તેને લખો પણ , હું ખરેખર તમને સૂત્રને યાદ રાખવાનો પ્રયાસ ન કરવા માટે પ્રોત્સાહિત કરું છું, તેના બદલે જરૂર મુજબ આ રેખાકૃતિ દોરો.", "n_reviews": 1, "start": 517.08, @@ -497,7 +497,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "હવે જો તમને લાગતું હોય કે ખેડૂત ગ્રંથપાલની જેમ જ પુરાવામાં ફિટ થવાની શક્યતા છે, તો પ્રમાણ બદલાતું નથી, જે સમજ માં આવે તેવું છે બરાબર?", "n_reviews": 1, "start": 577.64, @@ -623,7 +623,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "તેના બદલે, જો સહભાગીઓને કહેવામાં આવે કે આ વર્ણનને અનુરૂપ 100 લોકો છે, અને પછી તે 100 માંથી કેટલા બેંક ટેલર છે અને કેટલા બેંક ટેલર નારીવાદી ચળવળમાં સક્રિય છે તે અનુમાન કરવા માટે કહેવામાં આવે, તો કોઈ ભૂલ કરતું નથી.", "n_reviews": 1, "start": 694.96, @@ -658,7 +658,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "લોકો ઘણીવાર સંભાવના વિશે અનિશ્ચિતતાના અભ્યાસ તરીકે વિચારે છે, અને તે અલબત્ત વિજ્ઞાનમાં તે એવી રીતે લાગુ થાય છે, પરંતુ સંભવિતતાનું વાસ્તવિક ગણિત, જ્યાંથી તમામ સૂત્રો આવે છે, તે માત્ર પ્રમાણનું ગણિત છે, અને તે સંદર્ભમાં.ભૂમિતિ તરફ જવું અત્યંત મદદરૂપ છે.", "n_reviews": 1, "start": 745.22, diff --git a/2019/bayes-theorem/hindi/sentence_translations.json b/2019/bayes-theorem/hindi/sentence_translations.json index d0b16fb10..c948f674a 100644 --- a/2019/bayes-theorem/hindi/sentence_translations.json +++ b/2019/bayes-theorem/hindi/sentence_translations.json @@ -273,7 +273,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "हमारे द्वारा उपयोग की गई पहली प्रासंगिक संख्या को याद रखें, उस नए साक्ष्य पर विचार करने से पहले परिकल्पना की संभावना क्या है।", "n_reviews": 0, "start": 320.22, @@ -455,7 +455,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "हालाँकि आप इसे लिखते हैं, मैं वास्तव में आपको सूत्र को याद करने का प्रयास न करने के लिए प्रोत्साहित करता हूँ, बल्कि आवश्यकतानुसार इस आरेख को बनाने के लिए प्रोत्साहित करता हूँ।", "n_reviews": 0, "start": 517.08, @@ -497,7 +497,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "अब यदि आप सोचते हैं कि एक किसान भी लाइब्रेरियन के समान ही साक्ष्यों में फिट बैठता है, तो अनुपात नहीं बदलता है, जिसका कोई मतलब होना चाहिए, है ना?", "n_reviews": 0, "start": 577.64, @@ -623,7 +623,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "इसके बजाय, यदि प्रतिभागियों को बताया गया कि 100 लोग हैं जो इस विवरण में फिट बैठते हैं, और फिर यह अनुमान लगाने के लिए कहा गया कि उन 100 में से कितने बैंक टेलर हैं, और कितने बैंक टेलर नारीवादी आंदोलन में सक्रिय हैं, तो कोई भी गलती नहीं करता है।", "n_reviews": 0, "start": 694.96, @@ -658,7 +658,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "लोग अक्सर संभाव्यता को अनिश्चितता के अध्ययन के रूप में सोचते हैं, और निश्चित रूप से इसे विज्ञान में इसी तरह लागू किया जाता है, लेकिन संभाव्यता का वास्तविक गणित, जहां से सभी सूत्र आते हैं, केवल अनुपात का गणित है, और उस संदर्भ में ज्यामिति अत्यधिक सहायक है।", "n_reviews": 0, "start": 745.22, diff --git a/2019/bayes-theorem/indonesian/sentence_translations.json b/2019/bayes-theorem/indonesian/sentence_translations.json index 6ce023d2d..1ce541198 100644 --- a/2019/bayes-theorem/indonesian/sentence_translations.json +++ b/2019/bayes-theorem/indonesian/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "Ingat angka relevan pertama yang kami gunakan, probabilitas hipotesis tersebut berlaku sebelum mempertimbangkan bukti baru tersebut.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "Bagaimana pun Anda menulisnya, saya sebenarnya menganjurkan Anda untuk tidak mencoba menghafal rumusnya, melainkan menggambar diagram ini sesuai kebutuhan.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "Sekarang jika Anda berpikir seorang petani mempunyai kemungkinan yang sama dengan seorang pustakawan, maka proporsinya tidak akan berubah, dan ini masuk akal, bukan?", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "Sebaliknya, jika peserta diberitahu bahwa ada 100 orang yang sesuai dengan deskripsi ini, dan kemudian diminta memperkirakan berapa banyak dari 100 orang tersebut yang merupakan teller bank, dan berapa banyak teller bank yang aktif dalam gerakan feminis, tidak ada yang melakukan kesalahan.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "Orang sering menganggap probabilitas sebagai ilmu yang mempelajari ketidakpastian, dan tentu saja hal ini diterapkan dalam sains, namun perhitungan probabilitas yang sebenarnya, yang merupakan asal mula semua rumus, hanyalah perhitungan proporsi, dan dalam konteks tersebut beralih ke probabilitas. geometri sangat membantu.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/italian/sentence_translations.json b/2019/bayes-theorem/italian/sentence_translations.json index d71ab2d2f..6dbf2e59a 100644 --- a/2019/bayes-theorem/italian/sentence_translations.json +++ b/2019/bayes-theorem/italian/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "Ricorda il primo numero rilevante che abbiamo usato, la probabilità che l'ipotesi sia valida prima di considerare qualsiasi nuova prova.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "Comunque lo scrivi, in realtà ti incoraggio a non provare a memorizzare la formula, ma a disegnare invece questo diagramma secondo necessità.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "Ora, se pensi che un agricoltore abbia le stesse probabilità di soddisfare le prove di un bibliotecario, allora la proporzione non cambia, il che dovrebbe avere senso, giusto?", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "Invece, se ai partecipanti venisse detto che ci sono 100 persone che corrispondono a questa descrizione, e poi si chiedesse di stimare quanti di questi 100 sono cassieri di banca, e quanti sono cassieri di banca attivi nel movimento femminista, nessuno commetterebbe l’errore.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "Le persone spesso pensano alla probabilità come allo studio dell'incertezza, e questo è ovviamente il modo in cui viene applicata nella scienza, ma la matematica effettiva della probabilità, da cui provengono tutte le formule, è solo la matematica delle proporzioni, e in quel contesto rivolgersi a la geometria è estremamente utile.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/japanese/sentence_translations.json b/2019/bayes-theorem/japanese/sentence_translations.json index 96a619988..5e7728d61 100644 --- a/2019/bayes-theorem/japanese/sentence_translations.json +++ b/2019/bayes-theorem/japanese/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "新しい証拠を検討する前に、最初に使用した関連する数 値、つまり仮説が成立する確率を思い出してください。", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "どのように書いても、式を暗記しようとするのではな く、必要に応じてこの図を描くことをお勧めします。", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "ここで、農家が図書館員と同じくらい証拠に適合する可能性が高いと考 えるなら、その割合は変わらないということは、当然のことでしょう?", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "代わりに、参加者に、この説明に当てはまる人が 100 人いると伝え、その 100 人のうち何人が銀行窓口係であり、何人がフェミニスト運動に積極的 な銀行窓口係であるかを推定するよう依頼した場合、誰も間違いを犯しません。", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "確率は不確実性の研究であるとよく考えられており、もちろん それが科学での応用方法でもありますが、すべての式の由来 となる実際の確率の数学は単なる比率の数学であり、その文 脈では次のようになります。 幾何学は非常に役に立ちます。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/korean/sentence_translations.json b/2019/bayes-theorem/korean/sentence_translations.json index 73fb976b8..b14c780dc 100644 --- a/2019/bayes-theorem/korean/sentence_translations.json +++ b/2019/bayes-theorem/korean/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "우리가 사용한 첫 번째 관련 숫자, 즉 새로운 증거를 고려하기 전에 가설이 유지될 확률을 기억하십시오.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "어떻게 작성하든 공식을 외우려고 하지 말고 대신 필요에 따라 이 다이어그램을 그려 보시기 바랍니다.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "이제 농부가 사서와 마찬가지로 증거에 적합할 가능성이 높다고 생각한다면 그 비율은 변하지 않습니다. 그게 말이 되겠죠?", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "대신 참가자들에게 이 설명에 맞는 사람이 100명 있다는 말을 듣고 그 100명 중 은행원이 몇 명인지, 페미니스트 운동에 적극적으로 참여하는 은행원이 몇 명인지 추산해 보라고 하면 아무도 오류를 범하지 않습니다.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "사람들은 확률을 불확실성에 대한 연구라고 생각하는 경우가 많습니다. 물론 이것이 과학에 적용되는 방식이지만, 모든 공식의 근원이 되는 실제 확률 수학은 단지 비율의 수학일 뿐이며 그런 맥락에서 다음과 같이 변합니다. 기하학은 매우 도움이 됩니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/marathi/sentence_translations.json b/2019/bayes-theorem/marathi/sentence_translations.json index f1e9921c2..61043dba3 100644 --- a/2019/bayes-theorem/marathi/sentence_translations.json +++ b/2019/bayes-theorem/marathi/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "आम्ही वापरलेला पहिला संबंधित क्रमांक लक्षात ठेवा, त्या कोणत्याही नवीन पुराव्याचा विचार करण्यापूर्वी गृहीतकाची संभाव्यता.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "तुम्ही ते लिहिता तरी, मी तुम्हाला सूत्र लक्षात ठेवण्याचा प्रयत्न करू नका, तर त्याऐवजी आवश्यकतेनुसार हा आकृती काढण्यासाठी प्रोत्साहित करतो.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "आता जर तुम्हाला वाटत असेल की एखादा शेतकरी ग्रंथपाल म्हणून पुराव्यात बसण्याची शक्यता आहे, तर प्रमाण बदलत नाही, ज्याला अर्थ आहे, बरोबर?", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "त्याऐवजी, या वर्णनाशी जुळणारे 100 लोक आहेत असे सहभागींना सांगितले गेले आणि नंतर त्या 100 पैकी किती बँक टेलर आहेत आणि किती बँक टेलर स्त्रीवादी चळवळीत सक्रिय आहेत याचा अंदाज लावायला सांगितले तर कोणीही चूक करत नाही.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "लोक बहुधा संभाव्यतेचा अनिश्चिततेचा अभ्यास म्हणून विचार करतात, आणि अर्थातच ते विज्ञानात कसे लागू केले जाते, परंतु संभाव्यतेचे वास्तविक गणित, जिथे सर्व सूत्रे येतात, फक्त प्रमाणांचे गणित असते आणि त्या संदर्भात वळणे. भूमिती अत्यंत उपयुक्त आहे.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/persian/sentence_translations.json b/2019/bayes-theorem/persian/sentence_translations.json index 0d5fb3761..70c0daff3 100644 --- a/2019/bayes-theorem/persian/sentence_translations.json +++ b/2019/bayes-theorem/persian/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence. ", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence. ", "translatedText": "اولین عدد مربوطه ای را که استفاده کردیم، به خاطر بسپارید، احتمال وجود این فرضیه قبل از در نظر گرفتن هر یک از آن شواهد جدید. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", "translatedText": "هر طور که آن را بنویسید، من در واقع شما را تشویق می کنم که سعی نکنید فرمول را حفظ کنید، بلکه در صورت نیاز این نمودار را ترسیم کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", "translatedText": "حالا اگر فکر می کنید که یک کشاورز به اندازه یک کتابدار به احتمال زیاد شواهد را دارد، پس نسبت تغییر نمی کند، که باید منطقی باشد، درست است؟ و شواهد باورهای شما را تغییر نمی دهد. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error. ", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error. ", "translatedText": "در عوض، اگر به شرکت کنندگان گفته شود که 100 نفر هستند که با این توصیف مطابقت دارند، و سپس از آنها خواسته شود که تخمین بزنند که از این 100 عابر بانک، و چند نفر عابر بانک فعال در جنبش فمینیستی هستند، هیچ کس اشتباه نمی کند. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", "translatedText": "مردم اغلب در مورد احتمال به عنوان مطالعه عدم قطعیت فکر می کنند، و البته این روشی است که در علم به کار می رود، اما ریاضی واقعی احتمال، جایی که همه فرمول ها از آنجا می آیند، فقط ریاضی نسبت ها است، و در این زمینه به هندسه بسیار مفید است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/portuguese/sentence_translations.json b/2019/bayes-theorem/portuguese/sentence_translations.json index e4e733462..e790ff111 100644 --- a/2019/bayes-theorem/portuguese/sentence_translations.json +++ b/2019/bayes-theorem/portuguese/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "Lembre-se do primeiro número relevante que usamos, a probabilidade de a hipótese ser válida antes de considerar qualquer uma dessas novas evidências.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "Seja como for que você o escreva, eu realmente o encorajo a não tentar memorizar a fórmula, mas sim a desenhar este diagrama conforme necessário.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "Agora, se você acha que um agricultor tem tanta probabilidade de se enquadrar nas evidências quanto um bibliotecário, então a proporção não muda, o que deveria fazer sentido, certo?", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "Em vez disso, se fosse dito aos participantes que há 100 pessoas que se enquadram nesta descrição, e depois lhes fosse pedido que estimassem quantos desses 100 são caixas de banco, e quantos são caixas de banco activos no movimento feminista, ninguém cometeria o erro.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "As pessoas muitas vezes pensam na probabilidade como sendo o estudo da incerteza, e é claro que é assim que ela é aplicada na ciência, mas a verdadeira matemática da probabilidade, de onde vêm todas as fórmulas, é apenas a matemática das proporções e, nesse contexto, voltando-nos para a geometria é extremamente útil.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/russian/sentence_translations.json b/2019/bayes-theorem/russian/sentence_translations.json index 677d0231d..b2cf0dd70 100644 --- a/2019/bayes-theorem/russian/sentence_translations.json +++ b/2019/bayes-theorem/russian/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "Вспомните первое подходящее число, которое мы использовали, — вероятность того, что гипотеза справедлива до рассмотрения любого из этих новых доказательств.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "Как бы вы это ни написали, я на самом деле советую вам не пытаться запомнить формулу, а вместо этого рисовать эту диаграмму по мере необходимости.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "Теперь, если вы думаете, что фермер с такой же вероятностью соответствует доказательствам, как и библиотекарь, тогда пропорция не изменится, что должно иметь смысл, не так ли?", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "Вместо этого, если участникам сказать, что есть 100 человек, которые подходят под это описание, а затем попросить оценить, сколько из этих 100 являются банковскими кассирами и сколько банковских кассиров активно участвуют в феминистском движении, никто не допустит ошибки.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "Люди часто думают о вероятности как об изучении неопределенности, и именно так она и применяется в науке, но реальная математика вероятностей, откуда берутся все формулы, — это всего лишь математика пропорций, и в этом контексте обращение к геометрия чрезвычайно полезна.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/tamil/sentence_translations.json b/2019/bayes-theorem/tamil/sentence_translations.json index 0f288ee26..e0e116517 100644 --- a/2019/bayes-theorem/tamil/sentence_translations.json +++ b/2019/bayes-theorem/tamil/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence. ", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence. ", "translatedText": "நாங்கள் பயன்படுத்திய முதல் தொடர்புடைய எண்ணை நினைவில் கொள்ளுங்கள், அந்த புதிய சான்றுகளில் ஏதேனும் ஒன்றைக் கருத்தில் கொள்வதற்கு முன் கருதுகோள் வைத்திருக்கும் நிகழ்தகவு. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", "translatedText": "நீங்கள் எப்படி எழுதினாலும், சூத்திரத்தை மனப்பாடம் செய்ய முயற்சிக்க வேண்டாம், மாறாக இந்த வரைபடத்தை தேவைக்கேற்ப வரையுமாறு நான் உங்களை ஊக்குவிக்கிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", "translatedText": "இப்போது நீங்கள் ஒரு விவசாயிக்கு நூலகர் போன்ற சான்றுகள் பொருந்தக்கூடும் என்று நீங்கள் நினைத்தால், விகிதாச்சாரம் மாறாது, அது அர்த்தமுள்ளதாக இருக்க வேண்டும், இல்லையா? ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error. ", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error. ", "translatedText": "அதற்குப் பதிலாக, பங்கேற்பாளர்களிடம் இந்த விளக்கத்திற்குப் பொருந்தக்கூடிய 100 பேர் இருப்பதாகக் கூறப்பட்டால், அந்த 100 பேரில் எத்தனை பேர் வங்கிக் கணக்குத் தருபவர்கள் என்றும், பெண்ணிய இயக்கத்தில் செயல்படும் வங்கிச் சொல்பவர்கள் எத்தனை பேர் என்பதைக் கணக்கிடுமாறும் கேட்டால், யாரும் தவறில்லை. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", "translatedText": "நிகழ்தகவை நிச்சயமற்ற தன்மையின் ஆய்வு என்று மக்கள் அடிக்கடி நினைக்கிறார்கள், அது அறிவியலில் எவ்வாறு பயன்படுத்தப்படுகிறது, ஆனால் நிகழ்தகவின் உண்மையான கணிதம், எல்லா சூத்திரங்களும் எங்கிருந்து வருகின்றன என்பது விகிதாச்சாரத்தின் கணிதம் மட்டுமே, மேலும் அந்தச் சூழலில் மாறுகிறது. வடிவியல் மிகவும் பயனுள்ளதாக இருக்கிறது. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/telugu/sentence_translations.json b/2019/bayes-theorem/telugu/sentence_translations.json index 88382eba6..59d922730 100644 --- a/2019/bayes-theorem/telugu/sentence_translations.json +++ b/2019/bayes-theorem/telugu/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence. ", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence. ", "translatedText": "మేము ఉపయోగించిన మొదటి సంబంధిత సంఖ్యను గుర్తుంచుకోండి, ఆ కొత్త సాక్ష్యాలను పరిగణనలోకి తీసుకునే ముందు పరికల్పన కలిగి ఉన్న సంభావ్యత. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", "translatedText": "అయితే మీరు దీన్ని వ్రాసినా, సూత్రాన్ని గుర్తుంచుకోవడానికి ప్రయత్నించవద్దని నేను మిమ్మల్ని ప్రోత్సహిస్తున్నాను, బదులుగా అవసరమైన విధంగా ఈ రేఖాచిత్రాన్ని గీయండి. ", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", "translatedText": "ఇప్పుడు మీరు ఒక రైతు లైబ్రేరియన్‌గా సాక్ష్యాధారాలకు సరిపోయే అవకాశం ఉందని మీరు అనుకుంటే, నిష్పత్తి మారదు, ఇది అర్ధమే, సరియైనదా? ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error. ", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error. ", "translatedText": "బదులుగా, ఈ వివరణకు సరిపోయే 100 మంది వ్యక్తులు ఉన్నారని పాల్గొనేవారికి చెప్పబడి, ఆ 100 మందిలో ఎంత మంది బ్యాంక్ టెల్లర్లు ఉన్నారో మరియు స్త్రీవాద ఉద్యమంలో ఎంత మంది బ్యాంక్ టెల్లర్లు చురుకుగా ఉన్నారో అంచనా వేయమని అడిగితే, ఎవరూ తప్పు చేయరు. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", "translatedText": "ప్రజలు తరచుగా సంభావ్యత గురించి అనిశ్చితి యొక్క అధ్యయనం అని అనుకుంటారు, మరియు అది సైన్స్‌లో ఎలా వర్తింపజేయబడుతుంది, అయితే సంభావ్యత యొక్క వాస్తవ గణితము, అన్ని సూత్రాలు ఎక్కడ నుండి వచ్చాయి, ఇది కేవలం నిష్పత్తుల గణితమే మరియు ఆ సందర్భంలో మారుతుంది. జ్యామితి చాలా సహాయకారిగా ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/thai/sentence_translations.json b/2019/bayes-theorem/thai/sentence_translations.json index 43286b212..b8f17e76e 100644 --- a/2019/bayes-theorem/thai/sentence_translations.json +++ b/2019/bayes-theorem/thai/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence. ", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error. ", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/turkish/sentence_translations.json b/2019/bayes-theorem/turkish/sentence_translations.json index 6fdf04676..2eb007269 100644 --- a/2019/bayes-theorem/turkish/sentence_translations.json +++ b/2019/bayes-theorem/turkish/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "Kullandığımız ilk ilgili sayıyı, yani yeni kanıtlardan herhangi birini dikkate almadan önce hipotezin geçerli olma olasılığını hatırlayın.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "Nasıl yazarsanız yazın, aslında formülü ezberlememenizi, bunun yerine bu diyagramı gerektiği gibi çizmenizi tavsiye ederim.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "Şimdi eğer bir çiftçinin de bir kütüphaneci gibi kanıtlara uyma ihtimalinin yüksek olduğunu düşünüyorsanız o zaman oran değişmiyor, bu mantıklı olmalı, değil mi?", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "Bunun yerine, katılımcılara bu tanıma uyan 100 kişinin olduğu söylense ve bu 100 kişiden kaçının banka gişe memuru olduğunu ve kaçının feminist harekette aktif banka gişe memuru olduğunu tahmin etmeleri istense, kimse hata yapmaz.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "İnsanlar genellikle olasılığın belirsizliğin incelenmesi olduğunu düşünürler ve bilimde de bu şekilde uygulanır, ancak tüm formüllerin geldiği gerçek olasılık matematiği sadece oranların matematiğidir ve bu bağlamda geometri son derece faydalıdır.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/ukrainian/sentence_translations.json b/2019/bayes-theorem/ukrainian/sentence_translations.json index a3fd16efd..8f0461859 100644 --- a/2019/bayes-theorem/ukrainian/sentence_translations.json +++ b/2019/bayes-theorem/ukrainian/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "Запам'ятайте перше релевантне число, яке ми використали, ймовірність того, що гіпотеза справедлива, перш ніж розглядати будь-які з цих нових доказів.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "Як би ви це не писали, я насправді заохочую вас не намагатися запам’ятати формулу, а натомість намалювати цю діаграму за потреби.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "Тепер, якщо ви думаєте, що фермер з такою ж імовірністю відповідатиме доказам, як і бібліотекар, то пропорція не зміниться, що мало б мати сенс, чи не так?", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "Натомість, якщо учасникам сказали, що є 100 осіб, які відповідають цьому опису, а потім попросили оцінити, скільки з цих 100 банківських касирів і скільки банківських касирів є активними у феміністичному русі, ніхто не припустився помилки.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "Люди часто думають про ймовірність як про дослідження невизначеності, і, звичайно, це те, як це застосовується в науці, але фактична математика ймовірності, звідки походять усі формули, є просто математикою пропорцій, і в цьому контексті звертаючись до геометрія надзвичайно корисна.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/urdu/sentence_translations.json b/2019/bayes-theorem/urdu/sentence_translations.json index adaf2d2d6..369708468 100644 --- a/2019/bayes-theorem/urdu/sentence_translations.json +++ b/2019/bayes-theorem/urdu/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence. ", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence. ", "translatedText": "پہلے متعلقہ نمبر کو یاد رکھیں جو ہم نے استعمال کیا تھا، اس امکان کا جو مفروضہ اس نئے ثبوت میں سے کسی پر غور کرنے سے پہلے رکھتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed. ", "translatedText": "بہر حال آپ اسے لکھتے ہیں، میں درحقیقت آپ کی حوصلہ افزائی کرتا ہوں کہ فارمولہ کو یاد کرنے کی کوشش نہ کریں، بلکہ ضرورت کے مطابق اس خاکہ کو کھینچیں۔", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right? ", "translatedText": "اب اگر آپ کو لگتا ہے کہ ایک کسان کا لائبریرین کی طرح شواہد پر فٹ ہونے کا امکان ہے، تو تناسب تبدیل نہیں ہوتا، جس کا مطلب ہونا چاہیے، ٹھیک ہے؟ اور ثبوت آپ کے عقائد کو تبدیل نہیں کرتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error. ", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error. ", "translatedText": "اس کے بجائے، اگر شرکاء کو بتایا جائے کہ 100 لوگ ہیں جو اس تفصیل کے مطابق ہیں، اور پھر اندازہ لگانے کے لیے کہا جائے کہ ان 100 میں سے کتنے بینک ٹیلر ہیں، اور کتنے بینک ٹیلر ہیں جو حقوق نسواں کی تحریک میں سرگرم ہیں، کوئی بھی غلطی نہیں کرتا۔", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful. ", "translatedText": "لوگ اکثر امکان کے بارے میں سوچتے ہیں کہ یہ غیر یقینی صورتحال کا مطالعہ ہے، اور یقیناً اس کا سائنس میں اطلاق کیا جاتا ہے، لیکن امکان کی اصل ریاضی، جہاں سے تمام فارمولے آتے ہیں، صرف تناسب کی ریاضی ہے، اور اس تناظر میں جیومیٹری بہت مددگار ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/bayes-theorem/vietnamese/sentence_translations.json b/2019/bayes-theorem/vietnamese/sentence_translations.json index 0fc8d1b5e..08b95cc11 100644 --- a/2019/bayes-theorem/vietnamese/sentence_translations.json +++ b/2019/bayes-theorem/vietnamese/sentence_translations.json @@ -312,7 +312,7 @@ "end": 318.96 }, { - "input": "Remember the first relevant number we used, the probability that the hypothesis holds before considering any of that new evidence.", + "input": "Now remember the first relevant number we used, it was the probability that the hypothesis holds before considering any of that new evidence.", "translatedText": "Hãy nhớ con số liên quan đầu tiên mà chúng ta đã sử dụng, xác suất mà giả thuyết đó có trước khi xem xét bất kỳ bằng chứng mới nào.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 516.34 }, { - "input": "However you write it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", + "input": "However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.", "translatedText": "Dù bạn viết nó như thế nào, tôi thực sự khuyến khích bạn không nên cố gắng ghi nhớ công thức mà thay vào đó hãy vẽ sơ đồ này nếu cần.", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 576.94 }, { - "input": "Now if you think a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", + "input": "Now, if you happen to think that a farmer is just as likely to fit the evidence as a librarian, then the proportion doesn't change, which should make sense, right?", "translatedText": "Bây giờ nếu bạn cho rằng một người nông dân cũng có khả năng phù hợp với bằng chứng như một thủ thư, thì tỷ lệ đó sẽ không thay đổi, điều này có lý, phải không?", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 694.1 }, { - "input": "Instead, if participants were told that there are 100 people who fit this description, and then asked to estimate how many of those 100 are bank tellers, and how many are bank tellers active in the feminist movement, nobody makes the error.", + "input": "Instead, if participants were told that there are 100 people who fit this description, and then they're asked to estimate how many of those 100 are bank tellers, and how many of them are bank tellers who are active in the feminist movement, nobody makes the error.", "translatedText": "Thay vào đó, nếu người tham gia được thông báo rằng có 100 người phù hợp với mô tả này, sau đó được yêu cầu ước tính bao nhiêu trong số 100 người đó là giao dịch viên ngân hàng và bao nhiêu giao dịch viên ngân hàng hoạt động tích cực trong phong trào nữ quyền, thì không ai mắc lỗi cả.", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 744.04 }, { - "input": "People often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", + "input": "You see, people often think about probability as being the study of uncertainty, and that is of course how it's applied in science, but the actual math of probability, where all the formulas come from, is just the math of proportions, and in that context turning to geometry is exceedingly helpful.", "translatedText": "Mọi người thường nghĩ về xác suất như một nghiên cứu về sự không chắc chắn, và đó tất nhiên là cách nó được áp dụng trong khoa học, nhưng phép toán thực sự của xác suất, nguồn gốc của tất cả các công thức, chỉ là phép toán về tỷ lệ, và trong bối cảnh đó chuyển sang hình học là cực kỳ hữu ích.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/arabic/sentence_translations.json b/2019/clacks-solution/arabic/sentence_translations.json index 2e9b3a934..459cb4798 100644 --- a/2019/clacks-solution/arabic/sentence_translations.json +++ b/2019/clacks-solution/arabic/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall. ", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall. ", "translatedText": "لذا لاحظ أن هذا يعطينا نقطة واحدة أخرى فقط يمكننا الانتقال إليها، ويجب أن يكون من المنطقي أنه شيء حيث يصبح الإحداثي x أقل سلبية ويصبح الإحداثي y سالبًا، لأن ذلك يتوافق مع الكبير تتباطأ الكتلة قليلاً، بينما تتجه الكتلة الصغيرة باتجاه الحائط. ", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term. ", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term. ", "translatedText": "لكي نكون أكثر دقة، فإن توسيع متسلسلة تايلور لظل ثيتا يوضح أن هذا التقريب سيكون له حد خطأ مكعب فقط. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000. ", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth. ", "translatedText": "على سبيل المثال، يختلف ظل 1100 عن 1100 نفسه بمقدار 1,1,000,000. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta. ", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta. ", "translatedText": "على وجه التحديد، نظرًا لبعض هندسة الزوايا المنقوشة، فإن النقاط التي وصلنا إليها في هذه الدائرة متباعدة بشكل متساوٍ، مفصولة بزاوية نسميها 2 ثيتا. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/bengali/sentence_translations.json b/2019/clacks-solution/bengali/sentence_translations.json index bbd1d4b71..5148cafb8 100644 --- a/2019/clacks-solution/bengali/sentence_translations.json +++ b/2019/clacks-solution/bengali/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall. ", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall. ", "translatedText": "সুতরাং লক্ষ্য করুন, এটি আমাদের একটি এবং শুধুমাত্র একটি অন্য পয়েন্ট দেয় যেটিতে আমরা লাফ দিতে পারি, এবং এটি বোঝা উচিত যে এটি এমন কিছু যেখানে x-স্থানাঙ্কটি একটু কম ঋণাত্মক হয় এবং y-স্থানাঙ্কটি নেতিবাচক হয়ে যায়, যেহেতু এটি বড়টির সাথে মিলে যায় ব্লকটি একটু মন্থর হচ্ছে, যখন ছোট ব্লকটি দেয়ালের দিকে জুম করছে।", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term. ", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term. ", "translatedText": "আরও সুনির্দিষ্টভাবে বলতে গেলে, থিটার স্পর্শকটির টেলর সিরিজের প্রসারণ দেখায় যে এই অনুমানে শুধুমাত্র একটি ঘন ত্রুটির শব্দ থাকবে।", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000. ", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth. ", "translatedText": "উদাহরণস্বরূপ, 1,100 এর স্পর্শক 1,100 থেকে 1,100,000 এর ক্রম অনুসারে আলাদা।", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta. ", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta. ", "translatedText": "বিশেষত, কিছু খোদাই করা কোণ জ্যামিতির কারণে, আমরা এই বৃত্তের যে বিন্দুগুলিকে আঘাত করি সেগুলিকে আমরা 2 থিটা বলে একটি কোণ দ্বারা পৃথক করে সমানভাবে ব্যবধান করে থাকি।", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/chinese/sentence_translations.json b/2019/clacks-solution/chinese/sentence_translations.json index 1afc91cf4..7dbf84a1a 100644 --- a/2019/clacks-solution/chinese/sentence_translations.json +++ b/2019/clacks-solution/chinese/sentence_translations.json @@ -359,7 +359,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall.", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall.", "translatedText": "所以请注意，这给了我们一个且仅有的一个我们可以跳 转到的点，并且它应该是有意义的，即 x 坐标的 负值稍微小一些，而 y 坐标变为负值，因为它对应 于大方块稍微放慢了速度，而小方块则向墙壁飞去。", "model": "google_nmt", "from_community_srt": "所以 注意， 这给了我们一个而且只有一个 我们可以跳到。 它应该 理解它是什么东西 x坐标减少了一些负面因素 y坐标是负的， 因为它对应 到了我们的大块，", @@ -782,7 +782,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", "translatedText": "更准确地说，theta 正切的泰勒级 数展开式表明该近似值只有三次误差项。", "model": "google_nmt", "from_community_srt": "更确切地说， 泰勒系列 tan（theta）的扩展表明了这种近似 只会有一个立方误差项。", @@ -791,7 +791,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000.", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth.", "translatedText": "例如，1,100 的正切与 1,100 本身相差大约 1,1,000,000。", "model": "google_nmt", "from_community_srt": "因此对于 例如， tan（1/100）与1/100相差不同 大约1 / 1,000,000的东西。", @@ -836,7 +836,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta.", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta.", "translatedText": "具体来说，由于某些内切角几何形状，我们在该圆上碰到的 点均匀分布，以我们称为 2 theta 的角度隔开。", "model": "google_nmt", "from_community_srt": "具体来说， 由于一些内切角 几何， 我们击中这个圆圈的点 均匀地间隔开， 由角度分开 我们打电话给2 * theta。", diff --git a/2019/clacks-solution/czech/sentence_translations.json b/2019/clacks-solution/czech/sentence_translations.json index 3797592f2..2136d085f 100644 --- a/2019/clacks-solution/czech/sentence_translations.json +++ b/2019/clacks-solution/czech/sentence_translations.json @@ -383,7 +383,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "", "from_community_srt": "Tomu odpovídá tento výsek v diagramu, takže budeme bloky srážet tak dlouho,", "n_reviews": 0, diff --git a/2019/clacks-solution/english/captions.srt b/2019/clacks-solution/english/captions.srt index 8a5e729ec..826dc118d 100644 --- a/2019/clacks-solution/english/captions.srt +++ b/2019/clacks-solution/english/captions.srt @@ -371,12 +371,12 @@ until the velocity of that smaller block is both positive and smaller than the velocity of the big one, meaning they'll never touch again. 94 -00:05:57,860 --> 00:06:03,837 +00:05:57,860 --> 00:06:01,858 That corresponds to this triangular region in the upper right of the diagram, 95 -00:06:03,837 --> 00:06:05,140 -so in our region. +00:06:01,858 --> 00:06:05,140 +so in our process we keep bouncing until we land in that region. 96 00:06:07,420 --> 00:06:10,243 diff --git a/2019/clacks-solution/english/sentence_timings.json b/2019/clacks-solution/english/sentence_timings.json index 54639bdda..faee680e7 100644 --- a/2019/clacks-solution/english/sentence_timings.json +++ b/2019/clacks-solution/english/sentence_timings.json @@ -240,7 +240,7 @@ 357.12 ], [ - "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", 357.86, 365.14 ], diff --git a/2019/clacks-solution/english/transcript.txt b/2019/clacks-solution/english/transcript.txt index 7deace44a..de8e948b6 100644 --- a/2019/clacks-solution/english/transcript.txt +++ b/2019/clacks-solution/english/transcript.txt @@ -46,7 +46,7 @@ When the second block bounces off the wall, its speed stays the same, but it goe So in this diagram, that corresponds to reflecting about the x-axis, since the y-coordinate gets multiplied by negative 1. Then once more, the next collision corresponds to a jump along a line with slope negative square root of m1 over m2, since staying on such a line is what conservation of momentum looks like in this diagram. And from here you can fill in the rest for how the block collisions correspond to hopping around the circle in our picture, where we keep going like this, until the velocity of that smaller block is both positive and smaller than the velocity of the big one, meaning they'll never touch again. -That corresponds to this triangular region in the upper right of the diagram, so in our region. +That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region. What we've drawn here is called a phase diagram, which is a simple but powerful idea in math where you encode the state of some system, in this case the velocities of our sliding blocks, as a single point in some abstract space. What's powerful here is that it turns questions about dynamics into questions about geometry. In this case, the dynamical idea of all possible pairs of velocities that conserve energy corresponds to the geometric idea of a circle, and counting the total number of collisions turns into counting the total number of hops along these lines, alternating between vertical and diagonal. diff --git a/2019/clacks-solution/french/sentence_translations.json b/2019/clacks-solution/french/sentence_translations.json index 61abec59e..fdbeda232 100644 --- a/2019/clacks-solution/french/sentence_translations.json +++ b/2019/clacks-solution/french/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall.", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall.", "translatedText": "Alors remarquez, cela nous donne un et un seul autre point auquel nous pourrions accéder, et il devrait être logique que ce soit quelque chose où la coordonnée x devient un peu moins négative et la coordonnée y devient négative, puisque cela correspond au grand Le bloc ralentit un peu, tandis que le petit bloc fonce vers le mur.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", "translatedText": "Pour être plus précis, le développement en série de Taylor de la tangente de thêta montre que cette approximation n'aura qu'un terme d'erreur cubique.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000.", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth.", "translatedText": "Par exemple, la tangente de 1 100 diffère de 1 100 elle-même d’environ 1 1 000 000.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta.", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta.", "translatedText": "Plus précisément, en raison d'une certaine géométrie d'angle inscrit, les points que nous frappons de ce cercle sont espacés uniformément, séparés par un angle que nous appelons 2 thêta.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/german/sentence_translations.json b/2019/clacks-solution/german/sentence_translations.json index 5282cf576..d2280c595 100644 --- a/2019/clacks-solution/german/sentence_translations.json +++ b/2019/clacks-solution/german/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall. ", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall. ", "translatedText": "Das gibt uns einen und nur einen weiteren Punkt, zu dem wir springen könnten, und es sollte Sinn machen, dass es sich um einen Punkt handelt, bei dem die x-Koordinate etwas weniger negativ und die y-Koordinate negativ wird, da dies der leichten Verlangsamung des großen Blocks entspricht, während der kleine Block in Richtung Wand gestoßen wird. ", "model": "google_nmt", "n_reviews": 1, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term. ", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term. ", "translatedText": "Genauer gesagt zeigt die Taylorr-Entwicklung des Theta-Tangens, dass diese Näherung nur einen kubischen Fehlerterm aufweist. ", "model": "google_nmt", "n_reviews": 1, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000. ", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth. ", "translatedText": "Beispielsweise unterscheidet sich der Tangens von einem Hundertstel von einem Hundertstel selbst um etwas in der Größenordnung von einem Millionstel. ", "model": "google_nmt", "n_reviews": 1, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta. ", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta. ", "translatedText": "Insbesondere sind die Punkte, die wir auf diesem Kreis treffen, aufgrund des Kreiswinkelsatzes gleichmäßig voneinander entfernt und durch einen Winkel getrennt, den wir 2 Theta nennen. ", "model": "google_nmt", "n_reviews": 1, diff --git a/2019/clacks-solution/greek/sentence_translations.json b/2019/clacks-solution/greek/sentence_translations.json index fe6f19320..bc66604fd 100644 --- a/2019/clacks-solution/greek/sentence_translations.json +++ b/2019/clacks-solution/greek/sentence_translations.json @@ -382,7 +382,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "", "from_community_srt": "Αυτό αντιστοιχεί σε αυτό περιοχή του διαγράμματος, έτσι στη διαδικασία μας,", "n_reviews": 0, diff --git a/2019/clacks-solution/hebrew/sentence_translations.json b/2019/clacks-solution/hebrew/sentence_translations.json index c1a0ea7fb..4ae541806 100644 --- a/2019/clacks-solution/hebrew/sentence_translations.json +++ b/2019/clacks-solution/hebrew/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall.", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall.", "translatedText": "אז שימו לב, זה נותן לנו עוד נקודה אחת ויחידה שנוכל לקפוץ אליה, וזה אמור להיות הגיוני שזה משהו שבו קואורדינטת ה-x נהיית קצת פחות שלילית וקואורדינטת ה-y הופכת לשלילית, מכיוון שזה מתאים לקואורדינטת ה-x. בלוק מאט מעט, בעוד הבלוק הקטן מתקרב לכיוון הקיר.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", "translatedText": "ליתר דיוק, הרחבת סדרת טיילור של טנגנס של תטא מראה שלקירוב זה יהיה רק מונח שגיאה מעוקב.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000.", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth.", "translatedText": "לדוגמה, הטנגנס של 1,100 שונה מ-1,100 עצמו במשהו בסדר גודל של 1,1,000,000.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta.", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta.", "translatedText": "באופן ספציפי, בגלל גיאומטריית זווית מסויימת, הנקודות בהן פגענו במעגל הזה מפוזרות באופן שווה, מופרדות על ידי זווית שאנו מכנים 2 תטא.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/hindi/sentence_translations.json b/2019/clacks-solution/hindi/sentence_translations.json index d3edc3d5b..474a871ba 100644 --- a/2019/clacks-solution/hindi/sentence_translations.json +++ b/2019/clacks-solution/hindi/sentence_translations.json @@ -280,7 +280,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall.", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall.", "translatedText": "तो ध्यान दें, यह हमें एक और केवल एक अन्य बिंदु देता है जिस पर हम जा सकते हैं, और यह समझ में आना चाहिए कि यह कुछ ऐसा है जहां x-निर्देशांक थोड़ा कम नकारात्मक हो जाता है और y-निर्देशांक नकारात्मक हो जाता है, क्योंकि यह बड़े से मेल खाता है ब्लॉक थोड़ा धीमा हो रहा है, जबकि छोटा ब्लॉक दीवार की ओर बढ़ रहा है।", "n_reviews": 0, "start": 295.38, @@ -609,14 +609,14 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", "translatedText": "अधिक सटीक होने के लिए, थीटा के स्पर्शरेखा के टेलर श्रृंखला विस्तार से पता चलता है कि इस सन्निकटन में केवल एक घन त्रुटि शब्द होगा।", "n_reviews": 0, "start": 752.48, "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000.", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth.", "translatedText": "उदाहरण के लिए, 1,100 की स्पर्श रेखा 1,100 से 1,1,000,000 के क्रम में भिन्न होती है।", "n_reviews": 0, "start": 760.98, @@ -651,7 +651,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta.", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta.", "translatedText": "विशेष रूप से, कुछ अंकित कोण ज्यामिति के कारण, इस वृत्त के जिन बिंदुओं पर हम पहुँचते हैं वे समान दूरी पर होते हैं, एक कोण से अलग होते हैं जिसे हम 2 थीटा कहते हैं।", "n_reviews": 0, "start": 805.62, diff --git a/2019/clacks-solution/hungarian/sentence_translations.json b/2019/clacks-solution/hungarian/sentence_translations.json index 696293c86..2be711cb4 100644 --- a/2019/clacks-solution/hungarian/sentence_translations.json +++ b/2019/clacks-solution/hungarian/sentence_translations.json @@ -384,7 +384,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "Ez megfelel ennek a háromszög alakú régiónak az ábra jobb felső részén, tehát a mi régiónkban.", "model": "DeepL", "n_reviews": 0, diff --git a/2019/clacks-solution/indonesian/sentence_translations.json b/2019/clacks-solution/indonesian/sentence_translations.json index bdc136acf..b9c01e84c 100644 --- a/2019/clacks-solution/indonesian/sentence_translations.json +++ b/2019/clacks-solution/indonesian/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall.", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall.", "translatedText": "Jadi perhatikan, ini memberi kita satu-satunya titik lain yang bisa kita lompati, dan masuk akal bahwa ini adalah sesuatu yang koordinat x-nya menjadi sedikit kurang negatif dan koordinat y menjadi negatif, karena itu sesuai dengan besarnya blok melambat sedikit, sementara blok kecil meluncur ke arah dinding.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", "translatedText": "Lebih tepatnya, perluasan garis singgung theta deret Taylor menunjukkan bahwa perkiraan ini hanya mempunyai suku kesalahan kubik.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000.", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth.", "translatedText": "Misalnya, garis singgung 1.100 berbeda dari 1.100 itu sendiri dengan orde 1,1.000. 000.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta.", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta.", "translatedText": "Secara khusus, karena beberapa geometri sudut tertulis, titik-titik yang kita temui pada lingkaran ini diberi jarak yang sama, dipisahkan oleh sudut yang kita sebut 2 theta.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/italian/sentence_translations.json b/2019/clacks-solution/italian/sentence_translations.json index 8708baed3..a68cfbbcc 100644 --- a/2019/clacks-solution/italian/sentence_translations.json +++ b/2019/clacks-solution/italian/sentence_translations.json @@ -429,7 +429,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "Ciò corrisponde a questa regione triangolare in alto a destra del diagramma, quindi nella nostra regione.", "model": "DeepL", "from_community_srt": "Ciò corrisponde a questa regione triangolare del diagramma, così nel nostro processo si continua a rimbalzare fino a quando si arriva in quella regione.", diff --git a/2019/clacks-solution/japanese/sentence_translations.json b/2019/clacks-solution/japanese/sentence_translations.json index 3d73b72fa..15ce5fe06 100644 --- a/2019/clacks-solution/japanese/sentence_translations.json +++ b/2019/clacks-solution/japanese/sentence_translations.json @@ -353,7 +353,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall.", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall.", "translatedText": "注目してください。 これにより、ジャンプできる点がもう 1 つだけ得られ ます。 これは、X 座標の負の値が少し減り、Y 座標が負になる点である ことは理にかなっています。 これは、大きな点に対応するためです。 ブロック は少し速度を落とし、小さなブロックは壁に向かってズームアウトします。", "model": "google_nmt", "from_community_srt": "ここで注意。 これがジャンプの着地点を指すのは一回きりです。 そしてこれからx座標を少し大きな負の数になり y座標は負の数になるとわかるでしょう。 大ブロックが少し遅くなって小ブロックが壁に向かって去っていくように。", @@ -774,7 +774,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", "translatedText": "より正確には、シータのタンジェントのテイラー級数展開は、 この近似が 3 次誤差項のみを持つことを示しています。", "model": "google_nmt", "from_community_srt": "より正確にするならtan(θ)のテイラー級数展開からこの近似が3乗の誤差項しか持たないことが示される。", @@ -783,7 +783,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000.", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth.", "translatedText": "たとえば、1,100 のタンジェントは 1,100 そ のものとは 1,1,000,000 程度異なります。", "model": "google_nmt", "from_community_srt": "それで例えば tan(1/100) は1/100と1/100万のオーダーしか違わない。", @@ -828,7 +828,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta.", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta.", "translatedText": "具体的には、内接角の幾何学形状により、この円の点は均 等に配置され、2 θ と呼ばれる角度で分離されます。", "model": "google_nmt", "from_community_srt": "特に、円周角の幾何学によりこの縁に打つ各点は等間隔で空け 2＊θと呼んでいる角度だけ分断される。", diff --git a/2019/clacks-solution/korean/sentence_translations.json b/2019/clacks-solution/korean/sentence_translations.json index c87886289..b73a86dff 100644 --- a/2019/clacks-solution/korean/sentence_translations.json +++ b/2019/clacks-solution/korean/sentence_translations.json @@ -429,7 +429,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "이는 다이어그램의 오른쪽 상단에 있는 삼각형 영역에 해당하므로 우리 지역에서는 이 영역에 해당합니다.", "model": "DeepL", "from_community_srt": "이는 두 블록이 더이상 충돌하지 않는다는 것을 의미합니다. 이는 다시 좌표평면 상의 색칠된 영역을 나타내고,", diff --git a/2019/clacks-solution/marathi/sentence_translations.json b/2019/clacks-solution/marathi/sentence_translations.json index ce8237e81..a6efcba35 100644 --- a/2019/clacks-solution/marathi/sentence_translations.json +++ b/2019/clacks-solution/marathi/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall.", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall.", "translatedText": "तर लक्ष द्या, हे आम्हाला एक आणि फक्त एक दुसरा मुद्दा देते ज्यावर आपण उडी मारू शकतो आणि याचा अर्थ असा आहे की हे असे काहीतरी आहे जेथे x-कोऑर्डिनेट थोडा कमी नकारात्मक होतो आणि y-कोऑर्डिनेट नकारात्मक होतो, कारण ते मोठ्याशी संबंधित आहे ब्लॉक थोडा कमी होतो, तर छोटा ब्लॉक भिंतीकडे झूम बंद होतो.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", "translatedText": "अधिक तंतोतंत सांगायचे तर, थीटाच्या स्पर्शिकेचा टेलर मालिका विस्तार दर्शविते की या अंदाजेमध्ये फक्त एक घन त्रुटी संज्ञा असेल.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000.", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth.", "translatedText": "उदाहरणार्थ, 1,100 ची स्पर्शिका 1,100 पेक्षा 1,100,000 च्या क्रमाने भिन्न असते.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta.", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta.", "translatedText": "विशेषत:, काही कोरलेल्या कोन भूमितीमुळे, या वर्तुळाचे आपण जे बिंदू मारतो ते बिंदू समान रीतीने अंतरावर असतात, एका कोनाने वेगळे केले जातात ज्याला आपण 2 थीटा म्हणतो.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/persian/sentence_translations.json b/2019/clacks-solution/persian/sentence_translations.json index 3bf92d85c..21c50134a 100644 --- a/2019/clacks-solution/persian/sentence_translations.json +++ b/2019/clacks-solution/persian/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall. ", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall. ", "translatedText": "بنابراین توجه کنید، این به ما یک و تنها یک نکته دیگر می‌دهد که می‌توانیم به آن بپریم، و باید منطقی باشد که چیزی است که در آن مختصات x کمی منفی می‌شود و مختصات y منفی می‌شود، زیرا این مربوط به بزرگ است. بلوک کمی کند می شود، در حالی که بلوک کوچک به سمت دیوار بزرگ می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term. ", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term. ", "translatedText": "به عبارت دقیق‌تر، بسط مماس تتا سری تیلور نشان می‌دهد که این تقریب فقط یک خطای مکعبی خواهد داشت. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000. ", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth. ", "translatedText": "به عنوان مثال، مماس 1100 با خود 1100 با چیزی در مرتبه 1،1،000،000 متفاوت است. ", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta. ", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta. ", "translatedText": "به طور خاص، به دلیل هندسه زاویه محاط شده، نقاطی که به این دایره برخورد می کنیم به طور مساوی با زاویه ای که آن را 2 تتا می نامیم از هم جدا می شوند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/polish/sentence_translations.json b/2019/clacks-solution/polish/sentence_translations.json index 6d3c0ae37..1a3f4dca9 100644 --- a/2019/clacks-solution/polish/sentence_translations.json +++ b/2019/clacks-solution/polish/sentence_translations.json @@ -383,7 +383,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "", "from_community_srt": "że ​​już nigdy więcej się nie zderzą. To odpowiada temu regionowi wykresu, więc w naszym procesie skaczemy,", "n_reviews": 0, diff --git a/2019/clacks-solution/portuguese/sentence_translations.json b/2019/clacks-solution/portuguese/sentence_translations.json index 326f72b1d..cabb589fa 100644 --- a/2019/clacks-solution/portuguese/sentence_translations.json +++ b/2019/clacks-solution/portuguese/sentence_translations.json @@ -431,7 +431,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "Isso corresponde a esta região triangular no canto superior direito do diagrama, ou seja, na nossa região.", "model": "google_nmt", "from_community_srt": "assim nunca mais se enconstando Isso é representado por essa região triangular, então no processo,", diff --git a/2019/clacks-solution/romanian/sentence_translations.json b/2019/clacks-solution/romanian/sentence_translations.json index 113e2147d..99b3f3a95 100644 --- a/2019/clacks-solution/romanian/sentence_translations.json +++ b/2019/clacks-solution/romanian/sentence_translations.json @@ -380,7 +380,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "", "from_community_srt": "Asta corespunde cu regiunea asta a diagramei, deci in procesul nostru,", "n_reviews": 0, diff --git a/2019/clacks-solution/russian/sentence_translations.json b/2019/clacks-solution/russian/sentence_translations.json index 323dbe2f9..fa7b30ab4 100644 --- a/2019/clacks-solution/russian/sentence_translations.json +++ b/2019/clacks-solution/russian/sentence_translations.json @@ -383,7 +383,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "Это соответствует этой треугольной области в правом верхнем углу диаграммы, то есть в нашем регионе.", "from_community_srt": "что они никогда не встретятся. Это соответствует этому региону диаграммы, то есть в нашем процессе мы продолжаем прыжки,", "n_reviews": 0, diff --git a/2019/clacks-solution/serbian/sentence_translations.json b/2019/clacks-solution/serbian/sentence_translations.json index 235770035..817ea50cc 100644 --- a/2019/clacks-solution/serbian/sentence_translations.json +++ b/2019/clacks-solution/serbian/sentence_translations.json @@ -383,7 +383,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "", "from_community_srt": "To odgovara području trougla u gornjem desnom delu dijagrama, tako da ćemo se pomerati unutar kruga dok se ne nađemo u tom području.", "n_reviews": 0, diff --git a/2019/clacks-solution/spanish/sentence_translations.json b/2019/clacks-solution/spanish/sentence_translations.json index 9373832ae..733ac7d35 100644 --- a/2019/clacks-solution/spanish/sentence_translations.json +++ b/2019/clacks-solution/spanish/sentence_translations.json @@ -381,7 +381,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "Eso corresponde a esta región triangular en la parte superior derecha del diagrama, es decir, en nuestra región.", "from_community_srt": "Esto corresponde a esta región del diagrama, entonces en nuestro proceso,", "n_reviews": 0, diff --git a/2019/clacks-solution/tamil/sentence_translations.json b/2019/clacks-solution/tamil/sentence_translations.json index 47dd1aaa8..8ab6be90c 100644 --- a/2019/clacks-solution/tamil/sentence_translations.json +++ b/2019/clacks-solution/tamil/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall.", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall.", "translatedText": "எனவே கவனிக்கவும், இது நாம் குதிக்கக்கூடிய ஒரே ஒரு புள்ளியை நமக்குத் தருகிறது, மேலும் இது x-ஆயத்தொகை சிறிது எதிர்மறையாகி, y-ஆயத்தொகை எதிர்மறையாக மாறும், ஏனெனில் அது பெரியதுடன் ஒத்துப்போகிறது. தொகுதி சிறிது குறைகிறது, அதே சமயம் சிறிய தொகுதி சுவரை நோக்கி பெரிதாகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", "translatedText": "இன்னும் துல்லியமாகச் சொல்வதென்றால், தீட்டாவின் டேன்ஜென்ட்டின் டெய்லர் தொடர் விரிவாக்கம், இந்த தோராயமானது ஒரு கனசதுரப் பிழைச் சொல்லை மட்டுமே கொண்டிருக்கும் என்பதைக் காட்டுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000.", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth.", "translatedText": "எடுத்துக்காட்டாக, 1,100 இன் தொடுகோடு 1,100 இல் இருந்து 1,1,000,000 வரிசையில் வேறுபடுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta.", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta.", "translatedText": "குறிப்பாக, சில பொறிக்கப்பட்ட கோண வடிவவியலின் காரணமாக, இந்த வட்டத்தில் நாம் அடிக்கும் புள்ளிகள் சமமாக இடைவெளியில் உள்ளன, ஒரு கோணத்தால் பிரிக்கப்பட்டால் நாம் 2 தீட்டா என்று அழைக்கிறோம்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/telugu/sentence_translations.json b/2019/clacks-solution/telugu/sentence_translations.json index d34b3eb29..8e04fc6fb 100644 --- a/2019/clacks-solution/telugu/sentence_translations.json +++ b/2019/clacks-solution/telugu/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall.", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall.", "translatedText": "కాబట్టి గమనించండి, ఇది మనకు దూకగలిగే మరొక పాయింట్‌ను మాత్రమే ఇస్తుంది మరియు ఇది x-కోఆర్డినేట్ కొద్దిగా తక్కువ ప్రతికూలతను పొందుతుంది మరియు y-కోఆర్డినేట్ ప్రతికూలంగా మారుతుందని అర్థం చేసుకోవాలి, ఎందుకంటే అది పెద్దదానికి అనుగుణంగా ఉంటుంది. బ్లాక్ కొద్దిగా మందగిస్తుంది, అయితే చిన్న బ్లాక్ గోడ వైపు జూమ్ అవుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", "translatedText": "మరింత ఖచ్చితంగా చెప్పాలంటే, తీటా యొక్క టాంజెంట్ యొక్క టేలర్ శ్రేణి విస్తరణ ఈ ఉజ్జాయింపులో క్యూబిక్ ఎర్రర్ పదం మాత్రమే ఉంటుందని చూపిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000.", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth.", "translatedText": "ఉదాహరణకు, 1,100 యొక్క టాంజెంట్ 1,100 నుండి 1,1,000,000 క్రమంలో భిన్నంగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta.", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta.", "translatedText": "నిర్దిష్టంగా, కొన్ని లిఖించబడిన కోణ జ్యామితి కారణంగా, ఈ వృత్తంలో మనం కొట్టే పాయింట్లు సమానంగా ఖాళీగా ఉంటాయి, మనం 2 తీటా అని పిలుస్తున్న కోణంతో వేరు చేయబడతాయి.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/thai/sentence_translations.json b/2019/clacks-solution/thai/sentence_translations.json index 071fbc075..b68143dac 100644 --- a/2019/clacks-solution/thai/sentence_translations.json +++ b/2019/clacks-solution/thai/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall. ", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term. ", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000. ", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta. ", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/turkish/sentence_translations.json b/2019/clacks-solution/turkish/sentence_translations.json index cb2058674..4f68c7498 100644 --- a/2019/clacks-solution/turkish/sentence_translations.json +++ b/2019/clacks-solution/turkish/sentence_translations.json @@ -427,7 +427,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "Bu da diyagramın sağ üst köşesindeki üçgen bölgeye, yani bizim bölgemize denk geliyor.", "model": "DeepL", "from_community_srt": "Bu durum grafiğin sağ üstündeki üçgensel bölgeye karşılık geliyor. Bu demektir ki nokta o bölgeye ulaşana dek sıçramaya devam ediyor.", diff --git a/2019/clacks-solution/ukrainian/sentence_translations.json b/2019/clacks-solution/ukrainian/sentence_translations.json index d0dd774bc..5cfbafcad 100644 --- a/2019/clacks-solution/ukrainian/sentence_translations.json +++ b/2019/clacks-solution/ukrainian/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall.", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall.", "translatedText": "Отже, зверніть увагу, це дає нам одну і лише одну іншу точку, до якої ми можемо перейти, і має бути зрозуміло, що це щось, де координата x стає трохи менш від’ємною, а координата y стає від’ємною, оскільки це відповідає великій блок трохи сповільнюється, тоді як маленький блок віддаляється до стіни.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term.", "translatedText": "Якщо бути більш точним, розкладання тангенса тета в ряд Тейлора показує, що це наближення матиме лише кубічну помилку.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000.", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth.", "translatedText": "Наприклад, тангенс 1100 відрізняється від самого 1100 приблизно на 1 1 000 000.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta.", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta.", "translatedText": "Зокрема, через певну геометрію вписаного кута точки цього кола, на які ми потрапляємо, розташовані рівномірно, розділені кутом, який ми називаємо 2 тета.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/urdu/sentence_translations.json b/2019/clacks-solution/urdu/sentence_translations.json index 116386806..3e2e902e9 100644 --- a/2019/clacks-solution/urdu/sentence_translations.json +++ b/2019/clacks-solution/urdu/sentence_translations.json @@ -320,7 +320,7 @@ "end": 294.76 }, { - "input": "So notice, this gives us one and only one other point that we could jump to, and it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block slowing down a little, while the little block zooms off towards the wall. ", + "input": "So notice, this gives us one and only one other point that we could jump to. And it should make sense that it's something where the x-coordinate gets a little less negative and the y-coordinate becomes negative, since that corresponds to the big block, which is slowing down a little, while the little block zooms off towards the wall. ", "translatedText": "تو دھیان دیں، یہ ہمیں ایک اور صرف ایک اور نکتہ دیتا ہے جس پر ہم چھلانگ لگا سکتے ہیں، اور اسے یہ سمجھنا چاہیے کہ یہ وہ چیز ہے جہاں x-coordinate تھوڑا کم منفی ہو جاتا ہے اور y-coordinate منفی ہو جاتا ہے، کیونکہ یہ بڑے سے مساوی ہے۔بلاک تھوڑا سا سست ہوتا ہے، جبکہ چھوٹا بلاک دیوار کی طرف بڑھتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 751.78 }, { - "input": "To be more precise, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term. ", + "input": "To be more precise about it, the Taylor series expansion of tangent of theta shows that this approximation will have only a cubic error term. ", "translatedText": "زیادہ درست ہونے کے لیے، تھیٹا کے ٹینجنٹ کی ٹیلر سیریز کی توسیع سے پتہ چلتا ہے کہ اس قربت میں صرف ایک کیوبک غلطی کی اصطلاح ہوگی۔", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 760.32 }, { - "input": "For example, the tangent of 1,100 differs from 1,100 itself by something on the order of 1,1,000,000. ", + "input": "So for example, the tangent of 1 one hundredth differs from 1 one hundredth itself by something on the order of 1 one millionth. ", "translatedText": "مثال کے طور پر، 1،100 کا مماس خود 1،100 سے 1،100،000 کے آرڈر پر کسی چیز سے مختلف ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 804.96 }, { - "input": "Specifically, due to some inscribed angle geometry, the points we hit of this circle are spaced out evenly, separated by an angle we call 2 theta. ", + "input": "Specifically, due to some inscribed angle geometry, the points that we hit of this circle are spaced out evenly, separated by an angle we were calling 2 theta. ", "translatedText": "خاص طور پر، کچھ کندہ شدہ زاویہ جیومیٹری کی وجہ سے، اس دائرے کے ہم جن پوائنٹس کو مارتے ہیں وہ یکساں طور پر فاصلہ رکھتے ہیں، ایک زاویہ سے الگ کیا جاتا ہے جسے ہم 2 تھیٹا کہتے ہیں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-solution/vietnamese/sentence_translations.json b/2019/clacks-solution/vietnamese/sentence_translations.json index bdcaa9916..a1c32b29d 100644 --- a/2019/clacks-solution/vietnamese/sentence_translations.json +++ b/2019/clacks-solution/vietnamese/sentence_translations.json @@ -395,7 +395,7 @@ "end": 357.12 }, { - "input": "That corresponds to this triangular region in the upper right of the diagram, so in our region.", + "input": "That corresponds to this triangular region in the upper right of the diagram, so in our process we keep bouncing until we land in that region.", "translatedText": "Điều đó tương ứng với vùng hình tam giác này ở phía trên bên phải của sơ đồ, cũng như trong vùng của chúng ta.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks-via-light/arabic/sentence_translations.json b/2019/clacks-via-light/arabic/sentence_translations.json index 2d3bfbf52..d51c0fc87 100644 --- a/2019/clacks-via-light/arabic/sentence_translations.json +++ b/2019/clacks-via-light/arabic/sentence_translations.json @@ -394,7 +394,7 @@ "end": 352.7 }, { - "input": "Likewise, the rate of change for y is the square root of m2 times v2.", + "input": "Likewise, the rate of change for y is going to be the square root of m2 times v2.", "translatedText": "وبالمثل، فإن معدل التغير لـ y هو الجذر التربيعي لـ m2 في v2.", "model": "google_nmt", "from_community_srt": "معدل تغير d1 بالنسبة للزمن هو سرعة الجسم الكبير والذي سأسميه v1 وبالمثل، فإن معدل تغير y بالنسبة للزمن هو الجذر التربيعي لـ m1 مضروباً ب v2", @@ -666,7 +666,7 @@ "end": 611.78 }, { - "input": "Remember, the mass ratio of our blocks determines this angle theta in the analogy.", + "input": "Remember, the mass ratio of our blocks completely determines this angle theta in the analogy.", "translatedText": "تذكر أن نسبة الكتلة للمكعبات تحدد زاوية ثيتا هذه في القياس.", "model": "google_nmt", "from_community_srt": "تذكروا أن نسبة كتلتي الجسمين المدروسين تحدد الزاوية θ في الرسم البياني يمكنني الآن سماع بعضكم يتذمر:", diff --git a/2019/clacks-via-light/czech/sentence_translations.json b/2019/clacks-via-light/czech/sentence_translations.json index b0f70d95a..f599570d7 100644 --- a/2019/clacks-via-light/czech/sentence_translations.json +++ b/2019/clacks-via-light/czech/sentence_translations.json @@ -358,7 +358,7 @@ "end": 352.7 }, { - "input": "Likewise, the rate of change for y is the square root of m2 times v2.", + "input": "Likewise, the rate of change for y is going to be the square root of m2 times v2.", "translatedText": "", "from_community_srt": "Stejně tak, míra změny pro y je sqrt(m2)*v2.", "n_reviews": 0, @@ -604,7 +604,7 @@ "end": 611.78 }, { - "input": "Remember, the mass ratio of our blocks determines this angle theta in the analogy.", + "input": "Remember, the mass ratio of our blocks completely determines this angle theta in the analogy.", "translatedText": "", "from_community_srt": "vyjádřeno jako funkce úhlu théta? Pamatujme, že poměr hmotností našich bloků v tomto přirovnání určuje úhel théta.", "n_reviews": 0, diff --git a/2019/clacks-via-light/english/captions.srt b/2019/clacks-via-light/english/captions.srt index 1f62c861a..1feb2d17e 100644 --- a/2019/clacks-via-light/english/captions.srt +++ b/2019/clacks-via-light/english/captions.srt @@ -376,7 +376,7 @@ Let's call that v1. 95 00:05:53,680 --> 00:05:59,580 -Likewise, the rate of change for y is the square root of m2 times v2. +Likewise, the rate of change for y is going to be the square root of m2 times v2. 96 00:06:00,140 --> 00:06:05,240 @@ -635,250 +635,254 @@ meeting each other at some angle, let's say theta, how many times would that light bounce off of the mirrors as a function of that angle? 160 -00:10:12,920 --> 00:10:17,980 -Remember, the mass ratio of our blocks determines this angle theta in the analogy. +00:10:12,920 --> 00:10:15,640 +Remember, the mass ratio of our blocks completely 161 +00:10:15,640 --> 00:10:17,980 +determines this angle theta in the analogy. + +162 00:10:19,380 --> 00:10:21,633 Now I can hear some of you complaining, haven't -162 +163 00:10:21,633 --> 00:10:23,840 we just replaced one tricky setup with another? -163 +164 00:10:24,280 --> 00:10:26,920 This might make for a cute analogy, but how is it progress? -164 +165 00:10:27,640 --> 00:10:31,300 It's true that counting the number of light bounces is hard, -165 +166 00:10:31,300 --> 00:10:33,220 but now we have a helpful trick. -166 +167 00:10:33,740 --> 00:10:36,815 When the beam of light hits the mirror, instead of thinking of -167 +168 00:10:36,815 --> 00:10:40,623 that beam as reflected about the mirror, think of the beam as going straight, -168 +169 00:10:40,623 --> 00:10:43,260 while the whole world gets flipped through the mirror. -169 +170 00:10:43,920 --> 00:10:46,221 It's as if the beam is passing through a piece -170 +171 00:10:46,221 --> 00:10:48,620 of glass into an illusory looking glass universe. -171 +172 00:10:49,540 --> 00:10:51,060 Think of actual mirrors here. -172 +173 00:10:51,500 --> 00:10:54,953 This wire on the left will represent a laser beam coming into the mirror, -173 +174 00:10:54,953 --> 00:10:57,520 and the one on the right will represent its reflection. -174 +175 00:10:58,320 --> 00:11:01,706 The illusion is that the beam goes straight through the mirror, -175 +176 00:11:01,706 --> 00:11:05,040 as if passing through a window separating us from another room. -176 +177 00:11:05,720 --> 00:11:08,792 But notice, crucially, for this illusion to work, -177 +178 00:11:08,792 --> 00:11:12,480 the angle of incidence has to equal the angle of reflection. -178 +179 00:11:13,080 --> 00:11:18,240 Otherwise, the flipped copy of the reflected beam won't line up with the first part. -179 +180 00:11:19,000 --> 00:11:21,900 So all of that work we did, rescaling coordinates and futzing -180 +181 00:11:21,900 --> 00:11:24,520 through the momentum equations, was certainly necessary. -181 +182 00:11:25,000 --> 00:11:27,520 But now we get to enjoy the fruits of our labor. -182 +183 00:11:28,140 --> 00:11:31,844 Watch how this helps us elegantly solve the question of how many mirror bounces -183 +184 00:11:31,844 --> 00:11:35,780 there will be, which is also the question of how many block collisions there will be. -184 +185 00:11:39,000 --> 00:11:43,304 Every time the beam hits a mirror, don't think of the beam as getting reflected, -185 +186 00:11:43,304 --> 00:11:46,280 let it continue straight while the world gets reflected. -186 +187 00:11:47,000 --> 00:11:50,861 As this goes on, the illusion to the beam of light is that instead of -187 +188 00:11:50,861 --> 00:11:54,281 getting bounced around between two angled mirrors many times, -188 +189 00:11:54,281 --> 00:11:58,860 it's passing through a sequence of angled pieces of glass all the same angle apart. -189 +190 00:12:00,000 --> 00:12:04,538 Right now I'm showing you all of the reflected copies of the bouncing trajectory, -190 +191 00:12:04,538 --> 00:12:07,140 which I think has a very striking beauty to it. -191 +192 00:12:11,060 --> 00:12:13,854 But for a clearer view, let's just focus on the -192 +193 00:12:13,854 --> 00:12:16,940 original bouncing beam and the illusory straight one. -193 +194 00:12:17,640 --> 00:12:20,680 The question of counting bounces turns into a question -194 +195 00:12:20,680 --> 00:12:23,720 of how many pieces of glass this illusory beam crosses. -195 +196 00:12:24,280 --> 00:12:26,980 How many reflected copies of the world does it pass into? -196 +197 00:12:34,980 --> 00:12:38,159 Well, calling the angle between the mirrors theta, -197 +198 00:12:38,159 --> 00:12:42,959 the answer here is however many times you can add theta to itself before you -198 +199 00:12:42,959 --> 00:12:47,696 get more than halfway around a circle, which is to say before you add up to -199 +200 00:12:47,696 --> 00:12:49,380 more than pi total radians. -200 +201 00:12:51,780 --> 00:12:56,620 Written as a formula, the answer to this question is the floor of pi divided by theta. -201 +202 00:12:57,440 --> 00:12:58,680 So let's review. -202 +203 00:12:59,080 --> 00:13:02,937 We started by drawing a configuration space for our colliding blocks where -203 +204 00:13:02,937 --> 00:13:06,640 the x and the y coordinates represented the two distances from the wall. -204 +205 00:13:07,780 --> 00:13:10,751 This kind of looked like light bouncing between two mirrors, -205 +206 00:13:10,751 --> 00:13:14,744 but to make the analogy work properly we needed to rescale the coordinates by the -206 +207 00:13:14,744 --> 00:13:16,060 square roots of the masses. -207 +208 00:13:16,820 --> 00:13:20,440 This made it so that the slope of one of our lines was square -208 +209 00:13:20,440 --> 00:13:24,061 root of m2 divided by square root of m1, so the angle between -209 +210 00:13:24,061 --> 00:13:27,740 those bounding lines will be the inverse tangent of that slope. -210 +211 00:13:28,740 --> 00:13:32,220 To figure out how many bounces there are between two mirrors like this, -211 +212 00:13:32,220 --> 00:13:35,652 think of the illusion of the beam going straight through a sequence of -212 +213 00:13:35,652 --> 00:13:38,940 looking glass universes separated by a semi-circular fan of windows. -213 +214 00:13:39,540 --> 00:13:42,810 The answer then comes down to how many times the value -214 +215 00:13:42,810 --> 00:13:46,200 of this angle fits into 180 degrees, which is pi radians. -215 +216 00:13:47,100 --> 00:13:51,816 From here, to understand why exactly the digits of pi show up when the mass ratio is -216 +217 00:13:51,816 --> 00:13:56,700 a power of 100, is exactly what we did in the last video, so I won't repeat myself here. -217 +218 00:13:57,500 --> 00:14:02,194 And finally, as we reflect now on how absurd the initial appearance of pi seemed, -218 +219 00:14:02,194 --> 00:14:07,118 and on the two solutions we've now seen, and on how unexpectedly helpful it can be to -219 +220 00:14:07,118 --> 00:14:10,667 represent the state of your system with points in some space, -220 +221 00:14:10,667 --> 00:14:14,446 I leave you with this quote from the computer scientist Alan Kay, -221 +222 00:14:14,446 --> 00:14:17,080 A change in perspective is worth 80 IQ points. diff --git a/2019/clacks-via-light/english/sentence_timings.json b/2019/clacks-via-light/english/sentence_timings.json index 0ea71d755..23006b937 100644 --- a/2019/clacks-via-light/english/sentence_timings.json +++ b/2019/clacks-via-light/english/sentence_timings.json @@ -225,7 +225,7 @@ 352.7 ], [ - "Likewise, the rate of change for y is the square root of m2 times v2.", + "Likewise, the rate of change for y is going to be the square root of m2 times v2.", 353.68, 359.58 ], @@ -380,7 +380,7 @@ 611.78 ], [ - "Remember, the mass ratio of our blocks determines this angle theta in the analogy.", + "Remember, the mass ratio of our blocks completely determines this angle theta in the analogy.", 612.92, 617.98 ], diff --git a/2019/clacks-via-light/english/transcript.txt b/2019/clacks-via-light/english/transcript.txt index 801bb8120..268ed38af 100644 --- a/2019/clacks-via-light/english/transcript.txt +++ b/2019/clacks-via-light/english/transcript.txt @@ -43,7 +43,7 @@ x is the square root of m1 times d1, and the mass doesn't change, so it depends What's the rate at which d1 changes? Well, that's the velocity of the big block. Let's call that v1. -Likewise, the rate of change for y is the square root of m2 times v2. +Likewise, the rate of change for y is going to be the square root of m2 times v2. Now, notice what the magnitude of our little configuration space changing vector is. Using the Pythagorean theorem, it's the square root of the sum of each of these component rates of change squared, which is square root of m1 times v1 squared plus m2 times v2 squared. This inner expression should look awfully familiar, it's exactly twice the kinetic energy of our system. @@ -74,7 +74,7 @@ So our configuration point is bouncing off that horizontal line as if it was a m So step back a moment and think about what this means for our original question of counting block collisions and trying to understand why on earth pi would show up. We can translate it to a completely different question. If you shine a beam of light at a pair of mirrors, meeting each other at some angle, let's say theta, how many times would that light bounce off of the mirrors as a function of that angle? -Remember, the mass ratio of our blocks determines this angle theta in the analogy. +Remember, the mass ratio of our blocks completely determines this angle theta in the analogy. Now I can hear some of you complaining, haven't we just replaced one tricky setup with another? This might make for a cute analogy, but how is it progress? It's true that counting the number of light bounces is hard, but now we have a helpful trick. diff --git a/2019/clacks-via-light/hungarian/sentence_translations.json b/2019/clacks-via-light/hungarian/sentence_translations.json index 85e982b74..f70f66fc7 100644 --- a/2019/clacks-via-light/hungarian/sentence_translations.json +++ b/2019/clacks-via-light/hungarian/sentence_translations.json @@ -360,7 +360,7 @@ "end": 352.7 }, { - "input": "Likewise, the rate of change for y is the square root of m2 times v2.", + "input": "Likewise, the rate of change for y is going to be the square root of m2 times v2.", "translatedText": "Hasonlóképpen, az y változásának mértéke az m2 négyzetgyöke szorozva v2-vel.", "model": "DeepL", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 611.78 }, { - "input": "Remember, the mass ratio of our blocks determines this angle theta in the analogy.", + "input": "Remember, the mass ratio of our blocks completely determines this angle theta in the analogy.", "translatedText": "Ne feledjük, hogy a blokkjaink tömegaránya határozza meg ezt a theta szöget az analógiában.", "model": "DeepL", "n_reviews": 0, diff --git a/2019/clacks-via-light/italian/sentence_translations.json b/2019/clacks-via-light/italian/sentence_translations.json index 24bf8f82f..80f598ae5 100644 --- a/2019/clacks-via-light/italian/sentence_translations.json +++ b/2019/clacks-via-light/italian/sentence_translations.json @@ -404,7 +404,7 @@ "end": 352.7 }, { - "input": "Likewise, the rate of change for y is the square root of m2 times v2.", + "input": "Likewise, the rate of change for y is going to be the square root of m2 times v2.", "translatedText": "Allo stesso modo, il tasso di variazione di y è la radice quadrata di m2 per v2.", "model": "DeepL", "from_community_srt": "Allo stesso modo, il tasso di variazione per y è sqrt (m2) * v2.", @@ -681,7 +681,7 @@ "end": 611.78 }, { - "input": "Remember, the mass ratio of our blocks determines this angle theta in the analogy.", + "input": "Remember, the mass ratio of our blocks completely determines this angle theta in the analogy.", "translatedText": "Ricorda che il rapporto di massa dei nostri blocchi determina l'angolo theta nell'analogia.", "model": "DeepL", "from_community_srt": "in funzione di ciò angolo? Ricorda, il rapporto di massa dei nostri blocchi determina questo angolo theta nell'analogia.", diff --git a/2019/clacks-via-light/polish/sentence_translations.json b/2019/clacks-via-light/polish/sentence_translations.json index fb617e73d..00dcea4c5 100644 --- a/2019/clacks-via-light/polish/sentence_translations.json +++ b/2019/clacks-via-light/polish/sentence_translations.json @@ -360,7 +360,7 @@ "end": 352.7 }, { - "input": "Likewise, the rate of change for y is the square root of m2 times v2.", + "input": "Likewise, the rate of change for y is going to be the square root of m2 times v2.", "translatedText": "", "from_community_srt": "Podobnie szybkość zmian dla y wynosi sqrt(m₂)*v₂.", "n_reviews": 0, @@ -607,7 +607,7 @@ "end": 611.78 }, { - "input": "Remember, the mass ratio of our blocks determines this angle theta in the analogy.", + "input": "Remember, the mass ratio of our blocks completely determines this angle theta in the analogy.", "translatedText": "", "from_community_srt": "w zależności od tego kąta? Pamiętaj, że stosunek masy naszych bloków determinuje ten kąt 𝜃 w naszej analogii.", "n_reviews": 0, diff --git a/2019/clacks-via-light/portuguese/sentence_translations.json b/2019/clacks-via-light/portuguese/sentence_translations.json index 0964e2924..2b335bab7 100644 --- a/2019/clacks-via-light/portuguese/sentence_translations.json +++ b/2019/clacks-via-light/portuguese/sentence_translations.json @@ -405,7 +405,7 @@ "end": 352.7 }, { - "input": "Likewise, the rate of change for y is the square root of m2 times v2.", + "input": "Likewise, the rate of change for y is going to be the square root of m2 times v2.", "translatedText": "Da mesma forma, a taxa de variação de y é a raiz quadrada de m2 vezes v2.", "model": "google_nmt", "from_community_srt": "Da mesma forma, a taxa de mudança para y será a raiz quadrada de m2 vezes v2.", @@ -683,7 +683,7 @@ "end": 611.78 }, { - "input": "Remember, the mass ratio of our blocks determines this angle theta in the analogy.", + "input": "Remember, the mass ratio of our blocks completely determines this angle theta in the analogy.", "translatedText": "Lembre-se, a proporção de massa dos nossos blocos determina este ângulo teta na analogia.", "model": "google_nmt", "from_community_srt": "como uma função daquele ângulo? Lembre-se, a razão das massas dos nossos blocos determina completamente o ângulo theta na nossa analogia.", diff --git a/2019/clacks-via-light/vietnamese/sentence_translations.json b/2019/clacks-via-light/vietnamese/sentence_translations.json index d8f287395..420871cb6 100644 --- a/2019/clacks-via-light/vietnamese/sentence_translations.json +++ b/2019/clacks-via-light/vietnamese/sentence_translations.json @@ -404,7 +404,7 @@ "end": 352.7 }, { - "input": "Likewise, the rate of change for y is the square root of m2 times v2.", + "input": "Likewise, the rate of change for y is going to be the square root of m2 times v2.", "translatedText": "Tương tự, tốc độ thay đổi của y là căn bậc hai của m2 nhân v2.", "model": "google_nmt", "from_community_srt": "Tương tự, tốc độ thay đổi của y là sqrt (m2) * v2.", @@ -681,7 +681,7 @@ "end": 611.78 }, { - "input": "Remember, the mass ratio of our blocks determines this angle theta in the analogy.", + "input": "Remember, the mass ratio of our blocks completely determines this angle theta in the analogy.", "translatedText": "Hãy nhớ rằng, tỷ lệ khối lượng của các khối của chúng ta xác định góc theta này trong phép tương tự.", "model": "google_nmt", "from_community_srt": "như là một chức năng của điều đó góc? Hãy nhớ rằng, tỷ lệ khối lượng của các khối của chúng tôi xác định góc này theta trong tương tự.", diff --git a/2019/clacks/arabic/sentence_translations.json b/2019/clacks/arabic/sentence_translations.json index 6c9903b64..01b9d0a17 100644 --- a/2019/clacks/arabic/sentence_translations.json +++ b/2019/clacks/arabic/sentence_translations.json @@ -392,7 +392,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "شكرًا لك.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks/bengali/sentence_translations.json b/2019/clacks/bengali/sentence_translations.json index 8a9b6ab2b..f8c5151e6 100644 --- a/2019/clacks/bengali/sentence_translations.json +++ b/2019/clacks/bengali/sentence_translations.json @@ -128,7 +128,7 @@ "end": 108.16 }, { - "input": "Well, hang on, wait for it…wait for it… Okay, 314 clacks. ", + "input": "Well, actually, hang on. Wait for it. Wait for it. Okay, 314 clacks. ", "translatedText": "কিন্তু যদি আমরা কিছুক্ষন অপেক্ষা করি তাহলেই বোঝা যাবে আসলে এই সংখ্যা টি 314।", "model": "google_nmt", "n_reviews": 1, diff --git a/2019/clacks/bulgarian/sentence_translations.json b/2019/clacks/bulgarian/sentence_translations.json index 0dbb1310e..c98946ca4 100644 --- a/2019/clacks/bulgarian/sentence_translations.json +++ b/2019/clacks/bulgarian/sentence_translations.json @@ -358,7 +358,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "", "n_reviews": 0, "start": 301.62, diff --git a/2019/clacks/chinese/sentence_translations.json b/2019/clacks/chinese/sentence_translations.json index 711038877..aa72a79f3 100644 --- a/2019/clacks/chinese/sentence_translations.json +++ b/2019/clacks/chinese/sentence_translations.json @@ -144,7 +144,7 @@ "end": 108.16 }, { - "input": "Well, hang on, wait for it…wait for it… Okay, 314 clacks. ", + "input": "Well, actually, hang on. Wait for it. Wait for it. Okay, 314 clacks. ", "translatedText": "好吧，等一下，等一下……等 一下……好吧，314 声。", "model": "google_nmt", "from_community_srt": "等等...再等等...马上就好...好了， 314声脆响。", diff --git a/2019/clacks/czech/sentence_translations.json b/2019/clacks/czech/sentence_translations.json index cba9b46ee..e7526c635 100644 --- a/2019/clacks/czech/sentence_translations.json +++ b/2019/clacks/czech/sentence_translations.json @@ -356,7 +356,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "", "n_reviews": 0, "start": 301.62, diff --git a/2019/clacks/dutch/sentence_translations.json b/2019/clacks/dutch/sentence_translations.json index 9469e56e9..71515204b 100644 --- a/2019/clacks/dutch/sentence_translations.json +++ b/2019/clacks/dutch/sentence_translations.json @@ -356,7 +356,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "", "from_community_srt": "-=X=- Vertaling door GijsjeVids -=X=-", "n_reviews": 0, diff --git a/2019/clacks/english/captions.srt b/2019/clacks/english/captions.srt index 9bed112bf..239ab9073 100644 --- a/2019/clacks/english/captions.srt +++ b/2019/clacks/english/captions.srt @@ -119,11 +119,11 @@ all happening very rapidly at one point, adding up and all to 313 total collisio Well, actually, hang on. 31 -00:01:50,660 --> 00:01:50,040 +00:01:50,660 --> 00:01:51,480 Wait for it. 32 -00:01:50,660 --> 00:01:54,280 +00:01:51,480 --> 00:01:54,280 Wait for it. 33 @@ -324,5 +324,5 @@ It's a hard puzzle, so it never hurts to recruit some other smart minds to the t 82 00:05:01,620 --> 00:05:12,240 -Thank you. +Thanks for watching. I'll see you next time. Bye. diff --git a/2019/clacks/english/sentence_timings.json b/2019/clacks/english/sentence_timings.json index 9849eb08e..2a143c397 100644 --- a/2019/clacks/english/sentence_timings.json +++ b/2019/clacks/english/sentence_timings.json @@ -82,11 +82,11 @@ [ "Wait for it.", 110.66, - 110.04 + 111.48 ], [ "Wait for it.", - 110.66, + 111.48, 114.28 ], [ @@ -225,7 +225,7 @@ 291.64 ], [ - "Thank you.", + "Thanks for watching. I'll see you next time. Bye.", 301.62, 312.24 ] diff --git a/2019/clacks/english/transcript.txt b/2019/clacks/english/transcript.txt index 4d659fd5c..b5ebc3f13 100644 --- a/2019/clacks/english/transcript.txt +++ b/2019/clacks/english/transcript.txt @@ -43,4 +43,4 @@ In fact, you're going to see two separate methods, which are each as stunning an Delaying gratification though, I will make you wait until the next video to see what's going on. In the meantime, I highly encourage you to take a stab at it yourself, and be social about it. It's a hard puzzle, so it never hurts to recruit some other smart minds to the task. -Thank you. \ No newline at end of file +Thanks for watching. I'll see you next time. Bye. \ No newline at end of file diff --git a/2019/clacks/french/sentence_translations.json b/2019/clacks/french/sentence_translations.json index 7d4da3e26..ed1627cc4 100644 --- a/2019/clacks/french/sentence_translations.json +++ b/2019/clacks/french/sentence_translations.json @@ -356,7 +356,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "Merci.", "n_reviews": 0, "start": 301.62, diff --git a/2019/clacks/german/sentence_translations.json b/2019/clacks/german/sentence_translations.json index c0932719e..b0aca75d6 100644 --- a/2019/clacks/german/sentence_translations.json +++ b/2019/clacks/german/sentence_translations.json @@ -403,7 +403,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "Vielen Dank!", "model": "DeepL", "n_reviews": 0, diff --git a/2019/clacks/greek/sentence_translations.json b/2019/clacks/greek/sentence_translations.json index 67ee76943..58bee8e37 100644 --- a/2019/clacks/greek/sentence_translations.json +++ b/2019/clacks/greek/sentence_translations.json @@ -357,7 +357,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "", "n_reviews": 0, "start": 301.62, diff --git a/2019/clacks/hebrew/sentence_translations.json b/2019/clacks/hebrew/sentence_translations.json index 09ca28c2e..0f74dc2a5 100644 --- a/2019/clacks/hebrew/sentence_translations.json +++ b/2019/clacks/hebrew/sentence_translations.json @@ -402,7 +402,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "תודה.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks/hindi/sentence_translations.json b/2019/clacks/hindi/sentence_translations.json index 019dc9449..a81cfab1e 100644 --- a/2019/clacks/hindi/sentence_translations.json +++ b/2019/clacks/hindi/sentence_translations.json @@ -105,14 +105,14 @@ "end": 95.18 }, { - "input": "In that case, there would be quite a few more clacks, all happening very rapidly at one point, adding up to 313 total collisions.", + "input": "In that case, there would be quite a few more clacks, all happening very rapidly at one point, adding up and all to 313 total collisions.", "translatedText": "उस स्थिति में, कुछ और खड़खड़ाहटें होंगी, सभी एक बिंदु पर बहुत तेजी से घटित होंगी, जिससे कुल मिलाकर 313 टकराव होंगे।", "n_reviews": 0, "start": 95.86, "end": 108.16 }, { - "input": "Well, hang on, wait for it…wait for it… Okay, 314 clacks.", + "input": "Well, actually, hang on. Wait for it. Wait for it. Okay, 314 clacks.", "translatedText": "ठीक है, रुको, इसके लिए प्रतीक्षा करो...इसके लिए प्रतीक्षा करो... ठीक है, 314 क्लैक्स।", "n_reviews": 0, "start": 108.92, diff --git a/2019/clacks/hungarian/sentence_translations.json b/2019/clacks/hungarian/sentence_translations.json index cf0c2ffaa..550825be4 100644 --- a/2019/clacks/hungarian/sentence_translations.json +++ b/2019/clacks/hungarian/sentence_translations.json @@ -403,7 +403,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "Köszönöm.", "model": "DeepL", "n_reviews": 0, diff --git a/2019/clacks/indonesian/sentence_translations.json b/2019/clacks/indonesian/sentence_translations.json index 9635afc9d..e1457a697 100644 --- a/2019/clacks/indonesian/sentence_translations.json +++ b/2019/clacks/indonesian/sentence_translations.json @@ -400,7 +400,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "Terima kasih.", "model": "DeepL", "n_reviews": 0, diff --git a/2019/clacks/italian/sentence_translations.json b/2019/clacks/italian/sentence_translations.json index fa9954aa4..9f1778781 100644 --- a/2019/clacks/italian/sentence_translations.json +++ b/2019/clacks/italian/sentence_translations.json @@ -402,7 +402,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "Grazie.", "model": "DeepL", "from_community_srt": "È un problema difficile, quindi non fa mai male coinvolgere altre menti per affrontarlo insieme.", diff --git a/2019/clacks/japanese/sentence_translations.json b/2019/clacks/japanese/sentence_translations.json index a7b33bdf5..66eeab755 100644 --- a/2019/clacks/japanese/sentence_translations.json +++ b/2019/clacks/japanese/sentence_translations.json @@ -120,7 +120,7 @@ "end": 95.18 }, { - "input": "In that case, there would be quite a few more clacks, all happening very rapidly at one point, adding up to 313 total collisions.", + "input": "In that case, there would be quite a few more clacks, all happening very rapidly at one point, adding up and all to 313 total collisions.", "translatedText": "その場合、かなりの数のカチッという音が一度に非常に急速に発 生し、合計で 313 回の衝突が発生することになります。", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 108.16 }, { - "input": "Well, hang on, wait for it…wait for it… Okay, 314 clacks.", + "input": "Well, actually, hang on. Wait for it. Wait for it. Okay, 314 clacks.", "translatedText": "まあ、ちょっと待って…待って … さて、314 カチッ。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks/korean/sentence_translations.json b/2019/clacks/korean/sentence_translations.json index adab8f5ad..fb7a0964d 100644 --- a/2019/clacks/korean/sentence_translations.json +++ b/2019/clacks/korean/sentence_translations.json @@ -401,7 +401,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "감사합니다.", "model": "DeepL", "from_community_srt": "* 2019년 1월 20일에 해답이 공개됩니다. *", diff --git a/2019/clacks/marathi/sentence_translations.json b/2019/clacks/marathi/sentence_translations.json index 5f276fde6..fc28c9990 100644 --- a/2019/clacks/marathi/sentence_translations.json +++ b/2019/clacks/marathi/sentence_translations.json @@ -128,7 +128,7 @@ "end": 108.16 }, { - "input": "Well, hang on, wait for it…wait for it… Okay, 314 clacks. ", + "input": "Well, actually, hang on. Wait for it. Wait for it. Okay, 314 clacks. ", "translatedText": "बरं, थांबा, त्याची प्रतीक्षा करा... वाट पहा... ठीक आहे, ३१४ क्लॅक. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks/norwegian/sentence_translations.json b/2019/clacks/norwegian/sentence_translations.json index 0066bc232..5e7127d70 100644 --- a/2019/clacks/norwegian/sentence_translations.json +++ b/2019/clacks/norwegian/sentence_translations.json @@ -357,7 +357,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "", "n_reviews": 0, "start": 301.62, diff --git a/2019/clacks/persian/sentence_translations.json b/2019/clacks/persian/sentence_translations.json index 92fb9f8bb..148002e83 100644 --- a/2019/clacks/persian/sentence_translations.json +++ b/2019/clacks/persian/sentence_translations.json @@ -400,7 +400,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "متشکرم.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks/polish/sentence_translations.json b/2019/clacks/polish/sentence_translations.json index 487f46134..95b489d4e 100644 --- a/2019/clacks/polish/sentence_translations.json +++ b/2019/clacks/polish/sentence_translations.json @@ -356,7 +356,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "", "n_reviews": 0, "start": 301.62, diff --git a/2019/clacks/portuguese/sentence_translations.json b/2019/clacks/portuguese/sentence_translations.json index 2ca076d83..16c4744c0 100644 --- a/2019/clacks/portuguese/sentence_translations.json +++ b/2019/clacks/portuguese/sentence_translations.json @@ -401,7 +401,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "Obrigado.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks/russian/sentence_translations.json b/2019/clacks/russian/sentence_translations.json index 0a8a8bcd6..cb326dcd6 100644 --- a/2019/clacks/russian/sentence_translations.json +++ b/2019/clacks/russian/sentence_translations.json @@ -353,7 +353,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "Спасибо.", "n_reviews": 0, "start": 301.62, diff --git a/2019/clacks/serbian/sentence_translations.json b/2019/clacks/serbian/sentence_translations.json index a516b53ea..e66fc4e3d 100644 --- a/2019/clacks/serbian/sentence_translations.json +++ b/2019/clacks/serbian/sentence_translations.json @@ -358,7 +358,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "", "from_community_srt": "Rešenje će biti postavljeno 20. 1. 2019. za radoznale umove sa prostora Balkana preveo: Aleksandar Milinković", "n_reviews": 0, diff --git a/2019/clacks/spanish/sentence_translations.json b/2019/clacks/spanish/sentence_translations.json index 44b2517f0..405460340 100644 --- a/2019/clacks/spanish/sentence_translations.json +++ b/2019/clacks/spanish/sentence_translations.json @@ -356,7 +356,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "Gracias.", "n_reviews": 0, "start": 301.62, diff --git a/2019/clacks/tamil/sentence_translations.json b/2019/clacks/tamil/sentence_translations.json index cbfff6f9e..8af9011e8 100644 --- a/2019/clacks/tamil/sentence_translations.json +++ b/2019/clacks/tamil/sentence_translations.json @@ -128,7 +128,7 @@ "end": 108.16 }, { - "input": "Well, hang on, wait for it…wait for it… Okay, 314 clacks. ", + "input": "Well, actually, hang on. Wait for it. Wait for it. Okay, 314 clacks. ", "translatedText": "சரி, காத்திருங்கள், காத்திருங்கள்... காத்திருங்கள்... சரி, 314 கிளாக்ஸ். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks/telugu/sentence_translations.json b/2019/clacks/telugu/sentence_translations.json index fba681ef5..78e0f260f 100644 --- a/2019/clacks/telugu/sentence_translations.json +++ b/2019/clacks/telugu/sentence_translations.json @@ -128,7 +128,7 @@ "end": 108.16 }, { - "input": "Well, hang on, wait for it…wait for it… Okay, 314 clacks. ", + "input": "Well, actually, hang on. Wait for it. Wait for it. Okay, 314 clacks. ", "translatedText": "సరే, వేచి ఉండండి, దాని కోసం వేచి ఉండండి... దాని కోసం వేచి ఉండండి... సరే, 314 క్లాక్‌లు. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks/thai/sentence_translations.json b/2019/clacks/thai/sentence_translations.json index fa2c7aab7..c20ac1bc5 100644 --- a/2019/clacks/thai/sentence_translations.json +++ b/2019/clacks/thai/sentence_translations.json @@ -128,7 +128,7 @@ "end": 108.16 }, { - "input": "Well, hang on, wait for it…wait for it… Okay, 314 clacks. ", + "input": "Well, actually, hang on. Wait for it. Wait for it. Okay, 314 clacks. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks/turkish/sentence_translations.json b/2019/clacks/turkish/sentence_translations.json index ca6557d44..e8df8fcdc 100644 --- a/2019/clacks/turkish/sentence_translations.json +++ b/2019/clacks/turkish/sentence_translations.json @@ -403,7 +403,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "Teşekkür ederim.", "model": "DeepL", "n_reviews": 0, diff --git a/2019/clacks/ukrainian/sentence_translations.json b/2019/clacks/ukrainian/sentence_translations.json index 18021e584..e9c1a380b 100644 --- a/2019/clacks/ukrainian/sentence_translations.json +++ b/2019/clacks/ukrainian/sentence_translations.json @@ -128,7 +128,7 @@ "end": 108.16 }, { - "input": "Well, hang on, wait for it…wait for it… Okay, 314 clacks. ", + "input": "Well, actually, hang on. Wait for it. Wait for it. Okay, 314 clacks. ", "translatedText": "Ну, почекайте, зачекайте… зачекайте… Гаразд, 314 ударів. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks/urdu/sentence_translations.json b/2019/clacks/urdu/sentence_translations.json index 608d203f0..a4e963878 100644 --- a/2019/clacks/urdu/sentence_translations.json +++ b/2019/clacks/urdu/sentence_translations.json @@ -128,7 +128,7 @@ "end": 108.16 }, { - "input": "Well, hang on, wait for it…wait for it… Okay, 314 clacks. ", + "input": "Well, actually, hang on. Wait for it. Wait for it. Okay, 314 clacks. ", "translatedText": "ٹھیک ہے، انتظار کرو، اس کا انتظار کرو… اس کا انتظار کرو… ٹھیک ہے، 314 گھڑیاں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/clacks/vietnamese/sentence_translations.json b/2019/clacks/vietnamese/sentence_translations.json index d15a0fdc2..f1167d767 100644 --- a/2019/clacks/vietnamese/sentence_translations.json +++ b/2019/clacks/vietnamese/sentence_translations.json @@ -399,7 +399,7 @@ "end": 291.64 }, { - "input": "Thank you.", + "input": "Thanks for watching. I'll see you next time. Bye.", "translatedText": "Cảm ơn.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/arabic/sentence_translations.json b/2019/cramers-rule/arabic/sentence_translations.json index 3cf0f1061..9ad9b8df6 100644 --- a/2019/cramers-rule/arabic/sentence_translations.json +++ b/2019/cramers-rule/arabic/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "على سبيل المثال، يمكن أن يكون لديك متجهان يشيران عمومًا في نفس الاتجاه مع منتج نقطي موجب، ويتم فصلهما عن بعضهما البعض أثناء التحويل بطريقة تجعلهما في النهاية منتجًا نقطيًا سالبًا. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "إن التفكير في حالات أكثر عمومية وإقناع نفسك بنجاحها هو المكان الذي سيحدث فيه كل التعلم، أكثر بكثير من الاستماع إلى بعض الأشخاص على YouTube وهم يشرحون المنطق مرة أخرى. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/bengali/sentence_translations.json b/2019/cramers-rule/bengali/sentence_translations.json index 6972ad9e1..352983270 100644 --- a/2019/cramers-rule/bengali/sentence_translations.json +++ b/2019/cramers-rule/bengali/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "উদাহরণস্বরূপ, আপনার কাছে একটি ইতিবাচক ডট পণ্যের সাথে সাধারণত একই দিকে নির্দেশিত দুটি ভেক্টর থাকতে পারে, যেগুলি রূপান্তরের সময় একে অপরের থেকে এমনভাবে টানা হয় যাতে তাদের একটি নেতিবাচক ডট পণ্য থাকে।", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "এই জাতীয় আরও সাধারণ ক্ষেত্রে চিন্তা করা এবং নিজেকে বোঝানো যে এটি কাজ করে এবং কেন এটি কাজ করে যেখানে সমস্ত শিক্ষা সত্যিই ঘটে, YouTube-এ কিছু বন্ধুর কথা শোনার চেয়ে আরও অনেক কিছু আপনাকে আবার একই যুক্তির মধ্য দিয়ে নিয়ে যায়।", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/chinese/sentence_translations.json b/2019/cramers-rule/chinese/sentence_translations.json index e952be26f..83cad9c37 100644 --- a/2019/cramers-rule/chinese/sentence_translations.json +++ b/2019/cramers-rule/chinese/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "例如，您可能有两个通常指向同一方向且具 有正点积的向量，这两个向量在转换过程中彼此分 开，最终导致它们具有负点积。", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "思考像这样的更一般的案例，并说服自己它是有效的以及为 什么它有效，这是所有学习真正发生的地方，比听 YouT ube 上的某个人再次引导你进行相同的推理要重要得多。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/english/captions.srt b/2019/cramers-rule/english/captions.srt index 07d925260..5c5166dea 100644 --- a/2019/cramers-rule/english/captions.srt +++ b/2019/cramers-rule/english/captions.srt @@ -47,16 +47,16 @@ a little bit of dot products, and of course linear systems of equations, so be sure to watch the relevant videos on those topics if you're unfamiliar or rusty. 13 -00:00:51,020 --> 00:00:54,339 -But first I should say up front that this Cramer's rule is +00:00:51,020 --> 00:00:54,311 +But first I should say up front that this Cramer's rule is not 14 -00:00:54,339 --> 00:00:57,715 -not actually the best way for computing solutions to linear +00:00:54,311 --> 00:00:58,386 +actually the best way for computing solutions to linear systems of equations, 15 -00:00:57,715 --> 00:01:01,260 -systems of equations, Gaussian elimination for example will always be faster. +00:00:58,386 --> 00:01:01,260 +Gaussian elimination for example will always be faster. 16 00:01:01,980 --> 00:01:03,520 @@ -231,16 +231,16 @@ For most linear transformations the dot product before and after the transformation will look very different. 59 -00:04:00,800 --> 00:04:04,135 -For example you could have two vectors generally pointing in the same +00:04:00,800 --> 00:04:04,479 +For example, you could have two vectors generally pointing in the same direction 60 -00:04:04,135 --> 00:04:07,803 -direction with a positive dot product which get pulled apart from each other +00:04:04,479 --> 00:04:08,067 +with a positive dot product, which get pulled apart from each other during the 61 -00:04:07,803 --> 00:04:11,520 -during the transformation in such a way that they have a negative dot product. +00:04:08,067 --> 00:04:11,520 +transformation in such a way that they end up having a negative dot product. 62 00:04:12,220 --> 00:04:15,592 @@ -291,19 +291,19 @@ they correspond to rigid motion with no stretching or squishing or morphing. Solving a linear system with an orthonormal matrix is actually super easy. 74 -00:04:57,260 --> 00:05:01,095 +00:04:57,260 --> 00:05:01,005 Because dot products are preserved, taking the dot product between the 75 -00:05:01,095 --> 00:05:05,038 -output vector and all the columns of your matrix will be the same as taking the +00:05:01,005 --> 00:05:05,014 +output vector and all the columns of your matrix will be the same as taking 76 -00:05:05,038 --> 00:05:09,090 -dot product between the mystery input vector and all of the basis vectors, +00:05:05,014 --> 00:05:09,181 +the dot product between the mystery input vector and all of the basis vectors, 77 -00:05:09,090 --> 00:05:12,980 +00:05:09,181 --> 00:05:12,980 which is the same as just finding the coordinates of that mystery input. 78 @@ -699,26 +699,30 @@ and the mystery input vector, what happens to that volume after the transformati And what are the various ways you can compute that volume? 176 -00:11:28,340 --> 00:11:32,959 -Really, pause and think through the details of generalizing this to higher dimensions, +00:11:28,340 --> 00:11:31,286 +Really, pause and take a moment to think through the details 177 -00:11:32,959 --> 00:11:37,420 -finding an expression for each coordinate of the solution to a larger linear system. +00:11:31,286 --> 00:11:34,425 +of generalizing this to higher dimensions, finding an expression 178 +00:11:34,425 --> 00:11:37,420 +for each coordinate of the solution to a larger linear system. + +179 00:11:38,060 --> 00:11:41,495 Thinking through more general cases like this and convincing yourself that it -179 +180 00:11:41,495 --> 00:11:44,359 works and why it works is where all the learning really happens, -180 +181 00:11:44,359 --> 00:11:47,795 much more so than listening to some dude on YouTube walk you through the same -181 +182 00:11:47,795 --> 00:11:48,500 reasoning again. diff --git a/2019/cramers-rule/english/sentence_timings.json b/2019/cramers-rule/english/sentence_timings.json index 593f30afc..97afcbd2b 100644 --- a/2019/cramers-rule/english/sentence_timings.json +++ b/2019/cramers-rule/english/sentence_timings.json @@ -135,7 +135,7 @@ 240.24 ], [ - "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", 240.8, 251.52 ], @@ -420,7 +420,7 @@ 687.48 ], [ - "Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system.", + "Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system.", 688.34, 697.42 ], diff --git a/2019/cramers-rule/english/transcript.txt b/2019/cramers-rule/english/transcript.txt index 2840eb240..5b80dc52a 100644 --- a/2019/cramers-rule/english/transcript.txt +++ b/2019/cramers-rule/english/transcript.txt @@ -25,7 +25,7 @@ So maybe you hope that after the transformation the dot products with the transf That'd be fantastic because we know what the transformed version of each of those vectors are. There's just one problem with it, it's not at all true. For most linear transformations the dot product before and after the transformation will look very different. -For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product. +For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product. Likewise things that start off perpendicular with dot product 0, like the two basis vectors, quite often don't stay perpendicular to each other after the transformation, that is they don't preserve that 0 dot product. And looking at the example I just showed dot products certainly aren't preserved, they tend to get bigger since most vectors are getting stretched out. In fact, worthwhile side note here, transformations which do preserve dot products are special enough to have their own name, orthonormal transformations. @@ -82,5 +82,5 @@ Here, I'll give you a little bit of momentum. What we have is a known transformation given by some 3x3 matrix and a known output vector given by the right side of our linear system, and we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? -Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. +Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again. \ No newline at end of file diff --git a/2019/cramers-rule/french/sentence_translations.json b/2019/cramers-rule/french/sentence_translations.json index a1d54a932..2a1349a52 100644 --- a/2019/cramers-rule/french/sentence_translations.json +++ b/2019/cramers-rule/french/sentence_translations.json @@ -216,7 +216,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "Par exemple, vous pourriez avoir deux vecteurs pointant généralement dans la même direction avec un produit scalaire positif qui se séparent l'un de l'autre pendant la transformation de telle manière qu'ils ont un produit scalaire négatif.", "from_community_srt": "Par exemple, on peut prendre deux vecteurs ayant plus ou moins la même direction, avec un produit positif, qui s'écartent l'un de l'autre par la transformation, de telle façon qu'ils finissent par avoir un produit négatif.", "n_reviews": 0, @@ -671,7 +671,7 @@ "end": 687.48 }, { - "input": "Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system.", + "input": "Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system.", "translatedText": "Vraiment, faites une pause et réfléchissez aux détails de la généralisation de cela à des dimensions supérieures, en trouvant une expression pour chaque coordonnée de la solution à un système linéaire plus grand.", "from_community_srt": "Vraiment, faites pause et prenez un moment pour considérer tous les détails pour généraliser à d'autres dimensions, en trouvant une expression pour chaque coordonnée de la solution du système élargi.", "n_reviews": 0, diff --git a/2019/cramers-rule/german/sentence_translations.json b/2019/cramers-rule/german/sentence_translations.json index fa3cc69f7..ba845abdc 100644 --- a/2019/cramers-rule/german/sentence_translations.json +++ b/2019/cramers-rule/german/sentence_translations.json @@ -242,7 +242,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "Du könntest zum Beispiel zwei Vektoren haben, die im Allgemeinen in die gleiche Richtung zeigen und ein positives Punktprodukt haben, die aber während der Transformation so auseinandergezogen werden, dass sie ein negatives Punktprodukt haben.", "model": "DeepL", "from_community_srt": "Beispielsweise könnten Sie im Allgemeinen zwei Vektoren haben in die gleiche Richtung zeigen, mit einem positiven Punktprodukt, das von jedem weggezogen wird andere während der Transformation, in einem solchen Art und Weise, dass sie dann ein negatives Punktprodukt haben.", @@ -755,7 +755,7 @@ "end": 687.48 }, { - "input": "Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system.", + "input": "Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system.", "translatedText": "Halte wirklich inne und denke über die Details der Verallgemeinerung auf höhere Dimensionen nach, indem du einen Ausdruck für jede Koordinate der Lösung eines größeren linearen Systems findest.", "model": "DeepL", "from_community_srt": "Machen Sie wirklich eine Pause und nehmen Sie sich einen Moment Zeit zum Nachdenken die Details der Verallgemeinerung auf höher Maße; für jeden einen Ausdruck finden Koordinate der Lösung zu größeren linearen Systeme.", diff --git a/2019/cramers-rule/hebrew/sentence_translations.json b/2019/cramers-rule/hebrew/sentence_translations.json index bb64f070a..cb5e390d1 100644 --- a/2019/cramers-rule/hebrew/sentence_translations.json +++ b/2019/cramers-rule/hebrew/sentence_translations.json @@ -203,7 +203,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "לדוגמה, אתה יכול לקבל שני וקטורים המצביעים בדרך כלל לאותו כיוון, עם מכפלת נקודה חיובית, אשר נמשכים זה מזה במהלך הטרנספורמציה, באופן שלאחר מכן יש להם מכפלת נקודה שלילית. ", "n_reviews": 0, "start": 240.8, @@ -574,7 +574,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "לחשוב על מקרים כלליים יותר ולשכנע את עצמך שזה עובד זה המקום שבו כל הלמידה תתרחש, הרבה יותר מאשר להאזין לאיזה בחור ביוטיוב שעובר שוב על ההיגיון. ", "n_reviews": 0, "start": 665.1, diff --git a/2019/cramers-rule/hindi/sentence_translations.json b/2019/cramers-rule/hindi/sentence_translations.json index ecb15e043..f0b0fa659 100644 --- a/2019/cramers-rule/hindi/sentence_translations.json +++ b/2019/cramers-rule/hindi/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "उदाहरण के लिए, आपके पास एक सकारात्मक डॉट उत्पाद के साथ आम तौर पर एक ही दिशा में इंगित करने वाले दो वेक्टर हो सकते हैं, जो परिवर्तन के दौरान एक-दूसरे से इस तरह से अलग हो जाते हैं कि अंत में उनमें एक नकारात्मक डॉट उत्पाद बन जाता है।", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "इस तरह के अधिक सामान्य मामलों के बारे में सोचना और अपने आप को यह विश्वास दिलाना कि यह काम करता है और यह क्यों काम करता है, यहीं पर वास्तव में सारी सीख मिलती है, यूट्यूब पर किसी व्यक्ति को सुनने से कहीं अधिक, जो आपको फिर से उसी तर्क के बारे में बताता है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/hungarian/sentence_translations.json b/2019/cramers-rule/hungarian/sentence_translations.json index 9c73045fd..f01f95b62 100644 --- a/2019/cramers-rule/hungarian/sentence_translations.json +++ b/2019/cramers-rule/hungarian/sentence_translations.json @@ -216,7 +216,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "Például lehet két olyan vektor, amely általában ugyanabba az irányba mutat, pozitív ponttényezővel, de a transzformáció során úgy távolodnak el egymástól, hogy negatív ponttényezőt kapnak.", "model": "DeepL", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 687.48 }, { - "input": "Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system.", + "input": "Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system.", "translatedText": "Tényleg állj meg, és gondolkodj el a részleteken, hogy ezt magasabb dimenziókra általánosítva egy nagyobb lineáris rendszer megoldásának minden egyes koordinátájára találj egy kifejezést.", "model": "DeepL", "n_reviews": 0, diff --git a/2019/cramers-rule/indonesian/sentence_translations.json b/2019/cramers-rule/indonesian/sentence_translations.json index 7157b5500..524cac3bc 100644 --- a/2019/cramers-rule/indonesian/sentence_translations.json +++ b/2019/cramers-rule/indonesian/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "Misalnya, Anda dapat mempunyai dua buah vektor yang umumnya menunjuk ke arah yang sama dengan perkalian titik positif, yang akan terpisah satu sama lain selama transformasi sedemikian rupa sehingga menghasilkan perkalian titik negatif. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "Memikirkan kasus-kasus yang lebih umum seperti ini dan meyakinkan diri sendiri bahwa hal ini berhasil dan mengapa hal itu berhasil adalah saat di mana semua pembelajaran benar-benar terjadi, lebih dari sekadar mendengarkan beberapa pria di YouTube memandu Anda melalui alasan yang sama lagi. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/italian/sentence_translations.json b/2019/cramers-rule/italian/sentence_translations.json index fe45711c1..bc6d933f2 100644 --- a/2019/cramers-rule/italian/sentence_translations.json +++ b/2019/cramers-rule/italian/sentence_translations.json @@ -203,7 +203,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "Ad esempio, potresti avere due vettori che puntano generalmente nella stessa direzione, con un prodotto scalare positivo, che vengono allontanati l'uno dall'altro durante la trasformazione, in modo tale da avere poi un prodotto scalare negativo. ", "n_reviews": 0, "start": 240.8, @@ -574,7 +574,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "Pensare a casi più generali e convincersi che funziona è il luogo in cui avverrà tutto l'apprendimento, molto più che ascoltare qualche tizio su YouTube che ripercorre il ragionamento. ", "n_reviews": 0, "start": 665.1, diff --git a/2019/cramers-rule/japanese/sentence_translations.json b/2019/cramers-rule/japanese/sentence_translations.json index 643831959..b55d2c27f 100644 --- a/2019/cramers-rule/japanese/sentence_translations.json +++ b/2019/cramers-rule/japanese/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "たとえば、正の内積を持つ 2 つのベクトルが通常は同じ方 向を向いている場合、変換中にこれらのベクトルが互いに引き離されて 、最終的に負の内積になる場合があります。", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "このようなより一般的なケースを考えて、それが機能すること、そしてなぜ機能 するのかを自分に納得させることによって、すべての学習が実際に行われます。Y ouTube で同じ推論をもう一度説明するのを聞くよりもはるかに重要です。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/korean/sentence_translations.json b/2019/cramers-rule/korean/sentence_translations.json index 9c1d928cb..a7cce1e99 100644 --- a/2019/cramers-rule/korean/sentence_translations.json +++ b/2019/cramers-rule/korean/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "예를 들어, 일반적으로 양의 내적을 사용하여 동일한 방향을 가리키는 두 개의 벡터가 있을 수 있습니다. 이 벡터는 변환 중에 서로 떨어져서 결국 음의 내적을 갖게 됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "이와 같은 보다 일반적인 사례를 생각하고 그것이 작동하고 왜 작동하는지 스스로 확신하는 것이 모든 학습이 실제로 일어나는 곳입니다. YouTube에서 누군가의 말을 듣는 것보다 훨씬 더 동일한 추론을 다시 안내합니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/marathi/sentence_translations.json b/2019/cramers-rule/marathi/sentence_translations.json index fba0a09e5..824ba237c 100644 --- a/2019/cramers-rule/marathi/sentence_translations.json +++ b/2019/cramers-rule/marathi/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "उदाहरणार्थ, तुमच्याकडे पॉझिटिव्ह डॉट उत्पादनासह एकाच दिशेने दोन वेक्टर असू शकतात, जे परिवर्तनादरम्यान एकमेकांपासून अशा प्रकारे खेचले जातात की त्यांना नकारात्मक बिंदू उत्पादन मिळते. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "यासारख्या सामान्य प्रकरणांचा विचार करणे आणि ते कार्य करते आणि ते का कार्य करते हे स्वतःला पटवून देणे हे सर्व शिक्षण खरोखरच घडते तेथेच, YouTube वर काही मित्रांचं ऐकण्यापेक्षा तुम्हाला पुन्हा त्याच तर्काकडे नेले जाईल. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/persian/sentence_translations.json b/2019/cramers-rule/persian/sentence_translations.json index 78b9ac00d..3794694d9 100644 --- a/2019/cramers-rule/persian/sentence_translations.json +++ b/2019/cramers-rule/persian/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "به عنوان مثال، شما می توانید دو بردار داشته باشید که به طور کلی در یک جهت با یک نقطه مثبت نشان می دهند، که در طول تبدیل به گونه ای از یکدیگر جدا می شوند که در نهایت حاصل ضرب نقطه منفی دارند. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "فکر کردن به موارد کلی‌تر مانند این و متقاعد کردن خود به اینکه کار می‌کند و چرا کار می‌کند، جایی است که همه یادگیری‌ها واقعاً اتفاق می‌افتد، بسیار بیشتر از گوش دادن به برخی از دوستان در YouTube که شما را دوباره در همان استدلال راهنمایی می‌کند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/polish/sentence_translations.json b/2019/cramers-rule/polish/sentence_translations.json index 4856bb858..b2116ae91 100644 --- a/2019/cramers-rule/polish/sentence_translations.json +++ b/2019/cramers-rule/polish/sentence_translations.json @@ -215,7 +215,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "", "from_community_srt": "Na przykład, moglibyśmy mieć dwa wektory wskazujące ogólnie ten sam kierunek, z dodatnim iloczynem skalarnym, które są odciągane od siebie przy przekształceniu, w taki sposób, że mają potem ujemny iloczyn skalarny.", "n_reviews": 0, @@ -671,7 +671,7 @@ "end": 687.48 }, { - "input": "Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system.", + "input": "Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system.", "translatedText": "", "from_community_srt": "Serio, zatrzymaj film i poświęć moment na to, jak uogólnić ten pomysł na wyższe wymiary; czyli jak znajdować daną współrzędną rozwiązania układu równań liniowych.", "n_reviews": 0, diff --git a/2019/cramers-rule/portuguese/sentence_translations.json b/2019/cramers-rule/portuguese/sentence_translations.json index 9e956af95..02dfb6419 100644 --- a/2019/cramers-rule/portuguese/sentence_translations.json +++ b/2019/cramers-rule/portuguese/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "Por exemplo, você poderia ter dois vetores geralmente apontando na mesma direção com um produto escalar positivo, que são separados um do outro durante a transformação de tal forma que acabam tendo um produto escalar negativo. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "Pensar em casos mais gerais como esse e convencer-se de que funciona e por que funciona é onde todo o aprendizado realmente acontece, muito mais do que ouvir algum cara no YouTube explicar o mesmo raciocínio novamente. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/russian/sentence_translations.json b/2019/cramers-rule/russian/sentence_translations.json index 7822fc837..f43a80e37 100644 --- a/2019/cramers-rule/russian/sentence_translations.json +++ b/2019/cramers-rule/russian/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "Например, у вас могут быть два вектора, обычно указывающие в одном направлении с положительным скалярным произведением, которые во время преобразования отделяются друг от друга таким образом, что в конечном итоге имеют отрицательное скалярное произведение. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "Обдумывание более общих случаев, подобных этому, и убеждение себя в том, что это работает и почему это работает, — вот где действительно происходит все обучение, гораздо больше, чем слушать, как какой-то чувак на YouTube снова проводит вас через те же рассуждения. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/spanish/sentence_translations.json b/2019/cramers-rule/spanish/sentence_translations.json index 34a9a6671..b890051f7 100644 --- a/2019/cramers-rule/spanish/sentence_translations.json +++ b/2019/cramers-rule/spanish/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "Por ejemplo, podría tener dos vectores que generalmente apuntan en la misma dirección con un producto escalar positivo, que se separan entre sí durante la transformación de tal manera que terminan teniendo un producto escalar negativo. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "Pensar en casos más generales como este y convencerse de que funciona y de por qué funciona es donde realmente ocurre todo el aprendizaje, mucho más que escuchar a un tipo en YouTube explicarle el mismo razonamiento nuevamente. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/tamil/sentence_translations.json b/2019/cramers-rule/tamil/sentence_translations.json index bb62a968b..65c14f390 100644 --- a/2019/cramers-rule/tamil/sentence_translations.json +++ b/2019/cramers-rule/tamil/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "எடுத்துக்காட்டாக, நேர்மறை புள்ளி தயாரிப்புடன் பொதுவாக ஒரே திசையில் இரண்டு திசையன்களை நீங்கள் வைத்திருக்கலாம், அவை மாற்றத்தின் போது ஒருவருக்கொருவர் விலகி எதிர்மறை புள்ளி தயாரிப்புடன் முடிவடையும். ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "இது போன்ற பொதுவான நிகழ்வுகளைப் பற்றி சிந்தித்து, அது வேலை செய்கிறது மற்றும் ஏன் வேலை செய்கிறது என்று உங்களை நீங்களே நம்பிக் கொள்வது, உண்மையில் கற்றல் அனைத்தும் நடக்கும் இடத்திலேயே, YouTube இல் சில தோழர்களைக் கேட்பதை விட, மீண்டும் அதே காரணத்தை உங்களுக்குத் தெரிவிக்கும். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/telugu/sentence_translations.json b/2019/cramers-rule/telugu/sentence_translations.json index 0f8918d00..824a02561 100644 --- a/2019/cramers-rule/telugu/sentence_translations.json +++ b/2019/cramers-rule/telugu/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "ఉదాహరణకు, మీరు సానుకూల డాట్ ఉత్పత్తితో సాధారణంగా ఒకే దిశలో సూచించే రెండు వెక్టర్‌లను కలిగి ఉండవచ్చు, అవి రూపాంతరం సమయంలో ఒకదానికొకటి వేరుగా ఉంటాయి, తద్వారా అవి ప్రతికూల డాట్ ఉత్పత్తిని కలిగి ఉంటాయి. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "ఇలాంటి మరిన్ని సాధారణ కేసుల గురించి ఆలోచించడం మరియు ఇది పని చేస్తుందని మరియు ఎందుకు పని చేస్తుందని మిమ్మల్ని మీరు ఒప్పించుకోవడం అనేది నిజంగా నేర్చుకునే చోటే జరుగుతుంది, YouTubeలో కొంతమంది వాసి మాటలు వినడం కంటే మీరు మళ్లీ అదే తార్కికం ద్వారా మిమ్మల్ని నడిపిస్తారు. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/thai/sentence_translations.json b/2019/cramers-rule/thai/sentence_translations.json index e611fbe40..1d627aca3 100644 --- a/2019/cramers-rule/thai/sentence_translations.json +++ b/2019/cramers-rule/thai/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/turkish/sentence_translations.json b/2019/cramers-rule/turkish/sentence_translations.json index 6ec4334c6..46aee6219 100644 --- a/2019/cramers-rule/turkish/sentence_translations.json +++ b/2019/cramers-rule/turkish/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "Örneğin, pozitif bir nokta çarpımla genellikle aynı yöne işaret eden ve dönüşüm sırasında birbirlerinden negatif bir nokta çarpım elde edecek şekilde ayrılan iki vektörünüz olabilir. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "Bunun gibi daha genel vakalar üzerinde düşünmek ve bunun işe yaradığına ve neden işe yaradığına kendinizi ikna etmek, tüm öğrenmenin gerçekte gerçekleştiği yerdir; YouTube'daki bir adamı dinlemek size aynı mantık yürütmeyi tekrar anlatmaktan çok daha fazlasıdır. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/ukrainian/sentence_translations.json b/2019/cramers-rule/ukrainian/sentence_translations.json index 5e70e0ebc..9152b3974 100644 --- a/2019/cramers-rule/ukrainian/sentence_translations.json +++ b/2019/cramers-rule/ukrainian/sentence_translations.json @@ -203,7 +203,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "Наприклад, у вас можуть бути два вектори, які загалом вказують в одному напрямку, з додатним скалярним добутком, які віддаляються один від одного під час перетворення таким чином, що потім вони отримують від’ємний скалярний добуток. ", "n_reviews": 0, "start": 240.8, @@ -574,7 +574,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "Розмірковуючи про більш загальні випадки та переконавши себе, що це працює, це те, де все буде навчено, набагато більше, ніж слухати якогось чувака на YouTube, який знову розповідає про міркування. ", "n_reviews": 0, "start": 665.1, diff --git a/2019/cramers-rule/urdu/sentence_translations.json b/2019/cramers-rule/urdu/sentence_translations.json index 34c42468c..d196bb741 100644 --- a/2019/cramers-rule/urdu/sentence_translations.json +++ b/2019/cramers-rule/urdu/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "مثال کے طور پر، آپ کے پاس دو ویکٹر ہو سکتے ہیں جو عام طور پر ایک مثبت ڈاٹ پروڈکٹ کے ساتھ ایک ہی سمت میں اشارہ کرتے ہیں، جو تبدیلی کے دوران ایک دوسرے سے اس طرح کھینچے جاتے ہیں کہ ان کا اختتام منفی ڈاٹ پروڈکٹ ہوتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "اس طرح کے مزید عمومی معاملات کے بارے میں سوچنا اور اپنے آپ کو اس بات پر قائل کرنا کہ یہ کام کرتا ہے اور یہ کیوں کام کرتا ہے وہیں جہاں تمام سیکھنا واقعتاً ہوتا ہے، اس سے کہیں زیادہ کہ یوٹیوب پر کسی دوست کو سننا آپ کو دوبارہ اسی استدلال سے گزرتا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/cramers-rule/vietnamese/sentence_translations.json b/2019/cramers-rule/vietnamese/sentence_translations.json index 9352db178..58256d26d 100644 --- a/2019/cramers-rule/vietnamese/sentence_translations.json +++ b/2019/cramers-rule/vietnamese/sentence_translations.json @@ -232,7 +232,7 @@ "end": 240.24 }, { - "input": "For example you could have two vectors generally pointing in the same direction with a positive dot product which get pulled apart from each other during the transformation in such a way that they have a negative dot product.", + "input": "For example, you could have two vectors generally pointing in the same direction with a positive dot product, which get pulled apart from each other during the transformation in such a way that they end up having a negative dot product.", "translatedText": "Ví dụ: bạn có thể có hai vectơ thường chỉ cùng hướng với tích số chấm dương, chúng bị tách ra khỏi nhau trong quá trình biến đổi theo cách mà cuối cùng chúng có tích số chấm âm. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 664.68 }, { - "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", + "input": "nd we want to know what input lands on that output. And if you think of, say, the z-coordinate of that input vector as the volume of that special parallelepiped we were looking at earlier, spanned by i-hat, j-hat, and the mystery input vector, what happens to that volume after the transformation? And what are the various ways you can compute that volume? Really, pause and take a moment to think through the details of generalizing this to higher dimensions, finding an expression for each coordinate of the solution to a larger linear system. Thinking through more general cases like this and convincing yourself that it works and why it works is where all the learning really happens, much more so than listening to some dude on YouTube walk you through the same reasoning again.", "translatedText": "Suy nghĩ về những trường hợp tổng quát hơn như thế này và thuyết phục bản thân rằng nó hoạt động và lý do tại sao nó hoạt động là nơi mà tất cả quá trình học tập thực sự diễn ra, hơn là nghe một anh chàng nào đó trên YouTube hướng dẫn bạn lý do tương tự một lần nữa. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/arabic/sentence_translations.json b/2019/differential-equations/arabic/sentence_translations.json index 9cfc0e5e3..f2cf5b738 100644 --- a/2019/differential-equations/arabic/sentence_translations.json +++ b/2019/differential-equations/arabic/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "هذا واضح عندما أسميه مصطلح مقاومة الهواء، لكن تخيل أنك رأيت هذه المعادلات خارج السياق، دون أن تعلم أنها تصف البندول. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/bengali/sentence_translations.json b/2019/differential-equations/bengali/sentence_translations.json index ad088cd1c..e2a5d4477 100644 --- a/2019/differential-equations/bengali/sentence_translations.json +++ b/2019/differential-equations/bengali/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "এটা সুস্পষ্ট যখন আমি এটিকে বায়ু প্রতিরোধের শব্দ বলি, কিন্তু কল্পনা করুন যে আপনি এই সমীকরণগুলিকে প্রেক্ষাপটের বাইরে দেখেছেন, না জেনে যে তারা একটি পেন্ডুলাম বর্ণনা করেছে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/chinese/sentence_translations.json b/2019/differential-equations/chinese/sentence_translations.json index d6df003e8..46aeb4fee 100644 --- a/2019/differential-equations/chinese/sentence_translations.json +++ b/2019/differential-equations/chinese/sentence_translations.json @@ -1096,7 +1096,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "当我将其称为空气阻力项时，这是显而易见的，但想象一下您 在断章取义地看到这些方程，而不知道它们描述了一个钟摆。", "model": "google_nmt", "from_community_srt": "這樣感覺很清楚， 好像不怎麼樣。 但請好好想一下， 當你只盯著方程式， 沒有座標圖，", diff --git a/2019/differential-equations/czech/sentence_translations.json b/2019/differential-equations/czech/sentence_translations.json index df77adbd2..4f1e6ab26 100644 --- a/2019/differential-equations/czech/sentence_translations.json +++ b/2019/differential-equations/czech/sentence_translations.json @@ -974,7 +974,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "", "from_community_srt": "Kdybychom ty rovnice viděli mimo kontext, aniž bysme věděli,", "n_reviews": 0, diff --git a/2019/differential-equations/english/captions.srt b/2019/differential-equations/english/captions.srt index f8a1264df..0cee5e892 100644 --- a/2019/differential-equations/english/captions.srt +++ b/2019/differential-equations/english/captions.srt @@ -1015,638 +1015,642 @@ you can immediately see how this will result in trajectories that spiral inward which is to say the pendulum slows down faster. 255 -00:16:46,130 --> 00:16:48,921 +00:16:46,130 --> 00:16:48,741 That's obvious when I call it the air resistance term, 256 -00:16:48,921 --> 00:16:53,490 -but imagine you saw these equations out of context, not knowing they described a pendulum. +00:16:48,741 --> 00:16:51,448 +but imagine that you saw these equations out of context, 257 +00:16:51,448 --> 00:16:53,490 +not knowing that they described a pendulum. + +258 00:16:54,010 --> 00:16:58,300 It's not obvious just looking at them that increasing this value of mu -258 +259 00:16:58,300 --> 00:17:02,590 means the system as a whole tends towards some attracting state faster. -259 +260 00:17:03,390 --> 00:17:06,240 So getting some software to draw these vector fields for you -260 +261 00:17:06,240 --> 00:17:09,089 can be a great way to build an intuition for how they behave. -261 +262 00:17:09,930 --> 00:17:14,105 What's wonderful is that any system of ordinary differential equations can be -262 +263 00:17:14,105 --> 00:17:18,869 described by a vector field like this, so it's a very general way to get a feel for them. -263 +264 00:17:19,470 --> 00:17:22,089 Usually, though, they have many more dimensions. -264 +265 00:17:22,720 --> 00:17:25,563 For example, consider the famous three-body problem, -265 +266 00:17:25,563 --> 00:17:29,533 which is to predict how three masses in three-dimensional space evolve if -266 +267 00:17:29,533 --> 00:17:33,665 they act on each other with gravity, and if you know their initial positions -267 +268 00:17:33,665 --> 00:17:34,470 and velocities. -268 +269 00:17:35,290 --> 00:17:38,425 Each mass has three coordinates describing its position, -269 +270 00:17:38,425 --> 00:17:40,570 and three more describing its momentum. -270 +271 00:17:41,270 --> 00:17:44,386 So the system has 18 degrees of freedom in total, -271 +272 00:17:44,386 --> 00:17:47,690 and hence an 18-dimensional space of possible states. -272 +273 00:17:48,250 --> 00:17:49,970 It's a bizarre thought, isn't it? -273 +274 00:17:50,230 --> 00:17:54,978 A single point meandering through an 18-dimensional space that we cannot visualize, -274 +275 00:17:54,978 --> 00:17:59,273 obediently taking steps through time based on whatever vector it happens to -275 +276 00:17:59,273 --> 00:18:03,513 be sitting on from moment to moment, completely encoding the positions and -276 +277 00:18:03,513 --> 00:18:07,470 the momenta of the three masses we see in ordinary, physical 3D space. -277 +278 00:18:08,650 --> 00:18:11,852 In practice, you can reduce the number of dimensions here by taking -278 +279 00:18:11,852 --> 00:18:15,008 advantage of the symmetries of your setup, but the point that more -279 +280 00:18:15,008 --> 00:18:18,730 degrees of freedom results in higher dimensional state spaces remains the same. -280 +281 00:18:21,290 --> 00:18:24,530 In math, we often call a space like this a phase space. -281 +282 00:18:25,090 --> 00:18:28,904 You'll hear me use that term broadly for spaces encoding all kinds of -282 +283 00:18:28,904 --> 00:18:33,263 states of changing systems, but you should know that in the context of physics, -283 +284 00:18:33,263 --> 00:18:37,949 especially Hamiltonian mechanics, the term is often reserved for a more special case, -284 +285 00:18:37,949 --> 00:18:41,110 namely a space whose axes represent position and momentum. -285 +286 00:18:41,890 --> 00:18:45,563 So a physicist would agree that the 18-dimensional space describing the -286 +287 00:18:45,563 --> 00:18:49,543 three-body problem is a phase space, but they might ask that we make a couple -287 +288 00:18:49,543 --> 00:18:53,370 of modifications to our pendulum setup for it to properly deserve the term. -288 +289 00:18:54,250 --> 00:18:57,201 For those of you who just watched the block collision video, -289 +290 00:18:57,201 --> 00:19:00,830 the planes we worked with there would be called phase spaces by math folk, -290 +291 00:19:00,830 --> 00:19:03,250 though a physicist might prefer other terminology. -291 +292 00:19:03,890 --> 00:19:06,830 Just know that the specific meaning may depend on your context. -292 +293 00:19:07,870 --> 00:19:11,699 It may seem like a simple idea, depending on how well indoctrinated you -293 +294 00:19:11,699 --> 00:19:15,636 are to modern ways of thinking about math, but it's worth keeping in mind -294 +295 00:19:15,636 --> 00:19:19,625 that it took humanity quite a while to really embrace thinking of dynamics -295 +296 00:19:19,625 --> 00:19:23,190 spatially like this, especially when the dimensions get very large. -296 -00:19:23,890 --> 00:19:40,550 -In his book Chaos, the author James Glick describes phase space as, - 297 -00:19:40,550 --> 00:19:54,270 -"One of the most powerful inventions of modern science". +00:19:23,890 --> 00:19:31,936 +In his book Chaos, the author James Glick describes phase space as, 298 -00:19:54,270 --> 00:19:57,229 -One reason its powerful is that you can ask questions, +00:19:31,936 --> 00:19:38,681 +"One of the most powerful inventions of modern science". 299 -00:19:57,229 --> 00:20:01,910 -not just about a single initial condition but about a whole spectrum of initial states. +00:19:38,681 --> 00:19:45,189 +One reason its powerful is that you can ask questions, 300 -00:20:02,770 --> 00:20:05,830 -The collection of all possible trajectories is reminiscent of a moving fluid. +00:19:45,189 --> 00:19:55,602 +not just about a single initial condition but about a whole spectrum of initial states. 301 -00:20:05,830 --> 00:20:07,790 -So we call it phase flow. +00:19:55,602 --> 00:20:04,831 +The collection of all possible trajectories is reminiscent of a moving fluid. 302 +00:20:04,831 --> 00:20:07,790 +So we call it phase flow. + +303 00:20:07,790 --> 00:20:10,696 To take one example of why phase flow is a fruitful idea, -303 +304 00:20:10,696 --> 00:20:12,450 consider the question of stability. -304 +305 00:20:12,450 --> 00:20:17,305 The origin of our space corresponds to the pendulum standing still, -305 +306 00:20:17,305 --> 00:20:22,876 and so does this point over here, representing when the pendulum is perfectly -306 +307 00:20:22,876 --> 00:20:24,090 balanced upright. -307 +308 00:20:24,090 --> 00:20:25,787 These are the so-called fixed points of our system, -308 +309 00:20:25,787 --> 00:20:28,236 and one natural question to ask is whether or not they're stable, that is, -309 +310 00:20:28,236 --> 00:20:30,783 will tiny nudges to the system result in a state that tends back towards that -310 +311 00:20:30,783 --> 00:20:31,730 fixed point, or away from it? -311 +312 00:20:32,090 --> 00:20:38,027 Physical intuition for the pendulum makes the answer here kind of obvious, -312 +313 00:20:38,027 --> 00:20:43,648 but how would you think about stability just looking at the equations, -313 +314 00:20:43,648 --> 00:20:49,190 say if they arose in some completely different less intuitive context? -314 +315 00:20:49,190 --> 00:20:52,348 We'll go over how to compute the answers to questions like this in following videos, -315 +316 00:20:52,348 --> 00:20:54,950 and the intuition for the relevant computations are guided heavily by -316 +317 00:20:54,950 --> 00:20:57,588 the thought of looking at small regions in space around a fixed point, -317 +318 00:20:57,588 --> 00:20:59,670 and asking whether the flow tends to contract or expand. -318 +319 00:20:59,670 --> 00:21:00,390 And speaking of attraction and stability, let's take a brief side-step to talk about love. -319 +320 00:21:00,390 --> 00:21:04,113 The Strogatz quote that I mentioned earlier comes from a whimsical column in -320 +321 00:21:04,113 --> 00:21:07,112 the New York Times on the mathematics of modelling affection, -321 +322 00:21:07,112 --> 00:21:10,691 an example well worth pilfering to illustrate that we're not just talking -322 +323 00:21:10,691 --> 00:21:11,610 about physics here. -323 +324 00:21:11,610 --> 00:21:17,210 Imagine you've been flirting with someone, but there's been some frustrating -324 +325 00:21:17,210 --> 00:21:20,846 inconsistency to how mutual your affection seems, -325 +326 00:21:20,846 --> 00:21:26,156 and perhaps during a moment when you turn your attention towards physics -326 +327 00:21:26,156 --> 00:21:31,320 to keep your mind off the romantic turmoil, mulling over the broken-up -327 +328 00:21:31,320 --> 00:21:36,848 pendulum equations, you suddenly understand the on-again-off-again dynamics -328 +329 00:21:36,848 --> 00:21:38,230 of your flirtation. -329 +330 00:21:38,230 --> 00:21:41,222 You've noticed that your own affection tends to increase when your -330 +331 00:21:41,222 --> 00:21:44,350 companion seems interested in you, but decrease when they seem colder. -331 +332 00:21:46,450 --> 00:21:44,350 That is, the rate of change for your love is proportional to their feelings for you. -332 +333 00:21:46,450 --> 00:21:47,667 But this sweetheart of yours is precisely the opposite, -333 +334 00:21:47,667 --> 00:21:48,863 strangely attracted to you when you seem uninterested, -334 +335 00:21:48,863 --> 00:21:49,690 but turned off once you seem too keen. -335 +336 00:21:49,690 --> 00:21:50,506 The phase space for these equations looks very -336 +337 00:21:50,506 --> 00:21:51,410 similar to the center part of your pendulum diagram. -337 +338 00:21:51,790 --> 00:21:56,990 The two of you will go back and forth between affection and repulsion in an endless cycle. -338 +339 00:21:58,810 --> 00:22:02,224 A metaphor of pendulum swings in your feelings would not just be apt, -339 +340 00:22:02,224 --> 00:22:03,590 but mathematically verified. -340 +341 00:22:03,590 --> 00:22:07,400 In fact, if your partner's feelings were further slowed when they feel -341 +342 00:22:07,400 --> 00:22:11,372 themselves too in love, let's say out of a fear of being made vulnerable, -342 +343 00:22:11,372 --> 00:22:14,377 we'd have a term matching the friction in the pendulum, -343 +344 00:22:14,377 --> 00:22:18,510 and you too would be destined to an inward spiral towards mutual ambivalence. -344 +345 00:22:19,090 --> 00:22:20,310 I hear wedding bells already. -345 +346 00:22:21,310 --> 00:22:25,869 The point is that two very different-seeming laws of dynamics, one from physics, -346 +347 00:22:25,869 --> 00:22:30,484 involving a single variable, and another from, uh, chemistry, with two variables, -347 +348 00:22:30,484 --> 00:22:35,381 actually have a very similar structure, easier to recognize when you're looking at the -348 +349 00:22:35,381 --> 00:22:36,170 phase diagram. -349 +350 00:22:36,890 --> 00:22:39,737 Most notably, even though the equations are different, -350 +351 00:22:39,737 --> 00:22:42,999 for example there's no sine function in the romance equations, -351 +352 00:22:42,999 --> 00:22:46,210 the phase space exposes an underlying similarity nevertheless. -352 +353 00:22:47,230 --> 00:22:50,558 In other words, you're not just studying a pendulum right now, -353 +354 00:22:50,558 --> 00:22:55,050 the tactics you develop to study one case have a tendency to transfer to many others. -354 +355 00:22:57,450 --> 00:23:00,854 Okay, so phase diagrams are a nice way to build understanding, -355 +356 00:23:00,854 --> 00:23:04,150 but what about actually computing the answer to our equation? -356 +357 00:23:05,110 --> 00:23:09,062 One way to do this is to essentially simulate what the universe would do, -357 +358 00:23:09,062 --> 00:23:13,710 but using finite time steps instead of the infinitesimals and limits defining calculus. -358 +359 00:23:14,430 --> 00:23:18,333 The basic idea is that if you're at some point in this phase diagram, -359 +360 00:23:18,333 --> 00:23:22,850 take a step based on the vector you're sitting on for a small time step, delta t. -360 +361 00:23:22,850 --> 00:23:26,990 Specifically, take a step equal to delta t times that vector. -361 +362 00:23:27,850 --> 00:23:29,983 As a reminder, in drawing these vector fields, -362 +363 00:23:29,983 --> 00:23:33,750 the magnitude for each vector has been artificially scaled down to prevent clutter. -363 +364 00:23:34,910 --> 00:23:40,536 When you do this repeatedly, your final location will be an approximation of theta t, -364 +365 00:23:40,536 --> 00:23:43,350 where t is the sum of all those time steps. -365 +366 00:23:44,330 --> 00:23:46,900 If you think about what's being shown right now, though, -366 +367 00:23:46,900 --> 00:23:49,380 and what that would imply for the pendulum's movement, -367 +368 00:23:49,380 --> 00:23:51,770 you'd probably agree that this is grossly inaccurate. -368 +369 00:23:52,210 --> 00:23:56,710 But that's only because the time step delta t of 0.5 is way too big. -369 +370 00:23:57,210 --> 00:24:02,618 If we turned it down, say to 0.01, you can get a much more accurate approximation, -370 +371 00:24:02,618 --> 00:24:05,290 it just takes more repeated steps is all. -371 +372 00:24:05,870 --> 00:24:10,570 In this case, computing theta of 10 requires 1000 little steps. -372 +373 00:24:11,350 --> 00:24:15,277 Luckily, we live in a world with computers, so repeating a simple task 1000 -373 +374 00:24:15,277 --> 00:24:19,050 times is as simple as articulating that task with a programming language. -374 +375 00:24:19,590 --> 00:24:22,271 In fact, let's finish things off by writing a little -375 +376 00:24:22,271 --> 00:24:24,650 python program that computes theta of t for us. -376 +377 00:24:25,270 --> 00:24:28,650 What it has to do is make use of the differential equation, -377 +378 00:24:28,650 --> 00:24:33,270 which returns the second derivative of theta as a function of theta and theta dot. -378 +379 00:24:34,230 --> 00:24:37,673 You start off by defining two variables, theta and theta dot, -379 +380 00:24:37,673 --> 00:24:39,950 each in terms of some initial conditions. -380 +381 00:24:40,590 --> 00:24:43,581 In this case I'll have theta start at pi thirds, -381 +382 00:24:43,581 --> 00:24:46,390 which is 60 degrees, and theta dot start at 0. -382 +383 00:24:47,050 --> 00:24:52,053 Next, write a loop that corresponds to taking many little time steps -383 +384 00:24:52,053 --> 00:24:57,710 between 0 and time t, each of size delta t, which I'm setting here to be 0.01. -384 +385 00:24:58,570 --> 00:25:02,957 In each step of this loop, increase theta by theta dot times delta t, -385 +386 00:25:02,957 --> 00:25:06,592 and increase theta dot by theta double dot times delta t, -386 +387 00:25:06,592 --> 00:25:11,230 where theta double dot can be computed based on the differential equation. -387 +388 00:25:11,910 --> 00:25:15,650 After all these little time steps, simply return the value of theta. -388 +389 00:25:16,690 --> 00:25:19,410 This is called solving a differential equation numerically. -389 +390 00:25:20,050 --> 00:25:24,200 Numerical methods can get way more sophisticated and intricate than this to better -390 +391 00:25:24,200 --> 00:25:28,650 balance the tradeoff between accuracy and efficiency, but this loop gives the basic idea. -391 +392 00:25:30,530 --> 00:25:33,825 So even though it sucks that we can't always find exact solutions, -392 +393 00:25:33,825 --> 00:25:37,712 there are still meaningful ways to study differential equations in the face of -393 +394 00:25:37,712 --> 00:25:38,450 this inability. -394 +395 00:25:38,790 --> 00:25:42,958 In the following videos, we'll look at several methods for finding exact -395 +396 00:25:42,958 --> 00:25:47,412 solutions when it's possible, but one theme I'd like to focus on is how these -396 +397 00:25:47,412 --> 00:25:51,810 exact solutions can also help us to study the more general, unsolvable cases. -397 +398 00:25:52,790 --> 00:25:54,050 But it gets worse. -398 +399 00:25:54,690 --> 00:25:58,368 Just as there's a limit to how far exact analytic solutions can get us, -399 +400 00:25:58,368 --> 00:26:02,199 one of the great fields to have emerged in the last century, chaos theory, -400 +401 00:26:02,199 --> 00:26:06,439 has exposed that there are further limits on how well we can use these systems for -401 +402 00:26:06,439 --> 00:26:08,330 prediction with or without solutions. -402 +403 00:26:09,210 --> 00:26:14,405 Specifically, we know that for some systems, small variations to the initial conditions, -403 +404 00:26:14,405 --> 00:26:17,674 say the kind due to necessarily imperfect measurements, -404 +405 00:26:17,674 --> 00:26:20,010 result in wildly different trajectories. -405 +406 00:26:20,530 --> 00:26:23,270 We've even built some good understanding for why this happens. -406 +407 00:26:23,830 --> 00:26:28,270 The three-body problem, for example, is known to have seeds of chaos within it. -407 +408 00:26:28,870 --> 00:26:32,460 So looking back at the quote from earlier, it seems almost cruel of the -408 +409 00:26:32,460 --> 00:26:36,001 universe to fill its language with riddles that we either can't solve, -409 +410 00:26:36,001 --> 00:26:40,190 or where we know that any solution would be useless for long-term prediction anyway. -410 +411 00:26:40,670 --> 00:26:44,350 It is cruel, but then again it should also be reassuring. -411 +412 00:26:45,010 --> 00:26:49,032 It gives some hope that the complexity we see in the world around us can be studied -412 +413 00:26:49,032 --> 00:26:53,246 somewhere in this math, and that it's not hidden away in the mismatch between model and -413 +414 00:26:53,246 --> 00:26:53,630 reality. diff --git a/2019/differential-equations/english/sentence_timings.json b/2019/differential-equations/english/sentence_timings.json index 19b6b3ccb..63307dc08 100644 --- a/2019/differential-equations/english/sentence_timings.json +++ b/2019/differential-equations/english/sentence_timings.json @@ -610,7 +610,7 @@ 1005.59 ], [ - "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", 1006.13, 1013.49 ], @@ -695,23 +695,8 @@ 1163.19 ], [ - "In his book Chaos, the author James Glick describes phase space as, \"One of the most powerful inventions of modern science\".", + "In his book Chaos, the author James Glick describes phase space as, \"One of the most powerful inventions of modern science\". One reason its powerful is that you can ask questions, not just about a single initial condition but about a whole spectrum of initial states. The collection of all possible trajectories is reminiscent of a moving fluid. So we call it phase flow.", 1163.89, - 1194.27 - ], - [ - "One reason its powerful is that you can ask questions, not just about a single initial condition but about a whole spectrum of initial states.", - 1194.27, - 1201.91 - ], - [ - "The collection of all possible trajectories is reminiscent of a moving fluid.", - 1202.77, - 1205.83 - ], - [ - "So we call it phase flow.", - 1205.83, 1207.79 ], [ diff --git a/2019/differential-equations/english/transcript.txt b/2019/differential-equations/english/transcript.txt index 4c1f3dbf9..c8d68e649 100644 --- a/2019/differential-equations/english/transcript.txt +++ b/2019/differential-equations/english/transcript.txt @@ -120,7 +120,7 @@ For example, in regions where theta dot is quite high, the vectors guide the poi This corresponds to a pendulum with a high enough initial velocity that it fully rotates around several times before settling into a decaying back and forth. Having a little more fun? When I tweak this air resistance term, mu, say increasing it, you can immediately see how this will result in trajectories that spiral inward faster, which is to say the pendulum slows down faster. -That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum. +That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum. It's not obvious just looking at them that increasing this value of mu means the system as a whole tends towards some attracting state faster. So getting some software to draw these vector fields for you can be a great way to build an intuition for how they behave. What's wonderful is that any system of ordinary differential equations can be described by a vector field like this, so it's a very general way to get a feel for them. diff --git a/2019/differential-equations/french/sentence_translations.json b/2019/differential-equations/french/sentence_translations.json index d3212f959..456239a70 100644 --- a/2019/differential-equations/french/sentence_translations.json +++ b/2019/differential-equations/french/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "C'est évident lorsque j'appelle cela le terme de résistance de l'air, mais imaginez que vous ayez vu ces équations hors de leur contexte, sans savoir qu'elles décrivaient un pendule.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/german/sentence_translations.json b/2019/differential-equations/german/sentence_translations.json index 1830a864b..d4f13e24b 100644 --- a/2019/differential-equations/german/sentence_translations.json +++ b/2019/differential-equations/german/sentence_translations.json @@ -1097,7 +1097,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "Das ist klar, wenn ich den Begriff \"Luftwiderstand\" nenne, aber stell dir vor, du hast diese Gleichungen aus dem Zusammenhang gerissen und wusstest nicht, dass sie ein Pendel beschreiben.", "model": "DeepL", "from_community_srt": "Das ist offensichtlich, wenn ich es den Luftwiderstand nenne, aber stell dir vor, du hättest Gleichungen ohne Kontext gesehen, nicht wissend, dass sie ein Pendel beschrieben haben;", diff --git a/2019/differential-equations/greek/sentence_translations.json b/2019/differential-equations/greek/sentence_translations.json index 49f1825fd..af7f0bf38 100644 --- a/2019/differential-equations/greek/sentence_translations.json +++ b/2019/differential-equations/greek/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "", "from_community_srt": "Αυτό είναι προφανές όταν το ονομάζω τον όρο της αντίστασης του αέρα, αλλά φανταστείτε ότι βλέπατε αυτές τις εξισώσεις εκτός πλαισίου, χωρίς να γνωρίζετε ότι περιγράφουν το εκκρεμές.", "n_reviews": 0, diff --git a/2019/differential-equations/hebrew/sentence_translations.json b/2019/differential-equations/hebrew/sentence_translations.json index c19bde508..276995cde 100644 --- a/2019/differential-equations/hebrew/sentence_translations.json +++ b/2019/differential-equations/hebrew/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "זה ברור כשאני קורא לזה מונח התנגדות האוויר, אבל תארו לעצמכם שראיתם את המשוואות האלה מחוץ להקשרן, בלי לדעת שהן תיארו מטוטלת.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/hindi/sentence_translations.json b/2019/differential-equations/hindi/sentence_translations.json index ec55eb5e7..5e908f71c 100644 --- a/2019/differential-equations/hindi/sentence_translations.json +++ b/2019/differential-equations/hindi/sentence_translations.json @@ -854,7 +854,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "यह स्पष्ट है जब मैं इसे वायु प्रतिरोध शब्द कहता हूं, लेकिन कल्पना कीजिए कि आपने इन समीकरणों को संदर्भ से बाहर देखा, यह नहीं जानते हुए कि उन्होंने एक पेंडुलम का वर्णन किया है।", "n_reviews": 0, "start": 1006.13, diff --git a/2019/differential-equations/hungarian/sentence_translations.json b/2019/differential-equations/hungarian/sentence_translations.json index a715ef78c..05dfc829b 100644 --- a/2019/differential-equations/hungarian/sentence_translations.json +++ b/2019/differential-equations/hungarian/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "Ez nyilvánvaló, amikor a légellenállás kifejezésnek nevezem, de képzeld el, hogy ezeket az egyenleteket összefüggéseikből kiragadva látod, és nem tudod, hogy egy ingát írnak le.", "model": "DeepL", "n_reviews": 0, diff --git a/2019/differential-equations/indonesian/sentence_translations.json b/2019/differential-equations/indonesian/sentence_translations.json index a2aeb3db3..fdbf8c97a 100644 --- a/2019/differential-equations/indonesian/sentence_translations.json +++ b/2019/differential-equations/indonesian/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "Hal ini sudah jelas ketika saya menyebutnya istilah hambatan udara, tetapi bayangkan Anda melihat persamaan ini di luar konteks, tanpa mengetahui bahwa persamaan tersebut menggambarkan pendulum.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/italian/sentence_translations.json b/2019/differential-equations/italian/sentence_translations.json index 00188f4da..dba44a47d 100644 --- a/2019/differential-equations/italian/sentence_translations.json +++ b/2019/differential-equations/italian/sentence_translations.json @@ -1092,7 +1092,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "Questo è ovvio quando lo chiamo termine di resistenza dell'aria, ma immagina di vedere queste equazioni fuori dal contesto, senza sapere che descrivono un pendolo.", "model": "DeepL", "from_community_srt": "Immagine di vedere le equazioni fuori dal contesto non sapendo che descrivono un pendolo;", diff --git a/2019/differential-equations/japanese/sentence_translations.json b/2019/differential-equations/japanese/sentence_translations.json index 17b578b19..41388baad 100644 --- a/2019/differential-equations/japanese/sentence_translations.json +++ b/2019/differential-equations/japanese/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "これを空気抵抗項と呼ぶとそれは明白ですが、これらの方程式が振り子を説明して いることを知らずに、文脈を無視してこれらの方程式を見たと想像してください。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/korean/sentence_translations.json b/2019/differential-equations/korean/sentence_translations.json index b9704ffc6..098349a9c 100644 --- a/2019/differential-equations/korean/sentence_translations.json +++ b/2019/differential-equations/korean/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "내가 그것을 공기 저항 항이라고 부르면 그것은 분명합니다. 그러나 여러분이 이 방정식이 진자를 묘사한다는 것을 모르고 문맥에서 벗어나 이 방정식을 보았다고 상상해 보십시오.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/marathi/sentence_translations.json b/2019/differential-equations/marathi/sentence_translations.json index f6a5708ab..66d7282a0 100644 --- a/2019/differential-equations/marathi/sentence_translations.json +++ b/2019/differential-equations/marathi/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "जेव्हा मी याला वायु प्रतिरोधक संज्ञा म्हणतो तेव्हा ते स्पष्ट आहे, परंतु कल्पना करा की तुम्ही ही समीकरणे संदर्भाबाहेर पाहिली आहेत, हे माहित नसले की त्यांनी पेंडुलमचे वर्णन केले आहे.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/persian/sentence_translations.json b/2019/differential-equations/persian/sentence_translations.json index 66659aafd..86e773c5b 100644 --- a/2019/differential-equations/persian/sentence_translations.json +++ b/2019/differential-equations/persian/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "وقتی من آن را اصطلاح مقاومت هوا می نامم واضح است، اما تصور کنید که شما این معادلات را خارج از زمینه دیدید، بدون اینکه بدانید که آنها یک آونگ را توصیف می کنند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/polish/sentence_translations.json b/2019/differential-equations/polish/sentence_translations.json index cd61c5551..cb4f58dae 100644 --- a/2019/differential-equations/polish/sentence_translations.json +++ b/2019/differential-equations/polish/sentence_translations.json @@ -973,7 +973,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "", "from_community_srt": "że wahadło szybciej hamuje. Wyobraź sobie, że zobaczyłbyś te równania wyrwane z kontekstu, nie wiedząc, że opisują one wahadło;", "n_reviews": 0, diff --git a/2019/differential-equations/portuguese/sentence_translations.json b/2019/differential-equations/portuguese/sentence_translations.json index 7f2d6db8b..28058a470 100644 --- a/2019/differential-equations/portuguese/sentence_translations.json +++ b/2019/differential-equations/portuguese/sentence_translations.json @@ -1098,7 +1098,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "Isso é óbvio quando eu chamo isso de termo de resistência do ar, mas imagine que você viu essas equações fora de contexto, sem saber que elas descreviam um pêndulo.", "model": "google_nmt", "from_community_srt": "Isso é óbvio pois é o termo da resistência do ar. Mas imagine que você viu essas equações fora de contexto, não sabendo que elas descrevem um pêndulo;", diff --git a/2019/differential-equations/russian/sentence_translations.json b/2019/differential-equations/russian/sentence_translations.json index 5b95fd74a..cc835a48b 100644 --- a/2019/differential-equations/russian/sentence_translations.json +++ b/2019/differential-equations/russian/sentence_translations.json @@ -974,7 +974,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "Это очевидно, когда я называю это термином сопротивления воздуха, но представьте, что вы видели эти уравнения вне контекста, не зная, что они описывают маятник.", "from_community_srt": "И очевидно, это именно то, что мы называем сопротивлением воздуха. Но представьте, что вы увидели бы это в виде уравнений без контекста не зная того, как они описывают маятник.", "n_reviews": 0, diff --git a/2019/differential-equations/spanish/sentence_translations.json b/2019/differential-equations/spanish/sentence_translations.json index 6a245b56e..21ce9aacf 100644 --- a/2019/differential-equations/spanish/sentence_translations.json +++ b/2019/differential-equations/spanish/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "Eso es obvio cuando lo llamo término de resistencia del aire, pero imagina que ves estas ecuaciones fuera de contexto, sin saber que describen un péndulo.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/tamil/sentence_translations.json b/2019/differential-equations/tamil/sentence_translations.json index a68867490..88f717321 100644 --- a/2019/differential-equations/tamil/sentence_translations.json +++ b/2019/differential-equations/tamil/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "காற்று எதிர்ப்புச் சொல் என்று நான் அழைக்கும்போது அது தெளிவாகத் தெரிகிறது, ஆனால் இந்தச் சமன்பாடுகள் ஊசலை விவரித்தன என்பதை அறியாமல், சூழலுக்கு வெளியே நீங்கள் பார்த்தீர்கள் என்று கற்பனை செய்து பாருங்கள்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/telugu/sentence_translations.json b/2019/differential-equations/telugu/sentence_translations.json index e544ddec5..e3eb57ba4 100644 --- a/2019/differential-equations/telugu/sentence_translations.json +++ b/2019/differential-equations/telugu/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "నేను దీనిని వాయు నిరోధక పదం అని పిలిచినప్పుడు అది స్పష్టంగా ఉంది, కానీ మీరు ఈ సమీకరణాలను సందర్భోచితంగా చూశారని ఊహించుకోండి, అవి లోలకాన్ని వివరించాయని తెలియక.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/thai/sentence_translations.json b/2019/differential-equations/thai/sentence_translations.json index f85eb971a..f4c4c597c 100644 --- a/2019/differential-equations/thai/sentence_translations.json +++ b/2019/differential-equations/thai/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/turkish/sentence_translations.json b/2019/differential-equations/turkish/sentence_translations.json index 1e2ee8553..bbf5d15c7 100644 --- a/2019/differential-equations/turkish/sentence_translations.json +++ b/2019/differential-equations/turkish/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "Buna hava direnci terimi dediğimde bu çok açık, ancak bu denklemleri bir sarkacı tanımladıklarını bilmeden bağlam dışında gördüğünüzü hayal edin.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/ukrainian/sentence_translations.json b/2019/differential-equations/ukrainian/sentence_translations.json index 196a3a8c7..b49debaeb 100644 --- a/2019/differential-equations/ukrainian/sentence_translations.json +++ b/2019/differential-equations/ukrainian/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "Це очевидно, коли я називаю це терміном опору повітря, але уявіть, що ви побачили ці рівняння поза контекстом, не знаючи, що вони описують маятник. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/urdu/sentence_translations.json b/2019/differential-equations/urdu/sentence_translations.json index ff5d790b3..c4adfe980 100644 --- a/2019/differential-equations/urdu/sentence_translations.json +++ b/2019/differential-equations/urdu/sentence_translations.json @@ -976,7 +976,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "جب میں اسے ہوا کی مزاحمت کی اصطلاح کہتا ہوں تو یہ واضح ہے، لیکن تصور کریں کہ آپ نے ان مساوات کو سیاق و سباق سے ہٹ کر دیکھا، یہ نہیں جانتے کہ انہوں نے پینڈولم کو بیان کیا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/differential-equations/vietnamese/sentence_translations.json b/2019/differential-equations/vietnamese/sentence_translations.json index 3807b774b..11ec77545 100644 --- a/2019/differential-equations/vietnamese/sentence_translations.json +++ b/2019/differential-equations/vietnamese/sentence_translations.json @@ -1095,7 +1095,7 @@ "end": 1005.59 }, { - "input": "That's obvious when I call it the air resistance term, but imagine you saw these equations out of context, not knowing they described a pendulum.", + "input": "That's obvious when I call it the air resistance term, but imagine that you saw these equations out of context, not knowing that they described a pendulum.", "translatedText": "Điều đó là hiển nhiên khi tôi gọi nó là thuật ngữ lực cản không khí, nhưng hãy tưởng tượng bạn thấy những phương trình này không đúng ngữ cảnh, không biết rằng chúng mô tả một con lắc.", "model": "google_nmt", "from_community_srt": "Cũng dễ hiểu vì tôi đang gọi nó là độ cản của không khí, nhưng giả như bạn chỉ nhìn thấy một mình ptvp,", diff --git a/2019/fourier-series-montage/english/transcript.txt b/2019/fourier-series-montage/english/transcript.txt index e69de29bb..2d244a5eb 100644 --- a/2019/fourier-series-montage/english/transcript.txt +++ b/2019/fourier-series-montage/english/transcript.txt @@ -0,0 +1 @@ +MUSIC you you you you you you you you you you \ No newline at end of file diff --git a/2019/fourier-series/arabic/sentence_translations.json b/2019/fourier-series/arabic/sentence_translations.json index eae6aad42..79fff22c1 100644 --- a/2019/fourier-series/arabic/sentence_translations.json +++ b/2019/fourier-series/arabic/sentence_translations.json @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", "translatedText": "إذا قمت بإجراء أحد هذه التحليلات إلى متجهات دوارة لرسم ممل، فإن ما سيحدث هو أن المتجهات ذات التردد 1 وسالب 1 سيكون لها نفس الطول، وستكون انعكاسات أفقية لبعضها البعض. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. ", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you. ", "translatedText": "ما رأيته للتو، وهو تقسيم الدالة إلى مزيج من هذه الأسيات واستخدام ذلك لحل معادلة تفاضلية، يظهر مرارًا وتكرارًا بأشكال وأشكال مختلفة. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/bengali/sentence_translations.json b/2019/fourier-series/bengali/sentence_translations.json index edaaff0f1..63653b389 100644 --- a/2019/fourier-series/bengali/sentence_translations.json +++ b/2019/fourier-series/bengali/sentence_translations.json @@ -72,7 +72,7 @@ "end": 83.4 }, { - "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something we have the math to describe and control completely.", + "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something that we have the math to describe and to control completely.", "translatedText": "এবং প্রকৃতির অন্য কোথাও আপনি যে উদ্ভূত জটিলতা খুঁজে পান তার বিপরীতে, এটি এমন কিছু যা আমাদের সম্পূর্ণরূপে বর্ণনা এবং নিয়ন্ত্রণ করার গণিত রয়েছে।", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", "translatedText": "আপনি যদি বিরক্তিকর অঙ্কনের জন্য এই পচনগুলি ঘূর্ণায়মান ভেক্টরগুলির মধ্যে একটি করেন, তাহলে যা ঘটবে তা হল যে কম্পাঙ্ক 1 এবং ঋণাত্মক 1 সহ ভেক্টরগুলির দৈর্ঘ্য একই হবে এবং তারা একে অপরের অনুভূমিক প্রতিফলন হবে।", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms.", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you.", "translatedText": "আপনি এইমাত্র যা দেখেছেন, এই সূচকগুলির সংমিশ্রণ হিসাবে একটি ফাংশনকে ভেঙে ফেলা এবং একটি ডিফারেনশিয়াল সমীকরণ সমাধান করার জন্য এটি ব্যবহার করে, বিভিন্ন আকার এবং আকারে বারবার উঠে আসে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/chinese/sentence_translations.json b/2019/fourier-series/chinese/sentence_translations.json index bba78cc85..1e861d0b0 100644 --- a/2019/fourier-series/chinese/sentence_translations.json +++ b/2019/fourier-series/chinese/sentence_translations.json @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", "translatedText": "如果您将这些分解之一分解为旋转矢量以绘制无聊的 绘图，则会发生频率为 1 和负 1 的矢量将 具有相同的长度，并且它们将是彼此的水平反射。", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. ", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you. ", "translatedText": "您刚才看到的，将函数分解为这些指数 的组合，并用它来求解微分方程，会以 不同的形状和形式一次又一次地出现。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/english/captions.srt b/2019/fourier-series/english/captions.srt index 8073c28bb..00de1475f 100644 --- a/2019/fourier-series/english/captions.srt +++ b/2019/fourier-series/english/captions.srt @@ -891,12 +891,12 @@ For right now, think of each of them as starting pointing one unit to the right at the number 1. 224 -00:14:23,080 --> 00:14:27,971 +00:14:23,080 --> 00:14:27,559 The easiest vector to describe is the constant one, which stays at the number 1, 225 -00:14:27,971 --> 00:14:32,260 -never moving, or if you prefer, it's rotating just at a frequency of 0. +00:14:27,559 --> 00:14:32,260 +never moving, or if you prefer, it's quote-unquote rotating just at a frequency of 0. 226 00:14:33,100 --> 00:14:36,772 @@ -915,12 +915,12 @@ That 2 pi is there because as t goes from 0 to 1, it needs to cover a distance of 2 pi along the circle. 230 -00:14:47,700 --> 00:14:51,486 -Technically it's actually one cycle every 10 seconds so things aren't too dizzying, +00:14:47,700 --> 00:14:50,610 +Technically in what's being shown, it's actually one cycle every 10 seconds 231 -00:14:51,486 --> 00:14:53,560 -I'm slowing everything down by a factor of 10. +00:14:50,610 --> 00:14:53,560 +so things aren't too dizzying, I'm slowing everything down by a factor of 10. 232 00:14:55,320 --> 00:14:59,848 @@ -1435,16 +1435,16 @@ how to relate this more general computation with what you might see in a textboo describing Fourier series only in terms of real valued functions with sines and cosines. 360 -00:23:41,840 --> 00:23:45,052 +00:23:41,840 --> 00:23:44,893 By the way, if you're looking for more Fourier series content, 361 -00:23:45,052 --> 00:23:48,416 +00:23:44,893 --> 00:23:48,093 I highly recommend the videos by Mathologer and The Coding Train, 362 -00:23:48,416 --> 00:23:51,680 -and I'd also recommend this blog post, links in the description. +00:23:48,093 --> 00:23:51,680 +and I'd also recommend this blog post, links of course in the description. 363 00:23:53,620 --> 00:23:57,097 diff --git a/2019/fourier-series/english/sentence_timings.json b/2019/fourier-series/english/sentence_timings.json index e345bc9e6..e81b91950 100644 --- a/2019/fourier-series/english/sentence_timings.json +++ b/2019/fourier-series/english/sentence_timings.json @@ -475,7 +475,7 @@ 862.58 ], [ - "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's rotating just at a frequency of 0.", + "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's quote-unquote rotating just at a frequency of 0.", 863.08, 872.26 ], @@ -490,7 +490,7 @@ 886.44 ], [ - "Technically it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", + "Technically in what's being shown, it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", 887.7, 893.56 ], @@ -775,7 +775,7 @@ 1420.94 ], [ - "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links in the description.", + "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links of course in the description.", 1421.84, 1431.68 ], diff --git a/2019/fourier-series/english/transcript.txt b/2019/fourier-series/english/transcript.txt index 1608428ed..7fce22e01 100644 --- a/2019/fourier-series/english/transcript.txt +++ b/2019/fourier-series/english/transcript.txt @@ -93,10 +93,10 @@ For right now, if you want, you can think of e to the i t as a notational shorth You'll notice I'm being a little loose with language using the words vector and complex numbers somewhat interchangeably, in large part because thinking of complex numbers as little arrows makes the idea of adding a lot of them together easier to visualize. Alright, armed with the function e to the i times t, let's write down a formula for each of these rotating vectors we're working with. For right now, think of each of them as starting pointing one unit to the right at the number 1. -The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's rotating just at a frequency of 0. +The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's quote-unquote rotating just at a frequency of 0. Then there will be the vector rotating one cycle every second, which we write as e to the 2 pi i times t. That 2 pi is there because as t goes from 0 to 1, it needs to cover a distance of 2 pi along the circle. -Technically it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10. +Technically in what's being shown, it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10. We also have a vector rotating at one cycle per second in the other direction, e to the negative 2 pi i times t. Similarly, the one going two rotations per second is e to the 2 times 2 pi i times t, where that 2 times 2 pi in the exponent describes how much distance is covered in one second. And we go on like this over all integers, both positive and negative, with a general formula of e to the n times 2 pi times i t. @@ -153,7 +153,7 @@ And remember, each pair of vectors rotating in opposite directions corresponds t To find the coefficients, you would need to compute this integral, and for the ambitious viewers among you itching to work out some integrals by hand, this is one where you can actually do the calculus to get an exact answer, rather than just having a computer do it numerically for you. I'll leave it as an exercise to work this out, and to relate it back to the idea of cosine waves by pairing off the vectors that rotate in opposite directions. And for the even more ambitious, I'll leave another exercise up on the screen for how to relate this more general computation with what you might see in a textbook describing Fourier series only in terms of real valued functions with sines and cosines. -By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links in the description. +By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links of course in the description. So on the one hand, this concludes our discussion of the heat equation, which was a little window into the study of partial differential equations. But on the other hand, this Fourier-to-Fourier series is a first glimpse at a deeper idea. Exponential functions, including their generalization into complex numbers and even matrices, play a very important role for differential equations, especially when it comes to linear equations. diff --git a/2019/fourier-series/french/sentence_translations.json b/2019/fourier-series/french/sentence_translations.json index b83b84ccd..6a0c9b1ae 100644 --- a/2019/fourier-series/french/sentence_translations.json +++ b/2019/fourier-series/french/sentence_translations.json @@ -752,7 +752,7 @@ "end": 862.58 }, { - "input": "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's rotating just at a frequency of 0.", + "input": "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's quote-unquote rotating just at a frequency of 0.", "translatedText": "Le vecteur le plus simple à décrire est le vecteur constant, qui reste au chiffre 1, ne bouge jamais, ou si vous préférez, il tourne juste à une fréquence de 0.", "from_community_srt": "qui reste fixe en 1 Ou si tu préfères, qui «tourne» à une fréquence nulle Ensuite,", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 886.44 }, { - "input": "Technically it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", + "input": "Technically in what's being shown, it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", "translatedText": "Techniquement c'est en fait un cycle toutes les 10 secondes donc ça n'est pas trop vertigineux, je ralentis tout d'un facteur 10.", "from_community_srt": "ce qui est affiché c'est 1 tour/10 sec afin que ça ne soit pas trop étourdissant J'ai ralenti les animations d'un facteur 10", "n_reviews": 0, @@ -1225,7 +1225,7 @@ "end": 1420.94 }, { - "input": "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links in the description.", + "input": "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links of course in the description.", "translatedText": "À propos, si vous recherchez davantage de contenu sur la série de Fourier, je recommande fortement les vidéos de Mathologer et The Coding Train, et je recommanderais également cet article de blog, les liens dans la description.", "from_community_srt": "si vous voulez davantage de contenu sur les séries de Fourier, je vous recommande les vidéos de Mathologer et The Coding Train sur le sujet Je vous recommande aussi cette publication sur le blogue de Jezzamoon", "n_reviews": 0, diff --git a/2019/fourier-series/german/sentence_translations.json b/2019/fourier-series/german/sentence_translations.json index decd3f77b..60af3a43f 100644 --- a/2019/fourier-series/german/sentence_translations.json +++ b/2019/fourier-series/german/sentence_translations.json @@ -854,7 +854,7 @@ "end": 862.58 }, { - "input": "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's rotating just at a frequency of 0.", + "input": "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's quote-unquote rotating just at a frequency of 0.", "translatedText": "Am einfachsten zu beschreiben ist der konstante Vektor, der bei der Zahl 1 bleibt und sich nie bewegt. Wenn du willst, rotiert er einfach mit einer Frequenz von 0.", "model": "DeepL", "from_community_srt": "Am einfachsten lässt sich der konstante Vektor beschreiben, der unbeweglich auf die Zahl 1 zeigt. Oder er, wenn dir das lieber ist, \"rotiert\" mit der Frequenz 0.", @@ -881,7 +881,7 @@ "end": 886.44 }, { - "input": "Technically it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", + "input": "Technically in what's being shown, it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", "translatedText": "Technisch gesehen ist es ein Zyklus alle 10 Sekunden, damit es nicht zu schwindelerregend wird, verlangsame ich alles um den Faktor 10.", "model": "DeepL", "from_community_srt": "Die im Video gezeigte Frequenz ist 1 Umdrehung alle 10 Sekunden, damit die Animation nicht zu verwirrend wird -- ich habe alles um einen Faktor 10 verlangsamt.", @@ -1391,7 +1391,7 @@ "end": 1420.94 }, { - "input": "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links in the description.", + "input": "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links of course in the description.", "translatedText": "Übrigens: Wenn du nach weiteren Inhalten zu Fourier-Reihen suchst, empfehle ich dir die Videos von Mathologer und The Coding Train und auch diesen Blogbeitrag (Links in der Beschreibung).", "model": "DeepL", "from_community_srt": "Wenn ihr, nebenbei bemerkt, mehr über Fourierreihen sucht, dann empfehle ich euch dringend die Videos von Mathologer und The Coding Train,", diff --git a/2019/fourier-series/hebrew/sentence_translations.json b/2019/fourier-series/hebrew/sentence_translations.json index 8f022ba05..e5dfaa9fa 100644 --- a/2019/fourier-series/hebrew/sentence_translations.json +++ b/2019/fourier-series/hebrew/sentence_translations.json @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", "translatedText": "אם תבצע אחד מהפירוקים הללו לוקטורים מסתובבים עבור ציור משעמם, מה שיקרה הוא שלווקטורים עם תדר 1 ושלילי 1 יהיה אותו אורך, והם יהיו השתקפויות אופקיות זה של זה. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. ", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you. ", "translatedText": "מה שראית זה עתה, פירוק פונקציה כשילוב של האקספוננציאלים הללו ושימוש בזה כדי לפתור משוואת דיפרנציאלית, מופיע שוב ושוב בצורות וצורות שונות. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/hindi/sentence_translations.json b/2019/fourier-series/hindi/sentence_translations.json index e9facdc9b..917b8481d 100644 --- a/2019/fourier-series/hindi/sentence_translations.json +++ b/2019/fourier-series/hindi/sentence_translations.json @@ -63,7 +63,7 @@ "end": 83.4 }, { - "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something we have the math to describe and control completely.", + "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something that we have the math to describe and to control completely.", "translatedText": "और प्रकृति में अन्यत्र पाई जाने वाली अधिकांश उभरती जटिलताओं के विपरीत, यह कुछ ऐसा है जिसका वर्णन करने और पूरी तरह से नियंत्रित करने के लिए हमारे पास गणित है।", "n_reviews": 0, "start": 83.84, @@ -553,7 +553,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", "translatedText": "यदि आप एक उबाऊ रेखाचित्र के लिए घूर्णनशील सदिशों में इनमें से एक अपघटन करते हैं, तो क्या होगा कि आवृत्ति 1 और ऋणात्मक 1 वाले सदिशों की लंबाई समान होगी, और वे एक दूसरे के क्षैतिज प्रतिबिंब होंगे।", "n_reviews": 0, "start": 685.42, @@ -1113,7 +1113,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms.", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you.", "translatedText": "आपने अभी जो देखा, इन घातांकों के संयोजन के रूप में एक फ़ंक्शन को तोड़ना और एक अंतर समीकरण को हल करने के लिए उसका उपयोग करना, बार-बार विभिन्न आकारों और रूपों में सामने आता है।", "n_reviews": 0, "start": 1456.92, diff --git a/2019/fourier-series/hungarian/sentence_translations.json b/2019/fourier-series/hungarian/sentence_translations.json index b6c6f1566..d1c67197b 100644 --- a/2019/fourier-series/hungarian/sentence_translations.json +++ b/2019/fourier-series/hungarian/sentence_translations.json @@ -760,7 +760,7 @@ "end": 862.58 }, { - "input": "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's rotating just at a frequency of 0.", + "input": "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's quote-unquote rotating just at a frequency of 0.", "translatedText": "A legkönnyebben leírható vektor az állandó vektor, amely az 1-es számnál marad, soha nem mozog, vagy ha úgy tetszik, csak 0-s frekvenciával forog.", "model": "DeepL", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 886.44 }, { - "input": "Technically it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", + "input": "Technically in what's being shown, it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", "translatedText": "Gyakorlatilag ez valójában egy ciklus 10 másodpercenként, így a dolgok nem túl szédítőek, mindent tízszeresére lassítok.", "model": "DeepL", "n_reviews": 0, @@ -1240,7 +1240,7 @@ "end": 1420.94 }, { - "input": "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links in the description.", + "input": "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links of course in the description.", "translatedText": "Egyébként, ha további Fourier-sorozatos tartalmakat keresel, ajánlom figyelmedbe a Mathologer és a The Coding Train videóit, és ezt a blogbejegyzést is, linkek a leírásban.", "model": "DeepL", "n_reviews": 0, diff --git a/2019/fourier-series/indonesian/sentence_translations.json b/2019/fourier-series/indonesian/sentence_translations.json index a66328f21..f114e696b 100644 --- a/2019/fourier-series/indonesian/sentence_translations.json +++ b/2019/fourier-series/indonesian/sentence_translations.json @@ -72,7 +72,7 @@ "end": 83.4 }, { - "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something we have the math to describe and control completely.", + "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something that we have the math to describe and to control completely.", "translatedText": "Dan tidak seperti kebanyakan kompleksitas yang Anda temukan di tempat lain di alam, hal ini adalah sesuatu yang harus kita uraikan dan kendalikan secara matematis.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", "translatedText": "Jika Anda melakukan salah satu penguraian ini menjadi vektor-vektor berputar untuk gambar yang membosankan, yang akan terjadi adalah vektor-vektor dengan frekuensi 1 dan negatif 1 akan memiliki panjang yang sama, dan vektor-vektor tersebut akan merupakan refleksi horizontal satu sama lain.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms.", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you.", "translatedText": "Apa yang baru saja Anda lihat, memecah suatu fungsi sebagai kombinasi eksponensial ini dan menggunakannya untuk menyelesaikan persamaan diferensial, muncul berulang kali dalam bentuk dan bentuk yang berbeda.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/italian/sentence_translations.json b/2019/fourier-series/italian/sentence_translations.json index 4d9e5f522..6531f58cb 100644 --- a/2019/fourier-series/italian/sentence_translations.json +++ b/2019/fourier-series/italian/sentence_translations.json @@ -72,7 +72,7 @@ "end": 83.4 }, { - "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something we have the math to describe and control completely.", + "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something that we have the math to describe and to control completely.", "translatedText": "E a differenza di gran parte della complessità emergente che trovi altrove in natura, questo è qualcosa che possiamo descrivere e controllare completamente con i calcoli.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", "translatedText": "Se esegui una di queste scomposizioni in vettori rotanti per un disegno noioso, ciò che accadrà è che i vettori con frequenza 1 e negativa 1 avranno la stessa lunghezza e saranno riflessi orizzontali l'uno dell'altro.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms.", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you.", "translatedText": "Ciò che hai appena visto, scomporre una funzione come una combinazione di questi esponenziali e usarla per risolvere un'equazione differenziale, si ripresenta ancora e ancora in forme e forme diverse.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/japanese/sentence_translations.json b/2019/fourier-series/japanese/sentence_translations.json index 67c183566..aff628c27 100644 --- a/2019/fourier-series/japanese/sentence_translations.json +++ b/2019/fourier-series/japanese/sentence_translations.json @@ -72,7 +72,7 @@ "end": 83.4 }, { - "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something we have the math to describe and control completely.", + "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something that we have the math to describe and to control completely.", "translatedText": "そして、自然界の他の場所で見られる新たな複雑さの多くとは異な り、これは私たちが数学で記述し、完全に制御できるものです。", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", "translatedText": "退屈な描画のためにこれらの分解の 1 つを回転ベクトルに実行す ると、何が起こるかというと、周波数 1 と負の 1 のベクト ルは同じ長さになり、それらは互いに水平方向の反射になります。", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms.", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you.", "translatedText": "これらの指数関数の組み合わせとして関数を分解 し、それを使用して微分方程式を解くという今 見たことは、さまざまな形で何度も現れます。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/korean/sentence_translations.json b/2019/fourier-series/korean/sentence_translations.json index a4eebb7f8..69a52ec93 100644 --- a/2019/fourier-series/korean/sentence_translations.json +++ b/2019/fourier-series/korean/sentence_translations.json @@ -72,7 +72,7 @@ "end": 83.4 }, { - "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something we have the math to describe and control completely.", + "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something that we have the math to describe and to control completely.", "translatedText": "그리고 자연의 다른 곳에서 발견되는 많은 새로운 복잡성과는 달리 이것은 우리가 완전히 설명하고 제어할 수 있는 수학을 가지고 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", "translatedText": "지루한 그림을 위해 이러한 분해 중 하나를 회전 벡터로 수행하면 주파수 1과 -1의 벡터가 동일한 길이를 가지며 서로 수평 반사가 되는 일이 발생합니다.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms.", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you.", "translatedText": "방금 본 것, 이러한 지수의 조합으로 함수를 분해하고 이를 사용하여 미분 방정식을 푸는 것은 다양한 모양과 형태로 계속해서 나타납니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/marathi/sentence_translations.json b/2019/fourier-series/marathi/sentence_translations.json index 32ee03af7..082cfa24c 100644 --- a/2019/fourier-series/marathi/sentence_translations.json +++ b/2019/fourier-series/marathi/sentence_translations.json @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", "translatedText": "कंटाळवाण्या रेखांकनासाठी जर तुम्ही यापैकी एक विघटन वेक्टर फिरवत व्हेक्टरमध्ये केले तर काय होईल, वारंवारता 1 आणि ऋण 1 सह व्हेक्टरची लांबी समान असेल आणि ते एकमेकांचे क्षैतिज प्रतिबिंब असतील. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. ", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you. ", "translatedText": "तुम्ही नुकतेच जे पाहिले, या घातांकांचे संयोजन म्हणून फंक्शन तोडणे आणि ते विभेदक समीकरण सोडवण्यासाठी वापरणे, ते पुन्हा पुन्हा वेगवेगळ्या आकारांमध्ये आणि रूपांमध्ये समोर येते. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/persian/sentence_translations.json b/2019/fourier-series/persian/sentence_translations.json index a9100ac3c..c53b40913 100644 --- a/2019/fourier-series/persian/sentence_translations.json +++ b/2019/fourier-series/persian/sentence_translations.json @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", "translatedText": "اگر یکی از این تجزیه را به بردارهای دوار برای ترسیم خسته کننده انجام دهید، اتفاقی که می افتد این است که بردارهای با فرکانس 1 و منفی 1 طول یکسانی خواهند داشت و بازتاب های افقی یکدیگر خواهند بود. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. ", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you. ", "translatedText": "چیزی که دیدید، شکستن یک تابع به عنوان ترکیبی از این نمایی ها و استفاده از آن برای حل یک معادله دیفرانسیل، بارها و بارها به اشکال و اشکال مختلف ظاهر می شود. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/polish/sentence_translations.json b/2019/fourier-series/polish/sentence_translations.json index 88ca4ee78..4e1134f06 100644 --- a/2019/fourier-series/polish/sentence_translations.json +++ b/2019/fourier-series/polish/sentence_translations.json @@ -758,7 +758,7 @@ "end": 862.58 }, { - "input": "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's rotating just at a frequency of 0.", + "input": "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's quote-unquote rotating just at a frequency of 0.", "translatedText": "", "from_community_srt": "kończący się w 1. Najłatwiejszy w opisie wektor to wektor stały, który pozostaje w 1, nie zmienia położenia. Albo jeśli wolisz, dokonuje obrotu z prędkością 0.", "n_reviews": 0, @@ -782,7 +782,7 @@ "end": 886.44 }, { - "input": "Technically it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", + "input": "Technically in what's being shown, it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", "translatedText": "", "from_community_srt": "Z tego co zobaczyliśmy, to właściwie 1 cykl, co każde 10 sekund, więc nie jest to zbyt szalona podróż,", "n_reviews": 0, @@ -1236,7 +1236,7 @@ "end": 1420.94 }, { - "input": "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links in the description.", + "input": "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links of course in the description.", "translatedText": "", "from_community_srt": "A tak w ogóle, jeśli chciałbyś poznać więcej faktów o szeregach Fouriera, polecam filmy udostępnione przez Mathologer i The Coding Train,", "n_reviews": 0, diff --git a/2019/fourier-series/portuguese/sentence_translations.json b/2019/fourier-series/portuguese/sentence_translations.json index 6e3ec80d6..fb95e181c 100644 --- a/2019/fourier-series/portuguese/sentence_translations.json +++ b/2019/fourier-series/portuguese/sentence_translations.json @@ -72,7 +72,7 @@ "end": 83.4 }, { - "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something we have the math to describe and control completely.", + "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something that we have the math to describe and to control completely.", "translatedText": "E, ao contrário de grande parte da complexidade emergente que encontramos em outros lugares da natureza, isso é algo que temos matemática para descrever e controlar completamente.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", "translatedText": "Se você fizer uma dessas decomposições em vetores rotativos para um desenho chato, o que acontecerá é que os vetores com frequência 1 e negativo 1 terão o mesmo comprimento e serão reflexos horizontais um do outro.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms.", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you.", "translatedText": "O que você acabou de ver, decompondo uma função como uma combinação dessas exponenciais e usando isso para resolver uma equação diferencial, surge repetidas vezes em diferentes formatos e formas.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/russian/sentence_translations.json b/2019/fourier-series/russian/sentence_translations.json index 009b1fdb2..56adac353 100644 --- a/2019/fourier-series/russian/sentence_translations.json +++ b/2019/fourier-series/russian/sentence_translations.json @@ -759,7 +759,7 @@ "end": 862.58 }, { - "input": "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's rotating just at a frequency of 0.", + "input": "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's quote-unquote rotating just at a frequency of 0.", "translatedText": "Самый простой для описания вектор — это постоянный вектор, который остается под номером 1 и никогда не движется, или, если хотите, он вращается только с частотой 0.", "from_community_srt": "под номером 1. Самый простой вектор для описания это константа тот, который просто остается на № 1, никогда перемещение. Или, если хотите, он «вращается» с частотой 0.", "n_reviews": 0, @@ -783,7 +783,7 @@ "end": 886.44 }, { - "input": "Technically it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", + "input": "Technically in what's being shown, it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", "translatedText": "Технически это на самом деле один цикл каждые 10 секунд, так что это не так уж и головокружительно, я замедляю все в 10 раз.", "from_community_srt": "В том, что показано, это на самом деле 1 цикл каждые 10 секунд, чтобы вещи не слишком головокружительно,", "n_reviews": 0, @@ -1239,7 +1239,7 @@ "end": 1420.94 }, { - "input": "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links in the description.", + "input": "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links of course in the description.", "translatedText": "Кстати, если вы ищете больше контента из серии Фурье, я настоятельно рекомендую видеоролики Mathologer и The Coding Train, а также этот пост в блоге, ссылки в описании.", "from_community_srt": "Кстати, если вы ищете больше Фурье содержание серии, я очень рекомендую видео Матологом и Поездом Кодирования на тема,", "n_reviews": 0, diff --git a/2019/fourier-series/spanish/sentence_translations.json b/2019/fourier-series/spanish/sentence_translations.json index 16bfc2eae..f1f29766b 100644 --- a/2019/fourier-series/spanish/sentence_translations.json +++ b/2019/fourier-series/spanish/sentence_translations.json @@ -752,7 +752,7 @@ "end": 862.58 }, { - "input": "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's rotating just at a frequency of 0.", + "input": "The easiest vector to describe is the constant one, which stays at the number 1, never moving, or if you prefer, it's quote-unquote rotating just at a frequency of 0.", "translatedText": "El vector más fácil de describir es el constante, que permanece en el número 1, sin moverse nunca, o si lo prefieres, gira solo con una frecuencia de 0.", "from_community_srt": "El mas fácil de describir es el vector constante El cual se queda en el número 1. Sin moverse O si prefieren \"está rotando con frecuencia 0\" Luego viene el vector que da una vuelta por segundo", "n_reviews": 0, @@ -775,7 +775,7 @@ "end": 886.44 }, { - "input": "Technically it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", + "input": "Technically in what's being shown, it's actually one cycle every 10 seconds so things aren't too dizzying, I'm slowing everything down by a factor of 10.", "translatedText": "Técnicamente, en realidad es un ciclo cada 10 segundos para que las cosas no sean demasiado vertiginosas, estoy ralentizando todo en un factor de 10.", "from_community_srt": "Técnicamente lo que están viendo es una vuelta cada 10 segundos Para que sea menos vertiginoso", "n_reviews": 0, @@ -1225,7 +1225,7 @@ "end": 1420.94 }, { - "input": "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links in the description.", + "input": "By the way, if you're looking for more Fourier series content, I highly recommend the videos by Mathologer and The Coding Train, and I'd also recommend this blog post, links of course in the description.", "translatedText": "Por cierto, si está buscando más contenido de la serie Fourier, le recomiendo los videos de Mathologer y The Coding Train, y también recomendaría esta publicación de blog, enlaces en la descripción.", "from_community_srt": "para los que quieran ver mas contenido sobre las series de Fourier Recomiendo totalmente los videos de Mathologer y The Coding Train Y también recomiendo este blog Links, por supuesto,", "n_reviews": 0, diff --git a/2019/fourier-series/tamil/sentence_translations.json b/2019/fourier-series/tamil/sentence_translations.json index eb71d2d89..98fdd2fb5 100644 --- a/2019/fourier-series/tamil/sentence_translations.json +++ b/2019/fourier-series/tamil/sentence_translations.json @@ -72,7 +72,7 @@ "end": 83.4 }, { - "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something we have the math to describe and control completely.", + "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something that we have the math to describe and to control completely.", "translatedText": "மேலும் இயற்கையில் வேறு இடங்களில் நீங்கள் காணும் பல சிக்கலான சிக்கலானது போலல்லாமல், இது முழுவதுமாக விவரிக்கவும் கட்டுப்படுத்தவும் எங்களிடம் கணிதம் உள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", "translatedText": "இந்த சிதைவுகளில் ஒன்றைச் சுழலும் திசையன்களாக மாற்றினால், அதிர்வெண் 1 மற்றும் எதிர்மறை 1 கொண்ட திசையன்கள் ஒரே நீளத்தைக் கொண்டிருக்கும், மேலும் அவை ஒன்றோடொன்று கிடைமட்ட பிரதிபலிப்புகளாக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms.", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you.", "translatedText": "நீங்கள் இப்போது பார்த்தது, இந்த அதிவேகங்களின் கலவையாக ஒரு செயல்பாட்டை உடைத்து, ஒரு வேறுபட்ட சமன்பாட்டைத் தீர்க்க அதைப் பயன்படுத்தி, வெவ்வேறு வடிவங்களிலும் வடிவங்களிலும் மீண்டும் மீண்டும் வருகிறது.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/telugu/sentence_translations.json b/2019/fourier-series/telugu/sentence_translations.json index 28f0fd917..cbc219925 100644 --- a/2019/fourier-series/telugu/sentence_translations.json +++ b/2019/fourier-series/telugu/sentence_translations.json @@ -72,7 +72,7 @@ "end": 83.4 }, { - "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something we have the math to describe and control completely.", + "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something that we have the math to describe and to control completely.", "translatedText": "మరియు ప్రకృతిలో మరెక్కడా మీరు కనుగొనే చాలా సంక్లిష్టత వలె కాకుండా, ఇది పూర్తిగా వివరించడానికి మరియు నియంత్రించడానికి మాకు గణితాన్ని కలిగి ఉంది.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", "translatedText": "మీరు బోరింగ్ డ్రాయింగ్ కోసం ఈ విచ్ఛేదనలలో ఒకదానిని తిరిగే వెక్టార్‌లుగా చేస్తే, ఏమి జరుగుతుంది ఫ్రీక్వెన్సీ 1 మరియు నెగెటివ్ 1 ఉన్న వెక్టర్‌లు ఒకే పొడవును కలిగి ఉంటాయి మరియు అవి ఒకదానికొకటి సమాంతర ప్రతిబింబాలుగా ఉంటాయి.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms.", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you.", "translatedText": "మీరు ఇప్పుడే చూసినది, ఈ ఘాతాంకాల కలయికగా ఫంక్షన్‌ను విచ్ఛిన్నం చేసి, అవకలన సమీకరణాన్ని పరిష్కరించడానికి దాన్ని ఉపయోగించడం ద్వారా, విభిన్న ఆకారాలు మరియు రూపాల్లో మళ్లీ మళ్లీ కనిపిస్తుంది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/thai/sentence_translations.json b/2019/fourier-series/thai/sentence_translations.json index efbcab791..685a8dea9 100644 --- a/2019/fourier-series/thai/sentence_translations.json +++ b/2019/fourier-series/thai/sentence_translations.json @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", "translatedText": "นอกเหนือจากความทั่วไปมากขึ้นแล้ว ในมุมมองของฉัน การคำนวณจะสะอาดขึ้น และเข้าใจได้ง่ายขึ้นว่าเหตุใดจึงใช้งานได้จริง ที่สำคัญกว่านั้นคือเป็นการวางรากฐานที่ดีสำหรับแนวคิดที่จะเกิดขึ้นในภายหลังในซีรีส์ เช่น การแปลง Laplace และความสำคัญของฟังก์ชันเลขชี้กำลัง เรายังคงนึกถึงฟังก์ชันที่อินพุตเป็นจำนวนจริงในช่วงเวลาจำกัด เช่น ตั้งแต่ 0 ถึง 1 เพื่อความง่าย แต่ในขณะที่ฟังก์ชันบางอย่างเช่นฟังก์ชันอุณหภูมิจะมีเอาต์พุตบนเส้นจำนวนจริง มุมมองที่กว้างขึ้นนี้จะทำให้เอาต์พุตเดินไปได้ทุกที่ในระนาบเชิงซ้อน 2d คุณอาจนึกถึงฟังก์ชันดังกล่าวเหมือนภาพวาด โดยมีปลายดินสอลากตามจุดต่างๆ ในระนาบเชิงซ้อน เนื่องจากอินพุตมีช่วงตั้งแต่ 0 ถึง 1 และแทนที่จะให้คลื่นไซน์เป็นตัวสร้างพื้นฐาน อย่างที่คุณเห็นตอนเริ่มต้น เราจะเน้นที่การแยกฟังก์ชันเหล่านี้ออกเป็นผลรวมของเวกเตอร์เล็กๆ ทั้งหมดหมุนด้วยความถี่จำนวนเต็มคงที่ ฟังก์ชั่นที่มีเอาต์พุตเป็นจำนวนจริงนั้นเป็นภาพวาดที่น่าเบื่อจริงๆ ซึ่งเป็นภาพร่างดินสอแบบมิติเดียว คุณอาจไม่คุ้นเคยกับการคิดแบบนี้ เนื่องจากปกติแล้วเราจะเห็นภาพฟังก์ชันดังกล่าวด้วยกราฟ แต่ตอนนี้เส้นทางที่วาดจะอยู่ในพื้นที่เอาต์พุตเท่านั้น หากคุณแบ่งแยกองค์ประกอบอย่างใดอย่างหนึ่งให้เป็นเวกเตอร์ที่หมุนได้สำหรับการวาดภาพน่าเบื่อ สิ่งที่จะเกิดขึ้นคือเวกเตอร์ที่มีความถี่ 1 และลบ 1 จะมีความยาวเท่ากัน และพวกมันจะสะท้อนกันในแนวนอน เมื่อคุณดูผลรวมของสองตัวนี้ขณะที่มันหมุน ผลรวมนั้นจะคงที่บนเส้นจำนวนจริง และมันจะแกว่งเหมือนคลื่นไซน์ หากคุณไม่เคยเห็นมันมาก่อน นี่อาจเป็นวิธีที่แปลกมากในการคิดว่าคลื่นไซน์คืออะไร เนื่องจากเราคุ้นเคยกับการดูกราฟของมันมากกว่าที่จะดูผลลัพธ์เพียงอย่างเดียวที่เคลื่อนไปตามเส้นจำนวนจริง แต่ใน บริบทที่กว้างขึ้นของฟังก์ชันที่มีเอาต์พุตจำนวนเชิงซ้อน การแกว่งบนเส้นแนวนอนจะมีลักษณะเหมือนคลื่นไซน์ ในทำนองเดียวกัน คู่ของเวกเตอร์ที่หมุนด้วยความถี่ 2 และลบ 2 จะเพิ่มส่วนประกอบของคลื่นไซน์อีกตัวหนึ่ง และต่อๆ ไป โดยคลื่นไซน์ที่เรากำลังมองหาก่อนหน้านี้ตอนนี้สอดคล้องกับคู่ของเวกเตอร์ที่หมุนในทิศทางตรงกันข้าม ดังนั้นบริบทที่ฟูริเยร์ศึกษาแต่แรก โดยแบ่งฟังก์ชันมูลค่าจริงออกเป็นคลื่นไซน์ จึงเป็นกรณีพิเศษของแนวคิดทั่วไปเกี่ยวกับการวาดภาพ 2 มิติและการหมุนเวกเตอร์ และ ณ จุดนี้ บางทีคุณอาจไม่เชื่อฉันว่าการขยายมุมมองของเราไปสู่ฟังก์ชันที่ซับซ้อนทำให้สิ่งต่างๆ เข้าใจง่ายขึ้น แต่ทนกับฉันเถอะ มันคุ้มค่าจริงๆ ที่เพิ่มความพยายามในการดูภาพที่สมบูรณ์ยิ่งขึ้น และฉันคิดว่าคุณจะต้องพอใจ ด้วยความสะอาดของการคำนวณจริงในบริบทที่กว้างขึ้นนี้ คุณอาจสงสัยว่าทำไม หากเราแยกมันเป็นสองมิติ เราไม่แค่พูดถึงเวกเตอร์ 2 มิติเท่านั้น สแควร์รูทของลบเกี่ยวอะไรกับอะไร? ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. ", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/turkish/sentence_translations.json b/2019/fourier-series/turkish/sentence_translations.json index a2fadc0d8..80b2edd7d 100644 --- a/2019/fourier-series/turkish/sentence_translations.json +++ b/2019/fourier-series/turkish/sentence_translations.json @@ -72,7 +72,7 @@ "end": 83.4 }, { - "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something we have the math to describe and control completely.", + "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something that we have the math to describe and to control completely.", "translatedText": "Ve doğanın başka yerlerinde bulduğunuz yeni ortaya çıkan karmaşıklığın çoğundan farklı olarak, bu, tamamen tanımlayacak ve kontrol edecek matematiğe sahip olduğumuz bir şey.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", "translatedText": "Sıkıcı bir çizim için bu ayrıştırmalardan birini dönen vektörlere yaparsanız, frekansı 1 ve negatif 1 olan vektörler aynı uzunluğa sahip olacak ve birbirlerinin yatay yansımaları olacaklardır.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms.", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you.", "translatedText": "Az önce gördüğünüz şey, bir fonksiyonu bu üstel sayıların birleşimi olarak parçalamak ve bunu bir diferansiyel denklemi çözmek için kullanmak, farklı şekil ve formlarda tekrar tekrar ortaya çıkıyor.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/ukrainian/sentence_translations.json b/2019/fourier-series/ukrainian/sentence_translations.json index 69661a163..538146ba2 100644 --- a/2019/fourier-series/ukrainian/sentence_translations.json +++ b/2019/fourier-series/ukrainian/sentence_translations.json @@ -72,7 +72,7 @@ "end": 83.4 }, { - "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something we have the math to describe and control completely.", + "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something that we have the math to describe and to control completely.", "translatedText": "І на відміну від великої частини складності, що виникає деінде в природі, це те, що ми маємо математику, щоб описати та повністю контролювати.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", "translatedText": "Якщо ви зробите одне з цих розкладів на обертові вектори для нудного малюнка, станеться, що вектори з частотою 1 і мінус 1 матимуть однакову довжину, і вони будуть горизонтальними відображеннями один одного.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms.", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you.", "translatedText": "Те, що ви щойно бачили, розкладання функції як комбінації цих показників і використання цього для вирішення диференціального рівняння, з’являється знову і знову в різних формах і формах.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/urdu/sentence_translations.json b/2019/fourier-series/urdu/sentence_translations.json index 72cb9ce59..5ee81d174 100644 --- a/2019/fourier-series/urdu/sentence_translations.json +++ b/2019/fourier-series/urdu/sentence_translations.json @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other. ", "translatedText": "اگر آپ بورنگ ڈرائنگ کے لیے ان میں سے کسی ایک کو گھمانے والے ویکٹر میں کرتے ہیں، تو کیا ہوگا کہ فریکوئنسی 1 اور منفی 1 والے ویکٹر کی لمبائی ایک جیسی ہوگی، اور وہ ایک دوسرے کے افقی عکاس ہوں گے۔", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. ", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you. ", "translatedText": "جو آپ نے ابھی دیکھا ہے، ایک فنکشن کو ان ایکسپونینشلز کے مجموعہ کے طور پر توڑنا اور اسے ایک تفریق مساوات کو حل کرنے کے لیے استعمال کرنا، بار بار مختلف اشکال اور شکلوں میں سامنے آتا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/fourier-series/vietnamese/sentence_translations.json b/2019/fourier-series/vietnamese/sentence_translations.json index 9daa1d30d..b2ef362ff 100644 --- a/2019/fourier-series/vietnamese/sentence_translations.json +++ b/2019/fourier-series/vietnamese/sentence_translations.json @@ -72,7 +72,7 @@ "end": 83.4 }, { - "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something we have the math to describe and control completely.", + "input": "And unlike much of the emergent complexity you find elsewhere in nature, this is something that we have the math to describe and to control completely.", "translatedText": "Và không giống như phần lớn sự phức tạp xuất hiện mà bạn tìm thấy ở những nơi khác trong tự nhiên, đây là thứ mà chúng ta có toán học để mô tả và kiểm soát hoàn toàn.", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 680.82 }, { - "input": "If you do one of these decompositions into rotating vectors for a boring drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", + "input": "If you do one of these decompositions into rotating vectors for a boring one-dimensional drawing, what will happen is that the vectors with frequency 1 and negative 1 will have the same length, and they'll be horizontal reflections of each other.", "translatedText": "Nếu bạn thực hiện một trong những phép phân tách này thành các vectơ quay cho một bản vẽ nhàm chán, điều sẽ xảy ra là các vectơ có tần số 1 và âm 1 sẽ có cùng độ dài và chúng sẽ phản xạ theo chiều ngang của nhau.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1456.24 }, { - "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms.", + "input": "What you just saw, breaking down a function as a combination of these exponentials and using that to solve a differential equation, comes up again and again in different shapes and forms. Thank you.", "translatedText": "Những gì bạn vừa thấy, chia nhỏ một hàm dưới dạng tổ hợp của các số mũ này và sử dụng hàm đó để giải một phương trình vi phân, sẽ xuất hiện lặp đi lặp lại dưới nhiều hình dạng và dạng khác nhau.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/arabic/sentence_translations.json b/2019/heat-equation/arabic/sentence_translations.json index 2a2a62fa6..56e6e61eb 100644 --- a/2019/heat-equation/arabic/sentence_translations.json +++ b/2019/heat-equation/arabic/sentence_translations.json @@ -145,7 +145,7 @@ "end": 102.0 }, { - "input": "3.", + "input": "And number three,", "translatedText": "3.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/bengali/sentence_translations.json b/2019/heat-equation/bengali/sentence_translations.json index 4075ed4f7..188f1df11 100644 --- a/2019/heat-equation/bengali/sentence_translations.json +++ b/2019/heat-equation/bengali/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave. ", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave. ", "translatedText": "আপনি দেখতে পাচ্ছেন, x এর মান L পর্যন্ত বাড়ার সাথে সাথে আমাদের কোসাইন এক্সপ্রেশনের ইনপুট পাই পর্যন্ত যায়, যা একটি কোসাইন তরঙ্গের অর্ধেক সময়কাল।", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/chinese/sentence_translations.json b/2019/heat-equation/chinese/sentence_translations.json index d202a0865..9045b30d4 100644 --- a/2019/heat-equation/chinese/sentence_translations.json +++ b/2019/heat-equation/chinese/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "您会看到， 当 x 增加到值 L 时，余弦表达式的输入增加到 pi， 这是余弦波周期的一半。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/english/captions.srt b/2019/heat-equation/english/captions.srt index c43f862cf..15f4f1c16 100644 --- a/2019/heat-equation/english/captions.srt +++ b/2019/heat-equation/english/captions.srt @@ -108,7 +108,7 @@ If you know multiple solutions, the sum of these functions is also a solution. 28 00:01:42,580 --> 00:01:43,100 -3. +And number three, 29 00:01:43,400 --> 00:01:47,380 diff --git a/2019/heat-equation/english/sentence_timings.json b/2019/heat-equation/english/sentence_timings.json index 85340c8a9..c2bcf88fb 100644 --- a/2019/heat-equation/english/sentence_timings.json +++ b/2019/heat-equation/english/sentence_timings.json @@ -85,7 +85,7 @@ 102.0 ], [ - "3.", + "And number three,", 102.58, 103.1 ], diff --git a/2019/heat-equation/english/transcript.txt b/2019/heat-equation/english/transcript.txt index 835131ba6..076a2f119 100644 --- a/2019/heat-equation/english/transcript.txt +++ b/2019/heat-equation/english/transcript.txt @@ -15,7 +15,7 @@ We can think of his solution as being broken down into three fundamental observa Certain sine waves offer a really simple solution to this equation. 2. If you know multiple solutions, the sum of these functions is also a solution. -3. +And number three, Most surprisingly, any function can be expressed as a sum of sine waves. A pedantic mathematician might point out that there are some pathological exceptions, but basically any distribution you would come across in practice, including discontinuous ones, can be written as a sum of sine waves, potentially infinitely many. And if you've ever heard of Fourier series, you've at least heard of this last idea. diff --git a/2019/heat-equation/french/sentence_translations.json b/2019/heat-equation/french/sentence_translations.json index 2eabb99d7..742d27b5b 100644 --- a/2019/heat-equation/french/sentence_translations.json +++ b/2019/heat-equation/french/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "Vous voyez, à mesure que x augmente jusqu'à la valeur L, l'entrée de notre expression cosinusoïdale monte jusqu'à pi, qui correspond à la moitié de la période d'une onde cosinusoïdale.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/german/sentence_translations.json b/2019/heat-equation/german/sentence_translations.json index 00b42bdc8..a6b73e3b8 100644 --- a/2019/heat-equation/german/sentence_translations.json +++ b/2019/heat-equation/german/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "Sie sehen, wenn x bis zum Wert L ansteigt, steigt die Eingabe unseres Kosinusausdrucks auf pi, was der halben Periode einer Kosinuswelle entspricht.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/greek/sentence_translations.json b/2019/heat-equation/greek/sentence_translations.json index 72313be19..3b4ffe15d 100644 --- a/2019/heat-equation/greek/sentence_translations.json +++ b/2019/heat-equation/greek/sentence_translations.json @@ -133,7 +133,7 @@ "end": 102.0 }, { - "input": "3.", + "input": "And number three,", "translatedText": "", "from_community_srt": "και 3) περιέργως,", "n_reviews": 0, diff --git a/2019/heat-equation/hebrew/sentence_translations.json b/2019/heat-equation/hebrew/sentence_translations.json index bfa2e9832..01e2f9764 100644 --- a/2019/heat-equation/hebrew/sentence_translations.json +++ b/2019/heat-equation/hebrew/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "אתה רואה, כאשר x עולה עד לערך L, הקלט של ביטוי הקוסינוס שלנו עולה ל-pi, שהוא מחצית מהתקופה של גל קוסינוס.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/hindi/sentence_translations.json b/2019/heat-equation/hindi/sentence_translations.json index 25258e0f3..28f9aa177 100644 --- a/2019/heat-equation/hindi/sentence_translations.json +++ b/2019/heat-equation/hindi/sentence_translations.json @@ -602,7 +602,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "आप देखते हैं, जैसे-जैसे x मान L तक बढ़ता है, हमारी कोसाइन अभिव्यक्ति का इनपुट pi तक जाता है, जो कोसाइन तरंग की आधी अवधि है।", "n_reviews": 0, "start": 717.21, diff --git a/2019/heat-equation/hungarian/sentence_translations.json b/2019/heat-equation/hungarian/sentence_translations.json index 775f6f504..17da97ffa 100644 --- a/2019/heat-equation/hungarian/sentence_translations.json +++ b/2019/heat-equation/hungarian/sentence_translations.json @@ -136,7 +136,7 @@ "end": 102.0 }, { - "input": "3.", + "input": "And number three,", "translatedText": "3.", "model": "DeepL", "n_reviews": 0, diff --git a/2019/heat-equation/indonesian/sentence_translations.json b/2019/heat-equation/indonesian/sentence_translations.json index 3ad008b4c..178b6de26 100644 --- a/2019/heat-equation/indonesian/sentence_translations.json +++ b/2019/heat-equation/indonesian/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "Anda lihat, ketika x meningkat hingga nilai L, masukan ekspresi kosinus kita naik menjadi pi, yang merupakan setengah periode gelombang kosinus.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/italian/sentence_translations.json b/2019/heat-equation/italian/sentence_translations.json index 611978bf5..64ca4290f 100644 --- a/2019/heat-equation/italian/sentence_translations.json +++ b/2019/heat-equation/italian/sentence_translations.json @@ -152,7 +152,7 @@ "end": 102.0 }, { - "input": "3.", + "input": "And number three,", "translatedText": "3.", "model": "DeepL", "from_community_srt": "e 3. Sorprendentemente,", diff --git a/2019/heat-equation/japanese/sentence_translations.json b/2019/heat-equation/japanese/sentence_translations.json index ce2e50cf2..43e187fcd 100644 --- a/2019/heat-equation/japanese/sentence_translations.json +++ b/2019/heat-equation/japanese/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "x が値 L まで増加すると、コサイン式の入力はコサイン波の周期の半分であ る pi まで増加します。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/korean/sentence_translations.json b/2019/heat-equation/korean/sentence_translations.json index 03c8e1360..c1970a496 100644 --- a/2019/heat-equation/korean/sentence_translations.json +++ b/2019/heat-equation/korean/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "x가 L 값까지 증가함에 따라 코사인 표현식의 입력은 코사인파 주기의 절반인 pi까지 올라갑니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/marathi/sentence_translations.json b/2019/heat-equation/marathi/sentence_translations.json index 0e76b7f61..64dc1fad6 100644 --- a/2019/heat-equation/marathi/sentence_translations.json +++ b/2019/heat-equation/marathi/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "तुम्ही पहा, जसे x हे मूल्य L पर्यंत वाढते, आमच्या कोसाइन अभिव्यक्तीचे इनपुट pi पर्यंत जाते, जो कोसाइन वेव्हच्या अर्धा कालावधी असतो.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/persian/sentence_translations.json b/2019/heat-equation/persian/sentence_translations.json index 07d82b025..2e6e113e9 100644 --- a/2019/heat-equation/persian/sentence_translations.json +++ b/2019/heat-equation/persian/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave. ", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave. ", "translatedText": "می بینید، با افزایش x تا مقدار L، ورودی بیان کسینوس ما به عدد pi می رسد، که نصف دوره یک موج کسینوس است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/portuguese/sentence_translations.json b/2019/heat-equation/portuguese/sentence_translations.json index 7e188979c..f45da4ee6 100644 --- a/2019/heat-equation/portuguese/sentence_translations.json +++ b/2019/heat-equation/portuguese/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "Veja, à medida que x aumenta até o valor L, a entrada da nossa expressão cosseno sobe para pi, que é metade do período de uma onda cosseno.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/russian/sentence_translations.json b/2019/heat-equation/russian/sentence_translations.json index 12e20c8d4..2879510a7 100644 --- a/2019/heat-equation/russian/sentence_translations.json +++ b/2019/heat-equation/russian/sentence_translations.json @@ -134,7 +134,7 @@ "end": 102.0 }, { - "input": "3.", + "input": "And number three,", "translatedText": "3.", "n_reviews": 0, "start": 102.58, diff --git a/2019/heat-equation/spanish/sentence_translations.json b/2019/heat-equation/spanish/sentence_translations.json index fb0529a6d..3657e0140 100644 --- a/2019/heat-equation/spanish/sentence_translations.json +++ b/2019/heat-equation/spanish/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "Verá, a medida que x aumenta hasta el valor L, la entrada de nuestra expresión coseno sube a pi, que es la mitad del período de una onda coseno.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/tamil/sentence_translations.json b/2019/heat-equation/tamil/sentence_translations.json index 4e3a19f46..8ec73799f 100644 --- a/2019/heat-equation/tamil/sentence_translations.json +++ b/2019/heat-equation/tamil/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "நீங்கள் பார்க்கிறீர்கள், x மதிப்பு L வரை அதிகரிக்கும் போது, எங்கள் கொசைன் வெளிப்பாட்டின் உள்ளீடு pi வரை செல்கிறது, இது ஒரு கொசைன் அலையின் பாதி காலமாகும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/telugu/sentence_translations.json b/2019/heat-equation/telugu/sentence_translations.json index b88906a80..17a29611b 100644 --- a/2019/heat-equation/telugu/sentence_translations.json +++ b/2019/heat-equation/telugu/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "మీరు చూస్తారు, x విలువ L వరకు పెరిగినప్పుడు, మా కొసైన్ వ్యక్తీకరణ యొక్క ఇన్‌పుట్ pi వరకు వెళుతుంది, ఇది కొసైన్ వేవ్ యొక్క సగం వ్యవధి.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/thai/sentence_translations.json b/2019/heat-equation/thai/sentence_translations.json index f8687cea9..53c798907 100644 --- a/2019/heat-equation/thai/sentence_translations.json +++ b/2019/heat-equation/thai/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave. ", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/turkish/sentence_translations.json b/2019/heat-equation/turkish/sentence_translations.json index 3c9bcb506..9020d82c8 100644 --- a/2019/heat-equation/turkish/sentence_translations.json +++ b/2019/heat-equation/turkish/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "Görüyorsunuz, x, L değerine yükseldikçe, kosinüs ifademizin girdisi, kosinüs dalgasının periyodunun yarısı olan pi'ye kadar çıkıyor.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/ukrainian/sentence_translations.json b/2019/heat-equation/ukrainian/sentence_translations.json index dc359a5c1..eaf34a49b 100644 --- a/2019/heat-equation/ukrainian/sentence_translations.json +++ b/2019/heat-equation/ukrainian/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "Ви бачите, коли х збільшується до значення L, вхід нашого косинусного виразу зростає до пі, що становить половину періоду хвилі косинуса.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/urdu/sentence_translations.json b/2019/heat-equation/urdu/sentence_translations.json index a8174ebd5..c2ad90a9b 100644 --- a/2019/heat-equation/urdu/sentence_translations.json +++ b/2019/heat-equation/urdu/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave. ", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave. ", "translatedText": "آپ دیکھتے ہیں، جیسے جیسے x قدر L تک بڑھتا ہے، ہمارے کوزائن ایکسپریشن کا ان پٹ pi تک جاتا ہے، جو کہ کوزائن لہر کی نصف مدت ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/heat-equation/vietnamese/sentence_translations.json b/2019/heat-equation/vietnamese/sentence_translations.json index b5904d591..a4f00a7f9 100644 --- a/2019/heat-equation/vietnamese/sentence_translations.json +++ b/2019/heat-equation/vietnamese/sentence_translations.json @@ -688,7 +688,7 @@ "end": 715.41 }, { - "input": "You see, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", + "input": "You see that way, as x increases up to the value L, the input of our cosine expression goes up to pi, which is half the period of a cosine wave.", "translatedText": "Bạn thấy đấy, khi x tăng lên đến giá trị L, đầu vào của biểu thức cosine của chúng ta sẽ tăng lên pi, bằng một nửa chu kỳ của sóng cosin.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/arabic/sentence_translations.json b/2019/pdes/arabic/sentence_translations.json index 0c0fb93c9..9f39e49a8 100644 --- a/2019/pdes/arabic/sentence_translations.json +++ b/2019/pdes/arabic/sentence_translations.json @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book. ", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book ", "translatedText": "مرة أخرى، أعلم أن هذا يبدو كإعلان، لكنه ليس كذلك، أعتقد أنك ستستمتع بالكتاب. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/bengali/sentence_translations.json b/2019/pdes/bengali/sentence_translations.json index f1dff2c3d..d6c62ecc4 100644 --- a/2019/pdes/bengali/sentence_translations.json +++ b/2019/pdes/bengali/sentence_translations.json @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book. ", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book ", "translatedText": "আবার, আমি জানি যে এটি একটি বিজ্ঞাপনের মতো শোনাচ্ছে, কিন্তু এটি নয়, আমি মনে করি আপনি বইটি উপভোগ করবেন।", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/chinese/sentence_translations.json b/2019/pdes/chinese/sentence_translations.json index 97bf95af9..e732b7df6 100644 --- a/2019/pdes/chinese/sentence_translations.json +++ b/2019/pdes/chinese/sentence_translations.json @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book. ", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book ", "translatedText": "再说一次，我知道这听起来像广告，但事实 并非如此，我只是认为你会喜欢这本书。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/french/sentence_translations.json b/2019/pdes/french/sentence_translations.json index b62ac0b5d..fc4f3018f 100644 --- a/2019/pdes/french/sentence_translations.json +++ b/2019/pdes/french/sentence_translations.json @@ -512,7 +512,7 @@ "end": 523.66 }, { - "input": "When this difference is greater than zero, T2 will heat up, and the bigger the difference, the faster it heats up.", + "input": "When this difference is greater than zero, T2 will tend to heat up. And the bigger the difference, the faster it heats up.", "translatedText": "Lorsque cette différence est supérieure à zéro, T2 va chauffer, et plus la différence est grande, plus il chauffe vite.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 533.46 }, { - "input": "Likewise, if it's negative, T2 will cool down, at a rate proportional to that difference.", + "input": "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", "translatedText": "De même, s’il est négatif, T2 se refroidira, à un rythme proportionnel à cette différence.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book.", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book", "translatedText": "Encore une fois, je sais que cela ressemble à une publicité, mais ce n'est pas le cas, je pense simplement que vous apprécierez le livre.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/german/sentence_translations.json b/2019/pdes/german/sentence_translations.json index f9ab35796..83660f1e1 100644 --- a/2019/pdes/german/sentence_translations.json +++ b/2019/pdes/german/sentence_translations.json @@ -512,7 +512,7 @@ "end": 523.66 }, { - "input": "When this difference is greater than zero, T2 will heat up, and the bigger the difference, the faster it heats up.", + "input": "When this difference is greater than zero, T2 will tend to heat up. And the bigger the difference, the faster it heats up.", "translatedText": "Wenn diese Differenz größer als Null ist, erwärmt sich T2, und je größer die Differenz, desto schneller erwärmt es sich.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 533.46 }, { - "input": "Likewise, if it's negative, T2 will cool down, at a rate proportional to that difference.", + "input": "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", "translatedText": "Wenn es negativ ist, kühlt sich T2 ebenfalls mit einer Geschwindigkeit ab, die proportional zu dieser Differenz ist.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book.", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book", "translatedText": "Auch hier weiß ich, dass das wie eine Werbung klingt, aber das ist es nicht. Ich denke einfach, dass Ihnen das Buch gefallen wird.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/hebrew/sentence_translations.json b/2019/pdes/hebrew/sentence_translations.json index f874b781d..303bbe80b 100644 --- a/2019/pdes/hebrew/sentence_translations.json +++ b/2019/pdes/hebrew/sentence_translations.json @@ -512,7 +512,7 @@ "end": 523.66 }, { - "input": "When this difference is greater than zero, T2 will heat up, and the bigger the difference, the faster it heats up.", + "input": "When this difference is greater than zero, T2 will tend to heat up. And the bigger the difference, the faster it heats up.", "translatedText": "כאשר ההבדל הזה גדול מאפס, T2 יתחמם, וככל שההפרש גדול יותר, כך הוא מתחמם מהר יותר.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 533.46 }, { - "input": "Likewise, if it's negative, T2 will cool down, at a rate proportional to that difference.", + "input": "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", "translatedText": "באופן דומה, אם הוא שלילי, T2 יתקרר, בקצב פרופורציונלי להפרש הזה.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book.", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book", "translatedText": "שוב, אני יודע שזה נשמע כמו פרסומת, אבל זה לא, אני רק חושב שאתה תהנה מהספר.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/hindi/sentence_translations.json b/2019/pdes/hindi/sentence_translations.json index c50fbefa7..e9e271471 100644 --- a/2019/pdes/hindi/sentence_translations.json +++ b/2019/pdes/hindi/sentence_translations.json @@ -448,14 +448,14 @@ "end": 523.66 }, { - "input": "When this difference is greater than zero, T2 will heat up, and the bigger the difference, the faster it heats up.", + "input": "When this difference is greater than zero, T2 will tend to heat up. And the bigger the difference, the faster it heats up.", "translatedText": "जब यह अंतर शून्य से अधिक होता है, तो T2 गर्म हो जाएगा, और अंतर जितना बड़ा होगा, यह उतनी ही तेजी से गर्म होगा।", "n_reviews": 0, "start": 525.0, "end": 533.46 }, { - "input": "Likewise, if it's negative, T2 will cool down, at a rate proportional to that difference.", + "input": "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", "translatedText": "इसी तरह, यदि यह नकारात्मक है, तो T2 उस अंतर के समानुपाती दर से ठंडा हो जाएगा।", "n_reviews": 0, "start": 535.86, @@ -924,7 +924,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book.", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book", "translatedText": "फिर, मुझे पता है कि यह एक विज्ञापन जैसा लगता है, लेकिन ऐसा नहीं है, मुझे लगता है कि आप किताब का आनंद लेंगे।", "n_reviews": 0, "start": 1032.5, diff --git a/2019/pdes/indonesian/sentence_translations.json b/2019/pdes/indonesian/sentence_translations.json index c42d240b0..9c3444277 100644 --- a/2019/pdes/indonesian/sentence_translations.json +++ b/2019/pdes/indonesian/sentence_translations.json @@ -512,7 +512,7 @@ "end": 523.66 }, { - "input": "When this difference is greater than zero, T2 will heat up, and the bigger the difference, the faster it heats up.", + "input": "When this difference is greater than zero, T2 will tend to heat up. And the bigger the difference, the faster it heats up.", "translatedText": "Ketika perbedaan ini lebih besar dari nol, T2 akan memanas, dan semakin besar perbedaannya, semakin cepat pula pemanasannya.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 533.46 }, { - "input": "Likewise, if it's negative, T2 will cool down, at a rate proportional to that difference.", + "input": "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", "translatedText": "Demikian pula, jika negatif, T2 akan mendingin, dengan laju yang sebanding dengan perbedaan tersebut.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book.", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book", "translatedText": "Sekali lagi, saya tahu kedengarannya seperti sebuah iklan, tetapi sebenarnya tidak, saya hanya berpikir Anda akan menikmati bukunya.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/italian/sentence_translations.json b/2019/pdes/italian/sentence_translations.json index 3d5044078..023b3f7c0 100644 --- a/2019/pdes/italian/sentence_translations.json +++ b/2019/pdes/italian/sentence_translations.json @@ -512,7 +512,7 @@ "end": 523.66 }, { - "input": "When this difference is greater than zero, T2 will heat up, and the bigger the difference, the faster it heats up.", + "input": "When this difference is greater than zero, T2 will tend to heat up. And the bigger the difference, the faster it heats up.", "translatedText": "Quando questa differenza è maggiore di zero, T2 si riscalda e maggiore è la differenza, più velocemente si riscalda.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 533.46 }, { - "input": "Likewise, if it's negative, T2 will cool down, at a rate proportional to that difference.", + "input": "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", "translatedText": "Allo stesso modo, se è negativo, T2 si raffredderà, ad una velocità proporzionale a tale differenza.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book.", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book", "translatedText": "Ancora una volta, so che sembra una pubblicità, ma non lo è, penso solo che il libro ti piacerà.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/japanese/sentence_translations.json b/2019/pdes/japanese/sentence_translations.json index a95cc5de2..6a3ffbbe3 100644 --- a/2019/pdes/japanese/sentence_translations.json +++ b/2019/pdes/japanese/sentence_translations.json @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book. ", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book ", "translatedText": "繰り返しますが、広告のように聞こえるかもしれませんが、そ うではありません。この本を楽しんでいただけると思います。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/korean/sentence_translations.json b/2019/pdes/korean/sentence_translations.json index 4db9fe6f5..38452f2f2 100644 --- a/2019/pdes/korean/sentence_translations.json +++ b/2019/pdes/korean/sentence_translations.json @@ -512,7 +512,7 @@ "end": 523.66 }, { - "input": "When this difference is greater than zero, T2 will heat up, and the bigger the difference, the faster it heats up.", + "input": "When this difference is greater than zero, T2 will tend to heat up. And the bigger the difference, the faster it heats up.", "translatedText": "이 차이가 0보다 크면 T2가 가열되고, 차이가 클수록 더 빨리 가열됩니다.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 533.46 }, { - "input": "Likewise, if it's negative, T2 will cool down, at a rate proportional to that difference.", + "input": "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", "translatedText": "마찬가지로 음수이면 T2는 그 차이에 비례하는 속도로 냉각됩니다.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book.", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book", "translatedText": "다시 말하지만, 광고처럼 들리겠지만 그렇지 않습니다. 단지 여러분이 이 책을 좋아하실 것이라고 생각합니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/marathi/sentence_translations.json b/2019/pdes/marathi/sentence_translations.json index 689614d73..977b57004 100644 --- a/2019/pdes/marathi/sentence_translations.json +++ b/2019/pdes/marathi/sentence_translations.json @@ -512,7 +512,7 @@ "end": 523.66 }, { - "input": "When this difference is greater than zero, T2 will heat up, and the bigger the difference, the faster it heats up.", + "input": "When this difference is greater than zero, T2 will tend to heat up. And the bigger the difference, the faster it heats up.", "translatedText": "जेव्हा हा फरक शून्यापेक्षा जास्त असेल, तेव्हा T2 गरम होईल आणि जितका मोठा फरक असेल तितक्या लवकर तो गरम होईल.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 533.46 }, { - "input": "Likewise, if it's negative, T2 will cool down, at a rate proportional to that difference.", + "input": "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", "translatedText": "त्याचप्रमाणे, जर ते नकारात्मक असेल तर, T2 त्या फरकाच्या प्रमाणात, थंड होईल.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book.", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book", "translatedText": "पुन्हा, मला माहित आहे की ते जाहिरातीसारखे वाटत आहे, परंतु तसे नाही, मला वाटते की तुम्हाला पुस्तक आवडेल.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/persian/sentence_translations.json b/2019/pdes/persian/sentence_translations.json index 40fc56f00..b6d64115f 100644 --- a/2019/pdes/persian/sentence_translations.json +++ b/2019/pdes/persian/sentence_translations.json @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book. ", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book ", "translatedText": "باز هم، می دانم که به نظر می رسد یک تبلیغ است، اما اینطور نیست، فقط فکر می کنم از کتاب لذت خواهید برد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/tamil/sentence_translations.json b/2019/pdes/tamil/sentence_translations.json index 7b5e3d0ab..fca9498f0 100644 --- a/2019/pdes/tamil/sentence_translations.json +++ b/2019/pdes/tamil/sentence_translations.json @@ -512,7 +512,7 @@ "end": 523.66 }, { - "input": "When this difference is greater than zero, T2 will heat up, and the bigger the difference, the faster it heats up.", + "input": "When this difference is greater than zero, T2 will tend to heat up. And the bigger the difference, the faster it heats up.", "translatedText": "இந்த வேறுபாடு பூஜ்ஜியத்தை விட அதிகமாக இருக்கும்போது, T2 வெப்பமடையும், மேலும் பெரிய வித்தியாசம், வேகமாக வெப்பமடையும்.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 533.46 }, { - "input": "Likewise, if it's negative, T2 will cool down, at a rate proportional to that difference.", + "input": "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", "translatedText": "அதேபோல், அது எதிர்மறையாக இருந்தால், அந்த வித்தியாசத்திற்கு விகிதாசார விகிதத்தில் T2 குளிர்ச்சியடையும்.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book.", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book", "translatedText": "மீண்டும், இது ஒரு விளம்பரம் போல் தெரிகிறது, ஆனால் அது இல்லை, நீங்கள் புத்தகத்தை ரசிப்பீர்கள் என்று நினைக்கிறேன்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/telugu/sentence_translations.json b/2019/pdes/telugu/sentence_translations.json index ca293cc08..6cf1d96ef 100644 --- a/2019/pdes/telugu/sentence_translations.json +++ b/2019/pdes/telugu/sentence_translations.json @@ -512,7 +512,7 @@ "end": 523.66 }, { - "input": "When this difference is greater than zero, T2 will heat up, and the bigger the difference, the faster it heats up.", + "input": "When this difference is greater than zero, T2 will tend to heat up. And the bigger the difference, the faster it heats up.", "translatedText": "ఈ వ్యత్యాసం సున్నా కంటే ఎక్కువగా ఉన్నప్పుడు, T2 వేడెక్కుతుంది మరియు పెద్ద వ్యత్యాసం, వేగంగా వేడెక్కుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 533.46 }, { - "input": "Likewise, if it's negative, T2 will cool down, at a rate proportional to that difference.", + "input": "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", "translatedText": "అలాగే, అది ప్రతికూలంగా ఉంటే, T2 ఆ వ్యత్యాసానికి అనులోమానుపాతంలో చల్లబడుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book.", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book", "translatedText": "మళ్ళీ, అది ప్రకటన లాగా ఉందని నాకు తెలుసు, కానీ అది కాదు, మీరు పుస్తకాన్ని ఆనందిస్తారని నేను భావిస్తున్నాను.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/thai/sentence_translations.json b/2019/pdes/thai/sentence_translations.json index ce594d711..85dcaa44f 100644 --- a/2019/pdes/thai/sentence_translations.json +++ b/2019/pdes/thai/sentence_translations.json @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book. ", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/ukrainian/sentence_translations.json b/2019/pdes/ukrainian/sentence_translations.json index 0f1a7461c..a26418c8e 100644 --- a/2019/pdes/ukrainian/sentence_translations.json +++ b/2019/pdes/ukrainian/sentence_translations.json @@ -512,7 +512,7 @@ "end": 523.66 }, { - "input": "When this difference is greater than zero, T2 will heat up, and the bigger the difference, the faster it heats up.", + "input": "When this difference is greater than zero, T2 will tend to heat up. And the bigger the difference, the faster it heats up.", "translatedText": "Коли ця різниця більше нуля, Т2 нагріється, і чим більша різниця, тим швидше він нагрівається.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 533.46 }, { - "input": "Likewise, if it's negative, T2 will cool down, at a rate proportional to that difference.", + "input": "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", "translatedText": "Аналогічно, якщо він негативний, Т2 охолоне зі швидкістю, пропорційною цій різниці.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book.", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book", "translatedText": "Знову ж таки, я знаю, що це звучить як реклама, але це не так, я просто думаю, що книга вам сподобається.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/urdu/sentence_translations.json b/2019/pdes/urdu/sentence_translations.json index 6346961dc..5ae38bc82 100644 --- a/2019/pdes/urdu/sentence_translations.json +++ b/2019/pdes/urdu/sentence_translations.json @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book. ", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book ", "translatedText": "ایک بار پھر، میں جانتا ہوں کہ یہ ایک اشتہار کی طرح لگتا ہے، لیکن ایسا نہیں ہے، مجھے لگتا ہے کہ آپ کتاب سے لطف اندوز ہوں گے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/pdes/vietnamese/sentence_translations.json b/2019/pdes/vietnamese/sentence_translations.json index e71983c08..619489adc 100644 --- a/2019/pdes/vietnamese/sentence_translations.json +++ b/2019/pdes/vietnamese/sentence_translations.json @@ -512,7 +512,7 @@ "end": 523.66 }, { - "input": "When this difference is greater than zero, T2 will heat up, and the bigger the difference, the faster it heats up.", + "input": "When this difference is greater than zero, T2 will tend to heat up. And the bigger the difference, the faster it heats up.", "translatedText": "Khi chênh lệch này lớn hơn 0 thì T2 sẽ nóng lên, chênh lệch càng lớn thì nhiệt độ càng nhanh.", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 533.46 }, { - "input": "Likewise, if it's negative, T2 will cool down, at a rate proportional to that difference.", + "input": "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", "translatedText": "Tương tự, nếu nó âm, T2 sẽ hạ nhiệt với tốc độ tỷ lệ thuận với sự chênh lệch đó.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 1032.06 }, { - "input": "Again, I know that sounds like an ad, but it's not, I just think you'll enjoy the book.", + "input": "Again, I kinda know that sounds like an ad, but it's not, I just actually think you'll enjoy the book", "translatedText": "Một lần nữa, tôi biết điều đó nghe có vẻ giống một quảng cáo, nhưng không phải vậy, tôi chỉ nghĩ bạn sẽ thích cuốn sách này.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/arabic/sentence_translations.json b/2019/prime-spirals/arabic/sentence_translations.json index fb9357ea6..72ca57e33 100644 --- a/2019/prime-spirals/arabic/sentence_translations.json +++ b/2019/prime-spirals/arabic/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance. ", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance. ", "translatedText": "تقع النقطة 1,1 على مسافة 1 من نقطة الأصل، بزاوية قدرها 1 راديان، مما يعني أن هذا القوس هو نفس طول الخط الشعاعي، و2,2 لها ضعف تلك الزاوية، وضعف المسافة. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on. ", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on. ", "translatedText": "لاحظ كيف أن جميع مضاعفات العدد 6 تشكل ذراعًا واحدًا لهذه الدوامة، ثم الذراع التالي هو كل عدد صحيح أعلى من مضاعف 6 بواحد، ثم يشمل جميع الأرقام 2 فوق مضاعف 6، وهكذا. ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", "translatedText": "يوجد 2 باي راديان لكل دورة، لذا فإن اتخاذ 44 خطوة يعطي إجمالي 44 مقسومًا على 2 دورة باي، والتي تخرج بالكاد فوق 7 دورات كاملة. ", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively. ", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you. ", "translatedText": "وهذا يعني بالتأكيد أنك تتعلمها بشكل أكثر فعالية. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/bengali/sentence_translations.json b/2019/prime-spirals/bengali/sentence_translations.json index 869f38e44..73d02693e 100644 --- a/2019/prime-spirals/bengali/sentence_translations.json +++ b/2019/prime-spirals/bengali/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance. ", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance. ", "translatedText": "এটি করার কোন বাস্তব কারণ নেই, এটি নিখুঁতভাবে মজাদার, আমরা কেবলমাত্র ডেটা ভিজ্যুয়ালাইজেশনের খেলার মাঠে ঘুরে বেড়াচ্ছি, এবং এর অর্থ কী তা বোঝার জন্য, শুধুমাত্র মৌলিক সংখ্যার পরিবর্তে সমস্ত সম্পূর্ণ সংখ্যার দিকে তাকান৷ বিন্দু 1,1 উৎপত্তি থেকে 1 দূরে একটি কোণ 1 রেডিয়ান সহ, যার মানে এই চাপটি সেই রেডিয়াল রেখার সমান দৈর্ঘ্য এবং 2,2 এর কোণটির দ্বিগুণ এবং দূরত্বের দ্বিগুণ।", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on. ", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on. ", "translatedText": "লক্ষ্য করুন কিভাবে 6-এর সমস্ত গুণিতক এই সর্পিলটির একটি বাহু তৈরি করে, তারপরের পরেরটি হল প্রতিটি পূর্ণসংখ্যা যা 6-এর গুণিতকের উপরে একটি, এবং তারপর 6-এর গুণিতকের উপরে সমস্ত সংখ্যা 2 অন্তর্ভুক্ত করে।", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", "translatedText": "প্রতি ঘূর্ণনে 2টি পাই রেডিয়ান আছে, তাই 44টি পদক্ষেপ নিলে মোট 44টিকে 2 পাই ঘূর্ণন দ্বারা ভাগ করা হয়, যা 7টি পূর্ণ বাঁকের ঠিক উপরে বের হয়।", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively. ", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you. ", "translatedText": "এবং এর অর্থ প্রায় নিশ্চিতভাবেই আপনি এটি আরও কার্যকরভাবে শিখবেন।", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/chinese/sentence_translations.json b/2019/prime-spirals/chinese/sentence_translations.json index 6b8ae17c5..4dd63dcfc 100644 --- a/2019/prime-spirals/chinese/sentence_translations.json +++ b/2019/prime-spirals/chinese/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance. ", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance. ", "translatedText": "点 1,1 距原点的距离为 1，角度为 1 弧度 ，这意味着该弧与该径向线的长度相同，而 2,2 的角度是该角度的两倍，距离也是该距离的两倍。", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on. ", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on. ", "translatedText": "注意所有 6 的倍数如何形成这个螺旋的一个臂，然 后下一个是 6 的倍数以上 1 的每个整数，然后 包括 6 的倍数以上的所有数字 2，依此类推。", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", "translatedText": "每旋转 2 个 pi 弧度，因此走 44 步得到的总数是 44 除以 2 pi 旋转，结果刚好超过 7 个整圈。", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively. ", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you. ", "translatedText": "这几乎肯定意味着你可以更有效地学习它。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/english/captions.srt b/2019/prime-spirals/english/captions.srt index 56a7551a7..2e4a5dd4e 100644 --- a/2019/prime-spirals/english/captions.srt +++ b/2019/prime-spirals/english/captions.srt @@ -207,1118 +207,1126 @@ Now this is shocking, and beautiful, and you might think that it suggests some divine hidden symmetry within the primes. 53 -00:03:33,820 --> 00:03:38,110 -But to steady your expectations, I should say that the fact that the person asking +00:03:33,820 --> 00:03:36,601 +but to study your expectations, I should say that the fact 54 -00:03:38,110 --> 00:03:42,400 -this question jumped right into prime numbers makes the puzzle a little misleading. +00:03:36,601 --> 00:03:39,477 +that the person asking this question on math exchange jumped 55 +00:03:39,477 --> 00:03:42,400 +right into prime numbers makes the puzzle a little misleading. + +56 00:03:43,240 --> 00:03:48,521 If you look at all the whole numbers, not just the primes, -56 +57 00:03:48,521 --> 00:03:52,640 as you zoom out, you see very similar spirals. -57 +58 00:03:54,580 --> 00:03:57,885 They're much cleaner, and now there's 44 of them instead of 20, -58 +59 00:03:57,885 --> 00:04:01,294 but it means that the question of where the spirals come from is, -59 +60 00:04:01,294 --> 00:04:05,375 perhaps disappointingly, completely separate from the question of what happens -60 +61 00:04:05,375 --> 00:04:07,080 when we limit our view to primes. -61 +62 00:04:08,580 --> 00:04:12,420 But don't be too disappointed, because both these questions are still phenomenal puzzles. -62 +63 00:04:12,840 --> 00:04:15,302 There's a very satisfying answer for the spirals, -63 +64 00:04:15,302 --> 00:04:17,665 and even if the primes don't cause the spirals, -64 +65 00:04:17,665 --> 00:04:21,112 asking what goes on when you filter for those primes does lead you to -65 +66 00:04:21,112 --> 00:04:24,854 one of the most important theorems about the distribution of prime numbers, -66 +67 00:04:24,854 --> 00:04:27,120 known in number theory as Dirichlet's theorem. -67 +68 00:04:28,660 --> 00:04:31,460 To kick things off, let's zoom back in a little bit smaller. -68 +69 00:04:31,980 --> 00:04:35,640 Did you notice that as we were zooming out, there were these six little spirals? -69 +70 00:04:36,220 --> 00:04:39,780 This offers a good starting point to explain what's happening in the two larger patterns. -70 +71 00:04:40,440 --> 00:04:44,020 Notice how all the multiples of 6 form one arm of this spiral. -71 -00:04:46,020 --> 00:04:50,040 -Then the next one is every integer that's one above a multiple of 6. - 72 +00:04:46,020 --> 00:04:48,204 +then the next one is every integer that's one above a multiple of 6, + +73 +00:04:48,204 --> 00:04:50,040 +and then includes all the numbers 2 above a multiple of 6, + +74 00:04:52,180 --> 00:04:56,080 Then after that it includes all the numbers 2 above a multiple of 6, and so on. -73 +75 00:04:56,600 --> 00:04:57,260 Why is that? -74 +76 00:04:59,180 --> 00:05:04,128 Well, remember that each step forward in this sequence involves a turn of one radian, -75 +77 00:05:04,128 --> 00:05:07,753 so when you count up by 6, you've turned a total of 6 radians, -76 +78 00:05:07,753 --> 00:05:10,400 which is a little less than 2 pi, a full turn. -77 +79 00:05:10,960 --> 00:05:15,660 So every time you count up by 6, you've almost made a full turn, it's just a little less. -78 +80 00:05:16,560 --> 00:05:18,820 Another six steps, a slightly smaller angle. -79 +81 00:05:20,120 --> 00:05:24,160 Six more steps, smaller still, and so on, with this angle changing -80 +82 00:05:24,160 --> 00:05:28,140 gently enough that it gives the illusion of a single curving line. -81 +83 00:05:29,280 --> 00:05:34,300 When you limit the view to prime numbers, all but two of these spiral arms will go away. -82 +84 00:05:34,940 --> 00:05:38,520 And think about it, a prime number can't be a multiple of 6, -83 +85 00:05:38,520 --> 00:05:42,043 and it also can't be 2 above a multiple of 6 unless it's 2, -84 +86 00:05:42,043 --> 00:05:45,800 or 4 above a multiple of 6, since all of those are even numbers. -85 +87 00:05:46,400 --> 00:05:50,568 It also can't be 3 above a multiple of 6, unless it's the number 3 itself, -86 +88 00:05:50,568 --> 00:05:52,680 since all of those are divisible by 3. -87 +89 00:05:53,780 --> 00:05:57,440 So, at least at this smaller scale, nothing magical is going on. -88 +90 00:05:57,760 --> 00:05:59,951 And while we're in this simpler context, let me -89 +91 00:05:59,951 --> 00:06:02,280 introduce some terminology that mathematicians use. -90 +92 00:06:02,780 --> 00:06:06,251 Each one of these sequences, where you're counting up by 6, -91 +93 00:06:06,251 --> 00:06:08,740 is fancifully called a residue class mod 6. -92 +94 00:06:09,900 --> 00:06:14,383 The word residue here is sort of an overdramatic way of saying remainder, -93 +95 00:06:14,383 --> 00:06:18,140 and mod means something like where the thing you divide by is. -94 +96 00:06:18,720 --> 00:06:23,660 So, for example, 6 goes into 20 three times, and it leaves a remainder of 2. -95 +97 00:06:25,500 --> 00:06:28,400 So 20 has a residue of 2 mod 6. -96 +98 00:06:30,040 --> 00:06:33,810 Together with all the other numbers leaving a remainder of 2 when -97 +99 00:06:33,810 --> 00:06:37,580 the thing you divide by is 6, you have a full residue class mod 6. -98 +100 00:06:38,260 --> 00:06:41,675 I know that sounds like the world's most pretentious way of saying -99 +101 00:06:41,675 --> 00:06:45,193 everything 2 above a multiple of 6, but this is the standard jargon, -100 +102 00:06:45,193 --> 00:06:48,100 and it is actually handy to have some words for the idea. -101 +103 00:06:49,040 --> 00:06:52,986 So looking at our diagram, in the lingo, each of these spiral arms -102 +104 00:06:52,986 --> 00:06:56,814 corresponds to a residue class mod 6, and the reason we see them -103 +105 00:06:56,814 --> 00:07:00,820 is that 6 is close to 2 pi, turning 6 radians is almost a full turn. -104 +106 00:07:01,460 --> 00:07:05,826 And the reason we see only 2 of them when filtering for primes is that all prime -105 +107 00:07:05,826 --> 00:07:10,140 numbers are either 1 or 5 above a multiple of 6, with the exceptions of 2 and 3. -106 +108 00:07:11,060 --> 00:07:13,360 With all that as a warmup, let's think about the larger scale. -107 +109 00:07:13,360 --> 00:07:17,232 In the same way that 6 steps is close to a full turn, -108 +110 00:07:17,232 --> 00:07:21,320 taking 44 steps is very close to a whole number of turns. -109 +111 00:07:21,980 --> 00:07:23,220 Here, let's compute it. -110 +112 00:07:23,760 --> 00:07:26,320 There are 2 pi radians per rotation, right? -111 +113 00:07:26,900 --> 00:07:33,089 So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, -112 +114 00:07:33,089 --> 00:07:36,860 which comes out to be just barely above 7 full turns. -113 +115 00:07:37,620 --> 00:07:42,384 You could also write this by saying that 44 sevenths is a close approximation for 2 pi, -114 +116 00:07:42,384 --> 00:07:47,040 which some of you may better recognize as the famous 22 sevenths approximation for pi. -115 +117 00:07:48,320 --> 00:07:52,363 What this means is when you count up by multiples of 44 in the diagram, -116 +118 00:07:52,363 --> 00:07:56,800 each point has almost the same angle as the last one, just a little bit bigger. -117 +119 00:07:57,260 --> 00:08:00,390 So as you continue on with more and more, we get this -118 +120 00:08:00,390 --> 00:08:03,520 very gentle spiral as the angle increases very slowly. -119 +121 00:08:05,400 --> 00:08:09,676 Similarly, all the numbers 1 above a multiple of 44 make another spiral, -120 +122 00:08:09,676 --> 00:08:12,020 but rotated one radian counterclockwise. -121 +123 00:08:12,720 --> 00:08:15,951 Same for everything 2 above a multiple of 44, and so on, -122 +124 00:08:15,951 --> 00:08:18,220 eventually filling out the full diagram. -123 +125 00:08:19,320 --> 00:08:22,048 To phrase it with our fancier language, each of -124 +126 00:08:22,048 --> 00:08:24,720 these spiral arms shows a residue class mod 44. -125 +127 00:08:29,260 --> 00:08:32,799 And maybe now you can tell me what happens when we limit our view to prime numbers. -126 +128 00:08:33,840 --> 00:08:37,500 Primes cannot be a multiple of 44, so that arm won't be visible. -127 +129 00:08:38,059 --> 00:08:41,742 Nor can a prime be 2 above a multiple of 44, or 4 above, and so on, -128 +130 00:08:41,742 --> 00:08:45,100 since all those residue classes have nothing but even numbers. -129 +131 00:08:48,100 --> 00:08:52,287 Likewise, any multiples of 11 can't be prime, except for 11 itself, -130 +132 00:08:52,287 --> 00:08:56,537 so the spiral of numbers 11 above a multiple of 44 won't be visible, -131 +133 00:08:56,537 --> 00:09:00,540 and neither will the spiral of numbers 33 above a multiple of 44. -132 +134 00:09:01,220 --> 00:09:03,920 This is what gives the picture those Milky Way-seeming gaps. -133 +135 00:09:04,400 --> 00:09:07,179 Each spiral we're left with is a residue class -134 +136 00:09:07,179 --> 00:09:09,840 that doesn't share any prime factors with 44. -135 +137 00:09:10,740 --> 00:09:14,036 And within each one of those arms that we can't reject out of hand, -136 +138 00:09:14,036 --> 00:09:16,460 the prime numbers seem to be randomly distributed. -137 +139 00:09:17,060 --> 00:09:20,100 That's a fact I'd like you to tuck away, we'll return to it later. -138 +140 00:09:21,540 --> 00:09:25,060 This is another good chance to inject some of the jargon mathematicians use. -139 +141 00:09:25,600 --> 00:09:28,561 What we care about right here are all the numbers between -140 +142 00:09:28,561 --> 00:09:31,420 0 and 43 that don't share a prime factor with 44, right? -141 +143 00:09:31,760 --> 00:09:34,700 The ones that aren't even and aren't divisible by 11. -142 +144 00:09:35,880 --> 00:09:39,447 When two numbers don't share any factors like this, -143 +145 00:09:39,447 --> 00:09:42,740 we call them relatively prime, or also co-prime. -144 +146 00:09:43,420 --> 00:09:48,322 In this example you could count that there are 20 different numbers between 1 -145 +147 00:09:48,322 --> 00:09:53,350 and 44 that are co-prime to 44, and this is a fact that a number theorist would -146 +148 00:09:53,350 --> 00:09:56,304 compactly write by saying phi of 44 equals 20, -147 +149 00:09:56,304 --> 00:10:00,578 where the Greek letter phi here refers to Euler's totient function, -148 +150 00:10:00,578 --> 00:10:05,668 yet another needlessly fancy word, which is defined to be the number of integers -149 +151 00:10:05,668 --> 00:10:08,120 from 1 up to n which are co-prime to n. -150 +152 00:10:08,840 --> 00:10:11,380 It comes up enough that it's handy to have compact notation. -151 +153 00:10:12,080 --> 00:10:15,414 More obscurely, and I had never heard this before but I find it too -152 +154 00:10:15,414 --> 00:10:19,240 delightful not to tell, these numbers are sometimes called the totitives of n. -153 +155 00:10:19,240 --> 00:10:23,736 Back to the main thread, in short, what the user on math exchange was seeing -154 +156 00:10:23,736 --> 00:10:28,000 are two unrelated pieces of number theory but illustrated in one drawing. -155 +157 00:10:28,620 --> 00:10:33,809 The first is that 44 sevenths is a very close rational approximation for 2 pi, -156 +158 00:10:33,809 --> 00:10:38,540 which results in the residue classes mod 44 being cleanly separated out. -157 +159 00:10:39,820 --> 00:10:43,969 The second is that many of these residue classes contain zero prime numbers, -158 +160 00:10:43,969 --> 00:10:48,443 or sometimes just one, so they won't show up, but on the other hand primes do show -159 +161 00:10:48,443 --> 00:10:53,240 up enough in all 20 of the other residue classes that it makes these spiral arms visible. -160 +162 00:10:54,580 --> 00:10:58,040 And at this point maybe you can predict what's going on at the larger scale. -161 +163 00:10:59,460 --> 00:11:02,933 Just as 6 radians is vaguely close to a full turn, -162 +164 00:11:02,933 --> 00:11:06,134 and 44 radians is quite close to 7 full turns, -163 +165 00:11:06,134 --> 00:11:12,060 it just so happens that 710 radians is extremely close to a whole number of full turns. -164 +166 00:11:12,740 --> 00:11:15,658 Visually, you can see this by the fact that the point ends up -165 +167 00:11:15,658 --> 00:11:18,860 almost exactly on the x-axis, but it's more compelling analytically. -166 +168 00:11:19,700 --> 00:11:30,280 710 radians is 710 divided by 2 pi rotations, which works out to be 113.000095. -167 +169 00:11:31,380 --> 00:11:33,180 Some of you may have seen this in another form. -168 +170 00:11:33,600 --> 00:11:38,809 It's saying that 710 one hundred thirteenths is a close approximation for 2 pi, -169 +171 00:11:38,809 --> 00:11:44,344 which is more commonly seen in saying that 355 over 113 is a very good approximation -170 +172 00:11:44,344 --> 00:11:44,800 for pi. -171 +173 00:11:46,360 --> 00:11:50,373 If you want to understand where these rational approximations are coming from, -172 +174 00:11:50,373 --> 00:11:53,320 and what it means for one like this to be unusually good, -173 +175 00:11:53,320 --> 00:11:56,927 like way better than you would get for phi or e or square root of 2 or -174 +176 00:11:56,927 --> 00:12:01,500 other famous irrationals, I highly recommend taking a look at this great Mathologer video. -175 +177 00:12:02,520 --> 00:12:07,260 For our storyline though, it means that when you move forward by steps of 710, -176 +178 00:12:07,260 --> 00:12:11,579 the angle of each new point is almost exactly the same as the last one, -177 +179 00:12:11,579 --> 00:12:13,260 just microscopically bigger. -178 +180 00:12:15,600 --> 00:12:19,320 Even very far out, one of these sequences looks like a straight line. -179 +181 00:12:20,120 --> 00:12:25,380 And of course, the other residue classes, mod 710, also form these nearly straight lines. -180 +182 00:12:26,440 --> 00:12:29,287 710 is a big number though, so when all of them are on screen, -181 +183 00:12:29,287 --> 00:12:33,040 and there's only so many pixels on the screen, it's a little hard to make them out. -182 +184 00:12:34,800 --> 00:12:38,596 So in this case, it's actually easier to see when we limit the view to primes, -183 +185 00:12:38,596 --> 00:12:41,000 where you don't see many of those residue classes. -184 +186 00:12:41,760 --> 00:12:44,866 In reality, with a little further zooming, you can see -185 +187 00:12:44,866 --> 00:12:47,860 that there actually is a very gentle spiral to these. -186 +188 00:12:48,320 --> 00:12:52,370 But the fact that it takes so long to become prominent is a wonderful illustration, -187 +189 00:12:52,370 --> 00:12:56,566 maybe the best illustration I've ever seen, for just how good an approximation this is -188 +190 00:12:56,566 --> 00:12:57,000 for 2 pi. -189 +191 00:12:59,160 --> 00:13:02,419 Tying up the remaining loose thread here, if you want to understand what -190 +192 00:13:02,419 --> 00:13:06,080 happens when you filter for primes, it's entirely analogous to what we did before. -191 +193 00:13:06,560 --> 00:13:11,257 The factors of 710 are 71, 5, and 2, so if the remainder, -192 +194 00:13:11,257 --> 00:13:16,440 or residue, is divisible by any of those, then so is the number. -193 +195 00:13:17,280 --> 00:13:21,549 When you pull up all of the residue classes with odd numbers, -194 +196 00:13:21,549 --> 00:13:26,300 it looks like every other ray in the otherwise quite crowded picture. -195 +197 00:13:29,920 --> 00:13:34,348 And then of those that remain, these are the ones that are divisible by 5, -196 +198 00:13:34,348 --> 00:13:37,360 which are nice and evenly spaced at every 5th line. -197 +199 00:13:39,880 --> 00:13:43,429 Notice the fact that prime numbers never show up in any of these is what -198 +200 00:13:43,429 --> 00:13:47,320 explains the pattern of the lines we saw at the beginning coming in clumps of 4. -199 +201 00:13:48,100 --> 00:13:52,760 And moreover, of those remaining, these four residue classes are the ones that are -200 +202 00:13:52,760 --> 00:13:56,240 divisible by 71, so the primes aren't going to show up there, -201 +203 00:13:56,240 --> 00:14:01,013 and that's what explains why the clumps of 4 that we saw occasionally have a missing -202 +204 00:14:01,013 --> 00:14:02,080 tooth in your cone. -203 +205 00:14:04,420 --> 00:14:08,098 And if you were wondering where that number 280 came from, -204 +206 00:14:08,098 --> 00:14:13,086 it comes from counting how many of the numbers from 1 up to 710 don't share any -205 +207 00:14:13,086 --> 00:14:14,520 prime factors with 710. -206 +208 00:14:15,220 --> 00:14:17,580 These are the ones that we can't rule out for including -207 +209 00:14:17,580 --> 00:14:19,940 primes based on some obvious divisibility consideration. -208 +210 00:14:21,140 --> 00:14:25,312 This of course doesn't guarantee that any particular one will contain prime numbers, -209 +211 00:14:25,312 --> 00:14:28,061 but at least empirically when you look at this picture, -210 +212 00:14:28,061 --> 00:14:32,185 it actually seems like the primes are pretty evenly distributed among the remaining -211 +213 00:14:32,185 --> 00:14:33,560 classes, wouldn't you agree? -212 +214 00:14:38,220 --> 00:14:41,760 This last point is actually the most interesting observation of the whole deal. -213 +215 00:14:42,080 --> 00:14:45,500 It relates to a pretty deep fact in number theory, known as Dirichlet's theorem. -214 +216 00:14:46,220 --> 00:14:51,120 To take a simpler example than residue classes mod 710, think of those mod 10. -215 +217 00:14:51,940 --> 00:14:54,740 Because we write numbers in base 10, this is the same thing -216 +218 00:14:54,740 --> 00:14:57,400 as grouping numbers together by what their last digit is. -217 +219 00:14:57,400 --> 00:15:00,249 Everything whose last digit is 0 is a residue class, -218 +220 00:15:00,249 --> 00:15:03,960 everything whose last digit is 1 is another residue class, and so on. -219 +221 00:15:04,500 --> 00:15:07,657 Other than 2, prime numbers can't have an even number as their last digit, -220 +222 00:15:07,657 --> 00:15:08,920 since that means they're even. -221 +223 00:15:09,940 --> 00:15:12,820 And likewise, any prime bigger than 5 can't end in a 5. -222 +224 00:15:13,300 --> 00:15:15,183 There's nothing surprising there, that's one of the -223 +225 00:15:15,183 --> 00:15:17,320 first facts you observe when you learn about prime numbers. -224 +226 00:15:17,660 --> 00:15:22,080 Anything bigger than 5 has to end in either a 1, a 3, a 7, or a 9. -225 +227 00:15:22,590 --> 00:15:25,622 A much more nuanced question, though, is how exactly these -226 +228 00:15:25,622 --> 00:15:28,500 primes are divvied up among those remaining four groups. -227 +229 00:15:29,420 --> 00:15:33,219 Here, let's make a quick histogram, counting through each prime number, -228 +230 00:15:33,219 --> 00:15:37,862 where the bars are going to show what proportion of the primes we've seen so far have a -229 +231 00:15:37,862 --> 00:15:38,760 given last digit. -230 +232 00:15:39,380 --> 00:15:43,000 So in particular, the 2 and the 5 slots should go down to 0 over time. -231 +233 00:15:43,740 --> 00:15:47,620 What would you predict is going to happen as we move through more and more primes? -232 +234 00:15:52,780 --> 00:15:56,218 Well, as we get a lot of them, it seems like a pretty -233 +235 00:15:56,218 --> 00:15:59,720 even spread between these four classes, about 25% each. -234 +236 00:16:00,200 --> 00:16:02,020 And probably that's what you would expect. -235 +237 00:16:02,320 --> 00:16:04,695 After all, why would prime numbers show some sort -236 +238 00:16:04,695 --> 00:16:06,880 of preference for one last digit over another? -237 +239 00:16:07,480 --> 00:16:10,093 But primes aren't random, they are a definite sequence, -238 +240 00:16:10,093 --> 00:16:13,826 and they show patterns in other ways, and it's highly non-obvious how you would -239 +241 00:16:13,826 --> 00:16:15,040 prove something like this. -240 +242 00:16:15,500 --> 00:16:19,740 Or, for that matter, how do you rigorously phrase what it is you want to prove? -241 +243 00:16:20,490 --> 00:16:23,020 A mathematician might go about it something like this. -242 +244 00:16:23,580 --> 00:16:28,260 If you look at all the prime numbers less than some big number x, -243 +245 00:16:28,260 --> 00:16:33,578 and you consider what fraction of them are, say, 1 above a multiple of 10, -244 +246 00:16:33,578 --> 00:16:38,187 that fraction should approach 1 fourth as x approaches infinity, -245 +247 00:16:38,187 --> 00:16:43,860 and likewise for all of the other allowable residue classes, like 3 and 7 and 9. -246 +248 00:16:45,720 --> 00:16:47,480 Of course, there's nothing special about 10. -247 +249 00:16:47,480 --> 00:16:50,180 A similar fact should hold for any other number. -248 +250 00:16:50,740 --> 00:16:54,559 Considering our old friends the residue classes mod 44, for example, -249 +251 00:16:54,559 --> 00:16:59,320 let's make a similar histogram, showing what proportion of the primes show up in each -250 +252 00:16:59,320 --> 00:17:00,040 one of these. -251 +253 00:17:08,319 --> 00:17:12,329 Again, as time goes on, we see an even spread between the 20 different -252 +254 00:17:12,329 --> 00:17:16,508 allowable residue classes, which you can think of in terms of each spiral -253 +255 00:17:16,508 --> 00:17:21,140 arm from our diagram having about the same number of primes as each of the others. -254 +256 00:17:22,260 --> 00:17:25,940 Maybe that's what you'd expect, but this is a shockingly hard fact to prove. -255 +257 00:17:27,660 --> 00:17:31,344 The first man who cracked this puzzle was Dirichlet in 1837, -256 +258 00:17:31,344 --> 00:17:35,994 and it forms one of the crowning jewels at the foundation of modern analytic -257 +259 00:17:35,994 --> 00:17:36,840 number theory. -258 +260 00:17:37,960 --> 00:17:39,912 Histograms like these ones give a pretty good -259 +261 00:17:39,912 --> 00:17:42,120 illustration of what the theorem is actually saying. -260 +262 00:17:42,120 --> 00:17:45,120 Nevertheless, you might find it enlightening to see how it might -261 +263 00:17:45,120 --> 00:17:48,260 be written in a math text, with all the fancy jargon and everything. -262 +264 00:17:48,860 --> 00:17:51,720 It's essentially what we just saw for 10, but more general. -263 +265 00:17:52,260 --> 00:17:56,018 Again, you look at all the primes up to some bound x, -264 +266 00:17:56,018 --> 00:18:01,865 but instead of asking for what proportion of them have a residue of, say, 1 mod 10, -265 +267 00:18:01,865 --> 00:18:07,015 you ask what proportion have a residue of r mod n, where n is any number, -266 +268 00:18:07,015 --> 00:18:09,800 and r is anything that's co-primed to n. -267 +269 00:18:09,800 --> 00:18:13,280 Remember, that means it doesn't share any factors with n bigger than 1. -268 +270 00:18:14,180 --> 00:18:17,300 Instead of approaching 1 fourth as x goes to infinity, -269 +271 00:18:17,300 --> 00:18:19,967 that proportion goes to 1 divided by phi of n, -270 +272 00:18:19,967 --> 00:18:23,940 where phi is that special function I mentioned earlier that gives the -271 +273 00:18:23,940 --> 00:18:26,380 number of possible residues co-primed to n. -272 +274 00:18:27,560 --> 00:18:31,365 In case this is too clear for the reader, you might see it buried in more notation, -273 +275 00:18:31,365 --> 00:18:33,993 where this denominator and the numerator are both written -274 +276 00:18:33,993 --> 00:18:35,760 with a special prime-counting function. -275 +277 00:18:36,340 --> 00:18:40,585 The convention, rather confusingly, is to use the symbol pi for this function, -276 +278 00:18:40,585 --> 00:18:43,380 even though it's totally unrelated to the number pi. -277 +279 00:18:44,220 --> 00:18:47,252 In some contexts, when people refer to Dirichlet's theorem, -278 +280 00:18:47,252 --> 00:18:50,942 they refer to a much more modest statement, which is simply that each of -279 +281 00:18:50,942 --> 00:18:55,340 these residue classes that might have infinitely many primes does have infinitely many. -280 +282 00:18:56,180 --> 00:18:59,570 In order to prove this, what Dirichlet did was show that the primes -281 +283 00:18:59,570 --> 00:19:03,060 are just as dense in any one of these residue classes as in any other. -282 +284 00:19:04,220 --> 00:19:07,303 For example, imagine someone asks you to prove that there are -283 +285 00:19:07,303 --> 00:19:09,641 infinitely many primes ending in the number 1, -284 +286 00:19:09,641 --> 00:19:13,620 and the way you do it is by showing that a quarter of all the primes end in a 1. -285 +287 00:19:14,040 --> 00:19:16,792 Together with the fact that there are infinitely many primes, -286 +288 00:19:16,792 --> 00:19:19,679 which we've known since Euclid, this gives a stronger statement, -287 +289 00:19:19,679 --> 00:19:21,100 and a much more interesting one. -288 +290 00:19:22,880 --> 00:19:27,900 Now the proof, well, it's way more involved than would be reasonable to show here. -289 +291 00:19:27,900 --> 00:19:32,495 One interesting fact worth mentioning is that it relies heavily on complex analysis, -290 +292 00:19:32,495 --> 00:19:35,631 which is the study of doing calculus with functions whose -291 +293 00:19:35,631 --> 00:19:37,740 inputs and outputs are complex numbers. -292 +294 00:19:38,540 --> 00:19:40,100 Now that might seem weird, right? -293 +295 00:19:40,520 --> 00:19:44,253 Prime numbers seem wholly unrelated to the continuous world of calculus, -294 +296 00:19:44,253 --> 00:19:48,549 much less when complex numbers end up in the mix, but since the early 19th century, -295 +297 00:19:48,549 --> 00:19:52,846 this is absolutely par for the course when it comes to understanding how primes are -296 +298 00:19:52,846 --> 00:19:53,460 distributed. -297 +299 00:19:53,460 --> 00:19:56,440 And this isn't just antiquated technology either. -298 +300 00:19:56,780 --> 00:19:59,800 Understanding the distribution of primes in residue classes -299 +301 00:19:59,800 --> 00:20:02,720 like this continues to be relevant in modern research too. -300 +302 00:20:03,040 --> 00:20:06,298 Some of the recent breakthroughs on small gaps between primes, -301 +303 00:20:06,298 --> 00:20:09,194 edging towards that ever-elusive twin-prime conjecture, -302 +304 00:20:09,194 --> 00:20:12,917 have their basis in understanding how primes split up among these kinds -303 +305 00:20:12,917 --> 00:20:13,900 of residue classes. -304 +306 00:20:17,900 --> 00:20:21,460 Okay, looking back over the puzzle, I want to emphasize something. -305 +307 00:20:21,920 --> 00:20:26,412 The original bit of data visualization whimsy that led to these patterns, -306 +308 00:20:26,412 --> 00:20:28,720 well, it doesn't matter, no one cares. -307 +309 00:20:29,120 --> 00:20:32,364 There's nothing special about plotting p,p in polar coordinates, -308 +310 00:20:32,364 --> 00:20:36,207 and most of the initial mystery in these spirals resulted from the artifacts -309 +311 00:20:36,207 --> 00:20:40,100 that come from dealing with integer number of radians, which is kind of weird. -310 +312 00:20:40,860 --> 00:20:44,315 But on the other hand, this kind of play is clearly worth it if the -311 +313 00:20:44,315 --> 00:20:48,786 end result is a line of questions that leads you to something like Dirichlet's theorem, -312 +314 00:20:48,786 --> 00:20:52,291 which is important, especially if it inspires you to learn enough to -313 +315 00:20:52,291 --> 00:20:54,680 understand the tactics of the underlying proof. -314 +316 00:20:55,440 --> 00:20:56,900 No small task, by the way. -315 +317 00:20:58,460 --> 00:21:01,258 And this isn't a coincidence that a fairly random question -316 +318 00:21:01,258 --> 00:21:04,200 like this can lead you to an important and deep fact for math. -317 +319 00:21:04,740 --> 00:21:07,610 What it means for a piece of math to be important -318 +320 00:21:07,610 --> 00:21:10,480 and deep is that it connects to many other topics. -319 +321 00:21:11,100 --> 00:21:15,681 So even an arbitrary exploration of numbers, as long as it's not too arbitrary, -320 +322 00:21:15,681 --> 00:21:18,660 has a chance of stumbling into something meaningful. -321 +323 00:21:20,300 --> 00:21:24,121 Sure, you'll get a much more concentrated dosage of important facts by going -322 +324 00:21:24,121 --> 00:21:28,388 through a textbook or a course, and there will be many fewer uninteresting dead ends, -323 +325 00:21:28,388 --> 00:21:32,160 but there is something special about rediscovering these topics on your own. -324 +326 00:21:32,560 --> 00:21:37,042 If you effectively reinvent Euler's totient function before you've ever seen it defined, -325 +327 00:21:37,042 --> 00:21:40,971 or if you start wondering about rational approximations before learning about -326 +328 00:21:40,971 --> 00:21:45,152 continued fractions, or if you seriously explore how primes are divvied up between -327 +329 00:21:45,152 --> 00:21:48,225 residue classes before you've even heard the name Dirichlet, -328 +330 00:21:48,225 --> 00:21:51,952 then when you do learn those topics, you'll see them as familiar friends, -329 +331 00:21:51,952 --> 00:21:56,032 not as arbitrary definitions, and that will almost certainly mean that you learn -330 +332 00:21:56,032 --> 00:21:57,040 it more effectively. -331 +333 00:22:19,560 --> 00:22:19,880 Thank you. diff --git a/2019/prime-spirals/english/sentence_timings.json b/2019/prime-spirals/english/sentence_timings.json index 28151166e..c42de0fd5 100644 --- a/2019/prime-spirals/english/sentence_timings.json +++ b/2019/prime-spirals/english/sentence_timings.json @@ -120,7 +120,7 @@ 213.26 ], [ - "But to steady your expectations, I should say that the fact that the person asking this question jumped right into prime numbers makes the puzzle a little misleading.", + "but to study your expectations, I should say that the fact that the person asking this question on math exchange jumped right into prime numbers makes the puzzle a little misleading.", 213.82, 222.4 ], @@ -165,7 +165,7 @@ 284.02 ], [ - "Then the next one is every integer that's one above a multiple of 6.", + "then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6,", 286.02, 290.04 ], diff --git a/2019/prime-spirals/english/transcript.txt b/2019/prime-spirals/english/transcript.txt index 5e9fdfc16..eb0d4dcab 100644 --- a/2019/prime-spirals/english/transcript.txt +++ b/2019/prime-spirals/english/transcript.txt @@ -22,7 +22,7 @@ Where do these spirals come from, and why do we instead get straight lines at th If you wanted, you could ask a more quantitative question, and count that there are 20 total spirals, and then up at that larger scale, if you patiently went through each ray, you'd count up a total of 280. And so this adds a further mystery of where these numbers are coming from, and why they would arise from primes. Now this is shocking, and beautiful, and you might think that it suggests some divine hidden symmetry within the primes. -But to steady your expectations, I should say that the fact that the person asking this question jumped right into prime numbers makes the puzzle a little misleading. +but to study your expectations, I should say that the fact that the person asking this question on math exchange jumped right into prime numbers makes the puzzle a little misleading. If you look at all the whole numbers, not just the primes, as you zoom out, you see very similar spirals. They're much cleaner, and now there's 44 of them instead of 20, but it means that the question of where the spirals come from is, perhaps disappointingly, completely separate from the question of what happens when we limit our view to primes. But don't be too disappointed, because both these questions are still phenomenal puzzles. @@ -31,7 +31,7 @@ To kick things off, let's zoom back in a little bit smaller. Did you notice that as we were zooming out, there were these six little spirals? This offers a good starting point to explain what's happening in the two larger patterns. Notice how all the multiples of 6 form one arm of this spiral. -Then the next one is every integer that's one above a multiple of 6. +then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on. Why is that? Well, remember that each step forward in this sequence involves a turn of one radian, so when you count up by 6, you've turned a total of 6 radians, which is a little less than 2 pi, a full turn. diff --git a/2019/prime-spirals/french/sentence_translations.json b/2019/prime-spirals/french/sentence_translations.json index c8fb41a54..115540163 100644 --- a/2019/prime-spirals/french/sentence_translations.json +++ b/2019/prime-spirals/french/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "Le point 1,1 se trouve à une distance de 1 de l'origine, avec un angle de 1 radian, ce qui signifie que cet arc a la même longueur que cette ligne radiale, et 2,2 a le double de cet angle et le double de la distance.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "Remarquez comment tous les multiples de 6 forment un bras de cette spirale, puis le suivant est chaque entier supérieur à un multiple de 6, puis inclut tous les nombres 2 au-dessus d'un multiple de 6, et ainsi de suite.", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "Il y a 2 pi radians par rotation, donc prendre 44 pas donne un total de 44 divisé par 2 pi rotations, ce qui revient à peine au-dessus de 7 tours complets.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "Et cela signifie presque certainement que vous l’apprendrez plus efficacement.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/german/sentence_translations.json b/2019/prime-spirals/german/sentence_translations.json index f26ecb51c..b18db52e3 100644 --- a/2019/prime-spirals/german/sentence_translations.json +++ b/2019/prime-spirals/german/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "Der Punkt 1,1 liegt in einem Abstand von 1 vom Ursprung entfernt, mit einem Winkel von 1 Bogenmaß, was bedeutet, dass dieser Bogen die gleiche Länge wie diese Radiallinie hat und 2,2 den doppelten Winkel und den doppelten Abstand hat.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "Beachten Sie, dass alle Vielfachen von 6 einen Arm dieser Spirale bilden, der nächste Arm dann jede ganze Zahl ist, die eins über einem Vielfachen von 6 liegt, und dann alle Zahlen 2 über einem Vielfachen von 6 enthält, und so weiter.", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "Es gibt 2 Pi-Bogenmaße pro Umdrehung, sodass 44 Schritte insgesamt 44 geteilt durch 2 Pi-Umdrehungen ergeben, was knapp über 7 volle Umdrehungen ergibt.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "Und das wird mit ziemlicher Sicherheit bedeuten, dass Sie es effektiver lernen.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/hebrew/sentence_translations.json b/2019/prime-spirals/hebrew/sentence_translations.json index 610d3fb20..b5fc9fee7 100644 --- a/2019/prime-spirals/hebrew/sentence_translations.json +++ b/2019/prime-spirals/hebrew/sentence_translations.json @@ -215,7 +215,7 @@ "end": 213.26 }, { - "input": "But to steady your expectations, I should say that the fact that the person asking this question jumped right into prime numbers makes the puzzle a little misleading.", + "input": "but to study your expectations, I should say that the fact that the person asking this question on math exchange jumped right into prime numbers makes the puzzle a little misleading.", "translatedText": "אבל כדי לייצב את הציפיות שלך, עלי לומר שהעובדה שהאדם ששואל את השאלה הזו קפץ ישר למספרים ראשוניים הופכת את הפאזל למטעה מעט.", "model": "google_nmt", "from_community_srt": "עכשיו זה מדהים ויפה, ואתה עשוי לחשוב שזה מרמז על איזו סימטריה אלוהית נסתרת בתוך הראשוניים.", @@ -295,7 +295,7 @@ "end": 284.02 }, { - "input": "Then the next one is every integer that's one above a multiple of 6.", + "input": "then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6,", "translatedText": "אז הבא הוא כל מספר שלם שנמצא אחד מעל לכפולה של 6.", "model": "google_nmt", "from_community_srt": "שמתם לב שבזמן שהתרחקנו הופיעו שש הספירלות הקטנות האלה? זה מציע נקודת התחלה טובה כדי להסביר מה קורה בשני הדפוסים הגדולים יותר", diff --git a/2019/prime-spirals/hindi/sentence_translations.json b/2019/prime-spirals/hindi/sentence_translations.json index 728c81aff..f37fae1ee 100644 --- a/2019/prime-spirals/hindi/sentence_translations.json +++ b/2019/prime-spirals/hindi/sentence_translations.json @@ -56,7 +56,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "बिंदु 1,1 1 रेडियन के कोण के साथ, मूल से 1 दूरी पर स्थित है, जिसका अर्थ है कि यह चाप उस रेडियल रेखा के समान लंबाई है, और 2,2 में उस कोण का दोगुना और दूरी दोगुनी है।", "n_reviews": 0, "start": 78.88, @@ -203,7 +203,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "ध्यान दें कि कैसे 6 के सभी गुणज इस सर्पिल की एक भुजा बनाते हैं, फिर अगला प्रत्येक पूर्णांक होता है जो 6 के गुणज के ऊपर एक होता है, और फिर 6 के गुणज के ऊपर की सभी संख्याएँ 2 शामिल होती हैं, और इसी तरह।", "n_reviews": 0, "start": 280.44, @@ -357,7 +357,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "प्रति रोटेशन 2 पीआई रेडियन होते हैं, इसलिए 44 कदम उठाने पर कुल 44 को 2 पीआई रोटेशन से विभाजित किया जाता है, जो कि 7 पूर्ण मोड़ से बमुश्किल ऊपर आता है।", "n_reviews": 0, "start": 443.76, @@ -1064,7 +1064,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "और इसका लगभग निश्चित रूप से मतलब यह होगा कि आप इसे अधिक प्रभावी ढंग से सीखेंगे।", "n_reviews": 0, "start": 1314.12, diff --git a/2019/prime-spirals/hungarian/sentence_translations.json b/2019/prime-spirals/hungarian/sentence_translations.json index 0e61fc782..0eb80f3eb 100644 --- a/2019/prime-spirals/hungarian/sentence_translations.json +++ b/2019/prime-spirals/hungarian/sentence_translations.json @@ -192,7 +192,7 @@ "end": 213.26 }, { - "input": "But to steady your expectations, I should say that the fact that the person asking this question jumped right into prime numbers makes the puzzle a little misleading.", + "input": "but to study your expectations, I should say that the fact that the person asking this question on math exchange jumped right into prime numbers makes the puzzle a little misleading.", "translatedText": "De hogy eloszlassam az elvárásaitokat, azt kell mondanom, hogy az a tény, hogy a kérdést feltevő személy rögtön a prímszámokra ugrott, kissé félrevezetővé teszi a rejtvényt.", "model": "DeepL", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 284.02 }, { - "input": "Then the next one is every integer that's one above a multiple of 6.", + "input": "then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6,", "translatedText": "A következő minden olyan egész szám, amely eggyel több mint 6 többszöröse.", "model": "DeepL", "n_reviews": 0, diff --git a/2019/prime-spirals/indonesian/sentence_translations.json b/2019/prime-spirals/indonesian/sentence_translations.json index 8a59b131f..f73ee1938 100644 --- a/2019/prime-spirals/indonesian/sentence_translations.json +++ b/2019/prime-spirals/indonesian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "Titik 1,1 terletak pada jarak 1 dari titik asal, dengan sudut 1 radian, artinya busur ini sama panjangnya dengan garis radial tersebut, dan 2,2 mempunyai sudut dua kali lipat, dan jarak dua kali lipat.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "Perhatikan bagaimana semua kelipatan 6 membentuk satu lengan spiral ini, lalu yang berikutnya adalah setiap bilangan bulat yang satu di atas kelipatan 6, lalu mencakup semua angka 2 di atas kelipatan 6, dan seterusnya.", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "Ada 2 pi radian per putaran, jadi mengambil 44 langkah menghasilkan total 44 dibagi 2 putaran pi, yang hasilnya sedikit di atas 7 putaran penuh.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "Dan itu hampir pasti berarti Anda mempelajarinya dengan lebih efektif.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/italian/sentence_translations.json b/2019/prime-spirals/italian/sentence_translations.json index a3b167b36..0e0790bcf 100644 --- a/2019/prime-spirals/italian/sentence_translations.json +++ b/2019/prime-spirals/italian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "Il punto 1,1 si trova a distanza 1 dall'origine, con un angolo di 1 radiante, il che significa che questo arco ha la stessa lunghezza della linea radiale, e 2,2 ha il doppio di quell'angolo e il doppio della distanza.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "Nota come tutti i multipli di 6 formano un braccio di questa spirale, poi il successivo è ogni intero che è uno sopra un multiplo di 6, e poi include tutti i numeri 2 sopra un multiplo di 6, e così via.", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "Ci sono 2 pi radianti per rotazione, quindi facendo 44 passi si ottiene un totale di 44 diviso per 2 rotazioni pi greco, che risulta essere appena sopra i 7 giri completi.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "E questo quasi sicuramente significherà che lo imparerai in modo più efficace.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/japanese/sentence_translations.json b/2019/prime-spirals/japanese/sentence_translations.json index 9f32df588..4c15c174b 100644 --- a/2019/prime-spirals/japanese/sentence_translations.json +++ b/2019/prime-spirals/japanese/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance. ", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance. ", "translatedText": "点 1,1 は原点から 1 距離離れた位置にあり、角度は 1 ラジ アンです。これは、この円弧がその放射状の線と同じ長さであることを 意味し、2,2 はその 2 倍の角度と 2 倍の距離を持ちます。", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on. ", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on. ", "translatedText": "すべての 6 の倍数がこのスパイラルの 1 つのアームを形成し、次のアーム は 6 の倍数の 1 つ上のすべての整数であり、さらに 6 の倍数の 2 つ上のすべての数値が含まれる、というように続くことに注目してください。", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", "translatedText": "1 回転あたり 2 円周率ラジアンがあるため、44 ステップを実行すると、合計 44 を 2 円周率回転で割った値となり、完全に 7 回転をわずかに超える程度になります。", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively. ", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you. ", "translatedText": "そしてそれはほぼ確実に、より効果的に学習できることを意味します。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/korean/sentence_translations.json b/2019/prime-spirals/korean/sentence_translations.json index 8149480a1..5ce6610d9 100644 --- a/2019/prime-spirals/korean/sentence_translations.json +++ b/2019/prime-spirals/korean/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance. ", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance. ", "translatedText": "점 1,1은 원점에서 1만큼 떨어져 있고 각도는 1라디안입니다. 즉, 이 호는 방사형 선과 길이가 같고 2,2는 해당 각도의 두 배, 거리의 두 배를 가집니다. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on. ", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on. ", "translatedText": "6의 모든 배수가 어떻게 이 나선의 한 가지 팔을 형성하는지 주목하세요. 그 다음은 6의 배수보다 1보다 큰 모든 정수이고, 그런 다음 6의 배수보다 높은 모든 숫자 2를 포함하는 식입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", "translatedText": "회전당 2파이 라디안이 있으므로 44단계를 수행하면 총 44를 2파이 회전으로 나눈 값이 나오며 이는 7회전을 거의 초과하는 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively. ", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you. ", "translatedText": "그리고 그것은 거의 확실히 당신이 그것을 더 효과적으로 배운다는 것을 의미할 것입니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/marathi/sentence_translations.json b/2019/prime-spirals/marathi/sentence_translations.json index 09a0cab9a..515cb5c3b 100644 --- a/2019/prime-spirals/marathi/sentence_translations.json +++ b/2019/prime-spirals/marathi/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "बिंदू 1,1 मूळपासून 1 अंतरावर 1 रेडियनच्या कोनासह बसतो, याचा अर्थ हा कंस त्या रेडियल रेषेइतकाच लांबीचा आहे आणि 2,2 मध्ये त्या कोनाच्या दुप्पट आणि अंतराच्या दुप्पट आहे.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "6 चे सर्व गुणाकार या सर्पिलचा एक हात कसा बनवतात याकडे लक्ष द्या, नंतर पुढील प्रत्येक पूर्णांक 6 च्या गुणाकाराच्या वर एक पूर्णांक आहे, आणि नंतर 6 च्या गुणाकाराच्या वरील सर्व संख्या 2 समाविष्ट करते, आणि असेच.", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "प्रत्येक रोटेशनमध्ये 2 pi रेडियन असतात, त्यामुळे 44 पावले उचलल्याने एकूण 44 भागिले 2 pi रोटेशन मिळतात, जे केवळ 7 पूर्ण वळणांच्या वर येते.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "आणि याचा अर्थ नक्कीच असा होईल की तुम्ही ते अधिक प्रभावीपणे शिकाल.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/persian/sentence_translations.json b/2019/prime-spirals/persian/sentence_translations.json index 1d674cf8e..ea993edc1 100644 --- a/2019/prime-spirals/persian/sentence_translations.json +++ b/2019/prime-spirals/persian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance. ", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance. ", "translatedText": "نقطه 1،1 در فاصله 1 دورتر از مبدا قرار دارد، با زاویه 1 رادیان، به این معنی که این کمان به اندازه آن خط شعاعی است، و 2،2 دو برابر آن زاویه و دو برابر فاصله دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on. ", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on. ", "translatedText": "توجه کنید که چگونه همه مضرب های 6 یک بازوی این مارپیچ را تشکیل می دهند، سپس عدد بعدی هر عدد صحیحی است که یک بالای مضرب 6 است، و سپس شامل تمام اعداد 2 بالای مضرب 6 می شود و به همین ترتیب. ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", "translatedText": "در هر چرخش 2 رادیان پی وجود دارد، بنابراین با برداشتن 44 مرحله، مجموعاً 44 تقسیم بر 2 چرخش پی به دست می‌آید که به سختی بیش از 7 دور کامل است. ", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively. ", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you. ", "translatedText": "و این تقریباً به این معنی است که شما آن را به طور مؤثرتری یاد می گیرید. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/portuguese/sentence_translations.json b/2019/prime-spirals/portuguese/sentence_translations.json index 707fd390f..40faea52b 100644 --- a/2019/prime-spirals/portuguese/sentence_translations.json +++ b/2019/prime-spirals/portuguese/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "O ponto 1,1 fica a uma distância 1 da origem, com um ângulo de 1 radiano, o que significa que este arco tem o mesmo comprimento daquela linha radial, e 2,2 tem o dobro desse ângulo e o dobro da distância.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "Observe como todos os múltiplos de 6 formam um braço desta espiral, então o próximo é cada número inteiro que está acima de um múltiplo de 6, e então inclui todos os números 2 acima de um múltiplo de 6, e assim por diante.", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "Existem 2 pi radianos por rotação, portanto, dar 44 passos dá um total de 44 dividido por 2 rotações pi, o que resulta um pouco acima de 7 voltas completas.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "E isso quase certamente significará que você aprenderá de forma mais eficaz.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/russian/sentence_translations.json b/2019/prime-spirals/russian/sentence_translations.json index 3cd7cad3a..66cb37aa1 100644 --- a/2019/prime-spirals/russian/sentence_translations.json +++ b/2019/prime-spirals/russian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "Точка 1,1 расположена на расстоянии 1 от начала координат и имеет угол в 1 радиан, что означает, что эта дуга имеет ту же длину, что и радиальная линия, а точка 2,2 имеет вдвое больший угол и вдвое большее расстояние.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "Обратите внимание, как все числа, кратные 6, образуют одно плечо этой спирали, затем следующее — каждое целое число, кратное 6, а затем включает все числа, кратные 6, и так далее.", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "На один оборот приходится 2 радиана пи, поэтому выполнение 44 шагов дает в общей сложности 44, разделенные на 2 оборота пи, что составляет чуть больше 7 полных оборотов.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "И это почти наверняка будет означать, что вы освоите его более эффективно.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/spanish/sentence_translations.json b/2019/prime-spirals/spanish/sentence_translations.json index 571e4211c..142574733 100644 --- a/2019/prime-spirals/spanish/sentence_translations.json +++ b/2019/prime-spirals/spanish/sentence_translations.json @@ -192,7 +192,7 @@ "end": 213.26 }, { - "input": "But to steady your expectations, I should say that the fact that the person asking this question jumped right into prime numbers makes the puzzle a little misleading.", + "input": "but to study your expectations, I should say that the fact that the person asking this question on math exchange jumped right into prime numbers makes the puzzle a little misleading.", "translatedText": "Pero para estabilizar sus expectativas, debo decir que el hecho de que la persona que hizo esta pregunta saltó directamente a los números primos hace que el enigma sea un poco engañoso.", "from_community_srt": "Pero para afianzar tus expectativas, he de decir que, el echo de que la persona que hizo esta pregunta en Math Exchange saltó directo a los números primos, hace la pregunta un tanto engañosa.", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 284.02 }, { - "input": "Then the next one is every integer that's one above a multiple of 6.", + "input": "then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6,", "translatedText": "Luego, el siguiente es cada número entero que sea uno mayor que un múltiplo de 6.", "from_community_srt": "Luego la siguiente son todos los enteros 1 enseguida de un múltiplo de 6.", "n_reviews": 0, diff --git a/2019/prime-spirals/tamil/sentence_translations.json b/2019/prime-spirals/tamil/sentence_translations.json index 7398ae61e..e06dd99b7 100644 --- a/2019/prime-spirals/tamil/sentence_translations.json +++ b/2019/prime-spirals/tamil/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "புள்ளி 1,1 தோற்றத்திலிருந்து 1 தொலைவில், 1 ரேடியன் கோணத்துடன் அமர்ந்திருக்கிறது, அதாவது இந்த வளைவு அந்த ரேடியல் கோட்டின் அதே நீளம், மற்றும் 2,2 இரண்டு மடங்கு கோணம் மற்றும் இரண்டு மடங்கு தூரம் கொண்டது.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "6 இன் அனைத்து பெருக்கல்களும் இந்த சுழலின் ஒரு கையை எவ்வாறு உருவாக்குகின்றன என்பதைக் கவனியுங்கள், அடுத்தது 6 இன் பெருக்கத்திற்கு மேல் உள்ள ஒவ்வொரு முழு எண்ணாகும், பின்னர் 6 இன் பெருக்கத்திற்கு மேல் உள்ள அனைத்து எண்கள் 2 ஐயும் உள்ளடக்கியது மற்றும் பல.", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "ஒரு சுழற்சிக்கு 2 பை ரேடியன்கள் உள்ளன, எனவே 44 படிகளை எடுப்பது மொத்தம் 44 ஐ 2 பை சுழற்சிகளால் வகுக்கப்படுகிறது, இது 7 முழு திருப்பங்களுக்கு மேல் இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "நீங்கள் அதை மிகவும் திறம்பட கற்றுக்கொள்கிறீர்கள் என்று நிச்சயமாக அர்த்தம்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/telugu/sentence_translations.json b/2019/prime-spirals/telugu/sentence_translations.json index c1a9248e8..e393b0f67 100644 --- a/2019/prime-spirals/telugu/sentence_translations.json +++ b/2019/prime-spirals/telugu/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "పాయింట్ 1,1 మూలం నుండి 1 దూరంలో, 1 రేడియన్ కోణంతో ఉంటుంది, అంటే ఈ ఆర్క్ ఆ రేడియల్ రేఖకు సమానమైన పొడవు, మరియు 2,2 దాని రెండు రెట్లు కోణం మరియు రెండు రెట్లు దూరం కలిగి ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "6 యొక్క అన్ని గుణిజాలు ఈ స్పైరల్‌లో ఒక చేతిని ఎలా ఏర్పరుస్తాయో గమనించండి, తర్వాత వచ్చేది 6 యొక్క గుణకం పైన ఉన్న ప్రతి పూర్ణాంకం, ఆపై 6 యొక్క గుణకం పైన ఉన్న అన్ని సంఖ్యలు 2 మరియు మొదలైనవి ఉంటాయి.", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "ప్రతి భ్రమణానికి 2 పై రేడియన్‌లు ఉన్నాయి, కాబట్టి 44 దశలను తీసుకుంటే మొత్తం 44ని 2 పై భ్రమణాలతో భాగించబడుతుంది, ఇది కేవలం 7 పూర్తి మలుపుల కంటే ఎక్కువగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "మరియు మీరు దీన్ని మరింత ప్రభావవంతంగా నేర్చుకుంటారని దాదాపు ఖచ్చితంగా అర్థం అవుతుంది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/thai/sentence_translations.json b/2019/prime-spirals/thai/sentence_translations.json index 3b59ad0cf..9301643c6 100644 --- a/2019/prime-spirals/thai/sentence_translations.json +++ b/2019/prime-spirals/thai/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance. ", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance. ", "translatedText": "ฉันเห็นรูปแบบนี้เป็นครั้งแรกที่ฉันจะแสดงให้คุณเห็นในคำถามเกี่ยวกับ Math Stack Exchange ผู้ใช้ชื่อดวีมาร์กถาม และเกร็ก มาร์ตินเป็นผู้ตอบ ซึ่งเกี่ยวข้องกับการแจกแจงของจำนวนเฉพาะ พร้อมด้วยการประมาณเหตุผลของพาย คุณเห็นไหมว่าสิ่งที่ผู้ใช้ทำคือเล่นกับข้อมูลในพิกัดเชิงขั้ว ขอเตือนไว้ก่อนว่าเราทุกคนอยู่ในหน้าเดียวกัน นี่หมายถึงการติดป้ายกำกับจุดในพื้นที่ 2 มิติที่ไม่ใช่พิกัด xy ตามปกติ แต่ใช้ระยะห่างจากจุดกำเนิดแทน ซึ่งโดยทั่วไปเรียกว่า r สำหรับรัศมี ร่วมกับมุมที่เป็นเส้นรัศมี ทำด้วยแนวนอนหรือเรียกทั่วไปว่าทีต้า และตามจุดประสงค์ของเรา มุมนี้จะวัดเป็นเรเดียน ซึ่งโดยพื้นฐานหมายความว่ามุมของพายคือครึ่งวงกลม และ 2 พายคือวงกลมเต็มวง และสังเกตว่าพิกัดเชิงขั้วนั้นไม่ซ้ำกัน ในแง่ที่ว่าการบวก 2 ไพเข้ากับพิกัดที่สองนั้นจะไม่เปลี่ยนตำแหน่งที่ตัวเลขคู่นี้อ้างถึง รูปแบบที่เราจะดูที่จุดศูนย์กลางรอบจุดพล็อตโดยที่พิกัดทั้งสองนี้เป็นจำนวนเฉพาะที่กำหนด ไม่มีเหตุผลเชิงปฏิบัติที่จะทำเช่นนี้ มันสนุกจริงๆ เราแค่กำลังเล่นสนุกไปกับการแสดงข้อมูลเป็นภาพ และเพื่อให้เข้าใจว่ามันหมายถึงอะไร ให้ดูที่จำนวนเต็มทั้งหมด แทนที่จะดูเฉพาะจำนวนเฉพาะ จุด 1,1 อยู่ห่างจากจุดกำเนิด 1 โดยมีมุม 1 เรเดียน ซึ่งหมายความว่าส่วนโค้งนี้มีความยาวเท่ากับเส้นรัศมีนั้น และ 2,2 มีมุมนั้น 2 เท่า และระยะห่าง 2 เท่า และเพื่อให้ได้ 3,3 คุณจะต้องหมุนอีก 1 เรเดียน โดยมีมุมรวมที่ตอนนี้น้อยกว่าครึ่งรอบเล็กน้อย เนื่องจาก 3 น้อยกว่าพายเล็กน้อย และคุณก้าวห่างจากจุดกำเนิดไป 1 หน่วย ฉันต้องการให้คุณแน่ใจว่าชัดเจนว่ากำลังวางแผนอะไรอยู่ เพราะทุกสิ่งที่ตามมาขึ้นอยู่กับความเข้าใจ แต่ละก้าวไปข้างหน้าเปรียบเสมือนปลายเข็มนาฬิกา ซึ่งหมุนหนึ่งเรเดียนในแต่ละขีด น้อยกว่าหนึ่งในหกของรอบเล็กน้อย และเพิ่มขึ้นทีละหนึ่งหน่วยในแต่ละก้าว เมื่อคุณดำเนินการต่อ จุดเหล่านี้จะหมุนวนออกไปด้านนอก ก่อให้เกิดสิ่งที่เป็นที่รู้จักในธุรกิจนี้ว่าเป็นเกลียวแบบอาร์คิมีดีน ทีนี้ ถ้าคุณทำการเคลื่อนไหวตามอำเภอใจเพื่อกำจัดทุกอย่างยกเว้นจำนวนเฉพาะ ในตอนแรกมันจะดูสุ่มๆ ท้ายที่สุดแล้ว จำนวนเฉพาะนั้นขึ้นชื่อในเรื่องพฤติกรรมที่วุ่นวายและยากต่อการคาดเดา เมื่อคุณซูมออก สิ่งที่คุณเริ่มเห็นคือกังหันที่ดูเหมือนกาแล็กซีที่ชัดเจนมาก และสิ่งแปลกคือแขนบางส่วนดูเหมือนจะหายไป จากนั้นเมื่อขยายออกไปอีก เกลียวเหล่านั้นก็จะมีรูปแบบที่แตกต่างกันออกไป ซึ่งเป็นรังสีที่ชี้ออกไปด้านนอกที่แตกต่างกันมากมาย และรังสีเหล่านั้นดูเหมือนส่วนใหญ่จะออกเป็นกระจุกสี่อัน แต่ก็มีช่องว่างอยู่บ้าง เหมือนหวีที่ขาดฟัน แน่นอนว่าคำถามสำหรับคุณและฉันคือเกิดอะไรขึ้นที่นี่? ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on. ", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively. ", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/turkish/sentence_translations.json b/2019/prime-spirals/turkish/sentence_translations.json index e1713b9fb..49f7c1748 100644 --- a/2019/prime-spirals/turkish/sentence_translations.json +++ b/2019/prime-spirals/turkish/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "1,1 noktası orijinden 1 uzaklıkta ve 1 radyanlık bir açıyla yer alır, bu da yayın şu radyal çizgiyle aynı uzunlukta olduğu ve 2,2'nin bu açının iki katı ve mesafenin iki katı olduğu anlamına gelir.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "6'nın tüm katlarının bu spiralin bir kolunu nasıl oluşturduğuna, ardından bir sonraki kolun 6'nın bir katının üzerindeki her tam sayı olduğuna ve ardından 6'nın katının üzerindeki tüm 2 sayılarını içerdiğine ve bu şekilde devam ettiğine dikkat edin.", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "Dönüş başına 2 pi radyan vardır, yani 44 adım atmak toplam 44 bölü 2 pi dönüşü verir, bu da 7 tam dönüşün biraz üzerinde olduğu ortaya çıkar.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "Ve bu neredeyse kesinlikle onu daha etkili bir şekilde öğreneceğiniz anlamına gelecektir.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/ukrainian/sentence_translations.json b/2019/prime-spirals/ukrainian/sentence_translations.json index 09c93325c..610475f59 100644 --- a/2019/prime-spirals/ukrainian/sentence_translations.json +++ b/2019/prime-spirals/ukrainian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "Точка 1,1 знаходиться на відстані 1 від початку координат з кутом 1 радіан, що означає, що ця дуга має таку саму довжину, що й радіальна лінія, а 2,2 має вдвічі більший кут і вдвічі більшу відстань.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "Зверніть увагу на те, що всі числа, кратні 6, утворюють один рукав цієї спіралі, потім наступним є кожне ціле число, яке на одиницю вище кратного 6, а потім включає всі числа, що на 2 вище кратного 6, і так далі.", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "На один оберт припадає 2 пі-радіани, тому 44 кроки дають загалом 44, поділені на 2 пі-оберти, що трохи перевищує 7 повних обертів.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "І це майже напевно означатиме, що ви навчитеся цьому більш ефективно.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/urdu/sentence_translations.json b/2019/prime-spirals/urdu/sentence_translations.json index c075ebceb..bd27130ee 100644 --- a/2019/prime-spirals/urdu/sentence_translations.json +++ b/2019/prime-spirals/urdu/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance. ", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance. ", "translatedText": "نقطہ 1,1 اصل سے 1 کے فاصلے پر 1 ریڈین کے زاویہ کے ساتھ بیٹھتا ہے، جس کا مطلب ہے کہ یہ قوس اس ریڈیل لائن کی لمبائی کے برابر ہے، اور 2,2 میں اس زاویہ سے دوگنا، اور فاصلے سے دوگنا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on. ", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on. ", "translatedText": "دیکھیں کہ کس طرح 6 کے تمام ضرب اس سرپل کے ایک بازو کو بناتے ہیں، پھر اگلا ہر عدد عدد ہے جو 6 کے ضرب سے اوپر ہے، اور پھر 6 کے ضرب کے اوپر تمام اعداد 2 کو شامل کرتا ہے، وغیرہ۔", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns. ", "translatedText": "ہر گردش میں 2 pi ریڈینز ہوتے ہیں، لہذا 44 قدم اٹھانے سے کل 44 کو 2 pi گردشوں سے تقسیم کیا جاتا ہے، جو بمشکل 7 مکمل موڑ سے اوپر نکلتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively. ", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you. ", "translatedText": "اور اس کا تقریباً یقینی طور پر مطلب یہ ہوگا کہ آپ اسے زیادہ مؤثر طریقے سے سیکھیں گے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/prime-spirals/vietnamese/sentence_translations.json b/2019/prime-spirals/vietnamese/sentence_translations.json index 705ddfb3a..7273c5cc7 100644 --- a/2019/prime-spirals/vietnamese/sentence_translations.json +++ b/2019/prime-spirals/vietnamese/sentence_translations.json @@ -64,7 +64,7 @@ "end": 78.4 }, { - "input": "The point 1,1 sits a distance 1 away from the origin, with an angle of 1 radian, which means this arc is the same length as that radial line, and 2,2 has twice that angle, and twice the distance.", + "input": "The point 1,1 sets a distance 1 away from the origin, with an angle of 1 radian, which actually means this arc is the same length as that radial line. And then 2,2 has twice that angle, and twice the distance.", "translatedText": "Điểm 1,1 nằm cách điểm gốc một khoảng 1, với góc 1 radian, nghĩa là cung này có cùng độ dài với đường hướng tâm đó và 2,2 có góc gấp đôi góc đó và khoảng cách gấp đôi.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 279.78 }, { - "input": "Notice how all the multiples of 6 form one arm of this spiral, then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, and so on.", + "input": "Notice how all the multiples of 6 form one arm of this spiral. then the next one is every integer that's one above a multiple of 6, and then includes all the numbers 2 above a multiple of 6, Then after that it includes all the numbers 2 above a multiple of 6, and so on.", "translatedText": "Lưu ý cách tất cả các bội số của 6 tạo thành một nhánh của đường xoắn ốc này, sau đó nhánh tiếp theo là mọi số nguyên nằm trên bội số của 6 và sau đó bao gồm tất cả các số 2 ở trên bội số của 6, v. v.", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 443.22 }, { - "input": "There are 2 pi radians per rotation, so taking 44 steps gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", + "input": "There are 2 pi radians per rotation, right? So taking 44 steps, turning 44 radians, gives a total of 44 divided by 2 pi rotations, which comes out to be just barely above 7 full turns.", "translatedText": "Có 2 pi radian cho mỗi vòng quay, do đó, thực hiện 44 bước sẽ có tổng cộng 44 chia cho 2 vòng quay pi, kết quả là chỉ trên 7 vòng quay đầy đủ.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1313.56 }, { - "input": "And that will almost certainly mean that you learn it more effectively.", + "input": "and that will almost certainly mean that you learn it more effectively. Thank you.", "translatedText": "Và điều đó gần như chắc chắn có nghĩa là bạn học nó hiệu quả hơn.", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/qa4/english/captions.srt b/2019/qa4/english/captions.srt index 4eec70839..74ac6df22 100644 --- a/2019/qa4/english/captions.srt +++ b/2019/qa4/english/captions.srt @@ -591,7 +591,7 @@ What advice would you give to a math enthusiast suffering from anxiety disorder, clinical depression, and ADHD? 149 -00:08:41,439 --> 00:08:45,754 +00:08:41,440 --> 00:08:45,754 I'm not entirely sure how the word math enthusiast in the question changes the answer, 150 @@ -1019,286 +1019,294 @@ They can just be very separate things, and kind of separating the brand of those two can't hurt. 256 -00:14:42,660 --> 00:14:46,873 -Well, just last week, all of the internet has been very abuzz about a +00:14:42,660 --> 00:14:45,708 +What's something you think could have been discovered long before it was 257 -00:14:46,873 --> 00:14:51,206 -certain result that came from these three physicists studying neutrinos +00:14:45,708 --> 00:14:48,923 +actually discovered? Well, just last week, all of the internet has been very 258 -00:14:51,206 --> 00:14:55,480 -about eigenvectors and eigenvalues, which is a crazy fundamental thing. +00:14:48,923 --> 00:14:52,097 +abuzz about a certain result that came from these three physicists studying 259 +00:14:52,097 --> 00:14:55,480 +neutrinos about eigenvectors and eigenvalues, which is a crazy fundamental thing. + +260 00:14:55,480 --> 00:14:59,663 You kind of wouldn't imagine that there are any new things to be discovered -260 +261 00:14:59,663 --> 00:15:03,626 about computing eigenvectors or computing eigenvalues, because it's so, -261 +262 00:15:03,626 --> 00:15:07,700 well, it's kind of old and it's very, I don't know, routine at this point. -262 +263 00:15:08,340 --> 00:15:11,943 But they found what they thought was a result and they sent it to Terry Tao, -263 +264 00:15:11,943 --> 00:15:15,313 who actually responded and his initial thought was, this can't be true, -264 +265 00:15:15,313 --> 00:15:17,420 it would be in every textbook if it was true. -265 +266 00:15:17,420 --> 00:15:22,082 And then within two hours, I think he found three independent proofs of the thing, -266 +267 00:15:22,082 --> 00:15:27,082 and yeah, it's just a different way to compute eigenvectors that was discovered in 2019, -267 +268 00:15:27,082 --> 00:15:31,240 even though that totally could have been discovered hundreds of years ago. -268 +269 00:15:32,220 --> 00:15:33,680 Can we fix math on Wikipedia? -269 +270 00:15:34,100 --> 00:15:35,060 Really serious here. -270 +271 00:15:35,360 --> 00:15:39,885 I constantly go there after your vids for a bit of a deeper dive and learn nothing more, -271 +272 00:15:39,885 --> 00:15:40,140 ever. -272 +273 00:15:40,140 --> 00:15:43,640 Compared to almost any other topic in the natural sciences or physics, -273 +274 00:15:43,640 --> 00:15:47,140 where at least I get an outline of where to go next, it's such a shame. -274 +275 00:15:48,560 --> 00:15:52,137 So this question kind of reminds me of that classic trope where you've -275 +276 00:15:52,137 --> 00:15:55,815 got a girl and she's dating a boy and, you know, he's kind of a bad boy, -276 +277 00:15:55,815 --> 00:15:59,039 he does a couple things that are wrong that adds to his allure, -277 +278 00:15:59,039 --> 00:16:03,020 he's kind of sexy in that way, but she's thinking, oh, I can change him, right? -278 +279 00:16:03,020 --> 00:16:04,760 He's flawed, but I can fix him. -279 +280 00:16:05,300 --> 00:16:09,060 And everyone in her life is looking and saying, oh, honey, like, he's not going to change. -280 +281 00:16:09,200 --> 00:16:09,920 People don't change. -281 +282 00:16:10,000 --> 00:16:11,180 You have to find someone else. -282 +283 00:16:11,640 --> 00:16:15,165 In the same way, if I see someone trying to learn math from Wikipedia -283 +284 00:16:15,165 --> 00:16:18,540 and not use it as a reference, it's like, it's not going to change. -284 +285 00:16:18,760 --> 00:16:19,740 Don't try to change it. -285 +286 00:16:19,740 --> 00:16:21,100 You've got to find a different source. -286 +287 00:16:21,540 --> 00:16:24,437 There's lots of really great blogs that you can go to, -287 +288 00:16:24,437 --> 00:16:27,914 or Math Exchange and Quora are great in terms of people trying to -288 +289 00:16:27,914 --> 00:16:32,340 explain things in approachable ways, and don't forget about just good old textbooks. -289 +290 00:16:32,340 --> 00:16:36,343 In math, more so than a lot of fields, I think there's a strong contrast between -290 +291 00:16:36,343 --> 00:16:40,100 what makes good reference material and what makes good pedagogical material. -291 +292 00:16:40,520 --> 00:16:44,790 And a general rule of thumb, this is not universal, but general rule of thumb, -292 +293 00:16:44,790 --> 00:16:48,520 things that are single-authored, I believe, are better pedagogically. -293 +294 00:16:48,840 --> 00:16:52,968 And I suspect the reason for this is that when you want to explain a topic, -294 +295 00:16:52,968 --> 00:16:57,476 often the best route to making it understandable is to start off by being a little -295 +296 00:16:57,476 --> 00:16:58,020 bit wrong. -296 +297 00:16:58,020 --> 00:17:02,310 You explain kind of a simplified version of something that isn't entirely accurate, -297 +298 00:17:02,310 --> 00:17:04,200 but it's easier to get a foothold in. -298 +299 00:17:04,619 --> 00:17:08,503 Then once you have that foothold, you slowly carve away what's wrong about -299 +300 00:17:08,503 --> 00:17:12,180 it until you end up at what is entirely accurate, but more complicated. -300 +301 00:17:12,500 --> 00:17:14,839 But you've taken this path through incorrectness. -301 +302 00:17:15,540 --> 00:17:18,983 Now when you have multiple authors, I think the tendency is that -302 +303 00:17:18,983 --> 00:17:22,480 you sort of wipe away and edit away the things that are incorrect. -303 +304 00:17:22,480 --> 00:17:24,619 That's like the stable equilibrium that you reach. -304 +305 00:17:25,220 --> 00:17:28,474 So what you're left with is a source that's entirely factually correct, -305 +306 00:17:28,474 --> 00:17:30,960 but it's harder to get a foothold into for that reason. -306 +307 00:17:31,480 --> 00:17:33,631 And Wikipedia just represents the extreme of this, -307 +308 00:17:33,631 --> 00:17:36,752 but I also think you see it if you look at a textbook that has, you know, -308 +309 00:17:36,752 --> 00:17:37,680 three or four authors. -309 +310 00:17:37,960 --> 00:17:40,960 Again, there are exceptions, but I like that as a rule of thumb. -310 -00:17:45,159 --> 00:17:48,080 +311 +00:17:45,160 --> 00:17:48,080 Real quick, I want to tell you about two new items that have -311 +312 00:17:48,080 --> 00:17:51,000 been added to the 3Blue1Brown store for any math enthusiasts. -312 +313 00:17:51,660 --> 00:17:55,745 The first one, in the spirit of upping the level of formality on things, -313 +314 00:17:55,745 --> 00:17:57,480 is this knot theory themed tie. -314 +315 00:17:58,200 --> 00:18:02,100 So as you can see, the pattern includes a lot of different simple mathematical knots. -315 +316 00:18:02,100 --> 00:18:06,477 So almost any knot that you are likely to tie with your tie anyway is going to -316 +317 00:18:06,477 --> 00:18:10,800 be topologically equivalent to one of these, unless you just go totally crazy. -317 +318 00:18:11,400 --> 00:18:14,408 In sourcing this, we wanted to make sure that it was, you know, -318 +319 00:18:14,408 --> 00:18:17,840 a legitimately high quality tie, and I'm really happy with what we found. -319 +320 00:18:18,240 --> 00:18:22,771 Then as a supplement to the ties, I also got these vector field socks produced, -320 +321 00:18:22,771 --> 00:18:26,056 and what they represent is the phase space of a pendulum, -321 +322 00:18:26,056 --> 00:18:30,134 which some of you may know is most naturally represented on a cylinder, -322 +323 00:18:30,134 --> 00:18:31,720 hence printing it on a sock. -323 +324 00:18:31,720 --> 00:18:35,320 So the whole item is just sort of a subtle nod to that fact. -324 +325 00:18:35,940 --> 00:18:39,994 I believe DFTBA is going to do some kind of sale on Black Friday and Cyber Monday, -325 +326 00:18:39,994 --> 00:18:43,120 so if you're watching this before then, definitely check it out. -326 -00:18:43,860 --> 00:18:47,320 -And with that, I will see you all in the next, probably much more typical, video. +327 +00:18:43,860 --> 00:18:45,454 +And with that, I will see you all in the next, + +328 +00:18:45,454 --> 00:18:47,320 +probably much more typical, video. Thanks for watching! diff --git a/2019/qa4/english/sentence_timings.json b/2019/qa4/english/sentence_timings.json index 79bece302..2b3dc580b 100644 --- a/2019/qa4/english/sentence_timings.json +++ b/2019/qa4/english/sentence_timings.json @@ -615,7 +615,7 @@ 878.28 ], [ - "Well, just last week, all of the internet has been very abuzz about a certain result that came from these three physicists studying neutrinos about eigenvectors and eigenvalues, which is a crazy fundamental thing.", + "What's something you think could have been discovered long before it was actually discovered? Well, just last week, all of the internet has been very abuzz about a certain result that came from these three physicists studying neutrinos about eigenvectors and eigenvalues, which is a crazy fundamental thing.", 882.66, 895.48 ], @@ -795,7 +795,7 @@ 1123.12 ], [ - "And with that, I will see you all in the next, probably much more typical, video.", + "And with that, I will see you all in the next, probably much more typical, video. Thanks for watching!", 1123.86, 1127.32 ] diff --git a/2019/qa4/english/transcript.txt b/2019/qa4/english/transcript.txt index 3f2605e12..dd653ead0 100644 --- a/2019/qa4/english/transcript.txt +++ b/2019/qa4/english/transcript.txt @@ -121,7 +121,7 @@ I think it's just that when you're a parent and you're showing a lot of attentio So all of the signaling that probably came from young Richard Feynman's dad showing this deep attentiveness to questions about the physical world, about mathematical patterns, probably made it such that young Richard would spend a lot of his own time thinking about those things, because they just pattern match off of their parents. Another thing I might say is try to draw a distinction between school math and math, right? They can just be very separate things, and kind of separating the brand of those two can't hurt. -Well, just last week, all of the internet has been very abuzz about a certain result that came from these three physicists studying neutrinos about eigenvectors and eigenvalues, which is a crazy fundamental thing. +What's something you think could have been discovered long before it was actually discovered? Well, just last week, all of the internet has been very abuzz about a certain result that came from these three physicists studying neutrinos about eigenvectors and eigenvalues, which is a crazy fundamental thing. You kind of wouldn't imagine that there are any new things to be discovered about computing eigenvectors or computing eigenvalues, because it's so, well, it's kind of old and it's very, I don't know, routine at this point. But they found what they thought was a result and they sent it to Terry Tao, who actually responded and his initial thought was, this can't be true, it would be in every textbook if it was true. And then within two hours, I think he found three independent proofs of the thing, and yeah, it's just a different way to compute eigenvectors that was discovered in 2019, even though that totally could have been discovered hundreds of years ago. @@ -157,4 +157,4 @@ In sourcing this, we wanted to make sure that it was, you know, a legitimately h Then as a supplement to the ties, I also got these vector field socks produced, and what they represent is the phase space of a pendulum, which some of you may know is most naturally represented on a cylinder, hence printing it on a sock. So the whole item is just sort of a subtle nod to that fact. I believe DFTBA is going to do some kind of sale on Black Friday and Cyber Monday, so if you're watching this before then, definitely check it out. -And with that, I will see you all in the next, probably much more typical, video. \ No newline at end of file +And with that, I will see you all in the next, probably much more typical, video. Thanks for watching! \ No newline at end of file diff --git a/2019/qa4/spanish/sentence_translations.json b/2019/qa4/spanish/sentence_translations.json index 41c7e1383..9d0db364d 100644 --- a/2019/qa4/spanish/sentence_translations.json +++ b/2019/qa4/spanish/sentence_translations.json @@ -1095,7 +1095,7 @@ "end": 878.28 }, { - "input": "Well, just last week, all of the internet has been very abuzz about a certain result that came from these three physicists studying neutrinos about eigenvectors and eigenvalues, which is a crazy fundamental thing.", + "input": "What's something you think could have been discovered long before it was actually discovered? Well, just last week, all of the internet has been very abuzz about a certain result that came from these three physicists studying neutrinos about eigenvectors and eigenvalues, which is a crazy fundamental thing.", "translatedText": "Pues bien, la semana pasada, todo Internet ha estado muy alborotado acerca de cierto resultado que obtuvieron estos tres físicos que estudian los neutrinos acerca de los vectores propios y los valores propios, que es una cosa fundamental muy loca.", "model": "DeepL", "from_community_srt": "¿Qué crees que podría haber sido descubierto mucho antes de su descubrimiento? Justamente la semana pasada, todo Internet se ha ajetreado acerca de unos resultados descubiertos por tres físicos que estudiaban neutrinos, los vectores propios y los valores propios. Lo que es un elemento matemático muy fundamental.", @@ -1415,7 +1415,7 @@ "end": 1123.12 }, { - "input": "And with that, I will see you all in the next, probably much more typical, video.", + "input": "And with that, I will see you all in the next, probably much more typical, video. Thanks for watching!", "translatedText": "Y con esto, os veré a todos en el próximo vídeo, probablemente mucho más típico.", "model": "DeepL", "from_community_srt": "podréis estar atentos. Y con esto, probablemente os veré en el próximo y más típico vídeo.", diff --git a/2019/windmills/arabic/sentence_translations.json b/2019/windmills/arabic/sentence_translations.json index a05a80556..04b24ca86 100644 --- a/2019/windmills/arabic/sentence_translations.json +++ b/2019/windmills/arabic/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "المسابقة نفسها هي في الأساس اختبار، مقسم على يومين، مع ثلاثة أسئلة مطروحة على 4.5 ساعات كل يوم. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "إنها جميعها صعبة، بالطبع، إنها المنظمة البحرية الدولية (IMO)، لكن المشكلتين الأولى والرابعة يجب أن تكونا قابلتين للتنفيذ، والمشكلتان الثانية والخامسة صعبة، والمشكلتان الثالثة والسادسة يمكن أن تكونا وحشيتين. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/bengali/sentence_translations.json b/2019/windmills/bengali/sentence_translations.json index b04089ddf..f4d8f5b29 100644 --- a/2019/windmills/bengali/sentence_translations.json +++ b/2019/windmills/bengali/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "প্রতিযোগিতাটি নিজেই মূলত একটি পরীক্ষা, যা দুই দিনের মধ্যে বিভক্ত, 4টির উপরে তিনটি প্রশ্ন দেওয়া হয়েছে।প্রতিদিন 5 ঘন্টা।", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "এগুলি সবই কঠিন, অবশ্যই, এটি আইএমও, তবে সমস্যা এক এবং চারটি সম্ভব হওয়া উচিত, সমস্যা দুটি এবং পাঁচটি চ্যালেঞ্জিং এবং সমস্যা তিনটি এবং ছয়টি নৃশংস হতে পারে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/chinese/sentence_translations.json b/2019/windmills/chinese/sentence_translations.json index 9f4369258..6b572cf16 100644 --- a/2019/windmills/chinese/sentence_translations.json +++ b/2019/windmills/chinese/sentence_translations.json @@ -18,7 +18,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "比赛本身本质上是一次测试，分两天进行，其中 3 个问题超过 4 题。每天5小时。", "model": "google_nmt", "from_community_srt": "比赛本质上是一个考验， 分裂 两天， 给出了三个问题 每天四个半小时。", @@ -330,7 +330,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "当然，它们都很难，这是IMO，但问题一和问题四应该是可行的 ，问题二和问题五具有挑战性，问题三和问题六可能是残酷的。", "model": "google_nmt", "from_community_srt": "当然， 他们都很难， 但问题 1和4应该是可行的， 问题2和5 应该是挑战， 问题3和 6 ......他们可以是残酷的。", diff --git a/2019/windmills/english/captions.srt b/2019/windmills/english/captions.srt index eee6bdcea..7af718d47 100644 --- a/2019/windmills/english/captions.srt +++ b/2019/windmills/english/captions.srt @@ -399,12 +399,12 @@ So for an example like this, if you start with the line going through that troublesome middle point, what happens? 101 -00:06:16,080 --> 00:06:18,488 -Again, we're just playing around, perhaps moving your +00:06:16,080 --> 00:06:18,007 +And again, at this point we're just playing around, 102 -00:06:18,488 --> 00:06:21,120 -pencil among dots you've drawn on a piece of scratch paper. +00:06:18,007 --> 00:06:21,120 +perhaps moving your pencil among dots that you've drawn on a piece of scratch paper. 103 00:06:21,520 --> 00:06:24,980 @@ -935,10 +935,14 @@ a mathematical Aesop summarizing that the moral of the story is to seek quantiti which stay constant. 235 -00:15:26,460 --> 00:15:33,291 +00:15:26,460 --> 00:15:30,922 But some of you watching this will one day face a problem where finding an invariant 236 -00:15:33,291 --> 00:15:39,480 -reveals a slick solution, and you might even look like a genius for doing so. +00:15:30,922 --> 00:15:35,017 +reveals a slick solution, and you might even look like a genius for doing so. + +237 +00:15:35,017 --> 00:15:39,480 +If a made-up windmill prepares you for a real problem, who cares that it's a fiction? diff --git a/2019/windmills/english/sentence_timings.json b/2019/windmills/english/sentence_timings.json index 1d9d059ca..f144ff335 100644 --- a/2019/windmills/english/sentence_timings.json +++ b/2019/windmills/english/sentence_timings.json @@ -285,7 +285,7 @@ 375.24 ], [ - "Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper.", + "And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", 376.08, 381.12 ], @@ -615,7 +615,7 @@ 925.98 ], [ - "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so.", + "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction?", 926.46, 939.48 ] diff --git a/2019/windmills/english/transcript.txt b/2019/windmills/english/transcript.txt index cc693c6e4..f824ab629 100644 --- a/2019/windmills/english/transcript.txt +++ b/2019/windmills/english/transcript.txt @@ -55,7 +55,7 @@ The fourth point is where it gets interesting. In some places, your windmill will go around the four points just like it did with the triangle, but if we put it inside that triangle, it looks like our windmill never hits it. Looking back at the problem, it's asking you to show that for some starting position of the line, not any position, the process will hit all the points infinitely many times. So for an example like this, if you start with the line going through that troublesome middle point, what happens? -Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper. +And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper. You want to believe a result before you try too hard to prove it. Here you'd see that your windmill does indeed bounce off of all the points as it goes through a cycle, and it ends up back where it started. The worry you might have is that in some large sets of points, where some are kind of inside the others, you might be able to start off on the inside, but maybe something about this windmill process takes the line to the outside, where as time goes on to infinity it'll be blocked off from those inner points. @@ -121,4 +121,4 @@ As a student, it's easy to take for granted the definitions handed down to you, Terence Tao, one of the greatest modern mathematicians and the world's youngest IMO medalist, wrote that mathematical problems or puzzles are important to real mathematics, like solving real-life problems, just as fables, stories, and anecdotes are important to the young in understanding real life. Sure, these kinds of puzzles are contrived, but they carry lessons relevant to useful problems you may actually need to solve one day. Maybe it seems silly to liken this windmill puzzle to a fairy tale, a mathematical Aesop summarizing that the moral of the story is to seek quantities which stay constant. -But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. \ No newline at end of file +But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction? \ No newline at end of file diff --git a/2019/windmills/french/sentence_translations.json b/2019/windmills/french/sentence_translations.json index 72ddbcf99..ee05856ca 100644 --- a/2019/windmills/french/sentence_translations.json +++ b/2019/windmills/french/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "Le concours lui-même est essentiellement un test, réparti sur deux jours, avec trois questions posées sur quatre. 5 heures chaque jour. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "Ils sont tous difficiles, bien sûr, c'est l'OMI, mais les problèmes un et quatre devraient être réalisables, les problèmes deux et cinq sont difficiles et les problèmes trois et six peuvent être brutaux. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/german/sentence_translations.json b/2019/windmills/german/sentence_translations.json index 9ca67af4a..34612f22b 100644 --- a/2019/windmills/german/sentence_translations.json +++ b/2019/windmills/german/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "Der Wettbewerb selbst ist im Wesentlichen ein Test, der über zwei Tage verteilt ist und bei dem drei Fragen über vier gestellt werden. 5 Stunden pro Tag. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "Sie sind natürlich alle schwer, das ist meine Meinung, aber die Probleme eins und vier sollten machbar sein, die Probleme zwei und fünf sind eine Herausforderung und die Probleme drei und sechs können brutal sein. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/greek/sentence_translations.json b/2019/windmills/greek/sentence_translations.json index f5d0f23f5..6acfc60a8 100644 --- a/2019/windmills/greek/sentence_translations.json +++ b/2019/windmills/greek/sentence_translations.json @@ -454,7 +454,7 @@ "end": 375.24 }, { - "input": "Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper.", + "input": "And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", "translatedText": "", "from_community_srt": "τι θα συμβεί; Και πάλι, τώρα απλώς παίζουμε γυρνώντας μάλλον το μολύβι μας ανάμεσα σε τελείες που ζωγραφίσαμε σε ένα πρόχειρο χαρτί,", "n_reviews": 0, @@ -981,7 +981,7 @@ "end": 925.98 }, { - "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so.", + "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction?", "translatedText": "", "from_community_srt": "Κάποιοι από εσάς μια μέρα, θα συναντήσετε ένα πρόβλημα όπου το να βρείτε μία σταθερά θα αποκαλύψει μία όμορφη λύση, και πιθανό να νιώσετε ιδιοφυΐες που το πράξατε. Εάν ένας κατασκευασμένος ανεμόμυλος σας προετοιμάσει για ένα πραγματικό πρόβλημα, ποιος νοιάζεται αν είναι μυθιστόρημα;", "n_reviews": 0, diff --git a/2019/windmills/hebrew/sentence_translations.json b/2019/windmills/hebrew/sentence_translations.json index e01e7add4..847a93c44 100644 --- a/2019/windmills/hebrew/sentence_translations.json +++ b/2019/windmills/hebrew/sentence_translations.json @@ -511,7 +511,7 @@ "end": 375.24 }, { - "input": "Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper.", + "input": "And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", "translatedText": "שוב, אנחנו רק משחקים, אולי מעבירים את העיפרון שלך בין נקודות שציירת על פיסת נייר גירוד.", "model": "google_nmt", "from_community_srt": "מה קורה? ושוב, בשלב זה אתה אנחנו סתם מנסים, אולי תזיז את העיפרון שלך בין נקודות שציירת על נייר,", @@ -1103,7 +1103,7 @@ "end": 925.98 }, { - "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so.", + "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction?", "translatedText": "אבל חלק מכם הצופים בזה יתמודד יום אחד עם בעיה שבה מציאת אינוריאנט חושף פתרון חלק, ואולי אפילו תיראו כמו גאון על כך.", "model": "google_nmt", "from_community_srt": "אבל חלק מכם שצפו בזה יום אחד יתקלו בבעיה שבה מציאת שמורה (invariant) מגלה פיתרון חלקלק, ואולי אפילו תראו כמו גאון על כך. אם טחנת רוח מומצאת מכינה אותך לבעיה אמיתית,", diff --git a/2019/windmills/hindi/sentence_translations.json b/2019/windmills/hindi/sentence_translations.json index bd860bab0..a98271c78 100644 --- a/2019/windmills/hindi/sentence_translations.json +++ b/2019/windmills/hindi/sentence_translations.json @@ -14,7 +14,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day.", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day.", "translatedText": "प्रतियोगिता स्वयं मूलतः एक परीक्षा है, जो दो दिनों में विभाजित है, जिसमें 4 से अधिक तीन प्रश्न दिए गए हैं।हर दिन 5 घंटे.", "n_reviews": 0, "start": 26.72, @@ -259,7 +259,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal.", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal.", "translatedText": "बेशक, वे सभी कठिन हैं, यह आईएमओ है, लेकिन समस्याएं एक और चार व्यवहार्य होनी चाहिए, समस्याएं दो और पांच चुनौतीपूर्ण हैं, और समस्याएं तीन और छह क्रूर हो सकती हैं।", "n_reviews": 0, "start": 246.02, @@ -840,7 +840,7 @@ "end": 899.72 }, { - "input": ".", + "input": "problems or puzzles are important to rea", "translatedText": ".", "n_reviews": 0, "start": 899.72, diff --git a/2019/windmills/hungarian/sentence_translations.json b/2019/windmills/hungarian/sentence_translations.json index f07002e14..601aa09fc 100644 --- a/2019/windmills/hungarian/sentence_translations.json +++ b/2019/windmills/hungarian/sentence_translations.json @@ -456,7 +456,7 @@ "end": 375.24 }, { - "input": "Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper.", + "input": "And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", "translatedText": "Ismét csak játszadozunk, talán a ceruzát mozgatjuk a pontok között, amelyeket egy darab karcolt papírra rajzoltunk.", "model": "DeepL", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 925.98 }, { - "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so.", + "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction?", "translatedText": "De néhányan, akik ezt nézik, egy napon olyan problémával fognak szembesülni, ahol egy invariáns megtalálása egy ügyes megoldást mutat, és talán még zseninek is fognak tűnni, ha ezt megteszik.", "model": "DeepL", "n_reviews": 0, diff --git a/2019/windmills/indonesian/sentence_translations.json b/2019/windmills/indonesian/sentence_translations.json index 3d5e93508..e17d22c68 100644 --- a/2019/windmills/indonesian/sentence_translations.json +++ b/2019/windmills/indonesian/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "Kontes itu sendiri pada dasarnya adalah sebuah ujian, dibagi dalam dua hari, dengan tiga pertanyaan diberikan dalam 4 hari. 5 jam setiap hari. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "Semuanya sulit, tentu saja, begitulah IMO, tapi soal satu dan empat harusnya bisa diselesaikan, soal dua dan lima menantang, dan soal tiga dan enam bisa jadi brutal. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/italian/sentence_translations.json b/2019/windmills/italian/sentence_translations.json index d829d1d00..f9532017a 100644 --- a/2019/windmills/italian/sentence_translations.json +++ b/2019/windmills/italian/sentence_translations.json @@ -513,7 +513,7 @@ "end": 375.24 }, { - "input": "Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper.", + "input": "And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", "translatedText": "Anche in questo caso, stiamo solo giocando, magari spostando la matita tra i punti che hai disegnato su un foglio di carta.", "model": "DeepL", "from_community_srt": "Di nuovo, ora vi trovate solo a giocare con il rompicapo, magari muovendo la vostra matita tra i punti che avete disegnato su un foglio,", @@ -1105,7 +1105,7 @@ "end": 925.98 }, { - "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so.", + "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction?", "translatedText": "Ma qualcuno di voi che sta guardando questo articolo si troverà un giorno di fronte a un problema in cui la ricerca di un invariante rivela una soluzione intelligente, e potrebbe persino sembrare un genio per averlo fatto.", "model": "DeepL", "from_community_srt": "Ma alcuni di voi un giorno dovranno affrontare un problema dove trovare una grandezza invariante si rivelerà un'astuta soluzione, e potrete anche sembrare dei geni per aver fatto ciò. Se un mulino artificioso può prepararvi ad affrontare un problema reale, a chi interessa se è solo finzione?", diff --git a/2019/windmills/japanese/sentence_translations.json b/2019/windmills/japanese/sentence_translations.json index bb1f4a9ff..3b47d8177 100644 --- a/2019/windmills/japanese/sentence_translations.json +++ b/2019/windmills/japanese/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "コンテスト自体は基本的にテストであり、2 日間に分かれて 4 つの質問に対し て 3 つの質問が出題されます。毎日5時間。", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "もちろん、それらはすべて難しいですが、問題 1 と 4 は実行可能であるはずです が、問題 2 と 5 は困難で、問題 3 と 6 は過酷になる可能性があります。", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/korean/sentence_translations.json b/2019/windmills/korean/sentence_translations.json index a9ee0fda7..71c8186ac 100644 --- a/2019/windmills/korean/sentence_translations.json +++ b/2019/windmills/korean/sentence_translations.json @@ -508,7 +508,7 @@ "end": 375.24 }, { - "input": "Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper.", + "input": "And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", "translatedText": "다시 말하지만, 스크래치 페이퍼에 그린 점들 사이에서 연필을 움직이며 장난을 치는 것일 뿐입니다.", "model": "DeepL", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 925.98 }, { - "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so.", + "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction?", "translatedText": "하지만 이 글을 보고 있는 여러분 중 일부는 언젠가 불변수를 찾으면 멋진 해결책이 드러나는 문제에 직면하게 될 것이고, 심지어 그렇게 하면 천재처럼 보일 수도 있습니다.", "model": "DeepL", "from_community_srt": "수학자로 변신한 이솝이 이야기를 마치며 교훈으로 불변값을 찾는 것으로 요약한다니요! 하지만 이것을 보는 여러분 중 몇몇은 언젠가 불변값이 교묘히 숨겨진 문제에 직면하게 될 것이고, 또 이를 찾아내어 푼다면, 여러분들은 천재처럼 여겨질지 모르는 일입니다. 만약 만들어진 풍차 문제가 여러분에게 진짜 문제를 준비시킨다면, 그것이 허구라고 누가 신경이나 쓸까요?", diff --git a/2019/windmills/malay/sentence_translations.json b/2019/windmills/malay/sentence_translations.json index 7a469c70b..d00e256ec 100644 --- a/2019/windmills/malay/sentence_translations.json +++ b/2019/windmills/malay/sentence_translations.json @@ -455,7 +455,7 @@ "end": 375.24 }, { - "input": "Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper.", + "input": "And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", "translatedText": "", "from_community_srt": "apa yang berlaku? Dan sekali lagi, pada ketika ini anda sedang bermain sekitar, mungkin memindahkan pensil anda di antara titik-titik anda telah menarik pada kertas awal anda,", "n_reviews": 0, @@ -981,7 +981,7 @@ "end": 925.98 }, { - "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so.", + "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction?", "translatedText": "", "from_community_srt": "Tetapi sesetengah daripada anda menonton ini akan suatu hari nanti menghadapi masalah di mana mencari invarian mendedahkan penyelesaian yang licin, dan anda mungkin juga kelihatan seperti seorang jenius untuk berbuat demikian. Sekiranya kincir angin buatan disediakan untuk anda yang sebenar masalah, yang peduli bahawa ia adalah fiksyen?", "n_reviews": 0, diff --git a/2019/windmills/marathi/sentence_translations.json b/2019/windmills/marathi/sentence_translations.json index 6a6abc42f..6c8e0aece 100644 --- a/2019/windmills/marathi/sentence_translations.json +++ b/2019/windmills/marathi/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "स्पर्धा स्वतःच एक चाचणी आहे, दोन दिवसांत विभागली जाते, 4 पेक्षा जास्त तीन प्रश्न दिले जातात. दररोज 5 तास. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "ते सर्व कठीण आहेत, अर्थातच, ते IMO आहे, परंतु समस्या एक आणि चार करणे शक्य असले पाहिजे, समस्या दोन आणि पाच आव्हानात्मक आहेत आणि समस्या तीन आणि सहा क्रूर असू शकतात. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/persian/sentence_translations.json b/2019/windmills/persian/sentence_translations.json index dea713ba5..2e52380b8 100644 --- a/2019/windmills/persian/sentence_translations.json +++ b/2019/windmills/persian/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "خود مسابقه در اصل یک تست است که در دو روز تقسیم می شود و سه سوال در 4 سوال ارائه می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "همه آنها سخت هستند، البته، این IMO است، اما مشکلات یک و چهار باید قابل انجام باشند، مشکلات دو و پنج چالش برانگیز هستند، و مشکلات سه و شش می توانند وحشیانه باشند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/polish/sentence_translations.json b/2019/windmills/polish/sentence_translations.json index e2e6e0e8b..8507cbece 100644 --- a/2019/windmills/polish/sentence_translations.json +++ b/2019/windmills/polish/sentence_translations.json @@ -455,7 +455,7 @@ "end": 375.24 }, { - "input": "Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper.", + "input": "And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", "translatedText": "", "from_community_srt": "co się stanie? W tym momencie tylko sprawdzamy, być może ruszając ołówek pomiędzy kropkami, które narysowaliśmy w brudnopisie.", "n_reviews": 0, @@ -982,7 +982,7 @@ "end": 925.98 }, { - "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so.", + "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction?", "translatedText": "", "from_community_srt": "które zostają stałe. Ale niektórzy z was mogą pewnego dnia napotkać problem, w którym znalezienie niezmiennika pozwoli na błyskotliwe rozwiązanie i może nawet sprawi, że będziecie wyglądali jak geniusze. Jeżeli wymyślony wiatrak przygotuje was na prawdziwy problem, to kogo obchodzi, że to fikcja?", "n_reviews": 0, diff --git a/2019/windmills/portuguese/sentence_translations.json b/2019/windmills/portuguese/sentence_translations.json index c5ab5d682..534d08799 100644 --- a/2019/windmills/portuguese/sentence_translations.json +++ b/2019/windmills/portuguese/sentence_translations.json @@ -513,7 +513,7 @@ "end": 375.24 }, { - "input": "Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper.", + "input": "And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", "translatedText": "Novamente, estamos apenas brincando, talvez movendo o lápis entre os pontos que você desenhou em um pedaço de papel de rascunho.", "model": "google_nmt", "from_community_srt": "o que acontece? E novamente, neste momento você está apenas brincando, talvez movendo o lápis entre os pontos que você desenhou no papel de rascunho.", @@ -1105,7 +1105,7 @@ "end": 925.98 }, { - "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so.", + "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction?", "translatedText": "Mas alguns de vocês que estão assistindo isso um dia enfrentarão um problema em que encontrar um invariante revela uma solução inteligente, e vocês podem até parecer um gênio por fazer isso.", "model": "google_nmt", "from_community_srt": "Mas alguns de vocês assistindo a isso, um dia, vão enfrentar um problema em que encontrar um invariante revela uma boa solução, e você pode até parecer um gênio por fazer isso. Se um moinho de vento planejado te prepara para um verdadeiro problema, por que importaria se ele é ou não uma ficção?", diff --git a/2019/windmills/russian/sentence_translations.json b/2019/windmills/russian/sentence_translations.json index 7934099a5..995135776 100644 --- a/2019/windmills/russian/sentence_translations.json +++ b/2019/windmills/russian/sentence_translations.json @@ -455,7 +455,7 @@ "end": 375.24 }, { - "input": "Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper.", + "input": "And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", "translatedText": "Опять же, мы просто играем, например, водя карандашом по точкам, которые вы нарисовали на листе бумаги для заметок.", "from_community_srt": "что происходит? И снова, сейчас мы просто играемся, возможно, двигая карандашом по точкам на черновой бумаге -- вам надо поверить в ваш вывод,", "n_reviews": 0, @@ -981,7 +981,7 @@ "end": 925.98 }, { - "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so.", + "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction?", "translatedText": "Но некоторые из вас, наблюдающих за этим, однажды столкнутся с проблемой, когда нахождение инварианта дает хорошее решение, и вы можете даже выглядеть гением, делая это.", "from_community_srt": "которые остаются неизменными. Но некоторые из вас, кто смотрят это, однажды, столкнуться с задачей, где нахождение инварианта показывает красивое решение, и вас могут даже принять за гения за это. Если выдуманая мельница готовит вас к настоящим задачам, кого волнует,", "n_reviews": 0, diff --git a/2019/windmills/serbian/sentence_translations.json b/2019/windmills/serbian/sentence_translations.json index b0d54c342..0a229e6a8 100644 --- a/2019/windmills/serbian/sentence_translations.json +++ b/2019/windmills/serbian/sentence_translations.json @@ -456,7 +456,7 @@ "end": 375.24 }, { - "input": "Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper.", + "input": "And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", "translatedText": "", "from_community_srt": "шта се дешава? И опет, сада се још увек играте, можда померајући своју оловку између тачака које сте исцртали на нацрту,", "n_reviews": 0, @@ -982,7 +982,7 @@ "end": 925.98 }, { - "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so.", + "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction?", "translatedText": "", "from_community_srt": "Али, неки од вас који ово гледате, ће једног дана да се суоче са проблемом у којем је налажење непроменљиве величине открити ефективно решење, а можда ћете чак и изгледати као геније тада. Ако вас измишљена ветрењача припреми за прави проблем, кога је брига за то што је то фикција?", "n_reviews": 0, diff --git a/2019/windmills/spanish/sentence_translations.json b/2019/windmills/spanish/sentence_translations.json index 39f90db7b..ee50e8cd2 100644 --- a/2019/windmills/spanish/sentence_translations.json +++ b/2019/windmills/spanish/sentence_translations.json @@ -453,7 +453,7 @@ "end": 375.24 }, { - "input": "Again, we're just playing around, perhaps moving your pencil among dots you've drawn on a piece of scratch paper.", + "input": "And again, at this point we're just playing around, perhaps moving your pencil among dots that you've drawn on a piece of scratch paper.", "translatedText": "Nuevamente, simplemente estamos jugando, tal vez moviendo el lápiz entre los puntos que has dibujado en una hoja de papel borrador.", "from_community_srt": "de nuevo, en este momento solo estamos jugando tal vez moviendo tu lápiz entre puntos que dibujaste en un trozo de papel .", "n_reviews": 0, @@ -975,7 +975,7 @@ "end": 925.98 }, { - "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so.", + "input": "But some of you watching this will one day face a problem where finding an invariant reveals a slick solution, and you might even look like a genius for doing so. If a made-up windmill prepares you for a real problem, who cares that it's a fiction?", "translatedText": "Pero algunos de los que estén viendo esto algún día se enfrentarán a un problema en el que encontrar una invariante revela una solución ingeniosa, e incluso podrían parecer un genio por hacerlo.", "from_community_srt": "algun dia enfrentaran un problema donde encontrar un invariante revelara una solucion escurridiza y, podrias incluso parecer un genio por hacer eso si un molino imaginario te prepara para un problema real", "n_reviews": 0, diff --git a/2019/windmills/tamil/sentence_translations.json b/2019/windmills/tamil/sentence_translations.json index 804c33a52..512c3a216 100644 --- a/2019/windmills/tamil/sentence_translations.json +++ b/2019/windmills/tamil/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "போட்டியே அடிப்படையில் ஒரு சோதனை, இரண்டு நாட்களில் பிரிக்கப்பட்டு, 4க்கு மேல் மூன்று கேள்விகள் கொடுக்கப்பட்டுள்ளன. ஒவ்வொரு நாளும் 5 மணிநேரம். ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "அவை அனைத்தும் கடினமானவை, நிச்சயமாக, இது IMO தான், ஆனால் ஒன்று மற்றும் நான்கு சிக்கல்கள் செய்யக்கூடியதாக இருக்க வேண்டும், இரண்டு மற்றும் ஐந்து சிக்கல்கள் சவாலானவை, மேலும் மூன்று மற்றும் ஆறு சிக்கல்கள் மிருகத்தனமானவை. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/telugu/sentence_translations.json b/2019/windmills/telugu/sentence_translations.json index e409e2f20..d39da6678 100644 --- a/2019/windmills/telugu/sentence_translations.json +++ b/2019/windmills/telugu/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "పోటీ అనేది తప్పనిసరిగా ఒక పరీక్ష, రెండు రోజుల పాటు విభజించబడింది, 4 కంటే మూడు ప్రశ్నలు ఇవ్వబడ్డాయి. ప్రతి రోజు 5 గంటలు. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "అవన్నీ కష్టతరమైనవి, వాస్తవానికి, ఇది IMO, కానీ ఒకటి మరియు నాలుగు సమస్యలు చేయదగినవిగా ఉండాలి, రెండు మరియు ఐదు సమస్యలు సవాలుగా ఉంటాయి మరియు మూడు మరియు ఆరు సమస్యలు క్రూరంగా ఉంటాయి. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/thai/sentence_translations.json b/2019/windmills/thai/sentence_translations.json index 2a2b16ad7..969f4018b 100644 --- a/2019/windmills/thai/sentence_translations.json +++ b/2019/windmills/thai/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/turkish/sentence_translations.json b/2019/windmills/turkish/sentence_translations.json index 396df31d9..fff2b2721 100644 --- a/2019/windmills/turkish/sentence_translations.json +++ b/2019/windmills/turkish/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "Yarışmanın kendisi aslında iki güne bölünmüş ve 4 üzerinden üç sorunun verildiği bir testtir. Her gün 5 saat. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "Bunların hepsi zordur elbette, konu IMO'dur, ancak bir ve dördüncü problemler yapılabilir olmalıdır, ikinci ve beşinci problemler zorludur ve üçüncü ve altıncı problemler acımasız olabilir. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/ukrainian/sentence_translations.json b/2019/windmills/ukrainian/sentence_translations.json index 13e95f5be..ab9f2c8c9 100644 --- a/2019/windmills/ukrainian/sentence_translations.json +++ b/2019/windmills/ukrainian/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "Сам конкурс, по суті, є тестом, який розбитий на два дні, із трьома запитаннями, які даються на 4.5 годин кожного дня. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "Звичайно, вони всі важкі, це IMO, але проблеми один і чотири мають бути здійсненними, проблеми два і п’ять є складними, а проблеми три і шости можуть бути жорстокими. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/urdu/sentence_translations.json b/2019/windmills/urdu/sentence_translations.json index e95ce250d..3df7bfb97 100644 --- a/2019/windmills/urdu/sentence_translations.json +++ b/2019/windmills/urdu/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "مقابلہ بذات خود بنیادی طور پر ایک ٹیسٹ ہے، جو دو دنوں میں تقسیم ہوتا ہے، جس میں 4 سے زیادہ تین سوالات دیئے جاتے ہیں۔ہر دن 5 گھنٹے۔", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "یہ سب مشکل ہیں، یقیناً، یہ IMO ہے، لیکن مسائل ایک اور چار قابل عمل ہونے چاہئیں، مسائل دو اور پانچ چیلنجنگ ہیں، اور مسائل تین اور چھ وحشیانہ ہو سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2019/windmills/vietnamese/sentence_translations.json b/2019/windmills/vietnamese/sentence_translations.json index a6d226468..9447dff11 100644 --- a/2019/windmills/vietnamese/sentence_translations.json +++ b/2019/windmills/vietnamese/sentence_translations.json @@ -16,7 +16,7 @@ "end": 26.1 }, { - "input": "The contest itself is essentially a test, split over two days, with three questions given over 4.5 hours each day. ", + "input": "The contest itself is essentially a test, split over two days, with three questions given over four and a half hours each day. ", "translatedText": "Bản thân cuộc thi về cơ bản là một bài kiểm tra, được chia thành hai ngày, với ba câu hỏi trong tổng số 4 câu hỏi. 5 giờ mỗi ngày. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 245.42 }, { - "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable, problems two and five are challenging, and problems three and six can be brutal. ", + "input": "They're all hard, of course, it's the IMO, but problems one and four should be doable. Problems two and five, they're challenging. And problems three and six, well they can be brutal. ", "translatedText": "Tất nhiên, tất cả đều khó, đó là IMO, nhưng vấn đề một và bốn có thể thực hiện được, vấn đề hai và năm là thách thức, còn vấn đề ba và sáu có thể rất khắc nghiệt. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/better-bayes/english/captions.srt b/2020/better-bayes/english/captions.srt index eb0ba021c..ddb672cf1 100644 --- a/2020/better-bayes/english/captions.srt +++ b/2020/better-bayes/english/captions.srt @@ -83,1326 +83,1330 @@ To see what I'm talking about though, we should really start by spending some time a little more concretely, and just laying out what exactly this paradox is. 22 -00:01:24,020 --> 00:01:27,940 -Picture a thousand women and suppose that 1% of them have breast cancer. +00:01:24,020 --> 00:01:25,922 +1% of women have breast cancer Picture a thousand 23 +00:01:25,922 --> 00:01:27,940 +women and suppose that 1% of them have breast cancer. + +24 00:01:28,680 --> 00:01:31,959 And let's say they all undergo a certain breast cancer screening, -24 +25 00:01:31,959 --> 00:01:35,139 and that 9 of those with cancer correctly get positive results, -25 +26 00:01:35,139 --> 00:01:36,680 and there's one false negative. -26 +27 00:01:37,480 --> 00:01:41,046 And then suppose that among the remainder without cancer, -27 +28 00:01:41,046 --> 00:01:44,920 89 get false positives, and 901 correctly get negative results. -28 +29 00:01:45,720 --> 00:01:50,081 So if all you know about a woman is that she does the screening and she gets a positive -29 +30 00:01:50,081 --> 00:01:53,700 result, you don't have information about symptoms or anything like that, -30 +31 00:01:53,700 --> 00:01:57,764 you know that she's either one of these 9 true positives or one of these 89 false -31 +32 00:01:57,764 --> 00:01:58,260 positives. -32 +33 00:01:59,360 --> 00:02:03,750 So the probability that she's in the cancer group given the test -33 +34 00:02:03,750 --> 00:02:08,139 result is 9 divided by 9 plus 89, which is approximately 1 in 11. -34 +35 00:02:09,080 --> 00:02:13,685 In medical parlance, you would call this the positive predictive value of the test, -35 +36 00:02:13,685 --> 00:02:18,620 or PPV, the number of true positives divided by the total number of positive test results. -36 +37 00:02:18,620 --> 00:02:20,440 You can see where the name comes from. -37 +38 00:02:20,740 --> 00:02:25,360 To what extent does a positive test result actually predict that you have the disease? -38 +39 00:02:26,820 --> 00:02:30,067 Now, hopefully, as I've presented it this way where we're thinking -39 +40 00:02:30,067 --> 00:02:33,460 concretely about a sample population, all of this makes perfect sense. -40 +41 00:02:33,960 --> 00:02:37,136 But where it comes across as counterintuitive is if you just look -41 +42 00:02:37,136 --> 00:02:40,312 at the accuracy of the test, present it to people as a statistic, -42 +43 00:02:40,312 --> 00:02:43,200 and then ask them to make judgments about their test result. -43 +44 00:02:44,020 --> 00:02:46,260 Test accuracy is not actually one number, but two. -44 +45 00:02:46,260 --> 00:02:51,120 First, you ask how often is the test correct on those with the disease. -45 +46 00:02:51,700 --> 00:02:54,353 This is known as the test sensitivity, as in how -46 +47 00:02:54,353 --> 00:02:57,440 sensitive is it to detecting the presence of the disease. -47 +48 00:02:58,260 --> 00:03:01,260 In our example, test sensitivity is 9 in 10, or 90%. -48 +49 00:03:02,020 --> 00:03:06,680 And another way to say the same fact would be to say the false negative rate is 10%. -49 +50 00:03:06,680 --> 00:03:11,711 And then a separate, not necessarily related number is how often it's correct for those -50 +51 00:03:11,711 --> 00:03:15,199 without the disease, which is known as the test specificity, -51 +52 00:03:15,199 --> 00:03:18,801 as in are positive results caused specifically by the disease, -52 +53 00:03:18,801 --> 00:03:22,060 or are there confounding triggers giving false positives. -53 +54 00:03:23,080 --> 00:03:26,580 In our example, the specificity is about 91%. -54 +55 00:03:26,580 --> 00:03:31,660 Or another way to say the same fact would be to say the false positive rate is 9%. -55 +56 00:03:31,660 --> 00:03:36,760 So the paradox here is that in one sense, the test is over 90% accurate. -56 +57 00:03:37,020 --> 00:03:40,660 It gives correct results to over 90% of the patients who take it. -57 +58 00:03:40,660 --> 00:03:45,396 And yet, if you learn that someone gets a positive result without any added information, -58 +59 00:03:45,396 --> 00:03:49,600 there's actually only a 1 in 11 chance that that particular result is accurate. -59 +60 00:03:50,620 --> 00:03:54,837 This is a bit of a problem, because of all of the places for math to be counterintuitive, -60 +61 00:03:54,837 --> 00:03:57,180 medical tests are one area where it matters a lot. -61 +62 00:03:57,940 --> 00:04:02,370 In 2006 and 2007, the psychologist Gerd Gigerenzer gave a series of statistics -62 +63 00:04:02,370 --> 00:04:06,800 seminars to practicing gynecologists, and he opened with the following example. -63 +64 00:04:06,800 --> 00:04:11,740 A 50-year-old woman, no symptoms, participates in a routine mammography screening. -64 +65 00:04:12,280 --> 00:04:15,159 She tests positive, is alarmed, and wants to know from you -65 +66 00:04:15,159 --> 00:04:18,380 whether she has breast cancer for certain or what her chances are. -66 +67 00:04:18,880 --> 00:04:21,740 Apart from the screening result, you know nothing else about this woman. -67 +68 00:04:22,580 --> 00:04:26,427 In that seminar, the doctors were then told that the prevalence of -68 +69 00:04:26,427 --> 00:04:29,241 breast cancer for women of this age is about 1%, -69 +70 00:04:29,241 --> 00:04:34,180 and then to suppose that the test sensitivity is 90% and that its specificity was 91%. -70 +71 00:04:34,180 --> 00:04:36,281 You might notice these are exactly the same numbers -71 +72 00:04:36,281 --> 00:04:38,180 from the example that you and I just looked at. -72 +73 00:04:38,360 --> 00:04:39,440 This is where I got them. -73 +74 00:04:39,760 --> 00:04:42,600 So, having already thought it through, you and I know the answer. -74 +75 00:04:42,880 --> 00:04:43,840 It's about 1 in 11. -75 +76 00:04:44,600 --> 00:04:47,981 However, the doctors in this session were not primed with the suggestion to -76 +77 00:04:47,981 --> 00:04:51,540 picture a concrete sample of a thousand individuals, the way that you and I had. -77 +78 00:04:52,040 --> 00:04:53,340 All they saw were these numbers. -78 +79 00:04:54,140 --> 00:04:58,420 They were then asked, how many women who test positive actually have breast cancer? -79 +80 00:04:58,620 --> 00:04:59,740 What is the best answer? -80 +81 00:04:59,900 --> 00:05:01,680 And they were presented with these four choices. -81 +82 00:05:01,680 --> 00:05:05,321 In one of the sessions, over half the doctors present -82 +83 00:05:05,321 --> 00:05:09,300 said that the correct answer was 9 in 10, which is way off. -83 +84 00:05:10,020 --> 00:05:12,656 Only a fifth of them gave the correct answer, which is worse -84 +85 00:05:12,656 --> 00:05:15,380 than what it would have been if everybody had randomly guessed. -85 +86 00:05:16,660 --> 00:05:19,280 It might seem a little extreme to be calling this a paradox. -86 +87 00:05:19,760 --> 00:05:21,140 I mean, it's just a fact. -87 +88 00:05:21,260 --> 00:05:23,500 It's not something intrinsically self-contradictory. -88 +89 00:05:24,200 --> 00:05:28,216 But, as these seminars with Gigerenzer show, people, including doctors, -89 +90 00:05:28,216 --> 00:05:33,068 definitely find it counterintuitive that a test with high accuracy can give you such a -90 +91 00:05:33,068 --> 00:05:34,240 low predictive value. -91 +92 00:05:35,200 --> 00:05:40,098 We might call this a veridical paradox, which refers to facts that are provably true, -92 +93 00:05:40,098 --> 00:05:43,800 but which nevertheless can feel false when phrased a certain way. -93 +94 00:05:44,300 --> 00:05:46,648 It's sort of the softest form of a paradox, saying -94 +95 00:05:46,648 --> 00:05:48,720 more about human psychology than about logic. -95 +96 00:05:49,580 --> 00:05:51,980 The question is how we can combat this. -96 +97 00:05:53,800 --> 00:05:57,246 Where we're going with this, by the way, is that I want you to be able -97 +98 00:05:57,246 --> 00:06:00,693 to look at numbers like this and quickly estimate in your head that it -98 +99 00:06:00,693 --> 00:06:04,140 means the predictive value of a positive test should be around 1 in 11. -99 +100 00:06:04,760 --> 00:06:07,187 Or, if I changed things and asked, what if it -100 +101 00:06:07,187 --> 00:06:09,720 was 10% of the population who had breast cancer? -101 +102 00:06:10,120 --> 00:06:12,574 You should be able to quickly turn around and say -102 +103 00:06:12,574 --> 00:06:14,980 that the final answer would be a little over 50%. -103 +104 00:06:15,920 --> 00:06:18,504 Or, if I said imagine a really low prevalence, -104 +105 00:06:18,504 --> 00:06:21,088 something like 0.1% of patients having cancer, -105 +106 00:06:21,088 --> 00:06:25,871 you should again quickly estimate that the predictive value of the test is around 1 in -106 +107 00:06:25,871 --> 00:06:30,600 100, that 1 in 100 of those with positive test results in that case would have cancer. -107 +108 00:06:31,580 --> 00:06:35,240 Or, let's say we go back to the 1% prevalence, but I make the test more accurate. -108 +109 00:06:35,440 --> 00:06:38,400 I tell you to imagine the specificity is 99%. -109 +110 00:06:38,400 --> 00:06:41,018 There, you should be able to relatively quickly -110 +111 00:06:41,018 --> 00:06:43,800 estimate that the answer is a little less than 50%. -111 +112 00:06:44,320 --> 00:06:47,740 The hope is that you're doing all of this with minimal calculations in your head. -112 +113 00:06:48,540 --> 00:06:52,266 Now, the goals of quick calculations might feel very different from the goals of -113 +114 00:06:52,266 --> 00:06:54,935 addressing whatever misconception underlies this paradox, -114 +115 00:06:54,935 --> 00:06:56,500 but they actually go hand in hand. -115 +116 00:06:56,900 --> 00:06:57,680 Let me show you what I mean. -116 +117 00:06:58,460 --> 00:07:01,220 On the side of addressing misconceptions, what would you -117 +118 00:07:01,220 --> 00:07:03,980 tell to the people in that seminar who answered 9 and 10? -118 +119 00:07:04,480 --> 00:07:06,900 What fundamental misconception are they revealing? -119 +120 00:07:08,180 --> 00:07:11,699 What I might tell them is that in much the same way that you shouldn't think -120 +121 00:07:11,699 --> 00:07:14,898 of tests as telling you deterministically whether you have a disease, -121 +122 00:07:14,898 --> 00:07:18,600 you shouldn't even think of them as telling you your chances of having a disease. -122 +123 00:07:19,560 --> 00:07:24,460 Instead, the healthy view of what tests do is that they update your chances. -123 +124 00:07:26,040 --> 00:07:28,691 In our example, before taking the test, a patient's -124 +125 00:07:28,691 --> 00:07:30,680 chances of having cancer were 1 in 100. -125 +126 00:07:31,120 --> 00:07:33,640 In Bayesian terms, we call this the prior probability. -126 +127 00:07:34,380 --> 00:07:39,140 The effect of this test was to update that prior by almost an order of magnitude, -127 +128 00:07:39,140 --> 00:07:40,360 up to around 1 in 11. -128 +129 00:07:41,020 --> 00:07:44,820 The accuracy of a test is telling us about the strength of this updating. -129 +130 00:07:45,120 --> 00:07:46,740 It's not telling us a final answer. -130 +131 00:07:47,900 --> 00:07:49,640 What does this have to do with quick approximations? -131 +132 00:07:50,300 --> 00:07:54,783 Well, a key number for those approximations is something called the Bayes factor, -132 +133 00:07:54,783 --> 00:07:58,392 and the very act of defining this number serves to reinforce this -133 +134 00:07:58,392 --> 00:08:01,400 central lesson about reframing what it is the tests do. -134 +135 00:08:02,420 --> 00:08:05,593 You see, one of the things that makes test statistics so very confusing -135 +136 00:08:05,593 --> 00:08:08,900 is that there are at least 4 numbers that you'll hear associated with them. -136 +137 00:08:08,900 --> 00:08:12,341 For those with the disease, there's the sensitivity and the false negative rate, -137 +138 00:08:12,341 --> 00:08:15,783 and then for those without, there's the specificity and the false positive rate, -138 +139 00:08:15,783 --> 00:08:18,800 and none of these numbers actually tell you the thing you want to know. -139 +140 00:08:19,500 --> 00:08:22,533 Luckily, if you want to interpret a positive test result, -140 +141 00:08:22,533 --> 00:08:25,620 you can pull out just one number to focus on from all this. -141 +142 00:08:26,040 --> 00:08:28,600 Take the sensitivity divided by the false positive rate. -142 +143 00:08:29,160 --> 00:08:31,950 In other words, how much more likely are you to see -143 +144 00:08:31,950 --> 00:08:34,740 the positive test result with cancer versus without? -144 +145 00:08:34,740 --> 00:08:37,140 In our example, this number is 10. -145 +146 00:08:37,900 --> 00:08:41,720 This is the Bayes factor, also sometimes called the likelihood ratio. -146 +147 00:08:43,100 --> 00:08:46,023 A very handy rule of thumb is that to update a small prior, -147 +148 00:08:46,023 --> 00:08:50,020 or at least to approximate the answer, you simply multiply it by the Bayes factor. -148 +149 00:08:50,760 --> 00:08:53,056 So in our example, where the prior was 1 in 100, -149 +150 00:08:53,056 --> 00:08:56,195 you would estimate that the final answer should be around 1 in 10, -150 +151 00:08:56,195 --> 00:08:58,820 which is in fact slightly above the true correct answer. -151 +152 00:08:59,400 --> 00:09:03,229 So based on this rule of thumb, if I asked you what would happen if the -152 +153 00:09:03,229 --> 00:09:07,111 prior from our example was instead 1 in 1000, you could quickly estimate -153 +154 00:09:07,111 --> 00:09:11,420 that the effect of the test should be to update those chances to around 1 in 100. -154 +155 00:09:12,360 --> 00:09:15,720 And in fact, take a moment to check yourself by thinking through a sample population. -155 +156 00:09:16,700 --> 00:09:20,880 In this case, you might picture 10,000 patients where only 10 of them really have cancer. -156 +157 00:09:22,140 --> 00:09:24,880 And then based on that 90% sensitivity, we would -157 +158 00:09:24,880 --> 00:09:27,900 expect 9 of those cancer cases to give true positives. -158 +159 00:09:29,000 --> 00:09:32,285 And on the other side, a 91% specificity means that -159 +160 00:09:32,285 --> 00:09:35,760 9% of those without cancer are getting false positives. -160 +161 00:09:36,660 --> 00:09:40,198 So we'd expect 9% of the remaining patients, which is around 900, -161 +162 00:09:40,198 --> 00:09:41,860 to give false positive results. -162 +163 00:09:42,700 --> 00:09:45,705 Here, with such a low prevalence, the false positives -163 +164 00:09:45,705 --> 00:09:47,820 really do dominate the true positives. -164 +165 00:09:47,900 --> 00:09:52,460 So the probability that a randomly chosen positive case from this population -165 +166 00:09:52,460 --> 00:09:57,020 actually has cancer is only around 1%, just like the rule of thumb predicted. -166 +167 00:09:58,700 --> 00:10:01,920 Now, this rule of thumb clearly cannot work for higher priors. -167 +168 00:10:02,420 --> 00:10:05,110 For example, it would predict that a prior of -168 +169 00:10:05,110 --> 00:10:07,860 10% gets updated all the way to 100% certainty. -169 +170 00:10:08,360 --> 00:10:09,320 But that can't be right. -170 +171 00:10:10,020 --> 00:10:13,021 In fact, take a moment to think through what the answer should be, -171 +172 00:10:13,021 --> 00:10:14,500 again, using a sample population. -172 +173 00:10:15,060 --> 00:10:17,860 Maybe this time we picture 10 out of 100 having cancer. -173 +174 00:10:18,540 --> 00:10:21,282 Again, based on the 90% sensitivity of the test, -174 +175 00:10:21,282 --> 00:10:24,920 we'd expect 9 of those true cancer cases to get positive results. -175 +176 00:10:24,920 --> 00:10:26,600 But what about the false positives? -176 +177 00:10:26,980 --> 00:10:28,100 How many do we expect there? -177 +178 00:10:29,880 --> 00:10:31,740 About 9% of the remaining 90. -178 +179 00:10:32,120 --> 00:10:32,620 About 8. -179 +180 00:10:33,820 --> 00:10:37,370 So, upon seeing a positive test result, it tells you that you're -180 +181 00:10:37,370 --> 00:10:41,140 either one of these 9 true positives or one of the 8 false positives. -181 +182 00:10:41,860 --> 00:10:46,920 So this means the chances are a little over 50%, roughly 9 out of 17, or 53%. -182 +183 00:10:48,020 --> 00:10:51,201 At this point, having dared to dream that Bayesian updating could look -183 +184 00:10:51,201 --> 00:10:54,697 as simple as multiplication, you might tear down your hopes and pragmatically -184 +185 00:10:54,697 --> 00:10:57,700 acknowledge that sometimes life is just more complicated than that. -185 +186 00:10:59,920 --> 00:11:01,120 Except it's not. -186 +187 00:11:01,620 --> 00:11:05,259 This rule of thumb turns into a precise mathematical fact as long as we -187 +188 00:11:05,259 --> 00:11:09,000 shift away from talking about probabilities to instead talking about odds. -188 +189 00:11:10,320 --> 00:11:14,746 If you've ever heard someone talk about the chances of an event being 1 to 1 or 2 to 1, -189 +190 00:11:14,746 --> 00:11:17,060 things like that, you already know about odds. -190 +191 00:11:17,060 --> 00:11:20,173 With probability, we're taking the ratio of the number -191 +192 00:11:20,173 --> 00:11:23,060 of positive cases out of all possible cases, right? -192 +193 00:11:23,400 --> 00:11:25,280 Things like 1 in 5 or 1 in 10. -193 +194 00:11:25,880 --> 00:11:30,320 With odds, what you do is take the ratio of all positive cases to all negative cases. -194 +195 00:11:31,540 --> 00:11:34,926 You commonly see odds written with a colon to emphasize the distinction, -195 +196 00:11:34,926 --> 00:11:37,060 but it's still just a fraction, just a number. -196 +197 00:11:37,940 --> 00:11:42,520 So an event with a 50% probability would be described as having 1 to 1 odds. -197 +198 00:11:42,520 --> 00:11:46,160 A 10% probability is the same as 1 to 9 odds. -198 +199 00:11:46,760 --> 00:11:49,500 An 80% probability is the same as 4 to 1 odds. -199 +200 00:11:49,940 --> 00:11:50,460 You get the point. -200 +201 00:11:51,480 --> 00:11:52,400 It's the same information. -201 +202 00:11:52,740 --> 00:11:55,073 It still describes the chances of a random event, -202 +203 00:11:55,073 --> 00:11:58,340 but it's presented a little differently, like a different unit system. -203 +204 00:11:59,320 --> 00:12:03,680 Probabilities are constrained between 0 and 1, with even chances sitting at 0.5. -204 +205 00:12:04,800 --> 00:12:09,540 But odds range from 0 up to infinity, with even chances sitting at the number 1. -205 +206 00:12:11,880 --> 00:12:14,500 The beauty here is that a completely accurate, -206 +207 00:12:14,500 --> 00:12:18,179 not even approximating things way to frame Bayes' rule is to say, -207 +208 00:12:18,179 --> 00:12:22,360 express your prior using odds, and then just multiply by the Bayes' factor. -208 +209 00:12:23,440 --> 00:12:25,220 Think about what the prior odds are really saying. -209 +210 00:12:25,580 --> 00:12:29,260 It's the number of people with cancer divided by the number without it. -210 +211 00:12:29,700 --> 00:12:33,360 Here, let's just write that down as a normal fraction for a moment so we can multiply it. -211 +212 00:12:33,360 --> 00:12:36,709 When you filter down just to those with positive test results, -212 +213 00:12:36,709 --> 00:12:39,421 the number of people with cancer gets scaled down, -213 +214 00:12:39,421 --> 00:12:43,143 scaled down by the probability of seeing a positive test result given -214 +215 00:12:43,143 --> 00:12:44,420 that someone has cancer. -215 +216 00:12:45,120 --> 00:12:49,253 And then similarly, the number of people without cancer also gets scaled down, -216 +217 00:12:49,253 --> 00:12:53,440 this time by the probability of seeing a positive test result, but in that case. -217 +218 00:12:54,180 --> 00:12:58,603 So the ratio between these two counts, the new odds upon seeing the test, -218 +219 00:12:58,603 --> 00:13:02,667 looks just like the prior odds except multiplied by this term here, -219 +220 00:13:02,667 --> 00:13:04,760 which is exactly the Bayes' factor. -220 +221 00:13:07,800 --> 00:13:10,500 Look back at our example, where the Bayes' factor was 10. -221 +222 00:13:11,000 --> 00:13:14,263 And as a reminder, this came from the 90% sensitivity -222 +223 00:13:14,263 --> 00:13:16,560 divided by the 9% false positive rate. -223 +224 00:13:16,880 --> 00:13:20,740 How much more likely are you to see a positive result with cancer versus without? -224 +225 00:13:21,720 --> 00:13:25,940 If the prior is 1%, expressed as odds, this looks like 1 to 99. -225 +226 00:13:26,900 --> 00:13:29,809 So by our rule, this gets updated to 10 to 99, -226 +227 00:13:29,809 --> 00:13:33,400 which if you want you could convert back to a probability. -227 +228 00:13:33,660 --> 00:13:37,220 It would be 10 divided by 10 plus 99, or about 1 in 11. -228 +229 00:13:38,200 --> 00:13:42,230 If instead, the prior was 10%, which was the example that tripped up -229 +230 00:13:42,230 --> 00:13:46,260 our rule of thumb earlier, expressed as odds, this looks like 1 to 9. -230 +231 00:13:46,940 --> 00:13:49,690 By our simple rule, this gets updated to 10 to 9, -231 +232 00:13:49,690 --> 00:13:52,440 which you can already read off pretty intuitively. -232 +233 00:13:52,440 --> 00:13:55,660 It's a little above even chances, a little above 1 to 1. -233 +234 00:13:56,340 --> 00:13:58,840 If you prefer, you can convert it back to a probability. -234 +235 00:13:59,180 --> 00:14:03,280 You would write it as 10 out of 19, or about 53%. -235 +236 00:14:03,280 --> 00:14:05,543 And indeed, that is what we already found by thinking -236 +237 00:14:05,543 --> 00:14:07,220 things through with a sample population. -237 +238 00:14:08,300 --> 00:14:11,700 Let's say we go back to the 1% prevalence, but I make the test more accurate. -238 +239 00:14:12,060 --> 00:14:16,640 Now what if I told you to imagine that the false positive rate was only 1% instead of 9%? -239 +240 00:14:17,120 --> 00:14:20,520 What that would mean is that our Bayes factor is 90 instead of 10. -240 +241 00:14:20,840 --> 00:14:22,460 The test is doing more work for us. -241 +242 00:14:23,160 --> 00:14:27,399 In this case, with the more accurate test, it gets updated to 90 to 99, -242 +243 00:14:27,399 --> 00:14:31,580 which is a little less than even chances, something a little under 50%. -243 +244 00:14:31,580 --> 00:14:34,454 To be more precise, you could make the conversion -244 +245 00:14:34,454 --> 00:14:37,560 back to probability and work out that it's around 48%. -245 +246 00:14:37,560 --> 00:14:41,400 But honestly, if you're just going for a gut feel, it's fine to stick with the odds. -246 +247 00:14:42,220 --> 00:14:44,934 Do you see what I mean about how just defining this -247 +248 00:14:44,934 --> 00:14:47,440 number helps to combat potential misconceptions? -248 +249 00:14:48,240 --> 00:14:52,925 For anybody who's a little hasty in connecting test accuracy directly to your probability -249 +250 00:14:52,925 --> 00:14:57,558 of having a disease, it's worth emphasizing that you could administer the same test with -250 +251 00:14:57,558 --> 00:15:01,930 the same accuracy to multiple different patients who all get the same exact result, -251 +252 00:15:01,930 --> 00:15:04,377 but if they're coming from different contexts, -252 +253 00:15:04,377 --> 00:15:06,720 that result can mean wildly different things. -253 +254 00:15:06,720 --> 00:15:10,690 However, the one thing that does stay constant in every case -254 +255 00:15:10,690 --> 00:15:14,660 is the factor by which each patient's prior odds get updated. -255 +256 00:15:16,300 --> 00:15:20,061 And by the way, this whole time we've been using the prevalence of the disease, -256 +257 00:15:20,061 --> 00:15:23,024 which is the proportion of people in a population who have it, -257 +258 00:15:23,024 --> 00:15:26,880 as a substitute for the prior, the probability of having it before you see a test. -258 +259 00:15:27,520 --> 00:15:29,460 However, that's not necessarily the case. -259 +260 00:15:29,780 --> 00:15:32,493 If there are other known factors, things like symptoms, -260 +261 00:15:32,493 --> 00:15:35,789 or in the case of a contagious disease, things like known contacts, -261 +262 00:15:35,789 --> 00:15:39,860 those also factor into the prior, and they could potentially make a huge difference. -262 +263 00:15:40,760 --> 00:15:44,449 As another side note, so far we've only talked about positive test results, -263 +264 00:15:44,449 --> 00:15:47,460 but way more often you would be seeing a negative test result. -264 +265 00:15:48,100 --> 00:15:50,250 The logic there is completely the same, but the base -265 +266 00:15:50,250 --> 00:15:52,320 factor that you compute is going to look different. -266 +267 00:15:52,760 --> 00:15:55,825 Instead, you look at the probability of seeing this negative -267 +268 00:15:55,825 --> 00:15:58,640 test result with the disease versus without the disease. -268 +269 00:15:58,640 --> 00:16:02,805 So in our cancer example, this would have been the 10% false -269 +270 00:16:02,805 --> 00:16:07,040 negative rate divided by the 91% specificity, or about 1 in 9. -270 +271 00:16:07,780 --> 00:16:11,174 In other words, seeing a negative test result in that example -271 +272 00:16:11,174 --> 00:16:14,460 would reduce your prior odds by about an order of magnitude. -272 +273 00:16:15,900 --> 00:16:18,420 When you write it all out as a formula, here's how it looks. -273 +274 00:16:18,760 --> 00:16:22,949 It says your odds of having a disease given a test result equals your -274 +275 00:16:22,949 --> 00:16:26,960 odds before taking the test, the prior odds, times the base factor. -275 +276 00:16:26,960 --> 00:16:30,546 Now let's contrast this with the usual way Bayes' rule is written, -276 +277 00:16:30,546 --> 00:16:32,260 which is a bit more complicated. -277 +278 00:16:33,060 --> 00:16:36,002 In case you haven't seen it before, it's essentially just what we were -278 +279 00:16:36,002 --> 00:16:38,780 doing with sample populations, but you wrap it all up symbolically. -279 +280 00:16:39,500 --> 00:16:42,880 Remember how every time we were counting the number of true positives and -280 +281 00:16:42,880 --> 00:16:46,260 then dividing it by the sum of the true positives and the false positives? -281 +282 00:16:46,800 --> 00:16:50,317 We do just that, except instead of talking about absolute amounts, -282 +283 00:16:50,317 --> 00:16:52,260 we talk of each term as a proportion. -283 +284 00:16:52,260 --> 00:16:55,503 So the proportion of true positives in the population comes -284 +285 00:16:55,503 --> 00:16:58,746 from the prior probability of having the disease multiplied -285 +286 00:16:58,746 --> 00:17:02,260 by the probability of seeing a positive test result in that case. -286 +287 00:17:03,000 --> 00:17:05,932 Then we copy that term down again into the denominator, -287 +288 00:17:05,932 --> 00:17:09,283 and then the proportion of false positives comes from the prior -288 +289 00:17:09,283 --> 00:17:13,157 probability of not having the disease times the probability of a positive -289 +290 00:17:13,157 --> 00:17:14,099 test in that case. -290 +291 00:17:15,079 --> 00:17:18,049 If you want, you could also write this down with words instead of symbols, -291 +292 00:17:18,049 --> 00:17:20,859 if terms like sensitivity and false positive rate are more comfortable. -292 +293 00:17:21,380 --> 00:17:24,910 And this is one of those formulas where once you say it out loud it seems like a bit -293 +294 00:17:24,910 --> 00:17:28,400 much, but it really is no different from what we were doing with sample populations. -294 +295 00:17:29,220 --> 00:17:31,686 If you wanted to make the whole thing look simpler, -295 +296 00:17:31,686 --> 00:17:35,576 you often see this entire denominator written just as the probability of seeing a -296 +297 00:17:35,576 --> 00:17:37,000 positive test result, overall. -297 +298 00:17:37,980 --> 00:17:40,795 While that does make for a really elegant little expression, -298 +299 00:17:40,795 --> 00:17:44,118 if you intend to use this for calculations, it's a little disingenuous, -299 +300 00:17:44,118 --> 00:17:47,303 because in practice, every single time you do this you need to break -300 +301 00:17:47,303 --> 00:17:50,580 down that denominator into two separate parts, breaking down the cases. -301 +302 00:17:51,700 --> 00:17:53,928 So taking this more honest representation of it, -302 +303 00:17:53,928 --> 00:17:56,020 let's compare our two versions of Bayes' rule. -303 +304 00:17:56,820 --> 00:18:00,280 And again, maybe it looks nicer if we use the words sensitivity and false positive rate. -304 +305 00:18:00,660 --> 00:18:03,062 If nothing else, it helps emphasize which parts of the -305 +306 00:18:03,062 --> 00:18:05,640 formula are coming from statistics about the test accuracy. -306 +307 00:18:05,640 --> 00:18:09,083 I mean, this actually emphasizes one thing I really like about the framing with -307 +308 00:18:09,083 --> 00:18:12,440 odds and a Bayes' factor, which is that it cleanly factors out the parts that -308 +309 00:18:12,440 --> 00:18:15,840 have to do with the prior and the parts that have to do with the test accuracy. -309 +310 00:18:16,660 --> 00:18:20,200 But over in the usual formula, all of those are very intermingled together. -310 +311 00:18:20,580 --> 00:18:22,360 And this has a very practical benefit. -311 +312 00:18:22,480 --> 00:18:26,260 It's really nice if you want to swap out different priors and easily see their effects. -312 +313 00:18:26,600 --> 00:18:27,900 This is what we were doing earlier. -313 +314 00:18:28,420 --> 00:18:32,200 But with the other formula, to do that, you have to recompute everything each time. -314 +315 00:18:32,380 --> 00:18:35,360 You can't leverage a precomputed Bayes' factor the same way. -315 +316 00:18:35,960 --> 00:18:38,861 The odds framing also makes things really nice if you want to do -316 +317 00:18:38,861 --> 00:18:42,120 multiple different Bayesian updates based on multiple pieces of evidence. -317 +318 00:18:42,740 --> 00:18:44,860 For example, let's say you took not one test, but two. -318 +319 00:18:45,360 --> 00:18:48,540 Or you wanted to think about how the presence of symptoms plays into it. -319 +320 00:18:49,120 --> 00:18:52,374 For each piece of new evidence you see, you always ask the question, -320 +321 00:18:52,374 --> 00:18:56,620 how much more likely would you be to see that with the disease versus without the disease? -321 +322 00:18:57,240 --> 00:19:00,012 Each answer to that question gives you a new Bayes' factor, -322 +323 00:19:00,012 --> 00:19:02,000 a new thing that you multiply by your odds. -323 +324 00:19:02,880 --> 00:19:06,400 Beyond just making calculations easier, there's something I really like about -324 +325 00:19:06,400 --> 00:19:09,920 attaching a number to test accuracy that doesn't even look like a probability. -325 +326 00:19:10,740 --> 00:19:13,379 I mean, if you hear that a test has, for example, -326 +327 00:19:13,379 --> 00:19:17,340 a 9% false positive rate, that's just such a disastrously ambiguous phrase. -327 +328 00:19:17,780 --> 00:19:20,179 It's so easy to misinterpret it to mean there's a -328 +329 00:19:20,179 --> 00:19:22,580 9% chance that your positive test result is false. -329 +330 00:19:23,300 --> 00:19:26,609 But imagine if instead the number that we heard tacked on to test -330 +331 00:19:26,609 --> 00:19:30,320 results was that the Bayes' factor for a positive test result is, say, 10. -331 +332 00:19:30,820 --> 00:19:34,140 There's no room to confuse that for your probability of having a disease. -332 +333 00:19:34,640 --> 00:19:36,912 The entire framing of what a Bayes' factor is, -333 +334 00:19:36,912 --> 00:19:39,040 is that it's something that acts on a prior. -334 +335 00:19:39,500 --> 00:19:43,308 It forces your hand to acknowledge the prior as something that's separate entirely, -335 +336 00:19:43,308 --> 00:19:45,440 and highly necessary to drawing any conclusion. -336 +337 00:19:47,260 --> 00:19:50,740 All that said, the usual formula is definitely not without its merits. -337 +338 00:19:51,080 --> 00:19:53,983 If you view it not simply as something to plug numbers into, -338 +339 00:19:53,983 --> 00:19:58,172 but as an encapsulation of the sample population idea that we've been using throughout, -339 +340 00:19:58,172 --> 00:20:01,980 you could very easily argue that that's actually much better for your intuition. -340 +341 00:20:02,560 --> 00:20:05,718 After all, it's what we were routinely falling back on in order to check -341 +342 00:20:05,718 --> 00:20:09,180 ourselves that the Bayes' factor computation even made sense in the first place. -342 +343 00:20:11,600 --> 00:20:15,380 Like any design decision, there is no clear-cut objective best here. -343 +344 00:20:15,420 --> 00:20:18,502 But it's almost certainly the case that giving serious consideration -344 +345 00:20:18,502 --> 00:20:21,720 to that question will lead you to a better understanding of Bayes' rule. -345 +346 00:20:30,100 --> 00:20:32,906 Also, since we're on the topic of kind of paradoxical things, -346 +347 00:20:32,906 --> 00:20:36,120 a friend of mine, Matt Cook, recently wrote a book all about paradoxes. -347 +348 00:20:37,040 --> 00:20:39,450 I actually contributed a small chapter to it with thoughts -348 +349 00:20:39,450 --> 00:20:41,820 on the question of whether math is invented or discovered. -349 +350 00:20:42,300 --> 00:20:45,635 And the book as a whole is this really nice connection of thought-provoking -350 +351 00:20:45,635 --> 00:20:48,400 paradoxical things ranging from philosophy to math and physics. -351 +352 00:20:48,820 --> 00:20:51,040 You can, of course, find all the details in the description. -352 +353 00:20:58,100 --> 00:20:51,040 . diff --git a/2020/better-bayes/english/sentence_timings.json b/2020/better-bayes/english/sentence_timings.json index 4137edd57..0b3c8a0f0 100644 --- a/2020/better-bayes/english/sentence_timings.json +++ b/2020/better-bayes/english/sentence_timings.json @@ -50,7 +50,7 @@ 78.1 ], [ - "Picture a thousand women and suppose that 1% of them have breast cancer.", + "1% of women have breast cancer Picture a thousand women and suppose that 1% of them have breast cancer.", 84.02, 87.94 ], diff --git a/2020/better-bayes/english/transcript.txt b/2020/better-bayes/english/transcript.txt index 4f69ad890..2a44e4af5 100644 --- a/2020/better-bayes/english/transcript.txt +++ b/2020/better-bayes/english/transcript.txt @@ -8,7 +8,7 @@ I mean, maybe in the case of notation it's easy to see that different choices ar In this video, you and I will dig into this paradox, but instead of using it to talk about the usual version of Bayes' rule, I'd like to motivate an alternate version, an alternate design choice. Now, what's up on the screen now is a little bit abstract, which makes it difficult to justify that there really is a substantive difference here, especially when I haven't explained either one yet. To see what I'm talking about though, we should really start by spending some time a little more concretely, and just laying out what exactly this paradox is. -Picture a thousand women and suppose that 1% of them have breast cancer. +1% of women have breast cancer Picture a thousand women and suppose that 1% of them have breast cancer. And let's say they all undergo a certain breast cancer screening, and that 9 of those with cancer correctly get positive results, and there's one false negative. And then suppose that among the remainder without cancer, 89 get false positives, and 901 correctly get negative results. So if all you know about a woman is that she does the screening and she gets a positive result, you don't have information about symptoms or anything like that, you know that she's either one of these 9 true positives or one of these 89 false positives. diff --git a/2020/better-bayes/vietnamese/sentence_translations.json b/2020/better-bayes/vietnamese/sentence_translations.json index c17f858ea..4042b897e 100644 --- a/2020/better-bayes/vietnamese/sentence_translations.json +++ b/2020/better-bayes/vietnamese/sentence_translations.json @@ -1575,4 +1575,4 @@ "start": 1248.82, "end": 1251.04 } -] +] \ No newline at end of file diff --git a/2020/binomial-distributions/arabic/sentence_translations.json b/2020/binomial-distributions/arabic/sentence_translations.json index 9ffb0315c..850283b7c 100644 --- a/2020/binomial-distributions/arabic/sentence_translations.json +++ b/2020/binomial-distributions/arabic/sentence_translations.json @@ -209,7 +209,7 @@ "end": 158.66 }, { - "input": "This is something known as Laplace's rule of succession, and to understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", + "input": "This is something known as Laplace's rule of succession, it dates back to the 18th century, To understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", "translatedText": "هذا ما يُعرف بقاعدة لابلاس للخلافة، ولكي نفهم ما هي الافتراضات الكامنة وراء ذلك، وكيف أن تغيير تلك الافتراضات أو هدفك الأساسي يمكن أن يغير الاختيار الذي تقوم به، نحن حقًا بحاجة إلى مراجعة جميع الرياضيات.", "model": "google_nmt", "from_community_srt": "وهذا مايعرف بقاعدة لابلاس للنجاح وهي تعود الى القرن 18 و لنفهم اي فرضيات تحكم هذه القاعدة و مدى تاثير تغيير هذه الفرضيات او الاهداف على النتائج او على احتمالية الحصول على شئ ما", diff --git a/2020/binomial-distributions/english/captions.srt b/2020/binomial-distributions/english/captions.srt index 53455efa6..822ed77c8 100644 --- a/2020/binomial-distributions/english/captions.srt +++ b/2020/binomial-distributions/english/captions.srt @@ -135,7 +135,7 @@ you pretend that there were two more reviews, one that was positive and one that's negative. 35 -00:01:55,860 --> 00:02:00,479 +00:01:55,860 --> 00:02:00,480 In this case, that means you pretend that it's 11 out of 12, which would give 91.7%. 36 @@ -171,23 +171,23 @@ you get 187 out of 202, or 92.6%. So according to this rule, it would mean your best bet is to go with seller number 2. 44 -00:02:39,299 --> 00:02:42,604 +00:02:39,300 --> 00:02:42,247 This is something known as Laplace's rule of succession, 45 -00:02:42,604 --> 00:02:45,850 -and to understand what assumptions are underlying this, +00:02:42,247 --> 00:02:46,745 +it dates back to the 18th century, To understand what assumptions are underlying this, 46 -00:02:45,850 --> 00:02:50,951 -and how changing either those assumptions or your underlying goal can change the choice +00:02:46,745 --> 00:02:50,365 +and how changing either those assumptions or your underlying goal can 47 -00:02:50,951 --> 00:02:54,140 -you make, we really do need to go through all the math. +00:02:50,365 --> 00:02:54,140 +change the choice you make, we really do need to go through all the math. 48 -00:02:54,660 --> 00:02:56,359 +00:02:54,660 --> 00:02:56,360 So without further ado, let's dive in. 49 @@ -263,7 +263,7 @@ Quite a few of those, around 60% actually, give 10 out of 10. So the data we see seems perfectly plausible if the seller's true success rate was 95%. 67 -00:04:15,580 --> 00:04:18,039 +00:04:15,580 --> 00:04:18,040 Or maybe it's really 90%, or 99%. 68 @@ -383,7 +383,7 @@ that many times, building up a histogram to get some sense of what this distribu looks like. 97 -00:06:21,659 --> 00:06:24,435 +00:06:21,660 --> 00:06:24,435 Likewise, you could simulate sets of 50 reviews, 98 @@ -431,7 +431,7 @@ This first term is pronounced 50 choose 48, and it represents the total number of ways that you could take 50 slots and fill out 48 of them. 109 -00:07:09,099 --> 00:07:13,789 +00:07:09,100 --> 00:07:13,789 For example, maybe you start with 48 good reviews and end with 2 bad reviews, 110 @@ -567,7 +567,7 @@ When s is equal to 0.96, that value is as high as it's ever going to get. And this should kind of make sense, because when you look at that review of 96%, 143 -00:09:35,419 --> 00:09:39,319 +00:09:35,419 --> 00:09:39,320 it should be most likely if the true underlying success rate was 96%. 144 diff --git a/2020/binomial-distributions/english/sentence_timings.json b/2020/binomial-distributions/english/sentence_timings.json index 0cc214aab..93288c089 100644 --- a/2020/binomial-distributions/english/sentence_timings.json +++ b/2020/binomial-distributions/english/sentence_timings.json @@ -120,7 +120,7 @@ 158.66 ], [ - "This is something known as Laplace's rule of succession, and to understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", + "This is something known as Laplace's rule of succession, it dates back to the 18th century, To understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", 159.3, 174.14 ], diff --git a/2020/binomial-distributions/english/transcript.txt b/2020/binomial-distributions/english/transcript.txt index 08cfc4b1b..ff528c41e 100644 --- a/2020/binomial-distributions/english/transcript.txt +++ b/2020/binomial-distributions/english/transcript.txt @@ -22,7 +22,7 @@ So in the case of 50 reviews, where you have 48 positive and 2 negative, you pre That's your probability of having a good experience with the second seller. Playing the same game with our third seller who had 200 reviews, you get 187 out of 202, or 92.6%. So according to this rule, it would mean your best bet is to go with seller number 2. -This is something known as Laplace's rule of succession, and to understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math. +This is something known as Laplace's rule of succession, it dates back to the 18th century, To understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math. So without further ado, let's dive in. Step 1, how exactly are we modeling the situation, and what exactly is it that you want to optimize? One option is to think of each seller as producing random experiences that are either positive or negative, and that each seller has some kind of constant underlying probability of giving a good experience, what we're going to call the success rate, or S for short. diff --git a/2020/binomial-distributions/german/sentence_translations.json b/2020/binomial-distributions/german/sentence_translations.json index d2c20c4cf..e59765af5 100644 --- a/2020/binomial-distributions/german/sentence_translations.json +++ b/2020/binomial-distributions/german/sentence_translations.json @@ -213,7 +213,7 @@ "end": 158.66 }, { - "input": "This is something known as Laplace's rule of succession, and to understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", + "input": "This is something known as Laplace's rule of succession, it dates back to the 18th century, To understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", "translatedText": "Um zu verstehen, welche Annahmen dem zugrunde liegen und wie eine Änderung dieser Annahmen oder deines Ziels deine Entscheidung beeinflussen kann, müssen wir die ganze Mathematik durchgehen.", "model": "DeepL", "from_community_srt": "Dies ist bekannt als Laplace˙s Regel der Abfolge die aus dem 18. Jhr. stammt. Um zu verstehen welche Annahmen dem zugrundeliegen und wie das Ändern dieser Annahmen oder das Ändern des zugrunde liegenden Ziels die Wahl die du triffst verändern kann, müssen wir uns die ganze Mathematik dahinter ansehen Ohne jetzt noch herumzureden,", diff --git a/2020/binomial-distributions/hebrew/sentence_translations.json b/2020/binomial-distributions/hebrew/sentence_translations.json index 9c015ed66..34db908c0 100644 --- a/2020/binomial-distributions/hebrew/sentence_translations.json +++ b/2020/binomial-distributions/hebrew/sentence_translations.json @@ -212,7 +212,7 @@ "end": 158.66 }, { - "input": "This is something known as Laplace's rule of succession, and to understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", + "input": "This is something known as Laplace's rule of succession, it dates back to the 18th century, To understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", "translatedText": "זה משהו המכונה כלל הירושה של לפלס, וכדי להבין אילו הנחות יסוד עומדות בבסיס זה, וכיצד שינוי של ההנחות האלה או המטרה הבסיסית שלך יכול לשנות את הבחירה שאתה עושה, אנחנו באמת צריכים לעבור את כל המתמטיקה.", "model": "google_nmt", "from_community_srt": "הוא מתארך למאה ה18, ועל מנת להבין מה הן ההנחות המובילות לכלל הזה, וכיצד שינויהן או שינוי המטרה הסופית, עלול להוביל לשינוי ההחלטה בה תבחר, עלינו לראות את כל המתמטיקה שבדבר.", diff --git a/2020/binomial-distributions/korean/sentence_translations.json b/2020/binomial-distributions/korean/sentence_translations.json index 2bd506358..6ce78231c 100644 --- a/2020/binomial-distributions/korean/sentence_translations.json +++ b/2020/binomial-distributions/korean/sentence_translations.json @@ -205,7 +205,7 @@ "end": 158.66 }, { - "input": "This is something known as Laplace's rule of succession, and to understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", + "input": "This is something known as Laplace's rule of succession, it dates back to the 18th century, To understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", "translatedText": "이를 라플라스 승계의 법칙이라고 하는데, 그 기저에 어떤 가정이 있는지, 그리고 그러한 가정이나 기본 목표를 변경하면 선택이 어떻게 달라지는지 이해하려면 실제로 모든 수학을 살펴볼 필요가 있습니다.", "model": "DeepL", "from_community_srt": "무슨 가정이 내재되어 있는지 또 그 가정이 바뀌거나 궁극적 목적이 바뀌면 당신의 결정도 달라진다는 것을 이해하려면 수학을 빼놓을 수 없습니다 그럼 이제 본격적으로 시작할까요!", diff --git a/2020/binomial-distributions/portuguese/sentence_translations.json b/2020/binomial-distributions/portuguese/sentence_translations.json index da53119f1..4fe06bb02 100644 --- a/2020/binomial-distributions/portuguese/sentence_translations.json +++ b/2020/binomial-distributions/portuguese/sentence_translations.json @@ -211,7 +211,7 @@ "end": 158.66 }, { - "input": "This is something known as Laplace's rule of succession, and to understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", + "input": "This is something known as Laplace's rule of succession, it dates back to the 18th century, To understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", "translatedText": "Isto é algo conhecido como regra de sucessão de Laplace, e para compreender que pressupostos estão subjacentes a isto, e como a mudança desses pressupostos ou do seu objectivo subjacente pode mudar a escolha que faz, precisamos realmente de passar por toda a matemática.", "model": "google_nmt", "from_community_srt": "sua melhor opção é o segundo vendedor Isso é conhecido como a regra de sucessão de Laplace Que data antes do século XVIII (18) E para entender quais premissas estão por de trás disso E como mudando essas premissas ou seus objetos podem mudar a escolha que você faz", diff --git a/2020/binomial-distributions/turkish/sentence_translations.json b/2020/binomial-distributions/turkish/sentence_translations.json index 1d42e7532..f980ee096 100644 --- a/2020/binomial-distributions/turkish/sentence_translations.json +++ b/2020/binomial-distributions/turkish/sentence_translations.json @@ -214,7 +214,7 @@ "end": 158.66 }, { - "input": "This is something known as Laplace's rule of succession, and to understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", + "input": "This is something known as Laplace's rule of succession, it dates back to the 18th century, To understand what assumptions are underlying this, and how changing either those assumptions or your underlying goal can change the choice you make, we really do need to go through all the math.", "translatedText": "Bu, Laplace'ın ardışıklık kuralı olarak bilinen bir şeydir ve bunun altında hangi varsayımların yattığını ve bu varsayımları ya da temel hedefinizi değiştirmenin yaptığınız seçimi nasıl değiştirebileceğini anlamak için gerçekten tüm matematiği gözden geçirmemiz gerekir.", "model": "DeepL", "from_community_srt": "18. yüzyıla kadar dayanıyor Bunun altında yatan ön kabulleri, ve değişen ön kabuller yahut da altta yatan amaca göre yaptığınız seçimin nasıl değişebileceğini anlamak için matematikten yardım almamız gerekecek.", diff --git a/2020/chessboard-puzzle/arabic/sentence_translations.json b/2020/chessboard-puzzle/arabic/sentence_translations.json index aafc888ca..cbe171536 100644 --- a/2020/chessboard-puzzle/arabic/sentence_translations.json +++ b/2020/chessboard-puzzle/arabic/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares. ", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares. ", "translatedText": "كان أحدهما هو إثبات أن التحدي مستحيل إذا قمت بتغيير الإعداد قليلاً، ربما جعلها رقعة شطرنج 6 × 6، أو إزالة أحد المربعات. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/bengali/sentence_translations.json b/2020/chessboard-puzzle/bengali/sentence_translations.json index 6a9de4e80..58866069c 100644 --- a/2020/chessboard-puzzle/bengali/sentence_translations.json +++ b/2020/chessboard-puzzle/bengali/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares. ", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares. ", "translatedText": "একটি ছিল এটি প্রমাণ করা যে চ্যালেঞ্জটি অসম্ভব যদি আপনি সেটআপটি কিছুটা পরিবর্তন করেন, সম্ভবত এটিকে 6x6 চেসবোর্ড তৈরি করেন, বা স্কোয়ারগুলির একটি সরান।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/chinese/sentence_translations.json b/2020/chessboard-puzzle/chinese/sentence_translations.json index 50adb30ef..ad693ed52 100644 --- a/2020/chessboard-puzzle/chinese/sentence_translations.json +++ b/2020/chessboard-puzzle/chinese/sentence_translations.json @@ -117,7 +117,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares.", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares.", "translatedText": "其一是证明，如果稍微改变一下设置，比如将其变成 6x6 的棋盘，或者移除其中一个方格，那么挑战是不可能的。", "model": "google_nmt", "from_community_srt": "一个是证明这一挑战是不可能的 例如， 如果您稍微改变设置 如果棋盘是6x6， 或者如果您将其删除 它的正方形之一。", diff --git a/2020/chessboard-puzzle/english/captions.srt b/2020/chessboard-puzzle/english/captions.srt index 63e137bd9..087afbb19 100644 --- a/2020/chessboard-puzzle/english/captions.srt +++ b/2020/chessboard-puzzle/english/captions.srt @@ -775,16 +775,16 @@ not only why this will never work in three dimensions, but also why it can't work in any dimension that's not a power of two. 195 -00:12:30,500 --> 00:12:34,524 -The idea is that the symmetry in the property we're looking at will end up +00:12:30,500 --> 00:12:34,540 +The idea is that the symmetry in the property that we're looking at will end up 196 -00:12:34,524 --> 00:12:38,871 -implying that there have to be an equal number of red, green, and blue vertices, +00:12:34,540 --> 00:12:38,631 +implying that there have to be an equal number of red, green, and blue vertices. 197 -00:12:38,871 --> 00:12:42,520 -but that would mean there's 8 thirds of each, which is not possible. +00:12:38,631 --> 00:12:42,520 +But that would mean that there's eight-thirds of each, which is not possible. 198 00:12:43,440 --> 00:12:48,180 @@ -840,7 +840,7 @@ so that final tally has to be three times the total number of red corners. 211 00:13:40,420 --> 00:13:43,780 -So it's simple, find a coloring where 8 thirds of the corners are red. +So, you know, it's simple. Find a coloring where eight-thirds of the corners are red. 212 00:13:44,940 --> 00:13:45,540 @@ -1099,6 +1099,10 @@ not everyone is as interested in symmetrical ways to paint a 64-dimensional cube But reliable data transmission? 276 -00:18:11,760 --> 00:18:13,900 -Come on, I think we can all agree that's universally sexy. +00:18:11,760 --> 00:18:12,805 +Come on, I think we can all agree that that's universally sexy. + +277 +00:18:12,805 --> 00:18:13,900 +Come on, I think we can all agree that that's universally sexy. you diff --git a/2020/chessboard-puzzle/english/sentence_timings.json b/2020/chessboard-puzzle/english/sentence_timings.json index ad30f29d6..9ef15297e 100644 --- a/2020/chessboard-puzzle/english/sentence_timings.json +++ b/2020/chessboard-puzzle/english/sentence_timings.json @@ -470,7 +470,7 @@ 749.7 ], [ - "The idea is that the symmetry in the property we're looking at will end up implying that there have to be an equal number of red, green, and blue vertices, but that would mean there's 8 thirds of each, which is not possible.", + "The idea is that the symmetry in the property that we're looking at will end up implying that there have to be an equal number of red, green, and blue vertices. But that would mean that there's eight-thirds of each, which is not possible.", 750.5, 762.52 ], @@ -510,7 +510,7 @@ 819.38 ], [ - "So it's simple, find a coloring where 8 thirds of the corners are red.", + "So, you know, it's simple. Find a coloring where eight-thirds of the corners are red.", 820.42, 823.78 ], @@ -675,7 +675,7 @@ 1091.32 ], [ - "Come on, I think we can all agree that's universally sexy.", + "Come on, I think we can all agree that that's universally sexy. Come on, I think we can all agree that that's universally sexy. you", 1091.76, 1093.9 ] diff --git a/2020/chessboard-puzzle/english/transcript.txt b/2020/chessboard-puzzle/english/transcript.txt index c9b26261a..8a1813e8f 100644 --- a/2020/chessboard-puzzle/english/transcript.txt +++ b/2020/chessboard-puzzle/english/transcript.txt @@ -92,7 +92,7 @@ And then maybe we move to the red neighbor and say that the other two adjacencie But at least how I've drawn it here, you're stuck, you're unable to choose a correct color for the next two. Can you see why? What I'd like to share is a lovely little argument that explains not only why this will never work in three dimensions, but also why it can't work in any dimension that's not a power of two. -The idea is that the symmetry in the property we're looking at will end up implying that there have to be an equal number of red, green, and blue vertices, but that would mean there's 8 thirds of each, which is not possible. +The idea is that the symmetry in the property that we're looking at will end up implying that there have to be an equal number of red, green, and blue vertices. But that would mean that there's eight-thirds of each, which is not possible. And before I go on, pause and see if you can think of a way to solidify that intuition. It's a fun exercise in turning a vague instinct into a solid proof. Alright, you ready? @@ -100,7 +100,7 @@ One way to do this is to imagine a process where you go through each corner and So, each step here, we're looking at the three neighbors of a given vertex, counting up the red ones, and adding that to a total tally. For this specific coloring, that count came out to be 12, but if we had the property we wanted, every corner would have exactly one red neighbor, so that count should be 8. On the other hand, every red corner is counted exactly three times, once for each instance where it's somebody's neighbor, so that final tally has to be three times the total number of red corners. -So it's simple, find a coloring where 8 thirds of the corners are red. +So, you know, it's simple. Find a coloring where eight-thirds of the corners are red. Isn't that nice? Counting how many times some corner has a red neighbor is the same as counting how many times a red corner has some neighbor, and that's enough to get us a contradiction. What's also nice is that this argument immediately generalizes to higher dimensions. @@ -133,4 +133,4 @@ Before explaining it, he and I simply walk through what it looks like for us to And if you're curious about the connection with Hamming codes and error correction, I'm definitely game to make a video on that, just let me know in the comments. I've been told that as far as motivating puzzles go, not everyone is as interested in symmetrical ways to paint a 64-dimensional cube as I am. But reliable data transmission? -Come on, I think we can all agree that's universally sexy. \ No newline at end of file +Come on, I think we can all agree that that's universally sexy. Come on, I think we can all agree that that's universally sexy. you \ No newline at end of file diff --git a/2020/chessboard-puzzle/french/sentence_translations.json b/2020/chessboard-puzzle/french/sentence_translations.json index 997a06dd6..3a5df02e6 100644 --- a/2020/chessboard-puzzle/french/sentence_translations.json +++ b/2020/chessboard-puzzle/french/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares.", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares.", "translatedText": "L'une d'elles consistait à prouver que le défi est impossible si vous modifiez un peu la configuration, peut-être en en faisant un échiquier 6x6, ou en supprimant l'une des cases.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/german/sentence_translations.json b/2020/chessboard-puzzle/german/sentence_translations.json index 6a7b1e498..fe4ed36ca 100644 --- a/2020/chessboard-puzzle/german/sentence_translations.json +++ b/2020/chessboard-puzzle/german/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares.", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares.", "translatedText": "Eine bestand darin, zu beweisen, dass die Herausforderung unmöglich ist, wenn man den Aufbau ein wenig variiert, indem man vielleicht ein 6x6-Schachbrett daraus macht oder eines der Felder entfernt.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/hebrew/sentence_translations.json b/2020/chessboard-puzzle/hebrew/sentence_translations.json index 07a63800c..1f7e3dc49 100644 --- a/2020/chessboard-puzzle/hebrew/sentence_translations.json +++ b/2020/chessboard-puzzle/hebrew/sentence_translations.json @@ -91,7 +91,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares.", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares.", "translatedText": "אחד מהם היה להוכיח שהאתגר הוא בלתי אפשרי אם משנים מעט את ההגדרה, אולי הופכים אותו ללוח שחמט בגודל 6x6, או מסירים את אחד המשבצות.", "n_reviews": 0, "start": 107.4, diff --git a/2020/chessboard-puzzle/hindi/sentence_translations.json b/2020/chessboard-puzzle/hindi/sentence_translations.json index cfb6f3376..90ed7cdbb 100644 --- a/2020/chessboard-puzzle/hindi/sentence_translations.json +++ b/2020/chessboard-puzzle/hindi/sentence_translations.json @@ -842,7 +842,7 @@ "end": 749.7 }, { - "input": "The idea is that the symmetry in the property we're looking at will end up implying that there have to be an equal number of red, green, and blue vertices, but that would mean there's 8 thirds of each, which is not possible.", + "input": "The idea is that the symmetry in the property that we're looking at will end up implying that there have to be an equal number of red, green, and blue vertices. But that would mean that there's eight-thirds of each, which is not possible.", "translatedText": "विचार यह है कि जिस संपत्ति को हम देख रहे हैं उसमें समरूपता का अर्थ यह होगा कि लाल, हरे और नीले शीर्षों की समान संख्या होनी चाहिए, लेकिन इसका मतलब होगा कि प्रत्येक का 8 तिहाई हिस्सा होगा, जो संभव नहीं है।", "model": "google_nmt", "from_community_srt": "यह विचार है कि संपत्ति में समरूपता हम चाहते हैं कि एक समान होना चाहिए लाल, हरे और नीले कोने की संख्या। लेकिन इसका मतलब होगा कि प्रत्येक का 8/3,", @@ -913,7 +913,7 @@ "end": 819.38 }, { - "input": "So it's simple, find a coloring where 8 thirds of the corners are red.", + "input": "So, you know, it's simple. Find a coloring where eight-thirds of the corners are red.", "translatedText": "तो यह आसान है, एक ऐसा रंग ढूंढें जिसके 8 तिहाई कोने लाल हों।", "model": "google_nmt", "from_community_srt": "तो यह आसान है,", @@ -1209,7 +1209,7 @@ "end": 1091.32 }, { - "input": "Come on, I think we can all agree that's universally sexy.", + "input": "Come on, I think we can all agree that that's universally sexy. Come on, I think we can all agree that that's universally sexy. you", "translatedText": "चलिए, मुझे लगता है कि हम सभी सहमत हो सकते हैं कि यह सार्वभौमिक रूप से सेक्सी है।", "model": "google_nmt", "from_community_srt": "लेकिन विश्वसनीय डेटा ट्रांसमिशन? मुझे लगता है कि हम सभी सार्वभौमिक रूप से सहमत हो सकते हैं कामुक।", diff --git a/2020/chessboard-puzzle/hungarian/sentence_translations.json b/2020/chessboard-puzzle/hungarian/sentence_translations.json index 48cb0be5d..dbf81f647 100644 --- a/2020/chessboard-puzzle/hungarian/sentence_translations.json +++ b/2020/chessboard-puzzle/hungarian/sentence_translations.json @@ -752,7 +752,7 @@ "end": 749.7 }, { - "input": "The idea is that the symmetry in the property we're looking at will end up implying that there have to be an equal number of red, green, and blue vertices, but that would mean there's 8 thirds of each, which is not possible.", + "input": "The idea is that the symmetry in the property that we're looking at will end up implying that there have to be an equal number of red, green, and blue vertices. But that would mean that there's eight-thirds of each, which is not possible.", "translatedText": "Az ötlet az, hogy a szimmetria a vizsgált tulajdonságban azt jelenti, hogy a piros, zöld és kék csúcsoknak egyenlő számúnak kell lenniük, de ez azt jelentené, hogy mindegyikből 8 harmad van, ami nem lehetséges.", "model": "DeepL", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 819.38 }, { - "input": "So it's simple, find a coloring where 8 thirds of the corners are red.", + "input": "So, you know, it's simple. Find a coloring where eight-thirds of the corners are red.", "translatedText": "Egyszerű tehát, keress egy olyan színezést, ahol a sarkok 8 harmada piros.", "model": "DeepL", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1091.32 }, { - "input": "Come on, I think we can all agree that's universally sexy.", + "input": "Come on, I think we can all agree that that's universally sexy. Come on, I think we can all agree that that's universally sexy. you", "translatedText": "Ugyan már, azt hiszem, mindannyian egyetértünk abban, hogy ez egyetemesen szexi.", "model": "DeepL", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/indonesian/sentence_translations.json b/2020/chessboard-puzzle/indonesian/sentence_translations.json index 9f27679fe..506d0c6a1 100644 --- a/2020/chessboard-puzzle/indonesian/sentence_translations.json +++ b/2020/chessboard-puzzle/indonesian/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares.", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares.", "translatedText": "Salah satunya adalah membuktikan bahwa tantangan tersebut mustahil dilakukan jika Anda sedikit memvariasikan pengaturannya, mungkin menjadikannya papan catur 6x6, atau menghilangkan salah satu kotaknya.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/italian/sentence_translations.json b/2020/chessboard-puzzle/italian/sentence_translations.json index 7ed5eefb0..8da70d214 100644 --- a/2020/chessboard-puzzle/italian/sentence_translations.json +++ b/2020/chessboard-puzzle/italian/sentence_translations.json @@ -91,7 +91,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares.", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares.", "translatedText": "Il primo era dimostrare che la sfida è impossibile se si varia leggermente la configurazione, magari trasformandola in una scacchiera 6x6 o rimuovendo uno dei quadrati.", "n_reviews": 0, "start": 107.4, diff --git a/2020/chessboard-puzzle/japanese/sentence_translations.json b/2020/chessboard-puzzle/japanese/sentence_translations.json index 96bf432e3..04c47169e 100644 --- a/2020/chessboard-puzzle/japanese/sentence_translations.json +++ b/2020/chessboard-puzzle/japanese/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares.", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares.", "translatedText": "1 つは、設定を少し変えると、たとえば 6x6 のチェス盤にするか、正方 形の 1 つを削除するなど、挑戦が不可能であることを証明することでした。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/korean/sentence_translations.json b/2020/chessboard-puzzle/korean/sentence_translations.json index bbfcc0927..3456d4aa4 100644 --- a/2020/chessboard-puzzle/korean/sentence_translations.json +++ b/2020/chessboard-puzzle/korean/sentence_translations.json @@ -839,7 +839,7 @@ "end": 749.7 }, { - "input": "The idea is that the symmetry in the property we're looking at will end up implying that there have to be an equal number of red, green, and blue vertices, but that would mean there's 8 thirds of each, which is not possible.", + "input": "The idea is that the symmetry in the property that we're looking at will end up implying that there have to be an equal number of red, green, and blue vertices. But that would mean that there's eight-thirds of each, which is not possible.", "translatedText": "우리가 살펴보고 있는 속성의 대칭은 결국 빨간색, 녹색, 파란색 정점의 수가 같아야 한다는 것을 의미하지만, 이는 각각 3분의 8이 있어야 한다는 것을 의미하므로 불가능합니다.", "model": "DeepL", "from_community_srt": "이 특성의 대칭성 때문에 빨간색, 초록색, 파란색 꼭짓점의 개수가 같아야 합니다. 그러나 이는 각각 8/3개 씩의 꼭짓점을 가져야 한다는 결론이 나오게 하고,", @@ -909,7 +909,7 @@ "end": 819.38 }, { - "input": "So it's simple, find a coloring where 8 thirds of the corners are red.", + "input": "So, you know, it's simple. Find a coloring where eight-thirds of the corners are red.", "translatedText": "모서리의 3분의 8이 빨간색인 색상을 찾으면 됩니다.", "model": "DeepL", "from_community_srt": "그러면 우리는 이제 8/3개의 꼭짓점이 빨간색인 배열을 찾아야 하는 것이다.", @@ -1202,7 +1202,7 @@ "end": 1091.32 }, { - "input": "Come on, I think we can all agree that's universally sexy.", + "input": "Come on, I think we can all agree that that's universally sexy. Come on, I think we can all agree that that's universally sexy. you", "translatedText": "우리 모두는 그것이 보편적으로 섹시하다는 데 동의할 수 있을 것입니다.", "model": "DeepL", "from_community_srt": "이는 우리 모두가 (펀쿨)섹시하다고 동의할 수 있을 것입니다.", diff --git a/2020/chessboard-puzzle/marathi/sentence_translations.json b/2020/chessboard-puzzle/marathi/sentence_translations.json index f20320708..cfe8f1a92 100644 --- a/2020/chessboard-puzzle/marathi/sentence_translations.json +++ b/2020/chessboard-puzzle/marathi/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares. ", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares. ", "translatedText": "एक म्हणजे तुम्ही सेटअप थोडासा बदलल्यास आव्हान अशक्य आहे हे सिद्ध करणे, कदाचित ते 6x6 चेसबोर्ड बनवणे किंवा एक स्क्वेअर काढून टाकणे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/persian/sentence_translations.json b/2020/chessboard-puzzle/persian/sentence_translations.json index 1872837ca..bc61a04c7 100644 --- a/2020/chessboard-puzzle/persian/sentence_translations.json +++ b/2020/chessboard-puzzle/persian/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares. ", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares. ", "translatedText": "یکی از آن‌ها اثبات این بود که اگر تنظیمات را کمی تغییر دهید، شاید آن را به یک صفحه شطرنج 6x6 تبدیل کنید یا یکی از مربع‌ها را بردارید، این چالش غیرممکن است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/portuguese/sentence_translations.json b/2020/chessboard-puzzle/portuguese/sentence_translations.json index f208b18a3..972d295e2 100644 --- a/2020/chessboard-puzzle/portuguese/sentence_translations.json +++ b/2020/chessboard-puzzle/portuguese/sentence_translations.json @@ -840,7 +840,7 @@ "end": 749.7 }, { - "input": "The idea is that the symmetry in the property we're looking at will end up implying that there have to be an equal number of red, green, and blue vertices, but that would mean there's 8 thirds of each, which is not possible.", + "input": "The idea is that the symmetry in the property that we're looking at will end up implying that there have to be an equal number of red, green, and blue vertices. But that would mean that there's eight-thirds of each, which is not possible.", "translatedText": "A ideia é que a simetria na propriedade que estamos vendo acabará implicando que deve haver um número igual de vértices vermelhos, verdes e azuis, mas isso significaria que há 8 terços de cada, o que não é possível.", "model": "google_nmt", "from_community_srt": "A ideia é que a simetria na propriedade queremos implica que deveria haver um igual número de vértices vermelhos, verdes e azuis. Mas isso significaria 8/3 de cada,", @@ -911,7 +911,7 @@ "end": 819.38 }, { - "input": "So it's simple, find a coloring where 8 thirds of the corners are red.", + "input": "So, you know, it's simple. Find a coloring where eight-thirds of the corners are red.", "translatedText": "Então é simples, encontre uma coloração onde 8 terços dos cantos sejam vermelhos.", "model": "google_nmt", "from_community_srt": "Portanto, é simples,", @@ -1207,7 +1207,7 @@ "end": 1091.32 }, { - "input": "Come on, I think we can all agree that's universally sexy.", + "input": "Come on, I think we can all agree that that's universally sexy. Come on, I think we can all agree that that's universally sexy. you", "translatedText": "Qual é, acho que todos podemos concordar que isso é universalmente sexy.", "model": "google_nmt", "from_community_srt": "mas transmissão de dados confiável? Eu acho que todos nós podemos concordar que é universalmente sexy.", diff --git a/2020/chessboard-puzzle/russian/sentence_translations.json b/2020/chessboard-puzzle/russian/sentence_translations.json index 5669a4d9f..b781c9114 100644 --- a/2020/chessboard-puzzle/russian/sentence_translations.json +++ b/2020/chessboard-puzzle/russian/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares.", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares.", "translatedText": "Один из них заключался в том, чтобы доказать, что задача невыполнима, если немного изменить расстановку, например, сделать шахматную доску 6x6 или удалить одно из полей.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/spanish/sentence_translations.json b/2020/chessboard-puzzle/spanish/sentence_translations.json index 7ef2e8183..d559afc8f 100644 --- a/2020/chessboard-puzzle/spanish/sentence_translations.json +++ b/2020/chessboard-puzzle/spanish/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares.", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares.", "translatedText": "Una era demostrar que el desafío es imposible si varías un poco la configuración, tal vez convirtiéndolo en un tablero de ajedrez de 6x6 o eliminando uno de los cuadrados.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/tamil/sentence_translations.json b/2020/chessboard-puzzle/tamil/sentence_translations.json index 8df878fa5..d116c0baf 100644 --- a/2020/chessboard-puzzle/tamil/sentence_translations.json +++ b/2020/chessboard-puzzle/tamil/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares. ", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares. ", "translatedText": "ஒன்று, நீங்கள் அமைப்பை சிறிது மாற்றினால், அதை 6x6 சதுரங்கப் பலகையாக மாற்றினால் அல்லது சதுரங்களில் ஒன்றை அகற்றினால் சவால் சாத்தியமற்றது என்பதை நிரூபிப்பது. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/telugu/sentence_translations.json b/2020/chessboard-puzzle/telugu/sentence_translations.json index 6c07417cf..5a0fa1a70 100644 --- a/2020/chessboard-puzzle/telugu/sentence_translations.json +++ b/2020/chessboard-puzzle/telugu/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares. ", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares. ", "translatedText": "ఒకటి, మీరు సెటప్‌ను కొద్దిగా మార్చడం, బహుశా దానిని 6x6 చదరంగం బోర్డుగా మార్చడం లేదా చతురస్రాల్లో ఒకదాన్ని తీసివేస్తే సవాలు అసాధ్యం అని నిరూపించడం. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/thai/sentence_translations.json b/2020/chessboard-puzzle/thai/sentence_translations.json index 4f0deab3f..a7293a6dc 100644 --- a/2020/chessboard-puzzle/thai/sentence_translations.json +++ b/2020/chessboard-puzzle/thai/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares. ", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/turkish/sentence_translations.json b/2020/chessboard-puzzle/turkish/sentence_translations.json index 95ddee38d..3c6f470e8 100644 --- a/2020/chessboard-puzzle/turkish/sentence_translations.json +++ b/2020/chessboard-puzzle/turkish/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares.", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares.", "translatedText": "Bunlardan biri, düzeni biraz değiştirirseniz, örneğin 6x6'lık bir satranç tahtası yaparsanız veya karelerden birini çıkarırsanız, bu zorluğun imkansız olduğunu kanıtlamaktı.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/ukrainian/sentence_translations.json b/2020/chessboard-puzzle/ukrainian/sentence_translations.json index 77185ec96..59ef0a272 100644 --- a/2020/chessboard-puzzle/ukrainian/sentence_translations.json +++ b/2020/chessboard-puzzle/ukrainian/sentence_translations.json @@ -91,7 +91,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares.", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares.", "translatedText": "Одна з них полягала в тому, щоб довести, що завдання неможливе, якщо ви трохи зміните налаштування, можливо, зробивши його шаховою дошкою 6x6 або видаливши одне з квадратів.", "n_reviews": 0, "start": 107.4, diff --git a/2020/chessboard-puzzle/urdu/sentence_translations.json b/2020/chessboard-puzzle/urdu/sentence_translations.json index c9e83672c..0e7529278 100644 --- a/2020/chessboard-puzzle/urdu/sentence_translations.json +++ b/2020/chessboard-puzzle/urdu/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares. ", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares. ", "translatedText": "ایک یہ ثابت کرنا تھا کہ چیلنج ناممکن ہے اگر آپ سیٹ اپ میں تھوڑا سا فرق کرتے ہیں، ہو سکتا ہے اسے 6x6 کی بساط بنائیں، یا کسی ایک چوک کو ہٹا دیں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/chessboard-puzzle/vietnamese/sentence_translations.json b/2020/chessboard-puzzle/vietnamese/sentence_translations.json index 1729779b1..3e081ed45 100644 --- a/2020/chessboard-puzzle/vietnamese/sentence_translations.json +++ b/2020/chessboard-puzzle/vietnamese/sentence_translations.json @@ -104,7 +104,7 @@ "end": 106.72 }, { - "input": "One was to prove that the challenge is impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or removing one of the squares. ", + "input": "One was to prove that the challenge is actually impossible if you vary the setup a little bit, maybe making it a 6x6 chessboard, or maybe removing one of the squares. ", "translatedText": "Một là để chứng minh rằng thử thách này là không thể nếu bạn thay đổi cách sắp xếp một chút, có thể biến nó thành một bàn cờ 6x6 hoặc loại bỏ một trong các ô vuông. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/epidemic-simulations/albanian/sentence_translations.json b/2020/epidemic-simulations/albanian/sentence_translations.json index b3817d04a..0e0018563 100644 --- a/2020/epidemic-simulations/albanian/sentence_translations.json +++ b/2020/epidemic-simulations/albanian/sentence_translations.json @@ -908,7 +908,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "", "from_community_srt": "Për krahasim, vlerësohet se R0 për COVID-19 e ka vlerën pak më shumë se 2, që është afërsisht sa vlera mesatare e pritur e R0 gjatë pandemisë së gripit spanjoll gjatë vitit 1918.", "n_reviews": 0, @@ -1044,7 +1044,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "", "from_community_srt": "nëse masës së distancimit ia shtojmë edhe zvoglimin e vajtje-ardhjeve me faktorin 5 (5 herë më pak)? Ose nëse e zvoglojmë gjasën e infeksionit me edhe për 2 herë tjera, p.sh. duke e konsideruar se njerëzit praktikojnë një higjienë më të mirë? Mirë,", "n_reviews": 0, diff --git a/2020/epidemic-simulations/arabic/sentence_translations.json b/2020/epidemic-simulations/arabic/sentence_translations.json index db2df41a5..f8afc5c77 100644 --- a/2020/epidemic-simulations/arabic/sentence_translations.json +++ b/2020/epidemic-simulations/arabic/sentence_translations.json @@ -1009,7 +1009,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "للمقارنة، يُقدر أن يكون R0 لـCOVID-19 أعلى قليلاً من 2، وهو أيضًا قريب مما كان عليه متوسط التقدير لـ R0 خلال جائحة الأنفلونزا الإسبانية عام 1918.", "model": "google_nmt", "from_community_srt": "للمقارنة ، يتم تقدير R0 لـ COVID-19 ليكون أعلى بقليل من 2 ، وهو أيضًا موجود التقدير المتوسط ​​لـ R0 خلال 1918 الإسبانية وباء الإنفلونزا.", @@ -1159,7 +1159,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "بالإضافة إلى تأثير التباعد الاجتماعي، قمنا بتقليل وتيرة ذهاب الأشخاص إلى تلك النقطة المركزية، ربما بعامل 5، أو إذا قمنا بتخفيض احتمالية الإصابة بعامل آخر قدره 2، مما يعني أن الناس أصبحوا أكثر حذرًا عن غسل أيديهم وعدم لمس وجوههم.", "model": "google_nmt", "from_community_srt": "الآن ما تعتقد أنه سيكون أكثر فعالية ، إذا قللنا من التباعد الاجتماعي التردد الذي يذهب الناس إلى النقطة المركزية بمعامل 5؟ أو إذا قطعنا احتمالية الإصابة إلى أسفل بعامل آخر 2 ، على سبيل المثال المعنى الناس يمارسون نظافة أفضل؟", diff --git a/2020/epidemic-simulations/bengali/sentence_translations.json b/2020/epidemic-simulations/bengali/sentence_translations.json index 79c7043a4..749a01fb2 100644 --- a/2020/epidemic-simulations/bengali/sentence_translations.json +++ b/2020/epidemic-simulations/bengali/sentence_translations.json @@ -416,7 +416,7 @@ "end": 413.19 }, { - "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, would you predict it'll be? ", + "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, but where on the spectrum would you predict it'll be? ", "translatedText": "স্পষ্টতই এটি মোট কোয়ারেন্টাইন এবং কিছুই না করার মধ্যে কোথাও একটি ফলাফল হবে, আপনি কি ভবিষ্যদ্বাণী করবেন যে এটি হবে? ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 580.45 }, { - "input": "It also means that the most dangerous viruses are the ones that kill some part of the population while laying unnoticed and spreadable among others, or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", + "input": "It also means that the most dangerous viruses are the ones that kill some part of the population in the first place, while laying unnoticed and spreadable among others. Or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", "translatedText": "এর মানে হল যে সবচেয়ে বিপজ্জনক ভাইরাসগুলি হল সেইগুলি যেগুলি জনসংখ্যার কিছু অংশকে মেরে ফেলে এবং অন্যদের মধ্যে ছড়িয়ে পড়ে, বা আরও খারাপ, যদি তারা শেষ পর্যন্ত প্রাণঘাতী হওয়ার আগে সবার নজরে না পড়ে এবং ছড়িয়ে পড়ে।", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 640.27 }, { - "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel too close to their neighbor. ", + "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel just a little too close to their neighbor. ", "translatedText": "এই অ্যানিমেশনগুলিতে, আমি এটিকে মানুষের মধ্যে একটি বিকর্ষণকারী শক্তি হিসাবে প্রয়োগ করব, এবং যখন তারা তাদের প্রতিবেশীর খুব কাছাকাছি বোধ করবে তখন তাদের হলুদ আভা দিব।", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 738.41 }, { - "input": "Again, I'll emphasize that these are toy models, and I leave it to the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", + "input": "Again, I'll emphasize that these are toy models, and I leave it to the intelligence of the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", "translatedText": "আবার, আমি জোর দিয়ে বলব যে এগুলি খেলনা মডেল, এবং এই ছোট বিন্দুগুলির আচরণটি আপনার এবং আপনার জীবনের জন্য সামাজিক দূরত্বের অর্থ কী তা সঠিকভাবে প্রতিফলিত করে কিনা তা বিচার করার জন্য আমি এটি দর্শকের উপর ছেড়ে দিচ্ছি।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/epidemic-simulations/bulgarian/sentence_translations.json b/2020/epidemic-simulations/bulgarian/sentence_translations.json index 844a0db5b..666f8d882 100644 --- a/2020/epidemic-simulations/bulgarian/sentence_translations.json +++ b/2020/epidemic-simulations/bulgarian/sentence_translations.json @@ -905,7 +905,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "", "from_community_srt": "За сравнение, R_0 на COVID-19 е изчислен на малко над 2, което е близко до стойностите на пандемията от испански грип през 1918 г.", "n_reviews": 0, @@ -1039,7 +1039,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "", "from_community_srt": "Дали ако намалим пет пъти честотата на посещение на централната точка ще имаме положителен ефект Или ако намалим два пъти вероятността за заразяване в следствие на добра лична хигиена? Нека разиграем и двата варианта.", "n_reviews": 0, diff --git a/2020/epidemic-simulations/chinese/sentence_translations.json b/2020/epidemic-simulations/chinese/sentence_translations.json index ddf64a8b6..8b23a41ad 100644 --- a/2020/epidemic-simulations/chinese/sentence_translations.json +++ b/2020/epidemic-simulations/chinese/sentence_translations.json @@ -468,7 +468,7 @@ "end": 413.19 }, { - "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, would you predict it'll be? ", + "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, but where on the spectrum would you predict it'll be? ", "translatedText": "显然，这会产生介于完全隔离和无所作 为之间的结果，您预测会是这样吗？", "model": "google_nmt", "from_community_srt": "顯然， 這將在某處產生結果 在完全隔離和無所事事之間，", @@ -611,7 +611,7 @@ "end": 580.45 }, { - "input": "It also means that the most dangerous viruses are the ones that kill some part of the population while laying unnoticed and spreadable among others, or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", + "input": "It also means that the most dangerous viruses are the ones that kill some part of the population in the first place, while laying unnoticed and spreadable among others. Or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", "translatedText": "这也意味着，最危险的病毒是那些杀死部分人群，同时不被注 意到并在其他人群中传播的病毒，或者更糟糕的是，如果它 们在最终变得致命之前一直未被注意到并在每个人中传播。", "model": "google_nmt", "from_community_srt": "但是 它不會傳播。 這也意味著最危險的病毒 是殺死一部分人口的人， 同時不引起注意並在其中傳播 其他。 更糟糕的是， 如果他們仍然不被注意， 在致死之前在所有人中傳播。", @@ -665,7 +665,7 @@ "end": 640.27 }, { - "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel too close to their neighbor. ", + "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel just a little too close to their neighbor. ", "translatedText": "在这些动画中，我将其应用为人与人之间的排斥 力，并让他们在感觉离邻居太近时发出黄色光。", "model": "google_nmt", "from_community_srt": "在這些動畫中， 我將其作為 人與人之間的排斥力 當他們感到離得太近時會發黃光 給他們的鄰居。", @@ -789,7 +789,7 @@ "end": 738.41 }, { - "input": "Again, I'll emphasize that these are toy models, and I leave it to the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", + "input": "Again, I'll emphasize that these are toy models, and I leave it to the intelligence of the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", "translatedText": "我再次强调，这些都是玩具模型，我让观 众来判断这些小点的行为是否准确地反映 了社交距离对你和你的生活意味着什么。", "model": "google_nmt", "from_community_srt": "同樣， 我會強調這些是玩具模型， 我把它留給了 觀察者判斷這些點的行為 準確反映出社會距離 對你和你的生活都意味著 有人被完全隔離在家裡 不一定會受到隨機抖動的影響 他們的鄰居。", diff --git a/2020/epidemic-simulations/croatian/sentence_translations.json b/2020/epidemic-simulations/croatian/sentence_translations.json index b2930c981..7f4a984b1 100644 --- a/2020/epidemic-simulations/croatian/sentence_translations.json +++ b/2020/epidemic-simulations/croatian/sentence_translations.json @@ -910,7 +910,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "", "from_community_srt": "Za usporedbu, R0 za COVID-19 je procijenjen nešto malo iznad 2, što je također oko onoga što je srednja procjena R0 bila za vrijeme pandemije Španjolske gripe 1918.", "n_reviews": 0, @@ -1045,7 +1045,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "", "from_community_srt": "Što mislite da će biti učinkovitije? Ako na učinak društvenog udaljavanja smanjimo učestalost s kojom ljudi odlaze na tu središnju točku, možda za čimbenik 5? Ili ako srežemo vjerojatnost zaraze za dodatni čimbenik 2, što bi, primjerice, značilo da su ljudi super posebno oprezni oko pranja ruku i ne diranja lica.", "n_reviews": 0, diff --git a/2020/epidemic-simulations/czech/sentence_translations.json b/2020/epidemic-simulations/czech/sentence_translations.json index 3ff14af5a..db5f256cf 100644 --- a/2020/epidemic-simulations/czech/sentence_translations.json +++ b/2020/epidemic-simulations/czech/sentence_translations.json @@ -908,7 +908,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "", "from_community_srt": "že je na ústupu. Pro srovnání, hodnota R_0pro COVID-19 je odhadována na o něco víc, než 2, což je podobně jako odhad R_0 během pandemie Španělské chřipky v roce 1918 Obyčejná sezónní chřipka má pro srovnání R_0 kolem 1.3.", "n_reviews": 0, @@ -1043,7 +1043,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "", "from_community_srt": "Když k \"social distancingu\" snížíme dejme tomu pětinásobně frekvenci s kterou chodí lidé do centrální lokace? Nebo když snížíme dvakrát pravděpodobnost nákazy, dejme tomu jako důsledek lepší hygieny? Pojďme se na to podívat.", "n_reviews": 0, diff --git a/2020/epidemic-simulations/dutch/sentence_translations.json b/2020/epidemic-simulations/dutch/sentence_translations.json index 04a9e56a8..4b00c71ca 100644 --- a/2020/epidemic-simulations/dutch/sentence_translations.json +++ b/2020/epidemic-simulations/dutch/sentence_translations.json @@ -908,7 +908,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "", "from_community_srt": "Ter vergelijking, R_0 voor COVID-19 is geschat op iets meer dan 2, wat vergelijkbaar is met de gemiddelde schatting van R_0 tijdens de Spaanse griep pandemie van 1918.", "n_reviews": 0, @@ -1043,7 +1043,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "", "from_community_srt": "bovenop social distancing verlagen we de frequency waarmee mensen naar de centrale locatie gaan met een factor 5? Of we verlagen de kans op infectie met een factor 2, dat kan betekenen dat mensen beter op hun hygiëne letten? Nou,", "n_reviews": 0, diff --git a/2020/epidemic-simulations/english/captions.srt b/2020/epidemic-simulations/english/captions.srt index e9358ba0c..e530415b3 100644 --- a/2020/epidemic-simulations/english/captions.srt +++ b/2020/epidemic-simulations/english/captions.srt @@ -911,15 +911,15 @@ When it holds steady around 1, that's when a disease is called endemic, and less than 1 means it's on the decline. 229 -00:16:16,190 --> 00:16:20,707 -For comparison, R0 for COVID-19 is estimated to be a little above 2, +00:16:16,190 --> 00:16:20,759 +For comparison, R-naught for COVID-19 is estimated to be a little above 2, 230 -00:16:20,707 --> 00:16:25,878 -which is also around what the mean estimate for R0 was during the 1918 Spanish +00:16:20,759 --> 00:16:25,937 +which is also around what the mean estimate for R-naught was during the 1918 Spanish 231 -00:16:25,878 --> 00:16:26,730 +00:16:25,937 --> 00:16:26,730 flu pandemic. 232 @@ -1059,24 +1059,24 @@ Now let me ask you to make a prediction. What do you think will be more effective? 266 -00:18:52,630 --> 00:18:57,016 +00:18:52,630 --> 00:18:56,849 If on top of that social distancing effect, we decrease the frequency with 267 -00:18:57,016 --> 00:19:00,642 +00:18:56,849 --> 00:19:00,336 which people go to that central spot, maybe by a factor of 5, 268 -00:19:00,642 --> 00:19:04,852 +00:19:00,336 --> 00:19:04,387 or if we chop the probability of infection down by another factor of 2, 269 -00:19:04,852 --> 00:19:09,180 -meaning people are super extra cautious about washing their hands and not +00:19:04,387 --> 00:19:08,774 +for example meaning people are super extra cautious about washing their hands 270 -00:19:09,180 --> 00:19:10,350 -touching their face. +00:19:08,774 --> 00:19:10,350 +and not touching their face. 271 00:19:11,670 --> 00:19:16,479 diff --git a/2020/epidemic-simulations/english/sentence_timings.json b/2020/epidemic-simulations/english/sentence_timings.json index 3ba9d28a9..2ddb5421b 100644 --- a/2020/epidemic-simulations/english/sentence_timings.json +++ b/2020/epidemic-simulations/english/sentence_timings.json @@ -570,7 +570,7 @@ 975.43 ], [ - "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", 976.19, 986.73 ], @@ -655,7 +655,7 @@ 1132.11 ], [ - "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", 1132.63, 1150.35 ], diff --git a/2020/epidemic-simulations/english/transcript.txt b/2020/epidemic-simulations/english/transcript.txt index 6553c9e78..0a9f8c08b 100644 --- a/2020/epidemic-simulations/english/transcript.txt +++ b/2020/epidemic-simulations/english/transcript.txt @@ -112,7 +112,7 @@ When you double that radius, there's about 4 times as many people inside it to i When we chopped the infection rate in half, it hovered around the 1.3 to 1.7 range. While R is greater than 1, the infection is growing exponentially, and it's at that point that it's known as an epidemic. When it holds steady around 1, that's when a disease is called endemic, and less than 1 means it's on the decline. -For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic. +For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic. The seasonal flu by comparison is much lower, around 1.3. In the travel case, as soon as we turn on social distancing and shut down travel, you can see R quickly drop down from 2. There's a little bit of a lag between the change we make to the model and the value of this number, since it's calculated based on current infectious cases, which might have existed prior to the method being put in place. @@ -129,7 +129,7 @@ Wandering dots aside, let me show you how this graph compares to the control cas The peak of the infection curve is a little lower than the control, but in terms of the total number of cases, keeping that central location active really defeats the effects of otherwise social distancing. Now let me ask you to make a prediction. What do you think will be more effective? -If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face. +If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face. The simulation on the left requires our dots to very heavily alter their daily routines, whereas on the right our dots can mostly continue their usual habits, but are much more conscious of hygiene. They're actually nearly identical, which surprised me, given that one of them is a fivefold decrease and the other is twofold. I guess it goes to show that the effect of hygiene, which is maybe easier said than done, really goes a long way. diff --git a/2020/epidemic-simulations/estonian/sentence_translations.json b/2020/epidemic-simulations/estonian/sentence_translations.json index 9e090b46a..7e54cee92 100644 --- a/2020/epidemic-simulations/estonian/sentence_translations.json +++ b/2020/epidemic-simulations/estonian/sentence_translations.json @@ -908,7 +908,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "", "from_community_srt": "siis nakatumine väheneb. Võrdluseks - Covid19 puhul on R0 väärtuseks hinnanguliselt natuke üle 2, mis on sama nagu keskmine R0 hinnang 1918.", "n_reviews": 0, @@ -1043,7 +1043,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "", "from_community_srt": "kui lisaks sotsiaalsele eraldumisele me vähendaks keskuse külastamise sagedust näiteks 5 korda? Või kui me vähendaks nakatumise tõenäosust veel poole võrra, näiteks kui kõik inimesed hakkaksid hügieenireegleid täitma? Proovime mõlemad ära.", "n_reviews": 0, diff --git a/2020/epidemic-simulations/french/sentence_translations.json b/2020/epidemic-simulations/french/sentence_translations.json index 487646df9..fb7845a2e 100644 --- a/2020/epidemic-simulations/french/sentence_translations.json +++ b/2020/epidemic-simulations/french/sentence_translations.json @@ -1022,7 +1022,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "À titre de comparaison, le R0 pour COVID-19 est estimé à un peu plus de 2, ce qui correspond également à l'estimation moyenne du R0 lors de la pandémie de grippe espagnole de 1918.", "model": "DeepL", "from_community_srt": "Pour comparer, R0 pour le Covid-19 est estimé à un peu plus de 2 ce qui correspond aussi à l'estimation grossière de R0 pour la pandémie de grippe espagnole en 1918.", @@ -1175,7 +1175,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "Si, en plus de cet effet de distanciation sociale, nous diminuons la fréquence à laquelle les gens se rendent à cet endroit central, peut-être d'un facteur 5, ou si nous réduisons la probabilité d'infection d'un autre facteur 2, ce qui signifie que les gens sont très prudents en se lavant les mains et en ne se touchant pas le visage.", "model": "DeepL", "from_community_srt": "si, en plus de la distanciation sociale, on diminue la fréquence à laquelle les individus vont à ce lieu commun, peut-être par un facteur 5, ou si on divise de nouveau la probabilité de contagion par 2, comme si, par exemple, les individus étaient très vigilants au fait de se laver régulièrement les mains et de ne pas se toucher le visage ?", diff --git a/2020/epidemic-simulations/georgian/sentence_translations.json b/2020/epidemic-simulations/georgian/sentence_translations.json index e43b94d01..9576fef5e 100644 --- a/2020/epidemic-simulations/georgian/sentence_translations.json +++ b/2020/epidemic-simulations/georgian/sentence_translations.json @@ -908,7 +908,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "", "from_community_srt": "ინფექცია გაქრობის ფაზაშია. შესადარებლად, COVID-19-ის R0 არის ცოტათი მეტი ვიდრე 2, რაც დაახლოებით იგივეა, რაც ესპანური გრიპისა 1918 წელს.", "n_reviews": 0, @@ -1043,7 +1043,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "", "from_community_srt": "თუ სოციალურ დისტანცირებასთან ერთად ხუთჯერ შევამცირებთ სიხშირეს, რითაც ხალხი მიდის ცენტრალურ ადგილას? ან თუ კიდევ ორჯერ შევამცირებთ ინფიცირების ალბათობას, მაგალითად რაც შეიძლება ნიშნავდეს უკეთეს ჰიგიენას? კეთილი,", "n_reviews": 0, diff --git a/2020/epidemic-simulations/german/sentence_translations.json b/2020/epidemic-simulations/german/sentence_translations.json index b3f9fbceb..b136d12ae 100644 --- a/2020/epidemic-simulations/german/sentence_translations.json +++ b/2020/epidemic-simulations/german/sentence_translations.json @@ -1024,7 +1024,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "Zum Vergleich: Der R0-Wert für COVID-19 wird auf etwas über 2 geschätzt, was in etwa dem durchschnittlichen R0-Wert während der Spanischen Grippe-Pandemie 1918 entspricht.", "model": "DeepL", "from_community_srt": "Zum Vergleich: R0 für CoViD-19 wird auf etwas über 2 geschätzt. Was nahe dem Wert für R0 während der Pandemie der spanischen Grippe von 1918.", @@ -1176,7 +1176,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "Wenn wir zusätzlich zu diesem sozialen Distanzierungseffekt die Häufigkeit, mit der Menschen diesen zentralen Ort aufsuchen, vielleicht um den Faktor 5 verringern, oder wenn wir die Ansteckungswahrscheinlichkeit um einen weiteren Faktor 2 verringern, was bedeutet, dass die Menschen besonders vorsichtig sind, wenn sie sich die Hände waschen und ihr Gesicht nicht berühren.", "model": "DeepL", "from_community_srt": "Ein Absenken der Besuchsfrequenz des zentralen Ortes zusätzlich zum social distancing, Vielleicht etwa um den Faktor 5? Oder eine erneutes Senken des Infektionsrisikos im Kreis um den Faktor 2? Das könnte zum Beispiel bedeuten, dass die Leute jetzt besonders darauf achten, sich regelmäßig ihr Hände zu waschen und sich nicht ins Gesicht zu fassen.", diff --git a/2020/epidemic-simulations/greek/sentence_translations.json b/2020/epidemic-simulations/greek/sentence_translations.json index aed193237..5c34c01bb 100644 --- a/2020/epidemic-simulations/greek/sentence_translations.json +++ b/2020/epidemic-simulations/greek/sentence_translations.json @@ -909,7 +909,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "", "from_community_srt": "Κάνοντας τη σύγκριση, το R0 της νόσου COVID-19 εκτιμάται λίγο πάνω από το 2, ίδια τιμή περίπου με το R0 που εκτιμάται ότι είχε η πανδημία της «ισπανικής γρίππης» του 1918.", "n_reviews": 0, @@ -1044,7 +1044,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "", "from_community_srt": "Τι θα είχε καλύτερο αποτέλεσμα; Να εφαρμόσουμε μαζί με την κοινωνική απόσταση και ένα περιορισμό της συχνότητας επίσκεψης των ανθρώπων από και προς την κεντρική τοποθεσία, στο ένα πέμπτο; Ή να ελαττώσουμε την πιθανότητα μετάδοσης πάλι κατά το 1/2, π.χ. όταν οι άνθρωποι γίνονται εξαιρετικά προσεκτικοί με το πλύσιμο των χεριών και το άγγιγμα του προσώπου.", "n_reviews": 0, diff --git a/2020/epidemic-simulations/hebrew/sentence_translations.json b/2020/epidemic-simulations/hebrew/sentence_translations.json index 3bd3303a4..972b0a294 100644 --- a/2020/epidemic-simulations/hebrew/sentence_translations.json +++ b/2020/epidemic-simulations/hebrew/sentence_translations.json @@ -1024,7 +1024,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "לשם השוואה, R0 עבור COVID-19 מוערך בקצת מעל 2, וזה גם בערך מה שההערכה הממוצעת עבור R0 הייתה במהלך מגיפת השפעת הספרדית של 1918.", "model": "google_nmt", "from_community_srt": "לשם השוואה, הערכה של R0 עבור COVID-19 היא מעט מעל 2, שזה גם בערך האומדן הממוצע ל R0 במהלך מגיפת השפעת הספרדית ב -1918.", @@ -1176,7 +1176,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "אם בנוסף לאפקט ההתרחקות החברתי הזה, אנחנו מורידים את התדירות שבה אנשים הולכים לאותו מקום מרכזי, אולי בפקטור של 5, או אם נצמצם את ההסתברות להידבקות בפקטור נוסף של 2, כלומר אנשים הם זהירים במיוחד. על שטיפת ידיים ואי נגיעה בפניהם.", "model": "google_nmt", "from_community_srt": "אם על גבי \"הריחוק החברתי\" נפחית את התדירות בה אנשים הולכים למקום המרכזי פי 5? או אם נקטין את ההסתברות להדבקה שוב לחצי, למשל,", diff --git a/2020/epidemic-simulations/hindi/sentence_translations.json b/2020/epidemic-simulations/hindi/sentence_translations.json index fc4762d0c..00d2a593c 100644 --- a/2020/epidemic-simulations/hindi/sentence_translations.json +++ b/2020/epidemic-simulations/hindi/sentence_translations.json @@ -416,7 +416,7 @@ "end": 413.19 }, { - "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, would you predict it'll be? ", + "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, but where on the spectrum would you predict it'll be? ", "translatedText": "जाहिर तौर पर इसका परिणाम संपूर्ण संगरोध और कुछ न करने के बीच कहीं न कहीं होगा, क्या आप अनुमान लगा सकते हैं कि ऐसा होगा? ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 580.45 }, { - "input": "It also means that the most dangerous viruses are the ones that kill some part of the population while laying unnoticed and spreadable among others, or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", + "input": "It also means that the most dangerous viruses are the ones that kill some part of the population in the first place, while laying unnoticed and spreadable among others. Or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", "translatedText": "इसका मतलब यह भी है कि सबसे खतरनाक वायरस वे हैं जो आबादी के कुछ हिस्से को मार देते हैं, जबकि वे किसी का ध्यान नहीं जाते और दूसरों के बीच फैल जाते हैं, या इससे भी बदतर, अगर वे किसी का ध्यान नहीं जाते और अंततः घातक बनने से पहले सभी में फैल जाते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 640.27 }, { - "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel too close to their neighbor. ", + "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel just a little too close to their neighbor. ", "translatedText": "इन एनिमेशन में, मैं इसे लोगों के बीच एक प्रतिकारक शक्ति के रूप में लागू करूंगा, और जब वे अपने पड़ोसी के बहुत करीब महसूस करेंगे तो उन्हें पीले रंग में चमकाऊंगा।", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 738.41 }, { - "input": "Again, I'll emphasize that these are toy models, and I leave it to the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", + "input": "Again, I'll emphasize that these are toy models, and I leave it to the intelligence of the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", "translatedText": "फिर से, मैं इस बात पर जोर दूंगा कि ये खिलौना मॉडल हैं, और मैं इसे दर्शकों पर छोड़ता हूं कि वे निर्णय लें कि क्या इन छोटे बिंदुओं का व्यवहार सटीक रूप से दर्शाता है कि आपके और आपके जीवन के लिए सामाजिक दूरी का क्या मतलब होगा।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/epidemic-simulations/hungarian/sentence_translations.json b/2020/epidemic-simulations/hungarian/sentence_translations.json index fee7a1e6b..53ab5cfb2 100644 --- a/2020/epidemic-simulations/hungarian/sentence_translations.json +++ b/2020/epidemic-simulations/hungarian/sentence_translations.json @@ -1023,7 +1023,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "Összehasonlításképpen, a COVID-19 R0 becsült értéke valamivel 2 fölött van, ami szintén nagyjából megfelel az 1918-as spanyolnátha világjárvány idején becsült R0 átlagának.", "model": "DeepL", "from_community_srt": "hogy hanyatlik. Összehasonlításképp a COVID-19 R₀ értéke kicsivel 2 feletti, ami nagyjából megegyezik az 1918-as spanyolnátha-világjárvány becsült R₀ értékével.", @@ -1176,7 +1176,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "Ha ezen a társadalmi távolságtartó hatáson felül csökkentjük a központi hely felkeresésének gyakoriságát, talán 5-szeresére, vagy ha a fertőzés valószínűségét még 2-szeresére csökkentjük, vagyis az emberek extra óvatosak lesznek a kézmosás és az arcukhoz való hozzáérés terén.", "model": "DeepL", "from_community_srt": "ha a szociális távolságtartás mellett 5-ödére csökkentjük a központi helyre történő látogatások sűrűségét? Vagy ha ismét a felére csökkentjük a fertőzés esélyét, azt kifejezve, hogy az emberek higiénikusabbak? Nos,", diff --git a/2020/epidemic-simulations/indonesian/sentence_translations.json b/2020/epidemic-simulations/indonesian/sentence_translations.json index 53e58db95..9189592c0 100644 --- a/2020/epidemic-simulations/indonesian/sentence_translations.json +++ b/2020/epidemic-simulations/indonesian/sentence_translations.json @@ -1024,7 +1024,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "Sebagai perbandingan, R0 untuk COVID-19 diperkirakan sedikit di atas 2, yang juga berada di sekitar estimasi rata-rata untuk R0 selama pandemi flu Spanyol tahun 1918.", "model": "DeepL", "from_community_srt": "Sebagi berbandingan, R0 untuk Covid-19 diperkirakan sedikit di atas 2, yang artinya mirip dengan perkiraan R0 ketika pandemi flu spanyol 1918 terjadi.", @@ -1176,7 +1176,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "Jika di atas efek social distancing itu, kita mengurangi frekuensi orang pergi ke tempat pusat, mungkin dengan faktor 5, atau jika kita mengurangi kemungkinan penularan dengan faktor 2, yang berarti orang akan sangat berhati-hati dalam mencuci tangan dan tidak menyentuh wajah mereka.", "model": "DeepL", "from_community_srt": "ketika diterapkan jarak sosial dengan menurunkan frekuensi orang pergi ke pusat keramaian menjadi seper-limanya, atau kita memotong kemungkinan infeksi turun menjadi seper-duanya, misal, orang mulai menerapkan hidup bersih yang lebih baik? Mari kita mainkan keduanya.", diff --git a/2020/epidemic-simulations/italian/sentence_translations.json b/2020/epidemic-simulations/italian/sentence_translations.json index ee42c01fb..d6324017d 100644 --- a/2020/epidemic-simulations/italian/sentence_translations.json +++ b/2020/epidemic-simulations/italian/sentence_translations.json @@ -1023,7 +1023,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "A titolo di confronto, l'R0 per il COVID-19 è stimato a poco più di 2, che è anche circa la stima media dell'R0 durante la pandemia di influenza spagnola del 1918.", "model": "DeepL", "from_community_srt": "(\"ipodemia\"?) Per fare un confronto con la realtà, Ro per il COVID-19 è stato stimato avere un valore un po' al di sopra del 2, che è anche circa la media stimata per Ro durante la pandemia di influenza spagnola del 1918.", @@ -1175,7 +1175,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "Se oltre a questo effetto di allontanamento sociale, diminuiamo la frequenza con cui le persone si recano in quel punto centrale, magari di un fattore 5, o se riduciamo la probabilità di infezione di un altro fattore 2, significa che le persone sono super caute nel lavarsi le mani e nel non toccarsi il viso.", "model": "DeepL", "from_community_srt": "Cosa pensate che sarebbe più efficace se oltre al distanziamento sociale diminuissimo di un fattore 5 la frequenza con la quale le persone vanno al punto centrale? O se dimezzassimo ulteriormente la probabilità di infezione, per rappresentare il fatto che la gente migliora le proprie pratiche igieniche? Beh,", diff --git a/2020/epidemic-simulations/japanese/sentence_translations.json b/2020/epidemic-simulations/japanese/sentence_translations.json index 182141040..5d2609a94 100644 --- a/2020/epidemic-simulations/japanese/sentence_translations.json +++ b/2020/epidemic-simulations/japanese/sentence_translations.json @@ -440,7 +440,7 @@ "end": 413.19 }, { - "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, would you predict it'll be? ", + "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, but where on the spectrum would you predict it'll be? ", "translatedText": "明らかに、これは完全な隔離と何もしない の間のどこかの結果になると思いますか? ", "model": "google_nmt", "n_reviews": 0, @@ -575,7 +575,7 @@ "end": 580.45 }, { - "input": "It also means that the most dangerous viruses are the ones that kill some part of the population while laying unnoticed and spreadable among others, or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", + "input": "It also means that the most dangerous viruses are the ones that kill some part of the population in the first place, while laying unnoticed and spreadable among others. Or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", "translatedText": "また、最も危険なウイルスとは、人口の一部を死に至らしめながら、気づかれずに 蔓延し、他の人に蔓延するウイルス、あるいはさらに悪いことに、ウイルスが気付 かれずに全員に蔓延し、最終的に致死的になるウイルスであることも意味します。", "model": "google_nmt", "from_community_srt": "また最も危険なウイルスは人口の一部が致命傷となり 残りが無意識に広げられるようなものです もしくは、症状が出る前に広げられるようになり、徐々に症状が悪化するものも出てくるかもしれません", @@ -626,7 +626,7 @@ "end": 640.27 }, { - "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel too close to their neighbor. ", + "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel just a little too close to their neighbor. ", "translatedText": "これらのアニメーションでは、これを人々間の斥力として 適用し、隣人に近づきすぎると黄色に光るようにします。", "model": "google_nmt", "n_reviews": 0, @@ -741,7 +741,7 @@ "end": 738.41 }, { - "input": "Again, I'll emphasize that these are toy models, and I leave it to the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", + "input": "Again, I'll emphasize that these are toy models, and I leave it to the intelligence of the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", "translatedText": "繰り返しになりますが、これらはおもちゃのモデルであることを強調します。これらの小さな点の動作が、社会的距離があなたとあなたの人生にとって何 を意味するかを正確に反映しているかどうかの判断は、見る人に委ねます。", "model": "google_nmt", "from_community_srt": "たった10%人が指示を聞かないのが 感染症が続く元となってしまいます もちろん、これはただのモデルなのでどのようにこれを読み取って、これがどれ程現実に近い物か判断するのは、視聴者の皆様に任せます。", diff --git a/2020/epidemic-simulations/korean/sentence_translations.json b/2020/epidemic-simulations/korean/sentence_translations.json index 70364c709..3f0e58310 100644 --- a/2020/epidemic-simulations/korean/sentence_translations.json +++ b/2020/epidemic-simulations/korean/sentence_translations.json @@ -990,7 +990,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "비교를 위해 코로나19의 R0는 2를 약간 상회하는 것으로 추정되며, 이는 1918년 스페인 독감 대유행 당시 R0의 평균 추정치와 비슷한 수준입니다.", "model": "DeepL", "from_community_srt": "(…비유행병?) 비교하자면 코로나19의 R0값은 2보다 조금 높다고 추정되는데 이는 1918년 스페인독감 유행시기의 평균 R0 추정치에 가깝습니다.", @@ -1139,7 +1139,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "사회적 거리두기 효과에 더해 사람들이 그 중심 장소에 가는 빈도를 5배로 줄이거나 감염 확률을 2배로 줄인다면, 사람들이 손을 씻고 얼굴을 만지지 않도록 매우 조심하게 될 것입니다.", "model": "DeepL", "from_community_srt": "뭐가 더 효과적인 것 같습니까? 사회적 거리두기를 유지한 채, 중심지 방문빈도를 낮추기? 1/5정도로? 아니면 감염확률을 줄이기? 이번에는 1/2 정도로? 예를 들어 사람들이 정말 엄청 너무 조심스러워서", diff --git a/2020/epidemic-simulations/lithuanian/sentence_translations.json b/2020/epidemic-simulations/lithuanian/sentence_translations.json index 216be2192..151ecace5 100644 --- a/2020/epidemic-simulations/lithuanian/sentence_translations.json +++ b/2020/epidemic-simulations/lithuanian/sentence_translations.json @@ -907,7 +907,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "", "from_community_srt": "užsikrėtimų mažėja. Palyginimui, įvertintas COVID-19 R0 yra šiek tiek aukščiau 2, kuris taip pat yra vidutinis R0 įvertinimas per 1918 m. ispaniškojo gripo pandemiją.", "n_reviews": 0, @@ -1043,7 +1043,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "", "from_community_srt": "jei kartu su atstumo laikymusi, mes sumažinsime žmonių ejimo į centrinę vietą dažnį 5 kartais. Arba jei kirsime užsikrėtimo galimybę pusiau, pavyzdžiui manydami kad, žmonės gerina higieną? Išbandykime šiuos scenarijus.", "n_reviews": 0, diff --git a/2020/epidemic-simulations/marathi/sentence_translations.json b/2020/epidemic-simulations/marathi/sentence_translations.json index f47ec5636..398247a8e 100644 --- a/2020/epidemic-simulations/marathi/sentence_translations.json +++ b/2020/epidemic-simulations/marathi/sentence_translations.json @@ -416,7 +416,7 @@ "end": 413.19 }, { - "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, would you predict it'll be? ", + "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, but where on the spectrum would you predict it'll be? ", "translatedText": "साहजिकच याचा परिणाम एकूण अलग ठेवणे आणि काहीही न करण्याच्या दरम्यान कुठेतरी होईल, आपण ते होईल असे भाकीत कराल का? ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 580.45 }, { - "input": "It also means that the most dangerous viruses are the ones that kill some part of the population while laying unnoticed and spreadable among others, or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", + "input": "It also means that the most dangerous viruses are the ones that kill some part of the population in the first place, while laying unnoticed and spreadable among others. Or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", "translatedText": "याचा अर्थ असाही होतो की सर्वात धोकादायक विषाणू असे असतात जे लोकसंख्येच्या काही भागाला मारतात आणि इतरांमध्ये लक्ष न देता आणि पसरवता येतात किंवा त्याहूनही वाईट, जर ते अखेरीस प्राणघातक होण्याआधी सर्वांमध्ये लक्ष न दिलेले आणि पसरण्यासारखे राहिले तर. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 640.27 }, { - "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel too close to their neighbor. ", + "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel just a little too close to their neighbor. ", "translatedText": "या अॅनिमेशनमध्ये, मी ते लोकांमध्ये तिरस्करणीय शक्ती म्हणून लागू करेन आणि जेव्हा त्यांना त्यांच्या शेजाऱ्याच्या खूप जवळ वाटत असेल तेव्हा त्यांना पिवळ्या रंगाची चमक दाखवू. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 738.41 }, { - "input": "Again, I'll emphasize that these are toy models, and I leave it to the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", + "input": "Again, I'll emphasize that these are toy models, and I leave it to the intelligence of the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", "translatedText": "पुन्हा, मी यावर जोर देईन की हे खेळण्यांचे मॉडेल आहेत आणि या छोट्या ठिपक्यांचे वर्तन तुमच्यासाठी आणि तुमच्या जीवनासाठी सामाजिक अंतराचा अर्थ काय आहे हे अचूकपणे प्रतिबिंबित करते का ते मी दर्शकांवर सोडतो. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/epidemic-simulations/persian/sentence_translations.json b/2020/epidemic-simulations/persian/sentence_translations.json index 5ba756408..3e3871b59 100644 --- a/2020/epidemic-simulations/persian/sentence_translations.json +++ b/2020/epidemic-simulations/persian/sentence_translations.json @@ -1009,7 +1009,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "برای مقایسه، R0 برای COVID-19 کمی بالاتر از 2 تخمین زده می‌شود، که تقریباً همان چیزی است که میانگین تخمین R0 در طول همه‌گیری آنفولانزای اسپانیایی 1918 بود.", "model": "google_nmt", "from_community_srt": "به عنوان مثال، R0 برای کووید-۱۹ کمی بالاتر از ۲ تخمین زده میشود، که در حدود بیماری دنیاگیر آنفولانزای اسپانیایی 1918 است.", @@ -1159,7 +1159,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "اگر در کنار این اثر فاصله‌گذاری اجتماعی، تعداد دفعات رفتن افراد به آن نقطه مرکزی را کاهش دهیم، شاید تا 5 برابر، یا اگر احتمال ابتلا را با ضریب دیگری کاهش دهیم، به این معنی که افراد فوق‌العاده محتاط هستند. در مورد شستن دست ها و دست نزدن به صورتشان.", "model": "google_nmt", "from_community_srt": "حالا فکر میکنید کدامیک از این دو موثرتر میباشند؟ اگر علاوه بر فاصله اجتماعی، ما میزان رجوع مردم به مکانهای مرکزی را به یک پنجم کاهش دهیم. یا اینکه احتمال سرایت عفونت را با ضریب ۲ کاهش دهیم. مثل اینکه مردم به طور کاملا آگاهانه دستانشان را بشورند و به صورتشان دست نزنند.", diff --git a/2020/epidemic-simulations/polish/sentence_translations.json b/2020/epidemic-simulations/polish/sentence_translations.json index 3d23abf2d..977c00e91 100644 --- a/2020/epidemic-simulations/polish/sentence_translations.json +++ b/2020/epidemic-simulations/polish/sentence_translations.json @@ -909,7 +909,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "", "from_community_srt": "że ​​spada. Dla porównania szacuje się R0 dla COVID-19 być trochę powyżej 2, co też jest w pobliżu średniego oszacowania R0 podczas hiszpanki w 1918 roku.", "n_reviews": 0, @@ -1045,7 +1045,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "", "from_community_srt": "jeśli oprócz dystansu społecznego zmniejszamy się częstotliwość, z jaką ludzie idą do centralnego miejsca 5-krotnie? Czy jeśli zmniejszymy prawdopodobieństwo infekcji w dół o kolejny współczynnik 2, na przykład ludzie znaczenie poprawią higienę? Przeprowadźmy obie symulacje.", "n_reviews": 0, diff --git a/2020/epidemic-simulations/portuguese/sentence_translations.json b/2020/epidemic-simulations/portuguese/sentence_translations.json index 1f7aabc65..910c38a0c 100644 --- a/2020/epidemic-simulations/portuguese/sentence_translations.json +++ b/2020/epidemic-simulations/portuguese/sentence_translations.json @@ -1024,7 +1024,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "Para efeito de comparação, estima-se que o R0 para a COVID-19 esteja um pouco acima de 2, o que também é próximo do que foi a estimativa média para o R0 durante a pandemia de gripe espanhola de 1918.", "model": "google_nmt", "from_community_srt": "Para comparação, estima-se que R_0 para COVID-19 seja um pouco acima de 2, o que também é próximo da estimativa média para R_0 durante a pandemia de gripe espanhola de 1918.", @@ -1177,7 +1177,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "Se, além do efeito do distanciamento social, diminuirmos a frequência com que as pessoas vão para aquele local central, talvez por um fator de 5, ou se reduzirmos a probabilidade de infecção por outro fator de 2, o que significa que as pessoas são extremamente cautelosas. sobre lavar as mãos e não tocar no rosto.", "model": "google_nmt", "from_community_srt": "além do distanciamento social, diminuirmos a frequência com que as pessoas vão ao ponto central por um fator de 5? Ou se reduzirmos a probabilidade de infecção por um outro fator de 2, por exemplo,", diff --git a/2020/epidemic-simulations/romanian/sentence_translations.json b/2020/epidemic-simulations/romanian/sentence_translations.json index 867d846d6..caba35f56 100644 --- a/2020/epidemic-simulations/romanian/sentence_translations.json +++ b/2020/epidemic-simulations/romanian/sentence_translations.json @@ -908,7 +908,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "", "from_community_srt": "Pentru comparație, R0 pentru COVID-19 este estimat la puțin peste 2, care este și în jur media estimării pentru R0 în perioada pandemiei de gripă spaniolă din 1918.", "n_reviews": 0, @@ -1043,7 +1043,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "", "from_community_srt": "dacă pe deasupra distanțării sociale scădem frecvența cu care oamenii merg la punct central cu un factor de 5? Sau dacă reducem probabilitatea de infecție cu un alt factor de 2, de exemplu, prin practicarea unei igiene mai bune? Ei bine,", "n_reviews": 0, diff --git a/2020/epidemic-simulations/russian/sentence_translations.json b/2020/epidemic-simulations/russian/sentence_translations.json index a28fd044a..d4a24db19 100644 --- a/2020/epidemic-simulations/russian/sentence_translations.json +++ b/2020/epidemic-simulations/russian/sentence_translations.json @@ -1024,7 +1024,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "Для сравнения, R0 для COVID-19 оценивается чуть выше 2, что также примерно соответствует средней оценке R0 во время пандемии испанского гриппа 1918 года.", "model": "DeepL", "from_community_srt": "Для сравнения, R_0 для COVID-19 держится чуть выше 2. Такое же R_0 было у пандемии испанского гриппа в 1918 году.", @@ -1176,7 +1176,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "Если в дополнение к этому эффекту социального дистанцирования мы уменьшим частоту посещения центрального места, возможно, в 5 раз, или снизим вероятность заражения еще в 2 раза, то есть люди будут с особой осторожностью мыть руки и не прикасаться к лицу.", "model": "DeepL", "from_community_srt": "если в системе с социальной дистанцией: мы понизим частоту посещения центра в пять раз или если мы снизим шанс заражения в два раза, например,", diff --git a/2020/epidemic-simulations/spanish/sentence_translations.json b/2020/epidemic-simulations/spanish/sentence_translations.json index 916160c02..690a16fd3 100644 --- a/2020/epidemic-simulations/spanish/sentence_translations.json +++ b/2020/epidemic-simulations/spanish/sentence_translations.json @@ -1023,7 +1023,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "A modo de comparación, se estima que el R0 de COVID-19 es un poco superior a 2, que es también alrededor de lo que fue la estimación media del R0 durante la pandemia de gripe española de 1918.", "model": "DeepL", "from_community_srt": "significa que está a la baja. Para comparar, el R0 del COVID-19 se estima un poco por encima de 2, lo que también es el estimado medio de R0 durante la pandemia de la gripe española de 1918 La influenza de temporada,", @@ -1175,7 +1175,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "Si además de ese efecto de distanciamiento social, disminuimos la frecuencia con la que la gente acude a ese punto central, quizá en un factor de 5, o si reducimos la probabilidad de infección en otro factor de 2, lo que significa que la gente es súper precavida en cuanto a lavarse las manos y no tocarse la cara.", "model": "DeepL", "from_community_srt": "encima del distanciamiento social, disminuimos la frecuencia a la que la gente va al punto central por un factor de 5? ¿O si recortamos la probabilidad de infección por un factor de 2 por ejemplo, signficando que la gente tiene una mejor higiene? Bueno,", diff --git a/2020/epidemic-simulations/tamil/sentence_translations.json b/2020/epidemic-simulations/tamil/sentence_translations.json index 4004e70df..605dd3378 100644 --- a/2020/epidemic-simulations/tamil/sentence_translations.json +++ b/2020/epidemic-simulations/tamil/sentence_translations.json @@ -416,7 +416,7 @@ "end": 413.19 }, { - "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, would you predict it'll be? ", + "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, but where on the spectrum would you predict it'll be? ", "translatedText": "வெளிப்படையாக, இது மொத்த தனிமைப்படுத்தலுக்கும் எதுவும் செய்யாமல் இருப்பதற்கும் இடையில் ஒரு விளைவை ஏற்படுத்தும், அது இருக்கும் என்று நீங்கள் கணிப்பீர்களா? ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 580.45 }, { - "input": "It also means that the most dangerous viruses are the ones that kill some part of the population while laying unnoticed and spreadable among others, or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", + "input": "It also means that the most dangerous viruses are the ones that kill some part of the population in the first place, while laying unnoticed and spreadable among others. Or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", "translatedText": "மிகவும் ஆபத்தான வைரஸ்கள் மக்கள்தொகையில் ஒரு பகுதியைக் கொல்லும் போது அவை கவனிக்கப்படாமல் மற்றவர்களுக்கு பரவக்கூடியவை, அல்லது இன்னும் மோசமாக, அவை கவனிக்கப்படாமல் மற்றும் இறுதியில் மரணமடைவதற்கு முன்பு அனைவருக்கும் பரவக்கூடியதாக இருந்தால். ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 640.27 }, { - "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel too close to their neighbor. ", + "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel just a little too close to their neighbor. ", "translatedText": "இந்த அனிமேஷன்களில், நான் அதை மக்களிடையே விரட்டும் சக்தியாகப் பயன்படுத்துவேன், மேலும் அவர்கள் தங்கள் அண்டை வீட்டாருடன் மிகவும் நெருக்கமாக உணரும்போது அவர்கள் மஞ்சள் நிறத்தில் ஒளிரச் செய்வேன். ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 738.41 }, { - "input": "Again, I'll emphasize that these are toy models, and I leave it to the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", + "input": "Again, I'll emphasize that these are toy models, and I leave it to the intelligence of the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", "translatedText": "மீண்டும், இவை பொம்மை மாதிரிகள் என்பதை நான் வலியுறுத்துகிறேன், மேலும் இந்த சிறிய புள்ளிகளின் நடத்தை உங்களுக்கும் உங்கள் வாழ்க்கைக்கும் சமூக விலகல் என்ன என்பதை துல்லியமாக பிரதிபலிக்கிறதா என்பதை தீர்மானிக்க பார்வையாளருக்கு விட்டுவிடுகிறேன். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/epidemic-simulations/telugu/sentence_translations.json b/2020/epidemic-simulations/telugu/sentence_translations.json index 5fb5ec57b..71f6c2f45 100644 --- a/2020/epidemic-simulations/telugu/sentence_translations.json +++ b/2020/epidemic-simulations/telugu/sentence_translations.json @@ -416,7 +416,7 @@ "end": 413.19 }, { - "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, would you predict it'll be? ", + "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, but where on the spectrum would you predict it'll be? ", "translatedText": "సహజంగానే ఇది మొత్తం నిర్బంధం మరియు ఏమీ చేయకుండా మధ్య ఎక్కడో ఒక ఫలితాన్ని కలిగి ఉంటుంది, ఇది ఉంటుందని మీరు అంచనా వేస్తారా? ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 580.45 }, { - "input": "It also means that the most dangerous viruses are the ones that kill some part of the population while laying unnoticed and spreadable among others, or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", + "input": "It also means that the most dangerous viruses are the ones that kill some part of the population in the first place, while laying unnoticed and spreadable among others. Or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", "translatedText": "అత్యంత ప్రమాదకరమైన వైరస్‌లు జనాభాలో కొంత భాగాన్ని గుర్తించకుండా మరియు ఇతరులలో వ్యాప్తి చెందే సమయంలో చంపేస్తాయని లేదా చివరికి ప్రాణాంతకంగా మారడానికి ముందు ప్రతి ఒక్కరిలో గుర్తించబడకుండా మరియు వ్యాప్తి చెందితే ఇంకా అధ్వాన్నంగా ఉంటాయని కూడా దీని అర్థం. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 640.27 }, { - "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel too close to their neighbor. ", + "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel just a little too close to their neighbor. ", "translatedText": "ఈ యానిమేషన్‌లలో, నేను దానిని వ్యక్తుల మధ్య వికర్షణ శక్తిగా వర్తింపజేస్తాను మరియు వారు తమ పొరుగువారికి చాలా దగ్గరగా ఉన్నట్లు అనిపించినప్పుడు వారు పసుపు రంగులో మెరిసిపోతారు. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 738.41 }, { - "input": "Again, I'll emphasize that these are toy models, and I leave it to the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", + "input": "Again, I'll emphasize that these are toy models, and I leave it to the intelligence of the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", "translatedText": "మళ్ళీ, ఇవి బొమ్మల నమూనాలు అని నేను నొక్కిచెబుతున్నాను మరియు ఈ చిన్న చుక్కల ప్రవర్తన మీకు మరియు మీ జీవితానికి సామాజిక దూరం ఏమిటో ఖచ్చితంగా ప్రతిబింబిస్తుందో లేదో నిర్ధారించడానికి వీక్షకుడికి వదిలివేస్తాను. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/epidemic-simulations/thai/sentence_translations.json b/2020/epidemic-simulations/thai/sentence_translations.json index de80b21d7..e30c244a7 100644 --- a/2020/epidemic-simulations/thai/sentence_translations.json +++ b/2020/epidemic-simulations/thai/sentence_translations.json @@ -999,7 +999,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "สำหรับการเปรียบเทียบ R0 สำหรับโรคโควิด-19 คาดว่าจะสูงกว่า 2 เล็กน้อย ซึ่งก็ใกล้เคียงกับค่าประมาณเฉลี่ยของ R0 ในช่วงการระบาดใหญ่ของไข้หวัดใหญ่สเปนปี 1918 เช่นกัน", "model": "google_nmt", "from_community_srt": "และถ้าค่าต่ำกว่า 1 คือการที่โรคนั้นกำลังจะหายไป ถ้าจะลองเทียบดู ค่า R0 ของโรคโควิด-19 จะมากกว่า 2 เล็กน้อย", @@ -1149,7 +1149,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "หากนอกเหนือจากผลกระทบการเว้นระยะห่างทางสังคมแล้ว เราจะลดความถี่ที่ผู้คนไปยังจุดศูนย์กลางนั้น อาจลดลง 5 เท่า หรือถ้าเราลดความน่าจะเป็นในการติดเชื้อลงอีก 2 เท่า หมายความว่าผู้คนมีความระมัดระวังเป็นพิเศษเป็นพิเศษ เกี่ยวกับการล้างมือและไม่สัมผัสใบหน้า", "model": "google_nmt", "from_community_srt": "ระหว่างการลดความถี่ของการเดินทางไปศูนย์กลางชุมชน ลงจนเหลือประมาณ 1 ใน 5 หรือการลดโอกาสติดเชื้อลงเหลือครึ่งเดียว หรืออีกนัยหนึ่งก็คือ มีสุขอนามัยป้องกันตัวเองอย่างดี คอยล้างมือและไม่จับหน้าตัวเองอย่างจริงจัง", diff --git a/2020/epidemic-simulations/turkish/sentence_translations.json b/2020/epidemic-simulations/turkish/sentence_translations.json index 7303906b0..0f2f17b1c 100644 --- a/2020/epidemic-simulations/turkish/sentence_translations.json +++ b/2020/epidemic-simulations/turkish/sentence_translations.json @@ -1021,7 +1021,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "Karşılaştırma yapmak gerekirse, COVID-19 için R0'ın 2'nin biraz üzerinde olduğu tahmin edilmektedir ki bu da 1918 İspanyol gribi pandemisi sırasında R0 için yapılan ortalama tahminin civarındadır.", "model": "DeepL", "from_community_srt": "Örneğin, COVID-19 için R0 sayısının 2’nin biraz üzerinde olduğu hesaplanmaktadır, bu aynı zamanda 1918’deki İspanyol Gribi salgınının R0 değerine yakındır.", @@ -1173,7 +1173,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "Bu sosyal mesafe etkisine ek olarak, insanların o merkezi noktaya gitme sıklığını belki 5 kat azaltırsak veya enfeksiyon olasılığını 2 kat daha düşürürsek, yani insanlar ellerini yıkama ve yüzlerine dokunmama konusunda çok daha dikkatli olurlar.", "model": "DeepL", "from_community_srt": "sosyal mesafelendirmenin üzerine ortak noktayı ziyaret eden insanların sayısını beşte birine düşürsek sizce ne olur? Veya enfeksiyon olasılığını yarıya düşürsek, örneğin bu insanlar daha iyi hijyen uyguladıklarında gerçekleşir. Pekala,", diff --git a/2020/epidemic-simulations/ukrainian/sentence_translations.json b/2020/epidemic-simulations/ukrainian/sentence_translations.json index 1296671ca..15051f983 100644 --- a/2020/epidemic-simulations/ukrainian/sentence_translations.json +++ b/2020/epidemic-simulations/ukrainian/sentence_translations.json @@ -1023,7 +1023,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "Для порівняння, R0 для COVID-19 оцінюється трохи вище 2, що також приблизно відповідає середньому значенню R0 під час пандемії іспанського грипу 1918 року.", "model": "DeepL", "from_community_srt": "що хвороба на спаді. Для порівняння: R0 для COVID-19, за приблизними оцінками, є трохи більшим за 2, що приблизно збігається з середньою оцінкою R0 протягом пандемії \"іспанського\" грипу 1918 року.", @@ -1176,7 +1176,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "Якщо на додачу до ефекту соціального дистанціювання ми зменшимо частоту, з якою люди відвідують це центральне місце, можливо, у 5 разів, або зменшимо ймовірність зараження ще у 2 рази, тобто люди будуть дуже обережно мити руки і не торкатися свого обличчя.", "model": "DeepL", "from_community_srt": "якщо разом з дистанціюванням ми зменшимо частоту відвідування центру упʼятеро чи зменшимо імовірність зараження удвічі? Наприклад, це може відображати, як люди почали краще дотримуватися гігієни. Що ж,", diff --git a/2020/epidemic-simulations/urdu/sentence_translations.json b/2020/epidemic-simulations/urdu/sentence_translations.json index 6249b1cf0..c8effb728 100644 --- a/2020/epidemic-simulations/urdu/sentence_translations.json +++ b/2020/epidemic-simulations/urdu/sentence_translations.json @@ -416,7 +416,7 @@ "end": 413.19 }, { - "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, would you predict it'll be? ", + "input": "Obviously this will have a result somewhere between a total quarantine and doing nothing, but where on the spectrum would you predict it'll be? ", "translatedText": "ظاہر ہے کہ اس کا نتیجہ کل قرنطینہ اور کچھ نہ کرنے کے درمیان ہوگا، کیا آپ پیشین گوئی کریں گے کہ ایسا ہوگا؟ بیک وقت کیسز کی چوٹی کی تعداد صرف تھوڑی زیادہ ہے، لیکن ایک بہت لمبی دم ہے کیونکہ اسے ختم ہونے میں بہت زیادہ وقت لگتا ہے، جس کے نتیجے میں کل کیسز کی تعداد تقریباً دوگنی ہوتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 580.45 }, { - "input": "It also means that the most dangerous viruses are the ones that kill some part of the population while laying unnoticed and spreadable among others, or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", + "input": "It also means that the most dangerous viruses are the ones that kill some part of the population in the first place, while laying unnoticed and spreadable among others. Or worse yet, if they remain unnoticed and spreadable in everyone before eventually becoming lethal. ", "translatedText": "اس کا مطلب یہ بھی ہے کہ سب سے زیادہ خطرناک وائرس وہ ہوتے ہیں جو آبادی کے کچھ حصے کو مار ڈالتے ہیں جب کہ وہ دوسروں کے درمیان کسی کا دھیان نہیں رکھتے اور پھیلتے ہیں، یا اس سے بھی بدتر، اگر وہ کسی کا دھیان نہ رہے اور آخرکار مہلک بننے سے پہلے ہر کسی میں پھیلنے کے قابل رہے۔", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 640.27 }, { - "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel too close to their neighbor. ", + "input": "In these animations, I'll apply it as a repulsive force between people, and have them glow yellow when they feel just a little too close to their neighbor. ", "translatedText": "ان اینیمیشنز میں، میں اسے لوگوں کے درمیان ایک نفرت انگیز قوت کے طور پر لاگو کروں گا، اور جب وہ اپنے پڑوسی کے بہت قریب محسوس کریں گے تو انہیں پیلے رنگ میں چمکا دوں گا۔", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 738.41 }, { - "input": "Again, I'll emphasize that these are toy models, and I leave it to the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", + "input": "Again, I'll emphasize that these are toy models, and I leave it to the intelligence of the viewer to judge if the behavior of these little dots accurately reflects what social distancing would mean for you and your life. ", "translatedText": "ایک بار پھر، میں اس بات پر زور دوں گا کہ یہ کھلونوں کے ماڈل ہیں، اور میں یہ فیصلہ ناظرین پر چھوڑتا ہوں کہ آیا ان چھوٹے نقطوں کا برتاؤ درست طریقے سے اس بات کی عکاسی کرتا ہے کہ سماجی دوری آپ اور آپ کی زندگی کے لیے کیا معنی رکھتی ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/epidemic-simulations/vietnamese/sentence_translations.json b/2020/epidemic-simulations/vietnamese/sentence_translations.json index 878469aa9..d2733ce3b 100644 --- a/2020/epidemic-simulations/vietnamese/sentence_translations.json +++ b/2020/epidemic-simulations/vietnamese/sentence_translations.json @@ -1019,7 +1019,7 @@ "end": 975.43 }, { - "input": "For comparison, R0 for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R0 was during the 1918 Spanish flu pandemic.", + "input": "For comparison, R-naught for COVID-19 is estimated to be a little above 2, which is also around what the mean estimate for R-naught was during the 1918 Spanish flu pandemic.", "translatedText": "Để so sánh, R0 cho COVID-19 được ước tính cao hơn 2 một chút, cũng gần bằng ước tính trung bình của R0 trong đại dịch cúm Tây Ban Nha năm 1918.", "model": "google_nmt", "from_community_srt": "Để so sánh, ước tính R0 cho COVID-19 ở trên 2 một chút, cũng xấp xỉ ước tính trung bình cho R_0 trong năm 1918, đại dịch cúm Tây Ban Nha (Spanish flu).", @@ -1171,7 +1171,7 @@ "end": 1132.11 }, { - "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, meaning people are super extra cautious about washing their hands and not touching their face.", + "input": "If on top of that social distancing effect, we decrease the frequency with which people go to that central spot, maybe by a factor of 5, or if we chop the probability of infection down by another factor of 2, for example meaning people are super extra cautious about washing their hands and not touching their face.", "translatedText": "Nếu ngoài hiệu ứng giãn cách xã hội đó, chúng tôi giảm tần suất mọi người đến địa điểm trung tâm đó, có thể xuống hệ số 5 hoặc nếu chúng tôi giảm xác suất lây nhiễm xuống một hệ số khác là 2, nghĩa là mọi người phải cực kỳ thận trọng về việc rửa tay và không chạm vào mặt.", "model": "google_nmt", "from_community_srt": "với bên cạnh việc giữ khoảng cách xã hội, chúng ta giảm tần suất mọi người đi đến điểm trung tâm theo hệ số 5? Hoặc nếu chúng ta giảm xác suất nhiễm bệnh bởi hệ số khác là 2, ví dụ như mọi người giữ vệ sinh tốt hơn khi thường xuyên rửa tay và không chạm lên mặt", diff --git a/2020/exponential-and-epidemics/arabic/sentence_translations.json b/2020/exponential-and-epidemics/arabic/sentence_translations.json index 8959f0253..8e896001c 100644 --- a/2020/exponential-and-epidemics/arabic/sentence_translations.json +++ b/2020/exponential-and-epidemics/arabic/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "إذا كان عدد الحالات في يوم معين هو n، ونقول أن كل فرد مصاب بالفيروس يتعرض لأشخاص في يوم معين، وكل واحد من هؤلاء التعرض لديه احتمال p أن يصبح عدوى جديدة، فإن العدد الحالات الجديدة في يوم معين هي e ضرب p ضرب n.", "model": "google_nmt", "from_community_srt": "إذا كان عدد الحالات في يوم معين هو N ، و كل فرد يحمل الفيروس، في المتوسط ​​، يخالط E شخص في اليوم، وكل مخالطة لها احتمال p لتصبح عدوى، فإن عدد الإصابات الجديدة كل يوم هي E * p * N.", @@ -268,7 +268,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "ليس من الممكن أن يتم ذلك من خلال الأسي في البداية، ولكن في النهاية سيستقر بمجرد اقترابك من إجمالي حجم السكان، وهو ما تتوقعه.", "model": "google_nmt", "from_community_srt": "يستوي في نهاية المطاف عند الاقتراب من إجمالي عدد السكان ، كما هو منوقع.", diff --git a/2020/exponential-and-epidemics/bulgarian/sentence_translations.json b/2020/exponential-and-epidemics/bulgarian/sentence_translations.json index dcbbd2c80..c4236c0e7 100644 --- a/2020/exponential-and-epidemics/bulgarian/sentence_translations.json +++ b/2020/exponential-and-epidemics/bulgarian/sentence_translations.json @@ -56,7 +56,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "", "from_community_srt": "Ако броят на случаите в даден ден е N, и казваме всеки индивид с вируса средно е изложен на хора от дадена страна ден и всяка експозиция има вероятност p да се превърне в инфекция, броят на новите случаи всеки ден е E * p * N.", "n_reviews": 0, @@ -238,7 +238,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "", "from_community_srt": "но в крайна сметка нива при приближаване до общото размер на населението,", "n_reviews": 0, diff --git a/2020/exponential-and-epidemics/catalan/sentence_translations.json b/2020/exponential-and-epidemics/catalan/sentence_translations.json index d5851c31c..83621d983 100644 --- a/2020/exponential-and-epidemics/catalan/sentence_translations.json +++ b/2020/exponential-and-epidemics/catalan/sentence_translations.json @@ -56,7 +56,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "", "from_community_srt": "els casos existents. Si el nombre de casos en un dia determinat és N, i diem a cada individu amb el virus s’exposa, de mitjana, a les persones E en un determinat termini dia, i cada exposició té una probabilitat p de convertir-se en infecció, el nombre de nous casos cada dia és E * p * N.", "n_reviews": 0, @@ -239,7 +239,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "", "from_community_srt": "però en última instància, nivells aproximant-se al total mida de la població,", "n_reviews": 0, diff --git a/2020/exponential-and-epidemics/czech/sentence_translations.json b/2020/exponential-and-epidemics/czech/sentence_translations.json index 094e46013..faf32ce72 100644 --- a/2020/exponential-and-epidemics/czech/sentence_translations.json +++ b/2020/exponential-and-epidemics/czech/sentence_translations.json @@ -56,7 +56,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "", "from_community_srt": "Pokud je v daný den počet případů N a, řekněme, že každý nakažený jedinec je průměrně vystaven E (exposure) lidem každý den, a každé vystavení má pravděpodobnost p, že dojde k infikování, počet nových případů každý den je E*p*N.", "n_reviews": 0, @@ -239,7 +239,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "", "from_community_srt": "která je k nerozeznání od exponenciály na začátku ale ke konci se srovná s populací, jak bychom očekávali.", "n_reviews": 0, diff --git a/2020/exponential-and-epidemics/dutch/sentence_translations.json b/2020/exponential-and-epidemics/dutch/sentence_translations.json index 730216879..f2d87765f 100644 --- a/2020/exponential-and-epidemics/dutch/sentence_translations.json +++ b/2020/exponential-and-epidemics/dutch/sentence_translations.json @@ -56,7 +56,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "", "from_community_srt": "de bestaande gevallen zelf. Als het aantal gevallen op een dag N is, en we zeggen dat elk individu met het virus gemiddeld blootgesteld is aan E mensen op een dag, en elke blootstelling heeft een kans p om een infectie te worden, dan is het aantal nieuwe gevallen elke dag E*p*N.", "n_reviews": 0, @@ -239,7 +239,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "", "from_community_srt": "maar uiteindelijk vlak wordt bij het naderen van de totale bevolkingsgrootte,", "n_reviews": 0, diff --git a/2020/exponential-and-epidemics/english/captions.srt b/2020/exponential-and-epidemics/english/captions.srt index 6ead25cdf..25ee3d20e 100644 --- a/2020/exponential-and-epidemics/english/captions.srt +++ b/2020/exponential-and-epidemics/english/captions.srt @@ -67,19 +67,19 @@ Viruses are a textbook example of this kind of growth, because what causes new cases are the existing cases. 18 -00:01:09,340 --> 00:01:13,875 -If the number of cases on a given day is n, and we say that each +00:01:09,340 --> 00:01:14,404 +If the number of cases on a given day is n, and we say that each individual 19 -00:01:13,875 --> 00:01:18,411 -individual with the virus is exposed to e people on a given day, +00:01:14,404 --> 00:01:18,870 +with the virus is exposed to, on average, e people on a given day, 20 -00:01:18,411 --> 00:01:24,064 +00:01:18,870 --> 00:01:24,268 and each one of those exposures has a probability p of becoming a new infection, 21 -00:01:24,064 --> 00:01:28,600 +00:01:24,268 --> 00:01:28,600 then the number of new cases on a given day is e times p times n. 22 @@ -279,16 +279,16 @@ Including that factor, and then solving for how N grows, you get what's known in the model. 71 -00:04:55,680 --> 00:04:58,040 -It's not possible from an exponential at the beginning, +00:04:55,680 --> 00:04:57,970 +ss as a logistic curve, which is essentially indistinguishable from 72 -00:04:58,040 --> 00:05:01,413 -but ultimately it levels out once you're approaching the total population size, +00:04:57,970 --> 00:05:00,362 +an exponential at the beginning, but ultimately levels out once you're 73 -00:05:01,413 --> 00:05:02,720 -which is what you would expect. +00:05:00,362 --> 00:05:02,720 +approaching the total population size, which is what you would expect. 74 00:05:03,360 --> 00:05:06,540 diff --git a/2020/exponential-and-epidemics/english/sentence_timings.json b/2020/exponential-and-epidemics/english/sentence_timings.json index ee69bbc0c..0bdc9d239 100644 --- a/2020/exponential-and-epidemics/english/sentence_timings.json +++ b/2020/exponential-and-epidemics/english/sentence_timings.json @@ -35,7 +35,7 @@ 67.06 ], [ - "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", 69.34, 88.6 ], @@ -150,7 +150,7 @@ 295.68 ], [ - "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", 295.68, 302.72 ], diff --git a/2020/exponential-and-epidemics/english/transcript.txt b/2020/exponential-and-epidemics/english/transcript.txt index cb6e6064f..12da08165 100644 --- a/2020/exponential-and-epidemics/english/transcript.txt +++ b/2020/exponential-and-epidemics/english/transcript.txt @@ -5,7 +5,7 @@ Never one to waste an opportunity for a math lesson, I thought this might be a g Exponential growth means that as you go from one day to the next, it involves multiplying by some constant. In our data, the number of cases in each day tends to be a multiple of about 1.15 to 1.25 of the number of cases the previous day. Viruses are a textbook example of this kind of growth, because what causes new cases are the existing cases. -If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n. +If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n. The fact that n itself is a factor in its own change is what really makes things go fast, because if n gets big, it means the rate of growth itself is getting big. One way to think about this is that as you add the new cases to get the next day's growth, you can factor out the n, so it's just the same as multiplying by some constant that's bigger than 1. This is sometimes easier to see if we put the y-axis of our graph on a logarithmic scale, which means that each step of a fixed distance corresponds to multiplying by a certain factor, in this case each step is another power of 10. @@ -28,7 +28,7 @@ Even in the most perfectly pernicious model for a virus, which would be where ev In our equation, that would mean that the probability of an exposure becoming a new infection would have to include some kind of factor to account for the probability that someone you're exposed to is already infected. For a random shuffling model like this, that could mean including a factor like 1 minus the proportion of people in the world who are already infected. Including that factor, and then solving for how N grows, you get what's known in the model. -It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect. +ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect. True exponentials essentially never exist in the real world, every one of them is the start of a logistic curve. This point right here, where that logistic curve goes from curving upward to instead curving downward, is known as the inflection point. There, the number of new cases each day, represented by the slope of this curve, stops increasing and stays roughly constant before it starts decreasing. diff --git a/2020/exponential-and-epidemics/finnish/sentence_translations.json b/2020/exponential-and-epidemics/finnish/sentence_translations.json index 673b811de..80df48549 100644 --- a/2020/exponential-and-epidemics/finnish/sentence_translations.json +++ b/2020/exponential-and-epidemics/finnish/sentence_translations.json @@ -56,7 +56,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "", "from_community_srt": "Jos tapausten määrä jonain päivänä on N .. ja jokainen sairastunut altistaa E ihmistä joka päivä, sekä tartunnan todennäköisyys altistuksesta on p.. .. uusien tapausten määrä joka päivä on E * p * N. — Se,", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "", "from_community_srt": "jota on käytännössä vaikea erottaa exponentiaalisesta käyrästä alkupäässä. Se kuitenkin lopulta tasoittuu, kun sairastuneiden määrä lähestyy koko väestön määrää.", "n_reviews": 0, diff --git a/2020/exponential-and-epidemics/french/sentence_translations.json b/2020/exponential-and-epidemics/french/sentence_translations.json index 74e1fc664..23478e66f 100644 --- a/2020/exponential-and-epidemics/french/sentence_translations.json +++ b/2020/exponential-and-epidemics/french/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "Si le nombre de cas un jour donné est n, et que nous disons que chaque individu porteur du virus est exposé à e personnes un jour donné, et que chacune de ces expositions a une probabilité p de devenir une nouvelle infection, alors le nombre de nouveaux cas un jour donné est e fois p fois n.", "model": "DeepL", "from_community_srt": "Si le nombre de cas pour un jour donné est N, et qu'on dit que chaque individu atteint du virus va, en moyenne, exposer un nombre E de personnes par jour, et que chaque exposition a une probabilité p d'aboutir à une contamination, alors le nombre de nouveaux cas quotidien sera E*p*N.", @@ -268,7 +268,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "Ce n'est pas possible à partir d'une exponentielle au début, mais en fin de compte, cela se stabilise une fois que tu t'approches de la taille totale de la population, ce qui est ce à quoi tu t'attends.", "model": "DeepL", "from_community_srt": "qui est quasi identique à une exponentielle au début, mais finit par s'aplanir en approchant de la taille totale de la population,", diff --git a/2020/exponential-and-epidemics/german/sentence_translations.json b/2020/exponential-and-epidemics/german/sentence_translations.json index 1555b5d9b..669310f4b 100644 --- a/2020/exponential-and-epidemics/german/sentence_translations.json +++ b/2020/exponential-and-epidemics/german/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "Wenn die Anzahl der Fälle an einem bestimmten Tag n ist und wir sagen, dass jede Person mit dem Virus an einem bestimmten Tag e Personen ausgesetzt ist und jede dieser Expositionen eine Wahrscheinlichkeit p für eine Neuinfektion hat, dann ist die Anzahl der neuen Fälle an einem bestimmten Tag e mal p mal n.", "model": "DeepL", "from_community_srt": "Wenn die Anzahl der Fälle an einem Tag N beträgt, und wir sagen jedes Individuum mit dem Virus ist im Durchschnitt jeden Tag mit E Personen in Kontakt und jeder Kontakt hat eine Wahrscheinlichkeit p eine Infektion zu werden, dann ist die Anzahl der neuen Fälle E * p * N.", @@ -269,7 +269,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "Am Anfang ist es nicht möglich, von einem Exponentialwert auszugehen, aber letztendlich gleicht es sich aus, wenn du dich der Gesamtbevölkerungsgröße näherst, was ja auch zu erwarten ist.", "model": "DeepL", "from_community_srt": "die am Anfang kaum von einer exponentiellen Kurve zu unterscheiden ist, aber sich letztendlich der Anzahl der kompletten Weltbevölkerung annähert.", diff --git a/2020/exponential-and-epidemics/greek/sentence_translations.json b/2020/exponential-and-epidemics/greek/sentence_translations.json index 75498578a..6bd5d06a1 100644 --- a/2020/exponential-and-epidemics/greek/sentence_translations.json +++ b/2020/exponential-and-epidemics/greek/sentence_translations.json @@ -56,7 +56,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "", "from_community_srt": "Αν N είναι τα κρούσματα μιας δεδομένης ημέρας και πούμε ότι κάθε άτομο που έχει τον ιό, τον μεταδίδει κατά μέσο όρο σε Ε άτομα την δεδομένη ημέρα, με πιθανότητα p να προσβληθεί και να γίνει νέο κρούσμα, τότε ο αριθμός των νέων κρουσμάτων κάθε μέρα θα είναι E*p*N.", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "", "from_community_srt": "αλλά τελικά εξισορροπεί προσεγγίζοντας τον τελικό πληθυσμό με έναν αναμενόμενο τρόπο.", "n_reviews": 0, diff --git a/2020/exponential-and-epidemics/hebrew/sentence_translations.json b/2020/exponential-and-epidemics/hebrew/sentence_translations.json index 2e45cfba7..be67fc548 100644 --- a/2020/exponential-and-epidemics/hebrew/sentence_translations.json +++ b/2020/exponential-and-epidemics/hebrew/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "אם מספר המקרים ביום נתון הוא n, ואנו אומרים שכל אדם עם הנגיף נחשף ל-e אנשים ביום נתון, ולכל אחת מהחשיפות הללו יש סבירות p להפוך לזיהום חדש, אז המספר של מקרים חדשים ביום נתון הוא e כפול p כפול n.", "model": "google_nmt", "from_community_srt": "הם המקרים הקיימים. אם מספר המקרים ביום מסוים הוא N, ואנחנו אומרים שכל נשא/ית של הוירוס נחשפים, בממוצע, לכמות E של אנשים ביום, ולכל חשיפה יש הסתברות p להפוך להדבקה, מספר המקרים בכל יום הוא E*p*N.", @@ -270,7 +270,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "זה לא אפשרי מאקספוננציאלי בהתחלה, אבל בסופו של דבר זה מתיישר ברגע שאתה מתקרב לגודל האוכלוסייה הכולל, וזה מה שהיית מצפה.", "model": "google_nmt", "from_community_srt": "שבהתחלה היא זהה לגידול האקספוננציאלי, אבל לבסוף משתטחת ככל שמתקרבים לגודל הכולל של האוכלוסייה,", diff --git a/2020/exponential-and-epidemics/hungarian/sentence_translations.json b/2020/exponential-and-epidemics/hungarian/sentence_translations.json index a2707904e..a3e257972 100644 --- a/2020/exponential-and-epidemics/hungarian/sentence_translations.json +++ b/2020/exponential-and-epidemics/hungarian/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "Ha az esetek száma egy adott napon n, és azt mondjuk, hogy minden egyes vírussal fertőzött egyén egy adott napon e emberrel érintkezik, és minden egyes ilyen érintkezésnek p valószínűsége van arra, hogy új fertőzéssé válik, akkor az új esetek száma egy adott napon e-szor p-szor n. Az új esetek száma egy adott napon e-szor p-szor n.", "model": "DeepL", "from_community_srt": "Ha az esetek száma egy adott napon N, és azt mondjuk, hogy minden egyén vírussal rendelkezik egy adott embernél átlagosan E embereknek vannak kitéve napon, és minden expozíció valószínűsége p fertőzéské válás, az újak száma minden nap E * p * N.", @@ -270,7 +270,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "Az exponenciálisból az elején nem lehet, de végül is kiegyenlítődik, amint megközelítjük a teljes populáció méretét, amit elvárnánk.", "model": "DeepL", "from_community_srt": "de végül szintre lép, ha megközelíti az egészet népesség nagysága,", diff --git a/2020/exponential-and-epidemics/indonesian/sentence_translations.json b/2020/exponential-and-epidemics/indonesian/sentence_translations.json index fd93af7c7..cb3946ad4 100644 --- a/2020/exponential-and-epidemics/indonesian/sentence_translations.json +++ b/2020/exponential-and-epidemics/indonesian/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "Jika jumlah kasus pada hari tertentu adalah n, dan kita mengatakan bahwa setiap individu dengan virus terpapar pada e orang pada hari tertentu, dan setiap paparan tersebut memiliki probabilitas p untuk menjadi infeksi baru, maka jumlah kasus baru pada hari tertentu adalah e dikali p dikali n.", "model": "DeepL", "from_community_srt": "adalah dari kasus yang ada. Jika jumlah kasus pada hari tertentu adalah N, dan kami katakan setiap individu dengan virus, terpapar ke, secara rata-rata, E orang pada suatu hari tertentu, dan setiap paparan memiliki probabilitas p untuk menjadi infeksi baru, jumlah kasus baru setiap hari adalah E * p * N.", @@ -270,7 +270,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "Ini tidak mungkin dari eksponensial di awal, tetapi pada akhirnya akan mendatar setelah Anda mendekati jumlah total populasi, dan itulah yang Anda harapkan.", "model": "DeepL", "from_community_srt": "tetapi ada saatnya nilai tersebut akan mendekati total populasi, seperti yang Anda pikirkan.", diff --git a/2020/exponential-and-epidemics/italian/sentence_translations.json b/2020/exponential-and-epidemics/italian/sentence_translations.json index b95e00994..dd57fd582 100644 --- a/2020/exponential-and-epidemics/italian/sentence_translations.json +++ b/2020/exponential-and-epidemics/italian/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "Se il numero di casi in un dato giorno è n, e diciamo che ogni individuo con il virus è esposto a e persone in un dato giorno, e ognuna di queste esposizioni ha una probabilità p di diventare una nuova infezione, allora il numero di nuovi casi in un dato giorno è e per p per n.", "model": "DeepL", "from_community_srt": "sono i casi già esistenti. Quindi se il numero di casi in un qualsiasi giorno è N, e diciamo che ogni individuo col virus è, in media, esposto a E persone ogni giorno, ed ognuna di esse ha una probabilità p di ammalarsi, allora il numero di nuovi casi ogni giorno è E*p*N.", @@ -270,7 +270,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "All'inizio non è possibile da un esponenziale, ma alla fine si livella una volta che ci si avvicina alla dimensione totale della popolazione, come ci si aspetterebbe.", "model": "DeepL", "from_community_srt": "ma all'avvicinarsi al numero totale della popolazione rallenta,", diff --git a/2020/exponential-and-epidemics/korean/sentence_translations.json b/2020/exponential-and-epidemics/korean/sentence_translations.json index 22ed18145..45726c113 100644 --- a/2020/exponential-and-epidemics/korean/sentence_translations.json +++ b/2020/exponential-and-epidemics/korean/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "주어진 날의 확진자 수가 n이고, 바이러스에 감염된 각 개인이 주어진 날에 e명의 사람들에게 노출되고 각 노출이 새로운 감염이 될 확률이 p라고 가정하면, 주어진 날의 신규 확진자 수는 e 곱하기 p 곱하기 n이 됩니다.", "model": "DeepL", "from_community_srt": "그러니까 어느 날의 감염자 수를 N이라고 하고, 하루동안 감염자 한 명에게 노출된 평균 인원 수를 E, 그리고 감염자에 대한 노출이 전염으로 이어질 확률을 p라고 했을 때 신규 감염자의 수는 E*p*N입니다.", @@ -270,7 +270,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "처음에는 기하급수적으로 늘어날 수는 없지만, 궁극적으로 전체 인구 규모에 가까워지면 예상대로 평준화됩니다.", "model": "DeepL", "from_community_srt": "이는 우리가 예측한 대로 처음에는 지수적 증가와 구별하기 힘들지만, 결국에는 전체 인구 수에 가까워지는 모양입니다.", diff --git a/2020/exponential-and-epidemics/norwegian/sentence_translations.json b/2020/exponential-and-epidemics/norwegian/sentence_translations.json index c497382df..a0b413bd2 100644 --- a/2020/exponential-and-epidemics/norwegian/sentence_translations.json +++ b/2020/exponential-and-epidemics/norwegian/sentence_translations.json @@ -56,7 +56,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "", "from_community_srt": "Hvis man har N tilfeller på en gitt dag, og vi sier at hvert individ med virus er, gjennomsnittlig, i kontakt med E andre på en gitt dag, og hver kontakt har en sannsynlighet p for å lede til smitte, så er de nye tilfellene hver dag lik E*p*N.", "n_reviews": 0, @@ -239,7 +239,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "", "from_community_srt": "men jevner seg ut når du nærmer den den totale befolkningen slik man ville sett for seg.", "n_reviews": 0, diff --git a/2020/exponential-and-epidemics/persian/sentence_translations.json b/2020/exponential-and-epidemics/persian/sentence_translations.json index 561cfab34..481669540 100644 --- a/2020/exponential-and-epidemics/persian/sentence_translations.json +++ b/2020/exponential-and-epidemics/persian/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "اگر تعداد موارد در یک روز معین n باشد، و بگوییم که هر فرد مبتلا به ویروس در یک روز معین در معرض انفراد قرار گرفته است، و هر یک از آن قرار گرفتن در معرض احتمال p برای تبدیل شدن به یک عفونت جدید است، آنگاه تعداد موارد جدید در یک روز معین e ضربدر p ضربدر n است.", "model": "google_nmt", "from_community_srt": "اگه تعداد موارد در یه روز خاص N باشه، و فرض کنیم هر فرد آلوده به ویروس بطور متوسط با E نفر در یه روز در تماس باشه، و هر تماس با احتمال p ویروس رو منتقل کنه، خب تعداد موارد جدید در هر روز میشه E ضربدر p ضربدر N.", @@ -269,7 +269,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "در ابتدا امکان نمایی وجود ندارد، اما در نهایت زمانی که به کل جمعیت نزدیک می‌شوید، به سطح می‌رسد، چیزی که انتظار دارید.", "model": "google_nmt", "from_community_srt": "این منحنی در ابتداش از منحنی نمایی غیرقابل تشخیصه، ولی با نزدیک شدن به جمعیت کل، کم کم ثابت میشه، همونطور که انتظار میره.", diff --git a/2020/exponential-and-epidemics/polish/sentence_translations.json b/2020/exponential-and-epidemics/polish/sentence_translations.json index 9f25e84b7..6156ea571 100644 --- a/2020/exponential-and-epidemics/polish/sentence_translations.json +++ b/2020/exponential-and-epidemics/polish/sentence_translations.json @@ -56,7 +56,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "", "from_community_srt": "to istniejący zarażeni. Jeżeli liczba zarażonych danego dnia to N, a każda osoba zarażona jest, średnio, w kontakcie z E osobami każdego dnia, a szansa na zarażenie się wynosi p liczba nowych zarażeń wynosi E*p*N.", "n_reviews": 0, @@ -239,7 +239,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "", "from_community_srt": "która jest zasadniczo nierozróżnialna od wzrostu wykładniczego na początku, lecz w pewnym momencie spowalnia, kończąc przy łącznej liczbie ludności,", "n_reviews": 0, diff --git a/2020/exponential-and-epidemics/portuguese/sentence_translations.json b/2020/exponential-and-epidemics/portuguese/sentence_translations.json index d93caa690..f4811aeaa 100644 --- a/2020/exponential-and-epidemics/portuguese/sentence_translations.json +++ b/2020/exponential-and-epidemics/portuguese/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "Se o número de casos em um determinado dia for n, e dissermos que cada indivíduo com o vírus está exposto a e pessoas em um determinado dia, e cada uma dessas exposições tem probabilidade p de se tornar uma nova infecção, então o número de novos casos em um determinado dia é e vezes p vezes n.", "model": "google_nmt", "from_community_srt": "os casos existentes Se o número de casos de um dia é \"N\", e se cada indivíduo com o vírus expõe-se, em média, a \"E\" pessoas num certo dia e que cada uma destas exposições tem uma probabilidade \"p\" de se tornar numa nova infeção, o número de novos casos em cada dia é E×p×N.", @@ -270,7 +270,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "Não é possível a partir de um exponencial no início, mas no final das contas ele se estabiliza quando você se aproxima do tamanho total da população, que é o que você esperaria.", "model": "google_nmt", "from_community_srt": "que é essencialmente igual à exponencial no início mas entretanto equilibra-se assim que nos aproximamos do total da população,", diff --git a/2020/exponential-and-epidemics/romanian/sentence_translations.json b/2020/exponential-and-epidemics/romanian/sentence_translations.json index bececb8e8..a5ebb8e55 100644 --- a/2020/exponential-and-epidemics/romanian/sentence_translations.json +++ b/2020/exponential-and-epidemics/romanian/sentence_translations.json @@ -56,7 +56,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "", "from_community_srt": "cazurile existente. Dacă numărul de cazuri într-o zi este N, și noi spunem că fiecare individ infectat este, în medie, expus la E oameni într-o zi, și fiecare expunere are probabilitatea p să devină o infecție, numărul de cazuri noi în fiecare zi este E × p × N.", "n_reviews": 0, @@ -239,7 +239,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "", "from_community_srt": "care nu poate fi distinsă de una exponențială la început, dar până la urmă se nivelează când se apropie de numărul total al populației,", "n_reviews": 0, diff --git a/2020/exponential-and-epidemics/russian/sentence_translations.json b/2020/exponential-and-epidemics/russian/sentence_translations.json index 4f41c235a..e30c4c363 100644 --- a/2020/exponential-and-epidemics/russian/sentence_translations.json +++ b/2020/exponential-and-epidemics/russian/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "Если количество случаев заболевания в определенный день равно n, и мы говорим, что каждый человек с вирусом подвергается воздействию e людей в определенный день, и каждый из этих воздействий имеет вероятность p стать новым инфектором, то количество новых случаев заболевания в определенный день равно e, умноженное на p, умноженное на n.", "model": "DeepL", "from_community_srt": "Если число зараженных в какой-то день равно N, и, например, каждый человек, зараженный вирусом, в среднем контактирует с E людей в тот же день, каждый контакт имеет вероятность заражения p, то число новых зараженных каждый день это E * p * N.", @@ -270,7 +270,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "В начале это невозможно по экспоненте, но в конечном итоге она выравнивается, когда ты приближаешься к общему размеру популяции, чего и следовало ожидать.", "model": "DeepL", "from_community_srt": "которая вначале практически не отличается от экспоненциальной, но в конечном итоге выравнивается при приближении к общей численности населения,", diff --git a/2020/exponential-and-epidemics/serbian/sentence_translations.json b/2020/exponential-and-epidemics/serbian/sentence_translations.json index ca169ad9e..f8241f314 100644 --- a/2020/exponential-and-epidemics/serbian/sentence_translations.json +++ b/2020/exponential-and-epidemics/serbian/sentence_translations.json @@ -56,7 +56,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "", "from_community_srt": "tako da, ako je broj slučajeva u datom danu N, i kažemo da je pojedinac zaražen virusom izložen, u proseku, E broju ljudi toga dana, i svako izlaganje ima verovatnoću p da postane infekcija onda će broj novih slučajeva svakoga dana biti E*p*N.", "n_reviews": 0, @@ -238,7 +238,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "", "from_community_srt": "koja je u suštini nerazlučiva od eksponencijalne krive na početku ali se na kraju izjednači kada se približite broju čitave populacije,", "n_reviews": 0, diff --git a/2020/exponential-and-epidemics/slovenian/sentence_translations.json b/2020/exponential-and-epidemics/slovenian/sentence_translations.json index 7b8f1e1e9..db8743511 100644 --- a/2020/exponential-and-epidemics/slovenian/sentence_translations.json +++ b/2020/exponential-and-epidemics/slovenian/sentence_translations.json @@ -56,7 +56,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "", "from_community_srt": "Če je N število okuženih opazovanega dne, in se vsak okuženi posameznik v povprečju sreča z E osebami na dan, na katere prenese okužbo z verjetnostjo p, potem je dnevno število novih okužb enako E krat p krat N.", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "", "from_community_srt": "ki je na začetku ni mogoče ločiti od eksponentne, toda kasneje se izravna in približuje velikosti populacije,", "n_reviews": 0, diff --git a/2020/exponential-and-epidemics/spanish/sentence_translations.json b/2020/exponential-and-epidemics/spanish/sentence_translations.json index 8c04b2fb9..864a4f715 100644 --- a/2020/exponential-and-epidemics/spanish/sentence_translations.json +++ b/2020/exponential-and-epidemics/spanish/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "Si el número de casos en un día dado es n, y decimos que cada individuo con el virus se expone a e personas en un día dado, y cada una de esas exposiciones tiene una probabilidad p de convertirse en una nueva infección, entonces el número de nuevos casos en un día dado es e veces p veces n.", "model": "DeepL", "from_community_srt": "son los casos ya existentes. Si el número de casos en un día cualquiera es N, y decimos que cada individuo contagiado está, en promedio, expuesto a E personas en un día cualquiera y cada exposición tiene una probabilidad p de convertirse en infección, el número de nuevos casos cada día es E*p*N.", @@ -270,7 +270,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "No es posible a partir de una exponencial al principio, pero al final se nivela una vez que te acercas al tamaño total de la población, que es lo que cabría esperar.", "model": "DeepL", "from_community_srt": "pero al final se aplana al acercarse a la población total, que es lo que esperarías.", diff --git a/2020/exponential-and-epidemics/thai/sentence_translations.json b/2020/exponential-and-epidemics/thai/sentence_translations.json index 91a1df908..a4bb4019c 100644 --- a/2020/exponential-and-epidemics/thai/sentence_translations.json +++ b/2020/exponential-and-epidemics/thai/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "หากจำนวนเคสในวันที่กำหนดคือ n และเราบอกว่าผู้ติดเชื้อแต่ละคนสัมผัสกับผู้คนในวันที่กำหนด และการสัมผัสแต่ละครั้งมีความน่าจะเป็น p ที่จะกลายเป็นการติดเชื้อใหม่ ดังนั้นจำนวน ของผู้ป่วยรายใหม่ในวันที่กำหนดคือ e คูณ p คูณ n", "model": "google_nmt", "from_community_srt": "ถ้าจำนวนผู้ติดเชื้อของในวันๆนึงเป็น N และเราบอกว่าสำรับทุกๆคนที่มีไวรัส โดยเฉลี่ยติดต่อกับคน E คนในทุกๆวัน และการติดต่อแต่ละครั้งมีความน่าจะเป็น p ที่จะกลายเป็นผู้ติดเชื้อรายใหม่ จำนวนคนติดเชื้อในแต่ละวันคือ E*p*N ความจริงที่ว่าตัวเลข N เอง", @@ -266,7 +266,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "มันเป็นไปไม่ได้จากการเอ็กซ์โพเนนเชียลตั้งแต่เริ่มต้น แต่ท้ายที่สุดแล้วก็จะค่อยๆ ลดลงเมื่อคุณเข้าใกล้ขนาดประชากรทั้งหมด ซึ่งเป็นสิ่งที่คุณคาดหวัง", "model": "google_nmt", "from_community_srt": "เส้นโค้งโลจิสติก ซึ่งจะเหมือนกับการโตแบบทวีคูณในตอนแรก แต่ สุดท้ายแล้วระดับจะเท่ากันเมื่อเข้าใกล้ประชากรทั้งหมด อย่างที่คุณคาดไว้", diff --git a/2020/exponential-and-epidemics/turkish/sentence_translations.json b/2020/exponential-and-epidemics/turkish/sentence_translations.json index 8b289d9a2..5bbb050f6 100644 --- a/2020/exponential-and-epidemics/turkish/sentence_translations.json +++ b/2020/exponential-and-epidemics/turkish/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "Belirli bir gündeki vaka sayısı n ise ve virüs taşıyan her bireyin belirli bir günde e kişiye maruz kaldığını ve bu maruziyetlerin her birinin yeni bir enfeksiyon olma olasılığının p olduğunu söylersek, belirli bir gündeki yeni vaka sayısı e çarpı p çarpı n'dir.", "model": "DeepL", "from_community_srt": "Belirli bir günde vaka sayısı N ise, ve virüs olan her bireye ortalama olarak, belirli bir durumda E kişilere maruz kaldığında gün ve her maruz kalma olasılığı p var enfeksiyon olma, yeni vakalar her gün E * p * N'dir.", @@ -270,7 +270,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "Başlangıçta bir üstelden mümkün değildir, ancak nihayetinde toplam nüfus büyüklüğüne yaklaştığınızda, beklediğiniz gibi düzleşir.", "model": "DeepL", "from_community_srt": "ama sonuçta toplam yaklaştıkça seviyeler tahmin edebileceğiniz gibi.", diff --git a/2020/exponential-and-epidemics/vietnamese/sentence_translations.json b/2020/exponential-and-epidemics/vietnamese/sentence_translations.json index 26e463418..86f8ee23c 100644 --- a/2020/exponential-and-epidemics/vietnamese/sentence_translations.json +++ b/2020/exponential-and-epidemics/vietnamese/sentence_translations.json @@ -63,7 +63,7 @@ "end": 67.06 }, { - "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", + "input": "If the number of cases on a given day is n, and we say that each individual with the virus is exposed to, on average, e people on a given day, and each one of those exposures has a probability p of becoming a new infection, then the number of new cases on a given day is e times p times n.", "translatedText": "Nếu số ca nhiễm trong một ngày nhất định là n và chúng ta nói rằng mỗi cá nhân nhiễm vi-rút tiếp xúc với e người vào một ngày nhất định và mỗi trường hợp phơi nhiễm đó có xác suất p trở thành một ca nhiễm mới, thì số số ca nhiễm mới vào một ngày nhất định là e nhân p nhân n.", "model": "google_nmt", "from_community_srt": "Nếu số ca nhiễm của một ngày bất kì là N và ta nói rằng mỗi cá nhân nhiễm virus này trung bình tiếp xúc với E người mỗi ngày, và mỗi lần tiếp xúc có xác suất p để trở thành một ca nhiễm mới,", @@ -269,7 +269,7 @@ "end": 295.68 }, { - "input": "It's not possible from an exponential at the beginning, but ultimately it levels out once you're approaching the total population size, which is what you would expect.", + "input": "ss as a logistic curve, which is essentially indistinguishable from an exponential at the beginning, but ultimately levels out once you're approaching the total population size, which is what you would expect.", "translatedText": "Điều đó là không thể ngay từ đầu theo cấp số nhân, nhưng cuối cùng nó sẽ chững lại khi bạn đạt đến tổng quy mô dân số, đó là những gì bạn mong đợi.", "model": "google_nmt", "from_community_srt": "nhưng cuối cùng sẽ tăng đến khi chạm vào tổng kích cỡ dân số, như bạn đã trông chờ.", diff --git a/2020/groups-and-monsters/arabic/sentence_translations.json b/2020/groups-and-monsters/arabic/sentence_translations.json index 00763173c..b42c2762b 100644 --- a/2020/groups-and-monsters/arabic/sentence_translations.json +++ b/2020/groups-and-monsters/arabic/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "على النقيض من ذلك، إذا أعطينا هذه النقاط بعض البنية، ربما جعلها زوايا شكل سداسي، مع الأخذ في الاعتبار فقط التباديل الذي يحافظ على مدى بعد كل واحدة عن الأخرى، فإننا نحصل فقط على تماثلات ندفة الثلج الـ 12 التي رأيناها سابقًا. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "لا أحد يفهم حقًا سبب وجود المجموعات المتفرقة، والوحش على وجه الخصوص. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/bengali/sentence_translations.json b/2020/groups-and-monsters/bengali/sentence_translations.json index 42f341352..2231d9922 100644 --- a/2020/groups-and-monsters/bengali/sentence_translations.json +++ b/2020/groups-and-monsters/bengali/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "বিপরীতে, যদি আমরা এই পয়েন্টগুলিকে কিছু কাঠামো দিয়ে থাকি, হয়ত সেগুলিকে একটি ষড়ভুজের কোণে তৈরি করি এবং শুধুমাত্র সেই স্থানান্তরগুলি বিবেচনা করে যেগুলি একে অপরের থেকে কতটা দূরত্ব বজায় রাখে, আমরা কেবলমাত্র 12টি স্নোফ্লেক প্রতিসাম্য পাই যা আমরা আগে দেখেছি।", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "বিক্ষিপ্ত গোষ্ঠী এবং বিশেষ করে দানব কেন সেখানে আছে তা কেউ সত্যিই বুঝতে পারে না।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/chinese/sentence_translations.json b/2020/groups-and-monsters/chinese/sentence_translations.json index e6121da3a..206747a45 100644 --- a/2020/groups-and-monsters/chinese/sentence_translations.json +++ b/2020/groups-and-monsters/chinese/sentence_translations.json @@ -240,7 +240,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "相比之下，如果我们给这些点一些结构，也许让它们成为 六边形的角，并且只考虑保持每个点彼此之间距离的排列 ，我们只能得到我们之前看到的 12 种雪花对称性。", "model": "google_nmt", "n_reviews": 0, @@ -1264,7 +1264,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "没有人真正理解为什么会有这 些零星的群体，尤其是怪物。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/french/sentence_translations.json b/2020/groups-and-monsters/french/sentence_translations.json index e7ed07736..934f8646f 100644 --- a/2020/groups-and-monsters/french/sentence_translations.json +++ b/2020/groups-and-monsters/french/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "En revanche, si nous donnons une certaine structure à ces points, en en faisant peut-être les coins d’un hexagone et en considérant uniquement les permutations qui préservent la distance les uns des autres, nous n’obtenons que les 12 symétries de flocon de neige que nous avons vues plus tôt. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "Personne ne comprend vraiment pourquoi les groupes sporadiques, et le monstre en particulier, sont là. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/hebrew/sentence_translations.json b/2020/groups-and-monsters/hebrew/sentence_translations.json index 90ddf8cad..95079c73d 100644 --- a/2020/groups-and-monsters/hebrew/sentence_translations.json +++ b/2020/groups-and-monsters/hebrew/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "לעומת זאת, אם נתנו לנקודות האלה מבנה כלשהו, אולי הופכים אותן לפינות של משושה ורק בהתחשב בתמורות המשמרות את המרחק של כל אחת מהשנייה, נקבל רק את 12 הסימטריות של פתיתי השלג שראינו קודם לכן. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "אף אחד לא באמת מבין למה הקבוצות הספורדיות, והמפלצת בפרט, נמצאות שם. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/hindi/sentence_translations.json b/2020/groups-and-monsters/hindi/sentence_translations.json index 68ac5fd12..fff01d254 100644 --- a/2020/groups-and-monsters/hindi/sentence_translations.json +++ b/2020/groups-and-monsters/hindi/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "इसके विपरीत, यदि हमने इन बिंदुओं को कुछ संरचना दी है, शायद उन्हें एक षट्भुज के कोने बना दिया है और केवल उन क्रमपरिवर्तनों पर विचार किया है जो यह संरक्षित करते हैं कि प्रत्येक एक दूसरे से कितना दूर है, तो हमें केवल 12 बर्फ के टुकड़े की समरूपताएं मिलती हैं जो हमने पहले देखी थीं।", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "वास्तव में कोई नहीं समझता कि छिटपुट समूह और विशेष रूप से राक्षस वहां क्यों हैं।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/indonesian/sentence_translations.json b/2020/groups-and-monsters/indonesian/sentence_translations.json index d1599a5a2..d182b0241 100644 --- a/2020/groups-and-monsters/indonesian/sentence_translations.json +++ b/2020/groups-and-monsters/indonesian/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "Sebaliknya, jika kita memberi struktur pada titik-titik ini, mungkin menjadikannya sudut-sudut segi enam dan hanya mempertimbangkan permutasi yang mempertahankan seberapa jauh jarak satu sama lain, kita hanya mendapatkan 12 kesimetrian kepingan salju yang kita lihat sebelumnya. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "Tidak ada yang benar-benar mengerti mengapa kelompok sporadis, dan khususnya monster, ada di sana. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/italian/sentence_translations.json b/2020/groups-and-monsters/italian/sentence_translations.json index d06fd7e0b..9e7492449 100644 --- a/2020/groups-and-monsters/italian/sentence_translations.json +++ b/2020/groups-and-monsters/italian/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "Al contrario, se diamo una struttura a questi punti, magari trasformandoli negli angoli di un esagono e considerando solo le permutazioni che preservano la distanza tra l'uno e l'altro, otteniamo solo le 12 simmetrie del fiocco di neve che abbiamo visto prima. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "Nessuno capisce veramente perché i gruppi sporadici, e il mostro in particolare, siano lì. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/japanese/sentence_translations.json b/2020/groups-and-monsters/japanese/sentence_translations.json index 73b3cb003..671509cfe 100644 --- a/2020/groups-and-monsters/japanese/sentence_translations.json +++ b/2020/groups-and-monsters/japanese/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "対照的に、これらの点に何らかの構造を与え、おそらくそれらを六角形の 角にし、各点が他の点からどれだけ離れているかを維持する順列のみを考 慮すると、前に見た 12 個の雪の結晶の対称性のみが得られます。", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "散発的なグループ、特に怪物がなぜそこに存 在するのか、誰も本当に理解していません。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/korean/sentence_translations.json b/2020/groups-and-monsters/korean/sentence_translations.json index d5678f66f..54c98f312 100644 --- a/2020/groups-and-monsters/korean/sentence_translations.json +++ b/2020/groups-and-monsters/korean/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "대조적으로, 우리가 이 점들에 어떤 구조를 부여하고, 아마도 그것들을 육각형의 모서리로 만들고, 각 점들이 다른 점들로부터 얼마나 멀리 떨어져 있는지를 보존하는 순열만을 고려한다면, 우리는 앞서 본 12개의 눈송이 대칭만 얻을 수 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "산발적인 집단, 특히 괴물이 존재하는 이유를 실제로 이해하는 사람은 아무도 없습니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/marathi/sentence_translations.json b/2020/groups-and-monsters/marathi/sentence_translations.json index 9936e353f..ed7b1374b 100644 --- a/2020/groups-and-monsters/marathi/sentence_translations.json +++ b/2020/groups-and-monsters/marathi/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "याउलट, जर आपण या बिंदूंना काही रचना दिली, कदाचित त्यांना षटकोनीचे कोपरे बनवले आणि प्रत्येक एक एकमेकांपासून किती अंतरावर आहे हे जपणाऱ्या क्रमपरिवर्तनांचा विचार केला तर, आपण आधी पाहिलेल्या 12 स्नोफ्लेक सममिती आपल्याला मिळतात. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "तुरळक गट आणि विशेषतः अक्राळविक्राळ तेथे का आहेत हे कोणालाही खरोखर समजत नाही. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/persian/sentence_translations.json b/2020/groups-and-monsters/persian/sentence_translations.json index e09d4675f..eaf90e1c5 100644 --- a/2020/groups-and-monsters/persian/sentence_translations.json +++ b/2020/groups-and-monsters/persian/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "یا 720 در مقابل، اگر به این نقاط ساختاری بدهیم، شاید آنها را گوشه‌های یک شش ضلعی بسازیم و تنها با در نظر گرفتن جایگشت‌هایی که فاصله هر یک از دیگری را حفظ می‌کنند، تنها 12 تقارن دانه‌های برف را که قبلاً دیدیم به دست می‌آوریم. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "هیچ کس واقعاً نمی فهمد که چرا گروه های پراکنده، و به ویژه هیولا، آنجا هستند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/portuguese/sentence_translations.json b/2020/groups-and-monsters/portuguese/sentence_translations.json index 5b490f0c0..5feeab55c 100644 --- a/2020/groups-and-monsters/portuguese/sentence_translations.json +++ b/2020/groups-and-monsters/portuguese/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "Por outro lado, se dermos alguma estrutura a estes pontos, talvez tornando-os os cantos de um hexágono e apenas considerando as permutações que preservam a distância entre cada um dos outros, obteremos apenas as 12 simetrias de flocos de neve que vimos anteriormente. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "Ninguém realmente entende por que os grupos esporádicos, e o monstro em particular, estão ali. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/russian/sentence_translations.json b/2020/groups-and-monsters/russian/sentence_translations.json index 55aec8108..e26d74ba7 100644 --- a/2020/groups-and-monsters/russian/sentence_translations.json +++ b/2020/groups-and-monsters/russian/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "Напротив, если бы мы придали этим точкам некоторую структуру, возможно, сделав их углами шестиугольника и учитывая только перестановки, которые сохраняют расстояние друг от друга, мы получим только 12 симметрий снежинок, которые мы видели ранее. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "Никто толком не понимает, почему существуют спорадические группы и монстр в частности. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/tamil/sentence_translations.json b/2020/groups-and-monsters/tamil/sentence_translations.json index 164132696..97b8fedfa 100644 --- a/2020/groups-and-monsters/tamil/sentence_translations.json +++ b/2020/groups-and-monsters/tamil/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "இதற்கு நேர்மாறாக, இந்த புள்ளிகளுக்கு சில அமைப்பைக் கொடுத்தால், அவற்றை ஒரு அறுகோணத்தின் மூலைகளாக மாற்றி, ஒவ்வொன்றும் மற்றொன்றிலிருந்து எவ்வளவு தூரத்தில் உள்ளன என்பதைப் பாதுகாக்கும் வரிசைமாற்றங்களைக் கருத்தில் கொண்டால், நாம் முன்பு பார்த்த 12 ஸ்னோஃப்ளேக் சமச்சீர்நிலைகள் மட்டுமே கிடைக்கும். ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "ஆங்காங்கே குழுக்கள் மற்றும் குறிப்பாக அசுரன் ஏன் இருக்கிறார்கள் என்பது யாருக்கும் புரியவில்லை. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/telugu/sentence_translations.json b/2020/groups-and-monsters/telugu/sentence_translations.json index ba8e6504e..cab2472bb 100644 --- a/2020/groups-and-monsters/telugu/sentence_translations.json +++ b/2020/groups-and-monsters/telugu/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "దీనికి విరుద్ధంగా, మేము ఈ పాయింట్‌లకు కొంత నిర్మాణాన్ని అందించినట్లయితే, వాటిని షడ్భుజి యొక్క మూలలుగా చేసి, ప్రతి ఒక్కటి ఒకదానికొకటి ఎంత దూరంలో ఉందో సంరక్షించే ప్రస్తారణలను మాత్రమే పరిగణనలోకి తీసుకుంటే, మనం ఇంతకు ముందు చూసిన 12 స్నోఫ్లేక్ సమరూపతలను మాత్రమే పొందుతాము. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "అడపాదడపా సమూహాలు మరియు ముఖ్యంగా రాక్షసుడు ఎందుకు అక్కడ ఉన్నారో ఎవరికీ అర్థం కాలేదు. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/thai/sentence_translations.json b/2020/groups-and-monsters/thai/sentence_translations.json index 997fc6c28..e3ef7010b 100644 --- a/2020/groups-and-monsters/thai/sentence_translations.json +++ b/2020/groups-and-monsters/thai/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/ukrainian/sentence_translations.json b/2020/groups-and-monsters/ukrainian/sentence_translations.json index c5f554621..f173c27e7 100644 --- a/2020/groups-and-monsters/ukrainian/sentence_translations.json +++ b/2020/groups-and-monsters/ukrainian/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "Навпаки, якщо ми надамо цим точкам певну структуру, можливо, зробивши їх кутами шестикутника та враховуючи лише перестановки, які зберігають відстань одна від одної, ми отримаємо лише 12 симетрій сніжинок, які ми бачили раніше. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "Ніхто насправді не розуміє, навіщо існують спорадичні групи, зокрема монстр. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/urdu/sentence_translations.json b/2020/groups-and-monsters/urdu/sentence_translations.json index 2c65b48e3..2ba460745 100644 --- a/2020/groups-and-monsters/urdu/sentence_translations.json +++ b/2020/groups-and-monsters/urdu/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "اس کے برعکس، اگر ہم نے ان پوائنٹس کو کچھ ڈھانچہ دیا، تو شاید ان کو مسدس کے کونے بنا دیں اور صرف ان ترتیبوں پر غور کریں جو محفوظ رکھتے ہیں کہ ہر ایک دوسرے سے کتنا فاصلہ رکھتا ہے، ہمیں صرف 12 سنو فلیک کی ہم آہنگی ملتی ہے جو ہم نے پہلے دیکھی تھیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "کوئی بھی واقعتا یہ نہیں سمجھتا ہے کہ چھٹپٹ گروپس اور خاص طور پر عفریت وہاں کیوں ہیں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/groups-and-monsters/vietnamese/sentence_translations.json b/2020/groups-and-monsters/vietnamese/sentence_translations.json index 2fa35455b..9cc1e48da 100644 --- a/2020/groups-and-monsters/vietnamese/sentence_translations.json +++ b/2020/groups-and-monsters/vietnamese/sentence_translations.json @@ -248,7 +248,7 @@ "end": 224.64 }, { - "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, we only get the 12 snowflake symmetries we saw earlier. ", + "input": "By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutations that preserve how far apart each one is from the other, well then we only get the 12 snowflake symmetries we saw earlier. ", "translatedText": "Ngược lại, nếu chúng ta cho những điểm này một số cấu trúc, có thể biến chúng thành các góc của một hình lục giác và chỉ xem xét các hoán vị đảm bảo khoảng cách giữa mỗi điểm này với điểm kia, thì chúng ta chỉ nhận được 12 đối xứng bông tuyết mà chúng ta đã thấy trước đó. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1176.4 }, { - "input": "No one really understands why the sporadic groups, and the monster in particular, are there. ", + "input": "Why the Sporadic Groups? No one really understands why the sporadic groups, and the monster in particular, are there. ", "translatedText": "Không ai thực sự hiểu tại sao các nhóm lẻ tẻ và đặc biệt là quái vật lại ở đó. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/arabic/sentence_translations.json b/2020/hamming-codes-2/arabic/sentence_translations.json index f71703612..98c49f4a4 100644 --- a/2020/hamming-codes-2/arabic/sentence_translations.json +++ b/2020/hamming-codes-2/arabic/sentence_translations.json @@ -32,7 +32,7 @@ "end": 34.6 }, { - "input": "But as you start to think about actually implementing this, either in software or hardware, that framing may actually undersell how elegant these codes really are. ", + "input": "hat there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error. In that video, the goal was to make Hamming codes feel as hands-on and rediscoverable as possible. But as ", "translatedText": "ولكن عندما تبدأ في التفكير في تنفيذ هذا فعليًا، سواء في البرامج أو الأجهزة، فإن هذا الإطار قد يقلل في الواقع من مدى أناقة هذه الرموز حقًا. ", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. ", + "input": "you read out the answers to the four parity checks we did in the last video, all as ones and zeros instead of yeses and nos, it literally spells out ", "translatedText": "على سبيل المثال، الرقم 7 في النظام الثنائي يبدو مثل 0111، مما يعني أنه 4 زائد 2 زائد 1. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error. ", + "input": "century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day. The goal here is to give you a very thorough understanding of one of the earlie ", "translatedText": "لذا فإن قراءة نتائج عمليات التحقق الأربعة هذه من الأسفل إلى الأعلى توضح بالفعل موضع الخطأ. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 87.54 }, { - "input": "There's nothing special about the example 7, this works in general, and this makes the logic for implementing the whole scheme in hardware shockingly simple. ", + "input": "st examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity groups, and the second, and the third, but not the last. So reading the results of those four checks ", "translatedText": "لا يوجد شيء خاص في المثال 7، وهذا يعمل بشكل عام، وهذا يجعل منطق تنفيذ المخطط بأكمله في الأجهزة بسيطًا بشكل صادم. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 132.36 }, { - "input": "It's worth it, though. ", + "input": "0, let's write them all in binary, running from 0000 up to 1111. ask feels at the star ", "translatedText": "إنه يستحق ذلك، على الرغم من ذلك. ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 166.16 }, { - "input": "In other words, that second check is asking, hey, me again, if there's an error, is the second to last bit of that position a 1? ", + "input": "ly spelled words. Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. ", "translatedText": "بمعنى آخر، هذا الفحص الثاني يسألني مرة أخرى، إذا كان هناك خطأ، فهل الجزء الثاني قبل الأخير من هذا الموضع هو 1؟ ", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 188.74 }, { - "input": "Everything we did earlier is the same as answering these four questions, which in turn is the same as spelling out a position in binary. ", + "input": "everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary. ", "translatedText": "كل ما فعلناه سابقًا هو نفس الإجابة على هذه الأسئلة الأربعة، والتي بدورها هي نفس توضيح الموضع في النظام الثنائي. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 197.74 }, { - "input": "I hope this makes two things clearer. ", + "input": "It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where tha ", "translatedText": "آمل أن يجعل هذا شيئين أكثر وضوحا. ", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two. ", + "input": "t final bit is a 1. What we get is the first of our four parity groups, which means that you can interpret that first check as asking, hey, if there's an err ", "translatedText": "الأول هو كيفية التعميم بشكل منهجي على أحجام الكتل التي تكون أكبر من قوى اثنين. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 216.68 }, { - "input": "Those of you who watched the chessboard puzzle I did with Matt Parker might find all this exceedingly familiar. ", + "input": "maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. ", "translatedText": "أولئك منكم الذين شاهدوا لغز رقعة الشطرنج الذي قمت به مع مات باركر قد يجدون كل هذا مألوفًا للغاية. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 237.32 }, { - "input": "These are the positions whose binary representation has just a single bit turned on. ", + "input": "at goes on at position 0, but don't worry about that for now. The third parity check covers every position whose third to last bit is turned ", "translatedText": "هذه هي المواضع التي تم تشغيل تمثيلها الثنائي بمقدار بت واحد فقط. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 285.5 }, { - "input": "XOR, for those of you who don't know, stands for exclusive or. ", + "input": "f 1s in the message is an even number. So for example right now, that total number of 1s is If it takes more bits to describe each p ", "translatedText": "XOR، لأولئك منكم الذين لا يعرفون، يرمز إلى حصري أو. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. ", + "input": "osition, like six bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. that special bit to be a 1, making the count even. But if the block had already started off with a ", "translatedText": "عندما تأخذ XOR لبتين، فسوف يُرجع 1 إذا تم تشغيل أي من تلك البتات، ولكن ليس إذا تم تشغيل كليهما أو إيقاف تشغيلهما. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 302.98 }, { - "input": "As a math person, I prefer to think about it as addition mod 2. ", + "input": "it would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core ", "translatedText": "باعتباري متخصصًا في الرياضيات، أفضّل التفكير في الأمر على أنه تعديل الإضافة 2. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 306.76 }, { - "input": "We also commonly talk about the XOR of two different bit strings, which basically does this component by component. ", + "input": "logic, but solving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity bits are sitti ", "translatedText": "نتحدث أيضًا بشكل شائع عن XOR لسلسلتين مختلفتين من البتات، والتي تقوم بشكل أساسي بتنفيذ هذا المكون تلو الآخر. ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 322.48 }, { - "input": "If you open up some Python right now and apply the caret operation between two integers, this is what it's doing but to the bit representations of those numbers under the hood. ", + "input": "of two, for example 1, 2, 4, and 8. These are the positions whose binary representation has just a single bit turned on. d say the parity is 0 or 1, which is typically more helpful once you start doing math with the idea. And this special bit that the sender uses to con ", "translatedText": "إذا قمت بفتح بعض لغة Python الآن وقمت بتطبيق عملية علامة الإقحام بين عددين صحيحين، فهذا ما تفعله ولكن على تمثيلات البت لتلك الأرقام الموجودة أسفل الغطاء. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop. ", + "input": "trol the parity is called the parity bit. And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure tha ", "translatedText": "النقطة الأساسية بالنسبة لي ولكم هي أن أخذ XOR للعديد من سلاسل البت المختلفة هو وسيلة فعالة لحساب المحاكاة الساخرة لمجموعة من المجموعات المنفصلة، كما هو الحال مع الأعمدة، كل ذلك في ضربة واحدة. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense? ", + "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is ", "translatedText": "هل هذا منطقي؟ ", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. ", + "input": "turned on, but not if both are turned on or if both are turned off. Phrased differently, it's the parity of these two bits. full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. ", "translatedText": "وبالمثل، يقوم العمود التالي بحساب عدد المواضع الموجودة في مجموعة التكافؤ الثانية، والمواضع التي يكون البت الثاني قبل الأخير هو 1، والتي يتم تمييزها أيضًا، وما إلى ذلك. ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 423.96 }, { - "input": "And so you know where it goes from here. ", + "input": "e also commonly talk about the XOR of two different bit s ", "translatedText": "وهكذا تعرف إلى أين يتجه الأمر من هنا. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 469.36 }, { - "input": "You see, if you add a bit string together twice, it's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. ", + "input": "ey point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop. This gives us a rather snazzy way ", "translatedText": "كما ترى، إذا قمت بإضافة سلسلة صغيرة معًا مرتين، فهذا يعني عدم وجودها على الإطلاق، لأنه في هذا العالم 1 زائد 1 يساوي 0. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 477.94 }, { - "input": "So adding a copy of this position to the total sum has the same effect as we're moving it. ", + "input": "to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation. ", "translatedText": "لذا فإن إضافة نسخة من هذا الموضع إلى المجموع الإجمالي له نفس التأثير الذي نحركه به. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 484.3 }, { - "input": "And that effect, again, is that the total result at the bottom here spells out the position of the error. ", + "input": "Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the mes ", "translatedText": "وهذا التأثير، مرة أخرى، هو أن النتيجة الإجمالية في الأسفل توضح موضع الخطأ. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 490.7 }, { - "input": "To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all of the logic on the receiver's end. ", + "input": "sage bit is turned on to a 1, and then collect these positions into one big column and take the XOR. You can probably guess that the four bits sitting at the bottom as a resu ", "translatedText": "لتوضيح مدى روعة هذا الأمر، اسمحوا لي أن أعرض سطرًا واحدًا من كود بايثون الذي أشرت إليه من قبل، والذي سيلتقط تقريبًا كل المنطق في نهاية المتلقي. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. ", + "input": "lt are the same as the four parity checks we've come to know and love, but take a moment to actually think about why exactly. This last column, for example, is counting all of the positions whose last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. ", "translatedText": "سنبدأ بإنشاء مصفوفة عشوائية مكونة من 16 1 و0 لمحاكاة كتلة البيانات، وسأعطيها بتات الاسم، ولكن بالطبع من الناحية العملية سيكون هذا شيئًا نتلقاه من المرسل، وبدلاً من ذلك نظرًا لكونها عشوائية، فإنها ستحمل 11 بتة بيانات مع 5 بتات تكافؤ. ", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15. ", + "input": "ht half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit, so these 8 bits already have an even pari Likewise, the next column counts how many positions are in the second parity group, the positions whose second to las ", "translatedText": "لذا، إذا قمنا بعد ذلك بإنشاء قائمة تدور حول كل هذه الأزواج، الأزواج التي تبدو مثل i، ثم قمنا بسحب قيمة i فقط، فقط الفهرس، حسنًا، الأمر ليس مثيرًا، سنستعيد تلك المؤشرات من 0 إلى 15 . ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 552.66 }, { - "input": "In this case it looks like those positions are 0, 4, 6, 9, etc. ", + "input": "ve on the same thing we've been doing. but for right now we're going to assume ", "translatedText": "في هذه الحالة يبدو أن هذه المواضع هي 0، 4، 6، 9، إلخ. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 567.24 }, { - "input": "To do this in Python, let me first import a couple helpful functions. ", + "input": "The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 000 ", "translatedText": "للقيام بذلك في بايثون، اسمحوا لي أولاً باستيراد وظيفتين مفيدتين. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 578.7 }, { - "input": "This basically eats its way through the list, taking XORs along the way. ", + "input": "es us a really nice way to think about why these four resulting bits at the bottom directly spell out the pos ", "translatedText": "وهذا في الأساس يشق طريقه عبر القائمة، ويأخذ XORs على طول الطريق. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 582.68 }, { - "input": "If you prefer, you can explicitly write out that XOR function without having to import it from anywhere. ", + "input": "ition of an error. Let's say you detect an error among the odd columns, and among the right half. It necessarily means the error is somewhere in th ", "translatedText": "إذا كنت تفضل ذلك، يمكنك كتابة دالة XOR بوضوح دون الحاجة إلى استيرادها من أي مكان. ", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 630.22 }, { - "input": "Isn't that neat? ", + "input": "an error that changes a 1 to a 0. You see, if you add a bit string together twice, it's the same as ", "translatedText": "أليس هذا أنيق؟ ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 641.06 }, { - "input": "And there's nothing special about the size 16 here. ", + "input": "And that effect, again, is that the total result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me sh ", "translatedText": "وليس هناك شيء خاص بخصوص الحجم 16 هنا. ", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 649.86 }, { - "input": "Needless to say, there is more code to write here, like doing the meta parity check to detect 2-bit errors, but the idea is that almost all of the core logic from our scheme comes down to a single XOR reduction. ", + "input": "ferenced before, which will capture almost all of the logic on the receiver's end. We'll start by creating a random array of 16 ones and zeros to simulate the data block, and I'll go ahead and give it the name bits, but of course in practice this would be something that we're receiving f ", "translatedText": "وغني عن القول أن هناك المزيد من التعليمات البرمجية التي يجب كتابتها هنا، مثل إجراء فحص التكافؤ التعريفي لاكتشاف أخطاء 2 بت، ولكن الفكرة هي أن كل المنطق الأساسي تقريبًا من مخططنا يعود إلى تقليل XOR واحد. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 690.5 }, { - "input": "The first one is easiest to actually do by hand, and I think it does a better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the log of the size of the block, or in other words, it grows one bit at a time as the block size doubles. ", + "input": "l out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on. In this case it looks like those positions are 0, 4, 6, 9, etc. Remember, what ", "translatedText": "الأول هو الأسهل في الواقع للقيام به يدويًا، وأعتقد أنه يقوم بعمل أفضل في غرس الحدس الأساسي الكامن وراء كل هذا، وهو أن المعلومات المطلوبة لتحديد موقع خطأ واحد مرتبطة بسجل حجم الكتلة أو بعبارة أخرى، فإنه ينمو قطعة واحدة في كل مرة مع تضاعف حجم الكتلة. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need. ", + "input": "we want is to collect together all of those positions, the positions of the bits that are turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. ", "translatedText": "الحقيقة ذات الصلة هنا هي أن هذه المعلومات تتوافق بشكل مباشر مع مقدار التكرار الذي نحتاجه. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there. ", + "input": "looks like if we do this on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but you could write a function wh ", "translatedText": "على سبيل المثال، رأينا أنه مع 256 بت، فإنك تستخدم 3% فقط من تلك المساحة للتكرار، ويستمر الأمر في التحسن من هناك. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling. ", + "input": "ere the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a state where running th ", "translatedText": "ومع زيادة عدد البتات المتماثلة واحدة تلو الأخرى، يستمر حجم الكتلة في التضاعف. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 804.3 }, { - "input": "Also, in practice, errors tend to come in little bursts, which would totally ruin a single block, so one common tactic to help spread out a burst of errors across many different blocks is to interlace those blocks, like this, before they're sent out or stored. ", + "input": "imulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the positio ", "translatedText": "أيضًا، من الناحية العملية، تميل الأخطاء إلى الظهور على شكل دفعات صغيرة، مما قد يؤدي إلى تدمير كتلة واحدة تمامًا، لذا فإن أحد الأساليب الشائعة للمساعدة في نشر موجة من الأخطاء عبر العديد من الكتل المختلفة هو تشبيك تلك الكتل، مثل هذا، قبل أن يتم دمجها إرسالها أو تخزينها. ", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind. ", + "input": "is perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it ", "translatedText": "هناك حوالي ست مرات خلال هذا الكتاب يشير فيها إلى مقولة لويس باستور، الحظ يفضل العقل المستعد. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 906.82 }, { - "input": "Part of the reason that clever ideas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong turns, underselling just how vast the space of explorable possibilities is at the start of a problem solving process, all of that. ", + "input": ", with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing 1 out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The first one is easiest to actually do by hand, and I think it does a better job instilling the core intuition underlying all of this, ", "translatedText": "جزء من السبب الذي يجعل الأفكار الذكية تبدو سهلة بشكل خادع هو أننا لا نرى سوى النتيجة النهائية، وننظف ما كان فوضويًا، ولا نذكر أبدًا كل المنعطفات الخاطئة، ونقلل من مدى اتساع مساحة الإمكانيات القابلة للاستكشاف في بداية المشكلة. عملية الحل، كل ذلك. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 922.86 }, { - "input": "But this is true in general. ", + "input": "which is that the information required to locate a single error is relat ", "translatedText": "ولكن هذا صحيح بشكل عام. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 924.9 }, { - "input": "I think for some special inventions, there's a second, deeper reason that we underappreciate them. ", + "input": "ed to the log of the size of the block, or in other words, it grows one bit at a time as the block size doubles. The relevant fact here i ", "translatedText": "أعتقد أنه بالنسبة لبعض الاختراعات الخاصة، هناك سبب ثانٍ وأعمق لعدم تقديرنا لها. ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 938.64 }, { - "input": "This was essentially concurrent with when Hamming developed his algorithm. ", + "input": "block is even, just like a normal parity check. Now, if there's a single bit error, then ", "translatedText": "كان هذا متزامنًا بشكل أساسي مع قيام هامينج بتطوير خوارزميته. ", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory. ", + "input": "the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks. However, if there's two errors, then the overall parity is going to toggle back to be And then, by the way, there is this whole other way that you s ", "translatedText": "كانت هذه هي نفس الورقة التأسيسية التي أظهرت، إلى حد ما، أن التصحيح الفعال للأخطاء أمر ممكن دائمًا، بغض النظر عن مدى احتمالية قلب البتات، على الأقل من الناحية النظرية. ", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was. ", + "input": "multiply the message by one big matrix. It's kind of nice because it relates it to the broader family of linear codes, but I think that gives almost no intuition for where it comes from or how it scales. ", "translatedText": "بعد مرور عدة عقود، وفي هذه الأيام، الكثير منا منغمسون جدًا في التفكير في أجزاء ومعلومات، مما يجعل من السهل التغاضي عن مدى تميز طريقة التفكير هذه. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/bengali/sentence_translations.json b/2020/hamming-codes-2/bengali/sentence_translations.json index 73b1087f4..1fa77c899 100644 --- a/2020/hamming-codes-2/bengali/sentence_translations.json +++ b/2020/hamming-codes-2/bengali/sentence_translations.json @@ -32,7 +32,7 @@ "end": 34.6 }, { - "input": "But as you start to think about actually implementing this, either in software or hardware, that framing may actually undersell how elegant these codes really are. ", + "input": "hat there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error. In that video, the goal was to make Hamming codes feel as hands-on and rediscoverable as possible. But as ", "translatedText": "কিন্তু আপনি যখন সফ্টওয়্যার বা হার্ডওয়্যারে বাস্তবিকই এটি বাস্তবায়নের কথা ভাবতে শুরু করেন, সেই ফ্রেমিং আসলে এই কোডগুলি কতটা মার্জিত তা বোঝা যায়।", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. ", + "input": "you read out the answers to the four parity checks we did in the last video, all as ones and zeros instead of yeses and nos, it literally spells out ", "translatedText": "উদাহরণস্বরূপ, বাইনারিতে 7 নম্বরটি 0111-এর মতো দেখায়, মূলত বলছে যে এটি 4 যোগ 2 যোগ 1।", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error. ", + "input": "century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day. The goal here is to give you a very thorough understanding of one of the earlie ", "translatedText": "সুতরাং নীচে থেকে উপরে এই চারটি চেকের ফলাফল পড়া প্রকৃতপক্ষে ত্রুটির অবস্থানটি বানান করে।", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 87.54 }, { - "input": "There's nothing special about the example 7, this works in general, and this makes the logic for implementing the whole scheme in hardware shockingly simple. ", + "input": "st examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity groups, and the second, and the third, but not the last. So reading the results of those four checks ", "translatedText": "উদাহরণ 7 সম্পর্কে বিশেষ কিছু নেই, এটি সাধারণভাবে কাজ করে, এবং এটি হার্ডওয়্যারে পুরো স্কিমটি বাস্তবায়নের যুক্তিকে চমকপ্রদভাবে সহজ করে তোলে।", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 132.36 }, { - "input": "It's worth it, though. ", + "input": "0, let's write them all in binary, running from 0000 up to 1111. ask feels at the star ", "translatedText": "এটা মূল্য, যদিও. ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 166.16 }, { - "input": "In other words, that second check is asking, hey, me again, if there's an error, is the second to last bit of that position a 1? ", + "input": "ly spelled words. Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. ", "translatedText": "অন্য কথায়, যে দ্বিতীয় চেক জিজ্ঞাসা করা হয়, আরে, আমাকে আবার, যদি একটি ত্রুটি আছে, যে অবস্থানের শেষ বিট দ্বিতীয় একটি 1? ", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 188.74 }, { - "input": "Everything we did earlier is the same as answering these four questions, which in turn is the same as spelling out a position in binary. ", + "input": "everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary. ", "translatedText": "আমরা আগে যা করেছি তা এই চারটি প্রশ্নের উত্তর দেওয়ার মতোই, যা বাইনারিতে একটি অবস্থানের বানান করার মতোই।", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 197.74 }, { - "input": "I hope this makes two things clearer. ", + "input": "It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where tha ", "translatedText": "আমি আশা করি এটি দুটি জিনিস পরিষ্কার করে।", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two. ", + "input": "t final bit is a 1. What we get is the first of our four parity groups, which means that you can interpret that first check as asking, hey, if there's an err ", "translatedText": "প্রথমটি হল দুটির বড় শক্তির আকারগুলিকে ব্লক করার পদ্ধতিগতভাবে সাধারণীকরণ কিভাবে করা যায়।", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 216.68 }, { - "input": "Those of you who watched the chessboard puzzle I did with Matt Parker might find all this exceedingly familiar. ", + "input": "maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. ", "translatedText": "আপনারা যারা ম্যাট পার্কারের সাথে দাবাবোর্ডের ধাঁধা দেখেছেন তারা হয়তো এই সব অতি পরিচিত মনে করতে পারেন।", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 237.32 }, { - "input": "These are the positions whose binary representation has just a single bit turned on. ", + "input": "at goes on at position 0, but don't worry about that for now. The third parity check covers every position whose third to last bit is turned ", "translatedText": "এই হল সেই পজিশন যার বাইনারি রিপ্রেজেন্টেশন মাত্র একটি বিট চালু হয়েছে।", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 285.5 }, { - "input": "XOR, for those of you who don't know, stands for exclusive or. ", + "input": "f 1s in the message is an even number. So for example right now, that total number of 1s is If it takes more bits to describe each p ", "translatedText": "XOR, আপনারা যারা জানেন না তাদের জন্য, একচেটিয়া বা।", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. ", + "input": "osition, like six bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. that special bit to be a 1, making the count even. But if the block had already started off with a ", "translatedText": "যখন আপনি দুটি বিটের XOR নেন, তখন এটি একটি 1 ফেরত দেয় যদি সেই বিটের যেকোনো একটি চালু থাকে, তবে উভয়টি চালু বা বন্ধ থাকলে নয়।", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 302.98 }, { - "input": "As a math person, I prefer to think about it as addition mod 2. ", + "input": "it would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core ", "translatedText": "একজন গণিত ব্যক্তি হিসাবে, আমি এটিকে সংযোজন মোড 2 হিসাবে ভাবতে পছন্দ করি।", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 306.76 }, { - "input": "We also commonly talk about the XOR of two different bit strings, which basically does this component by component. ", + "input": "logic, but solving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity bits are sitti ", "translatedText": "এছাড়াও আমরা সাধারণত দুটি ভিন্ন বিট স্ট্রিং এর XOR সম্পর্কে কথা বলি, যা মূলত এই উপাদানটি কম্পোনেন্ট দ্বারা করে।", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 322.48 }, { - "input": "If you open up some Python right now and apply the caret operation between two integers, this is what it's doing but to the bit representations of those numbers under the hood. ", + "input": "of two, for example 1, 2, 4, and 8. These are the positions whose binary representation has just a single bit turned on. d say the parity is 0 or 1, which is typically more helpful once you start doing math with the idea. And this special bit that the sender uses to con ", "translatedText": "আপনি যদি এখনই কিছু পাইথন খুলেন এবং দুটি পূর্ণসংখ্যার মধ্যে ক্যারেট অপারেশনটি প্রয়োগ করেন, তবে হুডের নীচে সেই সংখ্যাগুলির বিট উপস্থাপনা ছাড়া এটিই করছে।", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop. ", + "input": "trol the parity is called the parity bit. And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure tha ", "translatedText": "আপনার এবং আমার জন্য মূল বিষয় হল যে অনেকগুলি ভিন্ন বিট স্ট্রিংগুলির XOR নেওয়া কার্যকরভাবে একগুচ্ছ পৃথক গোষ্ঠীর প্যারোডিগুলি গণনা করার একটি উপায়, যেমন কলামগুলির সাথে, সমস্ত এককভাবে পড়ে যায়৷ এটি আমাদের হ্যামিং কোড অ্যালগরিদম থেকে মাল্টিপল প্যারিটি চেক সম্পর্কে চিন্তা করার জন্য একটি সহজ উপায় দেয় কারণ সবগুলিকে এক একক অপারেশনে একসাথে প্যাকেজ করা হচ্ছে।", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense? ", + "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is ", "translatedText": "যে জানার জন্য? ", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. ", + "input": "turned on, but not if both are turned on or if both are turned off. Phrased differently, it's the parity of these two bits. full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. ", "translatedText": "একইভাবে, পরবর্তী কলামটি দ্বিতীয় প্যারিটি গ্রুপে কতগুলি অবস্থান রয়েছে তা গণনা করে, যে অবস্থানগুলির দ্বিতীয় থেকে শেষ বিটটি 1, এবং যেগুলিকে হাইলাইট করা হয়েছে, ইত্যাদি।", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 423.96 }, { - "input": "And so you know where it goes from here. ", + "input": "e also commonly talk about the XOR of two different bit s ", "translatedText": "এবং তাই আপনি জানেন যে এটি এখান থেকে কোথায় যায়।", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 469.36 }, { - "input": "You see, if you add a bit string together twice, it's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. ", + "input": "ey point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop. This gives us a rather snazzy way ", "translatedText": "আপনি দেখতে পাচ্ছেন, আপনি যদি দুইবার একসাথে কিছুটা স্ট্রিং যোগ করেন, তবে এটি সেখানে না থাকার মতই, মূলত কারণ এই পৃথিবীতে 1 যোগ 1 সমান 0।", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 477.94 }, { - "input": "So adding a copy of this position to the total sum has the same effect as we're moving it. ", + "input": "to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation. ", "translatedText": "সুতরাং মোট যোগফলের সাথে এই অবস্থানের একটি অনুলিপি যোগ করলে একই প্রভাব রয়েছে যেভাবে আমরা এটি স্থানান্তর করছি।", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 484.3 }, { - "input": "And that effect, again, is that the total result at the bottom here spells out the position of the error. ", + "input": "Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the mes ", "translatedText": "এবং সেই প্রভাব, আবার, এখানে নীচের মোট ফলাফলটি ত্রুটির অবস্থানকে বানান করে।", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 490.7 }, { - "input": "To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all of the logic on the receiver's end. ", + "input": "sage bit is turned on to a 1, and then collect these positions into one big column and take the XOR. You can probably guess that the four bits sitting at the bottom as a resu ", "translatedText": "এটি কতটা মার্জিত তা বোঝাতে, আমি আগে উল্লেখ করেছি পাইথন কোডের একটি লাইন দেখাই, যা রিসিভারের প্রায় সমস্ত যুক্তি ক্যাপচার করবে।", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. ", + "input": "lt are the same as the four parity checks we've come to know and love, but take a moment to actually think about why exactly. This last column, for example, is counting all of the positions whose last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. ", "translatedText": "আমরা ডাটা ব্লকের অনুকরণের জন্য 16 1s এবং 0s এর একটি এলোমেলো অ্যারে তৈরি করে শুরু করব, এবং আমি এটির নাম বিট দেব, তবে অবশ্যই বাস্তবে এটি এমন কিছু হবে যা আমরা একজন প্রেরকের কাছ থেকে পাচ্ছি এবং পরিবর্তে এলোমেলো হওয়ায় এটি 5 প্যারিটি বিটের সাথে 11টি ডেটা বিট বহন করবে।", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15. ", + "input": "ht half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit, so these 8 bits already have an even pari Likewise, the next column counts how many positions are in the second parity group, the positions whose second to las ", "translatedText": "তাই যদি আমরা একটি তালিকা তৈরি করি যা এই সমস্ত জোড়ার উপর লুপ করে, যে জোড়াগুলি i এর মত দেখায়, এবং তারপরে আমরা শুধু i এর মান, শুধু সূচকটি বের করে ফেলি, ভাল এটি এত উত্তেজনাপূর্ণ নয়, আমরা 0 থেকে 15 সূচকগুলি ফিরে পাই।. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 552.66 }, { - "input": "In this case it looks like those positions are 0, 4, 6, 9, etc. ", + "input": "ve on the same thing we've been doing. but for right now we're going to assume ", "translatedText": "এই ক্ষেত্রে মনে হচ্ছে সেই অবস্থানগুলি হল 0, 4, 6, 9, ইত্যাদি।", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 567.24 }, { - "input": "To do this in Python, let me first import a couple helpful functions. ", + "input": "The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 000 ", "translatedText": "পাইথনে এটি করতে, আমাকে প্রথমে কয়েকটি সহায়ক ফাংশন আমদানি করতে দিন।", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 578.7 }, { - "input": "This basically eats its way through the list, taking XORs along the way. ", + "input": "es us a really nice way to think about why these four resulting bits at the bottom directly spell out the pos ", "translatedText": "এটি মূলত তালিকার মাধ্যমে তার পথ খায়, পথ ধরে XORs গ্রহণ করে।", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 582.68 }, { - "input": "If you prefer, you can explicitly write out that XOR function without having to import it from anywhere. ", + "input": "ition of an error. Let's say you detect an error among the odd columns, and among the right half. It necessarily means the error is somewhere in th ", "translatedText": "আপনি যদি চান, আপনি স্পষ্টভাবে XOR ফাংশনটি কোথাও থেকে আমদানি না করেই লিখতে পারেন।", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 630.22 }, { - "input": "Isn't that neat? ", + "input": "an error that changes a 1 to a 0. You see, if you add a bit string together twice, it's the same as ", "translatedText": "ঝরঝরে তাই না? ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 641.06 }, { - "input": "And there's nothing special about the size 16 here. ", + "input": "And that effect, again, is that the total result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me sh ", "translatedText": "এবং এখানে 16 আকার সম্পর্কে বিশেষ কিছু নেই।", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 649.86 }, { - "input": "Needless to say, there is more code to write here, like doing the meta parity check to detect 2-bit errors, but the idea is that almost all of the core logic from our scheme comes down to a single XOR reduction. ", + "input": "ferenced before, which will capture almost all of the logic on the receiver's end. We'll start by creating a random array of 16 ones and zeros to simulate the data block, and I'll go ahead and give it the name bits, but of course in practice this would be something that we're receiving f ", "translatedText": "বলা বাহুল্য, এখানে লেখার জন্য আরও কোড আছে, যেমন 2-বিট ত্রুটি সনাক্ত করতে মেটা প্যারিটি চেক করা, কিন্তু ধারণাটি হল যে আমাদের স্কিম থেকে প্রায় সমস্ত মূল যুক্তি একটি একক XOR হ্রাসে নেমে আসে।", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 690.5 }, { - "input": "The first one is easiest to actually do by hand, and I think it does a better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the log of the size of the block, or in other words, it grows one bit at a time as the block size doubles. ", + "input": "l out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on. In this case it looks like those positions are 0, 4, 6, 9, etc. Remember, what ", "translatedText": "প্রথমটি আসলে হাত দ্বারা করা সবচেয়ে সহজ, এবং আমি মনে করি এটি এই সমস্তটির অন্তর্নিহিত মূল অন্তর্জ্ঞান স্থাপনের জন্য একটি ভাল কাজ করে, যা হল যে একটি একক ত্রুটি সনাক্ত করার জন্য প্রয়োজনীয় তথ্য ব্লকের আকারের লগের সাথে সম্পর্কিত।, বা অন্য কথায়, ব্লকের আকার দ্বিগুণ হওয়ার সাথে সাথে এটি এক সময়ে এক বিট বৃদ্ধি পায়।", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need. ", + "input": "we want is to collect together all of those positions, the positions of the bits that are turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. ", "translatedText": "এখানে প্রাসঙ্গিক সত্য হল যে তথ্যটি সরাসরি আমাদের কতটা অপ্রয়োজনীয়তার প্রয়োজন তার সাথে মিলে যায়।", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there. ", + "input": "looks like if we do this on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but you could write a function wh ", "translatedText": "উদাহরণস্বরূপ, আমরা দেখেছি যে 256 বিটের সাথে, আপনি অপ্রয়োজনীয়তার জন্য সেই স্থানের মাত্র 3% ব্যবহার করছেন এবং এটি সেখান থেকে আরও ভাল হতে থাকে।", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling. ", + "input": "ere the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a state where running th ", "translatedText": "একের পর এক প্যারিটি বিটের সংখ্যা বাড়ার সাথে সাথে ব্লকের আকার দ্বিগুণ হতে থাকে।", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 804.3 }, { - "input": "Also, in practice, errors tend to come in little bursts, which would totally ruin a single block, so one common tactic to help spread out a burst of errors across many different blocks is to interlace those blocks, like this, before they're sent out or stored. ", + "input": "imulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the positio ", "translatedText": "এছাড়াও, অভ্যাসগতভাবে, ত্রুটিগুলি সামান্য বিস্ফোরণে আসে, যা একটি একক ব্লককে সম্পূর্ণরূপে ধ্বংস করে দেয়, তাই বিভিন্ন ব্লক জুড়ে ত্রুটিগুলি ছড়িয়ে দিতে সাহায্য করার জন্য একটি সাধারণ কৌশল হল ব্লকগুলিকে এইভাবে ইন্টারলেস করা, সেগুলি হওয়ার আগে পাঠানো বা সঞ্চিত।", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind. ", + "input": "is perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it ", "translatedText": "এই বই জুড়ে প্রায় অর্ধ ডজন বার তিনি লুই পাস্তুরের উদ্ধৃতি উল্লেখ করেছেন, ভাগ্য একটি প্রস্তুত মনকে সমর্থন করে।", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 906.82 }, { - "input": "Part of the reason that clever ideas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong turns, underselling just how vast the space of explorable possibilities is at the start of a problem solving process, all of that. ", + "input": ", with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing 1 out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The first one is easiest to actually do by hand, and I think it does a better job instilling the core intuition underlying all of this, ", "translatedText": "চতুর ধারণাগুলিকে প্রতারণামূলকভাবে সহজ দেখায় তার একটি কারণ হল যে আমরা কেবল চূড়ান্ত ফলাফলটি দেখতে পাই, যা অগোছালো ছিল তা পরিষ্কার করে, সমস্ত ভুল বাঁকগুলির কথা কখনও উল্লেখ করি না, একটি সমস্যার শুরুতে অন্বেষণযোগ্য সম্ভাবনার জায়গাটি কতটা বিস্তৃত তা আন্ডারসেলিং করি।সমাধান প্রক্রিয়া, যে সব. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 922.86 }, { - "input": "But this is true in general. ", + "input": "which is that the information required to locate a single error is relat ", "translatedText": "কিন্তু এটি সাধারণভাবে সত্য।", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 924.9 }, { - "input": "I think for some special inventions, there's a second, deeper reason that we underappreciate them. ", + "input": "ed to the log of the size of the block, or in other words, it grows one bit at a time as the block size doubles. The relevant fact here i ", "translatedText": "আমি মনে করি কিছু বিশেষ উদ্ভাবনের জন্য, একটি দ্বিতীয়, গভীর কারণ রয়েছে যে আমরা তাদের কম মূল্যায়ন করি।", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 938.64 }, { - "input": "This was essentially concurrent with when Hamming developed his algorithm. ", + "input": "block is even, just like a normal parity check. Now, if there's a single bit error, then ", "translatedText": "হ্যামিং যখন তার অ্যালগরিদম তৈরি করেছিলেন তখন এটি মূলত একই সাথে ছিল।", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory. ", + "input": "the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks. However, if there's two errors, then the overall parity is going to toggle back to be And then, by the way, there is this whole other way that you s ", "translatedText": "এটি একই ভিত্তিগত কাগজ যা দেখিয়েছিল, একটি নির্দিষ্ট অর্থে, যে দক্ষ ত্রুটি সংশোধন সর্বদা সম্ভব, বিট ফ্লিপ হওয়ার সম্ভাবনা যতই উচ্চ হোক না কেন, অন্তত তত্ত্বে।", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was. ", + "input": "multiply the message by one big matrix. It's kind of nice because it relates it to the broader family of linear codes, but I think that gives almost no intuition for where it comes from or how it scales. ", "translatedText": "বেশ কয়েক দশক ধরে দ্রুত এগিয়ে যাওয়া, এবং এই দিনগুলিতে, আমাদের মধ্যে অনেকেই বিট এবং তথ্য সম্পর্কে এতটাই নিমগ্ন যে চিন্তার এই পদ্ধতিটি কতটা স্বতন্ত্র ছিল তা উপেক্ষা করা সহজ।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/chinese/sentence_translations.json b/2020/hamming-codes-2/chinese/sentence_translations.json index 952c28878..ed512d5fd 100644 --- a/2020/hamming-codes-2/chinese/sentence_translations.json +++ b/2020/hamming-codes-2/chinese/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "我们讨论的是汉明码，这是一种创建数据块的方法 ，其中大多数位携带有意义的消息，而其他一些 位则充当一种冗余，这样如果任何位被翻转，要么 是一条消息位或冗余位，该块中的任何内容，接 收器将能够识别出存在错误，以及如何修复它。", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "那里提出的基本思想是如何使用多个奇 偶校验来进行二进制搜索以找出错误。", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "例如，二进制中的数字 7 看起来像 011 1，本质上是说它是 4 加 2 加 1。", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "请注意位置 7 所在的位置，它确实影响我们的第一 个奇偶组，以及第二个和第三个，但不影响最后一个。", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "因此，从下到上读取这四次检查的 结果确实可以阐明错误的位置。", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "当我们将这些二进制标签放回它们的盒子时 ，让我强调它们与实际发送的数据不同。", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "它们只不过是一个概念标签，可以 帮助你我理解四个奇偶组的来源。", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "用二进制描述我们所看到的一切的优雅可能会因为我 们所看到的一切都以二进制描述的混乱而被削弱。", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "我们得到的是四个奇偶校验组中的第一个，这意 味着您可以将第一个检查解释为询问，嘿，如果 有错误，该错误位置的最后一位是否为 1？", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "等等。", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "第一个是如何系统地推广到大于 2 的幂的块大小。", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "这意味着这些奇偶校验位中的每一个都位 于四个奇偶校验组中的一个且仅一个内。", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "当您对两位进行异或时，如果其中一位打开，它将返回 1，但如果两者都打开或关闭，则不会返回 1。", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "换句话说，它是这两个位的奇偶校验。", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "这就像加法，但你永远不会携带。", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "对你和我来说，关键点在于，对许多不同的位 串进行异或运算实际上是一种计算一堆单独 组的模仿的方法，就像列一样，一举完成。", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "这为我们提供了一种相当时髦的方式来思考汉明码算法中 的多个奇偶校验，因为所有这些都被打包到一个操作中。", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "那有意义吗？", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "同样，下一列计算第二个奇偶校验组中有 多少个位置、倒数第二个位为 1 的位 置以及也突出显示的位置，依此类推。", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "现在，一旦我们有了这样的结果，这给了我们一个非常好的方 法来思考为什么底部的这四个结果位直接拼出错误的位置。", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "假设此块中的某些位从 0 切换到 1。", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "这意味着该位的位置现在将包含在总 XOR 中，这会将总和从 0 更改 为这个新包含的值，即错误的位置。", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "我们将首先创建一个由 16 个 1 和 0 组成的随 机数组来模拟数据块，我将给它命名位，但当然，在实践 中，这将是我们从发送方接收的内容，而不是如果是随机 的，它将携带 11 个数据位和 5 个奇偶校验位。", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "因此，如果我们创建一个列表，循环遍历所有这些对，看起来像 i 的对，然后我们只取出 i 值，只取出索引，好吧，这并 不是那么令人兴奋，我们只是取回那些索引 0 到 15。", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "但是，如果我们添加条件以仅执行此 if 位，即如果该位 是 1 而不是 0，那么它只会提取相应位打开的位置。", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "我们不会在这里这样做，但您可以编写一个函数，其中发送方 使用该二进制表示形式根据需要设置四个奇偶校验位，最终 使该块达到在完整位列表上运行这行代码的状态一个 0。", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "很酷的是，如果我们切换此列表中的任何一位，模拟噪声引起的 随机错误，那么如果您运行同一行代码，它就会打印出该错误。", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "你可以突然得到这个块，在上面运行这一行，它会 自动吐出错误的位置，如果没有错误则吐出 0。", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "现在，根据您对二进制、异或和软件的熟悉程度，您可能 会发现这种观点有点令人困惑，或者更加优雅和简单，以 至于您想知道为什么我们不从一开始就开始使用它-去。", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "宽松地说，当非常直接地在硬件中实现汉明码时，更 容易考虑多重奇偶校验的观点，而当在软件中从更高 的层次上实现汉明码时，最容易考虑异或的观点。", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "这里的相关事实是，该信息直 接对应于我们需要多少冗余。", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "然后，顺便说一句，有时您会看到汉明码的另 一种呈现方式，即您将消息乘以一个大矩阵。", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "说到扩展，您可能会注意到，当我们增加 块大小时，该方案的效率只会变得更好。", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "例如，我们看到，对于 256 位，您仅使用该空间 的 3% 进行冗余，并且从那里开始变得越来越好。", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "随着奇偶校验位的数量逐个增加，块大小不断加倍。", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "如果你把它发挥到极致，你可能会拥有一个具有 1 00 万位的块，实际上你会用奇偶校验来回答 2 0 个问题，而它只使用 21 个奇偶校验位。", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "当然，问题在于，对于较大的块，看到超过一或两个位错误 的概率会上升，而汉明码无法处理超出此范围的任何内容。", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "但这是另一个话题了。", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "汉明在他的《科学与工程的艺术》一书中非常坦 诚地讲述了他发现这段代码的过程是多么曲折。", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "在这本书中，他大约有六次引用了路易斯· 巴斯德的名言：幸运眷顾有准备的头脑。", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "但你不应该自欺欺人地认为它们实际上 是显而易见的，因为它们绝对不是。", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "这篇基础论文在某种意义上表明，无 论位翻转的概率有多高，有效的纠错 总是可能的，至少在理论上是这样。", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "快进几十年，如今，我们中的许多人都沉浸在对比特和 信息的思考中，很容易忽视这种思维方式的独特性。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/english/captions.srt b/2020/hamming-codes-2/english/captions.srt index ffb94be28..3188eedcc 100644 --- a/2020/hamming-codes-2/english/captions.srt +++ b/2020/hamming-codes-2/english/captions.srt @@ -1,1008 +1,1168 @@ 1 -00:00:00,000 --> 00:00:02,560 -I'm assuming that everybody here is coming from part 1. +00:00:00,000 --> 00:00:02,205 +Have you ever wondered how it's possible to scratch a CD or a DVD and 2 -00:00:03,060 --> 00:00:06,506 -We were talking about Hamming codes, a way to create a block of data +00:00:02,205 --> 00:00:03,686 +still have it play back whatever it's storing? 3 -00:00:06,506 --> 00:00:09,053 -where most of the bits carry a meaningful message, +00:00:03,686 --> 00:00:05,513 +The scratch really does affect the 1s and 0s on the disk, 4 -00:00:09,053 --> 00:00:11,450 -while a few others act as a kind of redundancy, +00:00:05,513 --> 00:00:06,900 +so it reads off different data from what was 5 -00:00:11,450 --> 00:00:15,746 -in such a way that if any bit gets flipped, either a message bit or a redundancy bit, +00:00:06,900 --> 00:00:09,618 +We were talking about Hamming codes, a way to create a block of data 6 -00:00:15,746 --> 00:00:19,342 -anything in this block, a receiver is going to be able to identify that +00:00:09,618 --> 00:00:11,627 +where most of the bits carry a meaningful message, 7 -00:00:19,342 --> 00:00:21,240 -there was an error, and how to fix it. +00:00:11,627 --> 00:00:13,518 +while a few others act as a kind of redundancy, 8 +00:00:13,518 --> 00:00:16,906 +in such a way that if any bit gets flipped, either a message bit or a redundancy bit, + +9 +00:00:16,906 --> 00:00:19,742 +anything in this block, a receiver is going to be able to identify that + +10 +00:00:19,742 --> 00:00:21,240 +there was an error, and how to fix it. + +11 00:00:21,880 --> 00:00:24,449 The basic idea presented there was how to use multiple -9 +12 00:00:24,449 --> 00:00:27,160 parity checks to binary search your way down to the error. -10 +13 00:00:28,980 --> 00:00:31,847 In that video, the goal was to make Hamming codes -11 +14 00:00:31,847 --> 00:00:34,600 feel as hands-on and rediscoverable as possible. -12 +15 00:00:35,180 --> 00:00:38,227 But as you start to think about actually implementing this, -13 +16 00:00:38,227 --> 00:00:42,291 either in software or hardware, that framing may actually undersell how elegant -14 +17 00:00:42,291 --> 00:00:43,460 these codes really are. -15 +18 00:00:43,920 --> 00:00:47,008 You might think that you need to write an algorithm that keeps -16 +19 00:00:47,008 --> 00:00:51,273 track of all the possible error locations and cuts that group in half with each check, -17 +20 00:00:51,273 --> 00:00:53,480 but it's actually way, way simpler than that. -18 +21 00:00:53,940 --> 00:00:58,254 If you read out the answers to the four parity checks we did in the last video, -19 +22 00:00:58,254 --> 00:01:00,843 all as ones and zeros instead of yeses and nos, -20 +23 00:01:00,843 --> 00:01:04,080 it literally spells out the position of the error in binary. -21 -00:01:04,780 --> 00:01:08,284 -For example, the number 7 in binary looks like 0111, +24 +00:01:04,780 --> 00:01:08,219 +For the better part of the last century, this field has been a really -22 -00:01:08,284 --> 00:01:11,260 -essentially saying that it's 4 plus 2 plus 1. +25 +00:01:08,219 --> 00:01:11,462 +rich source of surprisingly deep math that gets incorporated into -23 -00:01:12,540 --> 00:01:14,460 +26 +00:01:11,462 --> 00:01:14,951 +devices we use every day. The goal here is to give you a very thorough + +27 +00:01:14,951 --> 00:01:18,440 +understanding of one of the earliest examples, known as a Hamming code. + +28 +00:01:19,300 --> 00:01:19,760 And notice where the position 7 sits. -24 -00:01:14,840 --> 00:01:18,253 +29 +00:01:19,760 --> 00:01:20,739 It does affect the first of our parity groups, -25 -00:01:18,253 --> 00:01:21,740 +30 +00:01:20,739 --> 00:01:21,740 and the second, and the third, but not the last. -26 +31 00:01:22,220 --> 00:01:24,903 So reading the results of those four checks from bottom -27 +32 00:01:24,903 --> 00:01:27,540 to top indeed does spell out the position of the error. -28 +33 00:01:28,320 --> 00:01:31,140 There's nothing special about the example 7, this works in general. -29 +34 00:01:31,780 --> 00:01:35,820 This makes the logic for implementing the whole scheme in hardware shockingly simple. -30 +35 00:01:37,240 --> 00:01:40,302 Now if you want to see why this magic happens, -31 +36 00:01:40,302 --> 00:01:45,905 take these 16 index labels for our positions, but instead of writing them in base 10, -32 +37 00:01:45,905 --> 00:01:49,880 let's write them all in binary, running from 0000 up to 1111. -33 -00:01:50,560 --> 00:01:53,423 -As we put these binary labels back into their boxes, - -34 -00:01:53,423 --> 00:01:57,800 -let me emphasize that they are distinct from the data that's actually being sent. - -35 -00:01:58,320 --> 00:02:00,910 -They're nothing more than a conceptual label to help you - -36 -00:02:00,910 --> 00:02:03,500 -and me understand where the four parity groups came from. - -37 -00:02:04,140 --> 00:02:08,083 -The elegance of having everything we're looking at be described in binary is maybe - 38 -00:02:08,083 --> 00:02:12,360 -undercut by the confusion of having everything we're looking at being described in binary. +00:01:50,560 --> 00:01:54,199 +ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. 39 -00:02:13,020 --> 00:02:14,120 -It's worth it, though. +00:01:54,199 --> 00:01:57,838 +The basic principle of error correction is that in a vast space of all possible messages, 40 -00:02:14,800 --> 00:02:18,240 -Focus your attention just on that last bit of all of these labels. +00:01:57,838 --> 00:02:00,871 +only some subset are going to be considered valid messages. As an analogy, 41 -00:02:19,880 --> 00:02:23,220 -And then highlight the positions where that final bit is a 1. +00:02:00,871 --> 00:02:03,500 +think about correctly spelled words vs incorrectly spelled words. 42 -00:02:24,240 --> 00:02:27,354 -What we get is the first of our four parity groups, +00:02:04,140 --> 00:02:07,036 +Whenever a valid message gets altered, the receiver is responsible 43 -00:02:27,354 --> 00:02:31,128 -which means that you can interpret that first check as asking, +00:02:07,036 --> 00:02:09,847 +for correcting what they see back to the nearest valid neighbor, 44 -00:02:31,128 --> 00:02:35,740 -hey, if there's an error, is the final bit in the position of that error a 1? +00:02:09,847 --> 00:02:12,787 +as you might do with a typo. Coming up with a concrete algorithm to 45 -00:02:38,200 --> 00:02:40,924 -Similarly, if you focus on the second to last bit, +00:02:12,787 --> 00:02:16,160 +efficiently categorize messages like this, though, takes a certain cleverness. 46 -00:02:40,924 --> 00:02:43,595 -and highlight all the positions where that's a 1, +00:02:16,160 --> 00:02:19,547 +The elegance of having everything we're looking at be described in binary is maybe 47 -00:02:43,595 --> 00:02:46,160 -you get the second parity group from our scheme. +00:02:19,547 --> 00:02:23,220 +undercut by the confusion of having everything we're looking at being described in binary. 48 -00:02:46,740 --> 00:02:50,377 -In other words, that second check is asking, hey, me again, +00:02:24,240 --> 00:02:25,640 +It's worth it, though. 49 -00:02:50,377 --> 00:02:54,500 -if there's an error, is the second to last bit of that position a 1? +00:02:25,640 --> 00:02:32,320 +Focus your attention just on that last bit of all of these labels. 50 -00:02:55,760 --> 00:02:56,900 -And so on. +00:02:32,540 --> 00:02:33,920 +And then highlight the positions where that final bit is a 1. 51 -00:02:57,220 --> 00:03:02,683 -The third parity check covers every position whose third to last bit is turned on, +00:02:33,920 --> 00:02:38,654 +What we get is the first of our four parity groups, 52 -00:03:02,683 --> 00:03:05,975 -and the last one covers the last eight positions, +00:02:38,654 --> 00:02:44,389 +which means that you can interpret that first check as asking, 53 -00:03:05,975 --> 00:03:08,740 -those ones whose highest order bit is a 1. +00:02:44,389 --> 00:02:51,400 +hey, if there's an error, is the final bit in the position of that error a 1? 54 -00:03:09,740 --> 00:03:14,034 -Everything we did earlier is the same as answering these four questions, +00:02:51,660 --> 00:02:54,414 +4 special bits to come nicely packaged together, 55 -00:03:14,034 --> 00:03:17,740 -which in turn is the same as spelling out a position in binary. +00:02:54,414 --> 00:02:57,787 +maybe at the end or something like that, but as you'll see, 56 -00:03:19,620 --> 00:03:21,480 -I hope this makes two things clearer. +00:02:57,787 --> 00:03:01,835 +having them sit in positions which are powers of 2 allows for something 57 -00:03:22,040 --> 00:03:24,274 -The first is how to systematically generalize +00:03:01,835 --> 00:03:05,545 +that's really elegant by the end. It also might give you a little 58 -00:03:24,274 --> 00:03:26,460 -to block sizes that are bigger powers of two. +00:03:05,545 --> 00:03:07,120 +hint about how this scales f 59 -00:03:26,960 --> 00:03:31,937 -If it takes more bits to describe each position, like six bits to describe 64 spots, +00:03:07,120 --> 00:03:12,419 +or larger blocks. Also technically it ends up being only 11 bits of data, 60 -00:03:31,937 --> 00:03:36,680 -then each of those bits gives you one of the parity groups that we need to check. +00:03:12,419 --> 00:03:17,360 +you'll find there's a mild nuance for what goes on at position 0, but 61 -00:03:38,400 --> 00:03:40,682 -Those of you who watched the chessboard puzzle I did +00:03:17,360 --> 00:03:21,480 +don't worry about that for now. 62 -00:03:40,682 --> 00:03:43,180 -with Matt Parker might find all this exceedingly familiar. +00:03:22,040 --> 00:03:24,629 +The third parity check covers every position whose third to last bit is turned on, 63 -00:03:43,660 --> 00:03:46,742 -It's the same core logic, but solving a different problem, +00:03:24,629 --> 00:03:26,189 +and the last one covers the last eight positions, 64 -00:03:46,742 --> 00:03:48,780 -and applied to a 64-squared chessboard. +00:03:26,189 --> 00:03:27,500 +those ones whose highest order bit is a 1. 65 -00:03:49,880 --> 00:03:53,393 -The second thing I hope this makes clear is why our parity bits are +00:03:27,500 --> 00:03:31,274 +ame thing as sending a message just from the past to the future instead of from one 66 -00:03:53,393 --> 00:03:57,320 -sitting in the positions that are powers of two, for example 1, 2, 4, and 8. +00:03:31,274 --> 00:03:34,915 +place to another. So that's the setup, but before we can dive in we need to talk 67 -00:03:58,000 --> 00:04:03,000 -These are the positions whose binary representation has just a single bit turned on. +00:03:34,915 --> 00:03:38,780 +about a related idea which was fresh on Hamming's mind in the time of his discovery, a 68 -00:04:03,600 --> 00:04:06,530 -What that means is each of those parity bits sits +00:03:38,780 --> 00:03:38,780 +I hope this makes two things clearer. 69 -00:04:06,530 --> 00:04:09,460 -inside one and only one of the four parity groups. +00:03:38,780 --> 00:03:45,366 +The only job of this special bit is to make sure that the total number of 1s in 70 -00:04:12,040 --> 00:04:16,095 -You can also see this in larger examples, where no matter how big you get, +00:03:45,366 --> 00:03:52,200 +the message is an even number. So for example right now, that total number of 1s is 71 -00:04:16,095 --> 00:04:19,339 -each parity bit conveniently touches only one of the groups. +00:03:52,200 --> 00:03:58,703 +If it takes more bits to describe each position, like six bits to describe 64 spots, 72 -00:04:25,600 --> 00:04:29,156 -Once you understand that these parity checks that we've focused so much of +00:03:58,703 --> 00:04:04,900 +then each of those bits gives you one of the parity groups that we need to check. 73 -00:04:29,156 --> 00:04:32,618 -our time on are nothing more than a clever way to spell out the position +00:04:04,900 --> 00:04:07,135 +that special bit to be a 1, making the count even. 74 -00:04:32,618 --> 00:04:36,127 -of an error in binary, then we can draw a connection with a different way +00:04:07,135 --> 00:04:10,115 +But if the block had already started off with an even number of 1s, 75 -00:04:36,127 --> 00:04:40,062 -to think about hamming codes, one that is arguably a lot simpler and more elegant, +00:04:10,115 --> 00:04:13,359 +then this special bit would have been kept at a 0. This is pretty simple, 76 -00:04:40,062 --> 00:04:43,240 -and which can basically be written down with a single line of code. +00:04:13,359 --> 00:04:14,280 +deceptively simple, b 77 -00:04:43,660 --> 00:04:45,500 -It's based on the XOR function. +00:04:14,360 --> 00:04:22,030 +It's the same core logic, but solving a different problem, 78 -00:04:46,940 --> 00:04:50,220 -XOR, for those of you who don't know, stands for exclusive or. +00:04:22,030 --> 00:04:27,100 +and applied to a 64-squared chessboard. 79 -00:04:50,780 --> 00:04:54,961 -When you take the XOR of two bits, it's going to return a 1 if either one of +00:04:27,100 --> 00:04:29,026 +The second thing I hope this makes clear is why our parity bits are 80 -00:04:54,961 --> 00:04:59,360 -those bits is turned on, but not if both are turned on or if both are turned off. +00:04:29,026 --> 00:04:31,180 +sitting in the positions that are powers of two, for example 1, 2, 4, and 8. 81 -00:05:00,100 --> 00:05:02,980 -Phrased differently, it's the parity of these two bits. +00:04:31,180 --> 00:04:31,400 +These are the positions whose binary representation has just a single bit turned on. 82 -00:05:03,540 --> 00:05:06,760 -As a math person, I prefer to think about it as addition mod 2. +00:04:31,400 --> 00:04:34,636 +d say the parity is 0 or 1, which is typically more helpful once 83 -00:05:07,360 --> 00:05:10,849 -We also commonly talk about the XOR of two different bit strings, +00:04:34,636 --> 00:04:37,723 +you start doing math with the idea. And this special bit that 84 -00:05:10,849 --> 00:05:13,440 -which basically does this component by component. +00:04:37,723 --> 00:04:40,860 +the sender uses to control the parity is called the parity bit. 85 -00:05:13,680 --> 00:05:15,720 -It's like addition, but where you never carry. +00:04:40,860 --> 00:04:44,053 +And actually, we should be clear, if the receiver sees an odd parity, 86 -00:05:16,500 --> 00:05:19,516 -Again, the more mathematically inclined might prefer to +00:04:44,053 --> 00:04:47,977 +it doesn't necessarily mean there was just one error, there might have been 3 errors, 87 -00:05:19,516 --> 00:05:22,480 -think of this as adding two vectors and reducing mod 2. +00:04:47,977 --> 00:04:51,400 +or 5, or any other odd number, but they can know for sure that it wasn't 0. 88 -00:05:23,500 --> 00:05:26,715 -If you open up some Python right now, and you apply the caret +00:04:51,400 --> 00:04:55,157 +On the other hand, if there had been 2 errors, or any even number of errors, 89 -00:05:26,715 --> 00:05:29,672 -operation between two integers, this is what it's doing, +00:04:55,157 --> 00:04:58,866 +that final count of 1s would still be even, so the receiver can't have full 90 -00:05:29,672 --> 00:05:32,940 -but to the bit representations of those numbers under the hood. +00:04:58,866 --> 00:05:02,526 +confidence that an even count necessarily means the message is error-free. 91 -00:05:34,960 --> 00:05:39,098 -The key point for you and me is that taking the XOR of many different +00:05:02,526 --> 00:05:06,332 +You might complain that a message which gets messed up by only 2 bit flips is 92 -00:05:39,098 --> 00:05:44,301 -bit strings is effectively a way to compute the parities of a bunch of separate groups, +00:05:06,332 --> 00:05:09,748 +pretty weak, and you would be absolutely right. Keep in mind, though, 93 -00:05:44,301 --> 00:05:47,140 -like so with the columns, all in one fell swoop. +00:05:09,748 --> 00:05:13,554 +there is no method for error detection or correction that could give you 100% 94 -00:05:51,260 --> 00:05:54,951 -This gives us a rather snazzy way to think about the multiple parity checks from +00:05:13,554 --> 00:05:15,360 +confidence that the message you recei 95 -00:05:54,951 --> 00:05:58,780 -our Hamming code algorithm as all being packaged together into one single operation. +00:05:15,360 --> 00:05:19,260 +It's based on the XOR function. 96 -00:05:59,480 --> 00:06:02,180 -Though at first glance it does look very different. +00:05:19,260 --> 00:05:22,160 +XOR, for those of you who don't know, stands for exclusive or. 97 -00:06:02,820 --> 00:06:07,494 -Specifically, write down the 16 positions in binary, like we had before, +00:05:22,160 --> 00:05:30,074 +When you take the XOR of two bits, it's going to return a 1 if either one of 98 -00:06:07,494 --> 00:06:12,617 -and now highlight only the positions where the message bit is turned on to a 1, +00:05:30,074 --> 00:05:38,400 +those bits is turned on, but not if both are turned on or if both are turned off. 99 -00:06:12,617 --> 00:06:17,100 -and then collect these positions into one big column and take the XOR. +00:05:38,400 --> 00:05:45,380 +Phrased differently, it's the parity of these two bits. 100 -00:06:19,260 --> 00:06:22,623 -You can probably guess that the four bits sitting at the bottom as +00:05:45,900 --> 00:05:50,079 +full message down to a single bit, what they give us is a 101 -00:06:22,623 --> 00:06:26,489 -a result are the same as the four parity checks we've come to know and love, +00:05:50,079 --> 00:05:54,620 +powerful building block for more sophisticated schemes. For exa 102 -00:06:26,489 --> 00:06:29,200 -but take a moment to actually think about why exactly. +00:05:54,620 --> 00:05:56,490 +We also commonly talk about the XOR of two different bit strings, 103 -00:06:32,220 --> 00:06:37,107 -This last column, for example, is counting all of the positions whose last bit is a 1, +00:05:56,490 --> 00:05:57,880 +which basically does this component by component. 104 -00:06:37,107 --> 00:06:40,535 -but we're already limited only to the highlighted positions, +00:05:57,880 --> 00:06:03,200 +It's like addition, but where you never carry. 105 -00:06:40,535 --> 00:06:45,029 -so it's effectively counting how many highlighted positions came from the first +00:06:03,620 --> 00:06:07,787 +Again, the more mathematically inclined might prefer to 106 -00:06:45,029 --> 00:06:45,760 -parity group. +00:06:07,787 --> 00:06:11,880 +think of this as adding two vectors and reducing mod 2. 107 -00:06:46,240 --> 00:06:46,800 -Does that make sense? +00:06:11,880 --> 00:06:15,014 +If you open up some Python right now, and you apply the caret 108 -00:06:49,080 --> 00:06:54,382 -Likewise, the next column counts how many positions are in the second parity group, +00:06:15,014 --> 00:06:17,895 +operation between two integers, this is what it's doing, 109 -00:06:54,382 --> 00:07:00,000 -the positions whose second to last bit is a 1, and which are also highlighted, and so on. +00:06:17,895 --> 00:06:21,080 +but to the bit representations of those numbers under the hood. 110 -00:07:00,260 --> 00:07:03,960 -It's really just a small shift in perspective on the same thing we've been doing. +00:06:21,080 --> 00:06:23,839 +The key point for you and me is that taking the XOR of many different 111 -00:07:07,760 --> 00:07:09,600 -And so you know where it goes from here. +00:06:23,839 --> 00:06:27,307 +bit strings is effectively a way to compute the parities of a bunch of separate groups, 112 -00:07:10,000 --> 00:07:13,425 -The sender is responsible for toggling some of the special +00:06:27,307 --> 00:06:29,200 +like so with the columns, all in one fell swoop. 113 -00:07:13,425 --> 00:07:16,560 -parity bits to make sure the sum works out to be 0000. +00:06:32,220 --> 00:06:33,967 +This gives us a rather snazzy way to think about the multiple parity checks from 114 -00:07:19,040 --> 00:07:23,151 -Once we have it like this, this gives us a really nice way to think about why +00:06:33,967 --> 00:06:35,780 +our Hamming code algorithm as all being packaged together into one single operation. 115 -00:07:23,151 --> 00:07:27,580 -these four resulting bits at the bottom directly spell out the position of an error. +00:06:35,780 --> 00:06:38,400 +Though at first glance it does look very different. 116 -00:07:28,460 --> 00:07:31,860 -Let's say some bit in this block gets toggled from a 0 to a 1. +00:06:38,400 --> 00:06:42,243 +Specifically, write down the 16 positions in binary, like we had before, 117 -00:07:32,600 --> 00:07:36,246 -What that means is that the position of that bit is now going to +00:06:42,243 --> 00:06:46,454 +and now highlight only the positions where the message bit is turned on to a 1, 118 -00:07:36,246 --> 00:07:39,893 -be included in the total XOR, which changes the sum from being 0 +00:06:46,454 --> 00:06:50,140 +and then collect these positions into one big column and take the XOR. 119 -00:07:39,893 --> 00:07:43,820 -to instead being this newly included value, the position of the error. +00:06:50,140 --> 00:06:52,434 +You can probably guess that the four bits sitting at the bottom as 120 -00:07:44,460 --> 00:07:49,360 -Slightly less obviously, the same is true if there's an error that changes a 1 to a 0. +00:06:52,434 --> 00:06:55,070 +a result are the same as the four parity checks we've come to know and love, 121 -00:07:50,180 --> 00:07:52,820 -You see, if you add a bit string together twice, +00:06:55,070 --> 00:06:56,920 +but take a moment to actually think about why exactly. 122 -00:07:52,820 --> 00:07:56,646 -it's the same as not having it there at all, basically because in this +00:06:56,920 --> 00:07:02,861 +This last column, for example, is counting all of the positions whose last bit is a 1, 123 -00:07:56,646 --> 00:07:57,940 -world 1 plus 1 equals 0. +00:07:02,861 --> 00:07:07,028 +but we're already limited only to the highlighted positions, 124 -00:07:58,920 --> 00:08:04,300 -So adding a copy of this position to the total sum has the same effect as removing it. +00:07:07,028 --> 00:07:12,492 +so it's effectively counting how many highlighted positions came from the first 125 -00:08:05,160 --> 00:08:07,903 -And that effect, again, is that the total result at +00:07:12,492 --> 00:07:13,380 +parity group. 126 -00:08:07,903 --> 00:08:10,700 -the bottom here spells out the position of the error. +00:07:13,380 --> 00:07:19,005 +hat we'll do. The second check is among the 8 bits on the right half of the grid, 127 -00:08:13,040 --> 00:08:17,084 -To illustrate how elegant this is, let me show that one line of Python code I +00:07:19,005 --> 00:07:24,767 +at least as we've drawn it here. This time we might use position 2 as a parity bit, 128 -00:08:17,084 --> 00:08:21,440 -referenced before, which will capture almost all of the logic on the receiver's end. +00:07:24,767 --> 00:07:27,580 +so these 8 bits already have an even pari 129 -00:08:22,080 --> 00:08:26,428 -We'll start by creating a random array of 16 ones and zeros to simulate the data block, +00:07:28,460 --> 00:07:34,713 +Likewise, the next column counts how many positions are in the second parity group, 130 -00:08:26,428 --> 00:08:30,184 -and I'll go ahead and give it the name bits, but of course in practice this +00:07:34,713 --> 00:07:41,340 +the positions whose second to last bit is a 1, and which are also highlighted, and so on. 131 -00:08:30,184 --> 00:08:34,336 -would be something that we're receiving from a sender, and instead of being random, +00:07:41,340 --> 00:07:48,040 +It's really just a small shift in perspective on the same thing we've been doing. 132 -00:08:34,336 --> 00:08:37,400 -it would be carrying 11 data bits together with 5 parity bits. +00:07:48,040 --> 00:07:55,424 +but for right now we're going to assume that there's at most one error 133 -00:08:38,120 --> 00:08:42,559 -If I call the function enumerateBits, what it does is pair together each of +00:07:55,424 --> 00:08:02,600 +in the entire block. Things break down completely for more than that. 134 -00:08:42,559 --> 00:08:47,000 -those bits with a corresponding index, in this case running from 0 up to 15. +00:08:02,600 --> 00:08:06,505 +The sender is responsible for toggling some of the special 135 -00:08:48,180 --> 00:08:51,991 -So if we then create a list that loops over all of these pairs, +00:08:06,505 --> 00:08:10,080 +parity bits to make sure the sum works out to be 0000. 136 -00:08:51,991 --> 00:08:55,980 -pairs that look like i,bit, and then we pull out just the i value, +00:08:10,080 --> 00:08:17,273 +Once we have it like this, this gives us a really nice way to think about why 137 -00:08:55,980 --> 00:09:01,340 -just the index, well, it's not that exciting, we just get back those indices 0 through 15. +00:08:17,273 --> 00:08:25,020 +these four resulting bits at the bottom directly spell out the position of an error. 138 -00:09:01,680 --> 00:09:05,072 -But if we add on the condition to only do this if bit, +00:08:25,020 --> 00:08:30,753 +Let's say you detect an error among the odd columns, and among the right half. 139 -00:09:05,072 --> 00:09:10,501 -meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where +00:08:30,753 --> 00:08:35,399 +It necessarily means the error is somewhere in the last column. 140 -00:09:10,501 --> 00:09:12,660 -the corresponding bit is turned on. +00:08:35,399 --> 00:08:40,480 +If there was no error in the odd column but there was one in the right 141 -00:09:13,380 --> 00:09:17,960 -In this case it looks like those positions are 0, 4, 6, 9, etc. +00:08:41,039 --> 00:08:43,672 +What that means is that the position of that bit is now going to 142 -00:09:19,980 --> 00:09:23,533 -Remember, what we want is to collect together all of those positions, +00:08:43,672 --> 00:08:46,305 +be included in the total XOR, which changes the sum from being 0 143 -00:09:23,533 --> 00:09:27,240 -the positions of the bits that are turned on, and then XOR them together. +00:08:46,305 --> 00:08:49,140 +to instead being this newly included value, the position of the error. 144 -00:09:29,180 --> 00:09:33,220 -To do this in Python, let me first import a couple helpful functions. +00:08:49,140 --> 00:08:52,220 +Slightly less obviously, the same is true if there's an error that changes a 1 to a 0. 145 -00:09:33,900 --> 00:09:38,700 -That way we can call reduce() on this list, and use the XOR function to reduce it. +00:08:52,220 --> 00:08:55,561 +You see, if you add a bit string together twice, 146 -00:09:39,100 --> 00:09:42,680 -This basically eats its way through the list, taking XORs along the way. +00:08:55,561 --> 00:09:00,403 +it's the same as not having it there at all, basically because in this 147 -00:09:44,800 --> 00:09:47,164 -If you prefer, you can explicitly write out that XOR +00:09:00,403 --> 00:09:02,040 +world 1 plus 1 equals 0. 148 -00:09:47,164 --> 00:09:49,440 -function without having to import it from anywhere. +00:09:02,040 --> 00:09:12,660 +So adding a copy of this position to the total sum has the same effect as removing it. 149 -00:09:51,940 --> 00:09:57,417 -So at the moment, it looks like if we do this on our random block of 16 bits, +00:09:13,380 --> 00:09:18,986 +And that effect, again, is that the total result at 150 -00:09:57,417 --> 00:10:01,280 -it returns 9, which has the binary representation 1001. +00:09:18,986 --> 00:09:24,700 +the bottom here spells out the position of the error. 151 -00:10:01,980 --> 00:10:06,291 -We won't do it here, but you could write a function where the sender uses that +00:09:24,700 --> 00:09:28,802 +To illustrate how elegant this is, let me show that one line of Python code I 152 -00:10:06,291 --> 00:10:09,456 -binary representation to set the 4 parity bits as needed, +00:09:28,802 --> 00:09:33,220 +referenced before, which will capture almost all of the logic on the receiver's end. 153 -00:10:09,456 --> 00:10:13,822 -ultimately getting this block to a state where running this line of code on the +00:09:33,900 --> 00:09:38,311 +We'll start by creating a random array of 16 ones and zeros to simulate the data block, 154 -00:10:13,822 --> 00:10:15,460 -full list of bits returns a 0. +00:09:38,311 --> 00:09:42,121 +and I'll go ahead and give it the name bits, but of course in practice this 155 -00:10:16,080 --> 00:10:18,200 -This would be considered a well-prepared block. +00:09:42,121 --> 00:09:46,332 +would be something that we're receiving from a sender, and instead of being random, 156 -00:10:19,880 --> 00:10:24,099 -Now what's cool is that if we toggle any one of the bits in this list, +00:09:46,332 --> 00:09:49,440 +it would be carrying 11 data bits together with 5 parity bits. 157 -00:10:24,099 --> 00:10:28,734 -simulating a random error from noise, then if you run this same line of code, +00:09:51,940 --> 00:09:57,030 +If I call the function enumerateBits, what it does is pair together each of 158 -00:10:28,734 --> 00:10:30,220 -it prints out that error. +00:09:57,030 --> 00:10:02,120 +those bits with a corresponding index, in this case running from 0 up to 15. 159 -00:10:30,960 --> 00:10:31,520 -Isn't that neat? +00:10:02,120 --> 00:10:05,224 +So if we then create a list that loops over all of these pairs, 160 -00:10:31,820 --> 00:10:36,124 -You could get this block from out of the blue, run this single line on it, +00:10:05,224 --> 00:10:08,474 +pairs that look like i,bit, and then we pull out just the i value, 161 -00:10:36,124 --> 00:10:41,060 -and it'll automatically spit out the position of an error, or a 0 if there wasn't any. +00:10:08,474 --> 00:10:12,840 +just the index, well, it's not that exciting, we just get back those indices 0 through 15. 162 -00:10:42,500 --> 00:10:45,200 -And there's nothing special about the size 16 here. +00:10:12,840 --> 00:10:15,905 +But if we add on the condition to only do this if bit, 163 -00:10:45,400 --> 00:10:49,860 -The same line of code would work if you had a list of 256 bits. +00:10:15,905 --> 00:10:20,809 +meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where 164 -00:10:51,880 --> 00:10:54,751 -Needless to say, there is more code to write here, +00:10:20,809 --> 00:10:22,760 +the corresponding bit is turned on. 165 -00:10:54,751 --> 00:10:57,960 -like doing the meta parity check to detect 2-bit errors, +00:10:22,760 --> 00:10:28,480 +In this case it looks like those positions are 0, 4, 6, 9, etc. 166 -00:10:57,960 --> 00:11:02,014 -but the idea is that almost all of the core logic from our scheme comes +00:10:28,480 --> 00:10:34,638 +Remember, what we want is to collect together all of those positions, 167 -00:11:02,014 --> 00:11:03,760 -down to a single XOR reduction. +00:10:34,638 --> 00:10:41,060 +the positions of the bits that are turned on, and then XOR them together. 168 -00:11:06,120 --> 00:11:09,966 -Now depending on your comfort with binary and XORs and software in general, +00:10:42,500 --> 00:10:45,200 +To do this in Python, let me first import a couple helpful functions. 169 -00:11:09,966 --> 00:11:13,054 -you may either find this perspective a little bit confusing, +00:10:45,400 --> 00:10:57,140 +That way we can call reduce() on this list, and use the XOR function to reduce it. 170 -00:11:13,054 --> 00:11:17,205 -or so much more elegant and simple that you're wondering why we didn't just start +00:10:57,140 --> 00:10:58,620 +This basically eats its way through the list, taking XORs along the way. 171 -00:11:17,205 --> 00:11:18,420 -with it from the get-go. +00:10:58,620 --> 00:11:05,295 +ays let you pin down a specific location, no matter where they turn out to be. 172 -00:11:19,140 --> 00:11:22,895 -Loosely speaking, the multiple parity check perspective is easier to think about +00:11:05,295 --> 00:11:12,477 +In fact, the astute among you might even notice a connection between these questions 173 -00:11:22,895 --> 00:11:25,631 -when implementing Hamming codes in hardware very directly, +00:11:12,477 --> 00:11:19,660 +and binary counting. And if you do, again let me emphasize, pause, try for yourself t 174 -00:11:25,631 --> 00:11:29,201 -and the XOR perspective is easiest to think about when doing it in software, +00:11:19,660 --> 00:11:20,527 +So at the moment, it looks like if we do this on our random block of 16 bits, 175 -00:11:29,201 --> 00:11:30,500 -from kind of a higher level. +00:11:20,527 --> 00:11:21,140 +it returns 9, which has the binary representation 1001. 176 -00:11:31,360 --> 00:11:34,214 -The first one is easiest to actually do by hand, +00:11:21,140 --> 00:11:24,850 +We won't do it here, but you could write a function where the sender uses that 177 -00:11:34,214 --> 00:11:39,281 -and I think it does a better job instilling the core intuition underlying all of this, +00:11:24,850 --> 00:11:27,574 +binary representation to set the 4 parity bits as needed, 178 -00:11:39,281 --> 00:11:43,825 -which is that the information required to locate a single error is related to +00:11:27,574 --> 00:11:31,331 +ultimately getting this block to a state where running this line of code on the 179 -00:11:43,825 --> 00:11:46,912 -the log of the size of the block, or in other words, +00:11:31,331 --> 00:11:32,740 +full list of bits returns a 0. 180 -00:11:46,912 --> 00:11:50,000 -it grows one bit at a time as the block size doubles. +00:11:32,740 --> 00:11:36,486 +ed, well, you can just try it. Take a moment to think about how any error among 181 -00:11:51,020 --> 00:11:53,439 -The relevant fact here is that that information +00:11:36,486 --> 00:11:39,905 +these four special bits is going to be tracked down just like any other, 182 -00:11:53,439 --> 00:11:56,060 -directly corresponds to how much redundancy we need. +00:11:39,905 --> 00:11:44,120 +with the same group of four questions. It doesn't really matter, since at the end of the d 183 -00:11:56,660 --> 00:11:59,922 -That's really what runs against most people's knee-jerk reaction when +00:11:44,120 --> 00:11:47,506 +Now what's cool is that if we toggle any one of the bits in this list, 184 -00:11:59,922 --> 00:12:02,765 -they first think about making a message resilient to errors, +00:11:47,506 --> 00:11:51,227 +simulating a random error from noise, then if you run this same line of code, 185 -00:12:02,765 --> 00:12:06,540 -where usually copying the whole message is the first instinct that comes to mind. +00:11:51,227 --> 00:11:52,420 +it prints out that error. 186 -00:12:07,500 --> 00:12:10,793 -And then, by the way, there is this whole other way that you sometimes see +00:11:52,420 --> 00:11:52,420 +correction bits are just riding along. But protecting 187 -00:12:10,793 --> 00:12:14,000 -Hamming codes presented where you multiply the message by one big matrix. +00:11:52,420 --> 00:11:52,420 +those bits as well is something that natural 188 -00:12:14,670 --> 00:12:18,736 -It's kind of nice because it relates it to the broader family of linear codes, +00:11:52,420 --> 00:11:56,221 +You could get this block from out of the blue, run this single line on it, 189 -00:12:18,736 --> 00:12:23,060 -but I think that gives almost no intuition for where it comes from or how it scales. +00:11:56,221 --> 00:12:00,580 +and it'll automatically spit out the position of an error, or a 0 if there wasn't any. 190 -00:12:25,200 --> 00:12:28,180 -And speaking of scaling, you might notice that the efficiency +00:12:00,580 --> 00:12:02,022 +hat these questions are in just a minute or two. 191 -00:12:28,180 --> 00:12:31,160 -of this scheme only gets better as we increase the block size. +00:12:02,022 --> 00:12:04,052 +Hopefully this sketch is enough to appreciate the efficiency of what 192 -00:12:35,000 --> 00:12:38,915 -For example, we saw that with 256 bits, you're using only 3% of that +00:12:04,052 --> 00:12:04,700 +we're developing here. 193 -00:12:38,915 --> 00:12:42,660 -space for redundancy, and it just keeps getting better from there. +00:12:04,700 --> 00:12:06,101 +The first thing, except for those eight highlighted parity bits, 194 -00:12:43,300 --> 00:12:47,340 -As the number of parity bits grows one by one, the block size keeps doubling. +00:12:06,101 --> 00:12:08,020 +can be whatever you want it to be, carrying whatever message or data you want. The 8 bits 195 -00:12:49,000 --> 00:12:52,599 -And if you take that to an extreme, you could have a block with, +00:12:08,020 --> 00:12:11,811 +are redundant in the sense that they're completely determined by the rest of the message, 196 -00:12:52,599 --> 00:12:56,309 -say, a million bits, where you would quite literally be playing 20 +00:12:11,811 --> 00:12:15,434 +but it's in a much smarter way than simply copying the message as a whole. And still, 197 -00:12:56,309 --> 00:13:00,020 -questions with your parity checks, and it uses only 21 parity bits. +00:12:15,434 --> 00:12:19,226 +for so little given up, you would be able to identify and fix any single bit error. Well, 198 -00:13:00,740 --> 00:13:03,792 -And if you step back to think about looking at a million +00:12:19,226 --> 00:12:23,017 +almost. Okay, so the one problem here is that if none of the four parity checks detect an 199 -00:13:03,792 --> 00:13:07,060 -bits and locating a single error, that genuinely feels crazy. +00:12:23,017 --> 00:12:23,060 +e 200 -00:13:08,200 --> 00:13:11,098 -The problem, of course, is that with a larger block, +00:12:25,200 --> 00:12:30,416 +Now depending on your comfort with binary and XORs and software in general, 201 -00:13:11,098 --> 00:13:14,761 -the probability of seeing more than one or two bit errors goes up, +00:12:30,416 --> 00:12:34,603 +you may either find this perspective a little bit confusing, 202 -00:13:14,761 --> 00:13:17,660 -and Hamming codes do not handle anything beyond that. +00:12:34,603 --> 00:12:40,232 +or so much more elegant and simple that you're wondering why we didn't just start 203 -00:13:18,320 --> 00:13:21,234 -So in practice, what you'd want is to find the right size +00:12:40,232 --> 00:12:41,880 +with it from the get-go. 204 -00:13:21,234 --> 00:13:24,300 -so that the probability of too many bit flips isn't too high. +00:12:41,880 --> 00:12:44,315 +tended, then it either means there was no error at all, 205 -00:13:26,600 --> 00:13:29,549 -Also, in practice, errors tend to come in little bursts, +00:12:44,315 --> 00:12:47,752 +or it narrows us down into position 0. You see, with four yes or no questions, 206 -00:13:29,549 --> 00:13:31,620 -which would totally ruin a single block. +00:12:47,752 --> 00:12:50,014 +we have 16 possible outcomes for our parity checks, 207 -00:13:32,200 --> 00:13:36,535 -So one common tactic to help spread out a burst of errors across many different +00:12:50,014 --> 00:12:53,668 +and at first that feels perfect for pinpointing 1 out of 16 positions in the block, 208 -00:13:36,535 --> 00:13:40,980 -blocks is to interlace those blocks, like this, before they're sent out or stored. +00:12:53,668 --> 00:12:56,800 +but you also need to communicate a 17th outcome, the no error condition. 209 -00:13:45,580 --> 00:13:49,541 -Then again, a lot of this is rendered completely moot by more modern codes, +00:12:56,800 --> 00:12:59,112 +The first one is easiest to actually do by hand, 210 -00:13:49,541 --> 00:13:52,512 -like the much more commonly used Reed-Solomon algorithm, +00:12:59,112 --> 00:13:03,217 +and I think it does a better job instilling the core intuition underlying all of this, 211 -00:13:52,512 --> 00:13:56,943 -which handles burst errors particularly well, and it can be tuned to be resilient to +00:13:03,217 --> 00:13:06,898 +which is that the information required to locate a single error is related to 212 -00:13:56,943 --> 00:13:58,820 -a larger number of errors per block. +00:13:06,898 --> 00:13:09,399 +the log of the size of the block, or in other words, 213 -00:13:59,360 --> 00:14:01,340 -But that is a topic for another time. +00:13:09,399 --> 00:13:11,900 +it grows one bit at a time as the block size doubles. 214 -00:14:02,500 --> 00:14:05,349 -In his book The Art of Doing Science and Engineering, +00:13:11,900 --> 00:13:15,855 +The relevant fact here is that that information 215 -00:14:05,349 --> 00:14:09,940 -Hamming is wonderfully candid about just how meandering his discovery of this code was. +00:13:15,855 --> 00:13:20,140 +directly corresponds to how much redundancy we need. 216 -00:14:10,620 --> 00:14:14,296 -He first tried all sorts of different schemes involving organizing the bits +00:13:20,140 --> 00:13:24,955 +at 0th one so that the parity of the full block is even, just like a normal parity check. 217 -00:14:14,296 --> 00:14:17,780 -into parts of a higher dimensional lattice and strange things like this. +00:13:24,955 --> 00:13:29,718 +Now, if there's a single bit error, then the parity of the full block toggles to be odd, 218 -00:14:18,300 --> 00:14:22,657 -The idea that it might be possible to get parity checks to conspire in a way that spells +00:13:29,718 --> 00:13:34,213 +but we would catch that anyway thanks to the four error-correcting checks. However, 219 -00:14:22,657 --> 00:14:26,966 -out the position of an error only came to Hamming when he stepped back after a bunch of +00:13:34,213 --> 00:13:38,280 +if there's two errors, then the overall parity is going to toggle back to be 220 -00:14:26,966 --> 00:14:31,275 -other analysis and asked, okay, what is the most efficient I could conceivably be about +00:13:38,280 --> 00:13:43,377 +And then, by the way, there is this whole other way that you sometimes see 221 -00:14:31,275 --> 00:14:31,520 -this? +00:13:43,377 --> 00:13:48,340 +Hamming codes presented where you multiply the message by one big matrix. 222 -00:14:32,620 --> 00:14:36,867 -He was also candid about how important it was that parity checks were already on +00:13:48,340 --> 00:13:54,640 +It's kind of nice because it relates it to the broader family of linear codes, 223 -00:14:36,867 --> 00:14:41,220 -his mind, which would have been way less common back in the 1940s than it is today. +00:13:54,640 --> 00:14:01,340 +but I think that gives almost no intuition for where it comes from or how it scales. 224 -00:14:41,920 --> 00:14:45,045 -There are like half a dozen times throughout this book that he +00:14:02,500 --> 00:14:06,220 +And speaking of scaling, you might notice that the efficiency 225 -00:14:45,045 --> 00:14:48,220 -references the Louis Pasteur quote, luck favors a prepared mind. +00:14:06,220 --> 00:14:09,940 +of this scheme only gets better as we increase the block size. 226 -00:14:49,320 --> 00:14:52,216 -Clever ideas often look deceptively simple in hindsight, +00:14:10,620 --> 00:14:14,279 +For example, we saw that with 256 bits, you're using only 3% of that 227 -00:14:52,216 --> 00:14:54,300 -which makes them easy to underappreciate. +00:14:14,279 --> 00:14:17,780 +space for redundancy, and it just keeps getting better from there. 228 -00:14:54,960 --> 00:14:57,517 -Right now my honest hope is that Hamming codes, +00:14:18,300 --> 00:14:21,000 +As the number of parity bits grows one by one, the block size keeps doubling. 229 -00:14:57,517 --> 00:15:01,300 -or at least the possibility of such codes, feels almost obvious to you. +00:14:21,000 --> 00:14:24,436 +And if you take that to an extreme, you could have a block with, 230 -00:15:01,660 --> 00:15:05,352 -But you shouldn't fool yourself into thinking that they actually are obvious, +00:14:24,436 --> 00:14:27,978 +say, a million bits, where you would quite literally be playing 20 231 -00:15:05,352 --> 00:15:06,820 -because they definitely aren't. +00:14:27,978 --> 00:14:31,520 +questions with your parity checks, and it uses only 21 parity bits. 232 -00:15:07,880 --> 00:15:11,701 -Part of the reason that clever ideas look deceptively easy is that we only +00:14:32,620 --> 00:14:34,957 +ugh so you can check yourself. To set up a message, 233 -00:15:11,701 --> 00:15:14,503 -ever see the final result, cleaning up what was messy, +00:14:34,957 --> 00:14:38,149 +whether that's a literal message you're translating over space or some 234 -00:15:14,503 --> 00:15:18,172 -never mentioning all of the wrong turns, underselling just how vast the +00:14:38,149 --> 00:14:42,060 +data you want to store over time, the first step is to divide it up into 11-bit chunks. 235 -00:15:18,172 --> 00:15:22,248 -space of explorable possibilities is at the start of a problem solving process, +00:14:42,060 --> 00:14:45,515 +The problem, of course, is that with a larger block, 236 -00:15:22,248 --> 00:15:22,860 -all of that. +00:14:45,515 --> 00:14:49,884 +the probability of seeing more than one or two bit errors goes up, 237 +00:14:49,884 --> 00:14:53,340 +and Hamming codes do not handle anything beyond that. + +238 +00:14:53,340 --> 00:14:54,987 +So in practice, what you'd want is to find the right size + +239 +00:14:54,987 --> 00:14:56,720 +so that the probability of too many bit flips isn't too high. + +240 +00:14:56,720 --> 00:14:56,943 +Also, in practice, errors tend to come in little bursts, + +241 +00:14:56,943 --> 00:14:57,100 +which would totally ruin a single block. + +242 +00:14:57,240 --> 00:15:02,251 +ow has an even parity, meaning you can set that bit number 0, + +243 +00:15:02,251 --> 00:15:07,666 +the overarching parity bit, to be 0. So as this block is sent off, + +244 +00:15:07,666 --> 00:15:14,860 +the parity of the four special subsets and the block as a whole will all be even, or 0. A + +245 +00:15:14,860 --> 00:15:17,253 +Then again, a lot of this is rendered completely moot by more modern codes, + +246 +00:15:17,253 --> 00:15:19,048 +like the much more commonly used Reed-Solomon algorithm, + +247 +00:15:19,048 --> 00:15:21,726 +which handles burst errors particularly well, and it can be tuned to be resilient to + +248 +00:15:21,726 --> 00:15:22,860 +a larger number of errors per block. + +249 00:15:23,820 --> 00:15:24,900 +see that it's even, so any error that exists would have to be in an even column. + +250 +00:15:24,900 --> 00:15:27,596 +In his book The Art of Doing Science and Engineering, + +251 +00:15:27,596 --> 00:15:31,940 +Hamming is wonderfully candid about just how meandering his discovery of this code was. + +252 +00:15:31,940 --> 00:15:35,604 +is odd, giving us confidence that there was one flip and not two. If it's three or more, + +253 +00:15:35,604 --> 00:15:37,869 +all bets are off. After correcting that bit number 10, + +254 +00:15:37,869 --> 00:15:41,492 +pulling out the 11 bits that were not used for correction gives us the relevant segment + +255 +00:15:41,492 --> 00:15:44,910 +of the original message, which if you rewind and compare is indeed exactly what we + +256 +00:15:44,910 --> 00:15:45,940 +started the example with. + +257 +00:15:45,940 --> 00:15:51,879 +The idea that it might be possible to get parity checks to conspire in a way that spells + +258 +00:15:51,879 --> 00:15:57,753 +out the position of an error only came to Hamming when he stepped back after a bunch of + +259 +00:15:57,753 --> 00:16:03,626 +other analysis and asked, okay, what is the most efficient I could conceivably be about + +260 +00:16:03,626 --> 00:16:03,960 +this? + +261 +00:16:03,960 --> 00:16:09,541 +He was also candid about how important it was that parity checks were already on + +262 +00:16:09,541 --> 00:16:15,260 +his mind, which would have been way less common back in the 1940s than it is today. + +263 +00:16:15,260 --> 00:16:17,116 +a machine to point to the position of an error, + +264 +00:16:17,116 --> 00:16:19,436 +how to systematically scale it, and how we can frame all of + +265 +00:16:19,436 --> 00:16:22,260 +this as one single operation rather than multiple separate parity checks. + +266 +00:16:22,260 --> 00:16:22,260 +Clever ideas often look deceptively simple in hindsight, + +267 +00:16:22,260 --> 00:16:22,260 +which makes them easy to underappreciate. + +268 +00:16:22,260 --> 00:16:22,260 +Right now my honest hope is that Hamming codes, + +269 +00:16:22,260 --> 00:16:22,260 +or at least the possibility of such codes, feels almost obvious to you. + +270 +00:16:22,260 --> 00:16:22,260 +But you shouldn't fool yourself into thinking that they actually are obvious, + +271 +00:16:22,260 --> 00:16:22,260 +because they definitely aren't. + +272 +00:16:22,260 --> 00:16:22,260 +Part of the reason that clever ideas look deceptively easy is that we only + +273 +00:16:22,260 --> 00:16:22,260 +ever see the final result, cleaning up what was messy, + +274 +00:16:22,260 --> 00:16:22,260 +never mentioning all of the wrong turns, underselling just how vast the + +275 +00:16:22,260 --> 00:16:22,260 +space of explorable possibilities is at the start of a problem solving process, + +276 +00:16:22,260 --> 00:16:22,260 +all of that. + +277 +00:16:22,260 --> 00:16:22,260 But this is true in general. -238 -00:15:24,900 --> 00:15:27,784 +278 +00:16:22,260 --> 00:16:22,260 I think for some special inventions, there's a second, -239 -00:15:27,784 --> 00:15:30,040 +279 +00:16:22,260 --> 00:16:22,260 deeper reason that we underappreciate them. -240 -00:15:30,840 --> 00:15:34,612 +280 +00:16:22,260 --> 00:16:22,260 Thinking of information in terms of bits had only really coalesced into a -241 -00:15:34,612 --> 00:15:38,640 +281 +00:16:22,260 --> 00:16:22,260 full theory by 1948, with Claude Shannon's seminal paper on information theory. -242 -00:15:39,280 --> 00:15:42,540 +282 +00:16:22,260 --> 00:16:22,260 This was essentially concurrent with when Hamming developed his algorithm. -243 -00:15:43,300 --> 00:15:46,836 +283 +00:16:22,260 --> 00:16:22,260 This was the same foundational paper that showed, in a certain sense, -244 -00:15:46,836 --> 00:15:49,464 +284 +00:16:22,260 --> 00:16:22,260 that efficient error correction is always possible, -245 -00:15:49,464 --> 00:15:52,900 +285 +00:16:22,260 --> 00:16:22,260 no matter how high the probability of bit flips, at least in theory. -246 -00:15:53,700 --> 00:15:57,038 +286 +00:16:22,260 --> 00:16:22,260 Shannon and Hamming, by the way, shared an office in Bell Labs, -247 -00:15:57,038 --> 00:16:01,160 +287 +00:16:22,260 --> 00:16:22,260 despite working on very different things, which hardly seems coincidental here. -248 -00:16:02,380 --> 00:16:05,718 +288 +00:16:22,260 --> 00:16:22,260 Fast forward several decades, and these days, many of us are -249 -00:16:05,718 --> 00:16:09,056 +289 +00:16:22,260 --> 00:16:22,260 so immersed in thinking about bits and information that it's -250 -00:16:09,056 --> 00:16:12,340 +290 +00:16:22,260 --> 00:16:22,260 easy to overlook just how distinct this way of thinking was. -251 -00:16:13,100 --> 00:16:17,707 +291 +00:16:22,260 --> 00:16:22,260 Ironically, the ideas that most profoundly shape the ways that a future generation -252 -00:16:17,707 --> 00:16:22,260 +292 +00:16:22,260 --> 00:16:22,260 thinks will end up looking to that future generation simpler than they really are. diff --git a/2020/hamming-codes-2/english/sentence_timings.json b/2020/hamming-codes-2/english/sentence_timings.json index 033a00599..f9bec5afe 100644 --- a/2020/hamming-codes-2/english/sentence_timings.json +++ b/2020/hamming-codes-2/english/sentence_timings.json @@ -1,12 +1,12 @@ [ [ - "I'm assuming that everybody here is coming from part 1.", + "Have you ever wondered how it's possible to scratch a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was", 0.0, - 2.56 + 6.9 ], [ "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", - 3.06, + 6.9, 21.24 ], [ @@ -35,18 +35,18 @@ 64.08 ], [ - "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day. The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code.", 64.78, - 71.26 + 78.44 ], [ "And notice where the position 7 sits.", - 72.54, - 74.46 + 79.3, + 79.76 ], [ "It does affect the first of our parity groups, and the second, and the third, but not the last.", - 74.84, + 79.76, 81.74 ], [ @@ -70,523 +70,523 @@ 109.88 ], [ - "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some subset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", 110.56, - 117.8 + 123.5 ], [ - "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", - 118.32, - 123.5 + "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messages like this, though, takes a certain cleverness.", + 124.14, + 136.16 ], [ "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", - 124.14, - 132.36 + 136.16, + 143.22 ], [ "It's worth it, though.", - 133.02, - 134.12 + 144.24, + 145.64 ], [ "Focus your attention just on that last bit of all of these labels.", - 134.8, - 138.24 + 145.64, + 152.32 ], [ "And then highlight the positions where that final bit is a 1.", - 139.88, - 143.22 + 152.54, + 153.92 ], [ "What we get is the first of our four parity groups, which means that you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", - 144.24, - 155.74 + 153.92, + 171.4 ], [ - "Similarly, if you focus on the second to last bit, and highlight all the positions where that's a 1, you get the second parity group from our scheme.", - 158.2, - 166.16 + "4 special bits to come nicely packaged together, maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. It also might give you a little hint about how this scales f", + 171.66, + 187.12 ], [ - "In other words, that second check is asking, hey, me again, if there's an error, is the second to last bit of that position a 1?", - 166.74, - 174.5 + "or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but", + 187.12, + 197.36 ], [ - "And so on.", - 175.76, - 176.9 + "don't worry about that for now.", + 197.36, + 201.48 ], [ "The third parity check covers every position whose third to last bit is turned on, and the last one covers the last eight positions, those ones whose highest order bit is a 1.", - 177.22, - 188.74 + 202.04, + 207.5 ], [ - "Everything we did earlier is the same as answering these four questions, which in turn is the same as spelling out a position in binary.", - 189.74, - 197.74 + "ame thing as sending a message just from the past to the future instead of from one place to another. So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a", + 207.5, + 218.78 ], [ "I hope this makes two things clearer.", - 199.62, - 201.48 + 218.78, + 218.78 ], [ - "The first is how to systematically generalize to block sizes that are bigger powers of two.", - 202.04, - 206.46 + "The only job of this special bit is to make sure that the total number of 1s in the message is an even number. So for example right now, that total number of 1s is", + 218.78, + 232.2 ], [ "If it takes more bits to describe each position, like six bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check.", - 206.96, - 216.68 + 232.2, + 244.9 ], [ - "Those of you who watched the chessboard puzzle I did with Matt Parker might find all this exceedingly familiar.", - 218.4, - 223.18 + "that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b", + 244.9, + 254.28 ], [ "It's the same core logic, but solving a different problem, and applied to a 64-squared chessboard.", - 223.66, - 228.78 + 254.36, + 267.1 ], [ "The second thing I hope this makes clear is why our parity bits are sitting in the positions that are powers of two, for example 1, 2, 4, and 8.", - 229.88, - 237.32 + 267.1, + 271.18 ], [ "These are the positions whose binary representation has just a single bit turned on.", - 238.0, - 243.0 + 271.18, + 271.4 ], [ - "What that means is each of those parity bits sits inside one and only one of the four parity groups.", - 243.6, - 249.46 + "d say the parity is 0 or 1, which is typically more helpful once you start doing math with the idea. And this special bit that the sender uses to control the parity is called the parity bit.", + 271.4, + 280.86 ], [ - "You can also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups.", - 252.04, - 259.34 + "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + 280.86, + 291.4 ], [ - "Once you understand that these parity checks that we've focused so much of our time on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a different way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code.", - 265.6, - 283.24 + "On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind, though, there is no method for error detection or correction that could give you 100% confidence that the message you recei", + 291.4, + 315.36 ], [ "It's based on the XOR function.", - 283.66, - 285.5 + 315.36, + 319.26 ], [ "XOR, for those of you who don't know, stands for exclusive or.", - 286.94, - 290.22 + 319.26, + 322.16 ], [ "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or if both are turned off.", - 290.78, - 299.36 + 322.16, + 338.4 ], [ "Phrased differently, it's the parity of these two bits.", - 300.1, - 302.98 + 338.4, + 345.38 ], [ - "As a math person, I prefer to think about it as addition mod 2.", - 303.54, - 306.76 + "full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. For exa", + 345.9, + 354.62 ], [ "We also commonly talk about the XOR of two different bit strings, which basically does this component by component.", - 307.36, - 313.44 + 354.62, + 357.88 ], [ "It's like addition, but where you never carry.", - 313.68, - 315.72 + 357.88, + 363.2 ], [ "Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2.", - 316.5, - 322.48 + 363.62, + 371.88 ], [ "If you open up some Python right now, and you apply the caret operation between two integers, this is what it's doing, but to the bit representations of those numbers under the hood.", - 323.5, - 332.94 + 371.88, + 381.08 ], [ "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", - 334.96, - 347.14 + 381.08, + 389.2 ], [ "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", - 351.26, - 358.78 + 392.22, + 395.78 ], [ "Though at first glance it does look very different.", - 359.48, - 362.18 + 395.78, + 398.4 ], [ "Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the message bit is turned on to a 1, and then collect these positions into one big column and take the XOR.", - 362.82, - 377.1 + 398.4, + 410.14 ], [ "You can probably guess that the four bits sitting at the bottom as a result are the same as the four parity checks we've come to know and love, but take a moment to actually think about why exactly.", - 379.26, - 389.2 + 410.14, + 416.92 ], [ "This last column, for example, is counting all of the positions whose last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group.", - 392.22, - 405.76 + 416.92, + 433.38 ], [ - "Does that make sense?", - 406.24, - 406.8 + "hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit, so these 8 bits already have an even pari", + 433.38, + 447.58 ], [ "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", - 409.08, - 420.0 + 448.46, + 461.34 ], [ "It's really just a small shift in perspective on the same thing we've been doing.", - 420.26, - 423.96 + 461.34, + 468.04 ], [ - "And so you know where it goes from here.", - 427.76, - 429.6 + "but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that.", + 468.04, + 482.6 ], [ "The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", - 430.0, - 436.56 + 482.6, + 490.08 ], [ "Once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", - 439.04, - 447.58 + 490.08, + 505.02 ], [ - "Let's say some bit in this block gets toggled from a 0 to a 1.", - 448.46, - 451.86 + "Let's say you detect an error among the odd columns, and among the right half. It necessarily means the error is somewhere in the last column. If there was no error in the odd column but there was one in the right", + 505.02, + 520.48 ], [ "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", - 452.6, - 463.82 + 521.04, + 529.14 ], [ "Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", - 464.46, - 469.36 + 529.14, + 532.22 ], [ "You see, if you add a bit string together twice, it's the same as not having it there at all, basically because in this world 1 plus 1 equals 0.", - 470.18, - 477.94 + 532.22, + 542.04 ], [ "So adding a copy of this position to the total sum has the same effect as removing it.", - 478.92, - 484.3 + 542.04, + 552.66 ], [ "And that effect, again, is that the total result at the bottom here spells out the position of the error.", - 485.16, - 490.7 + 553.38, + 564.7 ], [ "To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all of the logic on the receiver's end.", - 493.04, - 501.44 + 564.7, + 573.22 ], [ "We'll start by creating a random array of 16 ones and zeros to simulate the data block, and I'll go ahead and give it the name bits, but of course in practice this would be something that we're receiving from a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits.", - 502.08, - 517.4 + 573.9, + 589.44 ], [ "If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", - 518.12, - 527.0 + 591.94, + 602.12 ], [ "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15.", - 528.18, - 541.34 + 602.12, + 612.84 ], [ "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", - 541.68, - 552.66 + 612.84, + 622.76 ], [ "In this case it looks like those positions are 0, 4, 6, 9, etc.", - 553.38, - 557.96 + 622.76, + 628.48 ], [ "Remember, what we want is to collect together all of those positions, the positions of the bits that are turned on, and then XOR them together.", - 559.98, - 567.24 + 628.48, + 641.06 ], [ "To do this in Python, let me first import a couple helpful functions.", - 569.18, - 573.22 + 642.5, + 645.2 ], [ "That way we can call reduce() on this list, and use the XOR function to reduce it.", - 573.9, - 578.7 + 645.4, + 657.14 ], [ "This basically eats its way through the list, taking XORs along the way.", - 579.1, - 582.68 + 657.14, + 658.62 ], [ - "If you prefer, you can explicitly write out that XOR function without having to import it from anywhere.", - 584.8, - 589.44 + "ays let you pin down a specific location, no matter where they turn out to be. In fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t", + 658.62, + 679.66 ], [ "So at the moment, it looks like if we do this on our random block of 16 bits, it returns 9, which has the binary representation 1001.", - 591.94, - 601.28 + 679.66, + 681.14 ], [ "We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", - 601.98, - 615.46 + 681.14, + 692.74 ], [ - "This would be considered a well-prepared block.", - 616.08, - 618.2 + "ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions. It doesn't really matter, since at the end of the d", + 692.74, + 704.12 ], [ "Now what's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", - 619.88, - 630.22 + 704.12, + 712.42 ], [ - "Isn't that neat?", - 630.96, - 631.52 + "correction bits are just riding along. But protecting those bits as well is something that natural", + 712.42, + 712.42 ], [ "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", - 631.82, - 641.06 + 712.42, + 720.58 ], [ - "And there's nothing special about the size 16 here.", - 642.5, - 645.2 + "hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + 720.58, + 724.7 ], [ - "The same line of code would work if you had a list of 256 bits.", - 645.4, - 649.86 + "The first thing, except for those eight highlighted parity bits, can be whatever you want it to be, carrying whatever message or data you want. The 8 bits", + 724.7, + 728.02 ], [ - "Needless to say, there is more code to write here, like doing the meta parity check to detect 2-bit errors, but the idea is that almost all of the core logic from our scheme comes down to a single XOR reduction.", - 651.88, - 663.76 + "are redundant in the sense that they're completely determined by the rest of the message, but it's in a much smarter way than simply copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almost. Okay, so the one problem here is that if none of the four parity checks detect an e", + 728.02, + 743.06 ], [ "Now depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", - 666.12, - 678.42 + 745.2, + 761.88 ], [ - "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", - 679.14, - 690.5 + "tended, then it either means there was no error at all, or it narrows us down into position 0. You see, with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing 1 out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition.", + 761.88, + 776.8 ], [ "The first one is easiest to actually do by hand, and I think it does a better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the log of the size of the block, or in other words, it grows one bit at a time as the block size doubles.", - 691.36, - 710.0 + 776.8, + 791.9 ], [ "The relevant fact here is that that information directly corresponds to how much redundancy we need.", - 711.02, - 716.06 + 791.9, + 800.14 ], [ - "That's really what runs against most people's knee-jerk reaction when they first think about making a message resilient to errors, where usually copying the whole message is the first instinct that comes to mind.", - 716.66, - 726.54 + "at 0th one so that the parity of the full block is even, just like a normal parity check. Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks. However, if there's two errors, then the overall parity is going to toggle back to be", + 800.14, + 818.28 ], [ "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented where you multiply the message by one big matrix.", - 727.5, - 734.0 + 818.28, + 828.34 ], [ "It's kind of nice because it relates it to the broader family of linear codes, but I think that gives almost no intuition for where it comes from or how it scales.", - 734.67, - 743.06 + 828.34, + 841.34 ], [ "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", - 745.2, - 751.16 + 842.5, + 849.94 ], [ "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", - 755.0, - 762.66 + 850.62, + 857.78 ], [ "As the number of parity bits grows one by one, the block size keeps doubling.", - 763.3, - 767.34 + 858.3, + 861.0 ], [ "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", - 769.0, - 780.02 + 861.0, + 871.52 ], [ - "And if you step back to think about looking at a million bits and locating a single error, that genuinely feels crazy.", - 780.74, - 787.06 + "ugh so you can check yourself. To set up a message, whether that's a literal message you're translating over space or some data you want to store over time, the first step is to divide it up into 11-bit chunks.", + 872.62, + 882.06 ], [ "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", - 788.2, - 797.66 + 882.06, + 893.34 ], [ "So in practice, what you'd want is to find the right size so that the probability of too many bit flips isn't too high.", - 798.32, - 804.3 + 893.34, + 896.72 ], [ "Also, in practice, errors tend to come in little bursts, which would totally ruin a single block.", - 806.6, - 811.62 + 896.72, + 897.1 ], [ - "So one common tactic to help spread out a burst of errors across many different blocks is to interlace those blocks, like this, before they're sent out or stored.", - 812.2, - 820.98 + "ow has an even parity, meaning you can set that bit number 0, the overarching parity bit, to be 0. So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0. A", + 897.24, + 914.86 ], [ "Then again, a lot of this is rendered completely moot by more modern codes, like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly well, and it can be tuned to be resilient to a larger number of errors per block.", - 825.58, - 838.82 + 914.86, + 922.86 ], [ - "But that is a topic for another time.", - 839.36, - 841.34 + "see that it's even, so any error that exists would have to be in an even column.", + 923.82, + 924.9 ], [ "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", - 842.5, - 849.94 + 924.9, + 931.94 ], [ - "He first tried all sorts of different schemes involving organizing the bits into parts of a higher dimensional lattice and strange things like this.", - 850.62, - 857.78 + "is odd, giving us confidence that there was one flip and not two. If it's three or more, all bets are off. After correcting that bit number 10, pulling out the 11 bits that were not used for correction gives us the relevant segment of the original message, which if you rewind and compare is indeed exactly what we started the example with.", + 931.94, + 945.94 ], [ "The idea that it might be possible to get parity checks to conspire in a way that spells out the position of an error only came to Hamming when he stepped back after a bunch of other analysis and asked, okay, what is the most efficient I could conceivably be about this?", - 858.3, - 871.52 + 945.94, + 963.96 ], [ "He was also candid about how important it was that parity checks were already on his mind, which would have been way less common back in the 1940s than it is today.", - 872.62, - 881.22 + 963.96, + 975.26 ], [ - "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", - 881.92, - 888.22 + "a machine to point to the position of an error, how to systematically scale it, and how we can frame all of this as one single operation rather than multiple separate parity checks.", + 975.26, + 982.26 ], [ "Clever ideas often look deceptively simple in hindsight, which makes them easy to underappreciate.", - 889.32, - 894.3 + 982.26, + 982.26 ], [ "Right now my honest hope is that Hamming codes, or at least the possibility of such codes, feels almost obvious to you.", - 894.96, - 901.3 + 982.26, + 982.26 ], [ "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", - 901.66, - 906.82 + 982.26, + 982.26 ], [ "Part of the reason that clever ideas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong turns, underselling just how vast the space of explorable possibilities is at the start of a problem solving process, all of that.", - 907.88, - 922.86 + 982.26, + 982.26 ], [ "But this is true in general.", - 923.82, - 924.9 + 982.26, + 982.26 ], [ "I think for some special inventions, there's a second, deeper reason that we underappreciate them.", - 924.9, - 930.04 + 982.26, + 982.26 ], [ "Thinking of information in terms of bits had only really coalesced into a full theory by 1948, with Claude Shannon's seminal paper on information theory.", - 930.84, - 938.64 + 982.26, + 982.26 ], [ "This was essentially concurrent with when Hamming developed his algorithm.", - 939.28, - 942.54 + 982.26, + 982.26 ], [ "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", - 943.3, - 952.9 + 982.26, + 982.26 ], [ "Shannon and Hamming, by the way, shared an office in Bell Labs, despite working on very different things, which hardly seems coincidental here.", - 953.7, - 961.16 + 982.26, + 982.26 ], [ "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", - 962.38, - 972.34 + 982.26, + 982.26 ], [ "Ironically, the ideas that most profoundly shape the ways that a future generation thinks will end up looking to that future generation simpler than they really are.", - 973.1, + 982.26, 982.26 ] ] \ No newline at end of file diff --git a/2020/hamming-codes-2/english/transcript.txt b/2020/hamming-codes-2/english/transcript.txt index d45d11d90..9ffeb3bc8 100644 --- a/2020/hamming-codes-2/english/transcript.txt +++ b/2020/hamming-codes-2/english/transcript.txt @@ -1,44 +1,44 @@ -I'm assuming that everybody here is coming from part 1. +Have you ever wondered how it's possible to scratch a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error. In that video, the goal was to make Hamming codes feel as hands-on and rediscoverable as possible. But as you start to think about actually implementing this, either in software or hardware, that framing may actually undersell how elegant these codes really are. You might think that you need to write an algorithm that keeps track of all the possible error locations and cuts that group in half with each check, but it's actually way, way simpler than that. If you read out the answers to the four parity checks we did in the last video, all as ones and zeros instead of yeses and nos, it literally spells out the position of the error in binary. -For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. +For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day. The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error. There's nothing special about the example 7, this works in general. This makes the logic for implementing the whole scheme in hardware shockingly simple. Now if you want to see why this magic happens, take these 16 index labels for our positions, but instead of writing them in base 10, let's write them all in binary, running from 0000 up to 1111. -As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent. -They're nothing more than a conceptual label to help you and me understand where the four parity groups came from. +ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some subset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words. +Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messages like this, though, takes a certain cleverness. The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is the first of our four parity groups, which means that you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1? -Similarly, if you focus on the second to last bit, and highlight all the positions where that's a 1, you get the second parity group from our scheme. -In other words, that second check is asking, hey, me again, if there's an error, is the second to last bit of that position a 1? -And so on. +4 special bits to come nicely packaged together, maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. It also might give you a little hint about how this scales f +or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but +don't worry about that for now. The third parity check covers every position whose third to last bit is turned on, and the last one covers the last eight positions, those ones whose highest order bit is a 1. -Everything we did earlier is the same as answering these four questions, which in turn is the same as spelling out a position in binary. +ame thing as sending a message just from the past to the future instead of from one place to another. So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer. -The first is how to systematically generalize to block sizes that are bigger powers of two. +The only job of this special bit is to make sure that the total number of 1s in the message is an even number. So for example right now, that total number of 1s is If it takes more bits to describe each position, like six bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. -Those of you who watched the chessboard puzzle I did with Matt Parker might find all this exceedingly familiar. +that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but solving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity bits are sitting in the positions that are powers of two, for example 1, 2, 4, and 8. These are the positions whose binary representation has just a single bit turned on. -What that means is each of those parity bits sits inside one and only one of the four parity groups. -You can also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. -Once you understand that these parity checks that we've focused so much of our time on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a different way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. +d say the parity is 0 or 1, which is typically more helpful once you start doing math with the idea. And this special bit that the sender uses to control the parity is called the parity bit. +And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. +On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind, though, there is no method for error detection or correction that could give you 100% confidence that the message you recei It's based on the XOR function. XOR, for those of you who don't know, stands for exclusive or. When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or if both are turned off. Phrased differently, it's the parity of these two bits. -As a math person, I prefer to think about it as addition mod 2. +full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. For exa We also commonly talk about the XOR of two different bit strings, which basically does this component by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. @@ -49,13 +49,13 @@ Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the message bit is turned on to a 1, and then collect these positions into one big column and take the XOR. You can probably guess that the four bits sitting at the bottom as a result are the same as the four parity checks we've come to know and love, but take a moment to actually think about why exactly. This last column, for example, is counting all of the positions whose last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. -Does that make sense? +hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit, so these 8 bits already have an even pari Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. -And so you know where it goes from here. +but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000. Once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error. -Let's say some bit in this block gets toggled from a 0 to a 1. +Let's say you detect an error among the odd columns, and among the right half. It necessarily means the error is somewhere in the last column. If there was no error in the odd column but there was one in the right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0. You see, if you add a bit string together twice, it's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. @@ -71,39 +71,39 @@ Remember, what we want is to collect together all of those positions, the positi To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it. This basically eats its way through the list, taking XORs along the way. -If you prefer, you can explicitly write out that XOR function without having to import it from anywhere. +ays let you pin down a specific location, no matter where they turn out to be. In fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it looks like if we do this on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. -This would be considered a well-prepared block. +ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. -Isn't that neat? +correction bits are just riding along. But protecting those bits as well is something that natural You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any. -And there's nothing special about the size 16 here. -The same line of code would work if you had a list of 256 bits. -Needless to say, there is more code to write here, like doing the meta parity check to detect 2-bit errors, but the idea is that almost all of the core logic from our scheme comes down to a single XOR reduction. +hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. +The first thing, except for those eight highlighted parity bits, can be whatever you want it to be, carrying whatever message or data you want. The 8 bits +are redundant in the sense that they're completely determined by the rest of the message, but it's in a much smarter way than simply copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almost. Okay, so the one problem here is that if none of the four parity checks detect an e Now depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. -Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level. +tended, then it either means there was no error at all, or it narrows us down into position 0. You see, with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing 1 out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The first one is easiest to actually do by hand, and I think it does a better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the log of the size of the block, or in other words, it grows one bit at a time as the block size doubles. The relevant fact here is that that information directly corresponds to how much redundancy we need. -That's really what runs against most people's knee-jerk reaction when they first think about making a message resilient to errors, where usually copying the whole message is the first instinct that comes to mind. +at 0th one so that the parity of the full block is even, just like a normal parity check. Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks. However, if there's two errors, then the overall parity is going to toggle back to be And then, by the way, there is this whole other way that you sometimes see Hamming codes presented where you multiply the message by one big matrix. It's kind of nice because it relates it to the broader family of linear codes, but I think that gives almost no intuition for where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there. As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits. -And if you step back to think about looking at a million bits and locating a single error, that genuinely feels crazy. +ugh so you can check yourself. To set up a message, whether that's a literal message you're translating over space or some data you want to store over time, the first step is to divide it up into 11-bit chunks. The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that. So in practice, what you'd want is to find the right size so that the probability of too many bit flips isn't too high. Also, in practice, errors tend to come in little bursts, which would totally ruin a single block. -So one common tactic to help spread out a burst of errors across many different blocks is to interlace those blocks, like this, before they're sent out or stored. +ow has an even parity, meaning you can set that bit number 0, the overarching parity bit, to be 0. So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0. A Then again, a lot of this is rendered completely moot by more modern codes, like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly well, and it can be tuned to be resilient to a larger number of errors per block. -But that is a topic for another time. +see that it's even, so any error that exists would have to be in an even column. In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was. -He first tried all sorts of different schemes involving organizing the bits into parts of a higher dimensional lattice and strange things like this. +is odd, giving us confidence that there was one flip and not two. If it's three or more, all bets are off. After correcting that bit number 10, pulling out the 11 bits that were not used for correction gives us the relevant segment of the original message, which if you rewind and compare is indeed exactly what we started the example with. The idea that it might be possible to get parity checks to conspire in a way that spells out the position of an error only came to Hamming when he stepped back after a bunch of other analysis and asked, okay, what is the most efficient I could conceivably be about this? He was also candid about how important it was that parity checks were already on his mind, which would have been way less common back in the 1940s than it is today. -There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind. +a machine to point to the position of an error, how to systematically scale it, and how we can frame all of this as one single operation rather than multiple separate parity checks. Clever ideas often look deceptively simple in hindsight, which makes them easy to underappreciate. Right now my honest hope is that Hamming codes, or at least the possibility of such codes, feels almost obvious to you. But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't. diff --git a/2020/hamming-codes-2/french/sentence_translations.json b/2020/hamming-codes-2/french/sentence_translations.json index 20c719ed0..6593b4017 100644 --- a/2020/hamming-codes-2/french/sentence_translations.json +++ b/2020/hamming-codes-2/french/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "Nous parlions des codes de Hamming, une façon de créer un bloc de données dont la plupart des bits portent un message significatif, tandis que quelques autres agissent comme une sorte de redondance, de telle sorte que si un bit est inversé, soit un message bit ou un bit de redondance, n'importe quoi dans ce bloc, un récepteur sera capable d'identifier qu'il y a eu une erreur et comment la corriger.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "L'idée de base présentée ici était de savoir comment utiliser plusieurs contrôles de parité pour effectuer une recherche binaire jusqu'à l'erreur.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "Par exemple, le nombre 7 en binaire ressemble à 0111, ce qui signifie essentiellement que c'est 4 plus 2 plus 1.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "Et remarquez où se situe la position 7, elle affecte le premier de nos groupes paritaires, ainsi que le deuxième et le troisième, mais pas le dernier.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "Ainsi, la lecture des résultats de ces quatre contrôles de bas en haut précise bien la position de l’erreur.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "Alors que nous remettons ces étiquettes binaires dans leurs boîtes, permettez-moi de souligner qu'elles sont distinctes des données réellement envoyées.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "Ce n'est rien de plus qu'une étiquette conceptuelle pour nous aider, vous et moi, à comprendre d'où viennent les quatre groupes paritaires.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "L'élégance d'avoir tout ce que nous regardons soit décrit en binaire est peut-être sapée par la confusion d'avoir tout ce que nous regardons étant décrit en binaire.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "Ce que nous obtenons est le premier de nos quatre groupes de parité, ce qui signifie que vous pouvez interpréter cette première vérification comme demandant : hé, s’il y a une erreur, le dernier bit à la position de cette erreur est-il un 1?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "Et ainsi de suite.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "La première est de savoir comment généraliser systématiquement à des tailles de blocs qui sont des puissances de deux plus grandes.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "Cela signifie que chacun de ces bits de parité se trouve dans un et un seul des quatre groupes de parité.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "Lorsque vous effectuez le XOR de deux bits, il renvoie un 1 si l'un de ces bits est activé, mais pas si les deux sont activés ou désactivés.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "Exprimé différemment, c'est la parité de ces deux bits.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "C'est comme une addition, mais où l'on ne porte jamais.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "Le point clé pour vous et moi est que prendre le XOR de nombreuses chaînes de bits différentes est effectivement un moyen de calculer les parodies d'un groupe de groupes séparés, comme c'est le cas avec les colonnes, d'un seul coup.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "Cela nous donne une façon plutôt élégante de considérer les multiples contrôles de parité de notre algorithme de code de Hamming comme étant tous regroupés en une seule opération.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "Cela a-t-il du sens?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "De même, la colonne suivante compte le nombre de positions dans le deuxième groupe de parité, les positions dont l'avant-dernier bit est un 1 et qui sont également mises en surbrillance, et ainsi de suite.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "Maintenant, une fois que nous avons cela, cela nous donne une très bonne façon de réfléchir à la raison pour laquelle ces quatre bits résultants en bas indiquent directement la position d'une erreur.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "Disons qu'un élément de ce bloc passe de 0 à 1.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "Cela signifie que la position de ce bit va maintenant être incluse dans le XOR total, ce qui fait passer la somme de 0 à cette valeur nouvellement incluse, la position de l'erreur.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "Nous allons commencer par créer un tableau aléatoire de 16 1 et 0 pour simuler le bloc de données, et je lui donnerai le nom des bits, mais bien sûr, en pratique, ce serait quelque chose que nous recevons d'un expéditeur, et au lieu de étant aléatoire, il transporterait 11 bits de données ainsi que 5 bits de parité.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "Donc, si nous créons ensuite une liste qui boucle sur toutes ces paires, des paires qui ressemblent à i, et que nous extrayons ensuite uniquement la valeur i, juste l'index, eh bien, ce n'est pas si excitant, nous récupérons simplement ces indices de 0 à 15.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "Mais si nous ajoutons la condition de ne faire cela que si bit, c'est-à-dire si ce bit est un 1 et non un 0, alors il extrait uniquement les positions où le bit correspondant est activé.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "Nous ne le ferons pas ici, mais vous pouvez écrire une fonction dans laquelle l'expéditeur utilise cette représentation binaire pour définir les quatre bits de parité selon les besoins, amenant finalement ce bloc à un état où l'exécution de cette ligne de code sur la liste complète des bits renvoie un 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "Ce qui est cool, c'est que si nous basculons l'un des bits de cette liste, simulant une erreur aléatoire due au bruit, alors si vous exécutez cette même ligne de code, cette erreur est affichée.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "Vous pouvez obtenir ce bloc à l'improviste, exécuter cette seule ligne dessus, et il crachera automatiquement la position d'une erreur, ou un 0 s'il n'y en avait pas.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "Maintenant, selon votre aisance avec les binaires, les XOR et les logiciels en général, vous pouvez soit trouver cette perspective un peu déroutante, soit tellement plus élégante et simple que vous vous demandez pourquoi nous ne l'avons pas commencé dès le début. -aller.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "En gros, la perspective du contrôle de parité multiple est plus facile à penser lors de l'implémentation très directe des codes de Hamming dans le matériel, et la perspective XOR est la plus facile à penser lorsqu'on l'effectue dans le logiciel, à partir d'un niveau supérieur.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "Le fait pertinent ici est que ces informations correspondent directement au niveau de redondance dont nous avons besoin.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "Et puis, en passant, il y a cette toute autre façon de voir parfois les codes de Hamming présentés, où vous multipliez le message par une grande matrice.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "Et en parlant de mise à l'échelle, vous remarquerez peut-être que l'efficacité de ce système ne fait que s'améliorer à mesure que nous augmentons la taille des blocs.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "Par exemple, nous avons vu qu'avec 256 bits, vous n'utilisez que 3 % de cet espace pour la redondance, et la situation ne cesse de s'améliorer à partir de là.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "À mesure que le nombre de bits de parité augmente un par un, la taille du bloc continue de doubler.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "Et si vous poussez cela à l'extrême, vous pourriez avoir un bloc avec, disons, un million de bits, dans lequel vous joueriez littéralement 20 questions avec vos contrôles de parité, et il n'utiliserait que 21 bits de parité.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "Le problème, bien sûr, est qu’avec un bloc plus grand, la probabilité de voir plus d’un ou deux bits d’erreur augmente, et les codes de Hamming ne gèrent rien d’autre au-delà.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "Mais c'est un sujet pour une autre fois.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "Dans son livre The Art of Doing Science and Engineering, Hamming est merveilleusement franc sur les méandres de sa découverte de ce code.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "Il y a environ une demi-douzaine de fois tout au long de ce livre où il fait référence à la citation de Louis Pasteur, la chance favorise un esprit préparé.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "Mais vous ne devriez pas vous tromper en pensant qu’ils sont en réalité évidents, car ils ne le sont certainement pas.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "Il s’agissait du même article fondateur qui montrait, dans un certain sens, qu’une correction d’erreur efficace est toujours possible, quelle que soit la probabilité de retournements de bits, du moins en théorie.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "Avance rapide de plusieurs décennies, et de nos jours, beaucoup d’entre nous sont tellement plongés dans la réflexion sur des éléments et des informations qu’il est facile d’oublier à quel point cette façon de penser était distincte.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/german/sentence_translations.json b/2020/hamming-codes-2/german/sentence_translations.json index 6526b5211..a38c3d561 100644 --- a/2020/hamming-codes-2/german/sentence_translations.json +++ b/2020/hamming-codes-2/german/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "Wir sprachen über Hamming-Codes, eine Möglichkeit, einen Datenblock zu erstellen, bei dem die meisten Bits eine sinnvolle Nachricht enthalten, während einige andere als eine Art Redundanz fungieren, so dass, wenn ein Bit umgedreht wird, entweder eine Nachricht entsteht Ob ein Bit oder ein Redundanzbit, irgendetwas in diesem Block, ein Empfänger kann erkennen, dass ein Fehler aufgetreten ist und wie er ihn beheben kann.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "Die dort vorgestellte Grundidee bestand darin, mithilfe mehrerer Paritätsprüfungen binär bis zum Fehler zu suchen.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "Beispielsweise sieht die Zahl 7 im Binärformat wie 0111 aus, was im Wesentlichen bedeutet, dass sie 4 plus 2 plus 1 ist.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "Und beachten Sie, wo sich die Position 7 befindet. Sie betrifft zwar die erste unserer Paritätsgruppen und die zweite und die dritte, aber nicht die letzte.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "Wenn man also die Ergebnisse dieser vier Prüfungen von unten nach oben liest, lässt sich tatsächlich die Position des Fehlers erkennen.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "Während wir diese binären Etiketten wieder in ihre Boxen stecken, möchte ich betonen, dass sie sich von den Daten unterscheiden, die tatsächlich gesendet werden.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "Sie sind nichts weiter als eine konzeptionelle Bezeichnung, die Ihnen und mir helfen soll, zu verstehen, woher die vier Paritätsgruppen stammen.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "Die Eleganz, alles, was wir betrachten, binär zu beschreiben, wird möglicherweise durch die Verwirrung untergraben, die entsteht, wenn alles, was wir betrachten, binär beschrieben wird.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "Was wir bekommen, ist die erste unserer vier Paritätsgruppen, was bedeutet, dass Sie diese erste Prüfung so interpretieren können, dass Sie fragen: „Hey, wenn es einen Fehler gibt, ist das letzte Bit an der Position dieses Fehlers eine 1?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "“ Und so weiter.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "Die erste besteht darin, wie man systematisch auf Blockgrößen verallgemeinert, die größere Zweierpotenzen sind.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "Das bedeutet, dass jedes dieser Paritätsbits innerhalb einer und nur einer der vier Paritätsgruppen liegt.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "Wenn Sie die XOR-Verknüpfung zweier Bits verwenden, wird eine 1 zurückgegeben, wenn eines dieser Bits aktiviert ist, nicht jedoch, wenn beide aktiviert oder deaktiviert sind.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "Anders ausgedrückt ist es die Parität dieser beiden Bits.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "Es ist wie eine Ergänzung, die man aber nie trägt.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "Der entscheidende Punkt für Sie und mich ist, dass die XOR-Verknüpfung vieler verschiedener Bitfolgen effektiv eine Möglichkeit ist, die Parodien einer Reihe separater Gruppen, wie etwa bei den Spalten, auf einen Schlag zu berechnen.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "Dies gibt uns eine ziemlich schicke Möglichkeit, uns vorzustellen, dass die mehreren Paritätsprüfungen unseres Hamming-Code-Algorithmus alle in einer einzigen Operation zusammengefasst sind.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "Ist das sinnvoll?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "Ebenso zählt die nächste Spalte, wie viele Positionen sich in der zweiten Paritätsgruppe befinden, die Positionen, deren vorletztes Bit eine 1 ist und die ebenfalls hervorgehoben sind, und so weiter.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "Wenn wir es nun so haben, können wir wirklich gut darüber nachdenken, warum diese vier resultierenden Bits unten direkt die Position eines Fehlers angeben.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "Nehmen wir an, ein Bit in diesem Block wird von 0 auf 1 umgeschaltet.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "Das bedeutet, dass die Position dieses Bits nun in die gesamte XOR-Verknüpfung einbezogen wird, wodurch sich die Summe von 0 in diesen neu einbezogenen Wert, die Position des Fehlers, ändert.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "Wir beginnen damit, ein zufälliges Array aus 16 Einsen und Nullen zu erstellen, um den Datenblock zu simulieren, und ich gebe ihm den Namen Bits, aber in der Praxis würden wir das natürlich von einem Absender erhalten, und statt dessen Da es zufällig ist, würde es 11 Datenbits zusammen mit 5 Paritätsbits übertragen.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "Wenn wir dann also eine Liste erstellen, die alle diese Paare durchläuft, Paare, die wie i aussehen, und dann nur den i-Wert, nur den Index, herausziehen, ist das nicht so aufregend, wir bekommen einfach die Indizes 0 bis 15 zurück.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "Aber wenn wir die Bedingung hinzufügen, dies nur zu tun, wenn das Bit eine 1 und keine 0 ist, dann werden nur die Positionen herausgezogen, an denen das entsprechende Bit aktiviert ist.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "Wir werden es hier nicht tun, aber Sie könnten eine Funktion schreiben, bei der der Absender diese binäre Darstellung verwendet, um die vier Paritätsbits nach Bedarf zu setzen und diesen Block letztendlich in einen Zustand zu bringen, in dem die Ausführung dieser Codezeile auf der vollständigen Liste der Bits zurückkehrt eine 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "Das Coole ist, dass, wenn wir eines der Bits in dieser Liste umschalten und so einen zufälligen Fehler durch Rauschen simulieren, dieser Fehler ausgegeben wird, wenn Sie dieselbe Codezeile ausführen.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "Sie könnten diesen Block aus heiterem Himmel bekommen, diese einzelne Zeile darauf ausführen und er würde automatisch die Position eines Fehlers ausspucken, oder eine 0, wenn es keinen gab.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "Nun, je nachdem, wie gut Sie mit Binär- und XOR-Funktionen und Software im Allgemeinen vertraut sind, finden Sie diese Perspektive möglicherweise etwas verwirrend oder so viel eleganter und einfacher, dass Sie sich fragen, warum wir nicht gleich von Anfang an damit begonnen haben -gehen.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "Vereinfacht gesagt lässt sich die Perspektive der Mehrfachparitätsprüfung leichter in Betracht ziehen, wenn man Hamming-Codes direkt in Hardware implementiert, und die XOR-Perspektive lässt sich am einfachsten in Betracht ziehen, wenn man sie in Software von einer höheren Ebene aus durchführt.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "Die relevante Tatsache hierbei ist, dass diese Informationen direkt mit der benötigten Redundanz korrespondieren.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "Und dann gibt es übrigens noch eine ganz andere Art und Weise, wie man manchmal Hamming-Codes präsentiert, bei denen man die Nachricht mit einer großen Matrix multipliziert.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "Apropos Skalierung: Sie werden vielleicht feststellen, dass die Effizienz dieses Schemas nur dann besser wird, wenn wir die Blockgröße erhöhen.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "Wir haben beispielsweise gesehen, dass Sie bei 256 Bit nur 3 % dieses Speicherplatzes für Redundanz nutzen, und von da an wird es immer besser.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "Wenn die Anzahl der Paritätsbits nach und nach zunimmt, verdoppelt sich die Blockgröße immer weiter.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "Und wenn Sie das auf die Spitze treiben, könnten Sie einen Block mit, sagen wir, einer Million Bits haben, in dem Sie mit Ihren Paritätsprüfungen im wahrsten Sinne des Wortes 20 Fragen abspielen würden, und der nur 21 Paritätsbits verwendet.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "Das Problem besteht natürlich darin, dass mit einem größeren Block die Wahrscheinlichkeit steigt, mehr als ein oder zwei Bitfehler zu sehen, und Hamming-Codes verarbeiten nichts darüber hinaus.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "Aber das ist ein Thema für ein anderes Mal.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "In seinem Buch „The Art of Doing Science and Engineering“ spricht Hamming wunderbar offen darüber, wie kompliziert seine Entdeckung dieses Codes war.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "In diesem Buch verweist er ungefähr ein halbes Dutzend Mal auf das Zitat von Louis Pasteur: „Glück begünstigt einen vorbereiteten Geist.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "Aber Sie sollten sich nicht der Illusion hingeben, dass sie tatsächlich offensichtlich sind, denn das ist definitiv nicht der Fall.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "Dabei handelte es sich um dieselbe Grundlagenarbeit, die in gewisser Weise zeigte, dass eine effiziente Fehlerkorrektur immer möglich ist, unabhängig davon, wie hoch die Wahrscheinlichkeit von Bit-Flips ist, zumindest theoretisch.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "Mehrere Jahrzehnte später sind viele von uns so sehr in das Nachdenken über Bits und Informationen vertieft, dass man leicht übersieht, wie unterschiedlich diese Denkweise war.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/hebrew/sentence_translations.json b/2020/hamming-codes-2/hebrew/sentence_translations.json index cddda9d5a..94099b980 100644 --- a/2020/hamming-codes-2/hebrew/sentence_translations.json +++ b/2020/hamming-codes-2/hebrew/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "דיברנו על קודי Hamming, דרך ליצור גוש נתונים שבו רוב הביטים נושאים מסר משמעותי, בעוד שכמה אחרים פועלים כסוג של יתירות, באופן כזה שאם ביט כלשהו יתהפך, או הודעה סיביות או סיביות יתירות, כל דבר בבלוק הזה, מקלט יוכל לזהות שהייתה שגיאה ואיך לתקן אותה.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "הרעיון הבסיסי שהוצג שם היה כיצד להשתמש בבדיקות זוגיות מרובות כדי לחפש בינארי בדרך למטה אל השגיאה.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "לדוגמה, המספר 7 בבינארי נראה כמו 0111, בעצם אומר שהוא 4 ועוד 2 ועוד 1.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "ושימו לב היכן יושבת עמדה 7, היא אכן משפיעה על הראשונה בקבוצות השוויוניות שלנו, והשנייה, והשלישית, אך לא האחרונה.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "אז קריאת התוצאות של ארבעת הבדיקות הללו מלמטה למעלה אכן מפרטת את מיקומו של השגיאה.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "כשאנחנו מחזירים את התוויות הבינאריות האלה לקופסאות שלהן, הרשו לי להדגיש שהן שונות מהנתונים שנשלחים בפועל.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "הם לא יותר מאשר תווית מושגית כדי לעזור לך ולי להבין מאיפה הגיעו ארבע קבוצות השוויון.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "האלגנטיות של זה שכל מה שאנחנו מסתכלים עליו יתואר בבינארי הוא אולי תחת הבלבול של זה שכל מה שאנחנו מסתכלים עליו מתואר בבינארי.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "מה שאנחנו מקבלים היא הראשונה מבין ארבע קבוצות השוויון שלנו, מה שאומר שאתה יכול לפרש את הסימון הראשון הזה כשואל, היי, אם יש שגיאה, האם הביט האחרון במיקום השגיאה הוא 1?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "וכולי.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "הראשון הוא איך להכליל באופן שיטתי לגדלי בלוקים שהם עצמות גדולות יותר של שניים.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "המשמעות היא שכל אחד מאותם סיביות זוגיות יושב בתוך אחת ויחידה מארבע קבוצות הזוגיות.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "כאשר אתה לוקח את ה-XOR של שני סיביות, זה יחזיר 1 אם אחד מהסיביות האלה מופעל, אבל לא אם שניהם מופעלים או כבויים.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "בניסוח שונה, זה השוויון של שני הביטים האלה.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "זה כמו תוספת, אבל איפה שאתה אף פעם לא נושא.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "נקודת המפתח עבורך ולי היא שלקיחת ה-XOR של מחרוזות סיביות שונות היא למעשה דרך לחשב את הפרודיות של חבורה של קבוצות נפרדות, כמו כך עם העמודות, הכל במכה אחת.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "זה נותן לנו דרך די מטופשת לחשוב על בדיקות השוויון המרובות מאלגוריתם קוד Hamming שלנו, כשהם ארוזים יחד לפעולה אחת.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "האם זה הגיוני?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "כמו כן, העמודה הבאה סופרת כמה עמדות יש בקבוצת השוויון השנייה, המיקומים שהביט השני אחרון שלהם הוא 1, ואשר גם הם מודגשים, וכן הלאה.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "עכשיו ברגע שיש לנו את זה ככה, זה נותן לנו דרך ממש נחמדה לחשוב מדוע ארבעת הביטים המתקבלים האלה בתחתית מאייתים ישירות את המיקום של שגיאה.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "נניח שחלק מהגוש הזה עובר מ-0 ל-1.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "מה שזה אומר הוא שהמיקום של הביט הזה ייכלל כעת ב-XOR הכולל, מה שמשנה את הסכום מ-0 לערך זה החדש שנכלל, מיקום השגיאה.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "נתחיל ביצירת מערך אקראי של 16 1 ו-0 כדי לדמות את בלוק הנתונים, ואני אתן לו את סיביות השם, אבל כמובן שבפועל זה יהיה משהו שאנחנו מקבלים מהשולח, ובמקום בהיותו אקראי הוא יוביל 11 סיביות נתונים יחד עם 5 סיביות זוגיות.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "אז אם אנחנו יוצרים רשימה שמסתובבת בלולאה על כל הזוגות האלה, זוגות שנראים כמו i, ואז נוציא רק את ערך i, רק את המדד, ובכן, זה לא כל כך מרגש, אנחנו פשוט מקבלים בחזרה את המדדים האלה 0 עד 15.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "אבל אם נוסיף את התנאי לעשות את זה רק אם ביט, כלומר אם הביט הזה הוא 1 ולא 0, ובכן, אז הוא שולף רק את המיקומים שבהם הביט המקביל מופעל.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "לא נעשה את זה כאן, אבל אתה יכול לכתוב פונקציה שבה השולח משתמש בייצוג הבינארי הזה כדי להגדיר את ארבעת סיביות הזוגיות לפי הצורך, ובסופו של דבר להביא את הבלוק הזה למצב שבו הפעלת שורת קוד זו ברשימת הביטים המלאה מחזירה א 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "מה שמגניב הוא שאם נחליף כל אחד מהסיביות ברשימה הזו, המדמה שגיאה אקראית מרעש, אז אם אתה מפעיל את אותה שורת קוד, הוא מדפיס את השגיאה הזו.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "אתה יכול לקבל את הבלוק הזה ישר, להריץ עליו את השורה הבודדת הזו, והוא ינוק אוטומטית את המיקום של שגיאה, או 0 אם לא הייתה כזו.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "כעת, בהתאם לנוחות שלך עם רכיבי בינארי ו-XOR ותוכנה באופן כללי, ייתכן שתמצא את הפרספקטיבה הזו קצת מבלבלת, או הרבה יותר אלגנטי ופשוט שאתה תוהה למה לא התחלנו איתה מההתחלה -ללכת.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "באופן רופף, קל יותר לחשוב על פרספקטיבה של בדיקת זוגיות מרובה בעת יישום קודי Hamming בחומרה באופן ישיר מאוד, ואת פרספקטיבה של XOR קל יותר לחשוב עליה כאשר עושים זאת בתוכנה, מרמה גבוהה יותר.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "העובדה הרלוונטית כאן היא שהמידע הזה מתאים ישירות לכמות היתירות שאנחנו צריכים.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "ואז, אגב, יש את כל הדרך האחרת הזו שלפעמים רואים את קודי האמינג מוצגים, שבה אתה מכפיל את המסר במטריצה אחת גדולה.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "ואם כבר מדברים על קנה מידה, אולי תשים לב שהיעילות של תכנית זו רק משתפרת ככל שאנו מגדילים את גודל הבלוק.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "לדוגמה, ראינו שעם 256 סיביות, אתה משתמש רק ב-3% מהשטח הזה עבור יתירות, ומשם זה רק הולך ומשתפר.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "ככל שמספר סיביות הזוגיות גדל בזה אחר זה, גודל הבלוק ממשיך להכפיל את עצמו.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "ואם אתה לוקח את זה לקיצוניות, אתה יכול לקבל בלוק עם, נגיד, מיליון ביטים, שבו אתה ממש משחק 20 שאלות עם בדיקות השוויון שלך, והוא משתמש רק ב-21 ביטים זוגיות.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "הבעיה, כמובן, היא שעם בלוק גדול יותר, ההסתברות לראות יותר משגיאת סיביות אחת או שתיים עולה, וקודי Hamming לא מטפלים בשום דבר מעבר לזה.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "אבל זה נושא לפעם אחרת.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "בספרו The Art of Doing Science and Engineering, האמינג הוא כנה להפליא לגבי מידת הפיתול של גילוי הקוד הזה.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "יש כמו חצי תריסר פעמים במהלך הספר הזה שהוא מתייחס לציטוט של לואי פסטר, המזל מעדיף מוח מוכן.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "אבל אתה לא צריך להטעות את עצמך לחשוב שהם בעצם ברורים, כי הם בהחלט לא.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "זה היה אותו מאמר יסוד שהראה, במובן מסוים, שתיקון שגיאות יעיל תמיד אפשרי, לא משנה כמה גבוהה ההסתברות להיפוך סיביות, לפחות בתיאוריה.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "הרץ קדימה כמה עשורים, ובימים אלה, רבים מאיתנו שקועים כל כך בחשיבה על פיסות ומידע שקל להתעלם עד כמה הייתה צורת החשיבה הזו מובחנת.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/hindi/sentence_translations.json b/2020/hamming-codes-2/hindi/sentence_translations.json index 38dffc2f0..f11faa861 100644 --- a/2020/hamming-codes-2/hindi/sentence_translations.json +++ b/2020/hamming-codes-2/hindi/sentence_translations.json @@ -7,14 +7,14 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "हम हैमिंग कोड के बारे में बात कर रहे थे, डेटा का एक ब्लॉक बनाने का एक तरीका जहां अधिकांश बिट्स एक सार्थक संदेश ले जाते हैं, जबकि कुछ अन्य एक प्रकार की अतिरेक के रूप में कार्य करते हैं, इस तरह से कि यदि कोई बिट फ़्लिप हो जाता है, तो या तो एक संदेश बिट या अतिरेक बिट, इस ब्लॉक में कुछ भी, रिसीवर यह पहचानने में सक्षम होगा कि कोई त्रुटि थी, और इसे कैसे ठीक किया जाए।", "n_reviews": 0, "start": 3.06, "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "वहां प्रस्तुत मूल विचार यह था कि त्रुटि तक पहुंचने के लिए बाइनरी खोज के लिए एकाधिक समता जांच का उपयोग कैसे किया जाए।", "n_reviews": 0, "start": 21.88, @@ -49,21 +49,21 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "उदाहरण के लिए, बाइनरी में संख्या 7 0111 की तरह दिखती है, अनिवार्य रूप से यह कहती है कि यह 4 प्लस 2 प्लस 1 है।", "n_reviews": 0, "start": 64.78, "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "और ध्यान दें कि स्थिति 7 कहाँ बैठती है, यह हमारे समता समूहों में से पहले, और दूसरे, और तीसरे को प्रभावित करती है, लेकिन अंतिम को नहीं।", "n_reviews": 0, "start": 72.54, "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "इसलिए नीचे से ऊपर तक उन चार जांचों के परिणामों को पढ़ने से वास्तव में त्रुटि की स्थिति का पता चल जाता है।", "n_reviews": 0, "start": 82.22, @@ -84,21 +84,21 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "जैसे ही हम इन बाइनरी लेबलों को उनके बक्सों में वापस डालते हैं, मैं इस बात पर जोर देना चाहता हूं कि वे वास्तव में भेजे जा रहे डेटा से अलग हैं।", "n_reviews": 0, "start": 110.56, "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "वे आपको और मुझे यह समझने में मदद करने के लिए एक वैचारिक लेबल से ज्यादा कुछ नहीं हैं कि चार समता समूह कहां से आए।", "n_reviews": 0, "start": 118.32, "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "हम जो कुछ भी देख रहे हैं उसे बाइनरी में वर्णित करने की सुंदरता शायद इस भ्रम के कारण कम हो गई है कि हम जो कुछ भी देख रहे हैं उसका वर्णन बाइनरी में किया जा रहा है।", "n_reviews": 0, "start": 124.14, @@ -119,7 +119,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "हमें जो मिलता है वह हमारे चार समता समूहों में से पहला है, जिसका अर्थ है कि आप उस पहले चेक की व्याख्या यह पूछ सकते हैं कि, अरे, यदि कोई त्रुटि है, तो क्या उस त्रुटि की स्थिति में अंतिम बिट 1 है?", "n_reviews": 0, "start": 144.24, @@ -140,7 +140,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "और इसी तरह।", "n_reviews": 0, "start": 175.76, @@ -168,7 +168,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "पहला यह है कि उन आकारों को ब्लॉक करने के लिए व्यवस्थित रूप से सामान्यीकरण कैसे किया जाए जो दो की बड़ी घात हैं।", "n_reviews": 0, "start": 202.04, @@ -210,7 +210,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "इसका मतलब यह है कि उनमें से प्रत्येक समता बिट चार समता समूहों में से एक और केवल एक के अंदर बैठता है।", "n_reviews": 0, "start": 243.6, @@ -245,14 +245,14 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "जब आप दो बिट्स का एक्सओआर लेते हैं, तो यदि इनमें से कोई एक बिट चालू है तो यह 1 लौटाएगा, लेकिन यदि दोनों चालू या बंद हैं तो नहीं।", "n_reviews": 0, "start": 290.78, "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "अलग-अलग शब्दों में, यह इन दो बिट्स की समता है।", "n_reviews": 0, "start": 300.1, @@ -273,7 +273,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "यह जोड़ की तरह है, लेकिन जहां आप कभी नहीं ले जाते।", "n_reviews": 0, "start": 313.68, @@ -294,14 +294,14 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "आपके और मेरे लिए मुख्य बिंदु यह है कि कई अलग-अलग बिट स्ट्रिंग्स का एक्सओआर लेना प्रभावी रूप से अलग-अलग समूहों के समूह की पैरोडी की गणना करने का एक तरीका है, जैसे कि कॉलम के साथ, सभी एक ही झटके में।", "n_reviews": 0, "start": 334.96, "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "यह हमें हमारे हैमिंग कोड एल्गोरिदम से कई समता जांचों के बारे में सोचने का एक आसान तरीका देता है क्योंकि सभी को एक ही ऑपरेशन में एक साथ पैक किया जाता है।", "n_reviews": 0, "start": 351.26, @@ -336,14 +336,14 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "समझ आया?", "n_reviews": 0, "start": 406.24, "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "इसी तरह, अगला कॉलम गिनता है कि दूसरे समता समूह में कितनी स्थितियाँ हैं, वे स्थितियाँ जिनका दूसरा से अंतिम बिट 1 है, और जिन्हें हाइलाइट भी किया गया है, इत्यादि।", "n_reviews": 0, "start": 409.08, @@ -371,21 +371,21 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "अब एक बार जब हमारे पास यह इस तरह हो जाता है, तो यह हमें यह सोचने का एक बहुत अच्छा तरीका देता है कि नीचे ये चार परिणामी बिट्स सीधे त्रुटि की स्थिति क्यों बताते हैं।", "n_reviews": 0, "start": 439.04, "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "मान लीजिए कि इस ब्लॉक में कुछ बिट 0 से 1 पर टॉगल हो जाता है।", "n_reviews": 0, "start": 448.46, "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "इसका मतलब यह है कि उस बिट की स्थिति अब कुल XOR में शामिल होने जा रही है, जो योग को 0 से बदलकर इस नए शामिल मूल्य, त्रुटि की स्थिति में बदल देती है।", "n_reviews": 0, "start": 452.6, @@ -427,7 +427,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "हम डेटा ब्लॉक को अनुकरण करने के लिए 16 1s और 0s की एक यादृच्छिक सरणी बनाकर शुरू करेंगे, और मैं इसे बिट्स नाम दूंगा, लेकिन व्यवहार में यह कुछ ऐसा होगा जो हम एक प्रेषक से प्राप्त कर रहे हैं, और इसके बजाय यादृच्छिक होने के कारण इसमें 5 समता बिट्स के साथ 11 डेटा बिट्स होंगे।", "n_reviews": 0, "start": 502.08, @@ -441,14 +441,14 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "इसलिए यदि हम एक सूची बनाते हैं जो इन सभी जोड़ियों पर लूप करती है, जो जोड़े जो i की तरह दिखते हैं, और फिर हम केवल i मान, केवल सूचकांक निकालते हैं, खैर यह उतना रोमांचक नहीं है, हम बस उन सूचकांकों को 0 से 15 तक वापस लाते हैं .", "n_reviews": 0, "start": 528.18, "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "लेकिन अगर हम केवल बिट होने पर ऐसा करने की शर्त जोड़ते हैं, जिसका अर्थ है कि यदि वह बिट 1 है और 0 नहीं है, तो यह केवल उन स्थितियों को बाहर निकालता है जहां संबंधित बिट चालू है।", "n_reviews": 0, "start": 541.68, @@ -504,7 +504,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "हम इसे यहां नहीं करेंगे, लेकिन आप एक फ़ंक्शन लिख सकते हैं जहां प्रेषक आवश्यकतानुसार चार समता बिट्स सेट करने के लिए उस बाइनरी प्रतिनिधित्व का उपयोग करता है, अंततः इस ब्लॉक को उस स्थिति में ले जाता है जहां बिट्स की पूरी सूची पर कोड की इस पंक्ति को चलाने पर रिटर्न मिलता है एक 0.", "n_reviews": 0, "start": 601.98, @@ -518,7 +518,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "अच्छी बात यह है कि यदि हम शोर से यादृच्छिक त्रुटि का अनुकरण करते हुए इस सूची में से किसी एक बिट को टॉगल करते हैं, तो यदि आप कोड की इसी पंक्ति को चलाते हैं, तो यह उस त्रुटि को प्रिंट करता है।", "n_reviews": 0, "start": 619.88, @@ -532,7 +532,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "आप इस ब्लॉक को अचानक से प्राप्त कर सकते हैं, इस पर इस एकल पंक्ति को चला सकते हैं, और यह स्वचालित रूप से किसी त्रुटि की स्थिति या यदि कोई त्रुटि नहीं है तो 0 बता देगा।", "n_reviews": 0, "start": 631.82, @@ -560,14 +560,14 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "अब, सामान्य रूप से बाइनरी और एक्सओआर और सॉफ़्टवेयर के साथ आपकी सुविधा के आधार पर, आपको या तो यह परिप्रेक्ष्य थोड़ा भ्रमित करने वाला लग सकता है, या इतना अधिक सुरुचिपूर्ण और सरल कि आप सोच रहे होंगे कि हमने शुरुआत से ही इसकी शुरुआत क्यों नहीं की -जाना।", "n_reviews": 0, "start": 666.12, "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "संक्षेप में कहें तो, हार्डवेयर में हैमिंग कोड को बहुत सीधे लागू करते समय मल्टीपल पैरिटी चेक परिप्रेक्ष्य के बारे में सोचना आसान होता है, और सॉफ़्टवेयर में इसे उच्च स्तर से करते समय XOR परिप्रेक्ष्य के बारे में सोचना सबसे आसान होता है।", "n_reviews": 0, "start": 679.14, @@ -581,7 +581,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "यहां प्रासंगिक तथ्य यह है कि वह जानकारी सीधे तौर पर इस बात से मेल खाती है कि हमें कितनी अतिरेक की आवश्यकता है।", "n_reviews": 0, "start": 711.02, @@ -595,7 +595,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "और फिर, वैसे, यह एक और तरीका है जिसमें आप कभी-कभी हैमिंग कोड प्रस्तुत करते हुए देखते हैं, जहां आप संदेश को एक बड़े मैट्रिक्स से गुणा करते हैं।", "n_reviews": 0, "start": 727.5, @@ -609,28 +609,28 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "और स्केलिंग की बात करते हुए, आप देख सकते हैं कि जैसे-जैसे हम ब्लॉक आकार बढ़ाते हैं, इस योजना की दक्षता बेहतर होती जाती है।", "n_reviews": 0, "start": 745.2, "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "उदाहरण के लिए, हमने देखा कि 256 बिट्स के साथ, आप उस स्थान का केवल 3% अतिरेक के लिए उपयोग कर रहे हैं, और यह वहां से बेहतर होता जा रहा है।", "n_reviews": 0, "start": 755.0, "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "जैसे-जैसे समता बिट्स की संख्या एक-एक करके बढ़ती जाती है, ब्लॉक का आकार दोगुना होता जाता है।", "n_reviews": 0, "start": 763.3, "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "और यदि आप इसे चरम सीमा तक ले जाते हैं, तो आपके पास एक मिलियन बिट्स वाला एक ब्लॉक हो सकता है, जहां आप वस्तुतः अपने समता जांच के साथ 20 प्रश्न खेल रहे होंगे, और यह केवल 21 समता बिट्स का उपयोग करता है।", "n_reviews": 0, "start": 769.0, @@ -644,7 +644,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "निस्संदेह, समस्या यह है कि बड़े ब्लॉक के साथ, एक या दो से अधिक बिट त्रुटियाँ देखने की संभावना बढ़ जाती है, और हैमिंग कोड इससे आगे कुछ भी संभाल नहीं पाते हैं।", "n_reviews": 0, "start": 788.2, @@ -672,14 +672,14 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "लेकिन यह अगली बार का विषय है।", "n_reviews": 0, "start": 839.36, "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "अपनी पुस्तक द आर्ट ऑफ डूइंग साइंस एंड इंजीनियरिंग में, हैमिंग ने आश्चर्यजनक रूप से स्पष्ट रूप से बताया है कि इस कोड की उनकी खोज कितनी टेढ़ी-मेढ़ी थी।", "n_reviews": 0, "start": 842.5, @@ -707,7 +707,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "इस पुस्तक में लगभग आधा दर्जन बार उन्होंने लुई पाश्चर के इस कथन का संदर्भ दिया है कि भाग्य तैयार दिमाग का साथ देता है।", "n_reviews": 0, "start": 881.92, @@ -728,7 +728,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "लेकिन आपको यह सोचकर खुद को मूर्ख नहीं बनाना चाहिए कि वे वास्तव में स्पष्ट हैं, क्योंकि वे निश्चित रूप से नहीं हैं।", "n_reviews": 0, "start": 901.66, @@ -770,7 +770,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "यह वही मूलभूत पेपर था जिसने एक निश्चित अर्थ में दिखाया कि कुशल त्रुटि सुधार हमेशा संभव है, चाहे बिट फ़्लिप की संभावना कितनी भी अधिक क्यों न हो, कम से कम सिद्धांत में।", "n_reviews": 0, "start": 943.3, @@ -784,7 +784,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "कई दशक तेजी से आगे बढ़े, और इन दिनों, हममें से कई लोग छोटी-छोटी बातों और सूचनाओं के बारे में सोचने में इतने डूबे हुए हैं कि इस बात को नजरअंदाज करना आसान है कि सोचने का यह तरीका कितना अलग था।", "n_reviews": 0, "start": 962.38, diff --git a/2020/hamming-codes-2/hungarian/sentence_translations.json b/2020/hamming-codes-2/hungarian/sentence_translations.json index 0c9759860..2b738ff8f 100644 --- a/2020/hamming-codes-2/hungarian/sentence_translations.json +++ b/2020/hamming-codes-2/hungarian/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "A Hamming-kódokról beszéltünk, egy olyan adatblokk létrehozásának módjáról, ahol a legtöbb bit egy értelmes üzenetet hordoz, míg néhány másik egyfajta redundanciaként működik, oly módon, hogy ha bármelyik bit átbillen, akár egy üzenetbit, akár egy redundanciabit, bármi ebben a blokkban, a vevő képes lesz azonosítani, hogy hiba történt, és hogyan kell kijavítani azt.", "model": "DeepL", "n_reviews": 1, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "Az ott bemutatott alapötlet az volt, hogy hogyan lehet a többszörös paritásellenőrzést egyfajta bináris keresésként használva eljutni a hibáig.", "model": "DeepL", "n_reviews": 1, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "Például a 7-es szám binárisan 0111-nek néz ki, ami lényegében azt jelenti, hogy 4 plusz 2 plusz 1.", "model": "DeepL", "n_reviews": 1, @@ -72,7 +72,7 @@ "end": 74.46 }, { - "input": "It does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "Hatással van az első csoport paritására, a második és a harmadikéra is, de az utolsóéra nem.", "model": "DeepL", "n_reviews": 1, @@ -80,7 +80,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "Tehát a négy ellenőrzés eredményeinek alulról felfelé történő összeolvasása valóban megadja a hiba helyét.", "model": "DeepL", "n_reviews": 1, @@ -112,7 +112,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "Ahogy ezeket a bináris címkéket visszatesszük a dobozukba, hadd hangsúlyozzam, hogy ezek különböznek a ténylegesen küldött adatoktól.", "model": "DeepL", "n_reviews": 1, @@ -120,7 +120,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "Ezek csak kitalált címkék, amelyek segítenek neked és nekem beazonosítani a négy paritáscsoportot.", "model": "DeepL", "n_reviews": 1, @@ -128,7 +128,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "Annak eleganciáját, hogy ezen értékeket binárisan kezelhetjük, talán aláássa az a zűrzavar, hogy igazából minden más is amit nézünk binárisan van leírva.", "model": "DeepL", "n_reviews": 1, @@ -144,7 +144,7 @@ "end": 134.12 }, { - "input": "Focus your attention just on that last bit of all of these labels.", + "input": ", takes a certain cleverness. The elegance of having everything we're looking at be described in binary is maybe undercu", "translatedText": "Először csak a címkék utolsó bitjére koncentrálj.", "model": "DeepL", "n_reviews": 1, @@ -152,7 +152,7 @@ "end": 138.24 }, { - "input": "And then highlight the positions where that final bit is a 1.", + "input": "t by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all o", "translatedText": "Majd emeld ki azokat a pozíciókat, ahol az utolsó bit 1.", "model": "DeepL", "n_reviews": 1, @@ -160,7 +160,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means that you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "f these labels. And then highlight the positions where that final bit is a 1. What we get is the first of our four parity groups, which means that you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", "translatedText": "A négy paritáscsoportunk közül így az elsőt kapjuk, ami azt jelenti, hogy az első ellenőrzést egy kérdésként értelmezhetjük: Ha van hiba, akkor a hely címkéjének utolsó bitje 1?", "model": "DeepL", "n_reviews": 1, @@ -184,7 +184,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "ch are powers of 2 allows for somethi", "translatedText": "És így tovább.", "model": "DeepL", "n_reviews": 1, @@ -264,7 +264,7 @@ "end": 243 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "Ez azt jelenti, hogy minden egyes paritásbit a négy paritáscsoport közül csak és kizárólag egyhez tartozik.", "model": "DeepL", "n_reviews": 1, @@ -304,7 +304,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or if both are turned off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but", "translatedText": "Ha két bit XOR-ját vesszük, akkor 1-et kapunk, ha valamelyik bit be van kapcsolva, de 0 lesz, ha mindkettő be van kapcsolva, vagy ha mindkettő ki van kapcsolva.", "model": "DeepL", "n_reviews": 1, @@ -312,7 +312,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "solving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "Másképp fogalmazva, ez a két bit paritását adja.", "model": "DeepL", "n_reviews": 1, @@ -336,7 +336,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "Olyan, mint az összeadás, csak itt nem viszünk tovább értékeket.", "model": "DeepL", "n_reviews": 1, @@ -360,7 +360,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "Számunkra ebből az a lényeges, hogy különböző bitsorozatok XOR-ját véve gyakorlatilag egy csomó különálló csoport paritását tudjuk kiszámítani, mint az oszlopok esetében, mindezt egy csapásra.", "model": "DeepL", "n_reviews": 1, @@ -368,7 +368,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "Ez egy elég elegáns módot ad arra, hogy a Hamming-kód algoritmusunk többszörös paritás-ellenőrzését egyetlen műveletbe csomagolva képzeljük el.", "model": "DeepL", "n_reviews": 1, @@ -408,7 +408,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "Ez így érthető?", "model": "DeepL", "n_reviews": 1, @@ -416,7 +416,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "Hasonlóképpen, a következő oszlop azt számolja, hogy hány olyan pozíció van a második paritáscsoportban, amelyek utolsó előtti bitje 1, emellett szintén kiemeltek. És így tovább.", "model": "DeepL", "n_reviews": 1, @@ -448,7 +448,7 @@ "end": 436.56 }, { - "input": "Once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "Ha idáig sikerült eljutnunk, akkor már könnyebben át tudjuk gondolni, hogy ez a négy bit az alján miért írja le közvetlenül a hiba helyét.", "model": "DeepL", "n_reviews": 1, @@ -456,7 +456,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "Tegyük fel, hogy ebben a blokkban egy bit 0-ról 1-re változik.", "model": "DeepL", "n_reviews": 1, @@ -464,7 +464,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "Ez azt jelenti, hogy a bit helyének információja most már benne lesz a teljes XOR-ban, ami megváltoztatja az összeget 0-ról az újonnan bevezetett hiba pozíciója értékre.", "model": "DeepL", "n_reviews": 1, @@ -488,7 +488,7 @@ "end": 477.94 }, { - "input": "So adding a copy of this position to the total sum has the same effect as removing it.", + "input": "ts sitting at the bottom as a result are the same as the four parity checks we've come to know and love, but take a moment t", "translatedText": "Tehát ezen pozíció másolatának hozzáadása a teljes összeghez ugyanolyan hatású, mintha eltávolítanánk.", "model": "DeepL", "n_reviews": 1, @@ -512,7 +512,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 ones and zeros to simulate the data block, and I'll go ahead and give it the name bits, but of course in practice this would be something that we're receiving from a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "Azzal kezdem, hogy létrehozok egy 16 egyesből vagy nullából álló véletlenszerű tömböt az adatblokk szimulálására, és elnevezem biteknek. Ez a gyakorlatban természetesen olyasmi lenne, amit egy feladótól kapunk, és ahelyett, hogy véletlenszerű lenne, 11 adatbitet hordozna 5 paritásbittel együtt.", "model": "DeepL", "n_reviews": 1, @@ -528,7 +528,7 @@ "end": 527 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "Tehát ha létrehozunk egy listát, amely végigmegy ezeken a párokon, az \"i vessző bit\" alakú párokon, és csak az i értéket vesszük ki, vagyis az indexet, nos, ez nem olyan izgalmas, csak a 0-tól 15-ig terjedő indexeket kapjuk vissza.", "model": "DeepL", "n_reviews": 1, @@ -536,7 +536,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "De ha hozzáadjuk azt a feltételt, hogy csak akkor tegyük ezt, \"ha bit\", vagyis ha az a bit 1 és nem 0, akkor csak azokat a pozíciókat nézi, ahol a megfelelő bit be van kapcsolva.", "model": "DeepL", "n_reviews": 1, @@ -552,7 +552,7 @@ "end": 557.96 }, { - "input": "Remember, what we want is to collect together all of those positions, the positions of the bits that are turned on, and then XOR them together.", + "input": "Once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error. Let's say you detect an error among the odd columns, and among the right half.", "translatedText": "Ne feledjük, hogy az összes pozíciót, a bekapcsolt bitek pozícióit akarjuk összegyűjteni, majd XOR-olni őket egymással.", "model": "DeepL", "n_reviews": 1, @@ -560,7 +560,7 @@ "end": 567.24 }, { - "input": "To do this in Python, let me first import a couple helpful functions.", + "input": "It necessarily means the error is somewhere in the last column. If there was no error in the odd column but", "translatedText": "Ehhez Pythonban először is hadd importáljak néhány hasznos függvényt.", "model": "DeepL", "n_reviews": 1, @@ -576,7 +576,7 @@ "end": 578.7 }, { - "input": "This basically eats its way through the list, taking XORs along the way.", + "input": "that means is that the position of that bit is now going to be included in the total XOR, which changes the s", "translatedText": "Ez lényegében végigrágja magát a listán, miközben XOR-okat végez.", "model": "DeepL", "n_reviews": 1, @@ -592,7 +592,7 @@ "end": 589.44 }, { - "input": "So at the moment, it looks like if we do this on our random block of 16 bits, it returns 9, which has the binary representation 1001.", + "input": "0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes", "translatedText": "Jelenleg tehát úgy néz ki, hogy ha ezt a 16 bites véletlenszerű blokkunkkal végezzük el, akkor 9-et kapunk vissza, ami az 1001-es bináris ábrázolást jelenti.", "model": "DeepL", "n_reviews": 1, @@ -616,7 +616,7 @@ "end": 618.2 }, { - "input": "Now what's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "o adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the total result at the bottom here spells out the position of the error.", "translatedText": "Most az a klassz, hogy ha a listában lévő bitek bármelyikét átkapcsoljuk egy véletlenszerű hibát szimulálva, és lefuttatjuk ugyanazt a kódsort, akkor kiírja a hiba helyét.", "model": "DeepL", "n_reviews": 1, @@ -664,7 +664,7 @@ "end": 663.76 }, { - "input": "Now depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "Attól függően, hogy mennyire mozogsz otthonosan a kettes számrendszer, a XOR-ok és úgy általában a szoftverek terén, most vagy egy kicsit zavarosnak találod ezt a perspektívát, vagy annyira elegánsnak és egyszerűnek, hogy azon tűnődsz miért nem ezzel a szemlélettel kezdtük a kezdetektől fogva.", "model": "DeepL", "n_reviews": 1, @@ -672,7 +672,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "Lazán fogalmazva, a többszörös paritásellenőrzés szemléletre könnyebb gondolni, amikor a Hamming-kódokat hardveresen, nagyon közvetlenül implementáljuk, a XOR szemléletre pedig akkor, amikor szoftveresen, egyfajta magasabb szintről csináljuk.", "model": "DeepL", "n_reviews": 1, @@ -688,7 +688,7 @@ "end": 710 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "A legfontosabb tény itt az, hogy ezen információ mérete közvetlenül megfeleltethető a szükséges redundancia mennyiségével.", "model": "DeepL", "n_reviews": 1, @@ -704,7 +704,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "És egyébként még van egy másik módszer is, amit néha a Hamming-kódok bemutatására használnak, amikor az üzenetet egy nagy mátrixszal szorozzák meg.", "model": "DeepL", "n_reviews": 1, @@ -720,7 +720,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "És ha már a skálázásnál tartunk, vegyük észre, hogy ennek a rendszernek a hatékonysága csak javul, ahogy növeljük a blokkméretet.", "model": "DeepL", "n_reviews": 1, @@ -728,7 +728,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "Láttuk például, hogy 256 bitnek csak 3%-át használjuk redundanciaként. Több bitnél pedig csak egyre jobb ez az arány.", "model": "DeepL", "n_reviews": 1, @@ -736,7 +736,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "Ahogy a paritásbitek számát egyesével növeljük, a blokk méretét megduplázhatjuk!", "model": "DeepL", "n_reviews": 1, @@ -744,7 +744,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "És ha ezt a végletekig fokozzuk, olyan blokkunk is lehet, mondjuk, egymillió bitből, ahol szó szerint 20 kérdéssel elvégezhető a teljes paritásellenőrzés, és csak 21 paritásbitet használunk.", "model": "DeepL", "n_reviews": 1, @@ -760,7 +760,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "A probléma persze az, hogy nagyobb blokkok esetén megnő a valószínűsége annak, hogy egynél több bithiba keletkezik, és a Hamming-kódok ezeket már nem képesek kezelni.", "model": "DeepL", "n_reviews": 1, @@ -800,7 +800,7 @@ "end": 838.82 }, { - "input": "But that is a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "De ez már egy másik videó témáját képezhetné.", "model": "DeepL", "n_reviews": 1, @@ -808,7 +808,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "A \"The Art of Doing Science and Engineering\" című könyvében Hamming csodálatosan őszintén beszél arról, hogy mennyire kanyargós úton jutott el a kód felfedezéséhez.", "model": "DeepL", "n_reviews": 1, @@ -840,7 +840,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "A könyvben mintegy féltucatszor hivatkozik Louis Pasteur idézetére, miszerint \"A szerencse csak a felkészült elmének kedvez\".", "model": "DeepL", "n_reviews": 1, @@ -864,7 +864,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "De ne áltasd magad azzal, hogy ezek valóban nyilvánvalóak, mert egyáltalán nem azok.", "model": "DeepL", "n_reviews": 1, @@ -912,7 +912,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "Ez volt az az írás, amely bizonyos értelemben elsőként megmutatta, hogy a hatékony hibajavítás mindig lehetséges, függetlenül attól, hogy mekkora a bithibák valószínűsége. Legalábbis elméletben.", "model": "DeepL", "n_reviews": 1, @@ -928,7 +928,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "Ugorjunk előre néhány évtizedet. Manapság annyira elterjedtté vált a bitekről és információról alkotott szemlélet, hogy könnyű figyelmen kívül hagyni, korábban ez mennyire nem így volt.", "model": "DeepL", "n_reviews": 1, @@ -943,4 +943,4 @@ "start": 973.1, "end": 982.26 } -] +] \ No newline at end of file diff --git a/2020/hamming-codes-2/indonesian/sentence_translations.json b/2020/hamming-codes-2/indonesian/sentence_translations.json index 6cc63c747..1d25f1576 100644 --- a/2020/hamming-codes-2/indonesian/sentence_translations.json +++ b/2020/hamming-codes-2/indonesian/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "Kita berbicara tentang kode Hamming, suatu cara untuk membuat blok data yang sebagian besar bitnya membawa pesan yang bermakna, sementara beberapa bit lainnya bertindak sebagai semacam redundansi, sedemikian rupa sehingga jika ada bit yang dibalik, maka akan muncul pesan. bit atau bit redundansi, apa pun di blok ini, penerima akan dapat mengidentifikasi kesalahan yang terjadi, dan cara memperbaikinya.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "Ide dasar yang disajikan di sana adalah bagaimana menggunakan beberapa pemeriksaan paritas untuk mencari biner hingga menemukan kesalahan.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "Misalnya, angka 7 dalam biner terlihat seperti 0111, yang pada dasarnya berarti 4 ditambah 2 ditambah 1.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "Dan perhatikan di mana posisi 7 berada, itu memang mempengaruhi kelompok paritas pertama kita, dan kelompok paritas kedua, dan ketiga, tapi bukan yang terakhir.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "Jadi membaca hasil keempat pemeriksaan tersebut dari bawah ke atas memang menunjukkan posisi kesalahannya.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "Saat kita mengembalikan label biner ini ke dalam kotaknya, izinkan saya menekankan bahwa label tersebut berbeda dari data yang sebenarnya dikirim.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "Itu tidak lebih dari sebuah label konseptual untuk membantu Anda dan saya memahami dari mana empat kelompok paritas itu berasal.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "Keanggunan dalam mendeskripsikan segala sesuatu yang kita lihat dalam biner mungkin dilemahkan oleh kebingungan karena segala sesuatu yang kita lihat dijelaskan dalam biner.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "Apa yang kita dapatkan adalah yang pertama dari empat kelompok paritas, yang berarti Anda dapat menafsirkan pemeriksaan pertama itu sebagai pertanyaan, hei, jika ada kesalahan, apakah bit terakhir pada posisi kesalahan itu adalah 1?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "Dan seterusnya.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "Yang pertama adalah bagaimana menggeneralisasi secara sistematis ke ukuran blok yang merupakan pangkat dua yang lebih besar.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "Artinya, masing-masing bit paritas tersebut berada di dalam satu dan hanya satu dari empat grup paritas.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "Saat Anda mengambil XOR dari dua bit, itu akan mengembalikan 1 jika salah satu dari bit tersebut diaktifkan, tetapi tidak jika keduanya diaktifkan atau dinonaktifkan.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "Dengan kata lain, ini adalah paritas dari dua bit ini.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "Ini seperti tambahan, tetapi Anda tidak pernah membawanya.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "Poin kuncinya bagi Anda dan saya adalah bahwa mengambil XOR dari banyak string bit yang berbeda secara efektif merupakan cara untuk menghitung parodi dari sekelompok grup terpisah, seperti halnya dengan kolom, semuanya dalam satu gerakan.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "Ini memberi kita cara yang agak menarik untuk memikirkan tentang beberapa pemeriksaan paritas dari algoritma kode Hamming karena semuanya dikemas bersama menjadi satu operasi tunggal.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "Apakah itu masuk akal?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "Demikian pula, kolom berikutnya menghitung berapa banyak posisi dalam grup paritas kedua, posisi yang bit kedua hingga terakhirnya adalah 1, dan yang juga disorot, dan seterusnya.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "Sekarang setelah kita memilikinya seperti ini, ini memberi kita cara yang sangat bagus untuk memikirkan mengapa keempat bit yang dihasilkan di bawah ini secara langsung menguraikan posisi kesalahan.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "Katakanlah beberapa bit di blok ini diubah dari 0 menjadi 1.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "Artinya adalah posisi bit tersebut sekarang akan dimasukkan dalam total XOR, yang mengubah jumlah dari 0 menjadi nilai yang baru dimasukkan, posisi kesalahan.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "Kita akan mulai dengan membuat array acak 16 1 dan 0 untuk mensimulasikan blok data, dan saya akan memberinya nama bit, tapi tentu saja dalam praktiknya ini akan menjadi sesuatu yang kita terima dari pengirim, dan bukannya karena acak, ia akan membawa 11 bit data bersama dengan 5 bit paritas.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "Jadi jika kita kemudian membuat daftar yang mengulang semua pasangan ini, pasangan yang terlihat seperti i, dan kemudian kita hanya mengeluarkan nilai i, hanya indeksnya, itu tidak terlalu menarik, kita hanya mendapatkan kembali indeks tersebut 0 sampai 15.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "Tetapi jika kita menambahkan kondisi untuk hanya melakukan ini jika bit, yang berarti jika bit tersebut adalah 1 dan bukan 0, maka bit tersebut hanya akan menarik keluar posisi di mana bit terkait diaktifkan.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "Kami tidak akan melakukannya di sini, tetapi Anda dapat menulis fungsi di mana pengirim menggunakan representasi biner tersebut untuk mengatur empat bit paritas sesuai kebutuhan, yang pada akhirnya membuat blok ini ke keadaan di mana menjalankan baris kode ini pada daftar bit lengkap akan kembali sebuah 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "Yang keren adalah jika kita mengganti salah satu bit dalam daftar ini, menyimulasikan kesalahan acak dari noise, lalu jika Anda menjalankan baris kode yang sama, kesalahan tersebut akan dicetak.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "Anda bisa mendapatkan blok ini secara tiba-tiba, menjalankan satu baris ini di atasnya, dan secara otomatis akan memunculkan posisi kesalahan, atau 0 jika tidak ada.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "Sekarang, tergantung pada kenyamanan Anda dengan biner dan XOR serta perangkat lunak secara umum, Anda mungkin menganggap perspektif ini sedikit membingungkan, atau jauh lebih elegan dan sederhana sehingga Anda bertanya-tanya mengapa kami tidak memulainya dari awal. -pergi.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "Secara longgar, perspektif pemeriksaan paritas berganda lebih mudah untuk dipikirkan ketika mengimplementasikan kode Hamming pada perangkat keras secara langsung, dan perspektif XOR paling mudah untuk dipikirkan ketika melakukannya dalam perangkat lunak, dari tingkat yang lebih tinggi.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "Fakta yang relevan di sini adalah bahwa informasi tersebut secara langsung berhubungan dengan seberapa banyak redundansi yang kita butuhkan.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "Lalu, ada cara lain yang terkadang Anda lihat menampilkan kode Hamming, yaitu mengalikan pesan dengan satu matriks besar.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "Dan berbicara tentang penskalaan, Anda mungkin memperhatikan bahwa efisiensi skema ini semakin baik seiring dengan peningkatan ukuran blok.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "Misalnya, kita melihat bahwa dengan 256 bit, Anda hanya menggunakan 3% dari ruang tersebut untuk redundansi, dan hal tersebut terus menjadi lebih baik dari sana.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "Ketika jumlah bit paritas bertambah satu per satu, ukuran blok terus berlipat ganda.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "Dan jika Anda menganggapnya ekstrem, Anda bisa memiliki blok dengan, katakanlah, satu juta bit, di mana Anda benar-benar akan memainkan 20 pertanyaan dengan pemeriksaan paritas Anda, dan blok tersebut hanya menggunakan 21 bit paritas.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "Masalahnya, tentu saja, dengan blok yang lebih besar, kemungkinan melihat lebih dari satu atau dua kesalahan bit akan meningkat, dan kode Hamming tidak dapat menangani apa pun selain itu.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "Tapi itu topik untuk lain waktu.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "Dalam bukunya The Art of Doing Science and Engineering, Hamming sangat berterus terang tentang betapa berliku-liku penemuan kode ini.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "Ada setengah lusin kali dalam buku ini dia merujuk pada kutipan Louis Pasteur, keberuntungan berpihak pada pikiran yang siap.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "Namun Anda tidak boleh membodohi diri sendiri dengan berpikir bahwa hal tersebut sebenarnya sudah jelas, karena sebenarnya tidak.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "Ini adalah makalah dasar yang sama yang menunjukkan, dalam arti tertentu, bahwa koreksi kesalahan yang efisien selalu mungkin dilakukan, tidak peduli seberapa tinggi kemungkinan pembalikan bit, setidaknya secara teori.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "Beberapa dekade kemudian, dan saat ini, banyak dari kita yang begitu tenggelam dalam pemikiran tentang hal-hal kecil dan informasi sehingga mudah untuk mengabaikan betapa berbedanya cara berpikir ini.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/italian/sentence_translations.json b/2020/hamming-codes-2/italian/sentence_translations.json index cb9b0b582..f9e46dd52 100644 --- a/2020/hamming-codes-2/italian/sentence_translations.json +++ b/2020/hamming-codes-2/italian/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "Stavamo parlando dei codici di Hamming, un modo per creare un blocco di dati in cui la maggior parte dei bit porta un messaggio significativo, mentre alcuni altri agiscono come una sorta di ridondanza, in modo tale che se qualche bit viene invertito, o un messaggio bit o un bit di ridondanza, qualsiasi cosa in questo blocco, un ricevitore sarà in grado di identificare che si è verificato un errore e come risolverlo.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "L'idea di base presentata era come utilizzare più controlli di parità per eseguire la ricerca binaria fino all'errore.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "Ad esempio, il numero 7 in binario assomiglia a 0111, il che significa essenzialmente che è 4 più 2 più 1.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "E notate dove si trova la posizione 7, influenza il primo dei nostri gruppi di parità, il secondo e il terzo, ma non l'ultimo.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "Quindi leggere i risultati di questi quattro controlli dal basso verso l’alto effettivamente spiega la posizione dell’errore.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "Mentre rimettiamo queste etichette binarie nelle loro scatole, lasciatemi sottolineare che sono distinte dai dati effettivamente inviati.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "Non sono altro che un'etichetta concettuale per aiutare te e me a capire da dove provengono i quattro gruppi di parità.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "L'eleganza di avere tutto ciò che stiamo guardando descritto in binario è forse indebolita dalla confusione di avere tutto ciò che stiamo guardando descritto in binario.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "Ciò che otteniamo è il primo dei nostri quattro gruppi di parità, il che significa che puoi interpretare il primo controllo come se chiedessi, ehi, se c'è un errore, l'ultimo bit nella posizione di quell'errore è 1?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "E così via.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "Il primo è come generalizzare sistematicamente alle dimensioni dei blocchi che sono potenze maggiori di due.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "Ciò significa che ciascuno di questi bit di parità si trova all'interno di uno e solo uno dei quattro gruppi di parità.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "Quando prendi lo XOR di due bit, restituirà 1 se uno di questi bit è attivato, ma non se entrambi sono attivati o disattivati.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "Detto diversamente, è la parità di questi due bit.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "È come un'addizione, ma dove non porti mai.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "Il punto chiave per te e me è che prendere lo XOR di molte stringhe di bit diverse è effettivamente un modo per calcolare le parodie di un gruppo di gruppi separati, come nel caso delle colonne, tutto in un colpo solo.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "Questo ci offre un modo piuttosto elegante di pensare ai controlli di parità multipli del nostro algoritmo di codice Hamming come se fossero tutti raggruppati insieme in un'unica operazione.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "Ha senso?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "Allo stesso modo, la colonna successiva conta quante posizioni ci sono nel secondo gruppo di parità, le posizioni il cui penultimo bit è 1 e che sono anch'esse evidenziate, e così via.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "Ora, una volta ottenuto questo risultato, questo ci dà un modo davvero carino di pensare al motivo per cui questi quattro bit risultanti in basso indicano direttamente la posizione di un errore.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "Diciamo che qualche bit in questo blocco viene commutato da 0 a 1.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "Ciò significa che la posizione di quel bit verrà ora inclusa nello XOR totale, che cambia la somma da 0 a questo nuovo valore incluso, la posizione dell'errore.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "Inizieremo creando un array casuale di 16 1 e 0 per simulare il blocco di dati, e gli darò i bit del nome, ma ovviamente in pratica questo sarebbe qualcosa che riceviamo da un mittente, e invece di essendo casuale trasporterebbe 11 bit di dati insieme a 5 bit di parità.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "Quindi, se poi creiamo un elenco che scorre su tutte queste coppie, coppie che assomigliano a i, e poi tiriamo fuori solo il valore i, solo l'indice, beh non è così eccitante, recuperiamo semplicemente quegli indici da 0 a 15.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "Ma se aggiungiamo la condizione di farlo solo se bit, ovvero se quel bit è un 1 e non uno 0, allora estrarrà solo le posizioni in cui il bit corrispondente è attivato.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "Non lo faremo qui, ma potresti scrivere una funzione in cui il mittente utilizza quella rappresentazione binaria per impostare i quattro bit di parità secondo necessità, portando infine questo blocco a uno stato in cui l'esecuzione di questa riga di codice sull'elenco completo dei bit restituisce uno 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "La cosa interessante è che se attiviamo uno qualsiasi dei bit in questo elenco, simulando un errore casuale dovuto al rumore, se esegui la stessa riga di codice, viene stampato quell'errore.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "Potresti prendere questo blocco all'improvviso, eseguire questa singola riga su di esso e sputerà automaticamente la posizione di un errore o uno 0 se non ce n'era.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "Ora, a seconda della tua dimestichezza con il binario, gli XOR e il software in generale, potresti trovare questa prospettiva un po' confusa, o molto più elegante e semplice da chiederti perché non l'abbiamo iniziata dall'inizio -andare.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "In parole povere, la prospettiva del controllo di parità multipla è più facile da pensare quando si implementano i codici Hamming nell'hardware in modo molto diretto, e la prospettiva XOR è più facile da pensare quando lo si fa nel software, da un livello più alto.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "Il fatto rilevante qui è che tali informazioni corrispondono direttamente alla quantità di ridondanza di cui abbiamo bisogno.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "E poi, a proposito, c'è tutto questo altro modo in cui a volte vedi presentati i codici Hamming, dove moltiplichi il messaggio per un'unica grande matrice.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "E parlando di ridimensionamento, potresti notare che l'efficienza di questo schema migliora solo quando aumentiamo la dimensione del blocco.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "Ad esempio, abbiamo visto che con 256 bit si utilizza solo il 3% di quello spazio per la ridondanza e da lì in poi le cose continuano a migliorare.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "Man mano che il numero di bit di parità cresce uno per uno, la dimensione del blocco continua a raddoppiare.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "E se lo porti all'estremo, potresti avere un blocco con, diciamo, un milione di bit, dove giocheresti letteralmente a 20 domande con i tuoi controlli di parità, e utilizza solo 21 bit di parità.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "Il problema, ovviamente, è che con un blocco più grande, la probabilità di vedere più di uno o due bit di errore aumenta, e i codici di Hamming non gestiscono nulla oltre a questo.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "Ma questo è un argomento per un'altra volta.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "Nel suo libro The Art of Doing Science and Engineering, Hamming è meravigliosamente sincero riguardo a quanto tortuosa sia stata la sua scoperta di questo codice.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "Ci sono circa una mezza dozzina di volte in questo libro in cui fa riferimento alla citazione di Louis Pasteur, la fortuna aiuta una mente preparata.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "Ma non dovresti illuderti pensando che in realtà siano ovvi, perché sicuramente non lo sono.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "Si trattava dello stesso documento fondamentale che mostrava, in un certo senso, che una correzione efficiente degli errori è sempre possibile, non importa quanto sia alta la probabilità di bit flip, almeno in teoria.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "Andiamo avanti velocemente di diversi decenni e, al giorno d'oggi, molti di noi sono così immersi nel pensare a frammenti e informazioni che è facile trascurare quanto fosse distinto questo modo di pensare.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/japanese/sentence_translations.json b/2020/hamming-codes-2/japanese/sentence_translations.json index 6fe5fc4e6..3e9709e8a 100644 --- a/2020/hamming-codes-2/japanese/sentence_translations.json +++ b/2020/hamming-codes-2/japanese/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "私たちはハミング コードについて話していました。 これは、ほとんどのビットが意味のあ るメッセージを運ぶデータ ブロックを作成する方法であり、他のいくつかのビットは一種 の冗長性として機能し、ビットが反転した場合、メッセージが送信されるか、メッセージが 送信されるかが決まります。 ビットや冗長ビットなど、このブロック内のあらゆるものによ って、受信機はエラーがあったことと、それを修正する方法を識別できるようになります。", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "そこで提示された基本的なアイデアは、複数のパリティ チェッ クを使用してエラーに至るまでバイナリ検索を行う方法でした。", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "たとえば、2 進数の 7 は 0111 のように見え、本質的 には 4 プラス 2 プラス 1 であることを示しています。", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "位置 7 がどこに位置するかに注目してください。 パリティ グループ の最初、2 番目、3 番目には影響しますが、最後には影響しません。", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "したがって、これら 4 つのチェックの結果を下か ら上に読むと、エラーの位置が明らかになります。", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "これらのバイナリ ラベルを箱に戻すときに、実際に送信 されるデータとは区別されることを強調しておきます。", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "これらは、4 つのパリティ グループがどこから来たの かを理解するのに役立つ概念的なラベルにすぎません。", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "私たちが見ているものすべてがバイナリで記述されることの優雅さは、おそらく、私たち が見ているものすべてがバイナリで記述されることの混乱によって損なわれるでしょう。", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "得られるのは 4 つのパリティ グループの最初のグループです。 つ まり、最初のチェックは、エラーがある場合、そのエラーの位置にある 最後のビットは 1 ですか? という質問であると解釈できます。", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "等々。", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "1 つ目は、2 の累乗より大きいブロック サイズに体系的に一般化する方法です。", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "これが意味するのは、これらのパリティ ビットはそれぞれ、4 つのパ リティ グループのうちの 1 つのみの中に存在するということです。", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "2 つのビットの XOR を計算すると、それらのビットのいずれかがオンで あれば 1 が返されますが、両方がオンまたはオフの場合は返されません。", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "別の言い方をすると、これはこれら 2 ビットのパリティです。", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "それは足し算のようなものですが、決して持ち歩かない場所です。", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "あなたと私にとって重要な点は、多くの異なるビット文字列の X OR をとることは、列の場合と同様に、多数の別々のグループの パロディを一度に計算する効果的な方法であるということです。", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "これにより、ハミング コード アルゴリズムからの複数のパリティ チェックがすべて 1 つの操作にパッケージ化されていると考える、かなり気の利いた方法が得られます。", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "それは理にかなっていますか？", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "同様に、次の列では、2 番目のパリティ グループに位 置がいくつあるか、最後から 2 番目のビットが 1 で、強調表示されている位置などがカウントされます。", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "このようにすると、結果として得られる下部の 4 つのビットが エラーの位置を直接表す理由を考える非常に良い方法になります。", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "このブロック内の一部のビットが 0 から 1 に切り替わるとします。", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "これが意味するのは、そのビットの位置が合計 XOR に含 まれることになり、合計が 0 から、代わりにこの新しく含 まれた値、つまりエラーの位置に変更されるということです。", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "まず、データ ブロックをシミュレートするために 16 個の 1 と 0 の ランダムな配列を作成し、それに bits という名前を付けますが、もちろ ん実際には、これは送信者から受信するものになります。 ランダムであるため、1 1 個のデータ ビットと 5 個のパリティ ビットを運ぶことになります。", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "したがって、これらすべてのペア (i に似たペア) をループするリストを作 成し、i の値だけ、インデックスだけを取り出すとします。 それほど面白いこ とではありません。 インデックス 0 から 15 が返されるだけです。", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "しかし、ビットの場合のみ、つまりそのビットが 0 ではなく 1 である場合にのみこれを実 行するという条件を追加すると、対応するビットがオンになっている位置のみが抽出されます。", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "ここではそれを行いませんが、送信者がそのバイナリ表現を使用して必要に応じて 4 つのパリティ ビットを設定し、最終的にこのブロックをビットの完全なリストに対して このコード行を実行すると返される状態にする関数を作成することもできます。 0。", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "素晴らしいのは、このリストのビットのいずれかを切り替えて、ノイズによるランダムなエ ラーをシミュレートし、同じコード行を実行すると、そのエラーが出力されることです。", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "このブロックを突然取得し、その上でこの 1 行を実行すると、エラーの位 置が自動的に出力され、エラーが存在しない場合は 0 が出力されます。", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "さて、バイナリ、XOR、およびソフトウェア全般に慣れているかどうかに応じて、こ の視点が少しわかりにくいと感じるか、またははるかにエレガントでシンプルなので、 なぜ最初からこの視点を始めなかったのかと不思議に思うかもしれません。 -行く。", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "大まかに言うと、複数のパリティ チェックの観点は、ハミング コードを ハードウェアで直接実装する場合に考えやすく、XOR の観点は、ソフト ウェアで実行する場合に、より高いレベルから考えるのが最も簡単です。", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "ここで重要な事実は、その情報が必要な冗長 性の量に直接対応しているということです。", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "そして、ところで、時々ハミング コードが表示されるのを目にすることがあります が、これとはまったく別の方法で、メッセージに 1 つの大きな行列を掛けます。", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "スケーリングについて言えば、ブロック サイズが増加するにつれ てこのスキームの効率が向上することに気づくかもしれません。", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "たとえば、256 ビットでは、冗長性のためにそのスペースの 3% の みが使用されており、そこからさらに改善され続けることがわかりました。", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "パリティ ビットの数が 1 つずつ増加すると、ブロック サイズは 2 倍になり続けます。", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "これを極端に解釈すると、たとえば 100 万ビットのブロック があり、文字通り 20 問のパリティ チェックを行うことにな り、使用するパリティ ビットは 21 ビットだけになります。", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "もちろん、問題は、ブロックが大きくなると、1 つまたは 2 つ以上のビット エラーが 発生する確率が高くなり、ハミング コードがそれを超えるものを処理できないことです。", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "しかし、それはまた別の機会にお話します。", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "ハミング氏は著書『The Art of Doing Science and Engineering』 の中で、このコードの発見がどれほど曲がりくねったものであったかについて、驚くほど率直に語っています。", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "この本の中で、ルイ・パスツールの名言「幸運は準備ができ た心に味方する」という言葉が何度も引用されています。", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "しかし、それらは決して明らかではないので、それら が実際には明白であると思い込む必要はありません。", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "これは、ビット反転の確率がどれほど高くても、少な くとも理論上は、ある意味、効率的なエラー訂正が 常に可能であることを示した同じ基礎論文でした。", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "数十年が経ち、最近では私たちの多くが断片や情報について考えることに没 頭しているため、この考え方がいかに独特であったかを見落としがちです。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/korean/sentence_translations.json b/2020/hamming-codes-2/korean/sentence_translations.json index f4bf4161a..f344589a6 100644 --- a/2020/hamming-codes-2/korean/sentence_translations.json +++ b/2020/hamming-codes-2/korean/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "우리는 대부분의 비트가 의미 있는 메시지를 전달하는 반면 다른 비트는 일종의 중복 역할을 하는 데이터 블록을 생성하는 방법인 해밍 코드에 대해 이야기하고 있었습니다. 비트 또는 중복 비트 등 이 블록에 있는 모든 항목을 통해 수신자는 오류가 있음을 식별하고 이를 수정하는 방법을 확인할 수 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "거기에 제시된 기본 아이디어는 다중 패리티 검사를 사용하여 오류까지 이진 검색하는 방법이었습니다.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "예를 들어, 이진수 7은 0111처럼 보입니다. 이는 본질적으로 4 더하기 2 더하기 1을 의미합니다.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "그리고 위치 7이 어디에 있는지 확인하세요. 이는 패리티 그룹 중 첫 번째, 두 번째, 세 번째에 영향을 주지만 마지막에는 영향을 미치지 않습니다.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "따라서 이 네 가지 검사 결과를 아래에서 위로 읽으면 실제로 오류의 위치를 알 수 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "이러한 바이너리 레이블을 상자에 다시 넣을 때 실제로 전송되는 데이터와 구별된다는 점을 강조하겠습니다.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "이는 여러분과 제가 네 개의 패리티 그룹이 어디에서 왔는지 이해하는 데 도움이 되는 개념적 레이블에 지나지 않습니다.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "우리가 보고 있는 모든 것을 이진법으로 기술하는 것의 우아함은 우리가 보고 있는 모든 것을 이진법으로 기술하는 것의 혼란으로 인해 약화될 수도 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "우리가 얻는 것은 네 개의 패리티 그룹 중 첫 번째입니다. 즉, 첫 번째 검사를 다음과 같이 묻는 것으로 해석할 수 있습니다. 오류가 있으면 해당 오류 위치의 마지막 비트가 1인가요?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "등등.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "첫 번째는 2의 거듭제곱보다 큰 블록 크기를 체계적으로 일반화하는 방법입니다.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "이는 각 패리티 비트가 4개의 패리티 그룹 중 하나에만 위치한다는 것을 의미합니다.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "두 비트의 XOR을 수행하면 해당 비트 중 하나가 켜져 있으면 1이 반환되지만 둘 다 켜져 있거나 꺼져 있으면 반환되지 않습니다.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "다르게 말하면, 이 두 비트의 패리티입니다.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "그것은 덧셈과 비슷하지만 결코 가지고 다니지 않습니다.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "여러분과 저에게 중요한 점은 다양한 비트 문자열의 XOR을 취하는 것이 열의 경우와 같이 여러 개별 그룹의 패러디를 한꺼번에 계산하는 효과적인 방법이라는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "이는 해밍 코드 알고리즘의 다중 패리티 검사를 모두 하나의 단일 작업으로 함께 패키지하는 것으로 생각하는 다소 멋진 방법을 제공합니다.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "말이 돼?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "마찬가지로 다음 열에서는 두 번째 패리티 그룹에 위치가 몇 개 있는지, 마지막 비트에서 두 번째 비트가 1이고 역시 강조 표시된 위치 등을 계산합니다.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "이제 이와 같은 결과가 나오면 하단에 있는 4개의 결과 비트가 오류 위치를 직접적으로 설명하는 이유를 생각할 수 있는 정말 좋은 방법을 제공합니다.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "이 블록의 일부 비트가 0에서 1로 전환된다고 가정해 보겠습니다.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "이는 해당 비트의 위치가 이제 전체 XOR에 포함되어 합계가 0에서 새로 포함된 값인 오류 위치로 변경된다는 의미입니다.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "데이터 블록을 시뮬레이션하기 위해 16개의 1과 0으로 구성된 임의의 배열을 생성하는 것부터 시작하고 여기에 이름 비트를 부여할 것입니다. 무작위이므로 5개의 패리티 비트와 함께 11개의 데이터 비트를 전달합니다.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "그래서 우리가 이 모든 쌍, 즉 i처럼 보이는 쌍을 반복하는 목록을 생성하고 i 값만 추출하고 인덱스만 추출하면 그다지 흥미롭지는 않습니다. 0부터 15까지의 인덱스만 다시 가져옵니다.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "하지만 만약 비트인 경우에만 이 작업을 수행한다는 조건을 추가하면, 즉 해당 비트가 1이고 0이 아닌 경우 해당 비트가 켜져 있는 위치만 꺼냅니다.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "여기서는 수행하지 않겠지만 송신자가 이진 표현을 사용하여 필요에 따라 4개의 패리티 비트를 설정하는 함수를 작성할 수 있습니다. 그러면 궁극적으로 이 블록을 전체 비트 목록에서 이 코드 줄을 실행하는 상태가 반환됩니다. 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "멋진 점은 이 목록의 비트 중 하나를 전환하여 노이즈로 인한 임의 오류를 시뮬레이션한 다음 동일한 코드 줄을 실행하면 해당 오류가 인쇄된다는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "갑자기 이 블록을 가져와서 이 한 줄을 실행하면 오류 위치가 자동으로 표시되고, 오류가 없으면 0이 표시됩니다.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "이제 바이너리, XOR 및 일반적인 소프트웨어에 대한 편안함에 따라 이 관점이 약간 혼란스러울 수도 있고 훨씬 더 우아하고 단순하여 왜 우리가 처음부터 시작하지 않았는지 궁금해할 수도 있습니다. -가다.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "대략적으로 말하면 다중 패리티 검사 관점은 하드웨어에서 해밍 코드를 직접 구현할 때 생각하기가 더 쉽고, XOR 관점은 소프트웨어에서 수행할 때 더 높은 수준에서 생각하기 가장 쉽습니다.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "여기서 관련 사실은 해당 정보가 필요한 중복 정도와 직접적으로 일치한다는 것입니다.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "그런데 때로는 해밍 코드가 표시되는 완전히 다른 방식이 있는데, 여기서 메시지에 하나의 큰 행렬을 곱합니다.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "확장에 관해 말하자면, 블록 크기를 늘릴수록 이 체계의 효율성이 더 좋아진다는 것을 알 수 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "예를 들어, 256비트의 경우 중복성을 위해 해당 공간의 3%만 사용하고 그 이후로 점점 더 좋아지는 것을 확인했습니다.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "패리티 비트 수가 하나씩 증가함에 따라 블록 크기는 계속 두 배로 늘어납니다.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "그리고 이를 극단적으로 받아들인다면, 예를 들어 백만 비트의 블록을 가질 수 있습니다. 여기서 패리티 검사로 문자 그대로 20개의 질문을 플레이하고 21개의 패리티 비트만 사용하게 됩니다.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "물론 문제는 블록이 클수록 비트 오류가 1~2개 이상 나올 확률이 높아지고, 해밍 코드는 그 이상은 처리하지 못한다는 점이다.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "그러나 그것은 다른 시간에 다룰 주제입니다.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "Hamming은 자신의 저서 The Art of Doing Science and Engineering에서 자신이 발견한 이 코드가 얼마나 의미심장한 일이었는지 놀랍도록 솔직하게 밝혔습니다.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "이 책 전체에 걸쳐 그는 루이 파스퇴르의 명언을 여섯 번이나 언급합니다. 행운은 준비된 마음을 선호합니다.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "하지만 그것들이 실제로 명백하다고 생각하도록 자신을 속여서는 안 됩니다. 왜냐하면 그것들은 확실히 그렇지 않기 때문입니다.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "이것은 적어도 이론적으로는 비트 플립 확률이 아무리 높더라도 어떤 의미에서는 효율적인 오류 수정이 항상 가능하다는 것을 보여주는 동일한 기본 논문이었습니다.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "수십 년이 지난 지금, 우리 중 많은 사람들이 비트와 정보에 대한 생각에 너무 몰두하여 이러한 사고 방식이 얼마나 뚜렷한지 간과하기 쉽습니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/marathi/sentence_translations.json b/2020/hamming-codes-2/marathi/sentence_translations.json index f049cfd70..73b834420 100644 --- a/2020/hamming-codes-2/marathi/sentence_translations.json +++ b/2020/hamming-codes-2/marathi/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "आम्ही हॅमिंग कोडबद्दल बोलत होतो, डेटाचा एक ब्लॉक तयार करण्याचा एक मार्ग जिथे बहुतेक बिट एक अर्थपूर्ण संदेश देतात, तर काही इतर एक प्रकारचा रिडंडंसी म्हणून काम करतात, अशा प्रकारे की जर काही बिट फ्लिप झाले तर एकतर संदेश बिट किंवा रिडंडंसी बिट, या ब्लॉकमधील काहीही, एक प्राप्तकर्ता त्रुटी आहे हे ओळखण्यास सक्षम असेल आणि त्याचे निराकरण कसे करावे.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "तेथे सादर केलेली मूलभूत कल्पना ही होती की त्रुटीकडे जाण्यासाठी बायनरी शोधण्यासाठी एकाधिक पॅरिटी चेक कसे वापरायचे.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "उदाहरणार्थ, बायनरी मधील 7 ही संख्या 0111 सारखी दिसते, मूलत: ते 4 अधिक 2 अधिक 1 असे म्हणते.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "आणि लक्षात घ्या की स्थान 7 कुठे बसते, ते आमच्या समता गटांपैकी पहिल्यावर, आणि दुसऱ्या आणि तिसऱ्याला प्रभावित करते, परंतु शेवटचे नाही.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "त्यामुळे तळापासून वरपर्यंत त्या चार तपासण्यांचे निकाल वाचून त्रुटीची स्थिती स्पष्ट होते.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "जसे आम्ही ही बायनरी लेबले त्यांच्या बॉक्समध्ये परत ठेवतो, मी यावर जोर देतो की ते प्रत्यक्षात पाठवलेल्या डेटापेक्षा वेगळे आहेत.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "ते चार समानता गट कुठून आले हे तुम्हाला आणि मला समजण्यास मदत करण्यासाठी एक संकल्पनात्मक लेबलपेक्षा अधिक काही नाही.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "बायनरीमध्ये वर्णन केल्या जाणाऱ्या प्रत्येक गोष्टीचे वर्णन आपण पाहत आहोत या संभ्रमामुळे कदाचित कमी होईल.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "आम्हाला जे मिळते ते आमच्या चार पॅरिटी गटांपैकी पहिले आहे, याचा अर्थ तुम्ही त्या पहिल्या चेकचा विचार म्हणून अर्थ लावू शकता, अरे, जर एखादी त्रुटी असेल, तर त्या त्रुटीच्या स्थितीतील अंतिम बिट 1 आहे का?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "वगैरे.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "प्रथम म्हणजे दोन मोठ्या शक्ती असलेल्या आकारांना ब्लॉक करण्यासाठी पद्धतशीरपणे सामान्यीकरण कसे करावे.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "याचा अर्थ असा आहे की त्या प्रत्येक पॅरिटी बिट्स चार पॅरिटी गटांपैकी फक्त एकामध्ये बसतात.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "जेव्हा तुम्ही दोन बिट्सचा XOR घेता, तेव्हा त्यापैकी एक बिट चालू असल्यास ते 1 परत करेल, परंतु दोन्ही चालू किंवा बंद असल्यास नाही.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "वेगळ्या पद्धतीने शब्दबद्ध केले तर ही या दोन बिट्सची समानता आहे.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "हे जोडण्यासारखे आहे, परंतु जिथे आपण कधीही वाहून नेत नाही.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "तुमच्यासाठी आणि माझ्यासाठी महत्त्वाचा मुद्दा असा आहे की अनेक भिन्न बिट स्ट्रिंग्सचे XOR घेणे हा वेगवेगळ्या गटांच्या विडंबनांची प्रभावीपणे गणना करण्याचा एक मार्ग आहे, जसे की स्तंभांप्रमाणेच, सर्व काही एकाच वेळी होते.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "हे आम्हाला आमच्या हॅमिंग कोड अल्गोरिदममधील एकाधिक समानता तपासण्यांबद्दल विचार करण्याचा एक अतिशय आकर्षक मार्ग देते कारण सर्व एकाच ऑपरेशनमध्ये एकत्रित केले जातात.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "त्याला काही अर्थ आहे का?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "त्याचप्रमाणे, पुढील स्तंभ दुसऱ्या पॅरिटी गटात किती पोझिशन्स आहेत याची मोजणी करतो, ज्या पोझिशन्सचा दुसरा ते शेवटचा बिट 1 आहे आणि ज्या हायलाइट केल्या आहेत, आणि असेच पुढे.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "आता एकदा आमच्याकडे असे झाले की, हे आम्हाला तळाशी असलेले हे चार परिणामी बिट्स थेट त्रुटीची स्थिती का स्पष्ट करतात याबद्दल विचार करण्याचा एक चांगला मार्ग देते.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "समजा या ब्लॉकमध्ये ० ते १ पर्यंत टॉगल केले आहे.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "याचा अर्थ असा आहे की त्या बिटची स्थिती आता एकूण XOR मध्ये समाविष्ट केली जाणार आहे, जी बेरीज 0 वरून बदलते त्याऐवजी हे नवीन समाविष्ट केलेले मूल्य, त्रुटीची स्थिती.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "आम्ही डेटा ब्लॉकचे नक्कल करण्यासाठी 16 1s आणि 0s चा यादृच्छिक अॅरे तयार करून सुरुवात करू आणि मी त्याला नावाचे बिट्स देईन, परंतु अर्थातच सराव मध्ये हे असे काहीतरी असेल जे आम्हाला प्रेषकाकडून प्राप्त होत असेल आणि त्याऐवजी यादृच्छिक असल्याने ते 5 पॅरिटी बिट्ससह 11 डेटा बिट्स घेऊन जाईल.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "मग जर आपण या सर्व जोड्या, i सारख्या दिसणार्‍या जोड्या लूप करणारी यादी तयार केली आणि मग आपण फक्त i मूल्य, फक्त निर्देशांक काढू, तर ते इतके रोमांचक नाही, तर आपल्याला ते निर्देशांक 0 ते 15 परत मिळतात.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "पण जर आपण हे फक्त बिट जर करण्याची अट जोडली, म्हणजे जर तो बिट 1 असेल तर 0 नसेल, तर तो फक्त त्या पोझिशन्स बाहेर काढतो जिथे संबंधित बिट चालू आहे.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "आम्ही ते येथे करणार नाही, परंतु तुम्ही एक फंक्शन लिहू शकता जिथे प्रेषक आवश्यकतेनुसार चार पॅरिटी बिट्स सेट करण्यासाठी बायनरी प्रतिनिधित्व वापरतो, शेवटी हा ब्लॉक अशा स्थितीत आणतो जिथे कोडची ही ओळ बिट्सच्या संपूर्ण यादीवर चालते. एक 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "काय छान आहे की जर आपण या यादीतील कोणत्याही एका बिट्सला टॉगल केले, नॉइजमधून यादृच्छिक त्रुटीचे नक्कल करून, जर तुम्ही कोडची हीच ओळ चालवली तर ती त्रुटी प्रिंट करते.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "तुम्ही हा ब्लॉक निळ्या रंगातून मिळवू शकता, त्यावर ही एकल ओळ चालवा आणि ती आपोआप त्रुटीची स्थिती बाहेर टाकेल, किंवा जर नसेल तर 0.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "आता, बायनरी आणि XORs आणि सर्वसाधारणपणे सॉफ्टवेअरसह तुमच्या सोयीनुसार, तुम्हाला एकतर हा दृष्टीकोन थोडा गोंधळात टाकणारा किंवा इतका अधिक मोहक आणि सोपा वाटू शकतो की, आम्ही सुरुवातीपासूनच याची सुरुवात का केली नाही याचा तुम्हाला प्रश्न पडतो.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "- जा हलक्या शब्दात सांगायचे तर, हार्डवेअरमध्ये हॅमिंग कोडची अंमलबजावणी करताना मल्टीपल पॅरिटी चेक दृष्टीकोन विचार करणे सोपे आहे, आणि XOR दृष्टीकोन हे सॉफ्टवेअरमध्ये करत असताना विचार करणे सर्वात सोपा आहे.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "येथे संबंधित वस्तुस्थिती अशी आहे की ती माहिती आपल्याला किती रिडंडन्सीची आवश्यकता आहे याच्याशी थेट संबंधित आहे.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "आणि मग, तसे, हा संपूर्ण दुसरा मार्ग आहे ज्यामध्ये आपण कधीकधी हॅमिंग कोड सादर केलेले दिसतो, जिथे आपण संदेश एका मोठ्या मॅट्रिक्सने गुणाकार करतो.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "आणि स्केलिंगबद्दल बोलताना, तुमच्या लक्षात येईल की या योजनेची कार्यक्षमता केवळ आम्ही ब्लॉक आकार वाढवतो तेव्हाच अधिक चांगली होते.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "उदाहरणार्थ, आम्ही पाहिले की 256 बिट्ससह, तुम्ही त्यातील केवळ 3% जागा रिडंडंसीसाठी वापरत आहात आणि ते तिथून चांगले होत आहे.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "पॅरिटी बिट्सची संख्या एक एक करून वाढत असताना, ब्लॉकचा आकार दुप्पट होत जातो.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "आणि जर तुम्ही ते टोकापर्यंत नेले, तर तुमच्याकडे एक दशलक्ष बिट्सचा ब्लॉक असू शकतो, जिथे तुम्ही तुमच्या पॅरिटी चेकसह अक्षरशः 20 प्रश्न खेळत असाल आणि ते फक्त 21 पॅरिटी बिट्स वापरते.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "अर्थातच, समस्या अशी आहे की मोठ्या ब्लॉकसह, एक किंवा दोन बिट त्रुटींपेक्षा जास्त दिसण्याची शक्यता वाढते आणि हॅमिंग कोड त्यापलीकडे काहीही हाताळत नाहीत.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "पण तो दुसर्‍या वेळी विषय आहे.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "त्याच्या द आर्ट ऑफ डुइंग सायन्स अँड इंजिनीअरिंग या पुस्तकात, हॅमिंगने या कोडचा शोध किती क्षुल्लक होता हे आश्चर्यकारकपणे स्पष्ट केले आहे.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "या पुस्तकात अर्धा डझन वेळा त्यांनी लुई पाश्चरच्या कोटाचा संदर्भ दिला आहे, नशीब तयार मनाला अनुकूल आहे.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "परंतु ते प्रत्यक्षात स्पष्ट आहेत असा विचार करून तुम्ही स्वत:ला फसवू नका, कारण ते नक्कीच नाहीत.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "हा तोच पायाभूत कागद होता ज्याने एका विशिष्ट अर्थाने दाखवून दिले की, किमान सिद्धांतानुसार, बिट फ्लिपची संभाव्यता कितीही उच्च असली तरीही कार्यक्षम त्रुटी सुधारणे नेहमीच शक्य असते.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "अनेक दशके फास्ट फॉरवर्ड, आणि आजकाल, आपल्यापैकी बरेच जण बिट्स आणि माहितीबद्दल विचार करण्यात इतके मग्न आहेत की विचार करण्याची ही पद्धत किती वेगळी होती याकडे दुर्लक्ष करणे सोपे आहे.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/persian/sentence_translations.json b/2020/hamming-codes-2/persian/sentence_translations.json index 589a3d2dd..33598f287 100644 --- a/2020/hamming-codes-2/persian/sentence_translations.json +++ b/2020/hamming-codes-2/persian/sentence_translations.json @@ -32,7 +32,7 @@ "end": 34.6 }, { - "input": "But as you start to think about actually implementing this, either in software or hardware, that framing may actually undersell how elegant these codes really are. ", + "input": "hat there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error. In that video, the goal was to make Hamming codes feel as hands-on and rediscoverable as possible. But as ", "translatedText": "اما وقتی شروع به فکر کردن در مورد اجرای واقعی این کدها، چه در نرم‌افزار یا سخت‌افزار، می‌کنید، این قاب‌بندی ممکن است واقعاً از ظریف بودن این کدها کم‌فروش کند. ", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. ", + "input": "you read out the answers to the four parity checks we did in the last video, all as ones and zeros instead of yeses and nos, it literally spells out ", "translatedText": "به عنوان مثال، عدد 7 در باینری شبیه 0111 است، که در اصل می گوید که 4 به علاوه 2 به علاوه 1 است. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error. ", + "input": "century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day. The goal here is to give you a very thorough understanding of one of the earlie ", "translatedText": "بنابراین خواندن نتایج آن چهار بررسی از پایین به بالا واقعاً موقعیت خطا را مشخص می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 87.54 }, { - "input": "There's nothing special about the example 7, this works in general, and this makes the logic for implementing the whole scheme in hardware shockingly simple. ", + "input": "st examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity groups, and the second, and the third, but not the last. So reading the results of those four checks ", "translatedText": "هیچ چیز خاصی در مورد مثال 7 وجود ندارد، این به طور کلی کار می کند، و این منطق اجرای کل طرح در سخت افزار را به طرز تکان دهنده ای ساده می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 132.36 }, { - "input": "It's worth it, though. ", + "input": "0, let's write them all in binary, running from 0000 up to 1111. ask feels at the star ", "translatedText": "هرچند ارزشش را دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 166.16 }, { - "input": "In other words, that second check is asking, hey, me again, if there's an error, is the second to last bit of that position a 1? ", + "input": "ly spelled words. Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. ", "translatedText": "به عبارت دیگر، آن چک دوم از شما می پرسد، دوباره سلام، اگر خطایی وجود دارد، آیا بیت دوم تا آخر آن موقعیت 1 است؟ و غیره. ", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 188.74 }, { - "input": "Everything we did earlier is the same as answering these four questions, which in turn is the same as spelling out a position in binary. ", + "input": "everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary. ", "translatedText": "هر کاری که قبلا انجام دادیم مانند پاسخ دادن به این چهار سوال است، که به نوبه خود مانند املای یک موقعیت در باینری است. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 197.74 }, { - "input": "I hope this makes two things clearer. ", + "input": "It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where tha ", "translatedText": "امیدوارم این دو چیز را واضح تر کند. ", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two. ", + "input": "t final bit is a 1. What we get is the first of our four parity groups, which means that you can interpret that first check as asking, hey, if there's an err ", "translatedText": "اولین مورد این است که چگونه به طور سیستماتیک به اندازه های بلوکی که قدرت های دو بزرگتر هستند تعمیم دهیم. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 216.68 }, { - "input": "Those of you who watched the chessboard puzzle I did with Matt Parker might find all this exceedingly familiar. ", + "input": "maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. ", "translatedText": "کسانی از شما که پازل صفحه شطرنج را که من با مت پارکر انجام دادم تماشا کردند، ممکن است همه اینها را بسیار آشنا بیابند. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 237.32 }, { - "input": "These are the positions whose binary representation has just a single bit turned on. ", + "input": "at goes on at position 0, but don't worry about that for now. The third parity check covers every position whose third to last bit is turned ", "translatedText": "اینها موقعیت هایی هستند که نمایش باینری آنها فقط یک بیت روشن است. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 285.5 }, { - "input": "XOR, for those of you who don't know, stands for exclusive or. ", + "input": "f 1s in the message is an even number. So for example right now, that total number of 1s is If it takes more bits to describe each p ", "translatedText": "XOR، برای کسانی از شما که نمی دانند، مخفف منحصر به فرد یا. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. ", + "input": "osition, like six bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. that special bit to be a 1, making the count even. But if the block had already started off with a ", "translatedText": "هنگامی که XOR دو بیت را می گیرید، اگر یکی از آن بیت ها روشن باشد، 1 برمی گردد، اما اگر هر دو روشن یا خاموش باشند، نه. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 302.98 }, { - "input": "As a math person, I prefer to think about it as addition mod 2. ", + "input": "it would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core ", "translatedText": "به عنوان یک فرد ریاضی، ترجیح می دهم در مورد آن به عنوان مد 2 فکر کنم. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 306.76 }, { - "input": "We also commonly talk about the XOR of two different bit strings, which basically does this component by component. ", + "input": "logic, but solving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity bits are sitti ", "translatedText": "ما همچنین معمولاً در مورد XOR دو رشته بیت مختلف صحبت می کنیم که اساساً این جزء به جزء انجام می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 322.48 }, { - "input": "If you open up some Python right now and apply the caret operation between two integers, this is what it's doing but to the bit representations of those numbers under the hood. ", + "input": "of two, for example 1, 2, 4, and 8. These are the positions whose binary representation has just a single bit turned on. d say the parity is 0 or 1, which is typically more helpful once you start doing math with the idea. And this special bit that the sender uses to con ", "translatedText": "اگر همین الان پایتون را باز کنید و عملیات caret را بین دو عدد صحیح اعمال کنید، این همان کاری است که انجام می‌دهد به جز نمایش بیت‌های آن اعداد زیر سرپوش. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop. ", + "input": "trol the parity is called the parity bit. And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure tha ", "translatedText": "نکته کلیدی برای من و شما این است که گرفتن XOR از بسیاری از رشته‌های بیتی مختلف، به طور موثر راهی برای محاسبه تقلید هجو دسته‌ای از گروه‌های جداگانه است، مانند ستون‌ها، همه در یک لحظه. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense? ", + "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. ", + "input": "turned on, but not if both are turned on or if both are turned off. Phrased differently, it's the parity of these two bits. full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. ", "translatedText": "فرقی میکنه؟ به همین ترتیب، ستون بعدی تعداد موقعیت‌های گروه برابری دوم، موقعیت‌هایی که بیت دوم تا آخر آنها 1 است و همچنین برجسته شده‌اند و غیره می‌شمارند. ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 423.96 }, { - "input": "And so you know where it goes from here. ", + "input": "e also commonly talk about the XOR of two different bit s ", "translatedText": "و بنابراین می دانید که از اینجا به کجا می رود. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 469.36 }, { - "input": "You see, if you add a bit string together twice, it's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. ", + "input": "ey point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop. This gives us a rather snazzy way ", "translatedText": "ببینید، اگر یک رشته بیت را دو بار با هم اضافه کنید، مثل این است که اصلاً آن را نداشته باشید، اساساً به این دلیل که در این دنیا 1 به اضافه 1 برابر با 0 است. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 477.94 }, { - "input": "So adding a copy of this position to the total sum has the same effect as we're moving it. ", + "input": "to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation. ", "translatedText": "بنابراین افزودن یک کپی از این موقعیت به مجموع کل همان اثری را دارد که ما آن را جابجا می کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 484.3 }, { - "input": "And that effect, again, is that the total result at the bottom here spells out the position of the error. ", + "input": "Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the mes ", "translatedText": "و آن اثر، دوباره، این است که نتیجه کل در پایین اینجا موقعیت خطا را مشخص می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 490.7 }, { - "input": "To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all of the logic on the receiver's end. ", + "input": "sage bit is turned on to a 1, and then collect these positions into one big column and take the XOR. You can probably guess that the four bits sitting at the bottom as a resu ", "translatedText": "برای نشان دادن زیبایی این موضوع، اجازه دهید یک خط از کد پایتون را که قبلاً به آن ارجاع دادم نشان دهم، که تقریباً تمام منطق انتهای گیرنده را نشان می‌دهد. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. ", + "input": "lt are the same as the four parity checks we've come to know and love, but take a moment to actually think about why exactly. This last column, for example, is counting all of the positions whose last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. ", "translatedText": "ما با ایجاد یک آرایه تصادفی از 16 1 و 0 برای شبیه سازی بلوک داده شروع می کنیم، و من بیت های نام را به آن می دهم، اما البته در عمل این چیزی است که ما از فرستنده دریافت می کنیم، و به جای تصادفی بودن، 11 بیت داده همراه با 5 بیت برابری را حمل می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15. ", + "input": "ht half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit, so these 8 bits already have an even pari Likewise, the next column counts how many positions are in the second parity group, the positions whose second to las ", "translatedText": "بنابراین، اگر لیستی ایجاد کنیم که روی همه این جفت‌ها حلقه بزند، جفت‌هایی که شبیه i هستند، و سپس فقط مقدار i، فقط شاخص را بیرون بیاوریم، خوب آنقدرها هم هیجان‌انگیز نیست، فقط آن شاخص‌ها را از 0 تا 15 برمی‌گردانیم. . ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 552.66 }, { - "input": "In this case it looks like those positions are 0, 4, 6, 9, etc. ", + "input": "ve on the same thing we've been doing. but for right now we're going to assume ", "translatedText": "در این حالت به نظر می رسد که آن موقعیت ها 0، 4، 6، 9 و غیره هستند. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 567.24 }, { - "input": "To do this in Python, let me first import a couple helpful functions. ", + "input": "The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 000 ", "translatedText": "برای انجام این کار در پایتون، اجازه دهید ابتدا چند تابع مفید را وارد کنم. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 578.7 }, { - "input": "This basically eats its way through the list, taking XORs along the way. ", + "input": "es us a really nice way to think about why these four resulting bits at the bottom directly spell out the pos ", "translatedText": "این اساساً راه خود را از طریق لیست می خورد و XOR ها را در طول راه می برد. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 582.68 }, { - "input": "If you prefer, you can explicitly write out that XOR function without having to import it from anywhere. ", + "input": "ition of an error. Let's say you detect an error among the odd columns, and among the right half. It necessarily means the error is somewhere in th ", "translatedText": "اگر ترجیح می دهید، می توانید به صراحت آن تابع XOR را بدون نیاز به وارد کردن آن از جایی بنویسید. ", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 630.22 }, { - "input": "Isn't that neat? ", + "input": "an error that changes a 1 to a 0. You see, if you add a bit string together twice, it's the same as ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 641.06 }, { - "input": "And there's nothing special about the size 16 here. ", + "input": "And that effect, again, is that the total result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me sh ", "translatedText": "و هیچ چیز خاصی در مورد سایز 16 در اینجا وجود ندارد. ", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 649.86 }, { - "input": "Needless to say, there is more code to write here, like doing the meta parity check to detect 2-bit errors, but the idea is that almost all of the core logic from our scheme comes down to a single XOR reduction. ", + "input": "ferenced before, which will capture almost all of the logic on the receiver's end. We'll start by creating a random array of 16 ones and zeros to simulate the data block, and I'll go ahead and give it the name bits, but of course in practice this would be something that we're receiving f ", "translatedText": "نیازی به گفتن نیست، کدهای بیشتری برای نوشتن در اینجا وجود دارد، مانند انجام بررسی متا برابری برای تشخیص خطاهای 2 بیتی، اما ایده این است که تقریباً تمام منطق اصلی طرح ما به یک کاهش XOR منتهی می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 690.5 }, { - "input": "The first one is easiest to actually do by hand, and I think it does a better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the log of the size of the block, or in other words, it grows one bit at a time as the block size doubles. ", + "input": "l out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on. In this case it looks like those positions are 0, 4, 6, 9, etc. Remember, what ", "translatedText": "اولین مورد ساده‌ترین کار است که واقعاً با دست انجام می‌شود، و من فکر می‌کنم با القای شهود اصلی زیربنای همه این‌ها کار بهتری انجام می‌دهد، یعنی اطلاعات مورد نیاز برای یافتن یک خطا مربوط به گزارش اندازه بلوک است. ، یا به عبارت دیگر، با دو برابر شدن اندازه بلوک، هر بار یک بیت رشد می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need. ", + "input": "we want is to collect together all of those positions, the positions of the bits that are turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. ", "translatedText": "واقعیت مرتبط در اینجا این است که آن اطلاعات به طور مستقیم با میزان افزونگی مورد نیاز ما مطابقت دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there. ", + "input": "looks like if we do this on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but you could write a function wh ", "translatedText": "به عنوان مثال، ما دیدیم که با 256 بیت، شما فقط از 3٪ از آن فضا برای افزونگی استفاده می کنید، و از آنجا به بعد بهتر می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling. ", + "input": "ere the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a state where running th ", "translatedText": "همانطور که تعداد بیت های برابری یکی یکی افزایش می یابد، اندازه بلوک دو برابر می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 804.3 }, { - "input": "Also, in practice, errors tend to come in little bursts, which would totally ruin a single block, so one common tactic to help spread out a burst of errors across many different blocks is to interlace those blocks, like this, before they're sent out or stored. ", + "input": "imulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the positio ", "translatedText": "همچنین، در عمل، خطاها معمولاً به صورت پشت سر هم ظاهر می‌شوند، که یک بلوک را کاملاً خراب می‌کند، بنابراین یک تاکتیک رایج برای کمک به گسترش خطاها در بسیاری از بلوک‌های مختلف این است که آن بلوک‌ها، مانند این، قبل از اینکه به هم بریزند ارسال یا ذخیره می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind. ", + "input": "is perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it ", "translatedText": "در سراسر این کتاب تقریباً دوازده بار است که او به نقل قول لویی پاستور اشاره می کند، شانس به نفع ذهن آماده است. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 906.82 }, { - "input": "Part of the reason that clever ideas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong turns, underselling just how vast the space of explorable possibilities is at the start of a problem solving process, all of that. ", + "input": ", with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing 1 out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The first one is easiest to actually do by hand, and I think it does a better job instilling the core intuition underlying all of this, ", "translatedText": "بخشی از دلیل اینکه ایده‌های هوشمندانه به طرز فریبنده‌ای آسان به نظر می‌رسند این است که ما فقط نتیجه نهایی را می‌بینیم، تمیز کردن چیزهایی که به هم ریخته بود، هرگز به همه چرخش‌های اشتباه اشاره نمی‌کنیم، و کم‌فروش بودن فضای احتمالات قابل کشف در ابتدای یک مشکل را کم می‌کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 922.86 }, { - "input": "But this is true in general. ", + "input": "which is that the information required to locate a single error is relat ", "translatedText": "فرآیند حل، همه اینها اما این به طور کلی درست است. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 924.9 }, { - "input": "I think for some special inventions, there's a second, deeper reason that we underappreciate them. ", + "input": "ed to the log of the size of the block, or in other words, it grows one bit at a time as the block size doubles. The relevant fact here i ", "translatedText": "من فکر می کنم برای برخی اختراعات خاص، دلیل دوم و عمیق تری وجود دارد که ما از آنها قدردانی نمی کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 938.64 }, { - "input": "This was essentially concurrent with when Hamming developed his algorithm. ", + "input": "block is even, just like a normal parity check. Now, if there's a single bit error, then ", "translatedText": "این اساساً با زمانی که هامینگ الگوریتم خود را توسعه داد همزمان بود. ", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory. ", + "input": "the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks. However, if there's two errors, then the overall parity is going to toggle back to be And then, by the way, there is this whole other way that you s ", "translatedText": "این همان مقاله بنیادی بود که نشان داد، به معنای خاصی، تصحیح خطای کارآمد همیشه امکان پذیر است، مهم نیست که احتمال تلنگر بیت چقدر بالا باشد، حداقل در تئوری. ", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was. ", + "input": "multiply the message by one big matrix. It's kind of nice because it relates it to the broader family of linear codes, but I think that gives almost no intuition for where it comes from or how it scales. ", "translatedText": "چندین دهه به سرعت به جلو، و این روزها، بسیاری از ما چنان غرق در فکر کردن در مورد بیت ها و اطلاعات هستیم که به راحتی می توان از تفاوت این طرز تفکر چشم پوشی کرد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/portuguese/sentence_translations.json b/2020/hamming-codes-2/portuguese/sentence_translations.json index 83676b7b0..4ef7d5ecd 100644 --- a/2020/hamming-codes-2/portuguese/sentence_translations.json +++ b/2020/hamming-codes-2/portuguese/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "Estávamos falando sobre códigos de Hamming, uma forma de criar um bloco de dados onde a maioria dos bits carrega uma mensagem significativa, enquanto alguns outros atuam como uma espécie de redundância, de tal forma que se algum bit for invertido, será uma mensagem bit ou bit de redundância, qualquer coisa neste bloco, um receptor será capaz de identificar que houve um erro e como corrigi-lo.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "A ideia básica apresentada foi como usar múltiplas verificações de paridade para pesquisar binariamente até o erro.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "Por exemplo, o número 7 em binário se parece com 0111, significando essencialmente que é 4 mais 2 mais 1.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "E observe onde fica a posição 7, ela afeta o primeiro de nossos grupos de paridade, e o segundo, e o terceiro, mas não o último.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "Portanto, ler os resultados dessas quatro verificações de baixo para cima realmente explica a posição do erro.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "Ao colocarmos esses rótulos binários de volta em suas caixas, deixe-me enfatizar que eles são distintos dos dados que estão sendo realmente enviados.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "Eles nada mais são do que um rótulo conceitual para ajudar você e eu a entender de onde vieram os quatro grupos de paridade.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "A elegância de ter tudo o que estamos vendo descrito em binário talvez seja prejudicada pela confusão de ter tudo o que estamos vendo descrito em binário.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "O que obtemos é o primeiro dos nossos quatro grupos de paridade, o que significa que você pode interpretar essa primeira verificação como uma pergunta: ei, se houver um erro, o bit final na posição desse erro é 1?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "E assim por diante.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "A primeira é como generalizar sistematicamente para tamanhos de bloco maiores que sejam potências de dois.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "O que isso significa é que cada um desses bits de paridade está dentro de um e apenas um dos quatro grupos de paridade.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "Quando você pega o XOR de dois bits, ele retornará 1 se um desses bits estiver ativado, mas não se ambos estiverem ativados ou desativados.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "Em outras palavras, é a paridade desses dois bits.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "É como uma adição, mas onde você nunca carrega.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "O ponto principal para você e para mim é que obter o XOR de muitas cadeias de bits diferentes é efetivamente uma maneira de calcular as paródias de vários grupos separados, como acontece com as colunas, tudo de uma só vez.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "Isso nos dá uma maneira bastante elegante de pensar sobre as múltiplas verificações de paridade de nosso algoritmo de código de Hamming como sendo todas agrupadas em uma única operação.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "Isso faz sentido?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "Da mesma forma, a próxima coluna conta quantas posições estão no segundo grupo de paridade, as posições cujo penúltimo bit é 1, e que também estão destacadas, e assim por diante.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "Agora, uma vez que temos isto, isto dá-nos uma maneira muito boa de pensar sobre porque é que estes quatro bits resultantes na parte inferior indicam diretamente a posição de um erro.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "Digamos que algum bit neste bloco seja alternado de 0 para 1.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "O que isso significa é que a posição desse bit agora será incluída no XOR total, o que muda a soma de 0 para esse valor recém-incluído, a posição do erro.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "Começaremos criando um array aleatório de 16 1s e 0s para simular o bloco de dados, e darei a ele o nome de bits, mas é claro que na prática isso seria algo que receberíamos de um remetente e, em vez de sendo aleatório, carregaria 11 bits de dados junto com 5 bits de paridade.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "Então, se criarmos uma lista que percorre todos esses pares, pares que se parecem com i, e então extrairmos apenas o valor i, apenas o índice, bem, não é tão emocionante, apenas recuperamos esses índices de 0 a 15.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "Mas se adicionarmos a condição de fazer isso apenas se for bit, ou seja, se esse bit for 1 e não 0, bem, então ele retira apenas as posições onde o bit correspondente está ativado.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "Não faremos isso aqui, mas você poderia escrever uma função onde o remetente usa essa representação binária para definir os quatro bits de paridade conforme necessário, levando esse bloco a um estado em que a execução dessa linha de código na lista completa de bits retorna um 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "O que é legal é que se alternarmos qualquer um dos bits desta lista, simulando um erro aleatório de ruído, se você executar essa mesma linha de código, esse erro será impresso.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "Você pode obter esse bloco do nada, executar esta única linha nele e ele exibirá automaticamente a posição de um erro ou um 0, se não houver nenhum.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "Agora, dependendo do seu conforto com binários, XORs e software em geral, você pode achar essa perspectiva um pouco confusa ou muito mais elegante e simples que você está se perguntando por que não começamos com ela desde o início. -ir.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "Falando livremente, a perspectiva de verificação de paridade múltipla é mais fácil de pensar ao implementar códigos de Hamming em hardware de maneira muito direta, e a perspectiva XOR é mais fácil de pensar ao fazê-lo em software, de um nível mais alto.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "O fato relevante aqui é que essa informação corresponde diretamente à quantidade de redundância necessária.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "E então, a propósito, há toda essa outra maneira que às vezes você vê os códigos de Hamming apresentados, onde você multiplica a mensagem por uma grande matriz.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "E por falar em dimensionamento, você deve notar que a eficiência desse esquema só melhora à medida que aumentamos o tamanho do bloco.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "Por exemplo, vimos que com 256 bits, você está usando apenas 3% desse espaço para redundância, e isso continua melhorando a partir daí.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "À medida que o número de bits de paridade aumenta um por um, o tamanho do bloco continua duplicando.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "E se você levar isso ao extremo, você poderia ter um bloco com, digamos, um milhão de bits, onde você estaria literalmente jogando 20 perguntas com suas verificações de paridade, e ele usaria apenas 21 bits de paridade.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "O problema, claro, é que com um bloco maior, a probabilidade de ver mais de um ou dois erros de bit aumenta, e os códigos de Hamming não lidam com nada além disso.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "Mas isso é assunto para outra hora.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "Em seu livro The Art of Doing Science and Engineering, Hamming é maravilhosamente sincero sobre o quão sinuosa foi sua descoberta desse código.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "Há cerca de meia dúzia de vezes ao longo deste livro que ele faz referência à citação de Louis Pasteur: a sorte favorece uma mente preparada.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "Mas você não deve se enganar pensando que eles são realmente óbvios, porque definitivamente não são.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "Este foi o mesmo artigo fundamental que mostrou, em certo sentido, que a correção eficiente de erros é sempre possível, não importa quão alta seja a probabilidade de inversões de bits, pelo menos em teoria.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "Avançando várias décadas, hoje em dia muitos de nós estamos tão imersos em pensar sobre bits e informações que é fácil ignorar o quão distinta era essa forma de pensar.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/russian/sentence_translations.json b/2020/hamming-codes-2/russian/sentence_translations.json index 3c8e6690d..e4436539b 100644 --- a/2020/hamming-codes-2/russian/sentence_translations.json +++ b/2020/hamming-codes-2/russian/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "Мы говорили о кодах Хэмминга, способе создания блока данных, в котором большинство битов несут значимое сообщение, а несколько других действуют как своего рода избыточность, таким образом, что если какой-либо бит переворачивается, либо сообщение бит или бит избыточности, что-либо в этом блоке, получатель сможет определить, что произошла ошибка и как ее исправить.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "Основная идея, представленная там, заключалась в том, как использовать несколько проверок четности для двоичного поиска пути к ошибке.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "Например, число 7 в двоичном формате выглядит как 0111, что, по сути, означает, что это 4 плюс 2 плюс 1.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "И обратите внимание, где находится позиция 7: она влияет и на первую из наших групп четности, и на вторую, и на третью, но не на последнюю.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "Таким образом, чтение результатов этих четырех проверок снизу вверх действительно определяет положение ошибки.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "Размещая эти двоичные метки обратно в коробки, позвольте мне подчеркнуть, что они отличаются от фактически отправляемых данных.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "Это не что иное, как концептуальный ярлык, который поможет вам и мне понять, откуда взялись четыре группы паритета.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "Элегантность того, что все, на что мы смотрим, описывается в двоичном формате, возможно, подрывается путаницей, связанной с тем, что все, на что мы смотрим, описывается в двоичном формате.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "То, что мы получаем, — это первая из наших четырех групп четности, что означает, что вы можете интерпретировать эту первую проверку как вопрос: «Эй, если есть ошибка, последний бит в позиции этой ошибки равен 1?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "» И так далее.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "Во-первых, как систематически обобщать размеры блоков, превышающие степени двойки.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "Это означает, что каждый из этих битов четности находится внутри одной и только одной из четырех групп четности.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "Когда вы выполняете операцию XOR двух битов, она возвращает 1, если один из этих битов включен, но не если оба включены или выключены.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "Другими словами, это четность этих двух битов.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "Это как дополнение, но куда не понесешь.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "Ключевым моментом для нас с вами является то, что выполнение XOR множества различных битовых строк фактически является способом вычислить пародии на кучу отдельных групп, как это происходит со столбцами, одним махом.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "Это дает нам довольно привлекательный способ представить множественные проверки четности нашего алгоритма кода Хэмминга как объединенные в одну операцию.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "Имеет ли это смысл?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "Аналогично, в следующем столбце подсчитывается количество позиций во второй группе четности, позиций, предпоследний бит которых равен 1, которые также выделены и т. д.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "Теперь, когда у нас это получилось, это дает нам действительно хороший способ задуматься о том, почему эти четыре результирующих бита внизу непосредственно определяют положение ошибки.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "Допустим, какой-то бит в этом блоке переключается с 0 на 1.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "Это означает, что позиция этого бита теперь будет включена в общее исключающее ИЛИ, что изменит сумму с 0 на новое включенное значение, позицию ошибки.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "Мы начнем с создания случайного массива из 16 единиц и нулей для имитации блока данных, и я дам ему биты имени, но, конечно, на практике это будет то, что мы получаем от отправителя, и вместо будучи случайным, он будет нести 11 бит данных вместе с 5 битами четности.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "Итак, если мы затем создадим список, который будет циклически перебирать все эти пары, пары, которые выглядят как i, а затем мы вытащим только значение i, только индекс, ну, это не так уж и интересно, мы просто вернем эти индексы от 0 до 15.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "Но если мы добавим условие делать это только в том случае, если бит, то есть, если этот бит равен 1, а не 0, то тогда будут выбраны только те позиции, где включен соответствующий бит.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "Мы не будем этого делать здесь, но вы можете написать функцию, в которой отправитель использует это двоичное представление для установки четырех битов четности по мере необходимости, в конечном итоге переводя этот блок в состояние, при котором выполнение этой строки кода для полного списка бит возвращает результат. 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "Что круто, так это то, что если мы переключим любой из битов в этом списке, имитируя случайную ошибку из-за шума, то если вы запустите ту же строку кода, она выведет эту ошибку.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "Вы можете получить этот блок из ниоткуда, запустить к нему эту единственную строку, и он автоматически выдаст позицию ошибки или 0, если ее не было.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "Теперь, в зависимости от вашего опыта работы с двоичными файлами, операциями XOR и программным обеспечением в целом, вы можете найти эту точку зрения либо немного запутанной, либо настолько более элегантной и простой, что вы задаетесь вопросом, почему мы просто не начали с нее с самого начала. -идти.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "Грубо говоря, о перспективе множественной проверки четности легче думать при прямой реализации кодов Хэмминга в аппаратном обеспечении, а о перспективе XOR легче всего думать, когда она выполняется в программном обеспечении, на более высоком уровне.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "Важным фактом здесь является то, что эта информация напрямую соответствует тому, какая избыточность нам нужна.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "И, кстати, есть совершенно другой способ представления кодов Хэмминга: вы умножаете сообщение на одну большую матрицу.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "Говоря о масштабировании, вы можете заметить, что эффективность этой схемы становится только выше по мере увеличения размера блока.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "Например, мы увидели, что при 256 битах вы используете только 3% этого пространства для избыточности, и с этого момента ситуация становится все лучше.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "По мере того как количество битов четности увеличивается один за другим, размер блока продолжает удваиваться.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "А если довести это до крайности, то у вас может получиться блок, скажем, в миллион битов, в котором вы буквально будете разыгрывать 20 вопросов с проверками на четность, и он использует только 21 бит четности.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "Проблема, конечно, в том, что при увеличении блока вероятность увидеть более одного или двух битовых ошибок возрастает, а коды Хэмминга ничего сверх этого не обрабатывают.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "Но это тема для другого раза.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "В своей книге «Искусство заниматься наукой и инженерией» Хэмминг удивительно откровенно рассказывает о том, насколько запутанным было его открытие этого кода.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "В этой книге он примерно полдюжины раз ссылается на цитату Луи Пастера: удача любит подготовленный ум.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "Но не стоит обманывать себя, думая, что они на самом деле очевидны, потому что это определенно не так.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "Это была та же основополагающая статья, которая в определенном смысле показала, что эффективное исправление ошибок всегда возможно, независимо от того, насколько высока вероятность переворота битов, по крайней мере теоретически.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "Перенесемся на несколько десятилетий вперед, и в наши дни многие из нас настолько погружены в размышления о битах и информации, что легко упустить из виду, насколько особенным был этот образ мышления.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/spanish/sentence_translations.json b/2020/hamming-codes-2/spanish/sentence_translations.json index 2e1d815fe..2cf2f48d9 100644 --- a/2020/hamming-codes-2/spanish/sentence_translations.json +++ b/2020/hamming-codes-2/spanish/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "Estábamos hablando de códigos Hamming, una forma de crear un bloque de datos donde la mayoría de los bits llevan un mensaje significativo, mientras que algunos otros actúan como una especie de redundancia, de tal manera que si se voltea algún bit, ya sea un mensaje bit o un bit de redundancia, cualquier cosa en este bloque, un receptor podrá identificar que hubo un error y cómo solucionarlo.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "La idea básica presentada allí fue cómo utilizar múltiples comprobaciones de paridad para realizar una búsqueda binaria hasta llegar al error.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "Por ejemplo, el número 7 en binario se parece a 0111, lo que básicamente dice que es 4 más 2 más 1.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "Y observen dónde se ubica la posición 7, afecta al primero de nuestros grupos de paridad, al segundo y al tercero, pero no al último.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "Entonces, leer los resultados de esas cuatro comprobaciones de abajo hacia arriba sí explica la posición del error.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "Mientras volvemos a colocar estas etiquetas binarias en sus cajas, permítanme enfatizar que son distintas de los datos que realmente se envían.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "No son más que una etiqueta conceptual para ayudarnos a usted y a mí a comprender de dónde provienen los cuatro grupos de paridad.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "La elegancia de que todo lo que estamos viendo se describa en binario tal vez se vea socavada por la confusión de que todo lo que estamos viendo se describa en binario.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "Lo que obtenemos es el primero de nuestros cuatro grupos de paridad, lo que significa que puedes interpretar esa primera verificación como si preguntaras, oye, si hay un error, ¿el último bit en la posición de ese error es 1?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "Etcétera.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "La primera es cómo generalizar sistemáticamente a tamaños de bloques que son potencias de dos mayores.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "Lo que eso significa es que cada uno de esos bits de paridad se encuentra dentro de uno y sólo uno de los cuatro grupos de paridad.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "Cuando tomas el XOR de dos bits, devolverá un 1 si cualquiera de esos bits está activado, pero no si ambos están activados o desactivados.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "Dicho de otra manera, es la paridad de estos dos bits.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "Es como una suma, pero donde nunca se lleva.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "El punto clave para usted y para mí es que tomar el XOR de muchas cadenas de bits diferentes es efectivamente una forma de calcular las parodias de un grupo de grupos separados, como ocurre con las columnas, todo de una sola vez.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "Esto nos da una forma bastante elegante de pensar en las múltiples comprobaciones de paridad de nuestro algoritmo de código Hamming como si estuvieran todas empaquetadas en una sola operación.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "¿Tiene sentido?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "Asimismo, la siguiente columna cuenta cuántas posiciones hay en el segundo grupo de paridad, las posiciones cuyo penúltimo bit es un 1, y cuáles también están resaltadas, y así sucesivamente.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "Ahora, una vez que lo tenemos así, nos da una muy buena manera de pensar por qué estos cuatro bits resultantes en la parte inferior explican directamente la posición de un error.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "Digamos que una parte de este bloque se cambia de 0 a 1.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "Lo que eso significa es que la posición de ese bit ahora se incluirá en el XOR total, lo que cambia la suma de 0 a ser este valor recién incluido, la posición del error.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "Comenzaremos creando una matriz aleatoria de 16 1 y 0 para simular el bloque de datos, y le daré el nombre de bits, pero, por supuesto, en la práctica esto sería algo que recibiríamos de un remitente, y en lugar de al ser aleatorio, transportaría 11 bits de datos junto con 5 bits de paridad.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "Entonces, si luego creamos una lista que recorre todos estos pares, pares que se parecen a i, y luego extraemos solo el valor de i, solo el índice, bueno, no es tan emocionante, simplemente recuperamos esos índices del 0 al 15.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "Pero si agregamos la condición de hacer esto solo si el bit, es decir, si ese bit es un 1 y no un 0, entonces extrae solo las posiciones donde el bit correspondiente está activado.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "No lo haremos aquí, pero podría escribir una función en la que el remitente use esa representación binaria para establecer los cuatro bits de paridad según sea necesario y, en última instancia, llevar este bloque a un estado en el que la ejecución de esta línea de código en la lista completa de bits devuelva un 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "Lo bueno es que si alternamos cualquiera de los bits en esta lista, simulando un error aleatorio debido al ruido, si ejecuta esta misma línea de código, imprimirá ese error.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "Podrías obtener este bloque de la nada, ejecutar esta única línea en él y automáticamente mostrará la posición de un error, o un 0 si no hubo ninguno.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "Ahora, dependiendo de su comodidad con los binarios, los XOR y el software en general, puede encontrar esta perspectiva un poco confusa o mucho más elegante y simple que se pregunte por qué no comenzamos con ella desde el principio. -ir.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "En términos generales, es más fácil pensar en la perspectiva de verificación de paridad múltiple cuando se implementan códigos Hamming en hardware de manera muy directa, y es más fácil pensar en la perspectiva XOR cuando se hace en software, desde un nivel superior.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "El hecho relevante aquí es que esa información corresponde directamente a cuánta redundancia necesitamos.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "Y luego, por cierto, existe otra forma completamente distinta en la que a veces se presentan los códigos Hamming, donde se multiplica el mensaje por una gran matriz.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "Y hablando de escalamiento, es posible que notes que la eficiencia de este esquema solo mejora a medida que aumentamos el tamaño del bloque.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "Por ejemplo, vimos que con 256 bits, se utiliza solo el 3% de ese espacio para redundancia, y a partir de ahí sigue mejorando.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "A medida que el número de bits de paridad crece uno por uno, el tamaño del bloque se sigue duplicando.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "Y si lleva eso al extremo, podría tener un bloque con, digamos, un millón de bits, donde literalmente estaría jugando 20 preguntas con sus comprobaciones de paridad, y utiliza sólo 21 bits de paridad.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "El problema, por supuesto, es que con un bloque más grande, la probabilidad de ver más de uno o dos errores de bit aumenta, y los códigos Hamming no manejan nada más allá de eso.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "Pero ese es un tema para otro momento.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "En su libro El arte de hacer ciencia e ingeniería, Hamming es maravillosamente sincero acerca de cuán sinuoso fue su descubrimiento de este código.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "Hay como media docena de veces a lo largo de este libro en las que hace referencia a la cita de Louis Pasteur: La suerte favorece a una mente preparada.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "Pero no deberías engañarte pensando que en realidad son obvios, porque definitivamente no lo son.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "Este fue el mismo artículo fundamental que demostró, en cierto sentido, que siempre es posible una corrección de errores eficiente, sin importar cuán alta sea la probabilidad de que se produzcan cambios de bit, al menos en teoría.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "Varias décadas después, hoy en día muchos de nosotros estamos tan inmersos en pensar en bits e información que es fácil pasar por alto cuán distinta era esta forma de pensar.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/tamil/sentence_translations.json b/2020/hamming-codes-2/tamil/sentence_translations.json index d28b1aa4f..7beb32dbf 100644 --- a/2020/hamming-codes-2/tamil/sentence_translations.json +++ b/2020/hamming-codes-2/tamil/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "நாங்கள் ஹேமிங் குறியீடுகளைப் பற்றி பேசிக் கொண்டிருந்தோம், பெரும்பாலான பிட்கள் அர்த்தமுள்ள செய்தியைக் கொண்டு செல்லும் தரவுத் தொகுதியை உருவாக்கும் ஒரு வழி பிட் அல்லது ஒரு பணிநீக்கம் பிட், இந்த பிளாக்கில் உள்ள எதையும், ஒரு பெறுநரால் பிழை இருப்பதை அடையாளம் காண முடியும், அதை எவ்வாறு சரிசெய்வது.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "அங்கு வழங்கப்பட்ட அடிப்படை யோசனையானது, பிழைக்கான உங்கள் வழியை பைனரி தேடுவதற்கு பல சமநிலை சரிபார்ப்புகளை எவ்வாறு பயன்படுத்துவது என்பதுதான்.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "எடுத்துக்காட்டாக, பைனரியில் உள்ள எண் 7 0111 போல் தெரிகிறது, இது 4 கூட்டல் 2 கூட்டல் 1 என்று கூறுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "மற்றும் நிலை 7 எங்கு அமர்ந்திருக்கிறது என்பதைக் கவனியுங்கள், இது எங்கள் சமத்துவக் குழுக்களில் முதலாவது மற்றும் இரண்டாவது மற்றும் மூன்றாவது, ஆனால் கடைசியாக அல்ல.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "எனவே அந்த நான்கு காசோலைகளின் முடிவுகளை கீழிருந்து மேல் வரை படிப்பது பிழையின் நிலையை வெளிப்படுத்துகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "இந்த பைனரி லேபிள்களை அவற்றின் பெட்டிகளில் மீண்டும் வைக்கும்போது, அவை உண்மையில் அனுப்பப்படும் தரவுகளிலிருந்து வேறுபட்டவை என்பதை வலியுறுத்துகிறேன்.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "நான்கு சமத்துவக் குழுக்கள் எங்கிருந்து வந்தன என்பதைப் புரிந்துகொள்ள உங்களுக்கும் எனக்கும் உதவும் ஒரு கருத்தியல் லேபிளைத் தவிர வேறொன்றுமில்லை.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "நாம் பார்க்கும் அனைத்தும் பைனரியில் விவரிக்கப்படுவதன் நேர்த்தியானது, நாம் பார்க்கும் அனைத்தும் பைனரியில் விவரிக்கப்பட வேண்டும் என்ற குழப்பத்தால் குறைக்கப்படலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "நாங்கள் பெறுவது எங்களின் நான்கு சமத்துவக் குழுக்களில் முதன்மையானது, அதாவது அந்த முதல் காசோலையைக் கேட்பது போல் நீங்கள் விளக்கலாம், ஏய், பிழை இருந்தால், அந்த பிழையின் நிலையில் இறுதி பிட் 1 உள்ளதா?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "மற்றும் பல.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "முதலாவதாக, இரண்டின் பெரிய சக்திகளைக் கொண்ட தொகுதி அளவுகளை முறையாகப் பொதுமைப்படுத்துவது எப்படி.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "இதன் பொருள் என்னவென்றால், அந்த ஒவ்வொரு பாரிட்டி பிட்களும் ஒன்றுக்குள் அமர்ந்து நான்கு சமத்துவக் குழுக்களில் ஒன்று மட்டுமே.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "நீங்கள் இரண்டு பிட்களின் XOR ஐ எடுக்கும்போது, அந்த பிட்களில் ஏதேனும் ஒன்று இயக்கப்பட்டிருந்தால் அது 1ஐத் தரும், ஆனால் இரண்டும் ஆன் அல்லது ஆஃப் செய்யப்பட்டிருந்தால் அல்ல.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "வித்தியாசமாக சொற்றொடர், இந்த இரண்டு பிட்களின் சமநிலை.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "இது கூட்டல் போன்றது, ஆனால் நீங்கள் எடுத்துச் செல்லவே இல்லை.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "உங்களுக்கும் எனக்கும் முக்கியமான விஷயம் என்னவென்றால், பல வேறுபட்ட பிட் சரங்களின் XORஐ எடுத்துக்கொள்வது, தனித்தனி குழுக்களின் கேலிக்கூத்துகளை கணக்கிடுவதற்கான ஒரு வழியாகும்.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "எங்கள் ஹேமிங் குறியீடு அல்காரிதத்தில் இருந்து பல சமநிலை சரிபார்ப்புகளைப் பற்றி சிந்திக்க இது எங்களுக்கு மிகவும் எளிமையான வழியை வழங்குகிறது, ஏனெனில் அனைத்தும் ஒன்றாக ஒரே செயல்பாட்டில் தொகுக்கப்படுகின்றன.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "அதில் ஏதாவது பொருளிருக்கிறதா? அதில் அர்த்தமிருக்கிறதா?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "அதேபோல, அடுத்த நெடுவரிசை, இரண்டாவது சமத்துவக் குழுவில் எத்தனை நிலைகள் உள்ளன, இரண்டாவது முதல் கடைசி பிட் வரையிலான நிலைகள் 1, மற்றும் அவையும் தனிப்படுத்தப்பட்டவை மற்றும் பலவற்றைக் கணக்கிடுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "இப்போது நாம் இதைப் பெற்றவுடன், கீழே உள்ள இந்த நான்கு பிட்கள் ஏன் பிழையின் நிலையை நேரடியாக உச்சரிக்கின்றன என்பதைப் பற்றி சிந்திக்க இது ஒரு நல்ல வழியை வழங்குகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "இந்த பிளாக்கில் சில பிட்கள் 0 இலிருந்து 1 ஆக மாறுகிறது என்று வைத்துக்கொள்வோம்.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "இதன் பொருள் என்னவென்றால், அந்த பிட்டின் நிலை இப்போது மொத்த XOR இல் சேர்க்கப்படும், இது கூட்டுத்தொகையை 0 என்பதிலிருந்து புதிதாக சேர்க்கப்பட்ட இந்த மதிப்பாக மாற்றுகிறது, பிழையின் நிலை.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "தரவுத் தொகுதியை உருவகப்படுத்த 16 1s மற்றும் 0s என்ற சீரற்ற வரிசையை உருவாக்குவதன் மூலம் தொடங்குவோம், நான் அதற்கு பெயர் பிட்களை தருகிறேன், ஆனால் நடைமுறையில் இது ஒரு அனுப்புநரிடமிருந்து நாம் பெறும் ஒன்று, அதற்கு பதிலாக சீரற்றதாக இருந்தால், அது 11 டேட்டா பிட்களையும் 5 பேரிட்டி பிட்களையும் கொண்டு செல்லும்.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "எனவே, இந்த ஜோடிகள் அனைத்தின் மீதும் சுழலும் ஒரு பட்டியலை உருவாக்கினால், i போல தோற்றமளிக்கும் ஜோடிகள், பின்னர் i மதிப்பை, குறியீட்டை மட்டும் வெளியே இழுத்தால், அது அவ்வளவு உற்சாகமாக இல்லை, அந்த குறியீடுகளை 0 முதல் 15 வரை திரும்பப் பெறுவோம்.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "ஆனால் பிட் என்றால் மட்டுமே இதைச் செய்ய வேண்டும் என்ற நிபந்தனையைச் சேர்த்தால், அதாவது அந்த பிட் 1 மற்றும் 0 அல்ல என்றால், அது தொடர்புடைய பிட் இயக்கப்பட்டிருக்கும் நிலைகளை மட்டும் இழுக்கிறது.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "நாங்கள் அதை இங்கே செய்ய மாட்டோம், ஆனால் அனுப்புநர் அந்த பைனரி பிரதிநிதித்துவத்தைப் பயன்படுத்தி நான்கு பேரிட்டி பிட்களை தேவைக்கேற்ப அமைக்கும் ஒரு செயல்பாட்டை நீங்கள் எழுதலாம், இறுதியில் இந்தத் தொகுதியை பிட்களின் முழுப் பட்டியலிலும் இந்த குறியீட்டு வரியை இயக்கும் நிலைக்கு கொண்டு செல்லலாம். ஒரு 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "நல்ல விஷயம் என்னவென்றால், இந்த பட்டியலில் உள்ள பிட்களில் ஏதேனும் ஒன்றை நாம் மாற்றினால், இரைச்சலில் இருந்து ஒரு சீரற்ற பிழையை உருவகப்படுத்தினால், நீங்கள் இதே குறியீட்டை இயக்கினால், அது அந்த பிழையை அச்சிடுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "இந்தத் தொகுதியை நீங்கள் நீல நிறத்தில் இருந்து பெறலாம், இந்த ஒற்றை வரியை அதில் இயக்கலாம், மேலும் அது பிழையின் நிலையை தானாகவே துப்பிவிடும், அல்லது எதுவும் இல்லை என்றால் 0.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "இப்போது, பைனரி மற்றும் XORகள் மற்றும் பொதுவாக மென்பொருளில் உள்ள உங்கள் வசதியைப் பொறுத்து, இந்த முன்னோக்கைக் கொஞ்சம் குழப்பமாகவோ அல்லது மிகவும் நேர்த்தியாகவும் எளிமையாகவும் நாம் ஏன் தொடங்கவில்லை என்று நீங்கள் ஆச்சரியப்படுகிறீர்கள். -போ.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "தளர்வாகச் சொன்னால், வன்பொருளில் ஹேமிங் குறியீடுகளை நேரடியாகச் செயல்படுத்தும் போது, மல்டிபிள் பேரிட்டி காசோலைக் கண்ணோட்டத்தைப் பற்றி சிந்திக்க எளிதானது, மேலும் XOR முன்னோக்கு மென்பொருளில், உயர் மட்டத்தில் இருந்து அதைச் செய்யும்போது சிந்திக்க எளிதானது.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "இங்கே பொருத்தமான உண்மை என்னவென்றால், அந்த தகவல் நமக்கு எவ்வளவு பணிநீக்கம் தேவை என்பதை நேரடியாக ஒத்துள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "பின்னர், இந்த முழு வேறு வழியும் உள்ளது, சில சமயங்களில் ஹேமிங் குறியீடுகள் வழங்கப்படுகின்றன, அங்கு நீங்கள் செய்தியை ஒரு பெரிய மேட்ரிக்ஸால் பெருக்குகிறீர்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "மற்றும் அளவிடுதல் பற்றி பேசுகையில், தொகுதி அளவை அதிகரிக்கும்போது மட்டுமே இந்த திட்டத்தின் செயல்திறன் சிறப்பாக இருக்கும் என்பதை நீங்கள் கவனிக்கலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "எடுத்துக்காட்டாக, 256 பிட்கள் மூலம், நீங்கள் பணிநீக்கத்திற்கு அந்த இடத்தில் 3% மட்டுமே பயன்படுத்துகிறீர்கள் என்பதை நாங்கள் பார்த்தோம், மேலும் அது அங்கிருந்து சிறப்பாக வருகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "சமநிலை பிட்களின் எண்ணிக்கை ஒவ்வொன்றாக வளரும்போது, தொகுதி அளவு இரட்டிப்பாகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "நீங்கள் அதை ஒரு தீவிரத்திற்கு எடுத்துக் கொண்டால், நீங்கள் ஒரு மில்லியன் பிட்கள் கொண்ட ஒரு தொகுதியை வைத்திருக்கலாம், அங்கு நீங்கள் உண்மையில் உங்கள் சமநிலை சரிபார்ப்புகளுடன் 20 கேள்விகளை விளையாடுவீர்கள், மேலும் அது 21 சமநிலை பிட்களை மட்டுமே பயன்படுத்துகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "பிரச்சனை, நிச்சயமாக, ஒரு பெரிய தொகுதியுடன், ஒன்று அல்லது இரண்டு பிட் பிழைகளை பார்க்கும் நிகழ்தகவு அதிகரிக்கிறது, மேலும் ஹேமிங் குறியீடுகள் அதைத் தாண்டி எதையும் கையாளாது.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "ஆனால் அது மற்றொரு நேரத்திற்கு ஒரு தலைப்பு.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "அவரது தி ஆர்ட் ஆஃப் டூயிங் சயின்ஸ் அண்ட் இன்ஜினியரிங் என்ற புத்தகத்தில், ஹாமிங் இந்த குறியீட்டின் கண்டுபிடிப்பு எவ்வளவு வளைந்திருந்தது என்பதைப் பற்றி அற்புதமாக நேர்மையாக கூறினார்.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "லூயிஸ் பாஸ்டர் மேற்கோள்களை அவர் குறிப்பிடும் இந்த புத்தகம் முழுவதும் அரை டஜன் முறைகள் உள்ளன, அதிர்ஷ்டம் தயாராக இருக்கும் மனதை ஆதரிக்கிறது.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "ஆனால் அவை உண்மையில் வெளிப்படையானவை என்று நினைத்து உங்களை நீங்களே ஏமாற்றிக் கொள்ளக்கூடாது, ஏனென்றால் அவை நிச்சயமாக இல்லை.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "ஒரு குறிப்பிட்ட அர்த்தத்தில், பிட் ஃபிளிப்புகளின் நிகழ்தகவு எவ்வளவு அதிகமாக இருந்தாலும், குறைந்தபட்சம் கோட்பாட்டில், திறமையான பிழை திருத்தம் எப்போதும் சாத்தியமாகும் என்பதைக் காட்டிய அதே அடிப்படைக் கட்டுரை இதுவாகும்.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "பல தசாப்தங்களாக வேகமாக முன்னேறி, இந்த நாட்களில், நம்மில் பலர் பிட்கள் மற்றும் தகவல்களைப் பற்றி சிந்திப்பதில் மூழ்கிவிட்டோம், இந்த சிந்தனை முறை எவ்வளவு வித்தியாசமானது என்பதைக் கவனிப்பது எளிது.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/telugu/sentence_translations.json b/2020/hamming-codes-2/telugu/sentence_translations.json index 0bd1a7f40..9ac9a55b3 100644 --- a/2020/hamming-codes-2/telugu/sentence_translations.json +++ b/2020/hamming-codes-2/telugu/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "మేము హామింగ్ కోడ్‌ల గురించి మాట్లాడుతున్నాము, చాలా బిట్‌లు అర్థవంతమైన సందేశాన్ని కలిగి ఉండే డేటా బ్లాక్‌ను సృష్టించే మార్గం, మరికొన్ని ఒక రకమైన రిడెండెన్సీగా పనిచేస్తాయి, ఆ విధంగా ఏదైనా బిట్ తిప్పబడితే, సందేశం బిట్ లేదా రిడెండెన్సీ బిట్, ఈ బ్లాక్‌లో ఏదైనా, రిసీవర్ లోపం ఉందని మరియు దాన్ని ఎలా పరిష్కరించాలో గుర్తించగలదు.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "బైనరీ శోధన కోసం బహుళ పారిటీ తనిఖీలను ఎలా ఉపయోగించాలి అనేది అక్కడ అందించబడిన ప్రాథమిక ఆలోచన.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "ఉదాహరణకు, బైనరీలో 7 సంఖ్య 0111 లాగా కనిపిస్తుంది, ముఖ్యంగా ఇది 4 ప్లస్ 2 ప్లస్ 1 అని చెబుతోంది.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "మరియు స్థానం 7 ఎక్కడ కూర్చుందో గమనించండి, ఇది మా సమానత్వ సమూహాలలో మొదటిదానిని ప్రభావితం చేస్తుంది మరియు రెండవది మరియు మూడవది, కానీ చివరిది కాదు.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "కాబట్టి ఆ నాలుగు చెక్‌ల ఫలితాలను దిగువ నుండి పైకి చదవడం నిజంగా లోపం యొక్క స్థితిని తెలియజేస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "మేము ఈ బైనరీ లేబుల్‌లను వాటి పెట్టెల్లోకి తిరిగి ఉంచినప్పుడు, అవి వాస్తవానికి పంపబడుతున్న డేటా నుండి విభిన్నంగా ఉన్నాయని నేను నొక్కిచెబుతున్నాను.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "నాలుగు సమానత్వ సమూహాలు ఎక్కడ నుండి వచ్చాయో అర్థం చేసుకోవడంలో మీకు మరియు నాకు సహాయం చేయడానికి అవి సంభావిత లేబుల్ తప్ప మరేమీ కాదు.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "మనం చూస్తున్న ప్రతిదీ బైనరీలో వర్ణించబడటం యొక్క సొగసైనది బహుశా మనం చూస్తున్న ప్రతిదాన్ని బైనరీలో వివరించడం వల్ల కలిగే గందరగోళం వల్ల తగ్గుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "మేము పొందేది మా నాలుగు సమానత్వ సమూహాలలో మొదటిది, అంటే మీరు ఆ మొదటి చెక్‌ని అడుగుతున్నట్లు అర్థం చేసుకోవచ్చు, హే, లోపం ఉన్నట్లయితే, ఆ లోపం యొక్క స్థానం 1గా ఉందా?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "మరియు అందువలన న.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "మొదటిది రెండు పెద్ద శక్తులు ఉండే పరిమాణాలను బ్లాక్ చేయడానికి క్రమపద్ధతిలో సాధారణీకరించడం ఎలా.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "దాని అర్థం ఏమిటంటే, ఆ పారిటీ బిట్‌లలో ప్రతి ఒక్కటి నాలుగు సమాన సమూహాలలో ఒకటి మాత్రమే ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "మీరు రెండు బిట్‌ల XORను తీసుకున్నప్పుడు, ఆ బిట్‌లలో ఒకదానిని ఆన్ చేసినట్లయితే అది 1ని తిరిగి ఇస్తుంది, కానీ రెండూ ఆన్ లేదా ఆఫ్ చేయబడితే కాదు.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "విభిన్నంగా పదబంధం, ఇది ఈ రెండు బిట్‌ల సమానత్వం.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "ఇది అదనంగా వంటిది, కానీ మీరు ఎక్కడికి తీసుకెళ్లలేరు.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "మీకు మరియు నాకు ముఖ్యమైన అంశం ఏమిటంటే, అనేక విభిన్న బిట్ స్ట్రింగ్‌ల యొక్క XOR తీసుకోవడం అనేది నిలువు వరుసల మాదిరిగానే, ప్రత్యేక సమూహాల యొక్క పేరడీలను గణించడానికి ఒక మార్గం.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "ఇది మా హామింగ్ కోడ్ అల్గారిథమ్ నుండి బహుళ పారిటీ చెక్‌ల గురించి ఆలోచించడానికి చాలా చురుకైన మార్గాన్ని అందిస్తుంది, ఎందుకంటే అన్నీ ఒకే ఆపరేషన్‌లో ప్యాక్ చేయబడతాయి.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "అది సమంజసమా?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "అదేవిధంగా, తదుపరి నిలువు వరుస రెండవ సమాన సమూహంలో ఎన్ని స్థానాలు ఉన్నాయి, రెండవ నుండి చివరి బిట్ 1 వరకు ఉన్న స్థానాలు మరియు హైలైట్ చేయబడినవి మరియు మొదలైనవి కూడా లెక్కించబడతాయి.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "ఇప్పుడు మనం దీన్ని ఇలా కలిగి ఉంటే, దిగువన ఉన్న ఈ నాలుగు ఫలిత బిట్‌లు నేరుగా లోపం యొక్క స్థానాన్ని ఎందుకు వివరిస్తాయి అనే దాని గురించి ఆలోచించడానికి ఇది మాకు నిజంగా మంచి మార్గాన్ని ఇస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "ఈ బ్లాక్‌లోని కొంత బిట్ 0 నుండి 1కి టోగుల్ చేయబడిందని అనుకుందాం.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "దీని అర్థం ఏమిటంటే, ఆ బిట్ యొక్క స్థానం ఇప్పుడు మొత్తం XORలో చేర్చబడుతుంది, ఇది మొత్తాన్ని 0 నుండి బదులుగా ఈ కొత్తగా చేర్చబడిన విలువ, లోపం యొక్క స్థానంగా మారుస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "మేము డేటా బ్లాక్‌ను అనుకరించడానికి 16 1సె మరియు 0 సె యాదృచ్ఛిక శ్రేణిని సృష్టించడం ద్వారా ప్రారంభిస్తాము మరియు నేను దానికి బిట్‌లను ఇస్తాను, అయితే ఆచరణలో ఇది మనం పంపినవారి నుండి స్వీకరించేదే అవుతుంది మరియు బదులుగా యాదృచ్ఛికంగా ఇది 5 పారిటీ బిట్‌లతో కలిపి 11 డేటా బిట్‌లను కలిగి ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "కాబట్టి మనం ఈ అన్ని జతలపై లూప్ చేసే జాబితాను సృష్టించినట్లయితే, i లాగా కనిపించే జంటలు, ఆపై మేము కేవలం i విలువను, కేవలం సూచికను తీసివేస్తే, అది అంత ఉత్తేజకరమైనది కాదు, మేము ఆ సూచికలను 0 నుండి 15 వరకు తిరిగి పొందుతాము.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "కానీ మనం దీన్ని బిట్ అయితే మాత్రమే చేయాలనే షరతును జోడిస్తే, అంటే ఆ బిట్ 1 మరియు 0 కాకపోతే, అది సంబంధిత బిట్ ఆన్ చేయబడిన స్థానాలను మాత్రమే బయటకు తీస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "మేము దీన్ని ఇక్కడ చేయము, కానీ పంపినవారు నాలుగు పారిటీ బిట్‌లను అవసరమైన విధంగా సెట్ చేయడానికి బైనరీ ప్రాతినిధ్యాన్ని ఉపయోగించే ఒక ఫంక్షన్‌ను మీరు వ్రాయవచ్చు, చివరికి ఈ బ్లాక్‌ని బిట్‌ల పూర్తి జాబితాలో ఈ లైన్ కోడ్‌ని అమలు చేసే స్థితికి చేరుకుంటుంది. ఒక 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "మంచి విషయం ఏమిటంటే, శబ్దం నుండి యాదృచ్ఛిక లోపాన్ని అనుకరిస్తూ, ఈ జాబితాలోని ఏదైనా బిట్‌లను మనం టోగుల్ చేస్తే, మీరు ఇదే లైన్ కోడ్‌ను అమలు చేస్తే, అది ఆ లోపాన్ని ముద్రిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "మీరు ఈ బ్లాక్‌ను నీలిరంగు నుండి పొందవచ్చు, దానిపై ఈ సింగిల్ లైన్‌ను అమలు చేయవచ్చు మరియు అది స్వయంచాలకంగా లోపం యొక్క స్థానం లేదా ఏదైనా లేకుంటే 0ని ఉమ్మివేస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "ఇప్పుడు, బైనరీ మరియు XORలు మరియు సాధారణంగా సాఫ్ట్‌వేర్‌తో మీ సౌకర్యాన్ని బట్టి, మీరు ఈ దృక్పథాన్ని కొంచెం గందరగోళంగా లేదా చాలా సొగసైన మరియు సరళంగా కనుగొనవచ్చు, మేము దీన్ని ఎందుకు ప్రారంభించలేదని మీరు ఆశ్చర్యపోతున్నారు. -వెళ్ళండి.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "వదులుగా చెప్పాలంటే, హార్డ్‌వేర్‌లో హామింగ్ కోడ్‌లను నేరుగా అమలు చేసేటప్పుడు బహుళ పారిటీ తనిఖీ దృక్పథం గురించి ఆలోచించడం సులభం, మరియు XOR దృక్పథాన్ని సాఫ్ట్‌వేర్‌లో చేసేటప్పుడు, ఉన్నత స్థాయి నుండి ఆలోచించడం చాలా సులభం.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "ఇక్కడ సంబంధిత వాస్తవం ఏమిటంటే, ఆ సమాచారం మనకు ఎంత రిడెండెన్సీ అవసరమో దానికి నేరుగా అనుగుణంగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "ఆపై, మార్గం ద్వారా, మీరు కొన్నిసార్లు హామింగ్ కోడ్‌లను ప్రదర్శించే ఈ మొత్తం ఇతర మార్గం ఉంది, ఇక్కడ మీరు సందేశాన్ని ఒక పెద్ద మ్యాట్రిక్స్ ద్వారా గుణిస్తారు.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "మరియు స్కేలింగ్ గురించి చెప్పాలంటే, మేము బ్లాక్ పరిమాణాన్ని పెంచుతున్నప్పుడు మాత్రమే ఈ పథకం యొక్క సామర్థ్యం మెరుగుపడుతుందని మీరు గమనించవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "ఉదాహరణకు, 256 బిట్‌లతో, మీరు రిడెండెన్సీ కోసం ఆ స్థలంలో 3% మాత్రమే ఉపయోగిస్తున్నారని మేము చూశాము మరియు అది అక్కడ నుండి మెరుగుపడుతోంది.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "పారిటీ బిట్‌ల సంఖ్య ఒక్కొక్కటిగా పెరుగుతున్న కొద్దీ, బ్లాక్ పరిమాణం రెట్టింపు అవుతూ ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "మరియు మీరు దానిని విపరీతంగా తీసుకుంటే, మీరు మిలియన్ బిట్‌లతో బ్లాక్‌ను కలిగి ఉండవచ్చు, ఇక్కడ మీరు మీ పారిటీ తనిఖీలతో అక్షరాలా 20 ప్రశ్నలను ప్లే చేస్తారు మరియు ఇది 21 పారిటీ బిట్‌లను మాత్రమే ఉపయోగిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "సమస్య ఏమిటంటే, పెద్ద బ్లాక్‌తో, ఒకటి లేదా రెండు కంటే ఎక్కువ బిట్ ఎర్రర్‌లను చూసే సంభావ్యత పెరుగుతుంది మరియు హామింగ్ కోడ్‌లు అంతకు మించి దేనినీ నిర్వహించవు.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "కానీ అది మరొక సారి చర్చనీయాంశం.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "అతని పుస్తకం ది ఆర్ట్ ఆఫ్ డూయింగ్ సైన్స్ అండ్ ఇంజినీరింగ్‌లో, హామింగ్ ఈ కోడ్‌ని తన కనిపెట్టడం ఎంత మెలికలు తిరిగిపోయిందో అద్భుతంగా చెప్పాడు.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "అతను లూయిస్ పాశ్చర్ కోట్‌ను ప్రస్తావించిన అరడజను సార్లు ఈ పుస్తకంలో ఉన్నాయి, అదృష్టం సిద్ధమైన మనస్సుకు అనుకూలంగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "అయితే అవి స్పష్టంగా కనిపిస్తున్నాయని భావించి మిమ్మల్ని మీరు మోసం చేసుకోకూడదు, ఎందుకంటే అవి ఖచ్చితంగా కావు.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "బిట్ ఫ్లిప్‌ల సంభావ్యత ఎంత ఎక్కువగా ఉన్నా, కనీసం థియరీలో అయినా సమర్థవంతమైన లోపాన్ని సరిదిద్దడం ఎల్లప్పుడూ సాధ్యమేనని ఒక నిర్దిష్ట కోణంలో చూపించిన అదే పునాది పేపర్.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "కొన్ని దశాబ్దాలుగా ఫాస్ట్ ఫార్వార్డ్, మరియు ఈ రోజుల్లో, మనలో చాలా మంది బిట్స్ మరియు సమాచారం గురించి ఆలోచించడంలో మునిగిపోయారు, ఈ ఆలోచనా విధానం ఎంత విభిన్నంగా ఉందో పట్టించుకోవడం సులభం.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/thai/sentence_translations.json b/2020/hamming-codes-2/thai/sentence_translations.json index 2c41571c4..9f666cba2 100644 --- a/2020/hamming-codes-2/thai/sentence_translations.json +++ b/2020/hamming-codes-2/thai/sentence_translations.json @@ -32,7 +32,7 @@ "end": 34.6 }, { - "input": "But as you start to think about actually implementing this, either in software or hardware, that framing may actually undersell how elegant these codes really are. ", + "input": "hat there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error. In that video, the goal was to make Hamming codes feel as hands-on and rediscoverable as possible. But as ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. ", + "input": "you read out the answers to the four parity checks we did in the last video, all as ones and zeros instead of yeses and nos, it literally spells out ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error. ", + "input": "century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day. The goal here is to give you a very thorough understanding of one of the earlie ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 87.54 }, { - "input": "There's nothing special about the example 7, this works in general, and this makes the logic for implementing the whole scheme in hardware shockingly simple. ", + "input": "st examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity groups, and the second, and the third, but not the last. So reading the results of those four checks ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 132.36 }, { - "input": "It's worth it, though. ", + "input": "0, let's write them all in binary, running from 0000 up to 1111. ask feels at the star ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 166.16 }, { - "input": "In other words, that second check is asking, hey, me again, if there's an error, is the second to last bit of that position a 1? ", + "input": "ly spelled words. Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 188.74 }, { - "input": "Everything we did earlier is the same as answering these four questions, which in turn is the same as spelling out a position in binary. ", + "input": "everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 197.74 }, { - "input": "I hope this makes two things clearer. ", + "input": "It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where tha ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two. ", + "input": "t final bit is a 1. What we get is the first of our four parity groups, which means that you can interpret that first check as asking, hey, if there's an err ", "translatedText": "ฉันเดาว่าทุกคนที่นี่มาจากภาค 1 เรากำลังพูดถึงโค้ด Hamming ซึ่งเป็นวิธีในการสร้างบล็อกข้อมูลที่บิตส่วนใหญ่มีข้อความที่มีความหมาย ในขณะที่บิตอื่นๆ บางส่วนทำหน้าที่เป็นการซ้ำซ้อน ในลักษณะที่ว่าหากบิตใด ๆ พลิกกลับ ข้อความก็อาจเป็นข้อความหนึ่งก็ได้ บิตหรือบิตสำรอง อะไรก็ตามในบล็อกนี้ ผู้รับจะสามารถระบุได้ว่ามีข้อผิดพลาด และวิธีการแก้ไข แนวคิดพื้นฐานที่นำเสนอคือวิธีใช้การตรวจสอบพาริตีหลายรายการเพื่อค้นหาแบบไบนารี่ไปจนถึงข้อผิดพลาด ในวิดีโอนั้นเป้าหมายคือการทำให้โค้ด Hamming รู้สึกเหมือนได้ลงมือปฏิบัติจริงและสามารถค้นพบใหม่ได้มากที่สุดเท่าที่จะเป็นไปได้ แต่เมื่อคุณเริ่มคิดถึงการนำสิ่งนี้ไปใช้จริง ทั้งในซอฟต์แวร์หรือฮาร์ดแวร์ เฟรมนั้นอาจตอกย้ำว่าโค้ดเหล่านี้สวยงามเพียงใด คุณอาจคิดว่าคุณจำเป็นต้องเขียนอัลกอริทึมที่ติดตามตำแหน่งข้อผิดพลาดที่เป็นไปได้ทั้งหมด และลดกลุ่มนั้นลงครึ่งหนึ่งในการตรวจสอบแต่ละครั้ง แต่จริงๆ แล้วเป็นวิธีที่ง่ายกว่านั้นมาก หากคุณอ่านคำตอบของการตรวจสอบความเท่าเทียมกันทั้งสี่ครั้งที่เราทำในวิดีโอที่แล้ว ทั้งหมดเป็น 1 และ 0 แทนที่จะเป็นใช่และไม่ใช่ มันจะสะกดตำแหน่งของข้อผิดพลาดในรูปแบบไบนารี่ ตัวอย่างเช่น เลข 7 ในไบนารี่ดูเหมือน 0111 โดยพื้นฐานแล้วบอกว่ามันคือ 4 บวก 2 บวก 1 และสังเกตว่าตำแหน่งที่ 7 อยู่ที่ใด มันจะส่งผลต่อกลุ่มพาริตีกลุ่มแรกของเรา และกลุ่มที่สองและกลุ่มที่สาม แต่ไม่ใช่กลุ่มสุดท้าย ดังนั้นการอ่านผลการตรวจสอบทั้งสี่ครั้งจากล่างขึ้นบนจะช่วยระบุตำแหน่งของข้อผิดพลาดได้อย่างแน่นอน ไม่มีอะไรพิเศษเกี่ยวกับตัวอย่างที่ 7 ซึ่งใช้งานได้โดยทั่วไป และทำให้ตรรกะในการใช้โครงร่างทั้งหมดในฮาร์ดแวร์เป็นเรื่องง่ายอย่างน่าตกใจ ตอนนี้ หากคุณต้องการดูว่าเหตุใดเหตุการณ์มหัศจรรย์นี้จึงเกิดขึ้น ให้นำป้ายกำกับดัชนีทั้ง 16 รายการสำหรับตำแหน่งของเรา แต่แทนที่จะเขียนเป็นฐาน 10 ให้เขียนทั้งหมดในรูปแบบไบนารี่ โดยเริ่มจาก 0000 ถึง 1111 ขณะที่เราใส่ป้ายกำกับไบนารี่เหล่านี้กลับเข้าไปในกล่อง ฉันขอย้ำว่าป้ายเหล่านี้แตกต่างจากข้อมูลที่ถูกส่งจริง สิ่งเหล่านี้เป็นเพียงป้ายกำกับแนวคิดที่จะช่วยให้คุณและฉันเข้าใจว่ากลุ่มความเท่าเทียมกันทั้งสี่มาจากไหน ความสง่างามของการมีทุกสิ่งที่เรากำลังดูถูกอธิบายในรูปแบบไบนารี่อาจถูกลดทอนลงด้วยความสับสนของการมีทุกสิ่งที่เรากำลังดูถูกอธิบายในรูปแบบไบนารี มันก็คุ้มค่านะ มุ่งความสนใจของคุณไปที่ส่วนสุดท้ายของป้ายกำกับเหล่านี้ทั้งหมด จากนั้นไฮไลต์ตำแหน่งที่บิตสุดท้ายคือ 1 สิ่งที่เราได้รับคือกลุ่มพาริตีกลุ่มแรกจากสี่กลุ่ม ซึ่งหมายความว่าคุณสามารถตีความการตรวจสอบครั้งแรกเป็นการถามว่า เฮ้ หากมีข้อผิดพลาด บิตสุดท้ายในตำแหน่งของข้อผิดพลาดนั้นเป็น 1 หรือไม่ ในทำนองเดียวกัน หากคุณเน้นที่บิตที่สองถึงบิตสุดท้าย และเน้นตำแหน่งทั้งหมดที่เป็น 1 คุณจะได้กลุ่มแพริตีที่สองจากแผนของเรา กล่าวอีกนัยหนึ่ง การตรวจสอบครั้งที่สองจะถามว่า เฮ้ ฉันอีกครั้ง หากมีข้อผิดพลาด บิตที่สองจากบิตสุดท้ายของตำแหน่งนั้นเป็น 1 หรือไม่ และอื่นๆ การตรวจสอบพาริตี้ครั้งที่สามครอบคลุมทุกตำแหน่งที่เปิดบิตที่สามถึงสุดท้าย และการตรวจสอบสุดท้ายครอบคลุมแปดตำแหน่งสุดท้าย ซึ่งบิตลำดับสูงสุดคือ 1 ทุกสิ่งที่เราทำก่อนหน้านี้เหมือนกับการตอบคำถามสี่ข้อนี้ ซึ่งในทางกลับกันก็เหมือนกับการสะกดตำแหน่งในรูปแบบไบนารี่ ฉันหวังว่านี่จะทำให้สองสิ่งชัดเจนขึ้น ประการแรกคือวิธีการสรุปอย่างเป็นระบบเกี่ยวกับขนาดบล็อกที่มีพลังมากกว่าสอง หากต้องใช้บิตมากกว่าในการอธิบายแต่ละตำแหน่ง เช่น หกบิตเพื่ออธิบาย 64 จุด แต่ละบิตเหล่านั้นจะให้หนึ่งในกลุ่มพาริตีที่เราต้องตรวจสอบ บรรดาผู้ที่ดูปริศนากระดานหมากรุกที่ฉันทำกับแมตต์ ปาร์กเกอร์อาจพบว่าทั้งหมดนี้คุ้นเคยอย่างยิ่ง มันเป็นตรรกะหลักเดียวกัน แต่แก้ปัญหาที่แตกต่าง และนำไปใช้กับกระดานหมากรุกขนาด 64 สี่เหลี่ยม สิ่งที่สองที่ฉันหวังว่าสิ่งนี้จะทำให้ชัดเจนคือเหตุใดแพริตีบิตของเราจึงนั่งอยู่ในตำแหน่งที่เป็นกำลังของสอง เช่น 1, 2, 4 และ 8 ตำแหน่งเหล่านี้คือตำแหน่งที่การแสดงไบนารี่เปิดขึ้นเพียงเล็กน้อย นั่นหมายความว่าแต่ละแพริตีบิตเหล่านั้นอยู่ภายในกลุ่มพาริตีเพียงกลุ่มเดียวจากสี่กลุ่มเท่านั้น คุณยังสามารถดูสิ่งนี้ได้ในตัวอย่างที่ใหญ่กว่า โดยไม่ว่าคุณจะได้ขนาดใหญ่แค่ไหน แต่ละบิตของพาริตีจะแตะเพียงกลุ่มใดกลุ่มหนึ่งได้อย่างสะดวก เมื่อคุณเข้าใจว่าการตรวจสอบความเท่าเทียมกันเหล่านี้ที่เราทุ่มเทเวลาส่วนใหญ่นั้นไม่มีอะไรมากไปกว่าวิธีที่ชาญฉลาดในการระบุตำแหน่งของข้อผิดพลาดในรูปแบบไบนารี่ จากนั้นเราก็สามารถเชื่อมโยงด้วยวิธีคิดที่แตกต่างออกไปเกี่ยวกับการแฮมมิง รหัส ซึ่งอาจจะเรียบง่ายกว่าและสวยงามกว่ามาก และโดยทั่วไปแล้วสามารถเขียนลงไปได้ด้วยโค้ดเพียงบรรทัดเดียว มันขึ้นอยู่กับฟังก์ชัน XOR XOR สำหรับคนที่ไม่รู้จัก ย่อมาจาก Exclusive or เมื่อคุณรับ XOR ของสองบิต มันจะคืนค่า 1 หากบิตใดบิตหนึ่งนั้นเปิดอยู่ แต่จะไม่ได้ถ้าทั้งสองเปิดหรือปิด หากใช้ถ้อยคำต่างกัน มันคือความเท่าเทียมกันของสองบิตนี้ ในฐานะคนคณิต ฉันชอบคิดว่ามันเป็นการบวก mod 2 โดยทั่วไปเรายังพูดถึง XOR ของสตริงบิตที่แตกต่างกันสองตัว ซึ่งโดยพื้นฐานแล้วจะทำส่วนประกอบนี้ทีละส่วนประกอบ มันเหมือนกับการเติม แต่ที่คุณไม่เคยพกติดตัว ขอย้ำอีกครั้งว่ายิ่งมีความโน้มเอียงทางคณิตศาสตร์มากขึ้นอาจชอบคิดว่านี่เป็นการเพิ่มเวกเตอร์สองตัวและลด mod 2 หากคุณเปิด Python ขึ้นมาตอนนี้และใช้การดำเนินการเครื่องหมายรูปหมวกระหว่างจำนวนเต็มสองตัว นี่คือสิ่งที่มันกำลังทำอยู่ ยกเว้นการแสดงบิตของตัวเลขเหล่านั้นภายใต้ประทุน ประเด็นสำคัญสำหรับคุณและฉันคือการรับ XOR ของสตริงบิตต่างๆ มากมายเป็นวิธีที่มีประสิทธิภาพในการคำนวณการล้อเลียนกลุ่มกลุ่มที่แยกจากกัน เช่นเดียวกับคอลัมน์ ทั้งหมดในคราวเดียว สิ่งนี้ทำให้เรามีวิธีที่ค่อนข้างเก๋ในการคิดเกี่ยวกับการตรวจสอบพาริตีหลายรายการจากอัลกอริธึมโค้ด Hamming ของเรา โดยที่ทั้งหมดนี้ถูกรวมเข้าด้วยกันเป็นการดำเนินการเดียว แม้ว่ามองแวบแรกจะดูแตกต่างออกไปมากก็ตาม เขียนตำแหน่งทั้ง 16 ตำแหน่งในรูปแบบไบนารี่อย่างที่เราเคยมีมาก่อน และตอนนี้เน้นตำแหน่งที่บิตข้อความเปิดเป็น 1 จากนั้นรวบรวมตำแหน่งเหล่านี้เป็นคอลัมน์ขนาดใหญ่คอลัมน์เดียวแล้วรับ XOR คุณอาจเดาได้ว่าผลลัพธ์ 4 บิตที่อยู่ด้านล่างนั้นเหมือนกับการตรวจสอบความเท่าเทียมกัน 4 รายการที่เรารู้จักและชื่นชอบ แต่ใช้เวลาสักครู่เพื่อคิดจริงๆ ว่าเหตุใดจึงเป็นเช่นนั้น ตัวอย่างเช่น คอลัมน์สุดท้ายนี้ กำลังนับตำแหน่งทั้งหมดที่มีบิตสุดท้ายเป็น 1 แต่เราจำกัดไว้เฉพาะตำแหน่งที่ไฮไลต์อยู่แล้ว ดังนั้นจึงเป็นการนับอย่างมีประสิทธิภาพว่าตำแหน่งที่ไฮไลต์กี่ตำแหน่งมาจากกลุ่มแพริตีกลุ่มแรก นั่นสมเหตุสมผลไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 216.68 }, { - "input": "Those of you who watched the chessboard puzzle I did with Matt Parker might find all this exceedingly familiar. ", + "input": "maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 237.32 }, { - "input": "These are the positions whose binary representation has just a single bit turned on. ", + "input": "at goes on at position 0, but don't worry about that for now. The third parity check covers every position whose third to last bit is turned ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 285.5 }, { - "input": "XOR, for those of you who don't know, stands for exclusive or. ", + "input": "f 1s in the message is an even number. So for example right now, that total number of 1s is If it takes more bits to describe each p ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. ", + "input": "osition, like six bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. that special bit to be a 1, making the count even. But if the block had already started off with a ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 302.98 }, { - "input": "As a math person, I prefer to think about it as addition mod 2. ", + "input": "it would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 306.76 }, { - "input": "We also commonly talk about the XOR of two different bit strings, which basically does this component by component. ", + "input": "logic, but solving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity bits are sitti ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 322.48 }, { - "input": "If you open up some Python right now and apply the caret operation between two integers, this is what it's doing but to the bit representations of those numbers under the hood. ", + "input": "of two, for example 1, 2, 4, and 8. These are the positions whose binary representation has just a single bit turned on. d say the parity is 0 or 1, which is typically more helpful once you start doing math with the idea. And this special bit that the sender uses to con ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop. ", + "input": "trol the parity is called the parity bit. And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure tha ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense? ", + "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. ", + "input": "turned on, but not if both are turned on or if both are turned off. Phrased differently, it's the parity of these two bits. full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 423.96 }, { - "input": "And so you know where it goes from here. ", + "input": "e also commonly talk about the XOR of two different bit s ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 469.36 }, { - "input": "You see, if you add a bit string together twice, it's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. ", + "input": "ey point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop. This gives us a rather snazzy way ", "translatedText": "ในทำนองเดียวกัน คอลัมน์ถัดไปจะนับจำนวนตำแหน่งที่อยู่ในกลุ่มพาริตีที่สอง ตำแหน่งที่บิตที่สองถึงสุดท้ายคือ 1 และตำแหน่งที่ถูกไฮไลต์ด้วย และอื่นๆ มันเป็นเพียงการเปลี่ยนแปลงเล็กๆ น้อยๆ ในมุมมองต่อสิ่งเดียวกันที่เราทำอยู่ แล้วคุณจะรู้ว่ามันไปจากที่นี่ที่ไหน ผู้ส่งมีหน้าที่รับผิดชอบในการสลับบิตพาริตีพิเศษบางส่วนเพื่อให้แน่ใจว่าผลรวมจะเป็น 0000 ตอนนี้เมื่อเรามีแบบนี้แล้ว นี่ทำให้เรามีวิธีที่ดีที่จะคิดว่าเหตุใดผลลัพธ์สี่บิตที่อยู่ด้านล่างจึงสะกดตำแหน่งของข้อผิดพลาดได้โดยตรง สมมติว่าบางส่วนในบล็อกนี้มีการสลับจาก 0 เป็น 1 ความหมายก็คือตอนนี้ตำแหน่งของบิตนั้นจะถูกรวมไว้ใน XOR ทั้งหมด ซึ่งเปลี่ยนผลรวมจาก 0 เป็นค่าที่รวมใหม่แทน ซึ่งเป็นตำแหน่งของข้อผิดพลาด เห็นได้ชัดว่าน้อยกว่าเล็กน้อย กรณีเดียวกันนี้จะเกิดขึ้นหากมีข้อผิดพลาดที่เปลี่ยน 1 เป็น 0 คุณจะเห็นว่า ถ้าคุณบวกสตริงบิตเข้าด้วยกันสองครั้ง มันก็เหมือนกับการไม่มีมันเลย โดยพื้นฐานแล้ว เพราะในโลกนี้ 1 บวก 1 เท่ากับ 0 ดังนั้นการเพิ่มสำเนาของตำแหน่งนี้เข้ากับผลรวมทั้งหมดจะมีผลเช่นเดียวกับที่เรากำลังย้ายตำแหน่ง และผลกระทบนั้นอีกครั้ง ก็คือผลลัพธ์รวมที่อยู่ด้านล่างสุดนี้ ระบุตำแหน่งของข้อผิดพลาด เพื่อแสดงให้เห็นว่าสิ่งนี้สวยงามเพียงใด ฉันขอแสดงโค้ด Python หนึ่งบรรทัดที่ฉันอ้างถึงก่อนหน้านี้ ซึ่งจะรวบรวมตรรกะเกือบทั้งหมดในส่วนท้ายของผู้รับ เราจะเริ่มต้นด้วยการสร้างอาร์เรย์สุ่ม 16 1 วินาทีและ 0 เพื่อจำลองบล็อกข้อมูล และฉันจะตั้งชื่อบิตให้กับมัน แต่แน่นอนว่าในทางปฏิบัติ นี่จะเป็นสิ่งที่เราได้รับจากผู้ส่ง และแทนที่จะเป็น ถ้าสุ่มก็จะบรรทุกข้อมูล 11 บิตพร้อมกับพาริตีบิต 5 บิต ถ้าฉันเรียกใช้ฟังก์ชัน enumerateBits สิ่งที่มันทำคือจับคู่แต่ละบิตเหล่านั้นกับดัชนีที่สอดคล้องกัน ในกรณีนี้คือทำงานตั้งแต่ 0 ถึง 15 ดังนั้นหากเราสร้างรายการที่วนซ้ำคู่เหล่านี้ทั้งหมด คู่ที่ดูเหมือน i แล้วเราดึงเฉพาะค่า i ออกมา แค่ดัชนี มันไม่น่าตื่นเต้นขนาดนั้น เราแค่นำดัชนีเหล่านั้นกลับมา 0 ถึง 15 . ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 477.94 }, { - "input": "So adding a copy of this position to the total sum has the same effect as we're moving it. ", + "input": "to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 484.3 }, { - "input": "And that effect, again, is that the total result at the bottom here spells out the position of the error. ", + "input": "Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the mes ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 490.7 }, { - "input": "To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all of the logic on the receiver's end. ", + "input": "sage bit is turned on to a 1, and then collect these positions into one big column and take the XOR. You can probably guess that the four bits sitting at the bottom as a resu ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. ", + "input": "lt are the same as the four parity checks we've come to know and love, but take a moment to actually think about why exactly. This last column, for example, is counting all of the positions whose last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15. ", + "input": "ht half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit, so these 8 bits already have an even pari Likewise, the next column counts how many positions are in the second parity group, the positions whose second to las ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 552.66 }, { - "input": "In this case it looks like those positions are 0, 4, 6, 9, etc. ", + "input": "ve on the same thing we've been doing. but for right now we're going to assume ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 567.24 }, { - "input": "To do this in Python, let me first import a couple helpful functions. ", + "input": "The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 000 ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 578.7 }, { - "input": "This basically eats its way through the list, taking XORs along the way. ", + "input": "es us a really nice way to think about why these four resulting bits at the bottom directly spell out the pos ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 582.68 }, { - "input": "If you prefer, you can explicitly write out that XOR function without having to import it from anywhere. ", + "input": "ition of an error. Let's say you detect an error among the odd columns, and among the right half. It necessarily means the error is somewhere in th ", "translatedText": "แต่ถ้าเราเพิ่มเงื่อนไขให้ทำสิ่งนี้เฉพาะ if bit ซึ่งหมายความว่าถ้าบิตนั้นเป็น 1 ไม่ใช่ 0 มันก็จะดึงเฉพาะตำแหน่งที่เปิดบิตที่เกี่ยวข้องออกมาเท่านั้น ในกรณีนี้ ดูเหมือนว่าตำแหน่งเหล่านั้นคือ 0, 4, 6, 9 เป็นต้น สิ่งที่เราต้องการคือการรวบรวมตำแหน่งทั้งหมดเหล่านั้น ตำแหน่งของบิตที่เปิดอยู่ จากนั้น XOR เข้าด้วยกัน หากต้องการทำสิ่งนี้ใน Python ก่อนอื่นให้ฉันนำเข้าฟังก์ชันที่เป็นประโยชน์สองสามรายการก่อน ด้วยวิธีนี้เราสามารถเรียกย่อ () ในรายการนี้ และใช้ฟังก์ชัน XOR เพื่อลดค่านั้น โดยพื้นฐานแล้วสิ่งนี้จะกินทางผ่านรายการโดยนำ XOR ไปพร้อมกัน หากต้องการ คุณสามารถเขียนฟังก์ชัน XOR นั้นอย่างชัดเจนโดยไม่ต้องนำเข้าจากที่ใดก็ได้ ดังนั้นในขณะนี้ ดูเหมือนว่าถ้าเราทำเช่นนี้กับบล็อกสุ่มของเราที่มีขนาด 16 บิต มันจะคืนค่า 9 ซึ่งมีเลขฐานสองแทน 1001 เราจะไม่ทำที่นี่ แต่คุณสามารถเขียนฟังก์ชันที่ผู้ส่งใช้การแทนค่าไบนารีนั้นเพื่อตั้งค่าบิตพาริตีทั้งสี่ตามต้องการ ในท้ายที่สุดทำให้บล็อกนี้อยู่ในสถานะที่รันโค้ดบรรทัดนี้ในรายการบิตทั้งหมดส่งคืน 0 นี่ถือเป็นบล็อกที่เตรียมไว้อย่างดี สิ่งที่ยอดเยี่ยมก็คือหากเราสลับบิตใดบิตหนึ่งในรายการนี้ จำลองข้อผิดพลาดแบบสุ่มจากสัญญาณรบกวน จากนั้นหากคุณเรียกใช้โค้ดบรรทัดเดียวกัน มันจะพิมพ์ข้อผิดพลาดนั้นออกมา นั่นไม่เรียบร้อยเหรอ? ", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 630.22 }, { - "input": "Isn't that neat? ", + "input": "an error that changes a 1 to a 0. You see, if you add a bit string together twice, it's the same as ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 641.06 }, { - "input": "And there's nothing special about the size 16 here. ", + "input": "And that effect, again, is that the total result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me sh ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 649.86 }, { - "input": "Needless to say, there is more code to write here, like doing the meta parity check to detect 2-bit errors, but the idea is that almost all of the core logic from our scheme comes down to a single XOR reduction. ", + "input": "ferenced before, which will capture almost all of the logic on the receiver's end. We'll start by creating a random array of 16 ones and zeros to simulate the data block, and I'll go ahead and give it the name bits, but of course in practice this would be something that we're receiving f ", "translatedText": "คุณสามารถดึงบล็อกนี้ขึ้นมาใหม่ รันบรรทัดเดียวบนบล็อกนั้น และมันจะแยกตำแหน่งของข้อผิดพลาดออกโดยอัตโนมัติ หรือจะเป็น 0 หากไม่มีเลย และไม่มีอะไรพิเศษเกี่ยวกับไซส์ 16 ที่นี่ โค้ดบรรทัดเดียวกันนี้จะใช้ได้ถ้าคุณมีรายการ เช่น 256 บิต ไม่จำเป็นต้องพูดว่า มีโค้ดให้เขียนมากกว่านี้ เช่น การตรวจสอบเมตาพาริตี้เพื่อตรวจจับข้อผิดพลาด 2 บิต แต่แนวคิดก็คือตรรกะหลักเกือบทั้งหมดจากโครงการของเราลดลงเหลือเพียงการลด XOR เพียงครั้งเดียว ตอนนี้ ขึ้นอยู่กับความสะดวกสบายของคุณกับไบนารี่และ XOR และซอฟต์แวร์โดยทั่วไป คุณอาจพบว่ามุมมองนี้สับสนเล็กน้อย หรือหรูหราและเรียบง่ายกว่านั้นมากจนคุณสงสัยว่าทำไมเราไม่เพียงแค่เริ่มต้นจากจุดเริ่มต้น -ไป. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 690.5 }, { - "input": "The first one is easiest to actually do by hand, and I think it does a better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the log of the size of the block, or in other words, it grows one bit at a time as the block size doubles. ", + "input": "l out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on. In this case it looks like those positions are 0, 4, 6, 9, etc. Remember, what ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need. ", + "input": "we want is to collect together all of those positions, the positions of the bits that are turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there. ", + "input": "looks like if we do this on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but you could write a function wh ", "translatedText": "พูดง่ายๆ ก็คือ มุมมองการตรวจสอบความเท่าเทียมกันหลายรายการจะคิดได้ง่ายกว่าเมื่อใช้โค้ด Hamming ในฮาร์ดแวร์โดยตรง และมุมมอง XOR นั้นคิดได้ง่ายที่สุดเมื่อทำในซอฟต์แวร์จากระดับที่สูงกว่า วิธีแรกนั้นง่ายที่สุดที่ทำด้วยมือจริง ๆ และฉันคิดว่ามันจะทำงานได้ดีกว่าโดยปลูกฝังสัญชาตญาณหลักที่เป็นรากฐานทั้งหมดนี้ ซึ่งก็คือข้อมูลที่จำเป็นต้องใช้ในการค้นหาข้อผิดพลาดเดียวนั้นเกี่ยวข้องกับบันทึกขนาดของบล็อก หรืออีกนัยหนึ่ง มันจะขยายทีละบิตเมื่อขนาดบล็อกเพิ่มขึ้นสองเท่า ข้อเท็จจริงที่เกี่ยวข้องในที่นี้คือข้อมูลนั้นสอดคล้องโดยตรงกับปริมาณความซ้ำซ้อนที่เราต้องการ นั่นคือสิ่งที่ขัดแย้งกับปฏิกิริยากระตุกเข่าของคนส่วนใหญ่เมื่อพวกเขาคิดถึงการสร้างข้อความที่ยืดหยุ่นต่อข้อผิดพลาด ซึ่งโดยปกติแล้วการคัดลอกข้อความทั้งหมดถือเป็นสัญชาตญาณแรกที่เข้ามาในใจ แล้ว ยังมีอีกวิธีหนึ่ง ที่บางครั้งคุณจะเห็นโค้ดแฮมมิงนำเสนอ โดยคุณคูณข้อความด้วยเมทริกซ์ขนาดใหญ่ตัวเดียว เป็นเรื่องดีเพราะมันเกี่ยวข้องกับกลุ่มโค้ดเชิงเส้นที่กว้างกว่า แต่ฉันคิดว่านั่นแทบไม่ได้ให้สัญชาตญาณเลยว่ามันมาจากไหนหรือปรับขนาดอย่างไร และเมื่อพูดถึงการปรับขนาด คุณอาจสังเกตเห็นว่าประสิทธิภาพของโครงร่างนี้จะดีขึ้นเมื่อเราเพิ่มขนาดบล็อกเท่านั้น ตัวอย่างเช่น เราเห็นว่าด้วย 256 บิต คุณใช้พื้นที่เพียง 3% ของพื้นที่นั้นในการสำรอง และมันจะดีขึ้นเรื่อยๆ จากจุดนั้น เมื่อจำนวนบิตพาริตีเพิ่มขึ้นทีละบิต ขนาดบล็อกก็จะเพิ่มขึ้นเป็นสองเท่า และถ้าคุณใช้มันสุดโต่ง คุณสามารถมีบล็อกที่มีหนึ่งล้านบิต ซึ่งคุณคงจะเล่นคำถาม 20 ข้อกับการตรวจสอบพาริตีของคุณ และมันใช้เพียง 21 พาริตีบิตเท่านั้น และถ้าคุณย้อนกลับไปคิดถึงการดูล้านบิตและค้นหาข้อผิดพลาดเพียงจุดเดียว นั่นคงรู้สึกบ้าไปแล้วจริงๆ แน่นอนว่าปัญหาก็คือ เมื่อมีบล็อกขนาดใหญ่ ความน่าจะเป็นที่จะเห็นข้อผิดพลาดมากกว่าหนึ่งหรือสองบิตก็จะเพิ่มขึ้น และโค้ด Hamming ก็ไม่สามารถจัดการอะไรนอกเหนือจากนั้นได้ ในทางปฏิบัติ สิ่งที่คุณต้องการคือการหาขนาดที่เหมาะสม เพื่อไม่ให้ความน่าจะเป็นที่จะพลิกบิตมากเกินไป นอกจากนี้ ในทางปฏิบัติ ข้อผิดพลาดมักจะเกิดขึ้นเป็นชุดเล็กๆ น้อยๆ ซึ่งจะทำให้บล็อกเดียวเสียหายโดยสิ้นเชิง ดังนั้นกลวิธีทั่วไปประการหนึ่งที่จะช่วยกระจายข้อผิดพลาดแบบต่อเนื่องไปยังบล็อกต่างๆ มากมายคือการสอดประสานบล็อกเหล่านั้น เช่นนี้ ก่อนที่จะ ส่งหรือเก็บไว้ อีกครั้ง สิ่งเหล่านี้ส่วนใหญ่ถูกโต้แย้งอย่างสมบูรณ์ด้วยโค้ดที่ทันสมัยกว่า เช่น อัลกอริธึม Reed-Solomon ที่ใช้กันทั่วไปมากกว่า ซึ่งจัดการข้อผิดพลาดแบบ Burst ได้ดีเป็นพิเศษ และสามารถปรับให้มีความยืดหยุ่นต่อจำนวนข้อผิดพลาดที่มากขึ้นต่อบล็อก . ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling. ", + "input": "ere the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a state where running th ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 804.3 }, { - "input": "Also, in practice, errors tend to come in little bursts, which would totally ruin a single block, so one common tactic to help spread out a burst of errors across many different blocks is to interlace those blocks, like this, before they're sent out or stored. ", + "input": "imulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the positio ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind. ", + "input": "is perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 906.82 }, { - "input": "Part of the reason that clever ideas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong turns, underselling just how vast the space of explorable possibilities is at the start of a problem solving process, all of that. ", + "input": ", with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing 1 out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The first one is easiest to actually do by hand, and I think it does a better job instilling the core intuition underlying all of this, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 922.86 }, { - "input": "But this is true in general. ", + "input": "which is that the information required to locate a single error is relat ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 924.9 }, { - "input": "I think for some special inventions, there's a second, deeper reason that we underappreciate them. ", + "input": "ed to the log of the size of the block, or in other words, it grows one bit at a time as the block size doubles. The relevant fact here i ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 938.64 }, { - "input": "This was essentially concurrent with when Hamming developed his algorithm. ", + "input": "block is even, just like a normal parity check. Now, if there's a single bit error, then ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory. ", + "input": "the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks. However, if there's two errors, then the overall parity is going to toggle back to be And then, by the way, there is this whole other way that you s ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was. ", + "input": "multiply the message by one big matrix. It's kind of nice because it relates it to the broader family of linear codes, but I think that gives almost no intuition for where it comes from or how it scales. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/turkish/sentence_translations.json b/2020/hamming-codes-2/turkish/sentence_translations.json index 30c70f42a..eeb3f8b35 100644 --- a/2020/hamming-codes-2/turkish/sentence_translations.json +++ b/2020/hamming-codes-2/turkish/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "Bitlerin çoğunun anlamlı bir mesaj taşıdığı, diğer birkaç parçanın ise bir tür artıklık işlevi gördüğü bir veri bloğu oluşturmanın bir yolu olan Hamming kodlarından bahsediyorduk; herhangi bir bit ters çevrildiğinde ya bir mesaj Bit veya artıklık biti, bu bloktaki herhangi bir şeyde, alıcı bir hata olduğunu ve bunun nasıl düzeltileceğini tanımlayabilecektir.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "Orada sunulan temel fikir, hataya giden yolda ikili arama yapmak için çoklu eşlik kontrolünün nasıl kullanılacağıydı.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "Örneğin, ikili sistemde 7 sayısı 0111'e benzer, aslında 4 artı 2 artı 1 olduğunu söyler.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "Ve 7. pozisyonun nerede olduğuna dikkat edin, bu durum eşitlik gruplarımızın birincisini, ikincisini ve üçüncüsünü etkiliyor, ancak sonuncusunu etkilemiyor.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "Yani bu dört kontrolün sonuçlarını aşağıdan yukarıya doğru okumak gerçekten de hatanın konumunu ortaya koyuyor.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "Bu ikili etiketleri kutularına geri koyarken, bunların gerçekte gönderilen verilerden farklı olduklarını vurgulamam gerekiyor.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "Bunlar sizin ve benim dört eşitlik grubunun nereden geldiğini anlamamıza yardımcı olacak kavramsal bir etiketten başka bir şey değil.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "Baktığımız her şeyin ikili olarak tanımlanmasının zarafeti, belki de baktığımız her şeyin ikili olarak tanımlanmasının yarattığı kafa karışıklığı nedeniyle gölgede kalıyor.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "Elde ettiğimiz şey, dört eşlik grubumuzun ilkidir; bu, ilk kontrolü şu soruyla yorumlayabileceğiniz anlamına gelir: hey, eğer bir hata varsa, bu hatanın konumundaki son bit 1 mi?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "Ve benzeri.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "Birincisi, ikinin daha büyük kuvvetleri olan blok boyutlarına sistematik olarak nasıl genelleştirme yapılacağıdır.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "Bunun anlamı, bu eşlik bitlerinin her birinin, dört eşlik grubundan yalnızca birinin içinde yer almasıdır.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "İki bitin XOR'unu aldığınızda, bu bitlerden herhangi biri açıksa 1 değerini döndürür, ancak her ikisi de açık veya kapalıysa bu sonuç değişmez.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "Farklı bir ifadeyle bu iki bitin paritesidir.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "Toplama gibi ama asla taşımadığınız yer.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "Sizin ve benim için kilit nokta, birçok farklı bit dizisinin XOR'unu almanın, sütunlarda olduğu gibi bir grup ayrı grubun parodilerini tek bir hamlede etkili bir şekilde hesaplamanın bir yolu olmasıdır.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "Bu bize, Hamming kod algoritmamızın çoklu eşlik kontrollerinin tek bir işlemde paketlenmesi hakkında düşünmemiz için oldukça şık bir yol sağlıyor.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "bu mantıklı mı?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "Benzer şekilde, bir sonraki sütun, ikinci eşlik grubunda kaç konumun bulunduğunu, ikinciden sondan biti 1 olan ve hangilerinin vurgulandığını vb. sayar.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "Şimdi bu şekilde elde ettiğimizde, bu bize, alttaki sonuçta ortaya çıkan dört bitin neden doğrudan bir hatanın konumunu gösterdiğini düşünmemiz için gerçekten güzel bir yol sağlıyor.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "Diyelim ki bu bloktaki bir bit 0'dan 1'e değiştirildi.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "Bunun anlamı, o bitin konumu artık toplam XOR'a dahil edilecek, bu da toplamı 0'dan yeni eklenen değere, yani hatanın konumuna dönüştürecek.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "Veri bloğunu simüle etmek için 16 1 ve 0'lardan oluşan rastgele bir dizi oluşturarak başlayacağız ve ona bitlerin adını vereceğim, ancak elbette pratikte bu, bir göndericiden aldığımız bir şey olacaktır ve bunun yerine rastgele olduğundan 5 eşlik bitiyle birlikte 11 veri biti taşıyor olacaktır.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "Yani eğer daha sonra tüm bu çiftleri, i'ye benzeyen çiftleri kapsayan bir liste oluşturursak ve sonra sadece i değerini, sadece indeksi çıkarırsak, bu o kadar da heyecan verici değil, sadece 0'dan 15'e kadar olan indeksleri geri alırız.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "Ancak bunu yalnızca if biti yapma koşulunu eklersek, yani bu bit 0 değil de 1 ise, o zaman yalnızca karşılık gelen bitin açık olduğu konumları çeker.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "Bunu burada yapmayacağız, ancak gönderenin, dört eşlik bitini gerektiği gibi ayarlamak için bu ikili gösterimi kullandığı ve sonuçta bu bloğu, bu kod satırını bitlerin tam listesinde çalıştırmanın geri döndüğü bir duruma getirdiği bir fonksiyon yazabilirsiniz. bir 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "İşin güzel yanı, bu listedeki bitlerden herhangi birini değiştirirsek, gürültüden kaynaklanan rastgele bir hatayı simüle edersek, o zaman aynı kod satırını çalıştırırsanız, o hatayı yazdırır.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "Bu bloğu birdenbire alabilir, üzerinde bu tek satırı çalıştırabilirsiniz ve otomatik olarak bir hatanın konumunu veya eğer yoksa 0'ı söyler.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "Şimdi, ikili ve XOR'lar ve genel olarak yazılım konusundaki rahatınıza bağlı olarak, bu bakış açısını ya biraz kafa karıştırıcı bulabilir ya da çok daha şık ve basit bulabilir ve neden en baştan bununla başlamadığımızı merak edebilirsiniz. -Gitmek.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "Kabaca konuşursak, çoklu eşlik kontrolü perspektifini, Hamming kodlarını donanıma doğrudan uygularken düşünmek daha kolaydır ve XOR perspektifini, bunu yazılımda yaparken, daha yüksek bir seviyeden düşünmek en kolayıdır.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "Buradaki ilgili gerçek şu ki, bu bilgi doğrudan ne kadar fazlalığa ihtiyacımız olduğuna karşılık geliyor.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "Ve bu arada, bazen Hamming kodlarının sunulduğunu gördüğünüz, mesajı büyük bir matrisle çarptığınız tamamen farklı bir yol daha var.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "Ölçeklendirmeden bahsetmişken, blok boyutunu artırdıkça bu planın verimliliğinin de arttığını fark edebilirsiniz.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "Örneğin, 256 bit ile bu alanın yalnızca %3'ünü yedeklilik için kullandığınızı ve bu noktadan sonra giderek daha iyi hale geldiğini gördük.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "Eşlik bitlerinin sayısı birer birer arttıkça blok boyutu da iki katına çıkar.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "Ve eğer bunu aşırıya götürürseniz, diyelim ki bir milyon bitlik bir bloğunuz olabilir, burada kelimenin tam anlamıyla eşlik kontrollerinizle 20 soru oynuyorsunuz ve bu blok yalnızca 21 eşlik biti kullanıyor.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "Sorun, elbette, daha büyük bir blokla, bir veya iki bitten daha fazla hata görme olasılığının artması ve Hamming kodlarının bunun ötesinde hiçbir şeyi ele almamasıdır.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "Ama bu başka bir zamanın konusu.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "Hamming, The Art of Doing Science and Engineering adlı kitabında bu kodu keşfetmesinin ne kadar dolambaçlı olduğunu son derece samimi bir şekilde anlatıyor.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "Bu kitap boyunca neredeyse yarım düzine kez Louis Pasteur'ün bir sözüne gönderme yapıyor; şans hazırlıklı bir zihinden yanadır.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "Ancak bunların aslında bariz olduğunu düşünerek kendinizi kandırmamalısınız çünkü kesinlikle öyle değiller.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "Bu, en azından teoride, bit kayması olasılığı ne kadar yüksek olursa olsun, bir anlamda etkili hata düzeltmenin her zaman mümkün olduğunu gösteren aynı temel makaleydi.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "Birkaç on yıl ileri saralım ve bu günlerde çoğumuz küçük parçalar ve bilgiler üzerine düşünmeye o kadar dalmış durumdayız ki, bu düşünce tarzının ne kadar farklı olduğunu gözden kaçırmak çok kolay.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/ukrainian/sentence_translations.json b/2020/hamming-codes-2/ukrainian/sentence_translations.json index 7c9c54af4..e9822e9fe 100644 --- a/2020/hamming-codes-2/ukrainian/sentence_translations.json +++ b/2020/hamming-codes-2/ukrainian/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "Ми говорили про коди Хеммінга, спосіб створення блоку даних, де більшість бітів несуть значуще повідомлення, тоді як кілька інших діють як свого роду надлишковість, таким чином, що якщо будь-який біт перевертається, або повідомлення біт або біт надлишковості, будь-що в цьому блоці, приймач зможе визначити, що була помилка, і як її виправити.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "Основна ідея, представлена там, полягала в тому, як використовувати кілька перевірок на парність для бінарного пошуку на шляху до помилки.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "Наприклад, число 7 у двійковій системі виглядає як 0111, по суті кажучи, що це 4 плюс 2 плюс 1.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "І зауважте, де знаходиться позиція 7, вона дійсно впливає на першу з наших груп парності, і на другу, і на третю, але не на останню.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "Отже, читання результатів цих чотирьох перевірок знизу вгору справді пояснює місце помилки.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "Коли ми повертаємо ці двійкові мітки назад у свої коробки, дозвольте мені підкреслити, що вони відрізняються від даних, які насправді надсилаються.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "Вони не що інше, як концептуальний ярлик, який допоможе вам і мені зрозуміти, звідки взялися чотири паритетні групи.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "Елегантність того, що все, на що ми дивимося, описується у двійковому форматі, можливо, підривається плутаниною, пов’язаною з тим, що все, на що ми дивимося, описується у двійковому форматі.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "Що ми отримуємо, це перша з наших чотирьох груп парності, що означає, що ви можете інтерпретувати цю першу перевірку як запитання: якщо є помилка, чи є останній біт у позиції цієї помилки 1?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "І так далі.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "По-перше, як систематично узагальнювати розміри блоків, які є більшими степенями двійки.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "Це означає, що кожен із цих бітів парності знаходиться в одній і лише одній із чотирьох груп парності.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "Коли ви використовуєте XOR двох бітів, він повертає 1, якщо один із цих бітів увімкнено, але не якщо обидва увімкнено чи вимкнено.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "Іншими словами, це парність цих двох бітів.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "Це як доповнення, але куди ніколи не понесеш.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "Ключовим моментом для нас із вами є те, що використання XOR багатьох різних бітових рядків є ефективним способом обчислення пародії на купу окремих груп, як і зі стовпцями, одним махом.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "Це дає нам досить хитрий спосіб уявити, що численні перевірки парності з нашого алгоритму коду Хеммінга об’єднані в одну операцію.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "Чи має це сенс?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "Подібним чином у наступному стовпці підраховується кількість позицій у другій групі парності, позиції, передостанній біт яких дорівнює 1, які також виділені тощо.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "Тепер, коли ми маємо це таким чином, це дає нам дійсно гарний спосіб подумати про те, чому ці чотири результуючих біта внизу безпосередньо вказують на місце помилки.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "Припустимо, якийсь біт у цьому блоці перемикається з 0 на 1.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "Це означає, що позиція цього біта тепер буде включена в загальне XOR, яке змінює суму з 0 на це нове включене значення, позицію помилки.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "Ми почнемо зі створення випадкового масиву з 16 1 і 0 для імітації блоку даних, і я дам йому біти назви, але, звичайно, на практиці це буде щось, що ми отримуємо від відправника, а замість будучи випадковим, він містив би 11 біт даних разом із 5 бітами парності.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "Отже, якщо ми створимо список, який циклічно перебирає всі ці пари, пари, які виглядають як i, а потім витягуємо лише значення i, лише індекс, це не так цікаво, ми просто повертаємо ті індекси від 0 до 15.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "Але якщо ми додамо умову робити це тільки якщо біт, тобто якщо цей біт є 1, а не 0, добре, тоді він вилучає лише ті позиції, де відповідний біт увімкнено.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "Ми не будемо цього робити тут, але ви можете написати функцію, у якій відправник використовує це двійкове представлення, щоб за потреби встановити чотири біти парності, зрештою переводячи цей блок у стан, коли виконання цього рядка коду з повним списком бітів повертає а 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "Цікаво те, що якщо ми перемикаємо будь-який із бітів у цьому списку, імітуючи випадкову помилку через шум, тоді, якщо ви запустите цей самий рядок коду, він виведе цю помилку.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "Ви можете отримати цей блок зненацька, запустити на ньому цей єдиний рядок, і він автоматично видасть позицію помилки або 0, якщо її не було.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "Тепер, залежно від вашого комфорту з двійковими кодами та XOR та програмним забезпеченням загалом, ви можете або вважати цю перспективу трохи заплутаною, або настільки більш елегантною та простою, що ви дивуєтеся, чому ми не почали з неї з самого початку -іди.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "Грубо кажучи, про перспективу множинної перевірки парності легше подумати, якщо реалізовувати коди Хеммінга в апаратному забезпеченні дуже безпосередньо, а про перспективу XOR найпростіше подумати, роблячи це в програмному забезпеченні, на більш високому рівні.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "Релевантним фактом тут є те, що ця інформація безпосередньо відповідає тому, скільки резервування нам потрібно.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "І, до речі, є цілий інший спосіб, у який ви іноді бачите представлені коди Хеммінга, коли ви множите повідомлення на одну велику матрицю.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "Говорячи про масштабування, ви можете помітити, що ефективність цієї схеми стає лише кращою, коли ми збільшуємо розмір блоку.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "Наприклад, ми бачили, що з 256 бітами ви використовуєте лише 3% цього простору для резервування, і з цього моменту все стає краще.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "Оскільки кількість бітів парності зростає один за одним, розмір блоку продовжує подвоюватися.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "І якщо ви доведете це до крайності, у вас може бути блок, скажімо, з мільйоном біт, де ви б буквально відтворювали 20 запитань із перевіркою парності, і він використовує лише 21 біт парності.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "Проблема, звісно, полягає в тому, що з більшим блоком зростає ймовірність побачити більше однієї або двох бітових помилок, а коди Хеммінга не обробляють нічого, крім цього.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "Але це тема іншого разу.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "У своїй книзі «Мистецтво займатися наукою та технікою» Хеммінг надзвичайно відверто розповідає про те, наскільки звивистим було його відкриття цього коду.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "У цій книзі він півдюжини разів посилається на цитату Луї Пастера: удача сприяє підготовленому розуму.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "Але ви не повинні обманювати себе, думаючи, що вони насправді очевидні, тому що це точно не так.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "Це був той самий основоположний документ, який показав, у певному сенсі, що ефективне виправлення помилок завжди можливе, незалежно від того, наскільки висока ймовірність перевертань бітів, принаймні в теорії.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "Перемотуємо вперед на кілька десятиліть, і сьогодні багато з нас настільки занурені в роздуми про біти та інформацію, що легко не помітити, наскільки відмінним був цей спосіб мислення.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/urdu/sentence_translations.json b/2020/hamming-codes-2/urdu/sentence_translations.json index dbe38f48e..75805b24b 100644 --- a/2020/hamming-codes-2/urdu/sentence_translations.json +++ b/2020/hamming-codes-2/urdu/sentence_translations.json @@ -32,7 +32,7 @@ "end": 34.6 }, { - "input": "But as you start to think about actually implementing this, either in software or hardware, that framing may actually undersell how elegant these codes really are. ", + "input": "hat there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error. In that video, the goal was to make Hamming codes feel as hands-on and rediscoverable as possible. But as ", "translatedText": "لیکن جیسے ہی آپ سافٹ ویئر یا ہارڈ ویئر میں اس کو عملی جامہ پہنانے کے بارے میں سوچنا شروع کر دیتے ہیں، وہ فریمنگ حقیقت میں اس بات کو کم کر سکتی ہے کہ یہ کوڈز واقعی کتنے خوبصورت ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. ", + "input": "you read out the answers to the four parity checks we did in the last video, all as ones and zeros instead of yeses and nos, it literally spells out ", "translatedText": "مثال کے طور پر، بائنری میں نمبر 7 0111 کی طرح لگتا ہے، بنیادی طور پر یہ کہہ رہا ہے کہ یہ 4 جمع 2 جمع 1 ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error. ", + "input": "century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day. The goal here is to give you a very thorough understanding of one of the earlie ", "translatedText": "لہذا نیچے سے اوپر تک ان چار چیکوں کے نتائج کو پڑھنا واقعی غلطی کی پوزیشن کو واضح کرتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 87.54 }, { - "input": "There's nothing special about the example 7, this works in general, and this makes the logic for implementing the whole scheme in hardware shockingly simple. ", + "input": "st examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity groups, and the second, and the third, but not the last. So reading the results of those four checks ", "translatedText": "مثال 7 کے بارے میں کچھ خاص نہیں ہے، یہ عام طور پر کام کرتا ہے، اور یہ ہارڈ ویئر میں پوری اسکیم کو لاگو کرنے کی منطق کو حیران کن حد تک آسان بنا دیتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 132.36 }, { - "input": "It's worth it, though. ", + "input": "0, let's write them all in binary, running from 0000 up to 1111. ask feels at the star ", "translatedText": "یہ اس کے قابل ہے، اگرچہ. ", "model": "google_nmt", "n_reviews": 0, @@ -152,7 +152,7 @@ "end": 166.16 }, { - "input": "In other words, that second check is asking, hey, me again, if there's an error, is the second to last bit of that position a 1? ", + "input": "ly spelled words. Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. ", "translatedText": "دوسرے لفظوں میں، وہ دوسرا چیک پوچھ رہا ہے، ارے، مجھے دوبارہ، اگر کوئی غلطی ہے، تو کیا اس پوزیشن کا دوسرا آخری حصہ 1 ہے؟ اور اسی طرح. ", "model": "google_nmt", "n_reviews": 0, @@ -176,7 +176,7 @@ "end": 188.74 }, { - "input": "Everything we did earlier is the same as answering these four questions, which in turn is the same as spelling out a position in binary. ", + "input": "everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary. ", "translatedText": "ہم نے پہلے جو کچھ کیا وہ ان چار سوالوں کے جوابات کے برابر ہے، جو کہ بائنری میں پوزیشن کو ہجے کرنے کے مترادف ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 197.74 }, { - "input": "I hope this makes two things clearer. ", + "input": "It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where tha ", "translatedText": "مجھے امید ہے کہ اس سے دو چیزیں واضح ہو جائیں گی۔", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two. ", + "input": "t final bit is a 1. What we get is the first of our four parity groups, which means that you can interpret that first check as asking, hey, if there's an err ", "translatedText": "پہلا یہ ہے کہ کس طرح منظم طریقے سے ان سائزوں کو بلاک کرنا ہے جو دو کی بڑی طاقتیں ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 216.68 }, { - "input": "Those of you who watched the chessboard puzzle I did with Matt Parker might find all this exceedingly familiar. ", + "input": "maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. ", "translatedText": "آپ میں سے وہ لوگ جنہوں نے میٹ پارکر کے ساتھ میں نے شطرنج کی تختی کی پہیلی دیکھی ہے شاید وہ یہ سب کچھ بہت زیادہ مانوس محسوس کریں۔", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 237.32 }, { - "input": "These are the positions whose binary representation has just a single bit turned on. ", + "input": "at goes on at position 0, but don't worry about that for now. The third parity check covers every position whose third to last bit is turned ", "translatedText": "یہ وہ پوزیشنیں ہیں جن کی بائنری نمائندگی صرف ایک بٹ آن ہوئی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 285.5 }, { - "input": "XOR, for those of you who don't know, stands for exclusive or. ", + "input": "f 1s in the message is an even number. So for example right now, that total number of 1s is If it takes more bits to describe each p ", "translatedText": "XOR، آپ میں سے ان لوگوں کے لیے جو نہیں جانتے، خصوصی یا کے لیے کھڑا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. ", + "input": "osition, like six bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. that special bit to be a 1, making the count even. But if the block had already started off with a ", "translatedText": "جب آپ دو بٹس کا XOR لیتے ہیں، تو یہ 1 واپس کرنے جا رہا ہے اگر ان میں سے کوئی ایک بٹس آن ہو، لیکن نہیں اگر دونوں آن یا آف ہوں۔", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 302.98 }, { - "input": "As a math person, I prefer to think about it as addition mod 2. ", + "input": "it would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core ", "translatedText": "ایک ریاضی کے فرد کے طور پر، میں اس کے بارے میں اضافی موڈ 2 کے طور پر سوچنا پسند کرتا ہوں۔", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 306.76 }, { - "input": "We also commonly talk about the XOR of two different bit strings, which basically does this component by component. ", + "input": "logic, but solving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity bits are sitti ", "translatedText": "ہم عام طور پر دو مختلف بٹ سٹرنگز کے XOR کے بارے میں بھی بات کرتے ہیں، جو بنیادی طور پر اس جزو کو جزو کے لحاظ سے کرتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 322.48 }, { - "input": "If you open up some Python right now and apply the caret operation between two integers, this is what it's doing but to the bit representations of those numbers under the hood. ", + "input": "of two, for example 1, 2, 4, and 8. These are the positions whose binary representation has just a single bit turned on. d say the parity is 0 or 1, which is typically more helpful once you start doing math with the idea. And this special bit that the sender uses to con ", "translatedText": "اگر آپ ابھی کچھ Python کھولتے ہیں اور کیریٹ آپریشن کو دو عدد کے درمیان لاگو کرتے ہیں، تو یہ وہی کام کر رہا ہے لیکن ہڈ کے نیچے ان نمبروں کی تھوڑا سا نمائندگی کرنے کے لئے۔", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop. ", + "input": "trol the parity is called the parity bit. And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure tha ", "translatedText": "آپ کے اور میرے لیے اہم نکتہ یہ ہے کہ بہت سے مختلف بٹ سٹرنگز کا XOR لینا مؤثر طریقے سے الگ الگ گروپس کی پیروڈیز کی گنتی کرنے کا ایک طریقہ ہے، جیسا کہ کالموں کے ساتھ، سبھی ایک ساتھ جھپٹ پڑے۔", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense? ", + "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. ", + "input": "turned on, but not if both are turned on or if both are turned off. Phrased differently, it's the parity of these two bits. full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. ", "translatedText": "کیا اسکا کوئ مطلب بنتا ہے؟ اسی طرح، اگلا کالم شمار کرتا ہے کہ دوسرے برابری گروپ میں کتنی پوزیشنیں ہیں، وہ پوزیشنیں جن کا دوسرا سے آخری بٹ 1 ہے، اور جن پر روشنی ڈالی گئی ہے، وغیرہ۔", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 423.96 }, { - "input": "And so you know where it goes from here. ", + "input": "e also commonly talk about the XOR of two different bit s ", "translatedText": "اور اس طرح آپ جانتے ہیں کہ یہ یہاں سے کہاں جاتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 469.36 }, { - "input": "You see, if you add a bit string together twice, it's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. ", + "input": "ey point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop. This gives us a rather snazzy way ", "translatedText": "آپ دیکھتے ہیں، اگر آپ دو بار ایک ساتھ تھوڑا سا سٹرنگ جوڑتے ہیں، تو یہ بالکل ایسا ہی ہے جیسا کہ اس کا وہاں بالکل نہ ہونا، بنیادی طور پر کیونکہ اس دنیا میں 1 جمع 1 برابر 0 ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 477.94 }, { - "input": "So adding a copy of this position to the total sum has the same effect as we're moving it. ", + "input": "to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation. ", "translatedText": "لہذا اس پوزیشن کی ایک کاپی کو کل رقم میں شامل کرنے کا وہی اثر ہوتا ہے جیسا کہ ہم اسے منتقل کر رہے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 484.3 }, { - "input": "And that effect, again, is that the total result at the bottom here spells out the position of the error. ", + "input": "Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the mes ", "translatedText": "اور وہ اثر، ایک بار پھر، یہ ہے کہ یہاں نیچے کا کل نتیجہ غلطی کی پوزیشن کو ظاہر کرتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 490.7 }, { - "input": "To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all of the logic on the receiver's end. ", + "input": "sage bit is turned on to a 1, and then collect these positions into one big column and take the XOR. You can probably guess that the four bits sitting at the bottom as a resu ", "translatedText": "یہ بتانے کے لیے کہ یہ کتنا خوبصورت ہے، میں دکھاتا ہوں کہ ازگر کے کوڈ کی ایک لائن جس کا میں نے پہلے حوالہ دیا تھا، جو وصول کنندہ کے سرے پر موجود تقریباً تمام منطق کو پکڑ لے گی۔", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. ", + "input": "lt are the same as the four parity checks we've come to know and love, but take a moment to actually think about why exactly. This last column, for example, is counting all of the positions whose last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. ", "translatedText": "ہم ڈیٹا بلاک کی نقل کرنے کے لیے 16 1s اور 0s کی بے ترتیب صف بنا کر شروع کریں گے، اور میں اسے نام کے بٹس دوں گا، لیکن یقیناً عملی طور پر یہ وہ چیز ہوگی جو ہم بھیجنے والے سے وصول کر رہے ہیں، اور اس کے بجائے بے ترتیب ہونے کی وجہ سے یہ 5 برابری بٹس کے ساتھ 11 ڈیٹا بٹس لے کر جائے گا۔", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15. ", + "input": "ht half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit, so these 8 bits already have an even pari Likewise, the next column counts how many positions are in the second parity group, the positions whose second to las ", "translatedText": "لہذا اگر ہم پھر ایک فہرست بناتے ہیں جو ان تمام جوڑوں پر نظر آتی ہے، جوڑے جو i کی طرح نظر آتے ہیں، اور پھر ہم صرف i ویلیو کو نکالتے ہیں، صرف انڈیکس، ٹھیک ہے یہ اتنا دلچسپ نہیں ہے، ہم صرف ان انڈیکس کو 0 سے 15 تک واپس کر دیتے ہیں۔. ", "model": "google_nmt", "n_reviews": 0, @@ -520,7 +520,7 @@ "end": 552.66 }, { - "input": "In this case it looks like those positions are 0, 4, 6, 9, etc. ", + "input": "ve on the same thing we've been doing. but for right now we're going to assume ", "translatedText": "اس معاملے میں ایسا لگتا ہے کہ وہ پوزیشنیں 0، 4، 6، 9، وغیرہ ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 567.24 }, { - "input": "To do this in Python, let me first import a couple helpful functions. ", + "input": "The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 000 ", "translatedText": "Python میں ایسا کرنے کے لیے، مجھے پہلے ایک دو مددگار فنکشنز درآمد کرنے دیں۔", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 578.7 }, { - "input": "This basically eats its way through the list, taking XORs along the way. ", + "input": "es us a really nice way to think about why these four resulting bits at the bottom directly spell out the pos ", "translatedText": "یہ بنیادی طور پر فہرست کے ذریعے اپنا راستہ کھاتا ہے، راستے میں XORs کو لے جاتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 582.68 }, { - "input": "If you prefer, you can explicitly write out that XOR function without having to import it from anywhere. ", + "input": "ition of an error. Let's say you detect an error among the odd columns, and among the right half. It necessarily means the error is somewhere in th ", "translatedText": "اگر آپ چاہیں تو، آپ واضح طور پر اس XOR فنکشن کو کہیں سے درآمد کیے بغیر لکھ سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 630.22 }, { - "input": "Isn't that neat? ", + "input": "an error that changes a 1 to a 0. You see, if you add a bit string together twice, it's the same as ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 641.06 }, { - "input": "And there's nothing special about the size 16 here. ", + "input": "And that effect, again, is that the total result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me sh ", "translatedText": "اور یہاں سائز 16 کے بارے میں کچھ خاص نہیں ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 649.86 }, { - "input": "Needless to say, there is more code to write here, like doing the meta parity check to detect 2-bit errors, but the idea is that almost all of the core logic from our scheme comes down to a single XOR reduction. ", + "input": "ferenced before, which will capture almost all of the logic on the receiver's end. We'll start by creating a random array of 16 ones and zeros to simulate the data block, and I'll go ahead and give it the name bits, but of course in practice this would be something that we're receiving f ", "translatedText": "یہ کہنے کی ضرورت نہیں کہ یہاں لکھنے کے لیے مزید کوڈ موجود ہیں، جیسے کہ 2 بٹ کی غلطیوں کا پتہ لگانے کے لیے میٹا پیریٹی چیک کرنا، لیکن خیال یہ ہے کہ ہماری اسکیم سے تقریباً تمام بنیادی منطق ایک XOR میں کمی پر آتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 690.5 }, { - "input": "The first one is easiest to actually do by hand, and I think it does a better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the log of the size of the block, or in other words, it grows one bit at a time as the block size doubles. ", + "input": "l out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on. In this case it looks like those positions are 0, 4, 6, 9, etc. Remember, what ", "translatedText": "پہلا اصل میں ہاتھ سے کرنا سب سے آسان ہے، اور مجھے لگتا ہے کہ یہ ان سب کے بنیادی وجدان کو پیدا کرنے میں ایک بہتر کام کرتا ہے، جو کہ ایک غلطی کا پتہ لگانے کے لیے درکار معلومات کا تعلق بلاک کے سائز کے لاگ سے ہے۔، یا دوسرے الفاظ میں، یہ ایک وقت میں تھوڑا سا بڑھتا ہے کیونکہ بلاک کا سائز دوگنا ہوتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need. ", + "input": "we want is to collect together all of those positions, the positions of the bits that are turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. ", "translatedText": "یہاں متعلقہ حقیقت یہ ہے کہ وہ معلومات براہ راست اس بات سے مطابقت رکھتی ہے کہ ہمیں کتنی بے کاری کی ضرورت ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there. ", + "input": "looks like if we do this on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but you could write a function wh ", "translatedText": "مثال کے طور پر، ہم نے دیکھا کہ 256 بٹس کے ساتھ، آپ اس جگہ کا صرف 3% فالتو پن کے لیے استعمال کر رہے ہیں، اور یہ وہاں سے بہتر ہوتا جا رہا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling. ", + "input": "ere the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a state where running th ", "translatedText": "جیسے جیسے برابری بٹس کی تعداد ایک ایک کرکے بڑھتی ہے، بلاک کا سائز دوگنا ہوتا رہتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 804.3 }, { - "input": "Also, in practice, errors tend to come in little bursts, which would totally ruin a single block, so one common tactic to help spread out a burst of errors across many different blocks is to interlace those blocks, like this, before they're sent out or stored. ", + "input": "imulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the positio ", "translatedText": "اس کے علاوہ، عملی طور پر، غلطیاں تھوڑی دیر میں آتی ہیں، جو ایک ہی بلاک کو مکمل طور پر برباد کر دیتی ہیں، لہذا بہت سے مختلف بلاکس میں غلطیوں کو پھیلانے میں مدد کرنے کے لیے ایک عام حربہ یہ ہے کہ ان بلاکس کو آپس میں جوڑنا، اس طرح، ان کے ہونے سے پہلے۔بھیج دیا گیا یا ذخیرہ کیا گیا۔", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind. ", + "input": "is perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it ", "translatedText": "اس پوری کتاب میں نصف درجن کے قریب بار ہے کہ وہ لوئس پاسچر کے اقتباس کا حوالہ دیتے ہیں، قسمت ایک تیار دماغ کی حمایت کرتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 906.82 }, { - "input": "Part of the reason that clever ideas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong turns, underselling just how vast the space of explorable possibilities is at the start of a problem solving process, all of that. ", + "input": ", with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing 1 out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The first one is easiest to actually do by hand, and I think it does a better job instilling the core intuition underlying all of this, ", "translatedText": "ہوشیار خیالات دھوکہ دہی سے آسان نظر آنے کی ایک وجہ یہ ہے کہ ہم صرف حتمی نتیجہ دیکھتے ہیں، جو گندا تھا اسے صاف کرتے ہیں، تمام غلط موڑ کا کبھی ذکر نہیں کرتے، اس بات کو کم کرتے ہیں کہ کسی مسئلے کے آغاز میں قابل دریافت امکانات کی جگہ کتنی وسیع ہے۔حل کرنے کا عمل، یہ سب۔", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 922.86 }, { - "input": "But this is true in general. ", + "input": "which is that the information required to locate a single error is relat ", "translatedText": "لیکن یہ عام طور پر سچ ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 924.9 }, { - "input": "I think for some special inventions, there's a second, deeper reason that we underappreciate them. ", + "input": "ed to the log of the size of the block, or in other words, it grows one bit at a time as the block size doubles. The relevant fact here i ", "translatedText": "میرے خیال میں کچھ خاص ایجادات کے لیے، ایک دوسری، گہری وجہ ہے کہ ہم ان کی قدر نہیں کرتے۔", "model": "google_nmt", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 938.64 }, { - "input": "This was essentially concurrent with when Hamming developed his algorithm. ", + "input": "block is even, just like a normal parity check. Now, if there's a single bit error, then ", "translatedText": "یہ بنیادی طور پر اس کے ساتھ ہم آہنگ تھا جب ہیمنگ نے اپنا الگورتھم تیار کیا۔", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory. ", + "input": "the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks. However, if there's two errors, then the overall parity is going to toggle back to be And then, by the way, there is this whole other way that you s ", "translatedText": "یہ وہی بنیادی کاغذ تھا جس نے ظاہر کیا، ایک خاص معنوں میں، غلطی کی مؤثر اصلاح ہمیشہ ممکن ہے، چاہے بٹ پلٹنے کا امکان کتنا ہی زیادہ ہو، کم از کم تھیوری میں۔", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was. ", + "input": "multiply the message by one big matrix. It's kind of nice because it relates it to the broader family of linear codes, but I think that gives almost no intuition for where it comes from or how it scales. ", "translatedText": "کئی دہائیوں سے تیزی سے آگے بڑھیں، اور ان دنوں، ہم میں سے بہت سے لوگ بٹس اور معلومات کے بارے میں سوچنے میں اتنے ڈوبے ہوئے ہیں کہ یہ نظر انداز کرنا آسان ہے کہ سوچنے کا یہ طریقہ کتنا الگ تھا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes-2/vietnamese/sentence_translations.json b/2020/hamming-codes-2/vietnamese/sentence_translations.json index 6919145a8..6dcb1772d 100644 --- a/2020/hamming-codes-2/vietnamese/sentence_translations.json +++ b/2020/hamming-codes-2/vietnamese/sentence_translations.json @@ -8,7 +8,7 @@ "end": 2.56 }, { - "input": "We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this block, a receiver is going to be able to identify that there was an error, and how to fix it.", + "input": "h a CD or a DVD and still have it play back whatever it's storing? The scratch really does affect the 1s and 0s on the disk, so it reads off different data from what was We were talking about Hamming codes, a way to create a block of data where most of the bits carry a meaningful message, while a few others act as a kind of redundancy, in such a way that if any bit gets flipped, either a message bit or a redundancy bit, anything in this b", "translatedText": "Chúng ta đang nói về mã Hamming, một cách để tạo ra một khối dữ liệu trong đó hầu hết các bit mang một thông điệp có ý nghĩa, trong khi một số bit khác hoạt động như một loại dự phòng, theo cách mà nếu bất kỳ bit nào bị đảo lộn, thì đó sẽ là một thông báo. bit hoặc bit dự phòng, bất kỳ thứ gì trong khối này, bộ thu sẽ có thể xác định rằng đã xảy ra lỗi và cách khắc phục.", "model": "google_nmt", "n_reviews": 0, @@ -16,7 +16,7 @@ "end": 21.24 }, { - "input": "The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", + "input": "lock, a receiver is going to be able to identify that there was an error, and how to fix it. The basic idea presented there was how to use multiple parity checks to binary search your way down to the error.", "translatedText": "Ý tưởng cơ bản được trình bày ở đó là cách sử dụng nhiều biện pháp kiểm tra chẵn lẻ để tìm kiếm nhị phân theo cách của bạn để tìm ra lỗi.", "model": "google_nmt", "n_reviews": 0, @@ -56,7 +56,7 @@ "end": 64.08 }, { - "input": "For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1.", + "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", "translatedText": "Ví dụ: số 7 trong hệ nhị phân trông giống như 0111, về cơ bản nó có nghĩa là 4 cộng 2 cộng 1.", "model": "google_nmt", "n_reviews": 0, @@ -64,7 +64,7 @@ "end": 71.26 }, { - "input": "And notice where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last.", + "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. And notice where the position 7 sits. It does affect the first of our parity", "translatedText": "Và hãy để ý xem vị trí số 7 nằm ở đâu, nó ảnh hưởng đến nhóm đầu tiên trong số các nhóm ngang bằng của chúng ta, nhóm thứ hai và nhóm thứ ba, nhưng không ảnh hưởng đến nhóm cuối cùng.", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 81.74 }, { - "input": "So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", + "input": "groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error.", "translatedText": "Vì vậy, việc đọc kết quả của bốn lần kiểm tra đó từ dưới lên trên thực sự sẽ chỉ ra vị trí của lỗi.", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 109.88 }, { - "input": "As we put these binary labels back into their boxes, let me emphasize that they are distinct from the data that's actually being sent.", + "input": "ask feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some su", "translatedText": "Khi chúng ta đặt các nhãn nhị phân này trở lại hộp của chúng, hãy để tôi nhấn mạnh rằng chúng khác biệt với dữ liệu thực sự được gửi.", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 117.8 }, { - "input": "They're nothing more than a conceptual label to help you and me understand where the four parity groups came from.", + "input": "bset are going to be considered valid messages. As an analogy, think about correctly spelled words vs incorrectly spelled words.", "translatedText": "Chúng không gì khác hơn là một nhãn hiệu khái niệm để giúp bạn và tôi hiểu bốn nhóm ngang bằng đến từ đâu.", "model": "google_nmt", "n_reviews": 0, @@ -112,7 +112,7 @@ "end": 123.5 }, { - "input": "The elegance of having everything we're looking at be described in binary is maybe undercut by the confusion of having everything we're looking at being described in binary.", + "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. Coming up with a concrete algorithm to efficiently categorize messa", "translatedText": "Sự sang trọng của việc mọi thứ chúng ta đang xem xét được mô tả ở dạng nhị phân có thể bị giảm bớt do sự nhầm lẫn khi mọi thứ chúng ta đang xem xét được mô tả ở dạng nhị phân.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 143.22 }, { - "input": "What we get is the first of our four parity groups, which means you can interpret that first check as asking, hey, if there's an error, is the final bit in the position of that error a 1?", + "input": "be undercut by the confusion of having everything we're looking at being described in binary. It's worth it, though. Focus your attention just on that last bit of all of these labels. And then highlight the positions where that final bit is a 1. What we get is", "translatedText": "Những gì chúng tôi nhận được là nhóm đầu tiên trong số bốn nhóm chẵn lẻ, có nghĩa là bạn có thể hiểu lần kiểm tra đầu tiên đó là hỏi, này, nếu có lỗi, bit cuối cùng ở vị trí của lỗi đó có phải là 1 không?", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 174.5 }, { - "input": "And so on.", + "input": "hat error a 1? 4 special bits to come nicely packaged together, maybe at the end o", "translatedText": "Và như thế.", "model": "google_nmt", "n_reviews": 0, @@ -192,7 +192,7 @@ "end": 201.48 }, { - "input": "The first is how to systematically generalize to block sizes that are bigger powers of two.", + "input": "this scales f or larger blocks. Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position", "translatedText": "Đầu tiên là làm thế nào để khái quát hóa một cách có hệ thống các kích thước khối có lũy thừa lớn hơn bằng 2.", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 243.0 }, { - "input": "What that means is each of those parity bits sits inside one and only one of the four parity groups.", + "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a I hope this makes two things clearer.", "translatedText": "Điều đó có nghĩa là mỗi bit chẵn lẻ đó nằm bên trong một và chỉ một trong bốn nhóm chẵn lẻ.", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 290.22 }, { - "input": "When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off.", + "input": "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, b It's the same core logic, but s", "translatedText": "Khi bạn lấy XOR của hai bit, nó sẽ trả về 1 nếu một trong hai bit đó được bật, nhưng không trả về nếu cả hai bit được bật hoặc tắt.", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 299.36 }, { - "input": "Phrased differently, it's the parity of these two bits.", + "input": "olving a different problem, and applied to a 64-squared chessboard. The second thing I hope this makes clear is why our parity", "translatedText": "Nói cách khác, đó là tính chẵn lẻ của hai bit này.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.44 }, { - "input": "It's like addition, but where you never carry.", + "input": "These are the positions whose binary representation has just a single bit turned on.", "translatedText": "Nó giống như sự bổ sung, nhưng bạn không bao giờ mang theo.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 332.94 }, { - "input": "The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + "input": "odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would", "translatedText": "Điểm mấu chốt đối với bạn và tôi là việc lấy XOR của nhiều chuỗi bit khác nhau thực sự là một cách để tính toán các bản nhại của một loạt các nhóm riêng biệt, giống như với các cột, tất cả trong một cú trượt.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 347.14 }, { - "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming code algorithm as all being packaged together into one single operation.", + "input": "still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. Keep in mind", "translatedText": "Điều này mang lại cho chúng ta một cách khá thú vị để suy nghĩ về nhiều lần kiểm tra tính chẵn lẻ từ thuật toán mã Hamming của chúng ta khi tất cả được gói gọn lại thành một thao tác duy nhất.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 405.76 }, { - "input": "Does that make sense?", + "input": "For exa We also commonly talk about the XOR of two different bit strings, which basically does this component", "translatedText": "Điều đó có ý nghĩa?", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.8 }, { - "input": "Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on.", + "input": "by component. It's like addition, but where you never carry. Again, the more mathematically inclined might prefer to think of this as adding two vectors and reducing mod 2. If you open up some Python right now, and you a", "translatedText": "Tương tự, cột tiếp theo đếm số lượng vị trí trong nhóm chẵn lẻ thứ hai, các vị trí có bit thứ hai đến bit cuối cùng là 1 và cũng được đánh dấu, v. v.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 436.56 }, { - "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the position of an error.", + "input": "ou and me is that taking the XOR of many different bit strings is effectively a way to compute the parities of a bunch of separate groups, like so with the columns, all in one fell swoop.", "translatedText": "Bây giờ khi chúng ta đã có nó như thế này, điều này mang lại cho chúng ta một cách thực sự hay để suy nghĩ về lý do tại sao bốn bit kết quả ở phía dưới này trực tiếp đánh vần vị trí của một lỗi.", "model": "google_nmt", "n_reviews": 0, @@ -432,7 +432,7 @@ "end": 447.58 }, { - "input": "Let's say some bit in this block gets toggled from a 0 to a 1.", + "input": "This gives us a rather snazzy way to think about the multiple parity checks from our Hamming", "translatedText": "Giả sử một số bit trong khối này được chuyển từ 0 sang 1.", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 451.86 }, { - "input": "What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error.", + "input": "code algorithm as all being packaged together into one single operation. Though at first glance it does look very different. Specifically, write down the 16 positions in binary, like we had before, and now highlight only the positions where the m", "translatedText": "Điều đó có nghĩa là vị trí của bit đó hiện sẽ được bao gồm trong tổng XOR, làm thay đổi tổng từ 0 thành thay vào đó là giá trị mới được đưa vào, vị trí của lỗi.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 501.44 }, { - "input": "We'll start by creating a random array of 16 1s and 0s to simulate the data block, and I'll give it the name bits, but of course in practice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits.", + "input": "se last bit is a 1, but we're already limited only to the highlighted positions, so it's effectively counting how many highlighted positions came from the first parity group. hat we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit,", "translatedText": "Chúng ta sẽ bắt đầu bằng cách tạo một mảng ngẫu nhiên gồm 16 số 1 và 0 để mô phỏng khối dữ liệu và tôi sẽ đặt tên cho nó là các bit, nhưng tất nhiên trong thực tế đây sẽ là thứ chúng tôi nhận được từ người gửi và thay vì ngẫu nhiên nó sẽ mang 11 bit dữ liệu cùng với 5 bit chẵn lẻ.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 527.0 }, { - "input": "So if we then create a list that loops over all of these pairs, pairs that look like i, and then we pull out just the i value, just the index, well it's not that exciting, we just get back those indices 0 through 15.", + "input": "ons are in the second parity group, the positions whose second to last bit is a 1, and which are also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. but for right now we're going to assume that there's at m", "translatedText": "Vì vậy, nếu sau đó chúng ta tạo một danh sách lặp lại tất cả các cặp này, các cặp trông giống i và sau đó chúng ta chỉ lấy ra giá trị i, chỉ số, thì điều đó không thú vị lắm, chúng ta chỉ lấy lại các chỉ số đó từ 0 đến 15.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 541.34 }, { - "input": "But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then it pulls out only the positions where the corresponding bit is turned on.", + "input": "ost one error in the entire block. Things break down completely for more than that. The sender is responsible for toggling some of the special parity bits to make sure the sum works out to be 0000.", "translatedText": "Nhưng nếu chúng ta thêm vào điều kiện chỉ thực hiện điều này nếu bit, nghĩa là nếu bit đó là 1 chứ không phải 0, thì nó chỉ lấy ra các vị trí mà bit tương ứng được bật.", "model": "google_nmt", "n_reviews": 0, @@ -576,7 +576,7 @@ "end": 601.28 }, { - "input": "We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0.", + "input": "he right What that means is that the position of that bit is now going to be included in the total XOR, which changes the sum from being 0 to instead being this newly included value, the position of the error. Slightly less obviously, the same is true if there's an error that changes a 1 to a 0.", "translatedText": "Chúng tôi sẽ không làm điều đó ở đây, nhưng bạn có thể viết một hàm trong đó người gửi sử dụng biểu diễn nhị phân đó để đặt bốn bit chẵn lẻ nếu cần, cuối cùng đưa khối này về trạng thái chạy dòng mã này trên danh sách đầy đủ các bit trả về một số 0.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 618.2 }, { - "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error.", + "input": "t's the same as not having it there at all, basically because in this world 1 plus 1 equals 0. So adding a copy of this position to the total sum has the same effect as removing it. And that effect, again, is that the tota", "translatedText": "Điều thú vị là nếu chúng ta chuyển đổi bất kỳ bit nào trong danh sách này, mô phỏng một lỗi ngẫu nhiên do nhiễu, thì nếu bạn chạy cùng dòng mã này, nó sẽ in ra lỗi đó.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 631.52 }, { - "input": "You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", + "input": "l result at the bottom here spells out the position of the error. To illustrate how elegant this is, let me show that one line of Python code I referenced before, which will capture almost all", "translatedText": "Bạn có thể lấy khối này bất ngờ, chạy dòng đơn này trên đó và nó sẽ tự động đưa ra vị trí của lỗi hoặc số 0 nếu không có.", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 663.76 }, { - "input": "Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go.", + "input": "rom a sender, and instead of being random, it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", "translatedText": "Bây giờ, tùy thuộc vào sự thoải mái của bạn với nhị phân, XOR và phần mềm nói chung, bạn có thể thấy quan điểm này hơi khó hiểu hoặc thanh lịch và đơn giản hơn nhiều đến mức bạn đang tự hỏi tại sao chúng ta không bắt đầu với nó ngay từ đầu. -đi.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 678.42 }, { - "input": "Loosely speaking, the multiple parity check perspective is easier to think about when implementing Hamming codes in hardware very directly, and the XOR perspective is easiest to think about when doing it in software, from kind of a higher level.", + "input": "So if we then create a list that loops over all of these pairs, pairs that look like i,bit, and then we pull out just the i value, just the index, well, it's not that exciting, we just get back those indices 0 through 15. But if we add on the condition to only do this if bit, meaning if that bit is a 1 and not a 0, well then", "translatedText": "Nói một cách lỏng lẻo, phối cảnh kiểm tra tính chẵn lẻ sẽ dễ nghĩ đến hơn khi triển khai mã Hamming trong phần cứng một cách trực tiếp và phối cảnh XOR là dễ nghĩ đến nhất khi thực hiện nó trong phần mềm, từ cấp độ cao hơn.", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 710.0 }, { - "input": "The relevant fact here is that that information directly corresponds to how much redundancy we need.", + "input": "e turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", "translatedText": "Thực tế có liên quan ở đây là thông tin đó tương ứng trực tiếp với mức độ dư thừa mà chúng ta cần.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 726.54 }, { - "input": "And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix.", + "input": "fact, the astute among you might even notice a connection between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself t So at the moment, it", "translatedText": "Và sau đó, nhân tiện, có một cách hoàn toàn khác mà đôi khi bạn thấy mã Hamming được trình bày, trong đó bạn nhân thông điệp với một ma trận lớn.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 743.06 }, { - "input": "And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size.", + "input": "turns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the 4 parity bits as needed, ultimately getting this block to a", "translatedText": "Và nói về việc chia tỷ lệ, bạn có thể nhận thấy rằng hiệu quả của sơ đồ này chỉ trở nên tốt hơn khi chúng tôi tăng kích thước khối.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 751.16 }, { - "input": "For example, we saw that with 256 bits, you're using only 3% of that space for redundancy, and it just keeps getting better from there.", + "input": "state where running this line of code on the full list of bits returns a 0. ed, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, w", "translatedText": "Ví dụ: chúng tôi thấy rằng với 256 bit, bạn chỉ sử dụng 3% dung lượng đó để dự phòng và nó sẽ ngày càng tốt hơn từ đó.", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 762.66 }, { - "input": "As the number of parity bits grows one by one, the block size keeps doubling.", + "input": "ith the same group of four questions. It doesn't really matter, since at the end of the d Now what's cool is that if we toggle any on", "translatedText": "Khi số lượng bit chẵn lẻ tăng lên từng cái một, kích thước khối tiếp tục tăng gấp đôi.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 767.34 }, { - "input": "And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", + "input": "e of the bits in this list, simulating a random error from noise, then if you run this same line of code, it prints out that error. correction bits are just riding along. But protecting those bits as well is something that natural You could get this block", "translatedText": "Và nếu bạn coi điều đó đến mức cực đoan, bạn có thể có một khối với một triệu bit, trong đó bạn thực sự sẽ chơi 20 câu hỏi với các kiểm tra chẵn lẻ của mình và nó chỉ sử dụng 21 bit chẵn lẻ.", "model": "google_nmt", "n_reviews": 0, @@ -736,7 +736,7 @@ "end": 787.06 }, { - "input": "The problem, of course, is that with a larger block, the probability of seeing more than one or two bit errors goes up, and Hamming codes do not handle anything beyond that.", + "input": "he position of an error, or a 0 if there wasn't any. hat these questions are in just a minute or two. Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. The first thing, except for those eight highlighted pari", "translatedText": "Tất nhiên, vấn đề là với khối lớn hơn, xác suất nhìn thấy nhiều hơn một hoặc hai bit lỗi sẽ tăng lên và mã Hamming không xử lý được bất kỳ điều gì ngoài điều đó.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 838.82 }, { - "input": "But that's a topic for another time.", + "input": "hecks detect an e Now depending on your comfort with binary and XORs and software in general, you may either fi", "translatedText": "Nhưng đó là một chủ đề cho một thời điểm khác.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 841.34 }, { - "input": "In his book The Art of Doing Science and Engineering, Hamming is wonderfully candid about just how meandering his discovery of this code was.", + "input": "nd this perspective a little bit confusing, or so much more elegant and simple that you're wondering why we didn't just start with it from the get-go. tended, then it either means there wa", "translatedText": "Trong cuốn sách Nghệ thuật thực hiện khoa học và kỹ thuật, Hamming đã thẳng thắn một cách tuyệt vời về việc khám phá ra mật mã này của ông đã quanh co như thế nào.", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 881.22 }, { - "input": "There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind.", + "input": "better job instilling the core intuition underlying all of this, which is that the information required to locate a single error is related to the", "translatedText": "Có khoảng nửa tá lần trong suốt cuốn sách này ông đề cập đến câu nói của Louis Pasteur, may mắn sẽ đến với một tâm trí đã chuẩn bị sẵn sàng.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 901.3 }, { - "input": "But you shouldn't fool yourself into thinking that they actually are obvious, because they definitely aren't.", + "input": "les. The relevant fact here is that that information directly corresponds to how much redundancy we need. at 0th one so that the parity of the full block is even, just like a normal parity check.", "translatedText": "Nhưng bạn không nên tự lừa dối mình rằng chúng thực sự hiển nhiên, bởi vì chúng chắc chắn không phải vậy.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 942.54 }, { - "input": "This was the same foundational paper that showed, in a certain sense, that efficient error correction is always possible, no matter how high the probability of bit flips, at least in theory.", + "input": "or where it comes from or how it scales. And speaking of scaling, you might notice that the efficiency of this scheme only gets better as we increase the block size. For example, we saw that with 256 bits, you're using only 3% of that space for redu", "translatedText": "Theo một nghĩa nào đó, đây cũng chính là bài báo nền tảng đã chỉ ra rằng luôn có thể sửa lỗi hiệu quả, bất kể xác suất lật bit cao đến đâu, ít nhất là trên lý thuyết.", "model": "google_nmt", "n_reviews": 0, @@ -896,7 +896,7 @@ "end": 961.16 }, { - "input": "Fast forward several decades, and these days, many of us are so immersed in thinking about bits and information that it's easy to overlook just how distinct this way of thinking was.", + "input": "As the number of parity bits grows one by one, the block size keeps doubling. And if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits.", "translatedText": "Nhiều thập kỷ trôi qua nhanh chóng, và ngày nay, nhiều người trong chúng ta quá đắm chìm trong việc suy nghĩ về các bit và thông tin đến mức chúng ta dễ dàng bỏ qua cách suy nghĩ này khác biệt như thế nào.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/arabic/sentence_translations.json b/2020/hamming-codes/arabic/sentence_translations.json index 7405ab559..436e6bca8 100644 --- a/2020/hamming-codes/arabic/sentence_translations.json +++ b/2020/hamming-codes/arabic/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit. ", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notice ", "translatedText": "والاستراتيجية البسيطة لتصحيح أي جزء يتم قلبه هي تخزين ثلاث نسخ من كل جزء. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9! ", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how ", "translatedText": "على سبيل المثال، باستخدام الطريقة التي ستتعرف عليها في هذا الفيديو، يمكنك تخزين بياناتك في كتل بحجم 256 بت، حيث تستخدم كل كتلة 9 بت، 9! ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want. ", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan ", "translatedText": "لتكون بمثابة نوع من التكرار، والـ 247 بت الأخرى حرة في حمل أي رسالة أو بيانات ذات معنى تريدها. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 112.66 }, { - "input": "And honestly, that feels like magic. ", + "input": "m e emphasize that they are distinct from the data that's actually being sent. They're noth ", "translatedText": "وبصراحة، هذا يبدو وكأنه سحر. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 122.86 }, { - "input": "We'll talk a little bit later about how this scales for blocks with different sizes. ", + "input": "that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to identify that there was an error and precisely where i ", "translatedText": "سنتحدث بعد قليل عن كيفية قياس ذلك للكتل ذات الأحجام المختلفة. ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 141.94 }, { - "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. ", + "input": "that there were two errors, though it won't know how to fix them. We'll talk a little bit later about how this scales for blocks with different sizes. where that's a 1, you get the second parity group from our scheme ", "translatedText": "الهدف هنا هو إعطاؤك فهمًا شاملاً لأحد أقدم الأمثلة، والمعروف باسم كود هامينغ. ", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 206.94 }, { - "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. ", + "input": "e scheme is going to be before I tell you. Also, if you want your understanding to get down to the hardware level, Ben Eater has made a video in conjunction with this one ", "translatedText": "عندما يتم تغيير رسالة صالحة، يكون المتلقي مسؤولاً عن تصحيح ما يراه إلى أقرب جار صالح، كما قد تفعل مع خطأ مطبعي. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. ", + "input": "st how impossible this task feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is tha ", "translatedText": "نظرًا لأن الإحباط هو بوتقة الاختراع، فقد سئم كثيرًا لدرجة أنه اخترع أول رمز لتصحيح الأخطاء في العالم. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 248.42 }, { - "input": "There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. ", + "input": "t in a vast space of all possible messages, only some subset are going to be considered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words. also see th ", "translatedText": "هناك العديد من الطرق المختلفة لتأطير رموز هامينج، ولكن كمرحلة أولى سنتناول الأمر بالطريقة التي فكر بها هامينج نفسه. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 255.38 }, { - "input": "Let's use an example that's simple, but not too simple, a block of 16 bits. ", + "input": "is in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. ", "translatedText": "دعونا نستخدم مثالاً بسيطًا، ولكن ليس بسيطًا للغاية، وهو كتلة مكونة من 16 بت. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 260.94 }, { - "input": "We'll number the positions of these bits from 0 up to 15. ", + "input": "Once you understand that these parity checks that we've focused so much of our time on are nothing ", "translatedText": "سنقوم بترقيم مواضع هذه البتات من 0 إلى 15. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 273.0 }, { - "input": "The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. ", + "input": "binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. ", "translatedText": "كلمة زائدة عن الحاجة هنا لا تعني ببساطة النسخ، ففي نهاية المطاف، هذه البتات الأربعة لا تمنحنا مساحة كافية لنسخ البيانات بشكل أعمى. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 287.28 }, { - "input": "You might expect these 4 special bits to come nicely packaged together, maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. ", + "input": "r. When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the programs he kept putting through it kept failing, because every now and then a ", "translatedText": "قد تتوقع أن تأتي هذه القطع الأربعة الخاصة مجمعة بشكل جيد معًا، ربما في النهاية أو شيء من هذا القبيل، ولكن كما سترى، فإن وضعها في مواضع تمثل قوى 2 يسمح بشيء أنيق حقًا في النهاية. ", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors. ", + "input": "'s first error correction code. There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. ", "translatedText": "مثل أي خوارزمية لتصحيح الأخطاء، سيتضمن ذلك لاعبين، مرسل مسؤول عن تعيين هذه البتات الأربعة الخاصة، ومتلقي مسؤول عن إجراء نوع ما من الفحص وتصحيح الأخطاء. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", + "input": "e so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy, not adding any new information, but adding resilience. ", "translatedText": "إذن هذا هو الإعداد، ولكن قبل أن نتمكن من التعمق في الأمر، نحتاج إلى التحدث عن فكرة ذات صلة كانت جديدة في ذهن هامينج في وقت اكتشافه، وهي طريقة تتيح لك اكتشاف أي أخطاء في البتات، ولكن ليس تصحيحها، كما هو معروف. في الأعمال التجارية باعتبارها التحقق من التكافؤ. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.82 }, { - "input": "The only job of this special bit is to make sure that the total number of 1s in the message is an even number. ", + "input": "that make sense? Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and w ", "translatedText": "الوظيفة الوحيدة لهذا البت الخاص هي التأكد من أن العدد الإجمالي للآحاد في الرسالة هو رقم زوجي. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.28 }, { - "input": "So for example right now, that total number of 1s is 7, that's odd, so the sender needs to flip that special bit to be a 1, making the count even. ", + "input": "hich a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it goes from here. The sender is responsible for toggling ", "translatedText": "على سبيل المثال، في الوقت الحالي، إجمالي عدد الآحاد هو 7، وهذا أمر غريب، لذلك يحتاج المرسل إلى قلب هذا البت الخاص ليصبح 1، مما يجعل العدد متساويًا. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd. ", + "input": "sitio Like any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. ", "translatedText": "لاحظ أنه إذا تم قلب أي جزء من هذه الرسالة، إما من 0 إلى 1 أو من 1 إلى 0، فإنه يغير العدد الإجمالي للآحاد من زوجي إلى فردي. ", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. ", + "input": "Of course, the words sender and receiver really refer to machines or software that's doing checks, and the idea of a message is meant really broadly, to include things like storage. After all, storing data is the same thing as sending a message, just from the past ", "translatedText": "لذلك، إذا كنت المتلقي، ونظرت إلى هذه الرسالة، ورأيت عددًا فرديًا من 1، فيمكنك التأكد من حدوث خطأ ما، على الرغم من أنه قد لا يكون لديك أي فكرة عن مكان حدوثه. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit. ", + "input": "his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", "translatedText": "وهذه البتة الخاصة التي يستخدمها المرسل للتحكم في التكافؤ تسمى بت التكافؤ. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 485.44 }, { - "input": "Instead, the goal is to come up with a scheme that's robust up to a certain maximum number of errors, or maybe to reduce the probability of a false positive like this. ", + "input": "en kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information. ctice t ", "translatedText": "بدلاً من ذلك، الهدف هو التوصل إلى مخطط قوي يصل إلى حد أقصى معين من الأخطاء، أو ربما لتقليل احتمالية حدوث نتيجة إيجابية كاذبة مثل هذا. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 495.38 }, { - "input": "Parity checks on their own are pretty weak, but by distilling the idea of change across a full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. ", + "input": "his would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running f ", "translatedText": "تعتبر عمليات التحقق من التكافؤ في حد ذاتها ضعيفة جدًا، ولكن من خلال استخلاص فكرة التغيير عبر رسالة كاملة وصولاً إلى جزء واحد، فإن ما يقدمونه لنا هو لبنة بناء قوية لمخططات أكثر تعقيدًا. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error. ", + "input": "rom 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of 1s is known as its parity. o collect together all of those positions, the positions of the bits that are ", "translatedText": "على سبيل المثال، بينما كان هامينج يبحث عن طريقة لتحديد مكان حدوث الخطأ، وليس فقط مكان حدوثه، كانت رؤيته الرئيسية هي أنه إذا قمت بتطبيق بعض عمليات التحقق من التكافؤ ليس على الرسالة الكاملة، ولكن على مجموعات فرعية معينة مختارة بعناية، فيمكنك أن تسأل سلسلة أكثر دقة من الأسئلة التي تحدد موقع أي خطأ بت واحد. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 590.68 }, { - "input": "The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", + "input": "his on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but ", "translatedText": "الفحص الثاني يكون من بين 8 بتات في النصف الأيمن من الشبكة، على الأقل كما رسمناها هنا. ", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.3 }, { - "input": "This time we might use position 2 as a parity bit, so these 8 bits already have an even parity, and the sender can feel good leaving that bit number 2 unchanged. ", + "input": "you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this ", "translatedText": "هذه المرة قد نستخدم الموضع 2 كبت تكافؤ، وبالتالي فإن هذه البتات الثمانية لها بالفعل تكافؤ متساوي، ويمكن أن يشعر المرسل بالارتياح عند ترك هذا البت رقم 2 دون تغيير. ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half. ", + "input": "simulating a random error from noise, then if you run this same line of code, it print s out that error. Isn't that neat? ", "translatedText": "وإلا فهذا يعني أنه لا يوجد خطأ، أو أن الخطأ موجود في مكان ما في النصف الأيسر. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit. ", + "input": "own to a single XOR reduction. Now, depending on your comfort with binary and XORs and software in general, you may eithe ", "translatedText": "سيكون هناك فحص للتكافؤ في الصفوف الفردية، باستخدام الموضع 4 كبت تكافؤ. ", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0. ", + "input": "r find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a wa ", "translatedText": "لذلك في هذا المثال، هذه المجموعة لديها بالفعل تكافؤ زوجي، لذا سيتم تعيين البت 4 على 0. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit. ", + "input": "y to identify where an error happened, not just that it happened, his key insight was that if you apply some parity chec ", "translatedText": "وأخيرًا، هناك فحص تكافؤ في الصفين السفليين، باستخدام الموضع 8 كبت تكافؤ. ", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 701.84 }, { - "input": "As an example, imagine that during the transmission there's an error at, say, position 3. ", + "input": "ion of any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no ", "translatedText": "على سبيل المثال، تخيل أنه أثناء الإرسال، حدث خطأ في الموضع 3 على سبيل المثال. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 720.54 }, { - "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. ", + "input": "here is that that information directly corresponds to how much redundancy we need. That's really what runs against most people's knee-jerk reaction Then, if an error is detected, it gives the receiv ", "translatedText": "وهذا يتيح للمتلقي تحديد الخطأ حتى الصف الأول، وهو ما يعني بالضرورة الموضع 3، حتى يتمكنوا من إصلاح الخطأ. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 727.52 }, { - "input": "You might enjoy taking a moment to convince yourself that the answers to these four questions really will always let you pin down a specific location, no matter where they turn out to be. ", + "input": "er a little more information about where specifically the error is, namely that it's in an odd position. ent to errors, where usually copying the whole message is the first instinct that comes to min ", "translatedText": "قد تستمتع بتخصيص بعض الوقت لإقناع نفسك بأن الإجابات على هذه الأسئلة الأربعة ستسمح لك دائمًا بتحديد موقع معين، بغض النظر عن مكان تواجدهم. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 743.06 }, { - "input": "And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spoil it. ", + "input": "then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix. It's kind of nice because it relates ", "translatedText": "وإذا قمت بذلك، دعني أؤكد مرة أخرى، توقف مؤقتًا، حاول بنفسك أن ترسم الارتباط قبل أن أفسده. ", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 773.1 }, { - "input": "But protecting those bits as well is something that naturally falls out of the scheme as a byproduct. ", + "input": "more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group. ", "translatedText": "لكن حماية تلك البتات أيضًا هو أمر يخرج بشكل طبيعي عن المخطط كمنتج ثانوي. ", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales. ", + "input": "Here let's just choose position 1. For the example shown, the pari ", "translatedText": "قد تستمتع أيضًا بتوقع كيفية قياس هذا الأمر. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 781.76 }, { - "input": "If we used a block of size 256 bits, for example, in order to pin down a location, you need only eight yes or no questions to binary search your way down to some specific spot. ", + "input": "ty of these 8 bits is currently odd, so the sender is responsible for toggling that parity bit, and now it's even. This is only 1 out of 4 parity checks that we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", "translatedText": "إذا استخدمنا كتلة بحجم 256 بت، على سبيل المثال، لتحديد موقع ما، فأنت تحتاج فقط إلى ثمانية أسئلة بنعم أو لا للبحث الثنائي في طريقك إلى مكان محدد. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 813.66 }, { - "input": "The first thing, except for those eight highlighted parity bits, can be whatever you want it to be, carrying whatever message or data you want. ", + "input": "the sender can feel good leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your pari ", "translatedText": "أول شيء، باستثناء تلك البتات الثمانية المتماثلة المميزة، يمكن أن تكون كما تريد، وتحمل أي رسالة أو بيانات تريدها. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 821.0 }, { - "input": "The 8 bits are redundant in the sense that they're completely determined by the rest of the message, but it's in a much smarter way than simply copying the message as a whole. ", + "input": "ty checks, and it uses only 21 parity bits. And if you step back to think about looking at a million bits and locating a single error, that genuinely feels crazy. The problem, Otherwise, it means either there's ", "translatedText": "تعتبر البتات الثمانية زائدة عن الحاجة، بمعنى أنه يتم تحديدها بالكامل من خلال بقية الرسالة، ولكنها بطريقة أكثر ذكاءً من مجرد نسخ الرسالة ككل. ", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost. ", + "input": "or the error is somewhere on the left half. ", "translatedText": "حسنًا تقريبًا. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0. ", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you ", "translatedText": "حسنًا، المشكلة الوحيدة هنا هي أنه إذا لم تكتشف أي من عمليات التحقق من التكافؤ وجود خطأ، مما يعني أن المجموعات الفرعية المحددة خصيصًا والمكونة من 8 بتات جميعها لها تماثلات زوجية، تمامًا كما قصد المرسل، فهذا يعني إما أنه لم يكن هناك خطأ على الإطلاق أو أنه يضيق بنا إلى الموضع 0. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 915.54 }, { - "input": "Here's how it works. ", + "input": "er block. But it also might simply mean there's no error at ", "translatedText": "وإليك كيف يعمل. ", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 952.7 }, { - "input": "Isn't that clever? ", + "input": "s today. There are like half a dozen times throughout this book that he refer ", "translatedText": "أليس هذا ذكيا؟ ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them. ", + "input": "ences the Louis Pasteur quote, luck favors a prepared mind. Cl In this case, it looks like the sender needs to turn that bit 8 on in order to give the group even parity. Part of the reason that clever ideas look deceptively ", "translatedText": "على الرغم من أننا لا نستطيع تصحيح تلك الأخطاء ذات البتتين، إلا أنه بمجرد إعادة البتة الصفرية المزعجة هذه إلى العمل، فإنها تتيح لنا اكتشافها. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 965.22 }, { - "input": "Technically speaking, you now have a full description of what a Hamming code does, at least for the example of a 16-bit block. ", + "input": "nal result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the transmission there's an error at, say, position 3. ", "translatedText": "من الناحية الفنية، لديك الآن وصف كامل لما تفعله كود هامينج، على الأقل بالنسبة لمثال كتلة 16 بت. ", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it! ", + "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position ", "translatedText": "المضي قدما، في الواقع القيام بذلك! ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block. ", + "input": "3, so they can fix the error. You might enjoy taking a moment to convince yourself that the answers to these four ", "translatedText": "دعونا نتوقف مؤقتًا ونحاول تجميع هذه الكتلة معًا. ", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1007.02 }, { - "input": "Okay, you ready? ", + "input": "questions really will always let you pin down ", "translatedText": "حسنًا، هل أنت مستعد؟ ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0. ", + "input": "n between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spo ", "translatedText": "أنت بحاجة إلى أن يكون لهذه المجموعة تكافؤ متساوي، وهو ما يحدث بالفعل، لذا يجب عليك تعيين بت التكافؤ هذا في الموضع 1 ليكون 0. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1037.88 }, { - "input": "The group after that starts with an odd parity, so again you should have set its parity bit to 1. ", + "input": "ts affected, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, ", "translatedText": "تبدأ المجموعة بعد ذلك بتكافؤ فردي، لذلك يجب عليك مرة أخرى ضبط بت التكافؤ الخاص بها على 1. ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1079.78 }, { - "input": "What I'm going to do is change either 0, 1, or 2 of the bits in that block, and then ask you to figure out what it is that I did. ", + "input": "y eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set the app ", "translatedText": "ما سأفعله هو تغيير إما 0 أو 1 أو 2 من البتات في تلك الكتلة، ثم أطلب منك معرفة ما فعلته. ", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1107.91 }, { - "input": "The next check gives us an odd number, telling us both that there's at least one error, and narrowing us down into this specific column. ", + "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever you want it to be, carrying whatever message or data you want. ", "translatedText": "الفحص التالي يعطينا رقمًا فرديًا، ويخبرنا بوجود خطأ واحد على الأقل، ويضيق نطاقنا في هذا العمود المحدد. ", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off. ", + "input": ". And still, for so little given up, you would be able to identify and fix any single bit error. Well, almost. Okay, so the one ", "translatedText": "إذا كان هناك ثلاثة أو أكثر، كل الرهانات ملغاة. ", "model": "google_nmt", "n_reviews": 0, @@ -1240,7 +1240,7 @@ "end": 1163.17 }, { - "input": "You see, what I haven't told you yet is just how elegant this algorithm really is, how simple it is to get a machine to point to the position of an error, how to systematically scale it, and how we can frame all of this as one single operation rather than multiple separate parity checks. ", + "input": "You see, with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing one out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The solution here is actually pretty simple. Just forget about that zeroth bit entirely. So when we do our four parity checks and ", "translatedText": "كما ترى، ما لم أخبرك به بعد هو مدى أناقة هذه الخوارزمية حقًا، ومدى بساطة جعل الآلة تشير إلى موضع الخطأ، وكيفية قياسه بشكل منهجي، وكيف يمكننا تأطير كل ذلك هذا كعملية واحدة بدلاً من عمليات التحقق من التكافؤ المنفصلة المتعددة. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/bengali/sentence_translations.json b/2020/hamming-codes/bengali/sentence_translations.json index 9ef714ff0..c97af3efb 100644 --- a/2020/hamming-codes/bengali/sentence_translations.json +++ b/2020/hamming-codes/bengali/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit. ", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notice ", "translatedText": "এবং ফ্লিপ করা যে কোনও বিট সংশোধন করার একটি সহজ কৌশল হ'ল প্রতিটি বিটের তিনটি কপি সংরক্ষণ করা।", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9! ", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how ", "translatedText": "উদাহরণস্বরূপ, এই ভিডিওটি সম্পর্কে আপনি যে পদ্ধতিটি শিখবেন তা ব্যবহার করে আপনি 256-বিট ব্লকে আপনার ডেটা সংরক্ষণ করতে পারেন, যেখানে প্রতিটি ব্লক 9 বিট ব্যবহার করে, 9! ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want. ", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan ", "translatedText": "এক ধরনের অপ্রয়োজনীয়তা হিসাবে কাজ করতে, এবং অন্যান্য 247 বিটগুলি আপনি যা চান তা অর্থপূর্ণ বার্তা বা ডেটা বহন করতে বিনামূল্যে।", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 112.66 }, { - "input": "And honestly, that feels like magic. ", + "input": "m e emphasize that they are distinct from the data that's actually being sent. They're noth ", "translatedText": "এবং সত্যই, যে জাদু মত মনে হয়. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 122.86 }, { - "input": "We'll talk a little bit later about how this scales for blocks with different sizes. ", + "input": "that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to identify that there was an error and precisely where i ", "translatedText": "বিভিন্ন আকারের ব্লকগুলির জন্য এটি কীভাবে স্কেল করে সে সম্পর্কে আমরা একটু পরে কথা বলব।", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 141.94 }, { - "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. ", + "input": "that there were two errors, though it won't know how to fix them. We'll talk a little bit later about how this scales for blocks with different sizes. where that's a 1, you get the second parity group from our scheme ", "translatedText": "এখানে লক্ষ্য হল আপনাকে একটি হ্যামিং কোড নামে পরিচিত প্রাচীনতম উদাহরণগুলির একটি খুব পুঙ্খানুপুঙ্খভাবে বোঝানো।", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 206.94 }, { - "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. ", + "input": "e scheme is going to be before I tell you. Also, if you want your understanding to get down to the hardware level, Ben Eater has made a video in conjunction with this one ", "translatedText": "যখনই একটি বৈধ বার্তা পরিবর্তন করা হয়, রিসিভার নিকটতম বৈধ প্রতিবেশীর কাছে যা দেখেন তা সংশোধন করার জন্য দায়ী, যেমন আপনি একটি টাইপোতে করতে পারেন।", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. ", + "input": "st how impossible this task feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is tha ", "translatedText": "হতাশা উদ্ভাবনের ক্রুসিবল হওয়ায় তিনি এতটাই বিরক্ত হয়েছিলেন যে তিনি বিশ্বের প্রথম ত্রুটি সংশোধন কোড আবিষ্কার করেছিলেন।", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 248.42 }, { - "input": "There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. ", + "input": "t in a vast space of all possible messages, only some subset are going to be considered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words. also see th ", "translatedText": "হ্যামিং কোডগুলি ফ্রেম করার বিভিন্ন উপায় রয়েছে, তবে প্রথম পাস হিসাবে আমরা এটির মধ্য দিয়ে যেতে যাচ্ছি যেভাবে হ্যামিং নিজেই সেগুলি সম্পর্কে ভেবেছিল।", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 255.38 }, { - "input": "Let's use an example that's simple, but not too simple, a block of 16 bits. ", + "input": "is in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. ", "translatedText": "আসুন একটি উদাহরণ ব্যবহার করি যা সহজ, কিন্তু খুব সহজ নয়, 16 বিটের একটি ব্লক।", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 260.94 }, { - "input": "We'll number the positions of these bits from 0 up to 15. ", + "input": "Once you understand that these parity checks that we've focused so much of our time on are nothing ", "translatedText": "আমরা এই বিটগুলির অবস্থানগুলি 0 থেকে 15 পর্যন্ত সংখ্যা করব।", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 273.0 }, { - "input": "The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. ", + "input": "binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. ", "translatedText": "এখানে অপ্রয়োজনীয় শব্দটি কেবল অনুলিপি বোঝায় না, সর্বোপরি, সেই 4 বিটগুলি আমাদের অন্ধভাবে ডেটা অনুলিপি করার জন্য যথেষ্ট জায়গা দেয় না।", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 287.28 }, { - "input": "You might expect these 4 special bits to come nicely packaged together, maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. ", + "input": "r. When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the programs he kept putting through it kept failing, because every now and then a ", "translatedText": "আপনি আশা করতে পারেন যে এই 4টি বিশেষ বিট একসাথে সুন্দরভাবে প্যাকেজ করা হবে, হয়তো শেষে বা এরকম কিছু, কিন্তু আপনি দেখতে পাবেন, তাদের এমন অবস্থানে বসতে হবে যা 2 এর ক্ষমতা সম্পন্ন এমন কিছুর জন্য অনুমতি দেয় যা শেষ পর্যন্ত সত্যিই মার্জিত।", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors. ", + "input": "'s first error correction code. There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. ", "translatedText": "যেকোনো ত্রুটি সংশোধন অ্যালগরিদমের মতো, এতে দুইজন খেলোয়াড় জড়িত থাকবে, একজন প্রেরক যিনি এই 4টি বিশেষ বিট সেট করার জন্য দায়ী, এবং একজন প্রাপক যিনি কিছু ধরণের পরীক্ষা এবং ত্রুটি সংশোধন করার জন্য দায়ী৷ অবশ্যই, প্রেরক এবং প্রাপক শব্দগুলি আসলেই মেশিন বা সফ্টওয়্যারকে বোঝায় যা সমস্ত পরীক্ষা করছে এবং একটি বার্তার ধারণাটি সত্যিই বিস্তৃতভাবে বোঝানো হয়েছে, স্টোরেজের মতো জিনিসগুলিকে অন্তর্ভুক্ত করার জন্য।", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", + "input": "e so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy, not adding any new information, but adding resilience. ", "translatedText": "সুতরাং এটি সেটআপ, তবে আমরা ডুব দেওয়ার আগে আমাদের একটি সম্পর্কিত ধারণা সম্পর্কে কথা বলতে হবে যা হ্যামিংয়ের আবিষ্কারের সময় তার মনে তাজা ছিল, একটি পদ্ধতি যা আপনাকে কোনও একক বিট ত্রুটি সনাক্ত করতে দেয়, তবে সেগুলি সংশোধন করতে পারে না, পরিচিত।একটি সমতা চেক হিসাবে ব্যবসা. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.82 }, { - "input": "The only job of this special bit is to make sure that the total number of 1s in the message is an even number. ", + "input": "that make sense? Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and w ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.28 }, { - "input": "So for example right now, that total number of 1s is 7, that's odd, so the sender needs to flip that special bit to be a 1, making the count even. ", + "input": "hich a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it goes from here. The sender is responsible for toggling ", "translatedText": "উদাহরণস্বরূপ, এই মুহূর্তে, 1s-এর মোট সংখ্যা হল 7, এটি বিজোড়, তাই প্রেরককে সেই বিশেষ বিটটিকে 1 হতে উল্টাতে হবে, গণনাকে সমান করে তুলতে হবে।", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd. ", + "input": "sitio Like any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. ", "translatedText": "লক্ষ্য করুন যদি এই বার্তার কোনো বিট ফ্লিপ করা হয়, হয় 0 থেকে 1 বা 1 থেকে 0, এটি 1s-এর মোট গণনাকে জোড় থেকে বিজোড় হতে পরিবর্তন করে।", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. ", + "input": "Of course, the words sender and receiver really refer to machines or software that's doing checks, and the idea of a message is meant really broadly, to include things like storage. After all, storing data is the same thing as sending a message, just from the past ", "translatedText": "সুতরাং আপনি যদি রিসিভার হন, আপনি এই বার্তাটি দেখেন, এবং আপনি 1 সেকেন্ডের একটি বিজোড় সংখ্যা দেখতে পান, আপনি নিশ্চিতভাবে জানতে পারেন যে কিছু ত্রুটি ঘটেছে, যদিও আপনার ধারণা নেই যে এটি কোথায় ছিল।", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit. ", + "input": "his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", "translatedText": "এবং এই বিশেষ বিট যা প্রেরক প্যারিটি নিয়ন্ত্রণ করতে ব্যবহার করে তাকে প্যারিটি বিট বলে।", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 485.44 }, { - "input": "Instead, the goal is to come up with a scheme that's robust up to a certain maximum number of errors, or maybe to reduce the probability of a false positive like this. ", + "input": "en kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information. ctice t ", "translatedText": "পরিবর্তে, লক্ষ্য হল এমন একটি স্কিম নিয়ে আসা যা একটি নির্দিষ্ট সর্বাধিক সংখ্যক ত্রুটির জন্য শক্তিশালী, অথবা এইরকম একটি মিথ্যা ইতিবাচক হওয়ার সম্ভাবনা কমাতে পারে।", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 495.38 }, { - "input": "Parity checks on their own are pretty weak, but by distilling the idea of change across a full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. ", + "input": "his would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running f ", "translatedText": "প্যারিটি চেকগুলি তাদের নিজস্বভাবে বেশ দুর্বল, কিন্তু একটি সম্পূর্ণ বার্তা জুড়ে পরিবর্তনের ধারণাটি একক বিট পর্যন্ত ছড়িয়ে দিয়ে, তারা আমাদের যা দেয় তা আরও পরিশীলিত স্কিমগুলির জন্য একটি শক্তিশালী বিল্ডিং ব্লক।", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error. ", + "input": "rom 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of 1s is known as its parity. o collect together all of those positions, the positions of the bits that are ", "translatedText": "উদাহরণস্বরূপ, যেহেতু হ্যামিং একটি ত্রুটি কোথায় ঘটেছে তা সনাক্ত করার একটি উপায় অনুসন্ধান করছিলেন, কেবল এটি ঘটেছিল তা নয়, তার মূল অন্তর্দৃষ্টি ছিল যে আপনি যদি কিছু প্যারিটি চেক সম্পূর্ণ বার্তায় নয়, তবে কিছু সাবধানে নির্বাচিত উপসেটগুলিতে প্রয়োগ করেন, আপনি জিজ্ঞাসা করতে পারেন প্রশ্নগুলির একটি আরও পরিমার্জিত সিরিজ যা যেকোনো একক বিট ত্রুটির অবস্থানকে পিন করে।", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 590.68 }, { - "input": "The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", + "input": "his on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but ", "translatedText": "এটি 4টি প্যারিটি চেকের মধ্যে মাত্র 1টি যা আমরা করব৷ দ্বিতীয় চেকটি গ্রিডের ডান অর্ধেকের 8 বিটের মধ্যে রয়েছে, অন্তত আমরা এটি এখানে আঁকছি।", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.3 }, { - "input": "This time we might use position 2 as a parity bit, so these 8 bits already have an even parity, and the sender can feel good leaving that bit number 2 unchanged. ", + "input": "you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this ", "translatedText": "এইবার আমরা একটি প্যারিটি বিট হিসাবে অবস্থান 2 ব্যবহার করতে পারি, তাই এই 8 বিটের ইতিমধ্যেই একটি সমান সমতা রয়েছে এবং প্রেরক সেই বিট নম্বর 2 অপরিবর্তিত রেখে ভাল বোধ করতে পারেন।", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half. ", + "input": "simulating a random error from noise, then if you run this same line of code, it print s out that error. Isn't that neat? ", "translatedText": "অন্যথায় এর মানে হয় কোন ত্রুটি নেই, বা ত্রুটিটি বাম অর্ধেকের কোথাও রয়েছে।", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit. ", + "input": "own to a single XOR reduction. Now, depending on your comfort with binary and XORs and software in general, you may eithe ", "translatedText": "একটি প্যারিটি বিট হিসাবে অবস্থান 4 ব্যবহার করে, বিজোড় সারি একটি সমতা চেক হতে যাচ্ছে. ", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0. ", + "input": "r find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a wa ", "translatedText": "সুতরাং এই উদাহরণে সেই গোষ্ঠীর ইতিমধ্যেই একটি সমান সমতা রয়েছে, তাই বিট 4 একটি 0 এ সেট করা হবে।", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit. ", + "input": "y to identify where an error happened, not just that it happened, his key insight was that if you apply some parity chec ", "translatedText": "এবং অবশেষে একটি প্যারিটি চেক আছে নীচের দুটি সারিতে, একটি প্যারিটি বিট হিসাবে অবস্থান 8 ব্যবহার করে।", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 701.84 }, { - "input": "As an example, imagine that during the transmission there's an error at, say, position 3. ", + "input": "ion of any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no ", "translatedText": "একটি উদাহরণ হিসাবে, কল্পনা করুন যে ট্রান্সমিশনের সময় অবস্থান 3 এ একটি ত্রুটি রয়েছে।", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 720.54 }, { - "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. ", + "input": "here is that that information directly corresponds to how much redundancy we need. That's really what runs against most people's knee-jerk reaction Then, if an error is detected, it gives the receiv ", "translatedText": "এবং এটি রিসিভারকে প্রথম সারি পর্যন্ত ত্রুটিটি চিহ্নিত করতে দেয়, যার অর্থ অগত্যা অবস্থান 3, যাতে তারা ত্রুটিটি ঠিক করতে পারে।", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 727.52 }, { - "input": "You might enjoy taking a moment to convince yourself that the answers to these four questions really will always let you pin down a specific location, no matter where they turn out to be. ", + "input": "er a little more information about where specifically the error is, namely that it's in an odd position. ent to errors, where usually copying the whole message is the first instinct that comes to min ", "translatedText": "আপনি নিজেকে সন্তুষ্ট করতে একটি মুহূর্ত উপভোগ করতে পারেন যে এই চারটি প্রশ্নের উত্তর সত্যিই সর্বদা আপনাকে একটি নির্দিষ্ট অবস্থানে পিন করতে দেয়, সেগুলি যেখানেই হোক না কেন।", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 743.06 }, { - "input": "And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spoil it. ", + "input": "then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix. It's kind of nice because it relates ", "translatedText": "এবং যদি আপনি করেন, আবার আমাকে জোর দিন, বিরতি দিন, আমি এটি নষ্ট করার আগে সংযোগটি আঁকতে চেষ্টা করুন।", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 773.1 }, { - "input": "But protecting those bits as well is something that naturally falls out of the scheme as a byproduct. ", + "input": "more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group. ", "translatedText": "তবে সেই বিটগুলিকেও রক্ষা করা এমন কিছু যা স্বাভাবিকভাবেই একটি উপজাত হিসাবে স্কিম থেকে বেরিয়ে আসে।", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales. ", + "input": "Here let's just choose position 1. For the example shown, the pari ", "translatedText": "আপনি এই দাঁড়িপাল্লা কিভাবে আন্দাজ উপভোগ করতে পারে. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 781.76 }, { - "input": "If we used a block of size 256 bits, for example, in order to pin down a location, you need only eight yes or no questions to binary search your way down to some specific spot. ", + "input": "ty of these 8 bits is currently odd, so the sender is responsible for toggling that parity bit, and now it's even. This is only 1 out of 4 parity checks that we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", "translatedText": "যদি আমরা 256 বিটের আকারের একটি ব্লক ব্যবহার করি, উদাহরণস্বরূপ, একটি অবস্থান পিন ডাউন করার জন্য, কোনো নির্দিষ্ট স্থানে বাইনারি অনুসন্ধান করার জন্য আপনার শুধুমাত্র আটটি হ্যাঁ বা না প্রশ্নের প্রয়োজন।", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 813.66 }, { - "input": "The first thing, except for those eight highlighted parity bits, can be whatever you want it to be, carrying whatever message or data you want. ", + "input": "the sender can feel good leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your pari ", "translatedText": "প্রথম জিনিস, সেই আটটি হাইলাইট করা প্যারিটি বিট ব্যতীত, আপনি যা চান তা হতে পারে, আপনি যা চান তা বার্তা বা ডেটা বহন করতে পারেন।", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 821.0 }, { - "input": "The 8 bits are redundant in the sense that they're completely determined by the rest of the message, but it's in a much smarter way than simply copying the message as a whole. ", + "input": "ty checks, and it uses only 21 parity bits. And if you step back to think about looking at a million bits and locating a single error, that genuinely feels crazy. The problem, Otherwise, it means either there's ", "translatedText": "8 বিট এই অর্থে অপ্রয়োজনীয় যে তারা সম্পূর্ণরূপে বার্তার বাকি অংশ দ্বারা নির্ধারিত হয়, তবে এটি সম্পূর্ণরূপে বার্তাটি অনুলিপি করার চেয়ে অনেক স্মার্ট উপায়ে।", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost. ", + "input": "or the error is somewhere on the left half. ", "translatedText": "ভাল প্রায়. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0. ", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you ", "translatedText": "ঠিক আছে, তাই এখানে একটি সমস্যা হল যে যদি চারটি প্যারিটি চেকের মধ্যে কোনোটিই একটি ত্রুটি সনাক্ত না করে, যার অর্থ হল 8 বিটের বিশেষভাবে নির্বাচিত উপসেটগুলির সমতুল্য রয়েছে, ঠিক প্রেরকের মতই, তাহলে এর মানে হয় কোনও ত্রুটি ছিল না , অথবা এটি আমাদের অবস্থান 0 এ সংকুচিত করে।", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 915.54 }, { - "input": "Here's how it works. ", + "input": "er block. But it also might simply mean there's no error at ", "translatedText": "এখানে কিভাবে এটা কাজ করে. ", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 952.7 }, { - "input": "Isn't that clever? ", + "input": "s today. There are like half a dozen times throughout this book that he refer ", "translatedText": "চতুর তাই না? ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them. ", + "input": "ences the Louis Pasteur quote, luck favors a prepared mind. Cl In this case, it looks like the sender needs to turn that bit 8 on in order to give the group even parity. Part of the reason that clever ideas look deceptively ", "translatedText": "যদিও আমরা সেই 2-বিট ত্রুটিগুলি সংশোধন করতে পারি না, কেবলমাত্র সেই একটি সামান্য বিরক্তিকর 0 তম বিটকে কাজে ফিরিয়ে দিয়ে, এটি আমাদের সেগুলি সনাক্ত করতে দেয়।", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 965.22 }, { - "input": "Technically speaking, you now have a full description of what a Hamming code does, at least for the example of a 16-bit block. ", + "input": "nal result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the transmission there's an error at, say, position 3. ", "translatedText": "প্রযুক্তিগতভাবে বলতে গেলে, আপনার কাছে এখন একটি হ্যামিং কোড কী করে তার সম্পূর্ণ বিবরণ রয়েছে, অন্তত একটি 16-বিট ব্লকের উদাহরণের জন্য।", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it! ", + "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position ", "translatedText": "এগিয়ে যান, আসলে এটা করতে! ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block. ", + "input": "3, so they can fix the error. You might enjoy taking a moment to convince yourself that the answers to these four ", "translatedText": "আসুন বিরতি দিন এবং এই ব্লকটি একসাথে রাখার চেষ্টা করি।", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1007.02 }, { - "input": "Okay, you ready? ", + "input": "questions really will always let you pin down ", "translatedText": "ঠিক আছে, আপনি প্রস্তুত? ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0. ", + "input": "n between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spo ", "translatedText": "আপনার এই গোষ্ঠীর একটি সমান সমতা থাকা দরকার, যা এটি ইতিমধ্যেই করে, তাই আপনার সেই প্যারিটি বিটটিকে 0 হিসাবে 1 অবস্থানে সেট করা উচিত ছিল।", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1037.88 }, { - "input": "The group after that starts with an odd parity, so again you should have set its parity bit to 1. ", + "input": "ts affected, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, ", "translatedText": "এর পরের গ্রুপটি একটি বিজোড় সমতা দিয়ে শুরু হয়, তাই আবার আপনার প্যারিটি বিট 1 এ সেট করা উচিত ছিল।", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1079.78 }, { - "input": "What I'm going to do is change either 0, 1, or 2 of the bits in that block, and then ask you to figure out what it is that I did. ", + "input": "y eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set the app ", "translatedText": "আমি যা করতে যাচ্ছি তা হল 0, 1, বা 2টি সেই ব্লকের বিটগুলিকে পরিবর্তন করুন এবং তারপরে আপনাকে জিজ্ঞাসা করতে চাই যে আমি কী করেছি।", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1107.91 }, { - "input": "The next check gives us an odd number, telling us both that there's at least one error, and narrowing us down into this specific column. ", + "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever you want it to be, carrying whatever message or data you want. ", "translatedText": "পরবর্তী চেকটি আমাদের একটি বিজোড় সংখ্যা দেয়, আমাদের উভয়কেই বলে যে অন্তত একটি ত্রুটি রয়েছে এবং আমাদের এই নির্দিষ্ট কলামে সংকুচিত করে।", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off. ", + "input": ". And still, for so little given up, you would be able to identify and fix any single bit error. Well, almost. Okay, so the one ", "translatedText": "যদি এটি তিন বা তার বেশি হয়, তাহলে সমস্ত বাজি বন্ধ হয়ে যাবে।", "model": "google_nmt", "n_reviews": 0, @@ -1240,7 +1240,7 @@ "end": 1163.17 }, { - "input": "You see, what I haven't told you yet is just how elegant this algorithm really is, how simple it is to get a machine to point to the position of an error, how to systematically scale it, and how we can frame all of this as one single operation rather than multiple separate parity checks. ", + "input": "You see, with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing one out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The solution here is actually pretty simple. Just forget about that zeroth bit entirely. So when we do our four parity checks and ", "translatedText": "আপনি দেখতে পাচ্ছেন, আমি আপনাকে এখনও যা বলিনি তা হল এই অ্যালগরিদমটি আসলে কতটা মার্জিত, একটি ত্রুটির অবস্থান নির্দেশ করার জন্য একটি মেশিন পাওয়া কতটা সহজ, কীভাবে এটিকে পদ্ধতিগতভাবে স্কেল করা যায় এবং কীভাবে আমরা সমস্ত কিছু ফ্রেম করতে পারি এটি একাধিক পৃথক প্যারিটি চেকের পরিবর্তে একটি একক অপারেশন হিসাবে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/chinese/sentence_translations.json b/2020/hamming-codes/chinese/sentence_translations.json index d0707a393..9820b75b0 100644 --- a/2020/hamming-codes/chinese/sentence_translations.json +++ b/2020/hamming-codes/chinese/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit. ", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notice ", "translatedText": "纠正任何被翻转的位的一个简单 策略是存储每个位的三个副本。", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9! ", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how ", "translatedText": "例如，使用您将在本视频中了解的方法，您可以将数据存储在 256 位块中，其中每个块使用 9 位，9！", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want. ", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan ", "translatedText": "作为一种冗余，其他 2 47 位可以自由地携带任何您想要的有意义的消息或数据。", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 112.66 }, { - "input": "And honestly, that feels like magic. ", + "input": "m e emphasize that they are distinct from the data that's actually being sent. They're noth ", "translatedText": "老实说，这感觉就像魔法一样。", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 122.86 }, { - "input": "We'll talk a little bit later about how this scales for blocks with different sizes. ", + "input": "that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to identify that there was an error and precisely where i ", "translatedText": "稍后我们将讨论如何针对不同大小的块进行缩放。", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 141.94 }, { - "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. ", + "input": "that there were two errors, though it won't know how to fix them. We'll talk a little bit later about how this scales for blocks with different sizes. where that's a 1, you get the second parity group from our scheme ", "translatedText": "这里的目标是让您非常彻底地了 解最早的示例之一，即汉明码。", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 206.94 }, { - "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. ", + "input": "e scheme is going to be before I tell you. Also, if you want your understanding to get down to the hardware level, Ben Eater has made a video in conjunction with this one ", "translatedText": "每当有效消息被更改时，接收者就有责任将他们看到的内容 纠正回最近的有效邻居，就像您可能会处理拼写错误一样。", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. ", + "input": "st how impossible this task feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is tha ", "translatedText": "挫折是发明的严峻考验，他受够了， 因此发明了世界上第一个纠错码。", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 248.42 }, { - "input": "There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. ", + "input": "t in a vast space of all possible messages, only some subset are going to be considered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words. also see th ", "translatedText": "构建汉明码的方法有很多种，但作为第一步， 我们将按照汉明本人对它们的看法来了解它。", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 255.38 }, { - "input": "Let's use an example that's simple, but not too simple, a block of 16 bits. ", + "input": "is in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. ", "translatedText": "让我们使用一个简单但又不太简单的例子，一个 16 位的块。", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 260.94 }, { - "input": "We'll number the positions of these bits from 0 up to 15. ", + "input": "Once you understand that these parity checks that we've focused so much of our time on are nothing ", "translatedText": "我们将这些位的位置从 0 到 15 进行编号。", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 273.0 }, { - "input": "The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. ", + "input": "binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. ", "translatedText": "这里的冗余一词并不简单地意味着复制，毕竟这4 位并没有给我们足够的空间来盲目地复制数据。", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 287.28 }, { - "input": "You might expect these 4 special bits to come nicely packaged together, maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. ", + "input": "r. When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the programs he kept putting through it kept failing, because every now and then a ", "translatedText": "您可能期望这 4 个特殊位能够很好地打包在一起， 也许是在最后或类似的地方，但正如您所看到的，让它 们位于 2 的幂的位置可以让最终变得非常优雅。", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors. ", + "input": "'s first error correction code. There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. ", "translatedText": "与任何纠错算法一样，这将涉及两个参与者：一 个负责设置这 4 个特殊位的发送者，以及 一个负责执行某种检查和纠正错误的接收者。", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", + "input": "e so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy, not adding any new information, but adding resilience. ", "translatedText": "这就是设置，但在我们深入讨论之前，我们需要讨 论一个相关的想法，这是汉明在发现时的新鲜想 法，这种方法可以让您检测到任何单个位错误， 但不能纠正它们，已知在业务中作为奇偶校验。", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.82 }, { - "input": "The only job of this special bit is to make sure that the total number of 1s in the message is an even number. ", + "input": "that make sense? Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and w ", "translatedText": "该特殊位的唯一作用是确保消 息中 1 的总数为偶数。", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.28 }, { - "input": "So for example right now, that total number of 1s is 7, that's odd, so the sender needs to flip that special bit to be a 1, making the count even. ", + "input": "hich a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it goes from here. The sender is responsible for toggling ", "translatedText": "例如，现在 1 的总数是 7，这是奇数，因此发 送方需要将该特殊位翻转为 1，使计数为偶数。", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd. ", + "input": "sitio Like any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. ", "translatedText": "请注意，如果此消息的任何位被翻转（从 0 变为 1 或从 1 变为 0），它会将 1 的总数从偶数更改为奇数。", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. ", + "input": "Of course, the words sender and receiver really refer to machines or software that's doing checks, and the idea of a message is meant really broadly, to include things like storage. After all, storing data is the same thing as sending a message, just from the past ", "translatedText": "因此，如果您是接收者，您查看此消息，并 且看到奇数个 1，您可以确定发生了某些 错误，即使您可能不知道错误发生在哪里。", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit. ", + "input": "his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", "translatedText": "发送方用来控制奇偶校验的 特殊位称为奇偶校验位。", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 485.44 }, { - "input": "Instead, the goal is to come up with a scheme that's robust up to a certain maximum number of errors, or maybe to reduce the probability of a false positive like this. ", + "input": "en kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information. ctice t ", "translatedText": "相反，我们的目标是提出一种在一定的最大错误数范 围内稳健的方案，或者减少像这样的误报的可能性。", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 495.38 }, { - "input": "Parity checks on their own are pretty weak, but by distilling the idea of change across a full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. ", + "input": "his would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running f ", "translatedText": "奇偶校验本身相当弱，但通过将整个消 息的变化思想提炼为单个位，它们为我 们提供了更复杂方案的强大构建块。", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error. ", + "input": "rom 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of 1s is known as its parity. o collect together all of those positions, the positions of the bits that are ", "translatedText": "例如，由于汉明正在寻找一种方法来识别错误发生的位置，而 不仅仅是错误发生的地方，他的主要见解是，如果您不对完整 消息而是对某些精心选择的子集应用一些奇偶校验检查，您可 以询问一系列更精确的问题，可以确定任何一位错误的位置。", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 590.68 }, { - "input": "The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", + "input": "his on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but ", "translatedText": "第二个检查位于网格右半部分的 8 位 中，至少我们在这里绘制的是这样的。", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.3 }, { - "input": "This time we might use position 2 as a parity bit, so these 8 bits already have an even parity, and the sender can feel good leaving that bit number 2 unchanged. ", + "input": "you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this ", "translatedText": "这次我们可能使用位置 2 作为奇偶校验位，因此这 8 位已经 具有偶校验，并且发送方可以感觉良好，保持该位号 2 不变。", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half. ", + "input": "simulating a random error from noise, then if you run this same line of code, it print s out that error. Isn't that neat? ", "translatedText": "否则，这意味着要么没有错误，要么错误位于左半部分。", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit. ", + "input": "own to a single XOR reduction. Now, depending on your comfort with binary and XORs and software in general, you may eithe ", "translatedText": "将使用位置 4 作为奇偶校验位对奇数行进行奇偶校验。", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0. ", + "input": "r find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a wa ", "translatedText": "因此，在此示例中，该组已经具有偶 校验，因此位 4 将设置为 0。", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit. ", + "input": "y to identify where an error happened, not just that it happened, his key insight was that if you apply some parity chec ", "translatedText": "最后，对底部两行进行奇偶校验， 使用位置 8 作为奇偶校验位。", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 701.84 }, { - "input": "As an example, imagine that during the transmission there's an error at, say, position 3. ", + "input": "ion of any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no ", "translatedText": "举个例子，假设在传输过程中，位置 3 处出现错误。", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 720.54 }, { - "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. ", + "input": "here is that that information directly corresponds to how much redundancy we need. That's really what runs against most people's knee-jerk reaction Then, if an error is detected, it gives the receiv ", "translatedText": "这可以让接收者精确定位到第一行（这必然意味着 位置 3）的错误，这样他们就可以修复错误。", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 727.52 }, { - "input": "You might enjoy taking a moment to convince yourself that the answers to these four questions really will always let you pin down a specific location, no matter where they turn out to be. ", + "input": "er a little more information about where specifically the error is, namely that it's in an odd position. ent to errors, where usually copying the whole message is the first instinct that comes to min ", "translatedText": "您可能会喜欢花点时间说服自己，这四个问题的答案确实 总是能让您确定一个特定的位置，无论它们最终在哪里。", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 743.06 }, { - "input": "And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spoil it. ", + "input": "then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix. It's kind of nice because it relates ", "translatedText": "如果你这样做了，请再次让我强调一下，暂停一 下，在我破坏之前尝试自己找出其中的联系。", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 773.1 }, { - "input": "But protecting those bits as well is something that naturally falls out of the scheme as a byproduct. ", + "input": "more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group. ", "translatedText": "但保护这些位也是 该计划的副产品。", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales. ", + "input": "Here let's just choose position 1. For the example shown, the pari ", "translatedText": "您可能还喜欢预测其规模如何。", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 781.76 }, { - "input": "If we used a block of size 256 bits, for example, in order to pin down a location, you need only eight yes or no questions to binary search your way down to some specific spot. ", + "input": "ty of these 8 bits is currently odd, so the sender is responsible for toggling that parity bit, and now it's even. This is only 1 out of 4 parity checks that we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", "translatedText": "例如，如果我们使用大小为 256 位的块，为了确定位置 ，您只需要八个是或否问题即可二进制搜索到某个特定位置。", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 813.66 }, { - "input": "The first thing, except for those eight highlighted parity bits, can be whatever you want it to be, carrying whatever message or data you want. ", + "input": "the sender can feel good leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your pari ", "translatedText": "第一件事，除了那八个突出显示的奇偶校验位之外，可以 是您想要的任何内容，携带您想要的任何消息或数据。", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 821.0 }, { - "input": "The 8 bits are redundant in the sense that they're completely determined by the rest of the message, but it's in a much smarter way than simply copying the message as a whole. ", + "input": "ty checks, and it uses only 21 parity bits. And if you step back to think about looking at a million bits and locating a single error, that genuinely feels crazy. The problem, Otherwise, it means either there's ", "translatedText": "8 位是冗余的，因为它们完全由消息的其余部分 决定，但它比简单地复制整个消息要聪明得多。", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost. ", + "input": "or the error is somewhere on the left half. ", "translatedText": "嗯，差不多了。", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0. ", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you ", "translatedText": "好的，所以这里的一个问题是，如果四个奇偶校验检查都没 有检测到错误，这意味着专门选择的 8 位子集都具有 偶数奇偶校验，就像发送者预期的那样，那么这要么意味着 根本没有错误，或者它会将我们的范围缩小到位置 0。", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 915.54 }, { - "input": "Here's how it works. ", + "input": "er block. But it also might simply mean there's no error at ", "translatedText": "这是它的工作原理。", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 952.7 }, { - "input": "Isn't that clever? ", + "input": "s today. There are like half a dozen times throughout this book that he refer ", "translatedText": "这不是很聪明吗？", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them. ", + "input": "ences the Louis Pasteur quote, luck favors a prepared mind. Cl In this case, it looks like the sender needs to turn that bit 8 on in order to give the group even parity. Part of the reason that clever ideas look deceptively ", "translatedText": "尽管我们无法纠正这些 2 位错误，但只需将那个有点麻 烦的第 0 位重新投入工作，我们就可以检测到它们。", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 965.22 }, { - "input": "Technically speaking, you now have a full description of what a Hamming code does, at least for the example of a 16-bit block. ", + "input": "nal result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the transmission there's an error at, say, position 3. ", "translatedText": "从技术上讲，您现在已经完整地描述了汉明码的 作用，至少对于 16 位块的示例是这样。", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it! ", + "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position ", "translatedText": "话不多说，实际行动起来吧！", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block. ", + "input": "3, so they can fix the error. You might enjoy taking a moment to convince yourself that the answers to these four ", "translatedText": "让我们暂停一下并尝试将这个块放在一起。", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1007.02 }, { - "input": "Okay, you ready? ", + "input": "questions really will always let you pin down ", "translatedText": "好吧，你准备好了吗？", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0. ", + "input": "n between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spo ", "translatedText": "您需要该组具有偶校验，它已经这样做了，因此您 应该将位置 1 中的奇偶校验位设置为 0。", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1037.88 }, { - "input": "The group after that starts with an odd parity, so again you should have set its parity bit to 1. ", + "input": "ts affected, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, ", "translatedText": "之后的组以奇数奇偶校验开始，因此您 应该再次将其奇偶校验位设置为 1。", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1079.78 }, { - "input": "What I'm going to do is change either 0, 1, or 2 of the bits in that block, and then ask you to figure out what it is that I did. ", + "input": "y eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set the app ", "translatedText": "我要做的是更改该块中的 0、1 或 2 位，然后要求您弄清楚我做了什么。", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1107.91 }, { - "input": "The next check gives us an odd number, telling us both that there's at least one error, and narrowing us down into this specific column. ", + "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever you want it to be, carrying whatever message or data you want. ", "translatedText": "下一次检查给我们一个奇数，告诉我们至少有一 个错误，并将我们的范围缩小到这一特定列。", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off. ", + "input": ". And still, for so little given up, you would be able to identify and fix any single bit error. Well, almost. Okay, so the one ", "translatedText": "如果是三个或更多，则所有赌注都失败了。", "model": "google_nmt", "n_reviews": 0, @@ -1240,7 +1240,7 @@ "end": 1163.17 }, { - "input": "You see, what I haven't told you yet is just how elegant this algorithm really is, how simple it is to get a machine to point to the position of an error, how to systematically scale it, and how we can frame all of this as one single operation rather than multiple separate parity checks. ", + "input": "You see, with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing one out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The solution here is actually pretty simple. Just forget about that zeroth bit entirely. So when we do our four parity checks and ", "translatedText": "你看，我还没有告诉你的是这个算法到底有多么优 雅，让机器指出错误的位置是多么简单，如何系 统地缩放它，以及我们如何构建所有错误这是一 个单一操作，而不是多个单独的奇偶校验检查。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/english/captions.srt b/2020/hamming-codes/english/captions.srt index ce26a6fdc..c3bd09add 100644 --- a/2020/hamming-codes/english/captions.srt +++ b/2020/hamming-codes/english/captions.srt @@ -47,1242 +47,1374 @@ Any file, whether it's a video, sound, text, code, image, whatever, is ultimately some sequence of 1s and 0s. 13 -00:00:50,680 --> 00:00:53,340 -And a simple strategy to correct any bit that gets +00:00:50,680 --> 00:00:53,026 +nstead of yeses and nos, it literally spells out the position of the error in binary. 14 -00:00:53,340 --> 00:00:56,000 -flipped would be to store three copies of each bit. +00:00:53,026 --> 00:00:54,472 +For example, the number 7 in binary looks like 0111, 15 -00:00:57,580 --> 00:01:00,843 -Then the machine reading this file could compare these three copies +00:00:54,472 --> 00:00:56,000 +essentially saying that it's 4 plus 2 plus 1. And notice 16 -00:01:00,843 --> 00:01:04,060 -and always take the best 2 out of 3 whenever there's a discrepancy. +00:00:57,580 --> 00:01:03,519 +where the position 7 sits, it does affect the first of our parity groups, and the second, 17 -00:01:07,160 --> 00:01:10,860 -But what that means is using two thirds of your space for redundancy. +00:01:03,519 --> 00:01:08,930 +and the third, but not the last. So reading the results of those four checks from 18 -00:01:11,480 --> 00:01:13,595 -And even then, for all of that space given up, +00:01:08,930 --> 00:01:14,540 +bottom to top indeed does spell out the position of the error. There's nothing specia 19 -00:01:13,595 --> 00:01:17,240 -there's no strong guarantee about what happens if more than one bit gets flipped. +00:01:14,540 --> 00:01:18,920 +But what that means is using two thirds of your space for redundancy. 20 -00:01:17,980 --> 00:01:20,836 -The much more interesting question is how to make it so that +00:01:18,920 --> 00:01:21,997 +And even then, for all of that space given up, 21 -00:01:20,836 --> 00:01:24,020 -errors can be corrected while giving up as little space as possible. +00:01:21,997 --> 00:01:27,300 +there's no strong guarantee about what happens if more than one bit gets flipped. 22 -00:01:24,520 --> 00:01:28,427 -For example, using the method you'll learn about this video, +00:01:27,600 --> 00:01:33,605 +The much more interesting question is how to make it so that 23 -00:01:28,427 --> 00:01:33,360 -you could store your data in 256-bit blocks, where each block uses 9 bits, 9! +00:01:33,605 --> 00:01:40,300 +errors can be corrected while giving up as little space as possible. 24 -00:01:33,760 --> 00:01:37,030 -to act as a kind of redundancy, and the other 247 bits are +00:01:40,900 --> 00:01:44,811 +c for implementing the whole scheme in hardware shockingly simple. 25 -00:01:37,030 --> 00:01:40,300 -free to carry whatever meaningful message or data you want. +00:01:44,811 --> 00:01:47,554 +Now if you want to see why this magic happens, 26 -00:01:40,900 --> 00:01:44,166 -And it will still be the case that if any bit gets flipped here, +00:01:47,554 --> 00:01:52,575 +take these 16 index labels for our positions, but instead of writing them in base 10, 27 -00:01:44,166 --> 00:01:46,578 -just by looking at this block and nothing more, +00:01:52,575 --> 00:01:56,194 +let's write them all in binary, running from 0000 up to 1111. 28 -00:01:46,578 --> 00:01:50,549 -a machine will be able to identify that there was an error and precisely where +00:01:56,194 --> 00:01:59,580 +As we put these binary labels back into their boxes, let m 29 -00:01:50,549 --> 00:01:52,660 -it was so that it knows how to correct it. +00:01:59,580 --> 00:02:04,765 +e emphasize that they are distinct from the data that's actually being sent. 30 -00:01:52,660 --> 00:01:54,620 -And honestly, that feels like magic. +00:02:04,765 --> 00:02:08,940 +They're nothing more than a conceptual label to help you and m 31 -00:01:55,440 --> 00:01:57,987 -And for this particular scheme, if two bits get flipped, +00:02:08,940 --> 00:02:11,490 +And it will still be the case that if any bit gets flipped here, 32 -00:01:57,987 --> 00:02:01,206 -the machine will at least be able to detect that there were two errors, +00:02:11,490 --> 00:02:13,373 +just by looking at this block and nothing more, 33 -00:02:01,206 --> 00:02:02,860 -though it won't know how to fix them. +00:02:13,373 --> 00:02:16,472 +a machine will be able to identify that there was an error and precisely where 34 -00:02:03,520 --> 00:02:06,900 -We'll talk a little bit later about how this scales for blocks with different sizes. +00:02:16,472 --> 00:02:18,120 +it was so that it knows how to correct it. 35 -00:02:07,860 --> 00:02:10,676 -Methods that let you correct errors like this are known, +00:02:18,120 --> 00:02:21,940 +And honestly, that feels like magic. 36 -00:02:10,676 --> 00:02:12,900 -reasonably enough, as error correction codes. +00:02:22,840 --> 00:02:24,838 +And for this particular scheme, if two bits get flipped, 37 -00:02:13,660 --> 00:02:17,877 -For the better part of the last century, this field has been a really rich source +00:02:24,838 --> 00:02:27,362 +the machine will at least be able to detect that there were two errors, 38 -00:02:17,877 --> 00:02:21,940 -of surprisingly deep math that gets incorporated into devices we use every day. +00:02:27,362 --> 00:02:28,660 +though it won't know how to fix them. 39 -00:02:22,840 --> 00:02:25,800 -The goal here is to give you a very thorough understanding +00:02:29,520 --> 00:02:28,660 +We'll talk a little bit later about how this scales for blocks with different sizes. 40 -00:02:25,800 --> 00:02:28,660 -of one of the earliest examples, known as a Hamming code. +00:02:29,520 --> 00:02:31,452 +where that's a 1, you get the second parity group from our scheme. 41 -00:02:29,520 --> 00:02:33,042 -And by the way, the way I'm thinking about the structure of this video is less +00:02:31,452 --> 00:02:33,183 +In other words, that second check is asking, hey, me again, 42 -00:02:33,042 --> 00:02:36,431 -about explaining it as directly as possible, and more a matter of prompting +00:02:33,183 --> 00:02:35,260 +if there's an error, is the second to last bit of that position a 1? And 43 -00:02:36,431 --> 00:02:39,820 -you to invent it for yourself, with a little gentle guidance here and there. +00:02:35,260 --> 00:02:39,530 +so on. The third parity check covers every position whose third to last bit is turned on, 44 -00:02:40,120 --> 00:02:43,909 -So when you feel like you see where it's going at some point, take that moment to pause, +00:02:39,530 --> 00:02:41,902 +and the last one covers the last eight positions, 45 -00:02:43,909 --> 00:02:46,720 -actively predict what the scheme is going to be before I tell you. +00:02:41,902 --> 00:02:44,560 +those ones whose highest order bit is a 1. Everything we 46 -00:02:47,240 --> 00:02:50,744 -Also, if you want your understanding to get down to the hardware level, +00:02:44,560 --> 00:02:48,394 +The goal here is to give you a very thorough understanding 47 -00:02:50,744 --> 00:02:54,394 -Ben Eater has made a video in conjunction with this one showing you how to +00:02:48,394 --> 00:02:52,100 +of one of the earliest examples, known as a Hamming code. 48 -00:02:54,394 --> 00:02:58,240 -actually implement Hamming codes on breadboards, which is extremely satisfying. +00:02:52,100 --> 00:02:56,240 +four questions, which in turn is the same as spelling out a position in binary. 49 -00:02:59,300 --> 00:03:03,058 -You should know, Hamming codes are not as widely used as more modern codes, +00:02:56,240 --> 00:03:00,018 +I hope this makes two things clearer. The first is how to systematically 50 -00:03:03,058 --> 00:03:06,471 -like the Reed-Solomon algorithm, but there is a certain magic to the +00:03:00,018 --> 00:03:02,968 +generalize to block sizes that are bigger powers of two. 51 -00:03:06,471 --> 00:03:09,785 -contrast between just how impossible this task feels at the start, +00:03:02,968 --> 00:03:07,367 +If it takes more bits to describe each position, like six bits to describe 64 spots, 52 -00:03:09,785 --> 00:03:13,000 -and how utterly reasonable it seems once you learn about Hamming. +00:03:07,367 --> 00:03:11,560 +then each of those bits gives you one of the parity groups that we need to check. 53 -00:03:13,720 --> 00:03:18,830 -The basic principle of error correction is that in a vast space of all possible messages, +00:03:11,560 --> 00:03:16,199 +So when you feel like you see where it's going at some point, take that moment to pause, 54 -00:03:18,830 --> 00:03:22,180 -only some subset are going to be considered valid messages. +00:03:16,199 --> 00:03:19,640 +actively predict what the scheme is going to be before I tell you. 55 -00:03:22,800 --> 00:03:26,940 -As an analogy, think about correctly spelled words versus incorrectly spelled words. +00:03:19,640 --> 00:03:24,692 +Also, if you want your understanding to get down to the hardware level, 56 -00:03:28,900 --> 00:03:33,225 -Whenever a valid message gets altered, the receiver is responsible for correcting +00:03:24,692 --> 00:03:29,956 +Ben Eater has made a video in conjunction with this one showing you how to 57 -00:03:33,225 --> 00:03:37,340 -what they see back to the nearest valid neighbor, as you might do with a typo. +00:03:29,956 --> 00:03:35,500 +actually implement Hamming codes on breadboards, which is extremely satisfying. 58 -00:03:38,220 --> 00:03:42,348 -Coming up with a concrete algorithm to efficiently categorize messages like this, +00:03:35,500 --> 00:03:39,560 +You should know, Hamming codes are not as widely used as more modern codes, 59 -00:03:42,348 --> 00:03:44,060 -though, take a certain cleverness. +00:03:39,560 --> 00:03:43,247 +like the Reed-Solomon algorithm, but there is a certain magic to the 60 -00:03:46,780 --> 00:03:51,431 -The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, +00:03:43,247 --> 00:03:46,827 +contrast between just how impossible this task feels at the start, 61 -00:03:51,431 --> 00:03:55,067 -and some of his work involved using a very big expensive punch card +00:03:46,827 --> 00:03:50,300 +and how utterly reasonable it seems once you learn about Hamming. 62 -00:03:55,067 --> 00:03:57,420 -computer that he had only limited access to. +00:03:50,300 --> 00:03:56,473 +The basic principle of error correction is that in a vast space of all possible messages, 63 -00:03:57,800 --> 00:04:00,247 -And the programs he kept putting through it kept failing, +00:03:56,473 --> 00:04:00,520 +only some subset are going to be considered valid messages. 64 -00:04:00,247 --> 00:04:02,400 -because every now and then a bit would get misread. +00:04:00,520 --> 00:04:03,560 +As an analogy, think about correctly spelled words versus incorrectly spelled words. 65 -00:04:03,120 --> 00:04:05,747 -Frustration being the crucible of invention, he got so fed +00:04:03,560 --> 00:04:07,423 +also see this in larger examples, where no matter how big you get, 66 -00:04:05,747 --> 00:04:08,420 -up that he invented the world's first error correction code. +00:04:07,423 --> 00:04:10,940 +each parity bit conveniently touches only one of the groups. 67 -00:04:09,060 --> 00:04:11,397 -There are many different ways to frame Hamming codes, +00:04:10,940 --> 00:04:15,784 +Once you understand that these parity checks that we've focused so much of our time 68 -00:04:11,397 --> 00:04:14,557 -but as a first pass we're going to go through it the way Hamming himself +00:04:15,784 --> 00:04:20,800 +on are nothing more than a clever way to spell out the position of an error in binary, 69 -00:04:14,557 --> 00:04:15,380 -thought about them. +00:04:20,800 --> 00:04:23,280 +then we can draw a connection with a differ 70 -00:04:16,519 --> 00:04:20,940 -Let's use an example that's simple, but not too simple, a block of 16 bits. +00:04:23,280 --> 00:04:25,731 +ent way to think about hamming codes, one that is arguably a lot simpler and 71 -00:04:21,820 --> 00:04:24,740 -We'll number the positions of these bits from 0 up to 15. +00:04:25,731 --> 00:04:28,120 +more elegant, and which can basically be written down with a single line of 72 -00:04:25,620 --> 00:04:29,359 -The actual data we want to store is only going to make up 12 of these bits, +00:04:28,120 --> 00:04:31,388 +code. It's based on the XOR function. XOR, for those of you who don't know, 73 -00:04:29,359 --> 00:04:33,000 -while 4 of the positions are going to be reserved as a kind of redundancy. +00:04:31,388 --> 00:04:33,968 +stands for exclusive or. When you take the XOR of two bits, 74 -00:04:33,900 --> 00:04:36,896 -The word redundant here doesn't simply mean copy, after all, +00:04:33,968 --> 00:04:36,849 +it's going to return a 1 if either one of those bits is turned on, 75 -00:04:36,896 --> 00:04:40,040 -those 4 bits don't give us enough room to blindly copy the data. +00:04:36,849 --> 00:04:39,300 +but not if both are turned on or off. Phrased differently 76 -00:04:40,720 --> 00:04:44,616 -Instead, they'll need to be a much more nuanced and clever kind of redundancy, +00:04:39,300 --> 00:04:43,546 +And the programs he kept putting through it kept failing, 77 -00:04:44,616 --> 00:04:47,280 -not adding any new information, but adding resilience. +00:04:43,546 --> 00:04:47,280 +because every now and then a bit would get misread. 78 -00:04:48,600 --> 00:04:51,947 -You might expect these 4 special bits to come nicely packaged together, +00:04:48,600 --> 00:04:50,870 +Frustration being the crucible of invention, he got so fed 79 -00:04:51,947 --> 00:04:54,737 -maybe at the end or something like that, but as you'll see, +00:04:50,870 --> 00:04:53,180 +up that he invented the world's first error correction code. 80 -00:04:54,737 --> 00:04:58,411 -having them sit in positions which are powers of 2 allows for something that's +00:04:53,180 --> 00:04:56,405 +There are many different ways to frame Hamming codes, 81 -00:04:58,411 --> 00:04:59,620 -really elegant by the end. +00:04:56,405 --> 00:05:00,765 +but as a first pass we're going to go through it the way Hamming himself 82 -00:05:00,200 --> 00:05:03,540 -It also might give you a little hint about how this scales for larger blocks. +00:05:00,765 --> 00:05:01,900 +thought about them. 83 -00:05:04,900 --> 00:05:07,915 -Also, technically it ends up being only 11 bits of data, +00:05:01,900 --> 00:05:08,060 +Let's use an example that's simple, but not too simple, a block of 16 bits. 84 -00:05:07,915 --> 00:05:11,408 -you'll find there's a mild nuance for what goes on at position 0, +00:05:08,400 --> 00:05:08,560 +We'll number the positions of these bits from 0 up to 15. 85 -00:05:11,408 --> 00:05:13,260 -but don't worry about that for now. +00:05:08,560 --> 00:05:12,251 +bit representations of those numbers under the hood. 86 -00:05:14,140 --> 00:05:17,940 -Like any error correction algorithm, this will involve two players, a sender, +00:05:12,251 --> 00:05:18,172 +The key point for you and me is that taking the XOR of many different bit strings is 87 -00:05:17,940 --> 00:05:21,496 -who's responsible for setting these 4 special bits, and then a receiver, +00:05:18,172 --> 00:05:23,256 +effectively a way to compute the parodies of a bunch of separate groups, 88 -00:05:21,496 --> 00:05:25,540 -who's responsible for performing some kind of check and then correcting the errors. +00:05:23,256 --> 00:05:26,600 +like so with the columns, all in one fell swoop. 89 -00:05:26,300 --> 00:05:28,957 -Of course, the words sender and receiver really refer to +00:05:26,780 --> 00:05:31,572 +The word redundant here doesn't simply mean copy, after all, 90 -00:05:28,957 --> 00:05:31,755 -machines or software that's doing checks, and the idea of a +00:05:31,572 --> 00:05:36,600 +those 4 bits don't give us enough room to blindly copy the data. 91 -00:05:31,755 --> 00:05:34,740 -message is meant really broadly, to include things like storage. +00:05:36,600 --> 00:05:37,419 +Instead, they'll need to be a much more nuanced and clever kind of redundancy, 92 -00:05:35,340 --> 00:05:38,345 -After all, storing data is the same thing as sending a message, +00:05:37,419 --> 00:05:37,980 +not adding any new information, but adding resilience. 93 -00:05:38,345 --> 00:05:41,680 -just from the past to the future, instead of from one place to another. +00:05:38,140 --> 00:05:44,402 +s, so it's effectively counting how many highlighted positions came from the first parity 94 -00:05:42,560 --> 00:05:45,020 -So that's the setup, but before we can dive in, +00:05:44,402 --> 00:05:50,524 +group. Does that make sense? Likewise, the next column counts how many positions are in 95 -00:05:45,020 --> 00:05:49,635 -we need to talk about a related idea which was fresh on Hamming's mind in the time of his +00:05:50,524 --> 00:05:56,300 +the second parity group, the positions whose second to last bit is a 1, and which a 96 -00:05:49,635 --> 00:05:54,249 -discovery, a method which lets you detect any single bit errors, but not to correct them, +00:05:56,880 --> 00:05:59,960 +re also highlighted, and so on. It's really just a small shift in 97 -00:05:54,249 --> 00:05:56,300 -known in the business as a parity check. +00:05:59,960 --> 00:06:02,200 +perspective on the same thing we've been doing. 98 -00:05:56,880 --> 00:06:00,449 -For a parity check, we separate out only one single bit that the sender +00:06:02,200 --> 00:06:06,120 +And so you know where it goes from here. The sender is responsible for toggling some 99 -00:06:00,449 --> 00:06:03,820 -is responsible for tuning, and the rest are free to carry a message. +00:06:06,120 --> 00:06:10,080 +of the special parity bits to make sure the sum works out to be 0000. 100 -00:06:04,880 --> 00:06:08,021 -The only job of this special bit is to make sure that +00:06:10,080 --> 00:06:14,153 +Now once we have it like this, this gives us a really nice way to think 101 -00:06:08,021 --> 00:06:11,280 -the total number of 1s in the message is an even number. +00:06:14,153 --> 00:06:18,680 +about why these four resulting bits at the bottom directly spell out the positio 102 -00:06:12,080 --> 00:06:15,750 -So for example right now, that total number of 1s is 7, that's odd, +00:06:19,000 --> 00:06:23,453 +Like any error correction algorithm, this will involve two players, a sender, 103 -00:06:15,750 --> 00:06:19,960 -so the sender needs to flip that special bit to be a 1, making the count even. +00:06:23,453 --> 00:06:27,621 +who's responsible for setting these 4 special bits, and then a receiver, 104 -00:06:20,800 --> 00:06:24,038 -But if the block had already started off with an even number of 1s, +00:06:27,621 --> 00:06:32,360 +who's responsible for performing some kind of check and then correcting the errors. 105 -00:06:24,038 --> 00:06:26,420 -then this special bit would have been kept at a 0. +00:06:32,360 --> 00:06:36,101 +Of course, the words sender and receiver really refer to 106 -00:06:27,340 --> 00:06:32,086 -This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill +00:06:36,101 --> 00:06:40,039 +machines or software that's doing checks, and the idea of a 107 -00:06:32,086 --> 00:06:36,780 -the idea of change anywhere in a message to be reflected in a single bit of information. +00:06:40,039 --> 00:06:44,240 +message is meant really broadly, to include things like storage. 108 -00:06:37,500 --> 00:06:42,536 -Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, +00:06:44,240 --> 00:06:45,330 +After all, storing data is the same thing as sending a message, 109 -00:06:42,536 --> 00:06:46,540 -it changes the total count of 1s from being even to being odd. +00:06:45,330 --> 00:06:46,540 +just from the past to the future, instead of from one place to another. 110 -00:06:47,980 --> 00:06:50,681 -So if you're the receiver, you look at this message, +00:06:47,980 --> 00:06:50,731 +So that's the setup, but before we can dive in, 111 -00:06:50,681 --> 00:06:55,013 -and you see an odd number of 1s, you can know for sure that some error has occurred, +00:06:50,731 --> 00:06:55,889 +we need to talk about a related idea which was fresh on Hamming's mind in the time of his 112 -00:06:55,013 --> 00:06:57,460 -even though you might have no idea where it was. +00:06:55,889 --> 00:07:01,047 +discovery, a method which lets you detect any single bit errors, but not to correct them, 113 -00:06:58,500 --> 00:07:00,795 -In the jargon, whether a group of bits has an +00:07:01,047 --> 00:07:03,340 +known in the business as a parity check. 114 -00:07:00,795 --> 00:07:03,340 -even or an odd number of 1s is known as its parity. +00:07:04,860 --> 00:07:07,050 +For a parity check, we separate out only one single bit that the sender 115 -00:07:04,860 --> 00:07:07,634 -You could also use numbers and say the parity is 0 or 1, +00:07:07,050 --> 00:07:09,120 +is responsible for tuning, and the rest are free to carry a message. 116 -00:07:07,634 --> 00:07:11,187 -which is typically more helpful once you start doing math with the idea, +00:07:09,120 --> 00:07:10,916 +The only job of this special bit is to make sure that 117 -00:07:11,187 --> 00:07:15,520 -and this special bit that the sender uses to control the parity is called the parity bit. +00:07:10,916 --> 00:07:12,780 +the total number of 1s in the message is an even number. 118 -00:07:17,560 --> 00:07:21,136 -And actually, we should be clear, if the receiver sees an odd parity, +00:07:12,780 --> 00:07:15,574 +So for example right now, that total number of 1s is 7, that's odd, 119 -00:07:21,136 --> 00:07:23,895 -it doesn't necessarily mean there was just one error, +00:07:15,574 --> 00:07:18,780 +so the sender needs to flip that special bit to be a 1, making the count even. 120 -00:07:23,895 --> 00:07:27,011 -there might have been 3 errors or 5 or any other odd number, +00:07:18,780 --> 00:07:22,433 +But if the block had already started off with an even number of 1s, 121 -00:07:27,011 --> 00:07:29,260 -but they can know for sure that it wasn't 0. +00:07:22,433 --> 00:07:25,120 +then this special bit would have been kept at a 0. 122 -00:07:29,980 --> 00:07:34,159 -On the other hand, if there had been 2 errors, or any even number of errors, +00:07:25,120 --> 00:07:30,037 +This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill 123 -00:07:34,159 --> 00:07:38,283 -that final count of 1s would still be even, so the receiver can't have full +00:07:30,037 --> 00:07:34,900 +the idea of change anywhere in a message to be reflected in a single bit of information. 124 -00:07:38,283 --> 00:07:42,300 -confidence that an even count necessarily means the message is error-free. +00:07:34,900 --> 00:07:37,917 +ctice this would be something we're receiving from a sender, 125 -00:07:42,840 --> 00:07:45,984 -You might complain that a message which gets messed up by only +00:07:37,917 --> 00:07:42,370 +and instead of being random it would be carrying 11 data bits together with 5 parity bits. 126 -00:07:45,984 --> 00:07:49,080 -2 bit flips is pretty weak, and you would be absolutely right. +00:07:42,370 --> 00:07:46,179 + If I call the function enumerateBits, what it does is pair together each of 127 -00:07:49,700 --> 00:07:54,326 -Keep in mind, though, there is no method for error detection or correction that could +00:07:46,179 --> 00:07:49,940 +those bits with a corresponding index, in this case running from 0 up to 15. 128 -00:07:54,326 --> 00:07:58,900 -give you 100% confidence that the message you receive is the one the sender intended. +00:07:49,940 --> 00:07:52,168 +So if you're the receiver, you look at this message, 129 -00:07:59,580 --> 00:08:02,358 -After all, enough random noise could always change one +00:07:52,168 --> 00:07:55,741 +and you see an odd number of 1s, you can know for sure that some error has occurred, 130 -00:08:02,358 --> 00:08:05,440 -valid message into another valid message just by pure chance. +00:07:55,741 --> 00:07:57,760 +even though you might have no idea where it was. 131 -00:08:06,240 --> 00:08:10,837 -Instead, the goal is to come up with a scheme that's robust up to a certain maximum +00:07:57,760 --> 00:07:59,903 +In the jargon, whether a group of bits has an 132 -00:08:10,837 --> 00:08:15,380 -number of errors, or maybe to reduce the probability of a false positive like this. +00:07:59,903 --> 00:08:02,280 +even or an odd number of 1s is known as its parity. 133 -00:08:16,260 --> 00:08:19,803 -Parity checks on their own are pretty weak, but by distilling the +00:08:02,280 --> 00:08:05,773 +o collect together all of those positions, the positions of the bits that are turned on, 134 -00:08:19,803 --> 00:08:22,971 -idea of change across a full message down to a single bit, +00:08:05,773 --> 00:08:07,736 +and then XOR them together. To do this in Python, 135 -00:08:22,971 --> 00:08:27,160 -what they give us is a powerful building block for more sophisticated schemes. +00:08:07,736 --> 00:08:09,620 +let me first import a couple helpful functions. 136 -00:08:27,940 --> 00:08:32,507 -For example, as Hamming was searching for a way to identify where an error happened, +00:08:09,620 --> 00:08:12,840 +That way we can call reduce() on this list, and use the XOR function to reduce it. 137 -00:08:32,507 --> 00:08:37,020 -not just that it happened, his key insight was that if you apply some parity checks +00:08:12,840 --> 00:08:15,878 +And actually, we should be clear, if the receiver sees an odd parity, 138 -00:08:37,020 --> 00:08:40,674 -not to the full message, but to certain carefully selected subsets, +00:08:15,878 --> 00:08:18,222 +it doesn't necessarily mean there was just one error, 139 -00:08:40,674 --> 00:08:45,026 -you can ask a more refined series of questions that pin down the location of any +00:08:18,222 --> 00:08:20,870 +there might have been 3 errors or 5 or any other odd number, 140 -00:08:45,026 --> 00:08:45,940 -single bit error. +00:08:20,870 --> 00:08:22,780 +but they can know for sure that it wasn't 0. 141 -00:08:46,680 --> 00:08:49,931 -The overall feeling is a bit like playing a game of 20 questions, +00:08:22,780 --> 00:08:26,036 +On the other hand, if there had been 2 errors, or any even number of errors, 142 -00:08:49,931 --> 00:08:53,380 -asking yes or no queries that chop the space of possibilities in half. +00:08:26,036 --> 00:08:29,250 +that final count of 1s would still be even, so the receiver can't have full 143 -00:08:54,160 --> 00:08:57,605 -For example, let's say we do a parity check just on these 8 bits, +00:08:29,250 --> 00:08:32,380 +confidence that an even count necessarily means the message is error-free. 144 -00:08:57,605 --> 00:08:59,380 -all of the odd numbered positions. +00:08:32,760 --> 00:08:36,253 +So at the moment it looks like if we do this on our random block of 16 bits, 145 -00:09:00,100 --> 00:09:04,275 -Then, if an error is detected, it gives the receiver a little more information +00:08:36,253 --> 00:08:39,746 +it returns 9, which has the binary representation 1001. We won't do it here, 146 -00:09:04,275 --> 00:09:08,240 -about where specifically the error is, namely that it's in an odd position. +00:08:39,746 --> 00:08:43,375 +but you could write a function where the sender uses that binary representation 147 -00:09:08,940 --> 00:09:13,787 -If no error is detected among those 8 bits, it either means there's no error at all, +00:08:43,375 --> 00:08:46,960 +to set the four parity bits as needed, ultimately getting this block to a state 148 -00:09:13,787 --> 00:09:16,240 -or it sits somewhere in the even positions. +00:08:46,960 --> 00:08:50,168 +where running this line of code on the full list of bits returns a 0. 149 -00:09:17,180 --> 00:09:21,253 -You might think that limiting a parity check to half the bits makes it less effective, +00:08:50,168 --> 00:08:52,367 +This would be considered a well-prepared block. 150 -00:09:21,253 --> 00:09:24,297 -but when it's done in conjunction with other well-chosen checks, +00:08:52,367 --> 00:08:55,438 +What's cool is that if we toggle any one of the bits in this list, 151 -00:09:24,297 --> 00:09:27,200 -it counter-intuitively gives us something a lot more powerful. +00:08:55,438 --> 00:08:59,380 +simulating a random error from noise, then if you run this same line of code, it print 152 -00:09:29,240 --> 00:09:31,788 -To actually set up that parity check, remember, +00:09:00,100 --> 00:09:05,112 +s out that error. Isn't that neat? You could get this block from out of the blue, 153 -00:09:31,788 --> 00:09:35,611 -it requires earmarking some special bit that has control for the parity +00:09:05,112 --> 00:09:10,430 +run this single line on it, and it'll automatically spit out the position of an error, 154 -00:09:35,611 --> 00:09:36,620 -of that full group. +00:09:10,430 --> 00:09:15,260 +or a 0 if there wasn't any. And there's nothing special about the size 16 here. 155 -00:09:37,480 --> 00:09:39,180 -Here let's just choose position 1. +00:09:15,260 --> 00:09:22,915 +Instead, the goal is to come up with a scheme that's robust up to a certain maximum 156 -00:09:39,720 --> 00:09:43,124 -For the example shown, the parity of these 8 bits is currently odd, +00:09:22,915 --> 00:09:30,480 +number of errors, or maybe to reduce the probability of a false positive like this. 157 -00:09:43,124 --> 00:09:46,980 -so the sender is responsible for toggling that parity bit, and now it's even. +00:09:30,480 --> 00:09:34,086 +a parity check to detect 2-bit errors, but the idea is that almost all of 158 -00:09:47,940 --> 00:09:50,680 -This is only 1 out of 4 parity checks that we'll do. +00:09:34,086 --> 00:09:37,693 +the core logic from our scheme comes down to a single XOR reduction. Now, 159 -00:09:50,920 --> 00:09:54,578 -The second check is among the 8 bits on the right half of the grid, +00:09:37,693 --> 00:09:41,202 +depending on your comfort with binary and XORs and software in general, 160 -00:09:54,578 --> 00:09:56,300 -at least as we've drawn it here. +00:09:41,202 --> 00:09:44,760 +you may either find this perspective a little bit confusing, or so much m 161 -00:09:56,680 --> 00:09:59,580 -This time we might use position 2 as a parity bit. +00:09:44,760 --> 00:09:49,403 +For example, as Hamming was searching for a way to identify where an error happened, 162 -00:10:00,020 --> 00:10:02,930 -So these 8 bits already have an even parity, and the +00:09:49,403 --> 00:09:53,991 +not just that it happened, his key insight was that if you apply some parity checks 163 -00:10:02,930 --> 00:10:06,060 -sender can feel good leaving that bit number 2 unchanged. +00:09:53,991 --> 00:09:57,706 +not to the full message, but to certain carefully selected subsets, 164 -00:10:07,020 --> 00:10:11,127 -Then on the other end, if the receiver checks the parity of this group and they find +00:09:57,706 --> 00:10:02,131 +you can ask a more refined series of questions that pin down the location of any 165 -00:10:11,127 --> 00:10:15,380 -that it's odd, they'll know that the error is somewhere among these 8 bits on the right. +00:10:02,131 --> 00:10:03,060 +single bit error. 166 -00:10:15,820 --> 00:10:20,580 -Otherwise, it means either there's no error, or the error is somewhere on the left half. +00:10:03,060 --> 00:10:08,718 +The overall feeling is a bit like playing a game of 20 questions, 167 -00:10:21,120 --> 00:10:23,849 -Or I guess there could have been two errors, but for right now we're +00:10:08,718 --> 00:10:14,720 +asking yes or no queries that chop the space of possibilities in half. 168 -00:10:23,849 --> 00:10:26,500 -going to assume that there's at most one error in the entire block. +00:10:14,720 --> 00:10:18,735 +of the size of the block, or in other words, it grows one bit at a time as the block size 169 -00:10:26,940 --> 00:10:28,740 -Things break down completely for more than that. +00:10:18,735 --> 00:10:22,751 +doubles. The relevant fact here is that that information directly corresponds to how much 170 -00:10:29,160 --> 00:10:32,004 -Here, before we look at the next two checks, take a moment to think +00:10:22,751 --> 00:10:26,500 +redundancy we need. That's really what runs against most people's knee-jerk reaction 171 -00:10:32,004 --> 00:10:35,100 -about what these first two allow us to do when you consider them together. +00:10:26,940 --> 00:10:28,694 +Then, if an error is detected, it gives the receiver a little more information 172 -00:10:35,800 --> 00:10:39,660 -Let's say you detect an error among the odd columns and among the right half. +00:10:28,694 --> 00:10:30,360 +about where specifically the error is, namely that it's in an odd position. 173 -00:10:40,200 --> 00:10:43,040 -It necessarily means the error is somewhere in the last column. +00:10:30,360 --> 00:10:34,531 +ent to errors, where usually copying the whole message is the first instinct that 174 -00:10:43,820 --> 00:10:47,413 -If there was no error in the odd column but there was one in the right half, +00:10:34,531 --> 00:10:38,397 +comes to mind. And then, by the way, there is this whole other way that you 175 -00:10:47,413 --> 00:10:49,700 -that tells you it's in the second to last column. +00:10:38,397 --> 00:10:42,924 +sometimes see Hamming codes presented, where you multiply the message by one big matrix. 176 -00:10:50,440 --> 00:10:53,823 -Likewise, if there is an error in the odd columns but not in the right half, +00:10:42,924 --> 00:10:46,740 +It's kind of nice because it relates it to the broader family of linear cod 177 -00:10:53,823 --> 00:10:56,020 -you know that it's somewhere in the second column. +00:10:46,740 --> 00:10:52,277 +You might think that limiting a parity check to half the bits makes it less effective, 178 -00:10:56,020 --> 00:10:59,340 -And then if neither of those two parity checks detects anything, +00:10:52,277 --> 00:10:56,414 +but when it's done in conjunction with other well-chosen checks, 179 -00:10:59,340 --> 00:11:03,120 -it means the only place that an error could be is in that leftmost column. +00:10:56,414 --> 00:11:00,360 +it counter-intuitively gives us something a lot more powerful. 180 -00:11:03,340 --> 00:11:06,120 -But it also might simply mean there's no error at all. +00:11:00,360 --> 00:11:03,398 +To actually set up that parity check, remember, 181 -00:11:06,300 --> 00:11:08,643 -Which is all a rather belabored way to say that +00:11:03,398 --> 00:11:07,957 +it requires earmarking some special bit that has control for the parity 182 -00:11:08,643 --> 00:11:10,840 -two parity checks let us pin down the column. +00:11:07,957 --> 00:11:09,160 +of that full group. 183 -00:11:11,480 --> 00:11:13,640 -From here, you can probably guess what follows. +00:11:09,160 --> 00:11:10,100 +Here let's just choose position 1. 184 -00:11:13,800 --> 00:11:16,140 -We do basically the same thing but for the rows. +00:11:10,100 --> 00:11:12,932 +For the example shown, the parity of these 8 bits is currently odd, 185 -00:11:16,440 --> 00:11:20,900 -There's going to be a parity check on the odd rows, using position 4 as a parity bit. +00:11:12,932 --> 00:11:16,140 +so the sender is responsible for toggling that parity bit, and now it's even. 186 -00:11:21,380 --> 00:11:25,820 -So in this example, that group already has an even parity, so bit 4 would be set to a 0. +00:11:16,440 --> 00:11:16,980 +This is only 1 out of 4 parity checks that we'll do. 187 -00:11:26,560 --> 00:11:29,798 -And finally, there's a parity check on the bottom two rows, +00:11:16,980 --> 00:11:17,456 +The second check is among the 8 bits on the right half of the grid, 188 -00:11:29,798 --> 00:11:31,580 -using position 8 as a parity bit. +00:11:17,456 --> 00:11:17,680 +at least as we've drawn it here. 189 -00:11:32,120 --> 00:11:34,470 -In this case, it looks like the sender needs to turn +00:11:17,680 --> 00:11:20,900 +This time we might use position 2 as a parity bit. 190 -00:11:34,470 --> 00:11:36,820 -that bit 8 on in order to give the group even parity. +00:11:21,380 --> 00:11:22,777 +So these 8 bits already have an even parity, and the 191 -00:11:37,700 --> 00:11:40,132 -Just as the first two checks let us pin down the column, +00:11:22,777 --> 00:11:24,280 +sender can feel good leaving that bit number 2 unchanged. 192 -00:11:40,132 --> 00:11:41,840 -these next two let you pin down the row. +00:11:24,280 --> 00:11:28,237 +d if you take that to an extreme, you could have a block with, say, a million bits, 193 -00:11:42,880 --> 00:11:47,540 -As an example, imagine that during the transmission there's an error at, say, position 3. +00:11:28,237 --> 00:11:32,053 +where you would quite literally be playing 20 questions with your parity checks, 194 -00:11:48,180 --> 00:11:52,164 -Well, this affects the first parity group, and it also affects the second parity group, +00:11:32,053 --> 00:11:35,916 +and it uses only 21 parity bits. And if you step back to think about looking at a 195 -00:11:52,164 --> 00:11:55,560 -so the receiver knows that there's an error somewhere in that right column. +00:11:35,916 --> 00:11:39,780 +million bits and locating a single error, that genuinely feels crazy. The problem, 196 -00:11:56,100 --> 00:12:00,540 -But it doesn't affect the third group, and it doesn't affect the fourth group. +00:11:40,040 --> 00:11:39,780 +Otherwise, it means either there's no error, or the error is somewhere on the left half. 197 -00:12:01,240 --> 00:12:04,501 -And that lets the receiver pinpoint the error up to the first row, +00:11:40,040 --> 00:11:46,635 +Or I guess there could have been two errors, but for right now we're 198 -00:12:04,501 --> 00:12:07,520 -which necessarily means position 3, so they can fix the error. +00:11:46,635 --> 00:11:53,040 +going to assume that there's at most one error in the entire block. 199 -00:12:08,580 --> 00:12:11,404 -You might enjoy taking a moment to convince yourself that the +00:11:53,040 --> 00:11:53,840 +Things break down completely for more than that. 200 -00:12:11,404 --> 00:12:15,459 -answers to these four questions really will always let you pin down a specific location, +00:11:53,840 --> 00:11:56,847 +Here, before we look at the next two checks, take a moment to think 201 -00:12:15,459 --> 00:12:17,100 -no matter where they turn out to be. +00:11:56,847 --> 00:12:00,120 +about what these first two allow us to do when you consider them together. 202 -00:12:17,720 --> 00:12:20,262 -In fact, the astute among you might even notice a +00:12:00,120 --> 00:12:04,460 +Let's say you detect an error among the odd columns and among the right half. 203 -00:12:20,262 --> 00:12:23,060 -connection between these questions and binary counting. +00:12:04,460 --> 00:12:04,460 +It necessarily means the error is somewhere in the last column. 204 -00:12:23,500 --> 00:12:26,105 -And if you do, again let me emphasize, pause, try +00:12:04,460 --> 00:12:05,841 +If there was no error in the odd column but there was one in the right half, 205 -00:12:26,105 --> 00:12:28,920 -for yourself to draw the connection before I spoil it. +00:12:05,841 --> 00:12:06,720 +that tells you it's in the second to last column. 206 -00:12:30,500 --> 00:12:34,569 -If you're wondering what happens if a parity bit itself gets affected, +00:12:06,720 --> 00:12:13,680 +Likewise, if there is an error in the odd columns but not in the right half, 207 -00:12:34,569 --> 00:12:36,060 -well, you can just try it. +00:12:13,680 --> 00:12:18,200 +you know that it's somewhere in the second column. 208 -00:12:36,440 --> 00:12:40,406 -Take a moment to think about how any error among these four special bits is going +00:12:18,660 --> 00:12:20,068 +like the much more commonly used Reed-Solomon algorithm, 209 -00:12:40,406 --> 00:12:44,180 -to be tracked down just like any other, with the same group of four questions. +00:12:20,068 --> 00:12:22,219 +which handles burst errors particularly well, and it can be tuned to be resilient to a 210 -00:12:47,060 --> 00:12:50,059 -It doesn't really matter, since at the end of the day what we want is to +00:12:22,219 --> 00:12:23,060 +larger number of errors per block. 211 -00:12:50,059 --> 00:12:53,100 -protect the message bits, the error correction bits are just riding along. +00:12:23,500 --> 00:12:31,540 +But it also might simply mean there's no error at all. 212 -00:12:53,600 --> 00:12:55,772 -But protecting those bits as well is something that +00:12:31,540 --> 00:12:32,665 +Which is all a rather belabored way to say that 213 -00:12:55,772 --> 00:12:57,820 -naturally falls out of the scheme as a byproduct. +00:12:32,665 --> 00:12:33,720 +two parity checks let us pin down the column. 214 -00:12:59,200 --> 00:13:01,760 -You might also enjoy anticipating how this scales. +00:12:33,720 --> 00:12:36,060 +From here, you can probably guess what follows. 215 -00:13:02,300 --> 00:13:07,242 -If we used a block of size 256 bits, for example, in order to pin down a location, +00:12:36,440 --> 00:12:43,530 +spire in a way that spells out the position of an error only came to Hamming when 216 -00:13:07,242 --> 00:13:12,482 -you need only eight yes or no questions to binary search your way down to some specific +00:12:43,530 --> 00:12:50,880 +he stepped back after a bunch of other analysis and asked, okay, what is the most eff 217 -00:13:12,482 --> 00:13:12,780 -spot. +00:12:50,880 --> 00:12:51,241 +icient I could conceivably be about this? He was 218 -00:13:15,640 --> 00:13:18,142 -And remember, each question requires giving up only +00:12:51,241 --> 00:12:51,640 +also candid about how important it was that parity che 219 -00:13:18,142 --> 00:13:20,500 -a single bit to set the appropriate parity check. +00:12:51,640 --> 00:12:54,760 +So in this example, that group already has an even parity, so bit 4 would be set to a 0. 220 -00:13:23,160 --> 00:13:26,053 -Some of you may already see it, but we'll talk later about the +00:12:54,760 --> 00:12:57,812 +it is today. There are like half a dozen times throughout this book that 221 -00:13:26,053 --> 00:13:29,360 -systematic way to find what these questions are in just a minute or two. +00:12:57,812 --> 00:13:00,740 +he references the Louis Pasteur quote, luck favors a prepared mind. Cl 222 -00:13:29,880 --> 00:13:31,790 -Hopefully this sketch is enough to appreciate +00:13:00,740 --> 00:13:05,680 +In this case, it looks like the sender needs to turn 223 -00:13:31,790 --> 00:13:33,660 -the efficiency of what we're developing here. +00:13:05,680 --> 00:13:10,620 +that bit 8 on in order to give the group even parity. 224 -00:13:33,660 --> 00:13:37,168 -Everything except for those eight highlighted parity bits can be +00:13:10,620 --> 00:13:14,458 +Part of the reason that clever ideas look deceptively easy is that we only ever 225 -00:13:37,168 --> 00:13:41,000 -whatever you want it to be, carrying whatever message or data you want. +00:13:14,458 --> 00:13:18,440 +see the final result, cleaning up what was messy, never mentioning all of the wrong 226 -00:13:41,720 --> 00:13:45,846 -The eight bits are redundant in the sense that they're completely determined by the rest +00:13:18,440 --> 00:13:29,360 +As an example, imagine that during the transmission there's an error at, say, position 3. 227 -00:13:45,846 --> 00:13:50,020 -of the message, but it's in a much smarter way than simply copying the message as a whole. +00:13:29,880 --> 00:13:34,306 +Well, this affects the first parity group, and it also affects the second parity group, 228 -00:13:53,600 --> 00:13:56,040 -And still, for so little given up, you would be +00:13:34,306 --> 00:13:38,080 +so the receiver knows that there's an error somewhere in that right column. 229 -00:13:56,040 --> 00:13:58,380 -able to identify and fix any single bit error. +00:13:38,080 --> 00:13:41,000 +But it doesn't affect the third group, and it doesn't affect the fourth group. 230 -00:13:59,200 --> 00:14:00,400 -Well, almost. +00:13:41,720 --> 00:13:45,667 +And that lets the receiver pinpoint the error up to the first row, 231 -00:14:00,960 --> 00:14:05,822 -Okay, so the one problem here is that if none of the four parity checks detect an error, +00:13:45,667 --> 00:13:49,320 +which necessarily means position 3, so they can fix the error. 232 -00:14:05,822 --> 00:14:10,303 -meaning that the specially selected subsets of eight bits all have even parities, +00:13:49,320 --> 00:13:54,207 +You might enjoy taking a moment to convince yourself that the 233 -00:14:10,303 --> 00:14:14,619 -just like the sender intended, then it either means there was no error at all, +00:13:54,207 --> 00:14:01,222 +answers to these four questions really will always let you pin down a specific location, 234 -00:14:14,619 --> 00:14:16,860 -or it narrows us down into position zero. +00:14:01,222 --> 00:14:04,060 +no matter where they turn out to be. 235 -00:14:17,740 --> 00:14:22,497 -You see, with four yes or no questions, we have 16 possible outcomes for our parity +00:14:04,060 --> 00:14:04,479 +In fact, the astute among you might even notice a 236 -00:14:22,497 --> 00:14:27,198 -checks, and at first that feels perfect for pinpointing one out of 16 positions in +00:14:04,479 --> 00:14:04,940 +connection between these questions and binary counting. 237 -00:14:27,198 --> 00:14:31,900 -the block, but you also need to communicate a 17th outcome, the no error condition. +00:14:04,940 --> 00:14:07,430 +And if you do, again let me emphasize, pause, try 238 -00:14:33,020 --> 00:14:34,860 -The solution here is actually pretty simple. +00:14:07,430 --> 00:14:10,120 +for yourself to draw the connection before I spoil it. 239 -00:14:35,280 --> 00:14:37,300 -Just forget about that zeroth bit entirely. +00:14:10,120 --> 00:14:10,412 +If you're wondering what happens if a parity bit itself gets affected, 240 -00:14:37,840 --> 00:14:41,250 -So when we do our four parity checks and we see that they're all even, +00:14:10,412 --> 00:14:10,520 +well, you can just try it. 241 -00:14:41,250 --> 00:14:43,460 -it unambiguously means that there is no error. +00:14:10,520 --> 00:14:15,409 +Take a moment to think about how any error among these four special bits is going 242 -00:14:44,240 --> 00:14:48,680 -What that means is rather than working with a 16-bit block, we work with a 15-bit block, +00:14:15,409 --> 00:14:20,060 +to be tracked down just like any other, with the same group of four questions. 243 -00:14:48,680 --> 00:14:52,671 -where 11 of the bits are free to carry a message and four of them are there for +00:14:20,060 --> 00:14:22,254 +It doesn't really matter, since at the end of the day what we want is to 244 -00:14:52,671 --> 00:14:53,220 -redundancy. +00:14:22,254 --> 00:14:24,480 +protect the message bits, the error correction bits are just riding along. 245 -00:14:53,780 --> 00:14:56,236 -And with that, we now have what people in the +00:14:24,480 --> 00:14:29,628 +But protecting those bits as well is something that 246 -00:14:56,236 --> 00:14:58,800 -business would refer to as a 15-11 Hamming code. +00:14:29,628 --> 00:14:34,480 +naturally falls out of the scheme as a byproduct. 247 -00:14:59,860 --> 00:15:03,153 -That said, it is nice to have a block size that's a clean power of two, +00:14:34,480 --> 00:14:34,860 +You might also enjoy anticipating how this scales. 248 -00:15:03,153 --> 00:15:05,806 -and there's a clever way that we can keep that zeroth bit +00:14:35,280 --> 00:14:37,166 +If we used a block of size 256 bits, for example, in order to pin down a location, 249 -00:15:05,806 --> 00:15:08,140 -around and get it to do a little extra work for us. +00:14:37,166 --> 00:14:39,166 +you need only eight yes or no questions to binary search your way down to some specific 250 -00:15:08,700 --> 00:15:11,577 -If we use it as a parity bit across the whole block, +00:14:39,166 --> 00:14:39,280 +spot. 251 -00:15:11,577 --> 00:15:15,540 -it lets us actually detect, even though we can't correct, two-bit errors. +00:14:39,280 --> 00:14:39,403 +And remember, each question requires giving up only 252 -00:15:16,160 --> 00:15:16,820 -Here's how it works. +00:14:39,403 --> 00:14:39,520 +a single bit to set the appropriate parity check. 253 -00:15:17,180 --> 00:15:19,947 -After setting those four special error correcting bits, +00:14:39,520 --> 00:14:41,200 +Some of you may already see it, but we'll talk later about the 254 -00:15:19,947 --> 00:15:23,358 -we set that zeroth one so that the parity of the full block is even, +00:14:41,200 --> 00:14:43,120 +systematic way to find what these questions are in just a minute or two. 255 -00:15:23,358 --> 00:15:24,940 -just like a normal parity check. +00:14:43,120 --> 00:14:43,291 +Hopefully this sketch is enough to appreciate 256 -00:15:25,700 --> 00:15:29,987 -Now, if there's a single bit error, then the parity of the full block toggles to be odd, +00:14:43,291 --> 00:14:43,460 +the efficiency of what we're developing here. 257 -00:15:29,987 --> 00:15:33,600 -but we would catch that anyway, thanks to the four error correcting checks. +00:14:44,240 --> 00:14:46,734 +Everything except for those eight highlighted parity bits can be 258 -00:15:34,160 --> 00:15:37,769 -However, if there's two errors, then the overall parity is going to toggle +00:14:46,734 --> 00:14:49,460 +whatever you want it to be, carrying whatever message or data you want. 259 -00:15:37,769 --> 00:15:41,330 -back to being even, but the receiver would still see that there's been at +00:14:49,460 --> 00:14:54,103 +The eight bits are redundant in the sense that they're completely determined by the rest 260 -00:15:41,330 --> 00:15:45,180 -least some error because of what's going on with those four usual parity checks. +00:14:54,103 --> 00:14:58,800 +of the message, but it's in a much smarter way than simply copying the message as a whole. 261 -00:15:45,180 --> 00:15:49,124 -So if they notice an even parity overall, but something non-zero happening +00:14:59,860 --> 00:15:01,463 +And still, for so little given up, you would be 262 -00:15:49,124 --> 00:15:52,700 -with the other checks, it tells them there were at least two errors. +00:15:01,463 --> 00:15:03,000 +able to identify and fix any single bit error. 263 -00:15:53,520 --> 00:15:54,000 -Isn't that clever? +00:15:03,000 --> 00:15:04,160 +Well, almost. 264 -00:15:54,300 --> 00:15:56,799 -Even though we can't correct those two-bit errors, +00:15:04,160 --> 00:15:07,640 +Okay, so the one problem here is that if none of the four parity checks detect an error, 265 -00:15:56,799 --> 00:16:00,132 -just by putting that one little bothersome zeroth bit back to work, +00:15:07,640 --> 00:15:10,847 +meaning that the specially selected subsets of eight bits all have even parities, 266 -00:16:00,132 --> 00:16:01,260 -it lets us detect them. +00:15:10,847 --> 00:15:13,936 +just like the sender intended, then it either means there was no error at all, 267 -00:16:02,260 --> 00:16:05,220 -This is pretty standard, it's known as an extended Hamming code. +00:15:13,936 --> 00:15:15,540 +or it narrows us down into position zero. 268 -00:16:06,540 --> 00:16:10,698 -Technically speaking, you now have a full description of what a Hamming code does, +00:15:16,160 --> 00:15:21,865 +You see, with four yes or no questions, we have 16 possible outcomes for our parity 269 -00:16:10,698 --> 00:16:14,255 -at least for the example of a 16-bit block, but I think you'll find it +00:15:21,865 --> 00:15:27,502 +checks, and at first that feels perfect for pinpointing one out of 16 positions in 270 -00:16:14,255 --> 00:16:17,963 -more satisfying to check your understanding and solidify everything up to +00:15:27,502 --> 00:15:33,140 +the block, but you also need to communicate a 17th outcome, the no error condition. 271 -00:16:17,963 --> 00:16:21,320 -this point by doing one full example from start to finish yourself. +00:15:33,140 --> 00:15:37,180 +The solution here is actually pretty simple. 272 -00:16:22,080 --> 00:16:24,300 -I'll step through it with you though so you can check yourself. +00:15:37,180 --> 00:15:41,620 +Just forget about that zeroth bit entirely. 273 -00:16:25,120 --> 00:16:29,638 -To set up a message, whether that's a literal message that you're translating over space, +00:15:41,620 --> 00:15:48,343 +So when we do our four parity checks and we see that they're all even, 274 -00:16:29,638 --> 00:16:31,998 -or some data that you want to store over time, +00:15:48,343 --> 00:15:52,700 +it unambiguously means that there is no error. 275 -00:16:31,998 --> 00:16:34,660 -the first step is to divide it up into 11-bit chunks. +00:15:53,520 --> 00:15:55,260 +What that means is rather than working with a 16-bit block, we work with a 15-bit block, 276 -00:16:35,580 --> 00:16:39,760 -Each chunk is going to get packaged into an error-resistant 16-bit block. +00:15:55,260 --> 00:15:56,824 +where 11 of the bits are free to carry a message and four of them are there for 277 -00:16:39,760 --> 00:16:43,220 -So let's take this one as an example and actually work it out. +00:15:56,824 --> 00:15:57,040 +redundancy. 278 -00:16:43,740 --> 00:16:44,940 -Go ahead, actually do it! +00:15:57,040 --> 00:16:01,042 +And with that, we now have what people in the 279 -00:16:45,220 --> 00:16:47,020 -Pause and try putting together this block. +00:16:01,042 --> 00:16:05,220 +business would refer to as a 15-11 Hamming code. 280 -00:16:52,720 --> 00:16:53,680 -Okay, you ready? +00:16:06,540 --> 00:16:08,433 +That said, it is nice to have a block size that's a clean power of two, 281 -00:16:54,240 --> 00:16:58,805 -Remember, position 0 along with the other powers of 2 are reserved for error correction +00:16:08,433 --> 00:16:09,958 +and there's a clever way that we can keep that zeroth bit 282 -00:16:58,805 --> 00:17:03,320 -duty, so you start by placing the message bits in all of the remaining spots, in order. +00:16:09,958 --> 00:16:11,300 +around and get it to do a little extra work for us. 283 -00:17:05,339 --> 00:17:08,947 -You need this group to have an even parity, which it already does, +00:16:11,300 --> 00:16:12,444 +If we use it as a parity bit across the whole block, 284 -00:17:08,947 --> 00:17:12,339 -so you should have set that parity bit in position 1 to be a 0. +00:16:12,444 --> 00:16:14,020 +it lets us actually detect, even though we can't correct, two-bit errors. 285 -00:17:13,020 --> 00:17:15,450 -The next group starts off with an odd parity, +00:16:14,020 --> 00:16:14,600 +Here's how it works. 286 -00:17:15,450 --> 00:17:17,880 -so you should have set its parity bit to be 1. +00:16:14,600 --> 00:16:17,874 +After setting those four special error correcting bits, 287 -00:17:19,160 --> 00:17:21,673 -The group after that starts with an odd parity, +00:16:17,874 --> 00:16:21,908 +we set that zeroth one so that the parity of the full block is even, 288 -00:17:21,673 --> 00:17:24,240 -so again you should have set its parity bit to 1. +00:16:21,908 --> 00:16:23,780 +just like a normal parity check. 289 -00:17:24,780 --> 00:17:27,764 -And the final group also has an odd parity, meaning +00:16:23,780 --> 00:16:26,417 +Now, if there's a single bit error, then the parity of the full block toggles to be odd, 290 -00:17:27,764 --> 00:17:30,060 -we set that bit in position 8 to be a 1. +00:16:26,417 --> 00:16:28,640 +but we would catch that anyway, thanks to the four error correcting checks. 291 -00:17:31,300 --> 00:17:35,411 -And then as the final step, the full block now has an even parity, +00:16:28,640 --> 00:16:32,281 +However, if there's two errors, then the overall parity is going to toggle 292 -00:17:35,411 --> 00:17:40,320 -meaning that you can set that bit number 0, the overarching parity bit, to be 0. +00:16:32,281 --> 00:16:35,875 +back to being even, but the receiver would still see that there's been at 293 -00:17:41,340 --> 00:17:44,885 -So as this block is sent off, the parity of the four special +00:16:35,875 --> 00:16:39,760 +least some error because of what's going on with those four usual parity checks. 294 -00:17:44,885 --> 00:17:48,140 -subsets and the block as a whole will all be even, or 0. +00:16:39,760 --> 00:16:43,567 +So if they notice an even parity overall, but something non-zero happening 295 -00:17:48,820 --> 00:17:52,180 -As the second part of the exercise, let's have you play the role of the receiver. +00:16:43,567 --> 00:16:47,020 +with the other checks, it tells them there were at least two errors. 296 -00:17:53,480 --> 00:17:56,820 -Of course, that would mean you don't already know what this message is. +00:16:52,720 --> 00:16:47,020 +Isn't that clever? 297 +00:16:52,720 --> 00:16:55,945 +Even though we can't correct those two-bit errors, + +298 +00:16:55,945 --> 00:17:00,245 +just by putting that one little bothersome zeroth bit back to work, + +299 +00:17:00,245 --> 00:17:01,700 +it lets us detect them. + +300 +00:17:01,700 --> 00:17:03,320 +This is pretty standard, it's known as an extended Hamming code. + +301 +00:17:05,339 --> 00:17:10,258 +Technically speaking, you now have a full description of what a Hamming code does, + +302 +00:17:10,258 --> 00:17:14,465 +at least for the example of a 16-bit block, but I think you'll find it + +303 +00:17:14,465 --> 00:17:18,849 +more satisfying to check your understanding and solidify everything up to + +304 +00:17:18,849 --> 00:17:22,819 +this point by doing one full example from start to finish yourself. + +305 +00:17:22,819 --> 00:17:22,819 +I'll step through it with you though so you can check yourself. + +306 +00:17:22,819 --> 00:17:31,820 +To set up a message, whether that's a literal message that you're translating over space, + +307 +00:17:31,820 --> 00:17:36,520 +or some data that you want to store over time, + +308 +00:17:36,520 --> 00:17:41,820 +the first step is to divide it up into 11-bit chunks. + +309 +00:17:41,820 --> 00:17:50,200 +Each chunk is going to get packaged into an error-resistant 16-bit block. + +310 +00:17:50,620 --> 00:17:54,320 +So let's take this one as an example and actually work it out. + +311 +00:17:54,320 --> 00:17:55,000 +Go ahead, actually do it! + +312 +00:17:55,000 --> 00:17:56,820 +Pause and try putting together this block. + +313 00:17:57,080 --> 00:17:59,780 +Okay, you ready? + +314 +00:18:00,020 --> 00:18:03,238 +Remember, position 0 along with the other powers of 2 are reserved for error correction + +315 +00:18:03,238 --> 00:18:06,420 +duty, so you start by placing the message bits in all of the remaining spots, in order. + +316 +00:18:06,420 --> 00:18:17,495 +You need this group to have an even parity, which it already does, + +317 +00:18:17,495 --> 00:18:27,910 +so you should have set that parity bit in position 1 to be a 0. + +318 +00:18:28,800 --> 00:18:27,910 +The next group starts off with an odd parity, + +319 +00:18:29,690 --> 00:18:28,800 +so you should have set its parity bit to be 1. + +320 +00:18:29,690 --> 00:18:33,322 +The group after that starts with an odd parity, + +321 +00:18:33,322 --> 00:18:37,030 +so again you should have set its parity bit to 1. + +322 +00:18:38,550 --> 00:18:40,381 +And the final group also has an odd parity, meaning + +323 +00:18:40,381 --> 00:18:41,790 +we set that bit in position 8 to be a 1. + +324 +00:18:42,650 --> 00:18:47,399 +And then as the final step, the full block now has an even parity, + +325 +00:18:47,399 --> 00:18:53,070 +meaning that you can set that bit number 0, the overarching parity bit, to be 0. + +326 +00:18:53,070 --> 00:18:56,667 +So as this block is sent off, the parity of the four special + +327 +00:18:56,667 --> 00:18:59,970 +subsets and the block as a whole will all be even, or 0. + +328 +00:19:01,310 --> 00:19:07,610 +As the second part of the exercise, let's have you play the role of the receiver. + +329 +00:19:07,610 --> 00:19:17,810 +Of course, that would mean you don't already know what this message is. + +330 +00:19:18,030 --> 00:19:24,910 Maybe some of you memorized it, but let's assume that you haven't. -298 -00:18:00,020 --> 00:18:04,568 +331 +00:19:24,910 --> 00:19:27,702 What I'm going to do is change either 0, 1, or 2 of the bits in that block, -299 -00:18:04,568 --> 00:18:07,740 +332 +00:19:27,702 --> 00:19:29,650 and then ask you to figure out what it is that I did. -300 -00:18:08,260 --> 00:18:10,810 +333 +00:19:29,650 --> 00:19:29,650 So again, pause and try working it out. -301 -00:18:18,790 --> 00:18:22,496 +334 +00:19:29,650 --> 00:19:33,348 Okay, so you as the receiver now check the first parity group, -302 -00:18:22,496 --> 00:18:27,027 +335 +00:19:33,348 --> 00:19:37,869 and you can see that it's even, so any error that exists would have to be in -303 -00:18:27,027 --> 00:18:27,910 +336 +00:19:37,869 --> 00:19:38,750 an even column. -304 -00:18:29,690 --> 00:18:34,439 +337 +00:19:39,430 --> 00:19:40,568 The next check gives us an odd number, telling us both that there's at least one error, -305 -00:18:34,439 --> 00:18:37,030 +338 +00:19:40,568 --> 00:19:41,190 and narrowing us down into this specific column. -306 -00:18:38,550 --> 00:18:41,790 +339 +00:19:41,190 --> 00:19:41,310 The third check is even, chopping down the possibilities even further. -307 -00:18:42,650 --> 00:18:46,969 +340 +00:19:41,310 --> 00:19:41,310 And the last parity check is odd, telling us there's an error somewhere in the bottom, -308 -00:18:46,969 --> 00:18:49,650 +341 +00:19:41,310 --> 00:19:41,310 which by now we can see must be in position number 10. -309 -00:18:51,490 --> 00:18:54,342 +342 +00:19:41,310 --> 00:19:41,310 What's more, the parity of the whole block is odd, -310 -00:18:54,342 --> 00:18:57,530 +343 +00:19:41,310 --> 00:19:41,310 giving us confidence that there was one flip and not two. -311 -00:18:58,070 --> 00:18:59,970 +344 +00:19:41,310 --> 00:19:41,310 If it's three or more, all bets are off. -312 -00:19:01,310 --> 00:19:05,520 +345 +00:19:41,310 --> 00:19:41,310 After correcting that bit number 10, pulling out the 11 bits that were not -313 -00:19:05,520 --> 00:19:09,730 +346 +00:19:41,310 --> 00:19:41,310 used for correction gives us the relevant segment of the original message, -314 -00:19:09,730 --> 00:19:14,390 +347 +00:19:41,310 --> 00:19:41,310 which if you rewind and compare is indeed exactly what we started the example with. -315 -00:19:15,710 --> 00:19:18,012 +348 +00:19:41,310 --> 00:19:41,310 And now that you know how to do all this by hand, -316 -00:19:18,012 --> 00:19:21,834 +349 +00:19:41,310 --> 00:19:41,310 I'd like to show you how you can carry out the core part of all of this logic with -317 -00:19:21,834 --> 00:19:23,170 +350 +00:19:41,310 --> 00:19:41,310 a single line of Python code. -318 -00:19:23,870 --> 00:19:28,158 +351 +00:19:41,310 --> 00:19:41,310 You see, what I haven't told you yet is just how elegant this algorithm really is, -319 -00:19:28,158 --> 00:19:31,878 +352 +00:19:41,310 --> 00:19:41,310 how simple it is to get a machine to point to the position of an error, -320 -00:19:31,878 --> 00:19:35,598 +353 +00:19:41,310 --> 00:19:41,310 how to systematically scale it, and how we can frame all of this as one -321 -00:19:35,598 --> 00:19:38,750 +354 +00:19:41,310 --> 00:19:41,310 single operation rather than multiple separate parity checks. -322 -00:19:39,430 --> 00:19:41,310 +355 +00:19:41,310 --> 00:19:41,310 To see what I mean, come join me in part two. diff --git a/2020/hamming-codes/english/sentence_timings.json b/2020/hamming-codes/english/sentence_timings.json index ce7c32bc8..7f7e7d72c 100644 --- a/2020/hamming-codes/english/sentence_timings.json +++ b/2020/hamming-codes/english/sentence_timings.json @@ -25,768 +25,768 @@ 50.38 ], [ - "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notice", 50.68, 56.0 ], [ - "Then the machine reading this file could compare these three copies and always take the best 2 out of 3 whenever there's a discrepancy.", + "where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error. There's nothing specia", 57.58, - 64.06 + 74.54 ], [ "But what that means is using two thirds of your space for redundancy.", - 67.16, - 70.86 + 74.54, + 78.92 ], [ "And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped.", - 71.48, - 77.24 + 78.92, + 87.3 ], [ "The much more interesting question is how to make it so that errors can be corrected while giving up as little space as possible.", - 77.98, - 84.02 + 87.6, + 100.3 ], [ - "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", - 84.52, - 93.36 + "c for implementing the whole scheme in hardware shockingly simple. Now if you want to see why this magic happens, take these 16 index labels for our positions, but instead of writing them in base 10, let's write them all in binary, running from 0000 up to 1111. As we put these binary labels back into their boxes, let m", + 100.9, + 119.58 ], [ - "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", - 93.76, - 100.3 + "e emphasize that they are distinct from the data that's actually being sent. They're nothing more than a conceptual label to help you and m", + 119.58, + 128.94 ], [ "And it will still be the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to identify that there was an error and precisely where it was so that it knows how to correct it.", - 100.9, - 112.66 + 128.94, + 138.12 ], [ "And honestly, that feels like magic.", - 112.66, - 114.62 + 138.12, + 141.94 ], [ "And for this particular scheme, if two bits get flipped, the machine will at least be able to detect that there were two errors, though it won't know how to fix them.", - 115.44, - 122.86 + 142.84, + 148.66 ], [ "We'll talk a little bit later about how this scales for blocks with different sizes.", - 123.52, - 126.9 + 149.52, + 148.66 ], [ - "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", - 127.86, - 132.9 + "where that's a 1, you get the second parity group from our scheme. In other words, that second check is asking, hey, me again, if there's an error, is the second to last bit of that position a 1? And", + 149.52, + 155.26 ], [ - "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", - 133.66, - 141.94 + "so on. The third parity check covers every position whose third to last bit is turned on, and the last one covers the last eight positions, those ones whose highest order bit is a 1. Everything we", + 155.26, + 164.56 ], [ "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code.", - 142.84, - 148.66 + 164.56, + 172.1 ], [ - "And by the way, the way I'm thinking about the structure of this video is less about explaining it as directly as possible, and more a matter of prompting you to invent it for yourself, with a little gentle guidance here and there.", - 149.52, - 159.82 + "four questions, which in turn is the same as spelling out a position in binary. I hope this makes two things clearer. The first is how to systematically generalize to block sizes that are bigger powers of two. If it takes more bits to describe each position, like six bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check.", + 172.1, + 191.56 ], [ "So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", - 160.12, - 166.72 + 191.56, + 199.64 ], [ "Also, if you want your understanding to get down to the hardware level, Ben Eater has made a video in conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying.", - 167.24, - 178.24 + 199.64, + 215.5 ], [ "You should know, Hamming codes are not as widely used as more modern codes, like the Reed-Solomon algorithm, but there is a certain magic to the contrast between just how impossible this task feels at the start, and how utterly reasonable it seems once you learn about Hamming.", - 179.3, - 193.0 + 215.5, + 230.3 ], [ "The basic principle of error correction is that in a vast space of all possible messages, only some subset are going to be considered valid messages.", - 193.72, - 202.18 + 230.3, + 240.52 ], [ "As an analogy, think about correctly spelled words versus incorrectly spelled words.", - 202.8, - 206.94 + 240.52, + 243.56 ], [ - "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo.", - 208.9, - 217.34 + "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ", + 243.56, + 263.28 ], [ - "Coming up with a concrete algorithm to efficiently categorize messages like this, though, take a certain cleverness.", - 218.22, - 224.06 + "ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of", + 263.28, + 268.12 ], [ - "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", - 226.78, - 237.42 + "code. It's based on the XOR function. XOR, for those of you who don't know, stands for exclusive or. When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently", + 268.12, + 279.3 ], [ "And the programs he kept putting through it kept failing, because every now and then a bit would get misread.", - 237.8, - 242.4 + 279.3, + 287.28 ], [ "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", - 243.12, - 248.42 + 288.6, + 293.18 ], [ "There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them.", - 249.06, - 255.38 + 293.18, + 301.9 ], [ "Let's use an example that's simple, but not too simple, a block of 16 bits.", - 256.52, - 260.94 + 301.9, + 308.06 ], [ "We'll number the positions of these bits from 0 up to 15.", - 261.82, - 264.74 + 308.4, + 308.56 ], [ - "The actual data we want to store is only going to make up 12 of these bits, while 4 of the positions are going to be reserved as a kind of redundancy.", - 265.62, - 273.0 + "bit representations of those numbers under the hood. The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop.", + 308.56, + 326.6 ], [ "The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data.", - 273.9, - 280.04 + 326.78, + 336.6 ], [ "Instead, they'll need to be a much more nuanced and clever kind of redundancy, not adding any new information, but adding resilience.", - 280.72, - 287.28 + 336.6, + 337.98 ], [ - "You might expect these 4 special bits to come nicely packaged together, maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end.", - 288.6, - 299.62 + "s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which a", + 338.14, + 356.3 ], [ - "It also might give you a little hint about how this scales for larger blocks.", - 300.2, - 303.54 + "re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it goes from here. The sender is responsible for toggling some", + 356.88, + 366.12 ], [ - "Also, technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", - 304.9, - 313.26 + "of the special parity bits to make sure the sum works out to be 0000. Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio", + 366.12, + 378.68 ], [ "Like any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors.", - 314.14, - 325.54 + 379.0, + 392.36 ], [ "Of course, the words sender and receiver really refer to machines or software that's doing checks, and the idea of a message is meant really broadly, to include things like storage.", - 326.3, - 334.74 + 392.36, + 404.24 ], [ "After all, storing data is the same thing as sending a message, just from the past to the future, instead of from one place to another.", - 335.34, - 341.68 + 404.24, + 406.54 ], [ "So that's the setup, but before we can dive in, we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", - 342.56, - 356.3 + 407.98, + 423.34 ], [ "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", - 356.88, - 363.82 + 424.86, + 429.12 ], [ "The only job of this special bit is to make sure that the total number of 1s in the message is an even number.", - 364.88, - 371.28 + 429.12, + 432.78 ], [ "So for example right now, that total number of 1s is 7, that's odd, so the sender needs to flip that special bit to be a 1, making the count even.", - 372.08, - 379.96 + 432.78, + 438.78 ], [ "But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0.", - 380.8, - 386.42 + 438.78, + 445.12 ], [ "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", - 387.34, - 396.78 + 445.12, + 454.9 ], [ - "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", - 397.5, - 406.54 + "ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15.", + 454.9, + 469.94 ], [ "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", - 407.98, - 417.46 + 469.94, + 477.76 ], [ "In the jargon, whether a group of bits has an even or an odd number of 1s is known as its parity.", - 418.5, - 423.34 + 477.76, + 482.28 ], [ - "You could also use numbers and say the parity is 0 or 1, which is typically more helpful once you start doing math with the idea, and this special bit that the sender uses to control the parity is called the parity bit.", - 424.86, - 435.52 + "o collect together all of those positions, the positions of the bits that are turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it.", + 482.28, + 492.84 ], [ "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors or 5 or any other odd number, but they can know for sure that it wasn't 0.", - 437.56, - 449.26 + 492.84, + 502.78 ], [ "On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free.", - 449.98, - 462.3 + 502.78, + 512.38 ], [ - "You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right.", - 462.84, - 469.08 + "So at the moment it looks like if we do this on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state", + 512.76, + 526.96 ], [ - "Keep in mind, though, there is no method for error detection or correction that could give you 100% confidence that the message you receive is the one the sender intended.", - 469.7, - 478.9 + "where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block. What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it print", + 526.96, + 539.38 ], [ - "After all, enough random noise could always change one valid message into another valid message just by pure chance.", - 479.58, - 485.44 + "s out that error. Isn't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any. And there's nothing special about the size 16 here.", + 540.1, + 555.26 ], [ "Instead, the goal is to come up with a scheme that's robust up to a certain maximum number of errors, or maybe to reduce the probability of a false positive like this.", - 486.24, - 495.38 + 555.26, + 570.48 ], [ - "Parity checks on their own are pretty weak, but by distilling the idea of change across a full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes.", - 496.26, - 507.16 + "a parity check to detect 2-bit errors, but the idea is that almost all of the core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much m", + 570.48, + 584.76 ], [ "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", - 507.94, - 525.94 + 584.76, + 603.06 ], [ "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", - 526.68, - 533.38 + 603.06, + 614.72 ], [ - "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", - 534.16, - 539.38 + "of the size of the block, or in other words, it grows one bit at a time as the block size doubles. The relevant fact here is that that information directly corresponds to how much redundancy we need. That's really what runs against most people's knee-jerk reaction", + 614.72, + 626.5 ], [ "Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", - 540.1, - 548.24 + 626.94, + 630.36 ], [ - "If no error is detected among those 8 bits, it either means there's no error at all, or it sits somewhere in the even positions.", - 548.94, - 556.24 + "ent to errors, where usually copying the whole message is the first instinct that comes to mind. And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix. It's kind of nice because it relates it to the broader family of linear cod", + 630.36, + 646.74 ], [ "You might think that limiting a parity check to half the bits makes it less effective, but when it's done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful.", - 557.18, - 567.2 + 646.74, + 660.36 ], [ "To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", - 569.24, - 576.62 + 660.36, + 669.16 ], [ "Here let's just choose position 1.", - 577.48, - 579.18 + 669.16, + 670.1 ], [ "For the example shown, the parity of these 8 bits is currently odd, so the sender is responsible for toggling that parity bit, and now it's even.", - 579.72, - 586.98 + 670.1, + 676.14 ], [ "This is only 1 out of 4 parity checks that we'll do.", - 587.94, - 590.68 + 676.44, + 676.98 ], [ "The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here.", - 590.92, - 596.3 + 676.98, + 677.68 ], [ "This time we might use position 2 as a parity bit.", - 596.68, - 599.58 + 677.68, + 680.9 ], [ "So these 8 bits already have an even parity, and the sender can feel good leaving that bit number 2 unchanged.", - 600.02, - 606.06 + 681.38, + 684.28 ], [ - "Then on the other end, if the receiver checks the parity of this group and they find that it's odd, they'll know that the error is somewhere among these 8 bits on the right.", - 607.02, - 615.38 + "d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits. And if you step back to think about looking at a million bits and locating a single error, that genuinely feels crazy. The problem,", + 684.28, + 699.78 ], [ "Otherwise, it means either there's no error, or the error is somewhere on the left half.", - 615.82, - 620.58 + 700.04, + 699.78 ], [ "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", - 621.12, - 626.5 + 700.04, + 713.04 ], [ "Things break down completely for more than that.", - 626.94, - 628.74 + 713.04, + 713.84 ], [ "Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you consider them together.", - 629.16, - 635.1 + 713.84, + 720.12 ], [ "Let's say you detect an error among the odd columns and among the right half.", - 635.8, - 639.66 + 720.12, + 724.46 ], [ "It necessarily means the error is somewhere in the last column.", - 640.2, - 643.04 + 724.46, + 724.46 ], [ "If there was no error in the odd column but there was one in the right half, that tells you it's in the second to last column.", - 643.82, - 649.7 + 724.46, + 726.72 ], [ "Likewise, if there is an error in the odd columns but not in the right half, you know that it's somewhere in the second column.", - 650.44, - 656.02 + 726.72, + 738.2 ], [ - "And then if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", - 656.02, - 663.12 + "like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly well, and it can be tuned to be resilient to a larger number of errors per block.", + 738.66, + 743.06 ], [ "But it also might simply mean there's no error at all.", - 663.34, - 666.12 + 743.5, + 751.54 ], [ "Which is all a rather belabored way to say that two parity checks let us pin down the column.", - 666.3, - 670.84 + 751.54, + 753.72 ], [ "From here, you can probably guess what follows.", - 671.48, - 673.64 + 753.72, + 756.06 ], [ - "We do basically the same thing but for the rows.", - 673.8, - 676.14 + "spire in a way that spells out the position of an error only came to Hamming when he stepped back after a bunch of other analysis and asked, okay, what is the most eff", + 756.44, + 770.88 ], [ - "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", - 676.44, - 680.9 + "icient I could conceivably be about this? He was also candid about how important it was that parity che", + 770.88, + 771.64 ], [ "So in this example, that group already has an even parity, so bit 4 would be set to a 0.", - 681.38, - 685.82 + 771.64, + 774.76 ], [ - "And finally, there's a parity check on the bottom two rows, using position 8 as a parity bit.", - 686.56, - 691.58 + "it is today. There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind. Cl", + 774.76, + 780.74 ], [ "In this case, it looks like the sender needs to turn that bit 8 on in order to give the group even parity.", - 692.12, - 696.82 + 780.74, + 790.62 ], [ - "Just as the first two checks let us pin down the column, these next two let you pin down the row.", - 697.7, - 701.84 + "Part of the reason that clever ideas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong", + 790.62, + 798.44 ], [ "As an example, imagine that during the transmission there's an error at, say, position 3.", - 702.88, - 707.54 + 798.44, + 809.36 ], [ "Well, this affects the first parity group, and it also affects the second parity group, so the receiver knows that there's an error somewhere in that right column.", - 708.18, - 715.56 + 809.88, + 818.08 ], [ "But it doesn't affect the third group, and it doesn't affect the fourth group.", - 716.1, - 720.54 + 818.08, + 821.0 ], [ "And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error.", - 721.24, - 727.52 + 821.72, + 829.32 ], [ "You might enjoy taking a moment to convince yourself that the answers to these four questions really will always let you pin down a specific location, no matter where they turn out to be.", - 728.58, - 737.1 + 829.32, + 844.06 ], [ "In fact, the astute among you might even notice a connection between these questions and binary counting.", - 737.72, - 743.06 + 844.06, + 844.94 ], [ "And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spoil it.", - 743.5, - 748.92 + 844.94, + 850.12 ], [ "If you're wondering what happens if a parity bit itself gets affected, well, you can just try it.", - 750.5, - 756.06 + 850.12, + 850.52 ], [ "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", - 756.44, - 764.18 + 850.52, + 860.06 ], [ "It doesn't really matter, since at the end of the day what we want is to protect the message bits, the error correction bits are just riding along.", - 767.06, - 773.1 + 860.06, + 864.48 ], [ "But protecting those bits as well is something that naturally falls out of the scheme as a byproduct.", - 773.6, - 777.82 + 864.48, + 874.48 ], [ "You might also enjoy anticipating how this scales.", - 779.2, - 781.76 + 874.48, + 874.86 ], [ "If we used a block of size 256 bits, for example, in order to pin down a location, you need only eight yes or no questions to binary search your way down to some specific spot.", - 782.3, - 792.78 + 875.28, + 879.28 ], [ "And remember, each question requires giving up only a single bit to set the appropriate parity check.", - 795.64, - 800.5 + 879.28, + 879.52 ], [ "Some of you may already see it, but we'll talk later about the systematic way to find what these questions are in just a minute or two.", - 803.16, - 809.36 + 879.52, + 883.12 ], [ "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", - 809.88, - 813.66 + 883.12, + 883.46 ], [ "Everything except for those eight highlighted parity bits can be whatever you want it to be, carrying whatever message or data you want.", - 813.66, - 821.0 + 884.24, + 889.46 ], [ "The eight bits are redundant in the sense that they're completely determined by the rest of the message, but it's in a much smarter way than simply copying the message as a whole.", - 821.72, - 830.02 + 889.46, + 898.8 ], [ "And still, for so little given up, you would be able to identify and fix any single bit error.", - 833.6, - 838.38 + 899.86, + 903.0 ], [ "Well, almost.", - 839.2, - 840.4 + 903.0, + 904.16 ], [ "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of eight bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position zero.", - 840.96, - 856.86 + 904.16, + 915.54 ], [ "You see, with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing one out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition.", - 857.74, - 871.9 + 916.16, + 933.14 ], [ "The solution here is actually pretty simple.", - 873.02, - 874.86 + 933.14, + 937.18 ], [ "Just forget about that zeroth bit entirely.", - 875.28, - 877.3 + 937.18, + 941.62 ], [ "So when we do our four parity checks and we see that they're all even, it unambiguously means that there is no error.", - 877.84, - 883.46 + 941.62, + 952.7 ], [ "What that means is rather than working with a 16-bit block, we work with a 15-bit block, where 11 of the bits are free to carry a message and four of them are there for redundancy.", - 884.24, - 893.22 + 953.52, + 957.04 ], [ "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", - 893.78, - 898.8 + 957.04, + 965.22 ], [ "That said, it is nice to have a block size that's a clean power of two, and there's a clever way that we can keep that zeroth bit around and get it to do a little extra work for us.", - 899.86, - 908.14 + 966.54, + 971.3 ], [ "If we use it as a parity bit across the whole block, it lets us actually detect, even though we can't correct, two-bit errors.", - 908.7, - 915.54 + 971.3, + 974.02 ], [ "Here's how it works.", - 916.16, - 916.82 + 974.02, + 974.6 ], [ "After setting those four special error correcting bits, we set that zeroth one so that the parity of the full block is even, just like a normal parity check.", - 917.18, - 924.94 + 974.6, + 983.78 ], [ "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway, thanks to the four error correcting checks.", - 925.7, - 933.6 + 983.78, + 988.64 ], [ "However, if there's two errors, then the overall parity is going to toggle back to being even, but the receiver would still see that there's been at least some error because of what's going on with those four usual parity checks.", - 934.16, - 945.18 + 988.64, + 999.76 ], [ "So if they notice an even parity overall, but something non-zero happening with the other checks, it tells them there were at least two errors.", - 945.18, - 952.7 + 999.76, + 1007.02 ], [ "Isn't that clever?", - 953.52, - 954.0 + 1012.72, + 1007.02 ], [ "Even though we can't correct those two-bit errors, just by putting that one little bothersome zeroth bit back to work, it lets us detect them.", - 954.3, - 961.26 + 1012.72, + 1021.7 ], [ "This is pretty standard, it's known as an extended Hamming code.", - 962.26, - 965.22 + 1021.7, + 1023.32 ], [ "Technically speaking, you now have a full description of what a Hamming code does, at least for the example of a 16-bit block, but I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", - 966.54, - 981.32 + 1025.34, + 1042.82 ], [ "I'll step through it with you though so you can check yourself.", - 982.08, - 984.3 + 1042.82, + 1042.82 ], [ "To set up a message, whether that's a literal message that you're translating over space, or some data that you want to store over time, the first step is to divide it up into 11-bit chunks.", - 985.12, - 994.66 + 1042.82, + 1061.82 ], [ "Each chunk is going to get packaged into an error-resistant 16-bit block.", - 995.58, - 999.76 + 1061.82, + 1070.2 ], [ "So let's take this one as an example and actually work it out.", - 999.76, - 1003.22 + 1070.62, + 1074.32 ], [ "Go ahead, actually do it!", - 1003.74, - 1004.94 + 1074.32, + 1075.0 ], [ "Pause and try putting together this block.", - 1005.22, - 1007.02 + 1075.0, + 1076.82 ], [ "Okay, you ready?", - 1012.72, - 1013.68 + 1077.08, + 1079.78 ], [ "Remember, position 0 along with the other powers of 2 are reserved for error correction duty, so you start by placing the message bits in all of the remaining spots, in order.", - 1014.24, - 1023.32 + 1080.02, + 1086.42 ], [ "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", - 1025.34, - 1032.34 + 1086.42, + 1107.91 ], [ "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", - 1033.02, - 1037.88 + 1109.69, + 1107.91 ], [ "The group after that starts with an odd parity, so again you should have set its parity bit to 1.", - 1039.16, - 1044.24 + 1109.69, + 1117.03 ], [ "And the final group also has an odd parity, meaning we set that bit in position 8 to be a 1.", - 1044.78, - 1050.06 + 1118.55, + 1121.79 ], [ "And then as the final step, the full block now has an even parity, meaning that you can set that bit number 0, the overarching parity bit, to be 0.", - 1051.3, - 1060.32 + 1122.65, + 1133.07 ], [ "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", - 1061.34, - 1068.14 + 1133.07, + 1139.97 ], [ "As the second part of the exercise, let's have you play the role of the receiver.", - 1068.82, - 1072.18 + 1141.31, + 1147.61 ], [ "Of course, that would mean you don't already know what this message is.", - 1073.48, - 1076.82 + 1147.61, + 1157.81 ], [ "Maybe some of you memorized it, but let's assume that you haven't.", - 1077.08, - 1079.78 + 1158.03, + 1164.91 ], [ "What I'm going to do is change either 0, 1, or 2 of the bits in that block, and then ask you to figure out what it is that I did.", - 1080.02, - 1087.74 + 1164.91, + 1169.65 ], [ "So again, pause and try working it out.", - 1088.26, - 1090.81 + 1169.65, + 1169.65 ], [ "Okay, so you as the receiver now check the first parity group, and you can see that it's even, so any error that exists would have to be in an even column.", - 1098.79, - 1107.91 + 1169.65, + 1178.75 ], [ "The next check gives us an odd number, telling us both that there's at least one error, and narrowing us down into this specific column.", - 1109.69, - 1117.03 + 1179.43, + 1181.19 ], [ "The third check is even, chopping down the possibilities even further.", - 1118.55, - 1121.79 + 1181.19, + 1181.31 ], [ "And the last parity check is odd, telling us there's an error somewhere in the bottom, which by now we can see must be in position number 10.", - 1122.65, - 1129.65 + 1181.31, + 1181.31 ], [ "What's more, the parity of the whole block is odd, giving us confidence that there was one flip and not two.", - 1131.49, - 1137.53 + 1181.31, + 1181.31 ], [ "If it's three or more, all bets are off.", - 1138.07, - 1139.97 + 1181.31, + 1181.31 ], [ "After correcting that bit number 10, pulling out the 11 bits that were not used for correction gives us the relevant segment of the original message, which if you rewind and compare is indeed exactly what we started the example with.", - 1141.31, - 1154.39 + 1181.31, + 1181.31 ], [ "And now that you know how to do all this by hand, I'd like to show you how you can carry out the core part of all of this logic with a single line of Python code.", - 1155.71, - 1163.17 + 1181.31, + 1181.31 ], [ "You see, what I haven't told you yet is just how elegant this algorithm really is, how simple it is to get a machine to point to the position of an error, how to systematically scale it, and how we can frame all of this as one single operation rather than multiple separate parity checks.", - 1163.87, - 1178.75 + 1181.31, + 1181.31 ], [ "To see what I mean, come join me in part two.", - 1179.43, + 1181.31, 1181.31 ] ] \ No newline at end of file diff --git a/2020/hamming-codes/english/transcript.txt b/2020/hamming-codes/english/transcript.txt index 2116fbe90..9d9e0e562 100644 --- a/2020/hamming-codes/english/transcript.txt +++ b/2020/hamming-codes/english/transcript.txt @@ -3,40 +3,40 @@ The scratch really does affect the 1s and 0s on the disk, so it reads off differ There is a whole pile of mathematical cleverness that allows us to store data, and just as importantly to transmit data, in a way that's resilient to errors. Well, actually it doesn't take that much cleverness to come up with a way to do this. Any file, whether it's a video, sound, text, code, image, whatever, is ultimately some sequence of 1s and 0s. -And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit. -Then the machine reading this file could compare these three copies and always take the best 2 out of 3 whenever there's a discrepancy. +nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notice +where the position 7 sits, it does affect the first of our parity groups, and the second, and the third, but not the last. So reading the results of those four checks from bottom to top indeed does spell out the position of the error. There's nothing specia But what that means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how to make it so that errors can be corrected while giving up as little space as possible. -For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9! -to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want. +c for implementing the whole scheme in hardware shockingly simple. Now if you want to see why this magic happens, take these 16 index labels for our positions, but instead of writing them in base 10, let's write them all in binary, running from 0000 up to 1111. As we put these binary labels back into their boxes, let m +e emphasize that they are distinct from the data that's actually being sent. They're nothing more than a conceptual label to help you and m And it will still be the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to identify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect that there were two errors, though it won't know how to fix them. We'll talk a little bit later about how this scales for blocks with different sizes. -Methods that let you correct errors like this are known, reasonably enough, as error correction codes. -For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day. +where that's a 1, you get the second parity group from our scheme. In other words, that second check is asking, hey, me again, if there's an error, is the second to last bit of that position a 1? And +so on. The third parity check covers every position whose third to last bit is turned on, and the last one covers the last eight positions, those ones whose highest order bit is a 1. Everything we The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. -And by the way, the way I'm thinking about the structure of this video is less about explaining it as directly as possible, and more a matter of prompting you to invent it for yourself, with a little gentle guidance here and there. +four questions, which in turn is the same as spelling out a position in binary. I hope this makes two things clearer. The first is how to systematically generalize to block sizes that are bigger powers of two. If it takes more bits to describe each position, like six bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you. Also, if you want your understanding to get down to the hardware level, Ben Eater has made a video in conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming codes are not as widely used as more modern codes, like the Reed-Solomon algorithm, but there is a certain magic to the contrast between just how impossible this task feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is that in a vast space of all possible messages, only some subset are going to be considered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words. -Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. -Coming up with a concrete algorithm to efficiently categorize messages like this, though, take a certain cleverness. -The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to. +also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ +ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of +code. It's based on the XOR function. XOR, for those of you who don't know, stands for exclusive or. When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number the positions of these bits from 0 up to 15. -The actual data we want to store is only going to make up 12 of these bits, while 4 of the positions are going to be reserved as a kind of redundancy. +bit representations of those numbers under the hood. The key point for you and me is that taking the XOR of many different bit strings is effectively a way to compute the parodies of a bunch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy, not adding any new information, but adding resilience. -You might expect these 4 special bits to come nicely packaged together, maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. -It also might give you a little hint about how this scales for larger blocks. -Also, technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now. +s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and which a +re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it goes from here. The sender is responsible for toggling some +of the special parity bits to make sure the sum works out to be 0000. Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing checks, and the idea of a message is meant really broadly, to include things like storage. After all, storing data is the same thing as sending a message, just from the past to the future, instead of from one place to another. @@ -46,22 +46,22 @@ The only job of this special bit is to make sure that the total number of 1s in So for example right now, that total number of 1s is 7, that's odd, so the sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information. -Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd. +ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of 1s is known as its parity. -You could also use numbers and say the parity is 0 or 1, which is typically more helpful once you start doing math with the idea, and this special bit that the sender uses to control the parity is called the parity bit. +o collect together all of those positions, the positions of the bits that are turned on, and then XOR them together. To do this in Python, let me first import a couple helpful functions. That way we can call reduce() on this list, and use the XOR function to reduce it. And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors or 5 or any other odd number, but they can know for sure that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count of 1s would still be even, so the receiver can't have full confidence that an even count necessarily means the message is error-free. -You might complain that a message which gets messed up by only 2 bit flips is pretty weak, and you would be absolutely right. -Keep in mind, though, there is no method for error detection or correction that could give you 100% confidence that the message you receive is the one the sender intended. -After all, enough random noise could always change one valid message into another valid message just by pure chance. +So at the moment it looks like if we do this on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state +where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block. What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if you run this same line of code, it print +s out that error. Isn't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any. And there's nothing special about the size 16 here. Instead, the goal is to come up with a scheme that's robust up to a certain maximum number of errors, or maybe to reduce the probability of a false positive like this. -Parity checks on their own are pretty weak, but by distilling the idea of change across a full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. +a parity check to detect 2-bit errors, but the idea is that almost all of the core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfort with binary and XORs and software in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. -For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions. +of the size of the block, or in other words, it grows one bit at a time as the block size doubles. The relevant fact here is that that information directly corresponds to how much redundancy we need. That's really what runs against most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position. -If no error is detected among those 8 bits, it either means there's no error at all, or it sits somewhere in the even positions. +ent to errors, where usually copying the whole message is the first instinct that comes to mind. And then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix. It's kind of nice because it relates it to the broader family of linear cod You might think that limiting a parity check to half the bits makes it less effective, but when it's done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group. Here let's just choose position 1. @@ -70,7 +70,7 @@ This is only 1 out of 4 parity checks that we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. This time we might use position 2 as a parity bit. So these 8 bits already have an even parity, and the sender can feel good leaving that bit number 2 unchanged. -Then on the other end, if the receiver checks the parity of this group and they find that it's odd, they'll know that the error is somewhere among these 8 bits on the right. +d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your parity checks, and it uses only 21 parity bits. And if you step back to think about looking at a million bits and locating a single error, that genuinely feels crazy. The problem, Otherwise, it means either there's no error, or the error is somewhere on the left half. Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. @@ -79,16 +79,16 @@ Let's say you detect an error among the odd columns and among the right half. It necessarily means the error is somewhere in the last column. If there was no error in the odd column but there was one in the right half, that tells you it's in the second to last column. Likewise, if there is an error in the odd columns but not in the right half, you know that it's somewhere in the second column. -And then if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column. +like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly well, and it can be tuned to be resilient to a larger number of errors per block. But it also might simply mean there's no error at all. Which is all a rather belabored way to say that two parity checks let us pin down the column. From here, you can probably guess what follows. -We do basically the same thing but for the rows. -There's going to be a parity check on the odd rows, using position 4 as a parity bit. +spire in a way that spells out the position of an error only came to Hamming when he stepped back after a bunch of other analysis and asked, okay, what is the most eff +icient I could conceivably be about this? He was also candid about how important it was that parity che So in this example, that group already has an even parity, so bit 4 would be set to a 0. -And finally, there's a parity check on the bottom two rows, using position 8 as a parity bit. +it is today. There are like half a dozen times throughout this book that he references the Louis Pasteur quote, luck favors a prepared mind. Cl In this case, it looks like the sender needs to turn that bit 8 on in order to give the group even parity. -Just as the first two checks let us pin down the column, these next two let you pin down the row. +Part of the reason that clever ideas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the transmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the second parity group, so the receiver knows that there's an error somewhere in that right column. But it doesn't affect the third group, and it doesn't affect the fourth group. diff --git a/2020/hamming-codes/french/sentence_translations.json b/2020/hamming-codes/french/sentence_translations.json index 710362cc6..4497f3387 100644 --- a/2020/hamming-codes/french/sentence_translations.json +++ b/2020/hamming-codes/french/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "Et une stratégie simple pour corriger tout bit retourné serait de stocker trois copies de chaque bit.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "Par exemple, en utilisant la méthode que vous découvrirez dans cette vidéo, vous pourriez stocker vos données dans des blocs de 256 bits, où chaque bloc utilise 9 bits, 9!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "pour agir comme une sorte de redondance, et les 247 autres bits sont libres de transporter le message ou les données significatifs que vous souhaitez.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "Les méthodes qui vous permettent de corriger de telles erreurs sont connues, à juste titre, sous le nom de codes de correction d'erreurs.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "Pendant la majeure partie du siècle dernier, ce domaine a été une source très riche de mathématiques étonnamment approfondies qui sont intégrées aux appareils que nous utilisons quotidiennement.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "L'histoire commence dans les années 1940, lorsque le jeune Richard Hamming travaillait pour les Bell Labs et qu'une partie de son travail impliquait l'utilisation d'un très gros ordinateur à carte perforée coûteux auquel il n'avait qu'un accès limité.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "La frustration étant le creuset de l'invention, il en eut tellement marre qu'il inventa le premier code correcteur d'erreurs au monde.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "Cela pourrait également vous donner un petit indice sur la manière dont cela s'adapte à des blocs plus grands.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "De plus, techniquement, il ne s'agit que de 11 bits de données, vous constaterez qu'il y a une légère nuance pour ce qui se passe à la position 0, mais ne vous inquiétez pas pour l'instant.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "Comme tout algorithme de correction d'erreurs, celui-ci impliquera deux joueurs, un expéditeur chargé de définir ces 4 bits spéciaux et un récepteur chargé d'effectuer une sorte de vérification et de correction des erreurs.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "Après tout, stocker des données revient à envoyer un message uniquement du passé vers le futur plutôt que d’un endroit à un autre.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "Voilà donc la configuration, mais avant de pouvoir plonger dans le vif du sujet, nous devons parler d'une idée connexe qui était fraîche dans l'esprit de Hamming au moment de sa découverte, une méthode qui vous permet de détecter des erreurs sur un seul bit, mais pas de les corriger, connu dans l'entreprise comme chèque de parité.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "Pour un contrôle de parité, nous séparons un seul bit que l'expéditeur est responsable du réglage, et les autres sont libres de transporter un message.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "C'est assez simple, d'une simplicité trompeuse, mais c'est une manière incroyablement élégante de distiller l'idée de changement n'importe où dans un message pour la refléter dans un seul élément d'information.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "Remarquez que si un bit de ce message est inversé, soit de 0 à 1, soit de 1 à 0, cela change le nombre total de 1 de pair à impair.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "Donc, si vous êtes le destinataire, que vous regardez ce message et que vous voyez un nombre impair de 1, vous pouvez être sûr qu'une erreur s'est produite, même si vous n'avez aucune idée de l'endroit où elle se trouve.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "Et ce bit spécial que l’expéditeur utilise pour contrôler la parité est appelé bit de parité.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "Et en fait, soyons clairs, si le récepteur voit une parité impaire, cela ne signifie pas nécessairement qu'il y a eu une seule erreur, il peut y avoir eu 3 erreurs, ou 5, ou tout autre nombre impair, mais il peut en être sûr. que ce n'était pas 0.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "Par exemple, alors que Hamming cherchait un moyen d'identifier l'endroit où une erreur s'est produite, et pas seulement le fait qu'elle s'est produite, son idée clé était que si vous appliquez des contrôles de parité non pas au message complet, mais à certains sous-ensembles soigneusement sélectionnés, vous pouvez demander une série de questions plus raffinées qui permettent de localiser l'emplacement de toute erreur sur un seul bit.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "Le sentiment général est un peu comme jouer à un jeu de 20 questions, poser des requêtes par oui ou par non qui divisent par deux l’espace des possibilités.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "Par exemple, disons que nous effectuons un contrôle de parité uniquement sur ces 8 bits, toutes les positions impaires.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "Ensuite, si une erreur est détectée, cela donne au récepteur un peu plus d'informations sur l'endroit précis où se trouve l'erreur, à savoir qu'elle se trouve dans une position étrange.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "Il ne s'agit que d'un contrôle de parité sur quatre que nous effectuerons.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "Sinon, cela signifie soit qu'il n'y a pas d'erreur, soit que l'erreur se situe quelque part sur la moitié gauche.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "Ou je suppose qu'il aurait pu y avoir deux erreurs, mais pour l'instant, nous allons supposer qu'il y a au plus une erreur dans tout le bloc.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "Les choses s'effondrent complètement pour plus que ça.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "Disons que vous détectez une erreur parmi les colonnes impaires et parmi la moitié droite.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "Et si aucun de ces deux contrôles de parité ne détecte quoi que ce soit, cela signifie que le seul endroit où une erreur pourrait se trouver est dans la colonne la plus à gauche.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "Mais cela peut aussi simplement signifier qu’il n’y a aucune erreur.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "Nous faisons essentiellement la même chose mais pour les lignes.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "Il y aura un contrôle de parité sur les lignes impaires, en utilisant la position 4 comme bit de parité.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "Ainsi, dans cet exemple, ce groupe a déjà une parité paire, donc le bit 4 serait défini sur 0.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "Et enfin, il y a un contrôle de parité sur les deux rangées du bas, en utilisant la position 8 comme bit de parité.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "Mais cela n’affecte pas le troisième groupe, ni le quatrième groupe.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "Prenez un moment pour réfléchir à la manière dont toute erreur parmi ces quatre éléments spéciaux sera détectée comme n'importe quelle autre, avec le même groupe de quatre questions.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "Vous pourriez également aimer anticiper l’évolution de cette situation.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "Espérons que cette esquisse soit suffisante pour apprécier l’efficacité de ce que nous développons ici.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "Enfin presque.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "D'accord, le seul problème ici est que si aucun des quatre contrôles de parité ne détecte une erreur, ce qui signifie que les sous-ensembles de 8 bits spécialement sélectionnés ont tous des parités paires, tout comme l'expéditeur le voulait, alors cela signifie qu'il n'y a eu aucune erreur du tout. , ou cela nous réduit à la position 0.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "Et avec cela, nous avons maintenant ce que les gens du secteur appelleraient un code de Hamming 15-11.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "Maintenant, s'il y a une erreur sur un seul bit, alors la parité du bloc complet devient impaire, mais nous la détecterions de toute façon grâce aux quatre contrôles de correction d'erreur.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "Même si nous ne pouvons pas corriger ces erreurs de 2 bits, simplement en remettant au travail ce petit bit 0 gênant, cela nous permet de les détecter.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "C'est assez standard, c'est ce qu'on appelle un code de Hamming étendu.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "Mais je pense que vous trouverez plus satisfaisant de vérifier votre compréhension et de tout consolider jusqu'à présent en faisant vous-même un exemple complet du début à la fin.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "Chaque morceau sera regroupé dans un bloc de 16 bits résistant aux erreurs.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "Prenons donc celui-ci comme exemple et résolvons-le réellement.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "Allez-y, faites-le!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "Faisons une pause et essayons de constituer ce bloc.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "Vous avez besoin que ce groupe ait une parité paire, ce qui est déjà le cas, vous devriez donc avoir défini ce bit de parité en position 1 pour qu'il soit 0.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "Le groupe suivant commence avec une parité impaire, vous auriez donc dû définir son bit de parité sur 1.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "Ainsi, lorsque ce bloc est envoyé, la parité des quatre sous-ensembles spéciaux et du bloc dans son ensemble sera paire, soit 0.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "Encore une fois, faites une pause et essayez de trouver une solution.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "S'il y en a trois ou plus, tous les paris sont ouverts.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/german/sentence_translations.json b/2020/hamming-codes/german/sentence_translations.json index da817b83f..3b7de4430 100644 --- a/2020/hamming-codes/german/sentence_translations.json +++ b/2020/hamming-codes/german/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "Und eine einfache Strategie zur Korrektur jedes umgedrehten Bits wäre, drei Kopien jedes Bits zu speichern.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "Mit der Methode, die Sie in diesem Video kennenlernen, könnten Sie beispielsweise Ihre Daten in 256-Bit-Blöcken speichern, wobei jeder Block 9 Bits verwendet, 9!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "um als eine Art Redundanz zu fungieren, und die anderen 247 Bits sind frei, um jede gewünschte aussagekräftige Nachricht oder Daten zu übertragen.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "Methoden, mit denen Sie solche Fehler korrigieren können, werden vernünftigerweise als Fehlerkorrekturcodes bezeichnet.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "Über den größten Teil des letzten Jahrhunderts hinweg war dieses Gebiet eine wirklich reichhaltige Quelle überraschend tiefgreifender Mathematik, die in die Geräte einfließt, die wir täglich verwenden.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "Die Geschichte beginnt in den 1940er Jahren, als der junge Richard Hamming für Bell Labs arbeitete und bei einigen seiner Arbeiten einen sehr großen, teuren Lochkartencomputer benutzte, zu dem er nur begrenzten Zugang hatte.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "Da Frustration der Schmelztiegel der Erfindungen war, hatte er die Nase voll und erfand den weltweit ersten Fehlerkorrekturcode.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "Es könnte Ihnen auch einen kleinen Hinweis darauf geben, wie sich dies auf größere Blöcke skalieren lässt.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "Auch technisch gesehen sind es am Ende nur 11 Datenbits. Sie werden feststellen, dass es eine leichte Nuance für das gibt, was an Position 0 passiert, aber machen Sie sich darüber im Moment keine Sorgen.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "Wie bei jedem Fehlerkorrekturalgorithmus sind auch hier zwei Spieler beteiligt: ein Sender, der für das Setzen dieser vier Spezialbits verantwortlich ist, und ein Empfänger, der dafür verantwortlich ist, eine Art Prüfung durchzuführen und die Fehler zu korrigieren.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "Schließlich ist das Speichern von Daten dasselbe wie das Versenden einer Nachricht nur von der Vergangenheit in die Zukunft und nicht von einem Ort zum anderen.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "Das ist also der Aufbau, aber bevor wir uns darauf einlassen können, müssen wir über eine verwandte Idee sprechen, die Hamming zum Zeitpunkt seiner Entdeckung noch frisch im Kopf hatte: eine Methode, mit der man einzelne Bitfehler erkennen, aber nicht korrigieren kann, wie man weiß im Geschäft als Paritätskontrolle.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "Bei einer Paritätsprüfung trennen wir nur ein einzelnes Bit heraus, für dessen Abstimmung der Absender verantwortlich ist, und der Rest kann eine Nachricht übertragen.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "Das ist ziemlich einfach, täuschend einfach, aber es ist eine unglaublich elegante Möglichkeit, die Idee der Veränderung irgendwo in einer Nachricht zu destillieren, um sie in einer einzigen Information widerzuspiegeln.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "Beachten Sie, dass sich die Gesamtzahl der Einsen von gerade auf ungerade ändert, wenn ein Bit dieser Nachricht umgedreht wird, entweder von 0 auf 1 oder von 1 auf 0.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "Wenn Sie also der Empfänger sind, sich diese Nachricht ansehen und eine ungerade Anzahl von Einsen sehen, können Sie mit Sicherheit wissen, dass ein Fehler aufgetreten ist, auch wenn Sie möglicherweise keine Ahnung haben, wo dieser Fehler aufgetreten ist.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "Und dieses spezielle Bit, das der Absender zur Steuerung der Parität verwendet, wird Paritätsbit genannt.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "Und eigentlich sollten wir uns darüber im Klaren sein: Wenn der Empfänger eine ungerade Parität sieht, bedeutet das nicht unbedingt, dass nur ein Fehler aufgetreten ist, es könnten auch 3 Fehler oder 5 oder eine andere ungerade Zahl gewesen sein, aber er kann es mit Sicherheit wissen dass es nicht 0 war.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "Als Hamming beispielsweise nach einer Möglichkeit suchte, herauszufinden, wo ein Fehler aufgetreten ist und nicht nur, dass er aufgetreten ist, war seine wichtigste Erkenntnis, dass man fragen kann, wenn man einige Paritätsprüfungen nicht auf die gesamte Nachricht, sondern auf bestimmte sorgfältig ausgewählte Teilmengen anwendet eine verfeinerte Reihe von Fragen, die den Ort eines einzelnen Bitfehlers bestimmen.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "Das Gesamtgefühl ist ein bisschen so, als würde man ein Spiel mit 20 Fragen spielen, bei dem Ja- oder Nein-Fragen gestellt werden, die den Raum der Möglichkeiten halbieren.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "Nehmen wir zum Beispiel an, wir führen eine Paritätsprüfung nur für diese 8 Bits durch, also alle ungeradzahligen Positionen.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "Wenn dann ein Fehler erkannt wird, erhält der Empfänger etwas mehr Informationen darüber, wo genau sich der Fehler befindet, nämlich dass er sich an einer ungeraden Position befindet.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "Dies ist nur eine von vier Paritätsprüfungen, die wir durchführen werden.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "Andernfalls liegt entweder kein Fehler vor oder der Fehler liegt irgendwo in der linken Hälfte.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "Oder ich denke, es könnte zwei Fehler gegeben haben, aber im Moment gehen wir davon aus, dass es höchstens einen Fehler im gesamten Block gibt.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "Darüber hinaus brechen die Dinge völlig zusammen.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "Nehmen wir an, Sie entdecken einen Fehler in den ungeraden Spalten und in der rechten Hälfte.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "Und wenn keine dieser beiden Paritätsprüfungen etwas erkennt, bedeutet dies, dass der einzige Ort, an dem ein Fehler auftreten könnte, die Spalte ganz links ist.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "Es könnte aber auch einfach bedeuten, dass überhaupt kein Fehler vorliegt.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "Wir machen im Grunde das Gleiche, außer für die Zeilen.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "In den ungeraden Zeilen wird eine Paritätsprüfung durchgeführt, wobei Position 4 als Paritätsbit verwendet wird.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "In diesem Beispiel hat diese Gruppe also bereits eine gerade Parität, sodass Bit 4 auf 0 gesetzt wäre.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "Und schließlich gibt es eine Paritätsprüfung in den unteren beiden Zeilen, wobei Position 8 als Paritätsbit verwendet wird.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "Aber es hat keine Auswirkungen auf die dritte Gruppe und auch nicht auf die vierte Gruppe.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "Nehmen Sie sich einen Moment Zeit, um darüber nachzudenken, wie jeder Fehler in diesen vier Spezialbits genau wie jeder andere mit derselben Gruppe von vier Fragen aufgespürt werden kann.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "Es könnte Ihnen auch Spaß machen, vorherzusehen, wie sich dies skalieren wird.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "Hoffentlich reicht diese Skizze aus, um die Effizienz dessen, was wir hier entwickeln, zu würdigen.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "Naja fast.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "Okay, das einzige Problem besteht hier darin, dass, wenn keine der vier Paritätsprüfungen einen Fehler erkennt, was bedeutet, dass die speziell ausgewählten Teilmengen von 8 Bits alle gerade Paritäten haben, genau wie der Absender es beabsichtigt hat, dann bedeutet das entweder, dass überhaupt kein Fehler aufgetreten ist , oder es schränkt uns auf Position 0 ein.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "Und damit haben wir jetzt das, was die Leute in der Branche als 15-11-Hamming-Code bezeichnen würden.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "Wenn es nun einen einzelnen Bitfehler gibt, wechselt die Parität des gesamten Blocks in den ungeraden Zustand, aber dank der vier fehlerkorrigierenden Prüfungen würden wir das trotzdem erkennen.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "Auch wenn wir diese 2-Bit-Fehler nicht korrigieren können, können wir sie einfach dadurch erkennen, dass wir das eine kleine lästige 0. Bit wieder in Betrieb nehmen.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "Das ist ziemlich normal und wird als erweiterter Hamming-Code bezeichnet.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "Ich denke jedoch, dass es für Sie befriedigender sein wird, Ihr Verständnis zu überprüfen und alles bis zu diesem Punkt zu festigen, indem Sie selbst ein vollständiges Beispiel von Anfang bis Ende durcharbeiten.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "Jeder Block wird in einen fehlerresistenten 16-Bit-Block verpackt.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "Nehmen wir also dieses als Beispiel und arbeiten wir es tatsächlich aus.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "Machen Sie es tatsächlich!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "Lassen Sie uns innehalten und versuchen, diesen Block zusammenzusetzen.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "Diese Gruppe muss über eine gerade Parität verfügen, was bereits der Fall ist. Daher sollten Sie das Paritätsbit in Position 1 auf 0 gesetzt haben.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "Die nächste Gruppe beginnt mit einer ungeraden Parität, daher sollten Sie ihr Paritätsbit auf 1 gesetzt haben.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "Wenn dieser Block gesendet wird, ist die Parität der vier speziellen Teilmengen und des Blocks als Ganzes gerade oder 0.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "Halten Sie also noch einmal inne und versuchen Sie, es herauszufinden.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "Bei drei oder mehr sind alle Wetten ungültig.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/hebrew/sentence_translations.json b/2020/hamming-codes/hebrew/sentence_translations.json index d34d7247c..7e9e95ced 100644 --- a/2020/hamming-codes/hebrew/sentence_translations.json +++ b/2020/hamming-codes/hebrew/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "ואסטרטגיה פשוטה לתיקון כל סיביות שמתהפכת תהיה לאחסן שלושה עותקים של כל סיביות.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "לדוגמה, באמצעות השיטה שתלמד על הסרטון הזה, תוכל לאחסן את הנתונים שלך בבלוקים של 256 סיביות, כאשר כל בלוק משתמש ב-9 סיביות, 9!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "לפעול כסוג של יתירות, ושאר 247 הסיביות חופשיות לשאת כל מסר או נתונים משמעותיים שתרצו.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "שיטות המאפשרות לך לתקן שגיאות כמו זו ידועות, באופן סביר, בתור קודי תיקון שגיאות.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "בחלקה הטוב יותר של המאה הקודמת, תחום זה היה מקור עשיר באמת למתמטיקה עמוקה להפתיע שמשולבת במכשירים שאנו משתמשים בהם מדי יום.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "הסיפור מתחיל בשנות ה-40, כאשר ריצ'רד האמינג צעיר עבד במעבדות בל, וחלק מעבודתו כללו שימוש במחשב כרטיס ניקוב יקר מאוד שהייתה לו רק גישה מוגבלת אליו.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "התסכול בהיותו כור ההמצאה, כל כך נמאס לו שהוא המציא את קוד תיקון השגיאות הראשון בעולם.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "זה גם עשוי לתת לך רמז קטן לגבי איך זה מתאים לבלוקים גדולים יותר.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "גם מבחינה טכנית זה בסופו של דבר רק 11 סיביות של נתונים, תגלה שיש ניואנס מתון למה שמתרחש בעמדה 0, אבל אל תדאג בקשר לזה לעת עתה.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "כמו כל אלגוריתם לתיקון שגיאות, זה יכלול שני שחקנים, שולח שאחראי על הגדרת 4 הביטים המיוחדים הללו, ומקלט שאחראי על ביצוע איזושהי בדיקה ותיקון השגיאות.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "אחרי הכל, אחסון נתונים הוא אותו דבר כמו שליחת מסר רק מהעבר לעתיד במקום ממקום אחד לאחר.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "אז זה ההגדרה, אבל לפני שנוכל לצלול פנימה אנחנו צריכים לדבר על רעיון קשור שהיה טרי בראשו של האמינג בזמן גילויו, שיטה המאפשרת לך לזהות שגיאות סיביות בודדות, אך לא לתקן אותן, ידועה בעסק כמחאה זוגית.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "לבדיקת זוגיות, אנו מפרידים רק ביט בודד אחד שהשולח אחראי לכוונון, והשאר חופשיים לשאת הודעה.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "זה די פשוט, פשוט מטעה, אבל זו דרך אלגנטית להפליא לזקק את הרעיון של שינוי בכל מקום במסר שישתקף בפיסת מידע אחת.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "שימו לב שאם חלק כלשהו מההודעה הזו מתהפך, מ-0 ל-1 או מ-1 ל-0, זה משנה את הספירה הכוללת של 1 שניות מזוגיות לא-זוגית.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "אז אם אתה המקלט, אתה מסתכל על ההודעה הזו, ותראה מספר אי זוגי של 1, אתה יכול לדעת בוודאות שהתרחשה שגיאה כלשהי, למרות שאולי אין לך מושג היכן היא הייתה.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "והביט המיוחד הזה שהשולח משתמש בו כדי לשלוט בזוגיות נקרא סיבית הזוגיות.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "ולמעשה, עלינו להיות ברורים, אם המקלט רואה זוגיות אי זוגית, זה לא בהכרח אומר שהייתה רק שגיאה אחת, אולי היו 3 שגיאות, או 5, או כל מספר אי זוגי אחר, אבל הם יכולים לדעת בוודאות שזה לא היה 0.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "לדוגמה, מכיוון שהאמינג חיפש דרך לזהות היכן קרתה שגיאה, לא רק שהיא קרתה, התובנה העיקרית שלו הייתה שאם תחיל בדיקות זוגיות לא על ההודעה המלאה, אלא על תת-קבוצות מסוימות שנבחרו בקפידה, תוכל לשאול סדרה מעודנת יותר של שאלות המציינת את המיקום של כל שגיאת סיביות בודדת.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "התחושה הכללית היא קצת כמו לשחק במשחק של 20 שאלות, לשאול שאילתות כן או לא שחותכות את מרחב האפשרויות לשניים.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "לדוגמה, נניח שאנו עושים בדיקת זוגיות רק על 8 הסיביות הללו, כל המיקומים האי-זוגיים.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "לאחר מכן, אם מזוהה שגיאה, זה נותן למקלט קצת יותר מידע על היכן בדיוק נמצאת השגיאה, כלומר שהוא נמצא במיקום מוזר.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "זה רק 1 מתוך 4 בדיקות זוגיות שנבצע.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "אחרת זה אומר שאין שגיאה, או שהשגיאה נמצאת איפשהו בחצי השמאלי.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "או שאני מניח שיכולות להיות שתי שגיאות, אבל כרגע אנחנו הולכים להניח שיש לכל היותר שגיאה אחת בכל הבלוק.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "דברים מתקלקלים לגמרי ליותר מזה.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "נניח שאתה מזהה שגיאה בין העמודות האי זוגיות, ובמחצית הימנית.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "ואם אף אחת משתי בדיקות הזוגיות האלה לא מזהה משהו, זה אומר שהמקום היחיד שיכול להיות שגיאה הוא בעמודה השמאלית ביותר.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "אבל זה יכול גם פשוט אומר שאין שגיאה בכלל.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "אנחנו עושים את אותו הדבר חוץ מהשורות.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "תהיה בדיקת זוגיות בשורות האי זוגיות, תוך שימוש במיקום 4 בתור סיביות זוגיות.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "אז בדוגמה הזו לקבוצה כבר יש זוגיות זוגית, אז ביט 4 יוגדר ל-0.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "ולבסוף יש בדיקת זוגיות בשתי השורות התחתונות, תוך שימוש במיקום 8 בתור סיביות זוגיות.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "אבל זה לא משפיע על הקבוצה השלישית, וזה לא משפיע על הקבוצה הרביעית.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "קחו רגע לחשוב כיצד כל שגיאה בין ארבעת הסיביות המיוחדות הללו תתחקה בדיוק כמו כל אחר, עם אותה קבוצה של ארבע שאלות.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "אולי תיהנו גם לצפות כיצד זה מתרחב.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "אני מקווה שהסקיצה הזו מספיקה כדי להעריך את היעילות של מה שאנחנו מפתחים כאן.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "ובכן, כמעט.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "אוקיי, אז הבעיה האחת כאן היא שאם אף אחת מארבעת בדיקות השוויון לא מזהה שגיאה, כלומר שלתת-הקבוצות שנבחרו במיוחד של 8 סיביות יש לכולם זוגיות זוגיות, בדיוק כמו שהשולח התכוון, אז זה אומר שלא הייתה שגיאה בכלל , או שהוא מצמצם אותנו למצב 0.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "ועם זה, יש לנו עכשיו את מה שאנשים בעסק יתייחסו אליו כקוד 15-11 Hamming.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "עכשיו, אם יש שגיאת סיביות בודדת, אז השוויון של הבלוק המלא משתנה להיות אי-זוגי, אבל היינו תופסים את זה בכל מקרה הודות לארבעת הבדיקות לתיקון השגיאות.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "למרות שאיננו יכולים לתקן את השגיאות של 2 סיביות אלה, רק על ידי החזרת הסיבית ה-0 הקטנה והמטרידה הזו לעבודה, היא מאפשרת לנו לזהות אותן.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "זה די סטנדרטי, זה ידוע בתור קוד Hamming מורחב.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "אבל אני חושב שתמצא את זה יותר מספק לבדוק את ההבנה שלך ולגבש הכל עד לנקודה זו על ידי ביצוע דוגמה אחת מלאה מההתחלה ועד הסוף בעצמך.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "כל נתח הולך להיות ארוז לתוך בלוק 16 סיביות עמיד בפני שגיאות.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "אז בואו ניקח את זה כדוגמה ולמעשה נסתדר.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "קדימה, באמת תעשה את זה!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "בואו נעצור וננסה להרכיב את הבלוק הזה.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "אתה צריך שתהיה לקבוצה הזו זוגיות זוגית, מה שהיא כבר עושה, אז היית צריך להגדיר את סיביות הזוגיות הזו במיקום 1 להיות 0.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "הקבוצה הבאה מתחילה עם זוגיות אי זוגית, אז היית צריך להגדיר את סיביות הזוגיות שלה להיות 1.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "אז כאשר הבלוק הזה נשלח, השוויון של ארבע תת-הקבוצות המיוחדות ושל הבלוק בכללותו יהיה זוגי או 0.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "אז שוב, עצור ונסה לפתור את זה.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "אם זה שלושה או יותר, כל ההימורים מושבתים.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/hindi/sentence_translations.json b/2020/hamming-codes/hindi/sentence_translations.json index 842aea6ae..57a71257d 100644 --- a/2020/hamming-codes/hindi/sentence_translations.json +++ b/2020/hamming-codes/hindi/sentence_translations.json @@ -35,7 +35,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "और जो भी बिट फ़्लिप हो जाता है उसे ठीक करने की एक सरल रणनीति यह होगी कि प्रत्येक बिट की तीन प्रतियां संग्रहीत की जाएं।", "n_reviews": 0, "start": 50.68, @@ -70,14 +70,14 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "उदाहरण के लिए, जिस विधि के बारे में आप इस वीडियो के बारे में जानेंगे, उसका उपयोग करके आप अपना डेटा 256-बिट ब्लॉक में संग्रहीत कर सकते हैं, जहां प्रत्येक ब्लॉक 9 बिट्स, 9 का उपयोग करता है!", "n_reviews": 0, "start": 84.52, "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "एक प्रकार की अतिरेक के रूप में कार्य करने के लिए, और अन्य 247 बिट्स आपके इच्छित सार्थक संदेश या डेटा को ले जाने के लिए स्वतंत्र हैं।", "n_reviews": 0, "start": 93.76, @@ -112,14 +112,14 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "वे विधियाँ जो आपको इस तरह की त्रुटियों को ठीक करने देती हैं, यथोचित रूप से त्रुटि सुधार कोड के रूप में जानी जाती हैं।", "n_reviews": 0, "start": 127.86, "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "पिछली शताब्दी के अधिकांश समय में, यह क्षेत्र आश्चर्यजनक रूप से गहन गणित का एक समृद्ध स्रोत रहा है जो हमारे द्वारा प्रतिदिन उपयोग किए जाने वाले उपकरणों में शामिल हो जाता है।", "n_reviews": 0, "start": 133.66, @@ -189,7 +189,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "कहानी 1940 के दशक में शुरू होती है, जब एक युवा रिचर्ड हैमिंग बेल लैब्स के लिए काम कर रहे थे, और उनके कुछ काम में एक बहुत बड़े महंगे पंच कार्ड कंप्यूटर का उपयोग करना शामिल था, जिस तक उनकी सीमित पहुंच थी।", "n_reviews": 0, "start": 226.78, @@ -203,7 +203,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "आविष्कार की भट्ठी निराशा से वह इतना तंग आ गया कि उसने दुनिया का पहला त्रुटि सुधार कोड का आविष्कार किया।", "n_reviews": 0, "start": 243.12, @@ -259,21 +259,21 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "यह आपको इस बारे में भी थोड़ा संकेत दे सकता है कि यह बड़े ब्लॉकों के लिए कैसा है।", "n_reviews": 0, "start": 300.2, "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "इसके अलावा तकनीकी रूप से यह केवल 11 बिट डेटा बनकर रह जाता है, आप पाएंगे कि स्थिति 0 पर जो होता है उसमें थोड़ी बारीकियां हैं, लेकिन अभी इसके बारे में चिंता न करें।", "n_reviews": 0, "start": 304.9, "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "किसी भी त्रुटि सुधार एल्गोरिदम की तरह, इसमें दो खिलाड़ी शामिल होंगे, एक प्रेषक जो इन 4 विशेष बिट्स को सेट करने के लिए जिम्मेदार है, और एक रिसीवर जो किसी प्रकार की जांच करने और त्रुटियों को ठीक करने के लिए जिम्मेदार है।", "n_reviews": 0, "start": 314.14, @@ -287,21 +287,21 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "आख़िरकार, डेटा संग्रहीत करना एक संदेश को एक स्थान से दूसरे स्थान के बजाय अतीत से भविष्य में भेजने जैसा ही है।", "n_reviews": 0, "start": 335.34, "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "तो यह सेटअप है, लेकिन इससे पहले कि हम आगे बढ़ें हमें एक संबंधित विचार के बारे में बात करने की ज़रूरत है जो हैमिंग के दिमाग में उनकी खोज के समय ताजा था, एक ऐसी विधि जो आपको किसी भी बिट त्रुटि का पता लगाने देती है, लेकिन उन्हें ठीक करने की नहीं, ज्ञात है व्यवसाय में समता जाँच के रूप में।", "n_reviews": 0, "start": 342.56, "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "समता जांच के लिए, हम केवल एक बिट को अलग करते हैं जिसे ट्यूनिंग के लिए प्रेषक जिम्मेदार है, और बाकी संदेश ले जाने के लिए स्वतंत्र हैं।", "n_reviews": 0, "start": 356.88, @@ -329,21 +329,21 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "यह बहुत ही सरल, भ्रामक रूप से सरल है, लेकिन यह किसी संदेश में कहीं भी परिवर्तन के विचार को जानकारी के एक टुकड़े में प्रतिबिंबित करने का एक अविश्वसनीय रूप से सुंदर तरीका है।", "n_reviews": 0, "start": 387.34, "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "ध्यान दें कि यदि इस संदेश का कोई भी अंश 0 से 1 या 1 से 0 तक फ़्लिप हो जाता है, तो यह 1 की कुल गिनती को सम से विषम में बदल देता है।", "n_reviews": 0, "start": 397.5, "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "इसलिए यदि आप रिसीवर हैं, आप इस संदेश को देखते हैं, और आपको 1 की एक विषम संख्या दिखाई देती है, तो आप निश्चित रूप से जान सकते हैं कि कुछ त्रुटि हुई है, भले ही आपको पता न हो कि यह कहां थी।", "n_reviews": 0, "start": 407.98, @@ -364,14 +364,14 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "और यह विशेष बिट जिसे प्रेषक समता को नियंत्रित करने के लिए उपयोग करता है, समता बिट कहलाता है।", "n_reviews": 0, "start": 431.22, "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "और वास्तव में, हमें स्पष्ट होना चाहिए, यदि रिसीवर एक विषम समता देखता है, तो इसका मतलब यह नहीं है कि केवल एक त्रुटि थी, 3 त्रुटियां, या 5, या कोई अन्य विषम संख्या हो सकती है, लेकिन वे निश्चित रूप से जान सकते हैं कि यह 0 नहीं था.", "n_reviews": 0, "start": 437.56, @@ -420,28 +420,28 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "उदाहरण के लिए, जब हैमिंग यह पहचानने का एक तरीका खोज रहा था कि त्रुटि कहाँ हुई है, न कि केवल यह कि यह घटित हुई है, तो उसकी मुख्य अंतर्दृष्टि यह थी कि यदि आप कुछ समता जाँचों को पूर्ण संदेश पर नहीं, बल्कि कुछ सावधानीपूर्वक चयनित उपसमूहों पर लागू करते हैं, तो आप पूछ सकते हैं प्रश्नों की एक अधिक परिष्कृत श्रृंखला जो किसी भी एक बिट त्रुटि के स्थान को इंगित करती है।", "n_reviews": 0, "start": 507.94, "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "कुल मिलाकर भावना कुछ-कुछ 20 प्रश्नों का खेल खेलने जैसा है, जिसमें हां या ना में ऐसे प्रश्न पूछे जाते हैं जो संभावनाओं की जगह को आधा कर देते हैं।", "n_reviews": 0, "start": 526.68, "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "उदाहरण के लिए, मान लें कि हम केवल इन 8 बिट्स, सभी विषम संख्या वाली स्थितियों पर समता जांच करते हैं।", "n_reviews": 0, "start": 534.16, "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "फिर यदि कोई त्रुटि पाई जाती है, तो यह रिसीवर को इस बारे में थोड़ी अधिक जानकारी देता है कि विशेष रूप से त्रुटि कहाँ है, अर्थात् यह एक विषम स्थिति में है।", "n_reviews": 0, "start": 540.1, @@ -483,7 +483,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "यह 4 समता जांचों में से केवल 1 है जो हम करेंगे।", "n_reviews": 0, "start": 587.94, @@ -511,21 +511,21 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "अन्यथा इसका मतलब है कि या तो कोई त्रुटि नहीं है, या त्रुटि बाएं आधे हिस्से में कहीं है।", "n_reviews": 0, "start": 615.82, "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "या मुझे लगता है कि दो त्रुटियाँ हो सकती थीं, लेकिन अभी हम यह मानेंगे कि पूरे ब्लॉक में अधिकतम एक त्रुटि है।", "n_reviews": 0, "start": 621.12, "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "इससे अधिक के लिए चीजें पूरी तरह से टूट जाती हैं।", "n_reviews": 0, "start": 626.94, @@ -539,7 +539,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "मान लीजिए कि आपको विषम स्तंभों और दाएँ आधे भाग के बीच एक त्रुटि का पता चलता है।", "n_reviews": 0, "start": 635.8, @@ -567,14 +567,14 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "और यदि उन दोनों समता जांचों में से कोई भी कुछ भी पता नहीं लगाता है, तो इसका मतलब है कि एकमात्र स्थान जहां त्रुटि हो सकती है वह सबसे बाएं कॉलम में है।", "n_reviews": 0, "start": 656.02, "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "लेकिन इसका सीधा मतलब यह भी हो सकता है कि कोई त्रुटि ही नहीं है।", "n_reviews": 0, "start": 663.34, @@ -595,28 +595,28 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "हम मूल रूप से वही काम करते हैं लेकिन पंक्तियों के लिए।", "n_reviews": 0, "start": 673.8, "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "समता बिट के रूप में स्थिति 4 का उपयोग करते हुए, विषम पंक्तियों पर समता जांच की जाएगी।", "n_reviews": 0, "start": 676.44, "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "तो इस उदाहरण में उस समूह में पहले से ही सम समता है, इसलिए बिट 4 को 0 पर सेट किया जाएगा।", "n_reviews": 0, "start": 681.38, "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "और अंत में समता बिट के रूप में स्थिति 8 का उपयोग करते हुए नीचे की दो पंक्तियों पर एक समता जांच होती है।", "n_reviews": 0, "start": 686.56, @@ -651,7 +651,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "लेकिन तीसरे समूह पर इसका कोई प्रभाव नहीं पड़ता, और चौथे समूह पर इसका कोई प्रभाव नहीं पड़ता।", "n_reviews": 0, "start": 716.1, @@ -693,7 +693,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "इस बारे में सोचने के लिए एक क्षण लें कि इन चार विशेष बिट्स के बीच किसी भी त्रुटि को किसी भी अन्य की तरह, चार प्रश्नों के एक ही समूह के साथ कैसे ट्रैक किया जाएगा।", "n_reviews": 0, "start": 756.44, @@ -714,7 +714,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "आपको यह अनुमान लगाने में भी आनंद आ सकता है कि इसका पैमाना कैसा होगा।", "n_reviews": 0, "start": 779.2, @@ -742,7 +742,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "उम्मीद है कि हम यहां जो विकसित कर रहे हैं उसकी दक्षता की सराहना करने के लिए यह स्केच पर्याप्त है।", "n_reviews": 0, "start": 809.88, @@ -770,14 +770,14 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "हां तकरीबन।", "n_reviews": 0, "start": 839.2, "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "ठीक है, तो यहां एक समस्या यह है कि यदि चार समता जांचों में से कोई भी त्रुटि का पता नहीं लगाता है, जिसका अर्थ है कि 8 बिट्स के विशेष रूप से चयनित उपसमुच्चय में सम समताएं हैं, जैसा कि प्रेषक ने चाहा था, तो इसका मतलब यह है कि कोई त्रुटि नहीं थी , या यह हमें स्थिति 0 तक सीमित कर देता है।", "n_reviews": 0, "start": 840.96, @@ -812,7 +812,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "और इसके साथ ही, अब हमारे पास वह है जिसे व्यवसाय के लोग 15-11 हैमिंग कोड के रूप में संदर्भित करेंगे।", "n_reviews": 0, "start": 893.78, @@ -847,7 +847,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "अब, यदि एक भी बिट त्रुटि है, तो पूर्ण ब्लॉक की समता विषम होने के लिए टॉगल हो जाती है, लेकिन चार त्रुटि-सुधार जांचों के लिए धन्यवाद, हम इसे वैसे भी पकड़ लेंगे।", "n_reviews": 0, "start": 925.7, @@ -875,14 +875,14 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "भले ही हम उन 2-बिट त्रुटियों को ठीक नहीं कर सकते हैं, बस उस एक छोटे से परेशान करने वाले 0वें बिट को काम पर वापस रखकर, यह हमें उनका पता लगाने देता है।", "n_reviews": 0, "start": 954.3, "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "यह काफी मानक है, इसे विस्तारित हैमिंग कोड के रूप में जाना जाता है।", "n_reviews": 0, "start": 962.26, @@ -896,7 +896,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "लेकिन मुझे लगता है कि आपको शुरू से अंत तक एक पूरा उदाहरण देकर अपनी समझ की जांच करना और इस बिंदु तक सब कुछ मजबूत करना अधिक संतोषजनक लगेगा।", "n_reviews": 0, "start": 972.88, @@ -917,28 +917,28 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "प्रत्येक भाग को त्रुटि-प्रतिरोधी 16-बिट ब्लॉक में पैक किया जाएगा।", "n_reviews": 0, "start": 995.58, "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "तो आइए इसे एक उदाहरण के रूप में लें और वास्तव में इस पर काम करें।", "n_reviews": 0, "start": 999.76, "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "आगे बढ़ो, वास्तव में यह करो!", "n_reviews": 0, "start": 1003.74, "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "आइए रुकें और इस ब्लॉक को एक साथ रखने का प्रयास करें।", "n_reviews": 0, "start": 1005.22, @@ -959,14 +959,14 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "आपको इस समूह में एक सम समता की आवश्यकता है, जो कि पहले से ही है, इसलिए आपको उस समता बिट को स्थिति 1 में 0 पर सेट करना चाहिए।", "n_reviews": 0, "start": 1025.34, "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "अगला समूह एक विषम समता के साथ शुरू होता है, इसलिए आपको इसका समता बिट 1 पर सेट करना चाहिए।", "n_reviews": 0, "start": 1033.02, @@ -994,7 +994,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "इसलिए जैसे ही यह ब्लॉक भेजा जाता है, चार विशेष उपसमुच्चय और संपूर्ण ब्लॉक की समता सम हो जाएगी, या 0.", "n_reviews": 0, "start": 1061.34, @@ -1022,7 +1022,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "तो फिर, रुकें और इस पर काम करने का प्रयास करें।", "n_reviews": 0, "start": 1088.26, @@ -1064,7 +1064,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "यदि यह तीन या अधिक है, तो सभी दांव बंद हो जाएंगे।", "n_reviews": 0, "start": 1138.07, diff --git a/2020/hamming-codes/hungarian/sentence_translations.json b/2020/hamming-codes/hungarian/sentence_translations.json index 78b91036c..698b1d356 100644 --- a/2020/hamming-codes/hungarian/sentence_translations.json +++ b/2020/hamming-codes/hungarian/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "Az átbillent bitek kijavítására egy egyszerű stratégia az lenne, ha minden bitből három másolatot tárolnánk.", "model": "DeepL", "n_reviews": 1, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "Például az a módszer, amit hamarosan bemutatok, 256 bites blokkokban tárolja az adatokat, ahol minden blokk csak kilenc bitet használ. 9-et!", "model": "DeepL", "n_reviews": 1, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "Ezek egyfajta redundanciaként működnek, a többi 247 bit pedig szabadon hordozhat bármilyen értelmes üzenetet vagy adatot.", "model": "DeepL", "n_reviews": 1, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "Az ehhez hasonló módszereket - talán nem meglepő módon - hibajavító kódoknak nevezik.", "model": "DeepL", "n_reviews": 1, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "Ez a terület meglepően komoly matematikai eredmények gazdag forrása volt a múlt században, melyek nagy része beépült az általunk nap mint nap használt eszközökbe.", "model": "DeepL", "n_reviews": 1, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "A sztorink hőse a fiatal Richard Hamming, aki 1940 körül a Bell Labs-nél dolgozott. Feladatai egy nagyon nagy és drága lyukkártyás számítógép használatát igényelték, amelyhez csak korlátozott hozzáférése volt.", "model": "DeepL", "n_reviews": 1, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "Mivel a frusztráció a találékonyság bölcsője, annyira elege lett, hogy feltalálta a világ első hibajavító kódját.", "model": "DeepL", "n_reviews": 1, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors.", + "input": "ion of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It's based on the XOR function. XOR, for t", "translatedText": "Mint minden hibajavító algoritmusban, ebben is két szereplő vesz részt: egy feladó, aki a 4 speciális bit beállításáért felelős, és egy vevő, aki valamilyen ellenőrzés elvégzéséért, és a hibák kijavításáért felelős.", "model": "DeepL", "n_reviews": 1, @@ -320,7 +320,7 @@ "end": 325.54 }, { - "input": "Of course, the words sender and receiver really refer to machines or software that's doing checks, and the idea of a message is meant really broadly, to include things like storage.", + "input": "hose of you who don't know, stands for exclusive or. When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased", "translatedText": "Valójában a feladó és címzett szavak az ellenőrzéseket végző gépre vagy szoftverre utalnak és az üzenet fogalma nagyon tágan értendő, hogy még a tárolást is magába foglalja.", "model": "DeepL", "n_reviews": 1, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "A paritásellenőrzéshez csak egyetlen bit szükséges, amely értékét a küldő állítja be a többi üzenetet hordozó bit alapján.", "model": "DeepL", "n_reviews": 1, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "Ez roppant egyszerű, de hihetetlenül elegáns módja annak, hogy az üzenetben bárhol végbemenő változás eredménye, egyetlen bitnyi információban tükröződjön.", "model": "DeepL", "n_reviews": 1, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "Figyeljük meg, hogy ha az üzenet bármelyik bitje felcserélődik, akár 0-ról 1-re, akár 1-ről 0-ra, akkor az 1-ek száma párosról páratlanra változik.", "model": "DeepL", "n_reviews": 1, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "Tehát ha a címzettként megnézed ezt az üzenetet, és páratlan számú 1-est látsz, akkor biztosan tudhatod, hogy valamilyen hiba történt, még ha fogalmad sincs, hogy hol volt az.", "model": "DeepL", "n_reviews": 1, @@ -408,7 +408,7 @@ "end": 423.34 }, { - "input": "You could also use numbers and say the parity is 0 or 1, which is typically more helpful once you start doing math with the idea, and this special bit that the sender uses to control the parity is called the parity bit.", + "input": "for toggling some of the special parity bits to make sure the sum works out to be 0000. Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "A 0 vagy 1 számokkal is jelölhetjük a paritást, ami akkor lesz nagyon hasznos, amint matekosan kezdünk ezzel foglalkozni. Ezt a speciális bitet pedig, amellyel a küldő beállítja a paritást, paritásbitnek nevezzük.", "model": "DeepL", "n_reviews": 1, @@ -416,7 +416,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors or 5 or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "Azzal érdemes tisztában lenni, ha a vevő páratlan paritást lát, az nem feltétlenül azt jelenti, hogy csak egy hiba volt, lehet, hogy 3 hiba volt, vagy 5, vagy bármilyen más páratlan szám, de biztosan tudhatjuk hogy nem 0.", "model": "DeepL", "n_reviews": 1, @@ -472,7 +472,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "Hamming kulcsfontosságú meglátása az volt, hogy ha a paritásellenőrzést nem a teljes üzenetre, hanem bizonyos gondosan kiválasztott részhalmazokra alkalmazza, akkor a hiba létezésének detekcióján felül a hiba helyét is könnyebben azonosítani lehet. Egy kifinomult kérdéssort tehetünk fel, amely meghatározza bármely bithiba helyét.", "model": "DeepL", "n_reviews": 1, @@ -480,7 +480,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "Úgy is fel lehet fogni, mint egy 20 kérdésből álló játékot, ahol eldöntendő kérdéseket teszünk fel, amelyek kettévágják a lehetőségek terét.", "model": "DeepL", "n_reviews": 1, @@ -488,7 +488,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you", "translatedText": "Tegyük fel, hogy csak ezen a 8 biten végzünk paritásellenőrzést, vagyis az összes páratlanul számozott pozícióban.", "model": "DeepL", "n_reviews": 1, @@ -496,7 +496,7 @@ "end": 539.38 }, { - "input": "Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "can know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "Ezután, ha a vevő hibát észlel, egy kicsit több információt kap arról, hogy pontosan hol van a hiba, nevezetesen, hogy egy páratlan pozícióban.", "model": "DeepL", "n_reviews": 1, @@ -544,7 +544,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "Ez csak 1 a 4 paritás-ellenőrzésből, amelyet elvégzünk.", "model": "DeepL", "n_reviews": 1, @@ -576,7 +576,7 @@ "end": 606.06 }, { - "input": "Then on the other end, if the receiver checks the parity of this group and they find that it's odd, they'll know that the error is somewhere among these 8 bits on the right.", + "input": "So at the moment it looks like if we do this on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but you could write a function where the s", "translatedText": "A másik oldalon, ha a vevő ellenőrzi ennek a csoportnak a paritását és azt találja, hogy az páratlan, akkor tudni fogja, hogy a hiba valahol a jobb oldali 8 bit között van.", "model": "DeepL", "n_reviews": 1, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "Vagy persze, lehetett volna két hiba is, de egyelőre feltételezzük, hogy az egész blokkban legfeljebb egy hiba van.", "model": "DeepL", "n_reviews": 1, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "Ennél több hibánál teljesen összeomlanak a dolgok.", "model": "DeepL", "n_reviews": 1, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "Tegyük fel, hogy a páratlan oszlopok között és a jobb oldali felénél észlelünk hibát.", "model": "DeepL", "n_reviews": 1, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And then if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "ust up to a certain maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "És ha a két paritásellenőrzés egyike sem észlel semmit, akkor ez azt jelenti, hogy az egyetlen hely, ahol hiba lehet, az a bal szélső oszlop.", "model": "DeepL", "n_reviews": 1, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "De könnyen lehet, hogy egyáltalán nincs hiba.", "model": "DeepL", "n_reviews": 1, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "Alapvetően ugyanezt tesszük, csak a sorokra.", "model": "DeepL", "n_reviews": 1, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity che", "translatedText": "Lesz egy paritásellenőrzés a páratlan sorokra, a 4-es pozíciót használva paritásbitként.", "model": "DeepL", "n_reviews": 1, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example, that group already has an even parity, so bit 4 would be set to a 0.", + "input": "cks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "Ebben a példában tehát a csoport már páros paritású, így a 4. bit 0-ra lesz állítva.", "model": "DeepL", "n_reviews": 1, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally, there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "Végül az alsó két sorra végzünk paritásellenőrzést, a 8-as pozíciót használva paritásbitként.", "model": "DeepL", "n_reviews": 1, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "De nem érinti a harmadik csoportot, és nem érinti a negyedik csoportot.", "model": "DeepL", "n_reviews": 1, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "Gondold végig, hogy miért lehetséges e négy különleges bit bármelyikében előforduló hibát ugyanannak a négy kérdésnek a segítségével megtalálni!", "model": "DeepL", "n_reviews": 1, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "Azt is izgalmasnak találhatod, hogy mindez hogyan skálázódik.", "model": "DeepL", "n_reviews": 1, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "that genuinely feels crazy. The problem, Otherwise, it means either there's n", "translatedText": "Nos, majdnem.", "model": "DeepL", "n_reviews": 1, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of eight bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position zero.", + "input": "o error, or the error is somewhere on the left half. Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "Az egyetlen probléma itt az, hogy ha a négy paritásellenőrzés egyike sem észlel hibát, ami azt jelenti, hogy a nyolc bit speciálisan kiválasztott részhalmazai mind páros paritásúak, ahogyan a feladó tervezte, akkor ez vagy azt jelenti, hogy egyáltalán nem volt hiba, vagy leszűkít minket a nulladik pozícióra.", "model": "DeepL", "n_reviews": 1, @@ -944,7 +944,7 @@ "end": 898.8 }, { - "input": "That said, it is nice to have a block size that's a clean power of two, and there's a clever way that we can keep that zeroth bit around and get it to do a little extra work for us.", + "input": "y well, and it can be tuned to be resilient to a larger number of errors per block. But it also might simply mean there's no error at all. Which is all a rather belabored way to say that two parity checks let us pin down the column. From here, you can probably guess what follows.", "translatedText": "Ennek ellenére jó, ha olyan a blokkméretünk, amely kettő hatványa, és meg tudjuk tartani a nulladik bitet úgy, hogy valami trükkös módon kis extra hasznot húzzunk belőle.", "model": "DeepL", "n_reviews": 1, @@ -960,7 +960,7 @@ "end": 915.54 }, { - "input": "Here's how it works.", + "input": "nly came to Hamming when he stepped back af", "translatedText": "Így működik.", "model": "DeepL", "n_reviews": 1, @@ -968,7 +968,7 @@ "end": 916.82 }, { - "input": "After setting those four special error correcting bits, we set that zeroth one so that the parity of the full block is even, just like a normal parity check.", + "input": "ter a bunch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So", "translatedText": "A négy speciális hibajavító bit beállítása után a nulladik bitet úgy állítjuk be, hogy a teljes blokk paritása páros legyen, akárcsak a többi esetben.", "model": "DeepL", "n_reviews": 1, @@ -1008,7 +1008,7 @@ "end": 954 }, { - "input": "Even though we can't correct those two-bit errors, just by putting that one little bothersome zeroth bit back to work, it lets us detect them.", + "input": "y ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the transmission there's an error at, say, position 3. Well, this affects t", "translatedText": "Még ha nem is tudjuk kijavítani ezeket a kétbites hibákat, a kis zavaró nulladik bit munkába állításával elértük, hogy legalább felismerjük őket.", "model": "DeepL", "n_reviews": 1, @@ -1024,7 +1024,7 @@ "end": 965.22 }, { - "input": "Technically speaking, you now have a full description of what a Hamming code does, at least for the example of a 16-bit block, but I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "econd parity group, so the receiver knows that there's an error somewhere in that right column. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "Most már teljes rálátásod van arra, hogy mit csinál egy Hamming-kód, legalábbis a 16 bites blokk példájában. De azt hiszem, úgy a legteljesebb, ha ellenőrzöd a megértésedet és megszilárdítasz mindent, amit eddig a pontig megértettél, egy teljes példa megoldásával az elejétől a végéig.", "model": "DeepL", "n_reviews": 1, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "Minden egyes darabot egy hibaálló 16 bites blokkba csomagolunk.", "model": "DeepL", "n_reviews": 1, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "Vegyük tehát ezt példaként, és dolgozzuk ki.", "model": "DeepL", "n_reviews": 1, @@ -1112,7 +1112,7 @@ "end": 1037.88 }, { - "input": "The group after that starts with an odd parity, so again you should have set its parity bit to 1.", + "input": "the message bits, the error correction bits are just riding along. But protecting those bits as well is somet", "translatedText": "Az ezt követő csoport páratlan paritással kezdődik, ezért a paritásbitet ismét 1-re kell állítani.", "model": "DeepL", "n_reviews": 1, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to s", "translatedText": "Így amikor ez a blokk elküldésre kerül, a négy speciális részhalmaz és a blokk egészének paritása mind páros, azaz 0 lesz.", "model": "DeepL", "n_reviews": 1, @@ -1152,7 +1152,7 @@ "end": 1072.18 }, { - "input": "Of course, that would mean you don't already know what this message is.", + "input": "ady see it, but we'll talk later about the systematic way to find what these questions", "translatedText": "Ez persze azt jelenti, hogy nem tudod, mi volt az eredeti üzenet.", "model": "DeepL", "n_reviews": 1, @@ -1176,7 +1176,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "nough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can", "translatedText": "Tehát ismét szünet, és próbáld meg kidolgozni.", "model": "DeepL", "n_reviews": 1, @@ -1192,7 +1192,7 @@ "end": 1107.91 }, { - "input": "The next check gives us an odd number, telling us both that there's at least one error, and narrowing us down into this specific column.", + "input": "The eight bits are redundant in the sense that they're completely determined by the rest of the message, but it's in a much smarter way than simply copy", "translatedText": "A következő ellenőrzés páratlan számot ad, ami egyrészt azt jelzi, hogy legalább egy hiba van, másrészt pedig leszűkíti a kört erre a konkrét oszlopra.", "model": "DeepL", "n_reviews": 1, @@ -1216,7 +1216,7 @@ "end": 1129.65 }, { - "input": "What's more, the parity of the whole block is odd, giving us confidence that there was one flip and not two.", + "input": "t if none of the four parity checks detect an error, meaning that the specially selected subsets of eight bits all have even parities, just like the sender intended, then", "translatedText": "Ráadásul az egész blokk paritása páratlan, így biztosak lehetünk benne, hogy csak egy bit hiba volt, nem kettő.", "model": "DeepL", "n_reviews": 1, @@ -1232,7 +1232,7 @@ "end": 1139.97 }, { - "input": "After correcting that bit number 10, pulling out the 11 bits that were not used for correction gives us the relevant segment of the original message, which if you rewind and compare is indeed exactly what we started the example with.", + "input": "as no error at all, or it narrows us down into position zero. You see, with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing one out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error cond", "translatedText": "Miután kijavítottuk a 10-es számú bitet, a korrekcióra nem használt 11 bit kihúzásával megkapjuk az eredeti üzenet megfelelő szegmensét, amely, ha visszatekerjük és összehasonlítjuk, valóban pontosan az, amivel a példát kezdtük.", "model": "DeepL", "n_reviews": 1, @@ -1248,7 +1248,7 @@ "end": 1163.17 }, { - "input": "You see, what I haven't told you yet is just how elegant this algorithm really is, how simple it is to get a machine to point to the position of an error, how to systematically scale it, and how we can frame all of this as one single operation rather than multiple separate parity checks.", + "input": "So when we do our four parity checks and we see that they're all even, it unambiguously means that there is no error. What that means is rather than working with a 16-bit block, we work with a 15-bit block, where 11 of the bits are free to carry a message and four of them are there for redundancy. And with that, we", "translatedText": "Amit még nem mutattam meg neked, az az, hogy mennyire elegáns ez az algoritmus, és milyen egyszerű rávenni egy gépet a hiba megtalálására. Hogyan lehet nagyobb adatmennyiségeket kezelni, és hogyan tudjuk mindezt több különálló paritás-ellenőrzés helyett egyetlen műveletként megvalósítani.", "model": "DeepL", "n_reviews": 1, @@ -1263,4 +1263,4 @@ "start": 1179.43, "end": 1181.31 } -] +] \ No newline at end of file diff --git a/2020/hamming-codes/indonesian/sentence_translations.json b/2020/hamming-codes/indonesian/sentence_translations.json index c07cbbd8b..e9d832908 100644 --- a/2020/hamming-codes/indonesian/sentence_translations.json +++ b/2020/hamming-codes/indonesian/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "Dan strategi sederhana untuk memperbaiki bit apa pun yang terbalik adalah dengan menyimpan tiga salinan setiap bit.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "Misalnya, dengan menggunakan metode yang akan Anda pelajari dalam video ini, Anda dapat menyimpan data dalam blok 256-bit, yang setiap bloknya menggunakan 9 bit, 9!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "untuk bertindak sebagai semacam redundansi, dan 247 bit lainnya bebas membawa pesan atau data bermakna apa pun yang Anda inginkan.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "Metode yang memungkinkan Anda memperbaiki kesalahan seperti ini dikenal cukup masuk akal sebagai kode koreksi kesalahan.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "Selama hampir satu abad terakhir, bidang ini telah menjadi sumber yang sangat kaya akan matematika mendalam yang dapat dimasukkan ke dalam perangkat yang kita gunakan setiap hari.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "Ceritanya dimulai pada tahun 1940-an, ketika Richard Hamming muda bekerja untuk Bell Labs, dan beberapa pekerjaannya melibatkan penggunaan komputer kartu punch yang sangat besar dan mahal yang aksesnya terbatas.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "Frustrasi menjadi wadah penemuan, dia menjadi sangat muak sehingga dia menemukan kode koreksi kesalahan pertama di dunia.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "Ini juga mungkin memberi Anda sedikit petunjuk tentang bagaimana skala ini untuk blok yang lebih besar.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "Secara teknis juga hanya berisi 11 bit data, Anda akan menemukan ada sedikit perbedaan pada apa yang terjadi di posisi 0, tapi jangan khawatir tentang itu untuk saat ini.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "Seperti algoritma koreksi kesalahan lainnya, ini akan melibatkan dua pemain, pengirim yang bertanggung jawab untuk mengatur 4 bit khusus ini, dan penerima yang bertanggung jawab untuk melakukan semacam pemeriksaan dan memperbaiki kesalahan.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "Lagi pula, menyimpan data sama saja dengan mengirim pesan dari masa lalu ke masa depan, bukan dari satu tempat ke tempat lain.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "Jadi begitulah pengaturannya, tapi sebelum kita mendalami lebih dalam, kita perlu membicarakan tentang ide terkait yang masih segar di benak Hamming pada saat penemuannya, sebuah metode yang memungkinkan Anda mendeteksi kesalahan bit apa pun, namun tidak memperbaikinya, dikenal dalam bisnis sebagai pemeriksaan paritas.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "Untuk pemeriksaan paritas, kami memisahkan hanya satu bit yang menjadi tanggung jawab pengirim untuk disetel, dan sisanya bebas untuk membawa pesan.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "Ini cukup sederhana, tampak sederhana, namun merupakan cara yang sangat elegan untuk menyaring gagasan perubahan di bagian mana pun dalam pesan agar tercermin dalam sedikit informasi.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "Perhatikan jika ada bagian dari pesan ini yang dibalik, baik dari 0 ke 1 atau 1 ke 0, ini akan mengubah jumlah total 1 dari genap menjadi ganjil.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "Jadi jika Anda adalah penerimanya, Anda melihat pesan ini, dan Anda melihat angka 1 yang ganjil, Anda dapat mengetahui dengan pasti bahwa telah terjadi kesalahan, meskipun Anda mungkin tidak tahu di mana kesalahan itu terjadi.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "Dan bit khusus yang digunakan pengirim untuk mengontrol paritas disebut bit paritas.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "Dan sebenarnya harus jelas, jika penerima melihat paritas ganjil, bukan berarti hanya ada satu kesalahan, mungkin ada 3 kesalahan, atau 5, atau angka ganjil lainnya, tapi mereka bisa tahu pasti bahwa itu bukan 0.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "Misalnya, saat Hamming sedang mencari cara untuk mengidentifikasi di mana kesalahan terjadi, bukan hanya kesalahan itu terjadi, wawasan utamanya adalah jika Anda menerapkan beberapa pemeriksaan paritas bukan pada keseluruhan pesan, namun pada subkumpulan tertentu yang dipilih dengan cermat, Anda dapat bertanya serangkaian pertanyaan yang lebih halus yang menjelaskan lokasi kesalahan bit tunggal.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "Perasaan keseluruhannya seperti memainkan permainan 20 pertanyaan, menanyakan pertanyaan ya atau tidak yang memotong ruang kemungkinan menjadi dua.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "Sebagai contoh, katakanlah kita melakukan pemeriksaan paritas hanya pada 8 bit ini, semua posisi bernomor ganjil.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "Kemudian jika terdeteksi adanya error, maka akan memberikan sedikit informasi lebih kepada penerima mengenai di mana tepatnya error tersebut berada, yaitu pada posisi ganjil.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "Ini hanya 1 dari 4 pemeriksaan paritas yang akan kami lakukan.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "Kalau tidak, berarti tidak ada kesalahan, atau kesalahannya ada di bagian kiri.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "Atau saya kira mungkin ada dua kesalahan, tapi untuk saat ini kita akan berasumsi bahwa paling banyak ada satu kesalahan di seluruh blok.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "Segala sesuatunya rusak total karena lebih dari itu.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "Katakanlah Anda mendeteksi kesalahan di antara kolom ganjil, dan di paruh kanan.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "Dan jika tidak satu pun dari kedua pemeriksaan paritas tersebut yang mendeteksi apa pun, berarti satu-satunya tempat terjadinya kesalahan adalah di kolom paling kiri tersebut.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "Tapi itu juga mungkin berarti tidak ada kesalahan sama sekali.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "Pada dasarnya kami melakukan hal yang sama tetapi untuk barisnya.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "Akan ada pemeriksaan paritas pada baris ganjil, menggunakan posisi 4 sebagai bit paritas.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "Jadi dalam contoh ini grup tersebut sudah memiliki paritas genap, jadi bit 4 akan disetel ke 0.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "Dan terakhir ada pemeriksaan paritas pada dua baris terbawah, menggunakan posisi 8 sebagai bit paritas.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "Tapi itu tidak mempengaruhi kelompok ketiga, dan tidak mempengaruhi kelompok keempat.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "Luangkan waktu sejenak untuk memikirkan bagaimana kesalahan apa pun di antara empat bit khusus ini akan dilacak seperti yang lainnya, dengan kelompok empat pertanyaan yang sama.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "Anda mungkin juga senang mengantisipasi bagaimana skalanya.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "Semoga sketsa ini cukup untuk mengapresiasi efisiensi dari apa yang kami kembangkan di sini.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "Hampir saja.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "Oke, jadi satu-satunya masalah di sini adalah jika tidak satu pun dari empat pemeriksaan paritas yang mendeteksi kesalahan, artinya subset 8 bit yang dipilih secara khusus semuanya memiliki paritas genap, seperti yang diinginkan pengirim, maka itu berarti tidak ada kesalahan sama sekali. , atau mempersempit kita ke posisi 0.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "Dan dengan itu, kami sekarang memiliki apa yang oleh orang-orang dalam bisnis ini disebut sebagai kode 15-11 Hamming.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "Sekarang, jika ada kesalahan bit tunggal, maka paritas blok penuh akan berubah menjadi ganjil, namun kita akan tetap menangkapnya berkat empat pemeriksaan koreksi kesalahan.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "Meskipun kami tidak dapat memperbaiki kesalahan 2-bit tersebut, hanya dengan mengembalikan satu bit ke-0 yang mengganggu itu, kami dapat mendeteksinya.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "Ini cukup standar, dikenal sebagai kode Hamming yang diperluas.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "Namun saya pikir Anda akan merasa lebih puas jika memeriksa pemahaman Anda dan memantapkan semuanya hingga saat ini dengan melakukan sendiri satu contoh lengkap dari awal hingga akhir.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "Setiap potongan akan dikemas ke dalam blok 16-bit yang tahan kesalahan.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "Jadi mari kita ambil yang ini sebagai contoh dan kerjakan.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "Silakan, lakukanlah!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "Mari kita berhenti sejenak dan mencoba menyusun blok ini.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "Anda memerlukan grup ini untuk memiliki paritas genap, yang sudah dimilikinya, jadi Anda harus menyetel bit paritas tersebut di posisi 1 menjadi 0.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "Grup berikutnya dimulai dengan paritas ganjil, jadi Anda harus menyetel bit paritasnya menjadi 1.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "Jadi ketika blok ini dilepaskan, paritas dari empat himpunan bagian khusus dan blok secara keseluruhan semuanya akan menjadi genap, atau 0.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "Jadi sekali lagi, jeda dan coba kerjakan.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "Jika jumlahnya tiga atau lebih, semua taruhan dibatalkan.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/italian/sentence_translations.json b/2020/hamming-codes/italian/sentence_translations.json index 57d42c333..21f3661a7 100644 --- a/2020/hamming-codes/italian/sentence_translations.json +++ b/2020/hamming-codes/italian/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "E una strategia semplice per correggere qualsiasi bit che viene invertito sarebbe quella di memorizzare tre copie di ciascun bit.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "Ad esempio, utilizzando il metodo che imparerai in questo video, potresti archiviare i tuoi dati in blocchi da 256 bit, dove ciascun blocco utilizza 9 bit, 9!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "per agire come una sorta di ridondanza e gli altri 247 bit sono liberi di trasportare qualunque messaggio o dato significativo desideri.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "I metodi che consentono di correggere errori come questo sono noti, abbastanza ragionevolmente, come codici di correzione degli errori.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "Per gran parte del secolo scorso, questo campo è stato una fonte davvero ricca di matematica sorprendentemente profonda che viene incorporata nei dispositivi che usiamo ogni giorno.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "La storia inizia negli anni '40, quando un giovane Richard Hamming lavorava per i Bell Labs e parte del suo lavoro prevedeva l'utilizzo di un computer a scheda perforata molto costoso e al quale aveva solo un accesso limitato.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "Essendo la frustrazione il crogiolo dell'invenzione, ne fu così stufo che inventò il primo codice di correzione degli errori al mondo.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "Potrebbe anche darti un piccolo suggerimento su come questo si adatta ai blocchi più grandi.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "Inoltre tecnicamente finiscono per essere solo 11 bit di dati, scoprirai che c'è una leggera sfumatura per ciò che accade nella posizione 0, ma per ora non preoccuparti di questo.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "Come ogni algoritmo di correzione degli errori, coinvolgerà due giocatori, un mittente responsabile dell'impostazione di questi 4 bit speciali e un destinatario responsabile dell'esecuzione di una sorta di controllo e della correzione degli errori.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "Dopotutto, archiviare dati è come inviare un messaggio solo dal passato al futuro invece che da un luogo a un altro.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "Questa è la configurazione, ma prima di approfondire dobbiamo parlare di un'idea correlata che era fresca nella mente di Hamming al momento della sua scoperta, un metodo che consente di rilevare eventuali errori di singoli bit, ma non di correggerli, noto nel settore come controllo di parità.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "Per un controllo di parità, separiamo solo un singolo bit della cui ottimizzazione il mittente è responsabile, mentre il resto è libero di trasportare un messaggio.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "È piuttosto semplice, ingannevolmente semplice, ma è un modo incredibilmente elegante per distillare l'idea di cambiamento ovunque in un messaggio per rifletterlo in un singolo frammento di informazione.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "Nota se qualsiasi parte di questo messaggio viene invertita, da 0 a 1 o da 1 a 0, cambia il conteggio totale di 1 da pari a dispari.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "Quindi, se sei il destinatario, guardi questo messaggio e vedi un numero dispari di 1, puoi sapere con certezza che si è verificato qualche errore, anche se potresti non avere idea di dove fosse.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "E questo bit speciale che il mittente utilizza per controllare la parità è chiamato bit di parità.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "E in realtà, dovremmo essere chiari, se il ricevitore vede una parità dispari, non significa necessariamente che c'è stato un solo errore, potrebbero esserci stati 3 errori, o 5, o qualsiasi altro numero dispari, ma possono saperlo con certezza che non era 0.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "Ad esempio, mentre Hamming stava cercando un modo per identificare dove si è verificato un errore, non solo che si è verificato, la sua intuizione chiave è stata che se si applicano alcuni controlli di parità non all'intero messaggio, ma a determinati sottoinsiemi accuratamente selezionati, è possibile chiedere una serie più raffinata di domande che individuano la posizione di ogni singolo errore bit.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "La sensazione generale è un po’ come giocare a un gioco di 20 domande, ponendo domande sì o no che dimezzano lo spazio delle possibilità.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "Ad esempio, supponiamo di eseguire un controllo di parità solo su questi 8 bit, tutte le posizioni dispari.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "Quindi, se viene rilevato un errore, fornisce al ricevitore qualche informazione in più su dove si trova specificamente l'errore, vale a dire che si trova in una posizione strana.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "Questo è solo 1 dei 4 controlli di parità che faremo.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "Altrimenti significa che non ci sono errori oppure che l'errore è da qualche parte nella metà sinistra.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "Oppure immagino che potrebbero esserci stati due errori, ma per ora assumeremo che ci sia al massimo un errore nell'intero blocco.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "Le cose si guastano completamente per qualcosa di più.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "Supponiamo che rilevi un errore tra le colonne dispari e nella metà destra.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "E se nessuno dei due controlli di parità rileva nulla, significa che l'unico posto in cui potrebbe trovarsi un errore è nella colonna più a sinistra.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "Ma potrebbe anche semplicemente significare che non è presente alcun errore.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "Facciamo sostanzialmente la stessa cosa, ma per le righe.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "Verrà effettuato un controllo di parità sulle righe dispari, utilizzando la posizione 4 come bit di parità.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "Quindi in questo esempio quel gruppo ha già una parità pari, quindi il bit 4 verrebbe impostato su 0.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "E infine c'è un controllo di parità sulle due righe inferiori, utilizzando la posizione 8 come bit di parità.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "Ma non influisce sul terzo gruppo e non influisce sul quarto gruppo.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "Prenditi un momento per pensare a come qualsiasi errore tra questi quattro bit speciali verrà rintracciato proprio come qualsiasi altro, con lo stesso gruppo di quattro domande.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "Potresti anche divertirti ad anticipare come tutto questo si espanderà.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "Speriamo che questo schizzo sia sufficiente per apprezzare l'efficienza di ciò che stiamo sviluppando qui.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "Be 'quasi.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "Ok, quindi l'unico problema qui è che se nessuno dei quattro controlli di parità rileva un errore, il che significa che i sottoinsiemi di 8 bit appositamente selezionati hanno tutti parità pari, proprio come previsto dal mittente, allora significa che non si è verificato alcun errore , oppure ci restringe alla posizione 0.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "E con questo, ora abbiamo quello che le persone del settore chiamerebbero codice Hamming 15-11.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "Ora, se c'è un errore di un solo bit, la parità dell'intero blocco diventa dispari, ma lo rileveremmo comunque grazie ai quattro controlli di correzione degli errori.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "Anche se non possiamo correggere quegli errori a 2 bit, semplicemente rimettendo in funzione quel piccolo e fastidioso bit 0, possiamo rilevarli.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "Questo è piuttosto standard, è noto come codice Hamming esteso.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "Ma penso che troverai più soddisfacente verificare la tua comprensione e consolidare tutto fino a questo punto facendo tu stesso un esempio completo dall'inizio alla fine.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "Ogni pezzo verrà impacchettato in un blocco a 16 bit resistente agli errori.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "Quindi prendiamo questo come esempio e risolviamolo davvero.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "Vai avanti, fallo davvero!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "Facciamo una pausa e proviamo a mettere insieme questo blocco.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "È necessario che questo gruppo abbia una parità pari, cosa che già ha, quindi dovresti impostare il bit di parità nella posizione 1 su 0.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "Il gruppo successivo inizia con una parità dispari, quindi dovresti impostare il bit di parità su 1.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "Quindi, quando questo blocco viene espulso, la parità dei quattro sottoinsiemi speciali e del blocco nel suo insieme sarà pari, ovvero 0.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "Quindi, ancora una volta, fai una pausa e prova a risolverlo.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "Se sono tre o più, tutte le scommesse vengono annullate.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/japanese/sentence_translations.json b/2020/hamming-codes/japanese/sentence_translations.json index 46939ef9d..72434dac8 100644 --- a/2020/hamming-codes/japanese/sentence_translations.json +++ b/2020/hamming-codes/japanese/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit. ", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notice ", "translatedText": "反転したビットを修正するための簡単な戦略は、 各ビットのコピーを 3 つ保存することです。", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9! ", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how ", "translatedText": "たとえば、このビデオで説明する方法を使用すると、データを 256 ビット ブロックに保 存できます。各ブロックは 9 ビットを使用します。", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want. ", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan ", "translatedText": "ある種の冗長性として機能し、残り の 247 ビットは、必要な意味のあるメッセージやデータを自由に伝送できます。", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 112.66 }, { - "input": "And honestly, that feels like magic. ", + "input": "m e emphasize that they are distinct from the data that's actually being sent. They're noth ", "translatedText": "正直に言うと、それは魔法のように感じます。", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 122.86 }, { - "input": "We'll talk a little bit later about how this scales for blocks with different sizes. ", + "input": "that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to identify that there was an error and precisely where i ", "translatedText": "さまざまなサイズのブロックに対してこれがどのように拡張されるかについては、後で少し説明します。", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 141.94 }, { - "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. ", + "input": "that there were two errors, though it won't know how to fix them. We'll talk a little bit later about how this scales for blocks with different sizes. where that's a 1, you get the second parity group from our scheme ", "translatedText": "ここでの目標は、ハミング コードとして知られる最 も初期の例の 1 つを徹底的に理解することです。", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 206.94 }, { - "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. ", + "input": "e scheme is going to be before I tell you. Also, if you want your understanding to get down to the hardware level, Ben Eater has made a video in conjunction with this one ", "translatedText": "有効なメッセージが変更されるたびに、受信者は、タイプミスの場合と同様に、 表示された内容を最も近い有効な隣接メッセージに修正する責任があります。", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. ", + "input": "st how impossible this task feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is tha ", "translatedText": "欲求不満は発明の坩堝であるため、彼はうんざ りして世界初の誤り訂正符号を発明しました。", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 248.42 }, { - "input": "There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. ", + "input": "t in a vast space of all possible messages, only some subset are going to be considered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words. also see th ", "translatedText": "ハミング コードを組み立てるにはさまざまな方法がありますが、最 初のパスとして、ハミング自身が考えた方法を試してみましょう。", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 255.38 }, { - "input": "Let's use an example that's simple, but not too simple, a block of 16 bits. ", + "input": "is in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. ", "translatedText": "シンプルではありますが、単純すぎない 16 ビットのブロックの例を使用してみましょう。", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 260.94 }, { - "input": "We'll number the positions of these bits from 0 up to 15. ", + "input": "Once you understand that these parity checks that we've focused so much of our time on are nothing ", "translatedText": "これらのビットの位置に 0 から 15 までの番号を付けます。", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 273.0 }, { - "input": "The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. ", + "input": "binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. ", "translatedText": "ここでの冗長という言葉は単にコピーを意味するものではありません。結局のところ、こ れらの 4 ビットではデータをやみくもにコピーするのに十分な余地がありません。", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 287.28 }, { - "input": "You might expect these 4 special bits to come nicely packaged together, maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. ", + "input": "r. When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the programs he kept putting through it kept failing, because every now and then a ", "translatedText": "これら 4 つの特別なビットが、おそらく最後などでうまくパッケージ化され ることを期待するかもしれませんが、ご覧のとおり、これらを 2 のべき乗 の位置に配置することで、最後までに非常にエレガントなものが得られます。", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors. ", + "input": "'s first error correction code. There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. ", "translatedText": "他のエラー修正アルゴリズムと同様に、これには 2 人のプレイヤーが関 与します。送信者はこれら 4 つの特別なビットを設定する責任を負い、 受信者は何らかのチェックを実行してエラーを修正する責任を負います。", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", + "input": "e so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy, not adding any new information, but adding resilience. ", "translatedText": "これがセットアップですが、本題に入る前に、ハミングが発見した当時に 彼の頭の中に新たにあった関連するアイデアについて話す必要がありま す。それは、単一ビットのエラーを検出できるが、修正はできないという 既知の方法です。ビジネスではパリティチェックとして使用されます。", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.82 }, { - "input": "The only job of this special bit is to make sure that the total number of 1s in the message is an even number. ", + "input": "that make sense? Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and w ", "translatedText": "この特別なビットの唯一の役割は、メッセージ内の 1 の合計数が偶数であることを確認することです。", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.28 }, { - "input": "So for example right now, that total number of 1s is 7, that's odd, so the sender needs to flip that special bit to be a 1, making the count even. ", + "input": "hich a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it goes from here. The sender is responsible for toggling ", "translatedText": "たとえば、現時点では 1 の合計数は 7 で、これは奇数であるため、送信者 はその特別なビットを 1 に反転してカウントを偶数にする必要があります。", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd. ", + "input": "sitio Like any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. ", "translatedText": "このメッセージのビットが 0 から 1、または 1 から 0 に反転 すると、1 の合計数が偶数から奇数に変化することに注意してください。", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. ", + "input": "Of course, the words sender and receiver really refer to machines or software that's doing checks, and the idea of a message is meant really broadly, to include things like storage. After all, storing data is the same thing as sending a message, just from the past ", "translatedText": "したがって、あなたが受信者である場合、このメッセージを見て、奇数 の 1 が表示されれば、たとえどこでエラーが発生したかは分からな くても、何らかのエラーが発生したことを確実に知ることができます。", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit. ", + "input": "his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", "translatedText": "送信側がパリティを制御するために使用するこの特 別なビットは、パリティ ビットと呼ばれます。", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 485.44 }, { - "input": "Instead, the goal is to come up with a scheme that's robust up to a certain maximum number of errors, or maybe to reduce the probability of a false positive like this. ", + "input": "en kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information. ctice t ", "translatedText": "代わりに、目標は、特定の最大エラー数まで堅牢なスキームを考え出 すこと、またはおそらくこのような誤検知の確率を減らすことです。", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 495.38 }, { - "input": "Parity checks on their own are pretty weak, but by distilling the idea of change across a full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. ", + "input": "his would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running f ", "translatedText": "パリティ チェック自体は非常に弱いですが、メッセージ全体に わたる変更のアイデアを単一のビットにまで絞り出すことで、よ り洗練されたスキームのための強力な構成要素を提供します。", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error. ", + "input": "rom 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of 1s is known as its parity. o collect together all of those positions, the positions of the bits that are ", "translatedText": "たとえば、ハミング氏は、エラーが発生したことだけでなく、どこでエラーが発生したかを 特定する方法を探していました。彼の重要な洞察は、メッセージ全体ではなく、慎重に選択 された特定のサブセットにパリティ チェックを適用すると、次のようにできるということ でした。より洗練された一連の質問により、単一ビット エラーの位置が特定されます。", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 590.68 }, { - "input": "The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", + "input": "his on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but ", "translatedText": "2 番目のチェックは、少なくともここで描画したよう に、グリッドの右半分の 8 ビットの中にあります。", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.3 }, { - "input": "This time we might use position 2 as a parity bit, so these 8 bits already have an even parity, and the sender can feel good leaving that bit number 2 unchanged. ", + "input": "you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this ", "translatedText": "今回は位置 2 をパリティ ビットとして使用する可能性があるため、これらの 8 ビットに はすでに偶数パリティがあり、送信者はビット番号 2 を変更しないままで問題ありません。", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half. ", + "input": "simulating a random error from noise, then if you run this same line of code, it print s out that error. Isn't that neat? ", "translatedText": "それ以外の場合は、エラーがないか、左半分のどこかにエラーがあることを意味します。", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit. ", + "input": "own to a single XOR reduction. Now, depending on your comfort with binary and XORs and software in general, you may eithe ", "translatedText": "位置 4 をパリティ ビットとして使用して、奇数行のパリティ チェックが行われます。", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0. ", + "input": "r find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a wa ", "translatedText": "したがって、この例では、そのグループはすでに偶数パリティ を持っているため、ビット 4 は 0 に設定されます。", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit. ", + "input": "y to identify where an error happened, not just that it happened, his key insight was that if you apply some parity chec ", "translatedText": "最後に、位置 8 をパリティ ビットとして使用して 、下位 2 行のパリティ チェックが行われます。", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 701.84 }, { - "input": "As an example, imagine that during the transmission there's an error at, say, position 3. ", + "input": "ion of any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no ", "translatedText": "例として、送信中に、たとえば位置 3 でエラーが発生したと想像してください。", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 720.54 }, { - "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. ", + "input": "here is that that information directly corresponds to how much redundancy we need. That's really what runs against most people's knee-jerk reaction Then, if an error is detected, it gives the receiv ", "translatedText": "これにより、受信側は最初の行 (必然的に位置 3 を意味しま す) までのエラーを特定できるため、エラーを修正できます。", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 727.52 }, { - "input": "You might enjoy taking a moment to convince yourself that the answers to these four questions really will always let you pin down a specific location, no matter where they turn out to be. ", + "input": "er a little more information about where specifically the error is, namely that it's in an odd position. ent to errors, where usually copying the whole message is the first instinct that comes to min ", "translatedText": "これら 4 つの質問への答えによって、たとえそれがどこであったとしても、常に特定の場所を突き止 めることができるということを、少し時間をかけて自分に納得させてみるのも楽しいかもしれません。", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 743.06 }, { - "input": "And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spoil it. ", + "input": "then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix. It's kind of nice because it relates ", "translatedText": "もしそうなら、私が台無しにする前に、もう一度強調して、 立ち止まって、自分でそのつながりを描いてみてください。", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 773.1 }, { - "input": "But protecting those bits as well is something that naturally falls out of the scheme as a byproduct. ", + "input": "more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group. ", "translatedText": "しかし、これらのビットも保護することは 、副産物として自然に計画から外れます。", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales. ", + "input": "Here let's just choose position 1. For the example shown, the pari ", "translatedText": "これがどのようにスケールされるかを予想するのも楽しいかもしれません。", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 781.76 }, { - "input": "If we used a block of size 256 bits, for example, in order to pin down a location, you need only eight yes or no questions to binary search your way down to some specific spot. ", + "input": "ty of these 8 bits is currently odd, so the sender is responsible for toggling that parity bit, and now it's even. This is only 1 out of 4 parity checks that we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", "translatedText": "たとえば、場所を特定するためにサイズ 256 ビットのブロックを使用した場合、特定の場所にた どり着くまでにバイナリ検索を行うには、8 つの「はい」または「いいえ」の質問だけが必要です。", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 813.66 }, { - "input": "The first thing, except for those eight highlighted parity bits, can be whatever you want it to be, carrying whatever message or data you want. ", + "input": "the sender can feel good leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your pari ", "translatedText": "最初のものは、強調表示されている 8 つのパリティ ビットを除いて、任意 のものにすることができ、必要なメッセージやデータを運ぶことができます。", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 821.0 }, { - "input": "The 8 bits are redundant in the sense that they're completely determined by the rest of the message, but it's in a much smarter way than simply copying the message as a whole. ", + "input": "ty checks, and it uses only 21 parity bits. And if you step back to think about looking at a million bits and locating a single error, that genuinely feels crazy. The problem, Otherwise, it means either there's ", "translatedText": "8 ビットはメッセージの残りの部分によって完全に決定されるという意味で冗 長ですが、メッセージ全体を単にコピーするよりもはるかに賢明な方法です。", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost. ", + "input": "or the error is somewhere on the left half. ", "translatedText": "よくほとんど。", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0. ", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you ", "translatedText": "さて、ここでの 1 つの問題は、4 つのパリティ チェックのいずれもエラーを 検出しなかった場合、つまり、送信者の意図どおり、特別に選択された 8 ビッ トのサブセットがすべて偶数パリティを持つことを意味し、それはエラーがまったく なかったことを意味するかどうかです。、または位置 0 に絞り込まれます。", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 915.54 }, { - "input": "Here's how it works. ", + "input": "er block. But it also might simply mean there's no error at ", "translatedText": "仕組みは次のとおりです。", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 952.7 }, { - "input": "Isn't that clever? ", + "input": "s today. There are like half a dozen times throughout this book that he refer ", "translatedText": "それは賢明ではないでしょうか？", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them. ", + "input": "ences the Louis Pasteur quote, luck favors a prepared mind. Cl In this case, it looks like the sender needs to turn that bit 8 on in order to give the group even parity. Part of the reason that clever ideas look deceptively ", "translatedText": "これらの 2 ビット エラーを修正することはできませんが、少し面 倒な 0 番目のビットを動作に戻すだけで、エラーを検出できます。", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 965.22 }, { - "input": "Technically speaking, you now have a full description of what a Hamming code does, at least for the example of a 16-bit block. ", + "input": "nal result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the transmission there's an error at, say, position 3. ", "translatedText": "技術的に言えば、少なくとも 16 ビット ブロックの例について は、ハミング コードが何を行うかについて完全に説明できました。", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it! ", + "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position ", "translatedText": "さあ、実際にやってみましょう！", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block. ", + "input": "3, so they can fix the error. You might enjoy taking a moment to convince yourself that the answers to these four ", "translatedText": "立ち止まってこのブロックを組み立ててみましょう。", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1007.02 }, { - "input": "Okay, you ready? ", + "input": "questions really will always let you pin down ", "translatedText": "はい、準備はできましたか？", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0. ", + "input": "n between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spo ", "translatedText": "このグループには偶数パリティが必要ですが、すでにそうされているため 、位置 1 のパリティ ビットを 0 に設定する必要があります。", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1037.88 }, { - "input": "The group after that starts with an odd parity, so again you should have set its parity bit to 1. ", + "input": "ts affected, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, ", "translatedText": "その後のグループは奇数パリティで始まるため、やはりそ のパリティ ビットを 1 に設定する必要があります。", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1079.78 }, { - "input": "What I'm going to do is change either 0, 1, or 2 of the bits in that block, and then ask you to figure out what it is that I did. ", + "input": "y eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set the app ", "translatedText": "私がやろうとしているのは、そのブロック内のビットの 0、1、また は 2 を変更して、私が何をしたのかを理解してもらうことです。", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1107.91 }, { - "input": "The next check gives us an odd number, telling us both that there's at least one error, and narrowing us down into this specific column. ", + "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever you want it to be, carrying whatever message or data you want. ", "translatedText": "次のチェックでは奇数が得られ、少なくとも 1 つのエラ ーがあることがわかり、この特定の列に絞り込まれます。", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off. ", + "input": ". And still, for so little given up, you would be able to identify and fix any single bit error. Well, almost. Okay, so the one ", "translatedText": "3 つ以上の場合、すべての賭けは無効になります。", "model": "google_nmt", "n_reviews": 0, @@ -1240,7 +1240,7 @@ "end": 1163.17 }, { - "input": "You see, what I haven't told you yet is just how elegant this algorithm really is, how simple it is to get a machine to point to the position of an error, how to systematically scale it, and how we can frame all of this as one single operation rather than multiple separate parity checks. ", + "input": "You see, with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing one out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The solution here is actually pretty simple. Just forget about that zeroth bit entirely. So when we do our four parity checks and ", "translatedText": "ご存知のとおり、私がまだお伝えしていないのは、このアルゴリズムが実際にどれほど洗練 されているか、マシンにエラーの位置を指示させるのがどれほど簡単であるか、体系的に スケールを調整する方法、そしてすべてをどのようにフレーム化できるかということです。これは、複数の個別のパリティ チェックではなく、1 つの操作として実行されます。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/korean/sentence_translations.json b/2020/hamming-codes/korean/sentence_translations.json index d6b1a0d3d..aefca37b8 100644 --- a/2020/hamming-codes/korean/sentence_translations.json +++ b/2020/hamming-codes/korean/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "뒤집힌 비트를 수정하기 위한 간단한 전략은 각 비트의 복사본을 세 개씩 저장하는 것입니다.", "model": "DeepL", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "예를 들어, 이 동영상에서 배우게 될 방법을 사용하면 각 블록이 9비트인 256비트 블록으로 데이터를 저장할 수 있습니다!", "model": "DeepL", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "를 사용하여 일종의 중복성 역할을 하고, 나머지 247 비트는 원하는 의미 있는 메시지나 데이터를 자유롭게 전달할 수 있습니다.", "model": "DeepL", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "이와 같은 오류를 수정할 수 있는 방법을 오류 수정 코드라고 합니다.", "model": "DeepL", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "지난 세기 동안 이 분야는 우리가 매일 사용하는 기기에 통합되는 놀랍도록 심도 있는 수학의 풍부한 원천이었습니다.", "model": "DeepL", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "이 이야기는 1940년대에 젊은 리처드 해밍이 벨 연구소에서 일하던 시절에 시작되며, 그의 작업 중 일부는 접근이 제한된 매우 크고 비싼 펀치 카드 컴퓨터를 사용하는 것이었습니다.", "model": "DeepL", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "좌절은 발명의 도가니라는 말이 있듯이, 그는 좌절에 지쳐 세계 최초의 오류 수정 코드를 발명했습니다.", "model": "DeepL", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "또한 더 큰 블록에 대해 어떻게 확장되는지에 대한 약간의 힌트를 얻을 수 있습니다.", "model": "DeepL", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "또한 기술적으로는 11비트 데이터에 불과하며, 위치 0에서 일어나는 일에 대해 약간의 뉘앙스가 있지만 지금은 걱정하지 않아도 됩니다.", "model": "DeepL", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "다른 오류 수정 알고리즘과 마찬가지로, 여기에는 이 4개의 특수 비트를 설정하는 발신자와 일종의 검사를 수행하고 오류를 수정하는 수신자, 두 명의 플레이어가 참여하게 됩니다.", "model": "DeepL", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "결국 데이터를 저장하는 것은 한 장소에서 다른 장소로 메시지를 보내는 것이 아니라 과거에서 미래로 메시지를 보내는 것과 같습니다.", "model": "DeepL", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "이것이 설정입니다만, 자세히 알아보기 전에 해밍이 발견할 당시 해밍의 머릿속에 떠오른 관련 아이디어, 즉 단일 비트 오류를 감지할 수는 있지만 수정할 수는 없는 방법, 업계에서는 패리티 체크라고 알려진 방법에 대해 이야기할 필요가 있습니다.", "model": "DeepL", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "패리티 검사를 위해 발신자가 조정할 책임이 있는 하나의 비트만 분리하고 나머지는 자유롭게 메시지를 전달할 수 있도록 합니다.", "model": "DeepL", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "이것은 매우 간단하고 믿을 수 없을 정도로 간단하지만, 메시지 어디에서나 변화의 아이디어를 추출하여 하나의 정보에 반영할 수 있는 매우 우아한 방법입니다.", "model": "DeepL", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "이 메시지에서 0에서 1 또는 1에서 0으로 조금이라도 뒤집히면 1의 총 개수가 짝수에서 홀수로 바뀝니다.", "model": "DeepL", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "따라서 수신자인 여러분이 이 메시지를 보고 홀수인 1이 표시되면 오류가 어디에서 발생했는지는 모르더라도 오류가 발생했음을 확실히 알 수 있습니다.", "model": "DeepL", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "발신자가 패리티를 제어하기 위해 사용하는 이 특수 비트를 패리티 비트라고 합니다.", "model": "DeepL", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "수신자가 홀수 패리티를 본다고 해서 반드시 오류가 하나만 있었다는 의미는 아니며, 오류가 3개나 5개 또는 다른 홀수가 있었을 수도 있지만 0이 아니라는 것은 확실히 알 수 있습니다.", "model": "DeepL", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "예를 들어, 해밍은 오류가 발생했다는 사실뿐만 아니라 오류가 발생한 위치를 파악하는 방법을 찾던 중, 전체 메시지가 아닌 신중하게 선택한 특정 하위 집합에 패리티 검사를 적용하면 단일 비트 오류의 위치를 파악하는 보다 정교한 일련의 질문을 할 수 있다는 사실을 발견했습니다.", "model": "DeepL", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "전체적인 느낌은 20개의 질문으로 구성된 게임을 하는 것과 비슷하며, 예 또는 아니오로 질문하여 가능성의 공간을 반으로 줄이는 것과 비슷합니다.", "model": "DeepL", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "예를 들어, 홀수 위치의 8비트에 대해서만 패리티 검사를 수행한다고 가정해 보겠습니다.", "model": "DeepL", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "그런 다음 오류가 감지되면 수신자에게 오류가 구체적으로 어디에 있는지, 즉 오류가 이상한 위치에 있다는 정보를 조금 더 알려줍니다.", "model": "DeepL", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "이것은 4번의 패리티 검사 중 1번만 수행합니다.", "model": "DeepL", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "그렇지 않으면 오류가 없거나 왼쪽 절반 어딘가에 오류가 있다는 뜻입니다.", "model": "DeepL", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "또는 두 개의 오류가 있었을 수도 있지만 지금은 전체 블록에 최대 하나의 오류가 있다고 가정하겠습니다.", "model": "DeepL", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "그 이상이면 모든 것이 완전히 무너집니다.", "model": "DeepL", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "홀수 열과 오른쪽 절반 열에서 오류를 감지했다고 가정해 보겠습니다.", "model": "DeepL", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "이 두 가지 패리티 검사에서 아무것도 감지되지 않으면 오류가 발생할 수 있는 유일한 위치가 가장 왼쪽 열에 있다는 뜻입니다.", "model": "DeepL", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "그러나 이는 단순히 오류가 전혀 없다는 의미일 수도 있습니다.", "model": "DeepL", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "기본적으로 동일한 작업을 수행하지만 행에 대해서는 다릅니다.", "model": "DeepL", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "위치 4를 패리티 비트로 사용하여 홀수 행에 대한 패리티 검사가 수행됩니다.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "따라서 이 예제에서 해당 그룹은 이미 짝수 패리티를 가지고 있으므로 비트 4는 0으로 설정됩니다.", "model": "DeepL", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "마지막으로 아래쪽 두 행에서 8번 위치를 패리티 비트로 사용하여 패리티 검사를 수행합니다.", "model": "DeepL", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "그러나 세 번째 그룹에는 영향을 미치지 않으며 네 번째 그룹에는 영향을 미치지 않습니다.", "model": "DeepL", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "이 네 가지 특수 비트 중 어떤 오류가 발생하면 다른 오류와 마찬가지로 어떻게 추적할 수 있을지 잠시 생각해 보세요.", "model": "DeepL", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "이 기능이 어떻게 확장될지 예상해보는 것도 재미있을 것입니다.", "model": "DeepL", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "이 스케치만으로도 우리가 개발 중인 제품의 효율성을 충분히 이해할 수 있기를 바랍니다.", "model": "DeepL", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "뭐, 거의요.", "model": "DeepL", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "여기서 한 가지 문제는 4개의 패리티 검사 중 어느 하나도 오류를 감지하지 못하면, 즉 특별히 선택된 8비트의 하위 집합이 모두 발신자의 의도대로 짝수 패리티를 가지면 오류가 전혀 없거나 0으로 좁혀진다는 것입니다.", "model": "DeepL", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "이를 통해 이제 업계에서는 15-11 해밍 코드라고 부르는 코드가 생겼습니다.", "model": "DeepL", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "이제 단일 비트 오류가 발생하면 전체 블록의 패리티가 홀수로 전환되지만, 네 가지 오류 수정 검사 덕분에 어쨌든 이를 포착할 수 있습니다.", "model": "DeepL", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "2비트 오류를 수정할 수는 없지만, 조금 귀찮은 0비트를 다시 작동시키는 것만으로도 오류를 감지할 수 있습니다.", "model": "DeepL", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "이는 확장된 해밍 코드라고 하는 매우 표준적인 코드입니다.", "model": "DeepL", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "하지만 하나의 예제를 처음부터 끝까지 직접 해보면서 이해도를 점검하고 여기까지의 내용을 다지는 것이 더 만족스러울 것입니다.", "model": "DeepL", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "각 청크는 오류 방지 16비트 블록으로 패키징됩니다.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "이 사례를 예로 들어 실제로 해결해 보겠습니다.", "model": "DeepL", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "어서, 실제로 해보세요!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "잠시 멈추고 이 블록을 조합해 보겠습니다.", "model": "DeepL", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "이 그룹은 이미 짝수 패리티를 가져야 하므로 위치 1의 패리티 비트를 0으로 설정했어야 합니다.", "model": "DeepL", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "다음 그룹은 홀수 패리티로 시작하므로 패리티 비트를 1로 설정해야 합니다.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "따라서 이 블록이 전송되면 4개의 특수 하위 집합과 블록 전체의 패리티는 모두 짝수, 즉 0이 됩니다.", "model": "DeepL", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "다시 한 번 잠시 멈춰서 문제를 해결해 보세요.", "model": "DeepL", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "3개 이상이면 모든 베팅이 종료됩니다.", "model": "DeepL", "n_reviews": 0, diff --git a/2020/hamming-codes/marathi/sentence_translations.json b/2020/hamming-codes/marathi/sentence_translations.json index 7bd1320ea..5d5a96230 100644 --- a/2020/hamming-codes/marathi/sentence_translations.json +++ b/2020/hamming-codes/marathi/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "आणि कोणताही बिट जो पलटला जातो तो दुरुस्त करण्यासाठी एक साधी रणनीती म्हणजे प्रत्येक बिटच्या तीन प्रती साठवणे.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "उदाहरणार्थ, तुम्ही या व्हिडिओबद्दल शिकणार असलेल्या पद्धतीचा वापर करून, तुम्ही तुमचा डेटा २५६-बिट ब्लॉक्समध्ये संचयित करू शकता, जिथे प्रत्येक ब्लॉक ९ बिट, ९!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "एक प्रकारचा रिडंडंसी म्हणून काम करण्यासाठी, आणि इतर 247 बिट तुम्हाला हवा तो अर्थपूर्ण संदेश किंवा डेटा घेऊन जाण्यासाठी मोकळे आहेत.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "तुम्हाला यासारख्या चुका दुरुस्त करू देणार्‍या पद्धती एरर सुधारणा कोड म्हणून ओळखल्या जातात.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "गेल्या शतकाच्या चांगल्या भागासाठी, हे क्षेत्र आश्चर्यकारकपणे सखोल गणिताचे खरोखर समृद्ध स्त्रोत आहे जे आम्ही दररोज वापरत असलेल्या उपकरणांमध्ये समाविष्ट केले जाते.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "कथा 1940 च्या दशकात सुरू होते, जेव्हा एक तरुण रिचर्ड हॅमिंग बेल लॅबसाठी काम करत होता, आणि त्याच्या काही कामांमध्ये एक खूप मोठा महाग पंच कार्ड संगणक वापरत होता ज्यामध्ये त्याला फक्त मर्यादित प्रवेश होता.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "नैराश्य हे शोधाचे टोक असल्याने तो इतका कंटाळला की त्याने जगातील पहिला त्रुटी सुधारणेचा कोड शोधून काढला.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "हे तुम्हाला मोठ्या ब्लॉक्ससाठी कसे मोजते याबद्दल थोडीशी सूचना देखील देऊ शकते.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "तसेच तांत्रिकदृष्ट्या ते फक्त 11 बिट डेटाचे आहे, तुम्हाला 0 च्या स्थानावर काय चालले आहे यासाठी एक सौम्य सूक्ष्मता आढळेल, परंतु आत्ता त्याबद्दल काळजी करू नका.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "कोणत्याही त्रुटी सुधारण्याच्या अल्गोरिदमप्रमाणे, यात दोन खेळाडूंचा समावेश असेल, एक प्रेषक जो या 4 विशेष बिट सेट करण्यासाठी जबाबदार आहे आणि एक प्राप्तकर्ता जो काही प्रकारची तपासणी करण्यासाठी आणि त्रुटी सुधारण्यासाठी जबाबदार आहे.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "शेवटी, डेटा संग्रहित करणे हे एका ठिकाणाहून दुसर्‍या ठिकाणाऐवजी भूतकाळापासून भविष्यात संदेश पाठविण्यासारखेच आहे.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "तर हा सेटअप आहे, परंतु आपण आत जाण्यापूर्वी आपल्याला हॅमिंगच्या शोधाच्या वेळी त्याच्या मनावर ताज्या असलेल्या एका संबंधित कल्पनेबद्दल बोलणे आवश्यक आहे, एक पद्धत जी आपल्याला कोणत्याही एकल त्रुटी शोधू देते, परंतु त्या सुधारू शकत नाही, ज्ञात आहे. समता तपासणी म्हणून व्यवसायात.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "समता तपासणीसाठी, आम्ही फक्त एकच बिट वेगळे करतो जो पाठवणारा ट्युनिंगसाठी जबाबदार असतो आणि बाकीचे संदेश पाठवण्यास मोकळे असतात.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "हे खूपच सोपे आहे, भ्रामकपणे सोपे आहे, परंतु संदेशामध्ये कुठेही बदल करण्याची कल्पना एका छोट्या माहितीमध्ये प्रतिबिंबित करण्याचा हा एक आश्चर्यकारकपणे मोहक मार्ग आहे.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "लक्षात घ्या की या संदेशाचा कोणताही भाग 0 ते 1 किंवा 1 ते 0 पर्यंत फ्लिप झाल्यास, तो 1s ची एकूण संख्या सम असण्यापासून विषममध्ये बदलतो.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "म्हणून जर तुम्ही प्राप्तकर्ता असाल, तर तुम्ही हा संदेश पाहाल, आणि तुम्हाला 1s ची विषम संख्या दिसली, तुम्ही निश्चितपणे समजू शकता की काही त्रुटी आली आहे, जरी तुम्हाला कदाचित तो कुठे होता याची कल्पना नसेल.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "आणि प्रेषक समता नियंत्रित करण्यासाठी वापरतो या विशेष बिटला पॅरिटी बिट म्हणतात.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "आणि प्रत्यक्षात, आपण हे स्पष्ट केले पाहिजे, जर प्राप्तकर्त्याला विषम समता दिसली, तर याचा अर्थ फक्त एकच त्रुटी होती असे नाही, 3 त्रुटी, किंवा 5, किंवा इतर कोणतीही विषम संख्या असू शकते, परंतु ते निश्चितपणे जाणून घेऊ शकतात. की ते 0 नव्हते.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "उदाहरणार्थ, हॅमिंग एरर कुठे घडली हे ओळखण्याचा मार्ग शोधत होता, फक्त ती घडलीच नाही, तर त्याची मुख्य माहिती अशी होती की जर तुम्ही काही पॅरिटी चेक पूर्ण मेसेजवर लागू केले नाही तर काही काळजीपूर्वक निवडलेल्या उपसंचांना, तुम्ही विचारू शकता. प्रश्नांची अधिक परिष्कृत मालिका जी कोणत्याही एका बिट त्रुटीचे स्थान पिन डाउन करते.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "एकंदरीत भावना 20 प्रश्नांचा गेम खेळण्यासारखी आहे, होय किंवा नाही प्रश्न विचारणे ज्यामुळे शक्यतांची जागा अर्धी कापली जाते.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "उदाहरणार्थ, समजा की आम्ही फक्त या 8 बिट्सवर समता तपासणी करतो, सर्व विषम क्रमांकित स्थानांवर.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "नंतर एखादी त्रुटी आढळल्यास, ती प्राप्तकर्त्याला विशेषतः त्रुटी कुठे आहे याबद्दल थोडी अधिक माहिती देते, म्हणजे ती विचित्र स्थितीत आहे.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "हे 4 पैकी फक्त 1 पॅरिटी चेक आहे जे आम्ही करू.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "अन्यथा याचा अर्थ एकतर कोणतीही त्रुटी नाही किंवा त्रुटी डाव्या अर्ध्या भागात कुठेतरी आहे.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "किंवा मला वाटते की दोन त्रुटी असू शकतात, परंतु आत्ता आम्ही असे गृहीत धरणार आहोत की संपूर्ण ब्लॉकमध्ये जास्तीत जास्त एक त्रुटी आहे.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "त्याहून अधिक गोष्टींसाठी गोष्टी पूर्णपणे खंडित होतात.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "समजा तुम्हाला विषम स्तंभांमध्ये आणि उजव्या अर्ध्यामध्ये त्रुटी आढळली आहे.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "आणि जर या दोन्ही पॅरिटी तपासण्यांपैकी काहीही आढळले नाही, तर याचा अर्थ एरर असू शकते अशी एकमेव जागा त्या सर्वात डाव्या स्तंभात आहे.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "परंतु याचा अर्थ असा असू शकतो की कोणतीही त्रुटी नाही.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "आम्ही मुळात तेच करतो पण पंक्तींसाठी.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "पॅरिटी बिट म्हणून पोझिशन 4 वापरून विषम पंक्तींवर समता तपासणी केली जाईल.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "तर या उदाहरणात त्या गटात आधीपासून सम समता आहे, त्यामुळे बिट 4 0 वर सेट केला जाईल.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "आणि शेवटी पॅरिटी बिट म्हणून स्थान 8 वापरून तळाच्या दोन ओळींवर पॅरिटी चेक आहे.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "परंतु त्याचा तिसऱ्या गटावर परिणाम होत नाही आणि चौथ्या गटावर त्याचा परिणाम होत नाही.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "या चार विशेष बिट्समधील कोणतीही त्रुटी इतर कोणत्याही प्रमाणेच, चार प्रश्नांच्या समान गटासह कशी शोधली जाईल याचा विचार करण्यासाठी थोडा वेळ घ्या.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "हे प्रमाण कसे वाढेल याचा अंदाज लावण्याचा तुम्हाला आनंदही वाटेल.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "आशा आहे की आम्ही येथे जे विकसित करत आहोत त्याच्या कार्यक्षमतेचे कौतुक करण्यासाठी हे स्केच पुरेसे आहे.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "बरं, जवळजवळ.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "ठीक आहे, तर इथे एक अडचण अशी आहे की जर चार पॅरिटी तपासण्यांपैकी एकही त्रुटी आढळली नाही, म्हणजे 8 बिट्सच्या विशेष निवडलेल्या उपसमूहांमध्ये समान समानता आहेत, जसे की प्रेषकाने अभिप्रेत आहे, तर याचा अर्थ असा होतो की कोणतीही त्रुटी नव्हती. , किंवा ते आम्हाला स्थान 0 मध्ये कमी करते.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "आणि त्यासह, आमच्याकडे आता व्यवसायातील लोक 15-11 हॅमिंग कोड म्हणून संबोधतात.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "आता, जर एक बिट एरर असेल, तर पूर्ण ब्लॉकची पॅरिटी विषम म्हणून टॉगल करते, परंतु तरीही आम्ही चार त्रुटी-सुधारित तपासण्यांमुळे ते पकडू.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "जरी आम्ही त्या 2-बिट चुका दुरुस्त करू शकत नसलो तरी, फक्त एक थोडा त्रासदायक 0 वी बिट कामावर ठेवून, ते आम्हाला त्या शोधू देते.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "हे खूपच मानक आहे, हे विस्तारित हॅमिंग कोड म्हणून ओळखले जाते.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "परंतु मला वाटते की तुम्हाला तुमची समज तपासणे अधिक समाधानकारक वाटेल आणि सुरुवातीपासून ते स्वतःला पूर्ण करण्यापर्यंत एक पूर्ण उदाहरण देऊन या टप्प्यापर्यंत सर्व काही दृढ करा.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "प्रत्येक भाग त्रुटी-प्रतिरोधक 16-बिट ब्लॉकमध्ये पॅकेज केला जाईल.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "चला तर मग हे एक उदाहरण म्हणून घेऊ आणि ते प्रत्यक्षात आणू.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "पुढे जा, प्रत्यक्षात ते करा!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "चला थांबा आणि हा ब्लॉक एकत्र ठेवण्याचा प्रयत्न करूया.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "तुम्‍हाला सम समता असण्‍यासाठी या गटाची आवश्‍यकता आहे, जी ते आधीपासून करत आहे, त्यामुळे तुम्‍ही ते पॅरिटी बिट स्‍थिती 1 मध्‍ये 0 असायला हवे.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "पुढचा गट विषम समतेने सुरू होतो, त्यामुळे तुम्ही त्याचा पॅरिटी बिट 1 वर सेट केला असावा.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "म्हणून हा ब्लॉक पाठवला जात असताना, चार विशेष उपसमूहांची समता आणि संपूर्ण ब्लॉक सर्व सम किंवा 0 असेल.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "म्हणून पुन्हा, विराम द्या आणि प्रयत्न करा.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "ते तीन किंवा अधिक असल्यास, सर्व बेट्स बंद आहेत.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/persian/sentence_translations.json b/2020/hamming-codes/persian/sentence_translations.json index c18c7c9a6..3a5ef275c 100644 --- a/2020/hamming-codes/persian/sentence_translations.json +++ b/2020/hamming-codes/persian/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit. ", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notice ", "translatedText": "و یک استراتژی ساده برای تصحیح هر بیتی که برگردانده می شود، ذخیره سه نسخه از هر بیت است. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9! ", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how ", "translatedText": "به عنوان مثال، با استفاده از روشی که در مورد این ویدیو خواهید آموخت، می توانید داده های خود را در بلوک های 256 بیتی ذخیره کنید، جایی که هر بلوک از 9 بیت استفاده می کند، 9 بیت! ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want. ", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan ", "translatedText": "به عنوان نوعی افزونگی عمل می کند، و 247 بیت دیگر آزاد هستند تا هر پیام یا داده معناداری را که می خواهید حمل کنند. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 112.66 }, { - "input": "And honestly, that feels like magic. ", + "input": "m e emphasize that they are distinct from the data that's actually being sent. They're noth ", "translatedText": "و صادقانه بگویم، این یک جادو است. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 122.86 }, { - "input": "We'll talk a little bit later about how this scales for blocks with different sizes. ", + "input": "that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to identify that there was an error and precisely where i ", "translatedText": "ما کمی بعد در مورد اینکه چگونه این مقیاس برای بلوک های با اندازه های مختلف صحبت خواهیم کرد. ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 141.94 }, { - "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. ", + "input": "that there were two errors, though it won't know how to fix them. We'll talk a little bit later about how this scales for blocks with different sizes. where that's a 1, you get the second parity group from our scheme ", "translatedText": "هدف در اینجا این است که به شما درک کاملی از یکی از اولین نمونه ها، معروف به کد همینگ، بدهد. ", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 206.94 }, { - "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. ", + "input": "e scheme is going to be before I tell you. Also, if you want your understanding to get down to the hardware level, Ben Eater has made a video in conjunction with this one ", "translatedText": "هر زمان که یک پیام معتبر تغییر می کند، گیرنده مسئول تصحیح آنچه می بیند به نزدیکترین همسایه معتبر برمی گردد، همانطور که ممکن است با یک اشتباه تایپی انجام دهید. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. ", + "input": "st how impossible this task feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is tha ", "translatedText": "ناامیدی که بوته اختراع بود، آنقدر خسته شد که اولین کد تصحیح خطا در جهان را اختراع کرد. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 248.42 }, { - "input": "There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. ", + "input": "t in a vast space of all possible messages, only some subset are going to be considered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words. also see th ", "translatedText": "راه‌های مختلفی برای قاب‌بندی کدهای همینگ وجود دارد، اما به عنوان اولین گذر، همان‌طور که خود همینگ درباره آن‌ها فکر می‌کرد، آن را مرور می‌کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 255.38 }, { - "input": "Let's use an example that's simple, but not too simple, a block of 16 bits. ", + "input": "is in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. ", "translatedText": "بیایید از یک مثال ساده، اما نه خیلی ساده، یک بلوک 16 بیتی استفاده کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 260.94 }, { - "input": "We'll number the positions of these bits from 0 up to 15. ", + "input": "Once you understand that these parity checks that we've focused so much of our time on are nothing ", "translatedText": "ما موقعیت های این بیت ها را از 0 تا 15 شماره گذاری می کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 273.0 }, { - "input": "The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. ", + "input": "binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. ", "translatedText": "کلمه زاید در اینجا به سادگی به معنای کپی نیست، بالاخره آن 4 بیت فضای کافی برای کپی کورکورانه داده ها را به ما نمی دهد. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 287.28 }, { - "input": "You might expect these 4 special bits to come nicely packaged together, maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. ", + "input": "r. When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the programs he kept putting through it kept failing, because every now and then a ", "translatedText": "ممکن است انتظار داشته باشید که این 4 بیت خاص به خوبی در کنار هم قرار گیرند، شاید در انتها یا چیزی شبیه به آن، اما همانطور که خواهید دید، نشستن آنها در موقعیت هایی که توان 2 هستند، امکان چیزی را فراهم می کند که در پایان واقعاً زیبا باشد. ", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors. ", + "input": "'s first error correction code. There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. ", "translatedText": "مانند هر الگوریتم تصحیح خطا، این شامل دو بازیکن است، یک فرستنده که مسئول تنظیم این 4 بیت ویژه است و یک گیرنده که مسئول انجام نوعی بررسی و تصحیح خطاها است. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", + "input": "e so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy, not adding any new information, but adding resilience. ", "translatedText": "بنابراین این تنظیمات است، اما قبل از اینکه بتوانیم وارد آن شویم، باید در مورد یک ایده مرتبط صحبت کنیم که در زمان کشف همینگ در ذهن او تازه بود، روشی که به شما امکان می‌دهد خطاهای تک بیتی را تشخیص دهید، اما آنها را اصلاح نکنید. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.82 }, { - "input": "The only job of this special bit is to make sure that the total number of 1s in the message is an even number. ", + "input": "that make sense? Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and w ", "translatedText": "تنها کار این بیت ویژه این است که مطمئن شود مجموع 1 های پیام یک عدد زوج است. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.28 }, { - "input": "So for example right now, that total number of 1s is 7, that's odd, so the sender needs to flip that special bit to be a 1, making the count even. ", + "input": "hich a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it goes from here. The sender is responsible for toggling ", "translatedText": "بنابراین برای مثال در حال حاضر، تعداد کل 1ها 7 است، که فرد است، بنابراین فرستنده باید آن بیت خاص را برگرداند تا عدد 1 باشد و تعداد را زوج کند. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd. ", + "input": "sitio Like any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. ", "translatedText": "توجه کنید که اگر هر بیتی از این پیام از 0 به 1 یا 1 به 0 برگردانده شود، تعداد کل 1 ها را از زوج به فرد تغییر می دهد. ", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. ", + "input": "Of course, the words sender and receiver really refer to machines or software that's doing checks, and the idea of a message is meant really broadly, to include things like storage. After all, storing data is the same thing as sending a message, just from the past ", "translatedText": "بنابراین اگر شما گیرنده هستید، به این پیام نگاه می کنید و عدد فرد 1 را می بینید، می توانید مطمئن باشید که برخی از خطاها رخ داده است، حتی اگر نمی دانید کجا بوده است. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit. ", + "input": "his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", "translatedText": "و این بیت ویژه ای که فرستنده برای کنترل برابری استفاده می کند بیت برابری نامیده می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 485.44 }, { - "input": "Instead, the goal is to come up with a scheme that's robust up to a certain maximum number of errors, or maybe to reduce the probability of a false positive like this. ", + "input": "en kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information. ctice t ", "translatedText": "در عوض، هدف ارائه طرحی است که تا حداکثر تعداد معینی از خطاها قوی باشد، یا شاید کاهش احتمال مثبت کاذب مانند این. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 495.38 }, { - "input": "Parity checks on their own are pretty weak, but by distilling the idea of change across a full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. ", + "input": "his would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running f ", "translatedText": "بررسی‌های برابری به خودی خود بسیار ضعیف هستند، اما با تقطیر ایده تغییر در یک پیام کامل تا یک بیت، چیزی که به ما می‌دهند یک بلوک ساختمانی قدرتمند برای طرح‌های پیچیده‌تر است. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error. ", + "input": "rom 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of 1s is known as its parity. o collect together all of those positions, the positions of the bits that are ", "translatedText": "به عنوان مثال، وقتی همینگ به دنبال راهی برای شناسایی محل وقوع یک خطا بود، نه فقط اینکه خطا رخ داده است، دیدگاه کلیدی او این بود که اگر برخی از بررسی‌های برابری را نه برای پیام کامل، بلکه برای زیر مجموعه‌هایی که با دقت انتخاب شده‌اند اعمال کنید، می‌توانید بپرسید. مجموعه‌ای دقیق‌تر از سوالات که محل هر خطای بیتی را مشخص می‌کند. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 590.68 }, { - "input": "The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", + "input": "his on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but ", "translatedText": "بررسی دوم در میان 8 بیت در نیمه سمت راست شبکه است، حداقل همانطور که ما آن را در اینجا ترسیم کردیم. ", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.3 }, { - "input": "This time we might use position 2 as a parity bit, so these 8 bits already have an even parity, and the sender can feel good leaving that bit number 2 unchanged. ", + "input": "you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this ", "translatedText": "این بار ممکن است از موقعیت 2 به عنوان بیت برابری استفاده کنیم، بنابراین این 8 بیت از قبل دارای برابری زوج هستند و فرستنده می تواند احساس خوبی داشته باشد که بیت شماره 2 را بدون تغییر باقی بگذارد. ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half. ", + "input": "simulating a random error from noise, then if you run this same line of code, it print s out that error. Isn't that neat? ", "translatedText": "در غیر این صورت به این معنی است که یا خطایی وجود ندارد، یا خطا جایی در نیمه چپ است. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit. ", + "input": "own to a single XOR reduction. Now, depending on your comfort with binary and XORs and software in general, you may eithe ", "translatedText": "با استفاده از موقعیت 4 به عنوان بیت برابری، یک بررسی برابری روی ردیف های فرد انجام می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0. ", + "input": "r find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a wa ", "translatedText": "بنابراین در این مثال آن گروه از قبل دارای برابری زوج است، بنابراین بیت 4 روی 0 تنظیم می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit. ", + "input": "y to identify where an error happened, not just that it happened, his key insight was that if you apply some parity chec ", "translatedText": "و در نهایت یک بررسی برابری در دو ردیف پایین، با استفاده از موقعیت 8 به عنوان بیت برابری وجود دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 701.84 }, { - "input": "As an example, imagine that during the transmission there's an error at, say, position 3. ", + "input": "ion of any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no ", "translatedText": "به عنوان مثال، تصور کنید که در حین انتقال، مثلاً در موقعیت 3 خطایی وجود دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 720.54 }, { - "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. ", + "input": "here is that that information directly corresponds to how much redundancy we need. That's really what runs against most people's knee-jerk reaction Then, if an error is detected, it gives the receiv ", "translatedText": "و این به گیرنده اجازه می دهد تا خطا را تا ردیف اول مشخص کند، که لزوماً به معنای موقعیت 3 است، بنابراین آنها می توانند خطا را برطرف کنند. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 727.52 }, { - "input": "You might enjoy taking a moment to convince yourself that the answers to these four questions really will always let you pin down a specific location, no matter where they turn out to be. ", + "input": "er a little more information about where specifically the error is, namely that it's in an odd position. ent to errors, where usually copying the whole message is the first instinct that comes to min ", "translatedText": "ممکن است از صرف لحظه ای لذت ببرید و خود را متقاعد کنید که پاسخ به این چهار سوال واقعاً همیشه به شما امکان می دهد مکان خاصی را مشخص کنید، مهم نیست که کجا هستند. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 743.06 }, { - "input": "And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spoil it. ", + "input": "then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix. It's kind of nice because it relates ", "translatedText": "و اگر این کار را کردید، مجدداً اجازه دهید تأکید کنم، مکث کنید، خودتان سعی کنید قبل از اینکه من آن را خراب کنم، ارتباط را ترسیم کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 773.1 }, { - "input": "But protecting those bits as well is something that naturally falls out of the scheme as a byproduct. ", + "input": "more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group. ", "translatedText": "اما محافظت از آن بیت ها نیز چیزی است که به طور طبیعی به عنوان یک محصول جانبی از این طرح خارج می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales. ", + "input": "Here let's just choose position 1. For the example shown, the pari ", "translatedText": "همچنین ممکن است از پیش بینی چگونگی این مقیاس لذت ببرید. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 781.76 }, { - "input": "If we used a block of size 256 bits, for example, in order to pin down a location, you need only eight yes or no questions to binary search your way down to some specific spot. ", + "input": "ty of these 8 bits is currently odd, so the sender is responsible for toggling that parity bit, and now it's even. This is only 1 out of 4 parity checks that we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", "translatedText": "اگر از یک بلوک با اندازه 256 بیت استفاده کنیم، برای مثال، برای مشخص کردن یک مکان، شما فقط به هشت سوال بله یا خیر نیاز دارید تا مسیر خود را به صورت دودویی در یک نقطه خاص جستجو کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 813.66 }, { - "input": "The first thing, except for those eight highlighted parity bits, can be whatever you want it to be, carrying whatever message or data you want. ", + "input": "the sender can feel good leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your pari ", "translatedText": "اولین چیز، به جز آن هشت بیت برابری برجسته، می‌تواند هر چیزی باشد که می‌خواهید باشد و هر پیام یا داده‌ای را که می‌خواهید حمل کند. ", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 821.0 }, { - "input": "The 8 bits are redundant in the sense that they're completely determined by the rest of the message, but it's in a much smarter way than simply copying the message as a whole. ", + "input": "ty checks, and it uses only 21 parity bits. And if you step back to think about looking at a million bits and locating a single error, that genuinely feels crazy. The problem, Otherwise, it means either there's ", "translatedText": "8 بیت زائد هستند به این معنا که به طور کامل توسط بقیه پیام تعیین می شوند، اما به روشی بسیار هوشمندانه تر از کپی کردن پیام به عنوان یک کل است. ", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost. ", + "input": "or the error is somewhere on the left half. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0. ", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you ", "translatedText": "خب تقریبا بسیار خوب، بنابراین یک مشکل اینجاست که اگر هیچ یک از چهار بررسی برابری خطایی را شناسایی نکرد، به این معنی که زیرمجموعه های انتخاب شده ویژه 8 بیت، همگی دارای برابری زوج هستند، دقیقاً همانطور که فرستنده در نظر گرفته است، یا به این معنی است که اصلا خطایی وجود نداشته است. ، یا ما را به موقعیت 0 محدود می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 915.54 }, { - "input": "Here's how it works. ", + "input": "er block. But it also might simply mean there's no error at ", "translatedText": "در اینجا نحوه عملکرد آن آمده است. ", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 952.7 }, { - "input": "Isn't that clever? ", + "input": "s today. There are like half a dozen times throughout this book that he refer ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them. ", + "input": "ences the Louis Pasteur quote, luck favors a prepared mind. Cl In this case, it looks like the sender needs to turn that bit 8 on in order to give the group even parity. Part of the reason that clever ideas look deceptively ", "translatedText": "این هوشمندانه نیست؟ حتی اگر نمی‌توانیم آن خطاهای 2 بیتی را اصلاح کنیم، فقط با برگرداندن آن یک بیت 0 آزاردهنده کوچک، به ما امکان می‌دهد آنها را شناسایی کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 965.22 }, { - "input": "Technically speaking, you now have a full description of what a Hamming code does, at least for the example of a 16-bit block. ", + "input": "nal result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the transmission there's an error at, say, position 3. ", "translatedText": "از نظر فنی، اکنون شرح کاملی از کاری که یک کد Hamming انجام می دهد، حداقل برای مثال یک بلوک 16 بیتی، دارید. ", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it! ", + "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position ", "translatedText": "برو، در واقع این کار را انجام بده! ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block. ", + "input": "3, so they can fix the error. You might enjoy taking a moment to convince yourself that the answers to these four ", "translatedText": "بیایید مکث کنیم و سعی کنیم این بلوک را کنار هم قرار دهیم. ", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1007.02 }, { - "input": "Okay, you ready? ", + "input": "questions really will always let you pin down ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0. ", + "input": "n between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spo ", "translatedText": "شما به این گروه نیاز دارید که یک برابری زوج داشته باشد، که قبلاً هم دارد، بنابراین باید آن بیت برابری را در موقعیت 1 روی 0 قرار دهید. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1037.88 }, { - "input": "The group after that starts with an odd parity, so again you should have set its parity bit to 1. ", + "input": "ts affected, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, ", "translatedText": "گروه بعد از آن با یک برابری فرد شروع می شود، بنابراین دوباره باید بیت برابری آن را 1 تنظیم می کردید. ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1079.78 }, { - "input": "What I'm going to do is change either 0, 1, or 2 of the bits in that block, and then ask you to figure out what it is that I did. ", + "input": "y eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set the app ", "translatedText": "کاری که می‌خواهم انجام دهم این است که 0، 1 یا 2 بیت‌های آن بلوک را تغییر دهم و سپس از شما بخواهم بفهمید که چه کاری انجام دادم. ", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1107.91 }, { - "input": "The next check gives us an odd number, telling us both that there's at least one error, and narrowing us down into this specific column. ", + "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever you want it to be, carrying whatever message or data you want. ", "translatedText": "بررسی بعدی یک عدد فرد به ما می دهد و به هر دوی ما می گوید که حداقل یک خطا وجود دارد و ما را به این ستون خاص محدود می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off. ", + "input": ". And still, for so little given up, you would be able to identify and fix any single bit error. Well, almost. Okay, so the one ", "translatedText": "اگر سه یا بیشتر باشد، همه شرط‌ها لغو می‌شوند. ", "model": "google_nmt", "n_reviews": 0, @@ -1240,7 +1240,7 @@ "end": 1163.17 }, { - "input": "You see, what I haven't told you yet is just how elegant this algorithm really is, how simple it is to get a machine to point to the position of an error, how to systematically scale it, and how we can frame all of this as one single operation rather than multiple separate parity checks. ", + "input": "You see, with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing one out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The solution here is actually pretty simple. Just forget about that zeroth bit entirely. So when we do our four parity checks and ", "translatedText": "می بینید، چیزی که من هنوز به شما نگفته ام این است که این الگوریتم واقعاً چقدر ظریف است، چقدر ساده است که یک ماشین را به موقعیت خطا نشان دهد، چگونه به طور سیستماتیک آن را مقیاس بندی کنیم، و چگونه می توانیم همه موارد را چارچوب بندی کنیم. این به عنوان یک عملیات واحد به جای چندین بررسی برابری جداگانه است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/portuguese/sentence_translations.json b/2020/hamming-codes/portuguese/sentence_translations.json index 954664262..f4f5b09ba 100644 --- a/2020/hamming-codes/portuguese/sentence_translations.json +++ b/2020/hamming-codes/portuguese/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "E uma estratégia simples para corrigir qualquer bit invertido seria armazenar três cópias de cada bit.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "Por exemplo, usando o método que você aprenderá neste vídeo, você poderia armazenar seus dados em blocos de 256 bits, onde cada bloco usa 9 bits, 9!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "para atuar como uma espécie de redundância, e os outros 247 bits são livres para transportar qualquer mensagem ou dado significativo que você desejar.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "Métodos que permitem corrigir erros como esse são conhecidos, razoavelmente, como códigos de correção de erros.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "Durante a maior parte do século passado, este campo tem sido uma fonte realmente rica de matemática surpreendentemente profunda que é incorporada aos dispositivos que usamos todos os dias.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "A história começa na década de 1940, quando um jovem Richard Hamming trabalhava para o Bell Labs, e parte de seu trabalho envolvia o uso de um computador de cartão perfurado muito grande e caro, ao qual ele tinha acesso limitado.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "Sendo a frustração o cadinho da invenção, ele ficou tão farto que inventou o primeiro código de correção de erros do mundo.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "Também pode lhe dar uma pequena dica sobre como isso é dimensionado para blocos maiores.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "Além disso, tecnicamente, acaba sendo apenas 11 bits de dados. Você descobrirá que há uma pequena nuance no que acontece na posição 0, mas não se preocupe com isso por enquanto.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "Como qualquer algoritmo de correção de erros, isso envolverá dois jogadores, um remetente responsável por definir esses 4 bits especiais e um receptor responsável por realizar algum tipo de verificação e corrigir os erros.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "Afinal, armazenar dados é a mesma coisa que enviar uma mensagem apenas do passado para o futuro, em vez de de um lugar para outro.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "Então essa é a configuração, mas antes de começarmos precisamos falar sobre uma ideia relacionada que estava fresca na mente de Hamming no momento de sua descoberta, um método que permite detectar quaisquer erros de bit único, mas não corrigi-los, conhecido no negócio como uma verificação de paridade.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "Para uma verificação de paridade, separamos apenas um bit que o remetente é responsável pelo ajuste, e o restante fica livre para transportar uma mensagem.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "Isso é muito simples, aparentemente simples, mas é uma maneira incrivelmente elegante de destilar a ideia de mudança em qualquer lugar de uma mensagem para ser refletida em uma única informação.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "Observe que se qualquer bit desta mensagem for invertido, seja de 0 para 1 ou de 1 para 0, isso altera a contagem total de 1s de par para ímpar.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "Portanto, se você é o receptor, olha para esta mensagem e vê um número ímpar de 1s, pode ter certeza de que ocorreu algum erro, mesmo que não tenha ideia de onde ele estava.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "E esse bit especial que o remetente usa para controlar a paridade é chamado de bit de paridade.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "E, na verdade, devemos ser claros, se o receptor vê uma paridade ímpar, isso não significa necessariamente que houve apenas um erro, pode ter havido 3 erros, ou 5, ou qualquer outro número ímpar, mas eles podem ter certeza que não era 0.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "Por exemplo, enquanto Hamming procurava uma maneira de identificar onde ocorreu um erro, não apenas que ocorreu, seu principal insight foi que, se você aplicar algumas verificações de paridade não à mensagem completa, mas a certos subconjuntos cuidadosamente selecionados, poderá perguntar uma série mais refinada de perguntas que determinam a localização de qualquer erro de bit único.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "A sensação geral é como jogar um jogo de 20 perguntas, fazendo perguntas sim ou não que cortam o espaço de possibilidades pela metade.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "Por exemplo, digamos que fazemos uma verificação de paridade apenas nesses 8 bits, todas as posições ímpares.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "Então, se um erro for detectado, ele fornece ao receptor um pouco mais de informação sobre onde especificamente está o erro, ou seja, que ele está em uma posição estranha.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "Esta é apenas 1 das 4 verificações de paridade que faremos.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "Caso contrário, significa que não há erro ou que o erro está em algum lugar na metade esquerda.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "Ou acho que poderia ter havido dois erros, mas por enquanto vamos assumir que há no máximo um erro em todo o bloco.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "As coisas quebram completamente por mais do que isso.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "Digamos que você detecte um erro entre as colunas ímpares e entre a metade direita.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "E se nenhuma dessas duas verificações de paridade detectar nada, significa que o único lugar onde um erro pode estar é na coluna mais à esquerda.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "Mas também pode significar simplesmente que não há erro algum.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "Fazemos basicamente a mesma coisa, mas para as linhas.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "Haverá uma verificação de paridade nas linhas ímpares, usando a posição 4 como bit de paridade.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "Portanto, neste exemplo, esse grupo já tem uma paridade par, então o bit 4 seria definido como 0.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "E finalmente há uma verificação de paridade nas duas linhas inferiores, usando a posição 8 como bit de paridade.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "Mas isso não afeta o terceiro grupo e não afeta o quarto grupo.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "Reserve um momento para pensar em como qualquer erro entre esses quatro bits especiais será rastreado como qualquer outro, com o mesmo grupo de quatro perguntas.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "Você também pode gostar de antecipar como isso será dimensionado.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "Esperamos que este esboço seja suficiente para avaliar a eficiência do que estamos desenvolvendo aqui.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "Bem, quase.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "Ok, então o único problema aqui é que se nenhuma das quatro verificações de paridade detectar um erro, o que significa que todos os subconjuntos especialmente selecionados de 8 bits têm paridades pares, exatamente como o remetente pretendia, então isso significa que não houve erro algum , ou nos restringe à posição 0.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "E com isso, agora temos o que as pessoas do ramo chamariam de código de Hamming 15-11.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "Agora, se houver um erro de bit único, a paridade do bloco completo será ímpar, mas detectaríamos isso de qualquer maneira, graças às quatro verificações de correção de erros.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "Mesmo que não possamos corrigir esses erros de 2 bits, apenas colocando aquele bit 0 incômodo de volta ao trabalho, isso nos permite detectá-los.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "Isso é bastante padrão, é conhecido como código de Hamming estendido.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "Mas acho que você achará mais satisfatório verificar seu entendimento e solidificar tudo até agora fazendo você mesmo um exemplo completo do início ao fim.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "Cada pedaço será empacotado em um bloco de 16 bits resistente a erros.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "Então, vamos tomar este como exemplo e realmente resolver isso.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "Vá em frente, realmente faça isso!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "Vamos fazer uma pausa e tentar montar este bloco.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "Você precisa que este grupo tenha uma paridade par, o que já acontece, então você deveria ter definido o bit de paridade na posição 1 como 0.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "O próximo grupo começa com uma paridade ímpar, então você deveria ter definido seu bit de paridade como 1.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "Assim, à medida que este bloco é enviado, a paridade dos quatro subconjuntos especiais e do bloco como um todo será par, ou 0.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "Então, novamente, faça uma pausa e tente resolver isso.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "Se forem três ou mais, todas as apostas serão canceladas.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/russian/sentence_translations.json b/2020/hamming-codes/russian/sentence_translations.json index 835b36d32..b8e72747d 100644 --- a/2020/hamming-codes/russian/sentence_translations.json +++ b/2020/hamming-codes/russian/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "А простой стратегией исправления любого перевернутого бита было бы сохранение трех копий каждого бита.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "Например, используя метод, который вы узнаете из этого видео, вы можете хранить свои данные в 256-битных блоках, где каждый блок использует 9 бит, 9!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "действовать как своего рода избыточность, а остальные 247 бит могут свободно переносить любое значимое сообщение или данные, которые вы хотите.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "Методы, позволяющие исправлять подобные ошибки, вполне обоснованно называются кодами исправления ошибок.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "На протяжении большей части прошлого столетия эта область была действительно богатым источником удивительно глубокой математики, которая внедрялась в устройства, которые мы используем каждый день.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "История начинается в 1940-х годах, когда молодой Ричард Хэмминг работал в Bell Labs, и часть его работы заключалась в использовании очень большого дорогого компьютера с перфокартами, к которому у него был лишь ограниченный доступ.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "Разочарование было горнилом изобретений, и ему это настолько надоело, что он изобрел первый в мире код исправления ошибок.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "Это также может дать вам небольшую подсказку о том, как это масштабируется для более крупных блоков.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "Кроме того, технически это всего лишь 11 бит данных, вы обнаружите небольшой нюанс в том, что происходит в позиции 0, но пока не беспокойтесь об этом.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "Как и в любом алгоритме исправления ошибок, в нем участвуют два игрока: отправитель, который отвечает за установку этих 4 специальных битов, и получатель, который отвечает за выполнение некоторой проверки и исправление ошибок.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "В конце концов, хранение данных — это то же самое, что отправка сообщения только из прошлого в будущее, а не из одного места в другое.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "Итак, установка такова, но прежде чем мы углубимся в нее, нам нужно поговорить о связанной идее, которая была свежа в голове Хэмминга во время его открытия, о методе, который позволяет обнаруживать любые однобитовые ошибки, но не исправлять их, известный в бизнесе как проверка паритета.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "Для проверки четности мы выделяем только один бит, за настройку которого отвечает отправитель, а остальные могут свободно переносить сообщение.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "Это довольно просто, обманчиво просто, но это невероятно элегантный способ выразить идею изменения в любом месте сообщения и отразить ее в одном бите информации.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "Обратите внимание: если какой-либо бит этого сообщения переворачивается с 0 на 1 или с 1 на 0, общее количество единиц меняется с четного на нечетное.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "Итак, если вы получатель, посмотрите на это сообщение и увидите нечетное число единиц, вы можете точно знать, что произошла какая-то ошибка, даже если вы понятия не имеете, где она была.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "И этот специальный бит, который отправитель использует для контроля четности, называется битом четности.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "И на самом деле, нам должно быть ясно: если получатель видит нечетную четность, это не обязательно означает, что была только одна ошибка, могло быть 3 ошибки, или 5, или любое другое нечетное число, но он может знать это наверняка. что это не 0.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "Например, когда Хэмминг искал способ определить, где произошла ошибка, а не только то, что она произошла, его ключевой вывод заключался в том, что если вы примените некоторые проверки четности не ко всему сообщению, а к определенным тщательно выбранным подмножествам, вы можете спросить более уточненная серия вопросов, позволяющая определить местонахождение любой единичной ошибки.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "Общее ощущение немного похоже на игру из 20 вопросов, когда вы задаете вопросы «да» или «нет», которые делят пространство возможностей пополам.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "Например, предположим, что мы выполняем проверку четности только для этих 8 бит, всех позиций с нечетными номерами.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "Затем, если ошибка обнаружена, он дает приемнику немного больше информации о том, где именно находится ошибка, а именно, что она находится в нечетном положении.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "Это только 1 из 4 проверок четности, которые мы проведем.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "В противном случае это означает, что либо ошибки нет, либо ошибка где-то в левой половине.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "Или я предполагаю, что ошибок могло быть две, но сейчас мы будем предполагать, что во всем блоке есть не более одной ошибки.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "Более того, все полностью ломается.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "Допустим, вы обнаружили ошибку среди нечетных столбцов и среди правой половины.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "И если ни одна из этих двух проверок четности ничего не обнаружит, это означает, что единственное место, где может быть ошибка, — это самый левый столбец.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "Но это также может означать, что ошибки вообще нет.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "Мы делаем в основном то же самое, но для строк.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "Будет произведена проверка четности нечетных строк с использованием позиции 4 в качестве бита четности.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "Итак, в этом примере эта группа уже имеет четность, поэтому бит 4 будет установлен в 0.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "И, наконец, в двух нижних строках выполняется проверка четности с использованием позиции 8 в качестве бита четности.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "Но это не влияет на третью группу и не влияет на четвертую группу.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "Найдите минутку и подумайте, как можно отследить любую ошибку среди этих четырех специальных битов, как и любую другую, с помощью одной и той же группы из четырех вопросов.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "Вам также может понравиться предвидеть, как это масштабируется.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "Надеюсь, этого эскиза будет достаточно, чтобы оценить эффективность того, что мы здесь разрабатываем.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "Ну, почти.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "Итак, единственная проблема здесь заключается в том, что если ни одна из четырех проверок четности не обнаруживает ошибку, а это означает, что все специально выбранные подмножества 8 бит имеют четность, как и предполагал отправитель, то это либо означает, что ошибки вообще не было. , или это сузит нас до позиции 0.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "И теперь у нас есть то, что люди в бизнесе называют кодом Хэмминга 15-11.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "Теперь, если есть ошибка в один бит, то четность полного блока переключается на нечетную, но мы все равно уловим это благодаря четырем проверкам, исправляющим ошибки.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "Несмотря на то, что мы не можем исправить эти 2-битные ошибки, просто вернув в работу этот маленький надоедливый 0-й бит, мы сможем их обнаружить.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "Это довольно стандартный код, известный как расширенный код Хэмминга.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "Но я думаю, вам будет приятнее проверить свое понимание и закрепить все до этого момента, выполнив один полный пример от начала до конца самостоятельно.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "Каждый фрагмент будет упакован в устойчивый к ошибкам 16-битный блок.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "Итак, давайте возьмем это в качестве примера и на самом деле разберемся.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "Давай, действительно сделай это!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "Давайте остановимся и попробуем собрать этот блок.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "Вам нужно, чтобы эта группа имела четную четность, которая уже есть, поэтому вам следует установить этот бит четности в позиции 1 равным 0.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "Следующая группа начинается с нечетной четности, поэтому вам следует установить ее бит четности равным 1.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "Таким образом, при отправке этого блока четность четырех специальных подмножеств и блока в целом будет четной или равной 0.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "Итак, еще раз сделайте паузу и попробуйте разобраться.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "Если их три или больше, все ставки аннулируются.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/spanish/sentence_translations.json b/2020/hamming-codes/spanish/sentence_translations.json index 5364c962c..6d8a1bc44 100644 --- a/2020/hamming-codes/spanish/sentence_translations.json +++ b/2020/hamming-codes/spanish/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "Y una estrategia sencilla para corregir cualquier bit que se invierta sería almacenar tres copias de cada bit.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "Por ejemplo, usando el método que se aprenderá en este video, se podrían almacenar datos en bloques de 256 bits, donde cada bloque usa 9 bits, ¡9!", "model": "google_nmt", "n_reviews": 1, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "para actuar como una especie de redundancia, y los otros 247 bits son libres para transportar cualquier mensaje o dato significativo que se desee.", "model": "google_nmt", "n_reviews": 1, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "Los métodos que permiten corregir errores como este se conocen, razonablemente, como códigos de corrección de errores.", "model": "google_nmt", "n_reviews": 1, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "Durante la mayor parte del siglo pasado, este campo ha sido una fuente realmente rica de matemáticas sorprendentemente profundas que se incorporan a los dispositivos que utilizamos todos los días.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "La historia comienza en la década de 1940, cuando un joven Richard Hamming trabajaba para los Laboratorios Bell, y parte de su trabajo implicaba el uso de una computadora de tarjeta perforada muy grande y costosa a la que tenía acceso limitado.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "Siendo la frustración el crisol de la invención, se hartó tanto que inventó el primer código de corrección de errores del mundo.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "También podría darle una pequeña pista sobre cómo se adapta esto a bloques más grandes.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "Además, técnicamente termina siendo solo 11 bits de datos, encontrarás que hay una leve diferencia en lo que sucede en la posición 0, pero no te preocupes por eso por ahora.", "model": "google_nmt", "n_reviews": 1, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "Como cualquier algoritmo de corrección de errores, esto involucrará a dos jugadores, un remitente que es responsable de configurar estos 4 bits especiales y un receptor que es responsable de realizar algún tipo de verificación y corregir los errores.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "Después de todo, almacenar datos es lo mismo que enviar un mensaje del pasado al futuro en lugar de de un lugar a otro.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "Así que esa es la idea, pero antes de que podamos profundizar necesitamos hablar sobre una idea relacionada que estaba fresca en la mente de Hamming en el momento de su descubrimiento, un método que permite detectar errores de un solo bit, pero no corregirlos, conocido en la industria como control de paridad.", "model": "google_nmt", "n_reviews": 1, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "Para un control de paridad, separamos solo un bit que el remitente es responsable de ajustar, y el resto son libres de transportar un mensaje.", "model": "google_nmt", "n_reviews": 1, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "Esto es bastante simple, engañosamente simple, pero es una forma increíblemente elegante de destilar la idea de cambio en cualquier parte de un mensaje para que se refleje en un solo bit de información.", "model": "google_nmt", "n_reviews": 1, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "Observe que si alguna parte de este mensaje se invierte, ya sea de 0 a 1 o de 1 a 0, cambia el recuento total de unos de par a impar.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "Entonces, si tú eres el receptor, miras el mensaje y ves un número impar de unos, puedes estar seguro de que se ha producido algún error, aunque no tengas idea de dónde estaba.", "model": "google_nmt", "n_reviews": 1, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "Y este bit especial que utiliza el remitente para controlar la paridad se llama bit de paridad.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "Y, de hecho, debemos ser claros: si el receptor ve una paridad impar, no significa necesariamente que hubo solo un error, puede haber habido 3 errores, o 5, o cualquier otro número impar, pero puede estar seguro de que no fueron 0.", "model": "google_nmt", "n_reviews": 1, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "Por ejemplo, mientras Hamming buscaba una manera de identificar dónde ocurrió un error, no solo que sucedió, su idea clave fue que si se aplican algunos controles de paridad no al mensaje completo, sino a ciertos subconjuntos cuidadosamente seleccionados, se pueden preguntar una serie más refinada de preguntas que determinan la ubicación de cualquier error de un solo bit.", "model": "google_nmt", "n_reviews": 1, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "La sensación general es un poco como jugar un juego de 20 preguntas, haciendo preguntas de sí o no que reducen el espacio de posibilidades a la mitad.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "Por ejemplo, digamos que hacemos un control de paridad solo en estos 8 bits, todas las posiciones impares.", "model": "google_nmt", "n_reviews": 1, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "Luego, si se detecta un error, le da al receptor un poco más de información sobre dónde está específicamente el error, es decir, que está en una posición impar.", "model": "google_nmt", "n_reviews": 1, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "Este es sólo 1 de los 4 controles de paridad que haremos.", "model": "google_nmt", "n_reviews": 1, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "De lo contrario, significa que no hay error o que el error está en algún lugar de la mitad izquierda.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "O supongo que podrían haber habido dos errores, pero por ahora vamos a asumir que hay como máximo un error en todo el bloque.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "Las cosas se rompen por completo por más que eso.", "model": "google_nmt", "n_reviews": 1, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "Digamos que detectas un error en las columnas impares y en la mitad derecha.", "model": "google_nmt", "n_reviews": 1, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "Y si ninguno de esos dos controles de paridad detecta algo, significa que el único lugar donde podría haber un error es en la columna más a la izquierda.", "model": "google_nmt", "n_reviews": 1, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "Pero también podría significar simplemente que no hay ningún error.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "Básicamente hacemos lo mismo pero para las filas.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "Habrá un control de paridad en las filas impares, usando la posición 4 como bit de paridad.", "model": "google_nmt", "n_reviews": 1, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "En este ejemplo, ese grupo ya tiene una paridad par, por lo que el bit 4 se establecería en 0.", "model": "google_nmt", "n_reviews": 1, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "Y finalmente hay un control de paridad en las dos filas inferiores, usando la posición 8 como bit de paridad.", "model": "google_nmt", "n_reviews": 1, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "Pero no afecta al tercer grupo ni al cuarto grupo.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "Tómate un momento para pensar en cómo localizar cualquier error entre estos cuatro bits especiales como cualquier otro, con el mismo grupo de cuatro preguntas.", "model": "google_nmt", "n_reviews": 1, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "También puedes disfrutar anticipando cómo evoluciona esto.", "model": "google_nmt", "n_reviews": 1, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "Ojalá este boceto sea suficiente para apreciar la eficiencia de lo que aquí estamos desarrollando.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "Bueno, casi.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "Bien, entonces el único problema aquí es que si ninguno de los cuatro controles de paridad detecta un error, es decir, que los subconjuntos de 8 bits especialmente seleccionados tienen paridades pares, tal como lo pretendía el remitente, entonces significa que no hubo ningún error o nos reduce a la posición 0.", "model": "google_nmt", "n_reviews": 1, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "Y con eso, ahora tenemos lo que la gente en la industria llamaría un código Hamming 15-11.", "model": "google_nmt", "n_reviews": 1, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "Ahora, si hay un error de un solo bit, entonces la paridad del bloque completo cambia a impar, pero lo detectaremos de todos modos gracias a los cuatro controles de corrección de errores.", "model": "google_nmt", "n_reviews": 1, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "Aunque no podemos corregir esos errores de 2 bits, simplemente volviendo a poner a funcionar a ese pequeño molesto bit 0, nos permite detectarlos.", "model": "google_nmt", "n_reviews": 1, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "Esto es bastante estándar, se conoce como código Hamming extendido.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "Pero creo que te resultará más satisfactorio comprobar tu comprensión y solidificar todo hasta este punto haciendo tú mismo un ejemplo completo de principio a fin.", "model": "google_nmt", "n_reviews": 1, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "Cada fragmento se empaquetará en un bloque de 16 bits resistente a errores.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "Así que tomemos este como ejemplo y resolvámoslo.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "¡Adelante, hazlo de verdad!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "Hagamos una pausa e intentemos armar este bloque.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "Necesitas que este grupo tenga una paridad par, la cual ya tiene, por lo que debería haber configurado ese bit de paridad en la posición 1 para que sea 0.", "model": "google_nmt", "n_reviews": 1, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "El siguiente grupo comienza con una paridad impar, por lo que deberías haber establecido su bit de paridad en 1.", "model": "google_nmt", "n_reviews": 1, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "Entonces, cuando este bloque sea enviado, la paridad de los cuatro subconjuntos especiales y el bloque en su conjunto serán pares o 0.", "model": "google_nmt", "n_reviews": 1, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "Así que nuevamente, haz una pausa e intenta resolverlo.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "Si son tres o más, no podremos saber que ocurrió.", "model": "google_nmt", "n_reviews": 1, diff --git a/2020/hamming-codes/tamil/sentence_translations.json b/2020/hamming-codes/tamil/sentence_translations.json index 64c684313..850c3084d 100644 --- a/2020/hamming-codes/tamil/sentence_translations.json +++ b/2020/hamming-codes/tamil/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "புரட்டப்படும் எந்த பிட்டையும் சரிசெய்வதற்கான எளிய உத்தி, ஒவ்வொரு பிட்டின் மூன்று பிரதிகளை சேமிப்பதாகும்.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "எடுத்துக்காட்டாக, இந்த வீடியோவைப் பற்றி நீங்கள் அறியும் முறையைப் பயன்படுத்தி, உங்கள் தரவை 256-பிட் தொகுதிகளில் சேமிக்கலாம், அங்கு ஒவ்வொரு தொகுதியும் 9 பிட்கள், 9!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "ஒரு வகையான பணிநீக்கமாக செயல்பட, மற்ற 247 பிட்கள் நீங்கள் விரும்பும் அர்த்தமுள்ள செய்தி அல்லது தரவை எடுத்துச் செல்ல இலவசம்.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "இது போன்ற பிழைகளை சரிசெய்ய உங்களை அனுமதிக்கும் முறைகள், போதுமான அளவு, பிழை திருத்தக் குறியீடுகளாக அறியப்படுகின்றன.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "கடந்த நூற்றாண்டின் சிறந்த பகுதிக்கு, இந்தத் துறையானது வியக்கத்தக்க ஆழமான கணிதத்தின் மிகவும் வளமான ஆதாரமாக இருந்து வருகிறது, இது நாம் தினமும் பயன்படுத்தும் சாதனங்களில் இணைக்கப்பட்டுள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "கதை 1940 களில் தொடங்குகிறது, ஒரு இளம் ரிச்சர்ட் ஹாமிங் பெல் லேப்ஸில் பணிபுரிந்தார், மேலும் அவரது சில வேலைகளில் அவர் குறைந்த அணுகல் மட்டுமே இருந்த மிகப் பெரிய விலையுயர்ந்த பஞ்ச் கார்டு கணினியைப் பயன்படுத்தினார்.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "விரக்தியே கண்டுபிடிப்பின் முக்கிய அம்சமாக இருப்பதால், அவர் மிகவும் சோர்வடைந்தார், அவர் உலகின் முதல் பிழை திருத்தக் குறியீட்டைக் கண்டுபிடித்தார்.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "பெரிய தொகுதிகளுக்கு இது எவ்வாறு அளவிடப்படுகிறது என்பது பற்றிய சிறிய குறிப்பை இது உங்களுக்கு வழங்கக்கூடும்.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "மேலும் தொழில்நுட்ப ரீதியாக இது 11 பிட் டேட்டாவாக மட்டுமே முடிவடைகிறது, நிலை 0 இல் என்ன நடக்கிறது என்பதற்கு லேசான நுணுக்கம் இருப்பதை நீங்கள் காண்பீர்கள், ஆனால் இப்போது அதைப் பற்றி கவலைப்பட வேண்டாம்.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "எந்தவொரு பிழை திருத்தும் அல்காரிதம் போலவே, இது இரண்டு பிளேயர்களை உள்ளடக்கும், இந்த 4 சிறப்பு பிட்களை அமைப்பதற்கு பொறுப்பான அனுப்புநர் மற்றும் சில வகையான சரிபார்ப்பு மற்றும் பிழைகளை சரிசெய்வதற்கு பொறுப்பான ஒரு பெறுநர்.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "எல்லாவற்றிற்கும் மேலாக, தரவை சேமிப்பது என்பது ஒரு இடத்திலிருந்து இன்னொரு இடத்திற்கு அனுப்புவதற்குப் பதிலாக கடந்த காலத்திலிருந்து எதிர்காலத்திற்கு ஒரு செய்தியை அனுப்புவது போன்றது.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "அதுதான் அமைப்பு, ஆனால் நாம் உள்ளே நுழைவதற்கு முன், ஹேமிங்கின் கண்டுபிடிப்பின் போது அவரது மனதில் புதியதாக இருந்த ஒரு தொடர்புடைய யோசனையைப் பற்றி பேச வேண்டும், இது எந்த ஒரு பிட் பிழைகளையும் கண்டறிய உங்களை அனுமதிக்கிறது, ஆனால் அவற்றை சரிசெய்ய முடியாது. வணிகத்தில் சமநிலை சரிபார்ப்பு.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "சமநிலைச் சரிபார்ப்பிற்கு, அனுப்புநரே ட்யூனிங்கிற்குப் பொறுப்பானவர் என்று ஒரே ஒரு பிட் மட்டுமே பிரித்தெடுக்கிறோம், மீதமுள்ளவர்கள் செய்தியை எடுத்துச் செல்ல இலவசம்.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "இது மிகவும் எளிமையானது, ஏமாற்றும் வகையில் எளிமையானது, ஆனால் ஒரு செய்தியில் எந்த இடத்திலும் மாற்றம் குறித்த யோசனையை ஒரு பிட் தகவலில் பிரதிபலிக்கும் வகையில் இது ஒரு நம்பமுடியாத நேர்த்தியான வழியாகும்.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "இந்தச் செய்தியின் ஏதேனும் பிட் 0 முதல் 1 அல்லது 1 முதல் 0 வரை புரட்டப்பட்டால், அது 1 வினாடிகளின் மொத்த எண்ணிக்கையை இரட்டைப்படையாக மாற்றுகிறது என்பதைக் கவனியுங்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "நீங்கள் பெறுபவராக இருந்தால், நீங்கள் இந்த செய்தியைப் பார்க்கிறீர்கள், நீங்கள் ஒற்றைப்படை எண் 1ஐக் கண்டால், அது எங்கே என்று உங்களுக்குத் தெரியாவிட்டாலும், சில பிழை ஏற்பட்டிருப்பதை நீங்கள் உறுதியாக அறிந்து கொள்ளலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "மேலும் இந்த ஸ்பெஷல் பிட் பாரிட்டியைக் கட்டுப்படுத்த அனுப்புபவர் பயன்படுத்தும் பாரிட்டி பிட் என்று அழைக்கப்படுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "உண்மையில், நாம் தெளிவாக இருக்க வேண்டும், ரிசீவர் ஒரு ஒற்றைப்படை சமநிலையைக் கண்டால், அது ஒரே ஒரு பிழை என்று அர்த்தமல்ல, 3 பிழைகள், அல்லது 5 அல்லது வேறு ஏதேனும் ஒற்றைப்படை எண் இருந்திருக்கலாம், ஆனால் அவர்கள் நிச்சயமாக அறிந்து கொள்ள முடியும். அது 0 இல்லை என்று.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "எடுத்துக்காட்டாக, ஹேமிங் ஒரு பிழை எங்கே நடந்தது என்பதைக் கண்டறியும் வழியைத் தேடிக்கொண்டிருந்தபோது, அது நிகழ்ந்தது மட்டுமல்ல, அவருடைய முக்கிய நுண்ணறிவு என்னவென்றால், நீங்கள் சில சமத்துவச் சரிபார்ப்புகளை முழுச் செய்திக்கு அல்ல, ஆனால் கவனமாகத் தேர்ந்தெடுக்கப்பட்ட சில துணைக்குழுக்களுக்குப் பயன்படுத்தினால், நீங்கள் கேட்கலாம். எந்த ஒரு பிட் பிழையின் இருப்பிடத்தைக் குறிக்கும் மேலும் சுத்திகரிக்கப்பட்ட தொடர் கேள்விகள்.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "ஒட்டுமொத்த உணர்வு 20 கேள்விகள் கொண்ட விளையாட்டை விளையாடுவது போன்றது, ஆம் அல்லது இல்லை என்று கேள்விகளைக் கேட்பது சாத்தியக்கூறுகளின் இடத்தை பாதியாக வெட்டுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "எடுத்துக்காட்டாக, இந்த 8 பிட்கள், ஒற்றைப்படை எண்கள் உள்ள எல்லா நிலைகளிலும் சரிபார்ப்பைச் செய்கிறோம் என்று வைத்துக் கொள்வோம்.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "ஒரு பிழை கண்டறியப்பட்டால், அது குறிப்பாக பிழை எங்குள்ளது, அதாவது அது ஒற்றைப்படை நிலையில் உள்ளது என்பது பற்றிய கூடுதல் தகவலை பெறுநருக்கு வழங்குகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "நாங்கள் செய்வோம் 4 சமநிலை சரிபார்ப்புகளில் இது 1 மட்டுமே.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "இல்லையெனில் பிழை இல்லை அல்லது பிழை இடது பாதியில் எங்கோ உள்ளது என்று அர்த்தம்.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "அல்லது இரண்டு பிழைகள் இருந்திருக்கலாம் என்று நினைக்கிறேன், ஆனால் இப்போதைக்கு முழுத் தொகுதியிலும் அதிகபட்சம் ஒரு பிழை இருப்பதாகக் கருதுவோம்.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "அதை விட விஷயங்கள் முற்றிலும் உடைந்து போகின்றன.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "ஒற்றைப்படை நெடுவரிசைகள் மற்றும் வலது பாதியில் ஒரு பிழையை நீங்கள் கண்டறிந்ததாக வைத்துக்கொள்வோம்.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "அந்த இரண்டு சமநிலை சரிபார்ப்புகளில் எதுவுமே எதையும் கண்டறியவில்லை என்றால், அந்த இடதுபுற நெடுவரிசையில் மட்டுமே பிழை இருக்க முடியும் என்று அர்த்தம்.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "ஆனால் அது எந்த பிழையும் இல்லை என்று அர்த்தம்.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "நாங்கள் அடிப்படையில் அதையே செய்கிறோம் ஆனால் வரிசைகளுக்கு.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "நிலை 4ஐ சமநிலை பிட்டாகப் பயன்படுத்தி ஒற்றைப்படை வரிசைகளில் சமநிலைச் சரிபார்ப்பு இருக்கப் போகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "எனவே இந்த எடுத்துக்காட்டில் அந்தக் குழு ஏற்கனவே சம சமநிலையைக் கொண்டுள்ளது, எனவே பிட் 4 0 ஆக அமைக்கப்படும்.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "இறுதியாக, கீழே உள்ள இரண்டு வரிசைகளில் சமநிலைச் சரிபார்ப்பு உள்ளது, நிலை 8 ஐ சமநிலை பிட்டாகப் பயன்படுத்துகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "ஆனால் இது மூன்றாவது குழுவை பாதிக்காது, நான்காவது குழுவை பாதிக்காது.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "இந்த நான்கு ஸ்பெஷல் பிட்களில் எந்தப் பிழையும் ஒரே மாதிரியான நான்கு கேள்விகளைக் கொண்ட பிறவற்றைப் போலவே எப்படிக் கண்காணிக்கப் போகிறது என்பதைப் பற்றி சிறிது சிந்தித்துப் பாருங்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "இது எப்படி இருக்கும் என்பதை நீங்கள் எதிர்பார்த்து மகிழலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "நாங்கள் இங்கு உருவாக்கி வருவதன் செயல்திறனைப் பாராட்ட இந்த ஓவியம் போதுமானது என நம்புகிறோம்.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "சரி, கிட்டத்தட்ட.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "சரி, இங்கு உள்ள ஒரு பிரச்சனை என்னவென்றால், நான்கு சமநிலை சரிபார்ப்புகளில் எதுவுமே பிழையைக் கண்டறியவில்லை என்றால், அதாவது 8 பிட்களின் சிறப்பாகத் தேர்ந்தெடுக்கப்பட்ட துணைக்குழுக்கள் அனைத்தும், அனுப்புநரின் நோக்கத்தைப் போலவே சமமான சமநிலைகளைக் கொண்டிருக்கின்றன என்று அர்த்தம். , அல்லது அது நம்மை நிலை 0 ஆகக் குறைக்கிறது.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "அதனுடன், வணிகத்தில் உள்ளவர்கள் 15-11 ஹேமிங் குறியீடு என்று குறிப்பிடுவது இப்போது எங்களிடம் உள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "இப்போது, ஒரு பிட் பிழை இருந்தால், முழுத் தொகுதியின் சமநிலை ஒற்றைப்படையாக மாறுகிறது, ஆனால் நான்கு பிழை-திருத்தும் சரிபார்ப்புகளுக்கு நன்றி.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "அந்த 2-பிட் பிழைகளை நம்மால் சரிசெய்ய முடியாவிட்டாலும், அந்த ஒரு சிறிய தொந்தரவான 0வது பிட்டை மீண்டும் வேலை செய்ய வைப்பதன் மூலம், அது அவற்றைக் கண்டறிய உதவுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "இது மிகவும் நிலையானது, இது நீட்டிக்கப்பட்ட ஹேமிங் குறியீடு என்று அழைக்கப்படுகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "ஆனால், உங்கள் புரிதலைச் சரிபார்த்து, தொடக்கத்தில் இருந்து உங்களை முடிக்கும் வரை ஒரு முழு உதாரணத்தைச் செய்வதன் மூலம் இது வரை அனைத்தையும் உறுதிப்படுத்துவது உங்களுக்கு மிகவும் திருப்திகரமாக இருக்கும் என்று நினைக்கிறேன்.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "ஒவ்வொரு துண்டையும் ஒரு பிழை-எதிர்ப்பு 16-பிட் தொகுதிக்குள் தொகுக்கப் போகிறது.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "எனவே இதை ஒரு உதாரணமாக எடுத்துக்கொள்வோம், உண்மையில் அதைச் செயல்படுத்துவோம்.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "மேலே செல்லுங்கள், உண்மையில் அதைச் செய்யுங்கள்!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "இடைநிறுத்தப்பட்டு இந்த தொகுதியை ஒன்றாக இணைக்க முயற்சிப்போம்.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "இந்தக் குழுவில் சம சமநிலை இருக்க வேண்டும், அது ஏற்கனவே உள்ளது, எனவே நீங்கள் அந்த சமநிலை பிட்டை நிலை 1 இல் 0 ஆக அமைத்திருக்க வேண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "அடுத்த குழு ஒற்றைப்படை சமநிலையுடன் தொடங்குகிறது, எனவே நீங்கள் அதன் சமநிலை பிட்டை 1 ஆக அமைத்திருக்க வேண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "எனவே இந்தத் தொகுதி அனுப்பப்பட்டதால், நான்கு சிறப்பு துணைக்குழுக்களின் சமநிலை மற்றும் ஒட்டுமொத்த தொகுதி அனைத்தும் சமமாக அல்லது 0 ஆக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "எனவே மீண்டும், இடைநிறுத்தி அதைச் செயல்படுத்த முயற்சிக்கவும்.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "இது மூன்று அல்லது அதற்கு மேற்பட்டதாக இருந்தால், அனைத்து சவால்களும் நிறுத்தப்படும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/telugu/sentence_translations.json b/2020/hamming-codes/telugu/sentence_translations.json index a0e1b1091..eef65adf8 100644 --- a/2020/hamming-codes/telugu/sentence_translations.json +++ b/2020/hamming-codes/telugu/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "మరియు తిప్పబడిన బిట్‌ను సరిచేయడానికి ఒక సాధారణ వ్యూహం ప్రతి బిట్ యొక్క మూడు కాపీలను నిల్వ చేయడం.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "ఉదాహరణకు, మీరు ఈ వీడియో గురించి నేర్చుకునే పద్ధతిని ఉపయోగించి, మీరు మీ డేటాను 256-బిట్ బ్లాక్‌లలో నిల్వ చేయవచ్చు, ఇక్కడ ప్రతి బ్లాక్ 9 బిట్‌లను ఉపయోగిస్తుంది, 9!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "ఒక రకమైన రిడెండెన్సీగా పని చేయడానికి మరియు ఇతర 247 బిట్‌లు మీకు కావలసిన అర్థవంతమైన సందేశం లేదా డేటాను ఉచితంగా తీసుకెళ్లవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "ఇలాంటి లోపాలను సరిదిద్దడానికి మిమ్మల్ని అనుమతించే పద్ధతులు దోష దిద్దుబాటు కోడ్‌లుగా సమంజసంగా తగినంతగా తెలుసు.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "గత శతాబ్దంలో చాలా కాలం పాటు, ఈ ఫీల్డ్ ఆశ్చర్యకరంగా లోతైన గణితానికి నిజంగా గొప్ప మూలంగా ఉంది, అది మనం ప్రతిరోజూ ఉపయోగించే పరికరాలలో చేర్చబడుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "1940వ దశకంలో రిచర్డ్ హామింగ్ అనే యువకుడు బెల్ ల్యాబ్స్‌లో పని చేస్తున్నప్పుడు కథ మొదలవుతుంది మరియు అతని పనిలో కొంత భాగం అతనికి పరిమితమైన యాక్సెస్ మాత్రమే ఉన్న చాలా పెద్ద ఖరీదైన పంచ్ కార్డ్ కంప్యూటర్‌ను ఉపయోగించడం జరిగింది.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "ఆవిష్కారానికి మూలమైన నిరాశ, అతను చాలా విసుగు చెందాడు, అతను ప్రపంచంలోని మొట్టమొదటి దోష సవరణ కోడ్‌ను కనుగొన్నాడు.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "పెద్ద బ్లాక్‌ల కోసం ఇది ఎలా స్కేల్ అవుతుందనే దాని గురించి ఇది మీకు చిన్న సూచనను కూడా ఇవ్వవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "సాంకేతికంగా ఇది కేవలం 11 బిట్‌ల డేటాగా ముగుస్తుంది, 0 స్థానం వద్ద ఏమి జరుగుతుందనే దాని కోసం మీరు స్వల్ప స్వల్పభేదాన్ని కనుగొంటారు, కానీ ప్రస్తుతానికి దాని గురించి చింతించకండి.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "ఏదైనా ఎర్రర్ కరెక్షన్ అల్గారిథమ్ లాగా, ఇది ఇద్దరు ప్లేయర్‌లను కలిగి ఉంటుంది, ఈ 4 ప్రత్యేక బిట్‌లను సెట్ చేయడానికి బాధ్యత వహించే పంపినవారు మరియు ఒక రకమైన తనిఖీని నిర్వహించడానికి మరియు లోపాలను సరిదిద్దడానికి బాధ్యత వహించే రిసీవర్.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "అన్నింటికంటే, డేటాను నిల్వ చేయడం అనేది ఒక ప్రదేశం నుండి మరొక ప్రదేశానికి బదులుగా గతం నుండి భవిష్యత్తుకు సందేశాన్ని పంపడం లాంటిదే.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "కాబట్టి అది సెటప్, కానీ మనం డైవ్ చేయడానికి ముందు హామింగ్ కనుగొన్న సమయంలో అతని మనస్సులో తాజాగా ఉన్న సంబంధిత ఆలోచన గురించి మాట్లాడాలి, ఇది ఏదైనా ఒక బిట్ లోపాలను గుర్తించడానికి మిమ్మల్ని అనుమతిస్తుంది, కానీ వాటిని సరిదిద్దడానికి కాదు. పారిటీ చెక్‌గా వ్యాపారంలో.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "సమానత్వ తనిఖీ కోసం, పంపినవారు ట్యూనింగ్‌కు బాధ్యత వహించే ఒకే ఒక్క బిట్‌ను మాత్రమే మేము వేరు చేస్తాము మరియు మిగిలిన వారు సందేశాన్ని తీసుకువెళ్లడానికి ఉచితం.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "ఇది చాలా సరళమైనది, మోసపూరితమైనది, కానీ సందేశంలో ఎక్కడైనా మార్పు చేయాలనే ఆలోచనను ఒకే బిట్ సమాచారంలో ప్రతిబింబించేలా చేయడానికి ఇది చాలా సొగసైన మార్గం.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "ఈ సందేశంలోని ఏదైనా బిట్ 0 నుండి 1కి లేదా 1 నుండి 0కి తిప్పబడితే, అది 1సె మొత్తం గణనను సరి నుండి బేసిగా మారుస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "మీరు రిసీవర్ అయితే, మీరు ఈ సందేశాన్ని చూడండి మరియు మీకు బేసి సంఖ్య 1లు కనిపిస్తే, అది ఎక్కడ ఉందో మీకు తెలియకపోయినప్పటికీ, కొంత లోపం సంభవించిందని మీరు ఖచ్చితంగా తెలుసుకోవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "మరియు పంపినవారు పారిటీని నియంత్రించడానికి ఉపయోగించే ఈ ప్రత్యేక బిట్‌ని పారిటీ బిట్ అంటారు.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "వాస్తవానికి, మనం స్పష్టంగా ఉండాలి, రిసీవర్ బేసి సమానత్వాన్ని చూసినట్లయితే, అది కేవలం ఒక లోపం మాత్రమే ఉందని అర్థం కాదు, 3 లోపాలు లేదా 5 లేదా మరేదైనా బేసి సంఖ్య ఉండవచ్చు, కానీ వారు ఖచ్చితంగా తెలుసుకోగలరు అది 0 కాదని.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "ఉదాహరణకు, హ్యామింగ్ లోపం ఎక్కడ జరిగిందో గుర్తించడానికి ఒక మార్గం కోసం వెతుకుతున్నందున, అది జరిగిందనే కాకుండా, అతని ముఖ్య అంతర్దృష్టి ఏమిటంటే, మీరు కొన్ని సమానత్వ తనిఖీలను పూర్తి సందేశానికి కాకుండా, జాగ్రత్తగా ఎంచుకున్న కొన్ని ఉపసమితులకు వర్తింపజేస్తే, మీరు అడగవచ్చు ఏదైనా ఒక బిట్ ఎర్రర్ ఉన్న లొకేషన్‌ను పిన్ చేసే మరింత శుద్ధి చేసిన ప్రశ్నల శ్రేణి.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "మొత్తం ఫీలింగ్ 20 ప్రశ్నల గేమ్‌ను ఆడటం లాంటిది, అవునా లేదా కాదు అనే ప్రశ్నలను అడగడం, అవకాశాల ఖాళీని సగానికి తగ్గించడం.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "ఉదాహరణకు, ఈ 8 బిట్‌లలో, అన్ని బేసి సంఖ్యల స్థానాలపై మాత్రమే మనం సమానత్వ తనిఖీ చేశామని అనుకుందాం.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "అప్పుడు ఒక లోపం గుర్తించబడితే, అది రిసీవర్‌కు నిర్దిష్టంగా లోపం ఎక్కడ ఉందో దాని గురించి కొంచెం ఎక్కువ సమాచారాన్ని అందిస్తుంది, అంటే అది బేసి స్థానంలో ఉంది.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "ఇది మేము చేసే 4 పారిటీ తనిఖీలలో 1 మాత్రమే.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "లేకుంటే లోపం లేదని అర్థం, లేదా లోపం ఎడమ భాగంలో ఎక్కడో ఉంది.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "లేదా రెండు లోపాలు ఉండవచ్చని నేను ఊహిస్తున్నాను, కానీ ప్రస్తుతానికి మేము మొత్తం బ్లాక్‌లో గరిష్టంగా ఒక లోపం ఉందని భావించబోతున్నాము.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "అంతకు మించి విషయాలు పూర్తిగా విచ్ఛిన్నమవుతాయి.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "మీరు బేసి నిలువు వరుసలలో మరియు కుడి సగం మధ్య లోపాన్ని గుర్తించారని అనుకుందాం.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "మరియు ఆ రెండు సమానత్వ తనిఖీలలో దేనినీ గుర్తించలేకపోతే, ఆ ఎడమవైపున ఉన్న నిలువు వరుసలో లోపం ఉన్న ఏకైక స్థలం అని అర్థం.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "కానీ ఇది కేవలం ఏ లోపం లేదని అర్థం కావచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "మేము ప్రాథమికంగా అదే పని చేస్తాము కానీ వరుసల కోసం.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "స్థానం 4ని సమాన బిట్‌గా ఉపయోగించి, బేసి అడ్డు వరుసలలో సమాన తనిఖీ జరగబోతోంది.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "కాబట్టి ఈ ఉదాహరణలో ఆ సమూహం ఇప్పటికే సమాన సమానత్వాన్ని కలిగి ఉంది, కాబట్టి బిట్ 4 0కి సెట్ చేయబడుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "మరియు చివరగా దిగువ రెండు అడ్డు వరుసలలో 8వ స్థానాన్ని సమాన బిట్‌గా ఉపయోగిస్తూ సమానత్వ తనిఖీ ఉంది.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "కానీ ఇది మూడవ సమూహాన్ని ప్రభావితం చేయదు మరియు ఇది నాల్గవ సమూహాన్ని ప్రభావితం చేయదు.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "ఈ నాలుగు ప్రత్యేక బిట్‌లలోని ఏదైనా లోపం, అదే నాలుగు ప్రశ్నల సమూహంతో ఇతర వాటిలాగే ఎలా ట్రాక్ చేయబడుతుందో ఆలోచించండి.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "ఇది ఎలా స్కేల్ అవుతుందో కూడా మీరు ఊహించి ఆనందించవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "మేము ఇక్కడ అభివృద్ధి చేస్తున్న వాటి సామర్థ్యాన్ని అభినందించడానికి ఈ స్కెచ్ సరిపోతుందని ఆశిస్తున్నాము.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "బాగా, దాదాపు.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "సరే, ఇక్కడ ఒక సమస్య ఏమిటంటే, నాలుగు పారిటీ తనిఖీలలో ఏదీ లోపాన్ని గుర్తించకపోతే, అంటే 8 బిట్‌ల యొక్క ప్రత్యేకంగా ఎంపిక చేయబడిన ఉపసమితులు అన్నింటికీ సమాన సమానతలను కలిగి ఉంటాయి, పంపినవారు ఉద్దేశించినట్లుగా, అప్పుడు ఎటువంటి లోపం లేదని అర్థం. , లేదా అది మనల్ని స్థానం 0కి తగ్గిస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "దానితో, వ్యాపారంలో వ్యక్తులు 15-11 హామింగ్ కోడ్‌గా సూచించే వాటిని ఇప్పుడు మేము కలిగి ఉన్నాము.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "ఇప్పుడు, ఒక బిట్ ఎర్రర్ ఉన్నట్లయితే, పూర్తి బ్లాక్ యొక్క సమానత్వం బేసిగా టోగుల్ అవుతుంది, అయితే నాలుగు ఎర్రర్-కరెక్టింగ్ చెక్‌ల కారణంగా మేము దానిని పట్టుకుంటాము.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "మేము ఆ 2-బిట్ లోపాలను సరిదిద్దలేనప్పటికీ, ఒక చిన్న ఇబ్బందికరమైన 0వ బిట్‌ను తిరిగి పనిలో ఉంచడం ద్వారా, అది వాటిని గుర్తించేలా చేస్తుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "ఇది చాలా ప్రామాణికమైనది, దీనిని పొడిగించిన హామింగ్ కోడ్ అని పిలుస్తారు.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "కానీ మీ అవగాహనను తనిఖీ చేయడం మరియు మొదటి నుండి పూర్తి చేయడానికి ఒక పూర్తి ఉదాహరణ చేయడం ద్వారా ఈ పాయింట్ వరకు ప్రతిదీ పటిష్టం చేయడం మీకు మరింత సంతృప్తికరంగా ఉంటుందని నేను భావిస్తున్నాను.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "ప్రతి భాగం ఎర్రర్-రెసిస్టెంట్ 16-బిట్ బ్లాక్‌లోకి ప్యాక్ చేయబడుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "కాబట్టి దీనిని ఒక ఉదాహరణగా తీసుకుందాం మరియు వాస్తవానికి దాన్ని పని చేద్దాం.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "ముందుకు సాగండి, వాస్తవానికి దీన్ని చేయండి!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "పాజ్ చేసి, ఈ బ్లాక్‌ని కలపడానికి ప్రయత్నిద్దాం.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "మీకు ఈ సమూహానికి సమాన సమానత్వం అవసరం, ఇది ఇప్పటికే ఉంది, కాబట్టి మీరు ఆ పారిటీ బిట్‌ను స్థానం 1లో 0గా సెట్ చేసి ఉండాలి.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "తదుపరి సమూహం బేసి సమానత్వంతో ప్రారంభమవుతుంది, కాబట్టి మీరు దాని సమాన బిట్‌ను 1కి సెట్ చేసి ఉండాలి.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "కాబట్టి ఈ బ్లాక్ పంపబడినందున, నాలుగు ప్రత్యేక ఉపసమితులు మరియు మొత్తం బ్లాక్ యొక్క సమానత్వం సమం లేదా 0 అవుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "కాబట్టి మళ్ళీ, పాజ్ చేసి, దాన్ని పని చేయడానికి ప్రయత్నించండి.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "ఇది మూడు లేదా అంతకంటే ఎక్కువ ఉంటే, అన్ని పందాలు నిలిపివేయబడతాయి.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/thai/sentence_translations.json b/2020/hamming-codes/thai/sentence_translations.json index 69c0d7334..619cb17a8 100644 --- a/2020/hamming-codes/thai/sentence_translations.json +++ b/2020/hamming-codes/thai/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit. ", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notice ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9! ", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want. ", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan ", "translatedText": "เพื่อทำหน้าที่เป็นความซ้ำซ้อน และอีก 247 บิตมีอิสระในการส่งข้อความหรือข้อมูลที่มีความหมายตามที่คุณต้องการ และยังคงเป็นกรณีที่หากมีการพลิกตรงนี้เพียงดูที่บล็อกนี้แล้วไม่มีอะไรเพิ่มเติมเครื่องก็จะสามารถระบุได้ว่ามีข้อผิดพลาดและอยู่ที่ไหนอย่างแม่นยำเพื่อให้รู้วิธีแก้ไข . ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 112.66 }, { - "input": "And honestly, that feels like magic. ", + "input": "m e emphasize that they are distinct from the data that's actually being sent. They're noth ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 122.86 }, { - "input": "We'll talk a little bit later about how this scales for blocks with different sizes. ", + "input": "that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to identify that there was an error and precisely where i ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 141.94 }, { - "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. ", + "input": "that there were two errors, though it won't know how to fix them. We'll talk a little bit later about how this scales for blocks with different sizes. where that's a 1, you get the second parity group from our scheme ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 206.94 }, { - "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. ", + "input": "e scheme is going to be before I tell you. Also, if you want your understanding to get down to the hardware level, Ben Eater has made a video in conjunction with this one ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. ", + "input": "st how impossible this task feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is tha ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 248.42 }, { - "input": "There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. ", + "input": "t in a vast space of all possible messages, only some subset are going to be considered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words. also see th ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 255.38 }, { - "input": "Let's use an example that's simple, but not too simple, a block of 16 bits. ", + "input": "is in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 260.94 }, { - "input": "We'll number the positions of these bits from 0 up to 15. ", + "input": "Once you understand that these parity checks that we've focused so much of our time on are nothing ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 273.0 }, { - "input": "The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. ", + "input": "binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 287.28 }, { - "input": "You might expect these 4 special bits to come nicely packaged together, maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. ", + "input": "r. When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the programs he kept putting through it kept failing, because every now and then a ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors. ", + "input": "'s first error correction code. There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", + "input": "e so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy, not adding any new information, but adding resilience. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.82 }, { - "input": "The only job of this special bit is to make sure that the total number of 1s in the message is an even number. ", + "input": "that make sense? Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and w ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.28 }, { - "input": "So for example right now, that total number of 1s is 7, that's odd, so the sender needs to flip that special bit to be a 1, making the count even. ", + "input": "hich a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it goes from here. The sender is responsible for toggling ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd. ", + "input": "sitio Like any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. ", + "input": "Of course, the words sender and receiver really refer to machines or software that's doing checks, and the idea of a message is meant really broadly, to include things like storage. After all, storing data is the same thing as sending a message, just from the past ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit. ", + "input": "his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 485.44 }, { - "input": "Instead, the goal is to come up with a scheme that's robust up to a certain maximum number of errors, or maybe to reduce the probability of a false positive like this. ", + "input": "en kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information. ctice t ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 495.38 }, { - "input": "Parity checks on their own are pretty weak, but by distilling the idea of change across a full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. ", + "input": "his would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running f ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error. ", + "input": "rom 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of 1s is known as its parity. o collect together all of those positions, the positions of the bits that are ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 590.68 }, { - "input": "The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", + "input": "his on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.3 }, { - "input": "This time we might use position 2 as a parity bit, so these 8 bits already have an even parity, and the sender can feel good leaving that bit number 2 unchanged. ", + "input": "you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half. ", + "input": "simulating a random error from noise, then if you run this same line of code, it print s out that error. Isn't that neat? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit. ", + "input": "own to a single XOR reduction. Now, depending on your comfort with binary and XORs and software in general, you may eithe ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0. ", + "input": "r find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a wa ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit. ", + "input": "y to identify where an error happened, not just that it happened, his key insight was that if you apply some parity chec ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 701.84 }, { - "input": "As an example, imagine that during the transmission there's an error at, say, position 3. ", + "input": "ion of any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 720.54 }, { - "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. ", + "input": "here is that that information directly corresponds to how much redundancy we need. That's really what runs against most people's knee-jerk reaction Then, if an error is detected, it gives the receiv ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 727.52 }, { - "input": "You might enjoy taking a moment to convince yourself that the answers to these four questions really will always let you pin down a specific location, no matter where they turn out to be. ", + "input": "er a little more information about where specifically the error is, namely that it's in an odd position. ent to errors, where usually copying the whole message is the first instinct that comes to min ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 743.06 }, { - "input": "And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spoil it. ", + "input": "then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix. It's kind of nice because it relates ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 773.1 }, { - "input": "But protecting those bits as well is something that naturally falls out of the scheme as a byproduct. ", + "input": "more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales. ", + "input": "Here let's just choose position 1. For the example shown, the pari ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 781.76 }, { - "input": "If we used a block of size 256 bits, for example, in order to pin down a location, you need only eight yes or no questions to binary search your way down to some specific spot. ", + "input": "ty of these 8 bits is currently odd, so the sender is responsible for toggling that parity bit, and now it's even. This is only 1 out of 4 parity checks that we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 813.66 }, { - "input": "The first thing, except for those eight highlighted parity bits, can be whatever you want it to be, carrying whatever message or data you want. ", + "input": "the sender can feel good leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your pari ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 821.0 }, { - "input": "The 8 bits are redundant in the sense that they're completely determined by the rest of the message, but it's in a much smarter way than simply copying the message as a whole. ", + "input": "ty checks, and it uses only 21 parity bits. And if you step back to think about looking at a million bits and locating a single error, that genuinely feels crazy. The problem, Otherwise, it means either there's ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost. ", + "input": "or the error is somewhere on the left half. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0. ", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 915.54 }, { - "input": "Here's how it works. ", + "input": "er block. But it also might simply mean there's no error at ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 952.7 }, { - "input": "Isn't that clever? ", + "input": "s today. There are like half a dozen times throughout this book that he refer ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them. ", + "input": "ences the Louis Pasteur quote, luck favors a prepared mind. Cl In this case, it looks like the sender needs to turn that bit 8 on in order to give the group even parity. Part of the reason that clever ideas look deceptively ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 965.22 }, { - "input": "Technically speaking, you now have a full description of what a Hamming code does, at least for the example of a 16-bit block. ", + "input": "nal result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the transmission there's an error at, say, position 3. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it! ", + "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block. ", + "input": "3, so they can fix the error. You might enjoy taking a moment to convince yourself that the answers to these four ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1007.02 }, { - "input": "Okay, you ready? ", + "input": "questions really will always let you pin down ", "translatedText": "ลองหยุดและลองประกอบบล็อกนี้ดู โอเค คุณพร้อมหรือยัง? ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0. ", + "input": "n between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spo ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1037.88 }, { - "input": "The group after that starts with an odd parity, so again you should have set its parity bit to 1. ", + "input": "ts affected, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1079.78 }, { - "input": "What I'm going to do is change either 0, 1, or 2 of the bits in that block, and then ask you to figure out what it is that I did. ", + "input": "y eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set the app ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1107.91 }, { - "input": "The next check gives us an odd number, telling us both that there's at least one error, and narrowing us down into this specific column. ", + "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever you want it to be, carrying whatever message or data you want. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off. ", + "input": ". And still, for so little given up, you would be able to identify and fix any single bit error. Well, almost. Okay, so the one ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1240,7 +1240,7 @@ "end": 1163.17 }, { - "input": "You see, what I haven't told you yet is just how elegant this algorithm really is, how simple it is to get a machine to point to the position of an error, how to systematically scale it, and how we can frame all of this as one single operation rather than multiple separate parity checks. ", + "input": "You see, with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing one out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The solution here is actually pretty simple. Just forget about that zeroth bit entirely. So when we do our four parity checks and ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/turkish/sentence_translations.json b/2020/hamming-codes/turkish/sentence_translations.json index 404d91c8b..6b848c853 100644 --- a/2020/hamming-codes/turkish/sentence_translations.json +++ b/2020/hamming-codes/turkish/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "Ve ters çevrilen herhangi bir biti düzeltmek için basit bir strateji, her bitin üç kopyasını saklamak olacaktır.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "Örneğin, bu videoda öğreneceğiniz yöntemi kullanarak verilerinizi 256 bitlik bloklarda saklayabilirsiniz; burada her blok 9 bit, 9!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "bir tür artıklık görevi görür ve diğer 247 bit, istediğiniz anlamlı mesajı veya veriyi taşımakta özgürdür.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "Bunun gibi hataları düzeltmenize izin veren yöntemler, makul olarak, hata düzeltme kodları olarak bilinir.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "Geçen yüzyılın büyük bir bölümünde bu alan, her gün kullandığımız cihazlara dahil edilen, şaşırtıcı derecede derin bir matematik açısından gerçekten zengin bir kaynak oldu.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "Hikaye 1940'larda, genç Richard Hamming'in Bell Laboratuvarları için çalıştığı ve bazı işlerinin sınırlı erişime sahip olduğu çok büyük, pahalı bir delikli kart bilgisayarını kullandığı zaman başlıyor.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "Hayal kırıklığı buluşun potası olduğundan o kadar bıktı ki dünyanın ilk hata düzeltme kodunu icat etti.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "Ayrıca, bunun daha büyük bloklar için nasıl ölçeklendiğine dair size küçük bir ipucu verebilir.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "Ayrıca teknik olarak sadece 11 bitlik bir veriden ibarettir, 0 konumunda olup bitenlerle ilgili hafif bir nüans olduğunu göreceksiniz, ancak şimdilik bunun için endişelenmeyin.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "Herhangi bir hata düzeltme algoritması gibi, bu da iki oyuncuyu içerecektir; bu 4 özel bitin ayarlanmasından sorumlu olan bir gönderici ve bir tür kontrolün gerçekleştirilmesinden ve hataların düzeltilmesinden sorumlu olan bir alıcı.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "Sonuçta veri depolamak, mesajın bir yerden başka bir yere değil, geçmişten geleceğe gönderilmesiyle aynı şeydir.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "Kurulum bu, ancak konuya dalmadan önce, keşfi sırasında Hamming'in aklında yeni olan ilgili bir fikir hakkında konuşmamız gerekiyor; herhangi bir tek bit hatasını tespit etmenize izin veren ancak bunları düzeltmenize izin vermeyen bir yöntem, bilinen iş dünyasında parite kontrolü olarak.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "Eşlik kontrolü için, gönderenin ayarlamadan sorumlu olduğu tek bir biti ayırıyoruz ve geri kalanı mesaj taşımakta özgür.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "Bu oldukça basit, yanıltıcı derecede basit, ancak bir mesajın herhangi bir yerindeki değişimin tek bir bilgi parçasına yansıtılacağı fikrini damıtmanın inanılmaz derecede zarif bir yolu.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "Bu mesajın herhangi bir bitinin 0'dan 1'e veya 1'den 0'a çevrilmesi durumunda, 1'lerin toplam sayısının çiftten teke değişmesine dikkat edin.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "Yani eğer alıcıysanız, bu mesaja baktığınızda ve tek sayıda 1'ler görüyorsanız, nerede olduğu hakkında hiçbir fikriniz olmasa bile bir hatanın meydana geldiğinden emin olabilirsiniz.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "Ve göndericinin pariteyi kontrol etmek için kullandığı bu özel bit, eşlik biti olarak adlandırılır.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "Ve aslında, açık olmalıyız ki, eğer alıcı tek bir parite görürse, bu sadece bir hata olduğu anlamına gelmez, 3 hata, 5 hata veya başka herhangi bir tek sayı olabilir, ancak kesin olarak bilebilir. 0 değildi.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "Örneğin, Hamming bir hatanın sadece meydana geldiğini değil, nerede meydana geldiğini de belirlemenin bir yolunu ararken, onun temel görüşü şuydu: Eğer bazı eşitlik kontrollerini mesajın tamamına değil de dikkatle seçilmiş belirli alt kümelere uygularsanız, şu soruyu sorabilirsiniz: herhangi bir bit hatasının yerini belirleyen daha rafine bir dizi soru.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "Genel his biraz 20 soruluk bir oyun oynamaya, olasılıklar alanını yarıya indiren evet veya hayır soruları sormaya benziyor.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "Örneğin, diyelim ki sadece bu 8 bit üzerinde, tek sayılı konumların tümü üzerinde bir eşlik kontrolü yaptığımızı varsayalım.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "Daha sonra bir hata tespit edilirse, alıcıya hatanın tam olarak nerede olduğu, yani tek bir konumda olduğu hakkında biraz daha bilgi verilir.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "Bu, yapacağımız 4 eşlik kontrolünden yalnızca 1'idir.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "Aksi takdirde bu, ya hata olmadığı ya da hatanın sol yarıda bir yerde olduğu anlamına gelir.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "Ya da sanırım iki hata olabilirdi ama şimdilik tüm blokta en fazla bir hata olduğunu varsayacağız.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "Bundan daha fazlası için işler tamamen bozulur.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "Diyelim ki tek sütunlar arasında ve sağ yarıda bir hata tespit ettiniz.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "Ve eğer bu iki eşlik kontrolünden hiçbiri bir şey tespit etmezse, bu, hatanın olabileceği tek yerin en soldaki sütun olduğu anlamına gelir.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "Ancak bu aynı zamanda hiçbir hatanın olmadığı anlamına da gelebilir.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "Temelde aynı şeyi yapıyoruz ancak satırlar için.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "Konum 4'ü eşlik biti olarak kullanarak tek satırlarda bir eşlik kontrolü yapılacak.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "Yani bu örnekte bu grup zaten çift eşlikli olduğundan bit 4, 0'a ayarlanacaktır.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "Ve son olarak, alt iki satırda, konum 8'i eşlik biti olarak kullanan bir eşlik kontrolü var.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "Ama üçüncü grubu etkilemediği gibi dördüncü grubu da etkilemez.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "Bir dakikanızı ayırıp bu dört özel parça arasındaki herhangi bir hatanın, tıpkı diğerleri gibi, dört sorudan oluşan aynı grupla nasıl bulunacağını düşünün.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "Bunun nasıl ölçekleneceğini tahmin etmekten de keyif alabilirsiniz.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "Umarım bu taslak, burada geliştirdiğimiz şeyin verimliliğini takdir etmek için yeterlidir.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "Neredeyse.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "Tamam, buradaki sorun şu ki, eğer dört eşlik kontrolünden hiçbiri bir hata tespit etmezse, yani özel olarak seçilmiş 8 bitlik alt kümelerin tümü, gönderenin amaçladığı gibi eşit eşliklere sahipse, o zaman bu da hiçbir hata olmadığı anlamına gelir. veya bizi 0 pozisyonuna daraltır.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "Ve bununla birlikte, artık sektördeki insanların 15-11 Hamming kodu olarak adlandıracağı şeye sahibiz.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "Şimdi, eğer tek bir bit hatası varsa, o zaman tam bloğun paritesi tek olacak şekilde değişir, ancak dört hata düzeltme kontrolü sayesinde bunu yine de yakalarız.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "Her ne kadar bu 2 bitlik hataları düzeltemesek de, o küçük can sıkıcı 0'ıncı biti tekrar devreye sokarak, onları tespit etmemizi sağlıyor.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "Bu oldukça standarttır, genişletilmiş Hamming kodu olarak bilinir.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "Ancak, baştan sona kendi başınıza tam bir örnek yaparak anlayışınızı kontrol etmeyi ve bu noktaya kadar her şeyi sağlamlaştırmayı daha tatmin edici bulacağınızı düşünüyorum.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "Her parça, hataya dayanıklı 16 bitlik bir blok halinde paketlenecek.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "Hadi bunu bir örnek olarak alalım ve gerçekten üzerinde çalışalım.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "Devam edin, gerçekten yapın!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "Durup bu bloğu bir araya getirmeyi deneyelim.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "Bu grubun çift eşlikli olması gerekir ki zaten öyledir, dolayısıyla 1 konumundaki eşlik bitini 0 olacak şekilde ayarlamanız gerekir.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "Bir sonraki grup tek bir eşlikle başlar, dolayısıyla eşlik bitini 1 olarak ayarlamanız gerekir.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "Yani bu blok gönderildiğinde, dört özel alt kümenin ve bir bütün olarak bloğun paritesi çift veya 0 olacaktır.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "Tekrar duraklatın ve çözmeye çalışın.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "Üç veya daha fazla ise tüm bahisler kapalıdır.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/ukrainian/sentence_translations.json b/2020/hamming-codes/ukrainian/sentence_translations.json index 129443ce8..f58856801 100644 --- a/2020/hamming-codes/ukrainian/sentence_translations.json +++ b/2020/hamming-codes/ukrainian/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "І простою стратегією виправлення будь-якого біта, який перевертається, було б зберігати три копії кожного біта.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "Наприклад, використовуючи метод, який ви дізнаєтеся про це відео, ви можете зберігати свої дані у 256-бітних блоках, де кожен блок використовує 9 біт, 9!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "щоб діяти як своєрідна надлишковість, а інші 247 бітів можуть вільно переносити будь-які значущі повідомлення або дані, які ви хочете.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "Методи, які дозволяють виправляти такі помилки, відомі, досить розумно, як коди виправлення помилок.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "Протягом більшої частини минулого століття ця сфера була справді багатим джерелом напрочуд глибокої математики, яка вбудовується в пристрої, якими ми користуємося щодня.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "Історія починається в 1940-х роках, коли молодий Річард Хеммінг працював у Bell Labs, і частина його роботи включала використання дуже великого дорогого комп’ютера з перфокартами, до якого він мав лише обмежений доступ.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "Розчарування, будучи горнилом винахідництва, йому так набридло, що він винайшов перший у світі код виправлення помилок.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "Це також може дати вам невелику підказку про те, як це масштабується для більших блоків.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "Крім того, технічно це лише 11 біт даних, ви побачите, що є легкий нюанс у тому, що відбувається в позиції 0, але не хвилюйтеся про це зараз.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "Як і в будь-якому іншому алгоритмі виправлення помилок, у цьому будуть задіяні два гравці: відправник, який відповідає за встановлення цих 4 спеціальних бітів, і одержувач, який відповідає за виконання певної перевірки та виправлення помилок.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "Зрештою, зберігання даних — це те саме, що відправляти повідомлення лише з минулого в майбутнє, а не з одного місця в інше.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "Отже, це налаштування, але перш ніж ми зможемо зануритися в це, нам потрібно поговорити про пов’язану ідею, яка була свіжою в голові Геммінга під час його відкриття, метод, який дозволяє виявляти будь-які однобітові помилки, але не виправляти їх, відомий у бізнесі як перевірка паритету.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "Для перевірки парності ми відокремлюємо лише один біт, за налаштування якого відповідає відправник, а решта можуть вільно передавати повідомлення.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "Це досить просто, оманливо просто, але це неймовірно елегантний спосіб дистилювати ідею зміни будь-де в повідомленні, щоб її відобразити в одному біті інформації.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "Зверніть увагу, якщо будь-який біт цього повідомлення перевертається з 0 на 1 або з 1 на 0, це змінює загальну кількість одиниць з парної на непарну.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "Отже, якщо ви одержувач, ви дивитеся на це повідомлення та бачите непарну кількість одиниць, ви можете точно знати, що сталася якась помилка, навіть якщо ви можете не знати, де вона була.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "І цей спеціальний біт, який відправник використовує для контролю парності, називається бітом парності.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "І насправді, ми маємо чітко уточнити: якщо отримувач бачить непарну парність, це не обов’язково означає, що була лише одна помилка, могло бути 3 помилки, або 5, або будь-яке інше непарне число, але вони можуть знати напевно що це не 0.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "Наприклад, коли Хеммінг шукав спосіб визначити, де сталася помилка, а не просто те, що вона сталася, його ключове розуміння полягало в тому, що якщо ви застосовуєте деякі перевірки парності не до всього повідомлення, а до певних ретельно відібраних підмножин, ви можете запитати більш витончену серію запитань, які визначають місце будь-якої окремої бітової помилки.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "Загальне відчуття трохи схоже на гру з 20 запитань, задаючи запити «так» або «ні», які розрізають простір можливостей навпіл.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "Наприклад, скажімо, ми виконуємо перевірку парності лише для цих 8 бітів, усіх позицій з непарними номерами.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "Потім, якщо виявлено помилку, це дає одержувачу трохи більше інформації про те, де саме є помилка, а саме, що він знаходиться в непарній позиції.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "Це лише 1 із 4 перевірок парності, які ми зробимо.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "Інакше це означає, що або помилки немає, або помилка десь на лівій половині.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "Або я припускаю, що могло бути дві помилки, але зараз ми будемо припускати, що у всьому блоці є щонайбільше одна помилка.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "Речі повністю ламаються для більшого.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "Припустімо, ви виявили помилку серед непарних стовпців і серед правої половини.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "І якщо жодна з цих двох перевірок парності нічого не виявляє, це означає, що єдине місце, де може бути помилка, це крайній лівий стовпець.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "Але це також може означати, що помилки взагалі немає.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "Ми робимо в основному те саме, але для рядків.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "У непарних рядках буде перевірено парність із використанням позиції 4 як біта парності.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "Отже, у цьому прикладі ця група вже має парний паритет, тому біт 4 буде встановлено на 0.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "І, нарешті, є перевірка парності в нижніх двох рядках, використовуючи позицію 8 як біт парності.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "Але це не впливає на третю групу, і не впливає на четверту групу.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "Знайдіть хвилинку, щоб подумати про те, як будь-яка помилка серед цих чотирьох спеціальних бітів буде відстежуватися так само, як і будь-яка інша, за допомогою тієї самої групи з чотирьох запитань.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "Вам також може бути цікаво передбачити, як це масштабується.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "Сподіваюся, цього ескізу достатньо, щоб оцінити ефективність того, що ми тут розробляємо.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "Ну, майже.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "Гаразд, одна проблема тут полягає в тому, що якщо жодна з чотирьох перевірок парності не виявляє помилку, тобто всі спеціально вибрані підмножини з 8 біт мають парні паритети, як і хотів відправник, то це означає, що помилки взагалі не було , або це звужує нас до позиції 0.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "І разом з цим тепер ми маємо те, що люди в бізнесі називали б кодом Хеммінга 15-11.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "Тепер, якщо є помилка з одним бітом, то парність повного блоку перемикається на непарність, але ми все одно вловимо це завдяки чотирьом перевіркам виправлення помилок.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "Навіть незважаючи на те, що ми не можемо виправити ці 2-бітові помилки, просто повернувши цей маленький набридливий 0-й біт до роботи, це дозволить нам їх виявити.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "Це досить стандартний код, він відомий як розширений код Хеммінга.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "Але я думаю, що вам буде приємніше перевірити ваше розуміння та закріпити все до цього моменту, виконавши один повний приклад від початку до кінця самостійно.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "Кожна частина буде упакована в стійкий до помилок 16-бітний блок.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "Отже, давайте візьмемо це як приклад і розберемося.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "Давай, справді зроби це!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "Давайте зупинимося і спробуємо скласти цей блок.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "Вам потрібно, щоб ця група мала рівний паритет, який вона вже має, тому ви повинні були встановити цей біт парності в позиції 1 на 0.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "Наступна група починається з непарної парності, тому вам слід було встановити її біт парності на 1.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "Отже, коли цей блок відправляється, парність чотирьох спеціальних підмножин і блоку в цілому буде парною, або 0.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "Тож знову зробіть паузу та спробуйте розібратися.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "Якщо їх три або більше, усі ставки скасовуються.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/urdu/sentence_translations.json b/2020/hamming-codes/urdu/sentence_translations.json index 762879799..8d142655d 100644 --- a/2020/hamming-codes/urdu/sentence_translations.json +++ b/2020/hamming-codes/urdu/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit. ", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notice ", "translatedText": "اور پلٹ جانے والے کسی بھی بٹ کو درست کرنے کی ایک سادہ حکمت عملی یہ ہوگی کہ ہر بٹ کی تین کاپیاں اسٹور کی جائیں۔", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9! ", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how ", "translatedText": "مثال کے طور پر، اس ویڈیو کے بارے میں جو طریقہ آپ سیکھیں گے اس کو استعمال کرتے ہوئے، آپ اپنا ڈیٹا 256 بٹ بلاکس میں محفوظ کر سکتے ہیں، جہاں ہر بلاک 9 بٹس استعمال کرتا ہے، 9! ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want. ", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan ", "translatedText": "ایک قسم کی فالتو پن کے طور پر کام کرنے کے لیے، اور دیگر 247 بٹس جو بھی بامعنی پیغام یا ڈیٹا آپ چاہتے ہیں لے جانے کے لیے آزاد ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 112.66 }, { - "input": "And honestly, that feels like magic. ", + "input": "m e emphasize that they are distinct from the data that's actually being sent. They're noth ", "translatedText": "اور ایمانداری سے، یہ جادو کی طرح محسوس ہوتا ہے. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 122.86 }, { - "input": "We'll talk a little bit later about how this scales for blocks with different sizes. ", + "input": "that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to identify that there was an error and precisely where i ", "translatedText": "ہم تھوڑی دیر بعد اس بارے میں بات کریں گے کہ یہ مختلف سائز والے بلاکس کے لیے کیسے پیمانہ بناتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 141.94 }, { - "input": "The goal here is to give you a very thorough understanding of one of the earliest examples, known as a Hamming code. ", + "input": "that there were two errors, though it won't know how to fix them. We'll talk a little bit later about how this scales for blocks with different sizes. where that's a 1, you get the second parity group from our scheme ", "translatedText": "یہاں مقصد یہ ہے کہ آپ کو ابتدائی مثالوں میں سے ایک، جسے ہیمنگ کوڈ کے نام سے جانا جاتا ہے، کی مکمل تفہیم فراہم کی جائے۔", "model": "google_nmt", "n_reviews": 0, @@ -200,7 +200,7 @@ "end": 206.94 }, { - "input": "Whenever a valid message gets altered, the receiver is responsible for correcting what they see back to the nearest valid neighbor, as you might do with a typo. ", + "input": "e scheme is going to be before I tell you. Also, if you want your understanding to get down to the hardware level, Ben Eater has made a video in conjunction with this one ", "translatedText": "جب بھی کسی درست پیغام میں ردوبدل کیا جاتا ہے، وصول کنندہ کی ذمہ داری ہے کہ وہ اسے درست کرے جو وہ قریب ترین درست پڑوسی کو دیکھتا ہے، جیسا کہ آپ ٹائپنگ کی غلطی کے ساتھ کر سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. ", + "input": "st how impossible this task feels at the start, and how utterly reasonable it seems once you learn about Hamming. The basic principle of error correction is tha ", "translatedText": "فرسٹریشن ایجاد کا سنگ میل ہونے کی وجہ سے وہ اتنا تنگ آ گیا کہ اس نے دنیا کا پہلا غلطی کو درست کرنے والا کوڈ ایجاد کیا۔", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 248.42 }, { - "input": "There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. ", + "input": "t in a vast space of all possible messages, only some subset are going to be considered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words. also see th ", "translatedText": "ہیمنگ کوڈز کو فریم کرنے کے بہت سے مختلف طریقے ہیں، لیکن پہلے پاس کے طور پر ہم اس سے گزریں گے جس طرح سے ہیمنگ نے خود ان کے بارے میں سوچا تھا۔", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 255.38 }, { - "input": "Let's use an example that's simple, but not too simple, a block of 16 bits. ", + "input": "is in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. ", "translatedText": "آئیے ایک مثال استعمال کرتے ہیں جو سادہ ہے، لیکن بہت آسان نہیں، 16 بٹس کا ایک بلاک۔", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 260.94 }, { - "input": "We'll number the positions of these bits from 0 up to 15. ", + "input": "Once you understand that these parity checks that we've focused so much of our time on are nothing ", "translatedText": "ہم ان بٹس کی پوزیشن کو 0 سے 15 تک نمبر دیں گے۔", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 273.0 }, { - "input": "The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. ", + "input": "binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. ", "translatedText": "یہاں فالتو لفظ کا مطلب صرف کاپی نہیں ہے، آخر کار، وہ 4 بٹس ہمیں اتنی گنجائش نہیں دیتے کہ ڈیٹا کو آنکھیں بند کرکے کاپی کر سکیں۔", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 287.28 }, { - "input": "You might expect these 4 special bits to come nicely packaged together, maybe at the end or something like that, but as you'll see, having them sit in positions which are powers of 2 allows for something that's really elegant by the end. ", + "input": "r. When you take the XOR of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the programs he kept putting through it kept failing, because every now and then a ", "translatedText": "آپ توقع کر سکتے ہیں کہ یہ 4 خصوصی بٹس اچھی طرح سے ایک ساتھ پیک کیے جائیں گے، شاید آخر میں یا اس طرح کی کوئی چیز، لیکن جیسا کہ آپ دیکھیں گے، ان کو پوزیشنوں پر بیٹھنے سے جو کہ 2 کی طاقتیں ہیں آخر تک ایسی چیز کی اجازت دیتی ہے جو واقعی خوبصورت ہو۔", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors. ", + "input": "'s first error correction code. There are many different ways to frame Hamming codes, but as a first pass we're going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. ", "translatedText": "کسی بھی غلطی کو درست کرنے والے الگورتھم کی طرح، اس میں دو کھلاڑی شامل ہوں گے، ایک بھیجنے والا جو ان 4 خصوصی بٹس کو ترتیب دینے کا ذمہ دار ہے، اور ایک وصول کنندہ جو کسی قسم کی جانچ پڑتال کرنے اور غلطیوں کو درست کرنے کا ذمہ دار ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", + "input": "e so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy, not adding any new information, but adding resilience. ", "translatedText": "تو یہ سیٹ اپ ہے، لیکن اس سے پہلے کہ ہم اس میں غوطہ لگا سکیں ہمیں ایک متعلقہ آئیڈیا کے بارے میں بات کرنے کی ضرورت ہے جو ہیمنگ کے ذہن میں اس کی دریافت کے وقت تازہ تھا، ایک ایسا طریقہ جو آپ کو کسی بھی چھوٹی غلطی کا پتہ لگانے دیتا ہے، لیکن ان کو درست کرنے کے لیے نہیں، معلوم ہے۔کاروبار میں برابری کی جانچ کے طور پر۔", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.82 }, { - "input": "The only job of this special bit is to make sure that the total number of 1s in the message is an even number. ", + "input": "that make sense? Likewise, the next column counts how many positions are in the second parity group, the positions whose second to last bit is a 1, and w ", "translatedText": "اس خاص بٹ کا واحد کام اس بات کو یقینی بنانا ہے کہ پیغام میں 1s کی کل تعداد ایک ہموار نمبر ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.28 }, { - "input": "So for example right now, that total number of 1s is 7, that's odd, so the sender needs to flip that special bit to be a 1, making the count even. ", + "input": "hich a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it goes from here. The sender is responsible for toggling ", "translatedText": "تو مثال کے طور پر اس وقت، 1s کی کل تعداد 7 ہے، یہ عجیب ہے، لہذا بھیجنے والے کو اس خاص بٹ کو 1 ہونے کے لیے پلٹنا ہوگا، جس سے گنتی برابر ہوجائے گی۔", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd. ", + "input": "sitio Like any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. ", "translatedText": "نوٹ کریں کہ اگر اس پیغام کا کوئی حصہ پلٹ جاتا ہے، یا تو 0 سے 1 یا 1 سے 0 تک، یہ 1s کی کل گنتی کو برابر سے طاق میں بدل دیتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. ", + "input": "Of course, the words sender and receiver really refer to machines or software that's doing checks, and the idea of a message is meant really broadly, to include things like storage. After all, storing data is the same thing as sending a message, just from the past ", "translatedText": "لہذا اگر آپ وصول کنندہ ہیں، آپ اس پیغام کو دیکھتے ہیں، اور آپ کو 1s کی ایک طاق تعداد نظر آتی ہے، آپ یقینی طور پر جان سکتے ہیں کہ کچھ خرابی واقع ہوئی ہے، حالانکہ آپ کو اندازہ نہیں ہوگا کہ یہ کہاں تھا۔", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit. ", + "input": "his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check. ", "translatedText": "اور یہ خاص بٹ جسے بھیجنے والا برابری کو کنٹرول کرنے کے لیے استعمال کرتا ہے اسے برابری بٹ کہا جاتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 485.44 }, { - "input": "Instead, the goal is to come up with a scheme that's robust up to a certain maximum number of errors, or maybe to reduce the probability of a false positive like this. ", + "input": "en kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information. ctice t ", "translatedText": "اس کے بجائے، مقصد ایک ایسی اسکیم کے ساتھ آنا ہے جو غلطیوں کی ایک خاص زیادہ سے زیادہ تعداد تک مضبوط ہو، یا شاید اس طرح کے غلط مثبت ہونے کے امکان کو کم کرنا۔", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 495.38 }, { - "input": "Parity checks on their own are pretty weak, but by distilling the idea of change across a full message down to a single bit, what they give us is a powerful building block for more sophisticated schemes. ", + "input": "his would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is pair together each of those bits with a corresponding index, in this case running f ", "translatedText": "اپنے طور پر برابری کی جانچ پڑتال کافی کمزور ہیں، لیکن ایک مکمل پیغام میں تبدیلی کے خیال کو ایک ہی جگہ تک پھیلانے سے، وہ ہمیں جو کچھ دیتے ہیں وہ زیادہ نفیس سکیموں کے لیے ایک طاقتور عمارت ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error. ", + "input": "rom 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of 1s is known as its parity. o collect together all of those positions, the positions of the bits that are ", "translatedText": "مثال کے طور پر، جیسا کہ ہیمنگ یہ شناخت کرنے کا طریقہ تلاش کر رہا تھا کہ غلطی کہاں ہوئی، نہ صرف یہ کہ یہ ہوا، اس کی اہم بصیرت یہ تھی کہ اگر آپ کچھ برابری کی جانچ پڑتال پورے پیغام پر نہیں، بلکہ احتیاط سے منتخب کردہ کچھ ذیلی سیٹوں پر کرتے ہیں، تو آپ پوچھ سکتے ہیں۔سوالات کی ایک مزید بہتر سیریز جو کسی ایک بٹ کی غلطی کے مقام کو پن کرتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 590.68 }, { - "input": "The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", + "input": "his on our random block of 16 bits, it returns 9, which has the binary representation 1001. We won't do it here, but ", "translatedText": "دوسرا چیک گرڈ کے دائیں نصف پر 8 بٹس میں سے ہے، کم از کم جیسا کہ ہم نے اسے یہاں کھینچا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -568,7 +568,7 @@ "end": 596.3 }, { - "input": "This time we might use position 2 as a parity bit, so these 8 bits already have an even parity, and the sender can feel good leaving that bit number 2 unchanged. ", + "input": "you could write a function where the sender uses that binary representation to set the four parity bits as needed, ultimately getting this block to a state where running this ", "translatedText": "اس بار ہم پوزیشن 2 کو برابری بٹ کے طور پر استعمال کر سکتے ہیں، اس لیے ان 8 بٹس میں پہلے سے ہی برابر برابری موجود ہے، اور بھیجنے والا اس بٹ نمبر 2 کو بغیر کسی تبدیلی کے چھوڑ کر اچھا محسوس کر سکتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half. ", + "input": "simulating a random error from noise, then if you run this same line of code, it print s out that error. Isn't that neat? ", "translatedText": "بصورت دیگر اس کا مطلب ہے کہ یا تو کوئی خرابی نہیں ہے، یا غلطی بائیں آدھے حصے میں کہیں ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit. ", + "input": "own to a single XOR reduction. Now, depending on your comfort with binary and XORs and software in general, you may eithe ", "translatedText": "پوزیشن 4 کو برابری بٹ کے طور پر استعمال کرتے ہوئے، طاق قطاروں پر برابری کی جانچ کی جائے گی۔", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0. ", + "input": "r find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a wa ", "translatedText": "تو اس مثال میں اس گروپ میں پہلے سے ہی برابر برابری ہے، لہذا بٹ 4 کو 0 پر سیٹ کیا جائے گا۔", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit. ", + "input": "y to identify where an error happened, not just that it happened, his key insight was that if you apply some parity chec ", "translatedText": "اور آخر میں نچلی دو قطاروں پر برابری کی جانچ پڑتال ہے، پوزیشن 8 کو برابری بٹ کے طور پر استعمال کرتے ہوئے۔", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 701.84 }, { - "input": "As an example, imagine that during the transmission there's an error at, say, position 3. ", + "input": "ion of any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no ", "translatedText": "مثال کے طور پر، تصور کریں کہ ٹرانسمیشن کے دوران پوزیشن 3 پر ایک خرابی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -752,7 +752,7 @@ "end": 720.54 }, { - "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. ", + "input": "here is that that information directly corresponds to how much redundancy we need. That's really what runs against most people's knee-jerk reaction Then, if an error is detected, it gives the receiv ", "translatedText": "اور یہ وصول کنندہ کو پہلی قطار تک غلطی کی نشاندہی کرنے دیتا ہے، جس کا لازمی مطلب ہے پوزیشن 3، تاکہ وہ غلطی کو ٹھیک کر سکیں۔", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 727.52 }, { - "input": "You might enjoy taking a moment to convince yourself that the answers to these four questions really will always let you pin down a specific location, no matter where they turn out to be. ", + "input": "er a little more information about where specifically the error is, namely that it's in an odd position. ent to errors, where usually copying the whole message is the first instinct that comes to min ", "translatedText": "آپ اپنے آپ کو یہ باور کرانے کے لیے ایک لمحہ نکال کر لطف اندوز ہو سکتے ہیں کہ ان چار سوالوں کے جوابات واقعی آپ کو ہمیشہ ایک مخصوص جگہ کو پن کرنے دیں گے، چاہے وہ کہیں بھی کیوں نہ ہوں۔", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 743.06 }, { - "input": "And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spoil it. ", + "input": "then, by the way, there is this whole other way that you sometimes see Hamming codes presented, where you multiply the message by one big matrix. It's kind of nice because it relates ", "translatedText": "اور اگر آپ ایسا کرتے ہیں، تو مجھے دوبارہ زور دینے دیں، توقف کریں، اپنے لیے کوشش کریں کہ اس سے پہلے کہ میں اسے خراب کروں۔", "model": "google_nmt", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 773.1 }, { - "input": "But protecting those bits as well is something that naturally falls out of the scheme as a byproduct. ", + "input": "more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group. ", "translatedText": "لیکن ان بٹس کی حفاظت بھی ایک ایسی چیز ہے جو قدرتی طور پر اسکیم سے بطور پروڈکٹ خارج ہوجاتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales. ", + "input": "Here let's just choose position 1. For the example shown, the pari ", "translatedText": "آپ یہ اندازہ لگا کر بھی لطف اندوز ہو سکتے ہیں کہ یہ ترازو کیسے ہوتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 781.76 }, { - "input": "If we used a block of size 256 bits, for example, in order to pin down a location, you need only eight yes or no questions to binary search your way down to some specific spot. ", + "input": "ty of these 8 bits is currently odd, so the sender is responsible for toggling that parity bit, and now it's even. This is only 1 out of 4 parity checks that we'll do. The second check is among the 8 bits on the right half of the grid, at least as we've drawn it here. ", "translatedText": "اگر ہم نے سائز 256 بٹس کا ایک بلاک استعمال کیا ہے، مثال کے طور پر، کسی مقام کو پن کرنے کے لیے، آپ کو کسی مخصوص جگہ پر بائنری تلاش کرنے کے لیے صرف آٹھ ہاں یا نہیں سوالات کی ضرورت ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 813.66 }, { - "input": "The first thing, except for those eight highlighted parity bits, can be whatever you want it to be, carrying whatever message or data you want. ", + "input": "the sender can feel good leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 questions with your pari ", "translatedText": "پہلی چیز، ان آٹھ ہائی لائٹ کردہ برابری بٹس کے علاوہ، جو بھی آپ چاہتے ہیں ہو سکتا ہے، جو بھی پیغام یا ڈیٹا آپ چاہتے ہیں لے کر جا سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 821.0 }, { - "input": "The 8 bits are redundant in the sense that they're completely determined by the rest of the message, but it's in a much smarter way than simply copying the message as a whole. ", + "input": "ty checks, and it uses only 21 parity bits. And if you step back to think about looking at a million bits and locating a single error, that genuinely feels crazy. The problem, Otherwise, it means either there's ", "translatedText": "8 بٹس اس لحاظ سے بے کار ہیں کہ ان کا بقیہ پیغام سے مکمل طور پر تعین کیا جاتا ہے، لیکن یہ پیغام کو مجموعی طور پر کاپی کرنے سے کہیں زیادہ ہوشیار طریقے سے ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost. ", + "input": "or the error is somewhere on the left half. ", "translatedText": "ٹھیک ہے، تقریبا. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0. ", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you ", "translatedText": "ٹھیک ہے، تو یہاں ایک مسئلہ یہ ہے کہ اگر چار برابری چیک میں سے کوئی بھی خرابی کا پتہ نہیں لگاتا، یعنی 8 بٹس کے خصوصی طور پر منتخب کردہ سب سیٹس میں برابری ہوتی ہے، بالکل اسی طرح جیسے بھیجنے والے کا ارادہ ہے، تو اس کا مطلب ہے کہ کوئی غلطی نہیں تھی۔، یا یہ ہمیں پوزیشن 0 تک محدود کر دیتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 915.54 }, { - "input": "Here's how it works. ", + "input": "er block. But it also might simply mean there's no error at ", "translatedText": "یہ کیسے کام کرتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -992,7 +992,7 @@ "end": 952.7 }, { - "input": "Isn't that clever? ", + "input": "s today. There are like half a dozen times throughout this book that he refer ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them. ", + "input": "ences the Louis Pasteur quote, luck favors a prepared mind. Cl In this case, it looks like the sender needs to turn that bit 8 on in order to give the group even parity. Part of the reason that clever ideas look deceptively ", "translatedText": "کیا یہ ہوشیار نہیں ہے؟ اگرچہ ہم ان 2 بٹ کی غلطیوں کو درست نہیں کر سکتے ہیں، صرف ایک چھوٹی سی پریشان کن 0 ویں بٹ کو کام پر ڈال کر، یہ ہمیں ان کا پتہ لگانے دیتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 965.22 }, { - "input": "Technically speaking, you now have a full description of what a Hamming code does, at least for the example of a 16-bit block. ", + "input": "nal result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the transmission there's an error at, say, position 3. ", "translatedText": "تکنیکی طور پر، اب آپ کے پاس ہیمنگ کوڈ کی مکمل تفصیل ہے، کم از کم 16 بٹ بلاک کی مثال کے طور پر۔", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it! ", + "input": "And that lets the receiver pinpoint the error up to the first row, which necessarily means position ", "translatedText": "آگے بڑھو، اصل میں یہ کرو! ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block. ", + "input": "3, so they can fix the error. You might enjoy taking a moment to convince yourself that the answers to these four ", "translatedText": "آئیے توقف کریں اور اس بلاک کو اکٹھا کرنے کی کوشش کریں۔", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 1007.02 }, { - "input": "Okay, you ready? ", + "input": "questions really will always let you pin down ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0. ", + "input": "n between these questions and binary counting. And if you do, again let me emphasize, pause, try for yourself to draw the connection before I spo ", "translatedText": "آپ کو اس گروپ کے پاس برابر برابری کی ضرورت ہے، جو یہ پہلے سے ہی کرتا ہے، لہذا آپ کو اس برابری بٹ کو پوزیشن 1 میں 0 پر سیٹ کرنا چاہیے تھا۔", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1037.88 }, { - "input": "The group after that starts with an odd parity, so again you should have set its parity bit to 1. ", + "input": "ts affected, well, you can just try it. Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, ", "translatedText": "اس کے بعد گروپ ایک عجیب برابری کے ساتھ شروع ہوتا ہے، لہذا آپ کو دوبارہ اس کی برابری بٹ کو 1 پر سیٹ کرنا چاہیے تھا۔", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1079.78 }, { - "input": "What I'm going to do is change either 0, 1, or 2 of the bits in that block, and then ask you to figure out what it is that I did. ", + "input": "y eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set the app ", "translatedText": "میں کیا کرنے جا رہا ہوں یا تو اس بلاک میں بٹس میں سے 0، 1، یا 2 کو تبدیل کریں، اور پھر آپ سے پوچھیں کہ میں نے کیا کیا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1107.91 }, { - "input": "The next check gives us an odd number, telling us both that there's at least one error, and narrowing us down into this specific column. ", + "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever you want it to be, carrying whatever message or data you want. ", "translatedText": "اگلا چیک ہمیں ایک طاق نمبر دیتا ہے، جو ہم دونوں کو بتاتا ہے کہ کم از کم ایک خرابی ہے، اور ہمیں اس مخصوص کالم تک محدود کر دیتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off. ", + "input": ". And still, for so little given up, you would be able to identify and fix any single bit error. Well, almost. Okay, so the one ", "translatedText": "اگر یہ تین یا اس سے زیادہ ہے تو، تمام شرطیں بند ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1240,7 +1240,7 @@ "end": 1163.17 }, { - "input": "You see, what I haven't told you yet is just how elegant this algorithm really is, how simple it is to get a machine to point to the position of an error, how to systematically scale it, and how we can frame all of this as one single operation rather than multiple separate parity checks. ", + "input": "You see, with four yes or no questions, we have 16 possible outcomes for our parity checks, and at first that feels perfect for pinpointing one out of 16 positions in the block, but you also need to communicate a 17th outcome, the no error condition. The solution here is actually pretty simple. Just forget about that zeroth bit entirely. So when we do our four parity checks and ", "translatedText": "آپ نے دیکھا کہ میں نے آپ کو ابھی تک جو کچھ نہیں بتایا وہ یہ ہے کہ یہ الگورتھم واقعی کتنا خوبصورت ہے، کسی مشین کو غلطی کی پوزیشن کی طرف اشارہ کرنے کے لیے حاصل کرنا کتنا آسان ہے، اسے منظم طریقے سے کیسے پیمانہ کیا جائے، اور ہم کس طرح ان تمام چیزوں کو ترتیب دے سکتے ہیں۔یہ متعدد الگ الگ برابری چیک کے بجائے ایک واحد آپریشن کے طور پر ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/hamming-codes/vietnamese/sentence_translations.json b/2020/hamming-codes/vietnamese/sentence_translations.json index 9e7f3ac15..83d3e8e8e 100644 --- a/2020/hamming-codes/vietnamese/sentence_translations.json +++ b/2020/hamming-codes/vietnamese/sentence_translations.json @@ -40,7 +40,7 @@ "end": 50.38 }, { - "input": "And a simple strategy to correct any bit that gets flipped would be to store three copies of each bit.", + "input": "nstead of yeses and nos, it literally spells out the position of the error in binary. For example, the number 7 in binary looks like 0111, essentially saying that it's 4 plus 2 plus 1. And notic", "translatedText": "Và một chiến lược đơn giản để sửa bất kỳ bit nào bị đảo lộn là lưu trữ ba bản sao của mỗi bit.", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 84.02 }, { - "input": "For example, using the method you'll learn about this video, you could store your data in 256-bit blocks, where each block uses 9 bits, 9!", + "input": "hat means is using two thirds of your space for redundancy. And even then, for all of that space given up, there's no strong guarantee about what happens if more than one bit gets flipped. The much more interesting question is how", "translatedText": "Ví dụ: bằng cách sử dụng phương pháp bạn sẽ tìm hiểu về video này, bạn có thể lưu trữ dữ liệu của mình trong các khối 256 bit, trong đó mỗi khối sử dụng 9 bit, 9!", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 93.36 }, { - "input": "to act as a kind of redundancy, and the other 247 bits are free to carry whatever meaningful message or data you want.", + "input": "to make it so that errors can be corrected while giving up as little space as possible. c for implementing the whole scheme in hardware shockingly simple. Now if you wan", "translatedText": "hoạt động như một loại dự phòng và 247 bit còn lại có thể tự do mang bất kỳ thông điệp hoặc dữ liệu có ý nghĩa nào bạn muốn.", "model": "google_nmt", "n_reviews": 0, @@ -128,7 +128,7 @@ "end": 126.9 }, { - "input": "Methods that let you correct errors like this are known, reasonably enough, as error correction codes.", + "input": "the case that if any bit gets flipped here, just by looking at this block and nothing more, a machine will be able to iden", "translatedText": "Các phương pháp cho phép bạn sửa các lỗi như thế này được biết đến một cách hợp lý dưới dạng mã sửa lỗi.", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 132.9 }, { - "input": "For the better part of the last century, this field has been a really rich source of surprisingly deep math that gets incorporated into devices we use every day.", + "input": "tify that there was an error and precisely where it was so that it knows how to correct it. And honestly, that feels like magic. And for this particular scheme, if two bits get flipped, the machine will at least be able to detect", "translatedText": "Trong phần lớn thế kỷ trước, lĩnh vực này thực sự là một nguồn toán học sâu sắc đáng kinh ngạc được tích hợp vào các thiết bị chúng ta sử dụng hàng ngày.", "model": "google_nmt", "n_reviews": 0, @@ -216,7 +216,7 @@ "end": 224.06 }, { - "input": "The story begins in the 1940s, when a young Richard Hamming was working for Bell Labs, and some of his work involved using a very big expensive punch card computer that he had only limited access to.", + "input": "bits to describe 64 spots, then each of those bits gives you one of the parity groups that we need to check. So when you feel like you see where it's going at some point, take that moment to pause, actively predict what the scheme is going to be before I tell you.", "translatedText": "Câu chuyện bắt đầu vào những năm 1940, khi chàng trai trẻ Richard Hamming đang làm việc cho Bell Labs, và một số công việc của anh liên quan đến việc sử dụng một chiếc máy tính thẻ đục lỗ rất đắt tiền mà anh chỉ có quyền truy cập hạn chế.", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 242.4 }, { - "input": "Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code.", + "input": "conjunction with this one showing you how to actually implement Hamming codes on breadboards, which is extremely satisfying. You should know, Hamming c", "translatedText": "Sự thất vọng là lò thử thách của phát minh, anh ta chán nản đến mức phát minh ra mã sửa lỗi đầu tiên trên thế giới.", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 299.62 }, { - "input": "It also might give you a little hint about how this scales for larger blocks.", + "input": "dered valid messages. As an analogy, think about correctly spelled words versus incorrectly spelled words.", "translatedText": "Nó cũng có thể cung cấp cho bạn một gợi ý nhỏ về cách tỷ lệ này đối với các khối lớn hơn.", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 303.54 }, { - "input": "Also technically it ends up being only 11 bits of data, you'll find there's a mild nuance for what goes on at position 0, but don't worry about that for now.", + "input": "also see this in larger examples, where no matter how big you get, each parity bit conveniently touches only one of the groups. Once you understand that these parity checks that we've focused so much of our time", "translatedText": "Ngoài ra, về mặt kỹ thuật, nó chỉ có 11 bit dữ liệu, bạn sẽ thấy có một sắc thái nhẹ cho những gì diễn ra ở vị trí 0, nhưng hiện tại đừng lo lắng về điều đó.", "model": "google_nmt", "n_reviews": 0, @@ -312,7 +312,7 @@ "end": 313.26 }, { - "input": "Like any error correction algorithm, this will involve two players, a sender who's responsible for setting these 4 special bits, and a receiver who's responsible for performing some kind of check and correcting the errors.", + "input": "on are nothing more than a clever way to spell out the position of an error in binary, then we can draw a connection with a differ ent way to think about hamming codes, one that is arguably a lot simpler and more elegant, and which can basically be written down with a single line of code. It", "translatedText": "Giống như bất kỳ thuật toán sửa lỗi nào, điều này sẽ có sự tham gia của hai người chơi, một người gửi chịu trách nhiệm thiết lập 4 bit đặc biệt này và một người nhận chịu trách nhiệm thực hiện một số loại kiểm tra và sửa lỗi.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 334.74 }, { - "input": "After all, storing data is the same thing as sending a message just from the past to the future instead of from one place to another.", + "input": "R of two bits, it's going to return a 1 if either one of those bits is turned on, but not if both are turned on or off. Phrased differently And the", "translatedText": "Xét cho cùng, việc lưu trữ dữ liệu cũng giống như gửi một tin nhắn từ quá khứ đến tương lai thay vì từ nơi này đến nơi khác.", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 341.68 }, { - "input": "So that's the setup, but before we can dive in we need to talk about a related idea which was fresh on Hamming's mind in the time of his discovery, a method which lets you detect any single bit errors, but not to correct them, known in the business as a parity check.", + "input": "programs he kept putting through it kept failing, because every now and then a bit would get misread. Frustration being the crucible of invention, he got so fed up that he invented the world's first error correction code. There are many different ways to frame Hamming codes, but as a first pass we", "translatedText": "Đó là thiết lập, nhưng trước khi đi sâu vào chúng ta cần nói về một ý tưởng liên quan mới mẻ trong đầu Hamming vào thời điểm ông phát hiện ra, một phương pháp cho phép bạn phát hiện bất kỳ lỗi bit nào nhưng không sửa chúng, đã biết trong kinh doanh như một sự kiểm tra tính chẵn lẻ.", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 356.3 }, { - "input": "For a parity check, we separate out only one single bit that the sender is responsible for tuning, and the rest are free to carry a message.", + "input": "'re going to go through it the way Hamming himself thought about them. Let's use an example that's simple, but not too simple, a block of 16 bits. We'll number t", "translatedText": "Để kiểm tra tính chẵn lẻ, chúng tôi chỉ tách ra một bit duy nhất mà người gửi chịu trách nhiệm điều chỉnh và phần còn lại được tự do mang tin nhắn.", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 386.42 }, { - "input": "This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of information.", + "input": "ch of separate groups, like so with the columns, all in one fell swoop. The word redundant here doesn't simply mean copy, after all, those 4 bits don't give us enough room to blindly copy the data. Instead, they'll need to be a much more nuanced and clever kind of redundancy,", "translatedText": "Điều này khá đơn giản, có vẻ đơn giản, nhưng đó là một cách cực kỳ tinh tế để chắt lọc ý tưởng về sự thay đổi ở bất kỳ đâu trong thông điệp để được phản ánh trong một thông tin duy nhất.", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 396.78 }, { - "input": "Notice if any bit of this message gets flipped, either from 0 to 1 or 1 to 0, it changes the total count of 1s from being even to being odd.", + "input": "not adding any new information, but adding resilience. s, so it's effectively counting how many highlighted positions came from the first parity group. Does that make sense? Likewise, the next column counts how many posi", "translatedText": "Lưu ý nếu bất kỳ bit nào của thông báo này bị đảo ngược, từ 0 thành 1 hoặc 1 thành 0, nó sẽ thay đổi tổng số 1 từ chẵn thành lẻ.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 406.54 }, { - "input": "So if you're the receiver, you look at this message, and you see an odd number of 1s, you can know for sure that some error has occurred, even though you might have no idea where it was.", + "input": "tions are in the second parity group, the positions whose second to last bit is a 1, and which a re also highlighted, and so on. It's really just a small shift in perspective on the same thing we've been doing. And so you know where it g", "translatedText": "Vì vậy, nếu bạn là người nhận, bạn nhìn vào tin nhắn này và thấy số lẻ là 1, bạn có thể biết chắc chắn rằng đã xảy ra lỗi nào đó, mặc dù bạn có thể không biết lỗi đó ở đâu.", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 430.72 }, { - "input": "And this special bit that the sender uses to control the parity is called the parity bit.", + "input": "Now once we have it like this, this gives us a really nice way to think about why these four resulting bits at the bottom directly spell out the positio Like", "translatedText": "Và bit đặc biệt này mà người gửi sử dụng để kiểm soát tính chẵn lẻ được gọi là bit chẵn lẻ.", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 435.52 }, { - "input": "And actually, we should be clear, if the receiver sees an odd parity, it doesn't necessarily mean there was just one error, there might have been 3 errors, or 5, or any other odd number, but they can know for sure that it wasn't 0.", + "input": "any error correction algorithm, this will involve two players, a sender, who's responsible for setting these 4 special bits, and then a receiver, who's responsible for performing some kind of check and then correcting the errors. Of course, the words sender and receiver really refer to machines or software that's doing chec", "translatedText": "Và trên thực tế, chúng ta nên nói rõ, nếu người nhận nhìn thấy một số chẵn lẻ lẻ, điều đó không nhất thiết có nghĩa là chỉ có một lỗi, có thể có 3 lỗi, hoặc 5 lỗi hoặc bất kỳ số lẻ nào khác, nhưng họ có thể biết chắc chắn. rằng nó không phải là 0.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 507.16 }, { - "input": "For example, as Hamming was searching for a way to identify where an error happened, not just that it happened, his key insight was that if you apply some parity checks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of any single bit error.", + "input": "he sender needs to flip that special bit to be a 1, making the count even. But if the block had already started off with an even number of 1s, then this special bit would have been kept at a 0. This is pretty simple, deceptively simple, but it's an incredibly elegant way to distill the idea of change anywhere in a message to be reflected in a single bit of", "translatedText": "Ví dụ, khi Hamming đang tìm cách xác định lỗi đã xảy ra ở đâu, không chỉ lỗi xảy ra, hiểu biết sâu sắc quan trọng của ông là nếu bạn áp dụng một số kiểm tra tính chẵn lẻ không phải cho toàn bộ thông báo mà cho một số tập hợp con được chọn cẩn thận, bạn có thể hỏi một loạt câu hỏi tinh tế hơn nhằm xác định vị trí của bất kỳ lỗi bit nào.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 525.94 }, { - "input": "The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half.", + "input": "information. ctice this would be something we're receiving from a sender, and instead of being random it would be carrying 11 data bits together with 5 parity bits. If I call the function enumerateBits, what it does is", "translatedText": "Cảm giác tổng thể giống như chơi một trò chơi gồm 20 câu hỏi, đặt các câu hỏi có hoặc không để cắt đôi không gian khả năng.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 533.38 }, { - "input": "For example, let's say we do a parity check just on these 8 bits, all of the odd numbered positions.", + "input": "pair together each of those bits with a corresponding index, in this case running from 0 up to 15. So if you're the receiver, you look at this message, and you see an odd number of 1s, you c", "translatedText": "Ví dụ: giả sử chúng tôi thực hiện kiểm tra tính chẵn lẻ chỉ trên 8 bit này, tất cả các vị trí được đánh số lẻ.", "model": "google_nmt", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 539.38 }, { - "input": "Then if an error is detected, it gives the receiver a little more information about where specifically the error is, namely that it's in an odd position.", + "input": "an know for sure that some error has occurred, even though you might have no idea where it was. In the jargon, whether a group of bits has an even or an odd number of", "translatedText": "Sau đó, nếu phát hiện thấy lỗi, nó sẽ cung cấp cho người nhận thêm một chút thông tin về lỗi cụ thể ở đâu, cụ thể là lỗi ở vị trí kỳ lạ.", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 586.98 }, { - "input": "This is only 1 out of 4 parity checks that we'll do.", + "input": "that it wasn't 0. On the other hand, if there had been 2 errors, or any even number of errors, that final count o", "translatedText": "Đây chỉ là 1 trong 4 lần kiểm tra tính chẵn lẻ mà chúng tôi sẽ thực hiện.", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 615.38 }, { - "input": "Otherwise it means either there's no error, or the error is somewhere on the left half.", + "input": "ary representation 1001. We won't do it here, but you could write a function where the sender uses that binary representation to set the four pari", "translatedText": "Ngược lại, điều đó có nghĩa là không có lỗi hoặc lỗi nằm ở đâu đó ở nửa bên trái.", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 620.58 }, { - "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block.", + "input": "ty bits as needed, ultimately getting this block to a state where running this line of code on the full list of bits returns a 0. This would be considered a well-prepared block.", "translatedText": "Hoặc tôi đoán có thể có hai lỗi, nhưng hiện tại chúng ta giả định rằng có nhiều nhất một lỗi trong toàn bộ khối.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 626.5 }, { - "input": "Things break down completely for more than that.", + "input": "What's cool is that if we toggle any one of the bits in this list, simulating a random error from noise, then if", "translatedText": "Mọi thứ bị phá vỡ hoàn toàn vì nhiều hơn thế.", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 635.1 }, { - "input": "Let's say you detect an error among the odd columns, and among the right half.", + "input": "n't that neat? You could get this block from out of the blue, run this single line on it, and it'll automatically spit out the position of an error, or a 0 if there wasn't any.", "translatedText": "Giả sử bạn phát hiện lỗi giữa các cột lẻ và giữa nửa bên phải.", "model": "google_nmt", "n_reviews": 0, @@ -648,7 +648,7 @@ "end": 656.02 }, { - "input": "And if neither of those two parity checks detects anything, it means the only place that an error could be is in that leftmost column.", + "input": "maximum number of errors, or maybe to reduce the probability of a false positive like this. a parity check to detect 2-bit errors, but the idea is that almost all of th", "translatedText": "Và nếu cả hai lần kiểm tra chẵn lẻ đó đều không phát hiện ra điều gì, điều đó có nghĩa là nơi duy nhất có thể xảy ra lỗi là ở cột ngoài cùng bên trái.", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 663.12 }, { - "input": "But it also might simply mean there's no error at all.", + "input": "e core logic from our scheme comes down to a single XOR reduction. Now, depending on your comfo", "translatedText": "Nhưng nó cũng có thể đơn giản có nghĩa là không có lỗi nào cả.", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 673.64 }, { - "input": "We do basically the same thing but for the rows.", + "input": "in general, you may either find this perspective a little bit confusing, or so much m For example, as Hamming was searching for a way to identify w", "translatedText": "Về cơ bản chúng tôi làm điều tương tự nhưng đối với các hàng.", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 676.14 }, { - "input": "There's going to be a parity check on the odd rows, using position 4 as a parity bit.", + "input": "here an error happened, not just that it happened, his key insight was that if you apply some parity chec", "translatedText": "Sẽ có kiểm tra tính chẵn lẻ trên các hàng lẻ, sử dụng vị trí 4 làm bit chẵn lẻ.", "model": "google_nmt", "n_reviews": 0, @@ -696,7 +696,7 @@ "end": 680.9 }, { - "input": "So in this example that group already has an even parity, so bit 4 would be set to a 0.", + "input": "ks not to the full message, but to certain carefully selected subsets, you can ask a more refined series of questions that pin down the location of", "translatedText": "Vì vậy, trong ví dụ này, nhóm đó đã có tính chẵn lẻ chẵn, vì vậy bit 4 sẽ được đặt thành 0.", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 685.82 }, { - "input": "And finally there's a parity check on the bottom two rows, using position 8 as a parity bit.", + "input": "any single bit error. The overall feeling is a bit like playing a game of 20 questions, asking yes or no queries that chop the space of possibilities in half. of the size", "translatedText": "Và cuối cùng là kiểm tra tính chẵn lẻ ở hai hàng dưới cùng, sử dụng vị trí 8 làm bit chẵn lẻ.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 715.56 }, { - "input": "But it doesn't affect the third group, and it doesn't affect the fourth group.", + "input": "most people's knee-jerk reaction Then, if an error is detected, it gives the receiver a little more information about where specifically the error is, na", "translatedText": "Nhưng nó không ảnh hưởng đến nhóm thứ ba, và nó không ảnh hưởng đến nhóm thứ tư.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 756.06 }, { - "input": "Take a moment to think about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", + "input": "'s done in conjunction with other well-chosen checks, it counter-intuitively gives us something a lot more powerful. To actually set up that parity check, remember, it requires earmarking some special bit that has control for the parity of that full group.", "translatedText": "Hãy dành một chút thời gian để suy nghĩ xem làm thế nào bất kỳ lỗi nào trong số bốn phần đặc biệt này sẽ được theo dõi giống như bất kỳ lỗi nào khác, với cùng một nhóm bốn câu hỏi.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 777.82 }, { - "input": "You might also enjoy anticipating how this scales.", + "input": "ntly odd, so the sender is responsible for toggling that parity bit, and now it's even.", "translatedText": "Bạn cũng có thể thích dự đoán quy mô này như thế nào.", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 809.36 }, { - "input": "Hopefully this sketch is enough to appreciate the efficiency of what we're developing here.", + "input": "od leaving that bit number 2 unchanged. d if you take that to an extreme, you could have a block with, say, a million bits, where you would quite literally be playing 20 qu", "translatedText": "Hy vọng bản phác thảo này đủ để đánh giá cao tính hiệu quả của những gì chúng tôi đang phát triển ở đây.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 838.38 }, { - "input": "Well, almost.", + "input": "or is somewhere on the left half.", "translatedText": "Vâng, gần như vậy.", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 840.4 }, { - "input": "Okay, so the one problem here is that if none of the four parity checks detect an error, meaning that the specially selected subsets of 8 bits all have even parities, just like the sender intended, then it either means there was no error at all, or it narrows us down into position 0.", + "input": "Or I guess there could have been two errors, but for right now we're going to assume that there's at most one error in the entire block. Things break down completely for more than that. Here, before we look at the next two checks, take a moment to think about what these first two allow us to do when you", "translatedText": "Được rồi, vấn đề ở đây là nếu không có kiểm tra chẵn lẻ nào trong số bốn kiểm tra chẵn lẻ phát hiện ra lỗi, nghĩa là các tập hợp con 8 bit được chọn đặc biệt đều có các số chẵn lẻ, giống như dự định của người gửi, thì điều đó có nghĩa là không có lỗi nào cả , hoặc nó thu hẹp chúng ta xuống vị trí 0.", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 893.22 }, { - "input": "And with that, we now have what people in the business would refer to as a 15-11 Hamming code.", + "input": "olumn. like the much more commonly used Reed-Solomon algorithm, which handles burst errors particularly", "translatedText": "Và cùng với đó, giờ đây chúng ta có thứ mà mọi người trong ngành gọi là mã Hamming 15-11.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 924.94 }, { - "input": "Now, if there's a single bit error, then the parity of the full block toggles to be odd, but we would catch that anyway thanks to the four error-correcting checks.", + "input": "nch of other analysis and asked, okay, what is the most eff icient I could conceivably be about this? He was also candid about how important it was that parity che So in this", "translatedText": "Bây giờ, nếu có một lỗi bit nào đó thì tính chẵn lẻ của toàn bộ khối sẽ chuyển thành số lẻ, nhưng dù sao thì chúng ta cũng sẽ phát hiện được điều đó nhờ bốn bước kiểm tra sửa lỗi.", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 954.0 }, { - "input": "Even though we can't correct those 2-bit errors, just by putting that one little bothersome 0th bit back to work, it lets us detect them.", + "input": "deas look deceptively easy is that we only ever see the final result, cleaning up what was messy, never mentioning all of the wrong As an example, imagine that during the tra", "translatedText": "Mặc dù chúng tôi không thể sửa các lỗi 2 bit đó, nhưng chỉ cần đưa bit 0 khó chịu đó hoạt động trở lại, nó sẽ cho phép chúng tôi phát hiện ra chúng.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 961.26 }, { - "input": "This is pretty standard, it's known as an extended Hamming code.", + "input": "nsmission there's an error at, say, position 3. Well, this affects the first parity group, and it also affects the s", "translatedText": "Đây là mã khá chuẩn, nó được gọi là mã Hamming mở rộng.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 972.88 }, { - "input": "But I think you'll find it more satisfying to check your understanding and solidify everything up to this point by doing one full example from start to finish yourself.", + "input": "mn. But it doesn't affect the third group, and it doesn't affect the fourth group. And that lets the receiver pinpoint the error up to the first row, which necessarily means position 3, so they can fix the error. You might enjoy taking a moment to convince yoursel", "translatedText": "Nhưng tôi nghĩ bạn sẽ thấy hài lòng hơn khi kiểm tra sự hiểu biết của mình và củng cố mọi thứ cho đến thời điểm này bằng cách tự mình làm một ví dụ đầy đủ từ đầu đến cuối.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 994.66 }, { - "input": "Each chunk is going to get packaged into an error-resistant 16-bit block.", + "input": "t, the astute among you might even notice a connection between these questions and binary counting. And if you do,", "translatedText": "Mỗi đoạn sẽ được đóng gói thành một khối 16 bit chống lỗi.", "model": "google_nmt", "n_reviews": 0, @@ -1056,7 +1056,7 @@ "end": 999.76 }, { - "input": "So let's take this one as an example and actually work it out.", + "input": "again let me emphasize, pause, try for yourself to draw the connection before I spoil it. If you're wondering what happens if a parity bit itself g", "translatedText": "Vì vậy, hãy lấy ví dụ này làm ví dụ và thực sự giải quyết nó.", "model": "google_nmt", "n_reviews": 0, @@ -1064,7 +1064,7 @@ "end": 1003.22 }, { - "input": "Go ahead, actually do it!", + "input": "ets affected, well, you can just try it. Take a moment to thi", "translatedText": "Hãy tiếp tục, thực sự làm điều đó!", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 1004.94 }, { - "input": "Let's pause and try putting together this block.", + "input": "nk about how any error among these four special bits is going to be tracked down just like any other, with the same group of four questions.", "translatedText": "Hãy tạm dừng và thử ghép khối này lại với nhau.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1023.32 }, { - "input": "You need this group to have an even parity, which it already does, so you should have set that parity bit in position 1 to be a 0.", + "input": "ll is something that naturally falls out of the scheme as a byproduct. You might also enjoy anticipating how this scales. If we used a block of size 256 bits, for example, in order to pin down a location, you need on", "translatedText": "Bạn cần nhóm này có tính chẵn lẻ, điều này đã có sẵn, vì vậy bạn nên đặt bit chẵn lẻ đó ở vị trí 1 là 0.", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 1032.34 }, { - "input": "The next group starts off with an odd parity, so you should have set its parity bit to be 1.", + "input": "ly eight yes or no questions to binary search your way down to some specific spot. And remember, each question requires giving up only a single bit to set", "translatedText": "Nhóm tiếp theo bắt đầu với số chẵn lẻ lẻ, vì vậy bạn nên đặt bit chẵn lẻ của nó là 1.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1060.32 }, { - "input": "So as this block is sent off, the parity of the four special subsets and the block as a whole will all be even, or 0.", + "input": "ully this sketch is enough to appreciate the efficiency of what we're developing here. Everything except for those eight highlighted parity bits can be whatever yo", "translatedText": "Vì vậy, khi khối này được gửi đi, tính chẵn lẻ của bốn tập hợp con đặc biệt và toàn bộ khối sẽ là số chẵn hoặc bằng 0.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1087.74 }, { - "input": "So again, pause and try working it out.", + "input": "copying the message as a whole. And still, for so little given up, you would be able to identify and fix any single bit error. Well, almos", "translatedText": "Vì vậy, một lần nữa, hãy tạm dừng và thử giải quyết nó.", "model": "google_nmt", "n_reviews": 0, @@ -1216,7 +1216,7 @@ "end": 1137.53 }, { - "input": "If it's three or more, all bets are off.", + "input": "outcomes for our parity checks, and at first that", "translatedText": "Nếu là ba hoặc nhiều hơn, tất cả cược sẽ bị hủy.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/arabic/sentence_translations.json b/2020/ldm-complex-numbers/arabic/sentence_translations.json index ced9173ab..6ca075495 100644 --- a/2020/ldm-complex-numbers/arabic/sentence_translations.json +++ b/2020/ldm-complex-numbers/arabic/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "أود أن أقول أنه إذا كان شيئًا مفيدًا بالفعل في أحد التطبيقات، فهو حقيقي مثل الكلمات، أليس كذلك؟ لن تصادف أبدًا كلمة مجردة مثل السعادة هناك، ولكنها تحتوي على نوع من الواقع في أذهاننا، وأشياء مثل الجذر التربيعي لاثنين، والذي لا يمكنك التعبير عنه ككسر، أو أشياء مثل الجذر التربيعي للسالب الذي لا يظهر بين الأعداد العادية الحقيقية، كما تعلمون، حتى لو بدت مختلفة قليلاً. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "أتذكر عندما كنت في المدرسة وتعلمنا صيغ الجمع هذه، إذا كنت تريد معرفة جيب التمام لمجموع زاويتين مختلفتين، كما تعلم، فهو نوع طويل من حيث جيب التمام وجيب الزاويتين الأصليتين ، هناك علامة الطرح التي من شأنها دائمًا أن تتعثر الأشخاص، إذا فعلت الشيء نفسه بالنسبة للعلامة، فإنها تبدو مشابهة ولكن هناك علامة زائد، وبدلاً من وجود cos-cos لديك cos-sin، إنه شيء عرضة للخطأ للغاية إذا كنت تحاول فقط حفظها كما هي. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "ومع ذلك، إذا تعاملت مع الأمر بأرقام معقدة، فلن يكون هذا أقل عرضة للخطأ فحسب، بل له معنى جميل جدًا وسيسقط على الفور. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "لذا، حتى لو كنت لا تؤمن بالضرورة بحقيقة الجذر التربيعي لسالب 1، عليك على الأقل أن تعترف أنه من المثير للاهتمام أنه يمكن أن يجعل أجزاء أخرى من الرياضيات مفيدة، وأن أجزاء الرياضيات الأخرى أكثر قليلاً مفهومة جدا. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "أي رقم تربيعه، إذا كان موجبًا، فسيظل موجبًا. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "لن أحصل على أي شيء سلبي أبدًا. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "ومع ذلك، إذا جاء عالم رياضيات وقال، لا، لا، إنه موجود، فقد قمنا بتعريفه بحيث يكون هذا هو الحال. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "أعتقد أن رد الفعل الآخر الذي يمكن أن يحدثه شخص ما هو، انتظر لحظة، هل يمكنك فعل ذلك؟ عندما تكون لديك مشكلة لا يمكنك حلها، يمكنك فقط أن تقول، لقد حددت الأشياء بحيث أصبح لدينا الآن حل سحري. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "لذا، إذا كنت غير مرتاح لهذا الأمر، فأنت بالتأكيد لست وحدك. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "في الواقع، صاغ رينيه ديكارت المصطلح الخيالي لهذه الأرقام باعتباره مهينًا. ", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "ومن ثم تمسكنا بذلك كتقليد وما زلنا نسميها أرقامًا خيالية، وهو أمر سخيف حقًا. ", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "الشيء الغريب الثاني الذي تفعله عندما تبدأ بالحديث عن الأعداد المركبة هو القول، ليس هناك مثل هذا العدد i، ولكننا سنعطيه مكانًا. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "في البداية، دعونا نتحدث فقط عن كيف أنه إذا كنت تضيف أرقامًا ثنائية الأبعاد مثل هذه، فإن القواعد تكون واضحة جدًا وتعمل بشكل أساسي مثل المتجهات، لأي منكم قد يكون على دراية بالمتجهات. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "إذا أخذت ذلك على محمل الجد واتبعته، نأمل أن تساعد حقيقة أنه أصبح مفيدًا في تبرير سبب قيامنا بأي من هذا. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "حسنًا، يبدو أن هناك أ، هناك ذهابًا وإيابًا بين الإجابات f وd، لذا f هي جميعها، مما يعني أن كل هذه الإجابات يجب اعتبارها حقيقية. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "ومن المثير للاهتمام، d هو الذي يقول أنه يجب عليك أن تفكر في 2 الجذر التربيعي لـ 2 وسالب 1، ولكن ليس اللانهاية، لذلك هناك مجموعة جيدة منكم الذين يرفضون اعتبار اللانهاية حقيقية، لكنهم مرتاحون جدًا لـ الجذر التربيعي لسالب 1، هذا رائع. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "حسنًا، يبدو أن لدينا مجموعة من الأشخاص الذين يشعرون بالارتياح تجاه الرقم السالب 1، ومجموعة كبيرة لا يشعرون بالارتياح مع اللانهاية، وهذا موضوع ليوم آخر، لا تقلق بشأنه، ثم هناك عدد من الأشخاص الذين هم نوعًا ما في تلك الأرضية الوسطى التي ربما لا تكون مرتاحة للغاية لفكرة أن سالب 1 قد يكون حقيقيًا. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "لذا، بالنسبة لسؤالنا الرياضي الأول، كنوع من الإحماء، أريد فقط أن أطلب منك إضافة هذين السؤالين. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "قبل أن أعلمك كيفية إضافتها، قم بالتخمين حول كيفية عملها، وآمل أن يكون الأمر واضحًا ومباشرًا، فالإضافة هي في الواقع الجزء الأقل إثارة للاهتمام في هذا، ولكن من الجيد معرفة متى أنت تتعلم عن الأعداد المركبة. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "إذا كنت تحرك أربع وحدات إلى اليمين ثم وحدة واحدة إلى الأعلى، وتريد إضافة فكرة تحريك وحدتين إلى اليسار ثم وحدتين إلى الأعلى، حسنًا، ما عليك سوى القيام بكل وحدة من هذه الوحدات في المرة الواحدة. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "الجزء الحقيقي سيكون تلك الأربعة على اليمين، ثم ناقص اثنين على اليسار. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "فهل هذا واحد أنا زائد اثنين أنا. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "مقدمة بسيطة وجميلة هنا لا يوجد في عملية الإضافة أي شيء معقد، وهو أمر رائع. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "لذا، بالنسبة للمتجهات، ليس هناك حقًا أي فكرة عن ضربهم للحصول على متجهين مرة أخرى، على الأقل عندما نكون في المستوى ثنائي الأبعاد. ", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "وهو في الأساس، لنفترض أن لدي النقطة ثلاثة، اثنان. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "إذا كان لدي نوع من شبكة الإحداثيات وذهبت إلى النقطة ذات الإحداثي x ثلاثة والإحداثي y اثنين، ما هو دوران 90 درجة لهذا؟ إذا قمت بتدويرها 90 درجة ودعونا نقول عكس اتجاه عقارب الساعة. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "عكس عقارب الساعه. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "الآن الشيء الجميل في هذا هو أنه يمكننا ببساطة قلب ورقتنا لمعرفة ذلك. ", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "تقول حسنًا إذا بدأت عند ثلاثة، اثنان ثم قمت بالتدوير 90 درجة عكس اتجاه عقارب الساعة، يمكنني قراءة ذلك الآن على أنه سالب اثنين في اتجاه x ثم ثلاثة في اتجاه y، إذا كنت قد قمت بتدوير المستوى بأكمله بهذه الطريقة . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "إذن ما فعلناه هنا هو أننا أخذنا ثلاثة، اثنين ثم حولناه إلى سالب اثنين، ثلاثة. ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "وهذا سيكون دوران 90 درجة. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "إذا أخذت زوجًا من الأرقام فاصلة ب حسنًا ثم قلت إلى أين سيدور إذا قمت بتدويره 90 درجة، فسينتهي الأمر بتبديل الإحداثيات با ثم جعل الرقم الأول سالبًا. ", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "لذلك كان هذا دورانًا آخر بمقدار 90 درجة. ", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "حسنًا، ما حدث هنا هو أننا جعلنا كلا الإحداثيين سلبيين وهذا مطمئن لأنه إذا أخذت نقطة ما وجلست عند نقطة ab ثم قمت بتدويرها بمقدار 90 درجة، فسيكون هذا هو دوراني الأولي بمقدار 90 درجة ثم 90 درجة أخرى، وهذا هو مثل تعفن 180 درجة - أوه لا، لقد ارتكبت هذا الخطأ. ", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "سيكون هذا هو نفس الدوران بمقدار 180 درجة والذي يجب أن يبدو هكذا، مع تجاهل المتجه الآخر الذي رسمته والذي يأخذ كلا الإحداثيات ويجعلهما سالبًا سالبًا سالبًا b حسنًا، لذلك فهذا يطمئن هذه العملية التي تقوم بدوران 90 درجة في الواقع يتصرف كما تتوقعه. ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "أوه، انظر، لقد قدم الكثير من الأشخاص إجابات جيدة جدًا. ", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "حسنًا، يبدو أن أغلبكم حصل على الإجابة الصحيحة وهي 2 زائد 3i. ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "جيد جدا جيد جدا. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "أجاب البعض منكم بالنفي 2 3 والذي أعتقد أنه مجرد تبديل سواء كنت تأخذ 4 ناقص 2 أو 2 ناقص 4 لذا فهذا أمر مفهوم تمامًا. ", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "لدينا 2 زائد 3 والتي ربما تكون مجرد إسقاط i لذلك أعتقد أنه ربما يشبه إلى حد كبير الأخطاء البسيطة والإدخال وأنت تعلم أن هذا يحدث لنا جميعًا خاصة في الاختبارات، أحيانًا تعرف ما هي الإجابة الصحيحة ولكن بعد ذلك إذا نسيت رمزًا أو قمت بتبديل رمزين، فهذا جيد جدًا. ", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "كلمات ستال ستال، كلمات تعلمون أنها تخبرني أنها تعمل ومع ذلك، فإن التقدم للأمام بطيء جدًا، لذا تعلمون إذا لم أرد أن أتحدث معهم بكلمة صارمة، يمكنكم يا رفاق مخاطبتهم على تويتر أيضًا بنفس الطريقة المكان الذي نطرح فيه الأسئلة ونقول فقط مرحبًا kamineter، ألا يمكنك جعل الأسئلة المباشرة تعمل بشكل أفضل قليلاً بالنسبة لنا؟ حسنًا، أعتقد أننا وصلنا أخيرًا إلى هناك. ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "الجميع جاهزون؟ آها! ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "رائع! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "سؤال بسيط جدًا أريدك أن تأخذ الرقم i وأريدك أن تضربه في 3 زائد 2i وعلى الرغم من أنني لم أتحدث حقًا عن قواعد الضرب، ما يمكنني قوله هو التظاهر بأنه يعمل تمامًا كما يحدث من أجل الأعداد العادية لديك أشياء مثل خاصية التوزيع حيث يمكنك توزيعها ثم السمة المميزة لـ i هي فكرة أن i تربيع هو سالب وهذا هو الشيء الخاص الوحيد الذي تحتاج إلى معرفته عن ذلك بخلاف ذلك فقط تعامل معها كما لو كان رقمًا عاديًا حسنًا ثم تابع المنتج. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/bengali/sentence_translations.json b/2020/ldm-complex-numbers/bengali/sentence_translations.json index e0284e224..abfa87ce3 100644 --- a/2020/ldm-complex-numbers/bengali/sentence_translations.json +++ b/2020/ldm-complex-numbers/bengali/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "আমি বলব যে যদি এটি এমন কিছু হয় যা আসলে একটি অ্যাপ্লিকেশনে দরকারী, তবে এটি শব্দের মতো বাস্তব, তাই না? ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "আপনি সেখানে সুখের মতো একটি বিমূর্ত শব্দের মধ্যে কখনই দৌড়াতে যাচ্ছেন না, তবে এটি আমাদের মনে এক ধরণের বাস্তবতা রয়েছে এবং দুটির বর্গমূলের মতো জিনিস, যা আপনি ভগ্নাংশ হিসাবে প্রকাশ করতে পারবেন না, বা এর মতো জিনিসগুলি ঋণাত্মক একটির বর্গমূল যা প্রকৃত স্বাভাবিক সংখ্যার মধ্যে দেখা যায় না, আপনি জানেন, যদিও সেগুলি একটু ভিন্ন মনে হতে পারে।", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "আমার মনে আছে যখন আমি স্কুলে ছিলাম এবং আমরা এই যোগ সূত্রগুলি শিখেছিলাম, আপনি যদি দুটি ভিন্ন কোণের যোগফলের কোসাইন জানতে চান, আপনি জানেন, এটি আসল দুটি কোণের কোসাইন এবং সাইনের পরিপ্রেক্ষিতে এই ধরনের দীর্ঘ জিনিস।, এই বিয়োগ চিহ্নটি সর্বদা লোকেদের ট্রিপ করবে, আপনি যদি চিহ্নটির জন্য একই করেন তবে এটি একই রকম দেখায় তবে একটি প্লাস চিহ্ন রয়েছে এবং আপনার cos-cos থাকার পরিবর্তে cos-sin আছে, এটি এমন কিছু যা খুব ত্রুটি-প্রবণ।যদি আপনি শুধু এটি মনে রাখার চেষ্টা করছেন।", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "যাইহোক, যদি আপনি জটিল সংখ্যার সাথে এটিতে আসেন, এটি কেবলমাত্র অনেক কম ত্রুটি-প্রবণ নয়, এটির একটি খুব সুন্দর অর্থ রয়েছে এবং এটি ঠিকই পড়ে যায়।", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "সুতরাং আপনি যদি নেতিবাচক 1 এর বর্গমূলের বাস্তবতায় অগত্যা বিশ্বাস না করেন তবে আপনাকে অন্তত স্বীকার করতে হবে যে এটি আকর্ষণীয় যে এটি গণিতের অন্যান্য অংশগুলিকে উপযোগী করে তুলতে পারে, গণিতের অন্যান্য অংশগুলিকে একটু বেশি করে তুলতে পারে।খুব বোধগম্য।", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "কিন্তু সূচনা বিন্দু খুব অদ্ভুত দেখাচ্ছে, ঠিক আছে? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "একটি হল, না সেখানে নেই, তাই না? ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "আপনি যে কোনো সংখ্যার বর্গ করেন, যদি এটি ধনাত্মক হয়, ভাল যে শুধু ইতিবাচক থাকে।", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "আমি কখনই নেতিবাচক কিছু পেতে যাচ্ছি না।", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "যাইহোক, যদি একজন গণিতবিদ এসে বলেন, ওহ না না এটি বিদ্যমান, আমরা এটিকে সংজ্ঞায়িত করেছি যাতে এটিই হয়।", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "যখন আপনার কোনো সমস্যা থাকে যা আপনি সমাধান করতে পারবেন না, আপনি শুধু বলতে পারেন, ওহ আমি জিনিসগুলিকে সংজ্ঞায়িত করেছি যাতে আমাদের কাছে এখন জাদুকরী সমাধান আছে।", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "সুতরাং আপনি যদি এটির সাথে অস্বস্তিকর হন তবে আপনি অবশ্যই একা নন।", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "প্রকৃতপক্ষে, রেনে ডেসকার্টেস এই সংখ্যাগুলির জন্য একটি অবমাননাকর হিসাবে কাল্পনিক শব্দটি তৈরি করেছিলেন।", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "এবং তারপরে আমরা এটিকে একটি সম্মেলন হিসাবে আটকে রেখেছি এবং আমরা এখনও তাদের কাল্পনিক সংখ্যা বলি, যা সত্যিকার অর্থে অযৌক্তিক।", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "দ্বিতীয় অদ্ভুত জিনিসটি যা আপনি করেন যখন আপনি জটিল সংখ্যার কথা বলা শুরু করেন তা হল, এইরকম একটি সংখ্যাই নেই, তবে আমরা এটিকে একটি বাড়ি দিতে যাচ্ছি।", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "এবং, ঠিক আছে, যদি আমরা আমাদের নম্বর সিস্টেমকে প্রসারিত করতে চাই, আমি এটি পেয়েছি, সম্ভবত এটি সেখানে কিছু ধরণের নম্বর রাখা দরকারী, কিন্তু কেন আমি, ডান? ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "একেবারে শুরুতে, আসুন শুধু এই বিষয়ে কথা বলি যে আপনি যদি দ্বি-মাত্রিক সংখ্যাগুলি যোগ করেন তবে নিয়মগুলি বেশ সহজবোধ্য এবং এটি মূলত ভেক্টরের মতোই কাজ করে, আপনার মধ্যে যারা ভেক্টরের সাথে পরিচিত হতে পারে তাদের জন্য।", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "আপনি যদি এটিকে বিশ্বাসের ভিত্তিতে নেন এবং আপনি অনুসরণ করেন, আশা করি যে এটি কার্যকরী হয়ে ওঠে তা ন্যায্যতা দিতে সাহায্য করে যে আমরা কেন এটি করছি।", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "উম, মনে হচ্ছে একটি আছে, f এবং d উত্তরের মধ্যে একটি সামনে এবং পিছনে আছে, তাই f হল তাদের সবকটি, বলছে যে এইগুলিকে বাস্তব হিসাবে বিবেচনা করা উচিত।", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "এবং মজার ব্যাপার হল, d হল এমন একটি যেটি বলে যে আপনার 2 এর বর্গমূল এবং ঋণাত্মক 1 বিবেচনা করা উচিত, কিন্তু অসীম নয়, তাই সেখানে আপনার মধ্যে একটি ভাল দল আছে যারা কেবল অসীমতাকে বাস্তব হিসাবে বিবেচনা করে প্রত্যাখ্যান করবে, তবে এটির সাথে খুব স্বাচ্ছন্দ্য বোধ করবে ঋণাত্মক 1 এর বর্গমূল, এটি দুর্দান্ত।", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "ঠিক আছে, তাই মনে হচ্ছে আমরা এমন কিছু লোক পেয়েছি যারা নেতিবাচক 1 এর সাথে স্বাচ্ছন্দ্য বোধ করে, একটি বড় দল অসীমতার সাথে অস্বস্তিকর, এটি অন্য দিনের জন্য একটি বিষয়, এটি নিয়ে চিন্তা করবেন না, এবং তারপরে অনেক লোক যারা নেতিবাচক 1 বাস্তব হতে পারে যে ধারণা সঙ্গে সুপার আরামদায়ক হচ্ছে না হতে পারে যে মধ্যমাঠে ধরনের. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "তাই আমাদের প্রথম অনেক বেশি গাণিতিক প্রশ্নের জন্য, একটি ওয়ার্ম-আপ হিসাবে, আমি আপনাকে এই দুটি যোগ করতে বলতে চাই।", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "আমি আপনাকে কীভাবে সেগুলি যোগ করতে হয় তা শিখিয়ে দেওয়ার আগে, এটি কীভাবে কাজ করতে পারে সে সম্পর্কে একটি অনুমান করুন এবং আমি আশা করি এটি বেশ সহজবোধ্য মনে হবে, সংযোজন আসলে এটির সবচেয়ে কম আকর্ষণীয় অংশ, তবে এটি হল, কখন এটি জানা ভাল জিনিস আপনি জটিল সংখ্যা সম্পর্কে শিখছেন।", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "আপনি যদি চারটি ইউনিট ডানদিকে এবং তারপরে একটি ইউনিট উপরে নিয়ে যাচ্ছেন, এবং আপনি দুটি ইউনিট বাম দিকে এবং তারপরে দুটি ইউনিট উপরে সরানোর ধারণা যোগ করতে চান, তবে আপনি একবারে সেইগুলির প্রত্যেকটিই করবেন।", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "আসল অংশটি হবে সেই চারটি ডানদিকে, তারপরে বিয়োগ দুইটি বামে।", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "তাই যে এক i যোগ দুই i. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "সংযোজনে আসলেই জটিল কিছু চলছে না, যা দুর্দান্ত।", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "সুতরাং ভেক্টরের সাথে, দুটি ভেক্টর ফিরে পাওয়ার জন্য তাদের গুণ করার কোনো ধারণা নেই, অন্তত যখন আমরা 2D সমতলে থাকি।", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "যা মূলত, ধরুন আমার পয়েন্ট তিন, দুই আছে।", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "যদি আমার কাছে কিছু ধরণের স্থানাঙ্ক গ্রিড থাকে এবং আমি x স্থানাঙ্ক তিন এবং y স্থানাঙ্ক দুই সহ বিন্দুতে যাই, এর 90 ডিগ্রি ঘূর্ণন কী? ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "ঘড়ির কাঁটার বিপরীত দিকে।", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "এখন কি এই সম্পর্কে সুদৃশ্য আমরা মূলত শুধু এটা বের করতে আমাদের কাগজ চালু করতে পারেন. ", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "আপনি বলেন ঠিক আছে যদি এটি তিন, দুই এ শুরু হয় এবং তারপর আমি ঘড়ির কাঁটার বিপরীত দিকে 90 ডিগ্রী ঘোরতাম, আমি এখন এটিকে পড়তে পারি ঋণাত্মক দুইটি x দিক এবং তারপরে তিনটি y দিকে, যদি আমি পুরো প্লেনটিকে এভাবে ঘোরতাম।. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "সুতরাং আমরা এখানে যা করেছি তা হল আমরা তিন, দুই নিয়েছি এবং তারপর আমরা এটিকে ঋণাত্মক দুই, তিনে রূপান্তর করেছি।", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "যে 90 ডিগ্রী ঘূর্ণন হতে যাচ্ছে. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "যদি আমি একজোড়া সংখ্যা নিয়েছিলাম একটি কমা b ঠিক আছে এবং তারপর আমি বলেছিলাম যে এটি কোথায় ঘুরবে যদি আমি এটিকে 90 ডিগ্রি ঘোরাই তবে এটি স্থানাঙ্কগুলি অদলবদল করে শেষ হবে এবং তারপরে প্রথমটিকে নেতিবাচক করে তুলবে।", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "তাই যে অন্য 90 ডিগ্রী ঘূর্ণন ছিল. ", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "আচ্ছা এখানে যা হয়েছে তা হল আমরা উভয় স্থানাঙ্ককে নেতিবাচক করেছি এবং এটি আশ্বস্ত কারণ আমি যদি ab এ বসে কিছু বিন্দু নিই এবং তারপরে আমি এটিকে 90 ডিগ্রি ঘোরাই তাহলে এটি হবে আমার প্রাথমিক 90 ডিগ্রি ঘূর্ণন এবং তারপরে আরও 90 ডিগ্রি ঘূর্ণন হবে same as 180 ডিগ্রী rot- ওহ না আমি এটা ভুল করেছি।", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "এটি একটি 180 ডিগ্রী ঘূর্ণনের মতোই হবে যা এইরকম হওয়া উচিত অন্য ভেক্টরটিকে উপেক্ষা করে যা আমি আঁকেছি যা কেবলমাত্র উভয় স্থানাঙ্ক নিচ্ছে এবং তাদের ঋণাত্মক ঋণাত্মক বা নেতিবাচক করে তুলছে ঠিক আছে যাতে এই অপারেশনটি 90 ডিগ্রি ঘূর্ণন করে আসলে এমন আচরণ করে যেমন আপনি এটি আশা করেন।", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "ওহ দেখুন অনেক লোক খুব ভাল উত্তর জমা দিয়েছে।", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "ঠিক আছে তাই মনে হচ্ছে আপনার বেশিরভাগই সঠিক উত্তর পেয়েছেন যা 2 প্লাস 3i।", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "খুব ভালো খুব ভালো।", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "আপনাদের মধ্যে কেউ কেউ নেতিবাচক 2 3 উত্তর দিয়েছেন যা আমি অনুমান করছি যে আপনি 4 বিয়োগ 2 বা 2 বিয়োগ 4 নিচ্ছেন কিনা তা অদলবদল করছে যাতে এটি সম্পূর্ণরূপে বোধগম্য।", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "আমরা 2 প্লাস 3 পেয়েছি যা হয়তো শুধু i থেকে বাদ যাচ্ছে তাই আমি মনে করি হয়তো অনেক সাধারণ ত্রুটি এবং এন্ট্রির মতো এবং আপনি জানেন যে আমাদের সকলের সাথে বিশেষ করে পরীক্ষায় ঘটে, কখনও কখনও আপনি জানেন সঠিক উত্তর কি কিন্তু তারপর আপনি একটি প্রতীক ভুলে যান বা আপনি দুটি অদলবদল করেন যাতে এটি খুব ভাল।", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "Stal stal শব্দের শব্দগুলি আপনি জানেন যে তারা আমাকে বলে যে এটি কাজ করছে এবং তবুও আমার পক্ষে এগিয়ে যাওয়া খুব ধীর তাই আপনি জানেন যদি আমি তাদের সাথে কড়া কথা না বলি তবে আপনি তাদের টুইটারেও একই সাথে যেতে পারেন যেখানে আমরা প্রশ্ন জিজ্ঞাসা করি এবং শুধু বলি হেই কামিনেটার আপনি কি লাইভ প্রশ্নগুলিকে আমাদের জন্য একটু ভাল কাজ করতে পারেন না? ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "ঠিক আছে আমি মনে করি আমরা অবশেষে সেখানে আছি।", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "সবাই প্রস্তুত? ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "আহা! ", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "বিস্ময়কর! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "খুব সহজ প্রশ্ন আমি চাই আপনি i সংখ্যাটি নিন এবং আমি চাই আপনি এটিকে 3 যোগ করে 2i দ্বারা গুণ করুন এবং যদিও আমি গুণনের নিয়ম সম্পর্কে সত্যিই কথা বলিনি আমি যা বলতে পারি তা হল ভান করা যেন এটি কাজ করে ঠিক যেমন এটি করে সাধারণ সংখ্যাগুলি আপনার কাছে বিতরণমূলক সম্পত্তির মতো জিনিস রয়েছে যেখানে আপনি এটিকে সর্বত্র বিতরণ করতে পারেন এবং তারপরে i এর সংজ্ঞায়িত বৈশিষ্ট্যটি হল এই ধারণাটি যে আমি বর্গক্ষেত্রটি নেতিবাচক একটি যা একমাত্র বিশেষ জিনিস যা আপনাকে কেবলমাত্র এটির সাথে ব্যবহার করতে হবে।যেমন এটি একটি স্বাভাবিক সংখ্যা ঠিক আছে এবং তারপর পণ্যটির সাথে এগিয়ে যান।বিস্ময়কর! ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/chinese/sentence_translations.json b/2020/ldm-complex-numbers/chinese/sentence_translations.json index ec6dbcffa..1b16bfda7 100644 --- a/2020/ldm-complex-numbers/chinese/sentence_translations.json +++ b/2020/ldm-complex-numbers/chinese/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "我想说，如果它是在 应用程序中真正有用的东西，那么它就像文字一样真实，对吗？", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "你永 远不会遇到像幸福这样的抽象词，但它在我们的头脑中有一种 现实，比如二的平方根之类的东西，你不能用分数来表达， 或者类似的东西你知道，负数的平方根不会出现在真正的正常 数中，即使它们看起来有点不同。", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "我不会假设你知 道它们是什么，它只是一个基本的入门知识，但让我们直接开始吧，好吗？", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "我记得当我在学校的时候，我们学习了这些 加法公式，如果你想知道两个不同角度之和的余弦，你知道，这是 一种很长的东西，用原始两个角度的余弦和正弦来表示，有一个减 号总是会让人绊倒，如果你对这个符号做同样的事情，它看起来很 相似，但有一个加号，而不是 cos-cos 而是 cos- sin，这是非常容易出错的东西如果你只是想按原样记住它。", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "然而，如果你用复数来计算，这不仅不容易出错，而且它有一个非 常美丽的含义，而且它会直接消失。", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "因此，即使你不一定相信 负 1 的平方根的真实性，你至少也必须承认，有趣的是 ，它可以使其他数学部分变得有用，其他数学部分多一点 也可以理解。", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "但起点看起来很 奇怪，好吗？", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "一是，不，没有 ，对吧？", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "任何你平方的数 字，如果它是正数，那么它就保持正数。", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "我永远不会得到任何负面的东西。", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "然而，如果一位数学家走过来说，哦，不，它存在， 我们已经定义了它，所以情况就是如此。", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "当你遇到无法解决的问题时，你可以说，哦，我已经定 义了一些东西，所以我们现在神奇地有了一个解决方案。", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "因此，如果您对此感到不舒服，那么您绝 对不是一个人。", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "事实上，勒内·笛卡尔为这些数字创造了虚数这个词，作 为贬义。", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "然后我们坚持将其作为惯例，我们仍然 称它们为虚数，这确实很荒谬。", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "当你开始谈论复数时，你要做的第二件奇怪的事情是说， 不只是这样一个数字 i，但我们要给它一个家。", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "而且，好吧，如果我们想扩 展我们的数字系统，我明白了，也许在上面放某种数字很有用，但为什么我，对吧？", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "首先， 我们来谈谈如果要像这样添加二维数字，规则非常 简单，并且对于任何可能熟悉向量的人来说，其操 作本质上与向量相同。", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "如果你相信这一点并遵循它，希望它变得有用这一事实有 助于证明我们这样做的理由。", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "嗯，看起来好像有一个，答案f和d之间有一个来回，所以f就是所有的， 也就是说这些都应该被认为是真实的。", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "有趣的是，d 表示你应该 考虑 2 的平方根和负 1，但不应该考虑无穷大，所以 有很多人会拒绝将无穷大视为真实的，但对无穷大感到非常满 意负 1 的平方根，太棒了。", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "好吧，看来 我们有一群人对负 1 感到满意，一大群人对无穷 大感到不舒服，这是另一天的话题，不用担心，然后 有很多人有点处于中间立场，可能不太接受负 1 可能是真实的想法。", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "因此，对于我们的第一个更多的数学问题，作为热身，我只想请您添加 这两个。", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "在我教你如何添加它们之前，先猜测一下它是如何工作 的，我希望它感觉很简单，加法实际上是其中最不有趣的部分 ，但事实上，知道什么时候添加是一件好事你正在学习复数。", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "如果您要向右移动四个单位，然后向上移动一个单位，并且您想添加向 左移动两个单位，然后向上移动两个单位的想法，那么您只需一次执行其中每个单 位即可。", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "真正的部分是右边的那四个， 然后减去左边的两个。", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "1 i 加2 i 也 是这样吗？", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "加法实际上并没有发生任何复杂的事情，这很 棒。", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "因此，对于向量，实际上没有任何将它们相乘来得到两个向量的概 念，至少当我们在 2D 平面中时是这样。", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "基本上，假设我有第三点、第二点。", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "如果我只有某种坐标网格，并且我转到 x 坐标为 3、 y 坐标为 2 的点，那么它的 90 度旋转是多少？", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "逆时针方向。", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "现在，这一点的可爱之处在于我们基本上可以通过翻阅论文来找出答案。", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "你说好吧，如果它从三、二开始，然后我逆时针旋转 90 度，我 现在可以将其读为 x 方向的负二，然后 y 方向的三，如果 我像这样旋转整个平面。", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "所以我们在这里所做的就是取三、二，然 后将其转换为负二、三。", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "这将是 90 度旋转。", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "如果我取一对数字 a 逗号 b 好吧，然后我说如果我将其旋转 90 度，它将旋转到哪里，最终将交换坐标 ba ，然后将第一 个坐标设为负数。", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "这又是一次 90 度旋转。", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "好吧，这里发生的事情是 我们刚刚将两个坐标都设为负值，这令人放心，因为如果我在 ab 处取某 个点，然后将其旋转 90 度，那么这将是我最初的 90 度旋转，然后 再旋转 90 度，这就是与 180 度旋转相同 - 哦不，我做错了。", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "这与 180 度旋转相同，看起来应该像这样忽略我绘制的 另一个向量，该向量只是取两个坐标并使它们为负负负 b 好吧，这样就可以放心执行 90 度旋转的操作实际上的行 为就像你所期望的那样。", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "哦，看起来很多人都提交了很好的答案。", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "好吧，看来你们大多数人都得到 了正确答案，即 2 加 3i。", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "非常好非常好。", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "你们中有些人的答案是否定的 2 3，我想这只是交换您是取 4 减 2 还是 2 减 4，所以这是完全可以理解的。", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "我们有 2 加 3，这可能只是去掉了 i，所以我想可能很多都是简单的错误和输入，你 知道这种情况发生在我们所有人身上，尤其是在测试中，有时你知道正确的答案是 什么，但然后你忘记了一个符号或者交换了两个符号，这样都很好。", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "你知道的，你 知道的，他们告诉我，它正在发挥作用，但我前进的速度非常慢，所以你知 道，如果我不打算对他们严厉的话，你们也可以在推特上对他们进行同样 的攻击。我们提出问题并只是说“嘿，kamineter”，你不能让现 场提问对我们来说更好一点吗？", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "好吧，我想我们终于到了。", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "大家准备好了吗？", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "啊哈！", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "精彩的！", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "非常简单的问题，我希望你取数字 i，我希望你将它乘 以 3 加 2i，尽管我还没有真正讨论过乘法的规则，但我 可以说的是假装它的运行方式就像它一样正常数你有像分配属 性这样的东西，你可以在其中分配它，然后 i 的定义特征是 i 的平方是负数，这是你需要知道的唯一特殊的事情，除 了只是对待它就像它是一个正常的数字一样，然后继续使用该 产品。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/english/captions.srt b/2020/ldm-complex-numbers/english/captions.srt index 7890e6228..b80a35e0b 100644 --- a/2020/ldm-complex-numbers/english/captions.srt +++ b/2020/ldm-complex-numbers/english/captions.srt @@ -32,7 +32,7 @@ just to poll the audience on seeing what you guys can consider to be, well, real 9 00:00:28,740 --> 00:00:30,880 -What do you exist when it comes to numbers? +What do you consider to exist when it comes to numbers? 10 00:00:31,760 --> 00:00:35,240 @@ -235,1046 +235,1270 @@ I'll tell you, for me personally, I feel like it's very silly to answer anything that's either that's not all of them or none of them. 60 -00:03:35,560 --> 00:03:40,363 -I can maybe understand if someone wants to treat infinity as something different +00:03:35,560 --> 00:03:39,706 +I can maybe understand if someone wants to treat infinity as something different, 61 -00:03:40,363 --> 00:03:45,166 -because it's ill-defined, there's lots of different things that that might mean, +00:03:39,706 --> 00:03:43,802 +because it's ill-defined. There's lots of different things that that might mean, 62 -00:03:45,166 --> 00:03:49,316 -but insofar as numbers exist at all, if you have a number of numbers, +00:03:43,802 --> 00:03:48,354 +but insofar as numbers exist at all, if you consider what a number is to be a real thing, 63 -00:03:49,316 --> 00:03:50,740 -you can't really answer. +00:03:48,354 --> 00:03:52,298 +then it would... oh man, I can't believe that we're stalling out on this one. 64 -00:03:50,740 --> 00:03:52,224 -So, for me, I think that the question is, well, man, +00:03:52,298 --> 00:03:55,889 +We had fixed it by live stream too, but I guess there's going to be an 65 -00:03:52,224 --> 00:03:53,680 -I can't believe that we're stalling out on this one. +00:03:55,889 --> 00:03:58,620 +oscillation between when it works and when it doesn't. 66 -00:03:53,760 --> 00:03:56,190 -We had fixed it by live stream, too, but I guess there's going +00:03:59,640 --> 00:04:02,234 +So, for me, I think that the question is, well, man, 67 -00:03:56,190 --> 00:03:58,620 -to be an oscillation between when it works and when it doesn't. +00:04:02,234 --> 00:04:04,780 +I can't believe that we're stalling out on this one. 68 -00:03:59,640 --> 00:04:03,425 -But for me personally, basically, anytime that you have a numerical +00:04:04,780 --> 00:04:06,374 +t has a kind of reality in our minds, and things like the square root of two, 69 -00:04:03,425 --> 00:04:07,600 -construct that's helpful in the real world, you know, I consider that real. +00:04:06,374 --> 00:04:07,560 +which you can't express as a fraction, or things like the 70 -00:04:07,680 --> 00:04:11,702 -I would say that if it's something that's actually useful in an application, +00:04:07,560 --> 00:04:08,480 +square root of negative one that don't show u 71 -00:04:11,702 --> 00:04:13,740 -then it is as real as words are, right? +00:04:08,480 --> 00:04:11,627 +But for me personally, basically, anytime that you have a numerical 72 -00:04:13,740 --> 00:04:17,057 -You're never going to run into an abstract word like happiness out there, +00:04:11,627 --> 00:04:15,100 +construct that's helpful in the real world, you know, I consider that real. 73 -00:04:17,057 --> 00:04:18,940 -but it has a kind of reality in our minds. +00:04:15,100 --> 00:04:17,728 +What I'd like to do for you today, basically, is show you the sense in 74 -00:04:19,660 --> 00:04:23,611 -And things like the square root of 2, which you can't express as a fraction, +00:04:17,728 --> 00:04:20,320 +which imaginary numbers are useful, the complex numbers are useful, an 75 -00:04:23,611 --> 00:04:27,819 -or things like the square root of negative 1 that don't show up among real normal +00:04:20,320 --> 00:04:22,705 +d from there maybe try to imbue them with a little more reality. 76 -00:04:27,819 --> 00:04:31,462 -numbers, you know, even if they might seem a little bit different, oh, +00:04:22,705 --> 00:04:25,678 +I won't assume that you know what they are yet, it's meant to be a basic primer, 77 -00:04:31,462 --> 00:04:32,540 -this is such a shame. +00:04:25,678 --> 00:04:28,467 +but let's just dive right in, okay? The end, by the way, the very end here, 78 -00:04:32,680 --> 00:04:37,020 -I'm genuinely curious to see what your answers are, but it's not showing up for me, +00:04:28,467 --> 00:04:30,780 +I want to talk about two different trigonometric functions, and 79 -00:04:37,020 --> 00:04:40,172 -which I suppose means we'll have to move on with the lesson, +00:04:31,500 --> 00:04:35,555 +And things like the square root of 2, which you can't express as a fraction, 80 -00:04:40,172 --> 00:04:44,720 -but this will presumably begin working by the end and we can maybe pull things up again. +00:04:35,555 --> 00:04:39,874 +or things like the square root of negative 1 that don't show up among real normal 81 -00:04:46,720 --> 00:04:50,040 -So let me go ahead and take that away. +00:04:39,874 --> 00:04:43,613 +numbers, you know, even if they might seem a little bit different, oh, 82 -00:04:51,260 --> 00:04:54,477 -What I'd like to do for you today, basically, is show you the sense +00:04:43,613 --> 00:04:44,720 +this is such a shame. 83 -00:04:54,477 --> 00:04:57,837 -in which imaginary numbers are useful, the complex numbers are useful, +00:04:46,720 --> 00:04:49,712 +ing that we're going to build to, two identities from trigonometry, 84 -00:04:57,837 --> 00:05:00,960 -and from there maybe try to imbue them with a little more reality. +00:04:49,712 --> 00:04:53,233 +and I understand that maybe, oh, these complicated identities from trigonometry 85 -00:05:01,260 --> 00:05:03,712 -I won't assume that you know what they are yet, +00:04:53,233 --> 00:04:56,402 +is not going to be the best way to lure some people into understanding, 86 -00:05:03,712 --> 00:05:06,880 -it's meant to be a basic primer, but let's just dive right in. +00:04:56,402 --> 00:04:59,880 +oh yeah, complex numbers, they're really useful, you're really going to love th 87 -00:05:07,040 --> 00:05:10,514 -Okay, the end, by the way, the very end here, I want to talk about +00:04:59,880 --> 00:05:06,880 +em. But I do think it's interesting that you can have a fact tha 88 -00:05:10,514 --> 00:05:13,781 -two different trigonometric functions, and this is kind of the +00:05:07,040 --> 00:05:10,808 +What I'd like to do for you today, basically, is show you the sense 89 -00:05:13,781 --> 00:05:17,360 -thing that we're going to build to, two identities from trigonometry. +00:05:10,808 --> 00:05:14,742 +in which imaginary numbers are useful, the complex numbers are useful, 90 -00:05:18,040 --> 00:05:21,780 -And I understand that maybe, oh, these complicated identities from trigonometry +00:05:14,742 --> 00:05:18,400 +and from there maybe try to imbue them with a little more reality. 91 -00:05:21,780 --> 00:05:25,567 -is not going to be the best way to lure some people into understanding, oh yeah, +00:05:18,400 --> 00:05:21,533 +I won't assume that you know what they are yet, 92 -00:05:25,567 --> 00:05:28,980 -complex numbers, they're really useful, you're really going to love them. +00:05:21,533 --> 00:05:25,580 +it's meant to be a basic primer, but let's just dive right in. 93 -00:05:29,160 --> 00:05:31,730 -But I do think it's interesting that you can have a fact that has +00:05:25,580 --> 00:05:28,032 +ou can have facts that are pretty hard to remember. 94 -00:05:31,730 --> 00:05:34,495 -nothing to do with complex numbers or the square root of negative one, +00:05:28,032 --> 00:05:31,428 +I remember when I was in school and we learned these addition formulas, 95 -00:05:34,495 --> 00:05:37,300 -it's just trigonometry, it's everything we were talking about last time. +00:05:31,428 --> 00:05:35,296 +that if you want to know the cosine of the sum of two different angles, you know, 96 -00:05:37,540 --> 00:05:39,520 -And you can have facts that are pretty hard to remember. +00:05:35,296 --> 00:05:36,900 +it's this kind of long thing in te 97 -00:05:39,640 --> 00:05:42,963 -I remember when I was in school and we learned these addition formulas, +00:05:36,900 --> 00:05:40,592 +And I understand that maybe, oh, these complicated identities from trigonometry 98 -00:05:42,963 --> 00:05:46,749 -that if you want to know the cosine of the sum of two different angles, you know, +00:05:40,592 --> 00:05:44,330 +is not going to be the best way to lure some people into understanding, oh yeah, 99 -00:05:46,749 --> 00:05:50,720 -it's this kind of long thing in terms of cosines and sines of the original two angles. +00:05:44,330 --> 00:05:47,700 +complex numbers, they're really useful, you're really going to love them. 100 -00:05:50,900 --> 00:05:53,360 -There's this minus sign that would always trip people up. +00:05:47,700 --> 00:05:50,870 +But I do think it's interesting that you can have a fact that has 101 -00:05:53,780 --> 00:05:56,350 -If you do the same for the sine, it looks similar, +00:05:50,870 --> 00:05:54,281 +nothing to do with complex numbers or the square root of negative one, 102 -00:05:56,350 --> 00:05:59,980 -but there's a plus sign, and instead of having cos cos, you have cosine. +00:05:54,281 --> 00:05:57,740 +it's just trigonometry, it's everything we were talking about last time. 103 -00:06:00,280 --> 00:06:03,620 -It's something that's very error-prone if you're just trying to memorize it as it is. +00:05:57,740 --> 00:06:00,460 +And you can have facts that are pretty hard to remember. 104 -00:06:03,980 --> 00:06:08,531 -However, if you come at it with complex numbers, this is not only much less error-prone, +00:06:00,460 --> 00:06:02,746 +I remember when I was in school and we learned these addition formulas, 105 -00:06:08,531 --> 00:06:11,600 -it has a very beautiful meaning and it just falls right out. +00:06:02,746 --> 00:06:05,349 +that if you want to know the cosine of the sum of two different angles, you know, 106 -00:06:12,100 --> 00:06:16,171 -So even if you don't necessarily believe in the reality of the square root of negative +00:06:05,349 --> 00:06:08,080 +it's this kind of long thing in terms of cosines and sines of the original two angles. 107 -00:06:16,171 --> 00:06:20,195 -one, you at the very least have to admit that it's interesting that it can make other +00:06:08,080 --> 00:06:11,107 +y who's going into serious math, they'll tell you that complex numbers are as 108 -00:06:20,195 --> 00:06:24,220 -pieces of math useful, that other pieces of math a little bit more understandable too. +00:06:11,107 --> 00:06:13,513 +real a part of their work and their life as real numbers are. 109 -00:06:25,220 --> 00:06:27,660 -And trigonometry is just the tip of the iceberg. +00:06:13,513 --> 00:06:16,696 +But the starting point looks very strange, okay? When you start introducing this, 110 -00:06:27,800 --> 00:06:32,035 -If you talk to anybody who's in engineering, anybody who's going into serious math, +00:06:16,696 --> 00:06:17,240 +the very first 111 -00:06:32,035 --> 00:06:35,060 -they'll tell you that complex numbers are as real a part of +00:06:17,240 --> 00:06:20,678 +thing you do is to say, assume that there's some number i so that 112 -00:06:35,060 --> 00:06:37,380 -their work and their life as real numbers are. +00:06:20,678 --> 00:06:24,169 +i squared is equal to negative 1. And I think to a lot of students 113 -00:06:37,520 --> 00:06:40,940 -But the starting point looks very strange, okay? +00:06:24,169 --> 00:06:27,660 +there's maybe one of two possible reactions that you can have here. 114 -00:06:41,480 --> 00:06:45,645 -When you start introducing this, the very first thing you do is to say, +00:06:27,800 --> 00:06:31,440 +It's something that's very error-prone if you're just trying to memorize it as it is. 115 -00:06:45,645 --> 00:06:50,100 -assume that there's some number i so that i squared is equal to negative one. +00:06:31,540 --> 00:06:38,743 +However, if you come at it with complex numbers, this is not only much less error-prone, 116 -00:06:50,840 --> 00:06:52,900 -And I think to a lot of students there's maybe one +00:06:38,743 --> 00:06:43,600 +it has a very beautiful meaning and it just falls right out. 117 -00:06:52,900 --> 00:06:54,880 -of two possible reactions that you can have here. +00:06:43,600 --> 00:06:47,389 +So even if you don't necessarily believe in the reality of the square root of negative 118 -00:06:54,880 --> 00:06:57,620 -One is, no there isn't, right? +00:06:47,389 --> 00:06:51,134 +one, you at the very least have to admit that it's interesting that it can make other 119 -00:06:57,720 --> 00:07:00,518 -Any time I square a number, even if it's negative, +00:06:51,134 --> 00:06:54,880 +pieces of math useful, that other pieces of math a little bit more understandable too. 120 -00:07:00,518 --> 00:07:03,427 -if I take negative five for example and I square it, +00:06:54,880 --> 00:07:01,097 +d says, oh no no it exists, we've defined it so that that's the case. 121 -00:07:03,427 --> 00:07:06,720 -well a negative times a negative is a positive, so I get 25. +00:07:01,097 --> 00:07:08,380 +I think the other reaction someone can have is, hang on a second, you can do that? 122 -00:07:07,760 --> 00:07:11,480 -Any number that you square, if it's positive, well that just stays positive. +00:07:08,380 --> 00:07:11,474 +If you talk to anybody who's in engineering, anybody who's going into serious math, 123 -00:07:12,160 --> 00:07:16,300 -So it seems like no matter what, when I'm squaring numbers I always get a positive number. +00:07:11,474 --> 00:07:13,685 +they'll tell you that complex numbers are as real a part of 124 -00:07:16,380 --> 00:07:17,860 -I'm never going to get anything negative. +00:07:13,685 --> 00:07:15,380 +their work and their life as real numbers are. 125 -00:07:18,480 --> 00:07:20,940 -So this does not exist, no such number. +00:07:15,380 --> 00:07:17,860 +But the starting point looks very strange, okay? 126 -00:07:22,280 --> 00:07:25,306 -However, if a mathematician comes and says, oh no it exists, +00:07:18,480 --> 00:07:22,607 +do when you start talking about complex numbers is to say, 127 -00:07:25,306 --> 00:07:27,340 -we've defined it so that that's the case. +00:07:22,607 --> 00:07:27,434 +there's not just such a number i, but we're going to give it a home. 128 -00:07:27,720 --> 00:07:32,380 -I think the other reaction someone can have is, hang on a second, you can do that? +00:07:27,434 --> 00:07:33,381 +Instead of the real number line, which you know all of these numbers we know when we 129 -00:07:32,860 --> 00:07:35,340 -When you have a problem that you can't solve you can just say, +00:07:33,381 --> 00:07:36,320 +square them, you can't get a negative, wha 130 -00:07:35,340 --> 00:07:37,860 -oh I've defined things so that we now magically have a solution. +00:07:36,320 --> 00:07:38,964 +t we do is say i lives in a different dimension. i lives perpendicularly, 131 -00:07:38,500 --> 00:07:42,029 -Okay, next time I'm having trouble with my homework and I don't know what the answer +00:07:38,964 --> 00:07:41,860 +there's one above and then there's one below, negative i, and you can have negati 132 -00:07:42,029 --> 00:07:45,600 -to x is, I will be like, let x be the value defined to be the answer to this question. +00:07:41,860 --> 00:07:45,431 +ve 2i, you scale it however you want. Essentially it's proposing that 133 -00:07:46,480 --> 00:07:49,460 -So if you're uncomfortable with this, you're definitely not alone. +00:07:45,431 --> 00:07:48,697 +numbers be two-dimensional and that i has a very specific home, 134 -00:07:50,060 --> 00:07:55,520 -In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. +00:07:48,697 --> 00:07:52,320 +one unit perpendicular, uh, perpendicularly above the real number line. 135 -00:07:55,820 --> 00:07:58,218 -It was meant to make fun of the fact that obviously there +00:07:52,320 --> 00:07:53,706 +Any time I square a number, even if it's negative, 136 -00:07:58,218 --> 00:08:00,700 -is no such answer and it shouldn't be taken as serious math. +00:07:53,706 --> 00:07:55,148 +if I take negative five for example and I square it, 137 -00:08:01,100 --> 00:08:04,662 -And then we stuck with that as a convention and we still call them imaginary numbers, +00:07:55,148 --> 00:07:56,780 +well a negative times a negative is a positive, so I get 25. 138 -00:08:04,662 --> 00:08:05,740 -which is genuinely absurd. +00:07:56,780 --> 00:08:02,680 +Any number that you square, if it's positive, well that just stays positive. 139 -00:08:06,260 --> 00:08:08,500 -But that's not the only weird assumption that we make. +00:08:02,680 --> 00:08:04,280 +So it seems like no matter what, when I'm squaring numbers I always get a positive number. 140 -00:08:08,740 --> 00:08:12,636 -The second weird thing that you do when you start talking about complex numbers +00:08:04,280 --> 00:08:09,960 +I'm never going to get anything negative. 141 -00:08:12,636 --> 00:08:16,240 -is to say, there's not just a number i, but we're going to give it a home. +00:08:09,960 --> 00:08:17,740 +and then you move in that perpendicular direction into the extension of our number s 142 -00:08:16,820 --> 00:08:19,314 -Instead of the real number line, which you know, +00:08:17,740 --> 00:08:24,177 +ystem, which again, you're kind of asking the students to take a lot on faith here 143 -00:08:19,314 --> 00:08:23,183 -all of these numbers we know when we square them, you can't get a negative, +00:08:24,177 --> 00:08:30,460 +that you're okay to do that, that you're allowed to just pretend that the numbers 144 -00:08:23,183 --> 00:08:25,780 -what we do is say i lives in a different dimension. +00:08:30,460 --> 00:08:33,000 +I think the other reaction someone can have is, hang on a second, you can do that? 145 -00:08:26,460 --> 00:08:28,040 -i lives perpendicularly. +00:08:33,000 --> 00:08:34,805 +When you have a problem that you can't solve you can just say, 146 -00:08:28,780 --> 00:08:33,600 -There's one above and then there's one below, negative i, and you can have negative 2i. +00:08:34,805 --> 00:08:36,640 +oh I've defined things so that we now magically have a solution. 147 -00:08:33,860 --> 00:08:35,080 -You scale it however you want. +00:08:36,640 --> 00:08:44,593 +Okay, next time I'm having trouble with my homework and I don't know what the answer 148 -00:08:35,780 --> 00:08:40,710 -Essentially it's proposing that numbers be two-dimensional and that i has a very +00:08:44,593 --> 00:08:52,640 +to x is, I will be like, let x be the value defined to be the answer to this question. 149 -00:08:40,710 --> 00:08:45,640 -specific home, one unit perpendicular perpendicularly above the real number line. +00:08:52,640 --> 00:08:57,460 +So if you're uncomfortable with this, you're definitely not alone. 150 -00:08:46,500 --> 00:08:50,500 -And okay, if we want to extend our number system, I get it, +00:08:57,460 --> 00:08:57,640 +In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. 151 -00:08:50,500 --> 00:08:55,300 -maybe it's useful to put some kind of number up there, but why i, right? +00:08:57,640 --> 00:08:59,753 +It was meant to make fun of the fact that obviously there 152 -00:08:55,660 --> 00:08:59,062 -Why not say infinity is the number that sits one unit above zero, +00:08:59,753 --> 00:09:01,940 +is no such answer and it shouldn't be taken as serious math. 153 -00:08:59,062 --> 00:09:02,310 -or one divided by zero, or any other problem that you couldn't +00:09:01,940 --> 00:09:04,642 +And then we stuck with that as a convention and we still call them imaginary numbers, 154 -00:09:02,310 --> 00:09:05,920 -solve before and you make up an answer to, why should that live there? +00:09:04,642 --> 00:09:05,460 +which is genuinely absurd. 155 -00:09:06,220 --> 00:09:10,580 -What on earth does the idea of a point one unit above the real number +00:09:05,460 --> 00:09:11,458 +like there's a, there's a back and forth between answers f and d, 156 -00:09:10,580 --> 00:09:14,940 -line in a separate dimension have to do with squaring to negative one? +00:09:11,458 --> 00:09:18,092 +so f is all of them, saying that all of these should be considered real. 157 -00:09:15,740 --> 00:09:17,280 -So I hope to answer this for you. +00:09:18,092 --> 00:09:22,000 +And interesting, d is the one that says you 158 -00:09:17,300 --> 00:09:20,969 -At the very beginning, let's just talk about how if you're adding numbers that +00:09:22,000 --> 00:09:23,641 +The second weird thing that you do when you start talking about complex numbers 159 -00:09:20,969 --> 00:09:24,871 -are two-dimensional like this, the rules are pretty straightforward and it operates +00:09:23,641 --> 00:09:25,160 +is to say, there's not just a number i, but we're going to give it a home. 160 -00:09:24,871 --> 00:09:28,680 -essentially the same as vectors for any of you who might be familiar with vectors. +00:09:25,160 --> 00:09:29,330 +Instead of the real number line, which you know, 161 -00:09:29,460 --> 00:09:34,327 -So let's say hypothetically I have a number, oh I don't know, +00:09:29,330 --> 00:09:35,799 +all of these numbers we know when we square them, you can't get a negative, 162 -00:09:34,327 --> 00:09:38,880 -let me draw one here that's going to be four plus i, okay? +00:09:35,799 --> 00:09:40,140 +what we do is say i lives in a different dimension. 163 -00:09:39,580 --> 00:09:44,522 -And then I'm going to take a second number and it's helpful to draw them as vectors, +00:09:40,140 --> 00:09:42,720 +i lives perpendicularly. 164 -00:09:44,522 --> 00:09:48,068 -kind of an arrow from the number zero, and this one is going +00:09:42,720 --> 00:09:48,800 +There's one above and then there's one below, negative i, and you can have negative 2i. 165 -00:09:48,068 --> 00:09:50,220 -to end up at negative two plus two i. +00:09:48,800 --> 00:09:51,420 +You scale it however you want. 166 -00:09:50,620 --> 00:09:54,288 -So what I'm saying is you take the real number negative two and then you move in +00:09:51,420 --> 00:09:54,650 +Essentially it's proposing that numbers be two-dimensional and that i has a very 167 -00:09:54,288 --> 00:09:57,458 -that perpendicular direction into the extension of our number system, +00:09:54,650 --> 00:09:57,880 +specific home, one unit perpendicular perpendicularly above the real number line. 168 -00:09:57,458 --> 00:10:01,080 -which again you're kind of asking the students to take a lot on faith here that +00:09:58,040 --> 00:10:00,912 +And okay, if we want to extend our number system, I get it, 169 -00:10:01,080 --> 00:10:04,884 -you're okay to do that, that you're allowed to just pretend that the numbers extend +00:10:00,912 --> 00:10:04,360 +maybe it's useful to put some kind of number up there, but why i, right? 170 -00:10:04,884 --> 00:10:05,700 -in this direction. +00:10:04,360 --> 00:10:05,892 +Why not say infinity is the number that sits one unit above zero, 171 -00:10:06,040 --> 00:10:08,956 -If you take that on faith and you follow, hopefully the fact that +00:10:05,892 --> 00:10:07,354 +or one divided by zero, or any other problem that you couldn't 172 -00:10:08,956 --> 00:10:11,740 -it becomes useful helps to justify why we're doing any of this. +00:10:07,354 --> 00:10:08,980 +solve before and you make up an answer to, why should that live there? 173 -00:10:12,200 --> 00:10:15,480 -So my question for you is simply what happens when we add these two numbers? +00:10:08,980 --> 00:10:10,360 +What on earth does the idea of a point one unit above the real number 174 -00:10:16,260 --> 00:10:19,045 -Now assuming that our question system has not broken down, +00:10:10,360 --> 00:10:11,740 +line in a separate dimension have to do with squaring to negative one? 175 -00:10:19,045 --> 00:10:22,161 -I should be able to do this as a proper poll and let me go ahead, +00:10:12,200 --> 00:10:12,660 +So I hope to answer this for you. 176 -00:10:22,161 --> 00:10:26,080 -I guess we can first check the previous poll, okay things seem to be working so we +00:10:12,660 --> 00:10:15,974 +At the very beginning, let's just talk about how if you're adding numbers that 177 -00:10:26,080 --> 00:10:29,526 -can take a little step back in the lesson so I'm just genuinely curious, +00:10:15,974 --> 00:10:19,499 +are two-dimensional like this, the rules are pretty straightforward and it operates 178 -00:10:29,526 --> 00:10:31,840 -I want to know how you guys answered on this one. +00:10:19,499 --> 00:10:22,940 +essentially the same as vectors for any of you who might be familiar with vectors. 179 -00:10:32,280 --> 00:10:35,796 -It looks like there's a there's a back and forth between answers f and d, +00:10:22,940 --> 00:10:27,829 +So I guess I can pull it up on the, just on the piece of paper, 180 -00:10:35,796 --> 00:10:39,218 -so f is all of them saying that all of these should be considered real, +00:10:27,829 --> 00:10:32,871 +and you can follow along at home, see what the addition might be. 181 -00:10:39,218 --> 00:10:43,304 -and interesting d is the one that says you should consider two square root of two and +00:10:32,871 --> 00:10:39,060 +It turns out to be relatively straightforward. If you're moving four units to the 182 -00:10:43,304 --> 00:10:47,439 -negative one but not infinity, so there's a good contingent of you out there who would +00:10:39,060 --> 00:10:41,976 +And then I'm going to take a second number and it's helpful to draw them as vectors, 183 -00:10:47,439 --> 00:10:51,573 -just reject infinity as being considered real but are very comfortable with the square +00:10:41,976 --> 00:10:44,070 +kind of an arrow from the number zero, and this one is going 184 -00:10:51,573 --> 00:10:55,137 -root of negative one, that's awesome, and then after that it looks like c, +00:10:44,070 --> 00:10:45,340 +to end up at negative two plus two i. 185 -00:10:55,137 --> 00:10:58,179 -people who reject the square root of negative one, fascinating, +00:10:45,340 --> 00:10:48,424 +So what I'm saying is you take the real number negative two and then you move in 186 -00:10:58,179 --> 00:11:02,075 -I actually would have thought that none of them would have come higher than that, +00:10:48,424 --> 00:10:51,089 +that perpendicular direction into the extension of our number system, 187 -00:11:02,075 --> 00:11:06,257 -none of them is much lower at a, okay so it looks like we've got a cohort of people who +00:10:51,089 --> 00:10:54,136 +which again you're kind of asking the students to take a lot on faith here that 188 -00:11:06,257 --> 00:11:10,201 -are comfortable with negative one, a large cohort are uncomfortable with infinity, +00:10:54,136 --> 00:10:57,334 +you're okay to do that, that you're allowed to just pretend that the numbers extend 189 -00:11:10,201 --> 00:11:12,768 -that's a topic for another day, don't worry about it, +00:10:57,334 --> 00:10:58,020 +in this direction. 190 -00:11:12,768 --> 00:11:16,807 -and then a number of people who are kind of in that middle ground of maybe not being +00:10:58,020 --> 00:10:59,801 +can get you something like it. But the rules end up being very different 191 -00:11:16,807 --> 00:11:19,896 -super comfortable with the idea that negative one might be real, +00:10:59,801 --> 00:11:01,290 +from that in the number system. You can't really do algebra. 192 -00:11:19,896 --> 00:11:22,700 -let's see if we can convince you of the difference of that. +00:11:01,290 --> 00:11:03,120 +You can't do things like assume that if two numbers multiply to make zero, 193 -00:11:23,420 --> 00:11:26,045 -So for our first much more mathematical question, +00:11:03,120 --> 00:11:03,560 +then one of them h 194 -00:11:26,045 --> 00:11:29,300 -as kind of a warm-up, I just want to ask you to add these two. +00:11:03,560 --> 00:11:03,828 +as to be zero. But complex numbers are going to 195 -00:11:29,540 --> 00:11:33,385 -Before I've taught you how to add them, make a guess at how it might work, +00:11:03,828 --> 00:11:04,080 +end up behaving much like the real numbers, s 196 -00:11:33,385 --> 00:11:35,898 -and I hope that it feels pretty straightforward, +00:11:04,080 --> 00:11:06,829 +Now assuming that our question system has not broken down, 197 -00:11:35,898 --> 00:11:39,384 -addition is actually the least interesting part of this, but it is, +00:11:06,829 --> 00:11:09,905 +I should be able to do this as a proper poll and let me go ahead, 198 -00:11:39,384 --> 00:11:42,974 -it's a good thing to know when you're learning about complex numbers, +00:11:09,905 --> 00:11:13,774 +I guess we can first check the previous poll, okay things seem to be working so we 199 -00:11:42,974 --> 00:11:46,820 -it's definitely one of those operations that you are going to need to know. +00:11:13,774 --> 00:11:17,176 +can take a little step back in the lesson so I'm just genuinely curious, 200 -00:11:47,840 --> 00:11:52,322 -Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, +00:11:17,176 --> 00:11:19,460 +I want to know how you guys answered on this one. 201 -00:11:52,322 --> 00:11:55,888 -it looks like the question is still not loading completely correctly, +00:11:19,460 --> 00:11:23,011 +It looks like there's a there's a back and forth between answers f and d, 202 -00:11:55,888 --> 00:11:59,454 -so I'm going to have a stern word with Cam and Ider behind the scenes +00:11:23,011 --> 00:11:26,466 +so f is all of them saying that all of these should be considered real, 203 -00:11:59,454 --> 00:12:03,019 -who have otherwise built such a beautiful, beautiful interface that's +00:11:26,466 --> 00:11:30,594 +and interesting d is the one that says you should consider two square root of two and 204 -00:12:03,019 --> 00:12:06,280 -helpful for this kind of back and forth between you guys and me. +00:11:30,594 --> 00:11:34,769 +negative one but not infinity, so there's a good contingent of you out there who would 205 -00:12:06,760 --> 00:12:09,680 -I'm just gonna, I'm gonna have a stern word with them behind the scenes, +00:11:34,769 --> 00:11:38,944 +just reject infinity as being considered real but are very comfortable with the square 206 -00:12:09,680 --> 00:12:12,640 -but in the meantime, let's go ahead and move forward with the lesson here. +00:11:38,944 --> 00:11:42,544 +root of negative one, that's awesome, and then after that it looks like c, 207 -00:12:13,460 --> 00:12:16,664 -So I guess I can pull it up on the, just on the piece of paper, +00:11:42,544 --> 00:11:45,615 +people who reject the square root of negative one, fascinating, 208 -00:12:16,664 --> 00:12:19,920 -and you can follow along at home, see what the addition might be. +00:11:45,615 --> 00:11:49,551 +I actually would have thought that none of them would have come higher than that, 209 -00:12:20,240 --> 00:12:22,100 -It turns out to be relatively straightforward. +00:11:49,551 --> 00:11:53,774 +none of them is much lower at a, okay so it looks like we've got a cohort of people who 210 -00:12:22,940 --> 00:12:26,063 -If you're moving four units to the right and then one unit up, +00:11:53,774 --> 00:11:57,757 +are comfortable with negative one, a large cohort are uncomfortable with infinity, 211 -00:12:26,063 --> 00:12:30,228 -and you want to add the idea of moving two units to the left and then two units up, +00:11:57,757 --> 00:12:00,349 +that's a topic for another day, don't worry about it, 212 -00:12:30,228 --> 00:12:32,460 -well you just do each of those one at a time. +00:12:00,349 --> 00:12:04,428 +and then a number of people who are kind of in that middle ground of maybe not being 213 -00:12:32,900 --> 00:12:34,240 -I'll go ahead and pull out black here. +00:12:04,428 --> 00:12:07,548 +super comfortable with the idea that negative one might be real, 214 -00:12:34,820 --> 00:12:39,744 -The real part is going to be those four to the right, then minus two to the left, +00:12:07,548 --> 00:12:10,380 +let's see if we can convince you of the difference of that. 215 -00:12:39,744 --> 00:12:44,669 -okay, straightforward enough, and then the imaginary part is going to be this one +00:12:10,380 --> 00:12:17,423 +So what we've done here is we've taken three, two and then we convert it to negative two, 216 -00:12:44,669 --> 00:12:50,015 -unit up and then these two units up, one plus two, times i, so is that one i plus two i, +00:12:17,423 --> 00:12:22,666 +three. Something which maybe in our original system you know looks 217 -00:12:50,015 --> 00:12:54,640 -and then when you work that out four minus two is two, one plus two is three. +00:12:22,666 --> 00:12:25,640 +like this negative two and then three. 218 -00:12:55,920 --> 00:12:57,580 -A nice simple introduction here. +00:12:25,640 --> 00:12:29,507 +Before I've taught you how to add them, make a guess at how it might work, 219 -00:12:57,860 --> 00:13:01,440 -Addition doesn't really have anything complicated going on, which is great. +00:12:29,507 --> 00:12:32,035 +and I hope that it feels pretty straightforward, 220 -00:13:01,520 --> 00:13:04,200 -That means that it's one fewer thing for us to worry about. +00:12:32,035 --> 00:12:35,541 +addition is actually the least interesting part of this, but it is, 221 -00:13:04,420 --> 00:13:07,100 -What is so complex about complex numbers after all? +00:12:35,541 --> 00:12:39,152 +it's a good thing to know when you're learning about complex numbers, 222 -00:13:07,640 --> 00:13:09,951 -Well where everything becomes interesting is when +00:12:39,152 --> 00:12:43,020 +it's definitely one of those operations that you are going to need to know. 223 -00:13:09,951 --> 00:13:11,940 -you try to multiply these numbers together. +00:12:43,020 --> 00:12:49,734 +Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, 224 -00:13:12,500 --> 00:13:15,406 -So with vectors, there's not really any notion of multiplying +00:12:49,734 --> 00:12:55,075 +it looks like the question is still not loading completely correctly, 225 -00:13:15,406 --> 00:13:18,500 -them to get two vectors back, at least when we're in the 2d plane. +00:12:55,075 --> 00:13:00,416 +so I'm going to have a stern word with Cam and Ider behind the scenes 226 -00:13:18,760 --> 00:13:21,517 -You have some notions like cross products and dot products that +00:13:00,416 --> 00:13:05,756 +who have otherwise built such a beautiful, beautiful interface that's 227 -00:13:21,517 --> 00:13:23,714 -in three dimensions can get you something like it, +00:13:05,756 --> 00:13:10,640 +helpful for this kind of back and forth between you guys and me. 228 -00:13:23,714 --> 00:13:26,860 -but the rules end up being very different from that in the number system. +00:13:10,640 --> 00:13:14,722 +nice gut check here is to say what happens when we do that twice what 229 -00:13:26,860 --> 00:13:27,920 -You can't really do algebra. +00:13:14,722 --> 00:13:18,570 +if we do that same very mechanistic operation again twice and I'm 230 -00:13:28,200 --> 00:13:32,550 -You can't do things like assume that if two numbers multiply to make zero, +00:13:18,570 --> 00:13:22,711 +going to go and take this I swap the two coordinates we get a negative 231 -00:13:32,550 --> 00:13:37,306 -then one of them has to be zero, but complex numbers are going to end up behaving +00:13:22,711 --> 00:13:26,560 +b but then that first one becomes negative. So that was another 90 232 -00:13:37,306 --> 00:13:41,134 -much like the real numbers, so rules from algebra can carry over, +00:13:26,560 --> 00:13:29,588 +degree rotation. Well what's happened here is we've just made both of the coordinates 233 -00:13:41,134 --> 00:13:45,542 -but to understand what that rotation rule is, oh no I'm giving things away, +00:13:29,588 --> 00:13:32,300 +negative and that's reassuring because if I take some point sitting at a b an 234 -00:13:45,542 --> 00:13:49,951 -what that multiplication rule is, I just want to ask you a simple question, +00:13:32,300 --> 00:13:37,400 +It turns out to be relatively straightforward. 235 -00:13:49,951 --> 00:13:53,489 -which is basically suppose I have the point three two, okay, +00:13:37,400 --> 00:13:41,423 +h is just taking both of the coordinates and making them negative negative a 236 -00:13:53,489 --> 00:13:57,201 -we're not even going to think of it as a complex number per se, +00:13:41,423 --> 00:13:45,290 +negative b okay so that's reassuring this operation that does a 90 degree 237 -00:13:57,201 --> 00:14:02,073 -if I just have some sort of coordinate grid and I go to the point with x coordinate +00:13:45,290 --> 00:13:49,731 +rotation actually behaves like you would expect it to. Now why am I asking you this? 238 -00:14:02,073 --> 00:14:05,960 -three and y coordinate two, what is the 90 degree rotation of this? +00:13:49,731 --> 00:13:53,860 +Well I'm being told that supposedly I'm allowed to ask you questions again so I 239 -00:14:05,960 --> 00:14:12,421 -If I rotate it 90 degrees and let's say counterclockwise, +00:13:53,860 --> 00:13:57,869 +'m going to have you do your very first complex product. 240 -00:14:12,421 --> 00:14:19,440 -counter, counter, jeez, writing is difficult, counterclockwise. +00:13:57,869 --> 00:14:01,668 +Oh look a lot of people did submit answers very good. 241 -00:14:25,280 --> 00:14:28,293 -Okay, now what's lovely about this is we can basically +00:14:01,668 --> 00:14:05,960 +Great let's let's grade the complex addition actually let's l 242 -00:14:28,293 --> 00:14:30,320 -just turn our paper to figure it out. +00:14:05,960 --> 00:14:12,404 +et's see if it is as straightforward a process as I was hoping it was and see 243 -00:14:30,460 --> 00:14:35,252 -We say okay if it started at three two and then I rotate 90 degrees counterclockwise, +00:14:12,404 --> 00:14:18,849 +how much explanation is demanded. Okay so it looks like a majority of you did 244 -00:14:35,252 --> 00:14:39,654 -I can just read that off now as being negative two in the x direction and then +00:14:18,849 --> 00:14:24,136 +get the correct answer which is 2 plus 3i. Very good very good. 245 -00:14:39,654 --> 00:14:43,500 -three in the y direction, if I had rotated the whole plane like that. +00:14:24,136 --> 00:14:30,581 +52 of you answered simply 2 which would have been the real part of the answer 246 -00:14:44,080 --> 00:14:49,044 -So what we've done here is we've taken three two and then we convert it to +00:14:30,581 --> 00:14:36,860 +so maybe just the fact that there's some vertical component and you need to 247 -00:14:49,044 --> 00:14:54,075 -negative two three, something which maybe in our original system, you know, +00:14:36,860 --> 00:14:38,100 +still add those 248 -00:14:54,075 --> 00:14:59,900 -looks like this, negative two and then three, that's going to be the 90 degree rotation. +00:14:38,100 --> 00:14:42,060 +vertical components or maybe those of you who answer 2 reject the reality of 249 -00:14:59,900 --> 00:15:02,447 -And what's nice here is that that rule is very +00:14:42,060 --> 00:14:46,020 +imaginary numbers so you just don't even acknowledge that vertical component. 250 -00:15:02,447 --> 00:15:05,320 -simple and it applies to any pair that we might have. +00:14:46,020 --> 00:14:52,820 +Addition doesn't really have anything complicated going on, which is great. 251 -00:15:05,460 --> 00:15:09,653 -If I took a pair of numbers a comma b, okay, and then I said where is +00:14:52,820 --> 00:14:57,880 +s just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 252 -00:15:09,653 --> 00:15:12,708 -that going to rotate to if I rotate it 90 degrees, +00:14:57,880 --> 00:14:59,900 +4 so that's completely understandable. We've got 2 plus 3 which is maybe just dropping 253 -00:15:12,708 --> 00:15:16,842 -it's going to end up by swapping the coordinates b a and then making +00:14:59,900 --> 00:14:59,900 +Well where everything becomes interesting is when 254 -00:15:16,842 --> 00:15:18,280 -that first one negative. +00:14:59,900 --> 00:14:59,900 +you try to multiply these numbers together. 255 -00:15:19,240 --> 00:15:20,460 -That's a 90 degree rotation. +00:14:59,900 --> 00:15:00,713 +So with vectors, there's not really any notion of multiplying 256 -00:15:20,640 --> 00:15:23,940 -And a nice gut check here is to say what happens when we do that twice? +00:15:00,713 --> 00:15:01,580 +them to get two vectors back, at least when we're in the 2d plane. 257 -00:15:24,200 --> 00:15:27,520 -What if we do that same very mechanistic operation again twice? +00:15:01,580 --> 00:15:04,665 +that we ask questions and just say hey kamineter can't you make the live questions 258 -00:15:27,520 --> 00:15:31,279 -And I'm going to go and take this, I swap the two coordinates, +00:15:04,665 --> 00:15:07,788 +work a little bit better for us? Okay I think we're finally there. Everybody ready? 259 -00:15:31,279 --> 00:15:34,980 -we get a negative b, but then that first one becomes negative. +00:15:07,788 --> 00:15:10,540 +Aha! Wonderful! Very simple question I want you to take the number i and I 260 -00:15:35,920 --> 00:15:37,640 -So that was another 90 degree rotation. +00:15:10,540 --> 00:15:17,660 +want you to multiply it by 3 plus 2i and even though I haven't really talked about 261 -00:15:38,480 --> 00:15:43,376 -Well what's happened here is we've just made both of the coordinates negative +00:15:17,660 --> 00:15:21,754 +You can't do things like assume that if two numbers multiply to make zero, 262 -00:15:43,376 --> 00:15:48,210 -and that's reassuring because if I take some point sitting at a b and then I +00:15:21,754 --> 00:15:26,231 +then one of them has to be zero, but complex numbers are going to end up behaving 263 -00:15:48,210 --> 00:15:52,541 -rotate it 90 degrees, so this will be my initial 90 degree rotation, +00:15:26,231 --> 00:15:29,834 +much like the real numbers, so rules from algebra can carry over, 264 -00:15:52,541 --> 00:15:57,375 -and then another 90 degrees that's the same as a 180 degree roto- oh no I've +00:15:29,834 --> 00:15:33,983 +but to understand what that rotation rule is, oh no I'm giving things away, 265 -00:15:57,375 --> 00:15:58,380 -done that wrong. +00:15:33,983 --> 00:15:38,132 +what that multiplication rule is, I just want to ask you a simple question, 266 -00:15:59,860 --> 00:16:04,386 -That will be the same as a 180 degree rotation which should look like this, +00:15:38,132 --> 00:15:41,462 +which is basically suppose I have the point three two, okay, 267 -00:16:04,386 --> 00:16:08,139 -ignore the other vector that I drew, which is just taking both +00:15:41,462 --> 00:15:44,956 +we're not even going to think of it as a complex number per se, 268 -00:16:08,139 --> 00:16:10,760 -of the coordinates and making them negative. +00:15:44,956 --> 00:15:49,542 +if I just have some sort of coordinate grid and I go to the point with x coordinate 269 -00:16:11,700 --> 00:16:13,840 -Negative a negative b, okay. +00:15:49,542 --> 00:15:53,200 +three and y coordinate two, what is the 90 degree rotation of this? 270 -00:16:15,060 --> 00:16:17,690 -So that's reassuring this operation that does a 90 degree +00:15:53,200 --> 00:15:55,682 +If I rotate it 90 degrees and let's say counterclockwise, 271 -00:16:17,690 --> 00:16:20,140 -rotation actually behaves like you would expect it to. +00:15:55,682 --> 00:15:58,380 +counter, counter, jeez, writing is difficult, counterclockwise. 272 -00:16:20,400 --> 00:16:21,760 -Now why am I asking you this? +00:15:59,860 --> 00:16:00,529 +Okay, now what's lovely about this is we can basically 273 -00:16:22,660 --> 00:16:26,650 -Well I'm being told that supposedly I'm allowed to ask you questions again, +00:16:00,529 --> 00:16:00,980 +just turn our paper to figure it out. 274 -00:16:26,650 --> 00:16:29,800 -so I'm going to have you do your very first complex product. +00:16:00,980 --> 00:16:04,734 +ons as you do it rather than sitting in passively watching this is genuinely 275 -00:16:30,200 --> 00:16:32,700 -Oh look a lot of people did submit answers, very good. +00:16:04,734 --> 00:16:08,196 +delightful to me. Okay this is this isn't necessarily a question I was 276 -00:16:34,200 --> 00:16:36,640 -Great let's grade the complex addition actually, +00:16:08,196 --> 00:16:11,950 +expecting to divide the audience necessarily so unsurprisingly it looks like 277 -00:16:36,640 --> 00:16:40,576 -let's see if it is as straightforward a process as I was hoping it was and see +00:16:11,950 --> 00:16:15,510 +we have a very strong majority in one direction hopefully in the correct 278 -00:16:40,576 --> 00:16:42,220 -how much explanation is demanded. +00:16:15,510 --> 00:16:19,460 +direction but if not that would that would heavily inform where the lesson should 279 -00:16:43,060 --> 00:16:47,764 -Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i, +00:16:19,460 --> 00:16:22,704 +So what we've done here is we've taken three two and then we convert it to 280 -00:16:47,764 --> 00:16:48,900 -very good, very good. +00:16:22,704 --> 00:16:25,992 +negative two three, something which maybe in our original system, you know, 281 -00:16:49,280 --> 00:16:53,086 -52 of you answered simply 2 which would have been the real part of the answer so +00:16:25,992 --> 00:16:29,800 +looks like this, negative two and then three, that's going to be the 90 degree rotation. 282 -00:16:53,086 --> 00:16:56,799 -maybe just the fact that there's some vertical component and you need to still +00:16:30,200 --> 00:16:34,405 +it looks like the majority of you answered negative two plus three i 283 -00:16:56,799 --> 00:17:00,464 -add those vertical components or maybe those of you who answered 2 reject the +00:16:34,405 --> 00:16:38,671 +which is absolutely correct absolutely correct so there's two ways to 284 -00:17:00,464 --> 00:17:04,599 -reality of imaginary numbers so you just don't even acknowledge that vertical component. +00:16:38,671 --> 00:16:43,120 +think about this okay one of them is to walk forward with the algebra and 285 -00:17:05,440 --> 00:17:10,942 -Some of you answered negative 2 3 which I guess is just making- that's just swapping +00:16:43,120 --> 00:16:46,315 +just do it a little bit mechanistically okay so if we pull ourselves up our sheet if 286 -00:17:10,942 --> 00:17:16,380 -up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. +00:16:46,315 --> 00:16:49,473 +we take i times three plus two i three plus two i it just distributes i times three 287 -00:17:17,599 --> 00:17:21,391 -We've got 2 plus 3 which is maybe just dropping off the i so I think maybe +00:16:49,473 --> 00:16:52,706 +is going to be three i i times two i is going to be two times i squared by definition 288 -00:17:21,391 --> 00:17:25,131 -a lot of like simple errors and entry and you know that happens to all of +00:16:52,706 --> 00:16:55,940 +i squared is negative one which means that our final answer is going to look like nega 289 -00:17:25,131 --> 00:17:28,872 -us especially on tests is sometimes you know what the right answer is but +00:16:55,940 --> 00:16:56,120 +That's a 90 degree rotation. 290 -00:17:28,872 --> 00:17:32,360 -then you you forget a symbol or you swap two so that's all very good. +00:16:56,120 --> 00:16:58,980 +like I said it looks like a majority of you correctly did that product now it's one 291 -00:17:32,880 --> 00:17:35,860 -Let's go ahead and try our very first product though like I said. +00:16:58,980 --> 00:17:01,875 +thing to just walk through it mechanistically it's another to step back and say what 292 -00:17:36,660 --> 00:17:38,681 -So here because I already talked through one of the +00:17:01,875 --> 00:17:04,599 +just happened geometrically right because what we just talked through was the fa 293 -00:17:38,681 --> 00:17:40,820 -questions we're going to go ahead and skip ahead of it. +00:17:05,440 --> 00:17:05,668 +ct that if you want to rotate numbers 90 degrees the rule 294 -00:17:40,820 --> 00:17:44,602 -We know how to rotate something like 3 comma 2 so I'm not +00:17:05,668 --> 00:17:05,900 +is to swap the two coordinates and then multiply that first 295 -00:17:44,602 --> 00:17:48,580 -even going to give you time to do that and properly grade it. +00:17:05,900 --> 00:17:11,271 +one by negative two well look at what's happened here we've got three 296 -00:17:50,620 --> 00:17:52,200 -Stal, stal, words, words. +00:17:11,271 --> 00:17:16,566 +and two those coordinates have been swapped two is now the real part 297 -00:17:53,000 --> 00:17:57,107 -You know they tell me that it's working and yet it's very slow for me to progress +00:17:16,566 --> 00:17:21,938 +three is the imaginary part but that two got multiplied by a negative 298 -00:17:57,107 --> 00:18:01,165 -forward so you know if I'm not going to have a stern word with them you guys can +00:17:21,938 --> 00:17:27,540 +one because i has this defining feature of squaring to become negative on 299 -00:18:01,165 --> 00:18:05,272 -go at them on twitter too under the same place that we ask questions and just say +00:17:27,540 --> 00:17:28,961 +e so that should give you some indication that okay multiplying 300 -00:18:05,272 --> 00:18:09,280 -hey Kamineter can't you make the live questions work a little bit better for us? +00:17:28,961 --> 00:17:30,360 +by i has this action of rotating things by 90 degrees maybe tha 301 -00:18:10,020 --> 00:18:12,160 -Okay I think we're finally there. +00:17:30,360 --> 00:17:34,075 +Well what's happened here is we've just made both of the coordinates negative 302 -00:18:13,540 --> 00:18:14,220 -Everybody ready? +00:17:34,075 --> 00:17:37,743 +and that's reassuring because if I take some point sitting at a b and then I 303 -00:18:14,600 --> 00:18:15,520 -Aha wonderful. +00:17:37,743 --> 00:17:41,030 +rotate it 90 degrees, so this will be my initial 90 degree rotation, 304 -00:18:17,040 --> 00:18:18,060 -Very simple question. +00:17:41,030 --> 00:17:44,697 +and then another 90 degrees that's the same as a 180 degree roto- oh no I've 305 -00:18:18,220 --> 00:18:22,133 -I want you to take the number i and I want you to multiply it by 3 plus 2i +00:17:44,697 --> 00:17:45,460 +done that wrong. 306 -00:18:22,133 --> 00:18:26,046 -and even though I haven't really talked about the rules for multiplication +00:17:45,460 --> 00:17:49,627 +have a number that behaves this way it gives you a computational mechanism for all 307 -00:18:26,046 --> 00:18:30,220 -what I can say is pretend like it operates just like it does for normal numbers. +00:17:49,627 --> 00:17:53,894 +of the other types of rotations that you might want to do that might not necessarily 308 -00:18:30,320 --> 00:18:34,284 -You've got things like the distributive property where you can distribute this +00:17:53,894 --> 00:17:58,011 +be 90 degrees and to show you why this works i'm going to go ahead and pull up an 309 -00:18:34,284 --> 00:18:38,700 -throughout and then the defining feature of i is this idea that i squared is negative 1. +00:17:58,011 --> 00:18:02,380 +animation so let's say we have any number z and in this case z is going to be let's see 310 -00:18:38,720 --> 00:18:41,040 -That's the only special thing you need to know about that. +00:18:02,380 --> 00:18:04,736 +where do i have it z is going to be at two uh plus i great and let's 311 -00:18:41,380 --> 00:18:43,777 -Other than that just treat it like it's a normal +00:18:04,736 --> 00:18:07,093 +say i want to understand what is multiplying by z due to every other 312 -00:18:43,777 --> 00:18:46,420 -number okay and then proceed forward with the product. +00:18:07,093 --> 00:18:09,280 +possible complex number well we can go one by one the very first 313 -00:18:47,020 --> 00:18:49,173 -Wonderful okay so it looks like we've got quite a +00:18:10,020 --> 00:18:14,183 +So that's reassuring this operation that does a 90 degree 314 -00:18:49,173 --> 00:18:51,500 -few of you coming in to answer which is always lovely. +00:18:14,183 --> 00:18:18,060 +rotation actually behaves like you would expect it to. 315 -00:18:51,860 --> 00:18:54,958 -Super exciting for me by the way just how many people are enthusiastic +00:18:18,220 --> 00:18:19,000 +ask what is z times one where does it take the number one well z times one is going to be 316 -00:18:54,958 --> 00:18:58,056 -about coming and like getting back to the fundamentals of math in this +00:18:19,000 --> 00:18:22,375 +Well I'm being told that supposedly I'm allowed to ask you questions again, 317 -00:18:58,056 --> 00:19:01,372 -lockdown and just you know we're gonna sit back for an hour and we're gonna +00:18:22,375 --> 00:18:25,040 +so I'm going to have you do your very first complex product. 318 -00:19:01,372 --> 00:19:04,470 -learn about complex numbers and we're actually gonna participate we're +00:18:25,040 --> 00:18:26,475 +tch that arrow up to the point where z is great a kind of 319 -00:19:04,470 --> 00:19:08,180 -actually gonna answer questions as you do rather than sitting and passively watching. +00:18:26,475 --> 00:18:28,060 +trivial fact even though it's trivial i'm actually going to take 320 -00:19:08,660 --> 01:22:10,370 +00:18:28,060 --> 00:18:32,900 +a moment to write that down just so that we can oh no no no that's for that's for + +321 +00:18:32,900 --> 00:18:37,621 +later that is randy don't you guys worry about him he'll be coming in in just a + +322 +00:18:37,621 --> 00:18:42,698 +moment so i just want to write down three crucial facts that are getting an influence + +323 +00:18:42,698 --> 00:18:47,420 +rotation three facts i'll call it three facts about multiplication the first two + +324 +00:18:47,420 --> 00:18:49,337 +Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i, + +325 +00:18:49,337 --> 00:18:49,800 +very good, very good. + +326 +00:18:49,800 --> 00:18:53,964 +52 of you answered simply 2 which would have been the real part of the answer so + +327 +00:18:53,964 --> 00:18:58,025 +maybe just the fact that there's some vertical component and you need to still + +328 +00:18:58,025 --> 00:19:02,035 +add those vertical components or maybe those of you who answered 2 reject the + +329 +00:19:02,035 --> 00:19:06,560 +reality of imaginary numbers so you just don't even acknowledge that vertical component. + +330 +00:19:06,560 --> 00:19:09,419 +hatever the 90 degree rotation point for z itself is okay so + +331 +00:19:09,419 --> 00:19:12,232 +two down infinitely many to go okay we know what it does to + +332 +00:19:12,232 --> 00:19:15,280 +one we know what it does to i let's see if we can understand what + +333 +00:19:15,280 --> 00:19:18,931 +z does to any other possible number well it turns out those two is really + +334 +00:19:18,931 --> 00:19:22,632 +all we need to work with if we have the distributive property so the third + +335 +00:19:22,632 --> 00:19:26,135 +fact that's going to look kind of innocuous is let's say i take this z + +336 +00:19:26,135 --> 00:19:29,540 +and i multiply it by c plus d times i where c and d are just any two + +337 +00:19:29,540 --> 00:19:33,340 +numbers okay well this is going to distribute so z times c i'm actually going + +338 +00:19:33,340 --> 00:19:37,430 +to write that a little differently i'm going to write it as c times z plus z times + +339 +00:19:37,430 --> 00:19:41,619 +di which again i'm going to write in kind of a funny order and write that as d times + +340 +00:19:41,619 --> 00:19:45,660 +z i now the idea here is well we know where z is we also know where z times i is s + +341 +00:19:45,660 --> 00:19:46,994 +o if we're just scaling them up by some other constants that + +342 +00:19:46,994 --> 00:19:48,460 +completely constrains where we need to go so let me go ahead and wr + +343 +00:19:48,460 --> 00:19:52,745 +ite this down with an example okay let's say that we go back here and i want to know + +344 +00:19:52,745 --> 00:19:56,980 +what multiplying by z does to anything i want to tell i want to convince you that it + +345 +00:19:56,980 --> 00:19:56,980 +Stal, stal, words, words. + +346 +00:19:56,980 --> 00:20:02,062 +n a way that keeps these lines parallel it keeps them evenly spaced keeps them + +347 +00:20:02,062 --> 00:20:07,530 +perpendicular to each other it applies this very constrained rule to the whole plane + +348 +00:20:07,530 --> 00:20:12,805 +and really just think through any one particular point for this let's say that we + +349 +00:20:12,805 --> 00:20:18,273 +have two times negative i okay so you move two units in the positive right direction + +350 +00:20:18,273 --> 00:20:23,548 +and then negative one unit in the vertical direction well after the product where + +351 +00:20:23,548 --> 00:20:28,823 +that's going to land has to be two times wherever z lands plus negative one times + +352 +00:20:28,823 --> 00:20:29,660 +wherever i la + +353 +00:20:29,660 --> 00:20:33,895 +nds okay and we see that right it's two times this + +354 +00:20:33,895 --> 00:20:38,380 +yellow vector and it'll be negative one times the gree + +355 +00:20:38,380 --> 00:20:39,080 +n vector so here even before you actually work out the product we could just read off + +356 +00:20:39,080 --> 00:20:40,180 +Aha wonderful. + +357 +00:20:40,180 --> 00:20:40,180 +Very simple question. + +358 +00:20:40,180 --> 00:20:45,314 +which says what is two plus i times two minus i and if you have notes + +359 +00:20:45,314 --> 00:20:50,669 +right now if you have a pencil and paper which i encourage you to always + +360 +00:20:50,669 --> 00:20:55,951 +come to class with i want you to try working it out do the first inside + +361 +00:20:55,951 --> 00:21:01,600 +outside last distribution property just to see mechanistically what number en + +362 +00:21:01,600 --> 00:21:07,060 +ds up popping out from this and then we'll try to see how that squares with the + +363 +00:21:07,060 --> 00:21:12,315 +geometric intuition so while you're doing that while you're working that out + +364 +00:21:12,315 --> 00:21:17,980 +hopefully on pencil and paper it looks like we've got a question from the audience + +365 +00:21:17,980 --> 00:21:23,440 +which is is i the same as i and j the vectors in physics great question actually + +366 +00:21:23,440 --> 00:21:23,700 +That's the only special thing you need to know about that. + +367 +00:21:23,700 --> 00:21:24,052 +Other than that just treat it like it's a normal + +368 +00:21:24,052 --> 00:21:24,440 +number okay and then proceed forward with the product. + +369 +00:21:24,440 --> 00:21:27,593 +Wonderful okay so it looks like we've got quite a + +370 +00:21:27,593 --> 00:21:31,000 +few of you coming in to answer which is always lovely. + +371 +00:21:31,000 --> 00:21:34,941 +Super exciting for me by the way just how many people are enthusiastic + +372 +00:21:34,941 --> 00:21:38,882 +about coming and like getting back to the fundamentals of math in this + +373 +00:21:38,882 --> 00:21:43,100 +lockdown and just you know we're gonna sit back for an hour and we're gonna + +374 +00:21:43,100 --> 00:21:47,041 +learn about complex numbers and we're actually gonna participate we're + +375 +00:21:47,041 --> 00:21:51,760 +actually gonna answer questions as you do rather than sitting and passively watching. + +376 +00:21:51,760 --> 01:22:10,370 This is genuinely delightful to me. diff --git a/2020/ldm-complex-numbers/english/sentence_timings.json b/2020/ldm-complex-numbers/english/sentence_timings.json index 6b3a21be7..1243dbd37 100644 --- a/2020/ldm-complex-numbers/english/sentence_timings.json +++ b/2020/ldm-complex-numbers/english/sentence_timings.json @@ -25,7 +25,7 @@ 28.68 ], [ - "What do you exist when it comes to numbers?", + "What do you consider to exist when it comes to numbers?", 28.74, 30.88 ], @@ -135,583 +135,583 @@ 215.26 ], [ - "I can maybe understand if someone wants to treat infinity as something different because it's ill-defined, there's lots of different things that that might mean, but insofar as numbers exist at all, if you have a number of numbers, you can't really answer.", + "I can maybe understand if someone wants to treat infinity as something different, because it's ill-defined. There's lots of different things that that might mean, but insofar as numbers exist at all, if you consider what a number is to be a real thing, then it would... oh man, I can't believe that we're stalling out on this one. We had fixed it by live stream too, but I guess there's going to be an oscillation between when it works and when it doesn't.", 215.56, - 230.74 + 238.62 ], [ "So, for me, I think that the question is, well, man, I can't believe that we're stalling out on this one.", - 230.74, - 233.68 + 239.64, + 244.78 ], [ - "We had fixed it by live stream, too, but I guess there's going to be an oscillation between when it works and when it doesn't.", - 233.76, - 238.62 + "t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show u", + 244.78, + 248.48 ], [ "But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real.", - 239.64, - 247.6 + 248.48, + 255.1 ], [ - "I would say that if it's something that's actually useful in an application, then it is as real as words are, right?", - 247.68, - 253.74 + "What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an", + 255.1, + 260.32 ], [ - "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds.", - 253.74, - 258.94 + "d from there maybe try to imbue them with a little more reality. I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? The end, by the way, the very end here, I want to talk about two different trigonometric functions, and", + 260.32, + 270.78 ], [ "And things like the square root of 2, which you can't express as a fraction, or things like the square root of negative 1 that don't show up among real normal numbers, you know, even if they might seem a little bit different, oh, this is such a shame.", - 259.66, - 272.54 - ], - [ - "I'm genuinely curious to see what your answers are, but it's not showing up for me, which I suppose means we'll have to move on with the lesson, but this will presumably begin working by the end and we can maybe pull things up again.", - 272.68, + 271.5, 284.72 ], [ - "So let me go ahead and take that away.", + "ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love th", 286.72, - 290.04 + 299.88 + ], + [ + "em. But I do think it's interesting that you can have a fact tha", + 299.88, + 306.88 ], [ "What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, and from there maybe try to imbue them with a little more reality.", - 291.26, - 300.96 + 307.04, + 318.4 ], [ "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in.", - 301.26, - 306.88 + 318.4, + 325.58 ], [ - "Okay, the end, by the way, the very end here, I want to talk about two different trigonometric functions, and this is kind of the thing that we're going to build to, two identities from trigonometry.", - 307.04, - 317.36 + "ou can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te", + 325.58, + 336.9 ], [ "And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them.", - 318.04, - 328.98 + 336.9, + 347.7 ], [ "But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just trigonometry, it's everything we were talking about last time.", - 329.16, - 337.3 + 347.7, + 357.74 ], [ "And you can have facts that are pretty hard to remember.", - 337.54, - 339.52 + 357.74, + 360.46 ], [ "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles.", - 339.64, - 350.72 + 360.46, + 368.08 ], [ - "There's this minus sign that would always trip people up.", - 350.9, - 353.36 + "y who's going into serious math, they'll tell you that complex numbers are as real a part of their work and their life as real numbers are. But the starting point looks very strange, okay? When you start introducing this, the very first", + 368.08, + 377.24 ], [ - "If you do the same for the sine, it looks similar, but there's a plus sign, and instead of having cos cos, you have cosine.", - 353.78, - 359.98 + "thing you do is to say, assume that there's some number i so that i squared is equal to negative 1. And I think to a lot of students there's maybe one of two possible reactions that you can have here.", + 377.24, + 387.66 ], [ "It's something that's very error-prone if you're just trying to memorize it as it is.", - 360.28, - 363.62 + 387.8, + 391.44 ], [ "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out.", - 363.98, - 371.6 + 391.54, + 403.6 ], [ "So even if you don't necessarily believe in the reality of the square root of negative one, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too.", - 372.1, - 384.22 + 403.6, + 414.88 ], [ - "And trigonometry is just the tip of the iceberg.", - 385.22, - 387.66 + "d says, oh no no it exists, we've defined it so that that's the case. I think the other reaction someone can have is, hang on a second, you can do that?", + 414.88, + 428.38 ], [ "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their work and their life as real numbers are.", - 387.8, - 397.38 + 428.38, + 435.38 ], [ "But the starting point looks very strange, okay?", - 397.52, - 400.94 + 435.38, + 437.86 ], [ - "When you start introducing this, the very first thing you do is to say, assume that there's some number i so that i squared is equal to negative one.", - 401.48, - 410.1 + "do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, wha", + 438.48, + 456.32 ], [ - "And I think to a lot of students there's maybe one of two possible reactions that you can have here.", - 410.84, - 414.88 + "t we do is say i lives in a different dimension. i lives perpendicularly, there's one above and then there's one below, negative i, and you can have negati", + 456.32, + 461.86 ], [ - "One is, no there isn't, right?", - 414.88, - 417.62 + "ve 2i, you scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line.", + 461.86, + 472.32 ], [ "Any time I square a number, even if it's negative, if I take negative five for example and I square it, well a negative times a negative is a positive, so I get 25.", - 417.72, - 426.72 + 472.32, + 476.78 ], [ "Any number that you square, if it's positive, well that just stays positive.", - 427.76, - 431.48 + 476.78, + 482.68 ], [ "So it seems like no matter what, when I'm squaring numbers I always get a positive number.", - 432.16, - 436.3 + 482.68, + 484.28 ], [ "I'm never going to get anything negative.", - 436.38, - 437.86 + 484.28, + 489.96 ], [ - "So this does not exist, no such number.", - 438.48, - 440.94 + "and then you move in that perpendicular direction into the extension of our number s", + 489.96, + 497.74 ], [ - "However, if a mathematician comes and says, oh no it exists, we've defined it so that that's the case.", - 442.28, - 447.34 + "ystem, which again, you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers", + 497.74, + 510.46 ], [ "I think the other reaction someone can have is, hang on a second, you can do that?", - 447.72, - 452.38 + 510.46, + 513.0 ], [ "When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution.", - 452.86, - 457.86 + 513.0, + 516.64 ], [ "Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question.", - 458.5, - 465.6 + 516.64, + 532.64 ], [ "So if you're uncomfortable with this, you're definitely not alone.", - 466.48, - 469.46 + 532.64, + 537.46 ], [ "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory.", - 470.06, - 475.52 + 537.46, + 537.64 ], [ "It was meant to make fun of the fact that obviously there is no such answer and it shouldn't be taken as serious math.", - 475.82, - 480.7 + 537.64, + 541.94 ], [ "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd.", - 481.1, - 485.74 + 541.94, + 545.46 ], [ - "But that's not the only weird assumption that we make.", - 486.26, - 488.5 + "like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. And interesting, d is the one that says you", + 545.46, + 562.0 ], [ "The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it a home.", - 488.74, - 496.24 + 562.0, + 565.16 ], [ "Instead of the real number line, which you know, all of these numbers we know when we square them, you can't get a negative, what we do is say i lives in a different dimension.", - 496.82, - 505.78 + 565.16, + 580.14 ], [ "i lives perpendicularly.", - 506.46, - 508.04 + 580.14, + 582.72 ], [ "There's one above and then there's one below, negative i, and you can have negative 2i.", - 508.78, - 513.6 + 582.72, + 588.8 ], [ "You scale it however you want.", - 513.86, - 515.08 + 588.8, + 591.42 ], [ "Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line.", - 515.78, - 525.64 + 591.42, + 597.88 ], [ "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right?", - 526.5, - 535.3 + 598.04, + 604.36 ], [ "Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there?", - 535.66, - 545.92 + 604.36, + 608.98 ], [ "What on earth does the idea of a point one unit above the real number line in a separate dimension have to do with squaring to negative one?", - 546.22, - 554.94 + 608.98, + 611.74 ], [ "So I hope to answer this for you.", - 555.74, - 557.28 + 612.2, + 612.66 ], [ "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors.", - 557.3, - 568.68 + 612.66, + 622.94 ], [ - "So let's say hypothetically I have a number, oh I don't know, let me draw one here that's going to be four plus i, okay?", - 569.46, - 578.88 + "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It turns out to be relatively straightforward. If you're moving four units to the", + 622.94, + 639.06 ], [ "And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i.", - 579.58, - 590.22 + 639.06, + 645.34 ], [ "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers extend in this direction.", - 590.62, - 605.7 + 645.34, + 658.02 ], [ - "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this.", - 606.04, - 611.74 + "can get you something like it. But the rules end up being very different from that in the number system. You can't really do algebra. You can't do things like assume that if two numbers multiply to make zero, then one of them h", + 658.02, + 663.56 ], [ - "So my question for you is simply what happens when we add these two numbers?", - 612.2, - 615.48 + "as to be zero. But complex numbers are going to end up behaving much like the real numbers, s", + 663.56, + 664.08 ], [ "Now assuming that our question system has not broken down, I should be able to do this as a proper poll and let me go ahead, I guess we can first check the previous poll, okay things seem to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one.", - 616.26, - 631.84 + 664.08, + 679.46 ], [ "It looks like there's a there's a back and forth between answers f and d, so f is all of them saying that all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesome, and then after that it looks like c, people who reject the square root of negative one, fascinating, I actually would have thought that none of them would have come higher than that, none of them is much lower at a, okay so it looks like we've got a cohort of people who are comfortable with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative one might be real, let's see if we can convince you of the difference of that.", - 632.28, - 682.7 + 679.46, + 730.38 ], [ - "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two.", - 683.42, - 689.3 + "So what we've done here is we've taken three, two and then we convert it to negative two, three. Something which maybe in our original system you know looks like this negative two and then three.", + 730.38, + 745.64 ], [ "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers, it's definitely one of those operations that you are going to need to know.", - 689.54, - 706.82 + 745.64, + 763.02 ], [ "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me.", - 707.84, - 726.28 + 763.02, + 790.64 ], [ - "I'm just gonna, I'm gonna have a stern word with them behind the scenes, but in the meantime, let's go ahead and move forward with the lesson here.", - 726.76, - 732.64 + "nice gut check here is to say what happens when we do that twice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then that first one becomes negative. So that was another 90", + 790.64, + 806.56 ], [ - "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be.", - 733.46, - 739.92 + "degree rotation. Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b an", + 806.56, + 812.3 ], [ "It turns out to be relatively straightforward.", - 740.24, - 742.1 + 812.3, + 817.4 ], [ - "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time.", - 742.94, - 752.46 + "h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. Now why am I asking you this? Well I'm being told that supposedly I'm allowed to ask you questions again so I", + 817.4, + 833.86 ], [ - "I'll go ahead and pull out black here.", - 752.9, - 754.24 + "'m going to have you do your very first complex product. Oh look a lot of people did submit answers very good. Great let's let's grade the complex addition actually let's l", + 833.86, + 845.96 ], [ - "The real part is going to be those four to the right, then minus two to the left, okay, straightforward enough, and then the imaginary part is going to be this one unit up and then these two units up, one plus two, times i, so is that one i plus two i, and then when you work that out four minus two is two, one plus two is three.", - 754.82, - 774.64 + "et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been the real part of the answer so maybe just the fact that there's some vertical component and you need to still add those", + 845.96, + 878.1 ], [ - "A nice simple introduction here.", - 775.92, - 777.58 + "vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component.", + 878.1, + 886.02 ], [ "Addition doesn't really have anything complicated going on, which is great.", - 777.86, - 781.44 + 886.02, + 892.82 ], [ - "That means that it's one fewer thing for us to worry about.", - 781.52, - 784.2 + "s just making that's just swapping up whether you're taking 4 minus 2 or 2 minus", + 892.82, + 897.88 ], [ - "What is so complex about complex numbers after all?", - 784.42, - 787.1 + "4 so that's completely understandable. We've got 2 plus 3 which is maybe just dropping", + 897.88, + 899.9 ], [ "Well where everything becomes interesting is when you try to multiply these numbers together.", - 787.64, - 791.94 + 899.9, + 899.9 ], [ "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2d plane.", - 792.5, - 798.5 + 899.9, + 901.58 ], [ - "You have some notions like cross products and dot products that in three dimensions can get you something like it, but the rules end up being very different from that in the number system.", - 798.76, - 806.86 + "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. Everybody ready? Aha! Wonderful! Very simple question I want you to take the number i and I", + 901.58, + 910.54 ], [ - "You can't really do algebra.", - 806.86, - 807.92 + "want you to multiply it by 3 plus 2i and even though I haven't really talked about", + 910.54, + 917.66 ], [ "You can't do things like assume that if two numbers multiply to make zero, then one of them has to be zero, but complex numbers are going to end up behaving much like the real numbers, so rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay, we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this?", - 808.2, - 845.96 + 917.66, + 953.2 ], [ "If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise.", - 845.96, - 859.44 + 953.2, + 958.38 ], [ "Okay, now what's lovely about this is we can basically just turn our paper to figure it out.", - 865.28, - 870.32 + 959.86, + 960.98 ], [ - "We say okay if it started at three two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that.", - 870.46, - 883.5 + "ons as you do it rather than sitting in passively watching this is genuinely delightful to me. Okay this is this isn't necessarily a question I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should", + 960.98, + 979.46 ], [ "So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, that's going to be the 90 degree rotation.", - 884.08, - 899.9 + 979.46, + 989.8 ], [ - "And what's nice here is that that rule is very simple and it applies to any pair that we might have.", - 899.9, - 905.32 + "it looks like the majority of you answered negative two plus three i which is absolutely correct absolutely correct so there's two ways to think about this okay one of them is to walk forward with the algebra and", + 990.2, + 1003.12 ], [ - "If I took a pair of numbers a comma b, okay, and then I said where is that going to rotate to if I rotate it 90 degrees, it's going to end up by swapping the coordinates b a and then making that first one negative.", - 905.46, - 918.28 + "just do it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega", + 1003.12, + 1015.94 ], [ "That's a 90 degree rotation.", - 919.24, - 920.46 + 1015.94, + 1016.12 ], [ - "And a nice gut check here is to say what happens when we do that twice?", - 920.64, - 923.94 + "like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa", + 1016.12, + 1024.6 ], [ - "What if we do that same very mechanistic operation again twice?", - 924.2, - 927.52 + "ct that if you want to rotate numbers 90 degrees the rule is to swap the two coordinates and then multiply that first", + 1025.44, + 1025.9 ], [ - "And I'm going to go and take this, I swap the two coordinates, we get a negative b, but then that first one becomes negative.", - 927.52, - 934.98 + "one by negative two well look at what's happened here we've got three and two those coordinates have been swapped two is now the real part three is the imaginary part but that two got multiplied by a negative one because i has this defining feature of squaring to become negative on", + 1025.9, + 1047.54 ], [ - "So that was another 90 degree rotation.", - 935.92, - 937.64 + "e so that should give you some indication that okay multiplying by i has this action of rotating things by 90 degrees maybe tha", + 1047.54, + 1050.36 ], [ "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees, so this will be my initial 90 degree rotation, and then another 90 degrees that's the same as a 180 degree roto- oh no I've done that wrong.", - 938.48, - 958.38 + 1050.36, + 1065.46 ], [ - "That will be the same as a 180 degree rotation which should look like this, ignore the other vector that I drew, which is just taking both of the coordinates and making them negative.", - 959.86, - 970.76 + "have a number that behaves this way it gives you a computational mechanism for all of the other types of rotations that you might want to do that might not necessarily be 90 degrees and to show you why this works i'm going to go ahead and pull up an animation so let's say we have any number z and in this case z is going to be let's see", + 1065.46, + 1082.38 ], [ - "Negative a negative b, okay.", - 971.7, - 973.84 + "where do i have it z is going to be at two uh plus i great and let's say i want to understand what is multiplying by z due to every other possible complex number well we can go one by one the very first", + 1082.38, + 1089.28 ], [ "So that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to.", - 975.06, - 980.14 + 1090.02, + 1098.06 ], [ - "Now why am I asking you this?", - 980.4, - 981.76 + "ask what is z times one where does it take the number one well z times one is going to be", + 1098.22, + 1099.0 ], [ "Well I'm being told that supposedly I'm allowed to ask you questions again, so I'm going to have you do your very first complex product.", - 982.66, - 989.8 + 1099.0, + 1105.04 ], [ - "Oh look a lot of people did submit answers, very good.", - 990.2, - 992.7 + "tch that arrow up to the point where z is great a kind of trivial fact even though it's trivial i'm actually going to take", + 1105.04, + 1108.06 ], [ - "Great let's grade the complex addition actually, let's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded.", - 994.2, - 1002.22 + "a moment to write that down just so that we can oh no no no that's for that's for later that is randy don't you guys worry about him he'll be coming in in just a moment so i just want to write down three crucial facts that are getting an influence rotation three facts i'll call it three facts about multiplication the first two", + 1108.06, + 1127.42 ], [ "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i, very good, very good.", - 1003.06, - 1008.9 + 1127.42, + 1129.8 ], [ "52 of you answered simply 2 which would have been the real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answered 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component.", - 1009.28, - 1024.6 + 1129.8, + 1146.56 ], [ - "Some of you answered negative 2 3 which I guess is just making- that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable.", - 1025.44, - 1036.38 + "hatever the 90 degree rotation point for z itself is okay so two down infinitely many to go okay we know what it does to one we know what it does to i let's see if we can understand what", + 1146.56, + 1155.28 ], [ - "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good.", - 1037.6, - 1052.36 + "z does to any other possible number well it turns out those two is really all we need to work with if we have the distributive property so the third fact that's going to look kind of innocuous is let's say i take this z and i multiply it by c plus d times i where c and d are just any two numbers okay well this is going to distribute so z times c i'm actually going", + 1155.28, + 1173.34 ], [ - "Let's go ahead and try our very first product though like I said.", - 1052.88, - 1055.86 + "to write that a little differently i'm going to write it as c times z plus z times di which again i'm going to write in kind of a funny order and write that as d times z i now the idea here is well we know where z is we also know where z times i is s", + 1173.34, + 1185.66 ], [ - "So here because I already talked through one of the questions we're going to go ahead and skip ahead of it.", - 1056.66, - 1060.82 + "o if we're just scaling them up by some other constants that completely constrains where we need to go so let me go ahead and wr", + 1185.66, + 1188.46 ], [ - "We know how to rotate something like 3 comma 2 so I'm not even going to give you time to do that and properly grade it.", - 1060.82, - 1068.58 + "ite this down with an example okay let's say that we go back here and i want to know what multiplying by z does to anything i want to tell i want to convince you that it", + 1188.46, + 1196.98 ], [ "Stal, stal, words, words.", - 1070.62, - 1072.2 + 1196.98, + 1196.98 ], [ - "You know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey Kamineter can't you make the live questions work a little bit better for us?", - 1073.0, - 1089.28 + "n a way that keeps these lines parallel it keeps them evenly spaced keeps them perpendicular to each other it applies this very constrained rule to the whole plane and really just think through any one particular point for this let's say that we have two times negative i okay so you move two units in the positive right direction and then negative one unit in the vertical direction well after the product where that's going to land has to be two times wherever z lands plus negative one times wherever i la", + 1196.98, + 1229.66 ], [ - "Okay I think we're finally there.", - 1090.02, - 1092.16 + "nds okay and we see that right it's two times this yellow vector and it'll be negative one times the gree", + 1229.66, + 1238.38 ], [ - "Everybody ready?", - 1093.54, - 1094.22 + "n vector so here even before you actually work out the product we could just read off", + 1238.38, + 1239.08 ], [ "Aha wonderful.", - 1094.6, - 1095.52 + 1239.08, + 1240.18 ], [ "Very simple question.", - 1097.04, - 1098.06 + 1240.18, + 1240.18 ], [ - "I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers.", - 1098.22, - 1110.22 + "which says what is two plus i times two minus i and if you have notes right now if you have a pencil and paper which i encourage you to always come to class with i want you to try working it out do the first inside outside last distribution property just to see mechanistically what number en", + 1240.18, + 1261.6 ], [ - "You've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative 1.", - 1110.32, - 1118.7 + "ds up popping out from this and then we'll try to see how that squares with the geometric intuition so while you're doing that while you're working that out hopefully on pencil and paper it looks like we've got a question from the audience which is is i the same as i and j the vectors in physics great question actually", + 1261.6, + 1283.44 ], [ "That's the only special thing you need to know about that.", - 1118.72, - 1121.04 + 1283.44, + 1283.7 ], [ "Other than that just treat it like it's a normal number okay and then proceed forward with the product.", - 1121.38, - 1126.42 + 1283.7, + 1284.44 ], [ "Wonderful okay so it looks like we've got quite a few of you coming in to answer which is always lovely.", - 1127.02, - 1131.5 + 1284.44, + 1291.0 ], [ "Super exciting for me by the way just how many people are enthusiastic about coming and like getting back to the fundamentals of math in this lockdown and just you know we're gonna sit back for an hour and we're gonna learn about complex numbers and we're actually gonna participate we're actually gonna answer questions as you do rather than sitting and passively watching.", - 1131.86, - 1148.18 + 1291.0, + 1311.76 ], [ "This is genuinely delightful to me.", - 1148.66, + 1311.76, 4930.37 ] ] \ No newline at end of file diff --git a/2020/ldm-complex-numbers/english/transcript.txt b/2020/ldm-complex-numbers/english/transcript.txt index fcb3d9a6c..110fdc2c0 100644 --- a/2020/ldm-complex-numbers/english/transcript.txt +++ b/2020/ldm-complex-numbers/english/transcript.txt @@ -3,7 +3,7 @@ It's incredibly fundamental to engineering, to mathematics itself, to quantum me We call them complex numbers. And worse than that, the things that bring about complex numbers we call imaginary numbers. And before we get into any of it, what I want to do is start with kind of a poll, just to poll the audience on seeing what you guys can consider to be, well, real. -What do you exist when it comes to numbers? +What do you consider to exist when it comes to numbers? So we've already been doing a couple polls in the warm-up animations, but as a serious poll of sorts, one that's actually going to help me see where you're coming from before we begin the lesson here, I just want to ask you a very simple question. Okay, so let's go ahead and pull it up here. Pull it on up. @@ -25,39 +25,39 @@ It seems like we've got three top contenders, and then three that are falling pr What do you consider to really exist when it comes to numbers? Now, I can imagine which ones might be the top two, but I'm very curious about the fact that there's three all kind of coinciding with each other there, and it looks like I'm getting a little bit of a delay before the reveal, so there's kind of this nice dramatic pause. I'll tell you, for me personally, I feel like it's very silly to answer anything that's either that's not all of them or none of them. -I can maybe understand if someone wants to treat infinity as something different because it's ill-defined, there's lots of different things that that might mean, but insofar as numbers exist at all, if you have a number of numbers, you can't really answer. +I can maybe understand if someone wants to treat infinity as something different, because it's ill-defined. There's lots of different things that that might mean, but insofar as numbers exist at all, if you consider what a number is to be a real thing, then it would... oh man, I can't believe that we're stalling out on this one. We had fixed it by live stream too, but I guess there's going to be an oscillation between when it works and when it doesn't. So, for me, I think that the question is, well, man, I can't believe that we're stalling out on this one. -We had fixed it by live stream, too, but I guess there's going to be an oscillation between when it works and when it doesn't. +t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. -I would say that if it's something that's actually useful in an application, then it is as real as words are, right? -You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds. +What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an +d from there maybe try to imbue them with a little more reality. I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? The end, by the way, the very end here, I want to talk about two different trigonometric functions, and And things like the square root of 2, which you can't express as a fraction, or things like the square root of negative 1 that don't show up among real normal numbers, you know, even if they might seem a little bit different, oh, this is such a shame. -I'm genuinely curious to see what your answers are, but it's not showing up for me, which I suppose means we'll have to move on with the lesson, but this will presumably begin working by the end and we can maybe pull things up again. -So let me go ahead and take that away. +ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love th +em. But I do think it's interesting that you can have a fact tha What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, and from there maybe try to imbue them with a little more reality. I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in. -Okay, the end, by the way, the very end here, I want to talk about two different trigonometric functions, and this is kind of the thing that we're going to build to, two identities from trigonometry. +ou can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just trigonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. -There's this minus sign that would always trip people up. -If you do the same for the sine, it looks similar, but there's a plus sign, and instead of having cos cos, you have cosine. +y who's going into serious math, they'll tell you that complex numbers are as real a part of their work and their life as real numbers are. But the starting point looks very strange, okay? When you start introducing this, the very first +thing you do is to say, assume that there's some number i so that i squared is equal to negative 1. And I think to a lot of students there's maybe one of two possible reactions that you can have here. It's something that's very error-prone if you're just trying to memorize it as it is. However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. So even if you don't necessarily believe in the reality of the square root of negative one, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. -And trigonometry is just the tip of the iceberg. +d says, oh no no it exists, we've defined it so that that's the case. I think the other reaction someone can have is, hang on a second, you can do that? If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their work and their life as real numbers are. But the starting point looks very strange, okay? -When you start introducing this, the very first thing you do is to say, assume that there's some number i so that i squared is equal to negative one. -And I think to a lot of students there's maybe one of two possible reactions that you can have here. -One is, no there isn't, right? +do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, wha +t we do is say i lives in a different dimension. i lives perpendicularly, there's one above and then there's one below, negative i, and you can have negati +ve 2i, you scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I take negative five for example and I square it, well a negative times a negative is a positive, so I get 25. Any number that you square, if it's positive, well that just stays positive. So it seems like no matter what, when I'm squaring numbers I always get a positive number. I'm never going to get anything negative. -So this does not exist, no such number. -However, if a mathematician comes and says, oh no it exists, we've defined it so that that's the case. +and then you move in that perpendicular direction into the extension of our number s +ystem, which again, you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers I think the other reaction someone can have is, hang on a second, you can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. @@ -65,7 +65,7 @@ So if you're uncomfortable with this, you're definitely not alone. In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. It was meant to make fun of the fact that obviously there is no such answer and it shouldn't be taken as serious math. And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. -But that's not the only weird assumption that we make. +like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. And interesting, d is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it a home. Instead of the real number line, which you know, all of these numbers we know when we square them, you can't get a negative, what we do is say i lives in a different dimension. i lives perpendicularly. @@ -77,65 +77,65 @@ Why not say infinity is the number that sits one unit above zero, or one divided What on earth does the idea of a point one unit above the real number line in a separate dimension have to do with squaring to negative one? So I hope to answer this for you. At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. -So let's say hypothetically I have a number, oh I don't know, let me draw one here that's going to be four plus i, okay? +So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It turns out to be relatively straightforward. If you're moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers extend in this direction. -If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. -So my question for you is simply what happens when we add these two numbers? +can get you something like it. But the rules end up being very different from that in the number system. You can't really do algebra. You can't do things like assume that if two numbers multiply to make zero, then one of them h +as to be zero. But complex numbers are going to end up behaving much like the real numbers, s Now assuming that our question system has not broken down, I should be able to do this as a proper poll and let me go ahead, I guess we can first check the previous poll, okay things seem to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of them saying that all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesome, and then after that it looks like c, people who reject the square root of negative one, fascinating, I actually would have thought that none of them would have come higher than that, none of them is much lower at a, okay so it looks like we've got a cohort of people who are comfortable with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative one might be real, let's see if we can convince you of the difference of that. -So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. +So what we've done here is we've taken three, two and then we convert it to negative two, three. Something which maybe in our original system you know looks like this negative two and then three. Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers, it's definitely one of those operations that you are going to need to know. Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. -I'm just gonna, I'm gonna have a stern word with them behind the scenes, but in the meantime, let's go ahead and move forward with the lesson here. -So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. +nice gut check here is to say what happens when we do that twice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then that first one becomes negative. So that was another 90 +degree rotation. Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b an It turns out to be relatively straightforward. -If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. -I'll go ahead and pull out black here. -The real part is going to be those four to the right, then minus two to the left, okay, straightforward enough, and then the imaginary part is going to be this one unit up and then these two units up, one plus two, times i, so is that one i plus two i, and then when you work that out four minus two is two, one plus two is three. -A nice simple introduction here. +h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. Now why am I asking you this? Well I'm being told that supposedly I'm allowed to ask you questions again so I +'m going to have you do your very first complex product. Oh look a lot of people did submit answers very good. Great let's let's grade the complex addition actually let's l +et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been the real part of the answer so maybe just the fact that there's some vertical component and you need to still add those +vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is great. -That means that it's one fewer thing for us to worry about. -What is so complex about complex numbers after all? +s just making that's just swapping up whether you're taking 4 minus 2 or 2 minus +4 so that's completely understandable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2d plane. -You have some notions like cross products and dot products that in three dimensions can get you something like it, but the rules end up being very different from that in the number system. -You can't really do algebra. +that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. Everybody ready? Aha! Wonderful! Very simple question I want you to take the number i and I +want you to multiply it by 3 plus 2i and even though I haven't really talked about You can't do things like assume that if two numbers multiply to make zero, then one of them has to be zero, but complex numbers are going to end up behaving much like the real numbers, so rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay, we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about this is we can basically just turn our paper to figure it out. -We say okay if it started at three two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. +ons as you do it rather than sitting in passively watching this is genuinely delightful to me. Okay this is this isn't necessarily a question I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, that's going to be the 90 degree rotation. -And what's nice here is that that rule is very simple and it applies to any pair that we might have. -If I took a pair of numbers a comma b, okay, and then I said where is that going to rotate to if I rotate it 90 degrees, it's going to end up by swapping the coordinates b a and then making that first one negative. +it looks like the majority of you answered negative two plus three i which is absolutely correct absolutely correct so there's two ways to think about this okay one of them is to walk forward with the algebra and +just do it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. -And a nice gut check here is to say what happens when we do that twice? -What if we do that same very mechanistic operation again twice? -And I'm going to go and take this, I swap the two coordinates, we get a negative b, but then that first one becomes negative. -So that was another 90 degree rotation. +like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa +ct that if you want to rotate numbers 90 degrees the rule is to swap the two coordinates and then multiply that first +one by negative two well look at what's happened here we've got three and two those coordinates have been swapped two is now the real part three is the imaginary part but that two got multiplied by a negative one because i has this defining feature of squaring to become negative on +e so that should give you some indication that okay multiplying by i has this action of rotating things by 90 degrees maybe tha Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees, so this will be my initial 90 degree rotation, and then another 90 degrees that's the same as a 180 degree roto- oh no I've done that wrong. -That will be the same as a 180 degree rotation which should look like this, ignore the other vector that I drew, which is just taking both of the coordinates and making them negative. -Negative a negative b, okay. +have a number that behaves this way it gives you a computational mechanism for all of the other types of rotations that you might want to do that might not necessarily be 90 degrees and to show you why this works i'm going to go ahead and pull up an animation so let's say we have any number z and in this case z is going to be let's see +where do i have it z is going to be at two uh plus i great and let's say i want to understand what is multiplying by z due to every other possible complex number well we can go one by one the very first So that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. -Now why am I asking you this? +ask what is z times one where does it take the number one well z times one is going to be Well I'm being told that supposedly I'm allowed to ask you questions again, so I'm going to have you do your very first complex product. -Oh look a lot of people did submit answers, very good. -Great let's grade the complex addition actually, let's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. +tch that arrow up to the point where z is great a kind of trivial fact even though it's trivial i'm actually going to take +a moment to write that down just so that we can oh no no no that's for that's for later that is randy don't you guys worry about him he'll be coming in in just a moment so i just want to write down three crucial facts that are getting an influence rotation three facts i'll call it three facts about multiplication the first two Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i, very good, very good. 52 of you answered simply 2 which would have been the real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answered 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. -Some of you answered negative 2 3 which I guess is just making- that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. -We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. -Let's go ahead and try our very first product though like I said. -So here because I already talked through one of the questions we're going to go ahead and skip ahead of it. -We know how to rotate something like 3 comma 2 so I'm not even going to give you time to do that and properly grade it. +hatever the 90 degree rotation point for z itself is okay so two down infinitely many to go okay we know what it does to one we know what it does to i let's see if we can understand what +z does to any other possible number well it turns out those two is really all we need to work with if we have the distributive property so the third fact that's going to look kind of innocuous is let's say i take this z and i multiply it by c plus d times i where c and d are just any two numbers okay well this is going to distribute so z times c i'm actually going +to write that a little differently i'm going to write it as c times z plus z times di which again i'm going to write in kind of a funny order and write that as d times z i now the idea here is well we know where z is we also know where z times i is s +o if we're just scaling them up by some other constants that completely constrains where we need to go so let me go ahead and wr +ite this down with an example okay let's say that we go back here and i want to know what multiplying by z does to anything i want to tell i want to convince you that it Stal, stal, words, words. -You know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey Kamineter can't you make the live questions work a little bit better for us? -Okay I think we're finally there. -Everybody ready? +n a way that keeps these lines parallel it keeps them evenly spaced keeps them perpendicular to each other it applies this very constrained rule to the whole plane and really just think through any one particular point for this let's say that we have two times negative i okay so you move two units in the positive right direction and then negative one unit in the vertical direction well after the product where that's going to land has to be two times wherever z lands plus negative one times wherever i la +nds okay and we see that right it's two times this yellow vector and it'll be negative one times the gree +n vector so here even before you actually work out the product we could just read off Aha wonderful. Very simple question. -I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers. -You've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative 1. +which says what is two plus i times two minus i and if you have notes right now if you have a pencil and paper which i encourage you to always come to class with i want you to try working it out do the first inside outside last distribution property just to see mechanistically what number en +ds up popping out from this and then we'll try to see how that squares with the geometric intuition so while you're doing that while you're working that out hopefully on pencil and paper it looks like we've got a question from the audience which is is i the same as i and j the vectors in physics great question actually That's the only special thing you need to know about that. Other than that just treat it like it's a normal number okay and then proceed forward with the product. Wonderful okay so it looks like we've got quite a few of you coming in to answer which is always lovely. diff --git a/2020/ldm-complex-numbers/french/sentence_translations.json b/2020/ldm-complex-numbers/french/sentence_translations.json index d5828a931..1f585d9da 100644 --- a/2020/ldm-complex-numbers/french/sentence_translations.json +++ b/2020/ldm-complex-numbers/french/sentence_translations.json @@ -312,7 +312,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right?", + "input": "stalling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the", "translatedText": "Je dirais que si c'est quelque chose qui est réellement utile dans une application, alors c'est aussi réel que les mots, n'est-ce pas?", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different.", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality.", "translatedText": "Vous ne rencontrerez jamais un mot abstrait comme bonheur, mais il a une sorte de réalité dans nos esprits, et des choses comme la racine carrée de deux, que vous ne pouvez pas exprimer sous forme de fraction, ou des choses comme le racine carrée de moins un qui n'apparaît pas parmi les nombres réels normaux, vous savez, même s'ils peuvent sembler un peu différents.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 270.78 }, { - "input": "Oh, this is such a shame.", + "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay?", "translatedText": "Oh, c'est vraiment dommage.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out.", + "input": "esting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just trigonometry, it's everything we were talking about la", "translatedText": "Cependant, si vous y arrivez avec des nombres complexes, non seulement cela est beaucoup moins sujet aux erreurs, mais cela a une très belle signification et cela tombe tout simplement.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too.", + "input": "st time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of", "translatedText": "Alors même si on ne croit pas forcément à la réalité de la racine carrée de moins 1, il faut au moins admettre que c'est intéressant que ça puisse rendre d'autres mathématiques utiles, que d'autres mathématiques un peu plus compréhensible aussi.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right?", + "input": "It's something that's very error-prone if you're just trying", "translatedText": "L'un l'est, non, il n'y en a pas, n'est-ce pas?", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive.", + "input": "ex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out", "translatedText": "N'importe quel nombre que vous mettez au carré, s'il est positif, eh bien, il reste positif.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative.", + "input": "So even if you don't necessarily believe in the reality of", "translatedText": "Je n'aurai jamais rien de négatif.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 437.86 }, { - "input": "So this does not exist, no such number.", + "input": "the square root of negative one, you at the very least have to admit that it's interesting that it can make o", "translatedText": "Cela n’existe donc pas, ce numéro n’existe pas.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case.", + "input": "ther pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case.", "translatedText": "Cependant, si un mathématicien vient et dit : oh non non, ça existe, nous l'avons défini pour que ce soit le cas.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution.", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wo", "translatedText": "Lorsque vous avez un problème que vous ne pouvez pas résoudre, vous pouvez simplement dire : « oh, j'ai défini les choses pour que nous ayons maintenant, comme par magie, une solution ».", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone.", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home.", "translatedText": "Donc, si cela vous met mal à l’aise, vous n’êtes certainement pas seul.", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory.", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative,", "translatedText": "En fait, René Descartes a inventé le terme imaginaire pour désigner ces nombres de manière péjorative.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right?", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you", "translatedText": "Et, d'accord, si nous voulons étendre notre système de numérotation, je comprends, c'est peut-être utile de mettre une sorte de numéro là-haut, mais pourquoi, n'est-ce pas?", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors.", + "input": "have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question.", "translatedText": "Au tout début, parlons simplement de la façon dont si vous ajoutez des nombres bidimensionnels comme celui-ci, les règles sont assez simples et cela fonctionne essentiellement de la même manière que les vecteurs, pour tous ceux d'entre vous qui sont familiers avec les vecteurs.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 662.3 }, { - "input": "None of them is much lower at a.", + "input": "one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to", "translatedText": "Aucun d'entre eux n'est beaucoup plus bas à a.", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real.", + "input": ", why should that live there? What on earth does the idea of a point one unit above the real number line in a separate dimension have to do with squaring to negative one? So I hope to answer this for you. At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be famili", "translatedText": "D'accord, il semblerait que nous ayons une cohorte de personnes qui sont à l'aise avec le négatif 1, une grande cohorte qui n'est pas à l'aise avec l'infini, c'est un sujet pour un autre jour, ne vous inquiétez pas, et puis un certain nombre de personnes qui sont en quelque sorte à mi-chemin et ne sont peut-être pas très à l'aise avec l'idée que le négatif 1 pourrait être réel.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two.", + "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It turns out to be r", "translatedText": "Donc, pour notre première question beaucoup plus mathématique, en guise d’échauffement, je veux juste vous demander d’ajouter ces deux-là.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 706.82 }, { - "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me.", + "input": "ative two plus two i. So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers extend in this direction. can get you something lik", "translatedText": "Malheureusement, et vous pouvez le constater au fait que je traîne et à ce que je dis ici, il semble que la question ne se charge toujours pas complètement correctement, donc je vais avoir un mot sévère avec Cam et Ider derrière le scènes qui ont par ailleurs construit une interface si belle, si belle, utile pour ce genre de va-et-vient entre vous et moi.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 726.28 }, { - "input": "I'm going to have a stern word with them behind the scenes, but in the meantime let's go ahead and move forward with the lesson here.", + "input": "e it. But the rules end up being very different from that in the number system. You can't really do algebra. You can't do things like assume that if two numbers multiply to make zero, then", "translatedText": "Je vais avoir un mot sévère avec eux dans les coulisses, mais en attendant, allons de l'avant et avançons avec la leçon ici.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 732.64 }, { - "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be.", + "input": "one of them h as to be zero. But complex numbers are going to end up behaving much like the real numbers, s Now assuming that our question system ha", "translatedText": "Donc je suppose que je peux le mettre sur le, juste sur le morceau de papier, et vous pourrez suivre à la maison, voir quel pourrait être l'ajout.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 739.92 }, { - "input": "It turns out to be relatively straightforward.", + "input": "s not broken down, I should be able to do this as a proper poll and let me go ahead, I guess we can first check the previous poll, okay things se", "translatedText": "Cela s’avère relativement simple.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time.", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of th", "translatedText": "Si vous déplacez quatre unités vers la droite, puis une unité vers le haut, et que vous souhaitez ajouter l'idée de déplacer deux unités vers la gauche, puis deux unités vers le haut, eh bien, vous faites simplement chacune de ces unités une à la fois.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left.", + "input": "red real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a g", "translatedText": "La vraie partie sera ces quatre à droite, puis moins deux à gauche.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 760.88 }, { - "input": "And then the imaginary part is going to be this one unit up and then these two units up, one plus two, times i.", + "input": "you out there who would just reject infinity as being considered real but are very comfortable with the square root of negative o", "translatedText": "Et puis la partie imaginaire sera cette unité en haut, puis ces deux unités en haut, une plus deux, multipliées par i.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great.", + "input": "are root of negative one, fascinating, I actually would have thought that none of them would have come higher than t", "translatedText": "Addition ne se passe pas vraiment quelque chose de compliqué, ce qui est génial.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 784.2 }, { - "input": "What is so complex about complex numbers after all?", + "input": "m is much lower at a, okay so it looks like we've got a cohort of people who are comfortable with negative one, a la", "translatedText": "Après tout, qu’y a-t-il de si complexe dans les nombres complexes?", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 787.1 }, { - "input": "Well where everything becomes interesting is when you try to multiply these numbers together.", + "input": "rge cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people w", "translatedText": "Eh bien, là où tout devient intéressant, c'est lorsque vous essayez de multiplier ces nombres entre eux.", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 803.68 }, { - "input": "But the rules end up being very different from that in the number system.", + "input": "mfortable with the idea that negative one might be real, let's see if we can convince you of the difference of t", "translatedText": "Mais les règles finissent par être très différentes de celles du système numérique.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 806.86 }, { - "input": "You can't really do algebra.", + "input": "hat. So what we've done here is we've taken three, two and then", "translatedText": "Vous ne pouvez pas vraiment faire d'algèbre.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 817.78 }, { - "input": "But to understand what that multiplication rule is, I just want to ask you a simple question.", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, a", "translatedText": "Mais pour comprendre quelle est cette règle de multiplication, je veux juste vous poser une question simple.", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 831.82 }, { - "input": "We're not even going to think of it as a complex number per se.", + "input": "part of this, but it is, it's a good thing to know when you're learning about complex numbers, it'", "translatedText": "Nous n’allons même pas le considérer comme un nombre complexe en soi.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this?", + "input": "s definitely one of those operations that you are going to need to know. Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is stil", "translatedText": "Si j'ai juste une sorte de grille de coordonnées et que je vais au point avec la coordonnée x trois et la coordonnée y deux, quelle est la rotation de 90 degrés de celle-ci?", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 859.44 }, { - "input": "Okay.", + "input": "built such a beautiful, beautiful inte", "translatedText": "D'accord.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out.", + "input": "rface that's helpful for this kind of back and forth between you guys and me. nice gut check here is", "translatedText": "Ce qui est bien, c'est que nous pouvons simplement tourner notre papier pour le comprendre.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three.", + "input": "e. So that was another 90 degree rotation. Well what's happened here is we've just made both of the coordinates negative and that's", "translatedText": "Donc, ce que nous avons fait ici, c'est prendre trois, deux, puis nous le convertissons en moins deux, trois.", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 890.68 }, { - "input": "Something which maybe in our original system you know looks like this negative two and then three.", + "input": "reassuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them", "translatedText": "Quelque chose qui, peut-être dans notre système original, ressemble à ce moins deux puis trois.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation.", + "input": "negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation", "translatedText": "Ce sera la rotation de 90 degrés.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 899.9 }, { - "input": "And what's nice here is that that rule is very simple and it applies to any pair that we might have.", + "input": "actually behaves like you would expect it to. Now why am I asking you this? Well I'm being told that supposedly I'm allowed to ask you questions again so I 'm going to ha", "translatedText": "Et ce qui est bien ici, c'est que cette règle est très simple et s'applique à n'importe quelle paire que nous pourrions avoir.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong.", + "input": "52 of you answered simply 2 which would have been the real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated goin", "translatedText": "Eh bien, ce qui s'est passé ici, c'est que nous venons de rendre les deux coordonnées négatives et c'est rassurant parce que si je prends un point assis en ab et que je le fais pivoter de 90 degrés, ce sera ma rotation initiale de 90 degrés, puis un autre de 90 degrés, c'est le pareil pour la pourriture à 180 degrés - oh non, j'ai mal fait.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 980.14 }, { - "input": "Now why am I asking you this?", + "input": "you try to multiply these numbers together. So with vectors, there's not really any notion", "translatedText": "Maintenant, pourquoi je te demande ça?", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 981.76 }, { - "input": "Well I'm being told that supposedly I'm allowed to ask you questions again so I'm going to have you do your very first complex product.", + "input": "of multiplying them to get two vectors back, at least when we're in the 2d plane. that we ask questions and just say hey kamineter can't you make the live questio", "translatedText": "Eh bien, on me dit que je suis censé être autorisé à vous poser à nouveau des questions, donc je vais vous demander de réaliser votre tout premier produit complexe.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i.", + "input": "Wonderful! Very simple question I want you to take the number i and I want you to multiply it by 3 pl", "translatedText": "D'accord, il semble que la majorité d'entre vous ait obtenu la bonne réponse, soit 2 plus 3i.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1006.98 }, { - "input": "Very good very good.", + "input": "us 2i and even though I haven't really talked about You can't do things like assume that if two numbers multiply to mak", "translatedText": "Très bien très bien.", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good.", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t", "translatedText": "Nous avons 2 plus 3, ce qui est peut-être juste une baisse du i, donc je pense que c'est peut-être un peu comme de simples erreurs et entrées et vous savez que cela nous arrive à tous, en particulier lors des tests, parfois vous savez quelle est la bonne réponse, mais ensuite tu oublies un symbole ou tu en échange deux donc c'est très bien.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1052.36 }, { - "input": "Let's go ahead and try our very first product though like I said so here because I already talked through one of the questions we're going to go ahead and skip ahead of it we know how to rotate something like 3 comma 2 so I'm not even going to give you time to do that and properly grade it.", + "input": "his is we can basically just turn our paper to figure it out. ons as you do it rather than sitting in passively watching this is genuinely delightful to me. Okay this is this isn't necessarily a question I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not", "translatedText": "Allons-y et essayons notre tout premier produit, mais comme je l'ai dit ici parce que j'ai déjà répondu à l'une des questions, nous allons aller de l'avant et passer devant, nous savons comment faire pivoter quelque chose comme 3 virgules 2, donc je suis je ne vous laisserai même pas le temps de le faire et de le noter correctement.", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us?", + "input": "that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three i which is absolutely correct absolutely correct so there's two w", "translatedText": "Stal stal mots mots vous savez, ils me disent que ça marche et pourtant c'est très lent pour moi de progresser donc vous savez que si je ne vais pas avoir un mot sévère avec eux, vous pouvez aussi les aborder sur Twitter sous le même endroit où nous posons des questions et disons simplement salut kamineter, ne pouvez-vous pas faire en sorte que les questions en direct fonctionnent un peu mieux pour nous?", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there.", + "input": "ays to think about this okay one of them is", "translatedText": "D'accord, je pense que nous y sommes enfin.", "model": "google_nmt", "n_reviews": 0, @@ -1200,7 +1200,7 @@ "end": 1094.22 }, { - "input": "Aha!", + "input": "e algebra and just do it a little bit mechanistically okay so", "translatedText": "Ah!", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1094.76 }, { - "input": "Wonderful!", + "input": "if we pull ourselves up", "translatedText": "Merveilleux!", "model": "google_nmt", "n_reviews": 0, @@ -1224,7 +1224,7 @@ "end": 1126.42 }, { - "input": "Wonderful!", + "input": "that if you want to rotate numbers 90 degrees", "translatedText": "Merveilleux!", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/german/sentence_translations.json b/2020/ldm-complex-numbers/german/sentence_translations.json index dfd9e82e3..9517ce2f3 100644 --- a/2020/ldm-complex-numbers/german/sentence_translations.json +++ b/2020/ldm-complex-numbers/german/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "Ich würde sagen, wenn es etwas ist, das in einer Anwendung tatsächlich nützlich ist, dann ist es genauso real wie Worte, oder? ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "Man wird da draußen nie auf ein abstraktes Wort wie Glück stoßen, aber es hat eine Art Realität in unserem Kopf und Dinge wie die Quadratwurzel aus zwei, die man nicht als Bruch ausdrücken kann, oder Dinge wie das Quadratwurzel aus der negativen Eins, die bei echten Normalzahlen nicht auftauchen, wissen Sie, auch wenn sie vielleicht etwas anders erscheinen. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "Ich gehe nicht davon aus, dass Sie schon wissen, was sie sind, es soll nur eine Grundeinführung sein, aber lasst uns einfach loslegen, okay? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "Ich erinnere mich, als ich in der Schule war und wir diese Additionsformeln lernten, dass, wenn man den Kosinus der Summe zweier verschiedener Winkel wissen möchte, es sich um eine so lange Sache handelt, ausgedrückt als Kosinus und Sinus der beiden ursprünglichen Winkel , es gibt dieses Minuszeichen, das die Leute immer zum Stolpern bringen würde, wenn man das Gleiche für das Zeichen macht, sieht es ähnlich aus, aber es gibt ein Pluszeichen, und statt cos-cos gibt es cos-sünde, das ist etwas, das sehr fehleranfällig ist wenn Sie nur versuchen, es sich so einzuprägen, wie es ist. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "Wenn man es jedoch mit komplexen Zahlen angeht, ist dies nicht nur viel weniger fehleranfällig, es hat auch eine sehr schöne Bedeutung und fällt einfach sofort auf. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "Auch wenn Sie also nicht unbedingt an die Realität der Quadratwurzel von minus 1 glauben, müssen Sie zumindest zugeben, dass es interessant ist, dass sie andere Teile der Mathematik nützlich machen kann, dass andere Teile der Mathematik ein bisschen mehr nützlich sind auch verständlich. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "Aber der Ausgangspunkt sieht sehr seltsam aus, okay? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "Eines ist, nein, das gibt es nicht, oder? ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "Jede Zahl, die man quadriert, wenn sie positiv ist, dann bleibt sie einfach positiv. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "Ich werde nie etwas Negatives bekommen. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "Wenn jedoch ein Mathematiker kommt und sagt: „Oh nein, nein, es existiert“, dann haben wir es so definiert, dass das der Fall ist. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "Wenn Ihnen das also unangenehm ist, sind Sie definitiv nicht allein. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "Tatsächlich hat Rene Descartes den Begriff „imaginär“ für diese Zahlen als abwertende Bezeichnung geprägt. ", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "Und dann sind wir bei dieser Konvention geblieben und nennen sie immer noch imaginäre Zahlen, was wirklich absurd ist. ", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "Die zweite seltsame Sache, die Sie machen, wenn Sie anfangen, über komplexe Zahlen zu sprechen, ist zu sagen: Es gibt nicht nur so eine Zahl i, sondern wir werden ihr ein Zuhause geben. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "Und, okay, wenn wir unser Zahlensystem erweitern wollen, verstehe ich, vielleicht ist es nützlich, da oben irgendeine Zahl anzugeben, aber warum ich, oder? ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "Lassen Sie uns gleich zu Beginn darüber sprechen, dass beim Addieren von zweidimensionalen Zahlen auf diese Weise die Regeln ziemlich einfach sind und im Wesentlichen genauso funktionieren wie bei Vektoren, für alle, die sich mit Vektoren auskennen. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "Wenn Sie das glauben und befolgen, hilft die Tatsache, dass es nützlich wird, hoffentlich dabei, zu rechtfertigen, warum wir das alles tun. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "Ähm, es sieht so aus, als gäbe es ein Hin und Her zwischen den Antworten f und d, also ist f alle davon, was bedeutet, dass diese alle als real angesehen werden sollten. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "Und interessanterweise ist d diejenige, die besagt, dass man 2 Quadratwurzel aus 2 und minus 1 in Betracht ziehen sollte, aber nicht die Unendlichkeit. Es gibt also eine ganze Menge von euch da draußen, die die Unendlichkeit einfach als real ablehnen würden, sich aber damit sehr wohl fühlen Quadratwurzel von minus 1, das ist großartig. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "Okay, es sieht so aus, als hätten wir eine Kohorte von Leuten, die sich mit der negativen 1 wohlfühlen, eine große Kohorte fühlt sich mit der Unendlichkeit unwohl, das ist ein Thema für einen anderen Tag, machen Sie sich darüber keine Sorgen, und dann eine Reihe von Leuten, die sich mit der Unendlichkeit nicht wohlfühlen Ich befinde mich irgendwie in der Mitte und fühle mich mit der Vorstellung, dass die negative 1 real sein könnte, vielleicht nicht besonders wohl. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "Für unsere erste, viel mathematischere Frage möchte ich Sie zum Aufwärmen nur bitten, diese beiden hinzuzufügen. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "Bevor ich Ihnen beigebracht habe, wie man sie hinzufügt, raten Sie mal, wie es funktionieren könnte, und ich hoffe, dass es sich ziemlich einfach anfühlt. Das Hinzufügen ist eigentlich der am wenigsten interessante Teil davon, aber es ist eine gute Sache, zu wissen, wann Sie lernen etwas über komplexe Zahlen. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "Wenn Sie vier Einheiten nach rechts und dann eine Einheit nach oben bewegen und die Idee hinzufügen möchten, zwei Einheiten nach links und dann zwei Einheiten nach oben zu verschieben, tun Sie dies einfach einzeln. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "Der reale Teil wird diese vier auf der rechten Seite sein, dann minus zwei auf der linken Seite. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "Also ist das eins i plus zwei i. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "Bei Addition gibt es eigentlich nichts Kompliziertes, was großartig ist. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "Bei Vektoren gibt es also nicht wirklich die Idee, sie zu multiplizieren, um wieder zwei Vektoren zu erhalten, zumindest wenn wir uns in der 2D-Ebene befinden. ", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "Das heißt im Grunde genommen, nehmen wir an, ich habe den Punkt drei, zwei. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "Wenn ich nur eine Art Koordinatengitter habe und zu dem Punkt mit der X-Koordinate drei und der Y-Koordinate zwei gehe, wie groß ist dann die 90-Grad-Drehung? ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "Gegen den Uhrzeigersinn. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "Das Tolle daran ist, dass wir einfach unsere Zeitung umdrehen können, um es herauszufinden. ", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "Sie sagen: Okay, wenn es bei drei, zwei angefangen hat und ich mich dann um 90 Grad gegen den Uhrzeigersinn drehe, kann ich das jetzt einfach als minus zwei in x-Richtung und dann drei in y-Richtung ablesen, wenn ich die gesamte Ebene so gedreht hätte . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "Was wir hier also gemacht haben, ist, drei, zwei zu nehmen und es dann in minus zwei, drei umzuwandeln. ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "Das wird die 90-Grad-Drehung sein. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "Wenn ich ein Zahlenpaar a Komma b nimm, okay, und dann sage ich, wohin soll das gedreht werden? ", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "Das war also eine weitere 90-Grad-Drehung. ", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "Nun, was hier passiert ist, ist, dass wir einfach beide Koordinaten negativ gemacht haben, und das ist beruhigend, denn wenn ich einen Punkt nehme, der bei ab sitzt, und ihn dann um 90 Grad drehe, ist dies meine anfängliche Drehung um 90 Grad und dann noch einmal um 90 Grad Das Gleiche wie 180-Grad-Rotation – oh nein, das habe ich falsch gemacht. ", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "Das ist das Gleiche wie eine 180-Grad-Drehung, die so aussehen sollte. Ignorieren Sie den anderen Vektor, den ich gezeichnet habe, der nur beide Koordinaten nimmt und sie negativ negativ a negativ b macht. Okay, das ist beruhigend für diese Operation, die eine 90-Grad-Drehung durchführt verhält sich tatsächlich so, wie man es erwarten würde. ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "Oh, sehen Sie, viele Leute haben sehr gute Antworten eingereicht. ", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "Okay, es sieht so aus, als hätten die meisten von Ihnen die richtige Antwort erhalten, nämlich 2 plus 3i. ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "Sehr gut sehr gut. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "Einige von Ihnen haben mit „2 3“ verneint geantwortet, was meiner Meinung nach nur darauf zurückzuführen ist, dass Sie einfach vertauschen, ob Sie 4 minus 2 oder 2 minus 4 nehmen, also ist das völlig verständlich. ", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "Wir haben 2 plus 3, was vielleicht einfach aus dem i herausfällt, also denke ich, dass es sich vielleicht um einfache Fehler und Eingaben handelt, und Sie wissen, dass uns allen vor allem bei Tests passiert, dass man manchmal weiß, was die richtige Antwort ist, aber dann Wenn man ein Symbol vergisst oder zwei vertauscht, ist das alles sehr gut. ", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "Stal-Stal-Wörter, Sie wissen, dass sie mir sagen, dass es funktioniert, und dennoch komme ich nur sehr langsam voran. Wenn ich also kein strenges Wort mit ihnen wechseln möchte, können Sie sie unter dem gleichen Link auch auf Twitter angreifen Ort, an dem wir Fragen stellen und einfach sagen: „Hey Kamineter“, kannst du die Live-Fragen nicht ein bisschen besser für uns gestalten? ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "Okay, ich denke, wir sind endlich da. ", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "Alle bereit? ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "Aha! ", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "Wunderbar! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "Sehr einfache Frage: Ich möchte, dass Sie die Zahl i nehmen und sie mit 3 plus 2i multiplizieren. Auch wenn ich nicht wirklich über die Regeln für die Multiplikation gesprochen habe, kann ich nur sagen, dass sie so funktioniert, als ob sie genauso funktioniert Bei normalen Zahlen gibt es Dinge wie die Verteilungseigenschaft, mit der man dies überall verteilen kann, und dann ist das bestimmende Merkmal von i die Idee, dass i quadriert negativ ist, das ist das einzig Besondere, was man darüber wissen muss, abgesehen davon, dass man es einfach behandelt als wäre es eine normale Nummer, okay, und fahren Sie dann mit dem Produkt fort. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/hebrew/sentence_translations.json b/2020/ldm-complex-numbers/hebrew/sentence_translations.json index d748a174e..275ebbb9a 100644 --- a/2020/ldm-complex-numbers/hebrew/sentence_translations.json +++ b/2020/ldm-complex-numbers/hebrew/sentence_translations.json @@ -273,21 +273,21 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right?", + "input": "stalling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the", "translatedText": "הייתי אומר שאם זה משהו שבאמת שימושי באפליקציה, אז הוא אמיתי כמו מילים, נכון?", "n_reviews": 0, "start": 247.68, "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different.", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality.", "translatedText": "לעולם לא תתקל במילה מופשטת כמו אושר בחוץ, אבל יש לה סוג של מציאות במוחנו, ודברים כמו השורש של שניים, שאתה לא יכול לבטא כשבר, או דברים כמו ה שורש ריבועי של שלילי אחד שלא מופיע בין מספרים נורמליים אמיתיים, אתה יודע, גם אם הם עשויים להיראות קצת שונים.", "n_reviews": 0, "start": 253.74, "end": 270.78 }, { - "input": "Oh, this is such a shame.", + "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay?", "translatedText": "הו, זה כל כך חבל.", "n_reviews": 0, "start": 271.5, @@ -343,14 +343,14 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out.", + "input": "esting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just trigonometry, it's everything we were talking about la", "translatedText": "עם זאת, אם אתה בא על זה עם מספרים מרוכבים, זה לא רק שהוא הרבה פחות מועד לשגיאות, יש לזה משמעות מאוד יפה וזה פשוט נופל החוצה.", "n_reviews": 0, "start": 363.98, "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too.", + "input": "st time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of", "translatedText": "אז גם אם אתה לא בהכרח מאמין במציאות של השורש הריבועי של שלילית 1, אתה לכל הפחות חייב להודות שזה מעניין שזה יכול להפוך חלקים אחרים של מתמטיקה שימושיים, שחלקים אחרים של מתמטיקה קצת יותר גם מובן.", "n_reviews": 0, "start": 372.1, @@ -392,7 +392,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right?", + "input": "It's something that's very error-prone if you're just trying", "translatedText": "האחת היא, לא אין, נכון?", "n_reviews": 0, "start": 414.88, @@ -406,7 +406,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive.", + "input": "ex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out", "translatedText": "כל מספר שאתה בריבוע, אם הוא חיובי, ובכן, זה פשוט נשאר חיובי.", "n_reviews": 0, "start": 427.76, @@ -420,21 +420,21 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative.", + "input": "So even if you don't necessarily believe in the reality of", "translatedText": "אני לעולם לא אקבל שום דבר שלילי.", "n_reviews": 0, "start": 436.38, "end": 437.86 }, { - "input": "So this does not exist, no such number.", + "input": "the square root of negative one, you at the very least have to admit that it's interesting that it can make o", "translatedText": "אז זה לא קיים, אין מספר כזה.", "n_reviews": 0, "start": 438.48, "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case.", + "input": "ther pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case.", "translatedText": "עם זאת, אם בא מתמטיקאי ואומר, הו לא לא זה קיים, הגדרנו את זה כך שזה המצב.", "n_reviews": 0, "start": 442.28, @@ -448,7 +448,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution.", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wo", "translatedText": "כשיש לך בעיה שאתה לא יכול לפתור, אתה יכול פשוט לומר, הו, הגדרתי דברים כך שעכשיו יש לנו פתרון בקסם.", "n_reviews": 0, "start": 452.86, @@ -462,14 +462,14 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone.", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home.", "translatedText": "אז אם אתה מרגיש לא בנוח עם זה, אתה בהחלט לא לבד.", "n_reviews": 0, "start": 466.48, "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory.", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative,", "translatedText": "למעשה, רנה דקארט טבע את המונח דמיוני למספרים אלה כגנאי.", "n_reviews": 0, "start": 470.06, @@ -525,7 +525,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right?", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you", "translatedText": "ובכן, אם אנחנו רוצים להרחיב את מערכת המספרים שלנו, אני מבין, אולי זה שימושי לשים שם איזה מספר, אבל למה אני, נכון?", "n_reviews": 0, "start": 526.5, @@ -553,7 +553,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors.", + "input": "have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question.", "translatedText": "בהתחלה, בוא נדבר רק על כך שאם אתה מוסיף מספרים שהם דו מימדיים כמו זה, הכללים הם די פשוטים וזה פועל בעצם כמו וקטורים, עבור כל אחד מכם שאולי מכיר וקטורים.", "n_reviews": 0, "start": 557.3, @@ -630,14 +630,14 @@ "end": 662.3 }, { - "input": "None of them is much lower at a.", + "input": "one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to", "translatedText": "אף אחד מהם לא נמוך בהרבה ב-a.", "n_reviews": 0, "start": 662.7, "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real.", + "input": ", why should that live there? What on earth does the idea of a point one unit above the real number line in a separate dimension have to do with squaring to negative one? So I hope to answer this for you. At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be famili", "translatedText": "אוקיי, אז נראה שיש לנו קבוצה של אנשים שנוח להם עם 1 שלילי, קבוצה גדולה לא מרגישה בנוח עם אינסוף, זה נושא ליום אחר, אל תדאג בקשר לזה, ואז מספר אנשים ש נמצאים בערך בנקודת הביניים הזו של אולי לא נוחים במיוחד עם הרעיון שהשלילי 1 עשוי להיות אמיתי.", "n_reviews": 0, "start": 665.1, @@ -651,7 +651,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two.", + "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It turns out to be r", "translatedText": "אז לשאלה הרבה יותר מתמטית הראשונה שלנו, כסוג של חימום, אני רק רוצה לבקש מכם להוסיף את שני אלה.", "n_reviews": 0, "start": 683.42, @@ -672,35 +672,35 @@ "end": 706.82 }, { - "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me.", + "input": "ative two plus two i. So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers extend in this direction. can get you something lik", "translatedText": "למרבה הצער, ואתה יכול להבין לפי העובדה שאני מתעכב ומה שאני אומר כאן, נראה שהשאלה עדיין לא נטענת לגמרי כהלכה, אז אני הולך לדבר מילה חמורה עם קאם ואידר מאחורי סצנות שבנו אחרת ממשק כל כך יפה ויפה שעוזר לסוג כזה של הלוך ושוב בינכם וביני.", "n_reviews": 0, "start": 707.84, "end": 726.28 }, { - "input": "I'm going to have a stern word with them behind the scenes, but in the meantime let's go ahead and move forward with the lesson here.", + "input": "e it. But the rules end up being very different from that in the number system. You can't really do algebra. You can't do things like assume that if two numbers multiply to make zero, then", "translatedText": "אני הולך לדבר איתם בדיבור חמור מאחורי הקלעים, אבל בינתיים בואו נתקדם ונתקדם עם השיעור כאן.", "n_reviews": 0, "start": 726.76, "end": 732.64 }, { - "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be.", + "input": "one of them h as to be zero. But complex numbers are going to end up behaving much like the real numbers, s Now assuming that our question system ha", "translatedText": "אז אני מניח שאני יכול למשוך את זה על הדף, רק על פיסת הנייר, ואתה יכול לעקוב אחרי הבית, לראות מה התוספת יכולה להיות.", "n_reviews": 0, "start": 733.46, "end": 739.92 }, { - "input": "It turns out to be relatively straightforward.", + "input": "s not broken down, I should be able to do this as a proper poll and let me go ahead, I guess we can first check the previous poll, okay things se", "translatedText": "מסתבר שזה פשוט יחסית.", "n_reviews": 0, "start": 740.24, "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time.", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of th", "translatedText": "אם אתה מעביר ארבע יחידות ימינה ואז יחידה אחת למעלה, ואתה רוצה להוסיף את הרעיון של הזזת שתי יחידות שמאלה ואז שתי יחידות למעלה, ובכן, פשוט תעשה כל אחת מהן בכל פעם.", "n_reviews": 0, "start": 742.94, @@ -714,7 +714,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left.", + "input": "red real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a g", "translatedText": "החלק האמיתי יהיה הארבעה מימין, ואז מינוס שניים משמאל.", "n_reviews": 0, "start": 754.82, @@ -728,7 +728,7 @@ "end": 760.88 }, { - "input": "And then the imaginary part is going to be this one unit up and then these two units up, one plus two, times i.", + "input": "you out there who would just reject infinity as being considered real but are very comfortable with the square root of negative o", "translatedText": "ואז החלק הדמיוני הולך להיות יחידה אחת למעלה ואז שתי היחידות האלה למעלה, אחת ועוד שתיים, כפול i.", "n_reviews": 0, "start": 761.16, @@ -756,7 +756,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great.", + "input": "are root of negative one, fascinating, I actually would have thought that none of them would have come higher than t", "translatedText": "הוספה לא באמת מתרחשת משהו מסובך, וזה נהדר.", "n_reviews": 0, "start": 777.86, @@ -770,14 +770,14 @@ "end": 784.2 }, { - "input": "What is so complex about complex numbers after all?", + "input": "m is much lower at a, okay so it looks like we've got a cohort of people who are comfortable with negative one, a la", "translatedText": "מה בכל זאת מורכב במספרים מרוכבים?", "n_reviews": 0, "start": 784.42, "end": 787.1 }, { - "input": "Well where everything becomes interesting is when you try to multiply these numbers together.", + "input": "rge cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people w", "translatedText": "ובכן, המקום בו הכל הופך למעניין הוא כאשר אתה מנסה להכפיל את המספרים הללו יחד.", "n_reviews": 0, "start": 787.64, @@ -798,14 +798,14 @@ "end": 803.68 }, { - "input": "But the rules end up being very different from that in the number system.", + "input": "mfortable with the idea that negative one might be real, let's see if we can convince you of the difference of t", "translatedText": "אבל הכללים בסופו של דבר שונים מאוד מזה שבמערכת המספרים.", "n_reviews": 0, "start": 803.68, "end": 806.86 }, { - "input": "You can't really do algebra.", + "input": "hat. So what we've done here is we've taken three, two and then", "translatedText": "אתה לא באמת יכול לעשות אלגברה.", "n_reviews": 0, "start": 806.86, @@ -826,7 +826,7 @@ "end": 817.78 }, { - "input": "But to understand what that multiplication rule is, I just want to ask you a simple question.", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, a", "translatedText": "אבל כדי להבין מהו כלל הכפל הזה, אני רק רוצה לשאול אותך שאלה פשוטה.", "n_reviews": 0, "start": 818.3, @@ -840,14 +840,14 @@ "end": 831.82 }, { - "input": "We're not even going to think of it as a complex number per se.", + "input": "part of this, but it is, it's a good thing to know when you're learning about complex numbers, it'", "translatedText": "אנחנו אפילו לא הולכים לחשוב על זה כעל מספר מרוכב כשלעצמו.", "n_reviews": 0, "start": 832.58, "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this?", + "input": "s definitely one of those operations that you are going to need to know. Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is stil", "translatedText": "אם יש לי רק איזושהי רשת קואורדינטות ואני הולך לנקודה עם קואורדינטת x שלוש וקואורדינטת y שתיים, מה הסיבוב של 90 מעלות של זה?", "n_reviews": 0, "start": 835.72, @@ -875,14 +875,14 @@ "end": 859.44 }, { - "input": "Okay.", + "input": "built such a beautiful, beautiful inte", "translatedText": "בסדר.", "n_reviews": 0, "start": 865.28, "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out.", + "input": "rface that's helpful for this kind of back and forth between you guys and me. nice gut check here is", "translatedText": "עכשיו מה שמקסים בזה הוא שבעצם אנחנו יכולים פשוט להפוך את העיתון שלנו כדי להבין את זה.", "n_reviews": 0, "start": 865.76, @@ -896,28 +896,28 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three.", + "input": "e. So that was another 90 degree rotation. Well what's happened here is we've just made both of the coordinates negative and that's", "translatedText": "אז מה שעשינו כאן זה שלקחנו שלוש, שתיים ואז נמיר את זה לשתיים, שלוש.", "n_reviews": 0, "start": 884.08, "end": 890.68 }, { - "input": "Something which maybe in our original system you know looks like this negative two and then three.", + "input": "reassuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them", "translatedText": "משהו שאולי במערכת המקורית שלנו אתה מכיר נראה כמו שתיים השליליות הזה ואז שלוש.", "n_reviews": 0, "start": 891.58, "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation.", + "input": "negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation", "translatedText": "זה הולך להיות הסיבוב של 90 מעלות.", "n_reviews": 0, "start": 898.1, "end": 899.9 }, { - "input": "And what's nice here is that that rule is very simple and it applies to any pair that we might have.", + "input": "actually behaves like you would expect it to. Now why am I asking you this? Well I'm being told that supposedly I'm allowed to ask you questions again so I 'm going to ha", "translatedText": "ומה שיפה כאן הוא שהכלל הזה מאוד פשוט והוא חל על כל זוג שיש לנו.", "n_reviews": 0, "start": 899.9, @@ -945,7 +945,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong.", + "input": "52 of you answered simply 2 which would have been the real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated goin", "translatedText": "ובכן, מה שקרה כאן זה שהפכנו את שתי הקואורדינטות לשליליות וזה מרגיע כי אם אני לוקח איזו נקודה לשבת ב-ab ואז אני מסובב אותה 90 מעלות אז זה יהיה הסיבוב הראשוני של 90 מעלות ואז עוד 90 מעלות זה ה- זהה לרקבון של 180 מעלות- הו לא עשיתי את זה לא נכון.", "n_reviews": 0, "start": 938.48, @@ -959,14 +959,14 @@ "end": 980.14 }, { - "input": "Now why am I asking you this?", + "input": "you try to multiply these numbers together. So with vectors, there's not really any notion", "translatedText": "עכשיו למה אני שואל אותך את זה?", "n_reviews": 0, "start": 980.4, "end": 981.76 }, { - "input": "Well I'm being told that supposedly I'm allowed to ask you questions again so I'm going to have you do your very first complex product.", + "input": "of multiplying them to get two vectors back, at least when we're in the 2d plane. that we ask questions and just say hey kamineter can't you make the live questio", "translatedText": "ובכן אומרים לי שלכאורה מותר לי לשאול אותך שאלות שוב אז אני אבקש ממך לעשות את המוצר המורכב הראשון שלך.", "n_reviews": 0, "start": 982.66, @@ -987,14 +987,14 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i.", + "input": "Wonderful! Very simple question I want you to take the number i and I want you to multiply it by 3 pl", "translatedText": "אוקיי אז נראה שרובכם אכן קיבלו את התשובה הנכונה שהיא 2 פלוס 3i.", "n_reviews": 0, "start": 1003.06, "end": 1006.98 }, { - "input": "Very good very good.", + "input": "us 2i and even though I haven't really talked about You can't do things like assume that if two numbers multiply to mak", "translatedText": "טוב מאוד, טוב מאוד.", "n_reviews": 0, "start": 1007.48, @@ -1015,28 +1015,28 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good.", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t", "translatedText": "יש לנו 2 פלוס 3 שזה אולי פשוט מוריד את ה-i אז אני חושב שאולי הרבה כמו שגיאות פשוטות וכניסה ואתה יודע שזה קורה לכולנו במיוחד במבחנים זה לפעמים אתה יודע מה התשובה הנכונה אבל אז אתה שוכח סמל או שאתה מחליף שניים אז הכל טוב מאוד.", "n_reviews": 0, "start": 1037.6, "end": 1052.36 }, { - "input": "Let's go ahead and try our very first product though like I said so here because I already talked through one of the questions we're going to go ahead and skip ahead of it we know how to rotate something like 3 comma 2 so I'm not even going to give you time to do that and properly grade it.", + "input": "his is we can basically just turn our paper to figure it out. ons as you do it rather than sitting in passively watching this is genuinely delightful to me. Okay this is this isn't necessarily a question I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not", "translatedText": "בוא נמשיך וננסה את המוצר הראשון שלנו אבל כמו שאמרתי כאן כי כבר דיברתי על אחת השאלות שאנחנו הולכים קדימה ומדלג לפניה אנחנו יודעים איך לסובב משהו כמו 3 פסיק 2 אז אני אפילו לא מתכוון לתת לך זמן לעשות את זה ולדרג את זה כמו שצריך.", "n_reviews": 0, "start": 1052.88, "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us?", + "input": "that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three i which is absolutely correct absolutely correct so there's two w", "translatedText": "סטאל מילים סטלית מילים שאתה יודע הם אומרים לי שזה עובד ובכל זאת זה מאוד איטי בשבילי להתקדם קדימה אז אתה יודע אם אני לא הולך לדבר איתם מילה חמורה אתה יכול ללכת אליהם גם בטוויטר תחת אותו מקום שאנחנו שואלים שאלות ופשוט אומרים היי קמינטר, אתה לא יכול לגרום לשאלות החיות לעבוד קצת יותר טוב עבורנו?", "n_reviews": 0, "start": 1070.62, "end": 1089.28 }, { - "input": "Okay I think we're finally there.", + "input": "ays to think about this okay one of them is", "translatedText": "אוקיי אני חושב שאנחנו סוף סוף שם.", "n_reviews": 0, "start": 1090.02, @@ -1050,14 +1050,14 @@ "end": 1094.22 }, { - "input": "Aha!", + "input": "e algebra and just do it a little bit mechanistically okay so", "translatedText": "אהה!", "n_reviews": 0, "start": 1094.6, "end": 1094.76 }, { - "input": "Wonderful!", + "input": "if we pull ourselves up", "translatedText": "נִפלָא!", "n_reviews": 0, "start": 1094.76, @@ -1071,7 +1071,7 @@ "end": 1126.42 }, { - "input": "Wonderful!", + "input": "that if you want to rotate numbers 90 degrees", "translatedText": "נִפלָא!", "n_reviews": 0, "start": 1127.02, diff --git a/2020/ldm-complex-numbers/hindi/sentence_translations.json b/2020/ldm-complex-numbers/hindi/sentence_translations.json index c39161829..52fd6232d 100644 --- a/2020/ldm-complex-numbers/hindi/sentence_translations.json +++ b/2020/ldm-complex-numbers/hindi/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "मैं कहूंगा कि यदि यह कुछ ऐसा है जो किसी एप्लिकेशन में वास्तव में उपयोगी है, तो यह उतना ही वास्तविक है जितना शब्द हैं, है ना? ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "आप कभी भी खुशी जैसे किसी अमूर्त शब्द से नहीं मिलेंगे, लेकिन यह हमारे दिमाग में एक तरह की वास्तविकता है, और दो के वर्गमूल जैसी चीजें हैं, जिन्हें आप अंश के रूप में व्यक्त नहीं कर सकते हैं, या जैसी चीजें नकारात्मक का वर्गमूल जो वास्तविक सामान्य संख्याओं के बीच दिखाई नहीं देता है, आप जानते हैं, भले ही वे थोड़े अलग दिखें।", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "मैं यह नहीं मानूंगा कि आप अभी तक जानते हैं कि वे क्या हैं, इसका मतलब एक बुनियादी प्राइमर है, लेकिन आइए इसमें गहराई से उतरें, ठीक है? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "मुझे याद है जब मैं स्कूल में था और हमने ये जोड़ सूत्र सीखे थे, कि यदि आप दो अलग-अलग कोणों के योग की कोज्या जानना चाहते हैं, तो आप जानते हैं, मूल दो कोणों की कोज्या और ज्या के संदर्भ में यह बहुत लंबी बात है।, यह ऋण चिह्न है जो हमेशा लोगों को परेशान करेगा, यदि आप चिह्न के लिए भी ऐसा ही करते हैं, तो यह समान दिखता है लेकिन इसमें एक प्लस चिह्न है, और आपके पास कॉस-कॉस होने के बजाय कॉस-सिन है, यह कुछ ऐसा है जो बहुत त्रुटि-प्रवण है यदि आप इसे वैसे ही याद करने का प्रयास कर रहे हैं जैसे यह है।", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "हालाँकि, यदि आप इसे जटिल संख्याओं के साथ देखते हैं, तो यह न केवल बहुत कम त्रुटि-प्रवण है, इसका एक बहुत ही सुंदर अर्थ है और यह तुरंत स्पष्ट हो जाता है।", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "इसलिए भले ही आप जरूरी तौर पर नकारात्मक 1 के वर्गमूल की वास्तविकता पर विश्वास न करें, आपको कम से कम यह स्वीकार करना होगा कि यह दिलचस्प है कि यह गणित के अन्य टुकड़ों को उपयोगी बना सकता है, कि गणित के अन्य टुकड़े थोड़ा और अधिक उपयोगी हो सकते हैं समझने योग्य भी. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "लेकिन शुरुआती बिंदु बहुत अजीब लगता है, ठीक है? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "एक है, नहीं, ऐसा नहीं है, है ना? ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "आप किसी भी संख्या का वर्ग करें, यदि वह धनात्मक है, तो वह धनात्मक ही रहेगी।", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "मुझे कभी भी कुछ भी नकारात्मक नहीं मिलेगा।", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "हालाँकि, अगर कोई गणितज्ञ आता है और कहता है, अरे नहीं, नहीं, इसका अस्तित्व है, हमने इसे परिभाषित किया है ताकि यही मामला हो।", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "जब आपके पास कोई ऐसी समस्या हो जिसे आप हल नहीं कर सकते, तो आप बस इतना कह सकते हैं, ओह, मैंने चीजों को परिभाषित कर दिया है ताकि अब हमारे पास जादुई रूप से एक समाधान हो।", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "इसलिए यदि आप इससे असहज हैं, तो आप निश्चित रूप से अकेले नहीं हैं।", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "वास्तव में, रेने डेसकार्टेस ने इन संख्याओं के लिए अपमानजनक शब्द काल्पनिक शब्द गढ़ा।", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "और फिर हम इसे एक परंपरा के रूप में मानते रहे और हम अभी भी उन्हें काल्पनिक संख्याएँ कहते हैं, जो वास्तव में बेतुका है।", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "दूसरी अजीब बात जो आप तब करते हैं जब आप जटिल संख्याओं के बारे में बात करना शुरू करते हैं, वह यह कहना है कि, ऐसी कोई संख्या नहीं है, लेकिन हम इसे एक घर देने जा रहे हैं।", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "और, ठीक है, यदि हम अपनी संख्या प्रणाली का विस्तार करना चाहते हैं, तो मैं समझ गया, हो सकता है कि वहां किसी प्रकार की संख्या डालना उपयोगी हो, लेकिन मैं क्यों, ठीक है? ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "शुरुआत में, चलिए बस इस बारे में बात करते हैं कि यदि आप ऐसी संख्याएं जोड़ रहे हैं जो इस तरह से द्वि-आयामी हैं, तो नियम बहुत सीधे हैं और यह मूल रूप से वैक्टर के समान ही संचालित होता है, आप में से किसी के लिए जो वैक्टर से परिचित हो सकता है।", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "यदि आप इसे विश्वास पर लेते हैं और आप इसका पालन करते हैं, तो उम्मीद है कि यह तथ्य उपयोगी हो जाता है कि यह उचित ठहराने में मदद करता है कि हम ऐसा क्यों कर रहे हैं।", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "उम, ऐसा लगता है कि उत्तर f और d के बीच आगे-पीछे होता है, इसलिए f ये सभी हैं, यह कहते हुए कि इन सभी को वास्तविक माना जाना चाहिए।", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "और दिलचस्प बात यह है कि डी वह है जो कहता है कि आपको 2 के 2 वर्गमूल और नकारात्मक 1 पर विचार करना चाहिए, लेकिन अनंत पर नहीं, इसलिए वहां आप में से एक अच्छा समूह है जो अनंत को वास्तविक मानने से इनकार कर देगा, लेकिन इसके साथ बहुत सहज हैं ऋणात्मक 1 का वर्गमूल, यह अद्भुत है।", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "ठीक है, तो ऐसा लगता है कि हमें ऐसे लोगों का एक समूह मिल गया है जो नकारात्मक 1 के साथ सहज हैं, एक बड़ा समूह अनंत के साथ असहज है, यह एक और दिन का विषय है, इसके बारे में चिंता न करें, और फिर ऐसे कई लोग हैं जो शायद वे इस विचार के साथ बहुत सहज नहीं हैं कि नकारात्मक 1 वास्तविक हो सकता है।", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "तो हमारे पहले बहुत अधिक गणितीय प्रश्न के लिए, वार्म-अप के रूप में, मैं आपसे बस इन दोनों को जोड़ने के लिए कहना चाहता हूं।", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "इससे पहले कि मैं आपको सिखाऊं कि उन्हें कैसे जोड़ना है, एक अनुमान लगाएं कि यह कैसे काम कर सकता है, और मुझे आशा है कि यह बहुत सीधा लगता है, जोड़ना वास्तव में इसका सबसे कम दिलचस्प हिस्सा है, लेकिन यह है, यह जानना एक अच्छी बात है कि कब आप सम्मिश्र संख्याओं के बारे में सीख रहे हैं।", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "यदि आप चार इकाइयों को दाईं ओर और फिर एक इकाई को ऊपर ले जा रहे हैं, और आप दो इकाइयों को बाईं ओर और फिर दो इकाइयों को ऊपर ले जाने का विचार जोड़ना चाहते हैं, तो आप बस उनमें से प्रत्येक को एक समय में करें।", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "वास्तविक भाग दाहिनी ओर के चार, फिर बायीं ओर शून्य से दो होंगे।", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "तो क्या वह एक मैं प्लस दो मैं है।", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "जोड़ में वास्तव में कुछ भी जटिल नहीं है, जो बहुत अच्छी बात है।", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "इसलिए सदिशों के मामले में, वास्तव में दो सदिशों को वापस पाने के लिए उन्हें गुणा करने की कोई धारणा नहीं है, कम से कम जब हम 2डी विमान में हों।", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "जो मूल रूप से है, मान लीजिए मेरे पास बिंदु तीन, दो है।", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "यदि मेरे पास किसी प्रकार का समन्वय ग्रिड है और मैं x निर्देशांक तीन और y निर्देशांक दो वाले बिंदु पर जाता हूं, तो इसका 90 डिग्री घूर्णन क्या है? ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "वामावर्त. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "अब इसके बारे में अच्छी बात यह है कि हम मूल रूप से इसका पता लगाने के लिए अपने पेपर को पलट सकते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "आप कहते हैं कि ठीक है अगर यह तीन, दो पर शुरू हुआ और फिर मैं 90 डिग्री वामावर्त घुमाता हूं, तो मैं इसे अब एक्स दिशा में नकारात्मक दो और फिर वाई दिशा में तीन के रूप में पढ़ सकता हूं, अगर मैंने पूरे विमान को उसी तरह घुमाया होता . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "तो हमने यहां जो किया है वह यह है कि हमने तीन, दो लिया है और फिर हम इसे नकारात्मक दो, तीन में बदल देते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "यह 90 डिग्री का घूर्णन होगा।", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "यदि मैंने संख्याओं की एक जोड़ी को अल्पविराम बी लिया और फिर मैंने कहा कि यह कहां घूमेगा यदि मैं इसे 90 डिग्री घुमाऊं तो यह निर्देशांक की अदला-बदली करके समाप्त हो जाएगा और फिर उस पहले वाले को नकारात्मक बना देगा।", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "तो यह एक और 90 डिग्री रोटेशन था।", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "खैर, यहां जो हुआ है वह यह है कि हमने दोनों निर्देशांक को नकारात्मक बना दिया है और यह आश्वस्त करने वाला है क्योंकि अगर मैं एब पर बैठकर कुछ बिंदु लेता हूं और फिर मैं इसे 90 डिग्री घुमाता हूं तो यह मेरा प्रारंभिक 90 डिग्री रोटेशन होगा और फिर एक और 90 डिग्री होगा।180 डिग्री सड़न के समान- अरे नहीं, मैंने यह गलत किया है।", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "यह 180 डिग्री रोटेशन के समान होगा जो इस तरह दिखना चाहिए, मेरे द्वारा बनाए गए अन्य वेक्टर को अनदेखा करें जो सिर्फ दोनों निर्देशांक ले रहा है और उन्हें नकारात्मक नकारात्मक बना रहा है, ठीक है, इसलिए यह इस ऑपरेशन को आश्वस्त करता है जो 90 डिग्री रोटेशन करता है वास्तव में वैसा ही व्यवहार करता है जैसा आप अपेक्षा करते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "ओह देखिए, बहुत से लोगों ने बहुत अच्छे उत्तर प्रस्तुत किए।", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "ठीक है तो ऐसा लगता है कि आपमें से अधिकांश को सही उत्तर मिल गया है जो कि 2 प्लस 3i है।", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "बहुत अच्छा बहुत अच्छा. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "आप में से कुछ ने नकारात्मक 2 3 का उत्तर दिया, जो मुझे लगता है कि बस बना रहा है, बस अदला-बदली कर रहा है चाहे आप 4 घटा 2 या 2 घटा 4 ले रहे हों, तो यह पूरी तरह से समझ में आता है।", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "हमें 2 प्लस 3 मिला है जो शायद सिर्फ i को हटा रहा है इसलिए मुझे लगता है कि शायद बहुत सारी साधारण त्रुटियां और प्रविष्टियां पसंद हैं और आप जानते हैं कि यह हम सभी के साथ होता है, खासकर परीक्षणों पर, कभी-कभी आपको पता होता है कि सही उत्तर क्या है लेकिन फिर आप एक प्रतीक भूल जाते हैं या आप दो प्रतीकों की अदला-बदली कर देते हैं तो यह सब बहुत अच्छा है।", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "स्टाल स्टाल शब्द शब्द आप जानते हैं कि वे मुझे बताते हैं कि यह काम कर रहा है और फिर भी मेरे लिए आगे बढ़ना बहुत धीमा है, इसलिए आप जानते हैं कि अगर मैं उनके साथ कठोर शब्द नहीं बोलूंगा तो आप लोग ट्विटर पर भी उसी के तहत उन पर जा सकते हैं वह स्थान जहाँ हम प्रश्न पूछते हैं और बस यही कहते हैं कि हे कामिनेटर क्या आप लाइव प्रश्नों को हमारे लिए थोड़ा बेहतर नहीं बना सकते? ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "ठीक है, मुझे लगता है कि हम आख़िरकार वहाँ हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "हर कोई तैयार है? ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "अहा! ", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "आश्चर्यजनक! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "बहुत सरल प्रश्न है, मैं चाहता हूं कि आप संख्या i लें और मैं चाहता हूं कि आप इसे 3 प्लस 2i से गुणा करें और हालांकि मैंने वास्तव में गुणन के नियमों के बारे में बात नहीं की है, मैं जो कह सकता हूं वह दिखावा है जैसे यह वैसे ही काम करता है जैसे यह करता है।सामान्य संख्या में आपके पास वितरण संपत्ति जैसी चीजें होती हैं जहां आप इसे वितरित कर सकते हैं और फिर आई की परिभाषित विशेषता यह विचार है कि मैंने चुकता नकारात्मक है, यही एकमात्र विशेष चीज है जिसके बारे में आपको इसके बारे में जानने की जरूरत है, इसके अलावा बस इसका इलाज करें जैसे कि यह एक सामान्य संख्या है ठीक है और फिर उत्पाद के साथ आगे बढ़ें।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/hungarian/sentence_translations.json b/2020/ldm-complex-numbers/hungarian/sentence_translations.json index e80541543..110c93c4d 100644 --- a/2020/ldm-complex-numbers/hungarian/sentence_translations.json +++ b/2020/ldm-complex-numbers/hungarian/sentence_translations.json @@ -273,21 +273,21 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right?", + "input": "stalling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the", "translatedText": "Azt mondanám, hogy ha valami hasznos egy alkalmazásban, akkor az olyan valóságos, mint a szavak, igaz?", "n_reviews": 0, "start": 247.68, "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different.", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality.", "translatedText": "Soha nem fogsz belefutni egy olyan elvont szóba, mint a boldogság, de van egyfajta valóság az elménkben, és olyan dolgok, mint a kettő négyzetgyöke, amit nem tudsz törtként kifejezni, vagy olyan dolgok, mint A negatív szám négyzetgyöke, amelyek nem jelennek meg a valós normál számok között, még akkor is, ha kissé eltérőnek tűnnek.", "n_reviews": 0, "start": 253.74, "end": 270.78 }, { - "input": "Oh, this is such a shame.", + "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay?", "translatedText": "Ó, ez milyen szégyen.", "n_reviews": 0, "start": 271.5, @@ -343,14 +343,14 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out.", + "input": "esting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just trigonometry, it's everything we were talking about la", "translatedText": "Ha azonban összetett számokkal érünk hozzá, ez nem csak, hogy sokkal kevésbé hibás, hanem nagyon szép jelentése van, és egyszerűen kiesik.", "n_reviews": 0, "start": 363.98, "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too.", + "input": "st time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of", "translatedText": "Tehát még ha nem is hiszel feltétlenül a negatív 1 négyzetgyökének valóságában, legalább el kell ismerned, hogy érdekes, hogy más matematikai darabokat hasznossá tehet, más matematikai darabokat pedig egy kicsit jobban. érthető is.", "n_reviews": 0, "start": 372.1, @@ -392,7 +392,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right?", + "input": "It's something that's very error-prone if you're just trying", "translatedText": "Az egyik az, nem, nem, igaz?", "n_reviews": 0, "start": 414.88, @@ -406,7 +406,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive.", + "input": "ex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out", "translatedText": "Bármely szám, amelyet négyzetre vet, ha pozitív, akkor az csak pozitív marad.", "n_reviews": 0, "start": 427.76, @@ -420,21 +420,21 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative.", + "input": "So even if you don't necessarily believe in the reality of", "translatedText": "Soha semmi negatívat nem fogok kapni.", "n_reviews": 0, "start": 436.38, "end": 437.86 }, { - "input": "So this does not exist, no such number.", + "input": "the square root of negative one, you at the very least have to admit that it's interesting that it can make o", "translatedText": "Tehát ez nem létezik, nincs ilyen szám.", "n_reviews": 0, "start": 438.48, "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case.", + "input": "ther pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case.", "translatedText": "Ha azonban jön egy matematikus, és azt mondja, ó, nem, nem létezik, akkor úgy határoztuk meg, hogy ez a helyzet.", "n_reviews": 0, "start": 442.28, @@ -448,7 +448,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution.", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wo", "translatedText": "Ha van egy problémád, amit nem tudsz megoldani, akkor csak azt mondhatod, hogy ó, úgy határoztam meg a dolgokat, hogy most varázsütésre meglegyen a megoldás.", "n_reviews": 0, "start": 452.86, @@ -462,14 +462,14 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone.", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home.", "translatedText": "Tehát ha ez kényelmetlenül érzi magát, biztosan nem vagy egyedül.", "n_reviews": 0, "start": 466.48, "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory.", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative,", "translatedText": "Valójában Rene Descartes a képzeletbeli kifejezést ezekre a számokra lekicsinylőnek találta.", "n_reviews": 0, "start": 470.06, @@ -525,7 +525,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right?", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you", "translatedText": "És oké, ha ki akarjuk terjeszteni a számrendszerünket, akkor értem, talán hasznos valami számot feltenni oda, de miért én, nem?", "n_reviews": 0, "start": 526.5, @@ -553,7 +553,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors.", + "input": "have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question.", "translatedText": "A legelején beszéljünk arról, hogy ha kétdimenziós számokat adunk össze, akkor a szabályok meglehetősen egyszerűek, és lényegében ugyanúgy működnek, mint a vektorok, bárki számára, aki esetleg ismeri a vektorokat.", "n_reviews": 0, "start": 557.3, @@ -630,14 +630,14 @@ "end": 662.3 }, { - "input": "None of them is much lower at a.", + "input": "one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to", "translatedText": "Egyik sem sokkal alacsonyabb a.", "n_reviews": 0, "start": 662.7, "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real.", + "input": ", why should that live there? What on earth does the idea of a point one unit above the real number line in a separate dimension have to do with squaring to negative one? So I hope to answer this for you. At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be famili", "translatedText": "Rendben, úgy tűnik, hogy van egy csoportunk, akiknek kényelmes az 1. negatív, egy nagy csoport kényelmetlen a végtelen miatt, ez egy másik nap témája, ne aggódj, majd néhány ember, aki Valahogy abban a középmezőnyben vannak, hogy talán nem érzik túlságosan megnyugtatónak a gondolatot, hogy a negatív 1 valós lehet.", "n_reviews": 0, "start": 665.1, @@ -651,7 +651,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two.", + "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It turns out to be r", "translatedText": "Tehát az első, sokkal matematikaibb kérdésünkhöz, mintegy bemelegítésképpen, csak arra szeretném kérni, hogy adja hozzá ezt a kettőt.", "n_reviews": 0, "start": 683.42, @@ -672,35 +672,35 @@ "end": 706.82 }, { - "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me.", + "input": "ative two plus two i. So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers extend in this direction. can get you something lik", "translatedText": "Sajnos, és abból, hogy elakadok, és amit itt mondok, jól látszik, hogy a kérdés még mindig nem töltődik be teljesen helyesen, ezért szigorú szót váltok Cam és Iderrel a háttérben. olyan jeleneteket, akik egyébként olyan gyönyörű, gyönyörű felületet építettek fel, ami hasznos az ilyen jellegű oda-vissza kapcsolathoz közted és köztem.", "n_reviews": 0, "start": 707.84, "end": 726.28 }, { - "input": "I'm going to have a stern word with them behind the scenes, but in the meantime let's go ahead and move forward with the lesson here.", + "input": "e it. But the rules end up being very different from that in the number system. You can't really do algebra. You can't do things like assume that if two numbers multiply to make zero, then", "translatedText": "A színfalak mögött szigorú szót váltok velük, de addig is menjünk tovább, és haladjunk tovább az itteni leckével.", "n_reviews": 0, "start": 726.76, "end": 732.64 }, { - "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be.", + "input": "one of them h as to be zero. But complex numbers are going to end up behaving much like the real numbers, s Now assuming that our question system ha", "translatedText": "Szóval azt hiszem, fel tudom húzni a papírra, és otthon követhetitek, megnézhetitek, mi lehet a kiegészítés.", "n_reviews": 0, "start": 733.46, "end": 739.92 }, { - "input": "It turns out to be relatively straightforward.", + "input": "s not broken down, I should be able to do this as a proper poll and let me go ahead, I guess we can first check the previous poll, okay things se", "translatedText": "Viszonylag egyértelműnek bizonyul.", "n_reviews": 0, "start": 740.24, "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time.", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of th", "translatedText": "Ha négy egységet jobbra, majd egy egységgel feljebb mozgat, és azt az ötletet szeretné hozzáadni, hogy két egységet balra, majd két egységet feljebb, akkor tegye meg ezeket egyenként.", "n_reviews": 0, "start": 742.94, @@ -714,7 +714,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left.", + "input": "red real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a g", "translatedText": "Az igazi rész az a négy lesz jobbra, majd mínusz kettő balra.", "n_reviews": 0, "start": 754.82, @@ -728,7 +728,7 @@ "end": 760.88 }, { - "input": "And then the imaginary part is going to be this one unit up and then these two units up, one plus two, times i.", + "input": "you out there who would just reject infinity as being considered real but are very comfortable with the square root of negative o", "translatedText": "És akkor a képzeletbeli rész ez egy egységgel feljebb, majd ez a két egység feljebb, egy plusz kettő, i-szer.", "n_reviews": 0, "start": 761.16, @@ -756,7 +756,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great.", + "input": "are root of negative one, fascinating, I actually would have thought that none of them would have come higher than t", "translatedText": "A kiegészítésben nem igazán történik semmi bonyolult, ami nagyszerű.", "n_reviews": 0, "start": 777.86, @@ -770,14 +770,14 @@ "end": 784.2 }, { - "input": "What is so complex about complex numbers after all?", + "input": "m is much lower at a, okay so it looks like we've got a cohort of people who are comfortable with negative one, a la", "translatedText": "Végül is mi olyan bonyolult a komplex számokban?", "n_reviews": 0, "start": 784.42, "end": 787.1 }, { - "input": "Well where everything becomes interesting is when you try to multiply these numbers together.", + "input": "rge cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people w", "translatedText": "Nos, minden akkor válik érdekessé, amikor megpróbálja összeszorozni ezeket a számokat.", "n_reviews": 0, "start": 787.64, @@ -798,14 +798,14 @@ "end": 803.68 }, { - "input": "But the rules end up being very different from that in the number system.", + "input": "mfortable with the idea that negative one might be real, let's see if we can convince you of the difference of t", "translatedText": "De a szabályok végül nagyon eltérnek a számrendszer szabályaitól.", "n_reviews": 0, "start": 803.68, "end": 806.86 }, { - "input": "You can't really do algebra.", + "input": "hat. So what we've done here is we've taken three, two and then", "translatedText": "Nem igazán tudsz algebrát csinálni.", "n_reviews": 0, "start": 806.86, @@ -826,7 +826,7 @@ "end": 817.78 }, { - "input": "But to understand what that multiplication rule is, I just want to ask you a simple question.", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, a", "translatedText": "De hogy megértsük, mi ez a szorzási szabály, csak egy egyszerű kérdést szeretnék feltenni.", "n_reviews": 0, "start": 818.3, @@ -840,14 +840,14 @@ "end": 831.82 }, { - "input": "We're not even going to think of it as a complex number per se.", + "input": "part of this, but it is, it's a good thing to know when you're learning about complex numbers, it'", "translatedText": "Nem is fogjuk önmagában komplex számnak tekinteni.", "n_reviews": 0, "start": 832.58, "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this?", + "input": "s definitely one of those operations that you are going to need to know. Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is stil", "translatedText": "Ha csak van valami koordináta rácsom, és arra a pontra megyek, ahol x koordináta három és y koordináta kettő, akkor ennek mekkora a 90 fokos elforgatása?", "n_reviews": 0, "start": 835.72, @@ -875,14 +875,14 @@ "end": 859.44 }, { - "input": "Okay.", + "input": "built such a beautiful, beautiful inte", "translatedText": "Oké.", "n_reviews": 0, "start": 865.28, "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out.", + "input": "rface that's helpful for this kind of back and forth between you guys and me. nice gut check here is", "translatedText": "Ami ebben a szép, az az, hogy alapvetően elfordíthatjuk a papírunkat, hogy rájöjjünk.", "n_reviews": 0, "start": 865.76, @@ -896,28 +896,28 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three.", + "input": "e. So that was another 90 degree rotation. Well what's happened here is we've just made both of the coordinates negative and that's", "translatedText": "Tehát itt azt csináltuk, hogy vettünk hármat, kettőt, majd negatív kettőre, háromra konvertáltuk.", "n_reviews": 0, "start": 884.08, "end": 890.68 }, { - "input": "Something which maybe in our original system you know looks like this negative two and then three.", + "input": "reassuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them", "translatedText": "Valami, amit a mi eredeti rendszerünkben ismersz, úgy néz ki, mint ez a negatív kettő, majd három.", "n_reviews": 0, "start": 891.58, "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation.", + "input": "negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation", "translatedText": "Ez lesz a 90 fokos elforgatás.", "n_reviews": 0, "start": 898.1, "end": 899.9 }, { - "input": "And what's nice here is that that rule is very simple and it applies to any pair that we might have.", + "input": "actually behaves like you would expect it to. Now why am I asking you this? Well I'm being told that supposedly I'm allowed to ask you questions again so I 'm going to ha", "translatedText": "És ami itt jó, az az, hogy ez a szabály nagyon egyszerű, és minden lehetséges párra vonatkozik.", "n_reviews": 0, "start": 899.9, @@ -945,7 +945,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong.", + "input": "52 of you answered simply 2 which would have been the real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated goin", "translatedText": "Nos, ami itt történt, az imént mindkét koordinátát negatívvá tettük, és ez megnyugtató, mert ha egy pontot az ab-nál ülve elforgatom 90 fokkal, így ez lesz a kezdeti 90 fokos elforgatásom, majd a további 90 fok, akkor az ugyanaz, mint a 180 fokos rothadás – ó, nem, ezt rosszul csináltam.", "n_reviews": 0, "start": 938.48, @@ -959,14 +959,14 @@ "end": 980.14 }, { - "input": "Now why am I asking you this?", + "input": "you try to multiply these numbers together. So with vectors, there's not really any notion", "translatedText": "Most miért kérdezem ezt?", "n_reviews": 0, "start": 980.4, "end": 981.76 }, { - "input": "Well I'm being told that supposedly I'm allowed to ask you questions again so I'm going to have you do your very first complex product.", + "input": "of multiplying them to get two vectors back, at least when we're in the 2d plane. that we ask questions and just say hey kamineter can't you make the live questio", "translatedText": "Nos, azt mondják nekem, hogy állítólag ismét feltehetek neked kérdéseket, ezért megkérlek, hogy készítsd el a legelső összetett termékedet.", "n_reviews": 0, "start": 982.66, @@ -987,14 +987,14 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i.", + "input": "Wonderful! Very simple question I want you to take the number i and I want you to multiply it by 3 pl", "translatedText": "Rendben, úgy tűnik, hogy a legtöbben megkapták a helyes választ, ami 2 plusz 3i.", "n_reviews": 0, "start": 1003.06, "end": 1006.98 }, { - "input": "Very good very good.", + "input": "us 2i and even though I haven't really talked about You can't do things like assume that if two numbers multiply to mak", "translatedText": "Nagyon jó nagyon jó.", "n_reviews": 0, "start": 1007.48, @@ -1015,28 +1015,28 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good.", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t", "translatedText": "Van 2 plusz 3, ami talán csak leesik az i-ről, így azt hiszem, talán sok az egyszerű hibák és bejegyzések, és tudod, hogy ez mindannyiunkkal megtörténik, különösen a teszteken, ha néha tudod, mi a helyes válasz, de akkor elfelejtesz egy szimbólumot, vagy felcserélsz kettőt, így minden nagyon jó.", "n_reviews": 0, "start": 1037.6, "end": 1052.36 }, { - "input": "Let's go ahead and try our very first product though like I said so here because I already talked through one of the questions we're going to go ahead and skip ahead of it we know how to rotate something like 3 comma 2 so I'm not even going to give you time to do that and properly grade it.", + "input": "his is we can basically just turn our paper to figure it out. ons as you do it rather than sitting in passively watching this is genuinely delightful to me. Okay this is this isn't necessarily a question I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not", "translatedText": "Próbáljuk ki a legelső termékünket, de ahogy itt is mondtam, mert már beszéltem az egyik kérdésről, amit tovább fogunk menni, és kihagyjuk azt, tudjuk, hogyan kell elforgatni valamit, például 3 vessző 2, szóval még csak nem is adok időt, hogy ezt megtegye és megfelelően osztályozza.", "n_reviews": 0, "start": 1052.88, "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us?", + "input": "that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three i which is absolutely correct absolutely correct so there's two w", "translatedText": "Tudod, hogy a szálló szavak azt mondják nekem, hogy működik, és mégis nagyon lassan haladok előre, így tudod, ha nem fogok szigorú szót váltani velük, akkor a twitteren is rájuk nézhetsz. hely, ahol kérdéseket teszünk fel, és csak azt mondjuk, hé, kamineter, nem tudnád egy kicsit jobban működni az élő kérdéseket?", "n_reviews": 0, "start": 1070.62, "end": 1089.28 }, { - "input": "Okay I think we're finally there.", + "input": "ays to think about this okay one of them is", "translatedText": "Oké, azt hiszem, végre ott vagyunk.", "n_reviews": 0, "start": 1090.02, @@ -1050,14 +1050,14 @@ "end": 1094.22 }, { - "input": "Aha!", + "input": "e algebra and just do it a little bit mechanistically okay so", "translatedText": "Aha!", "n_reviews": 0, "start": 1094.6, "end": 1094.76 }, { - "input": "Wonderful!", + "input": "if we pull ourselves up", "translatedText": "Csodálatos!", "n_reviews": 0, "start": 1094.76, @@ -1071,7 +1071,7 @@ "end": 1126.42 }, { - "input": "Wonderful!", + "input": "that if you want to rotate numbers 90 degrees", "translatedText": "Csodálatos!", "n_reviews": 0, "start": 1127.02, diff --git a/2020/ldm-complex-numbers/indonesian/sentence_translations.json b/2020/ldm-complex-numbers/indonesian/sentence_translations.json index 2258e48ee..b2d191fdb 100644 --- a/2020/ldm-complex-numbers/indonesian/sentence_translations.json +++ b/2020/ldm-complex-numbers/indonesian/sentence_translations.json @@ -312,7 +312,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right?", + "input": "stalling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the", "translatedText": "Menurut saya, jika itu adalah sesuatu yang benar-benar berguna dalam sebuah aplikasi, maka itu sama nyatanya dengan kata-kata, bukan?", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different.", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality.", "translatedText": "Anda tidak akan pernah menemukan kata abstrak seperti kebahagiaan di luar sana, namun kata tersebut memiliki semacam realitas dalam pikiran kita, dan hal-hal seperti akar kuadrat dari dua, yang tidak dapat Anda ungkapkan sebagai pecahan, atau hal-hal seperti akar kuadrat dari bilangan negatif yang tidak muncul di antara bilangan normal nyata, meskipun mungkin tampak sedikit berbeda.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 270.78 }, { - "input": "Oh, this is such a shame.", + "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay?", "translatedText": "Oh, sungguh memalukan.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out.", + "input": "esting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just trigonometry, it's everything we were talking about la", "translatedText": "Namun, jika Anda membahasnya dengan bilangan kompleks, hal ini tidak hanya tidak terlalu rawan kesalahan, tetapi juga memiliki arti yang sangat indah dan langsung gagal.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too.", + "input": "st time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of", "translatedText": "Jadi meskipun Anda belum tentu percaya pada realitas akar kuadrat dari negatif 1, setidaknya Anda harus mengakui bahwa menarik bahwa hal ini dapat membuat matematika lain berguna, bahwa matematika lain sedikit lebih bermanfaat. bisa dimengerti juga.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right?", + "input": "It's something that's very error-prone if you're just trying", "translatedText": "Salah satunya adalah, tidak, tidak ada, kan?", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive.", + "input": "ex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out", "translatedText": "Bilangan apa pun yang Anda kuadratkan, jika positif, tetap positif.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative.", + "input": "So even if you don't necessarily believe in the reality of", "translatedText": "Saya tidak akan pernah mendapatkan sesuatu yang negatif.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 437.86 }, { - "input": "So this does not exist, no such number.", + "input": "the square root of negative one, you at the very least have to admit that it's interesting that it can make o", "translatedText": "Jadi ini tidak ada, tidak ada nomor tersebut.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case.", + "input": "ther pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case.", "translatedText": "Namun, jika seorang ahli matematika datang dan berkata, oh tidak, itu ada, kami telah mendefinisikannya sedemikian rupa.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution.", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wo", "translatedText": "Saat kamu punya masalah yang tidak bisa kamu pecahkan, kamu bisa bilang, oh, aku sudah mendefinisikan semuanya sehingga secara ajaib kita sekarang punya solusinya.", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone.", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home.", "translatedText": "Jadi jika Anda merasa tidak nyaman dengan hal ini, Anda tidak sendirian.", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory.", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative,", "translatedText": "Faktanya, Rene Descartes menciptakan istilah imajiner untuk angka-angka ini sebagai sebuah penghinaan.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right?", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you", "translatedText": "Dan, oke, jika kita ingin memperluas sistem bilangan kita, saya mengerti, mungkin ada gunanya menaruh semacam bilangan di sana, tapi kenapa saya, kan?", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors.", + "input": "have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question.", "translatedText": "Pada awalnya, mari kita bahas bagaimana jika Anda menjumlahkan bilangan dua dimensi seperti ini, aturannya cukup mudah dan cara kerjanya pada dasarnya sama dengan vektor, bagi Anda yang mungkin familiar dengan vektor.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 662.3 }, { - "input": "None of them is much lower at a.", + "input": "one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to", "translatedText": "Tak satu pun dari mereka jauh lebih rendah pada a.", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real.", + "input": ", why should that live there? What on earth does the idea of a point one unit above the real number line in a separate dimension have to do with squaring to negative one? So I hope to answer this for you. At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be famili", "translatedText": "Oke, jadi sepertinya kita punya kelompok orang yang merasa nyaman dengan negatif 1, kelompok besar tidak nyaman dengan ketidakterbatasan, itu topik untuk lain hari, jangan khawatir, dan kemudian sejumlah orang yang berada di jalan tengah karena mungkin merasa tidak terlalu nyaman dengan gagasan bahwa angka negatif 1 mungkin nyata.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two.", + "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It turns out to be r", "translatedText": "Jadi untuk pertanyaan matematis pertama kita, sebagai pemanasan, saya hanya ingin meminta Anda menambahkan keduanya.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 706.82 }, { - "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me.", + "input": "ative two plus two i. So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers extend in this direction. can get you something lik", "translatedText": "Sayangnya, dan Anda dapat mengetahui dari fakta bahwa saya mengulur waktu dan apa yang saya katakan di sini, sepertinya pertanyaannya masih belum dimuat sepenuhnya dengan benar, jadi saya akan berbicara tegas dengan Cam dan Ider di belakang adegan yang sebaliknya telah membangun antarmuka yang begitu indah dan indah yang berguna untuk bolak-balik antara kalian dan saya.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 726.28 }, { - "input": "I'm going to have a stern word with them behind the scenes, but in the meantime let's go ahead and move forward with the lesson here.", + "input": "e it. But the rules end up being very different from that in the number system. You can't really do algebra. You can't do things like assume that if two numbers multiply to make zero, then", "translatedText": "Saya akan berbicara tegas dengan mereka di belakang layar, namun sementara itu mari kita lanjutkan dan lanjutkan dengan pelajaran di sini.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 732.64 }, { - "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be.", + "input": "one of them h as to be zero. But complex numbers are going to end up behaving much like the real numbers, s Now assuming that our question system ha", "translatedText": "Jadi saya kira saya bisa menariknya ke atas, hanya di selembar kertas, dan Anda bisa mengikutinya di rumah, melihat tambahan apa yang mungkin ada.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 739.92 }, { - "input": "It turns out to be relatively straightforward.", + "input": "s not broken down, I should be able to do this as a proper poll and let me go ahead, I guess we can first check the previous poll, okay things se", "translatedText": "Ternyata hal ini relatif mudah.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time.", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of th", "translatedText": "Jika Anda memindahkan empat unit ke kanan lalu satu unit ke atas, dan Anda ingin menambahkan ide untuk memindahkan dua unit ke kiri lalu dua unit ke atas, lakukan saja masing-masing satu unit dalam satu waktu.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left.", + "input": "red real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a g", "translatedText": "Bagian sebenarnya adalah empat yang ke kanan, lalu dikurangi dua ke kiri.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 760.88 }, { - "input": "And then the imaginary part is going to be this one unit up and then these two units up, one plus two, times i.", + "input": "you out there who would just reject infinity as being considered real but are very comfortable with the square root of negative o", "translatedText": "Dan kemudian bagian imajinernya adalah satu satuan ke atas dan kemudian dua satuan ke atas, satu tambah dua, dikalikan i.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great.", + "input": "are root of negative one, fascinating, I actually would have thought that none of them would have come higher than t", "translatedText": "Penambahan tidak menimbulkan sesuatu yang rumit, dan itu bagus.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 784.2 }, { - "input": "What is so complex about complex numbers after all?", + "input": "m is much lower at a, okay so it looks like we've got a cohort of people who are comfortable with negative one, a la", "translatedText": "Apa sih rumitnya bilangan kompleks?", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 787.1 }, { - "input": "Well where everything becomes interesting is when you try to multiply these numbers together.", + "input": "rge cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people w", "translatedText": "Yang membuat semuanya menjadi menarik adalah ketika Anda mencoba mengalikan angka-angka ini.", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 803.68 }, { - "input": "But the rules end up being very different from that in the number system.", + "input": "mfortable with the idea that negative one might be real, let's see if we can convince you of the difference of t", "translatedText": "Namun aturannya menjadi sangat berbeda dengan sistem bilangan.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 806.86 }, { - "input": "You can't really do algebra.", + "input": "hat. So what we've done here is we've taken three, two and then", "translatedText": "Anda tidak bisa mengerjakan aljabar.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 817.78 }, { - "input": "But to understand what that multiplication rule is, I just want to ask you a simple question.", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, a", "translatedText": "Namun untuk memahami apa itu aturan perkalian, saya hanya ingin menanyakan pertanyaan sederhana.", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 831.82 }, { - "input": "We're not even going to think of it as a complex number per se.", + "input": "part of this, but it is, it's a good thing to know when you're learning about complex numbers, it'", "translatedText": "Kami bahkan tidak akan menganggapnya sebagai bilangan kompleks.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this?", + "input": "s definitely one of those operations that you are going to need to know. Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is stil", "translatedText": "Jika saya hanya memiliki semacam kotak koordinat dan saya menuju ke titik dengan koordinat x tiga dan koordinat y dua, berapakah rotasi 90 derajatnya?", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 859.44 }, { - "input": "Okay.", + "input": "built such a beautiful, beautiful inte", "translatedText": "Oke.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out.", + "input": "rface that's helpful for this kind of back and forth between you guys and me. nice gut check here is", "translatedText": "Yang menarik dari hal ini adalah pada dasarnya kita bisa membalik kertas kita untuk mencari tahu.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three.", + "input": "e. So that was another 90 degree rotation. Well what's happened here is we've just made both of the coordinates negative and that's", "translatedText": "Jadi yang kita lakukan di sini adalah mengambil tiga, dua, lalu mengubahnya menjadi negatif dua, tiga.", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 890.68 }, { - "input": "Something which maybe in our original system you know looks like this negative two and then three.", + "input": "reassuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them", "translatedText": "Sesuatu yang mungkin dalam sistem asli kita, Anda tahu, terlihat seperti dua dan kemudian tiga negatif.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation.", + "input": "negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation", "translatedText": "Itu akan menjadi rotasi 90 derajat.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 899.9 }, { - "input": "And what's nice here is that that rule is very simple and it applies to any pair that we might have.", + "input": "actually behaves like you would expect it to. Now why am I asking you this? Well I'm being told that supposedly I'm allowed to ask you questions again so I 'm going to ha", "translatedText": "Dan yang menarik di sini adalah aturan itu sangat sederhana dan berlaku untuk pasangan mana pun yang mungkin kita miliki.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong.", + "input": "52 of you answered simply 2 which would have been the real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated goin", "translatedText": "Apa yang terjadi di sini adalah kita baru saja membuat kedua koordinat menjadi negatif dan itu meyakinkan karena jika saya mengambil suatu titik di ab dan kemudian saya memutarnya 90 derajat maka ini akan menjadi rotasi awal saya sebesar 90 derajat dan kemudian 90 derajat lagi itulah sama seperti busuk 180 derajat- oh tidak, saya salah melakukannya.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 980.14 }, { - "input": "Now why am I asking you this?", + "input": "you try to multiply these numbers together. So with vectors, there's not really any notion", "translatedText": "Sekarang kenapa aku menanyakan ini padamu?", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 981.76 }, { - "input": "Well I'm being told that supposedly I'm allowed to ask you questions again so I'm going to have you do your very first complex product.", + "input": "of multiplying them to get two vectors back, at least when we're in the 2d plane. that we ask questions and just say hey kamineter can't you make the live questio", "translatedText": "Ya, saya diberitahu bahwa seharusnya saya diizinkan untuk mengajukan pertanyaan lagi, jadi saya akan meminta Anda mengerjakan produk kompleks pertama Anda.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i.", + "input": "Wonderful! Very simple question I want you to take the number i and I want you to multiply it by 3 pl", "translatedText": "Oke jadi sepertinya sebagian besar dari Anda mendapatkan jawaban yang benar yaitu 2 tambah 3i.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1006.98 }, { - "input": "Very good very good.", + "input": "us 2i and even though I haven't really talked about You can't do things like assume that if two numbers multiply to mak", "translatedText": "Sangat bagus sangat bagus.", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good.", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t", "translatedText": "Kami punya 2 ditambah 3 yang mungkin hanya menghilangkan i jadi saya pikir mungkin banyak kesalahan sederhana dan entri dan Anda tahu itu terjadi pada kita semua terutama pada tes kadang-kadang Anda tahu apa jawaban yang benar tetapi kemudian Anda, Anda lupa satu simbol atau Anda menukar dua jadi itu semua sangat bagus.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1052.36 }, { - "input": "Let's go ahead and try our very first product though like I said so here because I already talked through one of the questions we're going to go ahead and skip ahead of it we know how to rotate something like 3 comma 2 so I'm not even going to give you time to do that and properly grade it.", + "input": "his is we can basically just turn our paper to figure it out. ons as you do it rather than sitting in passively watching this is genuinely delightful to me. Okay this is this isn't necessarily a question I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not", "translatedText": "Mari lanjutkan dan coba produk pertama kita, seperti yang saya katakan di sini, karena saya sudah membicarakan salah satu pertanyaan, kita akan lanjutkan dan lewati dulu, kita tahu cara memutar sesuatu seperti 3 koma 2, jadi saya bahkan tidak akan memberi Anda waktu untuk melakukan itu dan menilainya dengan benar.", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us?", + "input": "that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three i which is absolutely correct absolutely correct so there's two w", "translatedText": "Kata-kata stal stal kata-kata yang Anda tahu mereka memberi tahu saya bahwa itu berhasil, namun sangat lambat bagi saya untuk maju ke depan jadi Anda tahu jika saya tidak ingin berbicara tegas dengan mereka, kalian dapat membahasnya di twitter juga di bawah yang sama tempat kami mengajukan pertanyaan dan hanya mengatakan hai kamineter, tidak bisakah Anda membuat pertanyaan langsung berfungsi sedikit lebih baik untuk kami?", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there.", + "input": "ays to think about this okay one of them is", "translatedText": "Oke, saya pikir kita akhirnya sampai di sana.", "model": "google_nmt", "n_reviews": 0, @@ -1200,7 +1200,7 @@ "end": 1094.22 }, { - "input": "Aha!", + "input": "e algebra and just do it a little bit mechanistically okay so", "translatedText": "Ya!", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1094.76 }, { - "input": "Wonderful!", + "input": "if we pull ourselves up", "translatedText": "Luar biasa!", "model": "google_nmt", "n_reviews": 0, @@ -1224,7 +1224,7 @@ "end": 1126.42 }, { - "input": "Wonderful!", + "input": "that if you want to rotate numbers 90 degrees", "translatedText": "Luar biasa!", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/italian/sentence_translations.json b/2020/ldm-complex-numbers/italian/sentence_translations.json index b9246c622..f2e02970e 100644 --- a/2020/ldm-complex-numbers/italian/sentence_translations.json +++ b/2020/ldm-complex-numbers/italian/sentence_translations.json @@ -273,21 +273,21 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right?", + "input": "stalling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the", "translatedText": "Direi che se è qualcosa che è effettivamente utile in un'applicazione, allora è reale quanto lo sono le parole, giusto?", "n_reviews": 0, "start": 247.68, "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different.", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality.", "translatedText": "Non ti imbatterai mai in una parola astratta come felicità là fuori, ma ha una sorta di realtà nella nostra mente, e cose come la radice quadrata di due, che non puoi esprimere come frazione, o cose come la radice quadrata di quello negativo che non compaiono tra i numeri normali reali, sai, anche se potrebbero sembrare un po' diversi.", "n_reviews": 0, "start": 253.74, "end": 270.78 }, { - "input": "Oh, this is such a shame.", + "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay?", "translatedText": "Oh, è davvero un peccato.", "n_reviews": 0, "start": 271.5, @@ -343,14 +343,14 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out.", + "input": "esting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just trigonometry, it's everything we were talking about la", "translatedText": "Tuttavia, se ci si arriva con numeri complessi, questo non solo è molto meno soggetto a errori, ma ha un significato molto bello e cade proprio fuori.", "n_reviews": 0, "start": 363.98, "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too.", + "input": "st time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of", "translatedText": "Quindi, anche se non credi necessariamente nella realtà della radice quadrata di meno 1, devi almeno ammettere che è interessante che possa rendere utili altri pezzi di matematica, che altri pezzi di matematica un po' di più anche comprensibile.", "n_reviews": 0, "start": 372.1, @@ -392,7 +392,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right?", + "input": "It's something that's very error-prone if you're just trying", "translatedText": "Uno lo è, no non c'è, giusto?", "n_reviews": 0, "start": 414.88, @@ -406,7 +406,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive.", + "input": "ex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out", "translatedText": "Qualsiasi numero quadrato, se è positivo, beh rimane positivo.", "n_reviews": 0, "start": 427.76, @@ -420,21 +420,21 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative.", + "input": "So even if you don't necessarily believe in the reality of", "translatedText": "Non otterrò mai nulla di negativo.", "n_reviews": 0, "start": 436.38, "end": 437.86 }, { - "input": "So this does not exist, no such number.", + "input": "the square root of negative one, you at the very least have to admit that it's interesting that it can make o", "translatedText": "Quindi questo non esiste, nessun numero del genere.", "n_reviews": 0, "start": 438.48, "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case.", + "input": "ther pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case.", "translatedText": "Tuttavia, se un matematico viene e dice, oh no no, esiste, lo abbiamo definito in modo che sia così.", "n_reviews": 0, "start": 442.28, @@ -448,7 +448,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution.", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wo", "translatedText": "Quando hai un problema che non puoi risolvere, puoi semplicemente dire: oh, ho definito le cose in modo che ora magicamente abbiamo una soluzione.", "n_reviews": 0, "start": 452.86, @@ -462,14 +462,14 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone.", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home.", "translatedText": "Quindi, se ti senti a disagio con questo, sicuramente non sei solo.", "n_reviews": 0, "start": 466.48, "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory.", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative,", "translatedText": "In effetti, René Descartes ha coniato il termine immaginario per questi numeri in senso dispregiativo.", "n_reviews": 0, "start": 470.06, @@ -525,7 +525,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right?", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you", "translatedText": "E, okay, se vogliamo estendere il nostro sistema numerico, capisco, forse è utile mettere qualche tipo di numero lassù, ma perché io, giusto?", "n_reviews": 0, "start": 526.5, @@ -553,7 +553,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors.", + "input": "have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question.", "translatedText": "All'inizio, parliamo solo di come se stai sommando numeri bidimensionali come questo, le regole sono piuttosto semplici e funziona essenzialmente come i vettori, per chiunque di voi abbia familiarità con i vettori.", "n_reviews": 0, "start": 557.3, @@ -630,14 +630,14 @@ "end": 662.3 }, { - "input": "None of them is much lower at a.", + "input": "one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to", "translatedText": "Nessuno di loro è molto più basso di a.", "n_reviews": 0, "start": 662.7, "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real.", + "input": ", why should that live there? What on earth does the idea of a point one unit above the real number line in a separate dimension have to do with squaring to negative one? So I hope to answer this for you. At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be famili", "translatedText": "Ok, sembra che abbiamo un gruppo di persone che si sentono a proprio agio con il negativo 1, un vasto gruppo di persone che si sentono a disagio con l'infinito, questo è un argomento per un altro giorno, non preoccuparti, e poi un numero di persone che sono un po' in quella via di mezzo, forse non sono molto a loro agio con l'idea che il negativo 1 possa essere reale.", "n_reviews": 0, "start": 665.1, @@ -651,7 +651,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two.", + "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It turns out to be r", "translatedText": "Quindi per la nostra prima domanda molto più matematica, come una sorta di riscaldamento, voglio solo chiederti di aggiungere queste due.", "n_reviews": 0, "start": 683.42, @@ -672,35 +672,35 @@ "end": 706.82 }, { - "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me.", + "input": "ative two plus two i. So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers extend in this direction. can get you something lik", "translatedText": "Sfortunatamente, e puoi capire dal fatto che sto temporeggiando e da quello che sto dicendo qui, sembra che la domanda non venga ancora caricata completamente correttamente, quindi avrò una parola severa con Cam e Ider dietro il scene che altrimenti hanno costruito un'interfaccia così bella che è utile per questo tipo di avanti e indietro tra voi e me.", "n_reviews": 0, "start": 707.84, "end": 726.28 }, { - "input": "I'm going to have a stern word with them behind the scenes, but in the meantime let's go ahead and move forward with the lesson here.", + "input": "e it. But the rules end up being very different from that in the number system. You can't really do algebra. You can't do things like assume that if two numbers multiply to make zero, then", "translatedText": "Avrò una parola severa con loro dietro le quinte, ma nel frattempo andiamo avanti e andiamo avanti con la lezione qui.", "n_reviews": 0, "start": 726.76, "end": 732.64 }, { - "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be.", + "input": "one of them h as to be zero. But complex numbers are going to end up behaving much like the real numbers, s Now assuming that our question system ha", "translatedText": "Quindi immagino di poterlo tirare su, semplicemente sul pezzo di carta, e tu puoi seguirlo a casa, vedere quale potrebbe essere l'aggiunta.", "n_reviews": 0, "start": 733.46, "end": 739.92 }, { - "input": "It turns out to be relatively straightforward.", + "input": "s not broken down, I should be able to do this as a proper poll and let me go ahead, I guess we can first check the previous poll, okay things se", "translatedText": "Risulta essere relativamente semplice.", "n_reviews": 0, "start": 740.24, "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time.", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of th", "translatedText": "Se stai spostando quattro unità a destra e poi un'unità in alto, e vuoi aggiungere l'idea di spostare due unità a sinistra e poi due unità in alto, beh, basta farlo una alla volta.", "n_reviews": 0, "start": 742.94, @@ -714,7 +714,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left.", + "input": "red real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a g", "translatedText": "La parte reale saranno quei quattro a destra, poi meno due a sinistra.", "n_reviews": 0, "start": 754.82, @@ -728,7 +728,7 @@ "end": 760.88 }, { - "input": "And then the imaginary part is going to be this one unit up and then these two units up, one plus two, times i.", + "input": "you out there who would just reject infinity as being considered real but are very comfortable with the square root of negative o", "translatedText": "E poi la parte immaginaria sarà questa unità su e poi queste due unità su, uno più due, per i.", "n_reviews": 0, "start": 761.16, @@ -756,7 +756,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great.", + "input": "are root of negative one, fascinating, I actually would have thought that none of them would have come higher than t", "translatedText": "L'aggiunta non richiede nulla di complicato, il che è fantastico.", "n_reviews": 0, "start": 777.86, @@ -770,14 +770,14 @@ "end": 784.2 }, { - "input": "What is so complex about complex numbers after all?", + "input": "m is much lower at a, okay so it looks like we've got a cohort of people who are comfortable with negative one, a la", "translatedText": "Dopotutto, cosa c'è di così complesso nei numeri complessi?", "n_reviews": 0, "start": 784.42, "end": 787.1 }, { - "input": "Well where everything becomes interesting is when you try to multiply these numbers together.", + "input": "rge cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people w", "translatedText": "Ebbene, tutto diventa interessante quando provi a moltiplicare questi numeri insieme.", "n_reviews": 0, "start": 787.64, @@ -798,14 +798,14 @@ "end": 803.68 }, { - "input": "But the rules end up being very different from that in the number system.", + "input": "mfortable with the idea that negative one might be real, let's see if we can convince you of the difference of t", "translatedText": "Ma le regole finiscono per essere molto diverse da quelle del sistema numerico.", "n_reviews": 0, "start": 803.68, "end": 806.86 }, { - "input": "You can't really do algebra.", + "input": "hat. So what we've done here is we've taken three, two and then", "translatedText": "Non puoi davvero studiare l'algebra.", "n_reviews": 0, "start": 806.86, @@ -826,7 +826,7 @@ "end": 817.78 }, { - "input": "But to understand what that multiplication rule is, I just want to ask you a simple question.", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, a", "translatedText": "Ma per capire qual è la regola della moltiplicazione, voglio solo farti una semplice domanda.", "n_reviews": 0, "start": 818.3, @@ -840,14 +840,14 @@ "end": 831.82 }, { - "input": "We're not even going to think of it as a complex number per se.", + "input": "part of this, but it is, it's a good thing to know when you're learning about complex numbers, it'", "translatedText": "Non lo considereremo nemmeno un numero complesso di per sé.", "n_reviews": 0, "start": 832.58, "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this?", + "input": "s definitely one of those operations that you are going to need to know. Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is stil", "translatedText": "Se ho solo una specie di griglia di coordinate e vado al punto con la coordinata x tre e la coordinata y due, qual è la rotazione di 90 gradi di questa?", "n_reviews": 0, "start": 835.72, @@ -875,14 +875,14 @@ "end": 859.44 }, { - "input": "Okay.", + "input": "built such a beautiful, beautiful inte", "translatedText": "Va bene.", "n_reviews": 0, "start": 865.28, "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out.", + "input": "rface that's helpful for this kind of back and forth between you guys and me. nice gut check here is", "translatedText": "Ora, la cosa bella è che possiamo semplicemente girare il foglio per capirlo.", "n_reviews": 0, "start": 865.76, @@ -896,28 +896,28 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three.", + "input": "e. So that was another 90 degree rotation. Well what's happened here is we've just made both of the coordinates negative and that's", "translatedText": "Quindi quello che abbiamo fatto qui è stato prendere tre, due e poi convertirlo in meno due, tre.", "n_reviews": 0, "start": 884.08, "end": 890.68 }, { - "input": "Something which maybe in our original system you know looks like this negative two and then three.", + "input": "reassuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them", "translatedText": "Qualcosa che forse nel nostro sistema originale assomiglia a questo negativo due e poi tre.", "n_reviews": 0, "start": 891.58, "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation.", + "input": "negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation", "translatedText": "Questa sarà la rotazione di 90 gradi.", "n_reviews": 0, "start": 898.1, "end": 899.9 }, { - "input": "And what's nice here is that that rule is very simple and it applies to any pair that we might have.", + "input": "actually behaves like you would expect it to. Now why am I asking you this? Well I'm being told that supposedly I'm allowed to ask you questions again so I 'm going to ha", "translatedText": "E la cosa bella è che questa regola è molto semplice e si applica a qualsiasi coppia che potremmo avere.", "n_reviews": 0, "start": 899.9, @@ -945,7 +945,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong.", + "input": "52 of you answered simply 2 which would have been the real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated goin", "translatedText": "Bene, quello che è successo qui è che abbiamo reso entrambe le coordinate negative e questo è rassicurante perché se prendo un punto seduto in ab e poi lo ruoto di 90 gradi, questa sarà la mia rotazione iniziale di 90 gradi e poi altri 90 gradi, questo è il uguale a una rotazione di 180 gradi - oh no, ho sbagliato.", "n_reviews": 0, "start": 938.48, @@ -959,14 +959,14 @@ "end": 980.14 }, { - "input": "Now why am I asking you this?", + "input": "you try to multiply these numbers together. So with vectors, there's not really any notion", "translatedText": "Ora perché ti sto chiedendo questo?", "n_reviews": 0, "start": 980.4, "end": 981.76 }, { - "input": "Well I'm being told that supposedly I'm allowed to ask you questions again so I'm going to have you do your very first complex product.", + "input": "of multiplying them to get two vectors back, at least when we're in the 2d plane. that we ask questions and just say hey kamineter can't you make the live questio", "translatedText": "Beh, mi è stato detto che presumibilmente mi è permesso farti ancora delle domande, quindi ti farò realizzare il tuo primo vero prodotto complesso.", "n_reviews": 0, "start": 982.66, @@ -987,14 +987,14 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i.", + "input": "Wonderful! Very simple question I want you to take the number i and I want you to multiply it by 3 pl", "translatedText": "Ok, sembra che la maggior parte di voi abbia ottenuto la risposta corretta che è 2 più 3i.", "n_reviews": 0, "start": 1003.06, "end": 1006.98 }, { - "input": "Very good very good.", + "input": "us 2i and even though I haven't really talked about You can't do things like assume that if two numbers multiply to mak", "translatedText": "Molto buono molto buono.", "n_reviews": 0, "start": 1007.48, @@ -1015,28 +1015,28 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good.", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t", "translatedText": "Abbiamo 2 più 3 che forse sta semplicemente lasciando cadere la i quindi penso che forse mi piacciono molto i semplici errori e le immissioni e sai che succede a tutti noi, specialmente nei test, a volte sai qual è la risposta giusta ma poi se dimentichi un simbolo o ne scambi due, va tutto molto bene.", "n_reviews": 0, "start": 1037.6, "end": 1052.36 }, { - "input": "Let's go ahead and try our very first product though like I said so here because I already talked through one of the questions we're going to go ahead and skip ahead of it we know how to rotate something like 3 comma 2 so I'm not even going to give you time to do that and properly grade it.", + "input": "his is we can basically just turn our paper to figure it out. ons as you do it rather than sitting in passively watching this is genuinely delightful to me. Okay this is this isn't necessarily a question I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not", "translatedText": "Andiamo avanti e proviamo il nostro primissimo prodotto, come ho detto qui perché ho già parlato di una delle domande che andremo avanti e salteremo oltre, sappiamo come ruotare qualcosa come 3 virgola 2 quindi sono non ti darò nemmeno il tempo di farlo e valutarlo correttamente.", "n_reviews": 0, "start": 1052.88, "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us?", + "input": "that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three i which is absolutely correct absolutely correct so there's two w", "translatedText": "Stal stal Words parole che sapete mi dicono che funziona eppure è molto lento per me andare avanti quindi sapete che se non avrò una parola severa con loro voi ragazzi potete attaccarli anche su Twitter sotto lo stesso luogo in cui facciamo domande e diciamo semplicemente: "Ehi Kamineter", non puoi far funzionare un po' meglio le domande dal vivo per noi?", "n_reviews": 0, "start": 1070.62, "end": 1089.28 }, { - "input": "Okay I think we're finally there.", + "input": "ays to think about this okay one of them is", "translatedText": "Ok, penso che finalmente ci siamo.", "n_reviews": 0, "start": 1090.02, @@ -1050,14 +1050,14 @@ "end": 1094.22 }, { - "input": "Aha!", + "input": "e algebra and just do it a little bit mechanistically okay so", "translatedText": "Ah!", "n_reviews": 0, "start": 1094.6, "end": 1094.76 }, { - "input": "Wonderful!", + "input": "if we pull ourselves up", "translatedText": "Meraviglioso!", "n_reviews": 0, "start": 1094.76, @@ -1071,7 +1071,7 @@ "end": 1126.42 }, { - "input": "Wonderful!", + "input": "that if you want to rotate numbers 90 degrees", "translatedText": "Meraviglioso!", "n_reviews": 0, "start": 1127.02, diff --git a/2020/ldm-complex-numbers/japanese/sentence_translations.json b/2020/ldm-complex-numbers/japanese/sentence_translations.json index a4403ed54..0b902286a 100644 --- a/2020/ldm-complex-numbers/japanese/sentence_translations.json +++ b/2020/ldm-complex-numbers/japanese/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "実際にアプリケーション で役立つものであれば、それは言葉と同じくらいリアルだと思いますよね。", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "世の中 で幸福のような抽象的な言葉に出会うことは決してありませんが、それ は私たちの心の中にある種の現実性を持っており、分数では表現できな い 2 の平方根や、負の 1 の平方根は、たとえ少し違うように見 えても、実際の正規数には現れません。", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "基本的な入門書を目 的としているので、まだ内容を理解しているとは思いませんが、早速見ていきましょう。", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "私が学校にいたとき、加算の公式を習ったことを覚え ています。2 つの異なる角度の和の余弦を知りたい場合、元の 2 つの角度の 余弦と正弦を計算すると、これほど長いものになります。常に人々をつまずかせ るマイナス記号があります。この記号に対して同じことを行うと、見た目は似て いますが、プラス記号があり、cos-cos の代わりに cos-sin があり、非常に間違いが発生しやすいものです。そのまま覚えようとするなら。", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "ただし、複素数を扱う場合、これは間違いがはるかに少ないだけでなく、非常 に美しい意味を持ち、すぐに抜け落ちます。", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "したがって、たとえマイナス 1 の平方 根の現実を必ずしも信じていないとしても、少なくとも、それが他の数学の部分を役立つようにする可能 性があること、他の数学の部分がもう少し役立つことは興味深いことであることを認めなければなりま せん。も理解できます。", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "1つは、いいえ、あり ませんよね？", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "2 乗した数値 が正であれば、そのまま正のままです。", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "決してネガティブなことを受け取るつもりはありません。", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "しかし、数学者が来て、「いやいや、そんなものは存在しない」と言った としても、そうなるように私たちはそれを定義しました。", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "解決できない問題があるときは、「ああ、私は物事を定義したの で、魔法のように解決策が得られました」と言うことができます。", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "したがって、これに不快感を感じているのは、決してあな ただけではありません。", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "実際、ルネ・デカルトはこれらの数字を軽蔑する意味で「虚数」という用語を 作りました。", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "そして私たちはそれを慣例として守り続け、今でもそれらを 虚数と呼んでいますが、これは本当にばかげています。", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "複素数について話し始めるときに行う 2 番目の奇妙なことは、そのような数 i だ けが存在するわけではないが、それにホームを与えるつもりだということです。", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "それで、番号システムを拡張したいの なら、それはわかります。そこに何らかの番号を置くと便利かもしれませんが、なぜ私がそうするのでしょうか? ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "最初に、 ベクトルに詳しい人向けに、このように 2 次元の数値を加算する 場合、ルールは非常に単純で、基本的にベクトルと同じように動作す る方法について説明しましょう。", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "これを信じて従うのであれば、それが役立つという事実が、私たちがこのようなことをしている 理由を正当化するのに役立つことを願っています。", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "ええと、a があるようですが、答え f と d の間で行ったり来たりしているので、f はすべてであり、 これらはすべて本物であると考えるべきだと言っています。", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "そして興味深いのは、d は 2 の 2 平方根とマイナス 1 は考慮すべきであるが、無限大は考慮すべきではない ということです。そのため、無限大は現実であるとみなされることを拒否するもの の、マイナス 1 の平方根、すごいですね。", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "さて、負の 1 に慣れている人々の集団がいるように見えますが、無限大に不快感を抱いている大規模 な集団がいます。これについては別の日の話題です。心配しないでください。そして 、次のような人々が多数いるようです。負の 1 が現実であるという考えにあまり満 足していないのかもしれません。", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "そこで、ウォーミングアップのような意味で、より数学的な最初の質問として、これら 2 つを追加していただきた いと思います。", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "足し算の方法を教える前に、それがどのように機能するかを推測してくださ い。それが非常に簡単だと感じていただければ幸いです。足し算は実際にはこの中で最 も興味のない部分ですが、それは事実です。あなたは複素数について学んでいます。", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "4 ユニットを右に移動してから 1 ユニット上に移動し、2 ユニットを左に移動して から 2 ユニット上に移動するというアイデアを追加したい場合は、それらを一度に 1 つずつ実行す るだけです。", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "実際の部分は、右側の 4 つと、左側の 2 つを引いたものになります。", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "つまり、1 つの i に 2 つの i を加えたものになります。", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "加算には複雑な処理は何もなく、これは素晴らしいこと です。", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "したがって、ベクトルに関しては、少なくとも 2D 平面にいる場合、それらを乗算して 2 つのベクトルを戻すという概念は実際にはありません。", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "これは基本的に、ポイント 3、2 があると仮定します。", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "ある種の座標グリッドがあり、x 座標が 3、y 座標が 2 の点に行く場合、これの 90 度の回転は何になるでしょうか?90度回転させ て反時計回りにしてみましょう。", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "反時計回り。", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "これの素晴らしい点は、基本的に紙をめくるだけでそれを理解できることです。", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "3、2 から始めても大丈夫だと言うので、反時計回りに 90 度回転させます。平面全 体をそのように回転させた場合、x 方向にマイナス 2、次に y 方向にマイナス 3 になると読み取れます。。", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "ここで私たちが行ったことは、3、2 を取得し、それをマイ ナスの 2、3 に変換することです。", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "つまり90度回転することになります。", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "数値のペアをカンマ b として取り、それを 90 度回転するとどこに回転 するのかと言った場合、最終的には座標 ba を交換して、最初の座標を負にす ることになります。", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "つまり、さらに 90 度回転しました。", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "さて、ここで何が起こったかとい うと、両方の座標を負にしただけです。これで安心できます。ab に座っている点をとって、それを 90 度回転すると、これが最初の 90 度の回転となり、次にさらに 90 度回転すると、こ れが最初の 90 度の回転になります。180度回転と同じです。ああ、間違ってしまいました。", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "これは 180 度の回転と同じになり、次のようになります。両方の座標 を取得してそれらを負の値、負の値、負の値にするだけの他のベクトルは無 視してください。これで 90 度の回転を行う操作が安心できます。実 際には期待どおりに動作します。", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "ああ、たくさんの人がとても良い回答を提出してくれました。", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "さて、皆さんの大多数が 2 プラス 3i という正しい答えを得たようです。", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "とても良いとても良い。", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "皆さんの中には、マイナス 2 3 と答えた人もいますが、これは 4 マイナス 2 を取るか 2 マイナス 4 を取るかを入れ替えているだけだと思います。それは完全に理解できます。", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "2 と 3 が ありますが、これはおそらく i を落としているだけなので、おそらく単純なエラーや入力のようなものが 多いと思います。特にテストでは、正しい答えが何かを知っているときは誰にでも起こりますが、その後はシン ボルを 1 つ忘れたり、2 つを交換したりしても、それはすべて非常に良いことです。", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "スタルスタルの言 葉、彼らは私にそれがうまくいっていると言ってくれますが、それでも私が前に進むのは非常に遅い ので、私が彼らに厳しい言葉を言うつもりがないなら、あなたたちもツイッターで彼らに同じよう に攻撃してください。私たちが質問して、「ねえ、カミネーター」と言う場所です。ライブの質問を もう少しうまく機能させてくれませんか?さて、ついに到着したようです。", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "皆さん準備はできていますか？", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "ああ！", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "素晴らしい！", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "非常に単純な質問です。数値 i を取得して、それに 3 プラス 2i を掛けてください。掛け算のルールについてはあまり話していませんが、私が言えるの は、それが次の場合と同じように動作するかのように振る舞うことです。通常の数に は、これを全体に分配できる分配特性のようなものがあります。そして、i の決定 的な特徴は、i の 2 乗が負であるという考えです。それが、それを扱うこと以外 に知っておくべき唯一の特別なことです。通常の数値と同じなので、製品を進めてく ださい。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/korean/sentence_translations.json b/2020/ldm-complex-numbers/korean/sentence_translations.json index e15bb0359..0bf87cfc5 100644 --- a/2020/ldm-complex-numbers/korean/sentence_translations.json +++ b/2020/ldm-complex-numbers/korean/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "응용 프로그램에서 실제로 유용한 것이라면 말만큼이나 현실적이라고 말하고 싶습니다. 그렇죠? ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "행복과 같은 추상적인 단어는 결코 접하지 못할 것입니다. 그러나 그것은 우리 마음 속에 일종의 현실을 갖고 있습니다. 분수로 표현할 수 없는 2의 제곱근이나 실제 일반 숫자에는 나타나지 않는 음수의 제곱근입니다. 비록 조금 달라 보일 수도 있습니다. 아, 정말 부끄러운 일이에요. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "나는 당신이 그것이 무엇인지 아직 안다고 가정하지 않을 것입니다. 그것은 기본 입문서가 될 것이지만 바로 뛰어 들어 갑시다. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "제가 학교에 다닐 때 이런 덧셈 공식을 배웠던 기억이 납니다. 서로 다른 두 각도의 합에 대한 코사인을 알고 싶다면 원래 두 각도의 코사인과 사인을 계산하면 이렇게 긴 작업이 됩니다. , 항상 사람들을 놀라게 하는 빼기 기호가 있습니다. 기호에 대해 동일한 작업을 수행하면 비슷해 보이지만 더하기 기호가 있고, cos-cos 대신 cos-sin이 있으므로 오류가 발생하기 쉽습니다. 그냥 있는 그대로 외우려고 한다면요. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "그러나 복소수를 사용하면 오류가 발생할 가능성이 훨씬 낮을 뿐만 아니라 매우 아름다운 의미를 갖고 있어 바로 이해되지 않습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "따라서 여러분이 -1의 제곱근의 현실을 반드시 믿지 않더라도 최소한 그것이 다른 수학 부분을 유용하게 만들 수 있다는 것과 다른 수학 부분을 조금 더 유용하게 만들 수 있다는 것이 흥미롭다는 점을 인정해야 합니다. 이해도 된다. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "그런데 출발점이 아주 이상해 보이는군요. ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "하나는, 아니, 그렇지 않습니까? ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "제곱한 숫자가 양수이면 음수는 그대로 유지됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "나는 결코 부정적인 것을 얻지 못할 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "그러나 수학자가 와서 '아, 안돼, 그게 존재한다'고 말한다면, 우리는 그것이 사실이 되도록 정의한 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "해결할 수 없는 문제가 있을 때, '아, 내가 정의했으므로 이제 마법처럼 해결책이 생겼습니다'라고 말할 수 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "따라서 이것이 불편하다면 확실히 혼자가 아닙니다. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "사실 르네 데카르트(Rene Descartes)는 이러한 숫자에 대해 경멸적인 의미로 상상이라는 용어를 만들었습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "그리고 우리는 그것을 관례로 고수하고 여전히 그것을 허수라고 부르는데, 이는 정말로 터무니없는 일입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "복소수에 대해 이야기하기 시작할 때 하는 두 번째 이상한 일은 그런 숫자 i만 있는 것이 아니라 우리가 그것을 가정할 것이라고 말하는 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "그리고 좋아요, 숫자 시스템을 확장하고 싶다면 알겠습니다. 거기에 숫자를 입력하는 것이 유용할 수도 있지만 왜 그렇죠? ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "처음에는 벡터에 익숙한 분들을 위해 이렇게 2차원 숫자를 추가하는 경우 규칙이 매우 간단하고 기본적으로 벡터와 동일하게 작동하는 방법에 대해 이야기하겠습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "당신이 그것을 믿고 따른다면, 그것이 유용하다는 사실이 우리가 이 일을 하는 이유를 정당화하는 데 도움이 되기를 바랍니다. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "음, a가 있는 것 같은데요, 답변 f와 d 사이에 앞뒤가 있는 것 같으니, f는 전부이고, 이 모든 것이 진짜로 간주되어야 한다는 말입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "그리고 흥미롭게도, d는 2의 제곱근과 -1을 고려해야 하지만 무한대는 고려하지 말아야 한다고 말하는 것입니다. 따라서 무한대가 실수라고 간주되는 것을 거부하지만 -1의 제곱근, 정말 멋지군요. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "좋습니다. 마이너스 1을 선호하는 집단이 있는 것 같고, 큰 집단은 무한대를 불편하게 여기는 것 같습니다. 그건 나중에 다루겠습니다. 걱정하지 마세요. 마이너스 1이 진짜일 수도 있다는 생각이 그다지 편하지 않을 수도 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "그래서 우리의 첫 번째 훨씬 더 수학적 질문에 대해 일종의 준비운동으로 이 두 가지를 추가해 달라고 요청하고 싶습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "추가하는 방법을 가르치기 전에 어떻게 작동할지 추측해 보세요. 꽤 간단하게 느껴지기를 바랍니다. 덧셈은 사실 이 중에서 가장 흥미롭지 않은 부분입니다. 하지만 그렇습니다. 당신은 복소수에 대해 배우고 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "4개의 유닛을 오른쪽으로 이동한 다음 한 개의 유닛을 위로 이동하고, 두 개의 유닛을 왼쪽으로 이동한 다음 두 개의 유닛을 위로 이동하는 아이디어를 추가하려면 한 번에 하나씩 수행하면 됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "실제 부분은 오른쪽에 있는 4개, 왼쪽에 있는 마이너스 2개가 될 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "그래서 그것은 1 i 더하기 2 i 입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "덧셈에는 실제로 복잡한 일이 일어나지 않습니다. 정말 좋습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "따라서 벡터의 경우 적어도 2D 평면에 있을 때 두 개의 벡터를 다시 얻기 위해 벡터를 곱한다는 개념이 실제로 없습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "기본적으로 제가 포인트 3, 2를 갖고 있다고 가정해 보겠습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "일종의 좌표 격자가 있고 x 좌표 3과 y 좌표 2가 있는 지점으로 가면 이것의 90도 회전은 무엇입니까? ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "시계 반대 방향. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "이제 이것에 대한 멋진 점은 기본적으로 종이를 뒤집어서 알아낼 수 있다는 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "3, 2에서 시작한 다음 시계 반대 방향으로 90도 회전하면 괜찮다고 하셨습니다. 이제 x 방향으로 -2가 되고 y 방향으로 3이 되는 것으로 읽을 수 있습니다. 전체 평면을 그렇게 회전했다면 . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "그래서 여기서 우리가 한 것은 3, 2를 취하고 그것을 -2, 3으로 변환한 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "그러면 90도 회전이 됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "내가 한 쌍의 숫자 a 쉼표 b를 선택하고 90도 회전하면 회전할 위치를 말하면 좌표 ba를 교환한 다음 첫 번째 값을 음수로 만들게 됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "그래서 그것은 또 다른 90도 회전이었습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "여기서 일어난 일은 우리가 두 좌표를 모두 음수로 만들었기 때문에 안심이 됩니다. 왜냐하면 ab에 앉아 어떤 점을 취하고 그것을 90도 회전하면 이것이 나의 초기 90도 회전이 되고 또 다른 90도 회전이 되기 때문입니다. 180도 회전과 동일합니다. 아뇨, 제가 잘못했습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "이는 180도 회전과 같을 것입니다. 이렇게 보일 것입니다. 제가 그린 다른 벡터는 무시하세요. 두 좌표를 모두 가져와 음수 a 음수 b로 만드는 것입니다. 그러면 90도 회전을 수행하는 이 작업이 안심됩니다. 실제로는 예상한 대로 작동합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "아, 많은 사람들이 아주 좋은 답변을 제출했습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "여러분 중 대다수가 2 더하기 3i라는 정답을 얻은 것 같습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "아주 좋아요 아주 좋아요. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "여러분 중 일부는 마이너스 2 3이라고 대답했는데, 제 생각에는 4 빼기 2를 사용하든지 2 빼기 4를 사용하든지 바꾸는 것일 뿐이므로 완전히 이해할 수 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "2 더하기 3이 있는데 아마도 i가 떨어지는 것일 수도 있습니다. 따라서 단순한 오류나 입력과 같은 경우가 많을 것이라고 생각합니다. 특히 테스트에서 우리 모두에게 발생하는 현상은 때때로 정답이 무엇인지 알지만 그렇다면 기호 하나를 잊어버리거나 두 개를 바꿔서 모두 아주 좋습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "부실한 말들 그들이 나에게 그것이 효과가 있다고 말하지만 앞으로 나아가는 것이 매우 느리기 때문에 내가 그들에게 단호한 말을 하지 않을 경우 여러분은 트위터에서도 같은 아래에 있는 그들에게 갈 수 있습니다. 우리가 질문을 하고 그냥 "Hey kamineter"라고 말하는 곳에서 실시간 질문이 우리에게 좀 더 잘 작동하도록 만들어 줄 수 없나요? ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "좋아요, 드디어 도착한 것 같아요. ", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "다들 준비됐나요? ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "아하! ", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "아주 멋진! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "매우 간단한 질문입니다. 숫자 i를 선택하고 여기에 3 더하기 2i를 곱하라고 합니다. 비록 제가 곱셈의 규칙에 대해 실제로 이야기하지는 않았지만 내가 말할 수 있는 것은 그것이 마치 그것이 하는 것처럼 작동하는 것처럼 가장한다는 것입니다. 정규 숫자는 이것을 전체적으로 분배할 수 있는 분배 속성과 같은 것을 가지고 있으며 i의 정의 기능은 내가 제곱한 것이 음수라는 아이디어입니다. 이는 당신이 그것에 대해 알아야 할 유일한 특별한 것입니다. 정상적인 숫자인 것처럼 괜찮습니다. 그런 다음 제품을 계속 진행하세요. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/marathi/sentence_translations.json b/2020/ldm-complex-numbers/marathi/sentence_translations.json index 7a927abcb..9588fefd8 100644 --- a/2020/ldm-complex-numbers/marathi/sentence_translations.json +++ b/2020/ldm-complex-numbers/marathi/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "मी म्हणेन की जर ते अनुप्रयोगात खरोखर उपयुक्त असेल तर ते शब्दांइतकेच खरे आहे, बरोबर? ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "तुम्ही आनंदासारख्या अमूर्त शब्दात कधीच भाग घेणार नाही, पण त्यात एक प्रकारची वास्तविकता आपल्या मनात असते आणि दोनच्या वर्गमूळासारख्या गोष्टी, ज्याला तुम्ही अपूर्णांक म्हणून व्यक्त करू शकत नाही, किंवा यासारख्या गोष्टी. वास्तविक सामान्य संख्यांमध्ये न दिसणार्‍या ऋणाचे वर्गमूळ, तुम्हाला माहिती आहे, जरी ते थोडेसे वेगळे वाटत असले तरीही. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "मी असे गृहीत धरणार नाही की ते अद्याप काय आहेत हे तुम्हाला माहिती आहे, ते एक मूलभूत प्राइमर आहे, परंतु आपण बरोबर जाऊया, ठीक आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "मला आठवते की मी शाळेत होतो आणि आम्ही ही जोड सूत्रे शिकलो, की जर तुम्हाला दोन भिन्न कोनांच्या बेरजेची कोसाइन जाणून घ्यायची असेल, तर तुम्हाला माहिती आहे, मूळ दोन कोनांच्या कोसाइन आणि साइन्सच्या बाबतीत ही अशी लांबलचक गोष्ट आहे. , हे वजा चिन्ह आहे जे नेहमी लोकांना वर आणते, जर तुम्ही चिन्हासाठी असे केले तर ते सारखे दिसते परंतु एक अधिक चिन्ह आहे, आणि तुमच्याकडे cos-cos असण्याऐवजी cos-sin आहे, हे असे काहीतरी आहे जे खूप त्रुटी-प्रवण आहे जर तुम्ही ते जसे आहे तसे लक्षात ठेवण्याचा प्रयत्न करत असाल. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "तथापि, जर तुम्ही त्यावर जटिल संख्यांसह आलात, तर हे केवळ कमी त्रुटी-प्रवणच नाही, तर त्याचा खूप सुंदर अर्थ आहे आणि तो अगदी बरोबर येतो. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "त्यामुळे जरी तुमचा ऋण 1 च्या वर्गमूळाच्या वास्तविकतेवर विश्वास नसला तरीही, तुम्हाला हे मान्य करावेच लागेल की हे मनोरंजक आहे की ते गणिताचे इतर तुकडे उपयुक्त बनवू शकतात, गणिताचे इतर तुकडे थोडे अधिक. समजण्यासारखे देखील. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "पण सुरुवातीचा बिंदू खूप विचित्र दिसत आहे, ठीक आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "एक आहे, नाही तिथे नाही, बरोबर? ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "तुम्ही वर्ग कराल अशी कोणतीही संख्या, जर ती सकारात्मक असेल, तर ती फक्त सकारात्मकच राहते. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "मला कधीही नकारात्मक काहीही मिळणार नाही. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "तथापि, जर एखादा गणितज्ञ आला आणि म्हणाला, अरे नाही नाही ते अस्तित्वात आहे, आम्ही ते परिभाषित केले आहे जेणेकरून ते तसे आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "जेव्हा तुम्हाला एखादी समस्या येते जी तुम्ही सोडवू शकत नाही, तेव्हा तुम्ही फक्त असे म्हणू शकता, अरे मी गोष्टी परिभाषित केल्या आहेत जेणेकरून आमच्याकडे आता जादुई उपाय आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "त्यामुळे जर तुम्हाला यात अस्वस्थता वाटत असेल, तर तुम्ही नक्कीच एकटे नाही आहात. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "खरं तर, रेने डेकार्टेसने या संख्यांसाठी काल्पनिक हा शब्द अपमानास्पद म्हणून तयार केला. ", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "आणि मग आम्ही ते एक अधिवेशन म्हणून अडकले आणि तरीही आम्ही त्यांना काल्पनिक संख्या म्हणतो, जे खरोखरच हास्यास्पद आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "दुसरी विचित्र गोष्ट जी तुम्ही कॉम्प्लेक्स नंबर्सबद्दल बोलायला सुरुवात करता ते म्हणजे, फक्त अशी संख्या i नाही, पण आम्ही त्याला घर देणार आहोत. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "आणि, ठीक आहे, जर आपल्याला आमची संख्या प्रणाली वाढवायची असेल, तर मला ते समजले, कदाचित तेथे काही प्रकारची संख्या ठेवणे उपयुक्त आहे, परंतु मी का, बरोबर? ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "अगदी सुरुवातीस, जर तुम्ही अशा द्विमितीय संख्या जोडत असाल तर नियम अगदी सरळ आहेत आणि ते मूलत: व्हेक्टर प्रमाणेच कार्य करतात, तुमच्यापैकी कोणाला व्हेक्टर परिचित असतील याबद्दल बोलूया. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "जर तुम्ही ते विश्वासावर घेतले आणि तुम्ही त्याचे अनुसरण केले, तर आशा आहे की ते उपयुक्त ठरते या वस्तुस्थितीमुळे आम्ही यापैकी काहीही का करत आहोत याचे समर्थन करण्यात मदत होईल. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "अं, असे दिसते की, f आणि d या उत्तरांमध्‍ये अ, पुढे आणि मागे आहे, त्यामुळे f हे सर्व आहे, असे म्हणणे की या सर्वांना वास्तविक मानले पाहिजे. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "आणि मनोरंजक, d हे असे आहे की तुम्ही 2 चे 2 वर्गमूळ आणि ऋण 1 विचारात घेतले पाहिजे, परंतु अनंत नाही, म्हणून तुमच्यापैकी एक चांगला संघ आहे जो फक्त अनंताला वास्तविक मानले जात आहे म्हणून नाकारेल, परंतु ते खूप सोयीस्कर आहे. ऋण 1 चे वर्गमूळ, ते छान आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "ठीक आहे, त्यामुळे असे दिसते की आमच्याकडे नकारात्मक 1 सह सोयीस्कर असलेल्या लोकांचा समूह आहे, एक मोठा समूह अनंताने अस्वस्थ आहे, हा दुसर्‍या दिवसाचा विषय आहे, त्याची काळजी करू नका, आणि नंतर अनेक लोक जे नकारात्मक 1 वास्तविक असू शकते या कल्पनेने कदाचित जास्त सोयीस्कर नसण्याच्या त्या मध्यम मैदानात आहेत. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "तर आमच्या पहिल्या अधिक गणिताच्या प्रश्नासाठी, एक प्रकारचा सराव म्हणून, मी तुम्हाला हे दोन जोडण्यास सांगू इच्छितो. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "मी तुम्हाला ते कसे जोडायचे ते शिकवण्यापूर्वी, ते कसे कार्य करू शकते याचा अंदाज लावा आणि मला आशा आहे की ते अगदी सरळ वाटेल, जोडणे हा यातील सर्वात कमी मनोरंजक भाग आहे, परंतु हे जाणून घेणे चांगली गोष्ट आहे की तुम्ही जटिल संख्यांबद्दल शिकत आहात. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "जर तुम्ही चार युनिट्स उजवीकडे आणि नंतर एक युनिट वर हलवत असाल, आणि तुम्हाला दोन युनिट्स डावीकडे आणि नंतर दोन युनिट्स वर हलवण्याची कल्पना जोडायची असेल, तर तुम्ही फक्त त्या प्रत्येकी एका वेळी करा. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "खरा भाग उजवीकडे त्या चार, नंतर डावीकडे वजा दोन असा असणार आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "म्हणजे एक i अधिक दोन i. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "जोडण्यामध्ये खरोखर काहीही क्लिष्ट होत नाही, जे छान आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "त्यामुळे व्हेक्टरमध्ये, दोन व्हेक्टर परत मिळविण्यासाठी त्यांचा गुणाकार करण्याची कोणतीही कल्पना नाही, किमान जेव्हा आपण 2D विमानात असतो. ", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "जे मुळात आहे, समजा माझ्याकडे मुद्दा तीन, दोन आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "जर माझ्याकडे फक्त काही प्रकारचे समन्वय ग्रिड असेल आणि मी x समन्वय तीन आणि y समन्वय दोन सह बिंदूवर गेलो, तर याचे 90 अंश रोटेशन काय आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "घड्याळाच्या उलट दिशेने. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "आता यात काय सुंदर आहे की आपण मुळात ते शोधण्यासाठी आपला पेपर फिरवू शकतो. ", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "तुम्ही म्हणता ठीक आहे जर ते तीन, दोन वाजता सुरू झाले आणि मग मी घड्याळाच्या विरुद्ध दिशेने 90 अंश फिरवले, तर मी आता फक्त ते वाचू शकतो ऋण दोन x दिशेने आणि नंतर तीन y दिशेने, जर मी संपूर्ण विमान असे फिरवले असते . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "तर आपण येथे काय केले आहे की आपण तीन, दोन घेतले आहेत आणि नंतर आपण त्यास नकारात्मक दोन, तीन मध्ये रूपांतरित केले आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "ते 90 अंश फिरणार आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "जर मी संख्यांची जोडी घेतली तर स्वल्पविराम b ठीक आहे आणि मग मी म्हणालो की ते कुठे फिरणार आहे जर मी ते 90 अंश फिरवले तर ते कोऑर्डिनेट्स ba चे अदलाबदल करून समाप्त होईल आणि नंतर प्रथम नकारात्मक बनवेल. ", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "तर ते आणखी एक 90 अंश रोटेशन होते. ", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "बरं इथे काय झालं ते म्हणजे आम्ही दोन्ही निर्देशांक ऋणात्मक केले आहेत आणि ते आश्वासक आहे कारण जर मी ab वर बसून काही बिंदू घेतला आणि नंतर मी ते 90 अंश फिरवले तर हे माझे सुरुवातीचे 90 अंश रोटेशन असेल आणि नंतर आणखी 90 अंश असेल. 180 डिग्री रोट सारखे- अरे नाही मी ते चुकीचे केले आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "ते 180 डिग्री रोटेशन सारखेच असेल जे मी काढलेल्या इतर वेक्टरकडे दुर्लक्ष करा जे फक्त दोन्ही निर्देशांक घेत आहे आणि त्यांना ऋण नकारात्मक बनवत आहे आणि 90 डिग्री रोटेशन करते हे या ऑपरेशनला आश्वस्त करत आहे. प्रत्यक्षात तुम्ही अपेक्षा कराल तसे वागते. ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "अरे बघा बर्‍याच लोकांनी खूप छान उत्तरे सबमिट केली आहेत. ", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "ठीक आहे, असे दिसते की तुमच्यापैकी बहुतेकांना बरोबर उत्तर मिळाले आहे जे 2 अधिक 3i आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "खूप चांगले खूप चांगले. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "तुमच्यापैकी काहींनी 2 3 नकारात्मक उत्तर दिले जे मला वाटते की तुम्ही 4 उणे 2 किंवा 2 उणे 4 घेत आहात की नाही हे फक्त बदलत आहे जेणेकरून ते पूर्णपणे समजण्यासारखे आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "आम्हाला 2 अधिक 3 मिळाले आहेत जे कदाचित फक्त i सोडत आहे म्हणून मला वाटते की कदाचित बर्‍याच साध्या चुका आणि नोंदी आहेत आणि तुम्हाला माहित आहे की आपल्या सर्वांच्या बाबतीत असे घडते विशेषत: चाचण्यांमध्ये कधीकधी तुम्हाला माहित असते की योग्य उत्तर काय आहे परंतु नंतर तुम्ही एखादे चिन्ह विसरलात किंवा तुम्ही दोन अदलाबदल करता ते सर्व खूप चांगले आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "Stal stal शब्द तुम्हाला माहित आहेत ते मला सांगतात की ते काम करत आहे आणि तरीही माझ्यासाठी प्रगती करणे खूप मंद आहे म्हणून तुम्हाला माहित आहे की मी त्यांच्याशी कठोर शब्द बोलणार नाही तर तुम्ही लोक त्यांच्याकडे ट्विटरवर देखील जाऊ शकता. ज्या ठिकाणी आम्ही प्रश्न विचारतो आणि फक्त अहो कामिनेटर म्हणतो, तुम्ही थेट प्रश्न आमच्यासाठी थोडे चांगले काम करू शकत नाही का? ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "ठीक आहे मला वाटते की आम्ही शेवटी तिथे आहोत. ", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "सगळे तयार? ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "अहाहा! ", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "अप्रतिम! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "अगदी सोपा प्रश्न मला तुम्ही i ही संख्या घ्यायची आहे आणि तुम्ही ती 3 अधिक 2i ने गुणाकार करावी अशी माझी इच्छा आहे आणि जरी मी गुणाकाराच्या नियमांबद्दल खरच बोललो नसलो तरी मी जे म्हणू शकतो ते असे ढोंग करा जसे ते चालते तसे चालते. सामान्य संख्या तुमच्याकडे वितरणात्मक मालमत्तेसारख्या गोष्टी आहेत जिथे तुम्ही ते सर्वत्र वितरीत करू शकता आणि नंतर i चे परिभाषित वैशिष्ट्य म्हणजे ही कल्पना आहे की मी स्क्वेअर केला आहे ही नकारात्मक आहे ही एकमेव विशेष गोष्ट आहे जी तुम्हाला त्या व्यतिरिक्त जाणून घेणे आवश्यक आहे. जसे की ही एक सामान्य संख्या आहे ठीक आहे आणि नंतर उत्पादनासह पुढे जा. अप्रतिम! ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/persian/sentence_translations.json b/2020/ldm-complex-numbers/persian/sentence_translations.json index 6d2a5882f..b5a577443 100644 --- a/2020/ldm-complex-numbers/persian/sentence_translations.json +++ b/2020/ldm-complex-numbers/persian/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "من می‌توانم بگویم که اگر این چیزی است که در یک برنامه کاربردی مفید است، پس به اندازه کلمات واقعی است، درست است؟ شما هرگز با یک کلمه انتزاعی مانند شادی روبرو نخواهید شد، اما نوعی واقعیت در ذهن ما وجود دارد، و چیزهایی مانند جذر دو، که نمی توانید آن را به صورت کسری بیان کنید، یا چیزهایی مانند می دانید که ریشه دوم منفی یک که در بین اعداد عادی واقعی نشان داده نمی شود، حتی اگر ممکن است کمی متفاوت به نظر برسند. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "یادم می آید زمانی که در مدرسه بودم و این فرمول های جمع را یاد گرفتیم، که اگر می خواهید کسینوس مجموع دو زاویه مختلف را بدانید، می دانید، از نظر کسینوس و سینوس دو زاویه اصلی، این چیز طولانی است. ، این علامت منفی وجود دارد که همیشه افراد را به سمت بالا می برد، اگر همین کار را برای علامت انجام دهید، شبیه به نظر می رسد اما یک علامت مثبت وجود دارد، و به جای داشتن cos-cos شما cos-sin دارید، این چیزی است که بسیار مستعد خطا است. اگر فقط سعی می کنید آن را همانطور که هست حفظ کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "با این حال، اگر با اعداد مختلط به آن بپردازید، این نه تنها خطای بسیار کمتری دارد، بلکه معنای بسیار زیبایی دارد و به سادگی از بین می رود. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "بنابراین حتی اگر لزوماً به واقعی بودن جذر منفی 1 اعتقاد ندارید، حداقل باید اعتراف کنید که جالب است که می‌تواند سایر بخش‌های ریاضی را مفیدتر کند، و سایر بخش‌های ریاضی را کمی بیشتر می‌کند. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "هر عددی را که مربع می‌کنید، اگر مثبت باشد، فقط مثبت می‌ماند. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "من هرگز چیزی منفی دریافت نمی کنم. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "با این حال، اگر یک ریاضیدان بیاید و بگوید، نه، وجود دارد، ما آن را تعریف کرده‌ایم تا اینطور باشد. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "فکر می‌کنم واکنش دیگری که کسی می‌تواند داشته باشد این است که یک لحظه صبر کنید، می‌توانید این کار را انجام دهید؟ وقتی مشکلی دارید که نمی توانید آن را حل کنید، فقط می توانید بگویید، اوه، من چیزهایی را تعریف کردم تا ما اکنون به طور جادویی یک راه حل داشته باشیم. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "بنابراین اگر با این کار ناراحت هستید، قطعا تنها نیستید. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "در واقع، رنه دکارت اصطلاح خیالی را برای این اعداد به عنوان تحقیر آمیز ابداع کرد. ", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "و سپس ما به آن به عنوان یک قرارداد پایبند بودیم و هنوز آنها را اعداد خیالی می نامیم، که واقعاً پوچ است. ", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "دومین کار عجیبی که هنگام شروع صحبت در مورد اعداد مختلط انجام می دهید این است که می گویید، فقط یک عدد i وجود ندارد، بلکه ما به آن خانه می دهیم. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "در همان ابتدا، اجازه دهید فقط در مورد این صحبت کنیم که چگونه اگر اعداد دو بعدی را به این شکل اضافه می کنید، قوانین بسیار ساده هستند و اساساً مانند بردارها عمل می کند، برای هر یک از شما که ممکن است با بردارها آشنا باشد. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "اگر آن را بر اساس ایمان بپذیرید و از آن پیروی کنید، امیدواریم این واقعیت که مفید باشد به توجیه اینکه چرا ما هر یک از این کارها را انجام می دهیم کمک کند. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "و جالب اینجاست که d آن چیزی است که می گوید باید 2 جذر از 2 و منفی 1 را در نظر بگیرید، اما بی نهایت را نه، بنابراین گروه خوبی از شما وجود دارد که بی نهایت را به عنوان واقعی رد می کنید، اما با این موضوع بسیار راحت هستید. جذر منفی 1، عالی است. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "خوب، بنابراین به نظر می رسد که ما گروهی از افراد داریم که با منفی 1 راحت هستند، یک گروه بزرگ با بی نهایت ناراحت هستند، این موضوع برای یک روز دیگر است، نگران آن نباشید، و سپس تعدادی از افرادی که به نوعی در آن حد وسط هستند که شاید با این ایده که منفی 1 ممکن است واقعی باشد، خیلی راحت نیستند. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "بنابراین برای اولین سوال ریاضی بسیار بیشتر ما، به عنوان نوعی گرم کردن، فقط می خواهم از شما بخواهم که این دو را اضافه کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "قبل از اینکه به شما یاد بدهم چگونه آنها را اضافه کنید، حدس بزنید که چگونه ممکن است کار کند، و امیدوارم که کاملاً ساده به نظر برسد، افزودن در واقع کمترین جذابیت بخش این است، اما خوب است که بدانید چه زمانی شما در مورد اعداد مختلط یاد می گیرید. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "اگر چهار واحد را به سمت راست و سپس یک واحد به بالا حرکت می‌دهید، و می‌خواهید این ایده را اضافه کنید که دو واحد به سمت چپ و سپس دو واحد به بالا حرکت کنید، خوب شما فقط هر کدام را در یک زمان انجام دهید. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "قسمت واقعی آن چهار در سمت راست و سپس منهای دو به سمت چپ خواهد بود. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "یک i به اضافه دو i هم همینطور است. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "افزودن واقعاً هیچ چیز پیچیده ای ندارد، که عالی است. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "بنابراین، در مورد بردارها، حداقل زمانی که در صفحه دوبعدی هستیم، واقعاً تصوری از ضرب آنها برای برگرداندن دو بردار وجود ندارد. ", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "که اساساً است، فرض کنید من نقطه سه، دو را دارم. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "اگر من فقط نوعی شبکه مختصات داشته باشم و به نقطه ای با مختصات x سه و مختصات y دو بروم، چرخش 90 درجه این چیست؟ اگر آن را 90 درجه بچرخانم و بگوییم خلاف جهت عقربه های ساعت. ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "پادساعتگرد. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "اکنون آنچه در این مورد دوست‌داشتنی است این است که اساساً می‌توانیم کاغذ خود را بچرخانیم تا آن را بفهمیم. ", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "شما می گویید خوب است اگر از سه، دو شروع شود و سپس من 90 درجه در خلاف جهت عقربه های ساعت بچرخانم، من فقط می توانم آن را به عنوان منفی دو در جهت x و سپس سه در جهت y بخوانم، اگر کل هواپیما را به همین شکل می چرخانم. . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "بنابراین کاری که ما در اینجا انجام داده ایم این است که سه، دو را گرفته ایم و سپس آن را به منفی دو، سه تبدیل می کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "این چرخش 90 درجه است. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "اگر من یک جفت اعداد را یک کاما b خوب بگیرم و سپس بگویم که اگر 90 درجه بچرخانم به کجا می‌چرخد، با تعویض مختصات ba و سپس منفی کردن آن اول به پایان می‌رسد. ", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "پس این چرخش 90 درجه ای دیگر بود. ", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "خوب اتفاقی که در اینجا افتاده این است که ما هر دو مختصات را منفی کرده ایم و این اطمینان بخش است زیرا اگر من نقطه ای را در حالت ab بنشینم و سپس آن را 90 درجه بچرخانم، بنابراین این چرخش 90 درجه اولیه من خواهد بود و سپس 90 درجه دیگر این است. مانند چرخش 180 درجه - نه من این کار را اشتباه کردم. ", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "این همان چرخش 180 درجه ای خواهد بود که باید به این شکل به نظر برسد، بردار دیگری را که من رسم کردم نادیده می گیرد که فقط هر دو مختصات را می گیرد و آنها را منفی منفی می کند. در واقع طوری رفتار می کند که شما انتظار دارید. ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "اوه نگاه کنید بسیاری از مردم پاسخ های بسیار خوبی ارسال کردند. ", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "بسیار خوب پس به نظر می رسد اکثریت شما پاسخ صحیح را دریافت کرده اید که 2 به علاوه 3i است. ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "برخی از شما پاسخ منفی 2 3 را دادید که من حدس می‌زنم که فقط این را می‌سازید که آیا شما 4 منهای 2 یا 2 منهای 4 می‌گیرید، بنابراین کاملاً قابل درک است. ", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "ما 2 بعلاوه 3 داریم که شاید فقط i را کنار بگذاریم، بنابراین فکر می‌کنم شاید خطاهای ساده و ورودی بسیار زیاد باشد و می‌دانید که برای همه ما اتفاق می‌افتد، مخصوصاً در تست‌ها، گاهی اوقات می‌دانید پاسخ درست چیست، اما بعد شما یک نماد را فراموش می کنید یا دو نماد را عوض می کنید، بنابراین همه چیز بسیار خوب است. ", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "کلماتی را که می دانید به من می گویند کار می کند و با این حال پیشرفت به جلو برای من بسیار کند است، بنابراین می دانید اگر قرار نیست یک کلمه سخت با آنها صحبت کنم، می توانید در توییتر نیز به آنها مراجعه کنید. جایی که ما سوال می کنیم و فقط می گوییم هی کمینتر نمی توانید کاری کنید که سوالات زنده کمی برای ما بهتر عمل کنند؟ باشه فکر کنم بالاخره رسیدیم همه آماده اند؟ آها! ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "فوق العاده! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "سوال خیلی ساده من می‌خواهم عدد i را بگیرید و می‌خواهم آن را در 3 به علاوه 2i ضرب کنید، و با وجود اینکه من واقعاً در مورد قوانین ضرب صحبت نکرده‌ام، می‌توانم بگویم وانمود کنید که درست مانند آن عمل می‌کند. اعداد عادی شما چیزهایی مانند ویژگی توزیعی دارید که می توانید آن را در سراسر توزیع کنید و سپس مشخصه تعیین کننده i این است که من مربع آن منفی است و تنها چیز خاصی است که باید در مورد آن بدانید غیر از اینکه فقط با آن رفتار کنید. مثل اینکه یک عدد معمولی است و سپس محصول را به جلو ادامه دهید. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/portuguese/sentence_translations.json b/2020/ldm-complex-numbers/portuguese/sentence_translations.json index 0d04c95e1..dd0b7b1ae 100644 --- a/2020/ldm-complex-numbers/portuguese/sentence_translations.json +++ b/2020/ldm-complex-numbers/portuguese/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "Eu diria que se for algo realmente útil em uma aplicação, então é tão real quanto as palavras, certo? ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "Você nunca vai encontrar uma palavra abstrata como felicidade por aí, mas ela tem uma espécie de realidade em nossas mentes, e coisas como a raiz quadrada de dois, que você não pode expressar como uma fração, ou coisas como o raiz quadrada de menos um que não aparece entre os números normais reais, você sabe, mesmo que pareçam um pouco diferentes. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "Não vou presumir que você já sabe o que são, é para ser uma introdução básica, mas vamos começar, ok? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "Lembro-me de quando estava na escola e aprendemos essas fórmulas de adição, que se você quiser saber o cosseno da soma de dois ângulos diferentes, sabe, é esse tipo de coisa longa em termos de cossenos e senos dos dois ângulos originais , há esse sinal de menos que sempre confunde as pessoas, se você fizer o mesmo para o sinal, parece semelhante, mas há um sinal de mais, e em vez de ter cos-cos você tem cos-sin, é algo muito sujeito a erros se você está apenas tentando memorizá-lo como é. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "No entanto, se você chegar a isso com números complexos, isso não só é muito menos sujeito a erros, como também tem um significado muito bonito e simplesmente não dá certo. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "Portanto, mesmo que você não acredite necessariamente na realidade da raiz quadrada de menos 1, você pelo menos tem que admitir que é interessante que isso possa tornar úteis outras peças de matemática, que outras peças de matemática sejam um pouco mais úteis. compreensível também. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "Mas o ponto de partida parece muito estranho, ok? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "Uma é, não, não existe, certo? ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "Qualquer número que você eleva ao quadrado, se for positivo, bem, ele permanece positivo. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "Nunca vou conseguir nada negativo. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "No entanto, se um matemático vier e disser, ah, não, não, isso existe, nós definimos para que seja esse o caso. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "Quando você tem um problema que não consegue resolver, você pode simplesmente dizer: ah, eu defini as coisas para que agora tenhamos uma solução magicamente. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "Então, se você não se sente confortável com isso, definitivamente não está sozinho. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "Na verdade, René Descartes cunhou o termo imaginário para esses números como depreciativo. ", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "E então mantivemos isso como uma convenção e ainda os chamamos de números imaginários, o que é genuinamente absurdo. ", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "A segunda coisa estranha que você faz quando começa a falar sobre números complexos é dizer que não existe apenas esse número i, mas vamos dar-lhe um lar. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "E, ok, se quisermos estender nosso sistema numérico, entendi, talvez seja útil colocar algum tipo de número aí, mas por que eu, certo? ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "No início, vamos apenas falar sobre como, se você estiver adicionando números bidimensionais assim, as regras são bastante diretas e funcionam essencialmente da mesma forma que os vetores, para qualquer um de vocês que esteja familiarizado com vetores. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "Se você acreditar nisso e seguir, espero que o fato de que isso se torne útil ajude a justificar por que estamos fazendo isso. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "Hum, parece que há um vaivém entre as respostas f e d, então f são todas elas, dizendo que todas elas devem ser consideradas reais. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "E interessante, d é aquele que diz que você deve considerar 2 raiz quadrada de 2 e menos 1, mas não infinito, então há um bom contingente de vocês por aí que simplesmente rejeitariam o infinito como sendo considerado real, mas estão muito confortáveis com o raiz quadrada de menos 1, isso é incrível. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "Ok, parece que temos um grupo de pessoas que se sentem confortáveis com o negativo 1, um grande grupo que se sente desconfortável com o infinito, isso é um assunto para outro dia, não se preocupe com isso, e então um número de pessoas que estamos no meio-termo de talvez não estarmos muito confortáveis com a ideia de que o negativo 1 pode ser real. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "Então, para a nossa primeira questão muito mais matemática, como uma espécie de aquecimento, só quero pedir que você adicione essas duas. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "Antes de ensinar como adicioná-los, adivinhe como isso pode funcionar e espero que pareça bastante simples. A adição é na verdade a parte menos interessante disso, mas é, é bom saber quando você está aprendendo sobre números complexos. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "Se você estiver movendo quatro unidades para a direita e depois uma unidade para cima, e quiser adicionar a ideia de mover duas unidades para a esquerda e depois duas unidades para cima, bem, basta fazer cada uma delas, uma de cada vez. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "A parte real serão aqueles quatro à direita, depois menos dois à esquerda. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "Então é aquele um i mais dois i. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "A adição realmente não tem nada complicado acontecendo, o que é ótimo. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "Então, com vetores, não há realmente nenhuma noção de multiplicá-los para obter dois vetores de volta, pelo menos quando estamos no plano 2D. ", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "O que é basicamente, suponha que eu tenha o ponto três, dois. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "Se eu tiver algum tipo de grade de coordenadas e for para o ponto com coordenada x três e coordenada y dois, qual é a rotação de 90 graus disso? ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "Sentido anti-horário. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "Agora, o que é adorável nisso é que basicamente podemos virar nosso papel para descobrir. ", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "Você diz que tudo bem se começou em três, dois e então eu giro 90 graus no sentido anti-horário, posso apenas ler isso agora como sendo menos dois na direção x e depois três na direção y, se eu tivesse girado todo o plano assim . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "Então, o que fizemos aqui foi pegar três, dois e depois convertê-lo para menos dois, três. ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "Essa será a rotação de 90 graus. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "Se eu pegasse um par de números uma vírgula b ok e então eu dissesse para onde isso vai girar se eu girar 90 graus vai acabar trocando as coordenadas ba e então tornando a primeira negativa. ", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "Então essa foi outra rotação de 90 graus. ", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "Bem, o que aconteceu aqui é que acabamos de tornar ambas as coordenadas negativas e isso é reconfortante porque se eu pegar algum ponto sentado em ab e depois girá-lo 90 graus, então esta será minha rotação inicial de 90 graus e depois outros 90 graus, esse será o o mesmo que rotação de 180 graus - ah, não, fiz isso errado. ", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "Isso será o mesmo que uma rotação de 180 graus, que deve ficar assim, ignore o outro vetor que desenhei, que está apenas pegando as duas coordenadas e tornando-as negativas, negativas, a negativas, b, ok, isso é tranquilizador para esta operação que faz uma rotação de 90 graus realmente se comporta como você esperaria. ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "Olha, muitas pessoas enviaram respostas muito boas. ", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "Ok, parece que a maioria de vocês acertou a resposta, que é 2 mais 3i. ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "Muito bom, muito bom. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "Alguns de vocês responderam 2 3 negativo, o que eu acho que é apenas uma questão de trocar se você está pegando 4 menos 2 ou 2 menos 4, então isso é completamente compreensível. ", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "Temos 2 mais 3, que talvez esteja apenas eliminando o i, então acho que talvez sejam muitos erros simples e entradas e você sabe que isso acontece com todos nós, especialmente em testes, às vezes você sabe qual é a resposta certa, mas então você esquece um símbolo ou troca dois, então está tudo muito bom. ", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "Palavras obscenas, você sabe, elas me dizem que está funcionando, mas é muito lento para eu progredir, então vocês sabem que se eu não vou ter uma palavra severa com eles, vocês podem abordá-las no Twitter também sob o mesmo lugar onde fazemos perguntas e apenas dizemos ei kamineter, você não pode fazer as perguntas ao vivo funcionarem um pouco melhor para nós? ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "Ok, acho que finalmente chegamos lá. ", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "Todos prontos? ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "Ah! ", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "Maravilhoso! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "Pergunta muito simples, quero que você pegue o número i e quero que multiplique por 3 mais 2i e, embora eu realmente não tenha falado sobre as regras de multiplicação, o que posso dizer é fingir que funciona exatamente como funciona para números normais, você tem coisas como a propriedade distributiva onde você pode distribuir isso por toda parte e então a característica definidora de i é essa ideia de que i ao quadrado é negativo, essa é a única coisa especial que você precisa saber sobre isso, além de apenas tratá-lo como se fosse um número normal, ok e então prossiga com o produto. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/russian/sentence_translations.json b/2020/ldm-complex-numbers/russian/sentence_translations.json index 9603017e9..b033394f0 100644 --- a/2020/ldm-complex-numbers/russian/sentence_translations.json +++ b/2020/ldm-complex-numbers/russian/sentence_translations.json @@ -312,7 +312,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right?", + "input": "stalling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the", "translatedText": "Я бы сказал, что если это что-то действительно полезное в приложении, то это так же реально, как и слова, верно?", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different.", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality.", "translatedText": "Вы никогда не встретите такое абстрактное слово, как «счастье», но в нашем сознании оно имеет некую реальность и такие вещи, как квадратный корень из двух, который невозможно выразить дробью, или такие вещи, как квадратный корень из отрицательного, который не встречается среди реальных нормальных чисел, знаете ли, даже если они могут показаться немного разными.", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 270.78 }, { - "input": "Oh, this is such a shame.", + "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay?", "translatedText": "О, это такой позор.", "model": "google_nmt", "n_reviews": 0, @@ -392,7 +392,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out.", + "input": "esting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just trigonometry, it's everything we were talking about la", "translatedText": "Однако если вы подойдете к этому с комплексными числами, это не только гораздо менее подвержено ошибкам, но и имеет очень красивый смысл и просто выпадает из поля зрения.", "model": "google_nmt", "n_reviews": 0, @@ -400,7 +400,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too.", + "input": "st time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of", "translatedText": "Таким образом, даже если вы не обязательно верите в реальность квадратного корня из отрицательного 1, вы, по крайней мере, должны признать, что интересно то, что это может сделать другие части математики полезными, что другие части математики немного более полезны. тоже понятно.", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right?", + "input": "It's something that's very error-prone if you're just trying", "translatedText": "Во-первых, нет, не так, верно?", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive.", + "input": "ex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out", "translatedText": "Любое число, которое вы возводите в квадрат, если оно положительное, оно остается положительным.", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative.", + "input": "So even if you don't necessarily believe in the reality of", "translatedText": "Я никогда не получу ничего негативного.", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 437.86 }, { - "input": "So this does not exist, no such number.", + "input": "the square root of negative one, you at the very least have to admit that it's interesting that it can make o", "translatedText": "Так вот этого не существует, нет такого номера.", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case.", + "input": "ther pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case.", "translatedText": "Однако если математик приходит и говорит: «О нет, нет, оно существует», мы определили его так, что это так.", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution.", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wo", "translatedText": "Если у вас есть проблема, которую вы не можете решить, вы можете просто сказать: «О, я определил вещи, и теперь у нас волшебным образом есть решение».", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone.", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home.", "translatedText": "Так что, если вам это не нравится, вы определенно не одиноки.", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory.", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative,", "translatedText": "Фактически, Рене Декарт придумал термин «мнимые» для этих чисел как уничижительный.", "model": "google_nmt", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right?", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you", "translatedText": "И ладно, если мы хотим расширить нашу систему счисления, я понимаю, может быть, полезно поместить туда какое-нибудь число, но почему бы и нет, верно?", "model": "google_nmt", "n_reviews": 0, @@ -632,7 +632,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors.", + "input": "have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question.", "translatedText": "В самом начале давайте просто поговорим о том, что если вы складываете двумерные числа, правила довольно просты и действуют по существу так же, как и векторы, для любого из вас, кто знаком с векторами.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 662.3 }, { - "input": "None of them is much lower at a.", + "input": "one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to", "translatedText": "Ни один из них не намного ниже a.", "model": "google_nmt", "n_reviews": 0, @@ -728,7 +728,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real.", + "input": ", why should that live there? What on earth does the idea of a point one unit above the real number line in a separate dimension have to do with squaring to negative one? So I hope to answer this for you. At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be famili", "translatedText": "Итак, похоже, что у нас есть группа людей, которым комфортно иметь отрицательное значение 1, большая группа людей, которым не нравится бесконечность, это тема для другого дня, не беспокойтесь об этом, а также несколько людей, которые они находятся в некотором роде в той золотой середине, возможно, им не очень комфортно с идеей, что отрицательная единица может быть реальной.", "model": "google_nmt", "n_reviews": 0, @@ -744,7 +744,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two.", + "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It turns out to be r", "translatedText": "Итак, для нашего первого гораздо более математического вопроса, в качестве разминки, я просто хочу попросить вас добавить эти два.", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 706.82 }, { - "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me.", + "input": "ative two plus two i. So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers extend in this direction. can get you something lik", "translatedText": "К сожалению, и это видно по тому, что я застопорился, и по тому, что я здесь говорю, похоже, что вопрос все еще загружается не совсем корректно, поэтому я собираюсь сурово поговорить с Кэмом и Идером за сцены, которые в противном случае создали такой красивый, красивый интерфейс, который полезен для такого рода общения между вами, ребята, и мной.", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 726.28 }, { - "input": "I'm going to have a stern word with them behind the scenes, but in the meantime let's go ahead and move forward with the lesson here.", + "input": "e it. But the rules end up being very different from that in the number system. You can't really do algebra. You can't do things like assume that if two numbers multiply to make zero, then", "translatedText": "Я собираюсь поговорить с ними строго за кулисами, а пока давайте продолжим урок и продолжим его.", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 732.64 }, { - "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be.", + "input": "one of them h as to be zero. But complex numbers are going to end up behaving much like the real numbers, s Now assuming that our question system ha", "translatedText": "Так что, я думаю, я могу просмотреть это, просто на листе бумаги, и вы сможете следить за ним дома и посмотреть, какое может быть дополнение.", "model": "google_nmt", "n_reviews": 0, @@ -792,7 +792,7 @@ "end": 739.92 }, { - "input": "It turns out to be relatively straightforward.", + "input": "s not broken down, I should be able to do this as a proper poll and let me go ahead, I guess we can first check the previous poll, okay things se", "translatedText": "Оказывается, это относительно просто.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time.", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of th", "translatedText": "Если вы перемещаете четыре единицы вправо, а затем одну единицу вверх, и хотите добавить идею перемещения двух единиц влево, а затем на две единицы вверх, вы просто выполняете каждую из этих единиц по одному.", "model": "google_nmt", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left.", + "input": "red real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a g", "translatedText": "Реальная часть будет состоять из четырех справа и минус двух слева.", "model": "google_nmt", "n_reviews": 0, @@ -832,7 +832,7 @@ "end": 760.88 }, { - "input": "And then the imaginary part is going to be this one unit up and then these two units up, one plus two, times i.", + "input": "you out there who would just reject infinity as being considered real but are very comfortable with the square root of negative o", "translatedText": "И тогда мнимая часть будет равна одной единице вверх, а затем этим двум единицам вверх, один плюс два, умноженному на i.", "model": "google_nmt", "n_reviews": 0, @@ -864,7 +864,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great.", + "input": "are root of negative one, fascinating, I actually would have thought that none of them would have come higher than t", "translatedText": "В дополнении нет ничего сложного, и это здорово.", "model": "google_nmt", "n_reviews": 0, @@ -880,7 +880,7 @@ "end": 784.2 }, { - "input": "What is so complex about complex numbers after all?", + "input": "m is much lower at a, okay so it looks like we've got a cohort of people who are comfortable with negative one, a la", "translatedText": "Что же такого сложного в комплексных числах?", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 787.1 }, { - "input": "Well where everything becomes interesting is when you try to multiply these numbers together.", + "input": "rge cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people w", "translatedText": "Что ж, все становится интересным, когда вы пытаетесь умножить эти числа.", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 803.68 }, { - "input": "But the rules end up being very different from that in the number system.", + "input": "mfortable with the idea that negative one might be real, let's see if we can convince you of the difference of t", "translatedText": "Но правила в конечном итоге сильно отличаются от правил в системе счисления.", "model": "google_nmt", "n_reviews": 0, @@ -920,7 +920,7 @@ "end": 806.86 }, { - "input": "You can't really do algebra.", + "input": "hat. So what we've done here is we've taken three, two and then", "translatedText": "Ты действительно не умеешь заниматься алгеброй.", "model": "google_nmt", "n_reviews": 0, @@ -944,7 +944,7 @@ "end": 817.78 }, { - "input": "But to understand what that multiplication rule is, I just want to ask you a simple question.", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, a", "translatedText": "Но чтобы понять, что это за правило умножения, я просто хочу задать вам простой вопрос.", "model": "google_nmt", "n_reviews": 0, @@ -960,7 +960,7 @@ "end": 831.82 }, { - "input": "We're not even going to think of it as a complex number per se.", + "input": "part of this, but it is, it's a good thing to know when you're learning about complex numbers, it'", "translatedText": "Мы даже не собираемся думать об этом как о комплексном числе как таковом.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this?", + "input": "s definitely one of those operations that you are going to need to know. Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is stil", "translatedText": "Если у меня просто есть какая-то координатная сетка и я иду к точке с координатой x три и координатой y два, каков будет ее поворот на 90 градусов?", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 859.44 }, { - "input": "Okay.", + "input": "built such a beautiful, beautiful inte", "translatedText": "Хорошо.", "model": "google_nmt", "n_reviews": 0, @@ -1008,7 +1008,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out.", + "input": "rface that's helpful for this kind of back and forth between you guys and me. nice gut check here is", "translatedText": "Самое приятное в этом то, что мы можем просто перевернуть бумагу, чтобы понять это.", "model": "google_nmt", "n_reviews": 0, @@ -1024,7 +1024,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three.", + "input": "e. So that was another 90 degree rotation. Well what's happened here is we've just made both of the coordinates negative and that's", "translatedText": "Итак, мы здесь взяли три, два, а затем конвертировали его в минус два, три.", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 890.68 }, { - "input": "Something which maybe in our original system you know looks like this negative two and then three.", + "input": "reassuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them", "translatedText": "Что-то, что, возможно, в нашей исходной системе, как вы знаете, выглядит как минус два, а затем три.", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation.", + "input": "negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation", "translatedText": "Это будет поворот на 90 градусов.", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 899.9 }, { - "input": "And what's nice here is that that rule is very simple and it applies to any pair that we might have.", + "input": "actually behaves like you would expect it to. Now why am I asking you this? Well I'm being told that supposedly I'm allowed to ask you questions again so I 'm going to ha", "translatedText": "И что приятно, это правило очень простое и применимо к любой паре, которая у нас может быть.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong.", + "input": "52 of you answered simply 2 which would have been the real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated goin", "translatedText": "Что здесь произошло, так это то, что мы только что сделали обе координаты отрицательными, и это обнадеживает, потому что если я возьму какую-то точку, находящуюся в точке ab, а затем поверну ее на 90 градусов, то это будет мой первоначальный поворот на 90 градусов, а затем еще на 90 градусов, и это будет то же самое, что поворот на 180 градусов — о нет, я сделал это неправильно.", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 980.14 }, { - "input": "Now why am I asking you this?", + "input": "you try to multiply these numbers together. So with vectors, there's not really any notion", "translatedText": "Почему я спрашиваю тебя об этом?", "model": "google_nmt", "n_reviews": 0, @@ -1104,7 +1104,7 @@ "end": 981.76 }, { - "input": "Well I'm being told that supposedly I'm allowed to ask you questions again so I'm going to have you do your very first complex product.", + "input": "of multiplying them to get two vectors back, at least when we're in the 2d plane. that we ask questions and just say hey kamineter can't you make the live questio", "translatedText": "Ну, мне сказали, что якобы мне разрешено снова задавать вам вопросы, поэтому я собираюсь попросить вас сделать ваш самый первый сложный продукт.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i.", + "input": "Wonderful! Very simple question I want you to take the number i and I want you to multiply it by 3 pl", "translatedText": "Хорошо, похоже, что большинство из вас получили правильный ответ: 2 плюс 3i.", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1006.98 }, { - "input": "Very good very good.", + "input": "us 2i and even though I haven't really talked about You can't do things like assume that if two numbers multiply to mak", "translatedText": "Очень хорошо, очень хорошо.", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good.", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t", "translatedText": "У нас есть 2 плюс 3, что, возможно, просто выпадает из i, поэтому я думаю, может быть, много таких, как простые ошибки и записи, и вы знаете, что это случается со всеми нами, особенно на тестах, иногда вы знаете, какой правильный ответ, но потом вы забываете символ или меняете местами два, и это все очень хорошо.", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1052.36 }, { - "input": "Let's go ahead and try our very first product though like I said so here because I already talked through one of the questions we're going to go ahead and skip ahead of it we know how to rotate something like 3 comma 2 so I'm not even going to give you time to do that and properly grade it.", + "input": "his is we can basically just turn our paper to figure it out. ons as you do it rather than sitting in passively watching this is genuinely delightful to me. Okay this is this isn't necessarily a question I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not", "translatedText": "Давайте продолжим и попробуем наш самый первый продукт, хотя, как я уже сказал, потому что я уже ответил на один из вопросов, мы собираемся пойти дальше и пропустить его, мы знаем, как вращать что-то вроде 3 запятой 2, так что я даже не дам вам времени сделать это и правильно оценить.", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us?", + "input": "that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three i which is absolutely correct absolutely correct so there's two w", "translatedText": "Стал слова, слова, вы знаете, они говорят мне, что это работает, но все же я очень медленно продвигаюсь вперед, так что вы знаете, если я не собираюсь сказать им суровое слово, вы, ребята, тоже можете написать им в Твиттере под тем же место, где мы задаем вопросы и просто говорим: «Эй, каминтер, не можешь ли ты сделать так, чтобы вопросы в прямом эфире работали для нас немного лучше?", "model": "google_nmt", "n_reviews": 0, @@ -1184,7 +1184,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there.", + "input": "ays to think about this okay one of them is", "translatedText": "» Хорошо, я думаю, мы наконец-то пришли.", "model": "google_nmt", "n_reviews": 0, @@ -1200,7 +1200,7 @@ "end": 1094.22 }, { - "input": "Aha!", + "input": "e algebra and just do it a little bit mechanistically okay so", "translatedText": "Ага!", "model": "google_nmt", "n_reviews": 0, @@ -1208,7 +1208,7 @@ "end": 1094.76 }, { - "input": "Wonderful!", + "input": "if we pull ourselves up", "translatedText": "Замечательный!", "model": "google_nmt", "n_reviews": 0, @@ -1224,7 +1224,7 @@ "end": 1126.42 }, { - "input": "Wonderful!", + "input": "that if you want to rotate numbers 90 degrees", "translatedText": "Замечательный!", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/spanish/sentence_translations.json b/2020/ldm-complex-numbers/spanish/sentence_translations.json index 37d3f6305..f8c8ac39f 100644 --- a/2020/ldm-complex-numbers/spanish/sentence_translations.json +++ b/2020/ldm-complex-numbers/spanish/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "Yo diría que si es algo que realmente es útil en una aplicación, entonces es tan real como las palabras, ¿verdad? ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "Nunca te encontrarás con una palabra abstracta como felicidad, pero tiene una especie de realidad en nuestras mentes, y cosas como la raíz cuadrada de dos, que no se puede expresar como una fracción, o cosas como la raíz cuadrada de uno negativo que no aparecen entre los números normales reales, ya sabes, incluso si pueden parecer un poco diferentes. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "No asumiré que ya sabes cuáles son, está destinado a ser una introducción básica, pero profundicemos, ¿de acuerdo? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "Recuerdo cuando estaba en la escuela y aprendimos estas fórmulas de suma, que si quieres saber el coseno de la suma de dos ángulos diferentes, ya sabes, es algo largo en términos de cosenos y senos de los dos ángulos originales. , existe este signo menos que siempre haría tropezar a la gente, si haces lo mismo con el signo, se ve similar pero hay un signo más, y en lugar de tener cos-cos tienes cos-sin, es algo que es muy propenso a errores si solo estás tratando de memorizarlo tal como está. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "Sin embargo, si lo aborda con números complejos, esto no sólo es mucho menos propenso a errores, sino que tiene un significado muy hermoso y simplemente se cae. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "Entonces, incluso si no necesariamente crees en la realidad de la raíz cuadrada de menos 1, al menos tienes que admitir que es interesante que puede hacer que otras piezas matemáticas sean útiles, que otras piezas matemáticas sean un poco más útiles. comprensible también. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "Pero el punto de partida parece muy extraño, ¿vale? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "Una es, no, no la hay, ¿verdad? ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "Cualquier número que elevas al cuadrado, si es positivo, sigue siendo positivo. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "Nunca obtendré nada negativo. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "Sin embargo, si viene un matemático y dice, oh no, no, existe, lo hemos definido para que así sea. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "Cuando tengas un problema que no puedas resolver, simplemente puedes decir: "Oh, he definido las cosas para que ahora mágicamente tengamos una solución". ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "Entonces, si no se siente cómodo con esto, definitivamente no está solo. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "De hecho, René Descartes acuñó el término imaginario para estos números con carácter despectivo. ", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "Y luego nos quedamos con eso como convención y todavía los llamamos números imaginarios, lo cual es genuinamente absurdo. ", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "La segunda cosa extraña que haces cuando empiezas a hablar de números complejos es decir, no existe tal número i, pero vamos a darle un hogar. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "Y, está bien, si queremos ampliar nuestro sistema numérico, lo entiendo, tal vez sea útil poner algún tipo de número ahí arriba, pero ¿por qué yo, verdad? ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "Al principio, hablemos de cómo si sumas números bidimensionales como este, las reglas son bastante sencillas y funciona esencialmente igual que los vectores, para cualquiera de ustedes que esté familiarizado con los vectores. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "Si lo tomas con fe y lo sigues, es de esperar que el hecho de que resulte útil ayude a justificar por qué estamos haciendo todo esto. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "Um, parece que hay un ida y vuelta entre las respuestas f y d, por lo que f son todas, lo que dice que todas deben considerarse reales. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "E interesante, d es el que dice que debes considerar 2 raíz cuadrada de 2 y menos 1, pero no el infinito, por lo que hay un buen contingente de ustedes que simplemente rechazarían el infinito como si se considerara real, pero se sienten muy cómodos con la raíz cuadrada de menos 1, eso es asombroso. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "Bien, entonces parece que tenemos una cohorte de personas que se sienten cómodas con el 1 negativo, una cohorte grande que se siente incómoda con el infinito, ese es un tema para otro día, no te preocupes, y luego un número de personas que Están en ese punto medio de no sentirse muy cómodos con la idea de que el negativo 1 pueda ser real. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "Entonces, para nuestra primera pregunta mucho más matemática, como una especie de calentamiento, solo quiero pedirles que sumen estas dos. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "Antes de enseñarte cómo agregarlos, adivina cómo podría funcionar, y espero que te parezca bastante sencillo, la suma es en realidad la parte menos interesante de esto, pero lo es, es bueno saber cuándo estás aprendiendo sobre números complejos. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "Si estás moviendo cuatro unidades hacia la derecha y luego una unidad hacia arriba, y quieres agregar la idea de mover dos unidades hacia la izquierda y luego dos unidades hacia arriba, simplemente haz cada una de ellas una a la vez. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "La parte real serán esos cuatro a la derecha, luego menos dos a la izquierda. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "Así es ese uno i más dos i. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "La suma realmente no tiene nada complicado, lo cual es genial. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "Entonces, con los vectores, realmente no existe la noción de multiplicarlos para obtener dos vectores, al menos cuando estamos en el plano 2D. ", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "Básicamente, supongamos que tengo el punto tres, dos. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "Si solo tengo una especie de cuadrícula de coordenadas y voy al punto con la coordenada x tres y la coordenada y dos, ¿cuál es la rotación de 90 grados de esto? ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "En sentido anti-horario. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "Lo bueno de esto es que básicamente podemos girar nuestro papel para resolverlo. ", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "Dices que está bien, si comenzó en tres, dos y luego giré 90 grados en sentido antihorario, ahora puedo leerlo como menos dos en la dirección x y luego tres en la dirección y, si hubiera girado todo el plano así. . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "Entonces, lo que hemos hecho aquí es tomar tres, dos y luego lo convertimos a menos dos, tres. ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "Esa será la rotación de 90 grados. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "Si tomé un par de números, una coma b, está bien y luego dije hacia dónde va a rotar. Si lo giro 90 grados, terminará intercambiando las coordenadas ba y luego haciendo que la primera sea negativa. ", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "Entonces esa fue otra rotación de 90 grados. ", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "Bueno, lo que sucedió aquí es que acabamos de hacer que ambas coordenadas sean negativas y eso es tranquilizador porque si tomo algún punto sentado en ab y luego lo giro 90 grados, esta será mi rotación inicial de 90 grados y luego otros 90 grados, esa es la Lo mismo que la rotación de 180 grados. Oh, no, lo he hecho mal. ", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "Eso será lo mismo que una rotación de 180 grados que debería verse así: ignore el otro vector que dibujé, que simplemente toma ambas coordenadas y las convierte en negativas negativas a negativas b, está bien, eso es tranquilizador para esta operación que realiza una rotación de 90 grados. en realidad se comporta como cabría esperar. ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "Oh, mira, mucha gente envió respuestas muy buenas. ", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "Bien, parece que la mayoría de ustedes obtuvieron la respuesta correcta, que es 2 más 3i. ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "Muy bien, muy bien. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "Algunos de ustedes respondieron negativamente 2 3, lo cual supongo que es solo cambiar si están tomando 4 menos 2 o 2 menos 4, así que es completamente comprensible. ", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "Tenemos 2 más 3, lo que tal vez simplemente elimine la i, así que creo que tal vez haya muchos errores y entradas simples, y sabes, eso nos pasa a todos, especialmente en los exámenes, a veces sabes cuál es la respuesta correcta, pero luego Si olvidas un símbolo o intercambias dos, todo está muy bien. ", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "Stal stal palabras palabras, sabes, me dicen que está funcionando y, sin embargo, es muy lento para mí avanzar, así que sabes que si no voy a tener una palabra severa con ellos, también puedes atacarlos en Twitter bajo el mismo lugar donde hacemos preguntas y simplemente decimos hola kamineter, ¿no puedes hacer que las preguntas en vivo funcionen un poco mejor para nosotros? ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "Bien, creo que finalmente llegamos allí. ", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "¿Todos listos? ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "¡Ajá! ", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "¡Maravilloso! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "Pregunta muy simple quiero que tomes el número i y quiero que lo multipliques por 3 más 2i y aunque realmente no he hablado sobre las reglas de multiplicación lo que puedo decir es fingir que funciona igual que lo hace para En los números normales tienes cosas como la propiedad distributiva donde puedes distribuir esto y luego la característica definitoria de i es la idea de que i al cuadrado es negativo, eso es lo único especial que necesitas saber sobre eso aparte de eso, simplemente trátalo como si fuera un número normal, está bien y luego continúa con el producto. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/tamil/sentence_translations.json b/2020/ldm-complex-numbers/tamil/sentence_translations.json index d744ba1dc..362b650fa 100644 --- a/2020/ldm-complex-numbers/tamil/sentence_translations.json +++ b/2020/ldm-complex-numbers/tamil/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "இது ஒரு பயன்பாட்டில் உண்மையில் பயனுள்ள ஒன்று என்றால், அது வார்த்தைகளைப் போலவே உண்மையானது என்று நான் கூறுவேன், இல்லையா? ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "நீங்கள் ஒருபோதும் மகிழ்ச்சியைப் போன்ற ஒரு சுருக்கமான வார்த்தையில் ஓடப் போவதில்லை, ஆனால் அது எங்கள் மனதில் ஒரு வகையான யதார்த்தத்தைக் கொண்டுள்ளது, இரண்டின் வர்க்கமூலம் போன்ற விஷயங்கள், நீங்கள் ஒரு பின்னமாக வெளிப்படுத்த முடியாது, அல்லது போன்ற விஷயங்கள் நிஜ சாதாரண எண்களில் காட்டப்படாத எதிர்மறை ஒன்றின் வர்க்கமூலம், அவை சற்று வித்தியாசமாகத் தோன்றினாலும், உங்களுக்குத் தெரியும். ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "அவை என்னவென்று உங்களுக்கு இன்னும் தெரியும் என்று நான் கருதமாட்டேன், இது ஒரு அடிப்படை ப்ரைமராக இருக்க வேண்டும், ஆனால் நாம் சரியாக உள்ளே நுழைவோம், சரியா? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "இரண்டு வெவ்வேறு கோணங்களின் கூட்டுத்தொகையை நீங்கள் அறிய விரும்பினால், அசல் இரண்டு கோணங்களின் கோசைன்கள் மற்றும் சைன்களின் அடிப்படையில் இது இந்த வகையான நீண்ட விஷயம் என்று நான் பள்ளியில் இருந்தபோது எனக்கு நினைவிருக்கிறது. , இந்த மைனஸ் அடையாளம் உள்ளது, இது எப்போதும் மக்களைத் தூண்டிவிடும், நீங்கள் அதையே அடையாளமாகச் செய்தால், அது ஒரே மாதிரியாகத் தெரிகிறது ஆனால் ஒரு பிளஸ் அடையாளம் உள்ளது, மேலும் cos-cos என்பதற்குப் பதிலாக உங்களுக்கு cos-sin உள்ளது, இது மிகவும் பிழையான ஒன்று. நீங்கள் அதை அப்படியே மனப்பாடம் செய்ய முயற்சிக்கிறீர்கள் என்றால். ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "இருப்பினும், நீங்கள் சிக்கலான எண்களைக் கொண்டு வந்தால், இது மிகவும் குறைவான பிழையுடையது மட்டுமல்ல, இது மிகவும் அழகான அர்த்தத்தைக் கொண்டுள்ளது மற்றும் அது சரியாக விழும். ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "எனவே, எதிர்மறை 1 இன் வர்க்க மூலத்தின் உண்மைத்தன்மையை நீங்கள் நம்ப வேண்டிய அவசியமில்லை என்றாலும், மற்ற கணிதத் துண்டுகளை அது பயனுள்ளதாக்குவது சுவாரஸ்யமானது என்பதை நீங்கள் ஒப்புக் கொள்ள வேண்டும். புரிந்துகொள்ளக்கூடியதும் கூட. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "ஆனால் தொடக்கப் புள்ளி மிகவும் விசித்திரமாகத் தெரிகிறது, சரியா? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "ஒன்று, இல்லை இல்லை, இல்லையா? ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "நீங்கள் எந்த எண்ணை சதுரப்படுத்துகிறீர்களோ, அது நேர்மறையாக இருந்தால், அது நேர்மறையாகவே இருக்கும். ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "நான் ஒருபோதும் எதிர்மறையான எதையும் பெறப் போவதில்லை. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "இருப்பினும், ஒரு கணிதவியலாளர் வந்து, ஐயோ இல்லை, அது இல்லை என்று சொன்னால், நாங்கள் அதை வரையறுத்துள்ளோம், அது அப்படித்தான். ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "உங்களால் தீர்க்க முடியாத ஒரு பிரச்சனை உங்களுக்கு இருக்கும்போது, நீங்கள் சொல்லலாம், ஓ நான் விஷயங்களை வரையறுத்தேன், அதனால் நாங்கள் இப்போது மாயமாக ஒரு தீர்வைப் பெற்றுள்ளோம். ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "எனவே இது உங்களுக்கு சங்கடமாக இருந்தால், நீங்கள் நிச்சயமாக தனியாக இல்லை. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "உண்மையில், ரெனே டெஸ்கார்ட்ஸ் இந்த எண்களுக்கு கற்பனை என்ற சொல்லை இழிவானதாக உருவாக்கினார். ", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "பின்னர் நாங்கள் அதை ஒரு மாநாட்டாக ஒட்டிக்கொண்டோம், இன்னும் அவற்றை கற்பனை எண்கள் என்று அழைக்கிறோம், இது உண்மையிலேயே அபத்தமானது. ", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "நீங்கள் சிக்கலான எண்களைப் பற்றி பேசத் தொடங்கும் போது நீங்கள் செய்யும் இரண்டாவது வித்தியாசமான விஷயம் என்னவென்றால், அத்தகைய எண் ஐ மட்டும் இல்லை, ஆனால் நாங்கள் அதற்கு ஒரு வீட்டைக் கொடுக்கப் போகிறோம். ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "மேலும், சரி, நாம் நமது எண் அமைப்பை நீட்டிக்க விரும்பினால், நான் புரிந்துகொள்கிறேன், சில வகையான எண்ணை அங்கு வைப்பது பயனுள்ளதாக இருக்கும், ஆனால் நான் ஏன், சரியா? ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "ஆரம்பத்தில், நீங்கள் இரண்டு பரிமாண எண்களைச் சேர்த்தால், விதிகள் மிகவும் எளிமையானவை மற்றும் திசையன்களை நன்கு அறிந்த உங்களில் எவருக்கும் இது திசையன்களைப் போலவே செயல்படுகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "நீங்கள் அதை விசுவாசமாக எடுத்துக் கொண்டால், நீங்கள் பின்பற்றினால், இது பயனுள்ளதாக இருக்கும் என்பது நாம் ஏன் இதைச் செய்கிறோம் என்பதை நியாயப்படுத்த உதவுகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "அட, f மற்றும் d பதில்களுக்கு இடையில் முன்னும் பின்னுமாக ஒரு உள்ளது போல் தெரிகிறது, எனவே f என்பது அனைத்தும் உண்மையானதாக கருதப்பட வேண்டும் என்று கூறுகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "மேலும் சுவாரஸ்யமானது, d என்பது 2 இன் 2 வர்க்க மூலத்தையும் எதிர்மறை 1 ஐயும் கருத்தில் கொள்ள வேண்டும் என்று கூறுகிறது, ஆனால் முடிவிலி அல்ல, எனவே முடிவிலியை உண்மையானதாகக் கருதுவதை நிராகரிக்கும் ஒரு நல்ல குழு உங்களில் உள்ளது, ஆனால் மிகவும் வசதியானது. எதிர்மறை 1 இன் வர்க்கமூலம், அது அருமை. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "சரி, நெகட்டிவ் 1 இல் வசதியாக இருக்கும் ஒரு கூட்டத்தை நாங்கள் பெற்றுள்ளோம் போல் தெரிகிறது, ஒரு பெரிய கூட்டத்தினர் முடிவிலியில் அசௌகரியமாக இருக்கிறார்கள், அது மற்றொரு நாளுக்கான தலைப்பு, அதைப் பற்றி கவலைப்பட வேண்டாம், பின்னர் பலர் எதிர்மறை 1 உண்மையானதாக இருக்கலாம் என்ற எண்ணத்தில் மிகவும் வசதியாக இல்லாமல் இருக்கலாம். ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "எனவே எங்கள் முதல் மிகவும் கணித கேள்விக்கு, ஒரு சூடான அப் போன்ற, நான் இந்த இரண்டு சேர்க்க நீங்கள் கேட்க விரும்புகிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "அவற்றை எப்படிச் சேர்ப்பது என்று நான் உங்களுக்குக் கற்பிப்பதற்கு முன், அது எப்படிச் செயல்படும் என்று யூகிக்கவும், அது மிகவும் நேரடியானதாக இருக்கும் என்று நம்புகிறேன், சேர்த்தல் என்பது உண்மையில் இதில் மிகவும் குறைவான சுவாரஸ்யமான பகுதியாகும், ஆனால் அது எப்போது என்பதைத் தெரிந்துகொள்வது நல்லது. நீங்கள் சிக்கலான எண்களைப் பற்றி கற்றுக்கொள்கிறீர்கள். ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "நீங்கள் நான்கு யூனிட்களை வலப்புறம் நகர்த்தி ஒரு யூனிட்டை மேலே நகர்த்தினால், இரண்டு யூனிட்களை இடதுபுறமாகவும், இரண்டு யூனிட்கள் மேலேயும் நகர்த்த வேண்டும் என்ற எண்ணத்தைச் சேர்க்க விரும்பினால், ஒவ்வொன்றையும் ஒரே நேரத்தில் செய்யுங்கள். ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "உண்மையான பகுதி அந்த நான்கு வலப்புறமாகவும், பின்னர் இரண்டைக் கழித்து இடதுபுறமாகவும் இருக்கும். ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "அதுவும் ஒன்று நான் பிளஸ் டூ ஐ. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "கூட்டல் உண்மையில் சிக்கலான எதுவும் இல்லை, இது சிறந்தது. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "எனவே திசையன்களுடன், இரண்டு திசையன்களைத் திரும்பப் பெறுவதற்கு அவற்றைப் பெருக்குவது பற்றி எந்த எண்ணமும் இல்லை, குறைந்தபட்சம் நாம் 2D விமானத்தில் இருக்கும்போது. ", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "அடிப்படையில், என்னிடம் மூன்று, இரண்டு புள்ளி இருப்பதாக வைத்துக்கொள்வோம். ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "என்னிடம் ஒருவித ஒருங்கிணைப்பு கட்டம் இருந்தால், நான் x ஆய மூன்று மற்றும் y ஒருங்கிணைப்பு இரண்டுடன் புள்ளிக்குச் சென்றால், இதன் 90 டிகிரி சுழற்சி என்ன? ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "எதிரெதிர் திசையில். ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "இப்போது இதைப் பற்றிய அழகான விஷயம் என்னவென்றால், அதைக் கண்டுபிடிக்க நாம் அடிப்படையில் எங்கள் காகிதத்தைத் திருப்பலாம். ", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "மூன்று, இரண்டில் ஆரம்பித்து 90 டிகிரிக்கு எதிரெதிர் திசையில் சுழற்றினால் சரி என்று சொல்கிறீர்கள், நான் விமானம் முழுவதையும் அப்படி சுழற்றியிருந்தால், அதை இப்போது x திசையில் இரண்டு எதிர்மறையாகவும், பின்னர் y திசையில் மூன்று எனவும் படிக்கலாம். . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "எனவே நாம் இங்கே என்ன செய்தோம் என்றால், நாம் மூன்று, இரண்டை எடுத்து, அதை எதிர்மறையான இரண்டு, மூன்றாக மாற்றுகிறோம். ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "அது 90 டிகிரி சுழற்சியாக இருக்கும். ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "நான் ஒரு ஜோடி எண்களை கமா பி ஓகே எடுத்தால், அதை 90 டிகிரியில் சுழற்றினால் அது எங்கு சுழலும் என்று சொன்னேன், அது ஆயத்தொலைவுகளை மாற்றுவதன் மூலம் முடிவடையும், பின்னர் அந்த முதல் ஒன்றை எதிர்மறையாக மாற்றும். ", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "எனவே அது மற்றொரு 90 டிகிரி சுழற்சி. ", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "சரி, இங்கே என்ன நடந்தது என்றால், நாங்கள் இரண்டு ஆயத்தொலைவுகளையும் எதிர்மறையாக மாற்றியுள்ளோம், அது உறுதியளிக்கிறது, ஏனென்றால் நான் ab-ல் உட்கார்ந்து சில புள்ளிகளை எடுத்துக் கொண்டால், நான் அதை 90 டிகிரி சுழற்றினால், இது எனது ஆரம்ப 90 டிகிரி சுழற்சியாக இருக்கும், பின்னர் மற்றொரு 90 டிகிரி ஆகும். அதே 180 டிகிரி அழுகல்- ஓ நான் தவறு செய்துவிட்டேன். ", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "இது 180 டிகிரி சுழற்சியைப் போலவே இருக்கும், இது நான் வரைந்த மற்ற திசையனைப் புறக்கணிக்க வேண்டும், இது இரண்டு ஆயத்தொலைவுகளையும் எடுத்து அவற்றை எதிர்மறை எதிர்மறையாக ஆக்குகிறது. உண்மையில் நீங்கள் எதிர்பார்ப்பது போல் நடந்து கொள்கிறது. இப்போது நான் ஏன் இதை உங்களிடம் கேட்கிறேன்? ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "ஆஹா, நிறைய பேர் பதில்களை மிகவும் நன்றாகச் சமர்ப்பித்துள்ளனர். ", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "சரி, உங்களில் பெரும்பான்மையானவர்கள் 2 கூட்டல் 3i என்ற சரியான பதிலைப் பெற்றிருப்பது போல் தெரிகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "மிக நன்று மிக நன்று. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "உங்களில் சிலர் எதிர்மறை 2 3 க்கு பதிலளித்துள்ளனர், இது நீங்கள் 4 கழித்தல் 2 அல்லது 2 கழித்தல் 4 ஐ எடுத்துக்கொள்கிறீர்களா என்பதை மாற்றியமைக்கிறது என்று நான் நினைக்கிறேன், அது முற்றிலும் புரிந்துகொள்ளத்தக்கது. ", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "எங்களிடம் 2 பிளஸ் 3 கிடைத்துள்ளது, இது ஐயை விட்டுவிடலாம், எனவே எளிய பிழைகள் மற்றும் நுழைவு போன்ற பல இருக்கலாம் என்று நான் நினைக்கிறேன், குறிப்பாக சோதனைகளில் நம் அனைவருக்கும் இது நடக்கும் என்று உங்களுக்குத் தெரியும், சில சமயங்களில் சரியான பதில் என்னவென்று உங்களுக்குத் தெரியும். நீங்கள் ஒரு சின்னத்தை மறந்துவிட்டீர்கள் அல்லது இரண்டை மாற்றுகிறீர்கள், அது மிகவும் நல்லது. ", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "உங்களுக்குத் தெரிந்த ஸ்டால் ஸ்டால் வார்த்தைகள் அது வேலை செய்கிறது என்று அவர்கள் என்னிடம் கூறுகிறார்கள், இன்னும் நான் முன்னேறுவது மிகவும் மெதுவாக உள்ளது, எனவே நான் அவர்களிடம் கடுமையான வார்த்தைகளைச் சொல்லப் போவதில்லை என்றால், நீங்கள் ட்விட்டரிலும் அவற்றைப் பார்க்கலாம். நாங்கள் கேள்விகளைக் கேட்கும் இடத்தில், ஏய் கேமினேட்டர் என்று சொல்லுங்கள், நேரடிக் கேள்விகளை எங்களுக்குச் சிறிது சிறப்பாகச் செய்ய உங்களால் முடியவில்லையா? ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "சரி, நாங்கள் இறுதியாக வந்துவிட்டோம் என்று நினைக்கிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "அனைவரும் தயாரா? ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "ஆஹா! ", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "அற்புதம்! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "மிக எளிமையான கேள்வி நீங்கள் i எண்ணை எடுக்க வேண்டும், நீங்கள் அதை 3 கூட்டல் 2i ஆல் பெருக்க வேண்டும் என்று நான் விரும்புகிறேன், மேலும் நான் உண்மையில் பெருக்குவதற்கான விதிகளைப் பற்றி பேசவில்லை என்றாலும், அது செயல்படுவதைப் போல பாசாங்கு செய்ய வேண்டும். சாதாரண எண்களை நீங்கள் விநியோகிக்கக்கூடிய சொத்து போன்ற விஷயங்களைப் பெற்றுள்ளீர்கள், அதை நீங்கள் முழுவதுமாக விநியோகிக்கலாம், பின்னர் ஐயின் வரையறுக்கும் அம்சம் என்னவென்றால், நான் ஸ்கொயர் செய்தேன் என்பது எதிர்மறையானது என்பதுதான், அதைப் பற்றி நீங்கள் தெரிந்து கொள்ள வேண்டிய ஒரே சிறப்பு விஷயம். இது ஒரு சாதாரண எண்ணாக இருந்தாலும் சரி, பிறகு தயாரிப்பைத் தொடரவும். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/telugu/sentence_translations.json b/2020/ldm-complex-numbers/telugu/sentence_translations.json index db2e64112..d2eae145e 100644 --- a/2020/ldm-complex-numbers/telugu/sentence_translations.json +++ b/2020/ldm-complex-numbers/telugu/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "ఇది ఒక అప్లికేషన్‌లో నిజంగా ఉపయోగకరంగా ఉన్నట్లయితే, అది పదాల వలె నిజమైనదని నేను చెబుతాను, సరియైనదా? ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "మీరు అక్కడ సంతోషం వంటి నైరూప్య పదానికి ఎప్పటికీ వెళ్లరు, కానీ అది మన మనస్సులలో ఒక రకమైన వాస్తవికతను కలిగి ఉంటుంది మరియు మీరు భిన్నం వలె వ్యక్తీకరించలేని రెండు వర్గమూలం వంటి వాటిని కలిగి ఉంటుంది లేదా నిజమైన సాధారణ సంఖ్యల మధ్య చూపబడని ప్రతికూల ఒకటి యొక్క వర్గమూలం, అవి కొద్దిగా భిన్నంగా కనిపించినప్పటికీ, మీకు తెలుసు. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "అవి ఏమిటో మీకు ఇంకా తెలుసని నేను అనుకోను, ఇది ప్రాథమిక ప్రైమర్‌గా ఉద్దేశించబడింది, అయితే ఇప్పుడే డైవ్ చేద్దాం, సరేనా? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "నేను స్కూల్లో ఉన్నప్పుడు నాకు గుర్తుంది మరియు మేము ఈ సంకలన సూత్రాలను నేర్చుకున్నాము, మీరు రెండు వేర్వేరు కోణాల మొత్తం కొసైన్ తెలుసుకోవాలనుకుంటే, మీకు తెలుసా, అసలు రెండు కోణాల కొసైన్‌లు మరియు సైన్‌ల పరంగా ఇది చాలా పెద్ద విషయం. , ఈ మైనస్ గుర్తు ఉంది, ఇది ఎల్లప్పుడూ ప్రజలను కదిలిస్తుంది, మీరు సంకేతం కోసం అదే చేస్తే, అది సారూప్యంగా కనిపిస్తుంది కానీ ప్లస్ గుర్తు ఉంది మరియు కాస్-కాస్‌ని కలిగి ఉండటానికి బదులుగా మీకు కాస్-సిన్ ఉంది, ఇది చాలా ఎర్రర్‌కు గురయ్యే విషయం. మీరు దానిని గుర్తుంచుకోవడానికి ప్రయత్నిస్తుంటే. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "అయినప్పటికీ, మీరు సంక్లిష్ట సంఖ్యలతో వచ్చినట్లయితే, ఇది చాలా తక్కువ లోపం మాత్రమే కాదు, ఇది చాలా అందమైన అర్థాన్ని కలిగి ఉంటుంది మరియు ఇది సరిగ్గా బయటకు వస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "కాబట్టి మీరు నెగెటివ్ 1 యొక్క వర్గమూలం యొక్క వాస్తవికతను తప్పనిసరిగా విశ్వసించనప్పటికీ, ఇది ఇతర గణిత ముక్కలను ఉపయోగకరంగా చేయడం ఆసక్తికరంగా ఉందని మీరు కనీసం అంగీకరించాలి. అర్థమయ్యేది కూడా. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "కానీ ప్రారంభ స్థానం చాలా వింతగా ఉంది, సరేనా? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "ఒకటి, లేదు లేదు, సరియైనదా? ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "మీరు వర్గీకరించిన ఏదైనా సంఖ్య, అది సానుకూలంగా ఉంటే, అది సానుకూలంగానే ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "నేను ఎప్పుడూ ప్రతికూలంగా ఏమీ పొందను. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "అయితే, ఒక గణిత శాస్త్రజ్ఞుడు వచ్చి, అరెరే, అది ఉనికిలో లేదు అని చెబితే, మేము దానిని నిర్వచించాము కాబట్టి అలా ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "మీరు పరిష్కరించలేని సమస్య మీకు ఉన్నప్పుడు, మీరు చెప్పగలరు, ఓహ్ నేను విషయాలను నిర్వచించాను, తద్వారా ఇప్పుడు మనకు అద్భుతంగా పరిష్కారం ఉంది. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "కాబట్టి మీరు దీనితో అసౌకర్యంగా ఉంటే, మీరు ఖచ్చితంగా ఒంటరిగా లేరు. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "వాస్తవానికి, రెనే డెస్కార్టెస్ ఈ సంఖ్యల కోసం ఊహాత్మక పదాన్ని అవమానకరమైనదిగా ఉపయోగించారు. ", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "ఆపై మేము దానిని ఒక కన్వెన్షన్‌గా ఉంచాము మరియు మేము ఇప్పటికీ వాటిని ఊహాత్మక సంఖ్యలు అని పిలుస్తాము, ఇది అసంబద్ధమైనది. ", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "మీరు సంక్లిష్ట సంఖ్యల గురించి మాట్లాడటం ప్రారంభించినప్పుడు మీరు చేసే రెండవ విచిత్రమైన విషయం ఏమిటంటే, అటువంటి సంఖ్య i మాత్రమే లేదు, కానీ మేము దానికి ఇంటిని ఇవ్వబోతున్నాము. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "మరియు, సరే, మనం మన నంబర్ సిస్టమ్‌ను పొడిగించాలనుకుంటే, నేను అర్థం చేసుకున్నాను, అక్కడ ఒక రకమైన సంఖ్యను ఉంచడం ఉపయోగకరంగా ఉండవచ్చు, కానీ నేను ఎందుకు, సరియైనదా? ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "ప్రారంభంలో, మీరు ఇలా ద్విమితీయంగా ఉండే సంఖ్యలను ఎలా జోడిస్తే, నియమాలు చాలా సరళంగా ఉంటాయి మరియు ఇది వెక్టర్‌లతో బాగా తెలిసిన మీలో ఎవరికైనా వెక్టర్‌ల మాదిరిగానే పనిచేస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "మీరు దానిని విశ్వాసం మీద తీసుకుంటే మరియు మీరు అనుసరిస్తే, మేము వీటిలో దేనినైనా ఎందుకు చేస్తున్నామో సమర్ధించుకోవడానికి ఇది ఉపయోగకరంగా మారుతుందనే వాస్తవం ఆశాజనకంగా ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "అయ్యో, f మరియు d సమాధానాల మధ్య ఒక ముందుకు వెనుకకు ఉన్నట్లుగా ఉంది, కాబట్టి f అనేది వాటన్నింటిని, ఇవన్నీ వాస్తవమని భావించాలని చెప్పారు. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "మరియు ఆసక్తికరంగా, d అనేది మీరు 2 యొక్క 2 వర్గమూలాన్ని మరియు ప్రతికూల 1ని పరిగణించాలని చెబుతుంది, కానీ అనంతం కాదు, కాబట్టి మీలో ఒక మంచి బృందం ఉంది, వారు అనంతాన్ని నిజమైనదిగా పరిగణించి తిరస్కరించవచ్చు, కానీ చాలా సౌకర్యవంతంగా ఉంటారు. ప్రతికూల 1 వర్గమూలం, అది అద్భుతం. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "సరే, నెగెటివ్ 1తో సౌకర్యంగా ఉండే వ్యక్తుల సమూహం మాకు లభించినట్లు కనిపిస్తోంది, ఒక పెద్ద సమూహం అనంతంతో అసౌకర్యంగా ఉంది, అది మరొక రోజుకి సంబంధించిన అంశం, దాని గురించి చింతించకండి, ఆపై చాలా మంది వ్యక్తులు ప్రతికూల 1 నిజమైనది కావచ్చు అనే ఆలోచనతో చాలా సౌకర్యంగా ఉండకపోవచ్చు. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "కాబట్టి మా మొదటి చాలా ఎక్కువ గణిత ప్రశ్న కోసం, సన్నాహక రకంగా, ఈ రెండింటిని జోడించమని నేను మిమ్మల్ని అడగాలనుకుంటున్నాను. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "వాటిని ఎలా జోడించాలో నేను మీకు బోధించే ముందు, ఇది ఎలా పని చేస్తుందో ఊహించండి మరియు ఇది చాలా సూటిగా ఉంటుందని నేను ఆశిస్తున్నాను, అదనంగా నిజానికి ఇందులో అతి తక్కువ ఆసక్తికరమైన భాగం, అయితే ఇది ఎప్పుడు తెలుసుకోవడం మంచిది మీరు సంక్లిష్ట సంఖ్యల గురించి నేర్చుకుంటున్నారు. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "మీరు నాలుగు యూనిట్లను కుడి వైపుకు మరియు ఆపై ఒక యూనిట్ పైకి తరలిస్తుంటే, మీరు రెండు యూనిట్లను ఎడమ వైపుకు మరియు ఆపై రెండు యూనిట్లు పైకి తరలించాలనే ఆలోచనను జోడించాలనుకుంటే, మీరు వాటిలో ప్రతి ఒక్కటి ఒకేసారి చేయండి. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "అసలు భాగం కుడివైపు ఆ నాలుగు, ఎడమవైపు మైనస్ రెండు ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "అదే వన్ ఐ ప్లస్ టూ ఐ. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "అదనంగా నిజంగా సంక్లిష్టంగా ఏమీ జరగలేదు, ఇది చాలా బాగుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "కాబట్టి వెక్టర్స్‌తో, కనీసం మనం 2D ప్లేన్‌లో ఉన్నప్పుడు, రెండు వెక్టర్‌లను తిరిగి పొందడానికి వాటిని గుణించడం గురించి నిజంగా ఎలాంటి భావన లేదు. ", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "ప్రాథమికంగా, నాకు మూడు, రెండు పాయింట్లు ఉన్నాయని అనుకుందాం. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "నేను ఒక విధమైన కోఆర్డినేట్ గ్రిడ్‌ని కలిగి ఉంటే మరియు నేను x కోఆర్డినేట్ త్రీ మరియు y కోఆర్డినేట్ టూతో పాయింట్‌కి వెళితే, దీని యొక్క 90 డిగ్రీ రొటేషన్ ఏమిటి? ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "అపసవ్య దిశలో. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "ఇప్పుడు దీని గురించి మనోహరమైనది ఏమిటంటే, దానిని గుర్తించడానికి మనం ప్రాథమికంగా మన కాగితాన్ని తిప్పవచ్చు. ", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "ఇది మూడు, రెండు వద్ద ప్రారంభించి, ఆపై నేను అపసవ్య దిశలో 90 డిగ్రీలు తిప్పినా సరే, నేను మొత్తం విమానాన్ని అలా తిప్పి ఉంటే, నేను దానిని ఇప్పుడు x దిశలో రెండు ప్రతికూలంగా మరియు y దిశలో మూడు అని చదవగలను. . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "కాబట్టి మనం ఇక్కడ ఏమి చేసాము అంటే మనం మూడు, రెండు తీసుకున్నాము మరియు దానిని ప్రతికూలంగా రెండు, మూడుగా మారుస్తాము. ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "అది 90 డిగ్రీల రొటేషన్ అవుతుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "నేను ఒక జత సంఖ్యలను కామా బి ఓకే తీసుకున్నట్లయితే, నేను దానిని 90 డిగ్రీలు తిప్పితే అది ఎక్కడికి తిరుగుతుందో చెప్పాను, అది కోఆర్డినేట్‌లను మార్చుకోవడం ద్వారా ముగుస్తుంది, ఆపై మొదటిది ప్రతికూలంగా ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "కాబట్టి అది మరొక 90 డిగ్రీల భ్రమణం. ", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "సరే ఇక్కడ ఏమి జరిగిందంటే, మేము రెండు కోఆర్డినేట్‌లను ప్రతికూలంగా చేసాము మరియు అది భరోసానిస్తుంది ఎందుకంటే నేను ab వద్ద కూర్చొని కొంత పాయింట్ తీసుకొని దానిని 90 డిగ్రీలు తిప్పుతాను కాబట్టి ఇది నా ప్రారంభ 90 డిగ్రీలు మరియు మరొక 90 డిగ్రీలు అవుతుంది. అదే 180 డిగ్రీ రాట్- ఓహ్ నేను తప్పు చేసాను. ", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "ఇది 180 డిగ్రీల భ్రమణం వలె ఉంటుంది, ఇది నేను గీసిన ఇతర వెక్టార్‌ను విస్మరించండి, ఇది రెండు కోఆర్డినేట్‌లను తీసుకొని వాటిని ప్రతికూలంగా ప్రతికూలంగా చేస్తుంది b ఓకే కాబట్టి ఇది 90 డిగ్రీల భ్రమణాన్ని చేసే ఈ ఆపరేషన్‌కు భరోసా ఇస్తుంది నిజానికి మీరు ఊహించినట్లుగా ప్రవర్తిస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "ఓహ్ చూడండి చాలా మంది చాలా మంచి సమాధానాలు సమర్పించారు. ", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "సరే, మీలో చాలా మందికి 2 ప్లస్ 3i అనే సరైన సమాధానం వచ్చినట్లు కనిపిస్తోంది. ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "చాలా బాగుంది చాలా బాగుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "మీలో కొందరు నెగెటివ్ 2 3కి సమాధానమిచ్చారు, ఇది మీరు 4 మైనస్ 2 లేదా 2 మైనస్ 4 తీసుకుంటున్నారా అనేదానిని మార్చుకుంటున్నారని నేను ఊహిస్తున్నాను కాబట్టి అది పూర్తిగా అర్థమవుతుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "మేము 2 ప్లస్ 3ని పొందాము, అది బహుశా i నుండి పడిపోవచ్చు కాబట్టి నేను సాధారణ లోపాలు మరియు ప్రవేశం వంటివి చాలా ఉండవచ్చు అని నేను అనుకుంటున్నాను మరియు మనందరికీ ముఖ్యంగా పరీక్షలలో ఇది జరుగుతుందని మీకు తెలుసు, కొన్నిసార్లు సరైన సమాధానం ఏమిటో మీకు తెలుసు. మీరు ఒక చిహ్నాన్ని మరచిపోతారు లేదా మీరు రెండిటిని మార్చుకుంటారు కనుక ఇది చాలా బాగుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "స్టాల్ స్టాల్ పదాలు మీకు తెలిసిన పదాలు అది పనిచేస్తోందని వారు నాకు చెబుతారు మరియు నేను ముందుకు సాగడం చాలా నెమ్మదిగా ఉంది కాబట్టి నేను వారితో కఠినంగా మాట్లాడకపోతే మీకు తెలుసు, మీరు వాటిని ట్విట్టర్‌లో కూడా చూడవచ్చు. మేము ప్రశ్నలు అడిగే స్థలం మరియు హే కామినేటర్ అని చెప్పండి, మీరు ప్రత్యక్ష ప్రశ్నలు మాకు కొంచెం మెరుగ్గా పని చేయలేదా? ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "సరే మేము చివరకు అక్కడ ఉన్నామని నేను అనుకుంటున్నాను. ", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "అందరూ సిద్ధంగా ఉన్నారా? ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "ఆహా! ", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "అద్భుతం! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "చాలా సులభమైన ప్రశ్న మీరు i అనే సంఖ్యను తీసుకోవాలని నేను కోరుకుంటున్నాను మరియు మీరు దానిని 3 ప్లస్ 2iతో గుణించాలని నేను కోరుకుంటున్నాను మరియు గుణకార నియమాల గురించి నేను నిజంగా మాట్లాడనప్పటికీ, నేను చెప్పగలిగేది అది పనిచేసే విధంగా నటించడం మీరు పంపిణీ చేసే ఆస్తి వంటి సాధారణ సంఖ్యలను మీరు పొందారు, ఇక్కడ మీరు దీన్ని అంతటా పంపిణీ చేయవచ్చు, ఆపై i యొక్క నిర్వచించే లక్షణం నేను స్క్వేర్డ్ చేసిన ఈ ఆలోచన ప్రతికూలమైనది, ఇది మాత్రమే మీరు దాని గురించి తెలుసుకోవలసిన ఏకైక విషయం. ఇది సాధారణ సంఖ్య అయితే సరే, ఆపై ఉత్పత్తితో ముందుకు సాగండి. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/thai/sentence_translations.json b/2020/ldm-complex-numbers/thai/sentence_translations.json index 0885f4d1e..3adc250de 100644 --- a/2020/ldm-complex-numbers/thai/sentence_translations.json +++ b/2020/ldm-complex-numbers/thai/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "ท้ายที่สุด ท้ายที่สุด ผมอยากพูดถึงฟังก์ชันตรีโกณมิติที่แตกต่างกันสองฟังก์ชัน และนี่คือสิ่งที่เราจะสร้างขึ้นมา สองอัตลักษณ์จากตรีโกณมิติ และผมเข้าใจว่าบางที โอ้ อัตลักษณ์ที่ซับซ้อนจากตรีโกณมิติจะไม่ใช่วิธีที่ดีที่สุดในการหลอกคนให้เข้าใจ โอ้ ใช่แล้ว จำนวนเชิงซ้อน พวกมันมีประโยชน์จริงๆ คุณจะต้องชอบมันแน่ๆ แต่ฉันคิดว่ามันน่าสนใจที่คุณสามารถมีข้อเท็จจริงที่ไม่เกี่ยวข้องกับจำนวนเชิงซ้อนหรือรากที่สองของลบ 1 มันเป็นแค่ตรีโกณมิติ มันคือทุกสิ่งที่เราพูดถึงครั้งที่แล้ว และคุณสามารถมีข้อเท็จจริงที่ค่อนข้างยาก ที่จะจำ ฉันจำได้ว่าตอนที่ฉันยังเรียนอยู่ และเราเรียนสูตรการบวกเหล่านี้ ถ้าคุณอยากทราบโคไซน์ของผลรวมของมุมที่ต่างกันสองมุม คุณก็รู้ มันเป็นอะไรที่ยาวในรูปของโคไซน์และไซน์ของมุมสองมุมเดิม มีเครื่องหมายลบที่จะทำให้คนอื่นสะดุดเสมอ ถ้าคุณทำแบบเดียวกันกับเครื่องหมาย มันจะดูคล้ายกันแต่มีเครื่องหมายบวก และแทนที่จะมี cos-cos คุณมี cos-sin กลับเป็นสิ่งที่เกิดข้อผิดพลาดได้ง่ายมาก หากคุณแค่พยายามจดจำมันอย่างที่มันเป็น อย่างไรก็ตาม หากคุณใช้จำนวนเชิงซ้อน นี่ไม่เพียงแต่จะเกิดข้อผิดพลาดน้อยลงเท่านั้น แต่ยังมีความหมายที่สวยงามมาก และมันก็หลุดออกมาทันที ดังนั้นแม้ว่าคุณจะไม่เชื่อในความเป็นจริงของรากที่สองของลบ 1 แต่อย่างน้อยที่สุด คุณก็ต้องยอมรับว่ามันน่าสนใจที่มันทำให้คณิตศาสตร์ชิ้นอื่นมีประโยชน์ได้ ส่วนคณิตศาสตร์ชิ้นอื่นๆ มากกว่านี้อีกหน่อย เข้าใจได้เช่นกัน และตรีโกณมิติเป็นเพียงส่วนเล็กของภูเขาน้ำแข็ง หากคุณพูดคุยกับใครก็ตามที่อยู่ในสายวิศวกรรม ใครก็ตามที่กำลังสนใจคณิตศาสตร์อย่างจริงจัง พวกเขาจะบอกคุณว่าจำนวนเชิงซ้อนเป็นส่วนหนึ่งในชีวิตของพวกเขา และชีวิตของพวกเขาก็เหมือนกับตัวเลขจริง แต่จุดเริ่มต้นดูแปลกมากใช่ไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "เมื่อคุณมีปัญหาที่คุณไม่สามารถแก้ไขได้ คุณสามารถพูดได้ว่า โอ้ ฉันได้กำหนดสิ่งต่าง ๆ แล้ว เพื่อที่เราจะได้มีวิธีแก้ไขอย่างน่าอัศจรรย์ โอเค คราวหน้าผมมีปัญหากับการบ้าน และไม่รู้ว่าคำตอบของ x คืออะไร ผมจะบอกว่า ให้ x เป็นค่าที่กำหนดให้เป็นคำตอบของคำถามนี้ ดังนั้นหากคุณไม่สบายใจกับเรื่องนี้ คุณไม่ได้อยู่คนเดียวอย่างแน่นอน ในความเป็นจริง Rene Descartes ได้บัญญัติคำว่าจินตภาพสำหรับตัวเลขเหล่านี้ว่าเป็นคำที่เสื่อมเสีย มีจุดมุ่งหมายเพื่อล้อเลียนความจริงที่ว่าไม่มีคำตอบเช่นนั้น และไม่ควรถือเป็นคณิตศาสตร์จริงจัง แล้วเราก็ยึดถือสิ่งนั้นตามแบบแผน และเรายังคงเรียกพวกมันว่าจำนวนจินตภาพ ซึ่งไร้สาระจริงๆ แต่นั่นไม่ใช่ข้อสันนิษฐานแปลก ๆ เพียงอย่างเดียวที่เราทำ เรื่องประหลาดอย่างที่สองที่คุณทำเมื่อเริ่มพูดถึงจำนวนเชิงซ้อน คือการบอกว่า มันไม่ใช่แค่ตัวเลข i เท่านั้น แต่เราจะให้มันบ้านแทน แทนที่จะเป็นเส้นจำนวนจริง ซึ่งคุณรู้จักตัวเลขเหล่านี้ทั้งหมดที่เรารู้เมื่อเรายกกำลังสองมัน คุณไม่สามารถได้ค่าลบ สิ่งที่เราทำคือบอกว่าฉันอยู่ในมิติที่ต่างออกไป ผมอยู่ตั้งฉาก มีอันหนึ่งข้างบน แล้วก็ข้างล่าง ลบ i แล้วคุณก็มีลบ 2i ได้ คุณปรับขนาดมันได้ตามต้องการ โดยพื้นฐานแล้ว มันเสนอว่าตัวเลขเป็นสองมิติ และผมมีบ้านที่เฉพาะเจาะจงมาก หนึ่งหน่วยตั้งฉาก เอ่อ ตั้งฉากเหนือเส้นจำนวนจริง และ โอเค ถ้าเราอยากขยายระบบตัวเลข ผมเข้าใจ บางทีการใส่ตัวเลขอะไรสักอย่างบนนั้นอาจมีประโยชน์ แต่ทำไมต้องผม ใช่ไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "แนวคิดเรื่องจุดหนึ่งหน่วยเหนือเส้นจำนวนจริงในมิติที่แยกจากกัน เกี่ยวอะไรกับการยกกำลังสองถึงลบ ดังนั้นฉันหวังว่าจะตอบคำถามนี้ให้กับคุณ ในตอนเริ่มต้น เรามาคุยกันว่าถ้าคุณบวกตัวเลขสองมิติแบบนี้ กฎจะค่อนข้างตรงไปตรงมา และมันทำงานเหมือนกับเวกเตอร์ สำหรับพวกคุณที่อาจคุ้นเคยกับเวกเตอร์ สมมุติว่า ผมมีตัวเลข ไม่รู้สิ ขอผมวาดอันนึงตรงนี้ซึ่งจะเท่ากับ 4 บวก i โอเคไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "ตอนนี้ สมมติว่าระบบคำถามของเรายังไม่พัง ฉันควรจะทำเช่นนี้เป็นการสํารวจที่เหมาะสม ให้ฉันไปก่อน ฉันเดาว่าเราจะตรวจสอบการสำรวจความคิดเห็นครั้งก่อนๆ ได้ โอเค ดูเหมือนทุกอย่างจะได้ผล เราจะย้อนกลับไปในบทเรียนได้นิดหน่อย ฉันเลยอยากรู้จริงๆ ฉันอยากรู้ว่าพวกคุณเป็นยังไงบ้าง ตอบในอันนี้ อืม ดูเหมือนว่ามี a มีการกลับไปกลับมาระหว่างคำตอบ f กับ d ดังนั้น f คือทั้งหมดนั้น โดยบอกว่าทั้งหมดนี้ควรถือว่าเป็นจริง และน่าสนใจ d คืออันที่บอกว่าคุณควรพิจารณา 2 สแควร์รูทของ 2 และลบ 1 แต่ไม่ใช่อนันต์ จึงมีกลุ่มดีๆ ข้างนอกนั่นที่ปฏิเสธอนันต์ว่าถือว่ามีจริง แต่สบายใจกับ สแควร์รูทของลบ 1 เยี่ยมมาก แล้วหลังจากนั้น ดูเหมือน c คนที่ปฏิเสธรากที่สองของลบ 1 น่าทึ่งมาก ผมคงคิดว่าไม่มีใครสูงกว่านั้นแล้ว ไม่มีสิ่งใดที่ต่ำกว่ามากที่ โอเค ดูเหมือนว่าเรามีกลุ่มคนที่พอใจกับค่าลบ 1 กลุ่มใหญ่ไม่สบายใจกับค่าอนันต์ นั่นเป็นหัวข้อสำหรับวันอื่น ไม่ต้องกังวลกับมัน และยังมีอีกหลายคนที่ อยู่ตรงกลางว่าอาจจะไม่สบายใจกับแนวคิดที่ว่าลบ 1 อาจมีจริง มาดูกันว่าเราสามารถโน้มน้าวคุณถึงความแตกต่างนั้นได้หรือไม่ สำหรับคำถามทางคณิตศาสตร์ข้อแรกของเรา เพื่อเป็นการอุ่นเครื่อง ผมแค่อยากให้คุณบวกสองตัวนี้เข้าไป ก่อนที่ฉันจะสอนวิธีเพิ่มมัน ให้เดาก่อนว่ามันทำงานอย่างไร และหวังว่ามันจะค่อนข้างตรงไปตรงมา การบวกเป็นส่วนที่น่าสนใจน้อยที่สุด แต่ก็เป็นเรื่องดีที่จะรู้ว่าเมื่อใด คุณกำลังเรียนรู้เกี่ยวกับจำนวนเชิงซ้อน นี่เป็นหนึ่งในการดำเนินการที่คุณจะต้องรู้อย่างแน่นอน น่าเสียดายที่คุณสามารถบอกได้ว่าฉันกำลังถ่วงเวลาและสิ่งที่ฉันพูดที่นี่ ดูเหมือนว่าคำถามยังโหลดไม่ถูกต้อง ดังนั้นฉันจะใช้คำพูดที่รุนแรงกับ Cam และ Ider ที่อยู่เบื้องหลัง ฉากที่สร้างอินเทอร์เฟซที่สวยงามและสวยงาม ซึ่งมีประโยชน์สำหรับการพูดคุยไปมาระหว่างพวกคุณกับฉัน ฉันจะพูดจารุนแรงกับพวกเขาเบื้องหลัง แต่ระหว่างนี้เรามาเริ่มบทเรียนที่นี่กันดีกว่า ผมว่าผมดึงมันขึ้นมาบนกระดาษได้เลย แล้วคุณตามไปที่บ้านก็ได้ ดูว่ามีอะไรเพิ่มเติมบ้าง ปรากฎว่าค่อนข้างตรงไปตรงมา หากคุณเลื่อนสี่หน่วยไปทางขวา แล้วขึ้นหนึ่งหน่วย และคุณต้องการเพิ่มแนวคิดในการเลื่อนสองหน่วยไปทางซ้าย แล้วขึ้นสองหน่วย คุณก็แค่ทำทีละหน่วย ฉันจะไปข้างหน้าและดึงสีดำออกมาที่นี่ ส่วนจริงจะเป็นสี่อันทางขวา แล้วลบ 2 ทางซ้าย โอเค ตรงไปตรงมาพอแล้ว แล้วส่วนจินตภาพจะเป็นหนึ่งหน่วยขึ้น แล้วสองหน่วยนี้ขึ้น หนึ่งบวกสอง คูณ i นั่นก็คือ i บวก 2 i แล้วเมื่อคุณหามันออกมา สี่ลบสองได้สอง หนึ่งบวกสองได้สาม การแนะนำง่ายๆที่ดีที่นี่ นอกจากนี้ไม่มีอะไรซับซ้อนเกิดขึ้นซึ่งดีมาก นั่นหมายความว่ามีสิ่งหนึ่งที่เรากังวลน้อยลง อะไรที่ซับซ้อนเกี่ยวกับจำนวนเชิงซ้อนในที่สุด? ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "ทีนี้ สิ่งน่ารักเกี่ยวกับเรื่องนี้ก็คือ เราสามารถพลิกกระดาษเพื่อหามันได้ คุณบอกว่าโอเค ถ้ามันเริ่มที่สาม สอง แล้วฉันหมุน 90 องศาทวนเข็มนาฬิกา ตอนนี้ฉันอ่านออกว่าเป็นลบ 2 ในทิศทาง x และสามในทิศทาง y หากฉันหมุนทั้งระนาบแบบนั้น . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "และสิ่งที่ดีที่นี่คือกฎนั้นง่ายมาก และใช้ได้กับคู่เงินใดๆ ที่เราอาจมี ถ้าผมเอาตัวเลขคู่หนึ่งมาด้วยลูกน้ำ b โอเค แล้วผมบอกว่ามันจะหมุนไปทางไหน ถ้าเราหมุนมัน 90 องศา มันจะจบลงด้วยการสลับพิกัด ba แล้วทำให้อันแรกเป็นลบ นั่นคือการหมุน 90 องศา และการตรวจสัญชาตญาณที่ดีตรงนี้ คือบอกว่าจะเกิดอะไรขึ้นเมื่อเราทำสิ่งนั้นสองครั้ง จะเป็นอย่างไรถ้าเราดำเนินการแบบกลไกเดิม ๆ อีกครั้งสองครั้ง แล้วผมจะไปเอาอันนี้ ผมสลับพิกัดทั้งสองเราจะได้ ลบ b แต่อันแรกกลายเป็นลบ นั่นคือการหมุนอีก 90 องศา สิ่งที่เกิดขึ้นตรงนี้คือ เราเพิ่งทำให้พิกัดทั้งสองเป็นลบ และนั่นก็มั่นใจ เพราะหากผมนั่งที่ ab แล้วผมหมุนมัน 90 องศา นี่จะเป็นการหมุน 90 องศาเริ่มแรกของผม แล้วตามด้วยอีก 90 องศา นั่นคือ เช่นเดียวกับการเน่า 180 องศา- โอ้ไม่ ฉันทำผิดไปแล้ว นั่นจะเหมือนกับการหมุน 180 องศา ซึ่งควรมีลักษณะแบบนี้ โดยไม่สนใจเวกเตอร์อีกตัวที่ผมวาด ซึ่งแค่เอาพิกัดทั้งสองมาและทำให้พวกมันเป็นลบ ลบ เป็นลบ b โอเค เพื่อให้มั่นใจว่าการดำเนินการนี้จะหมุน 90 องศา จริงๆ แล้วจะมีพฤติกรรมเหมือนที่คุณคาดหวังไว้ ตอนนี้ทำไมฉันถึงถามคุณเรื่องนี้? ", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "อ๋อ! ", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "มหัศจรรย์! คำถามง่ายๆ มากๆ ฉันอยากให้คุณเอาเลข i และฉันต้องการให้คุณคูณมันด้วย 3 บวก 2i และถึงแม้ว่าฉันจะไม่ได้พูดถึงกฎของการคูณจริงๆ ก็ตาม สิ่งที่ฉันสามารถพูดได้คือแสร้งทำเป็นว่ามันทำงานเหมือนกับที่มันทำ ตัวเลขปกติ คุณจะได้สมบัติการแจกแจงที่คุณสามารถกระจายค่านี้ไปตลอด แล้วคุณลักษณะที่กำหนดของ i คือแนวคิดที่ว่า i กำลังสองนั้นเป็นค่าลบ นั่นคือสิ่งเดียวที่พิเศษที่คุณต้องรู้ นอกเหนือจากนั้นก็แค่ปฏิบัติต่อมัน เหมือนเป็นเลขปกติโอเคแล้วจึงดำเนินการกับสินค้าต่อไป มหัศจรรย์! ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/turkish/sentence_translations.json b/2020/ldm-complex-numbers/turkish/sentence_translations.json index cbcb064ca..e60a468c0 100644 --- a/2020/ldm-complex-numbers/turkish/sentence_translations.json +++ b/2020/ldm-complex-numbers/turkish/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "Bir uygulamada gerçekten yararlı olan bir şeyse, o zaman kelimeler kadar gerçek olduğunu söyleyebilirim, değil mi? ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "Orada hiçbir zaman mutluluk gibi soyut bir kelimeyle karşılaşmayacaksınız, ama onun zihinlerimizde bir tür gerçekliği var ve ikinin karekökü gibi, kesir olarak ifade edemediğiniz şeyler veya bunun gibi şeyler var. Negatif birin karekökü, gerçek normal sayılar arasında görünmeyen, biraz farklı görünseler bile. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "Henüz bunların ne olduğunu bildiğinizi varsaymıyorum, temel bir başlangıç olması gerekiyor, ama hadi hemen konuya girelim, tamam mı? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "Okuldayken bu toplama formüllerini öğrendiğimizi hatırlıyorum, eğer iki farklı açının toplamının kosinüsünü bilmek istiyorsanız, bu orijinal iki açının kosinüsleri ve sinüsleri cinsinden uzun bir şey. , insanları her zaman tuzağa düşüren bir eksi işareti var, eğer işaret için de aynısını yaparsanız, benzer görünüyor ama bir artı işareti var ve cos-cos-cos-sin yerine, bu hataya çok açık bir şey eğer onu olduğu gibi ezberlemeye çalışıyorsan. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "Bununla birlikte, eğer ona karmaşık sayılarla yaklaşırsanız, bu sadece daha az hataya açık olmakla kalmaz, aynı zamanda çok güzel bir anlamı vardır ve hemen ortaya çıkar. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "Yani, negatif 1'in karekökünün gerçekliğine mutlaka inanmasanız bile, en azından bunun ilginç olduğunu, bunun matematiğin diğer bölümlerini faydalı hale getirebileceğini, diğer matematik bölümlerini biraz daha faydalı hale getirebileceğini kabul etmelisiniz. da anlaşılabilir. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "Ama başlangıç noktası çok tuhaf görünüyor, tamam mı? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "Birincisi, hayır yok, değil mi? ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "Karesini aldığınız herhangi bir sayı, eğer pozitifse, o da pozitif kalır. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "Asla olumsuz bir şeyle karşılaşmayacağım. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "Ama bir matematikçi gelip, ah hayır öyle bir şey var derse, biz öyle tanımladık. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "Çözemediğiniz bir sorununuz olduğunda, şöyle diyebilirsiniz: "Her şeyi tanımladım, böylece artık sihirli bir şekilde bir çözümümüz var." Tamam, bir dahaki sefere ödevimde sorun yaşadığımda ve x'in cevabını bilmediğimde şöyle olacağım, bu sorunun cevabı olarak tanımlanan değer x olsun. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "Yani eğer bu durumdan rahatsızsanız kesinlikle yalnız değilsiniz. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "Aslında Rene Descartes bu sayılar için hayali terimini aşağılayıcı bir ifade olarak kullanmıştır. ", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "Ve sonra buna bir gelenek olarak bağlı kaldık ve onlara hala sanal sayılar diyoruz ki bu gerçekten saçma. ", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "Karmaşık sayılar hakkında konuşmaya başladığınızda yapacağınız ikinci tuhaf şey, böyle bir i sayısının olmadığını, ama ona bir yuva vereceğimizi söylemektir. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "Ve tamam, eğer sayı sistemimizi genişletmek istiyorsak, anlıyorum, oraya bir tür sayı koymak yararlı olabilir, ama neden ben, değil mi? ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "Başlangıçta, bunun gibi iki boyutlu sayıları topluyorsanız kuralların oldukça basit olduğundan ve vektörlere aşina olabilecek herhangi biriniz için aslında vektörlerle aynı şekilde işlediğinden bahsedelim. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "Buna güvenirseniz ve takip ederseniz, umarım bunun faydalı olacağı gerçeği, bunları neden yaptığımızı haklı çıkarmaya yardımcı olur. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "Görünüşe göre f ve d cevapları arasında bir ileri geri var, yani f hepsi, bunların hepsinin gerçek sayılması gerektiğini söylüyor. ", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "Ve ilginç, d, 2'nin karekökü ve eksi 1'i dikkate almanız gerektiğini, ancak sonsuzluğu düşünmemeniz gerektiğini söyleyen sayıdır, yani dışarıda, sonsuzluğu gerçek olarak kabul etmeyi reddedecek, ancak bu denklemden çok memnun olan iyi bir grup var. Negatif 1'in karekökü, bu harika. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "Tamam, öyle görünüyor ki negatif 1'den memnun olan bir grup insan var, sonsuzluktan rahatsız olan büyük bir grup da var, bu başka bir günün konusu, endişelenmeyin ve sonra da bir takım insanlar Negatif 1'in gerçek olabileceği fikri konusunda pek rahat olamama gibi bir orta noktadayız. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "İlk çok daha matematiksel sorumuz için, bir nevi ısınma amaçlı olarak, sizden sadece bu ikisini eklemenizi istiyorum. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "Bunları nasıl ekleyeceğinizi size öğretmeden önce, nasıl çalışabileceğine dair bir tahminde bulunun ve umarım oldukça basit gelir, toplama aslında bunun en az ilginç kısmıdır, ama öyle, ne zaman yapılacağını bilmek iyi bir şey. karmaşık sayıları öğreniyorsunuz. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "Dört birim sağa ve sonra bir birim yukarı hareket ettiriyorsanız ve iki birim sola ve ardından iki birim yukarı hareket etme fikrini de eklemek istiyorsanız, bunların her birini birer birer yapın. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "Gerçek kısım sağdaki dördü, sonra soldaki eksi iki olacak. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "Yani bu bir i artı iki i. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "Eklemede gerçekten karmaşık hiçbir şey olmuyor, bu harika. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "Yani vektörlerde, en azından 2 boyutlu düzlemdeyken, iki vektörü geri elde etmek için onları çarpmanın hiçbir fikri yoktur. ", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "Temel olarak, diyelim ki üç, iki noktasına sahibim. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "Eğer bir tür koordinat ızgaram varsa ve x koordinatı üç ve y koordinatı iki olan noktaya gidersem, bunun 90 derecelik dönüşü nedir? ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "Saat yönünün tersine. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "Tamam aşkım. Şimdi bunun güzel yanı, aslında bunu anlamak için kağıdımızı çevirebilmemiz. ", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "Tamam diyorsunuz, eğer üç, iki ile başlasaydı ve sonra saat yönünün tersine 90 derece dönseydim, şimdi bunu x yönünde eksi iki ve sonra y yönünde üç olarak okuyabilirdim, eğer tüm düzlemi bu şekilde döndürmüş olsaydım . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "Burada yaptığımız şey şu; üç, ikiyi aldık ve sonra bunu eksi iki, üçe çevirdik. ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "Bu 90 derecelik dönüş olacak. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "Eğer bir çift sayı alırsam a virgül b tamam ve sonra bunun nereye döneceğini söylersem, eğer onu 90 derece döndürürsem, ba koordinatlarının yer değiştirmesi ve ardından ilkinin negatif olmasıyla sonuçlanacaktır. ", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "Yani bu başka bir 90 derecelik dönüştü. ", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "Burada olan şu ki, her iki koordinatı da negatif yaptık ve bu güven verici çünkü ab noktasında otururken bir nokta alırsam ve sonra onu 90 derece döndürürsem, bu benim ilk 90 derecelik dönüşüm olur ve sonra başka bir 90 derecelik dönüş olur. 180 derece çürümeyle aynı - ah hayır, bunu yanlış yaptım. ", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "Bu, 180 derecelik bir dönüşle aynı olacaktır, bu şekilde görünmeli, çizdiğim diğer vektörü dikkate almayın, bu sadece her iki koordinatı da alıp onları negatif negatif a negatif b yapıyor tamam, bu da 90 derecelik dönüş yapan bu işleme güven veriyor aslında beklediğiniz gibi davranıyor. ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "Bakın, pek çok kişi çok iyi yanıtlar gönderdi. ", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "Tamam, öyle görünüyor ki çoğunuz 2 artı 3i olan doğru cevabı bulmuşsunuz. ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "Çok iyi çok iyi. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "Bazılarınız eksi 2 3 cevabını verdi, sanırım bu sadece 4 eksi 2 ya da 2 eksi 4'ün yerini değiştirmek anlamına geliyor, yani bu tamamen anlaşılabilir bir durum. ", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "Elimizde 2 artı 3 var, bu da belki sadece i'yi bırakıyor, bu yüzden sanırım basit hatalar ve girişler gibi pek çok şey var ve biliyorsunuz, özellikle testlerde hepimizin başına gelen bazen doğru cevabın ne olduğunu bilirsiniz ama sonra bir sembolü unutursunuz ya da ikisini değiştirirsiniz, bu çok iyi. ", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "Stal stal kelimeler, bildiğiniz kelimeler bana bunun işe yaradığını söylüyorlar ve yine de ilerlemem çok yavaş, bu yüzden biliyorsunuz, eğer onlara sert bir söz söylemeyeceksem, siz de onlara Twitter'da da aynı şekilde gidebilirsiniz. Soru sorduğumuz ve sadece merhaba kamineter diyebildiğimiz bir yer, canlı soruların bizim için biraz daha iyi çalışmasını sağlayamaz mısın? ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "Tamam sanırım sonunda geldik. ", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "Herkes hazır mı? ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "Aha! ", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "Müthiş! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "Çok basit bir soru, i sayısını almanızı ve onu 3 artı 2i ile çarpmanızı istiyorum ve çarpma kuralları hakkında pek konuşmamış olsam da, söyleyebileceğim şey, tıpkı şu şekilde çalışıyormuş gibi davranmak: normal sayılar, bunu her tarafa dağıtabileceğiniz dağılım özelliği gibi şeylere sahipsiniz ve sonra i'nin tanımlayıcı özelliği, i'nin karesinin negatif olduğu fikri, bu konuda bilmeniz gereken tek özel şey, bunun dışında onu ele almak normal bir sayıymış gibi tamam ve ardından ürüne devam edin. Müthiş! ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/ukrainian/sentence_translations.json b/2020/ldm-complex-numbers/ukrainian/sentence_translations.json index 0913e5df3..614671d1e 100644 --- a/2020/ldm-complex-numbers/ukrainian/sentence_translations.json +++ b/2020/ldm-complex-numbers/ukrainian/sentence_translations.json @@ -273,21 +273,21 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right?", + "input": "stalling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the", "translatedText": "Я б сказав, що якщо це щось справді корисне в програмі, то це так само реально, як і слова, чи не так?", "n_reviews": 0, "start": 247.68, "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different.", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality.", "translatedText": "Ви ніколи не зустрінете таке абстрактне слово, як щастя, але воно має певну реальність у нашій свідомості, і такі речі, як квадратний корінь з двох, який ви не можете виразити у вигляді дробу, або такі речі, як квадратний корінь з від’ємного, які не відображаються серед справжніх звичайних чисел, знаєте, навіть якщо вони можуть здаватися дещо іншими.", "n_reviews": 0, "start": 253.74, "end": 270.78 }, { - "input": "Oh, this is such a shame.", + "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay?", "translatedText": "О, це така ганьба.", "n_reviews": 0, "start": 271.5, @@ -343,14 +343,14 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out.", + "input": "esting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just trigonometry, it's everything we were talking about la", "translatedText": "Однак, якщо ви прийдете до цього з комплексними числами, це не тільки набагато менше схильне до помилок, воно має дуже красиве значення, і воно просто випадає.", "n_reviews": 0, "start": 363.98, "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too.", + "input": "st time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of", "translatedText": "Отже, навіть якщо ви не обов’язково вірите в реальність квадратного кореня з мінус 1, ви принаймні повинні визнати, що це цікаво, що це може зробити інші частини математики корисними, що інші частини математики трохи більше теж зрозуміло.", "n_reviews": 0, "start": 372.1, @@ -392,7 +392,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right?", + "input": "It's something that's very error-prone if you're just trying", "translatedText": "Одне, ні, немає, правда?", "n_reviews": 0, "start": 414.88, @@ -406,7 +406,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive.", + "input": "ex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out", "translatedText": "Будь-яке число, яке ви підносите до квадрату, якщо воно додатне, добре, воно залишається додатним.", "n_reviews": 0, "start": 427.76, @@ -420,21 +420,21 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative.", + "input": "So even if you don't necessarily believe in the reality of", "translatedText": "Я ніколи не отримаю нічого негативного.", "n_reviews": 0, "start": 436.38, "end": 437.86 }, { - "input": "So this does not exist, no such number.", + "input": "the square root of negative one, you at the very least have to admit that it's interesting that it can make o", "translatedText": "Отже, цього не існує, такого номера немає.", "n_reviews": 0, "start": 438.48, "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case.", + "input": "ther pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case.", "translatedText": "Однак, якщо приходить математик і каже: «Ні, ні, це існує», ми визначили це так, щоб це було так.", "n_reviews": 0, "start": 442.28, @@ -448,7 +448,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution.", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wo", "translatedText": "Коли у вас є проблема, яку ви не можете вирішити, ви можете просто сказати: о, я визначив речі, щоб ми тепер чарівним чином отримали рішення.", "n_reviews": 0, "start": 452.86, @@ -462,14 +462,14 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone.", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home.", "translatedText": "Тож якщо вам це незручно, ви точно не самотні.", "n_reviews": 0, "start": 466.48, "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory.", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative,", "translatedText": "Насправді Рене Декарт ввів термін уявне для цих чисел як принизливий.", "n_reviews": 0, "start": 470.06, @@ -525,7 +525,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right?", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you", "translatedText": "І, гаразд, якщо ми хочемо розширити нашу систему числення, я розумію, можливо, буде корисно розмістити там якесь число, але чому я, чи не так?", "n_reviews": 0, "start": 526.5, @@ -553,7 +553,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors.", + "input": "have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question.", "translatedText": "На самому початку давайте просто поговоримо про те, що якщо ви додаєте двовимірні числа, як це, правила досить прості, і вони діють, по суті, так само, як вектори, для будь-кого з вас, хто може бути знайомий з векторами.", "n_reviews": 0, "start": 557.3, @@ -630,14 +630,14 @@ "end": 662.3 }, { - "input": "None of them is much lower at a.", + "input": "one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to", "translatedText": "Жоден з них не є набагато нижчим за a.", "n_reviews": 0, "start": 662.7, "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real.", + "input": ", why should that live there? What on earth does the idea of a point one unit above the real number line in a separate dimension have to do with squaring to negative one? So I hope to answer this for you. At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be famili", "translatedText": "Гаразд, схоже, у нас є когорта людей, яких влаштовує мінус 1, велика когорта невлаштовує нескінченність, це тема на інший день, не хвилюйтеся про це, а потім ще кілька людей, які начебто в тому середньому становищі, можливо, не дуже комфортно з ідеєю, що мінус 1 може бути реальним.", "n_reviews": 0, "start": 665.1, @@ -651,7 +651,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two.", + "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It turns out to be r", "translatedText": "Тож для нашого першого набагато більш математичного запитання, як свого роду розминка, я просто хочу попросити вас додати ці два.", "n_reviews": 0, "start": 683.42, @@ -672,35 +672,35 @@ "end": 706.82 }, { - "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me.", + "input": "ative two plus two i. So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers extend in this direction. can get you something lik", "translatedText": "На жаль, і ви можете сказати з того факту, що я зволікаю, і те, що я тут говорю, схоже, що запитання все ще не завантажується повністю правильно, тому я збираюся суворо поговорити з Кемом і Айдером за сцени, які іншим чином створили такий прекрасний, гарний інтерфейс, який корисний для такого типу туди-сюди між вами і мною.", "n_reviews": 0, "start": 707.84, "end": 726.28 }, { - "input": "I'm going to have a stern word with them behind the scenes, but in the meantime let's go ahead and move forward with the lesson here.", + "input": "e it. But the rules end up being very different from that in the number system. You can't really do algebra. You can't do things like assume that if two numbers multiply to make zero, then", "translatedText": "Я збираюся поговорити з ними суворо за лаштунками, а тим часом давайте продовжимо урок.", "n_reviews": 0, "start": 726.76, "end": 732.64 }, { - "input": "So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be.", + "input": "one of them h as to be zero. But complex numbers are going to end up behaving much like the real numbers, s Now assuming that our question system ha", "translatedText": "Тож я думаю, що я можу витягнути це на, просто на аркуші паперу, і ви можете слідкувати за цим удома, щоб побачити, що може бути доповненням.", "n_reviews": 0, "start": 733.46, "end": 739.92 }, { - "input": "It turns out to be relatively straightforward.", + "input": "s not broken down, I should be able to do this as a proper poll and let me go ahead, I guess we can first check the previous poll, okay things se", "translatedText": "Виявляється, це відносно просто.", "n_reviews": 0, "start": 740.24, "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time.", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of th", "translatedText": "Якщо ви рухаєтесь на чотири одиниці праворуч, а потім на одну одиницю вгору, і ви хочете додати ідею переміщення двох одиниць ліворуч, а потім двох одиниць угору, ви просто виконуєте кожну з них по черзі.", "n_reviews": 0, "start": 742.94, @@ -714,7 +714,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left.", + "input": "red real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a g", "translatedText": "Дійсна частина складатиметься з чотирьох праворуч, а потім мінус двох зліва.", "n_reviews": 0, "start": 754.82, @@ -728,7 +728,7 @@ "end": 760.88 }, { - "input": "And then the imaginary part is going to be this one unit up and then these two units up, one plus two, times i.", + "input": "you out there who would just reject infinity as being considered real but are very comfortable with the square root of negative o", "translatedText": "І тоді уявна частина буде цією одиницею вгору, а потім цими двома одиницями вгору, один плюс два, помножити на i.", "n_reviews": 0, "start": 761.16, @@ -756,7 +756,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great.", + "input": "are root of negative one, fascinating, I actually would have thought that none of them would have come higher than t", "translatedText": "Додавання насправді не містить нічого складного, що чудово.", "n_reviews": 0, "start": 777.86, @@ -770,14 +770,14 @@ "end": 784.2 }, { - "input": "What is so complex about complex numbers after all?", + "input": "m is much lower at a, okay so it looks like we've got a cohort of people who are comfortable with negative one, a la", "translatedText": "Зрештою, що такого складного в комплексних числах?", "n_reviews": 0, "start": 784.42, "end": 787.1 }, { - "input": "Well where everything becomes interesting is when you try to multiply these numbers together.", + "input": "rge cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people w", "translatedText": "Все стає цікавим, коли ви намагаєтеся помножити ці числа.", "n_reviews": 0, "start": 787.64, @@ -798,14 +798,14 @@ "end": 803.68 }, { - "input": "But the rules end up being very different from that in the number system.", + "input": "mfortable with the idea that negative one might be real, let's see if we can convince you of the difference of t", "translatedText": "Але правила в кінцевому підсумку сильно відрізняються від правил у системі числення.", "n_reviews": 0, "start": 803.68, "end": 806.86 }, { - "input": "You can't really do algebra.", + "input": "hat. So what we've done here is we've taken three, two and then", "translatedText": "Ви не можете займатися алгеброю.", "n_reviews": 0, "start": 806.86, @@ -826,7 +826,7 @@ "end": 817.78 }, { - "input": "But to understand what that multiplication rule is, I just want to ask you a simple question.", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, a", "translatedText": "Але щоб зрозуміти, що таке правило множення, я просто хочу поставити вам просте запитання.", "n_reviews": 0, "start": 818.3, @@ -840,14 +840,14 @@ "end": 831.82 }, { - "input": "We're not even going to think of it as a complex number per se.", + "input": "part of this, but it is, it's a good thing to know when you're learning about complex numbers, it'", "translatedText": "Ми навіть не будемо розглядати це як комплексне число як таке.", "n_reviews": 0, "start": 832.58, "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this?", + "input": "s definitely one of those operations that you are going to need to know. Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks like the question is stil", "translatedText": "Якщо у мене просто є якась координатна сітка, і я йду до точки з координатою x три та координатою y два, чому дорівнює поворот на 90 градусів?", "n_reviews": 0, "start": 835.72, @@ -875,14 +875,14 @@ "end": 859.44 }, { - "input": "Okay.", + "input": "built such a beautiful, beautiful inte", "translatedText": "Гаразд.", "n_reviews": 0, "start": 865.28, "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out.", + "input": "rface that's helpful for this kind of back and forth between you guys and me. nice gut check here is", "translatedText": "Що чудово в цьому, ми можемо просто перевернути папір, щоб зрозуміти це.", "n_reviews": 0, "start": 865.76, @@ -896,28 +896,28 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three.", + "input": "e. So that was another 90 degree rotation. Well what's happened here is we've just made both of the coordinates negative and that's", "translatedText": "Отже, ми взяли три, два, а потім перетворили це на мінус два, три.", "n_reviews": 0, "start": 884.08, "end": 890.68 }, { - "input": "Something which maybe in our original system you know looks like this negative two and then three.", + "input": "reassuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them", "translatedText": "Щось, що, можливо, у нашій оригінальній системі, як ви знаєте, виглядає як мінус два, а потім три.", "n_reviews": 0, "start": 891.58, "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation.", + "input": "negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation", "translatedText": "Це буде поворот на 90 градусів.", "n_reviews": 0, "start": 898.1, "end": 899.9 }, { - "input": "And what's nice here is that that rule is very simple and it applies to any pair that we might have.", + "input": "actually behaves like you would expect it to. Now why am I asking you this? Well I'm being told that supposedly I'm allowed to ask you questions again so I 'm going to ha", "translatedText": "І що приємно тут, так це те, що це правило дуже просте і воно застосовується до будь-якої пари, яка у нас може бути.", "n_reviews": 0, "start": 899.9, @@ -945,7 +945,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong.", + "input": "52 of you answered simply 2 which would have been the real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated goin", "translatedText": "Ось що сталося: ми щойно зробили обидві координати від’ємними, і це заспокоює, тому що якщо я візьму якусь точку, сидячи на ab, і потім поверну її на 90 градусів, то це буде мій початковий поворот на 90 градусів, а потім ще на 90 градусів, це те саме, що поворот на 180 градусів – о ні, я зробив це неправильно.", "n_reviews": 0, "start": 938.48, @@ -959,14 +959,14 @@ "end": 980.14 }, { - "input": "Now why am I asking you this?", + "input": "you try to multiply these numbers together. So with vectors, there's not really any notion", "translatedText": "Тепер чому я вас про це питаю?", "n_reviews": 0, "start": 980.4, "end": 981.76 }, { - "input": "Well I'm being told that supposedly I'm allowed to ask you questions again so I'm going to have you do your very first complex product.", + "input": "of multiplying them to get two vectors back, at least when we're in the 2d plane. that we ask questions and just say hey kamineter can't you make the live questio", "translatedText": "Ну, мені сказали, що нібито мені дозволено ставити вам запитання ще раз, тож я попрошу вас зробити ваш перший складний продукт.", "n_reviews": 0, "start": 982.66, @@ -987,14 +987,14 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i.", + "input": "Wonderful! Very simple question I want you to take the number i and I want you to multiply it by 3 pl", "translatedText": "Гаразд, схоже, що більшість із вас отримали правильну відповідь, тобто 2 плюс 3i.", "n_reviews": 0, "start": 1003.06, "end": 1006.98 }, { - "input": "Very good very good.", + "input": "us 2i and even though I haven't really talked about You can't do things like assume that if two numbers multiply to mak", "translatedText": "Дуже добре дуже добре.", "n_reviews": 0, "start": 1007.48, @@ -1015,28 +1015,28 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good.", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t", "translatedText": "У нас є 2 плюс 3, що, можливо, просто відкидає i, тому я думаю, що, можливо, багато схоже на прості помилки та введення, і ви знаєте, що трапляється з усіма нами, особливо на тестах, іноді ви знаєте, яка правильна відповідь, але потім ви забули символ або поміняли два, так що все дуже добре.", "n_reviews": 0, "start": 1037.6, "end": 1052.36 }, { - "input": "Let's go ahead and try our very first product though like I said so here because I already talked through one of the questions we're going to go ahead and skip ahead of it we know how to rotate something like 3 comma 2 so I'm not even going to give you time to do that and properly grade it.", + "input": "his is we can basically just turn our paper to figure it out. ons as you do it rather than sitting in passively watching this is genuinely delightful to me. Okay this is this isn't necessarily a question I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not", "translatedText": "Давайте спробуємо наш найперший продукт, хоча, як я вже сказав, тому що я вже говорив про одне із запитань, яке ми збираємося йти вперед і пропустити його, ми знаємо, як обертати щось на кшталт 3 кому 2, тому я навіть не збираюся дати вам час, щоб зробити це та правильно оцінити це.", "n_reviews": 0, "start": 1052.88, "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us?", + "input": "that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three i which is absolutely correct absolutely correct so there's two w", "translatedText": "Stal stal words слова, які ви знаєте, вони кажуть мені, що це працює, але я дуже повільно просуваюсь вперед, тож ви знаєте, якщо я не збираюся говорити з ними суворо, ви, хлопці, також можете накинутись на них у Twitter під тим самим місце, де ми ставимо запитання та просто кажемо: привіт, камінтер, чи не можете ви зробити, щоб запитання в прямому ефірі працювали для нас трохи краще?", "n_reviews": 0, "start": 1070.62, "end": 1089.28 }, { - "input": "Okay I think we're finally there.", + "input": "ays to think about this okay one of them is", "translatedText": "Гаразд, я думаю, ми нарешті там.", "n_reviews": 0, "start": 1090.02, @@ -1050,14 +1050,14 @@ "end": 1094.22 }, { - "input": "Aha!", + "input": "e algebra and just do it a little bit mechanistically okay so", "translatedText": "Ага!", "n_reviews": 0, "start": 1094.6, "end": 1094.76 }, { - "input": "Wonderful!", + "input": "if we pull ourselves up", "translatedText": "Чудово!", "n_reviews": 0, "start": 1094.76, @@ -1071,7 +1071,7 @@ "end": 1126.42 }, { - "input": "Wonderful!", + "input": "that if you want to rotate numbers 90 degrees", "translatedText": "Чудово!", "n_reviews": 0, "start": 1127.02, diff --git a/2020/ldm-complex-numbers/urdu/sentence_translations.json b/2020/ldm-complex-numbers/urdu/sentence_translations.json index 6f48774b3..98fb9bb8a 100644 --- a/2020/ldm-complex-numbers/urdu/sentence_translations.json +++ b/2020/ldm-complex-numbers/urdu/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "میں کہوں گا کہ اگر یہ کوئی ایسی چیز ہے جو حقیقت میں کسی ایپلی کیشن میں کارآمد ہے، تو یہ اتنا ہی حقیقی ہے جتنا کہ الفاظ ہیں، ٹھیک ہے؟ آپ کبھی بھی کسی تجریدی لفظ میں نہیں جائیں گے جیسے خوشی، لیکن یہ ہمارے ذہنوں میں ایک طرح کی حقیقت ہے، اور دو کی مربع جڑ جیسی چیزیں، جنہیں آپ کسی جز کے طور پر بیان نہیں کر سکتے، یا ایسی چیزیں منفی کا مربع جڑ جو حقیقی عام اعداد میں ظاہر نہیں ہوتا ہے، آپ جانتے ہیں، چاہے وہ تھوڑا سا مختلف معلوم ہوں۔", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "مجھے یاد ہے جب میں اسکول میں تھا اور ہم نے یہ اضافی فارمولے سیکھے تھے، کہ اگر آپ دو مختلف زاویوں کے مجموعہ کا کوسائن جاننا چاہتے ہیں، تو آپ جانتے ہیں، اصل دو زاویوں کے cosines اور sine کے لحاظ سے یہ اس قسم کی لمبی چیز ہے۔یہ مائنس نشان ہے جو لوگوں کو ہمیشہ ٹرپ کرتا ہے، اگر آپ نشان کے لیے ایسا کرتے ہیں تو یہ ایک جیسا لگتا ہے لیکن اس میں ایک جمع کا نشان ہے، اور cos-cos ہونے کے بجائے آپ کے پاس cos-sin ہے، یہ ایسی چیز ہے جو بہت زیادہ غلطی کا شکار ہے۔اگر آپ اسے ویسے ہی حفظ کرنے کی کوشش کر رہے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "تاہم، اگر آپ اس پر پیچیدہ نمبروں کے ساتھ آتے ہیں، تو یہ نہ صرف بہت کم غلطی کا شکار ہے، بلکہ اس کا ایک بہت ہی خوبصورت معنی ہے اور یہ بالکل ٹھیک ہو جاتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "لہذا یہاں تک کہ اگر آپ لازمی طور پر منفی 1 کے مربع جڑ کی حقیقت پر یقین نہیں رکھتے ہیں، آپ کو کم از کم یہ تسلیم کرنا ہوگا کہ یہ دلچسپ ہے کہ یہ ریاضی کے دوسرے ٹکڑوں کو کارآمد بنا سکتا ہے، ریاضی کے دوسرے ٹکڑوں کو تھوڑا سا زیادہ۔بھی قابل فہم. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "کوئی بھی عدد جس کا آپ مربع کرتے ہیں، اگر یہ مثبت ہے، تو وہ صرف مثبت ہی رہتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "مجھے کبھی بھی کوئی منفی چیز نہیں ملے گی۔", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "تاہم، اگر کوئی ریاضی دان آتا ہے اور کہتا ہے، اوہ نہیں، یہ موجود نہیں ہے، تو ہم نے اس کی وضاحت کی ہے کہ ایسا ہی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "میرے خیال میں کوئی دوسرا ردعمل جو کسی کا ہو سکتا ہے وہ یہ ہے کہ ایک سیکنڈ کے لیے رکیں، آپ ایسا کر سکتے ہیں؟ جب آپ کے پاس کوئی مسئلہ ہے جسے آپ حل نہیں کر سکتے ہیں، تو آپ صرف یہ کہہ سکتے ہیں، اوہ میں نے چیزوں کی وضاحت کی ہے تاکہ اب ہمارے پاس جادوئی طور پر ایک حل ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "لہذا اگر آپ اس سے بے چین ہیں، تو آپ یقینی طور پر اکیلے نہیں ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "درحقیقت، رینے ڈیکارٹس نے ان نمبروں کے لیے تصوراتی اصطلاح کو توہین آمیز قرار دیا۔", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "اور پھر ہم ایک کنونشن کے طور پر اس کے ساتھ پھنس گئے اور ہم انہیں اب بھی خیالی نمبر کہتے ہیں، جو کہ حقیقت میں مضحکہ خیز ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "دوسری عجیب بات جو آپ کرتے ہیں جب آپ کمپلیکس نمبرز کے بارے میں بات کرنا شروع کرتے ہیں تو یہ کہنا ہے کہ صرف اتنا نمبر i نہیں ہے، بلکہ ہم اسے ایک گھر دینے جا رہے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "بالکل شروع میں، آئیے صرف اس بارے میں بات کرتے ہیں کہ اگر آپ ایسے نمبرز جوڑ رہے ہیں جو اس طرح دو جہتی ہیں، تو اصول کافی سیدھے ہیں اور یہ بنیادی طور پر ویکٹر کی طرح کام کرتا ہے، آپ میں سے کسی کے لیے بھی جو ویکٹر سے واقف ہو سکتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "اگر آپ اسے ایمان کی بنیاد پر لیتے ہیں اور آپ اس کی پیروی کرتے ہیں تو امید ہے کہ یہ کارآمد ثابت ہونے سے یہ ثابت کرنے میں مدد ملے گی کہ ہم اس میں سے کچھ کیوں کر رہے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "ام، ایسا لگتا ہے کہ جوابات f اور d کے درمیان a ہے، آگے پیچھے ہے، لہذا f ان سب کا ہے، یہ کہتے ہوئے کہ ان سب کو اصلی سمجھا جانا چاہئے۔", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "اور دلچسپ بات یہ ہے کہ d وہ ہے جو کہتا ہے کہ آپ کو 2 کی 2 مربع جڑ اور منفی 1 پر غور کرنا چاہیے، لیکن لامحدودیت پر نہیں، اس لیے آپ کا ایک اچھا دستہ وہاں موجود ہے جو صرف لامحدودیت کو حقیقی سمجھ کر مسترد کر دے گا، لیکن اس کے ساتھ بہت آرام دہ ہے۔منفی 1 کا مربع جڑ، یہ بہت اچھا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "ٹھیک ہے، ایسا لگتا ہے کہ ہمارے پاس ایسے لوگوں کا ایک گروپ ہے جو منفی 1 کے ساتھ آرام دہ ہیں، ایک بڑی جماعت لامحدودیت کے ساتھ بے چین ہے، یہ ایک دوسرے دن کا موضوع ہے، اس کے بارے میں فکر نہ کریں، اور پھر بہت سے لوگ جو اس طرح کے درمیانی میدان میں ہیں شاید اس خیال کے ساتھ انتہائی آرام دہ نہ ہوں کہ منفی 1 حقیقی ہو سکتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "تو ہمارے پہلے بہت زیادہ ریاضی کے سوال کے لیے، ایک وارم اپ کی طرح، میں آپ سے صرف یہ کہنا چاہتا ہوں کہ ان دونوں کو شامل کریں۔", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "اس سے پہلے کہ میں آپ کو ان کو شامل کرنے کا طریقہ سکھا دوں، اندازہ لگائیں کہ یہ کیسے کام کر سکتا ہے، اور مجھے امید ہے کہ یہ کافی سیدھا محسوس ہوتا ہے، اضافہ دراصل اس کا سب سے کم دلچسپ حصہ ہے، لیکن یہ جاننا اچھی بات ہے کہ کب آپ پیچیدہ نمبروں کے بارے میں سیکھ رہے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "اگر آپ چار یونٹوں کو دائیں اور پھر ایک یونٹ اوپر لے جا رہے ہیں، اور آپ دو یونٹوں کو بائیں طرف اور پھر دو یونٹوں کو اوپر منتقل کرنے کا خیال شامل کرنا چاہتے ہیں، تو ٹھیک ہے آپ ان میں سے ہر ایک کو ایک وقت میں کریں۔", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "اصل حصہ وہ چار دائیں طرف، پھر مائنس دو بائیں طرف ہوگا۔", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "تو کیا وہ ایک i جمع دو i۔", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "اضافے میں واقعی کچھ بھی پیچیدہ نہیں ہے، جو بہت اچھا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "لہذا ویکٹر کے ساتھ، دو ویکٹرز کو واپس حاصل کرنے کے لیے ان کو ضرب دینے کا واقعی کوئی تصور نہیں ہے، کم از کم جب ہم 2D جہاز میں ہوں۔", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "جو بنیادی طور پر ہے، فرض کریں کہ میرے پاس پوائنٹ تین، دو ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "اگر میرے پاس کسی قسم کا کوآرڈینیٹ گرڈ ہے اور میں x کوآرڈینیٹ تھری اور y کوآرڈینیٹ ٹو کے ساتھ پوائنٹ پر جاتا ہوں تو اس کا 90 ڈگری گردش کیا ہے؟ اگر میں اسے 90 ڈگری گھماتا ہوں اور آئیے کہتے ہیں کہ گھڑی کی سمت میں۔", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "گھڑی کے مخالف سمت میں۔", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "اب اس کے بارے میں کیا خوبصورت ہے کہ ہم بنیادی طور پر صرف اس کا پتہ لگانے کے لیے اپنے کاغذ کو تبدیل کر سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "آپ کہتے ہیں ٹھیک ہے اگر یہ تین، دو سے شروع ہوتا ہے اور پھر میں 90 ڈگری کو گھڑی کی مخالف سمت میں گھماتا ہوں، تو میں اسے ابھی پڑھ سکتا ہوں کہ منفی دو x سمت میں اور پھر تین y سمت میں، اگر میں نے پورے جہاز کو اس طرح گھمایا ہوتا۔. ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "تو ہم نے یہاں کیا کیا ہے ہم نے تین، دو لیے ہیں اور پھر ہم اسے منفی دو، تین میں تبدیل کرتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "یہ 90 ڈگری گردش ہونے والا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "اگر میں نے نمبروں کا ایک جوڑا لیا تو ایک کوما b ٹھیک ہے اور پھر میں نے کہا کہ وہ کہاں گھومے گا اگر میں اسے 90 ڈگری گھماؤں گا تو یہ کوآرڈینیٹس ba کو تبدیل کرکے ختم ہوجائے گا اور پھر اس کو پہلے منفی بنا دے گا۔", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "تو یہ ایک اور 90 ڈگری گردش تھی۔", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "ٹھیک ہے یہاں کیا ہوا ہے کہ ہم نے صرف دونوں نقاط کو منفی بنا دیا ہے اور یہ یقین دہانی ہے کیونکہ اگر میں ab پر بیٹھ کر کچھ نقطہ اٹھاتا ہوں اور پھر میں اسے 90 ڈگری گھماتا ہوں تو یہ میری ابتدائی 90 ڈگری گردش ہوگی اور پھر ایک اور 90 ڈگری جو کہ ہے اسی طرح 180 ڈگری روٹ- اوہ نہیں میں نے یہ غلط کیا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "یہ 180 ڈگری گردش کی طرح ہوگا جو اس طرح نظر آنا چاہئے دوسرے ویکٹر کو نظر انداز کریں جسے میں نے کھینچا ہے جو صرف دونوں کوآرڈینیٹ لے رہا ہے اور انہیں منفی منفی کو منفی کو منفی بنا رہا ہے ٹھیک ہے تاکہ یہ اس آپریشن کو یقینی بنائے جو 90 ڈگری گردش کرتا ہے۔درحقیقت ایسا برتاؤ کرتا ہے جیسے آپ اس کی توقع کریں گے۔", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "اوہ دیکھو بہت سارے لوگوں نے بہت اچھے جوابات جمع کروائے۔", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "ٹھیک ہے تو ایسا لگتا ہے کہ آپ میں سے اکثریت کو صحیح جواب ملا ہے جو کہ 2 جمع 3i ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "بہت اچھا بہت اچھا۔", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "آپ میں سے کچھ نے منفی 2 3 کا جواب دیا جو میرا اندازہ ہے کہ یہ صرف بدل رہا ہے چاہے آپ 4 مائنس 2 لے رہے ہیں یا 2 مائنس 4 لے رہے ہیں تاکہ یہ مکمل طور پر قابل فہم ہو۔", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "ہمارے پاس 2 جمع 3 ہے جو شاید صرف i کو چھوڑ رہا ہے لہذا میں سمجھتا ہوں کہ شاید بہت سی آسان غلطیاں اور اندراج اور آپ جانتے ہیں کہ یہ ہم سب کے ساتھ ہوتا ہے خاص طور پر ٹیسٹوں میں کبھی کبھی آپ جانتے ہیں کہ صحیح جواب کیا ہے لیکن پھر آپ ایک علامت بھول جاتے ہیں یا آپ دو کو تبدیل کرتے ہیں تو یہ سب بہت اچھا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "اسٹال اسٹال الفاظ جو آپ جانتے ہیں وہ مجھے بتاتے ہیں کہ یہ کام کر رہا ہے اور پھر بھی میرے لیے آگے بڑھنا بہت سست ہے لہذا آپ کو معلوم ہے کہ اگر میں ان کے ساتھ سخت الفاظ نہیں بولوں گا تو آپ لوگ ٹویٹر پر بھی اسی کے تحت ان پر جا سکتے ہیں۔وہ جگہ جہاں ہم سوال پوچھتے ہیں اور صرف کہتے ہیں ارے کامینےٹر کیا آپ لائیو سوالات کو ہمارے لیے تھوڑا بہتر نہیں بنا سکتے؟ ٹھیک ہے مجھے لگتا ہے کہ ہم آخر کار وہاں ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "سب تیار ہیں؟ آہا! ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "کمال ہے! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "بہت آسان سوال میں چاہتا ہوں کہ آپ نمبر i لیں اور میں چاہتا ہوں کہ آپ اسے 3 جمع 2i سے ضرب دیں اور اگرچہ میں نے ضرب کے اصولوں کے بارے میں حقیقت میں بات نہیں کی ہے جو میں کہہ سکتا ہوں کہ یہ دکھاوا ہے جیسے یہ کام کرتا ہے جیسا کہ یہ کرتا ہے۔نارمل نمبرز آپ کے پاس تقسیمی جائیداد جیسی چیزیں ہیں جہاں آپ اسے پوری طرح تقسیم کر سکتے ہیں اور پھر i کی وضاحتی خصوصیت یہ ہے کہ i اسکوائر منفی ہے جس کے بارے میں آپ کو صرف ایک خاص چیز جاننے کی ضرورت ہے اس کے علاوہ صرف اس کا علاج کریں۔جیسے کہ یہ ایک عام نمبر ہے ٹھیک ہے اور پھر پروڈکٹ کے ساتھ آگے بڑھیں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-complex-numbers/vietnamese/sentence_translations.json b/2020/ldm-complex-numbers/vietnamese/sentence_translations.json index ec34ad32a..878aaebae 100644 --- a/2020/ldm-complex-numbers/vietnamese/sentence_translations.json +++ b/2020/ldm-complex-numbers/vietnamese/sentence_translations.json @@ -272,7 +272,7 @@ "end": 247.6 }, { - "input": "I would say that if it's something that's actually useful in an application, then it is as real as words are, right? ", + "input": "talling out on this one. t has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the ", "translatedText": "Tôi có thể nói rằng nếu đó là thứ gì đó thực sự hữu ích trong một ứng dụng thì nó cũng chân thực như lời nói, phải không? ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 253.74 }, { - "input": "You're never going to run into an abstract word like happiness out there, but it has a kind of reality in our minds, and things like the square root of two, which you can't express as a fraction, or things like the square root of negative one that don't show up among real normal numbers, you know, even if they might seem a little bit different. ", + "input": "square root of negative one that don't show u But for me personally, basically, anytime that you have a numerical construct that's helpful in the real world, you know, I consider that real. What I'd like to do for you today, basically, is show you the sense in which imaginary numbers are useful, the complex numbers are useful, an d from there maybe try to imbue them with a little more reality. ", "translatedText": "Bạn sẽ không bao giờ gặp phải một từ trừu tượng như hạnh phúc ngoài kia, nhưng nó có một loại thực tế trong tâm trí chúng ta, và những thứ như căn bậc hai của 2, mà bạn không thể diễn đạt dưới dạng phân số, hoặc những thứ như căn bậc hai của số âm không xuất hiện giữa các số thực bình thường, bạn biết đấy, ngay cả khi chúng có vẻ hơi khác một chút. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 300.96 }, { - "input": "I won't assume that you know what they are yet, it's meant to be a basic primer, but let's just dive right in, okay? ", + "input": "oh, this is such a shame. ing that we're going to build to, two identities from trigonometry, and I understand that maybe, oh, these complicated identities from trigonometry is not going ", "translatedText": "Tôi sẽ không cho rằng bạn biết chúng là gì, nó chỉ là lớp sơn lót cơ bản, nhưng chúng ta hãy đi sâu vào ngay, được chứ? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 339.52 }, { - "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles, there's this minus sign that would always trip people up, if you do the same for the sign, it looks similar but there's a plus sign, and instead of having cos-cos you have cos-sin, it's something that's very error-prone if you're just trying to memorize it as it is. ", + "input": "I remember when I was in school and we learned these addition formulas, that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in te And I understand that maybe, oh, these complicated identities from trigonometry is not going to be the best way to lure some people into understanding, oh yeah, complex numbers, they're really useful, you're really going to love them. But I do think it's interesting that you can have a fact that has nothing to do with complex numbers or the square root of negative one, it's just tr ", "translatedText": "Tôi nhớ khi còn đi học và chúng ta đã học các công thức cộng này, nếu bạn muốn biết cosin của tổng hai góc khác nhau, bạn biết đấy, nó dài như thế này xét theo cos và sin của hai góc ban đầu , có một dấu trừ luôn khiến mọi người bối rối, nếu bạn làm tương tự với dấu đó, nó trông giống nhau nhưng có một dấu cộng, và thay vì có cos-cos bạn có cos-sin, đó là một thứ rất dễ xảy ra lỗi nếu bạn chỉ đang cố gắng ghi nhớ nó như nó vốn có. ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 363.62 }, { - "input": "However, if you come at it with complex numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", + "input": "igonometry, it's everything we were talking about last time. And you can have facts that are pretty hard to remember. I remember when I was in school and we learned these addition formulas ", "translatedText": "Tuy nhiên, nếu bạn tiếp cận nó với số phức, điều này không chỉ ít xảy ra lỗi hơn mà còn có một ý nghĩa rất đẹp và nó dễ hiểu. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 371.6 }, { - "input": "So even if you don't necessarily believe in the reality of the square root of negative 1, you at the very least have to admit that it's interesting that it can make other pieces of math useful, that other pieces of math a little bit more understandable too. ", + "input": ", that if you want to know the cosine of the sum of two different angles, you know, it's this kind of long thing in terms of cosines and sines of the original two angles. y who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "Vì vậy, ngay cả khi bạn không nhất thiết phải tin vào thực tế của căn bậc hai của âm 1, thì ít nhất bạn cũng phải thừa nhận rằng thật thú vị khi nó có thể làm cho các phần toán khác trở nên hữu ích, các phần toán khác đó hữu ích hơn một chút. cũng dễ hiểu thôi. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 397.38 }, { - "input": "But the starting point looks very strange, okay? ", + "input": "hat i squared is equal to negative 1. And I think to a lot of students there's maybe one of ", "translatedText": "Nhưng điểm xuất phát trông rất lạ phải không? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 414.88 }, { - "input": "One is, no there isn't, right? ", + "input": "s something that's very error-prone if you're just trying t ", "translatedText": "Một là, không có không, phải không? ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 426.72 }, { - "input": "Any number that you square, if it's positive, well that just stays positive. ", + "input": "numbers, this is not only much less error-prone, it has a very beautiful meaning and it just falls right out. ", "translatedText": "Bất kỳ số nào bạn bình phương, nếu nó dương, thì nó vẫn dương. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 436.3 }, { - "input": "I'm never going to get anything negative. ", + "input": "u at the very least have to admit that it's interesting th ", "translatedText": "Tôi sẽ không bao giờ nhận được bất cứ điều gì tiêu cực. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 440.94 }, { - "input": "However, if a mathematician comes and says, oh no no it exists, we've defined it so that that's the case. ", + "input": "her pieces of math useful, that other pieces of math a little bit more understandable too. d says, oh no no it exists, we've defined it so that that's the case. ", "translatedText": "Tuy nhiên, nếu một nhà toán học đến và nói, ồ không, nó tồn tại, chúng ta đã định nghĩa nó là như vậy. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 452.38 }, { - "input": "When you have a problem that you can't solve, you can just say, oh I've defined things so that we now magically have a solution. ", + "input": "If you talk to anybody who's in engineering, anybody who's going into serious math, they'll tell you that complex numbers are as real a part of their wor ", "translatedText": "Khi bạn gặp một vấn đề không thể giải quyết, bạn chỉ cần nói, ồ, tôi đã xác định mọi thứ để bây giờ chúng ta có giải pháp một cách kỳ diệu. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 465.6 }, { - "input": "So if you're uncomfortable with this, you're definitely not alone. ", + "input": "talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", "translatedText": "Vì vậy, nếu bạn không thoải mái với điều này, bạn chắc chắn không đơn độc. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 469.46 }, { - "input": "In fact, Rene Descartes coined the term imaginary for these numbers as a derogatory. ", + "input": "Instead of the real number line, which you know all of these numbers we know when we square them, you can't get a negative, ", "translatedText": "Trên thực tế, Rene Descartes đã đặt ra thuật ngữ tưởng tượng cho những con số này như một sự xúc phạm. ", "model": "google_nmt", "n_reviews": 0, @@ -512,7 +512,7 @@ "end": 480.7 }, { - "input": "And then we stuck with that as a convention and we still call them imaginary numbers, which is genuinely absurd. ", + "input": "and then there's one below, negative i, and you can have negati ve 2i, you scale it however you want. Essentially it's prop ", "translatedText": "Và sau đó chúng tôi sử dụng quy ước đó và vẫn gọi chúng là số ảo, điều này thực sự vô lý. ", "model": "google_nmt", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 488.5 }, { - "input": "The second weird thing that you do when you start talking about complex numbers is to say, there's not just such a number i, but we're going to give it a home. ", + "input": "be two-dimensional and that i has a very specific home, one unit perpendicular, uh, perpendicularly above the real number line. Any time I square a number, even if it's negative, if I tak ", "translatedText": "Điều kỳ lạ thứ hai mà bạn làm khi bắt đầu nói về số phức là nói rằng, không chỉ có số i như vậy, mà chúng ta sẽ đặt cho nó một ngôi nhà. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 525.64 }, { - "input": "And, okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? ", + "input": "ever going to get anything negative. and then you move in that perpendicular direction into the extension of our number s ystem, which again, you' ", "translatedText": "Và, được thôi, nếu chúng ta muốn mở rộng hệ thống số của mình, tôi hiểu rồi, có lẽ sẽ hữu ích nếu đặt một loại số nào đó lên đó, nhưng tại sao lại như vậy, phải không? ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 557.28 }, { - "input": "At the very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors, for any of you who might be familiar with vectors. ", + "input": "can do that? When you have a problem that you can't solve you can just say, oh I've defined things so that we now magically have a solution. Okay, next time I'm having trouble with my homework and I don't know what the answer to x is, I will be like, let x be the value defined to be the answer to this question. ", "translatedText": "Lúc đầu, chúng ta hãy nói về việc nếu bạn cộng các số hai chiều như thế này, thì các quy tắc khá đơn giản và về cơ bản nó hoạt động giống như vectơ, đối với bất kỳ ai trong số các bạn có thể quen thuộc với vectơ. ", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 605.7 }, { - "input": "If you take that on faith and you follow, hopefully the fact that it becomes useful helps to justify why we're doing any of this. ", + "input": "is the one that says you The second weird thing that you do when you start talking about complex numbers is to say, there's not just a number i, but we're going to give it ", "translatedText": "Nếu bạn tin vào điều đó và làm theo, hy vọng thực tế là nó sẽ trở nên hữu ích sẽ giúp giải thích lý do tại sao chúng tôi lại làm bất kỳ điều gì trong số này. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 631.84 }, { - "input": "Um, it looks like there's a, there's a back and forth between answers f and d, so f is all of them, saying that all of these should be considered real. ", + "input": "'s one below, negative i, and you can have negative 2i. You scale it however you want. Essentially it's proposing that numbers be two-dimensional and that i has a very specific home, one unit perpendicular perpendicularly above the real number line. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -664,7 +664,7 @@ "end": 640.98 }, { - "input": "And interesting, d is the one that says you should consider 2 square root of 2 and negative 1, but not infinity, so there's a good contingent of you out there who would just reject infinity as being considered real, but are very comfortable with the square root of negative 1, that's awesome. ", + "input": "And okay, if we want to extend our number system, I get it, maybe it's useful to put some kind of number up there, but why i, right? Why not say infinity is the number that sits one unit above zero, or one divided by zero, or any other problem that you couldn't solve before and you make up an answer to, why should that live there? ", "translatedText": "Và thật thú vị, d là mệnh đề nói rằng bạn nên xét 2 căn bậc hai của 2 và âm 1, nhưng không phải là vô cùng, vậy nên có một số lượng lớn các bạn ngoài kia sẽ bác bỏ việc coi vô cực là thực, nhưng lại rất thoải mái với căn bậc hai của âm 1, thật tuyệt vời. ", "model": "google_nmt", "n_reviews": 0, @@ -688,7 +688,7 @@ "end": 664.58 }, { - "input": "Okay, so it looks like we've got a cohort of people who are comfortable with negative 1, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle ground of maybe not being super comfortable with the idea that negative 1 might be real. ", + "input": "e very beginning, let's just talk about how if you're adding numbers that are two-dimensional like this, the rules are pretty straightforward and it operates essentially the same as vectors for any of you who might be familiar with vectors. So I guess I can pull it up on the, just on the piece of paper, and you can follow along at home, see what the addition might be. It tu ", "translatedText": "Được rồi, có vẻ như chúng ta có một nhóm người cảm thấy thoải mái với số âm 1, một nhóm lớn không thoải mái với số vô cực, đó là chủ đề cho ngày khác, đừng lo lắng về điều đó, và sau đó là một số người có lẽ họ đang ở giữa mức độ cảm thấy không thoải mái cho lắm với ý tưởng rằng âm 1 có thể là thật. ", "model": "google_nmt", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 682.7 }, { - "input": "So for our first much more mathematical question, as kind of a warm-up, I just want to ask you to add these two. ", + "input": "e moving four units to the And then I'm going to take a second number and it's helpful to draw them as vectors, kind of an arrow from the number zero, and this one is going to end up at negative two plus two i. ", "translatedText": "Vì vậy, đối với câu hỏi đầu tiên mang tính toán học hơn nhiều, như một phần khởi động, tôi chỉ muốn yêu cầu bạn cộng hai câu hỏi này. ", "model": "google_nmt", "n_reviews": 0, @@ -712,7 +712,7 @@ "end": 689.3 }, { - "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of this, but it is, it's a good thing to know when you're learning about complex numbers. ", + "input": "So what I'm saying is you take the real number negative two and then you move in that perpendicular direction into the extension of our number system, which again you're kind of asking the students to take a lot on faith here that you're okay to do that, that you're allowed to just pretend that the numbers ", "translatedText": "Trước khi tôi hướng dẫn bạn cách thêm chúng, hãy đoán xem nó hoạt động như thế nào và tôi hy vọng bạn thấy nó khá đơn giản, phép cộng thực sự là phần kém thú vị nhất trong phần này, nhưng đó là điều tốt nếu bạn biết khi nào bạn đang học về số phức. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 742.1 }, { - "input": "If you're moving four units to the right and then one unit up, and you want to add the idea of moving two units to the left and then two units up, well you just do each of those one at a time. ", + "input": "em to be working so we can take a little step back in the lesson so I'm just genuinely curious, I want to know how you guys answered on this one. It looks like there's a there's a back and forth between answers f and d, so f is all of the ", "translatedText": "Nếu bạn đang di chuyển bốn đơn vị sang phải rồi lên một đơn vị và bạn muốn thêm ý tưởng di chuyển hai đơn vị sang trái rồi hai đơn vị lên trên, thì bạn chỉ cần thực hiện từng đơn vị đó một lần. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 754.24 }, { - "input": "The real part is going to be those four to the right, then minus two to the left. ", + "input": "hat all of these should be considered real, and interesting d is the one that says you should consider two square root of two and negative one but not infinity, so there's a ", "translatedText": "Phần thực sẽ là bốn phần ở bên phải, rồi trừ hai ở bên trái. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 768.54 }, { - "input": "So is that one i plus two i. ", + "input": "ould just reject infinity as being considered real but are very comfortable with the square root of negative one, that's awesom ", "translatedText": "Vậy đó là một tôi cộng hai i. ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 777.58 }, { - "input": "Addition doesn't really have anything complicated going on, which is great. ", + "input": "re root of negative one, fascinating, I actually would have thought that none of them would have come higher than tha ", "translatedText": "Việc bổ sung thực sự không có gì phức tạp đang diễn ra, điều này thật tuyệt. ", "model": "google_nmt", "n_reviews": 0, @@ -856,7 +856,7 @@ "end": 791.94 }, { - "input": "So with vectors, there's not really any notion of multiplying them to get two vectors back, at least when we're in the 2D plane. ", + "input": "le with negative one, a large cohort are uncomfortable with infinity, that's a topic for another day, don't worry about it, and then a number of people who are kind of in that middle gr ", "translatedText": "Vì vậy, với vectơ, thực sự không có khái niệm nào về việc nhân chúng để có được hai vectơ, ít nhất là khi chúng ta ở trong mặt phẳng 2D. ", "model": "google_nmt", "n_reviews": 0, @@ -912,7 +912,7 @@ "end": 826.9 }, { - "input": "Which is basically, suppose I have the point three, two. ", + "input": "which maybe in our original system you know looks like this negative two and then three. ", "translatedText": "Về cơ bản, giả sử tôi có điểm ba, hai. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 835.72 }, { - "input": "If I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? ", + "input": "Before I've taught you how to add them, make a guess at how it might work, and I hope that it feels pretty straightforward, addition is actually the least interesting part of thi ", "translatedText": "Nếu tôi chỉ có một loại lưới tọa độ nào đó và tôi đi đến điểm có tọa độ x ba và tọa độ y hai, góc quay 90 độ của cái này là bao nhiêu? ", "model": "google_nmt", "n_reviews": 0, @@ -952,7 +952,7 @@ "end": 855.44 }, { - "input": "Counterclockwise. ", + "input": "numbers, it's definitely one of those operations that you are going to need to know. ", "translatedText": "Ngược chiều kim đồng hồ. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 865.76 }, { - "input": "Now what's lovely about this is we can basically just turn our paper to figure it out. ", + "input": "Unfortunately, and you can tell by the fact that I'm stalling and what I'm saying here, it looks li ", "translatedText": "Điều thú vị ở đây là về cơ bản chúng ta có thể lật bài để tìm ra nó. ", "model": "google_nmt", "n_reviews": 0, @@ -976,7 +976,7 @@ "end": 870.32 }, { - "input": "You say okay if it started at three, two and then I rotate 90 degrees counterclockwise, I can just read that off now as being negative two in the x direction and then three in the y direction, if I had rotated the whole plane like that. ", + "input": "ke the question is still not loading completely correctly, so I'm going to have a stern word with Cam and Ider behind the scenes who have otherwise built such a beautiful, beautiful interface that's helpful for this kind of back and forth between you guys and me. nice gut check here is to say what happens when we do that t ", "translatedText": "Bạn nói được nếu nó bắt đầu ở vị trí ba, hai và sau đó tôi xoay 90 độ ngược chiều kim đồng hồ, bây giờ tôi có thể đọc nó là âm hai theo hướng x và sau đó là ba theo hướng y, nếu tôi đã xoay toàn bộ mặt phẳng như thế . ", "model": "google_nmt", "n_reviews": 0, @@ -984,7 +984,7 @@ "end": 883.5 }, { - "input": "So what we've done here is we've taken three, two and then we convert it to negative two, three. ", + "input": "wice what if we do that same very mechanistic operation again twice and I'm going to go and take this I swap the two coordinates we get a negative b but then t ", "translatedText": "Vì vậy, những gì chúng ta đã làm ở đây là chúng ta lấy 3, 2 và sau đó chúng ta chuyển nó thành âm 2, 3. ", "model": "google_nmt", "n_reviews": 0, @@ -1000,7 +1000,7 @@ "end": 898.08 }, { - "input": "That's going to be the 90 degree rotation. ", + "input": "first one becomes negative. So that was another 90 degree rotation. ", "translatedText": "Đó sẽ là vòng quay 90 độ. ", "model": "google_nmt", "n_reviews": 0, @@ -1016,7 +1016,7 @@ "end": 905.32 }, { - "input": "If I took a pair of numbers a comma b okay and then I said where is that going to rotate to if I rotate it 90 degrees it's going to end up by swapping the coordinates b a and then making that first one negative. ", + "input": "assuring because if I take some point sitting at a b an It turns out to be relatively straightforward. h is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation tha ", "translatedText": "Nếu tôi lấy một cặp số a dấu phẩy b thì được và sau đó tôi nói nó sẽ quay đến đâu nếu tôi xoay nó 90 độ thì nó sẽ kết thúc bằng cách hoán đổi tọa độ ba và sau đó biến số đầu tiên thành số âm. ", "model": "google_nmt", "n_reviews": 0, @@ -1032,7 +1032,7 @@ "end": 934.98 }, { - "input": "So that was another 90 degree rotation. ", + "input": "Oh look a lot of people did submit answers very good. ", "translatedText": "Vậy đó là một góc quay 90 độ nữa. ", "model": "google_nmt", "n_reviews": 0, @@ -1040,7 +1040,7 @@ "end": 937.64 }, { - "input": "Well what's happened here is we've just made both of the coordinates negative and that's reassuring because if I take some point sitting at a b and then I rotate it 90 degrees so this will be my initial 90 degree rotation and then another 90 degrees that's the same as 180 degree rot- oh no I've done that wrong. ", + "input": "Great let's let's grade the complex addition actually let's l et's see if it is as straightforward a process as I was hoping it was and see how much explanation is demanded. Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. Very good very good. 52 of you answered simply 2 which would have been th ", "translatedText": "Chà, điều xảy ra ở đây là chúng ta vừa làm cho cả hai tọa độ đều âm và điều đó thật yên tâm vì nếu tôi lấy một điểm nào đó ở điểm ab và sau đó tôi xoay nó 90 độ thì đây sẽ là góc quay 90 độ ban đầu của tôi và sau đó là 90 độ nữa đó là giống như quay 180 độ- ồ không, tôi đã làm sai rồi. ", "model": "google_nmt", "n_reviews": 0, @@ -1048,7 +1048,7 @@ "end": 958.38 }, { - "input": "That will be the same as a 180 degree rotation which should look like this ignore the other vector that I drew which is just taking both of the coordinates and making them negative negative a negative b okay so that's reassuring this operation that does a 90 degree rotation actually behaves like you would expect it to. ", + "input": "e real part of the answer so maybe just the fact that there's some vertical component and you need to still add those vertical components or maybe those of you who answer 2 reject the reality of imaginary numbers so you just don't even acknowledge that vertical component. Addition doesn't really have anything complicated going on, which is g ", "translatedText": "Nó sẽ giống như một phép quay 180 độ trông như thế này, bỏ qua vectơ khác mà tôi đã vẽ, chỉ lấy cả hai tọa độ và làm cho chúng âm âm a âm b được rồi, vậy nên điều đó yên tâm rằng thao tác này thực hiện xoay 90 độ thực sự hoạt động như bạn mong đợi. ", "model": "google_nmt", "n_reviews": 0, @@ -1072,7 +1072,7 @@ "end": 989.8 }, { - "input": "Oh look a lot of people did submit answers very good. ", + "input": "andable. We've got 2 plus 3 which is maybe just dropping Well where everything becomes interesting is when you try to multiply these numbers together. ", "translatedText": "Oh nhìn rất nhiều người đã gửi câu trả lời rất tốt. ", "model": "google_nmt", "n_reviews": 0, @@ -1088,7 +1088,7 @@ "end": 1002.22 }, { - "input": "Okay so it looks like a majority of you did get the correct answer which is 2 plus 3i. ", + "input": "that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? Okay I think we're finally there. ", "translatedText": "Được rồi, có vẻ như đa số các bạn đã trả lời đúng là 2 cộng 3i. ", "model": "google_nmt", "n_reviews": 0, @@ -1096,7 +1096,7 @@ "end": 1006.98 }, { - "input": "Very good very good. ", + "input": "Everybody ready? Aha! Wonderful! Very simple question ", "translatedText": "Rất tốt, rất tốt. ", "model": "google_nmt", "n_reviews": 0, @@ -1112,7 +1112,7 @@ "end": 1024.6 }, { - "input": "Some of you answered negative 2 3 which I guess is just making that's just swapping up whether you're taking 4 minus 2 or 2 minus 4 so that's completely understandable. ", + "input": "rules from algebra can carry over, but to understand what that rotation rule is, oh no I'm giving things away, what that multiplication rule is, I just want to ask you a simple question, which is basically suppose I have the point three two, okay ", "translatedText": "Một số bạn trả lời âm 2 3 mà tôi đoán là chỉ đang hoán đổi xem bạn đang lấy 4 trừ 2 hay 2 trừ 4 nên điều đó hoàn toàn dễ hiểu. ", "model": "google_nmt", "n_reviews": 0, @@ -1120,7 +1120,7 @@ "end": 1036.38 }, { - "input": "We've got 2 plus 3 which is maybe just dropping off the i so I think maybe a lot of like simple errors and entry and you know that happens to all of us especially on tests is sometimes you know what the right answer is but then you you forget a symbol or you swap two so that's all very good. ", + "input": ", we're not even going to think of it as a complex number per se, if I just have some sort of coordinate grid and I go to the point with x coordinate three and y coordinate two, what is the 90 degree rotation of this? If I rotate it 90 degrees and let's say counterclockwise, counter, counter, jeez, writing is difficult, counterclockwise. Okay, now what's lovely about t ", "translatedText": "Chúng ta có 2 cộng 3, có lẽ là bỏ đi chữ i nên tôi nghĩ có lẽ có nhiều lỗi đơn giản và mục nhập và bạn biết điều đó xảy ra với tất cả chúng ta, đặc biệt là trong các bài kiểm tra, đôi khi bạn biết câu trả lời đúng là gì nhưng sau đó bạn quên một biểu tượng hoặc bạn đổi hai biểu tượng, thế là ổn. ", "model": "google_nmt", "n_reviews": 0, @@ -1136,7 +1136,7 @@ "end": 1068.58 }, { - "input": "Stal stal words words you know they tell me that it's working and yet it's very slow for me to progress forward so you know if I'm not going to have a stern word with them you guys can go at them on twitter too under the same place that we ask questions and just say hey kamineter can't you make the live questions work a little bit better for us? ", + "input": "I was expecting to divide the audience necessarily so unsurprisingly it looks like we have a very strong majority in one direction hopefully in the correct direction but if not that would that would heavily inform where the lesson should So what we've done here is we've taken three two and then we convert it to negative two three, something which maybe in our original system, you know, looks like this, negative two and then three, ", "translatedText": "Những lời nói cứng rắn mà bạn biết chúng nói với tôi rằng nó đang hoạt động nhưng tôi tiến về phía trước rất chậm nên bạn biết nếu tôi không có lời nói nghiêm khắc với họ thì các bạn cũng có thể truy cập chúng trên twitter theo cách tương tự nơi mà chúng tôi đặt câu hỏi và chỉ nói này kamineter, bạn có thể làm cho các câu hỏi trực tiếp hoạt động tốt hơn một chút cho chúng tôi không? ", "model": "google_nmt", "n_reviews": 0, @@ -1144,7 +1144,7 @@ "end": 1089.28 }, { - "input": "Okay I think we're finally there. ", + "input": "that's going to be the 90 degree rotation. it looks like the majority of you answered negative two plus three ", "translatedText": "Được rồi, tôi nghĩ cuối cùng chúng ta cũng đến nơi. ", "model": "google_nmt", "n_reviews": 0, @@ -1152,7 +1152,7 @@ "end": 1092.16 }, { - "input": "Everybody ready? ", + "input": "i which is absolutely correct absolutely correct so ", "translatedText": "Mọi người sẵn sàng? ", "model": "google_nmt", "n_reviews": 0, @@ -1160,7 +1160,7 @@ "end": 1094.22 }, { - "input": "Aha! ", + "input": "there's two ways to think about this okay ", "translatedText": "A ha! ", "model": "google_nmt", "n_reviews": 0, @@ -1168,7 +1168,7 @@ "end": 1094.76 }, { - "input": "Wonderful! ", + "input": "one of them is to walk forward with the algebra and just d ", "translatedText": "Tuyệt vời! ", "model": "google_nmt", "n_reviews": 0, @@ -1176,7 +1176,7 @@ "end": 1095.52 }, { - "input": "Very simple question I want you to take the number i and I want you to multiply it by 3 plus 2i and even though I haven't really talked about the rules for multiplication what I can say is pretend like it operates just like it does for normal numbers you've got things like the distributive property where you can distribute this throughout and then the defining feature of i is this idea that i squared is negative one that's the only special thing you need to know about that other than that just treat it like it's a normal number okay and then proceed forward with the product. ", + "input": "o it a little bit mechanistically okay so if we pull ourselves up our sheet if we take i times three plus two i three plus two i it just distributes i times three is going to be three i i times two i is going to be two times i squared by definition i squared is negative one which means that our final answer is going to look like nega That's a 90 degree rotation. like I said it looks like a majority of you correctly did that product now it's one thing to just walk through it mechanistically it's another to step back and say what just happened geometrically right because what we just talked through was the fa ct that if you want to rotate ", "translatedText": "Câu hỏi rất đơn giản Tôi muốn bạn lấy số i và tôi muốn bạn nhân nó với 3 cộng 2i và mặc dù tôi chưa thực sự nói về các quy tắc nhân nhưng tôi có thể nói là giả vờ như nó hoạt động giống như nó những số bình thường bạn có những thứ như thuộc tính phân phối nơi bạn có thể phân phối cái này xuyên suốt và sau đó đặc điểm xác định của i là ý tưởng rằng tôi bình phương là số âm đó là điều đặc biệt duy nhất bạn cần biết về nó ngoài việc chỉ xử lý nó giống như đó là một con số bình thường được rồi và sau đó tiếp tục với sản phẩm. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/arabic/sentence_translations.json b/2020/ldm-eulers-formula/arabic/sentence_translations.json index 398b2e773..53bbfea87 100644 --- a/2020/ldm-eulers-formula/arabic/sentence_translations.json +++ b/2020/ldm-eulers-formula/arabic/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "سنأخذ بضع دقائق للقيام بذلك. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "لذا فإن شكل وجود أمثلة محددة تساعدك على الفهم لن يكون رقمًا مثل e إلى i pi. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "نعم، نحن في الواقع نرى الناس يفكرون في الأمر ويفكرون، هل صحيح أن الخيار 3 هنا صحيح بالضرورة؟ وهم ينتقدون ذلك حقًا. ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f لـ x يساوي 0 لا يعمل مع f لـ سالب 1 يساوي 1 على f لـ 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f لـ x يساوي 0 لا يعمل مع f لـ سالب 1 يساوي 1 على f لـ 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "لذا أعتقد أن حقيقة أن f لـ سالب 1 هي 1 على f لـ 1 لا تعني أن الناتج سيكون 0 على الإطلاق. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "وتشمل الخيارات مضاعفات 3 أو مضاعفات 3 طالما أنها موجبة، أو الأعداد الصحيحة 1 أسفل مضاعف 4 أو تلك التي تقتصر عليها فقط كونها إيجابية. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "والواحد التالي يشير إلى اليمين، لكن قوته 1 على 24، وقد قلصوا مجموعة كاملة. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "من يعرف؟ والذي هو جزء وهمي حولها. 84. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "يبدو أن الجزء الحقيقي أعلى قليلاً. 5، لذلك. 54، أعتقد ذلك. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "والجزء الخيالي هو زعم ماذا. ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "وبعد ذلك، إذا كنت مستعدًا لطرح سؤال المكافأة، فهذا يتعلق بكل الرصيد الإضافي في العالم، برر، لترى ما إذا كان بإمكانك القول، هل ستظل هذه النتيجة صحيحة إذا كان x وy أرقامًا معقدة، وهل ستظل كذلك يكون صحيحا إذا كانوا المصفوفات؟ أعتقد أن هذا في الواقع تمرين مهم جدًا يجب القيام به لفهم متى ينطبق ذلك على أنواع المدخلات. ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "أعتقد أن المرة الأولى التي رأيتها فيها، كانت أثناء تعلم الأعداد المركبة. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/bengali/sentence_translations.json b/2020/ldm-eulers-formula/bengali/sentence_translations.json index 8ddb95768..9f72fd60f 100644 --- a/2020/ldm-eulers-formula/bengali/sentence_translations.json +++ b/2020/ldm-eulers-formula/bengali/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "আমরা এটি করতে কয়েক মিনিট সময় নিতে যাচ্ছি।", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "সুতরাং নির্দিষ্ট উদাহরণ থাকার ফর্ম যা আপনার বোঝার জন্য সাহায্য করে i পাই এর মতো e এর মতো একটি সংখ্যা হবে না।", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "হ্যাঁ, তাই আমরা আসলে দেখছি লোকেরা এটি সম্পর্কে চিন্তা করে এবং চিন্তা করে, হ্যাং অন, এটা কি এমন যে এখানে বিকল্প 3 অগত্যা সত্য? ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "x এর f এর সমান 0 এর সাথে নেতিবাচক 1 এর সমান 1 এর f এর সাথে কাজ করে না।", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "x এর f এর সমান 0 এর সাথে নেতিবাচক 1 এর সমান 1 এর f এর সাথে কাজ করে না।", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "তাই আমি অনুমান করি যে নেতিবাচক 1 এর f 1 এর উপর f 1 এর আউটপুট কখনই 0 হবে তা বোঝায় না।", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "এবং বিকল্পগুলির মধ্যে 3-এর গুণিতক বা 3-এর গুণিতকগুলি যতক্ষণ পর্যন্ত তারা ধনাত্মক, বা 4-এর গুণিতকের নীচে পূর্ণসংখ্যা 1 বা যেখানে আপনি কেবল ইতিবাচক হওয়ার জন্য সীমাবদ্ধ থাকেন।", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "পরেরটি ডানদিকে নির্দেশ করা হয়েছে, তবে এটির মাত্রা 24 এর উপরে 1, এবং তারা কেবল একটি পুরো গুচ্ছ সঙ্কুচিত করে।", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "কে জানত? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "আর যার কাল্পনিক অংশ চারিদিকে।84. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "দেখে মনে হচ্ছে আসল অংশটা একটু উপরে।5, তাই।54, আমি এটা বিশ্বাস করি।", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "এবং কাল্পনিক অংশ, এটা দাবি, কি,. 84? ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "এবং তারপরে আপনি যদি বোনাস প্রশ্নের জন্য প্রস্তুত হন, এটি বিশ্বের সমস্ত অতিরিক্ত ক্রেডিট এর জন্য, ন্যায্যতা দিন, দেখুন আপনি বলতে পারেন কিনা, x এবং y জটিল সংখ্যা হলে এই ফলাফলটি কি সত্য হবে, এবং এটি এখনও কি যদি তারা matrices হয় সত্য? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "আমি মনে করি যে আমি প্রথমবার এটি দেখেছি, এটি জটিল সংখ্যা সম্পর্কে শেখার মধ্যে ছিল।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/chinese/sentence_translations.json b/2020/ldm-eulers-formula/chinese/sentence_translations.json index 0ded20b5b..3decc2f03 100644 --- a/2020/ldm-eulers-formula/chinese/sentence_translations.json +++ b/2020/ldm-eulers-formula/chinese/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "我们将花费几分钟来完成此操作。", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "因此，帮助您理解的具体示例的形式不会是像 e 到 i pi 这样的数字。", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "是的，所以我们实际上看到人们在思考这个问题，并思考，等等，这里的选项 3 一定是正确的吗？", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "x 的 f 等于 0 不适用于负 1 的 f 等于 1 大于 1 的 f。", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "x 的 f 等于 0 不适用于负 1 的 f 等于 1 大于 1 的 f。", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "所以我猜负 1 的 f 是 1 大于 1 的 f 的事实并不意味着输出将永远是 0。", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "这些选项包括 3 的倍数或 3 的倍数，只要它们是正数 ，或者小于 4 的倍数的整数 1 或仅限于正数的整数。", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "下一个指向右边，但它的震级为 1 比 24，而且它们只是缩小了一大堆。", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "谁知道？ ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "其虚部约为 0.84。 ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "看起来真实的部分在上面一点。5，所以。54、我相信这一点。", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "它声称，虚部是什么。84？", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "然后，如果你要回答额外的问题，这是为了世界上所有的额外 学分，请证明，看看你是否可以说，如果 x 和 y 是复 数，这个结果仍然是正确的吗？如果它们是矩阵，则为真？", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "我想我第一次看到它是在学习复数时。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/english/captions.srt b/2020/ldm-eulers-formula/english/captions.srt index aacc73098..3f5e2525b 100644 --- a/2020/ldm-eulers-formula/english/captions.srt +++ b/2020/ldm-eulers-formula/english/captions.srt @@ -19,7 +19,7 @@ ending up with this lesson, I'm going to go ahead and show you what we're aiming for at the end, which is a certain visualization. 6 -00:00:13,719 --> 00:00:16,538 +00:00:13,720 --> 00:00:16,538 So, I don't expect you to necessarily understand this immediately, 7 @@ -511,7 +511,7 @@ I think the healthy reaction to have to this, okay, if you're just seeing this for the first time, the healthy question to ask is w t f, okay? 129 -00:07:26,799 --> 00:07:32,400 +00:07:26,800 --> 00:07:32,400 What is the function at play and how is it defined, okay? 130 @@ -579,7 +579,7 @@ Instead, what has emerged in math is that we use e to the x to be a shorthand for another function. 146 -00:08:24,099 --> 00:08:26,400 +00:08:24,100 --> 00:08:26,400 A function which I'm going to give the name exp. 147 @@ -1239,7 +1239,7 @@ that actually prepares you for understanding e to the i x as a generality a litt bit better. 311 -00:18:50,239 --> 00:18:55,740 +00:18:50,240 --> 00:18:55,740 So with all of that, answers are still rolling in and I don't want you to feel rushed. 312 @@ -1259,7 +1259,7 @@ that something like 2 or 3 or 1 is included in your answer that any function with this special property necessarily follows that. 316 -00:19:15,939 --> 00:19:23,880 +00:19:15,940 --> 00:19:23,880 Oh, let's see if we can grade it when the top answer is the year. 317 @@ -1395,7 +1395,7 @@ That's not the same as the fact that e to the x is a shorthand for this crazy po So that can sometimes get a little confusing, the exponent is playing two different roles. 350 -00:21:32,919 --> 00:21:36,340 +00:21:32,920 --> 00:21:36,340 And we might give x of 1 a special name, a shorthand. 351 @@ -1447,12 +1447,12 @@ And the key to solving this, it's a little bit tricky, is to think about what happens when we multiply that by itself. 363 -00:22:13,840 --> 00:22:19,890 -x of 1 half times x of 1 half, because of this property, has to satisfy, +00:22:13,840 --> 00:22:18,677 +Exp of one half times exp of one half, because of this property, 364 -00:22:19,890 --> 00:22:24,780 -or I should say has to equal, x of 1 half plus x of 1 half. +00:22:18,677 --> 00:22:24,780 +has to satisfy, or I should say has to equal exp of one half plus exp of one half. 365 00:22:25,020 --> 00:22:31,080 @@ -1579,7 +1579,7 @@ Someone correct me on Twitter if I'm wrong about that. But, very interesting. 396 -00:24:20,399 --> 00:24:23,020 +00:24:20,400 --> 00:24:23,020 So if we go back to our paper, let's see. 397 @@ -1639,7 +1639,7 @@ you can use the number 1 to access pretty much all the real numbers by adding it to itself or dividing as needed or negating. 411 -00:25:11,139 --> 00:25:13,975 +00:25:11,140 --> 00:25:13,975 And then if you have certain continuity restrictions that 412 @@ -1739,7 +1739,7 @@ talking about a constant raised to some kind of power. But what you have to understand is that it's being plugged into this polynomial. 436 -00:26:35,819 --> 00:26:39,600 +00:26:35,820 --> 00:26:39,600 So I understand why we have the convention of writing this as e to the x. 437 @@ -1791,7 +1791,7 @@ So that's defined. For which values of n does i to the power n equal negative i? 449 -00:27:24,879 --> 00:27:29,580 +00:27:24,880 --> 00:27:29,580 So the spirit of this question is to have people thinking deeply about powers of i. 450 @@ -1807,11 +1807,11 @@ So one that you do have to put a little mental energy into. So let's go ahead and take a question from the audience while I take a drink of water. 453 -00:27:48,879 --> 00:27:54,060 +00:27:48,880 --> 00:27:54,060 F of x equals zero doesn't work with f of negative one equals one over f of one. 454 -00:27:57,739 --> 00:27:59,500 +00:27:57,740 --> 00:27:59,500 Yeah, f of x equals zero. 455 @@ -1871,11 +1871,11 @@ Oh, okay. Yeah, okay. 469 -00:28:48,640 --> 00:28:48,640 +00:28:48,640 --> 00:28:48,940 Great. 470 -00:28:48,640 --> 00:28:49,540 +00:28:49,300 --> 00:28:49,540 Great. 471 @@ -2175,7 +2175,7 @@ into Python if we wanted to and see how things play there. But first, let me go ahead and visualize it for you. 545 -00:32:53,159 --> 00:32:57,233 +00:32:53,160 --> 00:32:57,233 So I'm going to, uh, let's see, that circle is already suggestively drawn, 546 @@ -2507,7 +2507,7 @@ Okay, does that check out? Yeah, I buy it that the length of this vector is just less than five, so 4.93. 628 -00:37:41,799 --> 00:37:44,380 +00:37:41,800 --> 00:37:44,380 And then the next one, because of the i cubed, 629 @@ -3103,7 +3103,7 @@ It's useful, it is genuinely useful, and it's important to know what properties carry over and which ones don't. 777 -00:47:45,839 --> 00:47:49,310 +00:47:45,840 --> 00:47:49,310 Now, I was talking a little about this lecture with a friend beforehand, 778 @@ -3307,7 +3307,7 @@ I love this. This is just like a real class. 828 -00:49:59,759 --> 00:50:00,520 +00:49:59,760 --> 00:50:00,520 I'm up here. 829 diff --git a/2020/ldm-eulers-formula/english/sentence_timings.json b/2020/ldm-eulers-formula/english/sentence_timings.json index f7dfa2139..010e62f44 100644 --- a/2020/ldm-eulers-formula/english/sentence_timings.json +++ b/2020/ldm-eulers-formula/english/sentence_timings.json @@ -1170,7 +1170,7 @@ 1333.32 ], [ - "x of 1 half times x of 1 half, because of this property, has to satisfy, or I should say has to equal, x of 1 half plus x of 1 half.", + "Exp of one half times exp of one half, because of this property, has to satisfy, or I should say has to equal exp of one half plus exp of one half.", 1333.84, 1344.78 ], @@ -1587,11 +1587,11 @@ [ "Great.", 1728.64, - 1728.64 + 1728.94 ], [ "Great.", - 1728.64, + 1729.3, 1729.54 ], [ diff --git a/2020/ldm-eulers-formula/english/transcript.txt b/2020/ldm-eulers-formula/english/transcript.txt index 07095679c..e1f4d9638 100644 --- a/2020/ldm-eulers-formula/english/transcript.txt +++ b/2020/ldm-eulers-formula/english/transcript.txt @@ -232,7 +232,7 @@ Okay, so simply by virtue of this property, we can see that x of 5 has to be wri Now a little bit trickier was the question about plugging in 1 half. Okay? And the key to solving this, it's a little bit tricky, is to think about what happens when we multiply that by itself. -x of 1 half times x of 1 half, because of this property, has to satisfy, or I should say has to equal, x of 1 half plus x of 1 half. +Exp of one half times exp of one half, because of this property, has to satisfy, or I should say has to equal exp of one half plus exp of one half. Which is of course x of 1, which let's say we're using the shorthand and we call it e. So what does that mean? x of 1 half has to be a number such that multiplying it by itself equals e. diff --git a/2020/ldm-eulers-formula/french/sentence_translations.json b/2020/ldm-eulers-formula/french/sentence_translations.json index d21ac3bc2..22aab639e 100644 --- a/2020/ldm-eulers-formula/french/sentence_translations.json +++ b/2020/ldm-eulers-formula/french/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "Nous allons prendre quelques minutes pour le faire. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "Ainsi, la forme d'avoir des exemples spécifiques qui facilitent votre compréhension ne serait pas un nombre comme e au i pi. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "Ouais, donc nous voyons en fait des gens y réfléchir et se demander, attendez, est-il vrai que l'option 3 ici est nécessairement vraie ? ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f de x est égal à 0 ne fonctionne pas avec f de moins 1 est égal à 1 sur f de 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f de x est égal à 0 ne fonctionne pas avec f de moins 1 est égal à 1 sur f de 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "Donc, je suppose que le fait que f de moins 1 soit 1 sur f de 1 n'implique pas que la sortie sera un jour 0. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "Et les options incluent des multiples de 3 ou des multiples de 3 tant qu'ils sont positifs, ou des entiers 1 inférieurs à un multiple de 4 ou ceux pour lesquels vous êtes limité à ce qu'ils soient simplement positifs. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "Le suivant est pointé vers la droite, mais il a une magnitude de 1 sur 24, et ils rétrécissent considérablement. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "Qui savait? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "Et dont la partie imaginaire est autour. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "On dirait que la vraie partie est un peu au dessus. 5, donc. 54, je le crois. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "Et la partie imaginaire est, prétendait-on, quoi. 84 ? ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "Et puis, si vous êtes partant pour la question bonus, c'est pour tout le crédit supplémentaire du monde, justifiez, voyez si vous pouvez dire, ce résultat sera-t-il toujours vrai si x et y sont des nombres complexes, et est-ce que cela sera toujours vrai ? être vrai si ce sont des matrices ? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "Je pense que la première fois que je l’ai vu, c’était en apprenant les nombres complexes. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/german/sentence_translations.json b/2020/ldm-eulers-formula/german/sentence_translations.json index bfc8259f3..b5690d9e6 100644 --- a/2020/ldm-eulers-formula/german/sentence_translations.json +++ b/2020/ldm-eulers-formula/german/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "Wir werden uns dafür ein paar Minuten Zeit nehmen. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "Die Form, konkrete Beispiele zu haben, die Ihr Verständnis erleichtern, wäre also nicht eine Zahl wie e zum i pi. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "Es sieht also so aus, als ob ein paar Leute sagen: „Oh, jetzt sind wir in der Zukunft, im Jahr 2023.“ Ja, wir sehen also tatsächlich Leute, die darüber nachdenken und sich fragen: „Moment mal, ist es so, dass Option 3 hier unbedingt wahr ist?“ Und sie stehen dem wirklich kritisch gegenüber. ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f von x gleich 0 funktioniert nicht mit f von minus 1 gleich 1 über f von 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f von x gleich 0 funktioniert nicht mit f von minus 1 gleich 1 über f von 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "Ich vermute also, dass die Tatsache, dass f von minus 1 1 über f von 1 ist, nicht bedeutet, dass die Ausgabe jemals 0 sein wird. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "Und die Optionen umfassen Vielfache von 3 oder Vielfache von 3, solange sie positiv sind, oder ganze Zahlen 1 unter einem Vielfachen von 4 oder solche, bei denen Sie darauf beschränkt sind, dass sie nur positiv sind. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "Der nächste ist nach rechts gerichtet, hat aber eine Stärke von 1 über 24, und sie schrumpfen einfach um ein Vielfaches. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "Wer wusste? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "Und dessen imaginärer Teil rund ist. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "Es sieht so aus, als ob der eigentliche Teil etwas weiter oben liegt. 5, also. 54, das glaube ich. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "Und der Imaginärteil sei, so hieß es, was. 84? ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "Und wenn Sie sich dann für die Bonusfrage interessieren, ist dies für alle Extrakredite der Welt, begründen Sie, ob Sie sagen können, ob dieses Ergebnis immer noch wahr ist, wenn x und y komplexe Zahlen sind, und wird es immer noch wahr sein wahr sein, wenn es sich um Matrizen handelt? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "Ich glaube, das erste Mal, dass ich es sah, war, als ich etwas über komplexe Zahlen lernte. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/hebrew/sentence_translations.json b/2020/ldm-eulers-formula/hebrew/sentence_translations.json index c30627da4..ce5fd1485 100644 --- a/2020/ldm-eulers-formula/hebrew/sentence_translations.json +++ b/2020/ldm-eulers-formula/hebrew/sentence_translations.json @@ -672,7 +672,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this.", + "input": "We're going to take, you know, a couple minutes to do this.", "translatedText": "אנחנו הולכים להקדיש כמה דקות לעשות את זה.", "n_reviews": 0, "start": 589.64, @@ -1134,7 +1134,7 @@ "end": 978.8 }, { - "input": "One of the options is that f of 5 is equal to f of 1 raised to the power 5.", + "input": "One of the options is that f of five is equal to f of one raised to the power five.", "translatedText": "אחת האפשרויות היא ש-f של 5 שווה ל-f של 1 שהועלה בחזקת 5.", "n_reviews": 0, "start": 979.66, @@ -1274,7 +1274,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi.", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi.", "translatedText": "אז הצורה של דוגמאות ספציפיות המסייעות להבנתך לא תהיה מספר כמו e ל-i pi.", "n_reviews": 0, "start": 1103.16, @@ -1379,7 +1379,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true?", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true?", "translatedText": "כן, אז אנחנו בעצם רואים אנשים חושבים על זה וחושבים, רגע, האם זה המצב שאופציה 3 כאן בהכרח נכונה?", "n_reviews": 0, "start": 1177.46, @@ -1876,7 +1876,7 @@ "end": 1539.72 }, { - "input": "We could, if we wanted, define the number e to simply be, where is this polynomial at the number 1?", + "input": "t, we could, we could if we wanted define the number e to simply be where is this polynomial at the number 1.", "translatedText": "נוכל, אם נרצה, להגדיר את המספר e להיות פשוט, היכן נמצא הפולינום הזה במספר 1?", "n_reviews": 0, "start": 1539.8, @@ -2030,7 +2030,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1.", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one.", "translatedText": "f של x שווה 0 לא עובד עם f של שלילי 1 שווה 1 על f של 1.", "n_reviews": 0, "start": 1668.88, @@ -2058,14 +2058,14 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1.", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one.", "translatedText": "f של x שווה 0 לא עובד עם f של שלילי 1 שווה 1 על f של 1.", "n_reviews": 0, "start": 1688.4, "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0.", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero.", "translatedText": "אז אני מניח שהעובדה ש-f של 1 שלילי הוא 1 על פני f של 1 לא מרמזת שהפלט אי פעם יהיה 0.", "n_reviews": 0, "start": 1697.12, @@ -2093,7 +2093,7 @@ "end": 1735.02 }, { - "input": "f of x plus 0 is the same as f of x times f of 0, which implies f of x equals 1.", + "input": "F of x plus zero is the same as f of x times f of zero, which implies f of x equals one.", "translatedText": "f של x פלוס 0 זהה ל-f של x כפול f של 0, מה שמרמז על f של x שווה ל-1.", "n_reviews": 0, "start": 1735.84, @@ -2198,7 +2198,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive.", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive.", "translatedText": "והאפשרויות כוללות כפולות של 3 או כפולות של 3 כל עוד הן חיוביות, או מספרים שלמים 1 מתחת לכפולה של 4 או כאלה שבהם אתה מוגבל רק שהם חיוביים.", "n_reviews": 0, "start": 1795.16, @@ -2310,7 +2310,7 @@ "end": 1906.58 }, { - "input": "If it's a power of 4, you're at 1.", + "input": "If it's a power of four, you're at one. If", "translatedText": "אם זה חזק של 4, אתה ב-1.", "n_reviews": 0, "start": 1906.74, @@ -2464,7 +2464,7 @@ "end": 2031.24 }, { - "input": "Then after that, i cubed is pointed straight down, and we divide it by 6.", + "input": "Okay? Then after that, i cubed is pointed straight down, and we divide it by six.", "translatedText": "ואז אחרי זה, i cubed מופנה ישר למטה, ואנחנו מחלקים אותו ב-6.", "n_reviews": 0, "start": 2031.78, @@ -2478,7 +2478,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch.", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch.", "translatedText": "הבא מופנה ימינה, אבל יש לו עוצמה 1 על 24, והם פשוט מכווצים חבורה שלמה.", "n_reviews": 0, "start": 2042.82, @@ -2583,7 +2583,7 @@ "end": 2105.24 }, { - "input": "54.", + "input": "point five four,", "translatedText": "54.", "n_reviews": 0, "start": 2105.24, @@ -2604,7 +2604,7 @@ "end": 2109.42 }, { - "input": "84.", + "input": "point eight four,", "translatedText": "84.", "n_reviews": 0, "start": 2109.42, @@ -2625,7 +2625,7 @@ "end": 2115.62 }, { - "input": "5, so.", + "input": "point five, so point five fo", "translatedText": "5, אז.", "n_reviews": 0, "start": 2115.62, @@ -2646,7 +2646,7 @@ "end": 2121.46 }, { - "input": "84?", + "input": "point eight four?", "translatedText": "84?", "n_reviews": 0, "start": 2121.58, @@ -3206,7 +3206,7 @@ "end": 2564.84 }, { - "input": "Partly because of how we defined it, you know, only doing so many terms, and partly because computers can't do infinite precision.", + "input": "r because, partly because of how we defined it, you know, only doing so many terms, and partly because computers can't do infinite precision.", "translatedText": "חלקית בגלל איך שהגדרנו את זה, אתה יודע, עושה רק כל כך הרבה מונחים, וחלקית בגלל שמחשבים לא יכולים לעשות דיוק אינסופי.", "n_reviews": 0, "start": 2564.84, @@ -3472,7 +3472,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices?", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices?", "translatedText": "ואז אם אתה בעד שאלת הבונוס, זה עבור כל הקרדיט הנוסף שבעולם, הצדק, תראה אם אתה יכול לומר, האם התוצאה הזו עדיין תהיה נכונה אם x ו-y הם מספרים מרוכבים, והאם היא עדיין נכונה להיות נכון אם הם מטריצות?", "n_reviews": 0, "start": 2829.02, @@ -3528,7 +3528,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers.", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers.", "translatedText": "אני חושב שהפעם הראשונה שאי פעם ראיתי את זה, זה היה בלמידה על מספרים מרוכבים.", "n_reviews": 0, "start": 2887.54, diff --git a/2020/ldm-eulers-formula/hindi/sentence_translations.json b/2020/ldm-eulers-formula/hindi/sentence_translations.json index 8134e7baa..c105f2b9d 100644 --- a/2020/ldm-eulers-formula/hindi/sentence_translations.json +++ b/2020/ldm-eulers-formula/hindi/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "ऐसा करने में हमें कुछ मिनट लगेंगे।", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "तो आपकी समझ में सहायता करने वाले विशिष्ट उदाहरणों का स्वरूप i pi के लिए e जैसी संख्या नहीं होगा।", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "हाँ, तो हम वास्तव में लोगों को इसके बारे में सोचते हुए देख रहे हैं और सोच रहे हैं, क्या ऐसा है कि यहाँ विकल्प 3 आवश्यक रूप से सत्य है? ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "x का f, 0 के बराबर है, नकारात्मक 1 के f के साथ काम नहीं करता है, 1 के f के मुकाबले 1 बराबर है।", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "x का f, 0 के बराबर है, नकारात्मक 1 के f के साथ काम नहीं करता है, 1 के f के मुकाबले 1 बराबर है।", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "तो मुझे लगता है कि तथ्य यह है कि नकारात्मक 1 का एफ, 1 के एफ पर 1 है, इसका मतलब यह नहीं है कि आउटपुट कभी भी 0 होगा।", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "और विकल्पों में 3 के गुणज या 3 के गुणज शामिल हैं जब तक वे सकारात्मक हैं, या 4 के गुणज के नीचे 1 पूर्णांक हैं या वे जहां आप केवल सकारात्मक होने तक ही सीमित हैं।", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "अगले को दाहिनी ओर इंगित किया गया है, लेकिन इसका परिमाण 24 से 1 है, और वे बस एक पूरे समूह को छोटा कर देते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "किसे पता था? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "और जिसका काल्पनिक भाग चारों ओर है. 84. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "ऐसा लगता है जैसे असली हिस्सा थोड़ा ऊपर है. 5, तो. 54, मुझे ऐसा विश्वास है।", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "और काल्पनिक भाग यह है, यह दावा किया गया, क्या,।84? ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "और फिर यदि आप बोनस प्रश्न के लिए तैयार हैं, तो यह दुनिया के सभी अतिरिक्त क्रेडिट के लिए है, औचित्य बताएं, देखें कि क्या आप कह सकते हैं, क्या यह परिणाम अभी भी सच होगा यदि x और y जटिल संख्याएं हैं, और क्या यह अभी भी सच होगा यदि वे आव्यूह हैं तो सत्य होंगे? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "मुझे लगता है कि पहली बार मैंने इसे जटिल संख्याओं के बारे में सीखते हुए देखा था।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/hungarian/sentence_translations.json b/2020/ldm-eulers-formula/hungarian/sentence_translations.json index dd16774ef..f4953393e 100644 --- a/2020/ldm-eulers-formula/hungarian/sentence_translations.json +++ b/2020/ldm-eulers-formula/hungarian/sentence_translations.json @@ -672,7 +672,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this.", + "input": "We're going to take, you know, a couple minutes to do this.", "translatedText": "Néhány percet szánunk erre.", "n_reviews": 0, "start": 589.64, @@ -1134,7 +1134,7 @@ "end": 978.8 }, { - "input": "One of the options is that f of 5 is equal to f of 1 raised to the power 5.", + "input": "One of the options is that f of five is equal to f of one raised to the power five.", "translatedText": "Az egyik lehetőség az, hogy az 5-ös f értéke egyenlő az 5-ös hatványra emelt 1-es f-jével.", "n_reviews": 0, "start": 979.66, @@ -1274,7 +1274,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi.", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi.", "translatedText": "Tehát a konkrét példák, amelyek segítik a megértést, nem olyan számok, mint az e az i pi-hez.", "n_reviews": 0, "start": 1103.16, @@ -1379,7 +1379,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true?", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true?", "translatedText": "Igen, tehát látjuk, hogy az emberek elgondolkodnak rajta, és azt gondolják, várjunk csak, vajon a 3. lehetőség szükségszerűen igaz?", "n_reviews": 0, "start": 1177.46, @@ -1876,7 +1876,7 @@ "end": 1539.72 }, { - "input": "We could, if we wanted, define the number e to simply be, where is this polynomial at the number 1?", + "input": "t, we could, we could if we wanted define the number e to simply be where is this polynomial at the number 1.", "translatedText": "Ha akarnánk, úgy definiálhatnánk az e számot, hogy egyszerűen legyen, hol van ez a polinom az 1-es számnál?", "n_reviews": 0, "start": 1539.8, @@ -2030,7 +2030,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1.", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one.", "translatedText": "x f egyenlő 0-val nem működik, ha az f negatív 1 egyenlő 1-gyel, az 1-es f-hez képest.", "n_reviews": 0, "start": 1668.88, @@ -2058,14 +2058,14 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1.", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one.", "translatedText": "x f egyenlő 0-val nem működik, ha az f negatív 1 egyenlő 1-gyel, az 1-es f-hez képest.", "n_reviews": 0, "start": 1688.4, "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0.", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero.", "translatedText": "Tehát azt hiszem, hogy az a tény, hogy a negatív 1 f értéke 1, szemben az 1-gyel, nem jelenti azt, hogy a kimenet valaha is 0 lesz.", "n_reviews": 0, "start": 1697.12, @@ -2093,7 +2093,7 @@ "end": 1735.02 }, { - "input": "f of x plus 0 is the same as f of x times f of 0, which implies f of x equals 1.", + "input": "F of x plus zero is the same as f of x times f of zero, which implies f of x equals one.", "translatedText": "Az x plusz 0 f-je megegyezik az x-ből származó f szorzata f-vel 0-val, ami azt jelenti, hogy x-ből f egyenlő 1-gyel.", "n_reviews": 0, "start": 1735.84, @@ -2198,7 +2198,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive.", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive.", "translatedText": "A lehetőségek közé tartoznak a 3 többszörösei vagy a 3 többszörösei, amennyiben pozitívak, vagy a 4 többszöröse alatti 1-es egész számok, vagy olyanok, ahol csak pozitívak.", "n_reviews": 0, "start": 1795.16, @@ -2310,7 +2310,7 @@ "end": 1906.58 }, { - "input": "If it's a power of 4, you're at 1.", + "input": "If it's a power of four, you're at one. If", "translatedText": "Ha ez 4-es hatvány, akkor 1-nél jár.", "n_reviews": 0, "start": 1906.74, @@ -2464,7 +2464,7 @@ "end": 2031.24 }, { - "input": "Then after that, i cubed is pointed straight down, and we divide it by 6.", + "input": "Okay? Then after that, i cubed is pointed straight down, and we divide it by six.", "translatedText": "Ezután az i cubed egyenesen lefelé mutat, és elosztjuk 6-tal.", "n_reviews": 0, "start": 2031.78, @@ -2478,7 +2478,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch.", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch.", "translatedText": "A következő jobbra mutat, de 1 magnitúdója van 24 felett, és csak egy csomót zsugorítanak.", "n_reviews": 0, "start": 2042.82, @@ -2583,7 +2583,7 @@ "end": 2105.24 }, { - "input": "54.", + "input": "point five four,", "translatedText": "54.", "n_reviews": 0, "start": 2105.24, @@ -2604,7 +2604,7 @@ "end": 2109.42 }, { - "input": "84.", + "input": "point eight four,", "translatedText": "84.", "n_reviews": 0, "start": 2109.42, @@ -2625,7 +2625,7 @@ "end": 2115.62 }, { - "input": "5, so.", + "input": "point five, so point five fo", "translatedText": "5, szóval.", "n_reviews": 0, "start": 2115.62, @@ -2646,7 +2646,7 @@ "end": 2121.46 }, { - "input": "84?", + "input": "point eight four?", "translatedText": "84?", "n_reviews": 0, "start": 2121.58, @@ -3206,7 +3206,7 @@ "end": 2564.84 }, { - "input": "Partly because of how we defined it, you know, only doing so many terms, and partly because computers can't do infinite precision.", + "input": "r because, partly because of how we defined it, you know, only doing so many terms, and partly because computers can't do infinite precision.", "translatedText": "Részben azért, mert mi definiáltuk, csak annyi kifejezést használunk, részben pedig azért, mert a számítógépek nem képesek végtelen pontosságra.", "n_reviews": 0, "start": 2564.84, @@ -3472,7 +3472,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices?", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices?", "translatedText": "És ha készen áll a bónuszkérdésre, ez a világ összes extra kreditjére vonatkozik, indokolja meg, hátha meg tudja mondani, hogy ez az eredmény akkor is igaz lesz, ha x és y komplex számok, és akkor is igazak legyenek, ha mátrixok?", "n_reviews": 0, "start": 2829.02, @@ -3528,7 +3528,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers.", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers.", "translatedText": "Azt hiszem, amikor először láttam, az összetett számok megismerése volt.", "n_reviews": 0, "start": 2887.54, diff --git a/2020/ldm-eulers-formula/indonesian/sentence_translations.json b/2020/ldm-eulers-formula/indonesian/sentence_translations.json index 6038a5ceb..46499a03a 100644 --- a/2020/ldm-eulers-formula/indonesian/sentence_translations.json +++ b/2020/ldm-eulers-formula/indonesian/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "Kami akan memerlukan waktu beberapa menit untuk melakukan ini. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "Jadi bentuk contoh spesifik yang membantu pemahaman Anda bukanlah angka seperti e hingga i pi. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "Ya, jadi kita sebenarnya melihat orang-orang memikirkannya dan berpikir, tunggu dulu, apakah opsi 3 di sini benar? ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f dari x sama dengan 0 tidak berfungsi dengan f negatif 1 sama dengan 1 di atas f dari 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f dari x sama dengan 0 tidak berfungsi dengan f negatif 1 sama dengan 1 di atas f dari 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "Jadi saya rasa fakta bahwa f dari negatif 1 adalah 1 di atas f dari 1 tidak berarti bahwa outputnya akan menjadi 0. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "Dan opsinya mencakup kelipatan 3 atau kelipatan 3 selama bilangan tersebut positif, atau bilangan bulat 1 di bawah kelipatan 4, atau bilangan bulat yang dibatasi hanya bilangan positif saja. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "Yang berikutnya mengarah ke kanan, tapi magnitudonya 1 per 24, dan menyusut sejumlah besar saja. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "Siapa yang tahu? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "Dan bagian imajiner siapa yang ada. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "Sepertinya bagian aslinya ada sedikit di atas. 5, jadi. 54, saya percaya itu. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "Dan bagian imajinernya adalah, katanya, apa. ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "Dan kemudian jika Anda siap untuk pertanyaan bonus, ini untuk semua kredit ekstra di dunia, benarkan, lihat apakah Anda dapat mengatakan, apakah hasil ini masih benar jika x dan y adalah bilangan kompleks, dan akankah tetap benar? benarkah jika itu matriks? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "Saya rasa pertama kali saya melihatnya adalah saat mempelajari bilangan kompleks. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/italian/sentence_translations.json b/2020/ldm-eulers-formula/italian/sentence_translations.json index d1bf8c888..90023a43f 100644 --- a/2020/ldm-eulers-formula/italian/sentence_translations.json +++ b/2020/ldm-eulers-formula/italian/sentence_translations.json @@ -672,7 +672,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this.", + "input": "We're going to take, you know, a couple minutes to do this.", "translatedText": "Ci prenderemo un paio di minuti per farlo.", "n_reviews": 0, "start": 589.64, @@ -1134,7 +1134,7 @@ "end": 978.8 }, { - "input": "One of the options is that f of 5 is equal to f of 1 raised to the power 5.", + "input": "One of the options is that f of five is equal to f of one raised to the power five.", "translatedText": "Una delle opzioni è che f(5) sia uguale a f(1 elevato a 5).", "n_reviews": 0, "start": 979.66, @@ -1274,7 +1274,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi.", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi.", "translatedText": "Quindi il modo per avere esempi specifici che aiutano la tua comprensione non sarebbe un numero come e al i pi greco.", "n_reviews": 0, "start": 1103.16, @@ -1379,7 +1379,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true?", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true?", "translatedText": "Sì, quindi stiamo effettivamente vedendo le persone pensarci e pensare, aspetta, è vero che l'opzione 3 qui è necessariamente vera?", "n_reviews": 0, "start": 1177.46, @@ -1876,7 +1876,7 @@ "end": 1539.72 }, { - "input": "We could, if we wanted, define the number e to simply be, where is this polynomial at the number 1?", + "input": "t, we could, we could if we wanted define the number e to simply be where is this polynomial at the number 1.", "translatedText": "Potremmo, se volessimo, definire il numero e semplicemente come: dov'è questo polinomio al numero 1?", "n_reviews": 0, "start": 1539.8, @@ -2030,7 +2030,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1.", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one.", "translatedText": "f(x=0) non funziona con f(meno 1 = 1 su f(1).", "n_reviews": 0, "start": 1668.88, @@ -2058,14 +2058,14 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1.", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one.", "translatedText": "f(x=0) non funziona con f(meno 1 = 1 su f(1).", "n_reviews": 0, "start": 1688.4, "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0.", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero.", "translatedText": "Quindi immagino che il fatto che f di meno 1 sia 1 su f di 1 non implica che l'output sarà mai 0.", "n_reviews": 0, "start": 1697.12, @@ -2093,7 +2093,7 @@ "end": 1735.02 }, { - "input": "f of x plus 0 is the same as f of x times f of 0, which implies f of x equals 1.", + "input": "F of x plus zero is the same as f of x times f of zero, which implies f of x equals one.", "translatedText": "f(x più 0) è uguale a f(x) moltiplicato f(0), il che implica che f(x) è uguale a 1.", "n_reviews": 0, "start": 1735.84, @@ -2198,7 +2198,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive.", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive.", "translatedText": "E le opzioni includono multipli di 3 o multipli di 3 purché siano positivi, o numeri interi da 1 inferiori a multipli di 4 o quelli in cui sei limitato a essere solo positivi.", "n_reviews": 0, "start": 1795.16, @@ -2310,7 +2310,7 @@ "end": 1906.58 }, { - "input": "If it's a power of 4, you're at 1.", + "input": "If it's a power of four, you're at one. If", "translatedText": "Se è una potenza di 4, sei a 1.", "n_reviews": 0, "start": 1906.74, @@ -2464,7 +2464,7 @@ "end": 2031.24 }, { - "input": "Then after that, i cubed is pointed straight down, and we divide it by 6.", + "input": "Okay? Then after that, i cubed is pointed straight down, and we divide it by six.", "translatedText": "Poi, il cubo è rivolto verso il basso e lo dividiamo per 6.", "n_reviews": 0, "start": 2031.78, @@ -2478,7 +2478,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch.", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch.", "translatedText": "Il prossimo è puntato a destra, ma ha magnitudo 1 su 24, e si restringono parecchio.", "n_reviews": 0, "start": 2042.82, @@ -2583,7 +2583,7 @@ "end": 2105.24 }, { - "input": "54.", + "input": "point five four,", "translatedText": "54.", "n_reviews": 0, "start": 2105.24, @@ -2604,7 +2604,7 @@ "end": 2109.42 }, { - "input": "84.", + "input": "point eight four,", "translatedText": "84.", "n_reviews": 0, "start": 2109.42, @@ -2625,7 +2625,7 @@ "end": 2115.62 }, { - "input": "5, so.", + "input": "point five, so point five fo", "translatedText": "5, quindi.", "n_reviews": 0, "start": 2115.62, @@ -2646,7 +2646,7 @@ "end": 2121.46 }, { - "input": "84?", + "input": "point eight four?", "translatedText": "84?", "n_reviews": 0, "start": 2121.58, @@ -3206,7 +3206,7 @@ "end": 2564.84 }, { - "input": "Partly because of how we defined it, you know, only doing so many terms, and partly because computers can't do infinite precision.", + "input": "r because, partly because of how we defined it, you know, only doing so many terms, and partly because computers can't do infinite precision.", "translatedText": "In parte a causa di come l'abbiamo definito, usando solo un certo numero di termini, e in parte perché i computer non possono ottenere una precisione infinita.", "n_reviews": 0, "start": 2564.84, @@ -3472,7 +3472,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices?", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices?", "translatedText": "E poi se sei pronto per la domanda bonus, questo è per tutto il credito extra del mondo, giustifica, vedi se puoi dire, questo risultato sarà ancora vero se xey sono numeri complessi, e lo sarà ancora essere vero se sono matrici?", "n_reviews": 0, "start": 2829.02, @@ -3528,7 +3528,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers.", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers.", "translatedText": "Penso che la prima volta che l'ho visto, è stato mentre imparavo i numeri complessi.", "n_reviews": 0, "start": 2887.54, diff --git a/2020/ldm-eulers-formula/japanese/sentence_translations.json b/2020/ldm-eulers-formula/japanese/sentence_translations.json index 4e0113055..5d0a275cf 100644 --- a/2020/ldm-eulers-formula/japanese/sentence_translations.json +++ b/2020/ldm-eulers-formula/japanese/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "これには数分かかります。", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "したがって、理解を助ける具体的な例を持つ形式は、i に対する e のような数字ではありません。", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "そうですね、実際に人々がそれについて考え、「ちょっと待って、ここでの選択肢 3 が必ずしも正しいということなのでしょうか?」と考えているのを私たちは目にしています。", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "x の f が 0 に等しいということは、負の 1 の f が 1 の f よりも 1 に等しい場合には機能しません。", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "x の f が 0 に等しいということは、負の 1 の f が 1 の f よりも 1 に等しい場合には機能しません。", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "したがって、負の 1 の f が 1 の f よりも 1 であるという事実は、出力が 0 になることを意味しないと思います。", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "また、オプションには、3 の倍数、または正である限り 3 の倍数、または 4 の倍 数より 1 つ下の整数、または正であることだけに制限されているものが含まれます。", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "次のものは右を向いていますが、マグニチュード 1 は 24 で、全体が縮小するだけです。", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "そして、その虚数部は誰の周りにありますか。", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "実物はもう少し上にあるようです。5、それで。54、私はそう信じています。", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "そして、虚数部は、何であると主張しました。", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "そして、おまけの質問をする気があるなら、これは世界中の余分な功績すべてに対するもの です。正当化してください。x と y が複素数の場合、この結果は依然として真にな りますか、そして、それでもそうなりますか?行列の場合は true になりますか? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "私が初めてこれを目にしたのは、複素数について学習している時だったと思います。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/korean/sentence_translations.json b/2020/ldm-eulers-formula/korean/sentence_translations.json index 7644c9f43..9ed09eaa7 100644 --- a/2020/ldm-eulers-formula/korean/sentence_translations.json +++ b/2020/ldm-eulers-formula/korean/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "이 작업을 수행하는 데 몇 분 정도 걸릴 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "따라서 이해를 돕기 위한 구체적인 예를 갖는 형식은 i 파이의 e와 같은 숫자가 아닐 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "예, 그래서 우리는 실제로 사람들이 그것에 대해 생각하고, 잠깐만요, 여기 옵션 3이 반드시 사실일까요?라고 생각하는 것을 보고 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f(x)는 0과 동일하며 f(-1)은 f(1) 나누기 1과 동일하므로 작동하지 않습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f(x)는 0과 동일하며 f(-1)은 f(1) 나누기 1과 동일하므로 작동하지 않습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "그래서 나는 f(-1)이 1/f(1)이라는 사실이 출력이 0이 될 것이라는 의미는 아니라고 생각합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "그리고 옵션에는 양수인 경우 3의 배수 또는 3의 배수, 4의 배수보다 작은 정수 1 또는 양수로만 제한되는 정수가 포함됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "다음 것은 오른쪽을 가리키고 있지만 크기는 24분의 1이고 전체가 축소됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "누가 알았 겠어? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "그리고 그의 가상 부분은 주변에 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "실제 부분은 조금 더 높은 것 같습니다. 5, 그래서. 54, 나는 그것을 믿는다. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "그리고 허수부는 무엇이라고 주장합니다. 84? ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "그런 다음 보너스 질문에 응했다면 이것은 세상의 모든 추가 크레딧에 대한 것입니다. 정당화하고 말할 수 있는지 확인하십시오. x와 y가 복소수인 경우에도 이 결과가 여전히 참일까요? 행렬이면 사실인가요? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "제가 처음 본 것은 복소수에 대해 배울 때였던 것 같습니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/marathi/sentence_translations.json b/2020/ldm-eulers-formula/marathi/sentence_translations.json index fe8c3468f..bd77f138d 100644 --- a/2020/ldm-eulers-formula/marathi/sentence_translations.json +++ b/2020/ldm-eulers-formula/marathi/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "आम्ही हे करण्यासाठी काही मिनिटे घेणार आहोत. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "त्यामुळे तुम्हाला समजण्यास मदत करणारी विशिष्ट उदाहरणे असण्याचे स्वरूप e ते i pi सारखी संख्या असणार नाही. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "होय, म्हणून आम्ही प्रत्यक्षात पाहत आहोत की लोक त्याबद्दल विचार करतात आणि विचार करतात, थांबतात, हे असे आहे की येथे पर्याय 3 आवश्यक आहे का? ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f चा x बरोबर 0 हे नकारात्मक 1 च्या f बरोबर 1 च्या f बरोबर कार्य करत नाही. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f चा x बरोबर 0 हे नकारात्मक 1 च्या f बरोबर 1 च्या f बरोबर कार्य करत नाही. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "तर माझा अंदाज आहे की नकारात्मक 1 चा f 1 च्या वर 1 आहे याचा अर्थ असा नाही की आउटपुट कधीही 0 असेल. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "आणि पर्यायांमध्ये 3 च्या गुणाकार किंवा 3 च्या गुणाकारांचा समावेश आहे जोपर्यंत ते सकारात्मक आहेत, किंवा 4 च्या गुणाकाराच्या खाली पूर्णांक 1 किंवा तुम्ही फक्त सकारात्मक असण्यापुरते मर्यादित आहात. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "पुढील उजवीकडे निर्देशित केले आहे, परंतु त्याची परिमाण 1 पेक्षा 24 आहे, आणि ते फक्त एक संपूर्ण घड संकुचित करतात. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "कोणाला माहित होते? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "आणि ज्याचा काल्पनिक भाग आजूबाजूला आहे. ८४. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "खरा भाग थोडा वरचा आहे असे दिसते. 5, त्यामुळे. 54, माझा विश्वास आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "आणि काल्पनिक भाग आहे, तो दावा केला आहे, काय,. ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "आणि मग तुम्ही बोनस प्रश्नासाठी तयार असाल तर, हे जगातील सर्व अतिरिक्त क्रेडिटसाठी आहे, समर्थन करा, तुम्ही म्हणू शकता का ते पहा, x आणि y जटिल संख्या असल्यास हा परिणाम अजूनही खरा असेल का, आणि तरीही ते मॅट्रिक्स असतील तर खरे असेल? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "मला असे वाटते की मी ते पहिल्यांदा पाहिले, ते जटिल संख्यांबद्दल शिकताना होते. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/persian/sentence_translations.json b/2020/ldm-eulers-formula/persian/sentence_translations.json index 87c4d430e..56c2d3c7e 100644 --- a/2020/ldm-eulers-formula/persian/sentence_translations.json +++ b/2020/ldm-eulers-formula/persian/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "برای انجام این کار چند دقیقه وقت می گذاریم. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "بنابراین شکل داشتن مثال های خاص که به درک شما کمک می کند، عددی مانند e تا i pi نخواهد بود. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "بله، بنابراین ما در واقع می بینیم که مردم در مورد آن فکر می کنند و فکر می کنند، صبر کنید، آیا این چنین است که گزینه 3 در اینجا لزوماً درست است؟ و آنها واقعاً در مورد آن انتقاد می کنند. ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f از x مساوی 0 با f از منفی 1 برابر با 1 بر f از 1 کار نمی کند. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f از x مساوی 0 با f از منفی 1 برابر با 1 بر f از 1 کار نمی کند. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "بنابراین حدس می‌زنم که این واقعیت که f از منفی 1 برابر است با 1 بر f از 1 به این معنی نیست که خروجی هرگز 0 خواهد بود. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "و گزینه ها شامل مضربی از 3 یا مضربی از 3 تا زمانی که مثبت هستند، یا اعداد صحیح 1 زیر مضربی از 4 یا مواردی هستند که شما محدود به مثبت بودن آنها هستید. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "قدر بعدی به سمت راست است، اما قدر آن 1 روی 24 است، و آنها فقط یک دسته کامل کوچک می کنند. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "چه کسی می دانست؟ و قسمت خیالی آن اطراف است. 84. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "به نظر می رسد که قسمت واقعی کمی بالاتر است. 5، بنابراین. 54، من این را باور دارم. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "و قسمت خیالی آن است که ادعا می کرد چه،. ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "و سپس اگر برای سوال جایزه آماده هستید، این برای تمام اعتبار اضافی در جهان است، توجیه کنید، ببینید آیا می توانید بگویید، آیا اگر x و y اعداد مختلط هستند، آیا این نتیجه همچنان درست است؟ درست است اگر آنها ماتریس هستند؟ در واقع فکر می‌کنم این تمرین بسیار مهمی است که باید انجام شود تا بفهمیم چه زمانی برای چه نوع ورودی‌هایی درست است. ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "فکر می کنم اولین باری که آن را دیدم، در یادگیری اعداد مختلط بود. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/portuguese/sentence_translations.json b/2020/ldm-eulers-formula/portuguese/sentence_translations.json index b3c1f4858..5f786c5fe 100644 --- a/2020/ldm-eulers-formula/portuguese/sentence_translations.json +++ b/2020/ldm-eulers-formula/portuguese/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "Vamos levar alguns minutos para fazer isso. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "Portanto, a forma de ter exemplos específicos que auxiliam na sua compreensão não seria um número como e elevado a i pi. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "Sim, na verdade estamos vendo as pessoas pensarem sobre isso e pensarem, espere, será que a opção 3 aqui é necessariamente verdadeira? ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f de x igual a 0 não funciona com f de menos 1 igual a 1 sobre f de 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f de x igual a 0 não funciona com f de menos 1 igual a 1 sobre f de 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "Então, acho que o fato de f de 1 negativo ser 1 sobre f de 1 não implica que a saída será 0. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "E as opções incluem múltiplos de 3 ou múltiplos de 3, desde que sejam positivos, ou números inteiros 1 abaixo de um múltiplo de 4 ou aqueles em que você está restrito a eles apenas serem positivos. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "O próximo está apontado para a direita, mas tem magnitude 1 sobre 24, e eles encolhem bastante. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "Quem sabia? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "E cuja parte imaginária está por aí. 84. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "Parece que a parte real está um pouco acima. 5, então. 54, eu acredito nisso. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "E a parte imaginária é, afirmou, o quê. 84? ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "E então, se você estiver pronto para a pergunta bônus, isso é para todo o crédito extra do mundo, justifique, veja se você consegue dizer, esse resultado ainda será verdadeiro se x e y forem números complexos, e ainda será ser verdade se forem matrizes? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "Acho que a primeira vez que vi isso foi aprendendo sobre números complexos. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/russian/sentence_translations.json b/2020/ldm-eulers-formula/russian/sentence_translations.json index 75bea8d82..d11084a2a 100644 --- a/2020/ldm-eulers-formula/russian/sentence_translations.json +++ b/2020/ldm-eulers-formula/russian/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "Нам понадобится пара минут, чтобы сделать это. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "Таким образом, форма предоставления конкретных примеров, которая поможет вашему пониманию, не будет числом, подобным e вместо i pi. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "Да, мы на самом деле видим, как люди думают об этом и думают: подождите, действительно ли вариант 3 здесь обязательно верен? ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f от x равно 0, не работает, если f от отрицательного 1 равно 1, а не f от 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f от x равно 0, не работает, если f от отрицательного 1 равно 1, а не f от 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "Итак, я предполагаю, что тот факт, что f от отрицательного 1 равно 1, а не f от 1, не означает, что результат когда-либо будет равен 0. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "И варианты включают числа, кратные 3 или кратные 3, если они положительны, или целые числа, кратные 1, кратные 4, или те, где вы ограничены только положительными значениями. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "Следующий направлен вправо, но у него звездная величина 1 больше 24, а они просто сжимаются целой кучей. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "Кто знал? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "И чья мнимая часть находится вокруг. 84. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "Похоже, настоящая часть находится чуть выше. 5, так. 54, я в это верю. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "А мнимая часть, как он утверждал, это то, что. 84? ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "А затем, если вы готовы ответить на бонусный вопрос, это за все дополнительные баллы в мире, обоснуйте, посмотрите, сможете ли вы сказать, будет ли этот результат по-прежнему верен, если x и y — комплексные числа, и будет ли он по-прежнему верен? правда, если это матрицы? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "Думаю, впервые я это увидел, когда изучал комплексные числа. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/spanish/sentence_translations.json b/2020/ldm-eulers-formula/spanish/sentence_translations.json index f8acd6d9a..0fc765ab3 100644 --- a/2020/ldm-eulers-formula/spanish/sentence_translations.json +++ b/2020/ldm-eulers-formula/spanish/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "Nos tomaremos un par de minutos para hacer esto. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "Entonces, la forma de tener ejemplos específicos que ayuden a su comprensión no sería un número como e elevado a i pi. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "Sí, en realidad estamos viendo a la gente pensar en ello y pensar, espera, ¿es cierto que la opción 3 aquí es necesariamente cierta? ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f de x es igual a 0 no funciona con f de 1 negativo es igual a 1 sobre f de 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f de x es igual a 0 no funciona con f de 1 negativo es igual a 1 sobre f de 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "Entonces, supongo que el hecho de que f de menos 1 sea 1 sobre f de 1 no implica que la salida alguna vez será 0. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "Y las opciones incluyen múltiplos de 3 o múltiplos de 3 siempre que sean positivos, o números enteros 1 por debajo de un múltiplo de 4 o aquellos en los que está restringido a que sean simplemente positivos. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "El siguiente apunta hacia la derecha, pero tiene magnitud 1 sobre 24, y simplemente se reducen muchísimo. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "¿Quien sabe? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "Y cuya parte imaginaria está alrededor. 84. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "Parece que la parte real está un poco arriba. 5, entonces. 54, eso creo. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "Y la parte imaginaria es, afirmó, qué. 84? ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "Y luego, si está preparado para la pregunta adicional, esto es para obtener todo el crédito adicional del mundo, justifique, vea si puede decir, ¿este resultado seguirá siendo cierto si x e y son números complejos? ¿Será cierto si son matrices? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "Creo que la primera vez que lo vi fue mientras aprendía sobre números complejos. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/tamil/sentence_translations.json b/2020/ldm-eulers-formula/tamil/sentence_translations.json index acad6249a..ea938bec3 100644 --- a/2020/ldm-eulers-formula/tamil/sentence_translations.json +++ b/2020/ldm-eulers-formula/tamil/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "இதைச் செய்ய நாங்கள் இரண்டு நிமிடங்கள் எடுத்துக்கொள்கிறோம். ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "எனவே உங்கள் புரிதலுக்கு உதவும் குறிப்பிட்ட உதாரணங்களைக் கொண்ட வடிவம் e க்கு i pi போன்ற எண்ணாக இருக்காது. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "ஆமாம், எனவே மக்கள் அதைப் பற்றி யோசிப்பதையும், யோசிப்பதையும் நாங்கள் உண்மையில் பார்க்கிறோம், காத்திருக்கவும், இங்கே விருப்பம் 3 அவசியம் உண்மையா? ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f இன் x சமம் 0 ஆனது எதிர்மறை 1 இன் f உடன் வேலை செய்யாது ஆம், x இன் f என்பது 0க்கு சமம். ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "f இன் x சமம் 0 ஆனது எதிர்மறை 1 இன் f உடன் வேலை செய்யாது எனவே எதிர்மறை 1 இன் f என்பது 1 இன் f க்கு மேல் 1 என்பது வெளியீடு எப்பொழுதும் 0 ஆக இருக்கும் என்பதைக் குறிக்காது என்று நினைக்கிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "மேலும் விருப்பங்களில் 3 இன் பெருக்கல்கள் அல்லது 3 இன் பெருக்கல்கள் நேர்மறையாக இருக்கும் வரை, அல்லது 4 இன் பெருக்கத்திற்குக் கீழே உள்ள முழு எண்கள் 1 அல்லது நேர்மறையாக இருக்குமாறு நீங்கள் கட்டுப்படுத்தப்பட்டவை ஆகியவை அடங்கும். ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "அடுத்தது வலதுபுறம் சுட்டிக்காட்டப்பட்டுள்ளது, ஆனால் அது 24க்கு மேல் 1 அளவைக் கொண்டுள்ளது, மேலும் அவை முழுக் கொத்துகளையும் சுருக்கிவிடும். ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "யாருக்கு தெரியும்? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "யாருடைய கற்பனை பகுதி சுற்றி உள்ளது. 84. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "உண்மையான பகுதி சற்று மேலே இருப்பது போல் தெரிகிறது. 5, அதனால். 54, நான் அதை நம்புகிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "மற்றும் கற்பனையான பகுதி, அது கூறியது, என்ன,. 84? ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "பின்னர் நீங்கள் போனஸ் கேள்விக்கு தயாராக இருந்தால், இது உலகில் உள்ள அனைத்து கூடுதல் கடன்களுக்கானது, நியாயப்படுத்துங்கள், நீங்கள் சொல்ல முடியுமா என்று பாருங்கள், x மற்றும் y ஆகியவை கலப்பு எண்களாக இருந்தால் இந்த முடிவு இன்னும் உண்மையாக இருக்குமா, அது இன்னும் இருக்குமா? அவை மெட்ரிக்குகள் என்றால் உண்மையா? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "நான் அதை முதன்முதலில் பார்த்தேன் என்று நினைக்கிறேன், அது கலப்பு எண்களைப் பற்றி கற்றுக்கொண்டது. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/telugu/sentence_translations.json b/2020/ldm-eulers-formula/telugu/sentence_translations.json index 96e13fd44..19c411f8d 100644 --- a/2020/ldm-eulers-formula/telugu/sentence_translations.json +++ b/2020/ldm-eulers-formula/telugu/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "దీన్ని చేయడానికి మేము రెండు నిమిషాలు తీసుకుంటాము. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "కాబట్టి మీ అవగాహనకు సహాయపడే నిర్దిష్ట ఉదాహరణలను కలిగి ఉన్న రూపం i piకి e వంటి సంఖ్య కాదు. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "అవును, కాబట్టి ప్రజలు దాని గురించి ఆలోచించడం మరియు ఆలోచించడం మనం చూస్తున్నాము, వేచి ఉండండి, ఇక్కడ ఎంపిక 3 తప్పనిసరిగా నిజమేనా? ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f యొక్క x సమానం 0 ప్రతికూల 1 యొక్క fతో పని చేయదు, 1 యొక్క f కంటే 1 ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f యొక్క x సమానం 0 ప్రతికూల 1 యొక్క fతో పని చేయదు, 1 యొక్క f కంటే 1 ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "కాబట్టి ప్రతికూల 1 యొక్క f అనేది 1 యొక్క f కంటే 1 అనే వాస్తవం అవుట్‌పుట్ ఎప్పుడూ 0గా ఉంటుందని సూచించదు. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "మరియు ఎంపికలలో 3 యొక్క గుణిజాలు లేదా 3 యొక్క గుణిజాలు సానుకూలంగా ఉన్నంత వరకు ఉంటాయి లేదా పూర్ణాంకాలు 1 కంటే 4 యొక్క గుణకారం కంటే తక్కువగా ఉంటాయి లేదా మీరు వాటిని సానుకూలంగా మాత్రమే పరిమితం చేసినవి ఉంటాయి. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "తదుపరిది కుడివైపుకి చూపబడింది, కానీ దాని పరిమాణం 1 కంటే 24 కంటే ఎక్కువగా ఉంటుంది మరియు అవి మొత్తం సమూహాన్ని కుదించాయి. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "ఎవరికి తెలుసు? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "మరియు ఎవరి ఊహాత్మక భాగం చుట్టూ ఉంది. 84. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "అసలు భాగం కొంచెం పైన ఉన్నట్లు కనిపిస్తోంది. 5, కాబట్టి. 54, నేను నమ్ముతున్నాను. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "మరియు ఊహాత్మక భాగం, అది దావా వేయబడింది, ఏమిటి,. 84? ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "ఆపై మీరు బోనస్ ప్రశ్న కోసం సిద్ధంగా ఉన్నట్లయితే, ఇది ప్రపంచంలోని అన్ని అదనపు క్రెడిట్‌ల కోసం, సమర్థించండి, మీరు చెప్పగలరో లేదో చూడండి, x మరియు y సంక్లిష్ట సంఖ్యలు అయితే ఈ ఫలితం ఇప్పటికీ నిజమవుతుందా మరియు ఇది ఇప్పటికీ ఉంటుందా అవి మాత్రికలు అయితే నిజమేనా? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "నేను దీన్ని మొదటిసారి చూసినట్లు అనుకుంటున్నాను, ఇది సంక్లిష్ట సంఖ్యల గురించి తెలుసుకోవడం. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/thai/sentence_translations.json b/2020/ldm-eulers-formula/thai/sentence_translations.json index 214736d47..3e30a39c4 100644 --- a/2020/ldm-eulers-formula/thai/sentence_translations.json +++ b/2020/ldm-eulers-formula/thai/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "ทฤษฎีบทกรณีพิเศษ เช่น e กำลัง i pi หรือกรณีทั่วไปมากกว่านั้น เช่น e กำลัง i x โอ้ นั่นเป็นคำถามที่ยอดเยี่ยมมาก ว่าแต่ใครอยากถามคำถามก็ไปที่ Twitter แล้วใช้แฮชแท็กล็อคดาวน์คณิตได้เลย สิ่งเหล่านี้จะถูกส่งต่อให้ฉัน ฉันคิดว่าวิธีที่ดีที่สุดในการเรียนรู้คือการมีตัวอย่างที่เฉพาะเจาะจง และปล่อยให้ตัวเองเข้าใจรูปแบบที่แสดงโดยตัวอย่างที่เฉพาะเจาะจงเหล่านั้น จากนั้นจึงสรุปมัน สำหรับอันเฉพาะเจาะจงที่คุณมีตรงนี้ ฉันคิดว่าถ้าคุณเห็น e กำลัง i ไพ นั่นไม่นับว่าเป็นตัวอย่างเฉพาะเจาะจงที่ดีที่อธิบายเรื่องทั่วไปได้ ยิ่งกว่านั้นคุณต้องใส่ตัวเลขเฉพาะลงในสูตร ในกรณีนี้ e กำลัง ix อย่างที่คุณเห็น มันมีทุกอย่างเกี่ยวกับการเคลื่อนที่แบบวงกลม และระหว่างบรรยายนี้กับอันถัดไป เราจะพูดถึงว่าทำไมความสัมพันธ์นั้นถึงอยู่ตรงนั้น ดังนั้นรูปแบบของการมีตัวอย่างเฉพาะเจาะจงที่ช่วยให้คุณเข้าใจได้จะไม่ใช่ตัวเลขเหมือน e กำลัง i pi มันจะเป็นการสร้างความสัมพันธ์กับการเคลื่อนที่แบบวงกลมในสถานการณ์อื่นๆ เช่น การเรียนฟิสิกส์ที่คุณมีเทเธอร์บอลที่มีแรงสู่ศูนย์กลางและมันกำลังโคจรอยู่ หรือเหมือนกับกลศาสตร์การโคจร อะไรก็ตามที่คุณเข้าใจธรรมชาติของการเคลื่อนที่แบบวงกลมจริงๆ นั่นช่วยให้คุณเข้าใจ e กำลัง ix ได้ดีขึ้นนิดหน่อย จากทั้งหมดนี้ คำตอบยังคงมีอยู่เรื่อยๆ และฉันไม่อยากให้คุณรู้สึกเร่งรีบ จริงๆ แล้วผมจะให้เวลาเพิ่มอีกสักหน่อยตรงนี้ เพราะจำไว้ว่า คุณต้องแน่ใจจริงๆ ว่าถ้าคุณบอกว่ามี 2 หรือ 3 หรือ 1 รวมอยู่ในคำตอบด้วย ฟังก์ชันใดๆ ที่มีคุณสมบัติพิเศษนี้จะต้องเป็นไปตามนั้น ตกลง? ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "ใครรู้บ้าง? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "และมีส่วนจินตภาพอยู่รอบๆ 84. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "และในแง่ของภาพลักษณ์ของเรา คุณก็รู้ว่ามันเยี่ยมยอด ดูเหมือนของจริงจะสูงกว่าเล็กน้อย 5 ดังนั้น 54 ผมเชื่ออย่างนั้น และส่วนจินตภาพคือ มันอ้างว่า อะไร 84? ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/turkish/sentence_translations.json b/2020/ldm-eulers-formula/turkish/sentence_translations.json index 972fe8462..7ea8e3ce1 100644 --- a/2020/ldm-eulers-formula/turkish/sentence_translations.json +++ b/2020/ldm-eulers-formula/turkish/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "Bunu yapmak için birkaç dakikanızı ayıracağız. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "Yani anlamanıza yardımcı olacak spesifik örneklere sahip olmanın şekli, e üzeri i pi gibi bir sayı olmayacaktır. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "Evet, aslında insanların bunun hakkında düşündüğünü ve şunu düşündüğünü görüyoruz, bir dakika, buradaki 3. seçenek mutlaka doğru mu? ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f(x) eşittir 0, f(negatif 1) eşittir 1 bölü f(1) ile çalışmıyor. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f(x) eşittir 0, f(negatif 1) eşittir 1 bölü f(1) ile çalışmıyor. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "Yani sanırım f(negatif 1) bölü f(1) olması çıktının hiçbir zaman 0 olacağı anlamına gelmiyor. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "Ve seçenekler, pozitif oldukları sürece 3'ün katlarını veya 3'ün katlarını veya 4'ün katının altındaki 1 tamsayılarını veya yalnızca pozitif olmalarıyla sınırlı olduğunuz durumları içerir. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "Bir sonraki sağa dönük ama büyüklüğü 1 bölü 24 ve bir sürü küçülüyorlar. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "Kim biliyordu? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "Ve kimin hayali kısmı etraftadır. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "Gerçek kısmı biraz yukarıda gibi görünüyor. 5 yani. 54, buna inanıyorum. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "Ve hayali kısmın ne olduğu iddia edildi. 84? ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "Ve eğer bonus soruya hazırsanız, bu dünyadaki tüm ekstra kredi içindir, gerekçelendirin, bakın şunu söyleyebilir misiniz, eğer x ve y karmaşık sayılarsa bu sonuç yine de doğru olacak mı ve yine de doğru olacak mı? bunlar matris ise doğru mudur? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "Sanırım onu ilk kez karmaşık sayıları öğrenirken gördüm. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/ukrainian/sentence_translations.json b/2020/ldm-eulers-formula/ukrainian/sentence_translations.json index 77e76305d..ff91e6bc7 100644 --- a/2020/ldm-eulers-formula/ukrainian/sentence_translations.json +++ b/2020/ldm-eulers-formula/ukrainian/sentence_translations.json @@ -672,7 +672,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this.", + "input": "We're going to take, you know, a couple minutes to do this.", "translatedText": "На це нам знадобиться кілька хвилин.", "n_reviews": 0, "start": 589.64, @@ -1134,7 +1134,7 @@ "end": 978.8 }, { - "input": "One of the options is that f of 5 is equal to f of 1 raised to the power 5.", + "input": "One of the options is that f of five is equal to f of one raised to the power five.", "translatedText": "Один із варіантів полягає в тому, що f від 5 дорівнює f від 1 у степені 5.", "n_reviews": 0, "start": 979.66, @@ -1274,7 +1274,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi.", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi.", "translatedText": "Таким чином, форма наявності конкретних прикладів, які допомагають вашому розумінню, не буде числом, як e до i pi.", "n_reviews": 0, "start": 1103.16, @@ -1379,7 +1379,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true?", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true?", "translatedText": "Так, ми насправді бачимо, як люди думають про це і думають: почекай, чи так варіант 3 тут обов’язково вірний?", "n_reviews": 0, "start": 1177.46, @@ -1876,7 +1876,7 @@ "end": 1539.72 }, { - "input": "We could, if we wanted, define the number e to simply be, where is this polynomial at the number 1?", + "input": "t, we could, we could if we wanted define the number e to simply be where is this polynomial at the number 1.", "translatedText": "Ми могли б, якби захотіли, визначити число e так: де знаходиться цей многочлен при числі 1?", "n_reviews": 0, "start": 1539.8, @@ -2030,7 +2030,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1.", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one.", "translatedText": "f від x дорівнює 0 не працює з f від’ємним 1 дорівнює 1 над f від 1.", "n_reviews": 0, "start": 1668.88, @@ -2058,14 +2058,14 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1.", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one.", "translatedText": "f від x дорівнює 0 не працює з f від’ємним 1 дорівнює 1 над f від 1.", "n_reviews": 0, "start": 1688.4, "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0.", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero.", "translatedText": "Тож я вважаю, що той факт, що f від мінус 1 дорівнює 1 на f від 1, не означає, що результат коли-небудь буде 0.", "n_reviews": 0, "start": 1697.12, @@ -2093,7 +2093,7 @@ "end": 1735.02 }, { - "input": "f of x plus 0 is the same as f of x times f of 0, which implies f of x equals 1.", + "input": "F of x plus zero is the same as f of x times f of zero, which implies f of x equals one.", "translatedText": "f від x плюс 0 те саме, що f від x, помножене на f від 0, що означає, що f від x дорівнює 1.", "n_reviews": 0, "start": 1735.84, @@ -2198,7 +2198,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive.", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive.", "translatedText": "І варіанти включають кратні 3 або кратні 3, якщо вони додатні, або цілі числа 1 менше кратного 4 або ті, де ви обмежені лише додатними.", "n_reviews": 0, "start": 1795.16, @@ -2310,7 +2310,7 @@ "end": 1906.58 }, { - "input": "If it's a power of 4, you're at 1.", + "input": "If it's a power of four, you're at one. If", "translatedText": "Якщо це ступінь 4, ви маєте 1.", "n_reviews": 0, "start": 1906.74, @@ -2464,7 +2464,7 @@ "end": 2031.24 }, { - "input": "Then after that, i cubed is pointed straight down, and we divide it by 6.", + "input": "Okay? Then after that, i cubed is pointed straight down, and we divide it by six.", "translatedText": "Після цього куб i спрямований прямо вниз, і ми ділимо його на 6.", "n_reviews": 0, "start": 2031.78, @@ -2478,7 +2478,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch.", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch.", "translatedText": "Наступний спрямований праворуч, але він має зоряну величину 1 на 24, і вони просто зменшуються на цілу купу.", "n_reviews": 0, "start": 2042.82, @@ -2583,7 +2583,7 @@ "end": 2105.24 }, { - "input": "54.", + "input": "point five four,", "translatedText": "54.", "n_reviews": 0, "start": 2105.24, @@ -2604,7 +2604,7 @@ "end": 2109.42 }, { - "input": "84.", + "input": "point eight four,", "translatedText": "84.", "n_reviews": 0, "start": 2109.42, @@ -2625,7 +2625,7 @@ "end": 2115.62 }, { - "input": "5, so.", + "input": "point five, so point five fo", "translatedText": "5, отже.", "n_reviews": 0, "start": 2115.62, @@ -2646,7 +2646,7 @@ "end": 2121.46 }, { - "input": "84?", + "input": "point eight four?", "translatedText": "84?", "n_reviews": 0, "start": 2121.58, @@ -3206,7 +3206,7 @@ "end": 2564.84 }, { - "input": "Partly because of how we defined it, you know, only doing so many terms, and partly because computers can't do infinite precision.", + "input": "r because, partly because of how we defined it, you know, only doing so many terms, and partly because computers can't do infinite precision.", "translatedText": "Частково через те, як ми це визначили, знаєте, використовуючи лише таку кількість термінів, а частково тому, що комп’ютери не можуть виконувати нескінченну точність.", "n_reviews": 0, "start": 2564.84, @@ -3472,7 +3472,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices?", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices?", "translatedText": "І тоді, якщо ви готові відповісти на бонусне запитання, це за всі додаткові кредити у світі, обґрунтуйте, подивіться, чи можете ви сказати, чи цей результат все ще буде істинним, якщо x і y є комплексними числами, і чи він усе ще бути правдою, якщо це матриці?", "n_reviews": 0, "start": 2829.02, @@ -3528,7 +3528,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers.", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers.", "translatedText": "Я думаю, що вперше я це побачив, коли вивчав комплексні числа.", "n_reviews": 0, "start": 2887.54, diff --git a/2020/ldm-eulers-formula/urdu/sentence_translations.json b/2020/ldm-eulers-formula/urdu/sentence_translations.json index dfa9b2c36..87209efc8 100644 --- a/2020/ldm-eulers-formula/urdu/sentence_translations.json +++ b/2020/ldm-eulers-formula/urdu/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "ہمیں ایسا کرنے میں چند منٹ لگیں گے۔", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "لہذا مخصوص مثالیں رکھنے کی شکل جو آپ کو سمجھنے میں مدد دیتی ہے i pi سے e کی طرح کا نمبر نہیں ہوگا۔", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "ہاں، تو ہم دراصل دیکھ رہے ہیں کہ لوگ اس کے بارے میں سوچتے ہیں اور سوچتے رہتے ہیں، کیا یہ معاملہ ہے کہ یہاں آپشن 3 ضروری طور پر درست ہے؟ اور وہ واقعی اس کے بارے میں تنقید کر رہے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f کا x مساوی ہے 0 منفی 1 کے f کے ساتھ کام نہیں کرتا ہے 1 سے زیادہ f 1 کے برابر ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f کا x مساوی ہے 0 منفی 1 کے f کے ساتھ کام نہیں کرتا ہے 1 سے زیادہ f 1 کے برابر ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "تو میرا اندازہ ہے کہ حقیقت یہ ہے کہ منفی 1 کا f 1 سے زیادہ f 1 کا یہ مطلب نہیں ہے کہ آؤٹ پٹ کبھی 0 ہو گا۔", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "اور اختیارات میں 3 کے ضرب یا 3 کے ضرب شامل ہیں جب تک کہ وہ مثبت ہوں، یا 4 کے ضرب سے نیچے 1 عدد یا وہ جہاں آپ صرف مثبت ہونے تک ان تک محدود ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "اگلا ایک دائیں طرف اشارہ کیا گیا ہے، لیکن اس کی شدت 1 سے زیادہ 24 ہے، اور وہ صرف ایک پورے گروپ کو سکڑتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "کون جانتا تھا؟ اور جس کا خیالی حصہ ارد گرد ہے۔84. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "ایسا لگتا ہے کہ اصلی حصہ تھوڑا اوپر ہے۔5، تو. 54، مجھے یقین ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "اور خیالی حصہ ہے، اس نے دعویٰ کیا، کیا،۔", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "اور پھر اگر آپ بونس کے سوال کے لیے تیار ہیں، تو یہ دنیا کے تمام اضافی کریڈٹ کے لیے ہے، جواز پیش کریں، دیکھیں کہ کیا آپ کہہ سکتے ہیں، کیا یہ نتیجہ اب بھی درست ہوگا اگر x اور y پیچیدہ اعداد ہیں، اور کیا یہ اب بھی ہے سچ ہو اگر وہ میٹرک ہیں؟ یہ دراصل میرے خیال میں یہ سمجھنے کے لیے ایک بہت اہم مشق ہے کہ یہ کس قسم کے ان پٹ کے لیے درست ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "میرے خیال میں پہلی بار جب میں نے اسے دیکھا تھا، یہ پیچیدہ نمبروں کے بارے میں سیکھنے میں تھا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-eulers-formula/vietnamese/sentence_translations.json b/2020/ldm-eulers-formula/vietnamese/sentence_translations.json index 86a4f1cb8..4553f1947 100644 --- a/2020/ldm-eulers-formula/vietnamese/sentence_translations.json +++ b/2020/ldm-eulers-formula/vietnamese/sentence_translations.json @@ -760,7 +760,7 @@ "end": 589.62 }, { - "input": "We're going to take a couple minutes to do this. ", + "input": "We're going to take, you know, a couple minutes to do this. ", "translatedText": "Chúng ta sẽ mất một vài phút để làm điều này. ", "model": "google_nmt", "n_reviews": 0, @@ -1456,7 +1456,7 @@ "end": 1102.94 }, { - "input": "So the form of having specific examples that aids your understanding wouldn't be a number like e to the i pi. ", + "input": "So the form of having specific examples that aids your understanding wouldn't be, you know, a number like e to the i pi. ", "translatedText": "Vì vậy, dạng có ví dụ cụ thể giúp bạn hiểu sẽ không phải là một số như e mũ i pi. ", "model": "google_nmt", "n_reviews": 0, @@ -1576,7 +1576,7 @@ "end": 1176.44 }, { - "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, is it the case that option 3 here is necessarily true? ", + "input": "Yeah, so we're actually seeing people think about it and thinking, hang on, you know, is it the case that option 3 here is necessarily true? ", "translatedText": "Vâng, vì vậy chúng tôi thực sự đang thấy mọi người nghĩ về điều đó và suy nghĩ, chờ đã, liệu phương án 3 ở đây có nhất thiết đúng không? ", "model": "google_nmt", "n_reviews": 0, @@ -2320,7 +2320,7 @@ "end": 1665.62 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f của x bằng 0 không đúng với f âm 1 bằng 1 trên f 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2352,7 +2352,7 @@ "end": 1688.22 }, { - "input": "f of x equals 0 doesn't work with f of negative 1 equals 1 over f of 1. ", + "input": "F of x equals zero doesn't work with f of negative one equals one over f of one. ", "translatedText": "f của x bằng 0 không đúng với f âm 1 bằng 1 trên f 1. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1693.38 }, { - "input": "So I guess that fact that f of negative 1 is 1 over f of 1 doesn't imply that the output will ever be 0. ", + "input": "So I guess that fact that f of negative one is one over f of one doesn't imply that the output will ever be zero. ", "translatedText": "Vì vậy, tôi đoán rằng thực tế là f của âm 1 bằng 1 trên f của 1 không có nghĩa là đầu ra sẽ luôn bằng 0. ", "model": "google_nmt", "n_reviews": 0, @@ -2512,7 +2512,7 @@ "end": 1794.4 }, { - "input": "And the options include multiples of 3 or multiples of 3 as long as they're positive, or integers 1 below a multiple of 4 or those where you're restricted to them just being positive. ", + "input": "And the options include multiples of three or multiples of three as long as they're positive, or integers one below a multiple of four or those where you're restricted to them just being positive. ", "translatedText": "Và các tùy chọn bao gồm bội số của 3 hoặc bội số của 3 miễn là chúng dương hoặc số nguyên 1 dưới bội số của 4 hoặc những số mà bạn bị hạn chế đối với chúng chỉ là số dương. ", "model": "google_nmt", "n_reviews": 0, @@ -2832,7 +2832,7 @@ "end": 2042.8 }, { - "input": "The next one is pointed to the right, but it has magnitude 1 over 24, and they just shrink a whole bunch. ", + "input": "The next one is pointed to the right, but it has magnitude one over twenty four, and they just shrink a whole bunch. ", "translatedText": "Cái tiếp theo hướng về bên phải, nhưng nó có độ lớn 1 trên 24, và chúng chỉ thu nhỏ lại cả đống. ", "model": "google_nmt", "n_reviews": 0, @@ -2952,7 +2952,7 @@ "end": 2105.5 }, { - "input": "Who knew? ", + "input": "five four, who knew? ", "translatedText": "Ai biết? ", "model": "google_nmt", "n_reviews": 0, @@ -2960,7 +2960,7 @@ "end": 2106.66 }, { - "input": "And whose imaginary part is around .84. ", + "input": "And whose imaginary part is around point eight four, ", "translatedText": "Và phần tưởng tượng của ai ở xung quanh. 84. ", "model": "google_nmt", "n_reviews": 0, @@ -2976,7 +2976,7 @@ "end": 2113.56 }, { - "input": "It looks like the real part is a little above .5, so .54, I believe that. ", + "input": "It looks like the real part is a little above point five, so point five four, I believe that. ", "translatedText": "Có vẻ như phần thật ở trên một chút. 5, vậy. 54, tôi tin điều đó. ", "model": "google_nmt", "n_reviews": 0, @@ -2984,7 +2984,7 @@ "end": 2118.46 }, { - "input": "And the imaginary part is, it claimed, what, .84? ", + "input": "And the imaginary part is, it claimed what, point eight four? ", "translatedText": "Và phần tưởng tượng là, nó tuyên bố, cái gì,. ", "model": "google_nmt", "n_reviews": 0, @@ -3896,7 +3896,7 @@ "end": 2828.06 }, { - "input": "And then if you're up for the bonus question, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers, and will it still be true if they're matrices? ", + "input": "And then if you're up for the bonus question, this is kind of the, this is for all of the extra credit in the world, justify, see if you can say, will this result still be true if x and y are complex numbers? And will it still be true if they're matrices? ", "translatedText": "Và sau đó, nếu bạn sẵn sàng trả lời câu hỏi thưởng, đây là dành cho tất cả khoản tín dụng bổ sung trên thế giới, hãy chứng minh, xem liệu bạn có thể nói liệu kết quả này có còn đúng không nếu x và y là số phức, và liệu nó có còn đúng không? có đúng nếu chúng là ma trận không? ", "model": "google_nmt", "n_reviews": 0, @@ -3960,7 +3960,7 @@ "end": 2887.4 }, { - "input": "I think the first time that I ever saw it, it was in learning about complex numbers. ", + "input": "I think the first time that I ever saw it, it was in, you know, learning about complex numbers. ", "translatedText": "Tôi nghĩ lần đầu tiên tôi nhìn thấy nó là khi học về số phức. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/arabic/sentence_translations.json b/2020/ldm-i-to-i/arabic/sentence_translations.json index 5822661a1..070cdaa03 100644 --- a/2020/ldm-i-to-i/arabic/sentence_translations.json +++ b/2020/ldm-i-to-i/arabic/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "لذا، إذا كنت تبدأ من الرقم 1، فإن سرعتك الأولية هي السير بشكل مستقيم نحو 0، وكلما مشيت إلى مستوى أقل، إذا كنت تجلس عند النصف 1، فستظل تسير نحو 0، ولكن الآن متجه السرعة الخاص بك سيكون سالب 1 مرة حيث أنت، وهو سالب 1 نصف. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "والسؤال المثير للاهتمام هو أنك تعرف هل هناك دالة واحدة فقط من هذا القبيل تبدو معقولة للكتابة عنها لأنك تعرف ما إذا كنا سنكتبها بالشكل i إلى x، ليس فقط يجب أن تلبي هذا، بل يجب أن تُرضيك أيضًا عندما تعرف متى نقوم بتوصيل الرقم الأول الذي نحصل عليه i (من المفترض i) بالقوة الأولى ولكننا نفكر في أن هذه الوظيفة يجب أن تكون i. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "إذن لدينا 5 باي i نصفين وهي قيمة أخرى بالتأكيد يمكننا التعويض عنها بـ x هنا وفقط لتوضيح ذلك بصريًا أكثر قليلاً إذا أردنا أن ننظر إلى الوراء في دائرتنا هنا حيث لدينا لحظة مشيت لفترة من الوقت تساوي نصفين باي وهو 1.57 ماذا لو بدلًا من ذلك أخذنا دورة كاملة أخرى وذهبنا إلى نصفين pi آخر لنصل إلى pi الذي تعلم أنه قد نسجل فيه نوعًا ما، حيث تكون قيمة e إلى pi i هي أننا نسير نصفين pi آخرين نمشي نصفين pi آخرين عند في هذه النقطة، سنكون قد قطعنا دائرة كاملة لنعود إلى الواحد ثم نسير لخمسة نصفين، وهو ما يساوي عدديًا حوالي 7.85 نعم، هذا بالتأكيد رقم آخر يجعلنا نتفوق على i وإذا أردنا أن نمر عبر عملية إعادة التعبير عن i إلى الأس i عن طريق الكتابة أولاً e إلى 5 pi نصفين i إلى الأس i هؤلاء i اضرب لتصبح سالبًا وسننظر إلى e إلى النصف السالب 5 pi وهو رقم مختلف تمامًا، يمكننا بالفعل حساب هذا، لست متأكدًا من الجزء العلوي من رأسي، ولكن دعونا نلقي نظرة على Desmos . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "85 وحدة زمنية وهي 5 باي نصفين وماذا يحدث عندما تتحلل؟ هذا الطول الذي يوصلك إلى رقم أصغر بكثير ولكن هذه ليست الإجابة الوحيدة التي يمكننا إدخالها، فلدينا أشخاص آخرون يأتون إلى هنا مع سالب 3 نصفين في i pi الذي تعرفه من حيث دائرة الوحدة؟ يمكننا أن نفكر في قول مرحبًا إذا كنت أرغب في الوصول إلى ذلك بدلاً من المشي 90 درجة باي نصفين راديان بهذه الطريقة ماذا لو مشيت 270 درجة في الاتجاه الآخر 3 باي نصفين راديان وهو ما ربما سأفكر فيه على أنه سلبي لأن التقليد هو عادةً ما يكون عكس اتجاه عقارب الساعة موجبًا، وهذه بالتأكيد طريقة أخرى للتعبير عن ذلك، وهذا من شأنه أن يعطينا إجابة مختلفة إذا كان لدينا e أس سالب 3 pi نصفين i الكل للقوة i، سنخوض نفس اللعبة الآن، حيث يتم إلغاء i تربيع بـ سالب موجود بالفعل، ولدينا نصفين موجبين 3pi، ومن الناحية العددية، فإن هذا يعطينا إجابة مختلفة تمامًا عما كان لدينا من قبل والذي إذا تابعنا وقلنا مرحبًا، ما هو e بالنسبة إلى 3pi وليس 3 o 3 pi النصفان 111 نقطة 3 1 نوع مختلف تمامًا من الأرقام عما رأيناه قبل 111 نقطة ماذا كان 111 نقطة 3 1 عظيم 111 نقطة 3 1 أو نحو ذلك ومرة أخرى فيما يتعلق بالحدس، ما قد تتساءل عنه هو افتراض أن لدينا هذا التدوير ديناميكي لكننا نتحرك إلى الوراء في الوقت المناسب ونرى كم من الوقت مضى ما يجب أن أكون عليه بحيث إذا لعبت الأشياء للأمام من هناك سأصل إلى الرقم واحد في حالتي الأولية وعليك العودة في الوقت المناسب 3 وحدات نصف باي وبعد ذلك إذا كنت ستترجم إلى ديناميكيات الاضمحلال وهو ما يفعله رفع العين في هذا السياق، فستقول إذا كنت أبدأ من الرقم واحد ولكني أريد الرجوع إلى الوراء في الوقت المناسب وأقول أين كان يجب أن أبدأ إذا أريد أن أتحلل بحيث ينتهي بي الأمر في المركز الأول؟ بعد 3 وحدات نصفية من الوقت، من الواضح أن الإجابة تبدأ عند حوالي مائة وأحد عشر لهذا النوع من الاضمحلال الأسي ويمكنك أن ترى إلى أين يتجه هذا حيث يوجد في الواقع العديد من القيم المختلفة التي يمكننا تعويضها بـ X إذا كنا التفكير في e إلى X على أنه أنا وقد أدخل الناس الكثير هنا، عذرًا، ألقي دبوسي على الأرض كما يفعل المرء الكلاسيكي للمركز الثالث 9 pi نصفين خيار رائع 1729 pi نصفين، أنتم جميعًا المفضلين لدي كثيرًا والكثير من خيارات مختلفة، عدد لا نهائي من القيم المختلفة التي تبدو مربكة بعض الشيء في البداية لأننا ننظر إلى تعبير يبدو أنك تعلم أنه سيكون هناك بعض العمليات الحسابية، فقط أدخل ذلك في الآلة الحاسبة الخاصة بي وأرى ما سيظهر ولدينا عدة اختلافات القيم لذلك إذن ما الذي يحدث هنا أليس كذلك؟ ماذا يحدث وأعتقد؟ هذا يختصر حقًا فكرة كيفية تفكيرنا في الأسي بشكل عام ولكن قبل ذلك أريد التأكيد على أن هذه ليست المرة الوحيدة في الرياضيات حيث نواجه نوعًا من الغموض حول كيفية تفسير شيء ما؟ أعتقد أنه إذا قلت شيئًا مثل ما هو الجذر التربيعي لـ 25؟ كما تعلم، أعتقد أن الكثير منا يقول حسنًا إنه خمسة ولكن إذا كنا نقول أنك تعرف ما هو الجذر التربيعي الذي يجب أن يكون رقم X بحيث تحصل على 25 عندما تقوم بتربيعه؟ حسنًا، هناك إجابتان مختلفتان لذلك من الذي يقول إن اتفاقياتنا يجب أن تكون أن دالة الجذر التربيعي موجبة تعطينا عددًا موجبًا بدلاً من؟ سالب خمسة إذن لدينا تعبير واحد يبدو وكأنه يريد أن يكون له عدة قيم مختلفة، صحيح ويمكن أن يحدث هذا بالفعل في سياق آخر حيث بدلاً من الجذور التربيعية ماذا لو كنت أسأل عن الجذر الرابع لشيء مثل 16؟ عادةً ما نفكر في هذا كرقم موجب اثنين. ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "نحن نعتمد تقليدًا عندما يكون هناك خيارات متعددة مثل هذا عندما يكون لديك دالة متعددة القيم. غالبًا ما نختار إحدى هذه القيم لتكون ما نعنيه عندما نريد ذلك تعامل معها كدالة كشيء له مدخل واحد ومخرج واحد بلغة أكثر روعة. ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "يأتي هذا طوال الوقت عندما نتعامل مع الأعداد المركبة، فكرة شيء ما كنوع من العمليات التي ترغب في الحصول على قيم متعددة، ستحتاج أحيانًا سماع عبارة فرع حيث تختار فرعا من وظيفة الجذر التربيعي؟ وهذا يعني أنك اخترت اتفاقية معينة؟ في الأعداد الحقيقية، يكون الأمر لطيفًا وسهلًا في بعض الأحيان لأنك تقول فقط اختر العدد الموجب ولكن لا توجد فكرة عن الأعداد المركبة التي تشبه الأعداد المركبة الموجبة عندما نريد أن نأخذ جذورًا تربيعية فقط لإعطاء مثال واحد لنفترض أننا أردنا أخذ المربع جذر أنا ونريد أن نعرف ماذا ينبغي أن يكون؟ لأن هناك العديد من الإجابات المختلفة، كما تعلمون، نحن نفكر مرة أخرى في هذا الدوران بزاوية 90 درجة، وإذا كنا نفكر فيه على أنه دوران بزاوية 90 درجة، يبدو الأمر كما لو أن الجذر التربيعي يجب أن يكون، كما تعلمون، هناك شيء يجلس بزاوية 45 درجة ربما هذا هو المربع جذر I الذي يمكننا كتابته بوضوح شديد كجذر 2 على 2 جذر 2 على 2 I هذا مجرد استخدام لعلم المثلثات ولكن إذا كنا نفكر في I بدلاً من ذلك على أنها دوران سلبي بمقدار 270 درجة، يبدو الأمر وكأن نصف ذلك يقوم بنصف تلك العملية يجب أن يقودنا إلى الجانب الآخر ربما الرقم الموجود هنا يجب أن يكون الجذر التربيعي لـ I وهذا في الواقع مجرد سالب ما رأيناه من قبل سالب الجذر 2 على 2 ناقص جذر 2 على 2 ضرب I الآن في سياق الحقيقي وظائف قيمة يمكننا أن نقول نعم، ما عليك سوى اختيار الجذر التربيعي ليكون الإجابة الإيجابية مهما كانت، ولكن أي منها تعتبر الإجابة الإيجابية؟ كما تعلم، ربما يبدو الأمر وكأنه ما يجب أن نعتبره هذا الرقم العلوي لأن إحداثياته لها أرقام موجبة ولكن مهما حاولت تعريف موجب بطريقة لطيفة هنا، فسيكون ذلك متسقًا كما تعلم على سبيل المثال، يجب أن يتضاعف رقمان موجبان دائمًا للحصول على موجب الرقم لن تكون قادرًا حقًا على القيام بذلك بالطريقة التي يمكنك بها التعامل مع الأعداد الحقيقية وفي الواقع هذه الظاهرة هنا حيث نتجذر هي في الواقع نفس الظاهرة التي كنا ننظر إليها للتو عندما كنا نتحدث عنها القيم المتعددة التي رفعتها إلى قوة I لأنني نسيت أنني رفعت إلى قوة دعني أسأل ما هو المقدار الذي قد يبدو عليه، إنه سؤال أبسط بكثير وهو أخذ 2 إلى الأس 1 إلى النصف حسنًا، ما هو 2 إلى النصف 1؟ وأعتقد أنك تقول جيدًا، نحن نعرف ما هو هذا، ونعرفه على أنه الجذر التربيعي لـ 2، كل شيء جيد وجيد، ولكن ماذا لو قلت دعونا نتعامل مع هذا بنفس الطريقة التي كنا نقترب بها من "أنا" إلى تعبير "أنا" أريد أولاً التعبير عن الأشياء كـ e للشيء الصحيح ثم سأرفع ذلك إلى النصف بضرب النصف 1 في الأس وأقول حسنًا، أعتقد أنه يمكنني فعل ذلك e إلى ما هو يساوي 2 حسنًا، هذا هو اللوغاريتم الطبيعي لـ 2، وهو ثابت يبلغ حوالي 0. ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "69 أو نحو ذلك إذا رفعنا e إلى تلك القوة فسنحصل على 2 لذا يمكن أن نفكر في هذا على أنه e للسجل الطبيعي لـ 2 ضرب 1 نصف وإذا أردت ذلك إذا كنت تفكر في e إلى x؟ أنت تعلم أن هذا قد يكون نوعًا من المبالغة في سياق الأعداد الحقيقية، ولكن إذا كنت تفكر في e إلى x كاختصار لوظيفة x هذه، فيمكنك توصيل القيمة 0. ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "69 ضرب 1 النصف الذي أعتقد أنه سيكون حوالي 0.345، شيء من هذا القبيل، قم بإدخال تلك القيمة الملموسة في كثير الحدود الخاص بك، وانظر ما الذي سيخرجه، وستخرج حوالي 1.414 جذر تربيعي لعدد حقيقي جميل لـ 2 ما تتوقعه ولكن إذا فعلنا نفس الشيء الذي كنا نفعله للتو مع I والاعتراف بأن هناك في الواقع إجابات متعددة مختلفة عندما نريد كتابة شيء ما كـ e لقوة يمكننا أيضًا كتابة هذا قد يبدو هذا مضحكًا، لكن يمكننا كتابته على الصورة e إلى اللوغاريتم الطبيعي لـ 2 زائد 2 pi I. هذا الأمر برمته مرفوع إلى النصف 1 مباشرة بعد كل هذه القيمة سوف تساوي يمكنك تقسيمها كما هي e إلى اللوغاريتم الطبيعي لـ 2 مضروبًا في e إلى 2 pi I هذا له تأثير تدوير الأشياء 360 درجة، لذا فهو يساوي 1 إذن نحن ننظر إلى 2 ضرب 1 رائع يبدو وكأنه استبدال صالح، ومع ذلك متى نحن نلعب نفس اللعبة المتمثلة في أخذ هذا ورفعه إلى قوة ومعالجة ذلك عن طريق ضرب الأس في الأس انظر إلى ما يحدث لدينا e للسجل الطبيعي لـ 2 ضرب 1 نصف زائد حسنًا، ما هو 2 pi I ضرب 1 نصف حسنًا، هذا سيكون pi مضروبًا في I. الآن هذا الجزء الأول e إلى اللوغاريتم الطبيعي لـ 2 ضرب 1 نصف والذي سينتهي به الأمر إلى الجذر التربيعي المألوف لـ 2، هذا جيد وجيد، لكننا سنضرب ذلك في e إلى the pi I صحيح ومشهور جدًا e إلى pi I هو سالب 1 لذا في هذه الحالة يبدو أنه يقترح أنه إذا كنا نحل هذا التعبير 2 إلى النصف 1 من خلال اللعب بالإجابات المختلفة، فيمكننا التعويض بشيء مثل e إلى X يساوي 1 نصف ما ننتهي إليه هو إجابة أخرى قد نكتبها تقليديًا كجذر تربيعي سلبي لـ 2 وهنا أعني أنه من المضحك بعض الشيء أن يكون هناك قيم متعددة للنظر إلى 2 إلى 1 نصف و لنفترض أن هذا لا يساوي شيئًا واحدًا ولكن بناءً على الاختيارات التي نتخذها، يمكن أن يساوي عدة أشياء مختلفة ولكن الشيئين اللذين قد يبدوان معقولين تمامًا إذا كان هناك أي شيء يساوي 2 إلى 1 في النصف، فيبدو أنه يجب أن يكون إيجابيًا الجذر التربيعي الذي نعرفه أو المتغير السلبي لذلك لا يبدو في الواقع مثل هذه المشكلة وفي الواقع يمكننا أن نلعب هذه اللعبة بشكل أكبر حيث دعني أطلب منك المزيد من الإجابات الإبداعية لهذا التعبير لأنه ربما يمكننا العثور على قوى مضحكة أخرى لشيء مثل 2 أس X عندما نبدأ في توصيل قيم مختلفة مختلفة لـ X بناءً على الاستبدال الذي نقوم به إذا كنا نلتزم بنفس القواعد التي كنا نستخدمها في تقييم I أس power I إذن هذه المرة يطرح السؤال أو يحدد أن أحد حلول المعادلة e إلى x يساوي 2 هو الرقم الحقيقي اللوغاريتم الطبيعي لـ 2 حسنًا، هذا هو الحل الذي نعرفه. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "هل يمكنك التفكير في واحدة أخرى وهل يمكنك كتابة واحدة أخرى؟ الإجابة على السؤال e لـ x يساوي 2 ومرة أخرى، نرحب بالإبداع، لذا سأمنحك لحظة صغيرة أخرى لذلك سأمضي قدمًا وأحصل على بعض الإجابات هنا إذا كان هذا مناسبًا لك، لست متأكدًا من مقدار الوقت الذي أستغرقه يتطلب الأمر بالضرورة إجراء إدخال رياضي اعتمادًا على الجهاز الذي تنظر إليه ولكن لا تشعر بالتوتر الشديد إذا كان ذلك قبل أن تتاح لك الفرصة للإجابة على السؤال الذي تريده في الإجابة التي تريد الإجابة عليها، لذلك يبدو الأمر كما يلي: لقد دخل 131 منكم إلى المتغير حيث نأخذ Ln of 2 ونضيف 2ii وأعتقد أنني أكتب هذا السؤال عن طريق الخطأ، مثل وضع علامة على إحدى الإجابات على أنها صحيحة بينما يوجد في الواقع عدد لا بأس به من الإجابات الصحيحة المختلفة، لذا فهذا الأمر يقع على عاتقي لحقيقة أنني لا أعرف إذا كان يبدو لأي منكم مثل أوه، إنه أحمر، لقد أخطأت في ذلك عندما أدخلت Ln of 2 plus 42. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi وهو بالطبع خيار رائع ولكن يمكنك أيضًا الحصول على شيء مثل 4 pi I بالإضافة إلى السجل الطبيعي لـ 2 أو 6 pi I أو أي عدد صحيح مضاعف لـ 2 pi I إذا أضفت أنه لا يؤثر على e إلى X لأنه يحتوي فقط على تأثير الضرب بـ e إلى 2 pi I وهو تأثير الضرب بـ 1 ومرة أخرى، فإن هذا له نوع من العواقب المضحكة حيث يبدو أنه ينتج نوعًا من النتائج المعقولة عندما نفعل ذلك كمثال آخر. يبدو أن التعبير المدخل الثاني الأكثر شيوعًا هو أننا قد نستبدل 2 لذا دعونا نعتقد أننا نفكر في 2 أس 14، حسنًا، كان هناك اقتراح باستبدال 2 بـ e إلى اللوغاريتم الطبيعي لـ 2 زائد 4 pi I حسنًا زائد 4 pi I ونرفع كل ذلك إلى 14 حسنًا، إذا كنت ستلعب نفس اللعبة، فستحصل على e إلى اللوغاريتم الطبيعي لـ 2 في 14، وسنضرب في e إلى pi I الآن الجزء الأول من ذلك سيكون الجذر الرابع الموجب المعتاد لـ 2، وهو ما نعنيه عندما نعوض بتعبير مثل الجذر الرابع لـ 2 في الآلة الحاسبة وهو رقم موجب صغير لطيف، ولكن بعد ذلك الجزء الثاني هو سالب 1، لذا يبدو أنه يقول أنت تعلم إذا أردنا تفسير 2 بهذه الطريقة المختلفة ورفعه إلى الرابع عشر، فأنت تعلم أنها ليست الإجابة المعتادة التي نحصل عليها ولكنها إجابة معقولة. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "كنا ننظر إلى نصفي باي مضروبًا في I، وبدلاً من الضرب في سالب 1 كنا نضرب في I، وهي إجابة صحيحة مرة أخرى، ويبدو أنها نتيجة معقولة لشيء مثل 2 أس 14، لذلك عندما تكون بالنظر إلى حقيقة أنني بالنسبة للقوة يبدو أن لدي قيمًا مختلفة متعددة لذلك، لدينا هذه الظاهرة المضحكة حيث يمكننا توصيل e بنصفي 5 pi I سالب 3 pi نصفي I ونحصل على ما يبدو وكأنه إجابات مختلفة تمامًا شيء صغير جدًا، شيء كبير جدًا، مختلف جدًا عن الإجابة الخامسة عشرة تقريبًا والإجابة الخامسة عشرة التي وجدناها من قبل هنا إنها نفس الظاهرة تمامًا كما هو الحال عندما تسأل شيئًا مثل ما هو 2 أس 14 وتعترف بوجود العديد من الحلول المختلفة بالفعل إلى التعبير X إلى الرابع يساوي 2 4 حلول مختلفة في الواقع وما تنظر إليه هو حقيقة أن هناك العديد من الحلول المختلفة إلى التعبير e إلى X يساوي نوعًا ما من القاعدة سواء كانت هذه القاعدة هي I سواء كانت تلك القاعدة هي 2 أيًا كان الأمر، وإحدى الطرق التي قد نفكر بها في هذا الأمر هي أنه عندما تتعامل مع أرقام حقيقية، فإن الأشياء تكون جميلة، والأشياء لطيفة، وهناك علاقات فردية. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "إنه أمر رائع حيث إذا أردنا التفكير في الدوال الأسية، دعني أغطي بعضًا من هذه الأشياء لدينا هذا ذهابًا وإيابًا حيث يمكنك اختيار التعبير عن أي أسي كأساس لـ X مثل 2 إلى X أو يمكنك التعبير نفس الأسي مثل X لـ R مضروبًا في X والذي تعلم أنه هو كثير الحدود الذي نشير إليه عندما نشير إليه ضمنيًا عندما نكتب شيئًا مثل e إلى X وهناك إرجاع جميل ذهابًا وإيابًا لأنه يمكنك فقط أخذ لوغاريتم طبيعي لـ B ويعطيك إجابة واحدة على افتراض أن B هو رقم موجب وهذا هو نفس القول أن X لـ R يساوي B لذا إحدى الطرق التي تحدثت بها عن هذا سابقًا في السلسلة هي أنه إذا كنت تنظر إلى عائلة من جميع الأسيات الممكنة، يمكننا كتابتها بالشكل X لـ R في X وتغيير ما هو R وهذا هو بالضبط نفس الشيء مثل كتابة e إلى R في X إذا كان هذا هو الشيء الذي يناسبك أكثر، لذا e إلى R مضروبًا في XX لـ R مضروبًا في X، هذا هو نفس الشيء الذي يمكن أن نفكر فيه حول تغيير ما هو ولكن من ناحية أخرى، إذا كنت تفكر في جميع الأسيات المحتملة كقاعدة ما، دعني أقوم بحساب أس X وسنفعل ذلك لتغيير ماهية تلك القاعدة في البداية، يبدو الأمر وكأنه نوع مختلف من التعبير يمكن التلاعب به، ولكنه مجرد طريقة أخرى للتعبير عن نفس العائلة وطريقة قد تفكر بها حول هذا الأمر، فكيف نفكر في أي قاعدة تتوافق معها إذا كنا نفكر بشكل أكثر تجريدًا مثل Exp of R مرات X وهناك سبب للقيام بذلك لأننا على وشك تطبيق هذا على الأعداد المركبة حيث سيبدو الأمر أكثر غرابة، لذا تابع معي هنا إذا بدلاً من النظر إلى تلك القاعدة، هناك شيء واحد يمكنني فعله وهو أن أقول ما هي القيمة؟ من EXP لـ R صحيح وهو في الأساس هذه الوظيفة، ولكن عندما ندخل واحدًا، فإن EXP لـ R مضروبًا في واحد إذا كنت تفضل التفكير في الأمر بهذه الطريقة ويتم تمثيل ذلك بخطنا الأخضر وما يمكنك رؤيته لا بأس به إذا كنت احصل على R هذا العامل أمام X في دالتي الأسية exp لـ RX لتكون صفر، ستة تسعة، وهو ما أعرفه حول اللوغاريتم الطبيعي لاثنين، ما الذي يعنيه هذا؟ X لواحد يساوي اثنين تقريبًا، وهذا يتوافق مع الدالة التي نكتبها عادةً في صورة اثنين أس X صحيح، حسنًا، وفي الأساس أثناء قيامي بالتغيير حول R، كما تعلم، يمكنني محاولة تغييرها إلى شيء ما بحيث يبدو كما يلي ثلاثة إذن حول نقطة واحدة واحدة تبدو تلك الأسية مثل ثلاثة والتي نكتبها عادةً على هيئة ثلاثة أس XI. ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "يمكن أن يكون لدي قيمة R مضروبة في X حيث ربما تكون R تساوي صفر فاصل ستة تسعة لكن يمكنني إزاحة ذلك لأسفل بمقدار اثنين pi I وهذا لا يغير الأساس الذي سيتوافق معه والذي سيظل يتوافق مع اثنين أو يمكن أن يكون قم بإزاحتها لأعلى بمقدار اثنين pi I لا يغير القاعدة التي تتوافق معها لأنه في كل تلك الحالات عندما نعوض X بواحد نحصل على نفس الشيء ولكن كل هذه القيم المختلفة لـ X هي وظائف متميزة هذا هو لماذا رأينا عدة قيم مختلفة لـ I للقوة I لأن I لـ X هي دالة غامضة في هذا السياق، سيكون من الواضح أنه لا لبس فيه إذا قررنا أي قيمة لـ R بحيث يكون ما نمثله هو EXP لـ R مضروبًا في X أي قيمة من ر. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "هل نختار بمجرد أن نختار واحدًا؟ إنها دالة لا لبس فيها ولكن في هذه المرحلة يبدو الأمر كما لو أن ما نريده هو التوقف عن التفكير في الأشياء من حيث قاعدة ما مرفوعة إلى القوة X ربما بمجرد أن نكون في سياق الأعداد المركبة يجب أن نكتب فقط كل هذه الأمثلة تمثل ضربًا ثابتًا X إذا لم يكن هناك أي سبب آخر يوضح ذلك تمامًا كيف يمكننا بالفعل توصيل الأرقام إذا أردنا إجراء عملية حسابية أو مجرد إجراء عمليات حسابية فوقها، فلدينا كثيرة الحدود اللانهائية اللطيفة التي لدينا قم بتوصيلها وسأقدم لك حجة أخرى مفادها أن هذه ربما تكون الطريقة الصحيحة للتفكير في الأُسيات بمجرد أن نتوسع في مجالات أخرى مثل الأعداد المركبة ولهذا فلنقم بعمل نسخة احتياطية العودة إلى جرس الباب، وصلت بعض الأشياء، نعود إلى الطريقة الأصلية التي نوسع بها فكرة الأسي ونفكر فقط في ما هو 2 إلى X. صحيح أننا نعرف كيف نفكر في هذا الأمر بالنسبة للأعداد الطبيعية. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "أنت تعرف شيئًا مثل 2 إلى 3 الضرب المتكرر كيف يتم تعليمك أولاً التفكير في شيء مثل 2 إلى X للكميات الكسرية أو للكميات السالبة وأشياء من هذا القبيل. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "يتم تعليمك عادةً أن 2 أس 1 يجب أن يكون شيئًا حيث تعرف إذا قمت بضربه في نفسه وهذا يتبع القواعد المعتادة التي تطبقها الأسية في حساب الأرقام حيث يمكننا إضافة أشياء في هذا الأس يجب أن أحصل على 2 إلى 1 لذا يجب أن يكون رقمًا ما عندما أضربه في نفسه أحصل على 2 وأنت تعلم في هذه المرحلة أن لديك خيارًا، ربما يكون إيجابيًا. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "ربما تكون سلبية ولكن إذا قررت دائمًا اتخاذ الاختيار الإيجابي، فستكون قادرًا على الحصول على دالة متصلة جيدة من نفس هذه الصفقة إذا سألنا عن الأرقام السالبة ما الذي يجب أن يكون 2 إلى السالب 1 جيدًا والذي يجب أن يكون شيئًا ما أين عندما أضربه بـ 2 إلى 1؟ إنه يجعلني 2 إلى 0 وهذا نوع من المبررات لتقليدنا بأن الأسس السالبة تبدو وكأنها نصف واحد ولكن ما يحدث هنا حقًا هو أننا نقول أيًا كان هذا فإنه يجب أن يكون نوعًا ما من الوظائف التي تلبي هذه الخاصية f لـ a plus b يساوي f a في f b وعلاوة على ذلك، فإن حقيقة أن الأساس هو 2 تخبرنا بشكل أساسي أنها ليست مجرد دالة من هذا القبيل، إنها دالة حيث عندما نعوض بـ 1 نحصل على 2 وكما تعلم قليلًا سؤال أسلوب التحقق من سلامة العقل لمعرفة ما إذا كنت تتابع بعض الآثار هنا، أريد أن أسألك ما الذي لن أسميه مثل الكرة اللينة، ولكن هذا ليس من المفترض أن يكون مثل سؤال عميق بشكل لا يصدق بالضرورة. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "إنه مجرد فحص إذا كنت تتبع فكرة البدء بشكل تجريدي بخصائص الدالة ثم نوع من استنتاج الطرق التي قد نرغب في كتابتها بناءً على تلك الخصائص إذا كانت f لـ x تلبي هذه الخاصية الأسية f of a plus b يساوي f a في f b لجميع المدخلات ويرضي أيضًا f لـ 1 يساوي 2 أي مما يلي صحيح وهو ما يعني أي مما يلي صحيح بالضرورة بغض النظر عن الوظيفة التي تبدأها مع وأولئك منكم الذين يتذكرون أي محاضرة كانت، أيًا كانت المحاضرة التي كنا نتحدث عنها حول كيفية تفسير ما تقوله صيغة أويلر حقًا، فقد طرحت سؤالاً حول هذا الأسلوب حيث أهملت شرطًا واحدًا، كما تعلمون أنني لم أكتب حقيقة أننا نريد التأكد من أن f لـ x غير صفرية في كل مكان، ثم تسبب ذلك في حدوث قدر من الالتباس وهو أمر رائع، احصل على ارتباك على الشاشة يحدث لنا جميعًا ولكن القصد منه كان في الأساس إظهار أن هذه الخاصية المجردة لـ الشيء الذي يحول الجمع إلى الضرب هو ما يكفي لجعلك ترغب في كتابة الدالة على أنها تساوي واحدًا مرفوعًا إلى نوع ما من القوة. هذا هو جوهر السؤال الآن لدينا بضعة أسئلة في الواقع حول أبراج الطاقة يبدو أن هذا قد ظهر هنا وهو أمر مرتبط بشكل كبير بالمرة الأخيرة. ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "دعونا نتوقف عن سؤال برج الطاقة للحظة واحدة حتى نتمكن أولاً من الحصول على شعور أعمق مثل ما الذي يجب أن تعنيه الأسية هنا؟ لأنه يمكننا أن نكون ما أريد أن أدعيه هو أنه يمكننا الإجابة عليه بعدة طرق مختلفة، لذلك إذا أعطيتني طريقة واحدة فقط، فسنتحدث عن أبراج الطاقة وبعد ذلك تمامًا كما يمكن تمثيل خط الأعداد بمقياس لوغاريتمي يجب أن يتم نفس الشيء بالنسبة لطائرة معقدة؟ ربما رسم خريطة للمستوى المعقد على أسطوانة لا نهائية في اللوغاريتمي. ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "نعم في الواقع، هناك تصور سأقوم به بعد قليل هنا حيث نفعل شيئًا مشابهًا تمامًا لذلك لأن ما سنفعله هو اللعب مع دوال أسية مختلفة X لـ R في X ولكننا سوف نغير قيمة R التي سيتم تمثيلها بنقطة صفراء صغيرة لذلك سنتحدث نوعًا ما من خلال هذا لن نرسم خريطة للمستوى بأكمله، ولكن مجرد بضع نقاط عينة من المحور الحقيقي والمحور التخيلي لكن الفكرة هي أنه عندما نتحرك حول ما هو هذا الثابت، سنكون قادرين على تصور الأشياء المختلفة التي يفعلها بالمستوى وعلى نحو فعال، يبدو الأمر كما لو أنه يحول المحور السيني إلى مقياس لوغاريتمي ثم يلتف المحور التخيلي على طول الدائرة، وبمجرد أن تصبح قيمة R خيالية، يتم تبديل دور تلك الأرقام الحقيقية التي يتم وضعها على الدائرة ويتم وضع الأرقام التخيلية على مقياس لوغاريتمي محور إيجابي، سؤال رائع جدًا، أعتقد أن كل ثلاثة منها هم نوع من القفز للأمام نحو المكان الذي أريد أن أذهب إليه ولكن من الجيد أن أرى هذا هو المكان الذي يفكر فيه الناس في هذا. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "دعنا نمضي قدمًا ونصنفها فقط، الفكرة هي أن هذه الخاصية لـ f لـ a plus B تنتهي في نهاية المطاف بالسماح لك بالتعبير عن الكثير من الأشياء المختلفة من حيث ما هو f لـ 1 وفقط لتوضيح ذلك؟ صراحة شيء مثل f لـ 5 هو نفس الشيء لـ f لـ 1 زائد 1 زائد 1 زائد 1 زائد 1 وهو نفس الشيء لـ f لـ 1 مضروبًا في نفسه 5 مرات بسبب هذه الخاصية والتي إذا كانت f لـ 1 هي 2 هي نفسها مثل 2 أس 5 ثم شيء مثل f من سالب 5 يجب أن يكون الأمر أنه عندما نضربه في f من 5 نحصل على ما هو f من 0 وليس من الواضح على الفور ما هو f من 0 ولكن يمكننا أن نقول ذلك f من 1 زائد 0 يساوي ما يساوي f من 1 ما يساوي f من 0 ولكن f من 1 يساوي 2 وبالتالي فإن هذا أيضًا يساوي 2، لذلك نقول 2 يساوي 2 ضرب شيء ما حسنًا، هذا شيء ما يجب أن يكون 1، لذا في هذا السياق، يضمن هذا أن f لسالب 5 هو 2 أس سالب 5، وهو 1 على 2 أس 5. يمكننا أن نكتب هذا صراحة على هيئة 2 أس سالب 5 وهو ما يعني أن هاتين الخاصيتين معًا تشكلان نريد حقًا أن نكتب الدالة على هيئة 2 إلى X لأن أي رقم حسابي نضعه فيه سوف يفي بهذه الخصائص، وسيبدو الأمر وكأنه نضرب في نفسه هذا العدد من المرات في أي رقم كسري نضعه فيه سوف يفي بهذه الخصائص التي أردناها وقد تتساءل هل هذا فريد من نوعه وفي سياق الدوال ذات القيمة الحقيقية سيكون بالفعل ولكن في سياق الدوال ذات القيمة المعقدة سيكون هناك العديد من هذه الدوال التي يمكننا كتابتها لهذه الدالة والتي هي ما كنا عليه بالنظر من قبل حيث يمكن أن يكون لدينا وظيفة محددة لتكون EX للسجل الطبيعي لـ 2 زائد 2 pi I كل ذلك مرات X حسنًا، اغفر الارتباك هنا، أنا متحمس للكتابة عن هذا وهذه في الواقع وظيفة مختلفة مثل يتضح مما يحدث إذا عوضت بـ X يساوي نصفًا رأينا سابقًا كيف أنه عندما عوضت بنصف فإن ما تحصل عليه هو الجذر التربيعي السالب لـ 2، ثم إذا عوضت بربع واحد فلن تحصل على الجذر الرابع لـ 2 لكنني أضرب الجذر الرابع لـ 2 لذا فهي دالة مختلفة ولكنها لا تزال تفي بهذه الخصائص وهذا يجعلنا نوعًا ما نرغب في كتابتها كـ 2 إلى X ويجعلها تشير إلى أن 2 إلى X ربما يكون رقمًا غامضًا القليل من التدوين وعلينا فقط أن نكتب كل شيء من حيث EXP لـ R مضروبًا في شيء ما ولكن قد تتساءل جيدًا كما تعلم ربما أننا لسنا مبدعين بما فيه الكفاية مع جميع الوظائف التي تلبي هذه الخاصية ربما يكون هناك غموض عندما نكتب EXP من R مضروبًا في شيء ما، وهناك قيم مختلفة لـ R يمكن أن تلعب دورًا، لكنني سأقوم فقط بوضع مطالبة صغيرة ومن ثم ربما أقدم رسمًا تخطيطيًا لما سيبدو عليه الدليل إذا أردت ذلك، دعنا لنفترض أن لديك دالة معقدة F، وأنها تلبي الخصائص التالية أولاً، ويمكنك الحصول على مشتق منها. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "إنها قابلة للتمييز مما يمنعها من أن تكون شيئًا غير متصل وفوضوي تمامًا. هذا يشبه أخذ بعض القيم العشوائية اعتمادًا على معرفتك لمدى أي مساحة متجهة لا أعرف الكميات الكسرية التي قد ترغب في التفكير فيها بطرق مجنونة. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "إنها وظيفة جميلة. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "هذا قابل للتفاضل ولا يساوي 0 في كل مكان لذا فقد غاب هذا الشرط عن ذهني ونسيت المحاضرة التي ستخصص له أو شيء من هذا القبيل ثم لديه هذه الخاصية المركزية التي تحول الجمع إلى الضرب إذا كان لديك مثل هذه الوظيفة فأنا أدعي ذلك هناك رقم فريد ربما يجب أن أحدده حقًا، حيث يوجد رقم مركب فريد R بحيث يمكنك كتابة F لـ X باعتبارها في الأساس هذه الدالة الأسية لـ R مضروبة في تلك القيمة X والتي تعرف أساسًا أنه إذا كان لديك X كدالة فهذا متعددة الحدود اللانهائية مع خصائص مشتقة لطيفة وكل ذلك إذا كان لديك هذا، فستحصل على كل الأسي الذي تريده بالمعنى العام المجرد تمامًا للكلمة الأسية بناءً على الخاصية التي يمكن أن نريدها منها ورسم الدليل سيكون تبدو هكذا إذا كنت تريد أن تنظر أولاً إلى مشتقة هذه القيمة التي نفترض أنها موجودة في كل مكان، أليس كذلك؟ وأنت تكتب بوضوح ما هو الحد الأقصى لذلك، وسأتحدث عنه بسرعة كبيرة هنا لأولئك الذين يريدون التوقف مؤقتًا والتفكير في التفاصيل، فلا تتردد في الخاصية المركزية التي لدينا والتي تتيح لنا توسيع F of X زائد مصطلح H لذلك نحن نعتقد أنك تعرف تغييرًا طفيفًا في الإخراج مقارنة بالتغيير في المدخلات التي تسببت في ذلك. ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "يمكننا تحليل F لـ X من التعبير بالكامل ويتم التعبير عن الحد بأكمله فقط من حيث H والذي إذا فكرت في ما يعنيه في سياق المشتقات وحقيقة أن F لـ 0 يساوي بالضرورة 1، هذا التعبير المحدد بالكامل هو مجرد بعض الثوابت ولكن بشكل أكثر تحديدًا، مهما كانت مشتقة الدالة عند 0، إذن لديك هذا الشيء المضحك حيث إذا كنت تعرف مشتقتها عند 0 فهذا يحدد مشتقتها في كل مكان وفي سياق الدوال الأسية، نأمل أن يكون هذا مألوفًا تمامًا لأنه كل ما نقوله حقًا هو أن مشتق الدالة الأسية متناسب مع نفسه وأن ثابت التناسب يساوي أيًا كان المشتق عند 0، كل هذا تمت صياغته بشكل تجريدي للغاية ولكن الغرض منه هو التأكيد على أنه ليست بالضرورة مجرد وظائف نفكر فيها بالفعل على أنها a للقوة X ولكنها فئة أكثر اتساعًا من الوظائف التي تلبي هذه الخاصية المجردة لتحويل الجمع إلى ضرب ولكن إذا كان لديك ذلك، فهذا يضمن في الواقع أن لديك أيضًا المشتقة الثانية، وبالنسبة لهذه المسألة، مشتقة ثالثة، وذلك لأن الدالة المشتقة متناسبة تمامًا مع نفسها، لذا لكي تأخذ المشتقة n، ما عليك سوى النظر إلى ثابت التناسب هذا ورفعه إلى الأس n ومن هنا يمكنك إجراء توسيع متسلسلة تايلور وقد أترك ذلك كنوع من الواجبات المنزلية المتقدمة لأولئك منكم الذين يشعرون بالارتياح مع متسلسلة تايلور في هذه الفكرة خاصة إذا كنت تريد مزج فكرة أي دالة قابلة للاشتقاق بمعنى الأعداد المركبة، وهي نوع من موضوع الكلية بالتأكيد، كما تعلم أنه يمكنك خلط الاستدلال هناك كما تريد ولكن الاستدلال الغامض مسموح به في سياق شخص يعرف فقط عن سلسلة تايلور ولا شيء آخر لأخذ هذه الفكرة وإلقاء نظرة على توسيع تايلور لـ F و نوع من تبرير فكرة أن هناك رقمًا مركبًا فريدًا بحيث يمكن بالضرورة كتابة وظيفتنا F بهذه الطريقة ومن ثم يكون الاتصال بالأسس العادية عندما يكون لديك مثل هذه القيمة R نحن نفعل بشكل أساسي ما نفعله في السياق المعقد للأعداد الحقيقية هو إذا نظرت إلى exp لتلك الوظيفة ذات القيمة R واكتبها كقاعدة. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "يمكننا تفسير ذلك على أنه لا يعني فقط exp لنصفي pi I مضروبًا في X، ولكن يمكننا أيضًا تفسيره على أنه يعني ex لـ 5 pi نصفين I مضروبًا في X وهذه دوال منفصلة وهناك عائلة لا حصر لها من الدوال المنفصلة التي نشعر أنه ينبغي علينا اكتبها بالشكل I إلى X لذا فإن التعبير I إلى I إلا إذا كنت قد اعتمدت معيارًا لما سيعنيه ذلك بالضرورة عندما تقول أن لديها عددًا لا نهائيًا من المخرجات، هناك طريقة أخرى للتفكير في ذلك وهي أن الدالة I إلى X مع الترميز الذي لدينا غامض بعض الشيء الآن مع كل ذلك، دعونا نبدأ في تصور بعض من هذا لأنني أعتقد أن هذا ممتع وأنت تعلم أنك تخبرني إذا كان هذا مرئيًا مفيدًا أو مرئيًا أكثر إرباكًا ولكن ما سنفعله هو إلقاء نظرة على هذه الدالة exp لـ R مرات X، وهي في الأساس طريقة أخرى لكتابة e إلى قوة X في الواقع أعتقد أنني قدمت رسمًا متحركًا مختلفًا في مرحلة ما حددت ذلك لأنني كنت أخطط للتخطيط للقيام بذلك، لذا دعني أوه، نعم، ها أنت تعود إلى نظام الملفات الخاص بي، عد إلى حيث من المفترض أن تكون، هل هناك شكوى لأن هناك العديد من الأشياء المختلفة، سيكون الأمر كما لو أن هناك أوه استبدله يظهر على الشاشة الأخرى انتظر لماذا هو نعم، حسنًا استبدل؟ ضع كل ما تراه هناك والآن نعود إلى كل ذلك، كل هذا فقط حتى أتمكن من كتابته بشكل جيد إذا كنت غير مرتاح للتفكير في الأمر على أنه EXP لـ R مضروبًا في X هذه كثيرة الحدود اللانهائية فقط في خلف رأسك e إلى R في X وسنتغير حول R لذلك سأتبع نقاط المحور التخيلي، وسأتبع نقاط المحور الحقيقي ودعونا نرى ما سيفعله هذا جيدًا هذا كله سريع نوعًا ما، لذا دعني أفكر فيه ببطء أكثر قليلًا، كل الأرقام السالبة أي شيء هذا رقم حقيقي سالب سوف يتم سحقه في النطاق بين 0 و 1 ما الذي يجب أن يكون منطقيًا بالنسبة إلى السالب؟ a إلى رقم حقيقي سالب هو شيء بين 0 و 1 ونحن نتتبع على وجه التحديد f من سالب 1 والذي سيظهر حول أي شيء 1 على e يساوي حوالي 30 0. ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "37 f من 1 تهبط على e كما هو متوقع، هذا ما يمثله 1 f من I سوف يستقر على راديان واحد حول دائرة الوحدة، ومن الممتع متابعة المحور التخيلي بأكمله هنا على طول المحور التخيلي هنا كيف يلتف المحور التخيلي حول دائرة وماذا يحدث عندما نقوم بتعديل قيمة R؟ هذا لا يحدد أننا نتحدث فقط عن دالة أسية، ولكن أي دالة أسية هناك توافق جيد بين كل الدوال الأسية. ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "قد نرغب في أن تقوم قيم R هنا بتمديد الأشياء بشكل مختلف، لذلك عندما نضعها على 2، فأنت تعلم أنها تمد المحور الحقيقي أكثر بكثير بحيث ينتهي f لـ 1 حول المكان الذي يكون فيه e تربيع أعلى قليلاً من 7 f من السالب 1 أقرب بكثير إلى 0 f من I هو دوران 2 راديان حول الدائرة f من سالب I هو دوران سالب 2 راديان وبالطبع يمكننا الوصول إلى الصيغة المفضلة لدينا والتي إذا كانت pi كانت لدينا كثابت قياس لدينا، إذن يتمدد المحور الحقيقي كثيرًا كما تعلم أن f لـ 1 يقع عند e إلى pi وهو قريب جدًا من 20 زائد pi وهو أمر ممتع دائمًا وf لـ سالب 1 قريب جدًا من 0 لذا فهو ممتد حقًا إلى هذا الحد الحقيقي المحور كما أنها تمد الأشياء في اتجاه دائرة الوحدة بحيث الوصول إلى f من I أو f من السالب أسير في منتصف الطريق حول الدائرة، لذا هذا كل شيء جيد وجيد الآن كيف نفكر في دالة مثل؟ 2 إلى X وهو ماذا؟ سنكتب أيضًا كـ X لـ X للسجل الطبيعي لـ 2 × X، لذلك نقوم بتحريك النقطة الصفراء التي تمثل قيمة R إلى حوالي 0. ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "69 لا يوجد حتى الآن جزء وهمي مجرد رقم حقيقي 0.69 أو نحو ذلك هذا هو اللوغاريتم الطبيعي لـ 2 حسنًا، يمكنك أن ترى أن f لـ 1 يهبط على 2 ولهذا السبب نريد أن نطلق على هذه الدالة 2 إلى X f لـ 1 نصف آسف في الواقع f لـ سالب 1 يهبط مباشرة على 1 نصف f لـ أنا، إنها جولة حول دائرة الوحدة على وجه التحديد ستكون 0.69 راديان حول دائرة الوحدة والآن يمكننا الاستمتاع بمزيد من المرح ونقول ماذا سيحدث إذا قمنا بتغيير هذا إلى 0.69 بدلًا من أن يكون اللوغاريتم الطبيعي لـ 2، اجعله I مضروبًا في اللوغاريتم الطبيعي لـ 2 بحيث نفكر حقًا في شيء قد يكون له أساس أسي له. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "2 حوالي الخمس ولكن هناك العديد من الدوال الأسية المختلفة التي من شأنها أن تتمتع بخاصية وضع f من 1 على الرقم I لذا إذا أردنا توسيع نطاقها إلى أبعد من ذلك، فلا أعتقد أنني قمت بتحريكها هنا ولكن إذا أردنا أن نأخذها تلك النقطة الصفراء وارفعها للأعلى حتى تصل إلى 5 نصفين في pi I ما الذي ستراه هو دائرة الوحدة؟ يتم تدويره حول نفسه بحيث تدور f السالبة f لـ 1 حول 2 pi راديان أخرى وتستقر في مكانها ولكنها ستمتد المحور الحقيقي أكثر بكثير وهو المعنى الذي يكون فيه ناتج آخر من I إلى I هو رقم أصغر بكثير كان حول ما كان عليه 0. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "0003 أو نحو ذلك ولكن يمكننا أيضًا أن نرى ما أعتقد أنه ممتع للغاية. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "ماذا يحدث إذا أخذنا في الاعتبار التعبيرات البديلة التي نريد تفسيرها على أنها 2 أس X صحيح؟ لذلك عندما يكون R رقمًا حقيقيًا بحتًا، فإن اللوغاريتم الطبيعي لـ 2 يبدو منطقيًا أنه عند تعويضه هنا، فإن التعبير الذي نحصل عليه هو ما نريد كتابته كـ 2 للقوة X ولكن ماذا لو بدأنا في تحريكه في الوضع التخيلي اتجاه؟ حسنًا، وما سأفعله أولًا هو أن أحركه للأعلى بوحدات pi I الآن، ما الذي يحدث هنا؟ لدينا X من R مضروبًا في X وR يساوي هذه القيمة، وهي اللوغاريتم الطبيعي لـ 2 زائد pi في I ما يعنيه ذلك هو أنه عندما نعوض بـ 1 f من 1 يكون عند سالب 2 لذا نريد كتابة هذه الدالة مثل سالب 2 أس سالب 2 وهذا في الواقع شيء كما تعلمون، إنه أمر بسيط ومخادع بعض الشيء عندما نكتب رقمًا سالبًا أس سالب 2 إلى الأس X لا يبدو في البداية هكذا بالضرورة، فهو يجلب لنا إلى الأعداد المركبة بأي طريقة ولكن بالطبع عندما نعوض حتى بقيمة مثل 1 نصف حيث نطلب نوعًا ما جذرًا تربيعيًا لسالب 2 ندرك أننا نريد كتابة ذلك في صورة مثل I مضروبًا في الجذر التربيعي من 2 ولكن إذا نظرت إلى هذه الدالة سالب 2 إلى القوة X في المجال المعقد الكامل الذي تتعامل معه، فإن ما تنظر إليه هو دالة تأخذ قيمة 1 إلى سالب 2 وإذا فعلت ذلك فماذا إنه ينطبق على بقية خط الأعداد الحقيقي، هل هو نوع من اللوالب إلى الخارج؟ لذلك نرى أن f لـ سالب 1 يقع عند سالب 1 نصف تقريبًا حيث تتوقع إذا اتبعت f لـ 1 half سيكون بالضبط على الخط التخيلي وf لـ 1 نصف سيكون الجذر التربيعي لـ 2 حسنًا، يا الماوس ليس حيث أريد أن يكون. ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "سيكون حول الجذر التربيعي لـ 2 ضرب I، ومع استمرارك في هذا يظهر لك كل قوى القيمة الحقيقية لسالب 2 إلى X، فهي بالضرورة تلتف حولها ولكن يمكننا أيضًا نقل قيمة R إلى أعلى والحصول عليها ما يصل إلى حوالي tau مرات I حوالي ستة فاصل اثنين وثمانية مرات I وفي هذا السياق، هذه دالة أخرى نرغب في كتابتها كشيء مثل 2 إلى X لأنه بالنسبة لأي عدد صحيح إلى عدد صحيح تقوم بتوصيله لـ X، فإنه سيتم يبدو مثل الضرب المتكرر، بل إنه يحتوي على قيم معقولة لأشياء مثل 1 نصف حيث يلفظ الجذر التربيعي السالب بدلاً من الجذر التربيعي الموجب، ولكن ما يفعله في الواقع هو التحول إلى المستوى حيث يضع كل شيء هو الحقيقي ينتهي خط الأعداد إلى أن يكون عبارة عن حلزوني ملفوف بإحكام شديد يدور ويدور بطريقة حلزونية بحيث تهبط f 1 مباشرة على الرقم 2، ومن هذا المنطلق يمكننا أن نقول أن 2 إلى X يتم تفسيرها بشكل معقول على أنها دالة أسية منفصلة عن تلك التي اعتدنا عليها تقليديًا، لذلك أعتقد أنه مع كل ذلك سأترك الأمور لهذا اليوم وسأترك لك بضعة أسئلة عالقة للتفكير فيها، حسنًا، إذا كنت تريد ذلك فكر في I بالنسبة إلى I باعتباره تعبيرًا متعدد القيم، هل يمكنك القول إننا نعتمد اتفاقية بشكل خيالي، ستقول أنك تختار فرعًا من دالة اللوغاريتم الطبيعي وربما هذا يقيدك في هذا الكائن e إلى سالب pi نصفين ولكن إذا قلت أن هذا النوع يريد أن يكون عددًا لا نهائيًا من القيم المختلفة مثل القيم المختلفة التي رأيناها، فما عدد القيم التي يريد 2 إلى الثلث أن تكون بنفس المعنى؟ حيث نقوم باستبدال 2 بخيارات مختلفة ومختلفة لـ e إلى X بحيث يكون e إلى X يساوي 2، كم عدد القيم المختلفة التي تريد أن تكون أو كم عدد القيم التي تريد 2 إلى 3؟ تريد صياغة العشرات بشكل مختلف عن كل ذلك، دعني أقول عن جميع الدوال الأسية F لـ X التي ترضي أوه، هل كتبتها في مكان ما f من X يرضي كل هذه الخصائص التي كتبتها، لذا إذا كانت ترضي الكل من هذه وإذا كانت f 1 تساوي 2، فكم عدد المخرجات المختلفة التي سنحصل عليها عندما نعوض X بـ 3 أعشار للخيارات المختلفة لأي وظيفة؟ هذا هو، وكم عدد المخرجات التي سنحصل عليها؟ بالنسبة لـ 2 إلى pi للوظائف المختلفة التي يمكن أن يمثلها 2 إلى X إذا كنا نفكر في 2 إلى X كنوع من الدالة الأسية بمعنى هذا النوع من الخصائص المجردة وإذا كنا نعم، إذا كنا لدينا فئة من هذه الوظائف المختلفة، ونريد توصيل pi، فهذا يجعلني أضحك لمجرد أنها إجابة مضحكة نوعًا ما تظهر عندما تحاول التفكير في الأمر، لذا فهذه هي الأسئلة التي سأترككم وأعتقد أن هذا هو ما تعرفونه. ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "كان سؤالي المركزي في التعامل مع محاضرة اليوم هو ما إذا كنت أريد أن تكون نوعًا من الوصف مثل هذه الخصائص المجردة للدوال الأسية، ومن الرائع بالنسبة لي أن أبدأ من تلك الخصائص المجردة لقد أصبحت مقيدًا بفكرة e إلى rx أو أكثر. ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "فقط كما تعلم أعتقد أنني كتبت بصراحة أكبر لـ r مضروبًا في x لقيم مختلفة لـ r التي تحبسك في هذا الحد ولكنها لا تحبسك بقدر ما تمتلك فكرة لا لبس فيها لما يجب أن يكون 2 أس x أقل بكثير من شيء مثل I أس x الخطر في ذلك بالطبع هو أن الناس في بعض الأحيان لا يحبون التجريد وأحيانًا لا يبدو الأمر سهلاً ولكن إذا كان هذا هو إذا كنت تعلم، فقط أخبرني، أعتقد أن هناك دائرة كاملة مثيرة للاهتمام من الأفكار التي تحيط بكل هذه الأشياء لتشمل أبراج الطاقة لأنه إذا كنت تريد التحدث فعليًا عن أبراج الطاقة كما فعلنا في المرة السابقة في سياق الأعداد المركبة أو حتى مع القواعد السلبية عليك أن تفكر في أشياء مثل هذه، لذلك كان السؤال الذي طرحناه على الشاشة نعم، ماذا يحدث إذا فعلنا هذا من أجلي بقوة أنا؟ المعايرة كما تعلم، دعنا نجرب ذلك، دعنا نمضي قدمًا ونجرب برج الطاقة حيث نرفع I إلى قوة معينة ونرى ما الذي سيخرج منه، لذلك لم نكن نخطط للقيام بذلك ولكننا نستطيع، يمكننا دائمًا اسحب بايثون وافعل بشكل أساسي ما كنا نفعله في المرة السابقة، وبالتالي فإن الطريقة التي سينجح بها هذا هي أننا بدأنا ببعض القيمة الأساسية ثم بنوع ما من النطاق، ما كنا نفعله كنا نأخذ وسنقوم بإعادة التعيين يجب أن تكون أيًا كانت القاعدة التي في هذه الحالة مرفوعة إلى قوة a يجب أن تكون حسنًا، رائع، لذلك سنفعل ذلك ثم سنقوم بطباعة قيمة دعنا نفعل هذا من أجل نعم، إنه رقم أكبر بكثير مثل 200 لذا يبدو أن ما يحدث هو أن هناك احتمال حدوث فوضى مع هذه الأشياء في بعض الأحيان. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "لقد فعلنا ذلك، دعني أستورد NumPy لذا لدي الدالة الأسية، دعني أذهب إلى نطاقنا الكبير كما كان لدينا من قبل بدلاً من كتابته كما تعلم شيئًا يشبه I أس X، سأكتبه كدالة أسية لثابت مختلف، وهو ثابت مختلف سأجعله أريد أن يكون 5 باي نصفين، لذلك سأقوم بنصف 5 باي، لذا فهو رقم مركب وله 5 باي نصفين الجزء التخيلي إذن هذا يساوي 5 باي إلى نصفين وماذا أفعل؟ أنا أقوم بتأسيس ذلك لذا أريد ضرب "a" في الداخل هناك، حسنًا؟ هذه في الأساس طريقة أخرى يمكنك من خلالها تفسير التعبير I إلى X ولحسن الحظ أنك ستقول أنك اخترت فرعًا مختلفًا من دالة السجل الطبيعي ولكنها وظيفة أخرى يمكننا تكرارها على نفسها ونرى ما يحدث وقد نحصل عليه نتيجة مختلفة مثيرة جدًا للاهتمام، حسنًا، لدينا بالفعل نتيجة مختلفة. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/bengali/sentence_translations.json b/2020/ldm-i-to-i/bengali/sentence_translations.json index 5fd7ffa5a..9f0af555a 100644 --- a/2020/ldm-i-to-i/bengali/sentence_translations.json +++ b/2020/ldm-i-to-i/bengali/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "সুতরাং আপনি যদি 1 নম্বর থেকে শুরু করেন, আপনার প্রাথমিক বেগ হল সোজা 0 এর দিকে হাঁটতে হবে এবং আপনি আরও কম হাঁটতে হাঁটতে, আপনি যদি 1 অর্ধে বসে থাকেন তবে আপনি এখনও 0 এর দিকে হাঁটতেন, কিন্তু এখন আপনার বেগ ভেক্টর আপনি যেখানে আছেন সেখানে 1 বার নেতিবাচক হবে, যা নেতিবাচক 1 অর্ধেক।", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "এবং একটি আকর্ষণীয় প্রশ্ন আপনি জানেন যে এই ধরনের একটি ফাংশন আছে যা এটির জন্য লিখতে যুক্তিসঙ্গত মনে করে কারণ আপনি জানেন যদি আমরা এটিকে x তে i হিসাবে লিখব তবে এটি কেবল এটিই সন্তুষ্ট করবে না এটি আপনাকে সন্তুষ্ট করতে হবে যখন আপনি জানেন আমরা এক নম্বরে প্লাগ ইন করি যা আমরা পাই সম্ভবত আমি পাওয়ার ওয়ানে তবে আমরা ভাবছি এই ফাংশনটি হওয়া উচিত।", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "সুতরাং আমরা 5 পাই i অর্ধেক দুর্দান্ত পেয়েছি যা একেবারেই আরেকটি মান যা আমরা এখানে x এর জন্য প্লাগ ইন করতে পারি এবং কেবলমাত্র এটিকে আরও একটু বানান করতে পারি যদি আমরা এখানে আমাদের বৃত্তের দিকে ফিরে তাকাই যেখানে আমরা এখানে আছি মুহূর্তটি পাই অর্ধাংশের সমান সময়ের জন্য হাঁটে যা 1।57 কি হবে যদি এর পরিবর্তে আমরা আরেকটা পূর্ণ বাঁক নিই এবং আমরা পাই এর জন্য অন্য পাই অর্ধেক যাই যা আপনি জানেন যে আমরা এক ধরণের রেকর্ড করতে পারি যেখানে e থেকে পাই i এর মান হল আমরা অন্য পাই অর্ধেক হাঁটতে পারি আমরা অন্য পাই অর্ধেক হাঁটতে পারি এই বিন্দুতে আমরা একটি পূর্ণ বৃত্তে গিয়ে আমাদেরকে একটিতে ফিরে আসতাম এবং তারপরে আমরা পাঁচটি পাই অর্ধেকের জন্য হাঁটতাম যা সংখ্যাগতভাবে প্রায় 7।85 হ্যাঁ, এটি একেবারেই অন্য একটি সংখ্যা যা আমাদেরকে i-এর উপরে নিয়ে যায় এবং যদি আমরা i কে পাওয়ার iতে পুনঃপ্রকাশ করার পুরো রিগমারোলের মধ্য দিয়ে যেতে হয় প্রথমে 5 পাই অর্ধেক i তে e লিখে পাওয়ার i সেই i এর নেতিবাচক হওয়ার জন্য গুণ করুন এবং আমরা নেতিবাচক 5 পাই অর্ধেক ই-এর দিকে তাকাব যা একটি খুব ভিন্ন সংখ্যা ঠিক আমরা আসলে এটি গণনা করতে পারি আমি আমার মাথার উপরের অংশে নিশ্চিত নই, তবে আসুন একটি ডেসমোস দেখে নেওয়া যাক . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "যে দীর্ঘ যা আপনাকে অনেক ছোট সংখ্যায় নিয়ে যায় তবে এটিই একমাত্র উত্তর নয় যে আমরা সঠিকভাবে প্রবেশ করতে পারি আমাদের কাছে এখানে অন্যান্য লোকেরা আসছে নেতিবাচক 3 অর্ধেক গুণ i pi নিয়ে যা আপনি একটি ইউনিট বৃত্তের পরিপ্রেক্ষিতে জানেন? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "আমরা 90 ডিগ্রি পাই অর্ধেক রেডিয়ানে হাঁটার চেয়ে আমি যদি আমি পেতে চাই তবে আমি 270 ডিগ্রি অন্য পথে হাঁটলে 3 পাই অর্ধেক রেডিয়ান যা হয়তো আমি নেতিবাচক হিসাবে মনে করব কারণ কনভেনশনটি হল সাধারণত ঘড়ির কাঁটার বিপরীত দিকে ইতিবাচক হয় এটি প্রকাশ করার একেবারে অন্য উপায় এবং এটি আমাদের কাছে একটি ভিন্ন উত্তর পাবে যদি আমরা ঋণাত্মক 3 পাই অর্ধেক ই থাকতাম i সমস্ত শক্তিতে আমি একই খেলার মধ্য দিয়ে যাই এখন i বর্গক্ষেত্র বাতিল করে নেতিবাচক যা ইতিমধ্যেই আছে, এবং আমাদের একটি ধনাত্মক 3 পাই অর্ধেক রয়েছে এবং সংখ্যাগতভাবে এটি আমাদের আগে যা ছিল তার থেকে আরও আলাদা উত্তর দেয় যা যদি আমরা অতিক্রম করি এবং আমরা বলি আরে, 3 পাইতে 3 o 3 পাই নয় কি? অর্ধেক 111 পয়েন্ট 3 1 সংখ্যাটি আমরা 111 পয়েন্টের আগে যা দেখেছিলাম তার থেকে খুব আলাদা ধরণের এটি কী ছিল 111 পয়েন্ট 3 1 দুর্দান্ত 111 পয়েন্ট 3 1 বা তাই এবং আবার অন্তর্দৃষ্টির পরিপ্রেক্ষিতে আপনি যা জিজ্ঞাসা করছেন সেখানে ধরুন আমাদের এটি ঘোরানো আছে গতিশীল কিন্তু আমরা সময়ের সাথে পিছনের দিকে চলে যাই আমরা দেখি সময়ের কত আগে আমাকে এমন হতে হবে যে আমি যদি সেখান থেকে জিনিসগুলি সামনে খেলি তাহলে আমি আমার প্রাথমিক শর্ত এক নম্বরে অবতরণ করব এবং আপনাকে 3 পাই অর্ধেক ইউনিটে ফিরে যেতে হবে এবং তারপরে আপনি যদি ক্ষয় গতিবিদ্যায় অনুবাদ করতেন যা চোখের দিকে উত্থাপন করা হচ্ছে এই প্রসঙ্গে আপনি বলবেন যদি আমি এক নম্বর থেকে শুরু করছি তবে আমি সময়ের সাথে পিছনে যেতে চাই এবং বলতে চাই যদি আমার কোথায় শুরু করা উচিত ছিল আমি এমন ক্ষয় করতে চাই যে আমি এক নম্বরে শেষ? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "সময়ের 3 পাই অর্ধেক একক পরে উত্তরটি স্পষ্টতই এই ধরণের সূচকীয় ক্ষয়ের জন্য প্রায় একশ এগারো থেকে শুরু হচ্ছে এবং আপনি দেখতে পাচ্ছেন যে এটি কোথায় যাচ্ছে যেখানে আসলে অসীমভাবে অনেকগুলি ভিন্ন মান রয়েছে যা আমরা যদি এক্স এর জন্য প্লাগ ইন করতে পারি ই টু দ্য এক্সকে আমি ভাবছি এবং লোকেরা এখানে অনেক বেশি প্রবেশ করেছে আমাকে মাফ করবেন আমার পিনটি মাটিতে ছুঁড়ে দেওয়ার জন্য একজন তৃতীয় স্থানের জন্য ক্লাসিক করেছেন 9 পাই অর্ধেক দুর্দান্ত পছন্দ 1729 পাই অর্ধেক আপনি আমার প্রিয় প্রচুর এবং প্রচুর বিভিন্ন বিকল্প অসীমভাবে অনেকগুলি ভিন্ন মান যা প্রথমে ডানদিকে কিছুটা বিরক্তিকর মনে হয় কারণ আমরা একটি অভিব্যক্তি দেখি যা মনে হয় আপনি জানেন যে এখানে কিছু গণনা হতে চলেছে আমি কেবল এটি আমার ক্যালকুলেটরে প্লাগ করি এবং দেখি কী পপ আউট হয় এবং আমরা একাধিক ভিন্ন পেয়েছি এটার জন্য মান তাই এখানে কি ঠিক হচ্ছে? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "16-এর চতুর্থ রুটটি 2 হওয়া উচিত এবং উত্তরটি ভালভাবে শেষ হয় আমরা একটি কনভেনশন গ্রহণ করি যখন এই ধরনের একাধিক বিকল্প থাকে যখন আপনার কাছে একটি বহু-মূল্যবান ফাংশন থাকে তখন আমরা প্রায়শই সেই মানগুলির মধ্যে একটিকে বেছে নিতে চাই যখন আমরা চাই ফ্যান্সিয়ার লিঙ্গোতে একটি একক ইনপুট এবং একটি একক আউটপুট সহ এটিকে একটি ফাংশন হিসাবে বিবেচনা করুন এটি সর্বদাই উঠে আসে যখন আমরা জটিল সংখ্যাগুলির সাথে কাজ করি একটি অপারেশন হিসাবে কিছু করার ধারণাটি আপনি কখনও কখনও একাধিক মান রাখতে চান বাক্যাংশ শাখা শুনুন যেখানে আপনি বর্গমূল ফাংশনের একটি শাখা চয়ন করেন? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "কারণ এখানে একাধিক ভিন্ন উত্তর আছে আপনি জানেন আমরা আবার ভাবি আমি আবার এই 90 ডিগ্রী ঘূর্ণন এবং যদি আমরা এটিকে 90 ডিগ্রী ঘূর্ণন হিসাবে ভাবি তবে মনে হয় বর্গমূল হওয়া উচিত আপনি 45 ডিগ্রী কোণে বসে কিছু জানেন সম্ভবত এটি বর্গক্ষেত্র আমি এর রুট যা আমরা খুব স্পষ্টভাবে লিখতে পারি রুট 2 ওভার 2 রুট 2 ওভার 2 আমি এটি শুধু ত্রিকোণমিতি ব্যবহার করছি কিন্তু যদি আমরা এর পরিবর্তে আমিকে একটি নেতিবাচক 270 ডিগ্রি ঘূর্ণন হিসাবে ভাবি তবে এটি সেই অপারেশনের অর্ধেক করার মতো মনে হয় প্রকৃতপক্ষে আমাদের অন্য দিকে পাওয়া উচিত হতে পারে যে সংখ্যাটি এখানে বসে আছে তা I এর বর্গমূল হওয়া উচিত এবং এটি আসলে নেতিবাচক মূল 2 এর আগে আমরা যা দেখেছি তার নেতিবাচক 2 ওভার 2 বিয়োগ মূল 2 ওভার 2 বার I এখন বাস্তব প্রসঙ্গে মানসম্পন্ন ফাংশন আমরা বলতে পারি হ্যাঁ শুধু বর্গমূল বেছে নিন যা ইতিবাচক উত্তরই হোক না কেন কিন্তু এর মধ্যে কোনটিকে আপনি ইতিবাচক উত্তর বিবেচনা করেন? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "এবং আমি মনে করি আপনি ভাল বলেছেন আমরা জানি এটি কি আমরা এটিকে 2 এর বর্গমূল হিসাবে সংজ্ঞায়িত করি সব ঠিক আছে এবং ভাল কিন্তু আমি যদি বলি যে আসুন এটিকে একইভাবে ব্যবহার করি যেভাবে আমরা আমাদের I এর কাছে I অভিব্যক্তি I এর কাছে যাচ্ছিলাম প্রথমে জিনিসগুলিকে সঠিক কিছুতে e হিসাবে প্রকাশ করতে চাই এবং তারপর আমি 1 অর্ধেককে সূচকে গুন করে 1 অর্ধেক বাড়াতে যাচ্ছি এবং আমি বলি ঠিক আছে, আমি অনুমান করতে পারি যে আমি যা করতে পারি তা ই করতে পারি 2 ভাল এর সমান এটি 2 এর স্বাভাবিক লগ এটি একটি ধ্রুবক যা প্রায় 0।69 বা তাই যদি আমরা eকে সেই শক্তিতে বাড়াই তাহলে আমরা 2 পাব তাই আমরা এটাকে e হিসাবে ভাবতে পারি স্বাভাবিক লগের 2 গুণ 1 অর্ধেক এবং আপনি যদি চান তাহলে আপনি x এর জন্য e ভাবছেন? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "আপনি জানেন যে এটি বাস্তব সংখ্যার পরিপ্রেক্ষিতে এক ধরণের ওভারকিল হতে পারে তবে আপনি যদি এই x ফাংশনের জন্য ই থেকে xকে শর্টহ্যান্ড হিসাবে ভাবছেন তবে আপনি মান 0 প্লাগ করতে পারেন।69 বার 1 অর্ধেক যা আমি অনুমান করি প্রায় 0 হবে।345 ইশ এরকম কিছু আপনি আপনার বহুপদীতে সেই কংক্রিট মানটি প্লাগ করুন দেখুন এটি কী আউটপুট করে এবং এটি 1 এর কাছাকাছি আউটপুট করবে।414 একটি চমৎকার বাস্তব সংখ্যা 2 এর বর্গমূল আপনি কি আশা করবেন কিন্তু আমরা যদি একই জিনিস করি যা আমরা শুধু I দিয়ে করছি এবং স্বীকার করছি যে আসলে একাধিক ভিন্ন উত্তর আছে যখন আমরা একটি শক্তিতে e হিসাবে কিছু লিখতে চাই আমরা এটিও লিখতে পারি এটি মজার মনে হতে পারে, কিন্তু আমরা এটিকে 2 প্লাস 2 পাই আই এর প্রাকৃতিক লগে ই হিসাবে লিখতে পারি যে পুরো জিনিসটি 1 অর্ধে উত্থাপিত হয় ডানদিকে এই সমস্ত মান সমান হয়ে আসবে আপনি এটিকে ভেঙে ফেলতে পারেন কারণ এটি ই এর সাথে 2 এর প্রাকৃতিক লগ ই দ্বারা 2 পাই I এর সাথে গুন করা হয়েছে এটিতে জিনিসগুলিকে 360 ডিগ্রী ঘোরানোর প্রভাব রয়েছে, তাই এটি ঠিক 1 এর সমান হতে চলেছে তাই আমরা 2 গুণ 1 দুর্দান্ত দেখছি যা একটি বৈধ প্রতিস্থাপনের মতো মনে হয় এবং এখনও যখন আমরা একই খেলা খেলি এটিকে একটি শক্তিতে নিয়ে যাওয়া এবং ঘাতকে সূচকে গুন করলে কী ঘটে তা দেখুন আমাদের 2 গুণ 1 অর্ধেক প্লাস এর স্বাভাবিক লগে ই আছে আচ্ছা, 2 পাই I গুণ 1 অর্ধেক কত? ভাল যে পাই বার হবে I এখন এই প্রথম অংশটি 2 গুণ 1 অর্ধের প্রাকৃতিক লগে যা শেষ হবে 2 এর পরিচিত বর্গমূল যা সবই ভাল এবং ভাল, তবে আমরা এটিকে ই দ্বারা গুণ করতে যাচ্ছি pi I ডান এবং বেশ বিখ্যাতভাবে e থেকে পাই আমি নেতিবাচক 1 তাই এই ক্ষেত্রে এটি প্রস্তাব করা হচ্ছে যে যদি আমরা এই অভিব্যক্তিটি 2 থেকে 1 অর্ধেক সমাধান করি তবে বিভিন্ন উত্তরের সাথে খেলার মাধ্যমে আমরা এমন কিছুর জন্য প্লাগ ইন করতে পারি ই এর সাথে X এর সমান 1 অর্ধেক যা দিয়ে আমরা শেষ করি তা হল আরেকটি উত্তর যা আমরা ঐতিহ্যগতভাবে 2 এর এই ঋণাত্মক বর্গমূল হিসাবে লিখতে পারি এবং এখানে আমি বলতে চাইছি 2 থেকে 1 অর্ধেক দেখার জন্য একাধিক মান থাকাটা একটু মজার।বলুন যে একটি জিনিস সমান নয় কিন্তু আমরা পছন্দের ভিত্তিতে তৈরি করি এটি একাধিক ভিন্ন জিনিসের সমান হতে পারে তবে দুটি জিনিস যা এটি বেশ যুক্তিসঙ্গত বলে মনে হতে পারে যদি 2 থেকে 1 অর্ধেক এমন কিছু হতে থাকে তবে মনে হয় এটি হয় ইতিবাচক হওয়া উচিত বর্গমূল যার সাথে আমরা পরিচিত বা এর নেতিবাচক রূপ যা আসলে এমন সমস্যা বলে মনে হয় না এবং প্রকৃতপক্ষে আমরা এই গেমটি আরও বেশি খেলতে পারি যেখানে আমি আপনাকে এই অভিব্যক্তিটির আরও সৃজনশীল উত্তর চাই কারণ হয়তো আমরা 2 থেকে পাওয়ার X-এর মতো কিছুর অন্যান্য মজার ক্ষমতা খুঁজে পেতে পারি যখন আমরা X-এর বিভিন্ন মান প্লাগ করা শুরু করি তাহলে আমরা কি প্রতিস্থাপন করব যদি আমরা একই নিয়ম মেনে চলছি যা আমরা I-এর মূল্যায়নে ব্যবহার করছিলাম power I তাই এইবার প্রশ্ন জিজ্ঞাসা করে বা এটি নির্দিষ্ট করে যে x সমান 2 এর সমীকরণ e এর একটি সমাধান হল আসল সংখ্যা 2 এর প্রাকৃতিক লগ ঠিক আছে যেটি আমরা জানি।", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "এটি বিরক্তিকর নয়, তবে আমরা আর কী করতে পারি তার তুলনায় এটি বিরক্তিকর প্রশ্নটির উত্তর ই x সমান 2 এবং আবার সৃজনশীলতাকে স্বাগত জানানো হয়েছে, তাই আমি আপনাকে সেই জন্য আরও একটি ছোট মুহূর্ত দেব II এগিয়ে যাব এবং এখানে কিছু উত্তর লক করব যদি এটি আপনার সাথে ঠিক থাকে তবে আমি নিশ্চিত নই এটি কতটা সময় অগত্যা আপনি কোন ডিভাইসটি দেখছেন তার উপর নির্ভর করে গণিত এন্ট্রি করতে লাগে তবে আপনি যে প্রশ্নটির উত্তর দিতে চান সেটির উত্তর দেওয়ার সুযোগ পাওয়ার আগে যদি খুব বেশি চাপ দেবেন না তাই মনে হচ্ছে আপনার মধ্যে 131 জন ভেরিয়েন্টে প্রবেশ করেছেন যেখানে আমরা 2 এর Ln নিই এবং আমরা 2ii যোগ করি এবং আমি অনুমান করি যে আমি এই প্রশ্নটি লিখছি ভুলভাবে একটি উত্তরকে সঠিক হিসাবে চিহ্নিত করা হয়েছে যখন বাস্তবে বেশ কয়েকটি ভিন্ন সঠিক রয়েছে তাই এটি আমার উপর।কারণ আমি জানি না এটা আপনাদের কারোর মত মনে হচ্ছে ওহ এটা লাল আপনি ভুল বুঝেছেন যখন আপনি 2 প্লাস 42 এর Ln এ প্রবেশ করেছেন।", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi যা অবশ্যই একটি দুর্দান্ত পছন্দ তবে আপনার কাছে 4 pi I প্লাস 2 বা 6 pi I এর প্রাকৃতিক লগ বা সত্যিই 2 pi I এর যেকোন পূর্ণসংখ্যা গুণফলের মতো কিছু থাকতে পারে যদি আপনি যোগ করেন যে এটি e-তে প্রভাব ফেলবে না X কারণ এটিতে ই দ্বারা 2 পাই I তে গুণ করার প্রভাব রয়েছে যা 1 দ্বারা গুণ করার প্রভাব এবং আবার এটি একটি মজার পরিণতি রয়েছে যেখানে এটি যুক্তিসঙ্গত ফলাফল আউটপুট বলে মনে হয় যখন আমরা এটি অন্য উদাহরণ হিসাবে এটি করি।মনে হচ্ছে সেখানে দ্বিতীয় সবচেয়ে সাধারণ প্রবেশ করা অভিব্যক্তিটি ছিল যে আমরা 2 প্রতিস্থাপন করতে পারি তাই আসুন মনে করি আমরা 1 4 এর শক্তিতে 2 এর কথা ভাবছি, ঠিক আছে একটি পরামর্শ ছিল যে আমরা 2 যোগ 4 এর স্বাভাবিক লগে 2 এর সাথে প্রতিস্থাপন করেছি pi I ঠিক আছে প্লাস 4 pi I এবং আমরা সেই সবগুলোকে 1 4th তে তুলে দেই যদি আপনি একই গেমটি খেলতেন তাহলে আপনি 2 গুণ 1 4ম এর স্বাভাবিক লগে e পাবেন, এবং আমরা e দিয়ে গুন করব pi I এখন এর প্রথম অংশটি 2-এর স্বাভাবিক ধনাত্মক চতুর্থ রুট হতে চলেছে, যখন আপনি 2-এর চতুর্থ মূলের মতো একটি ক্যালকুলেটরে একটি সুন্দর ছোট ধনাত্মক সংখ্যার মতো একটি এক্সপ্রেশন প্লাগ ইন করেন, তবে এই দ্বিতীয় অংশটি হল নেতিবাচক 1 তাই এটা বলে মনে হচ্ছে আপনি জানেন যদি আমরা 2কে এই ভিন্ন উপায়ে 1 4র্থে উত্থাপন করি আপনি জানেন যে এটি আমরা যে সাধারণ উত্তর পাই তা নয় তবে এটি একটি যুক্তিসঙ্গত উত্তর।", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "আমরা পাই অর্ধেক বার I দেখতাম এবং ঋণাত্মক 1 দ্বারা গুণ করার পরিবর্তে আমরা I দ্বারা গুণ করতাম যা আবার একটি বৈধ উত্তর এটি 2 থেকে 1 4 তম এর মতো কিছুর জন্য যুক্তিসঙ্গত আউটপুট বলে মনে হচ্ছে তাই আপনি যখন আমি যে ক্ষমতার দিকে তাকাতে আমার মনে হয় এর জন্য আমার কাছে একাধিক ভিন্ন মান আছে কিছু অতি ছোট কিছু অতি বড় সবগুলি 15ম আনুমানিক 15ম উত্তর থেকে খুব আলাদা যা আমরা এখানে আগে পেয়েছি এটা ঠিক একই ঘটনা যখন আপনি 2 থেকে 1 4ম এর মত কিছু জিজ্ঞাসা করছেন এবং স্বীকার করছেন যে আসলে একাধিক ভিন্ন সমাধান রয়েছে এক্সপ্রেশনে X থেকে 4 এর সমান 2 4টি ভিন্ন সমাধান বাস্তবে এবং আপনি যা দেখছেন তা হল যে একাধিক ভিন্ন সমাধান রয়েছে এক্সপ্রেশনের জন্য e-এর X এক ধরনের বেস সমান কিনা সেই বেসটি আমি কিনা সেই বেসটি 2 যাই হোক না কেন এবং একটি উপায় যা আমরা এই সম্পর্কে ভাবতে পারি তা হল যে আপনি যখন বাস্তব সংখ্যার সাথে কাজ করছেন তখন জিনিসগুলি কেবল সুন্দর জিনিসগুলি সুন্দর হয় এক-এক সম্পর্ক রয়েছে।", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "এটি দুর্দান্ত যেখানে আমরা যদি সূচকীয় ফাংশনগুলি সম্পর্কে চিন্তা করতে চাই তবে আমাকে এই জিনিসগুলির কিছু কভার করতে দিন আমাদের সামনে পিছনে এটি চমৎকার রয়েছে যেখানে আপনি যে কোনও সূচককে X-এর বেস হিসাবে প্রকাশ করতে বেছে নিতে পারেন যেমন 2 থেকে X বা আপনি প্রকাশ করতে পারেন R গুন X এর X এর মতো একই সূচক যা আপনি জানেন যে এটি বহুপদী যা আমরা উল্লেখ করি যখনই আমরা ই-এর মতো কিছু লিখি তখনই X-এর কাছে স্পষ্টভাবে উল্লেখ করি এবং সেখানে একটি সুন্দর পিছন পিছন রয়েছে কারণ আপনি কেবল B-এর একটি প্রাকৃতিক লগারিদম নিতে পারেন এবং এটি আপনাকে একটি উত্তর দেয় অনুমান করে যে B একটি ধনাত্মক সংখ্যা এবং এটি একই কথা বলে যে R এর X সমান B এর তাই একটি উপায় যা আমি সিরিজের আগে এই সম্পর্কে কথা বলেছি তা হল আপনি যদি দেখে থাকেন সমস্ত সম্ভাব্য সূচকের পরিবার ঠিক আছে আমরা সেগুলিকে R গুন X এর X হিসাবে লিখতে পারি এবং R যা হয় তা পরিবর্তন করতে পারি এবং এটি R টাইমস X-এ e লেখার মতোই একই জিনিস যদি এটি এমন কিছু হয় যা আপনি আরও স্বাচ্ছন্দ্যবোধ করেন তাই ই-তে R R বার X এর গুণগুলি XX এগুলি একই জিনিস যা আমরা পরিবর্তন করার বিষয়ে ভাবতে পারি কিন্তু অন্যদিকে আপনি যদি কিছু বেস হিসাবে সম্ভাব্য সমস্ত সূচকের কথা ভাবতেন আমাকে X এর শক্তির ভিত্তি করতে দিন এবং আমরা যাচ্ছি সেই ভিত্তিটি কী তা পরিবর্তন করতে প্রথমে মনে হয় এটি হেরফের করার জন্য এটি একটি ভিন্ন ধরণের অভিব্যক্তি, কিন্তু এটি একই পরিবারকে প্রকাশ করার অন্য একটি উপায় এবং একটি উপায় যা আপনি এটি সম্পর্কে ভাবতে পারেন এর জন্য আমরা কীভাবে ভাবি যে এটি কোন ভিত্তির সাথে মিলে যায় আমরা যদি আর টাইমস এক্স এর এক্সপের হিসাবে একটু বেশি বিমূর্তভাবে চিন্তা করি এবং আমি এটি করছি এমন একটি কারণ আছে কারণ আমরা এটি জটিল সংখ্যাগুলিতে প্রয়োগ করতে চলেছি যেখানে এটি অদ্ভুত দেখাবে তাই এখানে আমার সাথে অনুসরণ করুন যদি সেই ভিত্তির দিকে তাকানোর পরিবর্তে আমি একটি জিনিস করতে পারি তা হল মূল্য কী? ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "আমার কাছে R গুন X এর এক্সপ হতে পারে যেখানে হয়তো R হল শূন্য পয়েন্ট ছয় নাইন এর মত কিছু কিন্তু আমি সেটাকে দুই পাই I দ্বারা নিচে নামাতে পারি এবং এটি বেসকে পরিবর্তন করে না যে এটির সাথে মিলবে তা এখনও দুইটির সাথে মিলে যাবে বা এটি হতে পারে এটিকে দুই পাই আই দ্বারা উপরে স্থানান্তর করুন যা এটির সাথে সম্পর্কিত বেসটি পরিবর্তন করে না কারণ এই সমস্ত ক্ষেত্রে যখন আমরা X এর সমান প্লাগ ইন করি তখন আমরা একই জিনিস পাই তবে X এর বিভিন্ন মানের জন্য এই সবগুলি আলাদা ফাংশন এটি কেন আমরা I থেকে পাওয়ার I এর জন্য একাধিক ভিন্ন মান দেখেছি কারণ I থেকে X একটি দ্ব্যর্থক ফাংশন সেই প্রেক্ষাপটে এটি দ্ব্যর্থহীন হবে যদি আমরা সিদ্ধান্ত নিই যে R এর কোন মান যেমন আমরা যা উপস্থাপন করছি তা হল R গুন X এর কোন মান R এর আমরা কি বেছে নেওয়ার সাথে সাথেই বেছে নেব? ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "এটি একটি দ্ব্যর্থহীন ফাংশন কিন্তু সেই মুহুর্তে এটি মনে হয় যে আমরা যা চাই তা হল শক্তি X পর্যন্ত উত্থাপিত কিছু বেসের পরিপ্রেক্ষিতে জিনিসগুলি সম্পর্কে চিন্তা করা বন্ধ করা হয়ত যত তাড়াতাড়ি আমরা জটিল সংখ্যার প্রেক্ষাপটে আছি আমাদের কেবল লিখতে হবে এগুলি সবই কিছু ধ্রুবক সময়ের এক্স হিসাবে এক্স যদি অন্য কোন কারণে এটি স্ফটিক পরিষ্কার করে না যে আমরা যদি একটি গণনা করতে চাই বা এর উপরে গণিত করতে চাই তবে আমরা কীভাবে সংখ্যাগুলিকে প্লাগ করতে চাই তা আমরা এই চমৎকার অসীম বহুপদী পেয়েছি যা আমরা পেয়েছি এগুলিকে প্লাগ ইন করুন এবং আমি আপনার জন্য আরেকটি কেস তৈরি করব যে এটিই সম্ভবত এক্সপোনেনশিয়াল সম্পর্কে চিন্তা করার সঠিক উপায় আমরা যত তাড়াতাড়ি আমরা জটিল সংখ্যার মতো অন্যান্য ডোমেনগুলিতে প্রসারিত করছি এবং এর জন্য আসুন কেবল ব্যাক আপ করা যাক ডোরবেলে ফিরে কিছু জিনিস এসেছে আসল উপায়ে ফিরে যান যে আমরা সূচকের ধারণাকে প্রসারিত করি এবং শুধু মনে করি 2 থেকে X রাইট কি আমরা জানি প্রাকৃতিক সংখ্যার জন্য এটি সম্পর্কে কীভাবে ভাবতে হয়।", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "আপনি 2 থেকে 3 বারবার গুণনের মত কিছু জানেন কিভাবে এটি আপনাকে প্রথমে ভগ্নাংশের পরিমাণের জন্য 2 থেকে X বা ঋণাত্মক রাশি এবং এই জাতীয় জিনিসগুলির জন্য 2 থেকে X এর মতো কিছু সম্পর্কে ভাবতে শেখানো হয়।আমরা হব. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "আপনাকে সাধারণত শেখানো হয় যে 2 থেকে 1 অর্ধেক এমন কিছু হওয়া উচিত যেখানে আপনি জানেন যদি আমি নিজে থেকে এটিকে গুণ করি এবং এটি সাধারণ নিয়মগুলি অনুসরণ করে যা সূচকগুলি সংখ্যা গণনার সাথে করে যেখানে আমরা সেই সূচকে জিনিসগুলি যোগ করতে সক্ষম হই যেখানে আমার 2 পাওয়া উচিত 1 থেকে তাই এটি এমন কিছু সংখ্যা হওয়া উচিত যে আমি নিজে থেকে এটিকে গুন করলে আমি 2 পাই এবং আপনি জানেন যে সেই সময়ে আপনার একটি পছন্দ আছে, সম্ভবত এটি ইতিবাচক।", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "হতে পারে এটা নেতিবাচক কিন্তু আপনি যদি সবসময় ইতিবাচক পছন্দ করার সিদ্ধান্ত নেন তাহলে আপনি এই একই চুক্তি থেকে একটি সুন্দর একটানা ফাংশন পেতে সক্ষম হবেন যদি আমরা নেতিবাচক সংখ্যা সম্পর্কে জিজ্ঞাসা করি 2 থেকে নেতিবাচক 1 কি ভাল হওয়া উচিত যা কিছু হওয়া উচিত কোথায় যখন আমি এটাকে 2 দিয়ে 1 দিয়ে গুণ করি? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "এটি আমাকে 2 থেকে 0 পায় এবং এটি আমাদের কনভেনশনের জন্য যুক্তিযুক্ত যে নেতিবাচক সূচকগুলি 1 অর্ধের মতো দেখায় তবে এখানে আসলে যা চলছে তা হল আমরা বলছি যে যাই হোক না কেন এটি এমন কিছু ফাংশন হওয়া উচিত যা এই বৈশিষ্ট্যটিকে সন্তুষ্ট করে a প্লাস b এর সমান f এর f এর b এর গুন এবং অধিকন্তু যে বেসটি 2 মূলত আমাদের বলছে যে এটি এমন কোন ফাংশন নয় এটি এমন একটি ফাংশন যেখানে আমরা 1 প্লাগ ইন করলে আমরা 2 পাই এবং সামান্য হিসাবে আপনি জানেন স্যানিটি চেক স্টাইল প্রশ্ন আপনি এখানে কিছু প্রভাব সহ অনুসরণ করছেন কিনা তা দেখতে আমি আপনাকে জিজ্ঞাসা করতে চাই যে আমি এটিকে সফ্টবলের মতো বলব না, তবে এটি একটি অবিশ্বাস্যভাবে গভীর প্রশ্নের মতো হওয়া বোঝানো হয়নি অগত্যা আপনি যদি অনুসরণ করছেন তাহলে এটি একটি চেক করার মতো আরও অনেক কিছুর সাথে একটি ফাংশনের বৈশিষ্ট্যগুলি দিয়ে বিমূর্তভাবে শুরু করার ধারণা এবং তারপরে এমন ধরনের ডিডিউসিং উপায় যা আমরা সেই বৈশিষ্ট্যগুলির উপর ভিত্তি করে এটি লিখতে চাই যদি x এর f এই সূচকীয় সম্পত্তি f সন্তুষ্ট করে সমস্ত ইনপুটের জন্য একটি প্লাস b এর f এর একটি গুণের f এর b সমান এবং এটি 1 এর সমান 2 এর f এর সাথে 2 এর মধ্যে কোনটি সত্য যা বলতে হবে নিচের কোনটি অগত্যা সত্য তা আপনি যে ফাংশনটি শুরু করছেন তা বিবেচ্য নয় সাথে এবং আপনারা যারা মনে রাখবেন কোন বক্তৃতাটি ছিল এটি কোনটি নিয়ে আমরা কথা বলছিলাম কীভাবে ব্যাখ্যা করা যায় অয়লারের সূত্রটি আসলে কী বলছে আমি এই শৈলীর একটি প্রশ্ন জিজ্ঞাসা করেছি যেখানে আমি একটি শর্তকে অবহেলা করেছি, আপনি জানেন আমি লিখিনি সত্য যে আমরা নিশ্চিত করতে চাই যে x এর f সর্বত্র অশূন্য হয় এবং তারপরে এটি কিছু পরিমাণে বিভ্রান্তির সৃষ্টি করে যা দুর্দান্ত হয় পর্দায় বিভ্রান্তি পান যা আমাদের সকলের সাথে ঘটে তবে এর উদ্দেশ্যটি মূলত দেখানো ছিল যে এই বিমূর্ত সম্পত্তি এমন কিছু যা যোগকে গুণে পরিণত করে তা হল মূলত আপনি ফাংশনটি লিখতে চান যেটি এটিকে একরকম শক্তিতে উত্থাপিত করার মতো যা কিছুর সমান লিখতে চান এটি এই প্রশ্নের আত্মা এখন আমরা পাওয়ার টাওয়ার সম্পর্কে আসলে কয়েকটি প্রশ্ন পেয়েছি যেটা এখানে পপ আপ হয়েছে বলে মনে হচ্ছে যা গতবারের সাথে দারুণভাবে সংযুক্ত ছিল চলুন এক মুহুর্তের জন্য পাওয়ার টাওয়ার প্রশ্নটি বন্ধ করে দেওয়া যাক যাতে আমরা প্রথমে একটি গভীর অনুভূতির মতো অনুভব করি যে এখানে ব্যাখ্যার অর্থ কী হওয়া উচিত? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "কারণ আমি যা দাবি করতে চাই তা আমরা হতে পারি তা হল আমরা একাধিক ভিন্ন উপায়ে এর উত্তর দিতে পারি তাই আপনি যদি আমাকে শুধুমাত্র একটি দেন, আমরা পাওয়ার টাওয়ার সম্পর্কে কথা বলব এবং তারপরে যেমন একটি সংখ্যারেখাকে লগারিদমিক স্কেলে উপস্থাপন করা যেতে পারে একই কাজ একটি জটিল প্লেনের জন্য করা হবে? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "হ্যাঁ প্রকৃতপক্ষে, এখানে একটি ভিজ্যুয়ালাইজেশন রয়েছে যা আমি এখানে মাত্র এক মুহুর্তের মধ্যে পেতে যাচ্ছি যেখানে আমরা তার মতোই কিছু করি কারণ আমরা যা করব তা হল বিভিন্ন এক্সপোনেনশিয়াল ফাংশন X এর R গুন X এর সাথে খেলতে হবে কিন্তু আমরা R এর মান পরিবর্তন করতে যাচ্ছি যা একটি সামান্য হলুদ বিন্দু দ্বারা প্রতিনিধিত্ব করা হবে তাই আমরা এর মাধ্যমে কথা বলব এটি পুরো সমতলকে ম্যাপ করতে যাচ্ছে না, তবে বাস্তব অক্ষ এবং কাল্পনিক অক্ষ থেকে মাত্র কয়েকটি নমুনা পয়েন্ট কিন্তু ধারণাটি হল যে আমরা সেই ধ্রুবকটির চারপাশে চলাফেরা করার সাথে সাথে আমরা বিভিন্ন জিনিসগুলিকে কল্পনা করতে সক্ষম হব যা এটি সমতলে করে এবং কার্যকরীভাবে এটি x-অক্ষকে লগারিদমিক স্কেলে পরিণত করে এবং তারপরে মোড়ানো হয় একটি বৃত্ত বরাবর কাল্পনিক অক্ষ এবং তারপর R এর মানটি কাল্পনিক হওয়ার সাথে সাথে এটি সেই বাস্তব সংখ্যাগুলির ভূমিকাকে অদলবদল করে বৃত্তে রাখা হয় এবং কাল্পনিক সংখ্যাগুলি লগারিদমিক স্কেল করা ধনাত্মক অক্ষের উপর রাখা হয় তাই দুর্দান্ত প্রশ্ন যার তিনটিই আমি অনুমান করি আমি যেখানে যেতে চাই তার জন্য বন্দুক নিয়ে ঝাঁপিয়ে পড়ছি কিন্তু ভালো লাগছে যেখানে মানুষ এই বিষয়ে তাই ভাবছে।", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "স্পষ্টতই 5 এর f এর মত কিছু হল f এর 1 প্লাস 1 প্লাস 1 প্লাস 1 প্লাস 1 প্লাস 1 যা একই জিনিস যা 1 এর f এর সাথে 5 বার গুণ করলে এই সম্পত্তির কারণে যা 1 এর f 2 হলে একই 5 এর শক্তি 2 এবং তারপর নেতিবাচক 5 এর f এর মত কিছু এমন হওয়া উচিত যে যখন আমরা এটিকে 5 এর f দিয়ে গুন করি তখন আমরা 0 এর f যাই হোক না কেন তা পাব এবং 0 এর f কী তা অবিলম্বে পরিষ্কার নয় তবে আমরা বলতে পারি যে 1 এর f প্লাস 0 হল 1 এর f এর সমান যা 1 এর f এর 0 এর গুন তবে 1 এর f 2 এর সমান এবং তাই এটিও 2 এর সমান তাই আমরা বলছি 2 এর সমান 2 গুণ কিছু ভাল একটি 1 হতে হবে তাই এই প্রসঙ্গে এটি গ্যারান্টি দেয় যে নেতিবাচক 5 এর f হল 2 থেকে নেতিবাচক 5 এটি 1 ওভার 2 থেকে 5 তম আমরা স্পষ্টভাবে এটিকে 2 থেকে নেতিবাচক 5 হিসাবে লিখতে পারি যা বলতে হবে এই দুটি বৈশিষ্ট্য একসাথে তৈরি করে আমরা আসলেই ফাংশনটিকে X-এর সাথে 2 হিসাবে লিখতে চাই কারণ আমরা এতে যে কোনো গণনা সংখ্যা রাখি তা পূরণ করতে যাচ্ছে এটি মনে হচ্ছে যে সংখ্যাটি আমরা যে কোনো ভগ্নাংশ সংখ্যার মধ্যে রাখি তা এই বৈশিষ্ট্যগুলিকে সন্তুষ্ট করবে যা আমরা চেয়েছিলাম এবং আপনি ভাবতে পারেন যে এটি অনন্য এবং বাস্তব মূল্যবান ফাংশনগুলির পরিপ্রেক্ষিতে এটি আসলেই হবে কিন্তু জটিল মূল্যবান ফাংশনগুলির পরিপ্রেক্ষিতে এমন একাধিক ফাংশন থাকবে যা আমরা এটির জন্য লিখতে পারি যার মধ্যে আমরা কী ছিলাম আগে দেখছি যেখানে আমাদের 2 প্লাস 2 পাই এর প্রাকৃতিক লগের এক্সপেক্ট হওয়ার জন্য একটি ফাংশন সংজ্ঞায়িত করা যেতে পারে আমি সেই সব সময় X ঠিক আছে, এখানে ঢালুতা ক্ষমা করুন, আমি শুধু এই বিষয়ে লিখতে উত্তেজিত হয়েছি এবং এটি আসলে একটি ভিন্ন ফাংশন হিসাবে আপনি X এর সমান 1 অর্ধেক প্লাগ ইন করলে কি হবে তার প্রমাণ আমরা একটু আগে দেখেছি কিভাবে আপনি যখন 1 অর্ধেক প্লাগ ইন করেন তখন আপনি যা পান তা হল 2 এর ঋণাত্মক বর্গমূল এবং তারপর আপনি যদি 1 চতুর্থাংশে প্লাগ করেন তবে আপনি এর চতুর্থ মূল পাবেন না 2 কিন্তু আমি 2 এর চতুর্থ রুটকে গুণ করি তাই এটি একটি ভিন্ন ফাংশন কিন্তু এটি এখনও এই বৈশিষ্ট্যগুলিকে সন্তুষ্ট করে এবং এটি একধরনের আমাদের এটিকে X থেকে 2 হিসাবে লিখতে চায় এবং এটি প্রস্তাব করে যে সম্ভবত 2 থেকে X একটি অস্পষ্ট স্বরলিপির বিট এবং আমাদের শুধু R বারের এক্সপের পরিপ্রেক্ষিতে সবকিছু লিখতে হবে কিন্তু আপনি হয়তো আশ্চর্য হবেন যে আপনি হয়তো জানেন যে আমরা এই বৈশিষ্ট্যকে সন্তুষ্ট করে এমন সমস্ত ফাংশনগুলির সাথে যথেষ্ট সৃজনশীল হতে পারছি না হয়তো আমরা যখন exp লিখি তখন একটি অস্পষ্টতা থাকতে পারে R এর সময় কিছু এবং R এর বিভিন্ন মান রয়েছে যা কার্যকর হতে পারে তবে আমি বলছি আমি কেবল একটি ছোট দাবি রাখব এবং তারপরে আপনি চাইলে প্রমাণটি কেমন হবে তার স্কেচের মতো দিতে পারেন যা আপনি চান বলুন আপনার কিছু জটিল ফাংশন F আছে, এবং এটি প্রথমে নিম্নলিখিত বৈশিষ্ট্যগুলিকে সন্তুষ্ট করে আপনি এটির একটি ডেরিভেটিভ নিতে সক্ষম।", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "এটি পার্থক্যযোগ্য যা এটিকে এমন কিছু হওয়া থেকে আটকে রাখে যা আপনি সম্পূর্ণরূপে অগোছালো বিচ্ছিন্ন জিনিসটি জানেন যা কিছু এলোমেলো মান গ্রহণ করার মতো যা আপনার উপর নির্ভর করে আপনি যে ভেক্টর স্থানের স্প্যান জানেন তার উপর আমি ভগ্নাংশের পরিমাণ জানি না যা আপনি পাগল উপায়ে ভাবতে চান।", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "এটি একটি চমৎকার ফাংশন. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "এটি পার্থক্যযোগ্য এটি সর্বত্র 0 এর সমান নয় তাই এই শর্তটি আমার মনকে স্খলিত করেছে এবং আমি ভুলে গেছি কোন বক্তৃতা বক্তৃতা বা এই জাতীয় কিছুর জন্য এবং তারপরে এটির এই কেন্দ্রীয় বৈশিষ্ট্য রয়েছে যে এটি যোগকে গুণে পরিণত করে যদি আপনার এমন একটি ফাংশন থাকে তবে আমি দাবি করি যে একটি অনন্য সম্ভবত আমার সত্যিই উল্লেখ করা উচিত সেখানে একটি অনন্য কমপ্লেক্স সংখ্যা R আছে যাতে আপনি X এর F লিখতে পারেন মূলত R গুণের এই সূচকীয় ফাংশনটি X এর মান যা আপনি জানেন যে আপনার কাছে যদি একটি ফাংশন হিসাবে X থাকে তবে এটি চমৎকার ডেরিভেটিভ বৈশিষ্ট্য সহ অসীম বহুপদী এবং এর সবই যদি আপনার কাছে থাকে তবে আপনার কাছে এমন প্রতিটি সূচক আছে যা আপনি চান এমন একটি বিমূর্ত জেনেরিক অর্থে এক্সপোনেনশিয়াল শব্দটি শুধুমাত্র একটি সম্পত্তির উপর ভিত্তি করে যা আমরা এটি থেকে চাই এবং প্রমাণের স্কেচ হবে এরকম কিছু দেখুন যদি আপনি প্রথমে দেখতে চান এই মানটির ডেরিভেটিভ কী যা আমরা ধরে নিচ্ছি সর্বত্র বিদ্যমান, তাই না? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "আমরা সম্পূর্ণরূপে এক্সপ্রেশনের বাইরে X-এর F ফ্যাক্টর করতে পারি এবং সম্পূর্ণ সীমাটি শুধুমাত্র H এর পরিপ্রেক্ষিতে প্রকাশ করা হয় যা আপনি যদি ডেরিভেটিভের প্রসঙ্গে এটির অর্থ কী তা নিয়ে চিন্তা করেন এবং সত্য যে 0 এর F অপরিহার্যভাবে 1 এর সমান এই সম্পূর্ণ সীমাবদ্ধ অভিব্যক্তিটি শুধু কিছু ধ্রুবক কিন্তু আরো নির্দিষ্টভাবে এটা যাই হোক না কেন আমাদের ফাংশন 0 এর ডেরিভেটিভ তাই আপনার কাছে এই মজার জিনিসটি আছে যেখানে আপনি যদি 0 এর ডেরিভেটিভ জানেন যে এটি নির্ধারণ করে যে এটির ডেরিভেটিভ সর্বত্র কি এবং সূচকীয় ফাংশনের প্রেক্ষাপটে এটি আশা করা যায় বেশ পরিচিত কারণ আমরা সত্যিই যা বলছি তা হল একটি সূচকীয় ফাংশনের ডেরিভেটিভ হল নিজের সাথে সমানুপাতিক এবং সেই সমানুপাতিক ধ্রুবকটি 0-তে ডেরিভেটিভ যাই হোক না কেন সমান।অগত্যা শুধু ফাংশন নয় যেগুলিকে আমরা ইতিমধ্যেই পাওয়ার এক্স হিসাবে মনে করি তবে এটি একটি সম্ভাব্য অনেক বেশি বিস্তৃত ফাংশন যা কেবলমাত্র যোগকে গুণে পরিণত করার এই বিমূর্ত বৈশিষ্ট্যকে সন্তুষ্ট করে তবে যদি আপনার কাছে থাকে তবে এটি আসলে গ্যারান্টি দেয় যে আপনারও একটি আছে দ্বিতীয় ডেরিভেটিভ এবং সেক্ষেত্রে একটি তৃতীয় ডেরিভেটিভ এবং যেমন কারণ ডেরিভেটিভ ফাংশনটি নিজের সাথে সমানুপাতিক তাই nম ডেরিভেটিভ নেওয়ার জন্য আপনি কেবল সেই সমানুপাতিক ধ্রুবকটি দেখুন এবং এটিকে এন শক্তিতে বাড়ান এবং তারপরে এখান থেকে আপনি একটি করতে পারেন টেলর সিরিজের সম্প্রসারণ এবং আমি হয়ত এটাকে অগ্রসর হোমওয়ার্ক হিসেবে ছেড়ে দিতে পারি যারা টেলর সিরিজের সাথে এই ধারণায় স্বাচ্ছন্দ্যবোধ করেন, বিশেষ করে যদি আপনি জটিল সংখ্যার অর্থে পার্থক্যযোগ্য কোনো ডিফারেনশিয়াবল ফাংশনের ধারণাকে মিশ্রিত করতে চান, যা হল একটি নিশ্চিতভাবে কলেজের বিষয় আপনি জানেন যে আপনি সেখানে আপনার ইচ্ছামত যুক্তি মিশ্রিত করতে পারেন তবে অস্পষ্ট যুক্তি এমন একজনের প্রসঙ্গে অনুমোদিত যে শুধুমাত্র টেলর সিরিজ সম্পর্কে জানে এবং এই ধারণাটি গ্রহণ করার জন্য এবং এফ এবং এর জন্য টেলর সম্প্রসারণটি দেখার জন্য অন্য কিছু নয় এই ধারণাটিকে ন্যায়সঙ্গত করা যে একটি অনন্য জটিল সংখ্যা রয়েছে যেমন আমাদের ফাংশন F অগত্যা এভাবে লেখা যেতে পারে এবং তারপরে স্বাভাবিক সূচকের সাথে সংযোগ যখনই আপনার কাছে এমন একটি মান থাকে R আমরা মূলত বাস্তব সংখ্যার জটিল প্রসঙ্গে যা করি তা করি আপনি যদি সেই মানের R-এর ফাংশনটির এক্সপেক্ট দেখেন এবং এটিকে বেস হিসাবে লিখুন।", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "আমরা ব্যাখ্যা করতে পারি যে শুধুমাত্র পাই অর্ধেক I বার X এর এক্সপের অর্থ নয়, তবে আমরা এটিকে 5 পাই অর্ধেক I টাইমস এক্স এর এক্সপের অর্থও ব্যাখ্যা করতে পারি এবং এইগুলি পৃথক ফাংশন এবং আলাদা ফাংশনের একটি অসীম পরিবার রয়েছে যা আমাদের মনে হয় সেগুলিকে X-এর কাছে I হিসাবে লিখুন সুতরাং I-এর কাছে I অভিব্যক্তিটি যদি না আপনি একটি মান গ্রহণ না করেন তবে এর অর্থ কী হতে চলেছে যখন আপনি বলবেন এতে অসীমভাবে অনেকগুলি আউটপুট রয়েছে তা ভাবার আরেকটি উপায় হল যে ফাংশন I থেকে X আমাদের কাছে যে স্বরলিপি আছে তা একটু দ্ব্যর্থহীন আমরা যা করতে যাচ্ছি তা হল R টাইমস এক্স-এর এই ফাংশন এক্সপের দিকে তাকান, যা মূলত X এর শক্তিতে ই লেখার আরেকটি উপায় আমি মনে করি আমি মনে করি আমি কিছু সময়ে একটি ভিন্ন অ্যানিমেশন রেন্ডার করেছি যা নির্দিষ্ট করেছে কারণ আমি এটি করার পরিকল্পনা করার পরিকল্পনা করছিলাম তাই আমাকে ওহ হ্যাঁ সেখানে আপনি আমার ফাইল সিস্টেমে ফিরে এসেছেন যেখানে আপনার থাকার কথা সেখানে ফিরে যান সেখানে এটি অভিযোগ করছে কারণ সেখানে একাধিক ভিন্ন আছে এটি একটি মত হতে চলেছে ওহ প্রতিস্থাপন এটি অন্য পর্দায় দেখায় অপেক্ষা করুন কেন এটা হ্যাঁ, ঠিক আছে প্রতিস্থাপন? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "আপনি সেখানে যা দেখতে পাচ্ছেন তা রাখুন এবং এখন আমরা ফিরে যাই ওহ সেখানে আমরা সেই সবগুলিই কেবল যাতে আমি সুন্দরভাবে লিখতে পারতাম যদি আপনি এটিকে R বার X এই অসীম বহুপদীর শেষ হিসাবে ভাবতে অস্বস্তি বোধ করেন।আপনার মাথার পিছনে ই আর টাইমস এক্স এবং আমরা R এর চারপাশে পরিবর্তিত হতে যাচ্ছি তাই আমি কাল্পনিক অক্ষের বিন্দুগুলি অনুসরণ করব, এবং আমি বাস্তব অক্ষের বিন্দুগুলি অনুসরণ করব এবং দেখা যাক এটি কী করে এটা সব ধরনের দ্রুত তাই আমাকে একটু ধীরে ধীরে চিন্তা করতে দিন সবগুলো নেতিবাচক সংখ্যার কিছু একটা নেতিবাচক বাস্তব সংখ্যা 0 এবং 1 এর মধ্যে সীমার মধ্যে ছিটকে যাবে যা নেতিবাচক অর্থে ই বোঝাতে হবে? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "একটি নেতিবাচক বাস্তব সংখ্যা 0 এবং 1 এর মধ্যে কিছু এবং আমরা বিশেষভাবে নেতিবাচক 1 এর f ট্র্যাক করছি যা 1 ওভার e 30 0 এর কাছাকাছি যা কিছু দেখাবে।1-এর 37 f প্রত্যাশিত হিসাবে e-তে অবতরণ করে যে 1-এর exp হল আমি একক বৃত্তের চারপাশে একটি রেডিয়ান অবতরণ করব, এবং এখানে সম্পূর্ণ কাল্পনিক অক্ষ বরাবর অনুসরণ করা এক ধরনের মজার ব্যাপার যে কীভাবে কাল্পনিক অক্ষ একটি বৃত্তের চারপাশে মোড়ানো হয় এবং আমরা R-এর এই মানটিকে পরিবর্তন করার সাথে সাথে কী ঘটবে? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "আমরা এখানে R এর মান চাই এবং এটি জিনিসগুলিকে ভিন্নভাবে প্রসারিত করে তাই যখন আমরা এটিকে 2 পর্যন্ত রাখি তখন আপনি জানেন এটি বাস্তব অক্ষকে অনেক বেশি প্রসারিত করে যাতে 1 এর f এর চারপাশে শেষ হয় যেখানে e বর্গক্ষেত্রটি ঋণাত্মকের 7 f এর উপরে থাকে 1 হল I-এর 0 f-এর অনেক কাছাকাছি একটি 2 রেডিয়ান ঘূর্ণন হল f-এর বৃত্তের চারপাশে ঘূর্ণন হল ঋণাত্মক 2 রেডিয়ান ঘূর্ণন এবং অবশ্যই আমরা আমাদের প্রিয় সূত্রটি পেতে পারি যে যদি পাই যেটি আমাদের স্কেলিং ধ্রুবক হিসাবে থাকে তাহলে বাস্তব অক্ষটি অনেক বেশি প্রসারিত হয় আপনি জানেন যে 1-এর f e-তে বসে আছে পাই যা 20 প্লাস পাই এর খুব কাছাকাছি যা সবসময় মজাদার এবং নেতিবাচক 1 এর f 0 এর খুব কাছাকাছি তাই এটি সত্যিই প্রসারিত হয়েছে যে বাস্তব অক্ষ এবং এটি একক বৃত্তের দিক থেকে জিনিসগুলিকেও প্রসারিত করেছে যাতে আমি বৃত্তের চারপাশে অর্ধেক পথ হাঁটতে পারি বা নেতিবাচকের f তে পৌঁছাতে পারি, তাই এখন সব ঠিক আছে এবং ভাল আছে আমরা কীভাবে একটি ফাংশন সম্পর্কে চিন্তা করব? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "আমরা 2 গুণ X এর প্রাকৃতিক লগের X এর X হিসাবেও লিখব তাই আমরা আমাদের হলুদ বিন্দুকে R-এর মানকে 0-এর কাছাকাছি স্থানান্তরিত করব।69 এখনও কোন কাল্পনিক অংশ নয় শুধুমাত্র একটি বাস্তব সংখ্যা 0।69 বা তাই এটি 2 এর স্বাভাবিক লগ আপনি দেখতে পাচ্ছেন যে 1 এর f 2 এর উপর ল্যান্ড করে তাই আমরা এই ফাংশন 2 কে X f এর 1 অর্ধেক বলতে চাই আসলে দুঃখিত f নেতিবাচক 1 জমির ঠিক 1 অর্ধেক f এর উপর আমি ইউনিট বৃত্তের চারপাশে কিছু হাঁটছি খুব নির্দিষ্টভাবে এটি 0 হতে চলেছে।একক বৃত্তের চারপাশে 69 রেডিয়ান এবং এখন আমরা একটু বেশি মজা করতে পারতাম এবং বলতে পারি যদি আমরা এটিকে 0 এর পরিবর্তে পরিবর্তন করি তাহলে কী হবে।69 এর পরিবর্তে 2-এর প্রাকৃতিক লগ হওয়ার পরিবর্তে এটিকে 2-এর প্রাকৃতিক লগের গুণে পরিণত করুন যাতে আমরা সত্যিই এমন কিছুর কথা ভাবি যার একটি সূচকীয় ভিত্তি থাকতে পারে।", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "I কি পাওয়ার আমি এই ক্ষেত্রে এটি 0 এর কাছাকাছি ঠেলে দেয়।2 পঞ্চমাংশের কাছাকাছি কিন্তু অনেকগুলি ভিন্ন সূচকীয় ফাংশন রয়েছে যেগুলির মধ্যে 1 এর f বসানোর এই বৈশিষ্ট্যটি I সংখ্যার উপর থাকবে তাই যদি আমরা এটিকে আরও বেশি স্কেল করি তবে আমি মনে করি না যে আমি এটি এখানে অ্যানিমেটেড করেছি কিন্তু যদি আমরা নিতে পারি যে হলুদ বিন্দু এবং এটি 5 অর্ধেক বার পাই না হওয়া পর্যন্ত এটি বাড়াতে আমি কি দেখতে হবে যে একক বৃত্ত? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "নিজের চারপাশে ঘোরানো হয় যাতে 1-এর নেতিবাচক f-এর f আরও 2 পাই রেডিয়ানের চারপাশে ঘোরে এবং এটি যেখানে আছে সেখানে অবতরণ করবে কিন্তু এটি বাস্তব অক্ষকে অনেক বেশি প্রসারিত করবে যা এই অর্থে I থেকে I-এর আরেকটি আউটপুট একটি অনেক অনেক ছোট সংখ্যা এটি 0 এর কাছাকাছি ছিল।0003 বা তাই কিন্তু আমরা এটাও দেখতে পারি যেটা আমার কাছে বেশ মজার মনে হয় যদি আমরা বিকল্প এক্সপ্রেশনগুলোকে বিবেচনা করি যেটাকে আমরা পাওয়ার X থেকে 2 হিসেবে ব্যাখ্যা করতে চাই তাহলে কী হবে? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "আমাদের কাছে R এর X গুন X আছে এবং R এই মানের সমান, যা 2 প্লাস পাই গুন I এর প্রাকৃতিক লগ এর মানে হল যে আমরা যখন 1 এর 1 f প্লাগ ইন করি তখন ঋণাত্মক 2 থাকে তাই আমরা এই ফাংশনটি লিখতে চাই পাওয়ার X-এর কাছে ঋণাত্মক 2 ঠিক আছে এবং এটি আসলে এমন কিছু যা আপনি জানেন, এটি একটি সামান্য প্রতারণামূলকভাবে সহজ যখন আমরা একটি পাওয়ার এক্সে ঋণাত্মক সংখ্যা লিখি 2 পাওয়ার X-এর কাছে এটি প্রথমে এমন মনে হয় না অগত্যা এটি আমাদের নিয়ে আসে যেকোন উপায়ে জটিল সংখ্যার মধ্যে কিন্তু অবশ্যই যখন আমরা 1 অর্ধের মত একটি মান প্লাগ ইন করি যেখানে আমরা ঋণাত্মক 2 এর বর্গমূল চাইছি আমরা বুঝতে পারি যে আমরা এটিকে বর্গমূলের গুণের মতো কিছু লিখতে চাই অফ 2 কিন্তু আপনি যদি এই ফাংশনটি নেগেটিভ 2 থেকে পাওয়ার এক্সকে সম্পূর্ণ জটিল ডোমেনে দেখতে চান যে এটি আপনি যা দেখছেন তা নিয়ে কাজ করছে এমন একটি ফাংশন যা 1 থেকে নেতিবাচক 2 এর মান নেয় এবং যদি এটি করে তবে কী হবে? এটি প্রকৃত সংখ্যারেখার বাকি অংশের সাথে করে এটি কি এটিকে বাইরের দিকে সর্পিল করে? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "সুতরাং আমরা দেখতে পাচ্ছি যে নেতিবাচক 1-এর f নেতিবাচক 1 অর্ধে বসেছে যেখানে আপনি আশা করবেন যদি আপনি 1 অর্ধের f অনুসরণ করেন তবে এটি ঠিক কাল্পনিক লাইনে বসবে এবং 1 অর্ধের f হবে 2 এর বর্গমূল ভাল, আমার মাউস যেখানে আমি এটা হতে চাই না. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "এটি 2 গুণ I এর বর্গমূলের কাছাকাছি হবে এবং আপনি এটি চালিয়ে যাওয়ার সাথে সাথে আপনাকে X থেকে ঋণাত্মক 2 এর সমস্ত বাস্তব মানের শক্তি দেখাচ্ছে এটি অগত্যা চারপাশে সর্পিল হয় তবে আমরা আমাদের R এর মানকে আরও বেশি স্থানান্তর করতে পারি এবং এটি পেতে পারি।প্রায় tau বার পর্যন্ত I প্রায় ছয় পয়েন্ট দুই আট বার I এবং সেই প্রসঙ্গে এটি আরেকটি ফাংশন যা আমরা X থেকে 2 এর মতো কিছু লিখতে চাই কারণ যে কোনো পূর্ণ সংখ্যা থেকে পূর্ণ সংখ্যার জন্য আপনি X এর জন্য প্লাগ ইন করবেন বারবার গুণনের মত দেখায় এবং 1 অর্ধের মতো জিনিসগুলির জন্য এটির যুক্তিসঙ্গত মানও রয়েছে যেখানে এটি একটি ধনাত্মক বর্গমূলের পরিবর্তে ঋণাত্মক বর্গমূল বের করে দেয়, কিন্তু এটি আসলে যা করছে তা হল সমতলে একটি রূপান্তর যেখানে এটি সবকিছু রাখে সেটিই আসল সংখ্যা রেখাটি একটি খুব শক্তভাবে ক্ষতবিক্ষত সর্পিল হয়ে শেষ হয় যা চারপাশে যায় এবং এটি এমনভাবে সর্পিল হয় যে 1 এর f 2 নম্বরের ঠিক ভূমিতে আসে তাই এটি সেই অর্থে যে আমরা X কে 2 বলতে পারি এটিকে যুক্তিসঙ্গতভাবে ব্যাখ্যা করা হয় একটি আলাদা সূচকীয় ফাংশন যা আমরা ঐতিহ্যগতভাবে ব্যবহার করি তাই আমি মনে করি যে সমস্ত কিছু দিয়ে আমি আজকের জন্য জিনিসগুলি রেখে দেব এবং ঠিক আছে সম্পর্কে চিন্তা করার জন্য আমি আপনাকে কয়েকটি দীর্ঘস্থায়ী প্রশ্ন রেখে দেব, তাই আপনি যদি চান I to the I কে একটি বহু-মূল্যবান অভিব্যক্তি হিসাবে ভাবুন ঠিক আপনি বলতে পারেন আমরা একটি কনভেনশন গ্রহণ করেছি, আপনি বলতে পারেন যে আপনি প্রাকৃতিক লগারিদম ফাংশনের একটি শাখা বেছে নিন এবং সম্ভবত এটি আপনাকে এই ই-তে লক করে নেতিবাচক পাইতে অর্ধেক কিন্তু যদি আপনি বলেন যে এই ধরনের অসীম হতে চায় বিভিন্ন মান যেমন আমরা দেখেছি 2 থেকে 1 তৃতীয়াংশ একই অর্থে কতগুলি মান হতে চায়? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "10 তম সকলের মধ্যে ভিন্নভাবে বাক্যাংশ করতে চায় আমাকে X এর F এর সমস্ত সূচকীয় ফাংশনগুলির কথা বলতে দিন যা ওহকে সন্তুষ্ট করে আমি এটিকে X এর f কোথাও লিখেছি যা এই সমস্ত বৈশিষ্ট্যগুলিকে সন্তুষ্ট করে যা আমি লিখেছি তাই যদি এটি সমস্তকে সন্তুষ্ট করে এর মধ্যে এবং যদি 1 এর f 2 এর সমান হয় তাহলে আমরা কোন ফাংশনের জন্য বিভিন্ন বিকল্পের জন্য X সমান 3 10 তম প্লাগ ইন করলে আমরা কতগুলি ভিন্ন আউটপুট পাব? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "2 থেকে পাই বিভিন্ন ফাংশনের জন্য যা 2 থেকে X প্রতিনিধিত্ব করতে পারে যদি আমরা 2 থেকে Xকে কিছু ধরণের সূচকীয় ফাংশন হিসাবে ভাবি এই ধরণের বিমূর্ত বৈশিষ্ট্যগুলির অর্থে সূচকীয় এবং যদি আমরা হ্যাঁ, যদি আমরা যদি আমাদের এই ধরনের বিভিন্ন ফাংশনের একটি ক্লাস আছে, এবং আমরা পাই প্লাগ ইন করতে চাই এটি আমাকে হাসায় কারণ এটি এমন একটি মজার উত্তর যা আমি জানি যে আপনি এটি সম্পর্কে চিন্তা করার চেষ্টা করছেন তখন পপ আউট হয়ে যায় তাই এই প্রশ্নগুলি আমি আপনাকে দিয়ে চলে যাব এবং আমি মনে করি এটি আপনি জানেন যে আজকের বক্তৃতার কাছে আমার কেন্দ্রীয় প্রশ্নটি ছিল আমি কি এটাকে সূচকীয় ফাংশনগুলির এই বিমূর্ত বৈশিষ্ট্যগুলির মতো বর্ণনা করতে চেয়েছিলাম এবং এটি আমার কাছে দুর্দান্ত যে সেই বিমূর্ত বৈশিষ্ট্যগুলি থেকে শুরু করা আপনি e থেকে rx বা আরও বেশি ধারণার মধ্যে আটকে যাবেন শুধু আপনি জানেন আমি মনে করি r এর বিভিন্ন মানের জন্য r গুণের এক্স এর আরও সততার সাথে লিখিত যে এটি আপনাকে এতদূরে আটকে রাখে তবে এটি আপনাকে যতদূর পর্যন্ত তালাবদ্ধ করে না একটি দ্ব্যর্থহীন ধারণা কি 2 থেকে পাওয়ার x এর চেয়ে অনেক কম কিছু হওয়া উচিত যেমন আমি পাওয়ার x এর সাথে অবশ্যই ঝুঁকিটি হল যে কখনও কখনও লোকেরা বিমূর্ততা পছন্দ করে না এবং কখনও কখনও এটি কাছে পৌঁছানো যায় না কিন্তু যদি তা হয় যদি আপনি জানেন তবে আপনি আমাকে জানান আমার মনে হয় আমার মনে হয় চিন্তার একটি সম্পূর্ণ আকর্ষণীয় বৃত্ত রয়েছে যা পাওয়ার টাওয়ারগুলিকে অন্তর্ভুক্ত করার জন্য এই সমস্ত জিনিসকে ঘিরে রয়েছে কারণ আপনি যদি আসলে পাওয়ার টাওয়ারগুলি সম্পর্কে বলতে চান যেমনটি আমরা শেষবার জটিল সংখ্যার প্রসঙ্গে বলেছিলাম বা এমনকি নেতিবাচক ঘাঁটি নিয়েও আপনাকে এই জাতীয় জিনিসগুলির মধ্য দিয়ে ভাবতে হবে, তাই এটি একটি প্রশ্ন ছিল যে আমরা পর্দায় উঠেছিলাম হ্যাঁ, আমরা যদি আমি ক্ষমতার জন্য এটি করি তবে কী হবে? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "টাইট্রেশন আপনি জানেন আসুন আমরা শুধু এটি চেষ্টা করি আসুন শুধু এগিয়ে যাই এবং একটি পাওয়ার টাওয়ার চেষ্টা করি যেখানে আমরা একটি নির্দিষ্ট শক্তিতে আমাকে উত্থাপন করছি এবং দেখুন এর থেকে কী আসে, তাই এটি করার পরিকল্পনা ছিল না তবে আমরা সবসময় পারি পাইথনকে টেনে আনুন এবং মূলত আমরা গতবার যা করছিলাম তা করুন সুতরাং এটি যেভাবে কাজ করবে তা হল আমরা কিছু বেস ভ্যালু দিয়ে শুরু করছি এবং তারপরে কিছু ধরণের পরিসরের জন্য আমরা কী করছিলাম আমরা একটি নিচ্ছিলাম এবং আমরা পুনরায় বরাদ্দ করতে যাচ্ছি এটা যাই হোক না কেন এই ক্ষেত্রে আমি যে বেসটি উত্থাপন করেছি তা a এর শক্তি হওয়া উচিত ঠিক আছে, ঠাণ্ডা, তাই আমরা এটি করতে যাচ্ছি এবং তারপরে আমরা একটি এর মান প্রিন্ট করতে যাচ্ছি এর জন্য এটি করা যাক হ্যাঁ, এটি 200 এর মতো অনেক বড় সংখ্যা তাই মনে হচ্ছে যা ঘটবে তা মাঝে মাঝে এই জিনিসগুলির সাথে বিশৃঙ্খলার সম্ভাবনা রয়েছে।", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "আমি আসলে আমাদের কাছে আছে তাই আমাকে NumPy আমদানি করতে দিন তাই আমার কাছে সূচকীয় ফাংশন আছে আমাকে আমাদের বড় পরিসরের জন্য যেতে দিন যেমন আমাদের আগে ছিল এটি লেখার পরিবর্তে আপনি এমন কিছু জানেন যেটা আমি X এর শক্তির মতো আমি এটি লিখতে যাচ্ছি একটি ভিন্ন ধ্রুবক ডানের সূচকীয় ফাংশন হিসাবে একটি ভিন্ন ধ্রুবক যা আমি 5 পাই অর্ধেক করতে চাই, তাই আমি 5 পাই অর্ধেক বার করব তাই এটি একটি জটিল সংখ্যা এবং এটি 5 পাই অর্ধেক পেয়েছে কাল্পনিক অংশ তাই এটি 5 পাই অর্ধেক গুণ আমি এবং আমি কি করছি? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/chinese/sentence_translations.json b/2020/ldm-i-to-i/chinese/sentence_translations.json index c985707e5..c94071d44 100644 --- a/2020/ldm-i-to-i/chinese/sentence_translations.json +++ b/2020/ldm-i-to-i/chinese/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "因此，如果你从数字 1 开始，你的初始速度是径直走向 0，当你走得更低时，如果你坐在 1 的一半，那么你仍然会走 向 0，但现在你的速度向量将会是负 1 倍，即负 1 倍。", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "一个有趣的问题是，你知道是否只有一个这样的函数 感觉为此编写是合理的，因为你知道我们是否将它写 为 i 到 x 不仅应该满足这一点，还应该满足 你知道什么时候我们将得到的数字 i 代入 i 的 1 次幂，但我们认为这个函数应该是 i。", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "所以我们有 5 pi i 一 半很好，这绝对是我们可以在此处代入 x 的另一个值， 如果我们回顾一下我们在此处的圆，则可以更直观地阐明这一 点moment 行走的时间等于 pi 的一半，即 1。57 如果相反，我们再转一整圈，再走另一个 pi 半段 ，让我们到达 pi，你知道我们可能会记录一下，这就是 p i 的 e 值，我的值是，我们走另一个 pi 半段，我 们走另一个 pi 半段，在此时我们会绕一整圈回到 1， 然后我们走了 5 个 pi 的一半，数值约为 7。85 是的，这绝对是另一个让我们在 i 之上的数字 ，如果我们要通过首先将 e 写到 5 pi 一半 i 的 i 次幂来重新表达 i 的 i 次方的整个繁琐工作乘以变成负数，我们会看到 e 到 pi 的负 5 半，这是一个非常不同的 数字，我们实际上可以计算这个，我不确定我的头脑 ，但让我们看一下 Desmos 。", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "这么长的时间会让你得到一个小得多的数字但这不是 我们可以输入的唯一答案，我们还有其他人带着负 3 半乘以 i pi 进来，你知道单位圆吗？", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "我们可以认为，如果我想到达 I，而不是那样走 9 0 度 pi 半弧度，那么如果我以相反的方式走 270 度 3 pi 半弧度，这可能是负数， 因为约定是通常逆时针方向是正数 这绝对是另一种 表达方式，如果我们有 e 到 pi 的负 3 半数，这会给我们带来不同的答案 i 所有的力量 i 我们经历同样的游戏，现在 i 的平方用 a 取消负值已经存在了，我们有一个正的 3 pi 一半，从数字上看，这给我们带来了一个与之前的 答案甚至不同的答案，如果我们回顾一下，我们说嘿 ，e 与 3 pi 的关系是什么，而不是 3 o 3 pi将 111 点 3 1 与我们之前 看到的数字非常不同 111 点 这是什么 11 1 点 3 1 很棒 111 点 3 1 左右 再一次，就直觉而言，你可能会问，假设我们有这个 旋转动态 但是我们在时间上向后移动，我们看到多 久以前我必须成为这样，如果我从那里向前玩东西， 我将落在我的初始条件的第一名上，而你必须在时间 上倒退 3 pi 半个单位然后，如果你要翻译成 衰变动力学，这就是在这种情况下引起人们注意的事 情，你会说，如果我从第一开始，但我想及时倒退， 说如果我应该从哪里开始，我想堕落到成为第一吗？", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "在 3 个 pi 将时间单位减半之后，对于这种指数 衰减，答案显然从大约 111 开始。你可以看到这将 走向何方，实际上有无数个不同的值，我们可以将这些值 代入 X，如果我们认为 e 到 X 是我，人们在这 里输入了更多请原谅我将我的别针扔到地上，就像一个经 典的第三名一样 9 pi 一半很棒的选择 1729 pi 一半你们都是我最喜欢的很多很多不同的选项无 限多个不同的值，一开始感觉有点令人不安，因为我们看 一个表达式，似乎你知道会有一些计算，我只需将其插入 我的计算器，看看会弹出什么，我们有多个不同的值它的 值 那么这是怎么回事呢？", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "16 的第四个根应该是 2，答案最终是好的 当 有多个选项时，我们采用一种约定，就像这样，当 你有一个多值函数时，当我们想要时，我们通常只 选择其中一个值来表达我们的意思将其视为具有单 个输入和单个输出的函数，用更高级的术语来说， 当我们处理复数时，这种情况总是会出现，将某些 东西视为一种运算的想法有时想要拥有多个值听到 短语“分支” 您在哪里选择平方根函数的分支？", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "因为有多种不同的答案 你知道我们再次想到的是这 个 90 度旋转 如果我们将其视为 90 度旋 转，感觉平方根应该是 你知道有一个 45 度 角的东西 也许这就是平方I 的根，我们可以非常 明确地写为 root 2 over 2 ro ot 2 over 2 I 这只是使用三角学， 但如果我们将 I 视为负 270 度旋转，那 么感觉就像一半做了一半的操作实际上应该让我们到 达另一边也许坐在这里的数字应该是 I 的平方 根，这实际上只是我们之前看到的负数 负根 2 除以 2 减去根 2 除 2 乘以 I 现在 在真实的背景下我们可以说是的，只要选择平方根作 为肯定答案，但是您认为其中哪一个是肯定答案？", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "我认为你说得很好我们知道这是什么我们将其 定义为 2 的平方根一切都很好但是如果 我说让我们以与我们接近 I 到 I 表 达式 I 相同的方式来处理这个问题呢想 要首先将正确的东西表达为 e 然后我将通 过将 1 的一半乘以指数来将其提高到 1 一半我说好吧，我可以我想我可以将 e 表达为等于 2 好 这是 2 的自 然对数 这是一个大约为 0 的常数。69 左右 如果我们计算 e 的幂，我们 会得到 2，所以我们可以将其视为 e 的 2 乘以 1 的一半的自然对数，如 果您愿意的话，是否将 e 视为 x？", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "您知道这在实数背景下可能有点矫枉过正，但如果您将 x 的 e 视为该 x 函数的简写，则可以插入值 0。69 乘以 1 的一半，我猜大约是 0。第 34 5 章41 4 一个很好的 2 的实数平方根，正如你所期望的但 是如果我们做同样的事情，我们只是用 I 做，并且承 认当我们想要写出 e 的幂时实际上有多个不同的答案 ，我们也可以写这个这可能看起来很有趣，但我们可以将其 写为 2 加 2 pi 的自然对数 I 整个事情升 到 1 一半之后，这个值将等于您可以将其分解，因为 它是 e 的2 的自然对数乘以 e 得到 2 pi I 这个只有将物体旋转 360 度的效果，所以它就 等于 1 所以我们正在寻找 2 乘以 1 很棒，感 觉像是一个有效的替换，但是当我们玩同样的游戏，将其 乘以指数，然后将其乘以指数，看看会发生什么我们有 e 的自然对数 2 乘以 1 一半加上 那么，2 p i I 乘以 1 一半是多少那么这将是 pi 乘以 I 现在，第一部分 e 等于 2 乘以 1 的一 半的自然对数，最终将成为熟悉的 2 的平方根，这一切 都很好，但我们将把它乘以 e 来pi I 对，众所 周知，e 到 pi I 是负数 1 所以在这种情况 下，这似乎表明，如果我们正在解决这个表达式 2 的 1 一半 通过尝试不同的答案，我们可以插入类似的东 西e 等于 1 的一半，我们最终得到的是另一个答案 ，我们传统上可以将其写为 2 的负平方根，这里我的 意思是，它有多个值来查看 2 的一半，这有点有趣说这 不等于一件事，但根据我们所做的选择，它可能等于多个 不同的事情但这两件事看起来很合理如果有什么东西是 2 的一半是它似乎应该是积极的我们熟悉的平方根或它的 负变体实际上看起来并不是这样一个问题事实上我们可以 嗯我们可以进一步玩这个游戏让我问你对这个表达式更有 创意的答案因为也许我们可以找到其他有趣的幂，例如 2 的 X 次方，当我们开始插入 X 的各种不同值时 ，如果我们遵守与评估 I 的相同规则，则根据我们所 做的替换幂 I 所以这一次问题提出或者指定方程 e 到 x 的一个解等于 2 是实数 2 的自然对数 ，我们知道这一点。", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "回答问题 e 到 x 等于 2 的问题，再 次欢迎创造力，所以我会再给你一点时间，如 果你同意的话，我会继续在这里锁定一些答案 ，我不确定需要多长时间根据您正在查看的设 备，必然需要进行数学输入，但是如果在您有 机会进入您想要的问题进入您希望它回答的答 案之前，请不要太紧张所以它看起来像你们中的 131 人已经输入了我们取 2 的 L n 并添加 2ii 的变体，我想我在写这 个问题时错误地将其中一个答案标记为正确的 ，而实际上有很多不同的正确答案，所以这是 我的责任事实上，我不知道你们是否喜欢它， 哦，它是红色的，当您输入 2 加 42 的 Ln 时，您弄错了。", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi 这当然是一 个很好的选择但是你也可以有像 4 pi I 加上 2 或 6 pi I 的自然对数或者实际上是 2 pi I 的任何整数倍，如果你添加它不会影响 e 到X 因为它只是具有乘以 e 到 2 pi I 的 效果，这是乘以 1 的效果，这又产生了一种有趣的 结果，当我们作为另一个例子时，它似乎输出了一种合理 的结果。看起来是第二个最常见的输入表达式，我们可以 替换 2 所以我们认为我们正在考虑 2 的 1 次 方 4 次方，好吧，有人建议我们用 e 替换 2 的 2 加 4 的自然对数pi I 好的加上 4 pi I，我们将所有这些都提高到 1 4 次右边 ，如果你要玩同一个游戏，你会得到 e 到 2 乘以 1 4 次的自然对数，我们将乘以 e 得到pi I 现在，它的第一部分将是通常的正数 2 的四次 方根，当您将 2 的四次根这样的表达式插入计算器时 ，我们的意思是一个很好的小正数，但是第二部分是负 1 所以它似乎在说你知道我们是否以这种不同的方式解 释 2 将其提高到 1 4 你知道这不是我们得到 的通常答案，但这是一个合理的答案。", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "我们会看 pi 乘以 I 的一半，而不是乘以负 1 我们会乘以 I 这又是一个有效的答案，对于 2 到 1 的 4 次之类的东西来说，这似乎是一个合理的输出 所以当你看看 I 的幂 I 似乎有多个不同的值 对吧，我们有一个有趣的现象 ，我们可以将 e 插入 5 个 pi 一半 I 负 3 个 p i 一半 I 我们得到了看起来截然不同的答案一些超小的东西 一些超级大的东西 与我们之前在这里找到的 1 5th 大约 1 5th 答案非常不同 这与当你问诸如 2 到 1 4th 之类的问题并承认实际上有多种不同的解决方案时，现象完全相同对 于表达式 X 到第 4 个，实际上等于 2 4 个不同的解，而 您所看到的是存在多个不同解的事实 对于表达式 e 到 X 等 于某种基数 是否该基数是 I 是否该基数是2 无论它是什么，我 们可能会想到的一种方式是，当你处理实数时，事情都是可爱的，事 情是美好的，存在一对一的关系。", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "太棒了如果我 们想考虑指数函数，让我来介绍一些这些东西我们有这个很好的来回 ，您可以选择将任何指数表示为 X 的基数，例如 2 表示 X 或者您可以表示与 R 乘以 X 的 X 相同的指数，您知道 这就是我们引用的多项式每当我们将类似 e 的值写入 X 时隐 式引用，并且有一个可爱的来回，因为您可以取 B 的自然对数假 设 B 是一个正数，它会给你一个答案 这与说 R 的 X 等 于 B 是一样的 所以我在本系列前面讨论过这个问题的一种方式 是，如果你正在看所有可能的指数族，我们可以将它们写为 X o f R 乘以 X 并更改 R 的含义 这与将 e 写成 R 乘 以 X 完全相同（如果您更喜欢这样） 所以 e 写成 R R 乘以 X 的 XX 是同样的事情，我们可以考虑改变它是什么 但另一方面，如果你将所有可能的指数视为某个底数 让我以 X 的幂为底，我们就去改变那个基础是什么 乍一看，感觉这是一种 不同的表达方式，但它只是表达同一个家族的另一种方式 你可能会 想到这一点 对于我们如何思考它对应的基础如果我们更抽象地思考 一下 R 乘以 X 的 Exp，我这样做是有原因的，因为我们 要把它应用到复数上，而复数看起来会很奇怪，所以请跟着我一起做 ，如果我可以做的一件事不是看这个基础，而是说它的价值是什么？", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "我可以有 R 乘 X 的 exp，其中 R 可能类 似于零点六九 但我可以将其向下移动两个 pi I 这不会改变它 所对应的基数仍然对应于 2 或者它可以将其向上移动两个 pi I 这不会改变它对应的基数，因为在所有这些情况下，当我们插入 X 等于 1 时，我们会得到相同的结果，但是所有这些对于不同的 X 值都是不同的函数这是为什么我们看到 I 的 I 次方有多个不 同的值 因为 I 的 X 次方在这种情况下是一个不明确的函数，如 果我们决定 R 的哪个值，那么我们表示的是 R 乘以 X 的 exp 的值，那么它就会是明确的R 的。", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "这是一个明确的函数，但在这一点上，感觉也许我们想 要的是 a 停止用某个基数的 X 次方来思考事物 也许一旦我们处于复数的背景下，我们就应该这样写 它们都是一些常数乘以 X 的指数，如果没有其他 原因，它就很清楚了如果我们想要进行计算或只是在其 基础上进行数学运算，我们实际上如何插入数字，我们 已经得到了这个很好的无限多项式把它们插入，我会 为你做另一个例子，这可能是思考指数的正确方法 一 旦我们扩展到其他领域，比如复数，为此，让我们备份 Go回到门铃，有些事情回到了最初的方式，我们 扩展了求幂的概念，只需想想 X 右边的 2 是多 少，我们知道如何考虑自然数。", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "你知道诸如 2 的 3 重复乘法之类的东西，你是如何首先被教导思考诸如 2 的 X 之类的小数或负数之类的东西。 ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "你通常会被告知，2 的 1 一半应该是你知道的东西，如果我将它乘以它本身，这遵循指数计算数字的通常规则，我们可以在该指数中添加东西，我应该得到 2到 1，所以它应该是某个数字，当我将它乘以它本身时，我得到 2，你知道在这一点上你有一个选择，也许它是积极的。 ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "也许它是负数但 是如果你总是决定做出正选择如果我们询问负数你将能够从同 一交易中得到一个很好的连续函数2到负1应该是多少这应 该是这样的当我将它乘以 2 并得到 1 时，在哪里？", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "它让我 2 到 0 这就是我们约定的合理性，即负指 数看起来像 1 的一半 但这里真正发生的是我们说 无论这是什么它应该是某种函数满足这个属性 f a 加 b 等于 a 的 f 乘以 b 的 f 并 且事实上，基数是 2 基本上告诉我们，它不仅仅是 任何这样的函数 这是一个函数，当我们插入 1 时 ，我们得到 2 就像你知道的那样健全性检查风格的 问题，看看您是否遵循这里的一些含义我想问您什么是我 不会将其称为垒球，但这并不意味着像一个令人难以置 信的深刻问题一定。", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "这只是更多的检查，如果您遵循抽象 地从函数的属性开始的想法，然后根据这些属性推导我们可能 想要将其写下来的方式，如果 x 的 f 满足这个指数 属性 f对于所有输入，a 加 b 等于 a 的 f 乘 以 b 的 f 并且它还满足 1 的 f 等于 2 以下哪一项为真 这就是说以下哪一项必然为真 无论您要启 动哪个这样的函数和你们中那些还记得哪一堂课的人无论我 们在谈论哪一堂课如何解释欧拉公式真正说的是什么我问了一 个这种风格的问题，我忽略了一个条件，你知道我没有写下 来事实上，我们想要确保 x 的 f 在任何地方都不为零 ，然后这会导致一定程度的混淆，这很酷，在屏幕上出现我 们所有人都会遇到的混淆，但其目的是基本上表明，这个抽象 属性将加法变成乘法的东西足以基本上让你想要将函数写成 任何它等于的函数，就像一个提升到某种幂的函数这就是问题 的精神现在我们有几个关于功率塔的问题这似乎是在这里突 然出现的，这与上次有很大的联系。让我们暂时搁置电力塔问 题，以便我们首先更深入地了解幂运算在这里意味着什么？", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "因为因为我们可以成为我想要声明的，所以我们可以用多种不同的方 式来回答它所以如果你只给我一个，我们将讨论电力塔然后就像数轴 可以用对数标度表示一样对于复杂的平面也可以做同样的事情吗？", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "是的，事实上，我稍后会介绍一个可视化，我们会做一些与此非常相似的事情，因为我们要做的是使用不同的指数函数 X 的 R 乘以 X 但我们是将改变 R 的值，该值将由一个小黄点表示所以我们将讨论这个它不会映射整个平面，而只是来自实轴和虚轴的几个样本点但我们的想法是，当我们围绕这个常数移动时，我们将能够可视化它对平面所做的不同事情，实际上就像将 x 轴转换为对数刻度，然后包裹起来然后，一旦 R 的值变成虚数，它就会交换那些实数放在圆上的角色，而虚数放在对数标度的正轴上，所以我想这三个问题都是很好的问题有点急于去往我想去的地方，但很高兴看到人们在这方面有这样的想法。 ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "明确地，像 f of 5 这样的东西与 f of 1 plus 1 plus 1 plus 1 plu s 1 plus 1 是一样的，这与 1 的 f 乘以自身 5 次是一样的，因为这个属性，如果 1 的 f 是 2 是相同的2 的 5 次方，然后是负 5 的 f 应该是这样的情况，当我们将它乘以 5 的 f 时，我们得到 0 的 f 是什么，并且不能 立即清楚 0 的 f 是什么，但我们可以说1 的 f 加 0 等于 1 的 f 乘以 0 的 f ，但 是 1 的 f 等于 2 所以这也等于 2 所以我 们说 2 等于某物的 2 倍必须是 1，所以在这种 情况下，这保证了负 5 的 f 是 2 的负 5， 它是 1 超过 2 的 5 次我们可以明确地将其写为 2 的负 5，这就是说这两个属性一起使得我们真的 想将函数写为 2 到 X 因为我们放入其中的任何计 数都将满足它看起来会与自身相乘我们放入的任何小数的 次数将满足这些属性你可能想知道，在实值函数的上下文中 ，它实际上是唯一的，但在复杂值函数的上下文中，我们 可以为此编写多个这样的函数 f，其中一个就是我们想 要的看看之前我们可以将一个函数定义为 2 加 2 pi 的自然对数的 exp 的函数我一直都是 X 好 吧，请原谅这里的草率，我只是很兴奋地写下这个 这实 际上是一个不同的函数如果你代入 X 等于 1 一半 ，会发生什么就证明了我们之前看到过，当你代入 1 一半时，你得到的是 2 的负平方根，然后如果你代入 1 四分之一，你得到的不是 4 的四次方根2 但我 乘以 2 的四次方根，所以它是一个不同的函数 但它 仍然满足这些属性，这让我们想把它写成 2 到 X 这使得它表明也许 2 到 X 是一个不明确的函数一点 符号 我们应该用 R 乘以某个值的 exp 来写所 有内容，但你可能会想知道，也许我们对满足这个属性的 所有函数都没有足够的创造力 也许当我们写 exp 时存在歧义R 乘以某物，R 的不同值可能会发挥作用但 我只是要提出一点主张，然后也许会给出一个草图，说明 如果您想要的话，证明会是什么样子，让我们看看假设你 有一些复杂的函数 F，并且它首先满足以下属性，你就 可以对其求导。", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "它是可微分的，这只是让它不会成为一些你知道的完全 混乱的不连续的东西，就像根据你知道的任何向量空间的跨度来取一些随机值 ，我不知道你可能想以疯狂的方式思考小数。", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "这是一个很好的功能。", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "这是可微的 它并不等于 0 到处都是，所以我忘记了这个条件，我忘记 了哪个讲座或类似的东西，然后它有这个中心属性，它将加法 变成乘法 如果你有这样的函数，我声称有一个独特的也许我 真的应该指定存在一个独特的复数R，这样你就可以将X的F 写成基本上是R乘以值X的指数函数，你知道基本上是说如果你 有X作为一个函数具有很好的导数属性的无限多项式以及所有 这些，如果你拥有这个，你就拥有了你想要的每个指数，在一 个非常像指数词的抽象通用意义上，只是基于我们可能想要从 中得到的属性，并且证明的草图将如果您想首先看看我们假设 无处不在的这个值的导数是什么，那么看起来像这样，对吧？", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "我们可以将 X 的 F 完全从表达 式中分解出来，并且整个极限仅以 H 的形式表示 ，如果您考虑它在导数上下文中的含义以及 0 的 F 必然等于 1 的事实，那么整个极限表达式就 是只是一些常数，但更具体地说，它是我们的函数在 0 处的导数所以你会遇到这个有趣的事情，如果 你知道它在 0 处的导数，就决定了它的导数在哪里 在指数函数的背景下，这希望是非常熟悉的，因为 我们真正要说的是指数函数的导数与其自身成正比， 并且比例常数等于 0 处的导数，这都是非常抽象 的措辞，但其目的是强调它是不一定只是我们已经认为 的 X 次方的函数 但它是一类可能更广泛的函数 ，正好满足将加法转化为乘法的抽象属性 但如果你 有这个，它实际上保证你也有一个二阶导数 就此而 言，还有三阶导数，因为导数函数与自身成正比 因此 ，为了求 n 阶导数，您只需查看该比例常数并将 其提高到 n 次幂，然后从这里您可以做泰勒级数 展开，对于那些熟悉泰勒级数的人来说，我可能会将 其作为高级作业，特别是如果您想混合任何在复数意义 上可微的可微函数的想法，即绝对是大学主题你知道 你可以根据需要混合推理但是在只知道泰勒级数而没 有其他知识的人的背景下允许模糊推理接受这个想法 并看看 F 的泰勒展开式有点证明这样的想法： 存 在一个唯一的复数，这样我们的函数 F 必然可以 这样写 然后与正态指数的联系就是每当你有这样的 值 R 我们本质上就是在实数的复杂背景下所做的事 情如果你查看该值 R 的函数的 exp 并将其 写为基数。", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "我们可以将其解释为不仅 意味着 pi 的一半 I 乘以 X 的指数，而 且我们还可以将其解释为意味着 5 pi 一半的 I 乘 X 的指数，并且这些是单独的函数并 且有一个无限的单独函数族，感觉我们应该把它们写 成 I 到 X 所以表达式 I 到 I 除非 你已经采用了一个标准来说明它的必然含义 当你说 它有无限多个输出时，另一种思考方式是函数 I 到 X我们所拥有的符号有点模糊 现在，让我们 开始可视化其中的一些，因为我认为这很有趣 你知 道你告诉我这是一个有用的视觉效果还是一个更令人 困惑的视觉效果，但是我们要做的是看一下 R 乘以 X 的函数 exp ，这基本上是另一种写 e 的 X 次方的方法。事实上，我认为我在 某个时刻渲染了一个不同的动画，指定了因为我正打 算计划这样做，所以让我哦，是的，你回到我的文件 系统中，回到你应该在的地方，继续下去，它在抱 怨，因为有多个不同的，就像有一个哦，替换它显示 在另一个屏幕上等等，为什么是的，可以替换吗？", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "把你看到的任何东西放在那里现在我们回到哦， 我们所有的一切只是为了我可以很好地写出来如 果你不舒服地将它视为R乘以X这个无限多项式 的exp就在后脑勺 e 到 R 乘以 X， 我们将围绕 R 变化，所以我将跟随虚轴的 点，我将跟随实轴的点，让我们看看这会做什么 这一切都有点快，所以让我慢慢地想一下所有负 数任何东西这是一个负实数将被压缩到 0 到 1 之间的范围内哪个应该对负数有意义？", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a 到负实数是 0 到 1 之间的数字，我们专门跟踪负 1 的 f，它将出现在 e 上的 1 大约为 30 0 的地方。3 7 f of 1 落在 e 上， 正如预期的那样，这就是 1 的 exp 是 f 的 I 将落在单 位圆周围一弧度，沿着整个虚轴跟踪 虚轴如何围绕圆缠绕是很有趣的当我 们调整 R 的值时会发生什么？", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "我们可能需要 R 的值，它会以不同的方式拉伸事物，因此当我们将其达到 2 时，您知道它会将实轴拉伸得更多，因此 1 的 f 会在 e 平方稍微高于 7 f 的负值附近结束1 更接近 0 I 的 f 是 2 弧度 绕负 I 的圆 f 的旋转是 负 2 弧度旋转 当然我们可以得到我们最喜欢的公式，如 果那是我们作为缩放常数的 pi 那么实数轴被拉伸了很多你 知道 1 的 f 位于 pi 的 e 处，非常接近 20 加 pi 这总是很有趣，负 1 的 f 非常接近 0 ，所以它真的被拉伸了axis 而且它还在单位圆方向上拉伸 了东西，这样到达 I 的 f 或负 I 的 f 就绕着圆 走了一半，所以现在一切都很好我们会如何考虑这样的函数？", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "我们还可以将 X 写成 2 乘以 X 的自然对数 ，这样我们就可以将表示 R 值的黄点移动到 0 左右。69仍然没有虚部，只有实数0。69 左 右 这是 2 的自然对数，你可以看到 1 的 f 落在 2 上 这就是为什么我们要将这个函数 2 称为 1 half 的 X f 实际上，负数 1 的 f 落在 1 half f 上我绕 单位圆走一圈，非常具体地说，它将是 0。围绕单位圆 69 弧度 现在我们可以有更多乐趣并说出如果我们将其更改为而不是 0 会发生什么。69 不是 2 的自然对数，而是我乘以 2 的自然对数，这样我们就可 以真正考虑到可能有指数底数的东西。", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "I 的 I 次方是多少，在这种情况下，它会将其推至 0 左右。2 大约五 分之一 但是有许多不同的指数函数都具有将 1 的 f 放到数字 I 上的属性所以如果我 们要进一步扩大它我不认为我在这里有它的动画 但是如果我们采取那个黄点并将其升高，直到它达 到 pi I 的 5 倍，您会看到单位圆？", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "绕自身旋转，这样 1 的负 f 的 f 会绕另外 2 pi 弧度旋转并落在它所在的位置 但它会将实 轴拉伸得更多 这就是 I 到 I 的另一个输出的 意义一个小得多的数字，大约是 0。0003 左右 但是我们也可以看到我认为非常有趣的东西 如果我们考虑我们 想要解释为 2 的 X 次方的替代表达式，会发生什么？", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "我们有 X of R 乘以 X，R 等于这个值， 它是 2 加 pi 乘以 I 的自然对数 这意 味着当我们代入 1 时，1 的 f 为负 2， 所以我们要编写这个函数负 2 的 X 次方，这 实际上是你知道的事情，当我们写一个负数的 X 次 方时，它有点看似简单，负 2 的 X 次方乍一 看并不一定是这样的，它一定会给我们带来以任何方 式进入复数，但是当然，当我们插入像 1 hal f 这样的值时，我们有点要求负 2 的平方根， 我们意识到我们想要将其写成类似 I 乘以平方根的 形式2 但如果你在它处理的完整复杂域中查看这个 负 2 的 X 次方函数，你所看到的是一个取 1 的负 2 的值的函数 如果它这样做的话会发 生什么它与实数轴的其余部分的关系是向外螺旋吗？", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "因此，我们看到负 1 的 f 位于负 1 half 处 如果您 遵循 1 half 的 f ，大约会在您所期望的位置 它会恰 好位于虚线上，并且 1 half 的 f 将是 2 的平方根 好吧，我的鼠标不在我想要的位置。", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "它大约是 2 乘以 I 的平方根，当你继续往下看时，这将向你展示负 2 到 X 的所有实际值幂，它必然会螺旋式旋转但我们也可 以将 R 的值移得更高并得到它大约 tau 次 I 大约六点二八次 I，在这种情况下，这是另一个函数 ，我们希望将其写为 2 到 X 之类的函数，因为对 于您为 X 插入的任何整数到整数，它都会看起来像 是重复的乘法，它甚至对像 1 half 这样的东西 有某种合理的值，它输出负平方根而不是正平方根，但它 实际上做的是对平面的变换，它把所有东西都放在真实的 地方数轴最终成为一个非常紧密缠绕的螺旋，它绕着一圈 ，它只是以这样一种方式螺旋，即 1 的 f 正好落 在数字 2 上所以从这个意义上来说，我们可以说 2 到 X 是 合理地解释为一个独立的指数函数，与我 们传统上习惯的指数函数不同，所以我想，我会把所有的 事情留到今天，我只会留给你一些挥之不去的问题来思考 ，好吧，所以如果你想认为 I 到 I 是一个多值表 达式，对吧，你可以说我们采用了一种约定，想象一下， 你会说你选择自然对数函数的一个分支，也许这会将你锁 定在 e 到负 pi 的状态中但是如果你说这种想要 成为无限多个不同的值，就像我们看到的各种值一样，那 么 2 的三分之一想要有多少个值具有相同的意义？", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "第十个想要以不同的方式表述，让我说的是满足 X 的所有指数函数 F 哦，我是否将其 写在满足 X 的 f 的某个位置，所以如果 它满足所有这些属性，那么我已经写了其中， 如果 1 的 f 等于 2 对，当我们将 X 等于 3 十分之一插入到什么函数的各 种选项中时，我们会得到多少种不同的输出？", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "对于 2 到 pi 来说，2 到 X 可以表 示的各种函数，如果我们将 2 到 X 视为 某种指数函数 指数 在这些抽象属性的意义上 ，如果我们是的，如果我们如果我们有一个不同 的此类函数的类，我们想插入 pi 这让我笑 只是因为它是这样一个我知道当你试图思考它时会 弹出一个有趣的答案所以这些就是问题我会留给 你，我想这是你知道的我在接近今天的讲座时我 的核心问题是我是否希望它像指数函数的这些抽 象属性一样描述对我来说从这些抽象属性开始很 酷你被锁在了 e 到 rx 或更多的想法中 只是你知道我认为更诚实地写出 r 乘以 x 对于 r 的不同值的 exp 它它把你锁在 那么远但它并没有把你锁在有一个明确的概念， 即 2 的 x 次方应该远不像 I 的 x 次方那样。当然，这样做的风险是，有时人们 不喜欢抽象，有时它并不容易理解，但如果这是 如果你知道你只是让我知道我想我认为围绕所有这 些东西包括电力塔有一个完整有趣的思想圈，因 为如果你想像我们上次在复数背景下那样实际谈 论电力塔或者即使是负面的基础你也必须思考这 样的事情，所以这是我们在屏幕上提出的一个问 题是的，如果我们为我做这件事，会发生什么？", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "滴定你知道，让我们尝试一下，让我们继续尝试一个功率塔 ，我们将 I 提高到给定的功率，看看会产生什么，所以 它不打算这样做，但我们可以，我们总是可以拉起Pyth on并基本上做我们上次做的事情所以它的工作方式是我们 从一些基值开始，然后对于某种范围我们在做什么，我们 正在采取a，我们将重新分配它可以是任何基数，在本例中 是我提高到 a 的幂，应该可以，很酷，所以我们要这样 做，然后我们将打印 a 的值，让我们这样做是的，这是 一个更大的数字，比如 200 所以看起来发生的事情 是这些事情有时可能会造成混乱。", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "我实际上有所以 让我让我导入NumPy所以我有指 数函数让我去像我们以前一样的大范 围而不是像你知道的那样写它就像我 的X的幂我要写它作为不同常数的指 数函数，我要制作一个不同的常数， 我希望它是 5 pi 一半，所以 我会做 5 pi 一半乘以 I， 所以它是一个复数，它有 5 pi 一半作为虚部 那么这是 5 p i 的一半乘以 I，我在做什么？", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/english/captions.srt b/2020/ldm-i-to-i/english/captions.srt index 9ff356828..6220c7eed 100644 --- a/2020/ldm-i-to-i/english/captions.srt +++ b/2020/ldm-i-to-i/english/captions.srt @@ -315,16 +315,16 @@ And then that's the same as what was written, what was categorized as the third most common one, one half pi times i, just swapping the two numbers there. 80 -00:04:52,300 --> 00:04:55,335 -And I think, yeah, I think the one following that, +00:04:52,300 --> 00:04:57,189 +and I think, yeah, I think the one following that I'm guessing people either forgot to 81 -00:04:55,335 --> 00:04:58,431 -I'm guessing people either forgot to mention the i, +00:04:57,189 --> 00:05:02,191 +mention the i but let's just, let's just write that out and kind of see where this moves 82 -00:04:58,431 --> 00:05:02,360 -but let's just write that out and kind of see where this moves us. +00:05:02,191 --> 00:05:02,360 +us. 83 00:05:02,660 --> 00:05:03,720 @@ -695,3914 +695,3926 @@ rotate that vector 90 degrees, that gives you your velocity, and that's enough to tell you how to move, assuming you know where you're starting. 175 -00:10:41,040 --> 00:10:45,422 +00:10:41,040 --> 00:10:44,348 And of course we have to draw this out in the complex plane, 176 -00:10:45,422 --> 00:10:50,019 -the fact that we've rotated 90 degrees already brings us there, +00:10:44,348 --> 00:10:47,820 +the fact that we've rotated 90 degrees already brings us there. 177 -00:10:50,019 --> 00:10:55,766 -and I could draw out a bunch of different potential position vectors, you know, +00:10:47,820 --> 00:10:52,159 +And I could draw out a bunch of different potential position vectors, you know, 178 -00:10:55,766 --> 00:11:02,160 -and what it would tell you is, okay, I've got to sit there, rotate my vector 90 degrees, +00:10:52,159 --> 00:10:56,445 +suggestively maybe putting them on a circle and saying really imagine yourself 179 -00:11:02,160 --> 00:11:04,100 -that describes my velocity. +00:10:56,445 --> 00:11:00,187 +sitting at any one of those and following the rule for this dynamic. 180 -00:11:04,420 --> 00:11:07,635 -And it doesn't necessarily have to be any one of the points on that circle, +00:11:00,187 --> 00:11:05,015 +And what it would tell you is, okay, I've got to sit there, rotate my vector 90 degrees, 181 -00:11:07,635 --> 00:11:10,089 -any point in the plane, if you're following this dynamic, +00:11:05,015 --> 00:11:06,480 +that describes my velocity. 182 -00:11:10,089 --> 00:11:13,389 -you say rotate that vector 90 degrees, which if we're drawing a vector field, +00:11:06,480 --> 00:11:09,093 +And it doesn't necessarily have to be any one of the points on that circle, 183 -00:11:13,389 --> 00:11:15,420 -we often scale down, and it's all well and good. +00:11:09,093 --> 00:11:11,087 +any point in the plane, if you're following this dynamic, 184 +00:11:11,087 --> 00:11:13,769 +you say rotate that vector 90 degrees, which if we're drawing a vector field, + +185 +00:11:13,769 --> 00:11:15,420 +we often scale down, and it's all well and good. + +186 00:11:15,980 --> 00:11:20,513 And here you could phrase this question without even talking about exponentials or -185 +187 00:11:20,513 --> 00:11:25,101 complex numbers or anything like that, and in fact I'm going to go back to the quiz -186 +188 00:11:25,101 --> 00:11:29,579 just to really emphasize that Euler's formula and what it's claiming and then how -187 +189 00:11:29,579 --> 00:11:32,911 we apply it to expressions like i to the power i, it's very, -188 +190 00:11:32,911 --> 00:11:36,735 it's really intuitive as soon as we're putting some dynamics into it, -189 +191 00:11:36,735 --> 00:11:41,213 and we can ask questions removed from the idea of exponentials that actually have -190 +192 00:11:41,213 --> 00:11:43,180 the same substance in their answers. -191 +193 00:11:43,440 --> 00:11:49,544 So here I want you to imagine starting a walk from the point 1,0 on the coordinate plane, -192 +194 00:11:49,544 --> 00:11:55,444 in such a way that at all moments your velocity vector is a 90 degree counterclockwise -193 +195 00:11:55,444 --> 00:12:00,260 rotation of the vector drawn from 0,0, the origin, up to where you are. -194 +196 00:12:00,700 --> 00:12:04,680 After pi halves units of time, where will you be on the plane? -195 +197 00:12:05,620 --> 00:12:07,971 Okay, so kind of think about that, think about that -196 +198 00:12:07,971 --> 00:12:10,640 dynamic and where you'll end up after a little bit of time. -197 +199 00:12:19,880 --> 00:12:22,040 While you answer, let me see if we have any questions -198 +200 00:12:22,040 --> 00:12:23,560 that have popped in from the audience. -199 +201 00:12:24,380 --> 00:12:28,714 No questions at the moment, at least that I see, so if ever you do want to ask, -200 +202 00:12:28,714 --> 00:12:33,264 go to Twitter, use the hashtag lockdownmath, those will be forwarded to me and then -201 +203 00:12:33,264 --> 00:12:36,515 sometimes they'll be on the screen that I can pull up here, -202 +204 00:12:36,515 --> 00:12:38,520 and that always makes for a fun time. -203 +205 00:12:39,840 --> 00:12:42,774 For example, it seems like now we actually have one, -204 +206 00:12:42,774 --> 00:12:47,480 which is how about other solutions to the expression x equals i, such as 5 pi halves? -205 +207 00:12:47,780 --> 00:12:50,211 Oh, I'm so glad you asked, I'm so very glad you asked, -206 +208 00:12:50,211 --> 00:12:51,980 we will certainly be talking about that. -207 +209 00:12:52,960 --> 00:12:56,191 And if you just hold on for like one little moment, -208 +210 00:12:56,191 --> 00:13:01,101 that's exactly what we're going to get to, so I'll keep that on file for us to -209 +211 00:13:01,101 --> 00:13:02,780 pull up in a little moment. -210 +212 00:13:03,240 --> 00:13:06,672 But right now while we're just trying to get intuition around the first, -211 +213 00:13:06,672 --> 00:13:08,600 you know, there will be multiple answers. -212 +214 00:13:08,620 --> 00:13:10,840 The first answer we saw was e to the negative pi halves. -213 +215 00:13:12,080 --> 00:13:13,920 Let's consider this lake question. -214 +216 00:13:15,260 --> 00:13:18,286 So it seems like most of you correctly said that -215 +217 00:13:18,286 --> 00:13:20,880 this would get you at the coordinates 0,1. -216 +218 00:13:20,880 --> 00:13:24,100 It's essentially walking you a quarter of the way around. -217 +219 00:13:24,260 --> 00:13:25,320 I said lake question. -218 +220 00:13:25,420 --> 00:13:30,026 I'm sort of picturing in my head as a big circular lake that you're going for -219 +221 00:13:30,026 --> 00:13:34,751 a walk around, because when you have this rule for dynamics where your position -220 +222 00:13:34,751 --> 00:13:39,830 vector rotated by 90 degrees gives you your velocity vector, that is circular motion, -221 +223 00:13:39,830 --> 00:13:44,260 because the tangent line to a circle is always perpendicular to its radius. -222 +224 00:13:44,860 --> 00:13:48,032 So this rule of motion corresponds with walking around a circle, -223 +225 00:13:48,032 --> 00:13:50,620 and that's kind of the intuition for Euler's formula. -224 +226 00:13:50,880 --> 00:13:51,200 Right? -225 +227 00:13:51,760 --> 00:13:55,181 That you keep increasing that value of t, and this dynamical rule, -226 +228 00:13:55,181 --> 00:13:58,858 if e to the i t is going to behave according to the derivatives that we -227 +229 00:13:58,858 --> 00:14:02,740 expect of a function like e to the t, necessarily walks you around a circle. -228 +230 00:14:03,140 --> 00:14:07,314 So in particular, when we're wondering how long does it take to get to i, -229 +231 00:14:07,314 --> 00:14:11,037 you would basically say just wait for an amount of time equal to, -230 +232 00:14:11,037 --> 00:14:14,140 well, whatever the angle to get up there is, pi halves. -231 +233 00:14:14,760 --> 00:14:16,320 Okay, so in this case we're thinking pi halves. -232 +234 00:14:16,400 --> 00:14:17,980 Let me see if I can get it exactly on the dot. -233 +235 00:14:18,160 --> 00:14:19,000 Okay, 1.57. -234 +236 00:14:19,000 --> 00:14:21,736 We're thinking of pi halves as kind of an amount of -235 +237 00:14:21,736 --> 00:14:24,420 time for these dynamics to get you on the number i. -236 +238 00:14:25,180 --> 00:14:29,459 And that gives us the first half of our intuition if we want to go back to our formula, -237 +239 00:14:29,459 --> 00:14:32,669 and we're asking in what sense do two 90 degree rotations combine -238 +240 00:14:32,669 --> 00:14:34,420 to make e to the negative pi halves? -239 +241 00:14:37,320 --> 00:14:41,669 Once we're thinking of our dynamics here as each one of your velocity vectors -240 +242 00:14:41,669 --> 00:14:44,736 as a 90 degree rotation of your position, this base i, -241 +243 00:14:44,736 --> 00:14:49,086 the way that we're kind of thinking about it reading it off as something with -242 +244 00:14:49,086 --> 00:14:52,878 meaning rather than just a number, is that as a point on the plane, -243 +245 00:14:52,878 --> 00:14:57,339 it's where you get after traveling for pi halves units of time according to the -244 +246 00:14:57,339 --> 00:15:01,800 dynamics of this expression, according to the idea that your velocity is always -245 +247 00:15:01,800 --> 00:15:03,920 a 90 degree rotation of your position. -246 +248 00:15:04,440 --> 00:15:05,740 Okay, so that's awfully nice. -247 +249 00:15:05,960 --> 00:15:08,877 And in fact, there's already sort of two different i's at play here, -248 +250 00:15:08,877 --> 00:15:11,160 which, or I guess there's three different i's at play. -249 +251 00:15:11,160 --> 00:15:15,640 We have this base, which is describing the 90 degrees that we walk around the circle. -250 +252 00:15:15,720 --> 00:15:19,142 We have the i that's sitting here, which is describing -251 +253 00:15:19,142 --> 00:15:22,440 the rule of rotating your velocity vector 90 degrees. -252 +254 00:15:22,440 --> 00:15:25,333 But now we're going to introduce another i, which -253 +255 00:15:25,333 --> 00:15:28,980 essentially has this effect of changing what your dynamics are. -254 +256 00:15:29,440 --> 00:15:33,646 Because as we go from e to the i times t, and instead we, you know, -255 +257 00:15:33,646 --> 00:15:38,656 raise things to another power of i, which I'll just write with a little caret i, -256 +258 00:15:38,656 --> 00:15:43,914 if we take e to the i t and we alter what that expression is by raising it to the i, -257 +259 00:15:43,914 --> 00:15:46,080 what we get is e to the negative t. -258 +260 00:15:48,040 --> 00:15:50,020 Okay, so we have e to the negative t. -259 +261 00:15:51,000 --> 00:15:56,206 And if we try to interpret that with the same sort of dynamics that we -260 +262 00:15:56,206 --> 00:15:59,800 had above in terms of a velocity and a position, -261 +263 00:15:59,800 --> 00:16:05,226 what that's telling us is that the derivative of our new dynamic e to the -262 +264 00:16:05,226 --> 00:16:11,460 negative t is equal to, well now the constant sitting in front of t is negative one, -263 +265 00:16:11,460 --> 00:16:17,840 so our chain rule will have it be negative one times itself, times e to the negative t. -264 +266 00:16:18,000 --> 00:16:21,680 So whatever your position is, now your velocity is negative one times itself. -265 +267 00:16:22,100 --> 00:16:27,131 So the effect of raising to another power i, it's kind of like we took the dynamics, -266 +268 00:16:27,131 --> 00:16:31,511 we looked at every velocity vector and we said rotate another 90 degrees, -267 +269 00:16:31,511 --> 00:16:35,299 so that in this context it would actually be in the beginning a -268 +270 00:16:35,299 --> 00:16:38,200 velocity vector pointing backwards with one unit. -269 +271 00:16:38,680 --> 00:16:40,841 So if you're starting off at the number one, your -270 +272 00:16:40,841 --> 00:16:42,960 initial velocity is to walk straight toward zero. -271 +273 00:16:43,280 --> 00:16:46,424 And as you walk even lower, if you were sitting at one half, -272 +274 00:16:46,424 --> 00:16:49,930 then you would still be walking towards zero, but now your velocity -273 +275 00:16:49,930 --> 00:16:53,900 vector would be negative one times where you are, which is negative one half. -274 +276 00:16:54,640 --> 00:16:57,753 And what this means for the actual motion that this would imply, -275 +277 00:16:57,753 --> 00:17:01,490 you might kind of imagine it, I haven't animated this nicely yet or anything, -276 +278 00:17:01,490 --> 00:17:05,419 but if you look at this whole vector field and you ask each one of those vectors, -277 +279 00:17:05,419 --> 00:17:08,820 take wherever you are and rotate another 90 degrees, counterclockwise, -278 +280 00:17:08,820 --> 00:17:11,119 they would all be pointed in towards the origin. -279 +281 00:17:11,839 --> 00:17:16,055 So if you were to have a little dot move in such a way that its velocity -280 +282 00:17:16,055 --> 00:17:19,231 always matches whatever vector it's sitting on top of, -281 +283 00:17:19,231 --> 00:17:23,677 what you would end up with is something where with each time step it kind of -282 +284 00:17:23,677 --> 00:17:28,181 takes a step towards zero and it just with each of your time steps is walking -283 +285 00:17:28,181 --> 00:17:32,511 towards zero and each step has a size that gets smaller and smaller as you -284 +286 00:17:32,511 --> 00:17:33,840 actually approach zero. -285 +287 00:17:34,880 --> 00:17:37,980 And of course in practice, this would be a bunch of -286 +288 00:17:37,980 --> 00:17:41,140 infinitesimal steps rather than very concrete sizing. -287 +289 00:17:41,560 --> 00:17:46,199 You might be very exact about it if you wanted and say what we're looking at -288 +290 00:17:46,199 --> 00:17:50,598 is scaling down by some number just less than one and we're doing this n -289 +291 00:17:50,598 --> 00:17:55,780 different times and then we're gonna multiply that by however much time we're waiting. -290 +292 00:17:56,980 --> 00:17:58,620 And you take this as a limiting expression. -291 +293 00:17:59,960 --> 00:18:03,856 That's just what we talked about in the compound interest lecture if you're curious -292 +294 00:18:03,856 --> 00:18:07,660 and it's kind of the standard way to talk about e to powers as this sort of limit. -293 +295 00:18:07,980 --> 00:18:11,280 You could also have that t living on the inside here instead if you wanted. -294 +296 00:18:12,200 --> 00:18:17,070 But now if we think about our original point i, our base, what that meant, -295 +297 00:18:17,070 --> 00:18:22,850 it was saying look at our dynamics and it's the point you get to when you wait pi halves -296 +298 00:18:22,850 --> 00:18:23,760 units of time. -297 +299 00:18:24,660 --> 00:18:28,467 So the effect of raising to the i changes our dynamics in such a way that -298 +300 00:18:28,467 --> 00:18:32,480 instead of walking around a circle we're doing this kind of exponential decay. -299 +301 00:18:32,640 --> 00:18:39,400 We're moving towards zero at a slowing and slowing rate and the place that you end -300 +302 00:18:39,400 --> 00:18:46,160 up after pi halves units of time will be e to the negative pi halves around 0.2079. -301 +303 00:18:47,340 --> 00:18:50,679 So, you know, that's a little bit of intuition for in what sense do -302 +304 00:18:50,679 --> 00:18:54,020 two 90 degree rotations combine to get you this very specific value. -303 +305 00:18:54,200 --> 00:18:57,081 One of them changes your dynamics and the original -304 +306 00:18:57,081 --> 00:19:00,020 one came from the idea of walking pi halves radians. -305 +307 00:19:00,400 --> 00:19:02,800 So I think that's kind of satisfying in its own way because -306 +308 00:19:02,800 --> 00:19:06,360 otherwise when you look at the expression i to the i equals e to the negative pi halves, -307 +309 00:19:06,360 --> 00:19:07,840 it's like what does any of this mean? -308 +310 00:19:07,920 --> 00:19:11,600 This doesn't correspond to any, you know, thing that might see in the real world. -309 +311 00:19:11,600 --> 00:19:14,221 But thinking through the intuitions actually have you thinking -310 +312 00:19:14,221 --> 00:19:16,760 through things like what are the dynamics of circular motion? -311 +313 00:19:17,100 --> 00:19:18,660 What are the dynamics for exponential decay? -312 +314 00:19:18,980 --> 00:19:21,540 Things that come up in the real world all the time. -313 +315 00:19:22,220 --> 00:19:24,989 But there is a potentially more pressing question -314 +316 00:19:24,989 --> 00:19:27,980 which was raised and let's see if it's still up there. -315 +317 00:19:28,600 --> 00:19:31,147 Well, whoever asked it a little bit earlier, what -316 +318 00:19:31,147 --> 00:19:33,440 about other solutions to e to the x equals i? -317 +319 00:19:33,840 --> 00:19:36,622 I think that's that's actually spot on and in fact, -318 +320 00:19:36,622 --> 00:19:40,100 let me let me ask other people to contribute some solutions here. -319 +321 00:19:40,260 --> 00:19:42,200 So let's let's go to our quiz. -320 +322 00:19:42,780 --> 00:19:47,929 Let me go to question number three, which is going to be very similar to -321 +323 00:19:47,929 --> 00:19:53,289 a question that was already seen, which is to write a value of x other than -322 +324 00:19:53,289 --> 00:19:58,580 the one we just saw x equals pi halves times i so that e to the x equals i. -323 +325 00:19:58,900 --> 00:20:01,476 And feel free to be as creative as you want to try to -324 +326 00:20:01,476 --> 00:20:04,100 get an answer that other people haven't written, right? -325 +327 00:20:04,100 --> 00:20:07,077 Um just to really emphasize that there are actually quite a -326 +328 00:20:07,077 --> 00:20:10,004 few different options here and then in a moment we'll talk -327 +329 00:20:10,004 --> 00:20:13,280 about why there are quite a few options if it's not already clear. -328 +330 00:20:13,420 --> 00:20:16,802 But I'll give you a moment to think of which of the many possible values of -329 +331 00:20:16,802 --> 00:20:20,184 x you could choose is the one that you want to put your fingerprint on that -330 +332 00:20:20,184 --> 00:20:23,700 you want to be contributing to the live stats page that we're seeing right now. -331 +333 00:20:24,140 --> 00:20:25,120 I'll give you a little moment. -332 +334 00:20:42,980 --> 00:20:46,480 So, and as you're answering that we've got a question coming in -333 +335 00:20:46,480 --> 00:20:50,090 that says wouldn't it be more accurate to say that i to the power -334 +336 00:20:50,090 --> 00:20:53,700 i is i different 90 degree rotations, not two 90 degree rotations. -335 +337 00:20:53,800 --> 00:20:56,820 After all it's multiplying by i that's the rotation. -336 +338 00:20:57,700 --> 00:21:01,600 So if we stretch the meaning of words too thin, aren't we multiplying by i i times? -337 +339 00:21:01,600 --> 00:21:05,380 I mean, I think the most honest answer here is just no not at all. -338 +340 00:21:05,560 --> 00:21:08,697 No offense meant but that's just not what exponentiation actually -339 +341 00:21:08,697 --> 00:21:11,740 means as soon as we're extending to the idea of complex numbers. -340 +342 00:21:14,220 --> 00:21:17,520 I get the intent you kind of want to stretch the meaning of taking i to the i. -341 +343 00:21:17,620 --> 00:21:20,240 It's as if you're multiplying by itself i different times. -342 +344 00:21:20,540 --> 00:21:26,100 But I cannot think of a way that that actually like satisfactorily makes sense in my mind. -343 -00:21:26,679 --> 00:21:33,100 +345 +00:21:26,680 --> 00:21:33,100 What I do think makes some sense is to try to think of the function i to the x. -344 +346 00:21:34,540 --> 00:21:37,120 Say what properties do we want this function to have? -345 +347 00:21:37,460 --> 00:21:40,685 And in the context of counting numbers, you know, -346 +348 00:21:40,685 --> 00:21:44,813 if n and k are just things like 3 and 5, we know that it should -347 +349 00:21:44,813 --> 00:21:49,780 satisfy this idea of multiplying the outputs correspond to adding the inputs. -348 +350 00:21:50,160 --> 00:21:53,455 So we want to say if this is any kind of function, and I, you know, -349 +351 00:21:53,455 --> 00:21:56,606 I've said this in many different forms in many different places, -350 +352 00:21:56,606 --> 00:22:00,920 we want it to satisfy the property that when you add the inputs you multiply the outputs. -351 +353 00:22:01,500 --> 00:22:04,772 That to me is kind of what exponentiation is more so than -352 +354 00:22:04,772 --> 00:22:07,820 trying to stretch the idea of repeated multiplication. -353 +355 00:22:09,540 --> 00:22:13,632 So, it like, I mean, correct me if I'm wrong, if you can definitely, -354 +356 00:22:13,632 --> 00:22:18,674 if you can find some way where if you read this off in a stretching language context -355 +357 00:22:18,674 --> 00:22:23,301 saying we're taking a 90 degree rotation and applying that i different times, -356 +358 00:22:23,301 --> 00:22:28,402 that's not nonsense and that somehow like intuitively gets you to the answer e to the -357 +359 00:22:28,402 --> 00:22:30,360 negative pi halves, I'm all ears. -358 +360 00:22:30,680 --> 00:22:35,768 But I think the healthier relationship to have is to say we have this central property -359 +361 00:22:35,768 --> 00:22:40,740 for exponentiation, which definitely holds in the context of repeated multiplication. -360 +362 00:22:41,040 --> 00:22:44,718 But to extend beyond the places where repeated multiplication makes any sense, -361 +363 00:22:44,718 --> 00:22:48,676 because it just it just doesn't make sense when we're talking about something that's -362 +364 00:22:48,676 --> 00:22:52,680 not a counting number like i, or other crazy things that we exponentiate in math like -363 +365 00:22:52,680 --> 00:22:53,100 matrices. -364 -00:22:54,759 --> 00:22:58,123 +366 +00:22:54,760 --> 00:22:58,123 Focus on the property more so than what some people think of as -365 +367 00:22:58,123 --> 00:23:01,540 the origins of that property or the original intuition behind it. -366 +368 00:23:03,380 --> 00:23:05,512 And an interesting question will be, you know, -367 +369 00:23:05,512 --> 00:23:08,780 is there just one such function that feels reasonable to write for this? -368 +370 00:23:09,100 --> 00:23:13,707 Because, you know, if we're going to write it as i to the x not only should it -369 +371 00:23:13,707 --> 00:23:18,898 satisfy this, it should also satisfy, you know, when we plug in the number one we get i, -370 +372 00:23:18,898 --> 00:23:23,740 presumably i to the power one, however we're thinking of this function should be i. -371 +373 00:23:24,120 --> 00:23:26,880 Is there just one such function or are there multiple such functions? -372 +374 00:23:27,760 --> 00:23:30,045 That's what the quiz question is starting to get at and we've -373 +375 00:23:30,045 --> 00:23:32,220 got lots of different answers, which always makes me happy. -374 +376 00:23:32,380 --> 00:23:35,440 So let's see some of the variety that people have thrown in here. -375 +377 00:23:35,680 --> 00:23:38,220 So we've got five pi i halves. -376 +378 00:23:38,580 --> 00:23:43,920 Great, that absolutely is another value that we could plug in for x here. -377 +379 00:23:44,120 --> 00:23:47,321 And just to spell out that a little bit more visually, -378 +380 00:23:47,321 --> 00:23:52,559 if we were to look back at our circle here where we've at the moment walked for an amount -379 +381 00:23:52,559 --> 00:23:57,739 of time equal to pi halves, which is 1.57, what if instead we took another full turn and -380 +382 00:23:57,739 --> 00:24:02,511 we go another pi halves to get us to pi, which you know, we might kind of record, -381 +383 00:24:02,511 --> 00:24:04,840 that's where the e to the pi i value is. -382 +384 00:24:05,480 --> 00:24:08,703 We walk another pi halves, we walk another pi halves, -383 +385 00:24:08,703 --> 00:24:13,300 which at this point we would have gone a full circle getting us back to one, -384 +386 00:24:13,300 --> 00:24:17,420 and then we walk for five pi halves, which numerically is about 7.85. -385 +387 00:24:18,040 --> 00:24:21,680 Yeah, that absolutely is another number that gets us on top of i. -386 +388 00:24:22,220 --> 00:24:27,763 And if we were to go through the whole rigmarole of re-expressing i -387 +389 00:24:27,763 --> 00:24:33,795 to the power i by first writing e to the five pi halves i to the power i, -388 +390 00:24:33,795 --> 00:24:39,501 those i's multiply to become negative and we'd be looking at e to the -389 +391 00:24:39,501 --> 00:24:44,800 negative five pi halves, which is a very different number, right? -390 +392 00:24:44,800 --> 00:24:46,240 We can actually calculate this. -391 +393 00:24:46,820 --> 00:24:50,840 I'm not sure off the top of my head, but let's take a look at Desmos maybe. -392 +394 00:24:51,180 --> 00:24:52,240 Go over here. -393 +395 00:24:52,480 --> 00:24:53,220 Let's ask it. -394 +396 00:24:53,280 --> 00:24:57,340 What is e to the negative five pi halves? -395 +397 00:24:57,960 --> 00:24:59,800 0.000388. -396 +398 00:25:00,600 --> 00:25:01,900 Okay, 000388. -397 +399 00:25:03,300 --> 00:25:04,800 Much smaller number. -398 +400 00:25:05,420 --> 00:25:07,820 0.000388. -399 +401 00:25:08,720 --> 00:25:12,100 Which begs the question of okay i to the i, what are you? -400 +402 00:25:12,680 --> 00:25:12,780 Right? -401 +403 00:25:12,920 --> 00:25:16,220 Are you about a fifth like we saw before around 0.2? -402 +404 00:25:16,240 --> 00:25:18,200 Or are you this much smaller number? -403 +405 00:25:18,840 --> 00:25:21,776 And in terms of our intuition, it's basically a -404 +406 00:25:21,776 --> 00:25:25,080 question of how we're interpreting that base i, right? -405 +407 00:25:25,380 --> 00:25:29,133 Are we thinking of it as you wait about 1.57 units of time and then you -406 +408 00:25:29,133 --> 00:25:32,938 translate your dynamics into something that looks like decay rather than -407 +409 00:25:32,938 --> 00:25:36,900 spinning and see where you decay to after that amount of time, which is 0.2. -408 +410 00:25:37,300 --> 00:25:42,560 Or do you wait even longer for about 7.85 units of time, which is five pi halves? -409 +411 00:25:42,560 --> 00:25:45,037 And see what happens when you decay for that long, -410 +412 00:25:45,037 --> 00:25:46,980 which gets you to a much smaller number. -411 +413 00:25:48,180 --> 00:25:50,320 But that's not the only answer that we could enter, right? -412 +414 00:25:50,440 --> 00:25:54,540 We have other people coming in here with negative three halves times i pi. -413 +415 00:25:55,520 --> 00:26:01,287 Which, you know, in terms of a unit circle, we could think of as saying, -414 +416 00:26:01,287 --> 00:26:05,948 hey if I want to get to i, rather than walking 90 degrees, -415 +417 00:26:05,948 --> 00:26:11,400 pi halves radians that way, what if I walk 270 degrees the other way? -416 +418 00:26:11,960 --> 00:26:14,585 Three pi halves radians, which maybe I'll think of as negative -417 +419 00:26:14,585 --> 00:26:17,420 because the convention is usually that counterclockwise is positive. -418 +420 00:26:17,700 --> 00:26:19,900 That absolutely is another way to express it. -419 +421 00:26:19,980 --> 00:26:21,380 And that would get us a different answer. -420 +422 00:26:21,580 --> 00:26:25,013 If we had e to the negative three pi halves i, -421 +423 00:26:25,013 --> 00:26:28,520 all to the power i, we go through the same game. -422 +424 00:26:28,760 --> 00:26:32,946 Now the i squared cancels with the negative that's already there, -423 +425 00:26:32,946 --> 00:26:35,420 and we have a positive three pi halves. -424 +426 00:26:36,260 --> 00:26:42,958 And numerically this gets us an even different looking answer from what we had before, -425 +427 00:26:42,958 --> 00:26:47,808 which if we go over and we say hey, what is e to the three pi, -426 +428 00:26:47,808 --> 00:26:50,580 not three o, three pi halves 111.31. -427 +429 00:26:50,980 --> 00:26:54,160 Very different kind of number than what we saw before. -428 +430 00:26:56,040 --> 00:26:58,340 111 point, what was it? -429 +431 00:26:58,840 --> 00:26:59,760 111.31. -430 +432 00:27:00,640 --> 00:27:00,740 Great. -431 +433 00:27:01,240 --> 00:27:03,180 111.31 or so. -432 +434 00:27:04,260 --> 00:27:07,773 And again in terms of the intuition, what you might be asking there is, -433 +435 00:27:07,773 --> 00:27:11,140 suppose we have this rotating dynamic, but we move backwards in time. -434 +436 00:27:11,220 --> 00:27:13,703 We see how long ago in time would I have to be, -435 +437 00:27:13,703 --> 00:27:17,893 such that if I played things forward from there, I would land on the number one, -436 +438 00:27:17,893 --> 00:27:18,980 my initial condition. -437 +439 00:27:19,440 --> 00:27:22,140 And you have to go back in time three pi halves units. -438 +440 00:27:22,620 --> 00:27:25,586 And then if you were to translate to the decay dynamics, -439 +441 00:27:25,586 --> 00:27:28,553 which is what raising to the i is doing in this context, -440 +442 00:27:28,553 --> 00:27:32,717 you say if I'm starting at the number one, but I want to move backwards in time -441 +443 00:27:32,717 --> 00:27:36,985 and say where should I have started if I want to decay down such that I end up at -442 +444 00:27:36,985 --> 00:27:39,640 the number one after three pi halves units of time? -443 -00:27:40,260 --> 00:27:44,400 -The answer is evidently starting at around 111 for that kind of exponential decay. +445 +00:27:40,260 --> 00:27:42,498 +the answer is evidently starting at around a hundred -444 +446 +00:27:42,498 --> 00:27:44,400 +and eleven for that kind of exponential decay + +447 00:27:44,880 --> 00:27:48,823 And you can see where this is going, where there's actually infinitely many different -445 +448 00:27:48,823 --> 00:27:52,400 values that we could plug in for x if we're thinking of e to the x as being i. -446 +449 00:27:52,920 --> 00:27:55,340 And people have entered a lot more here. -447 +450 00:27:56,240 --> 00:27:59,360 Excuse me, throwing my pen onto the ground as one does. -448 -00:28:00,159 --> 00:28:01,600 +451 +00:28:00,160 --> 00:28:01,600 Classic for third place. -449 +452 00:28:02,280 --> 00:28:03,740 Nine pi halves, great choice. -450 +453 00:28:04,420 --> 00:28:06,680 1729 pi halves, y'all are my favorite. -451 +454 00:28:07,340 --> 00:28:10,521 Lots and lots of different options, infinitely many different values, -452 +455 00:28:10,521 --> 00:28:12,840 which feels a little disconcerting at first, right? -453 +456 00:28:12,840 --> 00:28:15,036 Because we look at an expression that seems like, -454 +457 00:28:15,036 --> 00:28:17,320 you know, there's just going to be some computation. -455 +458 00:28:17,460 --> 00:28:19,900 I just plug that into my calculator and see what pops out. -456 +459 00:28:20,280 --> 00:28:22,540 And we've got multiple different values for it. -457 +460 00:28:22,860 --> 00:28:24,300 So what's going on here, right? -458 +461 00:28:24,620 --> 00:28:25,400 What's going on? -459 +462 00:28:25,780 --> 00:28:30,648 And I think this really cuts to the idea of how we think about exponentials in general, -460 +463 00:28:30,648 --> 00:28:30,980 right? -461 +464 00:28:31,080 --> 00:28:34,752 But before that I do want to emphasize that this isn't the only time in math -462 +465 00:28:34,752 --> 00:28:38,520 where we come across a kind of ambiguity for how to interpret something, right? -463 +466 00:28:38,540 --> 00:28:42,280 Because if I say something like, what is the square root of 25? -464 +467 00:28:43,460 --> 00:28:46,500 You know, I think a lot of us say, well, it's five. -465 +468 00:28:47,200 --> 00:28:50,460 But if we are saying, you know, what should the square root be? -466 +469 00:28:50,560 --> 00:28:54,740 It should be some number x such that when you square it you get 25. -467 +470 00:28:55,520 --> 00:28:57,300 Well, there's two different answers to that. -468 +471 00:28:57,780 --> 00:29:01,757 Who's to say that our conventions should be that the square root function is positive, -469 +472 00:29:01,757 --> 00:29:04,180 gives us a positive number rather than negative five. -470 +473 00:29:04,500 --> 00:29:08,426 So we have one expression that seems like it wants to have multiple different values, -471 +474 00:29:08,426 --> 00:29:08,700 right? -472 +475 00:29:09,300 --> 00:29:13,706 And this could actually happen in another context where instead of square roots, -473 +476 00:29:13,706 --> 00:29:17,080 what if I was asking for the fourth root of something like 16? -474 +477 00:29:18,380 --> 00:29:21,380 Usually we would think of this as positive number two, right? -475 +478 00:29:21,500 --> 00:29:24,540 Two is a number such that when you multiply by itself four times you get 16. -476 +479 00:29:24,740 --> 00:29:26,360 Seems like a decent answer to a fourth root. -477 +480 00:29:27,020 --> 00:29:30,574 But if we're thinking of this as answering the question, -478 +481 00:29:30,574 --> 00:29:32,820 what number to the fourth equals 16? -479 +482 00:29:33,620 --> 00:29:35,580 There is another answer to this. -480 +483 00:29:35,980 --> 00:29:38,220 We could also say negative two. -481 +484 00:29:38,500 --> 00:29:41,840 That's a number that when you multiply by itself four times you get 16. -482 +485 00:29:42,140 --> 00:29:43,320 But there's another answer. -483 +486 00:29:43,640 --> 00:29:44,920 You could say two times i. -484 +487 00:29:45,420 --> 00:29:47,420 That seems valid, but there's another answer. -485 +488 00:29:47,420 --> 00:29:50,160 You could think negative two times i. -486 +489 00:29:50,620 --> 00:29:52,780 All four of these numbers satisfy that property. -487 +490 00:29:53,140 --> 00:29:57,640 So who's to say that the fourth root of 16 should be two? -488 +491 00:29:58,360 --> 00:30:01,500 And the answer ends up being, well, we adopt a convention. -489 +492 00:30:01,900 --> 00:30:05,601 When there's multiple options like this, when you have a multi-valued function, -490 +493 00:30:05,601 --> 00:30:09,117 we often just choose one of those values to be what we mean when we want to -491 +494 00:30:09,117 --> 00:30:12,680 treat it as a function, as something with a single input and a single output. -492 +495 00:30:13,460 --> 00:30:17,319 In fancier lingo, this comes up all the time when we're dealing with complex numbers, -493 +496 00:30:17,319 --> 00:30:20,820 the idea of something as an operation kind of wanting to have multiple values. -494 +497 00:30:21,040 --> 00:30:24,773 You'll sometimes hear the phrase branch, where you choose a branch of -495 +498 00:30:24,773 --> 00:30:28,720 the square root function, which is to say you choose a certain convention. -496 +499 00:30:29,960 --> 00:30:31,719 In real numbers, it's nice and easy sometimes -497 +500 00:30:31,719 --> 00:30:33,440 because you say just choose the positive one. -498 +501 00:30:33,440 --> 00:30:36,203 But there's no notion of which complex numbers are like the -499 +502 00:30:36,203 --> 00:30:38,920 positive complex numbers when we want to take square roots. -500 +503 00:30:39,300 --> 00:30:43,060 Just to give one example, let's say we wanted to take the square root of i. -501 +504 00:30:43,520 --> 00:30:45,740 And we want to know what should that be? -502 +505 00:30:46,360 --> 00:30:47,800 Because there's multiple different answers. -503 +506 00:30:48,920 --> 00:30:52,040 You know, we think of i again as this 90 degree rotation. -504 +507 00:30:52,320 --> 00:30:55,711 And if we were thinking of it as a 90 degree rotation, -505 +508 00:30:55,711 --> 00:31:01,260 it feels like the square root should be, you know, something sitting at a 45 degree angle. -506 +509 00:31:01,420 --> 00:31:05,075 Maybe that's the square root of i, which we could write -507 +510 00:31:05,075 --> 00:31:08,600 out very explicitly as root 2 over 2, root 2 over 2 i. -508 +511 00:31:09,160 --> 00:31:10,600 That's just using trigonometry. -509 -00:31:11,159 --> 00:31:16,095 +512 +00:31:11,160 --> 00:31:16,095 But if we were thinking of i instead as being a negative 270 degree rotation, -510 +513 00:31:16,095 --> 00:31:19,765 it feels like half of that, doing half of that operation, -511 +514 00:31:19,765 --> 00:31:22,360 should actually get us on the other side. -512 +515 00:31:23,200 --> 00:31:25,500 Maybe the number sitting down here should be the square root of i. -513 +516 00:31:26,160 --> 00:31:28,760 And that's actually just the negative of what we saw before. -514 +517 00:31:30,520 --> 00:31:34,680 Negative root 2 over 2 minus root 2 over 2 times i. -515 +518 00:31:35,240 --> 00:31:38,072 Now in the context of real valued functions, we can say, yeah, -516 +519 00:31:38,072 --> 00:31:41,040 just choose the square root to be whatever the positive answer is. -517 +520 00:31:41,180 --> 00:31:43,200 But which of these do you consider the positive answer? -518 +521 00:31:43,780 --> 00:31:46,087 You know, maybe it feels like we should consider this -519 +522 00:31:46,087 --> 00:31:48,480 upper one because its coordinates have positive numbers. -520 +523 00:31:48,520 --> 00:31:52,274 But however you try to define positive in a nice way here that's going to be consistent, -521 +524 00:31:52,274 --> 00:31:55,903 you know, for example, two positive numbers should always multiply to make a positive -522 +525 00:31:55,903 --> 00:31:59,700 number, you're not really going to be able to do it the way that you can for real numbers. -523 +526 00:32:00,980 --> 00:32:04,307 And in fact, this phenomenon here where we're taking roots is -524 +527 00:32:04,307 --> 00:32:07,796 actually the same as the phenomenon we were just looking at when -525 +528 00:32:07,796 --> 00:32:11,500 we were talking about multiple values for i raised to the power of i. -526 +529 00:32:11,700 --> 00:32:15,394 Because forget i raised to the power of i, let me ask what might -527 +530 00:32:15,394 --> 00:32:19,260 look like a much simpler question of taking 2 to the power one half. -528 +531 00:32:20,060 --> 00:32:21,380 Okay, what's 2 to the one half? -529 +532 00:32:22,280 --> 00:32:24,260 Yeah, I think you say, well, we know what this is, -530 +533 00:32:24,260 --> 00:32:26,980 we kind of define it to be the square root of 2, all is well and good. -531 +534 00:32:27,440 --> 00:32:30,095 But what if I said let's approach this the same way -532 +535 00:32:30,095 --> 00:32:32,700 that we were approaching our i to the i expression? -533 +536 00:32:32,700 --> 00:32:36,524 I want to first express things as e to the something, right, -534 +537 00:32:36,524 --> 00:32:41,351 and then I'm going to raise that to the one half by multiplying the one half -535 +538 00:32:41,351 --> 00:32:42,480 into the exponent. -536 +539 00:32:43,440 --> 00:32:45,880 And I say, well, okay, I can, I guess I can do that. -537 +540 00:32:46,620 --> 00:32:47,960 e to the what is equal to 2? -538 +541 00:32:48,480 --> 00:32:49,780 Well, that's the natural log of 2. -539 +542 00:32:50,580 --> 00:32:54,640 It's a constant which is around 0.69 or so. -540 +543 00:32:55,780 --> 00:32:57,820 If we raise e to that power, we'll get 2. -541 +544 00:32:58,160 --> 00:33:03,240 So we could be thinking of this as e to the natural log of 2 times one half. -542 +545 00:33:04,080 --> 00:33:08,629 And if you wanted to, if you were thinking of e to the x, you know, -543 +546 00:33:08,629 --> 00:33:12,844 this might be kind of overkill in the context of real numbers, -544 +547 00:33:12,844 --> 00:33:17,727 but if you were thinking of e to the x as shorthand for this x function, -545 +548 00:33:17,727 --> 00:33:23,681 you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, -546 +549 00:33:23,681 --> 00:33:25,020 something like that. -547 +550 00:33:25,420 --> 00:33:28,475 You plug in that very concrete value into your polynomial, -548 +551 00:33:28,475 --> 00:33:31,220 see what it outputs, and it will output around 1.415. -549 -00:33:33,199 --> 00:33:34,480 +552 +00:33:33,200 --> 00:33:34,480 So that's a nice real number. -550 +553 00:33:34,680 --> 00:33:36,040 Square root of 2, what you would expect. -551 +554 00:33:36,760 --> 00:33:39,264 But if we do the same thing we were just doing with i and -552 +555 00:33:39,264 --> 00:33:41,984 acknowledging that there's actually multiple different answers -553 +556 00:33:41,984 --> 00:33:45,180 when we want to write something as e to a power, we could also write this. -554 +557 00:33:45,980 --> 00:33:53,140 This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. -555 +558 00:33:54,020 --> 00:33:56,560 That whole thing raised to the one half. -556 -00:33:57,639 --> 00:33:58,040 +559 +00:33:57,640 --> 00:33:58,040 Right? -557 +560 00:33:58,580 --> 00:34:01,320 After all, this value will come to equal 2. -558 +561 00:34:01,320 --> 00:34:07,460 You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. -559 +562 00:34:08,080 --> 00:34:11,520 This one just has the effect of rotating things 360 degrees. -560 +563 00:34:11,600 --> 00:34:14,719 So it's just going to equal 1, so we're looking at 2 times 1. -561 +564 00:34:15,199 --> 00:34:16,880 Great, that feels like a valid substitution. -562 +565 00:34:18,080 --> 00:34:22,256 And yet when we play the same game of taking this and raising it to a power and -563 +566 00:34:22,256 --> 00:34:26,380 treating that by multiplying the power into the exponent, look at what happens. -564 +567 00:34:26,760 --> 00:34:34,880 We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? -565 +568 00:34:35,000 --> 00:34:36,239 Well, that will be pi times i. -566 +569 00:34:37,699 --> 00:34:41,501 Now this first part, e to the natural log of 2 times one half, -567 +570 00:34:41,501 --> 00:34:44,699 that will end up being the familiar square root of 2. -568 +571 00:34:45,120 --> 00:34:46,320 That's all well and good. -569 -00:34:46,840 --> 00:34:49,140 +572 +00:34:46,840 --> 00:34:49,139 But we're going to be multiplying that by e to the pi i. -570 +573 00:34:50,480 --> 00:34:50,600 Right? -571 +574 00:34:50,880 --> 00:34:54,100 And quite famously e to the pi i is negative 1. -572 +575 00:34:55,000 --> 00:35:00,163 So in this case, it seems to be suggesting that if we are solving this expression -573 +576 00:35:00,163 --> 00:35:05,453 2 to the one half by playing around with the different answers we could plug in for -574 +577 00:35:05,453 --> 00:35:10,680 something like e to the x equaling one half, what we end up with is another answer. -575 +578 00:35:10,800 --> 00:35:14,260 What we might traditionally write as this negative square root of 2. -576 +579 00:35:15,000 --> 00:35:18,400 And here, I mean it's a little funny for it to have multiple values -577 +580 00:35:18,400 --> 00:35:21,800 to look at 2 to the one half and say that's not equaling one thing, -578 +581 00:35:21,800 --> 00:35:25,300 but based on choices we make it could equal multiple different things. -579 +582 00:35:25,460 --> 00:35:27,460 But the two things that it could seem quite reasonable. -580 +583 00:35:27,940 --> 00:35:30,333 If there's going to be anything that 2 to the one half is, -581 +584 00:35:30,333 --> 00:35:33,863 it seems like it should either be the positive square root that we're familiar with or -582 +585 00:35:33,863 --> 00:35:35,040 the negative variant of that. -583 +586 00:35:35,320 --> 00:35:36,880 That doesn't actually seem like such a problem. -584 +587 00:35:37,480 --> 00:35:41,309 And in fact, we could we could play this game even further, -585 +588 00:35:41,309 --> 00:35:45,840 where let me ask you for even more creative answers to this expression. -586 +589 00:35:46,080 --> 00:35:50,578 Because maybe we can find other funny powers of something like 2 to the power x as -587 +590 00:35:50,578 --> 00:35:55,240 we start plugging in various different values of x based on what substitution we make. -588 +591 00:35:55,400 --> 00:36:00,320 If we're abiding by the same rules that we were using in evaluating i to the power i. -589 +592 00:36:01,000 --> 00:36:04,563 So this time the question asks or it specifies that one solution of -590 +593 00:36:04,563 --> 00:36:08,180 the equation e to the x equals 2 is the real number natural log of 2. -591 +594 00:36:08,480 --> 00:36:09,580 Okay, that one we know it. -592 +595 00:36:09,840 --> 00:36:13,100 It's not boring, but it's boring in comparison to what else we could do. -593 +596 00:36:13,740 --> 00:36:15,180 Can you think of another one? -594 +597 00:36:15,480 --> 00:36:20,400 Can you write some other answer to the question e to the x equals 2? -595 +598 00:36:21,280 --> 00:36:22,840 And again, creativity is welcomed. -596 +599 00:36:23,260 --> 00:36:24,980 So I will give you another little moment for that. -597 -00:36:51,799 --> 00:36:56,740 +600 +00:36:51,800 --> 00:36:56,740 All right, I will go ahead and lock in some answers here if that's all right with you. -598 +601 00:36:57,660 --> 00:36:59,916 I'm not sure how much time it necessarily takes to do -599 +602 00:36:59,916 --> 00:37:02,340 the math entry depending on what device you're looking at. -600 +603 00:37:02,740 --> 00:37:06,460 But don't be too stressed if it's before you got the chance to enter -601 +604 00:37:06,460 --> 00:37:10,180 the question that you want into the answer that you wanted to answer. -602 +605 00:37:12,040 --> 00:37:16,211 So it looks like 131 of you have entered the variant -603 +606 00:37:16,211 --> 00:37:19,360 where we take ln of 2 and we add 2 pi i. -604 +607 00:37:19,720 --> 00:37:23,066 And I guess I in writing this question mistakenly like marked one of the -605 +608 00:37:23,066 --> 00:37:26,780 answers as being correct when in fact there's quite a few different correct ones. -606 +609 00:37:27,420 --> 00:37:31,173 So that's on me for the fact that I don't know if it looks to any of you like, -607 +610 00:37:31,173 --> 00:37:34,024 oh, it's red you got it wrong when you entered ln of 2 plus -608 +611 00:37:34,024 --> 00:37:36,020 42 i pi which is of course a great choice. -609 +612 00:37:37,220 --> 00:37:42,100 But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. -610 +613 00:37:42,100 --> 00:37:48,600 Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. -611 +614 00:37:49,040 --> 00:37:51,420 Because it just has the effect of multiplying by e -612 +615 00:37:51,420 --> 00:37:53,940 to the 2 pi i which is the effect of multiplying by 1. -613 +616 00:37:54,560 --> 00:37:57,680 And again, this has kind of a funny consequence where it -614 +617 00:37:57,680 --> 00:38:00,800 seems to output kind of reasonable results when we do it. -615 +618 00:38:01,180 --> 00:38:04,320 As another example, it looks like the second most common -616 +619 00:38:04,320 --> 00:38:07,240 entered expression there was that we might replace 2. -617 +620 00:38:07,720 --> 00:38:09,980 So let's think we're thinking of 2 to the power of one fourth. -618 +621 00:38:10,640 --> 00:38:10,800 Okay. -619 +622 00:38:11,440 --> 00:38:19,280 There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. -620 +623 00:38:20,620 --> 00:38:20,720 Okay. -621 +624 00:38:22,220 --> 00:38:23,260 Plus 4 pi i. -622 +625 00:38:23,580 --> 00:38:27,460 And we raise all of that to the one fourth, right? -623 +626 00:38:27,940 --> 00:38:32,344 Well if you were to play the same game you would get e to the natural -624 +627 00:38:32,344 --> 00:38:36,560 log of 2 times one fourth and we'd be multiplying by e to the pi i. -625 +628 00:38:37,120 --> 00:38:42,280 Now the first part of that is going to be the usual positive fourth root of 2. -626 +629 00:38:42,380 --> 00:38:46,630 The thing we mean when you plug in an expression like fourth root of 2 into a calculator, -627 +630 00:38:46,630 --> 00:38:48,000 a nice small positive number. -628 +631 00:38:48,500 --> 00:38:50,460 But then this second part is negative 1. -629 +632 00:38:51,100 --> 00:38:55,288 So it seems to be saying, you know, if we were to interpret 2 in this different way, -630 +633 00:38:55,288 --> 00:38:58,539 raising it to the one fourth, you know, it's not the usual answer -631 +634 00:38:58,539 --> 00:39:00,560 that we get but it's a reasonable answer. -632 +635 00:39:00,640 --> 00:39:04,560 It's another number that when you raise it to the fourth power you get 2. -633 +636 00:39:05,500 --> 00:39:09,864 And if we had done this with even different values, if instead we had been using 2 pi i, -634 +637 00:39:09,864 --> 00:39:13,640 you know, it's kind of fun to think about how that would have changed things. -635 +638 00:39:14,060 --> 00:39:16,964 If instead of 4 pi i we had been adding 2 pi i, -636 +639 00:39:16,964 --> 00:39:21,080 well then over here we would have been looking at pi halves times i. -637 +640 00:39:21,620 --> 00:39:26,900 And instead of multiplying by negative 1 we would have instead been multiplying by i. -638 +641 00:39:27,400 --> 00:39:29,340 Which again is a valid answer. -639 +642 00:39:29,500 --> 00:39:32,540 It seems like a reasonable output for something like 2 to the one fourth. -640 +643 00:39:32,540 --> 00:39:37,623 So when you're looking at the fact that i to the power i seems to have multiple different -641 +644 00:39:37,623 --> 00:39:42,538 values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi -642 +645 00:39:42,538 --> 00:39:47,340 halves i, negative 3 pi halves i, and we get what seem like wildly different answers. -643 +646 00:39:47,600 --> 00:39:51,802 Something super small, something super big, all very different from the one fifth, -644 +647 00:39:51,802 --> 00:39:54,840 approximately one fifth answer that we found before up here. -645 +648 00:39:55,280 --> 00:39:58,515 It's exactly the same phenomenon as when you're asking something like -646 +649 00:39:58,515 --> 00:40:01,612 what's 2 to the one fourth and acknowledging that there's actually -647 +650 00:40:01,612 --> 00:40:04,940 multiple different solutions to the expression x to the fourth equals 2. -648 +651 00:40:05,440 --> 00:40:06,900 Four different solutions in fact. -649 +652 00:40:07,380 --> 00:40:10,905 And what you're looking at is the fact that there's multiple different -650 +653 00:40:10,905 --> 00:40:14,133 solutions to the expression e to the x equals some kind of base, -651 +654 00:40:14,133 --> 00:40:17,560 whether that base is i, whether that base is 2, whatever it might be. -652 +655 00:40:18,380 --> 00:40:21,390 And one way that we might think about this is that when -653 +656 00:40:21,390 --> 00:40:24,400 you're dealing with real numbers things are just lovely. -654 +657 00:40:25,020 --> 00:40:25,780 Things are nice. -655 +658 00:40:26,320 --> 00:40:27,480 There's one-to-one relationships. -656 +659 00:40:27,640 --> 00:40:28,500 You've got positives. -657 +660 00:40:28,500 --> 00:40:29,580 It's great. -658 +661 00:40:30,880 --> 00:40:33,948 Where if we want to think about exponential functions, -659 +662 00:40:33,948 --> 00:40:36,180 let me just cover some of this stuff up. -660 +663 00:40:38,180 --> 00:40:42,438 We have this nice back and forth where you can choose to express any -661 +664 00:40:42,438 --> 00:40:47,006 exponential as a base to x like 2 to the x or you could express that same -662 +665 00:40:47,006 --> 00:40:51,573 exponential as x of r times x which you know, that is the polynomial that -663 +666 00:40:51,573 --> 00:40:56,820 we refer to whenever implicitly refer to whenever we write something like e to the x. -664 +667 00:40:57,380 --> 00:41:00,648 And there's a lovely back and forth because you can just take a natural -665 +668 00:41:00,648 --> 00:41:04,280 logarithm of b and it gives you one answer assuming that b is a positive number. -666 +669 00:41:05,080 --> 00:41:08,080 And that's the same thing as saying that x of r is equal to b. -667 +670 00:41:08,740 --> 00:41:12,284 So one way that I've talked about this earlier in the series is -668 +671 00:41:12,284 --> 00:41:16,105 that if you were looking at the family of all possible exponentials, -669 +672 00:41:16,105 --> 00:41:19,760 right, we could write them as x of r times x and change what r is. -670 +673 00:41:20,020 --> 00:41:22,410 And this is exactly the same thing as writing e to the r -671 +674 00:41:22,410 --> 00:41:24,800 times x if that's something you're more comfortable with. -672 +675 00:41:25,360 --> 00:41:28,580 So e to the r times x x of r times x those are the same thing. -673 -00:41:28,780 --> 00:41:30,439 +676 +00:41:28,780 --> 00:41:30,440 We could think about changing what that is. -674 +677 00:41:31,620 --> 00:41:36,380 But on the other hand if you were to think about all possible exponentials as some base, -675 +678 00:41:36,380 --> 00:41:40,500 let me do base to the power of x and we're going to change what that base is. -676 +679 00:41:41,400 --> 00:41:44,028 At first it feels like that's a different kind of expression to -677 +680 00:41:44,028 --> 00:41:46,780 manipulate but it's just another way of expressing the same family. -678 +681 00:41:47,600 --> 00:41:52,465 Right, and a way that you might think about this for how do we think about what base -679 +682 00:41:52,465 --> 00:41:57,560 does it correspond to if we're thinking a little bit more abstractly as exp of r times x. -680 +683 00:41:57,620 --> 00:41:59,341 And there's a reason I'm doing this because we're about to -681 +684 00:41:59,341 --> 00:42:01,180 apply this to complex numbers where it's going to look weirder. -682 +685 00:42:01,480 --> 00:42:02,580 So follow through with me here. -683 +686 00:42:02,800 --> 00:42:08,014 If instead of looking at that base, one thing I could do is say what is the -684 +687 00:42:08,014 --> 00:42:13,640 value of exp of r, right, which is basically this function of when we plug in one. -685 +688 00:42:13,860 --> 00:42:17,520 So exp of r times one if you prefer to think of it that way. -686 +689 00:42:18,920 --> 00:42:21,560 And that is being represented by our green line. -687 -00:42:21,880 --> 00:42:28,151 -And what you can see is okay if I get r that a factor in front of x in my exponential +690 +00:42:21,880 --> 00:42:27,656 +nd what you could see is okay if I get R that Factor in front of X in my exponential -688 -00:42:28,151 --> 00:42:33,840 -function exp of r x to be 0.69, which I know is around the natural log of two. +691 +00:42:27,656 --> 00:42:33,636 +function exp of R X to be zero point six nine Which I know is around the natural log of -689 +692 +00:42:33,636 --> 00:42:33,840 +two + +693 00:42:34,260 --> 00:42:37,860 What this means is that exp of one is about two. -690 +694 00:42:38,580 --> 00:42:40,643 And so this corresponds with the function that -691 +695 00:42:40,643 --> 00:42:42,620 we would usually write as two to the power x. -692 +696 00:42:43,520 --> 00:42:43,620 Right. -693 +697 00:42:44,820 --> 00:42:47,710 Okay, and basically as I change around my r, you know, -694 +698 00:42:47,710 --> 00:42:51,180 I could try to change it to something so that it looks like three. -695 +699 00:42:51,360 --> 00:42:54,088 So around 1.1 that exponential looks like three, -696 +700 00:42:54,088 --> 00:42:57,040 which we would usually write as three to the power x. -697 +701 00:42:57,520 --> 00:43:00,458 I would like to argue that it's a little bit healthier to -698 +702 00:43:00,458 --> 00:43:03,600 think about varying this value r rather than varying the base. -699 +703 00:43:03,920 --> 00:43:08,629 And the main reason is that as soon as we get to complex contexts and we're -700 +704 00:43:08,629 --> 00:43:13,153 thinking of exponentiation, you have this overloading that goes on where -701 +705 00:43:13,153 --> 00:43:17,740 if we change around what sits in front of the x, that's all well and good. -702 +706 00:43:17,880 --> 00:43:22,100 -I could have exp of r times x where maybe r is something like 0.69. +I could have exp of R times X where maybe R is something like zero point six nine -703 +707 00:43:22,820 --> 00:43:25,120 But I could shift that down by 2 pi i. -704 +708 00:43:26,060 --> 00:43:28,380 And that doesn't change the base that it would correspond to. -705 +709 00:43:28,500 --> 00:43:29,820 That would still correspond to two. -706 +710 00:43:30,120 --> 00:43:31,820 Or it could shift it up by 2 pi i. -707 +711 00:43:31,940 --> 00:43:33,840 That doesn't change the base that it corresponds to. -708 +712 00:43:34,060 --> 00:43:38,020 Because in all of those cases when we plug in x equals one, we get the same thing. -709 +713 00:43:38,020 --> 00:43:42,620 However, all of these for different values of x are distinct functions. -710 +714 00:43:43,000 --> 00:43:46,580 This is why we saw multiple different values for i to the power i. -711 +715 00:43:46,900 --> 00:43:49,880 Because i to the x is an ambiguous function in that context. -712 +716 00:43:50,020 --> 00:43:54,770 It would be unambiguous if we decided which value of r such that what -713 +717 00:43:54,770 --> 00:43:59,520 we're representing is exp of r times x, which value of r do we choose? -714 +718 00:43:59,660 --> 00:44:02,160 As soon as we choose one, it's an unambiguous function. -715 +719 00:44:02,720 --> 00:44:06,508 But at that point it just feels like maybe what we want is to stop -716 +720 00:44:06,508 --> 00:44:10,240 thinking about things in terms of some base raised to the power x. -717 +721 00:44:10,680 --> 00:44:13,589 Maybe as soon as we're in the context of complex numbers, -718 +722 00:44:13,589 --> 00:44:16,700 we should just write them all as exp of some constant times x. -719 +723 00:44:17,040 --> 00:44:20,624 If for no other reason, it makes crystal clear how we actually plug in -720 +724 00:44:20,624 --> 00:44:24,260 numbers if we want to do a computation, or just to do math on top of it. -721 +725 00:44:24,420 --> 00:44:27,040 We've got this nice infinite polynomial that we plug them into. -722 +726 00:44:28,220 --> 00:44:33,069 And I'll make another case for you that this is maybe the the correct way to think about -723 +727 00:44:33,069 --> 00:44:37,920 exponentials, as soon as we're extending into other domains, things like complex numbers. -724 +728 00:44:38,700 --> 00:44:40,780 And for that, let's just let's just back up. -725 +729 00:44:41,400 --> 00:44:41,600 Go back. -726 +730 00:44:41,960 --> 00:44:43,600 Oh doorbell, something's arrived. -727 +731 00:44:44,400 --> 00:44:48,057 Go back to the original way that we extend the idea of -728 +732 00:44:48,057 --> 00:44:51,980 exponentiation and just think of like what is two to the x? -729 +733 00:44:53,120 --> 00:44:53,280 Right. -730 +734 00:44:53,720 --> 00:44:56,757 We know how to think about this for natural numbers, -731 +735 00:44:56,757 --> 00:45:00,540 you know something like two to the three, repeated multiplication. -732 +736 00:45:01,080 --> 00:45:04,344 How is it that you're first taught to think about something like two to -733 +737 00:45:04,344 --> 00:45:07,700 the x for fractional amounts or for negative amounts and things like that? -734 +738 00:45:08,220 --> 00:45:13,447 Well, you're usually taught that two to the one half should be something where you know -735 +739 00:45:13,447 --> 00:45:18,556 if I multiply it by itself and this follows the usual rules that exponentials do with -736 +740 00:45:18,556 --> 00:45:22,476 counting numbers where we're able to add things in that exponent, -737 +741 00:45:22,476 --> 00:45:24,140 I should get two to the one. -738 +742 00:45:24,700 --> 00:45:28,380 So it should be some number that when I multiply it by itself, I get two. -739 -00:45:29,339 --> 00:45:30,940 +743 +00:45:29,340 --> 00:45:30,940 And you know at that point you have a choice. -740 +744 00:45:31,000 --> 00:45:31,680 Maybe it's positive. -741 +745 00:45:31,780 --> 00:45:32,460 Maybe it's negative. -742 +746 00:45:33,700 --> 00:45:35,659 But if you always decide to make the positive choice, -743 +747 00:45:35,659 --> 00:45:38,200 you're going to be able to get a nice continuous function out of this. -744 +748 00:45:38,680 --> 00:45:41,840 Same deal if we ask about negative numbers, what should two to the negative one be? -745 +749 00:45:42,460 --> 00:45:45,768 Well, that should be something where when I multiply it by two to the one, -746 +750 00:45:45,768 --> 00:45:46,960 it gets me two to the zero. -747 +751 00:45:47,400 --> 00:45:50,284 And that's kind of the justification for our convention -748 +752 00:45:50,284 --> 00:45:52,500 that negative exponents look like one half. -749 +753 00:45:52,980 --> 00:45:57,711 But what's really going on here is we're saying whatever this is, -750 +754 00:45:57,711 --> 00:46:03,806 it should be some kind of function that satisfies this property f of a plus b equals -751 +755 00:46:03,806 --> 00:46:05,240 f of a times f of b. -752 +756 00:46:06,540 --> 00:46:09,523 And moreover the fact that the base is two is basically -753 +757 00:46:09,523 --> 00:46:12,080 telling us that it's not just any such function. -754 +758 00:46:12,200 --> 00:46:15,440 It's a function where when we plug in one we get two. -755 +759 00:46:16,500 --> 00:46:19,443 And just as a little, you know, sanity check style question to -756 +760 00:46:19,443 --> 00:46:22,480 see if you're following along with some of the implications here. -757 +761 00:46:23,240 --> 00:46:28,458 I want to ask you what is, I won't call it like a softball, but this is, -758 +762 00:46:28,458 --> 00:46:33,320 this isn't meant to be like an incredibly deep question necessarily. -759 +763 00:46:33,440 --> 00:46:36,688 It's just more of a check if you're following along with the idea of -760 +764 00:46:36,688 --> 00:46:39,842 abstractly starting with properties of a function and then kind of -761 +765 00:46:39,842 --> 00:46:43,420 deducing ways that we might want to write it down based on those properties. -762 +766 00:46:44,060 --> 00:46:49,388 If f of x satisfies this exponential property f of a plus b equals f of a times f of b -763 +767 00:46:49,388 --> 00:46:54,900 for all inputs, and it also satisfies f of one equals two, which of the following is true? -764 +768 00:46:55,600 --> 00:46:58,437 Which is to say which of the following is necessarily -765 +769 00:46:58,437 --> 00:47:01,380 true no matter which such function you're starting with. -766 +770 00:47:02,200 --> 00:47:04,580 And those of you who remember which lecture was it? -767 +771 00:47:04,660 --> 00:47:06,420 It's whichever one we were talking about how to -768 +772 00:47:06,420 --> 00:47:08,180 interpret what Euler's formula is really saying. -769 +773 00:47:08,480 --> 00:47:12,185 I asked a question of this style where I neglected a single condition, -770 +774 00:47:12,185 --> 00:47:15,838 you know, I didn't write down the fact that we want to make sure f of -771 +775 00:47:15,838 --> 00:47:19,700 x is non-zero everywhere and then that caused some amount of confutlement. -772 +776 00:47:19,860 --> 00:47:23,160 Which is cool, get confutlement on screen that happens to all of us. -773 +777 00:47:23,520 --> 00:47:28,548 But the the intent of it was to basically show that this abstract property of -774 +778 00:47:28,548 --> 00:47:33,962 something that turns addition into multiplication is uh is enough to basically make -775 +779 00:47:33,962 --> 00:47:39,700 you want to write the function as whatever it equals as one raised to some kind of power. -776 +780 00:47:40,080 --> 00:47:41,980 This is the the spirit of the question. -777 +781 00:47:42,590 --> 00:47:47,455 Um Now we've got a couple questions actually about power towers that -778 +782 00:47:47,455 --> 00:47:52,180 seem to have popped up here, which is great connected to last time. -779 +783 00:47:52,680 --> 00:47:57,073 Um, let's let's hold off on the power tower question for just a moment, -780 +784 00:47:57,073 --> 00:48:02,260 so that we first get like a deeper feel of like what exponentiation should mean here. -781 +785 00:48:02,640 --> 00:48:04,927 Um, because because we can be what I want to claim -782 +786 00:48:04,927 --> 00:48:07,260 is we can answer it in like multiple different ways. -783 +787 00:48:07,460 --> 00:48:09,880 So if you give me just a moment, we'll talk about power towers. -784 +788 00:48:09,880 --> 00:48:14,258 Uh, and then just as a number line can be represented in a logarithmic scale, -785 +789 00:48:14,258 --> 00:48:16,560 can the same be done for a complex plane? -786 +790 00:48:16,880 --> 00:48:20,560 Maybe mapping the complex plane onto an infinite cylinder in the logarithmic sense. -787 +791 00:48:20,940 --> 00:48:21,160 Yeah. -788 +792 00:48:21,400 --> 00:48:25,266 Yeah, uh, in fact, there's a visualization that I'm going to get to -789 +793 00:48:25,266 --> 00:48:29,020 in just a moment here where we do something quite similar to that. -790 +794 00:48:29,260 --> 00:48:34,037 Because what we'll do is play around with different exponential functions x of r times x, -791 +795 00:48:34,037 --> 00:48:37,382 but we're going to change that value of r which is going to be -792 +796 00:48:37,382 --> 00:48:39,240 represented by a little yellow dot. -793 +797 00:48:39,600 --> 00:48:40,600 So we'll kind of talk through this. -794 +798 00:48:40,820 --> 00:48:43,200 It's not going to map the whole plane, but just a couple -795 +799 00:48:43,200 --> 00:48:45,540 sample points from the real axis and the imaginary axis. -796 +800 00:48:45,900 --> 00:48:49,011 But the idea is that as we move around what that constant is, -797 +801 00:48:49,011 --> 00:48:52,725 we're going to be able to kind of visualize the different things that um, -798 +802 00:48:52,725 --> 00:48:53,780 it does to the plane. -799 +803 00:48:54,200 --> 00:48:57,036 And effectively, it's like it's turning the x-axis into a -800 +804 00:48:57,036 --> 00:49:00,460 logarithmic scale and then wrapping the imaginary axis along a circle. -801 +805 00:49:00,760 --> 00:49:04,700 And then as soon as that value of r becomes imaginary, it swaps the role of those. -802 +806 00:49:04,700 --> 00:49:07,582 Real numbers get put on the circle and imaginary -803 +807 00:49:07,582 --> 00:49:10,760 numbers get put on a logarithmic scaled positive axis. -804 +808 00:49:11,320 --> 00:49:12,140 So great question. -805 +809 00:49:12,580 --> 00:49:15,675 All three of which I guess are sort of jumping the gun ahead for where I want to go, -806 +810 00:49:15,675 --> 00:49:17,460 but nice to see that's where people are thinking. -807 +811 00:49:18,020 --> 00:49:20,000 So on this one, let's go ahead and just grade it. -808 +812 00:49:20,640 --> 00:49:25,853 The idea is that this property of f of a plus b ends up letting you -809 +813 00:49:25,853 --> 00:49:31,220 express a lot of different things purely in terms of what f of one is. -810 +814 00:49:31,600 --> 00:49:37,553 And just to spell that out very explicitly, something like f of five is the -811 +815 00:49:37,553 --> 00:49:42,254 same thing as f of one plus one plus one plus one plus one, -812 +816 00:49:42,254 --> 00:49:48,208 which is the same thing as f of one multiplied by itself five times because -813 +817 00:49:48,208 --> 00:49:49,540 of this property. -814 +818 00:49:51,460 --> 00:49:54,520 Which if f of one is two is the same as two to the power five. -815 +819 00:49:55,200 --> 00:49:59,869 And then something like f of negative five, it should be the case -816 +820 00:49:59,869 --> 00:50:04,680 that when we multiply it by f of five, we get whatever f of zero is. -817 +821 00:50:04,960 --> 00:50:08,723 And it's not immediately clear what f of zero is, -818 +822 00:50:08,723 --> 00:50:14,293 but we could say that f of one plus zero is equal to whatever f of one is -819 +823 00:50:14,293 --> 00:50:16,100 times what f of zero is. -820 +824 00:50:16,320 --> 00:50:20,160 But f of one is equal to two, and so this is also equal to two. -821 +825 00:50:20,460 --> 00:50:22,320 So we're saying two is equal to two times something. -822 +826 00:50:22,440 --> 00:50:23,740 Well that something has to be a one. -823 +827 00:50:24,660 --> 00:50:29,960 So in this context this guarantees that f of negative five is two to the negative five. -824 +828 00:50:30,120 --> 00:50:31,560 It's one over two to the fifth. -825 +829 00:50:32,100 --> 00:50:34,440 So we could explicitly write this as two to the negative five. -826 +830 00:50:34,860 --> 00:50:40,061 Which is all to say, these two properties together make us really want to write -827 +831 00:50:40,061 --> 00:50:44,873 the function as two to the x, because any counting number that we put in, -828 +832 00:50:44,873 --> 00:50:50,334 it's going to satisfy, it's going to look like two multiplied by itself that number -829 +833 00:50:50,334 --> 00:50:50,920 of times. -830 +834 00:50:51,140 --> 00:50:54,820 Any fractional number we put in, it's going to satisfy these properties that we wanted. -831 +835 00:50:54,820 --> 00:50:57,400 And you might wonder, is that unique? -832 +836 00:50:57,880 --> 00:51:01,020 And in the context of real valued functions, it actually would be. -833 +837 00:51:01,600 --> 00:51:03,739 But in the context of complex valued functions, -834 +838 00:51:03,739 --> 00:51:06,860 there would be multiple such functions f that we could write for this. -835 +839 00:51:07,120 --> 00:51:11,681 One of which is what we were looking at before, -836 +840 00:51:11,681 --> 00:51:19,284 where we could have a function defined to be exp of the natural log of two plus -837 +841 00:51:19,284 --> 00:51:22,040 two pi i all of that times x. -838 -00:51:24,899 --> 00:51:26,480 +842 +00:51:24,900 --> 00:51:26,480 Okay, forgive the sloppiness here. -839 +843 00:51:26,520 --> 00:51:28,200 I just get excited writing about this. -840 +844 00:51:28,920 --> 00:51:31,480 And this is actually a different function, as evidenced -841 +845 00:51:31,480 --> 00:51:33,720 by what happens if you plug in x equals one half. -842 +846 00:51:34,680 --> 00:51:36,822 Right, we saw a little bit earlier how when you plug in one half, -843 +847 00:51:36,822 --> 00:51:38,380 what you get is the negative square root of two. -844 +848 00:51:39,140 --> 00:51:42,814 And then if you plug in one fourth, you get not the fourth root of two, -845 +849 00:51:42,814 --> 00:51:44,600 but i times the fourth root of two. -846 +850 00:51:44,620 --> 00:51:48,002 So it is a different function, but it still satisfies these properties, -847 +851 00:51:48,002 --> 00:51:50,680 and it kind of makes us want to write it as two to the x. -848 +852 00:51:50,680 --> 00:51:54,877 And it makes it suggest that maybe two to the x is an ambiguous bit of notation, -849 +853 00:51:54,877 --> 00:51:58,660 and we should just write everything in terms of exp of r times something. -850 +854 00:51:59,380 --> 00:52:02,135 But you might wonder, well, you know, maybe we're just not being -851 +855 00:52:02,135 --> 00:52:05,060 creative enough with all of the functions that satisfy this property. -852 +856 00:52:05,200 --> 00:52:07,816 Maybe there's an ambiguity when we write exp of r times something, -853 +857 00:52:07,816 --> 00:52:10,160 and there's different values of r that could come into play. -854 +858 00:52:10,720 --> 00:52:13,120 Um, but I'm just going to put down a little claim, -855 +859 00:52:13,120 --> 00:52:16,933 and then maybe give like a sketch of what the proof would look like if you want, -856 +860 00:52:16,933 --> 00:52:19,663 which is that let's say you have some complex function f, -857 +861 00:52:19,663 --> 00:52:21,640 and it satisfies the following properties. -858 +862 00:52:22,040 --> 00:52:24,200 First, you're able to take a derivative of it. -859 +863 00:52:24,340 --> 00:52:28,260 It's differentiable, which just keeps it from being some, uh, you know, -860 +864 00:52:28,260 --> 00:52:33,107 totally messy discontinuous thing that's like taking on some random values depending on, -861 +865 00:52:33,107 --> 00:52:36,592 you know, the span of whatever vector space over, I don't know, -862 +866 00:52:36,592 --> 00:52:39,860 fractional amounts you might want to think of in crazy ways. -863 +867 00:52:39,860 --> 00:52:41,700 It's a nice function that's differentiable. -864 +868 00:52:42,160 --> 00:52:43,900 It's not equal to zero everywhere. -865 +869 00:52:44,620 --> 00:52:48,454 So the condition that sort of slipped my mind and I forget which lecture, -866 +870 00:52:48,454 --> 00:52:50,320 lecture four or something like that. -867 +871 00:52:50,400 --> 00:52:54,380 And then it has this central property that it turns addition into multiplication. -868 +872 00:52:54,880 --> 00:52:59,627 If you have such a function, I claim that there's a unique, -869 +873 00:52:59,627 --> 00:53:05,800 maybe I should really specify, there exists a unique complex number r so that -870 +874 00:53:05,800 --> 00:53:11,576 you could write f of x as basically being this exponential function of r -871 +875 00:53:11,576 --> 00:53:13,080 times that value x. -872 +876 00:53:13,620 --> 00:53:17,514 Which is, you know, basically saying that if you have exp as a function, -873 +877 00:53:17,514 --> 00:53:21,463 this infinite polynomial with nice derivative properties and all of that, -874 +878 00:53:21,463 --> 00:53:25,838 if you have this you have every exponential that you want in a very like abstract -875 +879 00:53:25,838 --> 00:53:30,640 generic sense of the word exponential just based on a property that we could want from it. -876 +880 00:53:31,180 --> 00:53:33,520 And the sketch of the proof would look something like this. -877 +881 00:53:34,400 --> 00:53:37,471 If you want to first look at what is the derivative of this value, -878 +882 00:53:37,471 --> 00:53:39,580 which we're assuming exists everywhere, right? -879 +883 00:53:39,960 --> 00:53:42,540 And you explicitly write out what the limit of that is. -880 +884 00:53:43,080 --> 00:53:45,006 I'll just talk through it very quickly here for those who -881 +885 00:53:45,006 --> 00:53:47,100 want to like pause and think through the details, feel free to. -882 +886 00:53:47,560 --> 00:53:53,560 The central property that we have lets us expand out the f of x plus h term. -883 +887 00:53:53,980 --> 00:53:56,048 So we're thinking, you know, a change, slight change to -884 +888 00:53:56,048 --> 00:53:58,080 the output over the change to the input that caused it. -885 +889 00:53:58,140 --> 00:54:00,720 That's what df dx unwraps to. -886 +890 00:54:01,380 --> 00:54:05,722 And because we can factor that out, we can factor f of x out of the -887 +891 00:54:05,722 --> 00:54:10,320 expression entirely and the whole limit is expressed only in terms of h. -888 +892 00:54:10,900 --> 00:54:15,418 Which if you think about what it means in the context of derivatives and the fact -889 +893 00:54:15,418 --> 00:54:19,991 that f of zero necessarily equals one, this whole limiting expression is just some -890 +894 00:54:19,991 --> 00:54:24,840 constant, but more specifically it's whatever the derivative of our function at zero is. -891 +895 00:54:25,320 --> 00:54:27,665 So you have this funny thing where if you know its derivative -892 +896 00:54:27,665 --> 00:54:29,860 at zero that determines what its derivative is everywhere. -893 +897 00:54:30,460 --> 00:54:35,057 And in the context of exponential functions this is hopefully quite familiar because all -894 +898 00:54:35,057 --> 00:54:39,654 that we're really saying is the derivative of an exponential function is proportional to -895 +899 00:54:39,654 --> 00:54:44,200 itself and that proportionality constant is equal to whatever the derivative at zero is. -896 +900 00:54:44,480 --> 00:54:48,258 This is all very abstractly phrased and such, but the purpose of it is to -897 +901 00:54:48,258 --> 00:54:52,139 emphasize that it's not necessarily just functions that we already think of -898 +902 00:54:52,139 --> 00:54:56,224 as a to the power x, but it is a potentially much more broad class of functions -899 +903 00:54:56,224 --> 00:55:00,360 that just satisfy this abstract property of turning addition into multiplication. -900 +904 00:55:01,320 --> 00:55:05,380 But if you have that, it actually guarantees that you also have a second derivative. -901 -00:55:06,259 --> 00:55:08,681 +905 +00:55:06,260 --> 00:55:08,681 And for that matter a third derivative and such because -902 +906 00:55:08,681 --> 00:55:11,060 the derivative function is just proportional to itself. -903 +907 00:55:11,980 --> 00:55:15,281 So in order to take the nth derivative you just look at -904 +908 00:55:15,281 --> 00:55:18,700 that proportionality constant and raise it to the power n. -905 +909 00:55:19,260 --> 00:55:24,205 And then from here you could do a Taylor series expansion and I might leave that as sort -906 +910 00:55:24,205 --> 00:55:29,205 of the advanced homework for those of you who are comfortable with Taylor series and that -907 +911 00:55:29,205 --> 00:55:34,039 idea especially if you want to intermix the idea of any differentiable function that's -908 +912 00:55:34,039 --> 00:55:39,040 differentiable in a sense of complex numbers, which is sort of a definitely college topic. -909 +913 00:55:41,100 --> 00:55:44,454 You know, you could intermix the reasoning there as you want, -910 +914 00:55:44,454 --> 00:55:49,052 but fuzzy reasoning is allowed in the context of someone who only knows about Taylor -911 +915 00:55:49,052 --> 00:55:53,921 series and nothing else to take this idea and look at the Taylor expansion for f and kind -912 +916 00:55:53,921 --> 00:55:58,574 of justify the idea that there's a unique complex number such that our function f can -913 +917 00:55:58,574 --> 00:56:00,360 necessarily be written like this. -914 +918 00:56:00,360 --> 00:56:05,580 And then the connection to normal exponentials is whenever you have such -915 +919 00:56:05,580 --> 00:56:10,586 a value r we do essentially what we do in the complex context of real -916 +920 00:56:10,586 --> 00:56:15,664 numbers is if you look at x of that function of that value r and write -917 +921 00:56:15,664 --> 00:56:21,100 that as a base it feels like you should be able to write that as b to the x. -918 +922 00:56:21,800 --> 00:56:26,689 But the whole the whole point here of course is that when we play this game and -919 +923 00:56:26,689 --> 00:56:31,701 you're trying to interpret something like i to the x that's an ambiguous function -920 +924 00:56:31,701 --> 00:56:36,468 because there's lots of different values of r we could interpret that to mean -921 +925 00:56:36,468 --> 00:56:41,296 not just exp of pi halves i times x but we could also interpret it to mean exp -922 +926 00:56:41,296 --> 00:56:45,941 of five pi halves i times x and these are separate functions and there's an -923 +927 00:56:45,941 --> 00:56:51,320 infinite family of separate functions that feel like we should write them as i to the x. -924 +928 00:56:51,600 --> 00:56:55,896 So the expression i to the i unless you've adopted a standard for what that's necessarily -925 +929 00:56:55,896 --> 00:57:00,049 going to mean when you say it has infinitely many outputs another way to think of that -926 +930 00:57:00,049 --> 00:57:04,060 is that the function i to the x with the notation we have is a little bit ambiguous. -927 +931 00:57:04,620 --> 00:57:07,029 Now with all of that, let's let's just start visualizing -928 +932 00:57:07,029 --> 00:57:08,720 some of this because I think that's fun. -929 +933 00:57:09,680 --> 00:57:14,948 And you know, you you tell me if this is if this is a helpful visual or a more -930 +934 00:57:14,948 --> 00:57:20,017 confusing visual but what we're going to do is look at this function exp of -931 +935 00:57:20,017 --> 00:57:25,220 r times x which is basically this is another way to write e to the power of x. -932 +936 00:57:25,540 --> 00:57:29,068 In fact, I think I I think I rendered a different animation at some -933 +937 00:57:29,068 --> 00:57:32,960 point that specified that because I was planning on planning on doing that. -934 +938 00:57:33,200 --> 00:57:35,266 So let me oh, yeah, there you are get back in my -935 +939 00:57:35,266 --> 00:57:37,460 file system get back to where you're supposed to be. -936 +940 00:57:41,460 --> 00:57:44,360 Get on in there is it complaining because there's multiple different? -937 +941 00:57:46,040 --> 00:57:50,200 It's going to be like there's a oh replace it shows up on the other screen. -938 +942 00:57:51,040 --> 00:57:51,900 Wait, why is it? -939 -00:57:52,160 --> 00:57:54,919 +943 +00:57:52,160 --> 00:57:54,920 Yeah, okay replace place whatever you see there. -940 +944 00:57:57,280 --> 00:58:02,178 And now we go back to oh there we go all of that all of that just so that I could have -941 +945 00:58:02,178 --> 00:58:06,963 nicely written out uh, if you're uncomfortable with thinking of it as exp of r times -942 +946 00:58:06,963 --> 00:58:11,805 x this infinite polynomial Just in the back of your head e to the r times x and we're -943 +947 00:58:11,805 --> 00:58:16,478 going to vary around r So i'm going to follow the points of the imaginary axis and -944 +948 00:58:16,478 --> 00:58:21,376 i'm going to follow the points of the real axis And uh, let's see what this does Well, -945 +949 00:58:21,376 --> 00:58:26,049 that's all kind of fast so let me think through it a little bit more slowly all of -946 +950 00:58:26,049 --> 00:58:30,834 the negative numbers Anything that's a negative real number is going to get squished -947 +951 00:58:30,834 --> 00:58:35,620 into the range between zero and one which should make sense e to the negative e to a -948 +952 00:58:35,620 --> 00:58:40,518 negative real number is something between zero and one and we're Specifically tracking -949 +953 00:58:40,518 --> 00:58:45,359 f of negative one which is going to show up around whatever one over e is around zero -950 +954 00:58:45,359 --> 00:58:50,089 point three seven f of one lands on e As expected that's what x of one is f of i Is -951 +955 00:58:50,089 --> 00:58:54,874 going to land one radian around the unit circle and it's kind of fun to follow along -952 +956 00:58:54,874 --> 00:58:59,660 the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? -953 +957 00:59:00,800 --> 00:59:05,634 And what happens as we tweak this value of r that's determining not just that we're -954 +958 00:59:05,634 --> 00:59:10,469 talking about an exponential function But which exponential function there's a nice -955 +959 00:59:10,469 --> 00:59:15,188 one-to-one correspondence between all the exponential functions we might want and -956 +960 00:59:15,188 --> 00:59:20,138 values of r here It stretches things differently So when we put it up to two You know -957 +961 00:59:20,138 --> 00:59:24,857 it stretches out the real axis a lot more so that f of one ends up around where e -958 +962 00:59:24,857 --> 00:59:29,749 squared is a little Above seven f of negative one is much closer to zero f of i Is a -959 +963 00:59:29,749 --> 00:59:34,814 two radian rotation around the circle f of negative i is a negative two radian rotation -960 +964 00:59:34,814 --> 00:59:39,706 And of course we can get to our favorite formula that If that were pi that we had as -961 +965 00:59:39,706 --> 00:59:44,598 our scaling constant then the real axis gets stretched out quite a lot you know f of -962 +966 00:59:44,598 --> 00:59:49,433 one is sitting off at e to the pi which is very close to 20 plus pi which is always -963 +967 00:59:49,433 --> 00:59:54,267 fun and f of negative one extremely close to zero so It's really stretched out that -964 +968 00:59:54,267 --> 00:59:58,987 real axis and it's also stretched out things in the Unit circle direction so that -965 +969 00:59:58,987 --> 01:00:03,764 getting to f of i or f of negative i walks halfway around the circle So that's all -966 +970 01:00:03,764 --> 01:00:04,800 well and good now. -967 +971 01:00:04,800 --> 01:00:06,200 How would we think about a function like? -968 +972 01:00:07,760 --> 01:00:09,820 two to the x Which is what? -969 +973 01:00:10,520 --> 01:00:15,257 We would also write as exp of Exp of the natural log of two times x So we -970 +974 01:00:15,257 --> 01:00:20,186 kind of move our yellow dot representing the value of r to around zero point -971 +975 01:00:20,186 --> 01:00:25,180 six nine still no imaginary part Just a real number zero point six nine or so. -972 +976 01:00:25,320 --> 01:00:29,820 That's the natural log of two Well, you can see that f of one lands on two, -973 +977 01:00:29,820 --> 01:00:34,557 which is why we want to call this function two to the x f of one half actually, -974 +978 01:00:34,557 --> 01:00:39,650 sorry f of negative one lands right on one half f of i It's some walk around the unit -975 +979 01:00:39,650 --> 01:00:44,742 circle very specifically it's going to be 0.69 radians around the unit circle And now -976 +980 01:00:44,742 --> 01:00:49,953 we could have a little bit more fun and say what would happen if we were to Change this -977 +981 01:00:49,953 --> 01:00:54,987 to instead of being 0.69 instead of being the natural log of two make it i times the -978 +982 01:00:54,987 --> 01:00:59,783 natural log of two So that we're really thinking of something that might have an -979 +983 01:00:59,783 --> 01:01:04,935 exponential base to it This would be what we might think of as two times i Raised to a -980 +984 01:01:04,935 --> 01:01:10,146 power Well, if we move that yellow dot which is representing r off of the real axis and -981 +985 01:01:10,146 --> 01:01:15,298 onto the imaginary axis it swaps the roles of the Teal dots and all of the maroon dots -982 +986 01:01:15,298 --> 01:01:20,390 in this context which remember came from being the positive and imaginary axes And it -983 +987 01:01:20,390 --> 01:01:25,424 should make sense that it swaps their roles because what does it mean if we take the -984 +988 01:01:25,424 --> 01:01:27,260 input space and we multiply it? -985 +989 01:01:27,260 --> 01:01:31,828 By i it means we're rotating that input space So everything that was the real number -986 +990 01:01:31,828 --> 01:01:36,182 axis turns into the imaginary axis and everything that was the imaginary axis Is -987 +991 01:01:36,182 --> 01:01:40,643 getting turned into the real axis So for us what that means is our new exponential -988 +992 01:01:40,643 --> 01:01:45,158 function where our value of r is now purely imaginary Takes all of the real numbers -989 +993 01:01:45,158 --> 01:01:49,834 and it just wraps them around a circle and it takes all the imaginary numbers And it's -990 +994 01:01:49,834 --> 01:01:52,897 putting them onto the real number line So in particular, -991 +995 01:01:52,897 --> 01:01:57,681 let's say we scale this thing up so that we're sitting at around pi halves times i Well, -992 +996 01:01:57,681 --> 01:01:59,240 what does that actually mean? -993 +997 01:01:59,240 --> 01:02:03,106 That means it takes the real number one to the value i Which is the -994 +998 01:02:03,106 --> 01:02:07,200 sense in which we want to write this function as i to the power x Right. -995 +999 01:02:07,200 --> 01:02:11,434 It just tempts us to write it not as this abstract looking thing X of r -996 +1000 01:02:11,434 --> 01:02:14,198 times x where r is equal to pi halves i no No, -997 +1001 01:02:14,198 --> 01:02:18,491 we just want to write it as i to the x even if that's a little ambiguous -998 +1002 01:02:18,491 --> 01:02:22,961 What that really means is just that the function we're dealing with outputs -999 +1003 01:02:22,961 --> 01:02:26,137 i at one and if it's an exponential function that Uh, -1000 +1004 01:02:26,137 --> 01:02:28,960 and we're asking what does it do to the value i? -1001 +1005 01:02:29,240 --> 01:02:30,800 What is i to the power i? -1002 +1006 01:02:31,300 --> 01:02:35,424 In this case, it shoves it to around uh, 0.2 around a fifth But there's many -1003 +1007 01:02:35,424 --> 01:02:39,603 different exponential functions that would have this property of putting f of -1004 +1008 01:02:39,603 --> 01:02:43,085 one onto the number i So if we were to scale it up even further, -1005 +1009 01:02:43,085 --> 01:02:47,210 I don't think I have it animated here But if we were to take that yellow dot -1006 +1010 01:02:47,210 --> 01:02:51,335 and raise it up until it got to five halves times pi i What you would see is -1007 +1011 01:02:51,335 --> 01:02:52,460 that the unit circle? -1008 +1012 01:02:53,060 --> 01:02:58,119 Uh is rotated around on itself so that f of negative f of one would rotate around another -1009 +1013 01:02:58,119 --> 01:03:02,953 two pi radians and land where it is But it would stretch out the real axis a lot more -1010 +1014 01:03:02,953 --> 01:03:07,900 Which was the sense in which another output of i to the i is a much much smaller number. -1011 +1015 01:03:08,100 --> 01:03:08,880 It was around. -1012 +1016 01:03:08,940 --> 01:03:09,340 What was it? -1013 +1017 01:03:09,440 --> 01:03:14,400 0.0003 or so But we can also see what I think is quite fun. -1014 +1018 01:03:14,400 --> 01:03:17,573 What happens if we consider Alternate expressions -1015 +1019 01:03:17,573 --> 01:03:21,000 that we want to interpret as two to the power x right? -1016 +1020 01:03:21,000 --> 01:03:24,961 So when r is a purely real number the natural log of two kind of makes sense -1017 +1021 01:03:24,961 --> 01:03:28,923 that when you plug it in here The expression we get is what we want to write -1018 +1022 01:03:28,923 --> 01:03:33,040 as two to the power x But what if we start moving it in the imaginary direction? -1019 +1023 01:03:33,500 --> 01:03:40,420 Okay And what i'll first do is i'll move it up by pi i units Now what's going on here? -1020 +1024 01:03:40,680 --> 01:03:43,940 We have x of r times x and r is equal to this value, -1021 +1025 01:03:43,940 --> 01:03:48,369 which is the natural log of two plus pi times i What that means is that -1022 +1026 01:03:48,369 --> 01:03:49,600 when we plug in one? -1023 +1027 01:03:50,160 --> 01:03:55,104 f of one is at negative two, so we want to write this function as negative two to the -1024 +1028 01:03:55,104 --> 01:03:58,553 power x right, and that's actually something that You know, -1025 +1029 01:03:58,553 --> 01:04:03,727 it's it's a little deceptively simple when we write a negative number to a power Negative -1026 +1030 01:04:03,727 --> 01:04:08,557 two To the power x it doesn't at first look like this necessarily it brings us into -1027 +1031 01:04:08,557 --> 01:04:13,616 the complex numbers in any way but of course when we plug in even a value like One half -1028 +1032 01:04:13,616 --> 01:04:18,675 Where we're kind of asking for a square root of negative two We we realize that we want -1029 +1033 01:04:18,675 --> 01:04:23,677 to write this as something like i times the square root of two But if you were to look -1030 +1034 01:04:23,677 --> 01:04:28,391 at this function negative two to the power x in the full complex domain that it's -1031 +1035 01:04:28,391 --> 01:04:33,565 dealing with What you're looking at is a function that takes the value of one to negative -1032 +1036 01:04:33,565 --> 01:04:38,567 two And if it does that what it does to the rest of the real number line is it kind of -1033 +1037 01:04:38,567 --> 01:04:39,660 spirals it outward? -1034 +1038 01:04:40,240 --> 01:04:45,208 So we see that f of negative one sits at negative one half About where you would expect -1035 +1039 01:04:45,208 --> 01:04:50,064 if you were to follow to f of one half It would sit exactly on the imaginary line and -1036 +1040 01:04:50,064 --> 01:04:54,920 f of one half would be square root of two Well, my mouse is not where I want it to be. -1037 +1041 01:04:54,920 --> 01:05:00,230 It would be around a square root of two times i and As you continue further on this is -1038 +1042 01:05:00,230 --> 01:05:05,663 showing you all of the real value powers of negative two to the x it necessarily spirals -1039 +1043 01:05:05,663 --> 01:05:10,851 around Um, but we could also move our value of r even higher and get it up to around -1040 +1044 01:05:10,851 --> 01:05:14,086 tau times i around 6.28 times i And in that context, -1041 +1045 01:05:14,086 --> 01:05:19,214 this is another function that we would want to write as something like two to the x -1042 +1046 01:05:19,214 --> 01:05:24,464 because For any whole number to whole number that you plug in for x it will look like -1043 +1047 01:05:24,464 --> 01:05:29,713 repeated multiplication And it even has kind of reasonable values for things like one -1044 +1048 01:05:29,713 --> 01:05:34,841 half where it spits out the negative square root instead of the positive square But -1045 +1049 01:05:34,841 --> 01:05:40,090 what it's actually doing is a transformation to the plane where it puts everything Uh -1046 +1050 01:05:40,090 --> 01:05:45,401 is the real number line ends up being a very tightly wound spiral That goes around and -1047 +1051 01:05:45,401 --> 01:05:50,650 it just spirals in such a way that f of one lands right on the number two so it is in -1048 +1052 01:05:50,650 --> 01:05:56,022 that sense that we could say, um two to the x is Is plausibly interpreted as a separate -1049 +1053 01:05:56,022 --> 01:06:01,149 exponential function from the one that we are traditionally used to so I think with -1050 +1054 01:06:01,149 --> 01:06:06,216 all of that I will um I will leave things for today and i'll just leave you with a -1051 +1055 01:06:06,216 --> 01:06:08,780 couple lingering questions to think about. -1052 +1056 01:06:09,300 --> 01:06:15,440 Okay, so If you want to think of i to the i as being a multi-valued expression, right? -1053 +1057 01:06:15,560 --> 01:06:20,650 You could you could say we adopt a convention Fancifully you'd say you choose a branch -1054 +1058 01:06:20,650 --> 01:06:25,566 of the natural logarithm function and maybe that locks you into this being e to the -1055 +1059 01:06:25,566 --> 01:06:30,540 negative pi halves But if you say this kind of wants to be infinitely many different -1056 +1060 01:06:30,540 --> 01:06:35,397 values like the various ones that we saw How many values does two to the one-third -1057 +1061 01:06:35,397 --> 01:06:40,195 want to be in the same sense where we are replacing two with various different uh -1058 +1062 01:06:40,195 --> 01:06:44,993 various different options for e to the x Such that e to the x equals two How many -1059 +1063 01:06:44,993 --> 01:06:50,026 different values does that want to be or how many values does two to the three-tenths -1060 +1064 01:06:50,026 --> 01:06:50,670 want to be? -1061 +1065 01:06:51,880 --> 01:06:57,771 Phrased differently of all of the uh, let me say of all of the exponential functions -1062 +1066 01:06:57,771 --> 01:07:03,455 So f of x which satisfy oh have I written it down somewhere f of x that satisfies -1063 +1067 01:07:03,455 --> 01:07:09,070 All of these properties that i've written so if it satisfies all of these um and -1064 +1068 01:07:09,070 --> 01:07:14,754 if f of one is equal to two Right, how many different outputs are we going to get -1065 +1069 01:07:14,754 --> 01:07:20,300 when we plug in x equals three-tenths for the various options for what function? -1066 +1070 01:07:20,300 --> 01:07:22,580 That is and how many outputs are we going to get? -1067 -01:07:23,859 --> 01:07:28,980 +1071 +01:07:23,860 --> 01:07:28,980 For two to the pi for the various functions that two to the x could represent If we're -1068 +1072 01:07:28,980 --> 01:07:34,100 thinking of two to the x as some kind of exponential function exponential in the sense -1069 +1073 01:07:34,100 --> 01:07:37,396 of these sort of abstract properties and if we uh yeah, -1070 +1074 01:07:37,396 --> 01:07:42,576 if we if we have a Class of different such functions and we want to plug in pi it makes -1071 +1075 01:07:42,576 --> 01:07:47,696 me laugh just because it's such a I don't know Kind of a funny answer that pops out As -1072 +1076 01:07:47,696 --> 01:07:49,580 you're trying to think about it. -1073 +1077 01:07:49,620 --> 01:07:54,143 So those are the questions that i'll leave you with and I think this is you know, -1074 +1078 01:07:54,143 --> 01:07:58,776 my my My central question in approaching today's lecture was whether I wanted to be -1075 +1079 01:07:58,776 --> 01:08:03,576 um kind of describing like these abstract properties of exponential functions and it's -1076 +1080 01:08:03,576 --> 01:08:08,264 just cool to me that Starting from those abstract properties you get locked into the -1077 +1081 01:08:08,264 --> 01:08:12,953 idea of e to the rx or more You know, I think more honestly written exp of r times x -1078 +1082 01:08:12,953 --> 01:08:17,697 for different values of r That it locks you in that far but it doesn't lock you in as -1079 +1083 01:08:17,697 --> 01:08:22,276 far as having an unambiguous Notion of what two to the power x should be much less -1080 +1084 01:08:22,276 --> 01:08:26,909 something like i to the power x The risk in that of course is that sometimes people -1081 +1085 01:08:26,909 --> 01:08:31,708 don't love abstraction and sometimes it doesn't come off as approachable But if that's -1082 +1086 01:08:31,708 --> 01:08:36,397 the case, you know, you just let me know I think I think there's a whole interesting -1083 +1087 01:08:36,397 --> 01:08:41,031 circle of thoughts that surrounds all of this stuff to include power towers because -1084 +1088 01:08:41,031 --> 01:08:45,775 if you want to Actually talk about power towers like we were last time in the context -1085 +1089 01:08:45,775 --> 01:08:50,519 of complex numbers or even with negative bases You have to be thinking through things -1086 +1090 01:08:50,519 --> 01:08:54,435 like this so, um, it was a question that we had up on screen Uh, yeah, -1087 +1091 01:08:54,435 --> 01:08:58,241 what happens if we do this for i to the power i titration, you know, -1088 +1092 01:08:58,241 --> 01:09:02,875 let's just try this Let's just go ahead and try a power tower where we're raising i -1089 +1093 01:09:02,875 --> 01:09:07,509 to a given power and see what uh, what pops out of it so I wasn't planning on doing -1090 +1094 01:09:07,509 --> 01:09:12,087 this but we can We can always pull up python and essentially do what we were doing -1091 +1095 01:09:12,087 --> 01:09:16,886 last time so the way that this would work Is we were starting off with some base value -1092 +1096 01:09:16,886 --> 01:09:19,700 and then for some kind of range What were we doing? -1093 +1097 01:09:20,520 --> 01:09:26,036 We were taking a and we're going to reassign it to be whatever The base -1094 +1098 01:09:26,036 --> 01:09:31,399 which in this case is i raised to the power of a should be Okay, cool. -1095 +1099 01:09:31,439 --> 01:09:37,115 So we're going to do that and then we're going to print off the value of a and let's just -1096 +1100 01:09:37,115 --> 01:09:42,790 do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens -1097 +1101 01:09:42,790 --> 01:09:48,213 is There's there's potential for chaos with these things like sometimes it's not that -1098 +1102 01:09:48,213 --> 01:09:53,762 you've landed on a stable A stable value or it's not even that you've diverged it could -1099 +1103 01:09:53,762 --> 01:09:59,374 be that you're bouncing between a cycle of values or That you're like literally bouncing -1100 +1104 01:09:59,374 --> 01:10:04,923 in a way that's um, it's not periodic or anything and it's actually chaotic I I suspect -1101 +1105 01:10:04,923 --> 01:10:10,472 that doesn't happen for i but it's a thing to potentially look out for It looks like it -1102 +1106 01:10:10,472 --> 01:10:15,769 does kind of stabilize um, maybe there's Some little subjection to numerical error, -1103 +1107 01:10:15,769 --> 01:10:21,255 but we stay pretty consistently around something with a real part of 0.43 and 0.36 Now -1104 +1108 01:10:21,255 --> 01:10:26,804 what I would want to emphasize though is this expression So let's set a back to b equal -1105 +1109 01:10:26,804 --> 01:10:32,480 to 1 this expression of taking i to the power of a remember That's a little bit ambiguous. -1106 +1110 01:10:32,480 --> 01:10:37,790 It depends on what choice of the function I we actually have so let me Let me -1107 +1111 01:10:37,790 --> 01:10:43,304 import NumPy so I have the exponential function Let me go For our big range like -1108 +1112 01:10:43,304 --> 01:10:46,708 we had before Rather than writing it as you know, -1109 +1113 01:10:46,708 --> 01:10:52,359 something that's like i to the power of x I'm going to write it as the exponential -1110 +1114 01:10:52,359 --> 01:10:58,009 function of a different constant right a different constant That i'm going to make -1111 +1115 01:10:58,009 --> 01:11:00,120 I want it to be five pi halves. -1112 +1116 01:11:00,120 --> 01:11:06,461 So i'll do five pi halves times i so it's a complex number And it's got five pi -1113 +1117 01:11:06,461 --> 01:11:13,040 halves as the imaginary part So this is five pi halves times i and what am I doing? -1114 +1118 01:11:13,320 --> 01:11:17,500 I'm exponentiating that So I want to multiply a onto the inside there. -1115 +1119 01:11:17,620 --> 01:11:21,897 Okay, this is basically another way that you could interpret the expression i to the -1116 +1120 01:11:21,897 --> 01:11:26,326 x Thankfully you would say you've chosen a different branch of the natural log function -1117 +1121 01:11:26,326 --> 01:11:30,604 But it is another Function which we could iterate on itself and see what happens and -1118 +1122 01:11:30,604 --> 01:11:33,574 we might get a different result Ah very interesting, okay, -1119 +1123 01:11:33,574 --> 01:11:37,701 so we actually have a different result It looks like what ends up happening is it -1120 +1124 01:11:37,701 --> 01:11:39,060 bounces between two values? -1121 +1125 01:11:40,140 --> 01:11:41,660 Oh, wow, is it period three? -1122 +1126 01:11:42,000 --> 01:11:44,180 That's interesting period three implies chaos. -1123 +1127 01:11:44,360 --> 01:11:49,477 So we've got seven point three five then zero then point nine nine So it looks like it -1124 +1128 01:11:49,477 --> 01:11:54,594 gets into the cycle of bouncing between three separate values even though um In theory -1125 +1129 01:11:54,594 --> 01:11:59,652 in both of those cases we were doing something that was i to the x iterated on itself -1126 +1130 01:11:59,652 --> 01:12:04,770 It has everything to do with what actual function you think i to the x is referring to -1127 +1131 01:12:04,770 --> 01:12:09,946 so in that sense The power tower question is ambiguous Usually you just choose it to be -1128 +1132 01:12:09,946 --> 01:12:15,240 the pi halves i variant but it's more fun to see that it can be multiple things All right. -1129 +1133 01:12:15,240 --> 01:12:17,560 We have addition multiplication exponentiation titration. -1130 +1134 01:12:17,640 --> 01:12:21,620 Can we think of something in between halfway between multiplication and exponentiation? -1131 +1135 01:12:22,940 --> 01:12:26,744 oh, that's an uh, I mean Spirit of it because each one of them it comes -1132 +1136 01:12:26,744 --> 01:12:30,548 down to like a discrete step of you're repeating the previous operation -1133 +1137 01:12:30,548 --> 01:12:34,141 But oftentimes in math when you have something initially defined in -1134 +1138 01:12:34,141 --> 01:12:37,840 terms of repetition Like exponentiation you can extend it beyond that. -1135 +1139 01:12:37,920 --> 01:12:40,294 I can't think of anything off the top of my head, -1136 +1140 01:12:40,294 --> 01:12:44,520 but that's an interesting question I don't know if that's been That's an extended notion. -1137 +1141 01:12:44,600 --> 01:12:49,003 I mean you have things like fractional derivatives you have things like fractional -1138 +1142 01:12:49,003 --> 01:12:51,816 and complex exponents So it doesn't seem outlandish, -1139 +1143 01:12:51,816 --> 01:12:56,007 but i'm not familiar with one myself And then lastly do complex numbers to the -1140 +1144 01:12:56,007 --> 01:13:00,199 power of complex number values arise in physics and if yes How does one decide -1141 +1145 01:13:00,199 --> 01:13:01,420 which values are valid? -1142 +1146 01:13:02,140 --> 01:13:05,100 so I can't think of if they necessary. -1143 +1147 01:13:05,100 --> 01:13:06,340 Oh, sorry scene switching. -1144 +1148 01:13:06,420 --> 01:13:09,938 I can't think of if they Come up in physics in like a direct -1145 +1149 01:13:09,938 --> 01:13:13,400 sense and this is probably just because i'm not a physicist. -1146 -01:13:13,400 --> 01:13:17,125 -So I uh, I would say like ask ask your neighborhood physicist and see see what - -1147 -01:13:17,125 --> 01:13:20,992 -they have to say I mean clearly the point of teaching the lesson here is that the - -1148 -01:13:20,992 --> 01:13:24,812 -thoughts that you have to go through to make sense out of It do build a stronger - -1149 -01:13:24,812 --> 01:13:28,727 -relationship with exponentials things like it's not just more natural to represent - 1150 -01:13:28,727 --> 01:13:31,226 -them as e to the rx Once you get to complex numbers, +01:13:13,400 --> 01:13:17,131 +icist so I Would say like ask ask your neighborhood physicist and see see what they 1151 -01:13:31,226 --> 01:13:35,093 -you kind of have to represent them that way Whereas previously in like physics or +01:13:17,131 --> 01:13:20,951 +have to say I mean clearly The point of teaching the lesson here is that the thoughts 1152 -01:13:35,093 --> 01:13:36,320 -other real world contexts. +01:13:20,951 --> 01:13:24,726 +that you have to go through to make sense out of it do build a stronger relationship + +1153 +01:13:24,726 --> 01:13:28,591 +with exponentials things like it's not just more natural to represent them as e to the + +1154 +01:13:28,591 --> 01:13:32,455 +rx Once you get to complex numbers you kind of have to represent them that way whereas + +1155 +01:13:32,455 --> 01:13:36,320 +previously in like physics or other real-world contexts It's something that's just nice diff --git a/2020/ldm-i-to-i/english/sentence_timings.json b/2020/ldm-i-to-i/english/sentence_timings.json index 179f8923f..1b69a7d3e 100644 --- a/2020/ldm-i-to-i/english/sentence_timings.json +++ b/2020/ldm-i-to-i/english/sentence_timings.json @@ -210,7 +210,7 @@ 291.64 ], [ - "And I think, yeah, I think the one following that, I'm guessing people either forgot to mention the i, but let's just write that out and kind of see where this moves us.", + "and I think, yeah, I think the one following that I'm guessing people either forgot to mention the i but let's just, let's just write that out and kind of see where this moves us.", 292.3, 302.36 ], @@ -420,13 +420,13 @@ 640.18 ], [ - "And of course we have to draw this out in the complex plane, the fact that we've rotated 90 degrees already brings us there, and I could draw out a bunch of different potential position vectors, you know, and what it would tell you is, okay, I've got to sit there, rotate my vector 90 degrees, that describes my velocity.", + "And of course we have to draw this out in the complex plane, the fact that we've rotated 90 degrees already brings us there. And I could draw out a bunch of different potential position vectors, you know, suggestively maybe putting them on a circle and saying really imagine yourself sitting at any one of those and following the rule for this dynamic. And what it would tell you is, okay, I've got to sit there, rotate my vector 90 degrees, that describes my velocity.", 641.04, - 664.1 + 666.48 ], [ "And it doesn't necessarily have to be any one of the points on that circle, any point in the plane, if you're following this dynamic, you say rotate that vector 90 degrees, which if we're drawing a vector field, we often scale down, and it's all well and good.", - 664.42, + 666.48, 675.42 ], [ @@ -1065,7 +1065,7 @@ 1659.64 ], [ - "The answer is evidently starting at around 111 for that kind of exponential decay.", + "the answer is evidently starting at around a hundred and eleven for that kind of exponential decay", 1660.26, 1664.4 ], @@ -1865,7 +1865,7 @@ 2541.56 ], [ - "And what you can see is okay if I get r that a factor in front of x in my exponential function exp of r x to be 0.69, which I know is around the natural log of two.", + "nd what you could see is okay if I get R that Factor in front of X in my exponential function exp of R X to be zero point six nine Which I know is around the natural log of two", 2541.88, 2553.84 ], @@ -1905,7 +1905,7 @@ 2597.74 ], [ - "I could have exp of r times x where maybe r is something like 0.69.", + "I could have exp of R times X where maybe R is something like zero point six nine", 2597.88, 2602.1 ], @@ -2770,7 +2770,7 @@ 4393.4 ], [ - "So I uh, I would say like ask ask your neighborhood physicist and see see what they have to say I mean clearly the point of teaching the lesson here is that the thoughts that you have to go through to make sense out of It do build a stronger relationship with exponentials things like it's not just more natural to represent them as e to the rx Once you get to complex numbers, you kind of have to represent them that way Whereas previously in like physics or other real world contexts.", + "icist so I Would say like ask ask your neighborhood physicist and see see what they have to say I mean clearly The point of teaching the lesson here is that the thoughts that you have to go through to make sense out of it do build a stronger relationship with exponentials things like it's not just more natural to represent them as e to the rx Once you get to complex numbers you kind of have to represent them that way whereas previously in like physics or other real-world contexts It's something that's just nice", 4393.4, 4416.32 ] diff --git a/2020/ldm-i-to-i/english/transcript.txt b/2020/ldm-i-to-i/english/transcript.txt index bd24846a2..71e9b3928 100644 --- a/2020/ldm-i-to-i/english/transcript.txt +++ b/2020/ldm-i-to-i/english/transcript.txt @@ -40,7 +40,7 @@ Let me go back, pull up people's answers. Got a little excited, a little trigger happy with some of my clicking around here. And the second most common answer is something that's going to actually help us move forward here, which was one half of i times pi. And then that's the same as what was written, what was categorized as the third most common one, one half pi times i, just swapping the two numbers there. -And I think, yeah, I think the one following that, I'm guessing people either forgot to mention the i, but let's just write that out and kind of see where this moves us. +and I think, yeah, I think the one following that I'm guessing people either forgot to mention the i but let's just, let's just write that out and kind of see where this moves us. One half i times pi. Now, if you're confused about where that came from, I think this is a good time to remind ourselves of Euler's formula and what it's really saying. It's one half pi times i, where again, computationally, this very literally means if we plug in pi over two for theta here, which would be something like 1.57 or so. @@ -82,7 +82,7 @@ The outer expression, is going to stay the same, and then the inner expression a And this gives us a literal way to read the equation in terms of dynamics, right? What we're saying is take whatever your position vector is, if you rotate that 90 degrees, you apply this action of i, that's going to give you the velocity vector. So wherever you're standing, draw a vector from zero, the origin, up to where you are, rotate that vector 90 degrees, that gives you your velocity, and that's enough to tell you how to move, assuming you know where you're starting. -And of course we have to draw this out in the complex plane, the fact that we've rotated 90 degrees already brings us there, and I could draw out a bunch of different potential position vectors, you know, and what it would tell you is, okay, I've got to sit there, rotate my vector 90 degrees, that describes my velocity. +And of course we have to draw this out in the complex plane, the fact that we've rotated 90 degrees already brings us there. And I could draw out a bunch of different potential position vectors, you know, suggestively maybe putting them on a circle and saying really imagine yourself sitting at any one of those and following the rule for this dynamic. And what it would tell you is, okay, I've got to sit there, rotate my vector 90 degrees, that describes my velocity. And it doesn't necessarily have to be any one of the points on that circle, any point in the plane, if you're following this dynamic, you say rotate that vector 90 degrees, which if we're drawing a vector field, we often scale down, and it's all well and good. And here you could phrase this question without even talking about exponentials or complex numbers or anything like that, and in fact I'm going to go back to the quiz just to really emphasize that Euler's formula and what it's claiming and then how we apply it to expressions like i to the power i, it's very, it's really intuitive as soon as we're putting some dynamics into it, and we can ask questions removed from the idea of exponentials that actually have the same substance in their answers. So here I want you to imagine starting a walk from the point 1,0 on the coordinate plane, in such a way that at all moments your velocity vector is a 90 degree counterclockwise rotation of the vector drawn from 0,0, the origin, up to where you are. @@ -211,7 +211,7 @@ And again in terms of the intuition, what you might be asking there is, suppose We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one after three pi halves units of time? -The answer is evidently starting at around 111 for that kind of exponential decay. +the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. @@ -371,7 +371,7 @@ So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value of exp of r, right, which is basically this function of when we plug in one. So exp of r times one if you prefer to think of it that way. And that is being represented by our green line. -And what you can see is okay if I get r that a factor in front of x in my exponential function exp of r x to be 0.69, which I know is around the natural log of two. +nd what you could see is okay if I get R that Factor in front of X in my exponential function exp of R X to be zero point six nine Which I know is around the natural log of two What this means is that exp of one is about two. And so this corresponds with the function that we would usually write as two to the power x. Right. @@ -379,7 +379,7 @@ Okay, and basically as I change around my r, you know, I could try to change it So around 1.1 that exponential looks like three, which we would usually write as three to the power x. I would like to argue that it's a little bit healthier to think about varying this value r rather than varying the base. And the main reason is that as soon as we get to complex contexts and we're thinking of exponentiation, you have this overloading that goes on where if we change around what sits in front of the x, that's all well and good. -I could have exp of r times x where maybe r is something like 0.69. +I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. @@ -552,4 +552,4 @@ I mean you have things like fractional derivatives you have things like fraction so I can't think of if they necessary. Oh, sorry scene switching. I can't think of if they Come up in physics in like a direct sense and this is probably just because i'm not a physicist. -So I uh, I would say like ask ask your neighborhood physicist and see see what they have to say I mean clearly the point of teaching the lesson here is that the thoughts that you have to go through to make sense out of It do build a stronger relationship with exponentials things like it's not just more natural to represent them as e to the rx Once you get to complex numbers, you kind of have to represent them that way Whereas previously in like physics or other real world contexts. \ No newline at end of file +icist so I Would say like ask ask your neighborhood physicist and see see what they have to say I mean clearly The point of teaching the lesson here is that the thoughts that you have to go through to make sense out of it do build a stronger relationship with exponentials things like it's not just more natural to represent them as e to the rx Once you get to complex numbers you kind of have to represent them that way whereas previously in like physics or other real-world contexts It's something that's just nice \ No newline at end of file diff --git a/2020/ldm-i-to-i/french/sentence_translations.json b/2020/ldm-i-to-i/french/sentence_translations.json index 57a996ada..a25756b9e 100644 --- a/2020/ldm-i-to-i/french/sentence_translations.json +++ b/2020/ldm-i-to-i/french/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "Donc, si vous commencez au chiffre 1, votre vitesse initiale est de marcher tout droit vers 0 et à mesure que vous marchez encore plus bas, si vous étiez assis à la moitié, alors vous marcheriez toujours vers 0, mais maintenant votre vecteur vitesse serait négatif 1 fois là où vous êtes, ce qui est moins 1 moitié. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "Et une question intéressante sera, vous savez, est-ce qu'il existe une seule fonction de ce type qui semble raisonnable à écrire pour cela, car vous savez, si nous allons l'écrire comme i dans x, non seulement elle devrait satisfaire cela, elle devrait également satisfaire, vous savez quand nous branchons le numéro un que nous obtenons i probablement i à celui d'alimentation, mais nous pensons que cette fonction devrait être i. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "Nous avons donc 5 pi i moitiés géniales, c'est absolument une autre valeur que nous pourrions brancher pour x ici et juste pour l'épeler un peu plus visuellement si nous regardions en arrière notre cercle ici où nous sommes au moment parcouru pendant une durée égale aux moitiés de pi qui est 1.57 et si à la place nous prenions un autre tour complet et que nous parcourions une autre moitié de pi pour nous amener à pi que vous savez, nous pourrions en quelque sorte enregistrer, c'est là que le e de la valeur pi i est que nous parcourons une autre moitié de pi nous parcourons une autre moitié de pi qui à à ce stade, nous aurions fait un cercle complet pour revenir à un, puis nous marcherons pendant cinq moitiés de pi, ce qui est numériquement environ 7.85 ouais, c'est absolument un autre nombre qui nous met au-dessus de i et si nous devions passer par tout le gala de réexprimer i à la puissance i en écrivant d'abord e aux 5 moitiés pi i à la puissance i ces i multipliez pour devenir négatif et nous regarderions e aux moitiés négatives de 5 pi, ce qui est un nombre très différent, nous pouvons en fait calculer cela, je ne suis pas sûr de mémoire, mais jetons un coup d'œil à un Desmos . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "Ce long qui vous amène à un nombre beaucoup plus petit. Mais ce n'est pas la seule réponse que nous pourrions entrer, car d'autres personnes viennent ici avec moins 3 moitiés fois i pi. Que connaissez-vous en termes de cercle unité ? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "Nous pourrions penser à dire hé si je veux arriver à moi plutôt que de marcher 90 degrés pi moitiés radians de cette façon, que se passe-t-il si je marche 270 degrés dans l'autre sens 3 pi moitiés radians, ce que je considérerai peut-être comme négatif parce que la convention est généralement, le sens inverse des aiguilles d'une montre est positif. C'est absolument une autre façon de l'exprimer et cela nous donnerait une réponse différente si nous avions e aux moitiés négatives de 3 pi i Tout à la puissance i nous passons par le même jeu maintenant le i au carré s'annule avec un négatif qui est déjà là, et nous avons des moitiés positives de 3 pi et numériquement, cela nous donne une réponse encore différente de ce que nous avions avant. Et si nous y allons et que nous disons hé, qu'est-ce que e au 3 pi pas 3 o 3 pi moitiés 111 point 3 1 type de nombre très différent de ce que nous avons vu avant 111 point qu'est-ce que c'était 111 point 3 1 génial 111 point 3 1 environ Et encore une fois, en termes d'intuition, ce que vous pourriez demander, c'est que supposons que nous ayons cette rotation dynamique Mais nous remontons dans le temps, nous voyons depuis combien de temps ce que je dois être. De telle sorte que si je faisais avancer les choses à partir de là, j'atterrirais au numéro un de ma condition initiale et vous devez remonter dans le temps 3 moitiés pi unités Et puis, si vous deviez traduire par la dynamique de désintégration. Ce que fait le fait de lever le regard dans ce contexte, vous dites si je commence par le numéro un. Mais je veux revenir en arrière dans le temps et dire Par où aurais-je dû commencer si Je veux décliner de telle sorte que je finisse au numéro un ? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "Après 3 unités de temps pi, la réponse commence évidemment à environ cent onze pour ce type de décroissance exponentielle. Et vous pouvez voir où cela va, où il y a en fait une infinité de valeurs différentes que nous pourrions brancher pour X si nous sommes en pensant à e au X comme étant moi et les gens sont entrés beaucoup plus ici Excusez-moi de jeter mon épingle par terre comme on le fait classique pour la troisième place 9 moitiés pi excellent choix 1729 moitiés pi vous êtes tous mes lots préférés et beaucoup de différentes options une infinité de valeurs différentes, ce qui semble un peu déconcertant au début, car nous regardons une expression On dirait que vous savez qu'il va juste y avoir un calcul Je branche simplement cela sur ma calculatrice et vois ce qui apparaît et nous avons plusieurs différentes valeurs pour cela Alors, que se passe-t-il ici, n'est-ce pas ? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "La quatrième racine de 16 devrait être 2 et la réponse finit par être bonne. Nous adoptons une convention lorsqu'il existe plusieurs options comme celle-ci lorsque vous avez une fonction à plusieurs valeurs. Nous choisissons souvent simplement l'une de ces valeurs pour être ce que nous entendons lorsque nous voulons traitez-le comme une fonction comme quelque chose avec une seule entrée et une seule sortie dans un jargon plus sophistiqué Cela revient tout le temps lorsque nous traitons de nombres complexes l'idée de quelque chose comme une opération en quelque sorte vouloir avoir plusieurs valeurs, vous aurez parfois entendre l'expression branche Où choisissez-vous une branche de la fonction racine carrée ? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "Parce qu'il y a plusieurs réponses différentes Vous savez, nous pensons encore une fois à cette rotation de 90 degrés Et si nous y pensions comme une rotation de 90 degrés, nous avons l'impression que la racine carrée devrait être Vous savez, quelque chose est assis à un angle de 45 degrés Peut-être que c'est le carré racine de I que nous pourrions écrire très explicitement comme racine 2 sur 2 racine 2 sur 2 I Cela utilise simplement la trigonométrie, mais si nous pensions plutôt à I comme étant une rotation négative de 270 degrés, nous avons l'impression que la moitié de cela fait la moitié de cette opération devrait en fait nous amener de l'autre côté Peut-être que le nombre assis ici devrait être la racine carrée de I et c'est en fait juste le négatif de ce que nous avons vu auparavant Racine négative de 2 sur 2 moins racine de 2 sur 2 fois I Maintenant, dans le contexte du réel fonctions valorisées, nous pouvons dire oui. Choisissez simplement la racine carrée comme étant la réponse positive, mais laquelle d'entre elles considérez-vous comme la réponse positive ? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "Et je pense que vous dites bien Nous savons ce que c'est, nous le définissons en quelque sorte comme étant la racine carrée de 2, tout va bien. Mais et si je disais, abordons cela de la même manière que nous approchions notre je de l'expression je je je veux d'abord exprimer les choses comme e à quelque chose de correct, puis je vais augmenter cela à la moitié en multipliant la moitié par l'exposant Et je dis d'accord, je peux, je suppose que je peux faire ça e à ce qui est égal à 2 et bien C'est le logarithme népérien de 2 C'est une constante qui est autour de 0.69 environ Si nous élevons e à cette puissance, nous obtiendrons 2, nous pourrions donc considérer cela comme e au logarithme naturel de 2 fois 1 demi et si vous le vouliez, si vous pensiez à e au x ? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "Vous savez que cela peut être un peu exagéré dans le contexte de nombres réels. Mais si vous pensiez à e en x comme raccourci pour cette fonction x, vous pourriez insérer la valeur 0.69 fois 1 moitié, ce qui, je suppose, serait d'environ 0.345 C'est quelque chose comme ça. Vous branchez cette valeur très concrète dans votre polynôme et voyez ce qu'elle produit, et elle produira environ 1.414 un joli nombre réel racine carrée de 2 ce à quoi on s'attendrait Mais si nous faisions la même chose que nous faisions avec I et en reconnaissant qu'il y a en fait plusieurs réponses différentes lorsque nous voulons écrire quelque chose comme e à une puissance, nous pourrions également écrire ceci Cela peut paraître drôle, mais nous pourrions l'écrire sous la forme e dans le logarithme naturel de 2 plus 2 pi. Tout cela est élevé à la moitié. Juste après tout, cette valeur deviendra égale à vous pourriez la décomposer car elle est e à la log naturel de 2 Multiplié par e jusqu'à 2 pi I Celui-ci a juste pour effet de faire pivoter les choses à 360 degrés, donc ça va juste être égal à 1 Nous regardons donc 2 fois 1 super, cela ressemble à une substitution valide et pourtant quand nous jouons au même jeu : prendre ceci et l'élever à une puissance et traiter cela en multipliant la puissance par l'exposant, regardez ce qui se passe Nous avons e au logarithme naturel de 2 fois 1 moitié plus Eh bien, qu'est-ce que 2 pi I fois 1 moitié eh bien, ce sera pi fois I Maintenant, cette première partie e au logarithme naturel de 2 fois 1 moitié qui finira par être la racine carrée familière de 2, c'est bien beau, mais nous allons multiplier cela par e pour le pi I est correct et, comme on le sait, e au pi I est négatif 1. Donc, dans ce cas, cela semble suggérer que si nous résolvons cette expression 2 à la moitié 1, en jouant avec les différentes réponses, nous pourrions brancher quelque chose comme e au X égal à 1 moitié, ce que nous obtenons est une autre réponse, ce que nous pourrions traditionnellement écrire comme cette racine carrée négative de 2 et ici, je veux dire, c'est un peu drôle qu'il y ait plusieurs valeurs pour regarder 2 à 1 moitié et disons que cela n'équivaut pas à une chose, mais en fonction des choix que nous faisons, cela pourrait être égal à plusieurs choses différentes. Mais les deux choses pourraient sembler tout à fait raisonnables. S'il doit y avoir quelque chose qui correspond à 2 pour 1, il semble que cela devrait être soit le positif. racine carrée que nous connaissons ou la variante négative de celle-ci qui ne semble pas vraiment être un tel problème Et en fait, nous pourrions euh, nous pourrions jouer à ce jeu encore plus loin et permettez-moi de vous demander des réponses encore plus créatives à cette expression parce que peut-être pouvons-nous trouver d'autres puissances amusantes de quelque chose comme 2 à la puissance X lorsque nous commençons à brancher différentes valeurs de X en fonction de la substitution que nous effectuons si nous respectons les mêmes règles que celles que nous utilisions pour évaluer I à la puissance I Donc cette fois la question demande ou elle précise qu'une solution de l'équation e au x est égal à 2 est le nombre réel Log naturel de 2 ok celui-là on le connaît. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "réponse à la question e au x est égal à 2 et encore une fois la créativité est la bienvenue, donc je vais vous donner encore un petit moment pour cela II. J'irai de l'avant et verrouillerai quelques réponses ici si cela vous convient, je ne sais pas combien de temps cela prendra il faut nécessairement faire la saisie mathématique en fonction de l'appareil que vous regardez, mais ne soyez pas trop stressé si c'est avant d'avoir eu la chance de répondre à la question que vous souhaitez, à la réponse à laquelle vous souhaitez qu'il réponde. 131 d'entre vous ont entré la variante où nous prenons Ln de 2 et nous ajoutons 2ii et je suppose que j'écris cette question Par erreur, j'ai marqué l'une des réponses comme étant correcte alors qu'en fait il y en a pas mal de bonnes différentes Donc c'est de ma faute pour le fait que je ne sais pas si cela ressemble à l'un d'entre vous oh C'est rouge, vous vous êtes trompé lorsque vous avez entré Ln de 2 plus 42. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi, ce qui est bien sûr un excellent choix. Mais vous pourriez aussi avoir quelque chose comme 4 pi I plus le logarithme naturel de 2 ou 6 pi I Ou vraiment n'importe quel multiple entier de 2 pi I si vous ajoutez que cela n'affecte pas e au X Parce que cela a juste pour effet de multiplier par e par 2 pi I Ce qui est l'effet de multiplier par 1 et encore une fois, cela a une sorte de conséquence amusante où cela semble produire des résultats raisonnables lorsque nous le faisons comme un autre exemple. on dirait que la deuxième expression la plus courante était que nous pourrions remplacer 2. Imaginons donc que nous pensons à 2 à la puissance 1 4ème, d'accord, il a été suggéré que nous remplaçons 2 par e au logarithme naturel de 2 plus 4 pi I Okay Plus 4 pi I et nous élevons tout cela au 1 4ème, et bien, si vous jouiez au même jeu, vous obtiendriez e Au logarithme naturel de 2 fois 1 4ème, et nous multiplierions par e pour le pi I Maintenant, la première partie va être la quatrième racine positive habituelle de 2, ce que nous voulons dire lorsque vous branchez une expression comme la quatrième racine de 2 dans une calculatrice, un joli petit nombre positif, mais alors cette deuxième partie est négatif 1 donc cela semble dire Vous savez, si nous devions interpréter 2 de cette manière différente en l'élevant au 1 4e Vous savez que ce n'est pas la réponse habituelle que nous obtenons mais c'est une réponse raisonnable. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "Nous aurions regardé pi divisé par deux par I et au lieu de multiplier par Négatif 1, nous aurions plutôt multiplié par I Ce qui encore une fois est une réponse valide, cela semble être un résultat raisonnable pour quelque chose comme 2 puissance 1 4. Donc, quand vous êtes en regardant le fait que je suis au pouvoir, je semble avoir plusieurs valeurs différentes pour cela, c'est vrai, nous avons ce drôle de phénomène où nous pourrions brancher e aux moitiés de 5 pi I, moitiés négatives de 3 pi I et nous obtenons ce qui semblait être des réponses très différentes quelque chose de super petit quelque chose de super grand, tous très différents du 1 5ème, environ 1 5ème réponse que nous avons trouvée auparavant ici. C'est exactement le même phénomène que lorsque vous demandez quelque chose comme combien vaut 2 sur 1 4 et que vous reconnaissez qu'il existe en fait plusieurs solutions différentes. à l'expression X au 4ème est égal à 2 4 solutions différentes en fait et ce que vous regardez, c'est le fait qu'il y a plusieurs solutions différentes À l'expression e au X est égal à une sorte de base si cette base est I si cette base est 2 Quoi qu'il en soit, une façon de penser à cela est que lorsque vous avez affaire à des chiffres réels, les choses sont tout simplement belles, les choses sont belles. Il existe des relations individuelles. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "C'est génial. Où si nous voulons réfléchir aux fonctions exponentielles, laissez-moi juste en parler. Nous avons ce joli va-et-vient où vous pouvez choisir d'exprimer n'importe quelle exponentielle comme base pour X comme 2 pour X. Ou vous pouvez exprimer cette même exponentielle que X de R fois X dont vous savez que c'est le polynôme auquel nous nous référons chaque fois que nous nous référons implicitement à chaque fois que nous écrivons quelque chose comme e sur X Et il y a un joli va-et-vient parce que vous pouvez simplement prendre un logarithme naturel de B Et cela vous donne une réponse en supposant que B est un nombre positif. Et c'est la même chose que de dire que X de R est égal à B. Donc, une façon dont j'en ai parlé plus tôt dans la série est que si vous regardiez le famille de toutes les exponentielles possibles, nous pourrions les écrire sous la forme X de R fois X et changer ce qu'est R. Et c'est exactement la même chose que d'écrire e dans le R fois X si c'est quelque chose avec lequel vous êtes plus à l'aise. Donc e dans le R fois XX de R fois X c'est la même chose que nous pourrions penser à changer ce que c'est Mais d'un autre côté, si vous deviez considérer toutes les exponentielles possibles comme une base Laissez-moi faire la base à la puissance de X et nous y allons changer ce qu'est cette base Au début, on a l'impression que c'est un type d'expression différent à manipuler, mais c'est juste une autre façon d'exprimer la même famille Et une façon dont vous pourriez penser à cela Car comment pensons-nous à quelle base correspond-elle si nous pensons un peu plus abstraitement à l'Exp de R fois X et il y a une raison pour laquelle je fais cela parce que nous sommes sur le point d'appliquer cela à des nombres complexes où cela va paraître plus étrange, alors suivez-moi ici si au lieu de regarder cette base, une chose que je pourrais faire est de dire quelle est la valeur ? ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "Je pourrais avoir une exp de R fois X où peut-être R est quelque chose comme zéro virgule six neuf Mais je pourrais décaler cela de deux pi I Et cela ne change pas la base à laquelle cela correspondrait qui correspondrait toujours à deux Ou cela pourrait décalez-le de deux pi I cela ne change pas la base à laquelle il correspond car dans tous ces cas Quand on branche X est égal à un, on obtient la même chose cependant Tous ces éléments pour différentes valeurs de X sont des fonctions distinctes C'est pourquoi nous avons vu plusieurs valeurs différentes pour I à la puissance I Parce que I à X est une fonction ambiguë dans ce contexte, il serait sans ambiguïté si nous décidions quelle valeur de R De telle sorte que ce que nous représentons est l'exp de R fois X quelle valeur de R. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "C'est une fonction sans ambiguïté mais à ce stade, nous avons peut-être l'impression que ce que nous voulons, c'est arrêter de penser aux choses en termes de base élevée à la puissance X. Peut-être que dès que nous sommes dans le contexte de nombres complexes, nous devrions simplement écrire tous comme exp de certains temps constants X si pour aucune autre raison cela rend très clair Comment nous connectons réellement des nombres si nous voulons faire un calcul ou simplement faire des mathématiques par-dessus, nous avons ce joli polynôme infini que nous branchez-les et je vais vous faire valoir à nouveau que c'est peut-être la bonne façon de penser aux exponentielles. Dès que nous étendons à d'autres domaines des choses comme les nombres complexes et pour cela, revenons en arrière Go retour à la sonnette, certaines choses sont arrivées, reviennent à la manière originale selon laquelle nous étendons l'idée d'exponentiation et pensons simplement à ce qui est 2 en X. À droite, nous savons comment y penser pour les nombres naturels. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "Vous savez quelque chose comme 2 sur 3 Multiplication répétée Comment se fait-il qu'on vous apprenne d'abord à penser à quelque chose comme 2 sur X pour des montants fractionnaires ou pour des montants négatifs et des choses comme ça. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "On vous apprend généralement que 2 à 1 moitié devrait être quelque chose où vous savez que si je le multiplie par lui-même et cela suit les règles habituelles que les exponentielles font avec le comptage des nombres où nous pouvons ajouter des choses dans cet exposant, je devrais obtenir 2 au 1 donc ça devrait être un nombre qui, lorsque je le multiplie par lui-même, j'obtiens 2 et vous savez qu'à ce stade, vous avez le choix, peut-être que c'est positif. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "Peut-être que c'est négatif Mais si vous décidez toujours de faire le choix positif Vous allez pouvoir obtenir une belle fonction continue de cette même affaire si nous posons des questions sur les nombres négatifs Que devrait bien être 2 au moins 1, cela devrait être quelque chose où quand je le multiplie par 2 jusqu'à 1 ? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "Cela me donne 2 à 0 et c'est en quelque sorte la justification de notre convention selon laquelle les exposants négatifs ressemblent à 1 moitié. Mais ce qui se passe réellement ici, c'est que nous disons que quoi que ce soit, cela devrait être une sorte de fonction qui satisfait cette propriété f de a plus b est égal à f de a fois f de b et de plus, le fait que la base soit 2 nous dit en gros que ce n'est pas n'importe quelle fonction de ce type. C'est une fonction où lorsque nous connectons 1, nous obtenons 2. Et juste un peu, vous savez. question de style contrôle de santé mentale pour voir si vous suivez certaines des implications ici. Je veux vous demander ce que c'est. Je ne l'appellerai pas comme une balle molle, mais ceci n'est pas censé être comme une question incroyablement profonde. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "nécessairement. C'est juste plus une vérification si vous suivez l'idée de commencer de manière abstraite par les propriétés d'une fonction, puis de déduire les façons dont nous pourrions vouloir l'écrire en fonction de ces propriétés Si f de x satisfait cette propriété exponentielle f de a plus b est égal à f de a fois f de b pour toutes les entrées Et cela satisfait également f de 1 est égal à 2 lequel des énoncés suivants est vrai C'est-à-dire lequel des énoncés suivants est nécessairement vrai Quelle que soit la fonction que vous démarrez avec et ceux d'entre vous qui se souviennent de quelle conférence il s'agissait C'est de celle dont nous parlions sur la façon d'interpréter ce que dit réellement la formule d'Euler J'ai posé une question de ce style où j'ai négligé une seule condition, vous savez que je n'ai pas écrit le fait que nous voulons nous assurer que f de x est différent de zéro partout et cela a provoqué une certaine confusion, ce qui est cool, une confusion à l'écran qui nous arrive à tous. Mais l'intention était essentiellement de montrer que cette propriété abstraite de quelque chose qui transforme l'addition en multiplication est suffisant pour vous donner envie d'écrire la fonction comme ce qu'elle équivaut à une sorte de puissance. C'est l'esprit de la question. Maintenant, nous avons quelques questions sur les tours de puissance. cela semble être apparu ici, ce qui est très lié à la dernière fois. Arrêtons-nous un instant sur la question de la tour de puissance afin que nous ayons d'abord une idée plus profonde de ce que l'exponentiation devrait signifier ici ? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "Parce que parce que nous pouvons être ce que je veux affirmer, c'est que nous pouvons y répondre de plusieurs manières différentes. Donc, si vous m'en donnez une seule, nous parlerons de tours de puissance. Et puis, tout comme une droite numérique peut être représentée sur une échelle logarithmique, faire la même chose pour un avion complexe ? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "Ouais, en fait, il y a une visualisation à laquelle je vais aborder dans un instant ici où nous faisons quelque chose d'assez similaire à cela. Parce que ce que nous allons faire, c'est jouer avec différentes fonctions exponentielles X de R fois X. Mais nous sommes Je vais changer cette valeur de R qui va être représentée par un petit point jaune. Nous allons donc en parler en quelque sorte. Cela ne va pas cartographier tout le plan, mais juste quelques points d'échantillonnage de l'axe réel et de l'axe imaginaire. Mais l'idée est qu'à mesure que nous nous déplaçons autour de cette constante, nous allons être en mesure de visualiser les différentes choses qu'elle fait à l'avion et en fait, c'est comme si nous transformions l'axe des x en une échelle logarithmique, puis l'enroulions l'axe imaginaire le long d'un cercle Et puis dès que cette valeur de R devient imaginaire, elle échange le rôle de ces nombres réels sont placés sur le cercle et les nombres imaginaires sont placés sur une échelle logarithmique Axe positif donc grande question dont les trois je suppose Je suis en quelque sorte en train de sauter le pas vers où je veux aller. Mais c'est agréable de voir que c'est là que les gens le pensent dans celui-ci. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "explicitement Quelque chose comme f de 5 est la même chose que f de 1 plus 1 plus 1 plus 1 plus 1 Ce qui est la même chose que f de 1 multiplié par lui-même 5 fois à cause de cette propriété Qui si f de 1 est 2 est le même comme 2 à la puissance 5 et ensuite quelque chose comme f de moins 5. Il devrait se produire que lorsque nous le multiplions par f de 5, nous obtenons ce qu'est f de 0 et on ne sait pas immédiatement ce qu'est f de 0 mais nous pourrions dire que f de 1 plus 0 est égal à ce que f de 1 est multiplié par ce que f de 0 est mais f de 1 est égal à 2 Et donc c'est aussi égal à 2 donc nous disons que 2 est égal à 2 fois quelque chose et bien que quelque chose doit être un 1, donc dans ce contexte, cela garantit que f de moins 5 est 2 à la puissance moins 5, c'est 1 sur 2 à la puissance 5. Nous pourrions explicitement écrire cela comme 2 à la puissance moins 5, ce qui revient à dire que ces deux propriétés forment ensemble nous voulons vraiment écrire la fonction comme 2 sur X Parce que tout nombre de comptage que nous y mettons va satisfaire Cela va ressembler à une multiplication par lui-même ce nombre de fois n'importe quel nombre fractionnaire que nous avons mis va satisfaire ces propriétés que nous voulions Et vous pourriez vous demander est-ce unique et dans le contexte de fonctions à valeurs réelles, ce serait en fait Mais dans le contexte de fonctions à valeurs complexes Il y aurait plusieurs fonctions de ce type f que nous pourrions écrire pour celle-ci, dont celle-ci est ce que nous étions en regardant avant Où nous pourrions avoir une fonction définie comme étant l'exp du log naturel de 2 plus 2 pi J'ai tout ce temps X D'accord, pardonnez la négligence ici, je suis juste excité d'écrire à ce sujet Et c'est en fait une fonction différente car en témoigne ce qui se passe si vous branchez X est égal à 1 moitié. Nous avons vu un peu plus tôt que lorsque vous branchez 1 moitié, vous obtenez la racine carrée négative de 2 et puis si vous branchez 1 quart, vous obtenez Pas la quatrième racine de 2 mais je multiplie la quatrième racine de 2 donc c'est une fonction différente Mais cela satisfait toujours ces propriétés et cela nous donne en quelque sorte envie de l'écrire comme 2 en X Et cela suggère que peut-être 2 en X est un terme ambigu un peu de notation Et nous devrions simplement tout écrire en termes d'exp de R multiplié par quelque chose, mais vous vous demandez peut-être bien. Vous savez, peut-être que nous ne sommes tout simplement pas assez créatifs avec toutes les fonctions qui satisfont cette propriété. Peut-être y a-t-il une ambiguïté lorsque nous écrivons exp de R multiplié par quelque chose et il y a différentes valeurs de R qui pourraient entrer en jeu. Mais je vais juste formuler une petite affirmation et ensuite peut-être donner un aperçu de ce à quoi ressemblerait la preuve si vous le souhaitez. disons que vous avez une fonction complexe F, et qu'elle satisfait d'abord les propriétés suivantes. Vous pouvez en prendre une dérivée. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "C'est différentiable, ce qui l'empêche simplement d'être une chose discontinue totalement désordonnée. C'est comme prendre des valeurs aléatoires en fonction de la durée de l'espace vectoriel que vous connaissez. Je ne connais pas les quantités fractionnaires auxquelles vous pourriez vouloir penser de manière folle. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "C'est une fonction intéressante. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "C'est différentiable Ce n'est pas égal à 0 partout donc la condition qui m'a en quelque sorte échappé et j'oublie pour quelle conférence je cours ou quelque chose comme ça et puis elle a cette propriété centrale qu'elle transforme l'addition en multiplication Si vous avez une telle fonction, je prétends que il y a un unique, peut-être que je devrais vraiment préciser qu'il existe un nombre complexe unique R afin que vous puissiez écrire F de X comme étant fondamentalement cette fonction exponentielle de R multipliée par cette valeur X. Vous savez, en gros, cela dit que si vous avez X comme fonction, cela polynôme infini avec de belles propriétés dérivées et tout ça si vous avez ceci, vous avez Chaque exponentielle que vous voulez dans un sens générique abstrait du mot exponentiel juste basé sur une propriété que nous pourrions en attendre et l'esquisse de la preuve serait ressemblez à ceci si vous voulez d'abord regarder quelle est la dérivée de cette valeur qui, nous supposons, existe partout, n'est-ce pas ? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "Nous pouvons factoriser F de X hors de l'expression entièrement et toute la limite est exprimée uniquement en termes de H. Ce qui si vous réfléchissez à ce que cela signifie dans le contexte des dérivées et au fait que F de 0 est nécessairement égal à 1. Toute cette expression limitative est juste une constante mais plus précisément, quelle que soit la dérivée de notre fonction à 0. Donc vous avez ce truc drôle où si vous connaissez sa dérivée à 0 qui détermine quelle est sa dérivée est partout Et dans le contexte des fonctions exponentielles, c'est, espérons-le, assez familier parce que tout ce que nous disons en réalité, c'est que la dérivée d'une fonction exponentielle est proportionnelle à elle-même et que la constante de proportionnalité est égale à la dérivée à 0. Tout cela est formulé de manière très abstraite et ainsi de suite, mais le but est de souligner que c'est pas nécessairement seulement des fonctions que nous considérons déjà comme une puissance X. Mais il s'agit d'une classe de fonctions potentiellement beaucoup plus large qui satisfont simplement à cette propriété abstraite de transformer l'addition en multiplication. Mais si vous avez cela, cela garantit en fait que vous avez également une dérivée seconde Et d'ailleurs une dérivée troisième et autres parce que la fonction dérivée est juste proportionnelle à elle-même. Donc, pour prendre la nième dérivée, il suffit de regarder cette constante de proportionnalité et de l'élever à la puissance n, puis à partir d'ici, vous pouvez faire un Expansion de la série Taylor et je pourrais laisser cela comme une sorte de devoir avancé pour ceux d'entre vous qui sont à l'aise avec la série Taylor dans cette idée, surtout si vous souhaitez mélanger l'idée de toute fonction différentiable qui est différentiable dans le sens de nombres complexes, ce qui est une sorte de sujet définitivement universitaire. Vous savez que vous pouvez mélanger le raisonnement comme vous le souhaitez. Mais le raisonnement flou est autorisé dans le contexte de quelqu'un qui ne connaît que les séries de Taylor et rien d'autre pour prendre cette idée et regarder l'expansion de Taylor pour F et justifie en quelque sorte l'idée qu'il existe un nombre complexe unique tel que notre fonction F peut nécessairement être écrite comme ceci. Et puis la connexion aux exponentielles normales se fait chaque fois que vous avez une telle valeur R. Nous faisons essentiellement ce que nous faisons dans le contexte complexe des nombres réels. c'est si vous regardez l'exp de cette fonction de cette valeur R et écrivez-la comme base. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "Nous pourrions interpréter cela comme signifiant non seulement l'exp des moitiés de pi I fois X, mais nous pourrions également l'interpréter comme signifiant l'exp de 5 moitiés de pi I fois X et Ce sont des fonctions distinctes Et il existe une famille infinie de fonctions distinctes qui semblent que nous devrions écrivez-les comme I au X Donc l'expression I au I à moins que vous n'ayez adopté une norme pour ce que cela va nécessairement signifier Quand vous dites qu'il a une infinité de sorties, une autre façon de penser à cela est que La fonction I au X avec la notation que nous avons est un peu ambiguë. Maintenant avec tout cela, commençons simplement à visualiser une partie de cela parce que je pense que c'est amusant Et vous savez, vous me dites si c'est si c'est un visuel utile ou un visuel plus déroutant mais ce que nous allons faire, c'est regarder cette fonction exp de R fois X, qui est fondamentalement une autre façon d'écrire e à la puissance X en fait, je pense que je pense avoir rendu une animation différente à un moment donné qui précisait que parce que j'avais prévu de faire ça, alors laissez-moi, oh ouais, vous voilà de retour dans mon système de fichiers, revenez là où vous êtes censé être. Allez-y, est-ce qu'il se plaint parce qu'il y en a plusieurs différents? Ce sera comme s'il y avait un Oh, remplacez, il apparaît sur l'autre écran. Attendez, pourquoi est-ce ouais, d'accord, remplacez ? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "Placez tout ce que vous voyez là-bas Et maintenant, nous revenons à oh voilà, nous avons tout cela juste pour que j'aurais pu bien l'écrire Si vous n'êtes pas à l'aise à l'idée de le considérer comme une exp de R fois X, ce polynôme infini Juste dans le l'arrière de ta tête e au R fois X et on va varier autour de R donc je vais suivre les points de l'axe imaginaire, et je vais suivre les points de l'axe réel et voyons ce que ça fait Eh bien c'est assez rapide alors laissez-moi y réfléchir un peu plus lentement, tous les nombres négatifs, tout ce qui est un nombre réel négatif va être écrasé dans la plage entre 0 et 1. Ce qui devrait avoir du sens e au négatif ? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a à un nombre réel négatif est quelque chose entre 0 et 1 et nous suivons spécifiquement f de négatif 1 qui apparaîtra autour de 1 sur e autour de 30 0.37 f de 1 atterrit sur e comme prévu, c'est ce que l'exp de 1 est f de I va atterrir d'un radian autour du cercle unité, et c'est plutôt amusant de suivre tout l'axe imaginaire ici comment l'axe imaginaire s'enroule autour d'un cercle et que se passe-t-il lorsque nous ajustons cette valeur de R ? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "Nous pourrions vouloir des valeurs de R ici. Cela étire les choses différemment, donc quand nous le mettons jusqu'à 2, vous savez, cela étend beaucoup plus l'axe réel de sorte que f de 1 finit autour de l'endroit où e au carré est un peu au-dessus de 7 f de négatif. 1 est beaucoup plus proche de 0 f de I est une rotation de 2 radians. La rotation autour du cercle f de négatif I est une rotation de moins 2 radians. Et bien sûr, nous pouvons revenir à notre formule préférée: si c'était pi, nous avions comme constante d'échelle alors l'axe réel s'étire beaucoup. Vous savez que f de 1 est assis en e par rapport au pi qui est très proche de 20 plus pi Ce qui est toujours amusant et f de moins 1 extrêmement proche de 0 donc c'est vraiment étiré ce réel axe Et il a également étiré les choses dans la direction du cercle unitaire de sorte que Arriver à f de I ou f de négatif I marche à mi-chemin autour du cercle, donc tout va bien maintenant. Comment penserions-nous à une fonction comme ? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "Nous écririons également comme X de X du logarithme naturel de 2 fois X donc nous déplaçons en quelque sorte notre point jaune représentant la valeur de R To autour de 0.69 toujours pas de partie imaginaire juste un vrai chiffre 0.69 environ C'est le log naturel de 2 et bien vous pouvez voir que f de 1 atterrit sur 2 C'est pourquoi nous voulons appeler cette fonction 2 au X f de 1 moitié en fait désolé f de moins 1 atterrit juste sur 1 moitié f de I C'est une promenade autour du cercle unitaire, très précisément, ça va être 0.69 radians autour du cercle unité et maintenant nous pourrions nous amuser un peu plus et dire ce qui se passerait si nous changeions cela à au lieu d'être 0.69 au lieu d'être le log naturel de 2, multipliez-le par le log naturel de 2 afin que nous pensions vraiment à quelque chose qui pourrait avoir une base exponentielle. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "Qu'est-ce que je suis à la puissance I dans ce cas, il le pousse à environ 0.2 autour d'un cinquième Mais il y a beaucoup de fonctions exponentielles différentes qui auraient cette propriété de mettre f de 1 sur le nombre I Donc si nous devions l'agrandir encore plus, je ne pense pas l'avoir animé ici Mais si nous devions prendre ce point jaune et soulevez-le jusqu'à ce qu'il atteigne 5 moitiés fois pi. Ce que vous verriez, c'est le cercle unité ? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "Est tourné sur lui-même de sorte que f de f négatif de 1 tournerait autour de 2 pi radians supplémentaires et atterrirait là où il se trouve. Mais cela étirerait beaucoup plus l'axe réel. Quel était le sens dans lequel une autre sortie de I vers I est un nombre beaucoup plus petit C'était autour de ce que c'était 0.0003 environ Mais nous pouvons aussi voir ce que je trouve assez amusant. Que se passe-t-il si nous considérons des expressions alternatives que nous voulons interpréter comme 2 à la puissance X, n'est-ce pas ? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "Nous avons X de R fois X et R est égal à cette valeur, qui est le logarithme népérien de 2 plus pi fois I. Cela signifie que lorsque nous connectons 1 f de 1 est à moins 2, nous voulons donc écrire cette fonction comme moins 2 à la puissance X, c'est vrai et c'est en fait quelque chose que vous savez, c'est un peu trompeusement simple quand on écrit un nombre négatif à une puissance Négatif 2 À la puissance X, ça ne ressemble pas nécessairement à ça au premier abord, ça nous amène dans les nombres complexes de quelque manière que ce soit, mais bien sûr, lorsque nous insérons même une valeur comme 1 moitié. Là où nous demandons en quelque sorte une racine carrée de moins 2, nous réalisons que nous voulons écrire cela comme quelque chose comme je multiplie la racine carrée de 2 Mais si vous regardiez cette fonction moins 2 à la puissance X dans le domaine complexe complet dont elle traite. Ce que vous regardez est une fonction qui prend la valeur de 1 à moins 2 Et si elle fait cela, qu'est-ce que cela a un effet sur le reste de la droite numérique réelle, est-ce en quelque sorte une spirale vers l'extérieur ? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "Nous voyons donc que f de moins 1 se situe à moins 1 moitié. À peu près là où vous vous attendriez si vous deviez suivre f de 1 moitié. Il se situerait exactement sur la ligne imaginaire et f de 1 moitié serait la racine carrée de 2. Eh bien, mon la souris n'est pas là où je veux qu'elle soit. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "Ce serait autour de la racine carrée de 2 fois I et à mesure que vous continuez plus loin, cela vous montre toutes les puissances de valeur réelle de moins 2 au X, cela tourne nécessairement en spirale. Mais nous pourrions également déplacer notre valeur de R encore plus haut et l'obtenir. jusqu'à environ tau fois I environ six virgule deux huit fois I et dans ce contexte, c'est une autre fonction que nous voudrions écrire comme quelque chose comme 2 en X car pour tout nombre entier à nombre entier que vous branchez pour X, cela le fera ça ressemble à une multiplication répétée Et il a même des valeurs raisonnables pour des choses comme 1 moitié où il crache la racine carrée négative au lieu d'une racine carrée positive, mais ce qu'il fait en réalité est une transformation vers le plan Où il met tout est le réel La droite numérique finit par être une spirale très serrée qui fait le tour et qui tourne en spirale de telle manière que f de 1 atterrit directement sur le nombre 2. C'est donc dans ce sens que nous pourrions dire que 2 au X est est plausiblement interprété comme une fonction exponentielle distincte de celle à laquelle nous sommes traditionnellement habitués. Donc je pense qu'avec tout cela, je vais laisser les choses pour aujourd'hui Et je vais juste vous laisser avec quelques questions persistantes auxquelles réfléchir, d'accord, donc si vous voulez pensez à I au I comme étant une expression à valeurs multiples, n'est-ce pas, pourriez-vous dire que nous adoptons une convention De manière fantaisiste, vous diriez que vous choisissez une branche de la fonction logarithme naturel Et peut-être que cela vous enferme dans cet être e au pi négatif moitiés Mais si vous dites que ce genre de veut avoir une infinité de valeurs différentes comme les différentes que nous avons vues, combien de valeurs 2 au tiers veulent-ils avoir dans le même sens ? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "Les dixièmes veulent être formulés différemment de toutes les fonctions exponentielles F de X qui satisfont, oh l'ai-je écrit quelque part f de X qui satisfait Toutes ces propriétés que j'ai écrites donc si elles satisfont toutes de ceux-ci et si f de 1 est égal à 2, combien de sorties différentes allons-nous obtenir lorsque nous branchons X est égal à 3 dixièmes pour les différentes options pour quelle fonction ? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "Pour 2 en pi pour les différentes fonctions que 2 en X pourrait représenter si nous considérons 2 en X comme une sorte de fonction exponentielle Exponentielle dans le sens de ce genre de propriétés abstraites et si nous oui, si nous si nous avons une classe de différentes fonctions de ce type, et nous voulons brancher pi, cela me fait rire. Juste parce que c'est tellement drôle, je connais une réponse qui apparaît lorsque vous essayez d'y penser, donc ce sont les questions qui Je vais vous laisser avec et je pense que c'est, vous savez, ma question centrale en abordant la conférence d'aujourd'hui était de savoir si je voulais que ce soit une sorte de description de ces propriétés abstraites des fonctions exponentielles. Et c'est juste cool pour moi qu'à partir de ces propriétés abstraites vous êtes enfermé dans l'idée de e au rx ou plus. Juste vous savez, je pense que c'est plus honnêtement écrit exp de r fois x pour différentes valeurs de r Cela vous enferme jusqu'à présent Mais cela ne vous enferme pas dans la mesure où une notion sans ambiguïté de ce que 2 à la puissance x devrait être beaucoup moins quelque chose comme I à la puissance x Le risque, bien sûr, est que parfois les gens n'aiment pas l'abstraction et parfois cela ne semble pas aussi accessible. Mais si c'est le cas au cas où vous savez, faites-le-moi savoir, je pense qu'il y a tout un cercle de pensées intéressant qui entoure tout cela pour inclure les tours de puissance parce que si vous voulez parler en fait de tours de puissance comme nous l'étions la dernière fois dans le contexte des nombres complexes ou même avec des bases négatives. Vous devez réfléchir à des choses comme ça, donc c'était une question que nous avions à l'écran. Ouais, que se passe-t-il si nous faisons cela pour moi au pouvoir de moi ? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "Le titrage, vous savez, essayons ça, essayons une tour de puissance où nous élevons I à une puissance donnée et voyons ce qui en ressort, donc ce n'était pas prévu de faire ça Mais nous pouvons, nous pouvons toujours ouvrez Python et faites essentiellement ce que nous faisions la dernière fois. Donc, la façon dont cela fonctionnerait est que nous commençons avec une valeur de base, puis avec une sorte de plage. Que faisions-nous, nous prenions un et nous allons réaffecter ce doit être n'importe quoi. La base qui dans ce cas est que j'ai élevée à la puissance a devrait être Ok, cool, donc nous allons faire ça et ensuite nous allons imprimer la valeur de a faisons juste ça pour Ouais, c'est un nombre beaucoup plus grand comme 200. Il semble donc que ce qui se passe soit qu'il y ait un potentiel de chaos avec ces choses, comme parfois. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "En fait, laissez-moi importer NumPy pour que la fonction exponentielle me laisse partir Pour notre grande gamme comme nous l'avions avant Plutôt que de l'écrire comme vous le savez, quelque chose qui me ressemble à la puissance X, je vais l'écrire comme fonction exponentielle d'une constante différente, c'est une constante différente que je vais créer. Je veux que ce soit des moitiés de 5 pi, donc je ferai 5 moitiés de pi fois I donc c'est un nombre complexe et il a 5 moitiés de pi comme partie imaginaire Donc c'est 5 pi moitié fois moi et qu'est-ce que je fais ? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/german/sentence_translations.json b/2020/ldm-i-to-i/german/sentence_translations.json index 89b9db67a..c25eb2992 100644 --- a/2020/ldm-i-to-i/german/sentence_translations.json +++ b/2020/ldm-i-to-i/german/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "Wenn Sie also bei der Zahl 1 beginnen, besteht Ihre Anfangsgeschwindigkeit darin, geradeaus in Richtung 0 zu gehen, und wenn Sie noch tiefer gehen, würden Sie, wenn Sie bei 1 Hälfte sitzen würden, immer noch in Richtung 0 gehen, aber jetzt Ihr Geschwindigkeitsvektor wäre dort, wo Sie sich befinden, 1 Mal negativ, also 1 Hälfte negativ. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "Und eine interessante Frage wird sein, ob es nur eine solche Funktion gibt, deren Schreiben sinnvoll erscheint, denn wenn wir sie als i an das x schreiben, sollte sie nicht nur dies erfüllen, sondern auch, Sie wissen schon, wann Wir stecken die Nummer eins, die wir bekommen, vermutlich i, in die Power eins, aber wir denken, dass diese Funktion i sein sollte. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "Wir haben also 5 Pi-Hälften, großartig, das ist absolut ein weiterer Wert, den wir hier für Moment, der für eine Zeitspanne gelaufen ist, die Pi-Hälften entspricht, also 1.57 Was wäre, wenn wir stattdessen eine weitere volle Umdrehung machen würden und weitere Pi-Hälften gehen würden, um zu Pi zu kommen, was, wie Sie wissen, wir vielleicht sozusagen aufzeichnen würden? Das ist, wo der e zum Pi-i-Wert ist: Wir gehen weitere Pi-Hälften, wir gehen weitere Pi-Hälften, was bei An diesem Punkt wären wir einen vollen Kreis gegangen und wären wieder bei eins angelangt, und dann hätten wir fünf Pi-Hälften zurückgelegt, was numerisch etwa 7 entspricht. 85 Ja, das ist auf jeden Fall eine weitere Zahl, die uns auf i bringt, und wenn wir den ganzen Trick durchgehen würden, i in die Potenz i umzuwandeln, indem wir zuerst e in die 5 Pi-Hälften von i in die Potenz i schreiben, dann wären das i Wenn wir multiplizieren, um negativ zu werden, würden wir e mit den negativen 5 Pi-Hälften vergleichen, was eine ganz andere Zahl ist. Richtig, wir können das tatsächlich berechnen. Ich bin mir auf Anhieb nicht sicher, aber werfen wir einen Blick auf einen Desmos . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "So lang, was zu einer viel kleineren Zahl führt. Aber das ist nicht die einzige Antwort, die wir richtig eingeben könnten. Es kommen auch andere Leute hierher mit negativen 3 Hälften mal i pi. Was wissen Sie in Bezug auf einen Einheitskreis? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "Wir könnten uns vorstellen, zu sagen: „Hey, wenn ich zu I kommen will, anstatt 90 Grad zu gehen, Pi halbiert das Bogenmaß in diese Richtung, was wäre, wenn ich 270 Grad in die andere Richtung gehe, 3 Pi halbiert das Bogenmaß, was ich vielleicht als negativ empfinde, weil die Konvention so ist.“ normalerweise ist gegen den Uhrzeigersinn positiv. Das ist auf jeden Fall eine andere Möglichkeit, es auszudrücken, und das würde uns eine andere Antwort geben, wenn wir e zu den negativen 3 Pi-Hälften i hätten. Hoch i, wir machen jetzt das gleiche Spiel, das i quadriert hebt sich mit a auf Negativ, das ist bereits da, und wir haben positive 3 Pi-Hälften und numerisch gesehen erhalten wir dadurch eine noch anders aussehende Antwort als zuvor. Wenn wir also darüber hinweggehen und sagen: „Hey, was ist e zu den 3 Pi, nicht zu 3 oder 3 Pi?“. halbiert 111 Punkt 3 1 eine ganz andere Art von Zahl als die, die wir vorher gesehen haben. 111 Punkt, was war das? 111 Punkt 3 1 großartig 111 Punkt 3 1 oder so. Und noch einmal in Bezug auf die Intuition, was Sie sich dort fragen könnten: Nehmen wir an, wir haben diese Rotation dynamisch Aber wenn wir uns in der Zeit rückwärts bewegen, sehen wir, wie lange in der Zeit zurückliegen muss, was ich sein muss. Wenn ich die Dinge von dort aus vorwärts spielen würde, würde ich auf der Nummer Eins landen, meinem Ausgangszustand, und Sie müssen in der Zeit um 3 Pi-Hälften-Einheiten zurückgehen Und wenn Sie es dann auf die Abklingdynamik übertragen würden, was in diesem Zusammenhang das Anheben ins Auge bedeutet, sagen Sie, wenn ich bei der Nummer eins anfange, aber ich möchte mich in der Zeit rückwärts bewegen und sagen: Wo hätte ich anfangen sollen, wenn? Ich möchte so verfallen, dass ich bei der Nummer eins lande? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "Nach 3 Pi-Hälften der Zeit beginnt die Antwort offensichtlich bei etwa einhundertelf für diese Art von exponentiellem Abfall. Und Sie können sehen, wohin das führt, wo es tatsächlich unendlich viele verschiedene Werte gibt, die wir für X einsetzen könnten, wenn wir es wären Ich stelle mir vor, dass e zum Verschiedene Optionen, unendlich viele verschiedene Werte, was zunächst ein wenig beunruhigend wirkt, weil wir uns einen Ausdruck ansehen. Das scheint, als wüssten Sie, dass es einfach eine Berechnung geben wird. Ich stecke das einfach in meinen Taschenrechner und schaue, was herausspringt, und wir haben mehrere verschiedene Werte dafür Also, was ist hier los, oder? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "Die vierte Wurzel aus 16 sollte 2 sein und die Antwort lautet am Ende gut. Wir übernehmen eine Konvention, wenn es mehrere Optionen wie diese gibt, wenn Sie eine mehrwertige Funktion haben. Oft wählen wir einfach einen dieser Werte als das aus, was wir meinen, wenn wir wollen Behandeln Sie es als eine Funktion, als etwas mit einer einzigen Eingabe und einer einzigen Ausgabe, in ausgefallenerem Jargon. Wenn wir es mit komplexen Zahlen zu tun haben, kommt das immer wieder vor. Die Idee, dass etwas eine Art Operation ist, die mehrere Werte haben möchte, kommt manchmal vor Hören Sie den Ausdruck Zweig. Wo wählen Sie einen Zweig der Quadratwurzelfunktion? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "Weil es mehrere unterschiedliche Antworten gibt. Wissen Sie, wir denken wieder an „I“, das ist diese 90-Grad-Drehung. Und wenn wir es uns als 90-Grad-Drehung vorstellen, fühlt es sich an, als ob die Quadratwurzel sein sollte. Sie wissen, dass etwas in einem 45-Grad-Winkel sitzt. Vielleicht ist das das Quadrat Wurzel von I, die wir ganz explizit als Wurzel 2 über 2 Wurzel 2 über 2 I schreiben könnten. Das ist nur trigonometrisch, aber wenn wir uns I stattdessen als negative 270-Grad-Rotation vorstellen würden, fühlt es sich an, als ob die Hälfte davon die Hälfte dieser Operation ausführt sollte uns eigentlich auf die andere Seite bringen. Vielleicht sollte die Zahl, die hier unten sitzt, die Quadratwurzel von I sein und das ist eigentlich nur das Negativ von dem, was wir vorher gesehen haben. Negative Wurzel 2 über 2 minus Wurzel 2 über 2 mal I Jetzt im realen Kontext Bei den geschätzten Funktionen können wir sagen: „Ja“ Wählen Sie einfach die Quadratwurzel als die positive Antwort, aber welche davon halten Sie für die positive Antwort? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "Und ich denke, Sie sagen: Nun ja, wir wissen, was das ist, wir definieren es sozusagen als die Quadratwurzel von 2, alles ist schön und gut. Aber was wäre, wenn ich sagen würde, gehen wir das auf die gleiche Weise an, wie wir unser Ich dem I-Ausdruck I näherten Ich möchte die Dinge zuerst als e für das Richtige ausdrücken und dann erhöhe ich das auf die 1 Hälfte, indem ich die 1 Hälfte mit dem Exponenten multipliziere. Und ich sage „Okay, ich kann, ich schätze, ich kann das e mit dem machen, was ist.“ gleich 2, also das ist der natürliche Logarithmus von 2. Es ist eine Konstante, die bei etwa 0 liegt. 69 oder so Wenn wir e mit dieser Potenz erhöhen, erhalten wir 2, also könnten wir uns das als e hoch zum natürlichen Logarithmus von 2 mal 1 Hälfte vorstellen, und wenn Sie wollten, wenn Sie an e hoch x denken würden? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "Sie wissen, dass dies im Zusammenhang mit reellen Zahlen etwas übertrieben sein könnte. Wenn Sie sich jedoch e für x als Abkürzung für diese x-Funktion vorstellen, könnten Sie den Wert 0 eingeben. 69 mal 1 Hälfte, was meiner Meinung nach ungefähr 0 sein würde. 345 Ist so etwas in der Art. Sie fügen diesen sehr konkreten Wert in Ihr Polynom ein und sehen, was es ausgibt, und es wird ungefähr 1 ausgeben. 414 ist eine schöne reelle Zahl, die Quadratwurzel von 2, was man erwarten würde. Aber wenn wir das Gleiche tun würden, was wir gerade mit I gemacht haben, und wenn wir anerkennen, dass es tatsächlich mehrere unterschiedliche Antworten gibt, wenn wir etwas als e in eine Potenz schreiben wollen, könnten wir auch dies schreiben Das mag lustig klingen, aber wir könnten es als e zum natürlichen Logarithmus von 2 plus 2 pi schreiben natürlicher Logarithmus von 2, multipliziert mit e zu 2 pi I. Dies hat nur den Effekt, dass sich die Dinge um 360 Grad drehen, also wird es einfach gleich 1 sein. Wir sehen uns also 2 mal 1 an, großartig, das fühlt sich wie eine gültige Substitution an und doch wann Wir spielen das gleiche Spiel: Wir nehmen dies, potenzieren es und behandeln das, indem wir die Potenz mit dem Exponenten multiplizieren. Schauen Sie sich an, was passiert. Wir haben e zum natürlichen Logarithmus von 2 mal 1 Hälfte plus. Nun, was ist 2 Pi I mal 1 Hälfte Nun, das wird Pi mal I sein. Nun dieser erste Teil e zum natürlichen Logarithmus von 2 mal 1 Hälfte, der am Ende die bekannte Quadratwurzel von 2 sein wird, das ist alles schön und gut, aber wir werden das mit e multiplizieren das pi I Richtig und bekanntlich ist e zum pi I negativ 1. In diesem Fall scheint es also darauf hinzudeuten, dass wir, wenn wir diesen Ausdruck 2 zur 1. Hälfte auflösen, indem wir mit den verschiedenen Antworten herumspielen, etwas wie hinzufügen könnten e zum Sagen wir, das ist nicht gleichbedeutend mit einer Sache, aber basierend auf den Entscheidungen, die wir treffen, könnte es mehreren verschiedenen Dingen entsprechen. Aber die beiden Dinge könnten durchaus vernünftig erscheinen. Wenn es irgendetwas geben wird, das 2 zu 1 Hälfte ist, dann scheint es so, als ob es entweder das Positive sein sollte Quadratwurzel, mit der wir vertraut sind, oder die negative Variante davon, die eigentlich kein solches Problem zu sein scheint. Und tatsächlich könnten wir, ähm, wir könnten dieses Spiel noch weiter spielen, wo ich Sie um noch kreativere Antworten auf diesen Ausdruck bitten möchte denn vielleicht können wir andere lustige Potenzen von etwa 2 hoch X finden, wenn wir beginnen, verschiedene Werte von Potenz I Diesmal wird also die Frage gestellt oder angegeben, dass eine Lösung der Gleichung e für x gleich 2 die reelle Zahl ist. Natürlicher Logarithmus von 2, ok, das wissen wir. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "Antwort auf die Frage e zum x gleich 2 und Wieder ist Kreativität willkommen, also gebe ich Ihnen noch einen kleinen Moment dafür. Ich werde hier einige Antworten eintragen, wenn das für Sie in Ordnung ist. Ich bin mir nicht sicher, wie viel Zeit es kostet Es ist notwendig, die mathematische Eingabe vorzunehmen, je nachdem, welches Gerät Sie gerade betrachten. Seien Sie jedoch nicht zu gestresst, wenn Sie noch nicht die Möglichkeit haben, die Frage, die Sie beantworten möchten, in die Antwort umzuwandeln, die Sie beantworten möchten. So sieht es aus 131 von Ihnen haben die Variante eingegeben, bei der wir Ln von 2 nehmen und 2ii addieren, und ich schätze, ich habe beim Schreiben dieser Frage fälschlicherweise eine der Antworten als richtig markiert, obwohl es tatsächlich eine ganze Reihe verschiedener richtiger Antworten gibt. Das liegt also an mir für die Tatsache, dass ich nicht weiß, ob es für einen von euch so aussieht: Oh, es ist rot, du hast es falsch verstanden, als du Ln von 2 plus 42 eingegeben hast. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi, was natürlich eine gute Wahl ist. Sie könnten aber auch so etwas wie 4 pi I plus den natürlichen Logarithmus von 2 oder 6 pi I oder wirklich jedes ganzzahlige Vielfache von 2 pi I verwenden, wenn Sie hinzufügen, dass es keinen Einfluss auf e hat X Weil es nur den Effekt hat, mit e zu 2 pi I zu multiplizieren. Das ist der Effekt der Multiplikation mit 1, und das hat wiederum eine komische Konsequenz, dass es scheinbar vernünftige Ergebnisse liefert, wenn wir es als weiteres Beispiel machen Es sieht so aus, als ob der am zweithäufigsten eingegebene Ausdruck dort war, dass wir 2 ersetzen könnten. Nehmen wir also an, wir denken an 2 hoch 1 4. Okay, es gab einen Vorschlag, dass wir 2 durch e zum natürlichen Logarithmus von 2 plus 4 ersetzen pi I Okay Plus 4 pi I und wir erhöhen das alles auf 1 4. Nun, wenn Sie das gleiche Spiel spielen würden, würden Sie e zum natürlichen Logarithmus von 2 mal 1 4 erhalten, und wir würden mit e multiplizieren Der Pi I. Nun wird der erste Teil davon die übliche positive vierte Wurzel aus 2 sein, was wir meinen, wenn Sie einen Ausdruck wie die vierte Wurzel aus 2 in einen Taschenrechner eingeben, eine schöne kleine positive Zahl, aber dann ist dieser zweite Teil Negativ 1, also scheint es zu heißen: „Wissen Sie, wenn wir 2 auf diese andere Weise interpretieren würden, indem wir es auf 1 4 erhöhen“. Sie wissen, dass es nicht die übliche Antwort ist, die wir bekommen, aber es ist eine vernünftige Antwort. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "Wir hätten Pi-Hälften mal I betrachtet und anstatt mit Negativ 1 zu multiplizieren, hätten wir stattdessen mit I multipliziert. Was wiederum eine gültige Antwort ist, scheint eine vernünftige Ausgabe für etwa 2 hoch 1 4 zu sein. Wenn Sie also sind Wenn man sich die Tatsache ansieht, dass I hoch I mehrere unterschiedliche Werte dafür zu haben scheint. Richtig, wir haben dieses lustige Phänomen, bei dem wir e mit den 5 Pi-Hälften I verbinden könnten. Negativ 3 Pi-Hälften I, und wir bekommen scheinbar völlig unterschiedliche Antworten Etwas sehr Kleines, etwas sehr Großes, alles ganz anders als die 1/5-Antwort, die wir zuvor hier oben gefunden haben. Es ist genau das gleiche Phänomen, als wenn man fragt, was 2 bis 1/4 ist, und anerkennt, dass es tatsächlich mehrere verschiedene Lösungen gibt zum Ausdruck 2 Was auch immer es sein mag und eine Möglichkeit, wie wir darüber nachdenken könnten, ist, dass, wenn man es mit reellen Zahlen zu tun hat, die Dinge einfach schön sind, die Dinge sind schön. Es gibt Eins-zu-eins-Beziehungen. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "Es ist großartig. Wenn wir über Exponentialfunktionen nachdenken wollen, lassen Sie mich einfach einige davon besprechen. Wir haben dieses nette Hin und Her, bei dem Sie wählen können, ob Sie eine beliebige Exponentialfunktion als Basis für X ausdrücken möchten, z. B. 2 für das die gleiche Exponentialfunktion wie X von R mal Und es gibt Ihnen eine Antwort unter der Annahme, dass B eine positive Zahl ist. Und das ist das Gleiche, als würde man sagen, dass X von R gleich B ist Familie aller möglichen Exponentialfunktionen, richtig, wir könnten sie als X von R mal X schreiben und ändern, was R ist. Und das ist genau das Gleiche, als würde man e zu R mal mal XX von R mal um zu ändern, was diese Basis ist. Zuerst fühlt es sich so an, als wäre das eine andere Art von Ausdruck, den man manipulieren kann, aber es ist nur eine andere Art, dieselbe Familie auszudrücken. Und eine Art und Weise, wie Sie darüber nachdenken könnten. Denn wie denken wir darüber nach, welcher Basis sie entspricht? Wenn wir etwas abstrakter denken als Exp von R mal Anstatt auf diese Basis zu schauen, könnte ich sagen, was ist der Wert? ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "Ich könnte exp von R mal Verschieben Sie es um zwei pi I nach oben, wodurch sich die Basis, der es entspricht, nicht ändert, denn in all diesen Fällen erhalten wir dasselbe, wenn wir Warum haben wir mehrere unterschiedliche Werte für I hoch I gesehen? Da I hoch X eine mehrdeutige Funktion in diesem Zusammenhang ist, wäre es eindeutig, wenn wir entscheiden würden, welcher Wert von R so ist, dass wir exp von R mal X darstellen, welcher Wert von R. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "Es ist eine eindeutige Funktion, aber an diesem Punkt fühlt es sich einfach so an, als ob das, was wir vielleicht wollen, darin besteht, nicht mehr über die Dinge auf einer Basis nachzudenken, die auf die Potenz X erhöht ist. Vielleicht sollten wir einfach schreiben, sobald wir uns im Kontext komplexer Zahlen befinden Sie alle als Exp einiger konstanter Zeiten Stecken Sie sie ein und ich werde Ihnen noch einmal beweisen, dass dies vielleicht die richtige Art ist, über Exponentialfunktionen nachzudenken. Sobald wir in andere Bereiche vordringen, Dinge wie komplexe Zahlen, und dafür lasst uns einfach ein Backup erstellen Zurück zur Türklingel. Einige Dinge sind angekommen. Gehen Sie zurück zum Original. So erweitern wir die Idee der Potenzierung und denken einfach darüber nach, was 2 rechts von X ist. Wir wissen, wie wir darüber für natürliche Zahlen nachdenken müssen. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "Sie kennen so etwas wie 2 hoch 3. Wiederholte Multiplikation. Wie kommt es, dass Ihnen zunächst beigebracht wird, über so etwas wie 2 hoch X für gebrochene Beträge oder für negative Beträge und ähnliches nachzudenken? Also. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "Normalerweise wird einem beigebracht, dass 2 bis 1 Hälfte etwas sein sollte, bei dem man weiß, wenn ich es mit sich selbst multipliziere, und dies folgt den üblichen Regeln, die Exponentialfunktionen beim Zählen von Zahlen anwenden, wobei wir in der Lage sind, Dinge in diesem Exponenten zu addieren, sollte ich 2 erhalten zur 1, also sollte es eine Zahl sein, die ich, wenn ich sie mit sich selbst multipliziere, 2 erhalte, und Sie wissen, dass Sie an diesem Punkt eine Wahl haben, vielleicht ist sie positiv. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "Vielleicht ist es negativ. Aber wenn Sie sich immer für die positive Wahl entscheiden, können Sie aus demselben Deal eine schöne kontinuierliche Funktion erhalten, wenn wir nach negativen Zahlen fragen. Was sollte 2 zur negativen 1 sein? Nun, das sollte etwas sein Wohin, wenn ich es mit 2 mit 1 multipliziere? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "Es bringt mir 2 zur 0 und das ist eine Art Rechtfertigung für unsere Konvention, dass negative Exponenten wie 1 Halbe aussehen. Aber was hier wirklich vor sich geht, ist, dass wir sagen, was auch immer das ist, es sollte eine Art Funktion sein, die diese Eigenschaft f von erfüllt a plus b ist gleich f von a mal f von b und außerdem sagt uns die Tatsache, dass die Basis 2 ist, im Grunde, dass es sich nicht um irgendeine solche Funktion handelt. Es ist eine Funktion, bei der wir, wenn wir 1 einsetzen, 2 erhalten. Und genauso wenig wissen Sie Überprüfen Sie den Stil Ihrer geistigen Gesundheit, um zu sehen, ob Sie einigen der Implikationen hier folgen. Ich möchte Sie fragen, was das ist. Ich werde es nicht als Softball bezeichnen, aber das soll nicht so sein. Eine unglaublich tiefgründige Frage Notwendig. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "Es ist eher eine Überprüfung, wenn Sie der Idee folgen, abstrakt mit den Eigenschaften einer Funktion zu beginnen und dann Wege abzuleiten, wie wir sie auf der Grundlage dieser Eigenschaften aufschreiben möchten. Wenn f von x diese Exponentialeigenschaft f erfüllt von a plus b ist gleich f von a mal f von b für alle Eingaben. Und es erfüllt auch f von 1 gleich 2, welche der folgenden Aussagen wahr ist. Das heißt, welche der folgenden Aussagen ist notwendigerweise wahr. Egal welche Funktion Sie starten mit und diejenigen unter Ihnen, die sich erinnern, um welche Vorlesung es sich handelte. Es kommt darauf an, über welche Vorlesung wir gesprochen haben, wie zu interpretieren ist, was Eulers Formel wirklich sagt. Ich habe eine Frage dieser Art gestellt, bei der ich eine einzige Bedingung vernachlässigt habe, Sie wissen, dass ich sie nicht aufgeschrieben habe Die Tatsache, dass wir sicherstellen wollen, dass f von Etwas, das eine Addition in eine Multiplikation umwandelt, reicht aus, um im Grunde den Wunsch zu wecken, die Funktion als das zu schreiben, was auch immer sie einer Potenz entspricht. Das ist der Sinn der Frage. Jetzt haben wir ein paar Fragen zu Krafttürmen Das scheint hier aufgetaucht zu sein, was einen tollen Zusammenhang mit dem letzten Mal hat. Lasst uns mit der Power-Tower-Frage einen Moment innehalten, damit wir zunächst ein tieferes Gefühl dafür bekommen, was Potenzierung hier bedeuten sollte? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "Denn weil wir das sein können, was ich behaupten möchte, können wir es auf mehrere verschiedene Arten beantworten. Wenn Sie mir also nur eine geben, sprechen wir über Stromtürme. Und dann genauso, wie eine Zahlenlinie in einer logarithmischen Skala dargestellt werden kann Kann man dasselbe auch für eine komplexe Ebene tun? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "Ja, tatsächlich gibt es eine Visualisierung, auf die ich gleich noch eingehen werde, in der wir etwas ganz Ähnliches tun. Denn was wir tun werden, ist, mit verschiedenen Exponentialfunktionen X von R mal X herumzuspielen. Aber das tun wir Ich werde den Wert von R ändern, der durch einen kleinen gelben Punkt dargestellt wird. Also werden wir darüber reden. Es wird nicht die gesamte Ebene abgebildet, sondern nur ein paar Beispielpunkte von der realen Achse und der imaginären Achse Aber die Idee ist, dass wir, während wir uns um diese Konstante bewegen, in der Lage sein werden, die verschiedenen Dinge, die sie mit der Ebene macht, irgendwie zu visualisieren, und im Grunde ist es so, als würde die x-Achse in eine logarithmische Skala umgewandelt und dann umbrochen die imaginäre Achse entlang eines Kreises. Und sobald dieser Wert von R imaginär wird, vertauscht er die Rolle dieser reellen Zahlen auf dem Kreis und imaginäre Zahlen auf einer logarithmisch skalierten positiven Achse, also eine tolle Frage, alle drei, denke ich sind sozusagen voreilig, um dorthin zu gelangen, wo ich hin will. Aber es ist schön zu sehen, dass die Leute hier so denken. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "Explizit so etwas wie f von 5 ist dasselbe wie f von 1 plus 1 plus 1 plus 1 plus 1, was aufgrund dieser Eigenschaft dasselbe ist wie f von 1, multipliziert mit sich selbst fünfmal, was dasselbe ist, wenn f von 1 gleich 2 ist als 2 hoch 5 und dann so etwas wie f von minus 5. Es sollte so sein, dass wir, wenn wir es mit f von 5 multiplizieren, erhalten, was auch immer f von 0 ist, und es ist nicht sofort klar, was f von 0 ist, aber wir könnten das sagen f von 1 plus 0 ist gleich dem, was auch immer f von 1 mal dem ist, was f von 0 ist, aber f von 1 ist gleich 2. Und das ist also auch gleich 2, also sagen wir, dass 2 gleich 2 mal etwas ist, also etwas muss eine 1 sein, also garantiert dies in diesem Zusammenhang, dass f von minus 5 2 zu minus 5 ist, also 1 über 2 zu 5. Wir könnten dies explizit als 2 zu minus 5 schreiben, was bedeutet, dass diese beiden Eigenschaften zusammengenommen ausmachen Wir möchten die Funktion wirklich als 2 auf das X schreiben, weil jede Zählzahl, die wir eingeben, diese Eigenschaften erfüllen wird Das wollten wir. Und Sie fragen sich vielleicht, ob das einzigartig ist und dass es im Zusammenhang mit reellwertigen Funktionen tatsächlich so wäre. Aber im Zusammenhang mit komplexwertigen Funktionen gäbe es mehrere solcher Funktionen, für die wir schreiben könnten. Eine davon waren wir Schauen wir uns vorher an, wo wir eine Funktion definieren könnten, die exp des natürlichen Logarithmus von 2 plus 2 pi ist Dies wird durch das bewiesen, was passiert, wenn Sie 2, aber ich multipliziere die vierte Wurzel von 2, also ist es eine andere Funktion. Aber es erfüllt immer noch diese Eigenschaften und es bringt uns irgendwie dazu, es als 2 zum X zu schreiben. Und es lässt darauf schließen, dass 2 zum X vielleicht eine Mehrdeutigkeit ist Ein bisschen Notation Und wir sollten einfach alles in Form von exp mal R mal etwas schreiben, aber Sie fragen sich vielleicht, vielleicht sind wir bei all den Funktionen, die diese Eigenschaft erfüllen, einfach nicht kreativ genug. Vielleicht gibt es eine Mehrdeutigkeit, wenn wir exp schreiben von R mal etwas und es gibt verschiedene Werte von R, die ins Spiel kommen könnten. Aber ich werde nur eine kleine Behauptung aufstellen und dann vielleicht eine Skizze geben, wie der Beweis aussehen würde, wenn Sie möchten Angenommen, Sie haben eine komplexe Funktion F, die zunächst die folgenden Eigenschaften erfüllt. Sie können eine Ableitung davon bilden. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "Es ist differenzierbar, was es nur davon abhält, irgendetwas völlig chaotisches, diskontinuierliches Ding zu sein. Das ist so, als würde man zufällige Werte annehmen, abhängig davon, dass man die Spanne eines beliebigen Vektorraums kennt. Ich kenne keine Bruchbeträge, die man sich vielleicht auf verrückte Weise vorstellen möchte. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "Es ist eine schöne Funktion. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "Das ist differenzierbar. Es ist nicht überall gleich 0, also ist mir die Bedingung irgendwie entfallen, und ich habe vergessen, für welche Vorlesung Vorlesung oder so etwas in der Art, und dann hat es diese zentrale Eigenschaft, dass es Addition in Multiplikation umwandelt. Wenn Sie eine solche Funktion haben, behaupte ich das Es gibt eine eindeutige Zahl. Vielleicht sollte ich wirklich angeben, dass es eine eindeutige komplexe Zahl R gibt, sodass Sie F von Unendliches Polynom mit schönen Ableitungseigenschaften und all das. Wenn Sie das haben, haben Sie jedes Exponential, das Sie wollen, in einer sehr abstrakten allgemeinen Bedeutung des Wortes Exponential, nur basierend auf einer Eigenschaft, die wir von ihm erwarten könnten, und der Skizze des Beweises Sehen Sie etwa so aus, wenn Sie sich zunächst ansehen möchten, was die Ableitung dieses Werts ist, von dem wir annehmen, dass er überall existiert, oder? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "Wir können F von Nur eine Konstante, aber genauer gesagt, es ist die Ableitung unserer Funktion bei 0. Sie haben also diese lustige Sache: Wenn Sie ihre Ableitung bei 0 kennen, bestimmt das, was ihre Ableitung überall ist. Und im Kontext von Exponentialfunktionen ist das hoffentlich ziemlich vertraut, weil Alles, was wir wirklich sagen, ist, dass die Ableitung einer Exponentialfunktion proportional zu sich selbst ist und dass die Proportionalitätskonstante gleich der Ableitung bei 0 ist. Das ist alles sehr abstrakt formuliert und so, aber der Zweck besteht darin, zu betonen, dass es so ist Es handelt sich nicht unbedingt nur um Funktionen, die wir uns bereits als hoch zweite Ableitung Und übrigens eine dritte Ableitung und so, weil die Ableitungsfunktion einfach proportional zu sich selbst ist. Um also die n-te Ableitung zu bilden, schauen Sie sich einfach diese Proportionalitätskonstante an und erhöhen sie auf die Potenz n, und dann könnten Sie von hier aus a tun Die Erweiterung der Taylor-Reihen und das überlasse ich vielleicht als eine Art Hausaufgabe für Fortgeschrittene für diejenigen unter Ihnen, die mit Taylor-Reihen in dieser Idee vertraut sind, insbesondere wenn Sie die Idee einer differenzierbaren Funktion vermischen möchten, die im Sinne komplexer Zahlen differenzierbar ist Eine Art definitives College-Thema. Sie wissen, dass Sie die Argumentation dort nach Belieben mischen können. Aber Fuzzy-Argumentation ist im Kontext von jemandem zulässig, der sich nur mit Taylor-Reihen und nichts anderem auskennt und diese Idee aufnimmt und sich die Taylor-Erweiterung für F und ansieht Begründen Sie irgendwie die Idee, dass es eine eindeutige komplexe Zahl gibt, so dass unsere Funktion F notwendigerweise so geschrieben werden kann. Und dann besteht die Verbindung zu normalen Exponentialzahlen immer dann, wenn Sie einen solchen Wert R haben. Wir tun im Wesentlichen das, was wir im komplexen Kontext reeller Zahlen tun ist, wenn Sie exp dieser Funktion mit diesem Wert R betrachten und dies als Basis schreiben. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "Wir könnten das so interpretieren, dass es nicht nur exp von Pi-Hälften I mal X bedeutet, sondern wir könnten es auch so interpretieren, dass exp von 5 Pi-Hälften I mal Schreiben Sie sie als I zum Mit der Notation, die wir haben, ist das ein wenig mehrdeutig. Nach alledem fangen wir einfach damit an, einiges davon zu visualisieren, weil ich denke, dass es Spaß macht. Und Sie wissen, dass Sie mir sagen, ob das hilfreich ist oder ob es ein verwirrenderes Bild ist, aber Was wir tun werden, ist, uns diese Funktion exp von R mal weil ich vorhatte, das zu planen, also lass mich oh ja, da bist du, geh zurück in mein Dateisystem, geh zurück dorthin, wo du sein solltest Oh ersetzen, es wird auf dem anderen Bildschirm angezeigt. Warte, warum ist es ja, okay, ersetzen? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "Platzieren Sie dort, was auch immer Sie sehen. Und jetzt gehen wir zurück zu „Oh, da haben wir das alles, nur damit ich es schön hätte aufschreiben können. Wenn Sie sich nicht wohl dabei fühlen, es sich als exp von R mal X vorzustellen, ist dieses unendliche Polynom einfach in der Hinterkopf e zum R mal Das geht alles ziemlich schnell, also lassen Sie mich das Ganze etwas langsamer durchdenken. Alle negativen Zahlen, irgendetwas, das eine negative reelle Zahl ist, wird in den Bereich zwischen 0 und 1 gequetscht. Was sollte einen Sinn ergeben, ins Negative? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a zu einer negativen reellen Zahl ist irgendetwas zwischen 0 und 1 und wir verfolgen speziell f von minus 1, das ungefähr bei 1 über e bei etwa 30 0 angezeigt wird. 37 f von 1 landet erwartungsgemäß auf e. Das ist exp von 1. f von I wird einen Bogenmaß um den Einheitskreis landen, und es macht Spaß, hier entlang der gesamten imaginären Achse zu verfolgen, wie die imaginäre Achse um einen Kreis gewickelt wird und Was passiert, wenn wir diesen Wert von R anpassen? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "Wir möchten hier vielleicht die Werte von R. Es dehnt die Dinge anders aus. Wenn wir es also auf 2 setzen, streckt es die reale Achse viel mehr, so dass f von 1 ungefähr dort landet, wo e zum Quadrat etwas über 7 f negativ ist 1 ist viel näher an 0 f von I ist eine 2-Radiant-Rotation um den Kreis f von negativem I ist eine negative 2-Radiant-Rotation Und natürlich können wir zu unserer Lieblingsformel kommen, dass wir, wenn das Pi wäre, unsere Skalierungskonstante hätten Die reale Achse wird ziemlich stark gedehnt. Sie wissen, dass f von 1 bei e zum Pi liegt, was sehr nahe bei 20 plus pi liegt, was immer Spaß macht, und f von negativ 1 extrem nahe bei 0, also ist es wirklich so real gedehnt Achse Und es werden die Dinge auch in Richtung des Einheitskreises gestreckt, so dass man, um zu f von I oder f von negativ I zu gelangen, die Hälfte des Kreises umrunden muss, also ist das jetzt alles schön und gut. Wie würden wir über eine Funktion wie denken? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "Wir würden auch als X von X des natürlichen Logarithmus von 2 mal 69 ist immer noch kein Imaginärteil, sondern nur eine reelle Zahl 0.69 oder so Das ist der natürliche Logarithmus von 2. Nun, Sie können sehen, dass f von 1 auf 2 landet. Deshalb wollen wir diese Funktion 2 zum I Es ist ein Spaziergang um den Einheitskreis, ganz konkret, dass es 0 sein wird. 69 Bogenmaß um den Einheitskreis und jetzt könnten wir etwas mehr Spaß haben und sagen, was passieren würde, wenn wir dies auf statt auf 0 ändern würden. 69 Anstatt der natürliche Logarithmus von 2 zu sein, multipliziere ich den natürlichen Logarithmus von 2, damit wir wirklich an etwas denken, das eine exponentielle Basis haben könnte. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "Was ist ich mit der Potenz, die ich in diesem Fall auf etwa 0 schiebe? 2 etwa ein Fünftel. Aber es gibt viele verschiedene Exponentialfunktionen, die diese Eigenschaft hätten, f von 1 auf die Zahl I zu setzen. Wenn wir es also noch weiter vergrößern würden, glaube ich nicht, dass ich es hier animiert habe. Aber wenn wir es nehmen würden diesen gelben Punkt und heben Sie ihn an, bis er 5 Hälften mal Pi I erreicht. Was Sie sehen würden, ist das der Einheitskreis? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "Wird um sich selbst gedreht, so dass sich f von negativ f von 1 um weitere 2 Pi im Bogenmaß dreht und dort landet, wo es ist. Aber es würde die reale Achse viel mehr ausdehnen. Das war der Sinn, in dem eine weitere Ausgabe von I zu I erfolgt eine viel, viel kleinere Zahl. Sie lag ungefähr bei 0.0003 oder so Aber wir können auch sehen, was meiner Meinung nach ziemlich lustig ist. Was passiert, wenn wir alternative Ausdrücke in Betracht ziehen, die wir als 2 hoch X interpretieren möchten, oder? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "Wir haben X von R mal als negativ 2 hoch in die komplexen Zahlen auf irgendeine Weise, aber natürlich, wenn wir auch nur einen Wert wie 1 halbe Zahl einsetzen. Wo wir sozusagen nach einer Quadratwurzel aus minus 2 fragen, wird uns klar, dass wir das so schreiben wollen wie etwa I mal die Quadratwurzel von 2 Aber wenn Sie sich diese Funktion negativ 2 hoch X in dem gesamten komplexen Bereich ansehen würden, mit dem sie es zu tun hat, dann sehen Sie eine Funktion, die den Wert 1 negativ 2 annimmt. Und wenn sie das tut, was? Es wirkt sich auf den Rest der reellen Zahlenlinie aus, ist es eine Art Spirale nach außen? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "Wir sehen also, dass f von minus 1 bei minus 1 Hälfte liegt. Ungefähr dort, wo man es erwarten würde, wenn man f von 1 Hälfte folgen würde. Es würde genau auf der imaginären Linie liegen und f von 1 Hälfte wäre die Quadratwurzel von 2. Nun, meine Güte Die Maus ist nicht dort, wo ich sie haben möchte. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "Es wäre ungefähr die Quadratwurzel von 2 mal I, und wenn Sie weitermachen, zeigt Ihnen dies alle realen Wertpotenzen von minus 2 zum bis etwa Tau mal I, etwa sechs Komma zwei acht mal I, und in diesem Zusammenhang ist dies eine weitere Funktion, die wir als etwa 2 an das Es sieht aus wie eine wiederholte Multiplikation. Und es gibt sogar einigermaßen vernünftige Werte für Dinge wie 1 Hälfte, wo es die negative Quadratwurzel anstelle einer positiven Quadratwurzel ausspuckt, aber was es tatsächlich tut, ist eine Transformation in die Ebene, in die alles eingefügt wird, ist das Reale Die Zahlenlinie ist am Ende eine sehr eng gewundene Spirale, die herumläuft und sich einfach so dreht, dass f von 1 genau auf der Zahl 2 landet. In diesem Sinne könnten wir also sagen, dass 2 zum X ist. Wird plausibel interpretiert als eine andere Exponentialfunktion als die, an die wir traditionell gewöhnt sind. Ich denke also, dass ich nach alledem die Dinge für heute aufheben werde. Und ich lasse Sie einfach mit ein paar offenen Fragen zurück, über die Sie nachdenken müssen, okay, also wenn Sie möchten Stellen Sie sich „I“ zu „I“ als einen mehrwertigen Ausdruck vor. Richtig, man könnte sagen, wir übernehmen eine Konvention. Fantasievoll würden Sie sagen, Sie wählen einen Zweig der natürlichen Logarithmusfunktion. Und vielleicht bindet Sie das daran, dass e zum negativen Pi ist Hälften Aber wenn Sie sagen, dass diese Art von unendlich vielen verschiedenen Werten sein möchte, wie die verschiedenen, die wir gesehen haben, wie viele Werte möchten 2 bis 1 Drittel im gleichen Sinne sein? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "Wo wir 2 durch verschiedene verschiedene Optionen für e zum X ersetzen, so dass e zum Zehntel wollen von allen, sagen wir mal, von allen Exponentialfunktionen F von davon und wenn f von 1 gleich 2 ist. Richtig, wie viele verschiedene Ausgaben erhalten wir, wenn wir X gleich 3 Zehntel für die verschiedenen Optionen für welche Funktion einsetzen? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "Für 2 hoch pi für die verschiedenen Funktionen, die 2 hoch X darstellen könnten, wenn wir uns 2 hoch X als eine Art Exponentialfunktion vorstellen Wir haben eine Klasse verschiedener solcher Funktionen, und wir möchten Pi einbinden. Das bringt mich zum Lachen. Nur weil es so eine lustige Antwort ist, die einem auffällt, wenn man darüber nachdenkt. Das sind also die Fragen, die gestellt werden Ich überlasse es Ihnen und ich denke, das ist meine zentrale Frage bei der Herangehensweise an die heutige Vorlesung war, ob ich wollte, dass sie so etwas wie diese abstrakten Eigenschaften von Exponentialfunktionen beschreibt. Und es ist einfach cool für mich, dass ich von diesen abstrakten Eigenschaften ausgehe man wird an die Idee von e zu rx oder mehr gebunden. Nur wissen Sie, ich glaube, ehrlicher geschriebenes exp von r mal x für verschiedene Werte von r eine eindeutige Vorstellung davon, was 2 hoch x sein sollte, geschweige denn so etwas wie I hoch x. Das Risiko dabei besteht natürlich darin, dass Menschen manchmal Abstraktion nicht mögen und manchmal wirkt sie nicht so zugänglich. Aber wenn das so ist Falls Sie es wissen, lassen Sie es mich einfach wissen. Ich denke, es gibt einen ganzen interessanten Gedankenkreis, der all diese Dinge umgibt, einschließlich Stromtürmen, denn wenn Sie tatsächlich über Stromtürme sprechen möchten, wie wir es letztes Mal im Kontext komplexer Zahlen getan haben oder sogar mit negativen Basen. Man muss solche Dinge durchdenken, also war es eine Frage, die wir auf dem Bildschirm hatten. Ja, was passiert, wenn wir das für mich mit der Potenz I tun? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "Titration, wissen Sie, lasst uns das einfach versuchen. Lasst uns einfach einen Power Tower ausprobieren, wo wir I auf eine bestimmte Potenz erhöhen und sehen, was dabei herausspringt. Das war also nicht geplant. Aber wir können, wir können immer Rufen Sie Python auf und machen Sie im Wesentlichen das, was wir letztes Mal getan haben. Das würde also so funktionieren, dass wir mit einem Basiswert beginnen und dann für eine Art Bereich. Was wir gemacht haben, wir haben einen genommen und ihn neu zugewiesen Es soll sein, was auch immer. Die Basis, die ich in diesem Fall auf die Potenz von a erhöht habe, sollte sein: Okay, cool, also machen wir das und dann drucken wir den Wert von a aus. Lasst uns das einfach machen Ja, es ist eine viel größere Zahl wie 200. Es sieht also so aus, als würde bei diesen Dingen manchmal Chaos entstehen. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "Ich habe es tatsächlich getan, also lassen Sie mich NumPy importieren, damit ich die Exponentialfunktion habe. Lassen Sie mich gehen. Für unseren großen Bereich, wie wir ihn zuvor hatten. Anstatt es zu schreiben, wie Sie wissen, etwas, das wie ich hoch X ist, werde ich es schreiben als Exponentialfunktion einer anderen Konstante oder einer anderen Konstante, die ich erstellen werde. Ich möchte, dass sie 5 Pi-Hälften hat, also mache ich 5 Pi-Hälften mal, also ist es eine komplexe Zahl und sie hat 5 Pi-Hälften als Imaginärer Teil Das sind also 5 Pi-Hälften mal I und was mache ich? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/hebrew/sentence_translations.json b/2020/ldm-i-to-i/hebrew/sentence_translations.json index 21bfb0865..2757a5fa8 100644 --- a/2020/ldm-i-to-i/hebrew/sentence_translations.json +++ b/2020/ldm-i-to-i/hebrew/sentence_translations.json @@ -49,7 +49,7 @@ "end": 63.7 }, { - "input": "And in fact if we go, let's not show where things are going too much here, if we go ahead and rewrite that base i in terms of e, it can help us make sense out of this expression.", + "input": "And in fact if we go, oh no, that's not sure where things are going too much here, if we go ahead and rewrite that base i in terms of e, it can help us make sense out of this expression.", "translatedText": "ולמעשה אם נלך, בואו לא נראה לאן הדברים הולכים יותר מדי כאן, אם נמשיך ונכתוב מחדש את הבסיס i במונחים של e, זה יכול לעזור לנו להבין את הביטוי הזה.", "n_reviews": 0, "start": 64.12, @@ -994,7 +994,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half.", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half.", "translatedText": "אז אם אתה מתחיל במספר 1, המהירות ההתחלתית שלך היא ללכת ישר לכיוון 0 וככל שאתה הולך אפילו נמוך יותר, אם היית יושב על 1 חצי, אז עדיין היית הולך לכיוון 0, אבל עכשיו וקטור המהירות שלך יהיה שלילי פי 1 איפה שאתה נמצא, וזה שלילי 1 חצי.", "n_reviews": 0, "start": 998.68, @@ -1302,7 +1302,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i.", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i.", "translatedText": "ושאלה מעניינת תהיה שאתה יודע היא האם יש רק פונקציה אחת כזו שמרגישה הגיוני לכתוב עבור זה כי אתה יודע אם נכתוב אותה כ-i ל-x לא רק שהיא צריכה לספק את זה היא גם צריכה לספק את יודעת מתי אנו מחברים את מספר 1 אנו מקבלים, ככל הנראה i, ל-power one אולם אנו חושבים שהפונקציה הזו צריכה להיות i.", "n_reviews": 0, "start": 1383.38, @@ -1323,7 +1323,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos.", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma", "translatedText": "אז יש לנו 5 חצאי pi i נהדר שזה בהחלט עוד ערך שנוכל לחבר ל-x כאן ורק כדי לאיית את זה קצת יותר ויזואלית אם היינו מסתכלים אחורה על המעגל שלנו כאן, שבו יש לנו את רגע הליכה במשך פרק זמן השווה לחצאי פאי שהוא 1.57 מה אם במקום זאת ניקח עוד סיבוב שלם ונלך עוד חצאי פאי כדי להביא אותנו ל-pi שאתה יודע שאולי נוכל להקליט ששם הוא הערך e ל-pi i, אנחנו הולכים עוד חצאי פאי אנחנו הולכים עוד חצאי פי שב- בשלב הזה היינו הולכים מעגל שלם ומחזירים אותנו לאחד ואז אנחנו הולכים במשך חמישה חצאי פאי שהם מספרית בערך 7.85 כן, זה בהחלט עוד מספר שמעלה אותנו על ה-i, ואם היינו עוברים את כל המהלך של ביטוי מחדש של i לעוצמה i על ידי כתיבה תחילה של e ל-5 חצאי ה-i i לעוצמה i אלה i's תכפילו כדי להפוך לשלילי והיינו מסתכלים על e ל-5 חצאי pi השליליים שזה מספר שונה מאוד נכון, אנחנו יכולים למעשה לחשב את זה אני לא בטוח מהראש שלי, אבל בואו נסתכל על Desmos .", "n_reviews": 0, "start": 1415.68, @@ -1337,7 +1337,7 @@ "end": 1493.22 }, { - "input": "What is e to the negative 5 pi halves 0.000388 Okay, 0.000388 much smaller number 0.000388 Which begs the question of okay i to the i what are you right?", + "input": "What is e to the negative five pi halves? 0.000388. Okay, 000388. Much smaller number. 0.000388. Which begs the question of okay i to the i, what are you? Right?", "translatedText": "מה זה e לשליליים של 5 פי חצאי 0.000388 אוקיי, 0.000388 מספר קטן בהרבה 0.000388 מה שמעלה את השאלה של בסדר אני ל-i מה אתה צודק?", "n_reviews": 0, "start": 1493.28, @@ -1358,21 +1358,21 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle?", + "input": "that long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle,", "translatedText": "זה ארוך שמביא אותך למספר הרבה יותר קטן אבל זו לא התשובה היחידה שיכולנו להזין, יש לנו אנשים אחרים שנכנסים לכאן עם שליליים של 3 חצאים כפול i pi. מה שאתה יודע במונחים של מעגל יחידה?", "n_reviews": 0, "start": 1544.74, "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one?", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one", "translatedText": "נוכל לחשוב על להגיד היי אם אני רוצה להגיע ל-I במקום ללכת 90 מעלות פאי חצאי רדיאנים ככה מה אם אני הולך 270 מעלות לכיוון השני 3 חצאי פי רדיאנים שאולי אני אחשוב עליהם כשלילי כי המוסכמה היא בדרך כלל שנגד כיוון השעון הוא חיובי. זו בהחלט דרך אחרת לבטא את זה וזה היה נותן לנו תשובה אחרת אם היה לנו e לשלילה של 3 חצאי pi i הכל בחזקת i אנחנו עוברים את אותו משחק עכשיו הריבוע i מבטל עם a שלילי זה כבר שם, ויש לנו 3 חצאי פי חיוביים ומבחינה מספרית זה נותן לנו תשובה שונה אפילו ממה שהיה לנו קודם. אם נעבור ונאמר היי, מה זה e ל-3 פי לא 3 או 3 פי חצי 111 נקודה 3 1 מספר שונה מאוד ממה שראינו לפני 111 נקודה מה זה היה 111 נקודה 3 1 נהדר 111 נקודה 3 1 או משהו כזה ושוב מבחינת האינטואיציה מה שאתה עשוי לשאול שם הוא נניח שיש לנו את זה מסתובב דינמי אבל אנחנו זזים אחורה בזמן אנחנו רואים לפני כמה זמן מה אני צריך להיות כזה שאם אשחק דברים קדימה משם הייתי נוחת על מספר אחד המצב ההתחלתי שלי ואתה צריך לחזור אחורה בזמן 3 חצאי פאי יחידות ואז אם היית מתרגם לדינמיקת הדעיכה וזה מה שהרמת העין עושה בהקשר הזה אתה אומר אם אני מתחיל במספר אחד אבל אני רוצה לזוז אחורה בזמן ולומר איפה הייתי צריך להתחיל אם אני רוצה להתפרק כך שאגיע למקום הראשון?", "n_reviews": 0, "start": 1559.26, "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right?", + "input": "after three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right?", "translatedText": "לאחר 3 חצאי פאי יחידות זמן, התשובה מתחילה ככל הנראה בסביבות מאה ואחת עשרה עבור סוג כזה של דעיכה אקספוננציאלית ותוכלו לראות לאן זה הולך כאשר יש למעשה אינסוף ערכים שונים שנוכל לחבר עבור X אם אנחנו חושבים על e ל-X כעל אני ואנשים נכנסנו הרבה יותר לכאן סליחה שאני זורק את הסיכה שלי על הקרקע שכן אחד עושה קלאסי למקום השלישי 9 חצאי פי בחירה מצוינת 1729 חצאי פי כולכם האהובים עליי המון המון אפשרויות שונות אינסוף ערכים שונים שמרגישים קצת מדאיג בהתחלה כי אנחנו מסתכלים על ביטוי שנראה כאילו אתה יודע שפשוט יהיה איזה חישוב אני פשוט מחבר את זה למחשבון שלי ואראה מה קופץ החוצה ויש לנו כמה דברים שונים ערכים לזה אז מה קורה כאן נכון?", "n_reviews": 0, "start": 1657.18, @@ -1442,7 +1442,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function?", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function,", "translatedText": "השורש הרביעי של 16 צריך להיות 2 והתשובה בסופו של דבר טובה. אנו מאמצים מוסכמה כאשר יש מספר אפשרויות כמו זו כאשר יש לך פונקציה רבת ערכים. לעתים קרובות אנו פשוט בוחרים באחד מהערכים האלה להיות מה שאנחנו מתכוונים כשאנחנו רוצים התייחס לזה כאל פונקציה כמשהו עם קלט בודד ופלט בודד בשפה מהודרת יותר זה עולה כל הזמן כשאנחנו עוסקים במספרים מרוכבים הרעיון של משהו כפעולה סוג של רצון שיהיו לך מספר ערכים לפעמים לשמוע את הביטוי ענף איפה אתה בוחר ענף של פונקציית השורש הריבועי?", "n_reviews": 0, "start": 1795.66, @@ -1463,7 +1463,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer?", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer?", "translatedText": "בגלל שיש מספר תשובות שונות אתה יודע שאנחנו חושבים על אני שוב זה סיבוב של 90 מעלות ואם היינו חושבים על זה כעל סיבוב של 90 מעלות זה מרגיש כאילו השורש הריבועי צריך להיות אתה יודע משהו שיושב בזווית של 45 מעלות אולי זה הריבוע שורש ה-I שנוכל לכתוב בצורה מאוד מפורשת כשורש 2 על 2 שורש 2 על 2 I זה רק באמצעות טריגונומטריה, אבל אם היינו חושבים על I במקום זאת כעל סיבוב שלילי של 270 מעלות, זה מרגיש כמו חצי מזה שעושה חצי מהפעולה הזו. בעצם צריך להביא אותנו לצד השני אולי המספר שיושב כאן צריך להיות השורש הריבועי של I וזה בעצם רק השלילי של מה שראינו קודם שורש שלילי 2 על 2 מינוס שורש 2 על פי 2 אני עכשיו בהקשר של אמיתי פונקציות מוערכות שנוכל לומר כן פשוט בחרו בשורש הריבועי להיות התשובה החיובית אשר תהיה, אבל איזו מהן אתם מחשיבים כתשובה החיובית?", "n_reviews": 0, "start": 1846.36, @@ -1477,14 +1477,14 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x?", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x,", "translatedText": "ואני חושב שאתה אומר טוב אנחנו יודעים מה זה אנחנו די מגדירים את זה כשורש מרובע של 2 הכל טוב ויפה אבל מה אם הייתי אומר בוא ניגש לזה באותו האופן שבו התקרבנו לאי שלנו לביטוי אני אני אני רוצה קודם כל לבטא דברים בתור e למשהו הנכון ואז אני אעלה את זה לחצי 1 על ידי הכפלת החצי 1 לתוך המעריך ואני אומר בסדר, אני יכול אני מניח שאני יכול לעשות את ה-e הזה למה שהוא שווה ל-2 טוב זה הלוג הטבעי של 2 זה קבוע שהוא סביב 0.69 או משהו כזה אם נעלה את e לחזק הזה נקבל 2 כדי שנוכל לחשוב על זה כעל e ללוג הטבעי של 2 כפול 1 חצי ואם היית רוצה אם היית חושב על e ל-x?", "n_reviews": 0, "start": 1942.28, "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it.", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it.", "translatedText": "אתה יודע שזה יכול להיות סוג של מוגזם בהקשר של מספרים אמיתיים אבל אם חשבת על e ל-x כקיצור של פונקציית x זו, אתה יכול לחבר את הערך 0.69 כפול מחצית, שלדעתי יהיה בערך 0.345 Ish משהו כזה אתה מחבר את הערך הקונקרטי הזה לפולינום שלך ראה מה הוא מוציא, והוא יוציא בערך 1.414 שורש ריבועי של מספר ממשי נחמד של 2 מה שהיית מצפה אבל אם נעשה את אותו הדבר שהיינו רק עושים עם I ומכירים שבעצם ישנן מספר תשובות שונות כאשר אנו רוצים לכתוב משהו כ-e לחזקה, נוכל גם לכתוב את זה זה אולי נראה מצחיק, אבל נוכל לכתוב את זה בתור e ללוג הטבעי של 2 פלוס 2 pi I כל העניין הזה מועלה לחצי 1. אחרי הכל הערך הזה יגיע להיות שווה לך יכול לפרק אותו כפי שהוא e ל- לוג טבעי של 2 כפול e ל-2 pi I זה פשוט משפיע על סיבוב דברים ב-360 מעלות, אז הוא פשוט ישתווה ל-1 אז אנחנו מסתכלים על 2 כפול 1 נהדר שמרגיש כמו תחליף חוקי ובכל זאת כאשר אנחנו משחקים באותו משחק של לקחת את זה ולהעלות אותו לחזקה ולטפל בזה על ידי הכפלת העוצמה לתוך המעריך תסתכל מה קורה יש לנו e ללוג הטבעי של 2 כפול 1 חצי פלוס ובכן, מה זה 2 פאי I כפול 1 חצי ובכן זה יהיה פי פעמים I עכשיו זה חלק ראשון e ללוג הטבעי של 2 כפול 1 חצי שבסופו של דבר יהיה השורש הריבועי המוכר של 2 זה הכל טוב ויפה, אבל אנחנו הולכים להכפיל את זה ב-e ל ה-pi I נכון ודי מפורסם e ל-pi I הוא שלילי 1 אז במקרה הזה נראה שזה מציע שאם אנחנו פותרים את הביטוי הזה 2 עד 1 חצי על ידי משחק עם התשובות השונות נוכל לחבר משהו כמו e ל-X שווה ל-1 חצי מה שבסופו של דבר אנחנו מקבלים היא תשובה נוספת מה שאנחנו יכולים לכתוב באופן מסורתי כשורש ריבועי שלילי זה של 2, וכאן אני מתכוון שזה קצת מצחיק שיש לו מספר ערכים להסתכל על 2 עד 1 וחצי תגיד שזה לא שווה דבר אחד אבל בהתבסס על בחירות שאנחנו עושים זה יכול להיות שווה למספר דברים שונים אבל שני הדברים שזה יכול להיראות הגיוני למדי אם יהיה משהו ש-2 ל-1 חצי זה נראה כאילו זה צריך להיות חיובי שורש ריבועי שאנחנו מכירים או הגרסה השלילי של זה שבעצם לא נראה כמו בעיה כזו ולמעשה נוכל לשחק את המשחק הזה עוד יותר, שם הרשו לי לבקש מכם תשובות יצירתיות עוד יותר לביטוי הזה כי אולי נוכל למצוא כוחות מצחיקים אחרים של משהו כמו 2 לחזק X כשאנחנו מתחילים לחבר ערכים שונים של X בהתבסס על ההחלפה שאנחנו עושים אם אנחנו מצייתים לאותם כללים שבהם השתמשנו בהערכת I ל- כוח I אז הפעם השאלה שואלת או שהיא מציינת שפתרון אחד של המשוואה e ל-x שווה 2 הוא המספר האמיתי הטבעי לוג של 2 אוקיי זה שאנחנו מכירים אותו.", "n_reviews": 0, "start": 1989.66, @@ -1498,14 +1498,14 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42.", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42", "translatedText": "תשובה לשאלה e ל-x שווה ל-2 ושוב יצירתיות מתקבלת בברכה, אז אני אתן לך עוד רגע קטן בשביל זה. אני אמשיך ותנעל כאן כמה תשובות אם זה בסדר איתך אני לא בטוח כמה זמן זה צריך בהכרח לעשות את הערך המתמטי בהתאם לאיזה מכשיר אתה מסתכל אבל אל תהיה לחוץ מדי אם זה לפני שהייתה לך ההזדמנות להיכנס לשאלה שאתה רוצה לתוך התשובה שאתה רוצה שהיא תענה אז זה נראה כמו 131 מכם נכנסו לגרסה שבה אנחנו לוקחים Ln של 2 ומוסיפים 2ii ואני מניח שאני כותב את השאלה הזו בטעות סימנתי את אחת התשובות כנכונות כשלמעשה יש לא מעט נכונות שונות אז זה תלוי בי על העובדה שאני לא יודע אם למישהו מכם זה נראה כמו הו זה אדום טעיתם כשהזנתם Ln של 2 פלוס 42.", "n_reviews": 0, "start": 2176.56, "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer.", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer.", "translatedText": "I pi שהיא כמובן בחירה מצוינת אבל אתה יכול גם לקבל משהו כמו 4 pi I פלוס הלוג הטבעי של 2 או 6 pi I או בעצם כל כפולה שלמה של 2 pi I אם תוסיף שזה לא משפיע על e ל- X כי יש לזה רק את ההשפעה של הכפלה ב-e ל-2 pi I שזה ההשפעה של הכפלה ב-1 ושוב יש לזה תוצאה מצחיקה שבה נראה שהוא מוציא סוג של תוצאות סבירות כשאנחנו עושים את זה כדוגמה נוספת. נראה שהביטוי השני הכי נפוץ שהוזן שם היה שאולי נחליף 2 אז בוא נניח שאנחנו חושבים על 2 בחזקת 1 4, אוקיי הייתה הצעה שהחלפנו את 2 ב-e ללוג הטבעי של 2 ועוד 4 pi I אוקיי פלוס 4 pi I ואנחנו מעלים את כל זה ל-1 4 טוב טוב אם הייתם משחקים באותו משחק הייתם מקבלים e ללוג הטבעי של 2 כפול 1 4, והיינו מכפילים ב-e ל ה-pi I עכשיו החלק הראשון של זה הולך להיות השורש הרביעי של 2 החיובי הרגיל, הדבר שאנו מתכוונים אליו כאשר אתה מחבר ביטוי כמו שורש רביעי של 2 למחשבון מספר חיובי קטן ויפה, אבל אז החלק השני הזה הוא שלילי 1 אז נראה שזה אומר שאתה יודע אם היינו מפרשים את 2 בצורה שונה זו ומעלה אותו ל-1 4 אתה יודע שזו לא התשובה הרגילה שאנחנו מקבלים אבל זו תשובה סבירה.", "n_reviews": 0, "start": 2253.76, @@ -1519,7 +1519,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships.", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships.", "translatedText": "היינו מסתכלים על חצאי פאי כפול I ובמקום להכפיל ב-1 שלילי היינו במקום להכפיל ב-I שזו שוב תשובה חוקית זה נראה כמו פלט סביר למשהו כמו 2 עד 1 4 אז כשאתה בהסתכלות על העובדה שאני בעוצמה, נראה שיש לי מספר ערכים שונים עבורו נכון יש לנו את התופעה המצחיקה הזו שבה נוכל לחבר e ל-5 חצאי ה-Pi I שלילי 3 חצאי ה-Pi I ונקבל מה שנראה כמו תשובות שונות בטירוף משהו סופר קטן משהו סופר גדול הכל מאוד שונה מהתשובה ה-1 5 בערך 1 5 שמצאנו כאן למעלה זו בדיוק אותה תופעה כמו כשאתה שואל משהו כמו מה זה 2 ל-1 4 והכרה שלמעשה יש מספר פתרונות שונים לביטוי X ל-4 שווה 2 4 פתרונות שונים למעשה ומה שאתה מסתכל עליו הוא העובדה שיש מספר פתרונות שונים לביטוי e ל-X שווה בסיס כלשהו אם הבסיס הזה הוא אני אם הבסיס הזה הוא 2 מה שזה לא יהיה ואחת הדרכים שבהן נוכל לחשוב על זה היא שכאשר אתה מתמודד עם מספרים אמיתיים דברים הם פשוט דברים נחמדים דברים נחמדים יש מערכות יחסים של אחד לאחד.", "n_reviews": 0, "start": 2358.92, @@ -1533,7 +1533,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value?", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value", "translatedText": "זה נהדר איפה אם אנחנו רוצים לחשוב על פונקציות אקספוננציאליות, הרשו לי רק לכסות חלק מהדברים האלה יש לנו את זה נחמד הלוך ושוב שבו אתה יכול לבחור לבטא כל מעריכי כבסיס ל-X כמו 2 ל-X או שתוכל לבטא אותו מעריכי כמו X של R כפול X שאתה יודע שזה הפולינום שאליו אנו מתייחסים בכל פעם שאנו מתייחסים באופן מרומז בכל פעם שאנו כותבים משהו כמו e ל-X ויש קטע מקסים הלוך ושוב כי אתה יכול פשוט לקחת לוגריתם טבעי של B וזה נותן לך תשובה אחת בהנחה ש-B הוא מספר חיובי וזה אותו דבר כמו להגיד ש-X של R שווה ל-B אז דרך אחת שדיברתי על זה קודם בסדרה היא שאם היית מסתכל על משפחה של כל האקספוננציאלים האפשריים נכון, נוכל לכתוב אותם כ-X של R כפול X ולשנות מה זה R וזה בדיוק אותו דבר כמו לכתוב e ל-R כפול X אם זה משהו שאתה יותר נוח איתו אז e ל-R כפול XX של R כפול X זה אותו הדבר שאנחנו יכולים לחשוב על שינוי מה זה אבל מצד שני אם היית חושב על כל האקספוננציאלים האפשריים כבסיס כלשהו תן לי לעשות בסיס בחזקת X ואנחנו הולכים לשנות את הבסיס הזה בהתחלה זה מרגיש כאילו זה סוג אחר של ביטוי לתמרן, אבל זה רק עוד דרך לבטא את אותה משפחה ודרך שאולי תחשוב על זה כי איך אנחנו חושבים על איזה בסיס זה מתאים אם אנחנו חושבים בצורה קצת יותר מופשטת כמו Exp של R כפול X ויש סיבה שאני עושה את זה כי אנחנו עומדים להחיל את זה על מספרים מרוכבים שבהם זה יראה מוזר יותר אז תמשיך איתי כאן אם במקום להסתכל על הבסיס הזה דבר אחד שאני יכול לעשות הוא לומר מה הערך?", "n_reviews": 0, "start": 2428.5, @@ -1554,7 +1554,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R.", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d", "translatedText": "יכול להיות ש-exp של R כפול X שבו אולי R הוא משהו כמו אפס נקודה שש תשע אבל אני יכול להזיז את זה למטה בשתי pi I וזה לא משנה את הבסיס שהוא יתאים לו שעדיין יתאים לשני או שזה יכול העבר אותו למעלה ב-2 pi I שלא משנה את הבסיס שהוא מתאים אליו מכיוון שבכל המקרים האלה כשאנחנו מחברים X שווה לאחד אנחנו מקבלים את אותו הדבר אולם כל אלה עבור ערכים שונים של X הם פונקציות נפרדות. מדוע ראינו מספר ערכים שונים עבור I בחזקת I מכיוון ש-I ל-X היא פונקציה דו-משמעית בהקשר זה, זה יהיה חד-משמעי אם נחליט איזה ערך של R כך שמה שאנו מייצגים הוא exp של R כפול X איזה ערך של ר.", "n_reviews": 0, "start": 2597.88, @@ -1568,14 +1568,14 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers.", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers,", "translatedText": "זו פונקציה חד משמעית אבל בשלב הזה זה פשוט מרגיש כאילו מה שאנחנו רוצים זה להפסיק לחשוב על דברים במונחים של איזה בסיס שהועלה לחזק X אולי ברגע שאנחנו בהקשר של מספרים מרוכבים אנחנו צריכים פשוט לכתוב כולם כ-exex של כמה פעמים קבועים X אם מסיבה אחרת זה מבהיר בצורה ברורה איך אנחנו בעצם מחברים מספרים אם אנחנו רוצים לעשות חישוב או סתם לעשות מתמטיקה על זה, יש לנו את הפולינום האינסופי הנחמד הזה שאנחנו תחבר אותם ואסביר לך שזהו אולי הדרך הנכונה לחשוב על אקספוננציאלים ברגע שאנחנו מתרחבים לתחומים אחרים דברים כמו מספרים מרוכבים ולשם כך בואו פשוט בוא נגבה עבור חזרה לפעמון הדלת כמה דברים הגיעו חזור לדרך המקורית שבה אנחנו מרחיבים את רעיון האקספונציה ופשוט חושבים על מה זה 2 ל-X ימינה אנחנו יודעים לחשוב על זה עבור מספרים טבעיים.", "n_reviews": 0, "start": 2640.66, "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that.", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that?", "translatedText": "אתה יודע משהו כמו 2 ל-3 כפל חוזר איך זה שמלמדים אותך לראשונה לחשוב על משהו כמו 2 ל-X עבור כמויות שברים או עבור כמויות שליליות ודברים כאלה.", "n_reviews": 0, "start": 2696.88, @@ -1589,35 +1589,35 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive.", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive.", "translatedText": "בדרך כלל מלמדים אותך שחצי 2 עד 1 צריך להיות משהו שבו אתה יודע אם אני מכפיל את זה בעצמו וזה עוקב אחר הכללים הרגילים שעושים אקספוננציאלים עם ספירת מספרים שבהם אנחנו יכולים להוסיף דברים באותו מעריך אני אמור לקבל 2 ל-1 אז זה צריך להיות מספר שכאשר אני מכפיל אותו בעצמו אני מקבל 2 ואתה יודע שבשלב הזה יש לך בחירה, אולי זה חיובי.", "n_reviews": 0, "start": 2708.3, "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1?", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the", "translatedText": "אולי זה שלילי אבל אם תמיד תחליט לעשות את הבחירה החיובית תוכל לקבל פונקציה רציפה נחמדה מאותה עסקה אם נשאל לגבי מספרים שליליים. איפה כשאני מכפיל את זה ב-2 ל-1?", "n_reviews": 0, "start": 2731.78, "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily.", + "input": "one, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily.", "translatedText": "זה מביא אותי ל-2 ל-0 וזו סוג של ההצדקה למוסכמה שלנו לפיה מעריכים שליליים נראים כמו חצי אבל מה שבאמת קורה כאן הוא שאנחנו אומרים מה שזה לא יהיה זה צריך להיות סוג של פונקציה שעונה על התכונה f של a פלוס b שווה f של כפול f של b ויתרה מכך העובדה שהבסיס הוא 2 בעצם אומרת לנו שזה לא סתם פונקציה כזו זו פונקציה שבה כשאנחנו מחברים 1 אנחנו מקבלים 2 ובדיוק כמו שאתה יודע שפיות בדוק שאלה בסגנון כדי לראות אם אתה עוקב יחד עם כמה מההשלכות כאן אני רוצה לשאול אותך מה זה אני לא אקרא לזה כמו כדור סופט, אבל זה לא אמור להיות כמו שאלה עמוקה להפליא בהכרח.", "n_reviews": 0, "start": 2744.96, "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here?", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here.", "translatedText": "זה פשוט יותר בדיקה אם אתה עוקב אחרי הרעיון להתחיל באופן מופשט עם מאפיינים של פונקציה ואז להסיק דרכים שאולי נרצה לכתוב אותה על סמך המאפיינים האלה אם f של x עונה על התכונה המעריכית הזו f של פלוס b שווה f של כפול f של b עבור כל התשומות וזה גם עומד ב-f של 1 שווה 2 מה מהבאים נכון, כלומר איזה מהבאים נכון בהכרח לא משנה איזו פונקציה כזו אתה מתחיל עם ואלו מכם שזוכרים איזו הרצאה זו הייתה על איזו הרצאה שדיברנו איך לפרש את מה שהנוסחה של אוילר אומרת באמת שאלתי שאלה בסגנון הזה שבו הזנחתי תנאי אחד, אתה יודע שלא רשמתי העובדה שאנחנו רוצים לוודא ש-f של x אינו אפס בכל מקום ואז זה גרם לכמות מסוימת של Confudlement וזה מגניב. משהו שהופך את החיבור לכפל הוא האם מספיק כדי בעצם לגרום לך לרצות לכתוב את הפונקציה כמו מה שהיא שווה כאחת שהועלתה לסוג של כוח. זו רוח השאלה עכשיו יש לנו כמה שאלות בעצם על מגדלי כוח שנדמה שצצו כאן, וזה מאוד קשור לפעם הקודמת. בואו נעצור את שאלת מגדל הכוח רק לרגע כדי שנקבל קודם כל תחושה עמוקה יותר של כמו מה אמורה להיות אקספונציאנציה כאן?", "n_reviews": 0, "start": 2793.44, "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane?", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane?", "translatedText": "כי בגלל שאנחנו יכולים להיות מה שאני רוצה לטעון זה שאנחנו יכולים לענות על זה בכמה דרכים שונות אז אם תיתן לי רק אחת, נדבר על מגדלי כוח ואז בדיוק כפי שניתן לייצג קו מספר בסולם לוגריתמי. לעשות את אותו הדבר עבור מטוס מורכב?", "n_reviews": 0, "start": 2882.64, @@ -1638,7 +1638,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one.", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one", "translatedText": "כן למעשה, יש הדמיה שאני הולך להגיע אליה עוד רגע כאן, שבה אנחנו עושים משהו די דומה לזה כי מה שנעשה זה לשחק עם פונקציות אקספוננציאליות שונות X של R כפול X אבל אנחנו הולך לשנות את הערך הזה של R שיוצג על ידי נקודה צהובה קטנה אז אנחנו קצת נדבר על זה זה לא הולך למפות את כל המישור, אלא רק כמה נקודות דגימה מהציר האמיתי ומהציר הדמיוני אבל הרעיון הוא שכשאנחנו נעים סביב מה הוא הקבוע הזה נוכל לדמיין את הדברים השונים שהוא עושה למישור וביעילות זה כאילו הוא הופך את ציר ה-X לסולם לוגריתמי ואז עוטף הציר הדמיוני לאורך מעגל ואז ברגע שהערך הזה של R הופך לדמיוני, הוא מחליף את התפקיד של המספרים הממשיים האלה מוצבים במעגל ומספרים דמיוניים מוצבים על ציר בקנה מידה לוגריתמי חיובי כל כך שאלה נהדרת ששלושתם אני מניח הם סוג של קופצים קדימה לאן אני רוצה להגיע אבל נחמד לראות ששם אנשים חושבים כך בקטע הזה.", "n_reviews": 0, "start": 2901.54, @@ -1652,14 +1652,14 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it.", + "input": "explicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you want, which is that let's say you have some complex function f, a", "translatedText": "במפורש משהו כמו f של 5 זה אותו דבר כמו f של 1 ועוד 1 פלוס 1 ועוד 1 פלוס 1 שזה אותו דבר כמו f של 1 כפול עצמו פי 5 בגלל התכונה הזו שאם f של 1 הוא 2 זהה בתור 2 בחזקת 5 ואז משהו כמו f של שלילי 5 זה צריך להיות המצב שכאשר נכפיל את זה ב-f של 5 נקבל כל מה ש-f מ-0 וזה לא ברור מיד מה זה f מ-0 אבל אפשר לומר ש f של 1 ועוד 0 שווה לכל מה ש-f מ-1 הוא כפול ממה ש-f מ-0 אבל f של 1 שווה ל-2 ולכן זה גם שווה ל-2 אז אנחנו אומרים ש-2 שווה ל-2 פעמים משהו טוב שמשהו חייב להיות 1 אז בהקשר זה זה מבטיח ש-f של השלילי 5 הוא 2 לשלילי 5 זה 1 על 2 עד ה-5. אנחנו יכולים לכתוב את זה במפורש כ-2 לשלילי 5 וזה הכל אומר ששני המאפיינים האלה ביחד יוצרים אנחנו באמת רוצים לכתוב את הפונקציה כ-2 ל-X כי כל מספר ספירה שנכניס בו יספק את זה הוא יראה כמו להכפיל בעצמו את המספר הזה של הפעמים שמספר שבר כלשהו שנכניס אותו יספק את התכונות האלה שרצינו ואתה עשוי לתהות האם ייחודי ובהקשר של פונקציות ערכיות אמיתיות זה באמת יהיה אבל בהקשר של פונקציות ערכיות מורכבות יהיו מספר פונקציות כאלה שנוכל לכתוב עבור זו מהן מה שהיינו מסתכלים על לפני איפה יכולנו להיות פונקציה מוגדרת להיות exp של הלוג הטבעי של 2 פלוס 2 pi I כל הפעמים X אוקיי, סלח לי על הרשלנות כאן, אני פשוט מתרגש לכתוב על זה וזו למעשה פונקציה אחרת כמו עדות מה קורה אם מחברים X שווה לחצי 1 ראינו קצת קודם איך כשאתה מחבר 1 חצי מה שאתה מקבל הוא השורש הריבועי השלילי של 2 ואז אם אתה מחבר 1 רביעית אתה מקבל לא את השורש הרביעי של 2 אבל אני כפול את השורש הרביעי של 2 אז זו פונקציה אחרת אבל זה עדיין עונה על המאפיינים האלה וזה קצת גורם לנו לרצות לכתוב את זה בתור 2 ל-X וזה גורם לזה להציע שאולי 2 ל-X הוא דו-משמעי קצת סימון ואנחנו צריכים פשוט לכתוב הכל במונחים של exp של R כפול משהו אבל אתה עשוי לתהות טוב אתה יודע אולי אנחנו פשוט לא מספיק יצירתיים עם כל הפונקציות שמספקות את המאפיין הזה אולי יש אי בהירות כשאנחנו כותבים exp של R כפול משהו ויש ערכים שונים של R שיכולים לבוא לידי ביטוי. נגיד שיש לך איזו פונקציה מורכבת F, והיא עונה על המאפיינים הבאים תחילה אתה יכול לקחת נגזרת שלה.", "n_reviews": 0, "start": 2974.0, "end": 3140.02 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways.", + "input": "nd it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I don't know, fractional amounts you might want to think of in crazy ways", "translatedText": "זה ניתן להבדיל, מה שפשוט מונע מזה להיות משהו שאתה מכיר, דבר לא רציף מבולגן לחלוטין זה כמו לקחת על עצמו כמה ערכים אקראיים, תלוי שאתה יודע את הטווח של כל מרחב וקטור מעל. אני לא יודע כמויות חלקיות שאולי תרצה לחשוב עליהן בדרכים מטורפות.", "n_reviews": 0, "start": 3140.12, @@ -1673,7 +1673,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right?", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right?", "translatedText": "זה ניתן להבדיל זה לא שווה ל-0 בכל מקום אז המצב שקצת חמק לי מהראש ואני שוכח לאיזו הרצאה הרצאה או משהו כזה ואז יש לו תכונה מרכזית שזה הופך את החיבור לכפל אם יש לך פונקציה כזו אני טוען ש יש ייחודי אולי אני באמת צריך לציין שיש מספר מורכב ייחודי R כך שתוכל לכתוב F של X בתור בעצם הפונקציה האקספוננציאלית הזו של R כפול הערך X מה שאתה יודע בעצם אומר שאם יש לך X כפונקציה זה פולינום אינסופי עם תכונות נגזרת נחמדות וכל זה אם יש לך את זה יש לך כל מעריכי שאתה רוצה במובן גנרי מופשט מאוד של המילה מעריכי רק על סמך תכונה שהיינו יכולים לרצות ממנו והסקיצה של ההוכחה תהיה תראה משהו כזה אם אתה רוצה להסתכל תחילה מהי הנגזרת של הערך הזה שאנו מניחים שקיים בכל מקום, נכון?", "n_reviews": 0, "start": 3160.84, @@ -1694,7 +1694,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base.", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base", "translatedText": "אנחנו יכולים לקדם את F של X מתוך הביטוי לחלוטין וכל הגבול מתבטא רק במונחים של H אשר אם חושבים על מה זה אומר בהקשר של נגזרות והעובדה ש-F של 0 שווה בהכרח ל-1 כל הביטוי המגביל הזה הוא רק איזשהו קבוע, אבל ליתר דיוק, זה לא משנה מה הנגזרת של הפונקציה שלנו ב-0 אז יש לך את הדבר המצחיק הזה שאם אתה יודע את הנגזרת שלו ב-0 שקובע מה הנגזרת שלו בכל מקום ובהקשר של פונקציות אקספוננציאליות זה מקווה שהוא די מוכר כי כל מה שאנחנו באמת אומרים זה הנגזרת של פונקציה מעריכית היא פרופורציונלית לעצמה ושקבוע המידתיות שווה לכל מה שהנגזרת ב-0 היא זה הכל מנוסח בצורה מאוד מופשטת וכאלה אבל אבל המטרה של זה היא להדגיש שזה לא בהכרח רק פונקציות שאנחנו כבר חושבים עליהן כעל חזקה X אבל זה מחלקה הרבה יותר רחבה של פונקציות שפשוט ממלאות את התכונה המופשטת של הפיכת חיבור לכפל אבל אם יש לך את זה זה בעצם מבטיח שיש לך גם נגזרת שנייה ולצורך העניין נגזרת שלישית וכאלה כי פונקציית הנגזרת היא רק פרופורציונלית לעצמה אז כדי לקחת את הנגזרת ה-n אתה פשוט מסתכל על קבוע המידתיות הזה ומעלה אותו לחזקה n ואז מכאן אתה יכול לעשות הרחבת סדרת טיילור ואולי אשאיר את זה כמעין שיעורי בית מתקדמים לאלו מכם שנוח להם עם סדרת טיילור ברעיון הזה, במיוחד אם אתם רוצים לערבב את הרעיון של כל פונקציה דיפרנציאלית שניתן להבדיל במובן של מספרים מרוכבים, כלומר סוג של נושא מכללה בהחלט אתה יודע שאתה יכול לערבב את ההיגיון שם כמו שאתה רוצה אבל נימוקים מטושטשים מותר בהקשר של מישהו שיודע רק על סדרת טיילור ולא שום דבר אחר לקחת את הרעיון הזה ולהסתכל על הרחבה של טיילור עבור F ו סוג של להצדיק את הרעיון שיש מספר מרוכב ייחודי כך שהפונקציה F שלנו בהכרח יכולה להיכתב כך ואז הקשר למעריכים נורמליים הוא בכל פעם שיש לך ערך כזה R אנחנו עושים בעצם מה שאנחנו עושים בהקשר המורכב של מספרים ממשיים זה אם אתה מסתכל על exp של הפונקציה של אותו ערך R וכתוב את זה כבסיס.", "n_reviews": 0, "start": 3243.7, @@ -1708,21 +1708,21 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace?", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace", "translatedText": "נוכל לפרש את זה כמשמעות לא רק exp של חצאי pi I כפול X, אלא נוכל גם לפרש את זה כמשמעות של exp של 5 חצאי pi I פעמים X ואלה פונקציות נפרדות ויש משפחה אינסופית של פונקציות נפרדות שמרגישות שאנחנו צריכים כתוב אותם בתור I ל-X אז הביטוי I ל-I אלא אם כן אימצת תקן למה זה בהכרח אומר כשאתה אומר שיש לו אינסוף פלטים דרך אחרת לחשוב על זה היא שהפונקציה I ל-X עם הסימון שיש לנו הוא קצת מעורפל עכשיו עם כל זה בוא נתחיל לדמיין חלק מזה כי אני חושב שזה כיף ואתה יודע שאתה אומר לי אם זה אם זה חזותי מועיל או חזותי יותר מבלבל אבל מה שאנחנו הולכים לעשות זה להסתכל על הפונקציה exp של R כפול X, שהיא בעצם זו דרך נוספת לכתוב e בחזקת X למעשה אני חושב שאני חושב שעשיתי אנימציה אחרת בשלב מסוים שציינה את זה כי תכננתי לעשות את זה אז תן לי אה כן הנה אתה תחזור למערכת הקבצים שלי תחזור למקום שבו אתה אמור להיות תמשיך שם זה מתלונן כי יש כמה דברים הו להחליף זה מופיע במסך השני רגע למה זה כן, בסדר להחליף?", "n_reviews": 0, "start": 3391.12, "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative?", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative", "translatedText": "מקם את כל מה שאתה רואה שם ועכשיו אנחנו חוזרים אל הו, שם כולנו את כל זה רק כדי שהייתי יכול לכתוב יפה אם אתה לא מרגיש בנוח לחשוב על זה כ-exp של R כפול X הפולינום האינסופי הזה רק ב- מאחורי הראש שלך e ל-R כפול X ואנו נשתנה סביב R אז אני הולך לעקוב אחר נקודות הציר הדמיוני, ואני הולך לעקוב אחר נקודות הציר האמיתי ובואו נראה מה זה עושה טוב זה הכל די מהיר אז תן לי לחשוב על זה קצת יותר לאט את כל המספרים השליליים כל דבר. זה מספר ממשי שלילי הולך להידחס לטווח שבין 0 ל-1 מה שאמור להיות הגיוני e לשלילה?", "n_reviews": 0, "start": 3472.82, "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R?", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r", "translatedText": "a למספר ממשי שלילי הוא משהו בין 0 ל-1, ובאופן ספציפי אנו עוקבים אחר f של 1 שלילי שעומד להופיע סביב כל מה ש-1 מעל e הוא בסביבות 30 0.37 f מתוך 1 נוחת על e כצפוי זה מה ש-ex של 1 הוא f של I הולך להנחית רדיאן אחד סביב מעגל היחידה, וזה די כיף לעקוב לאורך כל הציר הדמיוני כאן איך הציר הדמיוני נכרך סביב מעגל ומה קורה כשאנו מצווים את הערך הזה של R?", "n_reviews": 0, "start": 3511.78, @@ -1736,7 +1736,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like?", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like?", "translatedText": "אולי נרצה וערכים של R כאן זה מותח דברים אחרת אז כשאנחנו מעלים את זה ל-2 אתה יודע שזה מותח את הציר האמיתי הרבה יותר כך ש-f של 1 מסתיימת סביב המקום שבו e בריבוע הוא קצת מעל 7 f של שלילי 1 הוא הרבה יותר קרוב ל-0 f של I הוא סיבוב של 2 רדיאן סיבוב סביב המעגל f של שלילי I הוא סיבוב שלילי 2 רדיאן וכמובן שאנחנו יכולים להגיע לנוסחה האהובה עלינו שאם זה היה pi שהיה לנו בתור קבוע קנה מידה אז הציר האמיתי נמתח די הרבה אתה יודע ש-f של 1 יושב ב-e ל-pi שהוא קרוב מאוד ל-20 פלוס pi שזה תמיד כיף ו-f של שלילי 1 קרוב מאוד ל-0 אז זה באמת מתוח שהאמיתי ציר וזה גם מתח דברים בכיוון מעגל היחידה כך שמעבר ל-f של I או f של שלילי אני הולך באמצע המעגל, אז הכל טוב ויפה עכשיו איך נחשוב על פונקציה כמו?", "n_reviews": 0, "start": 3551.6, @@ -1750,7 +1750,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it.", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it", "translatedText": "נכתוב גם כ-X של X של הלוג הטבעי של 2 כפול X אז נעביר את הנקודה הצהובה שלנו שמייצגת את הערך של R ל-0 בערך.69 עדיין אין חלק דמיוני רק מספר אמיתי 0.69 בערך זה היומן הטבעי של 2 ובכן אתה יכול לראות ש-f של 1 נוחתת על 2 וזו הסיבה שאנחנו רוצים לקרוא לפונקציה הזו 2 ל-X f של 1 חצי למעשה סליחה f של שלילי 1 נוחתת בדיוק על 1 חצי f של אני זה קצת הליכה סביב מעגל היחידה באופן ספציפי זה יהיה 0.69 רדיאנים מסביב למעגל היחידה ועכשיו נוכל ליהנות קצת יותר ולומר מה יקרה אם נשנה את זה ל-0 במקום להיות 0.69 במקום להיות הלוג הטבעי של 2 הפוך אותו ל-I כפול את הלוג הטבעי של 2 כך שאנחנו באמת חושבים על משהו שעשוי להיות לו בסיס אקספוננציאלי.", "n_reviews": 0, "start": 3610.52, @@ -1778,14 +1778,14 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle?", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle?", "translatedText": "מה אני בעוצמה אני במקרה הזה זה דוחף אותו לסביבות 0.2 בסביבות החמישית אבל יש הרבה פונקציות אקספוננציאליות שונות שיש להן את התכונה הזו של לשים f של 1 על המספר I אז אם היינו מגדילים את זה עוד יותר אני לא חושב שיש לי את זה מונפש כאן אבל אם היינו לוקחים את הנקודה הצהובה הזו והעלו אותה עד שהיא הגיעה ל-5 חצאים כפול פי I מה שהייתם רואים זה מעגל היחידה?", "n_reviews": 0, "start": 3749.24, "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right?", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right?", "translatedText": "מסתובב סביב עצמו כך ש-f של f שלילי של 1 יסתובב סביב עוד 2 רדיאנים פאי וינחת היכן שהוא נמצא אבל הוא היה מותח את הציר האמיתי הרבה יותר, וזה היה המובן שבו פלט אחר של I ל-I הוא מספר הרבה יותר קטן זה היה בערך 0.0003 או משהו כזה אבל אנחנו יכולים גם לראות מה שלדעתי די כיף מה קורה אם ניקח בחשבון ביטויים חלופיים שאנחנו רוצים לפרש כ-2 בחזקת X נכון?", "n_reviews": 0, "start": 3773.06, @@ -1806,21 +1806,21 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward?", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward?", "translatedText": "יש לנו X של R כפול X ו-R שווה לערך הזה, שהוא הלוג הטבעי של 2 פלוס pi כפול I מה זה אומר שכאשר אנו מחברים 1 f מתוך 1 הוא שלילי 2 אז אנחנו רוצים לכתוב את הפונקציה הזו כשלילי 2 בחזקת X נכון וזה בעצם משהו שאתה יודע, זה קצת פשוט מטעה כשאנחנו כותבים מספר שלילי בחזקת שלילי 2 בחזקת X זה לא נראה ככה בהתחלה זה בהכרח מביא לנו למספרים המרוכבים בכל דרך, אבל כמובן שכאשר אנו מחברים אפילו ערך כמו 1 חצי איפה שאנחנו מבקשים שורש ריבועי של שלילי 2, אנחנו מבינים שאנחנו רוצים לכתוב את זה כמשהו כמו אני כפול השורש הריבועי מתוך 2 אבל אם היית מסתכל על הפונקציה הזו שלילית 2 בחזקת X בכל התחום המורכב שבו היא עוסקת מה שאתה מסתכל עליו הוא פונקציה שלוקחת את הערך של 1 לשלילי 2 ואם היא עושה את זה מה זה משפיע על שאר קו המספרים האמיתי האם זה סוג של ספירלה כלפי חוץ?", "n_reviews": 0, "start": 3820.68, "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be.", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be.", "translatedText": "אז אנחנו רואים ש-f של שלילי 1 יושב על שלילי 1 חצי בערך איפה שהיית מצפה אם היית עוקב עד f של 1 חצי זה היה יושב בדיוק על הקו הדמיוני ו-f של 1 חצי יהיה שורש ריבועי של 2 ובכן, שלי העכבר הוא לא איפה שאני רוצה שהוא יהיה.", "n_reviews": 0, "start": 3880.24, "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense?", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense", "translatedText": "זה יהיה סביב השורש הריבועי של 2 כפול I וככל שתמשיך הלאה זה מראה לך את כל כוחות הערך האמיתיים של 2 שלילי ל-X זה בהכרח מסתחרר סביבו אבל נוכל גם להזיז את הערך שלנו של R אפילו גבוה יותר ולקבל אותו עד סביבות טאו פעמים I בסביבות שש נקודה שתיים שמונה פעמים I ובהקשר זה זו פונקציה נוספת שהיינו רוצים לכתוב כמשהו כמו 2 ל-X כי עבור כל מספר שלם למספר שלם שתחבר עבור X נראה כמו כפל חוזר ויש לו אפילו סוג של ערכים סבירים לדברים כמו חצי שבו הוא יורק את השורש הריבועי השלילי במקום שורש ריבועי חיובי, אבל מה שהוא בעצם עושה זה טרנספורמציה למישור שבו הוא שם הכל הוא האמיתי קו המספרים בסופו של דבר הוא ספירלה כרוכה בחוזקה שמסתובבת והיא פשוט מתפתלת בצורה כזו ש-f של 1 נוחת ממש על הספרה 2 אז זה מהבחינה הזו שאנחנו יכולים לומר ש-2 ל-X הוא מתפרש באופן סביר כמו פונקציה אקספוננציאלית נפרדת מזו שהתרגלנו אליה באופן מסורתי אז אני חושב שעם כל זה אני אשאיר דברים להיום ואני רק אשאיר לך כמה שאלות מתמשכות לחשוב עליהן אוקיי, אז אם אתה רוצה תחשוב על אני אל האני כעל ביטוי רב-ערכי, נכון, אתה יכול לומר שאנו מאמצים מוסכמה בדמיון היית אומר שאתה בוחר ענף של פונקציית הלוגריתם הטבעי ואולי זה נועל אותך בהוויה הזו e ל-pi השלילי חצאים אבל אם אתה אומר שסוג כזה של רוצה להיות אינסוף ערכים שונים כמו הערכים השונים שראינו כמה ערכים 2 עד 1 שליש רוצה להיות באותו מובן?", "n_reviews": 0, "start": 3894.92, @@ -1834,7 +1834,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function?", + "input": "three-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function?", "translatedText": "העשיריות רוצות להיות מנוסחות אחרת מכל הפונקציות המעריכיות של F של X שמקיימות או רשמתי את זה איפשהו F של X שמקיים את כל המאפיינים האלה שכתבתי אז אם זה עונה על כולם מתוכם ואם f של 1 שווה ל-2 נכון כמה יציאות שונות נקבל כשנחבר X שווה ל-3 עשיריות עבור האפשרויות השונות לאיזו פונקציה?", "n_reviews": 0, "start": 4008.86, @@ -1848,14 +1848,14 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I?", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i", "translatedText": "עבור 2 ל-pi עבור הפונקציות השונות ש-2 ל-X יכול לייצג אם אנחנו חושבים על 2 ל-X כעל איזושהי פונקציה אקספוננציאלית אקספוננציאלית במובן של מאפיינים מופשטים מסוג זה ואם אנחנו כן, אם אנחנו אם יש לנו מחלקה של פונקציות שונות כאלה, ואנחנו רוצים לחבר את pi זה מצחיק אותי רק בגלל שזה כזה אני יודע תשובה מצחיקה שצצה כשאתה מנסה לחשוב על זה אז אלו השאלות ש אני אשאיר אותך ואני חושב שזה אתה יודע שלי השאלה המרכזית שלי בהתקרבות להרצאה של היום הייתה האם אני רוצה שהיא תתאר כמו המאפיינים המופשטים של פונקציות אקספוננציאליות וזה פשוט מגניב לי שמתחילים מהמאפיינים המופשטים האלה אתה ננעל ברעיון של e ל-rx או יותר רק אתה יודע, אני חושב שכתוב בצורה כנה יותר exp של r כפול x עבור ערכים שונים של r שזה נועל אותך כל כך רחוק אבל זה לא נועל אותך עד כמה רעיון חד משמעי של מה 2 בחזקת x צריך להיות הרבה פחות משהו כמו אני בחזקת x הסיכון בזה כמובן הוא שלפעמים אנשים לא אוהבים הפשטה ולפעמים זה לא נראה נגיש אבל אם זה במקרה שאתה יודע אתה רק תודיע לי אני חושב שאני חושב שיש מעגל מעניין של מחשבות שמקיף את כל הדברים האלה כדי לכלול מגדלי כוח כי אם אתה רוצה בעצם לדבר על מגדלי כוח כמו שהיינו בפעם הקודמת בהקשר של מספרים מרוכבים או אפילו עם בסיסים שליליים אתה צריך לחשוב על דברים כאלה, אז זו הייתה שאלה שעלתה לנו על המסך כן, מה יקרה אם נעשה את זה בשבילי בכוח אני?", "n_reviews": 0, "start": 4043.86, "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes.", + "input": "titration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes", "translatedText": "טיטרציה אתה יודע בוא פשוט ננסה את זה בוא פשוט נמשיך וננסה מגדל כוח שבו אנחנו מעלים את ה-I לעוצמה נתונה ונראה מה מה שיוצא מזה, אז זה לא היה תכנון לעשות את זה אבל אנחנו יכולים אנחנו תמיד יכולים תרים את Python ובעצם תעשה את מה שעשינו בפעם הקודמת אז הדרך שזה יעבוד היא שהתחלנו עם איזה ערך בסיס ואז עבור איזשהו טווח מה עשינו לקחנו a ואנחנו הולכים להקצות מחדש זה יהיה מה שלא יהיה. הבסיס שבמקרה הזה הוא אני מעלה לעוצמה של a צריך להיות בסדר, מגניב, אז אנחנו הולכים לעשות את זה ואז נדפיס את הערך של בוא נעשה את זה בשביל כן, זה מספר הרבה יותר גדול כמו 200 אז נראה שמה שקורה הוא שיש פוטנציאל לכאוס עם הדברים האלה כמו לפעמים.", "n_reviews": 0, "start": 4135.8, @@ -1869,7 +1869,7 @@ "end": 4201.64 }, { - "input": "That's it's not periodic or anything and it's actually chaotic I Suspect that doesn't happen for I but it's a thing to potentially look out for it looks like it does kind of stabilize Maybe there's some little subjection to numerical error But we stay pretty consistently around something with a real part of 0.43 and 0.36 Now what I would want to emphasize though is this expression So let's set a back to be equal to 1 this expression of taking I to the power of a remember That's a little bit ambiguous.", + "input": "that's um, it's not periodic or anything and it's actually chaotic I I suspect that doesn't happen for i but it's a thing to potentially look out for It looks like it does kind of stabilize um, maybe there's Some little subjection to numerical error, but we stay pretty consistently around something with a real part of 0.43 and 0.36 Now what I would want to emphasize though is this expression So let's set a back to b equal to 1 this expression of taking i to the power of a remember That's a little bit ambiguous.", "translatedText": "זה לא מחזורי או משהו וזה בעצם כאוטי. אני חושד שזה לא קורה לי אבל זה דבר שצריך לשים לב אליו זה נראה כאילו זה קצת מתייצב אולי יש איזושהי כפוף קטן לשגיאות מספריות אבל אנחנו נשארים די בעקביות בסביבה משהו עם חלק אמיתי של 0.43 ו-0.36 עכשיו, מה שהייתי רוצה להדגיש הוא הביטוי הזה אז בואו נציב גב כדי להיות שווה ל-1 הביטוי הזה של לקיחת אני לכוח של זכור. זה קצת מעורפל.", "n_reviews": 0, "start": 4201.64, @@ -1883,7 +1883,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing?", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing?", "translatedText": "למעשה יש לנו אז תן לי לייבא את NumPy אז יש לי את הפונקציה האקספוננציאלית תן לי ללכת לטווח הגדול שלנו כמו שהיה לנו קודם במקום לכתוב את זה כמו שאתה יודע משהו שהוא כמוני בחזקת X אני אכתוב את זה בתור הפונקציה המעריכית של קבוע שונה ממש קבוע שונה שאני אעשה אני רוצה שזה יהיה 5 חצאי פי, אז אני אעשה 5 חצאי פי כפול I אז זה מספר מרוכב ויש לו 5 חצאי פי חלק דמיוני אז זה 5 חצאי פאי כפול אני ומה אני עושה?", "n_reviews": 0, "start": 4234.12, diff --git a/2020/ldm-i-to-i/hindi/sentence_translations.json b/2020/ldm-i-to-i/hindi/sentence_translations.json index cc512214c..da9c73227 100644 --- a/2020/ldm-i-to-i/hindi/sentence_translations.json +++ b/2020/ldm-i-to-i/hindi/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "इसलिए यदि आप संख्या 1 से शुरू कर रहे हैं, तो आपका प्रारंभिक वेग सीधे 0 की ओर चलना है और जैसे-जैसे आप और भी नीचे चलते हैं, यदि आप 1 आधे पर बैठे होते, तो आप अभी भी 0 की ओर चल रहे होते, लेकिन अब आपका वेग वेक्टर जहां आप हैं वहां 1 गुना नकारात्मक होगा, जो 1 आधा नकारात्मक है।", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "और एक दिलचस्प सवाल यह है कि क्या आपको पता है कि क्या ऐसा कोई फ़ंक्शन है जिसके लिए लिखना उचित लगता है क्योंकि आप जानते हैं कि अगर हम इसे एक्स के रूप में लिखने जा रहे हैं तो न केवल इसे संतुष्ट करना चाहिए बल्कि आपको यह भी संतुष्ट करना चाहिए कि कब हम जो नंबर एक प्राप्त करते हैं उसे संभवतः i को पावर वन में प्लग करते हैं, हालांकि हम इस फ़ंक्शन के बारे में सोच रहे हैं कि यह i होना चाहिए।", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "तो हमारे पास 5 पीआई आई है जो शानदार है, यह बिल्कुल एक और मूल्य है जिसे हम यहां एक्स के लिए प्लग इन कर सकते हैं और इसे थोड़ा और अधिक स्पष्ट रूप से बता सकते हैं यदि हम यहां अपने सर्कल को देखें जहां हम हैं क्षण पाई के आधे भाग के बराबर समय तक चला जो कि 1 है।57 क्या होगा अगर इसके बजाय हमने एक और पूर्ण मोड़ लिया और हमें पीआई तक लाने के लिए हम एक और पीआई आधे हिस्से में चले गए, जिसे आप जानते हैं कि हम एक तरह का रिकॉर्ड कर सकते हैं, जहां ई से पीआई मान है, हम एक और पीआई आधे हिस्से में चलते हैं, हम एक और पीआई आधे हिस्से में चलते हैं जो कि इस बिंदु पर हम एक पूरा चक्कर लगा चुके होते हैं और हमें एक पर वापस लाते हैं और फिर हम पांच पाई हिस्सों तक चलते हैं जो संख्यात्मक रूप से लगभग 7 है।85 हाँ, यह बिल्कुल एक और संख्या है जो हमें i के शीर्ष पर ले जाती है और यदि हमें i को घात i तक पुनः व्यक्त करने की पूरी प्रक्रिया से गुजरना होता है तो सबसे पहले 5 pi हिस्सों में e लिखकर i को घात i पर i लिखना होता है।नकारात्मक बनने के लिए गुणा करें और हम ई को नकारात्मक 5 पाई के आधे हिस्से में देखेंगे जो कि एक बहुत ही अलग संख्या है, हम वास्तव में इसकी गणना कर सकते हैं, मैं अपने दिमाग के ऊपर से निश्चित नहीं हूं, लेकिन आइए एक डेस्मोस पर एक नजर डालें . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "वह लंबा जो आपको बहुत कम संख्या में ले जाता है लेकिन यह एकमात्र उत्तर नहीं है जिसे हम सही तरीके से दर्ज कर सकते हैं हमारे पास अन्य लोग भी हैं जो नकारात्मक 3 आधे गुना i pi के साथ यहां आ रहे हैं जिसे आप एक इकाई सर्कल के संदर्भ में जानते हैं? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "हम यह कहने के बारे में सोच सकते हैं कि अरे, अगर मैं 90 डिग्री पाई हाफ रेडियन पर चलने के बजाय I तक पहुंचना चाहता हूं तो क्या होगा अगर मैं 270 डिग्री पर चलता हूं तो दूसरी तरफ 3 पाई हाफ रेडियन पर चलता हूं, जिसे शायद मैं नकारात्मक मानूंगा क्योंकि परंपरा है आम तौर पर यह वामावर्त सकारात्मक है, यह बिल्कुल इसे व्यक्त करने का एक और तरीका है और इससे हमें एक अलग उत्तर मिलेगा यदि हमारे पास नकारात्मक 3 पीआई हिस्सों के लिए ई है मैं पूरी शक्ति से मैं उसी खेल से गुजरता हूं अब मैं वर्ग एक के साथ रद्द कर देता हूं नकारात्मक वह पहले से ही मौजूद है, और हमारे पास सकारात्मक 3 पीआई आधे हैं और संख्यात्मक रूप से यह हमें पहले की तुलना में एक अलग दिखने वाला उत्तर देता है, अगर हम आगे बढ़ते हैं और हम कहते हैं कि अरे, 3 पीआई में ई क्या है न कि 3 या 3 पीआई आधे भाग 111 दशमलव 3 1 उससे बहुत अलग प्रकार की संख्या जो हमने पहले देखी थी 111 बिंदु यह क्या थी 111 दशमलव 3 1 महान 111 दशमलव 3 1 या तो और फिर अंतर्ज्ञान के संदर्भ में आप जो पूछ रहे होंगे वह यह है कि मान लीजिए कि हमारे पास यह घूर्णनशील है गतिशील लेकिन हम समय में पीछे की ओर चलते हैं, हम देखते हैं कि समय में मुझे कितना समय पहले होना है, जैसे कि अगर मैं वहां से चीजों को आगे बढ़ाता हूं तो मैं अपनी प्रारंभिक स्थिति में नंबर एक पर पहुंच जाऊंगा और आपको समय में 3 पीआई आधा यूनिट पीछे जाना होगा और फिर यदि आपको क्षय की गतिशीलता का अनुवाद करना है जो कि इस संदर्भ में आंख उठाने का काम कर रहा है तो आप कहते हैं कि क्या मैं नंबर एक से शुरू कर रहा हूं लेकिन मैं समय में पीछे जाना चाहता हूं और कहना चाहता हूं कि मुझे कहां से शुरू करना चाहिए था मैं ऐसा पतन करना चाहता हूं कि नंबर एक पर आ जाऊं? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "समय की 3 पीआई आधी इकाइयों के बाद उत्तर स्पष्ट रूप से उस तरह के घातीय क्षय के लिए लगभग एक सौ ग्यारह से शुरू हो रहा है और आप देख सकते हैं कि यह कहां जा रहा है जहां वास्तव में अनंत रूप से कई अलग-अलग मूल्य हैं जिन्हें हम एक्स के लिए प्लग इन कर सकते हैं यदि हम हैं ई से एक्स तक के बारे में सोचते हुए मैंने और लोगों ने यहां बहुत अधिक प्रवेश किया है, क्षमा करें, मैंने अपना पिन जमीन पर फेंक दिया जैसे कि कोई तीसरे स्थान के लिए क्लासिक करता है, 9 पीआई हाफ बढ़िया विकल्प 1729 पीआई हाफ आप सभी मेरे पसंदीदा हैं, बहुत सारे और बहुत सारे अलग-अलग विकल्प, अनंत रूप से कई अलग-अलग मान, जो पहली बार में थोड़ा परेशान करने वाला लगता है क्योंकि हम एक अभिव्यक्ति को देखते हैं ऐसा लगता है जैसे आप जानते हैं कि बस कुछ गणना होने वाली है, मैं बस इसे अपने कैलकुलेटर में प्लग करता हूं और देखता हूं कि क्या निकलता है और हमें कई अलग-अलग मिलते हैं इसके लिए मूल्य तो यहाँ क्या चल रहा है? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "16 का चौथा मूल 2 होना चाहिए और उत्तर अच्छा हो जाता है हम एक परंपरा अपनाते हैं जब इस तरह के कई विकल्प होते हैं जब आपके पास एक बहु-मूल्यवान फ़ंक्शन होता है हम अक्सर उन मूल्यों में से एक को चुनते हैं जो हमारा मतलब है जब हम चाहते हैं इसे एक फ़ंक्शन के रूप में मानें जिसमें एक इनपुट और एक आउटपुट के साथ कट्टर भाषा में यह हर समय सामने आता है जब हम जटिल संख्याओं के साथ काम कर रहे होते हैं, एक ऑपरेशन के रूप में कुछ का विचार चाहते हैं कि आप कभी-कभी कई मान रखें वाक्यांश शाखा सुनें आप वर्गमूल फ़ंक्शन की शाखा कहां चुनते हैं? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "क्योंकि कई अलग-अलग उत्तर हैं, आप जानते हैं कि हम फिर से सोचते हैं कि यह 90 डिग्री का घूर्णन है और यदि हम इसे 90 डिग्री के घूर्णन के रूप में सोच रहे थे तो ऐसा लगता है कि वर्गमूल होना चाहिए।आप 45 डिग्री के कोण पर बैठकर कुछ जानते हैं, शायद वह वर्ग है I का मूल जिसे हम बहुत स्पष्ट रूप से मूल 2 बटा 2 मूल 2 बटा 2 I के रूप में लिख सकते हैं, यह सिर्फ त्रिकोणमिति का उपयोग कर रहा है, लेकिन अगर हम I के बारे में नकारात्मक 270 डिग्री रोटेशन के रूप में सोच रहे थे तो ऐसा लगता है कि इसका आधा हिस्सा उस ऑपरेशन का आधा हिस्सा कर रहा है वास्तव में हमें दूसरी तरफ ले जाना चाहिए, हो सकता है कि यहां नीचे बैठी संख्या I का वर्गमूल होनी चाहिए और यह वास्तव में जो हमने पहले देखा था उसका नकारात्मक है नकारात्मक मूल 2 बटा 2 घटा मूल 2 बटा 2 गुना I अब वास्तविक के संदर्भ में मूल्यवान कार्यों के आधार पर हम कह सकते हैं कि हाँ, जो भी सकारात्मक उत्तर हो, उसके लिए वर्गमूल चुनें, लेकिन आप इनमें से किसे सकारात्मक उत्तर मानते हैं? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "और मुझे लगता है कि आप ठीक कहते हैं, हम जानते हैं कि यह क्या है, हम इसे 2 के वर्गमूल के रूप में परिभाषित करते हैं, सब ठीक है और अच्छा है, लेकिन क्या होगा अगर मैंने कहा कि आइए इसे उसी तरह से देखें जैसे हम अपने I को I अभिव्यक्ति I तक ले जा रहे थे।पहले चीजों को ई के रूप में किसी सही चीज़ के रूप में व्यक्त करना चाहता हूं और फिर मैं 1 आधे को घातांक में गुणा करके इसे 1 आधे तक बढ़ाने जा रहा हूं और मैं कहता हूं ठीक है, मुझे लगता है कि मैं वह कर सकता हूं जो ई है।2 के बराबर, यह 2 का प्राकृतिक लघुगणक है, यह एक स्थिरांक है जो 0 के आसपास है।69 या तो यदि हम ई को उस घात तक बढ़ाते हैं तो हमें 2 मिलेगा, इसलिए हम इसे 2 गुना 1 आधे के प्राकृतिक लॉग के रूप में ई के रूप में सोच सकते हैं और यदि आप चाहते हैं कि क्या आप एक्स के लिए ई के बारे में सोच रहे थे? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "आप जानते हैं कि वास्तविक संख्याओं के संदर्भ में यह एक तरह से अतिश्योक्ति हो सकती है, लेकिन यदि आप इस x फ़ंक्शन के लिए ई से एक्स को शॉर्टहैंड के रूप में सोच रहे हैं तो आप मान 0 प्लग इन कर सकते हैं।69 गुना 1 आधा जो मुझे लगता है लगभग 0 होगा।345 कुछ इस तरह है कि आप उस ठोस मान को अपने बहुपद में प्लग करें, देखें कि यह क्या आउटपुट देता है, और यह 1 के आसपास आउटपुट करेगा।414 एक अच्छी वास्तविक संख्या 2 का वर्गमूल जो आप उम्मीद करेंगे लेकिन अगर हम वही करें जो हम सिर्फ I के साथ कर रहे थे और यह स्वीकार करते हुए कि वास्तव में कई अलग-अलग उत्तर हैं जब हम एक घात के लिए ई के रूप में कुछ लिखना चाहते हैं तो हम इसे भी लिख सकते हैं यह अजीब लग सकता है, लेकिन हम इसे 2 प्लस 2 पीआई के प्राकृतिक लॉग में ई के रूप में लिख सकते हैं I वह पूरी चीज़ 1 आधे तक बढ़ गई है, ठीक इसके बाद यह मान बराबर हो जाएगा आप इसे ई के रूप में तोड़ सकते हैं 2 के प्राकृतिक लघुगणक को ई से 2 पीआई तक गुणा करने पर इसमें चीजों को 360 डिग्री तक घुमाने का प्रभाव होता है, इसलिए यह सिर्फ 1 के बराबर होने वाला है इसलिए हम 2 गुना 1 महान को देख रहे हैं जो एक वैध प्रतिस्थापन की तरह लगता है और फिर भी जब हम इसे लेने और इसे एक घात तक बढ़ाने और घातांक को घातांक में गुणा करके उसका इलाज करने का एक ही खेल खेलते हैं, देखें कि क्या होता है, हमारे पास 2 गुना 1 आधा प्लस के प्राकृतिक लॉग में ई है, खैर, 2 पीआई क्या है I गुना 1 आधा खैर यह पीआई गुना होगा I अब यह पहला भाग ई 2 गुणा 1 आधे के प्राकृतिक लॉग के लिए है जो अंत में 2 का परिचित वर्गमूल होगा यह सब ठीक है और अच्छा है, लेकिन हम इसे ई से गुणा करने जा रहे हैं पीआई I सही है और काफी प्रसिद्ध ई से पीआई आई नकारात्मक 1 है, इसलिए इस मामले में यह सुझाव दिया जा रहा है कि यदि हम इस अभिव्यक्ति 2 से 1 आधे को हल कर रहे हैं तो अलग-अलग उत्तरों के साथ खेलकर हम कुछ इस तरह प्लग इन कर सकते हैं ई से एक्स के बराबर 1 आधा जो हम प्राप्त करते हैं वह एक और उत्तर है जिसे हम पारंपरिक रूप से 2 के इस नकारात्मक वर्गमूल के रूप में लिख सकते हैं और यहां मेरा मतलब है कि 2 से 1 आधे को देखने के लिए कई मान होना थोड़ा अजीब है और कहते हैं कि यह एक चीज़ की बराबरी नहीं कर रहा है, लेकिन हमारे द्वारा चुने गए विकल्पों के आधार पर यह कई अलग-अलग चीजों के बराबर हो सकता है, लेकिन दो चीजें जो काफी उचित लग सकती हैं, अगर ऐसा कुछ होने वाला है कि 2 से 1 आधा ऐसा है तो ऐसा लगता है कि यह या तो सकारात्मक होना चाहिए वर्गमूल जिससे हम परिचित हैं या इसका नकारात्मक संस्करण वास्तव में ऐसी कोई समस्या नहीं लगती है और वास्तव में हम उम्मीद कर सकते हैं कि हम इस खेल को आगे भी खेल सकते हैं, मैं आपसे इस अभिव्यक्ति के और भी अधिक रचनात्मक उत्तर मांगता हूँ क्योंकि हो सकता है कि हम घात X के लिए 2 जैसी किसी चीज़ की अन्य मज़ेदार शक्तियाँ पा सकें, क्योंकि हम X के विभिन्न अलग-अलग मानों को प्लग इन करना शुरू करते हैं, जो इस पर आधारित है कि हम क्या प्रतिस्थापन करते हैं यदि हम उन्हीं नियमों का पालन कर रहे हैं जिनका उपयोग हम I से I का मूल्यांकन करने में कर रहे थे।शक्ति I तो इस बार प्रश्न पूछता है या यह निर्दिष्ट करता है कि समीकरण ई से एक्स का एक समाधान 2 के बराबर है जो वास्तविक संख्या है 2 का प्राकृतिक लॉग ठीक है जिसे हम जानते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "प्रश्न का उत्तर ई से एक्स 2 के बराबर है और फिर से रचनात्मकता का स्वागत है, इसलिए मैं आपको इसके लिए एक और छोटा सा क्षण दूंगा II आगे बढ़ूंगा और यहां कुछ उत्तर लॉक करूंगा यदि यह आपके लिए ठीक है, मुझे यकीन नहीं है कि इसमें कितना समय लगेगा आप जिस डिवाइस को देख रहे हैं उसके आधार पर गणित प्रविष्टि करना आवश्यक है, लेकिन अगर आपको उस प्रश्न में प्रवेश करने का मौका मिलने से पहले बहुत तनावग्रस्त न हों जिसे आप उत्तर देना चाहते हैं तो ऐसा लगता है आप में से 131 ने वैरिएंट दर्ज किया है जहां हम 2 का एलएन लेते हैं और हम 2आई जोड़ते हैं और मुझे लगता है कि मैं यह प्रश्न लिख रहा हूं, गलती से एक उत्तर को सही के रूप में चिह्नित कर दिया, जबकि वास्तव में कुछ अलग-अलग सही हैं, तो यह मुझ पर है इस तथ्य के लिए कि मुझे नहीं पता कि क्या यह आप में से किसी को ऐसा लगता है जैसे ओह, यह लाल है, जब आपने 2 प्लस 42 के एलएन में प्रवेश किया तो आप गलत हो गए।", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "आई पीआई जो निश्चित रूप से एक बढ़िया विकल्प है लेकिन आपके पास 4 पीआई आई प्लस 2 या 6 पीआई आई का प्राकृतिक लॉग या वास्तव में 2 पीआई आई का कोई भी पूर्णांक एकाधिक भी हो सकता है यदि आप इसे जोड़ते हैं तो यह ई को प्रभावित नहीं करता है X क्योंकि इसमें केवल e से 2 pi I को गुणा करने का प्रभाव होता है, जो कि 1 से गुणा करने का प्रभाव होता है और फिर से इसका एक अजीब परिणाम होता है, जहां जब हम इसे दूसरे उदाहरण के रूप में करते हैं तो यह उचित परिणाम उत्पन्न करता प्रतीत होता है।ऐसा लगता है कि वहां दर्ज की गई दूसरी सबसे आम अभिव्यक्ति यह थी कि हम 2 को प्रतिस्थापित कर सकते हैं तो आइए सोचें कि हम 2 की घात 1 के 4 के बारे में सोच रहे हैं, ठीक है, एक सुझाव था कि हम 2 प्लस 4 के प्राकृतिक लॉग में 2 को ई से बदल दें।पीआई आई ओके प्लस 4 पीआई मैं और हम उस सब को दाहिनी ओर से 1/4 तक बढ़ाते हैं, यदि आप वही खेल खेलते हैं तो आपको 2 गुना 1/4 के प्राकृतिक लॉग में ई मिलेगा, और हम ई से गुणा करेंगे pi I अब इसका पहला भाग सामान्य सकारात्मक 2 का चौथा मूल होने जा रहा है, हमारा मतलब यह है कि जब आप 2 के चौथे मूल जैसे एक अभिव्यक्ति को एक कैलकुलेटर में एक अच्छी छोटी सकारात्मक संख्या में प्लग करते हैं, लेकिन फिर यह दूसरा भाग है नकारात्मक 1 तो ऐसा लगता है कि आप जानते हैं कि अगर हमें 2 की इस अलग तरीके से व्याख्या करनी है तो इसे 1 4 तक बढ़ा दें, आप जानते हैं कि यह सामान्य उत्तर नहीं है जो हमें मिलता है लेकिन यह एक उचित उत्तर है।", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "हम पाई को आधा गुणा I से देख रहे होते और ऋणात्मक 1 से गुणा करने के बजाय हम I से गुणा कर रहे होते जो फिर से एक वैध उत्तर है, यह 2 से 1 4th जैसी किसी चीज़ के लिए एक उचित आउटपुट की तरह लगता है इसलिए जब आप इस तथ्य को देखते हुए कि मुझे लगता है कि मेरे पास इसके लिए कई अलग-अलग मूल्य हैं, ठीक है, हमारे पास यह अजीब घटना है जहां हम ई को 5 पीआई हिस्सों में प्लग कर सकते हैं I नकारात्मक 3 पीआई हिस्सों I और हमें वह मिलता है जो बेतहाशा अलग-अलग उत्तरों की तरह लग रहा था कुछ बहुत छोटा, कुछ बहुत बड़ा, ये सभी 1/5वें लगभग 1/5वें उत्तर से बहुत अलग हैं जो हमें यहां पहले मिले थे, यह बिल्कुल वैसी ही घटना है जैसे जब आप कुछ पूछ रहे हों कि 2 से 1/4वां क्या है और यह स्वीकार करना कि वास्तव में कई अलग-अलग समाधान हैं अभिव्यक्ति 2 जो भी हो और एक तरीका जिससे हम इस बारे में सोच सकते हैं वह यह है कि जब आप वास्तविक संख्याओं के साथ काम कर रहे होते हैं तो चीजें बहुत प्यारी होती हैं, चीजें अच्छी होती हैं, एक-से-एक रिश्ते होते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "यह बहुत अच्छा है, यदि हम घातीय कार्यों के बारे में सोचना चाहते हैं, तो मुझे इनमें से कुछ चीजों को कवर करने दें, हमारे पास यह अच्छा आगे और पीछे है जहां आप एक्स के आधार के रूप में किसी भी घातांक को व्यक्त करना चुन सकते हैं जैसे कि 2 से एक्स या आप व्यक्त कर सकते हैं वही घातांक जो कि R के X से गुणा X के समान है, जिसे आप जानते हैं कि यह वह बहुपद है जिसे हम संदर्भित करते हैं, जब भी हम X के लिए e जैसा कुछ लिखते हैं, तो इसका स्पष्ट उल्लेख होता है और इसमें आगे और पीछे एक प्यारा सा स्थान होता है, क्योंकि आप B का प्राकृतिक लघुगणक ले सकते हैं।और यह आपको यह मानकर एक उत्तर देता है कि B एक धनात्मक संख्या है और यह वही बात है जो कहती है कि R का X, B के बराबर है इसलिए एक तरीका जिसके बारे में मैंने श्रृंखला में पहले बात की है वह यह है कि यदि आप सभी संभावित घातांकों का परिवार सही है, हम उन्हें R के X से X के रूप में लिख सकते हैं और R को बदल सकते हैं और यह बिल्कुल वही बात है जैसे कि E को R से X के रूप में लिखना, यदि ऐसा कुछ है जिसके साथ आप अधिक सहज हैं तो e से R R का XX गुणा उस आधार को बदलने के लिए पहले तो ऐसा लगता है कि हेरफेर करने के लिए यह एक अलग तरह की अभिव्यक्ति है, लेकिन यह उसी परिवार को व्यक्त करने का एक और तरीका है और एक तरीका है जिसके बारे में आप सोच सकते हैं कि हम इस बारे में कैसे सोचते हैं कि यह किस आधार से मेल खाता है यदि हम थोड़ा अधिक अमूर्त रूप से R के गुणा उस आधार को देखने के बजाय एक चीज़ जो मैं कर सकता हूँ वह यह है कि मूल्य क्या है? ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "मैं R को X से गुणा कर सकता हूं, जहां शायद R शून्य दशमलव छह नौ जैसा कुछ है, लेकिन मैं इसे दो पीआई I से नीचे स्थानांतरित कर सकता हूं और इससे आधार नहीं बदलता है कि यह उसके अनुरूप होगा फिर भी दो के अनुरूप होगा या यह हो सकता है इसे दो पीआई I द्वारा ऊपर ले जाएं, जिससे वह आधार नहीं बदलता है जिससे यह मेल खाता है क्योंकि उन सभी मामलों में जब हम एक्स को एक के बराबर प्लग इन करते हैं तो हमें एक ही चीज मिलती है, हालांकि एक्स के विभिन्न मूल्यों के लिए ये सभी अलग-अलग कार्य हैं।हमने I से घात I के लिए कई अलग-अलग मान क्यों देखे क्योंकि I से X उस संदर्भ में एक अस्पष्ट कार्य है, यह स्पष्ट होगा यदि हम तय करें कि R का कौन सा मान है, जैसे कि हम जो प्रतिनिधित्व कर रहे हैं वह R का क्स्प गुना X का मान है आर का. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "यह एक स्पष्ट कार्य है, लेकिन उस बिंदु पर ऐसा लगता है कि शायद हम जो चाहते हैं वह है घातांक तक बढ़ाए गए कुछ आधार के संदर्भ में चीजों के बारे में सोचना बंद करना।हो सकता है कि जैसे ही हम जटिल संख्याओं के संदर्भ में हों, हमें बस लिखना चाहिए उन सभी को कुछ स्थिर समय के एक्सप के रूप में एक्स यदि किसी अन्य कारण से यह बिल्कुल स्पष्ट नहीं होता है कि हम वास्तव में संख्याओं को कैसे प्लग करते हैं यदि हम एक गणना करना चाहते हैं या बस इसके शीर्ष पर गणित करना चाहते हैं तो हमें यह अच्छा अनंत बहुपद मिल गया है उन्हें प्लग इन करें और मैं आपके लिए एक और मामला बनाऊंगा कि यह शायद घातांक के बारे में सोचने का सही तरीका है जैसे ही हम अन्य डोमेन में जटिल संख्याओं जैसी चीजों का विस्तार कर रहे हैं और इसके लिए आइए बस बैक अप लें।वापस डोरबेल पर कुछ चीजें आ गई हैं, मूल तरीके पर वापस जाएं, जिससे हम घातांक के विचार का विस्तार करते हैं और बस एक्स राइट के लिए 2 क्या है, इसके बारे में सोचते हैं, हम जानते हैं कि प्राकृतिक संख्याओं के लिए इसके बारे में कैसे सोचना है।", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "आप 2 से 3 बार-बार गुणन जैसे कुछ जानते हैं कि यह कैसे होता है कि आपको सबसे पहले आंशिक मात्रा के लिए 2 से एक्स या नकारात्मक मात्रा और इस तरह की चीजों के बारे में सोचना सिखाया जाता है। ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "आपको आम तौर पर सिखाया जाता है कि 2 से 1 आधा कुछ ऐसा होना चाहिए जहां आप जान सकें कि क्या मैं इसे स्वयं से गुणा करता हूं और यह सामान्य नियमों का पालन करता है जो एक्सपोनेंशियल संख्याओं की गिनती के साथ करते हैं जहां हम उस एक्सपोनेंट में चीजें जोड़ने में सक्षम होते हैं, मुझे 2 मिलना चाहिए 1 से तो यह कुछ ऐसी संख्या होनी चाहिए कि जब मैं इसे अपने आप से गुणा करूं तो मुझे 2 मिले और आप जानते हैं कि उस समय आपके पास एक विकल्प है, शायद यह सकारात्मक है। ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "हो सकता है कि यह नकारात्मक हो, लेकिन यदि आप हमेशा सकारात्मक विकल्प चुनने का निर्णय लेते हैं तो आप इसी सौदे से एक अच्छा निरंतर कार्य प्राप्त करने में सक्षम होंगे यदि हम नकारात्मक संख्याओं के बारे में पूछें तो 2 से ऋणात्मक 1 क्या होना चाहिए, यह कुछ होना चाहिए जब मैं इसे 2 से 1 तक गुणा करता हूँ तो कहाँ होता है? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "यह मुझे 2 से 0 तक ले जाता है और यह हमारे सम्मेलन के लिए एक तरह का औचित्य है कि नकारात्मक घातांक 1 आधे की तरह दिखते हैं लेकिन यहां वास्तव में क्या हो रहा है हम कह रहे हैं कि यह जो कुछ भी है यह किसी प्रकार का कार्य होना चाहिए जो इस संपत्ति को संतुष्ट करता है ए प्लस बी, बी के एफ के बराबर एफ है और इसके अलावा यह तथ्य कि आधार 2 है, मूल रूप से हमें बता रहा है कि यह ऐसा कोई फ़ंक्शन नहीं है, यह एक फ़ंक्शन है जहां जब हम 1 प्लग इन करते हैं तो हमें 2 मिलते हैं और जैसा कि आप थोड़ा जानते हैं यह देखने के लिए कि क्या आप यहां कुछ निहितार्थों का अनुसरण कर रहे हैं, विवेक जांच शैली प्रश्न मैं आपसे पूछना चाहता हूं कि मैं इसे सॉफ्टबॉल की तरह नहीं कहूंगा, लेकिन इसका मतलब यह नहीं है कि यह एक अविश्वसनीय रूप से गहरा प्रश्न है अनिवार्य रूप से।", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "यदि आप किसी फ़ंक्शन के गुणों के साथ संक्षेप में शुरू करने और फिर उन गुणों के आधार पर इसे लिखने के तरीके को निकालने के विचार का अनुसरण कर रहे हैं तो यह एक जांच से अधिक है यदि x का f इस घातीय गुण f को संतुष्ट करता है सभी इनपुट के लिए a प्लस b का f का गुना f का b के बराबर होता है और यह 1 के बराबर 2 के f को भी संतुष्ट करता है, निम्नलिखित में से कौन सा सत्य है, इसका मतलब यह है कि निम्नलिखित में से कौन सा आवश्यक रूप से सत्य है, इससे कोई फर्क नहीं पड़ता कि आप कौन सा ऐसा फ़ंक्शन शुरू कर रहे हैं आप में से जिन लोगों को याद है कि वह कौन सा व्याख्यान था, यह वह है जिसके बारे में हम बात कर रहे थे कि यूलर का सूत्र वास्तव में क्या कह रहा है उसकी व्याख्या कैसे करें, मैंने इस शैली का एक प्रश्न पूछा था जहां मैंने एक भी शर्त की उपेक्षा की थी, आप जानते हैं कि मैंने इसे नहीं लिखा था तथ्य यह है कि हम यह सुनिश्चित करना चाहते हैं कि x का f हर जगह गैर-शून्य है और फिर इससे कुछ मात्रा में कन्फ्यूडलमेंट हुआ जो कि अच्छा है, स्क्रीन पर कन्फ्यूडलमेंट मिलता है जो हम सभी के साथ होता है लेकिन इसका इरादा मूल रूप से यह दिखाना था कि यह अमूर्त संपत्ति है कुछ ऐसा जो जोड़ को गुणन में बदल देता है, वह मूल रूप से आपको फ़ंक्शन को लिखने के लिए प्रेरित करने के लिए पर्याप्त है, जो कुछ भी उसके बराबर होता है जैसे कि किसी प्रकार की शक्ति तक बढ़ाया गया यह प्रश्न की भावना है अब हमारे पास वास्तव में बिजली टावरों के बारे में कुछ प्रश्न हैं ऐसा प्रतीत होता है कि यह यहाँ उभर कर आया है जो पिछली बार से जुड़ा हुआ है, आइए पावर टावर के प्रश्न को एक पल के लिए रोक दें ताकि हम पहले इस बात को गहराई से महसूस कर सकें कि यहाँ घातांक का क्या अर्थ होना चाहिए? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "क्योंकि हम वही हो सकते हैं जो मैं दावा करना चाहता हूं कि हम इसका उत्तर कई अलग-अलग तरीकों से दे सकते हैं, इसलिए यदि आप मुझे सिर्फ एक देते हैं, तो हम बिजली टावरों के बारे में बात करेंगे और फिर जिस तरह एक संख्या रेखा को लघुगणकीय पैमाने में दर्शाया जा सकता है।एक जटिल विमान के लिए भी ऐसा ही किया जाना चाहिए? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "हाँ वास्तव में, यहाँ एक दृश्य है जो मैं बस एक क्षण में प्राप्त करने जा रहा हूँ जहाँ हम कुछ ऐसा ही करते हैं क्योंकि हम जो करेंगे वह अलग-अलग घातीय कार्यों के साथ खेलेंगे आर के उस मान को बदलने जा रहा है जिसे एक छोटे से पीले बिंदु द्वारा दर्शाया जाएगा इसलिए हम इसके माध्यम से बात करेंगे, यह पूरे विमान को मैप नहीं करेगा, बल्कि वास्तविक अक्ष और काल्पनिक अक्ष से केवल कुछ नमूना बिंदुओं को मैप करेगा।लेकिन विचार यह है कि जैसे-जैसे हम उस स्थिरांक के चारों ओर घूमते हैं, हम विभिन्न चीजों की कल्पना करने में सक्षम होने जा रहे हैं जो यह विमान पर करता है और प्रभावी रूप से यह ऐसा है जैसे यह एक्स-अक्ष को लघुगणकीय पैमाने में बदल रहा है और फिर लपेट रहा है एक वृत्त के अनुदिश काल्पनिक अक्ष और फिर जैसे ही R का वह मान काल्पनिक हो जाता है, यह उन वास्तविक संख्याओं की भूमिका को बदल देता है, जिन्हें वृत्त पर डाल दिया जाता है और काल्पनिक संख्याओं को एक लघुगणकीय स्केल किए गए सकारात्मक अक्ष पर रख दिया जाता है, इसलिए मुझे लगता है कि ये तीनों ही बहुत अच्छे प्रश्न हैं मैं जहाँ जाना चाहता हूँ उसके लिए एक तरह से आगे बढ़ने की कोशिश कर रहे हैं, लेकिन यह देखकर अच्छा लगा कि लोग इस मामले में ऐसा सोच रहे हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "स्पष्ट रूप से 5 का f जैसा कुछ, 1 का f, प्लस 1, प्लस 1 प्लस 1 प्लस 1 के समान है, जो कि इस गुण के कारण 1 के f को अपने आप से 5 गुना गुणा करने के समान है, यदि 1 का f 2 है, तो यह समान है 2 की घात 5 और फिर ऋणात्मक 5 के f जैसा कुछ।यह मामला होना चाहिए कि जब हम इसे 5 के f से गुणा करते हैं तो हमें 0 का जो भी f होता है वह प्राप्त होता है और यह तुरंत स्पष्ट नहीं है कि 0 का f क्या है लेकिन हम ऐसा कह सकते हैं 1 का f और 0 बराबर है 1 का f, 0 के f का गुना है, लेकिन 1 का f, 2 के बराबर है और इसलिए यह भी 2 के बराबर है, इसलिए हम कह रहे हैं कि 2, 2 के बराबर है कुछ अच्छा है कि कुछ 1 होना चाहिए इसलिए इस संदर्भ में यह गारंटी देता है कि ऋणात्मक 5 का f, ऋणात्मक 5 का 2 है, यह 2 से 5वें के ऊपर 1 है, हम इसे स्पष्ट रूप से ऋणात्मक 5 के 2 के रूप में लिख सकते हैं, जिसका अर्थ है कि ये दोनों गुण मिलकर बनाते हैं हम वास्तव में फ़ंक्शन को 2 से X के रूप में लिखना चाहते हैं क्योंकि इसमें जो भी गिनती संख्या हम डालते हैं वह संतुष्ट हो जाएगी ऐसा लगता है कि यह अपने आप से गुणा करने जैसा प्रतीत होगा कि हम इसमें जो भी भिन्नात्मक संख्या डालते हैं वह इन गुणों को संतुष्ट करने वाली है जो हम चाहते थे और आपको आश्चर्य हो सकता है कि क्या यह अद्वितीय है और वास्तविक मूल्यवान कार्यों के संदर्भ में यह वास्तव में होगा लेकिन जटिल मूल्यवान कार्यों के संदर्भ में ऐसे कई फ़ंक्शन होंगे जिनके लिए हम लिख सकते हैं जिनमें से एक यह है कि हम क्या थे पहले देखने पर जहां हम एक फ़ंक्शन को 2 प्लस 2 पीआई के प्राकृतिक लॉग के एक्सप के रूप में परिभाषित कर सकते थे, मैं उस समय के सभी एक्स ठीक है, यहां लापरवाही को माफ कर दें, मैं बस इसके बारे में लिखने के लिए उत्साहित हूं और यह वास्तव में एक अलग फ़ंक्शन है इसका प्रमाण यह है कि यदि आप एक्स को 1 आधे के बराबर प्लग इन करते हैं तो क्या होता है, हमने थोड़ा पहले देखा था कि जब आप 1 आधे को प्लग इन करते हैं तो आपको 2 का नकारात्मक वर्गमूल मिलता है और फिर यदि आप 1 चौथाई को प्लग इन करते हैं तो आपको चौथा मूल नहीं मिलता है।2 लेकिन मैं 2 के चौथे मूल को गुणा करता हूं इसलिए यह एक अलग कार्य है लेकिन यह अभी भी इन गुणों को संतुष्ट करता है और यह हमें इसे 2 से एक्स के रूप में लिखना चाहता है और यह सुझाव देता है कि शायद 2 से एक्स एक अस्पष्ट है अंकन का अंश और हमें हर चीज़ को आर के क्स्प के संदर्भ में कुछ लिखना चाहिए, लेकिन आपको आश्चर्य हो सकता है कि आप जानते हैं कि शायद हम इस संपत्ति को संतुष्ट करने वाले सभी कार्यों के साथ पर्याप्त रूप से रचनात्मक नहीं हो रहे हैं, हो सकता है कि जब हम क्स्प लिखते हैं तो एक अस्पष्टता हो।R का समय कुछ और है और R के अलग-अलग मूल्य हैं जो चलन में आ सकते हैं लेकिन मैं बस एक छोटा सा दावा करने जा रहा हूं और फिर शायद एक स्केच की तरह दे सकता हूं कि यदि आप चाहें तो सबूत कैसा दिखेगा, वह कौन सा है आइए जानें मान लें कि आपके पास कुछ जटिल फ़ंक्शन F है, और यह पहले निम्नलिखित गुणों को संतुष्ट करता है आप इसका व्युत्पन्न लेने में सक्षम हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "यह अलग-अलग है जो इसे कुछ ऐसा होने से रोकता है जिसे आप पूरी तरह से गन्दा असंतत चीज़ के रूप में जानते हैं, यह कुछ यादृच्छिक मूल्यों को लेने जैसा है जो इस बात पर निर्भर करता है कि आप किसी भी वेक्टर स्पेस की अवधि को जानते हैं, मुझे भिन्नात्मक मात्राएं नहीं पता हैं जिनके बारे में आप पागल तरीकों से सोचना चाहते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "यह एक अच्छा कार्य है. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "यह अलग-अलग है, यह हर जगह 0 के बराबर नहीं है, इसलिए यह स्थिति मेरे दिमाग से फिसल गई और मैं भूल गया कि कौन सा व्याख्यान व्याख्यान या उसके जैसा कुछ है और फिर इसमें यह केंद्रीय गुण है कि यह जोड़ को गुणन में बदल देता है यदि आपके पास ऐसा कोई कार्य है तो मैं दावा करता हूं कि वहाँ एक अद्वितीय है शायद मुझे वास्तव में निर्दिष्ट करना चाहिए कि एक अद्वितीय जटिल संख्या आर मौजूद है ताकि आप एक्स के एफ को मूल रूप से आर के इस घातीय फ़ंक्शन के रूप में लिख सकें जो कि मूल्य एक्स का गुना है जिसे आप मूल रूप से कह रहे हैं कि यदि आपके पास एक्स एक फ़ंक्शन के रूप में है अच्छे व्युत्पन्न गुणों के साथ अनंत बहुपद और यह सब यदि आपके पास यह है तो आपके पास प्रत्येक घातांक है जिसे आप घातांक शब्द के एक बहुत ही अमूर्त सामान्य अर्थ में चाहते हैं, बस उस संपत्ति पर आधारित है जिसे हम इससे चाहते हैं और सबूत का स्केच होगा यदि आप पहले यह देखना चाहते हैं कि इस मान का व्युत्पन्न क्या है, जिसे हम हर जगह मौजूद मानते हैं, तो कुछ इस तरह देखें, है ना? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "हम एक्स में से एफ को पूरी तरह से अभिव्यक्ति से बाहर कर सकते हैं और पूरी सीमा केवल एच के संदर्भ में व्यक्त की जाती है, यदि आप सोचते हैं कि डेरिवेटिव के संदर्भ में इसका क्या मतलब है और तथ्य यह है कि 0 का एफ आवश्यक रूप से 1 के बराबर है, यह पूरी सीमित अभिव्यक्ति है बस कुछ स्थिर लेकिन अधिक विशेष रूप से यह 0 पर हमारे फ़ंक्शन का व्युत्पन्न जो भी है, तो आपके पास यह मज़ेदार चीज़ है जहां यदि आप 0 पर इसका व्युत्पन्न जानते हैं तो यह निर्धारित करता है कि इसका व्युत्पन्न हर जगह क्या है और घातीय कार्यों के संदर्भ में यह उम्मीद से काफी परिचित है क्योंकि वास्तव में हम जो कुछ कह रहे हैं वह यह है कि एक घातीय फ़ंक्शन का व्युत्पन्न स्वयं के लिए आनुपातिक है और आनुपातिकता स्थिरांक 0 पर जो भी व्युत्पन्न है उसके बराबर है, यह सब बहुत ही सारगर्भित रूप से व्यक्त किया गया है और ऐसा है लेकिन इसका उद्देश्य इस बात पर जोर देना है कि यह है जरूरी नहीं कि केवल वे कार्य ही हों जिन्हें हम पहले से ही घात दूसरा व्युत्पन्न और उस मामले के लिए एक तीसरा व्युत्पन्न और ऐसा इसलिए क्योंकि व्युत्पन्न फ़ंक्शन केवल स्वयं के लिए आनुपातिक है इसलिए n वें व्युत्पन्न लेने के लिए आप बस उस आनुपातिकता स्थिरांक को देखें और इसे घात n तक बढ़ाएं और फिर यहां से आप एक कर सकते हैं टेलर श्रृंखला का विस्तार और मैं इसे आपमें से उन लोगों के लिए उन्नत होमवर्क के रूप में छोड़ सकता हूं जो उस विचार में टेलर श्रृंखला के साथ सहज हैं, खासकर यदि आप किसी भी भिन्न फ़ंक्शन के विचार को मिश्रित करना चाहते हैं जो जटिल संख्याओं के अर्थ में भिन्न है, जो है निश्चित रूप से एक कॉलेज विषय की तरह, आप जानते हैं कि आप वहां तर्क को अपनी इच्छानुसार मिश्रित कर सकते हैं, लेकिन अस्पष्ट तर्क की अनुमति किसी ऐसे व्यक्ति के संदर्भ में दी जाती है जो केवल टेलर श्रृंखला के बारे में जानता है और इस विचार को लेने के लिए और कुछ नहीं और एफ के लिए टेलर विस्तार को देखता है।इस विचार को उचित ठहराएं कि एक अद्वितीय जटिल संख्या है जैसे कि हमारा फ़ंक्शन एफ आवश्यक रूप से इस तरह लिखा जा सकता है और फिर सामान्य घातांक का कनेक्शन तब होता है जब आपके पास ऐसा कोई मान होता है आर हम अनिवार्य रूप से वही करते हैं जो हम वास्तविक संख्याओं के जटिल संदर्भ में करते हैं यदि आप उस मान R के उस फ़ंक्शन के exp को देखते हैं और उसे आधार के रूप में लिखते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "हम इसकी व्याख्या कर सकते हैं कि न केवल पाई के आधे हिस्सों के विस्तार का अर्थ है I गुना उन्हें I से X के रूप में लिखें, इसलिए अभिव्यक्ति I से I तक, जब तक कि आपने इसके लिए एक मानक नहीं अपनाया है कि इसका अनिवार्य रूप से क्या मतलब होगा, जब आप कहते हैं कि इसमें असीमित रूप से कई आउटपुट हैं, तो इसके बारे में सोचने का एक और तरीका यह है कि फ़ंक्शन I से X तक हमारे पास जो संकेतन है वह थोड़ा अस्पष्ट है।अब इन सबके साथ, आइए इसमें से कुछ की कल्पना करना शुरू करें क्योंकि मुझे लगता है कि यह मजेदार है और आप जानते हैं कि आप मुझे बताएं कि क्या यह एक उपयोगी दृश्य है या अधिक भ्रमित करने वाला दृश्य है लेकिन हम जो करने जा रहे हैं वह आर गुना एक्स के इस फ़ंक्शन एक्सपी को देखना है, जो मूल रूप से एक्स की शक्ति के लिए ई लिखने का एक और तरीका है वास्तव में मुझे लगता है II लगता है कि मैंने कुछ बिंदु पर एक अलग एनीमेशन प्रस्तुत किया है जो निर्दिष्ट करता है क्योंकि मैं ऐसा करने की योजना बना रहा था, इसलिए मुझे जाने दीजिए, ओह हाँ, आप मेरे फ़ाइल सिस्टम में वापस आ गए हैं, जहाँ आपको होना चाहिए वहाँ वापस आ जाएँ, वहाँ जाएँ, क्या यह शिकायत कर रहा है क्योंकि वहाँ कई अलग-अलग चीजें हैं, यह ऐसा होने वाला है जैसे वहाँ एक है ओह बदलें यह दूसरी स्क्रीन पर दिखाई देता है रुको ऐसा क्यों है हाँ, ठीक है बदलो? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "आप जो कुछ भी देखते हैं उसे वहां रखें और अब हम वापस जाते हैं ओह वहां हम वह सब, ताकि मैं अच्छी तरह से लिख सकूं यदि आप इसे इस अनंत बहुपद के आर गुना एक्स के एक्सप के रूप में सोचने में असहज हैं तो बस में आपके सिर के पीछे ई से आर गुना एक्स और हम आर के आसपास भिन्न होने वाले हैं इसलिए मैं काल्पनिक अक्ष के बिंदुओं का पालन करने वाला हूं, और मैं वास्तविक अक्ष के बिंदुओं का पालन करने वाला हूं और आइए देखें कि यह क्या करता है खैर यह सब बहुत तेजी से हो रहा है, इसलिए मुझे इसके बारे में थोड़ा और धीरे-धीरे सोचने दीजिए, सभी नकारात्मक संख्याएं कुछ भी हैं, यह एक नकारात्मक वास्तविक संख्या है जो 0 और 1 के बीच की सीमा में सिमटने वाली है, जिसे नकारात्मक के लिए ई का अर्थ बनाना चाहिए? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "एक ऋणात्मक वास्तविक संख्या 0 और 1 के बीच है और हम विशेष रूप से ऋणात्मक 1 के f को ट्रैक कर रहे हैं जो कि 30 0 के आसपास e के ऊपर 1 दिखाई देगा।1 का 37 एफ ई पर उतरता है जैसा कि अपेक्षित है 1 का एक्सप एफ का एफ है, मैं यूनिट सर्कल के चारों ओर एक रेडियन लैंड करने वाला हूं, और यहां पूरे काल्पनिक अक्ष के साथ अनुसरण करना मजेदार है कि कैसे काल्पनिक अक्ष एक सर्कल के चारों ओर लपेटा जाता है और जब हम R के इस मान को बदलते हैं तो क्या होता है? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "हम यहां R का मान चाहते हैं और यह चीजों को अलग तरह से फैलाता है इसलिए जब हम इसे 2 तक रखते हैं तो आप जानते हैं कि यह वास्तविक अक्ष को बहुत अधिक फैलाता है ताकि 1 में से f उस स्थान पर समाप्त हो जाए जहां e का वर्ग नकारात्मक के 7 f से थोड़ा ऊपर है।1, I के 0 f के बहुत करीब है, एक 2 रेडियन है, ऋणात्मक I के वृत्त f के चारों ओर घूर्णन, एक ऋणात्मक 2 रेडियन घूर्णन है और निश्चित रूप से हम अपने पसंदीदा सूत्र तक पहुँच सकते हैं कि यदि वह pi होता जो हमारे स्केलिंग स्थिरांक के रूप में हमारे पास था, तो वास्तविक अक्ष काफ़ी खिंच जाता है, आप जानते हैं कि 1 का f, pi के e से दूर बैठा है, जो कि 20 प्लस pi के बहुत करीब है, जो कि हमेशा मजेदार होता है और नकारात्मक 1 का f, 0 के बेहद करीब होता है, इसलिए यह वास्तव में वास्तविक रूप से फैला हुआ है अक्ष और यह चीजों को इकाई वृत्त की दिशा में भी फैलाता है ताकि I या F के नकारात्मक तक पहुंचने पर I वृत्त के चारों ओर आधा घूम जाए, तो अब यह सब ठीक है और अच्छा है कि हम किसी फ़ंक्शन के बारे में कैसे सोचेंगे? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "हम 2 गुना X के प्राकृतिक लॉग के X को X के रूप में भी लिखेंगे, इसलिए हम R के मान का प्रतिनिधित्व करने वाले अपने पीले बिंदु को 0 के आसपास ले जाते हैं।69 अभी भी कोई काल्पनिक भाग नहीं है बस एक वास्तविक संख्या 0 है।69 या तो यह 2 का प्राकृतिक लॉग है, आप देख सकते हैं कि 1 का एफ 2 पर उतरता है, यही कारण है कि हम इस फ़ंक्शन 2 को 1 आधे के एक्स एफ पर कॉल करना चाहते हैं, वास्तव में नकारात्मक 1 का एफ, 1 आधे एफ पर उतरता है।मैं यूनिट सर्कल के चारों ओर कुछ घूम रहा हूं, विशेष रूप से यह 0 होने जा रहा है।यूनिट सर्कल के चारों ओर 69 रेडियन और अब हम थोड़ा और मजा कर सकते हैं और कह सकते हैं कि अगर हम इसे 0 के बजाय बदल दें तो क्या होगा।69 को 2 का प्राकृतिक लघुगणक बनाने के बजाय इसे 2 के प्राकृतिक लघुगणक का I गुना बनाएं ताकि हम वास्तव में किसी ऐसी चीज़ के बारे में सोच सकें जिसका एक घातांकीय आधार हो।", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "इस मामले में I की शक्ति I क्या है, यह इसे लगभग 0 पर धकेल देती है।2 पांचवें के आसपास लेकिन कई अलग-अलग घातीय कार्य हैं जिनमें संख्या I पर 1 का एफ लगाने की यह संपत्ति होगी इसलिए अगर हमें इसे और भी बढ़ाना है तो मुझे नहीं लगता कि मैंने इसे यहां एनिमेटेड किया है लेकिन अगर हमें लेना है वह पीला बिंदु और इसे तब तक ऊपर उठाएं जब तक कि यह pi I से 5 आधा गुना न हो जाए आप क्या देखेंगे कि इकाई चक्र क्या है? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "अपने चारों ओर घुमाया जाता है ताकि 1 के नकारात्मक एफ का एफ अन्य 2 पीआई रेडियन के चारों ओर घूम सके और जहां यह है वहां उतर सके लेकिन यह वास्तविक अक्ष को बहुत अधिक फैला देगा जो कि वह अर्थ था जिसमें आई से आई का एक और आउटपुट है बहुत छोटी संख्या यह 0 के आसपास थी।0003 या तो लेकिन हम यह भी देख सकते हैं कि जो मुझे लगता है वह काफी मजेदार है यदि हम वैकल्पिक अभिव्यक्तियों पर विचार करते हैं जिन्हें हम 2 से घात X के रूप में व्याख्या करना चाहते हैं तो क्या होता है? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "हमारे पास R का X गुना X है और R इस मान के बराबर है, जो 2 प्लस pi गुना I का प्राकृतिक लॉग है।इसका मतलब यह है कि जब हम 1 में से 1 f प्लग इन करते हैं तो 1 का f नकारात्मक 2 पर होता है, इसलिए हम इस फ़ंक्शन को लिखना चाहते हैं घात X के दाईं ओर ऋणात्मक 2 के रूप में और यह वास्तव में कुछ ऐसा है जिसे आप जानते हैं, यह थोड़ा भ्रामक रूप से सरल है जब हम घात X के लिए ऋणात्मक 2 लिखते हैं तो घात X के लिए यह पहली बार में आवश्यक रूप से ऐसा नहीं दिखता है कि यह हमें लाता है किसी भी तरह से जटिल संख्याओं में, लेकिन निश्चित रूप से जब हम 1 आधे जैसे मान को भी जोड़ते हैं, जहां हम नकारात्मक 2 के वर्गमूल के लिए पूछ रहे होते हैं तो हमें एहसास होता है कि हम इसे कुछ इस तरह लिखना चाहते हैं जैसे कि मैं वर्गमूल को गुणा करता हूं 2 में से, लेकिन यदि आप इस फ़ंक्शन को पूर्ण जटिल डोमेन में घात यह वास्तविक संख्या रेखा के बाकी हिस्सों के साथ क्या यह बाहर की ओर सर्पिल होता है? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "तो हम देखते हैं कि ऋणात्मक 1 का f ऋणात्मक 1 आधे पर बैठता है, यदि आप 1 आधे के f का अनुसरण करते हैं तो आप कहाँ उम्मीद करेंगे, यह बिल्कुल काल्पनिक रेखा पर बैठेगा और 1 आधे का f 2 का वर्गमूल होगा, ठीक है, मेरी माउस वह नहीं है जहाँ मैं चाहता हूँ।", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "यह 2 गुना I के वर्गमूल के आसपास होगा और जैसा कि आप आगे जारी रखते हैं, यह आपको नकारात्मक 2 से X की सभी वास्तविक मूल्य शक्तियां दिखा रहा है, यह आवश्यक रूप से चारों ओर घूमता है, लेकिन हम R के अपने मान को और भी अधिक बढ़ा सकते हैं और इसे प्राप्त कर सकते हैं लगभग ताऊ समय तक I लगभग छह दशमलव दो आठ बार I और उस संदर्भ में यह एक और फ़ंक्शन है जिसे हम X के लिए 2 की तरह कुछ लिखना चाहेंगे क्योंकि किसी भी पूर्ण संख्या से पूर्ण संख्या के लिए जिसे आप X के लिए प्लग इन करते हैं, यह होगा बार-बार गुणा करने जैसा दिखता है और यहां तक कि इसमें 1 आधे जैसी चीजों के लिए उचित मूल्य भी हैं जहां यह सकारात्मक वर्गमूल के बजाय नकारात्मक वर्गमूल निकालता है, लेकिन यह वास्तव में जो कर रहा है वह उस स्तर में परिवर्तन है जहां यह सब कुछ डालता है वह वास्तविक है संख्या रेखा अंत में एक बहुत कसकर लपेटा हुआ सर्पिल बन जाती है जो चारों ओर घूमता है और यह इस तरह से सर्पिल होता है कि 1 में से एफ सीधे संख्या 2 पर गिरता है, इसलिए यह उस अर्थ में है कि हम एक्स को 2 कह सकते हैं, इसकी प्रशंसनीय रूप से व्याख्या की गई है पारंपरिक रूप से हम जिस फ़ंक्शन के आदी हैं, उससे एक अलग घातीय फ़ंक्शन, इसलिए मुझे लगता है कि इन सबके साथ, मैं चीजों को आज के लिए छोड़ दूंगा और मैं आपको सोचने के लिए बस कुछ लंबित प्रश्नों के साथ छोड़ दूंगा, ठीक है, इसलिए यदि आप चाहते हैं I से I तक को एक बहु-मूल्यवान अभिव्यक्ति के रूप में सोचें, ठीक है, आप कह सकते हैं कि हम एक सम्मेलन को अपनाते हैं, काल्पनिक रूप से आप कहेंगे कि आप प्राकृतिक लघुगणक फ़ंक्शन की एक शाखा चुनते हैं और हो सकता है कि यह आपको इस ई से नकारात्मक pi में बंद कर दे।आधे लेकिन अगर आप कहते हैं कि इस प्रकार के विभिन्न मान अनंत रूप से भिन्न होना चाहते हैं, जैसे कि हमने जो विभिन्न मान देखे हैं, 2 से 1 तिहाई एक ही अर्थ में कितने मान होना चाहते हैं? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "मैं एक्स के सभी घातीय कार्यों एफ के बारे में कहना चाहता हूं कि क्या मैंने इसे कहीं लिखा है जो एक्स के सभी गुणों को संतुष्ट करता है जो मैंने लिखा है, इसलिए यदि यह सभी को संतुष्ट करता है इनमें से और यदि 1 का f, 2 के बराबर है, तो किस फ़ंक्शन के लिए विभिन्न विकल्पों के लिए X बराबर 3 10 वें प्लग इन करने पर हमें कितने अलग-अलग आउटपुट मिलेंगे? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "विभिन्न कार्यों के लिए 2 से पीआई के लिए जो 2 से एक्स का प्रतिनिधित्व कर सकता है यदि हम 2 से एक्स के बारे में किसी प्रकार के घातीय फ़ंक्शन के रूप में सोच रहे हैं तो इस प्रकार के अमूर्त गुणों के अर्थ में एक्सपोनेंशियल और यदि हम हाँ, यदि हम यदि हमारे पास ऐसे विभिन्न कार्यों की एक कक्षा है, और हम पीआई को प्लग इन करना चाहते हैं, यह मुझे हंसाता है सिर्फ इसलिए कि यह एक ऐसा है, मैं जानता हूं कि यह एक अजीब जवाब है जो तब सामने आता है जब आप इसके बारे में सोचने की कोशिश कर रहे होते हैं, इसलिए ये वे प्रश्न हैं जो मैं आपका साथ छोड़ता हूँ और मुझे लगता है कि यह आप जानते हैं कि आज के व्याख्यान में मेरा मुख्य प्रश्न यह था कि क्या मैं चाहता था कि यह घातीय कार्यों के इन अमूर्त गुणों की तरह वर्णन करे और यह मेरे लिए बहुत अच्छा है कि उन अमूर्त गुणों से शुरुआत करें आप ई से आरएक्स या अधिक के विचार में बंद हो जाते हैं, बस आप जानते हैं कि मुझे लगता है कि आर के विभिन्न मूल्यों के लिए आर गुना एक्स का अधिक ईमानदारी से लिखा गया एक्सप यह आपको उस दूर तक बंद कर देता है, लेकिन यह आपको जहां तक हो रहा है, में बंद नहीं करता है घातांक 2 से लेकर घातांक तक की एक स्पष्ट धारणा बहुत कम होनी चाहिए जैसे I से घातांक यदि आप जानते हैं तो बस मुझे बताएं, मुझे लगता है कि विचारों का एक पूरा दिलचस्प चक्र है जो इन सभी चीज़ों से घिरा हुआ है, जिसमें पावर टावर भी शामिल हैं क्योंकि यदि आप वास्तव में पावर टावरों के बारे में बात करना चाहते हैं जैसे कि हम पिछली बार जटिल संख्याओं के संदर्भ में थे।या यहां तक कि नकारात्मक आधारों के साथ भी आपको इस तरह की चीजों के बारे में सोचना होगा, इसलिए यह एक सवाल था जो हमने स्क्रीन पर उठाया था हां, अगर हम I की शक्ति के लिए I के लिए ऐसा करते हैं तो क्या होगा? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "अनुमापन आप जानते हैं आइए बस इसे आज़माएं आइए बस आगे बढ़ें और एक पावर टावर का प्रयास करें जहां हम I को किसी दिए गए पावर तक बढ़ा रहे हैं और देखें कि इससे क्या निकलता है, इसलिए ऐसा करने की कोई योजना नहीं थी लेकिन हम ऐसा हमेशा कर सकते हैं पायथन को ऊपर खींचें और अनिवार्य रूप से वही करें जो हम पिछली बार कर रहे थे, तो जिस तरह से यह काम करेगा वह यह है कि हम कुछ आधार मूल्य के साथ शुरू कर रहे थे और फिर कुछ प्रकार की सीमा के लिए हम क्या कर रहे थे हम एक ले रहे थे और हम फिर से असाइन करने जा रहे हैं यह कुछ भी हो, इस मामले में जो आधार है, उसे मैंने a की शक्ति तक बढ़ाया है, ठीक है, अच्छा है, इसलिए हम ऐसा करने जा रहे हैं और फिर हम a का मान प्रिंट करने जा रहे हैं, आइए बस इसके लिए ऐसा करें हाँ, यह 200 जैसी बहुत बड़ी संख्या है इसलिए ऐसा लगता है कि कभी-कभी इन चीज़ों से अराजकता की संभावना होती है।", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "मेरे पास वास्तव में है इसलिए मुझे NumPy आयात करने दीजिए ताकि मेरे पास घातीय कार्य हो, मुझे हमारी बड़ी रेंज के लिए जाने दें जैसा कि हमारे पास पहले था, इसे लिखने के बजाय क्योंकि आप कुछ जानते हैं जो कि एक्स की शक्ति के समान है, मैं इसे लिखने जा रहा हूं एक भिन्न स्थिरांक के घातीय फलन के रूप में, एक अलग स्थिरांक जो मैं बनाने जा रहा हूं, मैं चाहता हूं कि यह 5 पीआई आधा हो, इसलिए मैं 5 पीआई आधा बार करूंगा, इसलिए यह एक जटिल संख्या है और इसमें 5 पीआई आधा भाग है काल्पनिक भाग तो यह मेरे से 5 पाई का आधा गुना है और मैं क्या कर रहा हूँ? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/hungarian/sentence_translations.json b/2020/ldm-i-to-i/hungarian/sentence_translations.json index db8e001a4..bacc2b4a4 100644 --- a/2020/ldm-i-to-i/hungarian/sentence_translations.json +++ b/2020/ldm-i-to-i/hungarian/sentence_translations.json @@ -49,7 +49,7 @@ "end": 63.7 }, { - "input": "And in fact if we go, let's not show where things are going too much here, if we go ahead and rewrite that base i in terms of e, it can help us make sense out of this expression.", + "input": "And in fact if we go, oh no, that's not sure where things are going too much here, if we go ahead and rewrite that base i in terms of e, it can help us make sense out of this expression.", "translatedText": "Valójában ha megyünk, ne mutassuk meg túlságosan, hogy hol tartanak a dolgok, ha továbbmegyünk, és átírjuk azt az i-t e-re, az segíthet megérteni ezt a kifejezést.", "n_reviews": 0, "start": 64.12, @@ -994,7 +994,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half.", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half.", "translatedText": "Tehát ha az 1-es számról indul, akkor a kezdeti sebessége az, hogy egyenesen a 0 felé sétál, és amikor még lejjebb megy, ha az 1 felénél ülne, akkor továbbra is a 0 felé járna, de most a sebességvektor 1-szer negatív lenne, ahol most van, ami pedig 1-szer negatív.", "n_reviews": 0, "start": 998.68, @@ -1302,7 +1302,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i.", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i.", "translatedText": "És egy érdekes kérdés az lesz, hogy tudod-e, hogy van-e csak egy ilyen függvény, aminek ésszerűnek tűnik ezt írni, mert tudod, ha i-ként írjuk az x-hez, nem csak akkor kell megfelelnie, hanem azt is, hogy mikor bedugjuk az egyes számút kapunk i feltehetően i-t a power one-hez, de úgy gondoljuk, hogy ennek a függvénynek az i-nek kell lennie.", "n_reviews": 0, "start": 1383.38, @@ -1323,7 +1323,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos.", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma", "translatedText": "Szóval van 5 pi i felünk, ami nagyszerű, ez egy másik érték, amit ide beilleszthetünk x-hez, és csak azért, hogy ezt egy kicsit vizuálisabban kifejezzük, ha visszanéznénk a körünkre, ahol a a megtett pillanat a pi felével megegyező ideig, ami 1.57 mi lenne, ha ehelyett megtennénk még egy teljes kört, és megyünk még egy pi-felet, hogy eljussunk a pi-hez, amiről tudod, hogy felvehetjük, hogy ahol az e-t a pi i-hez érjük, sétálunk még egy pi-felet, és megyünk még egy pi-felet, amely ezen a ponton egy teljes kört megtettünk volna, hogy visszajussunk az egyikhez, majd öt pi felét sétálunk, ami számszerűen körülbelül 7.85 igen, ez egy másik szám, amivel az i tetejére jutunk, és ha végigmennénk az i-nek az i hatványba való újrakifejezésének egész trükkjén, először e-t írnánk az i hatványba az i-t. szorozzuk meg negatívvá, és az e-t nézzük a negatív 5 pi felére, ami nagyon eltérő szám, igaz, ezt ki tudjuk számítani, nem vagyok benne biztos, de nézzünk meg egy Desmos-t .", "n_reviews": 0, "start": 1415.68, @@ -1337,7 +1337,7 @@ "end": 1493.22 }, { - "input": "What is e to the negative 5 pi halves 0.000388 Okay, 0.000388 much smaller number 0.000388 Which begs the question of okay i to the i what are you right?", + "input": "What is e to the negative five pi halves? 0.000388. Okay, 000388. Much smaller number. 0.000388. Which begs the question of okay i to the i, what are you? Right?", "translatedText": "Mennyi e a negatív 5 pi fele 0-hoz.000388 Oké, 0.000388 sokkal kisebb szám 0.000388 Ami felveti azt a kérdést, hogy oké i-nek mi van igazad?", "n_reviews": 0, "start": 1493.28, @@ -1358,21 +1358,21 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle?", + "input": "that long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle,", "translatedText": "Ez a hosszú, amivel egy sokkal kisebb számhoz jutsz. De nem ez az egyetlen válasz, amit beírhatnánk. Mások is jönnek ide negatív 3-szoros i pi-vel. Amit egy egységkörben ismersz?", "n_reviews": 0, "start": 1544.74, "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one?", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one", "translatedText": "Gondolhatnánk úgy, hogy jé, ha el akarok érni, ahelyett, hogy 90 fokkal sétálnék pi fele radiánt, mi van, ha 270 fokot másfelé sétálok 3 pi fele radiánt, amit talán negatívnak fogok gondolni, mert az egyezmény Általában az óramutató járásával ellentétes irányú pozitív. Ez egy másik módja annak kifejezésére, és más választ kapnánk, ha e lenne a negatív 3 pi felére i Mind az i hatványra, ugyanazt a játékot megyünk végig most az i négyzet törlődik a negatív, ami már ott van, és van egy pozitív 3 pi felünk, és számszerűen ez még más kinézetű választ ad nekünk, mint amit korábban kaptunk. Ha átmegyünk és azt mondjuk, hé, mi az e a 3 pi-nek, nem pedig a 3 o 3 pi-nek. felez 111 pont 3 1 nagyon más típusú szám, mint amit korábban láttunk 111 pont mi volt az dinamikus De visszafelé haladunk az időben látjuk, hogy milyen régen az időben minek kell lennem Olyan, hogy ha onnan játszanék előre dolgokat, akkor az első helyen landolnék a kezdeti feltételem és vissza kell menned az időben 3 pi fél egység És akkor, ha lefordítanád a pusztulási dinamikára, amit a szemre emelés tesz ebben az összefüggésben, akkor azt mondod, ha az első helyen kezdem, de vissza akarok lépni az időben, és azt mondod, hol kellett volna elkezdenem, ha Úgy akarok leépülni, hogy az első helyen végezzem?", "n_reviews": 0, "start": 1559.26, "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right?", + "input": "after three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right?", "translatedText": "3 pi fele időegység után a válasz nyilvánvalóan száztizenegy körül kezdődik az ilyen exponenciális csökkenésre. És láthatja, hogy ez merre tart, ahol valójában végtelenül sok különböző érték van, amelyeket csatlakoztathatnánk X-hez, ha az e-t az X-re úgy gondolva, hogy én és az emberek sokkal többet léptünk be ide. Elnézést, ha a gombostűmet a földre dobom, mint a klasszikust a harmadik helyen. különböző opciók végtelenül sok különböző érték, ami elsőre kissé zavarba ejtő, mert egy kifejezést nézünk. Úgy tűnik, mintha tudnád, hogy csak lesz valami számítás, csak bedugom a számológépembe, és megnézem, mi jön ki, és van több különböző Értékek tehát mi folyik itt, igaz?", "n_reviews": 0, "start": 1657.18, @@ -1442,7 +1442,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function?", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function,", "translatedText": "A 16 negyedik gyökének 2-nek kell lennie, és a válasz jó lesz. Egy konvenciót alkalmazunk, amikor több lehetőség van, mint ez, ha többértékű függvényünk van. Gyakran csak kiválasztunk egyet ezek közül az értékek közül, hogy mire gondolunk, amikor azt szeretnénk. kezelje függvényként, mint valami egyetlen bemenettel és egyetlen kimenettel a legkedveltebb nyelvhasználatban Ez mindig felmerül, amikor összetett számokkal foglalkozunk, valaminek az ötlete, mint egy művelet, amely több értéket szeretne hallja az elágazás kifejezést Hol választja ki a négyzetgyök függvény ágát?", "n_reviews": 0, "start": 1795.66, @@ -1463,7 +1463,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer?", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer?", "translatedText": "Mert több különböző válasz létezik Tudod, hogy újra az én gondolatunkra gondolunk, ez a 90 fokos elforgatás. És ha 90 fokos elforgatásnak gondolnánk, akkor olyan érzés, mintha a négyzetgyöknek kellene lennie. Tudsz valamit, ami 45 fokos szögben ül. Talán ez a négyzet I gyökér, amit kifejezetten úgy írhatnánk ki, hogy 2 gyökér 2 felett 2 gyökér 2 I Ez csak trigonometria használata, de ha inkább az I-re gondolnánk, mint egy negatív 270 fokos elforgatásra, akkor úgy tűnik, hogy ennek a fele csinálja a művelet felét. Valójában a másik oldalra kellene kerülnünk. Talán az itt ülő számnak az I négyzetgyökének kell lennie, és ez valójában csak a negatívja annak, amit láttunk. Értékes függvények igent mondhatunk Csak válassza ki a négyzetgyököt, hogy a pozitív válasz legyen, de ezek közül melyiket tekinti pozitív válasznak?", "n_reviews": 0, "start": 1846.36, @@ -1477,14 +1477,14 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x?", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x,", "translatedText": "És azt hiszem, jól mondod Tudjuk, mi ez, úgy definiáljuk, hogy ez a 2 négyzetgyöke, minden jó és jó. De mi lenne, ha azt mondanám, közelítsük meg ezt ugyanúgy, ahogyan az én-nket az I kifejezéshez közelítettük először szeretném kifejezni a dolgokat e-vel a valami helyesre, majd ezt az 1 felére emelem úgy, hogy az 1 felét megszorzom a kitevővel. És azt mondom, oké, azt hiszem, meg tudom tenni, hogy az e egyenlő 2-vel. Ez a 2 természetes logója. Ez egy állandó, amely 0 körül van.69 vagy így Ha e-t emelünk erre a hatványra, akkor 2-t kapunk, így ezt úgy gondolhatnánk, mint e-t a 2-szer 1 felének természetes logjához, és ha akarod, ha e-re gondolsz az x-re?", "n_reviews": 0, "start": 1942.28, "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it.", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it.", "translatedText": "Tudja, hogy ez egyfajta túlzás a valós számokkal összefüggésben, de ha az e-t az x-re gondolja, mint ennek az x függvénynek a rövidítését, beillesztheti a 0 értéket.69-szer 1 fele, ami szerintem 0 körül lenne.345 Ish valami ilyesmit. Bedugja azt a nagyon konkrét értéket a polinomba, nézze meg, mit ad ki, és 1 körül fog kiadni.414 egy szép valós szám négyzetgyöke 2-nek, amit vársz. De ha ugyanazt tesszük, amit az I-nél tettünk, és elismerjük, hogy valójában több különböző válasz létezik, amikor valamit e-ként akarunk írni egy hatványra, akkor ezt is írhatjuk. Ez viccesnek tűnhet, de felírhatjuk e-ként a 2 plusz 2 pi I természetes logójába. Az egész dolog 1 felére emelve. Végül is ez az érték egyenlő lesz azzal, hogy te lebonthatod úgy, hogy e 2 természetes logója e-vel szorozva a 2 pi I-hez Ez csak az a hatása, hogy a dolgokat 360 fokkal elforgatja, tehát csak 1 lesz. Tehát 2-szer nézünk 1 nagyszerűt, ami érvényes helyettesítésnek tűnik, és mégis amikor Ugyanazt a játékot játsszuk, hogy ezt vegyük és hatványra emeljük, és ezt kezeljük úgy, hogy a hatványt megszorozzuk a kitevővel, nézzük meg, mi történik. Van e a természetes loghoz 2-szer 1 fele plusz Nos, mi 2 pi I-szer 1 fele Nos, ez a pi-szer I Most ez az első e rész a 2-szer 1 felének természetes logójához, ami végül a 2 ismerős négyzetgyöke lesz, ami jó és jó, de ezt megszorozzuk e-vel, hogy a pi I Helyes és híresen e a pi I-hez negatív 1 Tehát ebben az esetben ez azt sugallja, hogy ha ezt a kifejezést 2-től az 1 felére oldjuk meg. Ha játszunk a különböző válaszokkal, akkor valami olyasmire csatlakozhatunk, mint pl. e az 1 felével egyenlő X-re, amit végül kapunk, egy másik válasz, amit hagyományosan 2 negatív négyzetgyökének írhatunk, és itt úgy értem, hogy kicsit vicces, hogy több értéket kell nézni 2-től 1 feléig, és mondjuk, hogy ez nem egyenlő Egy dolog, de az általunk meghozott döntések alapján több különböző dologgal is egyenlő lehet. De a két dolog, ami egészen ésszerűnek tűnhet. négyzetgyök, amit ismerünk, vagy annak negatív változata, ami valójában nem tűnik olyan problémának, sőt, még tovább is játszhatnánk ezzel a játékkal, ahol még kreatívabb válaszokat kérhetek erre a kifejezésre mert talán találhatunk valami más vicces hatványt is, például 2-t az X hatványhoz, amikor elkezdjük az X különböző értékeit csatlakoztatni az alapján, hogy milyen helyettesítést végzünk, ha betartjuk ugyanazokat a szabályokat, amelyeket az I-nek az I-hez való kiértékelésénél használtunk. I. hatvány Tehát ezúttal a kérdés felteszi, vagy megadja, hogy az e egyenlet egyik megoldása x-re egyenlő 2-vel a valós szám 2 természetes logaritmusa ok, hogy melyiket ismerjük.", "n_reviews": 0, "start": 1989.66, @@ -1498,14 +1498,14 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42.", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42", "translatedText": "válasz arra a kérdésre, hogy e és x egyenlő 2, és ismét a kreativitás üdvözlendő, úgyhogy adok még egy kis pillanatot erre. II Megyek és bezárok néhány választ ide, ha ez rendben van veled, nem tudom, mennyi idő szükségszerűen meg kell tennie a matematikai bejegyzést attól függően, hogy milyen eszközt néz, de ne stresszelje túl magát, ha még azelőtt lehetősége volt rá, hogy az Into a kérdésre, amelyre azt a választ szeretné, hogy válaszoljon. Tehát úgy néz ki, Önök közül 131-en írták be azt a változatot, ahol a 2-es Ln-t vesszük, és hozzáadunk 2ii-t, és azt hiszem, én írom ezt a kérdést. Hibásan az egyik választ helyesnek jelöltem meg, holott valójában jó néhány különböző helyes van. azért, mert nem tudom, hogy valakinek úgy tűnik-e, hogy ó, piros, rosszul tetted, amikor az Ln 2 plusz 42 értéket adta meg.", "n_reviews": 0, "start": 2176.56, "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer.", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer.", "translatedText": "I pi, ami természetesen nagyszerű választás, de lehet valami olyasmi is, mint 4 pi I plusz 2 vagy 6 pi I természetes logója, vagy tényleg 2 pi I tetszőleges egész számú többszöröse, ha hozzátesszük, hogy ez nem befolyásolja e-t a X Mert ennek csak az a hatása, hogy e-vel megszorozzuk a 2 pi I-et, ami az 1-gyel való szorzás hatása, és ennek megint egyfajta vicces következménye van, amikor úgy tűnik, hogy ésszerű eredményeket ad, ha ezt egy másik példaként tesszük. úgy néz ki, hogy a második leggyakrabban beírt kifejezés az volt, hogy lecserélhetjük a 2-t. Tehát gondoljuk, hogy 2-re gondolunk 1 4. hatványára, oké volt egy javaslat, hogy a 2-t e-vel helyettesítsük a 2 plusz 4 természetes logójához. pi I Oké Plusz 4 pi I és mindezt az 1 4-re emeljük, ha ugyanazt a játékot játszaná, akkor e-t kapna a 2-szer 1 4-es természetes loghoz, és megszoroznánk e-vel, hogy a pi I Most ennek az első része a szokásos pozitív 2 negyedik gyöke lesz, amit akkor értünk, amikor egy olyan kifejezést, mint a 2 negyedik gyöke, bedugsz egy számológépbe egy szép kis pozitív számot, de akkor ez a második rész negatív 1, tehát úgy tűnik, hogy azt mondja Tudod, ha a 2-t másképpen értelmeznénk, és az 1-re emelnénk 4. Tudod, hogy ez nem a szokásos válasz, amit kapunk, de ésszerű válasz.", "n_reviews": 0, "start": 2253.76, @@ -1519,7 +1519,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships.", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships.", "translatedText": "Megnéztük volna a pi felét I-szer, és ahelyett, hogy a negatív 1-gyel szoroztuk volna, ehelyett szoroztuk volna az I-vel. Ez ismét egy érvényes válasz, ésszerű kimenetnek tűnik valamihez, például 2-től 1-hez. Ha megnézzük azt a tényt, hogy én a hatalomhoz képest úgy tűnik, hogy több különböző értékkel bírok. Igaz, van ez a vicces jelenség, amikor az e-t csatlakoztathatjuk az 5 pi-i felekhez I Negatív 3 pi-i I-hez, és vadul eltérő válaszokat kapunk. valami szuper kicsi valami szuper nagy minden nagyon különbözik az 1 5. hozzávetőlegesen 1 5. választól, amit korábban találtunk itt. Ez pontosan ugyanaz a jelenség, mint amikor olyasmiket kérdezel, mint a 2-től az 1 4-ig, és elismered, hogy valójában több különböző megoldás létezik. az X kifejezésre a 4-re egyenlő valójában 2 4 különböző megoldás, és amit nézel, az az a tény, hogy több különböző megoldás létezik. Az e kifejezésre az X egyenlő valamilyen bázissal, hogy ez az alap én vagyok-e 2 Bármi legyen is az, és az egyik módja annak, hogy gondoljunk erre, az az, hogy amikor valós számokkal van dolgunk, a dolgok csak szépek, szépek. Vannak egy-egy kapcsolatok.", "n_reviews": 0, "start": 2358.92, @@ -1533,7 +1533,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value?", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value", "translatedText": "Nagyszerű. Ha az exponenciális függvényekre akarunk gondolni, hadd fedjek le néhány dolgot. Van ez a szép oda-vissza, ahol kiválaszthatja, hogy bármilyen exponenciális kifejezést adjon X alapjaként, például 2-t az X-hez, vagy kifejezheti ugyanaz az exponenciális, mint az R X-szerese X, amelyről tudod, hogy ez az a polinom, amelyre hivatkozunk. Amikor implicit módon hivatkozunk, amikor valami olyasmit írunk, mint e az X-hez. És van egy szép oda-vissza, mert egyszerűen veheted B természetes logaritmusát. És ez egy választ ad, ha feltételezzük, hogy B egy pozitív szám, és ez ugyanaz, mintha azt mondanánk, hogy R-ből X egyenlő B-vel. Tehát az egyik módja annak, ahogy erről beszéltem a sorozat elején, hogy ha a Az összes lehetséges exponenciális családját úgy írhatjuk fel, hogy X-et R-szer X-ből, és megváltoztathatjuk, hogy R mit jelent. És ez pontosan ugyanaz, mintha e-t írnánk az R-szer X-re, ha ez valami kényelmesebb, tehát e-t az R-hez R-szer X-szer XX-szer ezek ugyanazok a dolgok, amiken elgondolkodhatnánk azon, hogy megváltoztatjuk azt, ami ez, de másrészt, ha az összes lehetséges exponenciálisra gondolnánk, mint valami bázisra, hadd csináljam meg az alapot X erejéig, és már megyünk is. megváltoztatni, hogy mi ez az alap Először úgy tűnik, hogy ez egy másfajta kifejezés, amelyet manipulálni lehet, de ez csak egy másik módja annak, hogy kifejezzük ugyanazt a családot, és egy módja annak, hogy gondold ezt. Ha egy kicsit elvontabban gondolkodunk, mint az R szorzata X, és megvan az oka annak, hogy ezt teszem, mert ezt a komplex számokra fogjuk alkalmazni, ahol furcsábban fog kinézni, ezért kövesse velem, ha ahelyett, hogy ezt a bázist nézném, azt mondhatnám, hogy mi az érték?", "n_reviews": 0, "start": 2428.5, @@ -1554,7 +1554,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R.", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d", "translatedText": "Lehetne R-szer exp-e X-nek, ahol talán R valami olyan, mint nulla pont hat kilenc, de eltolhatnám ezt lefelé két pi I-vel, és ez nem változtatja meg azt az alapot, aminek megfelelne, ami még mindig kettőnek felelne meg. tolja fel két pi I-vel, ami nem változtatja meg a megfelelő bázist, mert minden esetben, amikor csatlakoztatjuk az X-et egyenlő eggyel, ugyanazt kapjuk, azonban ezek mindegyike X különböző értékeire külön függvények. miért láttunk több különböző értéket I-nek az I hatványhoz Mivel I az X-hez egy kétértelmű függvény ebben a kontextusban, egyértelmű lenne, ha eldöntenénk, hogy R melyik értékét úgy képviseljük, hogy az általunk képviselt exp R szor X melyik érték R.", "n_reviews": 0, "start": 2597.88, @@ -1568,14 +1568,14 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers.", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers,", "translatedText": "Ez egy egyértelmű függvény, de azon a ponton úgy tűnik, hogy talán azt akarjuk, hogy ne gondolkodjunk a dolgokról X hatványra emelt bázisban. mindegyik X konstans idő exp-jeként, ha más okból kristálytiszta, hogy valójában hogyan csatlakoztatunk számokat, ha számolni akarunk, vagy csak matematikát akarunk csinálni, van ez a szép végtelen polinom, amit csatlakoztassa őket, és elmondok egy másik esetet, hogy talán ez a helyes módja az exponenciális gondolkodásnak Amint más tartományokra is kiterjedünk, például a komplex számokra, és ehhez menjünk csak vissza a csengőhöz néhány dolog visszanyúlik az eredeti Módszerre, hogy kiterjesztjük a hatványozás fogalmát, és csak gondoljunk arra, hogy mi a 2 az X-hez.", "n_reviews": 0, "start": 2640.66, "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that.", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that?", "translatedText": "Tudsz olyasmit, hogy 2 a 3-hoz. Ismételt szorzás Hogyan van az, hogy először megtanítják gondolkodni valamiről, mint 2-től X-hez tört összegekre vagy negatív összegekre és hasonlókra.", "n_reviews": 0, "start": 2696.88, @@ -1589,35 +1589,35 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive.", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive.", "translatedText": "Általában azt tanítják, hogy a 2-től az 1-hez olyan dolognak kell lennie, ahol tudod, ha megszorzom önmagával, és ez követi a szokásos szabályokat, amelyeket az exponenciálisok a számok számlálásakor tesznek, ahol összeadhatunk dolgokat abban a kitevőben, hogy 2-t kapjak. az 1-hez tehát valami szám kell, hogy ha megszorzom önmagával, akkor 2-t kapok, és tudod, hogy ezen a ponton van választásod, talán ez pozitív.", "n_reviews": 0, "start": 2708.3, "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1?", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the", "translatedText": "Lehet, hogy ez negatív. De ha mindig a pozitív döntés mellett dönt, akkor ebből az ügyletből szép folytonos függvényt fog tudni kihozni, ha negatív számokról kérdezünk. Mi legyen a 2-es a negatív 1-hez jó, hogy legyen valami hol ha megszorzom 2-vel 1-hez?", "n_reviews": 0, "start": 2731.78, "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily.", + "input": "one, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily.", "translatedText": "Ezzel 2-t 0-hoz kapok, és ez egyfajta igazolása annak a konvenciónknak, hogy a negatív kitevők 1-nek néznek ki. De valójában itt az történik, hogy azt mondjuk, bármi legyen is ez, valamiféle függvénynek kell lennie, amely kielégíti az f tulajdonságot. a plusz b egyenlő f-vel, szor fvel b-vel, és az a tény, hogy az alap 2, alapvetően azt mondja nekünk, hogy ez nem akármilyen függvény, hanem egy olyan függvény, ahol az 1-es csatlakoztatásakor 2-t kapunk. épelméjűség-ellenőrzési stílusú kérdés, hogy megtudd, követed-e az itt található implikációkat. Szeretném megkérdezni, hogy mi az, nem fogom softball-nak nevezni, de ez nem az a célja, hogy egy hihetetlenül mély kérdés szükségszerűen.", "n_reviews": 0, "start": 2744.96, "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here?", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here.", "translatedText": "Ez csak inkább egy ellenőrzés, ha követi azt az ötletet, hogy absztrakt módon kezdjük egy függvény tulajdonságaival, majd levezetjük azokat a módokat, amelyek alapján ezeket a tulajdonságokat leírhatjuk. Ha x-ből f kielégíti ezt az f exponenciális tulajdonságot. a plusz b egyenlő f-vel és f-vel b-vel minden bemenetre És azt is kielégíti, hogy 1-ből f egyenlő 2-vel, melyik igaz az alábbiak közül, ami azt jelenti, hogy az alábbiak közül melyik szükségszerűen igaz. és azok, akik emlékeznek, melyik előadás volt az, akármelyikről beszéltünk, hogyan kell értelmezni, mit mond valójában az Euler-képlet. Feltettem egy ilyen stílusú kérdést, ahol egyetlen feltételt figyelmen kívül hagytam, tudod, hogy nem írtam le az a tény, hogy meg akarjuk győződni arról, hogy x f-je mindenhol nem nulla, majd ez némi zavart okoz, ami jó, zavart kap a képernyőn, ami mindannyiunkkal megtörténik. De a szándék lényegében az volt, hogy megmutassa, hogy ez az absztrakt tulajdonság Valami, ami az összeadást szorzássá változtatja, az elég ahhoz, hogy a függvényt úgy írjuk le, mint amivel egyenlő, ha valamiféle hatványra emeljük. Ez a kérdés lényege Most van néhány kérdésünk az erőtornyokkal kapcsolatban úgy tűnik, hogy felbukkantak itt, ami nagyszerűen kapcsolódik a legutóbbi alkalomhoz. Maradjunk egy pillanatra az erőtorony kérdésénél, hogy először mélyebben megérezzük, mit jelent itt a hatványozás?", "n_reviews": 0, "start": 2793.44, "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane?", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane?", "translatedText": "Mert mivel lehetünk azok, amiket állítani akarok, többféleképpen is válaszolhatunk rá. Tehát ha csak egyet adsz meg, akkor az erőtornyokról fogunk beszélni. És akkor ahogy egy számegyenes logaritmikus skálán is ábrázolható. ugyanezt meg kell tenni egy összetett síkon is?", "n_reviews": 0, "start": 2882.64, @@ -1638,7 +1638,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one.", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one", "translatedText": "Igen, valójában van egy vizualizáció, amelyhez egy pillanat alatt eljutok, és valami egészen hasonlót csinálunk, mint ami ehhez hasonló, mert azt fogjuk tenni, hogy különböző exponenciális függvényekkel játszunk X R-szer X. De mi meg fogja változtatni az R értékét, amit egy kis sárga pont fog ábrázolni, szóval ezt végig fogjuk beszélni. Nem az egész síkot fogja leképezni, hanem csak néhány mintapontot a valós tengelytől és a képzeletbeli tengelytől De az ötlet az, hogy ahogy mozgatjuk azt az állandót, képesek leszünk valamilyen módon vizualizálni a különböző dolgokat, amelyeket a síkkal tesz, és gyakorlatilag olyan, mintha az x tengelyt logaritmikus skálává alakítaná, majd burkolná. a képzeletbeli tengely egy kör mentén És akkor amint ez az R értéke képzeletbelivé válik, felcseréli a valós számok szerepét a körre, és a képzeletbeli számok egy logaritmikus skálára kerülnek. Pozitív tengely olyan nagyszerű kérdés, ami szerintem valahogy előre ugranak a fegyverrel, hogy merre akarok menni, de jó látni, hogy az emberek erre gondolnak.", "n_reviews": 0, "start": 2901.54, @@ -1652,14 +1652,14 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it.", + "input": "explicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you want, which is that let's say you have some complex function f, a", "translatedText": "kifejezetten Valami olyasmi, mint az 5-ös f, ugyanaz, mint az 1-es f plusz 1 plusz 1 plusz 1 plusz 1, ami ugyanaz, mint az 1-es f 5-ször megszorozva önmagával ennek a tulajdonságnak köszönhetően. Melyik, ha 1-ből f értéke 2, akkor ugyanaz mint 2 az 5-ös hatványhoz, majd valami olyasmi, mint a negatív 5 f-je. Úgy kell lennie, hogy ha megszorozzuk 5-ből f-vel, akkor azt kapjuk, hogy 0 f-je mennyi, és nem azonnal világos, hogy 0 f-je mennyi, de azt mondhatjuk, hogy 1 f-je plusz 0 egyenlő azzal, amivel az 1-es f a 0-nak a szorzata, de az 1-es f értéke egyenlő a 2-vel, tehát ez is egyenlő 2-vel, tehát azt mondjuk, hogy 2 egyenlő valaminek a kétszeresével. 1-nek kell lennie, tehát ebben az összefüggésben ez garantálja, hogy a negatív 5 f értéke 2 a negatív 5-höz, ez 1-et jelent, mint 2-t az 5-höz. Ezt kifejezetten 2-ként írhatjuk a negatív 5-re, ami azt jelenti, hogy Ez a két tulajdonság együtt nagyon szeretnénk a függvényt 2-ként írni az X-be, mert bármilyen számláló szám, amit beletesszük, úgy fog kinézni, mintha megszoroznánk önmagával azt a hányszoros törtszámot, amit betesszük, kielégíti ezeket a tulajdonságokat amit szerettünk volna És csodálkozhatsz, hogy ez az egyedi és a valós értékű függvények kontextusában valójában az lenne, de összetett értékű függvények kontextusában több ilyen függvény is lenne, amit erre írhatnánk, amelyek közül az egyiket mi voltunk Előtte megnézve Hol lehetne egy függvényt úgy definiálni, hogy ez a 2 plusz 2 pi természetes logójának exp-je I mindannyiszor X Oké, bocsáss meg a hanyagságért, csak izgatott leszek, ha erről írok. És ez valójában egy másik függvény, mint azt bizonyítja, hogy mi történik, ha csatlakoztatja X egyenlő 1 felével. Kicsit korábban láttuk, hogy amikor bedugja az 1 felét, akkor megkapja a 2 negatív négyzetgyökét, majd ha bedugja 1 negyedét, akkor nem a negyedik gyökét kapja. 2, de megszorozom a 2 negyedik gyökét, tehát ez egy másik függvény, de még mindig kielégíti ezeket a tulajdonságokat, és ez arra késztet bennünket, hogy 2-ként írjuk az X-hez, és ez azt sugallja, hogy talán a 2 az X-hez kétértelmű. egy kis jelölés És mindent csak R-szeres exp-ben kellene írnunk, de jól elgondolkodhat. Tudod, talán nem vagyunk elég kreatívak az összes olyan funkcióval, amely kielégíti ezt a tulajdonságot. R-nek valamivel megszorul, és R-nek különböző értékei jöhetnek szóba. De én csak leírok egy kis állítást, és akkor talán egy vázlatot adok arról, hogyan nézne ki a bizonyíték, ha akarod. tegyük fel, hogy van valami összetett F függvényed, és az először a következő tulajdonságokat elégíti ki. Le tudod venni belőle a származékot.", "n_reviews": 0, "start": 2974.0, "end": 3140.02 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways.", + "input": "nd it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I don't know, fractional amounts you might want to think of in crazy ways", "translatedText": "Megkülönböztethető, ami csak megakadályozza, hogy legyen valami, amit ismersz, teljesen rendetlen, nem folytonos dolog. Ez olyan, mintha véletlenszerű értékeket vennél fel, attól függően, hogy tudod, hogy milyen vektortér hatótávolságáról van szó.", "n_reviews": 0, "start": 3140.12, @@ -1673,7 +1673,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right?", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right?", "translatedText": "Ez differenciálható Nem mindenhol egyenlő 0-val, így az a feltétel, ami kiment a fejemből, és elfelejtem, hogy melyik előadásra vagy valami hasonlóra, és akkor megvan az a központi tulajdonsága, hogy az összeadást szorzássá változtatja Ha van ilyen függvénye, azt állítom, hogy van egy egyedi, talán tényleg meg kellene adnom, létezik egy egyedi R komplex szám, hogy X-ből F-et úgy írhassuk, hogy ez R-nek ez az exponenciális függvénye és az X érték szorzata. végtelen polinom szép származékos tulajdonságokkal és mindezekkel, ha ez megvan, akkor minden exponenciális, amit akarsz, az exponenciális szó absztrakt általános értelmében, csak egy olyan tulajdonság alapján, amelyet elvárhatunk tőle, és a bizonyítási vázlat nézzen ki valahogy így, ha először azt akarja megnézni, hogy mi ennek az értéknek a származéka, amelyről feltételezzük, hogy mindenhol létezik, igaz?", "n_reviews": 0, "start": 3160.84, @@ -1694,7 +1694,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base.", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base", "translatedText": "Az X F-jét teljesen ki tudjuk számolni a kifejezésből, és az egész határértéket csak H-ban fejezzük ki. Ami, ha belegondolunk, mit jelent a deriváltok kontextusában, és arra, hogy a 0-nak F szükségszerűen egyenlő 1-gyel Ez az egész korlátozó kifejezés csak valami konstans, de pontosabban attól függ, hogy mi a 0-s függvényünk deriváltja. Szóval van ez a vicces dolog, ahol ha tudod a deriváltját 0-nál, az meghatározza, hogy mi a deriváltja mindenhol És az exponenciális függvényekkel kapcsolatban ez remélhetőleg elég ismerős, mert igazából csak annyit mondunk, hogy egy exponenciális függvény deriváltja önmagával arányos, és az arányossági állandó megegyezik a 0-s deriválttal, ez az egész nagyon elvont módon van megfogalmazva, és ilyenek, de a célja annak hangsúlyozása, hogy nem feltétlenül csak olyan függvények, amelyekről már az X hatványt a-nak gondoljuk, de ez a függvények potenciálisan sokkal szélesebb osztálya, amely éppen kielégíti azt az absztrakt tulajdonságot, hogy az összeadást szorzássá alakítja. De ha ez megvan, az valójában garantálja, hogy van egy második derivált És ami azt illeti, egy harmadik derivált és ilyenek, mert a derivált függvény csak önmagával arányos. Tehát az n-edik derivált meghatározásához csak nézd meg azt az arányossági állandót és emeld fel n hatványra, majd innen csinálhatsz egy A Taylor sorozat kibővítése, és ezt meghagyhatnám egyfajta haladó házi feladatnak azoknak, akik kényelmesek a Taylor sorozattal ebben az ötletben, különösen akkor, ha össze akarják keverni bármely olyan differenciálható függvény ötletét, amely komplex számok értelmében differenciálható. amolyan határozottan egyetemi téma. Tudod, hogy tetszés szerint keverhetitek az érvelést. De a homályos érvelés megengedett olyan személy kontextusában, aki csak a Taylor sorozatról tud, és semmi mást nem, hogy vegye ezt az ötletet, és nézze meg az F és a Taylor-kiterjesztést. mintegy igazolja azt az elképzelést, hogy van egy egyedi komplex szám, így az F függvényünk szükségszerűen felírható így. És akkor a normál exponenciálisokkal való kapcsolat akkor van, amikor van ilyen R értékünk, lényegében azt csináljuk, amit a valós számok komplex kontextusában. az, ha megnézed az adott R értékű függvény exp-jét, és ezt írod alapnak.", "n_reviews": 0, "start": 3243.7, @@ -1708,21 +1708,21 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace?", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace", "translatedText": "Értelmezhetjük ezt úgy is, hogy nem csak pi felének exp I-szer X-et, hanem úgy is értelmezhetjük, hogy 5 pi felének exp-jét I-szer X-ben, és ezek külön függvények. És van egy végtelen család különálló függvényeknek, amelyeknek úgy érezzük, hogy kellene. írd őket I-ként az X-re Tehát az I-t az I-re, hacsak nem fogadtál el egy szabványt arra vonatkozóan, hogy ez szükségszerűen mit fog jelenteni. a mi jelölésünk egy kicsit kétértelmű. Most mindezzel kezdjük el vizualizálni ezt, mert szerintem ez szórakoztató. És tudod, hogy megmondod, hogy ez hasznos-e, vagy inkább zavaró, de azt fogjuk tenni, hogy megnézzük ezt a függvényt exp R-szer X, ami lényegében ez egy másik módja annak, hogy e-t X hatványára írjunk, valójában azt hiszem, azt hiszem, valamikor egy másik animációt készítettem, amely meghatározta, hogy mert azt terveztem, hogy megcsinálom, szóval hadd jöjjön vissza a fájlrendszerembe, és térjen vissza oda, ahol lennie kellene. Menjen be, mert panaszkodik, mert több különböző dolog van. Ó, cserélje ki, megjelenik a másik képernyőn Várjon, miért van ez igen, rendben cserélje ki?", "n_reviews": 0, "start": 3391.12, "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative?", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative", "translatedText": "Helyezd el, amit látsz, és most visszamegyünk oda, mindannyian, mindezt csak azért, hogy szépen kiírhattam volna. hátul a fejed e az R X-hez, és R körül fogunk váltani, tehát követem a képzeletbeli tengely pontjait, és követem a valós tengely pontjait, és lássuk, mit csinál ez. Ez nagyon gyors, szóval hadd gondoljak végig egy kicsit lassabban az összes negatív számon bármit. Ez egy negatív valós szám a 0 és 1 közötti tartományba kerül. Melyik értelme e-nek a negatívhoz képest?", "n_reviews": 0, "start": 3472.82, "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R?", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r", "translatedText": "a negatív valós számhoz valami 0 és 1 között van, és konkrétan a negatív 1 f-jét követjük, amely bármely 1 körül fog megjelenni, ha e 30 0 körül van.Az 1-ből 37 f az e-n landol, mint várható, ennyi az 1-es exp, az I egy radiánnal fog leszállni az egységkör körül, és jó móka itt követni a teljes képzeletbeli tengely mentén, hogyan tekeredik körbe a képzeletbeli tengely és Mi történik, ha módosítjuk az R értékét?", "n_reviews": 0, "start": 3511.78, @@ -1736,7 +1736,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like?", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like?", "translatedText": "Lehet, hogy itt szeretnénk és értékei R-nek. Másképpen nyújtja a dolgokat, így ha 2-re tesszük, akkor sokkal jobban kinyújtja a valós tengelyt, így az 1-es f olyan hely köré kerül, ahol az e négyzet egy kicsit nagyobb, mint 7 f negatív 1 sokkal közelebb van az I 0-jához, f egy 2 radián. A negatív I f köre körüli elforgatás negatív 2 radiános forgás És természetesen eljuthatunk kedvenc képletünkhöz, hogy ha ez pi lenne, akkor a skálázási állandónk. a valódi tengely eléggé megnyúlik Tudod, hogy az 1-ből f az e-nél ül a pi-hez, ami nagyon közel van a 20-hoz plusz pi ami mindig szórakoztató, és a negatív 1 f-je nagyon közel van a 0-hoz, szóval tényleg ki van húzva, hogy igazi tengely És az egységkör irányában is ki van feszítve a dolgok úgy, hogy az I-ből f-ig vagy a negatív I f-ig eljutva a kör felét megkerüljük, szóval ez most jó. Hogyan gondolnánk egy függvényt?", "n_reviews": 0, "start": 3551.6, @@ -1750,7 +1750,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it.", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it", "translatedText": "Azt is felírnánk, hogy az X természetes logójának X-e X-szerese, így az R értékét jelző sárga pontunkat 0 körülire mozgatjuk.69 még mindig nincs képzeletbeli rész, csak egy valós szám 0.69 vagy így Ez a 2 természetes logója, jól látható, hogy 1 f a 2-re kerül, ezért szeretnénk ezt a függvényt 2-nek hívni az 1 felének X f-jére. I Ez egy kis séta az egységkör körül, egészen pontosan 0 lesz.69 radián az egységkör körül, és most egy kicsit jobban szórakozhatnánk, és elmondhatnánk, mi történne, ha ezt megváltoztatnánk a 0 helyett.69 ahelyett, hogy a 2 természetes logója lenne, tegyük szorozzuk meg a 2 természetes logóját, hogy valóban valamire gondoljunk, aminek exponenciális alapja lehet.", "n_reviews": 0, "start": 3610.52, @@ -1778,14 +1778,14 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle?", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle?", "translatedText": "Mi az I az I hatványhoz, ebben az esetben 0 körülire tolja.2 körülbelül egy ötödik. De sok különböző exponenciális függvény létezik, amelynek megvan ez a tulajdonsága, hogy az 1-es f-et az I számra helyezzük. Tehát ha még tovább növelnénk, nem hiszem, hogy itt animálnám, de ha azt vennénk azt a sárga pontot, és emelje fel addig, amíg 5-ször nem éri el a pi I Amit látnál, az az egységkör?", "n_reviews": 0, "start": 3749.24, "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right?", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right?", "translatedText": "Önmaga körül forog úgy, hogy az 1 negatív f-je további 2 pi radián körül forogjon, és ott érjen el, ahol van, de sokkal jobban kinyújtja a valós tengelyt. Milyen értelemben az I másik kimenete az I-hez Sokkal sokkal kisebb szám Körülbelül 0 volt.0003 vagy hasonló De láthatjuk azt is, ami szerintem elég szórakoztató. Mi történik, ha figyelembe vesszük az alternatív kifejezéseket, amelyeket 2-ként szeretnénk értelmezni az X hatványhoz igaz?", "n_reviews": 0, "start": 3773.06, @@ -1806,21 +1806,21 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward?", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward?", "translatedText": "R-ből X X-szorosa van, és R egyenlő ezzel az értékkel, ami 2 plusz pi természetes logója I-szer. Ez azt jelenti, hogy amikor csatlakoztatjuk az 1-et, az 1 f negatív 2-nél van, tehát ezt a függvényt akarjuk írni. negatív 2-ként az X hatványhoz jobbra, és ez valójában valami, amit Tudod, ez egy kicsit megtévesztően egyszerű, amikor negatív számot írunk egy hatványra Negatív 2 Az X hatványra elsőre nem így néz ki, feltétlenül ez hozza nekünk bármilyen módon beírjuk a komplex számokba, de természetesen amikor még olyan értéket is beillesztünk, mint az 1 fele, ahol a negatív 2 négyzetgyökét kérjük, rájövünk, hogy ezt úgy akarjuk beírni, hogy az I szorozza a négyzetgyököt 2-ből De ha ezt a 2-es negatív függvényt az X hatványhoz nézné abban a teljes komplex tartományban, amivel foglalkozik, amit néz, az egy olyan függvény, amely az 1-es értékét negatív 2-re veszi, és ha ezt teszi, akkor a valós számsor többi részéhez igazodik, valahogy kifelé spirálozza?", "n_reviews": 0, "start": 3820.68, "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be.", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be.", "translatedText": "Tehát azt látjuk, hogy a negatív 1 f a negatív 1 felénél Körülbelül ott, ahol elvárnád, ha követnéd az 1 felének f-jét. Pontosan a képzeletbeli vonalon ülne, és az 1 felének f a 2 négyzetgyöke lenne. az egér nem ott van, ahol szeretném.", "n_reviews": 0, "start": 3880.24, "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense?", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense", "translatedText": "Körülbelül az I 2-szeresének négyzetgyöke, és ahogy tovább folytatod, ez megmutatja a negatív 2 összes valós értékhatványát az X-hez képest, szükségszerűen körbe-körbe spirál, de az R értékünket még magasabbra is mozgathatjuk, és megkapjuk körülbelül tau szor I körül hat pont kettő nyolcszor I és ebben az összefüggésben ez egy másik függvény, amit valami 2-ként szeretnénk írni az X-hez, mert bármely egész számtól egész számig, amelyet X-hez csatlakoztatunk úgy néz ki, mint az ismételt szorzás És még vannak ésszerű értékei is olyan dolgokhoz, mint például az 1 fele, ahol a negatív négyzetgyököt köpi ki pozitív négyzetgyök helyett, de amit valójában csinál, az az a sík átalakítása, ahol mindent elhelyez, az az igazi A számsor végül egy nagyon szorosan tekercselt spirál lesz, ami körbe-körbe megy, és csak úgy spirál, hogy 1-ből f közvetlenül a 2-es számra kerül. Tehát ebben az értelemben azt mondhatjuk, hogy a 2-t X-re úgy értelmezzük. külön exponenciális függvény, mint amit hagyományosan szoktunk. Szóval azt gondolom, hogy mindezzel meg fogom hagyni a dolgokat a mai napra, és csak hagyok neked pár elgondolkodtató kérdést, oké, szóval ha szeretnéd úgy gondolja, hogy én az I-hez egy többértékű kifejezés, igaz, mondhatnád, hogy elfogadunk egy konvenciót. Fantasztikusan azt mondanád, hogy a természetes logaritmusfüggvény egy ágát választod, és ez talán bezárja ebbe az e-be a negatív pi-be. fele De ha azt mondod, hogy ez a fajta végtelenül sokféle érték akar lenni, mint a sokféle, amit láttunk. Hány érték akar lenni 2-1 harmad ugyanabban az értelemben?", "n_reviews": 0, "start": 3894.92, @@ -1834,7 +1834,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function?", + "input": "three-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function?", "translatedText": "Hadd mondjam el, hogy X összes F exponenciális függvényének 10-edik másképp fogalmazzák meg, ó, felírtam-e valahova X-ből az f-ét, amely kielégíti az összes leírt tulajdonságot, tehát ha kielégíti az összes ezek közül, és ha 1-ből f egyenlő 2-vel Igaz, hány különböző kimenetet fogunk kapni, ha csatlakoztatjuk X-et, akkor melyik függvény különböző opcióinál 3 10-ed?", "n_reviews": 0, "start": 4008.86, @@ -1848,14 +1848,14 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I?", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i", "translatedText": "2-re a pi-re a különféle függvényekre, amelyeket 2-re X-re vonatkoztathat, ha 2-t X-hez valamiféle exponenciális függvényként tekintünk, amely exponenciális az ilyen típusú absztrakt tulajdonságok értelmében, és ha igen, ha Van egy osztályunk különböző ilyen funkciókkal, és szeretnénk csatlakoztatni a pi-t, ez megnevettet, mert ez egy olyan vicces válasz, ami kipattan, amikor gondolkodni próbálsz, tehát ezek a kérdések, A mai előadáshoz közeledve a központi kérdésem az volt, hogy azt akartam-e, hogy az exponenciális függvények absztrakt tulajdonságaihoz hasonlóan írja le. bezárkózol abba az elképzelésbe, hogy e az rx-hez vagy több Csak azt tudod, hogy szerintem őszintébben leírva az r exp-ét x-szer az r különböző értékeire, hogy ez olyan messzire zárja, de nem zárja be annyira, hogy Egyértelmű fogalma arról, hogy mi 2 az x hatványhoz sokkal kevésbé kellene, hogy olyasvalami, mint én az x hatványhoz. Ebben persze az a kockázat, hogy az emberek néha nem szeretik az absztrakciót, és néha ez nem tűnik megközelíthetőnek. De ha ez ha tudod, csak szólj nekem, szerintem egy egész érdekes gondolatkör veszi körül. Mindezeket a dolgokat az erőtornyokat is bele kell foglalni, mert ha valójában az erőtornyokról akarsz beszélni, mint legutóbb, összetett számok kontextusában. vagy akár negatív alapokon is át kell gondolnod az ilyen dolgokat, szóval ez volt egy kérdés, ami a képernyőn volt.", "n_reviews": 0, "start": 4043.86, "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes.", + "input": "titration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes", "translatedText": "Titrálás, tudod, próbáljuk meg ezt, menjünk tovább és próbáljunk ki egy erőtornyot, ahol felemelünk egy adott teljesítményre, és meglátjuk, mi sül ki belőle, szóval nem tervezte ezt, de mindig megtehetjük. húzzuk fel a Python-t, és lényegében azt csináljuk, amit legutóbb csináltunk. Tehát ez úgy működne, hogy valamilyen alapértékkel kezdtük, majd valamilyen tartományhoz Mit csináltunk, vettünk egy, és át fogjuk rendelni hogy bármi legyen. Az alap, amit jelen esetben az a hatványra emeltem, legyen Oké, klassz, szóval ezt fogjuk tenni, majd kinyomtatjuk a tegyük csak ezt az értékét. Igen, ez egy sokkal nagyobb szám, mint például a 200. Szóval úgy tűnik, hogy ez történik. Lehetséges, hogy ezekkel a dolgokkal néha káosz alakulhat ki.", "n_reviews": 0, "start": 4135.8, @@ -1869,7 +1869,7 @@ "end": 4201.64 }, { - "input": "That's it's not periodic or anything and it's actually chaotic I Suspect that doesn't happen for I but it's a thing to potentially look out for it looks like it does kind of stabilize Maybe there's some little subjection to numerical error But we stay pretty consistently around something with a real part of 0.43 and 0.36 Now what I would want to emphasize though is this expression So let's set a back to be equal to 1 this expression of taking I to the power of a remember That's a little bit ambiguous.", + "input": "that's um, it's not periodic or anything and it's actually chaotic I I suspect that doesn't happen for i but it's a thing to potentially look out for It looks like it does kind of stabilize um, maybe there's Some little subjection to numerical error, but we stay pretty consistently around something with a real part of 0.43 and 0.36 Now what I would want to emphasize though is this expression So let's set a back to b equal to 1 this expression of taking i to the power of a remember That's a little bit ambiguous.", "translatedText": "Ez nem periodikus, vagy ilyesmi, és valójában kaotikus. Gyanítom, hogy ez nem történik meg velem, de érdemes odafigyelni, mert úgy tűnik, ez egyfajta stabilizálódhat. Talán van némi kitéve a numerikus hibának, de elég következetesen maradunk. valami 0 valós része.43 és 0.36 Most azonban szeretném hangsúlyozni ezt a kifejezést. Tehát tegyük a hátsót egyenlőnek 1-gyel.", "n_reviews": 0, "start": 4201.64, @@ -1883,7 +1883,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing?", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing?", "translatedText": "Hadd importáljam a NumPy-t, hogy meglegyen az exponenciális függvény, hadd menjek. A nagy tartományunkhoz, mint korábban. egy másik állandó exponenciális függvényeként jobbra a Másik állandó, amit elkészítek, azt akarom, hogy 5 pi fele legyen, tehát 5 pi felezést teszek meg az I-vel, tehát ez egy komplex szám, és 5 pi fele van, mint a képzeletbeli rész Tehát ez 5 pi felezési idő és mit csinálok?", "n_reviews": 0, "start": 4234.12, diff --git a/2020/ldm-i-to-i/indonesian/sentence_translations.json b/2020/ldm-i-to-i/indonesian/sentence_translations.json index b0bf77ebf..da430cdcc 100644 --- a/2020/ldm-i-to-i/indonesian/sentence_translations.json +++ b/2020/ldm-i-to-i/indonesian/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "Jadi jika Anda memulai dari angka 1, kecepatan awal Anda adalah berjalan lurus menuju 0 dan saat Anda berjalan lebih rendah lagi, jika Anda duduk di paruh 1, maka Anda masih berjalan menuju 0, tetapi sekarang vektor kecepatan Anda akan menjadi negatif 1 kali di mana Anda berada, yaitu negatif 1 setengahnya. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "Dan pertanyaan yang menarik adalah Anda tahu apakah hanya ada satu fungsi yang dirasa masuk akal untuk ditulis untuk ini karena Anda tahu jika kita akan menuliskannya sebagai i ke x tidak hanya harus memenuhi ini tetapi juga harus memuaskan Anda tahu kapan kita masukkan nomor satu yang kita dapatkan i mungkin i ke pangkat satu namun menurut kita fungsi ini seharusnya i. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "Jadi kita punya 5 pi i separuhnya bagus, itu benar-benar nilai lain yang bisa kita masukkan untuk x di sini dan untuk mengejanya sedikit lebih visual jika kita melihat kembali lingkaran kita di sini di mana kita berada di momen berjalan selama jangka waktu yang sama dengan separuh pi yaitu 1.57 bagaimana jika sebaliknya kita mengambil satu putaran penuh lagi dan kita melakukan separuh pi lagi untuk membawa kita ke pi yang mana Anda tahu kita mungkin semacam catatan di situlah nilai e ke pi i adalah kita berjalan di separuh pi lainnya kita berjalan di separuh pi lainnya yang di pada titik ini kita akan membuat satu lingkaran penuh dan membawa kita kembali ke satu dan kemudian kita berjalan sejauh lima bagian pi yang secara numerik sekitar 7.85 ya, itu benar-benar angka lain yang membuat kita berada di atas i dan jika kita harus melalui seluruh omong kosong untuk menyatakan kembali i pangkat i dengan terlebih dahulu menulis e ke bagian 5 pi pangkat i pangkat i itu i kalikan menjadi negatif dan kita akan melihat e ke bagian negatif 5 pi yang merupakan angka yang sangat berbeda kan, kita sebenarnya dapat menghitungnya. Saya tidak yakin, tapi mari kita lihat Desmos . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "Panjang itu yang membawa Anda ke angka yang jauh lebih kecil Tapi itu bukan satu-satunya jawaban yang bisa kita masukkan kan, ada orang lain yang masuk ke sini dengan negatif 3 setengah kali i pi Yang mana yang Anda ketahui dalam bentuk lingkaran satuan? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "Kita dapat berpikir untuk mengatakan hei jika saya ingin sampai ke saya daripada berjalan 90 derajat pi membagi dua radian seperti itu bagaimana jika saya berjalan 270 derajat ke arah lain 3 pi membagi dua radian yang mungkin saya anggap negatif karena konvensinya adalah biasanya berlawanan arah jarum jam adalah positif Itu benar-benar cara lain untuk mengekspresikannya dan itu akan memberi kita jawaban yang berbeda jika kita memiliki e ke bagian negatif 3 pi i Semua pangkat i kita melalui permainan yang sama sekarang i kuadrat dibatalkan dengan a negatif itu sudah ada, dan kita mempunyai 3 pi positif dan secara numerik ini memberi kita jawaban yang tampak berbeda dari apa yang kita miliki sebelumnya. Yang mana jika kita membahasnya dan kita berkata hei, apa yang dimaksud dengan e pada 3 pi, bukan 3 o 3 pi setengah 111 poin 3 1 jenis angka yang sangat berbeda dari apa yang kita lihat sebelumnya 111 poin apa itu 111 poin 3 1 bagus 111 poin 3 1 atau lebih Dan lagi dalam hal intuisi apa yang mungkin Anda tanyakan di sana adalah misalkan kita memiliki putaran ini dinamis Tapi kita bergerak mundur dalam waktu Kita melihat berapa lama waktu yang lalu apa yang harus saya lakukan. Sehingga jika saya memainkan sesuatu ke depan dari sana saya akan mendarat di nomor satu kondisi awal saya dan Anda harus kembali ke waktu 3 pi setengah unit Dan kemudian jika Anda menerjemahkan ke dalam dinamika peluruhan Apa yang dilakukan dalam konteks ini, Anda berkata jika saya memulai dari nomor satu Tapi saya ingin mundur ke belakang dalam waktu dan mengatakan Di mana saya harus memulai jika Saya ingin membusuk sedemikian rupa sehingga saya menjadi yang nomor satu? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "Setelah 3 pi membagi satuan waktu, jawabannya jelas mulai dari sekitar seratus sebelas untuk peluruhan eksponensial semacam itu. Dan Anda dapat melihat ke mana arahnya, dimana sebenarnya terdapat banyak sekali nilai berbeda yang dapat kita masukkan ke dalam X jika kita memikirkan e ke X sebagai I dan orang-orang telah memasukkan lebih banyak lagi di sini Maafkan saya melemparkan pin saya ke tanah seperti yang dilakukan klasik untuk tempat ketiga 9 bagian pi pilihan bagus 1729 bagian pi kalian semua adalah favorit saya banyak sekali opsi berbeda tak terhingga banyak nilai berbeda yang awalnya terasa sedikit membingungkan karena kita melihat sebuah ekspresi Sepertinya Anda tahu bahwa hanya akan ada perhitungan. Saya cukup memasukkannya ke dalam kalkulator saya dan melihat apa yang muncul dan kita mendapatkan beberapa nilai yang berbeda nilai untuk itu Jadi apa yang terjadi di sini kan? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "Akar keempat dari 16 harusnya 2 dan jawabannya akan baik. Kita mengadopsi konvensi ketika ada beberapa pilihan seperti ini ketika Anda memiliki fungsi multi-nilai. Kita sering hanya memilih salah satu dari nilai-nilai tersebut untuk menjadi apa yang kita maksud ketika kita ingin perlakukan itu sebagai fungsi sebagai sesuatu dengan masukan tunggal dan keluaran tunggal dalam istilah yang lebih menarik. Hal ini selalu muncul ketika kita berurusan dengan bilangan kompleks. Gagasan tentang sesuatu sebagai operasi semacam keinginan Untuk memiliki banyak nilai, kadang-kadang Anda akan melakukannya dengar ungkapan cabang Di mana Anda memilih cabang dari fungsi akar kuadrat? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "Karena ada beberapa jawaban berbeda Anda tahu, kita memikirkan I lagi adalah rotasi 90 derajat ini Dan jika kita menganggapnya sebagai rotasi 90 derajat rasanya seperti akar kuadrat seharusnya Anda tahu sesuatu yang berada pada sudut 45 derajat Mungkin itu kuadratnya akar dari I yang dapat kita tulis secara eksplisit sebagai akar 2 di atas 2 akar 2 di atas 2 I Itu hanya menggunakan trigonometri tetapi jika kita memikirkan I sebagai rotasi Negatif 270 derajat, rasanya seperti setengahnya melakukan setengah dari operasi itu seharusnya membawa kita ke sisi yang lain. Mungkin angka yang ada di sini adalah akar kuadrat dari I dan itu sebenarnya hanyalah negatif dari apa yang kita lihat sebelumnya Akar negatif 2 di atas 2 dikurangi akar 2 di atas 2 kali I Sekarang dalam konteks nyata fungsi bernilai yang bisa kita katakan ya. Pilih saja akar kuadratnya, apa pun jawaban positifnya, tetapi yang manakah yang menurut Anda merupakan jawaban positif? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "Dan saya pikir Anda mengatakannya dengan baik. Kita tahu apa ini, kita mendefinisikannya sebagai akar kuadrat dari 2, semuanya baik-baik saja. Tetapi bagaimana jika saya mengatakan mari kita melakukan pendekatan dengan cara yang sama seperti kita mendekati ekspresi I ke ekspresi I. ingin menyatakan hal-hal sebagai e terlebih dahulu pada sesuatu yang benar dan kemudian saya akan menaikkannya menjadi 1 setengah dengan mengalikan 1 setengahnya ke dalam eksponen dan saya berkata oke, saya bisa, saya rasa saya bisa melakukan itu e pada apa yang benar sama dengan 2 yah Itu logaritma natural dari 2 Konstanta yang ada di sekitar 0.69 atau lebih Jika kita menaikkan e ke pangkat itu, kita akan mendapatkan 2 sehingga kita dapat menganggapnya sebagai e dari logaritma natural dari 2 dikalikan 1 setengah dan jika Anda mau, apakah Anda sedang memikirkan e ke x? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "Anda tahu ini mungkin berlebihan dalam konteks bilangan real. Namun jika Anda menganggap e ke x sebagai singkatan untuk fungsi x ini, Anda dapat memasukkan nilai 0.69 kali 1 setengah yang menurut saya sekitar 0.345 Ish kira-kira seperti itu Anda memasukkan nilai yang sangat konkret itu ke dalam polinomial Anda, lihat apa yang dihasilkannya, dan hasilnya akan sekitar 1.414 a Bilangan real yang bagus akar kuadrat dari 2 apa yang Anda harapkan Tetapi jika kita melakukan hal yang sama seperti yang kita lakukan dengan I dan Mengakui bahwa sebenarnya ada beberapa jawaban yang berbeda ketika kita ingin menulis sesuatu sebagai e pangkat kita juga bisa menulis ini Ini mungkin tampak lucu, tapi kita bisa menuliskannya sebagai e ke logaritma natural dari 2 ditambah 2 pi I Semuanya dipangkatkan ke bagian 1 Tepat setelah semua nilai ini akan menjadi sama dengan Anda dapat memecahnya menjadi e ke logaritma natural dari 2 Dikalikan e dengan 2 pi I Yang ini hanya mempunyai efek memutar benda 360 derajat, jadi hasilnya sama dengan 1 Jadi kita lihat 2 dikalikan 1 besar yang terasa seperti substitusi yang valid namun ketika kita memainkan permainan yang sama yaitu Mengambil ini dan menaikkannya ke pangkat dan memperlakukannya dengan mengalikan pangkat ke dalam eksponen lihat apa yang terjadi Kita punya e ke log natural 2 kali 1 setengah ditambah Nah, berapakah 2 pi I dikali 1 setengah nah itu akan menjadi pi dikalikan I Sekarang bagian pertama e ke logaritma natural dari 2 dikalikan 1 setengahnya akan menjadi akar kuadrat dari 2 yang sudah dikenal, semuanya baik-baik saja, tapi kita akan mengalikannya dengan e menjadi pi I Benar dan cukup terkenal e ke pi I negatif 1 Jadi dalam hal ini tampaknya menyarankan bahwa jika kita menyelesaikan ungkapan ini 2 hingga 1 setengah Dengan bermain-main dengan jawaban yang berbeda kita dapat memasukkan sesuatu seperti e ke X sama dengan 1 setengah yang kita dapatkan adalah jawaban lain yang biasanya kita tulis sebagai akar kuadrat negatif dari 2 dan Di sini maksud saya, agak lucu jika memiliki banyak nilai untuk melihat 2 hingga 1 setengah dan mengatakan itu tidak sama dengan Satu hal tetapi berdasarkan pilihan yang kita buat, hal itu bisa sama dengan banyak hal yang berbeda Tetapi dua hal yang tampaknya cukup masuk akal Jika akan ada sesuatu yang 2 berbanding 1, sepertinya itu harusnya positif akar kuadrat yang kita kenal atau varian negatifnya yang sebenarnya tidak tampak seperti masalah Dan sebenarnya kita bisa um kita bisa memainkan permainan ini lebih jauh lagi dan izinkan saya meminta jawaban yang lebih kreatif untuk ungkapan ini karena mungkin kita dapat menemukan pangkat lucu lainnya seperti 2 pangkat X saat kita mulai memasukkan berbagai nilai X yang berbeda berdasarkan substitusi apa yang kita buat jika kita Mematuhi aturan yang sama yang kita gunakan dalam mengevaluasi I ke pangkat I Jadi kali ini pertanyaannya menanyakan atau menetapkan bahwa salah satu penyelesaian persamaan e terhadap x sama dengan 2 adalah bilangan real Logika natural dari 2 oke yang kita ketahui. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "jawaban atas pertanyaan e pada x sama dengan 2 dan Sekali lagi kreativitas disambut baik, jadi saya akan memberi Anda sedikit waktu lagi untuk itu II Akan melanjutkan dan mengunci beberapa jawaban di sini jika Anda tidak keberatan Saya tidak yakin berapa lama waktu yang dibutuhkan perlu melakukan entri matematika tergantung pada perangkat apa yang Anda lihat tetapi Jangan terlalu stres jika itu sebelum Anda mendapat kesempatan untuk memasukkan pertanyaan yang Anda inginkan ke dalam jawaban yang ingin Anda jawab Jadi sepertinya 131 dari Anda telah memasukkan varian di mana kita mengambil Ln dari 2 dan kita menambahkan 2ii dan saya kira saya sedang menulis pertanyaan ini Keliru seperti menandai salah satu jawaban sebagai benar padahal sebenarnya ada beberapa jawaban yang benar Jadi itu terserah saya karena faktanya saya tidak tahu apakah menurut Anda ada yang suka oh Warnanya merah, Anda salah saat memasukkan Ln 2 ditambah 42. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi yang tentu saja merupakan pilihan bagus Tapi Anda juga bisa memiliki sesuatu seperti 4 pi I ditambah logaritma natural dari 2 atau 6 pi I Atau kelipatan bilangan bulat apa pun dari 2 pi I jika Anda menambahkannya tidak mempengaruhi e ke X Karena ini hanya mempunyai efek perkalian dengan e ke 2 pi I Yang merupakan efek perkalian dengan 1 dan sekali lagi ini memiliki konsekuensi yang lucu dimana sepertinya menghasilkan hasil yang masuk akal. Hasil ketika kita melakukannya sebagai contoh lain Itu sepertinya ekspresi masukan kedua yang paling umum adalah kita mungkin mengganti 2 Jadi anggap saja kita sedang memikirkan 2 pangkat 1 4, oke ada saran agar kita mengganti 2 dengan e ke logaritma natural dari 2 ditambah 4 pi I Oke Ditambah 4 pi I dan kita naikkan semua itu Ke 1 ke-4 kan kalau kamu memainkan permainan yang sama kamu akan mendapat e Ke logaritma natural 2 kali 1 ke-4, dan kita akan mengalikannya dengan e untuk pi I Sekarang bagian pertama dari itu akan menjadi positif biasa Akar keempat dari 2 hal yang kami maksud ketika Anda memasukkan ekspresi seperti akar keempat dari 2 ke dalam kalkulator, sebuah bilangan Positif kecil yang bagus, tapi kemudian bagian kedua ini adalah negatif 1 jadi sepertinya mengatakan Anda tahu jika kami menafsirkan 2 dengan cara yang berbeda menaikkannya ke 1 4 Anda tahu itu bukan jawaban yang biasa kami dapatkan tetapi itu jawaban yang masuk akal. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "Kita akan melihat pi dibelah dua kali I dan alih-alih mengalikannya dengan Negatif 1, kita malah akan mengalikannya dengan I Yang lagi-lagi merupakan jawaban yang valid, sepertinya keluaran yang masuk akal untuk sesuatu seperti 2 hingga 1 ke-4 Jadi, saat Anda melihat fakta bahwa I pangkat Saya tampaknya memiliki beberapa nilai berbeda untuk itu Benar kita memiliki fenomena lucu ini di mana kita dapat menyambungkan e ke bagian 5 pi I Negatif 3 pi bagian I dan kita mendapatkan jawaban yang sepertinya sangat berbeda sesuatu yang sangat kecil sesuatu yang sangat besar semuanya sangat berbeda dari jawaban ke-15 kira-kira ke-15 yang kita temukan sebelumnya di sini. Fenomenanya persis sama dengan ketika Anda menanyakan sesuatu seperti apa yang ke-2 sampai ke-14 dan Mengakui bahwa sebenarnya ada beberapa solusi yang berbeda pada ekspresi X pada ke-4 sama dengan 2 4 solusi yang berbeda pada kenyataannya dan yang Anda lihat adalah fakta bahwa ada beberapa solusi yang berbeda Untuk ekspresi e pada X sama dengan suatu basis, apakah basis tersebut adalah I, apakah basis tersebut adalah 2 Apa pun itu dan salah satu cara kita memikirkan hal ini adalah ketika Anda berurusan dengan bilangan real, segala sesuatunya indah, semuanya menyenangkan. Ada hubungan satu lawan satu. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "Sangat bagus Dimana jika kita ingin memikirkan tentang fungsi eksponensial, izinkan saya membahas beberapa hal ini. Kita mempunyai bolak-balik yang bagus di mana Anda dapat memilih untuk menyatakan eksponensial apa pun sebagai basis ke X seperti 2 ke X Atau Anda dapat menyatakannya eksponensial yang sama dengan X dari R dikali X yang Anda tahu itu adalah polinomial yang kita rujuk Kapanpun secara implisit mengacu pada setiap kali kita menulis sesuatu seperti e ke X Dan ada bolak-balik yang indah karena Anda dapat mengambil logaritma natural dari B Dan ini memberi Anda satu jawaban dengan asumsi bahwa B adalah bilangan positif Dan itu sama saja dengan mengatakan bahwa X dari R sama dengan B. Jadi salah satu cara saya membicarakan hal ini di awal seri adalah jika Anda melihat keluarga dari semua eksponensial yang mungkin benar, kita dapat menuliskannya sebagai X dari R dikali X dan mengubah apa itu R. Dan ini sama persis dengan menulis e ke R dikali X jika itu adalah sesuatu yang membuat Anda lebih nyaman Jadi e ke R dikalikan XX dari R dikalikan dengan X, itu adalah hal yang sama yang dapat kita pikirkan untuk mengubahnya. Tetapi di sisi lain jika Anda memikirkan tentang semua kemungkinan eksponensial sebagai suatu basis Biarkan saya mengerjakan basis dengan pangkat X dan kita akan mulai untuk mengubah dasar itu Pada awalnya rasanya seperti itu adalah ekspresi yang berbeda untuk dimanipulasi, tapi itu hanyalah cara lain untuk mengekspresikan keluarga yang sama Dan cara yang mungkin Anda pikirkan tentang hal ini Untuk bagaimana kita memikirkan tentang dasar apa yang sesuai dengannya jika kita berpikir sedikit lebih abstrak sebagai Exp dari R dikali X dan ada alasan saya melakukan ini karena kita akan menerapkan ini pada bilangan kompleks yang akan terlihat lebih aneh, jadi ikuti saya di sini jika daripada melihat dasar itu, satu hal yang bisa saya lakukan adalah mengatakan apa nilainya? ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "Saya bisa mendapatkan exp sebesar R dikali X dimana mungkin R adalah nol koma enam sembilan Tapi saya bisa menggesernya ke bawah sebanyak dua pi I Dan itu tidak mengubah basis yang akan berkorespondensi dengan itu akan tetap berkorespondensi dengan dua Atau bisa juga menggesernya ke atas sebanyak dua pi I yang tidak mengubah basis yang sesuai karena dalam semua kasus tersebut Ketika kita memasukkan X sama dengan satu kita mendapatkan hal yang sama namun Semua ini untuk nilai X yang berbeda adalah fungsi yang berbeda Ini adalah mengapa kita melihat beberapa nilai yang berbeda untuk I pangkat I Karena I pangkat X adalah fungsi ambigu dalam konteks itu, maka akan menjadi jelas jika kita memutuskan nilai R yang mana sehingga apa yang kita wakili adalah exp dari R dikali X yang nilainya dari R. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "Ini adalah fungsi yang tidak ambigu tetapi pada saat itu rasanya mungkin yang kita inginkan adalah berhenti memikirkan sesuatu dalam bentuk bilangan yang dipangkatkan X Mungkin segera setelah kita berada dalam konteks bilangan kompleks Kita sebaiknya menulis saja semuanya sebagai exp dari beberapa waktu yang konstan X jika tanpa alasan lain menjadi jelas Bagaimana kita sebenarnya memasukkan angka jika kita ingin melakukan perhitungan atau hanya untuk melakukan matematika di atasnya kita punya polinomial tak terbatas yang bagus yang kita sambungkan ke dalam dan saya akan membuat kasus lain untuk Anda bahwa ini mungkin cara yang benar untuk berpikir tentang eksponensial Segera setelah kita memperluas ke domain lain hal-hal seperti bilangan kompleks dan untuk itu mari kita buat cadangannya. kembali ke bel pintu beberapa hal tiba kembali ke awal Cara kita memperluas gagasan eksponensial dan hanya memikirkan seperti apa 2 pada X Benar kita tahu bagaimana memikirkan hal ini untuk bilangan asli. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "Anda tahu sesuatu seperti 2 banding 3 Perkalian berulang Bagaimana pertama kali Anda diajarkan untuk memikirkan sesuatu seperti 2 banding X untuk jumlah pecahan atau Untuk jumlah negatif dan sejenisnya. Dengan baik. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "Anda biasanya diajari bahwa 2 banding 1 harus berupa sesuatu yang Anda tahu jika saya mengalikannya dengan dirinya sendiri dan Ini mengikuti aturan yang biasa dilakukan Eksponensial dengan menghitung angka di mana kita dapat menambahkan sesuatu dalam eksponen itu saya akan mendapatkan 2 ke 1 jadi itu harusnya suatu bilangan yang ketika saya mengalikannya dengan dirinya sendiri saya mendapatkan 2 dan Anda tahu pada saat itu Anda punya pilihan, mungkin itu positif. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "Mungkin itu negatif Tetapi jika Anda selalu memutuskan untuk membuat pilihan positif Anda akan bisa mendapatkan fungsi kontinu yang bagus dari kesepakatan yang sama ini jika kita bertanya tentang bilangan negatif Berapa seharusnya 2 ke negatif 1 itu seharusnya menjadi sesuatu dimana jika saya mengalikannya dengan 2 banding 1? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "Ini memberi saya 2 ke 0 dan itu semacam pembenaran untuk konvensi kita bahwa eksponen negatif terlihat seperti 1 setengah Tapi apa yang sebenarnya terjadi di sini adalah kita mengatakan apa pun ini, itu harus menjadi semacam fungsi Yang memenuhi properti ini f dari a ditambah b sama dengan f dari a dikali f dari b dan Terlebih lagi fakta bahwa basisnya adalah 2 pada dasarnya memberitahu kita bahwa itu bukan sembarang fungsi. Ini adalah fungsi di mana ketika kita memasukkan 1 kita mendapatkan 2 Dan sedikit saja, Anda tahu pertanyaan gaya pemeriksaan kewarasan untuk melihat apakah Anda mengikuti beberapa implikasinya di sini Saya ingin bertanya kepada Anda apa itu Saya tidak akan menyebutnya seperti softball, tapi ini tidak dimaksudkan seperti itu Pertanyaan yang sangat mendalam perlu. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "Ini lebih merupakan pemeriksaan jika Anda mengikuti Gagasan memulai secara abstrak dengan properti suatu fungsi dan kemudian menyimpulkan cara-cara yang mungkin ingin kita tuliskan berdasarkan properti tersebut. Jika f dari x memenuhi properti eksponensial ini f dari a ditambah b sama dengan f dari a dikalikan f dari b untuk semua masukan Dan juga memenuhi f dari 1 sama dengan 2 manakah dari pernyataan berikut yang benar Artinya, manakah dari berikut ini yang benar Tidak peduli fungsi mana yang Anda mulai dengan dan kalian yang ingat kuliah yang mana Itu yang mana pun yang kita bicarakan tentang bagaimana menafsirkan apa yang sebenarnya dikatakan rumus Euler Saya mengajukan pertanyaan dengan gaya ini di mana saya mengabaikan satu kondisi, Anda tahu saya tidak menuliskannya fakta bahwa kita ingin memastikan f dari x bukan nol di mana-mana dan kemudian itu menyebabkan sejumlah Kebingungan yang keren, dapatkan kebingungan di layar yang terjadi pada kita semua. Tapi tujuannya pada dasarnya adalah untuk menunjukkan bahwa properti abstrak ini sesuatu yang mengubah penjumlahan menjadi perkalian pada dasarnya sudah cukup untuk membuat Anda ingin menulis fungsinya sebagai apa pun yang dipangkatkan Ini adalah inti dari pertanyaan Sekarang kita punya beberapa pertanyaan sebenarnya tentang menara listrik yang tampaknya telah muncul di sini dan sangat bagus terkait dengan yang terakhir kali. Mari kita tunda sejenak pertanyaan tentang menara listrik agar kita dapat merasakan lebih dalam seperti Apa arti eksponensial di sini? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "Karena karena kita bisa menjadi apa yang ingin saya klaim adalah kita bisa menjawabnya dengan berbagai cara yang berbeda Jadi jika Anda memberi saya satu saja, kita akan bicara tentang menara listrik Dan seperti garis bilangan yang bisa direpresentasikan dalam skala logaritmik, maka kita bisa menjawabnya dengan berbagai cara. hal yang sama dilakukan untuk bidang yang kompleks? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "Ya Faktanya, ada visualisasi yang akan saya bahas sebentar lagi di sini di mana kita melakukan sesuatu yang sangat mirip dengan itu. Karena yang akan kita lakukan adalah bermain-main dengan fungsi eksponensial yang berbeda X dari R kali X Tapi kita akan mengubah nilai R yang akan diwakili oleh titik kuning kecil Jadi kita akan membahasnya. Ini tidak akan memetakan keseluruhan bidang, tetapi hanya beberapa titik sampel dari sumbu nyata dan sumbu imajiner Namun idenya adalah ketika kita bergerak di sekitar konstanta tersebut, kita akan dapat memvisualisasikan berbagai hal yang dilakukannya pada bidang tersebut dan secara efektif hal ini seperti mengubah sumbu x menjadi skala logaritmik dan kemudian membungkusnya. sumbu imajiner di sepanjang lingkaran Dan segera setelah nilai R menjadi imajiner, ia menukar peran bilangan Real yang dimasukkan ke dalam lingkaran dan bilangan imajiner dimasukkan ke dalam skala logaritmik. Sumbu positif jadi pertanyaan bagus ketiganya menurut saya seperti melompat ke depan menuju tempat yang ingin saya tuju. Tapi senang melihat ke sanalah orang-orang berpikir demikian dalam hal ini. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "secara eksplisit Sesuatu seperti f dari 5 sama dengan f dari 1 ditambah 1 ditambah 1 ditambah 1 ditambah 1 Yang sama dengan f dari 1 dikalikan sendiri 5 kali karena sifat ini Yang mana jika f dari 1 adalah 2 adalah sama sebagai 2 pangkat 5 dan kemudian sesuatu seperti f dari negatif 5 Seharusnya ketika kita mengalikannya dengan f dari 5 Kita mendapatkan berapa pun f dari 0 dan tidak segera jelas apa itu f dari 0 tetapi kita dapat mengatakan bahwa f dari 1 ditambah 0 sama dengan berapa pun f dari 1 dikalikan dengan f dari 0 tetapi f dari 1 sama dengan 2 Jadi ini juga sama dengan 2 jadi kita katakan 2 sama dengan 2 dikalikan sesuatu dan sesuatu itu haruslah 1 jadi dalam konteks ini jaminan bahwa f dari negatif 5 adalah 2 terhadap negatif 5, itu adalah 1 di atas 2 hingga tanggal 5. Kita dapat secara eksplisit menuliskannya sebagai 2 pada negatif 5 yang berarti kedua sifat ini digabungkan menjadi kita benar-benar ingin menuliskan fungsi tersebut sebagai 2 pada X Karena bilangan hitung apa pun yang kita masukkan ke dalamnya akan memuaskan Ini akan terlihat seperti mengalikan dengan dirinya sendiri berapa kali bilangan pecahan apa pun yang kita masukkan ke dalamnya akan memenuhi sifat-sifat ini yang kita inginkan Dan Anda mungkin bertanya-tanya apakah hal itu unik dan dalam konteks fungsi bernilai riil. Namun dalam konteks fungsi bernilai kompleks Akan ada beberapa fungsi f yang dapat kita tulis untuk fungsi ini, salah satunya adalah fungsi yang kami inginkan melihat sebelumnya Di mana kita dapat memiliki fungsi yang didefinisikan sebagai exp dari logaritma natural 2 ditambah 2 pi Saya sepanjang waktu X Oke, maafkan kecerobohan di sini, Saya hanya bersemangat menulis tentang ini Dan ini sebenarnya adalah fungsi yang berbeda sebagai dibuktikan dengan apa yang terjadi jika kamu memasukkan X sama dengan 1 setengahnya Kita telah melihat sedikit sebelumnya bagaimana ketika kamu memasukkan 1 setengah yang kamu dapatkan adalah akar kuadrat negatif dari 2 dan kemudian jika kamu memasukkan 1 yang keempat kamu mendapatkan Bukan akar kuadrat yang keempat dari 2 tapi saya mengalikan akar keempat dari 2 jadi fungsinya berbeda Tapi masih memenuhi sifat-sifat ini dan itu membuat kita ingin menuliskannya sebagai 2 ke X Dan itu memberi kesan bahwa mungkin 2 ke X adalah ambigu sedikit notasi Dan kita harus menulis semuanya dalam bentuk exp dari R kali sesuatu tetapi Anda mungkin bertanya-tanya Anda tahu mungkin kita tidak cukup kreatif dengan semua fungsi yang memenuhi properti ini Mungkin ada ambiguitas ketika kita menulis exp dari R dikalikan dengan sesuatu dan ada nilai-nilai R yang berbeda yang bisa ikut berperan Tapi saya Saya hanya akan memberikan sedikit klaim dan kemudian mungkin memberikan sketsa seperti apa buktinya jika Anda mau Yang mana mari kita katakanlah Anda memiliki fungsi kompleks F, dan fungsi tersebut memenuhi sifat-sifat berikut terlebih dahulu. Anda dapat mengambil turunannya. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "Itu dapat dibedakan sehingga membuatnya tidak menjadi sesuatu yang Anda tahu benar-benar berantakan, hal yang terputus-putus. Itu seperti mengambil beberapa nilai acak tergantung pada Anda mengetahui rentang ruang vektor apa pun. Saya tidak tahu jumlah pecahan yang mungkin ingin Anda pikirkan dengan cara yang gila. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "Ini fungsi yang bagus. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "Itu terdiferensiasi Tidak sama dengan 0 di mana-mana sehingga kondisi seperti itu terlintas dalam pikiran saya dan saya Lupa kuliah mana untuk kuliah atau semacamnya dan kemudian ia memiliki properti pusat yang mengubah penjumlahan menjadi perkalian Jika Anda memiliki fungsi seperti itu, saya klaim itu ada yang unik mungkin saya harus benar-benar menentukan ada bilangan Kompleks unik R sehingga Anda dapat menulis F dari X sebagai fungsi eksponensial R dikalikan nilai X Yang pada dasarnya Anda tahu mengatakan bahwa jika Anda memiliki X sebagai fungsi ini polinomial tak terbatas dengan sifat turunan yang bagus dan semua itu jika Anda memiliki ini, Anda memiliki Setiap eksponensial yang Anda inginkan dalam pengertian umum abstrak dari kata eksponensial hanya berdasarkan pada properti yang kita inginkan darinya dan sketsa buktinya akan lihat kira-kira seperti ini jika Anda ingin melihat dulu apa turunan dari nilai ini yang kita asumsikan ada di mana-mana, bukan? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "Kita dapat memfaktorkan F dari X seluruh ekspresi dan seluruh limit dinyatakan hanya dalam bentuk H Yang mana jika Anda memikirkan apa artinya dalam konteks turunan dan fakta bahwa F dari 0 harus sama dengan 1 Seluruh ekspresi pembatas ini adalah hanya sebuah konstanta tetapi lebih khusus lagi, berapa pun turunan dari fungsi kita di 0 Jadi ada hal lucu di mana jika Anda mengetahui turunannya di 0 itu menentukan turunannya di mana-mana. Dan dalam konteks fungsi eksponensial, semoga hal ini cukup familier karena semua yang sebenarnya kami katakan adalah turunan dari fungsi eksponensial adalah Proporsional terhadap dirinya sendiri dan konstanta proporsionalitas itu sama dengan berapa pun turunannya di 0. Ini semua diutarakan dengan sangat abstrak dan semacamnya, tetapi tujuannya adalah untuk menekankan bahwa itu adalah belum tentu hanya fungsi-fungsi yang sudah kita anggap sebagai pangkat X Tetapi ini adalah kelas fungsi yang berpotensi jauh lebih luas yang hanya memenuhi sifat abstrak mengubah penjumlahan menjadi perkalian. Tetapi jika Anda memilikinya, itu sebenarnya menjamin bahwa Anda juga memiliki turunan kedua Dan dalam hal ini turunan ketiga dan semacamnya karena turunan fungsinya sebanding dengan fungsi itu sendiri. Jadi untuk mengambil turunan ke-n, Anda cukup melihat konstanta proporsionalitas itu dan menaikkannya ke pangkat n dan kemudian dari sini Anda dapat melakukan a Perluasan deret Taylor dan saya mungkin akan meninggalkannya sebagai pekerjaan rumah tingkat lanjut bagi Anda yang merasa nyaman dengan deret Taylor dalam gagasan tersebut terutama jika Anda ingin mencampurkan gagasan fungsi Diferensiasi apa pun yang dapat terdiferensiasi dalam pengertian bilangan kompleks, yaitu semacam topik kuliah yang pasti Anda tahu, Anda dapat mencampurkan alasannya di sana sesuka Anda Tetapi penalaran fuzzy diperbolehkan dalam konteks seseorang yang hanya tahu tentang deret Taylor dan tidak ada yang lain untuk mengambil ide ini dan melihat ekspansi Taylor untuk F dan semacam membenarkan gagasan bahwa Ada bilangan kompleks unik sehingga fungsi kita F dapat ditulis seperti ini Dan kemudian koneksi ke eksponensial normal adalah kapan pun Anda memiliki nilai seperti itu R Pada dasarnya kita melakukan apa yang kita lakukan dalam konteks kompleks bilangan real adalah jika Anda melihat exp dari fungsi nilai R itu dan Tulis itu sebagai basis. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "Kita dapat mengartikannya bukan hanya exp dari bagian pi I dikali X, namun kita juga dapat mengartikannya sebagai exp dari 5 bagian pi I Kali X dan Ini adalah fungsi yang terpisah Dan ada rangkaian fungsi terpisah yang tak terhingga yang rasanya seperti kita harus melakukannya tuliskan sebagai I ke X Jadi ekspresi I ke I kecuali Anda telah mengadopsi standar untuk apa artinya. Ketika Anda mengatakan itu memiliki banyak keluaran yang tak terhingga, cara lain untuk memikirkannya adalah Fungsi I ke X dengan notasi yang kita miliki agak ambigu Sekarang dengan semua itu mari kita mulai memvisualisasikan beberapa di antaranya karena menurut saya itu menyenangkan Dan Anda tahu, Anda memberi tahu saya apakah ini visual yang bermanfaat atau visual yang lebih membingungkan tetapi apa yang akan kita lakukan adalah melihat fungsi ini exp dari R kali X, yang pada dasarnya ini adalah cara lain untuk menulis e pangkat X sebenarnya saya pikir saya pikir saya membuat animasi yang berbeda di beberapa titik yang menentukan itu karena aku berencana Berencana untuk melakukan hal itu jadi izinkan aku oh iya, itu kamu, kembali ke sistem fileku, kembali ke tempat yang seharusnya. Lanjutkan, di sana, dia mengeluh karena ada banyak hal yang berbeda, Ini akan menjadi seperti ada a Oh ganti itu muncul di layar lain Tunggu kenapa ya, oke ganti? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "Tempatkan apa pun yang Anda lihat di sana Dan sekarang kita kembali ke oh begitulah kita semua itu semua supaya saya bisa menuliskannya dengan baik. Jika Anda merasa tidak nyaman menganggapnya sebagai exp dari R kali X polinomial tak hingga ini Tepat di bagian belakang kepalamu e ke R kali X dan kita akan bervariasi di sekitar R jadi saya akan mengikuti titik sumbu imajiner, dan saya akan mengikuti titik sumbu nyata dan Mari kita lihat apa fungsinya. itu semua cepat jadi izinkan saya memikirkannya sedikit lebih lambat semua bilangan negatif apa pun Itu bilangan real negatif akan terjepit ke dalam kisaran antara 0 dan 1 Manakah yang masuk akal e ke negatif? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a ke bilangan real negatif adalah sesuatu antara 0 dan 1 dan kami secara khusus melacak f dari negatif 1 yang akan muncul di sekitar 1 di atas e sekitar 30 0.37 f dari 1 mendarat di e seperti yang diharapkan, itulah exp dari 1 adalah f dari I akan mendaratkan satu radian di sekitar lingkaran satuan, dan menyenangkan untuk mengikuti seluruh sumbu imajiner di sini bagaimana sumbu imajiner melilit lingkaran dan Apa yang terjadi saat kita mengubah nilai R ini? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "Kita mungkin ingin dan nilai-nilai R di sini Ini meregangkan sesuatu secara berbeda sehingga ketika kita menaikkannya menjadi 2 Anda tahu itu merentangkan sumbu nyata lebih jauh sehingga f dari 1 berakhir di sekitar tempat e kuadrat sedikit Di atas 7 f negatif 1 lebih dekat ke 0 f dari I adalah 2 radian Rotasi mengelilingi lingkaran f negatif I adalah rotasi negatif 2 radian Dan tentu saja kita dapat memperoleh rumus favorit kita bahwa jika itu adalah pi yang kita miliki sebagai konstanta penskalaan kita Maka sumbu sebenarnya terentang cukup banyak Anda tahu f dari 1 berada di e terhadap pi yang sangat dekat dengan 20 ditambah pi Yang selalu menyenangkan dan f dari negatif 1 sangat dekat dengan 0 sehingga benar-benar terentang senyata itu sumbu Dan itu juga merentangkan benda-benda ke arah lingkaran satuan sehingga Untuk mencapai f dari I atau f negatif saya berjalan setengah lingkaran, jadi semuanya baik-baik saja sekarang Bagaimana pendapat kita tentang fungsi seperti itu? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "Kita juga akan menulis sebagai X dari X log natural 2 kali X sehingga kita memindahkan titik kuning yang mewakili nilai R Ke sekitar 0.69 masih belum ada bagian imajiner hanya bilangan real 0.69 atau lebih Itulah logaritma natural dari 2. Anda dapat melihat bahwa f dari 1 mendarat di 2 Itulah sebabnya kita ingin memanggil fungsi ini 2 ke X f dari 1 setengah sebenarnya maaf f dari negatif 1 mendarat tepat di 1 setengah f dari I Ini semacam berjalan mengelilingi lingkaran satuan dengan sangat spesifik, hasilnya akan menjadi 0.69 radian di sekitar lingkaran satuan dan Sekarang kita bisa bersenang-senang lebih banyak dan mengatakan apa yang akan terjadi jika kita mengubahnya menjadi 0.69 daripada menjadi logaritma natural dari 2, buatlah I dikalikan dengan logaritma natural dari 2 sehingga kita benar-benar memikirkan Sesuatu yang mungkin mempunyai basis eksponensial. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "Berapakah I pangkat I dalam hal ini mendorongnya ke sekitar 0.2 sekitar seperlima Tapi ada banyak fungsi eksponensial berbeda yang memiliki sifat menempatkan f dari 1 ke bilangan I Jadi jika kita ingin memperbesarnya lebih jauh lagi, saya rasa saya tidak akan menganimasikannya di sini. Tapi jika kita mengambil titik kuning itu dan naikkan hingga menjadi 5 setengah kali pi I Apa yang akan kamu lihat adalah lingkaran satuan? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "Diputar pada dirinya sendiri sehingga f dari f negatif dari 1 akan berputar mengelilingi 2 pi radian lainnya dan mendarat di tempatnya. Namun ia akan merentangkan sumbu sebenarnya lebih jauh lagi. Artinya keluaran lain dari I ke I adalah angka yang jauh lebih kecil. Itu sekitar 0.0003 atau lebih Tapi kita juga bisa melihat apa yang menurut saya cukup menyenangkan Apa jadinya jika kita mempertimbangkan ekspresi alternatif yang ingin kita tafsirkan sebagai 2 pangkat X kan? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "Kita mempunyai X dari R dikali X dan R sama dengan nilai ini, yang merupakan logaritma natural dari 2 ditambah pi dikalikan I. Artinya adalah ketika kita memasukkan 1, f dari 1 bernilai negatif 2 sehingga kita ingin menulis fungsi ini sebagai negatif 2 pangkat X kan dan itu sebenarnya sesuatu yang Anda tahu, itu agak sederhana ketika kita menulis bilangan negatif pangkat Negatif 2 pangkat X pada awalnya tidak terlihat seperti ini tentu saja itu membawa kita ke dalam bilangan kompleks dengan cara apa pun, tetapi tentu saja ketika kita memasukkan nilai genap seperti 1 setengah. Di mana kita seperti meminta akar kuadrat dari negatif 2, kita menyadari bahwa kita ingin menuliskannya sebagai sesuatu seperti I kali akar kuadrat dari 2 Tetapi jika Anda melihat fungsi ini negatif 2 pangkat X dalam domain kompleks penuh yang ditanganinya. Apa yang Anda lihat adalah fungsi yang mengambil nilai 1 menjadi negatif 2 Dan jika ia melakukan hal itu, apa pengaruhnya terhadap sisa garis bilangan real apakah itu seperti spiral ke luar? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "Jadi kita melihat bahwa f dari negatif 1 berada pada setengah negatif 1 Kira-kira di mana yang Anda harapkan jika Anda mengikuti ke f dari 1 setengah Itu akan berada tepat di garis imajiner dan f dari 1 setengah adalah akar kuadrat dari 2 Ya ampun mouse tidak berada di tempat yang saya inginkan. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "Itu akan berada di sekitar akar kuadrat dari 2 kali I dan Saat Anda melanjutkan lebih jauh, ini menunjukkan kepada Anda semua pangkat nilai nyata dari negatif 2 ke X, itu pasti berputar. Tapi kita juga bisa menaikkan nilai R kita lebih tinggi lagi dan mendapatkannya hingga sekitar tau kali I sekitar enam koma dua delapan kali I dan dalam konteks itu ini adalah fungsi lain yang ingin kita tulis seperti 2 ke X karena Untuk bilangan bulat ke bilangan bulat apa pun yang Anda masukkan untuk X, itu akan terlihat seperti perkalian berulang Dan bahkan memiliki nilai yang masuk akal untuk hal-hal seperti 1 setengah yang mengeluarkan akar kuadrat negatif dan bukan akar kuadrat positif, tapi apa yang sebenarnya dilakukannya adalah transformasi ke bidang Tempat ia menempatkan semuanya adalah nyata garis bilangan berakhir menjadi spiral yang melingkar sangat erat dan berputar sedemikian rupa sehingga f dari 1 mendarat tepat di angka 2 Jadi dalam pengertian itulah kita dapat mengatakan 2 pada X adalah Diartikan secara masuk akal sebagai fungsi eksponensial yang terpisah dari fungsi yang biasa kita gunakan. Jadi saya pikir dengan semua itu saya akan meninggalkan semuanya untuk hari ini Dan saya akan meninggalkan Anda dengan beberapa pertanyaan yang masih ada untuk dipikirkan oke, jadi Jika Anda mau bayangkan I ke I sebagai ekspresi multi-nilai, kan, Anda bisa mengatakan kita mengadopsi sebuah konvensi. Anehnya Anda akan mengatakan Anda memilih cabang dari fungsi logaritma natural Dan mungkin itu mengunci Anda ke dalam makhluk ini e ke pi negatif setengah Tetapi jika Anda mengatakan keinginan seperti ini menjadi banyak nilai berbeda yang tak terhingga seperti berbagai nilai yang kita lihat Berapa banyak nilai yang ingin dimiliki oleh 2 hingga 1 ketiga dalam arti yang sama? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "Ke-10 ingin diutarakan secara berbeda dari semua izinkan saya mengatakan semua fungsi eksponensial F dari X yang memenuhi oh sudahkah saya menuliskannya di suatu tempat f dari X yang memenuhi Semua properti yang saya tulis ini jadi jika memenuhi semua dari jumlah tersebut dan jika f dari 1 sama dengan 2 Benar, berapa banyak keluaran berbeda yang akan kita peroleh ketika kita menyambungkan X sama dengan 3 per 10 untuk berbagai pilihan fungsi apa? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "Untuk 2 ke pi untuk berbagai fungsi yang dapat diwakili oleh 2 ke X jika kita menganggap 2 ke X sebagai semacam fungsi eksponensial Eksponensial dalam pengertian sifat abstrak semacam ini dan jika ya, jika kita kami memiliki Kelas dengan fungsi yang berbeda-beda, dan kami ingin menyambungkan pi, itu membuat saya tertawa. Hanya karena itu adalah jawaban lucu yang muncul saat Anda mencoba memikirkannya, jadi itulah pertanyaan yang Saya akan meninggalkan Anda dan saya pikir ini adalah Anda tahu Pertanyaan utama saya dalam mendekati kuliah hari ini adalah apakah saya menginginkannya. Jenis menggambarkan seperti sifat-sifat abstrak dari fungsi eksponensial Dan sangat keren bagi saya bahwa memulai dari sifat-sifat abstrak itu Anda terjebak dalam gagasan e ke rx atau lebih Asal tahu saja, saya pikir lebih jujur menulis exp dari r kali x untuk nilai r yang berbeda Itu mengunci Anda sejauh itu Tapi itu tidak mengunci Anda sejauh memiliki Gagasan yang jelas tentang apa yang 2 pangkat x seharusnya lebih kecil dari sesuatu seperti I pangkat x Tentu saja risikonya adalah kadang-kadang orang tidak menyukai abstraksi dan kadang-kadang itu tidak terlihat mudah didekati. Tetapi jika itu adalah jika Anda tahu, beri tahu saya saja Saya pikir saya pikir ada lingkaran pemikiran menarik yang melingkupi Semua hal ini termasuk menara listrik karena jika Anda ingin Sebenarnya berbicara tentang menara listrik seperti yang terakhir kali kita lakukan dalam konteks bilangan kompleks atau bahkan dengan basis negatif Anda harus memikirkan hal-hal seperti ini, jadi itu adalah pertanyaan yang muncul di layar Ya, apa yang terjadi jika kita melakukan ini untuk kekuatan saya? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "Titrasi lho, ayo kita coba ini, ayo kita coba menara listrik Di mana kita menaikkan I ke pangkat tertentu dan lihat apa yang muncul darinya, jadi ia tidak berencana melakukan ini Tapi kita bisa, kita selalu bisa tarik Python dan pada dasarnya melakukan apa yang kita lakukan terakhir kali Jadi cara agar ini berhasil adalah kita memulai dengan beberapa nilai dasar dan kemudian untuk beberapa jenis rentang. Apa yang kita lakukan, kita mengambil a dan kita akan menugaskan kembali itu menjadi apapun Basis yang dalam hal ini saya naikkan ke pangkat a harusnya Oke, keren, jadi kita akan melakukan itu dan kemudian kita akan mencetak nilai a mari kita lakukan ini untuk Ya, itu angka yang jauh lebih besar seperti 200 Jadi sepertinya yang terjadi adalah Ada potensi kekacauan dengan hal-hal seperti ini kadang-kadang. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "Saya sebenarnya punya jadi izinkan saya mengimpor NumPy jadi saya memiliki fungsi eksponensial biarkan saya pergi Untuk rentang besar kita seperti yang kita miliki sebelumnya Daripada menulisnya karena Anda tahu sesuatu yang seperti saya pangkat X Saya akan menulisnya sebagai fungsi eksponensial dari konstanta yang berbeda kan a Konstanta berbeda yang akan saya buat Saya ingin menjadi 5 bagian pi, jadi saya akan melakukan 5 bagian pi dikalikan I jadi itu bilangan kompleks dan memiliki 5 bagian pi sebagai bagian imajiner Jadi ini adalah 5 pi setengah kali saya dan apa yang saya lakukan? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/italian/sentence_translations.json b/2020/ldm-i-to-i/italian/sentence_translations.json index 48201c23c..614edc69b 100644 --- a/2020/ldm-i-to-i/italian/sentence_translations.json +++ b/2020/ldm-i-to-i/italian/sentence_translations.json @@ -49,7 +49,7 @@ "end": 63.7 }, { - "input": "And in fact if we go, let's not show where things are going too much here, if we go ahead and rewrite that base i in terms of e, it can help us make sense out of this expression.", + "input": "And in fact if we go, oh no, that's not sure where things are going too much here, if we go ahead and rewrite that base i in terms of e, it can help us make sense out of this expression.", "translatedText": "E infatti se andiamo, non mostriamo troppo dove stanno andando le cose qui, se andiamo avanti e riscriviamo quella base i in termini di e, può aiutarci a dare un senso a questa espressione.", "n_reviews": 0, "start": 64.12, @@ -994,7 +994,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half.", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half.", "translatedText": "Quindi, se inizi dal numero 1, la tua velocità iniziale è camminare dritto verso 0 e mentre cammini ancora più in basso, se fossi seduto a 1 metà, allora cammineresti comunque verso 0, ma ora il tuo vettore di velocità sarebbe negativo 1 volte il punto in cui ti trovi, che è negativo 1 metà.", "n_reviews": 0, "start": 998.68, @@ -1302,7 +1302,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i.", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i.", "translatedText": "E una domanda interessante sarà che sai se esiste solo una di queste funzioni che sembra ragionevole scrivere per questo perché sai che se lo scriveremo come i alla x non solo dovrebbe soddisfare questo ma dovrebbe anche soddisfare sai quando inseriamo il numero uno e otteniamo i, presumibilmente i, alla potenza uno, tuttavia pensiamo che questa funzione dovrebbe essere i.", "n_reviews": 0, "start": 1383.38, @@ -1323,7 +1323,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos.", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma", "translatedText": "Quindi abbiamo 5 pi i metà fantastico che è assolutamente un altro valore che potremmo inserire per x qui e solo per spiegarlo un po' più visivamente se dovessimo guardare indietro al nostro cerchio qui dove abbiamo il momento ha camminato per un periodo di tempo pari a pi metà, ovvero 1.57 e se invece facessimo un altro giro completo e facessimo un'altra metà pi per arrivare a pi che, sai, potremmo registrare quello è il valore e del pi i camminiamo per un'altra metà pi camminiamo per un'altra metà pi che a a questo punto avremmo fatto un giro completo riportandoci all'uno e poi cammineremo per cinque metà pi che numericamente sono circa 7.85 sì, questo è assolutamente un altro numero che ci porta in cima a i e se dovessimo affrontare tutta la trafila di riesprimere i alla potenza i scrivendo prima e alle 5 metà pi greco i alla potenza i quelle i moltiplichiamo per diventare negativo e guarderemo e alla metà negativa di 5 pi greco che è un numero molto diverso, giusto, possiamo effettivamente calcolarlo, non ne sono sicuro a mente, ma diamo un'occhiata a Desmos .", "n_reviews": 0, "start": 1415.68, @@ -1337,7 +1337,7 @@ "end": 1493.22 }, { - "input": "What is e to the negative 5 pi halves 0.000388 Okay, 0.000388 much smaller number 0.000388 Which begs the question of okay i to the i what are you right?", + "input": "What is e to the negative five pi halves? 0.000388. Okay, 000388. Much smaller number. 0.000388. Which begs the question of okay i to the i, what are you? Right?", "translatedText": "Quanto fa e sta alla meno 5 pi greco metà di 0.000388 Va bene, 0.000388 numero molto più piccolo 0.000388 Il che fa sorgere la domanda: okay i, cosa hai ragione?", "n_reviews": 0, "start": 1493.28, @@ -1358,21 +1358,21 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle?", + "input": "that long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle,", "translatedText": "Così lungo che ti porta a un numero molto più piccolo Ma questa non è l'unica risposta che potremmo inserire, abbiamo altre persone che vengono qui con 3 metà negative per i pi greco Che conosci in termini di cerchio unitario?", "n_reviews": 0, "start": 1544.74, "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one?", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one", "translatedText": "Potremmo pensare di dire ehi se voglio arrivare a I invece di camminare di 90 gradi pi dimezza i radianti in questo modo e se cammino di 270 gradi nell'altro modo 3 pi greco dimezza i radianti che forse considererò negativo perché la convenzione è di solito quello in senso antiorario è positivo Questo è assolutamente un altro modo per esprimerlo e questo ci darebbe una risposta diversa se avessimo e = 3 pi greco metà negativi i Tutti alla potenza i facciamo lo stesso gioco ora i al quadrato si cancella con a negativo che è già lì, e abbiamo 3 metà pi greco positive e numericamente questo ci dà una risposta dall'aspetto ancora diverso da quello che avevamo prima Che se andiamo oltre e diciamo ehi, quanto fa e al 3 pi greco non 3 o 3 pi greco metà 111 punto 3 1 tipo di numero molto diverso da quello che abbiamo visto prima 111 punto cos'era 111 punto 3 1 fantastico 111 punto 3 1 o giù di lì E ancora in termini di intuizione quello che potresti chiedere è supporre che abbiamo questa rotazione dinamico Ma andiamo indietro nel tempo vediamo quanto tempo fa nel tempo cosa devo essere Tale che se giocassi in avanti da lì arriverei al numero uno la mia condizione iniziale e devi tornare indietro nel tempo 3 pi metà unità E poi se dovessi tradurre la dinamica del decadimento Che è ciò che fa alzare lo sguardo in questo contesto dici se comincio dal numero uno Ma voglio andare indietro nel tempo e dire Da dove avrei dovuto iniziare se Voglio decadere in modo tale da finire al numero uno?", "n_reviews": 0, "start": 1559.26, "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right?", + "input": "after three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right?", "translatedText": "Dopo 3 pi greco metà unità di tempo la risposta evidentemente inizia intorno a centoundici per quel tipo di decadimento esponenziale E puoi vedere dove sta andando dove ci sono in realtà infiniti valori diversi che potremmo inserire per X se siamo pensando alla e alla X come se fossi io e la gente è entrata molto di più qui Scusate, lancio la mia spilla a terra come si fa in modo classico per il terzo posto 9 metà pi greco ottima scelta 1729 metà pi greco siete tutti i miei preferiti, tanti, tanti diverse opzioni un'infinità di valori diversi il che all'inizio sembra un po' sconcertante perché guardiamo un'espressione Sembra che tu sappia che ci saranno solo dei calcoli Lo inserisco nella calcolatrice e vedo cosa salta fuori e ne otterremo molti diversi valori per esso Quindi, cosa sta succedendo qui, giusto?", "n_reviews": 0, "start": 1657.18, @@ -1442,7 +1442,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function?", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function,", "translatedText": "La quarta radice di 16 dovrebbe essere 2 e la risposta alla fine sarà positiva Adottiamo una convenzione quando ci sono più opzioni come questa quando hai una funzione multivalore Spesso scegliamo semplicemente uno di questi valori perché sia ciò che intendiamo quando vogliamo trattalo come una funzione come qualcosa con un singolo input e un singolo output in un gergo più elaborato Questo emerge spesso quando abbiamo a che fare con numeri complessi l'idea di qualcosa come un'operazione come se volessi avere più valori a volte ascolta la frase ramo Dove scegli un ramo della funzione radice quadrata?", "n_reviews": 0, "start": 1795.66, @@ -1463,7 +1463,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer?", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer?", "translatedText": "Perché ci sono molteplici risposte diverse Sai che pensiamo che io sia di nuovo questa rotazione di 90 gradi E se la pensassimo come una rotazione di 90 gradi sembra che la radice quadrata dovrebbe essere Sai qualcosa seduto ad un angolo di 45 gradi Forse questo è il quadrato radice di I che potremmo scrivere in modo molto esplicito come radice 2 su 2 radice 2 su 2 I Si tratta solo di usare la trigonometria, ma se invece pensassimo a I come ad una rotazione negativa di 270 gradi, sembra che metà di ciò esegua metà di quell'operazione dovrebbe effettivamente portarci dall'altra parte Forse il numero seduto qui dovrebbe essere la radice quadrata di I e in realtà è solo il negativo di ciò che abbiamo visto prima Radice negativa 2 su 2 meno radice 2 su 2 volte I Ora nel contesto del reale funzioni con valore possiamo dire di sì Basta scegliere la radice quadrata come qualunque sia la risposta positiva, ma quale di queste consideri la risposta positiva?", "n_reviews": 0, "start": 1846.36, @@ -1477,14 +1477,14 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x?", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x,", "translatedText": "E penso che tu dica bene Sappiamo di cosa si tratta, in un certo senso lo definiamo come la radice quadrata di 2 tutto va bene e bene Ma cosa succederebbe se dicessi: affrontiamolo nello stesso modo in cui ci stavamo avvicinando al nostro io all'espressione io io voglio prima esprimere le cose come e per qualcosa di giusto e poi lo eleverò a 1 metà moltiplicando 1 metà per l'esponente E dico okay, posso, immagino di poterlo fare e per ciò che è uguale a 2 beh Questo è il logaritmo naturale di 2 È una costante che è intorno a 0.69 o giù di lì Se eleviamo e a quella potenza otterremo 2 quindi potremmo pensare a questo come e al logaritmo naturale di 2 per 1 metà e se volessi pensarci e alla x?", "n_reviews": 0, "start": 1942.28, "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it.", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it.", "translatedText": "Sai che questo potrebbe essere eccessivo nel contesto dei numeri reali Ma se pensassi a e-x come una scorciatoia per questa funzione x potresti inserire il valore 0.69 volte 1 metà che immagino sia circa 0.345 Qualcosa del genere. Inserisci quel valore molto concreto nel tuo polinomio per vedere cosa restituisce e il risultato sarà circa 1.414 un bel numero reale radice quadrata di 2 quello che ti aspetteresti Ma se facciamo la stessa cosa che stavamo facendo con I e riconoscendo che in realtà ci sono più risposte diverse quando vogliamo scrivere qualcosa come e elevato a una potenza potremmo anche scrivere questo Potrebbe sembrare divertente, ma potremmo scriverlo come e nel logaritmo naturale di 2 più 2 pi I Tutta quella cosa elevata a 1 metà Subito dopo tutto questo valore sarà uguale a potresti scomporlo come è e in logaritmo naturale di 2 Moltiplicato per e a 2 pi greco I Questo ha semplicemente l'effetto di ruotare le cose di 360 gradi, quindi sarà uguale a 1 Quindi stiamo guardando 2 per 1 grande che sembra una sostituzione valida eppure quando giochiamo allo stesso gioco di prendere questo ed elevarlo a potenza e trattarlo moltiplicando la potenza per l'esponente guarda cosa succede Abbiamo e al logaritmo naturale di 2 per 1 mezzo più Bene, quanto fa 2 pi greco I per 1 mezzo beh sarà pi greco per I Ora questa prima parte e corrisponde al logaritmo naturale di 2 per 1 metà che finirà per essere la familiare radice quadrata di 2 va tutto bene, ma lo moltiplicheremo per e per il pi I Giusto e abbastanza famoso e al pi I è negativo 1 Quindi in questo caso sembra suggerire che se stiamo risolvendo questa espressione 2 alla 1 metà Giocando con le diverse risposte potremmo inserire qualcosa come e alla X uguale a 1 metà ciò che otteniamo è un'altra risposta che potremmo tradizionalmente scrivere come radice quadrata negativa di 2 e qui intendo dire che è un po' strano avere più valori da considerare 2 alla 1 metà e diciamo che non è uguale Una cosa ma in base alle scelte che facciamo potrebbe equivalere a più cose diverse Ma le due cose potrebbero sembrare abbastanza ragionevoli Se ci sarà qualcosa che è 2 a 1 metà sembra che dovrebbe essere o Il positivo radice quadrata che ci è familiare o la variante negativa di ciò che in realtà non sembra un grosso problema E in effetti potremmo uhm potremmo giocare a questo gioco anche oltre dove permettetemi di chiedervi risposte ancora più creative a questa espressione perché forse possiamo trovare altre potenze divertenti di qualcosa come 2 alla potenza X quando iniziamo a inserire vari valori diversi di X in base alla sostituzione che facciamo se rispettiamo le stesse regole che stavamo usando nel valutare I alla potenza I Quindi questa volta la domanda chiede ovvero specifica che una soluzione dell'equazione e alla x uguale a 2 è il numero reale Log naturale di 2 ok quello lo sappiamo.", "n_reviews": 0, "start": 1989.66, @@ -1498,14 +1498,14 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42.", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42", "translatedText": "risposta alla domanda e alla x è uguale a 2 e ancora una volta la creatività è benvenuta, quindi ti darò un altro piccolo momento per quello II Andrò avanti e inserirò alcune risposte qui se per te va bene non sono sicuro di quanto tempo ci vorrà richiede necessariamente di eseguire i calcoli a seconda del dispositivo che stai guardando, ma non stressarti troppo se è prima che tu abbia la possibilità di inserire la domanda a cui vuoi rispondere a cui vuoi che risponda Quindi sembra 131 di voi hanno inserito la variante in cui prendiamo Ln di 2 e aggiungiamo 2ii e immagino di stare scrivendo questa domanda Erroneamente ho contrassegnato una delle risposte come corretta quando in realtà ce ne sono parecchie diverse corrette Quindi dipende da me per il fatto che non so se a qualcuno di voi sembra oh È rosso, avete sbagliato a inserire Ln di 2 più 42.", "n_reviews": 0, "start": 2176.56, "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer.", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer.", "translatedText": "I pi che ovviamente è un'ottima scelta Ma potresti anche avere qualcosa come 4 pi I più il logaritmo naturale di 2 o 6 pi I O in realtà qualsiasi multiplo intero di 2 pi I se aggiungi che non influisce su e X Perché ha semplicemente l'effetto di moltiplicare per e = 2 pi greco I Che è l'effetto di moltiplicare per 1 e ancora questo ha una conseguenza divertente in cui sembra produrre risultati ragionevoli quando lo facciamo come altro esempio It sembra che la seconda espressione più comune inserita sia che potremmo sostituire 2 Quindi pensiamo di pensare a 2 elevato a 1 4, okay c'è stato un suggerimento di sostituire 2 con e nel logaritmo naturale di 2 più 4 pi I Ok Più 4 pi I e rilanciamo tutto fino a 1 4, bene, se giocassi allo stesso gioco otterresti e Al logaritmo naturale di 2 per 1 4, e moltiplicheremmo per e per il pi greco I Ora la prima parte sarà la solita quarta radice positiva di 2 ciò che intendiamo quando inserisci un'espressione come radice quarta di 2 in una calcolatrice un bel piccolo numero positivo, ma poi questa seconda parte è negativo 1 quindi sembra dire Sai se interpretassimo 2 in questo modo diverso elevandolo a 1 4 Sai che non è la solita risposta che otteniamo ma è una risposta ragionevole.", "n_reviews": 0, "start": 2253.76, @@ -1519,7 +1519,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships.", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships.", "translatedText": "Avremmo guardato pi metà per I e invece di moltiplicare per Negativo 1 avremmo invece moltiplicato per I Che ancora una volta è una risposta valida sembra un risultato ragionevole per qualcosa come 2 alla 1 4 Quindi quando sei guardando il fatto che I alla potenza I sembra avere più valori diversi Giusto, abbiamo questo strano fenomeno in cui potremmo collegare e alle 5 metà pi greco I Negativo 3 metà pi greco I e otteniamo quelle che sembravano risposte completamente diverse qualcosa di super piccolo qualcosa di super grande tutto molto diverso dal 15 circa 1 5 risposta che abbiamo trovato prima quassù È esattamente lo stesso fenomeno di quando chiedi qualcosa come quanto fa 2 alla 1 4 e riconosci che in realtà ci sono più soluzioni diverse all'espressione X alla quarta equivale a 2 4 soluzioni diverse in effetti e quello che stai guardando è il fatto che ci sono più soluzioni diverse All'espressione e alla X equivale a una sorta di base se quella base è io se quella base è 2 Qualunque cosa possa essere e un modo in cui potremmo pensarci è che quando hai a che fare con numeri reali le cose sono semplicemente adorabili, le cose sono belle Ci sono relazioni uno a uno.", "n_reviews": 0, "start": 2358.92, @@ -1533,7 +1533,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value?", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value", "translatedText": "È fantastico Dove, se vogliamo pensare alle funzioni esponenziali, lasciami coprire alcune di queste cose Abbiamo questo simpatico avanti e indietro in cui puoi scegliere di esprimere qualsiasi esponenziale come base di X come 2 alla X Oppure potresti esprimere quello stesso esponenziale di X di R per X che sai che è il polinomio a cui ci riferiamo Whenever a cui ci riferiamo implicitamente ogni volta che scriviamo qualcosa come e su X E c'è un bellissimo avanti e indietro perché puoi semplicemente prendere un logaritmo naturale di B E ti dà una risposta presupponendo che B sia un numero positivo Ed è la stessa cosa che dire che X di R è uguale a B Quindi un modo di cui ho parlato in precedenza nella serie è che se stavi guardando il famiglia di tutti gli esponenziali possibili giusto, potremmo scriverli come X di R per X e cambiare il valore di R E questa è esattamente la stessa cosa che scrivere e su R per X se è qualcosa con cui ti senti più a tuo agio Quindi e su R per XX di R per X quelle sono la stessa cosa a cui potremmo pensare di cambiare ciò che è Ma d'altra parte se dovessi pensare a tutti i possibili esponenziali come una base Fammi fare la base alla potenza di X e andiamo per cambiare ciò che è quella base All'inizio sembra che sia un diverso tipo di espressione da manipolare, ma è solo un altro modo di esprimere la stessa famiglia E un modo in cui potresti pensare a questo Per come pensiamo a quale base corrisponde a se stiamo pensando in modo un po' più astratto come Esp di R per X e c'è una ragione per cui lo sto facendo perché stiamo per applicarlo a numeri complessi dove sembrerà più strano quindi seguimi qui se invece di guardare quella base, una cosa che potrei fare è dire qual è il valore?", "n_reviews": 0, "start": 2428.5, @@ -1554,7 +1554,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R.", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d", "translatedText": "Potrei avere l'espressione di R per X dove forse R è qualcosa come zero virgola sei nove Ma potrei spostarlo verso il basso di due pi greco I E questo non cambia la base a cui corrisponderebbe che corrisponderebbe comunque a due Oppure potrebbe spostalo verso l'alto di due pi greco I ciò non cambia la base a cui corrisponde perché in tutti questi casi Quando inseriamo X è uguale a uno otteniamo comunque la stessa cosa Tutte queste per valori diversi di X sono funzioni distinte Questa è perché abbiamo visto più valori diversi per I elevato alla potenza I Poiché I elevato a X è una funzione ambigua in quel contesto non sarebbe ambiguo se decidessimo quale valore di R Tale che ciò che stiamo rappresentando è espressione di R moltiplicato per X quale valore di R.", "n_reviews": 0, "start": 2597.88, @@ -1568,14 +1568,14 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers.", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers,", "translatedText": "È una funzione inequivocabile ma a quel punto sembra che forse quello che vogliamo è smettere di pensare alle cose in termini di una base elevata alla potenza X Forse non appena siamo nel contesto dei numeri complessi Dovremmo semplicemente scrivere tutti come espressione di alcuni tempi costanti X se non altro rende chiarissimo Come inseriamo effettivamente i numeri se vogliamo fare un calcolo o semplicemente farci dei calcoli abbiamo questo bel polinomio infinito che noi collegali e ti spiegherò ancora una volta che questo è forse il modo corretto di pensare agli esponenziali Non appena estenderemo ad altri domini cose come i numeri complessi e per questo facciamo semplicemente un backup Vai tornando al campanello, alcune cose sono arrivate tornano al modo originale in cui estendiamo l'idea di esponenziazione e pensiamo solo a quanto fa 2 alla X. Sappiamo come pensarci per i numeri naturali.", "n_reviews": 0, "start": 2640.66, "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that.", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that?", "translatedText": "Conosci qualcosa come 2 alla 3 Moltiplicazione ripetuta Com'è che ti hanno insegnato per la prima volta a pensare a qualcosa come 2 alla X per importi frazionari o per importi negativi e cose del genere.", "n_reviews": 0, "start": 2696.88, @@ -1589,35 +1589,35 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive.", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive.", "translatedText": "Di solito ti viene insegnato che 2 alla 1 metà dovrebbe essere qualcosa in cui sai che se lo moltiplico per se stesso e questo segue le solite regole che fanno gli esponenziali con il conteggio dei numeri in cui siamo in grado di aggiungere cose in quell'esponente dovrei ottenere 2 a 1 quindi dovrebbe essere un numero che quando lo moltiplico per se stesso ottengo 2 e sai che a quel punto hai una scelta, forse è positivo.", "n_reviews": 0, "start": 2708.3, "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1?", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the", "translatedText": "Forse è negativo Ma se decidi sempre di fare la scelta positiva Sarai in grado di ottenere una bella funzione continua da questo stesso affare se chiediamo dei numeri negativi Cosa dovrebbe essere 2 meno 1 beh, dovrebbe essere qualcosa dove quando lo moltiplico per 2 a 1?", "n_reviews": 0, "start": 2731.78, "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily.", + "input": "one, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily.", "translatedText": "Mi dà 2^0 e questaèuna specie di giustificazione per la nostra convenzione secondo cui gli esponenti negativi assomigliano a 1 metà Ma quello che sta realmente succedendo qui è che stiamo dicendo qualunque cosa sia dovrebbe essere una sorta di funzione Che soddisfa questa proprietà f di a più b è uguale a f di a per f di b e Inoltre il fatto che la base sia 2 ci dice fondamentalmente che non è una funzione qualsiasi È una funzione in cui quando inseriamo 1 otteniamo 2 E altrettanto un po' lo sai domanda stile controllo di integrità per vedere se stai seguendo alcune delle implicazioni qui Voglio chiederti cos'è Non lo chiamerò come un softball, ma questo non vuole essere una domanda incredibilmente profonda necessariamente.", "n_reviews": 0, "start": 2744.96, "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here?", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here.", "translatedText": "È solo più un controllo se stai seguendo L'idea di iniziare astrattamente con le proprietà di una funzione e poi dedurre i modi in cui potremmo volerla scrivere in base a quelle proprietà Se f(x) soddisfa questa proprietà esponenziale f di a più b è uguale a f di a per f di b per tutti gli input E soddisfa anche f di 1 uguale a 2 quale delle seguenti è vera Vale a dire quale delle seguenti è necessariamente vera Non importa quale funzione stai avviando con e quelli di voi che ricordano quale lezione era Qualunque fosse quella di cui stavamo parlando su come interpretare ciò che dice realmente la formula di Eulero Ho fatto una domanda di questo stile in cui ho trascurato una singola condizione, sai che non l'ho scritta il fatto che vogliamo assicurarci che f(x) sia diverso da zero ovunque e quindi ciò ha causato una certa quantità di confusione, il che è bello ottenere confusione sullo schermo che accade a tutti noi Ma l'intento era fondamentalmente mostrare che questa proprietà astratta di qualcosa che trasformi l'addizione in moltiplicazioneèÈsufficiente per farti venir voglia di scrivere la funzione come qualunque cosa sia elevata a una sorta di potenza Questoèlo spirito della domanda Ora abbiamo un paio di domande sulle torri energetiche che sembrano essere spuntati qui, il che è molto collegato all'ultima volta. Aspettiamo la questione della torre di potenza solo per un momento in modo da avere prima una sensazione più profonda di cosa dovrebbe significare l'esponenziazione qui?", "n_reviews": 0, "start": 2793.44, "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane?", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane?", "translatedText": "Perché poiché possiamo essere ciò che voglio affermare è che possiamo rispondere in molti modi diversi Quindi se me ne dai solo uno, parleremo di torri di potere E poi proprio come una linea numerica può essere rappresentata in una scala logaritmica si può si può fare la stessa cosa per un piano complesso?", "n_reviews": 0, "start": 2882.64, @@ -1638,7 +1638,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one.", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one", "translatedText": "Sì In effetti, c'è una visualizzazione a cui arriverò tra un momento qui in cui facciamo qualcosa di abbastanza simile a quello Perché quello che faremo è giocare con diverse funzioni esponenziali X di R per X Ma siamo cambieràil valore di R che saràrappresentato da un piccolo punto giallo Quindi ne parleremo Non mapperà l'intero piano, ma solo un paio di punti campione dall'asse reale e dall'asse immaginario Ma l'idea è che mentre ci muoviamo intorno a quale sia quella costante saremo in grado di visualizzare le diverse cose che fa all'aereo e in effetti è come se trasformasse l'asse x in una scala logaritmica e poi avvolgesse l'asse immaginario lungo un cerchio E poi non appena quel valore di R diventa immaginario scambia il ruolo di quei numeri reali vengono messi sul cerchio e i numeri immaginari vengono messi su un asse positivo in scala logaritmica quindi bella domanda tutti e tre i quali immagino sto in un certo senso anticipando i tempi per capire dove voglio arrivare, ma è bello vedere che è lì che la gente la pensa così in questo.", "n_reviews": 0, "start": 2901.54, @@ -1652,14 +1652,14 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it.", + "input": "explicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you want, which is that let's say you have some complex function f, a", "translatedText": "esplicitamente Qualcosa come f(5) è la stessa cosa di f(1 più 1 più 1 più 1 più 1 Che è la stessa cosa di f(1) moltiplicato per se stesso 5 volte a causa di questa proprietà Che se f(1 è 2 è la stessa cosa) come 2 elevato a 5 e poi qualcosa come f(meno 5) Dovrebbe essere il caso che quando lo moltiplichiamo per f(5) otteniamo qualunque cosa sia f(0) e non è immediatamente chiaro cosa sia f(0), ma potremmo dirlo f(1 più 0 è uguale a qualunque cosa f(1) sia per quello che f(0 è ma f(1) è uguale a 2 E quindi anche questo è uguale a 2 quindi stiamo dicendo 2 è uguale a 2 volte qualcosa beh quel qualcosa deve essere un 1 quindi in questo contesto questo garantisce che f di meno 5 è 2 alla meno 5 è 1 su 2 alla 5a Potremmo scrivere esplicitamente questo come 2 alla meno 5 che è tutto per dire Queste due proprietà insieme formano vogliamo davvero scrivere la funzione come 2 alla X Perché qualsiasi numero di conteggio che inseriamo soddisferà Sembra che moltiplichi per se stesso quel numero di volte qualsiasi numero frazionario che inseriamo soddisferà queste proprietà che volevamo E potresti chiederti è unico e nel contesto delle funzioni con valori reali in realtà lo sarebbe Ma nel contesto delle funzioni con valori complessi Ci sarebbero più funzioni f che potremmo scrivere per questa ed è ciò che eravamo guardando prima Dove potremmo avere una funzione definita come espressione del logaritmo naturale di 2 più 2 pi I tutte quelle volte X Ok, perdona la trascuratezza qui, sono semplicemente entusiasta di scriverne E questa è in realtà una funzione diversa come evidenziato da ciò che accade se inserisci X è uguale a 1 metà Abbiamo visto poco prima come quando inserisci 1 metà quello che ottieni è la radice quadrata negativa di 2 e poi se inserisci 1 quarto ottieni Non la radice quarta di 2 ma moltiplica la radice quarta di 2 quindi è una funzione diversa Ma soddisfa comunque queste proprietà e ci fa venire voglia di scriverlo come 2 alla X E suggerisce che forse 2 alla X è ambiguo un po' di notazione E dovremmo semplicemente scrivere tutto in termini di espressione di R per qualcosa ma potresti chiederti bene Sai forse non siamo abbastanza creativi con tutte le funzioni che soddisfano questa proprietà Forse c'è un'ambiguità quando scriviamo espressione di R per qualcosa e ci sono diversi valori di R che potrebbero entrare in gioco Ma farò solo una piccola affermazione e poi forse darò uno schizzo di come sarebbe la dimostrazione se vuoi Cioè, diciamo supponiamo che tu abbia una funzione complessa F, e che soddisfi prima le seguenti proprietà. Puoi ricavarne una derivata.", "n_reviews": 0, "start": 2974.0, "end": 3140.02 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways.", + "input": "nd it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I don't know, fractional amounts you might want to think of in crazy ways", "translatedText": "È differenziabile, il che gli impedisce di essere qualcosa che conosci, una cosa discontinua totalmente disordinata. È come assumere alcuni valori casuali a seconda che tu conosca l'intervallo di qualunque spazio vettoriale oltre, non conosco quantità frazionarie a cui potresti voler pensare in modi folli.", "n_reviews": 0, "start": 3140.12, @@ -1673,7 +1673,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right?", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right?", "translatedText": "È differenziabile Non è uguale a 0 ovunque, quindi la condizione che mi è sfuggita di mente e ho dimenticato per quale lezione o qualcosa del genere e poi ha questa proprietà centrale che trasforma l'addizione in moltiplicazione Se hai una funzione del genere, lo affermo c'è un unico forse dovrei davvero specificare che esiste un numero complesso unico R in modo che tu possa scrivere F di X come se fosse fondamentalmente questa funzione esponenziale di R moltiplicato per il valore X In pratica sai dire che se hai X come funzione questo polinomio infinito con belle proprietà derivate e tutto il resto se hai questo hai ogni esponenziale che vuoi in un senso molto simile astratto e generico della parola esponenziale basato solo su una proprietà che potremmo desiderare da esso e lo schizzo della dimostrazione sarebbe guarda qualcosa del genere se vuoi prima vedere qual è la derivata di questo valore che assumiamo esista ovunque, giusto?", "n_reviews": 0, "start": 3160.84, @@ -1694,7 +1694,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base.", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base", "translatedText": "Possiamo scomporre completamente F di X dall'espressione e l'intero limite è espresso solo in termini di H. Il che, se si pensa a cosa significa nel contesto delle derivate e al fatto che F di 0 è necessariamente uguale a 1 L'intera espressione limitante è solo una costante ma più specificatamente qualunque sia la derivata della nostra funzione a 0 Quindi hai questa cosa divertente per cui se conosci la sua derivata a 0 determina qual è la sua derivata ovunque E nel contesto delle funzioni esponenziali questo si spera sia abbastanza familiare perché tutto ciò che stiamo veramente dicendo è che la derivata di una funzione esponenziale è proporzionale a se stessa e che la costante di proporzionalità è uguale a qualunque sia la derivata a 0 tutto questo è espresso in modo molto astratto e così via, ma lo scopo è enfatizzare che è non necessariamente solo funzioni che già consideriamo come elevate alla potenza X Ma è una classe di funzioni potenzialmente molto più ampia che soddisfa semplicemente questa proprietà astratta di trasformare l'addizione in moltiplicazione Ma se lo hai in realtà garantisce di avere anche un derivata seconda E del resto una derivata terza e tale perché la funzione derivativa è semplicemente proporzionale a se stessa Quindi per prendere la derivata n-esima basta guardare Quella costante di proporzionalità ed elevarla alla potenza n e poi da qui Potresti fare a Espansione in serie di Taylor e potrei lasciarlo come una sorta di compito avanzato per quelli di voi che si sentono a proprio agio con le serie di Taylor in quell'idea, specialmente se volete mescolare l'idea di qualsiasi funzione differenziabile che sia differenziabile nel senso di numeri complessi, che è una specie di argomento decisamente universitario Sai che potresti mescolare il ragionamento come vuoi Ma il ragionamento fuzzy è consentito nel contesto di qualcuno che conosce solo le serie di Taylor e nient'altro per prendere questa idea e guardare l'espansione di Taylor per F e in un certo senso giustifica l'idea che esiste un numero complesso unico tale che la nostra funzione F può necessariamente essere scritta in questo modo E quindi la connessione agli esponenziali normali avviene ogni volta che hai un tale valore R Facciamo essenzialmente quello che facciamo nel contesto complesso dei numeri reali è se guardi l'espressione di quella funzione di quel valore R e scrivila come base.", "n_reviews": 0, "start": 3243.7, @@ -1708,21 +1708,21 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace?", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace", "translatedText": "Potremmo interpretarlo non solo nel senso di exp di pi metà I per X, ma potremmo anche interpretarlo nel senso di exp di 5 pi metà I per X e Queste sono funzioni separate E c'è una famiglia infinita di funzioni separate che sembra che dovremmo scrivili come I alla X Quindi l'espressione I alla I a meno che tu non abbia adottato uno standard per cosa significherà necessariamente Quando dici che ha infiniti risultati un altro modo di pensarci è La funzione I alla X con la notazione che abbiamo è un po' ambiguo Ora con tutto ciò iniziamo a visualizzarne alcune perché penso che sia divertente E sai che tu mi dici se è così se è un'immagine utile o un'immagine più confusa ma quello che faremo è guardare questa funzione exp di R per X, che è fondamentalmente questo è un altro modo di scrivere e elevato a X infatti penso di aver renderizzato un'animazione diversa ad un certo punto che specificava che perché stavo pianificando di farlo quindi lasciami oh sì eccoti tornare nel mio file system tornare dove dovresti essere Entra lì si sta lamentando perché ce ne sono molti diversi Sarà come se ci fosse un Oh sostituisci, appare sull'altro schermo Aspetta, perché sì, ok, sostituisci?", "n_reviews": 0, "start": 3391.12, "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative?", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative", "translatedText": "Metti lì tutto ciò che vedi E ora torniamo a oh ecco, tutto questo, tutto così, così avrei potuto scriverlo bene Se ti senti a disagio nel pensarlo come un'espressione di R per X questo polinomio infinito Proprio nel dietro la testa e alla R per X e varieremo attorno a R quindi seguirò i punti dell'asse immaginario e seguirò i punti dell'asse reale e vediamo cosa fa Bene è tutto abbastanza veloce quindi lasciami riflettere un po' più lentamente tutti i numeri negativi qualsiasi cosa Questo è un numero reale negativo verrà schiacciato nell'intervallo tra 0 e 1 Quale dovrebbe avere senso e al negativo?", "n_reviews": 0, "start": 3472.82, "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R?", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r", "translatedText": "a con un numero reale negativo è qualcosa tra 0 e 1 e stiamo monitorando specificamente f di 1 negativo che verrà visualizzato intorno a qualunque cosa 1 su e sia circa 30 0.37 f di 1 si ferma su e come previsto ecco cosa è l'exp di 1 f di I farà atterrare un radiante attorno al cerchio unitario, ed è piuttosto divertente seguire lungo l'intero asse immaginario qui come l'asse immaginario viene avvolto attorno a un cerchio e cosa succede quando modifichiamo questo valore di R?", "n_reviews": 0, "start": 3511.78, @@ -1736,7 +1736,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like?", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like?", "translatedText": "Potremmo volere i valori di R qui Allunga le cose in modo diverso quindi quando lo mettiamo a 2 Sai che allunga l'asse reale molto di più così che f di 1 finisce intorno a dove e al quadrato è leggermente superiore a 7 f di negativo 1 è molto più vicino a 0 f di I è una rotazione di 2 radianti La rotazione attorno al cerchio f di I negativo è una rotazione di 2 radianti negativi E ovviamente possiamo arrivare alla nostra formula preferita che se fosse pi greco avessimo come costante di scala Allora l'asse reale si allunga parecchio Sai che f di 1 è seduto su e al pi greco che è molto vicino a 20 più pi greco Il che è sempre divertente e f di meno 1 estremamente vicino a 0 quindi è davvero allungato così tanto asse E ha anche allungato le cose nella direzione del cerchio unitario in modo che Arrivare a f di I o f di negativo I cammina a metà del cerchio, quindi va tutto bene adesso Come penseremmo di una funzione del genere?", "n_reviews": 0, "start": 3551.6, @@ -1750,7 +1750,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it.", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it", "translatedText": "Scriviamo anche come X di X del logaritmo naturale di 2 volte X, quindi spostiamo il punto giallo che rappresenta il valore di R To attorno a 0.69 ancora nessuna parte immaginaria, solo un numero reale 0.69 o giù di lì Questo è il logaritmo naturale di 2 beh puoi vedere che f di 1 si posiziona su 2 Ecco perché vogliamo chiamare questa funzione 2 alla X f di 1 metà in realtà scusa f di meno 1 si posiziona esattamente su 1 metà f di I È una passeggiata attorno al cerchio unitario in modo molto specifico e sarà 0.69 radianti attorno al cerchio unitario e ora potremmo divertirci un po' di più e dire cosa accadrebbe se lo cambiassimo in invece di essere 0.69 invece di essere il logaritmo naturale di 2 rendilo moltiplicato per I per il logaritmo naturale di 2 così pensiamo davvero a qualcosa che potrebbe avere una base esponenziale.", "n_reviews": 0, "start": 3610.52, @@ -1778,14 +1778,14 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle?", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle?", "translatedText": "Qual è I elevato alla potenza I in questo caso lo sposta a circa 0.2 intorno a un quinto Ma ci sono molte diverse funzioni esponenziali che avrebbero questa proprietà di mettere f(1) sul numero I Quindi se dovessimo ingrandirlo ulteriormente non credo di averlo animato qui Ma se dovessimo prendere quel punto giallo e sollevalo finché non arriva a 5 metà per pi greco. Quello che vedresti è il cerchio unitario?", "n_reviews": 0, "start": 3749.24, "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right?", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right?", "translatedText": "Viene ruotato su se stesso in modo che f di meno f di 1 ruoti attorno ad altri 2 pi radianti e finisca dove si trova Ma allungherebbe molto di più l'asse reale Qual era il senso in cui un'altra uscita di I verso I è un numero molto molto più piccolo Era intorno a quanto era 0.0003 o giù di lì Ma possiamo anche vedere quello che penso sia piuttosto divertente. Cosa succede se consideriamo espressioni alternative che vogliamo interpretare come 2 elevato alla potenza X, giusto?", "n_reviews": 0, "start": 3773.06, @@ -1806,21 +1806,21 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward?", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward?", "translatedText": "Abbiamo X di R per X e R è uguale a questo valore, che è il logaritmo naturale di 2 più pi greco per I Ciò significa che quando inseriamo 1 f di 1 è a meno 2, quindi vogliamo scrivere questa funzione come meno 2 alla potenza X giusto e in realtà è qualcosa che sai, è un po' ingannevolmente semplice quando scriviamo un numero negativo a una potenza Negativo 2 alla potenza X a prima vista non sembra necessariamente che ci porti nei numeri complessi in alcun modo ma ovviamente quando inseriamo anche un valore come 1 metà Dove stiamo chiedendo la radice quadrata di meno 2 ci rendiamo conto che vogliamo scriverlo come qualcosa come I moltiplicato la radice quadrata di 2 Ma se guardassi questa funzione meno 2 elevato alla potenza X nell'intero dominio complesso con cui ha a che fare Quello che stai guardando è una funzione che porta il valore di 1 a meno 2 E se fa questo cosa lo fa con il resto della linea dei numeri reali, è una specie di spirale verso l'esterno?", "n_reviews": 0, "start": 3820.68, "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be.", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be.", "translatedText": "Quindi vediamo che f di -1 metà si trova a -1 metà Più o meno dove ti aspetteresti se seguissi f di 1 metà Si troverebbe esattamente sulla linea immaginaria e f di 1 metà sarebbe radice quadrata di 2 Bene, il mio il mouse non è dove voglio che sia.", "n_reviews": 0, "start": 3880.24, "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense?", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense", "translatedText": "Sarebbe intorno alla radice quadrata di 2 per I e mentre continui oltre questo ti mostra tutte le potenze del valore reale di meno 2 alla X necessariamente si muove a spirale Ma potremmo anche spostare il nostro valore di R ancora più in alto e ottenerlo fino a circa tau volte I circa sei virgola due otto volte I e in quel contesto questa è un'altra funzione che vorremmo scrivere come qualcosa come 2 alla X perché per qualsiasi numero intero o numero intero che inserisci per X lo farà sembra una moltiplicazione ripetuta E ha anche dei valori ragionevoli per cose come 1 metà in cui sputa la radice quadrata negativa invece di una radice quadrata positiva, ma ciò che in realtà sta facendo è una trasformazione nel piano Dove mette tutto è il reale la linea numerica finisce per essere una spirale avvolta molto strettamente che gira intorno e si muove a spirale in modo tale che f di 1 si ferma proprio sul numero 2 Quindi è in questo senso che potremmo dire che 2 alla X è plausibilmente interpretato come una funzione esponenziale separata da quella a cui siamo tradizionalmente abituati Quindi penso che con tutto ciò lascerò le cose per oggi E ti lascerò solo con un paio di domande persistenti a cui pensare, okay, quindi se vuoi pensa a I-I come un'espressione con più valori, giusto potresti, potresti dire che adottiamo una convenzione Fantasiosamente diresti di scegliere un ramo della funzione logaritmo naturale E forse questo ti blocca in questo essere e al pi negativo metà Ma se dici che questo tipo di vuole essere infiniti valori diversi come i vari che abbiamo visto Quanti valori vuole essere 2^1 terzo nello stesso senso?", "n_reviews": 0, "start": 3894.92, @@ -1834,7 +1834,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function?", + "input": "three-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function?", "translatedText": "I decimi vogliono essere espressi in modo diverso rispetto a tutte le funzioni esponenziali F di X che soddisfano oh, l'ho scritto da qualche parte f di X che soddisfa Tutte queste proprietà che ho scritto quindi se soddisfa tutte di questi e se f(1) è uguale a 2 Giusto, quanti output diversi otterremo quando inseriamo X uguale a 3 decimi per le varie opzioni per quale funzione?", "n_reviews": 0, "start": 4008.86, @@ -1848,14 +1848,14 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I?", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i", "translatedText": "Per 2 alla X per le varie funzioni che 2 alla X potrebbe rappresentare se pensiamo a 2 alla X come una sorta di funzione esponenziale Esponenziale nel senso di questo tipo di proprietà astratte e se sì, se se abbiamo una classe di diverse funzioni simili e vogliamo collegare pi mi fa ridere Solo perché è una risposta così divertente che ti viene fuori mentre cerchi di pensarci quindi queste sono le domande che Ti lascio con e penso che tu conosca il mio La mia domanda centrale nell'approccio alla lezione di oggi era se volevo che fosse una specie di descrizione di queste proprietà astratte delle funzioni esponenziali Ed è semplicemente fantastico per me partire da quelle proprietà astratte rimani bloccato nell'idea di e per rx o più Solo sai che penso che sia più onesto scrivere exp di r per x per diversi valori di r Che ti blocca fino a quel punto Ma non ti blocca per quanto riguarda avere una nozione inequivocabile di ciò che 2 alla potenza x dovrebbe essere molto meno qualcosa come I alla potenza x Il rischio in questo ovviamente è che a volte le persone non amano l'astrazione e a volte non sembra accessibile Ma se questo è il nel caso lo sai, fammi sapere. Penso che ci sia un intero circolo di pensieri interessante che circonda tutta questa roba per includere le torri elettriche perché se vuoi parlare davvero delle torri elettriche come abbiamo fatto l'ultima volta nel contesto dei numeri complessi o anche con basi negative Devi pensare a cose come questa, quindi era una domanda che avevamo sullo schermo Sì, cosa succede se lo facciamo per I al potere I?", "n_reviews": 0, "start": 4043.86, "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes.", + "input": "titration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes", "translatedText": "Titolazione, sai, proviamolo, andiamo avanti e proviamo una torre di potenza dove stiamo elevando I a una determinata potenza e vediamo cosa ne esce fuori, quindi non era in programma di farlo Ma possiamo, possiamo sempre richiama Python ed essenzialmente fai quello che stavamo facendo l'ultima volta Quindi il modo in cui funzionerebbeèstavamo iniziando con un valore base e poi per un qualche tipo di intervallo Cosa stavamo facendo stavamo prendendo un e lo riassegneremo deve essere qualunque La base che in questo caso è che ho elevato alla potenza di a dovrebbe essere Ok, bello, quindi lo faremo e poi stamperemo il valore di a facciamolo semplicemente per Sì, è un numero molto più grande come 200. Quindi sembra che quello che succede è che c'è il potenziale per il caos con queste cose, come a volte.", "n_reviews": 0, "start": 4135.8, @@ -1869,7 +1869,7 @@ "end": 4201.64 }, { - "input": "That's it's not periodic or anything and it's actually chaotic I Suspect that doesn't happen for I but it's a thing to potentially look out for it looks like it does kind of stabilize Maybe there's some little subjection to numerical error But we stay pretty consistently around something with a real part of 0.43 and 0.36 Now what I would want to emphasize though is this expression So let's set a back to be equal to 1 this expression of taking I to the power of a remember That's a little bit ambiguous.", + "input": "that's um, it's not periodic or anything and it's actually chaotic I I suspect that doesn't happen for i but it's a thing to potentially look out for It looks like it does kind of stabilize um, maybe there's Some little subjection to numerical error, but we stay pretty consistently around something with a real part of 0.43 and 0.36 Now what I would want to emphasize though is this expression So let's set a back to b equal to 1 this expression of taking i to the power of a remember That's a little bit ambiguous.", "translatedText": "Non è periodico o altro ed è in realtà caotico Sospetto che non succeda per me, ma è una cosa a cui prestare attenzione sembra che si stabilizzi Forse c'è qualche piccola soggettività all'errore numerico Ma rimaniamo abbastanza costantemente in giro qualcosa con una parte reale pari a 0.43 e 0.36 Ora quello che vorrei sottolineare però è questa espressione. Quindi impostiamo un valore uguale a 1, questa espressione di portare io al potere di un ricordo. È un po' ambiguo.", "n_reviews": 0, "start": 4201.64, @@ -1883,7 +1883,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing?", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing?", "translatedText": "In realtà l'abbiamo fatto, quindi lasciami importare NumPy, quindi ho la funzione esponenziale, lasciami andare Per il nostro ampio intervallo come avevamo prima Piuttosto che scriverlo come sai qualcosa che è come I alla potenza di X, lo scriverò come funzione esponenziale di una costante diversa a destra una costante diversa che creerò voglio che sia 5 metà pi greco, quindi farò 5 metà pi greco per I quindi è un numero complesso e ha 5 metà pi greco come parte immaginaria Quindi questo è 5 pi greco metà per I e cosa sto facendo?", "n_reviews": 0, "start": 4234.12, diff --git a/2020/ldm-i-to-i/japanese/sentence_translations.json b/2020/ldm-i-to-i/japanese/sentence_translations.json index 02d317102..86e99d138 100644 --- a/2020/ldm-i-to-i/japanese/sentence_translations.json +++ b/2020/ldm-i-to-i/japanese/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "したがって、数字 1 から始める場合、初速は 0 に向かってまっすぐに歩くことになります。そ して、さらに低く歩いていくと、1 の半分の位置に座っていれば、依然として 0 に向かって歩いていることに なりますが、今の速度ベクトルはあなたがいる場所ではマイナス 1 倍、つまりマイナス 1 半分になります。", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "そして、興味深い質問は、これを書くのに合理的と思われる関数が 1 つだけあるということです。なぜなら、それを x に i として 書く場合、これを満たす必要があるだけでなく、いつ満たす必要があ るかがわかっているからです。得られた数値の 1 を i の 1 乗に代入しますが、この関数は i であるべきだと考えています。", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "したがって、5 pi i alve s great が得られました。これは、ここで x に代入できるもう 1 つの値です。ここで円を振り返ってみると、それをもう少し視覚的に詳し く説明できます。円周率の半分、つまり 1 に等しい時間だけ歩いた瞬間。57 代わりに、もう 1 回転して、円周率に到達するために円周率の半分 をさらに進んだ場合はどうなるでしょうか。つまり、円周率 i の値に対する e がどこにあるかを記録できるかもしれません。円周率の半分をさらに歩 き、円周率の半分を歩きます。この時点では、一周して 1 に戻り、その後、 円周率を半分に 5 つ分歩きます。これは数値的には約 7 です。85 そう、これは絶対に i の上に立つもう 1 つの数字です 。そして、最初に 5 円周率の半分に e を書き込むことによ って、i の i 乗を再表現するという厳密な手順を通過する としたら、それらの i の i 乗は乗算して負になると、e を負の 5 π の半分にすることになります。これはまった く異なる数値です。実際にこれを計算できます。頭ではわかりま せんが、Desmos を見てみましょう。。", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "これだけ長いと、はるかに小さい数になりますが、入力できる答えはこ れだけではありません。ここには、i pi の 3 の半分を負の値 で計算する他の人もいます。単位円に関してはどれを知っていますか? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "I に行きたければ、90 度歩くのではなく、「ねえ」と言うと考 えることができます。円周率はラジアンの半分になります。逆に 270 度歩くとどうなりますか。3 円周率はラジアンが半分 になります。通常、反時計回りは正です それは絶対に別の表現方法 であり、e を負の 3 円周率の半分 i にすべて乗した場合は 、別の答えが得られます。今、同じゲームを経験します。i の 2 乗は a でキャンセルされます。負の値はすでにそこにあり、正 の 3 円周率の半分があり、数値的には、これまでに得たものと はさらに異なる答えが得られます。これを調べて、「3 o 3 p i ではなく、3 pi に対する e は何ですか」と言うと、半 分 111 ポイント 3 1 前に見たものとはまったく異なる種 類の数字 111 ポイント それは何でしたか 111 ポイント 3 1 素晴らしい 111 ポイント 3 1 かそこら そし て、もう一度、直感の観点から、これが回転していると仮定します。動的 しかし、私たちは時間を逆行して、どれだけ前に私がいなけ ればならないかを見て、そこから物事を進めていけば、最初の状態に 着地するでしょう、そしてあなたは時間を戻さなければなりません 3円周率は単位を半分にしますそして、もしあなたが減衰のダイナミ クスに翻訳するとしたら、この文脈で目の前に上げるということは何 をしているのですか、あなたは私がナンバーワンから始めているのな ら言うのですが、私は時間を逆行して言いたいのですが、もしそうな らどこから始めるべきでしたか？一番になるまで朽ち果てたいの? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "3 円周率が時間単位を半分にすると、この種の指数関数的減衰の答えは明 らかに 111 あたりから始まります。そして、これがどこに向かうのか がわかります。実際には、X に代入できるさまざまな値が無限に存在しま す。e から X までを I と考えて、ここにはもっと多くの人々が エントリーしています 3 位を決めるクラシックのようにピンを地面に投 げるのはごめんなさい 9 円周率の半分は素晴らしい選択です 1729 円周率の半分は皆さん私のお気に入りのたくさんですさまざまなオプショ ンが無限にあり、さまざまな値があり、式を見ているので最初は少し当惑し ますが、計算が行われることはわかっているようです。それを電卓に接続し て、何が表示されるかを確認すると、複数の異なる値が得られますでは、こ こで何が起こっているのでしょうか？", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "16 の 4 番目の根は 2 になるはずで、答えは最終的には適切 になります。多値関数がある場合、このように複数のオプションがある 場合は、規則を採用します。多くの場合、必要な場合は、それらの値の 1 つを選択するだけです。もっと凝った専門用語で言うと、単一 の入力と単一の出力を持つ関数として扱います これは、複素数を扱う ときに常に出てきます 何かを演算として考えるという考え方は、複数 の値を持ちたいと思うことがありますブランチというフレーズを聞いた ことがありますか? 平方根関数のブランチをどこで選択しますか? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "複数の異なる答えがあるからです ご存知のように、私たちは再びこの 90 度の回転について考えます そして、それを 90 度の回転と して考えると、平方根があるはずだと感じます 45 度の角度に何かが あるのはご存知でしょう たぶん、それが平方根ですI のルートは、 ルート 2 over 2 ルート 2 over 2 I として非常 に明示的に書き出すことができます。これは三角法を使用しているだけ ですが、代わりに I を負の 270 度の回転として考えると、その 半分がその演算の半分を行うように感じられます。実際には反対側に着 くはずです おそらくここに座っている数字は I の平方根であるはず です そしてそれは実際には前に見たもののマイナスにすぎません 負 のルート 2 を 2 からマイナスルート 2 を 2 倍掛けたもの です 今、実数のコンテキストで肯定的な答えになるように平方根を選 択するだけですが、これらのうちどれが肯定的な答えだと思いますか? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "そして、あなたはよく言ったと思います、これが何であるかを 私たちは知っています、私たちはそれを2の平方根であると定 義しています、すべてが順調で、良いですしかし、私が「I」 という表現にアプローチするのと同じ方法でこれにアプローチ しましょうと言ったらどうなるでしょう「I」まず、物事を正 しいものに対して e として表現したいと思います。そして 、半分に指数を掛けて、それを半分に上げます。そして、私は 、わかりました、できますと言います。2 に等しい それ は 2 の自然対数です。これは 0 付近の定数です。69 かそこら e をそのように累乗すると 2 になるので、これを e の 2 倍 1/2 の自然対数と考えることができます。また、必 要に応じて、e の x を考えていますか? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "実数のコンテキストでは、これはちょっとやりすぎかもしれませんが、この x 関数の省 略表現として x に e を考えている場合は、値 0 を代入することができます。69 倍 1 半分なので、約 0 になると思います。34 5 似たようなものです。その非常に具体的な値を多項式に代入する と、出力が何になるかを確認すると、1 付近が出力されます。41 4 a 期待どおりの 2 の素晴らしい実数ですが、I で行っていたこ とと同じことを行うと、実際には複数の異なる答えがあることを認識して、 何かを e の累乗として書きたい場合、次のように書くこともできます。これは面白いように思えるかもしれませんが、2 に 2 を加えた pi I の自然対数に対して e と書くことができます。その全体を 1 の半分に上げます。結局のところ、この値はすぐに に等しくなります。e を に分解することもできます。2 の自然対数に e を掛けて 2 円周率 I にします。これは単に 360 度回転させる効果があるた め、1 に等しくなります。つまり、有効な置換のように感じられる 1 の 2 倍を検討していますが、これを 1 乗して累乗し、その累乗を指 数に掛けて処理するという同じゲームをします。何が起こるかを見てくだ さい。e は 2 に 1 の半分を加えた自然対数になります。では、2 円周率の I に 1 の半分を掛けたものは何ですかこれは pi に I を掛けたものになります。この最初の部分 e は 2 と 1/2 の自然対数を掛けたものになります。最終的にはよく知られた 2 の平 方根になります。これは問題ありませんが、これに e を掛けて次のよう にします。円周率 I は正しく、よく知られていますが、円周率 I に 対して e は負の 1 です。したがって、この場合、この式 2 を 1 の半分に解く場合、さまざまな答えを試してみることで、次のような ものをプラグインできることを示唆しているようです。e から X が 1 の半分に等しいということは、私たちが伝統的に 2 の負の平方根 として書くことになる別の答えです。ここで私が言いたいのは、2 から 1 の半分を調べるのに複数の値があるのは少しおかしいということです。それは等しくないと言いますが、私たちが行う選択に基づいて、それは複数 の異なるものと等しくなる可能性がありますが、2つのことは非常に合理的 であるように思われます2が1の半分になる何かがある場合、それはどちら かであるべきであるように思われます肯定的私たちがよく知っている平方根 、または実際にはそれほど問題とは思えないその否定的な変形です そして 実際、私たちはこのゲームをさらに進めることができます そこで、この式 に対するさらに創造的な答えを求めさせてくださいなぜなら、I をべき 乗 I したがって、今回の質問は、x に等しい方程式 e の 1 つ の解が 2 の実数自然対数であることを尋ねるか、または指定しています 。これはわかっています。", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "質問に対する e に対する x の答えは 2 に等しく、 ここでも創造性は歓迎されますので、そのために少し時間を割 きます II よろしければ、ここでいくつかの答えを固定し ておきます。どのくらい時間がかかるかわかりません閲覧して いるデバイスに応じて、必ず数学入力を行う必要がありますが 、必要な質問に答えて、答えてほしい答えを得る前であれば、 あまりストレスを感じないでください。あなたのうち 131 人が、Ln を 2 として、2ii を追加するバリア ントを入力しました。そして、私がこの質問を書いていると思 います。実際には、かなりの数の異なる正しい答えがあるのに 、誤って答えの 1 つを正しいとマークしてしまったような ので、それは私の責任です皆さんの中に「ああ、赤です」と見 えるかどうかはわかりませんが、2 プラス 42 の Ln を入力したときに間違えました。", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi これはもちろん素 晴らしい選択ですが、4 pi I に 2 または 6 pi I の 自然対数を加えたものを使用することもできます。または、e に影響 を与えないことを追加すると、実際には 2 pi I の整数倍にな ります。X は 2 の pi に e を乗算する効果があるだけだ から I これは 1 を乗算する効果であり、これもまた、別の例と して実行すると、ある種の合理的な結果が出力されるように見える、あ る種の面白い結果をもたらします。2 番目によく入力される式のよ うです。2 を置き換えるというものでした。では、2 の 1 の 4 乗について考えてみましょう。2 プラス 4 の自然対数に対して 、2 を e に置き換えるという提案がありました。pi I O K Plus 4 pi I と私たちはすべてを 1 4 乗に上げ ます。同じゲームをプレイするとしたら、 e を 2 倍 1 4 乗 の自然対数にして、e を掛けて次のようにします。円周率 I さて 、その最初の部分は、通常の正の 2 の 4 乗根になります。これ は、2 の 4 乗根のような式を電卓に差し込むときに意味する、小 さな正の数ですが、この 2 番目の部分は次のようになります。マイナ ス 1 なので、こう言っているようです。2 をこの異なる方法で 1 の 4 乗に解釈するとわかります。これは、私たちが得られる通 常の答えではありませんが、合理的な答えです。", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "私たちは pi の 2 分の 1 と I を調べ、負の 1 を掛ける 代わりに I を掛けていたでしょう。これもまた有効な答えです。これは 2 を 1 の 4 分の 1 にするようなものに対する妥当な出力のように思えます。I の I 乗 という事実を見ると、それには複数の異なる値があるようです そうです、e を 5 つの 円周率の半分の I 負の 3 つの円周率の半分の I に接続できるという面白い現象が 発生しました。すると、大きく異なる答えのように見えるものが得られます何か超小さいもの 何か超大きいもの すべては、ここまでで見つけた 1 5 番目の約 1 5 番目の答え とは大きく異なります。これは、2 から 1 4 番目は何ですか、というような質問をし て、実際には複数の異なる解決策があることを認識しているときとまったく同じ現象です。式 X から 4 番目までは 2 4 の異なる解に等しく、あなたが見ているのは、複数の 異なる解があるという事実です 式 e から X までは、ある種の塩基と等しくなります その塩基が I であるかどうか、その塩基が2 それが何であれ、私たちが考えられる 1 つの方法は、実数を扱っているとき、物事はただ美しいものであるだけであり、そこには 1 対 1 の関係があるということです。", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "それは素晴らしい です 指数関数について考えたい場合は、このことのいくつかを説明しましょう 2 から X のように、任意の指数を X の底として表現することを選択できる素晴らしいやり取りが あります または、次のように表現することもできますR に X を掛けた X と同じ指数 関数で、これが私たちが参照する多項式であることはご存知のとおり、X に e のようなも のを書き込むたびに暗黙的に参照されます。そして、B の自然対数を取ることができるため、 美しい前後関係が生まれます。そして、B が正の数であると仮定すると、答えが 1 つ得 られます。そして、それは、R の X が B に等しいと言っているのと同じことです。し たがって、このシリーズの前半でこれについて説明した 1 つの方法は、すべての可能な指数 関数のファミリーです。R 掛ける X の X として書いて、R を変更することもできま す。そして、これは、あなたがより慣れ親しんでいるものであれば、e を R 掛ける X に書くのと全く同じことです。つまり、e を R にしますR の XX 倍 X 倍 こ れらは同じことです。それを変更することも考えられます。しかし一方で、考えられるすべての 指数関数を基底として考えるとしたら、基底を X 乗してみましょう。その塩基が何であるか を変更する 最初は、それは操作する別の種類の式のように感じますが、それは同じ家族を表 現する別の方法であり、これについて考えるかもしれない方法です どの塩基が対応するかをど のように考えるかについてExp of R x X としてもう少し抽象的に考えている場合 に、これを行うのには理由があります。これを複素数に適用しようとしていて、奇妙に見えるか らです。そのベースを見る代わりに私にできることは、その値が何であるかを言うことです。", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "R に X を掛けた exp を得ることができます。ここで、R はゼロ ポ イント 6 9 のようなものです。しかし、それを 2 π I だけ下にシフトすることもできま す。そして、それが対応する基数は変更されず、依然として 2 に対応します。あるいは、それも可能 ですそれを 2 つの pi I だけ上にシフトします。これは、対応する基数を変更しません。なぜ なら、これらすべての場合で、X が 1 に等しいとプラグインすると、同じ結果が得られます。ただ し、X の異なる値に対するこれらはすべて別個の関数です。これは次のとおりです。I の I 乗 に複数の異なる値が表示された理由 I to the X はその文脈ではあいまいな関数であるた め、R のどの値を決定すれば明確になるでしょう。つまり、私たちが表現しているのは、R に X を乗じた exp であり、どの値であるかがわかります。", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "これは明確な関数ですが、その時点では、おそらく私たちが望んでいるのは、X 乗した基数の観点から物事を考えるのをやめるということです。多分、複素 数のコンテキストに入ったらすぐに次のように書く必要があるでしょう。それら はすべて、X の定数倍の exp として計算されます。他に理由がなけれ ば、それが非常に明確になります。計算を実行したい場合、またはその上で 単に数学を実行したい場合に、実際に数値をどのように代入するか。この素晴ら しい無限多項式が得られます。それらを接続してください。これがおそらく指 数関数について考える正しい方法であるという別の主張をします。複素数など の他の領域に拡張したらすぐに、そのために Go をバックアップしましょう 。ドアベルに戻る いくつかのものが到着しました 元に戻ります べき乗の 考え方を拡張して、X の 2 が何であるかのように考えるだけです。自然数 についてこれを考える方法はわかっています。", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "あなたは 2 対 3 の ようなものを知っています 掛け算の繰り返し 小数の金額や負 の金額などについて、2 対 X のようなものを考えることを 最初に教えられるのはなぜですか。", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "通常、2 から 1 の半分は、それ自体を掛ければわかるものである べきだと教えられます。これは、指数関数が数値を 数えるときに行う通常の規則に従い、その指数に何 かを加算できる場合、2 が得られるはずです。1 に 1 を掛けると 2 になるような数字に なるはずです。その時点で選択肢があるのはわかり ますが、おそらくそれはプラスです。", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "もしかしたら、それはマ イナスかもしれない でも、常にプラスの選択をすると決めていれば、負の数について尋ねれば、 この同じ取引から素晴らしい連続関数を得ることができるでしょう マイナスの 1 に対する 2 はどうあるべきですか、それは何かであるはずです1に2を掛けるとどこになるでしょうか？", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "これで 2 から 0 が得られ、これが負の指数が 1 の半分のように見えると いう慣例の正当化のようなものです しかし、ここで実際に起こっているのは、これ が何であれ、このプロパティ f を満たすある種の関数であるべきだと言ってい るということですa プラス b は、a の f に b の f を掛けたもの に等しい さらに、基数が 2 であるという事実は、基本的に、これが単なる関数 ではないことを示しています 1 を差し込むと 2 が得られる関数ですここで の含意を理解しているかどうかを確認するための健全性チェック形式の質問です。こ れは何ですかと聞きたいのですが、ソフトボールのようなものとは言いませんが、こ れは、これはそのようなものであることを意図したものではありません、信じられ ないほど深い質問です必然的に。", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "これは、関数のプロパティから抽象的に開始し、そ れらのプロパティに基づいて関数を書き留める方法を推測するというアイデアに従って いるかどうかのチェックです。x の f がこの指数特性 f を満たす場合a に b を加えたものは、すべての入力に対して a の f と b の f を掛 けたものに等しい そして、f of 1 が 2 に等しいという条件も満たしま す。次のうちどれが真ですか。つまり、次のどれが必ず真になります。どのような関数 を開始しているかに関係なく、どの講義だったか覚えている人たちと オイラーの公 式が実際に何を言っているのかをどう解釈するかについて話し合っていたのはどっち でもいい 私は単一の条件を無視したこのスタイルの質問をしました ご存知の通り、 私は書き留めていませんでしたx の f がどこでも 0 でないことを確認した いという事実があり、それによってある程度の混乱が引き起こされましたが、これはク ールなことであり、私たち全員に起こる画面上の混乱を引き起こしますが、その目的 は基本的に、この抽象的な性質が足し算を掛け算に変えるものは、基本的に関数をある 種のべき乗に等しいものとして書きたくなるだけで十分です これが質問の精神です さて、実際に送電塔についていくつか質問がありますここで出てきたようですが、こ れは前回と大きく関係しています 送電塔の質問は少しだけ控えて、最初に次のよう なことをより深く感じてみましょう ここでのべき乗は何を意味するのでしょうか? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "なぜなら、私たちが主張したいのは、複数の異なる方法でそれに答えることができるという ことだから、1つだけ教えていただければ、送電塔について話しましょうそして、数直線が 対数スケールで表現できるのと同じように、複素平面でも同じことができるでしょうか? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "ええ 、実際、ここですぐに説明する視覚化があります。そこでは、これと非常に似たよ うなことを行います。なぜなら、私たちがやることは、R 倍 X のさまざまな 指数関数を試すことだからです。しかし、私たちは小さな黄色の点で表される R の値を変更します。それでは、これについて少し説明します。平面全体をマッ ピングするのではなく、実軸と虚軸からのいくつかのサンプル ポイントだけをマ ッピングします。しかし、アイデアは、その定数が何であるかを移動するにつれて 、それが平面に対して行うさまざまな動作を視覚化できるようになり、事実上、X 軸を対数スケールに変換してラップするようなものです。円に沿った虚数軸 そして R の値が虚数になるとすぐに、その役割が入れ替わります 実数は円上 に配置され、虚数は対数スケールの正の軸に配置されます 3 つすべてに大きな 疑問があります私が行きたい場所に向かって先を急ぐようなものだが、この作品 で人々がそう考えているのを見るのは嬉しいことだ。", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "明示的に f of 5 のようなものは、f of 1 プラス 1 プラス 1 プラス 1 プラス 1 と同じです この性質により、f of 1 を 5 回乗算したものと同じになります 1 の f が 2 の場合は同じで す2 の 5 乗として、次にマイナス 5 の f のようなものになります。これに 5 の f を掛けると、0 の f が何であれ得られますが、0 の f が何であるかはすぐにはわかりませんが、次のように言えます。f of 1 plus 0 は、f of 1 に f of 0 を掛けたもの と等しいですが、f of 1 は 2 に等しいので、これも 2 に等しい ので、2 は何かの 2 倍に等しいと言えます。は 1 でなければならない ので、このコンテキストでは、負の 5 の f は負の 5 に対して 2 で あることが保証されます。これは 2 から 5 までの 1 です。これを明 示的に負の 5 に対して 2 と書くこともできます。つまり、これら 2 つのプロパティを組み合わせると、私たちは、X に 2 という関数を書きたい のです。なぜなら、この関数に入力する任意の数え数値が満たされるからです。それは、その回数だけを掛け算するように見えるからです。任意の小数を入力し ます。これらのプロパティを満たします。私たちが望んでいたものです そしてあ なたは不思議に思うかもしれませんが、実数値関数のコンテキストでは実際にそ うなります しかし、複素数値関数のコンテキストでは、このような関数は複数 あるでしょう f この関数に対して書くことができるのは、そのうちの 1 つ です前に見たように、2 + 2 pi の自然対数の exp として関数を 定義できます。私はいつも X さて、ここでずさんなことを許してください、 私はこれについて書くことに興奮しているだけです、そしてこれは実際には次のよ うな別の関数ですX が 1/2 に等しい場合に何が起こるかによって証明さ れています。1/2 を差し込んだ場合に得られる値は 2 の負の平方根で あり、次に 1/4 を差し込んだ場合に得られる結果は の 4 乗根ではあり ません。2 ですが、2 の 4 番目の根を掛けているので、これは別の関 数です しかし、それでもこれらの特性を満たしており、ある意味、2 to the X と書きたくなるのです そして、おそらく 2 to the X があいまいであることを示唆していますちょっとした表記 そして、すべてを R 倍の exp という観点で書けばいいのですが、よく不思議に思うかもし れません もしかしたら、この特性を満たすすべての関数について十分な創造性が 足りていないだけなのかもしれません exp を書くときに曖昧さがあるかも しれませんR に何かを掛けると、R にはさまざまな値が関係してくる可能性 がありますが、私はちょっとした主張を述べてから、必要に応じて証明がどのよう になるかのスケッチのようなものを与えるつもりです。それはどれですかある複 雑な関数 F があり、それが最初に次の特性を満たしているとします。その導 関数を取得できます。", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "それは微分可能であり、完全に厄介な不連続なものにならないようにするだけです。それは、ベクトル空間の範囲を知っているかどうかに応じて、いくつかのランダムな値を取るようなものです。あなた がクレイジーな方法で考えたいかもしれない分数の量はわかりません。", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "素敵な機能ですね。", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "それは微分可能で す どこでも 0 に等しくないので、この条件が私の頭から抜け落ち、どの講義の 講義かそのようなことは忘れましたが、加算を乗算に変えるという中心的な性質があり ます そのような関数がある場合、私は主張します固有のものがあります。固有の複素 数 R が存在することを指定する必要があります。そうすれば、X の F を基 本的に値 X の R 倍の指数関数として書くことができます。基本的に、関数とし て X がある場合、このことを示しているのはどれですか。優れた微分特性を備え た無限多項式とそのすべてがあれば、指数関数という言葉の抽象的な一般的な意味に非 常によく似た、必要なすべての指数関数が得られます。これは、必要な特性に基づい ており、証明のスケッチは次のようになります。まず最初に、どこにでも存在すると仮 定しているこの値の導関数が何であるかを確認したい場合は、次のようになります。", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "X の F を式から完全に因数分解することができ 、極限全体は H に関してのみ表現されます。導関数の文脈でそれが何 を意味するのか、そして 0 の F が必ず 1 に等しいという事 実を考えると、この極限式全体は次のようになります。単なる定数ですが 、より具体的には、関数の 0 での導関数が何であれ、この面白いこと ができます。0 での導関数がわかれば、どこにでもその導関数が何であ るかが決まります。指数関数のコンテキストでは、これはよく知られて いると思います。私たちが本当に言いたいのは、指数関数の導関数はそれ 自体に比例し、その比例定数は 0 での導関数が何であるかに等しいと いうことです。これはすべて非常に抽象的な表現になっていますが、その 目的は、次のことを強調することです。必ずしも X 乗としてすでに 考えられている関数だけではありません しかし、それは、加算を乗算に 変えるというこの抽象的な性質を満たすだけの、潜在的により広範な関数 のクラスです しかし、それがあれば、実際には、二次導関数 そしてさ らに言えば、三次導関数などです。なぜなら、導関数はそれ自体に比例 するからです。したがって、n次導関数を取得するには、その比例定数を 見て、それをn乗して、ここから次のことを行うことができます。テイラ ー級数の展開については、テイラー級数の考え方に慣れている人、特に複 素数の意味で微分可能な微分関数の考え方を混ぜ合わせたい場合には、 高度な宿題として残しておくかもしれません。間違いなく大学のトピック のようなものです 好きなように推論を混ぜることができることはご存知 でしょう しかし、テイラー級数についてしか知らない人がこのアイデア を取り入れて F と固有の複素数が存在するため、関数 F は必然 的に次のように記述できるという考えが正当化されます。そして、正規指 数関数への接続は、そのような値 R があるときはいつでも行われます 。私たちは本質的に、実数の複雑なコンテキストで行うことを実行します 。これは、その値 R の関数の exp を見て、それをベースとし て書く場合です。", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "これは、円周率の半分の I 倍 X の exp を意味するだけでなく、円周率 5 の 5 分の 1 I 倍 X の exp を意味すると解釈することもできます。これらは別個の関数で す。そして、別個の関数の無限のファミリーがあり、次のように感じます。I to the X と書きます。つまり、I to the I という式は、 それが必然的に何を意味するのかについての標準を採用していない限り、無限に 多くの出力があると言うとき、別の考え方として、関数 I to the X というものがあります。私たちが持っている表記法は少しあいまいです さて、 以上のことをすべて理解した上で、これを視覚化してみましょう。それは楽しい と思うからです。そして、あなたは知っていますか、これが役立つビジュアルなの か、それともより混乱を招くビジュアルなのかを教えてください。私たちがやろ うとしているのは、この関数 exp の R 倍 X を見てみるということ です。これは基本的に、これは e の X 乗を書く別の方法です。実際、ある 時点で、それを指定した別のアニメーションをレンダリングしたと思います。な ぜなら、私はそうするつもりだったから、そうさせてください、そうそう、あな たは私のファイルシステムに戻って、あなたがいるはずの場所に戻ってください、 そこに行きなさい、それは複数の異なるものがあるので文句を言っているのです かそれは、あるようなものになるでしょうああ、置き換えると別の画面に表示さ れます待ってください、なぜそうなるのですか、わかりました置き換えますか？", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "目に見えるものは何でもそこに置きます そして今、私たちはああ 、そこに戻ります、私はそれをすべてうまく書き出すことができ たので、R倍Xのexpとして考えることに不快感がある場合は 、この無限多項式をちょうどあなたの頭の後ろを R 倍 X にして、R を中心に変化させます。それで、虚数軸の点をたど り、実数軸の点をたどります。そして、これが何をするか見てみ ましょうそれはとても速いので、もう少しゆっくり考えさせてく ださい すべての負の数 何でも それは負の実数です 0と1 の間の範囲に押しつぶされます どれが負に意味を持ちますか？", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "負の実数に対する a は 0 と 1 の間の値であり、特に負の 1 の f を追跡し ています。これは、e 上の 1 が 30 0 付近にあるもの付近に表示されます。予 想どおり、1 の 37 f が e に着地します。そ れが 1 の f の exp です。I の f は単 位円の周りに 1 ラジアン着地します。ここで仮想軸全体 に沿って追跡するのはちょっと楽しいです。仮想軸が円の 周りにどのように巻き付けられるかを見るのは楽しいです 。この R の値を微調整すると何が起こるでしょうか? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "ここで R の値が必要にな る可能性があります。これを 2 にすると、異なる方法で引き伸ばされます。実際の軸が さらに引き伸ばされるので、1 の f が e の 2 乗の少し上のあたりになります 。負の 7 f より上になります。1 は 0 にかなり近いです I の f は 2 ラジアンです 負の I の円 f の周りの回転は負の 2 ラジアンの回転です そしてもちろん、それがスケーリング定数として持っていた pi である場合のお気に入 りの式に到達することができます。実軸はかなり引き伸ばされます ご存知のとおり、1 の f は 20 プラス pi に非常に近い pi の e に位置しています。これ はいつも楽しいことですし、マイナス 1 の f は 0 に非常に近いので、実際の軸 は実際に引き伸ばされています。軸 また、単位円方向にも引き伸ばされているため、I の f または負の I の f に到達すると、円の半周を歩くことができるので、これ ですべてがうまくいきました。次のような関数を考えるにはどうすればよいでしょうか。", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "また、X を 2 倍した自然対数の X of X と 書き、R To の値を表す黄色の点を 0 付近に移動 します。69 にはまだ虚数部はなく、実数の 0 だけです。69 かそ こら これは 2 の自然対数です。まあ、1 の f が 2 に着地することがわかり ます。これが、この関数を 2 を 1 の半分の X f に呼び出したい理由です。実際には申し訳ありませんが、マイナス 1 の f は 1 の半分の f に着地しま す。I 単位円の周りを少し歩くと、0 になります。単位円の周りの 69 ラジアン。ここで、もう少し楽しんで、これを 0 の代わりに に変更するとどうなるかを考えてみます。69 は 2 の自然対数ではなく、2 の自然対数の I 倍になるので、指数関数的な基底を持つ 可能性のある何かを実際に考えることができます。", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "この場合、I の I 乗は何になりますか。これは約 0 に押し込まれます。2 は 5 分の 1 のあたりですが、数値 I に 1 の f を代入 する性質を持つさまざまな指数関数がたくさんあります。した がって、さらにスケールアップするとしたら、ここではアニメー ション化していないと思います。その黄色の点を円周率 I の 5 の半分になるまで上げます。見えるのは単位円です。", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "これは、1 の負の f の f がさらに 2 パイ ラジアンの周りを回転 して、その場所に着地するように、それ自体を中心に回転しますが、実際の軸は さらに引き伸ばされてしまいます。これは、I から I への別の出力が行わ れる意味でした。ずっと小さい数字 0 くらいでした。0003 くらい しかし、私がとても楽しいと思うこともわかります。2 の X 乗として解釈したい別の表現を検討するとどうなるでしょうか? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "R に X を掛けた X があり、R はこの値に等しくなります。これは 2 に pi を掛けた I の自然対数です。これが意味す るのは、1 を代入すると 1 の f はマイナス 2 になるため 、この関数を書きたいということです。負の 2 の X 乗として、 正しく、これは実際に何かです。ご存知のとおり、負の数を 1 乗す るときは、少し一見単純です。負の 2 の X 乗は、最初はこの ように見えませんが、必ずしもそれが私たちにもたらすものです。何ら かの方法で複素数に変換しますが、もちろん 1/2 のような値を代 入すると、負の 2 の平方根を求めるような場合、これを平方根の I 倍のようなものとして書きたいことがわかります。of 2 し かし、この関数のマイナス 2 の X 乗を、それが扱っている完全 な複素数ドメインで見た場合、あなたが見ているのは、1 の値をマイ ナス 2 にする関数です。それは実数直線の残りの部分にも影響を及 ぼしますが、それは外側に向かって螺旋を描いているのでしょうか？", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "したがって、負の 1 の f は負の 1 の半分にあることがわかりま す。1 の半分の f をたどるとどこが予想されるでしょうか。それ は正確に想像線上にあり、1 の半分の f は 2 の平方根になりま す。マウスが希望の場所にありません。", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "これは、I の 2 倍の平方根付 近になります。さらに続けると、これは X に対するマイナス 2 の 実数値累乗をすべて示しています。必然的に螺旋状になります。しかし、 R の値をさらに高くして、それを取得することもできます。タウ倍くら いまで I は 6 の 2 の 8 倍くらい I そしてその文脈で は、これは X に 2 のようなものとして書きたいもう 1 つの関 数です。なぜなら、X にプラグインする任意の整数から整数の場合、次 のようになります。掛け算を繰り返しているように見えます そして、正の 平方根ではなく負の平方根を吐き出す 1/2 などの妥当な値さえあり ますが、実際に行っているのは平面への変換であり、そこにすべてが置か れているのが本物です数直線は最終的には非常にきつく巻かれた螺旋とな り、一周して、1 の f が数値 2 にちょうど着地するように螺旋 を描きます。したがって、その意味で、X に対する 2 は次のよう に解釈できます。私たちが伝統的に慣れ親しんでいる指数関数とは別の指 数関数です それで、以上のことを踏まえて、今日のことはこれくらいにし ておきます。そして、考えておきたいいくつかの疑問を残しておきます。I から I は多値表現であると考えてください。慣例を採用すると 言うこともできます。空想的には、自然対数関数の分岐を選択すると言う でしょう。そしておそらく、それはあなたを e から負の pi に固 定してしまうかもしれません。半分 しかし、この種のものが、私たちが 見たさまざまな値のように、無限に多くの異なる値になりたいと言うなら 、同じ意味で、2 から 1/3 はいくつの値になりたいでしょうか? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "10 分の 1 は別の言い方をしたいのですが、満たす X の指数関数 F のすべてについて言わせてください。ああ、満 たす X の f のどこかに書き留めておきました。これらの プロパティはすべて私が書いたものなので、すべてを満たす場合 はこれらのうち、1 の f が 2 に等しい場合、どの関数 のさまざまなオプションに対して X が 10 分の 3 に 等しいとすると、いくつの異なる出力が得られるでしょうか? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "2 から pi については、2 から X をある種の指数関数と して考える場合、2 から X が表すことができるさまざまな関 数を表します。この種の抽象的なプロパティの意味で指数関数を考え ます。さまざまな関数のクラスがあり、pi を接続したいと思っ ています。それは私を笑わせます。それがとても面白いので、考えよ うとしているときに突然出てくる面白い答えを知っています。それ で、これらが質問ですこのままにしておきますが、これはご存知だと 思います 今日の講義に臨む上での私の中心的な質問は、指数関数 の抽象的な性質のようなものを記述したいかどうかということでし た そして、それらの抽象的な性質から始めるのは、私にとってクー ルですあなたは、rx 以上の e という考えに囚われてしまい ます。知っておいてください、r のさまざまな値について、r に x を掛けた exp をもっと正直に書くと、それはあなたを そこまで閉じ込めますが、それはあなたを閉じ込めるわけではありま せん2 の乗 x は、I の乗 x のようなものであるべきで はないという明確な概念 もちろん、それに伴うリスクは、人々が 抽象化を好まない場合があり、親しみにくいと思われることもありま すが、もしそれが知っているなら、私に知らせてください このす べてを囲む興味深い思考の輪があり、送電塔も含まれると思います。なぜなら、実際に送電塔について話したければ、前回のように複素 数の文脈で話しましょうまたはマイナスのベースでも このようなこ とはよく考えなければなりません、それでそれは私たちが画面上で 考えた質問でした ええ、これを I 乗したらどうなりますか? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "滴定、これを試してみましょう 先に進んで電力塔を試してみましょう I を所 定の電力まで上げて、そこから何が飛び出すか見てみましょう だから、これを行 う予定はありませんでしたが、いつでもできますPython を起動して、基本 的に前回行っていたことを実行します。つまり、これが機能する方法は、いくつ かの基本値から始めて、その後、ある種の範囲に対して行うということです。私た ちは何をしていたかを取得し、再代入するつもりですそれは何でもいいです この 場合、私が a 乗した基数は、OK、クールです。それで、それを実行してか ら、 a の値を出力します。それでは、これを実行しましょうええ、それは 2 00 のようなはるかに大きな数字です。つまり、何が起こるかというと、これら のことで混乱が生じる可能性があるようです。", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "私たちは実際に持っ ているので、NumPy をインポートさせてく ださい。指数関数があるので、行かせてください。以前のように、大きな範囲については、知ってい るように書くのではなく、I の X 乗のような ものを書きます。別の定数の指数関数として右 a 別の定数を作成します 円周率を 5 等分し たいので、円周率 5 を半分に掛けます。これ は複素数であり、円周率の 5 等分が得られます 。虚数部 これは 5 円周率の半分に私を掛け たものですが、私は何をしているのでしょうか? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/korean/sentence_translations.json b/2020/ldm-i-to-i/korean/sentence_translations.json index ec29dd58f..482c3bb5c 100644 --- a/2020/ldm-i-to-i/korean/sentence_translations.json +++ b/2020/ldm-i-to-i/korean/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "따라서 숫자 1에서 시작하는 경우 초기 속도는 0을 향해 직선으로 걷는 것이고, 더 낮게 걸을수록 1의 절반에 앉아 있으면 여전히 0을 향해 걷는 것이지만 이제 속도 벡터입니다. 당신이 있는 곳에서는 마이너스 1번이 될 것이고, 이는 마이너스 1/2이 됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "그리고 흥미로운 질문은 이것을 작성하는 것이 합리적이라고 생각되는 함수가 단 하나뿐이라는 것입니다. 왜냐하면 우리가 x에 i로 쓸 것인지를 알기 때문입니다. 그것이 이것을 만족해야 할 뿐만 아니라 언제 만족해야 하는지도 알 수 있습니다. 우리는 아마도 i를 1번 전원에 연결하지만 이 함수는 i여야 한다고 생각합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "그래서 우리는 5개의 파이를 얻었습니다. 이것은 우리가 여기에 x에 연결할 수 있는 또 다른 값입니다. 그리고 우리가 여기 있는 원을 다시 보면 시각적으로 좀 더 자세히 설명할 수 있습니다. 순간은 파이 반감(1)과 동일한 시간 동안 걸었습니다. 57 대신에 우리가 또 다른 완전한 회전을 하고 또 다른 파이 반쪽으로 가서 파이에 도달하면 어떻게 될까요? 우리가 기록할 수 있는 것은 파이 값에 대한 e가 우리가 또 다른 파이 반쪽으로 걷고 또 다른 파이 반쪽으로 걷는다는 것입니다. 이 시점에서 우리는 완전한 원을 그리며 다시 1로 돌아간 다음 수치적으로 약 7에 해당하는 5파이 반을 걸을 것입니다. 85 예, 그것은 확실히 우리를 i 위에 올려놓는 또 다른 숫자입니다. 그리고 만약 우리가 먼저 e를 5파이 반으로 나누어 i를 i제곱으로 써서 i의 거듭제곱을 다시 표현하는 모든 속임수를 겪게 된다면, 그 i's입니다. 곱하면 음수가 되고 우리는 e를 -5 파이 반으로 나누는 것을 보게 될 것입니다. 이는 매우 다른 숫자입니다. 실제로 계산할 수 있습니다. 머리로는 잘 모르겠지만 Desmos를 살펴보겠습니다. . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "훨씬 더 작은 수를 얻을 수 있는 길이입니다. 그러나 이것이 우리가 입력할 수 있는 유일한 답은 아닙니다. 다른 사람들이 여기에 -3 반 곱하기 i pi를 가지고 들어오고 있습니다. 단위원에 대해 알고 있는 것은 무엇입니까? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "90도 파이 반 라디안을 걷는 것보다 나에게 가고 싶으면 안녕이라고 말하는 것으로 생각할 수 있습니다. 파이 반 라디안을 90도 걷는 것 반대 방향으로 270도 걷는다면 어떨까요 3 파이 반 라디안 컨벤션은 다음과 같기 때문에 부정적으로 생각할 것입니다. 일반적으로 시계 반대 방향은 양수입니다. 그것은 완전히 표현하는 또 다른 방법이며, 만약 우리가 e의 마이너스 3파이 반쪽 i를 가지고 있다면 다른 대답을 얻게 될 것입니다. i를 거듭제곱하여 우리는 같은 게임을 진행합니다. 이제 i 제곱은 a로 취소됩니다. 음수는 이미 거기에 있고 양수 3파이 반쪽이 있습니다. 수치적으로 이것은 우리가 이전에 가졌던 것과 훨씬 다른 모양의 답을 얻습니다. 만약 우리가 가서 이봐, 3o 3파이가 아니라 e의 3파이는 무엇입니까? 반쪽 111 포인트 3 1 이전에 본 것과 매우 다른 종류의 숫자 111 포인트 그게 무엇이었나요 111 포인트 3 1 크다 111 포인트 3 1 정도 그리고 다시 직관의 관점에서 당신이 질문할 수 있는 것은 우리가 다음과 같이 회전한다고 가정해 보겠습니다. 역동적인 하지만 우리는 시간을 뒤로 이동하여 시간이 얼마나 오래 전이어야 하는지 알 수 있습니다. 그런 식으로 거기에서 앞으로 작업을 수행하면 첫 번째 초기 조건에 도달하게 되며 시간을 거슬러 3파이 반 단위로 돌아가야 합니다. 그런 다음 이 맥락에서 눈을 들어 올리는 것이 하는 붕괴 역학으로 번역한다면 제가 1위에서 시작한다고 말할 것입니다. 그러나 저는 시간을 거꾸로 거슬러 올라가서 다음과 같이 말하고 싶습니다. 그렇다면 어디서 시작해야 합니까? 나는 1위가 될 정도로 하락하고 싶나요? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "3파이가 시간 단위를 반으로 줄인 후에는 그러한 종류의 지수적 붕괴에 대한 답은 분명히 약 111에서 시작됩니다. E에서 X까지의 나로 생각하고 사람들은 여기에 훨씬 더 많이 들어갔습니다. 실례합니다. 클래식 3위를 위한 핀을 땅에 던졌습니다. 9 파이 반은 훌륭한 선택입니다. 1729 파이 반은 제가 가장 좋아하는 많고 많은 것입니다. 다양한 옵션 무한히 다양한 값이 있어 약간 당황스럽습니다. 표현식을 보기 때문에 처음에는 약간 당황스럽습니다. 계산이 있을 것이라는 것을 아시는 것 같군요. 계산기에 연결하고 무엇이 나타나는지 확인하면 여러 가지 다른 결과가 나타납니다. 그것에 대한 가치 그래서 여기서 무슨 일이 벌어지고 있는 거죠? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "16의 네 번째 근은 2가 되어야 하며 답은 잘 나옵니다. 다중 값 함수가 있을 때 이와 같은 여러 옵션이 있을 때 규칙을 채택합니다. 우리는 종종 해당 값 중 하나를 선택하여 의미하는 대로 사용합니다. 고급 용어로 단일 입력과 단일 출력을 갖는 함수로 취급합니다. 이것은 우리가 복소수를 다룰 때 항상 나타나는 개념입니다. 여러 값을 갖기를 원하는 일종의 연산이라는 아이디어입니다. 분기라는 문구를 들어보세요. 제곱근 함수의 분기를 선택하는 곳은 어디입니까? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "여러 가지 다른 답이 있기 때문에 우리는 다시 이 90도 회전을 생각합니다. 그리고 우리가 그것을 90도 회전으로 생각한다면 제곱근은 45도 각도로 앉아 있는 것처럼 느껴집니다. 아마 그게 정사각형일 것입니다 루트 2(루트 2/2 루트 2/2 I)로 매우 명시적으로 작성할 수 있는 루트 2는 삼각법을 사용하는 것뿐이지만, 내가 대신 음의 270도 회전이라고 생각한다면 작업의 절반을 수행하는 것의 절반처럼 느껴집니다. 아마도 여기 있는 숫자는 I의 제곱근이 되어야 할 것입니다. 그것은 실제로 우리가 이전에 본 것의 음수일 뿐입니다. 음의 루트 2 분의 2 빼기 루트 2 분의 2 I 이제 실제의 맥락에서 우리가 말할 수 있는 가치 함수는 긍정적인 대답이 무엇이든 제곱근을 선택하는 것입니다. 하지만 다음 중 어느 것이 긍정적인 대답이라고 생각하시나요? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "그리고 당신이 잘 말한 것 같아요. 우리는 이것이 무엇인지 알고 있습니다. 우리는 그것을 2의 제곱근으로 정의합니다. 모든 것은 괜찮고 좋습니다. 그러나 우리가 I에서 I 표현 I에 접근하는 것과 같은 방식으로 접근하자고 말하면 어떻게 될까요? 먼저 사물을 올바른 것에 대한 e로 표현하고 그런 다음 1/2을 지수로 곱하여 1/2로 올리겠습니다. 그리고 좋아요라고 대답합니다. e를 무엇인가로 할 수 있을 것 같아요 2와 같습니다. 이것은 2의 자연 로그입니다. 이것은 약 0인 상수입니다. 69 정도 e를 그 거듭제곱으로 올리면 2를 얻게 되므로 e를 2 곱하기 1/2의 자연 로그로 생각할 수 있습니다. 그리고 원한다면 e를 x로 생각하고 있었나요? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "이것은 실수의 맥락에서 다소 과잉일 수 있다는 것을 알고 있습니다. 그러나 e의 x를 이 x 함수의 약어로 생각하고 있다면 값 0을 연결할 수 있습니다. 69 곱하기 1/2이면 약 0이 될 것 같아요. 345 그런 것 같아요. 매우 구체적인 값을 다항식에 연결하면 출력되는 내용을 확인하면 약 1이 출력됩니다. 414 a 당신이 기대하는 멋진 실수 제곱근 2 하지만 우리가 I로 했던 것과 똑같은 일을 한다면 e를 거듭제곱으로 쓰고 싶을 때 실제로 여러 가지 다른 대답이 있다는 것을 인정하면 다음과 같이 쓸 수도 있습니다. 이것은 재미있어 보일지 모르지만 우리는 2 더하기 2pi I의 자연 로그에 e라고 쓸 수 있습니다. 그 모든 것을 1/2로 올린 직후에 이 값은 다음과 같게 될 것입니다. e를 나누면 다음과 같습니다. 자연 로그 2에 e를 2pi로 곱한 것입니다. 이것은 360도 회전하는 효과가 있으므로 1과 같을 것입니다. 그래서 우리는 유효한 대체처럼 느껴지는 2 곱하기 1 대단한 것을 보고 있습니다. 우리는 동일한 게임을 합니다. 이것을 취하여 거듭제곱하고 지수에 거듭제곱을 곱하여 처리합니다. 무슨 일이 일어나는지 보세요. e의 자연로그 2 곱하기 1/2 더하기 음, 2파이 I 곱하기 1/2는 얼마입니까? 음, 그것은 파이 곱하기 I 이제 이 첫 번째 부분 e는 2 곱하기 1/2의 자연로그로 끝나게 되고 결국 친숙한 2의 제곱근이 됩니다. 모두 괜찮지만 우리는 여기에 e를 곱하여 파이 I 맞고 아주 유명하게도 e의 파이 I는 음수 1입니다. 따라서 이 경우에는 이 표현식을 2의 1/2로 풀면 다른 답을 가지고 놀면서 다음과 같은 결과를 얻을 수 있다는 뜻인 것 같습니다. e의 X = 1/2로 우리가 얻게 되는 답은 우리가 전통적으로 음수 제곱근 2로 쓸 수 있는 또 다른 답입니다. 여기서는 2의 1/2을 보기 위해 여러 값을 갖는 것이 조금 웃기다는 뜻입니다. 한 가지는 같지 않다고 말하지만 우리가 하는 선택에 따르면 그것은 여러 가지 다른 것과 같을 수 있습니다. 그러나 두 가지가 상당히 합리적으로 보일 수 있습니다. 2의 1/2이 될 것이 있다면 그것은 둘 중 하나여야 할 것 같습니다. 긍정적인 것 우리에게 익숙한 제곱근이나 그 음의 변형은 실제로는 그런 문제처럼 보이지 않습니다. 그리고 사실 우리는 음 이 게임을 훨씬 더 깊이 있게 플레이할 수 있는데, 이 표현에 대한 더 창의적인 답을 묻고 싶습니다. 왜냐하면 우리가 I를 평가할 때 사용했던 것과 동일한 규칙을 따르면 우리가 어떤 대체를 하는지에 따라 X의 다양한 다른 값을 연결하기 시작할 때 2의 X 거듭제곱과 같은 다른 재미있는 거듭제곱을 찾을 수 있기 때문입니다. 거듭제곱 I 그래서 이번에는 질문이 묻거나 x가 2인 방정식 e의 해 중 하나가 실수 자연 로그 2라는 것을 지정합니다. 우리는 그것을 알고 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "e에 대한 x의 답은 2입니다. 다시 한 번 창의력을 발휘해 보세요. 이에 대해 잠시 시간을 더 드리겠습니다. II 괜찮다면 여기에 몇 가지 답변을 입력하겠습니다. 시간이 얼마나 걸릴지 잘 모르겠습니다. 보고 있는 장치에 따라 반드시 수학 항목을 수행해야 하지만 기회를 얻기 전이라면 너무 스트레스받지 마세요. 원하는 질문에 대답하고 싶은 대답을 입력하면 다음과 같습니다. 여러분 중 131명이 Ln을 2에 더하고 2ii를 더하는 변형을 입력했습니다. 저는 이 질문을 작성하고 있는 것 같습니다. 실수로 답변 중 하나를 정답으로 표시한 것 같은데 실제로는 꽤 다른 정답이 있습니다. 그래서 그건 제가 맡겠습니다. 여러분 중 누군가가 '아, 빨간색이야'라고 생각하실지 모르겠습니다. Ln of 2 + 42를 입력했을 때 틀렸습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi는 물론 훌륭한 선택입니다. 하지만 4pi I에 2 또는 6pi의 자연 로그를 더한 I와 같은 것을 가질 수도 있습니다. 또는 실제로는 2pi I의 정수배가 e에 영향을 주지 않는다는 점을 추가하면 X e를 2pi I에 곱하는 효과가 있기 때문에 이것은 1을 곱하는 효과이고 또 다른 예로 이 작업을 수행하면 합리적인 결과를 출력하는 것처럼 보이는 재미있는 결과가 있습니다. 두 번째로 가장 많이 입력된 표현식은 2를 대체할 수 있다는 것입니다. 그럼 2의 14승을 생각한다고 가정해 보겠습니다. 좋습니다. 2를 자연 로그 2 더하기 4의 e로 대체하자는 제안이 있었습니다. pi I 알았어 더하기 4 pi 나와 우리는 그 모든 것을 1/4로 올림, 만약 여러분이 같은 게임을 한다면 e를 2 곱하기 14번째의 자연 로그에 곱하고 e를 곱하면 됩니다. 파이 I 이제 첫 번째 부분은 일반적인 양의 2의 4근이 될 것입니다. 2의 4근과 같은 표현식을 계산기에 아주 작은 양수로 대입할 때 의미하는 것은 두 번째 부분은 다음과 같습니다. 마이너스 1이므로 다음과 같이 말하는 것 같습니다. 2를 이렇게 다른 방식으로 1의 4승으로 올리면 해석할 수 있습니다. 우리가 얻는 일반적인 대답은 아니지만 합리적인 대답이라는 것을 알 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "우리는 파이를 반으로 곱하고 I를 곱한 것을 보고 마이너스 1을 곱하는 대신 I를 곱했을 것입니다. 다시 한 번 유효한 답입니다. 2의 14승과 같은 합리적인 결과인 것 같습니다. 내가 거듭제곱에 대해 여러 가지 다른 값을 갖고 있는 것 같습니다. 맞습니다. e를 5파이 반쪽 I에 연결하면 이 재미있는 현상이 있습니다. 마이너스 3파이 반쪽 I에 우리는 전혀 다른 답을 얻게 됩니다. 아주 작은 것 아주 큰 것 모두 우리가 여기에서 전에 찾은 1/5과 대략 1/5의 답변과 매우 다릅니다. 2의 1/4제곱이 무엇인지 묻고 실제로 여러 가지 다른 해결책이 있다는 것을 인정할 때와 똑같은 현상입니다. X의 4승은 2와 같습니다. 실제로 4개의 서로 다른 해가 있고 여러분이 보고 있는 것은 여러 가지 해가 있다는 사실입니다. 표현식 e의 X는 어떤 종류의 밑과 같습니다. 그 밑이 I인지 아닌지. 2 그것이 무엇이든, 그리고 우리가 할 수 있는 한 가지 방법에 대해 생각해보세요. 실수를 다룰 때 모든 것은 사랑스럽고 좋은 것입니다. 일대일 관계가 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "지수 함수에 대해 생각하고 싶다면 이 중 일부를 다루겠습니다. 2의 X처럼 X의 밑수로 지수를 표현하도록 선택할 수 있는 멋진 앞뒤가 있습니다. 아니면 다음과 같이 표현할 수도 있습니다. X의 R 곱하기 X와 같은 지수는 우리가 X에 e와 같은 것을 쓸 때마다 암시적으로 참조할 때마다 참조하는 다항식입니다. 그리고 B의 자연 로그를 취할 수 있기 때문에 앞뒤로 멋진 현상이 있습니다. 그리고 그것은 B가 양수라고 가정할 때 하나의 답을 줍니다. 그리고 그것은 R의 X가 B와 같다고 말하는 것과 같습니다. 그래서 제가 시리즈 초반에 이것에 관해 이야기한 한 가지 방법은, 가능한 모든 지수의 집합 맞습니다. 우리는 그것을 R의 X 곱하기 X로 작성하고 R이 무엇인지 변경할 수 있습니다. 그리고 이것은 e를 R 곱하기 X에 쓰는 것과 똑같습니다. 그게 더 편하다면 e를 R에 쓰는 거죠. 곱하기 R의 XX 곱하기 X 그것은 우리가 그것이 무엇인지 바꾸는 것에 대해 생각할 수 있는 것과 같습니다. 그러나 반면에 가능한 모든 지수를 어떤 밑수로 생각한다면 X의 거듭제곱을 밑으로 합시다. 그 베이스가 무엇인지 바꾸려면 처음에는 조작하기 위한 다른 종류의 표현인 것처럼 느껴지지만 이는 동일한 가족을 표현하는 또 다른 방법일 뿐이며 이에 대해 생각할 수도 있는 방법입니다. 어떤 베이스가 해당하는지에 대해 어떻게 생각합니까? 우리가 Exp of R 곱하기 X로 좀 더 추상적으로 생각하고 있고 더 이상해 보일 복소수에 이것을 적용하려고 하기 때문에 내가 이것을 하는 이유가 있다면 여기에서 나와 함께 따라 가십시오. 그 기반을 보는 대신 내가 할 수 있는 한 가지는 가치가 무엇인지 말하는 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "나는 R 곱하기 X의 exp를 가질 수 있는데, 여기서 R은 0.69와 같을 것입니다. 그러나 나는 그것을 2파이만큼 아래로 이동할 수 있습니다. 그리고 그것은 그것이 대응할 베이스를 바꾸지 않습니다. 그것은 여전히 2에 대응할 것입니다. 아니면 그것은 가능합니다 2파이만큼 위로 이동합니다. 이는 해당하는 베이스를 변경하지 않습니다. 왜냐하면 모든 경우에 X를 1과 연결하면 동일한 결과를 얻지만 X의 다른 값에 대한 이들 모두는 고유한 함수입니다. I의 I에 대한 여러 다른 값을 본 이유는 I에 대한 X가 해당 맥락에서 모호한 함수이기 때문에 R의 값을 결정하면 모호하지 않을 것입니다. 즉, 우리가 표현하는 것은 R 곱하기 X 값이 되는 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "그것은 모호하지 않은 함수이지만 그 시점에서 우리가 원하는 것은 아마도 X 거듭제곱의 관점에서 사물에 대해 생각하는 것을 멈추는 것입니다. 아마도 우리가 복소수의 맥락에 있게 되자마자 우리는 그냥 써야 합니다 다른 이유 없이 명확하게 알 수 있는 경우 모든 상수 시간 X의 exp로 계산을 수행하거나 그 위에 수학을 수행하려는 경우 실제로 숫자를 연결하는 방법은 다음과 같은 멋진 무한 다항식을 얻습니다. 그것들을 연결하면 이것이 아마도 지수에 대해 생각하는 올바른 방법일지도 모른다는 또 다른 사례를 만들겠습니다. 우리가 복소수와 같은 것들을 다른 영역으로 확장하자마자 바로 백업합시다 Go 초인종으로 돌아가서 어떤 것들이 도착했습니다. 우리가 지수의 개념을 확장하고 2의 X 오른쪽과 같은 것을 생각하는 원래 방식으로 돌아갑니다. 우리는 이것을 자연수에 대해 어떻게 생각하는지 알고 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "여러분은 2의 3승과 같은 것을 알고 있습니다. 반복 곱셈 분수에 대해 2의 X와 같은 것을 생각하거나 음수에 대해 생각하는 방법을 처음으로 배운 것은 어떻습니까? 잘. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "여러분은 일반적으로 2의 1/2이 내가 그 자체로 곱하면 알 수 있는 것이 되어야 한다고 배웠습니다. 이것은 지수에 물건을 더할 수 있는 숫자를 계산하는 지수 함수의 일반적인 규칙을 따릅니다. 나는 2를 얻어야 합니다. 1로 곱하면 2가 되는 숫자가 되어야 하고 그 시점에서 선택의 여지가 있다는 걸 알잖아요. 어쩌면 양수일 수도 있죠. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "음수일 수도 있지만 항상 긍정적인 선택을 하기로 결정한다면 음수에 대해 묻는다면 동일한 거래에서 멋진 연속 함수를 얻을 수 있을 것입니다. 2의 음수 1이 무엇이 되어야 할까요? 2의 1승을 곱하면 어디에 있지? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "그것은 2의 0을 얻게 하고 그것은 음의 지수가 1/2처럼 보인다는 우리의 관례에 대한 일종의 정당화입니다. 그러나 여기서 실제로 일어나고 있는 것은 이것이 무엇이든 그것은 이 속성 f를 만족하는 일종의 함수여야 한다는 것입니다. a 더하기 b는 f의 a 곱하기 f의 b이고 게다가 밑이 2라는 사실은 기본적으로 이것이 단순한 함수가 아니라는 것을 말해줍니다. 1을 연결하면 2를 얻는 함수입니다. 그리고 조금 아시다시피 여기에 암시된 몇 가지 사항을 따라하고 있는지 확인하기 위한 온전한 확인 스타일 질문 무엇인지 묻고 싶습니다. 소프트볼처럼 부르지는 않겠지만 이것은 다음과 같이 의도된 것은 아닙니다. 엄청나게 깊은 질문 반드시. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "함수의 속성으로 추상적으로 시작한 다음 해당 속성을 기반으로 함수를 기록할 수 있는 추론 방식을 따르는 경우 f의 x가 이 지수 속성 f를 충족하는 경우 확인이 더 필요합니다. a 더하기 b는 모든 입력에 대해 f의 a 곱하기 f의 b와 같습니다. 또한 f의 1은 2와 같습니다. 다음 중 어느 것이 참인지는 다음 중 어느 것이 반드시 참인지를 의미합니다. 어떤 함수를 시작하든 상관 없습니다. 어느 강의였는지 기억하시는 분들과 함께 오일러의 공식이 실제로 말하는 것을 해석하는 방법에 대해 이야기하던 중이었습니다. 저는 조건 하나를 무시한 이런 스타일의 질문을 했는데요, 제가 적지 않았다는 걸 아시죠? 우리는 x의 f가 어디에서나 0이 아닌지 확인하고 싶고 이로 인해 우리 모두에게 발생하는 화면에 혼란을 일으키는 멋진 혼란을 야기했습니다. 그러나 그것의 의도는 기본적으로 이 추상 속성이 덧셈을 곱셈으로 바꾸는 것은 기본적으로 함수를 어떤 종류의 거듭제곱으로 승격한 것과 같은 것으로 작성하고 싶게 만드는 데 충분합니다. 이것이 질문의 핵심입니다. 이제 전력 타워에 대해 실제로 몇 가지 질문이 있습니다. 지난번과 큰 관련이 있는 여기에 나타난 것 같습니다. 전력 타워 질문을 잠시 보류하여 먼저 지수화가 여기서 무엇을 의미해야 하는지에 대한 더 깊은 느낌을 받도록 합시다. ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "제가 주장하고 싶은 것은 우리가 여러 가지 다른 방법으로 답할 수 있기 때문입니다. 따라서 하나만 주시면 송전탑에 대해 이야기하겠습니다. 그리고 수직선이 로그 눈금으로 표현될 수 있듯이 복잡한 평면에 대해서도 동일한 작업이 수행됩니까? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "예 사실, 여기서 잠시 후에 시각화할 내용이 있는데, 여기서 우리는 그것과 매우 유사한 작업을 수행합니다. 왜냐하면 우리가 할 일은 X/R 곱하기 X의 다양한 지수 함수를 가지고 노는 것이기 때문입니다. 하지만 우리는 작은 노란색 점으로 표시될 R 값을 변경하겠습니다. 따라서 이에 대해 좀 더 이야기하겠습니다. 평면 전체를 매핑하는 것이 아니라 실제 축과 가상 축에서 몇 개의 샘플 점만 매핑할 것입니다. 하지만 아이디어는 우리가 그 상수가 무엇인지 돌아다닐 때 평면에 미치는 다양한 일을 시각화할 수 있다는 것입니다. 효과적으로 x축을 로그 눈금으로 바꾼 다음 래핑하는 것과 같습니다. 원을 따라 허수 축 그리고 나서 R의 값이 허수가 되자마자 실수의 역할이 바뀌어 원에 배치되고 허수는 로그 스케일링된 양의 축에 배치됩니다. 세 가지 모두 좋은 질문입니다. 내가 가고 싶은 곳을 향해 총구를 곤두세우고 있는 것 같지만, 이 글에서 사람들이 그렇게 생각하고 있다는 걸 보니 반갑습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "명시적으로 f/5와 같은 것은 f/1 더하기 1 더하기 1 더하기 1 더하기 1과 같습니다. 이는 f/1에 그 자체를 5번 곱한 것과 같습니다. 이 속성 때문에 f/1이 2이면 같습니다. 2의 5승과 f의 -5와 같은 경우입니다. 여기에 f/5를 곱하면 f/0이 무엇이든 얻을 수 있고 f/0이 무엇인지 즉시 알 수는 없지만 다음과 같이 말할 수 있습니다. f/1 더하기 0은 f/1의 곱하기 f/0의 곱과 같습니다. 그러나 f/1은 2와 같습니다. 그리고 이것도 2와 같으므로 2는 어떤 것의 2배와 같다고 말하는 것입니다. 1이어야 하므로 이 문맥에서 이는 -5의 f가 2의 -5의 1 나누기 2의 5승임을 보장합니다. 우리는 이것을 명시적으로 2의 -5의 5승으로 쓸 수 있습니다. 즉, 이 두 속성을 함께 사용하면 우리는 X에 2라는 함수를 쓰고 싶습니다. 왜냐하면 우리가 입력하는 모든 계산 숫자는 만족할 것이기 때문입니다. 우리가 입력한 임의의 분수에 해당 숫자를 곱하는 것처럼 보일 것이기 때문에 이러한 속성을 충족할 것입니다. 그리고 당신은 그것이 고유한지 궁금할 것입니다. 실제 값 함수의 맥락에서는 실제로 그럴 것입니다. 그러나 복잡한 값 함수의 맥락에서는 우리가 작성할 수 있는 함수 f가 여러 개 있을 것입니다. 그 중 하나가 바로 우리였습니다. 이전을 살펴보겠습니다. 자연 로그 2 + 2pi의 exp로 정의된 함수를 가질 수 있는 곳은 어디입니까? 저는 항상 X입니다. 엉성한 점은 양해해 주세요. 이것에 관해 글을 쓰다 보니 정말 신이 나네요. 그리고 이것은 실제로 다음과 같은 다른 함수입니다. X를 2분의 1과 연결하면 어떻게 되는지에 의해 증명됩니다. 우리는 조금 전에 1의 2분의 1을 연결하면 얻는 것은 음의 제곱근 2이고, 4분의 1을 연결하면 4제곱근이 아닌 것을 얻는다는 것을 조금 앞서 보았습니다. 2 하지만 나는 2의 네 번째 근을 곱하므로 그것은 다른 함수입니다. 그러나 그것은 여전히 이러한 속성을 만족하고 그것은 일종의 X의 2라고 쓰고 싶게 만듭니다. 그리고 그것은 아마도 2의 X가 모호하다는 것을 암시합니다. 약간의 표기법 그리고 우리는 모든 것을 exp의 R 곱으로 표현해야 하지만 궁금할 수도 있습니다. 아마도 우리가 이 속성을 만족하는 모든 함수를 충분히 창의적이지 못하고 있다는 것을 알 것입니다. exp를 작성할 때 모호함이 있을 수도 있습니다. R 곱하기 뭔가가 있고 작용할 수 있는 R의 다른 값이 있습니다. 하지만 나는 단지 약간의 주장을 내려놓고 원할 경우 증명이 어떻게 보일지에 대한 스케치를 제공할 것입니다. 당신이 복잡한 함수 F를 가지고 있고, 그것이 먼저 다음 속성을 만족한다고 가정해 보세요. 당신은 그것의 미분을 취할 수 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "그것은 당신이 알고 있는 완전히 지저분하고 불연속적인 것이 되는 것을 방지하는 미분 가능입니다. 그것은 당신이 어떤 벡터 공간의 범위를 아는지에 따라 임의의 값을 취하는 것과 같습니다. 나는 당신이 미친 방식으로 생각하고 싶은 분수 금액을 모릅니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "좋은 기능이네요. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "그것은 미분 가능합니다. 모든 곳에서 0이 아니기 때문에 정신이 나간 조건이 어느 강의인지 잊어버리고 덧셈을 곱셈으로 바꾸는 핵심 속성이 있습니다. 그런 함수가 있다면 나는 다음과 같이 주장합니다. 고유한 것이 있을 수 있습니다. 고유한 복소수 R이 존재하도록 지정해야 합니다. 그러면 X의 F를 기본적으로 값 X의 R 배의 지수 함수로 작성할 수 있습니다. 기본적으로 X를 함수로 사용하면 다음과 같이 말할 수 있습니다. 좋은 파생 속성을 가진 무한 다항식과 그 모든 것을 가지고 있다면 당신은 우리가 원하는 속성과 증명의 스케치에 기초한 지수라는 단어의 추상적인 일반적인 의미에서 원하는 모든 지수를 갖게 됩니다. 우리가 어디에나 존재한다고 가정하는 이 값의 파생물이 무엇인지 먼저 보고 싶다면 다음과 같이 보세요. 그렇죠? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "우리는 표현식에서 X의 F를 완전히 인수분해할 수 있으며 전체 극한은 H의 관점에서만 표현됩니다. 도함수의 맥락에서 그것이 의미하는 바를 생각하고 0의 F는 반드시 1과 같다는 사실을 생각해 보면 이 전체 극한 표현식은 다음과 같습니다. 단지 몇 가지 상수일 뿐이지만 더 구체적으로 말하면 0에서 함수의 도함수는 무엇이든 됩니다. 따라서 0에서의 도함수를 안다면 어디에서나 도함수가 무엇인지 결정하는 재미있는 일이 있습니다. 그리고 지수 함수의 맥락에서 이는 매우 친숙하기를 바랍니다. 우리가 실제로 말하는 것은 지수 함수의 도함수는 그 자체에 비례하고 비례 상수는 0의 도함수와 동일하다는 것입니다. 이것은 모두 매우 추상적으로 표현되었지만 그 목적은 다음과 같다는 점을 강조하는 것입니다. 우리가 이미 X의 거듭제곱으로 생각하는 함수일 필요는 없습니다. 그러나 이것은 덧셈을 곱셈으로 바꾸는 추상적인 속성을 만족시키는 잠재적으로 훨씬 더 광범위한 함수 클래스입니다. 2차 도함수 그리고 그 문제에 대해서는 3차 도함수 등이 있습니다. 왜냐하면 도함수 함수는 자기 자신에게 비례하기 때문입니다. 따라서 n차 도함수를 구하려면 해당 비례 상수를 보고 n의 거듭제곱으로 올리면 됩니다. 그런 다음 여기에서 다음을 수행할 수 있습니다. 테일러 급수 확장은 테일러 급수에 익숙한 사람들을 위한 일종의 고급 숙제로 남겨두겠습니다. 특히 복소수의 의미에서 미분 가능한 미분 함수의 아이디어를 혼합하려는 경우에는 더욱 그렇습니다. 일종의 확실히 대학 주제입니다. 원하는 대로 추론을 혼합할 수 있다는 것을 알고 있습니다. 그러나 퍼지 추론은 Taylor 시리즈에 대해서만 알고 이 아이디어를 F에 대한 Taylor 확장을 살펴보고 다른 것이 없는 사람의 맥락에서 허용됩니다. 함수 F가 반드시 이렇게 작성될 수 있는 고유한 복소수가 있다는 생각을 정당화하는 것 같습니다. 그런 다음 일반적인 지수에 대한 연결은 이러한 값 R을 가질 때마다 발생합니다. 우리는 본질적으로 실수의 복소수 맥락에서 수행하는 작업을 수행합니다. 해당 값 R의 함수 exp를 보고 이를 기본으로 작성하는 경우입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "우리는 이것을 단지 파이 반쪽 I 곱하기 X를 의미하는 것이 아니라 5 파이 반쪽 I 곱하기 X를 의미하는 것으로 해석할 수도 있습니다. 이것들은 별도의 함수입니다. 그리고 우리가 해야 한다고 생각하는 별도의 함수의 무한한 계열이 있습니다. I to the X 그래서 그것이 반드시 의미하게 될 표준을 채택하지 않은 한 I to the I라는 표현은 무한히 많은 출력이 있다고 말할 때 그것을 다른 방식으로 생각하는 것입니다. I to the X 우리가 가지고 있는 표기법은 약간 모호합니다. 이제 모든 것을 고려하여 이것 중 일부를 시각화해 보겠습니다. 왜냐하면 제 생각에는 그것이 재미있을 것 같기 때문입니다. 그리고 이것이 유용한 시각적인지 아니면 더 혼란스러운 시각적인지 말해주실 것입니다. 우리가 할 일은 R 곱하기 X의 이 함수 exp를 보는 것입니다. 이것은 기본적으로 e를 X의 거듭제곱으로 쓰는 또 다른 방법입니다. 사실 제 생각엔 제가 특정 지점에서 다른 애니메이션을 렌더링한 것 같습니다. 왜냐하면 난 그럴 계획이었으니까 그렇게 할 계획이니까 잠깐만요 아 예 거기 내 파일 시스템으로 돌아가서 원래 있어야 할 곳으로 돌아가세요 들어가세요 거기에 여러 가지가 있기 때문에 불평하는 건가요? 아 교체하세요 다른 화면에 뜹니다 잠깐만요 왜 그렇죠 예 교체해요? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "거기에 보이는 것을 무엇이든 놓아두세요 그리고 이제 우리는 아 거기로 돌아갑니다 우리는 그 모든 것을 훌륭하게 쓸 수 있도록 R 곱하기 X 이 무한 다항식의 exp로 생각하는 것이 불편하다면 그냥 머리 뒤쪽 e를 R 곱하기 X로 바꾸고 R을 중심으로 변화하므로 가상 축의 점을 따라갈 것이고 실제 축의 점을 따라갈 것이고 이것이 무엇을 하는지 봅시다. 그것은 모두 빠른 일이므로 좀 더 천천히 생각해 보겠습니다. 모든 음수는 무엇이든 음수 실수는 0과 1 사이의 범위로 뭉개질 것입니다. e가 음수라는 것이 이해가 될까요? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a에 대한 음의 실수는 0과 1 사이의 값이며 우리는 특히 음수 1의 f를 추적하고 있는데 이는 1 나누기 e가 약 30 0인 경우에 나타날 것입니다. 37 f of 1은 예상대로 e에 착지합니다. exp of 1은 f of I입니다. 단위원 주위에 1라디안이 표시됩니다. 여기서 가상 축이 원을 어떻게 감싸는지 전체 가상 축을 따라 따라가는 것은 꽤 재미있습니다. 이 R 값을 조정하면 어떤 일이 발생합니까? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "우리는 여기에서 R의 값을 원할 수도 있습니다. 이것은 사물을 다르게 늘려서 2까지 올리면 실제 축을 훨씬 더 늘려서 f/1이 e 제곱이 약간 작은 곳 근처에서 끝나게 된다는 것을 알고 있습니다. 음수의 7f 이상 1은 0에 훨씬 더 가깝습니다. 실제 축은 꽤 많이 늘어납니다. f/1이 e에서 20 + pi에 매우 가까운 파이에 있다는 것을 알고 있습니다. 항상 재미있고 음수 1의 f는 0에 매우 가깝기 때문에 실제로는 늘어납니다. axis 그리고 그것은 또한 단위원 방향으로 뻗어서 f(I) 또는 f(음수)에 도달하면 원 주위를 반쯤 걷게 되므로 이제 모두 괜찮습니다. 다음과 같은 함수에 대해 어떻게 생각할까요? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "또한 2 곱하기 X의 자연 로그를 X의 X로 작성하여 R 값을 나타내는 노란색 점을 0 근처로 이동합니다. 69는 여전히 허수부가 없고 단지 실수 0입니다. 69 정도입니다. 2의 자연로그입니다. f/1이 2에 도달한다는 것을 알 수 있습니다. 이것이 바로 우리가 이 함수를 X의 f/1/2라고 부르고 싶은 이유입니다. 실제로 죄송합니다. 마이너스 1의 f/f는 바로 1/f에 도달합니다. I 그것은 단위원 주위를 산책하는 것입니다. 구체적으로 말하면 0이 될 것입니다. 단위원 주위에 69라디안이 있고 이제 좀 더 재미있게 이것을 0이 아닌 것으로 바꾸면 어떤 일이 일어날지 말할 수 있습니다. 69는 2의 자연 로그가 아닌 2의 자연 로그를 곱하여 지수 기반을 가질 수 있는 것을 실제로 생각하게 합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "I에 대한 I의 거듭제곱은 무엇입니까? 이 경우 약 0으로 밀어 넣습니다. 2 약 5분의 2 하지만 f를 1로 숫자 I에 대입하는 속성을 갖는 다양한 지수 함수가 있습니다. 따라서 더 확장하려면 여기에 애니메이션을 적용하지 않을 것 같습니다. 그 노란색 점을 반으로 5배 곱하기 파이가 될 때까지 올리세요. 여러분이 보게 될 것은 단위원이겠죠? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "1의 음수 f의 f가 또 다른 2파이 라디안을 중심으로 회전하고 있는 위치에 도달하도록 자체적으로 회전합니다. 하지만 이는 실제 축을 훨씬 더 많이 늘릴 것입니다. 이는 I에 대한 I의 또 다른 출력이 다음과 같다는 의미입니다. 훨씬 더 작은 숫자는 0 정도였습니다. 0003 정도 하지만 제가 생각하기에 상당히 재미있다는 것도 알 수 있습니다. 우리가 해석하고 싶은 대체 표현을 2의 X승으로 고려하면 어떻게 될까요? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "우리는 X의 R 곱하기 X를 갖고 R은 이 값과 같습니다. 이는 2의 자연 로그 더하기 파이 곱하기 I입니다. 이것이 의미하는 바는 우리가 1의 f/1을 연결하면 -2이므로 이 함수를 작성하고 싶다는 것입니다. 음수 2의 X제곱은 맞고 실제로는 아시다시피, 음수를 X제곱에 쓰면 조금 믿을 수 없을 정도로 간단합니다. 음수 2의 X제곱은 처음에는 이렇게 보이지 않습니다. 어떤 방식으로든 복소수에 넣습니다. 하지만 물론 1/2과 같은 값을 대입하면 -2의 제곱근을 구하는 경우에 우리는 이것을 제곱근 곱하기와 같이 쓰고 싶다는 것을 깨닫게 됩니다. of 2 하지만 만약 여러분이 이 함수를 본다면 그것이 다루고 있는 전체 복소 영역에서 -2의 X 거듭제곱 당신이 보고 있는 것은 1의 값을 -2로 취하는 함수이고 그것이 그렇게 한다면 무엇을 합니까? 실수선의 나머지 부분에도 적용됩니다. 바깥쪽으로 나선 모양인가요? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "그래서 우리는 f(-1)이 -1/2에 위치한다는 것을 알 수 있습니다. 여러분이 f(1/2)을 따른다면 여러분이 예상할 수 있는 위치는 정확히 허수선 위에 놓일 것이고 f(1/2)는 2의 제곱근이 될 것입니다. 마우스가 내가 원하는 곳에 있지 않습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "그것은 약 2 곱하기 I의 제곱근이 될 것이며 계속해서 계속하면 이것은 X에 대한 -2의 실제 가치의 힘을 모두 보여주고 있으며 이는 필연적으로 나선형으로 돌아갑니다. 그러나 R의 값을 더 높게 움직여서 얻을 수도 있습니다. 최대 타우 곱하기 I 약 6.2 8 곱하기 I 그리고 그 맥락에서 이것은 우리가 X에 2와 같이 쓰고 싶은 또 다른 함수입니다. 왜냐하면 X에 연결하는 정수 대 정수의 경우 반복된 곱셈처럼 보입니다. 그리고 양의 제곱근 대신에 음의 제곱근을 뱉어내는 1/2과 같은 것에 대해 일종의 합리적인 값도 가지고 있지만 실제로 하는 일은 모든 것을 놓는 평면으로의 변환입니다. 수직선은 결국 매우 촘촘하게 감겨진 나선형으로 돌아가며 나선형으로 회전하여 1의 f가 숫자 2에 바로 도달합니다. 따라서 그런 의미에서 X의 2는 다음과 같이 해석될 수 있습니다. 우리가 전통적으로 사용했던 함수와는 별도의 지수 함수입니다. 그래서 제 생각에는 오늘은 이쯤으로 두고 생각해볼 몇 가지 질문을 남겨두겠습니다. I에 대한 I를 다중 값 표현으로 생각하세요. 맞습니까? 우리가 관례를 채택한다고 말할 수 있습니까? 공상적으로 자연 로그 함수의 분기를 선택한다고 말할 수 있습니다. 그리고 어쩌면 이것이 음수 파이에 대한 e가 되도록 가두게 될 수도 있습니다. 반쪽 하지만 이런 종류가 우리가 본 다양한 값처럼 무한히 다양한 값이 되고 싶다고 하면 2의 1/3이 같은 의미로 되고 싶은 값은 몇 개일까요? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "10번째는 모든 것을 다르게 표현하고 싶습니다. 만족하는 모든 지수 함수 F(X)에 대해 말하겠습니다. 아, 제가 그것을 만족하는 f(X) 어딘가에 적어두었나요. 제가 작성한 이 모든 속성이 모두 만족한다면요. 이들 중 f(1)이 2와 같다면, 어떤 기능에 대한 다양한 옵션에 대해 X가 3/10과 같다고 연결하면 얼마나 많은 다른 출력을 얻게 될까요? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "2의 파이에 대한 다양한 함수에 대한 2의 X는 우리가 2의 X를 일종의 지수 함수로 생각한다면 나타낼 수 있습니다. 이런 종류의 추상적 속성의 의미에서 지수 함수이고 만약 그렇다면, 만약 우리가 우리는 다양한 기능의 클래스를 가지고 있고, 파이를 연결하려고 하면 웃게 됩니다. 그런 것 때문에 생각하려고 할 때 튀어나오는 재미있는 대답이 있다는 것을 알고 있으므로 이것이 바로 질문입니다. 나는 여러분에게 이것을 남길 것입니다. 오늘 강의에 접근하면서 제가 가장 중요하게 생각한 질문은 이것이 지수 함수의 추상 속성과 같은 종류의 설명이 되기를 원하는지였습니다. 그리고 이러한 추상 속성에서 시작하는 것이 나에게는 정말 멋집니다. 당신은 e를 rx 이상으로 생각하는 것에 갇히게 됩니다. 제 생각에는 r의 다른 값에 대해 r 곱하기 x의 exp를 더 솔직하게 작성했다고 생각합니다. 그것은 당신을 그 정도까지 가두지만, 2의 x제곱은 I의 x제곱과 같은 것이 훨씬 적어야 한다는 명확한 개념 물론 그에 따른 위험은 때때로 사람들이 추상화를 좋아하지 않고 때로는 접근하기 쉬운 것으로 나오지 않는다는 것입니다. 알고 계시다면 제게 알려주시기 바랍니다. 제 생각에는 이 모든 것에는 전력 타워가 포함됩니다. 왜냐하면 여러분이 원한다면 전력 타워에 대해 지난 번 복소수의 맥락에서 이야기했던 것처럼 실제로 전력 타워에 대해 이야기하기 때문입니다. 아니면 부정적인 근거로라도 이렇게 생각해야 하니까 화면에 나왔던 질문이었죠. 그래, I를 위해 I의 힘으로 이렇게 하면 어떻게 될까요? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "적정 알다시피 이것을 해보자 그냥 가서 전력 타워를 시험해 봅시다. 여기서 우리는 I를 주어진 전력으로 끌어올리고 거기에서 무엇이 나타나는지 확인합니다. 그래서 이렇게 할 계획은 아니었습니다. 하지만 우리는 항상 할 수 있습니다 Python을 끌어와 본질적으로 지난번에 했던 작업을 수행합니다. 이것이 작동하는 방식은 기본 값으로 시작한 다음 일종의 범위에 대해 우리가 하고 있던 작업을 a를 취하고 다시 할당하는 것입니다. it to be everything 이 경우 내가 a의 거듭제곱으로 끌어올린 베이스는 다음과 같습니다. 좋아요, 좋습니다. 그럼 그렇게 하고 a의 값을 인쇄해 보겠습니다. 예, 200이라는 훨씬 더 큰 숫자입니다. 따라서 일어나는 일은 때때로 이러한 일로 인해 혼란이 발생할 가능성이 있는 것 같습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "실제로 NumPy를 가져오도록 하겠습니다. 지수 함수가 있습니다. 이전처럼 큰 범위를 위해 저와 X의 거듭제곱을 아는 대로 작성하는 대신 작성하겠습니다. 다른 상수의 지수 함수죠. 제가 만들 다른 상수는 5파이 반이 되길 원하므로 5파이 반을 곱하겠습니다. 그래서 그것은 복소수이고 5파이 반이 됩니다. 허수부분 그러면 이것은 5파이 반곱하기 I이고 나는 무엇을 하고 있는 걸까요? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/marathi/sentence_translations.json b/2020/ldm-i-to-i/marathi/sentence_translations.json index dfe9c186f..50e6acdca 100644 --- a/2020/ldm-i-to-i/marathi/sentence_translations.json +++ b/2020/ldm-i-to-i/marathi/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "म्हणून जर तुम्ही क्रमांक 1 पासून सुरुवात करत असाल, तर तुमचा प्रारंभिक वेग 0 च्या दिशेने सरळ चालणे आहे आणि जसे तुम्ही आणखी कमी चाललात, जर तुम्ही 1 अर्ध्यावर बसला असाल, तर तुम्ही अजूनही 0 च्या दिशेने चालत असाल, परंतु आता तुमचा वेग वेक्टर आहे. तुम्ही जिथे आहात तिथे 1 पट नकारात्मक असेल, जे ऋण 1 अर्धा आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "आणि एक मनोरंजक प्रश्न तुम्हाला माहित असेल की असे फक्त एक फंक्शन आहे जे यासाठी लिहिण्यास वाजवी वाटते कारण तुम्हाला माहिती आहे की आम्ही ते i टू द x म्हणून लिहिणार आहोत तर केवळ हेच समाधानी नाही तर तुम्हाला हे देखील माहित आहे की कधी आम्ही पहिल्या क्रमांकावर प्लग इन करतो i शक्यतो i पॉवर वनमध्ये तथापि आम्ही विचार करत आहोत की हे फंक्शन i असावे. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "त्यामुळे आम्हाला 5 pi i अर्धे छान मिळाले आहेत हे दुसरे मूल्य आहे जे आम्ही येथे x साठी प्लग इन करू शकतो आणि जर आम्ही आमच्या वर्तुळात परत पाहिले तर ते थोडे अधिक स्पष्टपणे स्पष्ट करण्यासाठी क्षण पाई अर्ध्या भागाच्या बरोबरीने चालला जे 1 आहे. 57 त्याऐवजी आम्ही आणखी एक पूर्ण वळण घेतले आणि आम्ही आणखी एका पायच्या अर्ध्या भागावर गेलो तर आम्हाला pi कडे नेले पाहिजे जे तुम्हाला माहित आहे की आम्ही एक प्रकारची नोंद करू शकतो जिथे e ते pi i ची किंमत आहे, आम्ही आणखी एक pi अर्ध्या भागावर चालतो जे येथे आहे या बिंदूवर आपण पूर्ण वर्तुळात गेलो असतो आणि आपल्याला परत एकाकडे नेले असते आणि नंतर आपण पाच पाय अर्ध्या भागासाठी चालत असतो जे संख्यात्मकदृष्ट्या सुमारे 7 असते. 85 होय, ती आणखी एक संख्या आहे जी आपल्याला i च्या शीर्षस्थानी आणते आणि जर आपण i ला पुन्हा व्यक्त करण्याच्या संपूर्ण रिग्मारोलमधून जायचे असेल तर i प्रथम 5 pi अर्ध्या i वर e लिहून पॉवर i त्या i च्या नकारात्मक होण्यासाठी गुणाकार करा आणि आपण ई कडे ऋण 5 pi अर्ध्या भागाकडे पाहत आहोत जी खूप वेगळी संख्या आहे बरोबर आपण याची गणना करू शकतो मला माझ्या डोक्याच्या वरच्या बाजूस खात्री नाही, पण डेस्मॉस पाहूया . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "ती लांबलचक जी तुम्हाला खूप कमी संख्येपर्यंत पोहोचवते परंतु हे एकमेव उत्तर नाही जे आम्ही बरोबर प्रविष्ट करू शकतो आमच्याकडे इतर लोक नकारात्मक 3 अर्ध्या वेळा i pi सह येतात जे तुम्हाला एकक वर्तुळाच्या दृष्टीने माहित आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "अहो मला 90 अंश पाय अर्ध्या रेडियनवर चालण्यापेक्षा मला I गाठायचे असेल तर त्या मार्गाने मी 270 अंश चाललो तर 3 पाई अर्ध्या रेडियन जे कदाचित मला नकारात्मक वाटेल असे म्हणण्याचा आपण विचार करू शकतो कारण अधिवेशन आहे सामान्यतः ते घड्याळाच्या उलट दिशेने सकारात्मक असते ते व्यक्त करण्याचा आणखी एक मार्ग आहे आणि जर आमच्याकडे e ते ऋण 3 pi अर्धे असतील तर ते आम्हाला एक वेगळे उत्तर मिळेल i सर्व शक्ती i आम्ही त्याच गेममधून जातो आता i स्क्वेअर रद्द करतो ऋणात्मक जे आधीपासून आहे, आणि आपल्याकडे सकारात्मक 3 pi अर्धे आहेत आणि संख्यात्मकदृष्ट्या हे आपल्याला आधी जे होते त्यापेक्षा अगदी वेगळे दिसणारे उत्तर मिळते ज्यावर आपण गेलो आणि आपण म्हणतो अरे, 3 pi चे e काय आहे 3 o 3 pi नाही अर्धा भाग 111 बिंदू 3 1 111 बिंदूच्या आधी आपण पाहिलेल्या संख्येपेक्षा खूप वेगळ्या प्रकारची संख्या होती ती काय होती 111 बिंदू 3 1 उत्तम 111 बिंदू 3 1 किंवा असे आणि पुन्हा अंतर्ज्ञानाच्या दृष्टीने आपण काय विचारत असाल समजा आपल्याकडे हे फिरत आहे डायनॅमिक पण आपण वेळेत मागे सरकतो आपण पाहतो की किती काळापूर्वी मी काय असायला हवे ते असे की जर मी तेथून पुढे खेळलो तर मी माझ्या सुरुवातीच्या स्थितीत पहिल्या क्रमांकावर उतरेन आणि तुम्हाला वेळेत परत जावे लागेल 3 pi हाल्व्ह युनिट्स आणि मग जर तुम्ही क्षय गतीशीलतेचे भाषांतर कराल तर या संदर्भात डोळा वर आणणे हे काय करत आहे तुम्ही म्हणाल की मी पहिल्या क्रमांकावर आहे का, पण मला वेळेत मागे सरकायचे आहे आणि म्हणायचे आहे की मी कुठे सुरुवात केली असेल तर मला असे क्षीण करायचे आहे की मी पहिल्या क्रमांकावर जाईन? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "वेळेच्या 3 pi अर्ध्या एककांनंतर उत्तर स्पष्टपणे अशा प्रकारच्या घातांकीय क्षयसाठी सुमारे एकशे अकरा वाजता सुरू होत आहे आणि आपण पाहू शकता की हे कोठे जात आहे तेथे खरोखर असीम अनेक भिन्न मूल्ये आहेत जी आपण X साठी प्लग इन करू शकलो तर मी म्हणून ई टू द एक्सचा विचार करत आहे आणि लोक इथे खूप जास्त घुसले आहेत मला माफ करा माझा पिन जमिनीवर फेकून द्या कारण एक तृतीय स्थानासाठी क्लासिक करतो 9 pi अर्धे उत्तम निवड 1729 pi अर्धे तुम्ही सर्व माझे आवडते लॉट आणि बरेच आहेत भिन्न पर्याय असीमपणे अनेक भिन्न मूल्ये जे प्रथम उजवीकडे थोडे अस्वस्थ वाटतात कारण आम्ही एक अभिव्यक्ती पाहतो असे दिसते की तुम्हाला माहित आहे की तेथे फक्त काही गणना होणार आहे मी फक्त ते माझ्या कॅल्क्युलेटरमध्ये प्लग केले आहे आणि काय पॉप आउट होते ते पहा आणि आमच्याकडे अनेक भिन्न आहेत त्यासाठी मूल्ये तर इथे काय चालले आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "16 चे चौथे मूळ 2 असले पाहिजे आणि उत्तर चांगले राहून संपेल तेव्हा आम्ही एक नियम स्वीकारतो जेव्हा तुमच्याकडे बहु-मूल्य असलेले कार्य असते तेव्हा असे अनेक पर्याय असतात जेव्हा आम्हाला हवे असते तेव्हा आम्ही अनेकदा त्यापैकी एक मूल्य निवडतो जे आम्हाला करायचे आहे. फॅन्सियर लिंगोमध्ये एकच इनपुट आणि एकच आउटपुट असलेले काहीतरी फंक्शन म्हणून हाताळा जेव्हा आम्ही जटिल संख्यांशी व्यवहार करत असतो तेव्हा हे नेहमीच समोर येते जेव्हा एखादी गोष्ट अशी कल्पना असते की तुम्हाला अनेक मूल्ये असण्याची इच्छा असते. तुम्ही वर्गमूळ फंक्शनची शाखा कुठे निवडता? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "कारण अनेक भिन्न उत्तरे आहेत कारण तुम्हाला माहिती आहे की आम्ही पुन्हा मी विचार करतो हे 90 अंश रोटेशन आहे आणि जर आपण त्याचा 90 अंश रोटेशन म्हणून विचार करत असू तर असे वाटते की वर्गमूळ असावे 45 अंश कोनात बसलेले काहीतरी तुम्हाला माहित आहे कदाचित तो चौरस असेल I चे रूट जे आपण रूट 2 ओवर 2 रूट 2 ओव्हर 2 म्हणून अगदी स्पष्टपणे लिहू शकतो I हे फक्त त्रिकोणमिती वापरत आहे परंतु जर आपण त्याऐवजी मी नकारात्मक 270 अंश रोटेशन असा विचार करत असू तर असे वाटते की अर्ध्यापैकी अर्ध्या ऑपरेशनच्या अर्ध्या भागासारखे आहे प्रत्यक्षात आपल्याला दुसऱ्या बाजूला मिळायला हवे कदाचित येथे खाली बसलेली संख्या I चे वर्गमूळ असावे आणि हे खरेतर आपण नकारात्मक मूळ 2 पेक्षा 2 वजा मूळ 2 पेक्षा 2 वेळा I च्या संदर्भात जे पाहिले होते त्याचे फक्त ऋण आहे. व्हॅल्यूड फंक्शन्स आपण होय म्हणू शकतो की सकारात्मक उत्तर काहीही असले तरी फक्त वर्गमूळ निवडा पण यापैकी कोणते सकारात्मक उत्तर तुम्ही मानता? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "आणि मला वाटते की तुम्ही चांगले म्हणता आम्हाला माहित आहे की हे काय आहे ते आम्ही 2 चे वर्गमूळ म्हणून परिभाषित करतो सर्व चांगले आणि चांगले आहे परंतु मी काय म्हटले तर आपण याकडे त्याच प्रकारे संपर्क साधूया ज्या प्रकारे आपण आपल्या I कडे I अभिव्यक्ती I कडे जात होतो. प्रथम एखाद्या गोष्टीला e बरोबर गोष्टी व्यक्त करायच्या आहेत आणि मग मी 1 अर्ध्याला घातांकात गुणाकार करून ते 1 अर्ध्यापर्यंत वाढवणार आहे आणि मी म्हणतो ठीक आहे, मी अंदाज लावू शकतो की मी ते e वर करू शकतो. 2 च्या बरोबरी म्हणजे 2 चा नैसर्गिक लॉग हा एक स्थिरांक आहे जो 0 च्या आसपास आहे. 69 किंवा त्याप्रमाणे जर आपण e ला त्या पॉवरमध्ये वाढवले तर आपल्याला 2 मिळेल त्यामुळे आपण 2 गुणिले 1 अर्धा च्या नैसर्गिक लॉगसाठी e असा विचार करू शकतो आणि जर तुम्हाला हवे असेल तर तुम्ही e ते x चा विचार करत असाल? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "तुम्हाला माहित आहे की वास्तविक संख्यांच्या संदर्भात हे एक प्रकारचे ओव्हरकिल असू शकते परंतु जर तुम्ही या x फंक्शनसाठी ई ते x शॉर्टहँड म्हणून विचार करत असाल तर तुम्ही मूल्य 0 प्लग करू शकता. 69 गुणिले 1 अर्धा जो माझ्या अंदाजानुसार 0 च्या आसपास असेल. 345 इश असे काहीतरी तुम्ही तुमच्या बहुपदीमध्ये अगदी ठोस मूल्य प्लग करा ते काय आउटपुट करते ते पहा आणि ते 1 च्या आसपास आउटपुट करेल. 414 एक छान वास्तविक संख्या 2 चे वर्गमूळ तुम्हाला काय अपेक्षित आहे पण जर आपण तेच केले तर आपण फक्त I बरोबर करत होतो आणि कबूल करतो की जेव्हा आपल्याला ई टू पॉवर म्हणून काहीतरी लिहायचे असते तेव्हा आपण हे देखील लिहू शकतो. हे कदाचित मजेदार वाटेल, परंतु आम्ही ते 2 अधिक 2 pi I च्या नैसर्गिक लॉगवर e असे लिहू शकतो की संपूर्ण गोष्ट 1 अर्ध्यापर्यंत वाढली आहे उजवीकडे हे सर्व मूल्य समान होईल आपण ते खंडित करू शकता कारण ते e आहे. 2 चा नैसर्गिक लॉग 2 pi I ला e ने गुणाकार केला आहे याला फक्त गोष्टी 360 अंश फिरवण्याचा प्रभाव आहे, त्यामुळे ते फक्त 1 च्या बरोबरीचे होणार आहे म्हणून आम्ही 2 गुणिले 1 ग्रेट पाहत आहोत जे वैध प्रतिस्थापनासारखे वाटते आणि तरीही जेव्हा आपण हाच खेळ खेळतो आणि त्याला घात वाढवतो आणि घातांकामध्ये घात गुणाकार केल्याने काय होते ते पहा आपल्याकडे 2 गुणिले 1 अर्धा अधिक या नैसर्गिक लॉगमध्ये ई आहे ठीक आहे, 2 pi I गुणिले 1 अर्धा किती आहे बरं ते pi पट असेल I आता हा पहिला भाग e 2 गुणिले 1 अर्धा च्या नैसर्गिक लॉगचा जो 2 चे परिचित वर्गमूळ असेल ते सर्व चांगले आणि चांगले आहे, परंतु आपण त्याचा e ने गुणाकार करणार आहोत. pi I बरोबर आणि अगदी प्रसिद्धपणे e ते pi I ऋण 1 आहे त्यामुळे या प्रकरणात असे दिसते की जर आपण ही अभिव्यक्ती 2 ते 1 अर्धा सोडवत आहोत तर वेगवेगळ्या उत्तरांसह खेळून आपण काहीतरी प्लग इन करू शकतो. e बरोबर X बरोबर 1 अर्धा आपण ज्याचा शेवट करतो ते दुसरे उत्तर आहे जे आपण पारंपारिकपणे 2 चे ऋण वर्गमूळ म्हणून लिहू शकतो आणि येथे 2 ते 1 अर्ध्याकडे पाहण्यासाठी अनेक मूल्ये असणे हे थोडे मजेदार आहे. म्हणा की ती समान नाही एक गोष्ट परंतु आपण निवडलेल्या निवडींवर आधारित ती अनेक भिन्न गोष्टींच्या समान असू शकते परंतु दोन गोष्टी ज्या अगदी वाजवी वाटू शकतात जर 2 ते 1 अर्धा असे काहीही असेल तर असे दिसते की ते एकतर सकारात्मक असावे वर्गमूळ ज्याच्याशी आपण परिचित आहोत किंवा त्याचे नकारात्मक रूप जे प्रत्यक्षात अशी समस्या वाटत नाही आणि खरं तर आपण हा खेळ आणखी पुढे खेळू शकतो, जिथे मी तुम्हाला या अभिव्यक्तीची आणखी सर्जनशील उत्तरे विचारू. कारण कदाचित आम्ही 2 ते पॉवर X सारख्या इतर मजेशीर शक्ती शोधू शकतो कारण आम्ही X ची विविध मूल्ये जोडणे सुरू करतो तेव्हा आम्ही I चे मूल्यमापन करताना वापरत असलेल्या समान नियमांचे पालन करत असल्यास आम्ही कोणते पर्याय बनवतो यावर आधारित पॉवर I तर यावेळी प्रश्न विचारतो किंवा तो निर्दिष्ट करतो की e समीकरणाचे x बरोबरी 2 हे खरे संख्या 2 चा नैसर्गिक लॉग आहे जो आपल्याला माहित आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "e च्या प्रश्नाचे उत्तर x बरोबर 2 आणि पुन्हा सर्जनशीलतेचे स्वागत आहे, म्हणून मी तुम्हाला आणखी एक छोटासा क्षण देईन II पुढे जाऊन काही उत्तरे येथे लॉक करेन जर ते तुमच्यासाठी ठीक असेल तर मला खात्री नाही की किती वेळ लागेल अपरिहार्यपणे आपण कोणते उपकरण पहात आहात यावर अवलंबून गणिताची नोंद करणे आवश्यक आहे परंतु आपल्याला ज्या प्रश्नाचे उत्तर हवे आहे त्या प्रश्नात जाण्याची संधी मिळण्यापूर्वी जास्त ताण घेऊ नका, त्यामुळे असे दिसते तुमच्यापैकी 131 जणांनी व्हेरिएंटमध्ये प्रवेश केला आहे जिथे आपण 2 चा Ln घेतो आणि 2ii जोडतो आणि मला वाटते की मी हा प्रश्न लिहीत आहे असे चुकून एक उत्तर बरोबर आहे असे चिन्हांकित केले आहे, जेव्हा प्रत्यक्षात काही भिन्न बरोबर आहेत तर ते माझ्यावर आहे कारण मला माहित नाही की ते तुमच्यापैकी कोणाला दिसत आहे की नाही हे लाल आहे तुम्ही 2 अधिक 42 च्या Ln मध्ये प्रवेश केला तेव्हा तुम्हाला चूक झाली. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi अर्थातच एक उत्तम पर्याय आहे पण तुमच्याकडे 4 pi I अधिक 2 किंवा 6 pi I चा नैसर्गिक लॉग किंवा 2 pi I चा कोणताही पूर्णांक गुणक तुम्ही जोडल्यास त्याचा e वर परिणाम होत नाही असे काहीतरी असू शकते. X कारण त्याचा फक्त e ने 2 pi I चा गुणाकार करण्याचा प्रभाव आहे जो 1 ने गुणाकार करण्याचा परिणाम आहे आणि पुन्हा याचा एक प्रकारचा गमतीशीर परिणाम आहे जिथे आपण ते दुसरे उदाहरण म्हणून करतो तेव्हा ते वाजवी परिणाम देतात असे दिसते. असे दिसते की दुसरी सर्वात सामान्य प्रविष्ट केलेली अभिव्यक्ती होती की आपण 2 ची जागा घेऊ शकतो, तर आपण 1 4थ्या घात 2 चा विचार करत आहोत असे समजू, ठीक आहे, अशी सूचना होती की आपण 2 अधिक 4 च्या नैसर्गिक लॉगमध्ये 2 च्या जागी e घातला आहे. pi I ठीक आहे प्लस 4 pi I आणि आम्ही ते सर्व 1 4 व्या बरोबर वाढवतो, जर तुम्ही समान खेळ खेळत असाल तर तुम्हाला e 2 गुणिले 1 4 व्या नैसर्गिक लॉगमध्ये मिळेल आणि आम्ही e ने गुणाकार करू. pi I आता त्याचा पहिला भाग 2 चा नेहमीचा सकारात्मक चौथा रूट असेल ज्याचा अर्थ आपण 2 च्या चौथ्या रूट सारख्या अभिव्यक्तीला कॅल्क्युलेटरमध्ये प्लग इन करता एक छान लहान सकारात्मक संख्या, परंतु नंतर हा दुसरा भाग आहे नकारात्मक 1 म्हणून असे दिसते आहे की आपण 2 चा अर्थ या वेगळ्या पद्धतीने 1 4थ्यापर्यंत वाढवायचा आहे का, हे आपल्याला माहिती आहे हे आपल्याला मिळालेले नेहमीचे उत्तर नाही परंतु ते एक वाजवी उत्तर आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "आपण पाईच्या अर्ध्या वेळा I पाहत असू आणि ऋण 1 ने गुणाण्याऐवजी आपण I ने गुणाकार करत असू जे पुन्हा एक वैध उत्तर आहे ते 2 ते 1 4 व्या सारख्या गोष्टीसाठी वाजवी आउटपुटसारखे दिसते तेव्हा आपण या वस्तुस्थितीकडे पाहिल्यास, माझ्याकडे अनेक भिन्न मूल्ये आहेत असे दिसते, बरोबर आमच्याकडे ही मजेदार घटना आहे जिथे आम्ही 5 pi अर्ध्या भागांमध्ये प्लग इन करू शकतो I नकारात्मक 3 pi अर्ध्या I आणि आम्हाला अगदी भिन्न उत्तरे सारखी दिसत होती काहीतरी खूप लहान काहीतरी खूप मोठे हे सर्व 1 5 व्या अंदाजे 1 5 व्या उत्तरापेक्षा खूप वेगळे आहे जे आम्हाला येथे आधी सापडले आहे ही अगदी सारखीच घटना आहे जेव्हा तुम्ही 2 ते 1 4 व्या काय आहे असे काहीतरी विचारत आहात आणि हे मान्य करत आहात की प्रत्यक्षात अनेक भिन्न उपाय आहेत एक्स ते 4थ या अभिव्यक्तीमध्ये 2 4 भिन्न समाधाने आहेत आणि तुम्ही जे पाहत आहात ते हे आहे की अनेक भिन्न समाधाने आहेत या अभिव्यक्तीसाठी e ते X हा काही प्रकारचा आधार आहे की तो आधार मी आहे की नाही 2 ते काहीही असो आणि आपण याचा विचार करू शकतो असा एक मार्ग म्हणजे जेव्हा आपण वास्तविक संख्यांशी व्यवहार करता तेव्हा गोष्टी फक्त सुंदर असतात. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "हे छान आहे जिथे आम्हाला घातांकीय फंक्शन्सबद्दल विचार करायचा असेल, तर मला यापैकी काही गोष्टी कव्हर करू द्या आमच्याकडे हे खूप छान आहे जिथे तुम्ही X ला आधार म्हणून कोणतेही घातांक व्यक्त करू शकता जसे की 2 ते X किंवा तुम्ही व्यक्त करू शकता R गुणिले X चा X सारखाच घातांक जो आपल्याला माहित आहे की बहुपदी आहे ज्याचा आपण संदर्भ घेतो तेव्हा जेव्हा आपण X ला e सारखे काहीतरी लिहितो तेव्हा त्याचा संदर्भ घेतो आणि पुढे आणि पुढे एक सुंदर आहे कारण आपण फक्त B चा नैसर्गिक लॉगरिथम घेऊ शकता आणि हे तुम्हाला B ही धन संख्या आहे असे गृहीत धरून एक उत्तर देते आणि R चा X हा B च्या बरोबरीचा आहे असे म्हणण्यासारखीच गोष्ट आहे, म्हणून मी या मालिकेमध्ये याआधी बोललो होतो असा एक मार्ग म्हणजे जर तुम्ही पाहत असाल तर सर्व संभाव्य घातांकांचे कुटुंब बरोबर आम्ही त्यांना R गुणिले X चा X असे लिहू शकतो आणि R काय आहे ते बदलू शकतो आणि ही गोष्ट आर टाइम्स X ला ई लिहिण्यासारखीच आहे जर ती गोष्ट तुम्हाला अधिक सोयीस्कर वाटत असेल तर आर गुणिले X च्या गुणा XX या समान गोष्टी आहेत जे आपण बदलण्याचा विचार करू शकतो परंतु दुसरीकडे जर आपण सर्व संभाव्य घातांकांचा काही आधार म्हणून विचार करत असाल तर मला X च्या पॉवरचा आधार द्या आणि आपण पुढे जाऊ तो आधार काय आहे हे बदलण्यासाठी सुरुवातीला असे वाटते की ते हाताळण्यासाठी वेगळ्या प्रकारची अभिव्यक्ती आहे, परंतु त्याच कुटुंबाला व्यक्त करण्याचा हा आणखी एक मार्ग आहे आणि आपण याबद्दल विचार करू शकता असा एक मार्ग आहे ज्यासाठी ते कोणत्या आधाराशी संबंधित आहे याबद्दल आम्ही कसे विचार करतो जर आपण आर टाइम्स एक्सचा विस्तार म्हणून थोडा अधिक अमूर्तपणे विचार करत आहोत आणि मी हे करत आहे असे एक कारण आहे कारण आम्ही हे जटिल संख्यांवर लागू करणार आहोत जिथे ते विचित्र दिसत आहे, तर माझ्याबरोबर येथे अनुसरण करा त्या आधाराकडे पाहण्याऐवजी मी एक गोष्ट करू शकतो की मूल्य काय आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "माझ्याकडे R गुणा X चा एक्सप असू शकतो जिथे कदाचित R हे शून्य बिंदू सहा नऊ सारखे काहीतरी आहे परंतु मी ते दोन pi I ने खाली हलवू शकेन आणि ते आधार बदलत नाही की ते अद्याप दोनशी संबंधित असेल किंवा ते असू शकते ते दोन pi I ने वर हलवा जे त्याच्याशी संबंधित असलेला आधार बदलत नाही कारण या सर्व प्रकरणांमध्ये जेव्हा आपण X बरोबरीने प्लग इन करतो तेव्हा आपल्याला समान गोष्ट मिळते तथापि X च्या भिन्न मूल्यांसाठी ही सर्व भिन्न कार्ये आहेत. आम्ही I ते पॉवर I साठी अनेक भिन्न मूल्ये का पाहिली कारण I ते X हे एक संदिग्ध कार्य आहे त्या संदर्भात जर आपण R चे कोणते मूल्य ठरवले तर ते अस्पष्ट होईल जसे की आपण जे प्रतिनिधित्व करीत आहोत ते R गुणा X चे मूल्य आहे च्या आर. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "हे एक निःसंदिग्ध कार्य आहे परंतु त्या क्षणी असे वाटते की आपल्याला जे हवे आहे ते म्हणजे शक्ती X पर्यंत वाढलेल्या काही बेसच्या संदर्भात गोष्टींबद्दल विचार करणे थांबवणे कदाचित आपण जटिल संख्यांच्या संदर्भात असू तेव्हाच आपण फक्त लिहावे ते सर्व काही स्थिर काळ X च्या एक्सप म्हणून इतर कोणत्याही कारणास्तव स्पष्ट झाले नाही तर आपल्याला गणना करायची असेल किंवा फक्त गणित करायचे असेल तर आपण संख्या कशी जोडू शकतो हे आपल्याला हे छान अनंत बहुपदी मिळाले आहे त्यांना प्लग इन करा आणि मी तुमच्यासाठी आणखी एक केस तयार करेन की घातांकांबद्दल विचार करण्याचा हा कदाचित योग्य मार्ग आहे जसे की आम्ही जटिल संख्यांसारख्या इतर डोमेनमध्ये विस्तारित करतो आणि त्यासाठी आपण फक्त बॅकअप घेऊया. डोरबेलवर परत काही गोष्टी आल्या मूळ मार्गावर परत या की आम्ही घातांकाची कल्पना वाढवतो आणि फक्त 2 ते X उजवीकडे काय आहे याचा विचार करतो, नैसर्गिक संख्यांसाठी याचा विचार कसा करायचा हे आम्हाला माहित आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "तुम्हाला 2 ते 3 पुनरावृत्ती होणारे गुणाकार असे काहीतरी माहित आहे की तुम्हांला प्रथम 2 ते X या अपूर्णांक रकमेसाठी किंवा नकारात्मक राशी आणि अशा गोष्टींबद्दल विचार करायला शिकवले जाते. विहीर. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "तुम्हाला सहसा असे शिकवले जाते की 2 ते 1 अर्धा असा काहीतरी असावा जिथे मी स्वतः गुणाकार केला तर हे तुम्हाला माहीत आहे आणि हे नेहमीच्या नियमांचे पालन करते जे घातांक संख्या मोजण्यासाठी करतात जेथे आम्ही त्या घातांकामध्ये गोष्टी जोडू शकतो तेव्हा मला 2 मिळायला हवे. 1 ला, म्हणजे ती अशी काही संख्या असावी की जेव्हा मी त्याचा स्वतःहून गुणाकार करतो तेव्हा मला 2 मिळेल आणि तुम्हाला माहित आहे की त्या वेळी तुम्हाला एक पर्याय आहे, कदाचित तो सकारात्मक असेल. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "कदाचित ते नकारात्मक असेल पण जर तुम्ही नेहमी सकारात्मक निवड करण्याचे ठरवले तर तुम्ही या समान करारातून एक चांगले निरंतर कार्य मिळवू शकाल जर आम्ही नकारात्मक संख्यांबद्दल विचारले तर 2 ते ऋण 1 काय चांगले असले पाहिजे ते काहीतरी असावे जेव्हा मी त्याला 2 ने 1 ने गुणाकार करतो तेव्हा कुठे? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "हे मला 2 ते 0 मिळते आणि हे आमच्या अधिवेशनाचे औचित्य आहे की नकारात्मक घातांक 1 अर्ध्यासारखे दिसतात परंतु येथे खरोखर काय चालले आहे ते असे आहे की आम्ही असे म्हणत आहोत की हे जे काही आहे ते काही प्रकारचे कार्य असावे जे या गुणधर्माचे समाधान करते. a प्लस b हे b च्या f गुणिले f च्या बरोबरीचे आहे आणि शिवाय बेस 2 आहे ही वस्तुस्थिती मुळात आपल्याला सांगते की हे फक्त असे कोणतेही फंक्शन नाही हे एक फंक्शन आहे जिथे आपण 1 प्लग इन करतो तेव्हा आपल्याला 2 मिळतात आणि आपल्याला थोडेसे माहित आहे येथे काही परिणामांसह तुम्ही फॉलो करत आहात का हे पाहण्यासाठी सॅनिटी चेक स्टाईल प्रश्न मला तुम्हाला विचारायचे आहे की मी याला सॉफ्टबॉल असे काय म्हणणार नाही, परंतु हे आश्चर्यकारकपणे खोल प्रश्नासारखे आहे असे नाही. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "अपरिहार्यपणे आपण फंक्शनच्या गुणधर्मांसह अमूर्तपणे प्रारंभ करण्याच्या कल्पनेसह अनुसरण करत असल्यास आणि नंतर त्या गुणधर्मांच्या आधारावर आपण ते लिहू इच्छित असल्‍याचे प्रकार कमी करण्‍याच्या कल्पनेसह फॉलो करत असाल तर हे अधिक तपासण्यासारखे आहे जर x ची f ही घातांकीय गुणधर्म f समाधानी असेल तर सर्व इनपुट्ससाठी a प्लस b च्या f च्या गुणा f च्या b च्या बरोबरीचे आहे आणि ते f च्या 1 च्या बरोबरी 2 चे समाधान करते 2 खालीलपैकी कोणते खरे आहे असे म्हणायचे आहे की आपण असे कोणते कार्य सुरू करत आहात हे महत्त्वाचे नाही सोबत आणि तुमच्यापैकी ज्यांना ते कोणते व्याख्यान आठवते ते कोणते व्याख्यान आहे हे आम्ही जे बोलतोय ते यूलरचे सूत्र काय आहे याचा अर्थ कसा लावायचा याबद्दल मी बोलत होतो मी या शैलीचा एक प्रश्न विचारला जिथे मी एका अटीकडे दुर्लक्ष केले, तुम्हाला माहिती आहे की मी ते लिहिले नाही आपल्याला हे सुनिश्चित करायचे आहे की x चे सर्वत्र शून्य शून्य आहे आणि त्यामुळे काही प्रमाणात गोंधळ निर्माण झाला जो छान आहे स्क्रीनवर गोंधळ होतो जो आपल्या सर्वांच्या बाबतीत घडतो परंतु त्याचा हेतू मुळात हे दर्शविण्याचा होता की हा अमूर्त गुणधर्म एखादी गोष्ट जी जोडून गुणाकारात बदलते ते मूलत: तुम्हाला फंक्शन लिहिण्याची इच्छा निर्माण करण्यासाठी पुरेसे आहे जसे की ते एखाद्या प्रकारच्या शक्तीसाठी उठवलेले आहे. ते येथे पॉप अप झाले आहे असे दिसते जे मागील वेळेशी जोडलेले आहे, चला पॉवर टॉवर प्रश्नावर क्षणभर थांबू या जेणेकरून प्रथम आपल्याला असे समजू शकेल की येथे घातांकाचा अर्थ काय असावा? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "कारण मला जे म्हणायचे आहे ते आम्ही असू शकतो कारण आम्ही त्याचे उत्तर वेगवेगळ्या प्रकारे देऊ शकतो, म्हणून जर तुम्ही मला फक्त एक दिला, तर आम्ही पॉवर टॉवर्सबद्दल बोलू आणि नंतर लॉगरिदमिक स्केलमध्ये संख्या रेषा दर्शविली जाऊ शकते. जटिल विमानासाठीही असेच केले जाते? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "होय, खरं तर, एक व्हिज्युअलायझेशन आहे जे मी येथे फक्त एका क्षणात मिळवणार आहे जिथे आपण त्याच्यासारखेच काहीतरी करू कारण आपण काय करणार आहोत ते वेगवेगळ्या एक्सपोनेन्शिअल फंक्शन्स X च्या R गुणा X सह खेळू पण आपण आहोत R चे मूल्य बदलणार आहे जे थोडे पिवळे ठिपके द्वारे दर्शविले जाणार आहे म्हणून आपण याद्वारे चर्चा करू हे संपूर्ण विमानाचा नकाशा बनवणार नाही, तर वास्तविक अक्ष आणि काल्पनिक अक्षातून फक्त दोन नमुना बिंदू. पण कल्पना अशी आहे की जेव्हा आपण ते स्थिरांक काय आहे त्याभोवती फिरत राहिलो तेव्हा आपण त्या विमानात केलेल्या विविध गोष्टींची कल्पना करू शकतो आणि परिणामकारकपणे असे आहे की ते x-अक्षाचे लॉगरिदमिक स्केलमध्ये रूपांतर करत आहे आणि नंतर गुंडाळत आहे. वर्तुळाच्या बाजूने काल्पनिक अक्ष आणि मग R चे मूल्य काल्पनिक होताच ते त्या वास्तविक संख्यांच्या भूमिकेची अदलाबदल करते वर्तुळावर ठेवले जाते आणि काल्पनिक संख्या लॉगरिदमिक स्केल केलेल्या सकारात्मक अक्षावर ठेवल्या जातात त्यामुळे तिन्ही प्रश्न मला वाटतात मला जिथे जायचे आहे तिथे बंदुक उडी मारत आहे पण लोक या बाबतीत असा विचार करत आहेत हे पाहून आनंद झाला. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "स्पष्टपणे 5 चा f सारखी गोष्ट 1 अधिक 1 अधिक 1 अधिक 1 अधिक 1 अधिक 1 सारखीच गोष्ट आहे जी 1 च्या f सारखीच गोष्ट आहे 5 पटीने गुणाकार केला आहे कारण या गुणधर्मामुळे जे f 1 चा 2 असेल तर समान आहे 2 ते घात 5 आणि नंतर ऋण 5 चे f सारखे काहीतरी असे असावे की जेव्हा आपण त्याला 5 च्या f ने गुणतो तेव्हा आपल्याला 0 चा f जे काही आहे ते मिळते आणि 0 चा f काय आहे हे लगेच स्पष्ट होत नाही परंतु आपण असे म्हणू शकतो 1 अधिक 0 चा f 1 च्या बरोबरीचा असेल तर f 0 च्या कितीही पट असेल पण 1 चा f 2 च्या बरोबरीचा आहे आणि म्हणून हे 2 च्या बरोबरीचे आहे म्हणून आपण म्हणत आहोत 2 च्या 2 पट बरोबर काहीतरी आहे एक 1 असणे आवश्यक आहे म्हणून या संदर्भात हे हमी देते की ऋण 5 चा f 2 ते ऋण 5 ते 1 ओव्हर 2 ते 5 वी आम्ही हे स्पष्टपणे 2 ते ऋण 5 असे लिहू शकतो जे म्हणायचे आहे की हे दोन गुणधर्म मिळून बनतात. आम्हाला खरंच फंक्शन 2 ते X असे लिहायचे आहे कारण आम्ही त्यात ठेवलेल्या कोणत्याही मोजणीच्या संख्येचे समाधान होईल आणि ते स्वतःच गुणाकार केल्यासारखे दिसेल. जे आम्हाला हवे होते आणि तुम्हाला आश्चर्य वाटेल की ते अनन्य आहे आणि वास्तविक मूल्यवान फंक्शन्सच्या संदर्भात ते प्रत्यक्षात असेल परंतु जटिल मूल्यवान फंक्शन्सच्या संदर्भात अशी अनेक फंक्शन्स असतील जी आम्ही यासाठी लिहू शकतो ज्यापैकी आम्ही काय आहोत. 2 अधिक 2 pi च्या नैसर्गिक लॉगचे एक्सप म्हणून फंक्शन परिभाषित केले जाऊ शकते ते आधी पहा, मी त्या वेळेस X ठीक आहे, येथे आळशीपणा माफ करा, मला याबद्दल लिहिण्यास आनंद होतो आणि हे खरं तर एक वेगळे कार्य आहे जर तुम्ही X बरोबर 1 अर्धा प्लग इन केला तर काय होते याचा पुरावा आम्ही थोडे आधी पाहिले आहे की तुम्ही 1 अर्धा प्लग इन करता तेव्हा तुम्हाला 2 चे ऋण वर्गमूळ मिळते आणि नंतर तुम्ही 1 चौथ्यामध्ये प्लग केल्यास तुम्हाला त्याचे चौथे मूळ मिळत नाही 2 पण मी 2 च्या चौथ्या मूळच्या गुणाकार करतो म्हणून ते एक वेगळे कार्य आहे परंतु तरीही ते या गुणधर्मांचे समाधान करते आणि यामुळे आपल्याला ते 2 ते X असे लिहायचे आहे आणि हे असे सुचवते की कदाचित 2 ते X हे संदिग्ध आहे. bit of notation आणि आम्ही फक्त R times च्या exp नुसार सर्वकाही लिहायला हवे पण तुम्हाला आश्चर्य वाटेल कदाचित तुम्हाला माहित असेल की या गुणधर्माचे समाधान करणार्‍या सर्व फंक्शन्ससह आम्ही पुरेसे सर्जनशील नाही आहोत कदाचित आम्ही जेव्हा exp लिहितो तेव्हा कदाचित एक संदिग्धता असेल R च्या वेळा काहीतरी आणि R ची भिन्न मूल्ये आहेत जी प्रत्यक्षात येऊ शकतात परंतु मी फक्त एक छोटासा दावा करणार आहे आणि नंतर कदाचित तुम्हाला हवे असल्यास पुरावा कसा दिसेल याचे स्केच देऊ. म्हणा तुमच्याकडे काही जटिल फंक्शन F आहे, आणि ते प्रथम खालील गुणधर्मांचे समाधान करते तुम्ही त्याचे डेरिव्हेटिव्ह घेण्यास सक्षम आहात. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "हे वेगळे करण्यायोग्य आहे जे तुम्हाला पूर्णपणे गोंधळलेली अखंड गोष्ट माहित असण्यापासून ते ठेवते. ते काही यादृच्छिक मूल्ये घेण्यासारखे आहे ज्यावर तुम्हाला वेक्टर स्पेसचा कालावधी माहित आहे, मला माहित नाही की तुम्हांला विलक्षण मार्गांनी विचार करायचा असेल अशा अपूर्णांकाची रक्कम माहित नाही. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "हे एक छान कार्य आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "ते वेगळे करण्यायोग्य आहे ते सर्वत्र 0 च्या बरोबरीचे नाही त्यामुळे माझ्या मनाची स्थिती घसरली आणि मी कोणत्या व्याख्यानासाठी किंवा असे काहीतरी विसरलो आणि नंतर त्याचा हा मध्यवर्ती गुणधर्म आहे की ते गुणाकारात जोडते जर तुमच्याकडे असे कार्य असेल तर मी दावा करतो की तेथे एक अद्वितीय आहे कदाचित मी खरोखर निर्दिष्ट करू शकतो की तेथे एक अद्वितीय कॉम्प्लेक्स क्रमांक R आहे जेणेकरुन तुम्ही X चे F लिहू शकाल मूळत: R च्या गुणाकाराचे हे घातांक फंक्शन X च्या गुणाप्रमाणे आहे जे तुम्हाला माहित आहे की तुमच्याकडे X हे फंक्शन असेल तर हे छान व्युत्पन्न गुणधर्मांसह अनंत बहुपदी आणि हे सर्व तुमच्याकडे असेल तर तुमच्याकडे प्रत्येक घातांक आहे जे तुम्हाला घातांक शब्दाच्या अगदी अमूर्त जेनेरिक अर्थाने हवे आहे आणि त्या गुणधर्मावर आधारित आहे आणि पुराव्याचे स्केच. असे काहीतरी पहा जर तुम्हाला प्रथम या मूल्याचे व्युत्पन्न काय आहे हे पहायचे असेल जे आम्ही गृहीत धरत आहोत की सर्वत्र अस्तित्वात आहे, बरोबर? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "आपण संपूर्णपणे अभिव्यक्तीतून X चा F घटक काढू शकतो आणि संपूर्ण मर्यादा केवळ H च्या संदर्भात व्यक्त केली जाते ज्याचा अर्थ डेरिव्हेटिव्ह्जच्या संदर्भात काय आहे याचा विचार केल्यास आणि 0 चा F आवश्यकतेने 1 च्या बरोबरीचा आहे ही वस्तुस्थिती ही संपूर्ण मर्यादित अभिव्यक्ती आहे. फक्त काही स्थिर पण विशेष म्हणजे ० वरील आमच्या फंक्शनचे डेरिव्हेटिव्ह जे काही आहे ते आहे म्हणून तुमच्याकडे ही मजेदार गोष्ट आहे जिथे तुम्हाला 0 वर त्याचे डेरिव्हेटिव्ह माहित असल्यास ते सर्वत्र त्याचे व्युत्पन्न काय आहे हे ठरवते आणि घातांकीय कार्यांच्या संदर्भात हे आशेने परिचित आहे कारण घातांकीय फंक्शनचे व्युत्पन्न जे काही आपण म्हणत आहोत ते स्वतःचे प्रमाण आहे आणि ते प्रमाण स्थिरता 0 वर जे काही व्युत्पन्न आहे त्याच्या बरोबरीचे आहे हे सर्व अगदी अमूर्तपणे शब्दबद्ध केले आहे आणि असे आहे परंतु त्याचा हेतू यावर जोर देणे आहे की ते आहे आवश्यक नाही फक्त फंक्शन्स ज्यांचा आपण आधीच पॉवर X म्हणून विचार करतो परंतु हे फंक्शन्सचा एक संभाव्य अधिक विस्तृत वर्ग आहे जो केवळ या अमूर्त गुणधर्माला गुणाकारात रूपांतरित करण्याच्या अमूर्त गुणधर्माचे समाधान करतो परंतु जर तुमच्याकडे असेल तर ते हमी देते की तुमच्याकडे देखील आहे सेकंड डेरिव्हेटिव्ह आणि त्या बाबतीत एक तिसरा व्युत्पन्न आणि जसे की व्युत्पन्न फंक्शन स्वतःचे प्रमाण आहे म्हणून nवे व्युत्पन्न घेण्यासाठी तुम्ही फक्त ते प्रमाणिकता स्थिरांक पहा आणि त्यास n ची घात करा आणि मग येथून तुम्ही एक करू शकता. टेलर मालिकेचा विस्तार आणि तुमच्यापैकी ज्यांना त्या कल्पनेतील टेलर मालिकेसाठी सोयीस्कर आहे त्यांच्यासाठी मी कदाचित प्रगत गृहपाठ म्हणून सोडू शकतो, विशेषत: जर तुम्हाला जटिल संख्यांच्या अर्थाने भिन्नता असलेल्या कोणत्याही भिन्न कार्याची कल्पना एकत्र करायची असेल तर निश्चितपणे महाविद्यालयीन विषयाची क्रमवारी तुम्हाला माहिती आहे की तुम्ही तुमच्या इच्छेनुसार युक्तिवाद तेथे मिसळू शकता परंतु ज्याला फक्त टेलर मालिकेबद्दल माहिती आहे अशा व्यक्तीच्या संदर्भात अस्पष्ट तर्काला परवानगी आहे आणि ही कल्पना घेण्यासाठी आणि टेलरच्या विस्ताराकडे पहा आणि F आणि या कल्पनेचे समर्थन करण्यासाठी की एक अद्वितीय जटिल संख्या आहे जसे की आपले कार्य F असे लिहिले जाऊ शकते आणि नंतर सामान्य घातांकाशी जोडणे म्हणजे जेव्हा जेव्हा आपल्याकडे असे मूल्य असते तेव्हा आपण वास्तविक संख्यांच्या जटिल संदर्भात जे करतो ते आपण करतो जर तुम्ही त्या मूल्याच्या R च्या फंक्शनचा exp बघितला आणि त्याला आधार म्हणून लिहा. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "आम्ही याचा अर्थ लावू शकतो याचा अर्थ केवळ pi हाल्व्ह I गुणा X चा exp असाच नाही तर 5 pi हाल्व्ह I टाइम्स X चा exp असा अर्थ देखील लावू शकतो आणि ही स्वतंत्र फंक्शन्स आहेत आणि स्वतंत्र फंक्शन्सचे अनंत कुटुंब आहे जे आपल्याला असे वाटते. त्यांना मी X वर I असे लिहा त्यामुळे I to the अभिव्यक्ती जोपर्यंत तुम्ही एक मानक स्वीकारत नाही तोपर्यंत त्याचा अर्थ काय आहे हे तुम्ही म्हणता तेव्हा त्यात अमर्यादपणे बरेच आउटपुट आहेत याचा विचार करण्याचा दुसरा मार्ग म्हणजे फंक्शन I ते X आमच्याकडे असलेले नोटेशन थोडेसे संदिग्ध आहे आता या सर्वांसह चला यापैकी काही दृश्यमान करणे सुरू करूया कारण मला वाटते की ते मजेदार आहे आणि तुम्हाला माहिती आहे की हे एक उपयुक्त व्हिज्युअल आहे की अधिक गोंधळात टाकणारे दृश्य आहे का ते तुम्ही मला सांगा. आपण R टाइम्स X चे हे फंक्शन एक्सप पाहणार आहोत, जे मुळात X च्या पॉवरवर e लिहिण्याचा हा आणखी एक मार्ग आहे खरेतर मला वाटते की मला वाटते की मी काही ठिकाणी वेगळे अॅनिमेशन रेंडर केले आहे ज्याने ते निर्दिष्ट केले आहे कारण मी ते करण्याचे प्लॅनिंग करण्याचा विचार करत होतो त्यामुळे मला द्या, अरे हो, तुम्ही माझ्या फाईल सिस्टीममध्ये परत आला आहात जिथे तुम्ही असायला हवे होते तिथे परत जा. अरे बदला ते दुसर्‍या स्क्रीनवर दिसते थांबा का हो, ठीक आहे बदला? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "तुम्हाला जे काही दिसते ते तिथे ठेवा आणि आता आम्ही परत जाऊया आम्ही ते सर्व ते सर्व फक्त यासाठी की मी छान लिहू शकलो असतो जर तुम्हाला R टाइम्स X या अनंत बहुपदी या अनंत बहुपदाचा शेवटचा विचार करण्यास अस्वस्थ वाटत असेल तर तुमच्या डोक्याच्या मागील बाजूस आर टाइम्स X पर्यंत आणि आम्ही R च्या आसपास बदलणार आहोत म्हणून मी काल्पनिक अक्षाच्या बिंदूंचे अनुसरण करणार आहे आणि मी वास्तविक अक्षाच्या बिंदूंचे अनुसरण करणार आहे आणि हे काय करते ते पाहूया हा सर्व प्रकारचा वेगवान आहे म्हणून मला त्यावर थोडा हळू विचार करू द्या सर्व ऋण संख्या काहीही आहे ही ऋण वास्तविक संख्या 0 आणि 1 मधील श्रेणीमध्ये घुसली जाणार आहे, ज्याचा अर्थ ई ते नकारात्मक आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a ते ऋण वास्तविक संख्या 0 आणि 1 मधील काहीतरी आहे आणि आम्ही विशेषत: नकारात्मक 1 चा f ट्रॅक करत आहोत जे 1 ओव्हर e 30 0 च्या आसपास दिसत आहे. 1 चा 37 f अपेक्षेप्रमाणे e वर उतरतो म्हणजे 1 चा exp म्हणजे मी एकक वर्तुळाभोवती एक रेडियन उतरवणार आहे, आणि काल्पनिक अक्ष वर्तुळाभोवती कसा गुंडाळला जातो हे येथे संपूर्ण काल्पनिक अक्षासह अनुसरण करणे एक प्रकारची मजा आहे. आणि जेव्हा आपण R चे हे मूल्य बदलतो तेव्हा काय होते? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "आपल्याला येथे R ची मूल्ये हवी असतील आणि ती गोष्टी वेगळ्या पद्धतीने पसरवते म्हणून जेव्हा आपण ते 2 पर्यंत ठेवतो तेव्हा आपल्याला माहित आहे की तो खरा अक्ष खूप जास्त पसरतो जेणेकरून f चा 1 जेथे e स्क्वेअर नकारात्मकच्या 7 f वर थोडा आहे त्याभोवती संपतो. 1 हे I च्या 0 f च्या खूप जवळ आहे एक 2 रेडियन परिभ्रमण आहे f च्या वर्तुळाभोवती ऋण I एक ऋण 2 रेडियन रोटेशन आहे आणि अर्थातच आपण आपले आवडते सूत्र मिळवू शकतो की जर आपल्या स्केलिंग स्थिरांक म्हणून पाई असेल तर मग वास्तविक अक्ष खूप पसरलेला आहे तुम्हाला माहिती आहे की 1 चा f हा e वर pi च्या अगदी जवळ बसलेला आहे जो 20 अधिक pi च्या अगदी जवळ आहे जो नेहमीच मजेदार असतो आणि f ऋण 1 चा 0 च्या अगदी जवळ असतो त्यामुळे तो खराखुरा पसरलेला आहे axis आणि ते एकक वर्तुळाच्या दिशेने गोष्टी देखील पसरवल्या आहेत जेणेकरुन i चा f किंवा ऋणाचा f वर जाणे I वर्तुळाच्या अर्ध्या रस्त्याने फिरते, त्यामुळे आता सर्व काही चांगले आणि चांगले आहे अशा कार्याबद्दल आपण कसे विचार करू? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "आम्ही 2 गुणिले X च्या नैसर्गिक लॉगच्या X चा X म्हणून देखील लिहू जेणेकरून आम्ही R चे मूल्य 0 च्या आसपास दर्शविणारा पिवळा बिंदू हलवू. 69 अजूनही काल्पनिक भाग नाही फक्त वास्तविक संख्या 0.69 किंवा त्यामुळे 2 चा नैसर्गिक लॉग आहे तुम्ही पाहू शकता की 1 चा f 2 वर येतो आणि म्हणूनच आम्ही या फंक्शनला 2 ते X f च्या 1 अर्ध्या भागावर खरं तर क्षमस्व f च्या नकारात्मक 1 जमिनीच्या 1 अर्ध्या f वर म्हणू इच्छितो. मी एकक वर्तुळाच्या भोवती फिरत आहे विशेषतः ते 0 असेल. एकक वर्तुळाभोवती 69 रेडियन आहेत आणि आता आपण थोडे अधिक मजा करू शकतो आणि म्हणू शकतो की आपण हे 0 ऐवजी बदलले तर काय होईल. 2 चा नैसर्गिक लॉग असण्याऐवजी 69 हे 2 च्या नैसर्गिक लॉगच्या पटीने बनवा जेणेकरुन आपण खरोखरच अशा गोष्टीचा विचार करत आहोत ज्याचा घातांक आधार असेल. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "या प्रकरणात मी पॉवर काय आहे I ते 0 च्या आसपास हलवते. पाचव्याच्या आसपास 2 पण अनेक भिन्न घातांकीय फंक्शन्स आहेत ज्यात 1 चा f हा क्रमांक I वर ठेवण्याचा गुणधर्म असेल, म्हणून जर आपण ते आणखी वाढवायचे असेल तर मला वाटत नाही की ते येथे अॅनिमेटेड आहे परंतु जर आपण घ्यायचे असेल तर तो पिवळा बिंदू आणि तो वर करा जोपर्यंत तो 5 अर्ध्या गुणा पाई पर्यंत पोहोचतो I तुम्हाला एकक वर्तुळ काय दिसेल? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "स्वतःभोवती फिरवले जाते जेणेकरून 1 च्या नकारात्मक f चा f हा आणखी 2 pi रेडियन्सभोवती फिरेल आणि तो जिथे आहे तिथे उतरेल पण तो खरा अक्ष खूप जास्त पसरेल ज्या अर्थाने I ते I चे दुसरे आउटपुट आहे. खूप लहान संख्या ती 0 च्या आसपास होती. 0003 किंवा असे पण आपण हे देखील पाहू शकतो की मला जे खूप मजेदार वाटते ते आपण 2 ते पॉवर X बरोबर अर्थ लावू इच्छित असलेल्या वैकल्पिक अभिव्यक्तींचा विचार केल्यास काय होईल? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "आमच्याकडे R चा X गुणिले X आहे आणि R हे या मूल्याच्या बरोबरीचे आहे, जे 2 अधिक pi गुणा I चा नैसर्गिक लॉग आहे याचा अर्थ असा आहे की आपण प्लग इन करतो तेव्हा 1 मधील 1 f ऋण 2 वर असतो त्यामुळे आपल्याला हे फंक्शन लिहायचे आहे. पॉवर X साठी ऋण 2 बरोबर आहे आणि हे तुम्हाला माहीत आहे, हे थोडेसे फसवे सोपे आहे जेव्हा आपण पॉवर नेगेटिव्ह 2 पॉवर X ला ऋण संख्या लिहितो तेव्हा ते प्रथमतः असे दिसत नाही की ते आपल्याला आणते. कोणत्याही प्रकारे जटिल संख्यांमध्ये परंतु अर्थातच जेव्हा आपण 1 अर्धा सारखे मूल्य प्लग इन करतो जेथे आपण ऋण 2 चे वर्गमूळ विचारत असतो तेव्हा आपल्याला समजते की आपल्याला हे वर्गमूळाच्या गुणाप्रमाणे लिहायचे आहे. 2 चे 2 परंतु जर तुम्ही हे फंक्शन नकारात्मक 2 ते पॉवर X या संपूर्ण कॉम्प्लेक्स डोमेनमध्ये पाहत असाल तर ते कार्य करत आहे जे तुम्ही पाहत आहात ते फंक्शन आहे जे 1 ते ऋण 2 चे मूल्य घेते आणि जर ते केले तर काय होईल बाकीच्या वास्तविक संख्या रेषेशी ते बाहेरच्या दिशेने फिरते का? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "तर आपण पाहतो की ऋण 1 चा f हा ऋण 1 अर्ध्यावर बसतो, जर तुम्ही 1 अर्ध्या भागाच्या f चे अनुसरण केले तर ते काल्पनिक रेषेवर बसेल आणि 1 अर्ध्याचा f हा 2 चे वर्गमूळ असेल तर तुम्ही कुठे अपेक्षा कराल. मला पाहिजे तिथे उंदीर नाही. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "हे 2 पट I च्या वर्गमूळाच्या आसपास असेल आणि तुम्ही पुढे पुढे चालू ठेवत असताना तुम्हाला ऋण 2 ते X च्या सर्व वास्तविक मूल्य शक्ती दाखवत आहे, ते आवश्यकतेनुसार सर्पिल होते परंतु आम्ही आमचे R चे मूल्य आणखी वाढवू शकतो आणि ते मिळवू शकतो. सुमारे tau वेळा पर्यंत I सुमारे सहा गुण दोन आठ वेळा I आणि त्या संदर्भात हे आणखी एक कार्य आहे जे आपल्याला 2 ते X असे काहीतरी लिहायचे आहे कारण कोणत्याही पूर्ण संख्येसाठी पूर्ण संख्येसाठी ते X साठी प्लग इन केले जाईल. पुनरावृत्तीच्या गुणाकारांसारखे दिसते आणि 1 अर्ध्या सारख्या गोष्टींसाठी वाजवी मूल्ये देखील आहेत जिथे ते सकारात्मक वर्गमूळ ऐवजी नकारात्मक वर्गमूळ बाहेर टाकते, परंतु प्रत्यक्षात ते जे करत आहे ते एक समतल रूपांतर आहे जिथे ते सर्वकाही ठेवते ते वास्तविक आहे संख्या रेषा एक अतिशय घट्ट जखमेच्या सर्पिल बनून जाते जी भोवती फिरते आणि ती फक्त अशा प्रकारे सर्पिल होते की 1 चा f क्रमांक 2 वर येतो त्यामुळे या अर्थाने आपण 2 ला X असे म्हणू शकतो असे समजले जाते. एक वेगळे घातांक फंक्शन ज्याची आपल्याला पारंपारिकपणे सवय आहे, त्यामुळे मला वाटते की त्या सर्व गोष्टींसह मी आजसाठी गोष्टी सोडेन आणि मी तुम्हाला काही रेंगाळणारे प्रश्न सोडेन जे काही ठीक आहे याचा विचार करा, त्यामुळे तुम्हाला हवे असल्यास I to the I ही एक बहु-मौल्यवान अभिव्यक्ती आहे असा विचार करा, तुम्ही असे म्हणू शकता की आम्ही एक अधिवेशन स्वीकारले आहे, तुम्ही असे म्हणू शकता की तुम्ही नैसर्गिक लॉगॅरिथम फंक्शनची एक शाखा निवडाल आणि कदाचित ते तुम्हाला नकारात्मक पाईमध्ये अडकवेल. halves पण जर तुम्ही म्हणाल की या प्रकारची असीम अनेक भिन्न मूल्ये हवी आहेत जसे की आपण पाहिलेल्या विविध मूल्यांमध्ये 2 ते 1 तृतीयांश किती मूल्ये एकाच अर्थाने हवी आहेत? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "10वीला सर्व पैकी वेगळ्या पद्धतीने वाक्यरचना करायची आहे, मला X ची सर्व घातांकीय फंक्शन्स सांगू द्या जी ओह पूर्ण करतात, हे मी कुठेतरी X ची f ची पूर्तता करणारी फंक्शन्स लिहून ठेवली आहे जी मी लिहिली आहे की ते सर्व समाधानी असेल तर यापैकी आणि जर 1 चा f 2 च्या बरोबर असेल तर कोणत्या फंक्शनसाठी विविध पर्यायांसाठी X बरोबर 3 10वा प्लग इन केल्यावर आपल्याला किती भिन्न आउटपुट मिळतील? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "2 ते पाई साठी विविध फंक्शन्ससाठी जे 2 ते X दर्शवू शकतात जर आपण 2 ते X या प्रकारच्या अमूर्त गुणधर्मांच्या अर्थाने घातांकीय कार्य म्हणून विचार करत आहोत आणि जर आपण होय, जर आपण जर आमच्याकडे अशा प्रकारच्या विविध फंक्शन्सचा वर्ग आहे, आणि आम्हाला pi प्लग इन करायचे आहे ते मला हसायला लावते कारण मला असे एक मजेदार उत्तर माहित आहे जे तुम्ही त्याबद्दल विचार करण्याचा प्रयत्न करत असताना पॉप आउट होईल त्यामुळे ते प्रश्न आहेत मी तुम्हाला सोडून देईन आणि मला वाटते की तुम्हाला हे माहित आहे की आजच्या व्याख्यानाला भेट देताना माझा मुख्य प्रश्न होता की घातांकीय फंक्शन्सच्या या अमूर्त गुणधर्मांसारखे वर्णन करायचे आहे का आणि त्या अमूर्त गुणधर्मांपासून सुरुवात करणे माझ्यासाठी छान आहे. तुम्ही e to the rx किंवा त्याहून अधिक कल्पनेत अडकता फक्त तुम्हाला माहित आहे की r च्या वेगवेगळ्या मूल्यांसाठी r गुणा x चा अधिक प्रामाणिकपणे exp लिहिला आहे की ते तुम्हाला तितक्या दूर लॉक करते परंतु ते तुम्हाला इतके लॉक करत नाही. 2 ची पॉवर x किती कमी असावी याविषयीची एक अस्पष्ट कल्पना I ची पॉवर x सारखी काहीशी कमी असली पाहिजे असा धोका अर्थातच असा आहे की काहीवेळा लोकांना अमूर्तता आवडत नाही आणि काहीवेळा ते सहज शक्य होत नाही पण जर ते जर तुम्हाला माहित असेल तर तुम्ही फक्त मला कळवा मला वाटते मला वाटते की पॉवर टॉवर्सचा समावेश करण्यासाठी या सर्व गोष्टींभोवती विचारांचे एक संपूर्ण मनोरंजक वर्तुळ आहे कारण जर तुम्हाला पॉवर टॉवर्सबद्दल खरोखर बोलायचे असेल तर आम्ही मागील वेळी जटिल संख्यांच्या संदर्भात बोललो होतो. किंवा अगदी नकारात्मक आधारांसहही तुम्हाला अशा गोष्टींचा विचार करावा लागेल, म्हणून हा प्रश्न पडद्यावर पडला होता, होय, जर आपण I टू पॉवर I साठी हे केले तर काय होईल? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "टायट्रेशन तुम्हाला माहित आहे चला फक्त हे करून पाहू या पुढे जा आणि पॉवर टॉवर वापरून पाहू या जिथे आपण मला दिलेल्या शक्तीवर वाढवत आहोत आणि त्यातून काय बाहेर पडते ते पहा, म्हणून हे करण्याचे नियोजन नव्हते परंतु आपण नेहमी करू शकतो पायथन वर खेचून घ्या आणि मूलत: आम्ही गेल्या वेळी जे करत होतो तेच करा त्यामुळे हे कार्य करण्याचा मार्ग म्हणजे आम्ही काही बेस व्हॅल्यूने सुरुवात करत होतो आणि नंतर काही प्रकारच्या श्रेणीसाठी आम्ही काय करत होतो आम्ही एक घेत होतो आणि आम्ही पुन्हा नियुक्त करणार आहोत हे जे काही असेल ते असेल जे बेस या प्रकरणात मी a च्या पॉवरवर वाढवले आहे ते ठीक आहे, छान आहे, म्हणून आपण ते करणार आहोत आणि नंतर आपण मूल्य प्रिंट करणार आहोत चला फक्त हे करूया होय, ही 200 सारखी खूप मोठी संख्या आहे त्यामुळे असे दिसते की काय होते ते काहीवेळा या गोष्टींमुळे गोंधळ होण्याची शक्यता आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "माझ्याकडे खरोखर आहे म्हणून मला NumPy आयात करू द्या म्हणून माझ्याकडे घातांकीय कार्य आहे मला आमच्या मोठ्या श्रेणीसाठी जाऊ द्या जसे आमच्याकडे आधी होते ते लिहिण्यापेक्षा ते लिहिण्यापेक्षा तुम्हाला माहित आहे की मला X च्या पॉवर सारखे काहीतरी आहे मी ते लिहिणार आहे भिन्न स्थिर उजव्याचे घातांक कार्य म्हणून भिन्न स्थिरांक जे मी बनवणार आहे ते मला 5 pi अर्धे करायचे आहे, म्हणून मी 5 pi अर्ध्या वेळा करीन त्यामुळे ती एक जटिल संख्या आहे आणि तिला 5 pi अर्धे आहेत काल्पनिक भाग तर हा 5 pi अर्धा गुण आहे आणि मी काय करत आहे? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/persian/sentence_translations.json b/2020/ldm-i-to-i/persian/sentence_translations.json index a276aa54b..0d57b5524 100644 --- a/2020/ldm-i-to-i/persian/sentence_translations.json +++ b/2020/ldm-i-to-i/persian/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "بنابراین اگر از عدد 1 شروع می کنید، سرعت اولیه شما این است که مستقیم به سمت 0 راه بروید و وقتی حتی پایین تر راه می روید، اگر در نیمه 1 نشسته بودید، همچنان به سمت 0 می رفتید، اما اکنون بردار سرعت شما منفی 1 برابر جایی که شما هستید، که منفی 1 نیمه است. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "شهود اصلی پشت آن و یک سوال جالب این است که بدانید آیا فقط یک تابع وجود دارد که نوشتن برای آن منطقی به نظر می رسد زیرا می دانید که اگر آن را به صورت i در x بنویسیم نه تنها باید این را برآورده کند، بلکه باید آن را نیز برآورده کند. شماره یک را که می‌گیریم i احتمالاً i به پاور وصل می‌کنیم، اما فکر می‌کنیم که این تابع باید i باشد. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "بنابراین ما نصف 5 پی و عالی داریم که کاملاً مقدار دیگری است که می‌توانیم آن را برای x در اینجا وصل کنیم و فقط اگر بخواهیم به دایره خود در اینجا نگاه کنیم، آن را کمی بیشتر به صورت بصری توضیح دهیم. لحظه راه رفتن برای مدت زمانی برابر با نصف پی که برابر با 1 است. 57 چه می شود اگر به جای آن یک چرخش کامل دیگر را انجام دهیم و به نصف های دیگری برویم تا به pi برسیم که می دانی ممکن است به نوعی رکورد کنیم، جایی که e تا مقدار پی i این است که نصف های دیگر پی را راه می رویم، نیمه های دیگری را پی می رویم که در در این نقطه یک دایره کامل می‌پیماییم و به یک برمی‌گشتیم و سپس برای پنج نیمه پی راه می‌رویم که عددی حدود ۷ است. 85 بله، این قطعاً عدد دیگری است که ما را در بالای i قرار می دهد و اگر بخواهیم کل ماجرای بیان مجدد i به توان i را با نوشتن e به 5 پی نصف i به توان i آن i طی کنیم. ضرب کنید تا منفی شود و ما به e تا نیمه های منفی 5 پی نگاه می کنیم که عدد بسیار متفاوتی است، درست است که می توانیم این را محاسبه کنیم. . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "85 واحد زمان که 5 پی است نصف می شود و ببینید چه اتفاقی می افتد وقتی که شما پوسیده شوید؟ این طولانی است که شما را به عدد بسیار کوچک‌تری می‌رساند اما این تنها پاسخی نیست که می‌توانیم درست وارد کنیم، ما افراد دیگری را داریم که با منفی 3 نیمه ضرب i pi وارد اینجا می‌شوند که شما از نظر دایره واحد می‌شناسید؟ می‌توانیم فکر کنیم اگر می‌خواهم به من برسم به جای اینکه 90 درجه رادیان رادیان را 90 درجه راه بروم هی بگوییم چه می‌شود اگر 270 درجه راه بروم و از طرف دیگر 3 پی رادیان را نصف می‌کنم که شاید منفی به نظر می‌رسد زیرا قرارداد این است. معمولاً خلاف جهت عقربه‌های ساعت مثبت است که کاملاً راه دیگری برای بیان آن است و اگر e به نیمه‌های 3 عدد منفی i داشته باشیم، پاسخ متفاوتی به ما می‌دهد. منفی که در حال حاضر وجود دارد، و ما یک نیمه 3 پی مثبت داریم و از نظر عددی این به ما یک پاسخ حتی متفاوت از آنچه قبلا داشتیم می دهد که اگر از آن عبور کنیم و بگوییم هی، e به 3 pi چیست نه 3 o 3 pi. نصف می کند 111 نقطه 3 1 نوع بسیار متفاوت از عددی که قبلاً می دیدیم 111 نقطه چه بود 111 امتیاز 3 1 عالی 111 امتیاز 3 1 یا بیشتر پویا اما ما در زمان به عقب حرکت می کنیم و می بینیم که چند وقت پیش در زمان باید چه چیزی باشم به طوری که اگر از آنجا چیزها را به جلو بازی می کردم روی شماره یک شرایط اولیه ام قرار می گرفتم و شما باید به زمان 3 واحد نیمه برگردید. ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "و سپس اگر بخواهید دینامیک فروپاشی را ترجمه کنید که این همان کاری است که بالا بردن چشم در این زمینه انجام می دهد، می گویید اگر از شماره یک شروع می کنم، اما می خواهم در زمان به عقب بروم و بگویم اگر باید از کجا شروع می کردم من می‌خواهم آنقدر زوال کنم که در رده اول قرار بگیرم؟ پس از 3 عدد پی نصف واحد زمان، پاسخ آشکارا از حدود صد و یازده برای آن نوع فروپاشی نمایی شروع می‌شود و می‌توانید ببینید که این به کجا می‌رود، جایی که در واقع بی‌نهایت مقادیر مختلف وجود دارد که می‌توانیم برای X وصل کنیم. فکر می کنم e تا X به عنوان من و مردم خیلی بیشتر وارد اینجا شده اند. ببخشید که پینم را روی زمین می اندازم، همانطور که کلاسیک برای مقام سوم انجام می شود. گزینه های مختلف بی نهایت مقادیر مختلف که در ابتدا کمی نگران کننده به نظر می رسد زیرا ما به عبارتی نگاه می کنیم که به نظر می رسد شما می دانید که فقط مقداری محاسبات وجود دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "ما زمانی که چندین گزینه مانند این وجود دارد، وقتی یک تابع چند ارزشی دارید، یک قرارداد را اتخاذ می کنیم. آن را به‌عنوان تابعی به‌عنوان چیزی با یک ورودی و یک خروجی در زبان شیک‌تر در نظر بگیرید، وقتی با اعداد مختلط سر و کار داریم، ایده چیزی به‌عنوان عملیاتی که به نوعی می‌خواهد چندین مقدار داشته باشد، مطرح می‌شود. ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "ریشه I و ما می خواهیم بدانیم که چه چیزی باید باشد؟ چون چندین پاسخ مختلف وجود دارد شما می دانید که ما دوباره به من فکر می کنیم این چرخش 90 درجه است و اگر به آن به عنوان یک چرخش 90 درجه فکر می کنیم، احساس می کنیم که ریشه مربع باید باشد. ریشه I که می‌توانیم خیلی صریح بنویسیم ریشه 2 روی 2 ریشه 2 روی 2 I این فقط از مثلثات استفاده می‌کند، اما اگر به جای I به عنوان یک چرخش منفی 270 درجه فکر می‌کردیم، به نظر می‌رسد نیمی از آن نیمی از آن عملیات را انجام می‌دهد. در واقع باید ما را به طرف دیگر بکشاند شاید عددی که اینجا نشسته است باید جذر I باشد و این در واقع فقط منفی چیزی است که قبل از آن دیدیم. ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "و من فکر می کنم شما خوب می گویید ما می دانیم این چیست، ما به نوعی تعریف می کنیم که جذر 2 باشد همه چیز خوب و خوب است، اما اگر من می گفتم بیایید به این موضوع همان گونه نزدیک شویم که I خود را به بیان I نزدیک می کردیم. می خواهم ابتدا چیزها را به صورت e به چیزی درست بیان کنم و سپس با ضرب نصف 1 در توان آن را به نصف 1 برسانم و می گویم خوب، می توانم حدس بزنم که می توانم آن e را با آنچه هست انجام دهم برابر با 2 خوب این log طبیعی 2 است. این یک ثابت است که حدود 0 است. 69 یا بیشتر اگر e را به آن توان برسانیم، 2 می‌گیریم، بنابراین می‌توانیم این را به عنوان e به گزارش طبیعی 2 برابر 1 نیمه در نظر بگیریم و اگر می‌خواهید اگر به e تا x فکر می‌کردید؟ می‌دانید که این ممکن است در زمینه اعداد واقعی به نوعی زیاده‌روی باشد، اما اگر از e به x به عنوان مخفف این تابع x فکر می‌کنید، می‌توانید مقدار 0 را وارد کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "69 ضربدر 1 نیمه که حدس می زنم حدود 0 باشد. 345 چیزی شبیه به آن شما آن مقدار بسیار مشخص را به چند جمله ای خود وصل می کنید ببینید چه چیزی را خروجی می دهد و حدود 1 خروجی خواهد داشت. 414 یک عدد واقعی زیبا از جذر 2 چه انتظاری دارید، اما اگر همان کاری را انجام دهیم که فقط با I انجام می‌دادیم و اذعان می‌کردیم که وقتی می‌خواهیم چیزی به‌عنوان e به یک توان بنویسیم، در واقع چندین پاسخ متفاوت وجود دارد، می‌توانیم این را نیز بنویسیم. این ممکن است خنده دار به نظر برسد، اما ما می توانیم آن را به صورت e در گزارش طبیعی 2 به علاوه 2 pi I بنویسیم که کل چیز به نصف 1 افزایش یابد، درست بعد از این همه این مقدار برابر با شما می شود، می توانید آن را به عنوان e تجزیه کنید. گزارش طبیعی 2 ضرب در e به 2 پی I این یکی فقط اثر چرخش 360 درجه اشیاء را دارد، بنابراین فقط برابر 1 می شود بنابراین ما به 2 ضربدر 1 عالی نگاه می کنیم که به نظر می رسد یک جایگزین معتبر است و هنوز وقتی ما همان بازی گرفتن این و بالا بردن آن به توان و درمان آن را انجام می دهیم که با ضرب توان در نمای نمایی که چه اتفاقی می افتد، ما e را به گزارش طبیعی 2 ضربدر 1 نصف به اضافه خوب، 2 پی I ضربدر 1 نیمه داریم. خوب که پی ضربدر I خواهد بود اکنون این قسمت اول e به گزارش طبیعی 2 برابر 1 نیمه می رسد که در نهایت ریشه مربع آشنای 2 خواهد بود که همه چیز خوب و خوب است، اما ما آن را در e ضرب می کنیم تا pi I درست و کاملاً معروف است که e نسبت به pi I منفی است 1 بنابراین در این مورد به نظر می رسد نشان می دهد که اگر این عبارت 2 تا 1 را حل کنیم با بازی کردن با پاسخ های مختلف می توانیم چیزی شبیه به e به X برابر با 1 نصف چیزی است که ما با آن به پایان می رسیم، پاسخ دیگری است که به طور سنتی ممکن است به عنوان این جذر منفی 2 بنویسیم و در اینجا منظورم این است که برای آن کمی خنده دار است که مقادیر متعددی داشته باشد و به نصف 2 تا 1 نگاه کند. بگوییم که مساوی نیست یک چیز اما بر اساس انتخاب هایی که انجام می دهیم می تواند برابر با چندین چیز متفاوت باشد اما دو چیز که می تواند کاملاً معقول به نظر برسد اگر قرار باشد چیزی وجود داشته باشد که 2 به 1 نیمه باشد، به نظر می رسد که یا باید مثبت باشد. ریشه مربعی که با آن آشنا هستیم یا نوع منفی آن که در واقع چنین مشکلی به نظر نمی رسد و در واقع ما می توانیم این بازی را حتی بیشتر از این هم انجام دهیم و اجازه دهید از شما پاسخ های خلاقانه تری برای این عبارت بپرسم. زیرا شاید بتوانیم قدرت های خنده دار دیگری از چیزی مانند 2 به توان X را پیدا کنیم در حالی که شروع به وصل کردن مقادیر مختلف X بر اساس جایگزینی که انجام می دهیم در صورتی که از قوانین مشابهی پیروی می کنیم که در ارزیابی I به آن استفاده می کردیم. power I بنابراین این بار سوال می پرسد یا مشخص می کند که یکی از راه حل های معادله e به x برابر با 2 است، عدد واقعی Log طبیعی 2 ok است که یک ما آن را می دانیم. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "پاسخ به سوال e به x برابر با 2 است و باز هم از خلاقیت استقبال می شود، بنابراین من یک لحظه دیگر را برای آن به شما می دهم. لزوماً نیاز به انجام ورودی ریاضی بسته به دستگاهی است که به آن نگاه می کنید، اما اگر قبل از این است که فرصت پیدا کنید به سؤالی که می خواهید به پاسخی که می خواهید پاسخ دهد، خیلی استرس نداشته باشید، بنابراین به نظر می رسد 131 نفر از شما گزینه ای را وارد کرده اید که در آن Ln از 2 را می گیریم و 2ii را اضافه می کنیم و فکر می کنم من در حال نوشتن این سؤال هستم به اشتباه یکی از پاسخ ها را به عنوان صحیح علامت گذاری کرده ام در حالی که در واقع چند پاسخ صحیح متفاوت وجود دارد بنابراین این به من مربوط است. برای این واقعیت که من نمی‌دانم برای هیچ‌کدام از شما مانند oh It's red به نظر می‌رسد، وقتی Ln 2 به اضافه 42 را وارد کردید اشتباه متوجه شدید. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi که البته یک انتخاب عالی است، اما شما همچنین می توانید چیزی مانند 4 pi I به اضافه ورود به سیستم طبیعی 2 یا 6 pi I یا واقعاً هر مضرب صحیحی از 2 pi I داشته باشید، اگر اضافه کنید که e را تحت تاثیر قرار نمی دهد. X چون فقط اثر ضرب در e به 2 پی I را دارد که این اثر ضرب در 1 است و باز هم این یک نتیجه خنده دار دارد که به نظر می رسد زمانی که آن را به عنوان مثال دیگری انجام می دهیم نتایج معقولی به دست می دهد. به نظر می رسد دومین عبارت رایج وارد شده در آنجا این بود که ما ممکن است 2 را جایگزین کنیم، بنابراین بیایید فکر کنیم به 2 به توان 1 4 فکر می کنیم، بسیار خوب، پیشنهادی وجود داشت که ما 2 را با e به گزارش طبیعی 2 به اضافه 4 جایگزین کنیم. pi I Okay Plus 4 pi I و ما همه اینها را تا 1 و 4 به درستی بالا می بریم اگر همان بازی را انجام دهید، e به گزارش طبیعی 2 ضربدر 1 4 می گیریم، و ما در e ضرب می کنیم تا pi I حالا اولین قسمت از آن، همان ریشه چهارم مثبت معمولی 2 خواهد بود، چیزی که وقتی یک عبارتی مانند ریشه چهارم از 2 را به یک ماشین حساب وصل می‌کنید، یک عدد مثبت کوچک خوب به آن وصل می‌کنیم، اما این قسمت دوم است. منفی 1 بنابراین به نظر می رسد می گویید می دانید اگر قرار بود 2 را به این شکل متفاوت تفسیر کنیم و آن را به 1 و 4 برسانیم. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "ما به نصف پی در من نگاه می‌کردیم و به جای ضرب در منفی 1، در عوض در I ضرب می‌شدیم که باز هم پاسخ معتبری است، به نظر می‌رسد خروجی معقولی برای چیزی حدود 2 تا 1 4 باشد، بنابراین وقتی شما با نگاه کردن به این واقعیت که من به قدرت، به نظر می رسد چندین مقدار متفاوت برای آن دارم، درست است که ما این پدیده خنده دار را داریم که می توانیم e را به نیمه های 5 پی متصل کنیم. چیزی فوق‌العاده کوچک چیزی فوق‌العاده بزرگ همه بسیار متفاوت از پاسخ 1 5 تقریباً 1 5 که قبلاً در اینجا یافتیم دقیقاً همان پدیده ای است که وقتی از چیزی مانند 2 تا 1 4 می پرسید و تصدیق می کنید که در واقع چندین راه حل مختلف وجود دارد. برای عبارت X به 4 برابر است با 2 4 راه حل مختلف در واقع و آنچه شما به آن نگاه می کنید این واقعیت است که چندین راه حل مختلف وجود دارد. 2 هر چه که ممکن است باشد و یکی از راه هایی که ممکن است در مورد آن فکر کنیم این است که وقتی با اعداد واقعی سر و کار دارید، چیزهای دوست داشتنی هستند، خوب هستند، روابط یک به یک وجود دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "شما نکات مثبتی دارید خیلی خوب است اگر بخواهیم در مورد توابع نمایی فکر کنیم، اجازه دهید برخی از این موارد را پوشش دهم. همان نمایی مانند X از R ضربدر X که می دانید چند جمله ای است که ما به آن اشاره می کنیم. و با فرض اینکه B یک عدد مثبت است یک پاسخ به شما می دهد و این همان چیزی است که می گوییم X از R برابر با B است بنابراین یکی از راه هایی که من قبلاً در این سری صحبت کردم این است که اگر شما به خانواده همه نمایی های ممکن درست است که می توانیم آنها را به عنوان X از R ضربدر X بنویسیم و R را تغییر دهیم و این دقیقاً همان چیزی است که e را به R ضربدر X بنویسیم اگر این چیزی است که شما با آن راحت تر هستید بنابراین e به R برابر XX از R ضربدر X همان چیزی است که ما می‌توانیم در مورد تغییر آن چیزی که هست فکر کنیم، اما از طرف دیگر، اگر به همه نمایی‌های ممکن به‌عنوان یک پایه فکر کنید، اجازه دهید پایه‌ای را به توان X انجام دهم و می‌رویم. تغییر دادن آن پایه در ابتدا به نظر می رسد که این یک نوع بیان متفاوت برای دستکاری است، اما این فقط روش دیگری برای بیان همان خانواده است و راهی است که ممکن است در مورد آن فکر کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "من می‌توانم مقدار R ضربدر X را داشته باشم که شاید R چیزی شبیه به نقطه صفر شش نه باشد، اما می‌توانم آن را با دو پی I به پایین جابجا کنم و این مبنایی را که با آن مطابقت دارد تغییر نمی‌کند که همچنان با دو مطابقت دارد یا می‌تواند آن را با دو پی I به سمت بالا تغییر دهید که پایه مربوط به آن را تغییر نمی دهد زیرا در همه این موارد وقتی X را برابر یک وصل می کنیم یک چیز را دریافت می کنیم، اما همه اینها برای مقادیر مختلف X توابع متمایز هستند. چرا ما چندین مقدار مختلف را برای I به توان I دیدیم زیرا I به X یک تابع مبهم در آن زمینه است، اگر تصمیم بگیریم کدام مقدار از R را تعیین کنیم به طوری که آنچه را که ما نشان می‌دهیم expe از R برابر X کدام مقدار باشد، بدون ابهام است. از R. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "آیا به محض انتخاب یکی را انتخاب می کنیم؟ این یک تابع بدون ابهام است، اما در آن نقطه احساس می‌کنیم شاید چیزی که می‌خواهیم این است که از فکر کردن به چیزها بر حسب یک پایه به توان X، شاید به محض اینکه در متن اعداد مختلط قرار گرفتیم، فقط باید بنویسیم. همه آنها به عنوان انقضای چند بار ثابت X هستند، اگر بدون هیچ دلیل دیگری واضح است چگونه ما واقعاً اعداد را وصل می کنیم اگر بخواهیم یک محاسبات انجام دهیم یا فقط ریاضی را در بالای آن انجام دهیم، این چند جمله ای بی نهایت خوب را داریم که ما آنها را به آن وصل کنید و من یک مورد دیگر برای شما درست خواهم کرد که این شاید راه درستی برای فکر کردن در مورد نمایی باشد به محض اینکه ما در حال گسترش مواردی مانند اعداد مختلط به دامنه های دیگر هستیم و برای آن، اجازه دهید فقط از Go نسخه پشتیبان تهیه کنیم. بازگشت به زنگ درب برخی چیزها به راه اصلی باز می گردند که ما ایده قدرت را گسترش می دهیم و فقط به چیزی فکر می کنیم که 2 به سمت راست X است که می دانیم چگونه در مورد این برای اعداد طبیعی فکر کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "شما چیزی شبیه 2 تا 3 ضرب مکرر را می دانید چگونه است که برای اولین بار به شما یاد می دهند که به چیزی مانند 2 تا X برای مقادیر کسری یا برای مقادیر منفی و چیزهایی از این قبیل فکر کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "معمولاً به شما آموزش داده می شود که 2 تا 1 نصف باید چیزی باشد که می دانید اگر آن را در خودش ضرب کنم و این از قوانین معمولی پیروی می کند که نمایی با شمارش اعداد انجام می دهد، جایی که ما می توانیم چیزهایی را در آن توان اضافه کنیم که باید 2 را بدست آوریم. به 1 بنابراین باید عددی باشد که وقتی آن را در خودش ضرب می کنم 2 می شود و می دانید که در آن نقطه یک انتخاب دارید، شاید مثبت باشد. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "شاید منفی باشد اما اگر همیشه تصمیم به انتخاب مثبت داشته باشید، اگر در مورد اعداد منفی بپرسیم 2 تا منفی 1 چه چیزی باید باشد، می توانید یک تابع پیوسته خوب از همین معامله دریافت کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "کجا وقتی آن را در 2 ضرب کنم در 1؟ من را از 2 به 0 می رساند و این نوعی توجیه برای قرارداد ما است که نماهای منفی شبیه 1 نیمه هستند اما آنچه واقعاً در اینجا اتفاق می افتد این است که ما می گوییم هر چیزی که باشد باید نوعی تابع باشد که این ویژگی f را برآورده کند. a به علاوه b برابر است با f ضربدر f از b و علاوه بر این، این واقعیت که پایه 2 است، اساساً به ما می گوید که این فقط هر تابعی از این قبیل نیست، تابعی است که وقتی 1 را وصل می کنیم، 2 می گیریم و به همان اندازه که می دانید سوال سبک بررسی سلامت عقل برای اینکه ببینم آیا شما به همراه برخی از مفاهیم در اینجا دنبال می کنید یا خیر، می خواهم از شما بپرسم که چیست من آن را مانند توپ سافت بال نمی نامم، اما این به معنای این نیست که مانند یک سوال فوق العاده عمیق باشد. لزوما. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "اگر شما ایده شروع انتزاعی با ویژگی های یک تابع و سپس نوعی استنتاج روش هایی که ممکن است بخواهیم آن را بر اساس آن ویژگی ها بنویسیم اگر f از x این ویژگی نمایی f را برآورده کند، فقط یک بررسی بیشتر است. از a به علاوه b برابر است با f ضربدر f از b برای همه ورودی ها و همچنین f از 1 برابر 2 را برآورده می کند که کدام یک از موارد زیر صحیح است، بدون توجه به اینکه کدام تابع را شروع می کنید، کدام یک از موارد زیر لزوما درست است. با و آنهایی از شما که به یاد دارید کدام سخنرانی بود، هر کدام از آنها صحبت می‌کردیم که چگونه فرمول اویلر را واقعاً تفسیر کنیم، من یک سوال از این سبک پرسیدم که در آن یک شرط را نادیده گرفتم، می‌دانید که یادداشت نکردم این واقعیت که ما می‌خواهیم مطمئن شویم که f از x در همه جا غیر صفر است و سپس باعث شد مقداری سردرگمی که جالب است، در صفحه نمایش سردرگمی شود که برای همه ما اتفاق می‌افتد، اما هدف آن اساساً نشان دادن این بود که این ویژگی انتزاعی چیزی که جمع را به ضرب تبدیل می کند کافی است که اساساً شما را وادار کند که تابع را به عنوان هر چیزی که برابر است به عنوان یک نوع قدرت بنویسید این روح سؤال است. به نظر می رسد که در اینجا ظاهر شده است که بسیار به دفعه قبل مرتبط است. ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "بیایید فقط یک لحظه در مورد سؤال برج برق دست نگه داریم تا ابتدا احساس عمیق تری داشته باشیم که منظور از قدرت در اینجا چیست؟ چون ما می‌توانیم همان چیزی باشیم که می‌خواهم ادعا کنم این است که می‌توانیم به چندین روش مختلف به آن پاسخ دهیم، بنابراین اگر فقط یکی را به من بدهید، در مورد دکل‌های برق صحبت خواهیم کرد و سپس همانطور که یک خط عددی را می‌توان در مقیاس لگاریتمی نشان داد. ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "بله، در واقع، یک تجسم وجود دارد که من فقط در یک لحظه در اینجا به آن می رسم، جایی که ما کاری کاملاً مشابه آن را انجام می دهیم، زیرا کاری که ما انجام خواهیم داد این است که با توابع نمایی مختلف X از R ضربدر X بازی کنیم. مقدار R را تغییر می دهیم که با یک نقطه زرد کوچک نشان داده می شود، بنابراین ما به نوعی در این مورد صحبت خواهیم کرد. قرار نیست کل صفحه را ترسیم کنیم، بلکه فقط چند نقطه نمونه از محور واقعی و محور خیالی را ترسیم خواهیم کرد. اما ایده این است که همانطور که در اطراف چیزی که ثابت است حرکت می کنیم، می توانیم کارهای مختلفی را که با هواپیما انجام می دهد تجسم کنیم و عملاً مانند این است که محور x را به یک مقیاس لگاریتمی تبدیل می کنیم و سپس آن را پیچیده می کنیم. محور خیالی در امتداد یک دایره و سپس به محض اینکه آن مقدار از R خیالی شد، نقش آن اعداد واقعی را روی دایره قرار می‌دهند و اعداد خیالی روی یک محور مثبت لگاریتمی قرار می‌گیرند، خیلی سؤال بزرگی است که هر سه آن را حدس می‌زنم. به نوعی اسلحه را برای جایی که می‌خواهم به جلو می‌پرند، اما خوشحالم که می‌بینم در این یکی اینجاست که مردم چنین فکری می‌کنند. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "بیایید ادامه دهیم و فقط آن را درجه بندی کنیم، ایده این است که این ویژگی f به اضافه B به شما اجازه می دهد چیزهای مختلف زیادی را صرفاً بر اساس آنچه f از 1 است بیان کنید و فقط آن را خیلی املا کنید؟ به طور صریح چیزی شبیه f از 5 همان چیزی است که f از 1 به اضافه 1 به علاوه 1 به علاوه 1 به علاوه 1 که همان چیزی است که f از 1 ضرب در خودش 5 برابر به دلیل این خاصیت که اگر f از 1 برابر 2 باشد. به عنوان 2 به توان 5 و سپس چیزی شبیه f از منفی 5 باید اینطور باشد که وقتی آن را در f در 5 ضرب می کنیم هر چه f از 0 باشد به دست می آوریم و بلافاصله مشخص نیست که f از 0 چیست اما می توانیم بگوییم که f از 1 به علاوه 0 برابر است با هر f از 1 برابر است با f از 0 است اما f از 1 برابر با 2 است و بنابراین این نیز برابر با 2 است بنابراین ما می گوییم 2 برابر است با 2 برابر چیزی خوب آن چیزی باید 1 باشد، بنابراین در این زمینه، این تضمین می کند که f از منفی 5 2 به منفی 5 است، 1 بر 2 به 5 است. ما واقعاً می خواهیم تابع را به صورت 2 در X بنویسیم زیرا هر عدد شمارشی که در آن قرار می دهیم این ویژگی ها را برآورده می کند. که ما می خواستیم و ممکن است تعجب کنید که منحصر به فرد است و در زمینه توابع با ارزش واقعی واقعاً خواهد بود، اما در زمینه توابع با ارزش پیچیده، چندین تابع از این قبیل وجود خواهد داشت که می توانیم برای این یکی بنویسیم که همان چیزی است که ما بودیم. نگاهی به قبل از جایی که می‌توانیم یک تابع تعریف شده داشته باشیم که اکسپت از گزارش طبیعی 2 بعلاوه 2 پیکسل باشد من تمام آن زمان‌ها X بسیار خوب، شلختگی اینجا را ببخشید، من فقط هیجان زده می‌شوم که در مورد این مطلب بنویسم و این در واقع یک تابع متفاوت است. نشان می دهد که اگر X را وصل کنید نصف یک نصف شود چه اتفاقی می افتد ما کمی قبل دیدیم که چگونه وقتی 1 نصف را وصل می کنید، آن چیزی که به دست می آورید جذر منفی 2 است و سپس اگر 1 چهارم را وصل کنید، ریشه چهارم را دریافت نمی کنید. 2 اما من ریشه چهارم 2 را ضرب می کنم بنابراین یک تابع متفاوت است اما همچنان این ویژگی ها را برآورده می کند و به نوعی باعث می شود که آن را به صورت 2 به X بنویسیم و این باعث می شود که شاید 2 به X یک مبهم باشد. کمی نماد و ما فقط باید همه چیز را بر حسب مقدار R برابر چیزی بنویسیم، اما ممکن است تعجب کنید خوب می دانید که شاید ما با تمام توابعی که این ویژگی را برآورده می کند به اندازه کافی خلاق نیستیم شاید وقتی expe را می نویسیم ابهامی وجود دارد از R برابر چیزی است و مقادیر متفاوتی از R وجود دارد که می تواند وارد عمل شود، اما من فقط می خواهم ادعای کوچکی را مطرح کنم و سپس ممکن است مانند طرحی از اینکه اگر می خواهید اثبات چگونه به نظر می رسد ارائه کنم که این است که اجازه دهید فرض کنید تابع پیچیده ای F دارید، و ابتدا ویژگی های زیر را برآورده می کند. شما می توانید یک مشتق از آن بگیرید. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "قابل تمایز است که فقط آن را از چیزهایی که شما می شناسید ناپیوسته کاملاً نامرتب را حفظ می کند. این مانند گرفتن مقادیر تصادفی است بسته به اینکه می دانید دامنه هر فضای برداری بیش از مقادیر کسری را نمی دانم که ممکن است بخواهید به روش های دیوانه کننده ای به آن فکر کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "عملکرد خوبی است. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "این قابل تفکیک است همه جا با 0 برابر نیست، بنابراین شرایطی که به نوعی ذهنم را درگیر کرده است و فراموش می کنم برای کدام سخنرانی یا چیزی شبیه به آن و سپس این خاصیت مرکزی را دارد که جمع را به ضرب تبدیل می کند اگر چنین تابعی دارید ادعا می کنم که یک عدد منحصربه‌فرد وجود دارد، شاید واقعاً باید مشخص کنم که یک عدد مختلط منحصربه‌فرد R وجود دارد، بنابراین می‌توانید F از X را به‌عنوان این تابع نمایی از R برابر مقدار X بنویسید. چند جمله‌ای نامتناهی با ویژگی‌های مشتق خوب و همه آن‌ها، اگر این را داشته باشید، هر نمایی را که می‌خواهید به معنای کلی انتزاعی کلمه نمایی، فقط بر اساس خاصیتی که می‌توانیم از آن بخواهیم و طرح اثبات آن را دارید، دارید. ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "ما می توانیم F از X را به طور کامل از عبارت فاکتور بگیریم و کل حد فقط برحسب H بیان می شود که اگر به معنای آن در زمینه مشتقات فکر کنید و این واقعیت که F از 0 لزوماً برابر است با 1 این کل عبارت محدود کننده است. فقط مقداری ثابت، اما به طور خاص هر چه مشتق تابع ما در 0 باشد، بنابراین شما این چیز خنده دار را دارید که اگر مشتق آن را در 0 بدانید، مشخص می کند که مشتق آن در همه جا چیست و در زمینه توابع نمایی، امیدواریم این موضوع کاملاً آشنا باشد زیرا تمام آنچه که ما واقعاً می گوییم مشتق یک تابع نمایی است متناسب با خودش است و ثابت تناسب برابر است با هر مشتق در 0. نه لزوماً فقط توابعی که قبلاً به عنوان a به توان X فکر می کنیم، اما این یک کلاس بالقوه بسیار گسترده تر از توابع است که فقط این ویژگی انتزاعی تبدیل جمع به ضرب را برآورده می کند، اما اگر این را داشته باشید در واقع تضمین می کند که شما نیز یک مشتق دوم و برای این موضوع یک مشتق سوم و مانند آن چون تابع مشتق فقط با خودش متناسب است، بنابراین برای گرفتن مشتق n باید به آن ثابت تناسب نگاه کنید و آن را به توان n برسانید و سپس از اینجا می توانید یک کار انجام دهید. بسط سری تیلور و من ممکن است آن را به عنوان یک تکلیف پیشرفته برای کسانی از شما که در آن ایده با سری تیلور راحت هستید، بگذارم، به خصوص اگر می خواهید ایده هر تابع متمایزپذیر را که به معنای اعداد مختلط قابل تمایز است، با هم مخلوط کنید. کاملاً یک موضوع دانشگاهی است شما می دانید که می توانید استدلال را در آنجا هر طور که می خواهید با هم مخلوط کنید، اما استدلال فازی در زمینه کسی که فقط در مورد سری تیلور می داند و هیچ چیز دیگری اجازه دارد این ایده را بپذیرد و به بسط تیلور برای F و نگاه کند. به نوعی این ایده را توجیه کنید که یک عدد مختلط منحصربه‌فرد وجود دارد به طوری که تابع F ما را می‌توان لزوماً به این شکل نوشت و سپس اتصال به نمایی عادی زمانی است که شما چنین مقداری داشته باشید R اساساً همان کاری را انجام می‌دهیم که در بافت پیچیده اعداد واقعی انجام می‌دهیم. این است که اگر به exp آن تابع آن مقدار R نگاه کنید و آن را به عنوان پایه بنویسید. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "ما می‌توانیم آن را به معنای انقضای نیم‌های پی I برابر X، بلکه می‌توانیم آن را به معنای انقضای نیمه‌های 5 پی تفسیر کنیم. آنها را به عنوان I به X بنویسید بنابراین عبارت I به I مگر اینکه استانداردی را برای معنای لزوماً اتخاذ کرده باشید وقتی می گویید بی نهایت خروجی دارد راه دیگری برای فکر کردن به آن این است که تابع I به X با نمادی که ما داریم کمی مبهم است حالا با همه اینها بیایید فقط شروع به تجسم بخشی از این کنیم زیرا فکر می کنم سرگرم کننده است و شما می دانید که به من بگویید اگر این تصویری مفید است یا تصویری گیج کننده تر. کاری که می‌خواهیم انجام دهیم این است که به این تابع R برابر X نگاه کنیم، که اساساً این روش دیگری برای نوشتن e به توان X است، در واقع فکر می‌کنم من فکر می‌کنم در نقطه‌ای انیمیشن متفاوتی را ارائه کرده‌ام که مشخص می‌کند که چون من قصد داشتم این کار را برنامه ریزی کنم، پس اجازه دهید اوه بله، شما به سیستم فایل من برگردید و به جایی که باید باشید برگردید. وارد آنجا شوید که شکایت دارد زیرا چندین مورد متفاوت وجود دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "اوه جایگزین کنید، آن را در صفحه دیگر نشان می دهد صبر کنید چرا بله، خوب جایگزین کنید؟ هر چیزی را که می بینید در آنجا قرار دهید و اکنون ما به آه آن جا برمی گردیم، همه آن چیزها را فقط برای اینکه بتوانم به خوبی بنویسم. پشت سر شما e به R ضربدر X و ما حول R تغییر می کنیم، بنابراین من نقاط محور خیالی را دنبال می کنم، و نقاط محور واقعی را دنبال می کنم و بیایید ببینیم این چه کار می کند خوب این یک جورهایی سریع است، بنابراین اجازه دهید کمی آهسته تر به همه اعداد منفی فکر کنم. ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a به یک عدد واقعی منفی چیزی بین 0 و 1 است و ما به طور خاص f از منفی 1 را دنبال می کنیم که در اطراف هر چیزی که 1 روی e حدود 30 0 باشد نشان داده می شود. 37 f از 1 روی e فرود می آید که انتظار می رود این همان چیزی است که اکسپت 1 برابر با f از I است که یک رادیان به دور دایره واحد فرود می آید، و دنبال کردن در طول کل محور خیالی در اینجا جالب است که چگونه محور خیالی به دور یک دایره پیچیده می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "ممکن است ما R را بخواهیم و مقادیر را در اینجا بخواهیم، چیزها را به گونه‌ای متفاوت گسترش می‌دهد، بنابراین وقتی آن را تا 2 قرار می‌دهیم، می‌دانید که محور واقعی را بسیار بیشتر باز می‌کند، به طوری که f از 1 به اطراف جایی که مربع e کمی بالاتر از 7 f منفی است ختم می‌شود. 1 بسیار نزدیکتر به 0 f از I 2 رادیان است چرخش به دور دایره f منفی I یک چرخش منفی 2 رادیانی است و البته می توانیم به فرمول مورد علاقه خود برسیم که اگر پی بود که ثابت مقیاس خود را داشتیم سپس محور واقعی بسیار کشیده می شود شما می دانید که f از 1 در e به پی می نشیند که بسیار نزدیک به 20 به علاوه pi است که همیشه سرگرم کننده است و f از منفی 1 بسیار نزدیک به 0 است، بنابراین واقعاً به اندازه واقعی کشیده شده است. محور و همچنین چیزهایی را در جهت دایره واحد کشیده است به طوری که با رسیدن به f از I یا f منفی I در نیمه دور دایره راه می رود، بنابراین همه چیز خوب و خوب است. ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "2 به X که چیست؟ ما همچنین به عنوان X از X از گزارش طبیعی 2 برابر X می نویسیم، بنابراین نقطه زرد خود را که نشان دهنده مقدار R است به حدود 0 منتقل می کنیم. 69 هنوز قسمت خیالی نیست فقط یک عدد واقعی 0.69 یا بیشتر این همان گزارش طبیعی 2 است، شما می توانید ببینید که f از 1 روی 2 فرود می آید، به همین دلیل است که ما می خواهیم این تابع را 2 به X f از 1 نیمه صدا بزنیم، در واقع متأسفیم f منفی 1 درست روی 1 نیمه f فرود می آید. من کمی دور دایره واحد قدم بزنید، به طور خاص 0 می شود. 69 رادیان در اطراف دایره واحد و حالا می‌توانیم کمی بیشتر لذت ببریم و بگوییم اگر بخواهیم این را به جای 0 تغییر دهیم، چه اتفاقی می‌افتد. 69 به جای اینکه گزارش طبیعی 2 باشد، آن را من برابر با ثبت طبیعی 2 قرار دهید تا ما واقعاً به چیزی فکر کنیم که ممکن است پایه نمایی برای آن داشته باشد. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "2 در حدود یک پنجم اما بسیاری از توابع نمایی مختلف وجود دارند که دارای این ویژگی قرار دادن f از 1 بر روی عدد I هستند، بنابراین اگر بخواهیم آن را حتی بیشتر بزرگتر کنیم، فکر نمی‌کنم آن را در اینجا متحرک داشته باشم، اما اگر بخواهیم استفاده کنیم. آن نقطه زرد را بالا بیاورید تا به 5 نصف پی I برسد. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "به دور خودش می چرخد به طوری که f از منفی f از 1 حول 2 پی رادیان دیگر می چرخد و در جایی که هست فرود می آید، اما محور واقعی را خیلی بیشتر گسترش می دهد که به این معنا بود که خروجی دیگری از I به I است. یک عدد بسیار کوچکتر حدود 0 بود. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "حالا اینجا چه خبر است؟ ما X از R را ضربدر X داریم و R برابر با این مقدار است، که log طبیعی 2 به اضافه پی ضربدر I است. معنی آن این است که وقتی 1 f از 1 را وصل می کنیم منفی 2 است بنابراین می خواهیم این تابع را بنویسیم. به عنوان منفی 2 به توان X درست است و این در واقع چیزی است که می دانید، زمانی که یک عدد منفی را به توان X می نویسیم کمی فریبنده ساده است. به هر شکلی وارد اعداد مختلط می‌شویم، اما البته وقتی حتی مقداری مانند 1 نصف را وارد می‌کنیم، جایی که به نوعی جذر منفی 2 را درخواست می‌کنیم، متوجه می‌شویم که می‌خواهیم این را به صورت چیزی بنویسیم مانند I ضربدر جذر. از 2 اما اگر بخواهید به این تابع منفی 2 به توان X در دامنه کامل پیچیده ای که با آن سر و کار دارید نگاه کنید آنچه شما به آن نگاه می کنید تابعی است که مقدار 1 را به منفی 2 می گیرد و اگر این کار را انجام دهد چه با بقیه خط اعداد واقعی انجام می دهد آیا به نوعی آن را مارپیچی به سمت بیرون می کشد؟ بنابراین می بینیم که f از منفی 1 در نیمه منفی 1 قرار می گیرد در مورد جایی که انتظار دارید اگر از f از 1 نیمه پیروی کنید دقیقاً روی خط فرضی قرار می گیرد و f از 1 نیمه به صورت جذر 2 خواهد بود خب، من ماوس جایی نیست که من می خواهم. ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "حدوداً 2 برابر من خواهد بود و همانطور که ادامه می دهید، تمام قدرت های ارزش واقعی منفی 2 تا X را به شما نشان می دهد که لزوماً در اطراف مارپیچ می چرخد اما می توانیم مقدار R خود را حتی بالاتر ببریم و آن را بدست آوریم. تا حدوداً تاو ضربدر من حدود شش نقطه دو هشت برابر I و در این زمینه این تابع دیگری است که می‌خواهیم چیزی شبیه 2 به X بنویسیم زیرا برای هر عدد کامل به عدد کاملی که برای X وصل می‌کنید، این تابع خواهد بود. شبیه ضرب مکرر است و حتی مقادیر معقولی برای چیزهایی مانند 1 نصف دارد که در آن به جای جذر مثبت، جذر منفی را بیرون می اندازد، اما کاری که در واقع انجام می دهد تبدیلی به صفحه است که در آن همه چیز را قرار می دهد، واقعی است. خط اعداد یک مارپیچ بسیار محکم است که دور می‌چرخد و فقط مارپیچی می‌شود به گونه‌ای که f از 1 درست روی عدد 2 فرود می‌آید، بنابراین از این نظر است که می‌توانیم بگوییم 2 به X است که به طور منطقی تفسیر می‌شود. یک تابع نمایی مجزا از تابعی که ما به طور سنتی به آن عادت کرده ایم، بنابراین فکر می کنم با همه اینها همه چیز را برای امروز می گذارم و فقط چند سوال طولانی برای شما باقی می گذارم تا در مورد خوب فکر کنید، بنابراین اگر می خواهید من را به من به عنوان یک عبارت چند ارزشی در نظر بگیرید، درست می توانید بگویید که ما یک قرارداد را پذیرفته ایم به طرز خیال انگیزی می گویید شاخه ای از تابع لگاریتم طبیعی را انتخاب می کنید و شاید این شما را در این وجود e به عدد پی منفی قفل می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "من شما را ترک می کنم و فکر می کنم این است که می دانید سؤال اصلی من در مورد سخنرانی امروز این بود که آیا می خواهم به نوعی توصیف این ویژگی های انتزاعی توابع نمایی باشد و برای من جالب است که از آن ویژگی های انتزاعی شروع کنم. شما درگیر ایده e به rx یا بیشتر می شوید فقط می دانید که من فکر می کنم صادقانه تر از r ضربدر x برای مقادیر مختلف r نوشته شده است که شما را تا آنجا قفل می کند اما شما را تا آنجا که داشتن قفل نمی کند یک مفهوم بدون ابهام در مورد اینکه چه چیزی 2 به توان x باید بسیار کمتر باشد چیزی شبیه به I به توان x البته خطر این است که گاهی اوقات مردم انتزاع را دوست ندارند و گاهی اوقات آن را قابل دسترس نیست، اما اگر اینطور باشد اگر می دانید فقط به من اطلاع دهید فکر می کنم فکر می کنم یک دایره فکری جالب وجود دارد که همه این چیزها را احاطه کرده است که شامل دکل های برق می شود زیرا اگر می خواهید در واقع در مورد برج های برق صحبت کنید مانند دفعه قبل که در زمینه اعداد مختلط بودیم. ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "یا حتی با پایه های منفی شما باید به چیزهایی مانند این فکر کنید، بنابراین این یک سوال بود که ما روی صفحه نمایش داشتیم بله، اگر این کار را برای من به قدرت من انجام دهیم چه اتفاقی می افتد؟ تیتراژ می‌دانید بیایید فقط این را امتحان کنیم، بیایید ادامه دهیم و یک برج قدرت را امتحان کنیم که در آن ما من را به یک قدرت معین بالا می‌بریم و ببینیم چه چیزی از آن بیرون می‌آید، بنابراین قصد انجام این کار را نداشت، اما ما همیشه می‌توانیم پایتون را بالا بکشید و اساساً کاری را که دفعه قبل انجام می‌دادیم انجام دهید، بنابراین روش کار این است که ما با مقداری پایه شروع می‌کردیم و سپس برای نوعی محدوده کاری که انجام می‌دادیم، یک را می‌گرفتیم و می‌خواهیم دوباره اختصاص دهیم. باید هر چیزی که باشد پایه ای که در این مورد من به توان a افزایش دادم باید خوب باشد، جالب است، بنابراین ما این کار را انجام می دهیم و سپس مقدار یک اجازه دهید این کار را انجام دهیم را چاپ کنیم. بله، این عدد بسیار بزرگ‌تر است مانند 200، بنابراین به نظر می‌رسد که اتفاقی که می‌افتد این است که با این چیزهایی مانند گاهی اوقات، احتمال آشفتگی وجود دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "من در واقع داریم، بنابراین اجازه دهید NumPy را وارد کنم تا تابع نمایی را داشته باشم اجازه دهید برای محدوده بزرگی که قبلا داشتیم به جای اینکه آن را بنویسم، همانطور که می دانید چیزی شبیه من به توان X است، آن را می نویسم به عنوان تابع نمایی یک ثابت متفاوت راست ثابت متفاوتی که می خواهم آن را بسازم، می خواهم آن را 5 نصف پی باشد، بنابراین 5 نصف پی را برابر من انجام می دهم تا یک عدد مختلط باشد و 5 نصف پی به عنوان عدد داشته باشد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/portuguese/sentence_translations.json b/2020/ldm-i-to-i/portuguese/sentence_translations.json index 1b0b8c441..1bd85dfa9 100644 --- a/2020/ldm-i-to-i/portuguese/sentence_translations.json +++ b/2020/ldm-i-to-i/portuguese/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "Então, se você está começando no número 1, sua velocidade inicial é caminhar direto em direção a 0 e, à medida que você anda ainda mais baixo, se você estivesse sentado na metade 1, ainda estaria caminhando em direção a 0, mas agora seu vetor velocidade seria negativo 1 vezes onde você está, o que é negativo 1 metade. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "E uma pergunta interessante é que você sabe se existe apenas uma função que parece razoável escrever para isso, porque você sabe que se vamos escrevê-la como i elevado a x não apenas deve satisfazer isso, mas também deve satisfazer você sabe quando conectamos o número um e obtemos i, presumivelmente i elevado à potência um, mas estamos pensando que esta função deveria ser i. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "Então, temos 5 pi i metades ótimo, que é absolutamente outro valor que poderíamos inserir para x aqui e apenas para explicar isso um pouco mais visualmente se olharmos para trás, para nosso círculo aqui, onde estamos no momento caminhou por um período de tempo igual às metades de pi, que é 1.57 e se, em vez disso, dermos outra volta completa e fizermos mais metades de pi para nos levar a pi, o que você sabe, podemos registrar que é onde o e para o valor de pi i é, andamos outras metades de pi, andamos outras metades de pi, que em neste ponto teríamos feito um círculo completo nos levando de volta ao um e então caminhamos por cinco metades de pi que numericamente é cerca de 7.85 sim, esse é absolutamente outro número que nos coloca em cima de i e se fôssemos passar por toda a burocracia de reexpressar i elevado a i, primeiro escrevendo e elevado a 5 metades de pi i elevado a i aqueles i's multiplique para se tornar negativo e estaríamos olhando para e elevado a 5 metades de pi negativo, que é um número muito diferente, certo, podemos realmente calcular isso, não tenho certeza de cara, mas vamos dar uma olhada em um Desmos . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "Esse comprimento leva você a um número muito menor Mas essa não é a única resposta que poderíamos inserir corretamente, temos outras pessoas chegando aqui com menos 3 meios vezes i pi O que você conhece em termos de um círculo unitário? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "Poderíamos pensar em dizer ei, se eu quiser chegar a I, em vez de andar 90 graus pi metades radianos dessa maneira, e se eu andar 270 graus na outra direção 3 pi metades radianos, o que talvez eu considere negativo porque a convenção é normalmente, o sentido anti-horário é positivo Isso absolutamente é outra maneira de expressá-lo e isso nos daria uma resposta diferente se tivéssemos e elevado a 3 metades de pi negativo i Todos elevado a i passamos pelo mesmo jogo agora o i ao quadrado cancela com um negativo que já está lá, e temos 3 pi positivos e numericamente isso nos dá uma resposta ainda diferente do que tínhamos antes. Se passarmos e dissermos ei, o que é e elevado a 3 pi e não 3 o 3 pi metades 111 ponto 3 1 tipo de número muito diferente do que vimos antes 111 ponto o que era 111 ponto 3 1 ótimo 111 ponto 3 1 ou mais E novamente em termos de intuição o que você pode estar perguntando é: suponha que tenhamos essa rotação dinâmico Mas retrocedemos no tempo, vemos há quanto tempo o que eu tenho que ser De tal forma que se eu jogasse as coisas para frente a partir daí eu chegaria ao número um, minha condição inicial e Você tem que voltar no tempo 3 pi metades unidades E então, se você traduzisse para a dinâmica de decaimento Que é o que elevar ao olho está fazendo neste contexto, você diria se estou começando do número um Mas eu quero voltar no tempo e dizer Por onde eu deveria ter começado se Eu quero decair tanto que acabe no número um? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "Depois de 3 pi metades de unidades de tempo a resposta evidentemente começa em torno de cento e onze para esse tipo de decaimento exponencial E você pode ver para onde isso vai, onde na verdade há infinitos valores diferentes que poderíamos inserir para X se estivéssemos pensando em e elevado ao X como sendo eu e as pessoas entraram muito mais aqui Com licença, jogando meu alfinete no chão como alguém faz o clássico para o terceiro lugar 9 metades de pi ótima escolha 1729 metades de pi vocês são meus favoritos, muitos e muitos opções diferentes, infinitos valores diferentes, o que parece um pouco desconcertante à primeira vista porque olhamos para uma expressão Parece que você sabe que vai haver algum cálculo Eu apenas conecto isso na minha calculadora e vejo o que aparece e temos vários diferentes valores para isso Então, o que está acontecendo aqui, certo? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "A raiz quarta de 16 deve ser 2 e a resposta acaba sendo boa Adotamos uma convenção quando há múltiplas opções como esta quando você tem uma função com vários valores Muitas vezes apenas escolhemos um desses valores para ser o que queremos dizer quando queremos trate-o como uma função como algo com uma única entrada e uma única saída em uma linguagem mais sofisticada Isso surge o tempo todo quando estamos lidando com números complexos a idéia de algo como uma operação meio que querendo Ter vários valores você às vezes ouça a frase ramo Onde você escolhe um ramo da função raiz quadrada? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "Porque há várias respostas diferentes Você sabe que pensamos em I de novo é esta rotação de 90 graus E se estivéssemos pensando nisso como uma rotação de 90 graus parece que a raiz quadrada deveria ser Você sabe de algo sentado em um ângulo de 45 graus Talvez seja o quadrado raiz de I que poderíamos escrever explicitamente como raiz de 2 sobre 2 raiz de 2 sobre 2 I Isso é apenas usar trigonometria, mas se estivéssemos pensando em I como sendo uma rotação negativa de 270 graus, parece que metade disso seria metade dessa operação deveria realmente nos levar para o outro lado Talvez o número aqui deveria ser a raiz quadrada de I e isso é na verdade apenas o negativo do que vimos antes Raiz negativa de 2 sobre 2 menos raiz de 2 sobre 2 vezes I Agora no contexto do real funções avaliadas, podemos dizer sim. Basta escolher a raiz quadrada para ser qualquer que seja a resposta positiva, mas qual destas você considera a resposta positiva? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "E eu acho que você disse bem Nós sabemos o que é isso, nós meio que definimos como sendo a raiz quadrada de 2, tudo está muito bem Mas e se eu dissesse, vamos abordar isso da mesma maneira que estávamos aproximando nosso I para a expressão I I quero primeiro expressar as coisas como e elevado a algo certo e então vou aumentar isso para a metade 1 multiplicando a metade 1 pelo expoente E digo ok, posso, acho que posso fazer isso e elevado ao que é igual a 2, bem, esse é o logaritmo natural de 2. É uma constante que está em torno de 0.69 ou mais Se elevarmos e a essa potência obteremos 2, então poderíamos pensar nisso como e elevado ao logaritmo natural de 2 vezes 1 meio e se você quisesse, se estivesse pensando em e elevado a x? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "Você sabe que isso pode ser um exagero no contexto de números reais. Mas se você estivesse pensando em e elevado a x como uma abreviação para esta função x, você poderia inserir o valor 0.69 vezes 1 metade, o que acho que seria cerca de 0.345 É algo assim Você insere esse valor muito concreto em seu polinômio, vê o que ele produz e ele produzirá cerca de 1.414 um belo número real raiz quadrada de 2 o que você esperaria Mas se fizermos a mesma coisa que estávamos fazendo com I e Reconhecendo que na verdade existem várias respostas diferentes quando queremos escrever algo como e elevado a uma potência, também poderíamos escrever isso Isso pode parecer engraçado, mas poderíamos escrever como e elevado ao logaritmo natural de 2 mais 2 pi I Essa coisa toda elevada à metade 1 Logo depois de tudo esse valor será igual a você poderia dividi-lo como é e elevado a logaritmo natural de 2 multiplicado por e elevado a 2 pi I Este apenas tem o efeito de girar as coisas 360 graus, então será igual a 1 Então estamos olhando para 2 vezes 1 ótimo, isso parece uma substituição válida e ainda assim quando nós jogamos o mesmo jogo de pegar isso e elevá-lo a uma potência e tratar isso multiplicando a potência pelo expoente veja o que acontece Temos e elevado ao logaritmo natural de 2 vezes 1 meio mais Bem, quanto é 2 pi I vezes 1 meio bem, isso será pi vezes I Agora, esta primeira parte e elevado ao logaritmo natural de 2 vezes 1 meio que acabará sendo a familiar raiz quadrada de 2, está tudo muito bem, mas vamos multiplicar isso por e para o pi I Certo e bastante famoso e elevado a pi I é negativo 1 Então neste caso parece estar sugerindo que se estivermos resolvendo esta expressão 2 elevado a 1 metade Brincando com as diferentes respostas poderíamos inserir algo como e elevado a X igual a 1 metade do que obtemos é outra resposta que poderíamos tradicionalmente escrever como esta raiz quadrada negativa de 2 e aqui quero dizer que é um pouco engraçado ter vários valores para olhar para 2 elevado a 1 metade e dizer que isso não é igual Uma coisa, mas com base nas escolhas que fazemos, pode ser igual a várias coisas diferentes Mas as duas coisas que podem parecer bastante razoáveis Se vai haver algo que 2 elevado a 1 é, parece que deveria ser O positivo raiz quadrada com a qual estamos familiarizados ou a variante negativa disso que na verdade não parece um grande problema E, na verdade, poderíamos, hum, poderíamos jogar este jogo ainda mais longe, deixe-me pedir respostas ainda mais criativas para esta expressão porque talvez possamos encontrar outras potências engraçadas de algo como 2 elevado à potência X à medida que começamos a inserir vários valores diferentes de X com base na substituição que fazemos se estivermos seguindo as mesmas regras que estávamos usando na avaliação de I elevado a potência I Então desta vez a pergunta pergunta ou especifica que uma solução da equação e elevado a x é igual a 2 é o número real Log natural de 2 ok esse nós sabemos. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "resposta à pergunta e para x é igual a 2 e Novamente a criatividade é bem-vinda, então darei a você mais um pequeno momento para isso II Vou em frente e trarei algumas respostas aqui se estiver tudo bem para você, não tenho certeza de quanto tempo leva necessariamente leva para fazer a entrada matemática dependendo do dispositivo que você está olhando, mas não fique muito estressado se for antes de você ter a chance de entrar na pergunta que deseja, na resposta que deseja que ela responda. 131 de vocês inseriram a variante em que pegamos Ln de 2 e adicionamos 2ii e acho que estou escrevendo esta pergunta. Por engano, marquei uma das respostas como correta quando na verdade há algumas respostas corretas diferentes. pelo fato de que não sei se parece para algum de vocês, oh, é vermelho, você errou ao inserir Ln de 2 mais 42. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi que é obviamente uma ótima escolha Mas você também pode ter algo como 4 pi I mais o logaritmo natural de 2 ou 6 pi I Ou realmente qualquer múltiplo inteiro de 2 pi I se você adicionar que isso não afeta e ao X Porque tem apenas o efeito de multiplicar por e elevado a 2 pi I Que é o efeito de multiplicar por 1 e, novamente, isso tem uma consequência meio engraçada, onde parece produzir resultados razoáveis quando fazemos isso como outro exemplo. parece que a segunda expressão inserida mais comum era que poderíamos substituir 2 Então, vamos pensar que estamos pensando em 2 elevado a 1 4º, ok, houve uma sugestão de substituirmos 2 por e elevado ao logaritmo natural de 2 mais 4 pi I Ok mais 4 pi I e aumentamos tudo isso para 1 4, bem, se você jogasse o mesmo jogo, obteria e Para o logaritmo natural de 2 vezes 1 4, e estaríamos multiplicando por e para o pi I Agora, a primeira parte disso será a quarta raiz positiva de 2, o que queremos dizer quando você insere uma expressão como a quarta raiz de 2 em uma calculadora, um belo e pequeno número positivo, mas esta segunda parte é 1 negativo, então parece estar dizendo: Você sabe, se interpretássemos 2 dessa maneira diferente, elevando-o para 1 4. Você sabe que não é a resposta usual que obtemos, mas é uma resposta razoável. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "Estaríamos olhando para pi metades vezes I e em vez de multiplicar por Negativo 1, estaríamos multiplicando por I O que novamente é uma resposta válida, parece uma saída razoável para algo como 2 elevado a 1 4 Então, quando você estiver olhando para o fato de que I elevado à potência I parece ter vários valores diferentes para isso Certo, temos esse fenômeno engraçado onde poderíamos conectar e às 5 metades de pi I Negativo 3 metades de pi I e obtemos o que pareciam ser respostas totalmente diferentes algo super pequeno algo super grande, tudo muito diferente da 1 5ª aproximadamente 1 5ª resposta que encontramos antes aqui É exatamente o mesmo fenômeno de quando você pergunta algo como quanto é 2 elevado a 1 4 e Reconhecendo que na verdade existem várias soluções diferentes para a expressão X elevado a 4 é igual a 2 4 soluções diferentes na verdade e o que você está vendo é o fato de que existem múltiplas soluções diferentes Para a expressão e elevado a X é igual a algum tipo de base se essa base é I se essa base é 2 Seja lá o que for, e uma forma de pensarmos sobre isso é que, quando você está lidando com números reais, as coisas são simplesmente adoráveis, as coisas são legais. Existem relacionamentos um-para-um. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "É ótimo Onde, se quisermos pensar em funções exponenciais, deixe-me apenas cobrir algumas dessas coisas. Temos essas ótimas idas e vindas onde você pode escolher expressar qualquer exponencial como uma base para X, como 2 elevado a X Ou você pode expressar aquela mesma exponencial que X de R vezes X que você sabe que é o polinômio ao qual nos referimos Sempre que nos referimos implicitamente sempre que escrevemos algo como e elevado a X E há um lindo vaivém porque você pode simplesmente pegar um logaritmo natural de B E isso lhe dá uma resposta assumindo que B é um número positivo E isso é a mesma coisa que dizer que X de R é igual a B Então uma maneira como falei sobre isso anteriormente na série é que se você estivesse olhando para o família de todos os exponenciais possíveis, certo, poderíamos escrevê-los como X de R vezes X e mudar o que R é E isso é exatamente a mesma coisa que escrever e elevado a R vezes X se isso for algo com o qual você se sente mais confortável Então e elevado a R vezes XX de R vezes X essas são a mesma coisa que poderíamos pensar em mudar o que é Mas por outro lado, se você pensasse em todas as exponenciais possíveis como alguma base Deixe-me fazer a base elevada à potência de X e vamos mudar o que é essa base No início parece que é um tipo diferente de expressão para manipular, mas é apenas outra forma de expressar a mesma família E uma forma de pensar sobre isto Como é que pensamos sobre a que base corresponde se estivermos pensando um pouco mais abstratamente como Exp de R vezes X e há uma razão pela qual estou fazendo isso porque estamos prestes a aplicar isso a números complexos onde parecerá mais estranho, então continue comigo aqui se em vez de olhar para essa base, uma coisa que eu poderia fazer é dizer qual é o valor? ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "Eu poderia ter exp de R vezes X onde talvez R seja algo como zero vírgula seis nove Mas eu poderia diminuir isso em dois pi I E isso não muda a base a que corresponderia, que ainda corresponderia a dois Ou poderia desloque-o para cima em dois pi I isso não altera a base a que corresponde porque em todos esses casos Quando substituímos X é igual a um obtemos a mesma coisa No entanto, todas estas para valores diferentes de X são funções distintas Isto é por que vimos vários valores diferentes para I elevado a I Porque I elevado a X é uma função ambígua nesse contexto, seria inequívoco se decidíssemos qual valor de R De tal forma que o que estamos representando é exp de R vezes X qual valor de R. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "É uma função inequívoca, mas nesse ponto parece que talvez o que queremos é Parar de pensar nas coisas em termos de alguma base elevada à potência X Talvez assim que estivermos no contexto de números complexos Devíamos apenas escrever todos eles como exp de alguns tempos constantes X, se por nenhuma outra razão isso deixa claro como nós realmente inserimos os números se quisermos fazer um cálculo ou apenas fazer contas em cima dele, temos esse belo polinômio infinito que temos conecte-os e eu apresentarei outro caso para você de que esta talvez seja a maneira correta de pensar sobre exponenciais Assim que estivermos estendendo para outros domínios coisas como números complexos e para isso vamos apenas voltar Vá de volta à campainha, algumas coisas chegaram, volte ao original. A maneira como estendemos a ideia de exponenciação e apenas pensamos no que é 2 elevado a X. Certo, sabemos como pensar sobre isso para números naturais. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "Você conhece algo como 2 elevado a 3 Multiplicação repetida. Como é que você primeiro foi ensinado a pensar em algo como 2 elevado a X para valores fracionários ou para valores negativos e coisas assim. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "Geralmente você aprende que 2 elevado a 1 deve ser algo em que você sabe que se eu multiplicá-lo por si mesmo e isso segue as regras usuais que os exponenciais fazem com a contagem de números, onde podemos somar coisas nesse expoente, devo obter 2 elevado a 1, então deve ser algum número que quando multiplico por si mesmo obtenho 2 e você sabe que nesse ponto você tem uma escolha, talvez seja positivo. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "Talvez seja negativo Mas se você sempre decidir fazer a escolha positiva Você será capaz de obter uma boa função contínua com esse mesmo negócio se perguntarmos sobre números negativos O que deveria ser 2 elevado a 1 negativo, bem, isso deveria ser alguma coisa onde quando multiplico por 2 elevado a 1? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "Isso me dá 2 elevado a 0 e isso é uma espécie de justificativa para nossa convenção de que expoentes negativos se parecem com 1 meio Mas o que realmente está acontecendo aqui é que estamos dizendo que seja o que for, deveria ser algum tipo de função Que satisfaça esta propriedade f de a mais b é igual a f de a vezes f de b e Além disso, o fato de que a base é 2 está basicamente nos dizendo que não é apenas uma função qualquer É uma função onde quando inserimos 1 obtemos 2 E tão pouco você sabe pergunta sobre estilo de verificação de sanidade para ver se você está acompanhando algumas das implicações aqui. Quero perguntar o que é que não vou chamar de softball, mas não foi feito para ser assim Uma pergunta incrivelmente profunda necessariamente. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "É apenas mais uma verificação se você estiver seguindo a ideia de começar abstratamente com as propriedades de uma função e então deduzir maneiras pelas quais podemos querer escrevê-la com base nessas propriedades Se f de x satisfaz esta propriedade exponencial f de a mais b é igual a f de a vezes f de b para todas as entradas E também satisfaz f de 1 é igual a 2 qual das afirmações a seguir é verdadeira O que significa qual das afirmações a seguir é necessariamente verdadeira Não importa qual função você está iniciando com e aqueles de vocês que se lembram de qual palestra foi essa É qualquer uma que estávamos falando sobre como interpretar o que a fórmula de Euler realmente está dizendo Eu fiz uma pergunta desse estilo onde negligenciei uma única condição, vocês sabem que não escrevi o fato de que queremos ter certeza de que f de x é diferente de zero em todos os lugares e isso causou alguma confusão, o que é legal, colocar na tela a confusão que acontece com todos nós. Mas a intenção disso era basicamente mostrar que essa propriedade abstrata de algo que transforma adição em multiplicação éÉ suficiente para basicamente fazer você querer escrever a função como qualquer coisa que seja igual a um elevado a algum tipo de potência Este é o espírito da questão Agora temos algumas perguntas sobre torres de energia que parecem ter aparecido aqui, o que é ótimo relacionado à última vez. Vamos adiar a questão da torre de energia por apenas um momento, para que primeiro tenhamos uma sensação mais profunda do que exponenciação deveria significar aqui? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "Porque porque podemos ser o que eu quero afirmar é que podemos responder de várias maneiras diferentes Então, se você me der apenas uma, falaremos sobre torres de energia E então, assim como uma reta numérica pode ser representada em uma escala logarítmica, pode o mesmo pode ser feito para um plano complexo? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "Sim, na verdade, há uma visualização que irei abordar daqui a pouco, onde faremos algo bastante semelhante a isso. Porque o que faremos é brincar com diferentes funções exponenciais X de R vezes X Mas estamos vamos mudar o valor de R que será representado por um pequeno ponto amarelo Então vamos conversar sobre isso Não vamos mapear o plano inteiro, mas apenas alguns pontos de amostra do eixo real e do eixo imaginário Mas a ideia é que, à medida que nos movemos em torno do que é essa constante, seremos capazes de visualizar as diferentes coisas que ela faz no plano e, efetivamente, é como se ela transformasse o eixo x em uma escala logarítmica e depois envolvesse o eixo imaginário ao longo de um círculo E então, assim que esse valor de R se torna imaginário, ele troca o papel daqueles. Os números reais são colocados no círculo e os números imaginários são colocados em uma escala logarítmica. Eixo positivo, então, grande questão, todos os três, eu acho estão meio que se precipitando para saber onde eu quero ir. Mas é bom ver que é onde as pessoas estão pensando assim neste. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "explicitamente Algo como f de 5 é a mesma coisa que f de 1 mais 1 mais 1 mais 1 mais 1 Que é a mesma coisa que f de 1 multiplicado por si mesmo 5 vezes por causa desta propriedade Que se f de 1 for 2 é o mesmo como 2 elevado a 5 e então algo como f de menos 5 Deveria ser o caso de quando multiplicamos por f de 5 Obtemos o que quer que seja f de 0 e não está imediatamente claro o que é f de 0, mas poderíamos dizer que f de 1 mais 0 é igual a qualquer f de 1 vezes o que f de 0 é, mas f de 1 é igual a 2 E então isso também é igual a 2, então estamos dizendo que 2 é igual a 2 vezes alguma coisa, bem, essa alguma coisa tem que ser 1, então neste contexto isso garante que f de menos 5 é 2 elevado a menos 5 é 1 sobre 2 elevado a 5 Poderíamos escrever isso explicitamente como 2 elevado a menos 5, o que significa que essas duas propriedades juntas formam nós realmente queremos escrever a função como 2 elevado a X Porque qualquer número de contagem que colocarmos nele irá satisfazer Será parecido com multiplicar por si mesmo esse número de vezes qualquer número fracionário que colocarmos nele irá satisfazer essas propriedades que queríamos E você pode se perguntar se isso é único e no contexto de funções com valor real realmente seria Mas no contexto de funções com valor complexo Haveria múltiplas funções f que poderíamos escrever para esta, uma das quais é a que estávamos olhando antes Onde poderíamos ter uma função definida como exp do logaritmo natural de 2 mais 2 pi I tudo isso vezes X Ok, perdoe o desleixo aqui, fico animado escrevendo sobre isso E esta é na verdade uma função diferente como evidenciado pelo que acontece se você inserir X igual a 1 metade Vimos um pouco antes como quando você inserir 1 metade o que você obtém é a raiz quadrada negativa de 2 e então se você inserir 1 quarto você obtém Não a raiz quarta de 2 mas I vezes a raiz quarta de 2, então é uma função diferente Mas ainda satisfaz essas propriedades e meio que nos faz querer escrevê-lo como 2 elevado a X E isso faz sugerir que talvez 2 elevado a X seja ambíguo um pouco de notação E deveríamos apenas escrever tudo em termos de exp de R vezes alguma coisa, mas você pode estar se perguntando Você sabe que talvez não estejamos sendo criativos o suficiente com todas as funções que satisfazem esta propriedade Talvez haja uma ambiguidade quando escrevemos exp de R vezes alguma coisa e há diferentes valores de R que podem entrar em jogo Mas eu vou apenas fazer uma pequena afirmação e então talvez dar um esboço de como seria a prova se você quiser. digamos que você tenha alguma função complexa F, e ela satisfaça primeiro as seguintes propriedades. Você é capaz de derivar dela. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "É diferenciável, o que apenas evita que seja algo que você conhece, algo descontínuo totalmente confuso. É como assumir alguns valores aleatórios, dependendo de você conhecer a extensão de qualquer espaço vetorial, não sei quantidades fracionárias que você possa querer pensar de maneiras malucas. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "É uma função legal. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "Isso é diferenciável Não é igual a 0 em todos os lugares, então a condição que me passou pela cabeça e eu esqueci para qual palestra ou algo assim e então tem essa propriedade central que transforma adição em multiplicação Se você tiver essa função, eu afirmo que há um único, talvez eu deva realmente especificar que existe um número complexo único R para que você possa escrever F de X como sendo basicamente esta função exponencial de R vezes aquele valor X O que você sabe basicamente dizendo que se você tiver X como uma função isso polinômio infinito com boas propriedades derivadas e tudo isso se você tiver isso você tem Cada exponencial que você deseja em um sentido genérico abstrato da palavra exponencial apenas com base em uma propriedade que poderíamos querer dela e o esboço da prova seria veja algo assim se você quiser primeiro ver qual é a derivada desse valor que estamos assumindo que existe em todos os lugares, certo? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "Podemos fatorar F de X inteiramente da expressão e todo o limite é expresso apenas em termos de H. O que se você pensar sobre o que isso significa no contexto de derivadas e o fato de que F de 0 é necessariamente igual a 1 Toda essa expressão limitante é apenas alguma constante, mas mais especificamente é qualquer que seja a derivada da nossa função em 0 Então você tem essa coisa engraçada onde se você souber sua derivada em 0 isso determina qual é sua derivada em todos os lugares E no contexto de funções exponenciais isso é esperançosamente bastante familiar porque tudo o que estamos realmente dizendo é que a derivada de uma função exponencial é proporcional a si mesma e que a constante de proporcionalidade é igual a qualquer que seja a derivada em 0, tudo isso é muito abstrato e tal, mas o objetivo é enfatizar que é não necessariamente apenas funções que já consideramos como elevadas a X Mas é uma classe potencialmente muito mais ampla de funções que apenas satisfaz esta propriedade abstrata de transformar adição em multiplicação Mas se você tiver isso, na verdade garante que você também terá um segunda derivada E, nesse caso, uma terceira derivada e tal, porque a função derivada é proporcional a si mesma. Então, para obter a n-ésima derivada, basta olhar para essa constante de proporcionalidade e elevá-la à potência n e, a partir daqui, você poderia fazer um Expansão da série de Taylor e posso deixar isso como uma espécie de lição de casa avançada para aqueles que estão confortáveis com a série de Taylor nessa ideia, especialmente se quiserem misturar a ideia de qualquer função diferenciável que seja diferenciável no sentido de números complexos, que é uma espécie de tópico definitivamente universitário Você sabe que pode misturar o raciocínio como quiser Mas o raciocínio difuso é permitido no contexto de alguém que conhece apenas a série de Taylor e nada mais para pegar essa ideia e olhar para a expansão de Taylor para F e meio que justifica a ideia de que existe um número complexo único tal que nossa função F pode necessariamente ser escrita assim E então a conexão com exponenciais normais é sempre que você tiver tal valor R Fazemos essencialmente o que fazemos no contexto complexo de números reais é se você olhar para exp daquela função daquele valor R e escrever isso como base. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "Poderíamos interpretar isso como significando não apenas exp das metades de pi I vezes X, mas também poderíamos interpretá-lo como significando exp de 5 metades de pi I vezes X e Estas são funções separadas E há uma família infinita de funções separadas que parecemos que deveríamos escreva-os como I elevado a X Então a expressão I elevado a I a menos que você tenha adotado um padrão para o que isso necessariamente significará Quando você diz que tem infinitas saídas outra maneira de pensar nisso é que A função I elevado a X com a notação que temos é um pouco ambígua Agora com tudo isso vamos começar a visualizar um pouco disso porque eu acho isso divertido E você sabe, me diga se é um visual útil ou mais confuso, mas o que vamos fazer é olhar para esta função exp de R vezes X, que é Basicamente, esta é outra maneira de escrever e elevado à potência de X, na verdade, acho que renderizei uma animação diferente em algum ponto que especificou isso porque eu estava planejando Planejando fazer isso então deixe-me, ah, sim, aí está você, volte para o meu sistema de arquivos, volte para onde deveria estar Entre lá, ele está reclamando porque há vários diferentes Vai ser como se houvesse um Ah, substitua, ele aparece na outra tela. Espere, por que está, sim, ok, substitua? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "Coloque tudo o que você vê lá E agora voltamos para, ah, aí, nós todos isso tudo isso só para que eu pudesse ter escrito bem Se você não se sentir confortável em pensar nisso como exp de R vezes X este polinômio infinito Apenas no atrás da sua cabeça e elevado a R vezes X e vamos variar em torno de R então vou seguir os pontos do eixo imaginário, e vou seguir os pontos do eixo real e vamos ver o que isso faz Bem isso é tudo meio rápido, então deixe-me pensar um pouco mais devagar, todos os números negativos, qualquer coisa. Esse é um número real negativo e será comprimido no intervalo entre 0 e 1. O que deve fazer sentido e elevado ao negativo? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a para um número real negativo é algo entre 0 e 1 e estamos rastreando especificamente f de 1 negativo, que aparecerá em torno de qualquer 1 sobre e que esteja em torno de 30 0.37 f de 1 cai em e como esperado é isso que exp de 1 é f de I vai pousar um radiano em torno do círculo unitário, e é divertido acompanhar todo o eixo imaginário aqui como o eixo imaginário envolve um círculo e o que acontece quando ajustamos esse valor de R? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "Podemos querer valores de R aqui. Ele estica as coisas de maneira diferente, então quando colocamos isso em 2 Você sabe que ele estica o eixo real muito mais, de modo que f de 1 termina em torno de onde e ao quadrado está um pouco acima de 7 f de negativo 1 está muito mais próximo de 0 f de I é uma rotação de 2 radianos em torno do círculo f de negativo I é uma rotação negativa de 2 radianos E é claro que podemos chegar à nossa fórmula favorita que se fosse pi que tivéssemos como nossa constante de escala Então o eixo real fica bastante esticado Você sabe que f de 1 está em e elevado a pi que é muito próximo de 20 mais pi O que é sempre divertido e f de menos 1 extremamente próximo de 0 então é realmente esticado tão real eixo E também esticou as coisas na direção do círculo unitário para que Chegando a f de I ou f de negativo I caminhe até a metade do círculo, então está tudo muito bem agora. Como pensaríamos sobre uma função como? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "Também escreveríamos como X de X do logaritmo natural de 2 vezes X, então moveríamos nosso ponto amarelo representando o valor de R para cerca de 0.69 ainda não há parte imaginária, apenas um número real 0.69 ou mais Esse é o logaritmo natural de 2 bem, você pode ver que f de 1 cai em 2 É por isso que queremos chamar esta função de 2 elevado a X f de 1 metade, na verdade, desculpe, f de menos 1 cai exatamente em 1 metade f de É uma caminhada ao redor do círculo unitário, muito especificamente, será 0.69 radianos ao redor do círculo unitário e agora poderíamos nos divertir um pouco mais e dizer o que aconteceria se mudássemos isso para em vez de 0.69 em vez de ser o logaritmo natural de 2, faça I vezes o logaritmo natural de 2 para que estejamos realmente pensando em algo que possa ter uma base exponencial. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "Quanto sou eu elevado a I, neste caso, ele o empurra para cerca de 0.2 em torno de um quinto Mas há muitas funções exponenciais diferentes que teriam essa propriedade de colocar f de 1 no número I Então, se aumentássemos ainda mais, acho que não o tenho animado aqui Mas se fôssemos pegar aquele ponto amarelo e levante-o até chegar a 5 meios vezes pi I O que você veria é que o círculo unitário? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "É girado em torno de si mesmo de modo que f de f negativo de 1 giraria em torno de outros 2 pi radianos e pousaria onde está Mas esticaria muito mais o eixo real Qual foi o sentido em que outra saída de I para I é um número muito menor. Era em torno de 0.0003 ou mais Mas também podemos ver o que considero bastante divertido. O que acontece se considerarmos expressões alternativas que queremos interpretar como 2 elevado à potência X, certo? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "Temos X de R vezes X e R é igual a este valor, que é o logaritmo natural de 2 mais pi vezes I. O que isso significa é que quando substituímos 1 f de 1 está em menos 2, então queremos escrever esta função como negativo 2 elevado à potência X, e isso é realmente algo que você sabe, é um pouco enganosamente simples quando escrevemos um número negativo elevado a uma potência Negativo 2 elevado à potência X, a princípio não parece necessariamente assim, isso nos traz nos números complexos de qualquer forma mas claro que quando inserimos um valor como 1 meio Onde estamos a pedir uma raiz quadrada de menos 2 percebemos que queremos escrever isto como algo como I vezes a raiz quadrada de 2 Mas se você olhar para esta função menos 2 elevado a X no domínio complexo completo com o qual ela está lidando O que você está vendo é uma função que leva o valor de 1 a menos 2 E se isso acontecer, o que o que acontece com o resto da reta numérica real é uma espécie de espiral para fora? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "Então vemos que f de menos 1 fica em menos 1 meio Mais ou menos onde você esperaria se seguisse para f de 1 meio Ficaria exatamente na linha imaginária e f de 1 meio seria a raiz quadrada de 2 Bem, meu mouse não está onde eu quero que esteja. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "Seria em torno da raiz quadrada de 2 vezes I e à medida que você continua, isso mostra todas as potências de valor real de menos 2 elevado a X, necessariamente gira em torno Mas também poderíamos mover nosso valor de R ainda mais alto e obtê-lo até cerca de tau vezes I cerca de seis vírgula dois oito vezes I e nesse contexto esta é outra função que gostaríamos de escrever como algo como 2 elevado a X porque para qualquer número inteiro a número inteiro que você inserir em X ele irá parece uma multiplicação repetida E ainda tem valores razoáveis para coisas como 1 meio onde cospe a raiz quadrada negativa em vez de uma raiz quadrada positiva, mas o que na verdade está fazendo é uma transformação para o plano Onde ele coloca tudo é o real a reta numérica acaba sendo uma espiral muito bem enrolada que gira e gira de tal maneira que f de 1 cai exatamente no número 2. Portanto, é nesse sentido que poderíamos dizer que 2 elevado a X é É plausivelmente interpretado como uma função exponencial separada daquela com a qual estamos tradicionalmente acostumados Então acho que com tudo isso vou deixar as coisas por hoje E vou deixar vocês com algumas questões pendentes para pensarem, ok, então se você quiser pense em I elevado a I como sendo uma expressão de múltiplos valores, certo, você poderia dizer que adotamos uma convenção. Fantasiamente, você diria que escolheu um ramo da função logaritmo natural E talvez isso o prenda a este ser e elevado ao pi negativo metades Mas se você disser que isso quer ter infinitos valores diferentes como os vários que vimos Quantos valores 2 elevado a 1 terço quer ter no mesmo sentido? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "Os décimos querem ser formulados de forma diferente de todas, deixe-me dizer, de todas as funções exponenciais F de X que satisfazem, ah, escrevi em algum lugar f de X que satisfaz Todas essas propriedades que escrevi, então se satisfaz todas destes e se f de 1 for igual a 2 Certo, quantas saídas diferentes obteremos quando inserirmos X igual a 3 décimos para as várias opções para qual função? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "Para 2 elevado a pi para as várias funções que 2 elevado a X poderia representar se estivermos pensando em 2 elevado a X como algum tipo de função exponencial Exponencial no sentido desse tipo de propriedade abstrata e se sim, se nós se temos uma classe de funções diferentes e queremos conectar pi, isso me faz rir Só porque é uma resposta meio engraçada que aparece enquanto você tenta pensar sobre isso, então essas são as perguntas que Vou deixar vocês com isso e acho que vocês sabem disso. Minha questão central ao abordar a aula de hoje era se eu queria que ela fosse descrita como essas propriedades abstratas de funções exponenciais E é muito legal para mim começar com essas propriedades abstratas você fica preso na ideia de e elevado a rx ou mais Só você sabe que eu acho mais honestamente escrito exp de r vezes x para valores diferentes de r Que isso prende você nessa distância Mas não prende você tanto quanto ter uma noção inequívoca de que 2 elevado a x deveria ser muito menos algo como I elevado a x O risco nisso, claro, é que às vezes as pessoas não amam a abstração e às vezes ela não parece tão acessível Mas se esse for o caso você saiba, é só me avisar Eu acho que há todo um círculo interessante de pensamentos que cerca Todas essas coisas para incluir torres de energia, porque se você quiser Na verdade, fale sobre torres de energia como fizemos da última vez no contexto de números complexos ou mesmo com bases negativas Você tem que estar pensando em coisas assim, então foi uma pergunta que a gente colocou na tela Sim, o que acontece se fizermos isso por I elevado à potência I? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "Titulação, você sabe, vamos tentar isso, vamos em frente e tentar uma torre de energia Onde estamos elevando I a uma determinada potência e ver o que sai dela, então não estava planejando fazer isso Mas podemos, sempre podemos puxar o Python e essencialmente fazer o que estávamos fazendo da última vez Então a maneira como isso funcionaria seria começar com algum valor base e depois com algum tipo de intervalo O que estávamos fazendo estávamos pegando um e vamos reatribuir será qualquer coisa A base que neste caso é eu elevada à potência de a deveria ser Ok, legal, então vamos fazer isso e então vamos imprimir o valor de a, vamos fazer isso para Sim, é um número muito maior, como 200. Então parece que o que acontece é que há potencial para o caos com essas coisas, às vezes. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "Na verdade, deixe-me importar o NumPy para ter a função exponencial, deixe-me ir Para o nosso grande intervalo como tínhamos antes Em vez de escrevê-lo como você sabe, algo que é como eu elevado à potência de X, vou escrevê-lo como a função exponencial de uma constante diferente, certo, uma constante diferente que vou fazer, quero que sejam 5 metades de pi, então farei 5 metades de pi vezes I, então é um número complexo e tem 5 metades de pi como o parte imaginária Então isso é 5 pi meio vezes I e o que estou fazendo? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/russian/sentence_translations.json b/2020/ldm-i-to-i/russian/sentence_translations.json index 0051d1a11..c1b182e36 100644 --- a/2020/ldm-i-to-i/russian/sentence_translations.json +++ b/2020/ldm-i-to-i/russian/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "Итак, если вы начинаете с цифры 1, ваша начальная скорость — идти прямо к 0, а по мере того, как вы идете еще ниже, если бы вы сидели на 1 половине, вы все равно шли бы к 0, но теперь ваш вектор скорости будет отрицательным 1 раз там, где вы находитесь, что является отрицательным 1 раз. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "И интересный вопрос, вы знаете, есть ли только одна такая функция, которую кажется разумным написать для этого, потому что вы знаете, если мы собираемся записать ее как i в x, она не только должна удовлетворять этому, она также должна удовлетворять, вы знаете, когда мы подключаем число один, которое получаем i, предположительно i, к степени единицы, однако мы думаем, что эта функция должна быть i. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "Итак, у нас есть 5 пи и половинки, это здорово, это, безусловно, еще одно значение, которое мы могли бы подставить сюда для x и просто выразить это немного более наглядно, если бы мы оглянулись на наш круг здесь, где мы находимся в момент, пройденный за промежуток времени, равный половине числа пи, что равно 1.57 что, если вместо этого мы сделаем еще один полный оборот и пройдем еще одну половину числа пи, чтобы добраться до числа пи, которое, как вы знаете, мы могли бы как бы записать вот где значение e для значения числа пи - мы пройдем еще одну половину числа пи, мы пройдем еще одну половину числа числа пи, что в в этот момент мы прошли бы полный круг, вернувшись к единице, а затем прошли бы пять половин числа Пи, что численно составляет около 7.85 да, это еще одно число, которое выводит нас на вершину i, и если бы нам пришлось пройти через всю эту ерунду повторного выражения i в степени i, сначала записав e в половинки 5 пи, i в степень i, в степень i, умножьте, чтобы стать отрицательным, и мы будем смотреть на e до отрицательных половин 5 пи, что представляет собой совсем другое число, верно, мы действительно можем это вычислить, я не уверен, что это сразу у меня в голове, но давайте посмотрим на Desmos . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "Такое длинное число, которое приведет вас к гораздо меньшему числу. Но это не единственный ответ, который мы могли бы ввести, ведь сюда приходят и другие люди с отрицательными значениями 3 половинных чисел пи, которые вы знаете в терминах единичного круга? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "Мы могли бы подумать о том, чтобы сказать: «Эй, если я хочу добраться до I», а не идти на 90 градусов пополам пи радиан в ту сторону. Что, если я пойду на 270 градусов в другую сторону 3 половинки пи радиан, которые, возможно, я буду считать отрицательными, потому что соглашение таково: обычно это против часовой стрелки является положительным. Это абсолютно другой способ выразить это, и это дало бы нам другой ответ, если бы у нас было e для отрицательных 3-х половин i Все в степени i мы проходим ту же игру, теперь i в квадрате отменяется с помощью отрицательный результат уже есть, и у нас есть положительные половины 3 пи, и в численном отношении это дает нам ответ, который выглядит даже иначе, чем тот, который у нас был раньше. Что, если мы перейдём и скажем: эй, что такое е для 3 пи, а не 3 или 3 пи? половинки 111 очко 3 1 совсем другой вид числа, чем то, что мы видели раньше 111 очко что это было 111 очко 3 1 отлично 111 очко 3 1 или около того И снова с точки зрения интуиции вы можете спросить, предположим, что у нас есть это вращение динамика Но мы движемся назад во времени, мы видим, как давно во времени я должен быть Таким, что, если бы я играл вперед оттуда, я бы приземлился на номер один, мое начальное состояние, и вам придется вернуться во времени на 3 половинки пи. И затем, если бы вы перевели динамику распада. Что именно делает в этом контексте поднятие глаз, вы говорите, если бы я начал с номера один. Но я хочу переместиться назад во времени и сказать, с чего мне следует начать, если Я хочу скатиться вниз так, чтобы оказаться на первом месте? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "После 3-х половинных единиц времени ответ, очевидно, начинается примерно со ста одиннадцати для такого рода экспоненциального затухания. И вы можете видеть, к чему это идет, поскольку на самом деле существует бесконечно много разных значений, которые мы могли бы подставить для X, если мы думаю, что от е до X — это я, и люди сюда ввели гораздо больше. Извините, я бросаю булавку на землю, как это делают классика для третьего места. 9 половинок пи, отличный выбор. 1729 половинок пи, вы все мои любимые лоты-много разные варианты бесконечно много разных значений, что поначалу немного сбивает с толку, потому что мы смотрим на выражение Кажется, вы знаете, что будут какие-то вычисления. Я просто подключаю это к своему калькулятору и смотрю, что выскакивает, и у нас есть несколько разных ценности для этого. Так что же здесь происходит, верно? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "Четвертый корень из 16 должен быть 2, и ответ в конечном итоге будет хорошим. Мы принимаем соглашение, когда существует несколько вариантов, подобных этому, когда у вас есть многозначная функция. Мы часто просто выбираем одно из этих значений в качестве того, что мы имеем в виду, когда хотим относитесь к нему как к функции, как к чему-то с одним входом и одним выходом на более модном жаргоне. Когда мы имеем дело с комплексными числами, это постоянно возникает, идея чего-то как операции типа желания иметь несколько значений, которые вы иногда будете иметь. услышите фразу «ветвь», где вы выбираете ветвь функции квадратного корня? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "Потому что есть несколько разных ответов. Вы знаете, мы снова думаем об I, это поворот на 90 градусов. И если бы мы думали об этом как о повороте на 90 градусов, кажется, что квадратный корень должен быть Вы знаете, что-то расположено под углом 45 градусов Может быть, это квадрат корень I, который мы могли бы очень явно записать как корень 2 из 2, корень 2 из 2 I. Это просто использование тригонометрии, но если бы мы вместо этого думали о I как об отрицательном вращении на 270 градусов, это было бы похоже на половину этого, выполняющего половину этой операции. на самом деле должно вывести нас на другую сторону. Может быть, число, сидящее здесь, должно быть квадратным корнем из I, и это на самом деле просто отрицательный результат того, что мы видели раньше. Отрицательный корень 2 из 2 минус корень 2 из 2, умноженный на I Теперь в контексте реального значимые функции, мы можем сказать «да». Просто выберите квадратный корень, который будет положительным ответом, но какой из них вы считаете положительным ответом? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "И я думаю, вы хорошо говорите. Мы знаем, что это такое, мы как бы определяем это как квадратный корень из 2, все хорошо. Но что, если я скажу, давайте подойдем к этому так же, как мы подходили к нашему Я к выражению Я? хочу сначала выразить что-то как e для чего-то правильного, а затем я собираюсь возвести это в 1 половину, умножив 1 половину в показатель степени. И я говорю: хорошо, я могу, я думаю, я могу сделать это e для того, что есть равно 2, ну, это натуральный логарифм 2. Это константа, равная около 0.69 или около того. Если мы возведем е в эту степень, мы получим 2, так что мы могли бы думать об этом как об е в натуральном логарифме 2, умноженного на 1 половину, и если бы вы хотели, если бы вы думали об е в x? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "Вы знаете, что это может быть излишним в контексте действительных чисел. Но если бы вы думали, что e вместо x является сокращением для этой функции x, вы могли бы подставить значение 0.69 умножить на 1 половину, что, я думаю, будет около 0.345 Вот что-то вроде этого. Вы подставляете это очень конкретное значение в свой полином и смотрите, что он выводит, и на выходе получается около 1.414 Хороший квадратный корень из действительного числа из 2, что и следовало ожидать. Но если мы сделаем то же самое, что мы только что делали с I, и признавая, что на самом деле существует несколько разных ответов, когда мы хотим записать что-то в виде e в степени, мы могли бы также написать это Это может показаться забавным, но мы могли бы записать это как е в натуральный логарифм 2 плюс 2 пи. Все это возвести в половину. Сразу после того, как все это значение станет равным, вы можете разбить его на е натуральный логарифм 2, умноженный на е до 2 пи I. Этот эффект просто вращает предметы на 360 градусов, так что он просто будет равен 1. Итак, мы рассматриваем 2, умноженное на 1, отлично, что кажется правильной заменой, и все же, когда мы играем в одну и ту же игру: возьмем это и возведем в степень, а затем умножим степень на показатель степени и посмотрим, что произойдет. ну, это будет пи, умноженное на I. Теперь это первая часть e для натурального логарифма 2, умноженного на 1 половину, которая в конечном итоге будет знакомым квадратным корнем из 2, это все хорошо, но мы собираемся умножить это на e, чтобы Пи I Правильно, и, как известно, е к пи I отрицательно 1. Таким образом, в этом случае, похоже, предполагается, что если мы решаем это выражение 2 к 1 половине, играя с разными ответами, мы могли бы подключить что-то вроде e к X, равному 1 половине, то, что мы в итоге получим, - это еще один ответ, который мы могли бы традиционно записать как отрицательный квадратный корень из 2, и здесь я имею в виду, что немного забавно иметь несколько значений, чтобы посмотреть на 2 к 1 половине и скажем, это не равно одному, но в зависимости от сделанного нами выбора оно может равняться множеству разных вещей. Но две вещи, которые могут показаться вполне разумными. квадратный корень, с которым мы знакомы, или его отрицательный вариант, который на самом деле не кажется такой проблемой. И на самом деле мы могли бы, хм, мы могли бы сыграть в эту игру еще дальше, и позвольте мне попросить вас дать еще более творческие ответы на это выражение потому что, возможно, мы сможем найти другие забавные степени чего-то вроде 2 в степени X, когда мы начнем подставлять различные значения X в зависимости от того, какую замену мы делаем, если мы придерживаемся тех же правил, которые мы использовали при вычислении I в степень X. power I Итак, на этот раз вопрос задается или указывает, что одно решение уравнения e к x равно 2 - это действительное число. Натуральный логарифм 2, ок, мы это знаем. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "ответ на вопрос от е до х равен 2 и Опять же, творческий подход приветствуется, поэтому я дам вам еще немного времени для этого II. Я продолжу и закреплю здесь несколько ответов, если вы не против. Я не уверен, сколько времени это займет. обязательно потребуется выполнить математический ввод в зависимости от того, на какое устройство вы смотрите, но не слишком переживайте, если это произойдет до того, как у вас появится возможность ответить на вопрос, на который вы хотите, и на ответ, на который вы хотите, чтобы он ответил. Так выглядит 131 из вас ввели вариант, в котором мы берем Ln из 2 и прибавляем 2ii, и, кажется, я пишу этот вопрос. По ошибке отметил один из ответов как правильный, хотя на самом деле существует довольно много разных правильных ответов. Так что это моя вина. за тот факт, что я не знаю, кажется ли кому-нибудь из вас это таким: «О, это красное, вы ошиблись, когда ввели Ln из 2 плюс 42». ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi, что, конечно, является отличным выбором. Но вы также можете иметь что-то вроде 4 pi I плюс натуральный логарифм 2 или 6 pi I. Или вообще любое целое число, кратное 2 pi I, если вы добавите, что это не влияет на e к X Потому что это просто дает эффект умножения на е до 2 пи. I Это эффект умножения на 1, и опять же, это имеет своего рода забавное последствие, когда кажется, что мы выдаем разумные результаты, когда мы делаем это в качестве другого примера. похоже, вторым наиболее распространенным введенным выражением было то, что мы могли бы заменить 2. Итак, давайте представим, что мы думаем о 2 в 1 4-й степени, хорошо, было предложение заменить 2 на e в натуральном логарифме 2 плюс 4 пи I Хорошо, плюс 4 пи I, и мы возведем все это в 1 4-й, хорошо, если бы вы играли в ту же игру, вы получили бы е в натуральный логарифм 2, умноженный на 1 4-й, и мы бы умножили на е, чтобы Пи I. Первой частью этого будет обычный положительный корень четвертой степени из 2, который мы имеем в виду, когда вы подставляете в калькулятор выражение, подобное корню четвертой степени из 2, красивое маленькое положительное число, но тогда эта вторая часть отрицательный 1, так что кажется, что он говорит: «Знаете, если бы мы интерпретировали 2 по-другому, возводя его в 1/4», вы знаете, что мы получаем не обычный ответ, но это разумный ответ. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "Мы бы смотрели на половину числа Пи, умноженную на I, и вместо умножения на Отрицательную 1 мы умножали бы на I. Что снова является допустимым ответом, это кажется разумным результатом для чего-то вроде 2 в 1 4-м. Итак, когда вы глядя на тот факт, что I в степени, я, кажется, имею для этого несколько разных значений. Верно, у нас есть этот забавный феномен, когда мы можем подставить e к половинам 5 пи I Отрицательные половинки 3 пи, и мы получаем, казалось бы, совершенно разные ответы. что-то очень маленькое, что-то очень большое, все очень отличается от 15-го примерно 15-го ответа, который мы нашли здесь ранее. Это точно такой же феномен, как когда вы спрашиваете что-то вроде того, сколько 2 к 14-му, и признаете, что на самом деле существует несколько разных решений. к выражению X до 4-го равно 2 4 разных решения на самом деле, и вы смотрите на тот факт, что существует несколько разных решений. К выражению e до X равно какому-то основанию, является ли это основанием I, является ли это основание 2 Как бы то ни было, и один из способов думать об этом состоит в том, что когда вы имеете дело с действительными числами, все просто прекрасно, все прекрасно. Есть отношения один к одному. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "Это здорово. Если мы хотим подумать о экспоненциальных функциях, позвольте мне рассказать о некоторых из этих вещей. У нас есть отличная возможность туда и обратно, где вы можете выразить любую экспоненту как основу для X, например, 2 для X. Или вы можете выразить та же самая экспонента, что и X от R, умноженного на X, которая, как вы знаете, является многочленом, на который мы ссылаемся Всякий раз, когда неявно ссылаемся на всякий раз, когда мы записываем что-то вроде e в X. И это прекрасная возможность взад и вперед, потому что вы можете просто взять натуральный логарифм B И это дает вам один ответ, если предположить, что B является положительным числом. И это то же самое, что сказать, что X из R равно B. Один из способов, о котором я говорил об этом ранее в этой серии, заключается в том, что если вы смотрите на семейство всех возможных экспонент, верно, мы могли бы записать их как X из R, умноженного на X, и изменить то, что такое R. И это точно то же самое, что написать e для R, умноженного на X, если вам так удобнее. Итак, e для R. умножения XX на R и умножения на X — это то же самое, о чем мы могли бы подумать, чтобы изменить то, что есть. изменить то, что представляет собой эта основа. Сначала кажется, что это другой вид выражения, которым нужно манипулировать, но это просто еще один способ выразить ту же самую семью И способ, которым вы можете об этом подумать. Как мы думаем о том, какой базе она соответствует? если мы думаем немного более абстрактно, как Exp R, умноженный на X, и есть причина, по которой я это делаю, потому что мы собираемся применить это к комплексным числам, где это будет выглядеть более странно, так что продолжайте со мной здесь, если вместо того, чтобы смотреть на эту базу, я мог бы сказать, какова ее ценность? ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "Я мог бы получить выражение R, умноженное на X, где, возможно, R равно чему-то вроде ноль целых шестьдесят девять. Но я мог бы сдвинуть это значение на два пи вниз. И это не меняет базу, которой оно будет соответствовать, которая все равно будет соответствовать двум Или это может быть сдвиньте его вверх на два пи I, что не изменит базу, которой он соответствует, потому что во всех этих случаях, когда мы подключаем X, равное единице, мы получаем одно и то же, однако все это для разных значений X являются разными функциями. Это почему мы видели несколько разных значений I в степени I. Поскольку I в X является неоднозначной функцией в этом контексте, было бы однозначно, если бы мы решили, какое значение R. Так что то, что мы представляем, представляет собой выражение R, умноженное на X, какое значение Р. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "Это однозначная функция, но в этот момент кажется, что, возможно, мы хотим: Перестать думать о вещах с точки зрения некоторого основания, возведенного в степень X. Может быть, как только мы окажемся в контексте комплексных чисел, нам следует просто написать все они как выражение некоторого постоянного времени X, если по какой-либо другой причине это становится кристально ясным. Как мы на самом деле подставляем числа, если хотим выполнить вычисления или просто выполнить математические действия поверх них, у нас есть этот красивый бесконечный полином, который мы подключите их, и я приведу вам еще одно доказательство того, что это, возможно, правильный способ думать об экспонентах. Как только мы расширим возможности таких вещей, как комплексные числа, и для этого давайте просто сделаем резервную копию Go вернемся к дверному звонку, кое-что прибыло, вернемся к исходному способу, которым мы расширяем идею возведения в степень и просто думаем о том, что равно 2 к X. Правильно, мы знаем, как думать об этом для натуральных чисел. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "Вы знаете что-то вроде 2 до 3. Повторное умножение. Как получилось, что вас сначала учат думать о чем-то вроде 2 до X для дробных сумм или для отрицательных сумм и тому подобное. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "Обычно вас учат, что 2 к 1 половине должно быть чем-то таким, о чем вы знаете, если я умножу это само на себя, и это следует обычным правилам, которые экспоненты делают при подсчете чисел, где мы можем складывать вещи в этом показателе, я должен получить 2 до 1, так что это должно быть какое-то число, которое, когда я умножаю его само на себя, я получаю 2, и вы знаете, что в этот момент у вас есть выбор, может быть, он положительный. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "Возможно, оно отрицательное. Но если вы всегда решаете сделать положительный выбор, вы сможете получить хорошую непрерывную функцию из этой же сделки, если мы спросим об отрицательных числах. Что должно быть хорошо при соотношении 2 к отрицательному 1, это должно быть что-то где, когда я умножаю это на 2 до 1? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "Это дает мне 2 к 0, и это своего рода оправдание нашего соглашения о том, что отрицательные показатели выглядят как 1 половина. Но на самом деле здесь происходит то, что мы говорим, что что бы это ни было, это должна быть какая-то функция, которая удовлетворяет этому свойству f а плюс b равно f a, умноженному на f от b. Кроме того, тот факт, что основание равно 2, по сути, говорит нам, что это не просто такая функция. Это функция, в которую, когда мы подключаем 1, мы получаем 2. И так же немного, как вы знаете Вопрос в стиле проверки здравомыслия, чтобы узнать, следите ли вы за некоторыми из выводов здесь. Я хочу спросить вас, что я не буду называть это софтболом, но это не должно быть похоже на Невероятно глубокий вопрос обязательно. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "Это просто дополнительная проверка, если вы следуете идее абстрактного начала со свойств функции, а затем своего рода вывода способов, которыми мы могли бы захотеть записать ее на основе этих свойств. Если f от x удовлетворяет этому экспоненциальному свойству f из a плюс b равно f из a, умноженного на f из b для всех входных данных. И это также удовлетворяет тому, что f из 1 равно 2, что из следующего верно. То есть, что из следующего обязательно верно. Независимо от того, какую такую функцию вы запускаете с и те из вас, кто помнит, какая это была лекция. О какой бы ни была речь, о которой мы говорили, как интерпретировать то, что на самом деле говорит формула Эйлера. Я задал вопрос в таком стиле, где я пренебрег одним условием, вы знаете, я не записал тот факт, что мы хотим убедиться, что f от x везде не равно нулю, и тогда это вызвало некоторую путаницу, и это круто, получить путаницу на экране, которая случается со всеми нами. Но цель этого заключалась в том, чтобы, по сути, показать, что это абстрактное свойство чего-то, что превращает сложение в умножение. Этого достаточно, чтобы вы захотели написать функцию так, как бы она ни равнялась возведенной в какую-то степень. В этом суть вопроса. Теперь у нас есть пара вопросов о силовых башнях. кажется, это всплыло здесь, и это здорово связано с прошлым разом. Давайте на минутку отложим вопрос о силовой башне, чтобы сначала получить более глубокое представление о том, что здесь должно означать возведение в степень? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "Потому что, поскольку мы можем быть тем, кем я хочу утверждать, мы можем ответить на этот вопрос множеством разных способов. Итак, если вы дадите мне только один, мы поговорим о силовых башнях. то же самое можно сделать и для сложной плоскости? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "Да. На самом деле, есть визуализация, к которой я собираюсь перейти через мгновение, где мы делаем что-то очень похожее на это. Потому что мы будем играть с различными экспоненциальными функциями X из R, умноженными на X. Но мы собираюсь изменить значение R, которое будет представлено маленькой желтой точкой. Итак, мы как бы обсудим это. Это не будет отображать всю плоскость, а всего лишь пару точек выборки от реальной оси и воображаемой оси. Но идея состоит в том, что по мере того, как мы будем определять, что представляет собой эта константа, мы сможем как бы визуализировать различные вещи, которые она делает с плоскостью. По сути, это похоже на превращение оси X в логарифмическую шкалу, а затем перенос воображаемая ось вдоль круга И затем, как только это значение R становится мнимым, оно меняет роль этих чисел. Реальные числа помещаются в круг, а мнимые числа помещаются в логарифмический масштаб. Положительная ось, такой замечательный вопрос, все три из которых, я думаю, вроде как забегают вперед в поисках того, куда я хочу пойти. Но приятно видеть, что именно об этом люди думают в этом фильме. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "явно Что-то вроде f из 5 - это то же самое, что f из 1 плюс 1 плюс 1 плюс 1 плюс 1. Это то же самое, что f из 1, умноженное само на себя 5 раз из-за этого свойства. Что, если f из 1 равно 2, то же самое как 2 в пятой степени, а затем что-то вроде f отрицательного 5. Должно быть так, что когда мы умножаем это на f из 5, мы получаем то, что равно f из 0, и не сразу понятно, что такое f из 0, но мы могли бы сказать, что f от 1 плюс 0 равно любому значению f от 1, умноженному на f от 0, но f от 1 равно 2. Итак, это также равно 2, поэтому мы говорим, что 2 равно 2, умноженному на что-то, ну, что-то должно быть 1, поэтому в этом контексте это гарантирует, что f от отрицательных 5 равно 2 к отрицательным 5, это 1 больше 2 к 5-му. Мы могли бы явно записать это как 2 к отрицательным 5, и это все, что можно сказать. Эти два свойства вместе составляют нам действительно нужно записать функцию как 2 для X, потому что любое счетное число, которое мы в нее вставляем, будет удовлетворять этим свойствам. то, что мы хотели И вы можете задаться вопросом, является ли это уникальным, и в контексте вещественнозначных функций это действительно было бы Но в контексте комплекснозначных функций было бы несколько таких функций f, которые мы могли бы написать для этой, из которых мы и были смотрим ранее Где мы могли бы определить функцию, выраженную как выражение натурального логарифма 2 плюс 2 пи. Я все это время X Хорошо, простите за неряшливость, мне просто нравится писать об этом. И на самом деле это другая функция, как о чем свидетельствует то, что произойдет, если вы подставите X, равный 1 половине. Чуть раньше мы видели, что, когда вы подставляете 1 половину, вы получаете отрицательный квадратный корень из 2, а затем, если вы подставляете 1 четверть, вы получаете не корень четвертой степени из 2, но я умножаю корень четвертой степени из 2, так что это другая функция. Но она по-прежнему удовлетворяет этим свойствам, и это как бы заставляет нас записывать ее как 2 для X. И это заставляет предположить, что, возможно, 2 для X является неоднозначным немного обозначений И мы должны просто писать все в терминах exp, умноженного на R, но вы можете задаться вопросом: Знаете, может быть, мы просто недостаточно изобретательны со всеми функциями, которые удовлетворяют этому свойству. Может быть, когда мы пишем exp, возникает двусмысленность R, умноженное на что-то, и есть разные значения R, которые могут сыграть роль. Но я просто изложу небольшое утверждение, а затем, возможно, приведу набросок того, как будет выглядеть доказательство, если хотите. Допустим, у вас есть некоторая сложная функция F, и сначала она удовлетворяет следующим свойствам. Вы можете взять от нее производную. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "Это дифференцируемо, что просто не позволяет ему быть какой-то, как вы знаете, совершенно беспорядочной прерывистой вещью. Это все равно, что брать некоторые случайные значения в зависимости от того, знаете ли вы диапазон любого векторного пространства, над которым я не знаю дробных величин, о которых вы, возможно, захотите думать сумасшедшими способами. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "Это хорошая функция. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "Это дифференцируемо. Оно не везде равно 0, поэтому это условие как бы вылетело у меня из головы, и я забыл, для какой лекции лекция или что-то в этом роде, и тогда у него есть центральное свойство: оно превращает сложение в умножение. Если у вас есть такая функция, я утверждаю, что есть уникальное, может быть, мне действительно следует указать, что существует уникальное комплексное число R, чтобы вы могли записать F от X как, по сути, экспоненциальную функцию R, умноженную на это значение X. Что, как вы знаете, в основном говорит о том, что если у вас есть X как функция, это бесконечный многочлен с хорошими производными свойствами и все такое, если оно у вас есть, у вас есть каждая экспонента, которую вы хотите, в очень похожем абстрактном общем смысле слова «экспонента», просто основанная на свойстве, которое мы могли бы получить от нее, и набросок доказательства будет посмотрите что-то вроде этого, если вы хотите сначала посмотреть, какова производная этого значения, которое, как мы предполагаем, существует повсюду, верно? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "Мы можем полностью исключить F из X из выражения, и весь предел будет выражен только через H. Что, если вы подумаете о том, что это означает в контексте производных и тот факт, что F от 0 обязательно равно 1, все это ограничивающее выражение есть просто некоторая константа, а точнее, это любая производная нашей функции в 0. Итак, у вас есть забавная вещь: если вы знаете ее производную в 0, которая определяет, чем является ее производная повсюду. И в контексте экспоненциальных функций это, надеюсь, довольно знакомо, потому что все, что мы на самом деле говорим, это то, что производная экспоненциальной функции пропорциональна самой себе, и что константа пропорциональности равна любой производной в 0, это все очень абстрактно сформулировано и тому подобное, но цель этого - подчеркнуть, что это не обязательно просто функции, о которых мы уже думаем как о степени X. Но это потенциально гораздо более широкий класс функций, которые просто удовлетворяют этому абстрактному свойству превращения сложения в умножение. вторая производная И, если на то пошло, третья производная и тому подобное, потому что производная функция просто пропорциональна самой себе. Итак, чтобы взять n-ю производную, вы просто смотрите на эту константу пропорциональности и возводите ее в степень n, а затем отсюда вы можете сделать Расширение ряда Тейлора, и я мог бы оставить это как своего рода продвинутое домашнее задание для тех из вас, кому нравится эта идея с рядом Тейлора, особенно если вы хотите смешать идею любой дифференцируемой функции, которая дифференцируема в смысле комплексных чисел, что это своего рода определенно студенческая тема. Знаете, вы можете смешивать рассуждения там, как хотите. Но нечеткие рассуждения допускаются в контексте того, кто знает только о рядах Тейлора и больше ничего, чтобы взять эту идею и посмотреть на расширение Тейлора для F и своего рода обоснование идеи о том, что существует уникальное комплексное число, такое, что наша функция F обязательно может быть записана следующим образом. это если вы посмотрите на exp этой функции этого значения R и запишите это как основу. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "Мы могли бы интерпретировать это не просто как выражение половинок числа пи I, умноженное на X, но мы могли бы также интерпретировать это как выражение выражения половинок числа Пи I, умноженное на X, и это отдельные функции. И существует бесконечное семейство отдельных функций, которые, по нашему мнению, должны запишите их как I для X. Итак, выражение I для I, если только вы не приняли стандарт того, что это обязательно будет означать. Когда вы говорите, что у него бесконечно много выходов, другой способ подумать об этом состоит в том, что функция I для X с обозначением, которое у нас есть, немного двусмысленно. Теперь, учитывая все это, давайте просто начнем визуализировать кое-что из этого, потому что я думаю, что это весело. И вы знаете, вы говорите мне, если это полезное визуальное представление или более запутанное визуальное представление, но то, что мы собираемся сделать, это посмотреть на эту функцию exp R, умноженную на X, что по сути это еще один способ записать е в степень X на самом деле, я думаю, я думаю, что в какой-то момент я отобразил другую анимацию, которая указала, что потому что я планировал Планировать это сделать, так что позвольте мне, о да, вот и вы, вернитесь в мою файловую систему, вернитесь туда, где вы должны быть. Заходите, он жалуется, потому что есть несколько разных. Это будет похоже на то, что есть О, замените, оно появляется на другом экране. Подождите, почему да, хорошо, замените? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "Поместите туда все, что видите. И теперь мы возвращаемся к тому, что мы все это все это просто для того, чтобы я мог красиво записать. Если вам неудобно думать об этом как о выражении R, умноженного на X, этот бесконечный многочлен Просто в затылком e на R, умноженное на X, и мы будем меняться вокруг R, поэтому я буду следовать точкам воображаемой оси, и я буду следовать точкам реальной оси, и давайте посмотрим, что это даст. это все очень быстро, так что позвольте мне обдумать это немного медленнее, все отрицательные числа, все, что угодно. Это отрицательное действительное число будет сжато в диапазоне от 0 до 1. Что должно иметь смысл в отрицательном? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a для отрицательного действительного числа - это что-то между 0 и 1, и мы специально отслеживаем f отрицательного 1, которое будет отображаться вокруг любого значения 1 над e, равного 30 0.37 f из 1 приземляется на e, как и ожидалось, вот что такое exp of 1, f of I приземлится на один радиан вокруг единичного круга, и это довольно забавно проследить вдоль всей воображаемой оси здесь, как воображаемая ось оборачивается вокруг круга и что происходит, когда мы меняем это значение R? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "Нам могут понадобиться значения R здесь. Он растягивает вещи по-другому, поэтому, когда мы увеличиваем его до 2. Вы знаете, что он растягивает реальную ось намного больше, так что f из 1 оказывается там, где e в квадрате немного выше 7 f отрицательного значения. 1 намного ближе к 0, f I — это 2 радиана. Вращение вокруг круга f отрицательного I — отрицательное вращение на 2 радиана. реальная ось довольно сильно растягивается. Вы знаете, что f из 1 находится на расстоянии от e до числа пи, что очень близко к 20 плюс пи, что всегда весело, а f от отрицательного 1 очень близко к 0, так что это действительно растянуто. ось И это также растягивает объекты в направлении единичного круга, так что путь к f I или f отрицательного I проходит половину круга, так что теперь все в порядке. Как бы мы подумали о такой функции, как? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "Мы бы также написали как X из X натурального логарифма, умноженного на X, поэтому мы как бы перемещаем нашу желтую точку, представляющую значение R To, около 0.69 по-прежнему не является мнимой частью, а является действительным числом 0.69 или около того. Это натуральный логарифм 2. Вы можете видеть, что f из 1 попадает на 2. Вот почему мы хотим вызвать эту функцию 2 для X f из 1 половины, на самом деле извините f из отрицательных 1 приземляется прямо на 1 половину f из Это какой-то обход по единичному кругу, в частности, он будет равен 0.69 радиан вокруг единичного круга. Теперь мы могли бы немного повеселиться и сказать, что произойдет, если мы изменим это значение на 0, а не на 0.69 вместо того, чтобы быть натуральным логарифмом 2, умножьте его на натуральный логарифм 2, чтобы мы действительно думали о Нечто, что могло бы иметь экспоненциальную основу. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "Что я в степени я в данном случае его пихаю в районе 0.2 около одной пятой. Но есть много разных экспоненциальных функций, которые обладают свойством помещать f из 1 в число I. Итак, если бы мы увеличили его еще больше, я не думаю, что у меня есть здесь анимация. Но если бы мы взяли эту желтую точку и поднимайте ее вверх, пока она не достигнет 5 половинного числа пи. Что вы увидите, это единичный круг? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "Вращается вокруг себя так, что f отрицательного f, равного 1, вращается вокруг еще 2 пи радиан и приземляется там, где оно есть. Но это растягивает реальную ось намного больше. В этом смысле другой вывод I на I есть гораздо меньшее число. Это было примерно 0.0003 или около того. Но мы также можем увидеть то, что, на мой взгляд, довольно забавно. Что произойдет, если мы рассмотрим альтернативные выражения, которые мы хотим интерпретировать как 2 в степени X, верно? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "У нас есть X из R, умноженный на X, и R равно этому значению, которое представляет собой натуральный логарифм 2 плюс пи, умноженный на I. Это означает, что когда мы подключаем 1, f из 1 имеет отрицательное значение 2, поэтому мы хотим написать эту функцию как отрицательное 2 в степени X справа, и это на самом деле то, что Вы знаете, это немного обманчиво просто, когда мы записываем отрицательное число в степень X. Отрицательное 2 в степени X на первый взгляд это не обязательно выглядит так, как это приносит нам в комплексные числа любым способом, но, конечно, когда мы подставляем даже такое значение, как 1 половина. Когда мы как бы запрашиваем квадратный корень из минус 2, мы понимаем, что хотим записать это как что-то вроде I, умноженного на квадратный корень из 2 Но если бы вы посмотрели на эту функцию с отрицательным значением 2 в степени X в полной комплексной области, с которой она имеет дело. То, на что вы смотрите, — это функция, которая принимает значение 1 в отрицательное 2. то же самое происходит и с остальной частью вещественной числовой прямой, она как бы раскручивается по спирали наружу? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "Итак, мы видим, что f отрицательной 1 находится на отрицательной 1 половине. О том, чего бы вы ожидали, если бы следовали до f, равной 1 половине. Оно располагалось бы точно на воображаемой линии, а f 1 половины было бы квадратным корнем из 2. Ну, мой мышь не там, где я хочу. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "Это будет примерно квадратный корень из 2, умноженный на I, и по мере того, как вы продолжаете дальше, это показывает вам все действительные степени значений от отрицательных 2 до X, которые обязательно вращаются по спирали. Но мы также могли бы поднять наше значение R еще выше и получить его. примерно до тау раз I примерно до шести целых два восемь раз I, и в этом контексте это еще одна функция, которую мы хотели бы записать как что-то вроде 2 до X, потому что для любого целого числа, которое вы подставляете для X, это будет похоже на повторное умножение. И у него даже есть разумные значения для таких вещей, как 1 половина, где он выплевывает отрицательный квадратный корень вместо положительного квадратного корня, но на самом деле он делает преобразование в плоскость. Куда он помещает все, является реальным Числовая линия в конечном итоге представляет собой очень туго закрученную спираль, которая вращается по спирали и просто закручивается по спирали таким образом, что f из 1 попадает прямо на число 2. Таким образом, именно в этом смысле мы могли бы сказать, что 2 к X правдоподобно интерпретируется как отдельная экспоненциальная функция, отличная от той, к которой мы традиционно привыкли. Итак, я думаю, что со всем этим я оставлю дела на сегодня. И я просто оставлю вам пару насущных вопросов для размышления, хорошо, так что если вы хотите думайте о I как о многозначном выражении, верно, вы могли бы сказать, что мы принимаем соглашение. Причудливо вы бы сказали, что выбираете ветвь функции натурального логарифма. И, возможно, это запирает вас в этом существе e для отрицательного числа пи половинки. Но если вы скажете, что этот вид хочет иметь бесконечно много разных значений, подобных тем, которые мы видели. Сколько значений хотят иметь от 2 до 1 трети в одном и том же смысле? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "Десятые части хотят быть сформулированы по-другому для всех, скажем так, для всех экспоненциальных функций F от X, которые удовлетворяют, о, я где-то записал это f от X, которое удовлетворяет всем этим свойствам, которые я написал, так что, если она удовлетворяет всем из них, и если f из 1 равно 2. Сколько разных выходных данных мы получим, когда подключим X, равный 3 десятым, для различных вариантов какой функции? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "От 2 до числа Пи для различных функций, которые могли бы представляться от 2 до X, если мы думаем о 2 до X как о своего рода экспоненциальной функции. Экспонента в смысле такого рода абстрактных свойств, и если мы да, если мы если у нас есть класс различных таких функций, и мы хотим подключить число pi, это заставляет меня смеяться. Просто потому, что это такой забавный ответ, который всплывает, когда вы пытаетесь об этом подумать, так что это вопросы, которые Я оставлю вас, и я думаю, что это, вы знаете, мой главный вопрос при подготовке к сегодняшней лекции заключался в том, хочу ли я, чтобы она была своего рода описанием подобных абстрактных свойств экспоненциальных функций. И для меня это просто здорово, что я начинаю с этих абстрактных свойств. вы зацикливаетесь на идее от e до rx или более. Просто вы знаете, я думаю, что более честно написано выражение r, умноженное на x для разных значений r. Что это запирает вас настолько далеко, Но это не запирает вас настолько, насколько однозначное представление о том, что 2 в степени x должно быть гораздо меньше чем-то вроде I в степени x. Риск, конечно, заключается в том, что иногда люди не любят абстракцию, а иногда это не кажется доступным. Но если это Если вы знаете, просто дайте мне знать, я думаю, что есть целый интересный круг мыслей, который окружает все эти вещи, включая электровышки, потому что, если вы хотите На самом деле говорить о электростанциях, как мы говорили в прошлый раз, в контексте комплексных чисел или даже с отрицательными основаниями. Вы должны продумывать подобные вещи, поэтому на экране у нас возник вопрос. Да, что произойдет, если мы сделаем это для I в степени I? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "Титрование, вы знаете, давайте просто попробуем это, давайте просто попробуем энергетическую башню, где мы поднимаем I до заданной мощности и посмотрим, что из этого выйдет, поэтому он не планировал это делать Но мы можем, мы всегда можем поднимите Python и, по сути, сделайте то, что мы делали в прошлый раз. Итак, это будет работать так: мы начинаем с некоторого базового значения, а затем для некоторого диапазона. Что мы делали, мы брали и собираемся переназначить это должно быть что угодно. База, которую в данном случае я возвел в степень a, должна быть Хорошо, круто, поэтому мы собираемся сделать это, а затем распечатать значение a, давайте просто сделаем это для Да, это гораздо большее число, например 200. Похоже, что происходит следующее: иногда такие вещи могут привести к хаосу. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "У нас действительно есть, поэтому позвольте мне импортировать NumPy, чтобы у меня была экспоненциальная функция, отпустите меня Для нашего большого диапазона, как у нас было раньше. Вместо того, чтобы писать это, как вы знаете, что-то вроде I в степени X, я собираюсь написать это как экспоненциальная функция другой константы справа Другая константа, которую я собираюсь сделать, я хочу, чтобы она была равна 5 половин пи, поэтому я буду делать 5 половин пи раз I, так что это комплексное число, и у него есть 5 половин пи, как мнимая часть Итак, это умножено на 5 пи пополам, и что я делаю? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/spanish/sentence_translations.json b/2020/ldm-i-to-i/spanish/sentence_translations.json index a7f06f5b5..49f99556e 100644 --- a/2020/ldm-i-to-i/spanish/sentence_translations.json +++ b/2020/ldm-i-to-i/spanish/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "Entonces, si estás comenzando en el número 1, tu velocidad inicial es caminar directamente hacia 0 y a medida que caminas aún más abajo, si estuvieras sentado en 1 mitad, entonces todavía estarías caminando hacia 0, pero ahora tu vector de velocidad Sería negativo 1 veces donde estás, que es menos 1 mitad. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "Y una pregunta interesante será: ¿existe una función que parezca razonable escribir para esto? Porque sabes, si la vamos a escribir como i a la x, no solo debería satisfacer esto, sino que también debería satisfacer, ¿sabes cuándo? Conectamos el número uno y obtenemos i, presumiblemente i al de alimentación, sin embargo, estamos pensando que esta función debería ser i. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "Así que tenemos 5 pi i mitades, lo cual es absolutamente otro valor que podríamos sustituir por x aquí y simplemente para explicarlo un poco más visualmente si miráramos hacia atrás a nuestro círculo aquí donde tenemos el momento caminó durante una cantidad de tiempo igual a pi mitades, que es 1.57 ¿Qué pasaría si en lugar de eso diésemos otro giro completo y recorriéramos otras mitades de pi para llegar a pi, lo cual sabes que podríamos registrar ahí es donde la e al valor de pi i es que caminamos otras mitades de pi, caminamos otras mitades de pi que al En este punto, habríamos dado un círculo completo para volver a uno y luego caminamos durante cinco mitades de pi, que numéricamente son aproximadamente 7.85 sí, ese es absolutamente otro número que nos coloca encima de i y si tuviéramos que pasar por todo el galimatías de reexpresar i elevado a i escribiendo primero e a las 5 mitades de pi i elevado a i esas i multiplíquelo para volverse negativo y estaríamos viendo e elevado a las mitades negativas de 5 pi, que es un número muy diferente. En realidad, podemos calcular esto. No estoy seguro de mi cabeza, pero echemos un vistazo a un Desmos. . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "Ese largo, lo que te lleva a un número mucho más pequeño. Pero esa no es la única respuesta que podemos ingresar. Tenemos otras personas que vienen aquí con 3 mitades negativas por i pi. ¿Cuál conoces en términos de un círculo unitario? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "Podríamos pensar en decir oye si quiero llegar a I en lugar de caminar 90 grados pi mitades radianes de esa manera, ¿qué pasa si camino 270 grados en la otra dirección 3 pi mitades radianes, lo cual tal vez lo consideraré negativo porque la convención es? normalmente eso en el sentido contrario a las agujas del reloj es positivo. Esa es absolutamente otra forma de expresarlo y eso nos daría una respuesta diferente si tuviéramos e elevado a las 3 mitades negativas de pi i. Todo elevado a la potencia i. Pasamos por el mismo juego. Ahora i al cuadrado se cancela con a. negativo que ya está ahí, y tenemos 3 mitades pi positivas y numéricamente esto nos da una respuesta aún diferente a la que teníamos antes. Si vamos y decimos oye, ¿qué es e para los 3 pi y no 3 o 3 pi? mitades 111 punto 3 1 tipo de número muy diferente al que vimos antes 111 punto ¿qué era? 111 punto 3 1 genial 111 punto 3 1 más o menos Y nuevamente, en términos de intuición, lo que podrías estar preguntando es supongamos que tenemos esta rotación dinámico Pero retrocedemos en el tiempo y vemos cuánto tiempo hace en el tiempo lo que tengo que ser. De modo que si jugara las cosas hacia adelante desde allí, aterrizaría en el número uno, mi condición inicial, y tienes que retroceder en el tiempo 3 pi mitades de unidades. Y luego, si tuvieras que traducir a la dinámica de decadencia, que es lo que hace la elevación a la vista en este contexto, dirías si estoy empezando en el número uno. Pero quiero retroceder en el tiempo y decir ¿Dónde debería haber empezado si? ¿Quiero decaer hasta terminar en el número uno? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "Después de 3 pi mitades de unidades de tiempo, la respuesta evidentemente comienza en alrededor de ciento once para ese tipo de decaimiento exponencial. Y puedes ver hacia dónde va esto, donde en realidad hay infinitos valores diferentes que podríamos conectar para X si pensando en e a la X como yo y la gente ha entrado mucho más aquí Disculpe por tirar mi alfiler al suelo como se hace con el clásico para el tercer lugar 9 mitades pi gran elección 1729 mitades pi ustedes son mis montones favoritos opciones diferentes, infinitos valores diferentes, lo cual se siente un poco desconcertante al principio, porque miramos una expresión. Parece que sabes que habrá algunos cálculos. Simplemente conecto eso a mi calculadora y veo qué aparece y tenemos múltiples valores diferentes. valores para ello Entonces, ¿qué está pasando aquí, verdad? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "La raíz cuarta de 16 debería ser 2 y la respuesta termina siendo buena. Adoptamos una convención cuando hay múltiples opciones como esta cuando tienes una función multivalor. A menudo simplemente elegimos uno de esos valores para que sea lo que queremos decir cuando queremos trátelo como una función, como algo con una sola entrada y una sola salida en una jerga más elegante. Esto surge todo el tiempo cuando tratamos con números complejos, la idea de algo como una operación, como querer tener múltiples valores, a veces necesitarás escucha la frase rama ¿Dónde eliges una rama de la función de raíz cuadrada? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "Porque hay múltiples respuestas diferentes. Sabes, pensamos que I nuevamente es esta rotación de 90 grados. Y si pensáramos en ello como una rotación de 90 grados, se siente como si la raíz cuadrada debería ser. Sabes, algo que se encuentra en un ángulo de 45 grados. Tal vez ese sea el cuadrado. raíz de I que podríamos escribir muy explícitamente como raíz 2 sobre 2 raíz 2 sobre 2 I Eso es solo usar trigonometría, pero si pensáramos en I como una rotación negativa de 270 grados, se sentiría como si la mitad de eso hiciera la mitad de esa operación. debería llevarnos al otro lado Tal vez el número que está aquí debería ser la raíz cuadrada de I y eso en realidad es solo el negativo de lo que vimos antes Raíz negativa de 2 sobre 2 menos raíz de 2 sobre 2 por I Ahora, en el contexto real funciones valoradas podemos decir sí. Simplemente elige la raíz cuadrada como sea cual sea la respuesta positiva, pero ¿cuál de estas consideras que es la respuesta positiva? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "Y creo que dices bien. Sabemos qué es esto, lo definimos como la raíz cuadrada de 2, todo está muy bien, pero ¿y si dijera que abordemos esto de la misma manera que acercábamos nuestra I a la expresión I? Primero quiero expresar las cosas como e a algo correcto y luego voy a elevar eso a la mitad multiplicando la mitad por el exponente. Y digo, está bien, supongo que puedo hacer eso e a lo que es. igual a 2, bueno, ese es el logaritmo natural de 2. Es una constante que ronda 0.69 más o menos Si elevamos e a esa potencia obtendremos 2, por lo que podríamos pensar en esto como e al logaritmo natural de 2 por 1 mitad y, si quisieras, ¿estabas pensando en e a la x? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "Sabes que esto podría ser un poco excesivo en el contexto de los números reales. Pero si estuvieras pensando en e elevado a x como una abreviatura de esta función x, podrías sustituir el valor 0.69 por 1 mitad, lo que supongo que sería alrededor de 0.345 Ish algo así. Introduces ese valor muy concreto en tu polinomio y ves lo que genera, y generará alrededor de 1.414 un buen número real de raíz cuadrada de 2, lo que cabría esperar. Pero si hacemos lo mismo que estábamos haciendo con I y reconocemos que en realidad hay múltiples respuestas diferentes cuando queremos escribir algo como e elevado a una potencia, también podríamos escribir esto Esto puede parecer gracioso, pero podríamos escribirlo como e al logaritmo natural de 2 más 2 pi I Todo eso elevado a 1 mitad Justo después de todo, este valor será igual a, podrías descomponerlo como si fuera e al logaritmo natural de 2 Multiplicado por e elevado a 2 pi I Este solo tiene el efecto de rotar las cosas 360 grados, por lo que será igual a 1 Entonces estamos viendo 2 por 1 genial, eso se siente como una sustitución válida y, sin embargo, cuando jugamos el mismo juego de tomar esto y elevarlo a una potencia y tratarlo multiplicando la potencia en el exponente mira lo que sucede Tenemos e al logaritmo natural de 2 por 1 mitad más Bueno, ¿cuánto es 2 pi I por 1 mitad? bueno, eso será pi multiplicado por I. Ahora, esta primera parte e del logaritmo natural de 2 por 1 mitad terminará siendo la familiar raíz cuadrada de 2, eso está muy bien, pero lo multiplicaremos por e el pi I Correcto y muy famoso e al pi I es negativo 1 Entonces, en este caso parece sugerir que si estamos resolviendo esta expresión 2 elevado a 1 mitad. Jugando con las diferentes respuestas podríamos conectar algo como e a la X igual a 1 mitad lo que obtenemos es otra respuesta que tradicionalmente podríamos escribir como esta raíz cuadrada negativa de 2 y aquí quiero decir que es un poco divertido que tenga múltiples valores para mirar 2 elevado a 1 mitad y Digamos que eso no es igualar una cosa, pero según las elecciones que hagamos, podría igualar varias cosas diferentes. Pero las dos cosas que podrían parecer bastante razonables. Si va a haber algo que sea 2 a 1 mitad, parece que debería ser El positivo. La raíz cuadrada con la que estamos familiarizados o la variante negativa de eso que en realidad no parece ser un problema. Y de hecho, podríamos jugar este juego aún más, donde permítanme pedirles respuestas aún más creativas a esta expresión. porque tal vez podamos encontrar otros poderes divertidos de algo como 2 elevado a X a medida que comenzamos a conectar varios valores diferentes de X en función de la sustitución que hagamos si cumplimos con las mismas reglas que estábamos usando al evaluar I elevado a potencia I Entonces esta vez la pregunta pregunta o especifica que una solución de la ecuación e a la x es igual a 2 es el número real Logaritmo natural de 2 ok, eso lo sabemos. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "la respuesta a la pregunta e a la x es igual a 2 y nuevamente la creatividad es bienvenida, así que te daré otro momento para eso. Seguiré adelante y bloquearé algunas respuestas aquí si te parece bien. No estoy seguro de cuánto tiempo necesariamente es necesario hacer la entrada matemática dependiendo del dispositivo que estés mirando, pero no te estreses demasiado si es antes de que tengas la oportunidad de ingresar la pregunta que deseas y la respuesta que deseas que responda. Entonces parece 131 de ustedes han ingresado a la variante donde tomamos Ln de 2 y sumamos 2ii y creo que estoy escribiendo esta pregunta. Por error marqué una de las respuestas como correcta cuando en realidad hay bastantes respuestas correctas diferentes. Así que eso depende de mí. por el hecho de que no sé si a alguno de ustedes le parece oh. Es rojo, se equivocó al ingresar Ln de 2 más 42. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi, que por supuesto es una gran opción. Pero también podrías tener algo como 4 pi I más el registro natural de 2 o 6 pi I O realmente cualquier múltiplo entero de 2 pi I si agregas que no afecta a e X Porque simplemente tiene el efecto de multiplicar por e hasta 2 pi I, que es el efecto de multiplicar por 1 y, nuevamente, esto tiene una consecuencia curiosa en la que parece generar resultados razonables cuando lo hacemos como otro ejemplo. parece que la segunda expresión ingresada más común fue que podríamos reemplazar 2. Entonces pensemos que estamos pensando en 2 elevado a 1 4, está bien, hubo una sugerencia de que reemplazáramos 2 con e al logaritmo natural de 2 más 4. pi I Está bien Más 4 pi I y elevamos todo eso al 1 4 bien, si jugaras el mismo juego obtendrías e Al registro natural de 2 por 1 4, y estaríamos multiplicando por e para el pi I Ahora, la primera parte de eso será la cuarta raíz positiva habitual de 2, lo que queremos decir cuando ingresas una expresión como la cuarta raíz de 2 en una calculadora, un pequeño número positivo, pero luego esta segunda parte es negativo 1 entonces parece decir Sabes, si interpretáramos 2 de esta manera diferente elevándolo al 14 Sabes que no es la respuesta habitual que obtenemos, pero es una respuesta razonable. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "Habríamos estado mirando pi mitades por I y en lugar de multiplicar por Negativo 1 hubiéramos multiplicado por I. Lo cual nuevamente es una respuesta válida, parece un resultado razonable para algo como 2 elevado a 1 4 Entonces, cuando estás Mirando el hecho de que I a la potencia parece tener múltiples valores diferentes. Correcto, tenemos este fenómeno divertido en el que podemos conectar e a las mitades de 5 pi I, mitades negativas de 3 pi I y obtenemos lo que parecían respuestas tremendamente diferentes. algo súper pequeño algo súper grande todo muy diferente de la 15 aproximadamente 15 respuesta que encontramos antes aquí Es exactamente el mismo fenómeno que cuando preguntas algo como cuánto es 2 elevado a 14 y reconoces que en realidad hay múltiples soluciones diferentes a la expresión X al 4 es igual a 2 4 soluciones diferentes de hecho y lo que estás viendo es el hecho de que hay múltiples soluciones diferentes A la expresión e a la X es igual a algún tipo de base si esa base es I si esa base es 2 Sea lo que sea y una forma en que podríamos pensar en esto es que cuando se trata de números reales, las cosas son simplemente hermosas, las cosas son agradables. Hay relaciones uno a uno. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "Si queremos pensar en funciones exponenciales, déjame cubrir algunas de estas cosas. Tenemos este bonito ida y vuelta donde puedes elegir expresar cualquier exponencial como una base para X, como 2 elevado a X. O puedes expresar ese mismo exponencial como X de R multiplicado por X, que sabes que es el polinomio al que nos referimos Siempre que nos referimos implícitamente cada vez que escribimos algo como e a la X Y hay un hermoso vaivén porque puedes simplemente tomar un logaritmo natural de B Y te da una respuesta suponiendo que B es un número positivo. Y eso es lo mismo que decir que X de R es igual a B. Entonces, una forma en la que hablé de esto anteriormente en la serie es que si estuvieras mirando el familia de todos los exponenciales posibles, cierto, podríamos escribirlos como X de R multiplicado por X y cambiar lo que es R. Y esto es exactamente lo mismo que escribir e elevado a R multiplicado por X si eso es algo con lo que te sientes más cómodo Entonces, e elevado a R multiplicado por XX de R multiplicado por X, eso es lo mismo, podríamos pensar en cambiar lo que es. Pero, por otro lado, si pensaras en todos los exponenciales posibles como alguna base, déjame hacer una base elevada a la potencia de X y vamos. cambiar cuál es esa base. Al principio parece que es un tipo diferente de expresión para manipular, pero es solo otra forma de expresar la misma familia. Y una forma en la que podrías pensar sobre esto. ¿Cómo pensamos a qué base corresponde? a si estamos pensando de manera un poco más abstracta como Exp de R multiplicado por X y hay una razón por la que estoy haciendo esto porque estamos a punto de aplicar esto a números complejos donde se verá más extraño, así que síganme aquí si en lugar de mirar esa base una cosa que podría hacer es decir ¿cuál es el valor? ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "Podría tener exp de R multiplicado por X donde tal vez R sea algo así como cero coma seis nueve Pero podría reducir eso en dos pi I Y eso no cambia la base a la que correspondería que aún correspondería a dos O podría muévalo hacia arriba en dos pi I eso no cambia la base a la que corresponde porque en todos esos casos, cuando reemplazamos X es igual a uno, obtenemos lo mismo. Sin embargo, todos estos para diferentes valores de X son funciones distintas. Esto es por qué vimos múltiples valores diferentes para I elevado a I Debido a que I elevado a X es una función ambigua en ese contexto, no sería ambiguo si decidiéramos qué valor de R De modo que lo que estamos representando es exp de R multiplicado por X qué valor de r. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "Es una función inequívoca, pero en ese punto parece que tal vez lo que queremos es dejar de pensar en las cosas en términos de alguna base elevada a la potencia X. Tal vez tan pronto como estemos en el contexto de números complejos, deberíamos simplemente escribir todos ellos como exp de algunos tiempos constantes X, si no es por otra razón, deja muy claro cómo realmente conectamos números si queremos hacer un cálculo o simplemente hacer matemáticas, además tenemos este lindo polinomio infinito que conéctelos y le presentaré otro caso de que esta es quizás la forma correcta de pensar en exponenciales. Tan pronto como nos extendamos a otros dominios, cosas como números complejos y para eso simplemente retrocedamos. De regreso al timbre llegaron algunas cosas, volvamos a la forma original en la que extendemos la idea de exponenciación y simplemente pensamos en lo que es 2 elevado a la X. Bien, sabemos cómo pensar en esto para los números naturales. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "Sabes algo como 2 elevado a 3. Multiplicación repetida. ¿Cómo es que primero te enseñan a pensar en algo como 2 elevado a X para cantidades fraccionarias o para cantidades negativas y cosas así? ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "Por lo general, te enseñan que 2 elevado a 1 y medio debería ser algo que sepas que si lo multiplico por sí mismo y esto sigue las reglas habituales que hacen los exponenciales al contar números, donde podemos sumar cosas en ese exponente, debería obtener 2. al 1, entonces debería ser algún número que cuando lo multiplique por sí mismo obtenga 2 y sabes que en ese punto tienes una opción, tal vez sea positiva. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "Tal vez sea negativo, pero si siempre decides hacer la elección positiva, podrás obtener una buena función continua de este mismo trato si preguntamos acerca de los números negativos. ¿Qué debería ser 2 elevado a 1 negativo? Eso debería ser algo. donde cuando lo multiplico por 2 al 1? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "Esto me da 2 elevado a 0 y esa es una especie de justificación para nuestra convención de que los exponentes negativos parecen 1 mitad. Pero lo que realmente está pasando aquí es que estamos diciendo que, sea lo que sea, debería ser algún tipo de función que satisfaga esta propiedad f de a más b es igual a f de a multiplicado por f de by Además, el hecho de que la base sea 2 básicamente nos dice que no es una función cualquiera. Es una función en la que cuando conectamos 1 obtenemos 2. Y como un poco ya sabes Pregunta de estilo de verificación de cordura para ver si estás siguiendo algunas de las implicaciones aquí. Quiero preguntarte qué es. No lo llamaré softbol, pero esto no pretende ser una pregunta increíblemente profunda. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "necesariamente. Es más bien una verificación si sigues la idea de comenzar de manera abstracta con las propiedades de una función y luego deducir formas en las que podríamos querer escribirla en función de esas propiedades. Si f de x satisface esta propiedad exponencial f de a más b es igual a f de a multiplicado por f de b para todas las entradas Y también satisface f de 1 es igual a 2 cuál de las siguientes afirmaciones es verdadera Es decir, cuál de las siguientes afirmaciones es necesariamente cierta No importa cuál función esté iniciando con y aquellos de ustedes que recuerdan cuál fue la conferencia Es de cuál estábamos hablando cómo interpretar lo que realmente dice la fórmula de Euler Hice una pregunta de este estilo donde descuidé una sola condición, ya saben, no escribí el hecho de que queremos asegurarnos de que f de x sea distinto de cero en todas partes y eso causó cierta confusión, lo cual es genial, que aparezca confusión en la pantalla, algo que nos sucede a todos. Pero la intención era básicamente mostrar que esta propiedad abstracta de algo que convierte la suma en multiplicación es suficiente para que básicamente quieras escribir la función como lo que sea igual a uno elevado a algún tipo de potencia. Este es el espíritu de la pregunta. Ahora tenemos un par de preguntas sobre las torres de energía. eso parece haber aparecido aquí, lo cual está muy relacionado con la última vez. Dejemos de lado la pregunta sobre la torre de energía por un momento para que primero tengamos una sensación más profunda de ¿Qué debería significar aquí la exponenciación? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "Porque podemos ser lo que quiero afirmar, podemos responderlo de múltiples maneras diferentes. Entonces, si me das solo una, hablaremos sobre torres de energía. Y luego, así como una recta numérica se puede representar en una escala logarítmica, ¿Se puede hacer lo mismo con un plano complejo? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "Sí. De hecho, hay una visualización a la que llegaré en un momento donde hacemos algo bastante similar a eso. Porque lo que haremos es jugar con diferentes funciones exponenciales X de R multiplicado por X. Pero estamos vamos a cambiar ese valor de R que va a estar representado por un pequeño punto amarillo. Así que hablaremos de esto. No va a mapear todo el plano, sino solo un par de puntos de muestra del eje real y el eje imaginario. Pero la idea es que a medida que nos movemos alrededor de cuál es esa constante, podremos visualizar las diferentes cosas que le hace al avión y, efectivamente, es como si estuviera convirtiendo el eje x en una escala logarítmica y luego envolviéndolo. el eje imaginario a lo largo de un círculo Y luego, tan pronto como ese valor de R se vuelve imaginario, intercambia el papel de esos Los números reales se colocan en el círculo y los números imaginarios se colocan en un eje positivo de escala logarítmica, una gran pregunta, supongo que los tres Estoy avanzando hacia dónde quiero ir, pero es bueno ver que es ahí donde la gente piensa eso en este caso. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "explícitamente Algo como f de 5 es lo mismo que f de 1 más 1 más 1 más 1 más 1 Que es lo mismo que f de 1 multiplicado por sí mismo 5 veces debido a esta propiedad Que si f de 1 es 2 es lo mismo como 2 elevado a 5 y luego algo como f de negativo 5. Debería darse el caso de que cuando lo multiplicamos por f de 5 obtenemos lo que sea f de 0 y no está inmediatamente claro qué es f de 0, pero podríamos decir que f de 1 más 0 es igual a lo que sea que f de 1 sea multiplicado por lo que es f de 0, pero f de 1 es igual a 2. Entonces esto también es igual a 2, por lo que estamos diciendo que 2 es igual a 2 multiplicado por algo, bueno, ese algo. tiene que ser un 1, por lo que en este contexto esto garantiza que f de 5 negativo es 2 elevado a 5 negativo, es 1 sobre 2 elevado a 5. Podríamos escribir esto explícitamente como 2 elevado a 5 negativo, lo cual es todo para decir que estas dos propiedades juntas hacen Realmente queremos escribir la función como 2 a la X porque cualquier número de conteo que le pongamos va a satisfacer. Parecerá que multiplicar por sí mismo esa cantidad de veces cualquier número fraccionario que le pongamos va a satisfacer estas propiedades. que queríamos Y quizás te preguntes si es único y en el contexto de funciones con valores reales en realidad lo sería. Pero en el contexto de funciones con valores complejos Habría múltiples funciones de este tipo que podríamos escribir para esta, una de las cuales es la que éramos Mirando antes Donde podríamos tener una función definida como exp del registro natural de 2 más 2 pi Yo todas esas veces X Bueno, perdonen el descuido aquí, me emociona escribir sobre esto Y esta es en realidad una función diferente ya que evidenciado por lo que sucede si reemplazas X es igual a 1 mitad. Vimos un poco antes cómo cuando ingresas 1 mitad lo que obtienes es la raíz cuadrada negativa de 2 y luego, si reemplazas 1 cuarto, obtienes No la raíz cuarta de 2 pero multiplico la raíz cuarta de 2, por lo que es una función diferente. Pero aún satisface estas propiedades y en cierto modo nos hace querer escribirlo como 2 elevado a la X. Y sugiere que tal vez 2 elevado a la X es ambiguo. un poco de notación Y deberíamos escribir todo en términos de exp de R multiplicado por algo, pero quizás te lo preguntes. Sabes, tal vez no estemos siendo lo suficientemente creativos con todas las funciones que satisfacen esta propiedad. Tal vez haya una ambigüedad cuando escribimos exp. de R multiplicado por algo y hay diferentes valores de R que podrían entrar en juego. Pero voy a hacer una pequeña afirmación y luego tal vez dar un esbozo de cómo se vería la prueba si quieres. ¿Cuál es eso? Digamos que tienes alguna función compleja F, y primero satisface las siguientes propiedades. Puedes tomar una derivada de ella. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "Es diferenciable, lo que simplemente evita que sea algo discontinuo y totalmente desordenado. Es como tomar algunos valores aleatorios dependiendo de que conozcas el lapso de cualquier espacio vectorial sobre, no sé, cantidades fraccionarias en las que quizás quieras pensar de maneras locas. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "Es una buena función. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "Eso es diferenciable. No es igual a 0 en todas partes, así que la condición que se me olvidó y olvidé para qué conferencia o algo así y luego tiene esta propiedad central de que convierte la suma en multiplicación. Si tienes esa función, afirmo que hay un número único, tal vez realmente debería especificar que existe un número complejo único R para que puedas escribir F de X como básicamente esta función exponencial de R multiplicada por ese valor X. Lo cual es, básicamente, decir que si tienes X como función, esto polinomio infinito con buenas propiedades derivadas y todo eso, si tienes esto, tienes cada exponencial que quieras en un sentido genérico muy abstracto de la palabra exponencial, simplemente basado en una propiedad que podríamos desear de él y el bosquejo de la prueba sería Mire algo como esto si quiere ver primero cuál es la derivada de este valor que asumimos que existe en todas partes, ¿verdad? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "Podemos factorizar F de X de la expresión por completo y todo el límite se expresa sólo en términos de H. Lo cual, si piensas en lo que significa en el contexto de las derivadas y en el hecho de que F de 0 necesariamente es igual a 1, toda esta expresión limitante es solo una constante, pero más específicamente es cualquiera que sea la derivada de nuestra función en 0. Entonces tienes esta cosa curiosa en la que, si conoces su derivada en 0, eso determina cuál es su derivada en todas partes. Y en el contexto de funciones exponenciales, esto es de esperar que sea bastante familiar porque todo lo que realmente estamos diciendo es que la derivada de una función exponencial es proporcional a sí misma y esa constante de proporcionalidad es igual a cualquiera que sea la derivada en 0. Todo esto está redactado de manera muy abstracta y tal, pero el propósito es enfatizar que es no necesariamente son solo funciones que ya consideramos elevadas a la potencia X. Pero es una clase potencialmente mucho más amplia de funciones que simplemente satisfacen esta propiedad abstracta de convertir la suma en multiplicación. Pero si tienes eso, en realidad garantiza que también tienes una segunda derivada Y, de hecho, una tercera derivada y demás, porque la función derivada es simplemente proporcional a sí misma. Entonces, para tomar la enésima derivada, simplemente mira esa constante de proporcionalidad y la elevas a la potencia n y luego desde aquí podrías hacer una Expansión de la serie de Taylor y podría dejar eso como una especie de tarea avanzada para aquellos de ustedes que se sienten cómodos con la serie de Taylor en esa idea, especialmente si quieren entremezclar la idea de cualquier función diferenciable que sea diferenciable en el sentido de números complejos, que es una especie de tema definitivamente universitario. Sabes que puedes mezclar el razonamiento allí como quieras. Pero el razonamiento difuso está permitido en el contexto de alguien que solo conoce las series de Taylor y nada más para tomar esta idea y observar la expansión de Taylor para F y En cierto modo justifica la idea de que existe un número complejo único tal que nuestra función F necesariamente puede escribirse así. Y luego la conexión con los exponenciales normales es siempre que tenga ese valor R. Hacemos esencialmente lo que hacemos en el contexto complejo de los números reales. es si miras exp de esa función de ese valor R y lo escribes como base. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "Podríamos interpretar que eso significa no solo exp de pi mitades I por X, sino que también podríamos interpretarlo como exp de 5 pi mitades I por X y Estas son funciones separadas Y hay una familia infinita de funciones separadas que sentimos que deberíamos escríbalos como I a la X Entonces, la expresión I a la I, a menos que haya adoptado un estándar para lo que eso necesariamente significará. Cuando dice que tiene infinitas salidas, otra forma de pensarlo es que La función I a la X con la notación que tenemos es un poco ambigua. Ahora con todo eso, comencemos a visualizar algo de esto porque creo que es divertido. Y sabes, dime si esto es una imagen útil o una imagen más confusa, pero Lo que vamos a hacer es mirar esta función exp de R multiplicado por X, que es básicamente otra forma de escribir e elevado a X. De hecho, creo... creo que rendericé una animación diferente en algún momento que especificaba eso. porque estaba planeando Planeando hacer eso, así que déjame, oh sí, ahí estás, regresa a mi sistema de archivos, regresa a donde se supone que debes estar. Continúa ahí, ¿se está quejando porque hay múltiples diferentes? Será como si hubiera un Oh, reemplázalo, aparece en la otra pantalla. Espera, ¿por qué es así? ¿Está bien, reemplázalo? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "Coloque lo que vea allí. Y ahora volvemos a oh, allí tenemos todo eso, todo eso solo para poder escribirlo bien. Si no se siente cómodo pensando en ello como exp de R multiplicado por X este polinomio infinito Solo en el detrás de tu cabeza e a R multiplicado por X y vamos a variar alrededor de R, así que seguiré los puntos del eje imaginario y seguiré los puntos del eje real y veamos qué hace esto. Bueno Eso es todo bastante rápido, así que déjenme pensarlo un poco más lentamente. Todos los números negativos. Cualquier cosa. Ese es un número real negativo que quedará aplastado en el rango entre 0 y 1. ¿Cuál debería tener sentido e al negativo? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a a un número real negativo es algo entre 0 y 1 y estamos rastreando específicamente f de 1 negativo que aparecerá alrededor de cualquier 1 sobre e que sea alrededor de 30 0.37 f de 1 aterriza en e como se esperaba, eso es lo que exp de 1 es f de I aterrizará un radian alrededor del círculo unitario, y es divertido seguir a lo largo de todo el eje imaginario cómo el eje imaginario se enrolla alrededor de un círculo. y ¿Qué sucede cuando modificamos este valor de R? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "Es posible que queramos valores de R aquí. Estira las cosas de manera diferente, así que cuando lo ponemos hasta 2, sabes que estira mucho más el eje real, de modo que f de 1 termina alrededor de donde e al cuadrado está un poco por encima de 7 f de negativo. 1 está mucho más cerca de 0 f de I es una rotación de 2 radianes alrededor del círculo f de I negativo es una rotación de 2 radianes negativos Y, por supuesto, podemos llegar a nuestra fórmula favorita: si fuera pi, tendríamos como nuestra constante de escala Entonces el eje real se estira bastante. Sabes, f de 1 está ubicado en e del pi, que está muy cerca de 20 más pi, lo cual siempre es divertido y f de menos 1 está extremadamente cerca de 0, por lo que está realmente estirado. eje Y también se extienden las cosas en la dirección del círculo unitario de modo que Llegar a f de I o f de negativo I camina a la mitad del círculo, así que eso está muy bien ahora. ¿Cómo pensaríamos en una función como? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "También escribiríamos como X de X del registro natural de 2 veces X, por lo que movemos nuestro punto amarillo que representa el valor de R alrededor de 0.69 todavía no hay una parte imaginaria, solo un número real 0.69 más o menos Ese es el logaritmo natural de 2, bueno, puedes ver que f de 1 cae en 2. Por eso queremos llamar a esta función 2 a la X f de 1 mitad, en realidad lo siento, f de negativo 1 cae justo en 1 mitad f de I Es un paseo alrededor del círculo unitario, muy específicamente, será 0.69 radianes alrededor del círculo unitario y ahora podríamos divertirnos un poco más y decir qué pasaría si cambiáramos esto a en lugar de 0.69 en lugar de ser el registro natural de 2, hazlo I multiplicado por el registro natural de 2 para que realmente estemos pensando en algo que podría tener una base exponencial. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "¿Cuál es I elevado a la potencia? En este caso, lo lleva alrededor de 0.2 alrededor de una quinta Pero hay muchas funciones exponenciales diferentes que tendrían esta propiedad de poner f de 1 en el número I. Entonces, si lo ampliamos aún más, no creo que lo tenga animado aquí. Pero si tomáramos ese punto amarillo y levántelo hasta que llegue a 5 mitades de pi I. ¿Lo que verías es el círculo unitario? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "Se gira sobre sí mismo de modo que f de f negativo de 1 rotaría alrededor de otros 2 pi radianes y aterrizaría donde está, pero estiraría mucho más el eje real, que era el sentido en el que se produce otra salida de I a I. un número mucho más pequeño. Era alrededor de 0.0003 más o menos Pero también podemos ver lo que me parece bastante divertido. ¿Qué pasa si consideramos expresiones alternativas que queremos interpretar como 2 elevado a X, verdad? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "Tenemos X de R multiplicado por X y R es igual a este valor, que es el logaritmo natural de 2 más pi multiplicado por I. Lo que eso significa es que cuando conectamos 1, f de 1 está en 2 negativo, por lo que queremos escribir esta función. como negativo 2 elevado a X, cierto, y eso es en realidad algo que ya sabes, es un poco engañosamente simple cuando escribimos un número negativo elevado a una potencia. Negativo 2 elevado a X, al principio no se ve así, necesariamente nos trae. en los números complejos de cualquier manera, pero, por supuesto, cuando ingresamos incluso un valor como 1 mitad, donde estamos pidiendo una raíz cuadrada de menos 2, nos damos cuenta de que queremos escribir esto como algo así como I multiplicado por la raíz cuadrada. de 2 Pero si miraras esta función menos 2 elevado a X en el dominio complejo completo con el que está tratando Lo que estás viendo es una función que toma el valor de 1 elevado a menos 2 Y si hace eso, ¿qué lo que hace con el resto de la recta numérica real, ¿es una especie de espiral hacia afuera? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "Entonces vemos que f de menos 1 se ubica en la mitad de menos 1. Aproximadamente donde se esperaría si siguiera hasta f de 1 mitad. Se ubicaría exactamente en la línea imaginaria y f de 1 mitad sería la raíz cuadrada de 2. Bueno, mi El mouse no está donde quiero que esté. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "Sería alrededor de la raíz cuadrada de 2 veces I y, a medida que continúa, esto le muestra todas las potencias de valor real de menos 2 elevado a X, necesariamente gira en espiral. Pero también podríamos mover nuestro valor de R aún más alto y obtenerlo. hasta alrededor de tau veces I alrededor de seis coma dos ocho veces I y en ese contexto esta es otra función que querríamos escribir como algo así como 2 a la X porque para cualquier número entero a número entero que conectes para X, lo hará parece una multiplicación repetida e incluso tiene valores razonables para cosas como 1 mitad donde escupe la raíz cuadrada negativa en lugar de una raíz cuadrada positiva, pero lo que en realidad está haciendo es una transformación al plano donde pone todo es real. La recta numérica termina siendo una espiral muy apretada que da vueltas y simplemente gira en espiral de tal manera que f de 1 cae justo en el número 2. Entonces es en ese sentido que podríamos decir que 2 a la X se interpreta plausiblemente como una función exponencial separada de la que estamos acostumbrados tradicionalmente. Así que creo que con todo eso dejaré las cosas para hoy y solo los dejaré con un par de preguntas pendientes para que piensen, está bien, así que si quieren Piensa en I a I como una expresión de múltiples valores, ¿verdad?, podrías decir que adoptamos una convención. Fantásticamente dirías que eliges una rama de la función de logaritmo natural. Y tal vez eso te encierre en este ser e elevado a pi negativo. mitades Pero si dices que este tipo de quiere ser infinitos valores diferentes como los que vimos, ¿cuántos valores quiere ser 2 elevado a 1 tercio en el mismo sentido? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "Las décimas quieren expresarse de manera diferente de todas las décimas de las funciones exponenciales F de X que satisfacen oh, ¿lo he escrito en algún lugar f de X que satisface todas estas propiedades que he escrito, así que si satisface todas? de estos y si f de 1 es igual a 2. ¿Cuántas salidas diferentes obtendremos cuando conectemos X es igual a 3 décimos para las distintas opciones y para qué función? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "Para 2 elevado a pi para las diversas funciones que 2 elevado a X podría representar si pensamos en 2 elevado a X como algún tipo de función exponencial Exponencial en el sentido de este tipo de propiedades abstractas y si sí, si tenemos una clase de funciones diferentes y queremos conectar pi, me hace reír. Sólo porque es una respuesta divertida que aparece cuando intentas pensar en ella, así que esas son las preguntas que Los dejo con y creo que esta es mi pregunta central al abordar la conferencia de hoy fue si quería que fuera una especie de descripción como estas propiedades abstractas de funciones exponenciales. Y es genial para mí que partiendo de esas propiedades abstractas te quedas atrapado en la idea de e elevado a rx o más. Solo sabes, creo que es más honesto escribir exp de r multiplicado por x para diferentes valores de r. Eso te bloquea hasta ese punto, pero no te bloquea hasta el punto de tener una noción inequívoca de lo que 2 elevado a x debería ser mucho menos algo como I elevado a x. El riesgo de esto, por supuesto, es que a veces a la gente no le encanta la abstracción y a veces no parece accesible, pero si ese es el En caso de que sepas, házmelo saber. Creo que hay todo un interesante círculo de pensamientos que rodea todo esto para incluir las torres de energía, porque si quieres hablar en realidad sobre torres de energía como lo hicimos la última vez en el contexto de números complejos. o incluso con bases negativas. Tienes que pensar en cosas como esta, así que era una pregunta que teníamos en la pantalla. Sí, ¿qué pasa si hacemos esto por I elevado a I? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "Titulación, ya sabes, intentemos esto, sigamos adelante y probemos una torre de energía donde elevamos I a una potencia determinada y vemos qué sale de ella, así que no estaba planeando hacer esto. Pero siempre podemos. abra Python y esencialmente haga lo que estábamos haciendo la última vez. Entonces, la forma en que esto funcionaría es que comenzamos con algún valor base y luego para algún tipo de rango. ¿Qué estábamos haciendo? Estábamos tomando un y vamos a reasignarlo. debe ser lo que sea. La base que en este caso es la que elevé a la potencia de a debería ser Bien, genial, entonces vamos a hacer eso y luego vamos a imprimir el valor de a, hagamos esto por Sí, es un número mucho mayor como 200. Entonces parece que lo que sucede es que a veces hay potencial para el caos con estas cosas. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "De hecho, lo tenemos, así que déjame importar NumPy para tener la función exponencial, déjame ir. Para nuestro gran rango como lo teníamos antes. En lugar de escribirlo como sabes, algo que es como yo elevado a X, lo escribiré. como función exponencial de una constante diferente, una constante diferente que voy a hacer. Quiero que sean 5 mitades de pi, así que multiplicaré 5 mitades de pi, así que es un número complejo y tiene 5 mitades de pi como parte imaginaria Entonces esto es 5 pi mitades de I y ¿qué estoy haciendo? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/tamil/sentence_translations.json b/2020/ldm-i-to-i/tamil/sentence_translations.json index 0140e952d..6ef5c3010 100644 --- a/2020/ldm-i-to-i/tamil/sentence_translations.json +++ b/2020/ldm-i-to-i/tamil/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "எனவே நீங்கள் எண் 1 இல் தொடங்கினால், உங்கள் ஆரம்ப வேகம் நேராக 0 ஐ நோக்கி நடக்க வேண்டும், நீங்கள் இன்னும் குறைவாக நடக்க வேண்டும், நீங்கள் 1 பாதியில் அமர்ந்திருந்தால், நீங்கள் இன்னும் 0 ஐ நோக்கி நடப்பீர்கள், ஆனால் இப்போது உங்கள் திசைவேக திசையன் நீங்கள் இருக்கும் இடத்தில் 1 மடங்கு எதிர்மறையாக இருக்கும், அதாவது எதிர்மறை 1 பாதி. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "மேலும் ஒரு சுவாரசியமான கேள்வி என்னவென்றால், இது போன்ற ஒரு செயல்பாடு மட்டுமே இதற்கு எழுதுவது நியாயமானது என்று உங்களுக்குத் தெரியும், ஏனென்றால் நாங்கள் அதை i to the x என்று எழுதப் போகிறோமா என்பது உங்களுக்குத் தெரியும், இதைத் திருப்திப்படுத்துவது மட்டுமல்லாமல், அது எப்போது உங்களுக்குத் தெரியும். நாம் பெறும் எண்ணை நாம் இணைத்துள்ளோம், நான் மறைமுகமாக நான் பவர் ஒன்னாக இருக்க வேண்டும், ஆனால் இந்த செயல்பாடு i ஆக இருக்க வேண்டும் என்று நாங்கள் நினைக்கிறோம். ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "எனவே நாம் 5 pi ஐ பாதியாகப் பெற்றுள்ளோம், இது x க்கு நாம் செருகக்கூடிய மற்றொரு மதிப்பாகும், மேலும் இங்கே நாம் இருக்கும் வட்டத்தை திரும்பிப் பார்த்தால், அதை இன்னும் கொஞ்சம் பார்வைக்கு உச்சரிக்க வேண்டும். கணம் 1 ஆகும் பை பாதிகளுக்கு சமமான நேரத்திற்கு நடந்தார். 57 அதற்குப் பதிலாக நாம் மற்றொரு முழு திருப்பத்தை எடுத்துக்கொண்டு, மற்றொரு பை பாதிக்குச் சென்று, நம்மை பைக்கு அழைத்துச் சென்றால் என்ன செய்வது என்பது உங்களுக்குத் தெரியும், இது ஒரு வகையான பதிவுகளாக இருக்கலாம், அங்குதான் e முதல் pi ஐ மதிப்பிடுவது நாம் மற்றொரு pi பாதியை நடந்தால் மற்றொரு pi பாதியில் நடப்போம் இந்த புள்ளியில் நாம் ஒரு முழு வட்டத்திற்குச் சென்றிருப்போம், பின்னர் நாங்கள் ஐந்து பை பகுதிகளுக்கு நடப்போம், இது எண்களின் அடிப்படையில் சுமார் 7 ஆகும். 85 ஆம், இது முற்றிலும் ஐயின் மேல் நம்மைக் கொண்டு செல்லும் மற்றொரு எண் மற்றும் நான் சக்தியை மீண்டும் வெளிப்படுத்தும் முழு ரிக்மரோலையும் கடந்து செல்ல வேண்டுமானால், முதலில் e ஐ 5 பை பாதிகளுக்கு i என்று எழுதுவதன் மூலம் நான் அந்த சக்திக்கு எதிர்மறையாக மாற பெருக்கி, எதிர்மறையான 5 பை பகுதிகளை நாம் பார்க்கிறோம், இது மிகவும் வித்தியாசமான எண்ணாகும், இதை நாம் உண்மையில் கணக்கிடலாம், என் தலையின் உச்சியில் இருந்து உறுதியாக தெரியவில்லை, ஆனால் ஒரு டெஸ்மோஸைப் பார்ப்போம். . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "அந்த நீளம் உங்களை மிகக் குறைந்த எண்ணிக்கையில் கொண்டு செல்கிறது, ஆனால் அது மட்டும் பதில் இல்லை, நாங்கள் சரியாக உள்ளிட முடியும், மற்றவர்கள் எதிர்மறையான 3 அரை மடங்குகள் i pi உடன் இங்கு வருகிறார்கள், இது யூனிட் வட்டத்தின் அடிப்படையில் உங்களுக்குத் தெரியுமா? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "90 டிகிரி பை பாதி ரேடியன்களில் நடப்பதை விட, நான் ஐயை அடைய விரும்பினால் ஏய் என்று சொல்லலாம் என்று நாம் நினைக்கலாம். வழக்கமாக எதிரெதிர் திசையில் நேர்மறையாக இருக்கும், அது முற்றிலும் அதை வெளிப்படுத்த மற்றொரு வழி மற்றும் எதிர்மறையான 3 pi பாதிகளுக்கு e இருந்தால், அது நமக்கு வித்தியாசமான பதிலைப் பெறும் i அனைத்தும் சக்திக்கு நான் இப்போது அதே விளையாட்டின் மூலம் செல்கிறோம், இப்போது i ஸ்கொயர்டு ரத்து செய்யப்படுகிறது எதிர்மறையானது ஏற்கனவே உள்ளது, மேலும் எங்களிடம் நேர்மறை 3 பை பாதிகள் உள்ளன, மேலும் எண்ணியல் ரீதியாக இது நமக்கு முன்பு இருந்ததை விட வித்தியாசமான பதிலைப் பெறும் பாதிகள் 111 புள்ளி 3 1 111 புள்ளிக்கு முன் நாம் பார்த்ததை விட மிகவும் வித்தியாசமான எண் அது என்ன 111 புள்ளி 3 1 பெரிய 111 புள்ளி 3 1 அல்லது அதற்கு மேல் உள்ளுணர்வின் அடிப்படையில் மீண்டும் நீங்கள் என்ன கேட்கலாம் என்று வைத்துக்கொள்வோம். மாறும் ஆனால் நாம் காலப்போக்கில் பின்னோக்கி நகர்கிறோம். பின்னர் நீங்கள் சிதைவு இயக்கவியலுக்கு மொழிபெயர்ப்பதாக இருந்தால், இந்த சூழலில் கண்ணை உயர்த்துவது என்னவென்று நீங்கள் சொல்கிறீர்கள், நான் முதலிடத்தில் தொடங்குகிறேன் என்று சொல்கிறீர்கள், ஆனால் நான் நேரத்தைப் பின்னோக்கி நகர்த்த விரும்புகிறேன், நான் எங்கு தொடங்க வேண்டும் என்று சொல்ல வேண்டும். நான் முதலிடத்தை அடையும் வகையில் சிதைந்து போக வேண்டுமா? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "3 pi அரை அலகுகளுக்குப் பிறகு, அந்த வகையான அதிவேகச் சிதைவுக்கான பதில் நூற்று பதினொன்றில் தொடங்குகிறது, மேலும் இது எங்கு செல்கிறது என்பதை நீங்கள் பார்க்கலாம், உண்மையில் எண்ணற்ற பல்வேறு மதிப்புகள் உள்ளன, அதை நாம் X க்கு செருகலாம். e to X என்று நினைத்துக்கொண்டு நான் மற்றும் மக்கள் இங்கு இன்னும் நிறைய நுழைந்துவிட்டீர்கள், மன்னிக்கவும், மூன்றாம் இடத்திற்கு கிளாசிக் செய்வது போல் என் முள் தரையில் வீசியதை மன்னிக்கவும் 9 pi பாதிகள் சிறந்த தேர்வு 1729 pi பாதிகள் எல்லாம் எனக்கு மிகவும் பிடித்தவை வெவ்வேறு விருப்பங்கள் எண்ணற்ற பல வேறுபட்ட மதிப்புகள் முதலில் கொஞ்சம் குழப்பமாக இருக்கும், ஏனென்றால் ஒரு வெளிப்பாட்டைப் பார்க்கிறோம், சில கணக்கீடுகள் இருக்கும் என்று உங்களுக்குத் தெரியும் என்று தோன்றுகிறது, நான் அதை எனது கால்குலேட்டரில் செருகி, என்ன வெளிவருகிறது என்பதைப் பார்க்கிறோம், மேலும் பல வேறுபட்டவைகளைப் பெற்றுள்ளோம். அதற்கான மதிப்புகள் இங்கே என்ன நடக்கிறது? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "16 இன் நான்காவது ரூட் 2 ஆக இருக்க வேண்டும், பதில் நன்றாக இருக்கும் ஃபேன்சியர் லிங்கோவில் ஒற்றை உள்ளீடு மற்றும் ஒற்றை வெளியீட்டைக் கொண்ட ஒரு செயல்பாடாக இதைக் கருதுங்கள், இது எப்பொழுதும் சிக்கலான எண்களைக் கையாளும் போது, ஏதாவது ஒரு செயல்பாட்டின் வகையிலான எண்ணம், நீங்கள் சில நேரங்களில் பல மதிப்புகளைப் பெற விரும்புவீர்கள். கிளை என்ற சொற்றொடரை கேட்கவும், வர்க்க மூல செயல்பாட்டின் கிளையை நீங்கள் எங்கு தேர்வு செய்கிறீர்கள்? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "பலவிதமான பதில்கள் இருப்பதால், நாங்கள் மீண்டும் என்னைப் பற்றி நினைப்பது இந்த 90 டிகிரி சுழற்சி என்று உங்களுக்குத் தெரியும், இதை 90 டிகிரி சுழற்சி என்று நினைத்தால், வர்க்க மூலமானது 45 டிகிரி கோணத்தில் அமர்ந்திருப்பது உங்களுக்குத் தெரியும் ஒருவேளை அது சதுரமாக இருக்கலாம் ரூட் 2 ஓவர் 2 ரூட் 2 ஓவர் 2 என மிகத் தெளிவாக எழுதலாம் நான் இது முக்கோணவியலைப் பயன்படுத்துகிறது ஆனால் அதற்குப் பதிலாக நான் ஒரு எதிர்மறை 270 டிகிரி சுழற்சி என்று நினைத்தால், அந்தச் செயல்பாட்டின் பாதியைச் செய்வது போல் உணர்கிறேன். உண்மையில் நம்மை மறுபுறம் கொண்டு செல்ல வேண்டும் ஒருவேளை இங்கே அமர்ந்திருக்கும் எண் I இன் வர்க்க மூலமாக இருக்க வேண்டும், அது உண்மையில் எதிர்மறை ரூட் 2 க்கு 2 மைனஸ் ரூட் 2 க்கு 2 க்கு 2 முறை நான் இப்போது உண்மையான சூழலில் பார்த்தது மதிப்புள்ள செயல்பாடுகளை நாம் ஆம் என்று சொல்லலாம், நேர்மறை பதில் எதுவாக இருந்தாலும், வர்க்க மூலத்தைத் தேர்ந்தெடுக்கவும், ஆனால் இவற்றில் எது நேர்மறை பதிலைக் கருதுகிறீர்கள்? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "நீங்கள் நன்றாகச் சொன்னீர்கள் என்று நினைக்கிறேன், இது என்னவென்று எங்களுக்குத் தெரியும், 2 இன் வர்க்கமூலமாக அனைத்துமே நன்றாக இருக்கிறது மற்றும் நல்லது என்று வரையறுக்கிறோம், ஆனால் நான் என்ன சொன்னேன் என்றால், நாம் நமது I ஐ அணுகுவதைப் போலவே இதை அணுகுவோம் நான் முதலில் எதையாவது சரியானதை e ஆக வெளிப்படுத்த விரும்புகிறேன், பின்னர் நான் அதை 1 பாதியை அடுக்குடன் பெருக்கி 1 பாதியாக உயர்த்தப் போகிறேன், நான் சரி என்று சொல்கிறேன், நான் அதைச் செய்ய முடியும் என்று யூகிக்க முடியும். 2 கிணறுக்கு சமம் அது 2 இன் இயற்கையான பதிவு இது 0 ஐ சுற்றி இருக்கும் ஒரு மாறிலி. 69 அல்லது அதற்கு மேல் நாம் e ஐ அந்த சக்திக்கு உயர்த்தினால், நமக்கு 2 கிடைக்கும், எனவே இதை 2 பெருக்கல் 1 பாதியின் இயற்கையான பதிவில் e என்று நினைத்துக் கொள்ளலாம், நீங்கள் விரும்பினால் e to the x என்று நினைத்தால்? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "உண்மையான எண்களின் சூழலில் இது ஒரு வகையான ஓவர்கில் என்று உங்களுக்குத் தெரியும், ஆனால் இந்த x செயல்பாட்டிற்கான சுருக்கெழுத்து e to the x என்று நீங்கள் நினைத்தால், மதிப்பை 0-ஐ செருகலாம். 69 பெருக்கல் 1 பாதி, இது 0 ஆக இருக்கும் என்று நான் நினைக்கிறேன். 345 ஐஷ் அது போன்ற உறுதியான மதிப்பை உங்கள் பல்லுறுப்புக்கோவையில் செருகவும், அது என்ன வெளியிடுகிறது என்பதைப் பார்க்கவும், அது 1 ஐ சுற்றி வெளியிடும். 414 ஒரு நல்ல உண்மையான எண் 2 இன் வர்க்க மூலத்தை நீங்கள் எதிர்பார்ப்பது, ஆனால் நாங்கள் அதையே செய்தால் நான் மற்றும் ஒரு சக்திக்கு e என எதையாவது எழுத விரும்பினால் உண்மையில் பல வேறுபட்ட பதில்கள் உள்ளன என்பதை ஒப்புக்கொண்டால் இதையும் எழுதலாம். இது வேடிக்கையாகத் தோன்றலாம், ஆனால் 2 பிளஸ் 2 பையின் இயல்பான பதிவில் e என்று எழுதலாம் I அந்த முழு விஷயமும் 1 பாதிக்கு உயர்த்தப்பட்ட பிறகு, இந்த மதிப்பு உங்களுக்கு சமமாக வந்த பிறகு, அது e க்கு சமமாக வருவதால் அதை உடைக்கலாம். 2 இன் இயற்கைப் பதிவு 2 pi ஐ ஆல் பெருக்கப்படுகிறது, இது 360 டிகிரி சுழலும் விளைவைக் கொண்டுள்ளது, எனவே இது 1 க்கு சமமாக இருக்கும், எனவே நாங்கள் 2 முறை 1 ஐப் பார்க்கிறோம், அது சரியான மாற்றாக உணர்கிறது. இதை எடுத்து ஒரு சக்தியாக உயர்த்தி, சக்தியைப் பெருக்குவதன் மூலம், என்ன நடக்கிறது என்பதைப் பாருங்கள், 2 மடங்கு 1 பாதி கூட்டல் சரி, 2 pi I பெருக்கல் 1 பாதி என்ற இயற்கைப் பதிவை நாங்கள் விளையாடுகிறோம். அது pi முறையாக இருக்கும் நான் இப்போது இந்த முதல் பகுதி e 2 முறை 1 பாதியின் இயற்கையான பதிவில் 2 இன் பழக்கமான சதுர மூலமாக முடிவடையும், அது நன்றாக இருக்கிறது, ஆனால் நாம் அதை e மூலம் பெருக்கப் போகிறோம் pi ஐ ரைட் மற்றும் மிகவும் பிரபலமாக e க்கு pi ஐ எதிர்மறை 1 எனவே இந்த வழக்கில் நாம் இந்த வெளிப்பாடு 2 முதல் 1 பாதி வரை வெவ்வேறு பதில்களுடன் விளையாடுவதன் மூலம் நாம் ஏதாவது ஒன்றைச் செருகலாம் என்று பரிந்துரைக்கிறது. e க்கு X க்கு சமமான 1 பாதியில் நாம் முடிவடையும் மற்றொரு பதில், இந்த எதிர்மறை வர்க்கமூலமான 2 என்று நாம் பாரம்பரியமாக எழுதலாம் மற்றும் 2 முதல் 1 பாதி வரை பல மதிப்புகளைக் கொண்டிருப்பது கொஞ்சம் வேடிக்கையானது மற்றும் ஒரு விஷயத்தைச் சமன் செய்வது அல்ல, ஆனால் நாம் செய்யும் தேர்வுகளின் அடிப்படையில் அது பல வேறுபட்ட விஷயங்களுக்குச் சமமாக இருக்கும், ஆனால் இரண்டு விஷயங்கள் மிகவும் நியாயமானதாகத் தோன்றும் நமக்குத் தெரிந்த வர்க்கமூலம் அல்லது எதிர்மறையான மாறுபாடு உண்மையில் அப்படியொரு பிரச்சனையாகத் தெரியவில்லை, உண்மையில் நாம் இந்த விளையாட்டை இன்னும் அதிகமாக விளையாடலாம், இந்த வெளிப்பாட்டிற்கு இன்னும் ஆக்கப்பூர்வமான பதில்களைக் கேட்கிறேன் ஏனென்றால் I ஐ மதிப்பிடுவதில் நாம் பயன்படுத்திய அதே விதிகளை கடைபிடித்தால், நாம் என்ன மாற்றீடு செய்கிறோம் என்பதன் அடிப்படையில் X இன் பல்வேறு மதிப்புகளை இணைக்கத் தொடங்கும் போது, 2 க்கு பவர் X போன்றவற்றின் மற்ற வேடிக்கையான சக்திகளைக் காணலாம். சக்தி I எனவே இந்த முறை கேள்வி கேட்கிறது அல்லது அது e சமன்பாட்டின் ஒரு தீர்வு x க்கு சமம் 2 என்று குறிப்பிடுகிறது உண்மையான எண் 2 OK இன் இயற்கைப் பதிவு நமக்குத் தெரிந்த ஒன்று. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "e க்கு x சமம் 2 என்ற கேள்விக்கான பதில் மற்றும் மீண்டும் படைப்பாற்றல் வரவேற்கப்படுகிறது, அதனால் நான் உங்களுக்கு இன்னும் ஒரு சிறிய தருணத்தை தருகிறேன் II அது சரியாக இருந்தால் இங்கே சில பதில்களைப் பூட்டுகிறேன், எவ்வளவு நேரம் ஆகும் என்று எனக்குத் தெரியவில்லை நீங்கள் எந்த சாதனத்தைப் பார்க்கிறீர்கள் என்பதைப் பொறுத்து கணிதப் பதிவைச் செய்ய வேண்டிய அவசியம் உள்ளது, ஆனால் நீங்கள் விரும்பும் கேள்விக்கு நீங்கள் பதிலளிக்க விரும்பும் பதிலைப் பெறுவதற்கான வாய்ப்பு உங்களுக்குக் கிடைப்பதற்கு முன், மிகவும் அழுத்தமாக இருக்க வேண்டாம். உங்களில் 131 பேர் 2 இன் Ln ஐ எடுத்துக் கொண்டு 2ii ஐச் சேர்க்கும் மாறுபாட்டிற்குள் நுழைந்துவிட்டீர்கள், மேலும் சில வேறுபட்ட சரியான விடைகள் இருக்கும் போது, தவறுதலாக விடைகளில் ஒன்றை சரியானதாகக் குறிக்கப்பட்டது போல் நான் இந்தக் கேள்வியை எழுதுகிறேன் என்று நினைக்கிறேன். உங்களில் யாருக்காவது இது சிவப்பு நிறமாகத் தோன்றுகிறதா என்று எனக்குத் தெரியவில்லை. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi இது நிச்சயமாக ஒரு சிறந்த தேர்வாகும், ஆனால் நீங்கள் 4 pi I மற்றும் 2 அல்லது 6 pi I இன் இயற்கைப் பதிவு அல்லது உண்மையில் 2 pi I இன் ஏதேனும் முழு எண் பெருக்கல் I ஐப் பாதிக்காது என்று நீங்கள் சேர்க்கலாம். X ஏனெனில் இது e ஆல் பெருக்குவதால் 2 pi I க்கு 1 ஆல் பெருக்கினால் ஏற்படும் விளைவு மற்றும் மீண்டும் இது ஒரு வேடிக்கையான விளைவைக் கொண்டிருக்கிறது, இது மற்றொரு உதாரணம் செய்யும்போது நியாயமான முடிவுகளை வெளியிடுவது போல் தெரிகிறது. பொதுவாக உள்ளிடப்பட்ட இரண்டாவது வெளிப்பாடாகத் தெரிகிறது, நாம் 2 ஐ மாற்றலாம், எனவே 2 ஐ 1 4 வது சக்தியாகக் கருதுகிறோம் என்று நினைக்கலாம், சரி, 2 ஐ e உடன் 2 கூட்டல் 4 இன் இயற்கைப் பதிவுக்கு மாற்றலாம் என்று ஒரு பரிந்துரை இருந்தது. pi ஐ ஓகே பிளஸ் 4 பை ஐ மற்றும் நாங்கள் அதை 1 4 வது வலமாக உயர்த்துவோம், நீங்கள் அதே விளையாட்டை விளையாடினால், நீங்கள் e க்கு 2 முறை 1 4 வது இயற்கை பதிவு கிடைக்கும், மேலும் நாங்கள் e க்கு பெருக்குவோம் pi I இப்போது அதன் முதல் பகுதி 2 இன் வழக்கமான நேர்மறை நான்காவது மூலமாக இருக்கும், அதாவது 2 இன் நான்காவது ரூட் போன்ற ஒரு எக்ஸ்ப்ரெஷனை நீங்கள் ஒரு கால்குலேட்டரில் ஒரு நல்ல சிறிய நேர்மறை எண்ணைச் செருகினால், இந்த இரண்டாவது பகுதி எதிர்மறை 1 எனவே 2 ஐ வேறு விதமாக விளக்கினால் 1 4 க்கு உயர்த்தினால் உங்களுக்குத் தெரியும் என்று சொல்வது போல் தெரிகிறது, இது எங்களுக்குக் கிடைக்கும் வழக்கமான பதில் அல்ல, ஆனால் இது ஒரு நியாயமான பதில். ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "நாம் pi ஐ பாதி முறை ஐப் பார்த்துக் கொண்டிருப்போம், எதிர்மறை 1 ஆல் பெருக்குவதற்குப் பதிலாக I ஆல் பெருக்கி இருப்போம், இது மீண்டும் சரியான பதில் 2 முதல் 1 4 வது போன்ற ஏதாவது ஒரு நியாயமான வெளியீடு போல் தெரிகிறது. நான் அதிகாரத்திற்கு நான் பல வேறுபட்ட மதிப்புகளைக் கொண்டிருப்பதாகத் தெரிகிறது. நாங்கள் இங்கு முன்பு கண்டறிந்த 15வது தோராயமாக 15வது பதிலில் இருந்து மிகவும் வித்தியாசமான ஒன்று மிகச்சிறிய ஒன்று சூப்பர் பெரியது, நீங்கள் 2 முதல் 14வது வரை என்ன என்று கேட்கும்போதும், உண்மையில் பலவிதமான தீர்வுகள் உள்ளன என்பதை ஒப்புக்கொள்வதும் இதே நிகழ்வுதான். X முதல் 4வது வரையிலான வெளிப்பாடு உண்மையில் 2 4 வெவ்வேறு தீர்வுகளுக்குச் சமம் மற்றும் நீங்கள் பார்ப்பது என்னவென்றால், பல வேறுபட்ட தீர்வுகள் உள்ளன. 2 எதுவாக இருந்தாலும், இதைப் பற்றி நாம் சிந்திக்கக்கூடிய ஒரு வழி என்னவென்றால், நீங்கள் உண்மையான எண்களைக் கையாளும் போது விஷயங்கள் அழகாக இருக்கும். ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "அதிவேக செயல்பாடுகளைப் பற்றி நாம் சிந்திக்க விரும்பினால், இந்த விஷயங்களில் சிலவற்றைப் பற்றிச் சொல்கிறேன், இது நன்றாக இருக்கிறது X இன் R பெருக்கல் X போன்ற அதே அதிவேகமானது, நாம் குறிப்பிடும் பல்லுறுப்புக்கோவை ஆகும், இது நாம் X க்கு e போன்ற ஒன்றை எழுதும் போதெல்லாம் மறைமுகமாகக் குறிப்பிடும் போதெல்லாம், முன்னும் பின்னுமாக ஒரு அழகான முன்னும் பின்னுமாக உள்ளது, ஏனெனில் நீங்கள் B இன் இயற்கை மடக்கையை எடுக்கலாம். மேலும் இது B என்பது நேர்மறை எண் எனக் கருதி உங்களுக்கு ஒரு பதிலைத் தருகிறது, மேலும் X இன் R B க்கு சமம் என்று கூறுவதும் ஒன்றுதான். எனவே இந்தத் தொடரில் இதைப் பற்றி நான் முன்பு பேசிய ஒரு வழி, நீங்கள் பார்த்தால் சாத்தியமான அனைத்து எக்ஸ்போனென்ஷியல்களின் குடும்பம், அவற்றை R இன் X இன் X என எழுதலாம் மற்றும் R என்பதை மாற்றலாம், மேலும் இது e க்கு R டைம்ஸ் X என எழுதுவது போலவே இருக்கும், அது உங்களுக்கு So e to R உடன் மிகவும் வசதியாக இருந்தால். XX இன் R முறை X முறைகள் XX இன் முறைகள், அது என்ன என்பதை மாற்றுவது பற்றி நாம் சிந்திக்கலாம், ஆனால் மறுபுறம், சாத்தியமான அனைத்து அதிவேகங்களையும் சில அடிப்படையாக நீங்கள் நினைத்தால், X இன் சக்தியை அடிப்படையாகச் செய்ய அனுமதிக்கிறேன், நாங்கள் செல்கிறோம். அந்த அடிப்படையை மாற்றுவதற்கு முதலில் அது கையாள்வது வேறுவிதமான வெளிப்பாடு என்று தோன்றுகிறது, ஆனால் இது ஒரே குடும்பத்தை வெளிப்படுத்தும் மற்றொரு வழி மற்றும் இதைப் பற்றி நீங்கள் சிந்திக்கக்கூடிய ஒரு வழி, இது எந்த அடிப்படையில் ஒத்துப்போகிறது என்பதைப் பற்றி நாங்கள் எப்படி சிந்திக்கிறோம் நாம் இன்னும் கொஞ்சம் சுருக்கமாக, எக்ஸ் ஆஃப் ஆர் டைம்ஸ் எக்ஸ் என நினைத்துக் கொண்டிருந்தால், நான் இதைச் செய்வதற்கு ஒரு காரணம் இருக்கிறது, ஏனென்றால் சிக்கலான எண்களுக்கு இதைப் பயன்படுத்தப் போகிறோம், அது வித்தியாசமாகத் தோன்றும், எனவே என்னுடன் பின்தொடரவும் அந்த அடிப்படையைப் பார்ப்பதற்குப் பதிலாக நான் செய்யக்கூடிய ஒன்று மதிப்பு என்ன? ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "நான் R முறை X இன் எக்ஸ்ப்ஸைப் பெற்றிருக்கலாம், அங்கு R என்பது பூஜ்ஜியப் புள்ளி ஆறு ஒன்பது போல இருக்கலாம், ஆனால் நான் அதை இரண்டு pi ஐ ஆல் கீழே மாற்ற முடியும், அது இன்னும் இரண்டிற்கு ஒத்ததாக இருக்கும் அடிப்படையை மாற்றாது அல்லது அது முடியும் அதை இரண்டு pi ஐ ஆல் மாற்றவும், அது தொடர்புடைய அடிப்படையை மாற்றாது, ஏனென்றால் அந்த எல்லா நிகழ்வுகளிலும் நாம் X ஐச் சமமாகச் செருகும்போது, ஒரே பொருளைப் பெறுகிறோம், இருப்பினும் X இன் வெவ்வேறு மதிப்புகளுக்கு இவை அனைத்தும் தனித்துவமான செயல்பாடுகளாகும். I முதல் பவர் வரை பல வேறுபட்ட மதிப்புகளை நாம் ஏன் பார்த்தோம், ஏனெனில் I to X என்பது ஒரு தெளிவற்ற செயல்பாடு என்பதால், R இன் எந்த மதிப்பை நாம் தீர்மானித்தால் அது தெளிவாக இருக்கும் ஆர். ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "இது ஒரு தெளிவற்ற செயல்பாடு, ஆனால் அந்த நேரத்தில் நாம் விரும்புவது போல் உணர்கிறோம் X சக்திக்கு உயர்த்தப்பட்ட சில அடிப்படைகளின் அடிப்படையில் விஷயங்களைப் பற்றி சிந்திப்பதை நிறுத்துவது ஒருவேளை நாம் சிக்கலான எண்களின் சூழலில் இருக்கும்போது நாம் எழுத வேண்டும். அவை அனைத்தும் சில நிலையான நேரங்கள் X இன் காலாவதியாக இருந்தால், வேறு எந்த காரணமும் இல்லாமல், நாம் ஒரு கணக்கீடு செய்ய விரும்பினால் அல்லது அதன் மேல் கணிதத்தை செய்ய விரும்பினால், உண்மையில் எண்களை எவ்வாறு செருகுவது என்பது தெளிவாகிறது. அவற்றைச் செருகவும், இதுவே அதிவேகங்களைப் பற்றி சிந்திக்க சரியான வழி என்று உங்களுக்காக நான் மற்றொரு வழக்கை முன்வைக்கிறேன், சிக்கலான எண்கள் போன்ற பிற டொமைன்களில் விஷயங்களை விரிவுபடுத்தியவுடன், அதைக் காப்புப் பிரதி எடுப்போம். மீண்டும் கதவு மணிக்கு சில விஷயங்கள் வந்துவிட்டன, அதன் மூலம் நாம் அதிவேக எண்ணத்தை விரிவுபடுத்துகிறோம், மேலும் 2 முதல் X ரைட் வரை உள்ளதைப் போன்றே சிந்தித்துப் பாருங்கள், இயற்கை எண்களுக்கு இதைப் பற்றி எப்படிச் சிந்திக்க வேண்டும் என்பது நமக்குத் தெரியும். ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "2 முதல் 3 மீண்டும் மீண்டும் பெருக்கல் என்பது உங்களுக்குத் தெரியும், பின்ன அளவுகளுக்கு 2 முதல் X வரை அல்லது எதிர்மறைத் தொகைகள் மற்றும் அது போன்ற விஷயங்களைப் பற்றி சிந்திக்க முதலில் உங்களுக்குக் கற்பிக்கப்படுகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "சரி. 2 முதல் 1 பாதி வரை இருக்கும் என்று நீங்கள் வழக்கமாகக் கற்பிக்கப்படுகிறீர்கள், நான் அதைத் தானாகப் பெருக்கினால், அது உங்களுக்குத் தெரியும், இது எக்ஸ்போனென்ஷியல்ஸ் செய்யும் வழக்கமான விதிகளைப் பின்பற்றுகிறது. 1 க்கு சில எண்ணாக இருக்க வேண்டும், அதனால் நான் அதை பெருக்கும்போது எனக்கு 2 கிடைக்கும், அந்த நேரத்தில் உங்களுக்கு ஒரு தேர்வு உள்ளது என்பது உங்களுக்குத் தெரியும், ஒருவேளை அது நேர்மறையாக இருக்கலாம். ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "ஒருவேளை அது எதிர்மறையாக இருக்கலாம், ஆனால் நீங்கள் எப்போதும் நேர்மறைத் தேர்வு செய்ய முடிவு செய்தால், எதிர்மறை எண்களைப் பற்றி நாம் கேட்டால், இதே ஒப்பந்தத்தில் இருந்து ஒரு நல்ல தொடர்ச்சியான செயல்பாட்டைப் பெற முடியும் நான் அதை 2 முதல் 1 ஆல் பெருக்கும்போது எங்கே? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "இது என்னை 2 முதல் 0 வரை பெறுகிறது மற்றும் எதிர்மறை அடுக்குகள் 1 பாதியாக இருக்கும் என்பது எங்கள் மாநாட்டின் நியாயமாகும், ஆனால் உண்மையில் இங்கு என்ன நடக்கிறது, இது என்னவாக இருந்தாலும் அது ஒருவிதமான செயல்பாடாக இருக்க வேண்டும் என்று சொல்கிறோம். a plus b ஆனது b இன் ஒரு மடங்கு f க்கு சமம், மேலும் அடிப்படை 2 என்பது அடிப்படையில் நமக்குச் சொல்கிறது, இது அத்தகைய செயல்பாடு மட்டுமல்ல, 1ஐச் செருகும்போது நமக்கு 2 கிடைக்கும். இங்கே உள்ள சில தாக்கங்களை நீங்கள் பின்பற்றுகிறீர்களா என்பதைப் பார்ப்பதற்கான நல்லறிவு சோதனை பாணி கேள்வி என்னவென்று நான் உங்களிடம் கேட்க விரும்புகிறேன், நான் இதை ஒரு சாப்ட்பால் என்று அழைக்க மாட்டேன், ஆனால் இது நம்பமுடியாத ஆழமான கேள்வியைப் போல இருக்கக்கூடாது. அவசியம். ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "ஒரு செயல்பாட்டின் பண்புகளுடன் சுருக்கமாகத் தொடங்கி, பின்னர் ஒருவிதமான வழிகளை நீங்கள் பின்தொடர்ந்தால், அது ஒரு சோதனைக்கு மேலானது. அனைத்து உள்ளீடுகளுக்கும் a plus b இன் ஒரு மடங்கு f க்கு சமம் f க்கு சமம், மேலும் இது 1 க்கு சமம் 2 க்கு சமம் 2 பின்வருவனவற்றில் எது உண்மை என்பதை நீங்கள் எந்த செயல்பாட்டைத் தொடங்கினாலும், பின்வருவனவற்றில் எது உண்மை என்று கூறுவது இது எந்த விரிவுரை என்பதை நினைவில் வைத்திருப்பவர்கள், ஆய்லரின் சூத்திரம் உண்மையில் என்ன சொல்கிறது என்பதைப் பற்றி நாங்கள் பேசிக்கொண்டிருந்தோம், நான் இந்த பாணியைப் பற்றி ஒரு கேள்வியைக் கேட்டேன், அங்கு நான் ஒரு நிபந்தனையையும் புறக்கணித்தேன், நான் எழுதவில்லை என்பது உங்களுக்குத் தெரியும். எல்லா இடங்களிலும் x இன் பூஜ்ஜியமாக இல்லை என்பதை உறுதிப்படுத்த விரும்புகிறோம், பின்னர் அது சில குழப்பங்களை ஏற்படுத்தியது, இது நம் அனைவருக்கும் நடக்கும் திரையில் குழப்பத்தை ஏற்படுத்துகிறது, ஆனால் இதன் நோக்கம் அடிப்படையில் இந்த சுருக்கமான சொத்து கூட்டலைப் பெருக்கமாக மாற்றுவது போதுமானது. கடந்த முறையுடன் இணைக்கப்பட்டிருப்பது இங்கே தோன்றியதாகத் தெரிகிறது, மின் கோபுர கேள்வியை ஒரு கணம் நிறுத்திவிடுவோம், இதன் மூலம் முதலில் ஒரு ஆழமான உணர்வைப் பெறுவோம். ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "ஏனென்றால், நான் கூற விரும்புவது நாமாக இருக்க முடியும் என்பதால், அதற்கு பல வழிகளில் பதிலளிக்க முடியும். எனவே நீங்கள் எனக்கு ஒன்றைக் கொடுத்தால், நாங்கள் மின் கோபுரங்களைப் பற்றி பேசுவோம், பின்னர் ஒரு எண் கோட்டை மடக்கை அளவில் குறிப்பிடுவது போல. ஒரு சிக்கலான விமானத்திற்கும் இதைச் செய்யலாமா? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "ஆமாம், உண்மையில், நான் இங்கே ஒரு கணத்தில் பெறப் போகிறேன் என்று ஒரு காட்சிப்படுத்தல் உள்ளது, அங்கு நாம் அதைப் போலவே ஏதாவது ஒன்றைச் செய்கிறோம், ஏனென்றால் நாங்கள் என்ன செய்வோம் வெவ்வேறு அதிவேக செயல்பாடுகளான X இன் R மடங்கு X ஆனால் நாங்கள் R இன் மதிப்பை மாற்றப் போகிறது, இது ஒரு சிறிய மஞ்சள் புள்ளியால் குறிப்பிடப்படும், எனவே நாம் இதைப் பற்றி பேசுவோம், இது முழு விமானத்தையும் வரைபடமாக்கப் போவதில்லை, ஆனால் உண்மையான அச்சு மற்றும் கற்பனை அச்சில் இருந்து ஒரு ஜோடி மாதிரி புள்ளிகள் ஆனால் யோசனை என்னவென்றால், அந்த மாறிலி என்ன என்பதை நாம் சுற்றிச் செல்லும்போது, அது விமானத்திற்குச் செய்யும் வெவ்வேறு விஷயங்களைக் காட்சிப்படுத்த முடியும், மேலும் இது x- அச்சை மடக்கை அளவுகோலாக மாற்றி பின்னர் மடக்குவது போன்றது. ஒரு வட்டத்துடன் கற்பனை அச்சு பின்னர் R இன் அந்த மதிப்பு கற்பனையாக மாறியவுடன் அது அந்த உண்மையான எண்களின் பங்கை வட்டத்தில் மாற்றுகிறது மற்றும் கற்பனை எண்கள் ஒரு மடக்கை அளவிடப்பட்ட நேர்மறை அச்சில் வைக்கப்படும் மிகவும் பெரிய கேள்வி இவை மூன்றும் நான் யூகிக்கிறேன் நான் எங்கு செல்ல விரும்புகிறேனோ அங்கு துப்பாக்கியை முன்னோக்கி குதிக்க வேண்டும், ஆனால் மக்கள் இதில் அப்படி நினைக்கிறார்கள் என்று பார்க்க மகிழ்ச்சியாக இருக்கிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "வெளிப்படையாக 5 இன் எஃப் போன்றது 1 கூட்டல் 1 கூட்டல் 1 கூட்டல் 1 கூட்டல் 1 க்கு சமமான விஷயம், இது 1 இன் f க்கு சமமான விஷயம் 1 இன் 1 கூட்டல் 1 இந்த பண்பு காரணமாக 5 மடங்கு பெருக்கப்படுகிறது. சக்தி 5 க்கு 2 ஆகவும் பின்னர் எதிர்ம 5 ன் f போலவும் இருக்க வேண்டும், நாம் அதை f 5 ஆல் பெருக்கும்போது 0 இன் f என்ன என்பதைப் பெறுகிறோம், மேலும் 0 இன் f என்ன என்பது உடனடியாகத் தெரியவில்லை ஆனால் நாம் அதைச் சொல்லலாம். f இன் 1 கூட்டல் 0 என்பது 1 இன் 1 க்கு சமம் என்பது 0 இன் மடங்கு f என்பது 2 க்கு சமம், எனவே இதுவும் 2 க்கு சமம் எனவே 2 க்கு சமம் 2 மடங்கு என்று சொல்கிறோம். ஒரு 1 ஆக இருக்க வேண்டும், எனவே இந்த சூழலில் எதிர்மறை 5 இன் f 2 க்கு எதிர்மறை 5 க்கு இது 1 க்கு 2 முதல் 5 வது என்று உத்தரவாதம் அளிக்கிறது. இதை நாம் வெளிப்படையாக 2 முதல் எதிர்மறை 5 என்று எழுதலாம். நாம் உண்மையில் செயல்பாட்டினை 2 முதல் X வரை எழுத விரும்புகிறோம், ஏனென்றால் எந்த எண்ணும் எண்ணை அதில் வைக்கிறோமோ அது திருப்திகரமாக இருக்கும், அது எந்த ஒரு பின்ன எண்ணை உள்ளிடுகிறோமோ அந்த எண்ணிக்கையை தானாகப் பெருக்குவது போல் தெரிகிறது. நாங்கள் விரும்புவது மற்றும் நீங்கள் ஆச்சரியப்படலாம், அது தனித்துவமானது மற்றும் உண்மையான மதிப்புள்ள செயல்பாடுகளின் சூழலில் அது உண்மையில் இருக்கும் ஆனால் சிக்கலான மதிப்புமிக்க செயல்பாடுகளின் சூழலில் இதுபோன்ற பல செயல்பாடுகள் இருக்கும். 2 ப்ளஸ் 2 பையின் இயற்கையான பதிவின் எக்ஸ்பிரஸ் என வரையறுக்கப்பட்ட ஒரு செயல்பாட்டை நாம் எங்கே வைத்திருக்க முடியும் என்பதை முன்னரே பார்க்கிறோம் X சரி, இங்குள்ள அலட்சியத்தை மன்னியுங்கள், இதைப் பற்றி எழுதுவதில் நான் உற்சாகமாக இருக்கிறேன், இது உண்மையில் வேறு செயல்பாடு Xஐச் செருகினால் 1 பாதிக்கு சமம் என்ன ஆகும் என்பதற்குச் சான்றாக, 1 பாதியில் செருகினால், 2 இன் எதிர்மறை வர்க்கமூலம் எப்படி கிடைக்கும் என்பதை நாம் சற்று முன்பு பார்த்தோம், பிறகு 1ஐ நான்கில் செருகினால், நான்காவது ரூட் இல்லை 2 ஆனால் நான் 2 இன் நான்காவது மூலத்தை பெருக்குகிறேன், எனவே இது ஒரு வித்தியாசமான செயல்பாடாகும், ஆனால் இது இன்னும் இந்த பண்புகளை திருப்திப்படுத்துகிறது, மேலும் இது 2 முதல் X வரை அதை எழுத விரும்புகிறது, மேலும் இது 2 முதல் X வரை தெளிவற்றதாக இருக்கலாம் என்று பரிந்துரைக்கிறது. குறிப்புகளின் பிட் மற்றும் நாம் எல்லாவற்றையும் ஆர் முறைகளின் காலாவதியின் அடிப்படையில் எழுத வேண்டும், ஆனால் நீங்கள் நன்றாக ஆச்சரியப்படலாம், ஒருவேளை இந்த சொத்தை திருப்திப்படுத்தும் அனைத்து செயல்பாடுகளிலும் நாங்கள் போதுமான ஆக்கப்பூர்வமாக இருக்கவில்லை என்பது உங்களுக்குத் தெரியும். ஆர் முறைகளில் ஏதாவது ஒன்று மற்றும் R இன் வெவ்வேறு மதிப்புகள் செயல்பாட்டுக்கு வரலாம் ஆனால் நான் ஒரு சிறிய கூற்றைக் கீழே வைக்கப் போகிறேன், பின்னர் நீங்கள் விரும்பினால் ஆதாரம் எப்படி இருக்கும் என்பதை ஒரு ஓவியமாகத் தரலாம். உங்களிடம் சில சிக்கலான செயல்பாடு F இருப்பதாகக் கூறவும், அது பின்வரும் பண்புகளை முதலில் திருப்திப்படுத்துகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "இது முற்றிலும் குழப்பமான தொடர்ச்சியற்ற விஷயம் என்று உங்களுக்குத் தெரிந்த சிலவற்றில் இருந்து வேறுபடுத்தக்கூடியது, அதாவது நீங்கள் பைத்தியக்காரத்தனமான வழிகளில் சிந்திக்க விரும்பும் பகுதியளவு அளவு எனக்குத் தெரியாது. இது ஒரு நல்ல செயல்பாடு. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "அது வேறுபடுத்தக்கூடியது இது எல்லா இடங்களிலும் 0 க்கு சமமாக இல்லை, எனவே அந்த நிலை என் மனதில் நழுவியது மற்றும் நான் எந்த விரிவுரை அல்லது அது போன்ற ஏதாவது ஒரு விரிவுரையை மறந்துவிட்டேன், பின்னர் இந்த மையப் பண்பு உள்ளது, அது கூட்டலைப் பெருக்கமாக மாற்றுகிறது. ஒரு தனித்துவமான சிக்கலான எண் R உள்ளது என்பதை நான் உண்மையில் குறிப்பிட வேண்டும், எனவே நீங்கள் X இன் F ஐ எழுதலாம். இதன் மூலம் நீங்கள் X இன் மதிப்பை R மடங்குகளின் அதிவேக செயல்பாடு என்று எழுதலாம். எல்லையற்ற பல்லுறுப்புக்கோவை நல்ல வழித்தோன்றல் பண்புகள் மற்றும் இவை அனைத்தும் உங்களிடம் இருந்தால், நீங்கள் விரும்பும் ஒவ்வொரு அதிவேகமும் உள்ளது. எல்லா இடங்களிலும் உள்ளது என்று நாங்கள் கருதும் இந்த மதிப்பின் வழித்தோன்றல் என்ன என்பதை நீங்கள் முதலில் பார்க்க விரும்பினால் இதுபோன்ற ஒன்றைப் பாருங்கள், இல்லையா? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "எக்ஸ்ப்ரெஷனில் இருந்து எஃப் இன் எக்ஸ் காரணியை முழுவதுமாக வெளியேற்றலாம் மற்றும் முழு வரம்பும் எச் அடிப்படையில் மட்டுமே வெளிப்படுத்தப்படும், இது வழித்தோன்றல்களின் பின்னணியில் என்ன அர்த்தம் என்று நீங்கள் நினைத்தால், 0 இன் எஃப் அவசியம் 1க்கு சமம் என்பது இந்த முழு வரம்பு வெளிப்பாடு ஆகும். சில நிலையானது ஆனால் இன்னும் குறிப்பாக அது 0 இல் உள்ள எங்கள் செயல்பாட்டின் வழித்தோன்றல் எதுவாக இருந்தாலும், உங்களுக்கு இந்த வேடிக்கையான விஷயம் இருக்கிறது, அதன் வழித்தோன்றல் 0 இல் உங்களுக்குத் தெரிந்தால், எல்லா இடங்களிலும் அதன் வழித்தோன்றல் என்ன என்பதை தீர்மானிக்கிறது மற்றும் அதிவேக செயல்பாடுகளின் சூழலில் இது மிகவும் நன்கு தெரிந்திருக்கும். நாம் உண்மையில் கூறுவது ஒரு அதிவேக செயல்பாட்டின் வழித்தோன்றல் தனக்கு விகிதாசாரமாகும் மற்றும் விகிதாச்சார மாறிலி 0 இல் உள்ள வழித்தோன்றலுக்கு சமம், இவை அனைத்தும் மிகவும் சுருக்கமாக சொற்றொடராக உள்ளன, ஆனால் அதன் நோக்கம் அதை வலியுறுத்துவதுதான் பவர் X என நாம் ஏற்கனவே நினைக்கும் செயல்பாடுகள் மட்டும் அவசியமில்லை ஆனால் இது மிகவும் பரந்த அளவிலான செயல்பாடுகளாகும், இது கூட்டலைப் பெருக்கமாக மாற்றும் இந்த சுருக்கப் பண்புகளை திருப்திப்படுத்துகிறது. இரண்டாவது வழித்தோன்றல் மற்றும் அந்த விஷயத்திற்கு ஒரு மூன்றாவது வழித்தோன்றல் மற்றும் அது போன்றது ஏனெனில் வழித்தோன்றல் செயல்பாடு தனக்கு விகிதாசாரமாக உள்ளது, எனவே n வது வழித்தோன்றலை எடுக்க நீங்கள் அந்த விகிதாசார மாறிலியைப் பார்த்து அதை சக்தி n க்கு உயர்த்தலாம், பின்னர் இங்கிருந்து நீங்கள் ஒரு செய்யலாம் டெய்லர் தொடர் விரிவாக்கம் மற்றும் அந்த யோசனையில் டெய்லர் தொடருடன் வசதியாக இருக்கும் உங்களில் ஒரு மேம்பட்ட வீட்டுப்பாடமாக நான் அதை விட்டுவிடலாம், குறிப்பாக சிக்கலான எண்களின் அர்த்தத்தில் வேறுபடுத்தக்கூடிய எந்தவொரு வேறுபட்ட செயல்பாட்டின் யோசனையையும் நீங்கள் இணைக்க விரும்பினால். நிச்சயமாக ஒரு கல்லூரி தலைப்பு, நீங்கள் விரும்பியபடி அங்கு பகுத்தறிவை இணைக்கலாம் என்பது உங்களுக்குத் தெரியும், ஆனால் டெய்லர் தொடரைப் பற்றி மட்டுமே அறிந்த ஒருவரின் சூழலில் தெளிவற்ற பகுத்தறிவு அனுமதிக்கப்படுகிறது மற்றும் இந்த யோசனையை எடுத்துக்கொண்டு எஃப் மற்றும் டெய்லர் விரிவாக்கத்தைப் பார்க்க வேறு எதுவும் இல்லை. ஒரு தனித்துவமான கலப்பு எண் உள்ளது என்ற எண்ணத்தை நியாயப்படுத்துவது, எஃப் செயல்பாடு இப்படித்தான் எழுதப்பட வேண்டும், பின்னர் சாதாரண அதிவேகங்களுடனான இணைப்பு, அத்தகைய மதிப்பு இருக்கும்போதெல்லாம் R உண்மையான எண்களின் சிக்கலான சூழலில் நாம் என்ன செய்வோம். அந்த மதிப்பின் R இன் செயல்பாட்டின் எக்ஸ்ப்ஸைப் பார்த்து, அதை ஒரு அடிப்படையாக எழுதுங்கள். ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "pi பாதிகள் I பெருக்கல் X இன் எக்ஸ்ப்ஸ் மட்டும் அல்ல என்பதை நாம் விளக்கலாம், ஆனால் அதை 5 pi பாதிகளின் எக்ஸ்ப்ஸ் ஐ டைம்ஸ் எக்ஸ் என்றும் விளக்கலாம் மற்றும் இவை தனித்தனி செயல்பாடுகள் மற்றும் முடிவற்ற குடும்பம் தனித்தனி செயல்பாடுகள் உள்ளன. அவற்றை I to the X என எழுதுங்கள். எனவே I to the I என்ற வெளிப்பாடு, நீங்கள் ஒரு தரநிலையை ஏற்றுக் கொள்ளாத வரையில், அது அவசியம் என்னவாக இருக்கும் என்று நீங்கள் கூறும்போது அது எண்ணற்ற பல வெளியீடுகளைக் கொண்டுள்ளது என்று நினைக்கும் மற்றொரு வழி என்னவென்றால், I முதல் X வரையிலான செயல்பாடு எங்களிடம் உள்ள குறிப்பேடு சிறிது தெளிவற்றதாக உள்ளது, இப்போது இவை அனைத்தையும் கொண்டு சிலவற்றைக் காட்சிப்படுத்தத் தொடங்குவோம், ஏனென்றால் இது வேடிக்கையாக இருக்கிறது என்று நான் நினைக்கிறேன், மேலும் இது பயனுள்ள காட்சியா அல்லது குழப்பமான காட்சியா என்று நீங்கள் சொல்லலாம். நாம் என்ன செய்யப் போகிறோம் என்பது R டைம்ஸ் X இன் செயல்பாட்டைப் பார்க்க வேண்டும், இது அடிப்படையில் X இன் சக்திக்கு e ஐ எழுதுவதற்கான மற்றொரு வழியாகும். ஏனென்றால் நான் அதைச் செய்யத் திட்டமிட்டிருந்தேன், எனவே நீங்கள் எனது கோப்பு முறைமையில் திரும்பி வந்துவிட்டீர்கள், நீங்கள் இருக்க வேண்டிய இடத்திற்குத் திரும்புங்கள், அதில் பல வேறுபாடுகள் இருப்பதால் அது புகார் அளிக்கிறதா, அது இருப்பது போல் இருக்கும் ஓ அதை மாற்றவும் மற்ற திரையில் காண்பிக்கப்படும் காத்திருங்கள் அது ஏன் ஆம், சரி மாற்றுங்கள்? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "நீங்கள் எதைப் பார்க்கிறீர்களோ அதை அங்கே வைக்கவும், இப்போது நாங்கள் திரும்பிச் செல்கிறோம், அதையெல்லாம் நாங்கள் நன்றாக எழுத முடியும், அதனால் நீங்கள் அதை R காலத்தின் எக்ஸ்பிரஸ் என்று நினைத்து அசௌகரியமாக இருந்தால், இந்த எல்லையற்ற பல்லுறுப்புக்கோவை வெறும் உங்கள் தலையின் பின்புறம் e முதல் R முறை X வரை மற்றும் நாம் R ஐச் சுற்றி மாறுபடுவோம், எனவே நான் கற்பனை அச்சின் புள்ளிகளைப் பின்பற்றப் போகிறேன், நான் உண்மையான அச்சின் புள்ளிகளைப் பின்பற்றப் போகிறேன், இது என்ன செய்கிறது என்பதைப் பார்ப்போம் அது அனைத்து வகையான வேகமானது, எனவே எதிர்மறை எண்கள் அனைத்தையும் கொஞ்சம் மெதுவாக சிந்திக்கிறேன், அது ஒரு எதிர்மறை உண்மையான எண் 0 மற்றும் 1 இடையே உள்ள வரம்பிற்குள் squished ஆக போகிறது எது e க்கு எதிர்மறையாக இருக்க வேண்டும்? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "ஒரு எதிர்மறை உண்மையான எண் என்பது 0 மற்றும் 1 க்கு இடையில் உள்ள ஒன்று மற்றும் நாங்கள் குறிப்பாக எதிர்மறை 1 ஐ கண்காணித்து வருகிறோம், இது 1 க்கு மேல் e 30 0 ஐ சுற்றி காண்பிக்கும். e இல் 1 இல் 37 எஃப் எதிர்பார்க்கப்படுகிறது, அதுதான் 1 இன் எக்ஸ்எப் எஃப் என்பது யூனிட் வட்டத்தைச் சுற்றி ஒரு ரேடியனைச் சுற்றி வரப்போகிறது, மேலும் கற்பனை அச்சு ஒரு வட்டத்தைச் சுற்றி எப்படிச் சுற்றி வருகிறது என்பதை இங்கே முழு கற்பனை அச்சில் பின்பற்றுவது வேடிக்கையாக இருக்கிறது. R இன் இந்த மதிப்பை நாம் மாற்றினால் என்ன நடக்கும்? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "இங்கே R இன் மதிப்புகள் தேவைப்படலாம், அது விஷயங்களை வித்தியாசமாக நீட்டுகிறது, எனவே அதை 2 வரை வைக்கும் போது அது உண்மையான அச்சை இன்னும் அதிகமாக நீட்டுகிறது என்பது உங்களுக்குத் தெரியும், அதனால் 1 இன் f ஆனது e ஸ்கொயர் 7 fக்கு சற்று மேலே எதிர்மறையாக இருக்கும் இடத்தில் முடிவடைகிறது. 1 என்பது I இன் 0 f க்கு மிக அருகில் உள்ளது f எதிர்மறையின் வட்டத்தை சுற்றி 2 ரேடியன் சுழற்சி I எதிர்மறை 2 ரேடியன் சுழற்சி மற்றும் நிச்சயமாக நாம் நமக்கு பிடித்த சூத்திரத்தை பெறலாம், அது pi என்றால் அது நமது அளவிடுதல் மாறிலியாக இருந்தது. உண்மையான அச்சு நிறைய நீட்டிக்கப்படுகிறது என்பது உங்களுக்குத் தெரியும், 20 பிளஸ் பைக்கு மிக அருகில் இருக்கும் பையில் 1 இன் எஃப் அமர்ந்திருக்கிறது, இது எப்போதும் வேடிக்கையாகவும் எதிர்மறை 1 இன் எஃப் 0 க்கு மிக நெருக்கமாகவும் இருக்கும். அச்சு மற்றும் அது அலகு வட்டம் திசையில் விஷயங்களை நீட்டி அதனால் I அல்லது f எதிர்மறையின் f பெறுவது நான் வட்டத்தை சுற்றி பாதியில் நடப்பேன், அதனால் எல்லாம் நன்றாக இருக்கிறது இப்போது நாம் எப்படி ஒரு செயல்பாடு பற்றி யோசிக்க வேண்டும்? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "2 மடங்கு X இன் இயல்பான பதிவின் X இன் X ஆகவும் எழுதுவோம், எனவே R இன் மதிப்பைக் குறிக்கும் மஞ்சள் புள்ளியை 0 க்கு நகர்த்துவோம். 69 இன்னும் கற்பனையான பகுதி இல்லை உண்மையான எண் 0.69 அல்லது அதற்கு மேற்பட்டது 2 இன் இயற்கையான பதிவு, 1 இன் எஃப் 2 இல் இருப்பதை நீங்கள் பார்க்கலாம் அதனால்தான் இந்த செயல்பாட்டை 2 முதல் 1 பாதியின் X f என்று அழைக்க விரும்புகிறோம், உண்மையில் எதிர்மறை 1 நிலங்களை மன்னிக்கவும் 1 அரை f இல் நான் யூனிட் வட்டத்தைச் சுற்றி நடப்பது மிகவும் குறிப்பாக அது 0 ஆக இருக்கும். யூனிட் வட்டத்தைச் சுற்றி 69 ரேடியன்கள் மற்றும் இப்போது நாம் இன்னும் கொஞ்சம் வேடிக்கையாக இருக்க முடியும் மற்றும் இதை 0 ஆக மாற்றினால் என்ன நடக்கும் என்று சொல்லலாம். 69 2 இன் இயற்கைப் பதிவாக இருப்பதற்குப் பதிலாக, 2 இன் இயற்கைப் பதிவை I முறை ஆக்குங்கள், இதனால் நாம் உண்மையில் அதிவேகத் தளத்தைக் கொண்ட ஒன்றைப் பற்றி சிந்திக்கிறோம். ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "சக்திக்கு நான் என்றால் என்ன, இந்த விஷயத்தில் அது அதை 0 க்கு நகர்த்துகிறது. 2 ஐந்தில் ஒரு பங்கு ஆனால் பலவிதமான அதிவேக செயல்பாடுகள் உள்ளன, அவை 1 ஐ எண்ணின் மீது எஃப் போடும் பண்புகளைக் கொண்டிருக்கின்றன, எனவே நாம் அதை மேலும் அளவிட விரும்பினால், அதை இங்கே அனிமேஷன் செய்ததாக நான் நினைக்கவில்லை, ஆனால் நாம் எடுக்க வேண்டுமானால் அந்த மஞ்சள் புள்ளி மற்றும் அதை 5 அரை மடங்கு pi ஐ அடையும் வரை உயர்த்தவும் I அலகு வட்டத்தை நீங்கள் பார்ப்பது என்ன? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "1 இன் எதிர்ம f இன் மற்றொரு 2 pi ரேடியனைச் சுற்றி சுழன்று அது இருக்கும் இடத்தில் தரையிறங்கும் வகையில் அது தன்னைச் சுற்றியே சுழற்றப்படுகிறது. மிகவும் சிறிய எண் அது 0 என்பதை சுற்றி இருந்தது. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "எங்களிடம் X இன் R பெருக்கல் X மற்றும் R இந்த மதிப்புக்கு சமம், இது 2 கூட்டல் pi முறை I இன் இயற்கையான பதிவு ஆகும், இதன் பொருள் என்னவென்றால், நாம் 1 இல் 1 f ஐ செருகும்போது எதிர்மறை 2 இல் உள்ளது, எனவே இந்த செயல்பாட்டை எழுத விரும்புகிறோம். பவர் X வலதுபுறத்தில் எதிர்மறை 2 ஆக உள்ளது, அது உண்மையில் உங்களுக்குத் தெரிந்த ஒன்றுதான், எதிர்மறை எண்ணை ஒரு சக்திக்கு எதிர்மறை எண்ணை எழுதும் போது அது கொஞ்சம் ஏமாற்றும் எளிமையானது தான். எந்த வகையிலும் சிக்கலான எண்களுக்குள் ஆனால் நிச்சயமாக 1 பாதி போன்ற ஒரு மதிப்பை நாம் செருகும்போது, எதிர்மறை 2 இன் வர்க்க மூலத்தை நாம் கேட்கும் போது, இதை நான் வர்க்க மூலத்தை பெருக்குவது போல் எழுத விரும்புகிறோம். 2 இன் 2 ஆனால் இந்த செயல்பாடு எதிர்மறை 2 க்கு முழு சிக்கலான டொமைனில் உள்ள X ஐப் பார்த்தால், அது நீங்கள் எதைப் பார்க்கிறீர்களோ அதைக் கையாளும் ஒரு செயல்பாடு 1 இன் மதிப்பை எதிர்மறையாக 2 க்கு எடுக்கும். இது உண்மையான எண் கோட்டின் மற்ற பகுதிகளுக்குச் செய்கிறது, அது வெளிப்புறமாகச் சுழல்கிறதா? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "எனவே எதிர்மறை 1 இன் f எதிர்மறை 1 பாதியில் அமர்ந்திருப்பதைக் காண்கிறோம், நீங்கள் 1 பாதியின் f ஐப் பின்பற்றினால் நீங்கள் எதிர்பார்க்கும் இடத்தைப் பற்றி அது கற்பனைக் கோட்டில் சரியாக அமர்ந்திருக்கும் மற்றும் 1 பாதியின் f 2 இன் வர்க்க மூலமாக இருக்கும். நான் விரும்பும் இடத்தில் சுட்டி இல்லை. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "இது 2 மடங்கு I மற்றும் நீங்கள் மேலும் தொடரும்போது, இது உங்களுக்கு எதிர்மறை 2 முதல் X வரை உள்ள உண்மையான மதிப்பு சக்திகள் அனைத்தையும் உங்களுக்குக் காட்டுகிறது, அது அவசியம் சுற்றி வருகிறது, ஆனால் R இன் மதிப்பை இன்னும் அதிகமாக நகர்த்தி அதைப் பெறலாம். சுமார் tau முறை வரை நான் ஆறு புள்ளி இரண்டு எட்டு முறை நான் மற்றும் அந்த சூழலில் இது மற்றொரு செயல்பாடு ஆகும், இது X க்கு 2 என எழுத விரும்புகிறோம், ஏனெனில் நீங்கள் X க்காக செருகும் எந்த முழு எண்ணுக்கும் முழு எண்ணுக்கும் அது இருக்கும். மீண்டும் மீண்டும் பெருக்குவது போல் தோற்றமளிக்கிறது, மேலும் இது 1 பாதி போன்ற விஷயங்களுக்கு நியாயமான மதிப்புகளைக் கொண்டுள்ளது, அங்கு அது நேர்மறை சதுர மூலத்திற்குப் பதிலாக எதிர்மறை வர்க்க மூலத்தைத் துப்புகிறது, ஆனால் அது உண்மையில் செய்வது விமானத்திற்கு மாற்றத்தை ஏற்படுத்துகிறது, அது எல்லாவற்றையும் வைக்கிறது. எண் கோடு மிகவும் இறுக்கமாக காயம்பட்ட சுழல் என முடிவடைகிறது, மேலும் அது 1 இன் எஃப் எண் 2 இல் சரியாகச் செல்லும் வகையில் சுழல்கிறது, எனவே அந்த அர்த்தத்தில் நாம் 2 முதல் X என்று கூறலாம். நாம் பாரம்பரியமாகப் பழகிக் கொண்டிருக்கும் ஒரு தனி அதிவேகச் செயல்பாடு, அதனால் எல்லாவற்றோடும் நான் இன்றைக்கு விஷயங்களை விட்டுவிடுவேன் என்று நினைக்கிறேன், சரி பற்றி யோசிக்க இரண்டு நீண்ட கேள்விகளை உங்களிடம் விட்டு விடுகிறேன், எனவே நீங்கள் விரும்பினால் I to I என்பது பல மதிப்புள்ள வெளிப்பாடு என்று நீங்கள் நினைக்கலாம். பாதிகள் ஆனால் இந்த வகையானது நாம் பார்த்த பல்வேறு மதிப்புகளைப் போல எண்ணற்ற பல மதிப்புகளாக இருக்க வேண்டும் என்று நீங்கள் கூறினால், 2 முதல் 1 மூன்றில் எத்தனை மதிப்புகள் ஒரே அர்த்தத்தில் இருக்க வேண்டும்? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "10 வது எல்லாவற்றிலிருந்தும் வித்தியாசமாக ப்ரேஸ் செய்யப்பட வேண்டும், இது X இன் எக்ஸ்போனென்ஷியல் செயல்பாடுகள் அனைத்தையும் பற்றி சொல்கிறேன், ஓ, நான் எழுதிய இந்த பண்புகள் அனைத்தையும் திருப்திப்படுத்தும் X இன் எங்காவது அதை நான் எழுதியிருக்கிறேனா, அது அனைத்தையும் திருப்திப்படுத்தினால் இவற்றில் மற்றும் f இன் 1 என்பது 2 க்கு சமமாக இருந்தால், Xஐச் செருகும்போது எத்தனை வெவ்வேறு வெளியீடுகளைப் பெறப் போகிறோம், எந்தச் செயல்பாட்டிற்கான பல்வேறு விருப்பங்களுக்கு 3 10 க்கு சமம்? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "2 முதல் X வரையிலான பல்வேறு செயல்பாடுகளுக்கு 2 முதல் X வரை குறிப்பிடக்கூடிய பல்வேறு செயல்பாடுகளை நாம் 2 முதல் X வரை ஒருவித அதிவேகச் சார்பு என நினைத்துக்கொண்டால், இந்த வகையான சுருக்க பண்புகளின் பொருளில் எக்ஸ்போனன்சியல் மற்றும் நாம் ஆம் என்றால், நாம் என்றால் எங்களிடம் பல்வேறு வகையான செயல்பாடுகள் உள்ளன, மேலும் நாங்கள் பையை செருக விரும்புகிறோம், அது என்னைச் சிரிக்க வைக்கிறது, ஏனெனில் இது போன்ற ஒரு வேடிக்கையான பதில் எனக்குத் தெரியும், நீங்கள் அதைப் பற்றி சிந்திக்க முயற்சிக்கும்போது அது வெளிவருகிறது. நான் உங்களை விட்டுவிடுகிறேன், இன்றைய விரிவுரையை அணுகுவதில் எனது மையக் கேள்வி என்னவென்றால், அதிவேக செயல்பாடுகளின் இந்த சுருக்க பண்புகளைப் போலவே இது விவரிக்கப்பட வேண்டுமா என்பதுதான். r இன் வெவ்வேறு மதிப்புகளுக்கு r டைம்ஸ் x ஐ மிகவும் நேர்மையாக எழுதினேன் என்று நான் நினைக்கிறேன், அது உங்களை அவ்வளவு தூரத்தில் அடைத்துவிடாது. என்ன 2 க்கு சக்தி x ஐப் போல மிகக் குறைவாக இருக்க வேண்டும் x என்ற தெளிவற்ற கருத்து, நிச்சயமாக அதில் உள்ள ஆபத்து என்னவென்றால், சில நேரங்களில் மக்கள் சுருக்கத்தை விரும்புவதில்லை, சில சமயங்களில் அது அணுகக்கூடியதாக வராது, ஆனால் அதுதான் மின் கோபுரங்களைச் சேர்ப்பதற்காக இவை அனைத்தையும் சுற்றி ஒரு சுவாரஸ்யமான எண்ணங்களின் வட்டம் இருப்பதாக நான் நினைக்கிறேன், ஏனெனில் நீங்கள் உண்மையில் மின் கோபுரங்களைப் பற்றி பேச விரும்பினால், சிக்கலான எண்களின் சூழலில் நாம் கடந்த முறை இருந்தது போல் அல்லது எதிர்மறையான அடிப்படைகளுடன் கூட நீங்கள் இதைப் போன்ற விஷயங்களைச் சிந்திக்க வேண்டும், எனவே திரையில் எங்களுக்கு ஒரு கேள்வி இருந்தது ஆம், நான் சக்திக்கு இதை செய்தால் என்ன ஆகும்? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "உங்களுக்குத் தெரியும் டைட்ரேஷன் இதை முயற்சிப்போம், மேலே சென்று ஒரு மின் கோபுரத்தை முயற்சிப்போம், அங்கு நான் கொடுக்கப்பட்ட சக்திக்கு என்னை உயர்த்துகிறோம், அதில் என்ன தோன்றுகிறது என்பதைப் பார்க்கவும், எனவே இதைச் செய்யத் திட்டமிடவில்லை, ஆனால் நம்மால் எப்போதும் முடியும் பைத்தானை மேலே இழுத்து, கடைசியாக நாங்கள் என்ன செய்து கொண்டிருந்தோமோ அதைச் செய்யுங்கள், எனவே இது செயல்படும் வழி, சில அடிப்படை மதிப்புடன் தொடங்குகிறோம், பின்னர் சில வகையான வரம்பிற்கு நாங்கள் என்ன செய்து கொண்டிருந்தோம் என்பதை எடுத்துக் கொண்டோம், நாங்கள் மீண்டும் ஒதுக்கப் போகிறோம் அது எதுவாக இருந்தாலும், இந்த விஷயத்தில் நான் ஒரு சக்திக்கு உயர்த்தப்பட்ட அடிப்படை சரி, குளிர்ச்சியாக இருக்க வேண்டும், எனவே நாங்கள் அதைச் செய்யப் போகிறோம், அதன் மதிப்பை அச்சிடப் போகிறோம், இதற்காக இதைச் செய்வோம். ஆமாம், இது 200 போன்ற மிகப் பெரிய எண், அதனால் என்ன நடக்கிறது என்பது போல் சில நேரங்களில் இதுபோன்ற விஷயங்களில் குழப்பம் ஏற்பட வாய்ப்புள்ளது. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "நான் உண்மையில் NumPy ஐ இறக்குமதி செய்ய அனுமதிக்கிறேன், அதனால் என்னிடம் அதிவேக செயல்பாடு உள்ளது, அதை எழுதுவதை விட எங்கள் பெரிய வரம்பிற்கு என்னை விடுங்கள். வெவ்வேறு மாறிலியின் அதிவேக செயல்பாடாக ஒரு மாறுபட்ட மாறிலி 5 பை பாதியாக இருக்க வேண்டும் என்று நான் விரும்புகிறேன், எனவே நான் 5 பை பாதி முறை செய்வேன், எனவே இது ஒரு கலப்பு எண் மற்றும் இது 5 பை பாதிகளாக உள்ளது கற்பனையான பகுதி, இது 5 பை பாதி மடங்கு நான் மற்றும் நான் என்ன செய்கிறேன்? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/telugu/sentence_translations.json b/2020/ldm-i-to-i/telugu/sentence_translations.json index ad378a689..10a485147 100644 --- a/2020/ldm-i-to-i/telugu/sentence_translations.json +++ b/2020/ldm-i-to-i/telugu/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "కాబట్టి మీరు సంఖ్య 1 నుండి ప్రారంభిస్తే, మీ ప్రారంభ వేగం నేరుగా 0 వైపు నడవడం మరియు మీరు మరింత దిగువకు నడిచేటప్పుడు, మీరు 1 సగం వద్ద కూర్చుంటే, మీరు ఇప్పటికీ 0 వైపు నడుస్తూ ఉంటారు, కానీ ఇప్పుడు మీ వేగం వెక్టార్ మీరు ఎక్కడ ఉన్నారో 1 సార్లు ప్రతికూలంగా ఉంటుంది, ఇది ప్రతికూలంగా 1 సగం ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "మరియు ఒక ఆసక్తికరమైన ప్రశ్న ఏమిటంటే, దీని కోసం వ్రాయడం సహేతుకమైనదిగా భావించే అలాంటి ఒక ఫంక్షన్ మాత్రమే ఉందా అని మీకు తెలుసు, ఎందుకంటే మేము దానిని i to the x అని వ్రాస్తామో లేదో మీకు తెలుసు, ఇది దీన్ని సంతృప్తిపరచడమే కాదు, అది మీకు ఎప్పుడు సంతృప్తినిస్తుంది మనం పొందే నంబర్ వన్‌ని ప్లగ్ ఇన్ చేస్తాం నేను బహుశా పవర్ వన్‌కి అయితే ఈ ఫంక్షన్ ఐ అయి ఉండాలి అని ఆలోచిస్తున్నాము. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "కాబట్టి మనకు 5 pi i సగభాగాలు ఉన్నాయి, అది ఖచ్చితంగా మనం ఇక్కడ x కోసం ప్లగిన్ చేయగల మరొక విలువ మరియు మనం ఇక్కడ ఉన్న మన సర్కిల్‌ని తిరిగి చూస్తే కొంచెం దృశ్యమానంగా స్పెల్లింగ్ చేయడం. క్షణం 1 అయిన pi అర్ధభాగాలకు సమానమైన సమయం వరకు నడిచింది. 57 బదులుగా మనం మరొక పూర్తి మలుపు తీసుకున్నట్లయితే మరియు మనం piకి చేరుకోవడానికి మరొక pi హాల్వ్స్‌కు వెళితే, అది మనకు ఒక రకమైన రికార్డ్ కావచ్చునని మీకు తెలుసు, ఇక్కడే e నుండి pi నేను విలువను కలిగి ఉంటాము, మనం మరొక piని నడిస్తే, మేము మరొక pi సగం వరకు నడుస్తాము. ఈ పాయింట్‌లో మనం పూర్తి వృత్తానికి వెళ్లి ఒకదానికి తిరిగి వస్తాము, ఆపై మేము ఐదు పై భాగాలకు నడుస్తాము, ఇది సంఖ్యాపరంగా 7 ఉంటుంది. 85 అవును, అది ఖచ్చితంగా మరొక సంఖ్య ఐ పైన మనలను పొందుతుంది మరియు మనం ఐ శక్తికి iని తిరిగి వ్యక్తీకరించే మొత్తం రిగ్మారోల్ ద్వారా వెళ్ళాలంటే, మొదట e నుండి 5 pi అర్ధభాగాలకు i అని నేను శక్తికి i అని వ్రాయడం ద్వారా ప్రతికూలంగా మారడానికి గుణించండి మరియు మేము ప్రతికూల 5 pi అర్ధభాగాలను చూస్తాము, ఇది చాలా భిన్నమైన సంఖ్య, మేము దీన్ని నిజంగా లెక్కించగలము, నా తలపై నుండి నాకు ఖచ్చితంగా తెలియదు, కానీ డెస్మోస్‌ను చూద్దాం . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "ఆ పొడవు మిమ్మల్ని చాలా చిన్న సంఖ్యకు తీసుకువెళుతుంది కానీ మేము సరిగ్గా నమోదు చేయగల ఏకైక సమాధానం కాదు, మేము ఇతర వ్యక్తులు ప్రతికూలంగా 3 అర్ధ సార్లు i piతో ఇక్కడకు వస్తున్నాము, ఇది యూనిట్ సర్కిల్ పరంగా మీకు ఏది తెలుసు? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "నేను 90 డిగ్రీలు pi హాల్వ్స్ రేడియన్లు నడవడం కంటే నేను 270 డిగ్రీలు వేరే మార్గంలో నడిస్తే 3 pi సగం రేడియన్లు నడవాలంటే నేను హే అని చెప్పవచ్చు, ఎందుకంటే నేను ప్రతికూలంగా భావించవచ్చు ఎందుకంటే సమావేశం సాధారణంగా అపసవ్యదిశలో సానుకూలంగా ఉంటుంది, అది ఖచ్చితంగా దానిని వ్యక్తీకరించడానికి మరొక మార్గం మరియు మేము ప్రతికూల 3 pi అర్ధభాగాలకు eని కలిగి ఉంటే, అది మనకు భిన్నమైన సమాధానాన్ని పొందుతుంది i అన్నీ శక్తికి నేను అదే గేమ్‌కు వెళ్తాము, ఇప్పుడు i స్క్వేర్డ్ రద్దు చేయబడుతుంది ప్రతికూలత ఇప్పటికే ఉంది మరియు మనకు సానుకూల 3 pi సగభాగాలు ఉన్నాయి మరియు సంఖ్యాపరంగా ఇది మనకు ఇంతకు ముందు ఉన్నదానికి భిన్నంగా కనిపించే సమాధానాన్ని పొందుతుంది, మనం వెళ్లి హే అని చెబితే, 3 పైకి e అంటే 3 o 3 pi కాదు సగభాగాలు 111 పాయింట్ 3 1 111 పాయింట్‌కి ముందు మనం చూసిన దానికంటే చాలా భిన్నమైన సంఖ్య ఇది 111 పాయింట్ 3 1 గొప్ప 111 పాయింట్ 3 1 లేదా అంతకంటే ఎక్కువ మరియు మళ్లీ అంతర్ దృష్టి పరంగా మీరు ఏమి అడుగుతున్నారో మనం ఈ తిరుగుతున్నామని అనుకుందాం డైనమిక్ కానీ మనం సమయానికి వెనుకకు కదులుతాము, ఎంత కాలం క్రితం నేను ఎలా ఉండాలో మనం చూస్తాము, నేను అక్కడ నుండి ముందుకు సాగితే నేను మొదటి స్థానంలో ఉంటాను మరియు మీరు సమయానికి తిరిగి వెళ్ళాలి 3 pi సగం యూనిట్లు ఆపై మీరు క్షీణత డైనమిక్స్‌కు అనువదించినట్లయితే, ఈ సందర్భంలో కంటికి పెంచడం అంటే నేను మొదటి స్థానంలో ఉన్నాను అని మీరు అంటారు, కానీ నేను సమయానికి వెనుకకు వెళ్లాలనుకుంటున్నాను మరియు నేను ఎక్కడ ప్రారంభించాలో చెప్పాలనుకుంటున్నాను నేను మొదటి స్థానంలో నిలిచే విధంగా క్షీణించాలనుకుంటున్నారా? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "3 pi సగం యూనిట్ల సమయం తర్వాత, ఆ రకమైన ఘాతాంక క్షీణతకు సమాధానం స్పష్టంగా నూట పదకొండు నుండి మొదలవుతుంది మరియు వాస్తవానికి మనం X కోసం ప్లగిన్ చేయగల అనేక విభిన్న విలువలు ఉన్న చోట ఇది ఎక్కడికి వెళుతుందో మీరు చూడవచ్చు. e నుండి X వరకు నేను అని ఆలోచిస్తున్నాను మరియు ప్రజలు ఇక్కడ చాలా ఎక్కువ ప్రవేశించారు, మూడవ స్థానానికి క్లాసిక్‌గా నా పిన్‌ని నేలపైకి విసిరేస్తున్నాను క్షమించండి 9 pi విభజించటం గొప్ప ఎంపిక 1729 pi విభజించటం మీకు చాలా ఇష్టమైనవి మరియు చాలా ఉన్నాయి విభిన్న ఎంపికలు అనంతమైన అనేక విభిన్న విలువలు మొదట్లో కొంచెం అయోమయంగా అనిపిస్తాయి, ఎందుకంటే మేము ఒక వ్యక్తీకరణను చూస్తాము ఎందుకంటే కొంత గణన ఉండబోతోందని మీకు తెలిసినట్లుగా నేను దానిని నా కాలిక్యులేటర్‌లోకి ప్లగ్ చేసి, పాప్ అవుట్ అయ్యే వాటిని చూస్తాను మరియు మేము చాలా విభిన్నంగా ఉన్నాము దాని కోసం విలువలు కాబట్టి ఇక్కడ ఏమి జరుగుతోంది? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "16 యొక్క నాల్గవ మూలం 2 అయి ఉండాలి మరియు సమాధానం బాగానే ముగుస్తుంది, మీరు బహుళ-విలువ గల ఫంక్షన్‌ని కలిగి ఉన్నప్పుడు ఇలాంటి బహుళ ఎంపికలు ఉన్నప్పుడు మేము ఒక సమావేశాన్ని అనుసరిస్తాము ఫ్యాన్సీయర్ లింగోలో ఒకే ఇన్‌పుట్ మరియు ఒకే అవుట్‌పుట్‌తో దీన్ని ఫంక్షన్‌గా పరిగణించండి, మేము సంక్లిష్ట సంఖ్యలతో వ్యవహరిస్తున్నప్పుడు ఇది అన్ని సమయాలలో వస్తుంది, ఇది ఒక ఆపరేషన్ రకంగా ఏదో ఒక రకమైన ఆలోచనను కలిగి ఉంటుంది. బ్రాంచ్ అనే పదబంధాన్ని వినండి మీరు స్క్వేర్ రూట్ ఫంక్షన్ యొక్క శాఖను ఎక్కడ ఎంచుకుంటారు? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "అనేక విభిన్న సమాధానాలు ఉన్నందున, మేము మళ్లీ నేను ఈ 90 డిగ్రీల భ్రమణం అని అనుకుంటాము మరియు 90 డిగ్రీల భ్రమణం అని ఆలోచిస్తే, వర్గమూలం ఉండాలి అని మీకు తెలుసు, 45 డిగ్రీల కోణంలో కూర్చోవడం మీకు తెలుసా బహుశా అది చతురస్రం కావచ్చు I యొక్క మూలాన్ని రూట్ 2 ఓవర్ 2 రూట్ 2 ఓవర్ 2 అని చాలా స్పష్టంగా వ్రాయవచ్చు I అది కేవలం త్రికోణమితిని ఉపయోగిస్తోంది, అయితే నేను ఒక ప్రతికూల 270 డిగ్రీల భ్రమణంగా భావించినట్లయితే, అది ఆ ఆపరేషన్‌లో సగభాగం చేస్తున్నట్లు అనిపిస్తుంది. వాస్తవానికి మనల్ని మరొక వైపుకు తీసుకురావాలి బహుశా ఇక్కడ కూర్చున్న సంఖ్య I యొక్క వర్గమూలం అయి ఉండవచ్చు మరియు అది నిజానికి మనం చూసే ప్రతికూల మూలం 2 కంటే 2 మైనస్ రూట్ 2 కంటే 2 సార్లు I ఇప్పుడు వాస్తవ సందర్భంలో విలువైన విధులు మేము అవును అని చెప్పగలం, సానుకూల సమాధానం ఏది అయినా వర్గమూలాన్ని ఎంచుకోండి, అయితే వీటిలో దేనిని మీరు సానుకూల సమాధానంగా పరిగణిస్తారు? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "మరియు మీరు బాగా చెప్పారని నేను అనుకుంటున్నాను, ఇది ఏమిటో మాకు తెలుసు, దానిని 2 యొక్క వర్గమూలంగా నిర్వచించాము, అన్నీ బాగానే ఉన్నాయి మరియు మంచిది, అయితే నేను చెప్పినట్లయితే, మనం I వ్యక్తీకరణకు ఎలా చేరుకుంటున్నామో అదే విధంగా దీన్ని చేరుద్దాం. ముందుగా ఏదైనా సరైనదానికి e గా తెలియజేయాలనుకుంటున్నాను మరియు నేను దానిని 1 సగానికి ఘాతాంకంలోకి గుణించడం ద్వారా 1 సగానికి పెంచబోతున్నాను మరియు నేను సరే అని చెప్పాను, నేను దానిని చేయగలనని ఊహించగలను 2 బావికి సమానం అది 2 యొక్క సహజ లాగ్ ఇది 0 చుట్టూ ఉండే స్థిరాంకం. 69 లేదా అంతకంటే ఎక్కువ మేము eని ఆ శక్తికి పెంచినట్లయితే, మనకు 2 వస్తుంది కాబట్టి మేము దీనిని 2 సార్లు 1 సగం యొక్క సహజ లాగ్‌కు eగా భావించవచ్చు మరియు మీరు e నుండి x గురించి ఆలోచిస్తున్నట్లయితే మీరు కావాలనుకుంటే? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "వాస్తవ సంఖ్యల సందర్భంలో ఇది ఒక రకమైన ఓవర్‌కిల్ అని మీకు తెలుసు, అయితే మీరు ఈ x ఫంక్షన్‌కి సంక్షిప్తలిపిగా e నుండి xని అనుకుంటే మీరు 0 విలువను ప్లగ్ చేయవచ్చు. 69 సార్లు 1 సగం అంటే దాదాపు 0 ఉంటుందని నేను ఊహిస్తున్నాను. 345 ఇష్ అలాంటిదేదో మీరు మీ బహుపదిలో ఆ కాంక్రీట్ విలువను ప్లగ్ చేయండి, అది ఏమి అవుట్‌పుట్ చేస్తుందో చూడండి మరియు అది 1 చుట్టూ అవుట్‌పుట్ అవుతుంది. 414 మీరు ఆశించే 2 యొక్క మంచి వాస్తవ సంఖ్య వర్గమూలం, కానీ మనం అదే పనిని ఐతో చేస్తున్నట్లయితే మరియు శక్తికి eగా ఏదైనా వ్రాయాలనుకున్నప్పుడు వాస్తవానికి అనేక విభిన్న సమాధానాలు ఉన్నాయని అంగీకరిస్తే మనం దీన్ని కూడా వ్రాయవచ్చు. ఇది హాస్యాస్పదంగా అనిపించవచ్చు, కానీ మేము దానిని 2 ప్లస్ 2 pi యొక్క సహజ లాగ్‌కు e అని వ్రాయవచ్చు I ఆ మొత్తం 1 సగానికి పెంచబడిన తర్వాత ఈ విలువ మీకు సమానంగా వచ్చిన తర్వాత అది e కి ఉన్నందున దానిని విచ్ఛిన్నం చేయవచ్చు. 2 యొక్క సహజ లాగ్ 2 pi నుండి గుణించబడుతుంది, ఇది 360 డిగ్రీలు తిరిగే ప్రభావాన్ని కలిగి ఉంటుంది, కనుక ఇది కేవలం 1కి సమానం అవుతుంది కాబట్టి మేము 2 సార్లు 1 గొప్పగా చూస్తున్నాము, అది చెల్లుబాటు అయ్యే ప్రత్యామ్నాయంగా అనిపిస్తుంది మరియు ఇంకా ఎప్పుడు మేము దీన్ని తీసుకొని దానిని శక్తిగా పెంచడం మరియు శక్తిని గుణించడం ద్వారా ఏమి జరుగుతుందో పరిశీలించడం ద్వారా అదే గేమ్ ఆడతాము, మనం సహజంగా 2 సార్లు 1 సగం ప్లస్ 2 సార్లు 1 సగానికి 1 అర్ధాన్ని కలిగి ఉన్నాము. అది pi సార్లు అవుతుంది I ఇప్పుడు ఈ మొదటి భాగం e 2 సార్లు 1 సగం యొక్క సహజ లాగ్‌కి, ఇది 2 యొక్క సుపరిచితమైన స్క్వేర్ రూట్‌గా ముగుస్తుంది, అది బాగానే ఉంది, కానీ మేము దానిని e ద్వారా గుణించబోతున్నాము pi I రైట్ మరియు చాలా ప్రముఖంగా pi నుండి నేను ప్రతికూలం 1 కాబట్టి ఈ సందర్భంలో మనం ఈ వ్యక్తీకరణను 2 నుండి 1 సగం వరకు పరిష్కరిస్తున్నట్లయితే, విభిన్న సమాధానాలతో ఆడుకోవడం ద్వారా మనం ఇలాంటి వాటి కోసం ప్లగ్ ఇన్ చేయవచ్చు అని సూచిస్తున్నట్లు కనిపిస్తోంది. e నుండి Xకి సమానమైన 1 సగానికి మనం ముగిసేదానికి మరొక సమాధానం, మనం సాంప్రదాయకంగా 2 యొక్క ఈ ప్రతికూల వర్గమూలంగా వ్రాయవచ్చు మరియు ఇక్కడ నా ఉద్దేశ్యం 2 నుండి 1 సగం వరకు చూసేందుకు బహుళ విలువలను కలిగి ఉండటం కొంచెం ఫన్నీ మరియు ఒక విషయాన్ని సమం చేయడం కాదు అని చెప్పండి, కానీ మనం చేసే ఎంపికల ఆధారంగా అది అనేక విభిన్న విషయాలకు సమానంగా ఉంటుంది, కానీ రెండు విషయాలు చాలా సహేతుకంగా అనిపించవచ్చు, 2 నుండి 1 సగం వరకు ఏదైనా ఉంటే అది సానుకూలంగా ఉండాలి మనకు తెలిసిన వర్గమూలం లేదా దాని యొక్క ప్రతికూల రూపాంతరం వాస్తవానికి అలాంటి సమస్యగా అనిపించదు మరియు వాస్తవానికి మేము ఈ గేమ్‌ని మరింత ముందుకు ఆడగలము, ఇక్కడ ఈ వ్యక్తీకరణకు మరింత సృజనాత్మక సమాధానాల కోసం నేను మిమ్మల్ని అడుగుతాను ఎందుకంటే Iని మూల్యాంకనం చేయడంలో మనం ఉపయోగించిన అదే నియమాలకు కట్టుబడి ఉంటే మనం ఏ ప్రత్యామ్నాయం చేస్తాము అనే దాని ఆధారంగా X యొక్క వివిధ విభిన్న విలువలను ప్లగ్ చేయడం ప్రారంభించినప్పుడు పవర్ Xకి 2 వంటి ఇతర ఫన్నీ పవర్‌లను కనుగొనవచ్చు. పవర్ I కాబట్టి ఈసారి ప్రశ్న అడుగుతుంది లేదా ఇది సమీకరణం యొక్క ఒక పరిష్కారం e నుండి xకి సమానం 2 అని నిర్దేశిస్తుంది, అది మనకు తెలిసిన 2 సరే యొక్క వాస్తవ సంఖ్య సహజ లాగ్. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "ఇ ప్రశ్నకు సమాధానం xకి 2కి సమానం మరియు మళ్లీ సృజనాత్మకత స్వాగతం పలుకుతుంది, కాబట్టి దాని కోసం నేను మీకు మరో చిన్న క్షణం ఇస్తాను II మీతో సరిగ్గా ఉంటే ఇక్కడ కొన్ని సమాధానాలను లాక్ చేస్తాను, ఎంత సమయం అయిందో నాకు ఖచ్చితంగా తెలియదు మీరు ఏ పరికరాన్ని చూస్తున్నారు అనేదానిపై ఆధారపడి గణిత ప్రవేశాన్ని తప్పనిసరిగా చేయవలసి ఉంటుంది, అయితే మీరు సమాధానం ఇవ్వాలనుకుంటున్న ప్రశ్నకు సమాధానం ఇవ్వడానికి మీకు అవకాశం రాకముందే ఎక్కువ ఒత్తిడికి గురికాకండి కాబట్టి ఇది ఇలా కనిపిస్తుంది మీలో 131 మంది వేరియంట్‌లోకి ప్రవేశించారు, అక్కడ మేము 2 యొక్క Ln తీసుకుంటాము మరియు మేము 2iiని జోడిస్తాము మరియు నేను ఈ ప్రశ్నను వ్రాస్తాను మరియు వాస్తవానికి చాలా కొన్ని సరైన సమాధానాలు ఉన్నప్పుడు తప్పుగా గుర్తించబడిన సమాధానాలలో ఒకటి సరైనదిగా గుర్తించబడిందని నేను భావిస్తున్నాను కాబట్టి అది నాపై ఉంది ఇది మీలో ఎవరికైనా ఓహ్ రెడ్ ఈస్ రెడ్ లాగా కనిపిస్తుందో లేదో నాకు తెలియదు కాబట్టి మీరు Ln ఆఫ్ 2 ప్లస్ 42 ఎంటర్ చేసినప్పుడు తప్పుగా అర్థం చేసుకున్నారు. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi ఇది ఒక గొప్ప ఎంపిక అయితే మీరు 4 pi I మరియు 2 లేదా 6 pi I యొక్క సహజ లాగ్ లేదా నిజంగా 2 pi I యొక్క ఏదైనా పూర్ణాంకం గుణింతాన్ని కలిగి ఉండవచ్చు, అది eని ప్రభావితం చేయదని మీరు జోడిస్తే X ఎందుకంటే ఇది కేవలం 2 pi I నుండి గుణించడం యొక్క ప్రభావాన్ని కలిగి ఉంటుంది, ఇది 1 ద్వారా గుణించడం యొక్క ప్రభావం మరియు మళ్లీ ఇది ఒక రకమైన ఫన్నీ పర్యవసానంగా ఉంటుంది, ఇక్కడ మనం మరొక ఉదాహరణగా చేసినప్పుడు ఇది సహేతుకమైన ఫలితాలను ఇస్తుంది. రెండవ అత్యంత సాధారణంగా నమోదు చేయబడిన వ్యక్తీకరణ వలె కనిపిస్తుంది, మనం 2ని భర్తీ చేయవచ్చు కాబట్టి మనం 2ని 1 4వ శక్తికి మారుస్తున్నామని అనుకుందాం, సరే మనం 2ని 2 ప్లస్ 4 యొక్క సహజ లాగ్‌కి eతో భర్తీ చేయాలనే సూచన ఉంది pi I Okay Plus 4 pi I మరియు మేము వాటన్నింటినీ 1 4వ కుడివైపుకి పెంచుతాము, మీరు అదే గేమ్‌ను ఆడితే మీరు e 2 సార్లు 1 4వ సహజ లాగ్‌ను పొందుతారు మరియు మేము e ద్వారా గుణిస్తాము pi I ఇప్పుడు దాని యొక్క మొదటి భాగం 2 యొక్క సాధారణ సానుకూలమైన నాల్గవ మూలంగా ఉంటుంది, మీరు 2 యొక్క నాల్గవ మూలం వంటి వ్యక్తీకరణను కాలిక్యులేటర్‌లో చక్కని చిన్న సానుకూల సంఖ్యను ప్లగ్ చేసినప్పుడు మేము అర్థం చేసుకున్నాము, అయితే ఈ రెండవ భాగం ప్రతికూల 1 కాబట్టి మేము 2ని ఈ విధంగా వేరే విధంగా అర్థం చేసుకుంటే మీకు తెలుసా అని చెబుతున్నట్లుగా ఉంది, దానిని 1 4కి పెంచడం వల్ల ఇది మనకు వచ్చే సాధారణ సమాధానం కాదని మీకు తెలుసు, అయితే ఇది సహేతుకమైన సమాధానం. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "మేము pi సగభాగాల రెట్లు Iని చూస్తాము మరియు ప్రతికూల 1తో గుణించే బదులు Iతో గుణించాము, ఇది మళ్లీ చెల్లుబాటు అయ్యే సమాధానం, ఇది 2 నుండి 1 4వ వరకు సహేతుకమైన అవుట్‌పుట్‌గా కనిపిస్తుంది కాబట్టి మీరు ఉన్నప్పుడు నేను శక్తికి చాలా భిన్నమైన విలువలను కలిగి ఉన్నట్లుగా అనిపించే వాస్తవాన్ని చూస్తే, మనకు ఈ ఫన్నీ దృగ్విషయం ఉంది, ఇక్కడ మనం 5 pi విభజించటం I ప్రతికూలం 3 pi విభజించటం I మరియు మేము చాలా భిన్నమైన సమాధానాలను పొందుతాము. మేము ఇక్కడ ఇంతకు ముందు కనుగొన్న 1 5వ సుమారు 1 5వ సమాధానం నుండి చాలా పెద్దది చాలా చిన్నది చాలా భిన్నంగా ఉంటుంది, మీరు 2 నుండి 1 4వ వరకు ఏది అని అడుగుతున్నప్పుడు మరియు వాస్తవానికి అనేక విభిన్న పరిష్కారాలు ఉన్నాయని అంగీకరిస్తున్నప్పుడు ఇది సరిగ్గా అదే దృగ్విషయం. X నుండి 4వ వరకు ఉన్న వ్యక్తీకరణకు సమానం 2 4 విభిన్న పరిష్కారాలు నిజానికి మరియు మీరు చూస్తున్నది బహుళ భిన్నమైన పరిష్కారాలు ఉన్నాయి అనే వాస్తవం e నుండి X వ్యక్తీకరణకు ఒక రకమైన బేస్ సమానం ఆ బేస్ I అయినా ఆ బేస్ అయినా 2 అది ఏమైనా కావచ్చు మరియు మేము దీని గురించి ఆలోచించే ఒక మార్గం ఏమిటంటే, మీరు వాస్తవ సంఖ్యలతో వ్యవహరించేటప్పుడు విషయాలు కేవలం మనోహరమైనవి మాత్రమే ఉంటాయి ఒకరి నుండి ఒకరికి సంబంధాలు ఉన్నాయి. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "ఎక్స్‌పోనెన్షియల్ ఫంక్షన్‌ల గురించి మనం ఆలోచించాలనుకుంటే ఇది చాలా బాగుంది, ఈ విషయాలలో కొన్నింటిని కవర్ చేయనివ్వండి, మా దగ్గర ఈ చక్కని ముందుకు వెనుకకు ఉంది, ఇక్కడ మీరు ఏదైనా ఎక్స్‌పోనెన్షియల్‌ని 2 నుండి X వరకు బేస్‌గా ఎక్స్‌పోనెన్షియల్‌గా వ్యక్తీకరించడానికి ఎంచుకోవచ్చు లేదా మీరు వ్యక్తీకరించవచ్చు. X యొక్క R సార్లు X వలె అదే ఘాతాంకం, మేము సూచించే బహుపది అని మీకు తెలుసు, ఇది మేము X కి e లాంటిది వ్రాసినప్పుడల్లా పరోక్షంగా సూచించినప్పుడల్లా సూచించబడుతుంది మరియు మీరు B యొక్క సహజ సంవర్గమానాన్ని తీసుకోవచ్చు కాబట్టి ఒక అందమైన ముందుకు వెనుకకు ఉంది. మరియు ఇది B అనేది సానుకూల సంఖ్య అని ఊహిస్తూ మీకు ఒక సమాధానం ఇస్తుంది మరియు X యొక్క R B కి సమానం అని చెప్పడం అదే విషయం కాబట్టి నేను ఈ సిరీస్‌లో ఇంతకు ముందు మాట్లాడిన ఒక మార్గం ఏమిటంటే మీరు చూస్తున్నట్లయితే సాధ్యమయ్యే అన్ని ఎక్స్‌పోనెన్షియల్‌ల కుటుంబం మేము వాటిని R రెట్లు X యొక్క X అని వ్రాసి, R అంటే ఏమిటో మార్చవచ్చు మరియు మీరు So e to the Rతో మరింత సౌకర్యవంతంగా ఉన్నట్లయితే, R టైమ్‌ల Xకి eని వ్రాయడం కూడా ఇదే. R సార్లు X యొక్క XX సార్లు X అనేది అదే విషయాన్ని మార్చడం గురించి మేము ఆలోచించగలము, కానీ మరోవైపు మీరు సాధ్యమయ్యే అన్ని ఎక్స్‌పోనెన్షియల్‌ల గురించి కొంత బేస్‌గా ఆలోచిస్తే, X యొక్క శక్తిని బేస్ చేయనివ్వండి మరియు మేము వెళ్తున్నాము ఆ స్థావరాన్ని మార్చడం మొదట్లో అది మానిప్యులేట్ చేయడానికి భిన్నమైన వ్యక్తీకరణగా అనిపిస్తుంది, కానీ ఇది ఒకే కుటుంబాన్ని వ్యక్తీకరించడానికి మరొక మార్గం మరియు మీరు దీని గురించి ఆలోచించే మార్గం. మేము R సార్లు X యొక్క ఎక్స్‌ప్‌ట్రాక్ట్‌గా కొంచెం వియుక్తంగా ఆలోచిస్తున్నట్లయితే మరియు నేను దీన్ని చేయడానికి ఒక కారణం ఉంది, ఎందుకంటే మేము దీన్ని సంక్లిష్ట సంఖ్యలకు వర్తింపజేయబోతున్నాము, అక్కడ ఇది విచిత్రంగా కనిపిస్తుంది కాబట్టి ఇక్కడ నాతో అనుసరించండి ఆ ఆధారాన్ని చూసే బదులు నేను చేయగలిగినది విలువ ఏమిటి? ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "నేను R సార్లు Xని కలిగి ఉండవచ్చు, ఇక్కడ R అనేది సున్నా పాయింట్ ఆరు తొమ్మిది లాగా ఉండవచ్చు కానీ నేను దానిని రెండు pi ద్వారా క్రిందికి మార్చగలను I మరియు అది ఇప్పటికీ రెండుకి అనుగుణంగా ఉండే బేస్‌ను మార్చదు లేదా అది చేయగలదు రెండు pi I ద్వారా దాన్ని మార్చండి, అది దానికి అనుగుణంగా ఉండే బేస్‌ను మార్చదు ఎందుకంటే ఆ సందర్భాలలో అన్నింటిలో మనం Xని ప్లగ్ ఇన్ చేసినప్పుడు ఒకదానికి సమానం అయినప్పుడు మనం అదే విషయాన్ని పొందుతాము, అయితే X యొక్క విభిన్న విలువల కోసం ఇవన్నీ విభిన్నమైన విధులు ఇది I నుండి శక్తికి I కోసం అనేక విభిన్న విలువలను మనం ఎందుకు చూశాము ఎందుకంటే I to X అనేది ఒక అస్పష్టమైన ఫంక్షన్ కాబట్టి ఆ సందర్భంలో R యొక్క ఏ విలువను మనం నిర్ణయించుకున్నామో అది నిస్సందేహంగా ఉంటుంది, అంటే మనం ప్రాతినిధ్యం వహిస్తున్నది R సార్లు X ఏ విలువను సూచిస్తుంది R యొక్క మనం ఒకదాన్ని ఎంచుకున్న వెంటనే ఎంపిక చేసుకుంటామా? ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "ఇది నిస్సందేహమైన పని, కానీ ఆ సమయంలో అది మనకు కావలసినది అనిపిస్తుంది X శక్తికి పెరిగిన కొన్ని బేస్ పరంగా విషయాల గురించి ఆలోచించడం మానేయడం బహుశా మనం సంక్లిష్ట సంఖ్యల సందర్భంలో ఉన్న వెంటనే మనం వ్రాయాలి అవన్నీ కొన్ని స్థిరమైన సమయాల ఎక్స్‌కి ఎక్స్‌గా ఉంటే, మరే ఇతర కారణం లేకుండా, మనం గణన చేయాలనుకుంటే లేదా దాని పైన గణితాన్ని చేయాలనుకుంటే, మనం వాస్తవానికి సంఖ్యలను ఎలా ప్లగ్ ఇన్ చేయాలో స్పష్టంగా తెలియజేస్తుంది, మనకు ఈ అద్భుతమైన అనంతమైన బహుపది వచ్చింది వాటిని ప్లగ్ చేయండి మరియు ఘాతాంకాలను గురించి ఆలోచించడానికి ఇదే సరైన మార్గం అని నేను మీ కోసం మరొక సందర్భం చేస్తాను, కాంప్లెక్స్ నంబర్‌ల వంటి ఇతర డొమైన్‌లలోకి విస్తరింపజేయడంతోపాటు దాని కోసం మనం బ్యాకప్ చేద్దాం తిరిగి డోర్‌బెల్‌కి వచ్చిన కొన్ని విషయాలు అసలు మార్గానికి తిరిగి వెళ్లి, మేము ఎక్స్‌పోనెన్షియేషన్ ఆలోచనను విస్తరింపజేస్తాము మరియు సహజ సంఖ్యల కోసం దీని గురించి ఎలా ఆలోచించాలో మనకు తెలుసు X కుడికి 2 అని ఆలోచించండి. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "2 నుండి 3 పునరావృత గుణకారం ఎలా ఉంటుందో మీకు తెలుసు, పాక్షిక మొత్తాల కోసం లేదా ప్రతికూల మొత్తాలు మరియు అలాంటి వాటి కోసం 2 నుండి X వంటి వాటి గురించి ఆలోచించడం మీకు మొదట నేర్పించబడింది. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "బాగా. మీరు సాధారణంగా 2 నుండి 1 సగానికి గుణించవలసి ఉంటుందని మీకు బోధిస్తారు మరియు ఇది ఘాతాంకంలోని అంశాలను మనం జోడించగలిగే ఘాతాంకంలో 2 పొందవలసిన సంఖ్యలను లెక్కించేటప్పుడు ఎక్స్‌పోనెన్షియల్స్ చేసే సాధారణ నియమాలను అనుసరిస్తుంది 1కి అది కొంత సంఖ్య అయి ఉండాలి కాబట్టి నేను దానిని స్వయంగా గుణించినప్పుడు నాకు 2 వస్తుంది మరియు ఆ సమయంలో మీకు ఎంపిక ఉందని మీకు తెలుసు, బహుశా అది సానుకూలంగా ఉండవచ్చు. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "బహుశా ఇది ప్రతికూలంగా ఉండవచ్చు కానీ మీరు ఎల్లప్పుడూ సానుకూలంగా ఎంపిక చేసుకోవాలని నిర్ణయించుకుంటే, మేము ప్రతికూల సంఖ్యల గురించి అడిగితే మీరు ఇదే ఒప్పందం నుండి చక్కని నిరంతర పనితీరును పొందగలుగుతారు నేను దానిని 2 నుండి 1కి గుణించినప్పుడు ఎక్కడ? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "ఇది నాకు 2 నుండి 0కి వస్తుంది మరియు ప్రతికూల ఘాతాంకాలు 1 సగం లాగా ఉన్నాయని మా సంప్రదాయానికి ఇది ఒక రకమైన సమర్థన, కానీ ఇక్కడ నిజంగా ఏమి జరుగుతోంది, ఇది ఏదైనా సరే అది ఏదో ఒక రకమైన ఫంక్షన్‌గా ఉండాలి, అది ఈ ఆస్తిని సంతృప్తిపరుస్తుంది. a ప్లస్ b అనేది b యొక్క రెట్లు f యొక్క fకి సమానం మరియు అంతేకాకుండా బేస్ 2 అనేది ప్రాథమికంగా మనకు చెబుతోంది, ఇది అటువంటి ఫంక్షన్ మాత్రమే కాదని ఇది ఒక ఫంక్షన్ అని మనం 1ని ప్లగ్ ఇన్ చేసినప్పుడు మనకు 2 వస్తుంది మరియు మీకు కొంచెం తెలుసు. ఇక్కడ ఉన్న కొన్ని చిక్కులతో పాటు మీరు అనుసరిస్తున్నారో లేదో తెలుసుకోవడానికి తెలివిని తనిఖీ చేసే శైలి ప్రశ్న నేను మిమ్మల్ని అడగాలనుకుంటున్నాను, నేను దీనిని సాఫ్ట్‌బాల్ లాగా పిలవను, కానీ ఇది చాలా లోతైన ప్రశ్న వలె ఉద్దేశించబడలేదు తప్పనిసరిగా. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "మీరు ఫంక్షన్ యొక్క లక్షణాలతో వియుక్తంగా ప్రారంభించి, ఆపై x యొక్క f ఈ ఎక్స్‌పోనెన్షియల్ ప్రాపర్టీని సంతృప్తిపరిచినట్లయితే, ఆ లక్షణాల ఆధారంగా వ్రాయాలనుకునే మార్గాలను తగ్గించే ఆలోచనతో పాటుగా మీరు అనుసరిస్తున్నట్లయితే ఇది మరింత తనిఖీ అవుతుంది. అన్ని ఇన్‌పుట్‌ల కోసం a ప్లస్ b ఒక సార్లు f యొక్క fకి సమానం మరియు ఇది 1కి సమానం 2కి సమానం 2 కిందివాటిలో ఏది నిజమో మీరు ఏ ఫంక్షన్‌ను ప్రారంభించినా కింది వాటిలో ఏది నిజం అని చెప్పాలి ఇది ఏ ఉపన్యాసం అని మీలో గుర్తుంచుకునే వారు, ఆయులర్ సూత్రం నిజంగా ఏమి చెప్పాలో మేము మాట్లాడుతున్నాము, నేను ఈ శైలి గురించి ఒక ప్రశ్న అడిగాను, అక్కడ నేను ఒక షరతును విస్మరించాను, నేను వ్రాయలేదని మీకు తెలుసు x యొక్క f ప్రతిచోటా నాన్ జీరో అని నిర్ధారించుకోవాలనుకుంటున్నాము మరియు అది కొంత మొత్తంలో గందరగోళానికి కారణమైంది, ఇది మనందరికీ జరిగే స్క్రీన్‌పై గందరగోళాన్ని కలిగిస్తుంది, అయితే దీని ఉద్దేశ్యం ప్రాథమికంగా ఈ నైరూప్య ఆస్తిని చూపించడం. సంకలనాన్ని గుణకారంగా మార్చడం అంటే ప్రాథమికంగా మీరు ఫంక్షన్‌ను ఏదో ఒక రకమైన శక్తికి పెంచినట్లుగా వ్రాయాలని కోరుకునేలా చేస్తే సరిపోతుంది, ఇది ప్రశ్న యొక్క స్ఫూర్తి ఇప్పుడు పవర్ టవర్‌ల గురించి వాస్తవానికి రెండు ప్రశ్నలు వచ్చాయి. ఇక్కడ పాప్ అప్ అయినట్లు అనిపిస్తుంది, ఇది చివరిసారిగా కనెక్ట్ చేయబడినది, పవర్ టవర్ ప్రశ్నను ఒక్క క్షణం ఆపివేద్దాం, తద్వారా మనం మొదట లోతైన అనుభూతిని పొందుతాము, ఇక్కడ ఎక్స్‌పోనెన్షియేషన్ అంటే ఏమిటి? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "ఎందుకంటే నేను క్లెయిమ్ చేయాలనుకున్నది మనం కావచ్చు కాబట్టి మేము దానికి అనేక రకాలుగా సమాధానం చెప్పగలము కాబట్టి మీరు నాకు ఒక్కటి ఇస్తే, మేము పవర్ టవర్ల గురించి మాట్లాడుతాము మరియు ఆపై ఒక సంఖ్యా రేఖను లాగరిథమిక్ స్కేల్‌లో సూచించవచ్చు. ఒక క్లిష్టమైన విమానం కోసం అదే జరుగుతుంది? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "అవును నిజానికి, నేను ఇక్కడ ఒక క్షణంలో పొందబోతున్న ఒక విజువలైజేషన్ ఉంది, ఇక్కడ మనం దానికి సారూప్యమైన పనిని చేస్తాము, ఎందుకంటే మేము చేసేది వివిధ ఎక్స్‌పోనెన్షియల్ ఫంక్షన్‌లు X యొక్క R సార్లు Xతో ఆడటం కానీ మేము కొద్దిగా పసుపు చుక్క ద్వారా సూచించబడే R విలువను మార్చబోతున్నాం కాబట్టి మేము దీని ద్వారా మాట్లాడతాము ఇది మొత్తం విమానాన్ని మ్యాప్ చేయదు, కానీ వాస్తవ అక్షం మరియు ఊహాత్మక అక్షం నుండి కేవలం ఒక జంట నమూనా పాయింట్లు కానీ ఆలోచన ఏమిటంటే, ఆ స్థిరాంకం ఏమిటో మనం చుట్టూ తిరిగేటప్పుడు, అది విమానానికి చేసే విభిన్న విషయాలను మనం దృశ్యమానం చేయగలము మరియు ప్రభావవంతంగా ఇది x- అక్షాన్ని లాగరిథమిక్ స్కేల్‌గా మార్చడం మరియు ఆపై చుట్టడం వంటిది. ఒక వృత్తం వెంబడి ఊహాత్మక అక్షం మరియు R యొక్క ఆ విలువ ఊహాత్మకంగా మారిన వెంటనే అది ఆ వాస్తవ సంఖ్యల పాత్రను వృత్తం మీద ఉంచబడుతుంది మరియు ఊహాత్మక సంఖ్యలు సంవర్గమాన స్కేల్ చేసిన సానుకూల అక్షంపై ఉంచబడతాయి కాబట్టి ఈ మూడింటిని నేను ఊహించిన గొప్ప ప్రశ్న నేను ఎక్కడికి వెళ్లాలనుకుంటున్నానో దాని కోసం తుపాకీని ముందుకు దూకుతాను కానీ ప్రజలు ఈ విషయంలో అలా ఆలోచిస్తున్నట్లు చూడటం ఆనందంగా ఉంది. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "స్పష్టంగా 5 యొక్క f లాంటిది 1 ప్లస్ 1 ప్లస్ 1 ప్లస్ 1 ప్లస్ 1 యొక్క f లాగా ఉంటుంది, ఈ లక్షణం కారణంగా 1 యొక్క fతో సమానంగా 5 సార్లు గుణించబడుతుంది, ఇది 1 యొక్క f అయితే 2 అదే 2 నుండి పవర్ 5కి, ఆపై f యొక్క ప్రతికూలత 5 లాగా ఉండాలి, మనం దానిని f 5తో గుణించినప్పుడు 0 యొక్క f సంసారాన్ని పొందుతాము మరియు 0 యొక్క f అంటే ఏమిటో వెంటనే స్పష్టంగా తెలియదు కానీ మనం చెప్పగలం f యొక్క 1 ప్లస్ 0 అనేది 1 యొక్క f సంసారానికి సమానం అయితే f 0 యొక్క రెట్లు 0 అయితే 1 యొక్క f 2కి సమానం కాబట్టి ఇది కూడా 2కి సమానం కాబట్టి మనం 2 అంటే 2 రెట్లు ఏదో బాగా చెప్పాలి. ఒక 1 అయి ఉండాలి కాబట్టి ఈ సందర్భంలో, ప్రతికూల 5 యొక్క f 2 నుండి ప్రతికూల 5 వరకు 1 అని హామీ ఇస్తుంది, ఇది 2 నుండి 5 వ వరకు ఉంటుంది అని మేము దీన్ని స్పష్టంగా 2 నుండి ప్రతికూల 5 అని వ్రాయవచ్చు, అంటే ఈ రెండు లక్షణాలు కలిసి చేస్తాయి మేము నిజంగా ఫంక్షన్‌ను 2 నుండి X వరకు వ్రాయాలనుకుంటున్నాము, ఎందుకంటే మనం దానిలో ఉంచిన ఏదైనా లెక్కింపు సంఖ్య సంతృప్తి చెందుతుంది, అది మనం ఉంచిన ఏదైనా భిన్న సంఖ్యను దాని ద్వారా గుణించినట్లు కనిపిస్తుంది, ఈ లక్షణాలను సంతృప్తి పరుస్తుంది. మేము కోరుకున్నది మరియు మీరు ఆశ్చర్యపోవచ్చు మరియు నిజమైన విలువ కలిగిన ఫంక్షన్‌ల సందర్భంలో ఇది నిజంగా ఉంటుంది, కానీ సంక్లిష్టమైన విలువ కలిగిన ఫంక్షన్‌ల సందర్భంలో, అటువంటి అనేక ఫంక్షన్‌లు ఉంటాయి, వీటిలో మనం దీని కోసం వ్రాయగలము. 2 ప్లస్ 2 pi యొక్క సహజ లాగ్ యొక్క ఎక్స్‌ప్రెస్‌గా నిర్వచించబడిన ఫంక్షన్ ఎక్కడ ఉంటుందో ముందు చూడటం X సరే, ఇక్కడ అలసత్వాన్ని మన్నించండి, నేను దీని గురించి వ్రాయడానికి సంతోషిస్తున్నాను మరియు ఇది వాస్తవానికి భిన్నమైన ఫంక్షన్ మీరు Xని 1 సగానికి ప్లగ్ చేస్తే ఏమి జరుగుతుందనే దానికి రుజువు మీరు 1 సగానికి ప్లగ్ చేసినప్పుడు మీకు వచ్చేది 2 యొక్క ప్రతికూల వర్గమూలం మరియు మీరు 1ని ప్లగ్ చేస్తే మీకు నాల్గవ మూలం కాదు. 2 కానీ నేను 2 యొక్క నాల్గవ మూలాన్ని రెట్లు పెంచుతాను కాబట్టి ఇది వేరొక ఫంక్షన్ అయితే ఇది ఇప్పటికీ ఈ లక్షణాలను సంతృప్తిపరుస్తుంది మరియు ఇది ఒకరకంగా దీనిని 2 నుండి X వరకు వ్రాయాలని కోరుకునేలా చేస్తుంది మరియు ఇది బహుశా 2 నుండి X వరకు అస్పష్టంగా ఉంటుందని సూచించేలా చేస్తుంది. బిట్ ఆఫ్ సంజ్ఞామానం మరియు మేము ప్రతిదానిని R సమయాల గడువు ప్రకారం వ్రాయాలి, కానీ మీరు బాగా ఆశ్చర్యపోవచ్చు, ఈ ఆస్తిని సంతృప్తిపరిచే అన్ని ఫంక్షన్‌లతో మేము తగినంత సృజనాత్మకంగా లేము అని మీకు తెలుసా బహుశా మేము ఎక్స్‌ప్రెస్ వ్రాసేటప్పుడు అస్పష్టత ఉండవచ్చు R సమయాలలో ఏదో ఒకటి మరియు R యొక్క విభిన్న విలువలు అమలులోకి రావచ్చు కానీ నేను కొంచెం దావా వేయబోతున్నాను మరియు మీకు కావాలంటే రుజువు ఎలా ఉంటుందో స్కెచ్ లాగా ఇస్తాను. మీకు కొంత సంక్లిష్టమైన ఫంక్షన్ F ఉందని చెప్పండి మరియు ఇది ముందుగా కింది లక్షణాలను సంతృప్తిపరుస్తుంది, మీరు దాని నుండి ఉత్పన్నం తీసుకోగలరు. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "ఇది విభిన్నంగా ఉంటుంది, ఇది మీకు పూర్తిగా గజిబిజిగా నిరంతరాయంగా ఉన్న విషయం కాకుండా ఉంచుతుంది, ఇది మీకు తెలిసిన కొన్ని యాదృచ్ఛిక విలువలను తీసుకోవడం లాంటిది, వెక్టార్ స్థలం యొక్క వ్యవధి గురించి మీకు తెలుసు. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "ఇది మంచి ఫంక్షన్. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "అది భేదాత్మకం ఇది ప్రతిచోటా 0కి సమానం కాదు కాబట్టి ఆ పరిస్థితి నా మదిలోకి జారుకుంది మరియు నేను ఏ ఉపన్యాసం కోసం లేదా అలాంటిదేదో మర్చిపోతాను మరియు అది ఈ కేంద్ర ఆస్తిని కలిగి ఉంది, అది కూడికను గుణకారంగా మారుస్తుంది మీకు అలాంటి ఫంక్షన్ ఉంటే నేను దానిని క్లెయిమ్ చేస్తాను. ఒక ప్రత్యేకమైన కాంప్లెక్స్ సంఖ్య R ఉనికిలో ఉందని నేను నిజంగా పేర్కొనాలి, తద్వారా మీరు X యొక్క Fను ప్రాథమికంగా R రెట్లు ఈ ఎక్స్‌పోనెన్షియల్ ఫంక్షన్‌గా వ్రాయవచ్చు, దీని విలువ X అంటే మీకు X ఫంక్షన్‌గా ఉంటే ఇది ప్రాథమికంగా చెప్పబడింది. అద్భుతమైన ఉత్పన్న లక్షణాలతో అనంతమైన బహుపది మరియు ఇవన్నీ మీకు ఉంటే, మీరు కోరుకునే ప్రతి ఎక్స్‌పోనెన్షియల్‌ను కలిగి ఉంటారు, ఎక్స్‌పోనెన్షియల్ అనే పదం యొక్క నైరూప్య సాధారణ అర్థంలో మనం కోరుకునే ఆస్తి మరియు రుజువు యొక్క స్కెచ్ ఆధారంగా మీరు ముందుగా ఈ విలువ యొక్క ఉత్పన్నం ఏమిటో చూడాలనుకుంటే ఇలా చూడండి, ఇది అన్ని చోట్లా ఉందని మేము ఊహిస్తున్నాము, సరియైనదా? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "మేము ఎక్స్‌ప్రెషన్ నుండి పూర్తిగా F కారకం చేయవచ్చు మరియు మొత్తం పరిమితి H పరంగా మాత్రమే వ్యక్తీకరించబడుతుంది, మీరు ఉత్పన్నాల సందర్భంలో దాని అర్థం ఏమిటి మరియు 0 యొక్క F తప్పనిసరిగా 1కి సమానం అని ఆలోచిస్తే ఈ మొత్తం పరిమితి వ్యక్తీకరణ కేవలం కొంత స్థిరంగా ఉంటుంది కానీ మరింత ప్రత్యేకంగా ఇది 0 వద్ద ఉన్న మా ఫంక్షన్ యొక్క ఉత్పన్నం ఏదైనా కాబట్టి మీకు ఈ ఫన్నీ విషయం ఉంది, దాని ఉత్పన్నం 0 వద్ద మీకు తెలిస్తే, అది ప్రతిచోటా దాని ఉత్పన్నం ఏమిటో నిర్ణయిస్తుంది మరియు ఎక్స్‌పోనెన్షియల్ ఫంక్షన్‌ల సందర్భంలో ఇది ఆశాజనకంగా చాలా సుపరిచితం ఎందుకంటే మేము నిజంగా చెబుతున్నదంతా ఘాతాంక ఫంక్షన్ యొక్క ఉత్పన్నం దానికదే అనులోమానుపాతంలో ఉంటుంది మరియు అనుపాత స్థిరాంకం 0 వద్ద ఉన్న ఉత్పన్నానికి సమానంగా ఉంటుంది, ఇది చాలా వియుక్తంగా పదబంధంగా ఉంటుంది మరియు అలాంటిదే కానీ దాని యొక్క ఉద్దేశ్యం దానిని నొక్కి చెప్పడం మేము ఇప్పటికే పవర్ X గా భావించే ఫంక్షన్‌లు అవసరం లేదు, కానీ ఇది సంకలనాన్ని గుణకారంగా మార్చే ఈ నైరూప్య లక్షణాన్ని సంతృప్తిపరిచే మరింత విస్తృతమైన ఫంక్షన్‌ల తరగతి. రెండవ ఉత్పన్నం మరియు ఆ విషయానికి మూడవ ఉత్పన్నం మరియు అలాంటిది ఎందుకంటే డెరివేటివ్ ఫంక్షన్ దానికదే అనులోమానుపాతంలో ఉంటుంది కాబట్టి n వ ఉత్పన్నాన్ని తీసుకోవడానికి మీరు ఆ అనుపాత స్థిరాంకాన్ని చూసి దానిని పవర్ nకి పెంచండి మరియు ఇక్కడ నుండి మీరు ఒక చేయవచ్చు టేలర్ సిరీస్ విస్తరణ మరియు నేను ఆ ఆలోచనలో టేలర్ సిరీస్‌తో సౌకర్యంగా ఉన్న మీ కోసం అధునాతన హోంవర్క్‌గా వదిలివేస్తాను, ప్రత్యేకించి మీరు సంక్లిష్ట సంఖ్యల కోణంలో విభిన్నంగా ఉండే ఏదైనా డిఫరెన్షియబుల్ ఫంక్షన్ యొక్క ఆలోచనను మిక్స్ చేయాలనుకుంటే. ఖచ్చితంగా కాలేజీ టాపిక్ అని మీకు తెలుసు, మీకు కావలసిన విధంగా మీరు అక్కడ తార్కికతను మిక్స్ చేయగలరని మీకు తెలుసు, అయితే టేలర్ సిరీస్ గురించి మాత్రమే తెలిసిన వ్యక్తి యొక్క సందర్భంలో మసక తార్కికం అనుమతించబడుతుంది మరియు ఈ ఆలోచనను తీసుకోవడానికి మరియు F కోసం టేలర్ విస్తరణను చూడడానికి మరియు మా ఫంక్షన్ F తప్పనిసరిగా ఇలా వ్రాయబడేలా ఒక ప్రత్యేకమైన స్థూల సంఖ్య ఉందనే ఆలోచనను సమర్థించండి, ఆపై సాధారణ ఘాతాంకాలకు కనెక్షన్ మీకు అలాంటి విలువ ఉన్నప్పుడు R వాస్తవ సంఖ్యల సంక్లిష్ట సందర్భంలో మనం ఏమి చేస్తాము. మీరు ఆ విలువ R యొక్క ఆ ఫంక్షన్ యొక్క ఎక్స్‌ని చూస్తే మరియు దానిని బేస్‌గా వ్రాయండి. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "మేము pi అర్ధభాగాలు I సార్లు X యొక్క ఎక్స్‌ప్షన్ మాత్రమే కాదు, 5 pi హాల్వ్స్ I టైమ్స్ X యొక్క ఎక్స్‌ప్రెస్ అని కూడా అర్థం చేసుకోగలము మరియు ఇవి ప్రత్యేక ఫంక్షన్‌లు మరియు అనంతమైన ప్రత్యేక ఫంక్షన్‌లు ఉన్నాయి మరియు మనకు అనిపించే ప్రత్యేక ఫంక్షన్‌లు ఉన్నాయి. వాటిని I to the X అని వ్రాయండి కాబట్టి మీరు దాని అర్థం కోసం ఒక ప్రమాణాన్ని స్వీకరించనంత వరకు I to the I అనే వ్యక్తీకరణ తప్పనిసరిగా అర్థం కాబోతుంది అని మీరు చెప్పినప్పుడు అది అనంతమైన అనేక అవుట్‌పుట్‌లను కలిగి ఉందని ఆలోచించడానికి మరొక మార్గం ఏమిటంటే, I to X ఫంక్షన్ మన దగ్గర ఉన్న సంజ్ఞామానం కొంచెం అస్పష్టంగా ఉంది, ఇప్పుడు వీటన్నిటితో మనం వీటిలో కొన్నింటిని విజువలైజ్ చేయడం ప్రారంభిద్దాం ఎందుకంటే ఇది సరదాగా ఉంటుందని నేను భావిస్తున్నాను మరియు ఇది ఉపయోగకరమైన దృశ్యమా లేదా మరింత గందరగోళ దృశ్యమా అయితే మీరు నాకు చెప్పండి. మనం చేయబోయేది R టైమ్స్ X యొక్క ఈ ఫంక్షన్ ఎక్స్‌ప్రెస్‌ని చూడటం, ఇది ప్రాథమికంగా X యొక్క శక్తికి eని వ్రాయడానికి మరొక మార్గం, వాస్తవానికి నేను ఏదో ఒక సమయంలో వేరే యానిమేషన్‌ను రెండర్ చేశానని అనుకుంటున్నాను. నేను అలా చేయడానికి ప్లాన్ చేస్తున్నాను కాబట్టి మీరు నా ఫైల్ సిస్టమ్‌లోకి తిరిగి రావడానికి నన్ను అనుమతించండి, మీరు ఎక్కడ ఉండాలనుకుంటున్నారో అక్కడికి చేరుకోండి, అక్కడ చాలా విభిన్నమైనవి ఉన్నందున అది ఫిర్యాదు చేస్తుంది. ఓహ్ రీప్లేస్ చేయండి, ఇది ఇతర స్క్రీన్‌పై చూపబడుతుంది వేచి ఉండండి, ఇది ఎందుకు అవును, ఓకే రీప్లేస్ చేయండి? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "మీరు చూసే ప్రతిదాన్ని అక్కడ ఉంచండి మరియు ఇప్పుడు మేము అక్కడకు తిరిగి వెళ్తాము, మేము అన్నింటినీ చక్కగా వ్రాస్తాను కాబట్టి మీరు దానిని R టైమ్స్ ఎక్స్‌గా భావించడం అసౌకర్యంగా ఉంటే X ఈ అనంతమైన బహుపది మీ తల వెనుక ఇ నుండి R సార్లు X వరకు మరియు మేము R చుట్టూ మారుతూ ఉంటాము కాబట్టి నేను ఊహాత్మక అక్షం యొక్క పాయింట్లను అనుసరించబోతున్నాను మరియు నేను నిజమైన అక్షం యొక్క పాయింట్లను అనుసరించబోతున్నాను మరియు ఇది ఏమి చేస్తుందో చూద్దాం అదంతా వేగవంతమైనది కాబట్టి నేను దాని ద్వారా కొంచెం నెమ్మదిగా ఆలోచిద్దాం అన్ని ప్రతికూల సంఖ్యలు ఏదైనా అది ప్రతికూల వాస్తవ సంఖ్య 0 మరియు 1 మధ్య పరిధిలోకి స్క్విష్ చేయబడి ఉంటుంది, ఏది ప్రతికూలంగా ఉంటుంది? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a నుండి ప్రతికూల వాస్తవ సంఖ్య 0 మరియు 1 మధ్య ఉంటుంది మరియు మేము నిర్దిష్టంగా ప్రతికూల 1 యొక్క fని ట్రాక్ చేస్తున్నాము, ఇది 1 పైగా e 30 0 చుట్టూ చూపబడుతుంది. eలో 1లో 37 f ఆశించిన విధంగా ల్యాండ్ అవుతుంది అంటే 1 యొక్క ఎక్స్‌ఫ్ 1 f అంటే నేను యూనిట్ సర్కిల్ చుట్టూ ఒక రేడియన్‌ను ల్యాండ్ చేయబోతున్నాను మరియు ఊహాత్మక అక్షం ఒక వృత్తం చుట్టూ ఎలా చుట్టబడి ఉంటుందో ఇక్కడ మొత్తం ఊహాత్మక అక్షం వెంట అనుసరించడం సరదాగా ఉంటుంది. మరియు R యొక్క ఈ విలువను మనం సర్దుబాటు చేసినప్పుడు ఏమి జరుగుతుంది? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "మనకు కావలసిన మరియు R యొక్క విలువలు ఇక్కడ ఉండవచ్చు కాబట్టి మేము దానిని 2 వరకు ఉంచినప్పుడు ఇది విభిన్నంగా సాగుతుంది కాబట్టి అది వాస్తవ అక్షాన్ని చాలా ఎక్కువగా విస్తరించి ఉంటుందని మీకు తెలుసు కాబట్టి e స్క్వేర్డ్ 7 f ప్రతికూలంగా ఉన్న చోట f 1 ముగుస్తుంది. 1 అనేది I యొక్క 0 fకి చాలా దగ్గరగా ఉంటుంది f నెగెటివ్ వృత్తం చుట్టూ 2 రేడియన్ భ్రమణం I ప్రతికూల 2 రేడియన్ భ్రమణం మరియు వాస్తవానికి మనం మన స్కేలింగ్ స్థిరాంకం వలె కలిగి ఉన్న pi అయితే మనకు ఇష్టమైన సూత్రాన్ని పొందవచ్చు. నిజమైన అక్షం చాలా వరకు విస్తరించి ఉంటుంది అని మీకు తెలుసు, 20 ప్లస్ piకి చాలా దగ్గరగా ఉన్న 1 యొక్క ఎఫ్ e వద్ద కూర్చొని ఉంటుంది, ఇది ఎల్లప్పుడూ సరదాగా ఉంటుంది మరియు ప్రతికూల 1 యొక్క f 0కి చాలా దగ్గరగా ఉంటుంది కాబట్టి ఇది నిజంగా వాస్తవమైనదిగా విస్తరించింది అక్షం మరియు ఇది యూనిట్ సర్కిల్ దిశలో విషయాలను కూడా విస్తరించింది, తద్వారా I లేదా f యొక్క ప్రతికూలత యొక్క fకి చేరుకోవడం నేను సర్కిల్ చుట్టూ సగం వరకు నడుస్తాను, కాబట్టి ఇప్పుడు అంతా బాగానే ఉంది మరియు అలాంటి ఫంక్షన్ గురించి మనం ఎలా ఆలోచిస్తాము? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "మేము 2 సార్లు X యొక్క సహజ లాగ్‌లో X యొక్క X అని కూడా వ్రాస్తాము కాబట్టి మేము R విలువను సూచించే పసుపు చుక్కను 0 చుట్టూ మారుస్తాము. 69 ఇప్పటికీ ఊహాత్మక భాగం లేదు కేవలం వాస్తవ సంఖ్య 0.69 లేదా అది 2 యొక్క సహజ చిట్టా, మీరు 1 యొక్క f 2పై ల్యాండ్ అవుతుందని మీరు చూడవచ్చు, అందుకే మేము ఈ ఫంక్షన్‌ని 2 నుండి 1 సగం యొక్క X f అని పిలవాలనుకుంటున్నాము, వాస్తవానికి ప్రతికూల 1 ల్యాండ్‌లను 1 సగం fలో క్షమించండి. నేను యూనిట్ సర్కిల్ చుట్టూ కొంత నడకను చాలా ప్రత్యేకంగా అది 0 అవుతుంది. యూనిట్ సర్కిల్ చుట్టూ 69 రేడియన్‌లు మరియు ఇప్పుడు మనం కొంచెం ఆనందించవచ్చు మరియు దీనిని 0కి బదులుగా మార్చినట్లయితే ఏమి జరుగుతుందో చెప్పగలము. 69 2 యొక్క సహజ లాగ్ కాకుండా I రెట్లు 2 యొక్క సహజ లాగ్ చేయండి, తద్వారా మేము దానికి ఘాతాంక ఆధారాన్ని కలిగి ఉన్న దాని గురించి నిజంగా ఆలోచిస్తున్నాము. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "ఈ సందర్భంలో శక్తికి నేను అంటే ఏమిటి, అది దానిని 0కి పంపుతుంది. 2 ఐదవ వంతు చుట్టూ 2 కానీ అనేక విభిన్న ఘాతాంక విధులు ఉన్నాయి, ఇందులో f 1ని I అనే సంఖ్యపై ఉంచే లక్షణం ఉంటుంది కాబట్టి మనం దానిని మరింతగా పెంచినట్లయితే, నేను దానిని ఇక్కడ యానిమేట్ చేశానని అనుకోను కానీ మనం తీసుకుంటే ఆ పసుపు చుక్క మరియు దానిని 5 రెట్లు pi వచ్చే వరకు పైకి లేపండి I మీరు చూసేది యూనిట్ సర్కిల్? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "దాని చుట్టూ తిప్పబడుతుంది, తద్వారా 1 యొక్క ప్రతికూల f యొక్క f మరొక 2 pi రేడియన్ల చుట్టూ తిరుగుతుంది మరియు అది ఉన్న చోట ల్యాండ్ అవుతుంది, అయితే ఇది నిజమైన అక్షాన్ని చాలా ఎక్కువగా విస్తరించింది, ఇది I నుండి I నుండి మరొక అవుట్‌పుట్‌ని సూచిస్తుంది చాలా చిన్న సంఖ్య ఇది 0 చుట్టూ ఉండేది. 0003 లేదా అంతకంటే ఎక్కువ కానీ నేను చాలా సరదాగా భావించేదాన్ని కూడా మనం చూడగలం, మనం పవర్ X కుడికి 2గా అన్వయించాలనుకునే ప్రత్యామ్నాయ వ్యక్తీకరణలను పరిగణనలోకి తీసుకుంటే ఏమి జరుగుతుంది? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "మనకు X యొక్క R సార్లు X మరియు R ఈ విలువకు సమానం, ఇది 2 ప్లస్ pi సార్లు I యొక్క సహజ లాగ్, అంటే మనం 1 f యొక్క 1ని ప్లగ్ చేసినప్పుడు ప్రతికూల 2 వద్ద ఉంటుంది కాబట్టి మనం ఈ ఫంక్షన్‌ని వ్రాయాలనుకుంటున్నాము పవర్ X కుడికి నెగెటివ్ 2గా ఉంది మరియు అది నిజానికి మీకు తెలిసిన విషయమే, మనం నెగెటివ్ నెగిటివ్ 2కి నెగెటివ్ నంబర్‌ని వ్రాసినప్పుడు ఇది కొంచెం మోసపూరితమైనది, ఇది X పవర్‌కి మొదట అలా కనిపించదు, అది మనకు తెస్తుంది సంక్లిష్ట సంఖ్యలను ఏ విధంగానైనా చేర్చండి, అయితే మనం 1 సగం వంటి విలువను కూడా ప్లగ్ చేసినప్పుడు ప్రతికూల 2 యొక్క వర్గమూలాన్ని అడుగుతున్నప్పుడు మనం దీన్ని వర్గమూలాన్ని నేను రెట్లు చేసినట్లుగా వ్రాయాలనుకుంటున్నాము. 2 యొక్క 2 కానీ మీరు ఈ ఫంక్షన్ నెగెటివ్ 2 నుండి పవర్ X ని పూర్తి కాంప్లెక్స్ డొమైన్‌లో చూసినట్లయితే, అది మీరు చూస్తున్న దానితో వ్యవహరిస్తుంది, అది 1 నుండి నెగెటివ్ 2కి విలువను తీసుకునే ఫంక్షన్ మరియు అది చేస్తే ఏమి చేయాలి ఇది మిగిలిన వాస్తవ సంఖ్య రేఖకు చేస్తుంది, అది బయటికి స్పైరల్‌గా ఉందా? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "కాబట్టి ప్రతికూల 1 యొక్క f నెగిటివ్ 1 సగం వద్ద కూర్చుంటుందని మేము చూస్తాము, మీరు 1 సగానికి fని అనుసరించినట్లయితే మీరు ఎక్కడ ఆశించవచ్చో అది ఖచ్చితంగా ఊహాత్మక రేఖపై కూర్చుంటుంది మరియు 1 సగం యొక్క f 2 యొక్క వర్గమూలం అవుతుంది. మౌస్ నేను కోరుకున్న చోట లేదు. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "ఇది సుమారుగా 2 రెట్లు I మరియు మీరు కొనసాగిస్తున్నప్పుడు, ఇది మీకు ప్రతికూల 2 నుండి X వరకు ఉన్న అన్ని వాస్తవ విలువ శక్తులను చూపుతుంది, ఇది తప్పనిసరిగా చుట్టూ తిరుగుతుంది, అయితే మనం R విలువను మరింత ఎక్కువగా తరలించి, దాన్ని పొందవచ్చు. దాదాపు టౌ సమయాల వరకు నేను ఆరు పాయింట్ల వరకు రెండు ఎనిమిది సార్లు నేను మరియు ఆ సందర్భంలో ఇది మరొక ఫంక్షన్, ఇది మేము 2 నుండి X లాగా వ్రాయాలనుకుంటున్నాము ఎందుకంటే మీరు X కోసం ప్లగిన్ చేసే ఏదైనా పూర్తి సంఖ్యకు పూర్తి సంఖ్యకు ఇది ఉంటుంది. పునరావృత గుణకారం లాగా కనిపిస్తుంది మరియు ఇది 1 సగం వంటి వాటికి సహేతుకమైన విలువలను కూడా కలిగి ఉంటుంది, ఇక్కడ అది సానుకూల స్క్వేర్ రూట్‌కు బదులుగా ప్రతికూల వర్గమూలాన్ని ఉమ్మివేస్తుంది, కానీ వాస్తవానికి అది చేస్తున్నది విమానంలో పరివర్తన చెందుతుంది, అది ప్రతిదీ ఉంచుతుంది సంఖ్య రేఖ చాలా గట్టిగా గాయపడిన స్పైరల్‌గా ముగుస్తుంది మరియు అది 1 యొక్క f నేరుగా సంఖ్య 2పైకి వచ్చే విధంగా స్పైరల్ అవుతుంది కాబట్టి ఆ కోణంలో మనం 2 నుండి X అని చెప్పవచ్చు. మనం సాంప్రదాయకంగా ఉపయోగించిన దాని నుండి ఒక ప్రత్యేక ఎక్స్‌పోనెన్షియల్ ఫంక్షన్ కాబట్టి నేను ఈ రోజు కోసం విషయాలను వదిలివేస్తానని అనుకుంటున్నాను మరియు సరే గురించి ఆలోచించడానికి నేను మీకు కొన్ని ఆలస్యమైన ప్రశ్నలను వదిలివేస్తాను, కాబట్టి మీరు కావాలనుకుంటే I to I అనేది బహుళ-విలువైన వ్యక్తీకరణగా భావించండి. సగభాగాలు కానీ మీరు ఈ రకంగా మనం చూసిన వివిధ విలువల వలె అనంతమైన అనేక విభిన్న విలువలు కావాలని మీరు చెబితే, 2 నుండి 1 మూడవ వంతు వరకు ఎన్ని విలువలు ఉండాలి? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "10వ పదాలు అన్నింటికీ భిన్నమైన పదబంధాన్ని కలిగి ఉండాలనుకుంటున్నాను, ఇది X యొక్క అన్ని ఎక్స్‌పోనెన్షియల్ ఫంక్షన్‌ల గురించి చెప్పనివ్వండి, ఇది నేను వ్రాసిన ఈ లక్షణాలన్నింటిని సంతృప్తిపరిచే X యొక్క ఎక్కడైనా f వ్రాసి ఉన్నాను, అది అన్నింటినీ సంతృప్తిపరిచినట్లయితే వీటిలో మరియు 1 యొక్క f 2కి సమానం అయితే మనం Xని ప్లగ్ ఇన్ చేసినప్పుడు మనం ఎన్ని విభిన్న అవుట్‌పుట్‌లను పొందబోతున్నాం, ఏ ఫంక్షన్ కోసం వివిధ ఎంపికల కోసం 3 10వ వంతుకు సమానం? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "ఈ విధమైన నైరూప్య లక్షణాల అర్థంలో మనం 2 నుండి X వరకు 2 నుండి X వరకు ఒక రకమైన ఘాతాంక ఫంక్షన్‌గా ఆలోచిస్తున్నట్లయితే వివిధ ఫంక్షన్‌ల కోసం 2 నుండి pi వరకు సూచించవచ్చు మరియు మనం అవును అయితే, మనం మాకు అలాంటి విభిన్న ఫంక్షన్‌ల క్లాస్ ఉంది మరియు మేము పైని ప్లగ్ చేయాలనుకుంటున్నాము, అది నాకు నవ్వు తెప్పిస్తుంది ఎందుకంటే ఇది నాకు నవ్వు తెప్పిస్తుంది, మీరు దాని గురించి ఆలోచించడానికి ప్రయత్నిస్తున్నప్పుడు అది బయటకు వస్తుంది కాబట్టి అవి ఆ ప్రశ్నలు నేను మిమ్మల్ని వదిలివేస్తాను మరియు ఈ రోజు ఉపన్యాసానికి చేరుకోవడంలో నా ప్రధాన ప్రశ్న ఏమిటంటే, ఎక్స్‌పోనెన్షియల్ ఫంక్షన్‌ల యొక్క ఈ అబ్‌స్ట్రాక్ట్ ప్రాపర్టీస్‌ను ఇలా వివరించాలని నేను కోరుకున్నాను మరియు ఆ నైరూప్య లక్షణాల నుండి ప్రారంభించడం నాకు చాలా బాగుంది. మీరు e నుండి rx లేదా అంతకంటే ఎక్కువ అనే ఆలోచనలోకి లాక్కుపోతారు మీకు తెలుసా, r యొక్క విభిన్న విలువల కోసం r టైమ్స్ xని మరింత నిజాయితీగా వ్రాసినట్లు నేను భావిస్తున్నాను, అది మిమ్మల్ని అంత దూరం లాక్ చేస్తుంది కానీ అది మిమ్మల్ని లాక్ చేయదు 2 నుండి పవర్ x కి నేను శక్తికి x అన్నది చాలా తక్కువగా ఉండాలి అనే నిస్సందేహమైన భావన x అయితే దానిలోని ప్రమాదం ఏమిటంటే, కొన్నిసార్లు వ్యక్తులు సంగ్రహణను ఇష్టపడరు మరియు కొన్నిసార్లు ఇది చేరుకోదగినదిగా రాదు కానీ అది అలా అయితే పవర్ టవర్‌లను చేర్చడానికి వీటన్నింటి చుట్టూ ఆసక్తికరమైన ఆలోచనలు ఉన్నాయని నేను భావిస్తున్నాను, ఎందుకంటే మీరు పవర్ టవర్‌ల గురించి నిజంగా మాట్లాడాలనుకుంటే, కాంప్లెక్స్ నంబర్ల సందర్భంలో మేము చివరిసారిగా మాట్లాడాము. లేదా నెగెటివ్ బేస్‌లతో కూడా మీరు ఇలాంటి విషయాల గురించి ఆలోచిస్తూ ఉండాలి, కాబట్టి ఇది తెరపై మనకు ఎదురయ్యే ప్రశ్న అవును, నేను శక్తికి నేను ఇలా చేస్తే ఏమవుతుంది? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "టైట్రేషన్ మీకు తెలిసిన టైట్రేషన్ దీన్ని ప్రయత్నిద్దాం, ముందుకు వెళ్లి పవర్ టవర్‌ని ప్రయత్నిద్దాం, అక్కడ మనం ఇచ్చిన శక్తికి నన్ను పెంచుతున్నాము మరియు దాని నుండి ఏమి పాప్ అవుతుందో చూడండి, కాబట్టి ఇది దీన్ని చేయడానికి ప్లాన్ చేయలేదు, కానీ మనం ఎల్లప్పుడూ చేయగలము పైథాన్‌ని పైకి లాగండి మరియు ముఖ్యంగా మనం చివరిసారి ఏమి చేస్తున్నామో అదే చేయండి కాబట్టి ఇది పని చేసే మార్గం ఏమిటంటే, మేము కొంత బేస్ విలువతో ప్రారంభించాము, ఆపై ఒక రకమైన పరిధి కోసం మనం ఏమి చేస్తున్నాము మరియు మేము తిరిగి కేటాయించబోతున్నాము అది ఏమైనా అయి ఉండాలి, ఈ సందర్భంలో నేను a యొక్క శక్తికి పెంచబడినది సరే, కూల్‌గా ఉండాలి, కాబట్టి మనం అలా చేయబోతున్నాం మరియు ఆపై మనం దీని విలువను ప్రింట్ చేయబోతున్నాం దీని కోసం దీన్ని చేద్దాం. అవును, ఇది 200 వంటి చాలా పెద్ద సంఖ్య కాబట్టి కొన్నిసార్లు ఇలాంటి వాటితో గందరగోళం ఏర్పడే అవకాశం ఉన్నట్లు అనిపిస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "నిజానికి నా దగ్గర ఉంది కాబట్టి నాకు ఎక్స్‌పోనెన్షియల్ ఫంక్షన్ ఉంది కాబట్టి నేను ఎక్స్‌పోనెన్షియల్ ఫంక్షన్‌ని కలిగి ఉన్నాను కాబట్టి నేను దానిని వ్రాయడం కంటే ముందు మా పెద్ద పరిధి కోసం నన్ను వెళ్లనివ్వండి X శక్తికి నా లాంటిది మీకు తెలిసినట్లుగా నేను దానిని వ్రాయబోతున్నాను. వేరొక స్థిరాంకం కుడి యొక్క ఎక్స్‌పోనెన్షియల్ ఫంక్షన్‌గా ఒక విభిన్న స్థిరాంకం నేను దానిని 5 pi విభజించాలని కోరుకుంటున్నాను, కాబట్టి నేను 5 pi సగం సార్లు చేస్తాను కనుక ఇది సంక్లిష్ట సంఖ్య మరియు దీనికి 5 pi విభజించబడింది ఊహాత్మక భాగం కాబట్టి ఇది 5 pi సగం సార్లు నేను మరియు నేను ఏమి చేస్తున్నాను? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/thai/sentence_translations.json b/2020/ldm-i-to-i/thai/sentence_translations.json index 09f428bd3..d6debd492 100644 --- a/2020/ldm-i-to-i/thai/sentence_translations.json +++ b/2020/ldm-i-to-i/thai/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "เมื่อเราคิดถึงไดนามิกของเราตรงนี้ เป็นเวกเตอร์ความเร็วแต่ละตัว เหมือนกับการหมุนตำแหน่งของคุณ 90 องศา, ฐาน i นี้, วิธีที่เราคิดว่ามันอ่านว่ามันมีความหมายมากกว่าแค่ ตัวเลขก็คือ ณ จุดหนึ่งบนเครื่องบิน มันคือจุดที่คุณได้รับหลังจากเดินทางเป็นเวลา pi โดยจะลดหน่วยเวลาลงครึ่งหนึ่งตามไดนามิกของนิพจน์นี้ ตามแนวคิดที่ว่าความเร็วของคุณคือการหมุนตำแหน่งของคุณ 90 องศาเสมอ โอเค มันดีมากเลย และอันที่จริงแล้ว ฉันกำลังเล่นอยู่สองแบบที่แตกต่างกันอยู่แล้ว ซึ่งหรือฉันเดาว่าฉันกำลังเล่นอยู่สามแบบที่แตกต่างกัน เรามีฐานนี้ ซึ่งอธิบายมุม 90 องศาที่เราเดินไปรอบวงกลม เรามี i อยู่ตรงนี้ ซึ่งอธิบายกฎของการหมุนเวกเตอร์ความเร็ว 90 องศา แต่ตอนนี้เราจะแนะนำ i อีกตัวหนึ่ง ซึ่งโดยพื้นฐานแล้วมีผลกระทบต่อการเปลี่ยนแปลงไดนามิกของคุณ เพราะเมื่อเราไปจาก e ไป i คูณ t และเราแทน คุณก็รู้ ยกสิ่งต่างๆ ขึ้นมาอีกกำลังของ i ซึ่งผมจะเขียนด้วยเครื่องหมายรูปหมวกเล็กๆ i ถ้าเราเอา e ไปที่มัน แล้วเราเปลี่ยนสิ่งที่นั้น พจน์คือการยกมันขึ้นไปที่ i, สิ่งที่เราได้คือ e กำลังลบ t โอเค เรามี e กำลังลบ t และถ้าเราพยายามตีความมันด้วยไดนามิกแบบเดียวกับที่เรามีข้างต้น ในรูปของความเร็วและตำแหน่ง สิ่งที่บอกเราคืออนุพันธ์ของไดนามิกใหม่ e ถึง ลบ t เท่ากับ ทีนี้ค่าคงตัวที่อยู่หน้า t คือลบ 1 ดังนั้นกฎลูกโซ่เราจะให้มันเป็นลบ 1 คูณตัวเองคูณ e กำลังลบ t ไม่ว่าตำแหน่งของคุณจะเป็นเช่นไร ทีนี้ความเร็วจะเป็นลบ 1 คูณตัวมันเอง ดังนั้นผลของการเพิ่มกำลัง i ก็เหมือนกับเราใช้ไดนามิก เราดูเวกเตอร์ความเร็วทุกตัว และเราบอกว่าหมุนอีก 90 องศา เพื่อว่าในบริบทนี้ จริงๆ แล้วมันจะเป็นเวกเตอร์ความเร็วเริ่มต้นที่ชี้ไปข้างหลังด้วย หนึ่งหน่วย ดังนั้น หากคุณเริ่มจากเลข 1 ความเร็วเริ่มต้นของคุณคือเดินตรงไปยัง 0 และเมื่อคุณเดินต่ำลงไปอีก หากคุณนั่งอยู่ที่ 1 ครึ่ง คุณจะยังคงเดินไปสู่ 0 แต่ตอนนี้เวกเตอร์ความเร็วของคุณ จะเป็นลบ 1 คูณจุดที่คุณอยู่ ซึ่งก็คือลบ 1 ครึ่งหนึ่ง และสิ่งนี้มีความหมายต่อการเคลื่อนที่จริงตามที่มันบอกเป็นนัย, คุณอาจลองนึกภาพดู, ผมยังเคลื่อนไหวมันไม่ค่อยดีหรืออะไรเลย แต่ถ้าคุณดูที่สนามเวกเตอร์ทั้งหมดนี้ แล้วถามเวกเตอร์แต่ละตัวว่า เอา ไม่ว่าคุณจะอยู่ที่ไหนและหมุนทวนเข็มนาฬิกาอีก 90 องศา ทุกอย่างก็จะชี้ไปที่จุดกำเนิด ดังนั้น หากคุณให้จุดเล็กๆ เคลื่อนที่ในลักษณะที่ความเร็วของมันตรงกับเวกเตอร์ใดๆ ก็ตามที่มันอยู่ด้านบนเสมอ สิ่งที่คุณจะได้คือเมื่อแต่ละก้าวของเวลา มันจะก้าวเข้าหา 0 และ เพียงแต่ในแต่ละก้าวของคุณ คุณกำลังเดินไปที่ 0 และแต่ละก้าวจะมีขนาดที่เล็กลงเรื่อยๆ เมื่อคุณเข้าใกล้ 0 จริงๆ และแน่นอนว่าในทางปฏิบัติ นี่อาจเป็นขั้นตอนเล็กๆ น้อยๆ แทนที่จะเป็นการกำหนดขนาดที่เป็นรูปธรรมมาก คุณอาจแม่นยำมากถ้าคุณต้องการ และบอกว่าสิ่งที่เรากำลังดูอยู่คือลดขนาดลงด้วยจำนวนที่น้อยกว่า 1 แล้วเราทำสิ่งนี้ในเวลาต่างกัน แล้วเราจะคูณมันด้วยเวลาเท่าไร เรากำลังรออยู่ และคุณถือว่านี่เป็นนิพจน์จำกัด นั่นเป็นเพียงสิ่งที่เราพูดถึงในการบรรยายเรื่องดอกเบี้ยทบต้น หากคุณสงสัย และมันเป็นวิธีมาตรฐานในการพูดถึง e กำลัง ก็คือขีดจำกัดแบบนี้ คุณสามารถให้สิ่งนั้นอยู่ข้างในแทนได้ถ้าคุณต้องการ แต่ตอนนี้ ถ้าเราคิดถึงจุดเดิม i, ฐานของเรา, นั่นหมายความว่าอย่างไร, มันบอกว่าให้ดูที่ไดนามิกของเรา และมันคือจุดที่คุณจะได้เมื่อรอ ไพ ลดลงครึ่งหนึ่งหน่วยของเวลา ดังนั้นผลของการเพิ่มไปที่ i จะเปลี่ยนไดนามิกของเรา ในลักษณะที่แทนที่จะเดินไปรอบๆ วงกลม เรากำลังสลายแบบเอ็กซ์โปเนนเชียลแบบนี้ เรากำลังเคลื่อนไปสู่ 0 ด้วยอัตราการชะลอตัวและช้าลง และตำแหน่งที่คุณได้มาหลังจากพายลดลงครึ่งหนึ่ง หน่วยของเวลาจะเป็น e กำลังลบ พายครึ่งหนึ่งประมาณ 0 2079. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "นั่นคือสิ่งที่คำถามแบบทดสอบเริ่มได้รับ และเรามีคำตอบที่แตกต่างกันมากมายซึ่งทำให้ฉันมีความสุขเสมอ ดังนั้นเรามาดูความหลากหลายที่ผู้คนโยนไว้ที่นี่กัน เรามี 5 ไพ ครึ่งหนึ่งที่ยอดเยี่ยม ซึ่งเป็นอีกค่าหนึ่งที่เราแทนค่า x ตรงนี้ได้ และเพื่อให้ชัดเจนขึ้นอีกหน่อย ถ้าเรามองย้อนกลับไปที่วงกลมตรงนี้ โมเมนต์เดินเป็นระยะเวลาเท่ากับครึ่งหนึ่งของพายซึ่งก็คือ 1 57 จะเป็นอย่างไรถ้าเราเลี้ยวเต็มอีกครั้งแล้วเราไปครึ่งหนึ่งของพายอีกครึ่งหนึ่งเพื่อพาเราไปที่พาย ซึ่งคุณก็รู้ว่าเราอาจบันทึกได้ว่านั่นคือจุดที่ e กำลังหาค่า i คือเราเดินไปอีกครึ่งหนึ่งของพาย เราเดินไปอีกครึ่งหนึ่งของพายซึ่งที่ จุดนี้เราจะวนครบรอบเพื่อกลับมาที่จุดหนึ่ง จากนั้นเราจะเดินไปอีก 5 ครึ่งของพาย ซึ่งคิดเป็นตัวเลขประมาณ 7 85 ใช่แล้ว นั่นเป็นจำนวนอีกจำนวนหนึ่งที่ทำให้เราอยู่เหนือ i และถ้าเราต้องผ่านกฎเกณฑ์ทั้งหมดของการแสดง i ยกกำลัง i โดยการเขียน e กำลัง 5 ไพก่อน จะลด i ยกกำลัง i ลงครึ่งหนึ่ง คูณจนกลายเป็นลบ แล้วเราจะดูที่ e กำลังลบ 5 ไพครึ่งหนึ่ง ซึ่งเป็นจำนวนที่ต่างออกไปมาก เราคำนวณอันนี้ได้ ผมไม่แน่ใจเหมือนกัน แต่ลองดู Desmos กันดีกว่า . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "ความยาวนั้นทำให้คุณได้จำนวนที่น้อยกว่ามาก แต่นั่นไม่ใช่คำตอบเดียวที่เราตอบได้ เรามีคนอื่นเข้ามาที่นี่ด้วยลบ 3 ครึ่งคูณ i ไพ ซึ่งคุณรู้ในรูปของวงกลมหน่วยไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "เราคิดได้ว่าพูดว่า เฮ้ ถ้าผมอยากไปให้ถึง แทนที่จะเดิน 90 องศา ไพ ลดครึ่งหนึ่งของเรเดียนแบบนั้น แล้วถ้าผมเดิน 270 องศา ในทางกลับกัน 3 ไพ ลดครึ่งเรเดียน ซึ่งบางทีผมอาจมองว่าเป็นลบ เพราะแบบแผนคือ โดยปกติทวนเข็มนาฬิกานั้นเป็นบวก นั่นก็เป็นอีกวิธีหนึ่งในการแสดงมันออกมา และนั่นจะทำให้เราได้คำตอบที่แตกต่างออกไป หากเรามี e กำลังลบ 3 ไพ แบ่งครึ่ง i ทั้งหมดยกกำลัง เราจะผ่านเกมเดียวกันตอนนี้ i กำลังสองยกเลิกด้วย ลบที่มีอยู่แล้ว, และเรามีบวก 3 ไพครึ่งหนึ่ง และในเชิงตัวเลข นี่ทำให้เราได้คำตอบที่ดูแตกต่างไปจากที่เรามีก่อนหน้านี้ ซึ่งถ้าเรากลับไปแล้วบอกว่า เฮ้ e กำลัง 3 ไพคืออะไร ไม่ใช่ 3 o 3 ไพ แบ่งครึ่ง 111 จุด 3 1 ตัวเลขที่แตกต่างจากที่เราเห็นก่อน 111 จุดคืออะไร 111 จุด 3 1 เยี่ยมยอด 111 จุด 3 1 หรือประมาณนั้น และอีกครั้งในแง่ของสัญชาตญาณ สิ่งที่คุณอาจจะถาม สมมติว่าเรามีการหมุนนี้ ไดนามิก แต่เราย้อนเวลากลับไป เราเห็นว่าฉันต้องเป็นเช่นไร เมื่อนานมาแล้ว หากฉันเล่นอะไรไปข้างหน้าจากตรงนั้น ฉันจะตกสู่สภาวะเริ่มต้นของฉันเป็นอันดับหนึ่ง และคุณต้องย้อนเวลากลับไป 3 ไพ แบ่งครึ่งหน่วย แล้วถ้าคุณแปลเป็นไดนามิกของการเสื่อมสลาย ซึ่งเป็นสิ่งที่ยกตาขึ้นมาในบริบทนี้ คุณจะบอกว่า ถ้าผมกำลังเริ่มที่อันดับหนึ่ง แต่ผมอยากจะย้อนเวลากลับไปแล้วบอกว่า ผมควรจะเริ่มจากตรงไหนถ้า อยากทรุดโทรมจนมาจบที่อันดับหนึ่ง? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "หลังจาก 3 ไพ แบ่งครึ่งหน่วยของเวลา เห็นได้ชัดว่าคำตอบเริ่มต้นที่ประมาณ 111 สิบเอ็ดสำหรับการสลายตัวแบบเอกซ์โปเนนเชียลแบบนั้น และคุณจะเห็นได้ว่าสิ่งนี้จะไปในทิศทางไหน ซึ่งจริงๆ แล้วมีค่าต่างๆ มากมายนับไม่ถ้วนที่เราสามารถแทนค่า X ได้ถ้าเรา คิดว่า e ถึง X เป็นเหมือนฉัน และผู้คนเข้ามาที่นี่มากขึ้น ขอโทษนะที่ฉันโยนหมุดของฉันลงบนพื้น เหมือนอย่างที่เราทำแบบคลาสสิกสำหรับอันดับที่สาม 9 pi แบ่งครึ่ง ทางเลือกที่ดี 1,729 pi แบ่งครึ่ง คุณทุกคนเป็นคนโปรดของฉันและมากมาย ตัวเลือกที่แตกต่างกัน ค่าที่แตกต่างกันมากมายอย่างไม่สิ้นสุด ซึ่งรู้สึกอึดอัดเล็กน้อยในตอนแรกเพราะเราดูสำนวน ดูเหมือนว่าคุณรู้ว่ามันจะต้องมีการคำนวณบางอย่าง ฉันแค่เสียบมันเข้ากับเครื่องคิดเลขของฉันแล้วดูว่ามีอะไรปรากฏออกมา และเราได้หลายค่าที่แตกต่างกัน ค่านิยมของมัน แล้วเกิดอะไรขึ้นที่นี่ล่ะ? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "รากที่สี่ของ 16 ควรเป็น 2 และคำตอบจบลงด้วยดี เราใช้แบบแผนเมื่อมีหลายตัวเลือกเช่นนี้เมื่อคุณมีฟังก์ชันหลายค่า เรามักจะเลือกค่าใดค่าหนึ่งให้เป็นสิ่งที่เราหมายถึงเมื่อเราต้องการ ถือว่ามันเป็นฟังก์ชันเหมือนกับบางสิ่งที่มีอินพุตเดี่ยวและเอาต์พุตเดี่ยวในภาษาแปลก ๆ แบบนี้เกิดขึ้นตลอดเวลาเมื่อเราจัดการกับจำนวนเชิงซ้อน แนวคิดของบางสิ่งที่เป็นการดำเนินการแบบที่ต้องการ มีหลายค่า คุณอาจจะ ฟังวลีสาขา คุณจะเลือกสาขาของฟังก์ชันรากที่สองที่ไหน? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "เพราะมีคำตอบที่แตกต่างกันหลายคำตอบ คุณรู้ว่าเราคิดว่าฉันอีกครั้งคือการหมุน 90 องศา และถ้าเราคิดว่ามันเป็นการหมุน 90 องศา มันรู้สึกเหมือนรากที่สองควรเป็น คุณรู้อะไรบางอย่างนั่งอยู่ที่มุม 45 องศา บางทีนั่นอาจเป็นกำลังสอง รากของ I ซึ่งเราสามารถเขียนได้ชัดเจนมากว่า รูท 2 ส่วน 2 รูท 2 ส่วน 2 I นั่นก็แค่ใช้ตรีโกณมิติ แต่ถ้าเราคิดว่า I เป็นการหมุน 270 องศาที่เป็นลบ รู้สึกเหมือนครึ่งหนึ่งของการดำเนินการนั้นครึ่งหนึ่ง จริงๆ ควรพาเราไปอีกด้านหนึ่ง บางทีตัวเลขที่อยู่ตรงนี้ควรเป็นสแควร์รูทของ I และนั่นก็แค่ค่าลบของสิ่งที่เราเห็นก่อน รากลบ 2 ส่วน 2 ลบ รูท 2 ส่วน 2 คูณ I ตอนนี้ในบริบทของจำนวนจริง ฟังก์ชันค่าที่เราสามารถบอกได้ ใช่ แค่เลือกสแควร์รูทให้เป็นคำตอบที่เป็นบวก แต่คุณคิดว่าคำตอบใดเป็นคำตอบที่เป็นบวก? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "และฉันคิดว่าคุณพูดได้ดี เรารู้ว่านี่คืออะไร เรานิยามมันเป็นสแควร์รูทของ 2 ทั้งหมดได้ดี แต่แล้วถ้าฉันบอกว่า ลองเข้าใกล้นี่แบบเดียวกับที่เราเข้าใกล้ I ของเราถึงนิพจน์ I ต้องการแสดงสิ่งต่างๆ เป็น e ไปยังสิ่งที่ถูกต้องก่อน แล้วฉันจะยกมันให้เป็น 1 ครึ่งโดยคูณครึ่ง 1 เข้ากับเลขชี้กำลัง แล้วฉันก็บอกว่า โอเค ฉันทำได้ ฉันเดาว่าฉันจะทำ e นั้นกับสิ่งที่เป็นอยู่ได้ เท่ากับ 2 หลุม นั่นคือลอกธรรมชาติของ 2 เป็นค่าคงที่ซึ่งอยู่ที่ประมาณ 0 69 หรือประมาณนั้น ถ้าเรายก e ยกกำลังนั้น เราจะได้ 2 เราก็คิดถึงนี่เป็น e กำลังลอกธรรมชาติของ 2 คูณ 1 ครึ่ง และถ้าคุณอยากคิดหา e กำลัง x ล่ะ? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "คุณรู้ว่ามันอาจเกินความจำเป็นในบริบทของจำนวนจริง แต่ถ้าคุณคิดว่า e กำลัง x เป็นการจดชวเลขสำหรับฟังก์ชัน x นี้ คุณสามารถแทนค่า 0 ได้ 69 คูณ 1 ครึ่ง ซึ่งผมคิดว่าน่าจะประมาณ 0 บทที่ 345 ก็เป็นอย่างนั้นแหละ คุณเสียบค่าที่เป็นรูปธรรมนั้นเข้าไปในพหุนามของคุณ แล้วดูว่ามันออกมาเป็นค่าอะไร และมันจะออกมาประมาณ 1 414 รากที่สองจำนวนจริงที่ดีของ 2 สิ่งที่คุณคาดหวัง แต่ถ้าเราทำสิ่งเดียวกัน เราก็ทำกับ I และยอมรับว่าจริงๆ แล้วมีคำตอบที่แตกต่างกันหลายคำตอบ เมื่อเราต้องการเขียนบางอย่างเป็น e ยกกำลัง เราก็สามารถเขียนสิ่งนี้ได้เช่นกัน นี่อาจดูตลก แต่เราเขียนมันเป็น e กำลังลอกธรรมชาติของ 2 บวก 2 ไพ I ทั้งหมดนั่นยกขึ้นเป็นครึ่ง 1 ได้ แล้วค่านี่จะเท่ากับ คุณแยกมันออกได้ เพราะมันคือ e กำลัง ลอกธรรมชาติของ 2 คูณด้วย e กำลัง 2 ไพ I อันนี้มีผลกับการหมุนสิ่งต่างๆ 360 องศา ดังนั้นมันจะเท่ากับ 1 เราก็กำลังดูที่ 2 คูณ 1 ยิ่งใหญ่ ซึ่งให้ความรู้สึกเหมือนเป็นการทดแทนที่ถูกต้อง แต่เมื่อใด เราเล่นเกมเดียวกันกับการเอาสิ่งนี้มาเพิ่มกำลังและปฏิบัติต่อมันโดยการคูณกำลังเป็นเลขชี้กำลัง ดูสิ่งที่เกิดขึ้น เรามี e กำลังลอกธรรมชาติของ 2 คูณ 1 ครึ่งบวก เอาละ 2 ไพ I คูณ 1 เท่ากับเท่าไร นั่นก็คือ ไพ คูณ I ทีนี้ส่วนแรก e กำลังลอกธรรมชาติของ 2 คูณ 1 ครึ่งที่จะลงเอยเป็นสแควร์รูทที่คุ้นเคยของ 2 ก็ดีไปหมด แต่เราจะคูณมันด้วย e ไพ I ถูกและมีชื่อเสียงมากว่า e กำลัง pi I เป็นลบ 1 ดังนั้นในกรณีนี้ มันดูเหมือนว่าจะกำลังบอกว่า หากเรากำลังแก้นิพจน์นี้ 2 ต่อครึ่ง 1 โดยเล่นกับคำตอบต่างๆ เราก็สามารถแทนค่าบางอย่างเช่น e กำลัง X เท่ากับ 1 ครึ่งสิ่งที่เราได้คืออีกคำตอบหนึ่งที่เรามักจะเขียนเป็นค่ารากที่สองที่เป็นลบของ 2 และในที่นี้ผมหมายถึงว่ามันตลกนิดหน่อยที่มันมีหลายค่าเพื่อดูที่ 2 ต่อ 1 ครึ่งและ บอกว่าไม่เท่ากัน สิ่งหนึ่ง แต่ขึ้นอยู่กับสิ่งที่เราเลือกมันอาจเท่ากับหลายสิ่งที่แตกต่างกัน แต่สองสิ่งที่ดูเหมือนจะค่อนข้างสมเหตุสมผล ถ้ามีสิ่งใดที่ 2 ต่อ 1 ครึ่ง ก็ดูเหมือนว่ามันควรจะเป็นเชิงบวก สแควร์รูทที่เราคุ้นเคยหรือตัวแปรเชิงลบของอันที่จริง ๆ แล้วดูเหมือนจะไม่เป็นปัญหา และอันที่จริงเราทำได้ เอิ่ม เล่นเกมนี้ให้ดียิ่งขึ้นไปอีก โดยให้ฉันถามคุณถึงคำตอบที่สร้างสรรค์กว่านี้สำหรับสำนวนนี้ เพราะบางทีเราอาจพบกำลังตลกๆ ของบางอย่าง เช่น 2 ยกกำลัง X เมื่อเราเริ่มแทนค่าต่างๆ ของ X ตามการแทนที่ที่เราทำ หากเราปฏิบัติตามกฎเดียวกันกับที่เราใช้ในการประเมิน I ถึง ยกกำลัง I คราวนี้คำถามถามหรือระบุว่าคำตอบหนึ่งของสมการ e กำลัง x เท่ากับ 2 คือจำนวนจริง ลอกธรรมชาติของ 2 โอเค อันนั้นที่เรารู้ มันไม่น่าเบื่อ แต่มันน่าเบื่อเมื่อเทียบกับอย่างอื่นที่เราทำได้ คุณคิดอย่างอื่นได้ไหม แล้วเขียนอย่างอื่นได้ไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "เราจะเลือกทันทีที่เราเลือกอย่างใดอย่างหนึ่งหรือไม่? ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "มันเป็นฟังก์ชันที่ชัดเจน แต่ ณ จุดนั้น มันแค่รู้สึกเหมือนบางที สิ่งที่เราต้องการคือการ หยุดคิดเกี่ยวกับสิ่งต่างๆ ในรูปของฐานบางตัวที่ยกกำลัง X บางทีทันทีที่เราอยู่ในบริบทของจำนวนเชิงซ้อน เราควรเขียนเลย พวกมันทั้งหมดเป็น exp ของค่าคงที่บางค่า X หากไม่มีเหตุผลอื่น มันทำให้ชัดเจน เราจะแทนค่าตัวเลขได้อย่างไรหากเราต้องการคำนวณหรือแค่คิดเลข นอกเหนือจากนั้น เราได้พหุนามอนันต์ที่ดีที่เรา เสียบเข้าไปแล้วฉันจะสร้างกรณีใหม่ให้คุณว่านี่อาจเป็นวิธีที่ถูกต้องในการคิดเลขชี้กำลัง ทันทีที่เราขยายไปสู่โดเมนอื่นๆ เช่น จำนวนเชิงซ้อน และเพื่อสิ่งนั้น เรามาสำรองข้อมูลกันดีกว่า ไป กลับไปที่กริ่ง มีบางอย่างมาถึง กลับไปที่วิธีดั้งเดิมที่เราขยายแนวคิดเรื่องการยกกำลัง และลองคิดดูว่า 2 กำลัง X คืออะไร เรารู้วิธีคิดเกี่ยวกับจำนวนธรรมชาตินี้ คุณรู้บางอย่างเช่น การคูณซ้ำ 2 ถึง 3 ทำไมคุณถึงถูกสอนให้คิดถึงจำนวนเศษส่วน เช่น 2 ถึง X สำหรับจำนวนเศษส่วน หรือจำนวนลบ และอะไรทำนองนั้น ดี. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "ปกติคุณจะถูกสอนว่า 2 ต่อครึ่ง 1 ควรเป็นสิ่งที่คุณรู้ถ้าฉันคูณมันด้วยตัวเอง และสิ่งนี้เป็นไปตามกฎปกติที่เลขชี้กำลังทำกับการนับตัวเลข โดยที่เราสามารถบวกสิ่งต่างๆ ในเลขชี้กำลังนั้นได้ ฉันควรจะได้ 2 ยกกำลัง 1 ดังนั้นมันควรเป็นตัวเลขที่เมื่อฉันคูณมันด้วยตัวเอง ฉันจะได้ 2 และคุณก็รู้ว่า ณ จุดนั้น คุณมีทางเลือก บางทีมันอาจเป็นค่าบวก บางทีมันอาจเป็นลบ แต่ถ้าคุณตัดสินใจเลือกเชิงบวกเสมอ คุณจะได้ฟังก์ชันต่อเนื่องดีๆ จากดีลเดียวกันนี้ ถ้าเราถามเกี่ยวกับจำนวนลบ 2 ยกกำลังลบ 1 ควรมีค่าอะไรสักอย่าง เมื่อไหร่ฉันจะคูณมันด้วย 2 ถึง 1? ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "มันทำให้ฉันได้ 2 กำลัง 0 และนั่นคือเหตุผลแบบแผนของเราที่ว่าเลขชี้กำลังลบดูเหมือน 1 ครึ่ง แต่สิ่งที่เกิดขึ้นจริงๆ ตรงนี้คือเรากำลังบอกว่าไม่ว่านี่คืออะไรก็ตาม มันควรจะเป็นฟังก์ชันอะไรสักอย่าง ที่ทำให้คุณสมบัตินี้เป็นไปตาม f ของ a บวก b เท่ากับ f ของ a คูณ f ของ b และยิ่งกว่านั้น ความจริงที่ว่าฐานคือ 2 โดยพื้นฐานแล้วบอกเราว่ามันไม่ใช่แค่ฟังก์ชันใดๆ มันคือฟังก์ชันที่เมื่อเราเสียบ 1 เราจะได้ 2 และอย่างที่คุณทราบเพียงเล็กน้อย คำถามสไตล์การตรวจสอบสติเพื่อดูว่าคุณกำลังติดตามพร้อมกับความหมายบางอย่างที่นี่หรือไม่ ฉันต้องการถามคุณว่าอะไร ฉันจะไม่เรียกมันว่าซอฟต์บอล แต่นี่ไม่ได้หมายความว่าจะเป็นเหมือนคำถามที่ลึกซึ้งอย่างไม่น่าเชื่อ อย่างจำเป็น. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "มันเป็นแค่การตรวจสอบมากกว่าถ้าคุณทำตาม แนวคิดของการเริ่มต้นด้วยคุณสมบัติของฟังก์ชันเชิงนามธรรม แล้วหาวิธีอนุมานที่เราอาจต้องการเขียนมันตามคุณสมบัติเหล่านั้น ถ้า f ของ x เป็นไปตามคุณสมบัติเลขชี้กำลัง f ของ a บวก b เท่ากับ f ของ a คูณ f ของ b สำหรับอินพุตทั้งหมด และมันยังเป็นไปตาม f ของ 1 เท่ากับ 2 ข้อใดต่อไปนี้เป็นจริง ซึ่งก็คือบอกว่าข้อใดต่อไปนี้จำเป็นต้องเป็นจริง ไม่ว่าคุณจะเริ่มใช้ฟังก์ชันใดก็ตาม กับพวกคุณที่จำได้ว่าเป็นการบรรยายเรื่องไหน ไม่ว่าเราจะพูดถึงเรื่องไหนก็ตามที่จะตีความว่าสูตรของออยเลอร์พูดจริง ๆ ฉันถามคำถามแบบนี้โดยที่ฉันละเลยเงื่อนไขเดียวคุณก็รู้ว่าฉันไม่ได้เขียนลงไป ความจริงที่ว่าเราต้องการให้แน่ใจว่า f ของ x นั้นไม่เป็นศูนย์ทุกที่ และนั่นทำให้เกิดความสับสนจำนวนหนึ่งซึ่งเจ๋งมาก ทำให้เกิดความสับสนบนหน้าจอที่เกิดขึ้นกับเราทุกคน แต่จุดประสงค์ของมันคือการแสดงโดยพื้นฐานว่าคุณสมบัตินามธรรมของ สิ่งที่เปลี่ยนการบวกเป็นการคูณก็เพียงพอที่จะทำให้คุณต้องการเขียนฟังก์ชันเป็นอะไรก็ได้ที่เท่ากับการยกกำลังบางอย่าง นี่คือจิตวิญญาณของคำถาม ตอนนี้เรามีคำถามสองสามข้อเกี่ยวกับหอคอยไฟฟ้า ที่ดูเหมือนจะโผล่ขึ้นมาที่นี่ซึ่งมีความเชื่อมโยงอย่างมากกับครั้งที่แล้ว เรามาพักคำถามของหอคอยพลังงานกันสักครู่เพื่อที่เราจะได้รู้สึกลึกลงไปว่าเช่น การยกกำลังควรหมายถึงอะไรในที่นี้? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "เพราะเพราะว่าเราสามารถเป็นสิ่งที่ผมอยากอ้างได้ก็คือเราสามารถตอบมันได้หลายวิธี ดังนั้นถ้าคุณให้ผมแค่อันเดียว เราจะพูดถึงหอคอยพลัง แล้วเช่นเดียวกับเส้นจำนวนที่สามารถแสดงเป็นสเกลลอการิทึมได้ สิ่งเดียวกันนี้สามารถทำได้กับระนาบที่ซับซ้อนหรือไม่? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "อาจทำแผนที่ระนาบเชิงซ้อนเข้ากับทรงกระบอกอนันต์ในลอการิทึม นั่นใช่ ใช่ ที่จริง มีภาพที่ผมจะพูดถึงในอีกสักครู่นี้ ซึ่งเราทำบางอย่างที่ค่อนข้างคล้ายกัน เพราะสิ่งที่เราจะทำคือเล่นกับฟังก์ชันเลขชี้กำลังที่แตกต่างกัน X ของ R คูณ X แต่เรา จะเปลี่ยนค่าของ R ซึ่งจะแสดงด้วยจุดสีเหลืองเล็กๆ แทน เราจะพูดถึงเรื่องนี้กัน มันไม่ได้จะแมปทั้งระนาบ แต่เป็นเพียงจุดตัวอย่างสองสามจุดจากแกนจริงและแกนจินตภาพ แต่แนวคิดก็คือ เมื่อเราเคลื่อนที่ไปรอบๆ ค่าคงที่นั้น เราจะสามารถเห็นภาพสิ่งต่าง ๆ ที่มันทำกับระนาบ และอย่างมีประสิทธิภาพ มันเหมือนกับว่ามันเปลี่ยนแกน x ให้เป็นสเกลลอการิทึม แล้วห่อ แกนจินตภาพตามแนววงกลม และทันทีที่ค่า R นั้นกลายเป็นจินตภาพ มันจะสลับบทบาทของจำนวนจริงเหล่านั้นบนวงกลม และจำนวนจินตภาพถูกวางบนแกนบวกที่มีมาตราส่วนของลอการิทึม ดังนั้นคำถามที่ดีทั้งสามข้อที่ฉันเดา เหมือนกับว่ากำลังกระโดดไปข้างหน้าเพื่อไปยังที่ที่ฉันอยากไป แต่ก็ดีที่ได้เห็นผู้คนคิดเช่นนั้นในเรื่องนี้ ลองให้คะแนนมันดู ไอเดียก็คือว่าคุณสมบัติของ f ของ a บวก B จะทำให้คุณแสดงสิ่งที่ต่างกันออกมามากมายในรูปของ f ของ 1 คืออะไร และแค่สะกดออกมาเฉยๆ? ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "อย่างชัดเจน บางอย่างเช่น f ของ 5 ก็เหมือนกับ f ของ 1 บวก 1 บวก 1 บวก 1 บวก 1 ซึ่งก็เหมือนกับ f ของ 1 คูณด้วยตัวมันเอง 5 ครั้ง เพราะคุณสมบัตินี้ ซึ่งถ้า f ของ 1 เป็น 2 ก็เท่ากัน เป็น 2 ยกกำลัง 5 แล้วอะไรประมาณ f ของลบ 5 มันควรจะเป็นกรณีที่เมื่อเราคูณมันด้วย f ของ 5 เราได้อะไรก็ตามที่เป็น f ของ 0 และมันไม่ชัดเจนในทันทีว่า f ของ 0 คืออะไร แต่เราสามารถบอกได้ว่า f ของ 1 บวก 0 เท่ากับอะไรก็ตามที่ f ของ 1 คูณสิ่งที่ f ของ 0 เป็น แต่ f ของ 1 เท่ากับ 2 แล้วนี่ก็เท่ากับ 2 ด้วย เราก็เลยบอกว่า 2 เท่ากับ 2 คูณอะไรสักอย่าง ดีอะไรสักอย่าง ต้องเป็น 1 ดังนั้นในบริบทนี้ นี่รับประกันได้ว่า f ของลบ 5 คือ 2 ยกกำลังลบ 5 มันคือ 1 ส่วน 2 ยกกำลัง 5 เราสามารถเขียนอันนี้ให้ชัดเจนเป็น 2 ยกกำลังลบ 5 ซึ่งก็คือคุณสมบัติทั้งสองนี้รวมกัน เราอยากเขียนฟังก์ชันเป็น 2 ยกกำลัง X จริงๆ เพราะเลขนับใดๆ ที่เราใส่ลงไป มันจะออกมาเป็นที่น่าพอใจ มันจะดูเหมือนคูณด้วยตัวมันเองด้วยจำนวนครั้งของเศษส่วนใดๆ ที่เราใส่ลงไป ก็จะได้คุณสมบัติเหล่านี้ ที่เราต้องการ และคุณอาจสงสัยว่ามันมีเอกลักษณ์เฉพาะตัวและในบริบทของฟังก์ชันมูลค่าจริง จริงๆ แล้วมันจะเป็น แต่ในบริบทของฟังก์ชันมูลค่าเชิงซ้อน จะมีฟังก์ชันดังกล่าวหลายฟังก์ชัน f ที่เราสามารถเขียนให้กับฟังก์ชันอันมีค่านี้ ซึ่งก็คือสิ่งที่เราเป็น ดูก่อน โดยที่เรากำหนดให้ฟังก์ชันเป็น exp ของลอกธรรมชาติของ 2 บวก 2 ไพ I ทั้งหมดนั้นคูณ X โอเค ขอโทษที่เรื่องเลอะเทอะตรงนี้ ฉันแค่ตื่นเต้นที่จะเขียนเรื่องนี้ และนี่คือฟังก์ชันที่ต่างจากเดิม เห็นได้จากสิ่งที่เกิดขึ้นหากคุณเสียบ X เท่ากับ 1 ครึ่ง เราเห็นก่อนหน้านี้เล็กน้อยว่าเมื่อคุณเสียบ 1 ครึ่งหนึ่งสิ่งที่คุณได้คือรากที่สองที่เป็นลบของ 2 แล้วถ้าคุณเสียบ 1 ในสี่ คุณจะได้ ไม่ใช่รากที่สี่ของ 2 แต่ฉันคูณรากที่สี่ของ 2 ดังนั้นมันจึงเป็นฟังก์ชันอื่น แต่มันยังคงเป็นไปตามคุณสมบัติเหล่านี้ และมันทำให้เราอยากเขียนมันเป็น 2 กำลัง X และมันทำให้แนะนำว่าบางที 2 กำลัง X นั้นคลุมเครือ สัญกรณ์เล็กน้อย และเราควรเขียนทุกอย่างในรูปของ exp ของ R คูณบางสิ่งบางอย่าง แต่คุณอาจสงสัยดี คุณรู้ไหมบางทีเราอาจไม่สร้างสรรค์เพียงพอกับฟังก์ชันทั้งหมดที่ตรงตามเงื่อนไขนี้ บางทีอาจมีความคลุมเครือเมื่อเราเขียน exp ของ R คูณอะไรบางอย่าง และมีค่า R ที่แตกต่างกันที่สามารถเข้ามามีบทบาท แต่ฉัน ฉันแค่จะวางข้อเรียกร้องเล็กน้อย และจากนั้นอาจจะให้คร่าวๆ ว่าการพิสูจน์จะเป็นอย่างไรถ้าคุณต้องการ ซึ่งก็คือว่า บอกว่าคุณมีฟังก์ชันที่ซับซ้อน F และมีคุณสมบัติตรงตามคุณสมบัติต่อไปนี้ก่อน คุณสามารถหาอนุพันธ์ของมันได้ มันหาอนุพันธ์ได้ ซึ่งแค่กันไม่ให้เป็นสิ่งที่คุณรู้ว่ายุ่งวุ่นวายไม่ต่อเนื่อง นั่นเหมือนกับการหาค่าสุ่ม ขึ้นอยู่กับคุณทราบสแปนของสเปซเวกเตอร์ใดๆ ส่วน ผมไม่รู้จำนวนเศษส่วนที่คุณอาจต้องการคิดแบบแปลกๆ มันเป็นฟังก์ชั่นที่ดี นั่นหาอนุพันธ์ได้ มันไม่เท่ากับ 0 ทุกหนทุกแห่ง ดังนั้นเงื่อนไขที่เลื่อนลอยในใจและฉันลืมไปว่าบรรยายวิชาไหนหรืออะไรทำนองนั้น แล้วมันก็มีคุณสมบัติส่วนกลางที่เปลี่ยนการบวกเป็นการคูณ หากคุณมีฟังก์ชันดังกล่าว ฉันอ้างว่า มันมีเอกลักษณ์เฉพาะตัว บางทีฉันควรระบุจริงๆ ว่ามีจำนวนเชิงซ้อน R อยู่ ดังนั้นคุณจึงสามารถเขียน F ของ X โดยพื้นฐานแล้วเป็นฟังก์ชันเอ็กซ์โปเนนเชียลของ R คูณค่า X นั้น ซึ่งโดยทั่วไปแล้วคุณก็รู้อยู่แล้วว่าถ้าคุณมี X เป็นฟังก์ชันนี้ พหุนามอนันต์ที่มีคุณสมบัติอนุพันธ์ที่ดี และทั้งหมดนั้นถ้าคุณมีสิ่งนี้ คุณจะมีเลขชี้กำลังทุกอันที่คุณต้องการในความหมายทั่วไปที่เป็นนามธรรมของคำว่าเลขชี้กำลัง เพียงยึดตามคุณสมบัติที่เราต้องการจากมันและภาพร่างของหลักฐาน หน้าตาแบบนี้ถ้าคุณต้องการดูก่อนว่าอนุพันธ์ของค่านี้ซึ่งเราสมมุติว่ามีอยู่ทุกที่คืออะไร, จริงไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "และคุณเขียนให้ชัดเจนว่าขีดจำกัดของมันคืออะไร ฉันจะพูดถึงมันอย่างรวดเร็ว ที่นี่ สำหรับผู้ที่ต้องการหยุดชั่วคราวและคิดรายละเอียดอย่างละเอียด รู้สึกอิสระที่ทรัพย์สินส่วนกลางที่เราได้ให้เราขยาย F ของ X บวกเทอม H เรากำลังคิดว่าคุณทราบการเปลี่ยนแปลงเล็กน้อยในเอาต์พุต ส่วนการเปลี่ยนแปลงอินพุตที่ทำให้เกิดการเปลี่ยนแปลง นั่นคือสิ่งที่ DF DX แกะออกมา และเพราะว่าเราสามารถแยกส่วนนั้นออกได้ เราสามารถแยกตัวประกอบ F ของ X ออกจากนิพจน์ทั้งหมดได้ และลิมิตทั้งหมดจะแสดงในรูปของ H เท่านั้น ซึ่งถ้าคุณคิดว่ามันหมายถึงอะไรในบริบทของอนุพันธ์ และความจริงที่ว่า F ของ 0 จำเป็นต้องเท่ากับ 1 นิพจน์จำกัดทั้งหมดนี้ก็คือ แค่ค่าคงที่บางส่วน แต่เจาะจงกว่านั้นคืออนุพันธ์ของฟังก์ชันของเราที่ 0 จะเป็นเท่าใด คุณมีเรื่องตลกๆ ที่ถ้าคุณรู้อนุพันธ์ของมันที่ 0 ที่กำหนดว่าอนุพันธ์ของมันอยู่ทุกหนทุกแห่ง และในบริบทของฟังก์ชันเอ็กซ์โปเนนเชียล หวังว่าจะคุ้นเคยกันดีเพราะ ทั้งหมดที่เรากำลังพูดจริงๆ คืออนุพันธ์ของฟังก์ชันเอ็กซ์โพเนนเชียลคือสัดส่วนกับตัวมันเอง และค่าคงที่สัดส่วนนั้นเท่ากับอนุพันธ์ใดๆ ที่ 0 นี่คือการใช้วลีแบบนามธรรมทั้งหมด แต่จุดประสงค์ของมันคือเพื่อเน้นว่า ไม่จำเป็นต้องเป็นแค่ฟังก์ชันที่เราคิดว่า a ยกกำลัง X อยู่แล้ว แต่มันเป็นคลาสของฟังก์ชันที่กว้างกว่ามากซึ่งเป็นไปตามคุณสมบัตินามธรรมของการเปลี่ยนการบวกเป็นการคูณ แต่ถ้าคุณมี มันรับประกันได้ว่าคุณจะมี อนุพันธ์อันดับสอง และสำหรับเรื่องนั้น อนุพันธ์อันดับสาม และเช่นนั้น เพราะฟังก์ชันอนุพันธ์เป็นเพียงสัดส่วนกับตัวมันเอง ดังนั้นในการหาอนุพันธ์อันดับ n คุณแค่ดูที่ค่าคงที่สัดส่วนนั้นแล้วยกกำลัง n แล้วจากตรงนี้ คุณก็ทำ การขยายซีรีส์ Taylor และฉันอาจปล่อยให้มันเป็นการบ้านขั้นสูงสำหรับผู้ที่คุ้นเคยกับซีรีส์ Taylor ในแนวคิดนั้น โดยเฉพาะอย่างยิ่งถ้าคุณต้องการผสมผสานแนวคิดของฟังก์ชันอนุพันธ์ใด ๆ ที่สามารถหาอนุพันธ์ได้ในแง่ของจำนวนเชิงซ้อน ซึ่งก็คือ ประเภทของหัวข้อวิทยาลัยที่แน่นอน คุณรู้ว่าคุณสามารถผสมเหตุผลเข้าด้วยกันได้ตามที่คุณต้องการ แต่อนุญาตให้ใช้เหตุผลที่คลุมเครือในบริบทของคนที่รู้เกี่ยวกับซีรีส์ของ Taylor เท่านั้นและไม่มีอะไรอื่นใดที่จะนำแนวคิดนี้ไปใช้และดูที่การขยายตัวของ Taylor สำหรับ F และ เป็นการพิสูจน์ความคิดที่ว่า มีจำนวนเชิงซ้อนเฉพาะตัวจนฟังก์ชัน F ของเราสามารถเขียนได้แบบนี้ แล้วการเชื่อมต่อกับเลขชี้กำลังปกติคือเมื่อใดก็ตามที่คุณมีค่า R เช่นนั้น เราทำสิ่งที่เราทำในบริบทที่ซับซ้อนของจำนวนจริงเป็นหลัก คือถ้าคุณดูที่ exp ของฟังก์ชันนั้นของค่า R แล้วเขียนมันเป็นฐาน รู้สึกเหมือนว่าคุณควรจะเขียนมันเป็น B ถึง X ได้ แต่ประเด็นทั้งหมดตรงนี้ก็คือ เมื่อเราเล่นเกมนี้ และคุณกำลังพยายามตีความบางอย่างเช่น I ถึง X ซึ่งเป็นฟังก์ชันที่ไม่ชัดเจน เพราะมี ค่า R ที่แตกต่างกันมากมาย เราตีความได้ว่าไม่ใช่แค่ exp ของครึ่งไพ I คูณ X, แต่เรายังตีความได้ว่าหมายถึง exp ของครึ่งไพ 5 ส่วน I คูณ X และพวกนี้เป็นฟังก์ชันแยกกัน และมีฟังก์ชันแยกกันเป็นตระกูลอนันต์ที่ให้ความรู้สึกเหมือนเราควร เขียนมันเป็น I กำลัง X ดังนั้นนิพจน์ I กำลัง I เว้นแต่ว่าคุณได้นำมาตรฐานมาใช้กับสิ่งที่จำเป็นต้องหมายถึง เมื่อคุณบอกว่ามันมีเอาต์พุตมากมายอย่างไม่สิ้นสุด วิธีคิดอีกอย่างก็คือ ฟังก์ชัน I กำลัง X ด้วยสัญกรณ์ที่เรามีนั้นค่อนข้างคลุมเครือ ทีนี้มาเริ่มนึกภาพบางส่วนกันดีกว่า เพราะฉันคิดว่ามันสนุก และคุณก็รู้ว่าคุณบอกฉันว่านี่คือภาพที่เป็นประโยชน์หรือภาพที่น่าสับสนมากกว่า สิ่งที่เรากำลังจะทำคือดูที่ฟังก์ชัน exp ของ R คูณ X ซึ่งโดยพื้นฐานแล้ว นี่เป็นอีกวิธีหนึ่งในการเขียน e ยกกำลังของ X อันที่จริง ฉันคิดว่า ฉันคิดว่า ฉันเรนเดอร์แอนิเมชั่นที่แตกต่างออกไป ณ จุดหนึ่งที่ระบุเอาไว้ว่า เพราะฉันกำลังวางแผนที่จะทำสิ่งนั้น ให้ฉันเถอะ คุณกลับมาในระบบไฟล์ของฉันแล้ว กลับไปยังจุดที่คุณควรจะอยู่ เข้าไปข้างในนั้น มันบ่นเพราะมีหลายสิ่งที่แตกต่างกัน มันจะเหมือนกับว่ามี โอ้ แทนที่ มันปรากฏขึ้นบนหน้าจออื่น เดี๋ยวก่อน ทำไมมันใช่ โอเค แทนที่? ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "วางสิ่งที่คุณเห็นตรงนั้น แล้วเรากลับไปที่ โอ้ นั่น เราทั้งหมดนั้นทั้งหมด เพื่อที่ผมจะได้เขียนออกมาดีๆ ถ้าคุณไม่สบายใจที่จะคิดว่ามันเป็น exp ของ R คูณ X พหุนามอนันต์นี้ แค่ในรูป หลังศีรษะของคุณ e กำลัง R คูณ X แล้วเราจะแปรผันรอบๆ R ผมจะตามจุดของแกนจินตภาพ และผมจะตามจุดของแกนจริง แล้วลองดูว่านี่ทำอะไรได้บ้าง มันเร็วมาก ขอผมลองคิดดูช้าๆ หน่อยดีกว่าว่าจำนวนลบทั้งหมด นั่นคือจำนวนจริงลบจะโดนบีบให้อยู่ในช่วงระหว่าง 0 ถึง 1 ซึ่งควรเข้าใจได้ว่า e กำลังเป็นลบ? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "a กำลังเป็นจำนวนจริงลบคือค่าระหว่าง 0 ถึง 1 และเรากำลังติดตาม f ของลบ 1 โดยเฉพาะ ซึ่งจะแสดงรอบๆ อะไรก็ตามที่ 1 ส่วน e อยู่ที่ประมาณ 30 0 37 f ของ 1 ตกลงบน e ตามที่คาดไว้ นั่นคือค่า exp ของ 1 คือ f ของ I จะได้หนึ่งเรเดียนรอบวงกลมหน่วย และมันสนุกดีที่ได้ติดตามแกนจินตภาพทั้งหมดตรงนี้ว่าแกนจินตภาพพันรอบวงกลมอย่างไร และจะเกิดอะไรขึ้นเมื่อเราปรับแต่งค่า R นี้ นั่นไม่ใช่แค่การพิจารณาว่าเรากำลังพูดถึงฟังก์ชันเอ็กซ์โปเนนเชียล แต่ฟังก์ชันเอ็กซ์โปเนนเชียลตัวใด มีการโต้ตอบแบบหนึ่งต่อหนึ่งที่ดีระหว่างฟังก์ชันเอ็กซ์โปเนนเชียลทั้งหมด เราอาจต้องการและค่าของ R ตรงนี้ มันยืดสิ่งต่าง ๆ ออกไป ดังนั้นเมื่อเราใส่มันได้ถึง 2 คุณรู้ไหมว่ามันยืดแกนจริงออกไปมากขึ้น ดังนั้น f ของ 1 จะจบลงที่บริเวณที่ e กำลังสองมีค่ามากกว่า 7 f เล็กน้อยของค่าลบ 1 ใกล้กับ 0 มาก f ของ I คือ 2 เรเดียน การหมุนรอบวงกลม f ของลบ I คือลบ 2 การหมุนเรเดียน และแน่นอน เราจะได้สูตรโปรดของเราว่าถ้านั่นเป็นพายที่เรามีค่าคงที่สเกลแล้ว แกนจริงยืดออกค่อนข้างมาก คุณรู้ไหมว่า f ของ 1 กำลังนั่งอยู่ที่ e ถึง pi ซึ่งอยู่ใกล้กับ 20 บวก pi มาก ซึ่งสนุกเสมอ และ f ของลบ 1 ใกล้ 0 มาก ดังนั้นมันจึงยืดออกตามความเป็นจริงจริงๆ แกน และมันยังยืดสิ่งต่าง ๆ ในทิศทางวงกลมหน่วยด้วย เพื่อว่าไปถึง f ของ I หรือ f ของลบ ฉันจะเดินไปครึ่งทางรอบวงกลม แล้วตอนนี้ก็ดีแล้ว เราจะคิดอย่างไรกับฟังก์ชันแบบนี้? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "เรายังเขียนเป็น X ของ X ของลอกธรรมชาติของ 2 คูณ X เราก็เลยย้ายจุดสีเหลืองแทนค่า R ไปประมาณ 0 69 ยังไม่มีส่วนจินตภาพ มีเพียงเลข 0 จริงเท่านั้น 69 หรือประมาณนั้น นั่นคือลอกธรรมชาติของ 2 ทีนี้ คุณจะเห็นว่า f ของ 1 ตกลงบน 2 ซึ่งเป็นสาเหตุที่เราต้องการเรียกฟังก์ชันนี้ว่า 2 กำลัง X f ของ 1 ครึ่ง ขอโทษจริงๆ f ของลบ 1 ตกลงบน 1 ครึ่งหนึ่งของ f ของ ผม มันคือการเดินรอบๆ วงกลมหน่วย โดยเฉพาะมันจะเป็น 0 69 เรเดียนรอบวงกลมหน่วย ทีนี้เรามาสนุกกันอีกหน่อยแล้วบอกว่าจะเกิดอะไรขึ้นถ้าเราเปลี่ยนค่านี้เป็น 0. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "ฉันคืออะไร ยกกำลัง ในกรณีนี้ มันผลักมันไปที่ประมาณ 0 2 ประมาณหนึ่งในห้า แต่มีฟังก์ชันเอ็กซ์โปเนนเชียลต่างๆ มากมายที่จะมีคุณสมบัติในการใส่ f ของ 1 ลงบนเลข I ดังนั้นหากเราขยายมันให้มากขึ้นไปอีก ฉันไม่คิดว่ามันจะเคลื่อนไหวตรงนี้ แต่ถ้าเราจะเอา จุดสีเหลืองนั้นแล้วยกขึ้นจนได้ 5 ครึ่งคูณ pi I สิ่งที่คุณจะเห็นคือวงกลมหน่วย? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "ถูกหมุนรอบตัวเองเพื่อให้ f ของลบ f ของ 1 หมุนรอบอีก 2 pi เรเดียนและลงจอดตรงที่มันอยู่ แต่มันจะยืดแกนจริงออกไปอีกมาก ซึ่งเป็นความรู้สึกที่เอาต์พุตอีกอันหนึ่งของ I ไปยัง I คือ ตัวเลขที่น้อยกว่ามาก มันประมาณว่ามันคือ 0 0003 หรือประมาณนั้น แต่เราก็ยังเห็นว่าสิ่งที่ฉันคิดว่าสนุกดี จะเกิดอะไรขึ้นถ้าเราพิจารณานิพจน์ทางเลือกที่เราต้องการตีความว่าเป็น 2 ยกกำลัง X ใช่ไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "เรามี X ของ R คูณ X และ R เท่ากับค่านี้, ซึ่งก็คือลอกธรรมชาติของ 2 บวก ไพ คูณ I นั่นหมายความว่าเมื่อเราเสียบ 1 f ของ 1 ไว้ที่ลบ 2 เราก็เลยอยากเขียนฟังก์ชันนี้ เป็นลบ 2 ยกกำลัง X ใช่ และนั่นคือสิ่งที่คุณรู้ มันง่ายมากเมื่อเราเขียนจำนวนลบลงในยกกำลัง ลบ 2 ยกกำลัง X ในตอนแรกมันไม่ได้ดูเหมือนสิ่งนี้ จำเป็นต้องดึงเราขึ้นมา เป็นจำนวนเชิงซ้อนไม่ว่าทางใดก็ทางหนึ่ง แต่แน่นอน เมื่อเราแทนค่าเช่น 1 ครึ่งหนึ่ง โดยที่เราถามหารากที่สองของลบ 2 เรารู้ว่าเราอยากเขียนสิ่งนี้เหมือนกับ I คูณรากที่สอง ของ 2 แต่ถ้าคุณต้องดูฟังก์ชันนี้ ลบ 2 ยกกำลัง X ในโดเมนเชิงซ้อนทั้งหมดที่มันกำลังยุ่งอยู่ สิ่งที่คุณกำลังมองหาคือฟังก์ชันที่นำค่า 1 ไปเป็นลบ 2 และถ้ามันทำอย่างนั้น มันไปกระทบส่วนที่เหลือของเส้นจำนวนจริง มันหมุนวนออกไปข้างนอกหรือเปล่า? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "เราจะเห็นว่า f ของลบ 1 อยู่ที่ลบ 1 ครึ่ง ประมาณว่าคุณคาดหวังไว้ตรงไหนหากคุณติดตาม f ของ 1 ครึ่ง มันจะอยู่บนเส้นจินตภาพพอดี และ f ของ 1 ครึ่งหนึ่งจะเป็นสแควร์รูทของ 2 เมาส์ไม่ใช่ที่ที่ฉันอยากให้เป็น มันจะอยู่ที่ประมาณสแควร์รูทของ 2 คูณ I และเมื่อคุณทำต่อไป สิ่งนี้จะแสดงให้คุณเห็นถึงค่ากำลังที่แท้จริงของลบ 2 ถึง X มันจำเป็นต้องหมุนวนไปรอบๆ แต่เราก็สามารถขยับค่า R ให้สูงขึ้นไปอีกและได้มันมาด้วย จนถึงประมาณเทาคูณ I ประมาณหกจุดสองแปดคูณ I และในบริบทนั้น นี่เป็นอีกฟังก์ชันหนึ่งที่เราอยากเขียนเป็นประมาณ 2 ยกกำลัง X เพราะสำหรับจำนวนเต็มใดๆ ถึงจำนวนเต็มที่คุณเสียบเข้ากับ X มันจะ ดูเหมือนการคูณซ้ำๆ และมันยังมีค่าที่สมเหตุสมผลสำหรับสิ่งต่างๆ อย่าง 1 ครึ่งหนึ่ง โดยแยกรากที่สองที่เป็นลบออกมา แทนที่จะเป็นรากที่สองที่เป็นบวก แต่สิ่งที่มันทำจริงๆ คือการแปลงระนาบ โดยที่มันทำให้ทุกอย่างเป็นจริง เส้นจำนวนกลายเป็นเกลียวที่พันแน่นมากซึ่งพันรอบและหมุนวนในลักษณะที่ f ของ 1 ตกลงบนหมายเลข 2 ในแง่นั้นเองที่เราพูดได้ว่า 2 ถึง X คือ ถูกตีความได้อย่างน่าเชื่อถือว่า ฟังก์ชันเอกซ์โปเนนเชียลที่แยกจากฟังก์ชันที่เราคุ้นเคยกันดีอยู่แล้ว ดังนั้น ฉันคิดว่าทั้งหมดนี้ ฉันจะทิ้งสิ่งต่างๆ ไว้สำหรับวันนี้ และฉันจะทิ้งคำถามค้างคาสองสามข้อให้คุณคิดเกี่ยวกับ โอเค ดังนั้น หากคุณต้องการ คิดว่า I กับ I เป็นนิพจน์ที่มีหลายค่า คุณอาจพูดได้ว่าเราใช้แบบแผน ลองจินตนาการว่าคุณเลือกสาขาของฟังก์ชันลอการิทึมธรรมชาติ และบางทีนั่นอาจล็อคคุณไว้กับ e กำลังลบ pi แบ่งครึ่ง แต่ถ้าจะบอกว่าแบบนี้อยากให้มีค่าต่างกันมากมายอย่างอนันต์เหมือนค่าต่างๆ ที่เราเห็น ค่า 2 ต่อ 1 ใน 3 ต้องการให้อยู่ในความหมายเดียวกันกี่ค่า? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "อันดับ 10 ต้องการใช้วลีที่แตกต่างกันไปจากทั้งหมด ขอฉันพูดถึงฟังก์ชันเอ็กซ์โปเนนเชียล F ของ X ที่ตรงใจ โอ้ ฉันเขียนมันลงไปที่ไหนสักแห่งแล้วที่ f ของ X ที่ตรงตามคุณสมบัติทั้งหมดนี้ที่ฉันเขียน ดังนั้นถ้ามันตรงใจทั้งหมด ของพวกนี้ และถ้า f ของ 1 เท่ากับ 2 จริงไหม เราจะได้เอาต์พุตต่างกันกี่ตัวเมื่อเราเสียบ X เท่ากับ 3 ใน 10 สำหรับตัวเลือกต่างๆ ของฟังก์ชันอะไร? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "สำหรับ 2 ถึง ไพ สำหรับฟังก์ชันต่างๆ ที่ 2 ถึง X สามารถแทนได้ ถ้าเราคิดว่า 2 กำลัง X เป็นฟังก์ชันเอ็กซ์โปเนนเชียล เอ็กซ์โปเนนเชียลในแง่ของคุณสมบัตินามธรรมพวกนี้ และถ้าเราใช่ ถ้าเราถ้า เรามีคลาสของฟังก์ชันที่แตกต่างกัน และเราต้องการเสียบ pi มันทำให้ฉันหัวเราะ เพียงเพราะมันเป็นคำตอบตลกๆ ที่ปรากฏขึ้นเมื่อคุณพยายามคิด ดังนั้นคำถามเหล่านั้นก็คือ ฉันจะทิ้งคุณไว้และฉันคิดว่านี่คือคุณคงรู้ดีว่าคำถามหลักของฉันในการเข้าสู่การบรรยายวันนี้คือฉันอยากให้มันเป็นชนิดที่อธิบายเหมือนคุณสมบัติเชิงนามธรรมของฟังก์ชันเอ็กซ์โปเนนเชียลหรือไม่ และมันก็เจ๋งสำหรับฉันที่เริ่มจากคุณสมบัติเชิงนามธรรมเหล่านั้น คุณถูกขังอยู่ในแนวคิดของ e กำลัง rx หรือมากกว่า แค่คุณรู้ ฉันคิดว่า exp ของ r ที่เขียนอย่างตรงไปตรงมามากกว่า x สำหรับค่าต่างๆ ของ r ที่มันล็อคคุณไว้ไกลขนาดนั้น แต่มันไม่ได้ล็อคคุณไว้เท่าที่มี แนวคิดที่ชัดเจนว่า 2 ยกกำลัง x ควรน้อยกว่ามาก เช่น ยกกำลัง x ความเสี่ยงแน่นอนก็คือ บางครั้งคนไม่ชอบสิ่งที่เป็นนามธรรม และบางครั้งก็ไม่ได้ดูเหมือนเข้าถึงได้ แต่ถ้าเป็นอย่างนั้น ในกรณีที่คุณรู้ว่าคุณแค่บอกฉัน ฉันคิดว่า ฉันคิดว่ามีวงกลมความคิดที่น่าสนใจล้อมรอบ ทั้งหมดนี้รวมไปถึงหอคอยพลังงานด้วย เพราะถ้าคุณต้องการพูดคุยเกี่ยวกับหอคอยพลังงานจริง ๆ เหมือนเราครั้งที่แล้วในบริบทของจำนวนเชิงซ้อน หรือแม้แต่ฐานลบ คุณต้องคิดทบทวนเรื่องแบบนี้ จึงเป็นคำถามที่เราเจอบนหน้าจอ ใช่ จะเกิดอะไรขึ้นถ้าเราทำสิ่งนี้เพื่อฉันกับผู้มีอำนาจ? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/turkish/sentence_translations.json b/2020/ldm-i-to-i/turkish/sentence_translations.json index 411e79542..5ca8dbb2d 100644 --- a/2020/ldm-i-to-i/turkish/sentence_translations.json +++ b/2020/ldm-i-to-i/turkish/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "Yani 1 numaradan başlıyorsanız, ilk hızınız düz olarak 0'a doğru yürümek olacaktır ve daha da aşağıya doğru yürüdüğünüzde, eğer 1 numarada oturuyor olsaydınız o zaman hala 0'a doğru yürüyor olurdunuz, ama şimdi hız vektörünüz Bulunduğunuz yerin 1 katı eksi olur, yani eksi 1 yarısı. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "Ve ilginç bir soru şu olacak, bunun için yazmak makul hissettiren böyle bir fonksiyon var mı, çünkü bunu i üzeri x olarak yazacaksak, sadece bunu karşılaması değil, aynı zamanda ne zaman tatmin etmesi gerektiğini de biliyorsun. Aldığımız i sayısını muhtemelen i'nin üssüne takıyoruz ancak bu fonksiyonun i olması gerektiğini düşünüyoruz. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "Yani elimizde 5 pi i yarı büyük var, bu da kesinlikle x'i buraya koyabileceğimiz başka bir değer ve bunu biraz daha görsel olarak ifade etmek için, eğer geriye dönüp burada bulunduğumuz çembere bakarsak, Moment pi'nin yarısına eşit olan bir süre boyunca yürüdü, yani 1.57 bunun yerine başka bir tam dönüş yapsaydık ve pi'ye ulaşmak için başka bir pi yarısı daha gitseydik ki bunu biliyorsunuz ki bir tür kayıt yapabiliriz e üzeri pi i değeri burada başka bir pi yarısı yürürsek başka bir pi yarısı yürürüz ki bu da bu noktada tam bir daire çizerek bire geri dönerdik ve sonra sayısal olarak yaklaşık 7 olan beş pi yarısı boyunca yürürdük. 85 evet, bu kesinlikle bizi i'nin tepesine çıkaran başka bir sayıdır ve eğer i'nin üssü i'yi yeniden ifade etmenin tüm saçmalıklarını ilk önce e üzeri 5 pi'nin yarısı i üzeri i'yi yazarak tekrar ifade edersek negatif olmak için çarpın ve e üzeri negatif 5 pi'nin yarısına bakıyor olacağız ki bu çok farklı bir sayı değil mi bunu aslında hesaplayabiliriz, tam emin değilim ama hadi bir Desmos'a bakalım . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "Bu kadar uzun, bu sizi çok daha küçük bir sayıya götürür Ama girebileceğimiz tek cevap bu değil, buraya negatif 3 yarım çarpı i pi ile gelen başka insanlar da var. Bunu birim çember cinsinden biliyorsunuz? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "Şöyle demeyi düşünebiliriz, eğer I'e ulaşmak istersem 90 derece yürümek yerine pi yarım radyan bu şekilde ya 270 derece diğer yönde yürürsem 3 pi yarım radyan olur ki belki de bu negatif olarak düşünebilirim çünkü gelenek şöyledir genellikle saat yönünün tersine pozitiftir Bu kesinlikle bunu ifade etmenin başka bir yoludur ve eğer e üzeri eksi 3 pi yarısı i olsaydı bu bize farklı bir cevap verirdi i Tümü kuvvet i aynı oyunu oynuyoruz şimdi i kare a ile sadeleşiyor negatif bu zaten orada ve elimizde pozitif 3 pi yarım var ve sayısal olarak bu bize daha önce sahip olduğumuzdan daha da farklı görünen bir cevap veriyor. Bunun üzerine gidip hey dersek, e üzeri 3 pi nedir, 3 o 3 pi değil yarımlar 111 virgül 3 1 daha önce gördüğümüzden çok farklı türde bir sayı 111 virgül neydi o 111 virgül 3 1 harika 111 virgül 3 1 falan Ve yine sezgi açısından sorabileceğiniz şey şu: diyelim ki elimizde bu dönen bir şey var dinamik Ama zamanda geriye doğru gidiyoruz, zamanda ne kadar zaman önce ne olmam gerektiğini görüyoruz Öyle ki, eğer işleri oradan ileriye doğru oynasaydım, başlangıç durumum olan bir numaraya inerdim ve zamanda geriye gitmeniz gerekirdi 3 pi yarım birim Ve sonra, bozunma dinamiklerini tercüme edecek olsaydınız, bu bağlamda göze yükseltmenin yaptığı şey, bir numaradan mı başlıyorum dersiniz. Ama zamanda geriye doğru hareket etmek istiyorum ve şunu söylemek istiyorum: Nereden başlamalıydım? Bir numaraya kadar düşmek mi istiyorum? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "3 pi yarım birim zaman süresinden sonra, bu tür üstel bozunma için cevap açıkça yüz on bir civarında başlıyor. Ve bunun nereye gittiğini görebilirsiniz, aslında burada X'i yerine koyabileceğimiz sonsuz sayıda farklı değer var. e üzeri X'i ben olarak düşünüyorum ve insanlar buraya çok daha fazla girdiler Kusura bakmayın üçüncü sırayı almak için klasik olarak iğnemi yere fırlatıyorum 9 pi yarıları harika seçim 1729 pi yarıları hepiniz benim favorimsiniz çok çok farklı seçenekler sonsuz birçok farklı değer bu biraz rahatsız edici geliyor ilk başta rahatsız edici çünkü bir ifadeye bakıyoruz Sanki bir hesaplama olacağını biliyormuşsunuz gibi görünüyor Bunu hesap makineme taktım ve neyin ortaya çıktığını görüyorum ve elimizde birden fazla farklı ifade var bunun için değerler Peki burada neler oluyor değil mi? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "16'nın dördüncü kökü 2 olmalıdır ve cevap iyi sonuçlanır Çok değerli bir fonksiyonunuz olduğunda bunun gibi birden fazla seçenek olduğunda bir kural benimseriz. Genellikle istediğimiz zaman kastettiğimiz şey için bu değerlerden birini seçeriz. Daha süslü bir dilde bunu tek girdisi ve tek çıktısı olan bir fonksiyon olarak ele alın Karmaşık sayılarla uğraştığımızda bu her zaman ortaya çıkar, bir şeyin bir işlem türü olduğu fikri bazen birden fazla değere sahip olmayı istemektir. dal ifadesini duyun Karekök fonksiyonunun dalını nerede seçersiniz? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "Çünkü birden fazla farklı cevap var Biliyorsunuz, ben yine bu 90 derecelik dönüş olduğunu düşünüyoruz Ve bunu 90 derecelik bir dönüş olarak düşünürsek, karekökün olması gerektiği gibi geliyor. 45 derecelik bir açıyla oturan bir şey biliyorsunuz Belki bu karedir kök I bunu çok açık bir şekilde kök 2 bölü 2 kök 2 bölü 2 I şeklinde yazabiliriz. Bu sadece trigonometri kullanıyor ama I'yi 270 derecelik negatif bir dönüş olarak düşünürsek, bunun yarısı bu işlemin yarısını yapıyormuş gibi geliyor aslında bizi diğer tarafa götürmeli Belki de burada oturan sayı I'in karekökü olmalıdır ve bu aslında daha önce gördüğümüzün negatifidir Negatif kök 2 bölü 2 eksi kök 2 bölü 2 çarpı I Şimdi reel bağlamı içinde değerli fonksiyonlara evet diyebiliriz. Olumlu cevap ne olursa olsun karekökü seçin ama bunlardan hangisini olumlu cevap olarak değerlendiriyorsunuz? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "Ve sanırım iyi diyorsunuz. Bunun ne olduğunu biliyoruz, bunu 2'nin karekökü olarak tanımlıyoruz, her şey iyi ve iyi. Peki ya buna, I üzeri I ifadesine yaklaştığımız gibi yaklaşalım desem? İlk önce şeyleri e üzeri doğru olarak ifade etmek istiyorum ve sonra bunu 1 yarıyı üsle çarparak 1 yarıya çıkaracağım. Ve tamam diyorum, sanırım bunu e üzeri ne yapabilirim diyorum. 2'ye eşit yani bu 2'nin doğal logaritması. 0 civarında bir sabit. 69 ya da öylesine e üzerini bu kuvvete yükseltirsek 2 elde ederiz, yani bunu e üzeri 2 çarpı 1 yarımın doğal logaritması olarak düşünebiliriz ve eğer e üzeri x'i düşünüyor olsaydınız? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "Gerçek sayılar bağlamında bunun biraz abartı olabileceğini biliyorsunuz ama e üzeri x'i bu x fonksiyonunun kısaltması olarak düşünüyorsanız 0 değerini koyabilirsiniz. 69 çarpı 1 yarım ki sanırım 0 civarında olur. 345 Buna benzer bir şey mi? Bu çok somut değeri polinomunuza eklersiniz, çıktısının ne olduğunu görün ve çıktısı 1 civarında olacaktır. 414 a Güzel bir karekök 2 gerçek sayısı, beklediğiniz şey. Ama az önce I ile yaptığımızın aynısını yaparsak ve aslında birden fazla farklı yanıtın olduğunu kabul ederek, bir şeyi e üzeri bir kuvvet olarak yazmak istediğimizde bunu da yazabiliriz. Bu komik görünebilir, ama bunu e üzeri 2 artı 2 pi I'nin doğal logaritması olarak yazabiliriz. Bütün bunlar 1 yarıya yükseltilirse, bu değer size eşit olacak, bunu e üzeri olarak parçalayabiliriz. 2'nin doğal logaritması çarpı e üzeri 2 pi I Bu sadece nesneleri 360 derece döndürme etkisine sahip, yani sadece 1'e eşit olacak Yani 2 çarpı 1 harikaya bakıyoruz, bu geçerli bir ikame gibi geliyor ve yine de ne zaman? Aynı oyunu oynuyoruz: Bunu alıp bir kuvvete yükseltiyoruz ve kuvveti üsle çarparak buna davranıyoruz, ne olduğuna bakın Elimizde e üzeri 2 çarpı 1 yarımın doğal logaritması var Peki, 2 pi I çarpı 1 yarım nedir peki bu pi çarpı I olacak Şimdi bu ilk kısım e üzeri 2 çarpı 1 yarımın doğal logaritması, bu da 2'nin tanıdık Karekökü olacak, hepsi iyi ve güzel, ama bunu e üzeri ile çarpacağız. pi I Doğru ve meşhur e üzeri pi I negatif 1 Yani bu durumda, eğer bu ifadeyi 2 üzeri 1 yarımı çözüyorsak, farklı cevaplarla oynayarak şunun gibi bir şey için yerine koyabileceğimizi öneriyor gibi görünüyor e üzeri X eşittir 1 yarım, elde ettiğimiz sonuç, geleneksel olarak bu negatif karekök 2 olarak yazabileceğimiz başka bir cevaptır ve Burada demek istediğim, 2 üzeri 1 yarımına bakmak için birden fazla değere sahip olmak biraz komik ve bunun bir şeye eşit olmadığını söyleyelim ama yaptığımız seçimlere göre birden fazla farklı şeye eşit olabilir. Ama iki şey oldukça makul görünebilir. Eğer 2 üzeri 1 yarım olan bir şey varsa, bu da ikisinden biri olmalı gibi görünüyor Pozitif Aşina olduğumuz karekök veya bunun negatif varyantı, aslında o kadar da sorun gibi görünmüyor Ve aslında bu oyunu daha da ileri düzeyde oynayabiliriz, burada sizden Bu ifadeye daha yaratıcı cevaplar istememe izin verin çünkü eğer I üzerini değerlendirirken kullandığımız kuralların aynısına uyuyorsak, yapacağımız ikameye bağlı olarak X'in çeşitli farklı değerlerini yerine koymaya başladığımızda, belki 2 üzeri X gibi bir şeyin başka komik kuvvetlerini de bulabiliriz. üs I Yani bu sefer soru şunu soruyor veya e üzeri x eşittir 2 denkleminin bir çözümünün 2'nin doğal logaritması gerçek sayısı olduğunu biliyoruz, tamam bunu biliyoruz. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "e üzeri x eşittir 2 sorusunun cevabı Ve yine yaratıcılık memnuniyetle karşılanıyor, bu yüzden size bunun için küçük bir dakika daha vereceğim II Devam edip bazı cevapları buraya kilitleyeceğim, eğer sizin için uygunsa, ne kadar zaman harcadığından emin değilim Hangi cihaza baktığınıza bağlı olarak mutlaka matematik girişi yapmanız gerekir, ancak cevaplamak istediğiniz cevaba istediğiniz soruyu girme şansınız olmadan önce çok fazla strese girmeyin. 131'iniz Ln of 2'yi aldığımız ve 2ii'yi eklediğimiz varyanta girdiniz ve sanırım bu soruyu yazıyorum. Aslında pek çok farklı doğru cevap varken yanlışlıkla cevaplardan birini doğru olarak işaretlemiş gibiyim. Yani bu bana düşüyor. çünkü herhangi birinize "oh" gibi görünüp görünmediğini bilmiyorum. Kırmızı, 2 artı 42'nin Ln'sini girerken yanlış anladınız. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi ki bu elbette harika bir seçim. Ama aynı zamanda 4 pi I artı 2 veya 6 pi I'nin doğal logaritması gibi bir şeye de sahip olabilirsiniz. Veya gerçekten 2 pi I'nin herhangi bir tamsayı katı, eğer bunun e üzerini etkilemediğini eklerseniz. X Çünkü sadece e üzeri 2 pi I ile çarpma etkisi var. Bu da 1 ile çarpmanın etkisi ve yine bunun komik bir sonucu var, başka bir örnek olarak bunu yaptığımızda makul sonuçlar veriyor gibi görünüyor. Görünüşe göre en sık girilen ikinci ifade 2'nin yerine koyabileceğimizdi. Yani 2 üssü 1 4'ü düşündüğümüzü düşünelim, tamam 2'yi e üzeri 2 artı 4'ün doğal logaritması ile değiştirmemiz yönünde bir öneri vardı. pi I Tamam Artı 4 pi I ve bunların hepsini 1 4'e yükseltiyoruz, peki aynı oyunu oynasaydınız, e üzeri 2 çarpı 1 4'ün doğal logaritmasını elde ederdik ve e ile çarpardık. pi I Şimdi bunun ilk kısmı olağan pozitif 2'nin dördüncü kökü olacak, bu da 2'nin dördüncü kökü gibi bir ifadeyi hesap makinesine koyduğunuzda kastettiğimiz şey güzel, küçük bir Pozitif sayı, ama sonra bu ikinci kısım negatif 1 yani şöyle diyor gibi görünüyor. Biliyorsunuz, 2'yi bu farklı şekilde yorumlayıp 1 4'e yükseltirsek, bunun aldığımız olağan cevap olmadığını biliyorsunuz ama makul bir cevap. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "Pi'nin yarısı çarpı I'e bakıyor olurduk ve Negatif 1 ile çarpmak yerine I ile çarpıyor olurduk. Bu da yine geçerli bir cevap, 2 üzeri 1 4 gibi bir şey için makul bir çıktı gibi görünüyor. I üssü I'in bunun için birçok farklı değeri var gibi göründüğü gerçeğine baktığımızda, e'yi 5 pi yarısına I Negatif 3 pi yarısına bağlayabildiğimiz komik bir olguya sahibiz ve çılgınca farklı gibi görünen yanıtlar alıyoruz süper küçük bir şey süper büyük bir şey, hepsi daha önce burada bulduğumuz 1 5'inci yaklaşık 1 5'inci cevaptan çok farklı Bu, 2 üzeri 1 4'ün ne olduğunu sormanız ve aslında birden fazla farklı çözüm olduğunu kabul etmenizle tamamen aynı fenomendir. X üzeri 4 ifadesi 2'ye eşit aslında 4 farklı çözüm ve baktığınız şey birden fazla farklı çözümün olduğu gerçeği. e üzeri X ifadesi bir tür tabana eşittir bu taban I olup olmadığı bu taban olup olmadığı 2 Her ne olursa olsun ve bunu düşünmemizin bir yolu da şudur: Gerçek sayılarla uğraşırken her şey çok güzeldir, her şey güzeldir. Bire bir ilişkiler vardır. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "Bu harika. Eğer üstel fonksiyonlar hakkında düşünmek istiyorsak, bazı şeyleri özetleyeyim. Herhangi bir üstel sayıyı X'e taban olarak ifade etmeyi seçebileceğiniz, 2 üzeri X gibi güzel bir ileri geri hareketimiz var. X üzeri R çarpı X ile aynı üstel, bildiğiniz gibi bu bizim bahsettiğimiz polinomdur Ne zaman e üzeri X gibi bir şey yazsak örtülü olarak bahsettiğimiz polinom Ve çok güzel bir ileri geri var çünkü B'nin doğal logaritmasını alabilirsiniz Ve bu, B'nin pozitif bir sayı olduğunu varsayarak size bir cevap verir. Ve bu, X(R)'nin B'ye eşit olduğunu söylemekle aynı şeydir. Bu konuda daha önce seride bahsettiğim yollardan biri şu: tüm olası Üstellerin ailesi, evet bunları X üzeri R çarpı X olarak yazabilir ve R'nin ne olduğunu değiştirebiliriz. Ve bu, e üzeri R çarpı X yazmakla tamamen aynı şeydir, eğer bu daha rahat edeceğiniz bir şeyse, yani e üzeri R çarpı XX çarpı R çarpı X bunlar, bunun ne olduğunu değiştirme konusunda düşünebileceğimiz şeylerle aynı. Ama diğer yandan, olası tüm üstel sayıları bir taban olarak düşünecek olursanız, X'in kuvvetine göre bir taban yapayım ve gidiyoruz bu tabanın ne olduğunu değiştirmek İlk başta bu, manipüle edilecek farklı bir ifade türü gibi geliyor, ancak bu sadece aynı aileyi ifade etmenin başka bir yolu Ve bunun hakkında düşünebileceğiniz bir yol Hangi tabana karşılık geldiği hakkında nasıl düşünüyoruz? Eğer biraz daha soyut olarak Exp R çarpı X şeklinde düşünüyorsak ve bunu yapmamın bir nedeni varsa, çünkü bunu daha tuhaf görünecek karmaşık sayılara uygulamak üzereyiz, bu yüzden benimle buradan devam edin. Bu temele bakmak yerine yapabileceğim tek şey değerin ne olduğunu söylemek olabilir. ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "R çarpı X'in exp'sini alabilirdim, burada belki R sıfır virgül altı dokuz gibi bir şeydir Ama bunu iki pi I kadar aşağı kaydırabilirim Ve bu, bunun karşılık geleceği tabanı değiştirmez, bunun hâlâ ikiye karşılık gelmesi olabilir Veya olabilir onu iki pi I kadar yukarı kaydırırız, bu da karşılık geldiği tabanı değiştirmez çünkü tüm bu durumlarda X eşittir bir'i yerine koyduğumuzda aynı şeyi elde ederiz ancak bunların hepsi X'in farklı değerleri için farklı fonksiyonlardır. neden I üzeri kuvvet I için birden fazla farklı değer gördük Çünkü I üzeri X belirsiz bir fonksiyondur bu bağlamda, eğer R'nin hangi değerine karar verirsek açık olacaktır. Öyle ki temsil ettiğimiz şey exp R çarpı X hangi değerdir R.'nin Birini seçer seçmez mi seçiyoruz? ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "Bu kesin bir fonksiyon ama bu noktada sanki istediğimiz şey sanki bazı şeyleri X kuvvetine yükseltilmiş bazı tabanlar açısından düşünmeyi bırakmakmış gibi geliyor. Belki karmaşık sayılar bağlamına girer girmez sadece yazmalıyız bunların hepsi bazı sabit çarpı X'in ifadesi olarak, eğer başka bir nedenden dolayı açıkça ortaya çıkmıyorsa, sayıları gerçekte nasıl yerine koyarız, eğer bir hesaplama yapmak istiyorsak ya da sadece bunun üzerine sadece matematik yapmak istiyorsak, elimizde şu güzel sonsuz polinom var: bunları takın ve sizin için bunun üstel sayılar hakkında düşünmenin doğru yolu olabileceğine dair başka bir örnek sunacağım. Karmaşık sayılar gibi şeyleri diğer alanlara genişlettiğimizde ve bunun için haydi sadece yedekleyelim Go kapı ziline geri dönelim, bazı şeyler geldi orijinaline geri dönüyoruz. Üstel alma fikrini genişletiyoruz ve sadece 2 üzeri X'in ne olduğunu düşünüyoruz. Doğru, bunu doğal sayılar için nasıl düşüneceğimizi biliyoruz. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "2 üzeri 3 gibi bir şeyi biliyorsunuz. Tekrarlanan çarpma Nasıl oldu da size ilk önce kesirli miktarlar için 2 üzeri X gibi bir şey veya negatif miktarlar ve bunun gibi şeyler öğretildi? ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "Genellikle size 2 üzeri 1 yarının bir şey olması gerektiği öğretilir, eğer onu kendisi ile çarparsam bunu bilirsiniz ve bu, Üstellerin sayıları sayarken yaptığı olağan kuralları takip eder, burada o üsse bir şeyler ekleyebiliriz, 2 elde etmeliyim 1'e göre bir sayı olmalı, kendisiyle çarptığım zaman 2 elde ediyorum ve o noktada bir seçeneğiniz olduğunu biliyorsunuz, belki de pozitiftir. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "Belki negatiftir Ama her zaman pozitif seçimi yapmaya karar verirseniz, negatif sayılar hakkında soru sorarsak, aynı işlemden güzel bir sürekli fonksiyon elde edebileceksiniz. 2 üzeri negatif 1 ne olmalı, bu bir şey olmalı bunu 2 üzeri 1 ile çarptığım zaman nerede? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "Bu bana 2 üzeri 0'ı getiriyor ve bu, negatif üslerin 1 yarım gibi görünmesine ilişkin uzlaşımımızın bir tür gerekçesi. Ancak burada gerçekte olan şu ki, bu her ne ise, f'nin bu özelliğini karşılayan bir tür fonksiyon olması gerektiğini söylüyoruz. a artı b eşittir f a çarpı f b ve Üstelik tabanın 2 olduğu gerçeği bize bunun herhangi bir fonksiyon olmadığını söylüyor. 1'i yerine koyduğumuzda 2 elde ettiğimiz bir fonksiyon. Buradaki bazı çıkarımlarla birlikte takip edip etmediğinizi görmek için akıl sağlığı kontrolü tarzı soru Size ne olduğunu sormak istiyorum, buna softball gibi demeyeceğim, ama bu böyle olması anlamına gelmiyor İnanılmaz derecede derin bir soru mutlaka. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "Bu, soyut olarak bir fonksiyonun özellikleriyle başlama ve ardından bu özelliklere dayanarak bunu yazmak isteyebileceğimiz yolları çıkarma fikri ile birlikte hareket edip etmediğinizi kontrol etmek gibi bir şey. Eğer f(x) bu üstel özellik f'yi sağlıyorsa a artı b eşittir f a çarpı f b tüm girdiler için Ve aynı zamanda f (1 eşittir 2) aşağıdakilerden hangisinin doğru olduğunu da karşılar Bu, aşağıdakilerden hangisinin zorunlu olarak doğru olduğu anlamına gelir Hangi fonksiyonu başlatırsanız başlatın hatırlayanlar hangi dersti Euler formülünün gerçekte ne söylediğini nasıl yorumlayacağımızı konuşuyorduk. Tek bir koşulu ihmal ettiğim bu tarz bir soru sordum, biliyorsunuz yazmadım. f(x)'in her yerde sıfırdan farklı olduğundan emin olmak istememiz ve bunun bir miktar Karışıklığa neden olması ki bu da ekranda hepimizin başına gelen bir kafa karışıklığı yaratıyor. Ancak bunun amacı temel olarak f'nin bu soyut özelliğinin şunu göstermekti: Toplamayı çarpmaya dönüştüren bir şey Temel olarak fonksiyonu, bir çeşit güce yükseltilmiş olarak, neye eşitse o şekilde yazmak istemenizi sağlamak için yeterlidir. Sorunun ruhu budur. Şimdi aslında güç kuleleri hakkında birkaç sorumuz var. burada ortaya çıkmış gibi görünüyor ki bu geçen seferkiyle harika bir bağlantı. Güç kulesi sorusuna bir anlığına ara verelim, böylece önce şunun gibi daha derin bir fikir edinebiliriz: Burada üstel alma ne anlama gelmeli? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "Çünkü iddia etmek istediğim şey olabileceğimiz için, buna birçok farklı şekilde cevap verebiliriz. Yani bana sadece bir tane verirseniz, güç kuleleri hakkında konuşuruz. Ve sonra tıpkı bir sayı doğrusunun logaritmik ölçekte temsil edilebilmesi gibi. aynı şey karmaşık bir düzlem için de yapılabilir mi? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "Evet Aslında, birazdan buna oldukça benzer bir şey yapacağımız bir görselleştirme var. Çünkü yapacağımız şey farklı üstel fonksiyonlar X (R çarpı X) ile oynamak. Küçük sarı bir nokta ile temsil edilecek olan R'nin değerini değiştireceğiz Yani bunun üzerinden konuşacağız. Tüm düzlemi haritalandırmayacağız, sadece gerçek eksenden ve sanal eksenden birkaç örnek noktayı haritalandıracağız Ama fikir şu ki, bu sabitin ne olduğu etrafında hareket ettikçe, bunun düzleme yaptığı farklı şeyleri bir nevi görselleştirebileceğiz ve bu aslında x eksenini logaritmik ölçeğe çevirmek ve sonra sarmak gibi bir şey. bir daire boyunca hayali eksen Ve sonra R'nin değeri hayali hale gelir gelmez, çemberin üzerine konulan Gerçek sayıların ve sanal sayıların logaritmik ölçekli Pozitif eksenin üzerine konulan rollerini değiştirir, o kadar harika bir soru ki bunların üçü de sanırım Gitmek istediğim yere gitmek için bir nevi acele ediyorum ama bu sefer insanların böyle düşündüğünü görmek güzel. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "Açıkça f/5 gibi bir şey f (1 artı 1 artı 1 artı 1 artı 1) ile aynı şeydir. Bu da f (1)'in 5 kez kendisiyle çarpılmasıyla aynı şeydir çünkü bu özellikten dolayı f (1) 2 ise aynı olur 2 üzeri 5 ve ardından f(negatif) 5 gibi bir şey. Bunu f(5) ile çarptığımızda f(0) ne olursa olsun elde ederiz ve f(0)'ın ne olduğu hemen belli olmaz ama şunu söyleyebiliriz: f (1 artı 0) eşittir ne varsa f (1) çarpı f (0) ama f (1) eşittir 2 Ve bu da 2'ye eşit yani 2'nin 2 çarpı bir şeye eşit olduğunu söylüyoruz yani bir şey 1 olması gerekiyor, yani bu bağlamda bu f (eksi 5)'in 2 üzeri eksi 5 olduğunu garanti eder, bu da 1 bölü 2 üzeri 5'tir. Bunu açıkça 2 üzeri eksi 5 olarak yazabiliriz, yani bu iki özellik birlikte şunu ifade eder: Biz aslında fonksiyonu 2 üzeri X olarak yazmak istiyoruz. Çünkü içine koyduğumuz herhangi bir sayma sayısı şunu sağlayacak: Bu sayıyı kendisi ile çarpmak gibi görünecek, içine koyduğumuz herhangi bir kesirli sayı bu özellikleri sağlayacak. istediğimiz şey Ve bunun benzersiz olup olmadığını merak edebilirsiniz ve gerçek değerli fonksiyonlar bağlamında aslında öyle olur. Ama karmaşık değerli fonksiyonlar bağlamında bu tür birden fazla f fonksiyonu olabilir ve bunun için yazabiliriz ki bunlardan biri de buydu. 2 artı 2 pi'nin doğal logaritması olarak tanımlanan bir fonksiyonun nerede olabileceğine bakıyorum I tüm bu zamanlar X Tamam, buradaki özensizliği affedin, sadece bunun hakkında yazarken heyecanlanıyorum Ve bu aslında farklı bir fonksiyon X eşittir 1 yarımı yerine koyarsanız ne olacağını biraz önce gördük. 1 yarımı yerine koyarsanız 2'nin negatif karekökünü elde edersiniz ve sonra dörtte birini yerine koyarsanız dördüncü kökü elde edemezsiniz. 2 ama I çarpı 2'nin dördüncü kökü yani bu farklı bir fonksiyon Ama yine de bu özellikleri sağlıyor ve bu bize onu 2 üzeri X olarak yazmak istememize neden oluyor. Ve bu da 2 üzeri X'in belirsiz bir sayı olabileceğini akla getiriyor biraz gösterim Ve her şeyi exp R çarpı bir şey cinsinden yazmalıyız ama merak edebilirsiniz Belki bu özelliği sağlayan tüm fonksiyonlar konusunda yeterince yaratıcı değilizdir Belki exp yazarken bir belirsizlik vardır R çarpı bir şeyin ve R'nin farklı değerleri devreye girebilir. Ama ben sadece küçük bir iddia ortaya koyacağım ve sonra belki isterseniz kanıtın nasıl görüneceğine dair bir taslak sunacağım. Diyelim ki karmaşık bir F fonksiyonuna sahipsiniz ve bu fonksiyon öncelikle aşağıdaki özellikleri sağlıyor. Bunun bir türevini alabiliyorsunuz. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "Diferansiyellenebilir, bu da onun bildiğiniz tamamen dağınık, süreksiz bir şey olmasını engelliyor. Bu, hangi vektör uzayının yayılımını bildiğinize bağlı olarak bazı rastgele değerler almak gibi. Çılgın şekillerde düşünmek isteyebileceğiniz kesirli miktarları bilmiyorum. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "Güzel bir fonksiyon. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "Bu diferansiyellenebilir Her yerde 0'a eşit değil yani bu durum aklımdan uçup gitti ve hangi ders için ders verdiğimi ya da buna benzer bir şeyi unuttum ve sonra toplamayı çarpmaya çeviren merkezi bir özelliği var. Eğer böyle bir fonksiyonunuz varsa iddia ediyorum ki benzersiz bir şey var belki de gerçekten belirtmeliyim ki benzersiz bir R Karmaşık sayısı var, böylece F/X'i temel olarak R'nin bu üstel fonksiyonu çarpı X değeri olarak yazabilirsiniz. güzel türev özelliklerine sahip sonsuz polinom ve bunların hepsi, eğer buna sahipseniz, üstel kelimesinin çok benzer soyut genel anlamında istediğiniz her üstel, sadece ondan isteyebileceğimiz bir özelliğe dayalıdır ve ispat taslağı Öncelikle her yerde var olduğunu varsaydığımız bu değerin türevinin ne olduğuna bakmak istiyorsanız şöyle bir şeye bakın, değil mi? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "F/X'i ifadenin tamamen dışında tutabiliriz ve limitin tamamı yalnızca H cinsinden ifade edilir. Türevler bağlamında bunun ne anlama geldiğini ve F/0'ın zorunlu olarak 1'e eşit olduğu gerçeğini düşünürseniz, bu sınırlayıcı ifadenin tamamı sadece bir sabit ama daha spesifik olarak fonksiyonumuzun 0'daki türevi ne olursa olsun. Yani, eğer 0'daki türevini biliyorsanız, bu onun her yerde türevinin ne olduğunu belirlerse, bu komik bir şeyle karşı karşıyasınız. Üstel fonksiyonlar bağlamında bu umarım oldukça tanıdıktır çünkü aslında söylediğimiz tek şey üstel bir fonksiyonun türevinin kendisiyle orantılı olduğu ve orantı sabitinin 0'daki türevi ne olursa olsun ona eşit olduğu, bunların hepsi çok soyut bir şekilde ifade edilmiş ve böyle ama bunun amacı bunun olduğunu vurgulamak. ille de sadece zaten a üssü X olarak düşündüğümüz fonksiyonlar değil. Ancak toplamayı çarpmaya dönüştürme şeklindeki bu soyut özelliği karşılayan, potansiyel olarak çok daha geniş bir fonksiyonlar sınıfıdır. ikinci türev Ve bu konuda üçüncü bir türev ve böyle çünkü türev fonksiyonu kendisiyle orantılıdır Yani n'inci türevi almak için sadece bu orantı sabitine bakarsınız ve onu n kuvvetine yükseltirsiniz ve sonra buradan şunu yapabilirsiniz: Taylor serilerinin genişletilmesi ve bu fikirde Taylor serileri konusunda rahat olanlarınız için bunu ileri düzey bir ödev olarak bırakabilirim, özellikle de karmaşık sayılar anlamında diferansiyellenebilir herhangi bir Diferansiyellenebilir fonksiyon fikrini birbirine karıştırmak istiyorsanız, bir nevi kesinlikle üniversite konusu Biliyorsunuz, buradaki mantığı istediğiniz gibi karıştırabilirsiniz. Ancak bulanık akıl yürütmeye, yalnızca Taylor serileri hakkında bilgisi olan ve bu fikri alıp F ve için Taylor açılımına bakacak başka hiçbir şey bilmeyen biri bağlamında izin verilir. Bu, F fonksiyonumuzun zorunlu olarak bu şekilde yazılabileceği benzersiz bir karmaşık sayı olduğu fikrini haklı çıkarıyor. Ve sonra normal üstellerle bağlantı, böyle bir R değerine sahip olduğunuzda olur. Esasen, gerçek sayıların karmaşık bağlamında yaptığımız şeyi yaparız. bu R değerinin fonksiyonunun exp'sine bakarsanız ve bunu taban olarak yazarsanız. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "Bunu sadece pi yarılarının exp'si I çarpı X olarak değil, aynı zamanda 5 pi yarının exp'si I Çarpı X olarak da yorumlayabiliriz ve Bunlar ayrı fonksiyonlardır. bunları I üzeri X olarak yazın Yani, I üzeri I ifadesi, bunun zorunlu olarak ne anlama geleceğine dair bir standart benimsemediğiniz sürece, sonsuz sayıda çıktısı olduğunu söylediğinizde, bunu düşünmenin başka bir yolu şudur: I fonksiyonu üzeri X sahip olduğumuz notasyon biraz belirsiz Şimdi tüm bunlarla birlikte hadi bunların bir kısmını görselleştirmeye başlayalım çünkü bence bu eğlenceli Ve bunun yararlı bir görsel mi yoksa daha kafa karıştırıcı bir görsel mi olduğunu bana söyleyeceğinizi biliyorsunuz ama Yapacağımız şey, bu fonksiyona exp R çarpı X'e bakmak, ki bu Temelde bu, e üzeri X'in kuvvetini yazmanın başka bir yoludur, aslında sanırım bunu belirten bir noktada farklı bir animasyon oluşturdum. çünkü bunu yapmayı planlıyordum o yüzden izin ver bana ah evet işte, dosya sistemime geri dön, olman gereken yere geri dön, içeri gir, şikayet mi ediyor çünkü birden fazla farklı var. Ah değiştir diğer ekranda görünüyor Bekle neden evet, tamam değiştir? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "Gördüğünüz her şeyi oraya yerleştirin Ve şimdi geri dönüyoruz, işte buradayız, bunların hepsi güzelce yazabilmem için. Eğer bunu exp R çarpı X bu sonsuz polinom olarak düşünmekten rahatsızsanız. başınızın arkası e üzeri R çarpı X ve R etrafında değişeceğiz yani hayali eksenin noktalarını takip edeceğim ve gerçek eksenin noktalarını takip edeceğim ve bakalım bu ne yapacak? bunların hepsi çok hızlı bu yüzden biraz daha yavaş düşüneyim tüm negatif sayılar herhangi bir şey Bu negatif bir reel sayı 0 ile 1 aralığına sıkıştırılacak Hangisi e'nin negatife göre anlamlı olması gerekir? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a'nın negatif bir gerçek sayıya oranı 0 ile 1 arasında bir şeydir ve özellikle f'nin negatif 1'ini izliyoruz, bu da 1 bölü e'nin 30 0 civarında olduğu herhangi bir şey civarında ortaya çıkacak. 37 f/1, beklendiği gibi e'ye iniyor, bu, exp 1'in f'si, birim çemberin etrafına bir radyan indireceğim ve burada hayali eksenin bir daire etrafına nasıl sarıldığını tüm hayali eksen boyunca takip etmek oldukça eğlenceli. ve R'nin bu değerini değiştirdiğimizde ne olur? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "Burada R'nin değerlerini isteyebiliriz. Her şeyi farklı bir şekilde uzatır, yani 2'ye kadar koyduğumuzda, reel ekseni çok daha fazla uzattığını biliyorsunuz, böylece f/1, e karenin biraz üzerinde olduğu yerde biter, f negatif. 1, 0'a çok daha yakındır f I, 2 radyandır Negatif I çemberi etrafındaki dönme, negatif 2 radyanlık dönmedir. Ve tabii ki en sevdiğimiz formüle de ulaşabiliriz; eğer bu bizim ölçeklendirme sabitimiz olan pi ise O zaman gerçek eksen oldukça genişliyor Biliyorsunuz, f(1) e üzeri pi'de duruyor ki bu da 20 artı pi'ye çok yakın Bu her zaman eğlencelidir ve f(negatif 1) 0'a son derece yakın yani gerçekten de gerilmiş bu gerçek eksen Ve bu aynı zamanda birim çember yönünde de uzatılmış, böylece f (I) veya f (negatif) I'e ulaşırken çemberin yarısına kadar yürüyoruz, yani bu artık iyi ve güzel Şöyle bir fonksiyon hakkında nasıl düşünürüz? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "Ayrıca 2 çarpı X'in doğal logaritmasının X(X) olarak da yazabiliriz, böylece R'nin değerini temsil eden sarı noktamızı 0 civarına taşıyabiliriz. 69 hala hayali kısım yok, sadece gerçek sayı 0.69 ya da öylesine Bu 2'nin doğal logaritması, yani f'nin 1'in 2'ye geldiğini görebiliyorsunuz. Bu yüzden bu fonksiyona 2 üzeri X f 1'in yarısı demek istiyoruz, aslında özür dilerim f'nin negatif 1'i tam 1'in yarısı f'nin üzerine çıkıyor Birim çemberin etrafında biraz dolaşacağız, özellikle 0 olacak. Birim çemberin etrafındaki 69 radyan ve şimdi biraz daha eğlenebilir ve bunu 0 yerine değiştirirsek ne olacağını söyleyebiliriz. 69, 2'nin doğal logaritması olmak yerine, onu I ile 2'nin doğal logaritması haline getirin, böylece gerçekten üstel bir tabanı olabilecek bir şey düşünmüş oluruz. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "Bu durumda I'in kuvveti ne kadardır, onu 0 civarına iter. 2 yaklaşık beşte Ama f'yi 1'in I sayısına koyma özelliğine sahip birçok farklı üstel fonksiyon var. Yani eğer onu daha da büyüteceksek, onu burada canlandırdığımı sanmıyorum. o sarı noktayı pi I'in 5 katı olana kadar yukarı kaldırın. Birim çemberi göreceksiniz. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "Kendi etrafında döndürülür, böylece f(negatif) f(1) başka bir 2 pi radyan etrafında döner ve bulunduğu yere iner. Ama gerçek ekseni çok daha fazla uzatır. Bu, I'nin I'ye olan başka bir çıktısının anlamıydı. çok daha küçük bir sayı 0 civarındaydı. 0003 falan Ama bence oldukça eğlenceli olanı da görebiliriz. 2'nin üssü X olarak yorumlamak istediğimiz alternatif ifadeleri düşünürsek ne olur? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "Elimizde X (R) çarpı X ve R eşittir bu değer, bu da 2 artı pi çarpı I'nin doğal logaritması. Bu şu anlama geliyor: 1 f (1)'i negatif 2'ye yerleştirdiğimizde bu fonksiyonu yazmak istiyoruz eksi 2 üssü X olarak doğru ve bu aslında bildiğiniz bir şey, bir kuvvete negatif bir sayı yazdığımızda bu biraz yanıltıcı derecede basit. Negatif 2 üssü X'e ilk başta böyle görünmüyor, bu bize mutlaka getiriyor karmaşık sayılara herhangi bir şekilde ama elbette 1 yarım gibi bir değeri bile yerine koyduğumuzda, bir nevi negatif 2'nin karekökünü istediğimizde, bunu I çarpı karekök gibi bir şey olarak yazmak istediğimizi fark ederiz. / 2 Ama eğer bu fonksiyona - negatif 2 üzeri X kuvvetine bakarsanız, bunun uğraştığı tam karmaşık tanım kümesinde baktığınız şey, 1'in değerini negatif 2'ye alan bir fonksiyondur. gerçel sayı doğrusunun geri kalanına da öyle geliyor, bir nevi dışarıya doğru spiraller çiziyor mu? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "Yani, f(negatif 1)'in, (eksi 1 yarım) olduğunu görüyoruz. Yaklaşık olarak f (1 yarım)'ı takip ederseniz beklediğiniz yer Tam olarak hayali çizginin üzerinde durur ve f (1 yarım) 2'nin karekökü olur. Fare olmasını istediğim yerde değil. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "Bu, karekök 2 çarpı I civarında olacaktır ve siz daha da devam ettikçe, bu size negatif 2 üzeri X'in tüm gerçek değer kuvvetlerini gösteriyor, zorunlu olarak spiral şeklinde dönüyor. Ama aynı zamanda R değerimizi daha da yükseğe çıkarıp onu elde edebiliriz. yaklaşık tau çarpı I yaklaşık altı virgül iki sekiz çarpı I ve bu bağlamda bu, 2 üzeri X gibi bir şey olarak yazmak isteyeceğimiz başka bir fonksiyondur çünkü X yerine koyduğunuz herhangi bir tam sayıdan tam sayıya tekrarlanan çarpmaya benziyor Ve hatta 1 yarım gibi şeyler için makul değerleri var, burada pozitif bir Karekök yerine negatif karekökü tüketiyor, ama aslında yaptığı şey düzleme bir dönüşüm. Her şeyin gerçek olduğu yere koyuyor Sayı doğrusu çok sıkı sarılmış bir sarmal haline gelir ve f(1) tam 2 sayısının üzerine gelecek şekilde sarmal yapar. Yani bu anlamda 2 üzeri X diyebiliriz. geleneksel olarak alışık olduğumuzdan ayrı bir üstel fonksiyon Bu yüzden sanırım tüm bunlarla birlikte işleri bugüne bırakacağım Ve sizi düşünmeniz için birkaç kalıcı soruyla bırakacağım tamam, yani eğer isterseniz I üzeri I'yi çok değerli bir ifade olarak düşünün, değil mi bir kural benimsediğimizi söyleyebilirsiniz Hayal ürünü bir şekilde doğal logaritma fonksiyonunun bir dalını seçtiğinizi söylersiniz Ve belki bu sizi bu e üzeri negatif pi varlığına kilitler yarımlar Ama diyorsanız bu tür gördüğümüz çeşitli değerler gibi sonsuz sayıda farklı değer olmak istiyor 2 üzeri 1/3 aynı anlamda kaç değer olmak istiyor? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "10'uncular farklı şekilde ifade edilmek istiyor, şunu söyleyeyim, F(X)'in sağladığı tüm üstel fonksiyonlar için bunu bir yere yazdım mı? bunlardan ve eğer f(1) eşittir 2 Doğru, hangi fonksiyonun çeşitli seçenekleri için X eşittir 3(10)'u yerine koyduğumuzda kaç farklı çıktı elde edeceğiz? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "2 üzeri pi için, 2 üzeri X'in temsil edebileceği çeşitli fonksiyonlar için, eğer 2 üzeri X'i bu tür soyut özellikler anlamında üstel bir fonksiyon olarak düşünüyorsak ve eğer evet ise, eğer bu tür farklı işlevlerden oluşan bir Sınıfımız var ve pi'yi takmak istiyoruz, bu beni güldürüyor Sırf bunun hakkında düşünmeye çalışırken ortaya çıkan bir tür biliyorum komik cevap olduğu için, bunlar sorular Sizi baş başa bırakıyorum ve sanırım bu benim bugünkü derse yaklaşırken asıl sorum, bunun üstel fonksiyonların soyut özelliklerini tanımlamasını isteyip istemediğimdi. Ve bu soyut özelliklerden başlamak benim için çok hoş. e üzeri rx veya daha fazlası fikrine kilitlenirsiniz Sadece biliyorsunuz, r'nin farklı değerleri için exp r çarpı x'in daha dürüst yazılması gerektiğini düşünüyorum Bu sizi o kadar uzağa kilitler Ama sahip olduğunuz kadar sizi kilitlemez 2 üssü x'in çok daha az olması gerektiğine dair net bir Kavram I üssü x gibi bir şey Tabii ki buradaki risk, insanların bazen soyutlamayı sevmemesi ve bazen de ulaşılabilir görünmemesidir. Biliyorsanız bana haber verin sanırım, tüm bunları çevreleyen çok ilginç bir düşünce çemberi var, güç kulelerini de içeriyor çünkü aslında geçen seferki gibi karmaşık sayılar bağlamında güç kuleleri hakkında konuşmak istiyorsanız veya negatif temellerle bile Bunun gibi şeyleri düşünmeniz gerekiyor, bu yüzden ekranda sorduğumuz bir soruydu Evet, bunu I'in gücü için yaparsak ne olur? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "Bildiğiniz titrasyon haydi bunu deneyelim haydi devam edelim ve bir güç kulesi deneyelim Burada I'yi belirli bir güce yükseltiyoruz ve ondan ne çıkacağını görüyoruz, yani bunu yapmayı planlamıyordu Ama yapabiliriz her zaman yapabiliriz Python'u yukarı çekin ve aslında geçen sefer yaptığımızın aynısını yapın. Yani bunun işe yaraması için bazı temel değerlerle başlıyorduk ve sonra bir tür aralık için ne yapıyorduk, a alıyorduk ve yeniden atayacağız her ne olursa olsun Bu durumda a'nın kuvvetine yükselttiğim taban şu şekilde olmalıdır Tamam, harika, öyleyse bunu yapacağız ve sonra a'nın değerini yazdıracağız, hadi bunu şunun için yapalım Evet, 200 gibi çok daha büyük bir sayı. Yani bazen bu tür şeylerde kaos potansiyeli var gibi görünüyor. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "Aslında biz izin verdik, NumPy'yi içe aktarmama izin verin, bu yüzden üstel fonksiyona sahibim, bırakayım gideyim Daha önce olduğu gibi büyük aralığımız için Bunu, bildiğiniz gibi I'in X'in kuvvetine benzeyen bir şey yazmak yerine, yazacağım farklı bir sabitin üstel fonksiyonu olarak yapacağım farklı bir sabitin 5 pi yarısı olmasını istiyorum, yani 5 pi yarısı çarpı I yapacağım yani bu karmaşık bir sayı ve 5 pi yarısı var sanal kısım Yani bu 5 pi yarım çarpı I ve ne yapıyorum? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/ukrainian/sentence_translations.json b/2020/ldm-i-to-i/ukrainian/sentence_translations.json index 2e23e163e..eba6e71c3 100644 --- a/2020/ldm-i-to-i/ukrainian/sentence_translations.json +++ b/2020/ldm-i-to-i/ukrainian/sentence_translations.json @@ -49,7 +49,7 @@ "end": 63.7 }, { - "input": "And in fact if we go, let's not show where things are going too much here, if we go ahead and rewrite that base i in terms of e, it can help us make sense out of this expression.", + "input": "And in fact if we go, oh no, that's not sure where things are going too much here, if we go ahead and rewrite that base i in terms of e, it can help us make sense out of this expression.", "translatedText": "І насправді, якщо ми підемо, давайте не будемо показувати, куди все тут йде, якщо ми продовжимо і перепишемо цю основу i через e, це може допомогти нам зрозуміти цей вираз.", "n_reviews": 0, "start": 64.12, @@ -994,7 +994,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half.", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half.", "translatedText": "Отже, якщо ви починаєте з цифри 1, ваша початкова швидкість полягає в тому, щоб рухатися прямо до 0, а коли ви йдете ще нижче, якби ви сиділи на 1 половині, ви все одно рухалися б до 0, але тепер ваш вектор швидкості було б мінус 1, помножене на ваше місце, що дорівнює мінус 1 половині.", "n_reviews": 0, "start": 998.68, @@ -1302,7 +1302,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i.", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i.", "translatedText": "І цікавим питанням буде: чи знаєте ви, чи є лише одна така функція, яку здається доцільним написати для цього, тому що ви знаєте, якщо ми будемо писати це як i до x, вона не тільки повинна задовольняти це, вона також повинна задовольняти, ви знаєте, коли ми підключаємо номер один, ми отримуємо i, мабуть, i до ступеня один, однак ми думаємо, що ця функція має бути i.", "n_reviews": 0, "start": 1383.38, @@ -1323,7 +1323,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos.", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma", "translatedText": "Отже, у нас є 5 pi i половинок чудово, це абсолютно ще одне значення, яке ми можемо підключити для x тут, і просто щоб описати це трохи візуальніше, якщо ми поглянемо на наше коло тут, де ми маємо момент, пройдений протягом часу, що дорівнює половині пі, що дорівнює 1.57 що, якби натомість ми зробили ще один повний хід і зробили ще півпі, щоб отримати пі, яке, як ви знаєте, ми могли б записати, де значення e до pi i є ми йдемо ще півпі, ми йдемо ще півпі, що на у цій точці ми б пройшли повне коло, повернувшись до одиниці, а потім пройшли б п’ять півпі, що числово становить приблизно 7.85 так, це абсолютно ще одне число, яке веде нас на вершину i, і якби ми пройшли через всю лабузку повторного вираження i в степені i, спочатку записавши e до 5 пі половинок i в степені i, ці i помножити, щоб стати від’ємним, і ми дивимось на e до мінус 5 половин пі, що є зовсім іншим числом, правильно, ми можемо обчислити це, я не впевнений на розумі, але давайте подивимося на Desmos .", "n_reviews": 0, "start": 1415.68, @@ -1337,7 +1337,7 @@ "end": 1493.22 }, { - "input": "What is e to the negative 5 pi halves 0.000388 Okay, 0.000388 much smaller number 0.000388 Which begs the question of okay i to the i what are you right?", + "input": "What is e to the negative five pi halves? 0.000388. Okay, 000388. Much smaller number. 0.000388. Which begs the question of okay i to the i, what are you? Right?", "translatedText": "Чому дорівнює e до мінус 5 пі, половини 0.000388 Добре, 0.000388 набагато менше числа 0.000388 У зв’язку з цим виникає питання добре i до i, що ти правий?", "n_reviews": 0, "start": 1493.28, @@ -1358,21 +1358,21 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle?", + "input": "that long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle,", "translatedText": "Ця довжина приведе вас до набагато меншого числа. Але це не єдина відповідь, яку ми можемо ввести правильно, ми маємо інших людей, які приходять сюди з від’ємними 3 половинами, помноженими на i pi. Що ви знаєте в термінах одиничного кола?", "n_reviews": 0, "start": 1544.74, "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one?", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one", "translatedText": "Ми могли б подумати про те, щоб сказати «привіт», якщо я хочу дістатися до I, а не піти 90 градусів пі ділить радіани в той бік, що, якщо я пройду 270 градусів в інший бік 3 пі ділить радіани, що, можливо, я вважаю негативним, оскільки конвенція така зазвичай, що проти годинникової стрілки є додатним. Абсолютно – це ще один спосіб виразити це, і це дало б нам іншу відповідь, якби ми мали e до мінус 3 половинок пі i Все до степеня i ми проходимо ту саму гру, тепер квадрат i скорочується за допомогою a від’ємний, який уже є, і ми маємо додатні 3 половини пі, і чисельно це дає нам навіть іншу відповідь, ніж те, що ми мали раніше. Якщо ми переглянемо і скажемо, чому дорівнює e у 3 пі, а не 3 або 3 пі половини 111 точка 3 1 зовсім інше число, ніж те, що ми бачили раніше 111 точка що це було 111 точка 3 1 чудово 111 точка 3 1 або близько того динамічний Але ми рухаємося назад у часі, ми бачимо, як давно у часі, ким я повинен бути. Так що, якби я почав діяти з цього моменту, я б опинився на першому місці мого початкового стану, і вам потрібно повернутися в часі 3 пі половини одиниць І тоді, якщо ви перекладете на динаміку розпаду. Що саме робить піднесення до ока в цьому контексті, ви скажете, якщо я починаю з номер один, але я хочу переміститися назад у часі та сказати, з чого я мав почати, якщо Я хочу занепасти так, щоб опинитися на першому місці?", "n_reviews": 0, "start": 1559.26, "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right?", + "input": "after three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right?", "translatedText": "Після 3 пі половинок одиниць часу відповідь, очевидно, починається приблизно зі ста одинадцяти для такого роду експоненційного розпаду. Ви можете побачити, куди це йде, де насправді існує нескінченна кількість різних значень, які ми можемо підключити до X, якщо ми думаючи про e до X як про я, а люди ввели тут набагато більше. Вибачте, я кидаю шпильку на землю, як це роблять класичні для третього місця. 9 половинок пі чудовий вибір 1729 половинок пі ви всі мої улюблені багато-багато різні варіанти нескінченно багато різних значень, що спочатку викликає збентеження, тому що ми дивимося на вираз. Здається, ви знаєте, що будуть певні обчислення. Я просто вставляю це в свій калькулятор і дивлюся, що з’являється, і ми отримуємо кілька різних цінності для нього. Отже, що тут відбувається?", "n_reviews": 0, "start": 1657.18, @@ -1442,7 +1442,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function?", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function,", "translatedText": "Четвертий корінь із 16 має дорівнювати 2, і відповідь буде правильною. Ми приймаємо конвенцію, коли є кілька таких варіантів, коли у вас є багатозначна функція. Ми часто просто вибираємо одне з цих значень, щоб мати на увазі те, що ми маємо на увазі, коли хочемо сприймайте це як функцію як щось із одним входом і єдиним виходом на модному жаргоні. Це виникає весь час, коли ми маємо справу з комплексними числами. Ідея чогось як операції на кшталт бажання мати кілька значень, які ви іноді почути словосполучення розгалуження Де ви обираєте розгалуження функції квадратного кореня?", "n_reviews": 0, "start": 1795.66, @@ -1463,7 +1463,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer?", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer?", "translatedText": "Тому що є кілька різних відповідей. Ви знаєте, ми знову думаємо про це обертання на 90 градусів. І якби ми думали про це як про обертання на 90 градусів, здавалося б, що квадратний корінь має бути Знаєте, щось сидить під кутом 45 градусів. Можливо, це квадрат корінь I, який ми могли б дуже чітко записати як корінь 2 на 2 корінь 2 на 2 I Це просто використання тригонометрії, але якби ми думали про I замість цього як про негативне обертання на 270 градусів, то відчувалося б, що половина цього виконує половину цієї операції насправді має перевести нас на інший бік. Можливо, число, яке сидить тут, має бути квадратним коренем з I, і це насправді лише мінус того, що ми бачили раніше. Від’ємний корінь 2 на 2 мінус корінь 2 на 2, помножений на I Тепер у контексті реального значущі функції, ми можемо сказати, що так. Просто виберіть квадратний корінь як позитивну відповідь, але яку з них ви вважаєте позитивною?", "n_reviews": 0, "start": 1846.36, @@ -1477,14 +1477,14 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x?", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x,", "translatedText": "І я думаю, ви добре скажете. Ми знаємо, що це таке, ми начебто визначаємо це як квадратний корінь з 2, усе добре і добре. Але що, якби я сказав, що давайте підійдемо до цього так само, як ми підійшли до нашого I до виразу I хочу спочатку виразити речі як e до щось правильно, а потім я збираюся підняти це до 1 половини, помноживши 1 половину на експоненту. І я кажу добре, я можу, я думаю, я можу зробити це e до того, що є дорівнює 2, це натуральний логарифм 2. Це константа, яка дорівнює приблизно 0.69 або близько того. Якщо ми піднесемо e до цього степеня, ми отримаємо 2, тож ми могли б подумати про це як e до натурального логарифму 2, помноженого на 1 половину, і якщо ви хочете, якщо ви думали про e до x?", "n_reviews": 0, "start": 1942.28, "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it.", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it.", "translatedText": "Ви знаєте, що це може бути надмірним у контексті дійсних чисел. Але якщо ви думаєте про e до x як скорочення для цієї функції x, ви можете вставити значення 0.69 помножити на 1 половину, що, я думаю, буде приблизно 0.345 Іш щось подібне Ви вставляєте це дуже конкретне значення у свій поліном, бачите, що він виводить, і він виводить приблизно 1.414 Гарне дійсне число, квадратний корінь з 2, чого ви очікуєте. Але якщо ми зробимо те саме, що ми щойно робили з I, і визнаємо, що насправді є кілька різних відповідей, коли ми хочемо записати щось як e у степені, ми також можемо написати це Це може здатися кумедним, але ми могли б записати це як e до натурального логарифму 2 плюс 2 пі I Усе це підняли до 1 половини Відразу після того, як це значення дорівнюватиме, ви можете розбити це, оскільки це e до натуральний логарифм 2, помножений на e до 2 pi. Це просто має ефект обертання речей на 360 градусів, тож це просто дорівнюватиме 1 Отже, ми дивимося на 2 помножити на 1, чудово, що це виглядає як дійсна заміна, але коли ми граємо в ту ж саму гру: беремо це і зводимо до степеня і обробляємо це шляхом множення степеня на експоненту, дивимося, що відбувається. Ми маємо e до натурального логарифму 2 помножити на 1 половину плюс Ну, скільки буде 2 пі, помножити на 1 половину ну це буде пі, помножене на I. Тепер ця перша частина e до натурального логарифму 2 помножити на 1 половину, яка в кінцевому підсумку буде знайомим квадратним коренем з 2, це все добре, але ми збираємося помножити це на e на pi I Правильно, і досить добре відомо, що e до pi I дорівнює від’ємному 1. Отже, у цьому випадку, здається, це означає, що якщо ми розв’язуємо цей вираз 2 до 1 наполовину, граючи з різними відповідями, ми можемо підключити щось на зразок e до X, що дорівнює 1 половині, що ми отримуємо в кінцевому підсумку, це ще одна відповідь, яку ми могли б традиційно записати як цей від’ємний квадратний корінь з 2. Тут я маю на увазі, що це трохи смішно мати кілька значень для перегляду 2 до 1 половини і кажуть, що це не дорівнює одній речі, але на основі нашого вибору це може дорівнювати багатьом різним речам. Але дві речі, які це може здатися цілком розумним. Якщо буде щось, що 2 до 1 половини, здається, що це має бути або позитивним квадратний корінь, з яким ми знайомі, або негативний варіант цього, який насправді не здається такою проблемою. Насправді ми могли б грати в цю гру ще далі, де дозвольте мені попросити вас ще більше творчих відповідей на цей вираз тому що, можливо, ми зможемо знайти інші кумедні ступені чогось на кшталт 2 у степені X, коли ми почнемо підключати різні значення X залежно від того, яку заміну ми робимо, якщо ми дотримуємося тих самих правил, які ми використовували при оцінці I до ступінь I Отже, цього разу питання запитує або вказує, що один розв’язок рівняння e на x дорівнює 2 є дійсним числом Натуральний логарифм 2 добре, що ми знаємо його.", "n_reviews": 0, "start": 1989.66, @@ -1498,14 +1498,14 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42.", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42", "translatedText": "відповідь на запитання від e до x дорівнює 2. Знову ж таки, креативність вітається, тож я приділю вам ще одну хвилинку для цього. II Продовжую і зафіксую деякі відповіді тут, якщо вас це влаштовує, я не знаю, скільки часу обов’язково потрібно зробити математичний запис залежно від того, на який пристрій ви дивитесь, але не переживайте, якщо це станеться до того, як ви отримаєте можливість ввійти в запитання, яке ви хочете, у відповідь, на яку ви хочете, щоб він відповів. Отже, це виглядає так: 131 із вас ввели варіант, де ми беремо Ln із 2 і додаємо 2ii, і я думаю, що я пишу це запитання, помилково позначив одну з відповідей як правильну, хоча насправді є багато різних правильних. Тож це моя справа за той факт, що я не знаю, чи це виглядає для когось із вас, як о, це червоний, ви помилилися, коли ввели Ln 2 плюс 42.", "n_reviews": 0, "start": 2176.56, "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer.", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer.", "translatedText": "I pi, що, звичайно, є чудовим вибором, але ви також можете мати щось на зразок 4 pi I плюс натуральний логарифм 2 чи 6 pi I або будь-яке ціле число, кратне 2 pi I, якщо додати, що це не впливає на e на X Тому що це просто ефект множення на e до 2 pi I, що є ефектом множення на 1, і знову це має якийсь кумедний наслідок, коли, здається, виводить розумні результати, коли ми робимо це як інший приклад. схоже, другий найпоширеніший введений вираз полягав у тому, що ми могли б замінити 2. Отже, давайте подумаємо, що ми маємо на увазі 2 у степені 1 4-го, гаразд, була пропозиція замінити 2 на e на натуральний логарифм 2 плюс 4 pi I Гаразд Плюс 4 pi I і ми підносимо все це до 1 4-го праворуч добре, якби ви грали в ту саму гру, ви б отримали e До натурального логарифму 2 помножити на 1 4-е, і ми б помножили на e до Пі I Тепер перша частина цього буде звичайним додатним четвертим коренем з 2, що ми маємо на увазі, коли ви вставляєте в калькулятор такий вираз, як четвертий корінь з 2, гарне маленьке додатне число, але тоді ця друга частина є від'ємне 1, тож це, здається, говорить. Ви знаєте, якби ми тлумачили 2 іншим способом, підвищивши його до 1 4-го. Ви знаєте, що це не звичайна відповідь, яку ми отримуємо, але це розумна відповідь.", "n_reviews": 0, "start": 2253.76, @@ -1519,7 +1519,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships.", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships.", "translatedText": "Ми б дивилися на пі навпіл, помножене на I, і замість того, щоб помножити на мінус 1, ми натомість помножили б на I. Що знову ж таки є правильною відповіддю, здається розумним результатом для чогось на кшталт 2 до 1 4-го. Отже, коли ви дивлячись на той факт, що I у степені, я, здається, має кілька різних значень для цього. Правильно, у нас є це кумедне явище, де ми можемо підключити e до 5 половинок пі I, мінус 3 половинок pi, і ми отримуємо те, що здавалося б дико різними відповідями щось супер маленьке щось супер велике все дуже відрізняється від 1 5 приблизно 1 5 відповіді, яку ми знайшли раніше тут. Це точно таке ж явище, як коли ви запитуєте щось на кшталт 2 до 1 4 і визнаєте, що насправді існує кілька різних рішень до виразу X до 4-го фактично дорівнює 2 4 різним розв’язкам, і ви бачите той факт, що існує кілька різних розв’язків Для виразу e до X дорівнює певній основі, чи є ця основа I, чи ця основа є 2 Як би це не було, і один із способів, як ми можемо подумати про це, полягає в тому, що коли ви маєте справу з реальними числами, все просто приємно, все приємно Є стосунки один-на-один.", "n_reviews": 0, "start": 2358.92, @@ -1533,7 +1533,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value?", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value", "translatedText": "Це чудово. Якщо ми хочемо подумати про експоненціальні функції, дозвольте мені просто охопити деякі з цих речей. У нас є ось цей гарний вперед і назад, де ви можете виразити будь-яку експоненціалу як основу X, як 2 до X Або ви можете виразити той самий експоненціал, як X від R, помноженого на X, який, як ви знаєте, є поліномом, на який ми посилаємося. Кожного разу, коли ми неявно посилаємося на нього, коли ми пишемо щось на зразок e до X. І є чудовий крок вперед і назад, тому що ви можете просто взяти натуральний логарифм B І це дає вам одну відповідь, припускаючи, що B є додатним числом. І це те ж саме, що сказати, що X з R дорівнює B. Отже, один із способів, про який я говорив про це раніше в серії, полягає в тому, що якби ви дивилися на сімейство всіх можливих експоненціалів, правильно, ми могли б записати їх як X від R, помноженого на X, і змінити значення R. Це те саме, що записувати e на R, помножене на X, якщо це те, що вам зручніше. Отже, e на R помножити XX на R помножити на X це те саме, що ми могли б подумати про зміну того, що це таке. Але з іншого боку, якщо ви думаєте про всі можливі експоненти як про деяку основу. Дозвольте мені зробити підставу в степені X, і ми йдемо змінити те, що таке ця основа. Спочатку здається, що це інший вид виразу, яким можна маніпулювати, але це просто інший спосіб вираження тієї самої сім’ї. якщо ми думаємо трохи абстрактніше, як Exp на R, помножене на X, і є причина, чому я це роблю, тому що ми збираємося застосувати це до комплексних чисел, де це виглядатиме дивніше, тому дотримуйтесь зі мною тут, якщо замість того, щоб дивитися на цю базу, я міг би сказати, яка вартість?", "n_reviews": 0, "start": 2428.5, @@ -1554,7 +1554,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R.", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d", "translatedText": "Я міг би мати exp R, помножене на X, де, можливо, R є чимось на кшталт нуль кома шість дев’ять, але я міг би зрушити це вниз на два пі. І це не змінює основу, якій вона відповідатиме, але все одно відповідатиме двом Або це може зміщує його на два пі, що не змінює базу, якій він відповідає, тому що в усіх тих випадках, коли ми вставляємо X дорівнює одиниці, ми отримуємо те саме, однак усе це для різних значень X є різними функціями. Це чому ми бачили кілька різних значень для I у степені I. Оскільки I в X є неоднозначною функцією в цьому контексті, було б однозначно, якби ми вирішували, яке значення R. Таким чином, те, що ми представляємо, є exp від R, помноженого на X, яке значення Р.", "n_reviews": 0, "start": 2597.88, @@ -1568,14 +1568,14 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers.", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers,", "translatedText": "Це однозначна функція, але на даний момент здається, що, можливо, ми хочемо перестати думати про речі в термінах якоїсь основи, зведеної до степеня X. Можливо, як тільки ми опинимося в контексті комплексних чисел, нам слід просто написати їх усіх як вираз деякої константи, помноженої на X, якщо ні з якої іншої причини це стає кристально зрозумілим. Як ми фактично підключаємо числа, якщо ми хочемо зробити обчислення чи просто виконати математику поверх цього, у нас є цей гарний нескінченний поліном, який ми підключіть їх, і я ще раз доведу для вас, що це, можливо, правильний спосіб думати про експоненти. Щойно ми поширимося на інші домени, такі як комплексні числа, і для цього давайте просто створимо резервну копію Go повертаємось до дверного дзвінка, деякі речі прибули, повертаються до оригінального способу, яким ми розширюємо ідею піднесення до степеня та просто думаємо про те, що дорівнює 2 у Х Правильно, ми знаємо, як думати про це для натуральних чисел.", "n_reviews": 0, "start": 2640.66, "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that.", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that?", "translatedText": "Ви знаєте щось на зразок 2 до 3. Повторне множення. Як вийшло, що вас вперше навчили думати про щось на зразок 2 до X для дробових чисел або для від’ємних чисел тощо.", "n_reviews": 0, "start": 2696.88, @@ -1589,35 +1589,35 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive.", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive.", "translatedText": "Зазвичай вас вчать, що 2 до 1 половини має бути чимось, про що ви дізнаєтеся, якщо я помножу це на самого себе, і це відповідає звичайним правилам, які експоненти роблять із підрахунком чисел, де ми можемо додавати речі в цьому показнику, який я повинен отримати 2 до 1, отже, це має бути якесь число, яке, коли я множу на одне, отримую 2, і ви знаєте, що в цей момент у вас є вибір, можливо, він позитивний.", "n_reviews": 0, "start": 2708.3, "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1?", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the", "translatedText": "Можливо, воно негативне, але якщо ви завжди вирішуєте робити позитивний вибір, ви зможете отримати гарну безперервну функцію з цієї самої угоди, якщо ми запитаємо про від’ємні числа. Що має бути добре, що має бути щось 2 до мінус 1 де, коли я помножу це на 2 на 1?", "n_reviews": 0, "start": 2731.78, "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily.", + "input": "one, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily.", "translatedText": "Це дає мені 2 до 0, і це свого роду обґрунтування нашої конвенції, згідно з якою від’ємні експоненти виглядають як 1 половина. Але що насправді тут відбувається, ми говоримо, що як би це не було, це має бути якась функція, яка задовольняє цю властивість f a плюс b дорівнює f від a помножене на f від b. Крім того, той факт, що основа дорівнює 2, в основному говорить нам, що це не просто така функція. Це функція, де, коли ми підставляємо 1, ми отримуємо 2. питання в стилі перевірки розуму, щоб побачити, чи ви розумієте деякі наслідки тут. Я хочу запитати вас, що таке я не буду називати це софтболом, але це це не означає, що це неймовірно глибоке запитання обов'язково.", "n_reviews": 0, "start": 2744.96, "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here?", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here.", "translatedText": "Це просто перевірка, якщо ви дотримуєтеся ідеї абстрактно починати з властивостей функції, а потім виводити способи, як ми можемо захотіти записати це на основі цих властивостей. Якщо f від x задовольняє експоненціальну властивість f від a плюс b дорівнює f від a помножене на f від b для всіх вхідних даних. І також задовольняє, що f від 1 дорівнює 2, яке з наступного є істинним. Тобто, яке з наступного обов’язково є істинним. Незалежно від того, яку функцію ви запускаєте з тими з вас, хто пам’ятає, яка це була лекція. Незалежно від того, про яку ми говорили, як інтерпретувати те, що насправді говорить формула Ейлера. Я поставив запитання в такому стилі, де я знехтував однією умовою, ви знаєте, я не записав той факт, що ми хочемо переконатися, що f від x всюди не дорівнює нулю, а потім це спричинило певну плутанину, що круто отримати плутанину на екрані, яка трапляється з усіма нами. Але мета цього полягала в тому, щоб показати, що ця абстрактна властивість чогось, що перетворює додавання на множення, достатньо, щоб ви захотіли записати функцію як будь-яку її рівність як одиницю, зведену до якогось степеня. Це суть питання. Тепер у нас є пара запитань про силові вежі які, здається, з’явилися тут, і це чудово пов’язано з минулим разом. Давайте на мить затримаємося на питанні про енергетичну вежу, щоб спочатку глибше відчути подібне до того, що тут має означати піднесення до степеня?", "n_reviews": 0, "start": 2793.44, "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane?", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane?", "translatedText": "Оскільки ми можемо бути тим, що я хочу стверджувати, ми можемо відповісти на це декількома різними способами. Отже, якщо ви дасте мені лише один, ми поговоримо про електровежі. І тоді так само, як числову лінію можна представити в логарифмічному масштабі, можна те ж саме можна зробити для складної площини?", "n_reviews": 0, "start": 2882.64, @@ -1638,7 +1638,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one.", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one", "translatedText": "Так, справді, є візуалізація, до якої я збираюся перейти за мить, де ми робимо щось схоже на це, тому що ми будемо грати з різними експоненціальними функціями X від R, помножених на X. Але ми ми змінимо це значення R, яке буде представлено маленькою жовтою крапкою. Тож ми якось поговоримо про це. Це не збирається відображати всю площину, а лише кілька зразкових точок від реальної осі та уявної осі Але ідея полягає в тому, що коли ми рухаємося навколо того, якою є ця константа, ми зможемо начебто візуалізувати різні речі, які вона робить із площиною, і фактично це схоже на те, що вісь x перетворюється на логарифмічну шкалу, а потім обертається уявна вісь вздовж кола. А потім, як тільки це значення R стає уявним, воно змінює роль тих дійсних чисел, які поміщаються в коло, а уявні числа – на логарифмічно масштабовану додатну вісь, таке чудове питання, що всі три, я думаю, начебто стрибають вперед, куди я хочу піти, але приємно бачити, що люди так думають у цьому.", "n_reviews": 0, "start": 2901.54, @@ -1652,14 +1652,14 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it.", + "input": "explicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you want, which is that let's say you have some complex function f, a", "translatedText": "явно Щось на зразок f з 5 — це те ж саме, що й f з 1 плюс 1 плюс 1 плюс 1 плюс 1 Що те ж саме, що f з 1, помножене на себе у 5 разів через цю властивість. Якщо f з 1 дорівнює 2, це те саме як 2 у степені 5, а потім щось на кшталт f від мінус 5. Має бути так, що коли ми помножимо це на f = 5, ми отримаємо будь-яке значення f, рівне 0, і не відразу зрозуміло, що таке f = 0, але ми можемо сказати, що f від 1 плюс 0 дорівнює тому, що f від 1 помножене на f від 0, але f від 1 дорівнює 2. Отже, це також дорівнює 2, тож ми кажемо, що 2 дорівнює 2, помноженим на щось добре, що щось має бути 1, тому в цьому контексті це гарантує, що f від мінус 5 дорівнює 2 у від’ємному 5, це 1 на 2 у 5-му ступені. Ми могли б явно записати це як 2 у від’ємному 5, що означає, що ці дві властивості разом складають ми справді хочемо записати функцію як 2 до X, тому що будь-яке рахункове число, яке ми введемо, задовольнить це буде виглядати так, як помножити на себе стільки разів, скільки дробове число, яке ми введемо, задовольнить ці властивості що ми хотіли. І ви можете здивуватися, що це унікально в контексті функцій зі справжнім значенням. Але в контексті функцій зі складним значенням було б кілька таких функцій, f які ми могли б написати для цієї, одну з яких ми були дивлячись раніше Де ми могли б мати функцію, визначену як exp натурального логарифму 2 плюс 2 пі. Я все це помножив на X Гаразд, вибачте за неохайність, я просто в захваті, пишучи про це. І це насправді інша функція, оскільки підтверджується тим, що станеться, якщо ви підставите X дорівнює 1 половині Ми бачили трохи раніше, як коли ви підставите 1 половину, ви отримаєте від’ємний квадратний корінь з 2, а потім, якщо підставите 1, четверту частину, ви отримаєте не четвертий корінь з 2, але I помножене на четвертий корінь з 2, тому це інша функція, але вона все одно задовольняє ці властивості, і це змушує нас записати це як 2 до X, і це змушує припустити, що, можливо, 2 до X є неоднозначним трохи нотації І ми повинні просто записати все в термінах exp від R, помноженого на щось, але ви можете здивуватися. Ви знаєте, можливо, ми просто недостатньо креативні з усіма функціями, які задовольняють цю властивість. Можливо, є двозначність, коли ми пишемо exp R, помножене на щось, і існують різні значення R, які можуть вступити в гру. Але я просто напишу невелике твердження, а потім, можливо, наведу ескіз того, як виглядатиме доказ, якщо захочете. Скажімо, у вас є деяка складна функція F, і вона спочатку задовольняє такі властивості. Ви можете взяти її похідну.", "n_reviews": 0, "start": 2974.0, "end": 3140.02 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways.", + "input": "nd it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I don't know, fractional amounts you might want to think of in crazy ways", "translatedText": "Він диференційований, що просто не дозволяє йому бути якоюсь, як ви знаєте, абсолютно безладною переривчастою річчю. Це схоже на взяття деяких випадкових значень залежно від того, чи знаєте ви проміжок будь-якого векторного простору над я не знаю дробовими величинами, про які ви можете думати божевільними способами.", "n_reviews": 0, "start": 3140.12, @@ -1673,7 +1673,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right?", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right?", "translatedText": "Це диференційовано. Воно не скрізь дорівнює 0, тому умова, яка виникла з моєї пам’яті, і я забув, для якої лекції лекція або щось подібне, і тоді вона має цю центральну властивість, що вона перетворює додавання на множення. Якщо у вас є така функція, я стверджую, що є унікальне, можливо, мені варто вказати, що існує унікальне комплексне число R, щоб ви могли записати F від X як експоненціальну функцію від R, помноженого на значення X. Це означає, що якщо у вас є X як функція, це нескінченний поліном із гарними властивостями похідної, і все це, якщо у вас є це, ви маєте кожну експоненціалу, яку ви хочете, у дуже схожому на абстрактному загальному значенні слова експоненціальну, яка базується лише на властивості, яку ми можемо отримати від неї, і ескіз доказу буде подивіться приблизно так, якщо ви хочете спочатку подивитися, що є похідною цього значення, яке, як ми припускаємо, існує всюди, чи не так?", "n_reviews": 0, "start": 3160.84, @@ -1694,7 +1694,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base.", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base", "translatedText": "Ми можемо повністю вилучити F з X із виразу, і вся межа виражається лише через H. Що, якщо ви подумаєте, що це означає в контексті похідних і того факту, що F з 0 обов’язково дорівнює 1 Весь цей граничний вираз є просто якась константа, але точніше, це будь-яка похідна нашої функції при 0. Тож у вас є така кумедна штука, коли, якщо ви знаєте її похідну при 0, яка визначає її похідну скрізь. Сподіваюся, у контексті експоненціальних функцій це досить знайомо, тому що все, що ми насправді говоримо, це те, що похідна експоненціальної функції пропорційна самій собі, і ця константа пропорційності дорівнює будь-якій похідній при 0. Це все дуже абстрактно сформульовано і тому подібне, але мета цього полягає в тому, щоб підкреслити, що це не обов’язково лише функції, які ми вже вважаємо степенем X. Але це потенційно набагато ширший клас функцій, які просто задовольняють цю абстрактну властивість перетворення додавання на множення. Але якщо у вас є це, це фактично гарантує, що ви також маєте друга похідна, і, якщо на те пішло, третя похідна, тому що функція похідної пропорційна самій собі. Отже, щоб взяти n-ту похідну, ви просто дивитеся на цю константу пропорційності та підносите її до степеня n, а потім звідси ви можете зробити Розширення в ряд Тейлора, і я міг би залишити це як своєрідне домашнє завдання для тих з вас, хто добре розуміє ряди Тейлора в цій ідеї, особливо якщо ви хочете змішати ідею будь-якої диференційованої функції, яка є диференційованою в сенсі комплексних чисел, яка є начебто, безумовно, коледжської теми. Ви знаєте, ви можете змішувати міркування там як завгодно. Але нечіткі міркування допускаються в контексті тих, хто знає лише про ряди Тейлора і нічого іншого, щоб взяти цю ідею та подивитися на розширення Тейлора для F і начебто обґрунтуйте ідею, що існує унікальне комплексне число, яке нашу функцію F обов’язково можна записати так. І тоді зв’язок із нормальними експоненціалами є щоразу, коли у вас є таке значення R. Ми робимо те, що ми робимо в комплексному контексті дійсних чисел якщо ви подивіться на exp цієї функції з таким значенням R і запишете це як основу.", "n_reviews": 0, "start": 3243.7, @@ -1708,21 +1708,21 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace?", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace", "translatedText": "Ми могли б інтерпретувати це як значення не просто exp від pi половинок I помножити на X, але ми також могли б інтерпретувати це як значення exp від 5 pi половинок I помножити на X. Це окремі функції. І існує нескінченна група окремих функцій, які, здається, повинні запишіть їх як I до X Отже, вираз I до I, якщо ви не прийняли стандарт для того, що це обов’язково означатиме. Коли ви говорите, що він має нескінченно багато виходів, інший спосіб подумати про це: функція I до X з позначеннями, які ми маємо, трохи неоднозначно. Тепер, з усім цим, давайте просто почнемо візуалізувати щось із цього, тому що я думаю, що це весело. І ви знаєте, що ви скажете мені, чи це корисне це зображення чи більш заплутане, але те, що ми збираємося зробити, це поглянути на цю функцію exp від R, помноженого на X, яка, по суті, є ще одним способом записати e у степені X, насправді, я думаю, я думаю, що в якийсь момент я відтворив іншу анімацію, яка вказувала, що тому що я планував Планую це зробити, тож дозвольте мені о, так, ось ви повернетеся в мою файлову систему, поверніться туди, де ви повинні бути. О, замініть, це відображається на іншому екрані. Зачекайте, чому це так, добре, замініть?", "n_reviews": 0, "start": 3391.12, "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative?", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative", "translatedText": "Розмістіть туди все, що ви бачите. А тепер ми повертаємося до того, ми все це все це просто для того, щоб я міг гарно написати. Якщо вам незручно думати про це як про exp R, помножене на X, цей нескінченний поліном Просто в потилицю e до R, помноженого на X, і ми будемо варіювати навколо R, тому я буду слідувати точкам уявної осі та я буду слідувати точкам реальної осі, і давайте подивимось, що це дасть це все якось швидко, тому дозвольте мені поміркувати над цим трохи повільніше всі від’ємні числа будь-що Це від’ємне дійсне число потрапить у діапазон між 0 і 1. Що має мати значення e до від’ємного?", "n_reviews": 0, "start": 3472.82, "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R?", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r", "translatedText": "a до від’ємного дійсного числа є чимось між 0 і 1, і ми спеціально відстежуємо f від від’ємного 1, яке з’являтиметься навколо того, що 1 на e становить приблизно 30 0.37 f з 1 припадає на e як Очікувалося, що exp з 1 є f з I потрапить на один радіан навколо одиничного кола, і дуже цікаво стежити за всією уявною віссю, як уявна вісь обертається навколо кола і що відбувається, коли ми змінюємо це значення R?", "n_reviews": 0, "start": 3511.78, @@ -1736,7 +1736,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like?", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like?", "translatedText": "Ми можемо забажати значення R тут. Це розтягує речі по-іншому, тому, коли ми доводимо його до 2. Ви знаєте, це розтягує дійсну вісь набагато більше, так що f від 1 закінчується приблизно там, де e у квадраті трохи вище 7 f від’ємного 1 набагато ближче до 0 f від I є 2 радіани. Обертання навколо кола f від'ємного I є від'ємним 2 радіанами. І, звичайно, ми можемо отримати нашу улюблену формулу: якби це було пі, яке ми мали як константу масштабування, тоді реальна вісь стає досить розтягнутою. Ви знаєте, що f від 1 знаходиться на відстані від e до пі, що дуже близько до 20 плюс пі, що завжди весело, а f від мінус 1 надзвичайно близьке до 0, тому воно дійсно розтягнуте вісь І це також розтягнуті речі в напрямку одиничного кола так, що Дістатися до f від I або f від’ємного I проходить півдороги навколо кола, тож це все добре і добре. Як ми думаємо про таку функцію?", "n_reviews": 0, "start": 3551.6, @@ -1750,7 +1750,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it.", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it", "translatedText": "Ми також записали б як X від X натурального логарифму 2, помноженого на X, щоб перемістити нашу жовту крапку, що представляє значення R, навколо 0.69 все ще не уявна частина, а дійсне число 0.69 або близько того. Це природний логарифм 2, ви можете бачити, що f з 1 потрапляє на 2. Ось чому ми хочемо назвати цю функцію 2 до X f з 1 половини, насправді, вибачте, f з мінус 1 потрапляє прямо на 1 половину f I Це певна прогулянка по одиничному колу, точніше це буде 0.69 радіан навколо одиничного кола, і тепер ми могли б трохи повеселитися і сказати, що станеться, якщо ми змінимо це на 0 замість 0.69 замість того, щоб бути натуральним логарифмом 2, зробіть це на I, помножене на натуральний логарифм 2, щоб ми дійсно думали про щось, що може мати експоненціальну основу.", "n_reviews": 0, "start": 3610.52, @@ -1778,14 +1778,14 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle?", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle?", "translatedText": "Що я маю на увазі, я в цьому випадку підштовхує його приблизно до 0.2 навколо п’ятої. Але є багато різних експоненціальних функцій, які мають цю властивість додавати f = 1 до числа I. Отже, якщо б ми збільшили це ще більше, я не думаю, що це анімовано тут. Але якби ми взяли цю жовту крапку і піднімайте її, поки вона не досягне 5 половин, помножених на пі. Що ви побачите, це одиничне коло?", "n_reviews": 0, "start": 3749.24, "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right?", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right?", "translatedText": "Обертається навколо себе так, що f від’ємне f від 1 обертається навколо ще 2 пі радіан і приземляється там, де воно є. Але це розтягне справжню вісь набагато більше. Це означає, що інший вихід I до I є набагато менше число. Це було приблизно 0.0003 або близько того. Але ми також можемо побачити те, що, на мою думку, є досить цікавим. Що станеться, якщо ми розглянемо альтернативні вирази, які ми хочемо інтерпретувати як 2 у степені X, правильно?", "n_reviews": 0, "start": 3773.06, @@ -1806,21 +1806,21 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward?", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward?", "translatedText": "Ми маємо X від R, помноженого на X, і R дорівнює цьому значенню, яке є натуральним логарифмом 2 плюс пі, помноженим на I. Це означає, що коли ми підключаємо 1 f з 1, це мінус 2, тому ми хочемо написати цю функцію як від’ємне число 2 у степені X правильно, і це насправді те, що Ви знаєте, це трохи оманливо просто, коли ми записуємо від’ємне число в степені Від’єм 2 у степені X, це спочатку не виглядає так, обов’язково це приносить нам у комплексні числа будь-яким способом, але, звісно, коли ми вставляємо навіть таке значення, як 1 половина. Де ми ніби запитуємо квадратний корінь із мінус 2, ми розуміємо, що хочемо записати це як щось на зразок I, помножене на квадратний корінь з 2. Але якби ви подивилися на цю функцію від’ємне 2 у степені X у повній комплексній області, з якою вона має справу, ви бачите функцію, яка приймає значення від 1 до від’ємного 2. І якщо вона робить це, що це стосується решти дійсної числової прямої, чи вона як би закручується назовні?", "n_reviews": 0, "start": 3820.68, "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be.", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be.", "translatedText": "Таким чином, ми бачимо, що f від мінус 1 знаходиться на мінус 1 половині. Приблизно там, де ви очікували б, якби ви дотримувалися f від 1 половини. Це буде точно на уявній лінії, а f з 1 половини буде квадратним коренем з 2 Ну, мій миша не там, де я хочу, щоб вона була.", "n_reviews": 0, "start": 3880.24, "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense?", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense", "translatedText": "Це було б приблизно квадратний корінь з 2, помноженого на I, і, як ви продовжите далі, це показує вам усі реальні значення ступенів від’ємного 2 до X, воно обов’язково обертається навколо. Але ми також можемо підвищити наше значення R і отримати його до приблизно тау, помноженого на I, приблизно шість кома два, помножене на I, і в цьому контексті це ще одна функція, яку ми хотіли б записати як щось на зразок 2 до X, оскільки для будь-якого цілого числа до цілого числа, яке ви підключаєте для X, це буде виглядає як повторне множення. І він навіть має розумні значення для таких речей, як 1 половина, де він викидає від’ємний квадратний корінь замість додатного квадратного кореня, але те, що він насправді робить, це перетворення на площину, де він розміщує все, є реальним Числова лінія в кінцевому підсумку є дуже туго закрученою спіраллю, яка обертається навколо, і вона просто обертається по спіралі таким чином, що f з 1 потрапляє прямо на число 2. Тож у цьому сенсі ми можемо сказати, що 2 до X правдоподібно інтерпретується як окрема експоненціальна функція від тієї, до якої ми традиційно звикли. Тож я думаю, що з усім цим я залишу все на сьогодні, і я просто залишу вам пару тривалих запитань, щоб подумати, добре, тож якщо ви хочете подумайте про I до I як про багатозначний вираз, правильно, ви могли б сказати, що ми приймаємо конвенцію. Як не дивно, ви б сказали, що ви обираєте гілку функції натурального логарифма. І, можливо, це замикає вас у тому, що e до від’ємного пі половини. Але якщо ви скажете, що цей вид хоче мати нескінченну кількість різних значень, подібних до тих, які ми бачили. Скільки значень має мати від 2 до 1 третини в тому самому сенсі?", "n_reviews": 0, "start": 3894.92, @@ -1834,7 +1834,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function?", + "input": "three-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function?", "translatedText": "10-ті хочуть бути сформульованими по-іншому від усіх, дозвольте мені сказати, від усіх експоненціальних функцій F від X, які задовольняють о, я записав це десь f від X, що задовольняє Усі ці властивості, які я написав, отже, якщо це задовольняє всі з них, і якщо f з 1 дорівнює 2. Правильно, скільки різних виходів ми отримаємо, коли підключимо X дорівнює 3 10-м для різних варіантів для якої функції?", "n_reviews": 0, "start": 4008.86, @@ -1848,14 +1848,14 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I?", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i", "translatedText": "Для 2 до пі для різних функцій, які 2 до X може представляти, якщо ми думаємо про 2 до X як про якусь експоненціальну функцію, експоненціальну в сенсі цих абстрактних властивостей, і якщо ми так, якщо ми, якщо у нас є клас різних подібних функцій, і ми хочемо підключити pi, це змушує мене сміятися Просто тому, що це таке, я знаю, смішна відповідь, яка з’являється, коли ви намагаєтеся про це подумати, тож це питання, які Я залишу вас, і я думаю, що це ви знаєте моє моє головне питання, коли я підходив до сьогоднішньої лекції, чи хочу я, щоб вона була на кшталт опису цих абстрактних властивостей експоненціальних функцій. І для мене просто круто, що починаючи з цих абстрактних властивостей ви потрапляєте в ідею e до rx або більше. Просто ви знаєте, я думаю, що більш чесно записаний exp від r, помножений на x для різних значень r. Це блокує вас настільки далеко, але це не блокує вас настільки далеко, наскільки однозначне уявлення про те, що 2 у степені x повинно бути набагато менше, щось на кшталт I у степені x Ризик у цьому, звичайно, полягає в тому, що інколи люди не люблять абстракцію, а інколи це не видається таким доступним, але якщо це якщо ви знаєте, ви просто дайте мені знати, я думаю, я думаю, що існує цілий цікавий круг думок, який оточує все це, включаючи електровежі, тому що якщо ви хочете насправді говорити про електровежі, як ми минулого разу в контексті комплексних чисел або навіть з негативними основами. Ви повинні думати про такі речі, тому це було запитання, яке ми висвітлювали на екрані. Так, що станеться, якщо ми зробимо це для Я в степені Я?", "n_reviews": 0, "start": 4043.86, "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes.", + "input": "titration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes", "translatedText": "Титрування, ви знаєте, давайте просто спробуємо це, давайте просто спробуємо вежу потужності, де ми піднімаємо I до заданої степені та дивимося, що з цього випливає, тому ми не планували це робити, але ми можемо ми завжди можемо запустіть Python і, по суті, зробіть те, що ми робили минулого разу. Таким чином, це спрацювало б так: ми почали з деякого базового значення, а потім для певного діапазону. Що ми робили, ми брали і збираємося перепризначити це бути будь-яким База, яка в даному випадку є мною піднесеною до степеня a, має бути Гаразд, круто, ми збираємося це зробити, а потім ми збираємося надрукувати значення a, давайте просто зробимо це для Так, це набагато більше число, як-от 200. Тож здається, що відбувається таке. Іноді з цими речами існує потенціал хаосу.", "n_reviews": 0, "start": 4135.8, @@ -1869,7 +1869,7 @@ "end": 4201.64 }, { - "input": "That's it's not periodic or anything and it's actually chaotic I Suspect that doesn't happen for I but it's a thing to potentially look out for it looks like it does kind of stabilize Maybe there's some little subjection to numerical error But we stay pretty consistently around something with a real part of 0.43 and 0.36 Now what I would want to emphasize though is this expression So let's set a back to be equal to 1 this expression of taking I to the power of a remember That's a little bit ambiguous.", + "input": "that's um, it's not periodic or anything and it's actually chaotic I I suspect that doesn't happen for i but it's a thing to potentially look out for It looks like it does kind of stabilize um, maybe there's Some little subjection to numerical error, but we stay pretty consistently around something with a real part of 0.43 and 0.36 Now what I would want to emphasize though is this expression So let's set a back to b equal to 1 this expression of taking i to the power of a remember That's a little bit ambiguous.", "translatedText": "Це не періодично чи щось таке, і це насправді хаотично. Я підозрюю, що цього не відбувається з I, але це річ, на яку потенційно слід звернути увагу. Схоже, це стабілізується. Можливо, є невелика схильність до чисельної помилки, але ми залишаємося досить стабільними щось із дійсною частиною 0.43 і 0.36 Тепер, на чому я хотів би підкреслити, це цей вираз. Тож давайте встановимо, що назад дорівнює 1, цей вираз передає I у силу запам’ятовування Це трохи неоднозначно.", "n_reviews": 0, "start": 4201.64, @@ -1883,7 +1883,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing?", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing?", "translatedText": "У нас насправді є, тож дозвольте мені імпортувати NumPy, щоб у мене була експоненціальна функція, дозвольте мені піти Для нашого великого діапазону, як у нас було раніше. Замість того, щоб писати це, як ви знаєте, щось, що схоже на I у ступені X, я збираюся написати це як експоненціальна функція іншої константи праворуч. Інша константа, яку я збираюся створити. Я хочу, щоб це було 5 пі наполовину, тому я зроблю 5 пі наполовину, помножене на I, тож це комплексне число, і воно має 5 пі наполовину, як уявна частина Отже, це 5 пі, помножене на половини I, і що я роблю?", "n_reviews": 0, "start": 4234.12, diff --git a/2020/ldm-i-to-i/urdu/sentence_translations.json b/2020/ldm-i-to-i/urdu/sentence_translations.json index 9a279b687..2f39f163a 100644 --- a/2020/ldm-i-to-i/urdu/sentence_translations.json +++ b/2020/ldm-i-to-i/urdu/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "لہذا اگر آپ نمبر 1 سے شروع کر رہے ہیں، تو آپ کی ابتدائی رفتار سیدھی 0 کی طرف چلنا ہے اور جیسا کہ آپ اس سے بھی کم چلتے ہیں، اگر آپ 1 نصف پر بیٹھے تھے، تب بھی آپ 0 کی طرف چل رہے ہوں گے، لیکن اب آپ کی رفتار ویکٹر منفی 1 بار ہوگا جہاں آپ ہیں، جو منفی 1 نصف ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "اور ایک دلچسپ سوال یہ ہوگا کہ آپ جانتے ہیں کہ کیا صرف ایک ایسا فنکشن ہے جو اس کے لیے لکھنا مناسب محسوس کرتا ہے کیونکہ آپ جانتے ہیں کہ اگر ہم اسے i کے طور پر x لکھیں گے تو نہ صرف اسے مطمئن کرنا چاہیے بلکہ آپ کو یہ بھی معلوم ہونا چاہیے کہ کب ہم نمبر ایک میں پلگ ان کرتے ہیں جو ہم حاصل کرتے ہیں i غالباً i پاور ون میں تاہم ہم سوچ رہے ہیں کہ اس فنکشن کو i ہونا چاہیے۔", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "لہذا ہمارے پاس 5 pi i نصف ہے جو بالکل ایک اور قدر ہے جسے ہم یہاں x کے لیے پلگ ان کر سکتے ہیں اور صرف اس بات کو تھوڑا سا مزید بصری طور پر واضح کرنے کے لیے کہ اگر ہم یہاں اپنے دائرے کو واپس دیکھیں جہاں ہم ہیں لمحہ pi halves کے برابر وقت کے لیے چلتا ہے جو کہ 1 ہے۔57 کیا ہوگا اگر اس کے بجائے ہم ایک اور مکمل موڑ لیں اور ہم pi پر جانے کے لیے ایک اور pi halves جائیں جس کے بارے میں آپ جانتے ہیں کہ ہم ایک قسم کا ریکارڈ بنا سکتے ہیں جہاں e سے pi i کی قدر ہوتی ہے ہم ایک اور pi halves پر چلتے ہیں ہم ایک اور pi halves چلتے ہیں جس پر اس مقام پر ہم ایک مکمل دائرہ طے کر کے ہمیں واپس ایک پر لے جاتے اور پھر ہم پانچ پائی کے حصوں تک چلتے ہیں جو کہ عددی طور پر تقریباً 7 ہے۔85 جی ہاں، یہ بالکل ایک اور نمبر ہے جو ہمیں i کے اوپر لے جاتا ہے اور اگر ہم i کو دوبارہ ظاہر کرنے کے پورے رگمارول سے گزرتے ہیں تو i پہلے e لکھ کر 5 pi halves i کو پاور i وہ i's منفی بننے کے لیے ضرب کریں اور ہم ای کو منفی 5 پائی کے حصوں کی طرف دیکھ رہے ہوں گے جو کہ ایک بہت ہی مختلف نمبر ہے، ہم اصل میں اس کا حساب لگا سکتے ہیں، مجھے اپنے سر کے اوپر سے یقین نہیں ہے، لیکن آئیے ایک نظر ڈالتے ہیں ڈیسموس . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "2 یا کیا آپ تقریباً 7 کے لیے اس سے بھی زیادہ انتظار کرتے ہیں؟ وقت کی 85 اکائیاں جو کہ 5 pi نصف ہیں اور کیا دیکھتے ہیں جب آپ کے لیے سڑ جاتا ہے تو کیا ہوتا ہے؟ وہ لمبا جو آپ کو بہت کم نمبر پر لے جاتا ہے لیکن یہ واحد جواب نہیں ہے کہ ہم صحیح درج کر سکتے ہیں ہمارے پاس دوسرے لوگ یہاں منفی 3 نصف گنا i pi کے ساتھ آتے ہیں جسے آپ یونٹ کے دائرے کے لحاظ سے جانتے ہیں؟ ہم یہ کہنے کے بارے میں سوچ سکتے ہیں کہ ارے اگر میں 90 ڈگری پائ ہاف ریڈینز پر چلنے کے بجائے I تک پہنچنا چاہتا ہوں تو کیا ہوگا اگر میں 270 ڈگری دوسرے راستے پر چلوں تو 3 پائی ہاف ریڈین جو شاید میں منفی سمجھوں گا کیونکہ کنونشن ہے عام طور پر وہ گھڑی کی مخالف سمت میں مثبت ہوتا ہے یہ بالکل اس کے اظہار کا ایک اور طریقہ ہے اور یہ ہمیں ایک مختلف جواب ملے گا اگر ہمارے پاس e سے منفی 3 pi halves i تمام کی طاقت i ہم اسی گیم سے گزرتے ہیں اب i اسکوائر کینسل منفی ہے جو پہلے سے موجود ہے، اور ہمارے پاس مثبت 3 پائی کے حصے ہیں اور عددی طور پر یہ ہمیں اس سے بھی مختلف نظر آتا ہے جو ہمارے پاس پہلے موجود تھا جسے اگر ہم اوپر جائیں اور ہم کہیں کہ ارے، 3 pi کا e کیا ہے 3 o 3 pi نہیں نصف 111 پوائنٹ 3 1 اس سے بہت مختلف قسم کا نمبر جو ہم نے 111 پوائنٹ سے پہلے دیکھا تھا یہ کیا تھا 111 پوائنٹ 3 1 زبردست 111 پوائنٹ 3 1 یا اس سے زیادہ اور پھر وجدان کے لحاظ سے جو آپ پوچھ رہے ہوں گے فرض کریں کہ ہمارے پاس یہ گھوم رہا ہے متحرک لیکن ہم وقت کے ساتھ پیچھے کی طرف بڑھتے ہیں ہم دیکھتے ہیں کہ وقت میں کتنی دیر پہلے مجھے ایسا ہونا ہے کہ اگر میں وہاں سے چیزیں آگے کھیلتا ہوں تو میں اپنی ابتدائی حالت میں نمبر ایک پر اتروں گا اور آپ کو وقت کے ساتھ پیچھے جانا ہوگا 3 pi halves یونٹس اور پھر اگر آپ کو زوال کی حرکیات کا ترجمہ کرنا ہے تو اس تناظر میں آنکھ کو اٹھانا کیا کر رہا ہے آپ کہتے ہیں کہ اگر میں پہلے نمبر سے شروع کر رہا ہوں لیکن میں وقت کے ساتھ پیچھے ہٹنا چاہتا ہوں اور کہنا چاہتا ہوں کہ مجھے کہاں سے شروع کرنا چاہیے تھا؟ میں اس طرح نیچے گرنا چاہتا ہوں کہ میں پہلے نمبر پر آؤں؟ وقت کی 3 pi نصف اکائیوں کے بعد ظاہر ہے کہ جواب تقریباً ایک سو گیارہ سے شروع ہو رہا ہے اس قسم کے کفایتی زوال کے لیے اور آپ دیکھ سکتے ہیں کہ یہ کہاں جا رہا ہے جہاں درحقیقت لامحدود طور پر بہت سی مختلف اقدار ہیں جنہیں ہم X کے لیے پلگ ان کر سکتے ہیں اگر ہم ای ٹو دی ایکس کے بارے میں سوچنا کہ میں ہوں اور لوگ یہاں بہت زیادہ داخل ہوچکے ہیں مجھے معاف کیجئے میں اپنا پن زمین پر پھینک رہا ہوں جیسا کہ کوئی تیسرے نمبر کے لیے کلاسک کرتا ہے 9 pi halves بہترین انتخاب 1729 pi halves آپ سب میرے پسندیدہ لاٹ ہیں اور بہت سارے مختلف اختیارات لامحدود طور پر بہت سی مختلف اقدار جو پہلے دائیں طرف تھوڑی پریشان کن محسوس ہوتی ہیں کیونکہ ہم ایک اظہار کو دیکھتے ہیں جس سے لگتا ہے کہ آپ جانتے ہیں کہ ابھی کچھ حساب ہونے والا ہے میں اسے اپنے کیلکولیٹر میں لگاتا ہوں اور دیکھتا ہوں کہ کیا پاپ آؤٹ ہوتا ہے اور ہمارے پاس متعدد مختلف ہیں اس کے لیے اقدار تو یہاں کیا ہو رہا ہے؟ کیا ہو رہا ہے اور میں سوچتا ہوں؟ یہ واقعی اس خیال کو کم کرتا ہے کہ ہم عام طور پر ایکسپونینشل کے بارے میں کس طرح سوچتے ہیں لیکن اس سے پہلے میں اس بات پر زور دینا چاہتا ہوں کہ ریاضی میں یہ واحد وقت نہیں ہے جہاں ہم کسی چیز کی تشریح کرنے کے بارے میں ایک قسم کے ابہام کا سامنا کرتے ہیں؟ میں سوچتا ہوں کیونکہ اگر میں کچھ کہوں تو 25 کا مربع جڑ کیا ہے؟ آپ جانتے ہیں کہ میرے خیال میں ہم میں سے بہت سے لوگ ٹھیک کہتے ہیں کہ یہ پانچ ہے لیکن اگر ہم کہہ رہے ہیں کہ آپ جانتے ہیں کہ مربع جڑ کیا ہونا چاہیے یہ کچھ نمبر X ہونا چاہیے کہ جب آپ اس کا مربع کریں تو آپ کو 25 ملے؟ ٹھیک ہے اس کے دو مختلف جوابات ہیں کون کہتا ہے کہ ہمارے کنونشنز ہونا چاہئے کہ مربع جڑ فعل مثبت ہے ہمیں ایک مثبت نمبر دیتا ہے بجائے؟ منفی پانچ تو ہمارے پاس ایک اظہار ہے جو ایسا لگتا ہے کہ یہ ایک سے زیادہ مختلف اقدار کا حامل ہونا چاہتا ہے اور یہ حقیقت میں کسی اور سیاق و سباق میں ہوسکتا ہے جہاں مربع جڑوں کی بجائے اگر میں 16 جیسی کسی چیز کی چوتھی جڑ مانگ رہا ہوں تو کیا ہوگا؟ عام طور پر ہم اسے مثبت نمبر دو کے طور پر سوچیں گے دائیں دو ایک عدد ایسا ہے کہ جب آپ خود سے چار گنا ضرب کرتے ہیں تو آپ کو سولہ مل جاتا ہے ایک چوتھی جڑ کا معقول جواب لگتا ہے، لیکن اگر ہم اسے اس سوال کے جواب کے طور پر سوچ رہے ہیں تو کیا ہوگا؟ چوتھے کا نمبر 16 کے برابر ہے؟ اس کا ایک اور جواب بھی ہے ہم منفی دو بھی کہہ سکتے ہیں جو کہ ایک عدد ہے جسے آپ خود سے چار گنا ضرب دیتے ہیں تو آپ کو سولہ ملتے ہیں لیکن ایک اور جواب ہے کہ آپ دو بار کہہ سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "میں یہ درست معلوم ہوتا ہے، لیکن ایک اور جواب ہے جس کے بارے میں آپ دو بار منفی سوچ سکتے ہیں، میں ان چاروں نمبروں سے اس پراپرٹی کو مطمئن کرتا ہوں تو یہ کون کہے؟ 16 کی چوتھی جڑ 2 ہونی چاہیے اور جواب ٹھیک ہونے پر ختم ہوتا ہے ہم ایک کنونشن کو اپناتے ہیں جب اس طرح کے متعدد آپشن ہوتے ہیں جب آپ کے پاس کثیر قدر والا فنکشن ہوتا ہے تو ہم اکثر ان اقدار میں سے کسی ایک کا انتخاب کرتے ہیں جب ہم چاہتے ہیں کہ ہمارا کیا مطلب ہو۔", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "اور ہم جاننا چاہتے ہیں کہ وہ کیا ہونا چاہیے؟ کیونکہ اس کے متعدد مختلف جوابات ہیں آپ جانتے ہیں کہ ہم دوبارہ سوچتے ہیں کہ یہ 90 ڈگری گردش ہے اور اگر ہم اسے 90 ڈگری گردش کے طور پر سوچ رہے ہیں تو ایسا محسوس ہوتا ہے کہ مربع جڑ ہونا چاہئے آپ کو معلوم ہے کہ 45 ڈگری کے زاویے پر بیٹھی ہوئی کوئی چیز شاید یہ مربع ہے۔روٹ آف I جسے ہم روٹ 2 اوور 2 روٹ 2 اوور 2 کے طور پر بہت واضح طور پر لکھ سکتے ہیں I یہ صرف مثلثیات کا استعمال کر رہا ہے لیکن اگر ہم اس کے بجائے I کو منفی 270 ڈگری گردش کے طور پر سوچ رہے ہیں تو ایسا محسوس ہوتا ہے کہ اس کا آدھا حصہ اس آپریشن کا آدھا کر رہا ہے۔درحقیقت ہمیں دوسری طرف لے جانا چاہئے شاید یہاں بیٹھا ہوا نمبر I کا مربع جڑ ہونا چاہئے اور یہ اصل میں صرف منفی ہے جو ہم نے منفی جڑ 2 سے پہلے 2 سے زیادہ مائنس جڑ 2 سے زیادہ 2 بار دیکھا تھا I اب حقیقی کے تناظر میں قابل قدر افعال ہم کہہ سکتے ہیں ہاں صرف مربع جڑ کا انتخاب کریں جو بھی مثبت جواب ہو لیکن آپ ان میں سے کس کو مثبت جواب سمجھتے ہیں؟ آپ جانتے ہیں کہ شاید ایسا محسوس ہوتا ہے کہ ہمیں اس اوپری نمبر پر کیا غور کرنا چاہئے کیونکہ اس کے نقاط میں مثبت اعداد ہوتے ہیں لیکن تاہم آپ یہاں مثبت کو اچھے طریقے سے بیان کرنے کی کوشش کرتے ہیں جو آپ کو معلوم ہوگا کہ مثال کے طور پر دو مثبت اعداد کو مثبت بنانے کے لئے ہمیشہ ضرب کرنا چاہئے نمبر آپ واقعی اس طرح نہیں کر پائیں گے جیسا کہ آپ حقیقی نمبروں کے لیے کر سکتے ہیں اور درحقیقت یہ رجحان یہاں جہاں ہم جڑیں پکڑ رہے ہیں دراصل وہی ہے جیسا کہ ہم اس وقت دیکھ رہے تھے جب ہم بات کر رہے تھے۔", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "ایک سے زیادہ قدروں کے لیے میں نے I کی طاقت میں اضافہ کیا کیونکہ بھول جاتے ہیں کہ میں نے کی طاقت میں اضافہ کیا ہے میں مجھے یہ پوچھنے دیتا ہوں کہ زیادہ کیا نظر آتا ہے یہ ایک بہت آسان سوال ہے 2 کو طاقت 1 نصف تک لے جانا ٹھیک ہے 2 سے 1 نصف کیا ہے؟ اور مجھے لگتا ہے کہ آپ اچھی طرح سے کہتے ہیں ہم جانتے ہیں کہ یہ کیا ہے ہم اسے 2 کا مربع جڑ قرار دیتے ہیں سب ٹھیک ہے اور اچھا ہے لیکن کیا ہوگا اگر میں کہوں کہ آئیے اس کو اسی طرح دیکھیں جس طرح ہم اپنے I سے I اظہار I تک پہنچ رہے ہیں۔پہلے چیزوں کو e کے طور پر کسی چیز کو صحیح سے ظاہر کرنا چاہتا ہوں اور پھر میں اسے 1 نصف تک بڑھاتا ہوں اور 1 نصف کو ایکسپوننٹ میں ضرب دیتا ہوں اور میں کہتا ہوں ٹھیک ہے، میں اندازہ لگا سکتا ہوں کہ میں کیا کر سکتا ہوں 2 کے برابر ہے یہ 2 کا قدرتی لاگ ہے یہ ایک مستقل ہے جو 0 کے قریب ہے۔69 یا اس طرح اگر ہم e کو اس طاقت تک بڑھاتے ہیں تو ہمیں 2 ملے گا تو ہم اسے 2 گنا 1 نصف کے قدرتی لاگ میں e کے طور پر سوچ سکتے ہیں اور اگر آپ چاہتے ہیں کہ کیا آپ e کو x کے بارے میں سوچ رہے تھے؟ آپ جانتے ہیں کہ یہ حقیقی نمبروں کے تناظر میں ایک قسم کی حد سے زیادہ حد تک ہو سکتا ہے لیکن اگر آپ اس x فنکشن کے لیے شارٹ ہینڈ کے طور پر e to x کے بارے میں سوچ رہے تھے تو آپ قدر 0 میں پلگ ان کر سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "69 گنا 1 نصف جو میرے خیال میں 0 کے لگ بھگ ہوگا۔345 ایش کچھ اس طرح ہے کہ آپ اپنے کثیر نام میں اس بہت ہی ٹھوس قدر کو پلگ کرتے ہیں دیکھیں کہ یہ کیا نکلتا ہے، اور یہ 1 کے قریب آؤٹ پٹ کرے گا۔414 ایک عمدہ اصلی عدد مربع جڑ 2 جس کی آپ توقع کریں گے لیکن اگر ہم وہی کام کرتے ہیں جو ہم صرف I کے ساتھ کر رہے تھے اور یہ تسلیم کرتے ہوئے کہ جب ہم کسی طاقت کو e کے طور پر کچھ لکھنا چاہتے ہیں تو ہم اسے بھی لکھ سکتے ہیں۔یہ مضحکہ خیز لگ سکتا ہے، لیکن ہم اسے 2 جمع 2 pi I کے قدرتی لاگ میں e کے طور پر لکھ سکتے ہیں کہ پوری چیز کو 1 نصف تک بڑھا دیا گیا ہے، اس تمام قدر کے برابر ہونے کے بعد آپ اسے توڑ سکتے ہیں کیونکہ یہ e کی طرف ہے۔2 کا قدرتی لاگ e سے 2 pi I سے ضرب ہم اسے لینے اور اسے طاقت تک بڑھانے اور اس کا علاج کرنے کا ایک ہی کھیل کھیلتے ہیں کہ طاقت کو ایکسپوننٹ میں ضرب دینے سے دیکھیں کہ کیا ہوتا ہے ہمارے پاس 2 ضرب 1 نصف جمع کے قدرتی لاگ میں ای ہے ٹھیک ہے، 2 pi I گنا 1 نصف کیا ہے ٹھیک ہے یہ pi گنا ہوگا pi I دائیں اور کافی مشہور طور پر e سے pi I منفی 1 ہے لہذا اس معاملے میں یہ تجویز کیا جا رہا ہے کہ اگر ہم اس اظہار 2 سے 1 نصف کو حل کر رہے ہیں تو مختلف جوابات کے ساتھ کھیل کر ہم کچھ اس طرح کے لئے پلگ ان کر سکتے ہیں۔1 نصف کے برابر ہونے والے X کا ایک اور جواب ہے جسے ہم روایتی طور پر 2 کے اس منفی مربع جڑ کے طور پر لکھ سکتے ہیں اور یہاں میرا مطلب یہ ہے کہ اس کے لیے 2 سے 1 نصف کو دیکھنے کے لیے متعدد قدروں کا ہونا تھوڑا مضحکہ خیز ہے۔کہتے ہیں کہ یہ ایک چیز کے برابر نہیں ہے لیکن ہم جو انتخاب کرتے ہیں اس کی بنیاد پر ہم اسے متعدد مختلف چیزوں کے برابر کر سکتے ہیں لیکن دو چیزیں جو کہ یہ کافی معقول لگ سکتی ہیں اگر کوئی چیز 2 سے 1 نصف کے درمیان ہو تو ایسا لگتا ہے کہ اسے یا تو مثبت ہونا چاہیے مربع جڑ جس سے ہم واقف ہیں یا اس کی منفی قسم جو درحقیقت ایسا کوئی مسئلہ نہیں لگتا ہے اور درحقیقت ہم اس کھیل کو اور بھی کھیل سکتے ہیں جہاں میں آپ سے اس اظہار کے مزید تخلیقی جوابات طلب کرتا ہوں۔کیونکہ ہوسکتا ہے کہ ہم 2 سے پاور X جیسی کسی چیز کی دوسری مضحکہ خیز طاقتیں تلاش کر سکیں جب ہم X کی مختلف مختلف اقدار میں پلگ لگانا شروع کر دیتے ہیں اس بنیاد پر کہ ہم کیا متبادل بناتے ہیں اگر ہم انہی اصولوں کی پابندی کر رہے ہیں جو ہم I کو جانچنے میں استعمال کر رہے تھے۔پاور I تو اس بار سوال پوچھتا ہے یا یہ بتاتا ہے کہ x برابر 2 کی مساوات e کا ایک حل اصل نمبر 2 کا قدرتی لاگ ہے ٹھیک ہے جسے ہم جانتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "یہ بورنگ نہیں ہے، لیکن یہ بورنگ ہے اس کے مقابلے میں ہم اور کیا کر سکتے ہیں کیا آپ کسی اور کے بارے میں سوچ سکتے ہیں اور کیا آپ کوئی اور لکھ سکتے ہیں؟ سوال کا جواب e to the x 2 کے برابر ہے اور پھر تخلیقی صلاحیتوں کا خیرمقدم کیا گیا ہے، اس لیے میں آپ کو اس کے لیے ایک اور چھوٹا لمحہ دوں گا II آگے بڑھیں گے اور کچھ جوابات یہاں بند کریں گے اگر یہ آپ کے ساتھ ٹھیک ہے تو مجھے یقین نہیں ہے کہ یہ کتنا وقت ہے لازمی طور پر ریاضی کے اندراج کو اس بات پر منحصر ہے کہ آپ کس ڈیوائس کو دیکھ رہے ہیں لیکن زیادہ دباؤ نہ ڈالیں اگر آپ کو اس سوال میں داخل ہونے کا موقع ملنے سے پہلے ہے جس کا جواب آپ چاہتے ہیں کہ آپ اسے جواب دینا چاہتے ہیں تو ایسا لگتا ہے آپ میں سے 131 نے مختلف قسم میں داخل کیا ہے جہاں ہم 2 کا Ln لیتے ہیں اور ہم 2ii کا اضافہ کرتے ہیں اور میرا اندازہ ہے کہ میں اس سوال کو لکھ رہا ہوں غلطی سے ایک جواب کو درست کے طور پر نشان زد کیا گیا ہے جب کہ حقیقت میں کچھ مختلف درست ہیں تو یہ مجھ پر ہے۔اس حقیقت کے لیے کہ میں نہیں جانتا کہ آیا یہ آپ میں سے کسی کو لگتا ہے کہ اوہ یہ سرخ ہے جب آپ 2 جمع 42 کے Ln میں داخل ہوئے تو آپ کو غلط معلوم ہوا۔", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi جو یقیناً ایک بہترین انتخاب ہے لیکن آپ کے پاس 4 pi I کے علاوہ 2 یا 6 pi I کا قدرتی لاگ یا واقعی 2 pi I کا کوئی بھی عدد عدد ہو سکتا ہے اگر آپ یہ شامل کریں کہ اس سے e پر کوئی اثر نہیں پڑتا ہے۔X کیونکہ اس میں صرف e سے 2 pi I کو ضرب کرنے کا اثر ہوتا ہے جو کہ 1 سے ضرب کرنے کا اثر ہوتا ہے اور پھر اس کا ایک مضحکہ خیز نتیجہ ہوتا ہے جہاں ایسا لگتا ہے کہ جب ہم اسے ایک اور مثال کے طور پر کرتے ہیں تو یہ معقول نتائج برآمد کرتا ہے۔ایسا لگتا ہے کہ دوسرا سب سے عام درج کردہ اظہار یہ تھا کہ ہم 2 کی جگہ لے سکتے ہیں تو آئیے سوچتے ہیں کہ ہم 2 کو 1 4 کی طاقت سے سوچ رہے ہیں، ٹھیک ہے ایک تجویز تھی کہ ہم نے 2 کو 2 جمع 4 کے قدرتی لاگ میں e سے بدل دیا ہے۔pi I Okay Plus 4 pi I اور ہم اس سب کو 1 4th تک بڑھاتے ہیں اگر آپ ایک ہی گیم کھیلتے ہیں تو آپ کو 2 گنا 1 4th کے قدرتی لاگ میں e ملے گا، اور ہم e سے ضرب کریں گے۔pi I اب اس کا پہلا حصہ عام طور پر 2 کا مثبت چوتھا جڑ بننے جا رہا ہے جس سے ہمارا مطلب ہے جب آپ 2 کے چوتھے جڑ جیسے اظہار کو ایک کیلکولیٹر میں ایک چھوٹے سے مثبت نمبر میں لگاتے ہیں، لیکن پھر یہ دوسرا حصہ ہے منفی 1 تو ایسا لگتا ہے کہ آپ یہ کہہ رہے ہیں کہ کیا ہم 2 کو اس مختلف طریقے سے 1 4 ویں تک بڑھاتے ہیں آپ جانتے ہیں کہ یہ عام جواب نہیں ہے جو ہمیں ملتا ہے لیکن یہ ایک معقول جواب ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "ہم pi نصف اوقات I کو دیکھ رہے ہوں گے اور منفی 1 سے ضرب کرنے کے بجائے ہم I سے ضرب کر رہے ہوں گے جو ایک بار پھر ایک درست جواب ہے ایسا لگتا ہے کہ 2 سے 1 4 ویں جیسی کسی چیز کے لئے ایک معقول آؤٹ پٹ ہے لہذا جب آپ اس حقیقت کو دیکھتے ہوئے کہ میں طاقت کے لحاظ سے میرے پاس اس کے لیے متعدد مختلف اقدار ہیں، ٹھیک ہے ہمارے پاس یہ مضحکہ خیز واقعہ ہے جہاں ہم 5 pi halves I منفی 3 pi halves I میں پلگ ان کر سکتے ہیں اور ہمیں وہی ملتا ہے جو بالکل مختلف جوابات کی طرح لگتا تھا۔کچھ بہت چھوٹی چیز بہت بڑی سب کچھ 15ویں تقریباً 15ویں جواب سے بہت مختلف ہے جو ہم نے یہاں سے پہلے پایا تھا یہ بالکل وہی رجحان ہے جب آپ کچھ پوچھ رہے ہوں جیسے 2 سے 14ویں کیا ہے اور یہ تسلیم کرنا کہ حقیقت میں متعدد مختلف حل موجود ہیں۔ایکسپریشن ایکس سے چوتھے کے برابر ہے 2 4 مختلف حل درحقیقت اور جو آپ دیکھ رہے ہیں وہ یہ ہے کہ ایک سے زیادہ مختلف حل ہیں ایکسپریشن e سے X کسی قسم کی بنیاد کے برابر ہے چاہے وہ بنیاد میں ہو خواہ وہ بنیاد ہے 2 جو کچھ بھی ہو اور ایک طریقہ جس کے بارے میں ہم سوچ سکتے ہیں وہ یہ ہے کہ جب آپ حقیقی نمبروں کے ساتھ معاملہ کر رہے ہوتے ہیں تو چیزیں صرف خوبصورت ہوتی ہیں اچھی ہوتی ہیں ایک دوسرے کے ساتھ تعلقات ہوتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "یہ بہت اچھا ہے جہاں اگر ہم ایکسپونینشل فنکشنز کے بارے میں سوچنا چاہتے ہیں تو مجھے صرف اس میں سے کچھ کا احاطہ کرنے دیں ہمارے پاس یہ اچھا ہے جہاں آپ کسی بھی ایکسپونیشنل کو X کی بنیاد کے طور پر ظاہر کرنے کا انتخاب کر سکتے ہیں جیسے 2 سے X یا آپ اظہار کر سکتے ہیں۔وہی کفایتی X کے R اوقات X کے طور پر جو آپ جانتے ہیں کہ وہ کثیرالاضلاع ہے جس کا ہم حوالہ دیتے ہیں جب بھی ہم واضح طور پر حوالہ دیتے ہیں جب بھی ہم X کو e کی طرح کچھ لکھتے ہیں اور آگے پیچھے ایک خوبصورت ہے کیونکہ آپ صرف B کا قدرتی لاگرتھم لے سکتے ہیں۔اور یہ آپ کو یہ فرض کرتے ہوئے ایک جواب دیتا ہے کہ B ایک مثبت نمبر ہے اور یہ وہی چیز ہے جو کہ کہہ رہی ہے کہ R کا X B کے برابر ہے تو ایک طریقہ جس کے بارے میں میں نے سیریز میں پہلے بات کی ہے وہ یہ ہے کہ اگر آپ دیکھ رہے تھے۔تمام ممکنہ Exponentials کے خاندان کا حق ہے کہ ہم انہیں R اوقات X کے X کے طور پر لکھ سکتے ہیں اور R کو تبدیل کر سکتے ہیں اور یہ بالکل وہی چیز ہے جو کہ R ٹائم X کو e لکھنا ہے اگر یہ ایسی چیز ہے جس سے آپ زیادہ آرام دہ ہیں تو e سے R R اوقات X کے اوقات XX یہ وہی چیز ہیں جو ہم اسے تبدیل کرنے کے بارے میں سوچ سکتے ہیں لیکن دوسری طرف اگر آپ تمام ممکنہ اشاریہ جات کے بارے میں سوچتے ہیں جیسا کہ کچھ بنیاد مجھے کرنے دیں مجھے X کی طاقت کی بنیاد دیں اور ہم جا رہے ہیں اس بنیاد کو تبدیل کرنے کے لیے پہلے تو یہ محسوس ہوتا ہے کہ یہ ایک مختلف قسم کا اظہار ہے جوڑ توڑ کے لیے، لیکن یہ ایک ہی خاندان کے اظہار کا ایک اور طریقہ ہے اور ایک ایسا طریقہ جس کے بارے میں آپ سوچ سکتے ہیں کہ ہم اس کے بارے میں کیسے سوچیں گے کہ یہ کس بنیاد سے مطابقت رکھتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "میرے پاس R ٹائم ایکس کا ایکسپ ہو سکتا ہے جہاں شاید R صفر پوائنٹ سکس نائن کی طرح ہے لیکن میں اسے دو pi I سے نیچے منتقل کر سکتا ہوں اور اس سے اس بنیاد کو تبدیل نہیں کیا جائے گا کہ یہ اس کے مساوی ہوگا پھر بھی دو کے مساوی ہوگا یا یہ ہوسکتا ہے۔اسے دو pi I کے ذریعے اوپر منتقل کریں جو اس بنیاد کو تبدیل نہیں کرتا ہے جس سے یہ مطابقت رکھتا ہے کیونکہ ان تمام صورتوں میں جب ہم X کے برابر پلگ ان کرتے ہیں تو ہمیں ایک ہی چیز ملتی ہے تاہم یہ تمام X کی مختلف اقدار کے لیے الگ الگ فنکشنز ہیں۔ہم نے I ٹو پاور I کے لیے متعدد مختلف قدریں کیوں دیکھی ہیں کیونکہ I to the X اس تناظر میں ایک مبہم فعل ہے اگر ہم یہ فیصلہ کریں کہ R کی کون سی قدر اس طرح کہ ہم جس کی نمائندگی کر رہے ہیں وہ R اوقات X کی exp ہے جس کی قدر R کا کیا ہم جیسے ہی کسی ایک کا انتخاب کرتے ہیں؟ یہ ایک غیر مبہم فعل ہے لیکن اس وقت ایسا محسوس ہوتا ہے کہ شاید ہم کیا چاہتے ہیں ایک طاقت X تک اٹھائے گئے کچھ بیس کے لحاظ سے چیزوں کے بارے میں سوچنا بند کرنا ہے شاید جیسے ہی ہم پیچیدہ اعداد کے تناظر میں ہوں ہمیں صرف لکھنا چاہئے۔", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "یہ سب کچھ مستقل اوقات کے طور پر ایکس کے طور پر اگر کسی اور وجہ سے یہ واضح نہیں ہوتا ہے کہ اگر ہم ایک حساب کرنا چاہتے ہیں یا صرف اس کے اوپر ریاضی کرنا چاہتے ہیں تو ہمیں یہ بہت اچھا لامحدود کثیر الجہتی مل گیا ہے ان میں پلگ ان کریں اور میں آپ کے لیے ایک اور کیس بناؤں گا کہ شاید یہ ہے ایکسپونینشلز کے بارے میں سوچنے کا صحیح طریقہ جیسے ہی ہم پیچیدہ نمبروں جیسی چیزوں کو دوسرے ڈومینز میں بڑھا رہے ہیں اور اس کے لیے آئیے بس بیک اپ کرتے ہیں۔واپس دروازے کی گھنٹی پر کچھ چیزیں پہنچیں اصل طریقے پر واپس جائیں کہ ہم کفایت شعاری کے خیال کو بڑھاتے ہیں اور صرف اس طرح کے بارے میں سوچتے ہیں جیسے 2 سے X کے دائیں طرف ہم جانتے ہیں کہ قدرتی اعداد کے لیے اس کے بارے میں کیسے سوچنا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "آپ کچھ جانتے ہیں جیسے 2 سے 3 بار بار کی ضرب۔ٹھیک ہے. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "آپ کو عام طور پر سکھایا جاتا ہے کہ 2 سے 1 نصف ایسی چیز ہونی چاہیے جہاں آپ کو معلوم ہو کہ اگر میں اسے خود سے ضرب کرتا ہوں اور یہ معمول کے اصولوں کی پیروی کرتا ہے جو ایکسپونینشل نمبروں کی گنتی کے ساتھ کرتے ہیں جہاں ہم اس قابل ہوتے ہیں کہ اس ایکسپوننٹ میں چیزیں شامل کر سکیں مجھے 2 ملنا چاہیے۔1 کے لیے تو یہ کچھ عدد ہونا چاہیے کہ جب میں اسے خود سے ضرب دیتا ہوں تو مجھے 2 مل جاتا ہے اور آپ جانتے ہیں کہ اس وقت آپ کے پاس کوئی انتخاب ہے، شاید یہ مثبت ہو۔", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "ہوسکتا ہے کہ یہ منفی ہو لیکن اگر آپ ہمیشہ مثبت انتخاب کرنے کا فیصلہ کرتے ہیں تو آپ اسی ڈیل سے ایک اچھا مسلسل فنکشن حاصل کرنے کے قابل ہو جائیں گے اگر ہم منفی نمبروں کے بارے میں پوچھیں کہ 2 سے منفی 1 کو کیا ہونا چاہیے جو کچھ ہونا چاہیے کہاں جب میں اسے 2 سے 1 سے ضرب کرتا ہوں؟ یہ مجھے 2 سے 0 حاصل کرتا ہے اور یہ ہمارے کنونشن کے لئے ایک طرح کا جواز ہے کہ منفی ایکسپونینٹس 1 نصف کی طرح نظر آتے ہیں لیکن یہاں واقعی کیا ہو رہا ہے ہم یہ کہہ رہے ہیں کہ یہ کچھ بھی ہے یہ کسی قسم کا فنکشن ہونا چاہئے جو اس خاصیت کو پورا کرتا ہے۔a جمع b برابر ہے f کے گنا f کے b اور اس کے علاوہ حقیقت یہ ہے کہ بیس 2 ہے بنیادی طور پر ہمیں بتا رہا ہے کہ یہ صرف ایسا کوئی فنکشن نہیں ہے یہ ایک ایسا فنکشن ہے جہاں ہم 1 کو پلگ ان کرتے ہیں تو ہمیں 2 ملتا ہے اور جیسا کہ آپ تھوڑا جانتے ہیں سنٹی چیک اسٹائل سوال یہ دیکھنے کے لیے کہ آیا آپ یہاں کچھ مضمرات کے ساتھ پیروی کر رہے ہیں میں آپ سے پوچھنا چاہتا ہوں کہ میں اسے سافٹ بال کی طرح نہیں کہوں گا، لیکن اس کا مطلب یہ نہیں ہے کہ یہ ایک ناقابل یقین حد تک گہرا سوال ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "لازمی طور پر یہ صرف ایک جانچ کی بات ہے اگر آپ کسی فنکشن کی خصوصیات کے ساتھ خلاصہ شروع کرنے کے خیال کے ساتھ ساتھ پیروی کر رہے ہیں اور پھر اس طرح کے کٹوتی کے طریقے جو ہم اسے ان خصوصیات کی بنیاد پر لکھنا چاہیں گے اگر x کا f اس کفایتی خاصیت کو مطمئن کرتا ہے۔تمام ان پٹ کے لیے ایک جمع بی کے برابر f کے گنا f کے b کے برابر ہے اور یہ 1 کے برابر 2 کے f کو بھی پورا کرتا ہے مندرجہ ذیل میں سے کون سا سچ ہے جس کا کہنا ہے کہ مندرجہ ذیل میں سے کون سا سچ ہے اس سے کوئی فرق نہیں پڑتا ہے کہ آپ کون سا فنکشن شروع کر رہے ہیں آپ کے ساتھ اور آپ میں سے جن کو یاد ہے کہ کون سا لیکچر تھا یہ جو بھی ہے ہم اس بارے میں بات کر رہے تھے کہ یولر کے فارمولے کی تشریح کیسے کی جائے میں نے اس انداز کا ایک سوال پوچھا جہاں میں نے ایک شرط کو نظر انداز کیا، آپ جانتے ہیں کہ میں نے نہیں لکھا۔حقیقت یہ ہے کہ ہم اس بات کو یقینی بنانا چاہتے ہیں کہ x کا f ہر جگہ غیر صفر ہے اور پھر اس کی وجہ سے کچھ مقدار میں کنفیڈلمنٹ پیدا ہوا جو کہ ٹھنڈا ہے اسکرین پر کنفیڈلمنٹ ملتا ہے جو ہم سب کے ساتھ ہوتا ہے لیکن اس کا مقصد بنیادی طور پر یہ ظاہر کرنا تھا کہ اس کی خلاصہ جائیداد کوئی چیز جو اضافے کو ضرب میں بدل دیتی ہے بنیادی طور پر آپ کو اس فنکشن کو لکھنے کے لئے کافی ہے جو بھی اس کے برابر ہے جیسا کہ کسی قسم کی طاقت پر اٹھایا گیا ہے یہ سوال کی روح ہے اب ہمارے پاس پاور ٹاورز کے بارے میں اصل میں کچھ سوالات ہیں ایسا لگتا ہے کہ یہاں پاپ اپ ہوا ہے جو پچھلی بار سے بہت اچھا جڑا ہوا ہے آئیے صرف ایک لمحے کے لئے پاور ٹاور کے سوال کو روکتے ہیں تاکہ ہم سب سے پہلے ایک گہرا احساس حاصل کریں جیسے یہاں وضاحت کا کیا مطلب ہونا چاہئے؟ کیونکہ ہم وہی ہوسکتے ہیں جس کا میں دعویٰ کرنا چاہتا ہوں کہ ہم اس کا جواب متعدد مختلف طریقوں سے دے سکتے ہیں لہذا اگر آپ مجھے صرف ایک دیں تو ہم پاور ٹاورز کے بارے میں بات کریں گے اور پھر جس طرح ایک عدد لکیر کو لاگرتھمک پیمانے میں دکھایا جاسکتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "ہاں درحقیقت، ایک تصور ہے جو میں یہاں صرف ایک لمحے میں حاصل کرنے جا رہا ہوں جہاں ہم اس سے بالکل مماثل کچھ کرتے ہیں کیونکہ ہم جو کچھ کریں گے وہ مختلف ایکسپونیشنل فنکشنز کے ساتھ کھیلنا ہے جو کہ R اوقات X کے X لیکن ہم ہیں R کی اس قدر کو تبدیل کرنے جا رہا ہے جس کی نمائندگی ایک چھوٹے سے پیلے نقطے سے کی جائے گی تو ہم اس کے ذریعے بات کریں گے یہ پورے جہاز کا نقشہ نہیں بنائے گا، بلکہ حقیقی محور اور خیالی محور سے صرف چند نمونے پوائنٹس لیکن خیال یہ ہے کہ جب ہم اس کے ارد گرد گھومتے ہیں کہ وہ مستقل کیا ہے تو ہم مختلف چیزوں کا تصور کرنے کے قابل ہو جائیں گے جو یہ ہوائی جہاز کے ساتھ کرتا ہے اور مؤثر طور پر یہ اس طرح ہے جیسے یہ ایکس محور کو لوگاریتھمک پیمانے میں تبدیل کر رہا ہے اور پھر لپیٹ رہا ہے۔ایک دائرے کے ساتھ خیالی محور اور پھر جیسے ہی R کی وہ قدر خیالی بن جاتی ہے یہ ان حقیقی نمبروں کے کردار کو تبدیل کر دیتی ہے جو دائرے میں ڈالے جاتے ہیں اور خیالی اعداد ایک لاگرتھمک سکیلڈ مثبت محور پر رکھے جاتے ہیں اتنے زبردست سوال جن میں سے تینوں میرا اندازہ ہے میں جہاں جانا چاہتا ہوں اس کے لیے بندوق چھلانگ لگا رہا ہوں لیکن یہ دیکھ کر اچھا لگا کہ لوگ اس میں ایسا سوچ رہے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "آئیے آگے بڑھیں اور صرف اس کی درجہ بندی کریں یہ خیال یہ ہے کہ یہ پلس B کے f کی یہ خاصیت آپ کو بہت سی مختلف چیزوں کا خالصتاً اظہار کرنے دیتی ہے کہ f کا 1 کیا ہے اور صرف اس کے ہجے کرنے کے لیے؟ واضح طور پر 5 کا f ایک ہی چیز ہے جیسا کہ f 1 جمع 1 جمع 1 جمع 1 جمع 1 جو ایک ہی چیز ہے جو 1 کے f کو خود سے 5 بار ضرب دیا جاتا ہے اس خاصیت کی وجہ سے جو کہ اگر f 1 کا 2 ہے تو وہی ہے 2 کی طاقت 5 اور پھر منفی 5 کا f جیسا کچھ ایسا ہونا چاہئے کہ جب ہم اسے 5 کے f سے ضرب دیتے ہیں تو ہمیں 0 کا f جو بھی ملتا ہے اور یہ فوری طور پر واضح نہیں ہوتا ہے کہ 0 کا f کیا ہے لیکن ہم کہہ سکتے ہیں کہ 1 جمع 0 کا f برابر ہے جو بھی f 1 کے برابر ہے جو f 0 کا ہے لیکن 1 کا f 2 کے برابر ہے اور اس طرح یہ بھی 2 کے برابر ہے تو ہم کہہ رہے ہیں کہ 2 برابر ہے 2 گنا کچھ ٹھیک ہے ایک 1 ہونا ضروری ہے لہذا اس تناظر میں یہ اس بات کی ضمانت دیتا ہے کہ منفی 5 کا f 2 سے منفی 5 ہے یہ 1 سے زیادہ 2 سے 5 ویں ہے ہم اسے واضح طور پر 2 سے منفی 5 کے طور پر لکھ سکتے ہیں جس کا مطلب یہ ہے کہ یہ دونوں خصوصیات مل کر بناتے ہیں۔ہم واقعی میں فنکشن کو 2 کے طور پر X لکھنا چاہتے ہیں کیونکہ کوئی بھی گنتی نمبر جو ہم اس میں ڈالتے ہیں اسے پورا کرنے والا ہے ایسا لگتا ہے کہ یہ خود سے ضرب کرے گا اس تعداد کو جتنی بار ہم اس میں ڈالیں گے ان خصوصیات کو پورا کرے گا۔جو ہم چاہتے تھے اور آپ حیران ہوں گے کہ یہ انوکھا ہے اور حقیقی قابل قدر فنکشنز کے تناظر میں یہ حقیقت میں ہوگا لیکن پیچیدہ ویلیو فنکشنز کے تناظر میں اس طرح کے متعدد فنکشنز ہوں گے جن میں سے ایک کے لیے ہم لکھ سکتے ہیں کہ ہم کیا تھے۔اس سے پہلے دیکھ رہا ہوں کہ ہمارے پاس 2 پلس 2 pi کے قدرتی لاگ کو ختم کرنے کے لئے ایک فنکشن ہو سکتا ہے I اس وقت کے تمام اوقات X ٹھیک ہے، یہاں سستی کو معاف کر دیں، میں صرف اس کے بارے میں لکھنے میں پرجوش ہوں اور یہ اصل میں ایک مختلف فنکشن ہے اس بات کا ثبوت ہے کہ اگر آپ X کے برابر 1 نصف پلگ ان کرتے ہیں تو کیا ہوتا ہے ہم نے تھوڑا سا پہلے دیکھا کہ جب آپ 1 نصف میں پلگ کرتے ہیں تو آپ کو 2 کا منفی مربع جڑ حاصل ہوتا ہے اور پھر اگر آپ 1 چوتھے حصے میں پلگ ان کرتے ہیں تو آپ کو اس کی چوتھی جڑ نہیں ملتی ہے۔2 لیکن میں 2 کی چوتھی جڑ کو گنا کرتا ہوں لہذا یہ ایک مختلف فنکشن ہے لیکن یہ پھر بھی ان خصوصیات کو پورا کرتا ہے اور اس طرح ہم اسے 2 سے X لکھنا چاہتے ہیں اور یہ تجویز کرتا ہے کہ شاید 2 سے X ایک مبہم ہے۔bit of notation اور ہمیں صرف R times کے exp کے لحاظ سے سب کچھ لکھنا چاہئے لیکن آپ کو حیرت ہوگی کہ شاید آپ کو معلوم ہو کہ ہم ان تمام افعال کے ساتھ کافی تخلیقی نہیں ہو رہے ہیں جو اس خاصیت کو پورا کرتے ہیں ہو سکتا ہے کہ جب ہم exp لکھتے ہیں تو کوئی ابہام ہو سکتا ہے۔R کے اوقات کچھ اور R کی مختلف اقدار ہیں جو عمل میں آسکتی ہیں لیکن میں صرف ایک چھوٹا سا دعویٰ کرنے والا ہوں اور پھر شاید اس کا خاکہ پیش کروں کہ ثبوت کیسا نظر آئے گا اگر آپ چاہیں تو وہ کون سا ہے کہتے ہیں کہ آپ کے پاس کچھ پیچیدہ فنکشن F ہے، اور یہ پہلے درج ذیل خصوصیات کو پورا کرتا ہے آپ اس کا مشتق لینے کے قابل ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "یہ فرق کرنے والا ہے جو اسے کچھ ہونے سے روکتا ہے جسے آپ مکمل طور پر گندی منقطع چیز جانتے ہیں یہ کچھ بے ترتیب قدروں کو لینے کی طرح ہے اس پر منحصر ہے کہ آپ کو معلوم ہے کہ ویکٹر کی کسی بھی جگہ کا دورانیہ مجھے نہیں معلوم کہ کسری مقداروں کے بارے میں آپ پاگل طریقوں سے سوچنا چاہتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "یہ ایک اچھا فنکشن ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "یہ قابل تفریق ہے یہ ہر جگہ 0 کے برابر نہیں ہے لہذا اس حالت نے میرے دماغ کو پھسلایا اور میں بھول جاتا ہوں کہ کس لیکچر کے لیے یا اس طرح کی کوئی چیز ہے اور پھر اس میں یہ مرکزی خاصیت ہے کہ یہ اضافے کو ضرب میں بدل دیتا ہے اگر آپ کے پاس ایسا فعل ہے تو میں دعویٰ کرتا ہوں کہ ایک انوکھا ہو سکتا ہے کہ میں واقعی میں بتاؤں کہ وہاں ایک منفرد کمپلیکس نمبر R موجود ہے تاکہ آپ X کا F لکھ سکیں کیونکہ بنیادی طور پر R کے اس کفایتی فنکشن کی وجہ سے X کی قدر ہوتی ہے جسے آپ بنیادی طور پر یہ کہتے ہوئے جانتے ہیں کہ اگر آپ کے پاس بطور فنکشن X ہے تو یہ اچھی مشتق خصوصیات کے ساتھ لامحدود کثیر الجہتی اور یہ سب کچھ اگر آپ کے پاس یہ ہے تو آپ کے پاس ہر ایک اسپونینشل ہے جسے آپ ایک خاصیت کی بنیاد پر صرف ایک خاصیت کی بنیاد پر ایک تجریدی عام معنی میں چاہتے ہیں جو ہم اس سے چاہتے ہیں اور ثبوت کا خاکہ کچھ اس طرح دیکھیں اگر آپ پہلے یہ دیکھنا چاہتے ہیں کہ اس قدر کا مشتق کیا ہے جسے ہم فرض کر رہے ہیں کہ ہر جگہ موجود ہے، ٹھیک ہے؟ اور آپ واضح طور پر لکھیں کہ اس کی حد کیا ہے میں یہاں اس پر بہت جلد بات کروں گا ان لوگوں کے لیے جو توقف کو پسند کرنا چاہتے ہیں اور تفصیلات کے ذریعے سوچنا چاہتے ہیں مرکزی جائیداد کے بارے میں آزاد محسوس کریں جو ہم نے ہمیں X کے F کو بڑھانے کی اجازت دی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "ہم ایکسپریشن سے مکمل طور پر X کے F کو فیکٹر کر سکتے ہیں اور پوری حد صرف H کے لحاظ سے ظاہر کی جاتی ہے جس کے بارے میں اگر آپ سوچتے ہیں کہ مشتقات کے تناظر میں اس کا کیا مطلب ہے اور حقیقت یہ ہے کہ 0 کا F لازمی طور پر 1 کے برابر ہے یہ مکمل محدود اظہار ہے۔صرف کچھ مستقل لیکن خاص طور پر یہ جو کچھ بھی ہے 0 پر ہمارے فنکشن کا مشتق ہے تو آپ کے پاس یہ مضحکہ خیز چیز ہے جہاں اگر آپ 0 پر اس کا مشتق جانتے ہیں جو اس بات کا تعین کرتا ہے کہ اس کا مشتق ہر جگہ کیا ہے اور ایکسپونینشل افعال کے تناظر میں یہ امید ہے کہ کافی واقف ہے کیونکہ جو کچھ ہم واقعی کہہ رہے ہیں وہ ایک exponential فنکشن کا مشتق ہے وہ اپنے آپ کے لیے متناسب ہے اور یہ کہ تناسب مستقل برابر ہے جو بھی 0 پر مشتق ہے، یہ سب بہت خلاصہ الفاظ میں ہے اور ایسا ہے لیکن اس کا مقصد اس بات پر زور دینا ہے کہ یہ ہے ضروری نہیں کہ صرف فنکشنز ہوں جن کے بارے میں ہم پہلے ہی پاور X کے بارے میں سوچتے ہیں لیکن یہ فنکشنز کی ایک ممکنہ طور پر بہت زیادہ وسیع کلاس ہے جو اضافے کو ضرب میں تبدیل کرنے کی اس تجریدی خاصیت کو پورا کرتی ہے لیکن اگر آپ کے پاس ہے تو یہ حقیقت میں اس بات کی ضمانت دیتا ہے کہ آپ کے پاس بھی ہے دوسرا مشتق اور اس معاملے کے لئے تیسرا مشتق اور اس طرح کہ مشتق فعل صرف اپنے آپ کے متناسب ہے لہذا نواں مشتق لینے کے لئے آپ صرف اس متناسب مستقل کو دیکھیں اور اسے طاقت n تک بڑھا دیں اور پھر یہاں سے آپ ایک کر سکتے ہیں۔ٹیلر سیریز کی توسیع اور میں اسے آپ میں سے ان لوگوں کے لیے جدید ہوم ورک کے طور پر چھوڑ سکتا ہوں جو اس آئیڈیا میں ٹیلر سیریز سے راضی ہیں خاص طور پر اگر آپ کسی بھی ڈیفرینٹی ایبل فنکشن کے آئیڈیا کو آپس میں ملانا چاہتے ہیں جو پیچیدہ نمبروں کے لحاظ سے مختلف ہو، جو یقینی طور پر کالج کے موضوع کی طرح آپ جانتے ہیں کہ آپ اپنی مرضی کے مطابق وہاں استدلال کو آپس میں ملا سکتے ہیں لیکن مبہم استدلال کی اجازت کسی ایسے شخص کے تناظر میں ہے جو صرف ٹیلر سیریز کے بارے میں جانتا ہے اور اس خیال کو لینے کے لیے اور کچھ نہیں اور F اور کے لیے ٹیلر کی توسیع کو دیکھیں۔اس خیال کا جواز پیش کرتے ہیں کہ ایک منفرد کمپلیکس نمبر ہے جیسا کہ ہمارے فنکشن F کو لازمی طور پر اس طرح لکھا جا سکتا ہے اور پھر نارمل ایکسپونینشلز سے تعلق جب بھی آپ کے پاس اتنی قدر ہوتی ہے R ہم بنیادی طور پر وہی کرتے ہیں جو ہم حقیقی اعداد کے پیچیدہ تناظر میں کرتے ہیں۔اگر آپ اس قدر R کے اس فنکشن کے exp کو دیکھیں اور اسے بیس کے طور پر لکھیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "ہم اس کی تشریح کر سکتے ہیں اس کا مطلب صرف pi halves I times X کا ہی نہیں ہے، بلکہ ہم اس کا مطلب یہ بھی کر سکتے ہیں کہ 5 pi halves I Times X اور یہ الگ الگ فنکشنز ہیں اور الگ الگ فنکشنز کا ایک لامحدود کنبہ ہے جو ایسا لگتا ہے جیسے ہمیں کرنا چاہیے۔انہیں X میں I کے طور پر لکھیں لہذا اظہار I to the I جب تک کہ آپ نے اس کے لئے کوئی معیار نہیں اپنایا ہے جس کا مطلب ہونا ضروری ہے جب آپ کہتے ہیں کہ اس میں لامحدود طور پر بہت سارے آؤٹ پٹ ہیں اس کے بارے میں سوچنے کا ایک اور طریقہ یہ ہے کہ فنکشن I to the X ہمارے پاس موجود اشارے کے ساتھ تھوڑا سا مبہم ہے اب اس سب کے ساتھ آئیے اس میں سے کچھ کو تصور کرنا شروع کرتے ہیں کیونکہ مجھے لگتا ہے کہ یہ تفریحی ہے اور آپ جانتے ہیں کہ آپ مجھے بتائیں کہ آیا یہ مددگار بصری ہے یا زیادہ الجھا ہوا بصری لیکن ہم کیا کرنے جا رہے ہیں R times X کے اس فنکشن کو دیکھیں، جو کہ بنیادی طور پر یہ ہے X کی طاقت پر e لکھنے کا ایک اور طریقہ درحقیقت میرے خیال میں I سوچتا ہوں کہ میں نے کسی موقع پر ایک مختلف اینیمیشن پیش کی ہے جس میں اس کی وضاحت کی گئی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "کیونکہ میں ایسا کرنے کی منصوبہ بندی کرنے کا منصوبہ بنا رہا تھا تو مجھے اجازت دیں کہ آپ میرے فائل سسٹم میں واپس جا رہے ہیں جہاں آپ کو ہونا چاہیے تھا وہاں پہنچ جاؤ وہاں یہ شکایت کر رہا ہے کیونکہ وہاں بہت سے مختلف ہیں یہ ایسا ہونے والا ہے جیسے ایک ہے اوہ بدلیں یہ دوسری سکرین پر ظاہر ہوتا ہے انتظار کیوں ہے ہاں، ٹھیک ہے بدل دیں؟ جو کچھ بھی آپ وہاں دیکھتے ہیں اسے رکھیں اور اب ہم اوہ وہاں واپس چلے جاتے ہیں ہم وہ سب کچھ صرف اتنا ہے کہ میں اچھی طرح سے لکھ سکتا ہوں اگر آپ اسے R اوقات X کے exp کے طور پر سوچنے میں بے چینی محسوس کرتے ہیں تو اس لامحدود کثیر الثانی صرف میں آپ کے سر کے پیچھے ای R ٹائم X کی طرف اور ہم R کے ارد گرد مختلف ہونے والے ہیں لہذا میں خیالی محور کے پوائنٹس کی پیروی کرنے والا ہوں، اور میں حقیقی محور کے پوائنٹس کی پیروی کرنے والا ہوں اور آئیے دیکھتے ہیں کہ یہ کیا کرتا ہے یہ تمام قسم کی تیز رفتار ہے لہذا مجھے اس کے ذریعے تھوڑا سا اور آہستہ آہستہ سوچنے دیں تمام منفی نمبرز کچھ بھی یہ ہے کہ ایک منفی حقیقی نمبر 0 اور 1 کے درمیان کی حد میں ٹکرا جائے گا جس کا مطلب e کو منفی ہونا چاہئے؟ a سے منفی حقیقی نمبر 0 اور 1 کے درمیان ہے اور ہم خاص طور پر منفی 1 کے f کو ٹریک کر رہے ہیں جو 1 سے زیادہ e 30 0 کے ارد گرد ظاہر ہونے والا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "1 کا 37 ایف ای پر اترتا ہے جیسا کہ توقع کی جاتی ہے کہ 1 کا exp وہی ہے جو I یونٹ کے دائرے کے گرد ایک ریڈین اترنے والا ہے، اور یہاں پورے خیالی محور کے ساتھ چلنے میں ایک طرح کا مزہ آتا ہے کہ خیالی محور دائرے کے گرد کیسے لپٹا جاتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "ہم یہاں R کی قدریں چاہتے ہیں اور یہ چیزوں کو مختلف طریقے سے پھیلاتا ہے لہذا جب ہم اسے 2 تک رکھتے ہیں تو آپ جانتے ہیں کہ یہ اصلی محور کو بہت زیادہ پھیلا دیتا ہے تاکہ f کا 1 اس کے ارد گرد ختم ہوتا ہے جہاں e مربع منفی کے 7 f سے تھوڑا اوپر ہوتا ہے۔1 I کے 0 f کے بہت قریب ہے منفی کے دائرے f کے گرد 2 ریڈین گردش ہے I ایک منفی 2 ریڈین گردش ہے اور یقینا ہم اپنے پسندیدہ فارمولے تک پہنچ سکتے ہیں کہ اگر یہ pi ہوتا جو ہمارے پاس اپنی اسکیلنگ مستقل کے طور پر ہوتا تو پھر اصلی محور کافی حد تک پھیلا ہوا ہے آپ جانتے ہیں کہ f کا 1 e پر pi کے قریب بیٹھا ہے جو 20 جمع pi کے بہت قریب ہے جو ہمیشہ مزے کا ہوتا ہے اور منفی 1 کا f 0 کے بہت قریب ہوتا ہے لہذا یہ حقیقت میں پھیلا ہوا ہے کہ اصلی axis اور اس نے یونٹ کے دائرے کی سمت میں چیزوں کو بھی پھیلا دیا ہے تاکہ I کا f یا منفی I کا f تک پہنچ کر دائرے کے ارد گرد آدھے راستے پر چلیں، تو یہ سب ٹھیک اور اچھا ہے اب ہم کسی فنکشن کے بارے میں کیسے سوچیں گے؟ 2 سے X کیا ہے؟ ہم 2 بار X کے قدرتی لاگ کے X کے X کے طور پر بھی لکھیں گے لہذا ہم اپنے پیلے رنگ کے ڈاٹ کو منتقل کرتے ہیں جو R کی قدر کو 0 کے قریب ظاہر کرتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "69 اب بھی کوئی خیالی حصہ نہیں صرف ایک حقیقی نمبر 0۔69 یا اس طرح یہ 2 کا قدرتی لاگ ہے آپ دیکھ سکتے ہیں کہ 1 کا f 2 پر اترتا ہے اسی وجہ سے ہم اس فنکشن 2 کو X f کے 1 نصف پر کہنا چاہتے ہیں اصل میں معذرت کے ساتھ منفی 1 زمینوں کے 1 نصف ایف پر درست ہے I یہ یونٹ کے دائرے میں کچھ چہل قدمی کر رہا ہے خاص طور پر یہ 0 ہونے والا ہے۔یونٹ کے دائرے کے گرد 69 ریڈینز اور اب ہم تھوڑا سا مزید مزہ لے سکتے ہیں اور کہہ سکتے ہیں کہ کیا ہوگا اگر ہم اسے 0 ہونے کی بجائے تبدیل کر دیں۔69 کو 2 کا قدرتی لاگ بننے کے بجائے اسے 2 کے قدرتی لاگ سے گنا بناتا ہے تاکہ ہم واقعی کسی ایسی چیز کے بارے میں سوچ رہے ہوں جس کی اس کی بنیاد ہو سکتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "پانچویں کے ارد گرد 2 لیکن بہت سے مختلف ایکسپونینشل فنکشنز ہیں جن میں 1 کے f کو نمبر I پر ڈالنے کی یہ خاصیت ہوگی لہذا اگر ہم اسے مزید بڑھانا چاہیں تو مجھے نہیں لگتا کہ میں نے اسے یہاں اینیمیٹ کیا ہے لیکن اگر ہم لیں وہ پیلے رنگ کے نقطے اور اسے اوپر کریں یہاں تک کہ یہ 5 نصف گنا تک پہنچ جائے pi I آپ کیا دیکھیں گے کہ یونٹ کا دائرہ ہے؟ اپنے اردگرد گھمایا جاتا ہے تاکہ 1 کے منفی f کا f دوسرے 2 pi ریڈینز کے گرد گھومے اور جہاں یہ ہے وہاں اترے لیکن یہ اصلی محور کو بہت زیادہ پھیلا دے گا جس کا مطلب یہ تھا کہ I سے I کا ایک اور آؤٹ پٹ ہے ایک بہت چھوٹی تعداد یہ 0 کے آس پاس تھی۔", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "سمت؟ ٹھیک ہے، اور میں سب سے پہلے کیا کروں گا کہ میں اسے pi I یونٹس کے ذریعے اوپر لے جاؤں گا اب یہاں کیا ہو رہا ہے؟ ہمارے پاس R کا X X کا X ہے اور R اس قدر کے برابر ہے، جو کہ 2 پلس pi گنا I کا قدرتی لاگ ہے اس کا مطلب یہ ہے کہ جب ہم 1 f کا 1 پلگ ان کرتے ہیں تو منفی 2 ہوتا ہے تو ہم اس فنکشن کو لکھنا چاہتے ہیں۔پاور X کے لیے منفی 2 کے طور پر صحیح ہے اور یہ حقیقت میں وہ چیز ہے جسے آپ جانتے ہیں، یہ تھوڑا سا دھوکہ دہی سے آسان ہے جب ہم پاور X پر منفی نمبر لکھتے ہیں منفی 2 پاور X کے لیے یہ پہلے ایسا نہیں لگتا کہ ضروری ہے کہ یہ ہمیں لے آئے پیچیدہ اعداد میں کسی بھی طرح سے لیکن یقیناً جب ہم 1 نصف جیسی قدر بھی لگاتے ہیں جہاں ہم منفی 2 کا مربع جڑ مانگتے ہیں تو ہمیں احساس ہوتا ہے کہ ہم اسے اس طرح لکھنا چاہتے ہیں جیسے میں مربع جڑ کو گنا کرتا ہوں۔2 کا لیکن اگر آپ مکمل پیچیدہ ڈومین میں پاور X کے لیے اس فنکشن کو منفی 2 کو دیکھیں کہ یہ آپ جس چیز کو دیکھ رہے ہیں وہ ایک فنکشن ہے جو 1 کی قدر کو منفی 2 تک لے جاتا ہے اور اگر ایسا ہوتا ہے تو کیا ہوتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "یہ باقی حقیقی نمبر لائن کے ساتھ کرتا ہے کیا یہ اس طرح سے باہر کی طرف گھومتا ہے؟ تو ہم دیکھتے ہیں کہ منفی 1 کا f منفی 1 نصف پر بیٹھتا ہے جس کے بارے میں آپ توقع کریں گے کہ اگر آپ 1 نصف کے f کی پیروی کریں گے تو یہ بالکل خیالی لکیر پر بیٹھے گا اور 1 نصف کا f 2 کا مربع جڑ ہوگا۔ماؤس وہ جگہ نہیں ہے جہاں میں اسے ہونا چاہتا ہوں۔", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "یہ 2 گنا I کے مربع جڑ کے ارد گرد ہوگا اور جیسا کہ آپ اس پر آگے بڑھتے ہیں آپ کو منفی 2 سے X کی تمام حقیقی قدر کی طاقتیں دکھا رہا ہے یہ ضروری طور پر اس کے گرد گھومتا ہے لیکن ہم R کی اپنی قدر کو اس سے بھی زیادہ بڑھا سکتے ہیں اور اسے حاصل کر سکتے ہیں۔تقریباً ٹاؤ ٹائم تک I لگ بھگ چھ پوائنٹ دو آٹھ بار I اور اس تناظر میں یہ ایک اور فنکشن ہے جسے ہم 2 سے X کی طرح لکھنا چاہیں گے کیونکہ کسی بھی پورے نمبر سے پورے نمبر کے لیے جو آپ X کے لیے پلگ ان کریں گے بار بار ضرب کی طرح نظر آتا ہے اور اس میں 1 نصف جیسی چیزوں کے لیے بھی معقول قدریں ہوتی ہیں جہاں یہ مثبت مربع جڑ کی بجائے منفی مربع جڑ کو تھوک دیتا ہے، لیکن یہ اصل میں اس جہاز میں تبدیلی ہے جہاں یہ سب کچھ رکھتا ہے اصلی ہے۔نمبر لائن ایک بہت مضبوط زخم سرپل بن کر ختم ہوتی ہے جو گھومتی ہے اور یہ صرف اس طرح گھومتی ہے کہ 1 کا f دائیں نمبر 2 پر اترتا ہے لہذا یہ اس معنی میں ہے کہ ہم X کو 2 کہہ سکتے ہیں اس کی تعبیر واضح طور پر کی جاتی ہے۔اس سے ایک الگ exponential فنکشن جس کے ہم روایتی طور پر عادی ہیں، اس لیے میں سوچتا ہوں کہ ان سب کے ساتھ میں آج کے لیے چیزیں چھوڑ دوں گا اور میں آپ کے لیے صرف چند سوالات کے ساتھ چھوڑ دوں گا کہ ٹھیک ہے، اس لیے اگر آپ چاہتے ہیں I to the I کے بارے میں سوچیں کہ یہ ایک کثیر قدری اظہار ہے، آپ کہہ سکتے ہیں کہ ہم ایک کنونشن کو اپناتے ہیں، آپ یہ کہہ سکتے ہیں کہ آپ قدرتی لوگارتھم فنکشن کی ایک شاخ کا انتخاب کرتے ہیں اور ہوسکتا ہے کہ یہ آپ کو منفی pi کے وجود میں بند کردے۔", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "halves لیکن اگر آپ کہتے ہیں کہ اس قسم کی لامحدود بہت سی مختلف اقدار بننا چاہتی ہیں جیسے کہ ہم نے دیکھا کہ 2 سے 1 تہائی کتنی قدریں ایک ہی معنی میں ہونا چاہتے ہیں؟ جہاں ہم 2 کو e سے X کے لیے مختلف مختلف مختلف اختیارات کے ساتھ بدل رہے ہیں جیسے کہ e سے X برابر 2 کتنی مختلف اقدار بننا چاہتا ہے یا 2 سے 3 کی کتنی قدریں ہیں؟ 10 ویں کو تمام میں سے مختلف طریقے سے فقرے کرنے کی خواہش ہے، میں X کے تمام ایکسپینینشنل فنکشنز کے بارے میں کہوں جو اوہ کو پورا کرتے ہیں، کیا میں نے اسے X کا کہیں f لکھا ہوا ہے جو ان تمام خصوصیات کو پورا کرتا ہے جو میں نے لکھا ہے، اگر یہ سب کو مطمئن کرتا ہے ان میں سے اور اگر 1 کا f 2 کے برابر ہے تو ہمیں کتنے مختلف آؤٹ پٹ ملیں گے جب ہم X کے برابر 3 10ویں نمبر پر پلگ ان کریں گے تو کس فنکشن کے لیے مختلف آپشنز ہیں؟ یہ ہے اور ہم کتنے آؤٹ پٹ حاصل کرنے جا رہے ہیں؟ مختلف فنکشنز کے لیے 2 سے pi کے لیے جو 2 سے X کی نمائندگی کر سکتا ہے اگر ہم 2 سے X کو کسی قسم کے ایکسپونیشنل فنکشن کے طور پر سوچ رہے ہیں اس طرح کے تجریدی خصوصیات کے لحاظ سے اور اگر ہم ہاں، اگر ہم اگر ہمارے پاس اس طرح کے مختلف فنکشنز کی ایک کلاس ہے، اور ہم pi کو پلگ ان کرنا چاہتے ہیں اس سے مجھے ہنسی آتی ہے صرف اس وجہ سے کہ یہ ایک ایسا مضحکہ خیز جواب ہے جو آپ کے بارے میں سوچنے کی کوشش کرتے ہی سامنے آتا ہے تو یہ وہ سوالات ہیں جو میں آپ کو چھوڑ دوں گا اور مجھے لگتا ہے کہ آپ یہ جانتے ہیں کہ آج کے لیکچر تک پہنچنے میں میرا مرکزی سوال یہ تھا کہ کیا میں یہ چاہتا ہوں کہ اس طرح کی وضاحتی خصوصیات کی ان تجریدی خصوصیات کی طرح ہوں اور یہ میرے لئے بہت اچھا ہے کہ ان تجریدی خصوصیات سے شروع کرنا آپ e to the rx یا اس سے زیادہ کے خیال میں بند ہو جاتے ہیں بس آپ جانتے ہیں کہ میں سمجھتا ہوں کہ r کی مختلف اقدار کے لیے r ٹائم ایکس کی زیادہ ایمانداری سے لکھی گئی ہے کہ یہ آپ کو اس دور میں بند کر دیتا ہے لیکن یہ آپ کو اس حد تک بند نہیں کرتا جہاں تک ہے ایک غیر مبہم تصور کیا ہے کہ 2 کو پاور x سے بہت کم ہونا چاہئے جیسا کہ I to power x میں خطرہ یہ ہے کہ بعض اوقات لوگ تجرید کو پسند نہیں کرتے ہیں اور بعض اوقات یہ قابل رسائی نہیں ہوتا ہے لیکن اگر یہ ہے اگر آپ جانتے ہیں کہ آپ صرف مجھے بتائیں میرے خیال میں میرے خیال میں خیالات کا ایک پورا دلچسپ حلقہ ہے جو پاور ٹاورز کو شامل کرنے کے لیے ان تمام چیزوں کو گھیرے ہوئے ہے کیونکہ اگر آپ اصل میں پاور ٹاورز کے بارے میں بات کرنا چاہتے ہیں جیسا کہ ہم پچھلی بار پیچیدہ نمبروں کے تناظر میں تھے۔", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "یا یہاں تک کہ منفی بنیادوں کے ساتھ بھی آپ کو اس طرح کی چیزوں کے بارے میں سوچنا پڑتا ہے، تو یہ ایک سوال تھا جو ہم نے اسکرین پر اٹھایا تھا ہاں، اگر ہم I to the power I کے لیے ایسا کرتے ہیں تو کیا ہوگا؟ ٹائٹریشن آپ کو معلوم ہے آئیے صرف اس کی کوشش کریں آئیے آگے بڑھیں اور ایک پاور ٹاور کو آزمائیں جہاں ہم مجھے ایک دی گئی طاقت پر بڑھا رہے ہیں اور دیکھیں کہ اس سے کیا نکلتا ہے، لہذا یہ ایسا کرنے کی منصوبہ بندی نہیں کر رہا تھا لیکن ہم ہمیشہ کر سکتے ہیں پائتھون کو کھینچیں اور بنیادی طور پر وہی کریں جو ہم پچھلی بار کر رہے تھے تو جس طرح سے یہ کام کرے گا وہ یہ ہے کہ ہم کچھ بنیادی قیمت کے ساتھ شروع کر رہے تھے اور پھر کسی قسم کی حد کے لئے ہم کیا کر رہے تھے ہم ایک لے رہے تھے اور ہم دوبارہ تفویض کرنے جا رہے ہیں یہ جو کچھ بھی ہو اس معاملے میں جس بنیاد کو میں نے a کی طاقت تک بڑھایا ہے وہ ٹھیک ہے، ٹھنڈا ہونا چاہیے، تو ہم ایسا کرنے جا رہے ہیں اور پھر ہم اس کی قدر پرنٹ کرنے جا رہے ہیں ہاں، یہ 200 کی طرح ایک بہت بڑا نمبر ہے تو ایسا لگتا ہے کہ کیا ہوتا ہے ان چیزوں کے ساتھ افراتفری کا امکان ہے جیسے کبھی کبھی۔", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "میرے پاس اصل میں ہے لہذا مجھے NumPy کو درآمد کرنے دیں تاکہ میرے پاس ایکسپونینشل فنکشن ہے مجھے ہماری بڑی رینج کے لیے جانے دیں جیسا کہ ہمارے پاس پہلے تھا اسے لکھنے کے بجائے جیسا کہ آپ کچھ جانتے ہیں جیسا کہ میں X کی طاقت تک ہے میں اسے لکھنے والا ہوں۔ایک مختلف مستقل دائیں ایک مختلف مستقل کے ایکسپونینشنل فنکشن کے طور پر جسے میں بنانے والا ہوں میں اسے 5 pi نصف کرنا چاہتا ہوں، لہذا میں 5 pi نصف گنا کروں گا لہذا یہ ایک پیچیدہ نمبر ہے اور اس کے 5 pi نصف حصے ہیں خیالی حصہ تو یہ ہے 5 pi نصف اوقات I اور میں کیا کر رہا ہوں؟ میں اس کی وضاحت کر رہا ہوں اس لیے میں وہاں کے اندر سے ضرب لگانا چاہتا ہوں، ٹھیک ہے؟ یہ بنیادی طور پر ایک اور طریقہ ہے جس سے آپ ایکسپریشن I کو X کی تشریح کر سکتے ہیں شکر ہے کہ آپ کہیں گے کہ آپ نے قدرتی لاگ فنکشن کی ایک مختلف شاخ کا انتخاب کیا ہے لیکن یہ ایک اور فنکشن ہے جسے ہم خود اعادہ کر سکتے ہیں اور دیکھ سکتے ہیں کہ کیا ہوتا ہے اور ہمیں مل سکتا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-i-to-i/vietnamese/sentence_translations.json b/2020/ldm-i-to-i/vietnamese/sentence_translations.json index bdc6a41e2..601c16947 100644 --- a/2020/ldm-i-to-i/vietnamese/sentence_translations.json +++ b/2020/ldm-i-to-i/vietnamese/sentence_translations.json @@ -1120,7 +1120,7 @@ "end": 998.2 }, { - "input": "So if you're starting off at the number 1, your initial velocity is to walk straight towards 0 and as you walk even lower, if you were sitting at 1 half, then you would still be walking towards 0, but now your velocity vector would be negative 1 times where you are, which is negative 1 half. ", + "input": "So if you're starting off at the number one, your initial velocity is to walk straight toward zero. And as you walk even lower, if you were sitting at one half, then you would still be walking towards zero, but now your velocity vector would be negative one times where you are, which is negative one half. ", "translatedText": "Vì vậy, nếu bạn bắt đầu ở số 1, vận tốc ban đầu của bạn là đi thẳng về phía 0 và khi bạn đi thấp hơn nữa, nếu bạn đang ngồi ở số 1, thì bạn vẫn sẽ đi về phía 0, nhưng bây giờ vectơ vận tốc của bạn sẽ âm 1 lần nơi bạn ở, tức là âm 1 nửa. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 1381.54 }, { - "input": "And an interesting question will be you know is is there just one such function that feels reasonable to write for this because you know if we're gonna write it as i to the x not only should it satisfy this it should also satisfy you know when we plug in the number one we get i presumably i to the power one however we're thinking of this function should be i. ", + "input": "And an interesting question will be, you know, is there just one such function that feels reasonable to write for this? Because, you know, if we're going to write it as i to the x not only should it satisfy this, it should also satisfy, you know, when we plug in the number one we get i, presumably i to the power one, however we're thinking of this function should be i. ", "translatedText": "Và một câu hỏi thú vị mà bạn biết là liệu chỉ có một hàm như vậy mà bạn cảm thấy hợp lý để viết cho cái này bởi vì bạn biết liệu chúng ta có viết nó dưới dạng i cho x không chỉ nó phải thỏa mãn điều này mà còn thỏa mãn bạn biết khi nào chúng tôi cắm số một mà chúng tôi nhận được tôi có lẽ là tôi vào số nguồn tuy nhiên chúng tôi đang nghĩ chức năng này phải là i. ", "model": "google_nmt", "n_reviews": 0, @@ -1496,7 +1496,7 @@ "end": 1415.44 }, { - "input": "So we've got 5 pi i halves great that absolutely is another value that we could plug in for x here and just to spell out that a little bit more visually if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves which is 1.57 what if instead we took another full turn and we go another pi halves to get us to pi which you know we might kind of record that's where the e to the pi i value is we walk another pi halves we walk another pi halves which at this point we would have gone a full circle getting us back to one and then we walk for five pi halves which numerically is about 7.85 yeah, that absolutely is another number that gets us on top of i and if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the 5 pi halves i to the power i those i's multiply to become negative and we'd be looking at e to the negative 5 pi halves which is a very different number right we can actually calculate this I'm not sure off the top of my head, but let's take a look at a Desmos. ", + "input": "So we've got five pi i halves. Great, that absolutely is another value that we could plug in for x here. And just to spell out that a little bit more visually, if we were to look back at our circle here where we've at the moment walked for an amount of time equal to pi halves, which is 1.57, what if instead we took another full turn and we go another pi halves to get us to pi, which you know, we might kind of record, that's where the e to the pi i value is. We walk another pi halves, we walk another pi halves, which at this point we would have gone a full circle getting us back to one, and then we walk for five pi halves, which numerically is about 7.85. Yeah, that absolutely is another number that gets us on top of i. And if we were to go through the whole rigmarole of re-expressing i to the power i by first writing e to the five pi halves i to the power i, those i's multiply to become negative and we'd be looking at e to the negative five pi halves, which is a very different number, right? We can actually calculate this. I'm not sure off the top of my head, but let's take a look at Desmos ma ", "translatedText": "Vì vậy, chúng ta có 5 pi i một nửa tuyệt vời đó hoàn toàn là một giá trị khác mà chúng ta có thể thay vào cho x ở đây và chỉ để đánh vần nó một cách trực quan hơn một chút nếu chúng ta nhìn lại vòng tròn của chúng ta ở đây, nơi chúng ta có khoảnh khắc đã đi trong một khoảng thời gian bằng nửa số pi là 1.57 điều gì sẽ xảy ra nếu thay vào đó chúng ta thực hiện một lượt đầy đủ nữa và chúng ta đi thêm một nửa số pi khác để đến số pi mà bạn biết rằng chúng ta có thể ghi lại rằng giá trị của e đối với pi i là chúng ta đi theo một nửa pi khác chúng ta đi một nửa pi khác mà tại đó Lúc này chúng ta sẽ đi một vòng tròn để quay về số một và sau đó chúng ta đi bộ năm nửa số pi, số đó là khoảng 7.85 vâng, đó chắc chắn là một con số khác giúp chúng ta vượt lên trên i và nếu chúng ta thực hiện toàn bộ sự chặt chẽ của việc biểu diễn lại i lũy thừa i bằng cách trước tiên viết e mũ 5 pi nửa i lũy thừa i những số đó tôi nhân để trở thành số âm và chúng ta sẽ xét e với 5 nửa pi âm, đây là một con số rất khác phải không, chúng ta thực sự có thể tính được số này. Tôi không chắc chắn lắm, nhưng chúng ta hãy nhìn vào Desmos . ", "model": "google_nmt", "n_reviews": 0, @@ -1536,7 +1536,7 @@ "end": 1544.74 }, { - "input": "That long which gets you to a much smaller number But that's not the only answer that we could enter right we have other people coming in here with negative 3 halves times i pi Which you know in terms of a unit circle? ", + "input": "hat long, which gets you to a much smaller number. But that's not the only answer that we could enter, right? We have other people coming in here with negative three halves times i pi. Which, you know, in terms of a unit circle, ", "translatedText": "Độ dài đó sẽ đưa bạn đến một số nhỏ hơn nhiều Nhưng đó không phải là câu trả lời duy nhất mà chúng ta có thể nhập đúng, chúng ta có những người khác đến đây với âm 3 nửa nhân i pi Bạn biết gì về vòng tròn đơn vị? ", "model": "google_nmt", "n_reviews": 0, @@ -1544,7 +1544,7 @@ "end": 1557.36 }, { - "input": "We could think of as saying hey if I want to get to I rather than walking 90 degrees pi halves radians that way what if I walk 270 degrees the other way 3 pi halves radians which maybe I'll think of as negative because the convention is usually that counterclockwise is positive That absolutely is another way to express it and that would get us a different answer if we had e to the negative 3 pi halves i All to the power i we go through the same game now the i squared cancels with a negative that's already there, and we have a positive 3 pi halves and Numerically this gets us an even different looking answer from what we had before Which if we go over and we say hey, what is e to the 3 pi not 3 o 3 pi halves 111 point 3 1 very different kind of number than what we saw before 111 point what was it 111 point 3 1 great 111 point 3 1 or so And again in terms of the intuition what you might be asking there is suppose we have this rotating dynamic But we move backwards in time we see how long ago in time what I have to be Such that if I played things forward from there I would land on the number one my initial condition and You have to go back in time 3 pi halves units And then if you were to translate to the decay dynamics Which is what raising to the eye is doing in this context you say if I'm starting at the number one But I want to move backwards in time and say Where should I have started if I want to decay down such that I end up at the number one? ", + "input": "we could think of as saying, hey if I want to get to i, rather than walking 90 degrees, pi halves radians that way, what if I walk 270 degrees the other way? Three pi halves radians, which maybe I'll think of as negative because the convention is usually that counterclockwise is positive. That absolutely is another way to express it. And that would get us a different answer. If we had e to the negative three pi halves i, all to the power i, we go through the same game. Now the i squared cancels with the negative that's already there, and we have a positive three pi halves. And numerically this gets us an even different looking answer from what we had before, which if we go over and we say hey, what is e to the three pi, not three o, three pi halves 111.31. Very different kind of number than what we saw before. 111 point, what was it? 111.31. Great. 111.31 or so. And again in terms of the intuition, what you might be asking there is, suppose we have this rotating dynamic, but we move backwards in time. We see how long ago in time would I have to be, such that if I played things forward from there, I would land on the number one, my initial condition. And you have to go back in time three pi halves units. And then if you were to translate to the decay dynamics, which is what raising to the i is doing in this context, you say if I'm starting at the number one, but I want to move backwards in time and say where should I have started if I want to decay down such that I end up at the number one a ", "translatedText": "Chúng ta có thể nghĩ đến việc nói này nếu tôi muốn đến chỗ tôi thay vì đi bộ 90 độ pi nửa radian theo cách đó thì sao nếu tôi đi 270 độ theo cách khác 3 pi nửa radian mà có lẽ tôi sẽ coi là tiêu cực vì quy ước là thường thì ngược chiều kim đồng hồ là dương Đó hoàn toàn là một cách khác để diễn đạt nó và điều đó sẽ cho chúng ta một câu trả lời khác nếu chúng ta có e âm 3 nửa pi i Tất cả là lũy thừa i chúng ta chơi cùng một trò chơi bây giờ bình phương i hủy với a âm đã có sẵn ở đó, và chúng ta có 3 nửa dương và về mặt số học, điều này mang lại cho chúng ta một câu trả lời thậm chí còn khác với những gì chúng ta có trước đó. Nếu chúng ta đi qua và nói này, e mũ 3 pi là bao nhiêu chứ không phải 3 o 3 pi chia đôi 111 điểm 3 1 loại số rất khác so với những gì chúng ta thấy trước 111 điểm nó là gì 111 điểm 3 1 tuyệt vời 111 điểm 3 1 hoặc hơn Và một lần nữa xét về mặt trực giác, những gì bạn có thể hỏi, giả sử chúng ta có sự luân chuyển này năng động Nhưng chúng ta di chuyển ngược thời gian chúng ta thấy cách đây bao lâu thời gian tôi phải là gì Như vậy nếu tôi chơi mọi thứ về phía trước từ đó tôi sẽ đạt đến số một điều kiện ban đầu của tôi và Bạn phải quay ngược thời gian 3 pi nửa đơn vị Và sau đó, nếu bạn dịch sang động lực học phân rã Đó là những gì mà mắt ta nhìn thấy trong bối cảnh này thì bạn nói nếu tôi bắt đầu từ số một Nhưng tôi muốn quay ngược thời gian và nói Tôi nên bắt đầu từ đâu nếu Tôi muốn phân rã đến mức cuối cùng tôi đứng ở vị trí số một? ", "model": "google_nmt", "n_reviews": 0, @@ -1552,7 +1552,7 @@ "end": 1657.18 }, { - "input": "After 3 pi halves units of time the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going where there's actually infinitely many different values that we could Plug in for X if we're thinking of e to the X as being I and people have entered a lot more here Excuse me throwing my pin onto the ground as one does classic for third place 9 pi halves great choice 1729 pi halves y'all are my favorite lots and lots of different options infinitely many different values which feels a little Disconcerting at first right because we look at an expression That seems like you know there's just gonna be some computation I just plug that into my calculator and see what pops out and we've got multiple different values for it So what's going on here right? ", + "input": "fter three pi halves units of time? the answer is evidently starting at around a hundred and eleven for that kind of exponential decay And you can see where this is going, where there's actually infinitely many different values that we could plug in for x if we're thinking of e to the x as being i. And people have entered a lot more here. Excuse me, throwing my pen onto the ground as one does. Classic for third place. Nine pi halves, great choice. 1729 pi halves, y'all are my favorite. Lots and lots of different options, infinitely many different values, which feels a little disconcerting at first, right? Because we look at an expression that seems like, you know, there's just going to be some computation. I just plug that into my calculator and see what pops out. And we've got multiple different values for it. So what's going on here, right? ", "translatedText": "Sau 3 pi nửa đơn vị thời gian, câu trả lời rõ ràng là bắt đầu từ khoảng 111 cho kiểu phân rã theo cấp số nhân đó Và bạn có thể thấy điều này sẽ đi đến đâu khi thực sự có vô số giá trị khác nhau mà chúng ta có thể Thay vào cho X nếu chúng ta nghĩ về e đến X như tôi và mọi người đã tham gia nhiều hơn ở đây Xin lỗi, tôi đã ném chiếc ghim của mình xuống đất như một người làm cổ điển cho vị trí thứ ba 9 nửa pi sự lựa chọn tuyệt vời 1729 nửa pi các bạn đều là rất nhiều và rất nhiều yêu thích của tôi các tùy chọn khác nhau vô cùng nhiều giá trị khác nhau mà ban đầu cảm thấy hơi bối rối vì chúng ta nhìn vào một biểu thức Có vẻ như bạn biết rằng sẽ có một số phép tính Tôi chỉ cần cắm nó vào máy tính của mình và xem những gì hiện ra và chúng ta có nhiều lựa chọn khác nhau giá trị cho nó Vậy chuyện gì đang xảy ra ở đây phải không? ", "model": "google_nmt", "n_reviews": 0, @@ -1632,7 +1632,7 @@ "end": 1794.86 }, { - "input": "The fourth root of 16 should be 2 and the answer ends up being well We adopt a convention when there's multiple options like this when you have a multi-valued function We often just choose one of those values to be what we mean when we want to treat it as a function as something with a single input and a single output in fancier lingo This comes up all the time when we're dealing with complex numbers the idea of something as an operation kind of wanting To have multiple values you'll sometimes hear the phrase branch Where you choose a branch of the square root function? ", + "input": "the fourth root of 16 should be two? And the answer ends up being, well, we adopt a convention. When there's multiple options like this, when you have a multi-valued function, we often just choose one of those values to be what we mean when we want to treat it as a function, as something with a single input and a single output. In fancier lingo, this comes up all the time when we're dealing with complex numbers, the idea of something as an operation kind of wanting to have multiple values. You'll sometimes hear the phrase branch, where you choose a branch of the square root function, ", "translatedText": "Căn bậc 4 của 16 phải là 2 và kết quả cuối cùng là đúng. Chúng tôi áp dụng quy ước khi có nhiều lựa chọn như thế này khi bạn có một hàm đa giá trị. Chúng tôi thường chỉ chọn một trong những giá trị đó làm ý nghĩa khi chúng tôi muốn coi nó như một hàm như một thứ gì đó có một đầu vào và một đầu ra duy nhất theo thuật ngữ thông dụng hơn. Điều này luôn xuất hiện khi chúng ta xử lý các số phức ý tưởng về một thứ gì đó như một loại phép toán mong muốn Đôi khi bạn sẽ có nhiều giá trị nghe cụm từ nhánh Ở đâu bạn chọn một nhánh của hàm căn bậc hai? ", "model": "google_nmt", "n_reviews": 0, @@ -1656,7 +1656,7 @@ "end": 1845.74 }, { - "input": "Because there's multiple different answers You know we think of I again is this 90 degree rotation And if we were thinking of it as a 90 degree rotation it feels like the square root should be You know something sitting at a 45 degree angle Maybe that's the square root of I which we could write out very explicitly as root 2 over 2 root 2 over 2 I That's just using trigonometry but if we were thinking of I instead as being a Negative 270 degree rotation it feels like half of that doing half of that operation should actually get us on the other side Maybe the number sitting down here should be the square root of I and that's actually just the negative of what we saw before Negative root 2 over 2 minus root 2 over 2 times I Now in the context of real valued functions we can say yeah Just choose the square root to be whatever the positive answer is but which of these do you consider the positive answer? ", + "input": "Because there's multiple different answers. You know, we think of i again as this 90 degree rotation. And if we were thinking of it as a 90 degree rotation, it feels like the square root should be, you know, something sitting at a 45 degree angle. Maybe that's the square root of i, which we could write out very explicitly as root 2 over 2, root 2 over 2 i. That's just using trigonometry. But if we were thinking of i instead as being a negative 270 degree rotation, it feels like half of that, doing half of that operation, should actually get us on the other side. Maybe the number sitting down here should be the square root of i. And that's actually just the negative of what we saw before. Negative root 2 over 2 minus root 2 over 2 times i. Now in the context of real valued functions, we can say, yeah, just choose the square root to be whatever the positive answer is. But which of these do you consider the positive answer? ", "translatedText": "Bởi vì có nhiều câu trả lời khác nhau Bạn biết đấy, chúng ta lại nghĩ đến tôi là góc quay 90 độ này Và nếu chúng ta nghĩ về nó như một góc quay 90 độ thì nó có vẻ giống như căn bậc hai Bạn biết thứ gì đó nằm ở góc 45 độ Có lẽ đó là hình vuông gốc của I mà chúng ta có thể viết ra rất rõ ràng là căn 2 trên 2 căn 2 trên 2 I Đó chỉ là sử dụng lượng giác nhưng nếu thay vào đó chúng ta nghĩ về I như một phép quay 270 độ Âm thì có cảm giác như một nửa số đó thực hiện một nửa phép toán đó thực sự sẽ đưa chúng ta sang phía bên kia Có lẽ số ở đây phải là căn bậc hai của I và đó thực sự chỉ là số âm của những gì chúng ta đã thấy trước Căn âm 2 trên 2 trừ căn 2 trên 2 lần I Bây giờ trong bối cảnh thực các hàm có giá trị mà chúng ta có thể nói vâng Chỉ cần chọn căn bậc hai cho bất kỳ câu trả lời khẳng định nào nhưng bạn coi câu trả lời khẳng định nào trong số này? ", "model": "google_nmt", "n_reviews": 0, @@ -1672,7 +1672,7 @@ "end": 1941.38 }, { - "input": "And I think you say well We know what this is we kind of define it to be the square root of 2 all is well and good But what if I said let's approach this the same way that we were approaching our I to the I expression I want to first Express things as e to the something right and Then I'm going to raise that to the 1 half by multiplying the 1 half into the exponent And I say okay, I can I guess I can do that e to the what is equal to 2 well That's the natural log of 2 It's a constant which is around 0.69 or so If we raise e to that power we'll get 2 so we could be thinking of this as e to the natural log of 2 times 1 half and if you wanted to if you Were thinking of e to the x? ", + "input": "Yeah, I think you say, well, we know what this is, we kind of define it to be the square root of 2, all is well and good. But what if I said let's approach this the same way that we were approaching our i to the i expression? I want to first express things as e to the something, right, and then I'm going to raise that to the one half by multiplying the one half into the exponent. And I say, well, okay, I can, I guess I can do that. e to the what is equal to 2? Well, that's the natural log of 2. It's a constant which is around 0.69 or so. If we raise e to that power, we'll get 2. So we could be thinking of this as e to the natural log of 2 times one half. And if you wanted to, if you were thinking of e to the x, ", "translatedText": "Và tôi nghĩ bạn nói hay Chúng ta biết đây là gì chúng ta định nghĩa nó là căn bậc hai của 2 tất cả đều tốt và tốt Nhưng nếu tôi nói hãy tiếp cận điều này giống như cách chúng ta tiếp cận cái I của chúng ta với biểu thức I I Đầu tiên tôi muốn diễn đạt mọi thứ dưới dạng e thành cái gì đó đúng và sau đó tôi sẽ nâng nó lên nửa 1 bằng cách nhân nửa 1 với số mũ Và tôi nói được thôi, tôi có thể đoán là tôi có thể làm điều đó e với cái gì bằng 2 giếng Đó là log tự nhiên của 2 Đó là một hằng số gần bằng 0.69 hoặc hơn Nếu chúng ta nâng e lên lũy thừa đó, chúng ta sẽ nhận được 2 nên chúng ta có thể coi đây là e mũ log tự nhiên của 2 nhân 1 nửa và nếu bạn muốn, liệu bạn có đang nghĩ e mũ x không? ", "model": "google_nmt", "n_reviews": 0, @@ -1680,7 +1680,7 @@ "end": 1987.8 }, { - "input": "You know this might be kind of overkill in the context of real numbers But if you were thinking of e to the x as shorthand for this x function you could plug in the value 0.69 times 1 half which I guess would be around 0.345 Ish something like that You plug in that very concrete value into your polynomial see what it outputs, and it will output around 1.414 a Nice real number square root of 2 what you would expect But if we do the same thing we were just doing with I and Acknowledging that there's actually multiple different answers when we want to write something as e to a power we could also write this This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi I That whole thing raised to the 1 half Right after all this value will come to equal to you could break it down as it's e to the natural log of 2 Multiplied by e to the 2 pi I This one just has the effect of rotating things 360 degrees, so it's just going to equal 1 So we're looking at 2 times 1 great that feels like a valid substitution and yet when we play the same game of Taking this and raising it to a power and treating that by multiplying the power into the exponent look at what happens We have e to the natural log of 2 times 1 half plus Well, what's 2 pi I times 1 half well that will be pi times I Now this first part e to the natural log of 2 times 1 half that will end up being the familiar Square root of 2 that's all well and good, but we're going to be multiplying that by e to the pi I Right and quite famously e to the pi I is negative 1 So in this case it seems to be suggesting that if we are solving this expression 2 to the 1 half By playing around with the different answers we could plug in for something like e to the X equaling 1 half what we end up with is another answer what we might traditionally write as this negative square root of 2 and Here I mean it's a little funny for it to have multiple values to look at 2 to the 1 half and say that's not equaling One thing but based on choices we make it could equal multiple different things But the two things that it could seem quite reasonable If there's going to be anything that 2 to the 1 half is it seems like it should either be The positive square root that we're familiar with or the negative variant of that that doesn't actually seem like such a problem And in fact we could um we could play this game even further where let me ask you for even more creative answers to This expression because maybe we can find other funny powers of something like 2 to the power X as we start plugging in various different values of X based on what substitution we make if we're Abiding by the same rules that we were using in evaluating I to the power I So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number Natural log of 2 ok that one we know it. ", + "input": "you know, this might be kind of overkill in the context of real numbers, but if you were thinking of e to the x as shorthand for this x function, you could plug in the value 0.69 times one half, which I guess would be around 0.345ish, something like that. You plug in that very concrete value into your polynomial, see what it outputs, and it will output around 1.415. So that's a nice real number. Square root of 2, what you would expect. But if we do the same thing we were just doing with i and acknowledging that there's actually multiple different answers when we want to write something as e to a power, we could also write this. This might seem funny, but we could write it as e to the natural log of 2 plus 2 pi i. That whole thing raised to the one half. Right? After all, this value will come to equal 2. You could break it down as it's e to the natural log of 2 multiplied by e to the 2 pi i. This one just has the effect of rotating things 360 degrees. So it's just going to equal 1, so we're looking at 2 times 1. Great, that feels like a valid substitution. And yet when we play the same game of taking this and raising it to a power and treating that by multiplying the power into the exponent, look at what happens. We have e to the natural log of 2 times one half plus Well, what's 2 pi i times one half? Well, that will be pi times i. Now this first part, e to the natural log of 2 times one half, that will end up being the familiar square root of 2. That's all well and good. But we're going to be multiplying that by e to the pi i. Right? And quite famously e to the pi i is negative 1. So in this case, it seems to be suggesting that if we are solving this expression 2 to the one half by playing around with the different answers we could plug in for something like e to the x equaling one half, what we end up with is another answer. What we might traditionally write as this negative square root of 2. And here, I mean it's a little funny for it to have multiple values to look at 2 to the one half and say that's not equaling one thing, but based on choices we make it could equal multiple different things. But the two things that it could seem quite reasonable. If there's going to be anything that 2 to the one half is, it seems like it should either be the positive square root that we're familiar with or the negative variant of that. That doesn't actually seem like such a problem. And in fact, we could we could play this game even further, where let me ask you for even more creative answers to this expression. Because maybe we can find other funny powers of something like 2 to the power x as we start plugging in various different values of x based on what substitution we make. If we're abiding by the same rules that we were using in evaluating i to the power i. So this time the question asks or it specifies that one solution of the equation e to the x equals 2 is the real number natural log of 2. Okay, that one we know it. ", "translatedText": "Bạn biết điều này có thể hơi quá mức cần thiết trong bối cảnh số thực Nhưng nếu bạn đang nghĩ e mũ x là cách viết tắt của hàm x này thì bạn có thể thế giá trị 0 vào. 69 nhân 1 nửa mà tôi đoán là khoảng 0.345 Có vẻ như thế Bạn thay giá trị cụ thể đó vào đa thức xem nó cho ra kết quả như thế nào và nó sẽ cho ra khoảng 1.414 a Căn bậc hai số thực đẹp của 2 như bạn mong đợi Nhưng nếu chúng ta làm điều tương tự mà chúng ta vừa làm với I và Thừa nhận rằng thực sự có nhiều câu trả lời khác nhau khi chúng ta muốn viết một cái gì đó dưới dạng e lũy thừa thì chúng ta cũng có thể viết điều này Điều này có vẻ buồn cười, nhưng chúng ta có thể viết nó dưới dạng e theo log tự nhiên của 2 cộng 2 pi I Toàn bộ số đó được nâng lên thành 1 nửa Ngay sau đó, giá trị này sẽ bằng với bạn có thể chia nhỏ nó thành e đối với log tự nhiên của 2 Nhân với e với 2 pi I Cái này chỉ có tác dụng xoay mọi thứ 360 độ, nên nó sẽ bằng 1 Vì vậy, chúng ta đang xem 2 nhân 1 thật tuyệt, cảm giác như một sự thay thế hợp lệ và tuy nhiên khi chúng ta chơi cùng một trò chơi Lấy cái này và nâng nó lên lũy thừa và coi nó bằng cách nhân lũy thừa với số mũ nhìn xem điều gì xảy ra Chúng ta có e log tự nhiên của 2 nhân 1 nửa cộng Vâng, 2 pi I nhân 1 nửa à, đó sẽ là pi nhân I Bây giờ phần đầu tiên e của log tự nhiên 2 nhân 1 nửa sẽ trở thành căn bậc hai quen thuộc của 2, điều đó ổn thôi, nhưng chúng ta sẽ nhân nó với e pi I Đúng và khá nổi tiếng là e với pi I âm 1 Vì vậy, trong trường hợp này có vẻ như gợi ý rằng nếu chúng ta giải biểu thức 2 mũ 1 này Bằng cách thử nghiệm với các câu trả lời khác nhau, chúng ta có thể thay thế cho một cái gì đó như e mũ X bằng 1 nửa những gì chúng ta có được là một câu trả lời khác mà chúng ta có thể viết theo cách truyền thống là căn bậc hai âm của 2 và ở đây ý tôi là sẽ hơi buồn cười khi có nhiều giá trị khi xét 2 mũ 1 và nói rằng điều đó không bằng Một điều nhưng dựa trên những lựa chọn mà chúng ta đưa ra, nó có thể bằng nhiều thứ khác nhau Nhưng hai điều đó có vẻ khá hợp lý Nếu có bất cứ điều gì từ 2 đến 1 thì có vẻ như nó phải là Số dương căn bậc hai mà chúng ta quen thuộc hoặc biến thể phủ định của nó thực sự không phải là một vấn đề như vậy Và trên thực tế, chúng ta có thể ừm chúng ta có thể chơi trò chơi này thậm chí còn xa hơn nữa khi để tôi hỏi bạn những câu trả lời sáng tạo hơn nữa cho Biểu thức này bởi vì có lẽ chúng ta có thể tìm thấy những lũy thừa buồn cười khác của thứ gì đó như 2 lũy thừa X khi chúng ta bắt đầu thay các giá trị khác nhau của X dựa trên sự thay thế mà chúng ta thực hiện nếu chúng ta tuân thủ các quy tắc tương tự mà chúng ta đang sử dụng để đánh giá I với lũy thừa I Vậy lần này câu hỏi hay nó chỉ rõ một nghiệm của phương trình e x bằng 2 là số thực Logarit tự nhiên của 2 ok mà ta đã biết. ", "model": "google_nmt", "n_reviews": 0, @@ -1696,7 +1696,7 @@ "end": 2176.56 }, { - "input": "answer to the question e to the x equals 2 and Again creativity is welcomed, so I will give you another little moment for that I I Will go ahead and lock in some answers here if that's alright with you I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at but Don't be too stressed if it's before you got the chance to Into the question that you want into the answer that you want it to answer So it looks like 131 of you have entered the variant where we take Ln of 2 and we add 2ii and I guess I am in writing this question Mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones So that's on me for the fact that I don't know if it looks to any of you like oh It's red you got it wrong when you entered Ln of 2 plus 42. ", + "input": "answer to the question e to the x equals 2? And again, creativity is welcomed. So I will give you another little moment for that. All right, I will go ahead and lock in some answers here if that's all right with you. I'm not sure how much time it necessarily takes to do the math entry depending on what device you're looking at. But don't be too stressed if it's before you got the chance to enter the question that you want into the answer that you wanted to answer. So it looks like 131 of you have entered the variant where we take ln of 2 and we add 2 pi i. And I guess I in writing this question mistakenly like marked one of the answers as being correct when in fact there's quite a few different correct ones. So that's on me for the fact that I don't know if it looks to any of you like, oh, it's red you got it wrong when you entered ln of 2 plus 42 ", "translatedText": "câu trả lời cho câu hỏi e cho x bằng 2 và một lần nữa sự sáng tạo được hoan nghênh, vì vậy tôi sẽ cho bạn một chút thời gian nữa cho câu hỏi đó II Sẽ tiếp tục và chốt một số câu trả lời ở đây nếu bạn đồng ý. Tôi không chắc là mất bao nhiêu thời gian nhất thiết phải thực hiện mục nhập toán tùy thuộc vào thiết bị bạn đang xem nhưng Đừng quá căng thẳng nếu đó là trước khi bạn có cơ hội Nhập câu hỏi mà bạn muốn vào câu trả lời mà bạn muốn nó trả lời Vì vậy, có vẻ như 131 người trong số các bạn đã nhập biến thể trong đó chúng tôi lấy Ln bằng 2 và chúng tôi thêm 2ii và tôi đoán là tôi đang viết câu hỏi này. Đánh nhầm một trong các câu trả lời là đúng trong khi thực tế là có khá nhiều câu trả lời đúng khác nhau Vì vậy, đó là lỗi của tôi thực tế là tôi không biết có ai trong số các bạn thấy nó giống như vậy không. Nó màu đỏ, bạn đã nhầm khi nhập Ln của 2 cộng 42. ", "model": "google_nmt", "n_reviews": 0, @@ -1704,7 +1704,7 @@ "end": 2253.76 }, { - "input": "I pi which is of course a great choice But you could also have something like 4 pi I plus the natural log of 2 or 6 pi I Or really any integer multiple of 2 pi I if you add that it doesn't affect e to the X Because it just has the effect of multiplying by e to the 2 pi I Which is the effect of multiplying by 1 and again this has kind of a funny consequence where it seems to output kind of reasonable Results when we do it as another example It looks like the second most common entered expression there was that we might replace 2 So let's think we're thinking of 2 to the power of 1 4th, okay there was a suggestion that we replaced 2 with e to the natural log of 2 plus 4 pi I Okay Plus 4 pi I and we raise all of that To the 1 4th right well if you were to play the same game you would get e To the natural log of 2 times 1 4th, and we'd be multiplying by e to the pi I Now the first part of that is going to be the usual positive Fourth root of 2 the thing we mean when you plug in an expression like fourth root of 2 into a calculator a nice small Positive number, but then this second part is negative 1 so it seems to be saying You know if we were to interpret 2 in this different way raising it to the 1 4th You know it's not the usual answer that we get but it's a reasonable answer. ", + "input": "i pi which is of course a great choice. But you could also have something like 4 pi i plus the natural log of 2 or 6 pi i. Or really any integer multiple of 2 pi i if you add that it doesn't affect e to the x. Because it just has the effect of multiplying by e to the 2 pi i which is the effect of multiplying by 1. And again, this has kind of a funny consequence where it seems to output kind of reasonable results when we do it. As another example, it looks like the second most common entered expression there was that we might replace 2. So let's think we're thinking of 2 to the power of one fourth. Okay. There was a suggestion that we replace 2 with e to the natural log of 2 plus 4 pi i. Okay. Plus 4 pi i. And we raise all of that to the one fourth, right? Well if you were to play the same game you would get e to the natural log of 2 times one fourth and we'd be multiplying by e to the pi i. Now the first part of that is going to be the usual positive fourth root of 2. The thing we mean when you plug in an expression like fourth root of 2 into a calculator, a nice small positive number. But then this second part is negative 1. So it seems to be saying, you know, if we were to interpret 2 in this different way, raising it to the one fourth, you know, it's not the usual answer that we get but it's a reasonable answer. ", "translatedText": "I pi tất nhiên là một lựa chọn tuyệt vời Nhưng bạn cũng có thể có cái gì đó như 4 pi I cộng log tự nhiên của 2 hoặc 6 pi I Hoặc thực sự là bất kỳ bội số nguyên nào của 2 pi I nếu bạn thêm rằng nó không ảnh hưởng đến e đến X Bởi vì nó chỉ có tác dụng nhân với e thành 2 pi I Đó là tác dụng của việc nhân với 1 và một lần nữa, điều này có một hậu quả buồn cười là nó dường như tạo ra loại kết quả hợp lý khi chúng ta làm điều đó như một ví dụ khác. có vẻ như biểu thức được nhập phổ biến thứ hai là chúng ta có thể thay thế 2 Vì vậy, hãy nghĩ rằng chúng ta đang nghĩ đến 2 lũy thừa 1 4, được rồi, có gợi ý rằng chúng ta thay 2 bằng e thành log tự nhiên của 2 cộng 4 pi I Được rồi Cộng với 4 pi I và chúng ta sẽ nâng tất cả những thứ đó Đến số 1 4 nếu bạn chơi cùng một trò chơi thì bạn sẽ nhận được e Theo log tự nhiên của 2 nhân 1 4, và chúng ta sẽ nhân với e thành số pi I Bây giờ phần đầu tiên của nó sẽ là số dương căn bậc 4 của 2 điều chúng tôi muốn nói khi bạn thay một biểu thức như căn bậc 4 của 2 vào máy tính một số dương nhỏ đẹp, nhưng phần thứ hai này là âm 1 nên có vẻ như đang nói Bạn biết nếu chúng ta giải thích 2 theo cách khác này thì nâng nó lên số 1 4 Bạn biết đó không phải là câu trả lời thông thường mà chúng tôi nhận được nhưng đó là một câu trả lời hợp lý. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 2358.92 }, { - "input": "We would have been looking at pi halves times I and instead of multiplying by Negative 1 we would have instead been multiplying by I Which again is a valid answer it seems like a reasonable output for something like 2 to the 1 4th So when you're looking at the fact that I to the power I seems to have multiple different values for it Right we have this funny phenomenon where we could plug in e to the 5 pi halves I Negative 3 pi halves I and we get what seemed like wildly different answers something super small something super big all very different from the 1 5th approximately 1 5th answer that we found before up here It's exactly the same phenomenon as when you're asking something like what's 2 to the 1 4th and Acknowledging that there's actually multiple different solutions to the expression X to the 4th equals 2 4 different solutions in fact and what you're looking at is the fact that there's multiple different solutions To the expression e to the X equals some kind of base whether that base is I whether that base is 2 Whatever it might be and one way that we might Think about this is that when you're dealing with real numbers things are just lovely things are nice There's one-to-one relationships. ", + "input": "we would have been looking at pi halves times i. And instead of multiplying by negative 1 we would have instead been multiplying by i. Which again is a valid answer. It seems like a reasonable output for something like 2 to the one fourth. So when you're looking at the fact that i to the power i seems to have multiple different values for it, right, we have this funny phenomenon where we can plug in e to the 5 pi halves i, negative 3 pi halves i, and we get what seem like wildly different answers. Something super small, something super big, all very different from the one fifth, approximately one fifth answer that we found before up here. It's exactly the same phenomenon as when you're asking something like what's 2 to the one fourth and acknowledging that there's actually multiple different solutions to the expression x to the fourth equals 2. Four different solutions in fact. And what you're looking at is the fact that there's multiple different solutions to the expression e to the x equals some kind of base, whether that base is i, whether that base is 2, whatever it might be. And one way that we might think about this is that when you're dealing with real numbers things are just lovely. Things are nice. There's one-to-one relationships. ", "translatedText": "Lẽ ra chúng ta đã xét pi chia đôi với I và thay vì nhân với Âm 1, thay vào đó chúng ta sẽ nhân với I. Đây lại là một câu trả lời hợp lệ, có vẻ như là một kết quả đầu ra hợp lý cho một số như 2 mũ 1 Vì vậy, khi bạn nhìn vào thực tế là tôi với lũy thừa tôi dường như có nhiều giá trị khác nhau cho nó Đúng là chúng ta có một hiện tượng buồn cười khi chúng ta có thể thế e vào 5 nửa pi I Âm 3 nửa pi tôi và chúng ta nhận được những câu trả lời dường như cực kỳ khác nhau một cái gì đó siêu nhỏ một cái gì đó siêu lớn tất cả đều rất khác với câu trả lời thứ 1 5 khoảng 1 5 mà chúng tôi đã tìm thấy trước đây. Đó chính xác là hiện tượng giống như khi bạn hỏi điều gì đó như 2 đến 1 4 là bao nhiêu và thừa nhận rằng thực sự có nhiều giải pháp khác nhau trên thực tế, biểu thức X mũ 4 bằng 2 4 nghiệm khác nhau và điều bạn đang xem là thực tế là có nhiều nghiệm khác nhau. Biểu thức e mũ X bằng một loại cơ số nào đó cho dù cơ số đó có phải là I hay không cơ số đó là 2 Dù nó có thể là gì và một cách mà chúng ta có thể nghĩ về điều này là khi bạn xử lý các số thực, mọi thứ chỉ là những điều đáng yêu, những điều tốt đẹp. Có những mối quan hệ một đối một. ", "model": "google_nmt", "n_reviews": 0, @@ -1736,7 +1736,7 @@ "end": 2428.5 }, { - "input": "It's great Where if we want to think about exponential functions, let me just cover some of this stuff up We have this nice back and forth where you can choose to express any exponential as a base to X like 2 to the X Or you could express that same exponential as X of R times X which you know that is the polynomial that we refer to Whenever implicitly refer to whenever we write something like e to the X And there's a lovely back and forth because you can just take a natural logarithm of B And it gives you one answer assuming that B is a positive number And that's the same thing as saying that X of R is equal to B So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible Exponentials right we could write them as X of R times X and change what R is And this is exactly the same thing as writing e to the R times X if that's something you're more comfortable with So e to the R times X X of R times X those are the same thing we could think about changing what that is But on the other hand if you were to think about all possible exponentials as some base Let me do base to the power of X and we're going to change what that base is At first it feels like that's a different kind of expression to manipulate, but it's just another way of expressing the same family And a way that you might think about this For how do we think about what base does it correspond to if we're thinking a little bit more abstractly as Exp of R times X and there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look Weirder so follow through with me here if instead of looking at that base one thing I could do is say what is the value? ", + "input": "It's great. Where if we want to think about exponential functions, let me just cover some of this stuff up. We have this nice back and forth where you can choose to express any exponential as a base to x like 2 to the x or you could express that same exponential as x of r times x which you know, that is the polynomial that we refer to whenever implicitly refer to whenever we write something like e to the x. And there's a lovely back and forth because you can just take a natural logarithm of b and it gives you one answer assuming that b is a positive number. And that's the same thing as saying that x of r is equal to b. So one way that I've talked about this earlier in the series is that if you were looking at the family of all possible exponentials, right, we could write them as x of r times x and change what r is. And this is exactly the same thing as writing e to the r times x if that's something you're more comfortable with. So e to the r times x x of r times x those are the same thing. We could think about changing what that is. But on the other hand if you were to think about all possible exponentials as some base, let me do base to the power of x and we're going to change what that base is. At first it feels like that's a different kind of expression to manipulate but it's just another way of expressing the same family. Right, and a way that you might think about this for how do we think about what base does it correspond to if we're thinking a little bit more abstractly as exp of r times x. And there's a reason I'm doing this because we're about to apply this to complex numbers where it's going to look weirder. So follow through with me here. If instead of looking at that base, one thing I could do is say what is the value o ", "translatedText": "Thật tuyệt Nếu chúng ta muốn nghĩ về hàm số mũ, hãy để tôi trình bày một số nội dung này. Chúng ta có cái này qua lại rất hay, nơi bạn có thể chọn biểu thị bất kỳ số mũ nào làm cơ số cho X như 2 đến X Hoặc bạn có thể biểu thị cùng số mũ đó với X của R nhân X mà bạn biết đó là đa thức mà chúng ta đề cập đến Bất cứ khi nào đề cập ngầm đến bất cứ khi nào chúng ta viết một cái gì đó như e cho X Và có một sự đảo ngược đáng yêu bởi vì bạn chỉ cần lấy logarit tự nhiên của B Và nó cho bạn một câu trả lời giả định rằng B là một số dương Và điều đó cũng giống như nói rằng X của R bằng B Vì vậy, một cách mà tôi đã nói về điều này ở phần trước trong loạt bài này là nếu bạn nhìn vào họ tất cả các Số mũ có thể có đúng, chúng ta có thể viết chúng dưới dạng X(R nhân X và thay đổi R bằng gì Và điều này hoàn toàn giống với việc viết e thành R nhân X nếu đó là thứ bạn cảm thấy thoải mái hơn với Vậy e thành R nhân XX của R nhân X đó là những thứ tương tự mà chúng ta có thể nghĩ về việc thay đổi nó. Nhưng mặt khác, nếu bạn nghĩ về tất cả các số mũ có thể có như một cơ số nào đó Hãy để tôi tính cơ sở theo lũy thừa của X và chúng ta sẽ đi để thay đổi cơ sở đó là gì Lúc đầu, có vẻ như đó là một kiểu biểu đạt khác để thao tác, nhưng đó chỉ là một cách khác để thể hiện cùng một họ Và một cách mà bạn có thể nghĩ về điều này Để chúng ta nghĩ về cơ sở nào nó tương ứng đến nếu chúng ta suy nghĩ trừu tượng hơn một chút là Exp của R nhân X và có lý do tôi làm điều này bởi vì chúng ta sắp áp dụng điều này cho các số phức nơi nó sẽ trông kỳ lạ hơn vì vậy hãy cùng tôi làm theo ở đây nếu thay vì nhìn vào cơ sở đó, tôi có thể nói giá trị là bao nhiêu? ", "model": "google_nmt", "n_reviews": 0, @@ -1760,7 +1760,7 @@ "end": 2597.74 }, { - "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by two pi I And that doesn't change the base that it would correspond to that would still correspond to two Or it could shift it up by two pi I that doesn't change the base that it corresponds to because in all of those cases When we plug in X equals one we get the same thing however All of these for different values of X are distinct functions This is why we saw multiple different values for I to the power I Because I to the X is an ambiguous function in that context it would be unambiguous if we decided which value of R Such that what we're representing is exp of R times X which value of R. ", + "input": "I could have exp of R times X where maybe R is something like zero point six nine But I could shift that down by 2 pi i. And that doesn't change the base that it would correspond to. That would still correspond to two. Or it could shift it up by 2 pi i. That doesn't change the base that it corresponds to. Because in all of those cases when we plug in x equals one, we get the same thing. However, all of these for different values of x are distinct functions. This is why we saw multiple different values for i to the power i. Because i to the x is an ambiguous function in that context. It would be unambiguous if we decided which value of r such that what we're representing is exp of r times x, which value of r d ", "translatedText": "Tôi có thể có exp bằng R nhân X trong đó có thể R giống như số 0 điểm sáu chín Nhưng tôi có thể dịch nó xuống hai pi I Và điều đó không làm thay đổi cơ số mà nó sẽ tương ứng với cái đó vẫn sẽ tương ứng với hai Hoặc nó có thể dịch chuyển nó lên hai pi I mà không làm thay đổi cơ số tương ứng bởi vì trong tất cả các trường hợp đó Khi chúng ta cắm X bằng một thì chúng ta nhận được kết quả giống nhau tuy nhiên Tất cả những thứ này cho các giá trị khác nhau của X là các hàm riêng biệt Đây là tại sao chúng ta thấy nhiều giá trị khác nhau của I mũ I Bởi vì I mũ X là một hàm không rõ ràng trong bối cảnh đó, sẽ rõ ràng nếu chúng ta quyết định giá trị nào của R Sao cho cái chúng ta đang biểu thị là exp của R nhân X giá trị nào của R Chúng ta có chọn ngay khi chúng ta chọn một không? ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 2640.64 }, { - "input": "It's an unambiguous function but at that point it just feels like maybe what we want is a To stop thinking about things in terms of some base raised to the power X Maybe as soon as we're in the context of complex numbers We should just write them all as exp of some constant times X if for no other reason it makes crystal clear How we actually plug in numbers if we want to do a computation or just to do math on top of it we've got this nice infinite polynomial that we plug them into and I'll make another case for you that this is this is maybe the the correct way to think about exponentials As soon as we're extending into other domains things like complex numbers and for that let's just let's just back up Go back to doorbell some things arrived go back to the original Way that we extend the idea of exponentiation and just think of like what is 2 to the X Right we know how to think about this for natural numbers. ", + "input": "it's an unambiguous function. But at that point it just feels like maybe what we want is to stop thinking about things in terms of some base raised to the power x. Maybe as soon as we're in the context of complex numbers, we should just write them all as exp of some constant times x. If for no other reason, it makes crystal clear how we actually plug in numbers if we want to do a computation, or just to do math on top of it. We've got this nice infinite polynomial that we plug them into. And I'll make another case for you that this is maybe the the correct way to think about exponentials, as soon as we're extending into other domains, things like complex numbers. And for that, let's just let's just back up. Go back. Oh doorbell, something's arrived. Go back to the original way that we extend the idea of exponentiation and just think of like what is two to the x? Right. We know how to think about this for natural numbers, ", "translatedText": "Đó là một hàm rõ ràng nhưng tại thời điểm đó, có vẻ như điều chúng ta muốn là ngừng suy nghĩ về mọi thứ theo cơ số nào đó được nâng lên lũy thừa X Có lẽ ngay khi chúng ta ở trong bối cảnh của số phức Chúng ta chỉ nên viết tất cả chúng đều là exp của một số lần không đổi X nếu không vì lý do nào khác mà nó trở nên rõ ràng. Cách chúng ta thực sự cắm các số nếu chúng ta muốn thực hiện một phép tính hoặc chỉ để làm toán trên đó, chúng ta có đa thức vô hạn tuyệt vời này mà chúng ta cắm chúng vào và tôi sẽ đưa ra một trường hợp khác cho bạn rằng đây có thể là cách chính xác để nghĩ về số mũ Ngay khi chúng ta mở rộng sang các lĩnh vực khác, những thứ như số phức và vì điều đó chúng ta hãy sao lưu Bắt đầu quay lại chuông cửa một số điều đã đến quay trở lại Cách ban đầu là chúng ta mở rộng ý tưởng về lũy thừa và chỉ nghĩ như thế nào là 2 mũ X Đúng vậy, chúng ta biết cách nghĩ về điều này cho các số tự nhiên. ", "model": "google_nmt", "n_reviews": 0, @@ -1784,7 +1784,7 @@ "end": 2696.36 }, { - "input": "You know something like 2 to the 3 Repeated multiplication How is it that you're first taught to think about something like 2 to the X for fractional amounts or For negative amounts and things like that. ", + "input": "you know something like two to the three, repeated multiplication. How is it that you're first taught to think about something like two to the x for fractional amounts or for negative amounts and things like that? ", "translatedText": "Bạn biết những thứ như 2 mũ 3 Phép nhân lặp lại Làm sao lần đầu tiên bạn được dạy nghĩ về những thứ như 2 mũ X cho số phân số hoặc cho số âm và những thứ tương tự. Tốt. ", "model": "google_nmt", "n_reviews": 0, @@ -1800,7 +1800,7 @@ "end": 2708.26 }, { - "input": "You're usually taught that 2 to the 1 half should be something where you know if I multiply it by itself and This follows the usual rules that Exponentials do with counting numbers where we're able to add things in that exponent I should get 2 to the 1 so it should be some number that when I multiply it by itself I get 2 and You know at that point you have a choice, maybe it's positive. ", + "input": "you're usually taught that two to the one half should be something where you know if I multiply it by itself and this follows the usual rules that exponentials do with counting numbers where we're able to add things in that exponent, I should get two to the one. So it should be some number that when I multiply it by itself, I get two. And you know at that point you have a choice. Maybe it's positive. ", "translatedText": "Bạn thường được dạy rằng 2 mũ 1 sẽ là số mà bạn biết nếu tôi nhân nó với chính nó và Điều này tuân theo các quy tắc thông thường mà Hàm mũ thực hiện khi đếm các số mà chúng ta có thể cộng các số vào số mũ đó tôi sẽ nhận được 2 đến 1 nên nó phải là một số nào đó mà khi tôi nhân nó với chính nó, tôi nhận được 2 và Bạn biết đấy, tại thời điểm đó bạn có một lựa chọn, có thể đó là số dương. ", "model": "google_nmt", "n_reviews": 0, @@ -1808,7 +1808,7 @@ "end": 2731.68 }, { - "input": "Maybe it's negative But if you always decide to make the positive choice You're going to be able to get a nice continuous function out of this same deal if we ask about negative numbers What should 2 to the negative 1 be well that should be something where when I multiply it by 2 to the 1? ", + "input": "Maybe it's negative. But if you always decide to make the positive choice, you're going to be able to get a nice continuous function out of this. Same deal if we ask about negative numbers, what should two to the negative one be? Well, that should be something where when I multiply it by two to the o ", "translatedText": "Có thể nó là số âm Nhưng nếu bạn luôn quyết định đưa ra lựa chọn tích cực Bạn sẽ có thể có được một hàm số liên tục tốt đẹp từ cùng một thỏa thuận này nếu chúng ta hỏi về số âm 2 mũ âm 1 sẽ như thế nào thì đó sẽ là một cái gì đó ở đâu khi tôi nhân nó với 2 với 1? ", "model": "google_nmt", "n_reviews": 0, @@ -1816,7 +1816,7 @@ "end": 2744.96 }, { - "input": "It gets me 2 to the 0 and that's kind of the justification for our convention that negative exponents look like 1 half But what's really going on here is we're saying whatever this is it should be some kind of function That satisfies this property f of a plus b equals f of a times f of b and Moreover the fact that the base is 2 is basically telling us that it's not just any such function It's a function where when we plug in 1 we get 2 And just as a little you know sanity check style question to see if you're following along with some of the implications here I want to ask you what is I won't call it like a softball, but this is this isn't meant to be like An incredibly deep question necessarily. ", + "input": "ne, it gets me two to the zero. And that's kind of the justification for our convention that negative exponents look like one half. But what's really going on here is we're saying whatever this is, it should be some kind of function that satisfies this property f of a plus b equals f of a times f of b. And moreover the fact that the base is two is basically telling us that it's not just any such function. It's a function where when we plug in one we get two. And just as a little, you know, sanity check style question to see if you're following along with some of the implications here. I want to ask you what is, I won't call it like a softball, but this is, this isn't meant to be like an incredibly deep question necessarily. ", "translatedText": "Nó mang lại cho tôi 2 mũ 0 và đó là sự biện minh cho quy ước của chúng ta rằng số mũ âm trông giống như 1 nửa Nhưng điều thực sự đang diễn ra ở đây là chúng ta đang nói bất kể đây là gì nó phải là một loại hàm nào đó thỏa mãn tính chất f của a cộng b bằng f của a nhân f của b và Hơn nữa, cơ số bằng 2 về cơ bản cho chúng ta biết rằng nó không chỉ là một hàm số như vậy. Đó là một hàm mà khi chúng ta thay 1 vào thì chúng ta có 2 Và chỉ một chút thôi bạn biết đấy câu hỏi kiểu kiểm tra sự tỉnh táo để xem liệu bạn có đang làm theo một số hàm ý ở đây không Tôi muốn hỏi bạn cái gì Tôi sẽ không gọi nó giống như một quả bóng mềm, nhưng đây không phải là một câu hỏi cực kỳ sâu sắc tất yếu. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 2793.32 }, { - "input": "It's just more of a check if you're following along with The idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs And it also satisfies f of 1 equals 2 which of the following is true Which is to say which of the following is necessarily true No matter which such function you're starting with and those of you who remember which which lecture was it It's whichever one we were talking about how to interpret what Euler's formula is really saying I asked a question of this style where I neglected a single condition, you know I didn't write down the fact that we want to make sure f of x is nonzero everywhere and then that caused some amount of Confudlement which is cool get confudlement on screen that happens to all of us But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is Is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power This is the the spirit of the question Now we've got a couple questions actually about power towers that seem to have popped up here which is great connected to last time Let's let's hold off on the power tower question for just a moment so that we first get like a deeper feel of like What exponentiation should mean here? ", + "input": "It's just more of a check if you're following along with the idea of abstractly starting with properties of a function and then kind of deducing ways that we might want to write it down based on those properties. If f of x satisfies this exponential property f of a plus b equals f of a times f of b for all inputs, and it also satisfies f of one equals two, which of the following is true? Which is to say which of the following is necessarily true no matter which such function you're starting with. And those of you who remember which lecture was it? It's whichever one we were talking about how to interpret what Euler's formula is really saying. I asked a question of this style where I neglected a single condition, you know, I didn't write down the fact that we want to make sure f of x is non-zero everywhere and then that caused some amount of confutlement. Which is cool, get confutlement on screen that happens to all of us. But the the intent of it was to basically show that this abstract property of something that turns addition into multiplication is uh is enough to basically make you want to write the function as whatever it equals as one raised to some kind of power. This is the the spirit of the question. Um Now we've got a couple questions actually about power towers that seem to have popped up here, which is great connected to last time. Um, let's let's hold off on the power tower question for just a moment, so that we first get like a deeper feel of like what exponentiation should mean here. ", "translatedText": "Nó chỉ giống một cuộc kiểm tra xem bạn có đang theo dõi hay không Ý tưởng bắt đầu một cách trừu tượng với các thuộc tính của hàm và sau đó là các cách suy luận mà chúng ta có thể muốn viết nó ra dựa trên các thuộc tính đó Nếu f của x thỏa mãn thuộc tính hàm mũ f này của a cộng b bằng f của a nhân f của b với tất cả các đầu vào Và nó cũng thỏa mãn f của 1 bằng 2 điều nào sau đây là đúng Điều đó có nghĩa là điều nào sau đây nhất thiết phải đúng Cho dù bạn đang bắt đầu chức năng nào với và những ai còn nhớ đó là bài giảng nào Đó là bài giảng mà chúng ta đang nói về cách giải thích ý nghĩa thực sự của công thức Euler Tôi đã hỏi một câu hỏi theo phong cách này khi tôi bỏ qua một điều kiện duy nhất, bạn biết đấy, tôi đã không viết ra thực tế là chúng tôi muốn đảm bảo f(x) khác 0 ở mọi nơi và sau đó điều đó gây ra một số nhầm lẫn, điều thú vị là sự nhầm lẫn trên màn hình xảy ra với tất cả chúng ta. Nhưng mục đích của nó về cơ bản là chỉ ra rằng thuộc tính trừu tượng này của thứ gì đó biến phép cộng thành phép nhân là đủ để về cơ bản khiến bạn muốn viết hàm này bằng bất cứ giá trị nào nó tương đương với một dạng lũy thừa nào đó Đây chính là tinh thần của câu hỏi Bây giờ chúng ta thực sự có một vài câu hỏi về tháp điện điều đó dường như đã xuất hiện ở đây và có mối liên hệ tuyệt vời với lần trước Chúng ta hãy tạm dừng câu hỏi về tháp điện chỉ một lát để trước tiên chúng ta có cảm nhận sâu sắc hơn về việc lũy thừa có ý nghĩa gì ở đây? ", "model": "google_nmt", "n_reviews": 0, @@ -1832,7 +1832,7 @@ "end": 2882.26 }, { - "input": "Because because we can be what I want to claim is we can answer it in like multiple different ways So if you give me just one, we'll talk about power towers And then just as a number line can be represented in a logarithmic scale can the same done be done for a complex plane? ", + "input": "Um, because because we can be what I want to claim is we can answer it in like multiple different ways. So if you give me just a moment, we'll talk about power towers. Uh, and then just as a number line can be represented in a logarithmic scale, can the same be done for a complex plane? ", "translatedText": "Bởi vì chúng ta có thể là những gì tôi muốn khẳng định là chúng ta có thể trả lời nó theo nhiều cách khác nhau Vì vậy, nếu bạn chỉ cho tôi một cách, chúng ta sẽ nói về tháp điện Và cũng giống như một trục số có thể được biểu diễn theo thang logarit điều tương tự có được thực hiện đối với một mặt phẳng phức tạp không? ", "model": "google_nmt", "n_reviews": 0, @@ -1856,7 +1856,7 @@ "end": 2901.54 }, { - "input": "Yeah In fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that Because what we'll do is play around with different exponential functions X of R times X But we're going to change that value of R which is going to be represented by a little yellow dot So we'll kind of talk through this It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis But the idea is that as we move around what that constant is We're going to be able to kind of visualize the different things that it does to the plane and Effectively it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle And then as soon as that value of R becomes imaginary it swaps the role of those Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled Positive axis so great question all three of which I guess are sort of jumping the gun ahead for where I want to go But nice to see that's where people are thinking so in this one. ", + "input": "uh, in fact, there's a visualization that I'm going to get to in just a moment here where we do something quite similar to that. Because what we'll do is play around with different exponential functions x of r times x, but we're going to change that value of r which is going to be represented by a little yellow dot. So we'll kind of talk through this. It's not going to map the whole plane, but just a couple sample points from the real axis and the imaginary axis. But the idea is that as we move around what that constant is, we're going to be able to kind of visualize the different things that um, it does to the plane. And effectively, it's like it's turning the x-axis into a logarithmic scale and then wrapping the imaginary axis along a circle. And then as soon as that value of r becomes imaginary, it swaps the role of those. Real numbers get put on the circle and imaginary numbers get put on a logarithmic scaled positive axis. So great question. All three of which I guess are sort of jumping the gun ahead for where I want to go, but nice to see that's where people are thinking. So on this one, ", "translatedText": "Vâng Trên thực tế, có một hình dung mà tôi sắp trình bày ngay sau đây khi chúng ta làm điều gì đó khá giống với điều đó Bởi vì những gì chúng ta sẽ làm là thử nghiệm với các hàm số mũ khác nhau X của R nhân X Nhưng chúng ta sẽ thay đổi giá trị của R sẽ được biểu thị bằng một chấm nhỏ màu vàng Vì vậy, chúng ta sẽ nói về điều này. Nó sẽ không ánh xạ toàn bộ mặt phẳng, mà chỉ một vài điểm mẫu từ trục thực và trục ảo Nhưng ý tưởng là khi chúng ta di chuyển xung quanh hằng số đó là gì. Chúng ta sẽ có thể hình dung được những điều khác nhau mà nó tác động lên mặt phẳng và về mặt hiệu quả, nó giống như biến trục x thành thang logarit rồi gói lại trục ảo dọc theo một vòng tròn Và sau đó ngay khi giá trị của R đó trở thành ảo, nó sẽ hoán đổi vai trò của các số thực đó được đặt trên đường tròn và các số ảo được đặt trên trục dương có tỷ lệ logarit nên tôi đoán là cả ba câu hỏi hay Tôi đang lao về phía trước để đến nơi tôi muốn đến Nhưng thật vui khi thấy đó là nơi mọi người đang nghĩ như vậy trong phần này. ", "model": "google_nmt", "n_reviews": 0, @@ -1872,7 +1872,7 @@ "end": 2973.06 }, { - "input": "explicitly Something like f of 5 is the same thing as f of 1 plus 1 plus 1 plus 1 plus 1 Which is the same thing as f of 1 multiplied by itself 5 times because of this property Which if f of 1 is 2 is the same as 2 to the power 5 and then something like f of negative 5 It should be the case that when we multiply it by f of 5 We get whatever f of 0 is and it's not immediately clear what f of 0 is but we could say that f of 1 plus 0 is Equal to whatever f of 1 is times what f of 0 is but f of 1 is equal to 2 And so this is also equal to 2 so we're saying 2 is equal to 2 times something well that something has to be a 1 so in this context this guarantees that f of negative 5 is 2 to the negative 5 it's 1 over 2 to the 5th We could explicitly write this as 2 to the negative 5 which is all to say These two properties together make us really want to write the function as 2 to the X Because any counting number that we put in it's going to satisfy It's it's going to look like to multiply by itself that number of times any fractional number We put in it's going to satisfy these properties that we wanted And you might wonder is that unique and in the context of real valued functions it actually would be But in the context of complex valued functions There would be multiple such functions f that we could write for this one of which is what we were looking at before Where we could have a function defined to be exp of the natural log of 2 plus 2 pi I all of that times X Okay, forgive the sloppiness here, I just get excited writing about this And this is actually a different function as evidenced by what happens if you plug in X equals 1 half We saw a little bit earlier how when you plug in 1 half what you get is the negative square root of 2 and then if you plug in 1 fourth you get Not the fourth root of 2 but I times the fourth root of 2 so it is a different function But it still satisfies these properties and it kind of makes us want to write it as 2 to the X And it makes it suggest that maybe 2 to the X is an ambiguous bit of notation And we should just write everything in terms of exp of R times something but you might wonder well You know maybe we're just not being creative enough with all of the functions that satisfy this property Maybe there's an ambiguity when we write exp of R times something and there's different values of R that could come into play But I'm I'm just gonna put down a little claim and then maybe give like a sketch of what the proof would look like if you want Which is that let's say you have some complex function F, and it satisfies the following properties first You're able to take a derivative of it. ", + "input": "xplicitly, something like f of five is the same thing as f of one plus one plus one plus one plus one, which is the same thing as f of one multiplied by itself five times because of this property. Which if f of one is two is the same as two to the power five. And then something like f of negative five, it should be the case that when we multiply it by f of five, we get whatever f of zero is. And it's not immediately clear what f of zero is, but we could say that f of one plus zero is equal to whatever f of one is times what f of zero is. But f of one is equal to two, and so this is also equal to two. So we're saying two is equal to two times something. Well that something has to be a one. So in this context this guarantees that f of negative five is two to the negative five. It's one over two to the fifth. So we could explicitly write this as two to the negative five. Which is all to say, these two properties together make us really want to write the function as two to the x, because any counting number that we put in, it's going to satisfy, it's going to look like two multiplied by itself that number of times. Any fractional number we put in, it's going to satisfy these properties that we wanted. And you might wonder, is that unique? And in the context of real valued functions, it actually would be. But in the context of complex valued functions, there would be multiple such functions f that we could write for this. One of which is what we were looking at before, where we could have a function defined to be exp of the natural log of two plus two pi i all of that times x. Okay, forgive the sloppiness here. I just get excited writing about this. And this is actually a different function, as evidenced by what happens if you plug in x equals one half. Right, we saw a little bit earlier how when you plug in one half, what you get is the negative square root of two. And then if you plug in one fourth, you get not the fourth root of two, but i times the fourth root of two. So it is a different function, but it still satisfies these properties, and it kind of makes us want to write it as two to the x. And it makes it suggest that maybe two to the x is an ambiguous bit of notation, and we should just write everything in terms of exp of r times something. But you might wonder, well, you know, maybe we're just not being creative enough with all of the functions that satisfy this property. Maybe there's an ambiguity when we write exp of r times something, and there's different values of r that could come into play. Um, but I'm just going to put down a little claim, and then maybe give like a sketch of what the proof would look like if you wa ", "translatedText": "một cách rõ ràng Cái gì đó giống như f(5) giống như f(1+1+1 plus 1 plus 1) Tương đương với f(1) nhân với chính nó 5 lần vì tính chất này Nếu f(1) bằng 2 thì bằng nhau bằng 2 lũy thừa 5 và sau đó đại loại như f của âm 5. Sẽ xảy ra trường hợp khi chúng ta nhân nó với f của 5. Chúng ta nhận được f của 0 là bao nhiêu và không rõ ngay f của 0 là gì nhưng chúng ta có thể nói rằng f(1 cộng 0) bằng bất cứ giá trị nào f(1) nhân với f(0) nhưng f(1) bằng 2 Và vì vậy cái này cũng bằng 2 nên ta đang nói 2 bằng 2 nhân cái gì đó cái gì đó phải là 1 nên trong bối cảnh này điều này đảm bảo rằng f của âm 5 bằng 2 mũ âm 5 nó là 1 trên 2 mũ 5 Chúng ta có thể viết rõ ràng cái này là 2 mũ âm 5, tất cả để nói rằng Hai thuộc tính này cùng nhau tạo nên chúng tôi thực sự muốn viết hàm dưới dạng 2 cho X Bởi vì bất kỳ số đếm nào chúng tôi đặt vào sẽ thỏa mãn. Nó trông giống như được nhân với chính nó gấp số phân số bất kỳ mà chúng tôi đặt vào sẽ thỏa mãn các tính chất này mà chúng tôi muốn Và bạn có thể thắc mắc liệu nó có phải là duy nhất không và trong bối cảnh các hàm có giá trị thực thì nó thực sự sẽ như vậy Nhưng trong bối cảnh các hàm có giá trị phức tạp Sẽ có nhiều hàm như vậy nếu chúng ta có thể viết cho hàm này, một trong số đó là những gì chúng ta đã có xem trước Nơi chúng ta có thể có một hàm được xác định là exp của log tự nhiên của 2 cộng 2 pi Tôi luôn luôn X Được rồi, tha thứ cho sự cẩu thả ở đây, tôi rất hào hứng khi viết về điều này Và đây thực sự là một hàm khác như được chứng minh bằng điều gì sẽ xảy ra nếu bạn thế X bằng 1 nửa Chúng ta đã thấy trước đó một chút rằng khi bạn thế 1 nửa thì kết quả bạn nhận được là căn bậc hai âm của 2 và sau đó nếu bạn thay 1 phần tư bạn sẽ nhận được Không phải căn bậc bốn của 2 nhưng tôi nhân căn bậc 4 của 2 nên nó là một hàm khác Nhưng nó vẫn thỏa mãn các tính chất này và nó khiến chúng ta muốn viết nó là 2 cho X Và nó gợi ý rằng có thể 2 mũ X là một số không rõ ràng một chút ký hiệu Và chúng ta chỉ nên viết mọi thứ dưới dạng exp của R nhân một cái gì đó nhưng bạn có thể thắc mắc Bạn biết đấy có lẽ chúng ta chưa đủ sáng tạo với tất cả các hàm thỏa mãn tính chất này Có lẽ có sự mơ hồ khi chúng ta viết exp của R nhân thứ gì đó và có các giá trị khác nhau của R có thể phát huy tác dụng Nhưng tôi sẽ đưa ra một khẳng định nhỏ và sau đó có thể đưa ra một bản phác thảo về bằng chứng sẽ trông như thế nào nếu bạn muốn. giả sử bạn có một hàm phức F nào đó và nó thỏa mãn các tính chất sau trước tiên. Bạn có thể lấy đạo hàm của nó. ", "model": "google_nmt", "n_reviews": 0, @@ -1880,7 +1880,7 @@ "end": 3136.42 }, { - "input": "It's differentiable which just keeps it from being some you know totally messy discontinuous thing That's like taking on some random values depending on you know the span of whatever vector space over I don't know fractional amounts you might want to think of in crazy ways. ", + "input": "nt, which is that let's say you have some complex function f, and it satisfies the following properties. First, you're able to take a derivative of it. It's differentiable, which just keeps it from being some, uh, you know, totally messy discontinuous thing that's like taking on some random values depending on, you know, the span of whatever vector space over, I ", "translatedText": "Nó có khả vi phân giúp nó không trở thành một thứ mà bạn biết là hoàn toàn lộn xộn, không liên tục. Điều đó giống như nhận một số giá trị ngẫu nhiên tùy thuộc vào việc bạn biết khoảng của bất kỳ không gian vectơ nào trên Tôi không biết số lượng phân số mà bạn có thể muốn nghĩ theo những cách điên rồ. ", "model": "google_nmt", "n_reviews": 0, @@ -1888,7 +1888,7 @@ "end": 3157.06 }, { - "input": "It's a nice function. ", + "input": "don't know, fractional amounts you might want to think of in crazy ways. It's a nice function ", "translatedText": "Đó là một chức năng tốt đẹp. ", "model": "google_nmt", "n_reviews": 0, @@ -1896,7 +1896,7 @@ "end": 3160.84 }, { - "input": "That's differentiable It's not equal to 0 everywhere so the condition that sort of slipped my mind and I Forget which lecture lecture for or something like that and then it has this central property that it turns addition into multiplication If you have such a function I claim that there's a unique maybe I should really specify there exists a unique Complex number R so that you could write F of X as basically being this exponential function of R times that value X Which is you know basically saying that if you have X as a function this infinite polynomial with nice derivative properties and all of that if you have this you have Every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it and the sketch of the proof would look something like this if You want to first look at what is the derivative of this value which we're assuming exists everywhere, right? ", + "input": "that's differentiable. It's not equal to zero everywhere. So the condition that sort of slipped my mind and I forget which lecture, lecture four or something like that. And then it has this central property that it turns addition into multiplication. If you have such a function, I claim that there's a unique, maybe I should really specify, there exists a unique complex number r so that you could write f of x as basically being this exponential function of r times that value x. Which is, you know, basically saying that if you have exp as a function, this infinite polynomial with nice derivative properties and all of that, if you have this you have every exponential that you want in a very like abstract generic sense of the word exponential just based on a property that we could want from it. And the sketch of the proof would look something like this. If you want to first look at what is the derivative of this value, which we're assuming exists everywhere, right? ", "translatedText": "Đó là khả vi Nó không bằng 0 ở mọi nơi nên điều kiện đó đã trượt khỏi tâm trí tôi và tôi quên mất bài giảng nào hoặc thứ gì đó tương tự và sau đó nó có tính chất trung tâm là nó biến phép cộng thành phép nhân Nếu bạn có một hàm như vậy tôi khẳng định rằng có một điều duy nhất có lẽ tôi thực sự nên xác định rằng tồn tại một số phức R duy nhất để bạn có thể viết F của X về cơ bản là hàm số mũ của R nhân giá trị X đó. Về cơ bản bạn biết rằng nếu bạn có X là một hàm thì đây đa thức vô hạn với các thuộc tính đạo hàm đẹp và tất cả những thứ đó nếu bạn có cái này bạn có Mọi số mũ mà bạn muốn theo một nghĩa rất trừu tượng chung của từ số mũ chỉ dựa trên một thuộc tính mà chúng ta có thể muốn từ nó và bản phác thảo của bằng chứng sẽ trông giống như thế này nếu trước tiên bạn muốn xem đạo hàm của giá trị này mà chúng ta giả sử tồn tại ở mọi nơi là gì, phải không? ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 3243.32 }, { - "input": "We can factor F of X out of the expression entirely and the whole limit is expressed only in terms of H Which if you think about what it means in the context of derivatives and the fact that F of 0 necessarily equals 1 This whole limiting expression Is just some constant but more specifically it's whatever the derivative of our function at 0 is So you have this funny thing where if you know its derivative at 0 that determines what its derivative is everywhere And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is Proportional to itself and that proportionality constant is equal to whatever the derivative at 0 is this is all very Abstractly phrased and such but but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power X But it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication But if you have that it actually guarantees that you also have a second derivative And for that matter a third derivative and such because the derivative function is just proportional to itself So in order to take the nth derivative you just look at That proportionality constant and raise it to the power n and then from here You could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series in that idea especially if you want to intermix the idea of any Differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic You know you you could intermix the reasoning there as you want But fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for F and kind of justify the idea that There's a unique complex number such that our function F can necessarily be written like this And then the connection to normal exponentials is whenever you have such a value R We do essentially what we do in the complex context of real numbers is if you look at exp of that function of that value R and Write that as a base. ", + "input": "we can factor f of x out of the expression entirely and the whole limit is expressed only in terms of h. Which if you think about what it means in the context of derivatives and the fact that f of zero necessarily equals one, this whole limiting expression is just some constant, but more specifically it's whatever the derivative of our function at zero is. So you have this funny thing where if you know its derivative at zero that determines what its derivative is everywhere. And in the context of exponential functions this is hopefully quite familiar because all that we're really saying is the derivative of an exponential function is proportional to itself and that proportionality constant is equal to whatever the derivative at zero is. This is all very abstractly phrased and such, but the purpose of it is to emphasize that it's not necessarily just functions that we already think of as a to the power x, but it is a potentially much more broad class of functions that just satisfy this abstract property of turning addition into multiplication. But if you have that, it actually guarantees that you also have a second derivative. And for that matter a third derivative and such because the derivative function is just proportional to itself. So in order to take the nth derivative you just look at that proportionality constant and raise it to the power n. And then from here you could do a Taylor series expansion and I might leave that as sort of the advanced homework for those of you who are comfortable with Taylor series and that idea especially if you want to intermix the idea of any differentiable function that's differentiable in a sense of complex numbers, which is sort of a definitely college topic. You know, you could intermix the reasoning there as you want, but fuzzy reasoning is allowed in the context of someone who only knows about Taylor series and nothing else to take this idea and look at the Taylor expansion for f and kind of justify the idea that there's a unique complex number such that our function f can necessarily be written like this. And then the connection to normal exponentials is whenever you have such a value r we do essentially what we do in the complex context of real numbers is if you look at x of that function of that value r and write that as a base ", "translatedText": "Chúng ta có thể phân tích F của X ra khỏi biểu thức hoàn toàn và toàn bộ giới hạn chỉ được biểu thị theo H Mà nếu bạn nghĩ về ý nghĩa của nó trong bối cảnh đạo hàm và thực tế là F của 0 nhất thiết phải bằng 1 Toàn bộ biểu thức giới hạn này là chỉ là một hằng số nào đó nhưng cụ thể hơn nó là đạo hàm của hàm số tại 0. Vì vậy, bạn có một điều buồn cười là nếu bạn biết đạo hàm của nó tại 0 thì nó sẽ xác định đạo hàm của nó ở mọi nơi Và trong bối cảnh của hàm mũ, điều này hy vọng là khá quen thuộc bởi vì tất cả những gì chúng ta thực sự đang nói là đạo hàm của một hàm mũ tỷ lệ với chính nó và hằng số tỷ lệ đó bằng bất cứ đạo hàm nào tại 0, tất cả đều được diễn đạt rất trừu tượng và như vậy nhưng mục đích của nó là để nhấn mạnh rằng đó là không nhất thiết chỉ là các hàm mà chúng ta đã nghĩ là lũy thừa X Nhưng nó là một lớp hàm có khả năng rộng hơn nhiều, chỉ thỏa mãn tính chất trừu tượng này là biến phép cộng thành phép nhân. Nhưng nếu bạn có điều đó thì nó thực sự đảm bảo rằng bạn cũng có một đạo hàm bậc hai Và đối với vấn đề đó, đạo hàm bậc ba và như vậy bởi vì hàm đạo hàm chỉ tỷ lệ với chính nó Vì vậy, để lấy đạo hàm bậc n, bạn chỉ cần nhìn vào hằng số tỷ lệ đó và nâng nó lên lũy thừa n và sau đó từ đây Bạn có thể làm một Việc mở rộng chuỗi Taylor và tôi có thể coi đó là một loại bài tập về nhà nâng cao dành cho những ai cảm thấy thoải mái với chuỗi Taylor trong ý tưởng đó, đặc biệt nếu bạn muốn kết hợp ý tưởng về bất kỳ hàm khả vi nào có thể khả vi theo nghĩa số phức, đó là chắc chắn là một chủ đề đại học Bạn biết đấy, bạn có thể kết hợp lý luận ở đó theo ý muốn Nhưng lý luận mờ được cho phép trong bối cảnh của một người chỉ biết về chuỗi Taylor và không biết gì khác để lấy ý tưởng này và xem xét khai triển Taylor cho F và loại để biện minh cho ý tưởng rằng Có một số phức duy nhất sao cho hàm F của chúng ta nhất thiết có thể được viết như thế này Và khi đó mối liên hệ với số mũ thông thường là bất cứ khi nào bạn có một giá trị như vậy R Về cơ bản, chúng ta thực hiện những gì chúng ta làm trong bối cảnh phức tạp của số thực là nếu bạn nhìn vào exp của hàm đó với giá trị R đó và viết nó làm cơ số. ", "model": "google_nmt", "n_reviews": 0, @@ -1936,7 +1936,7 @@ "end": 3391.12 }, { - "input": "We could interpret that to mean not just exp of pi halves I times X, but we could also interpret it to mean exp of 5 pi halves I Times X and These are separate functions And there's an infinite family of separate functions that feel like we should write them as I to the X So the expression I to the I unless you've adopted a standard for what that's necessarily going to mean When you say it has infinitely many outputs another way to think of that is that The function I to the X with the notation we have is a little bit ambiguous Now with all of that let's let's just start visualizing some of this because I think that's fun And you know you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of R times X, which is Basically this is another way to write e to the power of X in fact I think I I think I rendered a different animation at some point that specified that because I was planning on Planning on doing that so let me oh yeah there you are get back in my file system get back to where you're supposed to be Get on in there is it complaining because there's multiple different It's gonna be like there's a Oh replace it shows up on the other screen Wait why is it yeah, okay replace? ", + "input": "we could interpret that to mean not just exp of pi halves i times x but we could also interpret it to mean exp of five pi halves i times x and these are separate functions and there's an infinite family of separate functions that feel like we should write them as i to the x. So the expression i to the i unless you've adopted a standard for what that's necessarily going to mean when you say it has infinitely many outputs another way to think of that is that the function i to the x with the notation we have is a little bit ambiguous. Now with all of that, let's let's just start visualizing some of this because I think that's fun. And you know, you you tell me if this is if this is a helpful visual or a more confusing visual but what we're going to do is look at this function exp of r times x which is basically this is another way to write e to the power of x. In fact, I think I I think I rendered a different animation at some point that specified that because I was planning on planning on doing that. So let me oh, yeah, there you are get back in my file system get back to where you're supposed to be. Get on in there is it complaining because there's multiple different? It's going to be like there's a oh replace it shows up on the other screen. Wait, why is it? Yeah, okay replace ", "translatedText": "Chúng ta có thể giải thích điều đó không chỉ có nghĩa là exp của nửa pi I nhân X, mà chúng ta cũng có thể hiểu nó có nghĩa là exp của 5 nửa pi I Nhân X và Đây là những hàm riêng biệt Và có một họ vô hạn các hàm riêng biệt mà chúng ta nên làm viết chúng là I cho X Vậy biểu thức I cho I trừ khi bạn đã áp dụng một tiêu chuẩn cho ý nghĩa nhất thiết của nó Khi bạn nói nó có vô số kết quả đầu ra, một cách khác để nghĩ về điều đó là Hàm I cho X với ký hiệu mà chúng ta có hơi mơ hồ Bây giờ với tất cả những điều đó, chúng ta hãy bắt đầu hình dung một số điều này bởi vì tôi nghĩ điều đó thật thú vị Và bạn biết đấy, bạn hãy cho tôi biết đây là đây là hình ảnh hữu ích hay hình ảnh khó hiểu hơn nhưng những gì chúng ta sắp làm là xét hàm exp của R nhân X, về cơ bản đây là một cách khác để viết e lũy thừa của X. Trên thực tế, tôi nghĩ rằng tôi đã tạo ra một hình ảnh động khác tại một số điểm đã chỉ định rằng bởi vì tôi đang lên kế hoạch Lập kế hoạch thực hiện việc đó nên hãy để tôi ồ vâng, bạn quay trở lại hệ thống tập tin của tôi, quay lại nơi bạn phải ở. Hãy vào đó, nó đang phàn nàn vì có nhiều thứ khác nhau. Nó sẽ giống như có một Ồ thay thế nó hiển thị trên màn hình khác Đợi tại sao lại vậy, được chứ thay thế? ", "model": "google_nmt", "n_reviews": 0, @@ -1944,7 +1944,7 @@ "end": 3472.82 }, { - "input": "Place whatever you see there And now we go back to oh there we all of that all of that just so that I could have nicely written out If you're uncomfortable with thinking of it as exp of R times X this infinite polynomial Just in the back of your head e to the R times X and we're gonna vary around R so I'm gonna follow the points of the imaginary axis, and I'm gonna follow the points of the real axis and Let's see what this does Well that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers anything That's a negative real number is going to get squished into the range between 0 and 1 Which should make sense e to the negative? ", + "input": "place whatever you see there. And now we go back to oh there we go all of that all of that just so that I could have nicely written out uh, if you're uncomfortable with thinking of it as exp of r times x this infinite polynomial Just in the back of your head e to the r times x and we're going to vary around r So i'm going to follow the points of the imaginary axis and i'm going to follow the points of the real axis And uh, let's see what this does Well, that's all kind of fast so let me think through it a little bit more slowly all of the negative numbers Anything that's a negative real number is going to get squished into the range between zero and one which should make sense e to the negative ", "translatedText": "Đặt bất cứ thứ gì bạn thấy ở đó Và bây giờ chúng ta quay trở lại ồ, chúng ta tất cả những thứ đó tất cả những thứ đó chỉ để tôi có thể viết ra một cách hay Nếu bạn không thoải mái khi nghĩ về nó như exp của R nhân X đa thức vô hạn này Chỉ trong phía sau đầu của bạn e đến R nhân X và chúng ta sẽ thay đổi xung quanh R vì vậy tôi sẽ theo dõi các điểm của trục ảo, và tôi sẽ theo dõi các điểm của trục thực và Hãy xem điều này làm được gì mọi thứ đều khá nhanh nên để tôi suy nghĩ chậm hơn một chút tất cả các số âm bất cứ điều gì Đó là một số thực âm sẽ bị ép vào khoảng từ 0 đến 1. Cái nào sẽ hợp e với số âm? ", "model": "google_nmt", "n_reviews": 0, @@ -1952,7 +1952,7 @@ "end": 3511.78 }, { - "input": "a to a negative real number is something between 0 and 1 and we're Specifically tracking f of negative 1 which is going to show up around whatever 1 over e is around 30 0.37 f of 1 lands on e as Expected that's what exp of 1 is f of I is gonna land one radian around the unit circle, and it's kind of fun to follow along the whole imaginary axis here how the imaginary axis gets wrapped around a circle and What happens as we tweak this value of R? ", + "input": "e to a negative real number is something between zero and one and we're Specifically tracking f of negative one which is going to show up around whatever one over e is around zero point three seven f of one lands on e As expected that's what x of one is f of i Is going to land one radian around the unit circle and it's kind of fun to follow along the whole imaginary axis here How the imaginary axis gets uh wrapped around a circle? And what happens as we tweak this value of r ", "translatedText": "a đến số thực âm là giá trị nằm trong khoảng từ 0 đến 1 và chúng tôi đang theo dõi cụ thể f của âm 1, số này sẽ hiển thị xung quanh số 1 trên e là khoảng 30 0.37 f của 1 chạm vào e như mong đợi đó là exp của 1 bằng f của I sẽ hạ cánh bằng một radian quanh vòng tròn đơn vị, và thật thú vị khi theo dõi dọc theo toàn bộ trục ảo ở đây cách trục ảo được quấn quanh một vòng tròn và Điều gì xảy ra khi chúng ta điều chỉnh giá trị R này? ", "model": "google_nmt", "n_reviews": 0, @@ -1968,7 +1968,7 @@ "end": 3551.6 }, { - "input": "We might want and values of R here It stretches things differently so when we put it up to 2 You know it stretches out the real axis a lot more so that f of 1 ends up around where e squared is a little Above 7 f of negative 1 is much closer to 0 f of I is a 2 radian Rotation around the circle f of negative I is a negative 2 radian rotation And of course we can get to our favorite formula that if that were pi that we had as our scaling constant Then the real axis gets stretched out quite a lot You know f of 1 is sitting off at e to the pi which is very close to 20 plus pi Which is always fun and f of negative 1 extremely close to 0 so it's really stretched out that real axis And it's also stretched out things in the unit circle direction so that Getting to f of I or f of negative I walks halfway around the circle, so that's all well and good now How would we think about a function like? ", + "input": "we might want and values of r here It stretches things differently So when we put it up to two You know it stretches out the real axis a lot more so that f of one ends up around where e squared is a little Above seven f of negative one is much closer to zero f of i Is a two radian rotation around the circle f of negative i is a negative two radian rotation And of course we can get to our favorite formula that If that were pi that we had as our scaling constant then the real axis gets stretched out quite a lot you know f of one is sitting off at e to the pi which is very close to 20 plus pi which is always fun and f of negative one extremely close to zero so It's really stretched out that real axis and it's also stretched out things in the Unit circle direction so that getting to f of i or f of negative i walks halfway around the circle So that's all well and good now. How would we think about a function like? ", "translatedText": "Chúng ta có thể muốn và các giá trị của R ở đây Nó kéo giãn mọi thứ theo cách khác nên khi chúng ta đặt nó lên 2 Bạn biết đấy, nó kéo dài trục thực nhiều hơn nên f(1) kết thúc ở xung quanh nơi e bình phương hơi lớn hơn 7 f âm 1 gần hơn nhiều với 0 f của I là 2 radian Xoay quanh đường tròn f âm I là một phép quay âm 2 radian Và tất nhiên chúng ta có thể có được công thức yêu thích của mình rằng nếu đó là số pi mà chúng ta có là hằng số tỷ lệ Khi đó trục thực bị kéo giãn ra khá nhiều Bạn biết đấy, f(1) đang nằm ở e với pi rất gần với 20 cộng pi Điều này luôn thú vị và f của âm 1 cực kỳ gần với 0 nên nó thực sự bị giãn ra đến mức thực trục Và nó cũng kéo dãn mọi thứ theo hướng vòng tròn đơn vị sao cho Để đạt được f của I hoặc f âm Tôi đi được nửa vòng tròn, vậy là bây giờ mọi thứ đều ổn và tốt. Chúng ta sẽ nghĩ về một hàm số như thế nào? ", "model": "google_nmt", "n_reviews": 0, @@ -1984,7 +1984,7 @@ "end": 3609.82 }, { - "input": "We would also write as X of X of the natural log of 2 times X so we kind of move our yellow dot representing the value of R To around 0.69 still no imaginary part just a real number 0.69 or so That's the natural log of 2 well you can see that f of 1 lands on 2 Which is why we want to call this function 2 to the X f of 1 half actually sorry f of negative 1 lands right on 1 half f of I It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle and Now we could have a little bit more fun and say what would happen if we were to change this to instead of being 0.69 instead of being the natural log of 2 make it I times the natural log of 2 so that we're really thinking of Something that might have an exponential base to it. ", + "input": "We would also write as exp of Exp of the natural log of two times x So we kind of move our yellow dot representing the value of r to around zero point six nine still no imaginary part Just a real number zero point six nine or so. That's the natural log of two Well, you can see that f of one lands on two, which is why we want to call this function two to the x f of one half actually, sorry f of negative one lands right on one half f of i It's some walk around the unit circle very specifically it's going to be 0.69 radians around the unit circle And now we could have a little bit more fun and say what would happen if we were to Change this to instead of being 0.69 instead of being the natural log of two make it i times the natural log of two So that we're really thinking of something that might have an exponential base to it ", "translatedText": "Chúng ta cũng sẽ viết là X của X của log tự nhiên của 2 nhân X nên chúng ta di chuyển dấu chấm màu vàng biểu thị giá trị của R Đến khoảng 0.69 vẫn không có phần ảo mà chỉ là số 0 thực. 69 hoặc hơn Đó là log tự nhiên của 2 giếng bạn có thể thấy rằng f của 1 tiếp cận 2 Đó là lý do tại sao chúng ta muốn gọi hàm số này là 2 mũ X f của 1 nửa thực sự xin lỗi f của âm 1 nằm ngay trên 1 nửa f của I Đó là một số bước đi vòng quanh vòng tròn đơn vị, cụ thể là nó sẽ bằng 0.69 radian xung quanh vòng tròn đơn vị và Bây giờ chúng ta có thể vui hơn một chút khi nói xem điều gì sẽ xảy ra nếu chúng ta thay đổi giá trị này thành thay vì bằng 0.69 thay vì là log tự nhiên của 2, hãy nhân nó với log tự nhiên của 2 để chúng ta thực sự đang nghĩ về Cái gì đó có thể có cơ số mũ của nó. ", "model": "google_nmt", "n_reviews": 0, @@ -2016,7 +2016,7 @@ "end": 3748.96 }, { - "input": "What is I to the power I in this case it shoves it to around 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of 1 onto the number I So if we were to scale it up even further I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to 5 halves times pi I What you would see is that the unit circle? ", + "input": "What is i to the power i? In this case, it shoves it to around uh, 0.2 around a fifth But there's many different exponential functions that would have this property of putting f of one onto the number i So if we were to scale it up even further, I don't think I have it animated here But if we were to take that yellow dot and raise it up until it got to five halves times pi i What you would see is that the unit circle? ", "translatedText": "I bằng bao nhiêu lũy thừa I trong trường hợp này nó đẩy nó về khoảng 0.2 khoảng 1/5 Nhưng có nhiều hàm số mũ khác nhau sẽ có tính chất đặt f(1) vào số I Vậy nếu chúng ta mở rộng nó hơn nữa thì tôi không nghĩ là tôi có nó hoạt hình ở đây Nhưng nếu chúng ta lấy chấm màu vàng đó và nâng nó lên cho đến khi nó bằng 5 nửa số pi I Những gì bạn sẽ thấy đó là vòng tròn đơn vị? ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 3772.46 }, { - "input": "Is rotated around on itself so that f of negative f of 1 would rotate around another 2 pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of I to the I is a much much smaller number It was around what was it 0.0003 or so But we can also see what I think is quite fun What happens if we consider alternate expressions that we want to interpret as 2 to the power X right? ", + "input": "Uh is rotated around on itself so that f of negative f of one would rotate around another two pi radians and land where it is But it would stretch out the real axis a lot more Which was the sense in which another output of i to the i is a much much smaller number. It was around. What was it? 0.0003 or so But we can also see what I think is quite fun. What happens if we consider Alternate expressions that we want to interpret as two to the power x right? ", "translatedText": "Được quay quanh chính nó sao cho f âm f(1) sẽ quay quanh 2 pi radian khác và hạ cánh ở vị trí của nó Nhưng nó sẽ kéo trục thực ra nhiều hơn Đó là ý nghĩa của một đầu ra khác của I đối với I là một con số nhỏ hơn nhiều. Nó xấp xỉ bằng 0.0003 hoặc hơn Nhưng chúng ta cũng có thể thấy điều mà tôi nghĩ là khá thú vị. Điều gì sẽ xảy ra nếu chúng ta xem xét các biểu thức thay thế mà chúng ta muốn hiểu là 2 lũy thừa X phải không? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 3820.42 }, { - "input": "We have X of R times X and R is equal to this value, which is the natural log of 2 plus pi times I What that means is that when we plug in 1 f of 1 is at negative 2 so we want to write this function as negative 2 to the power X right and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative 2 To the power X it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like 1 half Where we're kind of asking for a square root of negative 2 we realize that we want to write this as something like I times the square root of 2 But if you were to look at this function negative 2 to the power X in the full complex domain that it's dealing with What you're looking at is a function that takes the value of 1 to negative 2 And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", + "input": "We have x of r times x and r is equal to this value, which is the natural log of two plus pi times i What that means is that when we plug in one? f of one is at negative two, so we want to write this function as negative two to the power x right, and that's actually something that You know, it's it's a little deceptively simple when we write a negative number to a power Negative two To the power x it doesn't at first look like this necessarily it brings us into the complex numbers in any way but of course when we plug in even a value like One half Where we're kind of asking for a square root of negative two We we realize that we want to write this as something like i times the square root of two But if you were to look at this function negative two to the power x in the full complex domain that it's dealing with What you're looking at is a function that takes the value of one to negative two And if it does that what it does to the rest of the real number line is it kind of spirals it outward? ", "translatedText": "Chúng ta có X của R nhân X và R bằng giá trị này, là log tự nhiên của 2 cộng pi nhân I Điều đó có nghĩa là khi chúng ta thay 1 f của 1 vào âm 2 nên chúng ta muốn viết hàm này đúng là âm 2 lũy thừa X và đó thực sự là điều mà Bạn biết đấy, nó hơi đơn giản một chút khi chúng ta viết số âm lũy thừa Âm 2 lũy thừa X thoạt nhìn nó không nhất thiết phải như thế này. vào các số phức theo bất kỳ cách nào nhưng tất nhiên là khi chúng ta thế vào một giá trị chẵn như 1 nửa Khi chúng ta đang yêu cầu căn bậc hai của âm 2, chúng ta nhận ra rằng chúng ta muốn viết cái này giống như I nhân căn bậc hai của 2 Nhưng nếu bạn xét hàm này âm 2 lũy thừa X trong miền phức đầy đủ mà nó đang xử lý Cái bạn đang xem là một hàm nhận giá trị từ 1 đến âm 2 Và nếu nó thực hiện điều đó thì sao nó tác động đến phần còn lại của dãy số thực nó có dạng xoắn ốc hướng ra ngoài không? ", "model": "google_nmt", "n_reviews": 0, @@ -2056,7 +2056,7 @@ "end": 3879.66 }, { - "input": "So we see that f of negative 1 sits at negative 1 half About where you would expect if you were to follow to f of 1 half It would sit exactly on the imaginary line and f of 1 half would be square root of 2 Well, my mouse is not where I want it to be. ", + "input": "So we see that f of negative one sits at negative one half About where you would expect if you were to follow to f of one half It would sit exactly on the imaginary line and f of one half would be square root of two Well, my mouse is not where I want it to be. ", "translatedText": "Vì vậy, chúng ta thấy rằng f của âm 1 nằm ở nửa âm 1 Về vị trí mà bạn mong đợi nếu bạn theo f của 1 nửa Nó sẽ nằm chính xác trên đường tưởng tượng và f của 1 nửa sẽ là căn bậc hai của 2 Chà, tôi ơi chuột không ở nơi tôi muốn. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 3894.92 }, { - "input": "It would be around square root of 2 times I and As you continue further on this is showing you all of the real value powers of negative 2 to the X it necessarily spirals around But we could also move our value of R even higher and get it up to around tau times I around six point two eight times I and in that context this is another function that we would want to write as something like 2 to the X because For any whole number to whole number that you plug in for X it will look like repeated multiplication And it even has kind of reasonable values for things like 1 half where it spits out the negative square root instead of a positive Square root, but what it's actually doing is a transformation to the plane Where it puts everything is the real number line ends up being a very tightly wound Spiral that goes around and it just spirals in such a way that f of 1 lands right on the number 2 So it is in that sense that we could say 2 to the X is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to So I think with all of that I will I will leave things for today And I'll just leave you with a couple lingering questions to think about okay, so If you want to think of I to the I as being a multi-valued expression right you could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function And maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does 2 to the 1 third want to be in the same sense? ", + "input": "It would be around a square root of two times i and As you continue further on this is showing you all of the real value powers of negative two to the x it necessarily spirals around Um, but we could also move our value of r even higher and get it up to around tau times i around 6.28 times i And in that context, this is another function that we would want to write as something like two to the x because For any whole number to whole number that you plug in for x it will look like repeated multiplication And it even has kind of reasonable values for things like one half where it spits out the negative square root instead of the positive square But what it's actually doing is a transformation to the plane where it puts everything Uh is the real number line ends up being a very tightly wound spiral That goes around and it just spirals in such a way that f of one lands right on the number two so it is in that sense that we could say, um two to the x is Is plausibly interpreted as a separate exponential function from the one that we are traditionally used to so I think with all of that I will um I will leave things for today and i'll just leave you with a couple lingering questions to think about. Okay, so If you want to think of i to the i as being a multi-valued expression, right? You could you could say we adopt a convention Fancifully you'd say you choose a branch of the natural logarithm function and maybe that locks you into this being e to the negative pi halves But if you say this kind of wants to be infinitely many different values like the various ones that we saw How many values does two to the one-third want to be in the same sense ", "translatedText": "Nó sẽ xấp xỉ căn bậc hai của 2 nhân I và khi bạn tiếp tục đi xa hơn, điều này sẽ cho bạn thấy tất cả lũy thừa giá trị thực của âm 2 đối với X, nó nhất thiết phải xoắn ốc xung quanh Nhưng chúng ta cũng có thể di chuyển giá trị R của mình lên cao hơn nữa và đạt được nó lên đến khoảng tau lần tôi khoảng sáu phẩy hai tám lần tôi và trong bối cảnh đó, đây là một hàm khác mà chúng ta muốn viết dưới dạng 2 mũ X bởi vì với mọi số nguyên đến số nguyên mà bạn thế vào cho X thì nó sẽ trông giống như phép nhân lặp đi lặp lại Và nó thậm chí còn có các loại giá trị hợp lý cho những thứ như 1 nửa trong đó nó tạo ra căn bậc hai âm thay vì căn bậc hai dương, nhưng điều nó thực sự đang làm là một phép biến đổi sang mặt phẳng Nơi nó đặt mọi thứ là thực trục số cuối cùng trở thành một đường xoắn ốc quấn rất chặt và nó chỉ xoắn ốc sao cho f(1) nằm ngay trên số 2 Vì vậy, theo nghĩa đó, chúng ta có thể nói 2 với X là được hiểu một cách hợp lý là một hàm số mũ riêng biệt với hàm mà chúng ta thường sử dụng Vì vậy, tôi nghĩ với tất cả những điều đó tôi sẽ để lại mọi thứ cho ngày hôm nay Và tôi sẽ chỉ để lại cho bạn một vài câu hỏi còn sót lại để suy nghĩ, được thôi, vì vậy nếu bạn muốn hãy coi I đối với I như là một biểu thức đa giá trị đúng không bạn có thể nói rằng chúng ta áp dụng một quy ước Thật kỳ lạ là bạn sẽ nói rằng bạn chọn một nhánh của hàm logarit tự nhiên Và có thể điều đó sẽ khóa bạn vào bản thể này e đến số pi âm một nửa Nhưng nếu bạn nói loại này muốn có vô số giá trị khác nhau giống như những giá trị khác nhau mà chúng ta đã thấy Có bao nhiêu giá trị từ 2 đến 1 phần ba muốn có cùng một nghĩa? ", "model": "google_nmt", "n_reviews": 0, @@ -2080,7 +2080,7 @@ "end": 4008.86 }, { - "input": "10ths want to be Phrased differently of all of the let me say of all of the exponential functions F of X which satisfy oh have I written it down somewhere f of X that satisfies All of these properties that I've written so if it satisfies all of these and if f of 1 is equal to 2 Right how many different outputs are we going to get when we plug in X equals 3 10ths for the various options for what function? ", + "input": "hree-tenths want to be? Phrased differently of all of the uh, let me say of all of the exponential functions So f of x which satisfy oh have I written it down somewhere f of x that satisfies All of these properties that i've written so if it satisfies all of these um and if f of one is equal to two Right, how many different outputs are we going to get when we plug in x equals three-tenths for the various options for what function? ", "translatedText": "Số 10 muốn được diễn đạt khác với tất cả, hãy để tôi nói về tất cả các hàm số mũ F của X thỏa mãn ồ tôi đã viết nó ra đâu đó f của X thỏa mãn Tất cả các tính chất mà tôi đã viết như vậy nếu nó thỏa mãn tất cả trong số này và nếu f(1) bằng 2 Đúng thì chúng ta sẽ nhận được bao nhiêu đầu ra khác nhau khi cắm X bằng 3 phần 10 cho các tùy chọn khác nhau cho hàm số nào? ", "model": "google_nmt", "n_reviews": 0, @@ -2096,7 +2096,7 @@ "end": 4042.58 }, { - "input": "For 2 to the pi for the various functions that 2 to the X could represent if we're thinking of 2 to the X as some kind of exponential function Exponential in the sense of these sort of abstract properties and if we yeah, if we if we have a Class of different such functions, and we want to plug in pi it makes me laugh Just because it's such a I know kind of a funny answer that pops out as you're trying to think about it so those are the questions that I'll leave you with and I think this is you know my My central question in approaching today's lecture was whether I wanted it to be Kind of describing like these abstract properties of exponential functions And it's just cool to me that starting from those abstract properties you get locked into the idea of e to the rx or more Just you know I think more honestly written exp of r times x for different values of r That it locks you in that far But it doesn't lock you in as far as having an unambiguous Notion of what 2 to the power x should be much less something like I to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes It doesn't come off as approachable But if that's the case you know you just let me know I think I think there's a whole interesting circle of thoughts that surrounds All of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this, so it was a question that we had up on screen Yeah, what happens if we do this for I to the power I? ", + "input": "For two to the pi for the various functions that two to the x could represent If we're thinking of two to the x as some kind of exponential function exponential in the sense of these sort of abstract properties and if we uh yeah, if we if we have a Class of different such functions and we want to plug in pi it makes me laugh just because it's such a I don't know Kind of a funny answer that pops out As you're trying to think about it. So those are the questions that i'll leave you with and I think this is you know, my my My central question in approaching today's lecture was whether I wanted to be um kind of describing like these abstract properties of exponential functions and it's just cool to me that Starting from those abstract properties you get locked into the idea of e to the rx or more You know, I think more honestly written exp of r times x for different values of r That it locks you in that far but it doesn't lock you in as far as having an unambiguous Notion of what two to the power x should be much less something like i to the power x The risk in that of course is that sometimes people don't love abstraction and sometimes it doesn't come off as approachable But if that's the case, you know, you just let me know I think I think there's a whole interesting circle of thoughts that surrounds all of this stuff to include power towers because if you want to Actually talk about power towers like we were last time in the context of complex numbers or even with negative bases You have to be thinking through things like this so, um, it was a question that we had up on screen Uh, yeah, what happens if we do this for i to the power i t ", "translatedText": "Đối với 2 mũ pi cho các hàm khác nhau mà 2 mũ X có thể biểu thị nếu chúng ta coi 2 mũ X là một loại hàm số mũ nào đó Hàm mũ theo nghĩa của các loại thuộc tính trừu tượng này và nếu chúng ta đúng, nếu chúng ta nếu chúng tôi có một Lớp gồm các chức năng khác nhau và chúng tôi muốn cắm số pi vào, nó khiến tôi bật cười Chỉ vì đó là một câu trả lời tôi biết rất buồn cười bật ra khi bạn đang cố nghĩ về nó nên đó là những câu hỏi mà Tôi sẽ để lại cho bạn và tôi nghĩ đây là điều bạn biết Câu hỏi trọng tâm của tôi khi tiếp cận bài giảng hôm nay là liệu tôi có muốn nó diễn ra Kiểu mô tả giống như những thuộc tính trừu tượng này của hàm số mũ Và thật tuyệt vời đối với tôi rằng bắt đầu từ những thuộc tính trừu tượng đó bạn bị khóa vào ý tưởng về e đến rx hoặc hơn. Bạn biết đấy, tôi nghĩ exp được viết trung thực hơn của r nhân x cho các giá trị khác nhau của r Rằng nó khóa bạn ở mức đó Nhưng nó không khóa bạn đến mức có một Khái niệm rõ ràng về những gì 2 mũ x sẽ ít giống I mũ x Rủi ro ở chỗ đó tất nhiên là đôi khi mọi người không thích sự trừu tượng và đôi khi Nó không có vẻ dễ tiếp cận Nhưng nếu đó là trường hợp bạn biết bạn chỉ cần cho tôi biết Tôi nghĩ tôi nghĩ có cả một vòng tròn suy nghĩ thú vị bao quanh Tất cả những thứ này bao gồm cả tháp điện bởi vì nếu bạn muốn Thực sự nói về tháp điện như lần trước chúng ta đã nói trong bối cảnh số phức hoặc thậm chí với các cơ sở phủ định Bạn phải suy nghĩ thấu đáo những điều như thế này, vì vậy đó là một câu hỏi mà chúng tôi đưa ra trên màn hình Vâng, điều gì sẽ xảy ra nếu chúng ta làm điều này cho tôi với sức mạnh của tôi? ", "model": "google_nmt", "n_reviews": 0, @@ -2104,7 +2104,7 @@ "end": 4135.8 }, { - "input": "Titration you know let's just try this let's just go ahead and try a power tower Where we're raising I to a given power and see what what pops out of it, so it wasn't planning on doing this But we can we can always pull up Python and essentially do what we were doing last time So the way that this would work is we were starting off with some base value and then for some kind of range What were we doing we were taking a and we're going to reassign it to be whatever The base which in this case is I raised to the power of a should be Okay, cool, so we're going to do that and then we're going to print off the value of a let's just do this for Yeah, it's a much bigger number like 200 So it seems like what happens is There's potential for chaos with these things like sometimes. ", + "input": "itration, you know, let's just try this Let's just go ahead and try a power tower where we're raising i to a given power and see what uh, what pops out of it so I wasn't planning on doing this but we can We can always pull up python and essentially do what we were doing last time so the way that this would work Is we were starting off with some base value and then for some kind of range What were we doing? We were taking a and we're going to reassign it to be whatever The base which in this case is i raised to the power of a should be Okay, cool. So we're going to do that and then we're going to print off the value of a and let's just do this for Uh, yeah, it's a much bigger number like 200 uh So it seems like what happens is There's there's potential for chaos with these things like sometimes ", "translatedText": "Chuẩn độ bạn biết đấy, hãy thử cái này hãy tiếp tục và thử một tháp điện Nơi chúng ta đang nâng tôi lên một sức mạnh nhất định và xem điều gì bật ra từ nó, vì vậy nó không có kế hoạch làm điều này Nhưng chúng ta có thể, chúng ta luôn có thể lấy Python ra và về cơ bản làm những gì chúng ta đã làm lần trước Vậy cách thức hoạt động là chúng ta bắt đầu với một số giá trị cơ bản và sau đó với một loại phạm vi nào đó. Chúng ta đang làm gì, chúng ta đang lấy a và chúng ta sẽ gán lại nó là bất cứ cái gì Cơ số mà trong trường hợp này là tôi đã nâng lên lũy thừa của a sẽ là Được rồi, tuyệt, vì vậy chúng ta sẽ làm điều đó và sau đó chúng ta sẽ in ra giá trị của a hãy cứ làm điều này cho Vâng, đó là một con số lớn hơn nhiều, chẳng hạn như 200. Vì vậy, có vẻ như điều xảy ra là Đôi khi, những thứ này có khả năng xảy ra hỗn loạn. ", "model": "google_nmt", "n_reviews": 0, @@ -2136,7 +2136,7 @@ "end": 4234.12 }, { - "input": "I we actually have so let me let me import NumPy so I have the exponential function let me go For our big range like we had before Rather than writing it as you know something that's like I to the power of X I'm gonna write it as the exponential function of a different constant right a Different constant that I'm gonna make I want it to be 5 pi halves, so I'll do 5 pi halves times I so it's a complex number and It's got 5 pi halves as the imaginary part So this is 5 pi halves times I and what am I doing? ", + "input": "I we actually have so let me Let me import NumPy so I have the exponential function Let me go For our big range like we had before Rather than writing it as you know, something that's like i to the power of x I'm going to write it as the exponential function of a different constant right a different constant That i'm going to make I want it to be five pi halves. So i'll do five pi halves times i so it's a complex number And it's got five pi halves as the imaginary part So this is five pi halves times i and what am I doing? ", "translatedText": "Tôi thực sự có nên hãy để tôi nhập NumPy để tôi có hàm số mũ cho phép tôi đi Đối với phạm vi lớn của chúng tôi như chúng tôi đã có trước đây Thay vì viết nó như bạn biết điều gì đó giống như tôi với sức mạnh của X Tôi sẽ viết nó là hàm số mũ của một hằng số khác phải Một hằng số khác mà tôi sẽ tạo Tôi muốn nó bằng 5 nửa pi, vì vậy tôi sẽ tính 5 nửa pi nhân với tôi nên nó là một số phức và nó có 5 nửa pi là phần ảo Vậy đây là 5 nửa pi nhân tôi và tôi đang làm gì? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/arabic/sentence_translations.json b/2020/ldm-imaginary-interest/arabic/sentence_translations.json index 520e4a5e2..4b696fb22 100644 --- a/2020/ldm-imaginary-interest/arabic/sentence_translations.json +++ b/2020/ldm-imaginary-interest/arabic/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "هل تعتقد أن مثل هذه الصدفة تجعل الرياضيات جمالًا مثاليًا؟ يا لها من تغريدة مكتوبة بشكل شعري، والتي تبدو مناسبة جدًا لشخص لديه ملف تعريف لـ Yoda هناك. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "أنا شخصياً أعتقد أن أجمل الأشياء هي تلك التي لها روابط غير متوقعة. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "على سبيل المثال، إذا أردنا معرفة ما يحدث في مثال الفائدة الخاص بنا، عندما بدأنا بـ m من 0، والذي قد يساوي 100 دولار تقريبًا، ما نفعله هو أننا نركز فقط على مصطلح rt هذا، ونقول إننا نعرف ذلك إذا قمنا برفع n، فهذا سيقترب من ثابت خاص مرفوع إلى قوة rt، وربما نوصف هذا على نحو ملائم بأنه نمو مركب مستمر، إذا أخذنا تلك الخطوة الزمنية وتركناها تقترب من 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "إذن ما نقوله هو أنه إذا بدأت بسرعة عالية وإزاحة منخفضة، فإن كتلتك تتحرك بسرعة، ولكنها ليست بعيدة جدًا عن نقطة التوازن، حسنًا، ستزداد x، لأن هذا ما يعني أن تتحرك بسرعة. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/bengali/sentence_translations.json b/2020/ldm-imaginary-interest/bengali/sentence_translations.json index e9916c2f7..6f02a81ef 100644 --- a/2020/ldm-imaginary-interest/bengali/sentence_translations.json +++ b/2020/ldm-imaginary-interest/bengali/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "কি একটি কাব্যিকভাবে লেখা টুইট, যা সেখানে Yoda এর প্রোফাইল সহ কারো জন্য বেশ মানানসই বলে মনে হচ্ছে।", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "ব্যক্তিগতভাবে, আমি মনে করি সবচেয়ে সুন্দর জিনিসগুলি হল অপ্রত্যাশিত সংযোগগুলি।", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "সুতরাং উদাহরণস্বরূপ, যদি আমরা জানতে চাই যে আমাদের আগ্রহের উদাহরণে কী ঘটে, যখন আমরা 0 এর m দিয়ে শুরু করেছি, যা $100 এর মতো কিছু হতে পারে, আমরা যা করি তা হল আমরা শুধুমাত্র সেই rt টার্মের উপর ফোকাস করি, আমরা বলি যে আমরা জানি আমরা ক্র্যাঙ্ক আপ n, এটি rt এর শক্তিতে উত্থাপিত কিছু বিশেষ ধ্রুবকের কাছে যেতে চলেছে, এবং সম্ভবত এটিকে আমরা যথাযথভাবে ক্রমাগত চক্রবৃদ্ধি হিসাবে বর্ণনা করব, যদি আমরা সেই সময় পদক্ষেপটি গ্রহণ করি এবং আমরা এটিকে 0 এর কাছে যেতে দেই।", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "তাই আমরা যা বলছি তা হল আপনি যদি উচ্চ বেগ এবং কম স্থানচ্যুতি দিয়ে শুরু করেন, তাহলে আপনার ভর দ্রুত গতিতে চলেছে, তবে এটি ভারসাম্য বিন্দু থেকে খুব বেশি দূরে নয়, হ্যাঁ, x বাড়তে চলেছে, কারণ এটাই এর অর্থ দ্রুত চলমান।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/chinese/sentence_translations.json b/2020/ldm-imaginary-interest/chinese/sentence_translations.json index a90c4b394..8c8d1aaef 100644 --- a/2020/ldm-imaginary-interest/chinese/sentence_translations.json +++ b/2020/ldm-imaginary-interest/chinese/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "多么富有诗意的推文，对于其中有尤达简介的人来说，这似乎非常合适。", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "就我个人而言，我确实认为最美丽的事物是那些有着意想不到的联系的事物。", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "例如，如果我们想知道在我们的兴趣示例中发生了什 么，当我们从 m 为 0 开始时，可能是 10 0 美元，我们所做的就是只关注 rt 项，我们 说我们知道我们提高 n，这将接近某个特殊常数的 rt 次方，如果我们采取该时间步长并让它接近 0，也许我们会恰当地将其描述为连续复合增长。", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "所以我们要说的是，如果你以高速度和低位移开始，那么你的质量移动得很快，但离平衡点并没有那么远，是的，x 会增加，因为这就是这意味着要快速行动。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/english/captions.srt b/2020/ldm-imaginary-interest/english/captions.srt index 5c4d210e7..334fbf6de 100644 --- a/2020/ldm-imaginary-interest/english/captions.srt +++ b/2020/ldm-imaginary-interest/english/captions.srt @@ -256,7 +256,7 @@ What I want to gauge is what your gut reaction to this question is, okay? 65 00:03:39,680 --> 00:03:43,420 -So there is a correct answer, but just answer what you think it might be. +so there is a correct answer, but just answer what you, what you think it might be. 66 00:03:43,880 --> 00:03:48,160 @@ -327,3630 +327,3642 @@ And in general, by the way, while these answers are rolling in, I'll sometimes be pulling up questions from the audience, which you can ask via Twitter. 83 -00:04:48,660 --> 00:04:53,780 -So the link is in the description for where you can go to ask these sorts of questions. +00:04:48,660 --> 00:04:51,220 +So the link is in the description for, you know, 84 +00:04:51,220 --> 00:04:53,780 +where you can go to ask these sorts of questions. + +85 00:04:53,780 --> 00:05:09,085 And it looks like someone with the profile of Yoda asks, What a poetically written tweet, -85 +86 00:05:09,085 --> 00:05:21,160 which seems pretty fitting for someone with a profile of Yoda in there. -86 +87 00:05:21,700 --> 00:05:23,400 I mean, it's a subjective question. -87 +88 00:05:23,440 --> 00:05:25,771 Personally, I do think the most beautiful things -88 +89 00:05:25,771 --> 00:05:27,960 are the ones that have unexpected connections. -89 +90 00:05:28,600 --> 00:05:31,220 And I don't know if that's like a natural thing that humans -90 +91 00:05:31,220 --> 00:05:33,840 just love to see things that seemed unrelated come together. -91 -00:05:34,260 --> 00:05:37,846 -Humor seems to have this root, like a really good joke kind of takes you by surprise, - 92 -00:05:37,846 --> 00:05:41,600 -but when there's some kind of logical connection, it's not just surprise for its own sake. +00:05:34,260 --> 00:05:37,653 +Humor seems to have this root, like a really good joke kind of takes you by surprise. 93 +00:05:37,653 --> 00:05:40,021 +But when it's when there's some kind of logical connection, + +94 +00:05:40,021 --> 00:05:41,600 +it's not just surprise for its own sake. + +95 00:05:42,980 --> 00:05:46,756 And as the lesson goes on, if you have questions relevant to what we're talking about, -94 +96 00:05:46,756 --> 00:05:48,840 ask them there and we'll pull them up on screen. -95 +97 00:05:49,560 --> 00:05:52,446 So, like I said, there's technically a right answer to this question, -96 +98 00:05:52,446 --> 00:05:54,880 but I don't really want it to be treated as right or wrong. -97 +99 00:05:55,280 --> 00:05:57,418 The reason I'm asking this one is to get kind of -98 +100 00:05:57,418 --> 00:05:59,600 an opinion poll to know where people are starting. -99 +101 00:06:00,140 --> 00:06:03,360 So I'm going to go ahead and lock in the answers and see where people are. -100 +102 00:06:04,280 --> 00:06:04,300 Interesting! -101 +103 00:06:04,660 --> 00:06:08,262 This might be the first time in lockdown math history that the -102 +104 00:06:08,262 --> 00:06:11,980 plurality has not gotten what turns out to be the correct answer. -103 +105 00:06:12,620 --> 00:06:14,833 So most of you said you would have more money -104 +106 00:06:14,833 --> 00:06:17,240 with bank B and it would be by more than a dollar. -105 +107 00:06:17,900 --> 00:06:21,611 The second most common answer was to say you'd have more money -106 +108 00:06:21,611 --> 00:06:25,440 with bank B by less than a dollar, which turns out to be correct. -107 +109 00:06:25,540 --> 00:06:26,700 We're going to walk through that in a moment. -108 +110 00:06:27,480 --> 00:06:29,840 After that, more with bank A. -109 +111 00:06:30,500 --> 00:06:33,115 And I think the ones that I most empathize with, -110 +112 00:06:33,115 --> 00:06:37,491 if I look back at what maybe an initial response to questions like this would be, -111 +113 00:06:37,491 --> 00:06:40,800 if you haven't thought about interest, is everyone who said C. -112 +114 00:06:41,200 --> 00:06:43,736 You know, it's not obvious that 12% over a year -113 +115 00:06:43,736 --> 00:06:46,220 is going to be any different than 1% per month. -114 +116 00:06:46,220 --> 00:06:48,020 -You're like, 1% per month? +r month. Like one percent per month, -115 +117 00:06:48,120 --> 00:06:50,000 That should add up to 12% over the year. -116 +118 00:06:51,000 --> 00:06:54,100 And I think it's worth just thinking through exactly why that's not the case. -117 +119 00:06:54,800 --> 00:06:57,160 So, how do we think about this kind of problem? -118 +120 00:06:57,760 --> 00:07:01,080 I've gone ahead and pulled up a Desmos graph for us to play around with. -119 +121 00:07:01,280 --> 00:07:05,500 So we'll get rid of Burckardt's beautiful face and we'll head on over to Desmos. -120 +122 00:07:06,580 --> 00:07:09,565 And the way I have things configured, it's going to be showing us -121 +123 00:07:09,565 --> 00:07:12,460 the graph of what happens to your money, in this case in bank A. -122 +124 00:07:12,460 --> 00:07:15,687 So over the first year, at any point that you check in, -123 +125 00:07:15,687 --> 00:07:20,760 it's just got that $100 and it's not until the end of the year that it jumps up to $112. -124 +126 00:07:21,540 --> 00:07:24,922 And the way you might think about that is by saying, okay, -125 +127 00:07:24,922 --> 00:07:28,820 let's take our $100, at the end of the year we add 12% of that $100. -126 +128 00:07:29,520 --> 00:07:31,460 And sensibly enough, you get 112. -127 +129 00:07:33,120 --> 00:07:39,101 And a nice way to think about this, which seems simple and it seems kind of innocuous, -128 +130 00:07:39,101 --> 00:07:41,920 but this turns out to be a powerful idea. -129 +131 00:07:41,920 --> 00:07:44,785 We'll talk about more why I might want to emphasize that -130 +132 00:07:44,785 --> 00:07:47,600 this is more of a significant move than you might think. -131 +133 00:07:47,700 --> 00:07:50,520 The fact that the rate of change is proportional to the -132 +134 00:07:50,520 --> 00:07:53,340 thing that's changing means we can factor out this $100. -133 +135 00:07:53,800 --> 00:07:57,780 And we can just say, oh, that step that we take, we're multiplying it by a constant. -134 +136 00:07:57,920 --> 00:08:00,200 In this case, that constant would be 1.12. -135 +137 00:08:01,120 --> 00:08:05,015 So as you go from this step to the second year, at the end of that second year, -136 +138 00:08:05,015 --> 00:08:07,303 your account balance doesn't jump by just $12, -137 +139 00:08:07,303 --> 00:08:11,540 because it's also earning interest on the 12 extra dollars that you got that last year. -138 +140 00:08:11,540 --> 00:08:13,400 So it jumps up to $125. -139 +141 00:08:13,840 --> 00:08:15,280 It's a multiplicative amount. -140 +142 00:08:16,020 --> 00:08:20,664 And this becomes particularly noticeable as we zoom out and see what would happen -141 +143 00:08:20,664 --> 00:08:25,140 over the course of many years with such a fantastic interest rate in your bank. -142 +144 00:08:26,480 --> 00:08:30,080 It's not just growing like a straight line, it grows with this exponential curve. -143 +145 00:08:30,600 --> 00:08:33,312 And in fact, if you look at 10 years in, it looks -144 +146 00:08:33,312 --> 00:08:35,700 like the Y coordinate of that is around 310. -145 +147 00:08:35,700 --> 00:08:40,594 So you let your money sit in that savings account with 12% interest for 10 years, -146 +148 00:08:40,594 --> 00:08:43,460 and you'd end up with three times as much money. -147 +149 00:08:43,940 --> 00:08:44,400 Pretty interesting. -148 +150 00:08:45,660 --> 00:08:49,357 So now when we start thinking about having the interest accrue, -149 +151 00:08:49,357 --> 00:08:53,402 not just at the end of the year, but at various chunks on the way in, -150 +152 00:08:53,402 --> 00:08:55,540 let me just ask you another question. -151 +153 00:08:55,700 --> 00:08:58,078 I think this is a fun way to start things off, -152 +154 00:08:58,078 --> 00:09:02,480 is rather than having me go through the logic, have you guys think through the details. -153 -00:09:03,239 --> 00:09:06,621 +155 +00:09:03,240 --> 00:09:06,621 So before we jump to the case of doing a step every month, -154 +156 00:09:06,621 --> 00:09:10,060 let's just say there was two steps halfway through the year. -155 +157 00:09:10,880 --> 00:09:12,160 So what does the question ask us? -156 +158 00:09:12,460 --> 00:09:15,164 It says a bank offers to increase the money in your -157 +159 00:09:15,164 --> 00:09:17,920 savings account by 6% at the end of every six months. -158 +160 00:09:18,900 --> 00:09:22,100 Which of the following represents how much money will be in -159 +161 00:09:22,100 --> 00:09:25,460 your account if you put in $100 and then you wait for one year? -160 +162 00:09:25,940 --> 00:09:29,540 So there's two different six month periods that have passed. -161 +163 00:09:29,540 --> 00:09:33,080 Which of the following expressions shows how much you're going to have? -162 +164 00:09:34,080 --> 00:09:37,220 -So I'm going to give you a moment to think through the details of this. +All right, so I'm going to give you a moment to think through the details of this. -163 +165 00:09:37,600 --> 00:09:39,460 Give you a little pause and ponder music maybe. -164 +166 00:10:13,240 --> 00:10:17,173 Okay, as always, I'm probably going to grade this faster than is a reasonable -165 +167 00:10:17,173 --> 00:10:21,360 amount of time for someone who really wants to think through the details of things. -166 +168 00:10:21,420 --> 00:10:22,980 So never feel like you're being rushed. -167 +169 00:10:23,460 --> 00:10:25,820 Sometimes it's just that I want to move forward with the lesson. -168 +170 00:10:26,460 --> 00:10:29,100 And it seems like answers are rolling in a little bit more slowly now. -169 +171 00:10:29,100 --> 00:10:31,358 So I'm going to go ahead and lock this in, but as always, -170 +172 00:10:31,358 --> 00:10:34,280 feel free to pause and just think things through more yourself if you want. -171 +173 00:10:34,660 --> 00:10:35,780 We're going to explain it in a moment. -172 +174 00:10:36,560 --> 00:10:41,234 So the correct expression, which looks like 3,000 of you got, -173 +175 00:10:41,234 --> 00:10:45,080 is that it should be $100 times 1 plus 0.6 squared. -174 +176 00:10:45,880 --> 00:10:46,060 Alright? -175 +177 00:10:46,600 --> 00:10:48,100 Now let's think through why that might be the case. -176 -00:10:48,300 --> 00:10:53,728 +178 +00:10:48,300 --> 00:10:53,366 So if we head back over to our Desmos expression that we were working with, -177 -00:10:53,728 --> 00:10:58,157 -If instead of increasing by 12%, we're saying increase by 6%, +179 +00:10:53,366 --> 00:10:58,433 +if instead of increasing by 12 percent, we're saying increase by 6 percent, -178 -00:10:58,157 --> 00:11:02,300 +180 +00:10:58,433 --> 00:11:02,300 when you factor it out, it looks like multiplying by 1.06. -179 +181 00:11:03,680 --> 00:11:05,560 And then there's two ways you can think about this. -180 +182 00:11:05,600 --> 00:11:08,440 If you're already comfortable with the idea that increasing by 6% -181 +183 00:11:08,440 --> 00:11:11,280 is multiplying by this constant, you say, oh, we just square that. -182 +184 00:11:12,140 --> 00:11:14,763 And if you want to think through the details for why that's true, -183 +185 00:11:14,763 --> 00:11:17,586 if that's not something you're entirely comfortable with, you can say, -184 +186 00:11:17,586 --> 00:11:18,580 let's repeat the process. -185 +187 00:11:18,580 --> 00:11:21,920 So this gets us 106 after that first six month period. -186 +188 00:11:22,180 --> 00:11:25,893 Okay, we've got 106, then we're going to add 6%, -187 +189 00:11:25,893 --> 00:11:30,820 0.06 times that 106, and that's going to get us the final amount. -188 +190 00:11:31,220 --> 00:11:34,740 But this we can factor, because the 106 shows up in two places. -189 +191 00:11:34,860 --> 00:11:39,038 Because the rate of growth is proportional to itself, -190 +192 00:11:39,038 --> 00:11:42,520 it lets us factor this out as 106 times 1.06. -191 +193 00:11:43,360 --> 00:11:45,171 And then you might realize, oh wait, I'm just -192 +194 00:11:45,171 --> 00:11:47,180 multiplying by the same constant that I did before. -193 +195 00:11:47,180 --> 00:11:50,280 So I might as well have just gone and put that constant up there. -194 +196 00:11:50,660 --> 00:11:55,040 And because it's the same constant, I might as well have simply written that as a square. -195 +197 00:11:57,060 --> 00:12:00,537 So what that means is if you take these steps of percent increases many, -196 +198 00:12:00,537 --> 00:12:03,729 many different times, what it looks like is taking a number that's -197 +199 00:12:03,729 --> 00:12:06,540 a little above 1 and then raising it to some kind of power. -198 +200 00:12:06,980 --> 00:12:11,480 So to our original question, asking about bank B, which, what did it say? -199 +201 00:12:11,540 --> 00:12:14,880 It was going to be a 1% increase to your savings every month. -200 +202 00:12:14,880 --> 00:12:20,788 The way you might think about that is by saying we multiply by 1.01, -201 +203 00:12:20,788 --> 00:12:25,840 okay, by 1 plus that 1%, and then we do it after 12 months. -202 +204 00:12:26,480 --> 00:12:34,300 Okay, so instead of the, instead of the $112 that you would have had from bank A, woo hoo! -203 +205 00:12:34,460 --> 00:12:39,180 It looks like compounding more frequently got us an extra 68 cents in our account. -204 +206 00:12:39,380 --> 00:12:41,560 So, wonderful, right? -205 +207 00:12:41,560 --> 00:12:45,901 In our graph, if we wanted to see what that would look like, again, -206 +208 00:12:45,901 --> 00:12:49,859 there's some machinery under here that I'll show in a moment, -207 +209 00:12:49,859 --> 00:12:55,157 but if I just crank up this number n to 12, that's basically asking how many times -208 +210 00:12:55,157 --> 00:12:57,520 per year do I compound this interest? -209 +211 00:12:57,800 --> 00:13:01,640 Do I make a little step increase based on what the annual interest rate is? -210 -00:13:01,840 --> 00:13:07,549 -So if the annual interest rate was 12% and you're making steps each 1 12th of a month, +212 +00:13:01,840 --> 00:13:06,194 +So if the annual interest rate was 12 percent and you're making steps each one twelfth -211 -00:13:07,549 --> 00:13:10,700 -that means you increase by 1% each one of those. +213 +00:13:06,194 --> 00:13:10,700 +of a month, of a munch, of a month, that means you increase by 1 percent each one of them. -212 +214 00:13:10,700 --> 00:13:15,040 So what it looks like is a step function that's a lot more fine, okay? -213 +215 00:13:15,580 --> 00:13:18,762 And we can see how at the end of the one year, -214 +216 00:13:18,762 --> 00:13:23,300 the y coordinate of our graph is that $112 with the extra 68 cents. -215 +217 00:13:23,660 --> 00:13:25,540 Not quite a dollar more, but it is more. -216 +218 00:13:26,260 --> 00:13:30,751 And again, as we zoom out, you can see that it fits this nice exponential curve, -217 +219 00:13:30,751 --> 00:13:32,360 the power of compound growth. -218 -00:13:32,360 --> 00:13:36,925 +220 +00:13:32,360 --> 00:13:36,581 And if you're curious, you know, previously after 10 years, -219 -00:13:36,925 --> 00:13:41,490 -we would have had $310, I think it was, which we can verify +221 +00:13:36,581 --> 00:13:39,887 +we would have had 310 dollars, I think it was, -220 -00:13:41,490 --> 00:13:46,360 -for ourselves if I take that $100 times 1 plus 0.12 to the 10th. +222 +00:13:39,887 --> 00:13:44,601 +which we can verify for ourselves if I take that 100 dollars times + +223 +00:13:44,601 --> 00:13:46,360 +1 plus 0.12 to the tenth. -221 +224 00:13:46,360 --> 00:13:47,600 Alright, this says $310. -222 +225 00:13:48,340 --> 00:13:52,649 When we compound every month instead, and we wait for 120 months, -223 +226 00:13:52,649 --> 00:13:55,980 it looks like we've squeezed out an extra 20 bucks. -224 +227 00:13:56,200 --> 00:13:57,640 So, not bad. -225 +228 00:13:57,640 --> 00:14:01,223 The important part here though, is the idea that the frequency with -226 +229 00:14:01,223 --> 00:14:04,860 which you compound actually changes the amount, which is interesting. -227 +230 00:14:05,280 --> 00:14:10,103 And this is the original circle of thoughts that Bernoulli was thinking about -228 +231 00:14:10,103 --> 00:14:14,680 that leads to the origins of the constant e that we all now know and love. -229 +232 00:14:14,920 --> 00:14:17,440 Maybe the second most famous constant only next to pi. -230 +233 00:14:18,300 --> 00:14:21,423 So let's write out some of what we're doing here, but with some formulas, -231 +234 00:14:21,423 --> 00:14:24,420 and then see what we can start learning with respect to those formulas. -232 +235 00:14:24,420 --> 00:14:27,807 And this is going to help us understand the bizarre question of an -233 +236 00:14:27,807 --> 00:14:32,306 interest rate of negative, not even negative, I keep saying interest rate of negative 1, -234 +237 00:14:32,306 --> 00:14:34,480 interest rate of square root of negative 1. -235 +238 00:14:35,600 --> 00:14:41,880 Nobody should let the Fed watch this video, I think it might introduce some shaky ideas. -236 +239 00:14:43,840 --> 00:14:46,961 Alright, so the way we might think about this is let's say you start -237 +240 00:14:46,961 --> 00:14:50,037 off with some amount of money, okay, and then we're going to let it -238 +241 00:14:50,037 --> 00:14:53,520 accrue some interest over some long amount of time, that I'm going to call t. -239 +242 00:14:53,520 --> 00:14:58,903 And the idea is that we're chopping this amount of time into a bunch of little pieces, -240 +243 00:14:58,903 --> 00:15:03,420 okay, and with each one of these steps we're going to increase our money. -241 +244 00:15:03,440 --> 00:15:06,800 There's going to be some kind of change to m, a delta m. -242 +245 00:15:07,480 --> 00:15:10,680 And each of these steps, let's say it's got a length of delta t. -243 +246 00:15:11,460 --> 00:15:16,148 So maybe delta t is a 12th of a year, or maybe it's 1 365th of a year, -244 +247 00:15:16,148 --> 00:15:18,460 it depends on how finely we cut it. -245 +248 00:15:18,460 --> 00:15:21,212 And just as another variable to throw down here, -246 +249 00:15:21,212 --> 00:15:24,582 let's keep track of how many steps there are along the way, -247 +250 00:15:24,582 --> 00:15:26,380 if we were to count them all up. -248 +251 00:15:26,720 --> 00:15:30,132 And the total number would be the total amount of time -249 +252 00:15:30,132 --> 00:15:33,980 divided by the size of the time steps that we're taking, okay? -250 +253 00:15:34,800 --> 00:15:39,057 So what interest rates do is they say that the amount your money -251 +254 00:15:39,057 --> 00:15:42,267 changes is going to equal that interest rate, r, -252 +255 00:15:42,267 --> 00:15:47,180 and you might think of r as being something like the 0.12 from our example. -253 +256 00:15:47,180 --> 00:15:50,320 And then later on we're going to play around with making it imaginary. -254 +257 00:15:51,960 --> 00:15:55,911 But you're going to multiply it by the size of your time step, okay, -255 +258 00:15:55,911 --> 00:16:00,780 which you might think of as being something like 1.12 if we were compounding monthly. -256 +259 00:16:01,180 --> 00:16:06,003 Because simply by compounding monthly it doesn't mean you get to grow by 12% every month, -257 +260 00:16:06,003 --> 00:16:09,540 it means you grow by 1.12 of what the annual rate was every month. -258 +261 00:16:10,320 --> 00:16:12,946 And then what makes compound growth so powerful, -259 +262 00:16:12,946 --> 00:16:15,840 it's also proportional to the amount of money you had. -260 +263 00:16:15,840 --> 00:16:18,483 You know, if you had $100 it grows by a certain amount, -261 +264 00:16:18,483 --> 00:16:21,080 if you had $1000 in there it grows by 10 times as much. -262 +265 00:16:21,300 --> 00:16:23,000 The more you have, the more it grows. -263 +266 00:16:23,660 --> 00:16:27,520 Okay, so that in our example you might think of as being something like $100. -264 +267 00:16:28,760 --> 00:16:30,140 Alright, so what's the magic of this? -265 +268 00:16:30,640 --> 00:16:35,410 It looks innocuous at first, but it means that our -266 +269 00:16:35,410 --> 00:16:40,180 money changes by going to m plus r delta t times m. -267 -00:16:40,699 --> 00:16:46,631 +270 +00:16:40,700 --> 00:16:46,631 And because m shows up in both of these terms, it lets us factor things out, -268 +271 00:16:46,631 --> 00:16:50,560 so that m factors out and we have 1 plus r delta t. -269 +272 00:16:51,780 --> 00:16:55,391 And because this is the same constant that we're going to multiply for each -270 +273 00:16:55,391 --> 00:16:59,240 one of these little time steps, all you're doing is multiplying by this constant. -271 +274 00:16:59,980 --> 00:17:04,160 We can write an expression for how much money you're going to have after time t. -272 +275 00:17:04,760 --> 00:17:09,774 What it's going to look like is the amount of money you had to start off -273 +276 00:17:09,774 --> 00:17:14,720 with at time t equals 0 multiplied by this expression, 1 plus r delta t. -274 +277 00:17:15,880 --> 00:17:18,740 And I'll change this in a moment, but I'll keep writing it as r delta t. -275 +278 00:17:19,200 --> 00:17:24,560 But we multiply it by itself n times, where n is the number of steps that we have in here. -276 +279 00:17:25,220 --> 00:17:27,573 Okay, but there's a lot of different variables at play here, -277 +280 00:17:27,573 --> 00:17:29,040 so I want to start consolidating them. -278 +281 00:17:29,040 --> 00:17:33,298 And instead of writing delta t, I'm going to go ahead -279 +282 00:17:33,298 --> 00:17:37,320 and express delta t as the total time divided by n. -280 +283 00:17:37,860 --> 00:17:41,451 We look at the number of chunks we've cut our period into, -281 +284 00:17:41,451 --> 00:17:44,860 and we take the total amount of time and divide by that. -282 +285 00:17:45,780 --> 00:17:50,971 So what our expression will look like is the amount of money we -283 +286 00:17:50,971 --> 00:17:56,000 started with times 1 plus r times the total time divided by n. -284 +287 00:17:56,000 --> 00:17:59,420 So this t divided by n is playing the role of delta t. -285 +288 00:17:59,920 --> 00:18:02,140 And we raise that to the nth power. -286 +289 00:18:02,920 --> 00:18:07,257 Now this is actually a very interesting expression, no pun intended, -287 +290 00:18:07,257 --> 00:18:11,280 because we want to know how does the size of n influence things. -288 +291 00:18:11,560 --> 00:18:14,940 If I crank up n a lot, meaning we're chopping our interval into very fine -289 +292 00:18:14,940 --> 00:18:18,640 increments and compounding very frequently, what does that mean for our interest? -290 -00:18:19,939 --> 00:18:23,152 +293 +00:18:19,940 --> 00:18:23,152 And you might, if you were a pure mathematician, -291 +294 00:18:23,152 --> 00:18:28,528 try to abstract away all the ideas of the money here, the specific interest rate, -292 +295 00:18:28,528 --> 00:18:33,511 and just say what we're interested in is this function of x that looks like -293 +296 00:18:33,511 --> 00:18:34,560 1 plus 1 over n. -294 +297 00:18:34,960 --> 00:18:37,260 Or maybe I'll say 1 plus x over n. -295 +298 00:18:37,320 --> 00:18:40,420 So we want to be able to plug in some value for r t. -296 +299 00:18:41,000 --> 00:18:42,960 And we'll raise that to the power n. -297 +300 00:18:43,620 --> 00:18:47,040 But you're not just curious about this for any particular n. -298 +301 00:18:47,480 --> 00:18:51,280 What you want to know is what happens as you crank up the value of n. -299 +302 00:18:51,740 --> 00:18:56,333 And the expression that mathematicians write for this, we write lim for limit, -300 +303 00:18:56,333 --> 00:18:59,066 and then we write n with an arrow to infinity, -301 +304 00:18:59,066 --> 00:19:03,660 saying I want to crank up that n and see what happens to this particular value. -302 +305 00:19:03,660 --> 00:19:07,197 Now it's not obvious, I think, what happens to it, -303 +306 00:19:07,197 --> 00:19:12,469 and playing with my toy again, I just want to do another half-opinion poll, -304 +307 00:19:12,469 --> 00:19:16,840 half-real question about this one to see what your instinct is. -305 +308 00:19:16,840 --> 00:19:24,359 So let's say, pull up the question here, what happens to the value 100 -306 +309 00:19:24,359 --> 00:19:32,620 times 1 plus 0.12 over n to the power n as the value of n approaches infinity? -307 +310 00:19:34,020 --> 00:19:38,020 And in the back of your mind, you can think this isn't just a purely mathematical -308 +311 00:19:38,020 --> 00:19:41,680 expression, it has a very real meaning in the context of compound interest. -309 +312 00:19:42,280 --> 00:19:46,237 What we're basically saying is if you have that 12% interest rate, -310 +313 00:19:46,237 --> 00:19:51,081 and you want to wait one year, but you chop up that year into n different pieces, -311 +314 00:19:51,081 --> 00:19:56,103 maybe n is 12 if you compound every month, maybe n is 365 if you compound every day, -312 +315 00:19:56,103 --> 00:20:00,120 or maybe you want to compound every microsecond or every picosecond. -313 +316 00:20:00,120 --> 00:20:02,461 The question is, how much money can you get at the -314 +317 00:20:02,461 --> 00:20:04,940 end of the year simply by compounding more frequently? -315 +318 00:20:05,580 --> 00:20:09,960 If you'll remember, when we went from yearly to monthly, that gave us an extra 68 cents. -316 +319 00:20:11,200 --> 00:20:14,080 So the question asks, basically, how high can this go? -317 +320 00:20:14,620 --> 00:20:17,980 And now we have a very wide split, which is wonderful. -318 +321 00:20:19,420 --> 00:20:23,240 I'm going to give you guys a couple more moments to think about this. -319 +322 00:20:24,000 --> 00:20:26,691 And I don't want you to calculate it necessarily, -320 +323 00:20:26,691 --> 00:20:30,352 the spirit of this question is to see what your initial thought is, -321 +324 00:20:30,352 --> 00:20:32,560 kind of intuitively based on the numbers. -322 +325 00:20:32,920 --> 00:20:35,532 So even though technically there's a right answer to this one, -323 +326 00:20:35,532 --> 00:20:37,980 so I'll highlight it green, there's no right or wrong here. -324 +327 00:20:38,080 --> 00:20:40,680 The spirit of it is for me to know where people are. -325 +328 00:20:41,140 --> 00:20:43,640 I'll give you just a little bit of time on this. -326 +329 00:20:54,760 --> 00:20:57,240 And while I do, let's take a question from the audience. -327 +330 00:20:58,240 --> 00:21:02,919 Considering that imaginary unit is defined in terms of operations on the real numbers, -328 +331 00:21:02,919 --> 00:21:07,760 is there an analogous definition of quaternions in terms of operations on complex numbers? -329 +332 00:21:08,440 --> 00:21:10,500 Okay, so a bit of an advanced question. -330 +333 00:21:10,620 --> 00:21:15,660 I don't expect most people watching the series to necessarily know what quaternions are. -331 +334 00:21:16,000 --> 00:21:19,054 But it's a number system that instead of including two dimensions, -332 +335 00:21:19,054 --> 00:21:21,380 like the complex numbers, involves four dimensions. -333 +336 00:21:21,380 --> 00:21:27,580 The short answer to the question, I think this is okay to say, the short answer is no. -334 +337 00:21:28,120 --> 00:21:30,404 When we get the complex numbers it's because we -335 +338 00:21:30,404 --> 00:21:32,880 have an expression that we don't have a solution to. -336 +339 00:21:33,020 --> 00:21:34,600 You want to know the square root of negative one. -337 +340 00:21:34,920 --> 00:21:37,060 A solution to x squared equals negative one. -338 +341 00:21:37,560 --> 00:21:40,420 And by introducing a solution, you extend your number system. -339 +342 00:21:41,000 --> 00:21:42,440 Quaternions don't come about like that. -340 +343 00:21:43,000 --> 00:21:45,626 For any polynomial, once you have the complex numbers, -341 +344 00:21:45,626 --> 00:21:48,540 you get all the solutions that you could ever want from that. -342 +345 00:21:48,540 --> 00:21:51,356 So it's not going to be there's a problem that you wanted solved -343 +346 00:21:51,356 --> 00:21:54,000 in the sense of a solution to a polynomial and you extend it. -344 +347 00:21:54,480 --> 00:21:57,700 But where it does come from is the idea, at least historically, -345 +348 00:21:57,700 --> 00:22:00,820 is the idea of wanting to describe motion in three dimensions. -346 +349 00:22:01,740 --> 00:22:05,621 And complex numbers can describe things like rotation in two dimensions, -347 +350 00:22:05,621 --> 00:22:10,300 but they don't have enough degrees of freedom to describe rotations in three dimensions. -348 +351 00:22:10,660 --> 00:22:13,340 So in that sense it was a concrete problem they couldn't solve. -349 -00:22:14,179 --> 00:22:16,792 +352 +00:22:14,180 --> 00:22:16,792 But the construct that you do to add on top of -350 +353 00:22:16,792 --> 00:22:19,460 it doesn't look like a solution to a polynomial. -351 +354 00:22:20,120 --> 00:22:23,783 Kind of a more advanced question than the spirit of the lessons necessarily, -352 +355 00:22:23,783 --> 00:22:27,780 but obviously I love to just engage with wherever you are, whoever is watching this. -353 +356 00:22:28,500 --> 00:22:29,380 So back to our question. -354 +357 00:22:30,020 --> 00:22:31,080 I'll go ahead and grade it. -355 +358 00:22:31,220 --> 00:22:34,780 And again, technically there's a right answer, but don't view this as right or wrong. -356 +359 00:22:34,780 --> 00:22:42,187 It looks like 1670 of you correctly noted that this value would rise above 112, -357 +360 00:22:42,187 --> 00:22:44,040 but never above 113. -358 +361 00:22:44,600 --> 00:22:48,646 So maybe taking the instincts from the fact that when we went from yearly to -359 +362 00:22:48,646 --> 00:22:52,640 monthly we only got 68 cents and saying you can't get much better than that. -360 +363 00:22:53,760 --> 00:22:59,026 The next biggest answer, interesting, was people who said it would rise above 114, -361 +364 00:22:59,026 --> 00:23:01,120 but stay below some finite bound. -362 +365 00:23:01,120 --> 00:23:03,840 Okay, so having that instinct that it would stay finite, -363 +366 00:23:03,840 --> 00:23:06,800 but that it was going to get bigger than anything listed here. -364 +367 00:23:06,900 --> 00:23:13,240 And I think the most thought-provoking answer here is E. -365 +368 00:23:13,820 --> 00:23:15,580 The thought that maybe it should blow up to infinity. -366 +369 00:23:15,960 --> 00:23:18,260 I mean, we're letting n get as big as we want. -367 +370 00:23:18,960 --> 00:23:23,220 It doesn't just have to be down to the picosecond, it could be down to a plank length. -368 +371 00:23:23,320 --> 00:23:26,471 It could be down to the tiniest, the biggest number that you could think of, -369 +372 00:23:26,471 --> 00:23:28,600 so that our time steps are as tiny as you can think. -370 +373 00:23:29,120 --> 00:23:31,660 It's not at all obvious that this thing would remain bounded. -371 +374 00:23:31,980 --> 00:23:34,740 So we can play around with this a little bit if we want to. -372 +375 00:23:35,020 --> 00:23:37,027 This won't be a proof that it remains bounded, -373 +376 00:23:37,027 --> 00:23:40,060 but maybe it can give us a little bit of an instinct in that direction. -374 +377 00:23:40,440 --> 00:23:46,042 If I take 1 plus 0.12 over n, and I raise this to the power n, -375 +378 00:23:46,042 --> 00:23:49,600 and I say what happens as we increase n? -376 +379 00:23:49,760 --> 00:23:50,960 And 100, that's not big enough. -377 +380 00:23:51,240 --> 00:23:55,500 I want to let myself increase it to some huge number, you know, 100 million. -378 +381 00:23:55,500 --> 00:23:59,300 So the value I want you to watch is sitting right here, okay? -379 +382 00:24:00,000 --> 00:24:03,218 And remember that represents how much money you'll have -380 +383 00:24:03,218 --> 00:24:06,380 if you compound your 12% interest much more frequently. -381 +384 00:24:07,260 --> 00:24:13,840 Even as I'm cranking it up around 100 million, this value really stays stable around 1.12. -382 +385 00:24:14,440 --> 00:24:17,100 Here, let's multiply it by the $100 to make it more intuitive. -383 +386 00:24:17,100 --> 00:24:21,532 Around the $112.74, about 75 cents, it stays really -384 +387 00:24:21,532 --> 00:24:25,880 stable around that even as I'm increasing by a lot. -385 +388 00:24:26,480 --> 00:24:27,200 That's not a proof. -386 +389 00:24:27,820 --> 00:24:31,688 For all we know, as you venture into the realm of huge, huge numbers, -387 +390 00:24:31,688 --> 00:24:35,501 10 to the 300th, or grams constant, or whatever big number you want, -388 +391 00:24:35,501 --> 00:24:39,260 maybe this thing would slowly crawl up and eventually get unbounded. -389 +392 00:24:40,360 --> 00:24:42,360 But it's an interesting fact that it doesn't. -390 +393 00:24:42,360 --> 00:24:47,380 Now, in terms of the math behind this, we can do an interesting manipulation. -391 +394 00:24:48,140 --> 00:24:51,360 Because we have this expression that is a little funky to think about, right? -392 +395 00:24:51,360 --> 00:24:55,982 We've got 1 plus x over n that you might think of as a partial interest rate, -393 +396 00:24:55,982 --> 00:24:57,760 and we're raising it to the n. -394 +397 00:24:58,100 --> 00:25:06,400 I can do a little substitution and say, what if I define the value m to be n divided by x? -395 -00:25:06,979 --> 00:25:12,520 +398 +00:25:06,980 --> 00:25:12,520 And the reason for doing that would be so that I can write that inside without an x in it. -396 +399 00:25:13,040 --> 00:25:15,780 So x divided by n will be the same as 1 divided by m. -397 +400 00:25:17,100 --> 00:25:20,235 And now I'll be raising that to the power, well not n, -398 +401 00:25:20,235 --> 00:25:23,600 well no, the power n, but I want to write it in terms of m. -399 +402 00:25:24,140 --> 00:25:26,700 So it'll look like m times x. -400 +403 00:25:27,700 --> 00:25:31,555 And based on how exponents work, I can think of that just as -401 +404 00:25:31,555 --> 00:25:35,600 the quantity 1 plus 1 over m to the power n all raised to the x. -402 +405 00:25:35,600 --> 00:25:40,880 And I'm curious about what happens here as I crank the value of m towards infinity. -403 +406 00:25:42,160 --> 00:25:44,588 And Bernoulli was also interested in what happens -404 +407 00:25:44,588 --> 00:25:46,920 as we crank this value of m up towards infinity. -405 +408 00:25:47,620 --> 00:25:51,973 And empirically, what you'll find is if you just play around with this, -406 +409 00:25:51,973 --> 00:25:56,991 and if we go over and take out the evidence that we were dealing with a real world -407 +410 00:25:56,991 --> 00:26:01,767 problem, with things like that $100 starting value, or that 12% interest rate, -408 +411 00:26:01,767 --> 00:26:05,273 and we just have clean constants like 1 for each of them, -409 +412 00:26:05,273 --> 00:26:08,720 it looks like this value ends up to being around 2.71828. -410 +413 00:26:09,480 --> 00:26:14,120 And no matter how big you make n, it seems to never really get much different than that. -411 +414 00:26:14,560 --> 00:26:17,310 Again, not a proof, but very suggestive that this -412 +415 00:26:17,310 --> 00:26:19,620 is maybe a fundamental constant of nature. -413 +416 00:26:20,300 --> 00:26:23,187 And so this is the original definition of e, this is -414 +417 00:26:23,187 --> 00:26:26,020 the original way that people were thinking about it. -415 +418 00:26:26,060 --> 00:26:30,961 As we pop over here, and we say instead of writing this whole limit expression, -416 +419 00:26:30,961 --> 00:26:36,168 I'll use a different color, say in principle, this is going to approach some kind of -417 +420 00:26:36,168 --> 00:26:36,720 constant. -418 +421 00:26:37,700 --> 00:26:44,720 Let's call this constant e that empirically has a value around 2.71828. -419 +422 00:26:45,900 --> 00:26:49,160 And then we can write our whole thing as e to the x. -420 +423 00:26:50,020 --> 00:26:54,965 So for example, if we wanted to know what happens in our interest example, -421 +424 00:26:54,965 --> 00:26:59,053 when we started out with $0, not $0, started out with m of 0, -422 +425 00:26:59,053 --> 00:27:04,394 which might be something like $100, what we do is we just focus on that rt term, -423 +426 00:27:04,394 --> 00:27:10,065 we say we know that as we crank up n, this is going to approach some special constant -424 +427 00:27:10,065 --> 00:27:11,780 raised to the power of rt. -425 +428 00:27:12,400 --> 00:27:16,544 And maybe this we would aptly describe as continuously compounded growth, -426 +429 00:27:16,544 --> 00:27:19,400 if we take that time step and we let it approach 0. -427 +430 00:27:19,940 --> 00:27:23,960 And this was actually before Newton and Leibniz, this was before calculus existed. -428 +431 00:27:23,960 --> 00:27:27,825 There was this idea of trying to take these discrete steps of increasing -429 +432 00:27:27,825 --> 00:27:31,480 and make it approach 0 to get something a little bit more continuous. -430 +433 00:27:31,560 --> 00:27:33,920 So I think that's quite cool. -431 -00:27:35,459 --> 00:27:41,862 +434 +00:27:35,460 --> 00:27:41,862 And one thing that you might notice here is we've got this large expression -432 +435 00:27:41,862 --> 00:27:47,760 for basically a function of x that we now tend to write as e to the x. -433 +436 00:27:47,760 --> 00:27:51,375 If you watched the last lecture, you would have remembered that I said -434 +437 00:27:51,375 --> 00:27:55,041 the way things have come about in math now, whenever we see e to the x, -435 +438 00:27:55,041 --> 00:27:58,300 we actually think of it as a shorthand for a certain polynomial. -436 +439 00:27:59,040 --> 00:28:03,707 And the polynomial, I was calling it exp, which follows convention, -437 +440 00:28:03,707 --> 00:28:08,031 looks like 1 plus x plus x squared over 2 plus x cubed over 6, -438 +441 00:28:08,031 --> 00:28:14,140 and all of the terms look like x to the k divided by k factorial for some whole number k. -439 +442 00:28:14,140 --> 00:28:18,553 And I said this is what e to the x is a shorthand for, this is how you think about it, -440 +443 00:28:18,553 --> 00:28:22,460 especially when there's weird inputs, things like an imaginary interest rate. -441 +444 00:28:23,280 --> 00:28:25,833 And you might wonder, what on earth does this -442 +445 00:28:25,833 --> 00:28:28,720 polynomial have to do with this limiting expression? -443 +446 00:28:29,600 --> 00:28:34,028 And as a super extra credit challenge question for those of you who are really -444 +447 00:28:34,028 --> 00:28:37,784 into algebra and who are very comfortable with binomial expansion, -445 +448 00:28:37,784 --> 00:28:42,157 if you know what those terms mean, if you take this expression for some large -446 +449 00:28:42,157 --> 00:28:46,585 value of n and you expand it, you just expand it and you kind of simplify your -447 +450 00:28:46,585 --> 00:28:51,182 terms and then you make approximations based on what's true when n is very large, -448 +451 00:28:51,182 --> 00:28:55,723 what you would find is that the polynomial this expands to looks a lot like this -449 +452 00:28:55,723 --> 00:28:56,620 polynomial here. -450 +453 00:28:57,540 --> 00:29:02,612 And in fact, as this value of n gets bigger and bigger and approaches infinity, -451 +454 00:29:02,612 --> 00:29:06,100 this polynomial will get closer and closer to this one. -452 +455 00:29:06,100 --> 00:29:10,103 It's a challenging question, so I wouldn't expect everyone to necessarily be -453 +456 00:29:10,103 --> 00:29:14,160 able to just bang it out, but if you want to it's a very elucidating exercise. -454 -00:29:14,500 --> 00:29:18,842 -And there's also a lot of delicacy in rigorously proving the fact that what this +457 +00:29:14,500 --> 00:29:18,127 +And there's also a lot of delicacy in rigorously proving the fact that -455 -00:29:18,842 --> 00:29:23,614 -approaches will also approach this, or that both of them even stay finite and they don't +458 +00:29:18,127 --> 00:29:22,469 +what this approaches will also approach this, or that both of them even stay finite, -456 -00:29:23,614 --> 00:29:28,224 -blow up when you add more and more terms to the sum or when you crank up n higher and +459 +00:29:22,469 --> 00:29:26,505 +and they don't blow up when you, you know, add more and more terms to the sum, -457 -00:29:28,224 --> 00:29:28,600 -higher. +460 +00:29:26,505 --> 00:29:28,600 +or when you crank up n higher and higher. -458 +461 00:29:28,620 --> 00:29:30,891 There's a lot of delicacy there, but you can probably -459 +462 00:29:30,891 --> 00:29:33,120 get to a point where there's an intuitive connection. -460 +463 00:29:33,120 --> 00:29:37,951 And the reason that in math we tend to work with this polynomial instead of that limit, -461 +464 00:29:37,951 --> 00:29:41,300 it's basically easier for computations and easier for theory. -462 +465 00:29:41,820 --> 00:29:46,864 But the value of this other expression is that it ties us back to the idea of compound -463 +466 00:29:46,864 --> 00:29:51,850 interest and the idea of taking a quantity that changes a little bit based on its own -464 +467 00:29:51,850 --> 00:29:52,140 size. -465 +468 00:29:52,280 --> 00:29:56,175 You can kind of readily read this as saying we're going to multiply -466 +469 00:29:56,175 --> 00:29:59,900 by a certain constant that's a little above one many, many times. -467 +470 00:29:59,900 --> 00:30:04,081 And if you'll remember, the reason we're doing that is when the way that -468 +471 00:30:04,081 --> 00:30:08,262 you change is proportional to yourself, that lets you factor things such -469 +472 00:30:08,262 --> 00:30:12,100 that you're just multiplying by something a little bigger than one. -470 +473 00:30:13,160 --> 00:30:18,250 And when you do this approaching continuity, when you do it in a way where that time step -471 +474 00:30:18,250 --> 00:30:23,340 is getting smaller and smaller, this is when we start writing e to the power of something. -472 +475 00:30:23,340 --> 00:30:27,160 What it's suggesting is that you have this compound interest expression, -473 +476 00:30:27,160 --> 00:30:29,620 but we're letting that n tend towards infinity. -474 +477 00:30:29,800 --> 00:30:31,600 We're letting the time step go as small as we want. -475 -00:30:32,939 --> 00:30:36,280 +478 +00:30:32,940 --> 00:30:36,280 So, with all of that said, we can start having fun. -476 +479 00:30:36,580 --> 00:30:40,740 We can start thinking about our original question of an imaginary interest rate. -477 +480 00:30:42,160 --> 00:30:43,440 So how on earth is this going to work? -478 +481 00:30:44,500 --> 00:30:49,100 First of all, let's just say from the get-go that if we try plugging in r to -479 +482 00:30:49,100 --> 00:30:53,820 this expression where we're raising a constant e to a power, it makes no sense. -480 +483 00:30:54,260 --> 00:30:57,740 If r is equal to the square root of negative one. -481 +484 00:30:58,180 --> 00:31:02,220 You cannot multiply a constant by itself the square root of negative one times. -482 +485 00:31:02,400 --> 00:31:03,640 So that doesn't make sense. -483 +486 00:31:03,640 --> 00:31:07,281 But what does is if we go to this original expression, -484 +487 00:31:07,281 --> 00:31:12,180 the origins of e, and try to imagine plugging in something like i to that. -485 +488 00:31:12,380 --> 00:31:15,420 We know how to divide i by n, that's fine. -486 +489 00:31:15,720 --> 00:31:17,880 We know how to add it to one, that's fine. -487 +490 00:31:18,260 --> 00:31:20,680 We know how to take a complex number and multiply it by itself. -488 +491 00:31:20,820 --> 00:31:22,680 All of the operations there are quite fine. -489 +492 00:31:23,300 --> 00:31:25,660 Now let's draw it out to think about what it would look like. -490 -00:31:26,659 --> 00:31:30,200 +493 +00:31:26,660 --> 00:31:30,200 And the other thing I want to emphasize is I know this seems like utter nonsense. -491 +494 00:31:30,500 --> 00:31:34,640 We're talking about imaginary numbers in the context of interest rates. -492 +495 00:31:34,980 --> 00:31:39,780 But in a couple minutes, I really do hope to make this relevant to physics and -493 +496 00:31:39,780 --> 00:31:44,580 hope to make you see that this is not a totally nonsensical circle of thoughts. -494 -00:31:45,500 --> 00:31:51,099 -So, I've got my axes here, where this is going to be my real money, +497 +00:31:45,500 --> 00:31:49,695 +So, I've got my axes here. This is going to be my real, -495 -00:31:51,099 --> 00:31:54,640 -and this will be all of my imaginary money. +498 +00:31:49,695 --> 00:31:54,640 +this is my real money. And this will be all of my imaginary money. -496 +499 00:31:55,600 --> 00:32:02,897 What it means if your interest rate is i, is that the change to the money looks like -497 +500 00:32:02,897 --> 00:32:10,280 i times whatever the time step is, delta t, times whatever the money is to begin with. -498 +501 00:32:11,120 --> 00:32:15,197 Now, in, I believe it was lecture 3, we talked all about complex numbers, -499 +502 00:32:15,197 --> 00:32:19,385 and one of the fundamental facts was that when you multiply i by something, -500 +503 00:32:19,385 --> 00:32:21,700 it has the effect of a 90 degree rotation. -501 +504 00:32:22,340 --> 00:32:25,100 And we talked about this in terms of looking at the coordinates and -502 +505 00:32:25,100 --> 00:32:27,820 realizing that you just swap the coordinates and make 1 negative 1. -503 +506 00:32:28,200 --> 00:32:31,288 So this connection between the square root of 1 property and the -504 +507 00:32:31,288 --> 00:32:34,520 very mechanistic idea of taking a vector and rotating it 90 degrees. -505 +508 00:32:35,040 --> 00:32:38,757 Now what this means for us is that the change to your money is going to -506 +509 00:32:38,757 --> 00:32:42,112 be a little arrow that, you know, it's not increasing the money, -507 +510 00:32:42,112 --> 00:32:46,655 you're not growing, you're not decreasing the money, it's not a negative interest rate, -508 +511 00:32:46,655 --> 00:32:49,960 it's moving somehow perpendicular to where the money already is. -509 +512 00:32:50,240 --> 00:32:55,540 So after a little step in time, you end up with a little bit of imaginary money. -510 +513 00:32:55,900 --> 00:32:58,120 So you have what you had originally and you can play Monopoly. -511 +514 00:32:59,200 --> 00:33:03,460 And now from that point, when you do another time step, it does the same thing. -512 +515 00:33:03,460 --> 00:33:08,487 It takes the vector that you've newly landed on and it rotates it 90 degrees, -513 +516 00:33:08,487 --> 00:33:11,840 okay, and then you take a little step based on that. -514 +517 00:33:12,380 --> 00:33:15,600 And then again, you take a little step based on that. -515 +518 00:33:15,760 --> 00:33:20,507 Where at each point you're looking at what is your money number that has some real part -516 +519 00:33:20,507 --> 00:33:25,040 and some imaginary part, you rotate it 90 degrees and you add a little step on that. -517 +520 00:33:25,640 --> 00:33:27,720 And now the question is what happens to this? -518 +521 00:33:28,060 --> 00:33:29,400 This is the original question. -519 +522 00:33:29,500 --> 00:33:33,720 The bank offers you this interest rate, what's it going to do? -520 +523 00:33:35,540 --> 00:33:38,959 The correct answer, I think, is that you need to ask the bank for more -521 +524 00:33:38,959 --> 00:33:42,860 information because how frequently you compound it is going to make a difference. -522 +525 00:33:43,540 --> 00:33:46,914 So to illustrate this, let's say you compounded it annually, -523 +526 00:33:46,914 --> 00:33:51,505 meaning at the end of every year you take a big step that's based on this interest -524 +527 00:33:51,505 --> 00:33:52,280 rate multiple. -525 +528 00:33:52,280 --> 00:33:56,708 So for every dollar that you have, if you started out just putting -526 +529 00:33:56,708 --> 00:34:02,591 your real money into the bank, at the end of the year you would add i times that amount, -527 +530 00:34:02,591 --> 00:34:05,500 which is a 90 degree rotation of your money. -528 +531 00:34:05,760 --> 00:34:08,940 Again, I know this is utter nonsense, but follow along with me. -529 +532 00:34:09,420 --> 00:34:11,719 One, it's fun, and two, it leads to real physics. -530 +533 00:34:12,860 --> 00:34:13,760 So where does that get you? -531 +534 00:34:14,280 --> 00:34:16,880 Well, it gets you $1 plus i dollars. -532 -00:34:17,159 --> 00:34:19,689 -So if you had $100 in the bank, you end up with +535 +00:34:17,159 --> 00:34:19,525 +So if you had a hundred dollars in the bank, you end up -533 -00:34:19,689 --> 00:34:22,060 -$100 real dollars and $100 imaginary dollars. +536 +00:34:19,525 --> 00:34:22,060 +with a hundred real dollars and a hundred imaginary dollars. -534 +537 00:34:22,540 --> 00:34:25,402 But then, after the next year, you take another step, -535 +538 00:34:25,402 --> 00:34:28,636 where it takes that new money vector, rotates it 90 degrees, -536 +539 00:34:28,636 --> 00:34:30,280 and adds that to where you are. -537 +540 00:34:30,739 --> 00:34:34,219 So after two years, you come back to your bank and you say, how's my money doing? -538 +541 00:34:34,620 --> 00:34:36,880 And they say, good news, sir, there's twice as much of it. -539 +542 00:34:37,040 --> 00:34:38,900 You say, that's fantastic, it's only been two years. -540 +543 00:34:39,159 --> 00:34:41,280 And they say, bad news, it's all imaginary. -541 +544 00:34:41,280 --> 00:34:46,060 So, very good for your Monopoly game, not great for life logistics. -542 +545 00:34:46,659 --> 00:34:47,280 But it gets worse. -543 +546 00:34:47,679 --> 00:34:52,630 As you wait more time, and you add another 90 degree rotation of where you currently are, -544 +547 00:34:52,630 --> 00:34:55,600 you're going to step into the negative real territory. -545 +548 00:34:56,219 --> 00:35:00,614 So at this point, now you have negative 200 real dollars, -546 +549 00:35:00,614 --> 00:35:04,100 but you still have that 200 imaginary dollars. -547 +550 00:35:04,100 --> 00:35:09,280 So, good for your Monopoly game, devastating now for your actual life. -548 +551 00:35:09,880 --> 00:35:10,540 But it gets worse. -549 +552 00:35:11,040 --> 00:35:15,177 As we take another step, four years later, after you've invested your hard-earned money, -550 +553 00:35:15,177 --> 00:35:18,105 you've not even invested, you weren't even putting it at risk, -551 +554 00:35:18,105 --> 00:35:19,500 it was just a savings account. -552 +555 00:35:19,820 --> 00:35:20,660 And where do you end up? -553 +556 00:35:21,260 --> 00:35:24,380 For every dollar you put in, you would now have negative four dollars. -554 +557 00:35:24,720 --> 00:35:27,960 So you come to the bank and they say, well, you owe -555 +558 00:35:27,960 --> 00:35:31,200 us $400 for the $100 that you put in four years ago. -556 -00:35:31,879 --> 00:35:35,400 +559 +00:35:31,880 --> 00:35:35,400 And just as you're about to get outraged at them, they say, sir, we encourage you to wait. -557 +560 00:35:35,840 --> 00:35:37,640 We really do think this is going to work out for you. -558 +561 00:35:37,700 --> 00:35:40,341 If you just let your money sit, do its work, and -559 +562 00:35:40,341 --> 00:35:42,660 in the long run this will work out for you. -560 +563 00:35:42,880 --> 00:35:47,051 And they're not entirely wrong, because if you keep playing this game of -561 +564 00:35:47,051 --> 00:35:51,565 rotating 90 degrees and then adding that vector, after a total of eight years, -562 +565 00:35:51,565 --> 00:35:55,280 where you're going to end up is at 16 times your original amount. -563 +566 00:35:55,980 --> 00:35:56,680 Not too bad. -564 +567 00:35:56,680 --> 00:36:00,472 So if you're willing to put up with a lot of stress going in, -565 +568 00:36:00,472 --> 00:36:03,775 changing your monopoly game, changing your real life, -566 +569 00:36:03,775 --> 00:36:09,220 and you were willing to just hold for eight total years, you would have 16 times as much. -567 +570 00:36:09,620 --> 00:36:11,520 So that's pretty good, I would say. -568 +571 00:36:13,200 --> 00:36:18,260 We might call this the venture capitalist approach, where you put in your money, -569 +572 00:36:18,260 --> 00:36:23,320 you can't see it for a long time, and most of the time it's completely imaginary. -570 +573 00:36:23,680 --> 00:36:26,320 But every now and then you get a giant multiple at the end. -571 -00:36:27,319 --> 00:36:29,733 +574 +00:36:27,320 --> 00:36:29,733 Now before you excitedly do this, though, let's say -572 +575 00:36:29,733 --> 00:36:32,100 that your bank doesn't compound it annually, right? -573 +576 00:36:32,280 --> 00:36:35,560 And they say, ah, actually we compound your interest continuously. -574 +577 00:36:36,000 --> 00:36:39,196 Meaning that our time steps are not delta t of one year, -575 +578 00:36:39,196 --> 00:36:41,720 but it's delta t getting smaller and smaller. -576 +579 00:36:42,460 --> 00:36:45,181 Well what that means is that each one of your steps that's -577 +580 00:36:45,181 --> 00:36:48,180 perpendicular to where you are is really just a tiny little step. -578 +581 00:36:48,580 --> 00:36:51,640 And I'm showing with the arrows here what happens if you wait eight total years. -579 +582 00:36:51,640 --> 00:36:54,685 So you take a tiny step that's perpendicular to where you are, -580 +583 00:36:54,685 --> 00:36:58,360 another tiny step that's perpendicular to where you are, and you keep going. -581 +584 00:36:59,320 --> 00:37:03,377 But as that time step gets smaller and smaller, smaller and smaller and smaller, -582 +585 00:37:03,377 --> 00:37:06,834 to the point where it's genuinely continuously compounding interest, -583 +586 00:37:06,834 --> 00:37:09,840 what it means is that you're simply walking around a circle. -584 +587 00:37:10,700 --> 00:37:14,325 So if you were writing this out as an expression, you know, -585 +588 00:37:14,325 --> 00:37:18,011 it might seem halfway reasonable to look at the fact that we -586 +589 00:37:18,011 --> 00:37:22,000 were using e to the x as a shorthand for this limiting value here. -587 +590 00:37:22,320 --> 00:37:25,980 And in the case of real numbers that makes total sense when we plug in a value for x. -588 +591 00:37:26,600 --> 00:37:30,091 But what it means here is with our imaginary money, -589 +592 00:37:30,091 --> 00:37:35,800 we're writing it as e to the i, that was our interest rate, times the amount of time. -590 +593 00:37:36,420 --> 00:37:39,078 Nonsense if we think about repeated multiplication, -591 +594 00:37:39,078 --> 00:37:43,066 but in the context of compounding interest, all it really means is that after -592 +595 00:37:43,066 --> 00:37:46,798 taking a bunch of steps that are perpendicular to your current position, -593 +596 00:37:46,798 --> 00:37:50,530 and we do that continuously, so those time steps are just infinitesimal, -594 +597 00:37:50,530 --> 00:37:52,320 it has you walking around a circle. -595 -00:37:53,259 --> 00:37:56,880 +598 +00:37:53,260 --> 00:37:56,880 So, the bank offers you this, do you take it? -596 +599 00:37:57,560 --> 00:37:58,360 Well let's think it through. -597 +600 00:37:58,860 --> 00:38:01,440 What happens when you put in $100 to your savings account? -598 -00:38:01,839 --> 00:38:05,127 +601 +00:38:01,840 --> 00:38:05,127 Well if you wait a total of pi halves years, which is the -599 +602 00:38:05,127 --> 00:38:09,152 distance along a quarter of a circle, a little over a year and a half, -600 +603 00:38:09,152 --> 00:38:12,100 you come back into your account and what do you see? -601 +604 00:38:13,020 --> 00:38:18,300 Well, for each dollar you put in, you have an imaginary dollar. -602 +605 00:38:18,800 --> 00:38:19,280 Not great. -603 +606 00:38:20,140 --> 00:38:23,572 But you push through, you know, the banker told you that this will work out in the end, -604 +607 00:38:23,572 --> 00:38:24,860 so you say, okay, I'm gonna wait. -605 +608 00:38:24,860 --> 00:38:28,497 I didn't need it for that year and a half, life wasn't going too terribly, -606 +609 00:38:28,497 --> 00:38:32,669 so I'm just gonna let my money still sit in that savings account for another year and -607 +610 00:38:32,669 --> 00:38:33,300 a half or so. -608 +611 00:38:33,400 --> 00:38:36,620 And you wait a total of pi years, around 3.14 years. -609 +612 00:38:36,720 --> 00:38:39,380 You come back, and now it really doesn't look good. -610 +613 00:38:39,700 --> 00:38:44,180 Now, you've got negative one dollar for each dollar that you originally put in. -611 +614 00:38:44,600 --> 00:38:48,050 And just like before, you're infuriated at the bank, not as much as you were, -612 +615 00:38:48,050 --> 00:38:51,279 because it hasn't gone out to be negative four, but it's still negative, -613 +616 00:38:51,279 --> 00:38:52,120 and that's not fun. -614 +617 00:38:52,120 --> 00:38:54,300 And the bank says, hold on, hold on, sir, we think -615 +618 00:38:54,300 --> 00:38:56,480 this is actually gonna work out for you in the end. -616 +619 00:38:57,000 --> 00:39:01,216 You say, okay, I've seen this work out for my friends who got 16 times their money, -617 +620 00:39:01,216 --> 00:39:02,120 so I'll just hold. -618 +621 00:39:02,980 --> 00:39:05,629 And you do, you just don't touch the money in your savings account, -619 +622 00:39:05,629 --> 00:39:07,460 you wait another year and a half and you check. -620 +623 00:39:07,820 --> 00:39:11,280 And now, not only is it entirely imaginary, it's negative. -621 +624 00:39:11,640 --> 00:39:14,560 So you can't live your real life, and you also can't even play Monopoly well. -622 +625 00:39:14,560 --> 00:39:18,660 So, 4.7 years is a real low during this whole experience. -623 +626 00:39:19,260 --> 00:39:23,926 But if you're willing to stick it out, and wait a total of 6.28 years, 2 pi years, -624 +627 00:39:23,926 --> 00:39:28,200 you come back, and for all of your stress, you're back to where you started. -625 +628 00:39:28,500 --> 00:39:31,280 You just have one dollar for every dollar that you put in originally. -626 +629 00:39:32,100 --> 00:39:34,560 So, should you accept this interest rate from your bank? -627 +630 00:39:35,020 --> 00:39:37,274 Depends on how much emotional turmoil you want, -628 +631 00:39:37,274 --> 00:39:41,360 but if it's continuously compounding, certainly not, I think is the appropriate answer. -629 -00:39:43,339 --> 00:39:45,180 +632 +00:39:43,340 --> 00:39:45,180 Now, is this at all useful? -630 +633 00:39:45,460 --> 00:39:49,024 This is kind of fun, I think, to take an idea that started off only -631 +634 00:39:49,024 --> 00:39:52,013 relevant to real numbers, you know, a 12% interest rate, -632 +635 00:39:52,013 --> 00:39:55,840 or if you have negative interest rates, you can think of how that decays. -633 +636 00:39:56,200 --> 00:39:58,400 And say, well, what if we did something else to it? -634 +637 00:39:58,440 --> 00:40:02,920 What if each time step wasn't in the direction, but it was perpendicular to the direction? -635 +638 00:40:03,920 --> 00:40:07,173 And the answer is that this is actually incredibly relevant to -636 +639 00:40:07,173 --> 00:40:10,840 anything that involves what's called simple harmonic motion in physics. -637 +640 00:40:11,340 --> 00:40:13,660 And so a great example of this is a spring. -638 +641 00:40:14,340 --> 00:40:17,100 Let's say we wanted to understand the motion of a spring. -639 +642 00:40:18,380 --> 00:40:19,800 So let me pull this up here. -640 +643 00:40:22,220 --> 00:40:26,540 I think I have an actual spring sitting somewhere here, if it didn't roll off my table. -641 +644 00:40:27,680 --> 00:40:33,200 I had trouble finding any reasonable ones, so I tried to just extract one from my pen. -642 -00:40:33,839 --> 00:40:37,080 +645 +00:40:33,840 --> 00:40:37,080 Maybe not the greatest physics demo in the world, but I've got one. -643 +646 00:40:37,640 --> 00:40:40,118 So the idea here is that if I pull the spring, -644 +647 00:40:40,118 --> 00:40:43,600 it's pushing me back in the other direction against how I pull it. -645 +648 00:40:43,980 --> 00:40:47,700 And similarly, if I push on it, then it's pushing me in the other direction. -646 +649 00:40:47,700 --> 00:40:50,460 It's trying to get back to that equilibrium point. -647 +650 00:40:50,860 --> 00:40:53,943 And the way that it does this obeys, at least to a loose -648 +651 00:40:53,943 --> 00:40:57,460 approximation for a lot of springs, something called Hooke's law. -649 +652 00:40:58,560 --> 00:41:02,258 So if I draw out a spring, and let's imagine that -650 +653 00:41:02,258 --> 00:41:05,440 there's some mass sitting at the end of it. -651 +654 00:41:05,800 --> 00:41:07,940 Let's say that this is where it wants to be. -652 +655 00:41:08,020 --> 00:41:12,299 This is the equilibrium position, kind of like how my spring is -653 +656 00:41:12,299 --> 00:41:17,180 sitting here before it just rolls away with no forces acting on the mass. -654 +657 00:41:17,180 --> 00:41:22,771 But if you stretch it out, if I pull it, that same mass at the end, -655 +658 00:41:22,771 --> 00:41:28,034 this distance, what we might call the displacement of our mass, -656 +659 00:41:28,034 --> 00:41:33,380 ends up influencing how much force the spring is pulling back on. -657 +660 00:41:33,780 --> 00:41:37,500 Because intuitively we know that the spring is pulling back with some kind of force. -658 +661 00:41:38,080 --> 00:41:43,040 And what Hooke's law suggests is that it's a force proportional to that size of x. -659 +662 00:41:43,040 --> 00:41:45,884 And we use k as a proportionality constant and negative -660 +663 00:41:45,884 --> 00:41:48,780 to emphasize that it's pointed in the opposite direction. -661 +664 00:41:49,340 --> 00:41:53,817 So in this context, what we're saying is if we double the amount that we're pulling it, -662 +665 00:41:53,817 --> 00:41:56,260 it's going to pull us back with twice the force. -663 +666 00:41:56,380 --> 00:41:58,760 If we triple it, it pulls us back with three times the force. -664 +667 00:41:59,740 --> 00:42:04,160 As with any kind of physical models, this is only true to an approximation. -665 +668 00:42:04,460 --> 00:42:07,517 You know, if we pull the spring such that we just distort it entirely, -666 +669 00:42:07,517 --> 00:42:08,680 this certainly won't apply. -667 -00:42:09,339 --> 00:42:13,959 +670 +00:42:09,340 --> 00:42:13,959 And often with anything that involves the simple harmonic motion we're about to describe, -668 +671 00:42:13,959 --> 00:42:16,320 it tends to only be true for small variations. -669 +672 00:42:16,500 --> 00:42:19,200 And as you get larger variations, the model would have to change. -670 +673 00:42:19,860 --> 00:42:22,360 But this is a pretty good one, and it's very general. -671 +674 00:42:22,400 --> 00:42:24,180 It comes up in a lot of different circumstances. -672 +675 00:42:25,100 --> 00:42:28,800 Now force is mass times acceleration. -673 +676 00:42:29,560 --> 00:42:30,920 This is what force is. -674 +677 00:42:30,980 --> 00:42:33,100 This is what Newton's second law tells us. -675 +678 00:42:33,120 --> 00:42:34,620 It says, what is your mass? -676 +679 00:42:34,900 --> 00:42:36,680 I'm telling you how much you're going to accelerate. -677 +680 00:42:36,680 --> 00:42:39,044 Acceleration is how much your velocity changes, -678 +681 00:42:39,044 --> 00:42:41,360 velocity being how much the value of x changes. -679 +682 00:42:42,320 --> 00:42:46,833 Now this is quite interesting, because what it means is that acceleration, -680 +683 00:42:46,833 --> 00:42:50,986 which is influencing the displacement, kind of in a second order way -681 +684 00:42:50,986 --> 00:42:55,260 via how it influences velocity, is itself influenced by the value of x. -682 +685 00:42:56,400 --> 00:42:58,600 So I'm going to go ahead and ask you a quiz question here. -683 +686 00:42:58,860 --> 00:43:02,160 It's basically to have us think about what velocity -684 +687 00:43:02,160 --> 00:43:05,460 and acceleration really mean in a context like this. -685 +688 00:43:05,460 --> 00:43:07,360 So we'll pull up our quiz again. -686 +689 00:43:07,840 --> 00:43:12,780 We are going to go to the final live question for this lesson. -687 +690 00:43:14,140 --> 00:43:16,980 Which is a big one, but it's largely just stating what I just did. -688 +691 00:43:17,700 --> 00:43:22,200 A mass on a spring is pulled a distance x away from an equilibrium point. -689 +692 00:43:22,940 --> 00:43:27,040 If the spring obeys Hooke's law, the mass will experience a force of -690 +693 00:43:27,040 --> 00:43:31,140 f equals negative kx, x being that displacement, for some constant k. -691 +694 00:43:31,140 --> 00:43:34,347 Keep in mind, by Newton's second law, f equals ma, -692 +695 00:43:34,347 --> 00:43:37,240 where m is the mass and a is the acceleration. -693 -00:43:38,500 --> 00:43:44,688 -If the mass starts out with a displacement of x naught and a velocity of v naught, +696 +00:43:38,500 --> 00:43:44,604 +Okay, if the mass starts out with a displacement of x naught and a velocity of v naught, -694 -00:43:44,688 --> 00:43:49,684 -which of the following most clearly describes the changes of these +697 +00:43:44,604 --> 00:43:48,513 +which of the following most clearly describes the value, -695 -00:43:49,684 --> 00:43:53,040 -values after a small change in time, delta t? +698 +00:43:48,513 --> 00:43:53,040 +the changes of these values after a small change in time, delta t? -696 +699 00:43:53,620 --> 00:43:57,376 Okay, so we're going to let time play out for a tiny step in time, -697 +700 00:43:57,376 --> 00:44:02,253 and it wants us to know which of these four options best describes what that change to -698 +701 00:44:02,253 --> 00:44:07,074 the x value is going to look like, delta x, and what the change to the velocity value -699 +702 00:44:07,074 --> 00:44:08,420 will look like, delta v. -700 +703 00:44:09,160 --> 00:44:10,780 So I'll give you a little bit of time for this one. -701 +704 00:44:42,160 --> 00:44:46,683 While you're thinking about that, we've got another question in about quaternions, -702 +705 00:44:46,683 --> 00:44:51,480 not entirely relevant to the lesson, but certainly a very interesting idea and question. -703 +706 00:44:51,480 --> 00:44:56,112 So if you're enjoying this kind of side plot to the full lecture, -704 +707 00:44:56,112 --> 00:44:59,200 rotation in 2D requires one extra number, i. -705 +708 00:44:59,760 --> 00:45:02,631 So why is it that it requires three extra numbers, -706 +709 00:45:02,631 --> 00:45:05,560 i, j, k, and not just two, i and j, for 3D rotation? -707 +710 00:45:05,780 --> 00:45:08,757 Is there an equivalent way for rotation in four dimensions, -708 +711 00:45:08,757 --> 00:45:11,040 and how many extra numbers are required there? -709 +712 00:45:11,040 --> 00:45:12,220 Okay, awesome question. -710 +713 00:45:12,960 --> 00:45:16,835 So a way to think about this, in complex numbers or in two dimensions, -711 +714 00:45:16,835 --> 00:45:20,328 you only need one degree of freedom to describe rotation, okay, -712 +715 00:45:20,328 --> 00:45:23,440 you just need to describe the angle that you're rotating. -713 +716 00:45:24,240 --> 00:45:26,938 And so you might say, oh, why do we need a 2D number system in -714 +717 00:45:26,938 --> 00:45:29,680 order to describe something that only has one degree of freedom? -715 +718 00:45:30,240 --> 00:45:33,240 And the idea is that it's nice to live in a slightly bigger space so -716 +719 00:45:33,240 --> 00:45:36,196 that we can have a circle, something where you come back on itself, -717 +720 00:45:36,196 --> 00:45:39,240 and then we just constrain all of the rotating actions to that circle. -718 +721 00:45:39,240 --> 00:45:41,716 So the way that complex numbers describe rotation is -719 +722 00:45:41,716 --> 00:45:44,100 entirely based on the numbers that sit on a circle. -720 +723 00:45:44,240 --> 00:45:47,620 It's only one dimension's worth of numbers in some sense. -721 +724 00:45:47,980 --> 00:45:48,200 Awesome. -722 +725 00:45:49,100 --> 00:45:53,284 So for three-dimensional rotation, it's not just two extra degrees of freedom you have, -723 +726 00:45:53,284 --> 00:45:56,660 you actually have three degrees of freedom to describe any 3D rotation. -724 +727 00:45:56,860 --> 00:46:00,540 You might think of it as saying, choose an axis for your rotation, -725 +728 00:46:00,540 --> 00:46:05,100 a latitude and longitude that you're going to poke a hole through the entire Earth. -726 +729 00:46:05,100 --> 00:46:07,553 And then after you choose that latitude and longitude, -727 +730 00:46:07,553 --> 00:46:10,987 two degrees of freedom for your axis, you have another degree of freedom for -728 +731 00:46:10,987 --> 00:46:11,880 how much you rotate. -729 +732 00:46:12,400 --> 00:46:16,480 So it's a total of three degrees of freedom to describe rotation in three dimensions. -730 +733 00:46:17,300 --> 00:46:21,297 Now again, you could say, in principle, we should be able to do this with a -731 +734 00:46:21,297 --> 00:46:25,927 three-dimensional space, where somehow you associate the three degrees of freedom of 3D -732 +735 00:46:25,927 --> 00:46:28,978 space with the three degrees of freedom of your rotation, -733 +736 00:46:28,978 --> 00:46:32,555 which you kind of can with a thing called stereographic projection, -734 +737 00:46:32,555 --> 00:46:36,080 which is entirely what the video I did about quaternions was about. -735 +738 00:46:36,640 --> 00:46:38,822 But, in the same way that to describe rotations, -736 +739 00:46:38,822 --> 00:46:41,940 it's nice to have a circle, something that lives one dimension higher. -737 +740 00:46:42,300 --> 00:46:45,600 To describe 3D rotations, it's nice to have a hypersphere, -738 +741 00:46:45,600 --> 00:46:50,580 basically something that loops back on itself, but it has three degrees of freedom on it. -739 +742 00:46:50,680 --> 00:46:54,620 Very weird for us to think about, but that's sort of the degrees of freedom argument. -740 +743 00:46:54,620 --> 00:46:58,580 And then four dimensions, rotations would actually have ten degrees of freedom. -741 +744 00:46:58,760 --> 00:47:01,281 There's not a number system for it, but if you're trying to -742 +745 00:47:01,281 --> 00:47:03,677 constrain your matrices and understand things like that, -743 +746 00:47:03,677 --> 00:47:06,620 which again, this is, if none of this makes sense to you, don't worry. -744 +747 00:47:07,100 --> 00:47:11,680 It's not in the spirit of this particular lecture or the target audience for it. -745 +748 00:47:11,680 --> 00:47:12,820 But it's an interesting topic. -746 +749 00:47:12,980 --> 00:47:17,160 And if it's what you're chatting about on Twitter, I'm happy to chime in on that. -747 +750 00:47:17,160 --> 00:47:21,586 So, yeah, thinking about degrees of freedom and the fact that it's nice to live one -748 +751 00:47:21,586 --> 00:47:26,223 dimension higher to have, basically have a more interesting surface than flat Euclidean -749 +752 00:47:26,223 --> 00:47:26,540 space. -750 +753 00:47:27,280 --> 00:47:32,518 Okay, with that as our changing scenes abruptly from the side plot of quaternions, -751 +754 00:47:32,518 --> 00:47:36,369 we return back to our regular programming of physics and how -752 +755 00:47:36,369 --> 00:47:39,020 it's relevant to imaginary interest rates. -753 +756 00:47:39,020 --> 00:47:45,060 Locking in our answers here, it looks like most of you correctly answered A. -754 +757 00:47:46,060 --> 00:47:47,160 And what is A saying here? -755 +758 00:47:47,460 --> 00:47:51,889 It's saying that the change to x is whatever your velocity was -756 +759 00:47:51,889 --> 00:47:56,600 times the small step in time, which that's what velocity is, right? -757 +760 00:47:56,640 --> 00:47:58,080 It's saying how many meters per second. -758 +761 00:47:58,400 --> 00:48:00,620 What is the change in distance per unit time? -759 +762 00:48:00,700 --> 00:48:02,240 So you multiply it by the change in time. -760 +763 00:48:02,240 --> 00:48:05,269 And then the only tricky part is knowing that the -761 +764 00:48:05,269 --> 00:48:08,480 velocity changes based on acceleration times delta t. -762 +765 00:48:09,220 --> 00:48:13,460 And acceleration in this case, we can work out based on the various equations. -763 +766 00:48:14,300 --> 00:48:16,300 Not sure if you can hear, but there's some sirens in the background. -764 +767 00:48:16,900 --> 00:48:20,660 In general, whenever I'm recording, it's so annoying when there's background noise. -765 +768 00:48:21,500 --> 00:48:25,880 So one of the nice things about live is that I just have no option. -766 +769 00:48:26,120 --> 00:48:27,840 So background noise needs to be tolerated. -767 +770 00:48:28,580 --> 00:48:32,901 Anyway, if we want to write a pure expression for the acceleration, -768 +771 00:48:32,901 --> 00:48:37,160 what it looks like is, just rearranging, negative k over m times x. -769 +772 00:48:38,480 --> 00:48:42,760 Now for where we're going with this, it's going to be a little bit simpler. -770 +773 00:48:43,440 --> 00:48:45,320 This is a classic mathematician thing to do. -771 +774 00:48:45,360 --> 00:48:46,720 Ignore the like real numbers. -772 +775 00:48:47,000 --> 00:48:50,671 And just assume that we're working with whatever units we need to, -773 +776 00:48:50,671 --> 00:48:52,480 such that k over m is equal to 1. -774 -00:48:53,779 --> 00:48:57,580 +777 +00:48:53,780 --> 00:48:57,580 And then what we can say is that the acceleration is negative x. -775 +778 00:48:58,220 --> 00:49:02,040 And the reason I want to do that will hopefully become apparent in a moment. -776 +779 00:49:02,800 --> 00:49:05,171 Now the way I want you to think about this spring -777 +780 00:49:05,171 --> 00:49:07,400 setup is going to look a little weird at first. -778 +781 00:49:08,120 --> 00:49:11,092 I'm going to have you follow two different numbers, -779 +782 00:49:11,092 --> 00:49:15,380 which is basically the displacement of our mass sitting up here, what is x. -780 +783 00:49:15,380 --> 00:49:17,760 But then you also want to know the velocity. -781 +784 00:49:17,840 --> 00:49:20,610 So let's say it's not just up there holding still, -782 +785 00:49:20,610 --> 00:49:24,740 it was actually moving a little bit with some kind of velocity to the right. -783 +786 00:49:25,040 --> 00:49:27,900 You need to understand those two numbers to follow the system as a whole. -784 +787 00:49:28,640 --> 00:49:32,000 And I'm going to package those two numbers together -785 +788 00:49:32,000 --> 00:49:34,780 as a single point in two dimensional space. -786 +789 00:49:35,080 --> 00:49:39,556 So let's say I had some position, or some state I should say, -787 +790 00:49:39,556 --> 00:49:43,600 that has a small displacement but then a large velocity. -788 +791 00:49:43,600 --> 00:49:50,330 I'm going to think of that as simply being a point, or maybe I draw an arrow to it, -789 +792 00:49:50,330 --> 00:49:56,500 thinking of it as a point in two dimensional space, with coordinates x and v. -790 +793 00:49:57,760 --> 00:50:01,520 All I'm doing is packaging them together as a single point. -791 +794 00:50:01,520 --> 00:50:07,303 And then what we might write is that the change to our two coordinates, -792 +795 00:50:07,303 --> 00:50:12,042 the change to x, v, looks like, well, the way x changes is -793 +796 00:50:12,042 --> 00:50:16,300 based on your velocity times a little change in time. -794 +797 00:50:16,380 --> 00:50:17,860 That's what velocity is. -795 +798 00:50:18,360 --> 00:50:21,380 And the way velocity changes, same deal but with acceleration. -796 +799 00:50:21,860 --> 00:50:24,660 But acceleration is negative x for this system. -797 +800 00:50:25,100 --> 00:50:27,680 So it's going to be negative x times delta t. -798 +801 00:50:29,580 --> 00:50:34,446 So we have a value that changes, and in fact the way that it changes -799 +802 00:50:34,446 --> 00:50:38,960 is going to be based on a little vector perpendicular to itself. -800 +803 00:50:39,320 --> 00:50:42,300 Which is what happens when you swap the variables and you make one negative. -801 +804 00:50:43,320 --> 00:50:46,946 And let me just talk through this, not in the context of writing them -802 +805 00:50:46,946 --> 00:50:50,780 down as two separate coordinates, but in the context of imaginary numbers. -803 +806 00:50:50,780 --> 00:50:54,899 Which feels like a very weird thing to do, but just walk through with -804 +807 00:50:54,899 --> 00:50:58,960 me this process because it feels so elegant once you let it play out. -805 +808 00:50:59,220 --> 00:51:03,880 I'm going to package this pair of numbers as a single complex number where the -806 +809 00:51:03,880 --> 00:51:08,540 real part is the position of the spring and the imaginary part is the velocity. -807 +810 00:51:09,060 --> 00:51:10,760 Again, totally crazy thing to do. -808 +811 00:51:10,840 --> 00:51:15,720 If that feels weird or you don't understand it, that's fine, it's not a natural thing. -809 +812 00:51:15,720 --> 00:51:18,776 But if we play this out and we start walking through the math, -810 +813 00:51:18,776 --> 00:51:20,620 hopefully it starts to justify itself. -811 +814 00:51:21,020 --> 00:51:27,543 Because what I can write is that the change to this state, -812 +815 00:51:27,543 --> 00:51:34,840 this complex number, is actually equal to negative i times itself. -813 +816 00:51:35,620 --> 00:51:39,840 Times x plus i times v times delta t. -814 +817 00:51:40,600 --> 00:51:41,120 Why? -815 +818 00:51:41,760 --> 00:51:43,780 Well, let me expand this out. -816 +819 00:51:44,420 --> 00:51:51,060 When I take that negative i, negative i times x is going to be negative x times i. -817 +820 00:51:51,660 --> 00:51:55,880 Negative i times iv is going to be negative i squared times v. -818 +821 00:51:56,240 --> 00:51:59,320 i squared is negative 1, so that's just going to end up being v. -819 +822 00:52:00,940 --> 00:52:03,360 So it's that whole number times our delta t. -820 +823 00:52:03,880 --> 00:52:04,860 What is this saying? -821 +824 00:52:05,240 --> 00:52:08,769 It says the change to the real part corresponds to v, -822 +825 00:52:08,769 --> 00:52:11,580 the velocity, which is what we saw up here. -823 +826 00:52:11,680 --> 00:52:14,160 The change to the first coordinate is based on v. -824 +827 00:52:14,840 --> 00:52:20,000 And the change to the imaginary part is based on the negative of the real part. -825 +828 00:52:20,500 --> 00:52:22,854 The change to your velocity, the acceleration, -826 +829 00:52:22,854 --> 00:52:25,360 what corresponds to force, is based on negative x. -827 +830 00:52:25,360 --> 00:52:29,743 So this weird idea of Hooke's Law, where the force that the spring -828 +831 00:52:29,743 --> 00:52:34,191 is pulling on turns out to be in the same, against the direction of -829 +832 00:52:34,191 --> 00:52:37,266 displacement and proportional to displacement, -830 +833 00:52:37,266 --> 00:52:41,780 lends itself to a very strange but natural description of our system -831 +834 00:52:41,780 --> 00:52:43,220 with a complex number. -832 +835 00:52:43,720 --> 00:52:47,270 And I could write this even more compactly by saying, -833 +836 00:52:47,270 --> 00:52:50,360 let's describe our whole state with a number z. -834 +837 00:52:51,200 --> 00:52:59,580 The way that z changes is equal to negative i times itself times a small step in time. -835 +838 00:53:00,860 --> 00:53:04,612 Geometrically, what that looks like is taking wherever you started, -836 +839 00:53:04,612 --> 00:53:07,868 multiplying it by negative i, which rotates at 90 degrees, -837 +840 00:53:07,868 --> 00:53:11,180 so this is a 90 degree angle, and then taking a little step. -838 +841 00:53:11,680 --> 00:53:16,409 And then now you're at a new state, and you rotate that new state by 90 degrees, -839 +842 00:53:16,409 --> 00:53:21,080 scale it down by delta t, whatever your time step is, and you take another step. -840 +843 00:53:21,560 --> 00:53:22,500 And you take another step. -841 +844 00:53:22,940 --> 00:53:27,087 And you keep doing this, and just like what we saw earlier for compound -842 +845 00:53:27,087 --> 00:53:31,580 interest that was imaginary, what you end up doing is walking around a circle. -843 +846 00:53:32,260 --> 00:53:35,140 Now this is maybe the worst attempt at a circle I've ever drawn. -844 +847 00:53:35,480 --> 00:53:38,000 This should go out a little bit further, something like that. -845 +848 00:53:38,000 --> 00:53:40,000 Ignore my dotted lines there. -846 +849 00:53:41,480 --> 00:53:43,183 Now let's gut check to see if this intuitively -847 +850 00:53:43,183 --> 00:53:44,960 makes sense that that's what our spring would do. -848 +851 00:53:45,480 --> 00:53:47,880 Because this vertical axis is the velocity. -849 +852 00:53:49,120 --> 00:53:53,358 So what we're saying is if you start off with a high velocity and a low displacement, -850 +853 00:53:53,358 --> 00:53:57,793 so your spring, your mass is moving fast, but it's not that far away from the equilibrium -851 +854 00:53:57,793 --> 00:54:02,180 point, well yeah, x is going to increase, because that's what it means to be moving fast. -852 +855 00:54:02,580 --> 00:54:06,000 So the x component, the real component here, is getting bigger and bigger. -853 +856 00:54:06,500 --> 00:54:09,489 And in the meantime, the y component, the velocity is slowing down, -854 +857 00:54:09,489 --> 00:54:11,160 because the spring is pulling it back. -855 +858 00:54:11,720 --> 00:54:14,500 So our y component is getting smaller and smaller, -856 +859 00:54:14,500 --> 00:54:17,225 until we reach a point where there's no velocity, -857 +860 00:54:17,225 --> 00:54:20,060 and it's as far out in the x component as it can go. -858 +861 00:54:20,380 --> 00:54:23,011 So we've kind of swung out, and then it's as far out as it can go, -859 +862 00:54:23,011 --> 00:54:24,740 and then it's going to start turning around. -860 +863 00:54:25,100 --> 00:54:28,440 The x component is going to start decreasing, and the velocity is negative. -861 +864 00:54:29,060 --> 00:54:34,046 And in general, this is just going to walk around a circle in our complex plane, -862 +865 00:54:34,046 --> 00:54:37,740 where the use of a complex plane feels kind of bizarre here. -863 +866 00:54:38,120 --> 00:54:40,700 But maybe intuitively, that's not that crazy a thought. -864 +867 00:54:41,200 --> 00:54:45,407 Whatever piece of math is describing the physics of a spring that goes back and forth, -865 +868 00:54:45,407 --> 00:54:48,888 and critically I should specify this is a spring where we're not taking -866 +869 00:54:48,888 --> 00:54:52,322 into account friction, so it'll just oscillate back and forth forever, -867 +870 00:54:52,322 --> 00:54:54,740 that oscillation corresponds with circular motion. -868 +871 00:54:55,420 --> 00:55:00,220 And we can see this maybe even more concretely if we write out the math of it. -869 +872 00:55:00,500 --> 00:55:05,861 So what we just saw is that imaginary compound interest invites us to write the solution, -870 +873 00:55:05,861 --> 00:55:09,436 the place that we're going to be after an amount of time t, -871 +874 00:55:09,436 --> 00:55:14,321 as wherever we were in the beginning, times e to the power of that interest rate, -872 +875 00:55:14,321 --> 00:55:17,360 where in this case the interest rate is negative i. -873 +876 00:55:17,620 --> 00:55:21,751 The thing influencing how you change, it's not in the direction of where you are, -874 +877 00:55:21,751 --> 00:55:24,220 it's 90 degrees to where you are, times the time. -875 +878 00:55:24,920 --> 00:55:27,744 This is actually a reasonable way to write things, -876 +879 00:55:27,744 --> 00:55:32,340 and it mirrors how you might see things written in certain parts of actual physics. -877 +880 00:55:32,540 --> 00:55:36,274 This e to the i t term shows up any time you have this kind of oscillation, -878 +881 00:55:36,274 --> 00:55:40,696 and the oscillation often corresponds to forces that are proportional to the displacement -879 +882 00:55:40,696 --> 00:55:41,040 itself. -880 +883 00:55:41,280 --> 00:55:43,040 It all loops together, it's all connected. -881 +884 00:55:43,900 --> 00:55:50,323 Now if you wanted to make this, I don't know, connect with the kind of graphs and -882 +885 00:55:50,323 --> 00:55:56,355 functions that we see elsewhere, you could write this using Euler's formula, -883 +886 00:55:56,355 --> 00:56:03,327 writing the expansion of the fact that e to an imaginary constant walks around a circle, -884 +887 00:56:03,327 --> 00:56:09,360 as your initial state times the cosine of t, and minus i times the sine of t. -885 +888 00:56:10,840 --> 00:56:15,724 And what this is basically saying is if we were to independently graph -886 +889 00:56:15,724 --> 00:56:20,540 the displacements and the velocities, they would look like sine waves. -887 +890 00:56:21,760 --> 00:56:25,792 Okay, so I'm just going to draw out a couple different plots here, -888 +891 00:56:25,792 --> 00:56:31,089 where on each one I'm going to let the x-axis represent time that we're moving forward, -889 +892 00:56:31,089 --> 00:56:34,460 and in one of them we're going to plot the displacement. -890 +893 00:56:35,100 --> 00:56:38,760 Well let's say the displacement started off entirely at one and the velocity was zero. -891 +894 00:56:38,760 --> 00:56:44,399 What it would look like would just be this cosine of t, excuse me, -892 +895 00:56:44,399 --> 00:56:50,880 cosine of t real component of our weird complex number description of things. -893 +896 00:56:51,500 --> 00:56:55,345 But if that all feels weird, hopefully it feels more concrete when you -894 +897 00:56:55,345 --> 00:56:59,135 realize all that saying is that the position of the spring oscillates -895 +898 00:56:59,135 --> 00:57:03,414 back and forth in a nice little cosine wave, which matches physical intuition, -896 +899 00:57:03,414 --> 00:57:06,880 it kind of oscillates back and forth in this gentle smooth wave. -897 +900 00:57:07,420 --> 00:57:12,796 And then similarly the velocity, let's say it started off at zero, -898 +901 00:57:12,796 --> 00:57:19,297 what our Euler's formula is telling us is that this will end up being a negative -899 +902 00:57:19,297 --> 00:57:20,100 sine wave. -900 +903 00:57:22,380 --> 00:57:25,396 So it's going to start off at zero, but then it's going to go down, -901 +904 00:57:25,396 --> 00:57:28,368 basically because it started off, you know, as far as it could be, -902 +905 00:57:28,368 --> 00:57:30,852 and then it's going to point in the opposite direction, -903 +906 00:57:30,852 --> 00:57:32,760 bringing the mass back towards equilibrium. -904 +907 00:57:33,640 --> 00:57:35,798 And then the velocity will reach some minimum and -905 +908 00:57:35,798 --> 00:57:38,000 then it'll come back up until the velocity is zero. -906 +909 00:57:38,060 --> 00:57:40,740 This is when the spring goes all the way to the other end. -907 +910 00:57:41,220 --> 00:57:43,735 And then the velocity will be positive for a while -908 +911 00:57:43,735 --> 00:57:46,300 and it'll keep oscillating back and forth like that. -909 +912 00:57:46,740 --> 00:57:49,668 And I'm sure I haven't quite lined up the waves appropriately, -910 +913 00:57:49,668 --> 00:57:52,830 but the idea that each one of them is a sort of sinusoidal pattern, -911 +914 00:57:52,830 --> 00:57:56,316 and that they're out of sync with each other, hopefully lines up with some -912 +915 00:57:56,316 --> 00:57:57,200 physical reasoning. -913 +916 00:57:57,760 --> 00:58:02,788 And I just think it's beautiful that what you can see is happening is it is -914 +917 00:58:02,788 --> 00:58:07,751 two different shadows of circular motion in an abstract mathematical idea, -915 +918 00:58:07,751 --> 00:58:12,780 of packaging the two components as a real and imaginary part of some number. -916 +919 00:58:13,120 --> 00:58:16,040 Because this Hooke's law let us describe things at a 90 degree angle. -917 +920 00:58:16,580 --> 00:58:19,180 And all of that is actually deeper than you might think it is. -918 +921 00:58:19,180 --> 00:58:23,499 There's certain formalizations of classical Newtonian mechanics that -919 +922 00:58:23,499 --> 00:58:28,320 introduce complex numbers for pretty similar reasons to what's going on here. -920 +923 00:58:28,520 --> 00:58:31,180 It's all a bit advanced, I won't necessarily go into it. -921 +924 00:58:31,240 --> 00:58:34,936 But what I want to emphasize is that this original question we asked of -922 +925 00:58:34,936 --> 00:58:39,352 what happens if your bank offers you an imaginary interest rate isn't totally insane, -923 +926 00:58:39,352 --> 00:58:42,997 and in fact, at least it's insane for money, but it's not insane as an -924 +927 00:58:42,997 --> 00:58:46,540 abstract idea to pursue, and it corresponds to some very real things. -925 +928 00:58:47,040 --> 00:58:51,466 So with that, I'm going to go ahead and take one final question from the audience, -926 +929 00:58:51,466 --> 00:58:52,800 and then say my goodbyes. -927 +930 00:58:53,560 --> 00:58:55,680 So the last question we have for today. -928 +931 00:58:56,520 --> 00:59:00,543 Can we get some hints or thoughts on the last challenge from the previous lecture, -929 +932 00:59:00,543 --> 00:59:02,240 particularly exp of x for matrices? -930 +933 00:59:02,920 --> 00:59:03,180 Excellent. -931 +934 00:59:03,540 --> 00:59:08,000 That was a hard question, by the way, the homework that I left at the end of last lecture. -932 +935 00:59:08,000 --> 00:59:13,903 So remember, last lecture we were talking all about thinking of the exponential function -933 +936 00:59:13,903 --> 00:59:19,807 as this infinite polynomial, and hopefully today the fact that it came from our limiting -934 +937 00:59:19,807 --> 00:59:25,380 expression, I didn't show why, but we could call that homework number two for today. -935 +938 00:59:25,400 --> 00:59:30,054 If you're really ambitious, it's showing that this expression that comes -936 +939 00:59:30,054 --> 00:59:34,136 from compound interest ends up expanding to be this polynomial, -937 +940 00:59:34,136 --> 00:59:39,620 which is what we mean anytime we refer to e to the x in math, at least in modern math. -938 +941 00:59:40,600 --> 00:59:45,418 So I asked some questions on the homework that involved expanding this -939 +942 00:59:45,418 --> 00:59:50,237 out all around trying to understand the crucial equation that when you -940 +943 00:59:50,237 --> 00:59:55,260 add two numbers in the input, this is the same as multiplying two outputs. -941 +944 00:59:55,260 --> 00:59:56,280 Okay. -942 +945 00:59:56,800 --> 01:00:02,860 And then what we have on screen is asking about for matrices, what goes on here. -943 +946 01:00:03,260 --> 01:00:07,790 So the thing that again, this is kind of advanced and maybe I feel a little -944 +947 01:00:07,790 --> 01:00:12,320 bit bad about that if I didn't properly emphasize it in giving the homework. -945 +948 01:00:13,380 --> 01:00:18,454 The thing I want you to notice is when you go through parts one and two of -946 +949 01:00:18,454 --> 01:00:23,460 the homework where you're expanding it out, and especially in the case of -947 +950 01:00:23,460 --> 01:00:28,738 part two where you're expanding things out, to think critically about whether -948 +951 01:00:28,738 --> 01:00:33,880 the commutative property, x times y equals y times x, is needed or relevant. -949 +952 01:00:34,660 --> 01:00:39,040 Because often when we expand out binomial terms, it assumes that this is true. -950 +953 01:00:39,360 --> 01:00:44,148 And the reason that the math for expanding binomial terms looks like what it does, -951 +954 01:00:44,148 --> 01:00:47,610 the reason it works has everything to do with the fact that -952 +955 01:00:47,610 --> 01:00:49,860 order of multiplication doesn't matter. -953 +956 01:00:50,420 --> 01:00:57,400 This is particularly relevant in part two where we're expanding this exp of x plus y. -954 +957 01:00:57,760 --> 01:01:00,119 And when you do that, you're going to run into -955 +958 01:01:00,119 --> 01:01:02,680 these terms that look like x plus y to the power n. -956 +959 01:01:03,400 --> 01:01:07,902 And then the parts of that are going to end up looking -957 +960 01:01:07,902 --> 01:01:12,160 like n choose k where this is the binomial constant. -958 +961 01:01:12,160 --> 01:01:14,920 So that's a bit of prerequisite knowledge. -959 +962 01:01:14,920 --> 01:01:19,360 If you don't have the prerequisite knowledge, don't feel intimidated by these problems. -960 +963 01:01:19,440 --> 01:01:22,120 It just means that there's a little bit more to learn. -961 +964 01:01:22,940 --> 01:01:26,820 The only reason that this expansion would work is if you have the commutative property. -962 +965 01:01:27,840 --> 01:01:29,040 So it's very subtle. -963 +966 01:01:29,040 --> 01:01:32,716 But what this means is that the whole line of argumentation to show -964 +967 01:01:32,716 --> 01:01:36,555 this property doesn't work if the things that you're plugging in don't -965 +968 01:01:36,555 --> 01:01:40,340 commute with each other, where x times y is not necessarily y times x. -966 +969 01:01:40,340 --> 01:01:43,462 For complex numbers this is true, for real numbers this is true, -967 +970 01:01:43,462 --> 01:01:45,480 but for matrices that's actually not true. -968 +971 01:01:46,640 --> 01:01:49,344 And this is one of the reasons, by the way, I actually -969 +972 01:01:49,344 --> 01:01:52,000 think writing e to the x for this is a bad convention. -970 +973 01:01:52,120 --> 01:01:56,407 Because you have this bizarre situation where when you have matrices, -971 +974 01:01:56,407 --> 01:02:01,429 and this does come up, there's real math that's done where you are exponentiating -972 +975 01:02:01,429 --> 01:02:06,635 matrices, the notation inspires you to think, oh, I should be able to write this as, -973 +976 01:02:06,635 --> 01:02:11,780 you know, exponentiating a times exponentiating b should be exponentiating a plus b. -974 +977 01:02:12,340 --> 01:02:13,760 But this is actually not true. -975 +978 01:02:14,060 --> 01:02:16,260 This is not true in the case of matrices. -976 +979 01:02:16,720 --> 01:02:19,530 We talked a little about quaternions today in the side plot, -977 +980 01:02:19,530 --> 01:02:21,420 it's not true in the case of quaternions. -978 +981 01:02:21,860 --> 01:02:25,388 So this convention of writing it as e to the x is incredibly -979 +982 01:02:25,388 --> 01:02:30,074 confusing as soon as you start extending into the realms where it's most useful, -980 +983 01:02:30,074 --> 01:02:33,140 where you have this particularly powerful polynomial. -981 +984 01:02:34,240 --> 01:02:38,244 And I'm just, I don't know, I think that is a fact worth emphasizing, -982 +985 01:02:38,244 --> 01:02:43,165 which is maybe why I wanted to sneak it into the homework, only later realizing that, -983 +986 01:02:43,165 --> 01:02:47,742 you know, if honestly my intention is to introduce people to the idea just at a -984 +987 01:02:47,742 --> 01:02:51,804 high level that this e to the x function is more than you think it is, -985 +988 01:02:51,804 --> 01:02:56,496 maybe jumping straight to the crazy idea of matrices and expecting them to notice -986 +989 01:02:56,496 --> 01:03:01,187 the nuance of commutativity being relevant to the n choose k formula for binomial -987 +990 01:03:01,187 --> 01:03:01,760 expansion. -988 +991 01:03:02,060 --> 01:03:03,886 Like, actually there's kind of a lot of advanced -989 +992 01:03:03,886 --> 01:03:05,900 things that go in there if we're really dissecting it. diff --git a/2020/ldm-imaginary-interest/english/sentence_timings.json b/2020/ldm-imaginary-interest/english/sentence_timings.json index 9bf80a841..21ac2fd94 100644 --- a/2020/ldm-imaginary-interest/english/sentence_timings.json +++ b/2020/ldm-imaginary-interest/english/sentence_timings.json @@ -170,7 +170,7 @@ 219.6 ], [ - "So there is a correct answer, but just answer what you think it might be.", + "so there is a correct answer, but just answer what you, what you think it might be.", 219.68, 223.42 ], @@ -240,7 +240,7 @@ 287.24 ], [ - "So the link is in the description for where you can go to ask these sorts of questions.", + "So the link is in the description for, you know, where you can go to ask these sorts of questions.", 288.66, 293.78 ], @@ -265,7 +265,7 @@ 333.84 ], [ - "Humor seems to have this root, like a really good joke kind of takes you by surprise, but when there's some kind of logical connection, it's not just surprise for its own sake.", + "Humor seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it's not just surprise for its own sake.", 334.26, 341.6 ], @@ -330,7 +330,7 @@ 406.22 ], [ - "You're like, 1% per month?", + "r month. Like one percent per month,", 406.22, 408.02 ], @@ -485,7 +485,7 @@ 573.08 ], [ - "So I'm going to give you a moment to think through the details of this.", + "All right, so I'm going to give you a moment to think through the details of this.", 574.08, 577.22 ], @@ -540,7 +540,7 @@ 648.1 ], [ - "So if we head back over to our Desmos expression that we were working with, If instead of increasing by 12%, we're saying increase by 6%, when you factor it out, it looks like multiplying by 1.06.", + "So if we head back over to our Desmos expression that we were working with, if instead of increasing by 12 percent, we're saying increase by 6 percent, when you factor it out, it looks like multiplying by 1.06.", 648.3, 662.3 ], @@ -640,7 +640,7 @@ 781.64 ], [ - "So if the annual interest rate was 12% and you're making steps each 1 12th of a month, that means you increase by 1% each one of those.", + "So if the annual interest rate was 12 percent and you're making steps each one twelfth of a month, of a munch, of a month, that means you increase by 1 percent each one of them.", 781.84, 790.7 ], @@ -665,7 +665,7 @@ 812.36 ], [ - "And if you're curious, you know, previously after 10 years, we would have had $310, I think it was, which we can verify for ourselves if I take that $100 times 1 plus 0.12 to the 10th.", + "And if you're curious, you know, previously after 10 years, we would have had 310 dollars, I think it was, which we can verify for ourselves if I take that 100 dollars times 1 plus 0.12 to the tenth.", 812.36, 826.36 ], @@ -1345,7 +1345,7 @@ 1754.16 ], [ - "And there's also a lot of delicacy in rigorously proving the fact that what this approaches will also approach this, or that both of them even stay finite and they don't blow up when you add more and more terms to the sum or when you crank up n higher and higher.", + "And there's also a lot of delicacy in rigorously proving the fact that what this approaches will also approach this, or that both of them even stay finite, and they don't blow up when you, you know, add more and more terms to the sum, or when you crank up n higher and higher.", 1754.5, 1768.6 ], @@ -1470,7 +1470,7 @@ 1904.58 ], [ - "So, I've got my axes here, where this is going to be my real money, and this will be all of my imaginary money.", + "So, I've got my axes here. This is going to be my real, this is my real money. And this will be all of my imaginary money.", 1905.5, 1914.64 ], @@ -1580,7 +1580,7 @@ 2056.88 ], [ - "So if you had $100 in the bank, you end up with $100 real dollars and $100 imaginary dollars.", + "So if you had a hundred dollars in the bank, you end up with a hundred real dollars and a hundred imaginary dollars.", 2057.16, 2062.06 ], @@ -2095,7 +2095,7 @@ 2617.24 ], [ - "If the mass starts out with a displacement of x naught and a velocity of v naught, which of the following most clearly describes the changes of these values after a small change in time, delta t?", + "Okay, if the mass starts out with a displacement of x naught and a velocity of v naught, which of the following most clearly describes the value, the changes of these values after a small change in time, delta t?", 2618.5, 2633.04 ], diff --git a/2020/ldm-imaginary-interest/english/transcript.txt b/2020/ldm-imaginary-interest/english/transcript.txt index f5b5387ab..28d806241 100644 --- a/2020/ldm-imaginary-interest/english/transcript.txt +++ b/2020/ldm-imaginary-interest/english/transcript.txt @@ -32,7 +32,7 @@ Make sure we can build up some of the math of that, and then once we have the ma Okay, so as our first, I don't even want to call it a real question, it's still meant as an opinion poll. Don't necessarily think you have to do the computation. What I want to gauge is what your gut reaction to this question is, okay? -So there is a correct answer, but just answer what you think it might be. +so there is a correct answer, but just answer what you, what you think it might be. So the question asks, two banks offer different interest rates on your savings. Bank A will increase your savings by 12% every year, okay? Pretty good bank. @@ -46,12 +46,12 @@ Okay, so it seems like we have pretty strong consensus. I don't know if I would call that strong consensus actually. There's a clear winner, but there's definitely a lot of contention behind it. And in general, by the way, while these answers are rolling in, I'll sometimes be pulling up questions from the audience, which you can ask via Twitter. -So the link is in the description for where you can go to ask these sorts of questions. +So the link is in the description for, you know, where you can go to ask these sorts of questions. And it looks like someone with the profile of Yoda asks, What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. I mean, it's a subjective question. Personally, I do think the most beautiful things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. -Humor seems to have this root, like a really good joke kind of takes you by surprise, but when there's some kind of logical connection, it's not just surprise for its own sake. +Humor seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it's not just surprise for its own sake. And as the lesson goes on, if you have questions relevant to what we're talking about, ask them there and we'll pull them up on screen. So, like I said, there's technically a right answer to this question, but I don't really want it to be treated as right or wrong. The reason I'm asking this one is to get kind of an opinion poll to know where people are starting. @@ -64,7 +64,7 @@ We're going to walk through that in a moment. After that, more with bank A. And I think the ones that I most empathize with, if I look back at what maybe an initial response to questions like this would be, if you haven't thought about interest, is everyone who said C. You know, it's not obvious that 12% over a year is going to be any different than 1% per month. -You're like, 1% per month? +r month. Like one percent per month, That should add up to 12% over the year. And I think it's worth just thinking through exactly why that's not the case. So, how do we think about this kind of problem? @@ -95,7 +95,7 @@ It says a bank offers to increase the money in your savings account by 6% at the Which of the following represents how much money will be in your account if you put in $100 and then you wait for one year? So there's two different six month periods that have passed. Which of the following expressions shows how much you're going to have? -So I'm going to give you a moment to think through the details of this. +All right, so I'm going to give you a moment to think through the details of this. Give you a little pause and ponder music maybe. Okay, as always, I'm probably going to grade this faster than is a reasonable amount of time for someone who really wants to think through the details of things. So never feel like you're being rushed. @@ -106,7 +106,7 @@ We're going to explain it in a moment. So the correct expression, which looks like 3,000 of you got, is that it should be $100 times 1 plus 0.6 squared. Alright? Now let's think through why that might be the case. -So if we head back over to our Desmos expression that we were working with, If instead of increasing by 12%, we're saying increase by 6%, when you factor it out, it looks like multiplying by 1.06. +So if we head back over to our Desmos expression that we were working with, if instead of increasing by 12 percent, we're saying increase by 6 percent, when you factor it out, it looks like multiplying by 1.06. And then there's two ways you can think about this. If you're already comfortable with the idea that increasing by 6% is multiplying by this constant, you say, oh, we just square that. And if you want to think through the details for why that's true, if that's not something you're entirely comfortable with, you can say, let's repeat the process. @@ -126,12 +126,12 @@ It looks like compounding more frequently got us an extra 68 cents in our accoun So, wonderful, right? In our graph, if we wanted to see what that would look like, again, there's some machinery under here that I'll show in a moment, but if I just crank up this number n to 12, that's basically asking how many times per year do I compound this interest? Do I make a little step increase based on what the annual interest rate is? -So if the annual interest rate was 12% and you're making steps each 1 12th of a month, that means you increase by 1% each one of those. +So if the annual interest rate was 12 percent and you're making steps each one twelfth of a month, of a munch, of a month, that means you increase by 1 percent each one of them. So what it looks like is a step function that's a lot more fine, okay? And we can see how at the end of the one year, the y coordinate of our graph is that $112 with the extra 68 cents. Not quite a dollar more, but it is more. And again, as we zoom out, you can see that it fits this nice exponential curve, the power of compound growth. -And if you're curious, you know, previously after 10 years, we would have had $310, I think it was, which we can verify for ourselves if I take that $100 times 1 plus 0.12 to the 10th. +And if you're curious, you know, previously after 10 years, we would have had 310 dollars, I think it was, which we can verify for ourselves if I take that 100 dollars times 1 plus 0.12 to the tenth. Alright, this says $310. When we compound every month instead, and we wait for 120 months, it looks like we've squeezed out an extra 20 bucks. So, not bad. @@ -267,7 +267,7 @@ And you might wonder, what on earth does this polynomial have to do with this li And as a super extra credit challenge question for those of you who are really into algebra and who are very comfortable with binomial expansion, if you know what those terms mean, if you take this expression for some large value of n and you expand it, you just expand it and you kind of simplify your terms and then you make approximations based on what's true when n is very large, what you would find is that the polynomial this expands to looks a lot like this polynomial here. And in fact, as this value of n gets bigger and bigger and approaches infinity, this polynomial will get closer and closer to this one. It's a challenging question, so I wouldn't expect everyone to necessarily be able to just bang it out, but if you want to it's a very elucidating exercise. -And there's also a lot of delicacy in rigorously proving the fact that what this approaches will also approach this, or that both of them even stay finite and they don't blow up when you add more and more terms to the sum or when you crank up n higher and higher. +And there's also a lot of delicacy in rigorously proving the fact that what this approaches will also approach this, or that both of them even stay finite, and they don't blow up when you, you know, add more and more terms to the sum, or when you crank up n higher and higher. There's a lot of delicacy there, but you can probably get to a point where there's an intuitive connection. And the reason that in math we tend to work with this polynomial instead of that limit, it's basically easier for computations and easier for theory. But the value of this other expression is that it ties us back to the idea of compound interest and the idea of taking a quantity that changes a little bit based on its own size. @@ -292,7 +292,7 @@ Now let's draw it out to think about what it would look like. And the other thing I want to emphasize is I know this seems like utter nonsense. We're talking about imaginary numbers in the context of interest rates. But in a couple minutes, I really do hope to make this relevant to physics and hope to make you see that this is not a totally nonsensical circle of thoughts. -So, I've got my axes here, where this is going to be my real money, and this will be all of my imaginary money. +So, I've got my axes here. This is going to be my real, this is my real money. And this will be all of my imaginary money. What it means if your interest rate is i, is that the change to the money looks like i times whatever the time step is, delta t, times whatever the money is to begin with. Now, in, I believe it was lecture 3, we talked all about complex numbers, and one of the fundamental facts was that when you multiply i by something, it has the effect of a 90 degree rotation. And we talked about this in terms of looking at the coordinates and realizing that you just swap the coordinates and make 1 negative 1. @@ -314,7 +314,7 @@ Again, I know this is utter nonsense, but follow along with me. One, it's fun, and two, it leads to real physics. So where does that get you? Well, it gets you $1 plus i dollars. -So if you had $100 in the bank, you end up with $100 real dollars and $100 imaginary dollars. +So if you had a hundred dollars in the bank, you end up with a hundred real dollars and a hundred imaginary dollars. But then, after the next year, you take another step, where it takes that new money vector, rotates it 90 degrees, and adds that to where you are. So after two years, you come back to your bank and you say, how's my money doing? And they say, good news, sir, there's twice as much of it. @@ -417,7 +417,7 @@ Which is a big one, but it's largely just stating what I just did. A mass on a spring is pulled a distance x away from an equilibrium point. If the spring obeys Hooke's law, the mass will experience a force of f equals negative kx, x being that displacement, for some constant k. Keep in mind, by Newton's second law, f equals ma, where m is the mass and a is the acceleration. -If the mass starts out with a displacement of x naught and a velocity of v naught, which of the following most clearly describes the changes of these values after a small change in time, delta t? +Okay, if the mass starts out with a displacement of x naught and a velocity of v naught, which of the following most clearly describes the value, the changes of these values after a small change in time, delta t? Okay, so we're going to let time play out for a tiny step in time, and it wants us to know which of these four options best describes what that change to the x value is going to look like, delta x, and what the change to the velocity value will look like, delta v. So I'll give you a little bit of time for this one. While you're thinking about that, we've got another question in about quaternions, not entirely relevant to the lesson, but certainly a very interesting idea and question. diff --git a/2020/ldm-imaginary-interest/french/sentence_translations.json b/2020/ldm-imaginary-interest/french/sentence_translations.json index c42c1bba0..525521908 100644 --- a/2020/ldm-imaginary-interest/french/sentence_translations.json +++ b/2020/ldm-imaginary-interest/french/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "Quel tweet poétiquement écrit, qui semble plutôt approprié pour quelqu'un avec un profil de Yoda là-dedans. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "Personnellement, je pense que les plus belles choses sont celles qui ont des liens inattendus. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "Ainsi, par exemple, si nous voulions savoir ce qui se passe dans notre exemple d'intérêt, lorsque nous avons commencé avec m de 0, ce qui pourrait être quelque chose comme 100 $, ce que nous faisons est de nous concentrer uniquement sur ce terme rt, nous disons que nous savons que comme nous augmentons n, cela va se rapprocher d'une constante spéciale élevée à la puissance rt, et peut-être que nous décririons cela à juste titre comme une croissance composée en continu, si nous prenons ce pas de temps et le laissons s'approcher de 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "Donc ce que nous disons, c'est que si vous commencez avec une vitesse élevée et un faible déplacement, donc votre masse se déplace rapidement, mais elle n'est pas si loin du point d'équilibre, eh bien oui, x va augmenter, parce que c'est ce que cela signifie aller vite. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/german/sentence_translations.json b/2020/ldm-imaginary-interest/german/sentence_translations.json index 1cd974079..989f3be7c 100644 --- a/2020/ldm-imaginary-interest/german/sentence_translations.json +++ b/2020/ldm-imaginary-interest/german/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "Was für ein poetisch geschriebener Tweet, der für jemanden mit einem Profil von Yoda darin ziemlich passend zu sein scheint. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "Persönlich denke ich, dass die schönsten Dinge diejenigen sind, die unerwartete Verbindungen haben. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "Wenn wir zum Beispiel wissen wollten, was in unserem Zinsbeispiel passiert, wenn wir mit m von 0 beginnen, was etwa 100 $ sein könnte, konzentrieren wir uns einfach auf diesen RT-Begriff, wir sagen, wir wissen das als Wenn wir n hochdrehen, nähert sich dies einer speziellen Konstante, die auf die Potenz von RT erhöht wird, und vielleicht würden wir dies treffend als kontinuierlich zusammengesetztes Wachstum beschreiben, wenn wir diesen Zeitschritt machen und ihn gegen 0 nähern lassen. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "Wir sagen also: Wenn Sie mit einer hohen Geschwindigkeit und einer geringen Verschiebung beginnen, bewegt sich Ihre Masse also schnell, ist aber nicht so weit vom Gleichgewichtspunkt entfernt. Nun ja, x wird zunehmen, denn darum geht es es bedeutet, schnell voranzukommen. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/hindi/sentence_translations.json b/2020/ldm-imaginary-interest/hindi/sentence_translations.json index 3afd7a3ed..4dd9daa8a 100644 --- a/2020/ldm-imaginary-interest/hindi/sentence_translations.json +++ b/2020/ldm-imaginary-interest/hindi/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "क्या काव्यात्मक ढंग से लिखा गया ट्वीट है, जो योडा की प्रोफ़ाइल वाले किसी व्यक्ति के लिए बिल्कुल उपयुक्त लगता है।", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "व्यक्तिगत रूप से, मुझे लगता है कि सबसे खूबसूरत चीजें वे हैं जिनके अप्रत्याशित संबंध हैं।", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "उदाहरण के लिए, यदि हम यह जानना चाहते हैं कि हमारे हित उदाहरण में क्या होता है, जब हमने 0 के एम के साथ शुरुआत की, जो कि $100 जैसा कुछ हो सकता है, तो हम बस उस आरटी शब्द पर ध्यान केंद्रित करते हैं, हम कहते हैं कि हम इसे जानते हैं हम एन को क्रैंक करते हैं, यह आरटी की शक्ति तक बढ़ाए गए कुछ विशेष स्थिरांक के करीब पहुंचने वाला है, और शायद इसे हम लगातार चक्रवृद्धि वृद्धि के रूप में वर्णित करेंगे, अगर हम उस समय कदम उठाते हैं और हम इसे 0 के करीब आने देते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "तो हम जो कह रहे हैं वह यह है कि यदि आप उच्च वेग और कम विस्थापन के साथ शुरुआत करते हैं, तो आपका द्रव्यमान तेजी से आगे बढ़ रहा है, लेकिन यह संतुलन बिंदु से बहुत दूर नहीं है, हाँ, x बढ़ने वाला है, क्योंकि यही है इसका मतलब है तेजी से आगे बढ़ना. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/indonesian/sentence_translations.json b/2020/ldm-imaginary-interest/indonesian/sentence_translations.json index 3b532cc8a..8caa97817 100644 --- a/2020/ldm-imaginary-interest/indonesian/sentence_translations.json +++ b/2020/ldm-imaginary-interest/indonesian/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "Sungguh tweet yang ditulis secara puitis, yang sepertinya cocok untuk seseorang dengan profil Yoda di sana. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "Secara pribadi, menurutku hal terindah adalah hal yang memiliki hubungan tak terduga. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "Jadi misalnya, jika kita ingin tahu apa yang terjadi pada contoh minat kita, ketika kita memulai dengan m dari 0, yang mungkin bernilai $100, yang kita lakukan adalah fokus pada suku pertama tersebut, kita katakan bahwa kita mengetahuinya sebagai kita naikkan n, ini akan mendekati suatu konstanta khusus yang dipangkatkan rt, dan mungkin ini akan dengan tepat kita gambarkan sebagai pertumbuhan yang terus menerus, jika kita mengambil langkah waktu itu dan membiarkannya mendekati 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "Jadi yang kita maksudkan adalah jika Anda memulai dengan kecepatan tinggi dan perpindahan rendah, maka massa Anda bergerak cepat, tetapi tidak terlalu jauh dari titik kesetimbangan, ya, x akan bertambah, karena itulah yang terjadi. artinya bergerak cepat. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/japanese/sentence_translations.json b/2020/ldm-imaginary-interest/japanese/sentence_translations.json index 978b31fbd..9dba98593 100644 --- a/2020/ldm-imaginary-interest/japanese/sentence_translations.json +++ b/2020/ldm-imaginary-interest/japanese/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "なんて詩的に書かれたツイートだろう。ヨーダのプロフィールが載っている人にぴったりだと思う。", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "個人的には、最も美しいものは予期せぬつながりを持っているものだと思います。", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "たとえば、関心のある例で、m が 0 ($100 程度の可能性があ ります) から始めたときに何が起こるかを知りたい場合、私たちが行う ことは、その rt 項に焦点を当てることだけであり、次のようにわ かっていると言います。n をクランクアップすると、これは rt 乗した特別な定数に近づきます。そのタイム ステップを 0 に近づけ ると、おそらくこれを継続的な複利成長と表現するのが適切でしょう。", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "つまり、私たちが言いたいのは、高い速度と低い変位で開始すると、質量は速く移動しますが、平衡点からそれほど遠くない場合、そうですね、x は増加するでしょう、なぜならそれが原因だからです速く動くという意味です。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/korean/sentence_translations.json b/2020/ldm-imaginary-interest/korean/sentence_translations.json index 8a7a19f92..7adce56d1 100644 --- a/2020/ldm-imaginary-interest/korean/sentence_translations.json +++ b/2020/ldm-imaginary-interest/korean/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "요다의 프로필이 있는 사람에게 매우 어울리는 시적으로 작성된 트윗입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "개인적으로 가장 아름다운 것은 예상치 못한 연결이 있는 것이라고 생각합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "예를 들어, 관심 예제에서 어떤 일이 일어나는지 알고 싶다면 m=0($100 정도)으로 시작할 때 우리가 하는 일은 해당 rt 용어에만 집중하는 것입니다. 우리는 n을 높입니다. 이것은 rt의 거듭제곱으로 올려진 어떤 특별한 상수에 접근하게 될 것입니다. 만약 우리가 그 시간 간격을 두고 0에 접근하게 두면 이것을 연속적인 복합 성장이라고 적절하게 묘사할 수 있을 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "따라서 우리가 말하는 것은 높은 속도와 낮은 변위로 시작하여 질량이 빠르게 움직이지만 평형점에서 그리 멀지 않은 경우, 음, 예, x는 증가할 것이라는 것입니다. 빠르게 움직인다는 뜻이다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/marathi/sentence_translations.json b/2020/ldm-imaginary-interest/marathi/sentence_translations.json index 4aceeeef4..ae2491753 100644 --- a/2020/ldm-imaginary-interest/marathi/sentence_translations.json +++ b/2020/ldm-imaginary-interest/marathi/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "किती कवितेने लिहिलेले ट्विट आहे, जे तिथल्या योदाचे प्रोफाइल असलेल्या व्यक्तीसाठी अगदी योग्य वाटते. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "वैयक्तिकरित्या, मला असे वाटते की सर्वात सुंदर गोष्टी म्हणजे अनपेक्षित कनेक्शन आहेत. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "उदाहरणार्थ, जर आम्हाला हे जाणून घ्यायचे असेल की आमच्या स्वारस्याच्या उदाहरणात काय होते, जेव्हा आम्ही m 0 ने सुरुवात केली, जे कदाचित $100 सारखे असू शकते, आम्ही काय करतो आम्ही फक्त त्या rt टर्मवर लक्ष केंद्रित करतो, आम्ही म्हणतो की आम्हाला ते माहित आहे आम्ही क्रॅंक अप n, हे rt च्या सामर्थ्यापर्यंत वाढवलेल्या काही विशिष्ट स्थिरांकाशी संपर्क साधणार आहे, आणि कदाचित याला आम्ही योग्यरित्या सतत चक्रवृद्धी वाढ म्हणून वर्णन करू, जर आम्ही त्या वेळेचे पाऊल उचलले आणि आम्ही ते 0 च्या जवळ येऊ दिले. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "तर आम्ही असे म्हणत आहोत की जर तुम्ही उच्च वेग आणि कमी विस्थापनाने सुरुवात केली, तर तुमचे वस्तुमान वेगाने पुढे जात आहे, परंतु ते समतोल बिंदूपासून इतके दूर नाही, होय, x वाढणार आहे, कारण तेच आहे. याचा अर्थ वेगाने चालणे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/persian/sentence_translations.json b/2020/ldm-imaginary-interest/persian/sentence_translations.json index 94b069f82..ebcda74c1 100644 --- a/2020/ldm-imaginary-interest/persian/sentence_translations.json +++ b/2020/ldm-imaginary-interest/persian/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "آیا فکر می‌کنید که چنین خوش اخلاقی ریاضی را به زیبایی اساسی تبدیل می‌کند؟ چه توییت شاعرانه ای، که برای کسی که نمایه یودا در آنجا دارد بسیار مناسب به نظر می رسد. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "من شخصاً فکر می کنم زیباترین چیزها آنهایی هستند که ارتباطات غیرمنتظره ای دارند. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "بنابراین، برای مثال، اگر می‌خواهیم بدانیم در مثال علاقه ما چه اتفاقی می‌افتد، زمانی که کار را با m از 0 شروع کردیم، که ممکن است چیزی در حدود 100 دلار باشد، کاری که می‌کنیم این است که فقط روی آن عبارت rt تمرکز می‌کنیم، می‌گوییم که می‌دانیم به عنوان ما n را بالا می بریم، این به مقداری ثابت خاص نزدیک می شود که به توان rt افزایش یافته است، و شاید به درستی آن را به عنوان رشد مرکب پیوسته توصیف کنیم، اگر آن گام زمانی را برداریم و اجازه دهیم به 0 نزدیک شود. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "بنابراین آنچه ما می گوییم این است که اگر با سرعت بالا و جابجایی کم شروع کنید، بنابراین جرم شما به سرعت حرکت می کند، اما از نقطه تعادل فاصله زیادی ندارد، خوب بله، x افزایش می یابد، زیرا این همان چیزی است که یعنی سریع حرکت کردن بنابراین مؤلفه x، مؤلفه واقعی در اینجا، بزرگتر و بزرگتر می شود. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/portuguese/sentence_translations.json b/2020/ldm-imaginary-interest/portuguese/sentence_translations.json index fafc595fd..33c3455a7 100644 --- a/2020/ldm-imaginary-interest/portuguese/sentence_translations.json +++ b/2020/ldm-imaginary-interest/portuguese/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "Que tweet escrito poeticamente, que parece bastante adequado para alguém com um perfil de Yoda ali. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "Pessoalmente, acho que as coisas mais bonitas são aquelas que têm conexões inesperadas. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "Então, por exemplo, se quiséssemos saber o que acontece em nosso exemplo de interesse, quando começamos com m de 0, que pode ser algo como US$ 100, o que fazemos é focar apenas naquele termo rt, dizemos que sabemos que como Se aumentarmos n, isso se aproximará de alguma constante especial elevada à potência de rt, e talvez descrevêssemos isso apropriadamente como crescimento continuamente composto, se dermos esse intervalo de tempo e deixarmos que ele se aproxime de 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "Então o que estamos dizendo é que se você começar com uma velocidade alta e um deslocamento baixo, então sua massa está se movendo rápido, mas não está tão longe do ponto de equilíbrio, bem, sim, x vai aumentar, porque é isso que significa estar se movendo rapidamente. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/russian/sentence_translations.json b/2020/ldm-imaginary-interest/russian/sentence_translations.json index 6ffc22aa1..ed2b7c8da 100644 --- a/2020/ldm-imaginary-interest/russian/sentence_translations.json +++ b/2020/ldm-imaginary-interest/russian/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "Какой поэтически написанный твит, который кажется вполне подходящим для человека, у которого там есть профиль Йоды. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "Лично я считаю, что самые красивые вещи — это те, которые имеют неожиданные связи. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "Так, например, если мы хотим знать, что происходит в нашем примере с интересами, когда мы начали с m, равным 0, что может быть примерно 100 долларов, мы просто сосредотачиваемся на этом термине rt, мы говорим, что знаем это как если мы увеличим n, это приблизится к какой-то особой константе, возведенной в степень rt, и, возможно, мы бы точно описали это как непрерывно усугубляемый рост, если мы сделаем этот временной шаг и позволим ему приблизиться к 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "Итак, мы говорим, что если вы начинаете с высокой скорости и малого смещения, поэтому ваша масса движется быстро, но она не так уж далеко от точки равновесия, ну да, x будет увеличиваться, потому что это то, что это значит двигаться быстро. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/spanish/sentence_translations.json b/2020/ldm-imaginary-interest/spanish/sentence_translations.json index 2ef989dd6..7dbf1d671 100644 --- a/2020/ldm-imaginary-interest/spanish/sentence_translations.json +++ b/2020/ldm-imaginary-interest/spanish/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "Qué tweet tan poéticamente escrito, que parece bastante apropiado para alguien con un perfil de Yoda allí. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "Personalmente, creo que las cosas más bellas son aquellas que tienen conexiones inesperadas. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "Entonces, por ejemplo, si quisiéramos saber qué sucede en nuestro ejemplo de interés, cuando comenzamos con m de 0, que podría ser algo así como $100, lo que hacemos es simplemente centrarnos en ese término rt, decimos que sabemos eso como Si aumentamos n, esto se acercará a alguna constante especial elevada a la potencia de rt, y tal vez esto lo describiríamos apropiadamente como crecimiento continuamente compuesto, si tomamos ese paso de tiempo y dejamos que se acerque a 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "Entonces, lo que estamos diciendo es que si comienzas con una velocidad alta y un desplazamiento bajo, entonces tu masa se mueve rápido, pero no está tan lejos del punto de equilibrio, bueno, sí, x aumentará, porque eso es lo que significa moverse rápido. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/tamil/sentence_translations.json b/2020/ldm-imaginary-interest/tamil/sentence_translations.json index a80c286bf..8c874ecd7 100644 --- a/2020/ldm-imaginary-interest/tamil/sentence_translations.json +++ b/2020/ldm-imaginary-interest/tamil/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "என்ன ஒரு கவிதையாக எழுதப்பட்ட ட்வீட், யோடாவின் சுயவிவரத்தைக் கொண்ட ஒருவருக்கு மிகவும் பொருத்தமாகத் தெரிகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "தனிப்பட்ட முறையில், நான் மிகவும் அழகான விஷயங்கள் எதிர்பாராத இணைப்புகளைக் கொண்டவை என்று நினைக்கிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "எடுத்துக்காட்டாக, எங்கள் வட்டி உதாரணத்தில் என்ன நடக்கிறது என்பதை அறிய விரும்பினால், 0 இன் m இல் தொடங்கும் போது, அது $100 போன்றதாக இருக்கலாம், நாம் என்ன செய்வோம், அந்த RT காலத்தின் மீது மட்டுமே கவனம் செலுத்துகிறோம், அது நமக்குத் தெரியும் என்று சொல்கிறோம். நாம் n ஐ வளைக்கிறோம், இது RT இன் சக்திக்கு உயர்த்தப்பட்ட சில சிறப்பு மாறிலிகளை அணுகப் போகிறது, மேலும் நாம் அந்த நேரத்தை எடுத்து 0 ஐ அணுக அனுமதித்தால், இது தொடர்ச்சியான கூட்டு வளர்ச்சி என்று பொருத்தமாக விவரிக்கலாம். ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "எனவே நாங்கள் சொல்வது என்னவென்றால், நீங்கள் அதிக வேகம் மற்றும் குறைந்த இடப்பெயர்ச்சியுடன் தொடங்கினால், உங்கள் நிறை வேகமாக நகர்கிறது, ஆனால் அது சமநிலைப் புள்ளியிலிருந்து வெகு தொலைவில் இல்லை, ஆம், x அதிகரிக்கப் போகிறது, ஏனென்றால் அதுதான். வேகமாக நகர்வதைக் குறிக்கிறது. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/telugu/sentence_translations.json b/2020/ldm-imaginary-interest/telugu/sentence_translations.json index 3cbc19b49..bf5947f5b 100644 --- a/2020/ldm-imaginary-interest/telugu/sentence_translations.json +++ b/2020/ldm-imaginary-interest/telugu/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "ఎంత కవితాత్మకంగా వ్రాసిన ట్వీట్, అక్కడ యోడా యొక్క ప్రొఫైల్ ఉన్నవారికి చాలా సరిపోతుందని అనిపిస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "వ్యక్తిగతంగా, ఊహించని కనెక్షన్‌లను కలిగి ఉన్నవి చాలా అందమైనవి అని నేను అనుకుంటున్నాను. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "కాబట్టి ఉదాహరణకు, మన ఆసక్తి ఉదాహరణలో ఏమి జరుగుతుందో తెలుసుకోవాలనుకుంటే, మేము 0 యొక్క m తో ప్రారంభించినప్పుడు, అది $100 లాగా ఉండవచ్చు, మనం చేసేది కేవలం ఆ rt పదంపై దృష్టి పెట్టడమే, అది మనకు తెలుసు అని చెబుతాము మేము n పైకి క్రాంక్ చేస్తాము, ఇది rt యొక్క శక్తికి పెంచబడిన కొన్ని ప్రత్యేక స్థిరాంకాలను చేరుకోబోతోంది, మరియు మనం ఆ సమయ దశను తీసుకొని దానిని 0కి చేరుకోవడానికి అనుమతించినట్లయితే, దీనిని నిరంతరంగా సమ్మేళనం చేసిన వృద్ధిగా సముచితంగా వర్ణించవచ్చు. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "కాబట్టి మేము చెప్పేది ఏమిటంటే, మీరు అధిక వేగంతో మరియు తక్కువ స్థానభ్రంశంతో ప్రారంభిస్తే, మీ ద్రవ్యరాశి వేగంగా కదులుతోంది, కానీ అది సమతౌల్య బిందువు నుండి చాలా దూరంలో లేదు, అవును, x పెరుగుతుంది, ఎందుకంటే అదే వేగంగా కదలడం అని అర్థం. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/thai/sentence_translations.json b/2020/ldm-imaginary-interest/thai/sentence_translations.json index 31d51a009..763b3475e 100644 --- a/2020/ldm-imaginary-interest/thai/sentence_translations.json +++ b/2020/ldm-imaginary-interest/thai/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/turkish/sentence_translations.json b/2020/ldm-imaginary-interest/turkish/sentence_translations.json index 60c7efd70..e30502e1c 100644 --- a/2020/ldm-imaginary-interest/turkish/sentence_translations.json +++ b/2020/ldm-imaginary-interest/turkish/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "Ne kadar şiirsel bir şekilde yazılmış bir tweet, Yoda'nın profiline sahip biri için oldukça uygun görünüyor. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "Kişisel olarak en güzel şeylerin beklenmedik bağlantıları olan şeyler olduğunu düşünüyorum. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "Örneğin, faiz örneğimizde ne olacağını bilmek istersek, m 0 ile başladığımızda, ki bu 100$ gibi bir şey olabilir, yaptığımız şey sadece rt terimine odaklanmaktır, bunu bildiğimizi söyleriz. n'yi yükseltirsek, bu, rt'nin kuvvetine yükseltilmiş özel bir sabite yaklaşacaktır ve eğer o zaman adımını alırsak ve 0'a yaklaşmasına izin verirsek, belki bunu uygun bir şekilde sürekli bileşik büyüme olarak tanımlayabiliriz. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "Yani demek istediğimiz şu, eğer yüksek hız ve düşük yer değiştirmeyle başlarsanız, yani kütleniz hızlı hareket ediyor, ama denge noktasından o kadar da uzakta değil, yani evet, x artacak, çünkü bu hızlı hareket etmek anlamına gelir. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/urdu/sentence_translations.json b/2020/ldm-imaginary-interest/urdu/sentence_translations.json index 50cb0d710..3cd461971 100644 --- a/2020/ldm-imaginary-interest/urdu/sentence_translations.json +++ b/2020/ldm-imaginary-interest/urdu/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "کیا آپ کو لگتا ہے کہ اس طرح کی بے حسی ریاضی کو ایک بہترین خوبصورتی فراہم کرتی ہے؟ کیا شاعرانہ انداز میں لکھا گیا ٹویٹ، جو کسی ایسے شخص کے لیے موزوں لگتا ہے جس میں یوڈا کا پروفائل ہو۔", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "ذاتی طور پر، میں سمجھتا ہوں کہ سب سے خوبصورت چیزیں وہ ہیں جن کا غیر متوقع تعلق ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "تو مثال کے طور پر، اگر ہم یہ جاننا چاہتے ہیں کہ ہماری دلچسپی کی مثال میں کیا ہوتا ہے، جب ہم نے m 0 کے ساتھ آغاز کیا، جو کہ $100 جیسا ہو سکتا ہے، تو ہم کیا کرتے ہیں کہ ہم صرف اس rt اصطلاح پر توجہ مرکوز کرتے ہیں، ہم کہتے ہیں کہ ہم جانتے ہیں کہ جیسا کہ ہم n کرینک اپ کرتے ہیں، یہ rt کی طاقت تک اٹھائے گئے کچھ خاص مستقل تک پہنچنے والا ہے، اور ہوسکتا ہے کہ اسے ہم مناسب طریقے سے مسلسل مرکب ترقی کے طور پر بیان کریں، اگر ہم اس وقت کا قدم اٹھائیں اور ہم اسے 0 تک پہنچنے دیں۔", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "تو ہم جو کہہ رہے ہیں وہ یہ ہے کہ اگر آپ تیز رفتار اور کم نقل مکانی کے ساتھ آغاز کرتے ہیں، تو آپ کا ماس تیزی سے آگے بڑھ رہا ہے، لیکن یہ توازن کے نقطہ سے اتنا دور نہیں ہے، ٹھیک ہے، x بڑھنے والا ہے، کیونکہ یہی ہے اس کا مطلب ہے تیزی سے آگے بڑھنا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-imaginary-interest/vietnamese/sentence_translations.json b/2020/ldm-imaginary-interest/vietnamese/sentence_translations.json index c4c808fc0..e7a8e569b 100644 --- a/2020/ldm-imaginary-interest/vietnamese/sentence_translations.json +++ b/2020/ldm-imaginary-interest/vietnamese/sentence_translations.json @@ -320,7 +320,7 @@ "end": 325.34 }, { - "input": "What a poetically written tweet, which seems pretty fitting for someone with a profile of Yoda in there. ", + "input": "ul things are the ones that have unexpected connections. And I don't know if that's like a natural thing that humans just love to see things that seemed unrelated come together. ", "translatedText": "Thật là một dòng tweet đầy chất thơ, có vẻ khá phù hợp với một người có hồ sơ về Yoda trong đó. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 334.48 }, { - "input": "Personally, I do think the most beautiful things are the ones that have unexpected connections. ", + "input": "or seems to have this root, like a really good joke kind of takes you by surprise. But when it's when there's some kind of logical connection, it ", "translatedText": "Cá nhân tôi nghĩ những điều đẹp đẽ nhất là những điều có sự kết nối bất ngờ. ", "model": "google_nmt", "n_reviews": 0, @@ -2024,7 +2024,7 @@ "end": 1609.16 }, { - "input": "So for example, if we wanted to know what happens in our interest example, when we started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt, and maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", + "input": "So for example, if we wanted to know what happens in our interest example, when we started out with $0, not $0, started out with m of 0, which might be something like $100, what we do is we just focus on that rt term, we say we know that as we crank up n, this is going to approach some special constant raised to the power of rt. And maybe this we would aptly describe as continuously compounded growth, if we take that time step and we let it approach 0. ", "translatedText": "Vì vậy, ví dụ: nếu chúng ta muốn biết điều gì xảy ra trong ví dụ quan tâm của mình, khi chúng ta bắt đầu với m bằng 0, có thể là khoảng 100 đô la, điều chúng ta làm là chỉ tập trung vào số hạng rt đó, chúng ta nói rằng chúng ta biết điều đó như chúng ta tăng tốc n, cái này sẽ tiến tới một hằng số đặc biệt nào đó được nâng lên lũy thừa rt, và có lẽ điều này chúng ta sẽ mô tả một cách thích hợp là sự tăng trưởng gộp liên tục, nếu chúng ta lấy bước thời gian đó và để nó tiến tới 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4080,7 +4080,7 @@ "end": 3227.88 }, { - "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", + "input": "So what we're saying is if you start off with a high velocity and a low displacement, so your spring, your mass is moving fast, but it's not that far away from the equilibrium point, well yeah, x is going to increase, because that's what it means to be moving fast. ", "translatedText": "Vì vậy, điều chúng tôi đang nói là nếu bạn bắt đầu với vận tốc cao và độ dịch chuyển thấp, do đó khối lượng của bạn chuyển động nhanh, nhưng nó không cách xa điểm cân bằng đến thế, vâng, x sẽ tăng, bởi vì đó là điều nó có nghĩa là đang di chuyển nhanh chóng. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/arabic/sentence_translations.json b/2020/ldm-logarithms/arabic/sentence_translations.json index cce8a8a99..21cfd4fbc 100644 --- a/2020/ldm-logarithms/arabic/sentence_translations.json +++ b/2020/ldm-logarithms/arabic/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵موسيقى🎵 مرحبًا بكم مجددًا في Lockdown Math. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "سنتحدث اليوم عن اللوغاريتمات ونوع من دروس العودة إلى الأساسيات. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "وكما هو الحال دائمًا، في بداية الأمر، أريد فقط أن أحصل على فكرة عن مكان تواجد الجمهور الآن. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "لذلك، إذا كنت تستطيع الذهاب إلى 3b1b. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "لم أسمع عنهم من قبل أو لم أتعلم عنهم من قبل ب. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "لقد تعلمت عنها ولكن في بعض الأحيان أشعر بالارتباك من جميع الخصائص ج. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "أنا أفهمهم ولكني لا أعرف كيفية تعليمهم ود. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "أنا أفهمهم جيدًا ويمكنني تعليمهم بشكل مريح لشخص آخر لجعلهم يفهمون جيدًا أيضًا. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "لذلك، لدينا تقسيم جيد. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "كما قلت، القصد من ذلك هو إنشاء درس يمكنني توجيه الأشخاص إليه في المستقبل إذا لم يكونوا مرتاحين مع اللوغاريتمات وأريد أن أكون قادرًا على القول، أوه، هذا هو المكان الذي يمكنك الذهاب إليه كيف أفكر، كما تعلمون، كيف أعتقد أنه يمكنك التعامل مع الأمر بشكل حدسي. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "سجل 1000 × x يساوي 3 أضعاف سجل x وتذكر أننا نستخدم الاصطلاح القائل بأن الأساس 10 سجل b. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "سجل 1000 مرة x يساوي سجل x مكعب ج. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "سجل 1000 مرة x يساوي 3 أس سجل x وe. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "لا شيء مما سبق وتذكر كما قلت سابقًا، يجب أن نتوقع تمامًا أن جميع هؤلاء الأشخاص في البداية الذين قالوا إنهم يفهمون السجلات جيدًا سوف يجيبون على الفور، وسوف يجيبون بشكل صحيح ولكن إذا كنت شخص لا يفعل ذلك، لا تدع ذلك يخيفك عندما تنظر إلى مشكلة كهذه، ما أود أن أشجعك على القيام به هو مجرد توصيل قوى العدد 10 المختلفة والتفكير في فكرة أن وظيفة السجل تحسب عدد الأصفار، لذا سأمنحك بعض الوقت للتفكير في ذلك، لذا سأمضي قدمًا وأقوم بتقييمه، وكما هو الحال دائمًا، إذا كان ذلك أسرع مما يناسبك، فاعلم أن ذلك فقط لأنني أريد المضي قدمًا مع الدرس، في هذه الحالة تكون الإجابة الصحيحة هي سجل 1000 مرة x وهو نفس أخذ 3 بالإضافة إلى سجل x والآن دعونا نفكر في ذلك للحظة وكما قلت عندما بدأت للتو أعتقد أن أفضل ما يمكنك فعله معهم هو أن تكون مرتاحًا لتوصيل أرقام مختلفة وأفضل الأرقام التي يمكن توصيلها هي تلك التي لها بالفعل قوى العدد 10، لذلك إذا كنت تطلب شيئًا مثل سجل 1000 مرة × حسنًا، فأنا لا أفعل ذلك. لا أعرف، دعنا نعوض بشيء ما لـ x log 1000 في 100 حسنًا، نحن نعرف عدد الأصفار التي ستكون في الإجابة النهائية هنا جيدًا 1000 في 100 يساوي 100000، لدينا بالفعل هذه الفكرة بشكل بديهي وهي أنه عندما نضرب 2 من قوى العدد 10 نحن فقط نأخذ الأصفار، الثلاثة أصفار من ذلك الـ 1000 والصفرين من ذلك الـ 100 ونضعهم بجانب بعضهم البعض لذا يجب أن يكون إجمالي 5 أصفار ولكن إذا كنت لا تفكر حقًا في كيفية تحول الرقم فحسب ولكن لماذا اتضح الأمر بهذه الطريقة، كان هناك 3 أصفار من 1000 بالإضافة إلى صفرين من 100 والتي يمكننا كتابتها أيضًا بقول عدد الأصفار في 1000 بالإضافة إلى عدد الأصفار في 100، لذا فهذه فكرة اللوغاريتم حاصل ضرب شيئين هو مجموع لوغاريتمات هذين الشيئين في سياق قوى العدد 10، وهذا مجرد توصيل ما هو بالفعل فكرة بديهية للغاية بالنسبة للكثير منا إذا أخذت قوتين للعدد 10 وقمت بضربهما، خذ كل أصفارها واحشرها فوق بعضها البعض، وبالتالي فإن الطريقة التي كتبت بها الأشياء هنا تشير في الواقع إلى حقيقة أكثر عمومية والتي ستكون أول خاصية لدينا في اللوغاريتمات وهي أنه إذا أخذنا سجل A مضروبًا في B يساوي سجل A بالإضافة إلى سجل B الآن في أي وقت ترى فيه إحدى قواعد اللوغاريتم هذه إذا وجدت نفسك محدقًا بعينيك أو كنت مرتبكًا قليلاً بشأن كيفية تذكرها، فما عليك سوى إدخال الأمثلة أنا لا لزوم لها، أنا أقول هذا كثيرًا ولكن لأنني أعتقد أنه من السهل جدًا أن تنسى بمجرد أن تكون غارقًا في الجبر نفسه وتجلس في نوع ما من الاختبار وهو يحتوي على الكثير من الرموز لتذكير نفسك أنك بخير، قم فقط بإدخال بعض الأرقام، وهو أمر جيد للقيام به وغالبًا ما يكون طريقة رائعة للحصول على الحدس، لذا في هذه الحالة، بقول سجل A في B وتقسيمها، يمكننا فقط التفكير، أوه، ذلك لوغاريتم 100 في 1000 وهو 5، هناك 5 أصفار مقسمة من حيث عدد الأصفار في كل جزء معين، رائع، رائع لذا نحمل هذا الحدس إلى أبعد من ذلك، فلنجرب مسألة تدريب أخرى ومرة أخرى، إذا كنت تعرف ذلك، عظيم، ستكون قادرًا على الإجابة عليها بشكل جيد ولكن ربما تفكر، ليس فقط ما هي الإجابة ولكن كيف سأشرح هذه الإجابة لشخص ما أو كيف سأحاول حث الطالب على التوصل إلى هذه الإجابة بمفرده دون الحاجة إلى إخباري لهم ما هي الإجابة، إذًا هناك عضوان محتملان من الجمهور، هناك أولئك الذين يهتمون بالدرس نفسه، ثم أولئك الذين يهتمون بالدرس الوصفي، لذا فإن سؤالنا يطرح، مرة أخرى، أي مما يلي صحيح؟ أ. ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "سجل x إلى n يساوي n ضرب سجل x b. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "سجل x إلى n يساوي سجل x للأس n c. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "سجل x إلى n يساوي n بالإضافة إلى سجل x أو d. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "لذا فإن الإجابة الصحيحة هنا هي a، والتي يبدو أن 4000 منكم حصلوا على التهاني، بإخبارنا أن سجل x مرفوعًا للقوة n يساوي n مضروبًا في سجل x لذا، مرة أخرى، لنفترض أنك تحاول تدريس هذا لشخص ما أو إذا كنت تحاول فهم ما يعنيه ذلك بنفسك، فأعتقد أن أفضل مكان للبدء هو توصيل شيء ما، وفي هذه الحالة، بالنسبة إلى log of x مرفوعًا للقوة n، فلنجرب ذلك باستخدام 100 مرفوعًا للقوة 3 ويمكنك تجربتها مع آخرين لمعرفة ما إذا كانت الأنماط التي تقوم بها تعمل بالفعل ولكن إذا كنت تفكر في الأمر ليس من حيث مجرد رؤية الإجابة ولكن محاولة التفكير في سبب ظهور الإجابة بهذه الطريقة في بعض الأحيان يكون أحد الأمثلة مناسبًا لأن 100 مكعب، يمكننا التفكير في ذلك على أنه أخذ جيد، وهذا يعني 3 نسخ من 100، فأنا آخذ 3 نسخ من 100 وعندما أضرب كل ذلك وأفكر في السجل باعتباره حساب عدد الأصفار التي لدينا لنقل، أوه، سيكون رقمًا يحتوي على 6 أصفار فقط، وهذا ما يعنيه ضرب 100 في 100 في 100. يمكنني فقط التفكير في تجميع كل هذه الأصفار معًا للحصول على مليون، لذا سيكون هذا الرقم 6 ولكن إذا فكرنا في الواقع لماذا كان 6 ليس فقط عدد الأصفار داخل المليون الذي جاء منه 6 هو أنه كان لدينا 3 نسخ من تلك الـ 100 وكل واحدة من تلك الـ 100 كان بها صفرين مختلفين، وبهذه الطريقة يكون الأمر أكثر عمومية يمكنك التفكير في الأمر، حيث أنه بدلاً من أخذ 100 مكعب كنا ننظر إلى 1000 مكعب أو 1000 أس n أو x أس n، يمكنك التفكير في أنه مهما كانت قيمة n هي عدد النسخ التي كنا نضربها في مرات عدد البئر، دعونا نرى، ليس x مضروبًا في عدد الأصفار التي كانت في أي شيء استبدلناه بـ x والذي كان في هذه الحالة 100، لذلك إذا كنت قد أخذت شيئًا مثل سجل 10000 للأس n، فسيكون هذا هو نفسه مثل أخذ n نسخة من الـ 10,000 بعد حساب عدد الأصفار في كل واحد منها وهو 4، وبالتالي سيكون n ضرب 4 وبالطبع الخاصية العامة التي أجاب عنها معظمكم بشكل صحيح هي أن لديك هذا التأثير الصغير الجميل حيث عندما شاهد سجل شيء مرفوع إلى قوة تقفز قوة صغيرة أمامه ولديك فقط سجل لما كان في الداخل الآن أحد أهم النتائج المترتبة على ذلك لا أعرف إذا كنت ستسميه ضمنا أو إذا كنت تسميها إعادة صياغة للتعريف إذا كنت آخذ السجل وسأعيد التأكيد على أنه الأساس 10 من 10 أس n يمكننا أن نفكر نوعًا ما في ذلك n الصغير على أنه قفز إلى الأسفل الأمامي ويصبح n مضروبًا في السجل ذي الأساس 10 من 10 وهو بالطبع 1. ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "ومن السهل أيضًا أن تشعر بالإحباط بسبب المهمة التي بين يديك، ولكن قد ترغب في تذكير نفسك بأن سبب اهتمامنا بهذه الأنواع من الأشياء هو أن فهم قواعد اللوغاريتمات يساعدنا في إجراء العمليات الحسابية في سياقات تشبه نمو الفيروس حيث من يوم إلى آخر، ومن خطوة إلى أخرى، تميل الأشياء إلى النمو بشكل مضاعف، ويساعدك فهم قواعد اللوغاريتمات في الحصول على فكرة أفضل عن هذا النوع من الأشياء، لذا قبل أن نقدم مثالًا حقيقيًا لطيفًا لما يمكن أن يبدو عليه ذلك مثلًا، اسمحوا لي أن أقوم بطرح سؤال اختباري آخر في هذا السياق للسؤال عن خصائص اللوغاريتمات سؤالًا أخيرًا قبل أن ننتقل إلى مثال بسيط من العالم الحقيقي للتخلص مما كان لدينا هنا والآن، أي مما يلي صحيح؟ سجل a plus b هو نفس سجل a plus سجل b سجل a plus b يساوي سجل ضرب b سجل a plus b يساوي واحدًا مقسومًا على سجل زائد b أو سجل a زائد b يساوي واحدًا مقسومًا على سجل ضرب b أو لا شيء مما سبق آه، والآن ليس لدينا نفس القدر من الإجماع، أليس كذلك؟ مثير للاهتمام للغاية، لدينا سباق خيول بين اثنين، لذا سأعطيكم لحظة للتفكير مليًا في هذا بينما يجيب الناس، في الواقع لدي سؤال صغير للجمهور، لذا، كما تعلمون، كنت أتحدث فقط عن كيف يمكننا فكر في النمو المضاعفة وهذا لا يجب أن يكون فقط قوى العدد عشرة، بل يمكننا أيضًا أن نفعل شيئًا مثل قوى العدد ثلاثة حيث إذا كنت تنتقل من واحد إلى ثلاثة إلى تسعة إلى سبعة وعشرين إلى واحد وثمانين، الكل من بين هذه يمكننا القول أن لوغاريتم هذه الأرقام للأساس ثلاثة ينمو في خطوات صغيرة لطيفة، لذا فإن لوغاريتم واحد للأساس ثلاثة، وثلاثة أس ما يساوي واحدًا، فإن الإجابة هي صفر بشكل عام، فإن سجل واحد، بغض النظر عن الأساس، سوف يكون صفرًا log ثلاثة للأساس ثلاثة، وثلاثة أس ما يساوي ثلاثة يساوي واحدًا وبالمثل log تسعة للأساس ثلاثة يساوي اثنين آه، قد تتساءل ما هو سؤالي، لكنه سيساعد في استخلاص كل هذه الأشياء ومن أجل متعتي الخاصة هنا، اسمحوا لي أن أكتب لوغاريتم واحد إضافي للأساس ثلاثة لواحد وثمانين يساوي أربعة الآن، لقد سمعت ذلك ظاهريًا إذا سألت طفلًا، دعنا نقول في سن الخامسة أو السادسة تقريبًا ما هو الرقم الذي يقع في المنتصف بين واحد وتسعة قل ما هو الرقم الذي يقع في المنتصف، فغرائزهم حول كيفية الإجابة هي لوغاريتمية، بينما تميل غرائزنا إلى أن تكون أكثر خطية، لذلك غالبًا ما نفكر في واحد وتسعة، ويكون لديك مجموعة من الأرقام المتباعدة بشكل متساوٍ بينهما اثنين، ثلاثة، أربعة، خمسة، ستة ، سبعة، ثمانية وإذا ذهبت في منتصف المسافة بينهما، فسوف تصل إلى خمسة ولكن إذا كنت تفكر في النمو المضاعف حيث يمكنك الانتقال من واحد إلى تسعة، فالأمر لا يتعلق بإضافة مجموعة من الأشياء ولكنك "إننا ننمو بمقدار معين، ننمو بعامل ثلاثة، ثم ننمو بعامل آخر وهو ثلاثة، من المفترض أن الغريزة الطبيعية للطفل تتوافق مع قول ثلاثة ومن المفترض أن هذا يتوافق أيضًا مع ما إذا كان لديك علماء أنثروبولوجيا يدرسون المجتمعات التي لديها" لقد طوروا أنظمة المحاسبة والكتابة بنفس الطريقة التي تمتلكها المجتمعات الحديثة، وسوف يجيبون على ثلاثة أسئلة لذلك، سؤالي للجمهور إذا كان أي منكم يشاهد الآن لديه إمكانية الوصول إلى طفل صغير، دعنا نقول، في حدود خمس سنوات انظر إذا كان بإمكانك الذهاب واسألهم عن الرقم الذي يقع في المنتصف بين واحد وتسعة، وإذا استطعت، فأخبرنا على تويتر بما يقوله الطفل وما هي إجابته الفعلية لأنني لا أعرف السبب، أنا قليل فقط متشكك فيما إذا كان ذلك سينجح بالفعل من الناحية العملية، أفهم أن هذه ليست طريقة علمية فائقة للقيام بذلك، فأنا لا أطلب من الأشخاص الذين يشاهدون بثًا مباشرًا على YouTube إجراء استطلاع رأي لأطفالهم ثم نشر الإجابة على تويتر، ولكن من أجل مصلحتي سيكون الأمر مثيرًا للاهتمام لرؤية نوع من التحقق من صحة سؤالنا، هذا هو السؤال الأول الذي لا يبدو أنه يحظى بإجماع كبير في اتجاه واحد، فلنمضي قدمًا ونصنفه لنرى ما هي الإجابة الرائعة، حسنًا، إذن 2400 أجاب أحدكم بشكل صحيح أنه ليس مما سبق أن سجل a plus b لا يلبي أيًا من هذه الخصائص الرائعة وبشكل عام، إلا إذا كنا سنعمل مع أنواع معينة من التقديرات خاصة عندما يتم تشغيل اللوغاريتم الطبيعي قد نتحدث عن هذا في المرة القادمة، فإن إضافة مدخلات اللوغاريتم هو في الواقع إحساس غريب للغاية، إنه أمر غريب جدًا القيام به وللتعرف على هذه الغرابة، قم بتوصيل بعض قوى العدد عشرة إذا طلبت منك تسجيل علامة زائد ب ما قد تبدأ في التفكير فيه هو، حسنًا، دعني أدخل بعض الأمثلة مثل 10000 و100 وأسأل نفسي، إذا قمت بإجراء عملية العد الصفري لما هو موجود في هذا الإدخال، كم عدد الأصفار الموجودة فيه؟ لكن الأمر غريب، لأنه عندما نضيف 10,100 جيدًا، لم نعد عند قوة نظيفة للعشرة، حسنًا، لا بأس، كما تعلمون، غالبًا ما تأخذ لوغاريتمات لأشياء ليست من القوى النظيفة للعشرة ولكنها تصبح من الغريب جدًا أن نسأل كيف يمكنك التعبير عن ذلك من حيث لوغاريتم 100 الذي كان اثنين، ولوغاريتم 10000 الذي كان أربعة لأنه إذا نظرت إلى لوغاريتم 10100، فإنك تسأل عشرة أس ما يساوي 10100، قد تقول، لا أفعل. ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "لا أعلم، سيكون أعلى قليلاً من أربعة لأنه قريب نوعًا ما من 10000، لذا فإن أفضل ما يمكنك تخمينه هنا هو، أوه، سيكون هذا شيئًا يشبه سجل 10000 ولكن هذا يبدو وكأنه مجرد صدفة استنادًا إلى الرقمين اللذين وضعناهما، لا يوجد سبب منهجي لطيف للوصول إلى هناك، لذا ربما تخمن، أوه، إذا كان الرقمان a وb مختلفين تمامًا، فهذا قريب نوعًا ما من الحد الأقصى لهما ولكنه غريب جدًا والأهم من ذلك، من أجل الاختبار، إذا نظرت فقط إلى الخيارات التي يقدمها لك إذا جربت ذلك باستخدام أي أرقام معينة، ستجد أن أيًا منها لا يعمل فعليًا، لذا، كل شيء على ما يرام في بعض الأحيان تحصل على شيء ما يبدو أنها ستكون خاصية جميلة ولكنها لن تكون خاصية جميلة في نهاية المطاف وأعتقد أيضًا أن هذا مهم بدلاً من مجرد العثور على نفسك تعمل فقط مع مختلف، كما تعلمون، سجل a ضرب b أو سجل x إلى القوة n هذه الأشياء التي لها قاعدة لطيفة أحيانًا تكون في عالم الرياضيات، وتعمل على حل بعض المشكلات لديك تعبير لوغاريتمي وتقوم بإضافة أشياء في الإدخال وتريد أن تكون قادرًا على الإلمام بها حقيقة أن هذا أمر غريب نوعًا ما أنك لن تكون قادرًا على تبسيطه، ولكن إذا لم تفكر في ذلك قبل أن تتساءل، أوه، هل هناك مجرد صيغة لم أفكر فيها كما رأينا من قبل، مع كل ذلك، اسمحوا لي بالمضي قدمًا وتلقي بعض الأسئلة من الجمهور قبل أن ننتقل إلى نوع مختلف من الأمثلة، لذا يبدو أن أوما شيرما تسأل هل يمكن أن تكون القاعدة صفرًا؟ هذا سؤال مثير للاهتمام، هل يمكن أن تكون قاعدة اللوغاريتم صفرًا؟ حسنًا، فيما يتعلق بمثلثنا، قد نفكر في ذلك كما تعلم، صفر مرفوعًا إلى نوع ما من القوى x يساوي قيمة أخرى y، وهذا شيء يمكننا كتابته إما بقول صفر مرفوعًا لـ x يساوي y أو يمكننا كتابته نفس الشيء بقول أن log للأساس صفر لـ y يساوي x صفر أس ما يساوي x الآن المشكلة هنا هي أن صفر لأي شيء ينتهي به الأمر ليصبح صفرًا صحيحًا، لذلك إذا كنا سنفكر فقط في log للأساس صفر لـ y بالنسبة لأي إدخال آخر، كما تعلم، فأنت تريد إدخال شيء مثل واحد أو اثنين أو pi أي شيء قد تريده، فأنت تطرح السؤال صفر على ما يساوي واحدًا أو اثنين أو pi أو أي رقم قد يكون لديك هناك ولن تكون هناك إجابة، لذا في أحسن الأحوال يمكنك أن تحاول أن تقول نعم، سجل الصفر، إنها دالة صالحة تمامًا ويتم تعريفها فقط على الإدخال صفر ولكن حتى في هذه الحالة ستواجه مشكلة في محاولة فهم ما تريد هناك لأن قول صفر إلى ما يساوي صفر يشبه أي شيء ينطبق عليه، لذا ستكون ذراعك ملتوية خلف ظهرك ولكنك تريد تنفيذ ذلك ويتوافق مع حقيقة أن الدالة الأسية ذات الأساس صفر هي صفر تمامًا لا يرسم الأرقام بطريقة لطيفة من واحد إلى واحد على بعضها البعض، لذا فهذا سؤال رائع، هل يمكن أن يكون لديك سجل ذو قاعدة صفر الآن نعود إلى فكرة مكان ظهور هذه الأشياء في العالم الحقيقي، أحد الأمثلة التي أحبها نوعًا ما هو مقياس ريختر للزلازل، لذا فإن مقياس ريختر يعطينا تقديرًا كميًا لمدى قوة الزلزال ويمكن أن يتراوح من أرقام صغيرة جدًا إلى أرقام كبيرة جدًا مثل أعتقد أن أكبر زلزال تم قياسه على الإطلاق وهذا مجرد مخطط يأتي من ويكيبيديا كانت 9.5 ولتقدير مدى جنون هذا الأمر، من المفيد النظر إلى العلاقة بين ما تعنيه هذه الأرقام ثم شيء مثل الكمية المكافئة من مادة TNT، وهو نوع من قياس مقدار الطاقة الموجودة فيه ثم ما يمكننا أن نحاول القيام به هنا هو معرفة ما إذا كان بإمكاننا الحصول على تعبير لرقم مقياس ريختر من حيث كمية الطاقة ولماذا تكون اللوغاريتمات طريقة طبيعية لوصف ذلك، لذا فإن المفتاح الذي يجب التركيز عليه هو مقدار زيادة الأشياء بينما نخطو خطوات للأمام لذلك على سبيل المثال، إذا انتقلنا من اثنين جيدًا في هذه الحالة، فهذا لا يوضح لنا مكان الرقم ثلاثة، لذا ربما نفكر في اتخاذ خطوة من اثنين إلى أربعة وهو ما يشبه اتخاذ خطوتين، ماذا يفعل ذلك فيما يتعلق بـ حسنًا، يبدو أنها تأخذنا من طن متري واحد من مادة تي إن تي والتي أعتقد أنها قنبلة كبيرة من الحرب العالمية الثانية وتستغرق ما يصل إلى كيلو طن ألف مرة وهي قنبلة ذرية صغيرة، لذلك خطوتين فقط على مقياس ريختر، الانتقال من زلزال بقوة 2 إلى زلزال بقوة 4 يأخذنا من قنبلة كبيرة من الحرب العالمية الثانية إلى العصر النووي، لذا فإن هذا جدير بالملاحظة والخطوة النظيفة الأولى التي نخطوها هي الانتقال من 4 إلى 5 في على الأقل فيما يتعلق بما يوضحه لنا هذا الرسم البياني بشكل جيد ومن الواضح أن خطوة واحدة للأعلى من 4 إلى 5 تقابل الانتقال من 1 كيلوطن إلى 32 كيلوطن ومن الواضح أن هذا كان حجم القنبلة المدمرة للمدينة التي سقطت على ناجازاكي، لذا ربما يكون هذا واحدًا شيء يمكن أن يكون غير بديهي فيما يتعلق بالمقاييس اللوغاريتمية إذا كنت تسمع للتو في الأخبار الفرق بين أوه كان هناك زلزال بقوة 4.0 مقابل الزلزال الذي كان 5.0 من السهل التفكير في أن 4 و5 هما رقمان متشابهان إلى حد كبير ولكن من الواضح أنه من حيث كميات TNT التي تتوافق مع الضرب في 32 للانتقال من 1 إلى الذي يليه والانتقال من 2 إلى 4 كان من الواضح الضرب بحوالي ألف والوحيد السبب الأكبر هو أن مخططنا هنا لم يكن يُظهر الرقم 3، لذلك كنا نتخذ خطوتين ويمكنك التحقق بنفسك من أنه إذا اتخذت خطوة 32 ثم ضربتها في 32 أخرى، فهذا في الواقع قريب جدًا من الألف، لذا يبدو أن فكرة أن الخطوات المضافة على رقم ريختر تتوافق مع الخطوات المضاعفة في مادة تي إن تي تشير إلى وجود شيء لوغاريتمي يلعب هنا، ومن المثير للاهتمام بعض الشيء أن نستمر هنا ونقول إلى أي مدى ينمو هذا جزئيًا بسبب الظواهر العالمية واصفًا نعم، ليست مفاجأة كبيرة أننا عندما نخطو خطوة أخرى، فإنها تتضاعف بحوالي 32 مرة أخرى، لكن بضبط ذلك في حدسنا، فإن هذا هو الفرق بين 32 كيلو طنًا لقنبلة ذرية صغيرة ثم ميجا طن واحد والذي قد نفكر فيه على أنه ليس قنبلة ذرية صغيرة، قنبلة ناغازاكي الذرية التي أعتقد أنها 32 قنبلة ذرية من ناغازاكي بقوة ميغا طن واحد، ومن الواضح أن حجم الزلزال المسطح المزدوج الذي ضرب ولاية نيفادا بالولايات المتحدة الأمريكية عام 1994 لم أكن أعرف ما هو ذلك، شكرًا ويكيبيديا من حيث الترددات بالمناسبة. بحثت أيضًا عن تلك التي تكون أقل من اثنتين، والتي تحدث طوال الوقت، هناك حوالي 8000 منها يوميًا ولكن بمجرد أن نكون في عالم القنابل الذرية، أشياء مثل 3. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "من الواضح أن هذه الأحداث تحدث بشكل متكرر في مكان ما على الأرض، حيث يوجد حوالي 134 منها تحدث في مكان ما كل يوم، من كان يعلم؟ ولكن بينما نصبح أكثر كثافة في هذا النطاق 5 و 6 الذي كان أعلى بكثير من مقياس القنبلة الذرية، فنحن الآن عند حوالي 2 فقط في اليوم، وأنا متأكد من أن أحد الجيولوجيين يمكن أن يأتي ويشرح لماذا يجب علينا جميعًا " لا أشعر بالقلق الشديد بشأن حقيقة أن هناك اختلالين مكافئين لقنبلة ذرية في قشرة الأرض يحدث كل يوم، ولكن من النادر بشكل خاص أن يتركز هؤلاء في مكان ما مثل مدينة يعيش فيها الكثير من الناس الآن، فقط نتحقق من فكرتنا بأن كل خطوة تتضمن نموًا قدره 32، فلننظر إلى الشكل الذي تبدو عليه الخطوة من 6 إلى 7، وهنا تعطينا الكثير من الأمثلة بينهما ربما تعطي الوهم بأن هذه خطوة أكبر مما هي عليه في الواقع، وبالفعل هذا هو الفرق بين 1 ميجا طن و 32 ميغا طن، وهذا ضرب في 32. أحد الأشياء التي وجدتها أكثر إثارة للاهتمام في هذا الرسم البياني بالمناسبة هو النظر إلى المدى الذي يتعين علينا أن نقطعه قبل أن نصل إلى أكبر سلاح نووي تم اختباره فعليًا على الإطلاق، وكان ذلك في ذروة الحرب الباردة قنبلة القيصر التي كانت قوتها 50 ميغا طن، وأعتقد أنه كان لديهم بالفعل خطط أصلية لامتلاك قنبلة بقوة 100 ميغا طن لكنهم خفضوا من تلك الـ 50 ميغا طن، ونحن نتحدث عن البدء عند 32 كيلو طن من قنبلة ناجازاكي مضروبة في 32 للحصول على ميغا طن تتضاعف بـ 32 أخرى، لذلك نحن نتحدث عن ألف مرة قوة الانفجار الذي أنهى الحرب العالمية الثانية وما زلت لم تصل إلى 50 ميغا طن مما تستطيع البشرية القيام به ومن الواضح أن هذا هو زلزال جاوة في إندونيسيا، لذا 7 . 0 ليس أكبر قليلاً من 6.0، إنه أكبر بكثير والنقطة هنا بالطبع هي أنه عندما يكون لديك مقياس يمنحك زيادات مضاعفة، فمن الجدير أن نقدر أن ما يبدو كخطوات صغيرة يمكن أن يكون في الواقع خطوات ضخمة من حيث الطاقة الضمنية أو القيم المطلقة الضمنية هنا لذلك عندما نفكر في حقيقة أنه كان هناك 9 على الإطلاق. 5 يبدو هذا سخيفًا في الواقع نظرًا لأنه موجود فقط في 7. نطاق 0 الذي نتحدث فيه عن أكبر سلاح نووي حراري تم إطلاقه على الإطلاق وهذا يدل على مجال واحد تميل فيه اللوغاريتمات إلى الظهور، وهو عندما يريد البشر إنشاء مقياس لشيء يمثل تباينًا واسعًا للغاية في مدى ضخامة الأشياء سواء كان ذلك في حالة حجم الزلازل، يمكنك الحصول على أشياء مما يحدث طوال الوقت حول الأرض، بحجم قنبلة يدوية كبيرة وتريد أن يكون ذلك على مقياسك وشيئًا للتفكير فيه يتراوح على طول الطريق إلى أكبر اضطراب شهدناه في تاريخ البشرية، ومن أجل الحصول على ذلك بطريقة لا تقوم فقط بكتابة مجموعة كاملة من الأرقام المختلفة في أرقامك لحالة واحدة ومجموعة كاملة من الأرقام المختلفة، رقم أصغر من الأرقام لرقمك في حالة أخرى، من الجيد أن تأخذ اللوغاريتمات ثم تضع ذلك على مقياس واحد يسحق بشكل أساسي تلك الأرقام بين 0 و10، ترى شيئًا مشابهًا جدًا يحدث مع مقياس الديسيبل للموسيقى الذي يعمل في الواقع قليلاً بشكل مختلف بعض الشيء حيث في كل مرة تأخذ خطوة للأعلى بمقدار 10 ديسيبل تتوافق مع الضرب في 10، فبدلاً من خطوة 1 الضرب في 10، إنها خطوة 10 تضرب في 10، لذا فإن هذا النوع من العمليات الحسابية يجعل الأمر قليلاً سخيفة بعض الشيء ولكن الفكرة هي نفسها، وهي أنك إذا كنت تستمع إلى صوت بقوة 50 ديسيبل مقابل 60 ديسيبل، فسيكون أكثر هدوءًا من حيث الطاقة التي يتم نقلها والانتقال منها، ماذا ستكون، 60 إلى 70 أو 70 إلى 80 تلك الخطوات، من 60 إلى 80، التي تتضمن مضاعفة كمية الطاقة لكل مساحة مربعة بعامل 100، لذا في كل مرة ترى مقياسًا لوغاريتميًا، اعلم في ذهنك أن هذا يعني أن كل ما يشير إليه تحت الغطاء ينمو بمقدار كمية هائلة، وهذا هو السبب مرة أخرى في رأينا الكثير من المقاييس اللوغاريتمية المستخدمة لوصف تفشي فيروس كورونا، فكيف يمكنك وصف علاقة كهذه حيث في كل مرة تقوم فيها بزيادة رقم مقياس ريختر بمقدار 1، فإنك تضرب في 32 جيدًا، نحن يمكن أن أفكر في السجل ذو الأساس 32، يمكنني القول إذا أخذت السجل، سأتصل بـ r، رقم مقياس ريختر الذي قد أفكر فيه على أنه السجل ذو الأساس 32 والذي سيتوافق مع ، لا لا لا، أنا أفعل هذا بشكل خاطئ، هذا ليس هو الشيء الذي تم تسجيله، فنحن نأخذ قاعدة السجل 32 للرقم الكبير، لرقم TMT، وهو شيء كان مثل 1 ميجا طن، وهو 1 مليون طن قاعدة السجل 32، ينبغي أن تتوافق مع رقم مقياس ريختر ولكن قد يكون هناك نوع من الإزاحة، لذلك قد نقول أن هناك نوعًا من الثابت الذي نضيفه إلى رقم مقياس ريختر هذا وهذا التعبير هو نفسه تمامًا، أعذرني على الخروج عن هذا في الأسفل هناك هذا التعبير هو نفسه تمامًا قول 32 أس بعض الإزاحة مضروبًا في رقم مقياس ريختر الخاص بنا وهو نفس أخذ 32 إلى ذلك الإزاحة، والذي في حد ذاته مجرد ثابت كبير، مضروبًا في 32 رقم مقياس ريختر، لذا قد تفكر في هذا على أنه مجرد عدد ثابت مضروبًا في 32 أس الرقم الذي تراه، لذا فإن طريقة الكتابة هذه تؤكد حقًا على النمو الأسي له، فإذا كان هذا هو ما يتوافق مع مقدار TMT الذي تراه، كلما قمت بزيادة ذلك خطوة بخطوة، أنت تضرب في 32 ولكن هناك طريقة أخرى لإيصال نفس الحقيقة تمامًا وهي أخذ الأساس اللوغاريتمي 32 لأي مبلغ مناسب الآن، والشيء التالي الذي أريد التحدث عنه هو كيف أننا لا نحتاج دائمًا إلى ذلك تقلق بشأن كيفية حساب سجلات ذات قواعد مختلفة، فمن الغريب بعض الشيء هنا أننا كنا نتحدث عن سجل ذو قاعدة 32، وقد أشرت سابقًا إلى كيف يحب علماء الرياضيات حقًا أن يكون لديهم سجل ذو قاعدة e، ويحب علماء الكمبيوتر حقًا أن يكون لديهم سجل ذو قاعدة 2، وهو قد يكون لأغراض حسابية أو أيضًا للتفكير في كيفية نمو هذه الأشياء إذا كان لديك سجل واحد، وإذا كنت قادرًا على حساب نوع واحد من السجلات، سواء كان هذا هو الأساس 10 أو الأساس 2 أو الأساس e، فيمكنك حساب أي شيء آخر تقريبًا تريد الآن توجيه حدسنا في هذا الاتجاه، فلنعد إلى اختبارنا وننتقل إلى السؤال التالي وأعتقد أن هذا السؤال هو الأكثر، لا أعرف، هذا سؤال معقول إلى حد ما، يجب أن يكون لطيفًا هذا سيجعلنا مستعدين للترجمة من سياق الأساس 2 إلى سياق الأساس 10، كما أنه حدس جيد لفهم قوى العدد 2 أن يكون لدينا بشكل عام العلاقة التي تربطها بقوى العدد 10 لأنه هذا النوع الجميل من المصادفة بطبيعة الحال، هذين النوعين من الأشياء جيدان، ستفهم ما أعنيه، إنهما يلعبان بشكل جيد مع بعضهما البعض، لذلك يطرح سؤالنا، نظرًا لحقيقة أن 2 أس 10 يساوي 1024، 1024، وهو ما يعادل 1000 تقريبًا، لذا إذا كنت فضفاضة بعض الشيء مع أرقامك وأنت تقوم فقط بإجراء تقديرات تقريبية من 2 إلى 10، في الأساس 1000، أي مما يلي هو الأقرب إلى الصحة؟ السجل 2 من 10 يساوي 0 تقريبًا. ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "ليّن. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "ليس على الإطلاق قرار بالإجماع هنا. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "لكن السؤال كان يدور حول أيهما أقرب إلى الحقيقة، ودعونا نرى كيف يمكننا التفكير في هذا الأمر. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "لذا فهو يشير إلى أن لديك قوة 2، وهي 1024، وهي قريبة جدًا من قوة 10، حوالي 10 مكعبة. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "فماذا يعني هذا؟ إذا كان سجل 10 للأساس 2 يساوي x، فهذا هو نفس قول 2 أس x يساوي 10، أليس كذلك؟ إنها تطلب منا 2 إلى ما يساوي 10. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "لا يمكنك فعل ذلك مع كل وظيفة. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "يبدو أن الناس يعتقدون أنه يمكنك القيام بذلك باستخدام أي وظيفة، لكنك لا تستطيع ذلك. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "وما يعنيه ذلك هو أن x تساوي حوالي 10 أثلاث، حسنًا؟ وهذا رائع، لذا فإن لوغاريتم الأساس 2 من 10 يساوي حوالي 10 أثلاث. ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "حسنًا بما فيه الكفاية، ما رأيناه سابقًا هو أن اللوغاريتم للأساس 2 من 10، يمكننا أيضًا أن نقول اللوغاريتم للأساس 10 من 2 هو 1 فقط على هذا المقدار، 1 على x. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "3. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "ولأننا نقوم بالأشياء على السجلات، سأقوم بكتابتها بهذه الطريقة. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "وبالمثل، لوغاريتم المليون للأساس 2، حسنًا، دعونا نرى، إذا كان علينا أن نضرب 2 في نفسه حوالي 10 مرات لنصل إلى ألف، فيجب علينا أن نضربه في نفسه حوالي 20 مرة لنصل إلى مليون. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "إنه أصغر قليلاً ولكن هذا نوع من التقريب الجميل الذي يجب أن تضعه في ذهنك. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20، نقوم بتصغير الحجم بنفس المقدار. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30، نخفض الحجم بنفس المقدار. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "تمام؟ الآن هذا حدس يستحق التذكر. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "سأعطيك وقتًا مفيدًا في هذا الأمر لأنه ليس واضحًا إلا إذا كنت على دراية باللوغاريتمات، ويستحق الأمر التفكير فيه قليلًا. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "شكرا لك كارين. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/bengali/sentence_translations.json b/2020/ldm-logarithms/bengali/sentence_translations.json index dab9a4773..546a4688b 100644 --- a/2020/ldm-logarithms/bengali/sentence_translations.json +++ b/2020/ldm-logarithms/bengali/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Music🎵 লকডাউন গণিতে আবার স্বাগতম।আজ আমরা লগারিদম সম্পর্কে কথা বলতে যাচ্ছি এবং পাঠের বেসিক সাজানোর জন্য এক ধরণের ফিরে আসছি।", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "এবং বরাবরের মতো, জিনিসগুলি বন্ধ করার জন্য, আমি কেবল এই মুহূর্তে দর্শকরা কোথায় রয়েছে তা বুঝতে চাই।", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "সুতরাং, আপনি 3b1b যেতে পারেন. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "ক আমি তাদের সম্পর্কে আগে কখনও শুনিনি বা তাদের সম্পর্কে আগে কখনও শিখিনি।", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "আমি তাদের সম্পর্কে শিখেছি কিন্তু কখনও কখনও সমস্ত বৈশিষ্ট্য দ্বারা বিভ্রান্ত হয় গ. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "আমি তাদের বুঝতে পারি কিন্তু কিভাবে তাদের শেখাতে হবে জানি না এবং ঘ. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "আমি তাদের ভালোভাবে বুঝি এবং তাদেরও ভালোভাবে বোঝার জন্য অন্য কারো কাছে তাদের শেখাতে পারি।", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "সুতরাং, আমরা একটি ভাল বিভক্ত আছে. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "যেমন আমি বলেছি, এর উদ্দেশ্য হল একটি পাঠ তৈরি করা যা আমি ভবিষ্যতে লোকেদের নির্দেশ করতে পারি যদি তারা লগারিদমগুলির সাথে স্বাচ্ছন্দ্যবোধ না করে এবং আমি বলতে সক্ষম হতে চাই, ওহ, এখানে এমন একটি জায়গা যেখানে আপনি যেতে পারেন আমি কীভাবে ভাবি, আপনি জানেন, আমি কীভাবে মনে করি আপনি স্বজ্ঞাতভাবে এটির কাছে যেতে পারেন।", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "কারণ এই বিশেষ বক্তৃতাটি করার আগে আমি কয়েকটি শিক্ষক ফোরামের চারপাশে স্ক্রোল করছিলাম এবং যখন লোকেরা জিজ্ঞাসা করে যে উচ্চ বিদ্যালয়ের গণিতে পড়াতে সবচেয়ে কঠিন বিষয় কোনটি এই অর্থে যে শিক্ষার্থীরা এতে সবচেয়ে বেশি সমস্যায় পড়েছে বলে মনে হয়, লগারিদম সবচেয়ে বেশি একটি।সাধারণভাবে নির্দেশিত উত্তরগুলি যা আকর্ষণীয় এবং আমি অনুমান করতে পারি সম্ভবত এটি কারণ এই বৈশিষ্ট্যগুলির একটি টন রয়েছে যা আপনাকে শিখতে হবে যা আপনাকে জানতে হবে, তাই আমরা যেখানে যেতে যাচ্ছি তার থেকে যদি আমরা এড়িয়ে যাই তবে আপনি এই সমস্ত গাদা পেয়ে যাবেন নিয়মগুলি যেগুলি কেবল বীজগণিতের একগুচ্ছের মতো দেখায় যা মনে রাখা কঠিন এবং আপনার মাথায় জিনিসগুলি মিশ্রিত করা সহজ এবং আমি মনে করি যখন লোকেরা হাই স্কুলের গণিত কেমন ছিল এবং কী ছিল সে সম্পর্কে এই ধরণের দুঃস্বপ্নের স্মৃতি মনে করে লগারিদম তাদের জন্য করেছে, এটি প্রায়শই সেই নির্দিষ্ট সূত্রগুলি মনে আসে এবং আমি আজ যা করতে চাই তা হল একটির মাধ্যমে কথা বলার চেষ্টা, কীভাবে সেগুলি সম্পর্কে ভাবতে হয় তবে আপনি যদি কাউকে বীজগণিত শেখান তবে মেটা স্তরের উপরও পয়েন্ট জোর মূল্য? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "তাদের অন্তর্দৃষ্টিতে এটি তৈরি করার উপায় কী? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "ওহ, এটিতে 3টি শূন্য রয়েছে এক মিলিয়নের লগ কী? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "1000 বার x এর লগ x এর লগের 3 গুণের সমান এবং মনে রাখবেন যে আমরা কনভেনশনটি ব্যবহার করছি যে এটি বেস 10 লগ b।", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "1000 গুণের লগ x সমান x ঘনক c এর লগ।", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "1000 গুণ x এর লগ x এবং e এর লগের ঘাত 3 এর সমান।", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "উপরের কোনটিই নয় এবং মনে রাখবেন যেমন আমি আগে বলেছিলাম আমাদের সম্পূর্ণরূপে আশা করা উচিত যে শুরুতে যারা বলেছিল যে তারা লগগুলি ভালভাবে বোঝে তারা অবিলম্বে উত্তর দিতে চলেছে, তারা সঠিকভাবে উত্তর দেবে কিন্তু যদি আপনি যে কেউ তা করে না, আপনি যখন এইরকম একটি সমস্যা দেখছেন তখন আপনাকে ভয় দেখাতে দেবেন না আমি আপনাকে যা করতে উত্সাহিত করব তা হল 10 এর বিভিন্ন ক্ষমতা প্লাগ ইন করুন এবং ধারণার পরিপ্রেক্ষিতে চিন্তা করুন যে লগ ফাংশন শূন্যের সংখ্যা গণনা করে তাই আমি আপনাকে এটি সম্পর্কে চিন্তা করার জন্য একটু মুহূর্ত দেব যাতে আমি এগিয়ে যাই এবং এটিকে গ্রেড করতে পারি এবং সবসময়ের মতো যদি এটি আপনার স্বাচ্ছন্দ্যের চেয়ে দ্রুত হয় তবে এটি শুধুমাত্র কারণ আমি এগিয়ে যেতে চাই পাঠের সাথে তাই এই ক্ষেত্রে সঠিক উত্তরটি 1000 বার x এর লগ হিসাবে বেরিয়ে আসে এবং 3 প্লাস x এর লগ নেওয়ার সমান এবং এখন আসুন এক মুহুর্তের জন্য এটি সম্পর্কে চিন্তা করি এবং যেমন আমি বলেছিলাম আপনি যখন শুরু করছেন তাদের সাথে আমি মনে করি সবচেয়ে ভালো জিনিস হল বিভিন্ন নম্বরে প্লাগ করা আরামদায়ক হওয়া এবং প্লাগ-ইন করার জন্য সেরা নম্বরগুলি হল সেইগুলি যেগুলি ইতিমধ্যেই 10 এর শক্তি, তাই আপনি যদি 1000 বার x এর লগের মতো কিছু জিজ্ঞাসা করেন তবে আমি তা করব না জানি না, আসুন 1000 গুণ 100 এর x লগের জন্য কিছু প্লাগ ইন করা যাক ভাল আমরা জানি এখানে চূড়ান্ত উত্তরে কতটি শূন্য থাকবে 1000 গুণ 100 হল 100,000 আমাদের ইতিমধ্যেই স্বজ্ঞাতভাবে এই ধারণাটি রয়েছে যে যখন আমরা 10 এর 2টি গুন করি আমরা শুধু শূন্য নিচ্ছি, সেই 1000 থেকে 3টি শূন্য এবং সেই 100 থেকে 2টি শূন্য এবং আমরা তাদের একে অপরের পাশে রাখছি তাই এটি মোট 5টি শূন্য হওয়া উচিত কিন্তু আপনি যদি সত্যিই প্রতিফলিত না হন তবে সংখ্যাটি কীভাবে পরিণত হয়েছে আউট কিন্তু কেন এটা এইভাবে পরিণত হল যে 1000 থেকে 3টি শূন্য এবং সেই 100 থেকে 2টি শূন্য যা আমরা 1000-এ শূন্যের সংখ্যা এবং 100-এ শূন্যের সংখ্যা বলেও লিখতে পারি তাই এই ধারণা যে একটি লগারিদম দুটি জিনিসের গুণফল হল 10 এর ক্ষমতার প্রেক্ষাপটে সেই দুটি জিনিসের লগারিদমের যোগফল যা কেবল যোগাযোগ করে যা ইতিমধ্যেই আমাদের অনেকের জন্য একটি অতি স্বজ্ঞাত ধারণা যদি আপনি 10 এর 2টি গুন নেন এবং আপনি তাদের গুণ করেন তাদের সমস্ত শূন্য এবং একধরনের ক্র্যাম একে অপরের সাথে নিয়ে যান যাতে আমি এখানে যেভাবে জিনিসগুলি লিখেছি এটি আসলে কিছুটা সাধারণ সত্যের ইঙ্গিত দেয় যা আমাদের লগারিদমের প্রথম বৈশিষ্ট্য হতে চলেছে যা হ'ল যদি আমরা গ্রহণ করি A বার B এর লগ এটি A এর লগের সাথে B এর লগের সমান হয় এখন আপনি এই লগারিদমের নিয়মগুলির একটি দেখতে পেলে আপনি যদি নিজের চোখ কুঁচকে দেখতে পান বা আপনি এটিকে কীভাবে মনে রাখবেন তা নিয়ে আপনি কিছুটা বিভ্রান্ত হন শুধু উদাহরণগুলিতে প্লাগ করুন আমি অপ্রয়োজনীয় হচ্ছি, আমি এটি অনেক বলছি কিন্তু কারণ আমি মনে করি একবার আপনি বীজগণিতের মধ্যে ডুবে গেলে ভুলে যাওয়া খুব সহজ এবং আপনি এক ধরণের পরীক্ষায় বসে আছেন এবং এটিতে অনেকগুলি প্রতীক রয়েছে নিজেকে মনে করিয়ে দেওয়ার জন্য আপনি ঠিক আছে শুধু কিছু নম্বর প্লাগ করতে পারেন যেটি করা একটি চমৎকার জিনিস এবং প্রায়শই এটি অন্তর্দৃষ্টি অর্জনের একটি দুর্দান্ত উপায় তাই এই ক্ষেত্রে, A টাইমস B এর লগ বলা এবং এটিকে ভেঙে ফেলা আমরা শুধু ভাবতে পারি, ওহ, যে 100 গুণ 1000 এর লগ যা হল 5, এতে 5টি শূন্য রয়েছে প্রতিটি প্রদত্ত অংশে শূন্যের সংখ্যার পরিপ্রেক্ষিতে বিভক্ত হয় দুর্দান্ত, দুর্দান্ত তাই সেই অন্তর্দৃষ্টিকে আরও বহন করে চলুন আরেকটি অনুশীলন সমস্যা চেষ্টা করি এবং আবার, যদি আপনি এটি জানেন, দুর্দান্ত, আপনি এটির উত্তর দিতে সক্ষম হবেন তবে হয়ত ভাবুন, উত্তরটি কী তা নয় তবে আমি কীভাবে কাউকে এই উত্তরটি ব্যাখ্যা করব বা আমি কীভাবে একজন শিক্ষার্থীকে আমাকে না বলেই এই উত্তরে আসতে চেষ্টা করব তাদের উত্তর কি তাই সেখানে দুজন সম্ভাব্য শ্রোতা সদস্য আছেন যারা পাঠে আগ্রহী এবং তারপরে যারা মেটা পাঠে আগ্রহী তাই আমাদের প্রশ্ন আবার জিজ্ঞাসা করে, নিচের কোনটি সত্য? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "ক n থেকে x এর লগ x b এর n গুন লগের সমান।", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "n থেকে x-এর লগ n c-এর পাওয়ার x-এর লগের সমান।", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "n থেকে x এর লগ x বা d এর n যোগ লগের সমান।", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "তাই এখানে সঠিক উত্তরটি হল a, যা দেখে মনে হচ্ছে আপনার মধ্যে 4,000 জন অভিনন্দন পেয়েছেন, আমাদের বলছেন যে x এর লগ n পাওয়ার n x এর n গুন লগের সমান তাই, আবার বলি যে আপনি এটি শেখানোর চেষ্টা করছেন কারো কাছে বা আপনি যদি নিজেকে বোঝাতে চাচ্ছেন তাহলে আমি মনে করি শুরু করার জন্য একটি ভালো জায়গা হল কিছু প্লাগ ইন করা এবং এই ক্ষেত্রে, x-এর লগের জন্য পাওয়ার n আসুন শুধুমাত্র 100-এর পাওয়ার দিয়ে চেষ্টা করি 3 এবং আপনি যে নিদর্শনগুলি করছেন তা আসলে কাজ করে কিনা তা দেখার জন্য আপনি অন্যদের সাথে এটি চেষ্টা করতে পারেন তবে আপনি যদি উত্তরটি কী তা কেবল দেখার পরিপ্রেক্ষিতে না ভেবে উত্তরটি এমনভাবে পরিণত হয়েছিল কেন তা ভাবার চেষ্টা করছেন।কখনও কখনও একটি উদাহরণ কাজ করবে কারণ 100 ঘনক, আমরা এটিকে ভালভাবে নেওয়া হিসাবে ভাবতে পারি, এটি 100 এর 3 কপি আমি 100 এর 3 কপি নিচ্ছি এবং যখন আমি এটিকে গুণ করি এবং আমি লগকে শূন্যের সংখ্যা হিসাবে গণনা করি বলুন, ওহ, এটি এমন কিছু সংখ্যা হতে চলেছে যেটিতে মাত্র 6টি শূন্য রয়েছে যার অর্থ 100 গুণ 100 গুণ 100 নেওয়ার অর্থ আমি কেবল সেই সমস্ত শূন্যগুলিকে একত্রিত করে এক মিলিয়ন পেতে চিন্তা করতে পারি তাই এই সংখ্যাটি হতে চলেছে 6 কিন্তু আমরা যদি ভাবি আসলে কেন এটা 6 ছিল না শুধুমাত্র মিলিয়নের মধ্যে শূন্যের সংখ্যা যেখান থেকে 6 এসেছে তা হল আমাদের কাছে সেই 100টির 3টি কপি ছিল এবং সেই 100 টির প্রতিটিতে 2টি ভিন্ন শূন্য ছিল তাই এটি আরও সাধারণ আপনি এটি সম্পর্কে চিন্তা করতে পারেন যেখানে 100 ঘনক নেওয়ার পরিবর্তে যদি আমরা 1000 ঘনক বা 1000 কে n বা x শক্তি n এর দিকে তাকাতাম তাহলে আপনি ভাবতে পারেন যে n এর মান যা-ই হোক না কেন আমরা কপির সংখ্যা গুণ করছি।কূপের সংখ্যা, চলুন দেখি, এটা x এর শূন্যের সংখ্যার x গুণ নয় যা আমরা x এর প্রতিস্থাপিত করেছি যা এই ক্ষেত্রে 100 ছিল তাই এর পরিবর্তে আমি যদি 10,000 এর লগের মতো কিছু নিয়ে যেতাম তবে এটি একই হবে সেই 10,000-এর n কপি নেওয়ার সময় তাদের প্রতিটিতে শূন্যের সংখ্যা গণনা করা হয় যা 4 তাই এটি হবে n গুণ 4 এবং অবশ্যই সাধারণ সম্পত্তি যা আপনার বেশিরভাগই সঠিকভাবে উত্তর দিয়েছেন তা হল আপনার এই সুন্দর সামান্য প্রভাব রয়েছে যেখানে আপনি যখন একটি শক্তিতে উত্থাপিত কিছুর লগ দেখুন যে সামান্য শক্তি এটির সামনে নেমে আসে এবং আপনার ভিতরে যা ছিল তার লগ আছে এখন সম্ভবত সবচেয়ে গুরুত্বপূর্ণ প্রভাবগুলির মধ্যে একটি যা আমি জানি না আপনি এটিকে কল করবেন কিনা একটি অন্তর্নিহিততা বা যদি আপনি এটিকে সংজ্ঞাটির পুনঃবিবৃতি বলবেন যদি আমি লগ নিচ্ছি এবং আমি কেবলমাত্র 10 এর 10 এর শক্তির উপর আবার জোর দেবো n আমরা সেই সামান্য n কে নিচের দিকে ঝাপিয়ে পড়ার মতো ভাবতে পারি সামনে এবং এটি 10 এর লগ বেস 10 এর n বার হয়ে যায় যা অবশ্যই 1 এই অভিব্যক্তিটি আপনি মনে করতে পারেন হয় শেষে শূন্যের সংখ্যা গণনা করা বা আরও সাধারণভাবে এটি 10 কে জিজ্ঞাসা করছে যা 10 এর সমান এবং উত্তরটি কেবল 1 যা খুবই আশ্বস্ত কারণ আরেকটি উপায় যে আপনি ফিরে যেতে পারেন এবং এই মূল অভিব্যক্তিটি পড়তে পারেন তা হল 10 কে 10 যা 10 এর সমান n ওহ আচ্ছা উত্তরটি এখন ঠিক আছে প্রতিটি প্রদত্ত লগারিদম বৈশিষ্ট্যের সাথে যা আমাদের আছে তাই এই ক্ষেত্রে আমরা শুধু পাওয়ার n-এ x এর একটি লগ পাওয়া গেছে যে n সামনে হপিং সবসময় একটি মিরর ইমেজ সূচকীয় সম্পত্তি হতে যাচ্ছে এবং এটি অন্য একটি উপায় যা আমরা নিজেদেরকে এইগুলির জন্য কিছুটা অন্তর্দৃষ্টি পেতে সাহায্য করতে পারি তাই আমাকে কেবল আড়াল করতে দিন কিছু ভবিষ্যত প্রপার্টি যা আমরা এখানে পেতে যাচ্ছি তা লুকানোর চেষ্টা করুন আমরা কোথায় যাচ্ছি তা আমরা এইমাত্র n-এর সামনে কিছু বাড়াতে দেখেছি যা সূচকীয় সম্পত্তির সাথে সামঞ্জস্যপূর্ণ যে যদি আমি x-এ 10 গ্রহণ করি এবং বাড়াই শক্তি n এর পুরো জিনিসটি 10 থেকে n বার x নেওয়ার সমান এবং এটি আমাদেরকে অন্য একটি অন্তর্দৃষ্টিতে নিয়ে যায় যা আপনার লগারিদমের জন্য থাকতে পারে যা তারা এমন ধরণের সূচকের মতো যা ভিতরের বাইরে পরিণত হয়েছে এবং এখানে আমি যা বোঝাতে চাইছি যে জিনিসটি লগের ভিতরে বসে আছে যদি আমি একটি লগ নিচ্ছি তাহলে আপনার মনে করা উচিত যেটি সম্পূর্ণ বাইরের অভিব্যক্তি হিসাবে এমন কিছুর জন্য যা সূচকীয় এই ক্ষেত্রে ভিতরের একটি জিনিসটি x এর সাথে 10 এর সাথে মিলে যায় ফাংশনের আউটপুট যেখানে সম্পূর্ণ জিনিসটি নিজেই একটি লগের সাথে মিলে যায় এখানে ভিতরে ভিতরে যা আছে ঠিক 10 এর এক্সপোনেন্ট কত তাই এখানে আপনি যেখানেই লগ এক্সপ্রেশন দেখতে পাচ্ছেন আপনি ভাবছেন যে ডানদিকে একটি সূচকের ভূমিকা পালন করে সাইড এবং প্রতিবার যখন আপনি একটি এক্সপোনেনশিয়াল দেখেন পুরো 10 থেকে x এক্সপ্রেশনের ডান দিকের পুরো বাইরের উপাদানটি এমন কিছুর সাথে মিলে যায় যা লগগুলির একটির ভিতরে বসে আছে এবং আমরা এই ধারণার উপরে দেখেছি যে আমরা যখন গুণ করছি ভিতরের দিকে যেটা বাইরের দিকে যোগ করছে যদি লগ ধরনের টার্ন এক্সপোনেনশিয়াল ভিতরে থাকে যেটা আমাদের বলছে যে বাইরের দিকে গুন করলে ফাংশনের আউটপুট গুন করাটা ভিতরে যোগ করার সমান কারণ এই লগগুলির প্রতিটি যেমন লগ a এবং log b ডানদিকের অভিব্যক্তিতে x এবং y এর ভূমিকা পালন করছে তাই এর সাথে খেলা চালিয়ে যাওয়া যাক আসুন এর মধ্যে আরও কয়েকটি করি এবং দেখি এই শেষটির জন্য আমরা কতগুলি বৈশিষ্ট্য তৈরি করতে পারি, সূচকগুলিকে পরেরটি নীচে নামানোর বিষয়ে খুব সুন্দর চিন্তাভাবনা এমন কিছু যা তাদের কাছে কিছুটা অদ্ভুত লাগতে পারে যারা লগারিদমের সাথে অগত্যা পরিচিত নয় কিন্তু আবার, এটির জন্য কিছু অন্তর্দৃষ্টি অর্জনের জন্য কিছু সংখ্যা প্লাগ করুন এবং আমরা এটিকে কিছুটা দেব নিচের কোনটি সত্য? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "ঠিক আছে, যদি 10 ঘনক 1000 হয় তাহলে 10 এর সমান 1000 কে 1 তৃতীয়াংশে উন্নীত করলে এখানে সূচকের গুণক বিপরীতটি জড়িত থাকে এবং যেভাবে বের হয় তা হল এটি 1 কে 3 দ্বারা ভাগ করলে মনে হয় এবং যে 3টি 1000 এর লগ বেস 10 এর সাথে মিলে যায় এটি 1000 এর লগ বেস 10 দ্বারা 1 ভাগ করে তাই সাধারণভাবে, আপনি এই একক উদাহরণের উপর ভিত্তি করে অনুমান করতে পারেন যে যখন আমরা ভিতরের অংশের সাথে বেসটি অদলবদল করি তখন এটি 1 ভাগ নেওয়ার সাথে মিলে যায় সেখানে বাইরে কি আছে এবং আবার, আপনি সংশ্লিষ্ট সূচকীয় নিয়মের দিকে তাকানোর পরিপ্রেক্ষিতে এটি ভাবতে পারেন এখন আমার সুন্দর ছোট্ট লগ এবং সূচকের কী হয়েছে? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "বিস্ময়কর তাই, আবার কিছু জিনিস লুকিয়ে রাখি কিছু অন্যান্য বৈশিষ্ট্য যা আমরা এখানে পাব এবং আমি এটিকে একই ক্রমে রাখব যা আমি এখানে আগে রেখেছিলাম আমি ভাবছিলাম যে এটি আগে থেকে লেখা থাকলে আমাকে রাখতে পারে।স্বাভাবিকের চেয়ে কিছুটা পরিষ্কার কিন্তু হয়ত এর সাথে কাগজ কাটার এই অদ্ভুত খেলাটি এলোমেলোভাবে খেলার সাথে জড়িত তাই আমরা এইমাত্র যা পেয়েছি, a এর লগ বেস যদি আপনি সেগুলিকে অদলবদল করেন তবে এটি 1 দ্বারা ভাগ করার সমান সূচকীয় ভূমি হল যদি আপনি b কে কিছু শক্তিতে নিয়ে যান এবং বলেন যে এটি a এর সমান যে একই বিবৃতিটি বলছে যে a থেকে সেই শক্তির বিপরীতে আবার b সমান হয়, এটি একটি মুহূর্ত সময় নেওয়া এবং লগারিদমগুলিকে ঘোরানো জিনিস হিসাবে ভাবতে সহায়ক।a এর এক্সপ্রেশন লগ বেস b এর ভিতরে x এর ভূমিকা পালন করছে এবং b এর এক্সপ্রেশন লগ বেস a এর ভূমিকা পালন করছে যা a এর উপরে বসে এবং তারপরে প্রতিসমভাবে, x এর পাওয়ার x এর পুরো এক্সপ্রেশন b খেলছে বাম দিকে ভিতরের ভূমিকা, এটি a এবং সম্পূর্ণ অভিব্যক্তির ভূমিকা পালন করে, a থেকে কিছুর শক্তি ভূমিকা পালন করে যা লগ বেসের ভিতরে বসে আছে a যাতে আপনি কিছু উদাহরণ প্লাগ করে দেখতে পারেন এবং সূচকীয় নিয়মের সাথে মিল রেখে আমরা ইতিমধ্যেই তিনটি ভিন্ন লগারিদম নিয়মের মাধ্যমে চিন্তা করতে পারি যেগুলি যদি আপনি জানেন যে বীজগণিতের টুকরোগুলি মুখস্থ করার জন্য সেগুলি হস্তান্তর করা হয় তবে আপনি সেগুলি মুখস্থ করতে পারতেন তবে তাদের পক্ষে আপনার থেকে সরে যাওয়া খুব সহজ।মাথা এবং হাতের কাজ দেখে হতাশ হওয়াও খুব সহজ তবে আপনি নিজেকে মনে করিয়ে দিতে চাইতে পারেন যে আমরা এই ধরণের জিনিসগুলির বিষয়ে যত্নশীল হওয়ার কারণ হল লগারিদমের নিয়মগুলি বোঝা আমাদেরকে সেই প্রসঙ্গে গণিত করতে সাহায্য করে যেখানে এটি একটি ভাইরাসের মতো যেখানে বৃদ্ধি পাচ্ছে একদিন থেকে পরের দিন, এক ধাপ থেকে পরের দিকে, জিনিসগুলি গুণগতভাবে বৃদ্ধি পেতে থাকে লগারিদমের নিয়মগুলি বোঝার ফলে আপনি এই ধরণের জিনিসগুলির জন্য আরও ভাল অনুভূতি পেতে সাহায্য করে তাই আমরা এটি দেখতে কী হতে পারে তার একটি সুন্দর বাস্তব বিশ্বের উদাহরণ করার আগে এই শিরায় আমাকে আরও একটি কুইজ প্রশ্ন করতে দিন লগারিদমের বৈশিষ্ট্যগুলি সম্পর্কে জিজ্ঞাসা করার আগে আমরা একটি বাস্তব জগতের উদাহরণের কিছুটা রূপান্তর করার আগে আমাদের এখানে এবং এখন যা ছিল তা থেকে পরিত্রাণ পেতে নিচের কোনটি সত্য? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "a প্লাস b এর লগ a প্লাস b এর b লগের লগের সমান a প্লাস b এর b লগের লগের সমান a plus b এর b লগের লগ সমান b এর a প্লাস লগের লগ দ্বারা ভাগ করা হয় বা a প্লাস b এর লগ সমান হয় b এর লগের লগ দ্বারা ভাগ করলে বা উপরের ah এর কোনটিই নয়, এবং এখন আমাদের এতটা ঐক্যমত নেই, আমরা কি? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "খুব মজার, আমরা দুজনের মধ্যে ঘোড়ার দৌড় পেয়েছি তাই লোকেরা উত্তর দেওয়ার সময় আমি আপনাকে এই বিষয়ে চিন্তা করার জন্য একটি মুহূর্ত দেব, আসলে আমার শ্রোতাদের জন্য একটি ছোট প্রশ্ন আছে, আপনি জানেন, আমি শুধু কথা বলছিলাম আমরা কীভাবে গুণগত বৃদ্ধির পরিপ্রেক্ষিতে চিন্তা করুন এবং এটি কেবল দশের শক্তি হতে হবে না, আমরা তিনের শক্তির মতো কিছুও করতে পারি যেখানে আপনি যদি এক থেকে তিন থেকে নয় থেকে সাতাশ থেকে আশির দিকে যাচ্ছেন, সব এর মধ্যে আমরা বলতে পারি যে এই সংখ্যাগুলির তিনটির লগ বেসটি কেবলমাত্র ছোট ছোট ধাপে বৃদ্ধি পায় তাই লগ বেস তিনটির একটি, তিনটি যা একটি সমান, উত্তরটি শূন্য সাধারণভাবে একটির লগ, বেস যাই হোক না কেন, হবে তিনের মধ্যে শূন্য লগ বেস তিন, তিনের সমান তিনের সমান একইভাবে লগ বেস তিনের নয়টি দুই আহ, আপনি হয়তো ভাবতে পারেন যে আমার প্রশ্নটি কী, কিন্তু এটি আমার নিজের আনন্দের জন্য এই সবগুলি আঁকতে সাহায্য করবে এখানে, আমি আরও একটি লগ বেস লিখতে দিই, এখন আশি-এর মধ্যে তিন হল চার, আমি শুনেছি যে আপনি যদি কোনও শিশুকে জিজ্ঞাসা করেন, আসুন প্রায় পাঁচ বা ছয় বছর বয়সী বলি যে সংখ্যাটি এক থেকে নয়টির মধ্যে অর্ধেক হবে? বলুন কোন সংখ্যাটি কীভাবে উত্তর দিতে হবে তার জন্য তাদের সহজাত প্রবৃত্তি লগারিদমিক যেখানে আমাদের প্রবৃত্তিগুলি আরও রৈখিক হতে থাকে তাই আমরা প্রায়শই এক এবং নয়টি মনে করি, আপনি তাদের মধ্যে দুটি, তিন, চার, পাঁচ, ছয়টি সমানভাবে ব্যবধানযুক্ত সংখ্যার একটি গুচ্ছ পেয়েছেন , সাত, আট এবং আপনি যদি মাঝখানে ঠিক অর্ধেক পথ যান, আপনি পাঁচটিতে অবতরণ করবেন কিন্তু আপনি যদি গুণগত বৃদ্ধির পরিপ্রেক্ষিতে ভাবছেন যে এক থেকে নয়টি কোথায় পাওয়া যাবে, এটি একটি গুচ্ছ যোগ করার বিষয় নয় কিন্তু আপনি 'একটি নির্দিষ্ট পরিমাণে বৃদ্ধি পাচ্ছেন যা আপনি তিনটির একটি ফ্যাক্টর দ্বারা বৃদ্ধি পাচ্ছেন, তারপরে আপনি তিনটির একটি ফ্যাক্টর দ্বারা বৃদ্ধি পাচ্ছেন বলে অনুমিতভাবে, একটি বাচ্চার স্বাভাবিক প্রবৃত্তি তিনটি বলার সাথে লাইন আপ করে এবং অনুমিতভাবে এটিও যদি আপনার নৃতত্ত্ববিদরা অধ্যয়নরত সমাজগুলি নিয়ে থাকে।' আধুনিক সমাজে যেভাবে অ্যাকাউন্টিং সিস্টেম এবং লেখার বিকাশ করা হয়েছে সেভাবে তারা এর জন্য তিনটি উত্তর দেবে তাই, দর্শকদের জন্য আমার প্রশ্ন যদি আপনি এখনই দেখছেন তাদের মধ্যে যদি একটি ছোট শিশুর অ্যাক্সেস থাকে তবে ধরা যাক, পাঁচ বছরের মধ্যে বৃদ্ধ দেখুন যদি আপনি যেতে পারেন তাদের জিজ্ঞাসা করতে পারেন কোন সংখ্যাটি এক থেকে নয়টির মধ্যে অর্ধেক পথ এবং আপনি যদি পারেন তবে আমাদের টুইটারে জানান যে শিশুটি কী বলে তাদের আসল উত্তর কী কারণ আমি জানি না কেন, আমি সামান্যই এটা বাস্তবে বাস্তবে ঘটবে কিনা তা নিয়ে সন্দিহান আমি বুঝতে পারি এটি করার জন্য এটি একটি সুপার বৈজ্ঞানিক উপায় নয় আমি YouTube লাইভস্ট্রিম দেখার লোকদের তাদের নিজের সন্তানদের জরিপ করতে এবং তারপর উত্তরটি টুইট করতে বলছি না কিন্তু আমার নিজের স্বার্থে এটি আকর্ষণীয় হবে আমাদের প্রশ্নে কিছু ধরণের বৈধতা দেখতে এটিই প্রথম যেটির এক দিক থেকে বিশাল ঐকমত্য আছে বলে মনে হয় না আসুন আমরা এগিয়ে যাই এবং এটিকে গ্রেড করে দেখি উত্তরটি কী দুর্দান্ত হতে চলেছে, ঠিক আছে, তাই 2,400 আপনি সঠিকভাবে উত্তর দিয়েছেন যে উপরের কোনটি নয় যে a প্লাস b এর লগ এই সুন্দর বৈশিষ্ট্যগুলির কোনটিকেই সন্তুষ্ট করে না এবং সাধারণভাবে, যদি না আমরা নির্দিষ্ট ধরণের আনুমানিকতার সাথে কাজ করতে যাচ্ছি বিশেষ করে যখন প্রাকৃতিক লগ কার্যকর হয় আমরা হয়তো পরের বার লগারিদমের ইনপুট যোগ করার বিষয়টি নিয়ে কথা বলতে পারি এটি আসলে একটি খুব অদ্ভুত অনুভূতি এটি করা একটি খুব অদ্ভুত জিনিস এবং সেই অদ্ভুততা বোঝার জন্য, যদি আমি আপনাকে একটি প্লাস বি লগ করতে বলি তাহলে দশের কিছু শক্তি যোগ করুন আপনি কি ভাবতে শুরু করতে পারেন, ঠিক আছে, আমাকে 10,000 এবং 100 এর মত কিছু উদাহরণ যোগ করতে দিন এবং আমি নিজেকে জিজ্ঞাসা করি, যদি আমি এই শূন্য গণনা ফাংশনটি করি তাহলে সেই ইনপুটে কী আছে তাতে কতটি শূন্য আছে? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "এটি একটি আকর্ষণীয় প্রশ্ন ঠিক আছে, লগারিদমের ভিত্তি কি শূন্য হতে পারে? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "ঠিক আছে আমাদের ত্রিভুজের পরিপ্রেক্ষিতে আমরা ভাবতে পারি যে আপনি জানেন যে, শূন্য থেকে কিছু শক্তি x সমান অন্য কিছু মানের y এটি এমন কিছু যা আমরা শূন্যকে x এর সমান y বলে লিখতে পারি বা আমরা লিখতে পারি y এর লগ বেস শূন্য বলার দ্বারা একই জিনিস x শূন্যের সমান যা x এর সমান এখন এখানে সমস্যাটি হল যে শূন্য থেকে যেকোনো কিছু শূন্যের অধিকারে শেষ হয়, তাই যদি আমরা কেবল লগ বেস শূন্যের কথা ভাবি y অন্য কোন ইনপুটের জন্য y আপনি জানেন, আপনি এক বা দুই বা পাই এর মতো কিছু ইনপুট করতে চান যা আপনি চাইতে পারেন, আপনি প্রশ্ন শূন্য জিজ্ঞাসা করছেন যা এক বা দুই বা পাই বা আপনার কাছে যে সংখ্যা থাকতে পারে এবং একটি উত্তর হবে না তাই সর্বোত্তমভাবে আপনি বলার চেষ্টা করতে পারেন ওহ হ্যাঁ, শূন্যের লগ, এটি একটি সম্পূর্ণ বৈধ ফাংশন এটি শুধুমাত্র ইনপুট শূন্যে সংজ্ঞায়িত করা হয়েছে কিন্তু তারপরও আপনি যা চান তা ফিনাগল করতে চেষ্টা করতে সমস্যা হবে সেখানে কারণ শূন্যকে শূন্য বলা যা শূন্যের সমান তা যেকোন কিছুর ক্ষেত্রে প্রযোজ্য তাই আপনার বাহুটি আপনার পিঠের পিছনে বাঁকানো যাচ্ছে তবে আপনি সেই কাজটি করতে চান এবং এটি এই সত্যের সাথে মিলে যায় যে বেস শূন্যের সাথে সূচকীয় ফাংশনটি সম্পূর্ণরূপে শূন্য।একে অপরের সাথে একটি সুন্দর এক থেকে এক ফ্যাশনে সংখ্যাগুলি ম্যাপ করে না তাই এটি একটি দুর্দান্ত প্রশ্ন, আপনি কি এখন একটি লগ বেস শূন্য থাকতে পারেন যে এই জিনিসগুলি বাস্তব জগতে কোথায় আসে সেই ধারণায় ফিরে যেতে পারে একটি উদাহরণ যা আমি পছন্দ করি ভূমিকম্পের জন্য রিখটার স্কেল তাই রিখটার স্কেল আমাদের একটি ভূমিকম্প কতটা শক্তিশালী তার পরিমাপ দেয় এবং এটি খুব ছোট সংখ্যা থেকে খুব বড় সংখ্যা পর্যন্ত যেকোনো কিছু হতে পারে যেমন আমি মনে করি এখন পর্যন্ত পরিমাপ করা সবচেয়ে বড় ভূমিকম্প এবং এটি শুধুমাত্র একটি চার্ট যা থেকে এসেছে উইকিপিডিয়া ছিল একটি 9.5 এবং এই সংখ্যাগুলির অর্থ কী এবং তারপরে TNT এর সমতুল্য পরিমাণের মতো কিছু এর মধ্যে কতটা শক্তি রয়েছে তার একটি পরিমাপ এবং তারপরে আমরা এখানে কী করার চেষ্টা করতে পারি তার মধ্যে সম্পর্কটি দেখার জন্য এটি কতটা উন্মাদনাপূর্ণ তা উপলব্ধি করার জন্য আমরা শক্তির পরিমাণের পরিপ্রেক্ষিতে রিখটার স্কেল সংখ্যার জন্য একটি অভিব্যক্তি পেতে পারি কিনা এবং কেন লগারিদমগুলি এটি বর্ণনা করার জন্য একটি প্রাকৃতিক উপায় হবে তা দেখতে হবে, তাই ফোকাস করার চাবিকাঠি হল আমরা সামনের দিকে পদক্ষেপ নিচ্ছি যে জিনিসগুলি কতটা বৃদ্ধি পায় সুতরাং উদাহরণস্বরূপ, এই ক্ষেত্রে যদি আমরা দুটি ভাল থেকে যাই তবে এটি আমাদের দেখায় না যে তিনটি কোথায় তাই হয়তো আমরা দুই থেকে চার পর্যন্ত একটি পদক্ষেপ নেওয়ার কথা ভাবি যা একধরনের দুটি পদক্ষেপ নেওয়ার মতো।শক্তির পরিমাণ ভাল মনে হচ্ছে এটি আমাদের এক মেট্রিক টন TNT থেকে নেয় যা আমি দ্বিতীয় বিশ্বযুদ্ধের একটি বড় বোমা অনুমান করি এবং এটি আমাদেরকে এক হাজার গুণ বেশি কিলোটন পর্যন্ত নিয়ে যায় যা একটি ছোট অ্যাটম বোমা তাই মাত্র দুই ধাপ রিখটার স্কেলে 2 মাত্রার ভূমিকম্প থেকে 4 মাত্রার ভূমিকম্পে যাওয়া আমাদেরকে বৃহৎ বোমা থেকে দ্বিতীয় বিশ্বযুদ্ধ থেকে পারমাণবিক যুগ পর্যন্ত নিয়ে যায় যাতে এটি লক্ষণীয় এবং প্রথম পরিচ্ছন্ন পদক্ষেপ যা আমরা পাই তা হল 4 থেকে 5 এ অন্ততপক্ষে এই চার্টটি আমাদেরকে সুন্দরভাবে দেখায় এবং স্পষ্টতই 4 থেকে 5 পর্যন্ত একটি একক ধাপ 1 কিলোটন থেকে 32 কিলোটনে যাওয়ার সাথে মিলে যায় এবং এটি স্পষ্টতই নাগাসাকিতে অবতরণকারী শহর ধ্বংসকারী বোমাটির আকার ছিল তাই এটি সম্ভবত একটি লগারিদমিক স্কেল সম্পর্কে যে জিনিসটি বিপরীতমুখী হতে পারে যদি আপনি খবরে শুনছেন যে ওহ একটি 4 ছিল একটি ভূমিকম্প হয়েছিল।0 বনাম একটি ভূমিকম্প যা ছিল 5।0 এটা ভাবা সহজ যে হ্যাঁ 4 এবং 5 এগুলি বেশ অনুরূপ সংখ্যা কিন্তু স্পষ্টতই TNT রাশির পরিপ্রেক্ষিতে যা 32 দ্বারা গুন করে 1 থেকে পরের দিকে যেতে এবং 2 থেকে 4 যেতে স্পষ্টতই প্রায় হাজার দ্বারা গুণিত হচ্ছে এবং একমাত্র এর চেয়ে বড় কারণ হল এখানে আমাদের চার্ট দেখাচ্ছিল না 3 কি ছিল তাই আমরা দুটি ধাপ নিচ্ছি এবং আপনি নিজের জন্য যাচাই করতে পারেন যে আপনি যদি 32 এর একটি ধাপ নেন এবং তারপরে আপনি অন্য 32 দ্বারা গুণ করেন যা আসলে এক হাজারের কাছাকাছি তাই রিখটার সংখ্যার সংযোজনমূলক পদক্ষেপগুলি টিএনটি-তে গুণগত পদক্ষেপের সাথে মিলে যায় এমন ধারণা থেকে মনে হয় যে লগারিদমিক কিছু এখানে কাজ করছে এবং এটি এখানে চালিয়ে যাওয়া এবং বলা একটু আকর্ষণীয় যে এটি আংশিকভাবে বিশ্বের ঘটনাগুলির কারণে কতটা বৃদ্ধি পায় হ্যাঁ বর্ণনা করা একটি বিশাল আশ্চর্যের বিষয় নয় যে আমরা যখন আরও একটি পদক্ষেপ নিচ্ছি এটি আবার প্রায় 32 দ্বারা গুণিত হচ্ছে তবে এটি আমাদের অন্তর্দৃষ্টিতে লাগাম দিচ্ছে যে 32 কিলোটন একটি ছোট অ্যাটম বোমা এবং তারপরে একটি মেগাটনের মধ্যে পার্থক্য যাকে আমরা ছোট অ্যাটম বোমা হিসাবে ভাবতে পারি না, নাগাসাকি পরমাণু বোমা যা আমি অনুমান করি এটি একটি মেগাটনের জন্য নাগাসাকি পরমাণু বোমার 32টি যা স্পষ্টতই নেভাদা মার্কিন যুক্তরাষ্ট্রে 1994 সালের ডাবল স্ট্রিং সমতল ভূমিকম্পের মাত্রা ছিল আমি জানতাম না যে এটি কী ছিল, ধন্যবাদ উইকিপিডিয়াকে ফ্রিকোয়েন্সি অনুসারে আমি এগুলিও স্পষ্টতই দেখেছি যেগুলি দুটিরও কম, সেগুলি সর্বদা ঘটছে প্রতিদিন 8000 এর মতো কিন্তু যত তাড়াতাড়ি আমরা 3 এর মতো পরমাণু বোমার রাজ্যে আছি।5 এবং 4 এগুলি স্পষ্টতই পৃথিবীতে কোথাও না কোথাও প্রায়ই ঘটে থাকে প্রায় 134 এর মধ্যে প্রতিদিন কোথাও না কোথাও ঘটছে কে জানত? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "কিন্তু যেহেতু আমরা এই 5 এবং 6 রেঞ্জের মধ্যে আরও বেশি তীব্র হয়ে উঠছি যা পারমাণবিক বোমার স্কেলের উপরে ছিল এখন আমরা প্রতিদিন মাত্র 2 এর কাছাকাছি আছি এবং আমি নিশ্চিত যে একজন ভূতাত্ত্বিক এসে ব্যাখ্যা করতে পারেন কেন আমাদের সবার উচিত' এই বিষয়ে খুব চিন্তিত হবেন না যে পৃথিবীর ভূত্বকের জন্য দুটি পরমাণু বোমার সমতুল্য ব্যাঘাত প্রতিদিন ঘটছে তবে সম্ভবত এটি বিশেষত বিরল যে তাদের পক্ষে এমন একটি জায়গায় কেন্দ্রীভূত হওয়া এমন একটি শহরের মতো যেখানে প্রচুর লোক বাস করে এখন আমাদের ধারণা যাচাই করছে যে প্রতিটি পদক্ষেপ 32 এর বৃদ্ধি জড়িত, আসুন দেখি 6 থেকে 7 পর্যন্ত ধাপটি কেমন দেখায় এবং এখানে এটি আমাদেরকে আরও অনেক উদাহরণ দিচ্ছে এর মধ্যে হয়ত এই বিভ্রম দেয় যে এটি আসলে এর চেয়ে একটি বড় পদক্ষেপ এবং প্রকৃতপক্ষে এটি 1 মেগাটন এবং এর মধ্যে পার্থক্য 32 মেগাটন যাতে 32 দ্বারা গুণিত হয় এই চার্টে আমার কাছে সবচেয়ে আকর্ষণীয় জিনিসগুলির মধ্যে একটি ছিল তা হল সবচেয়ে বড় পারমাণবিক অস্ত্রে পৌঁছানোর আগে আমাদের কতদূর যেতে হবে যা আসলেই পরীক্ষা করা হয়েছে এটি ছিল শীতল যুদ্ধের উচ্চতা জার বোমাটি ছিল 50 মেগাটন এবং আমি বিশ্বাস করি যে তাদের প্রকৃতপক্ষে 100 মেগাটন বোমা রাখার মূল পরিকল্পনা ছিল কিন্তু তারা 50 মেগাটন থেকে নিচের কথা বলেছিল, আমরা সেই 32 কিলোটন নাগাসাকি বোমা থেকে শুরু করে 32 দ্বারা গুণ করে মেগাটন আরও 32 দ্বারা গুণিত হয় তাই আমরা দ্বিতীয় বিশ্বযুদ্ধের শেষ বিস্ফোরণের এক হাজার গুণ শক্তির কথা বলছি এবং আপনি এখনও মানবতার সক্ষমতার 50 মেগাটনে নন এবং এটি স্পষ্টতই ইন্দোনেশিয়ার জাভা ভূমিকম্প তাই 7 . 0 শুধুমাত্র 6 এর থেকে একটু বড় নয়।0, এটি অনেক বড় এবং এখানে অবশ্যই পয়েন্টটি হল যে যখন আপনার কাছে একটি স্কেল থাকে যা আপনাকে গুনগত বৃদ্ধি দেয় তখন এটি প্রশংসা করার মতো যে ছোট পদক্ষেপের মতো দেখতে আসলে শক্তির নিরিখে বিশাল পদক্ষেপ হতে পারে বা এখানে নিহিত পরম মান তাই যখন আমরা এই সত্যটি নিয়ে ভাবছি যে 9 ছিল।5 যেটি আসলে অযৌক্তিক বলে মনে হচ্ছে যে এটি শুধুমাত্র 7 এর মধ্যে রয়েছে।0 পরিসর যা আমরা এখন পর্যন্ত রাখা সবচেয়ে বড় থার্মোনিউক্লিয়ার অস্ত্রের কথা বলছি এবং এটি এমন একটি ক্ষেত্রের ইঙ্গিত যেখানে লগারিদমগুলি এটি সম্পর্কে আসে যখন মানুষ এমন কিছুর জন্য একটি স্কেল তৈরি করতে চায় যেটি কত বড় জিনিসগুলি করতে পারে তার মধ্যে বিশাল বিস্তৃত পার্থক্যের জন্য দায়ী তাই ভূমিকম্পের আকারের ক্ষেত্রে পৃথিবীর চারপাশে যা ঘটছে তা থেকে আপনি জিনিসগুলি পেতে পারেন, একটি বড় হ্যান্ড গ্রেনেডের আকার এবং আপনি চান যে এটি আপনার স্কেলে হোক এবং সমস্ত উপায়ে বাড়ানোর বিষয়ে চিন্তা করার মতো কিছু মানব ইতিহাসে আমরা দেখেছি সবচেয়ে বড় ব্যাঘাতের জন্য এবং এটি এমনভাবে করার জন্য যে আপনি শুধুমাত্র একটি ক্ষেত্রের জন্য আপনার সংখ্যায় বিভিন্ন অঙ্কের সম্পূর্ণ গুচ্ছ এবং অন্য একটি ছোট সংখ্যার জন্য সম্পূর্ণ গুচ্ছ লিখছেন না।অন্য ক্ষেত্রে আপনার নম্বরের অঙ্কগুলি লগারিদম নেওয়া ভাল এবং তারপরে কেবলমাত্র একটি একক স্কেলে রাখুন যা মূলত 0 এবং 10 এর মধ্যে সেই সংখ্যাগুলিকে স্কুইশ করে আপনি দেখতে পাচ্ছেন যে সংগীতের জন্য ডেসিবেল স্কেলের সাথে খুব অনুরূপ কিছু চলছে যা আসলে কিছুটা কাজ করে একটু ভিন্নভাবে যেখানে আপনি প্রতিবার 10 ডেসিবেলের একটি ধাপ বাড়ান যা 10 দ্বারা গুণ করার সাথে মিলে যায় তাই 1 এর একটি ধাপ 10 দ্বারা গুন করার পরিবর্তে, এটি 10 এর একটি ধাপ যা 10 দ্বারা গুণ করে তাই এই ধরণের গণিতকে সামান্য করে তোলে বিট স্ক্রু কিন্তু ধারণা একই, আপনি যদি 50 ডেসিবেল বনাম 60 ডেসিবেল শব্দ শুনছেন তবে এটি শক্তি সঞ্চারিত হচ্ছে এবং যেখান থেকে যাচ্ছে তার দিক থেকে এটি অনেক শান্ত, এটি কী হবে, 60 থেকে 70 বা 70 থেকে 80 এই ধাপগুলি, 60 থেকে 80 পর্যন্ত, যার মধ্যে প্রতি বর্গক্ষেত্রে শক্তির পরিমাণকে 100 এর গুণিতক দ্বারা গুণ করা জড়িত তাই আপনি যখনই লগারিদমিক স্কেল দেখবেন, তখন আপনার মনের মধ্যে জেনে রাখুন যে এটি হুডের নীচে যা উল্লেখ করছে তার অর্থ বৃদ্ধি পায়।একটি বিশাল পরিমাণ আবার এই কারণেই আমরা করোনাভাইরাস প্রাদুর্ভাবের বর্ণনা করার জন্য প্রচুর লগারিদমিক স্কেল দেখেছি তাই আপনি কীভাবে এমন একটি সম্পর্ককে বর্ণনা করতে পারেন যেখানে আপনি প্রতিবার রিখটার স্কেল সংখ্যা 1 দ্বারা বৃদ্ধি করার সাথে সাথে আপনি 32 দ্বারা গুণ করছেন, আমরা বেস 32 সহ একটি লগের পরিপ্রেক্ষিতে আমি বলতে পারি যদি আমি এর লগটি নিই, আমি কেবল r কে কল করতে যাচ্ছি, রিখটার স্কেলের জন্য নম্বরটি আমি এটিকে লগ বেস 32 হিসাবে ভাবতে পারি এবং এটি এর সাথে মিলে যাচ্ছে , না না না, আমি এই ভুল করছি যেটি লগ করা জিনিসটি নয় আমরা বড় সংখ্যার লগ বেস 32 নিই, টিএমটি নম্বরের, এমন কিছু যা 1 মেগাটনের মতো ছিল এটি 1 মিলিয়ন টন লগ বেস 32, এটি করা উচিত রিখটার স্কেল নম্বরের সাথে মিল আছে কিন্তু কিছু ধরনের অফসেট থাকতে পারে, তাই আমরা বলতে পারি যে কিছু ধ্রুবক s আছে যা আমরা এই রিখটার স্কেল নম্বরে যোগ করছি এবং এই অভিব্যক্তিটি ঠিক একই, ক্ষমা করবেন নীচে এই অভিব্যক্তিটি কিছু অফসেট বারের শক্তিকে 32 বলার মতোই আমাদের রিখটার স্কেল নম্বর যা সেই অফসেটে 32 নেওয়ার সমান, যা নিজেই কিছু বড় ধ্রুবক, রিখটার স্কেল নম্বরের 32 গুণ তাই আপনি এটিকে আপনি যে সংখ্যাটি দেখছেন তার শক্তির সাথে 32 বার হওয়া মাত্র কিছু ধ্রুবক বলে মনে করতে পারেন তাই এটি লেখার এই পদ্ধতিটি সত্যিই এটির সূচকীয় বৃদ্ধির উপর জোর দেয় যে এটি যদি আপনি দেখেন যে টিএমটি পরিমাণের সাথে সামঞ্জস্যপূর্ণ হয়, যেমন আপনি এটি বাড়াচ্ছেন r ধাপে ধাপে আপনি 32 দ্বারা গুণ করছেন কিন্তু সঠিক একই তথ্য যোগাযোগের আরেকটি উপায় হল লগ বেস 32 গ্রহণ করা যা সেই পরিমাণের ঠিক আছে এখন আমি যে বিষয়ে কথা বলতে চাই তা হল কিভাবে আমাদের সবসময় করতে হবে না বিভিন্ন ঘাঁটির লগগুলি কীভাবে গণনা করা যায় তা নিয়ে চিন্তা করুন এটি এখানে একটু অদ্ভুত যে আমরা লগ বেস 32 সম্পর্কে কথা বলছি, আমি আগে উল্লেখ করেছি যে গণিতবিদরা বেস সহ লগ থাকতে পছন্দ করেন এবং কম্পিউটার বিজ্ঞানীরা বেস 2 এর সাথে লগ রাখতে পছন্দ করেন এবং এটি গণনামূলক উদ্দেশ্যে বা আপনার যদি একটি লগ থাকে তবে এই জিনিসগুলি কীভাবে বৃদ্ধি পায় সে সম্পর্কে চিন্তা করার জন্য, আপনি যদি এক ধরণের লগ গণনা করতে সক্ষম হন, তা বেস 10, বেস 2, বেস ই হোক না কেন আপনি অন্য যেকোন কিছু গণনা করতে পারেন আপনি এখন সেই দিকে আমাদের অন্তর্দৃষ্টি পেতে চান, আসুন আমাদের কুইজে ফিরে যাই এবং পরবর্তী প্রশ্নে যাই এবং আমি বিশ্বাস করি যে এই প্রশ্নটি সবচেয়ে বেশি, আমি জানি না, এটি একটি অর্ধেক যুক্তিসঙ্গত প্রশ্ন, এটি সুন্দর হওয়া উচিত এটি আমাদেরকে বেস 2 প্রসঙ্গ থেকে বেস 10 প্রেক্ষাপটে অনুবাদ করার জন্য প্রস্তুত করতে যাচ্ছে এবং এটি 2 এর ক্ষমতাগুলি বোঝার জন্য একটি ভাল অন্তর্দৃষ্টিও যা সাধারণভাবে 10 এর ক্ষমতার সাথে এর সম্পর্ক রয়েছে কারণ এটি এই সুন্দর ধরণের কাকতালীয় ঘটনা।প্রকৃতি যে এই দুটি ধরণের ভাল আপনি দেখতে পাবেন আমি কি বলতে চাইছি, তারা একে অপরের সাথে সুন্দরভাবে খেলে তাই আমাদের প্রশ্ন জিজ্ঞাসা করে যে 2 থেকে 10 তম হল 1024, 1024, যা প্রায় 1000 তাই যদি আপনি একজন হয়ে থাকেন আপনার সংখ্যার সাথে কিছুটা আলগা এবং আপনি 2 থেকে 10 তম অনুমান করছেন, মূলত 1000, নিচের কোনটি সত্য হওয়ার সবচেয়ে কাছাকাছি? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "টেন্ডার এখানে সর্বসম্মত সিদ্ধান্ত নয়।", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "3333 পুনরাবৃত্তি কিন্তু প্রশ্নটি জিজ্ঞাসা করছিল যে কোনটি সত্য হওয়ার সবচেয়ে কাছাকাছি, এবং আসুন আমরা কীভাবে এটি সম্পর্কে ভাবতে পারি তা দেখা যাক।", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "সুতরাং এটি নির্দেশ করে যে আপনার 2 এর একটি শক্তি আছে, যা 1024, 10 এর শক্তির খুব কাছাকাছি, প্রায় 10 ঘনক।", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "তাহলে এর অর্থ কি? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "10 এর লগ বেস 2 যদি x এর সমান হয়, তাহলে x 2 কে 10 এর সমান বলার মত একই জিনিস, তাই না? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "এটা আমাদের জিজ্ঞাসা করছে 2 থেকে 10 এর সমান।", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "আপনি প্রতিটি ফাংশন সঙ্গে তা করতে পারবেন না. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "লোকেরা মনে করে যে আপনি যে কোনও ফাংশন দিয়ে এটি করতে পারেন তবে আপনি তা করতে পারবেন না।", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "এবং এর মানে কি যে x প্রায় 10 তৃতীয়াংশ, ঠিক আছে? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "এবং যথেষ্ট, আমরা আগে যা দেখেছি তা হল 10 এর লগ বেস 2, আমরা এটাও বলতে পারি যে 2 এর লগ বেস 10 সেই পরিমাণের উপরে 1, x এর উপরে 1।", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "এবং কারণ আমরা লগ এ জিনিষ করছি আমি শুধু যে ভাবে এটা লিখতে যাচ্ছি. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "একইভাবে এক মিলিয়নের লগ বেস 2, আচ্ছা দেখা যাক, হাজারে পৌঁছানোর জন্য যদি আমাদের 2 কে নিজে থেকে প্রায় 10 বার গুণ করতে হয়, তাহলে এক মিলিয়নে উঠতে আমাদের এটিকে প্রায় 20 গুণ করতে হবে।", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "এটি একটি সামান্য বিট ছোট কিন্তু এটি আপনার মনে আছে একটি চমৎকার অনুমান ধরনের. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20, আমরা একই পরিমাণ দ্বারা স্কেল নিচে. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, আমরা একই পরিমাণ দ্বারা স্কেল নিচে. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "ঠিক আছে? ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "এখন এটি মনে রাখার মতো একটি অন্তর্দৃষ্টি।", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "ঠিক আছে? ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "এবং তারপরে শুধুমাত্র একটি সম্পূর্ণ গাদা বিভিন্ন সম্ভাব্য উপায়ে লগ বেস C এর B বার লগ বেস C এর A এর সাথে একত্রিত করা।", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "আমি আপনাকে এই বিষয়ে একটি অর্থপূর্ণ সময় দেব কারণ আপনি লগারিদমের সাথে পরিচিত না হওয়া পর্যন্ত এটি স্পষ্ট নয় এবং এটি একটু চিন্তা করার মতো।", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "ধন্যবাদ কারেন।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/chinese/sentence_translations.json b/2020/ldm-logarithms/chinese/sentence_translations.json index 457b2f05a..b469a3cf7 100644 --- a/2020/ldm-logarithms/chinese/sentence_translations.json +++ b/2020/ldm-logarithms/chinese/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵音乐🎵欢迎回到锁定数 学。", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "今天我们将讨论对数以及回归基础知识的课程。", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "和往常一样，首先，我只想了解一下观众现在的处境。", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "所以，如果你可以去3b1b。", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "我以前从未听说过或从未了解过它们 b. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "我已经了解了它们，但有时对所有属性 c 感到困惑。", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "我理解他们但不知道如何教他们d。", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "我很了解它们，并且可以轻松地将它们教给其他人，让他们也很好地理解。", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "所以，我们有一个很好的分裂。", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "就像我说的，这样做的目的是创造一个教训，如果人们对对数感到不舒服，我可以在将来向他们指出，我 希望能够说，哦，这是一个你可以去的地方我是如何想的，你知道，我认为你可以如何直观地处理它。", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "因为在做这个特别的讲座之前，我浏览了几个教师论坛，当 人们问高中数学中最难教的主题是什么时，学生似乎最难 教这个主题，对数是最难教的主题之一。通常指出的答案 很有趣，我猜也许是因为有大量这些属性你最终必须学习 你知道的，所以如果我们跳到我们要去的地方，你就会得 到所有这些堆这些规则看起来就像一堆代数，很难记住， 而且很容易在你的头脑中混淆，我认为当人们对高中数学 是什么样子以及什么有这样的噩梦般的回忆时对数对他们 来说很重要，通常会想到那些特定的公式，今天我想做的 是尝试讨论一个公式，如何思考它们，但也只是在元层面 上，如果你教某人代数，什么是有哪些值得强调的地方？", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "有什么方法可以让它建立在他们的直觉中？", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "哦，上面有 3 个零，一 百万的对数是多少？", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "1000 乘以 x 的对数等于 3 乘以 x 的对数 ，请记住我们使用的约定是它以 10 为底的对数 b。", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "1000 乘以 x 的对数等于 x 的对数的立方 c. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "1000 乘以 x 的对数等于 3 x 和 e 的 对数次方。", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "以上都不是，请记住，就像我之 前所说的那样，我们应该完全期望所有 一开始就说自己很了解日志的人都会立 即回答，他们会正确回答，但如果您如 果有人不这样做，当你遇到这样的问 题时，不要让这吓倒你，我鼓励你做的 就是插入 10 的各种幂，并根据对 数函数的想法来思考计算零的数量，所 以我会给你一点时间来思考这个问题， 所以我会继续进行评分，并且一如既往 ，如果这比你满意的速度要快，请知道 这只是因为我想继续前进根据课程内容 ，在这种情况下，正确答案是 100 0 乘以 x 的对数，与 3 加上 x 的对数相同，现在让我们考虑一 下，就像我在刚开始时所说的那样对于 它们，我认为最好的办法就是轻松地插 入各种数字，而插入的最佳数字是已 经是 10 的幂的数字，所以如果你 问的是 1000 次的对数 x 之 类的东西，我不知道不知道，让我们代 入 1000 乘以 100 的 x 对数，我们知道最终答案中有多少个 零，1000 乘以 100 是 1 00,000，我们已经直观地知道， 当我们乘以 10 的 2 次方时我 们只是取零，1000 中的 3 个 零，100 中的 2 个零，我们将 它们放在一起，所以总共应该是 5 个零，但如果你真的不仅仅考虑数字是 如何变化的但为什么结果会是这样呢 ？它是 1000 中的 3 个零加 上 100 中的 2 个零，我们也 可以这样写：1000 中零的数量加 上 100 中零的数量，所以这个想 法是一个对数两个数的乘积就是这两个 数在 10 次方的情况下的对数之和 ，这只是传达了对于我们很多人来说已 经是一个超级直观的想法，如果你取 2 个 10 的次方并将它们相乘， 你就可以了取所有的零并将它们相互塞 满，所以我在这里写的方式实际上表明 了一个稍微更普遍的事实，这将是我们 对数的第一个属性，即如果我们取A 乘以 B 的对数，它等于 A 的 对数加上 B 的对数，现在每当您看 到这些对数规则之一时，如果您发现自 己眯着眼睛，或者您对如何记住它有点 困惑，只需插入示例即可我是多余的， 我说了很多次，但这是因为我认为一旦 你沉浸在代数本身中并且你正在进行某 种测试并且它只是有很多符号，那么很 容易忘记提醒自己，你可以插入一些数 字，这是一件好事，而且通常这是产生 直觉的好方法，所以在这种情况下，说 A 乘以 B 的日志并将其分解， 我们可以这样想，哦， 100乘以 1000的对数，即5，其中有5个零 ，根据每个给定部分中零的数量来分解 ，太棒了，太棒了，所以进一步发挥这 种直觉，让我们尝试另一个练习题，然 后再次尝试，如果你知道的话，太棒了 ，你可以很好地回答它，但也许会想， 不仅仅是答案是什么，而是我如何向某 人解释这个答案，或者我如何尝试让学 生自己得出这个答案，而无需我告诉他 们的答案是什么，所以有两个潜在的观 众，一部分人对课程本身感兴趣，另一 部分人对元课程感兴趣，所以我们的问 题再次提出，以下哪一项是正确的？", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "A。", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "x 对 n 的 log 等于 x b 的 log 的 n 倍。", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "x 对 n 的对数等于 x 对 n c 次方的对数。", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "x 与 n 的对数等于 n 加上 x 或 d 的对数 。", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "所以这里的正确答案是a，看起 来你们中有4,000人恭喜 ，告诉我们x的n次方对数等 于x的n倍对数，所以，再一次 ，假设您正在尝试教这个对于 某人，或者如果你想自己理解 它的含义，我认为一个很好的起 点是插入一些东西，在这种情 况下，对于 x 的 log 的 n 次方，让我们尝试使 用 100 次方3、你可以 尝试与其他模式一起尝试，看 看你正在做的模式是否真的有效 ，但如果你仔细思考，不是简 单地看到答案是什么，而是试 图思考为什么答案是这样的有时 一个例子就可以了，因为 1 00 的立方，我们可以认为 这是很好的，这是 100 的 3 个副本，我取 100 的 3 个副本，当我将所 有这些相乘时，我认为对数就是 计算零的数量，我们说，哦， 这将是一个只有 6 个零的 数字，这就是 100 乘以 100 乘以 100 的意 思 我可以想到将所有这些零 组合在一起得到一百万，所以这 个数字将是6 但如果我们实 际上思考为什么是 6，不仅 仅是 6 所来自的百万个零 的数量，而是我们有 3 个 100 个副本，而这 10 0 个中的每一个都有 2 个不同的零，这样它就更通用了 你可以这样想，如果我们不是 取 100 的立方，而是看 1000 的立方或 100 0 的 n 次方或 x 的 n 次方，你可以认为 n 的值就是我们乘以倍数的副本 数井的数量，让我们看看，它 不是 x 乘以我们用 x 代替的任何内容中的零数量，在 本例中是 100，所以如果 我取 10,000 的 l og 的 n 次方，这将是相 同的取这 10,000 个 副本的 n 个副本，计算每 个副本中零的数量，即 4，所 以这将是 n 乘以 4，当 然，你们大多数人正确回答的 一般属性是，当您看到某物的日 志升到了一个幂，在它前面几 乎没有幂跳下来，你就得到了 里面的日志，现在这可能是最重 要的含义之一，我不知道你是 否会称之为它一个暗示，或者 如果你称它为定义的重述，如果 我取对数，我只是重新强调它 是 10 的 n 次方的基 数，我们可以将那个小 n 视为向下跳跃前面，它变成以 10 为底的对数的 n 倍 ，当然是 1 这个表达式你 可以认为是计算末尾零的数量， 或者更一般地说，它要求 1 0 到等于 10 的数，答 案就是 1这是非常令人放心的 ，因为你可以返回并阅读这个 原始表达式的另一种方式是说 10 等于 10 的 n 哦，好吧，现在对于我们拥有 的每个给定的对数属性，答案 是不行的，所以在这种情况下我 们刚刚发现 x 的 n 次 方对数涉及 n 在前面跳跃 ，总是会存在镜像指数属性，这 是我们可以帮助自己对这些有 一点直觉的另一种方式，所以 让我掩盖一下我们将要到达的一 些未来属性尝试隐藏我们刚刚 发现的内容，将某些内容提升 到前面的 n，这对应于指数属 性，如果我将 10 代入 x 并提高整件事的 n 次 幂与 10 的 n 次 x 相同，这让我们对对数有另一 种直觉，它们有点像指数翻转 ，这就是我的意思如果我取 a 的对数，那么位于日志内部 的东西应该将其视为指数级的 东西的整个外部表达式，在这 种情况下，a 内部的东西对应 于 10 到 x函数的输出 ，而整个事情本身 a 的对 数对应于里面的内容，这里只是 10 的指数，所以无论你 在这里看到一个对数表达式， 你都应该认为它扮演了右边指数 的角色每当你看到一个指数时 ，整个 10 到 x 表达 式右侧的整个外部组件对应于位 于其中一个日志内部的东西， 我们在上面看到了这个想法， 当我们相乘时如果对数是从内到 外的转指数，那就告诉我们， 在外部相乘与函数的输出相乘 与在内部相加是一样的，因为每 个对数都像 log a 和 log b在右边的表达式 中扮演 x 和 y 的角色， 所以让我们继续玩，让我们再 做几个这样的操作，看看我们 可以对其中多少属性建立直觉， 所以最后一个，对于那些不一 定熟悉对数的人来说，指数跳 到下一个的想法可能有点奇怪， 但再次插入一些数字以获得一 些直觉，我们将给出一点请稍 等一下，以下哪项是正确的？", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "好吧，如果 10 的立方是 1000，这与说 10 等于 1000 的 1/ 3 是一样的，这里做逆运算 涉及指数的乘法逆元，结果 看起来就像 1 除以 3 3 对应于以 10 为底 的对数 1000，它是 1 除以以 1000 为底 的对数通过外面的情况，你可 以通过查看相应的指数规则 来思考这一点，现在我可爱的 小日志和指数发生了什么？", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "太棒了，所以，再次让我们隐藏我们将到达 这里的一些东西和其他一些属性，我将按照 我之前在这里的顺序保留它，我想预先写好 它可以让我保留比平时干净一点，但也许它 只是涉及玩这种奇怪的剪纸游戏，所以我 们刚刚发现，如果你交换它们，对 a 的 底 b 进行记录，这与除以 1 所对应 的值相同，从 a指数域是指如果你取 b 的某个幂并说它等于 a，这与说 a 的该幂的倒数再次等于 b 是相同的说 法，花点时间将对数视为转动事物是有帮助 的从内到外，a 的对数底 b 的表达式 扮演着 x 的角色，b 的对数底 a 的表达式扮演着 a 之上的角色，然后 对称地，整个表达式 b 的 x 次方扮 演着角色左边里面的角色，它扮演着 a 和整个表达式的角色，a 的力量扮演着 日志基 a 里面的角色，所以你可以看到 ，只需插入一些例子和通过将其与指数规则 相对应，我们已经可以思考三种不同的对数 规则，如果它们只是作为需要记忆的代数片 段流传下来，你知道，你可以记住它们， 但它们很容易从你的记忆中溜走。头，也很 容易对手头的任务感到沮丧，但你可能想提 醒自己，我们关心这类事情的原因是理解对 数规则可以帮助我们在像病毒一样生长的 环境中做数学。从一天到下一天，从一步到 下一步，事物往往会成倍增长，理解对数规 则可以帮助您更好地感受这类事物，所以在 我们做一个很好的现实世界示例来展示它 的外观之前就像让我再做一个这样的测验问 题，在我们过渡到一点现实世界的例子之前 最后一个询问对数的属性，摆脱我们此时此 地所拥有的东西，以下哪一项是正确的？", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "a 加 b 的对数与 a 加 b 的对数相同 a 加 b 的对数等于 a 的对数乘以 b 的对数 a 加 b 的对 数等于 1 除以 a 加 b 的对数或a 的 log 加 b 等于 1 除以 a 的 log 乘以 b 的 lo g 或者以上都不是啊，现在我们还没有那么多共识，不是吗？", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "非常有趣，我们有一场两人之间的赛马比赛 ，所以我会给你们一点时间来思考，而人 们正在回答，实际上我有一个小问题要问观 众，所以，你知道，我只是在谈论我们如 何从乘法增长的角度思考，这不仅仅需要十 的幂，我们还可以做类似三的幂的事情， 如果你从一到三到九到二十七到八十一， 所有其中，我们可以说，这些数字的对数以 三为底，只是以很好的小步长增长，因此 以一为底的三对数，三到等于一的数，答案 是零，一般来说，一的对数，无论底数如 何，都会零对数三之三，三到等于三的数 是一，类似地对数九之三是二啊，你可能想 知道我的问题是什么，但这将有助于将所 有这些都画出来，并且为了我自己的乐趣在 这里，让我再写出一个对数，以 3 为 底，81 等于 4，我听说，表面上如 果你问一个孩子，让我们说大约五六岁的孩 子，一到九之间的中间数是多少？说出中 间的数字 他们对于如何回答的本能是对数 的，而我们的本能往往更线性，所以我们 经常想到一和九，在它们之间有一堆均匀 分布的数字二、三、四、五、六，七，八， 如果你在中间走，你会落在五，但如果你 考虑乘法增长从一到九的位置，这不是添加 一堆东西的问题，而是你“成长到一定程 度，你成长了三倍，然后你又成长了三倍 。据说，孩子的自然本能与说三是一致的， 而且如果你有人类学家研究尚未发展的社 会，那么这也符合”没有像现代社会那样开 发会计系统和写作方式，他们会为此回答 三个，所以，我向观众提出的问题是，你们 现在观看的任何人是否可以接触到一个小 孩，比如说，在五年的范围内老看看你是 否可以去问他们一到九之间的数字是多少， 如果可以的话，请在 Twitter 上告诉我们孩子说了什么，他们的实际答案 是什么，因为我不知道为什么，我只是有 点怀疑这是否真的在实践中成功 我知道 这不是一种超级科学的方法 我并不是要求 观看 YouTube 直播的人们调查 自己的孩子，然后在推特上发布答案，但就 我个人而言，这会很有趣看到对我们的问 题的某种验证，这是第一个似乎在一个方 向上没有达成巨大共识的问题，让我们继续 对其进行评分，看看答案是否很好，好吧 ，所以 2,400你们中有谁正确地回答 了，a 加 b 的对数不满足任何这些 好的属性，并且一般来说，除非我们要使 用某些类型的近似值，特别是当自然对数发 挥作用时，以上都不是下次我们可能会讨 论这个，添加对数的输入实际上是一种非常 奇怪的感觉，这是一件非常奇怪的事情， 为了获得这种奇怪的感觉，如果我问你 a 加 b 的对数，请插入 10 的一 些幂你可能会开始想的是，好吧，让我插 入一些例子，比如 10,000 和 1 00，然后我问自己，如果我对输入中的 内容执行零计数功能，其中有多少个零？", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "这是一个有趣的问题，好吧 ，对数的底可以为零吗？", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "就我们的三角形而言，我们可能会认为 ，你知道，零的某种幂 x 等于某个 其他值 y，我们可以通过说零 x 等于 y 来写，或者我们可以写同 样的事情，y 的对数以 0 为底等 于 x 0 等于 x 的值，现在 这里的问题是零到任何值最终都为零， 所以如果我们只想考虑以 0 为底 的对数y 对于您知道的任何其他输入 y，您想要输入诸如一或二或 pi 之类的任何您可能想要的东西，您 要问的问题是零到什么等于一或二或 pi 或您可能拥有的任何数字并且 不会有答案，所以最多你可以尝试说哦 ，是的，零的对数，这是一个完全有效 的函数，它仅在输入零上定义，但即 使如此，你也很难尝试欺骗你想要的东 西因为说零等于零，就像任何东西都适 用于它一样，所以你的手臂会在背后 扭转，但是你想让它起作用，它对应于 以零为底的指数函数完全为零的事实 并没有以一种很好的一对一的方式将数 字相互映射，所以这是一个很好的问题 ，你现在可以有一个以零为底的对数 回到这些东西在现实世界中出现的想法 ，我有点喜欢的一个例子是地震的里 氏震级，因此里氏震级为我们提供了地 震强度的量化，它可以是从非常小的数 字到非常大的数字，就像我认为有史 以来测量到的最大地震一样，这只是来 自以下的图表维基百科是9分。5 为了理解这有多么疯狂，有必要看 看这些数字的含义和 TNT 的 等量（某种衡量其中有多少能量 的量度）之间的关系，然后我们可 以在这里尝试做什么看看我们是 否可以用能量的量来表达里氏震级 数，以及为什么对数是描述这一点 的自然方式，所以关注的关键是 当我们向前迈出一步时，事情会增 加多少例如，在这种情况下，如 果我们从两个井开始，它不会告诉 我们三个在哪里，所以也许我们会 考虑从两个到四个，这有点像采 取两步，这在以下方面有什么作用 看起来一公吨TNT（我猜是二 战时期的大型炸弹）需要我们消耗 的能量，而它需要我们达到千吨T NT（小型原子弹）的一千倍能 量，所以只需两步里氏震级从 2 级地震到 4 级地震，将我们 从二战的大型炸弹带入了核时代 ，所以这是值得注意的，我们迈出 的第一步是从 4 级地震到 5 级地震。至少就这张图表很好 地向我们展示的内容而言，显然从 4 到 5 的一步对应于从 1 千吨到 32 千吨，这显 然是落在长崎的城市毁灭性炸弹 的大小，所以这可能是一个如果你 只是在新闻中听到“哦，发生了一 场 4 级地震”之间的区别， 那么对数尺度可能会违反直觉。0 级地震与 5 级地震。0 很容易认为 4 和 5 是非常相 似的数字，但显然就 TNT 数量 而言，相当于乘以 32 从 1 到 下一个，从 2 到 4 显然乘以大 约一千，并且是唯一的之所以更大， 是因为我们的图表没有显示 3 是什 么，所以我们采取了两步，您可以自己 验证一下，如果您采取 32 的步长 ，然后乘以另一个 32，实际上非 常接近 1000，所以里氏数上的加 法步骤对应于 TNT 中的乘法步骤 的想法似乎表明这里有一些对数在起 作用，继续在这里并说它增长了多少， 部分是因为它所存在的世界现象，这有 点有趣。描述是的，当我们再迈出一 步时，它会再次乘以约 32，这并不 令人意外，但根据我们的直觉，这就是 32 千吨小型原子弹和 1 兆 吨（我们可能认为不是小型原子弹）之 间的区别，长崎原子弹，我猜是 32 枚长崎原子弹，重量为 1 兆吨， 这显然是 1994 年美国内华达 州双弦扁平地震的震级，我不知道那是 什么，顺便感谢维基百科的频率还查了 这些显然少于两个的，这种情况一直 在发生，每天大约有 8000 个， 但一旦我们进入原子弹领域，就会发生 3 个这样的事情。5 和 4 这些显然也经常发生在地球上的 某个地方，每天大约有 134 起这样的事情发生，谁知道呢？", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "但随着我们更加深入地进入远高于原子弹规模 的 5 和 6 范围，现在我们每天只有 2 次左右，我相信地质学家可以进来解释 为什么我们都应该这样做。不必超级担心这 样一个事实，即每天都会发生两次相当于原子 弹爆炸的地壳破坏，但对于那些集中在像现 在有很多人居住的城市这样的地方的情况来说 ，可能是特别罕见的，只是验证了我们的想 法，每一步涉及到 32 的增长，让我们 看看从 6 到 7 的步骤是什么样的，这 里给了我们更多的例子，可能会给人一种错 觉，认为这比实际的步骤更大，实际上这就是 1 兆吨和 1 兆吨之间的区别。3 2 兆吨，所以乘以 32 顺便说一下，我 在这张图表上发现最有趣的事情之一是看看 在我们达到有史以来实际测试过的最大核武 器之前我们还需要走多远，这是冷战的高峰期 沙皇炸弹的重量为 50 兆吨，我相信他 们最初计划拥有 100 兆吨的炸弹，但他 们说服自己放弃了 50 兆吨的炸弹，我 们说的是从 32 千吨的长崎炸弹乘以 3 2 开始百万吨再乘以 32，所以我们谈 论的是二战结束时爆炸强度的一千倍，但你 仍然没有达到人类能够承受的 50 兆吨， 这显然是印度尼西亚爪哇地震，所以 7 。0不仅仅是比6大一点点。0，它要大得多 ，当然，这里的要点是，当你有一个给你乘法增加的 比例时，值得欣赏的是，就隐含的能量或隐含的绝 对值而言，看起来很小的步骤实际上可能是巨大的 步骤所以当我们思考曾经有过 9 的事实时。5 这实际上看起来很荒谬，因为它只在 7 中。0范围，我们正在谈论 有史以来最大的热核武器，这表明了对数往往会出现的一个领域 ，即当人类想要为某种事物创建一个比例尺，以解释事物可以 有多大的巨大差异时因此，就地震规模而言，您可以从地球上 一直发生的事情中获得信息，如一枚大手榴弹的大小，您希望它 符合您的规模，并且需要考虑一直向上的范围这是我们在人类 历史上见过的最大的破坏，为了实现这一点，你不仅仅是在一 个案例的数字中写下一大堆不同的数字，而且还写了一大堆不同 的、较小的数字在另一种情况下，你的数字的位数，最好取对 数，然后将其放在一个单一的刻度上，基本上将这些数字压缩 在 0 到 10 之间，你会看到与音乐的分贝刻度非常相似 的情况，实际上它有点作用有点不同的是，每次你提高 10 分贝，相当于乘以 10，所以不是 1 乘以 10，而 是 10 分贝乘以 10，这样就有点数学化了有点奇怪，但 想法是一样的，如果你听的是 50 分贝与 60 分贝的 声音，就传输的能量和从 60 到 70 或 70 到 70 分贝的能量而言，会安静得多。80 这些步骤，从 60 到 80，涉及将每平方面积的能量乘以 100，因 此每次看到对数刻度时，请记住，这意味着它所指的内容都会 增长数量巨大，这又是为什么我们看到很多对数刻度用于描述冠 状病毒的爆发，所以你如何描述这样的关系，每次你将里氏刻 度数增加 1，你就乘以 32 好，我们可以用以 32 为底的对数来思考 我可以说，如果我取对数，我将调用 r， 里氏震级的数字 我可能会将其视为以 32 为底的对数， 这将对应于，不不不，我做错了，这不是记录的事情，我们采 用大数字的对数基数 32，TMT 数字，就像 1 兆吨 ，它是 100 万吨，对数基数 32，应该对应于里氏震级 数，但可能存在某种偏移，所以我们可能会说，我们添加到这 个里氏震级数中的某种常数 s 并且这个表达式是完全相同 的，请原谅我离开在底部，这个表达式与说 32 的某些偏移 量乘以我们的里氏震级数的幂完全相同，这与对该偏移量取 32 相同，偏移量本身只是一些大常数，乘以 32 的里 氏震级数，所以你可能会认为这只是某个常数乘以 32 的你 所看到的数字的幂，所以这种写法确实强调了它的指数增长， 如果这对应于你所看到的 TMT 数量，当你增加它时r 一步一步地乘以 32，但传达完全相同事实的另一种方式是取 以 32 为底的对数，现在可以了，现在我想谈的下一件事 是我们如何不必总是这样做担心如何计算不同基数的对数，这 里有点奇怪，我们谈论的是基数 32 的对数，我之前提到过 数学家真的很喜欢基数为 2 的对数，计算机科学家真的很 喜欢基数为 2 的对数，它事实证明，出于计算目的，或者 也为了思考这些东西如何增长，如果你有一个对数，如果你能够 计算一种类型的对数，无论是基数 10、基数 2、基数 e，你都可以计算几乎任何其他东西你现在想得到我们朝这个 方向的直觉，让我们回到我们的测验并转到下一个问题，我相信 这个问题是最多的，我不知道，这是一个半合理的问题，这应 该很好这只是让我们准备好从以 2 为基数的上下文转换为 以 10 为基数的上下文，并且对于理解 2 的幂与 10 的幂之间的一般关系也是一个很好的直觉，因为这是一种可 爱的巧合本质上，这两种很好，你会明白我的意思，它们彼此 配合得很好，所以我们的问题是这样问的，考虑到 2 的 1 0 次方是 1024, 1024，大约是 1000，所 以如果你是你的数字有点松散，你只是做 2 到 10 的 近似值，基本上是 1000，以下哪一项最接近真实情况？", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "投标。", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "这里根本没有一致的决定。", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "但问题是问哪一个最接近真实，让我们看看我们如何 思考这一点。", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "所以它指出你有 2 的幂，即 1024，非常 接近 10 的幂，大约是 10 的立方。", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "那么 这是什么意思？", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "如果以 10 为底的对数 2 等于 x，那么就等于 2 除以 x 等于 10，对吧？", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "它要求我们 2 等于 10。", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "你不能对每个函数都 这样做。", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "人们似乎认为你可以用任何函数来做到这一点，但你就是 做不到。", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "这意味着 x 约为 10 三分之二，好吗？", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "好吧，我们之前看到的是以 10 为底 的对数 2，我们也可以说以 2 为底的对数 10 只是 1 超过这个值， 1 超过 x。", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "伟大的。 ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "因为 我们在日志中做事，所以我就以这种方式编写它。", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "类似地，以一百万为底的对数 2， 让我们看看，如果我们必须将 2 乘以大约 10 次才能得到 1000，那么我们应该必须将它乘以大约 20 次才能得到一 百万。", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "它有 点小，但这是您心中的一个很好的近似值。", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "3 就是 3。20，我们缩小同样的数量。", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30，我们 缩小同样的数量。", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "好的？", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "现在，这是一个值得记住的直觉。", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "好的？ ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "然后就是一堆各种可能的方法来组合 B 的对数基数 C 乘以 A 的对数基数 C。", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "我会给你一个 有意义的时间来讨论这个问题，因为除非你已经熟悉 对数，否则它并不明显，并且值得稍微思考一下。", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "谢谢凯伦。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/english/captions.srt b/2020/ldm-logarithms/english/captions.srt index 8d18281d8..997f5d382 100644 --- a/2020/ldm-logarithms/english/captions.srt +++ b/2020/ldm-logarithms/english/captions.srt @@ -1099,2410 +1099,2474 @@ a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? 276 -00:38:10,160 --> 00:38:22,435 -very interesting, we've got a horse race between two so I will give you a moment to +00:38:10,160 --> 00:38:20,540 +very interesting we've got a horse race between two so I will give you a moment 277 -00:38:22,435 --> 00:38:34,417 -think this through Ally Freed So, you know, I was just talking about how we might +00:38:20,540 --> 00:38:31,309 +to think this through while people are answering actually I have a little question 278 -00:38:34,417 --> 00:38:46,546 -think in terms of multiplicative growth and that doesn't just have to be powers of +00:38:31,309 --> 00:38:41,948 +for the audience so you know I was just talking about how we might think in terms 279 -00:38:46,546 --> 00:38:58,675 -10 We could also do something like powers of 3 or if you're going from 1 to 3 to 9 +00:38:41,948 --> 00:38:52,329 +of multiplicative growth and that doesn't just have to be powers of 10 we could 280 -00:38:58,675 --> 00:39:10,366 -to 27 to 81 all of these We could say that the log base 3 of these numbers Just +00:38:52,329 --> 00:39:02,709 +also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 281 -00:39:10,366 --> 00:39:23,080 -grows in nice little steps So log base 3 of 1, 3 to the what equals 1, the answer is 0. +00:39:02,709 --> 00:39:13,089 +81 all of these we could say that the log base 3 of these numbers just grows in 282 +00:39:13,089 --> 00:39:23,080 +nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 + +283 00:39:23,260 --> 00:39:25,960 In general the log of 1 no matter the base will be 0. -283 +284 00:39:26,340 --> 00:39:31,363 Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log -284 +285 00:39:31,363 --> 00:39:36,045 base 3 of 9 is 2 You might wonder what my question is, -285 +286 00:39:36,045 --> 00:39:41,920 but it'll help to draw all of these out and For my own pleasure here. -286 +287 00:39:41,920 --> 00:39:44,780 Let me just write out one more log base 3 of 81 is 4. -287 +288 00:39:45,020 --> 00:39:49,760 Now I've heard that ostensibly if you ask a child's let's say -288 +289 00:39:49,760 --> 00:39:54,960 around like 5 or 6 years old What number is halfway between 1 and 9? -289 +290 00:39:55,780 --> 00:39:57,000 Okay, you say what number is halfway? -290 +291 00:39:59,760 --> 00:40:04,805 Their instincts for how to answer are Logarithmic whereas our instincts tend to be more -291 +292 00:40:04,805 --> 00:40:09,678 linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between -292 +293 00:40:09,678 --> 00:40:14,552 them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But -293 +294 00:40:14,552 --> 00:40:19,540 if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not -294 +295 00:40:19,540 --> 00:40:24,413 a matter of adding a bunch of things But you're growing by a certain amount you grow -295 +296 00:40:24,413 --> 00:40:29,573 by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct -296 +297 00:40:29,573 --> 00:40:34,734 lines up with saying 3 and supposedly this also lines up with If you have anthropologists -297 +298 00:40:34,734 --> 00:40:39,722 studying societies that haven't developed Counting systems and writing in the same way -298 +299 00:40:39,722 --> 00:40:44,595 that modern societies have they'll answer 3 for this So my question for the audience -299 +300 00:40:44,595 --> 00:40:49,469 if any of you watching right now have access to a small child Let's say in the range -300 +301 00:40:49,469 --> 00:40:54,342 of 5 years old See if you can go ask them What number is halfway between 1 and 9 and -301 +302 00:40:54,342 --> 00:40:59,387 if you can let us know on Twitter what the What the child says what their actual answer -302 +303 00:40:59,387 --> 00:40:59,560 is? -303 +304 00:41:00,060 --> 00:41:02,766 Because I I don't know why I'm just a little bit -304 +305 00:41:02,766 --> 00:41:05,860 skeptical of whether that Actually pans out in practice. -305 +306 00:41:05,940 --> 00:41:10,106 I understand this is not a super scientific way to do it asking people watching a -306 +307 00:41:10,106 --> 00:41:14,476 YouTube live stream to Survey their own children and then tweet the answer but for my -307 +308 00:41:14,476 --> 00:41:18,642 own sake it would be interesting to see some kind of validation there Back to our -308 +309 00:41:18,642 --> 00:41:19,100 question. -309 +310 00:41:19,320 --> 00:41:24,739 This is the first one that doesn't seem to have a huge Consensus in one direction, -310 +311 00:41:24,739 --> 00:41:30,158 but let's go ahead and grade it to see what the answer turns out to be Great okay, -311 +312 00:41:30,158 --> 00:41:35,774 so 2400 of you correctly answered that it's none of the above okay that log of a plus -312 +313 00:41:35,774 --> 00:41:41,454 B doesn't satisfy any of these nice properties And in general unless we're going to be -313 +314 00:41:41,454 --> 00:41:47,200 working with Certain kinds of approximations especially when the natural log comes into -314 +315 00:41:47,200 --> 00:41:52,751 play we might talk about this next time Adding the inputs of a logarithm is actually -315 +316 00:41:52,751 --> 00:41:58,627 a very weird sensation It's a very weird thing to do and to get a sense of that weirdness -316 +317 00:41:58,627 --> 00:42:04,177 plug in some powers of 10 If I ask you log of a plus B What you might start thinking -317 +318 00:42:04,177 --> 00:42:04,700 is okay? -318 +319 00:42:04,700 --> 00:42:11,591 Let me just plug in some examples like 10,000 and 100 and I asked myself if I -319 +320 00:42:11,591 --> 00:42:18,660 do this zero counting function of what's in that input how many zeros are in it? -320 +321 00:42:19,320 --> 00:42:22,342 But it's weird because when we add 10,100 well, -321 +322 00:42:22,342 --> 00:42:26,120 we're no longer at a clean power of 10 and okay That's fine. -322 +323 00:42:26,180 --> 00:42:32,387 You know often you're taking logarithms of things that aren't clean -323 +324 00:42:32,387 --> 00:42:38,686 powers of 10 but it becomes very strange to ask how you express this -324 +325 00:42:38,686 --> 00:42:44,620 in terms of log of 100 which was 2 and log of 10,000 Which was 4? -325 -00:42:45,860 --> 00:42:51,337 -Because if you look at log of 10,100 it's asking 10 to the what is equal to - 326 -00:42:51,337 --> 00:42:57,030 -10,100 You might say it's gonna be a little above 4 because it's kind of close +00:42:45,860 --> 00:42:51,388 +because if you look at log of 10,100 it's asking 10 to the what is equal to 10,100 327 -00:42:57,030 --> 00:43:02,580 -to 10,000 so the best you might Guess here is oh, this is gonna be something. +00:42:51,388 --> 00:42:56,984 +you might say, I don't know, it's going to be a little above 4 because it's kind of 328 +00:42:56,984 --> 00:43:02,580 +close to 10,000 so the best you might guess here is oh this is going to be something + +329 00:43:02,580 --> 00:43:07,406 That's kind of like The log of 10,000, but that just feels like a coincidence based on -329 +330 00:43:07,406 --> 00:43:12,122 the two numbers that we happen to put in There's not a nice systematic reason coming -330 +331 00:43:12,122 --> 00:43:17,116 there, so maybe you guess oh if the numbers A and B are very different It's kind of close -331 +332 00:43:17,116 --> 00:43:21,887 to Whatever the maximum of them is But it's very bizarre and most importantly for the -332 +333 00:43:21,887 --> 00:43:26,880 sake of the quiz If you just look at the options that it's giving you if you try this out -333 +334 00:43:26,880 --> 00:43:31,763 with any particular numbers You'll find that none of those actually work So all is good -334 +335 00:43:31,763 --> 00:43:36,423 sometimes you get something that looks like it's going to be a nice property But it -335 +336 00:43:36,423 --> 00:43:41,361 doesn't end up being a nice property, and I also think that's important rather than just -336 +337 00:43:41,361 --> 00:43:46,243 finding yourself Only working with the various You know log of A times B or log of X to -337 +338 00:43:46,243 --> 00:43:51,126 the power N these things that have a nice rule Sometimes you're out in the mathematical -338 +339 00:43:51,126 --> 00:43:55,786 wild you're working on some problem You have a logarithm expression And it's adding -339 +340 00:43:55,786 --> 00:44:00,779 things in the input and you want to be able to Have familiarity with the fact that that's -340 +341 00:44:00,779 --> 00:44:05,551 kind of weird that you're not going to be able to simplify But if you you know if you -341 +342 00:44:05,551 --> 00:44:10,377 hadn't thought about that before you might wonder Oh is there just some formula that I -342 +343 00:44:10,377 --> 00:44:15,315 haven't seen before So with all of that let me go ahead and take a couple questions from -343 +344 00:44:15,315 --> 00:44:20,087 the audience Before we transition to a different sort of example So it looks like Uma -344 +345 00:44:20,087 --> 00:44:22,140 Sherma asks Can can the base be zero? -345 +346 00:44:22,760 --> 00:44:24,420 That's an interesting question okay? -346 +347 00:44:24,440 --> 00:44:25,980 Can the base of a logarithm be zero? -347 +348 00:44:26,960 --> 00:44:32,816 Well in terms of our triangle we might think of that as saying You know zero to -348 +349 00:44:32,816 --> 00:44:38,746 some kind of power X is equal to some other value Y Right this is something that -349 +350 00:44:38,746 --> 00:44:44,383 we could write either by saying zero to the X equals Y Or we could write the -350 +351 00:44:44,383 --> 00:44:50,240 same thing by saying log base zero of Y Is equal to X zero to the what equals X? -351 +352 00:44:50,580 --> 00:44:55,940 Now the issue here is that zero to anything ends up being zero right so? -352 +353 00:44:56,260 --> 00:45:03,220 If we're just going to be thinking of log base zero of Y for any other input Y. -353 +354 00:45:03,500 --> 00:45:09,819 You know you want to input something like one or two or pi Anything you might -354 +355 00:45:09,819 --> 00:45:16,140 want you're asking the question zero to the what is equal to one or two or pi? -355 +356 00:45:16,460 --> 00:45:20,761 Or whatever number you might have there, and there's just not going -356 +357 00:45:20,761 --> 00:45:25,000 to be an answer so at best You could try to say oh yes log of zero. -357 +358 00:45:25,020 --> 00:45:28,732 It's a perfectly valid function It's only defined on the input zero, -358 +359 00:45:28,732 --> 00:45:31,960 but even then you'd have trouble Trying to finagle what you -359 +360 00:45:31,960 --> 00:45:34,920 want there because saying zero to the what equals zero. -360 +361 00:45:35,080 --> 00:45:39,965 It's like anything anything applies to it So your arm is going to be twisted -361 +362 00:45:39,965 --> 00:45:44,660 behind your back However you want to make that work and it corresponds to -362 +363 00:45:44,660 --> 00:45:49,354 the fact that the exponential function with base zero Is entirely zero it -363 +364 00:45:49,354 --> 00:45:54,240 doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? -364 +365 00:45:55,060 --> 00:45:59,468 So that's a great question can you have a log base zero now back to -365 +366 00:45:59,468 --> 00:46:03,683 the idea of where these things come up in the real world One one -366 +367 00:46:03,683 --> 00:46:08,027 example I kind of like is the Richter scale for earthquakes so the -367 +368 00:46:08,027 --> 00:46:13,020 Richter scale Gives us a quantification for how strong an earthquake is okay? -368 +369 00:46:13,080 --> 00:46:16,600 And it can be anything from very small numbers up to very large -369 +370 00:46:16,600 --> 00:46:20,285 numbers like I think the largest earthquake ever measured And this -370 +371 00:46:20,285 --> 00:46:24,080 is just a chart that comes from Wikipedia was a nine point five okay? -371 +372 00:46:24,240 --> 00:46:29,123 And to appreciate just how insane that is it's worth looking at the relationship -372 +373 00:46:29,123 --> 00:46:34,007 between What these numbers mean and then something like the equivalent amount of -373 +374 00:46:34,007 --> 00:46:38,830 TNT some sort of measure of how much energy there is in it And then what we can -374 +375 00:46:38,830 --> 00:46:43,774 try to do here is see if we can get an expression for the Richter scale number in -375 +376 00:46:43,774 --> 00:46:49,140 terms of the amount of energy and Why logarithms would be a natural way to describe this? -376 +377 00:46:49,860 --> 00:46:54,060 So the key to focus on is as we're taking steps forward how much do things increase? -377 -00:46:54,799 --> 00:46:58,520 +378 +00:46:54,800 --> 00:46:58,520 So for example if we go from two Well in this case it doesn't show us where -378 +379 00:46:58,520 --> 00:47:02,339 three is so maybe we think of Taking a step from two up to four which is kind -379 +380 00:47:02,339 --> 00:47:06,060 of like taking two steps What does that do in terms of the amount of energy? -380 +381 00:47:06,740 --> 00:47:11,363 Well it looks like it takes us from one metric ton of TNT Which is I guess a -381 +382 00:47:11,363 --> 00:47:16,107 large bomb from World War two and it takes us up to a kiloton a thousand times -382 +383 00:47:16,107 --> 00:47:20,972 as much Okay Which is a small a small atom bomb so just two steps on the Richter -383 +384 00:47:20,972 --> 00:47:25,656 scale Going from an earthquake of magnitude two to an earthquake of magnitude -384 +385 00:47:25,656 --> 00:47:30,340 four takes us from large bomb from World War two Up to the nuclear age, right? -385 +386 00:47:30,640 --> 00:47:35,397 so that is noteworthy and The first clean step that we get is going from four -386 +387 00:47:35,397 --> 00:47:40,276 to five at least in terms of what this chart is nicely showing us And evidently -387 +388 00:47:40,276 --> 00:47:44,911 a single step up from four to five Corresponds to going from one kiloton to -388 +389 00:47:44,911 --> 00:47:49,608 32 kilotons And that was evidently the size of the city destroying bomb that -389 +390 00:47:49,608 --> 00:47:54,426 land on Nagasaki So this is maybe one thing that can be counterintuitive about -390 +391 00:47:54,426 --> 00:47:59,122 logarithmic scales if you're just hearing in the news the difference between -391 +392 00:47:59,122 --> 00:48:03,880 oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. -392 +393 00:48:03,880 --> 00:48:08,995 It's easy to think yeah four and five Those are pretty similar numbers But evidently -393 +394 00:48:08,995 --> 00:48:14,110 in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the -394 +395 00:48:14,110 --> 00:48:19,166 next And going from two to four was evidently multiplying by about a thousand okay, -395 +396 00:48:19,166 --> 00:48:24,341 and the only reason that's bigger is because here our chart wasn't showing what three -396 +397 00:48:24,341 --> 00:48:29,336 was so we were taking two steps and You can verify for yourself that if you take a -397 +398 00:48:29,336 --> 00:48:34,271 step of 32, and then you multiply by another 32 That's actually pretty close to a -398 +399 00:48:34,271 --> 00:48:38,905 thousand So the idea that additive steps on the Richter number correspond to -399 +400 00:48:38,905 --> 00:48:44,141 multiplicative steps in the TNT Seems to suggest that something logarithmic is at play -400 +401 00:48:44,141 --> 00:48:49,316 here, and it's a little interesting to just keep going here and say How how much does -401 +402 00:48:49,316 --> 00:48:50,340 this grow partly? -402 +403 00:48:51,260 --> 00:48:53,260 Because of the the world phenomena. -403 +404 00:48:53,260 --> 00:48:53,980 It's describing. -404 +405 00:48:54,080 --> 00:48:58,465 Yes, not a huge surprise that as we take another step It's -405 +406 00:48:58,465 --> 00:49:03,520 multiplying by about 32 again But raining that in to our intuitions. -406 +407 00:49:03,640 --> 00:49:09,030 That's the difference between 32 kilotons a small atom bomb And then one megaton -407 +408 00:49:09,030 --> 00:49:14,353 which we might think of as not small atom bomb Nagasaki atom bomb Which I guess -408 +409 00:49:14,353 --> 00:49:19,743 is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude -409 +410 00:49:19,743 --> 00:49:25,400 of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. -410 +411 00:49:25,400 --> 00:49:29,952 Thanks Wikipedia in terms of frequencies by the way I Also looked these up -411 +412 00:49:29,952 --> 00:49:34,626 evidently ones that are less than two those happen all the time There's like -412 +413 00:49:34,626 --> 00:49:39,421 8,000 of those per day, but as soon as we're in the realm of atom bombs things -413 +414 00:49:39,421 --> 00:49:44,460 like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. -414 +415 00:49:44,460 --> 00:49:49,794 There's around 134 of those happening somewhere every day who knew But as we get even -415 +416 00:49:49,794 --> 00:49:55,314 more intense into this 5 and 6 range which you know we're well above the atom bomb scale -416 +417 00:49:55,314 --> 00:50:00,710 now we're only merely at around 2 per day and You know I'm sure that a geologist could -417 +418 00:50:00,710 --> 00:50:06,044 come in and explain why we all shouldn't be super worried about the fact that there's -418 +419 00:50:06,044 --> 00:50:10,944 two atom bomb equivalent Disruptions to the Earth's crust happening every day, -419 +420 00:50:10,944 --> 00:50:16,526 but presumably it's particularly rare for those to be concentrated on some Some spot like -420 +421 00:50:16,526 --> 00:50:21,923 a city where what lots of people live Now just verifying that I thought that each step -421 +422 00:50:21,923 --> 00:50:27,319 involves a growth of 32 Let's look at what the step from 6 up to 7 looks like and here -422 +423 00:50:27,319 --> 00:50:32,591 It's giving us lots more examples in between maybe giving the illusion that that's a -423 +424 00:50:32,591 --> 00:50:37,987 bigger step than it actually is and indeed That's the difference between 1 megaton and -424 +425 00:50:37,987 --> 00:50:43,383 32 megatons, so that's multiplying by 32 One of the things I found most interesting on -425 +426 00:50:43,383 --> 00:50:48,655 this chart by the way was Look at how far we have to go before we get to the largest -426 +427 00:50:48,655 --> 00:50:54,176 nuclear weapon that's ever actually been Tested This was height of the Cold War the Tsar -427 +428 00:50:54,176 --> 00:50:59,634 bomb That was 50 megatons, and I believe they actually had original plans to have a 100 -428 +429 00:50:59,634 --> 00:51:05,154 megaton bomb but talked themselves down from that 50 megatons we're talking start off at -429 +430 00:51:05,154 --> 00:51:10,736 that 32 kilotons of the Nagasaki bomb Multiply by 32 to get a megaton multiply by another -430 +431 00:51:10,736 --> 00:51:16,319 32 Right so we're talking about a thousand times the strength of the World War two ending -431 +432 00:51:16,319 --> 00:51:21,901 explosion And you're still not at the 50 megatons of what humanity is capable of And that -432 +433 00:51:21,901 --> 00:51:27,297 is evidently you know the Java earthquake of Indonesia so 7.0 is not just a little bit -433 +434 00:51:27,297 --> 00:51:28,290 bigger than 6.0. -434 +435 00:51:28,390 --> 00:51:33,230 It's a lot bigger and The point here of course is just that when you have a scale giving -435 +436 00:51:33,230 --> 00:51:37,854 you multiplicative increases It's worth appreciating that what look like small steps -436 +437 00:51:37,854 --> 00:51:42,695 Can actually be huge steps in terms of the energy implied or the absolute values implied -437 +438 00:51:42,695 --> 00:51:47,264 here So it I mean when we're thinking about the fact that there was ever a 9.5 That -438 +439 00:51:47,264 --> 00:51:51,887 actually seems absurd given that it's only in the 7.0 range that we're talking about -439 +440 00:51:51,887 --> 00:51:56,619 the largest thermonuclear weapon ever put out And this is indicative of one area where -440 +441 00:51:56,619 --> 00:52:01,188 logarithms tend to come about it's When humans want to create a scale for something -441 +442 00:52:01,188 --> 00:52:05,757 that accounts for a hugely wide variance in how big things can be So in the case of -442 +443 00:52:05,757 --> 00:52:10,489 size of earthquakes you can have things from what happens Just all the time around the -443 +444 00:52:10,489 --> 00:52:14,895 earth the size of a large hand grenade And you want that to be on your scale and -444 +445 00:52:14,895 --> 00:52:19,682 something to think about ranging all the way up to you know the largest disruption that -445 +446 00:52:19,682 --> 00:52:24,305 we've Seen in human history right and in order to have that in a way that you're not -446 +447 00:52:24,305 --> 00:52:29,091 just Writing a whole bunch of different digits in your numbers for one case and a whole -447 +448 00:52:29,091 --> 00:52:33,878 bunch of different a smaller number of digits For your number in another case It's nice -448 +449 00:52:33,878 --> 00:52:38,719 to take logarithms And then just put that on a single scale that basically puts squishes -449 +450 00:52:38,719 --> 00:52:43,505 those numbers between 0 and 10 You see something very similar going on with the decibel -450 +451 00:52:43,505 --> 00:52:48,183 scale for music that one actually Works a little bit differently where every time you -451 +452 00:52:48,183 --> 00:52:52,752 take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather -452 +453 00:52:52,752 --> 00:52:57,539 than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind -453 +454 00:52:57,539 --> 00:53:02,108 of makes the math of it a little bit screwy But the idea is the same that if you're -454 +455 00:53:02,108 --> 00:53:06,840 listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of -455 +456 00:53:06,840 --> 00:53:11,463 the energy being transmitted and going from you know What would it be 60 to 70 or 70 -456 +457 00:53:11,463 --> 00:53:11,790 to 80? -457 +458 00:53:12,790 --> 00:53:18,130 Those steps you know from 60 up to 80 that involves multiplying the amount of energy per -458 +459 00:53:18,130 --> 00:53:23,290 square area By a factor of 100 so every time you see a logarithmic scale Know in your -459 +460 00:53:23,290 --> 00:53:28,450 mind that that means whatever it's referring to under the hood grows by a huge amount -460 +461 00:53:28,450 --> 00:53:33,609 This is again why we saw a lot of logarithmic scales used to describe the coronavirus -461 +462 00:53:33,609 --> 00:53:39,010 outbreak so How might you describe a relationship like this where every time you grow the -462 +463 00:53:39,010 --> 00:53:42,430 Richter scale number by 1 you're multiplying by 32 Well, -463 +464 00:53:42,430 --> 00:53:45,190 we could think in terms of a log with base 32. -464 +465 00:53:48,210 --> 00:53:54,330 I Could say if I take the log of I'm just gonna call our the number for the Richter scale. -465 +466 00:53:54,710 --> 00:54:01,430 I might think of this as log base 32 and That's going to correspond to No, no, -466 +467 00:54:01,430 --> 00:54:07,894 no, I'm doing this wrong That's not the thing that's logged We take the log -467 +468 00:54:07,894 --> 00:54:14,359 base 32 of the big number of the the TNT number something that was like You -468 +469 00:54:14,359 --> 00:54:20,739 know one Gigaton or one megaton It's one million Tons The log base 32 that -469 +470 00:54:20,739 --> 00:54:27,970 should correspond to the Richter scale number But there might be some kind of offset. -470 +471 00:54:28,330 --> 00:54:31,835 So we might say that there's some kind of Constant s that -471 +472 00:54:31,835 --> 00:54:35,583 we're adding to this Richter scale number and this expression -472 +473 00:54:35,583 --> 00:54:39,270 is exactly the same Excuse me for going off the bottom there. -473 +474 00:54:39,330 --> 00:54:45,180 This expression is exactly the same as saying 32 to the power of some offset times our -474 +475 00:54:45,180 --> 00:54:50,762 Richter scale number, which is the same as taking you know 32 to that offset which -475 +476 00:54:50,762 --> 00:54:56,344 itself is just some big constant times 32 to the Richter scale number so you might -476 +477 00:54:56,344 --> 00:55:02,128 think of this as just being some constant Times 32 to the power of the number you see -477 +478 00:55:02,128 --> 00:55:07,912 So this way of writing it really emphasizes the exponential growth of it that if this -478 +479 00:55:07,912 --> 00:55:13,762 is what corresponds to the TNT amount that you see as You increase that R step by step -479 +480 00:55:13,762 --> 00:55:19,412 you're multiplying by 32 But another way of communicating the exact same fact is to -480 +481 00:55:19,412 --> 00:55:25,128 take the log base 32 of whatever that amount is Alright now the next thing I want to -481 +482 00:55:25,128 --> 00:55:30,979 talk about is how we don't always have to Worry about how to compute logs of different -482 +483 00:55:30,979 --> 00:55:36,830 bases and it's a little weird here that we were talking about log base 32 I referenced -483 +484 00:55:36,830 --> 00:55:42,546 earlier how mathematicians really like to have a log with base e computer scientists -484 +485 00:55:42,546 --> 00:55:48,263 really like to have a log with base 2 and it turns out for Computational purposes or -485 +486 00:55:48,263 --> 00:55:53,912 for also thinking about how these things grow if you have one Log if you're able to -486 +487 00:55:53,912 --> 00:55:59,561 compute one type of log whether that's base 10 base 2 base e you can compute pretty -487 +488 00:55:59,561 --> 00:56:05,143 much anything else That you want Okay now to get our intuitions in that direction, -488 +489 00:56:05,143 --> 00:56:11,128 let's turn back to our quiz and go to the next question and I believe that this question -489 +490 00:56:11,128 --> 00:56:12,810 is the most I don't know. -490 +491 00:56:12,890 --> 00:56:14,470 This is this is a halfway reasonable question. -491 +492 00:56:14,790 --> 00:56:20,340 This should be nice This is just going to get us prepared to translate from base 2 -492 +493 00:56:20,340 --> 00:56:26,025 contexts to base 10 contexts and it's also a good intuition for understanding powers -493 +494 00:56:26,025 --> 00:56:31,643 of 2 to have in general The relationship that it has with powers of 10 because it's -494 +495 00:56:31,643 --> 00:56:36,391 this lovely kind of coincidence of nature that these two sort of Well, -495 +496 00:56:36,391 --> 00:56:37,930 you'll see what I mean. -496 -00:56:37,970 --> 00:56:44,532 -They play nicely with each other So a question asks given the fact that 2 to the 10th - 497 -00:56:44,532 --> 00:56:49,187 -is 10 24 a thousand and 24 which is approximately 1000 okay, +00:56:37,970 --> 00:56:42,094 +they play nicely with each other so our question asks, 498 -00:56:49,187 --> 00:56:55,826 -so you can if you're being a little bit loose with your numbers and you're just making +00:56:42,094 --> 00:56:48,319 +given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, 499 -00:56:55,826 --> 00:57:02,465 -approximations 2 to the 10th Basically a thousand which of the following is closest to +00:56:48,319 --> 00:56:54,844 +so you can if you're being a little bit loose with your numbers and you're just making 500 -00:57:02,465 --> 00:57:09,257 -being true Log base 2 of 10 is approximately 0.3 Log base 2 of 10 is approximately sorry +00:56:54,844 --> 00:57:01,518 +approximations 2 to the 10th, basically 1000, which of the following is closest to being 501 -00:57:09,257 --> 00:57:15,895 -log base 10 of 2 is approximately 0.3 Log base 2 of 10 is approximately 1 third or log +00:57:01,518 --> 00:57:07,968 +true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, 502 -00:57:15,895 --> 00:57:22,687 -base 10 of 2 is approximately 1 third Hey, which of these is closest to being true based +00:57:07,968 --> 00:57:14,493 +log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log 503 -00:57:22,687 --> 00:57:29,326 -on the fact that 2 to the 10th is essentially a thousand I'll give you a little moment +00:57:14,493 --> 00:57:21,243 +base 10 of 2 is approximately 1 third okay, which of these is closest to being true based 504 -00:57:29,326 --> 00:57:36,041 -for that You Interesting that we've got kind of a split on this one so I'm wondering if +00:57:21,243 --> 00:57:25,142 +on the fact that 2 to the 10th is essentially 1000? 505 -00:57:36,041 --> 00:57:42,756 -they're gonna be numerically pretty similar or if they're gonna be conceptually similar +00:57:25,142 --> 00:57:31,817 +I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting 506 -00:57:42,756 --> 00:57:49,395 -Or if there's even a difference between those two So since answers keep rolling in I'm +00:57:31,817 --> 00:57:38,042 +that we've got kind of a split on this one so I'm wondering if they're going to be 507 -00:57:49,395 --> 00:57:55,958 -gonna give this give this a little bit more time so anyone at home watching Hopefully +00:57:38,042 --> 00:57:44,642 +numerically pretty similar or if they're going to be conceptually similar or if there's 508 -00:57:55,958 --> 00:58:02,521 -you already have a pencil and paper out to be noodling through these yourself That is +00:57:44,642 --> 00:57:49,891 +even a difference between those two so since answers keep rolling in, 509 -00:58:02,521 --> 00:58:09,160 -the spirit of the lectures that we're doing If you don't now is the time to take out a +00:57:49,891 --> 00:57:55,441 +I'm going to give this a little bit more time so anyone at home watching, 510 -00:58:09,160 --> 00:58:15,875 -pencil and paper and See if you can think this one through and write it out some of the +00:57:55,441 --> 00:58:02,116 +hopefully you already have a pencil and paper out to be noodling through these yourself, 511 -00:58:15,875 --> 00:58:22,590 -problems that we're gonna build to here Definitely will require pencil and paper so now +00:58:02,116 --> 00:58:07,066 +that is the spirit of the lectures that we're doing if you don't, 512 -00:58:22,590 --> 00:58:29,077 -is as good a time as any and if you're watching this in the future even if You can't +00:58:07,066 --> 00:58:13,740 +now is the time to take out a pencil and paper and see if you can think this one through 513 -00:58:29,077 --> 00:58:31,290 -participate in the live poll. +00:58:13,740 --> 00:58:20,340 +and write it out some of the problems that we're going to build to here definitely will 514 -00:58:31,330 --> 00:58:46,050 -I really do think it's a lot of fun to Kind of throw your own hat into the mix even +00:58:20,340 --> 00:58:26,940 +require pencil and paper so now is as good a time as any and if you're watching this in 515 -00:58:46,050 --> 00:59:00,420 -if it's not going to contribute to one of the numbers That you see growing on the +00:58:26,940 --> 00:58:31,290 +the future, even if you can't participate in the live poll 516 -00:59:00,420 --> 00:59:14,614 -screen I'll give you a little bit more time here as the answers seem to continue +00:58:31,330 --> 00:58:43,461 +I really do think it's a lot of fun to kind of throw your own hat into the mix even 517 -00:59:14,614 --> 00:59:29,334 -rolling in You You Okay, so I'll go ahead and grade it now and Let's see how people +00:58:43,461 --> 00:58:55,304 +if it's not going to contribute to one of the numbers that you see growing on the 518 -00:59:29,334 --> 00:59:43,529 -did on this one So the correct answer is B Which is that the log base 10 of 2 is +00:58:55,304 --> 00:59:07,002 +screen I'll give you a little bit more time here as the answers seem to continue 519 -00:59:43,529 --> 00:59:58,775 -approximately 0.3 and it looks like 1850 of you correctly got that so congratulations, +00:59:07,002 --> 00:59:19,134 +rolling in so now is the time to take out your pencil and paper and write it out so 520 -00:59:58,775 --> 01:00:12,970 -but the close contender not at all a Unanimous decision here looks like it was D. +00:59:19,134 --> 00:59:31,410 +now is the time to write your own paper and write it out so now is the time to write 521 -01:00:13,310 --> 01:00:18,343 -Which is that the log base 10 of 2 is around 1 third so that's good They're very +00:59:31,410 --> 00:59:43,686 +your own paper and write it out so now is the time to write your own paper and write 522 -01:00:18,343 --> 01:00:23,563 -numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 +00:59:43,686 --> 00:59:56,106 +it out so now is the time to write your own paper and write it out so now is the time 523 -01:00:23,563 --> 01:00:28,162 -repeating But the question was asking which one is closest to being true, +00:59:56,106 --> 01:00:06,794 +to write your own paper And then I'm ready to move on to the google Okay, 524 -01:00:28,162 --> 01:00:33,506 -and let's see how we can think about this so It points out that you have a power of 2 +01:00:06,794 --> 01:00:17,770 +so I'll go ahead and grade it now and let's see how people did on this one. 525 -01:00:33,506 --> 01:00:38,912 -which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we +01:00:17,770 --> 01:00:21,670 +So the correct answer is B, 526 -01:00:38,912 --> 01:00:44,319 -can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as +01:00:21,670 --> 01:00:26,093 +Which is that the log base 10 of 2 is around 1 third so that's good They're very 527 -01:00:44,319 --> 01:00:49,663 -we saw earlier those are just the reciprocals of each other So what does this mean if +01:00:26,093 --> 01:00:30,680 +numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 528 -01:00:49,663 --> 01:00:51,590 -log base 2 of 10 is equal to X? +01:00:30,680 --> 01:00:34,721 +repeating But the question was asking which one is closest to being true, 529 -01:00:51,770 --> 01:00:56,070 -That's the same thing as saying 2 to the X is equal to 10 right. +01:00:34,721 --> 01:00:39,418 +and let's see how we can think about this so It points out that you have a power of 2 530 +01:00:39,418 --> 01:00:44,169 +which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we + +531 +01:00:44,169 --> 01:00:48,920 +can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as + +532 +01:00:48,920 --> 01:00:53,617 +we saw earlier those are just the reciprocals of each other So what does this mean if + +533 +01:00:53,617 --> 01:00:55,310 +log base 2 of 10 is equal to X? + +534 +01:00:55,310 --> 01:00:56,070 +That's the same thing as saying 2 to the X is equal to 10 right. + +535 01:00:56,070 --> 01:01:00,864 It's asking us 2 to the what equals 10 So what we have here is an -531 +536 01:01:00,864 --> 01:01:05,949 expression 10 cubed is approximately equal to 2 to the 10th so what I -532 +537 01:01:05,949 --> 01:01:11,470 might write out is we know that 2 to the 10th Instead of writing it as a 10. -533 +538 01:01:11,710 --> 01:01:16,072 I'm going to write that 10 as 2 to the X Where X is the -534 +539 01:01:16,072 --> 01:01:20,590 number such that 2 to the X is approximate is equal to 10? -535 +540 01:01:21,530 --> 01:01:25,963 So if that cubed is the same as 2 to the 10th This is I'll just write out -536 +541 01:01:25,963 --> 01:01:30,457 the full details the same as saying 2 to the 3 X is equal to 2 to the 10th -537 +542 01:01:30,457 --> 01:01:33,452 and Exponentiation is a nice one-to-one function, -538 +543 01:01:33,452 --> 01:01:37,886 so it's okay to just Say whatever is going on in the input if the outputs -539 +544 01:01:37,886 --> 01:01:42,379 are the same the inputs must also be the same You can't do that with every -540 +545 01:01:42,379 --> 01:01:46,992 function people seem to think you can do that with any function But you just -541 +546 01:01:46,992 --> 01:01:49,030 can't and what that means is that? -542 +547 01:01:50,270 --> 01:01:58,800 X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 -543 +548 01:01:58,800 --> 01:02:07,330 thirds, so if we looked at our answers though That's not actually any of the options. -544 +549 01:02:07,330 --> 01:02:13,607 We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it -545 +550 01:02:13,607 --> 01:02:20,262 looks like instead we should try to re-express this as log base 10 of 2 and Well enough -546 +551 01:02:20,262 --> 01:02:26,917 what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just -547 +552 01:02:26,917 --> 01:02:33,346 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal -548 +553 01:02:33,346 --> 01:02:40,001 to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what -549 +554 01:02:40,001 --> 01:02:46,430 we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? -550 +555 01:02:46,430 --> 01:02:50,941 Which is 0.3 great So this is kind of a nice constant to think -551 +556 01:02:50,941 --> 01:02:55,452 about because there's this wonderful pattern that happens when -552 +557 01:02:55,452 --> 01:03:00,250 we're looking at powers of 2 so if I ask What is the log base 2 of? -553 +558 01:03:01,430 --> 01:03:06,752 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal -554 +559 01:03:06,752 --> 01:03:12,011 to a thousand and because we're doing things at logs I'm just going to be writing -555 +560 01:03:12,011 --> 01:03:17,205 it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of -556 +561 01:03:17,205 --> 01:03:22,271 a million Well, let's see if we have to multiply 2 by itself about 10 times to -557 +562 01:03:22,271 --> 01:03:27,466 get to a thousand we should have to multiply it by itself around 20 times to get -558 +563 01:03:27,466 --> 01:03:32,468 up to a million and Indeed log base 2 of a million is approximately 20 It's a -559 +564 01:03:32,468 --> 01:03:37,726 little bit smaller, but this is kind of a nice approximation to have in your mind -560 +565 01:03:37,726 --> 01:03:42,728 And then similarly you'll see why I'm writing out this as a pattern in just a -561 +566 01:03:42,728 --> 01:03:47,666 moment if we wanted to go up to a billion Saying how many times do I have to -562 +567 01:03:47,666 --> 01:03:53,117 multiply 2 by itself to get to a billion This is about 30 And any computer scientist -563 +568 01:03:53,117 --> 01:03:58,312 out there who's thought about you know just how much as a kilobyte or a megabyte -564 +569 01:03:58,312 --> 01:03:59,210 or a gigabyte? -565 -01:03:59,290 --> 01:04:04,333 -They'll be familiar with the idea that Powers of 2 are nice and close to +570 +01:03:59,290 --> 01:04:03,388 +they'll be familiar with the idea that powers of 2 are nice and close to -566 -01:04:04,333 --> 01:04:09,307 -these powers of 10 or more specifically powers of a thousand now What I +571 +01:04:03,388 --> 01:04:06,644 +these powers of 10. Or more specifically, powers of 1000. -567 -01:04:09,307 --> 01:04:14,350 -want to do is just write all of the same things with log base 10 Not not +572 +01:04:06,644 --> 01:04:10,967 +Now what I want to do is just write all of the same things with log base 10, -568 -01:04:14,350 --> 01:04:19,670 -approximately equal to this is actually equal to 3 log base 10 of a thousand. +573 +01:04:10,967 --> 01:04:14,167 +not approximately equal to, this is actually equal to 3. -569 +574 +01:04:14,167 --> 01:04:17,873 +Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, + +575 +01:04:17,873 --> 01:04:19,670 +what's log base 10 of a million? + +576 01:04:19,950 --> 01:04:25,268 It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's -570 +577 01:04:25,268 --> 01:04:30,651 counting the number of zeros it ends up being about 6 and log base 10 of a billion -571 +578 01:04:30,651 --> 01:04:36,358 Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of -572 +579 01:04:36,358 --> 01:04:41,936 this out is to just emphasize an interesting pattern here Which is we're just growing -573 +580 01:04:41,936 --> 01:04:47,513 by these increments right as we go from a thousand to a million to a billion with log -574 +581 01:04:47,513 --> 01:04:53,026 base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 -575 +582 01:04:53,026 --> 01:04:58,798 we're stepping up by these increments of 3 So there's this nice relationship and in fact -576 +583 01:04:58,798 --> 01:05:04,636 for all of them to go from log base 2 to log base 10 It seems like we're just multiplying -577 +584 01:05:04,636 --> 01:05:05,090 by 0.3. -578 +585 01:05:05,230 --> 01:05:11,076 So 10 times 0.3 is 3 20 we scale down by that same amount 30 we -579 +586 01:05:11,076 --> 01:05:16,649 scale down by that same amount Okay now this is an intuition -580 +587 01:05:16,649 --> 01:05:22,770 worth remembering if you have Your numbers described with one base. -581 +588 01:05:23,110 --> 01:05:26,663 It's basically the same as describing them with another base, -582 +589 01:05:26,663 --> 01:05:31,306 but there's some rescaling constant Okay, and then the next question is going to -583 +590 01:05:31,306 --> 01:05:36,005 start getting us at that direction But it's going to be framed in a way that just -584 +591 01:05:36,005 --> 01:05:40,476 looks like a whole pile of algebra And again, I will encourage you to plug in -585 +592 01:05:40,476 --> 01:05:45,118 numbers if you want to to gain a little intuition for it So as our third to last -586 +593 01:05:45,118 --> 01:05:49,761 question, this will be a long lecture We have which of the following is true and -587 +594 01:05:49,761 --> 01:05:54,403 then just a whole pile of Various possible ways to combine log base C of B times -588 +595 01:05:54,403 --> 01:05:58,874 log base C of A Does that equal log base B log base B of A and rather than me -589 +596 01:05:58,874 --> 01:06:00,250 reading them out to you? -590 +597 01:06:00,410 --> 01:06:07,138 I'll just let you look through them plug in some numbers I'll give you I'll -591 +598 01:06:07,138 --> 01:06:13,600 give you a meaningful time on this one because it's not it's not obvious -592 +599 01:06:13,600 --> 01:06:20,594 unless you're already familiar with logarithms and It's worth thinking through -593 +600 01:06:20,594 --> 01:06:27,322 a little bit You We have an outstanding question from the audience Which is -594 +601 01:06:27,322 --> 01:06:34,050 does the bar length of the pole use some kind of log function and healthily? -595 +602 01:06:34,710 --> 01:06:40,187 It looks like Ben Eater has gone ahead and directly answered in the form of the code -596 +603 01:06:40,187 --> 01:06:45,664 involved where the chart max is mapped up is the raising 2 to the power of a Ceiling -597 +604 01:06:45,664 --> 01:06:51,013 of a log base 2 of the maximum attempt count which I think is to say unraveling If -598 +605 01:06:51,013 --> 01:06:56,619 you're looking at the maximum number I'm not I'm not great at Vanna whiting this thing -599 +606 01:06:56,619 --> 01:07:02,290 if you look at the maximum number in our poll It's asking what's the log base 2 of that? -600 +607 01:07:03,070 --> 01:07:13,539 so as it crosses different powers of 2 then that rescales it and Yes, -601 +608 01:07:13,539 --> 01:07:25,952 yes is the answer what a fantastically apropos question Thank You Karen All right, -602 +609 01:07:25,952 --> 01:07:37,917 so answers are still rolling in and I think like I said I just want to give you -603 +610 01:07:37,917 --> 01:07:50,331 some more time to think this through because it's looks like a big pile of algebra -604 +611 01:07:50,331 --> 01:08:02,595 plug in some numbers to see what seems to work well and See which answer fits You -605 +612 01:08:02,595 --> 01:08:14,560 You You Okay, so even if you are still thinking about it I'm gonna go ahead and -606 +613 01:08:14,560 --> 01:08:27,273 grade it here and then start talking about Why it's true and then also why we should -607 +614 01:08:27,273 --> 01:08:39,986 care why this is an operation that actually tells us something so the correct answer -608 +615 01:08:39,986 --> 01:08:52,549 which it looks like around 1700 of you got congratulations is Log base C of B times -609 +616 01:08:52,549 --> 01:09:06,010 log base B of A is equal to log base C of A great Now that's just a big ol pile of things. -610 +617 01:09:06,149 --> 01:09:07,210 Why would that be true? -611 +618 01:09:07,330 --> 01:09:08,390 How do we think about it? -612 +619 01:09:08,750 --> 01:09:11,468 Now phase one, like I said, we might plug in an example, -613 +620 01:09:11,468 --> 01:09:14,330 but let's try to actually think about why the example holds. -614 +621 01:09:15,050 --> 01:09:20,237 So let me Pull out we've got this as one more of the log rules This is just -615 +622 01:09:20,237 --> 01:09:25,562 repeating the end what the correct answer turned out to be where we've got an -616 +623 01:09:25,562 --> 01:09:30,954 expression for Log base C of B log base B of A and this ends up being log base -617 +624 01:09:30,954 --> 01:09:36,415 C of A It's gotten rid of the B's which is kind of interesting so Some examples -618 +625 01:09:36,415 --> 01:09:41,330 you might plug in here would be things like let's use a different color. -619 +626 01:09:41,330 --> 01:09:46,770 Let's use green Instead of C. -620 +627 01:09:46,970 --> 01:09:52,251 I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking -621 +628 01:09:52,251 --> 01:09:57,601 how many times does 10 go into a 100 in a multiplicative sense how many times -622 +629 01:09:57,601 --> 01:10:03,020 do I multiply 10 by itself to get to 100 where the answer is 2 and then log of -623 +630 01:10:03,020 --> 01:10:08,438 100 of Let's plug in another power of 10 It'll be nice if it's also a power of -624 +631 01:10:08,438 --> 01:10:13,788 100 So I'll do a million So This one is asking 10 to 100 to the what equals a -625 +632 01:10:13,788 --> 01:10:19,070 million How many times do I multiply a hundred by itself to get to a million? -626 +633 01:10:19,550 --> 01:10:21,410 How many times does a hundred go into a million? -627 +634 01:10:22,270 --> 01:10:27,519 Phrasing the same thing 10 different ways now the claim is that this is -628 +635 01:10:27,519 --> 01:10:32,840 the same thing as taking log base 10 of a million That if I ask how many -629 +636 01:10:32,840 --> 01:10:38,090 times does 10 go into 100 and how many times does 100 go into a million? -630 +637 01:10:38,090 --> 01:10:44,688 Multiplying those should give me the answer to how many times 10 goes into a million -631 +638 01:10:44,688 --> 01:10:51,209 now just checking the numbers this certainly works 10 goes into a hundred two times -632 +639 01:10:51,209 --> 01:10:57,419 100 goes into a million three times in a multiplicative sense in that a hundred -633 +640 01:10:57,419 --> 01:11:03,630 cubed is equal to a million and Indeed how many times does 10 go into a million? -634 +641 01:11:04,650 --> 01:11:09,301 well six Now we could think of this property in terms of the corresponding -635 +642 01:11:09,301 --> 01:11:14,077 exponent rule which is going to look a little bit stranger But it's actually -636 +643 01:11:14,077 --> 01:11:18,853 just saying the entirely the same thing So here we're if we have a base of C -637 +644 01:11:18,853 --> 01:11:24,001 and a base of B And we're trying to relate those to each other the whole statement -638 +645 01:11:24,001 --> 01:11:29,150 is equal to saying that um suppose that B to the X is equal to a Got some number B. -639 +646 01:11:29,170 --> 01:11:31,768 You raise it to some number X and equals a suppose -640 +647 01:11:31,768 --> 01:11:34,010 It's also the case that C to the Y equals B. -641 +648 01:11:34,410 --> 01:11:39,067 Those two together are the same as saying C to the XY equals a Now that's kind -642 +649 01:11:39,067 --> 01:11:43,782 of a mouthful to say out loud But if you plug in some numbers to translate what -643 +650 01:11:43,782 --> 01:11:48,381 all of that is really saying in the context of the example we just did Saying -644 +651 01:11:48,381 --> 01:11:52,979 if you can write a hundred is ten squared and if you can write a million as a -645 +652 01:11:52,979 --> 01:11:57,812 hundred cubed Well that lets you write a million in terms of a power of ten Okay, -646 +653 01:11:57,812 --> 01:12:02,470 so sort of asking this question How many times does one number go into another? -647 -01:12:03,530 --> 01:12:09,072 -But letting you layer it on top of each other Now if we rearrange that expression we get - -648 -01:12:09,072 --> 01:12:14,615 -what is probably the second most important of all of our log rules The most important is - -649 -01:12:14,615 --> 01:12:20,034 -this top one that when you multiply the inputs you add the outputs But the second most - -650 -01:12:20,034 --> 01:12:25,577 -important which is known as the change of base formula Lets us rate that if you want the - -651 -01:12:25,577 --> 01:12:31,058 -log base B of some value a Then for whatever C you want it doesn't actually matter what - -652 -01:12:31,058 --> 01:12:36,663 -log you have in your pocket if you use that other log And you take the log C of a divided - -653 -01:12:36,663 --> 01:12:40,088 -by the log C of B that gives you log base B of a Okay, - 654 -01:12:40,088 --> 01:12:45,631 -so just as an example here what this would look like is let's say I wanted to be able to +01:12:03,530 --> 01:12:08,683 +but letting you layer it on top of each other. Now if we rearrange that expression, 655 -01:12:45,631 --> 01:12:51,174 -compute log base 100 of things I just really want to it's not a button on my calculator, +01:12:08,683 --> 01:12:13,285 +we get what is probably the second most important of all of our log rules. 656 -01:12:51,174 --> 01:12:56,717 -but I would love to be able to do it and I want to do it in With an input like a million +01:12:13,285 --> 01:12:17,642 +The most important is this top one, that when you multiply the inputs, 657 -01:12:56,717 --> 01:13:02,198 -well even if I don't have the log base 100 button on my calculator what I can do is say +01:12:17,642 --> 01:12:20,832 +you add the outputs. But the second most important, 658 -01:13:02,198 --> 01:13:07,554 -I'll use the log base 10 button and Evaluate what's on the inside here which at least +01:12:20,832 --> 01:12:26,354 +which is known as the change of base formula, lets us write that if you want the log base 659 -01:13:07,554 --> 01:13:13,035 -positionally it's kind of above the 100 it has a higher altitude as we write it so This +01:12:26,354 --> 01:12:29,361 +b of some value a, then for whatever c you want, 660 -01:13:13,035 --> 01:13:18,266 -can line up with the notation a little bit that it sits on the numerator and on the +01:12:29,361 --> 01:12:34,760 +it doesn't actually matter what log you have in your pocket. If you use that other log, 661 -01:13:18,266 --> 01:13:23,685 -bottom I use the log base 10 button that's in my calculator on the base on the 100 and +01:12:34,760 --> 01:12:40,159 +and you take the log c of a, divided by the log c of b, that gives you log base b of a. 662 -01:13:23,685 --> 01:13:29,166 -Then I can evaluate both of those and it'll give me the answer in this case it gets you +01:12:40,159 --> 01:12:43,534 +So just as an example here, what this would look like, 663 -01:13:29,166 --> 01:13:34,584 -6 divided by 2 which will be 3 and If we really just think through what this is saying +01:12:43,534 --> 01:12:49,056 +is let's say I wanted to be able to compute log base 100 of things. I just really want to. 664 -01:13:34,584 --> 01:13:40,189 -I know I've said it many different times But it's a it's a convoluted enough way to write +01:12:49,056 --> 01:12:53,658 + It's not a button on my calculator, but I would love to be able to do it. 665 -01:13:40,189 --> 01:13:45,359 -things but an intuitive enough fact that I think just coming at it from a bunch of +01:12:53,658 --> 01:12:56,726 +And I want to do it with an input like a million. 666 -01:13:45,359 --> 01:13:50,777 -different angles can be important because like I said This is probably the second most +01:12:56,726 --> 01:13:00,898 +Well even if I don't have the log base 100 button on my calculator, 667 -01:13:50,777 --> 01:13:55,697 -important log rule We're asking how many times does 100 go into a million in a +01:13:00,898 --> 01:13:06,236 +what I can do is say I'll use the log base 10 button and evaluate what's on the inside 668 -01:13:55,697 --> 01:13:59,310 -multiplicative sense how many times we multiply by itself? +01:13:06,236 --> 01:13:10,040 +here, which at least positionally it's kind of above the 100. 669 +01:13:10,040 --> 01:13:15,439 +It has a higher altitude as we write it. So this can line up with the notation a little + +670 +01:13:15,439 --> 01:13:18,814 +bit, that it sits on the numerator. And on the bottom, + +671 +01:13:18,814 --> 01:13:23,600 +I use the log base 10 button that's in my calculator on the base, on the 100. + +672 +01:13:23,600 --> 01:13:27,772 +And then I can evaluate both of those and it'll give me the answer. + +673 +01:13:27,772 --> 01:13:31,331 +In this case it gets you 6 divided by 2, which will be 3. + +674 +01:13:31,331 --> 01:13:34,828 +And if we really just think through what this is saying, + +675 +01:13:34,828 --> 01:13:39,982 +I know I've said it many different times, but it's a convoluted enough way to write + +676 +01:13:39,982 --> 01:13:45,136 +things, but an intuitive enough fact that I think just coming at it from a bunch of + +677 +01:13:45,136 --> 01:13:48,572 +different angles can be important. Because like I said, + +678 +01:13:48,572 --> 01:13:51,824 +this is probably the second most important log rule. + +679 +01:13:51,824 --> 01:13:56,917 +We're asking how many times does 100 go into a million? In a multiplicative sense, + +680 +01:13:56,917 --> 01:13:59,310 +how many times do I multiply by itself? + +681 01:13:59,910 --> 01:14:03,090 But division is asking that same question in an additive -670 +682 01:14:03,090 --> 01:14:06,550 sense if I say how many times does the log of 100 go into the? -671 +683 01:14:06,570 --> 01:14:09,773 Log of a million that's what division means it's saying how many -672 +684 01:14:09,773 --> 01:14:12,830 times do I add this bottom number to itself to get to the top? -673 +685 01:14:13,590 --> 01:14:17,748 But anything additive in the logarithm realm is the same as anything multiplicative -674 +686 01:14:17,748 --> 01:14:21,609 in terms of what's inside the parentheses So both of the left-handed side and -675 +687 01:14:21,609 --> 01:14:25,470 the right-hand side are just saying how many times does 100 go into a million? -676 -01:14:25,550 --> 01:14:29,815 -But going about that in different ways So this is extremely nice because it +688 +01:14:25,550 --> 01:14:29,783 +but going about that in different ways. So this is extremely nice because it -677 -01:14:29,815 --> 01:14:34,081 -actually lets us compute things next time we're going to talk all about the +689 +01:14:29,783 --> 01:14:34,016 +actually lets us compute things. Next time we're going to talk all about the -678 -01:14:34,081 --> 01:14:38,458 -natural logarithm, which is log base e often written ln and Turns out this is +690 +01:14:34,016 --> 01:14:38,030 +natural logarithm, which is log base e, often written ln. And turns out, -679 -01:14:38,458 --> 01:14:43,004 -much easier to compute There's nice math behind it such that if you want to come +691 +01:14:38,030 --> 01:14:42,538 +this is much easier to compute. There's nice math behind it such that if you want -680 -01:14:43,004 --> 01:14:47,494 -up with an algorithm that your calculator can use It's actually a lot easier to +692 +01:14:42,538 --> 01:14:45,782 +to come up with an algorithm that your calculator can use, -681 -01:14:47,494 --> 01:14:52,490 -think of log base e of numbers So anytime that you need to go to some kind of calculator. +693 +01:14:45,782 --> 01:14:49,191 +it's actually a lot easier to think of log base e of numbers. -682 +694 +01:14:49,191 --> 01:14:52,490 +So any time that you need to, go to some kind of calculator. + +695 01:14:52,530 --> 01:14:57,861 I don't know let's say we popped over to something like Desmos Always happy to have -683 +696 01:14:57,861 --> 01:15:03,255 as a friend, and you wanted to compute log base 10 of Some number you know let's say -684 +697 01:15:03,255 --> 01:15:08,650 we're doing log base 10 of 57 and it looks like that's you know it should make sense. -685 +698 01:15:08,690 --> 01:15:14,510 It's between 1 and 2 because 57 was between 10 and 100 What's going on under the hood? -686 +699 01:15:14,510 --> 01:15:16,110 How is it actually figuring this out? -687 +700 01:15:16,110 --> 01:15:20,564 It's going to use somewhere in there a change of base formula Which is going -688 +701 01:15:20,564 --> 01:15:25,077 to be that the natural log of 57 divided by the natural log of 10 is the same -689 +702 01:15:25,077 --> 01:15:29,590 thing These are two different ways of writing it so if you know one logarithm. -690 +703 01:15:29,590 --> 01:15:32,443 You know all of the logarithms okay, and Well, -691 +704 01:15:32,443 --> 01:15:37,724 let's just use that fact to answer one more of our quiz questions And this will be the -692 +705 01:15:37,724 --> 01:15:43,188 second to last quiz question, so thank you all for sticking it through I think we will be -693 +706 01:15:43,188 --> 01:15:48,652 I Think you'll be pleased by the last question because the last question will actually be -694 +707 01:15:48,652 --> 01:15:54,055 like a fun Problem-solving puzzle II thing and it'll involve a lot of what we've used up -695 +708 01:15:54,055 --> 01:15:59,336 to this point kind of a culminating thing So before that though just to make sure that -696 +709 01:15:59,336 --> 01:16:02,190 we've got the instincts of Change of base down. -697 +710 01:16:02,430 --> 01:16:03,670 What do we have? -698 +711 01:16:04,310 --> 01:16:08,838 Use the approximation that log base 2 of 10 is around 10 thirds so -699 +712 01:16:08,838 --> 01:16:13,570 using that approximation Which of the following is approximately true? -700 +713 01:16:13,570 --> 01:16:18,768 Log base 2 of X is about 10 thirds of log base 10 of X Log base 2 of X is log -701 +714 01:16:18,768 --> 01:16:23,966 base 10 of 10 thirds of X log base 2 of X is log base 10 of X to the power 10 -702 +715 01:16:23,966 --> 01:16:29,164 thirds or Log base 2 of X is 10 thirds times the log base 10 of X and finally -703 +716 01:16:29,164 --> 01:16:34,429 none of the above So I'll give you a moment to think about that you might want -704 +717 01:16:34,429 --> 01:16:39,693 to think to the chart that we were drawing earlier and thinking about log base -705 +718 01:16:39,693 --> 01:16:44,691 2 and log base 10 as we're looking at powers of 1000 and how each of those -706 +719 01:16:44,691 --> 01:16:49,890 grows that can leverage some of The intuition but I'll let you think about it. -707 +720 01:16:50,030 --> 01:17:03,164 Stop talking You While answers are rolling in it looks like a number of -708 +721 01:17:03,164 --> 01:17:17,028 people have been asking on Twitter about Basically how complex numbers play -709 +722 01:17:17,028 --> 01:17:30,710 into this so you know we've got Jamil asking What if the base is imaginary? -710 -01:17:31,250 --> 01:17:35,728 -We've got Kelcon asking what about complex bases since e to the X is walking around +723 +01:17:31,250 --> 01:17:33,891 +We've got Kalkan asking, what about complex bases? -711 -01:17:35,728 --> 01:17:40,419 -the spiral wouldn't it hit every complex number Well it won't hit every complex number, +724 +01:17:33,891 --> 01:17:38,449 +Wouldn't z to the x is walking around the spiral? Wouldn't it hit every complex number? -712 -01:17:40,419 --> 01:17:44,845 -but your instinct that it's hitting multiple things is pretty spot-on Doesn't make +725 +01:17:38,449 --> 01:17:42,903 +Well, it won't hit every complex number. But your instinct that it's hitting multiple -713 -01:17:44,845 --> 01:17:49,430 -sense to talk about logs with imaginary numbers by Nitya so to all of these questions. +726 +01:17:42,903 --> 01:17:47,410 +things is pretty spot on. Doesn't make sense to talk about logs with imaginary numbers -714 +727 +01:17:47,410 --> 01:17:49,430 +by Nitya. So to all of these questions, + +728 01:17:49,430 --> 01:17:53,886 It's actually a very Nuanced question the short answer is yes -715 +729 01:17:53,886 --> 01:17:58,270 complex logarithms absolutely exist But each one of them has -716 +730 01:17:58,270 --> 01:18:02,870 multiple different outputs so a good a good analogy here is how? -717 +731 01:18:03,250 --> 01:18:08,056 If we have the square root function if I ask something like the square root of -718 +732 01:18:08,056 --> 01:18:12,558 5 You know we have the convention that you always do the positive amount, -719 +733 01:18:12,558 --> 01:18:17,425 but that doesn't quite feel honest It feels like the right answer is to specify -720 +734 01:18:17,425 --> 01:18:22,171 that there's two different outputs for the square root function two solutions -721 +735 01:18:22,171 --> 01:18:26,794 to x squared equals 25 And this is actually true in complex numbers as well -722 +736 01:18:26,794 --> 01:18:31,601 one of the things we talked about in a complex number lecture Was that you can -723 +737 01:18:31,601 --> 01:18:36,590 take the square root of I and you actually get root 2 over 2 plus root 2 over 2 I? -724 +738 01:18:36,590 --> 01:18:41,277 But that there's two solutions you can do plus or minus this value and so you could -725 +739 01:18:41,277 --> 01:18:44,179 say that the square root function Isn't a function, -726 +740 01:18:44,179 --> 01:18:48,812 but it's a multi-valued function It always has two different outputs now something -727 +741 01:18:48,812 --> 01:18:53,276 funky happens when we have exponents at play So if you're just like someone who -728 +742 01:18:53,276 --> 01:18:55,230 hasn't seen this stuff don't worry. -729 +743 01:18:55,330 --> 01:18:58,851 We talked about it in previous lectures I'm obviously jumping around on the -730 +744 01:18:58,851 --> 01:19:02,234 complexity level a lot here where if in lecture 5 I'm talking about Like -731 +745 01:19:02,234 --> 01:19:05,802 Euler's formula with complex numbers and compound interest and like how that -732 +746 01:19:05,802 --> 01:19:09,370 plays into physics and then lecture 6 We're back to the basics of logarithms. -733 +747 01:19:09,690 --> 01:19:16,020 I acknowledge that might be a little bit jarring to potential audience members, -734 +748 01:19:16,020 --> 01:19:23,142 but just continuing on with the answer if you note that e to the 2 pi times you know some -735 +749 01:19:23,142 --> 01:19:29,869 number times I This basically walks you around a circle so that the output will Walk -736 +750 01:19:29,869 --> 01:19:36,753 around a circle and just keep repeating as n goes from 0 up to 1 It'll walk around one -737 +751 01:19:36,753 --> 01:19:43,638 cycle and end up back where you started as n goes from 1 to 2 You'll end up back where -738 +752 01:19:43,638 --> 01:19:50,602 you started so for example e to the 0 sits here e to the I pi Is at negative 1 but e to -739 +753 01:19:50,602 --> 01:19:51,710 the 2 pi I is? -740 +754 01:19:53,030 --> 01:20:02,196 also at 1 same with Excuse me same with e to the 4 pi I that also Equals 1 -741 +755 01:20:02,196 --> 01:20:11,730 so in general if you wanted to ask something like what is the log base e of 1? -742 +756 01:20:12,890 --> 01:20:18,123 You know on the one hand we want to say the log of Log base anything of 1 should -743 +757 01:20:18,123 --> 01:20:23,291 be 0 because anything to the power 0 equals that 1 But if we're letting complex -744 +758 01:20:23,291 --> 01:20:28,589 numbers into the mix you would have to honestly say well 2 pi I is another pretty -745 +759 01:20:28,589 --> 01:20:33,822 good answer to this question because e to the 2 pi I also equals 1 and Same with -746 +760 01:20:33,822 --> 01:20:39,055 4 pi I and you could even go in the negative direction and in general n times pi -747 +761 01:20:39,055 --> 01:20:44,353 Times 2 times I kind of wrote that in a weird order for any integer in Feels like -748 +762 01:20:44,353 --> 01:20:49,521 a valid answer to this question So there's a couple ways that you can deal with -749 +763 01:20:49,521 --> 01:20:54,690 that in math and before we get back to our usual lesson Just on logarithm rules. -750 +764 01:20:54,810 --> 01:20:55,670 This is interesting enough. -751 +765 01:20:55,670 --> 01:21:01,160 I kind of want to pull it up Let's see what if we have complex logarithm -752 +766 01:21:01,160 --> 01:21:06,349 uh Great Wikipedia always to our aid probably it'll have a nice like -753 +767 01:21:06,349 --> 01:21:11,990 visual of some kind for us look such fancy color diagrams What I want okay. -754 +768 01:21:12,010 --> 01:21:13,230 This is what I want wonderful. -755 +769 01:21:14,110 --> 01:21:19,484 Let's zoom out a little bit Great, so there's this notion of what's called a Riemann -756 +770 01:21:19,484 --> 01:21:24,668 surface That's basically trying to capture the idea that you have a function with -757 +771 01:21:24,668 --> 01:21:30,043 multiple outputs and intuitively Maybe you can understand what it's getting at where -758 +772 01:21:30,043 --> 01:21:35,607 the input would be something on the XY plane and the output There's just many different -759 +773 01:21:35,607 --> 01:21:41,234 outputs sitting there so when you have a complex logarithm you have to account for that, -760 +774 01:21:41,234 --> 01:21:46,861 but it's used It's actually a very useful idea To do but it takes a lot more nuance than -761 +775 01:21:46,861 --> 01:21:52,552 than you might expect So with all of that hopefully that helped At least partially answer -762 +776 01:21:52,552 --> 01:21:57,863 some of people's questions same same by the way with Logarithms with based negative -763 +777 01:21:57,863 --> 01:22:03,300 numbers because if you're asking like negative 1 to the X and really noodling on what -764 +778 01:22:03,300 --> 01:22:07,283 that means it's It gets you into the realm of Complex numbers, -765 +779 01:22:07,283 --> 01:22:12,784 so you have to deal with the same multiple output idea now answers are rolling in more -766 +780 01:22:12,784 --> 01:22:18,158 slowly so this seems like a fine time to grade things and the answer turns out to be -767 +781 01:22:18,158 --> 01:22:23,849 You rescale it if you want to go from log base 2 of something to log base 10 of something -768 +782 01:22:23,849 --> 01:22:29,413 It involves rescaling and one way you could think about this is with the change of base -769 +783 01:22:29,413 --> 01:22:34,914 formula, so let's say we have Let's get rid of our e stuff Let's say you have Log base -770 +784 01:22:34,914 --> 01:22:40,351 2 of X, but we want to write it in terms of log base 10 We can write that as log base -771 +785 01:22:40,351 --> 01:22:45,409 10 of X Divided by log base 10 of 2 saying how many times does 2 go into X in a -772 +786 01:22:45,409 --> 01:22:50,973 multiplicative sense is The same as saying how many times does the log of 2 go into the -773 +787 01:22:50,973 --> 01:22:53,250 log of X in an additive sense, okay? -774 -01:22:53,929 --> 01:22:58,710 +788 +01:22:53,930 --> 01:22:58,710 And Well, what is 1 divided by the log base 10 of 2? -775 +789 01:22:59,030 --> 01:23:05,821 so we're going to keep our log base 10 of X out here and From what we found earlier -776 +790 01:23:05,821 --> 01:23:12,290 We found that log base 10 of 2 was approximately 3 tenths was approximately 0.3. -777 +791 01:23:12,630 --> 01:23:15,993 So when we divide by it that should get us 10 thirds Okay, -778 +792 01:23:15,993 --> 01:23:21,010 and this lines up with what we were looking at a little bit earlier with powers of 1000. -779 +793 01:23:21,230 --> 01:23:21,850 Let's see. -780 +794 01:23:21,850 --> 01:23:22,590 -Where was it? +where was it? Where was it? -781 -01:23:25,370 --> 01:23:29,745 -Great So when we're converting from log base 10 of something up to log base 2 You +795 +01:23:25,370 --> 01:23:29,573 +Great. So when we're converting from log base 10 of something up to log base 2, -782 -01:23:29,745 --> 01:23:34,120 -know here we were thinking of multiplying the top by 0.3 to get to the bottom But +796 +01:23:29,573 --> 01:23:33,935 +you know here we were thinking of multiplying the top by 0.3 to get to the bottom. -783 -01:23:34,120 --> 01:23:38,334 -you could also think of multiplying by 10 thirds to get to the top Anytime you +797 +01:23:33,935 --> 01:23:37,719 +But you could also think of multiplying by 10 thirds to get to the top. -784 -01:23:38,334 --> 01:23:42,549 -have log base 10 of some number you just rescale it and you have log base 2 of +798 +01:23:37,719 --> 01:23:41,975 +Anytime you have log base 10 of some number you just rescale it and you have log -785 -01:23:42,549 --> 01:23:46,764 -that number and the rescaling Constant comes from the log base 2 of 10 or vice +799 +01:23:41,975 --> 01:23:46,337 +base 2 of that number. And the rescaling constant comes from the log base 2 of 10. -786 -01:23:46,764 --> 01:23:50,019 -versa log base 10 of 2 So that's change of base like I said, +800 +01:23:46,337 --> 01:23:49,437 +Or vice versa, log base 10 of 2. So that's change of base. -787 -01:23:50,019 --> 01:23:54,394 -it's very important It lets you put everything into a nice universal language and +801 +01:23:49,437 --> 01:23:53,746 +Like I said it's very important. It lets you put everything into a nice universal -788 -01:23:54,394 --> 01:23:58,556 -that should be all of the hint that you need for the last question Which is a +802 +01:23:53,746 --> 01:23:58,002 +language. And that should be all of the hint that you need for the last question -789 -01:23:58,556 --> 01:24:00,210 -challenge question on this one. +803 +01:23:58,002 --> 01:24:00,210 +which is a challenge question on this one. -790 +804 01:24:00,230 --> 01:24:02,930 So if anyone's been watching and they've been like I know logarithms. -791 +805 01:24:02,990 --> 01:24:07,739 I've got this completely down Let me pull up the last question which -792 +806 01:24:07,739 --> 01:24:12,764 came up on let's see It's not the AMC, but it's whatever the predecessor -793 +807 01:24:12,764 --> 01:24:17,651 to the AMC was I believe So it's you know, it's gonna involve a little -794 +808 01:24:17,651 --> 01:24:22,470 cleverness and manipulation We've got this long Sequence of fractions. -795 +809 01:24:22,530 --> 01:24:27,869 Okay, you take 1 divided by the log base 2 of a hundred factorial And remember -796 +810 01:24:27,869 --> 01:24:33,343 100 factorial is a hundred times 99 times 98 on the way all the way down to 1 so -797 +811 01:24:33,343 --> 01:24:38,480 1 divided by the log base 2 of that plus 1 divided by the log base 3 of 100 -798 +812 01:24:38,480 --> 01:24:44,022 factorial Plus 1 divided by log base 4 of 100 factorial on and on and on up until -799 +813 01:24:44,022 --> 01:24:49,970 1 divided by the log base 100 of 100 factorial So this looks rather intimidating, right? -800 +814 01:24:51,330 --> 01:24:56,256 Certainly adding fractions is never fun adding 100 fractions seems even worse Dealing -801 +815 01:24:56,256 --> 01:25:01,240 with a bunch of logs of different bases seems like a pain and the factorial is playing -802 +816 01:25:01,240 --> 01:25:06,110 into this So I'm just gonna give you you know, given that this is the wind down time. -803 +817 01:25:06,110 --> 01:25:23,427 I'm gonna give you Two or three minutes to start thinking about this and if you don't -804 +818 01:25:23,427 --> 01:25:37,724 get it, it's totally fine We're gonna walk through what the answer is, -805 +819 01:25:37,724 --> 01:25:55,243 but I'm just gonna let people think about this final challenge question before we call -806 +820 01:25:55,243 --> 01:26:12,560 it a day You You You You You So I'm gonna give you a little bit more time on this one -807 +821 01:26:12,560 --> 01:26:30,079 because it's definitely It's definitely fun to work out and I think if you know how to -808 +822 01:26:30,079 --> 01:26:47,597 start then great But if you don't know how to start just letting yourself kind of work -809 +823 01:26:47,597 --> 01:27:04,512 with a couple of the different rules that we've worked out before Change of base it -810 +824 01:27:04,512 --> 01:27:22,031 comes into play if you'd like to use that And just kind of keep manipulating and if it -811 +825 01:27:22,031 --> 01:27:39,952 feels like you're getting something that's a little too messy see if you come at it from -812 +826 01:27:39,952 --> 01:27:57,471 a different angle and Because answers are rolling in a little bit more slowly now What -813 +827 01:27:57,471 --> 01:28:14,587 I'm going to do is just start to describe the explanation here and then come back to -814 +828 01:28:14,587 --> 01:28:20,830 grade it in just a moment here. -815 +829 01:28:21,250 --> 01:28:26,054 So the expression that we have Looks like I just started writing it out while you guys -816 +830 01:28:26,054 --> 01:28:30,858 were working on it one divided by log base two of a hundred factorial Plus one divided -817 +831 01:28:30,858 --> 01:28:35,552 by log base three of a hundred factorial and before you even start the fact that the -818 +832 01:28:35,552 --> 01:28:40,191 thing on the inside of The log involves a big product of stuff should actually feel -819 +833 01:28:40,191 --> 01:28:44,885 good because logs like to take in things that look like products because of the most -820 +834 01:28:44,885 --> 01:28:49,690 important property that we have which Is the idea that it turns products into addition? -821 +835 01:28:50,310 --> 01:28:53,832 Okay, so right away, you know that you're probably going to use that second -822 +836 01:28:53,832 --> 01:28:57,309 thing You notice is that it's uncomfortable to have all of these different -823 +837 01:28:57,309 --> 01:29:00,553 bases log base 2 of something log base 3 of something log base 100 of -824 +838 01:29:00,553 --> 01:29:04,030 something so Translating it all into the common language should be helpful. -825 +839 01:29:04,210 --> 01:29:05,750 How does change of base work? -826 +840 01:29:06,270 --> 01:29:10,650 Well asking how many times does to go into a hundred factorial is the same as asking? -827 +841 01:29:11,070 --> 01:29:15,310 How many times does the log of to divide into the log of a hundred factorial? -828 +842 01:29:15,750 --> 01:29:21,710 It doesn't even matter what logarithm you use this log could be base 10 base e base -829 +843 01:29:21,710 --> 01:29:27,457 2 doesn't matter this change of base formula Still holds now what that means for -830 +844 01:29:27,457 --> 01:29:33,205 our expression is instead of taking one divided by log base 2 of 100 factorial I -831 +845 01:29:33,205 --> 01:29:39,307 could write that as log of 2 Divided by so let me draw a little dividing line between -832 +846 01:29:39,307 --> 01:29:45,410 what I'm doing here log of 2 divided by log of 100 factorial Okay, and then similarly? -833 +847 01:29:46,310 --> 01:29:52,658 log of 3 divided by log of 100 factorial so all I'm doing here is taking The reciprocal -834 +848 01:29:52,658 --> 01:29:59,079 so instead of taking log of 100 factorial over log of 2 I have inverted it log of 2 over -835 +849 01:29:59,079 --> 01:30:05,500 100 factorial because that's what this reciprocal is doing so with that as the beginning -836 +850 01:30:05,500 --> 01:30:11,993 I'm gonna go ahead and just grade this lock in the answers and See how see how everyone's -837 +851 01:30:11,993 --> 01:30:18,486 doing So it looks like awesome around 1796 we always we always draw a little bit north of -838 +852 01:30:18,486 --> 01:30:24,113 Ramanujan's number around 1800 of you Answered that it's one which is correct -839 +853 01:30:24,113 --> 01:30:30,534 congratulations 70 of you answered 100 which we can maybe see where where that our error -840 +854 01:30:30,534 --> 01:30:36,811 would have come from Those of you that answered zero that's interesting it would imply -841 +855 01:30:36,811 --> 01:30:42,943 that somehow you have cancellation at play Because these are all positive numbers so -842 +856 01:30:42,943 --> 01:30:49,292 thinking one of them was negative So you could probably gut check against the idea that -843 +857 01:30:49,292 --> 01:30:55,713 it would be zero 41 of you want me to come up with a number fun numerical fact about 69, -844 +858 01:30:55,713 --> 01:30:59,104 but I won't 28 of you said 2 and I think yeah, -845 +859 01:30:59,104 --> 01:31:05,597 I think that's I Think the the difference between 1 and 100 maybe would be interesting to -846 +860 01:31:05,597 --> 01:31:11,441 try to analyze, but if we go back to our answer I Realize it may be a little bit -847 +861 01:31:11,441 --> 01:31:17,501 confusing how I've heard this this isn't this is not a big fraction this was just a -848 +862 01:31:17,501 --> 01:31:23,777 dividing line between Not true ironically trying not to confuse my fractions with each -849 +863 01:31:23,777 --> 01:31:30,270 other so rewriting our expression up on the top here as we add all of these things and we -850 +864 01:31:30,270 --> 01:31:36,691 can continue up until log of 100 all divided by log of 100 factorial The key is that now -851 +865 01:31:36,691 --> 01:31:42,968 all of the denominators are the same so we can add the numerators No more plus sign is -852 +866 01:31:42,968 --> 01:31:49,389 just going to equal something that top is going to look like log of 2 Plus log of 3 plus -853 +867 01:31:49,389 --> 01:31:55,810 on and on up to log of 100 all divided by the log of 100 factorial and it seems like the -854 +868 01:31:55,810 --> 01:32:02,303 only little contention among those answering the question is whether this should simplify -855 +869 01:32:02,303 --> 01:32:08,724 to be 1 or should it simplify to be a hundred and Really the way we can think about this -856 +870 01:32:08,724 --> 01:32:15,072 is to just break down that bottom part in terms of what a factorial means I'll go ahead -857 +871 01:32:15,072 --> 01:32:20,772 and do this on the bottom part since I didn't manage my paper real estate very -858 +872 01:32:20,772 --> 01:32:27,121 effectively here the log of 100 times 99 times 98 on and on times 2 times 1 is the same -859 +873 01:32:27,121 --> 01:32:33,469 as adding all of them and You can probably see this mostly cancels out with the top the -860 +874 01:32:33,469 --> 01:32:39,962 only question you might have though Is that in the factorial can we keep multiplying down -861 +875 01:32:39,962 --> 01:32:41,550 until we get that one? -862 +876 01:32:43,010 --> 01:32:45,920 Which doesn't really make a difference, but before you think -863 +877 01:32:45,920 --> 01:32:49,070 about it too much you might wonder Hang on you know on the bottom. -864 -01:32:49,610 --> 01:32:54,140 -I'm multiplying everything, and I'm adding this log of 1 but on the top From what we +878 +01:32:49,610 --> 01:32:52,614 +I'm multiplying everything and I'm adding this log of 1, -865 -01:32:54,140 --> 01:32:58,778 -found before I never saw that log of 1 does that make any difference and the answer is +879 +01:32:52,614 --> 01:32:56,197 +but on the top from what we found before I never saw that log of 1, -866 -01:32:58,778 --> 01:33:03,362 -no Because log of 1 is saying 10 to the power of what equals 1 and the answer is 0 So +880 +01:32:56,197 --> 01:32:58,991 +does that make any difference? And the answer is no, -867 -01:33:03,362 --> 01:33:08,160 -in fact Taking the log of 100 factorial is the same as adding the logs of all the numbers +881 +01:32:58,991 --> 01:33:03,312 +because log of 1 is saying 10 to the power of what equals 1, and the answer is 0. -868 -01:33:08,160 --> 01:33:12,158 -from 2 up to 100 and it simplifies Down to 1 so to those of you who got it +882 +01:33:03,312 --> 01:33:07,845 +So in fact, taking the log of 100 factorial is the same as adding the logs of all the -869 -01:33:12,158 --> 01:33:16,955 -congratulations to those of you who feel like you maybe have better intuitions for Change +883 +01:33:07,845 --> 01:33:12,324 +numbers from 2 up to 100 and it simplifies down to 1. So to those of you who got it, -870 -01:33:16,955 --> 01:33:21,593 -of base formulas for the fact that logarithms turn multiplication into addition That's +884 +01:33:12,324 --> 01:33:16,752 +congratulations. To those of you who feel like you maybe have better intuitions for -871 -01:33:21,593 --> 01:33:26,124 -my hope in the next time with this as a foundation of logarithms that can be pointed +885 +01:33:16,752 --> 01:33:21,389 +change of base formulas for the fact that logarithms turn multiplication into addition, -872 -01:33:26,124 --> 01:33:30,655 -back to What I would love to talk about is what's known as the natural logarithm Log +886 +01:33:21,389 --> 01:33:25,922 +that's my hope. In the next time, with this as a foundation of logarithms that can be -873 -01:33:30,655 --> 01:33:35,346 -base e and try to give an instinct for why that's something that we care about okay Why +887 +01:33:25,922 --> 01:33:30,138 +pointed back to, what I would love to talk about is what's known as the natural -874 -01:33:35,346 --> 01:33:39,983 -is it that mathematicians just seem to really love the number e sitting in the base of +888 +01:33:30,138 --> 01:33:34,829 +logarithm, log base e. And try to give an instinct for why that's something that we care -875 -01:33:39,983 --> 01:33:40,890 -their logarithms? +889 +01:33:34,829 --> 01:33:39,361 +about. Why is it that mathematicians just seem to really love the number e sitting in -876 +890 +01:33:39,361 --> 01:33:40,890 +the base of their logarithms? + +891 01:33:41,190 --> 01:33:42,170 What does that buy for them? -877 -01:33:42,590 --> 01:33:49,910 -Why does it show up in nature? +892 +01:33:42,590 --> 01:33:46,171 +Why does it show up in nature? So that will happen on Friday at the + +893 +01:33:46,171 --> 01:33:49,910 +same time as this lecture, and I look forward to seeing everyone there. diff --git a/2020/ldm-logarithms/english/sentence_timings.json b/2020/ldm-logarithms/english/sentence_timings.json index 91447f6d4..e2ab2b27b 100644 --- a/2020/ldm-logarithms/english/sentence_timings.json +++ b/2020/ldm-logarithms/english/sentence_timings.json @@ -60,7 +60,7 @@ 2289.64 ], [ - "very interesting, we've got a horse race between two so I will give you a moment to think this through Ally Freed So, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 We could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these We could say that the log base 3 of these numbers Just grows in nice little steps So log base 3 of 1, 3 to the what equals 1, the answer is 0.", + "very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0", 2290.16, 2363.08 ], @@ -125,7 +125,7 @@ 2564.62 ], [ - "Because if you look at log of 10,100 it's asking 10 to the what is equal to 10,100 You might say it's gonna be a little above 4 because it's kind of close to 10,000 so the best you might Guess here is oh, this is gonna be something.", + "because if you look at log of 10,100 it's asking 10 to the what is equal to 10,100 you might say, I don't know, it's going to be a little above 4 because it's kind of close to 10,000 so the best you might guess here is oh this is going to be something", 2565.86, 2582.58 ], @@ -290,23 +290,23 @@ 3397.93 ], [ - "They play nicely with each other So a question asks given the fact that 2 to the 10th is 10 24 a thousand and 24 which is approximately 1000 okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th Basically a thousand which of the following is closest to being true Log base 2 of 10 is approximately 0.3 Log base 2 of 10 is approximately sorry log base 10 of 2 is approximately 0.3 Log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third Hey, which of these is closest to being true based on the fact that 2 to the 10th is essentially a thousand I'll give you a little moment for that You Interesting that we've got kind of a split on this one so I'm wondering if they're gonna be numerically pretty similar or if they're gonna be conceptually similar Or if there's even a difference between those two So since answers keep rolling in I'm gonna give this give this a little bit more time so anyone at home watching Hopefully you already have a pencil and paper out to be noodling through these yourself That is the spirit of the lectures that we're doing If you don't now is the time to take out a pencil and paper and See if you can think this one through and write it out some of the problems that we're gonna build to here Definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future even if You can't participate in the live poll.", + "they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference between those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll", 3397.97, 3511.29 ], [ - "I really do think it's a lot of fun to Kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers That you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in You You Okay, so I'll go ahead and grade it now and Let's see how people did on this one So the correct answer is B Which is that the log base 10 of 2 is approximately 0.3 and it looks like 1850 of you correctly got that so congratulations, but the close contender not at all a Unanimous decision here looks like it was D.", + "I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B,", 3511.33, - 3612.97 + 3621.67 ], [ "Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X?", - 3613.31, - 3651.59 + 3621.67, + 3655.31 ], [ "That's the same thing as saying 2 to the X is equal to 10 right.", - 3651.77, + 3655.31, 3656.07 ], [ @@ -345,7 +345,7 @@ 3839.21 ], [ - "They'll be familiar with the idea that Powers of 2 are nice and close to these powers of 10 or more specifically powers of a thousand now What I want to do is just write all of the same things with log base 10 Not not approximately equal to this is actually equal to 3 log base 10 of a thousand.", + "they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million?", 3839.29, 3859.67 ], @@ -440,7 +440,7 @@ 4322.47 ], [ - "But letting you layer it on top of each other Now if we rearrange that expression we get what is probably the second most important of all of our log rules The most important is this top one that when you multiply the inputs you add the outputs But the second most important which is known as the change of base formula Lets us rate that if you want the log base B of some value a Then for whatever C you want it doesn't actually matter what log you have in your pocket if you use that other log And you take the log C of a divided by the log C of B that gives you log base B of a Okay, so just as an example here what this would look like is let's say I wanted to be able to compute log base 100 of things I just really want to it's not a button on my calculator, but I would love to be able to do it and I want to do it in With an input like a million well even if I don't have the log base 100 button on my calculator what I can do is say I'll use the log base 10 button and Evaluate what's on the inside here which at least positionally it's kind of above the 100 it has a higher altitude as we write it so This can line up with the notation a little bit that it sits on the numerator and on the bottom I use the log base 10 button that's in my calculator on the base on the 100 and Then I can evaluate both of those and it'll give me the answer in this case it gets you 6 divided by 2 which will be 3 and If we really just think through what this is saying I know I've said it many different times But it's a it's a convoluted enough way to write things but an intuitive enough fact that I think just coming at it from a bunch of different angles can be important because like I said This is probably the second most important log rule We're asking how many times does 100 go into a million in a multiplicative sense how many times we multiply by itself?", + "but letting you layer it on top of each other. Now if we rearrange that expression, we get what is probably the second most important of all of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if you want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. If you use that other log, and you take the log c of a, divided by the log c of b, that gives you log base b of a. So just as an example here, what this would look like, is let's say I wanted to be able to compute log base 100 of things. I just really want to. It's not a button on my calculator, but I would love to be able to do it. And I want to do it with an input like a million. Well even if I don't have the log base 100 button on my calculator, what I can do is say I'll use the log base 10 button and evaluate what's on the inside here, which at least positionally it's kind of above the 100. It has a higher altitude as we write it. So this can line up with the notation a little bit, that it sits on the numerator. And on the bottom, I use the log base 10 button that's in my calculator on the base, on the 100. And then I can evaluate both of those and it'll give me the answer. In this case it gets you 6 divided by 2, which will be 3. And if we really just think through what this is saying, I know I've said it many different times, but it's a convoluted enough way to write things, but an intuitive enough fact that I think just coming at it from a bunch of different angles can be important. Because like I said, this is probably the second most important log rule. We're asking how many times does 100 go into a million? In a multiplicative sense, how many times do I multiply by itself?", 4323.53, 4439.31 ], @@ -460,7 +460,7 @@ 4465.47 ], [ - "But going about that in different ways So this is extremely nice because it actually lets us compute things next time we're going to talk all about the natural logarithm, which is log base e often written ln and Turns out this is much easier to compute There's nice math behind it such that if you want to come up with an algorithm that your calculator can use It's actually a lot easier to think of log base e of numbers So anytime that you need to go to some kind of calculator.", + "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is log base e, often written ln. And turns out, this is much easier to compute. There's nice math behind it such that if you want to come up with an algorithm that your calculator can use, it's actually a lot easier to think of log base e of numbers. So any time that you need to, go to some kind of calculator.", 4465.55, 4492.49 ], @@ -510,7 +510,7 @@ 4650.71 ], [ - "We've got Kelcon asking what about complex bases since e to the X is walking around the spiral wouldn't it hit every complex number Well it won't hit every complex number, but your instinct that it's hitting multiple things is pretty spot-on Doesn't make sense to talk about logs with imaginary numbers by Nitya so to all of these questions.", + "We've got Kalkan asking, what about complex bases? Wouldn't z to the x is walking around the spiral? Wouldn't it hit every complex number? Well, it won't hit every complex number. But your instinct that it's hitting multiple things is pretty spot on. Doesn't make sense to talk about logs with imaginary numbers by Nitya. So to all of these questions,", 4651.25, 4669.43 ], @@ -590,12 +590,12 @@ 5001.85 ], [ - "Where was it?", + "where was it? Where was it?", 5001.85, 5002.59 ], [ - "Great So when we're converting from log base 10 of something up to log base 2 You know here we were thinking of multiplying the top by 0.3 to get to the bottom But you could also think of multiplying by 10 thirds to get to the top Anytime you have log base 10 of some number you just rescale it and you have log base 2 of that number and the rescaling Constant comes from the log base 2 of 10 or vice versa log base 10 of 2 So that's change of base like I said, it's very important It lets you put everything into a nice universal language and that should be all of the hint that you need for the last question Which is a challenge question on this one.", + "Great. So when we're converting from log base 10 of something up to log base 2, you know here we were thinking of multiplying the top by 0.3 to get to the bottom. But you could also think of multiplying by 10 thirds to get to the top. Anytime you have log base 10 of some number you just rescale it and you have log base 2 of that number. And the rescaling constant comes from the log base 2 of 10. Or vice versa, log base 10 of 2. So that's change of base. Like I said it's very important. It lets you put everything into a nice universal language. And that should be all of the hint that you need for the last question which is a challenge question on this one.", 5005.37, 5040.21 ], @@ -665,7 +665,7 @@ 5569.07 ], [ - "I'm multiplying everything, and I'm adding this log of 1 but on the top From what we found before I never saw that log of 1 does that make any difference and the answer is no Because log of 1 is saying 10 to the power of what equals 1 and the answer is 0 So in fact Taking the log of 100 factorial is the same as adding the logs of all the numbers from 2 up to 100 and it simplifies Down to 1 so to those of you who got it congratulations to those of you who feel like you maybe have better intuitions for Change of base formulas for the fact that logarithms turn multiplication into addition That's my hope in the next time with this as a foundation of logarithms that can be pointed back to What I would love to talk about is what's known as the natural logarithm Log base e and try to give an instinct for why that's something that we care about okay Why is it that mathematicians just seem to really love the number e sitting in the base of their logarithms?", + "I'm multiplying everything and I'm adding this log of 1, but on the top from what we found before I never saw that log of 1, does that make any difference? And the answer is no, because log of 1 is saying 10 to the power of what equals 1, and the answer is 0. So in fact, taking the log of 100 factorial is the same as adding the logs of all the numbers from 2 up to 100 and it simplifies down to 1. So to those of you who got it, congratulations. To those of you who feel like you maybe have better intuitions for change of base formulas for the fact that logarithms turn multiplication into addition, that's my hope. In the next time, with this as a foundation of logarithms that can be pointed back to, what I would love to talk about is what's known as the natural logarithm, log base e. And try to give an instinct for why that's something that we care about. Why is it that mathematicians just seem to really love the number e sitting in the base of their logarithms?", 5569.61, 5620.89 ], @@ -675,7 +675,7 @@ 5622.17 ], [ - "Why does it show up in nature?", + "Why does it show up in nature? So that will happen on Friday at the same time as this lecture, and I look forward to seeing everyone there.", 5622.59, 5629.91 ] diff --git a/2020/ldm-logarithms/english/transcript.txt b/2020/ldm-logarithms/english/transcript.txt index 66aa96f18..56d0389ff 100644 --- a/2020/ldm-logarithms/english/transcript.txt +++ b/2020/ldm-logarithms/english/transcript.txt @@ -10,7 +10,7 @@ maybe you think of drawing the little triangle saying something like we know 100 well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? -very interesting, we've got a horse race between two so I will give you a moment to think this through Ally Freed So, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 We could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these We could say that the log base 3 of these numbers Just grows in nice little steps So log base 3 of 1, 3 to the what equals 1, the answer is 0. +very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. @@ -23,7 +23,7 @@ This is the first one that doesn't seem to have a huge Consensus in one directio Let me just plug in some examples like 10,000 and 100 and I asked myself if I do this zero counting function of what's in that input how many zeros are in it? But it's weird because when we add 10,100 well, we're no longer at a clean power of 10 and okay That's fine. You know often you're taking logarithms of things that aren't clean powers of 10 but it becomes very strange to ask how you express this in terms of log of 100 which was 2 and log of 10,000 Which was 4? -Because if you look at log of 10,100 it's asking 10 to the what is equal to 10,100 You might say it's gonna be a little above 4 because it's kind of close to 10,000 so the best you might Guess here is oh, this is gonna be something. +because if you look at log of 10,100 it's asking 10 to the what is equal to 10,100 you might say, I don't know, it's going to be a little above 4 because it's kind of close to 10,000 so the best you might guess here is oh this is going to be something That's kind of like The log of 10,000, but that just feels like a coincidence based on the two numbers that we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Sometimes you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? @@ -56,8 +56,8 @@ So we might say that there's some kind of Constant s that we're adding to this R This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. -They play nicely with each other So a question asks given the fact that 2 to the 10th is 10 24 a thousand and 24 which is approximately 1000 okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th Basically a thousand which of the following is closest to being true Log base 2 of 10 is approximately 0.3 Log base 2 of 10 is approximately sorry log base 10 of 2 is approximately 0.3 Log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third Hey, which of these is closest to being true based on the fact that 2 to the 10th is essentially a thousand I'll give you a little moment for that You Interesting that we've got kind of a split on this one so I'm wondering if they're gonna be numerically pretty similar or if they're gonna be conceptually similar Or if there's even a difference between those two So since answers keep rolling in I'm gonna give this give this a little bit more time so anyone at home watching Hopefully you already have a pencil and paper out to be noodling through these yourself That is the spirit of the lectures that we're doing If you don't now is the time to take out a pencil and paper and See if you can think this one through and write it out some of the problems that we're gonna build to here Definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future even if You can't participate in the live poll. -I really do think it's a lot of fun to Kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers That you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in You You Okay, so I'll go ahead and grade it now and Let's see how people did on this one So the correct answer is B Which is that the log base 10 of 2 is approximately 0.3 and it looks like 1850 of you correctly got that so congratulations, but the close contender not at all a Unanimous decision here looks like it was D. +they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference between those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll +I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. @@ -67,7 +67,7 @@ X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is abou We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? -They'll be familiar with the idea that Powers of 2 are nice and close to these powers of 10 or more specifically powers of a thousand now What I want to do is just write all of the same things with log base 10 Not not approximately equal to this is actually equal to 3 log base 10 of a thousand. +they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? @@ -86,11 +86,11 @@ Multiplying those should give me the answer to how many times 10 goes into a mil well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying the entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each other the whole statement is equal to saying that um suppose that B to the X is equal to a Got some number B. You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say out loud But if you plug in some numbers to translate what all of that is really saying in the context of the example we just did Saying if you can write a hundred is ten squared and if you can write a million as a hundred cubed Well that lets you write a million in terms of a power of ten Okay, so sort of asking this question How many times does one number go into another? -But letting you layer it on top of each other Now if we rearrange that expression we get what is probably the second most important of all of our log rules The most important is this top one that when you multiply the inputs you add the outputs But the second most important which is known as the change of base formula Lets us rate that if you want the log base B of some value a Then for whatever C you want it doesn't actually matter what log you have in your pocket if you use that other log And you take the log C of a divided by the log C of B that gives you log base B of a Okay, so just as an example here what this would look like is let's say I wanted to be able to compute log base 100 of things I just really want to it's not a button on my calculator, but I would love to be able to do it and I want to do it in With an input like a million well even if I don't have the log base 100 button on my calculator what I can do is say I'll use the log base 10 button and Evaluate what's on the inside here which at least positionally it's kind of above the 100 it has a higher altitude as we write it so This can line up with the notation a little bit that it sits on the numerator and on the bottom I use the log base 10 button that's in my calculator on the base on the 100 and Then I can evaluate both of those and it'll give me the answer in this case it gets you 6 divided by 2 which will be 3 and If we really just think through what this is saying I know I've said it many different times But it's a it's a convoluted enough way to write things but an intuitive enough fact that I think just coming at it from a bunch of different angles can be important because like I said This is probably the second most important log rule We're asking how many times does 100 go into a million in a multiplicative sense how many times we multiply by itself? +but letting you layer it on top of each other. Now if we rearrange that expression, we get what is probably the second most important of all of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if you want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. If you use that other log, and you take the log c of a, divided by the log c of b, that gives you log base b of a. So just as an example here, what this would look like, is let's say I wanted to be able to compute log base 100 of things. I just really want to. It's not a button on my calculator, but I would love to be able to do it. And I want to do it with an input like a million. Well even if I don't have the log base 100 button on my calculator, what I can do is say I'll use the log base 10 button and evaluate what's on the inside here, which at least positionally it's kind of above the 100. It has a higher altitude as we write it. So this can line up with the notation a little bit, that it sits on the numerator. And on the bottom, I use the log base 10 button that's in my calculator on the base, on the 100. And then I can evaluate both of those and it'll give me the answer. In this case it gets you 6 divided by 2, which will be 3. And if we really just think through what this is saying, I know I've said it many different times, but it's a convoluted enough way to write things, but an intuitive enough fact that I think just coming at it from a bunch of different angles can be important. Because like I said, this is probably the second most important log rule. We're asking how many times does 100 go into a million? In a multiplicative sense, how many times do I multiply by itself? But division is asking that same question in an additive sense if I say how many times does the log of 100 go into the? Log of a million that's what division means it's saying how many times do I add this bottom number to itself to get to the top? But anything additive in the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the left-handed side and the right-hand side are just saying how many times does 100 go into a million? -But going about that in different ways So this is extremely nice because it actually lets us compute things next time we're going to talk all about the natural logarithm, which is log base e often written ln and Turns out this is much easier to compute There's nice math behind it such that if you want to come up with an algorithm that your calculator can use It's actually a lot easier to think of log base e of numbers So anytime that you need to go to some kind of calculator. +but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is log base e, often written ln. And turns out, this is much easier to compute. There's nice math behind it such that if you want to come up with an algorithm that your calculator can use, it's actually a lot easier to think of log base e of numbers. So any time that you need to, go to some kind of calculator. I don't know let's say we popped over to something like Desmos Always happy to have as a friend, and you wanted to compute log base 10 of Some number you know let's say we're doing log base 10 of 57 and it looks like that's you know it should make sense. It's between 1 and 2 because 57 was between 10 and 100 What's going on under the hood? How is it actually figuring this out? @@ -100,7 +100,7 @@ What do we have? Use the approximation that log base 2 of 10 is around 10 thirds so using that approximation Which of the following is approximately true? Log base 2 of X is about 10 thirds of log base 10 of X Log base 2 of X is log base 10 of 10 thirds of X log base 2 of X is log base 10 of X to the power 10 thirds or Log base 2 of X is 10 thirds times the log base 10 of X and finally none of the above So I'll give you a moment to think about that you might want to think to the chart that we were drawing earlier and thinking about log base 2 and log base 10 as we're looking at powers of 1000 and how each of those grows that can leverage some of The intuition but I'll let you think about it. Stop talking You While answers are rolling in it looks like a number of people have been asking on Twitter about Basically how complex numbers play into this so you know we've got Jamil asking What if the base is imaginary? -We've got Kelcon asking what about complex bases since e to the X is walking around the spiral wouldn't it hit every complex number Well it won't hit every complex number, but your instinct that it's hitting multiple things is pretty spot-on Doesn't make sense to talk about logs with imaginary numbers by Nitya so to all of these questions. +We've got Kalkan asking, what about complex bases? Wouldn't z to the x is walking around the spiral? Wouldn't it hit every complex number? Well, it won't hit every complex number. But your instinct that it's hitting multiple things is pretty spot on. Doesn't make sense to talk about logs with imaginary numbers by Nitya. So to all of these questions, It's actually a very Nuanced question the short answer is yes complex logarithms absolutely exist But each one of them has multiple different outputs so a good a good analogy here is how? If we have the square root function if I ask something like the square root of 5 You know we have the convention that you always do the positive amount, but that doesn't quite feel honest It feels like the right answer is to specify that there's two different outputs for the square root function two solutions to x squared equals 25 And this is actually true in complex numbers as well one of the things we talked about in a complex number lecture Was that you can take the square root of I and you actually get root 2 over 2 plus root 2 over 2 I? But that there's two solutions you can do plus or minus this value and so you could say that the square root function Isn't a function, but it's a multi-valued function It always has two different outputs now something funky happens when we have exponents at play So if you're just like someone who hasn't seen this stuff don't worry. @@ -116,8 +116,8 @@ And Well, what is 1 divided by the log base 10 of 2? so we're going to keep our log base 10 of X out here and From what we found earlier We found that log base 10 of 2 was approximately 3 tenths was approximately 0.3. So when we divide by it that should get us 10 thirds Okay, and this lines up with what we were looking at a little bit earlier with powers of 1000. Let's see. -Where was it? -Great So when we're converting from log base 10 of something up to log base 2 You know here we were thinking of multiplying the top by 0.3 to get to the bottom But you could also think of multiplying by 10 thirds to get to the top Anytime you have log base 10 of some number you just rescale it and you have log base 2 of that number and the rescaling Constant comes from the log base 2 of 10 or vice versa log base 10 of 2 So that's change of base like I said, it's very important It lets you put everything into a nice universal language and that should be all of the hint that you need for the last question Which is a challenge question on this one. +where was it? Where was it? +Great. So when we're converting from log base 10 of something up to log base 2, you know here we were thinking of multiplying the top by 0.3 to get to the bottom. But you could also think of multiplying by 10 thirds to get to the top. Anytime you have log base 10 of some number you just rescale it and you have log base 2 of that number. And the rescaling constant comes from the log base 2 of 10. Or vice versa, log base 10 of 2. So that's change of base. Like I said it's very important. It lets you put everything into a nice universal language. And that should be all of the hint that you need for the last question which is a challenge question on this one. So if anyone's been watching and they've been like I know logarithms. I've got this completely down Let me pull up the last question which came up on let's see It's not the AMC, but it's whatever the predecessor to the AMC was I believe So it's you know, it's gonna involve a little cleverness and manipulation We've got this long Sequence of fractions. Okay, you take 1 divided by the log base 2 of a hundred factorial And remember 100 factorial is a hundred times 99 times 98 on the way all the way down to 1 so 1 divided by the log base 2 of that plus 1 divided by the log base 3 of 100 factorial Plus 1 divided by log base 4 of 100 factorial on and on and on up until 1 divided by the log base 100 of 100 factorial So this looks rather intimidating, right? @@ -131,6 +131,6 @@ How many times does the log of to divide into the log of a hundred factorial? It doesn't even matter what logarithm you use this log could be base 10 base e base 2 doesn't matter this change of base formula Still holds now what that means for our expression is instead of taking one divided by log base 2 of 100 factorial I could write that as log of 2 Divided by so let me draw a little dividing line between what I'm doing here log of 2 divided by log of 100 factorial Okay, and then similarly? log of 3 divided by log of 100 factorial so all I'm doing here is taking The reciprocal so instead of taking log of 100 factorial over log of 2 I have inverted it log of 2 over 100 factorial because that's what this reciprocal is doing so with that as the beginning I'm gonna go ahead and just grade this lock in the answers and See how see how everyone's doing So it looks like awesome around 1796 we always we always draw a little bit north of Ramanujan's number around 1800 of you Answered that it's one which is correct congratulations 70 of you answered 100 which we can maybe see where where that our error would have come from Those of you that answered zero that's interesting it would imply that somehow you have cancellation at play Because these are all positive numbers so thinking one of them was negative So you could probably gut check against the idea that it would be zero 41 of you want me to come up with a number fun numerical fact about 69, but I won't 28 of you said 2 and I think yeah, I think that's I Think the the difference between 1 and 100 maybe would be interesting to try to analyze, but if we go back to our answer I Realize it may be a little bit confusing how I've heard this this isn't this is not a big fraction this was just a dividing line between Not true ironically trying not to confuse my fractions with each other so rewriting our expression up on the top here as we add all of these things and we can continue up until log of 100 all divided by log of 100 factorial The key is that now all of the denominators are the same so we can add the numerators No more plus sign is just going to equal something that top is going to look like log of 2 Plus log of 3 plus on and on up to log of 100 all divided by the log of 100 factorial and it seems like the only little contention among those answering the question is whether this should simplify to be 1 or should it simplify to be a hundred and Really the way we can think about this is to just break down that bottom part in terms of what a factorial means I'll go ahead and do this on the bottom part since I didn't manage my paper real estate very effectively here the log of 100 times 99 times 98 on and on times 2 times 1 is the same as adding all of them and You can probably see this mostly cancels out with the top the only question you might have though Is that in the factorial can we keep multiplying down until we get that one? Which doesn't really make a difference, but before you think about it too much you might wonder Hang on you know on the bottom. -I'm multiplying everything, and I'm adding this log of 1 but on the top From what we found before I never saw that log of 1 does that make any difference and the answer is no Because log of 1 is saying 10 to the power of what equals 1 and the answer is 0 So in fact Taking the log of 100 factorial is the same as adding the logs of all the numbers from 2 up to 100 and it simplifies Down to 1 so to those of you who got it congratulations to those of you who feel like you maybe have better intuitions for Change of base formulas for the fact that logarithms turn multiplication into addition That's my hope in the next time with this as a foundation of logarithms that can be pointed back to What I would love to talk about is what's known as the natural logarithm Log base e and try to give an instinct for why that's something that we care about okay Why is it that mathematicians just seem to really love the number e sitting in the base of their logarithms? +I'm multiplying everything and I'm adding this log of 1, but on the top from what we found before I never saw that log of 1, does that make any difference? And the answer is no, because log of 1 is saying 10 to the power of what equals 1, and the answer is 0. So in fact, taking the log of 100 factorial is the same as adding the logs of all the numbers from 2 up to 100 and it simplifies down to 1. So to those of you who got it, congratulations. To those of you who feel like you maybe have better intuitions for change of base formulas for the fact that logarithms turn multiplication into addition, that's my hope. In the next time, with this as a foundation of logarithms that can be pointed back to, what I would love to talk about is what's known as the natural logarithm, log base e. And try to give an instinct for why that's something that we care about. Why is it that mathematicians just seem to really love the number e sitting in the base of their logarithms? What does that buy for them? -Why does it show up in nature?. \ No newline at end of file +Why does it show up in nature? So that will happen on Friday at the same time as this lecture, and I look forward to seeing everyone there. \ No newline at end of file diff --git a/2020/ldm-logarithms/french/sentence_translations.json b/2020/ldm-logarithms/french/sentence_translations.json index 1bb4bc31e..6bfb7ebe8 100644 --- a/2020/ldm-logarithms/french/sentence_translations.json +++ b/2020/ldm-logarithms/french/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Musique🎵 Bienvenue à Lockdown Math. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "Aujourd'hui, nous allons parler de logarithmes et d'une sorte de leçon de retour aux bases. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "Et comme toujours, pour commencer, je veux juste avoir une idée de la situation actuelle du public. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "Donc, si vous pouvez passer à 3b1b. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "Je n'en ai jamais entendu parler auparavant ou je n'en ai jamais entendu parler auparavant b. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "J'en ai entendu parler, mais je suis parfois confus par toutes leurs propriétés. c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "Je les comprends mais je ne saurais pas comment leur enseigner et d. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "Je les comprends bien et je pourrais facilement les enseigner à quelqu'un d'autre pour bien leur faire comprendre aussi. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "Nous avons donc une bonne répartition. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "Comme je l'ai dit, l'intention est de créer une leçon vers laquelle je peux indiquer aux gens à l'avenir s'ils ne sont tout simplement pas à l'aise avec les logarithmes et je veux pouvoir dire, oh, voici un endroit où vous pouvez aller. comment je pense, vous savez, comment je pense que vous pourriez l'aborder intuitivement. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "Parce que j'ai parcouru quelques forums d'enseignants avant de faire ce cours en particulier et que lorsque les gens me demandent quel est le sujet le plus difficile à enseigner en mathématiques au lycée dans le sens où les étudiants semblent avoir le plus de problèmes avec cela, les logarithmes sont l'un des sujets les plus difficiles à enseigner. réponses communément indiquées, ce qui est intéressant et je peux supposer que c'est peut-être parce qu'il y a une tonne de ces propriétés que vous finissez par devoir apprendre, vous savez, donc si nous sautons avant où nous allons aller, vous avez toutes ces piles de des règles qui ressemblent à un tas d'algèbre qui peuvent être difficiles à retenir et faciles à mélanger dans votre tête et je pense que quand les gens ont, vous savez, ce genre de souvenirs cauchemardesques de ce qu'étaient les mathématiques au lycée et de ce que que les logarithmes ont fait pour eux, ce sont souvent ces formules particulières qui me viennent à l'esprit et ce que je veux faire aujourd'hui, c'est essayer d'en parler, comment y penser, mais aussi juste au niveau méta, si vous enseignez l'algèbre à quelqu'un, que sont-ils ? les points à souligner ? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "Quelle est la manière de l’intégrer dans leurs intuitions ? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "oh, il y a 3 zéros dessus, c'est quoi un log d'un million ? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "le log de 1000 fois x est égal à 3 fois le log de x et rappelez-vous que nous utilisons la convention selon laquelle il s'agit du log b en base 10. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "le log de 1 000 fois x est égal au log de x au cube c. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "log de 1000 fois x est égal à 3 à la puissance log de x et e. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "aucune de ces réponses et rappelez-vous que, comme je l'ai dit plus tôt, nous devons nous attendre à ce que toutes ces personnes au début qui ont déclaré bien comprendre les journaux répondront immédiatement, elles répondront correctement, mais si vous êtes quelqu'un qui ne le fait pas, ne vous laissez pas intimider lorsque vous examinez un problème comme celui-ci, ce que je vous encourage à faire est simplement de brancher diverses puissances de 10 et de penser en termes d'idée que la fonction de journalisation compte le nombre de zéros donc je vais vous donner un petit moment pour y réfléchir donc je vais aller de l'avant et le noter et comme toujours si c'est plus rapide que ce avec quoi vous êtes à l'aise, sachez que c'est uniquement parce que je veux aller de l'avant avec la leçon donc dans ce cas, la bonne réponse s'avère être un journal de 1000 fois x équivaut à prendre 3 plus le journal de x et maintenant réfléchissons-y un instant et comme je l'ai dit quand vous commencez tout juste avec eux, je pense que la meilleure chose à faire est simplement d'être à l'aise en branchant différents nombres et les meilleurs nombres à brancher sont ceux qui sont déjà des puissances de 10, donc si vous demandez quelque chose comme un journal de 1000 fois x eh bien, je ne le fais pas. Je ne sais pas, branchons simplement quelque chose pour x log de 1000 fois 100 et bien nous savons combien de zéros il y aura dans la réponse finale ici et bien 1000 fois 100 fait 100 000 nous avons déjà intuitivement cette idée que lorsque nous multiplions 2 puissances de 10 nous prenons juste les zéros, les 3 zéros de ces 1000, les 2 zéros de ces 100 et nous les mettons les uns à côté des autres donc cela devrait être 5 zéros au total, mais si vous réfléchissez vraiment non seulement à la façon dont le nombre a tourné mais pourquoi cela s'est-il passé ainsi, c'étaient les 3 zéros de ces 1000 plus les 2 zéros de ces 100 que l'on pourrait aussi écrire en disant le nombre de zéros dans 1000 plus le nombre de zéros dans 100 donc cette idée qu'un logarithme du produit de deux choses est la somme des logarithmes de ces deux choses dans le contexte de puissances de 10, cela ne fait que communiquer ce qui est déjà une idée super intuitive pour beaucoup d'entre nous si vous prenez 2 puissances de 10 et que vous les multipliez. prenez tous leurs zéros et placez-les les uns sur les autres, donc la façon dont j'ai écrit les choses ici indique en fait un fait légèrement plus général qui sera notre toute première propriété des logarithmes, c'est-à-dire que si nous prenons le log de A multiplié par B, cela équivaut au log de A plus le log de B maintenant, chaque fois que vous voyez l'une de ces règles de logarithme, si vous plissez les yeux ou si vous ne savez pas comment vous en souvenir, branchez simplement des exemples Je suis redondant, je le dis souvent mais c'est parce que je pense que c'est très facile d'oublier une fois qu'on est submergé par l'algèbre elle-même et qu'on est assis sur une sorte de test et qu'il y a juste beaucoup de symboles. pour vous rappeler que vous pouvez simplement insérer quelques chiffres, c'est une bonne chose à faire et c'est souvent un excellent moyen de donner de l'intuition, donc dans ce cas, en disant le journal de A fois B et en le décomposant, nous pourrions simplement penser, oh, que log de 100 fois 1000 qui fait 5, il y a 5 zéros dedans se décompose en termes de nombre de zéros dans chaque partie donnée super, merveilleux donc en poussant cette intuition plus loin, essayons un autre problème pratique et encore, si vous le savez, super, vous serez en mesure d'y répondre correctement, mais réfléchissez peut-être non seulement à quelle est la réponse, mais comment pourrais-je expliquer cette réponse à quelqu'un ou comment pourrais-je essayer d'amener un élève à parvenir à cette réponse par lui-même sans que j'aie à le dire leur quelle est la réponse, donc il y a deux membres potentiels du public : ceux qui sont intéressés par la leçon elle-même et ensuite ceux qui sont intéressés par la méta-leçon, donc notre question demande, encore une fois, laquelle des affirmations suivantes est vraie ? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "un. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "le log de x au n est égal à n fois le log de x b. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "le log de x à la n est égal au log de x à la puissance n c. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "le log de x au n est égal à n plus le log de x ou d. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "donc la bonne réponse ici est a, et il semble que 4 000 d'entre vous ont reçu des félicitations, nous disant que le log de x à la puissance n est égal à n fois le log de x donc, encore une fois, disons que vous essayez d'enseigner cela à quelqu'un ou si vous essayez de comprendre vous-même ce que cela signifie, je pense qu'un bon point de départ est de brancher quelque chose et dans ce cas, pour le journal de x à la puissance n, essayons simplement avec 100 à la puissance 3 et vous pouvez l'essayer avec d'autres pour voir si les modèles que vous faites fonctionnent réellement, mais si vous y réfléchissez non pas simplement en voyant quelle est la réponse, mais en essayant de réfléchir à la raison pour laquelle la réponse s'est avérée ainsi. parfois un exemple suffira parce que 100 au cube, nous pouvons considérer cela comme une bonne prise, cela fait 3 copies de 100, je prends 3 copies de 100 et quand je multiplie tout cela et je pense que le journal compte le nombre de zéros que nous dis, oh, ça va être un nombre qui aura juste 6 zéros, c'est ce que cela signifie de prendre 100 fois 100 fois 100, je peux juste penser à regrouper tous ces zéros pour obtenir un million donc ce nombre va être 6 mais si nous pensons en fait pourquoi était-ce 6 non seulement c'est le nombre de zéros à l'intérieur du million d'où vient ce 6, c'est que nous avions 3 copies de ces 100 et chacun de ces 100 avait 2 zéros différents donc de cette façon c'est un résultat plus général façon dont vous pouvez y penser, si au lieu de prendre 100 au cube nous regardions 1000 au cube ou 1000 puissance n ou x à la puissance n vous pouvez penser que quelle que soit la valeur de n était le nombre de copies que nous multipliions en fois le nombre de eh bien, voyons, ce n'est pas x fois le nombre de zéros qui étaient dans tout ce que nous avons remplacé par x qui dans ce cas était 100 donc si à la place j'avais pris quelque chose comme log de 10 000 à la puissance n ce serait pareil en prenant n copies de ces 10 000 en comptant le nombre de zéros dans chacun d'eux qui est 4, ce serait donc n fois 4 et bien sûr, la propriété générale à laquelle la plupart d'entre vous ont répondu correctement est que vous avez ce joli petit effet où lorsque vous voyez le journal de quelque chose élevé à une puissance, cette petite puissance saute devant lui et vous avez juste un journal de ce qu'il y avait à l'intérieur maintenant, l'une des implications peut-être les plus importantes de cela, je ne sais pas si vous l'appelleriez une implication ou si vous appelleriez cela une reformulation de la définition si je prends le journal et je vais juste souligner à nouveau que c'est une base 10 sur 10 à la puissance n, nous pouvons en quelque sorte penser à ce petit n comme sautant dans devant et cela devient n fois le log de base 10 de 10 qui est bien sûr 1, cette expression que vous pouvez considérer comme comptant le nombre de zéros à la fin ou plus généralement, elle demande 10 à ce qui est égal à 10 et la réponse est simplement 1 ce qui est très rassurant car une autre façon de revenir en arrière et de simplement lire cette expression originale est de dire 10 à ce qui est égal à 10 à n oh eh bien, la réponse est non ok maintenant avec chaque propriété de logarithme donnée que nous avons, donc dans ce cas, nous Je viens de trouver un log de x à la puissance n implique que n sautant devant, il y aura toujours une propriété exponentielle d'image miroir et c'est une autre façon dont nous pouvons nous aider à avoir un peu d'intuition pour ceux-ci, alors laissez-moi juste couvrir certaines des propriétés futures auxquelles nous allons arriver ici essaient de cacher où nous allons ce que nous venons de trouver en élevant quelque chose au n qui saute devant, cela correspond à la propriété exponentielle selon laquelle si je prends 10 en x et que j'augmente tout cela à la puissance n, c'est la même chose que de prendre 10 à la puissance n fois x et cela nous amène à une autre intuition que vous pourriez avoir pour les logarithmes, c'est-à-dire qu'ils sont en quelque sorte comme une exponentiation retournée et voici ce que je veux dire par que la chose qui se trouve à l'intérieur du journal, si je prends le journal d'un, vous devriez considérer cela comme l'expression extérieure de quelque chose qui est exponentiel dans ce cas, le a, la chose à l'intérieur correspond à 10 au x le sortie de la fonction alors que le tout lui-même, le journal de a correspond à ce qui se trouve à l'intérieur ici, quel est l'exposant du 10, donc partout où vous voyez une expression de journal ici, vous devriez penser qu'elle joue le rôle d'un exposant à droite côté et chaque fois que vous voyez une exponentielle, le 10 entier de l'expression x, tout le composant extérieur du côté droit qui correspond à quelque chose qui se trouve à l'intérieur de l'un des journaux et nous avons vu cela ci-dessus, l'idée que lorsque nous multiplions à l'intérieur, cela s'ajoute à l'extérieur, et bien, si les journaux tournent en quelque sorte les exponentielles à l'envers, cela nous dit que multiplier à l'extérieur, multiplier les sorties de la fonction équivaut à ajouter à l'intérieur, car chacun de ces journaux comme le journal a et le journal b joue le rôle du x et du y dans l'expression de droite, alors continuons à jouer, faisons-en quelques autres et voyons pour combien de ces propriétés nous pouvons construire une intuition pour cette dernière, très sympa de penser aux exposants descendant dans le suivant est quelque chose qui peut paraître un peu bizarre à ceux qui ne sont pas nécessairement familiers avec les logarithmes mais encore une fois, branchez quelques nombres pour avoir une certaine intuition et nous allons lui en donner un peu plus un moment pour déterminer lequel des énoncés suivants est vrai ? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "eh bien, si 10 au cube équivaut à 1000, cela revient à dire que 10 est égal à 1000 élevé au tiers, faire l'inverse ici implique l'inverse multiplicatif de l'exposant et la façon dont cela se déroule est que cela ressemble à 1 divisé par 3 et que 3 correspond au log de base 10 de 1000 c'est 1 divisé par le log de base 10 de 1000 donc plus généralement, vous pourriez deviner à partir de cet exemple unique que lorsqu'on échange la base avec ce qu'il y a à l'intérieur cela correspond à prendre 1 divisé par ce qu'il y a à l'extérieur ici et encore, vous pouvez réfléchir à cela en regardant la règle exponentielle correspondante. Maintenant, qu'est-il arrivé à mon joli petit journal et à mes exponentielles ? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "merveilleux donc, encore une fois, cachons où certaines des choses certaines des autres propriétés que nous aborderons ici et je les garderai dans le même ordre que je l'avais avant ici. Je pensais que l'avoir pré-écrit pourrait me garder un peu plus propre que d'habitude mais peut-être que cela implique simplement de jouer à ce jeu étrange de découpe de papier, donc ce que nous venons de trouver, enregistrez la base b de a si vous les échangez, c'est la même chose que de diviser par 1 ce à quoi cela correspond, sur un le terrain exponentiel, c'est que si vous prenez b à une certaine puissance et dites que cela est égal à a, c'est la même déclaration que dire que a à l'inverse de cette puissance est égal à b à nouveau, il est plutôt utile de prendre un moment et de penser aux logarithmes comme faisant tourner les choses à l'envers, l'expression log base b de a joue le rôle de ce x et l'expression log base a de b joue le rôle de tout ce qui se trouve au-dessus de a puis symétriquement, toute l'expression b à la puissance x joue le rôle de l'intérieur à gauche, il joue le rôle du a et l'expression entière, a à la puissance de quelque chose joue le rôle de ce qui se trouve à l'intérieur de la base du journal a donc vous pouvez voir, simplement en branchant quelques exemples et en le faisant correspondre aux règles exponentielles, nous pouvons déjà penser à trois règles de logarithme différentes qui, si elles étaient simplement transmises comme des morceaux d'algèbre à mémoriser, vous savez, vous pourriez les mémoriser mais il est très facile pour elles de glisser hors de votre tête et il est également très facile d'être frustré par la tâche à accomplir, mais vous voudrez peut-être vous rappeler que la raison pour laquelle nous nous soucions de ce genre de choses est que comprendre les règles des logarithmes nous aide à faire des mathématiques dans des contextes où c'est comme un virus qui se développe là où d'un jour à l'autre, d'une étape à l'autre, les choses ont tendance à se développer de manière multiplicative, comprendre les règles des logarithmes vous aide à avoir une meilleure idée de ce genre de choses, donc avant de donner un bel exemple concret de ce à quoi cela peut ressembler comme laissez-moi juste poser une dernière question de quiz dans cette veine pour poser une dernière question sur les propriétés des logarithmes avant de passer à un petit exemple du monde réel, débarrassez-vous de ce que nous avions ici et maintenant, lequel des énoncés suivants est vrai ? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "le log de a plus b est le même que le log de a plus le log de b le log de a plus b est égal au log de a fois le log de b le log de a plus b est égal à un divisé par le log de a plus le log de b ou le log de a plus b est égal à un divisé par le log de a multiplié par le log de b ou aucun des éléments ci-dessus ah, et maintenant nous n'avons pas autant de consensus, n'est-ce pas ? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "très intéressant, nous avons une course de chevaux entre deux donc je vais vous donner un moment pour y réfléchir pendant que les gens répondent, en fait j'ai une petite question pour le public donc, vous savez, je parlais juste de la façon dont nous pourrions pensez en termes de croissance multiplicative et cela ne doit pas nécessairement être des puissances de dix, nous pourrions aussi faire quelque chose comme des puissances de trois où si vous passez de un à trois à neuf à vingt-sept à quatre-vingt-un, tout parmi ceux-ci, nous pourrions dire que le log de base trois de ces nombres augmente simplement par petites étapes, donc log de base trois de un, trois jusqu'à ce qui est égal à un, la réponse est zéro en général, le log de un, quelle que soit la base, sera être zéro log base trois sur trois, trois à ce qui est égal à trois est un de la même manière log base trois sur neuf est deux ah, vous vous demandez peut-être quelle est ma question, mais cela m'aidera à tirer tout cela et pour mon propre plaisir ici, permettez-moi juste d'écrire un autre journal de base trois sur quatre-vingt-un fait quatre maintenant, j'ai entendu dire que si vous demandez à un enfant, disons vers cinq ou six ans, quel nombre est à mi-chemin entre un et neuf dire quel nombre est à mi-chemin, leur instinct pour savoir comment répondre est logarithmique alors que nos instincts ont tendance à être plus linéaires donc nous pensons souvent un et neuf, vous avez un tas de nombres régulièrement espacés entre eux deux, trois, quatre, cinq, six , sept, huit et si vous allez à mi-chemin entre les deux, vous atterrirez sur cinq, mais si vous réfléchissez en termes de croissance multiplicative, où passer de un à neuf, il ne s'agit pas d'ajouter un tas de choses mais vous 'vous grandissez d'un certain montant, vous grandissez d'un facteur de trois, puis vous grandissez d'un autre facteur de trois, soi-disant, l'instinct naturel d'un enfant s'aligne avec le fait de dire trois et soi-disant cela s'aligne également avec si vous avez des anthropologues qui étudient des sociétés qui ont' Je n'ai pas développé les systèmes de comptabilité et d'écriture de la même manière que les sociétés modernes, ils répondront à trois pour cela, donc, ma question pour le public, si l'un d'entre vous qui nous regarde en ce moment a accès à un petit enfant, disons, dans un délai de cinq ans. vieux, vois si tu peux aller leur demander quel nombre est à mi-chemin entre un et neuf et si tu peux, dis-nous sur Twitter ce que dit l'enfant quelle est sa vraie réponse parce que je ne sais pas pourquoi, je suis juste un petit peu Je suis sceptique quant à savoir si cela se concrétise réellement dans la pratique. Je comprends que ce n'est pas une façon super scientifique de le faire. Je ne demande pas aux gens qui regardent un livestream YouTube d'interroger leurs propres enfants, puis de tweeter la réponse, mais pour mon propre bien, ce serait intéressant. pour voir une sorte de validation de notre question, c'est la première qui ne semble pas avoir un énorme consensus dans une direction, allons-y et notons-la pour voir quelle est la réponse qui s'avère excellente, d'accord, donc 2 400 d'entre vous ont répondu correctement que ce n'est rien de ce qui précède, que le log de a plus b ne satisfait aucune de ces belles propriétés et en général, à moins que nous ne travaillions avec certains types d'approximations, en particulier lorsque le log naturel entre en jeu nous pourrions en parler la prochaine fois, ajouter les entrées d'un logarithme est en fait une sensation très étrange, c'est une chose très étrange à faire et pour avoir une idée de cette bizarrerie, branchez des puissances de dix si je vous demande le journal de a plus b ce que vous pourriez commencer à penser, c'est, d'accord, laissez-moi juste brancher quelques exemples comme 10 000 et 100 et je me demande, si je fais cette fonction de comptage de zéros de ce qu'il y a dans cette entrée, combien de zéros y a-t-il ? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "c'est une question intéressante, d'accord, la base d'un logarithme peut-elle être nulle ? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "eh bien, en termes de notre triangle, nous pourrions penser que cela signifie que vous savez, zéro à une sorte de puissance x est égal à une autre valeur y, c'est quelque chose que nous pourrions écrire soit en disant zéro à x est égal à y, soit nous pourrions écrire la même chose en disant que le log de base zéro de y est égal à x zéro à ce qui est égal à x maintenant le problème ici est que zéro à n'importe quoi finit par être zéro, c'est vrai, donc si nous pensons simplement au log de base zéro de y pour toute autre entrée y vous savez, vous voulez saisir quelque chose comme un ou deux ou pi tout ce que vous voulez, vous posez la question zéro à ce qui est égal à un ou deux ou pi ou quel que soit le nombre que vous pourriez avoir là et il n'y aura tout simplement pas de réponse, donc au mieux vous pourriez essayer de dire oh oui, log de zéro, c'est une fonction parfaitement valide, elle n'est définie que sur l'entrée zéro mais même dans ce cas, vous auriez du mal à essayer de trouver ce que vous voulez là parce que dire zéro à ce qui est égal à zéro c'est comme si tout s'appliquait à ça donc ton bras va être tordu derrière ton dos cependant tu veux que ça marche et ça correspond au fait que la fonction exponentielle de base zéro est entièrement nulle ça ne mappe pas les nombres de manière agréable les uns sur les autres, c'est donc une excellente question, pouvez-vous avoir une base de journal zéro maintenant pour revenir à l'idée de l'endroit où ces choses surviennent dans le monde réel, un exemple que j'aime bien est l'échelle de Richter pour les tremblements de terre, donc l'échelle de Richter nous donne une quantification de la force d'un tremblement de terre et cela peut aller de très petits nombres à de très grands nombres comme, je pense, le plus grand tremblement de terre jamais mesuré et ceci n'est qu'un graphique qui vient de Wikipédia avait un 9.5 et pour comprendre à quel point cela est insensé, cela vaut la peine d'examiner la relation entre ce que signifient ces chiffres et ensuite quelque chose comme la quantité équivalente de TNT, une sorte de mesure de la quantité d'énergie qu'il contient et ensuite ce que nous pouvons essayer de faire ici est de voir si nous pouvons obtenir une expression pour le nombre de l'échelle de Richter en termes de quantité d'énergie et pourquoi les logarithmes seraient un moyen naturel de décrire cela, donc la clé sur laquelle se concentrer est, à mesure que nous progressons, dans quelle mesure les choses augmentent donc par exemple, si nous passons de deux dans ce cas, cela ne nous montre pas où se trouve trois, alors peut-être pensons-nous faire un pas de deux à quatre, ce qui revient un peu à faire deux pas, qu'est-ce que cela fait en termes de quantité d'énergie, on dirait que cela nous prend d'une tonne métrique de TNT qui est, je suppose, une grosse bombe de la Seconde Guerre mondiale et cela nous prend jusqu'à un kilotonne mille fois plus, ce qui est une petite bombe atomique, donc juste deux étapes sur l'échelle de Richter, passer d'un tremblement de terre de magnitude 2 à un tremblement de terre de magnitude 4 nous fait passer de la grosse bombe de la Seconde Guerre mondiale à l'ère nucléaire, donc c'est remarquable et le premier pas propre que nous obtenons passe de 4 à 5 à du moins en termes de ce que ce graphique nous montre bien et évidemment, un seul pas de 4 à 5 correspond à passer de 1 kilotonne à 32 kilotonnes et c'était évidemment la taille de la bombe destructrice de ville qui a atterri sur Nagasaki, donc c'est peut-être un chose qui peut être contre-intuitive à propos des échelles logarithmiques si vous entendez simplement dans les informations la différence entre oh, il y a eu un tremblement de terre et un 4.0 contre un tremblement de terre qui était un 5.0 c'est facile de penser ouais 4 et 5 ce sont des nombres assez similaires mais évidemment en termes de montants de TNT cela correspond à multiplier par 32 pour passer de 1 au suivant et passer de 2 à 4 était évidemment multiplié par environ mille et le seul La raison pour laquelle c'est plus grand est qu'ici, notre graphique ne montrait pas ce qu'était 3, donc nous faisions deux pas et vous pouvez vérifier par vous-même que si vous faites un pas de 32 et que vous multipliez ensuite par 32 supplémentaires, cela est en fait assez proche de mille, donc l'idée que les étapes additives du nombre de Richter correspondent aux étapes multiplicatives du TNT semble suggérer que quelque chose de logarithmique est en jeu ici et il est un peu intéressant de continuer ici et de dire à quel point cela augmente en partie à cause des phénomènes mondiaux dans lesquels il se produit. décrivant oui, ce n'est pas une énorme surprise qu'à mesure que nous faisons un pas de plus, cela se multiplie à nouveau par environ 32, mais en maîtrisant cela dans nos intuitions, c'est la différence entre 32 kilotonnes pour une petite bombe atomique et puis une mégatonne que nous pourrions considérer comme n'étant pas une petite bombe atomique, Bombe atomique de Nagasaki qui, je suppose, représente 32 bombes atomiques de Nagasaki pour une mégatonne, ce qui est évidemment la magnitude du tremblement de terre plat à double chaîne au Nevada, aux États-Unis, en 1994. Je ne savais pas ce que c'était, merci Wikipedia en termes de fréquences, d'ailleurs. J'ai aussi regardé ceux-ci, évidemment ceux qui sont inférieurs à deux, ceux-ci arrivent tout le temps, il y en a environ 8 000 par jour, mais dès que nous sommes dans le domaine des bombes atomiques, des choses comme 3.5 et 4, cela se produit évidemment aussi assez fréquemment quelque part sur la terre, il y en a environ 134 qui se produisent quelque part chaque jour, qui savait ? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "mais à mesure que nous devenons encore plus intenses dans cette fourchette de 5 et 6 qui étaient bien au-dessus de l'échelle de la bombe atomique, nous n'en sommes plus qu'à environ 2 par jour et je suis sûr qu'un géologue pourrait venir et expliquer pourquoi nous devrions tous le faire. Je ne serais pas très inquiet du fait qu'il y ait chaque jour deux perturbations équivalentes à une bombe atomique sur la croûte terrestre, mais il est vraisemblablement particulièrement rare que celles-ci soient concentrées dans un endroit comme une ville où vivent maintenant de nombreuses personnes, vérifiant simplement notre pensée que chaque étape implique une croissance de 32, regardons à quoi ressemble le pas de 6 à 7 et ici cela nous donne beaucoup plus d'exemples entre les deux, donnant peut-être l'illusion que c'est un pas plus grand qu'il ne l'est en réalité et en effet c'est la différence entre 1 mégatonne et 32 mégatonnes, ce qui multiplie par 32. L'une des choses que j'ai trouvées les plus intéressantes sur ce graphique était d'ailleurs de voir jusqu'où nous devons aller avant d'arriver à la plus grande arme nucléaire jamais testée. C'était le plus fort de la guerre froide. la bombe du Tsar qui faisait 50 mégatonnes et je crois qu'ils avaient en fait prévu au départ d'avoir une bombe de 100 mégatonnes, mais ils se sont dissuadés de ces 50 mégatonnes, nous parlons de commencer avec les 32 kilotonnes de la bombe de Nagasaki, multipliées par 32 pour obtenir une mégatonne multipliée par 32 supplémentaires, nous parlons donc de mille fois la force de l'explosion de fin de la Seconde Guerre mondiale et vous n'êtes toujours pas aux 50 mégatonnes de ce dont l'humanité est capable et c'est évidemment le tremblement de terre de Java en Indonésie, donc 7 . 0 n’est pas juste un peu plus grand que 6.0, c'est beaucoup plus grand et le point ici bien sûr est simplement que lorsque vous avez une échelle vous donnant des augmentations multiplicatives, il vaut la peine de comprendre que ce qui ressemble à de petits pas peut en réalité être d'énormes pas en termes d'énergie implicite ou de valeurs absolues implicites ici. alors quand on pense au fait qu'il y a déjà eu un 9.5, cela semble en fait absurde étant donné que ce n'est que dans le 7.0, nous parlons de la plus grande arme thermonucléaire jamais utilisée et cela est révélateur d'un domaine dans lequel les logarithmes ont tendance à apparaître, c'est lorsque les humains veulent créer une échelle pour quelque chose qui représente une variation extrêmement large dans la taille des choses. Il en est ainsi dans le cas de la taille des tremblements de terre, vous pouvez avoir des choses de ce qui se passe tout le temps autour de la Terre, de la taille d'une grosse grenade à main et vous voulez que cela soit à votre échelle et quelque chose à penser allant jusqu'en haut. à la plus grande perturbation que nous ayons vue dans l'histoire de l'humanité et pour que cela ne se contente pas d'écrire tout un tas de chiffres différents dans vos chiffres pour un cas et tout un tas de chiffres différents, un nombre plus petit de chiffres pour votre numéro dans un autre cas, c'est bien de prendre des logarithmes et ensuite de simplement les mettre sur une seule échelle qui écrase ces nombres entre 0 et 10, vous voyez quelque chose de très similaire se produire avec l'échelle de décibels pour la musique sur laquelle on travaille en fait un peu un peu différemment où à chaque fois que vous augmentez de 10 décibels cela correspond à une multiplication par 10 donc plutôt qu'un pas de 1 multiplié par 10, c'est un pas de 10 qui se multiplie par 10 donc ça rend un peu le calcul c'est un peu fou mais l'idée est la même, que si vous écoutez un son de 50 décibels contre 60 décibels, c'est beaucoup plus silencieux en termes d'énergie transmise et allant de, qu'est-ce que ce serait, 60 à 70 ou 70 à 80 ces étapes, de 60 à 80, qui consistent à multiplier la quantité d'énergie par surface carrée par un facteur de 100, donc chaque fois que vous voyez une échelle logarithmique, sachez dans votre esprit que cela signifie que tout ce à quoi elle fait référence sous le capot grandit de une quantité énorme, c'est encore une fois la raison pour laquelle nous avons vu beaucoup d'échelles logarithmiques utilisées pour décrire l'épidémie de coronavirus, alors comment pourriez-vous décrire une relation comme celle-ci où chaque fois que vous augmentez le nombre de l'échelle de Richter de 1, vous multipliez par 32 eh bien, nous pourrais penser en termes d'un journal avec une base 32, je pourrais dire que si je prends le journal de, je vais juste appeler r, le nombre de l'échelle de Richter, je pourrais penser à cela comme un journal de base 32 et cela va correspondre à , non non non, je fais ça mal, ce n'est pas ce qui est enregistré, nous prenons la base logarithmique 32 du grand nombre, du nombre TMT, quelque chose qui était comme 1 mégatonne, c'est 1 million de tonnes la base logarithmique 32, ça devrait correspondent au numéro de l'échelle de Richter mais il peut y avoir une sorte de décalage, donc nous pourrions dire qu'il y a une sorte de constante s que nous ajoutons à ce numéro de l'échelle de Richter et cette expression est exactement la même, excusez-moi de m'écarter du numéro de l'échelle de Richter. en bas, cette expression équivaut exactement à dire 32 à la puissance d'un certain décalage multiplié par notre numéro d'échelle de Richter, ce qui équivaut à prendre 32 à ce décalage, qui lui-même n'est qu'une grande constante, multiplié par 32 au numéro d'échelle de Richter, donc vous pourrait penser que cela est simplement une constante de 32 à la puissance du nombre que vous voyez, donc cette façon d'écrire met vraiment l'accent sur sa croissance exponentielle. Si c'est ce qui correspond au montant de TMT que vous voyez, à mesure que vous l'augmentez étape par étape, vous multipliez par 32, mais une autre façon de communiquer exactement le même fait est de prendre la base logarithmique de 32, quel que soit ce montant, c'est bien maintenant, la prochaine chose dont je veux parler est de savoir comment nous ne sommes pas toujours obligés de le faire. vous inquiétez de la façon de calculer les journaux de différentes bases, c'est un peu bizarre ici que nous parlions du journal de base 32, j'ai mentionné plus tôt à quel point les mathématiciens aiment vraiment avoir un journal avec la base et les informaticiens aiment vraiment avoir un journal avec la base 2 et cela s'avère à des fins de calcul ou pour réfléchir également à la façon dont ces choses se développent si vous avez un seul journal, si vous êtes capable de calculer un type de journal, que ce soit en base 10, en base 2, en base e, vous pouvez calculer à peu près tout ce qui vous voulez maintenant orienter nos intuitions dans cette direction, revenons à notre quiz et passons à la question suivante et je crois que cette question est la plus, je ne sais pas, c'est une question à moitié raisonnable, ça devrait être sympa cela va juste nous préparer à traduire du contexte de base 2 au contexte de base 10 et c'est aussi une bonne intuition pour comprendre les puissances de 2 d'avoir en général la relation qu'il entretient avec les puissances de 10 parce que c'est cette belle sorte de coïncidence de nature, ces deux-là, vous verrez bien ce que je veux dire, ils jouent bien les uns avec les autres, donc notre question se pose, étant donné que 2 puissance 10 vaut 1024, 1024, ce qui est environ 1000, donc si vous êtes un un peu lâche avec vos chiffres et vous faites juste des approximations de 2 au 10, essentiellement 1000, lequel des énoncés suivants est le plus proche d'être vrai ? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "tendre. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "Ce n’est pas du tout une décision unanime ici. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "Mais la question était de savoir laquelle est la plus proche de la vérité, et voyons comment nous pouvons y réfléchir. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "Cela indique donc que vous avez une puissance de 2, soit 1024, terriblement proche d'une puissance de 10, environ 10 au cube. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "Qu'est-ce que cela signifie? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "Si le log de base 2 de 10 est égal à x, cela revient à dire que 2 à x est égal à 10, n'est-ce pas ? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "Cela nous demande 2 à ce qui est égal à 10. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "Vous ne pouvez pas faire cela avec toutes les fonctions. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "Les gens semblent penser que vous pouvez faire cela avec n’importe quelle fonction, mais ce n’est tout simplement pas possible. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "Et ce que cela signifie, c'est que x est d'environ 10 tiers, d'accord ? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "Et bien, ce que nous avons vu plus tôt, c'est que le log base 2 sur 10, nous pourrions aussi dire que le log base 10 sur 2 est juste 1 sur ce montant, 1 sur x. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "Super. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "Et parce que nous faisons des choses dans les journaux, je vais juste l'écrire de cette façon. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "De même, log de base 2 d'un million, eh bien, voyons, si nous devons multiplier 2 par lui-même environ 10 fois pour arriver à mille, nous devrions le multiplier par lui-même environ 20 fois pour obtenir un million. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "C'est un peu plus petit mais c'est une bonne approximation à garder à l'esprit. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20, nous réduisons du même montant. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, nous réduisons du même montant. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "D'accord? ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "C’est une intuition qui mérite d’être rappelée. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "D'accord? ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "Et puis tout un tas de différentes manières possibles de combiner le journal de base C de B multiplié par le journal de base C de A. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "Je vais vous donner un moment significatif sur celui-ci, car ce n'est pas évident à moins que vous ne soyez déjà familier avec les logarithmes, et cela vaut la peine d'y réfléchir un peu. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "Merci Karen. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/german/sentence_translations.json b/2020/ldm-logarithms/german/sentence_translations.json index 46bc7866b..1bc6ac538 100644 --- a/2020/ldm-logarithms/german/sentence_translations.json +++ b/2020/ldm-logarithms/german/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Musik🎵 Willkommen zurück bei Lockdown Math. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "Heute werden wir über Logarithmen sprechen und so eine Art Zurück-zu-den-Grundlagen-Lektion. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "Und wie immer möchte ich zum Auftakt einfach ein Gefühl dafür bekommen, wo sich das Publikum gerade befindet. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "Wenn Sie also zu 3b1b gehen können. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "Ich habe noch nie zuvor von ihnen gehört oder noch nie etwas davon erfahren. b. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "Ich habe davon erfahren, bin aber manchmal durch all die Eigenschaften verwirrt. c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "Ich verstehe sie, wüsste aber nicht, wie ich sie unterrichten soll und d. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "Ich verstehe sie gut und könnte sie problemlos jemand anderem beibringen, damit sie es auch gut verstehen. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "Wir haben also eine gute Aufteilung. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "Wie ich schon sagte, die Absicht dabei ist, eine Lektion zu schaffen, auf die ich Menschen in Zukunft hinweisen kann, wenn sie mit Logarithmen einfach nicht vertraut sind, und ich möchte sagen können: „Oh, hier ist ein Ort, an den man sich wenden kann.“ wie ich denke, wissen Sie, wie ich denke, dass man es intuitiv angehen könnte. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "Da ich vor dieser speziellen Vorlesung in einigen Lehrerforen gestöbert habe und die Leute fragen, welches Thema am schwierigsten zu lehren ist, weil die Schüler damit am meisten Probleme zu haben scheinen, sind Logarithmen eines der am häufigsten behandelten Themen Häufig angegebene Antworten sind interessant, und ich vermute, dass es vielleicht daran liegt, dass es eine Menge dieser Eigenschaften gibt, die man am Ende lernen muss. Wenn wir also überspringen, wo wir hingehen, haben Sie all diese Berge davon Regeln, die einfach wie ein Haufen Algebra aussehen, die man sich nur schwer merken kann und die im Kopf leicht durcheinander kommen, und ich denke, wenn die Leute, wissen Sie, diese alptraumhaften Erinnerungen daran haben, wie Mathematik in der High School war und was Logarithmen haben für sie funktioniert, es sind oft diese bestimmten Formeln, die einem in den Sinn kommen, und was ich heute tun möchte, ist, durch eine davon zu sprechen, wie man über sie denkt, aber auch nur auf der Metaebene, wenn man jemandem Algebra beibringt, was sind das? Welche Punkte sind hervorzuheben? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "Wie können sie es in ihre Intuition integrieren? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "Oh, es stehen 3 Nullen drauf. Was ist eine logarithmische Million? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "Der Logarithmus von 1000 mal x ist gleich dem Dreifachen des Logarithmus von x und denken Sie daran, dass wir die Konvention verwenden, dass es sich um die Basis 10 log b handelt. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "Logarithmus von 1000 mal x ist gleich Logarithmus von x kubiert c. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "log von 1000 mal x entspricht 3 hoch log von x und e. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "Keines der oben genannten Dinge und denken Sie daran, wie ich bereits sagte, wir sollten voll und ganz damit rechnen, dass all die Leute am Anfang, die sagten, dass sie Protokolle gut verstehen, sofort antworten werden, sie werden richtig antworten, aber wenn Sie es sind Lassen Sie sich davon nicht einschüchtern, wenn Sie ein Problem wie dieses betrachten. Ich möchte Sie dazu ermutigen, einfach verschiedene Zehnerpotenzen einzustecken und im Sinne der Protokollfunktion zu denken zählt die Anzahl der Nullen, also gebe ich Ihnen einen Moment Zeit, darüber nachzudenken, also werde ich mit der Bewertung fortfahren und wie immer, wenn das schneller ist, als Sie möchten, wissen Sie, dass es nur daran liegt, dass ich weitermachen möchte Mit der Lektion ergibt sich also in diesem Fall die richtige Antwort als Logarithmus von 1000 mal x. Das ist das Gleiche, als würde man 3 plus den Logarithmus von Ich denke, das Beste, was man mit ihnen tun kann, ist, einfach verschiedene Zahlen einzugeben, und die besten Zahlen, die man einstecken kann, sind diejenigen, die bereits Zehnerpotenzen sind. Wenn Sie also nach so etwas wie dem Logarithmus von 1000 mal x fragen, dann tue ich das nicht. Weiß ich nicht, fügen wir einfach etwas für x log von 1000 mal 100 ein. Nun, wir wissen, wie viele Nullen in der endgültigen Antwort hier enthalten sein werden. Nun, 1000 mal 100 ist 100.000. Wir haben bereits intuitiv die Vorstellung, dass wir zwei Zehnerpotenzen multiplizieren Wir nehmen einfach die Nullen, die 3 Nullen von dieser 1000, die 2 Nullen von dieser 100 und setzen sie nebeneinander, so dass es insgesamt 5 Nullen sein sollten, aber wenn man wirklich darüber nachdenkt, nicht nur darüber, wie sich die Zahl entwickelt hat Aber warum ist es so gekommen, dass es die 3 Nullen von 1000 plus die 2 Nullen von 100 waren, was wir auch schreiben könnten, indem wir die Anzahl der Nullen in 1000 plus die Anzahl der Nullen in 100 sagen, also ist diese Idee ein Logarithmus Das Produkt zweier Dinge ist die Summe der Logarithmen dieser beiden Dinge im Kontext von Zehnerpotenzen. Das vermittelt nur, was für viele von uns bereits eine sehr intuitive Idee ist, wenn man zwei Zehnerpotenzen nimmt und sie einfach multipliziert Nehmen Sie alle ihre Nullen und stopfen Sie sie sozusagen aufeinander, sodass die Art und Weise, wie ich die Dinge hier geschrieben habe, tatsächlich auf eine etwas allgemeinere Tatsache hinweist, die unsere allererste Eigenschaft von Logarithmen sein wird, nämlich dass, wenn wir die nehmen Logarithmus von A mal B, es entspricht dem Logarithmus von A plus dem Logarithmus von B. Wenn Sie jetzt eine dieser Logarithmusregeln sehen, wenn Sie die Augen zusammenkneifen oder ein wenig verwirrt sind, wie Sie sich das merken sollen, fügen Sie einfach Beispiele ein Ich bin überflüssig, ich sage das oft, aber das liegt daran, dass ich denke, dass man es sehr leicht vergisst, wenn man in der Algebra selbst versunken ist und an einer Art Test sitzt und es nur viele Symbole gibt Um sich selbst daran zu erinnern, dass es in Ordnung ist, einfach ein paar Zahlen einzugeben, ist das eine gute Sache und oft ist es eine gute Möglichkeit, der Intuition freien Lauf zu lassen. Wenn wir also in diesem Fall „Log von A mal B“ sagen und es auseinanderbrechen, könnten wir einfach denken: „Oh, das.“ Logarithmus von 100 mal 1000, was 5 ist, es gibt 5 Nullen darin, aufgeteilt nach der Anzahl der Nullen in jedem gegebenen Teil. Großartig, wunderbar. Um diese Intuition weiterzuführen, versuchen wir es mit einer anderen Übungsaufgabe und noch einmal, wenn Sie es wissen, großartig. Sie können es gut beantworten, aber denken Sie vielleicht nicht nur darüber nach, was die Antwort ist, sondern auch, wie ich diese Antwort jemandem erklären würde oder wie ich versuchen würde, einen Schüler dazu zu bringen, selbst zu dieser Antwort zu kommen, ohne dass ich es sagen muss Was ist die Antwort? Es gibt also zwei potenzielle Zuschauer: diejenigen, die sich für die Lektion selbst interessieren, und diejenigen, die sich für die Meta-Lektion interessieren. Unsere Frage lautet also noch einmal: Welche der folgenden Aussagen ist wahr? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "A. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "log von x zu n ist gleich dem n-fachen log von x b. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "Der Logarithmus von x hoch n ist gleich dem Logarithmus von x hoch n c. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "Der Logarithmus von x zu n ist gleich n plus Logarithmus von x oder d. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "Die richtige Antwort hier ist also a. Es sieht so aus, als hätten 4.000 von Ihnen Glückwünsche erhalten und uns mitgeteilt, dass der logarithmische Wert von an jemanden weitergeben oder wenn Sie versuchen, selbst herauszufinden, was es bedeutet, denke ich, dass ein guter Anfang darin besteht, etwas anzuschließen, und in diesem Fall versuchen wir es für den Logarithmus von x hoch n einfach mit 100 hoch 3 und Sie könnten es mit anderen ausprobieren, um zu sehen, ob die Muster, die Sie verwenden, tatsächlich funktionieren, aber wenn Sie darüber nachdenken, geht es Ihnen nicht nur darum, die Antwort zu sehen, sondern auch darum, warum die Antwort so ausgefallen ist Manchmal genügt ein Beispiel, weil 100 gewürfelt ist. Wir können uns das so vorstellen, dass wir 3 Kopien von 100 nehmen Sagen wir, oh, es wird eine Zahl sein, die nur 6 Nullen enthält. Das bedeutet, 100 mal 100 mal 100 zu nehmen. Ich kann mir einfach vorstellen, alle diese Nullen zusammenzufassen, um eine Million zu erhalten, also wird diese Zahl sein 6, aber wenn wir darüber nachdenken, warum es eigentlich 6 war, ist das nicht nur die Anzahl der Nullen innerhalb der Million, aus der diese 6 stammt, sondern dass wir 3 Kopien dieser 100 hatten und jede dieser 100 zwei verschiedene Nullen hatte, also ist es allgemeiner So können Sie sich das vorstellen: Wenn wir statt 100 Kubikmeter 1000 Kubikmeter oder 1000 hoch n oder x hoch n nehmen würden, können Sie sich vorstellen, dass der Wert von n immer die Anzahl der Kopien ist, die wir multiplizieren Die Zahl von nun, mal sehen, sie ist nicht das x-fache der Zahl der Nullen, die in dem enthalten waren, was wir für x ersetzt haben, was in diesem Fall 100 war. Wenn ich also stattdessen etwa den Logarithmus von 10.000 hoch n genommen hätte, wäre das dasselbe Wenn man von diesen 10.000 n Kopien nimmt und dabei die Anzahl der Nullen in jeder von ihnen zählt, also 4, dann wäre es n mal 4 und natürlich ist die allgemeine Eigenschaft, die die meisten von Ihnen richtig beantwortet haben, dass Sie diesen schönen kleinen Effekt haben, wo wann Sie sind Sehen Sie sich das Protokoll von etwas an, das zu einer Macht erhoben wurde, bei der eine kleine Kraft davor herabhüpft, und Sie haben nur ein Protokoll von dem, was sich darin befand. Das ist eine der vielleicht wichtigsten Implikationen davon. Ich weiß nicht, ob Sie das so nennen würden eine Implikation, oder wenn man es eine Neuformulierung der Definition nennen würde, wenn ich log nehme und ich nur noch einmal betone, dass es sich um die Basis 10 von 10 hoch n handelt, können wir uns dieses kleine n sozusagen als Hineinspringen vorstellen vorne und es wird das n-fache der logarithmischen Basis 10 von 10, was natürlich 1 ist. Diesen Ausdruck können Sie sich entweder so vorstellen, dass er die Anzahl der Nullen am Ende zählt, oder allgemeiner, er fragt 10 nach dem, was 10 ergibt, und die Antwort ist einfach 1 Das ist sehr beruhigend, denn eine andere Möglichkeit, zurückzugehen und diesen ursprünglichen Ausdruck einfach zu lesen, besteht darin, 10 zu sagen, was 10 zu n entspricht. Na ja, die Antwort ist jetzt n ok, mit jeder gegebenen Logarithmuseigenschaft, die wir haben, also in diesem Fall wir Ich habe gerade herausgefunden, dass ein Logarithmus von x hoch n bedeutet, dass es beim Vorwärtsspringen immer eine spiegelbildliche Exponentialeigenschaft geben wird, und das ist eine weitere Möglichkeit, wie wir dazu beitragen können, uns ein wenig Intuition für diese zu verschaffen, also lassen Sie mich das einfach vertuschen Einige der zukünftigen Eigenschaften, zu denen wir hier kommen werden, versuchen zu verbergen, wohin wir gehen, was wir gerade gefunden haben, indem wir etwas auf das n erhöhen, das davor springt. Dies entspricht der Exponentialeigenschaft, wenn ich 10 auf das x nehme und erhöhe Diese ganze Sache hoch n, das ist das Gleiche, als würde man 10 hoch n mal x nehmen, und das bringt uns zu einer anderen Intuition, die man möglicherweise für Logarithmen hat, nämlich dass sie sozusagen wie eine Potenzierung auf den Kopf gestellt sind, und hier ist, was ich damit meine Dass das Ding, das sich auf der Innenseite des Protokolls befindet, wenn ich das Protokoll von a nehme, Sie sich das als den gesamten äußeren Ausdruck für etwas vorstellen sollten, das in diesem Fall exponentiell ist. Das a, das Ding im Inneren, entspricht 10 dem x the Ausgabe der Funktion, während das Ganze selbst, der Logarithmus von a, dem entspricht, was sich hier im Inneren befindet, genau das, was der Exponent der 10 ist. Wo immer Sie also hier einen Logarithmusausdruck sehen, sollten Sie denken, dass dieser auf der rechten Seite die Rolle eines Exponenten spielt Seite und jedes Mal, wenn Sie eine Exponentialfunktion sehen, sehen Sie die gesamte 10 bis zum Auf der Innenseite addiert sich das auf der Außenseite. Nun, wenn Protokolle Exponentialfunktionen umdrehen, bedeutet das, dass das Multiplizieren auf der Außenseite und die Multiplikation der Ausgaben der Funktion dasselbe ist wie das Addieren auf der Innenseite, weil jedes dieser Protokolle wie Log a und Log B ist spielt im Ausdruck auf der rechten Seite die Rolle von Sehr schön, wenn man sich vorstellt, dass Exponenten beim nächsten nach unten hüpfen, könnte es für diejenigen, die sich nicht unbedingt mit Logarithmen auskennen, etwas seltsam aussehen, aber auch hier gilt: Geben Sie ein paar Zahlen ein, um eine gewisse Intuition dafür zu entwickeln, und wir werden es ein wenig erklären Noch einen Moment Zeit, um herauszufinden, welche der folgenden Aussagen wahr ist? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "Bei der Umkehrung handelt es sich hier um die multiplikative Umkehrung des Exponenten, und das Ergebnis ist, dass es wie 1 geteilt durch 3 aussieht und dass 3 der logarithmischen Basis 10 von 1000 entspricht, also 1 dividiert durch die logarithmische Basis 10 von 1000, sodass Sie anhand dieses einzelnen Beispiels allgemeiner vermuten könnten, dass das Vertauschen der Basis mit dem, was sich im Inneren befindet, einer Division durch 1 entspricht Anhand dessen, was dort und da draußen ist, können Sie dies anhand der entsprechenden Exponentialregel durchdenken. Was ist nun mit meinem schönen kleinen Protokoll und meinen Exponentialfunktionen passiert? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "Wunderbar, also lasst uns noch einmal verstecken, wo einige der Dinge, einige der anderen Eigenschaften, auf die wir hier eingehen, und ich werde es in der gleichen Reihenfolge behalten, in der ich es vorher hier hatte. Ich dachte, dass es mir helfen könnte, wenn ich es vorab geschrieben hätte ein bisschen sauberer als sonst, aber vielleicht geht es einfach darum, dieses seltsame Spiel des Scherenschnitts zu spielen und herumzuschlurfen, also haben wir gerade herausgefunden, dass die Basis b von a protokolliert wird, wenn man diese vertauscht, das ist das Gleiche, als würde man durch 1 dividieren, was dies entspricht, von an Exponentialland: Wenn man b zu einer Potenz addiert und sagt, dass das gleich a ist, ist das die gleiche Aussage, als würde man sagen, dass a hoch zur Umkehrung dieser Potenz wiederum b ist. Es ist irgendwie hilfreich, sich einen Moment Zeit zu nehmen und sich die Logarithmen als Dinge zu vorstellen, die Dinge drehen Von innen nach außen spielt der Ausdruck logarithmische Basis b von a die Rolle dieses x und der Ausdruck logarithmische Basis a von b spielt die Rolle dessen, was über a sitzt, und dann spielt symmetrisch der gesamte Ausdruck b hoch x Die Rolle des Inneren auf der linken Seite, es spielt die Rolle des A und des gesamten Ausdrucks, a hoch von etwas spielt die Rolle dessen, was sich im Inneren der Holzbasis befindet, a, so dass Sie es sehen können, indem Sie einfach einige Beispiele einfügen und Indem wir es den Exponentialregeln zuordnen, können wir bereits drei verschiedene Logarithmusregeln durchdenken, die, wenn sie nur als Teile der Algebra zum Auswendiglernen weitergegeben würden, Sie wissen, Sie könnten sie sich merken, aber es ist sehr leicht, dass sie Ihnen irgendwie entgleiten Kopf und es ist auch sehr leicht, von der anstehenden Aufgabe frustriert zu werden, aber Sie sollten sich vielleicht daran erinnern, dass der Grund, warum wir uns für solche Dinge interessieren, darin besteht, dass das Verständnis der Logarithmenregeln uns hilft, Mathematik in Kontexten durchzuführen, in denen es wie ein Virus ist, der wo wächst Von einem Tag auf den anderen, von einem Schritt zum nächsten neigen die Dinge dazu, multiplikativ zu wachsen. Das Verständnis der Logarithmenregeln hilft Ihnen, ein besseres Gefühl für solche Dinge zu bekommen. Bevor wir also ein schönes Beispiel aus der Praxis machen, wie das aussehen kann Lassen Sie mich zum Beispiel noch eine weitere Quizfrage in diesem Sinne stellen, um nach den Eigenschaften von Logarithmen zu fragen. Eine letzte Frage, bevor wir zu einem kleinen Beispiel aus der realen Welt übergehen: Entfernen Sie das, was wir hier und jetzt hatten. Welche der folgenden Aussagen ist wahr? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "Der Logarithmus von a plus b ist derselbe wie der Logarithmus von a plus der Logarithmus von b. Der Logarithmus von a plus b ist gleich dem Logarithmus von a mal dem Logarithmus von b Der Logarithmus von a plus b ist gleich eins dividiert durch den Logarithmus von a mal den Logarithmus von b oder nichts davon. Ah, und jetzt sind wir uns doch nicht so sehr einig, oder? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "Sehr interessant, wir haben ein Pferderennen zwischen zwei Teilnehmern, also gebe ich Ihnen einen Moment Zeit zum Nachdenken, während die Leute antworten. Eigentlich habe ich eine kleine Frage an das Publikum, also habe ich nur darüber gesprochen, wie wir das machen könnten Denken Sie an multiplikatives Wachstum, und das müssen nicht nur Zehnerpotenzen sein, wir könnten auch so etwas wie Dreierpotenzen machen, wenn man von eins über drei über neun über siebenundzwanzig bis einundachtzig geht, alles Von diesen könnten wir sagen, dass die logarithmische Basis drei dieser Zahlen einfach in netten kleinen Schritten wächst, also logarithmische Basis drei von eins, drei bis zu dem, was eins ist, die Antwort ist Null, im Allgemeinen wird der Logarithmus von eins, unabhängig von der Basis, Null sein sei Null, Logarithmus zur Basis drei von drei, drei hoch zu dem, was gleich drei ist, ist eins, und der Logarithmus zur Basis drei von neun ist zwei. Ah, Sie fragen sich vielleicht, was meine Frage ist, aber es wird hilfreich sein, all dies zu meinem eigenen Vergnügen herauszustellen Hier, lassen Sie mich einfach noch eine logarithmische Basis drei von einundachtzig aufschreiben, die jetzt vier ist. Ich habe angeblich gehört, wenn Sie ein Kind, sagen wir etwa fünf oder sechs Jahre alt, fragen, welche Zahl auf halbem Weg zwischen eins und neun liegt Sagen wir, welche Zahl auf halbem Weg ist, sind ihre Instinkte für die Antwort logarithmisch, während unsere Instinkte eher linear sind, sodass wir oft denken, eins und neun, da haben wir einen Haufen gleichmäßig verteilter Zahlen zwischen zwei, drei, vier, fünf, sechs , sieben, acht und wenn Sie genau in der Mitte dazwischen gehen, landen Sie bei fünf, aber wenn Sie im Sinne des multiplikativen Wachstums darüber nachdenken, wo Sie von eins auf neun kommen, geht es nicht darum, ein paar Dinge zu addieren, sondern Sie „Wenn man um einen bestimmten Betrag wächst, wächst man um den Faktor drei, dann wächst man um einen weiteren Faktor drei. Angeblich stimmt der natürliche Instinkt eines Kindes damit überein, drei zu sagen, und angeblich stimmt das auch damit überein, wenn Anthropologen Gesellschaften untersuchen, die haben.“ Da wir Buchhaltungssysteme und das Schreiben nicht auf die gleiche Art und Weise entwickelt haben wie moderne Gesellschaften, werden sie dafür drei Antworten geben. Meine Frage an das Publikum, falls einer von Ihnen, der gerade zuschaut, Zugang zu einem kleinen Kind hat, sagen wir, im Alter von etwa fünf Jahren Alter, schau, ob du sie fragen kannst, welche Zahl in der Mitte zwischen eins und neun liegt, und wenn du kannst, lass uns auf Twitter wissen, was das Kind sagt und wie seine tatsächliche Antwort lautet, denn ich weiß nicht warum, ich bin nur ein bisschen Ich bin skeptisch, ob sich das in der Praxis tatsächlich bewähren wird. Mir ist klar, dass dies kein besonders wissenschaftlicher Weg ist. Ich bitte die Leute, die sich einen YouTube-Livestream ansehen, nicht, ihre eigenen Kinder zu befragen und dann die Antwort zu twittern, aber für mich selbst wäre es interessant Um dort eine Art Bestätigung für unsere Frage zu sehen, ist dies die erste Frage, bei der es in einer Richtung keinen großen Konsens zu geben scheint. Lassen Sie uns weitermachen und sie bewerten, um zu sehen, welche Antwort sich als großartig herausstellt, okay, also 2.400 von Ihnen haben richtig geantwortet, dass es keines der oben genannten Dinge ist, dass der Logarithmus von a plus b keine dieser schönen Eigenschaften erfüllt, und im Allgemeinen, es sei denn, wir arbeiten mit bestimmten Arten von Näherungen, insbesondere wenn der natürliche Logarithmus ins Spiel kommt Vielleicht reden wir das nächste Mal darüber, dass das Addieren der Eingaben eines Logarithmus tatsächlich ein sehr seltsames Gefühl ist. ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "Das ist eine interessante Frage. Kann die Basis eines Logarithmus Null sein? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "Nun, in Bezug auf unser Dreieck könnten wir uns das so vorstellen, dass wir sagen: Null hoch zu einer Potenz x ist gleich einem anderen Wert y. Das ist etwas, das wir schreiben könnten, indem wir entweder sagen, dass null x gleich y ist, oder wir könnten schreiben Dasselbe gilt, wenn man sagt, dass die logarithmische Basis null von y gleich x ist, null zu dem, was gleich x ist. Das Problem hier ist nun, dass null zu irgendetwas letztendlich null ist, richtig, wenn wir also nur an die logarithmische Basis null von denken y für jede andere Eingabe y Sie wissen, Sie möchten so etwas wie eins oder zwei oder pi eingeben, was auch immer Sie wollen, Sie stellen die Frage Null zu dem, was gleich eins oder zwei oder pi ist oder welche Zahl auch immer Sie dort haben und es wird einfach keine Antwort geben, also könnten Sie bestenfalls versuchen zu sagen: Oh ja, Logarithmus von Null, es ist eine vollkommen gültige Funktion, die nur auf der Eingabe Null definiert ist, aber selbst dann würden Sie Schwierigkeiten haben, herauszufinden, was Sie wollen Denn wenn man Null zu dem sagt, was gleich Null ist, ist es so, als ob alles darauf zutrifft, sodass Ihr Arm hinter Ihrem Rücken verdreht wird, wie auch immer Sie das erreichen möchten, und es entspricht der Tatsache, dass die Exponentialfunktion mit der Basis Null vollständig Null ist ordnet Zahlen nicht in einer schönen Eins-zu-Eins-Manier einander zu, also ist das eine gute Frage. Können Sie eine logarithmische Basis Null haben? Zurück zu der Idee, wo diese Dinge in der realen Welt auftauchen. Ein Beispiel, das mir irgendwie gefällt, ist die Richterskala für Erdbeben, also gibt uns die Richterskala eine Quantifizierung dafür, wie stark ein Erdbeben ist, und sie kann alles von sehr kleinen bis zu sehr großen Zahlen annehmen, wie ich glaube, das größte Erdbeben, das jemals gemessen wurde, und dies ist nur eine Tabelle, die von stammt Wikipedia war eine 9.5 und um zu verstehen, wie verrückt das ist, lohnt es sich, einen Blick auf die Beziehung zwischen dem, was diese Zahlen bedeuten, und dann so etwas wie der entsprechenden TNT-Menge zu werfen, einer Art Maß dafür, wie viel Energie darin enthalten ist, und dann, was wir hier versuchen können Wir wollen sehen, ob wir einen Ausdruck für die Zahl auf der Richterskala in Bezug auf die Energiemenge finden können und warum Logarithmen eine natürliche Art wären, dies zu beschreiben. Der Schlüssel, auf den wir uns also konzentrieren sollten, ist, wie stark sich die Dinge erhöhen, wenn wir Fortschritte machen Wenn wir also zum Beispiel von zwei ausgehen, zeigt uns das in diesem Fall nicht, wo drei ist. Vielleicht denken wir also darüber nach, einen Schritt von zwei auf vier zu machen, was so etwas wie zwei Schritte ist. Was bewirkt das im Hinblick auf die? Nun, es sieht so aus, als würden wir aus einer Tonne TNT, was meiner Meinung nach eine große Bombe aus dem Zweiten Weltkrieg ist, bis zu einer Kilotonne tausendmal mehr Energie verbrauchen, was einer kleinen Atombombe entspricht, also nur zwei Schritte Auf der Richterskala führt uns der Übergang von einem Erdbeben der Stärke 2 zu einem Erdbeben der Stärke 4 von einer großen Bombe aus dem Zweiten Weltkrieg bis zum Atomzeitalter, das ist also bemerkenswert und der erste saubere Schritt, den wir bekommen, ist der Übergang von 4 auf 5 Zumindest im Hinblick auf das, was uns diese Tabelle schön zeigt, und offensichtlich entspricht ein einziger Schritt nach oben von 4 auf 5 einem Anstieg von 1 Kilotonne auf 32 Kilotonnen, und das war offensichtlich die Größe der stadtzerstörenden Bombe, die auf Nagasaki landete, also ist dies vielleicht eine Etwas, das bei logarithmischen Skalen kontraintuitiv sein kann, wenn Sie in den Nachrichten nur den Unterschied zwischen „Oh, es gab ein Erdbeben mit der Stärke 4“ hören. 0 im Vergleich zu einem Erdbeben, das 5 war. 0 Es ist leicht zu denken, ja, 4 und 5, das sind ziemlich ähnliche Zahlen, aber in Bezug auf TNT-Beträge entspricht das offensichtlich einer Multiplikation mit 32, um von 1 zur nächsten zu gelangen, und von 2 auf 4 zu gehen, war offensichtlich eine Multiplikation mit etwa tausend und die einzige Der Grund dafür, dass es größer ist, liegt darin, dass unser Diagramm hier nicht die 3 anzeigte, also haben wir zwei Schritte gemacht und Sie können selbst überprüfen, dass, wenn Sie einen Schritt von 32 machen und dann mit weiteren 32 multiplizieren, das tatsächlich ziemlich nahe an Tausend liegt, also Die Idee, dass additive Schritte auf der Richter-Zahl multiplikativen Schritten im TNT entsprechen, scheint darauf hinzudeuten, dass hier etwas Logarithmisches im Spiel ist, und es ist ein wenig interessant, hier einfach weiterzumachen und zu sagen, wie stark dies wächst, teilweise aufgrund der Weltphänomene, um die es sich handelt Ich beschreibe ja, es ist keine große Überraschung, dass es sich, wenn wir einen weiteren Schritt machen, wieder um etwa 32 vervielfacht, aber um es unserer Intuition anzupassen, ist das der Unterschied zwischen 32 Kilotonnen einer kleinen Atombombe und dann einer Megatonne, die wir uns vielleicht nicht als kleine Atombombe vorstellen könnten. Nagasaki-Atombombe, bei der es sich meiner Meinung nach um 32 der Nagasaki-Atombomben für eine Megatonne handelt, was offensichtlich der Stärke des Double-String-Flat-Erdbebens in Nevada, USA, 1994 entspricht. Ich wusste nicht, was das war, danke übrigens Wikipedia für die Frequenzen Ich habe auch diese nachgeschlagen, offensichtlich sind es weniger als zwei, die kommen ständig vor, es sind ungefähr 8000 davon pro Tag, aber sobald wir uns im Bereich der Atombomben befinden, sind es Dinge wie 3.5 und 4: Diese passieren offenbar auch recht häufig irgendwo auf der Erde. Etwa 134 davon passieren jeden Tag irgendwo. Wer hätte das gedacht? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "Aber da wir noch intensiver in den Bereich von 5 und 6 vordringen, die deutlich über der Atombombenskala lagen, sind wir jetzt nur noch bei etwa 2 pro Tag und ich bin sicher, dass ein Geologe vorbeikommen und erklären könnte, warum wir das nicht tun sollten. Ich muss mir keine allzu großen Sorgen darüber machen, dass es jeden Tag zu zwei Störungen der Erdkruste kommt, die einer Atombombe entsprechen, aber vermutlich ist es besonders selten, dass sich diese auf einen Ort wie eine Stadt konzentrieren, in der viele Menschen leben, die jetzt bei jedem Schritt nur unseren Gedanken überprüfen beinhaltet ein Wachstum von 32. Schauen wir uns an, wie der Schritt von 6 auf 7 aussieht, und hier werden uns viel mehr Beispiele dazwischen gegeben, was vielleicht die Illusion erweckt, dass das ein größerer Schritt ist, als er tatsächlich ist, und tatsächlich ist das der Unterschied zwischen 1 Megatonne und 32 Megatonnen, also multipliziert mit 32. Eines der Dinge, die ich an dieser Grafik übrigens am interessantesten fand, war die Frage, wie weit wir gehen müssen, bevor wir zur größten jemals tatsächlich getesteten Atomwaffe gelangen. Das war der Höhepunkt des Kalten Krieges die Zarenbombe, die 50 Megatonnen hatte, und ich glaube, sie hatten ursprünglich Pläne, eine 100-Megatonnen-Bombe zu haben, haben sich aber von diesen 50 Megatonnen heruntergeredet. Wir reden hier von 32 Kilotonnen der Nagasaki-Bombe, multiplizieren Sie sie mit 32, um eine zu erhalten Megatonnen mit weiteren 32 multiplizieren, wir sprechen also von der tausendfachen Stärke der Explosion am Ende des Zweiten Weltkriegs, und Sie sind immer noch nicht bei den 50 Megatonnen dessen, wozu die Menschheit fähig ist, und das ist offensichtlich das Java-Erdbeben in Indonesien, also 7 . 0 ist nicht nur ein bisschen größer als 6.0, es ist viel größer und der Punkt hier ist natürlich nur, dass man bei einer Skala, die multiplikative Steigerungen liefert, bedenken sollte, dass scheinbar kleine Schritte tatsächlich große Schritte sein können, was die implizierte Energie oder die hier implizierten absoluten Werte betrifft Wenn wir also daran denken, dass es jemals eine 9 gab. 5, was eigentlich absurd erscheint, wenn man bedenkt, dass es nur in der 7 steht. 0-Bereich sprechen wir von der größten thermonuklearen Waffe, die je entwickelt wurde, und dies weist auf einen Bereich hin, in dem Logarithmen häufig auftreten, nämlich dann, wenn Menschen eine Skala für etwas erstellen wollen, die eine enorm große Varianz in der Größe von Dingen berücksichtigt Im Falle der Größe von Erdbeben können Sie also Dinge von der Größe einer großen Handgranate haben, die ständig auf der Erde passieren, und Sie möchten, dass das auf Ihrer Skala liegt und dass Sie über etwas nachdenken können, das sich bis ganz nach oben erstreckt auf die größte Störung, die wir in der Geschichte der Menschheit gesehen haben, und zwar so, dass Sie nicht nur eine ganze Reihe verschiedener Ziffern für einen Fall und eine ganze Reihe verschiedener, kleinerer Zahlen in Ihre Zahlen schreiben In einem anderen Fall ist es schön, Logarithmen zu nehmen und das dann einfach auf eine einzige Skala zu übertragen, die diese Zahlen praktisch zwischen 0 und 10 quetscht. Sie sehen etwas sehr Ähnliches mit der Dezibelskala für Musik, die tatsächlich ein wenig funktioniert Etwas anders: Jedes Mal, wenn Sie einen Schritt um 10 Dezibel machen, entspricht das einer Multiplikation mit 10, also ist es kein Schritt von 1, der mit 10 multipliziert wird, sondern ein Schritt von 10, der mit 10 multipliziert wird. Das macht die Mathematik also ein wenig einfacher Etwas verrückt, aber die Idee ist dieselbe: Wenn man einen Ton hört, der 50 Dezibel gegenüber 60 Dezibel hat, ist er viel leiser, was die übertragene Energie und den Übergang von 60 zu 70 oder 70 zu angeht 80 dieser Schritte, von 60 bis 80, das bedeutet, die Energiemenge pro Quadratfläche mit dem Faktor 100 zu multiplizieren. Wenn Sie also eine logarithmische Skala sehen, wissen Sie im Kopf, dass das bedeutet, dass das, worauf es sich unter der Haube bezieht, umso größer wird Eine riesige Menge, das ist wiederum der Grund, warum wir viele logarithmische Skalen gesehen haben, die zur Beschreibung des Coronavirus-Ausbruchs verwendet wurden. Wie könnte man also eine Beziehung wie diese beschreiben, bei der man jedes Mal, wenn man die Zahl auf der Richterskala um 1 erhöht, mit 32 multipliziert? Nun ja, wir Ich könnte mir einen Logarithmus mit der Basis 32 vorstellen. Ich könnte sagen, wenn ich den Logarithmus nehme, nenne ich einfach r, die Zahl für die Richterskala. Ich könnte mir das als Logarithmus zur Basis 32 vorstellen, und das wird entsprechen , nein nein nein, ich mache das falsch, das ist nicht das, was protokolliert wird. Wir nehmen die logarithmische Basis 32 der großen Zahl, der TMT-Zahl, etwa 1 Megatonne, also 1 Million Tonnen, die logarithmische Basis 32, das sollte entsprechen der Zahl auf der Richterskala, aber es kann eine Art Versatz geben, also könnten wir sagen, dass es eine Art Konstante s gibt, die wir zu dieser Zahl auf der Richterskala hinzufügen, und dieser Ausdruck ist genau derselbe, entschuldigen Sie, dass ich da so abgewichen bin Unten ist dieser Ausdruck genau das Gleiche, als würde man sagen: 32 hoch mit einem Offset multipliziert mit unserer Zahl auf der Richterskala, was dasselbe ist, als würde man 32 mit diesem Offset multiplizieren, der selbst nur eine große Konstante ist, mal 32 mit der Zahl auf der Richterskala Man könnte sich vorstellen, dass dies einfach nur eine Konstante multipliziert mit der Potenz 32 der Zahl ist, die man sieht. Diese Schreibweise unterstreicht also wirklich das exponentielle Wachstum davon, wenn dies dem TMT-Betrag entspricht, den man sieht, wenn man diesen erhöht Schritt für Schritt multiplizieren Sie mit 32, aber eine andere Möglichkeit, genau die gleiche Tatsache zu kommunizieren, besteht darin, die logarithmische Basis 32 von dem zu nehmen, was auch immer dieser Betrag ist. Als nächstes möchte ich darüber sprechen, dass wir das nicht immer tun müssen Machen Sie sich Gedanken darüber, wie man Protokolle mit verschiedenen Basen berechnet. Es ist etwas seltsam, dass wir hier über Protokolle zur Basis 32 gesprochen haben. Ich habe vorhin darauf hingewiesen, dass Mathematiker gerne Protokolle mit Basis haben und Informatiker gerne Protokolle mit Basis 2 haben Es stellt sich heraus, dass es zu Rechenzwecken dient oder auch darüber nachdenkt, wie diese Dinge wachsen, wenn Sie ein Protokoll haben. Wenn Sie in der Lage sind, einen Protokolltyp zu berechnen, sei es Basis 10, Basis 2, Basis e, können Sie so ziemlich alles andere berechnen Wenn Sie jetzt unsere Intuitionen in diese Richtung lenken möchten, kehren wir zu unserem Quiz zurück und gehen zur nächsten Frage über. Ich glaube, dass diese Frage die am meisten ist, ich weiß nicht, das ist eine halbwegs vernünftige Frage, das sollte nett sein Dies bereitet uns nur darauf vor, vom Kontext der Basis 2 in den Kontext der Basis 10 zu übersetzen, und es ist auch eine gute Intuition für das Verständnis von Zweierpotenzen, im Allgemeinen die Beziehung zu haben, die es mit Zehnerpotenzen hat, weil es sich um eine schöne Art von Zufall handelt Es liegt in der Natur dieser beiden, Sie werden sehen, was ich meine, sie harmonieren gut miteinander, daher lautet unsere Frage angesichts der Tatsache, dass 2 hoch 10 1024, 1024 ist, was ungefähr 1000 ist Sie sind etwas locker mit Ihren Zahlen und machen nur Annäherungen von 2 bis 10, im Grunde genommen 1000. Welche der folgenden Aussagen kommt der Wahrheit am nächsten? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "zart. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "Keine einstimmige Entscheidung hier. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "Aber die Frage war, welche davon der Wahrheit am nächsten kommt, und mal sehen, wie wir darüber nachdenken können. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "Es weist also darauf hin, dass Sie eine Potenz von 2 haben, die 1024 ist, was einer Potenz von 10, etwa 10 kubisch, sehr nahe kommt. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "Was bedeutet das also? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "Wenn der Logarithmus zur Basis 2 von 10 gleich x ist, ist das dasselbe, als würde man sagen, dass 2 zu x gleich 10 ist, oder? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "Es verlangt von uns 2 hoch, was 10 entspricht. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "Das kann man nicht mit jeder Funktion machen. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "Die Leute scheinen zu denken, dass man das mit jeder Funktion machen kann, aber das ist einfach nicht möglich. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "Und das bedeutet, dass x etwa 10 Drittel beträgt, okay? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "Und gut genug, was wir vorhin gesehen haben, ist, dass die logarithmische Basis 2 von 10, wir könnten auch sagen, die logarithmische Basis 10 von 2 ist nur 1 über diesem Betrag, 1 über x. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "Großartig. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "Und weil wir die Dinge im Protokoll machen, werde ich es einfach so schreiben. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "Ähnlich logarithmisch zur Basis 2 einer Million. Nun mal sehen, wenn wir 2 etwa 10 Mal mit sich selbst multiplizieren müssen, um auf Tausend zu kommen, sollten wir es etwa 20 Mal mit sich selbst multiplizieren müssen, um auf eine Million zu kommen. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "Es ist etwas kleiner, aber das ist eine nette Annäherung, die man sich merken sollte. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20, wir skalieren um denselben Betrag herunter. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, wir skalieren um denselben Betrag herunter. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "Okay? ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "Das ist eine Intuition, an die man sich erinnern sollte. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "Okay? ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "Und dann noch eine ganze Reihe verschiedener Möglichkeiten, die logarithmische Basis C von B mal die logarithmische Basis C von A zu kombinieren. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "Ich möchte Ihnen hierauf einen aussagekräftigen Abschnitt geben, da es nicht offensichtlich ist, es sei denn, Sie sind bereits mit Logarithmen vertraut, und es lohnt sich, ein wenig darüber nachzudenken. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "Danke Karen. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/hebrew/sentence_translations.json b/2020/ldm-logarithms/hebrew/sentence_translations.json index 8caf97dcd..290e014ef 100644 --- a/2020/ldm-logarithms/hebrew/sentence_translations.json +++ b/2020/ldm-logarithms/hebrew/sentence_translations.json @@ -1,13 +1,13 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math.", + "input": "... you you you you you you you you you you you you you you you you you you you you you you you you you", "translatedText": "🎵מוזיקה🎵 ברוכים הבאים חזרה למתמטיקה של Lockdown.", "n_reviews": 0, "start": 0.0, "end": 691.84 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson.", + "input": "it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the ove", "translatedText": "היום אנחנו הולכים לדבר על לוגריתמים וסוג של שיעור חזרה ליסודות.", "n_reviews": 0, "start": 720.0, @@ -28,7 +28,7 @@ "end": 742.7 }, { - "input": "Because I have a couple suspicions, but I think doing a live poll to see where everyone is might be helpful.", + "input": "'re adding 5000 instead use a y axis where each step is multiplicative so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by 10 and wh", "translatedText": "כי יש לי כמה חשדות, אבל אני חושב שביצוע סקר חי כדי לראות היכן כולם נמצאים עשוי לעזור.", "n_reviews": 0, "start": 742.92, @@ -42,7 +42,7 @@ "end": 759.16 }, { - "input": "co.", + "input": "y axis is now plotting not the total number of cases but the log", "translatedText": "שיתוף.", "n_reviews": 0, "start": 759.16, @@ -63,7 +63,7 @@ "end": 770.84 }, { - "input": "a.", + "input": "would do and, you know, it's a little bit", "translatedText": "א.", "n_reviews": 0, "start": 770.84, @@ -77,7 +77,7 @@ "end": 774.0 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c.", + "input": "ive model to say oh it's going to grow exactly exponentially but in the early phases of something like this that is what it is so I kind of fast forward in the animation I m", "translatedText": "למדתי עליהם אבל לפעמים מתבלבל מכל המאפיינים ג.", "n_reviews": 0, "start": 774.0, @@ -133,7 +133,7 @@ "end": 864.84 }, { - "input": "What's the way to get it built in their intuitions?", + "input": "that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications", "translatedText": "מהי הדרך לבנות את זה באינטואיציות שלהם?", "n_reviews": 0, "start": 864.84, @@ -175,14 +175,14 @@ "end": 1063.5 }, { - "input": "and we see this sitting around early March or so and of course this is because this is when the corona outbreak was really starting to kick into high gear and everyone wanted to understand exponential growth and a common way that exponential growth is plotted is with what's known as a logarithmic scale so I actually made a video about this and in it I was creating some animations and wanted to illustrate this idea of exponential growth and the main idea here, I'll go ahead and skip back to a different animation is if you're tracking the numbers, in this case this was the number of recorded cases of COVID-19 outside of mainland China in the months leading up to March you could just track what the absolute number is but the pattern that you'll find is that as you go from one day to the next, you tend to be increasing multiplicatively it's a little bit like earlier, we were seeing the powers of 10 one step to the next, you're multiplying by some amount the way that the virus was growing was very similar from one day to the next, you're multiplying not quite by a constant but in this case, for this sequence of days, it was around 1.2 in that region, you're multiplying by something so when you're plotting this, it ends up looking like this classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the overall pattern is so a common trick is to say, instead of looking at this y-axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we're adding 5,000 instead use a y-axis where each step is multiplicative so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by 10 and what you can say is the y-axis is now plotting not the total number of cases but the logarithm of the total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend would do and it's a little bit of a naive model to say, oh it's going to grow exactly exponentially but in the early phases of something like this, that is what it is so I kind of fast-forward in the animation I made for that video and what's interesting is if back then, I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said, when is that line going to cross a million?", + "input": "eally liked this idea of making explicit how we have three totally different notations for the same exact fact one of them you're using relative positions of the numbers one of them we introduce a new symbol this radical and one of them we introduce a new word, log so these three syntactically different ways to communicate the same idea seemed wrong and so I made this video about an alternate possible notation and while I don't necessarily think that oh we should teach logarithms with this triangle because convention is what it is so it's better to start getting people used to the usual expression what I do like about it and starting off with it is when you see and think about this triangle it's really emphasizing that what the log wants to be is that exponent every time that you see log of some value you should think in your mind okay whatever this number is it really wants to be an exponent it wants to be an exponent and we'll see more of what that means as we go on okay so every time you see a log it wants to be an exponent this value three and more specifically it should be an exponent sitting on top of whatever that base is now in terms of convention for the first part of this video I'm just going to be using log without a base written on it to be the shorthand for log base 10 because log base 10 will be the most intuitive thing out there you should know that often in math the convention instead is that log without anything might mean log base e there's also another notation for that ln for natural log we're going to talk all about the natural log next time so don't worry too much about that right now and there's also yet another convention often if you're in a computer science setting log without any added sugar to indicate what it is defaults to meaning log base 2 so this can sometimes be a source of confusion but it basically depends on what discipline you're in in math, not moth, math people really like a base of e we'll see why next lecture in, I don't know, I'll say engineering but really it's anything where you want good intuition with our normal base 10 number system log means log base 10 and if you're curious often in computer science settings log base 2 comes up all the time so like I said, in the back of your mind if you're trying to think of some of these properties just resting on the idea that log counts the number of zeros at the end of a number that can get you a really far way so we're going to start going thro", "translatedText": "ואנחנו רואים את זה בסביבות תחילת מרץ בערך וזה כמובן בגלל שזהו כאשר התפרצות הקורונה באמת התחילה להיכנס להילוך גבוה וכולם רצו להבין את הצמיחה האקספוננציאלית והדרך הנפוצה שבה מתווים צמיחה מעריכית היא עם מה שידוע כקנה מידה לוגריתמי אז למעשה הכנתי סרטון על זה ובו יצרתי כמה אנימציות ורציתי להמחיש את הרעיון הזה של צמיחה אקספוננציאלית והרעיון המרכזי כאן, אני אמשיך ואדלג בחזרה לאנימציה אחרת הוא אם אתה אני עוקב אחר המספרים, במקרה הזה זה היה מספר המקרים שתועדו של COVID-19 מחוץ ליבשת סין בחודשים שקדמו למרץ, אתה יכול פשוט לעקוב אחר מה המספר המוחלט אבל הדפוס שתמצא הוא ככל שאתה עובר מיום אחד למשנהו, אתה נוטה להגדיל באופן מכפלי זה קצת כמו קודם, ראינו את הכוחות של 10 צעד אחד למשנהו, אתה מכפיל בכמות מסוימת את האופן שבו הנגיף גדל היה דומה מאוד מיום אחד למשנהו, אתה מכפיל לא ממש בקבוע אבל במקרה הזה, עבור רצף הימים הזה, זה היה בערך 1.2 באזור הזה, אתה מכפיל במשהו אז כשאתה מתווה את זה, זה בסופו של דבר נראה כמו העקומה האקספוננציאלית הקלאסית הזו שמתעקלת כלפי מעלה ולפעמים אני יכול להקשות לראות לאן זה הולך או מה התבנית הכוללת כל כך טריק נפוץ הוא לומר, במקום להסתכל על ציר ה-y הזה שגדל באופן ליניארי, שכן כאן אני עובר מ-5k ל-10k, 10k ל-15k, 15k ל-20k כל שלב הוא תוסף, אנחנו מוסיפים 5,000 במקום להשתמש ב- ציר y שבו כל שלב הוא כפל, אז אתה עובר מ-10 ל-100, 100 ל-1000, 1000 ל-10, 10,000 כל אלה הם עליות על ידי הכפלה ב-10 ומה שאתה יכול לומר הוא שציר ה-y כעת מתווה לא המספר הכולל של המקרים אבל הלוגריתם של המספר הכולל של המקרים, וזה למעשה מקל על הראייה על העלילה אם אתה רוצה להקרין מה המגמה הזו תעשה וזה קצת מודל נאיבי לומר, אה, זה הולך לגדול בדיוק אקספוננציאלי אבל בשלבים המוקדמים של משהו כזה, זה מה שזה אז אני די מהר קדימה באנימציה שעשיתי עבור הסרטון הזה ומה שמעניין זה אם אז, אני חושב שפרסמתי את זה ב-6 במרץ, אם רק מצאת קו הכי מתאים ומתחת אותו ואמרת, מתי הקו הזה יחצה מיליון?", "n_reviews": 0, "start": 1063.5, "end": 1210.12 }, { - "input": "which because the y-axis is growing with multiplicative steps each time that you step up, you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know, when you understand logarithmic scales, it actually didn't seem that far it was only 30 days away if you naively just drew out that line and in fact, fast-forward to around April 5th, which is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day, I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully, since then, the growth has stopped being exponential so if you look at it on a logarithmic plot, instead of going up in a straight line, it starts to taper off but, point being, any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined, okay?", + "input": "ugh a couple of these properties and I want to do this just with a set of practiced examples so we'll transition away from the poll and this time to the first proper question and the question asks you which of the following is true a. the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. log of 1000 times x equals log of x cubed c. log of 1000 times x equals 3 plus the log of x d. log of 1000 times x equals 3 to the power of log of x and e. none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that great ok so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in t", "translatedText": "שבגלל שציר ה-y גדל עם שלבים מכפלים בכל פעם שאתה עולה, אתה מכפיל ב-10 אז גם אם זה אולי נראה כאילו 20,000 המקרים בערך שהיו אז זה רחוק מאוד ממיליון שאתה יודע, כשאתה מבין סולמות לוגריתמיים, זה למעשה לא נראה כל כך רחוק, זה היה רק 30 יום משם אם אתה רק מותח בתמימות את הקו הזה ולמעשה, מהר קדימה לסביבות ה-5 באפריל, שזה הזמן שבו זה היה צופה שנגיע ל- מיליון מקרים מחוץ לסין זה פחות או יותר היום שזה קרה, אני חושב פלוס מינוס ביום, אני לא זוכר בדיוק אבל זה היה בדיוק בשכונה הזו כי אני זוכר שחשבתי וואו זה היה סוג של מודל נאיבי לסרטון אפילו להשתמש וזה מזעזע שזה התאים כל כך בדיוק למרבה המזל, מאז, הצמיחה הפסיקה להיות אקספוננציאלית, כך שאם אתה מסתכל על זה על מגרש לוגריתמי, במקום לעלות בקו ישר, הוא מתחיל להצטמצם, אבל, נקודתית, בכל פעם שאתה נתקל במשהו בטבע או אפילו במבנה מעשה ידי אדם שבו מה שטבעי לחשוב עליו הוא לוגריתמים של עליות כפליות באים כדי לעזור לך אז בואו נמשיך ונחשוב מה הם בעצם, איך הם מוגדרים, בסדר?", "n_reviews": 0, "start": 1210.12, @@ -238,7 +238,7 @@ "end": 2014.88 }, { - "input": "a.", + "input": "case, the correct answer of the choices we hav", "translatedText": "א.", "n_reviews": 0, "start": 2014.88, @@ -301,7 +301,7 @@ "end": 2567.02 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true?", + "input": "f 10,100 it's asking 10 to the what is equal to 10,100 you might say, I don't know, it's going to be a little above 4 because it's kind of close to 10,000 so the best you might guess here is oh this is going to be something That's kind of like The log of 10,000, but that just feels like a coincidence based on the two numbers that we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Sometimes you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y.", "translatedText": "נפלא אז, שוב בוא נסתיר איפה חלק מהדברים שחלק מהנכסים האחרים שנגיע אליהם כאן ואני אשמור את זה באותו סדר שהיה לי את זה קודם כאן חשבתי שכתוב מראש יכול לשמור עליי קצת יותר נקי מהרגיל אבל אולי זה רק כרוך במשחק המוזר הזה של חיתוך נייר ודשדש, אז מה שמצאנו עכשיו, רישום בסיס b של a אם אתה מחליף אותם, זה אותו דבר כמו לחלק ב-1 למה שזה מתאים, ארץ אקספוננציאלית היא שאם אתה לוקח את b לעוצמה כלשהי ואומר שזה שווה ל-a זה אותה אמירה כמו לומר ש-a להיפך של החזקה הזו שווה שוב ל-b, זה קצת מועיל לקחת רגע ולחשוב על הלוגריתמים כאל הפיכת דברים מבפנים החוצה הביטוי log base b של a משחק את התפקיד של אותו x והביטוי log base a של b משחק את התפקיד של כל מה שיושב על ה-a ואז באופן סימטרי, כל הביטוי b בחזקת x משחק את התפקיד של הפנים בצד שמאל, זה משחק את התפקיד של ה-a ואת הביטוי כולו, a בכוחו של משהו משחק את התפקיד של מה שיושב בתוך בסיס היומן א כדי שתוכל לראות, רק על ידי חיבור של כמה דוגמאות ו על ידי התאמתו לכללים האקספוננציאליים נוכל כבר לחשוב דרך שלושה כללי לוגריתם שונים, שאם הם היו מועברים רק כחלקי אלגברה שיש לשנן, אתה יודע, אתה יכול לשנן אותם, אבל קל להם מאוד לחמוק החוצה. ראש וזה גם קל מאוד להיות מתוסכל מהמשימה שלפנינו, אבל אולי כדאי להזכיר לעצמך שהסיבה שאכפת לנו מדברים מהסוג הזה היא הבנת חוקי הלוגריתמים עוזרת לנו לעשות מתמטיקה בהקשרים שבהם זה כמו וירוס שצומח איפה מיום אחד למשנהו, מצעד אחד למשנהו, דברים נוטים לגדול בצורה מכפלה הבנה של חוקי הלוגריתמים עוזרת לך לקבל תחושה טובה יותר של דברים מהסוג הזה, אז לפני שנעשה דוגמה נחמדה בעולם האמיתי של איך זה יכול להיראות כמו הרשו לי פשוט לעשות עוד שאלת חידון ברוח זו כדי לשאול על מאפיינים של לוגריתמים אחת אחרונה לפני שנעבור לדוגמא קטנה בעולם האמיתי להיפטר ממה שהיה לנו כאן ועכשיו, מה מהבאים נכון?", "n_reviews": 0, "start": 2567.02, @@ -336,7 +336,7 @@ "end": 3053.77 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew?", + "input": "bomb That was 50 megatons, and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons we're talking start off at that 32 kilotons of the Nagasaki bomb Multiply by 32 to get a megaton multiply by another 32 Right so we're talking about a thousand times the strength of the World War two ending explosion And you're still not at the 50 megatons of what humanity is capable of And that is evidently you know the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0. It's a lot bigger and The point here of course is just that when you have a scale giving you multiplicative increases It's worth appreciating that what look like small steps Can actually be huge steps in terms of the energy implied or the absolute values implied here So it I mean when we're thinking about the fact that there was ever a 9.5 That actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out And this is indicative of one area where logarithms tend to come about it's When humans want to create a scale for something that accounts for a hugely wide variance in how big things can be So in the case of size of earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you want that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next ques", "translatedText": "ובכן במונחים של המשולש שלנו אנחנו יכולים לחשוב על זה כאילו אתה יודע, אפס לסוג כלשהו של חזקה x שווה לערך אחר y זה משהו שנוכל לכתוב או על ידי אמירת אפס ל-x שווה y או שנוכל לכתוב אותו דבר על ידי אמירת בסיס יומן אפס של y שווה ל-x אפס למה ששווה x עכשיו הבעיה כאן היא שאפס לכל דבר בסופו של דבר הוא אפס נכון, אז אם אנחנו רק הולכים לחשוב על בסיס יומן אפס של y עבור כל קלט אחר אתה יודע, אתה רוצה להזין משהו כמו אחד או שניים או פי כל מה שתרצה, אתה שואל את השאלה אפס למה שווה לאחד או שניים או פי או כל מספר שיש לך שם ופשוט לא תהיה תשובה אז במקרה הטוב אתה יכול לנסות להגיד אה כן, לוג של אפס, זו פונקציה חוקית לחלוטין שהיא מוגדרת רק על הקלט אפס אבל גם אז תהיה לך בעיה לנסות לסיים את מה שאתה רוצה שם כי אם אומרים אפס למה ששווה לאפס זה כאילו כל דבר חל עליו אז הזרוע שלך תהיה מפותלת מאחורי הגב שלך בכל אופן שאתה רוצה לגרום לזה לעבוד וזה מתאים לעובדה שהפונקציה המעריכית עם בסיס אפס היא אפס לחלוטין לא ממפה מספרים בצורה יפה אחד לאחד זה לזה אז זו שאלה מצוינת, האם אתה יכול לקבל בסיס יומן אפס עכשיו בחזרה לרעיון של איפה הדברים האלה באים בעולם האמיתי דוגמה אחת שאני די אוהב היא סולם ריכטר לרעידות אדמה אז סולם ריכטר נותן לנו כימות לכמה חזקה רעידת אדמה והיא יכולה להיות כל דבר, החל ממספרים קטנים מאוד ועד למספרים גדולים מאוד, כמו שלדעתי רעידת האדמה הגדולה ביותר שנמדדה אי פעם וזה רק תרשים שמגיע מ ויקיפדיה הייתה 9.5 וכדי להעריך עד כמה זה מטורף, כדאי להסתכל על הקשר בין המשמעות של המספרים האלה ואז משהו כמו הכמות המקבילה של TNT איזושהי מדד של כמה אנרגיה יש בו ואז מה אנחנו יכולים לנסות לעשות כאן הוא נראה אם נוכל לקבל ביטוי למספר סולם ריכטר במונחים של כמות האנרגיה ומדוע לוגריתמים יהיו דרך טבעית לתאר זאת, כך שהמפתח להתמקד בו הוא כשאנחנו צועדים צעדים קדימה, כמה דברים גדלים אז למשל, אם נעבור משתי טובות במקרה הזה זה לא מראה לנו איפה שלוש זה אז אולי אנחנו חושבים לקחת צעד משניים עד ארבע שזה בערך כמו לקחת שני שלבים מה זה עושה מבחינת כמות אנרגיה טוב זה נראה כאילו זה לוקח אותנו מטון מטרי אחד של TNT שזה אני מניח פצצה גדולה ממלחמת העולם השנייה וזה לוקח אותנו עד קילוטון פי אלף שזה פצצת אטום קטנה אז רק שני צעדים בסולם ריכטר עובר מרעידת אדמה בעוצמה 2 לרעידת אדמה בעוצמה 4 לוקח אותנו מפצצה גדולה ממלחמת העולם השנייה ועד לעידן הגרעיני אז זה ראוי לציון והצעד הנקי הראשון שנקבל הוא מעבר מ-4 ל-5 בשעה לפחות במונחים של מה שהתרשים הזה מראה לנו בצורה יפה, וברור שעלייה בודדת מ-4 ל-5 מקבילה למעבר מ-1 קילוטון ל-32 קילוטון וזה כנראה היה בגודל של הפצצה ההורסת של העיר שנחתה על נגסאקי אז זה אולי אחד דבר שיכול להיות מנוגד לאינטואיציה לגבי סולמות לוגריתמיים אם אתה רק שומע בחדשות את ההבדל בין הו, הייתה רעידת אדמה שהייתה 4.0 לעומת רעידת אדמה שהייתה 5.0 קל לחשוב שכן 4 ו-5 אלו מספרים די דומים אבל ברור שמבחינת כמויות TNT שמתאים להכפלה ב-32 כדי להגיע מ-1 ל-1 הבא והמעבר מ-2 ל-4 היה כנראה הכפלה בערך באלף והיחיד הסיבה שהיא גדולה יותר היא כי כאן התרשים שלנו לא הראה מה זה 3 אז עשינו שני צעדים ואתה יכול לוודא בעצמך שאם אתה לוקח צעד של 32 ואז אתה מכפיל בעוד 32 זה בעצם די קרוב לאלף אז נראה שהרעיון שצעדים מתווספים במספר ריכטר תואמים לצעדים כפליים ב-TNT מרמז שמשהו לוגריתמי משחק כאן, וזה קצת מעניין להמשיך כאן ולהגיד כמה זה גדל חלקית בגלל התופעות בעולם. לתאר כן לא הפתעה ענקית שכאשר אנחנו עושים עוד צעד זה מתכפיל שוב ב-32 בערך אבל מרסן את זה לאינטואיציות שלנו זה ההבדל בין 32 קילוטון פצצת אטום קטנה ואז מגהטון אחד שאולי נוכל לחשוב עליו כפצצת אטום לא קטנה, פצצת אטום של נגסאקי שאני מניח שהיא 32 מפצצות האטום של נגסאקי עבור מגהטון אחד שזה ללא ספק עוצמת רעידת האדמה השטוחה במיתר כפול בנבאדה ארה"ב 1994 לא ידעתי מה זה, תודה ויקיפדיה מבחינת תדרים אגב. גם חיפשו את אלה בבירור כאלה שהם פחות משניים, אלה קורים כל הזמן יש בערך 8000 כאלה ביום אבל ברגע שאנחנו בממלכת פצצות האטום דברים כמו 3.5 ו-4 אלה מתרחשים כנראה לעתים קרובות למדי איפשהו על פני כדור הארץ, יש בערך 134 כאלה שקורים איפשהו כל יום מי ידע?", "n_reviews": 0, "start": 3053.77, @@ -350,7 +350,7 @@ "end": 3901.15 }, { - "input": "log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000?", + "input": "the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship", "translatedText": "log base 2 מתוך 10 הוא בערך 0.3 יומן בסיס 2 מתוך 10 הוא בערך, סליחה, בסיס יומן 10 מתוך 2 הוא בערך 0.3 יומן בסיס 2 מתוך 10 הוא בערך 1 שליש או בסיס יומן 10 מתוך 2 הוא בערך 1 שליש מה מאלה הכי קרוב להיות נכון בהתבסס על העובדה ש-2 עד 10 הוא בעצם 1000?", "n_reviews": 0, "start": 3901.15, @@ -364,14 +364,14 @@ "end": 4025.43 }, { - "input": "tender.", + "input": "attempt count which I think is to say unraveling If you're looking at the maximum number I'm not I'm", "translatedText": "מִכרָז.", "n_reviews": 0, "start": 4025.43, "end": 4029.59 }, { - "input": "Not at all a unanimous decision here.", + "input": "not great at Vanna whiting this thing if you look at the maximum number in our poll It's asking what's the log base 2 of that?", "translatedText": "בכלל לא החלטה פה אחד.", "n_reviews": 0, "start": 4029.59, @@ -385,7 +385,7 @@ "end": 4038.31 }, { - "input": "So that's good, they're very numerically similar, right?", + "input": "ent powers of 2 then that rescales it and Yes, yes is the answer what a fantastically apropos questio", "translatedText": "אז זה טוב, הם מאוד דומים מבחינה מספרית, נכון?", "n_reviews": 0, "start": 4038.31, @@ -413,35 +413,35 @@ "end": 4124.69 }, { - "input": "And the question is how we can leverage this to understand something like log base 2 of 10, or log base 10 of 2.", + "input": "u some more time to think this through because it's looks like a big pile of algebra plug in some numbers to see what seems to work well and See which answer fits You You You Okay, so eve", "translatedText": "והשאלה היא איך נוכל למנף את זה כדי להבין משהו כמו בסיס יומן 2 מתוך 10, או בסיס יומן 10 מתוך 2.", "n_reviews": 0, "start": 4124.69, "end": 4137.67 }, { - "input": "As we saw earlier, those are just the reciprocals of each other.", + "input": "n if you are still thinking about it I'm gonna go ahead and grade it here and then start talking about Why it's true and then also why we should", "translatedText": "כפי שראינו קודם, אלה רק ההדדיות אחד של השני.", "n_reviews": 0, "start": 4137.67, "end": 4146.01 }, { - "input": "So what does this mean?", + "input": "care why this is an operation that actually tells", "translatedText": "אז מה זה אומר?", "n_reviews": 0, "start": 4146.15, "end": 4147.95 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right?", + "input": "us something so the correct answer which it looks like around 1700 of you got congratulations is Log base C of B times log base B of A is equal to log base", "translatedText": "אם בסיס יומן 2 מתוך 10 שווה ל-x, זה אותו דבר כמו להגיד ש-2 ל-x שווה ל-10, נכון?", "n_reviews": 0, "start": 4147.95, "end": 4157.99 }, { - "input": "It's asking us 2 to the what equals 10.", + "input": "C of A great Now that's just a big ol pile of things. Why would that be true?", "translatedText": "זה שואל אותנו 2 עד מה ששווה 10.", "n_reviews": 0, "start": 4157.99, @@ -483,7 +483,7 @@ "end": 4213.53 }, { - "input": "People seem to think you can do that with any function, but you just can't.", + "input": "here would be things like let's use a different color. Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 o", "translatedText": "נראה שאנשים חושבים שאתה יכול לעשות את זה עם כל פונקציה, אבל אתה פשוט לא יכול.", "n_reviews": 0, "start": 4213.53, @@ -518,14 +518,14 @@ "end": 4229.95 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x.", + "input": "t's plug in another power of 10 It'll be nice if it's also a power of 100 So I'll do a million So This one is asking 10 to 100 to the what equals a million How many times do I multiply a hundred by itself to get to a million?", "translatedText": "ובכן, מה שראינו קודם הוא שבסיס יומן 2 מתוך 10, נוכל גם לומר שבסיס יומן 10 מתוך 2 הוא רק 1 מעל הכמות הזו, 1 על x.", "n_reviews": 0, "start": 4229.95, "end": 4234.87 }, { - "input": "And you can see this pretty easily by writing 2 is equal to 10 to the 1 over x.", + "input": "How many times does a hundred go into a million? Phrasing the same thing 10 different ways now the claim is that this is", "translatedText": "ואתה יכול לראות את זה די בקלות על ידי כתיבת 2 שווה ל-10 ל-1 מעל x.", "n_reviews": 0, "start": 4234.87, @@ -546,7 +546,7 @@ "end": 4245.93 }, { - "input": "Great.", + "input": "ng log base 10 of a million That if I ask how many times d", "translatedText": "גדול.", "n_reviews": 0, "start": 4245.93, @@ -560,7 +560,7 @@ "end": 4266.57 }, { - "input": "So if I ask what is the log base 2 of a thousand, like we just saw, it's approximately the case that 2 to the power 10 is equal to a thousand.", + "input": "ve me the answer to how many times 10 goes into a million now just checking the numbers this certainly works 10 goes into a hundred two times 100 goes into a million three times in a multiplicative sense in that a hundr", "translatedText": "אז אם אני שואל מהו בסיס היומן 2 מתוך אלף, כמו שראינו זה עתה, זה בערך המקרה ש-2 בחזקת 10 שווה לאלף.", "n_reviews": 0, "start": 4266.57, @@ -574,14 +574,14 @@ "end": 4285.29 }, { - "input": "Log 2 of a thousand is approximately 10.", + "input": "ll six Now we could think of this property in terms of the corresponding exponent rule which is going to look a l", "translatedText": "יומן 2 מתוך אלף הוא בערך 10.", "n_reviews": 0, "start": 4285.29, "end": 4288.75 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million.", + "input": "ittle bit stranger But it's actually just saying the entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each other the whole statement is equal to saying that um suppose that B to the X is equal to a Got some", "translatedText": "באופן דומה יומן בסיס 2 מתוך מיליון, ובכן בוא נראה, אם אנחנו צריכים להכפיל 2 בעצמו בערך פי 10 כדי להגיע לאלף, אנחנו צריכים להכפיל אותו בעצמו בערך פי 20 כדי להגיע למיליון.", "n_reviews": 0, "start": 4288.75, @@ -623,7 +623,7 @@ "end": 4344.51 }, { - "input": "Log base 10 of a thousand is equal to 3.", + "input": "g you layer it on top of each other. Now if we rearrange that expression, we get what is probably the second most important of all of our", "translatedText": "בסיס יומן 10 מתוך אלף שווה ל-3.", "n_reviews": 0, "start": 4344.51, @@ -637,14 +637,14 @@ "end": 4353.53 }, { - "input": "It's counting the number of zeros, it ends up being about 6.", + "input": "n you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if you want", "translatedText": "זה סופר את מספר האפסים, בסופו של דבר זה בערך 6.", "n_reviews": 0, "start": 4353.53, "end": 4358.35 }, { - "input": "And log base 10 of a billion, counting the number of zeros, it ends up being 9.", + "input": "the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. If", "translatedText": "ובסיס יומן 10 מתוך מיליארד, בספירת מספר האפסים, זה בסופו של דבר יהיה 9.", "n_reviews": 0, "start": 4358.35, @@ -679,35 +679,35 @@ "end": 4411.57 }, { - "input": "Okay?", + "input": "say I'll use the log base 10 button and evaluat", "translatedText": "בסדר?", "n_reviews": 0, "start": 4411.57, "end": 4417.25 }, { - "input": "Now this is an intuition worth remembering.", + "input": "e what's on the inside here, which at least positionally it's kind of above the 100.", "translatedText": "עכשיו זו אינטואיציה שכדאי לזכור.", "n_reviews": 0, "start": 4417.25, "end": 4417.93 }, { - "input": "If you have your numbers described with one base, it's basically the same as describing them with another base, but there's some rescaling constant.", + "input": "It has a higher altitude as we write it. So this can line up with the notation a little bit, that it sits on the numerator. And on the bottom, I use the log base 10 button that's in my calculator on th", "translatedText": "אם המספרים שלך מתוארים עם בסיס אחד, זה בעצם זהה לתיאור אותם עם בסיס אחר, אבל יש קבוע שינוי קנה מידה.", "n_reviews": 0, "start": 4417.93, "end": 4428.15 }, { - "input": "Okay?", + "input": "e base, on the 100.", "translatedText": "בסדר?", "n_reviews": 0, "start": 4428.15, "end": 4429.21 }, { - "input": "And then the next question is going to start getting us at that direction, but it's going to be framed in a way that just looks like a whole pile of algebra, and again I will encourage you to plug in numbers if you want to to gain a little intuition for it.", + "input": "And then I can evaluate both of those and it'll give me the answer. In this case it gets you 6 divided by 2, which will be 3. And if we really just think through what this is saying, I know I've said it many different times, but it's a convoluted enough way to write things, but an intuitive enough", "translatedText": "ואז השאלה הבאה תתחיל להביא אותנו לכיוון הזה, אבל היא תוסגר בצורה שפשוט נראית כמו ערימה שלמה של אלגברה, ושוב אמליץ לך לחבר מספרים אם תרצה להרוויח קצת אינטואיציה לזה.", "n_reviews": 0, "start": 4429.21, @@ -728,7 +728,7 @@ "end": 4452.11 }, { - "input": "Does that equal log base B of A?", + "input": "Because like I said, this is probably the second most important log rule. We're", "translatedText": "האם זה שווה בסיס לוג B של A?", "n_reviews": 0, "start": 4452.11, @@ -763,7 +763,7 @@ "end": 4481.63 }, { - "input": "If you're looking at the maximum number, I'm not great at Vanna Whiting this thing, if you look at the maximum number in our poll, it's asking what's the log base 2 of that, so as it crosses different powers of 2 then that rescales it, and yes, yes is the answer.", + "input": "But anything additive in the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the left-handed side and the right-hand side are just saying how many times does 100 go into a million? but going about that in different ways. So this is extremely nice", "translatedText": "אם אתה מסתכל על המספר המקסימלי, אני לא טוב בוואנה וויטינג הדבר הזה, אם אתה מסתכל על המספר המקסימלי בסקר שלנו, זה שואל מה בסיס היומן 2 של זה, כך שהוא חוצה חזקות שונות של 2 ואז זה משנה את קנה המידה שלו, וכן, כן היא התשובה.", "n_reviews": 0, "start": 4481.63, @@ -777,14 +777,14 @@ "end": 4490.09 }, { - "input": "Thank You Karen.", + "input": "Next time we're going to talk", "translatedText": "תודה לך קרן.", "n_reviews": 0, "start": 4490.09, "end": 4490.95 }, { - "input": "All right so answers are still rolling in, and I think like I said I just want to give you some more time to think this through because it looks like a big pile of algebra.", + "input": "all about the natural logarithm, which is log base e, often written ln. And turns out, this is much easier to compute. There's nice math behind it such that if you want to come up with an algorithm that your calculator can use, it's actually a lot easier to think of l", "translatedText": "בסדר, אז התשובות עדיין מתגלגלות, ואני חושב שכמו שאמרתי אני רק רוצה לתת לך עוד זמן לחשוב על זה כי זה נראה כמו ערימה גדולה של אלגברה.", "n_reviews": 0, "start": 4490.95, diff --git a/2020/ldm-logarithms/hindi/sentence_translations.json b/2020/ldm-logarithms/hindi/sentence_translations.json index 08a7dd3f0..4dafc4151 100644 --- a/2020/ldm-logarithms/hindi/sentence_translations.json +++ b/2020/ldm-logarithms/hindi/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵संगीत🎵 लॉकडाउन मठ में आपका पुनः स्वागत है।", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "आज हम लघुगणक और मूल पाठ की ओर लौटने के बारे में बात करने जा रहे हैं।", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "और हमेशा की तरह, चीजों को शुरू करने के लिए, मैं बस यह जानना चाहता हूं कि दर्शक इस समय कहां हैं।", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "तो, यदि आप 3बी1बी पर जा सकते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "मैंने उनके बारे में पहले कभी नहीं सुना था या बी से पहले उनके बारे में कभी नहीं सीखा था।", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "मैंने उनके बारे में सीखा है लेकिन कभी-कभी सभी संपत्तियों से भ्रमित हो जाता हूं।", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "मैं उन्हें समझता हूं लेकिन यह नहीं जानता कि उन्हें कैसे सिखाया जाए और डी।", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "मैं उन्हें अच्छी तरह से समझता हूं और आराम से उन्हें किसी और को पढ़ा सकता हूं ताकि वे भी अच्छी तरह से समझ सकें।", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "तो, हमें एक अच्छा विभाजन मिल गया है।", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "जैसा कि मैंने कहा, इसका इरादा एक सबक तैयार करना है जिससे मैं भविष्य में लोगों को इंगित कर सकूं अगर वे लघुगणक के साथ सहज नहीं हैं और मैं यह कहने में सक्षम होना चाहता हूं, ओह, यहां एक जगह है जहां आप जा सकते हैं मैं कैसे सोचता हूं, आप जानते हैं, मैं कैसे सोचता हूं कि आप इसे सहजता से देख सकते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "क्योंकि मैं इस विशेष व्याख्यान को करने से पहले कुछ शिक्षक मंचों पर स्क्रॉल कर रहा था और जब लोग पूछते थे कि हाई स्कूल गणित में पढ़ाने के लिए सबसे कठिन विषय क्या है, इस अर्थ में कि छात्रों को इससे सबसे अधिक परेशानी होती है, लघुगणक सबसे अधिक में से एक है आम तौर पर बताए गए उत्तर दिलचस्प हैं और मैं अनुमान लगा सकता हूं कि शायद ऐसा इसलिए है क्योंकि इनमें से बहुत सारी संपत्तियां हैं जिन्हें आपको अंततः सीखना होगा, आप जानते हैं, इसलिए यदि हम आगे बढ़ जाते हैं जहां हम जाने वाले हैं तो आपको ये सभी ढेर मिल जाएंगे नियम जो बीजगणित के एक समूह की तरह दिखते हैं जिन्हें याद रखना कठिन हो सकता है और आपके दिमाग में चीजों को मिश्रण करना आसान हो सकता है और मुझे लगता है कि जब लोगों के पास होता है, तो आप जानते हैं, हाई स्कूल गणित कैसा था और क्या था, इस तरह की दुःस्वप्न वाली यादें लघुगणक ने उनके लिए किया, यह अक्सर वे विशेष सूत्र होते हैं जो दिमाग में आते हैं और आज मैं जो करना चाहता हूं वह एक के माध्यम से बात करने का प्रयास करना है, उनके बारे में कैसे सोचना है लेकिन मेटा स्तर पर भी अगर आप किसी को बीजगणित पढ़ा रहे हैं, तो क्या हैं जोर देने लायक बिंदु? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "इसे उनके अंतर्ज्ञान में स्थापित करने का क्या तरीका है? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "ओह, इसमें 3 शून्य हैं, दस लाख का लॉग क्या है? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "1000 गुना x का लघुगणक, x के लघुगणक के 3 गुना के बराबर है और याद रखें कि हम इस परिपाटी का उपयोग कर रहे हैं कि इसका आधार 10 लघुगणक b है।", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "1000 गुना x का लघुगणक, x घन के लघुगणक के बराबर है।", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "1000 गुना x का लघुगणक x और e के लघुगणक की घात 3 के बराबर है।", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "उपरोक्त में से कोई भी नहीं और याद रखें जैसा कि मैंने पहले कहा था, हमें पूरी उम्मीद करनी चाहिए कि शुरुआत में वे सभी लोग जिन्होंने कहा था कि वे लॉग को अच्छी तरह समझते हैं, वे तुरंत उत्तर देंगे, वे सही उत्तर देंगे, लेकिन यदि आप जो ऐसा नहीं करता है, जब आप इस तरह की समस्या को देख रहे हों तो उसे आपको भयभीत न करने दें, मैं आपको जो करने के लिए प्रोत्साहित करूंगा वह बस 10 की विभिन्न शक्तियों को प्लग इन करना है और इस विचार के संदर्भ में सोचना है कि लॉग फ़ंक्शन शून्य की संख्या गिनता है इसलिए मैं आपको इसके बारे में सोचने के लिए थोड़ा समय दूंगा ताकि मैं आगे बढ़ूं और इसे ग्रेड कर सकूं और हमेशा की तरह यदि यह आपकी सुविधा से तेज है तो जान लें कि यह केवल इसलिए है क्योंकि मैं आगे बढ़ना चाहता हूं पाठ के साथ, इस मामले में सही उत्तर 1000 गुना x का लघुगणक निकलता है, जो x के लघुगणक को 3 प्लस लेने के समान है और अब आइए एक पल के लिए इसके बारे में सोचें और जैसा कि मैंने तब कहा था जब आप अभी शुरुआत कर रहे थे उनके साथ मुझे लगता है कि सबसे अच्छी बात यह है कि विभिन्न नंबरों को सहजता से प्लग इन किया जाए और प्लग इन करने के लिए सबसे अच्छे नंबर वे हैं जो पहले से ही 10 की घात वाले हों, इसलिए यदि आप 1000 गुना x के लॉग जैसा कुछ पूछ रहे हैं तो मैं नहीं जानता।' पता नहीं, चलो बस 1000 गुणा 100 के एक्स लॉग के लिए कुछ प्लग इन करें, हम जानते हैं कि यहां अंतिम उत्तर में कितने शून्य होंगे, 1000 गुणा 100 100,000 है, हमारे पास पहले से ही सहज रूप से यह विचार है कि जब हम 10 की 2 शक्तियों को गुणा करते हैं हम केवल शून्य ले रहे हैं, उस 1000 में से 3 शून्य, उस 100 में से 2 शून्य और हम उन्हें एक-दूसरे के बगल में रख रहे हैं, इसलिए यह कुल 5 शून्य होना चाहिए, लेकिन यदि आप वास्तव में केवल इस पर विचार नहीं करते हैं कि संख्या कैसे बदल गई बाहर लेकिन ऐसा क्यों हुआ, यह उस 1000 में से 3 शून्य और उस 100 में से 2 शून्य थे, जिसे हम 1000 में शून्य की संख्या और 100 में शून्य की संख्या कहकर भी लिख सकते थे, इसलिए यह विचार है कि एक लघुगणक दो चीज़ों का गुणनफल 10 की घातों के संदर्भ में उन दो चीज़ों के लघुगणक का योग है, जो कि हममें से बहुत से लोगों के लिए पहले से ही एक सुपर सहज विचार को संप्रेषित कर रहा है यदि आप 10 की 2 घातें लेते हैं और आप उन्हें गुणा करते हैं तो आप बस उनके सभी शून्य लीजिए और उन्हें एक-दूसरे के ऊपर रख दीजिए, इसलिए जिस तरह से मैंने यहां चीजें लिखी हैं, वह वास्तव में थोड़ा अधिक सामान्य तथ्य का संकेत है जो लघुगणक की हमारी पहली संपत्ति होने जा रही है, जो कि यदि हम लेते हैं A का लघुगणक गुना B, यह A के लघुगणक और B के लघुगणक के बराबर होता है, अब जब भी आप इन लघुगणक नियमों में से किसी एक को देखते हैं, यदि आप स्वयं को अपनी आँखें सिकोड़ते हुए पाते हैं या आप थोड़ा भ्रमित हैं कि इसे कैसे याद रखें तो बस उदाहरणों को प्लग इन करें मैं निरर्थक हो रहा हूं, मैं यह बहुत कुछ कह रहा हूं लेकिन ऐसा इसलिए है क्योंकि मुझे लगता है कि एक बार जब आप बीजगणित में फंस जाते हैं और आप किसी प्रकार की परीक्षा में बैठे होते हैं और इसमें बहुत सारे प्रतीक होते हैं तो इसे भूलना बहुत आसान होता है अपने आप को याद दिलाने के लिए कि आप कुछ संख्याओं को प्लग इन करने के लिए ठीक हैं, यह एक अच्छी बात है और अक्सर यह अंतर्ज्ञान उत्पन्न करने का एक शानदार तरीका है, इसलिए इस मामले में, ए टाइम्स बी का लॉग कहें और इसे अलग-अलग तोड़कर हम बस सोच सकते हैं, ओह, वह 100 गुना 1000 का लॉग जो 5 है, इसमें 5 शून्य हैं, प्रत्येक दिए गए भाग में शून्य की संख्या के संदर्भ में विभाजित है, बढ़िया, अद्भुत इसलिए उस अंतर्ज्ञान को आगे बढ़ाते हुए आइए एक और अभ्यास समस्या का प्रयास करें और फिर से, यदि आप इसे जानते हैं, बढ़िया, आप इसका उत्तर ठीक से दे पाएंगे, लेकिन शायद सोचिए, सिर्फ यह नहीं कि उत्तर क्या है, बल्कि यह भी कि मैं यह उत्तर किसी को कैसे समझाऊंगा या मैं किसी छात्र को मेरे बताए बिना ही इस उत्तर पर आने के लिए कैसे प्रेरित करूंगा? उनका उत्तर क्या है, इसलिए दो संभावित श्रोता सदस्य हैं, एक वे हैं जो पाठ में रुचि रखते हैं और फिर वे जो मेटा पाठ में रुचि रखते हैं, इसलिए हमारा प्रश्न फिर से पूछता है, निम्नलिखित में से कौन सा सत्य है? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "एक।", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "x से n का लघुगणक, x b के लघुगणक के n गुना के बराबर है।", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "x से n का लघुगणक x की घात n c के लघुगणक के बराबर है।", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "x से n का लघुगणक n प्लस x या d के लघुगणक के बराबर है।", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "तो यहां सही उत्तर ए है, ऐसा लगता है कि आपमें से 4,000 लोगों को बधाई मिली है, हमें बता रहे हैं कि x की घात n का लघुगणक, x के लघुगणक के n गुना के बराबर है, इसलिए, फिर से, मान लें कि आप इसे सिखाने का प्रयास कर रहे हैं किसी के लिए या यदि आप स्वयं यह समझने का प्रयास कर रहे हैं कि इसका क्या अर्थ है तो मुझे लगता है कि शुरुआत करने के लिए एक अच्छी जगह कुछ प्लग इन करना है और इस मामले में, x की शक्ति के लॉग के लिए आइए इसे 100 की शक्ति के साथ आज़माएँ 3 और आप यह देखने के लिए इसे अन्य लोगों के साथ आज़मा सकते हैं कि क्या आप जो पैटर्न बना रहे हैं वह वास्तव में काम करता है, लेकिन यदि आप इसे केवल यह देखने के संदर्भ में नहीं सोच रहे हैं कि उत्तर क्या है, बल्कि यह सोचने का प्रयास कर रहे हैं कि उत्तर इस तरह क्यों निकला कभी-कभी एक उदाहरण काम करेगा क्योंकि 100 घन, हम इसे अच्छी तरह से लेने के रूप में सोच सकते हैं, यह 100 की 3 प्रतियां हैं, मैं 100 की 3 प्रतियां ले रहा हूं और जब मैं उन सभी को गुणा करता हूं और मैं शून्य की संख्या की गिनती के रूप में लॉग के बारे में सोचता हूं तो हम कहो, ओह, यह कुछ ऐसी संख्या होने जा रही है जिस पर केवल 6 शून्य हैं, 100 गुना 100 गुना 100 लेने का यही मतलब है, मैं बस उन सभी शून्यों को एक साथ समूहित करने के बारे में सोच सकता हूं ताकि एक मिलियन प्राप्त हो सके, इसलिए यह संख्या होने जा रही है 6 लेकिन अगर हम वास्तव में सोचते हैं कि यह 6 क्यों था, न केवल यह कि लाखों के अंदर शून्य की संख्या कहां से आई, इसका मतलब यह है कि हमारे पास उस 100 की 3 प्रतियां थीं और उन 100 में से प्रत्येक में 2 अलग-अलग शून्य थे, इस तरह यह अधिक सामान्य है आप इसके बारे में सोच सकते हैं कि यदि 100 घन लेने के बजाय हम 1000 घन या 1000 को n या x की घात n पर देख रहे थे, तो आप सोच सकते हैं कि यह जो भी हो, n का वह मान प्रतियों की संख्या थी जिसे हम कई बार गुणा कर रहे थे ठीक है, चलो देखते हैं, यह शून्य की संख्या का x गुना नहीं है जो कि हमने x के लिए जो कुछ भी प्रतिस्थापित किया था, वह इस मामले में 100 था, इसलिए अगर इसके बजाय मैंने घात n के लिए 10,000 के लॉग जैसा कुछ लिया होता तो यह वही होता उस 10,000 की n प्रतियां लेने पर उनमें से प्रत्येक में शून्य की संख्या गिनने पर जो 4 है, इसलिए यह n गुना 4 होगा और निश्चित रूप से सामान्य संपत्ति जिसका आप में से अधिकांश ने सही उत्तर दिया है वह यह है कि आपके पास यह प्यारा सा प्रभाव है जहां आप जब किसी चीज के लॉग को इतनी शक्ति तक उठाए हुए देखें कि उसके सामने छोटी सी शक्ति नीचे गिरती है और आपके पास अंदर जो कुछ था उसका लॉग है, अब शायद इसके सबसे महत्वपूर्ण निहितार्थों में से एक है, मुझे नहीं पता कि आप इसे कॉल करेंगे या नहीं एक निहितार्थ या यदि आप इसे परिभाषा का पुनर्कथन कहेंगे यदि मैं लॉग ले रहा हूं और मैं बस इसके आधार 10 को 10 की शक्ति पर फिर से जोर दूंगा n हम उस छोटे n के बारे में सोच सकते हैं जैसे कि नीचे की ओर जाना सामने और यह 10 में से लॉग बेस 10 का एन गुना हो जाता है जो निश्चित रूप से 1 है, इस अभिव्यक्ति को आप या तो अंत में शून्य की संख्या की गिनती के रूप में सोच सकते हैं या अधिक सामान्यतः यह 10 को 10 के बराबर पूछ रहा है और उत्तर बस 1 है जो बहुत आश्वस्त करने वाला है क्योंकि एक और तरीका जिससे आप वापस जा सकते हैं और बस इस मूल अभिव्यक्ति को पढ़ सकते हैं, वह है 10 को 10 कहना जो कि 10 के बराबर है, ओह ठीक है, उत्तर अब हर दिए गए लघुगणक गुण के साथ ठीक है जो हमारे पास है, इसलिए इस मामले में हम अभी-अभी x से घात n का एक लॉग मिला है, जिसमें n शामिल है, सामने कूदने पर हमेशा एक दर्पण छवि घातीय संपत्ति होती है और यह एक और तरीका है जिससे हम खुद को इनके लिए थोड़ा सा अंतर्ज्ञान प्राप्त करने में मदद कर सकते हैं, इसलिए मुझे बस इसे कवर करने दें भविष्य की कुछ संपत्तियाँ जो हम यहाँ प्राप्त करने जा रहे हैं, उन्हें छिपाने की कोशिश करें कि हम कहाँ जा रहे हैं, हमने अभी-अभी जो कुछ पाया है वह एन में कुछ बढ़ा रहा है जो सामने कूदता है यह घातीय संपत्ति से मेल खाता है कि अगर मैं एक्स में 10 लेता हूं और बढ़ाता हूं घात n के लिए यह पूरी चीज़ 10 से n गुना x लेने के समान है और यह हमें एक और अंतर्ज्ञान तक ले जाती है जो आपके पास लघुगणक के लिए हो सकता है, जो कि वे एक प्रकार के घातांक की तरह हैं जो अंदर से बाहर की ओर निकले हुए हैं और यहां मेरा मतलब यही है यदि मैं लॉग का लॉग ले रहा हूं तो लॉग के अंदर की चीज़ को आपको इस मामले में घातांकीय किसी चीज़ के लिए संपूर्ण बाहरी अभिव्यक्ति के रूप में सोचना चाहिए, अंदर की चीज़ 10 से x से मेल खाती है फ़ंक्शन का आउटपुट, जबकि संपूर्ण चीज़ का लॉग स्वयं यहां के अंदर जो कुछ है उससे मेल खाता है, बस 10 का प्रतिपादक क्या है, इसलिए जहां भी आप यहां एक लॉग अभिव्यक्ति देखते हैं, आपको यह सोचना चाहिए कि यह दाईं ओर एक प्रतिपादक की भूमिका निभाता है पक्ष और हर बार जब आप x अभिव्यक्ति के लिए संपूर्ण 10 का घातांक देखते हैं तो दाहिनी ओर संपूर्ण बाहरी घटक जो कि किसी एक लॉग के अंदर बैठे किसी चीज़ से मेल खाता है और हमने इसे इस विचार के ऊपर देखा कि जब हम गुणा कर रहे होते हैं अंदर की तरफ जो बाहर की तरफ अच्छी तरह से जोड़ रहा है, अगर लॉग अंदर से बाहर की ओर मुड़ते हैं, तो यह हमें बता रहा है कि फ़ंक्शन के आउटपुट को बाहर से गुणा करना अंदर से जोड़ने के समान है क्योंकि इनमें से प्रत्येक लॉग जैसे लॉग ए और लॉग बी दाहिनी ओर अभिव्यक्ति में x और y की भूमिका निभा रहा है, तो आइए खेलते रहें, आइए इनमें से कुछ और करें और देखें कि हम इनमें से कितने गुणों के लिए अंतर्ज्ञान बना सकते हैं, इसलिए यह आखिरी वाला है, घातांक के अगले भाग में कूदने के बारे में बहुत अच्छी सोच कुछ ऐसी है जो उन लोगों के लिए थोड़ी अजीब लग सकती है जो आवश्यक रूप से लघुगणक से परिचित नहीं हैं, लेकिन फिर, इसके लिए कुछ अंतर्ज्ञान प्राप्त करने के लिए कुछ संख्याओं को प्लग इन करें और हम इसे थोड़ा सा देंगे निम्नलिखित में से कौन सा सत्य है? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "ठीक है, यदि 10 का घन 1000 है, तो यह कहने के समान है कि 10, 1000 के बराबर है, जिसे 1 तिहाई तक बढ़ा दिया गया है, यहां व्युत्क्रम करने से घातांक का गुणन व्युत्क्रम शामिल होता है और जिस तरह से पता चलता है वह यह है कि 1 को 3 से विभाजित करने जैसा दिखता है और वह 3 1000 के लॉग बेस 10 से मेल खाता है, यह 1 1000 के लॉग बेस 10 से विभाजित है, इसलिए अधिक सामान्यतः, आप इस एकल उदाहरण के आधार पर अनुमान लगा सकते हैं कि जब हम आधार को अंदर की तरफ से स्वैप करते हैं तो यह 1 को विभाजित करने के अनुरूप होता है बार-बार बाहर क्या है, आप इसके बारे में संबंधित घातीय नियम को देखकर सोच सकते हैं कि अब मेरे प्यारे छोटे लॉग और घातांक का क्या हुआ? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "अद्भुत तो, चलो फिर से कुछ चीजों को छिपाते हैं, कुछ अन्य संपत्तियों को जो हमें यहां मिलेंगी और मैं इसे उसी क्रम में रखूंगा, मेरे पास यह पहले था, मैं सोच रहा था कि इसे पहले से लिखा होने से मैं इसे रख सकता हूं सामान्य से थोड़ा अधिक साफ-सुथरा, लेकिन हो सकता है कि इसमें पेपर कटिंग और इधर-उधर इधर-उधर घुमाने का यह अजीब खेल खेलना शामिल हो, इसलिए हमने अभी जो पाया, ए का लॉग बेस बी यदि आप उन्हें स्वैप करते हैं, तो यह 1 से विभाजित करने के समान है, जो इससे मेल खाता है, ए से।घातीय भूमि यह है कि यदि आप b को किसी घात तक ले जाते हैं और कहते हैं कि वह a के बराबर है, तो यह वही कथन है जो कहता है कि उस घात का व्युत्क्रम b फिर से b के बराबर है, एक क्षण लेना और लघुगणक को चीजों को मोड़ने के रूप में सोचना मददगार है अंदर से a का एक्सप्रेशन लॉग बेस b उस x की भूमिका निभा रहा है और b का एक्सप्रेशन लॉग बेस a, a के शीर्ष पर जो कुछ भी बैठता है उसकी भूमिका निभा रहा है और फिर सममित रूप से, संपूर्ण एक्सप्रेशन b से घात x की भूमिका निभा रहा है बाईं ओर अंदर की भूमिका, यह ए और संपूर्ण अभिव्यक्ति की भूमिका निभाती है, ए किसी चीज़ की शक्ति के लिए लॉग बेस ए के अंदर क्या बैठा है इसकी भूमिका निभाता है ताकि आप देख सकें, बस कुछ उदाहरणों को प्लग इन करके और इसे घातांकीय नियमों के अनुरूप बनाकर हम पहले से ही तीन अलग-अलग लघुगणक नियमों के बारे में सोच सकते हैं, जिन्हें अगर आप याद रखने के लिए बीजगणित के टुकड़ों के रूप में सौंप देते हैं, तो आप उन्हें याद कर सकते हैं, लेकिन उनके लिए आपके दिमाग से बाहर निकलना बहुत आसान है सिर और हाथ में काम से निराश होना भी बहुत आसान है, लेकिन आप खुद को याद दिलाना चाहेंगे कि हम इस प्रकार की चीजों की परवाह करते हैं क्योंकि लघुगणक के नियमों को समझना हमें उन संदर्भों में गणित करने में मदद करता है जहां यह एक वायरस की तरह बढ़ रहा है एक दिन से दूसरे दिन, एक कदम से अगले कदम तक, चीजें कई गुना बढ़ती रहती हैं, लघुगणक के नियमों को समझने से आपको उस तरह की चीजों को बेहतर ढंग से समझने में मदद मिलती है, इसलिए इससे पहले कि हम वास्तविक दुनिया का एक अच्छा उदाहरण लें कि वह क्या दिख सकता है।जैसे कि मुझे इस क्रम में एक और प्रश्नोत्तरी प्रश्न पूछने दें, इससे पहले कि हम वास्तविक दुनिया के एक छोटे से उदाहरण में संक्रमण करें, लघुगणक के गुणों के बारे में एक आखिरी बार पूछें, जो हमारे पास यहां और अभी था, उससे छुटकारा पाएं, निम्नलिखित में से कौन सा सत्य है? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "ए प्लस बी का लॉग ए प्लस बी के लॉग के समान है ए प्लस बी का लॉग ए प्लस बी के लॉग के बराबर है ए प्लस बी का लॉग ए प्लस बी के लॉग से विभाजित एक के बराबर है बी का लॉग या ए प्लस बी का लॉग, ए के लॉग के लॉग से विभाजित एक के बराबर है, बी का लॉग या उपरोक्त में से कोई नहीं, और अब हमारे पास उतनी आम सहमति नहीं है, क्या हमारे पास है? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "बहुत दिलचस्प है, हमारे पास दो लोगों के बीच एक घुड़दौड़ है, इसलिए जब लोग जवाब दे रहे हों तो मैं आपको यह सोचने के लिए कुछ समय दूंगा, वास्तव में मेरे पास दर्शकों के लिए एक छोटा सा प्रश्न है, इसलिए, आप जानते हैं, मैं सिर्फ इस बारे में बात कर रहा था कि हम कैसे हो सकते हैं गुणात्मक वृद्धि के संदर्भ में सोचें और यह केवल दस की घातों तक ही सीमित नहीं है, हम तीन की घातों जैसा कुछ भी कर सकते हैं, जहां यदि आप एक से तीन से नौ से सत्ताईस से इक्यासी तक जा रहे हैं, तो सभी इनमें से हम कह सकते हैं कि इनमें से तीन संख्याओं का लघुगणक बस छोटे-छोटे चरणों में बढ़ता है, इसलिए एक का आधार तीन, एक के बराबर तीन का लघुगणक, सामान्य तौर पर उत्तर शून्य है, एक का लघुगणक, चाहे आधार कोई भी हो, होगा तीन का शून्य लॉग बेस तीन, तीन का बराबर तीन एक है इसी तरह नौ का लॉग बेस तीन दो है आह, आपको आश्चर्य हो सकता है कि मेरा प्रश्न क्या है, लेकिन यह इन सभी को बाहर निकालने में मदद करेगा और मेरी अपनी खुशी के लिए यहां, मुझे बस एक और लॉग आधार लिखना है इक्यासी में से तीन अब चार है, मैंने सुना है कि जाहिरा तौर पर यदि आप एक बच्चे से पूछते हैं, मान लीजिए कि पांच या छह साल के बच्चे के पास एक और नौ के बीच कौन सी संख्या आधी है कहें कि कौन सी संख्या आधी है, उत्तर देने के तरीके के बारे में उनकी प्रवृत्ति लघुगणकीय होती है, जबकि हमारी प्रवृत्ति अधिक रैखिक होती है, इसलिए हम अक्सर एक और नौ के बारे में सोचते हैं, आपके पास दो, तीन, चार, पांच, छह के बीच समान दूरी वाली संख्याओं का एक समूह है।, सात, आठ और यदि आप ठीक बीच में आधे रास्ते पर जाते हैं, तो आप पांच पर पहुंच जाएंगे, लेकिन यदि आप गुणात्मक वृद्धि के संदर्भ में सोच रहे हैं कि एक से नौ तक कहां पहुंचा जाए, तो यह चीजों का एक समूह जोड़ने का मामला नहीं है, बल्कि आप 'एक निश्चित मात्रा में आप बढ़ रहे हैं, आप तीन के कारक से बढ़ रहे हैं, फिर आप तीन के दूसरे कारक से बढ़ रहे हैं, माना जाता है कि एक बच्चे की प्राकृतिक प्रवृत्ति तीन कहने के अनुरूप होती है और कथित तौर पर यह भी इसी के अनुरूप होती है यदि आपके पास ऐसे समाजों का अध्ययन करने वाले मानवविज्ञानी हैं जो ' टी ने आधुनिक समाजों की तरह ही लेखांकन और लेखन प्रणालियाँ विकसित की हैं, वे इसके लिए तीन उत्तर देंगे, इसलिए दर्शकों के लिए मेरा प्रश्न है कि क्या आप में से कोई भी अभी देख रहा है, मान लीजिए, पांच साल की सीमा में एक छोटे बच्चे तक पहुंच है।देखिए, क्या आप उनसे पूछ सकते हैं कि एक और नौ के बीच कौन सी संख्या आधी है और यदि आप कर सकते हैं, तो हमें ट्विटर पर बताएं कि बच्चा क्या कहता है, उनका वास्तविक उत्तर क्या है क्योंकि मुझे नहीं पता क्यों, मैं बस थोड़ा सा हूं मुझे इस बात पर संदेह है कि क्या यह वास्तव में व्यवहार में आएगा, मैं समझता हूं कि ऐसा करने का यह कोई सुपर वैज्ञानिक तरीका नहीं है, मैं यूट्यूब लाइवस्ट्रीम देखने वाले लोगों से अपने बच्चों का सर्वेक्षण करने और फिर उत्तर ट्वीट करने के लिए नहीं कह रहा हूं, लेकिन मेरे लिए यह दिलचस्प होगा हमारे प्रश्न पर किसी प्रकार की मान्यता देखने के लिए यह पहला प्रश्न है जिस पर एक दिशा में बहुत बड़ी सहमति नहीं है, आइए आगे बढ़ें और इसे ग्रेड करें और देखें कि क्या उत्तर बहुत बढ़िया आता है, ठीक है, तो 2,400 आपमें से लोगों ने सही उत्तर दिया है कि उपरोक्त में से कोई भी नहीं है कि ए प्लस बी का लॉग इन अच्छे गुणों में से किसी को भी संतुष्ट नहीं करता है और सामान्य तौर पर, जब तक कि हम कुछ विशेष प्रकार के अनुमानों के साथ काम नहीं कर रहे हैं, खासकर जब प्राकृतिक लॉग खेल में आता है हम अगली बार इसके बारे में बात कर सकते हैं, लघुगणक के इनपुट को जोड़ना वास्तव में एक बहुत ही अजीब अनुभूति है, यह करना बहुत ही अजीब बात है और उस अजीबता का एहसास पाने के लिए, यदि मैं आपसे ए प्लस बी का लॉग पूछूं तो दस की कुछ शक्तियों को प्लग इन करें।आप शायद यह सोचना शुरू कर देंगे कि, ठीक है, मुझे बस 10,000 और 100 जैसे कुछ उदाहरण प्लग इन करने दीजिए और मैं खुद से पूछूंगा, अगर मैं यह शून्य गिनती फ़ंक्शन करता हूं कि उस इनपुट में क्या है तो इसमें कितने शून्य हैं? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "यह एक दिलचस्प सवाल है, ठीक है, क्या लघुगणक का आधार शून्य हो सकता है? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "ठीक है, हमारे त्रिकोण के संदर्भ में हम ऐसा सोच सकते हैं कि आप जानते हैं, किसी प्रकार की शक्ति के लिए शून्य x किसी अन्य मान y के बराबर है, यह कुछ ऐसा है जिसे हम या तो यह कहकर लिख सकते हैं कि x बराबर y है या हम लिख सकते हैं वही बात यह है कि y का लॉग बेस शून्य x शून्य के बराबर है जो x के बराबर है अब यहां मुद्दा यह है कि किसी भी चीज का शून्य अंत में शून्य ही होता है, इसलिए यदि हम सिर्फ लॉग बेस शून्य के बारे में सोच रहे हैं आप किसी भी अन्य इनपुट के लिए जानते हैं, आप कुछ इनपुट करना चाहते हैं जैसे कि एक या दो या पीआई कुछ भी जो आप चाहते हैं, आप प्रश्न शून्य से पूछ रहे हैं कि एक या दो या पीआई के बराबर क्या है या जो भी संख्या आपके पास हो सकती है और इसका कोई उत्तर नहीं मिलने वाला है, इसलिए अधिक से अधिक आप यह कहने का प्रयास कर सकते हैं कि अरे हाँ, शून्य का लॉग, यह एक पूरी तरह से मान्य फ़ंक्शन है, इसे केवल इनपुट शून्य पर परिभाषित किया गया है, लेकिन फिर भी आप जो चाहते हैं उसे अंतिम रूप देने में आपको परेशानी होगी क्योंकि जो शून्य के बराबर होता है उसे शून्य कहना ऐसा है जैसे कि कुछ भी उस पर लागू होता है, इसलिए आपकी बांह आपकी पीठ के पीछे मुड़ जाएगी, हालांकि आप वह काम करना चाहते हैं और यह इस तथ्य से मेल खाता है कि आधार शून्य के साथ घातीय कार्य पूरी तरह से शून्य है।संख्याओं को एक-से-एक तरीके से एक-दूसरे पर मैप नहीं करता है, इसलिए यह एक अच्छा सवाल है, क्या आपके पास लॉग बेस शून्य हो सकता है, अब इस विचार पर वापस जाएं कि वास्तविक दुनिया में ये चीजें कहां आती हैं, एक उदाहरण जो मुझे पसंद है वह है भूकंपों के लिए रिक्टर स्केल इसलिए रिक्टर स्केल हमें यह बताता है कि भूकंप कितना शक्तिशाली है और यह बहुत छोटी संख्या से लेकर बहुत बड़ी संख्या तक कुछ भी हो सकता है, जैसे मुझे लगता है कि अब तक का सबसे बड़ा भूकंप मापा गया है और यह सिर्फ एक चार्ट है जो इससे आता है विकिपीडिया 9 था।5 और यह समझने के लिए कि यह कितना पागलपन है, इन संख्याओं के अर्थ के बीच के संबंध को देखना उचित है और फिर टीएनटी की समतुल्य मात्रा जैसी किसी चीज़ से पता चलता है कि इसमें कितनी ऊर्जा है और फिर हम यहां क्या करने का प्रयास कर सकते हैं यह देखना है कि क्या हम ऊर्जा की मात्रा के संदर्भ में रिक्टर स्केल संख्या के लिए एक अभिव्यक्ति प्राप्त कर सकते हैं और लघुगणक इसका वर्णन करने का एक प्राकृतिक तरीका क्यों होगा, इसलिए ध्यान केंद्रित करने की कुंजी यह है कि जैसे-जैसे हम कदम आगे बढ़ा रहे हैं चीजें कितनी बढ़ती हैं इसलिए उदाहरण के लिए यदि हम इस मामले में दो से आगे बढ़ते हैं तो यह हमें नहीं दिखाता है कि तीन कहां है इसलिए हो सकता है कि हम दो से चार तक एक कदम उठाने के बारे में सोचें जो कि दो कदम उठाने जैसा है, इससे क्या होता है ऊर्जा की मात्रा, ऐसा लगता है कि यह हमें एक मीट्रिक टन टीएनटी से लेती है, जो मुझे लगता है कि द्वितीय विश्व युद्ध का एक बड़ा बम है और यह हमें एक हजार गुना अधिक एक किलोटन तक ले जाती है, जो कि एक छोटा परमाणु बम है, इसलिए बस दो कदम रिक्टर पैमाने पर 2 तीव्रता के भूकंप से 4 तीव्रता के भूकंप तक जाना हमें द्वितीय विश्व युद्ध के बड़े बम से लेकर परमाणु युग तक ले जाता है, इसलिए यह उल्लेखनीय है और पहला स्वच्छ कदम जो हमें मिलता है वह 4 से 5 तक जा रहा है।कम से कम उस संदर्भ में जो यह चार्ट हमें अच्छी तरह से दिखा रहा है और जाहिर तौर पर 4 से 5 तक एक कदम ऊपर जाना 1 किलोटन से 32 किलोटन तक जाने के बराबर है और यह जाहिर तौर पर शहर को नष्ट करने वाले बम का आकार था जो नागासाकी पर गिरा था, इसलिए यह शायद एक है यदि आप केवल समाचारों में सुन रहे हैं कि ओह, एक भूकंप था जो 4 था, के बीच का अंतर लघुगणकीय पैमानों के बारे में प्रतिकूल हो सकता है।0 बनाम एक भूकंप जो 5 था।0 यह सोचना आसान है कि हाँ 4 और 5 वे काफी समान संख्याएँ हैं लेकिन जाहिर तौर पर टीएनटी मात्रा के संदर्भ में जो 1 से अगले तक पहुंचने के लिए 32 से गुणा करने के अनुरूप है और 2 से 4 तक जाना जाहिर तौर पर लगभग एक हजार से गुणा करना था और एकमात्र इसका बड़ा कारण यह है कि यहां हमारा चार्ट यह नहीं दिखा रहा था कि 3 क्या है इसलिए हम दो कदम उठा रहे थे और आप स्वयं सत्यापित कर सकते हैं कि यदि आप 32 का एक कदम उठाते हैं और फिर आप अन्य 32 से गुणा करते हैं तो यह वास्तव में एक हजार के काफी करीब है।यह विचार कि रिक्टर संख्या पर योगात्मक चरण टीएनटी में गुणक चरणों के अनुरूप हैं, ऐसा प्रतीत होता है कि यहां कुछ लघुगणक चल रहा है और यहां चलते रहना और यह कहना थोड़ा दिलचस्प है कि यह आंशिक रूप से विश्व की घटनाओं के कारण कितना बढ़ता है।यह वर्णन करना कोई बहुत बड़ा आश्चर्य नहीं है कि जैसे-जैसे हम एक और कदम उठाते हैं, यह फिर से लगभग 32 से बढ़ जाता है, लेकिन हम अपने अंतर्ज्ञान पर लगाम लगाते हैं कि 32 किलोटन एक छोटे परमाणु बम और फिर एक मेगाटन के बीच का अंतर है जिसे हम छोटे परमाणु बम के रूप में नहीं सोच सकते हैं, नागासाकी परमाणु बम जो मुझे लगता है कि एक मेगाटन के लिए नागासाकी परमाणु बमों में से 32 हैं, जो स्पष्ट रूप से नेवादा यूएसए 1994 में डबल स्ट्रिंग फ्लैट भूकंप की तीव्रता है, मुझे नहीं पता था कि वह क्या था, वैसे मैं आवृत्तियों के संदर्भ में विकिपीडिया को धन्यवाद देता हूं इन्हें भी देखा जो स्पष्ट रूप से दो से कम हैं, ये हर समय होते हैं, प्रति दिन इनकी संख्या लगभग 8000 होती है लेकिन जैसे ही हम परमाणु बम के दायरे में आते हैं तो 3 जैसी चीजें होती हैं।5 और 4 वे स्पष्ट रूप से पृथ्वी पर कहीं न कहीं अक्सर घटित होते हैं, उनमें से लगभग 134 हर दिन कहीं न कहीं घटित होते हैं, कौन जानता था? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "लेकिन जैसे-जैसे हम इस 5 और 6 रेंज में और अधिक गहन होते जा रहे हैं, जो परमाणु बम पैमाने से काफी ऊपर थे, अब हम केवल प्रति दिन लगभग 2 पर हैं और मुझे यकीन है कि एक भूविज्ञानी आ सकता है और समझा सकता है कि हम सभी को ऐसा क्यों करना चाहिए।' इस तथ्य के बारे में अत्यधिक चिंतित न हों कि पृथ्वी की पपड़ी में हर दिन दो परमाणु बम के बराबर व्यवधान हो रहे हैं, लेकिन संभवतः यह विशेष रूप से दुर्लभ है कि वे किसी शहर जैसे किसी स्थान पर केंद्रित हों जहां अब बहुत सारे लोग रहते हैं, बस हमारे विचार की पुष्टि हो रही है कि प्रत्येक कदम 32 की वृद्धि शामिल है आइए देखें कि 6 से 7 तक का चरण कैसा दिखता है और यहां यह हमें बीच-बीच में कई और उदाहरण दे रहा है, शायद यह भ्रम दे रहा है कि यह वास्तव में जितना बड़ा कदम है उससे कहीं अधिक बड़ा है और वास्तव में 1 मेगाटन और के बीच यही अंतर है।32 मेगाटन, इसलिए इसे 32 से गुणा किया जा रहा है, इस चार्ट पर मुझे जो चीजें सबसे दिलचस्प लगीं उनमें से एक यह थी कि सबसे बड़े परमाणु हथियार तक पहुंचने से पहले हमें कितनी दूर जाना होगा जिसका वास्तव में परीक्षण किया गया है, यह शीत युद्ध की चरम सीमा थी ज़ार बम जो 50 मेगाटन का था और मेरा मानना है कि वास्तव में उनके पास 100 मेगाटन बम रखने की मूल योजना थी, लेकिन उन्होंने खुद को 50 मेगाटन से कम करने की बात की, हम 32 किलोटन नागासाकी बम को 32 से गुणा करके शुरू करने की बात कर रहे हैं।मेगाटन को अन्य 32 से गुणा करें, इसलिए हम द्वितीय विश्व युद्ध के अंत के विस्फोट की एक हजार गुना ताकत के बारे में बात कर रहे हैं और आप अभी भी 50 मेगाटन पर नहीं हैं जो मानवता सक्षम है और यह स्पष्ट रूप से इंडोनेशिया का जावा भूकंप है इसलिए 7 . 0, 6 से थोड़ा सा भी बड़ा नहीं है।0, यह बहुत बड़ा है और यहां मुद्दा सिर्फ इतना है कि जब आपके पास एक पैमाना होता है जो आपको गुणात्मक वृद्धि देता है तो यह सराहना के लायक है कि छोटे कदमों की तरह दिखने वाले वास्तव में निहित ऊर्जा या यहां निहित पूर्ण मूल्यों के संदर्भ में बड़े कदम हो सकते हैं तो जब हम इस तथ्य के बारे में सोच रहे हैं कि कभी 9 भी था।5 जो वास्तव में बेतुका लगता है क्योंकि यह केवल 7 में है।0 रेंज के बारे में हम अब तक के सबसे बड़े थर्मोन्यूक्लियर हथियार के बारे में बात कर रहे हैं और यह एक ऐसे क्षेत्र का संकेत है जहां लघुगणक आते हैं, यह तब होता है जब मनुष्य किसी चीज के लिए एक पैमाना बनाना चाहते हैं जो कि बड़ी चीजों में बड़े पैमाने पर भिन्नता का कारण बनता है।भूकंप के आकार के मामले में, आपके पास पृथ्वी के चारों ओर हर समय होने वाली घटनाओं से लेकर एक बड़े हथगोले के आकार की चीजें हो सकती हैं और आप चाहते हैं कि यह आपके पैमाने पर हो और सभी तरह से सोचने के लिए कुछ हो।यह सबसे बड़ा व्यवधान है जो हमने मानव इतिहास में देखा है और इसे इस तरह से करने के लिए कि आप केवल एक मामले के लिए अपनी संख्याओं में विभिन्न अंकों का एक पूरा समूह नहीं लिख रहे हैं और विभिन्न, एक छोटी संख्या का एक पूरा समूह लिख रहे हैं।किसी अन्य मामले में, आपके नंबर के लिए अंकों का लघुगणक लेना अच्छा होता है और फिर उसे केवल एक पैमाने पर रखना होता है जो मूल रूप से 0 और 10 के बीच उन संख्याओं को निचोड़ता है, आप देखते हैं कि संगीत के लिए डेसीबल पैमाने के साथ कुछ ऐसा ही हो रहा है जो वास्तव में थोड़ा काम करता है थोड़ा अलग है जहां हर बार जब आप 10 डेसिबल का एक कदम उठाते हैं जो 10 से गुणा करने के अनुरूप होता है, तो 1 के चरण को 10 से गुणा करने के बजाय, यह 10 का एक कदम होता है जिसे 10 से गुणा किया जाता है इसलिए इस तरह से इसका गणित थोड़ा सा हो जाता है थोड़ा गड़बड़ है लेकिन विचार वही है, कि यदि आप 50 डेसिबल बनाम 60 डेसिबल की ध्वनि सुन रहे हैं तो यह ऊर्जा के संचारित होने और वहां से जाने के मामले में बहुत शांत है, यह क्या होगी, 60 से 70 या 70 से 60 से 80 तक वे चरण, जिसमें प्रति वर्ग क्षेत्र में ऊर्जा की मात्रा को 100 के कारक से गुणा करना शामिल है, इसलिए हर बार जब आप एक लघुगणकीय पैमाना देखें, तो अपने दिमाग में जानें कि इसका मतलब है कि हुड के नीचे जो कुछ भी संदर्भित किया जा रहा है वह बढ़ता है।एक बड़ी राशि, यही कारण है कि हमने कोरोनोवायरस प्रकोप का वर्णन करने के लिए बहुत सारे लघुगणकीय पैमानों का उपयोग किया है, तो आप इस तरह के संबंध का वर्णन कैसे कर सकते हैं जहां हर बार जब आप रिक्टर स्केल संख्या को 1 से बढ़ाते हैं, तो आप 32 से गुणा कर रहे होते हैं, हम आधार 32 के साथ एक लॉग के संदर्भ में सोच सकता हूं, मैं कह सकता हूं कि अगर मैं लॉग लेता हूं, तो मैं बस आर को कॉल करने जा रहा हूं, रिक्टर पैमाने के लिए संख्या मैं इसे लॉग बेस 32 के रूप में सोच सकता हूं और यह इसके अनुरूप होगा , नहीं, नहीं, मैं यह गलत कर रहा हूं, यह वह चीज नहीं है जो लॉग की गई है, हम बड़ी संख्या का लॉग बेस 32 लेते हैं, टीएमटी नंबर का, कुछ ऐसा जो 1 मेगाटन जैसा था, यह 1 मिलियन टन है लॉग बेस 32, जो होना चाहिए रिक्टर स्केल संख्या के अनुरूप है लेकिन इसमें किसी प्रकार की ऑफसेट हो सकती है, इसलिए हम कह सकते हैं कि कुछ प्रकार का स्थिरांक है जिसे हम इस रिक्टर स्केल संख्या में जोड़ रहे हैं और यह अभिव्यक्ति बिल्कुल वैसी ही है, मुझे इससे दूर जाने के लिए क्षमा करें नीचे यह अभिव्यक्ति बिल्कुल हमारे रिक्टर स्केल संख्या के कुछ ऑफसेट समय की घात को 32 कहने के समान है, जो कि उस ऑफसेट पर 32 लेने के समान है, जो स्वयं कुछ बड़ा स्थिरांक है, रिक्टर स्केल संख्या का 32 गुना है, इसलिए आप इसे आप जो संख्या देख रहे हैं उसकी घात का कुछ स्थिर गुणा 32 के रूप में सोच सकते हैं, इसलिए इसे लिखने का यह तरीका वास्तव में इसकी घातीय वृद्धि पर जोर देता है कि यदि यह टीएमटी मात्रा से मेल खाता है जो आप देखते हैं, जैसे-जैसे आप इसे बढ़ाते हैं चरण दर चरण आप 32 से गुणा कर रहे हैं लेकिन ठीक उसी तथ्य को संप्रेषित करने का दूसरा तरीका यह है कि जो भी राशि ठीक है उसका लॉग बेस 32 लें अब अगली बात जो मैं बात करना चाहता हूं वह यह है कि हमें हमेशा ऐसा नहीं करना पड़ता है विभिन्न आधारों के लॉग की गणना कैसे करें, इसके बारे में चिंता करें, यहां यह थोड़ा अजीब है कि हम लॉग बेस 32 के बारे में बात कर रहे थे, मैंने पहले उल्लेख किया था कि कैसे गणितज्ञ वास्तव में बेस के साथ लॉग रखना पसंद करते हैं और कंप्यूटर वैज्ञानिक वास्तव में बेस 2 के साथ लॉग रखना पसंद करते हैं और यह कम्प्यूटेशनल प्रयोजनों के लिए या यह सोचने के लिए कि यदि आपके पास एक लॉग है तो ये चीजें कैसे बढ़ती हैं, यदि आप एक प्रकार के लॉग की गणना करने में सक्षम हैं, चाहे वह आधार 10, आधार 2, आधार ई हो, तो आप लगभग किसी भी चीज़ की गणना कर सकते हैं अब आप हमारे अंतर्ज्ञान को उस दिशा में ले जाना चाहते हैं, आइए हमारी प्रश्नोत्तरी पर वापस जाएं और अगले प्रश्न पर जाएं और मेरा मानना है कि यह प्रश्न सबसे अधिक है, मुझे नहीं पता, यह आधा-अधूरा उचित प्रश्न है, यह अच्छा होना चाहिए यह हमें बेस 2 संदर्भ से बेस 10 संदर्भ में अनुवाद करने के लिए तैयार करने जा रहा है और यह 2 की शक्तियों को समझने के लिए एक अच्छा अंतर्ज्ञान भी है जो सामान्य रूप से 10 की शक्तियों के साथ संबंध रखता है क्योंकि यह इस प्रकार का सुंदर संयोग है प्रकृति यह है कि ये दोनों अच्छी तरह से आप देखेंगे कि मेरा क्या मतलब है, वे एक-दूसरे के साथ अच्छी तरह से खेलते हैं इसलिए हमारा प्रश्न पूछता है, इस तथ्य को देखते हुए कि 2 से 10 तक 1024, 1024 है, जो लगभग 1000 है, इसलिए यदि आप एक हैं आपकी संख्याओं में थोड़ी ढील है और आप केवल 2 से 10वें तक का अनुमान लगा रहे हैं, मूल रूप से 1000, निम्नलिखित में से कौन सा सत्य होने के सबसे करीब है? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "नाज़ुक।", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "यहां बिल्कुल भी सर्वसम्मत निर्णय नहीं है।", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "लेकिन सवाल यह पूछ रहा था कि कौन सा सत्य होने के सबसे करीब है, और आइए देखें कि हम इस बारे में कैसे सोच सकते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "तो यह इंगित करता है कि आपके पास 2 की शक्ति है, जो 1024 है, 10 की शक्ति के बहुत करीब, लगभग 10 घन।", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "अच्छा तो इसका क्या मतलब है? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "यदि 10 का लघुगणक आधार 2 x के बराबर है, तो यह वैसा ही है जैसे कि 2 को x कहना 10 के बराबर है, है ना? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "यह हमसे 2 पूछ रहा है कि 10 क्या होता है।", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "आप हर फ़ंक्शन के साथ ऐसा नहीं कर सकते. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "लोगों को लगता है कि आप किसी भी फ़ंक्शन के साथ ऐसा कर सकते हैं, लेकिन आप ऐसा नहीं कर सकते।", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "और इसका मतलब यह है कि x लगभग 10 तिहाई है, ठीक है? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "और ठीक है, जो हमने पहले देखा वह यह है कि 10 का लॉग बेस 2, हम यह भी कह सकते हैं कि 2 का लॉग बेस 10 उस राशि से सिर्फ 1 अधिक है, 1 x से अधिक है।", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "3. महान।", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "और क्योंकि हम लॉग पर चीजें कर रहे हैं, मैं इसे उसी तरह से लिखूंगा।", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "इसी प्रकार, एक मिलियन के आधार पर 2 लॉग करें, आइए देखें, यदि हमें एक हजार तक पहुंचने के लिए 2 को लगभग 10 बार गुणा करना है, तो हमें एक मिलियन तक पहुंचने के लिए इसे लगभग 20 बार गुणा करना होगा।", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "यह थोड़ा छोटा है लेकिन आपके दिमाग में यह एक अच्छा अनुमान है।", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20, हम उतनी ही मात्रा में कमी करते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, हम उतनी ही मात्रा में कमी करते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "ठीक है? ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "अब यह स्मरण रखने योग्य अंतर्ज्ञान है।", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "ठीक है? ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "और फिर बी के लॉग बेस सी को ए के लॉग बेस सी से संयोजित करने के विभिन्न संभावित तरीकों का एक पूरा ढेर।", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "मैं आपको इस पर सार्थक समय दूंगा क्योंकि यह तब तक स्पष्ट नहीं है जब तक कि आप पहले से ही लघुगणक से परिचित न हों, और इस पर थोड़ा विचार करना उचित है।", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "धन्यवाद करेन. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/hungarian/sentence_translations.json b/2020/ldm-logarithms/hungarian/sentence_translations.json index d21141cea..17d6de8a1 100644 --- a/2020/ldm-logarithms/hungarian/sentence_translations.json +++ b/2020/ldm-logarithms/hungarian/sentence_translations.json @@ -1,13 +1,13 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math.", + "input": "... you you you you you you you you you you you you you you you you you you you you you you you you you", "translatedText": "🎵Zene🎵 Üdvözöljük újra a Lockdown Math.", "n_reviews": 0, "start": 0.0, "end": 691.84 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson.", + "input": "it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the ove", "translatedText": "Ma a logaritmusokról fogunk beszélni, és egyfajta visszalépés az alapokhoz.", "n_reviews": 0, "start": 720.0, @@ -28,7 +28,7 @@ "end": 742.7 }, { - "input": "Because I have a couple suspicions, but I think doing a live poll to see where everyone is might be helpful.", + "input": "'re adding 5000 instead use a y axis where each step is multiplicative so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by 10 and wh", "translatedText": "Mert van egy-két gyanúm, de azt hiszem, hasznos lehet egy élő szavazás, hogy megnézzem, hol van mindenki.", "n_reviews": 0, "start": 742.92, @@ -42,7 +42,7 @@ "end": 759.16 }, { - "input": "co.", + "input": "y axis is now plotting not the total number of cases but the log", "translatedText": "co.", "n_reviews": 0, "start": 759.16, @@ -63,7 +63,7 @@ "end": 770.84 }, { - "input": "a.", + "input": "would do and, you know, it's a little bit", "translatedText": "a.", "n_reviews": 0, "start": 770.84, @@ -77,7 +77,7 @@ "end": 774.0 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c.", + "input": "ive model to say oh it's going to grow exactly exponentially but in the early phases of something like this that is what it is so I kind of fast forward in the animation I m", "translatedText": "Tanultam róluk, de néha összezavarodok az összes tulajdonságtól c.", "n_reviews": 0, "start": 774.0, @@ -133,7 +133,7 @@ "end": 864.84 }, { - "input": "What's the way to get it built in their intuitions?", + "input": "that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications", "translatedText": "Mi a módja annak, hogy ezt beépítsék az intuícióikba?", "n_reviews": 0, "start": 864.84, @@ -175,14 +175,14 @@ "end": 1063.5 }, { - "input": "and we see this sitting around early March or so and of course this is because this is when the corona outbreak was really starting to kick into high gear and everyone wanted to understand exponential growth and a common way that exponential growth is plotted is with what's known as a logarithmic scale so I actually made a video about this and in it I was creating some animations and wanted to illustrate this idea of exponential growth and the main idea here, I'll go ahead and skip back to a different animation is if you're tracking the numbers, in this case this was the number of recorded cases of COVID-19 outside of mainland China in the months leading up to March you could just track what the absolute number is but the pattern that you'll find is that as you go from one day to the next, you tend to be increasing multiplicatively it's a little bit like earlier, we were seeing the powers of 10 one step to the next, you're multiplying by some amount the way that the virus was growing was very similar from one day to the next, you're multiplying not quite by a constant but in this case, for this sequence of days, it was around 1.2 in that region, you're multiplying by something so when you're plotting this, it ends up looking like this classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the overall pattern is so a common trick is to say, instead of looking at this y-axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we're adding 5,000 instead use a y-axis where each step is multiplicative so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by 10 and what you can say is the y-axis is now plotting not the total number of cases but the logarithm of the total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend would do and it's a little bit of a naive model to say, oh it's going to grow exactly exponentially but in the early phases of something like this, that is what it is so I kind of fast-forward in the animation I made for that video and what's interesting is if back then, I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said, when is that line going to cross a million?", + "input": "eally liked this idea of making explicit how we have three totally different notations for the same exact fact one of them you're using relative positions of the numbers one of them we introduce a new symbol this radical and one of them we introduce a new word, log so these three syntactically different ways to communicate the same idea seemed wrong and so I made this video about an alternate possible notation and while I don't necessarily think that oh we should teach logarithms with this triangle because convention is what it is so it's better to start getting people used to the usual expression what I do like about it and starting off with it is when you see and think about this triangle it's really emphasizing that what the log wants to be is that exponent every time that you see log of some value you should think in your mind okay whatever this number is it really wants to be an exponent it wants to be an exponent and we'll see more of what that means as we go on okay so every time you see a log it wants to be an exponent this value three and more specifically it should be an exponent sitting on top of whatever that base is now in terms of convention for the first part of this video I'm just going to be using log without a base written on it to be the shorthand for log base 10 because log base 10 will be the most intuitive thing out there you should know that often in math the convention instead is that log without anything might mean log base e there's also another notation for that ln for natural log we're going to talk all about the natural log next time so don't worry too much about that right now and there's also yet another convention often if you're in a computer science setting log without any added sugar to indicate what it is defaults to meaning log base 2 so this can sometimes be a source of confusion but it basically depends on what discipline you're in in math, not moth, math people really like a base of e we'll see why next lecture in, I don't know, I'll say engineering but really it's anything where you want good intuition with our normal base 10 number system log means log base 10 and if you're curious often in computer science settings log base 2 comes up all the time so like I said, in the back of your mind if you're trying to think of some of these properties just resting on the idea that log counts the number of zeros at the end of a number that can get you a really far way so we're going to start going thro", "translatedText": "és ezt látjuk körülbelül március elején, és persze ez azért van, mert a koronajárvány ekkor kezdett igazán nagy sebességbe lendülni, és mindenki meg akarta érteni az exponenciális növekedést, és az exponenciális növekedés általános ábrázolási módja az ismert dolgokkal mint logaritmikus skála, ezért valójában készítettem egy videót erről, és abban készítettem néhány animációt, és szerettem volna illusztrálni az exponenciális növekedés gondolatát, és a fő gondolatot itt, és visszatérek egy másik animációhoz, ha nyomon követjük a számokat, ebben az esetben ez volt a regisztrált COVID-19 megbetegedések száma Kínán kívül a márciust megelőző hónapokban, csak nyomon lehet követni, mi az abszolút szám, de a minta az, hogy ahogy egyik napról a másikra haladsz, hajlamos vagy többszörösen növekedni, ez egy kicsit olyan, mint korábban, láttuk a 10 erejét egyik lépésről a másikra, te megsokszorozod valamivel a vírus növekedésének módját nagyon hasonló volt egyik napról a másikra, nem egészen konstanssal szorozod meg, de ebben az esetben ebben a napsorozatban 1 körül volt.2 abban a régióban, szorozsz valamivel, így amikor ezt ábrázolod, a végén úgy néz ki, mint ez a klasszikus exponenciális görbe, amely felfelé görbül, és néha megnehezítem, hogy lássam, merre halad, vagy mi az általános minta. egy gyakori trükk az, hogy ahelyett, hogy ezt az y-tengelyt néznénk, amely lineárisan növekszik, ahogy itt 5 000-ról 10 000-ra, 10 000-ról 15 000-ra megyek, 15 000-ról 20 000-ra minden lépés additív, hozzáadunk 5000-et, helyette használjunk egy az y-tengely, ahol minden lépés szorzóképes, tehát 10-ről 100-ra megy, 100-ról 1000-re, 1000-ről 10-re, 10 000-re, ezek mindegyike növekszik, ha megszorozzuk 10-zel, és azt mondhatjuk, hogy az y-tengely most nem ábrázol az esetek teljes száma, de az esetek teljes számának logaritmusa, és ez valójában megkönnyíti a megjelenítést egy rajzon, ha ki akarja vetíteni, hogy ez a tendencia mit csinál, és ez egy kicsit naiv modell, ó, pontosan exponenciálisan fog növekedni, de valami ilyesmi korai szakaszában ez az, ami így van, szóval kicsit előretekertem az animációt, amit ehhez a videóhoz készítettem, és az az érdekes, hogy akkoriban azt hiszem, közzétettem. március 6-án, ha megtaláltad a legmegfelelőbb sort, és kinyújtod, és azt mondod, mikor lépi át az a vonal egy milliót?", "n_reviews": 0, "start": 1063.5, "end": 1210.12 }, { - "input": "which because the y-axis is growing with multiplicative steps each time that you step up, you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know, when you understand logarithmic scales, it actually didn't seem that far it was only 30 days away if you naively just drew out that line and in fact, fast-forward to around April 5th, which is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day, I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully, since then, the growth has stopped being exponential so if you look at it on a logarithmic plot, instead of going up in a straight line, it starts to taper off but, point being, any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined, okay?", + "input": "ugh a couple of these properties and I want to do this just with a set of practiced examples so we'll transition away from the poll and this time to the first proper question and the question asks you which of the following is true a. the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. log of 1000 times x equals log of x cubed c. log of 1000 times x equals 3 plus the log of x d. log of 1000 times x equals 3 to the power of log of x and e. none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that great ok so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in t", "translatedText": "ami, mivel az y tengely szorzós lépésekkel növekszik minden alkalommal, amikor feljebb lép, 10-zel szoroz, így még ha úgy is tűnhet, hogy az akkori 20 000 eset még akkor is nagyon messze van a milliótól, Ha megérted a logaritmikus skálákat, akkor valójában nem tűnt olyan messze, hogy már csak 30 nap van hátra, ha naivan csak kihúztad ezt a vonalat, és valójában gyorsan előrepörgöttél április 5-e körül, amikor is ez azt jelezte volna, hogy elérjük a millió eset Kínán kívül, ez nagyjából az a nap, amikor megtörtént, azt hiszem, plusz vagy mínusz egy nap, nem emlékszem pontosan, de pont azon a környéken volt, mert emlékszem, hogy azt gondoltam, hú, ez egy naiv modell volt a videó számára. Használata, és megdöbbentő, hogy szerencsére ilyen pontosan megegyezett, azóta a növekedés nem volt exponenciális, így ha logaritmikus diagramon nézzük, ahelyett, hogy egyenesen felfelé haladna, elkezd elvékonyodni, de a lényeg, bármikor, amikor találkozik valamivel a természetben vagy akár egy ember alkotta konstrukcióban, ahol a természetes, hogy a szorzónövekedésre gondolunk, a logaritmusok segítségére vannak, tehát menjünk tovább, és gondoljuk át, mik is ezek valójában, hogyan definiálják őket. oké?", "n_reviews": 0, "start": 1210.12, @@ -238,7 +238,7 @@ "end": 2014.88 }, { - "input": "a.", + "input": "case, the correct answer of the choices we hav", "translatedText": "a.", "n_reviews": 0, "start": 2014.88, @@ -301,7 +301,7 @@ "end": 2567.02 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true?", + "input": "f 10,100 it's asking 10 to the what is equal to 10,100 you might say, I don't know, it's going to be a little above 4 because it's kind of close to 10,000 so the best you might guess here is oh this is going to be something That's kind of like The log of 10,000, but that just feels like a coincidence based on the two numbers that we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Sometimes you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y.", "translatedText": "csodálatos szóval, ismét rejtsük el, hol van néhány dolog, néhány más tulajdonság, amihez itt eljutunk, és ugyanabban a sorrendben fogom tartani, mint korábban itt volt, arra gondoltam, hogy ha előre megírják, az megtarthat egy kicsit tisztább, mint általában, de lehet, hogy csak ezt a furcsa papírvágási játékot kell játszani, szóval amit most találtunk, a napló b alapja, ha ezeket felcseréli, az ugyanaz, mint ha elosztjuk 1-gyel, aminek ez felel meg. Az exponenciális föld az, ha felveszed b-t valamilyen hatványra, és azt mondod, hogy ez egyenlő a-val, ez ugyanaz az állítás, mintha azt mondaná, hogy a hatvány inverze ismét egyenlő b-vel, hasznos lehet egy pillanatra úgy gondolni, hogy a logaritmusok megváltoztatják a dolgokat. kifelé az a log bázis b kifejezése az x szerepét játssza, és a log bázis a b kifejezés azt a szerepét játssza, ami az a tetején van, majd szimmetrikusan, a teljes b kifejezés az x hatványra játszik a belső szerepe a bal oldalon, az a szerepét és az egész kifejezést játssza, a valami erejéig azt a szerepét tölti be, ami a rönk alapjában van a, így láthatja, csak néhány példát csatlakoztatva és az exponenciális szabályoknak való megfeleltetésével máris átgondolhatunk három különböző logaritmusszabályt, amelyeket ha csak mint memorizálandó algebra darabokat adnánk át, akkor meg tudnád őket jegyezni, de nagyon könnyen kicsúszhatnak a nagyon könnyen elkeseredhet az adott feladat miatt, de érdemes emlékeztetnie magát arra, hogy az ok, amiért törődünk az efféle dolgokkal, az az, hogy a logaritmusok szabályainak megértése segít nekünk matematikailag végezni olyan helyzetekben, ahol olyan, mint egy vírus. egyik napról a másikra, egyik lépésről a másikra a dolgok hajlamosak multiplikatív módon növekedni, a logaritmusok szabályainak megértése segít jobban átérezni az ilyen dolgokat, így mielőtt egy szép valós példát mutatnánk be, hogyan nézhet ki. Hadd tegyek még egy kvízkérdést ebben a szellemben, hogy utoljára kérdezzek a logaritmusok tulajdonságairól, mielőtt egy kicsit áttérnénk egy való világbeli példára, hogy megszabaduljunk attól, ami itt és most volt, az alábbiak közül melyik igaz?", "n_reviews": 0, "start": 2567.02, @@ -336,7 +336,7 @@ "end": 3053.77 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew?", + "input": "bomb That was 50 megatons, and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons we're talking start off at that 32 kilotons of the Nagasaki bomb Multiply by 32 to get a megaton multiply by another 32 Right so we're talking about a thousand times the strength of the World War two ending explosion And you're still not at the 50 megatons of what humanity is capable of And that is evidently you know the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0. It's a lot bigger and The point here of course is just that when you have a scale giving you multiplicative increases It's worth appreciating that what look like small steps Can actually be huge steps in terms of the energy implied or the absolute values implied here So it I mean when we're thinking about the fact that there was ever a 9.5 That actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out And this is indicative of one area where logarithms tend to come about it's When humans want to create a scale for something that accounts for a hugely wide variance in how big things can be So in the case of size of earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you want that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next ques", "translatedText": "nos a mi háromszögünkre úgy gondolhatnánk, hogy tudod, nulla valamilyen x hatványhoz egyenlő valamilyen más y értékkel. Ugyanezt mondjuk, ha azt mondjuk, hogy az y naplózási bázis nulla egyenlő x nullával az x nullával, most itt az a probléma, hogy nulla bármihez az a végeredmény, hogy nulla, tehát ha csak a naplózási bázis nullára gondolunk. y bármilyen más y bemenetre, amit ismer, valami olyasmit szeretne megadni, mint egy, kettő vagy pi, amit csak akar, felteszi a kérdést nullától a mi egyenlő eggyel vagy kettővel vagy pi-vel vagy bármilyen számmal, ami ott van. és egyszerűen nem lesz válasz, így a legjobb esetben megpróbálhatod azt mondani, hogy igen, a nulla naplója, ez egy tökéletesen érvényes függvény, csak a nulla bemeneten van definiálva, de még akkor is gondot okoz, ha megpróbálod eldönteni, hogy mit akarsz. ott, mert ha nullát mondunk a nullára, az olyan, mintha bármi vonatkozik rá, tehát a karja a háta mögé csavarodik, bárhogyan is működni akarja, és ez megfelel annak a ténynek, hogy az exponenciális függvény nulla alapértékkel teljesen nulla. nem képezi le szépen a számokat egymásra, szóval ez egy nagyszerű kérdés, lehet-e egy naplózási alap nulla, most visszatérve ahhoz az ötlethez, hogy ezek a dolgok hol jönnek elő a való világban. Egy példa, amit szeretek a Richter-skála a földrengésekhez, tehát a Richter-skála számszerűsíti a földrengések erősségét, és bármi lehet a nagyon kis számoktól egészen a nagyon nagy számokig, például szerintem a valaha mért legnagyobb földrengés, és ez csak egy diagram, amely A Wikipédia 9-es volt.5 és hogy értékeljük, hogy ez mennyire őrült, érdemes megnézni az összefüggést aközött, amit ezek a számok jelentenek, és utána valami hasonló mennyiségű TNT valamiféle mértéke annak, hogy mennyi energia van benne, és mit próbálhatunk meg itt csinálni. Nézzük meg, hogy kaphatunk-e kifejezést a Richter-skála számára az energia mennyiségére vonatkozóan, és miért lenne ennek természetes leírása a logaritmus, így a legfontosabb, hogy összpontosítsunk, amikor lépéseket teszünk, mennyivel növekednek a dolgok így például ha a kettőről jól megyünk ebben az esetben, az nem mutatja meg, hogy hol van a három, szóval talán arra gondolunk, hogy kettőről négyre lépünk, ami olyan, mintha két lépést tennénk, mit tesz ez a Az energia mennyisége jól néz ki, mint egy metrikus tonna TNT-ből, ami azt hiszem, egy nagy bomba a második világháborúból, és akár egy kilotonnát is ezerszer annyi, ami egy kis atombomba, tehát csak két lépés A Richter-skála szerint a 2-es erősségű földrengéstől a 4-es erősségű földrengésig a második világháborútól a nagy bombától egészen a nukleáris korszakig tartunk. legalábbis abból a szempontból, amit ez a diagram szépen mutat nekünk, és nyilvánvalóan egyetlen lépés 4-ről 5-re feljebb 1 kilotonnáról 32 kilotonnára való fellépésnek felel meg, és ez nyilvánvalóan akkora volt, mint a városromboló bomba, amely Nagaszakira szállt, szóval ez talán egy dolog, ami ellentétes lehet a logaritmikus skálákkal kapcsolatban, ha csak a hírekben hallod a különbséget aközött, hogy ó, volt egy 4-es földrengés.0 egy 5-ös földrengéssel szemben.0 könnyű azt gondolni, igen, 4 és 5 ezek nagyon hasonló számok, de nyilvánvalóan a TNT mennyiségét tekintve, ami azt jelenti, hogy 32-vel megszorozzuk, hogy 1-ről a következőre jussunk, és ha 2-ről 4-re lépünk, az nyilvánvalóan körülbelül ezerrel szoroz, és az egyetlen Ennek az az oka, hogy ez nagyobb, mert itt a diagramunk nem azt mutatta, hogy mi a 3, ezért két lépést tettünk, és te magad is ellenőrizheted, hogy ha megtesz egy lépést 32-vel, majd megszorozod egy másik 32-vel, az valójában elég közel ezer. az az elképzelés, hogy a Richter-szám additív lépései megfelelnek a TNT multiplikatív lépéseinek, azt sugallja, hogy itt valami logaritmikus játszik, és egy kicsit érdekes, hogy folytassuk itt, és elmondjuk, mennyivel nő ez részben a világ jelenségei miatt. ha leírjuk, igen, nem óriási meglepetés, hogy ahogy teszünk egy újabb lépést, ez ismét 32-vel szoroz, de megérzéseink szerint ez a különbség a 32 kilotonnás kis atombomba és az egy megatonna között, amit nem kis atombombának gondolhatunk. Nagaszaki atombomba, ami azt hiszem, 32 Nagaszaki atombomba egy megatonna, ami nyilvánvalóan akkora, mint a kettős szálú lapos földrengés Nevadában, USA 1994. Nem tudtam, mi az, köszönet a Wikipédiának a frekvenciák tekintetében. Nyilvánvalóan megkereste ezeket a kettőnél kisebbeket is, ezek állandóan előfordulnak, napi 8000 ilyen van, de amint az atombombák birodalmába kerülünk, olyanok, mint a 3.5. és 4. ezek nyilvánvalóan szintén gyakran megtörténnek valahol a földön, ezek közül naponta körülbelül 134 történik valahol, aki tudta?", "n_reviews": 0, "start": 3053.77, @@ -350,7 +350,7 @@ "end": 3901.15 }, { - "input": "log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000?", + "input": "the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship", "translatedText": "10-ből 2. log-alap körülbelül 0.3 log-alap 10-ből 2 megközelítőleg, sajnáljuk, 2-ből a 10-es log-alap körülbelül 0.3 log-alap 10-ből 2 körülbelül 1 harmada vagy 10 2-ből körülbelül 1 harmada ezek közül melyik áll a legközelebb az igazhoz annak alapján, hogy a 2-től a 10-ig lényegében 1000?", "n_reviews": 0, "start": 3901.15, @@ -364,14 +364,14 @@ "end": 4025.43 }, { - "input": "tender.", + "input": "attempt count which I think is to say unraveling If you're looking at the maximum number I'm not I'm", "translatedText": "pályázati kiírás.", "n_reviews": 0, "start": 4025.43, "end": 4029.59 }, { - "input": "Not at all a unanimous decision here.", + "input": "not great at Vanna whiting this thing if you look at the maximum number in our poll It's asking what's the log base 2 of that?", "translatedText": "Itt egyáltalán nem egyhangú döntés.", "n_reviews": 0, "start": 4029.59, @@ -385,7 +385,7 @@ "end": 4038.31 }, { - "input": "So that's good, they're very numerically similar, right?", + "input": "ent powers of 2 then that rescales it and Yes, yes is the answer what a fantastically apropos questio", "translatedText": "Szóval ez jó, szám szerint nagyon hasonlóak, nem?", "n_reviews": 0, "start": 4038.31, @@ -413,35 +413,35 @@ "end": 4124.69 }, { - "input": "And the question is how we can leverage this to understand something like log base 2 of 10, or log base 10 of 2.", + "input": "u some more time to think this through because it's looks like a big pile of algebra plug in some numbers to see what seems to work well and See which answer fits You You You Okay, so eve", "translatedText": "A kérdés pedig az, hogy hogyan tudjuk ezt kihasználni, hogy megértsünk valamit, mint például a 2/10-es naplóalap, vagy a 2-ből 10-es naplóalap.", "n_reviews": 0, "start": 4124.69, "end": 4137.67 }, { - "input": "As we saw earlier, those are just the reciprocals of each other.", + "input": "n if you are still thinking about it I'm gonna go ahead and grade it here and then start talking about Why it's true and then also why we should", "translatedText": "Amint korábban láttuk, ezek csak egymás kölcsönösségei.", "n_reviews": 0, "start": 4137.67, "end": 4146.01 }, { - "input": "So what does this mean?", + "input": "care why this is an operation that actually tells", "translatedText": "Szóval mit jelent ez?", "n_reviews": 0, "start": 4146.15, "end": 4147.95 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right?", + "input": "us something so the correct answer which it looks like around 1700 of you got congratulations is Log base C of B times log base B of A is equal to log base", "translatedText": "Ha a 10-es 2. log-alap egyenlő x-szel, az ugyanaz, mintha azt mondanánk, hogy az x-re 2 egyenlő 10-el, igaz?", "n_reviews": 0, "start": 4147.95, "end": 4157.99 }, { - "input": "It's asking us 2 to the what equals 10.", + "input": "C of A great Now that's just a big ol pile of things. Why would that be true?", "translatedText": "2-t kér tőlünk a mi egyenlő 10-hez.", "n_reviews": 0, "start": 4157.99, @@ -483,7 +483,7 @@ "end": 4213.53 }, { - "input": "People seem to think you can do that with any function, but you just can't.", + "input": "here would be things like let's use a different color. Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 o", "translatedText": "Úgy tűnik, az emberek azt hiszik, hogy ezt bármilyen funkcióval megteheti, de egyszerűen nem.", "n_reviews": 0, "start": 4213.53, @@ -518,14 +518,14 @@ "end": 4229.95 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x.", + "input": "t's plug in another power of 10 It'll be nice if it's also a power of 100 So I'll do a million So This one is asking 10 to 100 to the what equals a million How many times do I multiply a hundred by itself to get to a million?", "translatedText": "És elég jól, amit korábban láttunk, az az, hogy a 10-ből 2-es naplóalap, mondhatnánk azt is, hogy a 2-ből 10-es naplóalap csak 1-gyel haladja meg ezt az összeget, 1-gyel az x-hez képest.", "n_reviews": 0, "start": 4229.95, "end": 4234.87 }, { - "input": "And you can see this pretty easily by writing 2 is equal to 10 to the 1 over x.", + "input": "How many times does a hundred go into a million? Phrasing the same thing 10 different ways now the claim is that this is", "translatedText": "És ezt elég könnyen beláthatod, ha azt írod, hogy a 2 egyenlő 10-zel az 1-gyel x fölé.", "n_reviews": 0, "start": 4234.87, @@ -546,7 +546,7 @@ "end": 4245.93 }, { - "input": "Great.", + "input": "ng log base 10 of a million That if I ask how many times d", "translatedText": "Nagy.", "n_reviews": 0, "start": 4245.93, @@ -560,7 +560,7 @@ "end": 4266.57 }, { - "input": "So if I ask what is the log base 2 of a thousand, like we just saw, it's approximately the case that 2 to the power 10 is equal to a thousand.", + "input": "ve me the answer to how many times 10 goes into a million now just checking the numbers this certainly works 10 goes into a hundred two times 100 goes into a million three times in a multiplicative sense in that a hundr", "translatedText": "Tehát ha azt kérdezem, hogy mi az ezres 2-es rönk alapja, ahogy az imént láttuk, akkor körülbelül az a helyzet, hogy a 2 a 10-es hatványhoz egyenlő ezerrel.", "n_reviews": 0, "start": 4266.57, @@ -574,14 +574,14 @@ "end": 4285.29 }, { - "input": "Log 2 of a thousand is approximately 10.", + "input": "ll six Now we could think of this property in terms of the corresponding exponent rule which is going to look a l", "translatedText": "A 2. napló ezerből körülbelül 10.", "n_reviews": 0, "start": 4285.29, "end": 4288.75 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million.", + "input": "ittle bit stranger But it's actually just saying the entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each other the whole statement is equal to saying that um suppose that B to the X is equal to a Got some", "translatedText": "Hasonlóképpen naplózza a milliós 2-es bázist, hát lássuk, ha körülbelül 10-szer kell megszoroznunk a 2-t önmagával, hogy ezresre jussunk, akkor körülbelül 20-szor kell megszoroznunk önmagával, hogy milliót kapjunk.", "n_reviews": 0, "start": 4288.75, @@ -623,7 +623,7 @@ "end": 4344.51 }, { - "input": "Log base 10 of a thousand is equal to 3.", + "input": "g you layer it on top of each other. Now if we rearrange that expression, we get what is probably the second most important of all of our", "translatedText": "Ezerből 10 rönk alap egyenlő 3-mal.", "n_reviews": 0, "start": 4344.51, @@ -637,14 +637,14 @@ "end": 4353.53 }, { - "input": "It's counting the number of zeros, it ends up being about 6.", + "input": "n you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if you want", "translatedText": "A nullák számát számolja, és végül körülbelül 6 lesz.", "n_reviews": 0, "start": 4353.53, "end": 4358.35 }, { - "input": "And log base 10 of a billion, counting the number of zeros, it ends up being 9.", + "input": "the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. If", "translatedText": "És egymilliárdból 10 naplóalap, a nullák számát számolva végül 9 lesz.", "n_reviews": 0, "start": 4358.35, @@ -679,35 +679,35 @@ "end": 4411.57 }, { - "input": "Okay?", + "input": "say I'll use the log base 10 button and evaluat", "translatedText": "Oké?", "n_reviews": 0, "start": 4411.57, "end": 4417.25 }, { - "input": "Now this is an intuition worth remembering.", + "input": "e what's on the inside here, which at least positionally it's kind of above the 100.", "translatedText": "Ez egy olyan intuíció, amelyet érdemes megjegyezni.", "n_reviews": 0, "start": 4417.25, "end": 4417.93 }, { - "input": "If you have your numbers described with one base, it's basically the same as describing them with another base, but there's some rescaling constant.", + "input": "It has a higher altitude as we write it. So this can line up with the notation a little bit, that it sits on the numerator. And on the bottom, I use the log base 10 button that's in my calculator on th", "translatedText": "Ha a számokat egy alappal írjuk le, akkor az alapvetően ugyanaz, mint egy másik alappal, de van némi átskálázási állandó.", "n_reviews": 0, "start": 4417.93, "end": 4428.15 }, { - "input": "Okay?", + "input": "e base, on the 100.", "translatedText": "Oké?", "n_reviews": 0, "start": 4428.15, "end": 4429.21 }, { - "input": "And then the next question is going to start getting us at that direction, but it's going to be framed in a way that just looks like a whole pile of algebra, and again I will encourage you to plug in numbers if you want to to gain a little intuition for it.", + "input": "And then I can evaluate both of those and it'll give me the answer. In this case it gets you 6 divided by 2, which will be 3. And if we really just think through what this is saying, I know I've said it many different times, but it's a convoluted enough way to write things, but an intuitive enough", "translatedText": "És akkor a következő kérdés ebbe az irányba fog elvezetni minket, de úgy fog megfogalmazni, hogy csak úgy néz ki, mint egy egész halom algebra, és ismét arra biztatlak, hogy illesszen be számokat, ha nyerni akar. egy kis intuíció ahhoz.", "n_reviews": 0, "start": 4429.21, @@ -728,7 +728,7 @@ "end": 4452.11 }, { - "input": "Does that equal log base B of A?", + "input": "Because like I said, this is probably the second most important log rule. We're", "translatedText": "Ez egyenlő az A logaritmikus bázisával?", "n_reviews": 0, "start": 4452.11, @@ -763,7 +763,7 @@ "end": 4481.63 }, { - "input": "If you're looking at the maximum number, I'm not great at Vanna Whiting this thing, if you look at the maximum number in our poll, it's asking what's the log base 2 of that, so as it crosses different powers of 2 then that rescales it, and yes, yes is the answer.", + "input": "But anything additive in the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the left-handed side and the right-hand side are just saying how many times does 100 go into a million? but going about that in different ways. So this is extremely nice", "translatedText": "Ha a maximális számot nézi, nem vagyok jó Vanna Whitingnek ebben a dologban, ha megnézi a szavazásunkban szereplő maximális számot, akkor azt kérdezi, hogy ennek mekkora a 2-es naplózási alapja, tehát keresztezi a 2 különböző hatványait. akkor ez átméretezi, és igen, igen a válasz.", "n_reviews": 0, "start": 4481.63, @@ -777,14 +777,14 @@ "end": 4490.09 }, { - "input": "Thank You Karen.", + "input": "Next time we're going to talk", "translatedText": "Köszönöm Karen.", "n_reviews": 0, "start": 4490.09, "end": 4490.95 }, { - "input": "All right so answers are still rolling in, and I think like I said I just want to give you some more time to think this through because it looks like a big pile of algebra.", + "input": "all about the natural logarithm, which is log base e, often written ln. And turns out, this is much easier to compute. There's nice math behind it such that if you want to come up with an algorithm that your calculator can use, it's actually a lot easier to think of l", "translatedText": "Rendben, a válaszok még mindig érkeznek, és úgy gondolom, ahogy mondtam, csak szeretnék még egy kis időt adni, hogy végiggondold ezt, mert úgy néz ki, mint egy nagy halom algebra.", "n_reviews": 0, "start": 4490.95, diff --git a/2020/ldm-logarithms/indonesian/sentence_translations.json b/2020/ldm-logarithms/indonesian/sentence_translations.json index 3849269da..29ffb63c2 100644 --- a/2020/ldm-logarithms/indonesian/sentence_translations.json +++ b/2020/ldm-logarithms/indonesian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Musik🎵 Selamat datang kembali di Lockdown Math. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "Hari ini kita akan berbicara tentang logaritma dan pelajaran kembali ke dasar. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "Dan seperti biasa, sebagai permulaan, saya hanya ingin mengetahui posisi penonton saat ini. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "Jadi, kalau bisa ke 3b1b. bersama. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "Saya belum pernah mendengarnya sebelumnya atau belum pernah mempelajarinya sebelumnya b. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "Saya telah mempelajarinya tetapi terkadang bingung dengan semua propertinya c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "Saya memahaminya tetapi tidak tahu cara mengajarnya dan d. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "Saya memahaminya dengan baik dan dapat dengan nyaman mengajarkannya kepada orang lain agar mereka juga memahaminya dengan baik. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "Jadi, kita punya perpecahan yang bagus. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "Seperti yang saya katakan, tujuannya adalah untuk menciptakan pelajaran yang bisa saya tunjukkan kepada orang-orang di masa depan jika mereka merasa tidak nyaman dengan logaritma dan saya ingin bisa berkata, oh, inilah tempat yang bisa Anda datangi. bagaimana menurut saya, Anda tahu, bagaimana menurut saya Anda dapat melakukan pendekatan secara intuitif. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "Karena saya menelusuri beberapa forum guru sebelum melakukan kuliah khusus ini dan ketika orang bertanya topik apa yang paling sulit untuk diajarkan dalam matematika sekolah menengah dalam arti bahwa siswa tampaknya paling mengalami kesulitan dengan topik tersebut, logaritma adalah salah satu yang paling banyak. jawaban yang umum ditunjukkan yang menarik dan saya dapat menebak mungkin itu karena ada banyak sekali properti yang akhirnya harus Anda pelajari, Anda tahu, jadi jika kita melompati ke mana kita akan pergi, Anda memiliki semua tumpukan ini peraturan yang terlihat seperti sekumpulan aljabar yang sulit untuk diingat dan mudah untuk membingungkan banyak hal di kepala Anda dan saya pikir ketika orang-orang memiliki, Anda tahu, kenangan buruk tentang seperti apa matematika di sekolah menengah dan apa itu logaritma bermanfaat bagi mereka, sering kali rumus-rumus khusus tersebut terlintas dalam pikiran dan apa yang ingin saya lakukan hari ini adalah mencoba membahasnya, bagaimana memikirkannya tetapi juga pada tingkat meta jika Anda mengajari seseorang aljabar, apa itu poin-poin yang perlu ditekankan? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "Apa cara untuk membangun intuisi mereka? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "oh, ada 3 angka nol di situ, berapa log satu juta? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "log 1000 kali x sama dengan 3 kali log x dan ingat kita menggunakan ketentuan bahwa log tersebut berbasis 10 b. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "log 1000 kali x sama dengan log x pangkat tiga c. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "log 1000 kali x sama dengan 3 pangkat log x dan e. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "tidak satu pun dari hal di atas dan ingat seperti yang saya katakan sebelumnya, kita harus berharap sepenuhnya bahwa semua orang di awal yang mengatakan bahwa mereka memahami log dengan baik, mereka akan segera menjawab, mereka akan menjawab dengan benar, tetapi jika Anda seseorang yang tidak, jangan biarkan hal itu mengintimidasi Anda ketika Anda sedang melihat masalah seperti ini. Apa yang saya sarankan agar Anda lakukan hanyalah memasukkan berbagai pangkat 10 dan memikirkan gagasan bahwa fungsi log menghitung jumlah angka nol jadi saya akan memberi Anda sedikit waktu untuk memikirkannya jadi saya akan melanjutkan dan menilainya dan seperti biasa jika itu lebih cepat dari apa yang Anda rasa nyaman, ketahuilah bahwa itu hanya karena saya ingin melanjutkan ke depan dengan pelajaran jadi dalam hal ini jawaban yang benar adalah logaritma 1000 kali x sama dengan mengambil 3 ditambah logaritma x dan sekarang mari kita pikirkan sejenak dan seperti yang saya katakan ketika Anda baru memulai dengan mereka saya pikir hal terbaik untuk dilakukan adalah merasa nyaman memasukkan berbagai angka dan angka terbaik untuk dimasukkan adalah angka yang sudah pangkat 10 jadi jika Anda menanyakan sesuatu seperti log 1000 kali x baiklah saya tidak' Tidak tahu, mari kita masukkan sesuatu untuk x log 1000 kali 100 ya kita tahu berapa banyak angka nol yang akan ada di jawaban akhir di sini 1000 kali 100 adalah 100.000 kita sudah secara intuitif memiliki gagasan bahwa ketika kita mengalikan 2 pangkat 10 kita hanya mengambil angka nolnya, 3 angka nol dari 1000 itu, 2 angka nol dari 100 itu dan kita letakkan di samping satu sama lain sehingga seharusnya menjadi 5 angka nol total tetapi jika Anda benar-benar merenungkan bukan hanya bagaimana angkanya berubah keluar tapi kenapa ternyata seperti itu 3 angka nol dari 1000 itu ditambah 2 angka nol dari 100 itu yang juga bisa kita tulis dengan mengatakan banyaknya angka nol dalam 1000 ditambah banyaknya angka nol dalam 100 jadi ini idenya bahwa logaritma hasil perkalian dua benda adalah jumlah logaritma kedua benda tersebut dalam konteks pangkat 10. Itu hanya mengkomunikasikan apa yang sudah menjadi ide yang sangat intuitif bagi banyak dari kita jika Anda mengambil 2 pangkat 10 dan mengalikannya. mengambil semua angka nolnya dan menjejalkannya satu sama lain sehingga cara saya menuliskannya di sini sebenarnya menunjukkan fakta yang sedikit lebih umum yang akan menjadi properti logaritma pertama kita yaitu jika kita mengambil log dari A dikali B sama dengan log dari A ditambah log dari B sekarang kapan saja Anda melihat salah satu dari aturan logaritma ini jika Anda mendapati diri Anda menyipitkan mata atau Anda sedikit bingung bagaimana cara mengingatnya, cukup masukkan contoh Saya berlebihan, saya sering mengatakan ini, tetapi itu karena menurut saya sangat mudah untuk melupakan begitu Anda tenggelam dalam aljabar itu sendiri dan Anda sedang mengerjakan semacam ujian dan itu hanya mendapat banyak simbol. untuk mengingatkan diri sendiri bahwa Anda boleh saja memasukkan beberapa angka, itu adalah hal yang baik untuk dilakukan dan sering kali merupakan cara yang bagus untuk menghasilkan intuisi, jadi dalam hal ini, dengan mengucapkan log A dikali B dan memecahnya, kita hanya bisa berpikir, oh, itu log 100 kali 1000 yaitu 5, ada 5 angka nol di dalamnya dipecah berdasarkan jumlah angka nol di setiap bagian yang diberikan bagus, bagus sekali jadi bawa intuisi itu lebih jauh mari kita coba soal latihan lain dan lagi, jika Anda mengetahuinya, bagus, Anda akan dapat menjawabnya dengan baik tetapi mungkin berpikir, bukan hanya apa jawabannya tetapi bagaimana saya menjelaskan jawaban ini kepada seseorang atau bagaimana saya mencoba membuat siswa sampai pada jawaban ini sendiri tanpa saya harus memberi tahu mereka apa jawabannya jadi ada dua calon penonton, ada yang tertarik dengan pelajaran itu sendiri dan kemudian ada yang tertarik dengan meta pelajaran sehingga pertanyaan kita bertanya, sekali lagi, manakah dari berikut ini yang benar? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "A. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "log dari x ke n sama dengan n kali log dari x b. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "log dari x ke n sama dengan log dari x pangkat n c. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "log dari x ke n sama dengan n ditambah log dari x atau d. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "jadi jawaban yang benar di sini adalah a, yang sepertinya 4.000 dari Anda mendapat ucapan selamat, memberi tahu kami bahwa log x pangkat n sama dengan n kali log x jadi, sekali lagi, katakanlah Anda mencoba mengajarkan ini kepada seseorang atau jika Anda mencoba memahami artinya sendiri. Saya pikir tempat yang baik untuk memulai adalah mencolokkan sesuatu dan dalam hal ini, untuk log x pangkat n mari kita coba dengan 100 pangkat 3 dan Anda dapat mencobanya dengan pola lain untuk melihat apakah pola yang Anda lakukan benar-benar berhasil, tetapi jika Anda memikirkannya dengan matang, bukan sekadar melihat jawabannya, tetapi mencoba memikirkan mengapa jawabannya berubah seperti itu kadang-kadang satu contoh bisa digunakan karena 100 pangkat tiga, kita dapat menganggapnya sebagai pengambilan yang baik, itu berarti 3 salinan dari 100. Saya mengambil 3 salinan dari 100 dan ketika saya mengalikan semuanya dan saya menganggap log sebagai menghitung jumlah nol yang kita katakanlah, oh, itu akan menjadi suatu bilangan yang hanya memiliki 6 angka nol, itulah artinya jika dikalikan 100 dikalikan 100 dikalikan 100 Saya hanya berpikir untuk mengelompokkan semua angka nol itu untuk mendapatkan satu juta, jadi bilangan ini akan menjadi 6 tapi kalau dipikir sebenarnya kenapa 6 bukan hanya itu jumlah angka nol di dalam satuan dari mana angka 6 itu berasal adalah kita punya 3 salinan dari 100 itu dan masing-masing dari 100 itu punya 2 angka nol yang berbeda jadi itu lebih umum cara Anda dapat memikirkannya di mana jika alih-alih mengambil 100 pangkat tiga kita melihat 1000 pangkat tiga atau 1000 pangkat n atau x pangkat n Anda dapat berpikir bahwa berapa pun nilai n adalah jumlah salinan yang kita kalikan jumlah dari nah, mari kita lihat, ini bukan x kali jumlah angka nol yang ada pada berapa pun kita mengganti x yang dalam hal ini adalah 100 jadi jika saya mengambil log 10.000 ke pangkat n, hasilnya akan sama seperti mengambil n salinan dari 10.000 itu dengan menghitung jumlah nol di masing-masingnya yaitu 4 sehingga akan menjadi n kali 4 dan tentu saja properti umum yang sebagian besar dari Anda jawab dengan benar adalah bahwa Anda memiliki efek kecil yang indah ini ketika Anda lihat log dari sesuatu yang dinaikkan menjadi kekuatan yang kekuatan kecilnya melompat ke depannya dan Anda hanya memiliki log dari apa yang ada di dalamnya sekarang salah satu implikasi yang mungkin paling penting dari itu. Saya tidak tahu apakah Anda akan menyebutnya sebuah implikasi atau jika Anda menyebutnya pernyataan ulang definisi jika saya mengambil log dan saya hanya akan menekankan kembali basis 10 dari 10 pangkat n kita dapat menganggap n kecil itu sebagai melompat ke bawah depan dan itu menjadi n kali basis log 10 dari 10 yang tentu saja 1 ekspresi ini dapat Anda anggap menghitung jumlah nol di akhir atau lebih umum lagi menanyakan 10 pada apa yang sama dengan 10 dan jawabannya hanyalah 1 yang sangat meyakinkan karena cara lain yang bisa Anda lakukan untuk kembali dan membaca ungkapan asli ini adalah dengan mengatakan 10 pada apa yang sama dengan 10 pada n oh baiklah jawabannya adalah n oke sekarang dengan setiap properti logaritma yang kita miliki jadi dalam hal ini kita baru saja menemukan satu log x pangkat n melibatkan n melompat di depan akan selalu ada properti eksponensial bayangan cermin dan itu cara lain yang dapat kita bantu untuk mendapatkan sedikit intuisi untuk ini jadi izinkan saya menutupinya beberapa properti masa depan yang akan kita dapatkan di sini mencoba menyembunyikan ke mana kita pergi apa yang baru saja kita temukan menaikkan sesuatu ke n yang melompat di depan ini sesuai dengan properti eksponensial yang jika saya ambil 10 ke x dan naikkan semua itu dipangkatkan n itu sama dengan memperparah 10 n kali x dan ini membawa kita ke intuisi lain yang mungkin Anda miliki untuk logaritma yang mana mereka seperti eksponen yang dibalik dan inilah yang saya maksud dengan bahwa benda yang ada di dalam log jika saya mengambil log a, Anda harus menganggapnya sebagai keseluruhan ekspresi luar untuk sesuatu yang eksponensial dalam hal ini a benda di dalam sama dengan 10 x x output dari fungsi sedangkan keseluruhan log dari a sesuai dengan apa yang ada di dalam di sini, berapa eksponen dari 10 jadi di mana pun Anda melihat ekspresi log di sini, Anda harus berpikir bahwa memainkan peran eksponen di sebelah kanan sisi dan setiap kali Anda melihat eksponensial seluruh ekspresi 10 x, seluruh komponen luar di sisi kanan yang sesuai dengan sesuatu yang ada di dalam salah satu log dan kita melihat ini di atas gagasan bahwa ketika kita mengalikan di dalam yang menjumlahkan di luar dengan baik jika jenis log mengubah eksponensial dalam ke luar yang memberi tahu kita bahwa mengalikan di luar mengalikan keluaran fungsi sama dengan menjumlahkan di dalam karena masing-masing log ini seperti log a dan log b sedang memainkan peran x dan y dalam ekspresi di sebelah kanan jadi dengan itu mari kita terus bermain mari kita lakukan beberapa hal lagi dan lihat berapa banyak dari sifat-sifat ini yang dapat kita bangun intuisinya sehingga yang terakhir ini, pemikiran yang sangat bagus tentang eksponen yang melompat ke bawah yang berikutnya adalah sesuatu yang mungkin terlihat sedikit aneh bagi mereka yang belum terbiasa dengan logaritma tetapi sekali lagi, masukkan beberapa angka untuk mendapatkan intuisi dan kami akan memberikannya sedikit tunggu sebentar untuk mengetahui manakah dari berikut ini yang benar? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "nah, kalau 10 pangkat tiga sama dengan 1000, itu sama saja dengan mengatakan 10 sama dengan 1000 dipangkatkan ke 1. Melakukan invers di sini melibatkan invers perkalian eksponen dan cara kerjanya adalah seperti 1 dibagi 3 dan bahwa 3 sama dengan basis log 10 dari 1000, maka 1 dibagi dengan basis log 10 dari 1000 jadi lebih umum, Anda mungkin menebak berdasarkan contoh ini bahwa ketika kita menukar basis dengan apa yang ada di dalamnya, maka itu sama dengan mengambil 1 dibagi dengan melihat apa yang ada di luar sana dan lagi, Anda dapat memikirkannya baik-baik dengan melihat aturan eksponensial yang sesuai. Sekarang apa yang terjadi dengan log kecil dan eksponensial saya yang cantik? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "luar biasa jadi, sekali lagi mari kita sembunyikan di mana beberapa hal beberapa properti lain yang akan kita bahas di sini dan saya akan menyimpannya dalam urutan yang sama dengan yang saya miliki sebelumnya di sini Saya berpikir bahwa memilikinya yang telah ditulis sebelumnya dapat mempertahankan saya sedikit lebih bersih dari biasanya tapi mungkin ini hanya melibatkan permainan aneh memotong kertas sambil menyeret-nyeret jadi apa yang baru saja kita temukan, log base b dari a jika Anda menukarnya, itu sama saja dengan membagi dengan 1 yang berhubungan dengan ini, off an tanah eksponensial adalah jika Anda membawa b ke suatu pangkat dan mengatakan bahwa itu sama dengan a, itu adalah pernyataan yang sama dengan mengatakan bahwa a dengan kebalikan dari pangkat itu sama dengan b lagi, ada baiknya untuk mengambil waktu sejenak dan memikirkan logaritma sebagai pembalikan sesuatu dalam ke luar ekspresi basis log b dari a memainkan peran x itu dan ekspresi basis log a dari b memainkan peran apa pun yang berada di atas a dan kemudian secara simetris, seluruh ekspresi b pangkat x dimainkan peran bagian dalam di sebelah kiri, ia memainkan peran a dan keseluruhan ekspresi, a untuk kekuatan sesuatu memainkan peran apa yang ada di dalam basis log a sehingga Anda dapat melihatnya, hanya dengan memasukkan beberapa contoh dan dengan menghubungkannya dengan aturan eksponensial kita sudah bisa memikirkan tiga aturan logaritma berbeda yang jika aturan-aturan tersebut hanya diturunkan sebagai potongan-potongan aljabar untuk dihafal, Anda tahu, Anda bisa menghafalnya tetapi sangat mudah bagi aturan-aturan tersebut untuk keluar dari ingatan Anda. kepala dan juga sangat mudah untuk merasa frustrasi dengan tugas yang ada tetapi Anda mungkin ingin mengingatkan diri sendiri bahwa alasan kita peduli dengan hal-hal semacam ini adalah memahami aturan logaritma membantu kita melakukan matematika dalam konteks di mana itu seperti virus yang tumbuh di mana dari satu hari ke hari berikutnya, dari satu langkah ke langkah berikutnya, segalanya cenderung tumbuh secara multiplikatif, memahami aturan logaritma membantu Anda lebih memahami hal-hal semacam itu, jadi sebelum kita membuat contoh dunia nyata yang bagus tentang apa yang terlihat seperti izinkan saya mengerjakan satu pertanyaan kuis lagi dalam nada ini untuk menanyakan tentang sifat-sifat logaritma yang terakhir sebelum kita beralih ke sedikit contoh dunia nyata, singkirkan apa yang kita miliki di sini dan saat ini, manakah dari berikut ini yang benar? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "log a plus b sama dengan log a plus log b log a plus b sama dengan log a dikalikan log b log a plus b sama dengan satu dibagi log a plus log b atau log dari a ditambah b sama dengan satu dibagi log dari a dikalikan log dari b atau tidak satu pun dari ah di atas, dan sekarang kita tidak memiliki banyak konsensus, bukan? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "sangat menarik, kita ada pacuan kuda antara dua orang jadi saya akan memberi Anda waktu sejenak untuk memikirkan hal ini sementara orang-orang menjawab, sebenarnya saya punya sedikit pertanyaan untuk penonton jadi, Anda tahu, saya baru saja berbicara tentang bagaimana kita bisa pikirkan dalam hal pertumbuhan multiplikatif dan itu tidak hanya harus berupa pangkat sepuluh, kita juga bisa melakukan sesuatu seperti pangkat tiga dimana jika Anda beralih dari satu ke tiga ke sembilan ke dua puluh tujuh ke delapan puluh satu, semuanya dari sini kita dapat mengatakan bahwa basis log tiga dari angka-angka ini hanya tumbuh dalam langkah-langkah kecil yang bagus sehingga basis log tiga dari satu, tiga ke apa yang sama dengan satu, jawabannya adalah nol secara umum log dari satu, tidak peduli basisnya, akan menjadi nol basis log tiga dari tiga, tiga dengan apa yang sama dengan tiga adalah satu sama halnya dengan basis log tiga dari sembilan adalah dua ah, Anda mungkin bertanya-tanya apa pertanyaan saya, tetapi akan membantu untuk menggambarkan semua ini dan untuk kesenangan saya sendiri di sini, izinkan saya menuliskan satu log base lagi tiga dari delapan puluh satu adalah empat sekarang, saya pernah mendengar bahwa jika Anda bertanya kepada seorang anak, katakanlah sekitar lima atau enam tahun berapa angka yang berada di tengah-tengah antara satu dan sembilan Anda katakan angka berapa di tengah-tengah naluri mereka untuk cara menjawab adalah logaritmik sedangkan naluri kita cenderung lebih linier jadi kita sering berpikir satu dan sembilan, Anda punya banyak angka yang berjarak sama di antara keduanya dua, tiga, empat, lima, enam , tujuh, delapan dan jika Anda melangkah tepat di tengah-tengahnya, Anda akan mendapatkan lima, tetapi jika Anda berpikir dalam istilah pertumbuhan multiplikatif untuk mendapatkan dari satu hingga sembilan, itu bukan masalah menambahkan banyak hal, melainkan Anda 'tumbuh dalam jumlah tertentu, Anda tumbuh dengan faktor tiga, lalu Anda tumbuh dengan faktor tiga lainnya. Seharusnya, naluri alami seorang anak sejalan dengan mengatakan tiga dan seharusnya ini juga sejalan dengan jika Anda memiliki antropolog yang mempelajari masyarakat yang belum' Saya tidak mengembangkan sistem akuntansi dan penulisan dengan cara yang sama seperti masyarakat modern. Mereka akan menjawab tiga hal ini, jadi, pertanyaan saya untuk penonton, apakah ada di antara Anda yang menonton saat ini yang memiliki akses terhadap anak kecil, katakanlah, dalam rentang lima tahun lihat apakah Anda boleh bertanya kepada mereka berapa angka yang berada di tengah-tengah antara satu dan sembilan dan jika Anda bisa, beri tahu kami di Twitter apa yang anak itu katakan, apa jawaban sebenarnya karena saya tidak tahu mengapa, saya hanya sedikit skeptis apakah hal ini benar-benar berjalan dengan baik dalam praktiknya. Saya memahami bahwa ini bukanlah cara yang sangat ilmiah untuk melakukannya. Saya tidak meminta orang-orang yang menonton siaran langsung YouTube untuk mensurvei anak-anak mereka sendiri dan kemudian men-tweet jawabannya, tetapi demi kepentingan saya sendiri, hal ini akan menarik untuk melihat semacam validasi di sana kembali ke pertanyaan kita ini adalah pertanyaan pertama yang sepertinya tidak memiliki konsensus besar dalam satu arah mari kita lanjutkan dan nilai untuk melihat apa jawabannya ternyata bagus, oke, jadi 2.400 salah satu dari Anda menjawab dengan benar bahwa tidak ada satu pun di atas yang log dari a plus b tidak memenuhi salah satu properti bagus ini dan secara umum, kecuali kita akan bekerja dengan jenis perkiraan tertentu terutama ketika log natural mulai berlaku. kita mungkin membicarakan hal ini lain kali menambahkan masukan logaritma sebenarnya merupakan sensasi yang sangat aneh. Ini adalah hal yang sangat aneh untuk dilakukan dan untuk memahami keanehan itu, masukkan pangkat sepuluh jika saya meminta Anda log plus b apa yang mungkin Anda mulai pikirkan adalah, oke, izinkan saya memasukkan beberapa contoh seperti 10.000 dan 100 dan saya bertanya pada diri sendiri, jika saya melakukan fungsi penghitungan nol ini, apa yang ada di masukan itu, berapa banyak angka nol di dalamnya? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "itu pertanyaan yang menarik oke, apakah basis logaritma bisa nol? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "baik dalam segitiga kita, kita mungkin menganggapnya seperti yang Anda tahu, nol pangkat x sama dengan nilai lain y ini adalah sesuatu yang bisa kita tulis dengan mengatakan nol pada x sama dengan y atau kita bisa menulis hal yang sama dengan mengatakan log basis nol dari y sama dengan x nol dengan apa yang sama dengan x sekarang masalahnya di sini adalah bahwa nol untuk apa pun akhirnya menjadi nol, jadi jika kita hanya memikirkan basis log nol dari y untuk masukan lainnya y Anda tahu, Anda ingin memasukkan sesuatu seperti satu atau dua atau pi apa pun yang Anda inginkan, Anda menanyakan pertanyaan nol hingga berapa yang sama dengan satu atau dua atau pi atau berapa pun bilangan yang mungkin Anda miliki di sana dan tidak akan ada jawaban jadi paling banter Anda bisa mencoba mengatakan oh ya, log nol, ini adalah fungsi yang benar-benar valid, itu hanya ditentukan pada input nol tetapi meskipun demikian Anda akan kesulitan mencoba menyelesaikan apa yang Anda ingin di sana karena mengatakan nol pada apa yang sama dengan nol itu seperti segala sesuatu berlaku padanya sehingga lengan Anda akan diputar ke belakang, bagaimanapun Anda ingin membuatnya berfungsi dan itu sesuai dengan fakta bahwa fungsi eksponensial dengan basis nol sepenuhnya nol itu tidak memetakan angka-angka dengan cara yang bagus satu ke satu satu sama lain jadi itu pertanyaan yang bagus, bisakah Anda memiliki basis log nol sekarang kembali ke gagasan di mana hal-hal ini muncul di dunia nyata, salah satu contoh yang saya suka adalah skala Richter untuk gempa bumi sehingga skala Richter memberi kita kuantifikasi seberapa kuat suatu gempa dan bisa berupa apa saja, mulai dari angka yang sangat kecil hingga angka yang sangat besar seperti menurut saya gempa terbesar yang pernah diukur dan ini hanyalah sebuah grafik yang berasal dari Wikipedia adalah peringkat 9.5 dan untuk menghargai betapa gilanya hal ini, ada baiknya melihat hubungan antara arti angka-angka ini dan kemudian sesuatu seperti jumlah setara TNT, semacam ukuran berapa banyak energi yang ada di dalamnya dan kemudian apa yang dapat kita coba lakukan di sini kita lihat apakah kita bisa mendapatkan ekspresi untuk angka skala Richter dalam kaitannya dengan jumlah energi dan mengapa logaritma merupakan cara alami untuk menggambarkan hal ini sehingga kunci untuk fokus adalah saat kita mengambil langkah ke depan, seberapa besar peningkatannya jadi misalnya jika kita melangkah dari dua sumur dalam hal ini tidak menunjukkan di mana letak tiga jadi mungkin kita berpikir untuk mengambil langkah dari dua ke empat yang seperti mengambil dua langkah, apa fungsinya dalam kaitannya dengan jumlah energinya sepertinya dibutuhkan dari satu metrik ton TNT yang menurut saya merupakan bom besar dari Perang Dunia II dan dibutuhkan hingga satu kiloton seribu kali lebih banyak yang merupakan bom atom kecil jadi hanya dua langkah dalam skala Richter, mulai dari gempa bumi berkekuatan 2 hingga gempa berkekuatan 4 skala Richter membawa kita dari bom besar dari Perang Dunia II hingga zaman nuklir sehingga hal ini patut diperhatikan dan langkah bersih pertama yang kita dapatkan adalah beralih dari 4 menjadi 5 pada setidaknya dalam hal apa yang ditunjukkan dengan baik oleh bagan ini kepada kita dan ternyata satu langkah naik dari 4 ke 5 sama dengan peningkatan dari 1 kiloton menjadi 32 kiloton dan itu jelas merupakan ukuran bom penghancur kota yang mendarat di Nagasaki jadi ini mungkin salah satunya hal yang mungkin berlawanan dengan skala logaritmik jika Anda hanya mendengar di berita perbedaan antara oh ada gempa berkekuatan 4.0 versus gempa berkekuatan 5.0 mudah untuk berpikir ya 4 dan 5 itu adalah angka yang sangat mirip tetapi ternyata dalam hal jumlah TNT yang sesuai dengan mengalikan dengan 32 untuk berpindah dari 1 ke berikutnya dan beralih dari 2 ke 4 ternyata mengalikan sekitar seribu dan satu-satunya Alasannya lebih besar adalah karena di sini bagan kita tidak menunjukkan angka 3 jadi kita mengambil dua langkah dan Anda dapat memverifikasi sendiri bahwa jika Anda mengambil langkah 32 dan kemudian Anda mengalikannya dengan 32 lagi, itu sebenarnya mendekati seribu jadi gagasan bahwa langkah-langkah aditif pada bilangan Richter sesuai dengan langkah-langkah perkalian di TNT tampaknya menunjukkan bahwa sesuatu yang logaritmik sedang berperan di sini dan sedikit menarik untuk terus melanjutkan di sini dan mengatakan seberapa besar pertumbuhannya sebagian karena fenomena dunia itu. menggambarkan ya, bukan sebuah kejutan besar bahwa saat kita mengambil langkah berikutnya, jumlahnya akan berlipat ganda menjadi sekitar 32 lagi, tetapi dengan mengekang intuisi kita, itulah perbedaan antara 32 kiloton sebuah bom atom kecil dan kemudian satu megaton yang mungkin kita anggap bukan bom atom kecil, Bom atom Nagasaki yang menurut saya adalah 32 bom atom Nagasaki untuk satu megaton yang ternyata sebesar gempa datar string ganda di Nevada AS tahun 1994. Saya tidak tahu apa itu, terima kasih Wikipedia dalam hal frekuensi ngomong-ngomong. juga mencari yang ini ternyata jumlahnya kurang dari dua, yang terjadi sepanjang waktu ada sekitar 8000 per hari tetapi begitu kita berada di dunia bom atom, hal-hal seperti 3.5 dan 4 itu ternyata juga cukup sering terjadi di suatu tempat di bumi ini, ada sekitar 134 di antaranya terjadi di suatu tempat setiap hari, siapa tahu? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "namun ketika kita semakin intens memasuki kisaran 5 dan 6 yang jauh di atas skala bom atom, kini kita hanya berada pada kisaran 2 bom atom per hari dan saya yakin seorang ahli geologi dapat datang dan menjelaskan mengapa kita semua harus melakukan hal tersebut. Kita tidak terlalu khawatir dengan fakta bahwa ada dua gangguan yang setara dengan bom atom terhadap kerak bumi yang terjadi setiap hari, tetapi mungkin sangat jarang gangguan tersebut terkonsentrasi di suatu tempat seperti kota yang dihuni banyak orang, hanya untuk memverifikasi pemikiran kita bahwa setiap langkah melibatkan pertumbuhan sebesar 32 mari kita lihat seperti apa langkah dari 6 ke 7 dan ini memberi kita lebih banyak contoh di antaranya mungkin memberikan ilusi bahwa itu adalah langkah yang lebih besar daripada yang sebenarnya dan memang itulah perbedaan antara 1 megaton dan 1 megaton. 32 megaton jadi itu dikalikan dengan 32. Salah satu hal yang menurut saya paling menarik dalam tabel ini adalah melihat sejauh mana kita harus melangkah sebelum kita sampai pada senjata nuklir terbesar yang pernah diuji. Ini adalah puncak perang dingin bom Tsar yang berbobot 50 megaton dan saya yakin mereka sebenarnya punya rencana awal untuk membuat bom berkekuatan 100 megaton namun mereka menolak rencana 50 megaton tersebut, kita mulai dengan 32 kiloton bom Nagasaki dikalikan 32 untuk mendapatkan megaton kalikan dengan 32 lagi jadi kita berbicara tentang kekuatan seribu kali lipat dari ledakan yang mengakhiri Perang Dunia II dan Anda masih belum mencapai 50 megaton dari kemampuan umat manusia dan itu jelas merupakan gempa bumi Jawa di Indonesia jadi 7 . 0 tidak hanya sedikit lebih besar dari 6.0, ini jauh lebih besar dan intinya di sini tentu saja ketika Anda memiliki skala yang memberi Anda peningkatan multiplikatif, ada baiknya untuk menghargai bahwa apa yang tampak seperti langkah kecil sebenarnya bisa menjadi langkah besar dalam hal energi yang tersirat atau nilai absolut yang tersirat di sini. jadi ketika kita memikirkan fakta bahwa angka 9 pernah ada. 5 yang sebenarnya terkesan tidak masuk akal mengingat hanya ada di angka 7.0 kisaran yang kita bicarakan tentang senjata termonuklir terbesar yang pernah dikeluarkan dan ini merupakan indikasi dari satu area di mana logaritma cenderung muncul, yaitu ketika manusia ingin membuat skala untuk sesuatu yang menjelaskan variasi yang sangat luas dalam seberapa besar hal tersebut dapat terjadi. jadi dalam hal ukuran gempa bumi, Anda dapat memperoleh sesuatu dari apa yang terjadi sepanjang waktu di sekitar Bumi, seukuran granat tangan yang besar dan Anda menginginkannya sesuai skala Anda dan sesuatu yang perlu dipikirkan mulai dari skala ke atas. ke gangguan terbesar yang pernah kita lihat dalam sejarah umat manusia dan untuk mewujudkannya sedemikian rupa sehingga Anda tidak hanya menuliskan sejumlah digit berbeda dalam angka-angka Anda untuk satu kasus dan sejumlah angka berbeda, angka yang lebih kecil digit untuk nomor Anda dalam kasus lain, bagus untuk mengambil logaritma dan kemudian meletakkannya pada satu skala yang pada dasarnya menekan angka-angka itu antara 0 dan 10 Anda melihat sesuatu yang sangat mirip terjadi dengan skala desibel untuk musik yang benar-benar berfungsi sedikit sedikit berbeda di mana setiap kali Anda mengambil langkah 10 desibel yang setara dengan mengalikan dengan 10 jadi daripada langkah 1 mengalikan dengan 10, itu adalah langkah 10 yang dikalikan dengan 10 sehingga membuat perhitungannya menjadi sedikit agak aneh tetapi idenya sama, bahwa jika Anda mendengarkan suara yang berkekuatan 50 desibel versus 60 desibel, suaranya jauh lebih senyap dalam hal energi yang ditransmisikan dan dihasilkan, apa jadinya, 60 hingga 70 atau 70 hingga 80 langkah-langkah tersebut, dari 60 hingga 80, yang melibatkan mengalikan jumlah energi per luas persegi dengan faktor 100 sehingga setiap kali Anda melihat skala logaritmik, ketahuilah dalam pikiran Anda bahwa itu berarti apa pun yang dimaksud di balik terpal akan bertambah sebesar jumlah yang sangat besar, sekali lagi ini adalah alasan mengapa kita melihat banyak skala logaritmik yang digunakan untuk menggambarkan wabah virus corona, jadi bagaimana Anda menggambarkan hubungan seperti ini di mana setiap kali Anda menambah angka skala Richter dengan 1, Anda mengalikannya dengan 32, ya, kita dapat membayangkan log dengan basis 32. Saya dapat mengatakan jika saya mengambil log, saya hanya akan memanggil r, angka untuk skala Richter. Saya mungkin menganggap ini sebagai log basis 32 dan itu akan sesuai dengan , tidak tidak tidak, aku melakukan kesalahan ini bukan itu yang dicatat kita ambil basis log 32 dari angka besar, dari angka TMT, kira-kira 1 megaton itu 1 juta ton basis log 32, itu seharusnya sesuai dengan angka skala Richter tetapi mungkin ada semacam offset, jadi kita dapat mengatakan bahwa ada semacam konstanta s yang kita tambahkan ke angka skala Richter ini dan ungkapan ini persis sama, maafkan saya karena keluar dari angka tersebut di bawah sana ungkapan ini persis sama dengan mengatakan 32 pangkat beberapa offset dikalikan dengan angka skala Richter kita yang sama dengan mengambil 32 untuk offset tersebut, yang mana itu sendiri hanyalah suatu konstanta besar, dikalikan 32 dengan angka skala Richter sehingga Anda mungkin menganggap ini hanya beberapa kali konstan 32 pangkat dari angka yang Anda lihat, jadi cara penulisan ini benar-benar menekankan pertumbuhan eksponensialnya sehingga jika ini sesuai dengan jumlah TMT yang Anda lihat, saat Anda meningkatkannya selangkah demi selangkah Anda mengalikannya dengan 32 tetapi cara lain untuk mengkomunikasikan fakta yang sama adalah dengan mengambil basis log 32 berapa pun jumlahnya, oke. Sekarang hal berikutnya yang ingin saya bicarakan adalah bagaimana kita tidak selalu harus melakukannya khawatir tentang cara menghitung log dengan basis yang berbeda, agak aneh di sini kita berbicara tentang log basis 32, saya referensikan sebelumnya bagaimana matematikawan sangat suka memiliki log dengan basis e ilmuwan komputer sangat suka memiliki log dengan basis 2 dan itu ternyata untuk tujuan komputasi atau untuk memikirkan bagaimana hal-hal ini berkembang jika Anda memiliki satu log, jika Anda dapat menghitung satu jenis log, apakah itu basis 10, basis 2, basis e Anda dapat menghitung hampir semua hal lain yang Anda sekarang ingin mengarahkan intuisi kita ke arah itu, mari kita kembali ke kuis kita dan melanjutkan ke pertanyaan berikutnya dan saya yakin pertanyaan ini adalah yang paling banyak, saya tidak tahu, ini adalah pertanyaan yang setengah masuk akal, ini pasti bagus ini hanya akan membuat kita siap untuk menerjemahkan dari konteks basis 2 ke konteks basis 10 dan ini juga merupakan intuisi yang baik untuk memahami pangkat 2 untuk memiliki secara umum hubungan yang dimilikinya dengan pangkat 10 karena ini adalah jenis kebetulan yang indah. sifat kedua jenis ini baik-baik saja Anda akan mengerti maksud saya, mereka bermain baik satu sama lain jadi pertanyaan kita bertanya, mengingat fakta bahwa 2 sampai 10 adalah 1024, 1024, yaitu kira-kira 1000 jadi jika Anda seorang sedikit longgar dengan angka-angka Anda dan Anda hanya membuat perkiraan 2 sampai 10, pada dasarnya 1000, manakah dari berikut ini yang paling mendekati kebenaran? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "lembut. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "Sama sekali bukan keputusan bulat di sini. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "Namun pertanyaannya adalah menanyakan mana yang paling mendekati kebenaran, dan mari kita lihat bagaimana kita bisa memikirkan hal ini. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "Jadi ini menunjukkan bahwa Anda mempunyai pangkat 2, yaitu 1024, sangat mendekati pangkat 10, sekitar 10 pangkat tiga. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "Jadi apa artinya ini? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "Jika logaritma basis 2 dari 10 sama dengan x, itu sama saja dengan mengatakan 2 pada x sama dengan 10, bukan? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "Ini menanyakan kita 2 pada apa yang sama dengan 10. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "Anda tidak dapat melakukan itu pada setiap fungsi. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "Orang-orang sepertinya mengira Anda bisa melakukan itu dengan fungsi apa pun, tapi ternyata tidak. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "Dan artinya x itu sekitar 10 pertiganya ya? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "Dan cukup baik, apa yang kita lihat sebelumnya adalah basis log 2 dari 10, kita juga bisa mengatakan basis log 10 dari 2 hanya 1 di atas jumlah tersebut, 1 di atas x. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "3. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "Dan karena kita melakukan sesuatu di log, saya hanya akan menulisnya dengan cara itu. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "Demikian pula dengan basis log 2 dari satu juta, mari kita lihat, jika kita harus mengalikan 2 dengan dirinya sendiri sekitar 10 kali untuk mendapatkan seribu, kita harus mengalikannya dengan dirinya sendiri sekitar 20 kali untuk mendapatkan satu juta. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "Ini sedikit lebih kecil tetapi ini adalah perkiraan yang bagus untuk diingat. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20, kami menurunkannya dengan jumlah yang sama. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, kami menguranginya dengan jumlah yang sama. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "Oke? ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "Ini adalah intuisi yang perlu diingat. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "Oke? ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "Dan kemudian tumpukan berbagai cara yang mungkin untuk menggabungkan basis log C dari B dengan basis log C dari A. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "Saya akan memberi Anda waktu yang bermakna untuk hal ini karena hal ini tidak jelas kecuali Anda sudah familiar dengan logaritma, dan ada baiknya Anda memikirkannya sedikit. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/italian/sentence_translations.json b/2020/ldm-logarithms/italian/sentence_translations.json index 72a677670..e61493985 100644 --- a/2020/ldm-logarithms/italian/sentence_translations.json +++ b/2020/ldm-logarithms/italian/sentence_translations.json @@ -1,13 +1,13 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math.", + "input": "... you you you you you you you you you you you you you you you you you you you you you you you you you", "translatedText": "🎵Music🎵 Bentornati a Lockdown Math.", "n_reviews": 0, "start": 0.0, "end": 691.84 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson.", + "input": "it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the ove", "translatedText": "Oggi parleremo di logaritmi e di una sorta di lezione di ritorno alle basi.", "n_reviews": 0, "start": 720.0, @@ -28,7 +28,7 @@ "end": 742.7 }, { - "input": "Because I have a couple suspicions, but I think doing a live poll to see where everyone is might be helpful.", + "input": "'re adding 5000 instead use a y axis where each step is multiplicative so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by 10 and wh", "translatedText": "Perché ho un paio di sospetti, ma penso che fare un sondaggio dal vivo per vedere dove si trovano tutti potrebbe essere utile.", "n_reviews": 0, "start": 742.92, @@ -42,7 +42,7 @@ "end": 759.16 }, { - "input": "co.", + "input": "y axis is now plotting not the total number of cases but the log", "translatedText": "co.", "n_reviews": 0, "start": 759.16, @@ -63,7 +63,7 @@ "end": 770.84 }, { - "input": "a.", + "input": "would do and, you know, it's a little bit", "translatedText": "UN.", "n_reviews": 0, "start": 770.84, @@ -77,7 +77,7 @@ "end": 774.0 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c.", + "input": "ive model to say oh it's going to grow exactly exponentially but in the early phases of something like this that is what it is so I kind of fast forward in the animation I m", "translatedText": "Li ho conosciuti ma a volte mi sento confuso da tutte le proprietà c.", "n_reviews": 0, "start": 774.0, @@ -133,7 +133,7 @@ "end": 864.84 }, { - "input": "What's the way to get it built in their intuitions?", + "input": "that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications", "translatedText": "Qual è il modo per farlo costruire nelle loro intuizioni?", "n_reviews": 0, "start": 864.84, @@ -175,14 +175,14 @@ "end": 1063.5 }, { - "input": "and we see this sitting around early March or so and of course this is because this is when the corona outbreak was really starting to kick into high gear and everyone wanted to understand exponential growth and a common way that exponential growth is plotted is with what's known as a logarithmic scale so I actually made a video about this and in it I was creating some animations and wanted to illustrate this idea of exponential growth and the main idea here, I'll go ahead and skip back to a different animation is if you're tracking the numbers, in this case this was the number of recorded cases of COVID-19 outside of mainland China in the months leading up to March you could just track what the absolute number is but the pattern that you'll find is that as you go from one day to the next, you tend to be increasing multiplicatively it's a little bit like earlier, we were seeing the powers of 10 one step to the next, you're multiplying by some amount the way that the virus was growing was very similar from one day to the next, you're multiplying not quite by a constant but in this case, for this sequence of days, it was around 1.2 in that region, you're multiplying by something so when you're plotting this, it ends up looking like this classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the overall pattern is so a common trick is to say, instead of looking at this y-axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we're adding 5,000 instead use a y-axis where each step is multiplicative so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by 10 and what you can say is the y-axis is now plotting not the total number of cases but the logarithm of the total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend would do and it's a little bit of a naive model to say, oh it's going to grow exactly exponentially but in the early phases of something like this, that is what it is so I kind of fast-forward in the animation I made for that video and what's interesting is if back then, I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said, when is that line going to cross a million?", + "input": "eally liked this idea of making explicit how we have three totally different notations for the same exact fact one of them you're using relative positions of the numbers one of them we introduce a new symbol this radical and one of them we introduce a new word, log so these three syntactically different ways to communicate the same idea seemed wrong and so I made this video about an alternate possible notation and while I don't necessarily think that oh we should teach logarithms with this triangle because convention is what it is so it's better to start getting people used to the usual expression what I do like about it and starting off with it is when you see and think about this triangle it's really emphasizing that what the log wants to be is that exponent every time that you see log of some value you should think in your mind okay whatever this number is it really wants to be an exponent it wants to be an exponent and we'll see more of what that means as we go on okay so every time you see a log it wants to be an exponent this value three and more specifically it should be an exponent sitting on top of whatever that base is now in terms of convention for the first part of this video I'm just going to be using log without a base written on it to be the shorthand for log base 10 because log base 10 will be the most intuitive thing out there you should know that often in math the convention instead is that log without anything might mean log base e there's also another notation for that ln for natural log we're going to talk all about the natural log next time so don't worry too much about that right now and there's also yet another convention often if you're in a computer science setting log without any added sugar to indicate what it is defaults to meaning log base 2 so this can sometimes be a source of confusion but it basically depends on what discipline you're in in math, not moth, math people really like a base of e we'll see why next lecture in, I don't know, I'll say engineering but really it's anything where you want good intuition with our normal base 10 number system log means log base 10 and if you're curious often in computer science settings log base 2 comes up all the time so like I said, in the back of your mind if you're trying to think of some of these properties just resting on the idea that log counts the number of zeros at the end of a number that can get you a really far way so we're going to start going thro", "translatedText": "e lo vediamo intorno all'inizio di marzo o giù di lì e ovviamente questo è perché è quando l'epidemia di corona stava davvero iniziando a prendere il sopravvento e tutti volevano capire la crescita esponenziale e un modo comune in cui viene tracciata la crescita esponenziale è con ciò che è noto come scala logaritmica, quindi in realtà ho realizzato un video a riguardo e in esso stavo creando alcune animazioni e volevo illustrare questa idea di crescita esponenziale e l'idea principale qui, andrò avanti e tornerò a un'animazione diversa se tu stai monitorando i numeri, in questo caso questo era il numero di casi registrati di COVID-19 al di fuori della Cina continentale nei mesi precedenti a marzo potresti semplicemente monitorare qual è il numero assoluto, ma lo schema che troverai è questo man mano che passi da un giorno all'altro, tendi ad aumentare in modo moltiplicativo, è un po' come prima, vedevamo le potenze di 10 un passo dopo l'altro, ti stai moltiplicando per una certa quantità nel modo in cui il virus stava crescendo era molto simile da un giorno all'altro, non stai moltiplicando proprio per una costante ma in questo caso, per questa sequenza di giorni, era circa 1.2 in quella regione, stai moltiplicando per qualcosa quindi quando lo stai tracciando, finisce per assomigliare a questa classica curva esponenziale che curva verso l'alto e a volte posso rendere difficile vedere dove sta andando o qual è lo schema generale così un trucco comune è dire, invece di guardare questo asse y che aumenta linearmente come qui sto andando da 5k a 10k, da 10k a 15k, da 15k a 20k ogni passo è additivo, stiamo aggiungendo 5.000 invece usa a asse y in cui ogni passaggio è moltiplicativo, quindi vai da 10 a 100, da 100 a 1000, da 1000 a 10, 10.000 tutti questi sono aumenti moltiplicando per 10 e quello che puoi dire è che l'asse y ora non sta tracciando il numero totale di casi ma il logaritmo del numero totale di casi e questo in realtà rende più facile vedere su un grafico se si volesse proiettare cosa farebbe quella tendenza ed è un modello un po' ingenuo da dire, oh crescerà in modo esattamente esponenziale, ma nelle prime fasi di qualcosa del genere, questo è ciò che è, quindi avanzo velocemente nell'animazione che ho realizzato per quel video e la cosa interessante è che se allora, penso di averlo pubblicato il 6 marzo se hai appena trovato una linea di adattamento migliore e l'hai allungata e hai detto, quando quella linea supererà il milione?", "n_reviews": 0, "start": 1063.5, "end": 1210.12 }, { - "input": "which because the y-axis is growing with multiplicative steps each time that you step up, you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know, when you understand logarithmic scales, it actually didn't seem that far it was only 30 days away if you naively just drew out that line and in fact, fast-forward to around April 5th, which is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day, I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully, since then, the growth has stopped being exponential so if you look at it on a logarithmic plot, instead of going up in a straight line, it starts to taper off but, point being, any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined, okay?", + "input": "ugh a couple of these properties and I want to do this just with a set of practiced examples so we'll transition away from the poll and this time to the first proper question and the question asks you which of the following is true a. the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. log of 1000 times x equals log of x cubed c. log of 1000 times x equals 3 plus the log of x d. log of 1000 times x equals 3 to the power of log of x and e. none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that great ok so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in t", "translatedText": "che poiché l'asse y cresce con passi moltiplicativi ogni volta che fai un passo avanti, stai moltiplicando per 10 quindi anche se potrebbero sembrare 20.000 casi o giù di lì di allora sono molto lontani da un milione, sai, quando capisci le scale logaritmiche, in realtà non sembrava così lontano, mancavano solo 30 giorni se ingenuamente tracciavi quella linea e, di fatto, avanzavi velocemente fino al 5 aprile circa, che è quando ciò avrebbe previsto che avremmo raggiunto un milioni di casi fuori dalla Cina è più o meno il giorno in cui è successo, credo più o meno un giorno, non ricordo esattamente ma era proprio in quel quartiere perché ricordo di aver pensato wow, era una specie di modello ingenuo anche per il video uso ed è scioccante che corrisponda così esattamente, per fortuna, da allora la crescita ha smesso di essere esponenziale quindi se la guardi su un grafico logaritmico, invece di salire in linea retta, inizia a diminuire ma, il punto è, ogni volta che ti imbatti in qualcosa in natura o anche in un costrutto creato dall'uomo in cui ciò a cui è naturale pensare sono gli aumenti moltiplicativi, i logaritmi entrano in gioco per aiutarti, quindi andiamo avanti e pensiamo a cosa sono realmente, come vengono definiti, Va bene?", "n_reviews": 0, "start": 1210.12, @@ -238,7 +238,7 @@ "end": 2014.88 }, { - "input": "a.", + "input": "case, the correct answer of the choices we hav", "translatedText": "UN.", "n_reviews": 0, "start": 2014.88, @@ -301,7 +301,7 @@ "end": 2567.02 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true?", + "input": "f 10,100 it's asking 10 to the what is equal to 10,100 you might say, I don't know, it's going to be a little above 4 because it's kind of close to 10,000 so the best you might guess here is oh this is going to be something That's kind of like The log of 10,000, but that just feels like a coincidence based on the two numbers that we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Sometimes you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y.", "translatedText": "meraviglioso quindi, ancora una volta nascondiamo alcune delle cose, alcune delle altre proprietà a cui arriveremo qui e le terrò nello stesso ordine in cui le avevo prima, qui pensavo che averlo pre-scritto avrebbe potuto trattenermi un po' più pulito del solito ma forse implica semplicemente giocare a questo strano gioco di tagliare la carta trascinandosi qua e là, quindi quello che abbiamo appena trovato, log base b di a se li scambi, è come dividere per 1 ciò a cui corrisponde, da a terra esponenziale è se prendi b a una certa potenza e dici che è uguale ad a è la stessa affermazione che dire che a all'inverso di quella potenza è uguale di nuovo a b, è utile prendersi un momento e pensare ai logaritmi come a cambiare le cose al contrario l'espressione log base b di a sta giocando il ruolo di quell'x e l'espressione log base a di b sta giocando il ruolo di qualunque cosa si trovi sopra a e poi simmetricamente, sta giocando l'intera espressione b elevato alla potenza x il ruolo dell'interno a sinistra, gioca il ruolo di a e l'intera espressione, a alla potenza di qualcosa gioca il ruolo di ciò che si trova all'interno della base del tronco a così puoi vedere, semplicemente inserendo alcuni esempi e facendolo corrispondere alle regole esponenziali possiamo già pensare a tre diverse regole del logaritmo che se fossero semplicemente tramandate come pezzi di algebra da memorizzare, sai, potresti memorizzarle ma è molto facile che ti scivolino fuori testa ed è anche molto facile sentirsi frustrati dal compito da svolgere, ma potresti voler ricordare a te stesso che il motivo per cui ci preoccupiamo di questo genere di cose è che comprendere le regole dei logaritmi ci aiuta a fare matematica in contesti in cui è come un virus che cresce dove da un giorno all'altro, da un passo all'altro, le cose tendono a crescere in modo moltiplicativo comprendere le regole dei logaritmi ti aiuta a farti un'idea migliore di quel tipo di cose quindi prima di fare un bell'esempio nel mondo reale di come può apparire ad esempio, lasciami fare un'altra domanda a quiz in questo senso per chiedere informazioni sulle proprietà dei logaritmi, un'ultima prima di passare a un piccolo esempio del mondo reale, sbarazzarci di ciò che avevamo qui e ora, quale delle seguenti affermazioni è vera?", "n_reviews": 0, "start": 2567.02, @@ -336,7 +336,7 @@ "end": 3053.77 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew?", + "input": "bomb That was 50 megatons, and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons we're talking start off at that 32 kilotons of the Nagasaki bomb Multiply by 32 to get a megaton multiply by another 32 Right so we're talking about a thousand times the strength of the World War two ending explosion And you're still not at the 50 megatons of what humanity is capable of And that is evidently you know the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0. It's a lot bigger and The point here of course is just that when you have a scale giving you multiplicative increases It's worth appreciating that what look like small steps Can actually be huge steps in terms of the energy implied or the absolute values implied here So it I mean when we're thinking about the fact that there was ever a 9.5 That actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out And this is indicative of one area where logarithms tend to come about it's When humans want to create a scale for something that accounts for a hugely wide variance in how big things can be So in the case of size of earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you want that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next ques", "translatedText": "beh, nei termini del nostro triangolo potremmo pensare che come dire, sai, zero su qualche tipo di potenza x è uguale a qualche altro valore y questo è qualcosa che potremmo scrivere dicendo che zero su x è uguale a y oppure potremmo scrivere la stessa cosa dicendo che log base zero di y è uguale a x zero a ciò che è uguale a x ora il problema qui è che zero rispetto a qualsiasi cosa finisce per essere zero, giusto, quindi se pensiamo solo a log base zero di y per qualsiasi altro input, sai, vuoi inserire qualcosa come uno o due o pi greco qualunque cosa tu voglia, stai ponendo la domanda da zero a quanto è uguale a uno o due o pi greco o qualunque numero tu possa avere lì e semplicemente non ci sarà una risposta quindi nella migliore delle ipotesi potresti provare a dire oh sì, logaritmo di zero, è una funzione perfettamente valida è definita solo sull'input zero ma anche in questo caso avresti difficoltà a provare a capire cosa vuoi lì perché dire zero a ciò che è uguale a zero è come se qualsiasi cosa si applicasse ad esso quindi il tuo braccio sarà girato dietro la schiena comunque tu voglia farlo funzionare e corrisponde al fatto che la funzione esponenziale con base zero è interamente zero non mappa i numeri uno a uno l'uno sull'altro, quindi è un'ottima domanda, puoi avere un logaritmo in base zero ora tornando all'idea di dove queste cose emergono nel mondo reale, un esempio che mi piace è la scala Richter per i terremoti quindi la scala Richter ci dà una quantificazione della forza di un terremoto e può essere qualsiasi cosa, da numeri molto piccoli fino a numeri molto grandi come penso il più grande terremoto mai misurato e questo è solo un grafico che proviene da Wikipedia era un 9.5 e per apprezzare quanto sia folle vale la pena guardare la relazione tra il significato di questi numeri e qualcosa come la quantità equivalente di TNT, una sorta di misura di quanta energia c'è in esso e quindi cosa possiamo provare a fare qui è vedere se riusciamo a ottenere un'espressione per il numero della scala Richter in termini di quantità di energia e perché i logaritmi sarebbero un modo naturale per descriverlo, quindi la chiave su cui concentrarsi è mentre stiamo facendo passi avanti quanto aumentano le cose quindi per esempio se andiamo da due beh in questo caso non ci mostra dove si trova tre quindi forse pensiamo di fare un passo da due a quattro che è un po' come fare due passi che cosa fa in termini di quantità di energia beh, sembra che ci occorre da una tonnellata di TNT che è immagino una grande bomba della Seconda Guerra Mondiale e ci impiega fino a un chilotone mille volte di più che è una piccola bomba atomica quindi solo due passi sulla scala Richter passando da un terremoto di magnitudo 2 a un terremoto di magnitudo 4 ci porta dalla grande bomba della seconda guerra mondiale fino all'era nucleare quindi è degno di nota e il primo passo pulito che otteniamo è andare da 4 a 5 a almeno in termini di ciò che questo grafico ci mostra bene ed evidentemente un singolo passo da 4 a 5 corrisponde a passare da 1 kiloton a 32 kiloton e questa era evidentemente la dimensione della bomba che distrusse la città che sbarcò su Nagasaki quindi questa è forse una cosa che può essere controintuitiva sulle scale logaritmiche se senti al telegiornale la differenza tra oh c'è stato un terremoto che era un 4.0 contro un terremoto che era 5.0 è facile pensare sì 4 e 5 sono numeri abbastanza simili ma evidentemente in termini di importi di TNT ciò corrisponde a moltiplicare per 32 per passare da 1 a successivo e passare da 2 a 4 evidentemente significava moltiplicare per circa mille e l'unico il motivo per cui è più grande è perché qui il nostro grafico non mostrava cosa fosse 3 quindi stavamo facendo due passi e puoi verificare tu stesso che se fai un passo di 32 e poi moltiplichi per un altro 32 in realtà è abbastanza vicino a mille quindi l'idea che i passi additivi sul numero Richter corrispondano a passi moltiplicativi nel TNT sembra suggerire che qui sia in gioco qualcosa di logaritmico ed è un po' interessante continuare qui e dire quanto cresce, in parte a causa dei fenomeni mondiali che sta succedendo descrivendo sì, non una grande sorpresa che mentre facciamo un altro passo si moltiplichi di nuovo per circa 32 ma tenendo a freno le nostre intuizioni questa è la differenza tra 32 kilotoni di una piccola bomba atomica e poi un megatone che potremmo considerare non una piccola bomba atomica, Bomba atomica di Nagasaki che immagino siano 32 delle bombe atomiche di Nagasaki per un megaton che è evidentemente la magnitudo del terremoto piatto a doppia corda nel Nevada USA 1994 Non sapevo cosa fosse, grazie Wikipedia in termini di frequenze tra l'altro ho cercato anche questi, evidentemente quelli che sono meno di due, quelli accadono continuamente, ce ne sono circa 8000 al giorno ma non appena siamo nel regno delle bombe atomiche cose come 3.5 e 4 quelli evidentemente accadono anche abbastanza frequentemente da qualche parte sulla terra ce ne sono circa 134 che accadono da qualche parte ogni giorno chi lo sapeva?", "n_reviews": 0, "start": 3053.77, @@ -350,7 +350,7 @@ "end": 3901.15 }, { - "input": "log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000?", + "input": "the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship", "translatedText": "logaritmo in base 2 di 10 è circa 0.3 logaritmo in base 2 di 10 è approssimativamente, scusate, logaritmo in base 10 di 2 è approssimativamente 0.3 logaritmo in base 2 di 10 è circa 1 terzo o logaritmo in base 10 di 2 è circa 1 terzo quale di questi è più vicino ad essere vero in base al fatto che 2^10 è essenzialmente 1000?", "n_reviews": 0, "start": 3901.15, @@ -364,14 +364,14 @@ "end": 4025.43 }, { - "input": "tender.", + "input": "attempt count which I think is to say unraveling If you're looking at the maximum number I'm not I'm", "translatedText": "tenero.", "n_reviews": 0, "start": 4025.43, "end": 4029.59 }, { - "input": "Not at all a unanimous decision here.", + "input": "not great at Vanna whiting this thing if you look at the maximum number in our poll It's asking what's the log base 2 of that?", "translatedText": "Qui non si tratta affatto di una decisione unanime.", "n_reviews": 0, "start": 4029.59, @@ -385,7 +385,7 @@ "end": 4038.31 }, { - "input": "So that's good, they're very numerically similar, right?", + "input": "ent powers of 2 then that rescales it and Yes, yes is the answer what a fantastically apropos questio", "translatedText": "Quindi va bene, sono numericamente molto simili, giusto?", "n_reviews": 0, "start": 4038.31, @@ -413,35 +413,35 @@ "end": 4124.69 }, { - "input": "And the question is how we can leverage this to understand something like log base 2 of 10, or log base 10 of 2.", + "input": "u some more time to think this through because it's looks like a big pile of algebra plug in some numbers to see what seems to work well and See which answer fits You You You Okay, so eve", "translatedText": "E la domanda è come possiamo sfruttare questo per capire qualcosa come logaritmo in base 2 di 10, o logaritmo in base 10 di 2.", "n_reviews": 0, "start": 4124.69, "end": 4137.67 }, { - "input": "As we saw earlier, those are just the reciprocals of each other.", + "input": "n if you are still thinking about it I'm gonna go ahead and grade it here and then start talking about Why it's true and then also why we should", "translatedText": "Come abbiamo visto prima, questi sono solo i reciproci l'uno dell'altro.", "n_reviews": 0, "start": 4137.67, "end": 4146.01 }, { - "input": "So what does this mean?", + "input": "care why this is an operation that actually tells", "translatedText": "Che cosa significa questo?", "n_reviews": 0, "start": 4146.15, "end": 4147.95 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right?", + "input": "us something so the correct answer which it looks like around 1700 of you got congratulations is Log base C of B times log base B of A is equal to log base", "translatedText": "Se logaritmo in base 2 di 10 è uguale a x, è come dire 2^x è uguale a 10, giusto?", "n_reviews": 0, "start": 4147.95, "end": 4157.99 }, { - "input": "It's asking us 2 to the what equals 10.", + "input": "C of A great Now that's just a big ol pile of things. Why would that be true?", "translatedText": "Ci chiede 2 alla cosa uguale a 10.", "n_reviews": 0, "start": 4157.99, @@ -483,7 +483,7 @@ "end": 4213.53 }, { - "input": "People seem to think you can do that with any function, but you just can't.", + "input": "here would be things like let's use a different color. Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 o", "translatedText": "Le persone sembrano pensare che tu possa farlo con qualsiasi funzione, ma semplicemente non puoi.", "n_reviews": 0, "start": 4213.53, @@ -518,14 +518,14 @@ "end": 4229.95 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x.", + "input": "t's plug in another power of 10 It'll be nice if it's also a power of 100 So I'll do a million So This one is asking 10 to 100 to the what equals a million How many times do I multiply a hundred by itself to get to a million?", "translatedText": "E abbastanza bene, quello che abbiamo visto prima è che logaritmo in base 2 di 10, potremmo anche dire logaritmo in base 10 di 2 è solo 1 su quella quantità, 1 su x.", "n_reviews": 0, "start": 4229.95, "end": 4234.87 }, { - "input": "And you can see this pretty easily by writing 2 is equal to 10 to the 1 over x.", + "input": "How many times does a hundred go into a million? Phrasing the same thing 10 different ways now the claim is that this is", "translatedText": "E puoi vederlo abbastanza facilmente scrivendo 2 è uguale a 10^1 su x.", "n_reviews": 0, "start": 4234.87, @@ -546,7 +546,7 @@ "end": 4245.93 }, { - "input": "Great.", + "input": "ng log base 10 of a million That if I ask how many times d", "translatedText": "Grande.", "n_reviews": 0, "start": 4245.93, @@ -560,7 +560,7 @@ "end": 4266.57 }, { - "input": "So if I ask what is the log base 2 of a thousand, like we just saw, it's approximately the case that 2 to the power 10 is equal to a thousand.", + "input": "ve me the answer to how many times 10 goes into a million now just checking the numbers this certainly works 10 goes into a hundred two times 100 goes into a million three times in a multiplicative sense in that a hundr", "translatedText": "Quindi se chiedo qual è il logaritmo in base 2 di mille, come abbiamo appena visto, è più o meno il caso che 2 elevato a 10 sia uguale a mille.", "n_reviews": 0, "start": 4266.57, @@ -574,14 +574,14 @@ "end": 4285.29 }, { - "input": "Log 2 of a thousand is approximately 10.", + "input": "ll six Now we could think of this property in terms of the corresponding exponent rule which is going to look a l", "translatedText": "Il log 2 su mille è circa 10.", "n_reviews": 0, "start": 4285.29, "end": 4288.75 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million.", + "input": "ittle bit stranger But it's actually just saying the entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each other the whole statement is equal to saying that um suppose that B to the X is equal to a Got some", "translatedText": "Allo stesso modo logaritmo in base 2 di un milione, beh vediamo, se dobbiamo moltiplicare 2 per se stesso circa 10 volte per arrivare a mille, dovremmo moltiplicarlo per se stesso circa 20 volte per arrivare a un milione.", "n_reviews": 0, "start": 4288.75, @@ -623,7 +623,7 @@ "end": 4344.51 }, { - "input": "Log base 10 of a thousand is equal to 3.", + "input": "g you layer it on top of each other. Now if we rearrange that expression, we get what is probably the second most important of all of our", "translatedText": "Logaritmo in base 10 di mille è uguale a 3.", "n_reviews": 0, "start": 4344.51, @@ -637,14 +637,14 @@ "end": 4353.53 }, { - "input": "It's counting the number of zeros, it ends up being about 6.", + "input": "n you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if you want", "translatedText": "Contando il numero di zeri, alla fine risulta essere circa 6.", "n_reviews": 0, "start": 4353.53, "end": 4358.35 }, { - "input": "And log base 10 of a billion, counting the number of zeros, it ends up being 9.", + "input": "the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. If", "translatedText": "E logaritmo in base 10 di un miliardo, contando il numero di zeri, finisce per essere 9.", "n_reviews": 0, "start": 4358.35, @@ -679,35 +679,35 @@ "end": 4411.57 }, { - "input": "Okay?", + "input": "say I'll use the log base 10 button and evaluat", "translatedText": "Va bene?", "n_reviews": 0, "start": 4411.57, "end": 4417.25 }, { - "input": "Now this is an intuition worth remembering.", + "input": "e what's on the inside here, which at least positionally it's kind of above the 100.", "translatedText": "Ecco, questa è un'intuizione che vale la pena ricordare.", "n_reviews": 0, "start": 4417.25, "end": 4417.93 }, { - "input": "If you have your numbers described with one base, it's basically the same as describing them with another base, but there's some rescaling constant.", + "input": "It has a higher altitude as we write it. So this can line up with the notation a little bit, that it sits on the numerator. And on the bottom, I use the log base 10 button that's in my calculator on th", "translatedText": "Se i tuoi numeri vengono descritti con una base, è praticamente come descriverli con un'altra base, ma c'è una costante di ridimensionamento.", "n_reviews": 0, "start": 4417.93, "end": 4428.15 }, { - "input": "Okay?", + "input": "e base, on the 100.", "translatedText": "Va bene?", "n_reviews": 0, "start": 4428.15, "end": 4429.21 }, { - "input": "And then the next question is going to start getting us at that direction, but it's going to be framed in a way that just looks like a whole pile of algebra, and again I will encourage you to plug in numbers if you want to to gain a little intuition for it.", + "input": "And then I can evaluate both of those and it'll give me the answer. In this case it gets you 6 divided by 2, which will be 3. And if we really just think through what this is saying, I know I've said it many different times, but it's a convoluted enough way to write things, but an intuitive enough", "translatedText": "E poi la prossima domanda inizierà a portarci in quella direzione, ma sarà strutturata in un modo che assomiglierà a un intero mucchio di algebra, e ancora una volta ti incoraggio a inserire i numeri se vuoi guadagnare un po' di intuizione per questo.", "n_reviews": 0, "start": 4429.21, @@ -728,7 +728,7 @@ "end": 4452.11 }, { - "input": "Does that equal log base B of A?", + "input": "Because like I said, this is probably the second most important log rule. We're", "translatedText": "È uguale al logaritmo base B di A?", "n_reviews": 0, "start": 4452.11, @@ -763,7 +763,7 @@ "end": 4481.63 }, { - "input": "If you're looking at the maximum number, I'm not great at Vanna Whiting this thing, if you look at the maximum number in our poll, it's asking what's the log base 2 of that, so as it crosses different powers of 2 then that rescales it, and yes, yes is the answer.", + "input": "But anything additive in the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the left-handed side and the right-hand side are just saying how many times does 100 go into a million? but going about that in different ways. So this is extremely nice", "translatedText": "Se stai guardando il numero massimo, non sono bravo con Vanna Whiting questa cosa, se guardi il numero massimo nel nostro sondaggio, ti chiede qual è il logaritmo in base 2, così come incrocia diverse potenze di 2 allora questo lo ridimensiona, e sì, sì è la risposta.", "n_reviews": 0, "start": 4481.63, @@ -777,14 +777,14 @@ "end": 4490.09 }, { - "input": "Thank You Karen.", + "input": "Next time we're going to talk", "translatedText": "Grazie Karen.", "n_reviews": 0, "start": 4490.09, "end": 4490.95 }, { - "input": "All right so answers are still rolling in, and I think like I said I just want to give you some more time to think this through because it looks like a big pile of algebra.", + "input": "all about the natural logarithm, which is log base e, often written ln. And turns out, this is much easier to compute. There's nice math behind it such that if you want to come up with an algorithm that your calculator can use, it's actually a lot easier to think of l", "translatedText": "Va bene, quindi le risposte stanno ancora arrivando, e penso che, come ho detto, voglio solo darti un po' più di tempo per pensarci perché sembra un grosso mucchio di algebra.", "n_reviews": 0, "start": 4490.95, diff --git a/2020/ldm-logarithms/japanese/sentence_translations.json b/2020/ldm-logarithms/japanese/sentence_translations.json index bf56de9ba..ce3646201 100644 --- a/2020/ldm-logarithms/japanese/sentence_translations.json +++ b/2020/ldm-logarithms/japanese/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵音楽🎵 ロックダウンの数学へよう こそ。", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "今日は対数について、そして基本に戻るようなレッスンについて話します。", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "そしていつものように、まず最初に、聴衆が今どのような位置にいるのかを把握したいと思います。", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "ということで、3b1bまで行けたら。", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "ａ．私はこれまでそれらについて聞いたことも、以前にそれらについて学んだこともありませんでした。", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "それらについて学習しましたが、すべての特性に時々混乱することがあります。c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "私はそれらを理解していますが、どのように教えればよいのかわかりません 。", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "私は彼らのことをよく理解しているので、他の人に彼らにもよく理解してもらうために快適に教えることができます。", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "ということで、うまく分かれました。", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "先ほども言いましたが、この目的は、将来、対数に慣れていない人に向けて、「ああ、ここがおすすめです」と言えるようにす るためのレッスンを作成することです。私がどのように考えているか、どのように直感的にアプローチできると思いますか。", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "なぜなら、この特定の講義を行う前に、私はいくつかの教師フォーラムをスクロールして いたので、生徒が最も苦手としているように見える高校数学で教えるのが最も難しいトピ ックは何かと人々が尋ねたとき、対数は最も重要なトピックの 1 つであるからです 。一般的に示されている答えは興味深いもので、おそらく、最終的に学ばなければならな いこれらのプロパティが大量にあるためだと推測できます。代数の塊のように見えるルー ルは、覚えるのが難しく、頭の中でごちゃ混ぜになりやすいものです。高校の数学とは 何だったのか、そしてどんなものだったのかという悪夢のような思い出を人々が抱くとき 、私は思います。対数は彼らのために役に立ちました。それらの特定の公式が頭に浮かぶ ことはよくあります。今日私がやりたいのは、それらについてどのように考えるかとい うことについて話してみることですが、誰かに代数を教えている場合、それが何であるか というメタレベルについても話してみることです。強調する価値のある点は何ですか？", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "それを彼らの直感に組み込む方法は何でしょうか？", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "ああ、ゼロが 3 つあります。", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "x の 1000 倍の対数は、x の対数の 3 倍に等しく、底 10 の対数 b であるという規則を使用していることを思い出してください。", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "1000 倍の対数 x は、x の対数の 3 乗に等しい c. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "x の 1000 倍の対数は、x と e の対数の 3 乗に 相当します。", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "上記のいずれでもないし、先ほども言った ように、最初にログをよく理解していると言っていた 人たちは全員、すぐに答えてくれるだろう、正しく答 えてくれるだろうと十分に期待する必要があることを 覚えておいてください。そうでない人も、このような 問題を見ているときに怖がらないでください。私があ なたに勧めたいのは、さまざまな 10 のべき乗を 当てはめて、ログ関数が次のような考え方をすると いう考え方に基づいて考えることです。ゼロの数を数 えますので、少し考える時間を与えますので、先に進 んで採点します。いつものように、それがあなたが満 足しているものよりも速い場合は、それは私が先に進 みたいだけであることを知ってくださいこの場合、正 解は x の 1000 回の対数になります。これ は 3 に x の対数を加えたものと同じです。で は、それについて少し考えてみましょう。始めたばか りのときに言いました。彼らにとって最善のことは 、さまざまな数値を快適に入力することだと思います 。そして、接続するのに最適な数値は、すでに 10 のべき乗になっている数値です。つまり、1000 回 x の対数のようなことを求めているのであれ ば、私はそうしません。わかりません、100 倍の x 対数に何かを代入してみましょう ここで、最 終的な答えにゼロが何個含まれるかはわかっています 1000 倍 100 は 100,000 です 10 の 2 乗を掛けると、という考えがすで に直感的にありますゼロを取り出しているだけです、 その 1000 から 3 つのゼロ、その 100 から 2 つのゼロ、そしてそれらを並べて配置し ているので、合計 5 つのゼロになるはずですが、 数値がどのように変化したかだけでなく、本当に反映 しているのであれば出ましたが、なぜそのようになっ たのですか。1000 の 3 つのゼロと 100 の 2 つのゼロです。これは、1000 のゼロ の数と 100 のゼロの数を足したものと書くこ ともできるので、この考えは対数です2 つのものの 積の 2 つのものの対数の合計は、10 のべき乗 のコンテキストでのそれら 2 つのものの対数の合 計です。これは、10 の 2 つのべき乗を計算し 、それらを乗算すると、多くの人にとってすでに超直 感的なアイデアであることを伝えるだけです。すべて のゼロを取り、それらを互いに詰め込むので、ここで 私が物事を書いた方法は、実際には、対数の最初の性 質となるもう少し一般的な事実を示しています。A の対数 B を掛けたものは、A の対数と B の対数を足したものに等しくなります。これらの対 数規則のいずれかを目にするたびに、目を細めたり、 覚え方に少し混乱したりする場合は、例を入力してく ださい。冗長です。これを何度も言いますが、代数自 体に夢中になって、ある種のテストを受けて、ただ記 号がたくさんあるだけだと、忘れてしまうのは非常に 簡単だと思うからです。自分に思い出させるために、 数値を入力するだけで大丈夫です。それは良いこと ですし、多くの場合、直感を導き出すのに最適な方法 です。この場合、A 掛ける B の対数を言って、 それを分解すると、ああ、あれだと考えることができ ます。100 掛ける 1000 の対数は 5 で、ゼロが 5 つあります。指定された各部分のゼ ロの数で分割されます。素晴らしい、素晴らしいです 。その直感をさらに信じて、別の練習問題をもう一度 試してみましょう。知っているなら、素晴らしいです 。あなたはそれにうまく答えることができるでしょ うが、答えが何であるかだけでなく、この答えを誰か にどのように説明するか、または私が言わなくても生 徒が自分でこの答えにたどり着くようにするにはどう すればよいかを考えてみてください。答えは何ですか 。したがって、潜在的な聴衆は 2 人います。レッ スン自体に興味がある人、そしてメタ レッスンに興 味がある人です。したがって、私たちの質問は、次の どちらが正しいかということをもう一度尋ねます。", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "ａ．x の n に対する対数は、x b の対数の n 倍に等しい。", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "x の n に対する対数は、x の n 乗の対数 c に等しい。", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "x の n に対する対数は、n に x または d の対数を加えたものに等しくなりま す。", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "ここでの正解は a です。4,000 人がお祝いの言葉を受け取ったようです 。x の n 乗の対数は x の対数の n 倍に等しいと教えてもらいました 。それで、もう一度、これを教えようとし ているとしましょう。誰かに伝えるか、 あるいは自分自身でその意味を理解しよう としているなら、何かを接続することか ら始めるのが良いと思います。この場合 、x の n 乗の対数について、100 の乗で試してみましょう。3 そし て、他のパターンで試して、やっているパ ターンが実際に機能するかどうかを確認 することもできますが、単純に答えを見る という観点からではなく、なぜ答えがそ のようになったのかを考えようとする場合 は、場合によっては 1 つの例で十分 です。なぜなら、100 の 3 乗は 十分に得られると考えることができます。それは 100 の 3 つのコピーで す。私は 100 の 3 つのコピーを 取り、それをすべて掛け算すると、lo g はゼロの数を数えると考えることがで きます。言ってください、ああ、それは ゼロが 6 つある数字になるでしょう 、つまり 100 かける 100 かけ る 100 を意味します これらのゼ ロをすべてグループ化して 100 万を 得ることが考えられるので、この数字は 次のようになりますしかし、実際に考えて みると、なぜ 6 だったのかというと 、100 万の中のゼロの数だけではなく 、その 6 の由来は、その 100 のコピーが 3 つあり、それらの 1 00 のそれぞれに 2 つの異なるゼロ があるため、より一般的になります。100 の 3 乗の代わりに 1000 の 3 乗、または 1000 の n 乗または x の n 乗を調べた場 合、n の値は何倍にもなるコピー数で あると考えることができます。の数は、 x に代入したゼロの数の x 倍ではあ りません。この場合は 100 なので 、代わりに 10,000 の n 乗の ようなものを計算しても、これは同じに なります。その 10,000 個のコピ ーを n 枚取り、それぞれの 0 の 数を数えると 4 になるので、n の 4 倍になります。そしてもちろん、ほ とんどの人が正しく答えた一般的な特性 は、この素敵な小さな効果があるというこ とです。何かがその前でわずかな力で飛 び降りるまで上昇したログを見てください 。そうすれば、内側に何があったのかの ログが得られます。それがおそらく最も重 要な意味の1つです。それをそう呼んで いいのかわかりませんが、含意、あるい は定義の言い換えと呼ぶなら、もし私がロ グをとっていて、それが 10 の底 10 の n 乗であることをもう一度強 調しておきますが、その小さな n は 、ある種の中でホップダウンしていると考 えることができます。この式は、末尾の ゼロの数を数えているか、より一般的には 10 に等しいものを 10 と求め ており、答えは単純に 1 ですこれは 非常に安心できます。なぜなら、戻ってこ の元の式を読むこともできる別の方法は 、10 と n が 10 に等しいと言 っているからです。まあ、答えは、与え られたすべての対数特性で OK です。この場合、次のようになります。x の n 乗の対数を 1 つ見つけまし た。n が前にホッピングすることを意味 します。常に鏡像の指数特性が存在しま す。これは、これらについて少し直感的に 理解するのに役立つもう 1 つの方法 です。ここにたどり着く予定の将来のプロ パティのいくつかは、どこに行くのかを 隠そうとします。今見つけたものを前にホ ップする n に何かを上げます。これ は、x に 10 を加えて上げた場合 の指数特性に対応します。この全体の n 乗は、10 の n 倍 x と同じ です。そして、これにより、対数に対して 抱くかもしれない別の直観が得られます 。対数は、べき乗を裏返しにしたようなも のです。これが私が言いたいことです。ログの内側にあるものは、a のログを 取得する場合、それを指数関数的な何かの 外側の式全体として考える必要がありま す。この場合、a 内側にあるものは 1 0 から x に対応します。関数の出 力自体は、a の対数全体が内部にあるも のに対応しており、10 の指数に相当 するため、ここで対数式が表示される場合 は、それが右側の指数の役割を果たして いると考える必要があります。指数関数 を見るたびに、10 から x までの式 全体が右側の外側の要素全体になります 。これは、ログの 1 つの内側にあるも のに対応します。これは、乗算するとき のアイデアの上で見られました。内側では 外側で足し算をしているので、ログが指 数関数的に裏返しになっている場合は、 外側で乗算して関数の出力を乗算すること は、内側で足し算するのと同じであるこ とがわかります。なぜなら、これらのログ はそれぞれ log a と log b のようになっているからです。は右側 の式の x と y の役割を果たして いるので、これで遊びを続けましょう。こ れらをさらにいくつか実行して、直感を 構築できるこれらのプロパティの数を確 認しましょう。それでは、これが最後のも のです。指数が次の指数に飛び降りると いう非常に素晴らしい考え方は、対数に必 ずしも慣れていない人にとっては少し奇 妙に見えるかもしれませんが、もう一度、 いくつかの数値を入力して直観を得るこ とができます。次のどれが真実ですか? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "そうですね、10 の 3 乗が 1000 である 場合、それは 10 が 1000 の 3 分の 1 に等しいと言っているのと同じことです。ここ で逆算を行うと、指数の逆乗が必要になり、結果は 1 割る 3 のようになります。そして、3 は 1000 の対数底 10 に対応します。これ は、1 を 1000 の対数底 10 で割ったも のです。したがって、より一般的には、この 1 つの例に基づいて、底を内側にあるものと交換すると 、1 を割ることに相当すると推測できるでしょう 。外側に何があるかによって、対応する指数則を見て 、このことをよく考えてみてください。私の愛らし い小さな丸太と指数関数はどうなったでしょうか? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "素晴らしいので、ここでまた他のプロパティのいくつかをどこ に隠しましょう。ここで前に持っていたのと同じ順序で保持 します。事前に書き込んでおけば、私を保つことができると 思っていましたいつもより少しきれいですが、おそらくそれ は、紙を切るという奇妙なゲームをシャッフルしてプレイす るだけかもしれません。つまり、私たちが今見つけたもの、 aの底bの対数を交換すると、これが対応するものを1で割 ることと同じです。指数関数的土地とは、b を何乗して、 それが a に等しいと言った場合、それは a のその逆 乗が再び b に等しいと言っているのと同じことです。少 し時間をとって、対数が物事を回転させるものであると考え ると役に立ちます。裏返しに、a の式 log ベース b が x の役割を果たし、b の式 log ベース a が a の上にあるものとして役割を果たし、その後対 称的に、式全体の b の x 乗が役割を果たします。左 側の内側の役割、それは a の役割と式全体を果たします 。a の何かの力は、ログ ベース a の内部にあるも のの役割を果たします。これで、いくつかの例を差し込むだ けでわかります。それを指数規則に対応させることで、すで に 3 つの異なる対数規則を考えることができます。これ らの対数規則が暗記すべき代数の一部として伝えられただけ であれば、暗記することはできますが、非常に簡単に自分の 記憶から抜け落ちてしまいます。また、目の前の仕事にイラ イラしてしまいがちですが、私たちがこの種のことを気にす る理由は、対数の法則を理解することで、ウイルスが増殖す るような状況で数学を行うのに役立つからであることを思い 出したほうがよいかもしれません。ある日から次の日、ある ステップから次のステップへと、物事は乗算的に成長する傾 向があります。対数の法則を理解すると、その種のことをよ りよく理解できるようになります。その前に、それがどのよ うなものになるかを示す実際の例を見てみましょう。現実世 界の例に少し移る前に、対数の性質について尋ねるために、 この流れでもう 1 つクイズを出させてください。今ここ で持っていたものを取り除き、次のうちどれが真実ですか? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "a プラス b の対数は、a プラス b の対数と同じです。a プラス b の対数は、a の対数に b の対数を掛けたものに等しいです。a プラス b の対数は、a の対数に b の対数を加えたもので割ったものに等しい、ま たはa と b の対数は、a の対数×b の対数で割ったものに等しいか、また は上記のどれでもない、ああ、今ではそれほど多くの合意が得られていませんね。", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "非常に興味深いですね、二人の間で競馬が行われているので、 皆さんが答えている間、少し時間を割いて考えさせていただ きます。実は、聴衆にちょっとした質問があるのですが、つ まり、私はちょうど私たちがどうするかについて話していたの です。乗法的な成長の観点から考えてください。それは 1 0 の累乗である必要はありません。1 から 3、9、2 7、81 まで進む場合、すべての 3 の累乗のようなこと もできます。これらのうち、これらの数値の対数の底 3 は、少しずつ増加するだけであると言えます。したがって、 1 の底 3 を対数にし、3 を 1 に等しいものにす ると、答えは一般にゼロになります。底に関係なく、1 の対 数は次のようになります。ゼロであること 3 の 3 の 対数、3 に等しいものは 1 同様に、9 の対数 3 は 2 ああ、私の質問は何なのかと思われるかもしれません が、これらすべてを引き出すと、私自身の楽しみになります 。ここで、もう 1 つだけ書いてみましょう。81 の底 3 は現在 4 です。表向きは、子供に尋ねると、5 歳か 6 歳くらいだと思いますが、1 と 9 の中間の数 字は何ですかと聞いたことがあります。どの数字が半分であ るかを言う 彼らの答えに対する本能は対数的ですが、私た ちの本能はより線形である傾向があるため、私たちはよく1と 9を考えます、それらの間には等間隔の数字がたくさんあり ます、2、3、4、5、6 、7、8、そしてその中間を右 に進むと、5 に到達します。しかし、乗法的な成長の観点 から 1 から 9 までどこに到達するかを考えている場合 、それはたくさんのものを追加するという問題ではありませ ん。「ある一定の量だけ成長すると、3倍に成長し、その 後、さらに3倍に成長します。おそらく、子供の自然な本能は 3という言葉と一致します。おそらく、これは、成長してい ない社会を研究している人類学者がいる場合にも一致します 。」現代社会と同じように会計システムや文章を開発したの です。彼らはこれについて 3 つ答えるでしょう。それでは 、視聴者への私の質問ですが、今ご覧になっている方の中に 、たとえば 5 歳以内の小さな子供と接することができる 人がいるかどうかを教えてください。古い、1 と 9 の中 間の数字は何なのかを尋ねることができるかどうか、できれ ば Twitter でその子の発言を教えてください。実 際の答えは何なのか、理由はわかりません。私はほんの少し だからです。それが実際にうまくいくかどうかは懐疑的です これが超科学的な方法ではないことは理解しています Yo uTubeのライブストリームを見ている人たちに自分の子 供たちにアンケートをとってその答えをツイートするように求 めているわけではありませんが、私自身のためにもそれは興 味深いでしょう私たちの質問に戻って、ある種の検証を確認 するために、これは、一方向で大きなコンセンサスが得られ そうにない最初の質問です。先に進み、答えが素晴らしいこと が判明するために採点しましょう、わかりました、それで 2,400あなたの中で、a プラス b の対数がこれら の優れた特性を満たさないということは上記のどれでもない、 そして一般的には、特に自然対数が機能する場合に特定の種 類の近似を扱うつもりでない限り、と正しく答えました。次 回これについて話すかもしれませんが、対数の入力を追加す るのは実際には非常に奇妙な感覚です。それは非常に奇妙なこ とです。その奇妙な感覚を得るには、a プラス b の対 数を尋ねる場合は 10 の累乗を代入してください。あな たが考え始めるのは、「わかりました。10,000 と 1 00 のような例をいくつか入力してみましょう。その入力 に含まれるものをゼロ カウントする関数を実行すると、そ の中にゼロはいくつ含まれるでしょうか?」と自問します。", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "興味深い質問ですね。対数の底は ゼロになることがありますか? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "三角形に関して言えば、これは、ゼロの x の累 乗は他の値 y に等しい、ということを言ってい ると考えるかもしれません。これは、x のゼロが y に等しいと書くか、または次のように書くこ とができます。y の対数ゼロが x に等しい と言うのと同じことです。ここで問題となるのは 、ゼロから何でも結局はゼロになるということで す。他の入力の場合は y ご存知のとおり、1 または 2 または pi のような何かを入力し たい場合は、0 から 1 または 2 または pi またはそこにある可能性のある数字に等しい ものまでの質問をしています。答えは見つからない ので、せいぜい「そうそう、ゼロの対数、これは 完全に有効な関数です。入力ゼロに対してのみ定義 されています。しかし、それでも、必要なものを解 決するのは難しいでしょう」と言うことができます 。ゼロに等しいものに対してゼロと言うのは、何で も当てはまるようなものなので、腕を後ろでねじる ことになりますが、それを機能させたいのですが 、それは底がゼロの指数関数が完全にゼロであると いう事実に対応しています。数値を相互に適切に 1 対 1 でマッピングしないので、それは素晴 らしい質問です。ゼロを底とする対数を考えてもら えますか。現実世界でこれらのことがどこで発生す るかについての考えに戻りますが、私が気に入っ ている例の 1 つは次のとおりです。地震のリヒ ター スケール リヒター スケールは、地震の強 さを定量化します。非常に小さな数値から、これま でに測定された最大の地震のように非常に大きな数 値まであります。これは、以下から得られた単なる グラフです。ウィキペディアは9でした。そして 、それがどれほど狂気であるかを理解するには、こ れらの数字が意味することと、それに含まれるエネ ルギーの量を示すある種の尺度であるTNTの等 量のようなものと、ここで何ができるかを検討する 価値があります。エネルギー量の観点からリヒター スケールの数値を表現できるかどうか、また、こ れを説明するには対数が自然な方法である理由を確 認してください。そのため、焦点を当てるべき鍵は 、前進するときに物事がどれだけ増加するかです 。たとえば、この場合、2 から 4 に進むと、 どこが 3 なのかがわかりません。そのため、 2 から 4 にステップアップすることを考えま す。これは、2 つのステップを踏むのと同じよう なものです。エネルギー量はそうですね、第二次 世界大戦の大型爆弾である1トンのTNTから私た ちがかかるようですが、それはその1000倍の1 キロトンまでかかります。これは小型の原子爆弾 なので、わずか2ステップですリヒタースケールで マグニチュード 2 の地震からマグニチュード 4 の地震に至るということは、第二次世界大戦 の大型爆弾から核時代に至るまでのことなので、こ れは注目に値することであり、私たちが得られる最 初のきれいなステップは、マグニチュード 4 からマグニチュード 5 になることです。少なく とも、このグラフが私たちにうまく示していること に関して言えば、明らかに 4 から 5 への 1 段階の上昇は 1 キロトンから 32 キ ロトンへの増加に相当し、それは明らかに長崎に 着弾した都市破壊型爆弾の規模であったので、これ はおそらく 1 つです。ニュースで聞いただけだ と、対数スケールについて直観に反するかもしれ ません。ああ、震度 4 の地震がありました。0 対 5 の地震。0 、そう思うのは簡単ですが、4と5はかなり似た数 字ですが、明らかにTNTの量で言えば、1から次 の値に移動するために32を乗算することに相当し 、2から4に進むのは明らかに約1000倍であり、 唯一の値です。これが大きい理由は、ここではチャ ートに 3 が示されていないため、2 つのステ ップを実行していて、32 のステップを実行して からさらに 32 を掛けると、実際には 1,0 00 にかなり近い値になることが自分で確認でき ます。リヒター数の加法ステップが TNT の乗 法ステップに対応するという考えは、ここで対数的 な何かが働いていることを示唆しているように思え ます。ここで話を続けて、これがどれだけ増加する かを言うのは少し興味深いです。一部には世界的な 現象が原因です。「はい、もう一歩踏み出すと再 び約32倍になるのはそれほど驚くことではありませ ん。しかし、それが小型原子爆弾の32キロトンと 、小型原子爆弾ではないと考えられる1メガトンの 差であるということを私たちの直観に抑え込みます 。長崎原子爆弾は、1メガトンに32発の長崎原子 爆弾だと思います。これは明らかに1994年に米 国ネバダ州で起きた二重弦平坦地震の規模に相当し ます。私はそれが何であるか分かりませんでした、 ちなみに周波数に関してはウィキペディアに感謝し ますまた、これらを調べてみると、明らかに2個未 満のものがあり、それらは常に発生しており、1日 に8000個ほどありますが、原子爆弾の領域に入 るとすぐに3個ほどになります。5 と 4 それ らも明らかに地球上のどこかでかなり頻繁に起こっ ており、毎日どこかで約 134 件のそれらが 起こっていることを誰が知っていたでしょうか？", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "しかし、原子爆弾の規模をはるかに上回っていたこの5と6 の範囲にさらに激しさを増しており、今では1日あたりわず か約2にすぎません。地質学者が来て、なぜ誰もがすべきか を説明してくれると確信しています。」原爆2発分に匹敵す る地殻破壊が毎日起こっているという事実をそれほど心配す る必要はないが、おそらくそれが多くの人が住んでいる都 市のような場所に集中するのは特に稀であり、今は各段階で の私たちの考えを検証しているところだ。32 の増加が 含まれます。6 から 7 までのステップがどのようなも のかを見てみましょう。ここでは、その間にさらに多くの例 が示されています。おそらく、それが実際よりも大きなス テップであるかのような錯覚を与えています。実際、それが 1 メガトンと 1 メガトンの違いです。32メガト ンなので、32倍になります ちなみに、このグラフで私が 最も興味深いと思ったことの1つは、これまでに実際に実験 された最大の核兵器に到達するまでに、どれだけ遠くまで行 かなければならないかを見てください、これは冷戦の真っ 最中でしたツァーリの爆弾は50メガトンで、実際には10 0メガトンの爆弾を開発する当初の計画があったと思います が、その50メガトンから自分たちを説得しました。私たち は、長崎の爆弾の32キロトンに32を掛けると、メガトン にさらに 32 を掛けると、第二次世界大戦終結の爆発 の強さの 1,000 倍について話していることになりま すが、人類の能力の 50 メガトンにはまだ達しておらず 、それは明らかにインドネシアのジャワ地震ですので、7 。0 は 6 より少し大きいだけではありません。0、それははるかに大きく 、ここでのポイントは、もちろん、乗法的な増加を与えるスケールがある場合、 小さなステップのように見えるものが、ここで暗示されるエネルギーまたは絶対値 の観点から見ると、実際には大きなステップである可能性があることを評価する価 値があるということです。かつて 9 が存在したという事実を考えるとき。5 は 7 にしか含まれていないことを考えると、実際には不合理に思えます。射程距離 0 は、これまでに発 売された最大の熱核兵器について話しているものであり、これは対数が発生する傾向があ る領域の 1 つを示しています。それは、人間が何かのスケールを作成したい場合に、物 事がどれだけ大きなものになるかについて非常に幅広い差異を考慮する必要がある場合で す。地震の規模については、地球上で常に起こっていること、大きな手榴弾ほどの規模の ものを想定することができますが、それを自分の規模に合わせて、さらに広範囲にわたるこ とについて考えたいと考えます。人類史上最大の混乱に直面しており、それを実現するに は、単に 1 つのケースに対して多数の異なる桁を数字に書き込むだけではなく、多数 の異なるより小さい数字を書き込む必要があります。別のケースでは、数字の桁数を計算 して、それを基本的に 0 から 10 までの数値を押しつぶす 1 つのスケールに当 てはめると、音楽のデシベル スケールでも非常によく似たことが起こっていることが分 かります。これは、実際には少し機能します。少し異なりますが、10 デシベルずつス テップアップするたびに 10 を乗算することに相当するため、1 ステップに 10 を乗算するのではなく、10 ステップで 10 を乗算することになるため、計算が少 し難しくなります。ちょっとおかしな話ですが、考え方は同じです。50 デシベルの音 と 60 デシベルの音を聞いている場合、エネルギーの伝達と発信の点ではるかに静か になります。60 デシベルから 70 または 70 デシベルになるとどうでしょうか 。60 から 80 までの 80 段階では、平方面積あたりのエネルギー量を 1 00 倍する必要があります。対数目盛りを見るたびに、それが内部で何を意味している のかを頭の中で理解してください。膨大な量です これが、コロナウイルスの発生を説明す るために多くの対数スケールが使用されている理由でもあります。では、リヒタースケー ルの数値が 1 増えるたびに、32 が乗算されるこのような関係をどのように説明す ればよいでしょうか。基数 32 の対数の観点から考えることができます。対数を取得す る場合は、リヒター スケールの数値 r を呼び出します。これを基数 32 の対数 として考えると、次の値に対応すると言えます。いいえ、いいえ、私はこれを間違ってい ます、それは記録されたものではありません、TMT 数値の大きな数字の対数底 32 を取得します、1 メガトンのようなものです、それは 100 万トンです、対数底 32、それははずですはリヒター スケール番号に対応しますが、何らかのオフセットが ある可能性があります。したがって、このリヒター スケール番号に追加しているある種 の定数 s があると言えます。この式はまったく同じです。話が逸れてしまい申し訳あり ませんが、一番下にあるこの式は、32 のオフセット乗にリヒター スケール数値を乗 じたものとまったく同じです。これは、そのオフセットに 32 を乗じることと同じで あり、それ自体は単なる大きな定数であり、リヒター スケール数値に 32 を乗じた ものです。これは、表示される数字の 32 乗の定数倍であると考えるかもしれません。そのため、この書き方では、これが表示される TMT 量に対応するものである場合、 それを増やすにつれて指数関数的に増加することを強調しています。ステップバイステッ プで 32 を掛けていますが、まったく同じ事実を伝えるもう 1 つの方法は、32 を底とする対数を取得することです。その量は問題ありません。次に私が話したいのは、 常にそうする必要はないということです。さまざまな基数の対数を計算する方法について 心配します。ここで対数 32 について話しているのは少し奇妙です。数学者は基数 32 の対数を本当に好むのと、コンピューター科学者は基数 2 の対数を本当に好むこ とを以前に言及しました。これは、計算目的、あるいは、ログが 1 つある場合に、こ れらのものがどのように成長するかを考えるためであることがわかりました。1 つのタ イプのログを計算できれば、それが底 10、底 2、底 e であっても、その他のほと んどすべてのものを計算できます。あなたは今、その方向への直感を知りたいのですが、 クイズに戻って次の質問に行きましょう。この質問が一番だと思います、わかりません、 これは中途半端に妥当な質問です、これはいいはずですこれは、基数 2 のコンテキス トから基数 10 のコンテキストに変換する準備を整えるだけであり、2 のべき乗が 10 のべき乗と一般的に持つ関係を理解するための優れた直観でもあります。これは、 このような素敵な偶然の一致であるためです。この 2 つの性質は、私が言いたいこと はわかると思いますが、これら 2 つは互いにうまく機能します。そこで、2 の 10 の位が 1024、1024、つまり約 1000 であるという事実を考慮して、私 たちの質問が尋ねられます。数値に少し余裕があり、2 から 10 の位まで、基本的 には 1000 まで近似しているだけですが、次のうちどれが真実に最も近いですか? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "入札。", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "ここでは全会一致の決定ではありません。", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "しかし、問題はどれが真実に最も近いかを尋ねることでした。これについてどのように考えることが できるかを見てみましょう。", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "つまり、2 の累乗、つまり 1024 は 10 の累乗 (約 10 の 3 乗) に非常に近いということになります。", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "では、 これは何を意味するのでしょうか？", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "10 の底 2 の対数が x に等しい場合、それは x の 2 が 1 0 に等しいと言っているのと同じことですよね?10 に等しいものを 2 にするよう求めています。", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "すべての関数でそれを行うこと はできません。", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "どの関数でもそれができると思われているようですが、それは不 可能です。", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "それが意味するのは、x は約 3 分の 10 ということです。", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "先ほど見たのは、10 の底 2 の対数で すが、2 の底 10 の対数は、その量を 1 上回るだけ、つまり x を 1 上回 るとも言えます。", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "ログ でやっているので、そのように書くつもりです。", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "同様に、100 万の底 2 の対数を 計算します。1,000 になるまでに 2 を約 10 回掛ける必要が ある場合、100 万になるまでに 2 を約 20 回掛ける必要があり ます。", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "それは少 し小さいですが、これは頭の中に入れておくと良い近似値です。", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "3は3です。20 では、同じ量だけスケールダウンします。", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30 では、同じ 量だけスケールダウンします。", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "わかった？これは覚えておく価値のある直観です。", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "これについては 有意義な時間を割きます。なぜなら、対数に慣れていないと 明らかではないからです。少し考えてみる価値はあります。", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "ありがとうカレン。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/korean/sentence_translations.json b/2020/ldm-logarithms/korean/sentence_translations.json index 2d96628f4..e0d562bb6 100644 --- a/2020/ldm-logarithms/korean/sentence_translations.json +++ b/2020/ldm-logarithms/korean/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Music🎵 Lockdown Math에 다시 오신 것을 환영합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "오늘 우리는 로그에 대해 이야기하고 기본으로 돌아가는 수업에 대해 이야기할 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "늘 그렇듯, 시작하면서 저는 청중이 지금 어디에 있는지 알고 싶습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "그래서 3b1b로 갈 수 있다면. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "나는 그들에 대해 이전에 들어본 적도 없고, 이전에 배운 적도 없습니다. b. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "나는 그것들에 대해 배웠지만 때때로 모든 속성에 대해 혼란스러워합니다. c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "나는 그것들을 이해하지만 어떻게 가르쳐야 할지 모른다. 그리고 d. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "나는 그것을 잘 이해하고 다른 사람에게도 편안하게 가르쳐서 그들이 잘 이해할 수 있도록 할 수 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "그래서 우리는 잘 헤어졌습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "내가 말했듯이, 이것의 목적은 사람들이 로그에 익숙하지 않고 미래에 사람들에게 지적할 수 있는 교훈을 만드는 것입니다. 그리고 나는 이렇게 말할 수 있기를 원합니다. 제 생각에는 여러분이 직관적으로 접근할 수 있다고 생각합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "왜냐하면 제가 이 특정 강의를 하기 전에 몇 개의 교사 포럼을 스크롤하고 있었고 사람들이 고등학교 수학에서 가르치기 가장 어려운 주제가 무엇인지 물었을 때 학생들이 가장 어려움을 겪는 것 같다는 의미에서 로그는 가장 어려운 주제 중 하나였습니다. 일반적으로 나타나는 답변은 흥미롭고 아마도 여러분이 배워야 할 속성이 너무 많기 때문일 것입니다. 따라서 우리가 갈 곳보다 앞서 건너뛰면 이 많은 속성을 갖게 될 것입니다. 대수학처럼 보이는 규칙은 기억하기 힘들고 머릿속에서 복잡하게 얽히기 쉬우며, 사람들이 고등학교 수학이 어땠는지, 어땠는지에 대한 악몽 같은 기억을 갖고 있는 것 같아요. 로그가 그들에게 도움이 되었는데, 그것은 종종 마음에 떠오르는 특정 공식입니다. 오늘 제가 하고 싶은 것은 공식을 통해 그것에 대해 어떻게 생각하는지에 대해 이야기하는 것뿐 아니라 누군가에게 대수학을 가르치고 있다면 메타 수준에서도 이야기하는 것입니다. 강조할 만한 점은? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "직관에 따라 구축하는 방법은 무엇입니까? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "아, 거기에 0이 3개 있어요. 백만의 로그는 얼마죠? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "x의 1000배 로그는 x 로그의 3배와 같습니다. 그리고 우리는 그것이 밑수 10 log b라는 관례를 사용하고 있다는 것을 기억하세요. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "1000배 x의 로그는 x의 세제곱의 로그와 같습니다. c. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "x의 1000배의 로그는 3의 로그 x와 e의 거듭제곱과 같습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "위의 어느 것도 없으며 앞서 말했듯이 우리는 처음에 로그를 잘 이해한다고 말한 모든 사람들이 즉시 대답할 것이고 올바르게 대답할 것이라고 완전히 기대해야 합니다. 그렇지 않은 사람은 이와 같은 문제를 볼 때 위협하지 마십시오. 제가 여러분에게 권하고 싶은 것은 다양한 10의 거듭제곱을 연결하고 로그 함수가 다음과 같은 개념으로 생각하는 것입니다. 0의 개수를 세므로 그것에 대해 잠시 생각할 시간을 드리고 등급을 매기겠습니다. 항상 그렇듯이 그것이 여러분이 편한 것보다 빠르다면 그것은 단지 앞으로 나아가고 싶기 때문이라는 것을 알아주세요 수업을 통해 이 경우 정답은 x의 1000배 로그가 나옵니다. 이는 3에 x의 로그를 더한 것과 같습니다. 이제 그것에 대해 잠시 생각해 보도록 하겠습니다. 제가 막 시작할 때 말씀드린 것처럼요. 그들과 함께 할 수 있는 가장 좋은 일은 편안하게 다양한 숫자를 연결하는 것이고 연결하기 가장 좋은 숫자는 이미 10의 거듭제곱인 숫자이므로 1000배의 로그 x 음과 같은 것을 묻는다면 나는 그렇지 않습니다. 글쎄요, x 로그 1000 곱하기 100에 뭔가를 연결해 봅시다. 여기 최종 답에 0이 몇 개가 나올지 알고 있습니다. 1000 곱하기 100은 100,000입니다. 우리는 이미 직관적으로 10의 2승을 곱한다는 생각을 갖고 있습니다. 1000에서 0 3개, 100에서 0 2개를 가져와 서로 옆에 배치하면 총 0이 5개가 되어야 하지만 숫자가 어떻게 변했는지만 생각하는 것이 아니라 실제로 생각해 보면 그런데 왜 그렇게 밝혀진 걸까요? 1000의 0 3개 더하기 100의 0 2개입니다. 1000의 0 수 더하기 100의 0 수를 더해 이 아이디어를 로그로 작성할 수도 있습니다. 두 가지의 곱은 10의 거듭제곱 맥락에서 두 가지의 로그의 합입니다. 이는 10의 2제곱을 취하고 이를 곱하면 이미 많은 사람들에게 매우 직관적인 아이디어가 무엇인지 전달하는 것입니다. 0을 모두 취하고 서로 밀어넣는 식으로 여기에 쓴 방식은 실제로 로그의 첫 번째 속성이 될 약간 더 일반적인 사실을 나타냅니다. A의 로그 곱하기 B는 A의 로그 더하기 B의 로그와 같습니다. 이제 이러한 로그 규칙 중 하나를 볼 때마다 눈을 가늘게 뜨고 있거나 기억하는 방법이 약간 혼란스러우면 예를 연결하기만 하면 됩니다. 중복되는 말이네요. 많이 말하지만 일단 대수학 자체에 푹 빠져서 일종의 시험을 치르고 기호가 너무 많으면 잊어버리기 쉽다고 생각하기 때문입니다. 숫자를 대입해도 괜찮다는 점을 기억하세요. 이는 좋은 일이고 종종 직관력을 발휘하는 좋은 방법이므로 이 경우 로그 A 곱하기 B를 말하고 이를 분리하면 다음과 같이 생각할 수 있습니다. 로그 100 곱하기 1000은 5입니다. 0이 5개 있습니다. 주어진 각 부분의 0 개수로 나누어집니다. 훌륭하네요. 직관을 더 발전시켜서 또 다른 연습 문제를 시도해 보도록 하겠습니다. 알고 계시다면, 훌륭합니다. 당신은 대답을 잘 할 수 있을 것입니다. 하지만 대답이 무엇인지 뿐만 아니라 이 대답을 다른 사람에게 어떻게 설명할 것인지, 내가 말하지 않고도 학생이 스스로 이 대답에 도달하도록 어떻게 노력할 것인지 생각할 수도 있습니다. 대답은 무엇입니까? 두 명의 잠재 청중이 있습니다. 수업 자체에 관심이 있는 사람과 메타 수업에 관심이 있는 사람이 있습니다. 그래서 우리의 질문은 다시 묻습니다. 다음 중 어느 것이 사실인가요? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "ㅏ. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "x의 n에 대한 로그는 x의 로그 n 곱하기 b와 같습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "x의 n승 로그는 x의 n승 로그와 같습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "x의 n에 대한 로그는 n에 x 또는 d의 로그를 더한 것과 같습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "여기 정답은 a입니다. 여러분 중 4,000명이 축하를 받은 것 같습니다. 로그 x의 n제곱은 n 곱하기 x의 로그와 같습니다. 다시 한 번 여러분이 이것을 가르치려고 한다고 가정해 보겠습니다. 누군가에게 또는 그것이 무엇을 의미하는지 이해하려고 한다면 시작하기 좋은 곳은 무언가를 연결하는 것입니다. 이 경우 로그 x의 n제곱에 대해 100의 거듭제곱으로 시도해 보겠습니다. 3 그리고 당신이 하고 있는 패턴이 실제로 작동하는지 확인하기 위해 다른 패턴과 함께 시도해 볼 수도 있지만, 단순히 대답이 무엇인지 보는 것이 아니라 왜 대답이 그런 식으로 나타났는지 생각하는 측면에서 철저하게 생각한다면 때로는 한 가지 예가 100의 세제곱이기 때문에 가능합니다. 우리는 그것을 잘 취하는 것으로 생각할 수 있습니다. 그것은 100의 3개의 복사본입니다. 나는 100의 3개의 복사본을 취하고 모든 것을 곱할 때 로그를 우리가 0의 수를 세는 것으로 생각합니다. 예를 들어, 아, 0이 6개만 있는 숫자가 될 것입니다. 즉 100 곱하기 100 곱하기 100을 취한다는 의미입니다. 그 0을 모두 그룹화하여 백만 개가 되는 것을 생각해 보면 이 숫자는 다음과 같습니다. 6 하지만 실제로 왜 6이었는지 생각해 보면 6이 나온 백만 안에 있는 0의 개수는 100의 복사본이 3개 있고 100개 각각에 2개의 서로 다른 0이 있으므로 더 일반적입니다. 100의 세제곱을 취하는 대신에 1000의 세제곱 또는 1000의 n승 또는 x의 n승을 보고 있다면 n의 값이 우리가 번 곱한 복사본 수라고 생각할 수 있습니다. 음, 보자, 그것은 우리가 x를 대체한 무엇이든에 있던 0의 수의 x배가 아니며 이 경우에는 100이었습니다. 따라서 대신에 로그 10,000과 같은 것을 n의 거듭제곱으로 취했다면 이것은 동일할 것입니다. 10,000개의 복사본을 n개 가져가서 각각의 0의 수를 4로 세면 n 곱하기 4가 됩니다. 물론 여러분 대부분이 올바르게 대답한 일반적인 속성은 다음과 같은 사랑스러운 작은 효과가 있다는 것입니다. 작은 힘이 그 앞으로 뛰어내리는 힘으로 올라간 어떤 것의 기록을 보세요. 그러면 내부에 무엇이 있었는지에 대한 기록이 있을 뿐입니다. 아마도 그것의 가장 중요한 의미 중 하나일 것입니다. 그것을 뭐라고 부를지 모르겠습니다. 의미 또는 만약 내가 로그를 취한다면 그것을 정의의 재진술이라고 부르고 그것이 10의 n승이라는 점을 다시 강조하겠습니다. 우리는 그 작은 n을 뛰어내리는 것으로 생각할 수 있습니다. 앞부분은 로그 베이스 10/10의 n 배가 됩니다. 이는 물론 1입니다. 이 표현은 끝에 있는 0의 수를 세는 것으로 생각할 수 있거나 더 일반적으로 10과 동일한 것에 10을 묻는 것이고 대답은 간단히 1입니다. 이는 매우 안심이 됩니다. 왜냐하면 여러분이 돌아가서 이 원래 표현식을 읽을 수 있는 또 다른 방법은 10의 10과 n의 10과 같다고 말하는 것이기 때문입니다. 아 글쎄, 우리가 가지고 있는 모든 주어진 로그 속성에 대해 대답은 이제 아니오입니다. 이 경우에 우리는 방금 x의 n제곱 로그 하나를 찾았습니다. n이 앞쪽으로 뛰어오르는 것과 관련이 있습니다. 거울 이미지 지수 속성은 항상 거울 이미지 지수 속성이 될 것입니다. 이는 우리가 이에 대해 약간의 직관을 얻는 데 도움이 될 수 있는 또 다른 방법이므로 그냥 덮어 두겠습니다. 여기서 우리가 얻게 될 미래의 속성 중 일부는 우리가 가고 있는 곳을 숨기려고 합니다. 방금 발견한 n으로 뭔가를 올리면서 앞으로 도약하는 것은 지수 속성에 해당합니다. 즉, 10을 x로 올리고 올리면 이 모든 것을 n의 거듭제곱은 10의 n 곱하기 x를 취하는 것과 같습니다. 이것은 로그에 대해 가질 수 있는 또 다른 직관을 갖게 합니다. 로그는 뒤집어진 지수와 같으며 이것이 제가 의미하는 바입니다. 내가 a의 로그를 취하고 있다면 로그 안쪽에 있는 것은 이 경우 지수적인 것에 대한 전체 외부 표현으로 생각해야 합니다. 내부에 있는 것은 10의 x에 해당합니다. 함수의 출력인 반면 a의 로그 전체는 여기 내부에 있는 것과 일치합니다. 10의 지수는 무엇입니까? 여기서 로그 표현식을 볼 때마다 오른쪽의 지수 역할을 한다고 생각해야 합니다. 측면과 x의 전체 10의 지수를 볼 때마다 로그 중 하나의 내부에 있는 항목에 해당하는 오른쪽의 전체 외부 구성요소가 표시됩니다. 우리는 이것을 곱할 때 아이디어 위에서 보았습니다. 로그가 내부적으로 회전하는 지수라면 내부에 외부에 추가하는 것입니다. 이는 외부에서 함수의 출력을 곱하는 것이 내부에 추가하는 것과 동일하다는 것을 말해줍니다. 왜냐하면 이러한 각 로그는 로그 a 및 로그 b와 같기 때문입니다. 오른쪽 표현식에서 x와 y의 역할을 합니다. 계속해서 이 중 몇 가지를 더 수행하고 직관을 구축할 수 있는 속성이 얼마나 많은지 살펴보겠습니다. 마지막 속성은 지수가 다음으로 뛰어내린다는 아주 좋은 생각은 로그에 익숙하지 않은 사람들에게는 약간 이상하게 보일 수 있지만 다시 직관을 얻기 위해 숫자를 연결하면 조금 알려 드리겠습니다. 다음 중 어느 것이 사실인가요? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "글쎄요, 10의 세제곱이 1000이라면 이는 10이 1000을 1/3로 올린 것과 같습니다. 여기에서 역을 하면 지수의 곱셈 역원이 포함되고 전개 방식은 1을 3으로 나눈 것처럼 보입니다. 3은 1000의 로그 밑수 10에 해당합니다. 이는 1을 1000의 로그 밑수 10으로 나눈 값입니다. 따라서 더 일반적으로 이 단일 예를 기반으로 밑수를 내부에 있는 것과 교환하면 1을 나누는 것에 해당한다고 추측할 수 있습니다. 외부에 무엇이 있는지에 따라 해당 지수 규칙을 살펴보는 관점에서 이것을 통해 생각할 수 있습니다. 이제 내 사랑스러운 작은 로그와 지수에는 무슨 일이 일어났습니까? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "훌륭합니다. 다시 한 번 여기에서 다룰 다른 속성 중 일부를 숨기고 여기에서 이전에 했던 것과 동일한 순서로 유지하겠습니다. 미리 작성하면 계속 유지할 수 있다고 생각했습니다. 평소보다 조금 더 깨끗하지만 어쩌면 종이를 섞는 이상한 게임을 하는 것일 수도 있습니다. 방금 찾은 것은 a의 로그 베이스 b를 바꾸면 이것이 해당하는 것을 1로 나누는 것과 같습니다. 지수 토지는 b를 어떤 거듭제곱으로 취하고 그것이 a와 같다고 말한다면 a의 역수는 다시 b와 같다고 말하는 것과 같습니다. 잠시 시간을 내어 로그를 상황을 바꾸는 것으로 생각하는 것이 도움이 됩니다. 내부적으로 a의 로그 베이스 b 표현식은 x의 역할을 하고 b의 로그 베이스 a 표현식은 a 위에 있는 역할을 수행하고 대칭적으로 전체 표현식 b의 x 거듭제곱이 재생됩니다. 왼쪽의 내부 역할은 a의 역할과 전체 표현의 역할을 합니다. a의 힘은 로그 베이스 a 내부에 있는 역할을 합니다. 예를 연결하면 알 수 있습니다. 이를 지수 규칙에 대응시킴으로써 우리는 이미 세 가지 다른 로그 규칙을 통해 생각할 수 있습니다. 만약 그것들이 암기할 대수학의 조각으로 전달된다면 여러분은 그것을 암기할 수 있지만 여러분의 학습에서 빠져나가는 것은 매우 쉽습니다. 당면한 작업 때문에 좌절하기가 매우 쉽습니다. 하지만 우리가 이런 종류의 일에 관심을 갖는 이유는 로그의 규칙을 이해하는 것이 바이러스가 성장하는 것과 같은 맥락에서 수학을 수행하는 데 도움이 되기 때문이라는 점을 스스로 상기하고 싶을 수도 있습니다. 하루에서 다음 단계로, 한 단계에서 다음 단계로, 일이 곱셈적으로 커지는 경향이 있습니다. 로그의 규칙을 이해하면 그런 종류의 것에 대해 더 나은 느낌을 얻는 데 도움이 되므로 그것이 어떻게 보일 수 있는지에 대한 멋진 실제 예를 보기 전에 예를 들어, 실제 사례로 전환하기 전에 로그의 속성에 대해 마지막 질문을 하기 위해 이 맥락에서 퀴즈 질문을 하나 더 하도록 하겠습니다. 지금 여기에서 있었던 것을 제거하려면 다음 중 어느 것이 사실인가요? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "a 더하기 b의 로그는 a 더하기 b의 로그와 같습니다. a 더하기 b의 로그는 a의 로그 곱하기 b의 로그와 같습니다. a의 로그 더하기 b는 1을 a의 로그 더하기 b의 로그로 나눈 것과 같습니다. a의 로그 더하기 b는 1을 로그 a의 로그 곱하기 b의 로그로 나눈 것과 같거나 위의 어느 것도 아닙니다. 아, 그리고 이제 우리는 그다지 합의를 이루지 못했습니다. 그렇죠? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "매우 흥미롭습니다. 두 사람 사이에 경마가 있어서 사람들이 대답하는 동안 잠시 생각해 보겠습니다. 실제로 청중에게 질문이 조금 있습니다. 아시다시피 저는 우리가 어떻게 할 수 있는지에 대해 이야기하고 있었습니다. 곱셈의 관점에서 생각해 보세요. 꼭 10의 거듭제곱일 필요는 없습니다. 3의 거듭제곱과 같은 것도 할 수 있습니다. 여기서 1에서 3, 9, 27, 81이 되면 모두 이 중 우리는 이 숫자 중 로그 베이스 3이 멋진 작은 단계로 증가한다고 말할 수 있습니다. 따라서 로그 베이스 3/1, 3 대 1, 대답은 일반적으로 0입니다. 베이스에 관계없이 1의 로그는 0 로그 밑 3/3, 3의 3과 같은 것은 1입니다. 마찬가지로 로그 밑 9/3은 2입니다. 제 질문이 무엇인지 궁금하실 수도 있지만 이 모든 것을 알아내는 데 도움이 될 것입니다. 여기, 로그 베이스를 하나 더 쓰겠습니다. 이제 81 중 3은 4입니다. 표면상으로는 어린아이에게 물어보면 5살이나 6살 정도의 숫자가 1과 9 사이의 중간이라고 가정하겠습니다. 숫자의 중간이 무엇인지 말해보세요. 대답하는 방법에 대한 본능은 대수적이지만 우리의 본능은 더 선형적인 경향이 있으므로 종종 1과 9라고 생각합니다. 그 사이에는 2, 3, 4, 5, 6 등의 간격으로 일정한 숫자가 많이 있습니다. , 7, 8 그리고 그 중간에 오른쪽으로 가면 5에 착지하게 되지만 곱셈 성장 측면에서 1에서 9까지 어디에서 갈지 생각한다면 많은 것을 추가하는 것이 문제가 아니지만 '어느 정도 성장하면 3배로 성장하고 또 3배로 성장합니다. 어린이의 타고난 본능은 '3'이라고 말하는 것과 일치하며, 인류학자가 사회를 연구하는 경우에도 마찬가지입니다.' 현대 사회와 같은 방식으로 회계 시스템과 글쓰기를 개발하지 않았으므로 이에 대해 세 가지로 대답할 것입니다. 그래서 지금 시청하고 있는 여러분 중에 어린 아이에게 접근할 수 있는 사람이 있는지 청중에게 묻는 질문입니다. 예를 들어 5년 범위에서 가서 1과 9 사이의 중간 숫자가 무엇인지 물어볼 수 있는지 알아보고 가능하다면 아이가 말하는 내용과 실제 대답이 무엇인지 트위터로 알려주세요. 이유는 모르겠습니다. 그것이 실제로 실제로 효과가 있을지에 대해 회의적입니다. 저는 이것이 매우 과학적인 방법이 아니라는 것을 이해합니다. YouTube 라이브 스트리밍을 시청하는 사람들에게 자신의 자녀를 설문조사한 다음 답을 트윗하도록 요청하는 것은 아니지만 저 자신을 위해서라면 흥미로울 것입니다. 우리 질문에 대한 어떤 종류의 검증을 보려면 이것이 한 방향으로 큰 합의가 없는 것 같은 첫 번째 질문입니다. 계속해서 점수를 매겨 답변이 훌륭하다는 것을 확인하겠습니다. 좋습니다. 2,400 여러분 중 a + b의 로그가 이러한 좋은 속성을 만족하지 않는 것은 위의 어느 것도 아니라고 올바르게 대답했습니다. 특히 자연 로그가 작용할 때 특정 종류의 근사치를 사용하지 않는 한 일반적으로 그렇습니다. 다음에 이것에 대해 이야기할 수도 있습니다. 로그의 입력을 추가하는 것은 실제로 매우 이상한 느낌입니다. 매우 이상한 일입니다. 그 이상한 느낌을 얻으려면 제가 여러분에게 a 더하기 b의 로그를 묻는다면 10의 거듭제곱을 연결하세요. 10,000과 100과 같은 몇 가지 예를 연결하고 입력에 무엇이 있는지 0 계산 기능을 수행하면 그 안에 0이 몇 개 있는지 스스로에게 물어볼 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "흥미로운 질문이군요. 로그의 밑이 0이 될 수 있나요? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "음, 삼각형의 관점에서 우리는 0의 어떤 종류의 거듭제곱 x가 다른 값 y와 같다고 말하는 것으로 생각할 수 있습니다. 이것은 x의 0이 y와 같다고 쓸 수도 있고 다음과 같이 쓸 수도 있습니다. y의 로그 밑수 0이 x와 같다고 말하는 것도 마찬가지입니다. 0은 x와 같습니다. 여기서 문제는 0에 대한 모든 것이 결국 0이 된다는 것입니다. 따라서 우리가 로그 밑수 0을 생각한다면 y 다른 입력에 대해 y 아시다시피, 1이나 2 또는 pi와 같은 것을 입력하고 싶을 때, 1이나 2, pi 또는 거기에 있을 수 있는 숫자에 대해 0이라는 질문을 하는 것입니다. 답이 없을 테니 기껏해야 아 예, 0의 로그입니다. 완벽하게 유효한 함수입니다. 입력 0에서만 정의되지만 그래도 원하는 것을 마무리하는 데 어려움을 겪을 수 있습니다. 거기에서 0을 0과 같다고 말하는 것은 무엇이든 적용되는 것과 같기 때문에 팔이 등 뒤로 비틀어질 것입니다. 그러나 당신이 그 일을 하고 싶고 그것은 밑이 0인 지수 함수가 완전히 0이라는 사실에 해당합니다. 숫자를 일대일 방식으로 서로 매핑하지 않으니 좋은 질문이군요. 이제 이러한 것들이 현실 세계에서 어디에서 나타나는지에 대한 아이디어로 돌아가서 로그 베이스 0을 가질 수 있습니까? 제가 좋아하는 한 가지 예는 다음과 같습니다. 지진에 대한 리히터 척도입니다. 리히터 척도는 지진이 얼마나 강한지에 대한 정량화를 제공하며 아주 작은 숫자부터 아주 큰 숫자까지 될 수 있습니다. 지금까지 측정된 가장 큰 지진이라고 생각합니다. 이것은 다음에서 나온 차트일 뿐입니다. 위키피디아는 9였습니다. 5 그리고 그것이 얼마나 미친 것인지 이해하려면 이 숫자가 의미하는 것과 동등한 양의 TNT와 같은 것 사이의 관계를 살펴볼 가치가 있습니다. 그 안에 얼마나 많은 에너지가 있는지 측정한 다음 여기서 우리가 시도할 수 있는 것이 무엇입니까? 에너지 양 측면에서 리히터 척도 수에 대한 표현을 얻을 수 있는지, 로그가 이를 설명하는 자연스러운 방법인 이유를 알아보는 것입니다. 따라서 초점을 맞춰야 할 핵심은 사물이 얼마나 증가하는지 앞으로 나아가는 것입니다. 예를 들어, 이 경우 2개 우물에서 가면 3개가 어디에 있는지 알려주지 않으므로 2개에서 4개까지 한 단계 더 나아가는 것을 생각할 수 있습니다. 이는 두 단계를 밟는 것과 비슷합니다. 에너지의 양은 1미터톤의 TNT에서 필요한 것으로 보입니다. 이는 제2차 세계 대전의 대형 폭탄인 것 같습니다. 그리고 작은 원자 폭탄인 경우 최대 1천 배나 더 많은 킬로톤이 필요하므로 두 단계만 거치면 됩니다. 진도 2의 지진에서 진도 4의 지진으로 가는 리히터 규모의 지진은 제2차 세계 대전의 대형 폭탄에서 핵 시대까지 우리를 데려가므로 주목할 만하며 우리가 얻는 첫 번째 깨끗한 단계는 4에서 5로 가는 것입니다. 적어도 이 차트가 우리에게 잘 보여주는 점은 분명히 4에서 5로 한 단계 올라가는 것은 1킬로톤에서 32킬로톤으로 가는 것에 해당하고 그것은 분명히 나가사키에 떨어진 도시 파괴 폭탄의 크기였기 때문에 이것은 아마도 하나일 것입니다. 뉴스에서 진도 4의 지진이 발생했습니다. 0 대 5의 지진. 0 생각하기 쉽습니다. 4와 5는 꽤 비슷한 숫자이지만 분명히 TNT 양으로 볼 때 1에서 다음으로 이동하기 위해 32를 곱하고 2에서 4로 가는 것은 분명히 약 1000을 곱한 것과 같습니다. 더 큰 이유는 여기 차트에 3이 무엇인지 표시되지 않았기 때문입니다. 그래서 우리는 두 단계를 밟고 있었고 32 단계를 밟은 다음 또 다른 32를 곱하면 실제로는 거의 1000에 가깝다는 것을 직접 확인할 수 있습니다. 리히터 수의 덧셈 단계가 TNT의 곱셈 단계에 해당한다는 생각은 여기에 대수적인 무언가가 작용하고 있음을 암시하는 것 같습니다. 여기서 계속 진행하여 부분적으로 세계 현상으로 인해 이것이 얼마나 증가하는지 말하는 것은 약간 흥미롭습니다. 예, 우리가 다음 단계를 밟을 때 약 32배가 다시 증가한다는 사실은 크게 놀라운 일이 아니지만 직관에 따르면 32킬로톤의 작은 원자 폭탄과 1메가톤의 작은 원자 폭탄이 아닌 것으로 생각할 수 있습니다. 내가 추측하는 나가사키 원자폭탄은 1메가톤당 나가사키 원자폭탄 중 32개로, 이는 분명히 1994년 미국 네바다주에서 발생한 이중현평지진의 크기입니다. 그게 무엇인지 몰랐습니다. 주파수 측면에서 Wikipedia에 감사드립니다. 또한 이것들을 찾아보니 분명히 2보다 작은 것들이 있었고 하루에 8000개 정도의 그런 것들이 항상 발생했지만 우리가 원자폭탄의 영역에 들어가자마자 3과 같은 것들이 있었습니다. 5번과 4번은 분명히 지구상 어딘가에서 꽤 자주 일어나는 일입니다. 매일 어딘가에서 약 134번의 일이 일어나고 있다는 걸 누가 알았을까요? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "하지만 우리가 원자 폭탄 규모보다 훨씬 높은 이 5와 6 범위에 더욱 집중함에 따라 이제 우리는 하루에 약 2개에 불과하며 지질학자가 들어와서 우리 모두가 왜 그렇게 해야 하는지 설명할 수 있을 것이라고 확신합니다. 매일 지각에 두 번의 원자폭탄에 상응하는 파괴가 발생한다는 사실에 대해 크게 걱정하지 마세요. 그러나 아마도 현재 많은 사람들이 살고 있는 도시와 같은 장소에 집중되는 경우는 거의 없을 것입니다. 32의 성장이 포함됩니다. 6에서 7까지의 단계가 어떻게 보이는지 살펴보겠습니다. 여기서는 사이에 더 많은 예를 제공하고 있습니다. 아마도 실제보다 더 큰 단계라는 환상을 줄 수 있으며 실제로 이것이 1메가톤과 1메가톤 사이의 차이입니다. 32메가톤이므로 32를 곱하면 이 차트에서 가장 흥미로워요 50메가톤이었던 차르 폭탄과 나는 그들이 실제로 100메가톤 폭탄을 보유하려는 원래 계획을 가지고 있다고 생각하지만 그 50메가톤에 대해 스스로를 반대했습니다. 우리는 나가사키 폭탄의 32킬로톤에서 32를 곱하여 메가톤에 32를 곱하면 제2차 세계대전이 끝난 폭발 강도의 1000배에 대해 이야기하고 있지만 아직 인류가 할 수 있는 50메가톤에 도달하지 못했습니다. 이는 분명히 인도네시아 자바 지진이므로 7 . 0은 6보다 조금 더 큰 것이 아닙니다. 0, 훨씬 더 크고 물론 여기서 중요한 점은 곱셈 증가를 제공하는 척도가 있을 때 작은 단계처럼 보이는 것이 실제로 내포된 에너지 또는 여기에 내포된 절대값 측면에서 큰 단계가 될 수 있다는 점을 인식할 가치가 있다는 것입니다. 그래서 우리가 9가 있었다는 사실을 생각할 때. 5는 7에만 있다는 점을 고려하면 실제로 터무니없어 보입니다. 0 범위는 지금까지 나온 것 중 가장 큰 열핵 무기에 대해 이야기하고 있으며 이는 로그가 발생하는 경향이 있는 영역을 나타냅니다. 인간은 얼마나 큰 일이 가능한지에 대한 엄청나게 넓은 차이를 설명하는 무언가에 대한 척도를 만들고 싶어할 때입니다. 지진 규모의 경우 지구 주위에서 항상 일어나는 일, 즉 큰 수류탄의 크기에 대한 정보를 얻을 수 있으며, 그것이 귀하의 규모에 맞춰지고 모든 범위에 걸쳐 생각할 수 있기를 원합니다. 우리가 인류 역사상 본 가장 큰 혼란에 대해 그리고 이를 위해서는 한 가지 경우에 대해 숫자에 여러 숫자를 쓰는 것이 아니라 더 작은 숫자를 사용하는 방식으로 이루어집니다. 다른 경우에는 로그를 취한 다음 기본적으로 0에서 10 사이의 숫자를 압축하는 단일 스케일에 두는 것이 좋습니다. 음악의 데시벨 스케일에서도 실제로 약간 작동하는 것과 매우 유사한 일이 일어나는 것을 볼 수 있습니다. 조금 다르게 말하면 10데시벨씩 올라갈 때마다 10을 곱하는 것과 같습니다. 따라서 1을 10으로 곱하는 것이 아니라 10을 곱하는 것이 10을 곱하는 것이므로 계산이 조금 더 쉬워집니다. 약간 이상하지만 아이디어는 동일합니다. 50데시벨 대 60데시벨의 소리를 듣는 경우 에너지가 전달되고 전달되는 측면에서 훨씬 더 조용합니다. 60에서 70 또는 70에서 70데시벨이 될 것입니다. 60에서 80까지 80개의 단계는 평방 면적당 에너지 양을 100배로 곱하여 로그 눈금을 볼 때마다 그것이 내부적으로 말하는 모든 것이 다음과 같이 증가한다는 것을 마음속으로 아십시오. 이것이 바로 우리가 코로나바이러스 발생을 설명하는 데 사용되는 많은 로그 척도를 본 이유입니다. 리히터 척도를 1씩 늘릴 때마다 32를 곱하는 관계를 어떻게 설명할 수 있을까요? 밑이 32인 로그의 관점에서 생각할 수 있습니다. 로그를 취하면 r이라고 부르겠습니다. 리히터 척도에 대한 숫자입니다. 나는 이것을 밑이 32인 로그로 생각할 수 있으며 이는 다음과 같습니다. , 아냐 아냐 아냐, 내가 잘못하고 있는 거야 그건 기록된 것이 아니야 우리는 TMT 번호의 큰 숫자인 로그 베이스 32를 취하는데, 1메가톤 정도였으니 로그 베이스 32가 100만 톤이 되어야 해. 리히터 규모 숫자에 해당하지만 일종의 오프셋이 있을 수 있으므로 이 리히터 규모 숫자에 추가하는 일종의 상수 s가 있다고 말할 수 있으며 이 표현은 정확히 동일합니다. 아래쪽에 있는 이 표현은 32의 오프셋 곱하기 리히터 척도 수를 말하는 것과 똑같습니다. 이는 그 오프셋에 32를 곱하는 것과 같습니다. 그 자체는 일종의 큰 상수이고, 리히터 척도 숫자에 32를 곱한 것입니다. 이것을 단지 상수 곱하기 32의 숫자의 거듭제곱이라고 생각할 수도 있습니다. 따라서 이 쓰기 방식은 기하급수적인 성장을 강조합니다. 만약 이것이 여러분이 보는 TMT 양에 해당한다면, 증가함에 따라 단계적으로 32를 곱하지만 똑같은 사실을 전달하는 또 다른 방법은 그 양이 무엇이든 로그 베이스 32를 취하는 것입니다. 이제 제가 다음으로 이야기하고 싶은 것은 우리가 항상 그럴 필요는 없다는 것입니다. 서로 다른 베이스의 로그를 계산하는 방법에 대해 걱정하세요. 여기서 우리가 로그 베이스 32에 대해 이야기하고 있다는 것이 조금 이상합니다. 앞서 수학자들이 베이스 e로 로그를 갖는 것을 얼마나 좋아하는지 언급했습니다. 컴퓨터 과학자들은 정말로 베이스 2로 로그를 갖는 것을 좋아합니다. 계산 목적으로 또는 하나의 로그가 있는 경우 이러한 것들이 어떻게 성장하는지 생각하기 위해 밝혀졌습니다. 기본 10, 기본 2, 기본 e 등 한 가지 유형의 로그를 계산할 수 있다면 거의 모든 것을 계산할 수 있습니다. 이제 그 방향으로 우리의 직관을 얻으려면 퀴즈로 돌아가서 다음 질문으로 넘어가겠습니다. 저는 이 질문이 가장 좋다고 생각합니다. 잘 모르겠습니다. 이것은 절반 정도 합리적인 질문입니다. 좋을 것입니다. 이것은 단지 2진수 문맥에서 10진수 문맥으로 변환할 준비를 하게 해 줄 것이며, 2의 거듭제곱이 일반적으로 10의 거듭제곱과 갖는 관계를 이해하는 데 좋은 직관이 될 것입니다. 이 두 가지 종류는 내 말이 무슨 뜻인지 알 수 있을 것입니다. 그들은 서로 잘 어울리므로 우리의 질문은 2의 10제곱이 1024, 1024이고 이는 대략 1000이라는 사실을 고려하면 그렇습니다. 숫자가 약간 느슨하고 대략 2의 10승, 기본적으로 1000을 계산하고 있습니다. 다음 중 사실에 가장 가까운 것은 무엇입니까? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "부드러운. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "여기서 만장일치로 결정된 것은 아닙니다. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "하지만 문제는 어느 것이 사실에 가장 가까운지 묻는 것이었습니다. 이에 대해 어떻게 생각할 수 있는지 살펴보겠습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "따라서 이는 2의 거듭제곱, 즉 1024를 가지고 있다는 것을 의미합니다. 이는 10의 거듭제곱, 약 10의 세제곱에 매우 가깝습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "그렇다면 이것은 무엇을 의미합니까? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "만약 10의 로그 밑수 2가 x와 같다면, 이는 2의 x가 10과 같다고 말하는 것과 같습니다. 그렇죠? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "그것은 우리에게 2의 10이 무엇인지를 묻고 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "모든 기능에서 그렇게 할 수는 없습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "사람들은 어떤 기능을 사용해도 그렇게 할 수 있다고 생각하는 것 같지만 실제로는 그렇지 않습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "그리고 그것이 의미하는 바는 x가 약 10/3이라는 것입니다. 그렇죠? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "그리고 앞서 우리가 본 것은 로그 베이스 2/10이고, 로그 베이스 10/2는 그 양의 1분의 1, x의 1분의 1이라고 말할 수도 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "삼. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "그리고 우리는 로그에서 작업을 하고 있기 때문에 나는 그런 식으로 기록할 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "마찬가지로 백만분의 2의 로그 밑을 봅시다. 1,000이 되기 위해 2를 10번 곱해야 한다면, 백만이 되기 위해서는 20번 정도 곱해야 합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "조금 더 작지만 이것은 마음속에 간직할 수 있는 좋은 근사치입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "3은 3이다. 20, 우리는 같은 양만큼 축소합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, 우리는 같은 양만큼 축소합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "좋아요? ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "이제 이것은 기억할 가치가 있는 직관입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "그리고 B의 로그 베이스 C와 A의 로그 베이스 C를 결합하는 다양한 가능한 방법의 전체 더미입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "이미 로그에 익숙하지 않은 이상 명확하지 않고 조금 생각해 볼 가치가 있기 때문에 이에 대해 의미 있는 시간을 드리겠습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/marathi/sentence_translations.json b/2020/ldm-logarithms/marathi/sentence_translations.json index ee040b412..e8e31197a 100644 --- a/2020/ldm-logarithms/marathi/sentence_translations.json +++ b/2020/ldm-logarithms/marathi/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Music🎵 Lockdown Math मध्ये तुमचे परत स्वागत आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "आज आपण लॉगरिदम आणि मूलभूत गोष्टींबद्दलच्या धड्याबद्दल बोलणार आहोत. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "आणि नेहमीप्रमाणे, गोष्टी सुरू करण्यासाठी, मला फक्त प्रेक्षक सध्या कुठे आहेत हे जाणून घ्यायचे आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "तर, जर तुम्ही 3b1b वर जाऊ शकता. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "a मी त्यांच्याबद्दल यापूर्वी कधीही ऐकले नाही किंवा त्यांच्याबद्दल कधीही शिकले नाही. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "मी त्यांच्याबद्दल शिकलो आहे परंतु कधीकधी सर्व गुणधर्मांमुळे गोंधळून जातो c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "मी त्यांना समजतो पण त्यांना कसे शिकवावे हे मला माहीत नाही आणि डी. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "मी त्यांना चांगले समजतो आणि त्यांना नीट समजावे म्हणून त्यांना आरामात इतर कोणाला तरी शिकवू शकतो. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "तर, आम्हाला चांगले विभाजन मिळाले आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "मी म्हटल्याप्रमाणे, यामागचा हेतू एक धडा तयार करणे हा आहे की मी भविष्यात लोकांना सूचित करू शकेन जर त्यांना लॉगरिदम आवडत नसतील आणि मला असे म्हणायचे आहे की, अरे, येथे एक ठिकाण आहे जिथे तुम्ही जाऊ शकता मी कसे विचार करतो, तुम्हाला माहिती आहे, मला वाटते की तुम्ही याकडे अंतर्ज्ञानाने कसे पोहोचू शकता. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "कारण हे विशिष्ट व्याख्यान करण्यापूर्वी मी दोन शिक्षक मंचांभोवती स्क्रोल करत होतो आणि जेव्हा लोक विचारतात की हायस्कूल गणितामध्ये शिकवण्यासाठी सर्वात कठीण विषय कोणता आहे या अर्थाने विद्यार्थ्यांना याचा सर्वात जास्त त्रास होतो असे दिसते, लॉगरिदम सर्वात जास्त आहे. सामान्यत: सूचित केलेली उत्तरे जी मनोरंजक आहेत आणि मी अंदाज लावू शकतो की कदाचित असे बरेच गुणधर्म आहेत जे तुम्हाला माहित असणे आवश्यक आहे, म्हणून आम्ही जिथे जाणार आहोत त्यापासून पुढे गेलो तर तुम्हाला हे सर्व ढीग मिळतील. नियम जे बीजगणिताच्या गुच्छासारखे दिसतात जे लक्षात ठेवणे कठीण आणि आपल्या डोक्यात गोष्टी मिसळणे सोपे असू शकते आणि मला वाटते जेव्हा लोकांकडे हायस्कूलचे गणित कसे होते आणि काय होते याबद्दल अशा भयानक आठवणी असतात. लॉगरिदम त्यांच्यासाठी केले, बहुतेकदा ती विशिष्ट सूत्रे मनात येत असतात आणि आज मला काय करायचे आहे ते एकाद्वारे बोलण्याचा प्रयत्न करा, त्यांचा विचार कसा करायचा परंतु फक्त मेटा लेव्हलवर जर तुम्ही एखाद्याला बीजगणित शिकवत असाल तर, काय आहेत जोर देण्यासारखे मुद्दे? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "त्यांच्या अंतर्मनात ते तयार करण्याचा मार्ग काय आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "ओह, त्यावर ३ शून्य आहेत लाखाचा लॉग काय? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "1000 गुणा x चा लॉग x च्या लॉगच्या 3 पट आहे आणि लक्षात ठेवा की आम्ही हे नियम वापरत आहोत की ते बेस 10 लॉग b आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "1000 गुणा x चा लॉग x क्यूबड c च्या लॉगच्या बरोबरीचा. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "1000 गुणा x चा लॉग x आणि e च्या लॉगच्या घाताच्या 3 च्या बरोबरीचा आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "वरीलपैकी काहीही नाही आणि मी आधी म्हटल्याप्रमाणे लक्षात ठेवा की आपण पूर्ण अपेक्षा केली पाहिजे की सुरुवातीला ज्या लोकांनी सांगितले की त्यांना लॉग चांगले समजतात ते लगेच उत्तर देतील, ते बरोबर उत्तर देतील परंतु जर तुम्ही जो कोणी करत नाही, तो तुम्हाला घाबरू देऊ नका जेव्हा तुम्ही यासारखी समस्या पाहत असाल तेव्हा मी तुम्हाला फक्त 10 च्या विविध पॉवर्समध्ये प्लग इन करा आणि लॉग फंक्शनच्या कल्पनेनुसार विचार करा. शून्यांची संख्या मोजते म्हणून मी तुम्हाला त्याबद्दल विचार करण्यासाठी थोडा वेळ देईन जेणेकरून मी पुढे जाईन आणि ते श्रेणीबद्ध करेन आणि नेहमीप्रमाणे जर ते तुम्हाला सोयीस्कर आहे त्यापेक्षा ते अधिक वेगवान असेल हे माहित आहे की ते फक्त कारण मला पुढे जायचे आहे धड्यासह म्हणून या प्रकरणात योग्य उत्तर 1000 गुणा x चे लॉग म्हणून बाहेर येते हे 3 अधिक x चा लॉग घेण्यासारखे आहे आणि आता आपण त्याबद्दल क्षणभर विचार करूया आणि जसे मी म्हटल्याप्रमाणे आपण नुकतेच प्रारंभ करत आहात त्यांच्याबरोबर मला वाटते की सर्वोत्कृष्ट गोष्ट म्हणजे विविध संख्यांमध्ये प्लग इन करणे सोयीस्कर आहे आणि प्लग इन करण्यासाठी सर्वोत्तम संख्या म्हणजे आधीपासून 10 पॉवर आहेत, म्हणून जर तुम्ही 1000 पट x लॉग सारखे काहीतरी विचारत असाल तर मी करू शकत नाही. माहित नाही, चला 1000 गुणिले 100 च्या x लॉगसाठी काहीतरी प्लग इन करू या, 1000 गुणिले 100 म्हणजे 100,000 च्या अंतिम उत्तरात किती शून्य असतील हे आपल्याला माहीत आहे. आम्ही फक्त शून्य घेत आहोत, त्या 1000 मधील 3 शून्य आणि त्या 100 मधील 2 शून्य आणि आम्ही त्यांना एकमेकांच्या पुढे ठेवत आहोत म्हणून ते एकूण 5 शून्य असले पाहिजे परंतु जर तुम्ही खरोखर संख्या कशी वळली यावर विचार केला नाही तर बाहेर पण असे का निघाले ते त्या 1000 मधील 3 शून्य अधिक त्या 100 मधील 2 शून्य होते जे आपण 1000 मधील शून्य संख्या अधिक 100 मधील शून्य संख्या सांगून देखील लिहू शकतो म्हणून ही कल्पना आहे की लॉगरिदम दोन गोष्टींच्या गुणाकाराची बेरीज 10 च्या शक्तींच्या संदर्भात त्या दोन गोष्टींच्या लॉगॅरिथमची बेरीज आहे जी आपण 10 च्या 2 पॉवर्स घेतल्यास आणि आपण त्यांना फक्त गुणाकार केल्यास आपल्यापैकी बर्‍याच जणांसाठी आधीपासूनच एक सुपर अंतर्ज्ञानी कल्पना आहे. त्यांचे सर्व शून्य घ्या आणि त्यांना एकमेकांवर घट्ट करा म्हणजे मी येथे ज्या प्रकारे गोष्टी लिहिल्या आहेत ते प्रत्यक्षात थोड्या अधिक सामान्य वस्तुस्थितीचे सूचक आहे जे लॉगरिदमची आमची पहिली मालमत्ता असेल जे म्हणजे जर आपण A गुणा B चा लॉग तो A चा लॉग अधिक B च्या लॉगच्या बरोबरीचा आहे आता आपण या लॉगॅरिथम नियमांपैकी एखादा नियम पाहिल्यास आपण डोळे वटारत आहात किंवा आपण ते कसे लक्षात ठेवावे याबद्दल थोडेसे गोंधळलेले असाल तर फक्त उदाहरणे प्लग करा मी निरर्थक आहे, मी हे खूप बोलतोय पण कारण मला वाटते की एकदा तुम्ही बीजगणितात बुडून गेलात की विसरणे खूप सोपे आहे आणि तुम्ही एका प्रकारच्या परीक्षेला बसला आहात आणि त्यात बरीच चिन्हे आहेत. स्वतःला स्मरण करून देण्यासाठी तुम्हाला काही संख्या प्लग इन करणे ठीक आहे ही एक चांगली गोष्ट आहे आणि बर्‍याचदा अंतर्ज्ञान मिळवण्याचा हा एक चांगला मार्ग आहे, म्हणून या प्रकरणात, ए टाइम्स बी चे लॉग म्हणणे आणि ते वेगळे करणे आपण फक्त विचार करू शकतो, अरे, ते 100 गुणिले 1000 चा लॉग जो 5 आहे, त्यामध्ये 5 शून्य आहेत प्रत्येक दिलेल्या भागातील शून्यांच्या संख्येच्या संदर्भात खंडित होतो, खूप छान, अप्रतिम त्यामुळे ती अंतर्ज्ञान पुढे नेण्यासाठी आपण आणखी एक सराव समस्या वापरून पाहू आणि पुन्हा, जर तुम्हाला माहित असेल तर, छान, तुम्ही याचे उत्तर चांगले देऊ शकाल परंतु कदाचित विचार करा, फक्त उत्तर काय आहे असे नाही तर मी हे उत्तर कोणाला कसे समजावून सांगू किंवा मला सांगावयाशिवाय विद्यार्थ्याला स्वतःहून हे उत्तर कसे मिळवावे यासाठी मी प्रयत्न करू. त्यांचे उत्तर काय आहे म्हणून तेथे दोन संभाव्य प्रेक्षक सदस्य आहेत ज्यांना धड्यातच रस आहे आणि नंतर ज्यांना मेटा धड्यात रस आहे, त्यामुळे आमचा प्रश्न पुन्हा विचारतो, खालीलपैकी कोणते खरे आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "a x ते n चा लॉग x b च्या n गुणिले लॉगच्या बरोबरीचा आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "n ला x चा लॉग n c च्या पॉवर x च्या लॉगच्या बरोबरीचा आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "x चा लॉग n बरोबर n अधिक x किंवा d चा लॉग आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "तर येथे योग्य उत्तर a आहे, जे असे दिसते की तुमच्यापैकी 4,000 लोकांनी अभिनंदन केले आहे, आम्हाला सांगत आहे की x चा लॉग n ची पॉवर n x च्या n गुणा लॉग बरोबर आहे, म्हणून, आपण हे शिकवण्याचा प्रयत्न करत आहात असे पुन्हा म्हणूया एखाद्याला किंवा तुम्ही स्वतःला याचा अर्थ समजून घेण्याचा प्रयत्न करत असाल तर मला वाटते की प्रारंभ करण्यासाठी एक चांगली जागा म्हणजे काहीतरी प्लग इन करणे आणि या प्रकरणात, x च्या पॉवरच्या लॉगसाठी n चला 100 पॉवरसह प्रयत्न करूया 3 आणि तुम्ही करत असलेले नमुने प्रत्यक्षात काम करतात की नाही हे पाहण्यासाठी तुम्ही इतरांसोबत प्रयत्न करू शकता परंतु जर तुम्ही उत्तर काय आहे ते फक्त पाहण्याच्या दृष्टीने नाही तर उत्तर असे का आले याचा विचार करण्याचा प्रयत्न करत असाल तर काहीवेळा एक उदाहरण होईल कारण 100 क्यूब केलेले, आपण 100 च्या 3 प्रती घेतो असे समजू शकतो, मी 100 च्या 3 प्रती घेतो आणि जेव्हा मी त्या सर्वांचा गुणाकार करतो आणि मी लॉगचा विचार करतो तेव्हा आपण शून्यांची संख्या मोजतो. म्हणा, अरे, ही अशी काही संख्या असणार आहे ज्यावर फक्त 6 शून्य आहेत म्हणजे 100 गुणिले 100 गुणिले 100 घेणे म्हणजे मी फक्त त्या सर्व शून्यांना एकत्र करून एक दशलक्ष मिळवण्याचा विचार करू शकतो, त्यामुळे ही संख्या असेल 6 पण जर आपण खरे तर विचार केला तर 6 दशलक्षातील शून्यांची संख्या इतकीच का नव्हती जिथून 6 आले ते म्हणजे आपल्याकडे त्या 100 च्या 3 प्रती आहेत आणि त्या 100 पैकी प्रत्येकामध्ये 2 भिन्न शून्य आहेत त्यामुळे ते अधिक सामान्य आहे. आपण त्याबद्दल विचार करू शकता, जेथे 100 घन घेण्याऐवजी आपण 1000 घन किंवा 1000 n कडे किंवा x पॉवर n कडे पाहत असू तर आपण विचार करू शकता की n चे मूल्य जे काही आहे ते आपण वेळाने गुणाकार करत असलेल्या प्रतींची संख्या आहे. विहिरीची संख्या, बघूया, आपण x च्या बदली केलेल्या शून्याच्या संख्येच्या x पट नाही जे या प्रकरणात 100 होते, म्हणून जर त्याऐवजी मी 10,000 च्या लॉग सारखे काहीतरी पॉवर n घेतले असते तर हे समान असते त्या 10,000 च्या n प्रती घेतल्यास त्या प्रत्येकातील शून्यांची संख्या 4 आहे म्हणून ती n गुणिले 4 होईल आणि अर्थातच तुमच्यापैकी बहुतेकांनी बरोबर उत्तर दिलेली सामान्य मालमत्ता ही आहे की तुमचा हा सुंदर थोडा प्रभाव आहे जेव्हा तुम्ही एखाद्या सामर्थ्यापर्यंत उंचावलेल्या एखाद्या वस्तूचा लॉग पहा आणि त्याच्या समोर थोडी शक्ती खाली येते आणि तुमच्याकडे फक्त आतमध्ये काय आहे याचा लॉग आहे त्याचा कदाचित सर्वात महत्त्वाचा परिणाम म्हणजे मला माहित नाही की तुम्ही त्याला कॉल कराल की नाही एक तात्पर्य किंवा जर मी लॉग घेत असेन आणि मी फक्त 10 पैकी 10 च्या बळावर पुन्हा जोर देईन तर आपण त्याला व्याख्येची पुनरावृत्ती म्हणू इच्छितो आणि आपण त्या लहान n चा विचार करू शकतो. समोर आहे आणि तो 10 च्या n पटीने लॉग बेस 10 बनतो जो अर्थातच 1 आहे ही अभिव्यक्ती आपण विचार करू शकता एकतर शेवटी शून्यांची संख्या मोजणे किंवा अधिक सामान्यतः ते 10 ला 10 च्या बरोबरीचे विचारत आहे आणि उत्तर फक्त 1 आहे जे खूप आश्वासक आहे कारण तुम्ही परत जाऊन फक्त ही मूळ अभिव्यक्ती वाचू शकता असा दुसरा मार्ग म्हणजे 10 ला 10 च्या बरोबरीचे n n ओह बरं उत्तर आहे n आता आमच्याकडे दिलेल्या प्रत्येक लॉगरिथम गुणधर्मासह या प्रकरणात आम्ही फक्त पॉवर n साठी x चा एक लॉग सापडला n समोर उभं राहणं यात नेहमीच मिरर इमेज एक्सपोनेन्शिअल प्रॉपर्टी असेल आणि हा आणखी एक मार्ग आहे ज्याद्वारे आपण स्वतःला या गोष्टींसाठी थोडी अंतर्ज्ञान प्राप्त करण्यास मदत करू शकतो म्हणून मी फक्त लपवू दे भविष्यातील काही गुणधर्म जे आम्ही येथे मिळवणार आहोत ते लपवण्याचा प्रयत्न करा आम्ही कुठे जात आहोत जे आम्हाला आत्ताच समोरच्या n वर काहीतरी वाढवताना आढळले हे घातांकीय गुणधर्माशी संबंधित आहे जे मी x ला 10 घेतले आणि वाढवल्यास पॉवर n साठी ती संपूर्ण गोष्ट म्हणजे 10 ला n वेळा x घेण्यासारखेच आहे आणि यामुळे आपल्याला लॉगरिदमसाठी आणखी एक अंतर्ज्ञान मिळू शकेल जे ते घातांक आतून बाहेर काढल्यासारखे आहेत आणि मला काय म्हणायचे आहे ते येथे आहे की लॉगच्या आतील बाजूस बसलेली गोष्ट जर मी लॉग घेत असेल तर तुम्ही याचा विचार करत असाल की घातांक असलेल्या एखाद्या गोष्टीसाठी संपूर्ण बाह्य अभिव्यक्ती आहे या प्रकरणात आतील बाजू 10 ते x च्याशी संबंधित आहे फंक्शनचे आउटपुट, तर संपूर्ण गोष्टीचा लॉग स्वतः आतल्या बाजूला काय आहे याच्याशी संबंधित आहे फक्त 10 चा घातांक किती आहे, म्हणून जिथे जिथे तुम्हाला लॉग एक्सप्रेशन दिसेल तिथे उजवीकडे घातांकाची भूमिका बजावते असा विचार केला पाहिजे बाजू आणि प्रत्येक वेळी जेव्हा तुम्ही घातांक पहाल तेव्हा संपूर्ण 10 ते x अभिव्यक्ती उजव्या बाजूला संपूर्ण बाह्य घटक जो एका लॉगच्या आतील बाजूस बसलेल्या एखाद्या गोष्टीशी सुसंगत आहे आणि जेव्हा आपण गुणाकार करतो तेव्हा आम्ही हे वर पाहिले आतील बाजूने बाहेरील बाजूने जोडले जात आहे जर लॉग्सच्या आतील बाहेरील घातांकाचे प्रकार जे आपल्याला सांगत आहेत की फंक्शनच्या आउटपुटचा बाहेरून गुणाकार करणे हे आतील बाजूस जोडण्यासारखेच आहे कारण यातील प्रत्येक लॉग जसे की लॉग a आणि log b उजवीकडील अभिव्यक्तीमध्ये x आणि y ची भूमिका बजावत आहे, म्हणून आपण खेळत राहू या यापैकी आणखी काही करूया आणि या शेवटच्यासाठी यापैकी किती गुणधर्म तयार करू शकतो ते पाहूया, घातांकांचा पुढील भाग खाली उतरवण्याचा खूप छान विचार म्हणजे लॉगरिदमशी परिचित नसलेल्यांना थोडेसे विचित्र वाटू शकते परंतु पुन्हा, त्यासाठी काही अंतर्ज्ञान मिळविण्यासाठी काही संख्या प्लग इन करा आणि आम्ही ते थोडेसे देऊ. पुढीलपैकी कोणते सत्य आहे ते वर काढण्यासाठी आणखी एक क्षण? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "बरं, जर 10 क्यूबड 1000 असेल तर 10 म्हणजे 1000 बरोबर 1 तृतीयांश असे म्हटल्यासारखे आहे की येथे व्युत्क्रम केला तर घातांकाचा गुणाकार व्युत्क्रम समाविष्ट होतो आणि तो बाहेर पडण्याचा मार्ग म्हणजे 1 ला 3 ने भाग घेतल्यासारखे दिसते आणि 3 हे 1000 मधील लॉग बेस 10 शी संबंधित आहे ते 1000 च्या लॉग बेस 10 ने 1 ने भागले आहे त्यामुळे अधिक सामान्यपणे, तुम्ही या एकाच उदाहरणावर आधारित असा अंदाज लावू शकता की जेव्हा आपण बेसची आतील बाजूने अदलाबदल करतो तेव्हा ते 1 विभाजित घेण्याशी संबंधित असते बाहेरील काय आहे यावरून आणि पुन्हा, तुम्ही संबंधित घातांक नियम बघून याचा विचार करू शकता आता माझ्या सुंदर छोट्या लॉग आणि घातांकांचे काय झाले? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "अद्भूत आहे म्हणून, पुन्हा काही गोष्टी कुठे लपवूया काही इतर गुणधर्म ज्या आपण येथे मिळवू आणि मी ते त्याच क्रमाने ठेवू ज्या माझ्याकडे पूर्वी होत्या त्याच क्रमाने मी विचार करत होतो की ते आधीच लिहून ठेवल्यास मला ठेवता येईल. नेहमीपेक्षा थोडेसे स्वच्छ पण कदाचित त्यात फक्त पेपर कटिंगचा हा विचित्र खेळ खेळणे समाविष्ट आहे, त्यामुळे आम्हाला जे सापडले आहे, a चा लॉग बेस बी जर तुम्ही त्या अदलाबदल केला, तर हे 1 ने विभाजित करण्यासारखेच आहे, ज्याच्याशी संबंधित आहे. घातांकीय जमीन म्हणजे जर तुम्ही b ला काही बळावर नेले आणि असे म्हणता की ते a च्या बरोबरीचे समान विधान आहे की a ते त्या बळाच्या व्यस्ततेच्या बरोबरीने पुन्हा b समान आहे, तर थोडा वेळ काढणे आणि लॉगरिदमला गोष्टी वळवल्यासारखे समजणे उपयुक्त आहे आत a चा लॉग बेस b हा x ची भूमिका बजावत आहे आणि b चा लॉग बेस a हा a च्या वर बसलेल्या गोष्टीची भूमिका बजावत आहे आणि नंतर सममितीने, x ची संपूर्ण अभिव्यक्ती b खेळत आहे डावीकडील आतील बाजूची भूमिका, ती a आणि संपूर्ण अभिव्यक्तीची भूमिका बजावते, a ते एखाद्या गोष्टीची शक्ती लॉग बेसच्या आत काय बसले आहे याची भूमिका बजावते आणि काही उदाहरणे जोडून तुम्ही पाहू शकता. घातांकीय नियमांशी जुळवून घेऊन आपण तीन वेगवेगळ्या लॉगॅरिथम नियमांद्वारे आधीच विचार करू शकतो जे आपल्याला माहित असलेल्या लक्षात ठेवण्यासाठी बीजगणिताचे तुकडे म्हणून दिले असल्यास, आपण ते लक्षात ठेवू शकता परंतु त्यांच्यासाठी आपल्यापासून दूर जाणे खूप सोपे आहे. डोके आणि हातातील कामामुळे निराश होणे देखील खूप सोपे आहे परंतु आपण स्वत: ला आठवण करून देऊ इच्छित असाल की या प्रकारच्या गोष्टींबद्दल आपल्याला काळजी आहे याचे कारण म्हणजे लॉगरिदमचे नियम समजून घेणे हे आम्हाला अशा संदर्भांमध्ये गणित करण्यास मदत करते जिथे तो विषाणू वाढतो आहे. एका दिवसापासून दुसर्‍या दिवसापर्यंत, एका पायरीपासून दुसर्‍या टप्प्यापर्यंत, गोष्टी गुणाकाराने वाढतात लॉगरिदमचे नियम समजून घेणे आपल्याला अशा प्रकारच्या सामग्रीबद्दल अधिक चांगले अनुभव घेण्यास मदत करते, म्हणून आम्ही ते काय दिसू शकते याचे एक छान वास्तविक जग उदाहरण करण्यापूर्वी जसे की मी या शिरामध्ये आणखी एक प्रश्नमंजुषा प्रश्न करू दे की लॉगरिदमच्या गुणधर्मांबद्दल विचारण्यासाठी एक शेवटचा प्रश्न विचारू या आधी आपण एका वास्तविक जगाच्या उदाहरणाकडे जाण्यापूर्वी आपल्याकडे येथे आणि आता काय होते ते काढून टाका, खालीलपैकी कोणते खरे आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "a प्लस b चा लॉग सारखा आहे a plus b च्या b log चा लॉग a plus b च्या b log च्या logo च्या बरोबरीचा आहे a plus b च्या b लॉग च्या टाइम लॉग च्या लॉग च्या बरोबर a plus b च्या लॉग च्या लॉग ने भागलेल्या b च्या बरोबर आहे किंवा a plus b चा लॉग हा b च्या गुणानुक्रमाच्या लॉगने भागिले एक बरोबर आहे किंवा वरीलपैकी काहीही नाही, आणि आता आमच्यात तितकी एकमत नाही, का? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "अतिशय मनोरंजक, आमची दोघांमध्ये घोड्यांची शर्यत आहे, त्यामुळे लोक उत्तरे देत असताना मी तुम्हाला यावर विचार करण्यासाठी एक क्षण देईन, खरं तर माझ्याकडे प्रेक्षकांसाठी एक छोटासा प्रश्न आहे, त्यामुळे तुम्हाला माहिती आहे, मी फक्त आम्ही कसे करू शकतो याबद्दल बोलत होतो गुणाकार वाढीच्या संदर्भात विचार करा आणि ते फक्त दहाच्या शक्ती असणे आवश्यक नाही, आपण तीनच्या शक्तीसारखे काहीतरी देखील करू शकतो जिथे आपण एक ते तीन ते नऊ ते सत्तावीस ते ऐंशी पर्यंत जात असाल तर, सर्व यापैकी आपण असे म्हणू शकतो की या संख्यांचा लॉग बेस तीन फक्त छान छोट्या पायऱ्यांमध्ये वाढतो म्हणून लॉग बेस तीनपैकी एक, तीन ते एक म्हणजे काय, उत्तर शून्य आहे सर्वसाधारणपणे एकाचा लॉग, बेस काहीही असो. शून्य लॉग बेस तीनपैकी तीन, तीन ते तीन म्हणजे तीन म्हणजे एक समान लॉग बेस तीन पैकी नऊ म्हणजे दोन आह, तुम्हाला कदाचित आश्चर्य वाटेल की माझा प्रश्न काय आहे, परंतु हे सर्व बाहेर काढण्यात आणि माझ्या स्वतःच्या आनंदासाठी मदत करेल इथे, मी आता आणखी एक लॉग बेस लिहू दे, ऐंशी-एके पैकी तीन आता चार आहे, मी ऐकले आहे की जर तुम्ही एखाद्या मुलाला विचारले तर, पाच किंवा सहा वर्षांच्या जवळपास सांगूया की एक आणि नऊ मधील अर्धी संख्या किती आहे? कोणती संख्या अर्धी आहे ते सांगा उत्तर कसे द्यायचे याची त्यांची अंतःप्रेरणा लॉगरिदमिक आहे तर आमची अंतःप्रेरणा अधिक रेषीय असते म्हणून आम्ही सहसा एक आणि नऊ विचार करतो, तुम्हाला त्यांच्यामध्ये दोन, तीन, चार, पाच, सहा अशा समान अंतर असलेल्या संख्यांचा समूह आहे , सात, आठ आणि जर तुम्ही मध्यभागी बरोबर गेलात, तर तुम्ही पाचवर उतराल पण जर तुम्ही गुणाकार वाढीच्या दृष्टीने विचार करत असाल की एक ते नऊ पर्यंत कुठे जायचे असेल, तर त्यात काही गोष्टींचा गुच्छ जोडण्याची गरज नाही पण तुम्ही 'विशिष्ट प्रमाणात तुम्ही तीनच्या घटकाने वाढता, नंतर तुम्ही तीनच्या दुसर्‍या घटकाने वाढता, असे मानले जाते की, लहान मुलाची नैसर्गिक प्रवृत्ती तीन म्हणण्याशी जुळते आणि कदाचित तुमच्याकडे मानववंशशास्त्रज्ञ आहेत ज्या समाजाचा अभ्यास करत असतील तर हे देखील याच्याशी जुळते' आधुनिक समाजात ज्या प्रकारे लेखा प्रणाली आणि लेखन विकसित केले आहे त्याप्रमाणे ते यासाठी तीन उत्तरे देतील, म्हणून, तुमच्यापैकी कोणीही सध्या पाहत असलेल्या प्रेक्षकांसाठी माझा प्रश्न आहे, जर पाच वर्षांच्या श्रेणीत लहान मुलाला प्रवेश असेल तर समजा. म्हातारे बघा तुम्ही त्यांना जाऊन विचारू शकता की एक ते नऊ मधील अर्धा क्रमांक कोणता आहे आणि जर तुम्हाला शक्य असेल तर आम्हाला Twitter वर कळू द्या की मूल काय म्हणते त्यांचे खरे उत्तर काय आहे कारण मला माहित नाही का, मी थोडासा आहे प्रत्यक्ष व्यवहारात ते बाहेर पडते की नाही याबद्दल साशंक आहे मला समजते की हे करण्याचा हा एक सुपर सायंटिफिक मार्ग नाही मी YouTube लाइव्ह स्ट्रीम पाहणाऱ्या लोकांना त्यांच्या स्वतःच्या मुलांचे सर्वेक्षण करण्यासाठी आणि नंतर उत्तर ट्विट करण्यास सांगत नाही पण माझ्या स्वतःच्या फायद्यासाठी ते मनोरंजक असेल आमच्या प्रश्नावर काही प्रकारचे प्रमाणीकरण पाहण्यासाठी हा पहिला प्रश्न आहे ज्यामध्ये एका दिशेने एकमत होत नाही असे वाटत नाही, चला पुढे जाऊ आणि त्याचे उत्तर काय चांगले आहे हे पाहण्यासाठी ते ग्रेड करू, ठीक आहे, म्हणून 2,400 तुमच्यापैकी बरोबर उत्तर दिले आहे की वरीलपैकी काहीही नाही की a प्लस b चा लॉग यापैकी कोणत्याही चांगल्या गुणधर्मांची पूर्तता करत नाही आणि सर्वसाधारणपणे, जोपर्यंत आम्ही विशिष्ट प्रकारच्या अंदाजे काम करत नाही तोपर्यंत, विशेषत: जेव्हा नैसर्गिक लॉग लागू होतो. आपण पुढील वेळी याविषयी बोलू शकतो लॉगरिदमचे इनपुट जोडणे ही खरोखर एक अतिशय विचित्र संवेदना आहे ही खूप विचित्र गोष्ट आहे आणि त्या विचित्रपणाची जाणीव करून घेण्यासाठी, जर मी तुम्हाला एक प्लस बी लॉग विचारले तर दहाचे काही पॉवर प्लग करा. तुम्ही काय विचार करू शकता, ठीक आहे, मला फक्त 10,000 आणि 100 सारखी काही उदाहरणे जोडू द्या आणि मी स्वतःला विचारतो, जर मी हे शून्य मोजण्याचे कार्य केले तर त्या इनपुटमध्ये किती शून्य आहेत? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "हा एक मनोरंजक प्रश्न आहे ठीक आहे, लॉगरिदमचा आधार शून्य असू शकतो का? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "आपल्या त्रिकोणाच्या संदर्भात आपण असा विचार करू शकतो की आपल्याला माहित आहे की, शून्य ते काही प्रकारच्या शक्ती x समान आहे इतर काही मूल्य y हे असे आहे जे आपण एकतर शून्य ला x बरोबर y असे लिहू शकतो किंवा आपण लिहू शकतो y चा लॉग बेस शून्य म्हणजे x शून्य बरोबर x बरोबर काय असे सांगून तीच गोष्ट आता येथे मुद्दा असा आहे की शून्य ते कोणत्याही गोष्टीचा शेवट शून्य बरोबर होतो, म्हणून जर आपण लॉग बेस शून्याचा विचार करणार आहोत तुम्हाला माहीत असलेल्या इतर कोणत्याही इनपुटसाठी y, तुम्हाला एक किंवा दोन किंवा pi सारखे काहीतरी इनपुट करायचे आहे, तुम्हाला हवे असलेले काहीही, तुम्ही शून्य हा प्रश्न विचारत आहात की एक किंवा दोन किंवा pi किंवा तुमच्याकडे कोणतीही संख्या असेल. आणि असे उत्तर मिळणार नाही म्हणून तुम्ही ओह होय, शून्याचा लॉग म्हणण्याचा प्रयत्न करू शकता, हे एक पूर्णपणे वैध कार्य आहे ते फक्त इनपुट शून्यावर परिभाषित केले आहे परंतु तरीही तुम्हाला काय हवे आहे ते पूर्ण करण्याचा प्रयत्न करताना तुम्हाला त्रास होईल. तेथे कारण शून्याला शून्य म्हणणे हे कोणत्याही गोष्टीला लागू होते, त्यामुळे तुमचा हात तुमच्या पाठीमागे फिरवला जाईल, तथापि तुम्हाला ते काम करायचे आहे आणि ते बेस शून्यासह घातांकीय कार्य पूर्णपणे शून्य आहे या वस्तुस्थितीशी जुळते. एकमेकांवर छानपैकी एक-एक पद्धतीने संख्या मॅप करत नाही, त्यामुळे हा एक चांगला प्रश्न आहे, या गोष्टी वास्तविक जगात कोठे येतात या कल्पनेवर आता तुमच्याकडे लॉग बेस शून्य असू शकतो का मला एक प्रकारचं उदाहरण आवडलं. भूकंपासाठी रिश्टर स्केल त्यामुळे रिश्टर स्केल आपल्याला भूकंप किती तीव्र आहे याचे प्रमाण देते आणि ते अगदी लहान संख्येपासून ते खूप मोठ्या संख्येपर्यंत काहीही असू शकते जसे की मला वाटते की आतापर्यंतचा सर्वात मोठा भूकंप आहे आणि हा फक्त एक चार्ट आहे विकिपीडिया 9 होता. 5 आणि किती वेडेपणाचे आहे याचे कौतुक करण्यासाठी या संख्यांचा अर्थ काय आहे आणि नंतर TNT च्या समतुल्य रकमेमध्ये किती ऊर्जा आहे याचे काही प्रकार आणि मग आपण येथे काय करण्याचा प्रयत्न करू शकतो यामधील संबंध पाहण्यासारखे आहे. आपल्याला उर्जेच्या प्रमाणात रिश्टर स्केल क्रमांकासाठी अभिव्यक्ती मिळू शकते का हे पाहणे आणि लॉगरिदम हे वर्णन करण्याचा एक नैसर्गिक मार्ग का आहे, त्यामुळे गोष्टी किती वाढतात यावर लक्ष केंद्रित करणे महत्त्वाचे आहे उदाहरणार्थ, जर आपण या प्रकरणात दोन विहिरीवरून गेलो तर ते आपल्याला दर्शवत नाही की तीन कुठे आहेत त्यामुळे कदाचित आपण दोन ते चार पर्यंत एक पाऊल उचलण्याचा विचार करतो जे दोन पावले उचलण्यासारखे आहे ते काय करते एक मेट्रिक टन टीएनटी मधून आपल्याला ऊर्जा मिळते असे दिसते, जे मला वाटते की दुसऱ्या महायुद्धातील एक मोठा बॉम्ब आहे आणि तो आपल्याला एका लहान अणू बॉम्बपेक्षा हजार पटीने किलोटनपर्यंत नेतो, त्यामुळे फक्त दोन पावले रिश्टर स्केलवर 2 तीव्रतेच्या भूकंपापासून 4 तीव्रतेच्या भूकंपाकडे जाताना आपल्याला मोठ्या बॉम्बपासून दुसऱ्या महायुद्धापासून आण्विक युगापर्यंत नेले जाते जेणेकरुन ते लक्षात घेण्याजोगे आहे आणि आपल्याला मिळालेली पहिली स्वच्छ पायरी 4 ते 5 वाजता आहे. कमीत कमी हा तक्ता आपल्याला छान दाखवत आहे आणि स्पष्टपणे 4 ते 5 ची एक पायरी 1 किलोटन ते 32 किलोटन पर्यंत जाण्याशी संबंधित आहे आणि हे स्पष्टपणे नागासाकीवर उतरलेल्या शहराचा नाश करणार्‍या बॉम्बचा आकार होता, त्यामुळे हे कदाचित एक असेल. जर तुम्ही बातम्यांमध्ये ऐकत असाल तर लॉगॅरिदमिक स्केल बद्दल विरोधाभासी असू शकते अरे 4 चा भूकंप झाला होता. 0 विरुद्ध भूकंप जो 5 होता. 0 हा विचार करणे सोपे आहे होय 4 आणि 5 या अगदी समान संख्या आहेत परंतु स्पष्टपणे TNT रकमेच्या संदर्भात जे 32 ने गुणाकार करण्याशी संबंधित आहे 1 वरून पुढील आणि 2 ते 4 वर जाणे हे स्पष्टपणे एक हजाराने गुणाकार होते आणि फक्त याचे कारण मोठे आहे कारण येथे आमचा चार्ट 3 म्हणजे काय हे दाखवत नव्हता म्हणून आम्ही दोन पावले टाकत होतो आणि तुम्ही स्वतःच हे सत्यापित करू शकता की जर तुम्ही 32 चे एक पाऊल उचलले आणि नंतर तुम्ही दुसर्या 32 ने गुणाकार केला तर ते एक हजाराच्या जवळपास आहे. रिश्टर क्रमांकावरील अॅडिटीव्ह पायऱ्या TNT मधील गुणाकार पायऱ्यांशी संबंधित आहेत ही कल्पना सूचित करते की येथे काहीतरी लॉगरिदमिक आहे आणि फक्त येथे जात राहणे थोडे मनोरंजक आहे आणि हे सांगणे थोडे मनोरंजक आहे की जागतिक घटनांमुळे हे अंशतः किती वाढते आहे. होय हे वर्णन करणे फार मोठे आश्चर्य नाही की आपण दुसरे पाऊल टाकल्यावर ते पुन्हा 32 ने गुणाकार करत आहे परंतु आपल्या अंतर्ज्ञानानुसार 32 किलोटन एक छोटा अणुबॉम्ब आणि नंतर एक मेगाटन यातील फरक आहे ज्याला आपण लहान अणुबॉम्ब नाही असे समजू शकतो, नागासाकी अणुबॉम्ब जो एका मेगाटनसाठी नागासाकी अणुबॉम्बपैकी 32 अणुबॉम्ब आहे जो स्पष्टपणे नेवाडा यूएसए 1994 मध्ये दुहेरी स्ट्रिंग फ्लॅट भूकंपाची तीव्रता आहे हे मला माहित नव्हते, ते काय होते, विकिपीडिया धन्यवाद. हे देखील स्पष्टपणे पाहिले जे दोन पेक्षा कमी आहेत, ते सर्व वेळ घडतात त्यापैकी दररोज 8000 सारखे असतात परंतु जसे आपण अणुबॉम्बच्या क्षेत्रात आहोत 3 सारख्या गोष्टी. 5 आणि 4 ते देखील स्पष्टपणे पृथ्वीवर कुठेतरी वारंवार घडतात, त्यापैकी सुमारे 134 दररोज कुठेतरी घडत असतात कोणाला माहित आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "पण अणुबॉम्बच्या प्रमाणापेक्षा जास्त असलेल्या या 5 आणि 6 श्रेणीमध्ये आपण आणखी तीव्र होत गेलो आहोत आणि आता आपण दररोज फक्त 2 च्या आसपास आहोत आणि मला खात्री आहे की भूगर्भशास्त्रज्ञ येऊन आपण सर्वांनी असे का केले पाहिजे हे स्पष्ट केले पाहिजे. पृथ्वीच्या कवचाला दोन अणुबॉम्ब समतुल्य व्यत्यय दररोज घडत आहेत या वस्तुस्थितीबद्दल जास्त काळजी करू नका परंतु बहुधा अशा शहरासारख्या एखाद्या ठिकाणी लक्ष केंद्रित करणे विशेषतः दुर्मिळ आहे जिथे बरेच लोक राहतात आता फक्त प्रत्येक पाऊल आपल्या विचाराची पडताळणी करत आहे. 32 च्या वाढीचा समावेश आहे चला 6 ते 7 पर्यंतची पायरी कशी दिसते ते पाहूया आणि येथे ते आपल्याला या दरम्यान आणखी बरीच उदाहरणे देत आहे कदाचित हा भ्रम देत आहे की ती प्रत्यक्षात आहे त्यापेक्षा मोठी पायरी आहे आणि खरंच हाच फरक आहे 1 मेगाटन आणि 32 मेगाटन म्हणजे 32 ने गुणाकार करणे म्हणजे या चार्टवर मला सर्वात मनोरंजक वाटलेली एक गोष्ट म्हणजे आतापर्यंत चाचणी झालेल्या सर्वात मोठ्या अण्वस्त्रापर्यंत पोहोचण्याआधी आपल्याला किती लांब जावे लागेल हे पाहणे ही शीतयुद्धाची उंची होती. झार बॉम्ब जो 50 मेगाटनचा होता आणि मला विश्वास आहे की 100 मेगाटन बॉम्ब ठेवण्याची त्यांची मूळ योजना होती परंतु त्यांनी त्या 50 मेगाटन बॉम्ब वरून खाली बोलले, आम्ही त्या 32 किलोटन बॉम्बच्या 32 ने गुणाकार करण्यासाठी 32 ने गुणाकार करतो. मेगाटन दुसर्‍या 32 ने गुणाकार करतो म्हणून आम्ही दुसर्‍या महायुद्धाच्या समाप्तीच्या स्फोटाच्या एक हजार पट शक्तीबद्दल बोलत आहोत आणि आपण अद्याप मानवतेच्या क्षमतेच्या 50 मेगाटनवर नाही आहोत आणि हे स्पष्टपणे इंडोनेशियातील जावा भूकंप आहे. . 0 हे 6 पेक्षा थोडेसे मोठे नाही. 0, हे खूप मोठे आहे आणि इथे मुद्दा फक्त इतकाच आहे की जेव्हा तुमच्याकडे गुणाकार वाढवणारे स्केल असते तेव्हा हे कौतुक करण्यासारखे आहे की लहान पायऱ्यांसारखे दिसणारे हे प्रत्यक्षात निहित उर्जेच्या किंवा निरपेक्ष मूल्यांच्या दृष्टीने खूप मोठे पाऊल असू शकतात. म्हणून जेव्हा आपण या वस्तुस्थितीबद्दल विचार करत असतो की 9 होते. 5 हे केवळ 7 मध्‍ये आहे हे दिलेल्‍याने त्‍याला अवास्तव वाटते. 0 श्रेणी जी आम्ही आतापर्यंतच्या सर्वात मोठ्या थर्मोन्यूक्लियर शस्त्राविषयी बोलत आहोत आणि हे एका क्षेत्राचे सूचक आहे जिथे लॉगरिदम येतात तेव्हा मानवांना एखाद्या गोष्टीसाठी स्केल तयार करायचा असतो ज्यामध्ये मोठ्या गोष्टी किती मोठ्या प्रमाणात बदलू शकतात. भूकंपाच्या आकाराच्या बाबतीत, पृथ्वीभोवती सतत घडणाऱ्या गोष्टी, मोठ्या हँड ग्रेनेडचा आकार आणि तुम्हाला ते तुमच्या स्केलवर हवे असते आणि त्या सर्व गोष्टींवर विचार करण्यासारखे काहीतरी असू शकते. मानवी इतिहासात आपण पाहिलेला सर्वात मोठा व्यत्यय आणि तो अशा प्रकारे होण्यासाठी की आपण फक्त एका प्रकरणासाठी आपल्या संख्येमध्ये भिन्न अंकांचा संपूर्ण गुच्छ लिहित नाही आणि संपूर्ण गुच्छ भिन्न, लहान संख्येसाठी दुसर्‍या प्रकरणात तुमच्या नंबरसाठी अंकांचे लॉगरिदम घेणे चांगले आहे आणि नंतर ते एका स्केलवर ठेवा जे मुळात 0 आणि 10 च्या दरम्यान त्या संख्येला स्क्वेश करते, तुम्हाला संगीतासाठी डेसिबल स्केलसह काहीतरी समान चाललेले दिसते जे खरोखर थोडेसे कार्य करते. जरा वेगळ्या पद्धतीने जिथे तुम्ही प्रत्येक वेळी 10 डेसिबलची एक पायरी चढता जी 10 ने गुणाकार करण्याशी संबंधित असेल तर 1 ची पायरी 10 ने गुणाकार करण्यापेक्षा, ती 10 ची पायरी आहे जी 10 ने गुणाकार करते जेणेकरून अशा प्रकारचे गणित थोडेसे बनते जरा विचित्र पण कल्पना सारखीच आहे, जर तुम्ही ५० डेसिबल विरुद्ध ६० डेसिबल असा आवाज ऐकत असाल तर तो ऊर्जा प्रसारित होण्याच्या आणि त्यातून जाणार्‍याच्या दृष्टीने खूपच शांत आहे, तो काय असेल, ६० ते ७० किंवा ७० ते 80 त्या पायऱ्या, 60 ते 80 पर्यंत, ज्यामध्ये प्रति चौरस क्षेत्रफळातील ऊर्जेचे प्रमाण 100 च्या घटकाने गुणाकार करणे समाविष्ट आहे, म्हणून प्रत्येक वेळी जेव्हा तुम्ही लॉगरिदमिक स्केल पहाल तेव्हा तुमच्या मनात हे जाणून घ्या की त्याचा अर्थ जे काही आहे ते हुडच्या खाली वाढते. हीच खूप मोठी रक्कम आहे का आम्ही कोरोनाव्हायरसच्या उद्रेकाचे वर्णन करण्यासाठी वापरल्या जाणार्‍या अनेक लॉगरिदमिक स्केल पाहिल्या, त्यामुळे तुम्ही अशा नातेसंबंधाचे वर्णन कसे करू शकता जिथे तुम्ही प्रत्येक वेळी रिश्टर स्केल संख्या 1 ने वाढवता तेव्हा तुम्ही 32 ने गुणाकार करता, आम्ही बेस 32 सह लॉगच्या संदर्भात विचार करू शकतो, जर मी लॉग घेतला तर मी म्हणू शकतो की मी फक्त r ला कॉल करणार आहे, रिश्टर स्केलसाठी मी याला लॉग बेस 32 मानू शकतो आणि तो त्याच्याशी संबंधित असेल , नाही नाही नाही, मी हे चुकीचे करत आहे ती लॉग केलेली गोष्ट नाही आम्ही मोठ्या संख्येचा लॉग बेस 32 घेतो, टीएमटी नंबरचा, काहीतरी जे 1 मेगाटन सारखे होते ते 1 दशलक्ष टन लॉग बेस 32, ते पाहिजे रिश्टर स्केल क्रमांकाशी संबंधित आहे परंतु काही प्रकारचे ऑफसेट असू शकते, म्हणून आम्ही असे म्हणू शकतो की या रिश्टर स्केल क्रमांकामध्ये काही प्रकारचे स्थिरांक जोडत आहोत आणि ही अभिव्यक्ती अगदी सारखीच आहे, मला माफ करा खाली ही अभिव्यक्ती काही ऑफसेट वेळेच्या पॉवरला 32 म्हणण्यासारखीच आहे, जी आमच्या रिश्टर स्केल नंबरला 32 घेण्यासारखे आहे, जे स्वतःच काही मोठे स्थिरांक आहे, रिश्टर स्केल क्रमांकाला 32 गुणाकार आहे. आपण पहात असलेल्या संख्येच्या पॉवरच्या 32 गुणानुरूप काही स्थिर गुणाप्रमाणे याचा विचार केला जाऊ शकतो, म्हणून हे लिहिण्याचा हा मार्ग खरोखरच त्याच्या घातांकीय वाढीवर भर देतो की जर हे आपण पहात असलेल्या TMT रकमेशी संबंधित असेल, जसे आपण ते वाढवत आहात. r स्टेप बाय स्टेप तुम्ही 32 ने गुणाकार करत आहात पण नेमकी हीच वस्तुस्थिती कळवण्याचा आणखी एक मार्ग म्हणजे त्या रकमेचा लॉग बेस 32 घ्यायचा आहे. वेगवेगळ्या बेसच्या लॉगची गणना कशी करायची याबद्दल काळजी करा हे थोडे विचित्र आहे की आपण लॉग बेस 32 बद्दल बोलत होतो, मी याआधी उल्लेख केला आहे की गणितज्ञांना बेससह लॉग असणे कसे आवडते आणि संगणक शास्त्रज्ञांना बेस 2 सह लॉग असणे खरोखर आवडते आणि ते संगणकीय हेतूंसाठी किंवा तुमच्याकडे एक लॉग असल्यास या गोष्टी कशा वाढतात याचा विचार करण्यासाठी, जर तुम्ही एका प्रकारच्या लॉगची गणना करू शकत असाल, मग ते बेस 10, बेस 2, बेस आणि तुम्ही इतर कोणत्याही गोष्टीची गणना करू शकता. तुम्हाला आता आमचे अंतर्ज्ञान त्या दिशेने मिळवायचे आहे, चला आमच्या प्रश्नमंजुषाकडे वळू आणि पुढील प्रश्नाकडे जाऊ आणि मला विश्वास आहे की हा प्रश्न सर्वात जास्त आहे, मला माहित नाही, हा अर्धवट वाजवी प्रश्न आहे, हा छान असावा हे आपल्याला बेस 2 संदर्भातून बेस 10 संदर्भामध्ये भाषांतरित करण्यास तयार करणार आहे आणि 2 च्या शक्तींना 10 च्या शक्तींशी असलेले संबंध समजून घेण्यासाठी देखील एक चांगली अंतर्ज्ञान आहे कारण हा एक सुंदर प्रकारचा योगायोग आहे. निसर्ग की या दोन प्रकारच्या चांगल्याप्रकारे तुम्हाला मला काय म्हणायचे आहे ते दिसेल, ते एकमेकांशी छान खेळतात म्हणून आमचा प्रश्न विचारतो की, 2 ते 10 वी 1024, 1024 आहे, जे अंदाजे 1000 आहे, जर तुम्ही एक असाल तर तुमची संख्या थोडी कमी आहे आणि तुम्ही फक्त 2 ते 10 वी, मुळात 1000 अंदाजे बनवत आहात, खालीलपैकी कोणते सत्य असण्याच्या सर्वात जवळ आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "निविदा येथे सर्वानुमते निर्णय नाही. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "पण प्रश्न विचारत होता की कोणते सत्य असण्याच्या सर्वात जवळ आहे आणि आपण याचा विचार कसा करू शकतो ते पाहूया. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "तर हे सूचित करते की तुमच्याकडे 2 ची पॉवर आहे, जी 1024 आहे, 10 च्या पॉवरच्या अगदी जवळ आहे, सुमारे 10 घन. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "मग याचा अर्थ काय? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "जर 10 चा लॉग बेस 2 हा x च्या बरोबरीचा असेल, तर 2 ला x 10 च्या बरोबरीचे म्हणण्यासारखेच आहे, बरोबर? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "ते आम्हाला 2 ते 10 च्या बरोबरीचे विचारत आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "तुम्ही प्रत्येक फंक्शनसह असे करू शकत नाही. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "लोकांना असे वाटते की तुम्ही ते कोणत्याही फंक्शनसह करू शकता, परंतु तुम्ही करू शकत नाही. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "आणि याचा अर्थ काय आहे की x सुमारे 10 तृतीयांश आहे, ठीक आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "आणि पुरेसे आहे, आपण आधी पाहिलेला लॉग बेस 2 पैकी 10, आपण असेही म्हणू शकतो की 2 पैकी लॉग बेस 10 त्या रकमेपेक्षा फक्त 1 आहे, x पेक्षा 1. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "आणि आम्ही लॉगमध्ये गोष्टी करत असल्यामुळे मी ते अशा प्रकारे लिहित आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "त्याचप्रमाणे दशलक्षचा लॉग बेस 2, बरं पाहू, हजारापर्यंत पोहोचण्यासाठी 2 ला स्वतःहून 10 पटीने गुणाकार करावा लागतो, तर एक दशलक्षपर्यंत जाण्यासाठी आपल्याला 20 पटीने गुणाकार करावा लागेल. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "हे थोडेसे लहान आहे परंतु तुमच्या मनात हे एक चांगले अंदाज आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20, आम्ही त्याच रकमेने कमी करतो. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, आम्ही त्याच रकमेने कमी करतो. ठीक आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "आता हे लक्षात ठेवण्यासारखे एक अंतर्ज्ञान आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "ठीक आहे? ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "आणि नंतर लॉग बेस C च्या B वेळा लॉग बेस C च्या A चे संयोजन करण्यासाठी विविध संभाव्य मार्गांचा एक संपूर्ण ढीग. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "मी तुम्हाला यावर एक अर्थपूर्ण वेळ देईन कारण जोपर्यंत तुम्ही लॉगरिदमशी परिचित नसता तोपर्यंत हे स्पष्ट नाही आणि थोडासा विचार करणे योग्य आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/persian/sentence_translations.json b/2020/ldm-logarithms/persian/sentence_translations.json index daef854d6..882fc901a 100644 --- a/2020/ldm-logarithms/persian/sentence_translations.json +++ b/2020/ldm-logarithms/persian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵موسیقی🎵 به Lockdown Math خوش آمدید. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "امروز قرار است در مورد لگاریتم ها و نوعی بازگشت به اصول اولیه صحبت کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "و مثل همیشه، برای شروع کار، فقط می‌خواهم درک کنم که مخاطب در حال حاضر در کجاست. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "بنابراین، اگر می توانید به 3b1b بروید. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "من قبلاً در مورد آنها چیزی نشنیده ام یا قبلاً هرگز در مورد آنها چیزی نشنیده ام. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "من در مورد آنها یاد گرفته ام، اما گاهی اوقات با همه ویژگی ها گیج می شوم. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "من آنها را درک می کنم اما نمی دانم چگونه به آنها آموزش دهم و d. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "من آنها را به خوبی درک می کنم و می توانم به راحتی آنها را به دیگری آموزش دهم تا آنها را نیز به خوبی درک کنند. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "بنابراین، ما یک تقسیم خوب داریم. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "همانطور که گفتم، هدف از این کار ایجاد درسی است که بتوانم در آینده به مردم اشاره کنم، اگر آنها با لگاریتم راحت نیستند و می‌خواهم بتوانم بگویم، اوه، اینجا جایی است که می‌توانید به آن بروید. چگونه فکر می کنم، می دانید، من فکر می کنم چگونه می توانید به طور شهودی به آن نزدیک شوید. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "از آنجایی که من قبل از انجام این سخنرانی خاص در چند تالار گفتمان معلمان می چرخیدم و وقتی مردم می پرسند سخت ترین مبحث برای تدریس در ریاضی دبیرستان چیست به این معنا که به نظر می رسد دانش آموزان بیشترین مشکل را با آن دارند، لگاریتم یکی از مهمترین آنهاست. پاسخ‌هایی که معمولاً نشان داده می‌شوند جالب است و من می‌توانم حدس بزنم شاید به این دلیل است که تعداد زیادی از این ویژگی‌ها وجود دارد که در نهایت باید بدانید که می‌دانید، بنابراین اگر از جایی که قرار است برویم بگذریم، همه این انبوهی از آنها را خواهید داشت. قوانینی که فقط شبیه یک دسته جبر به نظر می رسند که به خاطر سپردن آنها سخت است و به راحتی می توان چیزها را در ذهن تان قاطی کرد و من فکر می کنم وقتی مردم، می دانید، این نوع خاطرات کابوس وار از اینکه ریاضی دبیرستان چگونه بوده و چگونه بوده است، دارند. لگاریتم‌ها برای آن‌ها انجام می‌دهند، اغلب آن فرمول‌های خاصی به ذهنم می‌آیند و کاری که من امروز می‌خواهم انجام دهم این است که سعی کنم از طریق یکی صحبت کنم، چگونه در مورد آنها فکر کنم، اما همچنین فقط در سطح متا اگر به کسی جبر را آموزش می‌دهید، چه چیزی وجود دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "این افزایش‌های 1 را می‌توان به این صورت در نظر گرفت که می‌گوییم، هوم، وقتی عددی را به لاگ وصل می‌کنم، اگر آن عدد به‌صورت توان 10 باشد، من فقط تعداد صفرها را می‌شمارم که لگ 1000 چیست؟ اوه، 3 صفر روی آن وجود دارد که رقم یک میلیون چیست؟ اوه، 6 صفر روی آن است، این تابع شمارش صفر است و من فکر می‌کنم شما می‌توانید از نظر درک ویژگی‌های مختلف لگاریتم‌ها، فقط با فکر کردن در پشت ذهنتان، خیلی به خودتان دست پیدا کنید، خوب، وقتی توان 10 را وصل کنم. ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "log 1000 برابر x برابر است با 3 برابر log x و به یاد داشته باشید که ما از این قرارداد استفاده می کنیم که پایه 10 log b است. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "log 1000 برابر x برابر است با لگ x مکعب c. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "log 1000 برابر x برابر 3 به توان log x و e است. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "هیچ یک از موارد بالا را ندارید و به یاد داشته باشید همانطور که قبلاً گفتم ما باید کاملاً انتظار داشته باشیم که همه آن افرادی که در ابتدا گفتند که گزارش ها را به خوبی درک می کنند بلافاصله پاسخ خواهند داد، آنها به درستی پاسخ خواهند داد، اما اگر شما کسی که این کار را نمی کند، اجازه ندهید وقتی به مشکلی مانند این نگاه می کنید، شما را بترساند، آنچه من شما را تشویق می کنم انجام دهید این است که فقط قدرت های مختلف 10 را وصل کنید و به این ایده فکر کنید که عملکرد log تعداد صفرها را می‌شمارد، بنابراین من کمی به شما فرصت می‌دهم تا در مورد آن فکر کنید، بنابراین ادامه می‌دهم و آن را درجه‌بندی می‌کنم و مثل همیشه اگر سریع‌تر از چیزی است که شما با آن راحت هستید، بدانید که این فقط به این دلیل است که می‌خواهم به جلو ادامه دهم. با درس، بنابراین در این مورد پاسخ صحیح به صورت log 1000 برابر x برابر است با گرفتن 3 به اضافه log x و حالا بیایید یک لحظه به آن فکر کنیم و همانطور که گفتم وقتی تازه شروع کرده اید. با آنها فکر می کنم بهترین کار این است که به راحتی اعداد مختلف را به برق وصل کنید و بهترین اعداد برای وصل اعدادی هستند که در حال حاضر توان های 10 دارند، بنابراین اگر چیزی مانند log 1000 برابر x خوب می خواهید من نمی خواهم. نمی دانیم، بیایید فقط چیزی را برای x log 1000 ضربدر 100 وصل کنیم، خوب می دانیم که در پاسخ نهایی چند صفر خواهد بود، خوب 1000 ضربدر 100 می شود 100000، ما به طور شهودی این ایده را داریم که وقتی 2 توان 10 را ضرب می کنیم ما فقط صفرها را می گیریم، 3 صفر از آن 1000، 2 صفر از آن 100 و آنها را در کنار یکدیگر قرار می دهیم، بنابراین باید 5 صفر در مجموع باشد، اما اگر واقعاً فقط به این فکر نکنید که این عدد چگونه تغییر کرده است. اما چرا اینطور شد که 3 صفر از آن 1000 به اضافه 2 صفر از آن 100 بود که می توانیم با گفتن تعداد صفر در 1000 به اضافه تعداد صفرها در 100 بنویسیم، بنابراین این ایده که یک لگاریتم است. حاصلضرب دو چیز، مجموع لگاریتم های آن دو چیز در زمینه توان های 10 است که اگر 2 توان 10 را بگیرید و آنها را ضرب کنید، فقط چیزی را که در حال حاضر برای بسیاری از ما یک ایده فوق العاده شهودی است، به شما منتقل می کند. تمام صفرهای آنها را بگیرید و به نوعی آنها را روی یکدیگر بچسبانید، بنابراین روشی که من در اینجا نوشتم، در واقع نشان دهنده یک واقعیت کلی تر است که اولین خاصیت لگاریتم ما خواهد بود، این است که اگر log از A ضربدر B برابر است با Log A به اضافه Log B اکنون هر زمان که یکی از این قوانین لگاریتمی را مشاهده کردید، اگر دیدید که چشمانتان را چروک می کنید یا کمی گیج شده اید که چگونه آن را به خاطر بسپارید، فقط مثال هایی را وارد کنید. من زیاده‌روی می‌کنم، این را زیاد می‌گویم، اما به این دلیل است که فکر می‌کنم وقتی در خود جبر غوطه‌ور شده‌اید و در یک نوع آزمون نشسته‌اید و نمادهای زیادی دارد فراموش کردن بسیار آسان است. برای اینکه به خود یادآوری کنید نمی‌توانید فقط برخی از اعداد را وصل کنید که کار خوبی است و اغلب این یک راه عالی برای به دست آوردن شهود است، بنابراین در این مورد، گفتن Log از A ضربدر B و جدا کردن آن، می‌توانیم فکر کنیم، اوه، که لاگ 100 ضربدر 1000 که 5 می شود، 5 صفر در آن وجود دارد که از نظر تعداد صفر در هر قسمت داده شده، بسیار عالی، فوق العاده است، بنابراین با ادامه این شهود، اجازه دهید یک مشکل تمرین دیگری را امتحان کنیم و دوباره، اگر می دانید، عالی است. شما می توانید به خوبی به آن پاسخ دهید، اما شاید فکر کنید، نه فقط پاسخ چیست، بلکه چگونه می توانم این پاسخ را برای کسی توضیح دهم یا چگونه می توانم تلاش کنم دانش آموزی به تنهایی به این پاسخ برسد بدون اینکه من به آن بگویم. پاسخ آنها چیست، بنابراین دو مخاطب بالقوه وجود دارد، کسانی که به خود درس علاقه مند هستند و سپس کسانی که به درس متا علاقه مند هستند، بنابراین سؤال ما دوباره می پرسد، کدام یک از موارد زیر درست است؟ آ. ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "log x به n برابر است با n ضرب log x b. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "log x به n برابر است با log x به توان n c. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "log x به n برابر است با n به اضافه log x یا d. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "بنابراین، پاسخ صحیح در اینجا a است، که به نظر می رسد 4000 نفر از شما تبریک گفته اند، به ما می گویند که log از x به توان n برابر است با n ضربدر log x، بنابراین، دوباره، بیایید بگوییم که شما سعی دارید این را آموزش دهید. به کسی یا اگر می‌خواهید به معنای آن دست پیدا کنید، فکر می‌کنم نقطه خوبی برای شروع این است که چیزی را به برق وصل کنید و در این مورد، برای ورود x به پاور n، اجازه دهید آن را با 100 به پاور امتحان کنیم. 3 و می‌توانید آن را با سایرین امتحان کنید تا ببینید آیا الگوهایی که انجام می‌دهید واقعاً کار می‌کنند یا خیر، اما آیا به این فکر می‌کنید که صرفاً ببینید چه پاسخی چیست، بلکه سعی می‌کنید به این فکر کنید که چرا پاسخ به این شکل بوده است. گاهی اوقات یک مثال این کار را انجام می دهد زیرا 100 مکعب، ما می توانیم آن را به عنوان گرفتن خوب در نظر بگیریم، این 3 کپی از 100 است، من 3 کپی از 100 را می گیرم و وقتی همه آن را ضرب می کنم و فکر می کنم وارد حساب کاربری می شود که تعداد صفرهایی را که می شماریم. بگو، اوه، این عددی خواهد بود که فقط 6 صفر روی آن باشد، این یعنی 100 ضربدر 100 ضربدر 100، من فقط می توانم به این فکر کنم که همه آن صفرها را با هم گروه کنم تا یک میلیون بدست بیاورم، بنابراین این عدد می شود 6 اما اگر واقعاً فکر کنیم که چرا 6 بود نه فقط این تعداد صفرهای درون میلیونی است که آن 6 از آنجا آمده است، این است که ما 3 کپی از آن 100 داشتیم و هر یک از آن 100 دارای 2 صفر متفاوت بود، بنابراین این یک عدد کلی تر است. می توانید در مورد آن فکر کنید که اگر به جای 100 مکعب به 1000 مکعب یا 1000 به n یا x به توان n نگاه می کردیم، می توانید فکر کنید که هر مقداری است که آن مقدار n تعداد کپی هایی است که در ضرب ضرب می کنیم. تعداد چاه، بیایید ببینیم، این x برابر تعداد صفرهای موجود در هر چیزی نیست که ما جایگزین x که در این مورد 100 بود، بنابراین اگر به جای آن چیزی شبیه log 10000 را به توان n می گرفتم، این یکسان بود. مانند گرفتن n کپی از آن 10000 با شمردن تعداد صفرهای هر یک از آنها که 4 است، بنابراین n ضربدر 4 می شود و البته ویژگی کلی که اکثر شما به درستی پاسخ دادید این است که شما این اثر کوچک دوست داشتنی را دارید که در زمانی که سیاهه چیزی را ببینید که به قدرتی رسیده است که قدرت کمی از جلوی آن پایین می‌آید و شما فقط گزارشی از آنچه در داخل وجود داشت دارید، اکنون یکی از شاید مهم‌ترین پیامدهای آن است که نمی‌دانم اسمش را بگذارید یا نه یک مفهوم یا اگر شما آن را بیان مجدد تعریف بنامید اگر من log را می گیرم و دوباره تاکید می کنم که پایه 10 از 10 به توان n است، ما می توانیم آن n کوچک را به عنوان پایین پریدن در نظر بگیریم. جلو و n ضربدر پایه log 10 از 10 می شود که البته 1 است این عبارت شما می توانید به عنوان شمارش تعداد صفرها در انتها یا به طور کلی تر از 10 بپرسید تا برابر 10 شود و پاسخ به سادگی 1 است. که بسیار اطمینان‌بخش است زیرا راه دیگری که می‌توانید به عقب برگردید و فقط این عبارت اصلی را بخوانید این است که بگویید 10 به 10 برابر n اوه خوب، اکنون با هر خاصیت لگاریتمی که داریم، پاسخ n ok است، بنابراین در این مورد ما فقط یک گزارش از x به توان n پیدا کردم که شامل این است که n پریدن از جلو همیشه یک ویژگی نمایی تصویر آینه ای وجود خواهد داشت و این راه دیگری است که می توانیم به خودمان کمک کنیم تا کمی شهود برای این موارد بدست آوریم، بنابراین اجازه دهید من فقط آن را بپوشانم. برخی از ویژگی‌های آینده‌ای که قرار است به آن‌ها برسیم، سعی می‌کنند جایی را که می‌رویم پنهان کنیم. این کل چیز به توان n برابر است با گرفتن 10 به n ضربدر x و این ما را به شهود دیگری می‌رساند که ممکن است برای لگاریتم‌ها داشته باشید، این است که آنها به نوعی مانند توانی هستند که از داخل به بیرون تبدیل شده‌اند و در اینجا منظور من از لگاریتم است. که چیزی که در داخل لاگ نشسته است، اگر من log از a را بگیرم، باید آن را به عنوان کل عبارت بیرونی برای چیزی که نمایی است در نظر بگیرید، در این مورد a چیزی در داخل با 10 به x مطابقت دارد. خروجی تابع در حالی که کل چیز خود گزارش a مطابق با آنچه در داخل است در اینجا دقیقاً برابر است با نمایی از 10، بنابراین هر جا که یک عبارت log را در اینجا می بینید باید فکر کنید که نقش یک توان در سمت راست را بازی می کند. طرف و هر بار که شما یک نمایی کل 10 تا عبارت x می بینید کل جزء بیرونی در سمت راست که مربوط به چیزی است که در داخل یکی از لاگ ها قرار دارد و ما این ایده را در بالا دیدیم که وقتی در حال ضرب هستیم در داخل که در حال جمع کردن در بیرون چاه است، اگر لاگ ها به نوعی نمایی را به داخل می چرخانند، به ما می گوید که ضرب در خارج، ضرب خروجی های تابع مانند جمع کردن در داخل است، زیرا هر یک از این گزارش ها مانند log a و log b هستند. در حال ایفای نقش x و y در عبارت سمت راست است، پس بیایید به بازی ادامه دهیم، بیایید چند مورد دیگر را انجام دهیم و ببینیم چند مورد از این ویژگی‌ها را می‌توانیم برای این مورد آخر شهود ایجاد کنیم، فکر کردن بسیار خوب به پرش نماهای بعدی چیزی است که ممکن است برای کسانی که لزوماً با لگاریتم آشنا نیستند کمی عجیب به نظر برسد، اما دوباره، برخی از اعداد را وصل کنید تا شهودی برای آن به دست آورید و ما کمی آن را ارائه خواهیم کرد. ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "و اینکه 3 مربوط به پایه log 10 از 1000 است، 1 تقسیم بر پایه log 10 از 1000 است بنابراین به طور کلی تر، ممکن است بر اساس همین مثال حدس بزنید که وقتی پایه را با آنچه در داخل است عوض می کنیم، با تقسیم 1 مطابقت دارد. با توجه به آنچه در بیرون وجود دارد و دوباره، می توانید این را از نظر نگاه کردن به قاعده نمایی مربوطه در نظر بگیرید، حالا چه اتفاقی برای سیاهه کوچک و نمایی های دوست داشتنی من افتاد؟ فوق العاده است، دوباره اجازه دهید برخی از چیزها را پنهان کنیم که برخی از ویژگی های دیگر را به اینجا خواهیم رساند و من آن را به همان ترتیبی که قبلاً در اینجا داشتم نگه می دارم. ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "داشتم فکر می کردم که داشتن آن از قبل می تواند مرا نگه دارد. کمی تمیزتر از حد معمول است، اما شاید فقط شامل انجام این بازی عجیب و غریب برش کاغذ باشد، بنابراین چیزی که ما پیدا کردیم، اگر آنها را عوض کنید، پایه b از a را وارد کنید، مانند تقسیم بر 1 چیزی است که با آن مطابقت دارد. زمین نمایی این است که اگر b را به مقداری توان بگیرید و بگویید که برابر با a است، همان عبارتی است که بگوییم a به معکوس آن توان دوباره برابر b است، به نوعی مفید است که لحظه ای وقت بگذارید و لگاریتم ها را به عنوان چرخش اشیا در نظر بگیرید. از داخل عبارت log پایه b از a نقش آن x را بازی می کند و عبارت log پایه a از b نقش هر چیزی را بازی می کند که بالای a قرار می گیرد و سپس به طور متقارن، کل عبارت b به توان x در حال بازی است. نقش داخل در سمت چپ، نقش a و کل عبارت را بازی می‌کند، a به قدرت چیزی نقش چیزی را بازی می‌کند که درون پایه سیاهه‌ای a نشسته است، بنابراین می‌توانید ببینید، فقط با وصل کردن چند مثال و با تطبیق آن با قوانین نمایی، می‌توانیم از طریق سه قانون لگاریتمی مختلف فکر کنیم که اگر آنها فقط به عنوان قطعات جبر برای به خاطر سپردن تحویل داده می‌شدند، می‌توانید آنها را حفظ کنید، اما برای آنها بسیار آسان است که به نوعی از شما خارج شوند. سر و کار بسیار آسانی است که از کار در دست ناامید شوید، اما ممکن است بخواهید به خود یادآوری کنید که دلیل اهمیت ما به این چیزها این است که درک قوانین لگاریتم به ما کمک می کند تا در زمینه هایی که مانند ویروس رشد می کند، ریاضیات را انجام دهیم. از یک روز به روز دیگر، از یک مرحله به مرحله دیگر، همه چیز به طور مضاعف رشد می کند. درک قوانین لگاریتم به شما کمک می کند تا احساس بهتری نسبت به این نوع چیزها داشته باشید، بنابراین قبل از اینکه ما یک مثال خوب در دنیای واقعی از آنچه می تواند به نظر برسد انجام دهیم. ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "اجازه دهید فقط یک سوال مسابقه دیگر در این راستا انجام دهم تا آخرین مورد را در مورد ویژگی‌های لگاریتم بپرسم قبل از اینکه کمی به یک مثال دنیای واقعی تبدیل شویم و از آنچه در اینجا و اکنون داشتیم خلاص شویم، کدام یک از موارد زیر درست است؟ log یک بعلاوه b همان log یک log بعلاوه b است log یک بعلاوه b برابر است با log بار log از b log یک بعلاوه b برابر است با یک تقسیم بر log یک log به علاوه b یا log یک بعلاوه b برابر است با یک تقسیم بر log بار log b یا هیچ یک از ah های بالا، و حالا به این اندازه اجماع نداریم، درست است؟ خیلی جالب است، ما یک مسابقه اسب دوانی داریم، بنابراین من به شما فرصت می‌دهم تا زمانی که مردم در حال پاسخگویی هستند، به این موضوع فکر کنید، در واقع من یک سوال کوچک از مخاطبان دارم، بنابراین، می‌دانید، من فقط در مورد اینکه چگونه ممکن است صحبت کنیم از نظر رشد ضربی فکر کنید و این فقط نباید قدرت های ده باشد، ما همچنین می توانیم کاری شبیه به توان های سه انجام دهیم که اگر از یک به سه به نه به بیست و هفت به هشتاد و یک بروید، همه از اینها می توانیم بگوییم که پایه لاگ سه از این اعداد فقط در مراحل کوچک خوب رشد می کند، بنابراین پایه سه از یک را وارد کنید، سه را به مقدار یک وارد کنید، پاسخ صفر است به طور کلی لاگ یک، بدون توجه به پایه، صفر است. صفر log پایه سه از سه، سه برابر با سه است یک به طور مشابه، پایه سه از نه دو آه است، ممکن است تعجب کنید که سوال من چیست، اما این کمک خواهد کرد که همه اینها را برای خودم به تصویر بکشم. در اینجا، اجازه دهید من فقط یک گزارش دیگر را بنویسم، پایه سه از هشتاد و یک، اکنون چهار است، من شنیده ام که ظاهراً اگر از یک کودک بپرسید، بیایید بگوییم حدوداً پنج یا شش ساله چه عددی در نیمه راه بین یک و نه شما است. بگویید چه عددی در نیمه راه است غرایز آنها برای پاسخگویی لگاریتمی است در حالی که غرایز ما بیشتر خطی هستند بنابراین ما اغلب یک و نه فکر می کنیم، شما تعداد زیادی اعداد با فاصله مساوی بین آنها دو، سه، چهار، پنج، شش دارید. ، هفت، هشت، و اگر درست در نیمه راه بروید، به پنج می رسید، اما اگر به رشد ضربی فکر می کنید که از یک به نه می توانید به کجا برسید، موضوع اضافه کردن یک سری چیزها نیست، بلکه شما "با مقدار مشخصی رشد می کنید، سه ضریب رشد می کنید، سپس ظاهراً سه ضریب دیگر رشد می کنید، غریزه طبیعی یک کودک با گفتن سه مطابقت دارد و ظاهراً اگر انسان شناسانی داشته باشید که جوامعی را مطالعه کرده اند، مطابقت دارد." سیستم‌های حسابداری و نوشتن را به همان روشی که جوامع مدرن دارند توسعه داده‌اند، آنها به سه مورد برای این پاسخ خواهند داد، سؤال من از مخاطبان است که آیا هر یک از شما که در حال حاضر تماشا می‌کنید به یک کودک کوچک دسترسی دارید، مثلاً در محدوده پنج سال. قدیمی ببینید آیا می توانید بروید از آنها بپرسید که چه عددی بین یک و نه است و اگر می توانید، در توییتر به ما بگویید که کودک چه می گوید پاسخ واقعی آنها چیست زیرا من نمی دانم چرا، من فقط کمی هستم در مورد اینکه آیا واقعاً در عمل جواب می‌دهد یا نه، می‌دانم که این یک روش فوق‌العاده علمی برای انجام آن نیست، من از افرادی که یک پخش زنده یوتیوب را تماشا می‌کنند نمی‌خواهم از فرزندان خود نظرسنجی کنند و سپس پاسخ را توییت کنند، اما به خاطر خودم جالب است. برای دیدن نوعی اعتبار سنجی در آنجا به سوال ما، این اولین موردی است که به نظر نمی رسد در یک جهت اتفاق نظر زیادی داشته باشد، بیایید جلوتر برویم و آن را درجه بندی کنیم تا ببینیم جواب عالی است، خوب، بنابراین 2400 از شما به درستی پاسخ دادید که هیچ یک از موارد بالا نیست که log از a به علاوه b هیچ یک از این ویژگی های خوب را برآورده نمی کند و به طور کلی، مگر اینکه قرار باشد با انواع خاصی از تقریب ها کار کنیم، به خصوص زمانی که گزارش طبیعی وارد بازی شود. ممکن است دفعه بعد در مورد این موضوع صحبت کنیم که افزودن ورودی های لگاریتم در واقع احساس بسیار عجیبی است، این کار بسیار عجیبی است و برای درک آن عجیب و غریب، اگر از شما بپرسم یک بعلاوه b را وارد کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "این واقعیتی است که شما نمی توانید آن را ساده کنید، اما اگر می دانید، اگر قبلاً به آن فکر نکرده بودید، ممکن است تعجب کنید، اوه، آیا فقط فرمولی وجود دارد که من آن را انجام نداده ام پیش از این دیده شده است، بنابراین با تمام این موارد، اجازه دهید پیش از این که به نمونه‌ای متفاوت بپردازیم، چند سوال از مخاطبان بپرسم، بنابراین به نظر می‌رسد که اوما شرما می‌پرسد آیا مبنای صفر است؟ این یک سوال جالب است، آیا پایه لگاریتم می تواند صفر باشد؟ خوب از نظر مثلث ما ممکن است به این فکر کنیم که می دانید، صفر به نوعی توان x برابر با مقدار دیگری y است، این چیزی است که می توانیم با گفتن صفر به x برابر y بنویسیم یا می توانیم بنویسیم. ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "همان چیزی که با گفتن log پایه صفر از y برابر با x صفر برابر با x است، اکنون مسئله اینجاست که صفر به هر چیزی در نهایت به صفر می رسد درست است، بنابراین اگر ما فقط به پایه log صفر فکر می کنیم. y برای هر ورودی دیگری که می‌دانید، می‌خواهید چیزی مانند یک یا دو یا پی هر چیزی را که می‌خواهید وارد کنید، سؤال صفر را می‌پرسید تا برابر یک یا دو یا پی یا هر عددی که ممکن است در آنجا داشته باشید. و پاسخی وجود نخواهد داشت، بنابراین در بهترین حالت می توانید سعی کنید بگویید اوه بله، ورود به سیستم صفر، این یک تابع کاملاً معتبر است و فقط در ورودی صفر تعریف شده است، اما حتی در این صورت هم برای تلاش برای حل کردن آنچه می خواهید مشکل خواهید داشت. در آنجا چون گفتن صفر به چیزی که برابر با صفر است، مثل این است که هر چیزی در مورد آن صدق می کند، بنابراین بازوی شما به هر حال که می خواهید این کار را انجام دهید، پشت سر شما می پیچد و با این واقعیت مطابقت دارد که تابع نمایی با پایه صفر کاملاً صفر است. اعداد را به شکلی زیبا از یک به یک بر روی یکدیگر نگاشت نمی کند، بنابراین این یک سوال عالی است، آیا می توانید یک پایه گزارش صفر داشته باشید اکنون به این ایده که این چیزها در دنیای واقعی کجا می آیند، یک مثالی که من دوست دارم این است مقیاس ریشتر برای زمین لرزه ها بنابراین مقیاس ریشتر به ما یک کمیت برای قدرت یک زلزله می دهد و می تواند هر چیزی باشد از اعداد بسیار کوچک تا اعداد بسیار بزرگ مانند من فکر می کنم بزرگترین زلزله اندازه گیری شده است و این فقط یک نمودار است که از ویکی پدیا 9 بود. 5 و برای درک این که چقدر دیوانه کننده است، ارزش دارد به رابطه بین معنای این اعداد و سپس چیزی شبیه به مقدار معادل TNT که نوعی اندازه گیری میزان انرژی در آن است و سپس آنچه می توانیم در اینجا انجام دهیم، نگاه کنیم. این است که ببینیم آیا می‌توانیم بیانی برای عدد مقیاس ریشتر بر حسب مقدار انرژی به دست آوریم و چرا لگاریتم‌ها راهی طبیعی برای توصیف این موضوع هستند، بنابراین نکته کلیدی برای تمرکز بر روی این است که در حال برداشتن گام‌هایی رو به جلو هستیم که چقدر چیزها افزایش می‌یابند. به عنوان مثال، اگر در این مورد از دو چاه برویم، به ما نشان نمی‌دهد که سه کجاست، بنابراین شاید به فکر برداشتن یک گام از دو به چهار باشیم که به نوعی مانند برداشتن دو مرحله است که از نظر مقدار انرژی خوب به نظر می رسد که ما را از یک تن متریک TNT می گیرد که حدس می زنم یک بمب بزرگ از جنگ جهانی دوم است و ما را تا یک کیلوتن هزار برابر می برد که یک بمب اتمی کوچک است، بنابراین فقط دو قدم در مقیاس ریشتر رفتن از یک زلزله 2 ریشتری به زلزله 4 ریشتری ما را از بمب بزرگ از جنگ جهانی دوم تا عصر هسته ای می برد به طوری که قابل توجه است و اولین قدم تمیزی که می گیریم از 4 به 5 در حداقل از نظر آنچه که این نمودار به خوبی به ما نشان می دهد و ظاهراً یک پله از 4 به 5 به بالا رفتن از 1 کیلوتن به 32 کیلوتن مربوط می شود و ظاهراً به اندازه بمب تخریب کننده شهر بود که در ناکازاکی فرود آمد، بنابراین این شاید یکی باشد. چیزی که می تواند در مورد مقیاس های لگاریتمی غیر منطقی باشد، اگر فقط در اخبار می شنوید که تفاوت بین زلزله ای بود که 4 بود. 0 در مقابل زلزله ای که 5 بود. 0 به راحتی می توان فکر کرد که بله 4 و 5 اعداد تقریباً مشابهی هستند، اما آشکارا از نظر مقادیر TNT که مربوط به ضرب در 32 برای رسیدن از 1 به بعدی و رفتن از 2 به 4 است، به وضوح در حدود هزار ضرب می شود و تنها دلیل اینکه بزرگتر است به این دلیل است که در اینجا نمودار ما عدد 3 را نشان نمی‌دهد، بنابراین ما دو قدم برداشته‌ایم و می‌توانید خودتان تأیید کنید که اگر یک گام از 32 بردارید و سپس در 32 دیگر ضرب کنید، در واقع تقریباً به هزار نزدیک می‌شود. به نظر می رسد این ایده که پله های افزایشی روی عدد ریشتر با گام های ضربی در TNT مطابقت دارد نشان می دهد که چیزی لگاریتمی در اینجا در حال بازی است و کمی جالب است که فقط به اینجا ادامه دهیم و بگوییم که این تا حدی به دلیل پدیده های جهانی چقدر رشد می کند. توضیح دادن بله تعجب بزرگی نیست که با برداشتن یک قدم دیگر دوباره در حدود 32 ضرب می شود، اما در شهود ما این تفاوت بین 32 کیلوتنی یک بمب اتمی کوچک و سپس یک مگاتون است که ممکن است به عنوان بمب اتم کوچک فکر کنیم، بمب اتمی ناکازاکی که حدس می‌زنم 32 بمب اتمی ناکازاکی برای یک مگاتون باشد که به وضوح به بزرگی زلزله دو رشته‌ای در نوادا ایالات متحده آمریکا 1994 است. همچنین این موارد را که ظاهراً کمتر از دو عدد هستند، بررسی کردیم، این موارد همیشه اتفاق می‌افتند، تقریباً 8000 مورد در روز وجود دارد، اما به محض اینکه در قلمرو بمب‌های اتمی قرار گرفتیم مواردی مانند 3 وجود دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "5 و 4 آنهایی که ظاهراً در جایی روی زمین نیز اغلب اتفاق می‌افتند، حدود 134 نفر از آن‌هایی که هر روز در جایی اتفاق می‌افتند، چه کسی می‌دانست؟ اما از آنجایی که ما به این محدوده 5 و 6 که بسیار بالاتر از مقیاس بمب اتمی بودند، شدیدتر می شویم، اکنون فقط در حدود 2 در روز هستیم و من مطمئن هستم که یک زمین شناس می تواند وارد شود و توضیح دهد که چرا همه ما باید این کار را انجام دهیم. نگران این واقعیت نباشید که هر روز دو اختلال معادل بمب اتم در پوسته زمین اتفاق می‌افتد، اما احتمالاً به ندرت اتفاق می‌افتد که آن‌ها در نقطه‌ای مانند شهری که در آن افراد زیادی زندگی می‌کنند، تمرکز کنند و فقط فکر ما را تأیید کنند که هر مرحله شامل رشد 32 است، بیایید ببینیم که گام از 6 تا 7 چگونه به نظر می رسد و در اینجا مثال های بیشتری را در این بین به ما می دهد، شاید این توهم ایجاد کند که این گام بزرگتر از آنچه هست است و در واقع این تفاوت بین 1 مگاتون و 32 مگاتون یعنی ضربدر 32 یکی از چیزهایی که من در این نمودار جالب‌ترین چیزها را دیدم این بود که ببینیم چقدر باید پیش برویم تا به بزرگترین سلاح هسته‌ای برسیم که واقعاً آزمایش شده است، اوج جنگ سرد بود. بمب تزار که 50 مگاتن بود و من فکر می کنم آنها در واقع برنامه های اولیه برای داشتن یک بمب 100 مگاتنی داشتند، اما خودشان از آن 50 مگاتن کم کردند. مگاتون را در 32 دیگر ضرب کنید، بنابراین ما در مورد قدرت هزار برابر انفجار پایان جنگ جهانی دوم صحبت می کنیم و شما هنوز به 50 مگاتن آن چیزی که بشریت قادر به انجام آن است نیستید و آن آشکارا زلزله جاوه اندونزی است. . 0 فقط کمی بزرگتر از 6 نیست. 0، بسیار بزرگتر است و البته نکته اینجاست که وقتی مقیاسی دارید که افزایش های ضربی را به شما می دهد، ارزش این را دارد که آنچه مانند گام های کوچک به نظر می رسد در واقع می تواند گام های بزرگی از نظر انرژی ضمنی یا مقادیر مطلق ذکر شده در اینجا باشد. بنابراین وقتی به این واقعیت فکر می کنیم که همیشه 9 وجود داشته است. 5 که در واقع پوچ به نظر می رسد با توجه به اینکه فقط در 7 است. برد 0 که ما در مورد بزرگترین سلاح گرما هسته‌ای صحبت می‌کنیم و این نشان‌دهنده حوزه‌ای است که لگاریتم‌ها معمولاً به وجود می‌آیند، زمانی است که انسان‌ها می‌خواهند مقیاسی برای چیزی ایجاد کنند که واریانس بسیار گسترده‌ای را در میزان بزرگی چیزها ایجاد کند. در مورد اندازه زمین لرزه ها می توانید چیزهایی از آنچه که همیشه در اطراف زمین اتفاق می افتد، به اندازه یک نارنجک دستی بزرگ داشته باشید و می خواهید که در مقیاس شما باشد و چیزی برای فکر کردن در مورد دامنه آن از تمام راه ها بالا باشد. به بزرگترین اختلالی که در تاریخ بشر دیده‌ایم و برای ایجاد آن به نحوی که شما فقط یک دسته کامل از ارقام مختلف را در اعداد خود برای یک مورد و یک دسته کامل از رقم‌های مختلف و یک عدد کوچکتر نمی‌نویسید. از ارقام برای عدد شما در مورد دیگری خوب است که لگاریتمی بگیرید و سپس فقط آن را روی یک مقیاس قرار دهید که اساساً آن اعداد را بین 0 تا 10 قرار می دهد، شما می بینید که چیزی بسیار مشابه با مقیاس دسی بل برای موسیقی در حال وقوع است که در واقع یک مقدار کمی کار می کند. کمی متفاوت است، جایی که هر بار که شما یک پله از 10 دسی بل بالا می برید که مربوط به ضرب در 10 است به جای گام 1 ضرب در 10، گامی از 10 است که در 10 ضرب می شود، به طوری که یک گام ریاضی آن را کمی می کند. کمی پیچیده است، اما ایده یکسان است، که اگر به صدایی 50 دسی بل در مقابل 60 دسی بل گوش می دهید، از نظر انرژی که منتقل می شود و از آن می رود، بسیار آرام تر است، 60 تا 70 یا 70 به 80 مرحله، از 60 تا 80، که شامل ضرب مقدار انرژی در هر سطح مربع در ضریب 100 می شود، بنابراین هر بار که مقیاس لگاریتمی را می بینید، در ذهن خود بدانید که این بدان معناست که هر چیزی که زیر کاپوت به آن اشاره می کند رشد می کند. به همین دلیل است که ما شاهد استفاده از مقیاس‌های لگاریتمی زیادی برای توصیف شیوع ویروس کرونا بودیم، بنابراین چگونه می‌توانید رابطه‌ای مانند این را توصیف کنید که در آن هر بار که عدد مقیاس ریشتر را 1 افزایش می‌دهید، به خوبی در 32 ضرب می‌شوید. می توانم بر حسب سیاهه ای با پایه 32 فکر کنم، می توانم بگویم اگر گزارش را بگیرم، فقط با r تماس می گیرم، شماره ای را برای مقیاس ریشتر ممکن است به عنوان log پایه 32 در نظر بگیرم و این با آن مطابقت دارد. ، نه نه نه، من این کار را اشتباه انجام می دهم که چیزی نیست که ثبت شده است، ما پایه log 32 عدد بزرگ را می گیریم، از عدد TMT، چیزی که شبیه 1 مگاتون بود، 1 میلیون تن است، پایه 32، که باید با عدد مقیاس ریشتر مطابقت دارد، اما ممکن است نوعی افست وجود داشته باشد، بنابراین ممکن است بگوییم که نوعی ثابت s وجود دارد که ما به این عدد مقیاس ریشتر اضافه می کنیم و این عبارت دقیقاً یکسان است، ببخشید که از این عدد خارج می شوم. در پایین این عبارت دقیقاً مشابه این است که می‌گوییم 32 به توان چند افست ضربدر عدد مقیاس ریشتر ما، که برابر است با گرفتن 32 به آن آفست، که خود فقط مقداری ثابت بزرگ است، ضربدر 32 به عدد مقیاس ریشتر. ممکن است تصور کنید که این فقط چند بار ثابت 32 نسبت به توان عددی است که می بینید، بنابراین این شیوه نوشتن واقعاً بر رشد تصاعدی آن تأکید می کند که اگر این چیزی است که با مقدار TMT که می بینید مطابقت دارد، همانطور که آن را افزایش می دهید. r گام به گام شما در 32 ضرب می کنید، اما راه دیگری برای برقراری ارتباط دقیقاً یکسان این است که از هر مقداری که آن مقدار درست است، پایه log 32 را بگیرید، نکته بعدی که می خواهم در مورد آن صحبت کنم این است که چگونه همیشه مجبور نیستیم. نگران نحوه محاسبه لاگ‌های پایه‌های مختلف هستید، اینجا کمی عجیب است که ما در مورد log پایه 32 صحبت می‌کردیم، قبلاً اشاره کردم که چگونه ریاضی‌دانان واقعاً دوست دارند یک گزارش با پایه و پایه داشته باشند دانشمندان رایانه واقعاً دوست دارند یک گزارش با پایه 2 داشته باشند و آن را برای مقاصد محاسباتی یا برای فکر کردن در مورد اینکه چگونه این چیزها رشد می کنند اگر یک گزارش داشته باشید، اگر بتوانید یک نوع گزارش را محاسبه کنید، چه پایه 10، پایه 2، پایه e باشد، می توانید تقریباً هر چیز دیگری را محاسبه کنید. اکنون می خواهید شهودمان را به آن سمت ببریم، بیایید به مسابقه خود برگردیم و به سؤال بعدی برویم و من معتقدم که این سؤال از همه بیشتر است، نمی دانم، این یک سؤال نیمه منطقی است، این باید خوب باشد این فقط ما را آماده می کند تا از زمینه پایه 2 به زمینه پایه 10 ترجمه کنیم و همچنین شهود خوبی برای درک قدرت های 2 است که به طور کلی رابطه ای را که با توان های 10 دارد داشته باشیم زیرا این یک نوع تصادف دوست داشتنی است. طبیعت که این دو به خوبی منظور من را متوجه خواهید شد، آنها به خوبی با یکدیگر بازی می کنند، بنابراین سوال ما می پرسد، با توجه به این واقعیت که 2 تا 10 1024 است، 1024، که تقریباً 1000 است، بنابراین اگر شما یک با اعداد خود کمی شل شوید و فقط تقریب های 2 تا 10 را انجام می دهید، در واقع 1000، کدام یک از موارد زیر به درستی نزدیکتر است؟ پایه گزارش 2 از 10 تقریباً 0 است. ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "مناقصه در اینجا اصلاً یک تصمیم متفق القول نیست. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "اما سوال این بود که کدام یک به درستی نزدیکتر است، و بیایید ببینیم چگونه می توانیم در مورد این فکر کنیم. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "بنابراین نشان می دهد که شما توان 2 دارید، یعنی 1024، به شدت نزدیک به توان 10، حدود 10 مکعب. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "خوب این به چه معنا است؟ اگر log پایه 2 از 10 برابر با x باشد، این همان چیزی است که بگوییم 2 به x برابر با 10 است، درست است؟ از ما 2 تا برابر 10 می خواهد. ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "شما نمی توانید این کار را با هر عملکردی انجام دهید. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "به نظر می رسد مردم فکر می کنند که شما می توانید این کار را با هر عملکردی انجام دهید، اما شما نمی توانید. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "و این به این معنی است که x حدود 10 سوم است، خوب؟ که بسیار عالی است، بنابراین لاگ پایه 2 از 10 حدود 10 سوم است. ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "و به اندازه کافی خوب، آنچه قبلاً دیدیم این بود که پایه log 2 از 10، همچنین می‌توانیم بگوییم پایه log 10 از 2 فقط 1 بیش از این مقدار است، 1 بر x. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "و از آنجایی که ما کارهایی را در لاگ انجام می دهیم، من فقط آن را به این شکل می نویسم. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "به طور مشابه پایه 2 از یک میلیون را وارد کنید، خوب ببینیم، اگر باید 2 را در خودش حدود 10 برابر ضرب کنیم تا به هزار برسیم، باید آن را در حدود 20 برابر در خودش ضرب کنیم تا به یک میلیون برسیم. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "کمی کوچکتر است اما این یک نوع تقریب خوب است که در ذهن شما وجود دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "3 می شود 3.20، ما به همان میزان کاهش می دهیم. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30، ما به همان میزان کاهش می دهیم. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "باشه؟ اکنون این شهودی است که ارزش یادآوری را دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "من در این مورد به شما زمان معنی‌داری می‌دهم، زیرا واضح نیست مگر اینکه قبلاً با لگاریتم‌ها آشنا باشید و ارزش آن را دارد که کمی فکر کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/portuguese/sentence_translations.json b/2020/ldm-logarithms/portuguese/sentence_translations.json index 2d985135f..b780cadad 100644 --- a/2020/ldm-logarithms/portuguese/sentence_translations.json +++ b/2020/ldm-logarithms/portuguese/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Música🎵 Bem-vindo de volta ao Lockdown Math. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "Hoje vamos falar sobre logaritmos e uma espécie de lição de volta ao básico. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "E como sempre, para começar, só quero ter uma ideia de onde o público está agora. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "Então, se você puder ir para 3b1b. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "Nunca ouvi falar deles antes ou nunca aprendi sobre eles antes b. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "Aprendi sobre eles, mas às vezes fico confuso com todas as propriedades c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "Eu os entendo, mas não saberia como ensiná-los e d. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "Eu os entendo bem e poderia ensiná-los confortavelmente a outra pessoa para que eles também entendessem bem. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "Então, temos uma boa divisão. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "Como eu disse, a intenção disso é criar uma lição que eu possa indicar às pessoas no futuro, caso elas simplesmente não se sintam confortáveis com logaritmos e quero poder dizer, ah, aqui está um lugar onde você pode ir. como eu acho, você sabe, como eu acho que você poderia abordar isso intuitivamente. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "Porque eu estava navegando em alguns fóruns de professores antes de fazer esta palestra em particular e quando as pessoas perguntam qual é o tópico mais difícil de ensinar matemática no ensino médio, no sentido de que os alunos parecem ter mais problemas com isso, logaritmos é um dos mais respostas comumente indicadas, o que é interessante e posso adivinhar que talvez seja porque há uma tonelada dessas propriedades que você acaba tendo que aprender, sabe, então, se pularmos para onde vamos, você terá todas essas pilhas de regras que parecem um monte de álgebra que podem ser difíceis de lembrar e fáceis de misturar as coisas na sua cabeça e eu acho que quando as pessoas têm, você sabe, esse tipo de lembranças de pesadelo de como era a matemática do ensino médio e como logaritmos fizeram por eles, muitas vezes são essas fórmulas específicas que vêm à mente e o que eu quero fazer hoje é tentar falar sobre uma delas, como pensar sobre elas, mas também apenas no meta nível de se você está ensinando álgebra a alguém, o que são os pontos que vale a pena enfatizar? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "Qual é a maneira de incorporá-lo em suas intuições? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "ah, tem 3 zeros, qual é o log de um milhão? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "o log de 1000 vezes x é igual a 3 vezes o log de x e lembre-se que estamos usando a convenção de que é base 10 log b. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "log de 1000 vezes x é igual a log de x ao cubo c. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "log de 1000 vezes x é igual a 3 elevado à potência de log de x e e. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "nenhuma das opções acima e lembre-se, como eu disse antes, devemos esperar que todas aquelas pessoas no início que disseram que entendem bem os registros responderão imediatamente, responderão corretamente, mas se você estiver alguém que não o faz, não deixe que isso o intimide quando você estiver olhando para um problema como este, o que eu encorajaria você a fazer é apenas conectar várias potências de 10 e pensar em termos da ideia de que o log funciona conta o número de zeros, então vou lhe dar um tempinho para pensar sobre isso, então irei em frente e avaliarei e, como sempre, se for mais rápido do que você se sente confortável, saiba que é apenas porque quero prosseguir. com a lição, então neste caso a resposta correta é log de 1000 vezes x é o mesmo que pegar 3 mais o log de x e agora vamos pensar sobre isso por um momento e como eu disse quando você estava apenas começando com eles, acho que a melhor coisa a fazer é ficar confortável conectando vários números e os melhores números para inserir são aqueles que já são potências de 10, então se você está perguntando algo como log de 1000 vezes x bem, eu não' não sei, vamos apenas inserir algo para x log de 1000 vezes 100 bem, sabemos quantos zeros haverá na resposta final aqui bem, 1000 vezes 100 é 100.000 já temos intuitivamente essa ideia de que quando multiplicamos 2 potências de 10 estamos apenas pegando os zeros, os 3 zeros daquele 1000, os 2 zeros daquele 100 e os colocamos um ao lado do outro, então deve haver 5 zeros no total, mas se você realmente refletir não apenas sobre como o número virou mas por que aconteceu assim foram os 3 zeros daquele 1000 mais os 2 zeros daquele 100 que também poderíamos escrever dizendo o número de zeros em 1000 mais o número de zeros em 100 então esta ideia de que um logaritmo do produto de duas coisas é a soma dos logaritmos dessas duas coisas no contexto de potências de 10, isso apenas comunica o que já é uma ideia superintuitiva para muitos de nós, se você pegar 2 potências de 10 e multiplicá-las, você apenas pegue todos os seus zeros e coloque-os uns sobre os outros, então a maneira como escrevi as coisas aqui é na verdade indicativa de um fato um pouco mais geral que será nossa primeira propriedade dos logaritmos, que é que se pegarmos o log de A vezes B é igual ao log de A mais o log de B agora, sempre que você vir uma dessas regras de logaritmo, se estiver semicerrando os olhos ou se estiver um pouco confuso sobre como lembrá-la, basta inserir exemplos Estou sendo redundante, estou dizendo muito isso, mas é porque acho que é muito fácil esquecer quando você está atolado na própria álgebra e está fazendo algum tipo de teste e ele só tem muitos símbolos para lembrar a si mesmo que você está bem em apenas inserir alguns números, isso é uma boa coisa a fazer e muitas vezes é uma ótima maneira de produzir intuição, então, neste caso, dizendo log de A vezes B e separando-o, poderíamos apenas pensar, ah, isso log de 100 vezes 1000, que é 5, há 5 zeros nele, dividido em termos do número de zeros em cada parte, ótimo, maravilhoso, então, levando essa intuição adiante, vamos tentar outro problema prático e, novamente, se você souber, ótimo, você será capaz de responder bem, mas talvez pense, não apenas qual é a resposta, mas como eu explicaria essa resposta a alguém ou como tentaria fazer com que um aluno chegasse a essa resposta por conta própria, sem que eu tivesse que dizer qual é a resposta para que haja dois membros potenciais do público: há aqueles que estão interessados na lição em si e depois aqueles que estão interessados na meta-lição, então nossa pergunta pergunta, novamente, qual das afirmações a seguir é verdadeira? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "a. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "log de x elevado a n é igual a n vezes log de x b. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "log de x elevado a n é igual a log de x elevado a n c. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "log de x elevado a n é igual a n mais log de x ou d. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "então a resposta correta aqui é a, e parece que 4.000 de vocês receberam os parabéns, nos dizendo que log de x elevado a n é igual a n vezes log de x então, novamente, digamos que você está tentando ensinar isso para alguém ou se você está tentando entender o que isso significa, acho que um bom lugar para começar é conectar algo e, neste caso, para log de x elevado a n, vamos tentar com 100 elevado a 3 e você pode tentar com outros para ver se os padrões que você está fazendo realmente funcionam, mas se você está pensando nisso, não em termos de simplesmente ver qual é a resposta, mas tentando pensar por que a resposta acabou dessa maneira às vezes um exemplo serve porque 100 ao cubo, podemos pensar nisso como pegar bem, são 3 cópias de 100, estou pegando 3 cópias de 100 e quando multiplico tudo isso e penso em log como contar o número de zeros que digamos, ah, será um número que tem apenas 6 zeros, é isso que significa pegar 100 vezes 100 vezes 100. Posso apenas pensar em agrupar todos esses zeros para obter um milhão, então esse número será 6, mas se pensarmos, na verdade, por que era 6, não apenas esse é o número de zeros dentro do milhão de onde veio esse 6, é que tínhamos 3 cópias desse 100 e cada um desses 100 tinha 2 zeros diferentes, então dessa forma é mais geral maneira que você pode pensar sobre isso, se em vez de pegar 100 ao cubo estivéssemos olhando para 1000 ao cubo ou 1000 elevado a n ou x elevado a n você pode pensar que é qualquer que seja o valor de n era o número de cópias que estávamos multiplicando em vezes o número de bem, vamos ver, não é x vezes o número de zeros que estavam em tudo o que substituímos por x que neste caso era 100 então se em vez disso eu tivesse levado algo como log de 10.000 elevado a n isso seria o mesmo como tirar n cópias desses 10.000 contando o número de zeros em cada um deles que é 4, então seria n vezes 4 e, claro, a propriedade geral que a maioria de vocês respondeu corretamente é que você tem esse pequeno e adorável efeito onde quando você veja o registro de algo elevado a uma potência que um pequeno poder desce na frente dele e você apenas tem o registro do que estava dentro agora, uma das implicações talvez mais importantes disso, não sei se você chamaria isso uma implicação ou se você chamar isso de uma reformulação da definição se estou pegando log e vou apenas enfatizar novamente que é base 10 de 10 elevado a n, podemos pensar nesse pequeno n como um salto para baixo frente e se torna n vezes o log de base 10 de 10, que é claro 1. Esta expressão você pode pensar como contar o número de zeros no final ou, mais geralmente, perguntar 10 elevado a quanto é igual a 10 e a resposta é simplesmente 1 o que é muito reconfortante porque outra maneira de você voltar e apenas ler esta expressão original é dizer 10 elevado a quanto é igual a 10 elevado a n, bem, a resposta está ok agora com cada propriedade de logaritmo que temos, então neste caso nós acabei de encontrar um logaritmo de x elevado a n envolve que n pulando na frente sempre haverá uma propriedade exponencial de imagem espelhada e essa é outra maneira de ajudarmos a obter um pouco de intuição para isso, então deixe-me apenas encobrir algumas das propriedades futuras que veremos aqui tentam esconder para onde estamos indo o que acabamos de descobrir elevando algo a n que salta na frente, isso corresponde à propriedade exponencial de que se eu levar 10 a x e aumentar aquela coisa toda elevada à potência n é o mesmo que levar 10 elevado a n vezes x e isso nos leva a outra intuição que você pode ter para logaritmos, que é como se fossem uma exponenciação virada do avesso e aqui está o que quero dizer com que a coisa que está dentro do log, se estou pegando o log de a, você deveria pensar nisso como toda a expressão externa para algo que é exponencial, neste caso, a a coisa que está dentro corresponde a 10 elevado a x the saída da função enquanto a coisa toda em si o logaritmo de a corresponde ao que está dentro aqui apenas qual é o expoente de 10 então sempre que você vir uma expressão logarítmica aqui você deve estar pensando que ela desempenha o papel de um expoente à direita lado e toda vez que você vê um exponencial, todo o 10 elevado à expressão x, todo o componente externo no lado direito que corresponde a algo que está dentro de um dos logs e vimos isso acima da ideia de que quando estamos multiplicando por dentro, isso é somar por fora, bem, se os logs virarem as exponenciais do avesso, isso nos diz que multiplicar por fora, multiplicar as saídas da função é o mesmo que somar por dentro, porque cada um desses logs, como log a e log b está desempenhando o papel de xey na expressão à direita, então vamos continuar jogando, vamos fazer mais algumas dessas e ver quantas dessas propriedades podemos construir uma intuição, então esta última, pensar muito bem em expoentes pulando para o próximo é algo que pode parecer um pouco estranho para aqueles que não estão necessariamente familiarizados com logaritmos, mas, novamente, insira alguns números para obter alguma intuição e daremos um pouco mais um momento para descobrir qual das afirmações a seguir é verdadeira? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "bem, se 10 ao cubo é 1.000, isso é a mesma coisa que dizer que 10 é igual a 1.000 elevado a 1 terço, fazer o inverso aqui envolve o inverso multiplicativo do expoente e o resultado é que parece 1 dividido por 3 e que 3 corresponde ao log de base 10 de 1000 é 1 dividido pelo log de base 10 de 1000 então, de forma mais geral, você pode adivinhar com base neste único exemplo que quando trocamos a base pelo que está dentro, isso corresponde a pegar 1 dividido pelo que está lá fora e novamente, você pode pensar nisso em termos de olhar para a regra exponencial correspondente. Agora, o que aconteceu com meu adorável pequeno registro e exponenciais? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "maravilhoso, então, mais uma vez, vamos esconder onde estão algumas das coisas, algumas das outras propriedades que veremos aqui e vou mantê-las na mesma ordem que as tinha antes aqui. Eu estava pensando que tê-las pré-escritas poderia me manter um pouco mais limpo do que o normal, mas talvez envolva apenas jogar esse jogo estranho de cortar papel e embaralhar, então o que acabamos de encontrar, log na base b de a, se você trocá-los, é o mesmo que dividir por 1, o que isso corresponde, fora de um terra exponencial é se você elevar b a alguma potência e disser que isso é igual a a, é a mesma afirmação que dizer que a elevado ao inverso dessa potência é igual a b novamente, é útil parar um momento e pensar nos logaritmos como coisas que giram de dentro para fora, a expressão log base b de a está desempenhando o papel daquele x e a expressão log base a de b está desempenhando o papel de tudo o que está no topo de a e então simetricamente, toda a expressão b elevada à potência x está desempenhando o papel do interior à esquerda, ele desempenha o papel do a e de toda a expressão, a elevado à potência de algo desempenha o papel do que está dentro da base do log a para que você possa ver, apenas inserindo alguns exemplos e correspondendo-o às regras exponenciais, já podemos pensar em três regras de logaritmo diferentes que, se fossem apenas transmitidas como peças de álgebra para serem memorizadas, você sabe, você poderia memorizá-las, mas é muito fácil para elas escaparem do seu cabeça e também é muito fácil ficar frustrado com a tarefa em questão, mas você pode querer se lembrar que a razão pela qual nos preocupamos com esse tipo de coisa é que entender as regras dos logaritmos nos ajuda a fazer contas em contextos onde é como um vírus crescendo onde de um dia para o outro, de um passo para o outro, as coisas tendem a crescer multiplicativamente, entender as regras dos logaritmos ajuda você a ter uma noção melhor desse tipo de coisa, então antes de fazermos um bom exemplo do mundo real do que isso pode parecer tipo, deixe-me fazer mais uma pergunta nesse sentido para perguntar sobre as propriedades dos logaritmos, uma última antes de fazermos a transição para um exemplo do mundo real, nos livrarmos do que tínhamos aqui e agora, qual das afirmações a seguir é verdadeira? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "log de a mais b é igual a log de a mais log de b log de a mais b é igual a log de a vezes log de b log de a mais b é igual a um dividido por log de a mais log de b ou log de a mais b é igual a um dividido por log de a vezes log de b ou nenhuma das opções acima ah, e agora não temos tanto consenso, não é? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "muito interessante, temos uma corrida de cavalos entre dois, então darei a vocês um momento para pensarem sobre isso enquanto as pessoas respondem. Na verdade, tenho uma pequena pergunta para o público, então, você sabe, eu estava falando sobre como poderíamos pense em termos de crescimento multiplicativo e isso não precisa ser apenas potência de dez, também poderíamos fazer algo como potências de três, onde se você for de um para três para nove para vinte e sete para oitenta e um, tudo destes, poderíamos dizer que o log de base três desses números apenas cresce em pequenos passos, então log de base três de um, três elevado a um, a resposta é zero, em geral, o log de um, não importa a base, será seja zero, log de base três de três, três elevado a três é um, da mesma forma, log de base três de nove é dois, ah, você pode estar se perguntando qual é a minha pergunta, mas ajudará a extrair tudo isso e para meu próprio prazer aqui, deixe-me escrever mais um log de base três de oitenta e um é quatro agora, ouvi dizer que se você perguntar a uma criança, digamos, por volta dos cinco ou seis anos de idade, qual número está no meio do caminho entre um e nove você diga qual número está na metade, seus instintos de como responder são logarítmicos, enquanto nossos instintos tendem a ser mais lineares, então muitas vezes pensamos em um e nove, você tem um monte de números espaçados uniformemente entre eles dois, três, quatro, cinco, seis , sete, oito e se você for direto no meio do caminho, chegará ao cinco, mas se estiver pensando em termos de crescimento multiplicativo onde ir de um a nove, não é uma questão de adicionar um monte de coisas, mas você estamos crescendo em uma certa quantidade, você cresce em um fator de três, então você cresce em outro fator de três, supostamente, o instinto natural de uma criança se alinha com dizer três e, supostamente, isso também se alinha se você tiver antropólogos estudando sociedades que não' desenvolveram sistemas de contabilidade e escrita da mesma forma que as sociedades modernas, eles responderão três para isso, então, minha pergunta para o público se algum de vocês que está assistindo agora tem acesso a uma criança pequena, digamos, na faixa de cinco anos velho, veja se você pode perguntar qual é o número que está no meio do caminho entre um e nove e, se puder, diga-nos no Twitter o que a criança diz, qual é a resposta real, porque não sei por que, só estou um pouco cético sobre se isso realmente acontece na prática, eu entendo que esta não é uma maneira supercientífica de fazer isso, não estou pedindo às pessoas que assistem a uma transmissão ao vivo no YouTube que pesquisem seus próprios filhos e depois twittem a resposta, mas para meu próprio bem, seria interessante para ver algum tipo de validação em relação à nossa pergunta, esta é a primeira que não parece ter um grande consenso em uma direção, vamos em frente e avaliar para ver qual a resposta é ótima, ok, então 2.400 de vocês responderam corretamente que não é nenhuma das opções acima que o logaritmo de a mais b não satisfaz nenhuma dessas boas propriedades e, em geral, a menos que trabalhemos com certos tipos de aproximações, especialmente quando o logaritmo natural entra em jogo poderíamos falar sobre isso na próxima vez que adicionar as entradas de um logaritmo é na verdade uma sensação muito estranha, é uma coisa muito estranha de se fazer e para ter uma noção dessa estranheza, insira algumas potências de dez se eu lhe perguntar o log de a mais b o que você pode começar a pensar é, ok, deixe-me inserir alguns exemplos como 10.000 e 100 e me pergunto: se eu fizer essa função de contagem de zeros do que há nessa entrada, quantos zeros há nela? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "essa é uma pergunta interessante, ok, a base de um logaritmo pode ser zero? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "bem, em termos do nosso triângulo, podemos pensar nisso como se estivéssemos dizendo, você sabe, zero elevado a algum tipo de potência x é igual a algum outro valor y, isso é algo que poderíamos escrever dizendo que zero elevado a x é igual a y ou poderíamos escrever a mesma coisa dizendo que log de base zero de y é igual a x zero elevado a quanto é igual a x agora o problema aqui é que zero elevado a qualquer coisa acaba sendo zero, certo, então se vamos pensar apenas em log de base zero de y para qualquer outra entrada y você sabe, você deseja inserir algo como um ou dois ou pi o que quiser, você está fazendo a pergunta zero elevado a o que é igual a um ou dois ou pi ou qualquer número que você possa ter lá e simplesmente não haverá uma resposta, então na melhor das hipóteses você poderia tentar dizer ah, sim, log de zero, é uma função perfeitamente válida, só é definida na entrada zero, mas mesmo assim você teria problemas para tentar conseguir o que deseja aí porque dizer zero ao que é igual a zero é como se qualquer coisa se aplicasse a ele, então seu braço vai ficar torcido nas costas da maneira que você quiser fazer isso funcionar e isso corresponde ao fato de que a função exponencial com base zero é inteiramente zero não mapeia números de uma maneira agradável um para um, então essa é uma ótima pergunta, você pode ter um log de base zero agora, de volta à ideia de onde essas coisas surgem no mundo real, um exemplo que eu gosto é a escala Richter para terremotos, então a escala Richter nos dá uma quantificação de quão forte é um terremoto e pode ser qualquer coisa, desde números muito pequenos até números muito grandes, como acho que o maior terremoto já medido e este é apenas um gráfico que vem de A Wikipédia foi nota 9.5 e para avaliar o quão insano isso é, vale a pena olhar para a relação entre o que esses números significam e então algo como a quantidade equivalente de TNT, algum tipo de medida de quanta energia existe nele e então o que podemos tentar fazer aqui é ver se podemos obter uma expressão para o número da escala Richter em termos de quantidade de energia e por que os logaritmos seriam uma maneira natural de descrever isso, então a chave para focar é, à medida que avançamos, quanto as coisas aumentam então, por exemplo, se formos de dois bem, neste caso, isso não nos mostra onde está o três, então talvez pensemos em dar um passo de dois para quatro, o que é como dar dois passos, o que isso faz em termos de quantidade de energia bem, parece que nos leva de uma tonelada métrica de TNT, que é, eu acho, uma grande bomba da Segunda Guerra Mundial e nos leva até um quiloton mil vezes mais, o que é uma pequena bomba atômica, então apenas dois passos na escala Richter, passar de um terremoto de magnitude 2 para um terremoto de magnitude 4 nos leva de uma grande bomba da Segunda Guerra Mundial até a era nuclear, então isso é digno de nota e o primeiro passo limpo que damos é passar de 4 para 5 em pelo menos em termos do que este gráfico está nos mostrando e, evidentemente, um único passo acima de 4 para 5 corresponde a passar de 1 quiloton para 32 quilotons e esse era evidentemente o tamanho da bomba destruidora da cidade que caiu em Nagasaki, então este é talvez um coisa que pode ser contra-intuitiva sobre escalas logarítmicas se você estiver apenas ouvindo no noticiário a diferença entre ah, houve um terremoto que foi 4.0 versus um terremoto que foi 5.0, é fácil pensar que sim, 4 e 5, esses são números bastante semelhantes, mas evidentemente em termos de quantidades de TNT que corresponde a multiplicar por 32 para ir de 1 para o próximo e ir de 2 para 4 era evidentemente multiplicar por cerca de mil e o único a razão pela qual é maior é porque aqui nosso gráfico não estava mostrando quanto era 3, então estávamos dando dois passos e você pode verificar por si mesmo que se você der um passo de 32 e depois multiplicar por outro 32 isso é na verdade bem próximo de mil, então a ideia de que passos aditivos no número de Richter correspondem a passos multiplicativos no TNT parece sugerir que algo logarítmico está em jogo aqui e é um pouco interessante continuar aqui e dizer quanto isso cresce em parte por causa dos fenômenos mundiais que é descrevendo sim, não é uma grande surpresa que, à medida que damos outro passo, ele esteja se multiplicando por cerca de 32 novamente, mas controlando isso em nossas intuições, essa é a diferença entre 32 quilotons de uma pequena bomba atômica e um megaton que poderíamos considerar como uma bomba atômica não pequena, Bomba atômica de Nagasaki, que eu acho que são 32 das bombas atômicas de Nagasaki por um megaton que é evidentemente a magnitude do terremoto plano de corda dupla em Nevada, EUA, 1994. Eu não sabia o que era isso, obrigado Wikipedia em termos de frequências, a propósito, eu também procurei esses, evidentemente, aqueles que são menos de dois, acontecem o tempo todo, há cerca de 8.000 deles por dia, mas assim que estamos no reino das bombas atômicas, coisas como 3.5 e 4, evidentemente, eles também acontecem com bastante frequência em algum lugar do mundo, há cerca de 134 deles acontecendo em algum lugar todos os dias, quem sabia? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "mas à medida que nos tornamos ainda mais intensos nesta faixa de 5 e 6, que estavam bem acima da escala da bomba atômica, agora estamos apenas em torno de 2 por dia e tenho certeza de que um geólogo poderia entrar e explicar por que todos nós deveríamos ' Não estamos muito preocupados com o fato de que há duas perturbações equivalentes a bombas atômicas na crosta terrestre acontecendo todos os dias, mas presumivelmente é particularmente raro que elas estejam concentradas em algum lugar como uma cidade onde muitas pessoas vivem agora, apenas verificando nosso pensamento de que cada passo envolve um crescimento de 32, vamos ver como é o passo de 6 a 7 e aqui está nos dando muitos mais exemplos, talvez dando a ilusão de que esse é um passo maior do que realmente é e, de fato, essa é a diferença entre 1 megaton e 32 megatons, então isso é multiplicado por 32. A propósito, uma das coisas que achei mais interessante neste gráfico foi ver até onde precisamos ir antes de chegarmos à maior arma nuclear já testada, este foi o auge da guerra fria a bomba do czar que tinha 50 megatons e acredito que eles realmente tinham planos originais de ter uma bomba de 100 megatons, mas se acalmaram a partir desses 50 megatons, estamos falando de começar com os 32 quilotons da bomba de Nagasaki multiplicados por 32 para obter um megaton multiplique por outros 32, então estamos falando de mil vezes a força da explosão que encerrou a Segunda Guerra Mundial e você ainda não chegou aos 50 megatons do que a humanidade é capaz e isso é evidentemente o terremoto de Java na Indonésia, então 7 . 0 não é apenas um pouco maior que 6.0, é muito maior e o ponto aqui, claro, é que quando você tem uma escala que fornece aumentos multiplicativos, vale a pena perceber que o que parecem pequenos passos podem na verdade ser passos enormes em termos de energia implícita ou dos valores absolutos implícitos aqui então, quando pensamos no fato de que alguma vez existiu um 9.5, isso na verdade parece absurdo, visto que está apenas no 7.0 que estamos falando sobre a maior arma termonuclear já lançada e isso é indicativo de uma área onde os logaritmos tendem a surgir é quando os humanos querem criar uma escala para algo que leva em conta uma variação enormemente ampla em quão grandes as coisas podem seja assim, no caso do tamanho dos terremotos, você pode ter coisas do que acontece o tempo todo ao redor da Terra, do tamanho de uma granada de mão grande e você quer que isso esteja na sua escala e algo em que pensar, indo até o fim para a maior perturbação que já vimos na história da humanidade e para que isso aconteça de uma forma que você não esteja apenas escrevendo um monte de dígitos diferentes em seus números para um caso e um monte de diferentes, um número menor de dígitos para o seu número, em outro caso, é bom pegar logaritmos e depois colocá-los em uma única escala que basicamente comprime esses números entre 0 e 10, você vê algo muito semelhante acontecendo com a escala de decibéis para música, que realmente funciona um pouco um pouco diferente, onde cada vez que você aumenta 10 decibéis que corresponde à multiplicação por 10, então, em vez de um passo de 1 multiplicando por 10, é um passo de 10 que multiplica por 10, então isso torna a matemática um pouco um pouco estranho, mas a ideia é a mesma, que se você estiver ouvindo um som de 50 decibéis versus 60 decibéis, é muito mais silencioso em termos de energia sendo transmitida e indo de, o que seria, 60 para 70 ou 70 para 80 dessas etapas, de 60 a 80, que envolvem multiplicar a quantidade de energia por área quadrada por um fator de 100, então toda vez que você vir uma escala logarítmica, saiba em sua mente que isso significa que o que quer que esteja se referindo sob o capô cresce uma quantidade enorme, é novamente por isso que vimos muitas escalas logarítmicas usadas para descrever o surto de coronavírus, então como você pode descrever um relacionamento como este, onde cada vez que você aumenta o número da escala Richter em 1, você está multiplicando por 32, bem, nós poderia pensar em termos de um logaritmo com base 32, eu poderia dizer que se eu pegar o logaritmo de, vou apenas chamar r, o número da escala Richter, posso pensar nisso como logaritmo de base 32 e isso vai corresponder a , não, não, não, estou fazendo errado, não é isso que está registrado, pegamos o log de base 32 do grande número, do número TMT, algo que era como 1 megaton, é 1 milhão de toneladas, o log de base 32, isso deveria correspondem ao número da escala Richter, mas pode haver algum tipo de deslocamento, então podemos dizer que há algum tipo de constante s que estamos adicionando a esse número da escala Richter e esta expressão é exatamente a mesma, desculpe-me por sair do lá embaixo, esta expressão é exatamente o mesmo que dizer 32 elevado a algum deslocamento vezes nosso número da escala Richter, que é o mesmo que levar 32 a esse deslocamento, que em si é apenas uma grande constante, vezes 32 elevado ao número da escala Richter, então você você pode pensar nisso como sendo apenas uma constante vezes 32 elevado à potência do número que você vê, então esta forma de escrevê-lo realmente enfatiza o crescimento exponencial disso, se isso é o que corresponde à quantidade de TMT que você vê, conforme você aumenta isso passo a passo, você está multiplicando por 32, mas outra maneira de comunicar exatamente o mesmo fato é pegar o logaritmo de base 32 de qualquer que seja esse valor, agora a próxima coisa que quero falar é como nem sempre precisamos preocupe-se em como calcular logaritmos de bases diferentes, é um pouco estranho aqui que estávamos falando sobre logaritmo de base 32, mencionei anteriormente como os matemáticos realmente gostam de ter um logaritmo com base e os cientistas da computação realmente gostam de ter um logaritmo com base 2 e isso acontece para fins computacionais ou também para pensar em como essas coisas crescem se você tiver um log, se você for capaz de calcular um tipo de log, seja base 10, base 2, base e, você pode calcular praticamente qualquer outra coisa que você quer agora levar nossas intuições nessa direção, vamos voltar ao nosso quiz e passar para a próxima pergunta e acredito que essa pergunta é a mais, não sei, essa é uma pergunta meio razoável, isso deve ser legal isso apenas nos preparará para traduzir do contexto de base 2 para o contexto de base 10 e também é uma boa intuição para entender potências de 2 ter em geral a relação que tem com potências de 10 porque é esse tipo adorável de coincidência de natureza, esses dois, bem, você verá o que quero dizer, eles combinam bem um com o outro, então nossa pergunta é, dado o fato de que 2 elevado a 10 é 1024, 1024, que é aproximadamente 1000, então se você está sendo um um pouco solto com seus números e você está apenas fazendo aproximações de 2 elevado a 10, basicamente 1000, qual das afirmações a seguir está mais próxima de ser verdade? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "macio. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "Não é uma decisão unânime aqui. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "Mas a questão era saber qual delas está mais próxima de ser verdade, e vamos ver como podemos pensar sobre isso. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "Então isso indica que você tem uma potência de 2, que é 1024, muito próximo de uma potência de 10, cerca de 10 ao cubo. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "Então o que isso quer dizer? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "Se log de base 2 de 10 é igual a x, é a mesma coisa que dizer que 2 elevado a x é igual a 10, certo? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "Está nos perguntando 2 elevado a quanto é igual a 10. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "Você não pode fazer isso com todas as funções. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "As pessoas parecem pensar que você pode fazer isso com qualquer função, mas simplesmente não consegue. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "E o que isso significa é que x é cerca de 10 terços, ok? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "E muito bem, o que vimos anteriormente é que o log de base 2 de 10, também poderíamos dizer que o log de base 10 de 2 é apenas 1 sobre esse valor, 1 sobre x. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "Ótimo. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "E como estamos fazendo coisas em logs, vou escrever dessa maneira. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "Da mesma forma, log de base 2 de um milhão, bem, vamos ver, se tivermos que multiplicar 2 por ele mesmo cerca de 10 vezes para chegar a mil, deveríamos ter que multiplicá-lo por si mesmo cerca de 20 vezes para chegar a um milhão. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "É um pouco menor, mas é uma boa aproximação para se ter em mente. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20, reduzimos na mesma quantidade. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, reduzimos na mesma quantidade. OK? ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "Agora, esta é uma intuição que vale a pena lembrar. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "OK? ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "E então apenas uma pilha inteira de várias maneiras possíveis de combinar log base C de B vezes log base C de A. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "Darei a você um tempo significativo sobre isso porque não é óbvio, a menos que você já esteja familiarizado com logaritmos, e vale a pena pensar um pouco. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/russian/sentence_translations.json b/2020/ldm-logarithms/russian/sentence_translations.json index 059b89b66..a650dbd8b 100644 --- a/2020/ldm-logarithms/russian/sentence_translations.json +++ b/2020/ldm-logarithms/russian/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Музыка🎵 Добро пожаловать обратно в Lockdown Math. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "Сегодня мы поговорим о логарифмах и своего рода уроке возвращения к основам. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "И, как всегда, для начала я просто хочу понять, где сейчас находится аудитория. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "Итак, если вы можете перейти к 3b1b. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "Я никогда раньше о них не слышал или никогда не узнавал о них раньше b. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "Я узнал о них, но иногда меня путают все свойства c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "Я понимаю их, но не знаю, как их учить, и d. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "Я хорошо их понимаю и мог бы с легкостью научить им кого-нибудь еще, чтобы он тоже хорошо понял. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "Итак, у нас хорошее разделение. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "Как я уже сказал, цель этого состоит в том, чтобы создать урок, на который я смогу указать людям в будущем, если они просто не разбираются в логарифмах, и я хочу иметь возможность сказать: «О, вот место, куда вы можете пойти». как я думаю, знаете, как я думаю, что вы могли бы подойти к этому интуитивно. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "Поскольку перед тем, как прочитать эту конкретную лекцию, я просмотрел пару форумов учителей, и когда люди спрашивают, какую тему труднее всего преподавать в средней школе в том смысле, что у учеников с ней больше всего проблем, логарифмы — одна из самых сложных тем. обычно указанные ответы, что интересно, и я могу предположить, что, возможно, это потому, что есть масса этих свойств, которые вам в конечном итоге придется изучить, вы знаете, поэтому, если мы пропустим то, к чему мы собираемся идти, у вас есть все эти кучи правила, которые выглядят как набор алгебры, которые трудно запомнить, но которые легко перепутать в голове, и я думаю, что когда у людей возникают, знаете ли, такие кошмарные воспоминания о том, какой была математика в средней школе и что логарифмы, часто на ум приходят именно эти формулы, и сегодня я хочу попытаться обсудить одну из них, как о них думать, но также и просто на метауровне: если вы учите кого-то алгебре, что такое моменты, на которые стоит обратить внимание? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "Каким образом это можно внедрить в их интуицию? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "ой, там 3 нуля, что такое журнал миллиона? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "журнал 1000 раз по x равен 3-кратному журналу x, и помните, что мы используем соглашение, согласно которому это журнал b по основанию 10. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "log 1000 умножения x равен логарифму x в кубе c. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "log 1000 раз x равен 3 в степени log x и e. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "ничего из вышеперечисленного, и помните, как я уже говорил ранее, мы должны полностью ожидать, что все те люди, которые вначале сказали, что хорошо понимают журналы, будут отвечать немедленно, они будут отвечать правильно, но если вы кто-то, кто этого не делает, не позволяйте этому запугать вас, когда вы смотрите на такую проблему, я бы посоветовал вам просто подключить различные степени 10 и думать с точки зрения идеи, что функция журнала подсчитывает количество нулей, поэтому я дам вам немного времени подумать об этом, поэтому я продолжу и оценю его, и, как всегда, если это быстрее, чем вам удобно, знайте, что это только потому, что я хочу продолжить с уроком, поэтому в этом случае правильный ответ будет равен логарифму 1000, умноженному на х, это то же самое, что взять 3 плюс логарифм х, и теперь давайте на мгновение подумаем об этом, и, как я уже говорил, когда вы только начали с ними, я думаю, лучше всего просто удобно вставлять различные числа, и лучше всего вставлять числа, которые уже являются степенями 10, поэтому, если вы спрашиваете что-то вроде журнала 1000 раз x, я не знаю не знаю, давайте просто подставим что-нибудь для x log 1000 умножить на 100, ну, мы знаем, сколько нулей будет в окончательном ответе, ну, 1000 умножить на 100 будет 100 000, у нас уже интуитивно есть идея, что когда мы умножаем 2 степени 10 мы просто берем нули, 3 нуля из этой 1000, 2 нуля из этой 100 и ставим их рядом друг с другом, так что всего должно получиться 5 нулей, но если вы действительно задумаетесь не только о том, как повернулось число но почему так получилось, что это были 3 нуля из этой 1000 плюс 2 нуля из этой 100, которые мы могли бы также записать, сказав количество нулей в 1000 плюс количество нулей в 100, так что эта идея, что логарифм произведения двух чисел — это сумма логарифмов этих двух чисел в контексте степеней 10, это просто передает то, что уже является суперинтуитивной идеей для многих из нас, если вы возьмете 2 степени 10 и умножите их, вы просто возьмем все их нули и как бы напишем их друг на друга, так что то, как я здесь все написал, на самом деле указывает на несколько более общий факт, который будет нашим самым первым свойством логарифмов, а именно: если мы возьмем Логарифм A, умноженный на B, равен логарифму A плюс логарифм B, теперь каждый раз, когда вы видите одно из этих правил логарифма, если вы щуритесь или немного смущены тем, как его запомнить, просто подключите примеры Я излишне, я говорю это много, но это потому, что я думаю, что это очень легко забыть, когда ты погряз в самой алгебре и сидишь на каком-то тесте, а в нем просто много символов чтобы напомнить себе, что с вами все в порядке, достаточно просто подставить несколько чисел, это неплохо, и часто это отличный способ проявить интуицию, поэтому в этом случае, произнося логарифм A, умноженный на B, и разбивая его на части, мы могли бы просто подумать: о, это журнал 100 умножить на 1000, что равно 5, в нем 5 нулей, он разбивается по количеству нулей в каждой данной части, отлично, замечательно, так что развивая эту интуицию дальше, давайте попробуем еще одну практическую задачу и снова, если вы это знаете, отлично, вы сможете ответить на него нормально, но, возможно, подумайте не только о том, каков ответ, но и о том, как бы я объяснил этот ответ кому-то или как бы я попытался заставить студента прийти к этому ответу самостоятельно, без моего указания Каков им ответ? Итак, есть два потенциальных члена аудитории: те, кто заинтересован в самом уроке, а затем те, кто заинтересован в мета-уроке, поэтому наш вопрос снова задается вопросом, что из следующего верно? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "а. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "log от x до n равен n, умноженному на log x b. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "log x в n равен log x в степени n c. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "log от x до n равен n плюс log от x или d. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "Итак, правильный ответ здесь - a, и похоже, что 4000 из вас получили поздравления, сообщив нам, что log x в степени n равен n, умноженному на log x, так что, опять же, предположим, что вы пытаетесь научить этому кому-то, или если вы сами пытаетесь понять, что это значит, я думаю, что хорошее место для начала — это подключить что-то, и в этом случае, для журнала x в степени n, давайте просто попробуем это со 100 в степени 3, и вы можете попробовать его с другими, чтобы увидеть, работают ли шаблоны, которые вы используете, на самом деле, но если вы продумываете это не с точки зрения простого видения ответа, а с точки зрения попытки подумать, почему ответ оказался таким иногда подойдет один пример, потому что 100 в кубе, мы можем думать об этом как о взятии ну, это 3 копии 100. Я беру 3 копии 100, и когда я все это умножаю, я думаю, что журнал - это подсчет количества нулей, которые мы скажем, о, это будет какое-то число, в котором всего 6 нулей, вот что значит взять 100 раз 100 раз 100. Я могу просто подумать о том, чтобы сгруппировать все эти нули вместе, чтобы получить миллион, так что это число будет 6, но если мы на самом деле подумаем, почему это было 6, а не просто количество нулей внутри миллиона, откуда взялась эта 6, то у нас было 3 копии этой 100, и каждая из этих 100 имела 2 разных нуля, так что это более общий вы можете подумать об этом, где, если бы вместо того, чтобы брать 100 в кубе, мы смотрели на 1000 в кубе или 1000 в n или x в степени n, вы можете думать, что это независимо от того, какое значение n было количеством копий, которые мы умножали в разы число ну, давайте посмотрим, это не x, умноженное на количество нулей, которые были в том, что мы подставили вместо x, которое в данном случае было 100, поэтому, если бы вместо этого я взял что-то вроде log 10 000 в степени n, это было бы то же самое если взять n копий этих 10 000 и подсчитать количество нулей в каждом из них, которое равно 4, то это будет n раз 4, и, конечно, общее свойство, на которое большинство из вас правильно ответили, заключается в том, что у вас есть этот прекрасный маленький эффект, когда вы посмотрите журнал чего-то, возведенного в степень, перед которой прыгает маленькая сила, и вы просто получаете журнал того, что было внутри, теперь одно из, возможно, самых важных последствий этого, я не знаю, можно ли это назвать импликация или, если бы вы назвали это повторением определения, если я беру журнал и просто еще раз подчеркиваю, что это основание 10 из 10 в степени n, мы можем как бы думать об этом маленьком n как о прыжке вниз спереди, и он становится n-кратным логарифмом по базе 10 из 10, что, конечно же, равно 1. Это выражение можно рассматривать либо как подсчет количества нулей в конце, либо, в более общем плане, оно запрашивает 10 для того, что равно 10, и ответ просто 1 это очень обнадеживает, потому что вы можете вернуться назад и просто прочитать это исходное выражение, сказав 10 тому, что равно 10, нет, ну ладно, теперь ответ - нет, ок, с каждым заданным свойством логарифма, которое у нас есть, поэтому в этом случае мы только что нашел один логарифм x в степени n, предполагает, что n прыгает вперед, всегда будет экспоненциальное свойство зеркального отображения, и это еще один способ, которым мы можем помочь себе немного интуиции для них, так что позвольте мне просто прикрыть некоторые из будущих свойств, к которым мы здесь собираемся, пытаются скрыть, куда мы идем, что мы только что нашли, поднимая что-то до n, которое прыгает вперед, это соответствует экспоненциальному свойству, что если я возьму 10 до x и подниму все это в степени n, это то же самое, что перевести 10 в n, умноженное на x, и это подводит нас к еще одному интуитивному выводу, который может возникнуть у вас в отношении логарифмов, а именно, они как бы возведены в степень, вывернутое наизнанку, и вот что я имею в виду под что вещь, находящаяся внутри бревна, если я беру журнал a, вы должны думать об этом как о целом внешнем выражении для чего-то экспоненциального, в данном случае a вещь внутри соответствует 10 к x вывод функции, тогда как все это само по себе, журнал а соответствует тому, что находится внутри, вот что такое показатель степени 10, поэтому, где бы вы ни видели здесь логарифмическое выражение, вы должны думать, что это играет роль показателя степени справа и каждый раз, когда вы видите экспоненту, все выражение от 10 до x, весь внешний компонент с правой стороны, который соответствует чему-то, что находится внутри одного из бревен, и мы видели это выше идеи о том, что когда мы умножаем внутри это сложение снаружи, ну, если журналы как бы выворачивают экспоненту наизнанку, это говорит нам, что умножение снаружи, умножение выходных данных функции, это то же самое, что сложение внутри, потому что каждый из этих журналов, например log a и log b играет роль x и y в выражении справа, так что давайте продолжим игру, давайте просто сделаем еще пару таких и посмотрим, для скольких из этих свойств мы можем построить интуицию, поэтому этот последний, очень приятно думать о том, что показатели степени прыгают вниз к следующему, это может показаться немного странным для тех, кто не обязательно знаком с логарифмами, но опять же, подставьте несколько чисел, чтобы получить некоторую интуицию, и мы немного дадим этому больше момента, чтобы подтянуться, что из следующего верно? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "ну, если 10 в кубе равно 1000, это то же самое, что сказать, что 10 равно 1000, возведенному в 1 треть. Выполнение обратного действия здесь включает в себя мультипликативную обратную экспоненту, и в результате получается, что это выглядит как 1, разделенный на 3. и что 3 соответствует логарифмической базе 10 из 1000, это 1, разделенная на логарифмическую базу 10 из 1000, так что в более общем плане вы можете догадаться, основываясь на этом единственном примере, что когда мы меняем базу на то, что внутри, это соответствует делению 1 по тому, что находится снаружи, вы можете обдумать это с точки зрения рассмотрения соответствующего экспоненциального правила, а что случилось с моим прекрасным маленьким журналом и экспонентами? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "замечательно, давайте еще раз спрячем некоторые вещи, некоторые другие свойства, до которых мы доберемся здесь, и я оставлю их в том же порядке, в котором они были у меня раньше. Я думал, что, если их заранее написать, это поможет мне сохранить немного чище, чем обычно, но, возможно, это просто включает в себя игру в эту странную игру с перетасовкой бумаги, поэтому то, что мы только что нашли, база журнала b из a, если вы поменяете их местами, это то же самое, что разделить на 1 то, что это соответствует, от экспоненциальная земля - это если вы возведете b в некоторую степень и скажете, что это равно a, это то же самое, что сказать, что a в обратной степени этой степени снова равно b, полезно воспользоваться моментом и подумать о логарифмах как о повороте вещей наизнанку база журнала выражений b из a играет роль этого x, а база журнала выражений a из b играет роль того, что находится поверх a, а затем симметрично все выражение b в степени x играет роль внутренней части слева, она играет роль a и всего выражения, a в зависимости от чего-то играет роль того, что находится внутри базы журнала a, чтобы вы могли видеть, просто подключив несколько примеров и сопоставляя его с экспоненциальными правилами, мы уже можем придумать три разных правила логарифмирования, которые, если бы они были просто переданы как части алгебры для запоминания, вы знаете, вы могли бы их запомнить, но им очень легко как бы выскользнуть из вашего сознания. Кроме того, очень легко расстроиться из-за поставленной задачи, но вы, возможно, захотите напомнить себе, что причина, по которой мы заботимся о такого рода вещах, заключается в том, что понимание правил логарифмирования помогает нам заниматься математикой в контекстах, где это похоже на рост вируса, где изо дня в день, от шага к шагу все имеет тенденцию расти мультипликативно. Понимание правил логарифмирования помогает вам лучше разобраться в такого рода вещах, поэтому, прежде чем мы приведем хороший реальный пример того, как это может выглядеть. например, позвольте мне задать еще один вопрос викторины в этом духе, чтобы задать последний вопрос о свойствах логарифмов, прежде чем мы перейдем к небольшому примеру из реального мира: избавьтесь от того, что у нас было здесь и сейчас, что из следующего верно? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "log a плюс b равен логарифму a плюс log b log a плюс b равен log a, умноженному на log b log a плюс b равен единице, разделенной на log a плюс log b или log a плюс b равен единице, разделенной на log a, умноженной на log b, или ничего из вышеперечисленного, ах, и теперь у нас нет такого консенсуса, не так ли? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "очень интересно, у нас есть скачки между двумя, так что я дам вам минутку подумать, пока люди отвечают, вообще-то у меня есть небольшой вопрос к аудитории, так что, вы знаете, я просто говорил о том, как мы могли бы думайте с точки зрения мультипликативного роста, и это не обязательно должны быть степени десяти, мы могли бы также сделать что-то вроде степеней трех, где, если вы переходите от одного к трем, к девяти, к двадцати семи, к восьмидесяти одному, все из них мы могли бы сказать, что логарифмическая база трех этих чисел просто растет небольшими небольшими шагами, поэтому логарифмическая база три равна единице, три до того, что равно единице, ответ равен нулю, в общем случае логарифм единицы, независимо от базы, будет быть нулем, логарифм по основанию три из трех, три к тому, что равно трем, равно одному, аналогично логарифм по основанию три из девяти, два ах, вы можете задаться вопросом, в чем мой вопрос, но это поможет вытянуть все это и для моего собственного удовольствия вот, позвольте мне просто выписать еще один логарифм по основанию три из восьмидесяти одного теперь четыре, я слышал, что якобы если вы спросите ребенка, скажем, лет пяти или шести, какое число находится на полпути между одним и девятью, вы скажите, какое число находится на полпути, их инстинкты, как ответить, логарифмические, тогда как наши инстинкты, как правило, более линейны, поэтому мы часто думаем один и девять, у вас есть куча равномерно расположенных чисел между ними два, три, четыре, пять, шесть , семь, восемь, и если вы пойдете прямо на полпути между ними, вы получите пять, но если вы думаете с точки зрения мультипликативного роста, как перейти от одного к девяти, это не вопрос добавления кучи вещей, но вы вырастете на определенную величину, вы вырастете в три раза, затем вы вырастете еще в три раза, предположительно, естественный инстинкт ребенка соответствует слову «три», и предположительно это также соответствует тому, если у вас есть антропологи, изучающие общества, которые имеют t разработали системы бухгалтерского учета и письменной форме так же, как в современном обществе, они ответят на это три, поэтому мой вопрос к аудитории, есть ли у кого-нибудь из вас, кто смотрит прямо сейчас, доступ к маленькому ребенку, скажем, в пределах пяти лет старый, посмотри, можешь ли ты пойти спросить их, какое число находится посередине между одним и девятью, и если сможешь, дайте нам знать в Твиттере, что говорит ребенок, каков его фактический ответ, потому что я не знаю почему, я просто немного скептически отношусь к тому, сработает ли это на практике. Я понимаю, что это не супернаучный способ сделать это. Я не прошу людей, которые смотрят прямую трансляцию на YouTube, опросить своих собственных детей, а затем написать ответ в Твиттере, но ради меня самого это было бы интересно чтобы увидеть какое-то подтверждение обратному нашему вопросу, это первый вопрос, который, похоже, не имеет большого консенсуса в одном направлении, давайте продолжим и оценим его, чтобы увидеть, какой ответ окажется отличным, хорошо, итак 2400 из вас правильно ответили, что это не что-то из вышеперечисленного, что логарифм a плюс b не удовлетворяет ни одному из этих замечательных свойств, и в целом, если мы не собираемся работать с определенными видами приближений, особенно когда в игру вступает натуральный логарифм мы могли бы поговорить об этом в следующий раз, когда сложение входных данных логарифма на самом деле очень странное ощущение, это очень странная вещь, и чтобы почувствовать эту странность, подключите несколько степеней десяти, если я попрошу вас записать плюс b вы можете подумать: хорошо, позвольте мне просто подключить несколько примеров, таких как 10 000 и 100, и я спрошу себя, если я выполню эту функцию подсчета нулей того, что находится в этом входном файле, сколько в нем нулей? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "это интересный вопрос, окей, может ли основание логарифма быть нулем? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "ну, с точки зрения нашего треугольника мы могли бы думать об этом как о том, что нуль в какой-то степени x равен некоторому другому значению y, это то, что мы могли бы написать, либо сказав, что ноль для x равен y, или мы могли бы написать то же самое можно сказать и о том, что нулевой логарифм y равен x ноль тому, что равно x, теперь проблема в том, что ноль любого значения в конечном итоге оказывается равным нулю, так что, если мы просто собираемся думать о нулевом логарифме y для любого другого ввода. y, вы знаете, вы хотите ввести что-то вроде одного или двух или пи, что угодно, вы задаете вопрос от нуля до того, что равно одному, двум, пи или любому другому числу, которое у вас там может быть. и ответа просто не будет, поэтому в лучшем случае вы можете попытаться сказать: о да, журнал нуля, это совершенно допустимая функция, она определяется только для входного нуля, но даже тогда у вас возникнут проблемы с попыткой найти то, что вы хотите там, потому что, если сказать ноль тому, что равно нулю, к нему применимо все, что угодно, поэтому ваша рука будет закручена за спиной, однако вы хотите, чтобы это работало, и это соответствует тому факту, что экспоненциальная функция с нулевым основанием полностью равна нулю. не сопоставляет числа друг с другом в порядке один к одному, так что это отличный вопрос, можете ли вы получить логарифм с нулевым основанием, а теперь вернемся к идее о том, где эти вещи возникают в реальном мире. Один пример, который мне нравится, это шкала Рихтера для землетрясений, поэтому шкала Рихтера дает нам количественную оценку силы землетрясения, и это может быть что угодно, от очень небольших чисел до очень больших чисел, как я думаю, самое сильное землетрясение, когда-либо измеренное, и это всего лишь диаграмма, взятая из Википедия получила 9.5, и чтобы оценить, насколько это безумие, стоит посмотреть на взаимосвязь между тем, что означают эти цифры, а затем что-то вроде эквивалентного количества тротила, своего рода меры того, сколько энергии в нем есть, а затем то, что мы можем попытаться здесь сделать. Посмотрим, сможем ли мы получить выражение для числа по шкале Рихтера через количество энергии и почему логарифмы будут естественным способом описания этого, поэтому ключом, на котором следует сосредоточиться, является то, насколько мы делаем шаги вперед, насколько все увеличивается. так, например, если мы идем от двух колодцев, в этом случае это не показывает нам, где находится три, поэтому, возможно, мы подумаем о том, чтобы сделать шаг от двух до четырех, что похоже на два шага, что это дает с точки зрения количество энергии, ну, похоже, что оно забирает у нас одну метрическую тонну тротила, что, я думаю, является большой бомбой времен Второй мировой войны, и это требует у нас до килотонны в тысячу раз больше, что представляет собой небольшую атомную бомбу, так что всего два шага По шкале Рихтера переход от землетрясения магнитудой 2 к землетрясению магнитудой 4 переносит нас от большой бомбы времен Второй мировой войны к ядерному веку, так что это примечательно, и первый чистый шаг, который мы получаем, - это переход от 4 до 5 баллов по крайней мере, с точки зрения того, что хорошо показывает нам эта диаграмма, и, очевидно, один шаг вверх от 4 до 5 соответствует переходу от 1 килотонны до 32 килотонн, и это, очевидно, был размер бомбы, разрушающей город, которая упала на Нагасаки, так что это, возможно, один вещь, которая может показаться нелогичной в отношении логарифмических шкал, если вы просто слышите в новостях разницу между «о, было землетрясение силой 4». 0 по сравнению с землетрясением, которое было 5.0 легко подумать, да, 4 и 5, это очень похожие числа, но, очевидно, с точки зрения количества тротила это соответствует умножению на 32, чтобы перейти от 1 к следующему, и переходу от 2 к 4, очевидно, было умножением примерно на тысячу, и единственное Причина, по которой это больше, заключается в том, что здесь наша диаграмма не показывала, что такое 3, поэтому мы сделали два шага, и вы можете сами убедиться, что если вы сделаете шаг в 32, а затем умножите еще на 32, это на самом деле довольно близко к тысяче, так что идея о том, что аддитивные шаги числа Рихтера соответствуют мультипликативным шагам в TNT, по-видимому, предполагает, что здесь играет роль что-то логарифмическое, и немного интересно просто продолжать говорить здесь и говорить, насколько это растет, отчасти из-за мировых явлений, которые это описывая да, не является большим сюрпризом то, что, когда мы делаем еще один шаг, оно снова умножается примерно на 32, но, учитывая нашу интуицию, это разница между 32 килотоннами небольшой атомной бомбы и затем одной мегатонной, которую мы могли бы считать не маленькой атомной бомбой, Атомная бомба Нагасаки, которая, я думаю, равна 32 атомным бомбам Нагасаки по одной мегатонне, что, очевидно, соответствует магнитуде плоского землетрясения с двойной струной в Неваде, США, 1994 год. Я не знал, что это было, кстати, спасибо Википедии с точки зрения частот. также посмотрел, очевидно, те, которых меньше двух, такие случаются постоянно, их около 8000 в день, но как только мы попадаем в царство атомных бомб, таких вещей вроде 3.5 и 4, очевидно, тоже происходят довольно часто где-то на земле, около 134 таких случается где-то каждый день, кто знал? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "но по мере того, как мы становимся еще более интенсивными в диапазоне 5 и 6, которые были намного выше шкалы атомной бомбы, теперь мы всего лишь около 2 в день, и я уверен, что геолог мог бы прийти и объяснить, почему нам всем следует Не стоит слишком беспокоиться о том факте, что каждый день происходят разрушения земной коры, эквивалентные двум атомным бомбам, но, по-видимому, особенно редко они концентрируются в каком-то месте, например, в городе, где сейчас живет много людей, просто проверяя нашу мысль о том, что каждый шаг предполагает рост на 32, давайте посмотрим, как выглядит шаг от 6 до 7, и здесь он дает нам гораздо больше примеров между ними, возможно, создает иллюзию, что это больший шаг, чем есть на самом деле, и действительно, в этом разница между 1 мегатонной и 32 мегатонны, так что это умножение на 32. Одна из вещей, которые мне показались наиболее интересными на этой диаграмме, кстати, это то, как далеко нам нужно пройти, прежде чем мы доберемся до самого большого ядерного оружия, которое когда-либо было испытано. Это был разгар холодной войны. Царь-бомба, мощность которой составляла 50 мегатонн, и я полагаю, что у них действительно были первоначальные планы иметь бомбу мощностью 100 мегатонн, но они отговорили себя от этих 50 мегатонн, мы говорим, начните с 32 килотонн бомбы Нагасаки, умножьте на 32, чтобы получить мегатонну умножьте еще на 32, так что мы говорим о том, что сила взрыва, завершившегося Второй мировой войной, в тысячу раз превышает силу, а вы все еще не достигли 50 мегатонн, на которые способно человечество, и это, очевидно, землетрясение на острове Ява в Индонезии, так что 7 . 0 — это не просто немного больше, чем 6.0, он намного больше, и дело здесь, конечно, в том, что когда у вас есть шкала, дающая мультипликативное увеличение, стоит понимать, что то, что выглядит как маленькие шаги, на самом деле может быть огромными шагами с точки зрения подразумеваемой энергии или подразумеваемых здесь абсолютных значений. поэтому, когда мы думаем о том факте, что когда-либо была цифра 9.5, что на самом деле кажется абсурдным, учитывая, что оно есть только в 7.0, мы говорим о самом большом термоядерном оружии, когда-либо созданном, и это указывает на одну область, где логарифмы имеют тенденцию возникать, а именно, когда люди хотят создать шкалу для чего-то, что объясняет чрезвычайно широкий разброс в том, насколько большие вещи могут быть так, что в случае с землетрясениями вы можете иметь вещи из того, что происходит постоянно вокруг Земли, размером с большую ручную гранату, и вы хотите, чтобы это было в вашем масштабе и что-то, о чем стоит подумать, в диапазоне до самого верха. к крупнейшему нарушению, которое мы видели в истории человечества, и для того, чтобы добиться этого таким образом, чтобы вы не просто записывали в свои числа целую кучу разных цифр для одного случая и целую кучу разных, меньших чисел. цифр для вашего номера, в другом случае хорошо взять логарифмы, а затем просто поместить это в единую шкалу, которая в основном сжимает эти числа между 0 и 10, вы видите что-то очень похожее происходит со шкалой децибел для музыки, которая на самом деле немного работает немного по-другому: каждый раз, когда вы делаете шаг вверх на 10 децибел, что соответствует умножению на 10, то вместо шага 1, умноженного на 10, это шаг 10, который умножается на 10, так что это немного усложняет математику. немного странно, но идея та же самая: если вы слушаете звук громкостью 50 децибел против 60 децибел, он намного тише с точки зрения передаваемой и переходной энергии, от 60 до 70 или от 70 до 80 — это шаги, от 60 до 80, которые включают в себя умножение количества энергии на квадратную площадь в 100 раз, поэтому каждый раз, когда вы видите логарифмическую шкалу, знайте, что это означает, что все, о чем идет речь, под капотом увеличивается на огромное количество, вот почему мы увидели множество логарифмических шкал, используемых для описания вспышки коронавируса. Как бы вы могли описать такую зависимость, где каждый раз, когда вы увеличиваете число по шкале Рихтера на 1, вы умножаете на 32, ну, мы мог бы думать в терминах журнала с основанием 32. Я мог бы сказать, что если я возьму журнал, я просто назову r, число для шкалы Рихтера. Я мог бы думать об этом как о логарифме по основанию 32, и это будет соответствовать , нет, нет, я делаю неправильно, это не то, что записывается, мы берем логарифм по основанию 32 от большого числа, числа ТМТ, что-то вроде 1 мегатонны, это 1 миллион тонн, логарифм по основанию 32, это должно соответствуют числу по шкале Рихтера, но может быть какое-то смещение, поэтому мы могли бы сказать, что есть какая-то константа s, которую мы добавляем к этому числу по шкале Рихтера, и это выражение точно такое же, извините за отступление от внизу это выражение точно то же самое, что сказать 32 в степени некоторого смещения, умноженное на наше число по шкале Рихтера, что то же самое, что взять 32 в это смещение, которое само по себе является некой большой константой, умноженное на 32 на число по шкале Рихтера, так что вы можно подумать об этом как о некоторой константе, умноженной на 32 в степени числа, которое вы видите, поэтому такой способ записи действительно подчеркивает его экспоненциальный рост, что, если это то, что соответствует количеству TMT, которое вы видите, по мере того, как вы увеличиваете это r шаг за шагом вы умножаете на 32, но другой способ сообщить тот же самый факт — взять логарифмическую базу 32 для любой этой суммы, это нормально, теперь следующее, о чем я хочу поговорить, это то, что нам не всегда нужно беспокойтесь о том, как вычислять журналы с разными базами, немного странно, что мы говорили о журнале с базой 32, ранее я упоминал, что математикам очень нравится иметь журнал с базой e, ученым-компьютерщикам действительно нравится иметь журнал с базой 2, и это Оказывается, для вычислительных целей или для размышлений о том, как эти вещи растут, если у вас есть один журнал, если вы можете вычислить один тип журнала, будь то основание 10, основание 2, основание e, вы можете вычислить практически все остальное, что теперь вы хотите направить нашу интуицию в этом направлении, давайте вернемся к нашему тесту и перейдем к следующему вопросу, и я считаю, что этот вопрос самый лучший, я не знаю, это наполовину разумный вопрос, это должно быть хорошо это просто подготовит нас к переводу из контекста по основанию 2 в контекст по основанию 10, и это также хорошая интуиция для понимания степени 2, чтобы иметь в целом связь, которую она имеет со степенями 10, потому что это прекрасный вид совпадения природа, что эти двое вроде как хорошо понимают, что я имею в виду, они прекрасно взаимодействуют друг с другом, поэтому возникает наш вопрос, учитывая тот факт, что от 2 до 10 равно 1024, 1024, что примерно равно 1000, поэтому, если вы вы немного ошибаетесь в своих цифрах и просто делаете приближения от 2 до 10, в основном 1000, что из следующего ближе всего к истине? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "нежный. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "Здесь совсем не единогласное решение. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "Но вопрос заключался в том, какое из них ближе всего к истине, и давайте посмотрим, как мы можем думать об этом. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "Итак, это указывает на то, что у вас есть степень 2, которая равна 1024, очень близка к степени 10, примерно 10 в кубе. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "Так что же это значит? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "Если логарифмическое основание 2 из 10 равно x, это то же самое, что сказать, что 2 по x равно 10, верно? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "Он просит нас 2 к тому, что равно 10. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "Вы не можете сделать это с каждой функцией. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "Кажется, люди думают, что это можно сделать с любой функцией, но вы просто не можете. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "И это означает, что х составляет около 10 третей, ясно? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "И действительно, то, что мы видели ранее, это то, что логарифм по основанию 2 из 10, мы также могли бы сказать, что логарифм по основанию 10 из 2 равен всего лишь 1 больше этой суммы, 1 больше x. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "Большой. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "И поскольку мы что-то делаем в журналах, я буду писать именно так. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "Аналогичным образом запишите основание миллиона 2. Давайте посмотрим: если нам нужно умножить 2 само на себя примерно 10 раз, чтобы получить тысячу, нам придется умножить его на себя примерно 20 раз, чтобы получить миллион. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "Он немного меньше, но это своего рода хорошее приближение, которое стоит иметь в виду. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20, мы уменьшаем на ту же величину. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, мы уменьшаем на ту же величину. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "Хорошо? ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "Это интуиция, которую стоит запомнить. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "Хорошо? ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "А потом просто целая куча различных возможных способов объединить базу C из B, умноженную на базу C из A. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "Я уделю вам этому немало времени, потому что это не очевидно, если вы уже не знакомы с логарифмами, и стоит немного подумать. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "Спасибо, Карен. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/spanish/sentence_translations.json b/2020/ldm-logarithms/spanish/sentence_translations.json index 9b54256ae..7e8c7a65c 100644 --- a/2020/ldm-logarithms/spanish/sentence_translations.json +++ b/2020/ldm-logarithms/spanish/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Música🎵 Bienvenido de nuevo a Lockdown Math. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "Hoy vamos a hablar de logaritmos y de una especie de lección de regreso a lo básico. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "Y como siempre, para empezar, sólo quiero tener una idea de dónde se encuentra la audiencia en este momento. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "Entonces, si puedes, ve a 3b1b. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "Nunca había oído hablar de ellos antes ni había aprendido nada de ellos antes b. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "He aprendido sobre ellos pero a veces me confunden todas las propiedades c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "Los entiendo pero no sabría enseñarlos y d. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "Los entiendo bien y podría enseñárselos cómodamente a otra persona para que él también los entienda bien. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "Entonces, tenemos una buena división. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "Como dije, la intención de esto es crear una lección que pueda señalar a las personas en el futuro si simplemente no se sienten cómodas con los logaritmos y quiero poder decir, oh, aquí hay un lugar al que pueden acudir. cómo pienso, ya sabes, cómo creo que podrías abordarlo de forma intuitiva. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "Debido a que estuve navegando por un par de foros de profesores antes de hacer esta conferencia en particular y cuando la gente pregunta cuál es el tema más difícil de enseñar en matemáticas en la escuela secundaria en el sentido de que los estudiantes parecen tener más problemas con él, los logaritmos son uno de los más respuestas comúnmente indicadas, lo cual es interesante y puedo suponer que tal vez sea porque hay un montón de estas propiedades que terminas teniendo que aprender, ya sabes, así que si nos saltamos el lugar al que vamos a ir, tendrás todos estos montones de reglas que parecen un montón de álgebra que pueden ser difíciles de recordar y fáciles de mezclar en la cabeza y creo que cuando la gente tiene, ya sabes, este tipo de recuerdos de pesadilla de cómo eran las matemáticas en la escuela secundaria y cómo logaritmos hicieron por ellos, a menudo me vienen a la mente esas fórmulas particulares y lo que quiero hacer hoy es tratar de hablar sobre una, cómo pensar en ellas, pero también en el meta nivel de si le estás enseñando álgebra a alguien, ¿cuáles son? ¿Cuáles son los puntos que vale la pena destacar? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "¿Cuál es la manera de conseguir que se construya en sus intuiciones? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "Oh, tiene 3 ceros. ¿Cuál es el logaritmo de un millón? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "el log de 1000 por x es igual a 3 veces el log de x y recuerda que estamos usando la convención de que es log b en base 10. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "log de 1000 por x es igual a log de x al cubo c. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "log de 1000 por x es igual a 3 elevado a la potencia de log de x y e. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "nada de lo anterior y recuerde, como dije antes, debemos esperar que todas esas personas al principio que dijeron que entienden bien los registros responderán de inmediato, responderán correctamente, pero si alguien que no lo haga, no deje que eso lo intimide cuando esté analizando un problema como este, lo que le invito a hacer es simplemente conectar varias potencias de 10 y pensar en términos de la idea de que la función de registro cuenta el número de ceros, así que te daré un momento para pensar en eso, así que seguiré adelante y lo calificaré y, como siempre, si eso es más rápido de lo que te hace sentir cómodo, debes saber que es solo porque quiero seguir adelante. con la lección entonces en este caso la respuesta correcta resulta ser log de 1000 x x es lo mismo que tomar 3 más el log de x y ahora pensemos en eso por un momento y como dije cuando recién estás comenzando con ellos creo que lo mejor que puedes hacer es sentirte cómodo ingresando varios números y los mejores números para ingresar son los que ya son potencias de 10, así que si preguntas algo como un registro de 1000 veces x bueno, no lo hago. No lo sé, simplemente conectemos algo para x log de 1000 por 100. Bueno, sabemos cuántos ceros habrá en la respuesta final. Bueno, 1.000 por 100 es 100.000, ya tenemos intuitivamente esta idea de que cuando multiplicamos 2 potencias de 10. Solo estamos tomando los ceros, los 3 ceros de ese 1000, los 2 ceros de ese 100 y los estamos poniendo uno al lado del otro, por lo que deberían ser 5 ceros en total, pero si realmente reflexionas no solo sobre cómo giró el número. pero ¿por qué resultó de esa manera? Fueron los 3 ceros de ese 1000 más los 2 ceros de ese 100 que también podríamos escribir diciendo el número de ceros en 1000 más el número de ceros en 100, entonces esta idea de que un logaritmo del producto de dos cosas es la suma de los logaritmos de esas dos cosas en el contexto de potencias de 10, eso simplemente comunica lo que ya es una idea súper intuitiva para muchos de nosotros, si tomas 2 potencias de 10 y las multiplicas, simplemente tomamos todos sus ceros y los amontonamos unos sobre otros, de modo que la forma en que he escrito las cosas aquí en realidad es indicativo de un hecho ligeramente más general que será nuestra primera propiedad de los logaritmos, que es que si tomamos el log de A multiplicado por B es igual al log de A más el log de B ahora, cada vez que veas una de estas reglas de logaritmos, si entrecierras los ojos o estás un poco confundido sobre cómo recordarlo, simplemente ingresa ejemplos Estoy siendo redundante, digo esto mucho, pero es porque creo que es muy fácil olvidar una vez que estás inundado en el álgebra misma y estás sentado en algún tipo de prueba y solo tiene muchos símbolos. para recordar que está bien simplemente ingresar algunos números, eso es algo bueno y, a menudo, es una excelente manera de generar intuición, por lo que en este caso, al decir log de A multiplicado por B y separarlo, podríamos pensar, oh, eso registro de 100 por 1000, que es 5, hay 5 ceros, se divide en términos del número de ceros en cada parte dada. Genial, maravilloso, así que llevando esa intuición más lejos, intentemos con otro problema de práctica y nuevamente, si lo sabes, genial. Podrás responderla bien, pero tal vez pienses, no solo cuál es la respuesta, sino cómo le explicaría esta respuesta a alguien o cómo intentaría que un estudiante llegue a esta respuesta por sí solo sin que yo tenga que decírselo. Entonces, hay dos miembros potenciales de la audiencia: aquellos que están interesados en la lección en sí y luego aquellos que están interesados en la metalección, por lo que nuestra pregunta es, nuevamente, ¿cuál de las siguientes afirmaciones es cierta? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "a. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "log de x elevado a n es igual a n veces log de x b. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "log de x elevado a n es igual a log de x elevado a n c. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "log de x elevado a n es igual a n más log de x o d. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "entonces la respuesta correcta aquí es a, y parece que 4000 de ustedes recibieron felicitaciones, diciéndonos que log de x elevado a n es igual a n veces log de x entonces, nuevamente, digamos que están tratando de enseñar esto a alguien o si estás tratando de entender lo que significa, creo que un buen lugar para comenzar es enchufar algo y en este caso, para log de x elevado a n, intentémoslo con 100 elevado a 3 y podrías probarlo con otros para ver si los patrones que estás haciendo realmente funcionan, pero si lo piensas detenidamente no en términos de simplemente ver cuál es la respuesta, sino de tratar de pensar por qué la respuesta resultó de esa manera. A veces un ejemplo sirve porque 100 al cubo, podemos pensar que eso es tomar bien, son 3 copias de 100. Estoy tomando 3 copias de 100 y cuando multiplico todo eso, pienso que log es contar el número de ceros. digamos, oh, va a ser un número que solo tendrá 6 ceros, eso es lo que significa tomar 100 por 100 por 100. Se me ocurre agrupar todos esos ceros para obtener un millón, por lo que este número será 6 pero si pensamos en realidad por qué era 6, no solo ese es el número de ceros dentro del millón de donde vino ese 6, es que teníamos 3 copias de ese 100 y cada uno de esos 100 tenía 2 ceros diferentes, de esa manera es más general. Puedes pensar en ello si en lugar de tomar 100 al cubo estuviéramos mirando 1000 al cubo o 1000 elevado a n o x elevado a n, puedes pensar que es cualquiera que ese valor de n fuera el número de copias que estábamos multiplicando por tiempos. el número de bueno, veamos, no es x multiplicado por el número de ceros que había en lo que sustituimos por x, que en este caso era 100, así que si en lugar de eso hubiera tomado algo como log de 10,000 elevado a n, esto sería lo mismo. como tomar n copias de esos 10,000 contando el número de ceros en cada una de ellas, que es 4, por lo que sería n veces 4 y, por supuesto, la propiedad general que la mayoría de ustedes respondió correctamente es que tiene este pequeño y encantador efecto donde cuando ve el registro de algo elevado a una potencia, ese pequeño poder salta frente a él y solo tienes un registro de lo que había en el interior, ahora una de las implicaciones quizás más importantes de eso, no sé si lo llamarías. una implicación o si lo llamarías una reformulación de la definición si estoy tomando log y simplemente volveré a enfatizar que es base 10 de 10 a la potencia n, podemos pensar en esa pequeña n como si saltara hacia abajo frente y se convierte en n veces el logaritmo en base 10 de 10, que es, por supuesto, 1. Esta expresión se puede considerar como contar el número de ceros al final o, más generalmente, pedir 10 a lo que es igual a 10 y la respuesta es simplemente 1. lo cual es muy tranquilizador porque otra manera de regresar y simplemente leer esta expresión original es decir 10 elevado a lo que es igual a 10 elevado a n oh bueno, la respuesta es n ahora con cada propiedad de logaritmo dada que tenemos, así que en este caso Acabo de encontrar un logaritmo de x elevado a n implica que n saltando al frente siempre habrá una propiedad exponencial de imagen especular y esa es otra forma en la que podemos ayudarnos a tener un poco de intuición para esto, así que déjenme taparlo. algunas de las propiedades futuras a las que llegaremos aquí intentan ocultar hacia dónde vamos lo que acabamos de encontrar elevando algo a la n que salta al frente esto corresponde a la propiedad exponencial que si tomo 10 a la x y elevo Todo eso elevado a la potencia n es lo mismo que tomar 10 elevado a n por x y esto nos lleva a otra intuición que podrías tener para los logaritmos, que es como una exponenciación al revés y esto es lo que quiero decir con que la cosa que se encuentra en el interior del registro, si estoy tomando el registro de a, debería pensar en eso como la expresión externa completa para algo que es exponencial, en este caso la a, la cosa en el interior corresponde a 10 elevado a x la salida de la función, mientras que todo en sí, el registro de a corresponde a lo que hay en el interior aquí, justo cuál es el exponente del 10, por lo que dondequiera que vea una expresión logarítmica aquí debería pensar que desempeña el papel de un exponente a la derecha. lado y cada vez que ves un exponencial, el 10 completo a la expresión x, todo el componente externo en el lado derecho que corresponde a algo que está ubicado en el interior de uno de los troncos y vimos esto arriba de la idea de que cuando multiplicamos en el interior eso es sumar en el exterior bueno, si los registros se vuelven exponenciales de adentro hacia afuera, eso nos dice que multiplicar en el exterior multiplicar las salidas de la función es lo mismo que sumar en el interior porque a cada uno de estos registros les gusta log a y log b está desempeñando el papel de x e y en la expresión de la derecha, así que sigamos jugando, hagamos un par más de estas y veamos para cuántas de estas propiedades podemos desarrollar una intuición, así que esta última, Muy bien pensar en exponentes que saltan hacia abajo es algo que puede parecer un poco extraño para aquellos que no están necesariamente familiarizados con los logaritmos, pero nuevamente, ingrese algunos números para ganar algo de intuición y le daremos un poco. ¿Más un momento para sacar cuál de las siguientes afirmaciones es verdadera? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "bueno, si 10 al cubo es 1000, eso es lo mismo que decir que 10 es igual a 1000 elevado a 1 tercio. Hacer lo inverso aquí implica el inverso multiplicativo del exponente y la forma en que resulta es que parece 1 dividido por 3. y que 3 corresponde al logaritmo en base 10 de 1000, es 1 dividido por el logaritmo en base 10 de 1000, por lo que, en términos más generales, puedes adivinar, basándose en este único ejemplo, que cuando intercambiamos la base con lo que hay en el interior, corresponde a tomar 1 dividido. por lo que hay afuera una y otra vez, puedes pensar en esto en términos de observar la regla exponencial correspondiente. Ahora, ¿qué pasó con mi pequeño y encantador logaritmo y mis exponenciales? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "maravilloso así que, de nuevo, ocultemos algunas de las cosas, algunas de las otras propiedades a las que llegaremos aquí y las mantendré en el mismo orden en que las tenía antes. Estaba pensando que tenerlas escritas previamente podría mantenerme un poco más limpio de lo habitual, pero tal vez solo implique jugar este extraño juego de cortar papel barajando, así que lo que acabamos de encontrar, registra la base b de a, si los intercambias, es lo mismo que dividir por 1 lo que corresponde a, de un La tierra exponencial es que si elevas b a alguna potencia y dices que es igual a a, es lo mismo que decir que a elevado a la inversa de esa potencia es igual a b nuevamente, es útil tomarse un momento y pensar en los logaritmos como cosas que giran. De adentro hacia afuera, la expresión log base b de a está desempeñando el papel de esa x y la expresión log base a de b está desempeñando el papel de cualquier cosa que se encuentre encima de a y luego, simétricamente, toda la expresión b elevada a x está desempeñando el papel del interior a la izquierda, juega el papel de a y la expresión completa, a al poder de algo juega el papel de lo que está dentro de la base del tronco a para que puedas verlo, simplemente conectando algunos ejemplos y al correlacionarlo con las reglas exponenciales, ya podemos pensar en tres reglas de logaritmos diferentes que, si se transmitieran simplemente como piezas de álgebra para memorizar, ya sabes, podrías memorizarlas, pero es muy fácil que se te escapen de la memoria. cabeza y también es muy fácil frustrarse por la tarea en cuestión, pero tal vez quieras recordar que la razón por la que nos preocupamos por este tipo de cosas es que comprender las reglas de los logaritmos nos ayuda a hacer matemáticas en contextos donde es como un virus creciendo donde De un día para otro, de un paso al siguiente, las cosas tienden a crecer multiplicativamente. Comprender las reglas de los logaritmos te ayuda a tener una mejor idea de ese tipo de cosas, así que antes de que veamos un buen ejemplo del mundo real de cómo puede verse. Permítanme hacer una pregunta más en este sentido para preguntar sobre las propiedades de los logaritmos. Una última antes de pasar a un pequeño ejemplo del mundo real. Deshacernos de lo que teníamos aquí y ahora. ¿Cuál de las siguientes afirmaciones es cierta? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "log de a más b es lo mismo que log de a más log de b log de a más b es igual a log de a multiplicado por log de b log de a más b es igual a uno dividido por log de a más log de b o log de a más b es igual a uno dividido por log de a por log de b o ninguna de las anteriores ah, y ahora no tenemos tanto consenso, ¿verdad? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "muy interesante, tenemos una carrera de caballos entre dos, así que les daré un momento para pensar en esto mientras la gente responde. De hecho, tengo una pequeña pregunta para la audiencia, así que, ya saben, solo estaba hablando de cómo podríamos Piense en términos de crecimiento multiplicativo y eso no tiene que ser solo potencias de diez, también podríamos hacer algo como potencias de tres donde si vas de uno a tres a nueve a veintisiete a ochenta y uno, todos de estos, podríamos decir que el logaritmo en base tres de estos números simplemente crece en pequeños pasos, por lo que el logaritmo en base tres de uno, tres elevado a lo que es igual a uno, la respuesta es cero, en general, el logaritmo de uno, sin importar la base, ser cero logaritmo en base tres de tres, tres elevado a lo que es igual a tres es uno de manera similar logaritmo en base tres de nueve es dos ah, quizás te preguntes cuál es mi pregunta, pero me ayudará a extraer todo esto y para mi propio placer aquí, déjame escribir un logaritmo más en base tres de ochenta y uno es cuatro ahora, he escuchado eso aparentemente si le preguntas a un niño, digamos que tiene cinco o seis años, qué número está a medio camino entre uno y nueve. dicen qué número está a la mitad, sus instintos sobre cómo responder son logarítmicos, mientras que nuestros instintos tienden a ser más lineales, por lo que a menudo pensamos en uno y nueve, tienes un montón de números espaciados uniformemente entre ellos dos, tres, cuatro, cinco, seis. , siete, ocho y si vas justo a la mitad del camino, llegarás a cinco, pero si estás pensando en términos de crecimiento multiplicativo dónde llegar del uno al nueve, no es cuestión de sumar un montón de cosas, sino que "Estás creciendo en una cierta cantidad, creces en un factor de tres, luego creces en otro factor de tres, supuestamente, el instinto natural de un niño se alinea con decir tres y supuestamente esto también se alinea con antropólogos que estudian sociedades que tienen". Desarrollamos sistemas de contabilidad y escritura de la misma manera que lo han hecho las sociedades modernas. Responderán a tres preguntas. Mi pregunta para la audiencia: si alguno de ustedes que está mirando ahora mismo tiene acceso a un niño pequeño, digamos, en el rango de cinco años. viejo a ver si puedes ir a preguntarles qué número está a medio camino entre el uno y el nueve y si puedes, cuéntanos en Twitter qué dice el niño cuál es su respuesta real porque no sé por qué, solo estoy un poco Soy escéptico sobre si eso realmente funciona en la práctica. Entiendo que esta no es una forma súper científica de hacerlo. No estoy pidiendo a las personas que miran una transmisión en vivo de YouTube que encuesten a sus propios hijos y luego tuiteen la respuesta, pero por mi propio bien, sería interesante. para ver algún tipo de validación, volviendo a nuestra pregunta, esta es la primera que no parece tener un gran consenso en una dirección, sigamos adelante y califiquemos para ver cuál resulta ser la respuesta excelente, está bien, entonces 2,400 de ustedes respondieron correctamente que no es nada de lo anterior que el registro de a más b no satisface ninguna de estas buenas propiedades y, en general, a menos que vayamos a trabajar con ciertos tipos de aproximaciones, especialmente cuando el registro natural entra en juego. podríamos hablar de esto la próxima vez agregar las entradas de un logaritmo es en realidad una sensación muy extraña, es algo muy extraño de hacer y para tener una idea de esa rareza, inserte algunas potencias de diez si le pido el registro de a más b Lo que podrían empezar a pensar es, está bien, déjenme introducir algunos ejemplos como 10,000 y 100 y me pregunto, si hago esta función de conteo de ceros de lo que hay en esa entrada, ¿cuántos ceros hay en ella? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "Esa es una pregunta interesante, ¿puede la base de un logaritmo ser cero? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "bueno, en términos de nuestro triángulo, podríamos pensar en eso como decir, ya sabes, cero elevado a algún tipo de potencia x es igual a algún otro valor y esto es algo que podríamos escribir diciendo que cero elevado a x es igual a y o podríamos escribir lo mismo al decir log base cero de y es igual a x cero a lo que es igual a x ahora el problema aquí es que cero para cualquier cosa termina siendo cero, así que si solo vamos a pensar en log base cero de y para cualquier otra entrada y ya sabes, quieres ingresar algo como uno o dos o pi, cualquier cosa que quieras, estás haciendo la pregunta cero a lo que es igual a uno o dos o pi o cualquier número que puedas tener ahí y simplemente no habrá una respuesta, así que en el mejor de los casos podrías intentar decir oh sí, log de cero, es una función perfectamente válida, solo está definida en la entrada cero, pero incluso entonces tendrías problemas para intentar encontrar lo que quieres. ahí porque decir cero a lo que es igual a cero es como si todo se aplicara a ello, por lo que tu brazo estará torcido detrás de tu espalda, sin embargo quieres que eso funcione y corresponde al hecho de que la función exponencial con base cero es completamente cero. no asigna números uno a uno entre sí, así que esa es una gran pregunta, ¿puedes tener un logaritmo en base cero ahora? Volviendo a la idea de dónde surgen estas cosas en el mundo real, un ejemplo que me gusta es la escala de Richter para terremotos, por lo que la escala de Richter nos da una cuantificación de qué tan fuerte es un terremoto y puede ser cualquier cosa, desde números muy pequeños hasta números muy grandes, como creo que es el terremoto más grande jamás medido y este es solo un gráfico que proviene de Wikipedia fue un 9.5 y para apreciar cuán loco es eso, vale la pena observar la relación entre lo que significan estos números y luego algo así como la cantidad equivalente de TNT, algún tipo de medida de cuánta energía hay en él y luego lo que podemos intentar hacer aquí. Es ver si podemos obtener una expresión para el número de la escala de Richter en términos de la cantidad de energía y por qué los logaritmos serían una forma natural de describir esto, por lo que la clave en la que debemos centrarnos es, a medida que avanzamos, en cuánto aumentan las cosas. Entonces, por ejemplo, si pasamos de dos, en este caso no nos muestra dónde está tres, así que tal vez pensemos en dar un paso de dos a cuatro, que es algo así como dar dos pasos, ¿qué efecto tiene eso en términos de Bueno, parece que nos lleva desde una tonelada métrica de TNT, que es, supongo, una bomba grande de la Segunda Guerra Mundial, hasta un kilotón, mil veces más, que es una bomba atómica pequeña, así que solo dos pasos. en la escala de Richter pasar de un terremoto de magnitud 2 a un terremoto de magnitud 4 nos lleva desde la gran bomba de la Segunda Guerra Mundial hasta la era nuclear, eso es digno de mención y el primer paso limpio que obtenemos es pasar de 4 a 5 en al menos en términos de lo que este gráfico nos muestra muy bien y evidentemente un solo paso hacia arriba de 4 a 5 corresponde a pasar de 1 kilotón a 32 kilotones y ese era evidentemente el tamaño de la bomba destructora de la ciudad que aterrizó en Nagasaki, así que esta es quizás una Algo que puede ser contradictorio acerca de las escalas logarítmicas si solo escuchas en las noticias la diferencia entre oh, hubo un terremoto y fue de 4.0 versus un terremoto que fue un 5.0, es fácil pensar que sí, 4 y 5, esos son números bastante similares, pero evidentemente en términos de cantidades de TNT, eso corresponde a multiplicar por 32 para pasar del 1 al siguiente y pasar del 2 al 4 evidentemente fue multiplicar por aproximadamente mil y el único La razón por la que es más grande es porque aquí nuestro gráfico no mostraba lo que era 3, así que tomamos dos pasos y puedes verificar por ti mismo que si das un paso de 32 y luego lo multiplicas por otro 32, eso en realidad es bastante cercano a mil. La idea de que los pasos aditivos en el número de Richter corresponden a pasos multiplicativos en el TNT parece sugerir que algo logarítmico está en juego aquí y es un poco interesante continuar aquí y decir cuánto crece esto, en parte debido a los fenómenos mundiales que Describir que sí, no es una gran sorpresa que a medida que damos un paso más se multiplique por aproximadamente 32 nuevamente, pero controlando eso en nuestras intuiciones, esa es la diferencia entre 32 kilotones de una bomba atómica pequeña y luego un megatón que podríamos considerar una bomba atómica no pequeña. Bomba atómica de Nagasaki, que supongo que son 32 de las bombas atómicas de Nagasaki por un megatón, que evidentemente es la magnitud del terremoto plano de doble cadena en Nevada, EE. UU., 1994. No sabía qué era eso, gracias Wikipedia en términos de frecuencias, por cierto. También busqué estos, evidentemente, que son menos de dos, suceden todo el tiempo, hay como 8000 por día, pero tan pronto como estamos en el ámbito de las bombas atómicas, cosas como 3.5 y 4, evidentemente también suceden con bastante frecuencia en algún lugar de la Tierra; hay alrededor de 134 de esos que suceden en algún lugar todos los días, ¿quién lo sabía? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "pero a medida que nos volvemos aún más intensos en este rango de 5 y 6, que estaban muy por encima de la escala de la bomba atómica, ahora solo estamos en alrededor de 2 por día y estoy seguro de que un geólogo podría venir y explicar por qué todos deberíamos hacerlo. No debemos preocuparnos mucho por el hecho de que cada día se producen dos perturbaciones equivalentes a las de una bomba atómica en la corteza terrestre, pero presumiblemente es particularmente raro que se concentren en algún lugar como una ciudad donde ahora vive mucha gente, simplemente verificando nuestra idea de que cada paso implica un crecimiento de 32, veamos cómo se ve el paso de 6 a 7 y aquí nos da muchos más ejemplos intermedios, tal vez dando la ilusión de que es un paso más grande de lo que realmente es y, de hecho, esa es la diferencia entre 1 megatón y 32 megatones, eso es multiplicarlo por 32. Por cierto, una de las cosas que encontré más interesantes en este gráfico fue ver qué tan lejos tenemos que llegar antes de llegar al arma nuclear más grande que jamás se haya probado. Esto fue el apogeo de la guerra fría. la bomba del Zar era de 50 megatones y creo que en realidad tenían planes originales de tener una bomba de 100 megatones, pero se convencieron de bajar esos 50 megatones, estamos hablando de comenzar con esos 32 kilotones de la bomba de Nagasaki, multiplicarlos por 32 para obtener una megatones se multiplican por otros 32, así que estamos hablando de mil veces la fuerza de la explosión que puso fin a la Segunda Guerra Mundial y todavía no estamos en los 50 megatones de lo que la humanidad es capaz de hacer y eso es evidentemente el terremoto de Java en Indonesia, por lo que 7 . 0 no es sólo un poquito más grande que 6.0, es mucho más grande y el punto aquí, por supuesto, es que cuando tienes una escala que te da aumentos multiplicativos, vale la pena apreciar que lo que parecen pequeños pasos en realidad pueden ser pasos enormes en términos de la energía implicada o los valores absolutos implicados aquí. Entonces, cuando pensamos en el hecho de que alguna vez hubo un 9.5 eso en realidad parece absurdo dado que solo está en el 7.0, estamos hablando del arma termonuclear más grande jamás lanzada y esto es indicativo de un área donde los logaritmos tienden a aparecer: cuando los humanos quieren crear una escala para algo que represente una variación enormemente amplia en el tamaño de las cosas. Sea así, en el caso del tamaño de los terremotos, puedes tener cosas de lo que sucede todo el tiempo alrededor de la Tierra, el tamaño de una granada de mano grande y quieres que eso esté en tu escala y algo en qué pensar, que vaya hasta arriba. a la mayor alteración que hemos visto en la historia de la humanidad y para lograrlo de manera que no estés simplemente escribiendo un montón de dígitos diferentes en tus números para un caso y un montón de números diferentes, un número más pequeño de dígitos para tu número, en otro caso, es bueno tomar logaritmos y luego simplemente ponerlos en una escala única que básicamente aplasta esos números entre 0 y 10. Ves que sucede algo muy similar con la escala de decibelios para la música, que en realidad funciona un poco. un poco diferente donde cada vez que das un paso hacia arriba de 10 decibeles eso corresponde a multiplicar por 10, por lo que en lugar de un paso de 1 multiplicado por 10, es un paso de 10 que se multiplica por 10, por lo que eso hace que las matemáticas sean un poco un poco loco, pero la idea es la misma, que si estás escuchando un sonido de 50 decibelios versus 60 decibeles, es mucho más silencioso en términos de la energía que se transmite y pasa de, ¿cuál sería? 60 a 70 o 70 a 80 esos pasos, de 60 a 80, que implican multiplicar la cantidad de energía por área cuadrada por un factor de 100, de modo que cada vez que veas una escala logarítmica, sepas mentalmente que eso significa que cualquier cosa a la que se refiere debajo del capó crece en En gran medida, esta es nuevamente la razón por la que vimos muchas escalas logarítmicas utilizadas para describir el brote de coronavirus. Entonces, ¿cómo podrías describir una relación como esta en la que cada vez que aumentas el número de la escala de Richter en 1, lo multiplicas por 32? Podría pensar en términos de un registro con base 32. Podría decir que si tomo el registro de, simplemente voy a llamar a r, el número de la escala de Richter. Podría pensar en esto como un registro en base 32 y eso corresponderá a , no no no, estoy haciendo esto mal eso no es lo que se registra tomamos el registro base 32 del número grande, del número TMT, algo que era como 1 megatón es 1 millón de toneladas el registro base 32, eso debería corresponden al número de la escala de Richter pero puede haber algún tipo de compensación, por lo que podríamos decir que hay algún tipo de constante s que estamos sumando a este número de la escala de Richter y esta expresión es exactamente la misma, disculpe por salirme del Abajo, esta expresión es exactamente lo mismo que decir 32 elevado a la potencia de algún desplazamiento multiplicado por nuestro número de la escala Richter, que es lo mismo que tomar 32 a ese desplazamiento, que en sí mismo es solo una gran constante, multiplicado por 32 elevado al número de la escala Richter, así que Podría pensar que esto es simplemente una constante multiplicada por 32 elevado a la potencia del número que ve, por lo que esta forma de escribir realmente enfatiza el crecimiento exponencial del mismo. Si esto es lo que corresponde a la cantidad de TMT que ve, a medida que aumenta eso. Paso a paso estás multiplicando por 32, pero otra forma de comunicar exactamente el mismo hecho es tomar el logaritmo en base 32 de cualquier cantidad que sea. Ahora lo siguiente de lo que quiero hablar es de cómo no siempre tenemos que hacerlo. preocuparse por cómo calcular registros de diferentes bases, es un poco extraño que estuviéramos hablando de log en base 32. Anteriormente mencioné cómo a los matemáticos realmente les gusta tener un registro con base e, a los informáticos realmente les gusta tener un registro con base 2 y resulta para fines computacionales o también para pensar en cómo crecen estas cosas si tienes un registro, si eres capaz de calcular un tipo de registro, ya sea base 10, base 2, base e, puedes calcular prácticamente cualquier otra cosa que Ahora quieres llevar nuestras intuiciones en esa dirección, volvamos a nuestro cuestionario y pasemos a la siguiente pregunta. Creo que esta pregunta es la más, no lo sé, esta es una pregunta medio razonable, debería ser agradable. esto simplemente nos preparará para traducir del contexto de base 2 al contexto de base 10 y también es una buena intuición para entender que las potencias de 2 tienen en general la relación que tiene con las potencias de 10 porque es este encantador tipo de coincidencia de naturaleza, estos dos, bueno, verás lo que quiero decir, combinan muy bien entre sí, por lo que nuestra pregunta es, dado el hecho de que 2 elevado a 10 es 1024, 1024, que es aproximadamente 1000, por lo que si estás siendo un un poco flojo con tus números y solo estás haciendo aproximaciones de 2 elevado a 10, básicamente 1000, ¿cuál de las siguientes afirmaciones se acerca más a ser cierta? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "licitación. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "En este caso no se trata en absoluto de una decisión unánime. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "Pero la pregunta era cuál es la más cercana a ser cierta, y veamos cómo podemos pensar en esto. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "Entonces señala que tienes una potencia de 2, que es 1024, muy cerca de una potencia de 10, aproximadamente 10 al cubo. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "Entonces, ¿qué significa esto? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "Si log en base 2 de 10 es igual a x, eso es lo mismo que decir 2 elevado a x es igual a 10, ¿verdad? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "Nos está pidiendo 2 elevado a lo que es igual a 10. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "No puedes hacer eso con todas las funciones. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "La gente parece pensar que puedes hacer eso con cualquier función, pero simplemente no puedes. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "Y lo que eso significa es que x es aproximadamente 10 tercios, ¿vale? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "Y bueno, lo que vimos antes es que log en base 2 de 10, también podríamos decir que log en base 10 de 2 es solo 1 sobre esa cantidad, 1 sobre x. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "Excelente. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "Y como estamos haciendo cosas en los registros, lo escribiré de esa manera. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "De manera similar, log en base 2 de un millón, bueno, veamos, si tenemos que multiplicar 2 por sí mismo unas 10 veces para llegar a mil, deberíamos tener que multiplicarlo por sí mismo unas 20 veces para llegar a un millón. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "Es un poco más pequeño, pero es una buena aproximación para tener en cuenta. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20, reducimos la misma cantidad. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, reducimos la misma cantidad. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "¿Bueno? ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "Ésta es una intuición que vale la pena recordar. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "¿Bueno? ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "Y luego solo un montón de formas posibles de combinar log base C de B multiplicado log base C de A. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "Le daré un tiempo significativo en este caso porque no es obvio a menos que ya esté familiarizado con los logaritmos, y vale la pena pensar un poco. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "Gracias karen. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/tamil/sentence_translations.json b/2020/ldm-logarithms/tamil/sentence_translations.json index aa87d7e89..38db43366 100644 --- a/2020/ldm-logarithms/tamil/sentence_translations.json +++ b/2020/ldm-logarithms/tamil/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵இசை🎵 லாக்டவுன் கணிதத்திற்கு மீண்டும் வரவேற்கிறோம். இன்று நாம் மடக்கைகளைப் பற்றி பேசப் போகிறோம் மற்றும் பாடத்தின் அடிப்படைகளுக்குத் திரும்புவோம். ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "எப்பொழுதும் போல, விஷயங்களைத் தொடங்க, பார்வையாளர்கள் இப்போது எங்கே இருக்கிறார்கள் என்பதை உணர விரும்புகிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "எனவே, நீங்கள் 3b1b க்கு செல்ல முடியுமானால். ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "நான் அவர்களைப் பற்றி இதற்கு முன் கேள்விப்பட்டதில்லை அல்லது அவர்களைப் பற்றி அறியவே இல்லை பி. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "நான் அவற்றைப் பற்றி கற்றுக்கொண்டேன் ஆனால் சில நேரங்களில் எல்லா பண்புகளாலும் குழப்பமடைகிறேன் c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "நான் அவர்களை புரிந்துகொள்கிறேன் ஆனால் அவர்களுக்கு எப்படி கற்பிப்பது என்று தெரியவில்லை. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "நான் அவர்களை நன்றாகப் புரிந்துகொள்கிறேன், அவர்களுக்கும் நன்றாகப் புரியவைக்க வசதியாக வேறு ஒருவருக்குக் கற்பிக்க முடியும். ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "எனவே, எங்களுக்கு ஒரு நல்ல பிளவு உள்ளது. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "நான் சொன்னது போல், இதன் நோக்கம், எதிர்காலத்தில் மக்கள் மடக்கைகளில் வசதியாக இல்லாவிட்டால், நான் அவர்களுக்குச் சுட்டிக்காட்டக்கூடிய ஒரு பாடத்தை உருவாக்குவதாகும், மேலும் நான் சொல்ல விரும்புகிறேன், ஓ, இதோ நீங்கள் செல்லக்கூடிய இடம் நான் எப்படி நினைக்கிறேன், உங்களுக்குத் தெரியும், நீங்கள் அதை எப்படி உள்ளுணர்வாக அணுகலாம் என்று நான் நினைக்கிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "இந்தக் குறிப்பிட்ட விரிவுரையைச் செய்வதற்கு முன் நான் இரண்டு ஆசிரியர் மன்றங்களைச் சுற்றிக் கொண்டிருந்ததால், உயர்நிலைப் பள்ளிக் கணிதத்தில் மாணவர்களுக்குக் கற்பிக்க கடினமான தலைப்பு எது என்று மக்கள் கேட்கும்போது, அதில் மாணவர்கள் மிகவும் சிரமப்படுவதாகத் தோன்றினால், மடக்கைகள் மிகவும் ஒன்றாகும். பொதுவாகக் குறிப்பிடப்படும் பதில்கள் சுவாரஸ்யமானவை மற்றும் நான் யூகிக்க முடியும், ஏனெனில் இந்த பண்புகள் ஒரு டன் உங்களுக்குத் தெரிந்திருக்க வேண்டும், எனவே நாங்கள் எங்கு செல்லப் போகிறோம் என்பதைத் தவிர்த்தால், இந்த குவியல்கள் அனைத்தும் உங்களிடம் உள்ளன. இயற்கணிதம் போல் இருக்கும் விதிகள் நினைவில் கொள்ள கடினமாகவும் எளிதாகவும் உங்கள் தலையில் விஷயங்களைக் கலக்கலாம், மேலும் உயர்நிலைப் பள்ளிக் கணிதம் எப்படி இருந்தது, எப்படி இருந்தது என்பதைப் பற்றிய இந்த மாதிரியான பயங்கரமான நினைவுகள் மக்களுக்கு இருக்கும் போது நான் நினைக்கிறேன். மடக்கைகள் அவர்களுக்காகச் செய்தன, அது பெரும்பாலும் அந்த குறிப்பிட்ட சூத்திரங்கள் நினைவுக்கு வருகின்றன, இன்று நான் செய்ய விரும்புவது ஒன்றைப் பற்றி பேச முயற்சிப்பது, அவற்றைப் பற்றி எப்படி சிந்திக்க வேண்டும், ஆனால் நீங்கள் யாருக்காவது இயற்கணிதம் கற்பிக்கிறீர்களா, என்ன என்ற மெட்டா மட்டத்தில் வலியுறுத்த வேண்டிய புள்ளிகள்? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "அதை அவர்களின் உள்ளுணர்வில் கட்டமைக்க என்ன வழி? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "ஓ, அதில் 3 பூஜ்ஜியங்கள் உள்ளன, ஒரு மில்லியனின் பதிவு என்ன? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "1000 மடங்கு x இன் பதிவானது x இன் பதிவின் 3 மடங்குக்கு சமம் மேலும் இது அடிப்படை 10 பதிவு b என்ற மரபைப் பயன்படுத்துகிறோம் என்பதை நினைவில் கொள்க. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "1000 மடங்கு x பதிவு x கனசதுர சியின் பதிவிற்கு சமம். ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "1000 மடங்கு x இன் பதிவு x மற்றும் e இன் பதிவின் சக்திக்கு 3 சமம். ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "மேலே கூறப்பட்டவை எதுவுமில்லை, நான் முன்பு கூறியது போல் நினைவில் வைத்துக்கொள்ளுங்கள், ஆரம்பத்தில் தங்களுக்குப் பதிவுகள் நன்றாகப் புரியும் என்று சொன்னவர்கள் அனைவரும் உடனடியாகப் பதிலளிப்பார்கள், அவர்கள் சரியாகப் பதிலளிப்பார்கள் ஆனால் நீங்கள் இருந்தால் அவ்வாறு செய்யாத ஒருவர், இது போன்ற ஒரு பிரச்சனையை நீங்கள் பார்க்கும்போது அது உங்களை பயமுறுத்த வேண்டாம், நான் உங்களை ஊக்குவிப்பது என்னவென்றால், 10 இன் பல்வேறு சக்திகளை செருகவும், பதிவு செயல்பாடு என்ற எண்ணத்தின் அடிப்படையில் சிந்திக்கவும் பூஜ்ஜியங்களின் எண்ணிக்கையை எண்ணுகிறது, எனவே அதைப் பற்றி சிந்திக்க உங்களுக்கு சிறிது நேரம் தருகிறேன், எனவே நான் மேலே சென்று தரம் தருகிறேன், எப்போதும் போல் நீங்கள் வசதியாக இருப்பதை விட இது வேகமாக இருந்தால், அது நான் முன்னோக்கி தொடர விரும்புவதால் மட்டுமே என்பதை அறிந்து கொள்ளுங்கள் பாடத்துடன், இந்த விஷயத்தில் சரியான பதில் 1000 மடங்கு x பதிவாகும், 3 ஐயும் x இன் பதிவையும் எடுத்துக்கொள்வதற்கு சமம், இப்போது அதைப் பற்றி ஒரு கணம் யோசிப்போம், நீங்கள் தொடங்கும் போது நான் சொன்னது போல் அவர்களுடன் நான் செய்ய வேண்டிய சிறந்த விஷயம், பல்வேறு எண்களை சொருகுவதற்கு வசதியாக இருப்பது மற்றும் செருகுவதற்கான சிறந்த எண்கள் ஏற்கனவே 10 இன் சக்தியாக உள்ளன, எனவே நீங்கள் 1000 மடங்கு பதிவு போன்ற ஒன்றைக் கேட்கிறீர்கள் என்றால் நான் செய்ய மாட்டேன் 1000 பெருக்கல் 100 இன் x பதிவில் எதையாவது செருகுவோம், இங்கே இறுதி விடையில் 1000 பெருக்கல் 100 என்பது 100,000 என்பது எவ்வளவு பூஜ்ஜியங்கள் என்று நமக்குத் தெரியும். நாங்கள் பூஜ்ஜியங்களை எடுத்துக்கொள்கிறோம், அந்த 1000 இலிருந்து 3 பூஜ்ஜியங்கள் அந்த 100 இல் இருந்து 2 பூஜ்ஜியங்கள் மற்றும் நாங்கள் அவற்றை ஒருவருக்கொருவர் அடுத்ததாக வைக்கிறோம், எனவே அது 5 மொத்த பூஜ்ஜியங்களாக இருக்க வேண்டும், ஆனால் நீங்கள் உண்மையில் எண்ணினால், எண் எப்படி மாறியது என்பதை மட்டும் சிந்திப்பதில்லை. ஆனால் அது ஏன் அப்படி மாறியது அந்த 1000 இலிருந்து 3 பூஜ்ஜியங்கள் கூட்டல் அந்த 100 இலிருந்து 2 பூஜ்ஜியங்கள், இதை 1000 இல் உள்ள பூஜ்ஜியங்களின் எண்ணிக்கையையும் 100 இல் உள்ள பூஜ்ஜியங்களின் எண்ணிக்கையையும் சேர்த்து எழுதலாம், எனவே இந்த யோசனை ஒரு மடக்கை இரண்டு விஷயங்களின் பலன் என்பது 10 இன் அதிகாரங்களின் சூழலில் அந்த இரண்டு விஷயங்களின் மடக்கைகளின் கூட்டுத்தொகையாகும். அவற்றின் பூஜ்ஜியங்கள் அனைத்தையும் எடுத்து, அவற்றை ஒன்றோடொன்று இழுத்துவிடுங்கள். A முறை B இன் பதிவு இது A இன் பதிவையும் B இன் பதிவையும் சமமாகும் நான் தேவையற்றவனாக இருக்கிறேன், நான் இதை அதிகமாகச் சொல்கிறேன், ஆனால் நீங்கள் அல்ஜீப்ராவில் மூழ்கி, நீங்கள் ஒருவித சோதனையில் அமர்ந்தால், அதை மறப்பது மிகவும் எளிதானது என்று நான் நினைப்பதால் தான். சில எண்களைச் செருகுவது நல்லது என்பதை நீங்களே நினைவுபடுத்திக்கொள்ள, இது ஒரு நல்ல விஷயம் மற்றும் பெரும்பாலும் உள்ளுணர்வுக்கு இது ஒரு சிறந்த வழியாகும், எனவே இந்த விஷயத்தில், A முறை B இன் பதிவைச் சொல்லி, அதை உடைத்து நாம் சிந்திக்கலாம், ஓ, என்று 100 பெருக்கல் 1000 இன் பதிவு, அதாவது 5, அதில் 5 பூஜ்ஜியங்கள் உள்ளன, கொடுக்கப்பட்ட ஒவ்வொரு பகுதியிலும் உள்ள பூஜ்ஜியங்களின் எண்ணிக்கையின் அடிப்படையில் உடைகிறது, அற்புதம், அந்த உள்ளுணர்வை மேலும் சுமந்துகொண்டு மற்றொரு நடைமுறை சிக்கலை முயற்சிப்போம், உங்களுக்குத் தெரிந்தால், சிறந்தது, உங்களால் நன்றாக பதில் சொல்ல முடியும் ஆனால் யோசிக்கலாம், என்ன பதில் என்று மட்டும் அல்ல, ஆனால் இந்த பதிலை ஒருவருக்கு எப்படி விளக்குவது அல்லது நான் சொல்லாமலேயே ஒரு மாணவனை இந்த பதிலுக்கு தாங்களாகவே வர வைப்பது எப்படி? அவர்களுக்கு என்ன பதில் இருக்கிறது, எனவே பாடத்தில் ஆர்வமுள்ளவர்கள் மற்றும் மெட்டா பாடத்தில் ஆர்வமுள்ளவர்கள் இரண்டு சாத்தியமான பார்வையாளர்கள் உள்ளனர், எனவே எங்கள் கேள்வி மீண்டும் கேட்கிறது, பின்வருவனவற்றில் எது உண்மை? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "அ. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "x இன் பதிவு n க்கு சமம் x b இன் n மடங்கு பதிவு. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "n க்கு x இன் பதிவு சமம் x இன் பதிவிற்கு n c. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "n க்கு x இன் பதிவு சமம் n கூட்டல் x அல்லது d இன் பதிவு. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "எனவே இங்கே சரியான பதில் a, இது உங்களில் 4,000 பேருக்கு வாழ்த்துகள் கிடைத்தது போல் தெரிகிறது, x இன் பவர் n என்பது x இன் n மடங்கு பதிவுக்கு சமம் என்று எங்களிடம் சொல்லி, மீண்டும், நீங்கள் இதை கற்பிக்க முயற்சிக்கிறீர்கள் என்று சொல்லலாம். யாரிடமாவது அல்லது அதன் அர்த்தம் என்ன என்பதை நீங்களே புரிந்து கொள்ள முயற்சிக்கிறீர்கள் என்றால், தொடங்குவதற்கு ஒரு சிறந்த இடம் ஏதோ ஒன்றைச் செருகுவதாக நான் நினைக்கிறேன், இந்த விஷயத்தில், சக்திக்கு x இன் பதிவு n க்கு 100 உடன் முயற்சிப்போம் 3 மற்றும் நீங்கள் செய்யும் முறைகள் உண்மையில் செயல்படுகிறதா என்பதைப் பார்க்க மற்றவர்களுடன் முயற்சி செய்யலாம், ஆனால் நீங்கள் யோசித்தால், பதில் என்னவென்று பார்க்காமல், பதில் ஏன் அப்படி மாறியது என்று சிந்திக்க முயற்சிக்கிறீர்கள். சில சமயங்களில் ஒரு உதாரணம் 100 கனசதுரமானது, நாம் அதை நன்றாக எடுத்துக் கொள்ளலாம் என்று நினைக்கலாம், அது 100 இன் 3 பிரதிகள் நான் 100 இன் 3 நகல்களை எடுத்துக்கொள்கிறேன், அதையெல்லாம் பெருக்கும்போது பூஜ்ஜியங்களின் எண்ணிக்கையை எண்ணுவது போல் பதிவு செய்ய வேண்டும் என்று நினைக்கிறேன். சொல்லுங்கள், ஓ, அது 6 பூஜ்ஜியங்களைக் கொண்ட சில எண்ணாக இருக்கும், அதுதான் 100 பெருக்கல் 100 பெருக்கல் 100 என்று எடுத்துக் கொள்வது என்றால், அந்த பூஜ்ஜியங்கள் அனைத்தையும் ஒன்றாகக் கூட்டி ஒரு மில்லியனைப் பெறுவது பற்றி நான் யோசிக்கிறேன், அதனால் இந்த எண் இருக்கும். 6 ஆனால் உண்மையில் அது ஏன் 6 என்று நாம் நினைத்தால், அது மில்லியனுக்குள்ளான பூஜ்ஜியங்களின் எண்ணிக்கையில் அந்த 6 எங்கிருந்து வந்தது என்றால், அந்த 100 இன் 3 பிரதிகள் எங்களிடம் இருந்தன, அந்த 100 இல் ஒவ்வொன்றும் 2 வெவ்வேறு பூஜ்ஜியங்களைக் கொண்டிருந்தன, எனவே இது மிகவும் பொதுவானது. 100 கனசதுரத்தை எடுத்துக்கொள்வதற்குப் பதிலாக 1000 கனசதுரத்தையோ அல்லது 1000 க்கு n அல்லது x க்கு சக்தி n ஐப் பார்க்கிறோம் என்றால், n இன் மதிப்பு என்னவாக இருக்கும் என்று நீங்கள் நினைக்கலாம். கிணற்றின் எண்ணிக்கை, பார்ப்போம், இது x க்கு மாற்றாக இருக்கும் பூஜ்ஜியங்களின் எண்ணிக்கையை விட x மடங்கு இல்லை, இந்த விஷயத்தில் 100 ஆக இருந்தது, அதற்கு பதிலாக நான் 10,000 பதிவு போன்ற ஒன்றை சக்திக்கு எடுத்திருந்தால் இதுவே இருக்கும். அந்த 10,000 இன் n நகல்களை எடுத்து, அவை ஒவ்வொன்றிலும் உள்ள பூஜ்ஜியங்களின் எண்ணிக்கையை 4 ஆக எண்ணினால், அது n பெருக்கல் 4 ஆக இருக்கும், நிச்சயமாக உங்களில் பெரும்பாலானோர் சரியாகப் பதிலளித்த பொதுவான சொத்து என்னவென்றால், நீங்கள் இந்த அழகான சிறிய விளைவைக் கொண்டிருக்கிறீர்கள். ஒரு சக்தியாக உயர்த்தப்பட்ட ஏதோவொன்றின் பதிவைக் காண்க, அதன் முன் சிறிய சக்தி கீழே விழுகிறது, இப்போது உள்ளே இருந்ததைப் பற்றிய பதிவு உங்களிடம் உள்ளது, அதன் மிக முக்கியமான தாக்கங்களில் ஒன்று நீங்கள் அதை அழைப்பீர்களா என்று எனக்குத் தெரியவில்லை ஒரு உட்குறிப்பு அல்லது நான் பதிவை எடுத்துக்கொண்டால் அதை வரையறையின் மறுபரிசீலனை என்று நீங்கள் கூறினால், அதன் அடிப்படை 10ல் 10ஐ மீண்டும் வலியுறுத்துகிறேன். அந்த சிறிய n பற்றி நாம் சிந்திக்கலாம். முன் மற்றும் அது 10 இல் 10 இன் பதிவு அடிப்படையாக n மடங்கு ஆகிறது, இது நிச்சயமாக 1 இந்த வெளிப்பாடு முடிவில் உள்ள பூஜ்ஜியங்களின் எண்ணிக்கையை எண்ணுவதாக நீங்கள் நினைக்கலாம் அல்லது பொதுவாக 10 க்கு சமமான 10 ஐக் கேட்கிறது மற்றும் பதில் வெறுமனே 1 ஆகும் இது மிகவும் உறுதியளிக்கிறது, ஏனென்றால் நீங்கள் திரும்பிச் சென்று இந்த அசல் வெளிப்பாட்டை படிக்கக்கூடிய மற்றொரு வழி 10 க்கு 10 க்கு சமமான 10 ஐக் கூறுவது சரி, பதில் இப்போது எங்களிடம் உள்ள ஒவ்வொரு மடக்கைப் பண்புக்கும் சரி. சக்தி n க்கு x இன் ஒரு பதிவைக் கண்டறிந்தது, n முன்னால் துள்ளல் எப்போதும் கண்ணாடிப் பட அதிவேகச் சொத்து இருக்கும், மேலும் இது ஒரு சிறிய உள்ளுணர்வைப் பெற உதவும் மற்றொரு வழி, எனவே நான் மறைக்கிறேன் நாம் இங்கு வரவிருக்கும் சில எதிர்கால பண்புகள், நாம் எங்கு செல்கிறோம் என்பதை மறைக்க முயற்சி செய்கிறோம். சக்தி n க்கு முழு விஷயமும் 10 ஐ n முறை x க்கு எடுத்துச் செல்வது போன்றது, இது மடக்கைகளுக்கு நீங்கள் வைத்திருக்கக்கூடிய மற்றொரு உள்ளுணர்விற்கு நம்மை அழைத்துச் செல்கிறது, அதாவது அவை ஒரு வகையான எக்ஸ்போனென்ஷியேஷன் போன்றது, இங்கே நான் என்ன சொல்கிறேன் பதிவின் உட்பகுதியில் அமர்ந்திருக்கும் விஷயம், நான் ஒரு பதிவை எடுத்துக் கொண்டால், இந்த விஷயத்தில் அதிவேகமாக இருக்கும் ஒரு பொருளின் முழு வெளிப்புற வெளிப்பாடு என்று நீங்கள் நினைக்க வேண்டும். செயல்பாட்டின் வெளியீடு, முழு விஷயமும் ஒரு இன் பதிவேட்டின் உள்ளே உள்ளதை ஒத்திருக்கிறது, 10 இன் அடுக்கு என்ன, எனவே நீங்கள் எங்கு ஒரு பதிவு வெளிப்பாட்டைக் கண்டாலும் வலதுபுறத்தில் ஒரு அடுக்குப் பாத்திரத்தை வகிக்கிறது என்று நீங்கள் நினைக்க வேண்டும். பக்கம் மற்றும் ஒவ்வொரு முறையும் நீங்கள் ஒரு அதிவேகத்தை பார்க்கும் போது முழு 10 முதல் x வெளிப்பாடு வரையிலான முழு வெளிப்புற கூறுகளையும் வலது பக்கம் இருக்கும், அது ஒரு பதிவின் உட்புறத்தில் அமர்ந்திருக்கும் ஒன்றை ஒத்திருக்கிறது, மேலும் நாம் பெருக்கும் போது இதை மேலே பார்த்தோம். உள்ளே இருக்கும், அது வெளியில் நன்றாகச் சேர்க்கிறது. பதிவுகள் உள்ளே அதிவேகங்களைத் திருப்பினால், அது செயல்பாட்டின் வெளியீடுகளை வெளியில் பெருக்குவது உள்ளே சேர்ப்பதற்கு சமம் என்று நமக்குச் சொல்கிறது, ஏனெனில் இந்த பதிவுகள் ஒவ்வொன்றும் log a மற்றும் log b போன்றவை. வலதுபுறத்தில் உள்ள வெளிப்பாட்டில் x மற்றும் y ஆகியவற்றின் பாத்திரத்தை வகிக்கிறது, அதனுடன் தொடர்ந்து விளையாடுவோம், இவற்றில் இன்னும் சிலவற்றைச் செய்வோம், மேலும் இந்த பண்புகளில் எத்தனை உள்ளுணர்வை உருவாக்க முடியும் என்பதைப் பார்ப்போம். அதிவேகங்கள் அடுத்ததைத் தாண்டுவதைப் பற்றிய நல்ல சிந்தனை, மடக்கைகளைப் பற்றித் தெரியாதவர்களுக்கு கொஞ்சம் வித்தியாசமாகத் தோன்றலாம், ஆனால் மீண்டும், சில உள்ளுணர்வைப் பெற சில எண்களைச் செருகவும், நாங்கள் அதைக் கொஞ்சம் தருவோம் பின்வருவனவற்றில் எது உண்மை என்பதை மேலே இழுக்க இன்னும் ஒரு கணம்? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "சரி, 10 கனசதுரம் 1000 எனில், 10 என்பது 1000க்கு சமம் என்று சொல்வது 1 மூன்றில் 1-க்கு உயர்த்தப்பட்டது, இங்கே தலைகீழாகச் செய்வது அதிவேகத்தின் பெருக்கல் தலைகீழாக இருக்கும், மேலும் வெளியேறும் விதம் 1 ஆல் வகுக்கப்படுவது போல் தெரிகிறது. மற்றும் 3 என்பது 1000 இன் 10 இன் லாக் பேஸுடன் ஒத்துப்போகிறது, அது 1 என்பது 1000 இன் 10 இன் லாக் பேஸால் வகுக்கப்படுகிறது, எனவே பொதுவாக, இந்த ஒற்றை எடுத்துக்காட்டின் அடிப்படையில் நீங்கள் யூகிக்கலாம், உள்ளே உள்ளதை வைத்து அடிப்படையை மாற்றும்போது அது 1 வகுக்கப்படுவதற்கு ஒத்ததாக இருக்கும். வெளியில் என்ன இருக்கிறது என்பதன் மூலம் மீண்டும் மீண்டும், அதனுடன் தொடர்புடைய அதிவேக விதியைப் பார்க்கும்போது, எனது அழகான சிறிய பதிவு மற்றும் அதிவேகங்களுக்கு என்ன ஆனது? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "அற்புதம் எனவே, மீண்டும் சில விஷயங்களை மறைப்போம், மற்ற சில சொத்துக்கள் இங்கு கிடைக்கும், நான் அதை இங்கே முன்பு வைத்திருந்த அதே வரிசையில் வைப்பேன், அதை முன்பே எழுதினால் என்னை வைத்திருக்க முடியும் என்று நினைத்தேன் வழக்கத்தை விட கொஞ்சம் தூய்மையானது, ஆனால் இது இந்த வித்தியாசமான பேப்பர் கட்டிங் ஷஃபிங் விளையாட்டை விளையாடுவதை உள்ளடக்கியிருக்கலாம், எனவே நாங்கள் இப்போது கண்டுபிடித்தது, a இன் லாக் பேஸ் b ஐ நீங்கள் மாற்றினால், இது 1 ஆல் வகுக்கப்படுவதற்கு சமம். அதிவேக நிலம் என்பது நீங்கள் ஒரு சக்திக்கு சமம் என்று சொன்னால், அந்த சக்தியின் தலைகீழ் a மீண்டும் bக்கு சமம் என்று சொல்வது போல் அதே கூற்று தான், ஒரு கணம் எடுத்து மடக்கைகளை திருப்புவது என்று நினைப்பது உதவியாக இருக்கும். உள்ளே a இன் வெளிப்பாடு பதிவு அடிப்படை b என்பது அந்த x இன் பாத்திரத்தை வகிக்கிறது மற்றும் b இன் வெளிப்பாடு பதிவு தளம் a வின் மேல் அமர்ந்திருக்கும் பாத்திரத்தை வகிக்கிறது, பின்னர் சமச்சீராக, முழு வெளிப்பாடு b முதல் x சக்தி வரை விளையாடுகிறது இடதுபுறத்தில் உள்ள உட்புறத்தின் பங்கு, அது a மற்றும் முழு வெளிப்பாட்டின் பாத்திரத்தை வகிக்கிறது, ஏதோவொன்றின் சக்திக்கு a என்பது பதிவுத் தளத்தின் உள்ளே அமர்ந்திருக்கும் பாத்திரத்தை வகிக்கிறது, எனவே நீங்கள் பார்க்க முடியும், சில எடுத்துக்காட்டுகளை செருகுவதன் மூலம் மற்றும் அதிவேக விதிகளுக்கு இணங்குவதன் மூலம் நாம் ஏற்கனவே மூன்று வெவ்வேறு மடக்கை விதிகள் மூலம் சிந்திக்க முடியும், அவை உங்களுக்கு மனப்பாடம் செய்ய இயற்கணிதத்தின் துண்டுகளாக வழங்கப்பட்டால், நீங்கள் அவற்றை மனப்பாடம் செய்யலாம், ஆனால் அவை உங்களிடமிருந்து நழுவுவது மிகவும் எளிதானது. தலை மற்றும் கையில் இருக்கும் பணியால் விரக்தியடைவது மிகவும் எளிதானது, ஆனால் இதுபோன்ற விஷயங்களில் நாங்கள் அக்கறை கொள்வதற்கான காரணம் மடக்கைகளின் விதிகளைப் புரிந்துகொள்வதுதான் என்பதை நீங்கள் நினைவுபடுத்த விரும்பலாம். ஒரு நாளிலிருந்து அடுத்த நாள், ஒரு படியில் இருந்து அடுத்த படி வரை, மடக்கைகளின் விதிகளைப் புரிந்துகொள்வது, அந்த மாதிரியான விஷயங்களுக்கு சிறந்த உணர்வைப் பெற உதவுகிறது, எனவே அது என்னவாக இருக்கும் என்பதற்கு ஒரு நல்ல நிஜ உலக உதாரணத்தைச் செய்வதற்கு முன் இந்த நரம்பில் இன்னும் ஒரு வினாடி வினாக் கேள்வியைச் செய்து, மடக்கைகளின் பண்புகளைப் பற்றிக் கேட்கிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "a plus b இன் பதிவேடு, a plus b இன் b பதிவின் ஒரு கூட்டல் பதிவின் பதிவிற்கு சமம். a plus b இன் log ஆனது, b இன் நேரப் பதிவின் பதிவால் வகுத்தால் ஒன்றுக்கு சமம் அல்லது மேலே உள்ள ah எதுவுமில்லை, இப்போது நமக்கு அவ்வளவு ஒருமித்த கருத்து இல்லை, இல்லையா? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "மிகவும் சுவாரஸ்யமானது, நாங்கள் இருவருக்கும் இடையே ஒரு குதிரைப் பந்தயம் உள்ளது, எனவே மக்கள் பதிலளிக்கும் போது இதைப் பற்றி சிந்திக்க நான் உங்களுக்கு ஒரு கணம் தருகிறேன், உண்மையில் பார்வையாளர்களிடம் எனக்கு ஒரு சிறிய கேள்வி உள்ளது, உங்களுக்குத் தெரியும், நாங்கள் எப்படி இருக்க முடியும் என்பதைப் பற்றி நான் பேசிக் கொண்டிருந்தேன். பெருக்கல் வளர்ச்சியின் அடிப்படையில் சிந்தித்துப் பாருங்கள், அது வெறும் பத்தின் சக்திகளாக இருக்க வேண்டிய அவசியமில்லை, மூன்றின் சக்திகள் போன்றவற்றை நாங்கள் செய்யலாம், அங்கு நீங்கள் ஒன்றிலிருந்து மூன்றிலிருந்து ஒன்பது முதல் இருபத்தேழு முதல் எண்பத்தி ஒன்று வரை சென்றால், அனைத்தும் இவற்றில் இந்த எண்களின் பதிவு அடிப்படை மூன்றானது நல்ல சிறிய படிகளில் வளர்கிறது என்று கூறலாம், எனவே ஒன்றின் அடிப்படை மூன்றையும், ஒன்றுக்கு சமமான மூன்றையும் உள்நுழைக, பதில் பொதுவாக பூஜ்ஜியமாக இருக்கும் ஒன்றின் பதிவு, அடிப்படை எதுவாக இருந்தாலும், மூன்றில் பூஜ்ஜிய பதிவு அடிப்படை மூன்றாக இரு, மூன்றிற்கு சமம் மூன்றில் ஒன்று இதேபோல் ஒன்பதில் மூன்று அடிப்படை இரண்டு ஆ, என் கேள்வி என்ன என்று நீங்கள் ஆச்சரியப்படலாம், ஆனால் இவை அனைத்தையும் வரையவும் எனது சொந்த மகிழ்ச்சிக்காகவும் இது உதவும் இங்கே, எண்பத்தி ஒன்றின் மூன்றில் ஒரு பதிவு அடிப்படையை இப்போது நான் எழுதுகிறேன், நீங்கள் ஒரு குழந்தையிடம் கேட்டால், ஐந்து அல்லது ஆறு வயதுடையவர்களில் ஒன்றிலிருந்து ஒன்பதுக்கு இடையில் என்ன எண் என்று சொல்லலாம் என்று நான் கேள்விப்பட்டேன். பதிலளிப்பதற்கான அவர்களின் உள்ளுணர்வின் பாதியில் எந்த எண் உள்ளது என்பதைச் சொல்லுங்கள், அதேசமயம் நமது உள்ளுணர்வுகள் நேரியல் சார்ந்ததாக இருக்கும், எனவே ஒன்று மற்றும் ஒன்பது என்று நாங்கள் அடிக்கடி நினைக்கிறோம், அவற்றுக்கிடையே இரண்டு, மூன்று, நான்கு, ஐந்து, ஆறு என்ற சம இடைவெளி எண்கள் உள்ளன. , ஏழு, எட்டு மற்றும் இடையில் பாதியிலேயே சென்றால், ஐந்தில் இறங்குவீர்கள் ஆனால் ஒன்றிலிருந்து ஒன்பது வரை எங்கு செல்வது என்று பெருக்கல் வளர்ச்சியின் அடிப்படையில் நீங்கள் யோசித்தால், நிறைய விஷயங்களைச் சேர்ப்பது ஒரு விஷயம் அல்ல, ஆனால் நீங்கள் 'ஒரு குறிப்பிட்ட அளவு மூலம் நீங்கள் மூன்று காரணிகளால் வளர்கிறீர்கள், பின்னர் நீங்கள் மூன்று காரணிகளால் வளர்கிறீர்கள், பின்னர் நீங்கள் மூன்று காரணிகளால் வளர்கிறீர்கள், ஒரு குழந்தையின் இயல்பான உள்ளுணர்வு மூன்று என்று கூறுகிறது. கணக்கியல் முறைமைகள் மற்றும் எழுத்து முறைகளை உருவாக்கியது போல் நவீன சமூகங்கள் இதற்கு மூன்றில் பதில் அளிப்பார்கள், இப்போது பார்க்கும் உங்களில் யாருக்காவது ஒரு சிறு குழந்தை கிடைக்குமா என்று பார்வையாளர்களுக்கான எனது கேள்வி ஐந்தாண்டுகளுக்குள் சொல்லலாம். பழையது ஒன்றுக்கும் ஒன்பதுக்கும் இடைப்பட்ட எண் என்ன என்று அவர்களிடம் கேட்க முடியுமா என்று பாருங்கள், உங்களால் முடிந்தால், குழந்தை அவர்களின் உண்மையான பதில் என்ன என்று ட்விட்டரில் எங்களுக்குத் தெரியப்படுத்துங்கள், ஏனென்றால் ஏன் என்று எனக்குத் தெரியவில்லை, நான் கொஞ்சம் தான் இது நடைமுறையில் நடக்கிறதா என்ற சந்தேகம் எனக்குப் புரிகிறது, இது ஒரு சூப்பர் அறிவியல் வழி அல்ல என்பதை நான் புரிந்துகொள்கிறேன், யூடியூப் லைவ்ஸ்ட்ரீமைப் பார்க்கும் நபர்களிடம் தங்கள் குழந்தைகளை ஆய்வு செய்து, பதிலை ட்வீட் செய்யுமாறு நான் கேட்கவில்லை, ஆனால் எனது சொந்த நலனுக்காக அது சுவாரஸ்யமாக இருக்கும் எங்கள் கேள்விக்கு சில வகையான சரிபார்ப்பைக் காண, இது ஒரு திசையில் பெரிய ஒருமித்த கருத்து இல்லை என்று தோன்றிய முதல் விஷயம் இது தான், பதில் என்னவாக இருக்கும் என்பதைப் பார்க்க, சரி, 2,400 மேலே உள்ளவைகளில் எதுவுமே இல்லை என்று நீங்கள் சரியாகப் பதிலளித்தீர்கள். அடுத்த முறை இதைப் பற்றி பேசலாம் மடக்கையின் உள்ளீடுகளைச் சேர்ப்பது உண்மையில் மிகவும் வித்தியாசமான உணர்வு, இது மிகவும் வித்தியாசமான விஷயம் மற்றும் அந்த வித்தியாசத்தை உணர, நான் உங்களிடம் ஒரு பிளஸ் பி பதிவைக் கேட்டால், பத்தில் சில சக்திகளைச் செருகவும். நீங்கள் சிந்திக்கத் தொடங்குவது என்னவென்றால், சரி, 10,000 மற்றும் 100 போன்ற சில உதாரணங்களைச் செருகுகிறேன், நான் என்னையே கேட்டுக்கொள்கிறேன், அந்த உள்ளீட்டில் உள்ளவற்றின் பூஜ்ஜிய எண்ணும் செயல்பாட்டைச் செய்தால், அதில் எத்தனை பூஜ்ஜியங்கள் உள்ளன? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "இது ஒரு சுவாரஸ்யமான கேள்வி, சரி, மடக்கையின் அடிப்பகுதி பூஜ்ஜியமாக இருக்க முடியுமா? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "நமது முக்கோணத்தைப் பொறுத்தவரை, உங்களுக்குத் தெரியும், பூஜ்ஜியத்திற்கு சில சக்தி x என்பது வேறு சில மதிப்புக்கு சமம் y என்று நாம் நினைக்கலாம், இது x க்கு பூஜ்ஜியத்தை y என்று சொல்லி எழுதலாம் அல்லது எழுதலாம். y இன் பதிவு அடிப்படை பூஜ்ஜியம் x க்கு சமம் x பூஜ்ஜியத்திற்கு சமம் என்று சொல்வதன் மூலம் இப்போது இங்கே பிரச்சினை என்னவென்றால், எதற்கும் பூஜ்ஜியம் சரியாக பூஜ்ஜியமாக முடிவடைகிறது, எனவே நாம் பதிவு அடிப்படை பூஜ்ஜியத்தைப் பற்றி சிந்திக்கப் போகிறோம். y உங்களுக்குத் தெரிந்த வேறு எந்த உள்ளீட்டிற்கும், நீங்கள் ஒன்று அல்லது இரண்டு அல்லது pi போன்றவற்றை உள்ளிட விரும்புகிறீர்கள், ஒன்று அல்லது இரண்டு அல்லது pi அல்லது உங்களிடம் உள்ள எந்த எண்ணுக்குச் சமம் என்ற கேள்வியை பூஜ்ஜியமாகக் கேட்கிறீர்கள். மற்றும் ஒரு பதில் இருக்கப்போவதில்லை, எனவே நீங்கள் ஓ, ஆம், பூஜ்ஜியத்தின் பதிவு என்று சொல்ல முயற்சி செய்யலாம், இது ஒரு சரியான செயல்பாடாகும், இது உள்ளீடு பூஜ்ஜியத்தில் மட்டுமே வரையறுக்கப்படுகிறது, ஆனால் நீங்கள் விரும்புவதை முடிக்க முயற்சிப்பதில் சிக்கல் இருக்கும். பூஜ்ஜியத்திற்குச் சமமானதை பூஜ்ஜியமாகக் கூறுவது அதற்குப் பொருந்தும், எனவே உங்கள் கை உங்கள் முதுகுக்குப் பின்னால் முறுக்கப்பட்டிருக்கும், ஆனால் நீங்கள் அதைச் செய்ய விரும்புகிறீர்கள், மேலும் இது அடிப்படை பூஜ்ஜியத்துடன் கூடிய அதிவேக செயல்பாடு முற்றிலும் பூஜ்ஜியமாகும் என்பதற்கு ஒத்திருக்கிறது. எண்களை ஒன்றுக்கு ஒன்றுக்கு ஒரு பாணியில் நன்றாக வரைபடமாக்கவில்லை, அதனால் இது ஒரு பெரிய கேள்வி, நிஜ உலகில் இந்த விஷயங்கள் எங்கே வருகின்றன என்ற எண்ணத்திற்கு இப்போது ஒரு பதிவு அடிப்படை பூஜ்ஜியத்தை வைத்திருக்க முடியுமா? நிலநடுக்கங்களுக்கான ரிக்டர் அளவுகோல், நிலநடுக்கம் எவ்வளவு வலிமையானது என்பதற்கான அளவீட்டை ரிக்டர் அளவுகோல் நமக்குத் தருகிறது, மேலும் இது மிகச் சிறிய எண்கள் முதல் மிகப் பெரிய எண்கள் வரை எதுவாகவும் இருக்கலாம், இது இதுவரை அளவிடப்பட்ட நிலநடுக்கங்களில் மிகப்பெரிய நிலநடுக்கம் என்று நான் நினைக்கிறேன், இது ஒரு விளக்கப்படம் மட்டுமே விக்கிபீடியா 9 ஆக இருந்தது. 5 மற்றும் அது எவ்வளவு பைத்தியக்காரத்தனமானது என்பதைப் புரிந்துகொள்வது, இந்த எண்கள் எதைக் குறிக்கின்றன என்பதற்கும், TNT க்கு சமமான அளவு போன்றவற்றுக்கும் இடையே உள்ள உறவைப் பார்ப்பது மதிப்புக்குரியது. ஆற்றலின் அளவின் அடிப்படையில் ரிக்டர் அளவுகோல் எண்ணுக்கு ஒரு வெளிப்பாட்டைப் பெற முடியுமா என்பதைப் பார்க்கவும், ஏன் மடக்கைகள் இதை விவரிக்க இயற்கையான வழியாகும், எனவே கவனம் செலுத்த வேண்டிய முக்கிய விஷயம் என்னவென்றால், விஷயங்கள் எவ்வளவு அதிகரிக்கும் எடுத்துக்காட்டாக, நாம் இரண்டிலிருந்து நன்றாகச் சென்றால், இந்த விஷயத்தில் மூன்று எங்கே என்று அது நமக்குக் காட்டாது, எனவே இரண்டிலிருந்து நான்கு வரை ஒரு படி எடுக்கலாம் என்று நினைக்கலாம், இது இரண்டு படிகள் எடுப்பது போன்றது. ஒரு மெட்ரிக் டன் TNT இலிருந்து நம்மை எடுத்துக்கொள்வது போல் தெரிகிறது, இது இரண்டாம் உலகப் போரின் பெரிய வெடிகுண்டு என்று நான் நினைக்கிறேன், மேலும் இது ஒரு சிறிய அணுகுண்டை விட ஆயிரம் மடங்கு அதிகமாக ஒரு கிலோடன் வரை நம்மை அழைத்துச் செல்கிறது, எனவே இரண்டு படிகள் ரிக்டர் அளவுகோலில், ரிக்டர் அளவுகோலில் 2-ல் இருந்து 4-ம் அளவு நிலநடுக்கம் வரை, பெரிய வெடிகுண்டிலிருந்து இரண்டாம் உலகப் போரில் இருந்து அணுசக்தி யுகம் வரை நம்மை அழைத்துச் செல்கிறது. குறைந்த பட்சம், இந்த விளக்கப்படம் நமக்கு அழகாகக் காட்டுவது மற்றும் 4 முதல் 5 வரை ஒரு படி மேலே சென்றால், 1 கிலோட்டனில் இருந்து 32 கிலோடன்கள் வரை செல்வதற்கு ஒத்திருக்கிறது, மேலும் நாகசாகியில் தரையிறங்கிய வெடிகுண்டை அழிக்கும் நகரத்தின் அளவு இதுவாக இருக்கலாம். 4 என்ற அளவில் நிலநடுக்கம் ஏற்பட்டது என்பதற்கு இடையே உள்ள வித்தியாசத்தை நீங்கள் செய்திகளில் கேட்கிறீர்கள் என்றால் மடக்கை அளவுகோல்களுக்கு எதிர்மறையான விஷயம். 0 மற்றும் நிலநடுக்கம் 5 ஆக இருந்தது. 0 யோசிப்பது எளிது ஆம் 4 மற்றும் 5 இவை மிகவும் ஒத்த எண்கள் ஆனால் வெளிப்படையாக TNT அளவுகளின் அடிப்படையில் 32 ஆல் பெருக்க 1 இலிருந்து அடுத்ததாக வருவதற்கும், 2 முதல் 4 வரை செல்வது, ஆயிரத்தால் மட்டுமே பெருக்கப்படுகிறது. பெரியதாக இருப்பதற்கான காரணம் என்னவென்றால், இங்கே எங்கள் விளக்கப்படம் 3 என்ன என்பதைக் காட்டவில்லை, எனவே நாங்கள் இரண்டு படிகளை எடுத்துக்கொண்டோம், நீங்கள் 32 இல் ஒரு படி எடுத்து மற்றொரு 32 ஆல் பெருக்கினால் அது உண்மையில் ஆயிரத்திற்கு மிக அருகில் இருக்கும் என்பதை நீங்களே சரிபார்க்கலாம். ரிக்டர் எண்ணில் உள்ள கூட்டல் படிகள் TNTயில் உள்ள பெருக்கல் படிகளுடன் ஒத்துப்போகின்றன என்ற எண்ணம் இங்கே ஏதோ மடக்கை விளையாடுகிறது என்று கூறுவது போல் தெரிகிறது, மேலும் இங்கு தொடர்ந்து சென்று உலக நிகழ்வுகளின் காரணமாக இது எவ்வளவு வளர்ச்சி அடைகிறது என்று சொல்வது கொஞ்சம் சுவாரஸ்யமாக உள்ளது. ஆம் என்று விவரிப்பதில் பெரிய ஆச்சரியம் இல்லை, நாம் இன்னொரு அடியை எடுத்து வைக்கும்போது அது மீண்டும் சுமார் 32 ஆல் பெருக்கப்படுகிறது, ஆனால் அதை நம் உள்ளுணர்வில் கட்டுப்படுத்துகிறது, இது 32 கிலோடன்கள் ஒரு சிறிய அணுகுண்டு மற்றும் பின்னர் ஒரு மெகாட்டான் சிறிய அணுகுண்டு என்று நாம் நினைக்கலாம். நாகசாகி அணுகுண்டு ஒரு மெகாடனுக்கான நாகசாகி அணுகுண்டுகளில் 32 என்று நான் யூகிக்கிறேன், இது அமெரிக்காவில் 1994 ஆம் ஆண்டு நெவாடாவில் இரட்டை சரம் கொண்ட தட்டையான நிலநடுக்கத்தின் அளவு என்று எனக்குத் தெரியவில்லை, அது என்னவென்று எனக்குத் தெரியவில்லை, அதிர்வெண்களின் அடிப்படையில் விக்கிபீடியாவிற்கு நன்றி. இவை இரண்டுக்கும் குறைவானவை என்பதைத் தெளிவாகப் பார்த்தோம், அவை எல்லா நேரத்திலும் நிகழ்கின்றன, அவை ஒரு நாளைக்கு 8000 ஆகும், ஆனால் நாம் அணுகுண்டுகளின் உலகில் 3 போன்ற விஷயங்கள். 5 மற்றும் 4 இவை பூமியில் எங்கோ அடிக்கடி நிகழும் நிகழ்வுகளில் 134 நிகழ்வுகள் ஒவ்வொரு நாளும் எங்காவது நடக்கின்றன என்பது யாருக்குத் தெரியும்? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "ஆனால் அணுகுண்டு அளவை விட அதிகமாக இருந்த இந்த 5 மற்றும் 6 வரம்பில் நாம் இன்னும் தீவிரமடைந்து வருவதால், இப்போது நாம் ஒரு நாளைக்கு சுமார் 2 மட்டுமே இருக்கிறோம், ஒரு புவியியலாளர் உள்ளே வந்து ஏன் நாம் அனைவரும் செய்யக்கூடாது என்பதை விளக்க முடியும் என்று நான் நம்புகிறேன். ஒவ்வொரு நாளும் பூமியின் மேலோட்டத்தில் இரண்டு அணுகுண்டுக்கு சமமான இடையூறுகள் ஏற்படுகின்றன என்பதைப் பற்றி கவலைப்பட வேண்டாம், ஆனால் ஒவ்வொரு அடியிலும் நம் எண்ணத்தை சரிபார்த்து, இப்போது ஏராளமான மக்கள் வசிக்கும் நகரம் போன்ற ஏதாவது ஒரு இடத்தில் குவிந்திருப்பது மிகவும் அரிது. 32 இன் வளர்ச்சியை உள்ளடக்கியது, 6 முதல் 7 வரையிலான படி எப்படி இருக்கும் என்பதைப் பார்ப்போம், இங்கே அது நமக்கு நிறைய எடுத்துக்காட்டுகளைத் தருகிறது. 32 மெகாடான்கள், அது 32 ஆல் பெருக்கப்படுகிறது, இந்த விளக்கப்படத்தில் எனக்கு மிகவும் சுவாரசியமான விஷயம் என்னவென்றால், இதுவரை சோதிக்கப்பட்ட மிகப்பெரிய அணு ஆயுதத்தை அடைவதற்கு முன்பு நாம் எவ்வளவு தூரம் செல்ல வேண்டும் என்பதைப் பார்ப்பது பனிப்போரின் உச்சம். ஜார் வெடிகுண்டு 50 மெகா டன் இருந்தது, அவர்கள் உண்மையில் 100 மெகாடன் வெடிகுண்டு வைத்திருக்கும் அசல் திட்டத்தை வைத்திருந்தார்கள் என்று நான் நம்புகிறேன், ஆனால் அந்த 50 மெகாடன்களில் இருந்து தங்களைத் தாங்களே குறைத்துக் கொண்டோம், நாங்கள் பேசுகிறோம், நாகசாகி குண்டின் 32 கிலோடன்கள் 32 ஆல் பெருக்கப்பட வேண்டும். மெகாடன் மற்றொரு 32 ஆல் பெருக்கினால், இரண்டாம் உலகப் போரை விட ஆயிரம் மடங்கு வலிமையைப் பற்றி நாங்கள் பேசுகிறோம், மேலும் நீங்கள் இன்னும் 50 மெகாடன்களில் மனிதகுலத்தின் திறனைப் பெறவில்லை, அது இந்தோனேசியாவின் ஜாவா நிலநடுக்கம் ஆகும். . 0 என்பது 6 ஐ விட சற்று பெரியது அல்ல. 0, இது மிகவும் பெரியது மற்றும் இங்கே முக்கிய விஷயம் என்னவென்றால், உங்களிடம் ஒரு அளவு இருந்தால், உங்களுக்கு பெருக்கல் அதிகரிக்கும் போது, சிறிய படிகள் போல தோற்றமளிக்கும் சக்தியின் அடிப்படையில் அல்லது இங்கே குறிப்பிடப்பட்டுள்ள முழுமையான மதிப்புகளின் அடிப்படையில் உண்மையில் பெரிய படிகளாக இருக்கும் என்பதைப் பாராட்டுவது மதிப்பு. எனவே எப்போதாவது ஒரு 9 இருந்தது என்ற உண்மையைப் பற்றி நாம் சிந்திக்கும்போது. 5 உண்மையில் அபத்தமாகத் தெரிகிறது, அது 7 இல் மட்டுமே உள்ளது. 0 வரம்பில் நாம் இதுவரை வெளியிடப்பட்ட மிகப்பெரிய தெர்மோநியூக்ளியர் ஆயுதத்தைப் பற்றி பேசுகிறோம், இது மடக்கைகள் வரக்கூடிய ஒரு பகுதியைக் குறிக்கிறது. நிலநடுக்கங்களின் அளவைப் பொறுத்தவரை, பூமியைச் சுற்றி எல்லா நேரங்களிலும் என்ன நடக்கிறது, ஒரு பெரிய கைக்குண்டின் அளவு போன்ற விஷயங்களை நீங்கள் பெறலாம், மேலும் அது உங்கள் அளவில் இருக்க வேண்டும் மற்றும் எல்லா வழிகளிலும் வருவதைப் பற்றி சிந்திக்க வேண்டும் மனித வரலாற்றில் நாங்கள் கண்ட மிகப் பெரிய இடையூறு மற்றும் அதைக் கொண்டிருப்பதற்காக, நீங்கள் உங்கள் எண்களில் வெவ்வேறு இலக்கங்களின் மொத்தக் கூட்டத்தையும், வெவ்வேறு, சிறிய எண்ணின் மொத்தக் கூட்டத்தையும் எழுதவில்லை. மற்றொரு சந்தர்ப்பத்தில் உங்கள் எண்ணுக்கான இலக்கங்களின் எண்ணிக்கையை மடக்கைகளை எடுத்து, பின்னர் அந்த எண்களை 0 மற்றும் 10 க்கு இடையில் ஒரே அளவில் வைப்பது நல்லது சற்று வித்தியாசமாக ஒவ்வொரு முறையும் 10 டெசிபல்களின் படி மேலே செல்லும் போது 10 ஆல் பெருக்குவதற்கு ஒத்ததாக இருக்கும், எனவே 1 இன் படியை 10 ஆல் பெருக்குவதை விட, இது 10 இன் ஒரு படி 10 ஆல் பெருக்குகிறது. 60 டெசிபல்களுக்கு எதிராக 50 டெசிபல்களின் ஒலியை நீங்கள் கேட்கிறீர்கள் என்றால், அது கடத்தப்படும் மற்றும் செல்லும் ஆற்றலின் அடிப்படையில் மிகவும் அமைதியாக இருக்கும், அது 60 முதல் 70 அல்லது 70 வரை இருக்கும். 80 அந்த படிகள், 60 முதல் 80 வரை, இது ஒரு சதுரப் பகுதிக்கு ஆற்றலின் அளவை 100 காரணியால் பெருக்குவதை உள்ளடக்குகிறது, எனவே ஒவ்வொரு முறையும் நீங்கள் மடக்கை அளவைப் பார்க்கும் போது, அது பேட்டைக்கு அடியில் எதைக் குறிப்பிடுகிறதோ அது வளரும் என்பதை உங்கள் மனதில் அறிந்து கொள்ளுங்கள். ஒரு பெரிய தொகை அதனால்தான் கொரோனா வைரஸ் வெடிப்பை விவரிக்கப் பயன்படுத்தப்படும் மடக்கை அளவுகோல்களை நாங்கள் பார்த்தோம், எனவே ஒவ்வொரு முறையும் நீங்கள் ரிக்டர் அளவுகோல் எண்ணை 1 ஆல் வளர்க்கும்போது, நீங்கள் 32 ஆல் பெருக்குகிறீர்கள், இது போன்ற உறவை நீங்கள் எவ்வாறு விவரிக்கலாம். அடிப்படை 32 கொண்ட பதிவின் அடிப்படையில் சிந்திக்க முடியும், நான் பதிவை எடுத்துக் கொண்டால், நான் r ஐ அழைக்கப் போகிறேன், ரிக்டர் அளவுகோலுக்கான எண்ணை நான் பதிவு அடிப்படை 32 என்று நினைக்கலாம், அது தொடர்புடையதாக இருக்கும் , இல்லை இல்லை இல்லை, நான் இந்த தவறு செய்கிறேன் அது உள்நுழைந்த விஷயம் அல்ல, பெரிய எண்ணின் பதிவு அடிப்படை 32 ஐ எடுத்துக்கொள்கிறோம், TMT எண்ணின் 1 மெகாடன் போல இருந்தது, அது 1 மில்லியன் டன்கள் பதிவு அடிப்படை 32, அது வேண்டும் ரிக்டர் அளவுகோல் எண்ணுடன் தொடர்புடையது ஆனால் சில வகையான ஆஃப்செட் இருக்கலாம், எனவே இந்த ரிக்டர் அளவுகோலில் நாம் சேர்க்கும் நிலையான கள் உள்ளன என்று கூறலாம், மேலும் இந்த வெளிப்பாடு சரியாகவே உள்ளது, வெளியேறுவதற்கு மன்னிக்கவும் கீழே இந்த வெளிப்பாடு சில ஆஃப்செட் நேரங்களின் சக்திக்கு 32 என்று சொல்வது போலவே இருக்கிறது, இது ரிக்டர் அளவுகோல் எண்ணுக்கு 32 ஐ எடுத்துக்கொள்வதற்கு சமம், அதுவே சில பெரிய மாறிலி ஆகும், ரிக்டர் அளவுகோல் எண்ணுக்கு 32 மடங்கு நீங்கள் பார்க்கும் எண்ணின் சக்திக்கு இது சில நிலையான நேரங்கள் 32 என்று நினைக்கலாம், எனவே இந்த எழுதும் முறை அதன் அதிவேக வளர்ச்சியை வலியுறுத்துகிறது. நீங்கள் படிப்படியாக 32 ஆல் பெருக்குகிறீர்கள், ஆனால் அதே உண்மையைத் தெரிவிப்பதற்கான மற்றொரு வழி என்னவென்றால், அந்தத் தொகையின் பதிவு அடிப்படை 32 ஐ எடுத்துக்கொள்வது, இப்போது நான் அடுத்ததாக பேச விரும்புவது, நாம் எப்போதுமே செய்ய வேண்டியதில்லை வெவ்வேறு தளங்களின் பதிவுகளை எவ்வாறு கணக்கிடுவது என்பதைப் பற்றி கவலைப்படுவது இங்கே நாம் லாக் பேஸ் 32 பற்றி பேசுவது கொஞ்சம் வித்தியாசமானது, கணிதவியலாளர்கள் அடிப்படை மற்றும் கணினி விஞ்ஞானிகள் உண்மையில் அடிப்படை 2 உடன் ஒரு பதிவை வைத்திருக்க விரும்புகிறார்கள் என்பதை நான் முன்பே குறிப்பிட்டேன். கணக்கீட்டு நோக்கங்களுக்காக அல்லது உங்களிடம் ஒரு பதிவு இருந்தால் இந்த விஷயங்கள் எவ்வாறு வளரும் என்பதைப் பற்றி சிந்திக்கவும், நீங்கள் ஒரு வகை பதிவைக் கணக்கிட முடிந்தால், அது அடிப்படை 10, அடிப்படை 2, அடிப்படை மற்றும் நீங்கள் வேறு எதையும் கணக்கிடலாம். நீங்கள் இப்போது எங்கள் உள்ளுணர்வை அந்த திசையில் பெற விரும்புகிறீர்கள், நமது வினாடி வினாவுக்குத் திரும்புவோம், அடுத்த கேள்விக்குச் செல்வோம், இந்தக் கேள்வியே மிக அதிகம் என்று நான் நம்புகிறேன், எனக்குத் தெரியாது, இது ஒரு பாதி நியாயமான கேள்வி, இது நன்றாக இருக்க வேண்டும் அடிப்படை 2 சூழலில் இருந்து அடிப்படை 10 சூழலுக்கு மொழிபெயர்க்க இது நம்மைத் தயார்படுத்தப் போகிறது, மேலும் 2 இன் சக்திகளைப் புரிந்துகொள்வதற்கான ஒரு நல்ல உள்ளுணர்வு, பொதுவாக 10 இன் சக்திகளுடன் அது கொண்டிருக்கும் உறவைப் புரிந்துகொள்வது, ஏனெனில் இது இந்த அழகான தற்செயல் நிகழ்வு. இந்த இரண்டு வகைகளும் நான் சொல்வதை நீங்கள் பார்ப்பீர்கள். உங்கள் எண்களுடன் சிறிது தளர்வானது மற்றும் நீங்கள் தோராயமான 2 முதல் 10வது வரை, அடிப்படையில் 1000 வரை செய்கிறீர்கள், பின்வருவனவற்றில் எது உண்மை என்பதற்கு மிக அருகில் உள்ளது? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "ஒப்பந்தம். ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "இங்கே ஒருமித்த முடிவு இல்லை. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "ஆனால் எது உண்மை என்பதற்கு மிக நெருக்கமானது என்று கேள்வி கேட்கப்பட்டது, இதைப் பற்றி நாம் எவ்வாறு சிந்திக்கலாம் என்பதைப் பார்ப்போம். ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "எனவே, உங்களிடம் 2-ன் சக்தி உள்ளது, அதாவது 1024, 10-க்கு மிக அருகாமையில், 10 கனசதுரத்தில் உள்ளது. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "எனவே இதன் அர்த்தம் என்ன? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "பதிவு அடிப்படை 2 இல் 10 x க்கு சமம் என்றால், x க்கு 2 ஐ 10 க்கு சமம் என்று சொல்வது ஒன்றுதான், இல்லையா? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "இது 10க்கு சமம் 2 என்று கேட்கிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "ஒவ்வொரு செயல்பாட்டிலும் இதைச் செய்ய முடியாது. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "எந்தவொரு செயல்பாட்டிலும் நீங்கள் அதைச் செய்ய முடியும் என்று மக்கள் நினைக்கிறார்கள், ஆனால் உங்களால் முடியாது. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "அதன் அர்த்தம் என்னவென்றால், x என்பது மூன்றில் 10 பங்கு, சரியா? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "மற்றும் போதுமானது, நாம் முன்பு பார்த்தது என்னவென்றால், 10 இன் லாக் பேஸ் 2, லாக் பேஸ் 10 இன் 2 என்பது அந்தத் தொகைக்கு மேல் 1, xக்கு மேல் 1 என்றும் சொல்லலாம். ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "நன்று. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "நாங்கள் பதிவுகளில் விஷயங்களைச் செய்வதால் நான் அதை அந்த வழியில் எழுதப் போகிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "இதேபோல் ஒரு மில்லியனில் லாக் பேஸ் 2, சரி பார்ப்போம், ஆயிரத்தை பெற 2 ஐ 10 மடங்கு பெருக்க வேண்டும் என்றால், ஒரு மில்லியனைப் பெற அதை சுமார் 20 மடங்கு பெருக்க வேண்டும். ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "இது கொஞ்சம் சிறியது ஆனால் இது உங்கள் மனதில் இருக்கும் ஒரு நல்ல தோராயமாகும். ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "3 என்பது 3.20, அதே அளவு குறைக்கிறோம். ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, அதே அளவு குறைக்கிறோம். சரி? ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "இப்போது இது நினைவில் கொள்ள வேண்டிய ஒரு உள்ளுணர்வு. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "பின்னர், பதிவு அடிப்படை C இன் B நேரங்களின் பதிவு அடிப்படை C இன் A ஐ இணைக்க பல்வேறு சாத்தியமான வழிகளின் முழுக் குவியல். ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "இதைப் பற்றி நான் உங்களுக்கு ஒரு அர்த்தமுள்ள நேரத்தைத் தருகிறேன், ஏனென்றால் நீங்கள் ஏற்கனவே மடக்கைகளை நன்கு அறிந்திருந்தால் தவிர இது வெளிப்படையாகத் தெரியவில்லை, மேலும் இது கொஞ்சம் சிந்திக்கத் தகுந்தது. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "நன்றி கரேன். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/telugu/sentence_translations.json b/2020/ldm-logarithms/telugu/sentence_translations.json index 6d6db284f..c297c55b0 100644 --- a/2020/ldm-logarithms/telugu/sentence_translations.json +++ b/2020/ldm-logarithms/telugu/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵సంగీతం🎵 లాక్‌డౌన్ మఠానికి తిరిగి స్వాగతం. ఈ రోజు మనం లాగరిథమ్‌లు మరియు బేసిక్స్ విధమైన పాఠం గురించి మాట్లాడబోతున్నాం. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "మరియు ఎప్పటిలాగే, విషయాలను ప్రారంభించేందుకు, ప్రస్తుతం ప్రేక్షకులు ఎక్కడ ఉన్నారో నేను అర్థం చేసుకోవాలనుకుంటున్నాను. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "కాబట్టి, మీరు 3b1bకి వెళ్లగలిగితే. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "నేను వారి గురించి ఇంతకు ముందెన్నడూ వినలేదు లేదా ఇంతకు ముందు వాటి గురించి నేర్చుకోలేదు బి. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "నేను వాటి గురించి తెలుసుకున్నాను కానీ కొన్నిసార్లు అన్ని లక్షణాలతో గందరగోళానికి గురవుతాను c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "నేను వాటిని అర్థం చేసుకున్నాను కానీ వారికి ఎలా నేర్పించాలో తెలియదు మరియు డి. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "నేను వారిని బాగా అర్థం చేసుకున్నాను మరియు వారికి కూడా బాగా అర్థం చేసుకునేలా మరొకరికి హాయిగా నేర్పించగలను. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "కాబట్టి, మాకు మంచి విభజన వచ్చింది. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "నేను చెప్పినట్లు, భవిష్యత్తులో వ్యక్తులు లాగరిథమ్‌లతో సౌకర్యంగా లేకుంటే నేను వారికి సూచించగలిగే పాఠాన్ని సృష్టించడం దీని ఉద్దేశం మరియు నేను చెప్పాలనుకుంటున్నాను, ఓహ్, ఇక్కడ మీరు వెళ్లగలిగే స్థలం ఉంది నేను ఎలా అనుకుంటున్నాను, మీకు తెలుసా, మీరు దానిని అకారణంగా ఎలా చేరుకోవచ్చు అని నేను అనుకుంటున్నాను. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "ఈ ప్రత్యేక ఉపన్యాసం చేయడానికి ముందు నేను రెండు టీచర్ ఫోరమ్‌ల చుట్టూ స్క్రోల్ చేస్తున్నాను మరియు విద్యార్థులు హైస్కూల్ గణితంలో బోధించడానికి కష్టతరమైన అంశం ఏమిటి అని అడిగినప్పుడు, విద్యార్థులు ఎక్కువగా ఇబ్బంది పడుతున్నట్లు అనిపించినప్పుడు, లాగరిథమ్‌లు చాలా ఎక్కువ. సాధారణంగా సూచించబడిన సమాధానాలు ఆసక్తికరంగా ఉంటాయి మరియు నేను ఊహించగలను, ఎందుకంటే ఈ లక్షణాలలో టన్నుల కొద్దీ మీరు తెలుసుకోవలసిన అవసరం ఉంది, కాబట్టి మేము ఎక్కడికి వెళ్లబోతున్నామో ముందుగా దాటవేస్తే మీకు ఈ పైల్స్ అన్నీ లభిస్తాయి. బీజగణితం యొక్క సమూహం వలె కనిపించే నియమాలు గుర్తుంచుకోవడం కష్టం మరియు మీ తలలో విషయాలను కలపడం సులభం మరియు ప్రజలు హైస్కూల్ గణితం ఎలా ఉండేదో మరియు ఎలా ఉండేదో ఈ విధమైన పీడకలల జ్ఞాపకాలను కలిగి ఉన్నారని నేను భావిస్తున్నాను. సంవర్గమానం వారి కోసం చేసింది, ఇది తరచుగా ఆ నిర్దిష్ట సూత్రాలు గుర్తుకు వస్తాయి మరియు ఈ రోజు నేను ఏమి చేయాలనుకుంటున్నాను, వాటి గురించి ఎలా ఆలోచించాలి, కానీ మీరు ఎవరికైనా బీజగణితాన్ని బోధిస్తున్నారా అనే మెటా స్థాయిలో కూడా ఈ రోజు నేను చేయాలనుకుంటున్నాను నొక్కి చెప్పవలసిన అంశాలు? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "వారి అంతర్ దృష్టిలో దానిని నిర్మించడానికి మార్గం ఏమిటి? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "ఓహ్, దానిపై 3 సున్నాలు ఉన్నాయి, మిలియన్ల లాగ్ ఏమిటి? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "1000 రెట్లు x యొక్క లాగ్ x యొక్క లాగ్‌కి 3 రెట్లు సమానం మరియు ఇది బేస్ 10 లాగ్ b అని మేము సంప్రదాయాన్ని ఉపయోగిస్తున్నామని గుర్తుంచుకోండి. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "1000 రెట్ల లాగ్ x x క్యూబ్డ్ సి లాగ్‌కి సమానం. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "1000 సార్లు x యొక్క లాగ్ x మరియు e యొక్క లాగ్ యొక్క శక్తికి 3 సమానం. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "పైన పేర్కొన్న వాటిలో ఏదీ లేదు మరియు నేను ఇంతకు ముందు చెప్పినట్లుగా గుర్తుంచుకోండి, మొదట్లో లాగ్‌లను బాగా అర్థం చేసుకున్న వ్యక్తులందరూ వెంటనే సమాధానం ఇస్తారు, వారు సరిగ్గా సమాధానం ఇస్తారు కానీ మీరు అయితే అలా చేయని వ్యక్తి, మీరు ఇలాంటి సమస్యను చూస్తున్నప్పుడు అది మిమ్మల్ని భయపెట్టనివ్వకండి సున్నాల సంఖ్యను గణిస్తుంది కాబట్టి దాని గురించి ఆలోచించడానికి నేను మీకు కొంచెం సమయం ఇస్తాను కాబట్టి నేను ముందుకు వెళ్లి దానిని గ్రేడ్ చేస్తాను మరియు ఎప్పటిలాగే మీరు సౌకర్యవంతంగా ఉన్నదాని కంటే వేగంగా ఉంటే అది నేను ముందుకు సాగాలనుకుంటున్నాను కాబట్టి మాత్రమే అని తెలుసుకోండి పాఠంతో కాబట్టి ఈ సందర్భంలో సరైన సమాధానం 1000 రెట్ల లాగ్‌గా వస్తుంది x 3ని కలిపి x లాగ్‌ని తీసుకుంటే సమానం మరియు ఇప్పుడు దాని గురించి ఒక్క క్షణం ఆలోచిద్దాం మరియు మీరు ఇప్పుడే ప్రారంభించినప్పుడు నేను చెప్పినట్లుగా వారితో నేను చేయవలసిన ఉత్తమమైన విషయం ఏమిటంటే, వివిధ సంఖ్యలను ప్లగ్ చేయడం సౌకర్యంగా ఉండటమే అని నేను భావిస్తున్నాను మరియు ప్లగ్ ఇన్ చేయడానికి ఉత్తమమైన నంబర్‌లు ఇప్పటికే 10 పవర్‌లను కలిగి ఉంటాయి కాబట్టి మీరు 1000 రెట్ల లాగ్ లాంటివి అడుగుతున్నట్లయితే x నేను చేయను' తెలియదు, 1000 రెట్లు 100 యొక్క x లాగ్ కోసం ఏదైనా ప్లగ్ చేద్దాం, ఇక్కడ తుది సమాధానంలో ఎన్ని సున్నాలు ఉండబోతున్నాయో మనకు బాగా తెలుసు 1000 సార్లు 100 అంటే 100,000 మనం 10 యొక్క 2 శక్తులను గుణించినప్పుడు మనకు అకారణంగా ఈ ఆలోచన ఉంది. మేము కేవలం సున్నాలను తీసుకుంటున్నాము, ఆ 1000 నుండి 3 సున్నాలు ఆ 100 నుండి 2 సున్నాలు మరియు మేము వాటిని ఒకదానికొకటి ఉంచుతున్నాము కాబట్టి అది మొత్తం 5 సున్నాలుగా ఉండాలి, కానీ మీరు నిజంగా ప్రతిబింబిస్తే, సంఖ్య ఎలా మారిపోయింది ఎందుకు బయటకు వచ్చింది కానీ అది ఆ 1000 నుండి 3 సున్నాలు మరియు ఆ 100 నుండి 2 సున్నాలు అని మేము 1000లోని సున్నాల సంఖ్యను కలిపి 100లోని సున్నాల సంఖ్యను చెప్పడం ద్వారా కూడా వ్రాయవచ్చు కాబట్టి ఈ ఆలోచన ఒక లాగరిథమ్ రెండు వస్తువుల ఉత్పత్తి యొక్క సంవర్గమానాల మొత్తం 10 యొక్క శక్తుల సందర్భంలో ఉంటుంది, మీరు 10 యొక్క 2 శక్తులను తీసుకుంటే, మీరు వాటిని గుణించినట్లయితే, మనలో చాలా మందికి ఇది ఇప్పటికే ఒక సూపర్ సహజమైన ఆలోచన ఏమిటో తెలియజేస్తుంది. వాటి సున్నాలన్నింటినీ తీసుకుని, వాటిని ఒకదానిపై మరొకటి క్రామ్ చేయండి, కాబట్టి నేను ఇక్కడ విషయాలను వ్రాసిన విధానం వాస్తవానికి కొంచెం ఎక్కువ సాధారణ వాస్తవాన్ని సూచిస్తుంది, ఇది లాగరిథమ్‌ల యొక్క మా మొదటి ఆస్తి అవుతుంది, అంటే మనం తీసుకుంటే A సార్లు B యొక్క లాగ్ ఇప్పుడు A యొక్క లాగ్ మరియు B యొక్క లాగ్‌కి సమానం నేను అనవసరంగా ఉన్నాను, నేను దీన్ని చాలా చెబుతున్నాను కానీ మీరు బీజగణితంలో మునిగిపోయి, మీరు ఏదో ఒక రకమైన పరీక్షలో కూర్చున్నప్పుడు దాన్ని మర్చిపోవడం చాలా సులభం అని నేను భావిస్తున్నాను మరియు దీనికి చాలా చిహ్నాలు ఉన్నాయి మీకు గుర్తు చేసుకోవడానికి మీరు కొన్ని సంఖ్యలను ప్లగ్ చేయడం మంచిది మరియు తరచుగా ఇది అంతర్ దృష్టిని అందించడానికి గొప్ప మార్గం కాబట్టి ఈ సందర్భంలో, A టైమ్స్ B యొక్క లాగ్‌ని చెప్పి, దానిని విడదీయడం ద్వారా మనం ఆలోచించవచ్చు, ఓహ్ 100 సార్లు 1000 యొక్క చిట్టా 5, దానిలో 5 సున్నాలు ఉన్నాయి, ఇచ్చిన ప్రతి భాగంలోని సున్నాల సంఖ్య పరంగా విడదీయడం చాలా బాగుంది, అద్భుతమైనది కాబట్టి ఆ అంతర్ దృష్టిని మరింత ముందుకు తీసుకువెళ్లి మరొక అభ్యాస సమస్యను ప్రయత్నిద్దాం, మీకు తెలిస్తే, గొప్పది, మీరు దీనికి చక్కగా సమాధానం చెప్పగలరు, కానీ ఆలోచించండి, సమాధానం ఏమిటో మాత్రమే కాదు, నేను ఈ సమాధానాన్ని ఎవరికైనా ఎలా వివరిస్తాను లేదా నేను చెప్పాల్సిన అవసరం లేకుండా ఒక విద్యార్థి స్వయంగా ఈ సమాధానానికి వచ్చేలా ఎలా ప్రయత్నిస్తాను వారికి సమాధానం ఏమిటి కాబట్టి ఇద్దరు సంభావ్య ప్రేక్షకులు ఉన్నారు కాబట్టి పాఠంపై ఆసక్తి ఉన్నవారు మరియు మెటా పాఠంపై ఆసక్తి ఉన్నవారు ఉన్నారు కాబట్టి మా ప్రశ్న మళ్లీ అడుగుతుంది, ఈ క్రింది వాటిలో ఏది నిజం? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "a. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "n నుండి x యొక్క లాగ్ x b యొక్క n సార్లు లాగ్‌కు సమానం. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "n నుండి x లాగ్, పవర్ n cకి x లాగ్‌కి సమానం. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "n నుండి x లాగ్ సమానం n ప్లస్ x లేదా d లాగ్. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "కాబట్టి ఇక్కడ సరైన సమాధానం a, ఇది మీలో 4,000 మంది అభినందనలు పొందినట్లు కనిపిస్తోంది, x యొక్క లాగ్ పవర్ n కు సమానం అని మాకు చెబుతుంది కాబట్టి x యొక్క n సార్లు లాగ్‌కి సమానం కాబట్టి, మీరు దీన్ని నేర్పడానికి ప్రయత్నిస్తున్నారని చెప్పండి ఎవరికైనా లేదా మీరు మీరే అర్థం చేసుకోవడానికి ప్రయత్నిస్తున్నట్లయితే, ప్రారంభించడానికి ఒక మంచి ప్రదేశం ఏదైనా ప్లగ్ చేయడం అని నేను భావిస్తున్నాను మరియు ఈ సందర్భంలో, శక్తికి x యొక్క లాగ్ n కోసం 100 పవర్‌తో ప్రయత్నిద్దాం 3 మరియు మీరు చేస్తున్న నమూనాలు నిజంగా పనిచేస్తాయో లేదో తెలుసుకోవడానికి మీరు దీన్ని ఇతరులతో ప్రయత్నించవచ్చు, కానీ మీరు దాని గురించి ఆలోచిస్తుంటే, సమాధానం ఏమిటో చూడటం గురించి కాకుండా సమాధానం ఎందుకు అలా వచ్చిందో ఆలోచించడానికి ప్రయత్నిస్తుంది. కొన్నిసార్లు ఒక ఉదాహరణ సరిపోతుంది ఎందుకంటే 100 క్యూబ్డ్, అది బాగా తీసుకుంటుందని మనం అనుకోవచ్చు, అది 100కి 3 కాపీలు నేను 100కి 3 కాపీలు తీసుకుంటున్నాను మరియు నేను అన్నింటినీ గుణించినప్పుడు మరియు సున్నాల సంఖ్యను లెక్కించినట్లుగా లాగ్ అనుకుంటున్నాను చెప్పండి, ఓహ్, అది కేవలం 6 సున్నాలను కలిగి ఉన్న కొంత సంఖ్య అవుతుంది, అంటే 100 సార్లు 100 సార్లు 100 తీసుకోవడం అంటే, మిలియన్‌ని పొందడానికి నేను ఆ సున్నాలను అన్నింటినీ ఒకదానితో ఒకటి సమూహపరచడం గురించి ఆలోచించగలను కాబట్టి ఈ సంఖ్య ఉంటుంది 6 కానీ వాస్తవానికి అది 6 ఎందుకు అని మనం ఆలోచిస్తే, అది మిలియన్ లోపల ఉన్న సున్నాల సంఖ్య మాత్రమే కాదు, ఆ 6 ఎక్కడ నుండి వచ్చింది అంటే మన దగ్గర ఆ 100 యొక్క 3 కాపీలు ఉన్నాయి మరియు ఆ 100లో ప్రతి ఒక్కటి 2 వేర్వేరు సున్నాలను కలిగి ఉంటాయి కాబట్టి ఇది మరింత సాధారణమైనది. 100 క్యూబ్‌లను తీసుకునే బదులు మనం 1000 క్యూబ్‌లు లేదా 1000 నుండి n లేదా x నుండి పవర్ n వరకు చూస్తున్నట్లయితే, అది n యొక్క విలువ ఏదైతేనేం అని మీరు ఆలోచించవచ్చు. బావుల సంఖ్య, చూద్దాం, మనం xకి ప్రత్యామ్నాయంగా ఉన్న సున్నాల సంఖ్య కంటే x రెట్లు కాదు, ఈ సందర్భంలో 100 ఉంది కాబట్టి బదులుగా నేను 10,000 లాగ్ వంటి దానిని శక్తికి తీసుకున్నట్లయితే ఇది అదే అవుతుంది. ఆ 10,000 యొక్క n కాపీలను తీసుకుంటే, వాటిలో ప్రతి దానిలోని సున్నాల సంఖ్యను 4గా లెక్కిస్తే, అది n రెట్లు 4 అవుతుంది మరియు మీలో చాలామంది సరిగ్గా సమాధానం ఇచ్చిన సాధారణ లక్షణం ఏమిటంటే, మీరు ఎక్కడ ఈ మనోహరమైన ప్రభావాన్ని కలిగి ఉంటారు కొద్దిగా శక్తి దాని ముందు కిందకి దూసుకెళ్లే శక్తికి పెరిగిన లాగ్‌ను చూడండి మరియు మీరు ఇప్పుడు లోపల ఉన్న వాటి లాగ్‌ని కలిగి ఉన్నారు, దాని యొక్క అతి ముఖ్యమైన చిక్కులలో ఒకటి మీరు దీన్ని పిలుస్తారో లేదో నాకు తెలియదు ఒక అంతరార్థం లేదా మీరు దానిని డెఫినిషన్ యొక్క పునఃస్థాపన అని పిలిస్తే, నేను లాగ్‌ని తీసుకుంటే మరియు నేను దాని 10కి 10 ఆధారాన్ని తిరిగి నొక్కి చెబుతాను n ఆ చిన్న n గురించి మనం ఆలోచించవచ్చు ముందు మరియు ఇది 10కి 10కి n రెట్లు లాగ్ బేస్ అవుతుంది, అంటే 1 ఈ వ్యక్తీకరణ మీరు చివరలో ఉన్న సున్నాల సంఖ్యను లెక్కించినట్లుగా భావించవచ్చు లేదా సాధారణంగా ఇది 10కి సమానమైన దానికి 10ని అడుగుతుంది మరియు సమాధానం కేవలం 1 ఇది చాలా భరోసానిస్తుంది ఎందుకంటే మీరు వెనుకకు వెళ్లి ఈ అసలైన వ్యక్తీకరణను చదవగలిగే మరొక మార్గం 10కి 10కి సమానం అంటే n ఓహ్, సమాధానం n ఓకే అని చెప్పవచ్చు, మేము కలిగి ఉన్న ప్రతి లాగరిథమ్ ప్రాపర్టీతో ఈ సందర్భంలో మేము శక్తి nకి x యొక్క ఒక లాగ్‌ని కనుగొన్నారు, n ముందు దూకడం ఎల్లప్పుడూ మిర్రర్ ఇమేజ్ ఎక్స్‌పోనెన్షియల్ ప్రాపర్టీగా ఉంటుంది మరియు దీని కోసం మనం కొంచెం అంతర్ దృష్టిని పొందడంలో సహాయపడే మరొక మార్గం, కాబట్టి నన్ను కప్పిపుచ్చుకోనివ్వండి మనం ఇక్కడ పొందబోయే కొన్ని భవిష్యత్తు లక్షణాలు మనం ఎక్కడికి వెళ్తున్నామో దాచడానికి ప్రయత్నించండి ఆ మొత్తం 10 నుండి n సార్లు xకి తీసుకెళ్తుంది మరియు ఇది లాగరిథమ్‌ల కోసం మీరు కలిగి ఉండే మరొక అంతర్ దృష్టికి మమ్మల్ని తీసుకువెళుతుంది, అంటే అవి ఒక రకమైన ఎక్స్‌పోనెన్షియేషన్ లాగా ఉంటాయి మరియు నా ఉద్దేశ్యం ఇక్కడ ఉంది నేను లాగ్ యొక్క లాగ్‌ను తీసుకుంటే, లాగ్ లోపలి భాగంలో కూర్చున్న విషయం మీరు ఈ సందర్భంలో ఘాతాంకమైన దాని కోసం మొత్తం బాహ్య వ్యక్తీకరణగా భావించాలి, ఈ సందర్భంలో a లోపల ఉన్న విషయం 10 నుండి x దికి అనుగుణంగా ఉంటుంది ఫంక్షన్ యొక్క అవుట్‌పుట్, అయితే a యొక్క మొత్తం లాగ్ లోపల ఉన్న దానికి అనుగుణంగా ఉంటుంది, అయితే 10 యొక్క ఘాతాంకం ఏమిటి కాబట్టి మీరు ఇక్కడ లాగ్ ఎక్స్‌ప్రెషన్‌ని ఎక్కడ చూసినా కుడి వైపున ఘాతాంకం పాత్ర పోషిస్తుందని మీరు ఆలోచిస్తూ ఉండాలి. వైపు మరియు ప్రతిసారీ మీరు ఎక్స్‌ప్రెషన్ మొత్తం 10 నుండి x ఎక్స్‌ప్రెషన్‌ని చూసిన ప్రతిసారీ లాగ్‌లలో ఒకదాని లోపలి భాగంలో కూర్చొని ఉన్నదానికి అనుగుణంగా కుడి వైపున ఉన్న మొత్తం బయటి భాగం మరియు మేము గుణించేటప్పుడు ఈ ఆలోచన పైన చూసాము లాగ్‌లు ఘాతాంకాలను లోపలికి మార్చినట్లయితే లోపలి భాగంలో బాగా జోడిస్తుంది, ఇది ఫంక్షన్ యొక్క అవుట్‌పుట్‌లను వెలుపల గుణించడం అనేది లోపలికి జోడించడం వలె ఉంటుంది ఎందుకంటే ఈ లాగ్‌లలో ప్రతి ఒక్కటి లాగ్ a మరియు లాగ్ బి వంటివి కుడి వైపున ఉన్న వ్యక్తీకరణలో x మరియు y పాత్రను పోషిస్తోంది కాబట్టి దానితో మనం ఆడుతూనే ఉంటాము, వీటిలో కొన్నింటిని మాత్రమే చేద్దాం మరియు వీటిలో ఎన్ని లక్షణాల కోసం మనం అంతర్ దృష్టిని నిర్మించగలమో చూద్దాం, ఈ చివరిది, ఘాతాంకాలు తరువాతి స్థానానికి దూకడం గురించి ఆలోచించడం అనేది లాగరిథమ్‌ల గురించి ఖచ్చితంగా తెలియని వారికి కొంచెం విచిత్రంగా అనిపించవచ్చు, కానీ దాని కోసం కొంత అంతర్ దృష్టిని పొందడానికి కొన్ని సంఖ్యలను ప్లగ్ చేయండి మరియు మేము దానిని కొద్దిగా ఇస్తాము కింది వాటిలో ఏది నిజం? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "సరే, 10 క్యూబ్‌లు 1000 అయితే, 10ని 1000కి సమానం అని చెప్పడం 1 థర్డ్‌కు పెంచడం వంటిది ఇక్కడ విలోమం చేయడంలో ఘాతాంకం యొక్క గుణకార విలోమం ఉంటుంది మరియు అది 1ని 3తో విభజించినట్లుగా కనిపిస్తుంది. మరియు 3 అనేది లాగ్ బేస్ 10కి 1000కి అనుగుణంగా ఉంటుంది, ఇది 1ని లాగ్ బేస్ 10తో 1000తో భాగించబడుతుంది కాబట్టి సాధారణంగా, మీరు ఈ ఒక్క ఉదాహరణ ఆధారంగా ఊహించవచ్చు, మనం లోపల ఉన్న దానితో బేస్‌ను మార్చుకున్నప్పుడు అది 1 విభజించబడిందని తీసుకుంటుంది. బయట ఉన్నవాటిని బట్టి మళ్లీ మళ్లీ, సంబంధిత ఎక్స్‌పోనెన్షియల్ రూల్‌ని చూడటం ద్వారా మీరు దీన్ని ఆలోచించవచ్చు, ఇప్పుడు నా అందమైన చిన్న లాగ్ మరియు ఎక్స్‌పోనెన్షియల్‌లకు ఏమి జరిగింది? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "అద్భుతం కాబట్టి, మళ్ళీ మనం ఇక్కడకు వచ్చే కొన్ని ఇతర ఆస్తులను ఎక్కడ దాచిపెడతాము మరియు నేను ఇక్కడ ముందు కలిగి ఉన్న అదే క్రమంలో ఉంచుతాను, ఇది ముందే వ్రాసి ఉంచడం వల్ల నన్ను ఉంచవచ్చని నేను ఆలోచిస్తున్నాను సాధారణం కంటే కొంచెం శుభ్రంగా ఉంటుంది, అయితే ఇది ఈ విచిత్రమైన పేపర్ కటింగ్ షఫుల్ గేమ్‌ను ఆడటం వల్ల మనం ఇప్పుడే కనుగొన్నాము, లాగ్ బేస్ బిని మీరు మార్చుకుంటే, ఇది 1తో భాగించినట్లే ఉంటుంది ఎక్స్‌పోనెన్షియల్ ల్యాండ్ అంటే మీరు బిని కొంత పవర్‌కి తీసుకుని, అది ఎకి సమానం అని చెబితే, ఆ పవర్ యొక్క విలోమానికి a మళ్లీ బికి సమానం అని చెప్పినట్లే అదే స్టేట్‌మెంట్, కొంత సమయం తీసుకొని లాగరిథమ్‌లను టర్నింగ్ టర్నింగ్ అని ఆలోచించడం ఒక రకంగా సహాయపడుతుంది. a యొక్క ఎక్స్‌ప్రెషన్ లాగ్ బేస్ b లోపల ఆ x పాత్రను పోషిస్తోంది మరియు b యొక్క ఎక్స్‌ప్రెషన్ లాగ్ బేస్ a పైన కూర్చున్న దాని పాత్రను పోషిస్తోంది మరియు తర్వాత సౌష్టవంగా, శక్తి x వరకు b మొత్తం వ్యక్తీకరణ ప్లే అవుతోంది. ఎడమవైపు లోపలి పాత్ర, ఇది a మరియు మొత్తం వ్యక్తీకరణ పాత్రను పోషిస్తుంది, ఏదో యొక్క శక్తికి a అనేది లాగ్ బేస్ లోపల కూర్చున్న పాత్రను పోషిస్తుంది a కాబట్టి మీరు కొన్ని ఉదాహరణలను ప్లగ్ చేయడం ద్వారా చూడవచ్చు మరియు ఘాతాంక నియమాలకు అనుగుణంగా మేము ఇప్పటికే మూడు వేర్వేరు సంవర్గమాన నియమాల ద్వారా ఆలోచించవచ్చు, వాటిని గుర్తుంచుకోవడానికి బీజగణితం ముక్కలుగా అందజేస్తే, మీరు వాటిని గుర్తుంచుకోవచ్చు, కానీ అవి మీ నుండి జారిపోవడం చాలా సులభం. తల మరియు చేతిలో ఉన్న పనిని చూసి నిరుత్సాహపడటం కూడా చాలా సులభం, అయితే మేము ఈ విధమైన విషయాల పట్ల శ్రద్ధ వహించడానికి కారణం లాగరిథమ్‌ల నియమాలను అర్థం చేసుకోవడం వల్ల అది వైరస్ లాగా పెరుగుతున్న సందర్భాల్లో గణితాన్ని చేయడంలో మాకు సహాయపడుతుందని మీరు గుర్తుంచుకోవాలి. ఒక రోజు నుండి మరొక దశకు, ఒక దశ నుండి మరొక దశకు, సంవర్గమానాల నియమాలను అర్థం చేసుకోవడం ద్వారా విషయాలు గుణకారంగా పెరుగుతాయి కాబట్టి ఆ రకమైన విషయాల కోసం మంచి అనుభూతిని పొందడంలో మీకు సహాయపడుతుంది కాబట్టి మేము కనిపించే దానికి ఒక చక్కని వాస్తవ ప్రపంచ ఉదాహరణను చేస్తాము లాగారిథమ్‌ల లక్షణాల గురించి అడగడానికి ఈ పంథాలో మరో క్విజ్ ప్రశ్నను అడగనివ్వండి, మనం కొంచెం వాస్తవ ప్రపంచ ఉదాహరణకి మారే ముందు మనం ఇక్కడ మరియు ఇప్పుడు ఉన్నవాటిని వదిలించుకోండి, కింది వాటిలో ఏది నిజం? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "a plus b యొక్క చిట్టా ఒక ప్లస్ b లాగ్ యొక్క ఒక ప్లస్ లాగ్ యొక్క లాగ్ వలె ఉంటుంది. ఒక ప్లస్ b యొక్క లాగ్ బి యొక్క టైమ్ లాగ్ యొక్క లాగ్ ద్వారా భాగించబడిన ఒకదానికి సమానం లేదా పై ఏదీ లేదు ah, మరియు ఇప్పుడు మనకు అంత ఏకాభిప్రాయం లేదు, లేదా? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "చాలా ఆసక్తికరంగా, మేము ఇద్దరి మధ్య గుర్రపు పందెం కలిగి ఉన్నాము కాబట్టి ప్రజలు సమాధానం ఇస్తున్నప్పుడు నేను మీకు కొంత సమయం ఇస్తాను, వాస్తవానికి ప్రేక్షకుల కోసం నా దగ్గర ఒక చిన్న ప్రశ్న ఉంది కాబట్టి, మీకు తెలుసా, మనం ఎలా ఉండవచ్చనే దాని గురించి నేను మాట్లాడుతున్నాను గుణకార వృద్ధి పరంగా ఆలోచించండి మరియు అది కేవలం పది శక్తులుగా ఉండవలసిన అవసరం లేదు, మీరు ఒకటి నుండి మూడు నుండి తొమ్మిది నుండి ఇరవై ఏడు నుండి ఎనభై ఒకటి వరకు వెళుతున్నట్లయితే, మేము మూడు శక్తులు వంటి వాటిని కూడా చేయవచ్చు. వీటిలో లాగ్ బేస్ మూడు ఈ సంఖ్యల చిన్న చిన్న దశల్లో పెరుగుతాయని మనం చెప్పగలం, కాబట్టి ఒకదానిలో మూడు బేస్, మూడింటిని ఒకదానికి సమానం చేయడానికి లాగ్ చేయండి, సమాధానం సాధారణంగా ఒకదాని లాగ్, ఆధారంతో సంబంధం లేకుండా సున్నా అవుతుంది. సున్నా లాగ్ బేస్ మూడు ఆఫ్ త్రీ, త్రీ ఈక్వల్స్ త్రీ ఒకటి అదే లాగ్ బేస్ త్రీ ఆఫ్ నైన్ రెండు ఆహ్, మీరు నా ప్రశ్న ఏమిటని ఆశ్చర్యపోవచ్చు, కానీ వీటన్నింటిని గీయడానికి మరియు నా స్వంత ఆనందం కోసం ఇది సహాయపడుతుంది ఇక్కడ, నేను ఇప్పుడు ఎనభై ఒకటిలో మూడు లాగ్ బేస్ వ్రాద్దామనుకుంటున్నాను, మీరు ఒక పిల్లవాడిని అడిగితే, నేను విన్నాను, మీరు ఒక పిల్లవాడిని అడిగితే, ఐదు లేదా ఆరు సంవత్సరాల వయస్సులో మీరు ఒకటి మరియు తొమ్మిది మధ్య ఏ సంఖ్య సగం అని చెప్పండి సమాధానం ఎలా చెప్పాలో వారి ప్రవృత్తులు సగానికి సగం అని చెప్పండి, అయితే మన ప్రవృత్తులు మరింత సరళంగా ఉంటాయి కాబట్టి మేము తరచుగా ఒకటి మరియు తొమ్మిది అని అనుకుంటాము, మీరు వాటి మధ్య రెండు, మూడు, నాలుగు, ఐదు, ఆరు సమాన ఖాళీ సంఖ్యల సమూహాన్ని పొందారు. , ఏడు, ఎనిమిది మరియు మీరు మధ్యలో సగానికి వెళితే, మీరు ఐదులో అడుగుపెడతారు, కానీ మీరు గుణకార వృద్ధి పరంగా ఒకటి నుండి తొమ్మిది వరకు ఎక్కడికి వెళ్లాలి అని ఆలోచిస్తుంటే, అది కొన్ని విషయాలను జోడించడం కాదు, కానీ మీరు 'ఒక నిర్దిష్ట మొత్తంలో మీరు మూడు రెట్లు పెరుగుతారు, ఆ తర్వాత మీరు మరో మూడు కారకాలతో పెరుగుతారు, పిల్లల సహజ ప్రవృత్తి మూడు అని చెప్పవచ్చు మరియు మీరు స్వర్గధామమైన సమాజాలను అధ్యయనం చేసే మానవ శాస్త్రవేత్తలను కలిగి ఉంటే ఇది కూడా వర్తిస్తుంది' అకౌంటింగ్ సిస్టమ్‌లను అభివృద్ధి చేసి, ఆధునిక సమాజాలు రూపొందించిన విధంగానే రాయడం ద్వారా వారు దీనికి మూడింటికి సమాధానం ఇస్తారు కాబట్టి, ప్రస్తుతం చూస్తున్న మీలో ఎవరికైనా ఐదేళ్ల వ్యవధిలో చిన్న పిల్లలకు యాక్సెస్ ఉందా అని ప్రేక్షకులకు నా ప్రశ్న. ఒకటి మరియు తొమ్మిది మధ్య సగం సంఖ్య అని మీరు వారిని అడగవచ్చో లేదో చూడండి మరియు మీకు వీలైతే, పిల్లవాడు వారి అసలు సమాధానం ఏమిటో ట్విట్టర్‌లో మాకు తెలియజేయండి ఎందుకంటే నాకు ఎందుకు తెలియదు, నేను కొంచెం ఉన్నాను ఇది నిజంగా ఆచరణలో సాగుతుందా అనే సందేహం ఉంది, ఇది చేయడానికి ఇది సూపర్ సైంటిఫిక్ మార్గం కాదని నేను అర్థం చేసుకున్నాను, YouTube లైవ్‌స్ట్రీమ్‌ని చూస్తున్న వ్యక్తులను వారి స్వంత పిల్లలను సర్వే చేసి, ఆపై సమాధానాన్ని ట్వీట్ చేయమని నేను అడగడం లేదు కానీ నా ప్రయోజనాల కోసం ఇది ఆసక్తికరంగా ఉంటుంది మా ప్రశ్నకు తిరిగి ఒక రకమైన ధృవీకరణను చూడటానికి, ఒక దిశలో పెద్దగా ఏకాభిప్రాయం లేనటువంటి మొదటిది ఇదే అని చెప్పవచ్చు పైన పేర్కొన్న వాటిలో ఏ ప్లస్ b యొక్క లాగ్ ఈ మంచి లక్షణాలలో దేనినీ సంతృప్తిపరచదని మరియు సాధారణంగా, మేము కొన్ని రకాల ఉజ్జాయింపులతో ప్రత్యేకించి సహజ లాగ్ అమలులోకి వచ్చినప్పుడు తప్ప పని చేయబోతున్నామని మీరు సరిగ్గా సమాధానం ఇచ్చారు మేము దీని గురించి తదుపరిసారి మాట్లాడవచ్చు ఒక లాగరిథమ్ యొక్క ఇన్‌పుట్‌లను జోడించడం నిజానికి చాలా విచిత్రమైన సంచలనం, ఇది చాలా విచిత్రమైన పని మరియు ఆ విచిత్రతను అర్థం చేసుకోవడానికి, నేను మిమ్మల్ని ప్లస్ బి యొక్క లాగ్‌ను అడిగితే పదిలోని కొన్ని పవర్‌లను ప్లగ్ చేయండి మీరు ఆలోచించడం ప్రారంభించవచ్చు, సరే, 10,000 మరియు 100 వంటి కొన్ని ఉదాహరణలను ప్లగ్ చేయనివ్వండి మరియు నేను ఈ ఇన్‌పుట్‌లో ఉన్నవాటికి జీరో కౌంటింగ్ ఫంక్షన్ చేస్తే, అందులో ఎన్ని సున్నాలు ఉన్నాయి? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "అది ఒక ఆసక్తికరమైన ప్రశ్న సరే, సంవర్గమానం యొక్క ఆధారం సున్నా కాగలదా? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "మన త్రిభుజం పరంగా మీకు తెలిసినట్లుగా మనం భావించవచ్చు, సున్నాకి ఒకరకమైన శక్తికి x కొంత ఇతర విలువకు సమానం y ఇది మనం xకి సున్నా అని చెప్పడం ద్వారా వ్రాయవచ్చు లేదా మనం వ్రాయవచ్చు అదే విషయం y యొక్క లాగ్ బేస్ జీరో x సున్నాకి సమానం అని చెప్పడం ద్వారా ఇప్పుడు ఇక్కడ సమస్య ఏమిటంటే, దేనికైనా సున్నా సున్నాగా ముగుస్తుంది, కాబట్టి మనం లాగ్ బేస్ జీరో గురించి ఆలోచిస్తున్నాము. y మీకు తెలిసిన ఏదైనా ఇతర ఇన్‌పుట్ కోసం, మీరు ఒకటి లేదా రెండు లేదా pi వంటి ఏదైనా ఇన్‌పుట్ చేయాలనుకుంటున్నారు, మీరు ఒకటి లేదా రెండు లేదా piకి సమానం లేదా మీరు అక్కడ కలిగి ఉన్న సంఖ్యకు సున్నా అనే ప్రశ్నను అడుగుతున్నారు. మరియు సమాధానం ఉండదు కాబట్టి మీరు ఓహ్ అవును, సున్నా యొక్క లాగ్ అని చెప్పడానికి ప్రయత్నించవచ్చు, ఇది ఖచ్చితంగా చెల్లుబాటు అయ్యే ఫంక్షన్, ఇది ఇన్‌పుట్ సున్నాపై మాత్రమే నిర్వచించబడుతుంది, అయితే మీరు కోరుకున్నదాన్ని ఫినాగల్ చేయడానికి ప్రయత్నించడంలో మీకు ఇబ్బంది ఉంటుంది. ఎందుకంటే సున్నాకి సమానమైన దానికి సున్నా అని చెప్పడం దానికి ఏదైనా వర్తిస్తుంది కాబట్టి మీ చేయి మీ వెనుకకు వంగి ఉంటుంది, అయితే మీరు ఆ పనిని చేయాలనుకుంటున్నారు మరియు ఇది బేస్ జీరోతో ఎక్స్‌పోనెన్షియల్ ఫంక్షన్ పూర్తిగా సున్నా అనే దానికి అనుగుణంగా ఉంటుంది. సంఖ్యలను ఒకదానికొకటి ఒకదానితో ఒకటి చక్కగా మ్యాప్ చేయడం లేదు కాబట్టి ఇది ఒక గొప్ప ప్రశ్న, వాస్తవ ప్రపంచంలో ఈ విషయాలు ఎక్కడికి వస్తాయనే ఆలోచనకు మీరు ఇప్పుడు లాగ్ బేస్ జీరోని కలిగి ఉండగలరా, నేను ఇష్టపడే ఒక ఉదాహరణ భూకంపాలకు సంబంధించిన రిక్టర్ స్కేల్ కాబట్టి భూకంపం ఎంత బలంగా ఉంటుందో రిక్టర్ స్కేల్ మనకు పరిమాణాన్ని ఇస్తుంది మరియు ఇది చాలా చిన్న సంఖ్యల నుండి చాలా పెద్ద సంఖ్యల వరకు ఏదైనా కావచ్చు, ఇది ఇప్పటివరకు కొలిచిన అతిపెద్ద భూకంపం అని నేను భావిస్తున్నాను మరియు ఇది కేవలం ఒక చార్ట్ మాత్రమే వికీపీడియా 9.5 మరియు అది ఎంత పిచ్చిగా ఉందో అర్థం చేసుకోవడానికి, ఈ సంఖ్యల మధ్య సంబంధాన్ని చూడటం విలువైనది మరియు TNTకి సమానమైన మొత్తం వంటిది, దానిలో ఎంత శక్తి ఉంది మరియు మనం ఇక్కడ ఏమి చేయడానికి ప్రయత్నించవచ్చు. శక్తి పరిమాణం పరంగా రిక్టర్ స్కేల్ సంఖ్యకు వ్యక్తీకరణను మనం పొందగలమా మరియు సంవర్గమానాలు దీనిని వివరించడానికి సహజమైన మార్గంగా ఎందుకు ఉండగలదో చూడండి, కాబట్టి మనం విషయాలు ఎంతవరకు పెరుగుతాయో మనం అడుగులు వేస్తున్నప్పుడు దృష్టి సారించాల్సిన కీలకం కాబట్టి ఉదాహరణకు, ఈ సందర్భంలో మనం రెండింటి నుండి బాగా వెళితే అది మూడు ఎక్కడ ఉందో అది మనకు చూపించదు కాబట్టి మనం రెండు నుండి నాలుగు వరకు ఒక అడుగు వేయాలని అనుకుంటాము, ఇది రెండు అడుగులు వేయడం లాంటిది, అది పరంగా ఏమి చేస్తుంది ఇది ఒక మెట్రిక్ టన్ను TNT నుండి మనల్ని తీసుకున్నట్లు కనిపిస్తోంది, అంటే రెండవ ప్రపంచ యుద్ధం నాటి పెద్ద బాంబు అని నేను ఊహిస్తున్నాను మరియు ఇది ఒక చిన్న అణు బాంబు కంటే వెయ్యి రెట్లు ఎక్కువ కిలోటన్‌ను తీసుకుంటుంది కాబట్టి కేవలం రెండు అడుగులు రిక్టర్ స్కేల్‌లో భూకంపం 2 నుండి 4 తీవ్రత వరకు భూకంపం వరకు వెళుతుంది, ఇది రెండవ ప్రపంచ యుద్ధం నుండి అణు యుగం వరకు పెద్ద బాంబు నుండి మనలను తీసుకువెళుతుంది కాబట్టి ఇది గమనించదగినది మరియు మనకు లభించే మొదటి శుభ్రమైన దశ 4 నుండి 5 వరకు కనీసం ఈ చార్ట్ మనకు చక్కగా చూపుతున్న దాని పరంగా మరియు స్పష్టంగా 4 నుండి 5 వరకు ఒక్క మెట్టు 1 కిలోటన్ నుండి 32 కిలోటన్‌లకు వెళ్లడానికి అనుగుణంగా ఉంటుంది మరియు ఇది నాగసాకిపై పడిన బాంబును నాశనం చేసే నగరం యొక్క పరిమాణం అని స్పష్టంగా తెలుస్తుంది కాబట్టి ఇది బహుశా ఒకటి కావచ్చు. భూకంపం 4 వచ్చిందని మీరు వార్తల్లో వింటున్నట్లయితే, సంవర్గమాన ప్రమాణాల గురించి ప్రతికూలంగా ఉంటుంది. 0 వర్సెస్ భూకంపం 5.0 అవును 4 మరియు 5 చాలా సారూప్యమైన సంఖ్యలు అని అనుకోవడం చాలా సులభం, అయితే స్పష్టంగా TNT మొత్తాల పరంగా 32తో గుణించి 1 నుండి తదుపరిదానికి మరియు 2 నుండి 4కి వెళ్లడం అనేది దాదాపు వెయ్యి మరియు ఏకైక సంఖ్యతో గుణించబడుతోంది. దానికి కారణం పెద్దది ఎందుకంటే ఇక్కడ మా చార్ట్ 3 ఏమిటో చూపడం లేదు కాబట్టి మేము రెండు అడుగులు వేస్తున్నాము మరియు మీరు 32 అడుగులు వేసి మరో 32తో గుణిస్తే అది దాదాపు వెయ్యికి దగ్గరగా ఉంటుందని మీరే ధృవీకరించుకోవచ్చు. రిక్టర్ సంఖ్యపై సంకలిత దశలు TNTలోని గుణకార దశలకు అనుగుణంగా ఉంటాయి అనే ఆలోచన ఇక్కడ లాగరిథమిక్ ఏదో ప్లే అవుతుందని సూచించినట్లు అనిపిస్తుంది మరియు ఇక్కడ కొనసాగడం మరియు ప్రపంచ దృగ్విషయం కారణంగా ఇది ఎంతవరకు పెరుగుతుందో చెప్పడం కొంచెం ఆసక్తికరంగా ఉంది. అవును అని వర్ణించడం పెద్ద ఆశ్చర్యం కాదు, మనం మరో అడుగు వేస్తున్నప్పుడు అది మళ్లీ 32 ద్వారా గుణించబడుతోంది, కానీ మన అంతర్ దృష్టిలో 32 కిలోటన్లు చిన్న అణు బాంబు మరియు ఒక మెగాటన్ మధ్య వ్యత్యాసం చిన్న అణు బాంబు కాదు. నాగసాకి అటామ్ బాంబ్ ఒక మెగాటన్ కోసం నాగసాకి అటామ్ బాంబ్‌లలో 32 అని నేను ఊహిస్తున్నాను, ఇది స్పష్టంగా చెప్పాలంటే నెవాడా USAలో డబుల్ స్ట్రింగ్ ఫ్లాట్ భూకంపం యొక్క పరిమాణం 1994 నాకు అది ఏమిటో తెలియదు, పౌనఃపున్యాల పరంగా వికీపీడియాకు ధన్యవాదాలు వీటిని కూడా స్పష్టంగా చూసారు, రెండు కంటే తక్కువ ఉన్నవి, అవి అన్ని సమయాలలో జరుగుతాయి రోజుకు 8000 వంటివి ఉన్నాయి, అయితే మనం అణు బాంబుల రంగంలోకి వచ్చిన వెంటనే 3 వంటి విషయాలు. 5 మరియు 4 భూమ్మీద ఎక్కడో ఒకచోట చాలా తరచుగా జరుగుతుంటాయి, ప్రతిరోజూ ఎక్కడో 134 జరుగుతున్నాయి ఎవరికి తెలుసు? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "కానీ మనం ఈ 5 మరియు 6 శ్రేణిలో అటామ్ బాంబ్ స్కేల్ కంటే చాలా ఎక్కువగా ఉన్నందున ఇప్పుడు మనం కేవలం రోజుకు 2 మాత్రమే ఉన్నాము మరియు ఒక భూగర్భ శాస్త్రవేత్త వచ్చి మనమందరం ఎందుకు చేయకూడదో వివరించగలడని నేను ఖచ్చితంగా అనుకుంటున్నాను. భూమి యొక్క క్రస్ట్‌కు ప్రతిరోజూ రెండు అటామ్ బాంబ్ సమానమైన అంతరాయాలు జరుగుతున్నాయనే వాస్తవం గురించి చాలా ఆందోళన చెందకండి, కానీ ప్రతి అడుగు అని మన ఆలోచనను ధృవీకరిస్తూ చాలా మంది ప్రజలు నివసిస్తున్న నగరం వంటి ఏదో ఒక ప్రదేశంలో కేంద్రీకృతమై ఉండటం చాలా అరుదు. 32 పెరుగుదలను కలిగి ఉంటుంది, 6 నుండి 7 వరకు ఉన్న దశ ఎలా ఉంటుందో చూద్దాం మరియు ఇక్కడ ఇది మాకు చాలా ఎక్కువ ఉదాహరణలను అందిస్తోంది, ఇది వాస్తవానికి కంటే పెద్ద అడుగు అని భ్రమ కలిగించవచ్చు మరియు వాస్తవానికి ఇది 1 మెగాటన్ మరియు మధ్య వ్యత్యాసం 32 మెగాటన్లు కాబట్టి 32తో గుణించడం అనేది ఈ చార్ట్‌లో నాకు చాలా ఆసక్తికరంగా అనిపించిన వాటిలో ఒకటి, వాస్తవానికి పరీక్షించబడిన అతిపెద్ద అణ్వాయుధాన్ని పొందడానికి ముందు మనం ఎంత దూరం వెళ్లాలి అనేది చూడండి, ఇది ప్రచ్ఛన్న యుద్ధం యొక్క ఎత్తు. జార్ బాంబు 50 మెగాటన్లు మరియు వాస్తవానికి వారు 100 మెగాటన్ బాంబును కలిగి ఉండటానికి అసలు ప్రణాళికలు కలిగి ఉన్నారని నేను నమ్ముతున్నాను, అయితే ఆ 50 మెగాటన్‌ల నుండి తమను తాము తగ్గించుకున్నామని, మేము మాట్లాడుతున్నాము, నాగసాకి బాంబు యొక్క 32 కిలోటన్లు 32తో గుణించడం ప్రారంభించండి మెగాటన్ మరో 32తో గుణించబడుతుంది కాబట్టి మేము రెండవ ప్రపంచ యుద్ధం ముగిసిన పేలుడు యొక్క వెయ్యి రెట్లు బలం గురించి మాట్లాడుతున్నాము మరియు మీరు ఇప్పటికీ మానవాళి సామర్థ్యంలో 50 మెగాటన్‌ల వద్ద లేరు మరియు అది స్పష్టంగా ఇండోనేషియాలోని జావా భూకంపం కాబట్టి 7 . 0 6 కంటే కొంచెం పెద్దది కాదు. 0, ఇది చాలా పెద్దది మరియు ఇక్కడ ఉన్న విషయం ఏమిటంటే, మీకు గుణకారాన్ని పెంచే స్కేల్‌ని కలిగి ఉన్నప్పుడు, చిన్న దశల వలె కనిపించేవి నిజానికి ఇక్కడ సూచించబడిన శక్తి లేదా సంపూర్ణ విలువల పరంగా భారీ దశలుగా ఉండవచ్చని అభినందించడం విలువైనదే. కాబట్టి మనం ఎప్పుడూ 9 అనే వాస్తవం గురించి ఆలోచిస్తున్నప్పుడు. 5 నిజానికి 7లో మాత్రమే ఉన్నందున అసంబద్ధంగా అనిపిస్తుంది. 0 శ్రేణి మేము ఇప్పటివరకు బయటపెట్టిన అతిపెద్ద థర్మోన్యూక్లియర్ ఆయుధం గురించి మాట్లాడుతున్నాము మరియు ఇది లాగరిథమ్‌లు వచ్చే ఒక ప్రాంతాన్ని సూచిస్తుంది, మానవులు దేనికైనా ఒక స్కేల్‌ను సృష్టించాలనుకున్నప్పుడు, అది ఎంత పెద్ద విషయాలు చేయగలదు అనే దానిలో చాలా విస్తృతమైన వ్యత్యాసానికి కారణమవుతుంది. భూకంపాల పరిమాణం విషయంలో మీరు భూమి చుట్టూ అన్ని సమయాలలో ఏమి జరుగుతుందో, ఒక పెద్ద హ్యాండ్ గ్రెనేడ్ పరిమాణంలో విషయాలను పొందవచ్చు మరియు అది మీ స్కేల్‌లో ఉండాలని మీరు కోరుకుంటారు మరియు అన్ని విధాలుగా శ్రేణి గురించి ఆలోచించాల్సిన అవసరం ఉంది. మానవ చరిత్రలో మేము చూసిన అతి పెద్ద అంతరాయానికి మరియు మీరు ఒక సందర్భంలో మీ సంఖ్యలలో వేర్వేరు అంకెలతో కూడిన మొత్తం సమూహాన్ని మరియు విభిన్నమైన, చిన్న సంఖ్యల యొక్క మొత్తం సమూహాన్ని మాత్రమే వ్రాయడం లేదు. మరొక సందర్భంలో మీ సంఖ్యకు సంబంధించిన అంకెలను లాగరిథమ్‌లను తీసుకొని, ఆ సంఖ్యలను ప్రాథమికంగా 0 మరియు 10 మధ్య స్క్విష్ చేసే ఒకే స్కేల్‌లో ఉంచడం ఆనందంగా ఉంది, సంగీతం కోసం డెసిబెల్ స్కేల్‌తో చాలా సారూప్యమైనదేదో మీరు చూస్తారు. కొంచెం భిన్నంగా మీరు ప్రతిసారీ 10 డెసిబెల్‌ల మెట్టు పైకి తీసుకెళ్తే అది 10తో గుణించటానికి అనుగుణంగా ఉంటుంది, కాబట్టి 1 యొక్క ఒక దశను 10తో గుణించడం కంటే, ఇది 10 ద్వారా గుణించే 10 యొక్క ఒక దశ, ఆ రకంగా దాని గణితాన్ని కొద్దిగా చేస్తుంది. కొంచెం స్క్రూ కానీ ఆలోచన ఒకటే, మీరు 50 డెసిబెల్స్ మరియు 60 డెసిబెల్స్ ఉన్న శబ్దాన్ని వింటున్నట్లయితే, అది ప్రసారం చేయబడే మరియు దాని నుండి వెళ్ళే శక్తి పరంగా చాలా నిశ్శబ్దంగా ఉంటుంది, అది 60 నుండి 70 లేదా 70 వరకు 80 ఆ దశలు, 60 నుండి 80 వరకు, ప్రతి చదరపు ప్రాంతానికి శక్తిని 100 కారకంతో గుణించడం ఉంటుంది, కాబట్టి మీరు లాగరిథమిక్ స్కేల్‌ని చూసిన ప్రతిసారీ, హుడ్ కింద సూచించేదంతా పెరుగుతుందని మీ మనస్సులో తెలుసుకోండి. కరోనావైరస్ వ్యాప్తిని వివరించడానికి మేము చాలా లాగరిథమిక్ స్కేల్‌లను ఎందుకు చూశాము, కాబట్టి మీరు రిక్టర్ స్కేల్ సంఖ్యను 1 ద్వారా పెంచిన ప్రతిసారీ, మీరు 32 ద్వారా గుణించబడుతున్నప్పుడు ఇలాంటి సంబంధాన్ని మీరు ఎలా వర్ణించవచ్చు? బేస్ 32తో ఉన్న లాగ్ పరంగా ఆలోచించగలను, నేను లాగ్‌ను తీసుకుంటే చెప్పగలను, నేను rకి కాల్ చేయబోతున్నాను, రిక్టర్ స్కేల్ కోసం నేను దీన్ని లాగ్ బేస్ 32గా భావించవచ్చు మరియు దానికి అనుగుణంగా ఉంటుంది , కాదు కాదు కాదు, నేను ఈ తప్పు చేస్తున్నాను అది లాగ్ చేయబడిన విషయం కాదు, మేము పెద్ద సంఖ్య యొక్క లాగ్ బేస్ 32ని తీసుకుంటాము, TMT సంఖ్య, 1 మెగాటన్ లాగా ఉండేది, అది 1 మిలియన్ టన్నుల లాగ్ బేస్ 32, అది చేయాలి రిక్టర్ స్కేల్ సంఖ్యకు అనుగుణంగా ఉంటుంది, కానీ కొన్ని రకాల ఆఫ్‌సెట్ ఉండవచ్చు, కాబట్టి మేము ఈ రిక్టర్ స్కేల్ నంబర్‌కి జోడిస్తున్నాము మరియు ఈ వ్యక్తీకరణ సరిగ్గా అదే విధంగా ఉందని మేము చెప్పవచ్చు, తప్పుకున్నందుకు నన్ను క్షమించండి దిగువన, ఈ వ్యక్తీకరణ ఖచ్చితంగా కొంత ఆఫ్‌సెట్ సమయాల శక్తికి 32 అని చెప్పడానికి సమానంగా ఉంటుంది, ఇది ఆ ఆఫ్‌సెట్‌కు 32ని తీసుకున్నట్లే ఉంటుంది, ఇది కొంత పెద్ద స్థిరాంకం, రిక్టర్ స్కేల్ సంఖ్యకు రెట్లు 32 కాబట్టి మీరు మీరు చూసే సంఖ్య యొక్క శక్తికి ఇది కొన్ని స్థిరమైన సమయాలు 32 అని అనుకోవచ్చు, కాబట్టి ఇది వ్రాసే విధానం దాని యొక్క ఘాతాంక పెరుగుదలను నొక్కి చెబుతుంది, ఇది మీరు చూసే TMT మొత్తానికి అనుగుణంగా ఉంటే, మీరు దానిని పెంచినప్పుడు మీరు అంచెలంచెలుగా 32తో గుణిస్తున్నారు, అయితే అదే వాస్తవాన్ని తెలియజేయడానికి మరొక మార్గం ఏమిటంటే, ఆ మొత్తంలో లాగ్ బేస్ 32ని తీసుకోవడం మంచిది, ఇప్పుడు నేను మాట్లాడాలనుకుంటున్న తదుపరి విషయం ఏమిటంటే మనం ఎల్లప్పుడూ ఎలా చేయనవసరం లేదు వేర్వేరు స్థావరాల లాగ్‌లను ఎలా గణించాలనే దాని గురించి చింతించండి, ఇక్కడ మనం లాగ్ బేస్ 32 గురించి మాట్లాడుకోవడం కొంచెం విచిత్రంగా ఉంది, గణిత శాస్త్రజ్ఞులు నిజంగా బేస్‌తో లాగ్‌ను కలిగి ఉండాలనుకుంటున్నారు మరియు కంప్యూటర్ శాస్త్రవేత్తలు నిజంగా బేస్ 2తో లాగ్‌ను కలిగి ఉండటానికి ఇష్టపడతారు అని నేను ఇంతకు ముందు ప్రస్తావించాను. గణన ప్రయోజనాల కోసం లేదా మీ వద్ద ఒక లాగ్ ఉంటే ఈ విషయాలు ఎలా పెరుగుతాయి అనే దాని గురించి ఆలోచించడం కోసం, మీరు ఒక రకమైన లాగ్‌ని గణించగలిగితే, అది బేస్ 10, బేస్ 2, బేస్ మరియు మీరు చాలా మరేదైనా గణించవచ్చు మీరు ఇప్పుడు మన అంతర్ దృష్టిని ఆ దిశలో పొందాలనుకుంటున్నారు, మన క్విజ్‌కి తిరిగి వెళ్లి తదుపరి ప్రశ్నకు వెళ్దాం మరియు ఈ ప్రశ్న చాలా ఎక్కువ అని నేను నమ్ముతున్నాను, నాకు తెలియదు, ఇది సగం సహేతుకమైన ప్రశ్న, ఇది బాగుంది ఇది బేస్ 2 సందర్భం నుండి బేస్ 10 సందర్భానికి అనువదించడానికి మమ్మల్ని సిద్ధం చేయబోతోంది మరియు 2 యొక్క శక్తులను అర్థం చేసుకోవడానికి 10 శక్తులతో ఉన్న సంబంధాన్ని సాధారణంగా కలిగి ఉండటానికి ఇది మంచి అంతర్ దృష్టి ఎందుకంటే ఇది యాదృచ్చికం యొక్క ఈ అందమైన రకం. ప్రకృతిలో ఈ రెండు రకాలుగా మీరు నా ఉద్దేశ్యం ఏమిటో చూస్తారు, అవి ఒకదానితో ఒకటి చక్కగా ఆడుకుంటాయి కాబట్టి మా ప్రశ్న అడిగేది, 2 నుండి 10వ తేదీ వరకు 1024, 1024, అంటే దాదాపు 1000 కాబట్టి మీరు ఒక వ్యక్తి అయితే మీ సంఖ్యలతో కొంచెం వదులుగా ఉంది మరియు మీరు కేవలం 2 నుండి 10వ వరకు ఉజ్జాయింపులు చేస్తున్నారు, ప్రాథమికంగా 1000, కింది వాటిలో ఏది నిజం కావడానికి దగ్గరగా ఉంటుంది? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "టెండర్. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "ఇక్కడ ఏకగ్రీవ నిర్ణయం కాదు. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "కానీ ఏది నిజం అనే ప్రశ్నకు దగ్గరగా ఉంది మరియు దీని గురించి మనం ఎలా ఆలోచించవచ్చో చూద్దాం. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "కాబట్టి మీకు 2 శక్తి ఉందని, అది 1024 అని, 10 క్యూబ్‌ల శక్తికి చాలా దగ్గరగా ఉందని ఇది సూచిస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "కాబట్టి దీని అర్థం ఏమిటి? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "10లో లాగ్ బేస్ 2 xకి సమానం అయితే, xకి 2 అంటే 10కి సమానం అని చెప్పాలి, సరియైనదా? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "ఇది 10కి సమానం అని మనల్ని 2 అడుగుతోంది. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "మీరు ప్రతి ఫంక్షన్‌తో అలా చేయలేరు. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "మీరు ఏదైనా ఫంక్షన్‌తో దీన్ని చేయగలరని ప్రజలు అనుకుంటున్నారు, కానీ మీరు చేయలేరు. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "మరియు దాని అర్థం ఏమిటంటే x 10 వంతులు, సరేనా? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "మరియు తగినంత, మనం ఇంతకు ముందు చూసినది ఏమిటంటే, లాగ్ బేస్ 2 ఆఫ్ 10, మేము లాగ్ బేస్ 10 ఆఫ్ 2 అనేది ఆ మొత్తంపై కేవలం 1, x కంటే 1 అని కూడా చెప్పవచ్చు. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "గొప్ప. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "మరియు మేము లాగ్‌లలో పనులు చేస్తున్నందున నేను దానిని ఆ విధంగా వ్రాయబోతున్నాను. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "అదేవిధంగా లాగ్ బేస్ 2 ఆఫ్ ఎ మిలియన్, సరే చూద్దాం, వెయ్యికి చేరుకోవడానికి మనం 2ని స్వయంగా 10 రెట్లు గుణించవలసి వస్తే, మిలియన్ వరకు పొందడానికి మనం దానిని దాదాపు 20 రెట్లు గుణించాలి. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "ఇది కొంచెం చిన్నది, కానీ ఇది మీ మనస్సులో ఉండే ఒక మంచి ఉజ్జాయింపు. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "3 అంటే 3.20, మేము అదే మొత్తాన్ని తగ్గించాము. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, మేము అదే మొత్తాన్ని తగ్గించాము. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "సరే? ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "ఇప్పుడు ఇది గుర్తుంచుకోవలసిన అంతర్ దృష్టి. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "సరే? ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "ఆపై లాగ్ బేస్ C యొక్క B సార్లు లాగ్ బేస్ C యొక్క Aని కలపడానికి సాధ్యమయ్యే వివిధ మార్గాల మొత్తం పైల్. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "మీకు ఇప్పటికే లాగరిథమ్‌లతో పరిచయం ఉంటే తప్ప ఇది స్పష్టంగా కనిపించదు మరియు కొంచెం ఆలోచించడం విలువైనది కాబట్టి నేను దీని గురించి మీకు అర్ధవంతమైన సమయాన్ని ఇస్తాను. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/thai/sentence_translations.json b/2020/ldm-logarithms/thai/sentence_translations.json index 6b1c5957f..e61e6263d 100644 --- a/2020/ldm-logarithms/thai/sentence_translations.json +++ b/2020/ldm-logarithms/thai/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Music🎵 ยินดีต้อนรับกลับสู่ Lockdown Math วันนี้เราจะมาพูดถึงลอการิทึม และการย้อนกลับไปสู่บทเรียนพื้นฐาน จุดประสงค์มีไว้สำหรับผู้ที่อาจไม่คุ้นเคยกับลอการิทึมหรือผู้ที่เคยเห็นกฎบางข้อแต่ยังสับสนกับกฎเหล่านั้น และเช่นเคย เพื่อเริ่มต้นสิ่งต่างๆ ฉันแค่อยากจะเข้าใจว่าผู้ชมอยู่ที่ไหนในขณะนี้ เนื่องจากฉันมีข้อสงสัยอยู่ 2-3 ข้อ แต่ฉันคิดว่าการทำโพลสดเพื่อดูว่าทุกคนอยู่ที่ไหนอาจเป็นประโยชน์ได้ แล้ว, ถ้าคุณไป 3b1b ได้ ร่วม Live ซึ่งคุณจะพบลิงก์ในคำอธิบาย และคุณยังสามารถคลิกลิงก์ที่ปรากฏบนหน้าจอได้ที่นี่ คุณจะพบแบบสำรวจ และในขณะนี้จะถามว่าอะไรอธิบายความสัมพันธ์ของคุณกับลอการิทึมได้ดีที่สุด? ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "ฉันไม่เคยได้ยินเกี่ยวกับพวกเขามาก่อนหรือไม่เคยเรียนรู้เกี่ยวกับพวกเขามาก่อนข. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "ฉันได้เรียนรู้เกี่ยวกับพวกเขาแล้ว แต่บางครั้งก็สับสนกับคุณสมบัติทั้งหมดค. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "ฉันเข้าใจพวกเขาแต่ไม่รู้จะสอนพวกเขาอย่างไรและง. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "ฉันเข้าใจพวกเขาดีและสามารถสอนพวกเขาให้คนอื่นเข้าใจได้ดีเช่นกัน ดังนั้นเราจึงมีการแบ่งแยกที่ดี นี่เป็นสิ่งที่ดีและน่าสนใจ อย่างที่ฉันบอกไป จุดประสงค์ของสิ่งนี้คือการสร้างบทเรียนที่ฉันสามารถชี้แนะผู้คนได้ในอนาคต หากพวกเขาไม่คุ้นเคยกับลอการิทึม และฉันอยากจะพูดว่า โอ้ นี่คือที่ที่คุณสามารถไปได้ ฉันคิดอย่างไร ฉันคิดว่าคุณสามารถเข้าใกล้มันอย่างสังหรณ์ใจได้อย่างไร และสำหรับผู้ที่อยู่ในค่าย สมมติว่า เข้าใจมันแต่ไม่รู้ว่าจะสอนมันอย่างไร ฉันคิดว่ามีคำถามเมตาที่น่าสนใจเกี่ยวกับลอการิทึม เนื่องจากฉันกำลังเลื่อนดูฟอรัมของครูสองสามฟอรัมก่อนที่จะทำการบรรยายนี้ และเมื่อมีคนถามว่าหัวข้อใดที่ยากที่สุดในการสอนวิชาคณิตศาสตร์ระดับมัธยมศึกษาตอนปลายในแง่ที่ว่านักเรียนดูเหมือนจะมีปัญหากับเรื่องนี้มากที่สุด ลอการิทึมจึงเป็นหนึ่งในหัวข้อที่สำคัญที่สุด คำตอบที่ระบุโดยทั่วไปซึ่งน่าสนใจ และฉันเดาได้เลยว่าอาจเป็นเพราะว่ามีคุณสมบัติมากมายที่คุณต้องเรียนรู้ ดังนั้นหากเราข้ามไปข้างหน้าว่าเรากำลังจะไปที่ไหน คุณก็จะได้คำตอบมากมาย กฎที่ดูเหมือนพีชคณิตกลุ่มหนึ่งที่จำยากและผสมสิ่งต่าง ๆ ไว้ในหัวได้ง่าย และฉันคิดว่าเมื่อผู้คนมีความทรงจำชวนฝันร้าย ๆ เหล่านี้ว่าคณิตศาสตร์มัธยมปลายเป็นอย่างไรและอะไร ลอการิทึมทำเพื่อพวกเขา มันมักจะนึกถึงสูตรเฉพาะเหล่านั้นขึ้นมา และสิ่งที่ผมอยากทำวันนี้คือพยายามพูดถึงมัน วิธีคิดเกี่ยวกับมัน แต่ยังรวมถึงระดับเมตาด้วยว่าถ้าคุณสอนพีชคณิตใครสักคน ประเด็นที่ควรเน้น? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "มีวิธีใดที่จะสร้างมันขึ้นมาจากสัญชาตญาณของพวกเขา? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "โอ้ มันมีเลขศูนย์ 3 ตัวอยู่ บันทึกของล้านเป็นเท่าไหร่? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "คำตอบที่ถูกต้องตรงนี้คือ a ซึ่งดูเหมือนพวกคุณ 4,000 คนจะแสดงความยินดี โดยบอกเราว่าลอกของ x ยกกำลัง n เท่ากับ n คูณลอกของ x เหมือนเดิม สมมุติว่าคุณกำลังพยายามสอนเรื่องนี้ สำหรับใครบางคนหรือถ้าคุณกำลังพยายามทำความเข้าใจกับความหมายของตัวเอง ฉันคิดว่าจุดเริ่มต้นที่ดีคือการเสียบปลั๊กอะไรบางอย่างเข้าไป และในกรณีนี้ สำหรับล็อกของ x ยกกำลัง n มาลองใช้ 100 ยกกำลังกัน 3 และคุณสามารถลองใช้กับรูปแบบอื่นเพื่อดูว่ารูปแบบที่คุณกำลังทำอยู่นั้นใช้งานได้จริงหรือไม่ แต่หากคุณกำลังคิดทบทวน ไม่ใช่เพียงเพื่อดูว่าคำตอบคืออะไร แต่พยายามคิดว่าเหตุใดคำตอบจึงกลายเป็นเช่นนั้น บางครั้งตัวอย่างหนึ่งจะทำได้เพราะ 100 กำลังสาม เราคิดได้ว่ากำลังดี นั่นคือ 3 ชุดของ 100 ฉันจะเอา 3 ชุดของ 100 และเมื่อฉันคูณทั้งหมดนั้นออก แล้วฉันคิดว่าบันทึกเป็นการนับจำนวนศูนย์ที่เรา บอกว่า โอ้ มันจะเป็นตัวเลขที่มีศูนย์ 6 ตัวอยู่ นั่นคือความหมายของการเอา 100 คูณ 100 คูณ 100 ฉันแค่คิดจะรวมศูนย์ทั้งหมดเข้าด้วยกันเพื่อให้ได้หนึ่งล้าน แล้วตัวเลขนี้จะเท่ากับ 6 แต่ถ้าเราคิดจริง ๆ แล้วทำไมถึงเป็น 6 ไม่ใช่แค่จำนวนศูนย์ในล้านที่ 6 นั้นมาจาก เรามีสำเนา 3 ชุดจาก 100 ตัวนั้น และแต่ละชุดใน 100 ตัวนั้นมีศูนย์ต่างกัน 2 ตัว ดังนั้น วิธีนี้จะเป็นแบบทั่วไปมากกว่า วิธีที่คุณคิดได้ โดยที่ถ้าแทนที่จะเอา 100 ลูกบาศก์ เราดูที่ 1,000 ลูกบาศก์หรือ 1,000 ยกกำลัง n หรือ x ยกกำลัง n คุณคงคิดได้ว่าค่าของ n ก็คือจำนวนสำเนาที่เราคูณด้วยหน่วยเท่า จำนวน ทีนี้ ลองดู มันไม่ใช่ x คูณจำนวนศูนย์ที่อยู่ในสิ่งที่เราแทน x ซึ่งในกรณีนี้คือ 100 ดังนั้นหากผมเอาบางอย่างเช่นล็อก 10,000 ยกกำลัง n นี่ก็จะเหมือนเดิม จากการคัดลอก n ชุดของ 10,000 ตัวนั้นโดยนับจำนวนศูนย์ในแต่ละตัวซึ่งเป็น 4 ดังนั้นมันจะเป็น n คูณ 4 และแน่นอนว่าคุณสมบัติทั่วไปที่พวกคุณส่วนใหญ่ตอบถูกคือคุณมีผลเล็กๆ น้อยๆ ที่น่ารักนี้ โดยที่เมื่อคุณ ดูบันทึกของบางสิ่งที่ถูกยกขึ้นเป็นพลังซึ่งมีพลังเพียงเล็กน้อยกระโดดลงมาข้างหน้า และคุณก็มีบันทึกของสิ่งที่อยู่ข้างในในตอนนี้ หนึ่งในผลกระทบที่สำคัญที่สุดที่อาจเกิดขึ้น ฉันไม่รู้ว่าคุณจะเรียกมันว่าอะไร ความหมายโดยนัย หรือถ้าคุณเรียกมันว่าเป็นการย้ำนิยาม ถ้าผมเอาบันทึก และผมจะเน้นย้ำอีกครั้งว่ามันเป็นฐาน 10 ของ 10 ยกกำลัง เราสามารถคิดว่า n ตัวเล็กๆ นั้นกระโดดลงมาได้ ข้างหน้า และมันกลายเป็น n คูณฐานล็อก 10 ของ 10 ซึ่งแน่นอนว่าเป็น 1 นิพจน์นี้ที่คุณคิดได้ว่าเป็นการนับจำนวนศูนย์ที่ส่วนท้ายหรือมากกว่านั้น โดยทั่วไปแล้วจะถาม 10 ถึงจำนวนที่เท่ากับ 10 และคำตอบก็คือ 1 ซึ่งมั่นใจมากเพราะอีกวิธีหนึ่งที่คุณสามารถย้อนกลับไปอ่านนิพจน์เดิมได้คือบอกว่า 10 กำลังเท่ากับ 10 กำลัง n โอ้ เอาละ คำตอบตอนนี้ไม่เป็นไรกับคุณสมบัติลอการิทึมทุกตัวที่เรามีในกรณีนี้ เพิ่งพบบันทึกหนึ่งของ x ยกกำลัง n เกี่ยวข้องกับการที่ n กระโดดไปข้างหน้า มันจะมีคุณสมบัติเอ็กซ์โพเนนเชียลของภาพสะท้อนเสมอ และนั่นเป็นอีกวิธีหนึ่งที่เราจะช่วยให้เข้าใจสัญชาตญาณตัวเองได้นิดหน่อย ขอผมปกปิดไว้ก่อน คุณสมบัติบางอย่างในอนาคตที่เราจะได้มาถึงตรงนี้ พยายามซ่อนจุดที่เรากำลังจะไป สิ่งที่เราเพิ่งพบ การยกบางอย่างขึ้นเป็น n ที่กระโดดไปข้างหน้า นี่สอดคล้องกับคุณสมบัติเอ็กซ์โพเนนเชียลที่ถ้าฉันนำ 10 ยกกำลัง x แล้วเพิ่ม สิ่งทั้งหมดนั้นยกกำลัง n ก็เหมือนกับการนำ 10 ยกกำลัง n คูณ x และนี่ทำให้เราได้สัญชาตญาณอีกอย่างที่คุณอาจมีสำหรับลอการิทึม ซึ่งก็คือ พวกมันเหมือนกับการยกกำลังกลับด้าน และนี่คือสิ่งที่ฉันหมายถึง ว่าสิ่งที่นั่งอยู่ด้านในของบันทึกถ้าฉันกำลังเอาบันทึกของ a คุณควรคิดว่าเป็นนิพจน์ภายนอกทั้งหมดสำหรับสิ่งที่เป็นเอกซ์โปเนนเชียลในกรณีนี้ a สิ่งที่อยู่ด้านในตรงกับ 10 กำลัง x เอาท์พุตของฟังก์ชัน ในขณะที่ตัวมันเองทั้งหมด บันทึกของ a สอดคล้องกับสิ่งที่อยู่ข้างในตรงนี้ แค่เลขชี้กำลังของ 10 คืออะไร ดังนั้นเมื่อใดก็ตามที่คุณเห็นนิพจน์บันทึกที่นี่ คุณควรคิดว่ามีบทบาทเป็นเลขชี้กำลังทางด้านขวา ด้านข้าง และทุกครั้งที่คุณเห็นนิพจน์เอ็กซ์โพเนนเชียลของนิพจน์ 10 ทั้งหมดถึง x องค์ประกอบภายนอกทั้งหมดทางด้านขวาซึ่งสอดคล้องกับสิ่งที่นั่งอยู่ด้านในของบันทึกรายการใดรายการหนึ่ง และเราเห็นสิ่งนี้เหนือแนวคิดที่ว่าเมื่อเราคูณ ด้านในที่บวกกับด้านนอก ถ้าบันทึกการกลับด้านแบบเอ็กซ์โปเนนเชียลกลับด้านใน นั่นบอกเราว่าการคูณด้านนอกเพื่อคูณผลลัพธ์ของฟังก์ชันจะเหมือนกับการเพิ่มด้านใน เพราะแต่ละบันทึกเหล่านี้ เช่น log a และ log b กำลังเล่นบทบาทของ x และ y ในพจน์ทางขวา งั้นลองเล่นต่อไป ลองทำอีกสองสามอย่างแล้วดูว่ามีคุณสมบัติกี่อย่างที่เราสามารถสร้างสัญชาตญาณได้ อันสุดท้ายนี้ การคิดเลขยกกำลังที่กระโดดลงไปในอันถัดไปเป็นสิ่งที่ดีมาก เป็นสิ่งที่อาจดูแปลกนิดหน่อยสำหรับผู้ที่ไม่คุ้นเคยกับลอการิทึม แต่ขอย้ำอีกครั้งว่า เสียบตัวเลขเข้าไปเพื่อให้ได้สัญชาตญาณ แล้วเราจะเล่าให้ฟังสักหน่อย ขอเวลาสักครู่เพื่อดึงข้อมูลข้อใดต่อไปนี้เป็นจริง? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "ถ้า 10 ลูกบาศก์เท่ากับ 1,000 นั่นก็เหมือนกับการบอกว่า 10 เท่ากับ 1,000 ยกกำลัง 1 ใน 3 การทำอินเวอร์สตรงนี้เกี่ยวข้องกับค่าผกผันการคูณของเลขชี้กำลัง และวิธีแพนออกคือดูเหมือนว่า 1 หารด้วย 3 และ 3 สอดคล้องกับฐานบันทึก 10 ของ 1,000 มันคือ 1 หารด้วยฐานบันทึก 10 ของ 1,000 โดยทั่วไปแล้ว คุณอาจเดาตามตัวอย่างเดียวนี้ว่าเมื่อเราสลับฐานกับสิ่งที่อยู่ด้านใน มันจะสอดคล้องกับการแบ่ง 1 จากสิ่งที่อยู่ข้างนอกนั่นและอีกครั้ง คุณสามารถคิดผ่านในแง่ของการดูกฎเลขชี้กำลังที่สอดคล้องกัน ตอนนี้เกิดอะไรขึ้นกับบันทึกและเลขชี้กำลังเล็กๆ น้อยๆ ของฉัน? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "เยี่ยมมาก เรามาซ่อนคุณสมบัติอื่นๆ ที่เราไปถึงที่นี่กันดีกว่า และฉันจะเก็บมันไว้ในลำดับเดียวกับที่ฉันเคยมีก่อนหน้านี้ ฉันคิดว่าการเขียนไว้ล่วงหน้าจะทำให้ฉัน สะอาดกว่าปกตินิดหน่อย แต่บางทีมันก็แค่เล่นเกมแปลกๆ ของการตัดกระดาษ สับไปมา ดังนั้นสิ่งที่เราเพิ่งพบ บันทึกฐาน b ของ a ถ้าคุณสลับมัน มันก็เหมือนกับการหารด้วย 1 ว่าสิ่งนี้สอดคล้องกับอะไร ปิด พื้นที่เอ็กซ์โพเนนเชียลคือถ้าคุณหา b ยกกำลังแล้วบอกว่านั่นเท่ากับ a นั่นเป็นข้อความเดียวกับที่บอกว่า a กำลังผกผันของกำลังนั้นเท่ากับ b อีกครั้ง การใช้เวลาสักครู่แล้วคิดว่าลอการิทึมเป็นการเปลี่ยนสิ่งต่างๆ ด้านในออก ฐานบันทึกนิพจน์ b ของ a กำลังเล่นบทบาทของ x นั้น และฐานบันทึกนิพจน์ a ของ b กำลังเล่นบทบาทของอะไรก็ตามที่อยู่บน a แล้วจึงสมมาตร นิพจน์ทั้งหมด b กำลัง x กำลังเล่นอยู่ บทบาทของด้านในทางด้านซ้าย มีบทบาทเป็น a และนิพจน์ทั้งหมด a ยกกำลังของบางสิ่งบางอย่าง มีบทบาทเป็นสิ่งที่นั่งอยู่ในฐานล็อก a เพื่อให้คุณมองเห็นได้ เพียงแค่เสียบตัวอย่างบางส่วนแล้ว โดยการทำให้มันสอดคล้องกับกฎเลขชี้กำลัง เราก็สามารถคิดผ่านกฎลอการิทึมที่แตกต่างกันสามกฎได้ ซึ่งถ้าพวกมันถูกส่งต่อเป็นพีชคณิตที่จะจำ คุณก็รู้ คุณก็จำมันได้ แต่มันง่ายมากสำหรับกฎเหล่านั้นที่จะหลุดออกไปจากคุณ หัว และมันง่ายมากที่จะหงุดหงิดกับงานที่ทำอยู่ แต่คุณอาจต้องการเตือนตัวเองว่าเหตุผลที่เราสนใจเรื่องพวกนี้ คือการเข้าใจกฎของลอการิทึมช่วยให้เราคำนวณคณิตศาสตร์ในบริบทที่มันเหมือนกับไวรัสที่เติบโตใน จากวันหนึ่งไปสู่อีกขั้น จากขั้นหนึ่งไปอีกขั้น สิ่งต่างๆ มีแนวโน้มที่จะเพิ่มขึ้นแบบทวีคูณ การทำความเข้าใจกฎของลอการิทึมจะช่วยให้คุณรู้สึกดีขึ้นกับสิ่งประเภทนั้น ก่อนที่เราจะยกตัวอย่างที่ดีในโลกแห่งความเป็นจริงของสิ่งที่สามารถดูได้ เช่น ขอผมถามคำถามอีกข้อในเรื่องนี้เพื่อถามเกี่ยวกับคุณสมบัติของลอการิทึมข้อสุดท้าย ก่อนที่เราจะเปลี่ยนไปใช้ตัวอย่างเล็กๆ น้อยๆ ในโลกแห่งความเป็นจริง กำจัดสิ่งที่เรามีที่นี่และตอนนี้ ข้อใดต่อไปนี้เป็นจริง log ของ a บวก b เหมือนกับ log ของ a บวก log ของ b log ของ a บวก b เท่ากับ log ของ a คูณ log ของ b log ของ a บวก b เท่ากับ 1 หารด้วย log ของ a plus log ของ b หรือ log ของ a บวก b เท่ากับ 1 หารด้วย log ของ a คูณ log ของ b หรือไม่ตรงกับข้อใดเลย และตอนนี้ เราไม่มีมติมากนักใช่ไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "น่าสนใจมาก เรามีการแข่งม้าระหว่างสองคน ดังนั้นฉันจะให้เวลาคุณคิดในขณะที่คนอื่นกำลังตอบคำถาม จริงๆ แล้วฉันมีคำถามเล็กๆ น้อยๆ สำหรับผู้ฟัง คุณก็รู้ ฉันแค่กำลังพูดถึงว่าเราจะทำอย่างไร คิดในแง่ของการเติบโตแบบทวีคูณ และนั่นไม่ใช่แค่ต้องยกกำลัง 10 เท่านั้น เรายังทำอะไรสักอย่างยกกำลัง 3 ได้ด้วย โดยถ้าคุณเปลี่ยนจากหนึ่งเป็นสามเป็นเก้าเป็นยี่สิบเจ็ดเป็นแปดสิบเอ็ด ทั้งหมด ในจำนวนนี้ เราสามารถพูดได้ว่าล็อกฐานสามของจำนวนเหล่านี้เพิ่มขึ้นทีละขั้นเล็กๆ ที่ดี ดังนั้นลอกฐานสามของหนึ่ง สามคูณกับสิ่งที่เท่ากับหนึ่ง คำตอบคือศูนย์ โดยทั่วไปล็อกของหนึ่ง ไม่ว่าฐานใดก็ตาม เป็นฐานล็อกเป็นศูนย์ 3 จาก 3, 3 เท่ากับ 3 เป็น 1 ล็อกฐาน 3 จาก 9 เท่ากับ 2 อ่า คุณอาจสงสัยว่าคำถามของฉันคืออะไร แต่มันจะช่วยดึงสิ่งเหล่านี้ออกมาทั้งหมด และเพื่อความสุขของฉันเอง ตรงนี้ ขอผมเขียนบันทึกฐานอีกสักหนึ่งฐาน 3 ของ 81 เป็น 4 ในตอนนี้ ผมได้ยินมาว่าถ้าคุณถามเด็ก สมมุติว่าอายุราวๆ 5 หรือ 6 ขวบ ตัวเลขใดอยู่กึ่งกลางระหว่าง 1 ถึง 9 ขวบ บอกว่าจำนวนใดที่อยู่ครึ่งทาง สัญชาตญาณในการตอบคือลอการิทึม ในขณะที่สัญชาตญาณของเรามีแนวโน้มที่จะเป็นเส้นตรงมากกว่า ดังนั้นเรามักจะคิดว่าหนึ่งและเก้า คุณมีตัวเลขที่เว้นระยะเท่ากันจำนวนหนึ่งระหว่างตัวเลข 2, สาม, สี่, ห้า, หก เจ็ด แปด และถ้าคุณไปครึ่งทาง คุณจะไปถึงห้า แต่ถ้าคุณคิดในแง่ของการเติบโตแบบทวีคูณที่จะได้จากหนึ่งถึงเก้า มันไม่ใช่เรื่องของการเพิ่มสิ่งต่างๆ มากมาย แต่คุณ 'เติบโตขึ้นตามจำนวนหนึ่ง คุณเติบโตขึ้นสามเท่า จากนั้นคุณก็เติบโตขึ้นอีกสามเท่า สัญชาตญาณตามธรรมชาติของเด็กสอดคล้องกับคำว่าสาม และนี่ก็สอดคล้องกับถ้าคุณมีนักมานุษยวิทยาที่กำลังศึกษาสังคมที่ยังไม่ได้' ไม่ได้พัฒนาระบบบัญชีและการเขียนแบบเดียวกับที่สังคมสมัยใหม่มี พวกเขาจะตอบสามข้อสำหรับเรื่องนี้ ดังนั้น คำถามของฉันสำหรับผู้ฟังว่ามีใครที่ดูอยู่ตอนนี้สามารถเข้าถึงเด็กเล็กได้ สมมติว่าในช่วงห้าปี ดูว่าคุณสามารถไปถามพวกเขาได้ไหมว่าตัวเลขใดอยู่กึ่งกลางระหว่างหนึ่งถึงเก้า และถ้าคุณทำได้ บอกเราบน Twitter ว่าเด็กพูดว่าอะไรคำตอบที่แท้จริงของพวกเขาคืออะไร เพราะฉันไม่รู้ว่าทำไม ฉันแค่เพียงเล็กน้อยเท่านั้น สงสัยว่าในทางปฏิบัติจริงหรือไม่ ฉันเข้าใจว่านี่ไม่ใช่วิธีทางวิทยาศาสตร์ที่ยอดเยี่ยม ฉันไม่ได้ขอให้คนที่ดูสตรีมสดของ YouTube สำรวจลูก ๆ ของตัวเองแล้วทวีตคำตอบ แต่เพื่อประโยชน์ของฉันเอง มันจะน่าสนใจ เพื่อดูการตรวจสอบความถูกต้องบางประการในคำถามของเรา นี่เป็นข้อแรกที่ดูเหมือนจะไม่มีความเห็นพ้องต้องกันมากนักในทิศทางเดียว ลองให้คะแนนเพื่อดูว่าคำตอบนั้นยอดเยี่ยมแค่ไหน โอเค ดังนั้น 2,400 พวกคุณตอบถูกแล้วว่าไม่ใช่ข้อใดข้างต้นที่บันทึกของ a บวก b ไม่เป็นไปตามคุณสมบัติดีๆ เหล่านี้ และโดยทั่วไป เว้นแต่ว่าเราจะทำงานกับการประมาณค่าบางประเภท โดยเฉพาะอย่างยิ่งเมื่อบันทึกธรรมชาติเข้ามามีบทบาท เราอาจพูดถึงเรื่องนี้ในครั้งต่อไปการเพิ่มอินพุตของลอการิทึมเป็นความรู้สึกที่แปลกมาก มันเป็นสิ่งที่แปลกมากที่จะทำ และเพื่อให้เข้าใจถึงความแปลกประหลาดนั้น ให้เสียบเลขยกกำลัง 10 ลงไปถ้าฉันถามคุณถึงลอกของ a บวก b สิ่งที่คุณอาจเริ่มคิดคือ โอเค ขอผมแทนตัวอย่าง เช่น 10,000 และ 100 แล้วถามตัวเองว่า ถ้าผมใช้ฟังก์ชันการนับศูนย์ของสิ่งที่อยู่ในอินพุตนั้น มีศูนย์กี่ตัวในนั้น? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "ในรูปสามเหลี่ยมของเรา เราอาจคิดว่าเป็นการบอกว่าคุณก็รู้, 0 กำลัง x เท่ากับค่าอื่น y นี่คือสิ่งที่เราเขียนได้ โดยบอกว่า 0 กำลัง x เท่ากับ y หรือเราเขียนก็ได้ แบบเดียวกันโดยบอกว่าล็อกฐาน 0 ของ y เท่ากับ x 0 เท่ากับ x ทีนี้ ปัญหาตรงนี้คือ 0 ต่ออะไรก็กลายเป็น 0 ใช่ไหม แล้วถ้าเราจะคิดถึงล็อกฐาน 0 ของ y สำหรับการป้อนข้อมูลอื่นๆ ที่คุณรู้จัก คุณต้องการป้อนบางอย่างเช่น หนึ่งหรือสองหรือ pi อะไรก็ได้ที่คุณต้องการ คุณกำลังถามคำถามเป็นศูนย์ถึงสิ่งที่เท่ากับหนึ่งหรือสองหรือ pi หรือตัวเลขอะไรก็ตามที่คุณอาจมี และจะไม่มีคำตอบ ดังนั้นอย่างดีที่สุดคุณสามารถลองพูดว่า โอ้ ใช่ บันทึกของศูนย์ มันเป็นฟังก์ชันที่ถูกต้องสมบูรณ์ มันถูกกำหนดไว้ที่อินพุตศูนย์เท่านั้น แต่ถึงอย่างนั้นคุณก็จะมีปัญหาในการพยายามสรุปสิ่งที่คุณต้องการ ที่นั่นเนื่องจากการบอกศูนย์ถึงสิ่งที่เท่ากับศูนย์ มันเหมือนกับว่ามีอะไรใช้กับแขนของคุณ ดังนั้นแขนของคุณจะถูกบิดไปด้านหลัง แต่คุณอยากจะทำให้มันได้ผล และมันก็สอดคล้องกับความจริงที่ว่าฟังก์ชันเลขชี้กำลังที่มีฐานเป็นศูนย์นั้นเป็นศูนย์ทั้งหมด ไม่ได้แมปตัวเลขแบบหนึ่งต่อหนึ่งเข้าด้วยกัน นั่นเป็นคำถามที่ดี คุณช่วยมีฐานบันทึกเป็นศูนย์ได้ไหม กลับมาที่แนวคิดว่าสิ่งเหล่านี้เกิดขึ้นได้อย่างไรในโลกแห่งความเป็นจริง ตัวอย่างหนึ่งที่ฉันชอบคือ มาตราริกเตอร์ของแผ่นดินไหว ดังนั้น มาตราริกเตอร์ทำให้เราทราบถึงความแรงของแผ่นดินไหว และสามารถเป็นอะไรก็ได้ตั้งแต่ตัวเลขเล็กๆ น้อยๆ ไปจนถึงตัวเลขที่มีขนาดใหญ่มาก เหมือนที่ผมคิดว่าแผ่นดินไหวครั้งใหญ่ที่สุดที่เคยวัดได้ และนี่เป็นเพียงแผนภูมิที่มาจาก วิกิพีเดียคือ 9 5 และเพื่อชื่นชมว่ามันบ้าขนาดไหน ก็คุ้มค่าที่จะดูความสัมพันธ์ระหว่างความหมายของตัวเลขเหล่านี้ กับปริมาณทีเอ็นทีที่เทียบเท่ากัน ซึ่งเป็นการวัดปริมาณพลังงานในนั้น แล้วเราจะลองทำอะไรได้บ้าง คือดูว่าเราสามารถหานิพจน์สำหรับเลขมาตราริกเตอร์ในแง่ของปริมาณพลังงานได้หรือไม่ และเหตุใดลอการิทึมจึงเป็นวิธีธรรมชาติในการอธิบายสิ่งนี้ ดังนั้นกุญแจสำคัญที่ต้องมุ่งเน้นคือในขณะที่เรากำลังก้าวไปข้างหน้า สิ่งต่างๆ เพิ่มขึ้นเท่าใด ตัวอย่างเช่น ถ้าเราไปจากสองหลุมในกรณีนี้ มันจะไม่แสดงให้เราเห็นว่าสามอยู่ที่ไหน บางทีเราคิดว่าการก้าวจากสองถึงสี่ ซึ่งเหมือนกับการก้าวสองก้าว มันทำอะไรในแง่ของ ปริมาณพลังงาน ดูเหมือนว่ามันจะพาเราไปจากทีเอ็นทีหนึ่งเมตริกตัน ซึ่งฉันเดาว่าเป็นระเบิดขนาดใหญ่จากสงครามโลกครั้งที่สอง และมันพาเราไปมากกว่าหนึ่งกิโลตันมากกว่าพันเท่าซึ่งเป็นระเบิดปรมาณูขนาดเล็ก ดังนั้นแค่สองก้าวเท่านั้น ในระดับริกเตอร์ตั้งแต่แผ่นดินไหวขนาด 2 ไปจนถึงแผ่นดินไหวขนาด 4 นำเราจากระเบิดขนาดใหญ่ตั้งแต่สงครามโลกครั้งที่สองจนถึงยุคนิวเคลียร์ นั่นคือสิ่งที่น่าสังเกต และขั้นตอนแรกที่สะอาดที่เราได้รับคือไปจาก 4 เป็น 5 ที่ อย่างน้อยที่สุดในแง่ของสิ่งที่แผนภูมินี้แสดงให้เราเห็นอย่างชัดเจน และเห็นได้ชัดว่าก้าวเดียวเพิ่มขึ้นจาก 4 เป็น 5 สอดคล้องกับการเพิ่มจาก 1 กิโลตันเป็น 32 กิโลตัน และนั่นเห็นได้ชัดว่ามีขนาดเท่ากับระเบิดทำลายเมืองที่ตกลงบนนางาซากิ ดังนั้นนี่อาจเป็นหนึ่ง สิ่งที่ขัดกับสัญชาตญาณเกี่ยวกับมาตราส่วนลอการิทึม หากคุณเพิ่งได้ยินข่าวถึงความแตกต่างระหว่างแผ่นดินไหวขนาด 4 0 เทียบกับแผ่นดินไหวที่ระดับ 5 0 มันง่ายที่จะคิด ใช่ 4 และ 5 นั้นเป็นตัวเลขที่ค่อนข้างคล้ายกัน แต่เห็นได้ชัดในแง่ของจำนวนทีเอ็นทีที่สอดคล้องกับการคูณด้วย 32 เพื่อให้ได้จาก 1 ไปอีกจำนวนหนึ่ง และการเปลี่ยนจาก 2 ไปเป็น 4 เห็นได้ชัดว่าคูณด้วยประมาณหนึ่งพันและเป็นอย่างเดียว เหตุผลที่ใหญ่กว่านั้นคือเพราะที่นี่แผนภูมิของเราไม่ได้แสดงว่า 3 คืออะไร เราจึงดำเนินการสองขั้นตอน และคุณสามารถตรวจสอบด้วยตัวคุณเองว่า ถ้าคุณก้าวไปเป็น 32 แล้วคุณคูณด้วยอีก 32 ซึ่งจริงๆ แล้วเกือบจะเป็นพัน แนวคิดที่ว่าขั้นตอนการบวกบนเลขริกเตอร์สอดคล้องกับขั้นตอนการคูณใน TNT ดูเหมือนจะบ่งบอกว่ามีบางอย่างเกี่ยวกับลอการิทึมกำลังเกิดขึ้นที่นี่ และน่าสนใจนิดหน่อยที่จะดำเนินการต่อที่นี่และบอกว่าสิ่งนี้เพิ่มขึ้นเท่าใด ส่วนหนึ่งเนื่องจากปรากฏการณ์ของโลก การอธิบายว่าใช่ไม่ใช่เรื่องน่าแปลกใจเลยที่เมื่อเราก้าวไปอีกขั้นหนึ่ง มันจะคูณด้วย 32 อีกครั้ง แต่ควบคุมมันตามสัญชาตญาณของเรา นั่นคือความแตกต่างระหว่างระเบิดปรมาณูขนาดเล็ก 32 กิโลตันกับหนึ่งเมกะตันซึ่งเราอาจคิดว่าไม่ใช่ระเบิดปรมาณูขนาดเล็ก ระเบิดปรมาณูนางาซากิ ซึ่งฉันเดาว่าเป็นระเบิดปรมาณูนางาซากิ 32 ลูกต่อหนึ่งเมกะตัน ซึ่งเห็นได้ชัดว่ามีขนาดเท่าแผ่นดินไหวแบนสายคู่ในรัฐเนวาดา สหรัฐอเมริกา ปี 1994 ฉันไม่รู้ว่ามันคืออะไร ขอบคุณวิกิพีเดียในแง่ของความถี่ ยังดูเห็นได้ชัดว่ามีอันที่น้อยกว่าสองอัน ซึ่งเกิดขึ้นตลอดเวลาจะมีประมาณ 8,000 อันต่อวัน แต่ทันทีที่เราอยู่ในขอบเขตของระเบิดปรมาณู สิ่งต่างๆ เช่น 3 เห็นได้ชัดว่าเหตุการณ์ที่ 5 และ 4 เกิดขึ้นค่อนข้างบ่อยที่ไหนสักแห่งบนโลก โดยมีประมาณ 134 ครั้งเกิดขึ้นที่ไหนสักแห่งทุกวัน ใครจะรู้? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "แต่เมื่อเรามีความเข้มข้นมากขึ้นในช่วง 5 และ 6 นี้ ซึ่งสูงกว่าระดับระเบิดปรมาณูมาก ตอนนี้เราอยู่ที่ประมาณ 2 ต่อวันเท่านั้น และผมแน่ใจว่านักธรณีวิทยาจะเข้ามาอธิบายว่าทำไมเราทุกคนจึงควร" อย่ากังวลอย่างยิ่งกับความจริงที่ว่ามีการหยุดชะงักของเปลือกโลกเทียบเท่ากับระเบิดปรมาณู 2 ลูกที่เกิดขึ้นทุกวัน แต่คงเป็นเรื่องยากโดยเฉพาะอย่างยิ่งสำหรับผู้ที่จะมุ่งความสนใจไปที่จุดใดจุดหนึ่งเช่นเมืองที่มีผู้คนจำนวนมากอาศัยอยู่ในขณะนี้ เพียงตรวจสอบความคิดของเราว่าแต่ละขั้นตอน เกี่ยวข้องกับการเพิ่มขึ้นเป็น 32 ลองดูว่าขั้นตอนจาก 6 ถึง 7 เป็นอย่างไร และที่นี่ให้ตัวอย่างอื่นๆ มากมายในระหว่างนั้น บางทีอาจให้ภาพลวงตาว่านั่นเป็นขั้นตอนที่ใหญ่กว่าที่เป็นจริง และนั่นคือความแตกต่างระหว่าง 1 เมกะตันกับ 32 เมกะตัน ซึ่งคูณด้วย 32 สิ่งหนึ่งที่ฉันพบว่าน่าสนใจที่สุดในแผนภูมินี้คือการดูว่าเราต้องไปได้ไกลแค่ไหนก่อนที่จะไปถึงอาวุธนิวเคลียร์ที่ใหญ่ที่สุด ที่เคยทดสอบจริง นี่คือจุดสูงสุดของสงครามเย็น ระเบิดซาร์ที่มีขนาด 50 เมกะตัน และฉันเชื่อว่าจริงๆ แล้วพวกเขามีแผนเดิมที่จะมีระเบิดขนาด 100 เมกะตัน แต่คุยกันเองจาก 50 เมกะตันนั้น เรากำลังพูดถึงการเริ่มต้นที่ 32 กิโลตันของระเบิดนางาซากินั้นคูณด้วย 32 เพื่อให้ได้ เมกะตันคูณด้วยอีก 32 ดังนั้นเรากำลังพูดถึงความแข็งแกร่งของการระเบิดในสงครามโลกครั้งที่สองเป็นพันเท่า และคุณยังไม่ถึง 50 เมกะตันของสิ่งที่มนุษยชาติสามารถทำได้ และเห็นได้ชัดว่าแผ่นดินไหวชวาในอินโดนีเซีย ดังนั้น 7 . 0 ไม่ใช่แค่ใหญ่กว่า 6 นิดหน่อย 0 มันใหญ่กว่ามากและประเด็นตรงนี้ก็คือ เมื่อคุณมีสเกลที่ให้การเพิ่มขึ้นแบบคูณ ก็คุ้มค่าที่จะซาบซึ้งว่าสิ่งที่ดูเหมือนก้าวเล็กๆ จริงๆ แล้วอาจเป็นก้าวที่ยิ่งใหญ่ในแง่ของพลังงานโดยนัยหรือค่าสัมบูรณ์ที่บอกเป็นนัยตรงนี้ เมื่อเราคิดถึงข้อเท็จจริงที่ว่าเคยมี 9 5 ที่ดูไร้สาระจริงๆ เพราะมีแค่ 7 เท่านั้น 0 ที่เรากำลังพูดถึงอาวุธแสนสาหัสที่ใหญ่ที่สุดเท่าที่เคยมีมา และนี่เป็นตัวบ่งชี้ถึงพื้นที่หนึ่งที่ลอการิทึมมีแนวโน้มที่จะเกิดขึ้น คือการที่มนุษย์ต้องการสร้างมาตราส่วนสำหรับบางสิ่งที่คำนึงถึงความแปรปรวนในวงกว้างอย่างมหาศาลว่าสิ่งใหญ่ๆ สามารถเกิดขึ้นได้อย่างไร ในกรณีของขนาดของแผ่นดินไหว คุณสามารถมีสิ่งต่าง ๆ จากสิ่งที่เกิดขึ้นตลอดเวลารอบโลก ขนาดของระเบิดมือขนาดใหญ่ และคุณต้องการให้มันอยู่ในขนาดของคุณและบางสิ่งบางอย่างที่ต้องคำนึงถึงตั้งแต่ต้นจนจบ ไปสู่การหยุดชะงักครั้งใหญ่ที่สุดที่เราเคยเห็นในประวัติศาสตร์ของมนุษย์ และเพื่อที่จะให้เกิดสิ่งนั้น ในลักษณะที่คุณไม่เพียงแค่เขียนตัวเลขหลายๆ หลักในตัวเลขของคุณสำหรับกรณีเดียว และอีกหลายๆ หลักที่แตกต่างกันด้วยจำนวนที่น้อยลง ของตัวเลขของคุณ ในอีกกรณีหนึ่ง เป็นการดีที่จะใช้ลอการิทึม แล้วใส่มันลงในสเกลเดียว โดยพื้นฐานแล้วจะบีบตัวเลขเหล่านั้นระหว่าง 0 ถึง 10 คุณจะเห็นสิ่งที่คล้ายกันมากเกิดขึ้นกับสเกลเดซิเบลสำหรับดนตรีที่ใช้งานได้จริงเพียงเล็กน้อย แตกต่างออกไปเล็กน้อย โดยที่ทุกครั้งที่คุณเพิ่มระดับ 10 เดซิเบลซึ่งสอดคล้องกับการคูณด้วย 10 ดังนั้น แทนที่จะเป็นขั้นที่ 1 คูณด้วย 10 กลับกลายเป็นขั้นตอนที่ 10 ซึ่งคูณด้วย 10 ดังนั้นแบบนั้นจะทำให้คณิตศาสตร์น้อยลง ค่อนข้างเพี้ยนแต่แนวคิดก็เหมือนกัน นั่นคือถ้าคุณกำลังฟังเสียงที่มีความดัง 50 เดซิเบล กับ 60 เดซิเบล มันจะเงียบกว่ามากในแง่ของพลังงานที่ถูกส่งและไป มันจะเป็น 60 ถึง 70 หรือ 70 ถึง 80 ขั้นตอนเหล่านั้น จาก 60 ถึง 80 ที่เกี่ยวข้องกับการคูณปริมาณพลังงานต่อพื้นที่ตารางด้วยตัวคูณ 100 ดังนั้นทุกครั้งที่คุณเห็นมาตราส่วนลอการิทึม ให้รู้ในใจว่านั่นหมายถึงอะไรก็ตามที่อ้างอิงถึงภายใต้ประทุนจะเติบโตขึ้น จำนวนมาก นี่เป็นอีกครั้งว่าทำไมเราจึงเห็นมาตราส่วนลอการิทึมจำนวนมากที่ใช้อธิบายการระบาดของไวรัสโคโรนา แล้วคุณจะอธิบายความสัมพันธ์แบบนี้ได้อย่างไร โดยทุกครั้งที่คุณเพิ่มมาตราส่วนริกเตอร์ขึ้น 1 คุณจะคูณด้วย 32 แล้วเรา คิดในแง่ของบันทึกที่มีฐาน 32 ผมบอกได้เลยว่าถ้าผมเอาบันทึกของ ผมจะเรียก r ตัวเลขสำหรับมาตราริกเตอร์ ผมอาจคิดว่านี่เป็นบันทึกฐาน 32 และนั่นจะสอดคล้องกับ ไม่ ไม่ ไม่ ฉันกำลังทำผิด นั่นไม่ใช่สิ่งที่ถูกบันทึกไว้ เราใช้ล็อกฐาน 32 ของเลขใหญ่ ของเลข TMT บางอย่างที่ประมาณ 1 เมกะตัน ก็คือ 1 ล้านตันของล็อกฐาน 32 ที่ควร ตรงกับเลขมาตราริกเตอร์ แต่อาจมีค่าออฟเซ็ตอยู่บ้าง ดังนั้นเราจึงอาจบอกว่ามีค่าคงที่บางค่าที่เราบวกเข้ากับเลขมาตราริกเตอร์ และนิพจน์นี้ก็เหมือนกันทุกประการ ขอโทษที่นอกเรื่อง ข้างล่างนั่น นิพจน์นี้เหมือนกับการบอกว่า 32 กำลังของออฟเซ็ตบางค่าคูณเลขมาตราริกเตอร์ของเรา ซึ่งเหมือนกับการนำ 32 ไปยังออฟเซ็ตนั้น ซึ่งตัวมันเองเป็นเพียงค่าคงที่ใหญ่ค่าหนึ่ง คูณ 32 กับเลขมาตราริกเตอร์ ดังนั้นคุณ อาจคิดว่านี่เป็นแค่ค่าคงที่คูณ 32 ยกกำลังของตัวเลขที่คุณเห็น ดังนั้นการเขียนแบบนี้จึงเน้นการเติบโตแบบเอ็กซ์โปเนนเชียลจริงๆ ว่าถ้านี่คือสิ่งที่สอดคล้องกับจำนวน TMT ที่คุณเห็น เมื่อคุณเพิ่มค่านั้น คุณกำลังคูณด้วย 32 ทีละขั้นตอน แต่วิธีอื่นในการสื่อสารข้อเท็จจริงเดียวกันเป๊ะๆ ก็คือเอาลอกฐาน 32 ของจำนวนเท่าใดก็ได้ที่เป็นไร ทีนี้สิ่งต่อไปที่ผมอยากพูดถึงคือเราไม่จำเป็นต้องทำเสมอไป กังวลเกี่ยวกับวิธีคำนวณบันทึกของฐานต่างๆ มันแปลกนิดหน่อยที่นี่ที่เรากำลังพูดถึงบันทึกฐาน 32 ฉันเคยกล่าวไว้ก่อนหน้านี้ว่านักคณิตศาสตร์ชอบที่จะมีบันทึกที่มีฐาน นักวิทยาศาสตร์คอมพิวเตอร์ ชอบที่จะมีบันทึกที่มีฐาน 2 จริงๆ และมัน ปรากฏว่ามีจุดประสงค์ในการคำนวณหรือคิดดูว่าสิ่งเหล่านี้จะเติบโตได้อย่างไรถ้าคุณมีบันทึกเดียว หากคุณสามารถคำนวณบันทึกประเภทใดประเภทหนึ่งได้ ไม่ว่าจะเป็นฐาน 10 ฐาน 2 ฐาน e คุณสามารถคำนวณอย่างอื่นได้แทบทั้งนั้น ตอนนี้คุณต้องการได้สัญชาตญาณไปในทิศทางนั้น ลองย้อนกลับไปที่แบบทดสอบของเราแล้วไปที่คำถามถัดไป และผมเชื่อว่าคำถามนี้ตรงที่สุด ไม่รู้สิ นี่เป็นคำถามที่สมเหตุสมผลครึ่งทาง ก็น่าจะดี นี่เป็นเพียงการช่วยให้เราเตรียมพร้อมที่จะแปลจากบริบทฐาน 2 เป็นบริบทฐาน 10 และยังเป็นสัญชาตญาณที่ดีในการทำความเข้าใจเลขยกกำลัง 2 โดยทั่วไปความสัมพันธ์ที่มีกับเลขยกกำลัง 10 เพราะมันเป็นเรื่องบังเอิญที่น่ารักของ โดยธรรมชาติแล้วทั้งสองประเภทนี้คุณจะเห็นว่าฉันหมายถึงอะไร พวกเขาเล่นกันได้ดี ดังนั้นคำถามของเราจึงถาม เมื่อพิจารณาจากข้อเท็จจริงที่ว่า 2 กำลัง 10 คือ 1,024, 1,024 ซึ่งมีค่าประมาณ 1,000 ดังนั้นหากคุณเป็น หลวมๆ กับตัวเลขของคุณนิดหน่อย แล้วคุณก็แค่ประมาณ 2 ถึง 10 หรือหลักๆ คือ 1,000 ข้อใดต่อไปนี้ใกล้เคียงกับความเป็นจริงมากที่สุด ฐานบันทึก 2 ของ 10 มีค่าประมาณ 0 3 ฐานบันทึก 2 ของ 10 มีค่าประมาณ ขออภัย ฐานบันทึก 10 ของ 2 มีค่าประมาณ 0 3 ฐานล็อก 2 ของ 10 มีค่าประมาณ 1 ใน 3 หรือฐานล็อก 10 ของ 2 มีค่าประมาณ 1 ใน 3 ฐานล็อกใดที่ใกล้เคียงที่สุดเมื่อพิจารณาจากข้อเท็จจริงที่ว่า 2 ยกกำลัง 10 เป็น 1,000 ฉันจะให้เวลาคุณสักครู่เพื่อดูสิ่งที่น่าสนใจว่าเราได้แยกส่วนกันในเรื่องนี้ ดังนั้นฉันสงสัยว่าพวกเขาจะค่อนข้างคล้ายกันในเชิงตัวเลข หรือจะคล้ายกันในเชิงแนวคิด หรือถ้า ทั้งสองมีความแตกต่างกันด้วยซ้ำ ดังนั้นเมื่อคำตอบมีเข้ามาเรื่อยๆ ฉันจะให้เวลาเพิ่มอีกนิด เพื่อให้ใครก็ตามที่อยู่ที่บ้านดูอยู่ หวังว่าคุณคงมีดินสอและกระดาษออกมาสำหรับใช้เขียนสิ่งเหล่านี้ด้วยตัวเองแล้ว นั่นคือ จิตวิญญาณของการบรรยายที่เรากำลังทำอยู่ถ้าคุณไม่ทำ ตอนนี้เป็นเวลาที่จะหยิบดินสอและกระดาษออกมาดูว่าคุณสามารถคิดทบทวนและเขียนปัญหาบางส่วนที่เรากำลังจะเผชิญได้หรือไม่ การสร้างที่นี่ต้องใช้ดินสอและกระดาษอย่างแน่นอน ดังนั้นตอนนี้จึงเป็นช่วงเวลาที่ดีพอๆ กัน และหากคุณดูรายการนี้ในอนาคต แม้ว่าคุณจะไม่สามารถเข้าร่วมการสำรวจความคิดเห็นแบบสดได้ ฉันคิดว่ามันสนุกมาก ลองโยนหมวกของคุณเองเข้าไปผสม แม้ว่ามันจะไม่ส่งผลต่อตัวเลขที่คุณเห็นเพิ่มขึ้นบนหน้าจอ ฉันจะให้เวลาคุณมากขึ้นอีกหน่อยที่นี่ เพราะคำตอบดูเหมือนจะยังคงดำเนินต่อไป ดังนั้นตอนนี้ก็คือ ถึงเวลาหยิบดินสอและกระดาษออกมาแล้วดูว่าคุณสามารถคิดเรื่องนี้ผ่านและเขียนออกมาได้หรือไม่ ดังนั้นตอนนี้ก็เป็นเวลาที่ดีพอ ๆ กัน และถ้าคุณคิดเรื่องนี้ผ่านได้ คุณก็สามารถคิดผ่านและเขียนมันออกมาได้ ตอนนี้เป็นช่วงเวลาที่ดีพอๆ กัน และหากคุณสามารถคิดเรื่องนี้ผ่านแล้วทำ Move ของคุณได้เลย ฉันจะให้คะแนนตอนนี้แล้วมาดูกันว่าผู้คนทำอย่างไรกับเรื่องนี้ ดังนั้นคำตอบที่ถูกต้องคือ B ซึ่ง คือว่าฐานบันทึก 10 จาก 2 ยินดีต้อนรับ! ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "อ่อนโยน. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "แต่คำถามคือถามว่าข้อไหนใกล้เคียงกับความเป็นจริงมากที่สุด มาดูกันว่าเราจะคิดอย่างไรเกี่ยวกับเรื่องนี้ มันชี้ให้เห็นว่าคุณมีกำลัง 2 ซึ่งก็คือ 1,024 ใกล้เคียงกับกำลัง 10 อย่างมาก หรือประมาณ 10 ลูกบาศก์ และคำถามคือเราจะใช้ประโยชน์จากสิ่งนี้เพื่อทำความเข้าใจบางอย่าง เช่น บันทึกฐาน 2 จาก 10 หรือบันทึกฐาน 10 จาก 2 ได้อย่างไร ดังที่เราเห็นก่อนหน้านี้ สิ่งเหล่านี้เป็นเพียงส่วนกลับของกันและกัน แล้วนี่หมายความว่าอะไร? ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "ถ้าลอกฐาน 2 ของ 10 เท่ากับ x นั่นก็เหมือนกับการบอกว่า 2 กำลัง x เท่ากับ 10 จริงไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "มันถามเรา 2 เท่ากับ 10. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "และการยกกำลังเป็นฟังก์ชันแบบหนึ่งต่อหนึ่งที่ดี ดังนั้นจึงเป็นเรื่องปกติที่จะพูดอะไรก็ตามที่เกิดขึ้นในอินพุต ถ้าเอาต์พุตเหมือนกัน อินพุตก็ต้องเหมือนกันด้วย คุณไม่สามารถทำได้กับทุกฟังก์ชั่น ดูเหมือนว่าผู้คนจะคิดว่าคุณสามารถทำเช่นนั้นกับฟังก์ชันใดก็ได้ แต่คุณก็ทำไม่ได้ นั่นหมายความว่า x มีค่าประมาณ 10 ส่วน 3 ใช่ไหม? ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "ซึ่งเยี่ยมมาก บันทึกฐาน 2 ของ 10 มีค่าประมาณ 10 ใน 3 ดังนั้น หากเราดูคำตอบของเรา นั่นไม่ใช่ตัวเลือกใดๆ เลย เรามีหลายอย่างที่ขอให้บันทึกฐาน 2 จาก 10 อยู่ที่ประมาณ 0 3 หรือ 1 ใน 3 ดูเหมือนว่าเราควรพยายามเขียนนี่เป็นบันทึกฐาน 10 ของ 2 แทน และเพียงพอแล้ว สิ่งที่เราเห็นก่อนหน้านี้คือล็อกฐาน 2 ของ 10 เรายังบอกได้ว่าล็อกฐาน 10 ของ 2 เป็นเพียง 1 ส่วนจำนวนนั้น 1 ส่วน x และคุณเห็นมันได้ง่ายๆ โดยเขียน 2 เท่ากับ 10 ยกกำลัง 1 ส่วน x ถ้าเราถาม 10 ว่าเท่ากับ 2 คำตอบคือ 1 ส่วนสิ่งที่เราเพิ่งได้มา. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "นี่เป็นค่าคงที่ที่ดีที่ต้องคิด เพราะมีรูปแบบมหัศจรรย์ที่เกิดขึ้นเมื่อเราดูกำลังของ 2 แล้วถ้าผมถามว่าลอกฐาน 2 ของพันเป็นเท่าไหร่ อย่างที่เราเพิ่งเห็น มันก็ประมาณว่า 2 ยกกำลัง 10 เท่ากับพัน และเนื่องจากเรากำลังทำสิ่งต่าง ๆ บนบันทึก ฉันจึงจะเขียนมันด้วยวิธีนั้น บันทึก 2 ของพันมีค่าประมาณ 10 ในทำนองเดียวกัน บันทึกฐาน 2 ของล้าน ลองดูว่า ถ้าเราคูณ 2 ด้วยตัวมันเองประมาณ 10 เท่าจึงจะได้พัน เราควรจะต้องคูณด้วยตัวมันเองประมาณ 20 เท่าจึงจะได้หนึ่งล้าน และแท้จริงแล้ว ฐานล็อก 2 ของล้านมีค่าประมาณ 20 มันเล็กกว่านิดหน่อย แต่นี่เป็นการประมาณที่ดีในใจคุณ แล้วในทำนองเดียวกัน คุณจะเห็นว่าทำไมผมจึงเขียนสิ่งนี้เป็นรูปแบบในเวลาสั้นๆ ถ้าเราอยากจะขึ้นไปถึงพันล้าน โดยบอกว่าฉันต้องคูณ 2 ด้วยตัวเองกี่ครั้งถึงจะได้พันล้าน นี่ก็ประมาณ 30 และนักวิทยาศาสตร์คอมพิวเตอร์คนไหนที่คิดอยู่ว่า 1 กิโลไบต์ เมกะไบต์ หรือ 1 กิกะไบต์ มีค่าเท่าไหร่ พวกเขาจะคุ้นเคยกับแนวคิดที่ว่า เลขยกกำลัง 2 นั้นดี และใกล้เคียงกับเลขยกกำลัง 10 หรือมากกว่านั้น โดยเฉพาะพลังนับพัน ตอนนี้สิ่งที่ผมอยากทำคือเขียนสิ่งเดียวกันทั้งหมดด้วยล็อกฐาน 10 ไม่เท่ากับโดยประมาณ นี่เท่ากับ 3 จริงๆ ฐานล็อก 10 ของพันเท่ากับ 3 ล็อกฐาน 10 คุณก็บอกฉันหน่อย ล็อกฐาน 10 ของล้านเป็นเท่าไหร่? ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "3 คือ 3 20 เราลดขนาดลงด้วยจำนวนที่เท่ากัน 30 เราก็ลดขนาดลงด้วยจำนวนที่เท่ากัน ตกลง? ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "และแทนที่จะให้ฉันอ่านมันให้คุณฟัง ฉันจะให้คุณดูมัน แทนตัวเลขลงไป ฉันจะให้เวลาคุณอย่างมีความหมายกับอันนี้ เพราะมันไม่ชัดเจน เว้นแต่คุณจะคุ้นเคยกับลอการิทึมอยู่แล้ว และมันก็คุ้มค่าที่จะคิดทบทวนสักหน่อย เรามีคำถามที่โดดเด่นจากผู้ฟังว่า ความยาวของแท่งไม้ของเสาใช้ฟังก์ชันบันทึกบางประเภทหรือไม่? ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/turkish/sentence_translations.json b/2020/ldm-logarithms/turkish/sentence_translations.json index 1ef0d6678..3f43c40b4 100644 --- a/2020/ldm-logarithms/turkish/sentence_translations.json +++ b/2020/ldm-logarithms/turkish/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Müzik🎵 Lockdown Math'a tekrar hoş geldiniz. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "Bugün logaritmalardan ve bir nevi temel bilgilere dönüş dersinden bahsedeceğiz. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "Ve her zaman olduğu gibi, işleri başlatmak için seyircilerin şu anda nerede olduğuna dair bir fikir edinmek istiyorum. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "Yani, eğer 3b1b'ye gidebilirseniz. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "Onları daha önce hiç duymadım veya daha önce hiç öğrenmedim b. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "Bunları öğrendim ama bazen tüm özellikler kafamı karıştırıyor c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "Onları anlıyorum ama nasıl öğreteceğimi bilmiyorum ve d. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "Onları iyi anlıyorum ve onların da iyi anlamasını sağlamak için rahatlıkla başka birine öğretebilirim. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "Yani iyi bir ayrım yaptık. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "Söylediğim gibi bunun amacı, gelecekte logaritma konusunda rahat olmayan insanlara yönlendirebileceğim bir ders yaratmak ve şunu söyleyebilmek istiyorum: ah, işte gidebileceğiniz bir yer nasıl düşünüyorum, bilirsiniz, buna sezgisel olarak nasıl yaklaşabileceğinizi düşünüyorum. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "Çünkü bu dersi vermeden önce birkaç öğretmen forumunda geziniyordum ve insanlar lise matematiğinde öğretilmesi en zor konunun ne olduğunu sorduğunda, yani öğrencilerin bu konuda en çok sorun yaşadığını düşündüğümde logaritma en çok sorulan konulardan biri. ilginç olan ortak olarak belirtilen cevaplar ve sanırım bunun nedeni, sonunda öğrenmek zorunda kalacağınız bu özelliklerden bir ton olmasıdır, yani gideceğimiz yerin ilerisine atlarsak, bir sürü yığınla karşı karşıya kalırsınız. Hatırlanması zor ve kafanızda bazı şeyleri karıştırması kolay bir grup cebir gibi görünen kurallar ve sanırım insanlar, bilirsiniz, lise matematiğinin nasıl bir şey olduğuna ve neye benzediğine dair bu tür kabus gibi hatıralara sahipler. logaritmalar onlar için işe yaradı, genellikle aklıma bu belirli formüller geliyor ve bugün yapmak istediğim şey bunlardan biri aracılığıyla konuşmaya çalışmak, bunlar hakkında nasıl düşüneceğimi ama aynı zamanda meta düzeyde de eğer birine cebir öğretiyorsanız, bu formüller nelerdir? vurgulamaya değer noktalar? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "Bunu sezgilerine yerleştirmenin yolu nedir? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "ah, üzerinde 3 sıfır var, bir milyonun logaritması nedir? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "1000 çarpı x'in log'u, log x'in 3 katına eşittir ve bunun 10 log b tabanı olduğu kuralını kullandığımızı unutmayın. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "1000 çarpı x'in log'u eşittir log x'in küpü c. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "1000 çarpı x'in log'u, 3 üzeri log x ve e'nin kuvvetine eşittir. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "yukarıdakilerin hiçbiri ve daha önce de söylediğim gibi, başlangıçta günlükleri iyi anladıklarını söyleyen tüm bu kişilerin hemen yanıt vermesini, doğru yanıt vermesini tam olarak beklemeliyiz, ancak eğer siz bunu yapmayan biri, bunun gibi bir problemle karşılaştığınızda bunun sizi korkutmasına izin vermeyin, yapmanızı tavsiye edeceğim şey sadece 10'un çeşitli kuvvetlerini yerine koymanız ve log fonksiyonunun ne olduğu fikri açısından düşünmenizdir. sıfırların sayısını sayıyor, bu yüzden size bunun hakkında düşünmeniz için biraz zaman vereceğim, bu yüzden devam edip not vereceğim ve her zaman olduğu gibi, eğer bu sizin rahat ettiğinizden daha hızlıysa, bunun sadece ileriye doğru ilerlemek istediğim için olduğunu bilin. derste, yani bu durumda doğru cevap log 1000 çarpı x olarak çıkıyor, 3 artı log x'i almakla aynı ve şimdi bunu bir anlığına düşünelim ve daha yeni başladığınızda söylediğim gibi Onlarla yapılacak en iyi şeyin çeşitli sayıları rahatça girmek olduğunu düşünüyorum ve takılacak en iyi sayılar zaten 10'un kuvvetleri olan sayılardır, yani eğer log 1000 çarpı x gibi bir şey soruyorsanız pekala sormuyorum' Bilmiyorum, hadi x log 1000 çarpı 100 için bir şey koyalım, son cevapta kaç sıfır olacağını biliyoruz, burada 1000 çarpı 100 eşittir 100.000, sezgisel olarak 10'un 2 üssünü çarptığımızda bu fikre zaten sahibiz. biz sadece sıfırları alıyoruz, şu 1000'den 3 sıfır, şu 100'den 2 sıfır ve bunları yan yana koyuyoruz yani toplam 5 sıfır olmalı ama gerçekten sadece sayının nasıl döndüğünü düşünmezseniz peki ama neden bu şekilde ortaya çıktı, 1000'deki 3 sıfır artı 100'deki 2 sıfır, bunu 1000'deki sıfırların sayısını artı 100'deki sıfırların sayısını söyleyerek de yazabiliriz, yani bu fikir bir logaritmadır iki şeyin çarpımının toplamı, bu iki şeyin 10'un kuvvetleri bağlamında logaritmasının toplamıdır, bu da çoğumuz için zaten süper sezgisel bir fikir olan şeyi ifade eder, eğer 10'un 2 kuvvetini alıp bunları çarparsanız, sadece onların tüm sıfırlarını alın ve onları bir şekilde üst üste sıkıştırın, böylece buraya yazma şeklim aslında biraz daha genel bir gerçeğin göstergesidir ki bu bizim logaritmanın ilk özelliği olacak, yani eğer A'nın log'u çarpı B, A'nın log'u artı B'nin log'u şimdi bu logaritma kurallarından birini gördüğünüzde, gözlerinizi kısarak gözlerinizi kısarak bulursanız veya nasıl hatırlayacağınız konusunda biraz kafanız karışırsa, sadece örnekleri ekleyin Gereksiz konuşuyorum, bunu çok söylüyorum ama bunun nedeni bence cebirin içinde boğulduğunuzda ve bir tür testte oturduğunuzda ve içinde bir sürü sembol bulunduğunda bunu unutmanın çok kolay olduğunu düşünüyorum. kendinize sadece bazı sayıları yerine koymanın sorun olmayacağını hatırlatmak için bu iyi bir şeydir ve çoğu zaman sezgi elde etmenin harika bir yoludur, bu durumda, A çarpı B'nin logaritmasını söyleyip onu parçalara ayırarak şunu düşünebiliriz: ah, şunu düşünebiliriz: 100 çarpı 1000'in logaritması, yani 5, içinde 5 sıfır var, her bir kısımdaki sıfırların sayısına göre ayrılıyor harika, harika bu sezgiyi daha da ileriye taşıyarak başka bir pratik problem deneyelim ve yine, eğer biliyorsanız, harika, güzel cevap verebileceksiniz ama belki sadece cevabın ne olduğunu değil, bu cevabı birine nasıl açıklayacağımı veya bir öğrencinin benim söylememe gerek kalmadan bu cevaba kendi başına gelmesini nasıl sağlayacağımı düşünebilirsiniz. onlara cevabın ne olduğunu yani iki potansiyel izleyici var, dersin kendisiyle ilgilenenler ve meta dersle ilgilenenler var, yani sorumuz yine aşağıdakilerden hangisinin doğru olduğunu soruyor? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "A. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "log x üzeri n eşittir n çarpı log x b. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "log x üzeri n eşittir log x üzeri n c. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "log x üzeri n eşittir n artı log x veya d. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "yani buradaki doğru cevap a, öyle görünüyor ki 4000'iniz tebrik aldınız, log x üzeri n'nin n çarpı log x'e eşit olduğunu söylediniz, yani yine diyelim ki bunu öğretmeye çalışıyorsunuz Birisi için ya da bunun ne anlama geldiğini kendiniz anlamaya çalışıyorsanız, bence başlamak için iyi bir yer bir şeyi takmak ve bu durumda, x'in logu n'nin kuvveti için 100'ün kuvvetiyle deneyelim 3 ve yaptığınız kalıpların gerçekten işe yarayıp yaramadığını görmek için bunu diğerleriyle deneyebilirsiniz, ancak bunu yalnızca cevabın ne olduğunu görmek açısından değil, cevabın neden bu şekilde ortaya çıktığını düşünmeye çalışmak açısından derinlemesine düşünüyorsanız bazen bir örnek işe yarar çünkü 100'ün küpü, bunu iyi almak olarak düşünebiliriz, bu 100'ün 3 kopyası. 100'ün 3 kopyasını alıyorum ve bunların hepsini çarptığım zaman log'u sıfırların sayısını saymak olarak düşünüyorum. Diyelim ki, üzerinde sadece 6 sıfır bulunan bir sayı olacak, bu 100 çarpı 100 çarpı 100 almanın anlamıdır. Tüm bu sıfırları bir araya toplayıp bir milyon elde etmeyi düşünebilirim, yani bu sayı şöyle olacak: 6 ama aslında neden 6 olduğunu düşünürsek, bu sadece milyon içindeki sıfırların sayısı değil, bu 6'nın geldiği yer, elimizde bu 100'ün 3 kopyası vardı ve bu 100'ün her birinde 2 farklı sıfır vardı, bu şekilde daha genel bir ifade olur. 100 küp almak yerine 1000 küp veya 1000 üzeri n veya x üzeri n'ye bakıyor olsaydık, bunun n'nin değerinin ne olursa olsun, çarpım yaptığımız kopya sayısı olduğunu düşünebilirsiniz. bakalım, x çarpı, x'in yerine koyduğumuz şeydeki sıfır sayısı değil ki bu durumda 100'dü, bunun yerine log 10.000'in n kuvveti gibi bir şey alsaydım bu aynı olurdu bu 10.000'in n kopyasını alarak her birindeki sıfırları sayarsak, yani 4 olur, yani n çarpı 4 olur ve elbette çoğunuzun doğru yanıtladığı genel özellik, bu sevimli küçük etkiye sahip olduğunuzdur. küçük bir gücün onun önüne atladığı bir güce yükseltilmiş bir şeyin kütüğünü görüyorsunuz ve elinizde sadece içeride ne olduğuna dair bir kayıt var şimdi bunun belki de en önemli çıkarımlarından biri buna denir mi bilmiyorum bir çıkarım ya da buna tanımın yeniden ifade edilmesi diyorsanız, eğer log alıyorum ve bunun 10'un 10 üssü n olduğunu yeniden vurgulayacağım, bu küçük n'nin aşağı indiğini düşünebiliriz. ön ve n çarpı log tabanı 10/10 olur ki bu da elbette 1'dir. Bu ifadeyi ya sondaki sıfır sayısını saymak olarak düşünebilirsiniz ya da daha genel olarak 10 üzeri 10'a eşit olanı sormaktır ve cevap basitçe 1'dir. bu çok güven verici çünkü geri dönüp bu orijinal ifadeyi okumanın başka bir yolu da 10 üzeri 10 üzeri 10'a eşit demek, yani cevap hayır, elimizdeki her logaritma özelliğiyle bu durumda bu durumda x üssü n'nin bir logunu buldum, n'nin öne atlamasını içerir, her zaman bir ayna görüntüsü üstel özelliği olacaktır ve bu, bunlar hakkında biraz sezgi kazanmamıza yardımcı olabileceğimiz başka bir yoldur, o yüzden üzerini örteyim buraya ulaşacağımız gelecekteki bazı özellikler nereye gittiğimizi saklamaya çalışıyor az önce bulduğumuz şeyi öne atlayan n'ye yükselterek bu üstel özelliğe karşılık geliyor, eğer 10'u x'e alıp yükseltirsem tüm bunların n üssü bu, 10 üzeri n çarpı x'i almakla aynı şey ve bu bizi logaritmalar için sahip olabileceğiniz başka bir sezgiye götürüyor ki bunlar bir nevi ters çevrilmiş üstel sayı gibiler ve işte şunu kastediyorum eğer kütüğün iç kısmında duran şey, eğer a'nın logaritmasını alıyorsam bunu üstel bir şeyin dış ifadesinin tamamı olarak düşünmeliyim, bu durumda a içteki şey 10 üzeri x'e karşılık gelir fonksiyonun çıktısı, oysa her şeyin kendisi a'nın logaritması içeride olana karşılık geliyor, burada sadece 10'un üssü nedir, yani burada log ifadesini gördüğünüz her yerde sağdaki üs rolünü oynadığını düşünüyor olmalısınız tarafta ve 10 üzeri x'in tamamındaki bir üstel ifadeyi her gördüğünüzde, sağ taraftaki tüm dış bileşen, kütüklerden birinin iç kısmında duran bir şeye karşılık gelir ve bunu çarpma işlemi sırasında fikrin üstünde gördük. içeride bu, dışarıdaki toplamadır, eğer günlükler bir tür üstel sayıları tersyüz ederse, bu bize fonksiyonun çıktılarını dışarıda çarpmanın içeride toplamayla aynı olduğunu söylüyor çünkü bu günlüklerin her biri log a ve log b gibi sağdaki ifadede x ve y'nin rolünü oynuyor yani bununla oynamaya devam edelim, bunlardan birkaç tane daha yapalım ve bu özelliklerden kaç tanesi için bir sezgi oluşturabileceğimizi görelim, yani bu sonuncusu için, Bir sonraki sayıya atlayan üslü sayıların çok güzel düşünülmesi, logaritmaya aşina olmayanlar için biraz tuhaf görünebilir, ancak yine de, bazı sezgileri kazanmak için bazı sayıları yerine koyun ve biraz verelim. Aşağıdakilerden hangisi doğrudur? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "peki, eğer 10'un küpü 1000 ise bu, 10 eşittir 1000'in 1/3'üne eşit demekle aynı şeydir, burada tersini yapmak üssün çarpımsal tersini gerektirir ve sonuç, 1 bölü 3 gibi görünmesidir. ve 3, 1000'in log tabanı 10'a karşılık gelir, bu 1'in 1000'in log tabanı 10'a bölümüdür, yani daha genel olarak, bu tek örneğe dayanarak tabanı içtekiyle değiştirdiğimizde bunun 1'e bölünmesine karşılık geldiğini tahmin edebilirsiniz. Dışarıda ne olduğuna bakılırsa, bunu karşılık gelen üstel kurala bakarak düşünebilirsiniz, şimdi benim sevimli küçük günlüğüme ve üstel sayılarıma ne oldu? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "harika o yüzden yine bazı şeyleri nereye saklayalım buraya ulaşacağımız diğer özelliklerden bazıları ve onu daha önce aldığım sırayla tutacağım burada önceden yazılmış olmasının beni tutabileceğini düşünüyordum normalden biraz daha temiz ama belki de bu tuhaf kağıt kesme oyununu karıştırmayı içeriyor, yani az önce bulduğumuz şey, a'nın b tabanını loglayın, eğer bunları değiştirirseniz, bu, bunun neye karşılık geldiğini 1'e bölmekle aynıdır. üstel alan, eğer b'nin bir kuvvetini alırsanız ve bunun a'ya eşit olduğunu söylerseniz bu, a üzeri bu kuvvetin tersinin b'ye eşit olduğunu söylemekle aynı ifadedir, bir dakikanızı ayırıp logaritmaların bazı şeyleri döndürmek olduğunu düşünmek biraz yararlı olacaktır. tersten logaritmik b tabanı ifadesi x'in rolünü oynuyor ve logaritmik a tabanı b ifadesi a'nın üzerinde ne varsa onun rolünü oynuyor ve simetrik olarak, b üzeri x ifadesinin tamamı oynuyor solda iç kısmın rolü, a rolünü oynuyor ve tüm ifade, a üzeri bir şeyin kuvveti log tabanının içinde oturan şeyin rolünü oynuyor a böylece sadece bazı örnekleri takarak görebilirsiniz ve bunu üstel kurallara karşılık getirerek zaten üç farklı logaritma kuralını düşünebiliriz; bunlar ezberlenecek cebir parçaları olarak aktarılsaydı, biliyorsunuz, onları ezberleyebilirsiniz ama bunların aklınızdan çıkması çok kolaydır. Elinizdeki görev yüzünden hayal kırıklığına uğramak da çok kolaydır, ancak bu tür şeyleri önemsememizin nedeninin, logaritma kurallarını anlamanın, bir virüsün büyüdüğü bağlamlarda matematik yapmamıza yardımcı olması olduğunu kendinize hatırlatmak isteyebilirsiniz. bir günden diğerine, bir adımdan diğerine, şeyler katlanarak büyüme eğilimindedir logaritma kurallarını anlamak bu tür şeyler hakkında daha iyi bir fikir edinmenize yardımcı olur, bu yüzden bunun neye benzeyebileceğine dair güzel bir gerçek dünya örneği yapmadan önce Bu bağlamda logaritmanın özelliklerini sormak için bir sınav sorusu daha yapmama izin verin, gerçek dünya örneğine geçmeden önce son bir soru burada ve şimdi sahip olduklarımızdan kurtulun, aşağıdakilerden hangisi doğrudur? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "log a artı b eşittir log a artı log b'nin log a artı b eşittir log a çarpı log a artı b eşittir bir bölü log a artı log b veya log a artı b eşittir bir bölü log a çarpı log b veya yukarıdakilerin hiçbiri ah ve şimdi o kadar da fikir birliğine sahip değiliz, değil mi? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "çok ilginç, ikisi arasında bir at yarışımız var bu yüzden insanlar cevap verirken size bunu düşünmeniz için biraz zaman vereceğim, aslında izleyicilere küçük bir sorum var o yüzden biliyorsunuz, sadece nasıl yarışabileceğimizden bahsediyordum çarpımsal büyüme açısından düşünün ve bu sadece onun kuvvetleri olmak zorunda değil, aynı zamanda üçün kuvvetleri gibi bir şey de yapabiliriz; burada birden üçe, dokuza, yirmi yediden seksen bire gidiyorsanız, hepsi Bunlardan, bu sayıların logaritmik üç tabanının güzel küçük adımlarla büyüdüğünü söyleyebiliriz, dolayısıyla log üç tabanı bir, üç üzeri bir eşittir, cevap genel olarak sıfırdır, taban ne olursa olsun birin logu, sıfır log üç üssü üç, üç üssü üç eşittir bir eşittir aynı şekilde log üç üssü dokuz eşittir iki ah, sorumun ne olduğunu merak edebilirsiniz, ama bunların hepsini ortaya çıkarmama yardımcı olacak ve kendi zevkim için burada, bir logaritmik taban daha yazayım, üç bölü seksen bir eşittir dört şimdi, duydum ki, eğer bir çocuğa, diyelim ki beş ya da altı yaşında bir çocuğa, bir ile dokuz arasındaki sayının ortasında ne olduğunu sorarsanız, hangi sayının yarı yolda olduğunu söyleyin nasıl cevap vereceklerine dair içgüdüleri logaritmiktir, oysa bizim içgüdülerimiz daha doğrusal olma eğilimindedir, bu nedenle genellikle bir ve dokuzu düşünürüz, aralarında eşit aralıklı bir grup sayı vardır iki, üç, dört, beş, altı , yedi, sekiz ve aradaki yolun yarısına giderseniz beşe ulaşırsınız ancak çarpımsal büyüme açısından birden dokuza kadar nereye gideceğinizi düşünüyorsanız, mesele bir sürü şey eklemek değil, siz 'Belirli bir miktarda büyüdüğünüzde üç kat büyürsünüz, sonra bir üç kat daha büyürsünüz, sözde bir çocuğun doğal içgüdüsü üç demekle aynı doğrultudadır ve sözüm ona bu, eğer toplumları inceleyen antropologlarınız varsa da aynı doğrultudadır' Muhasebe sistemlerini ve yazımı modern toplumlarda olduğu gibi geliştirmediler, buna üç cevap verecekler, dolayısıyla izleyicilere sorum şu anda izleyen herhangi birinizin, diyelim ki beş yıl içinde küçük bir çocuğa erişimi var mı? Bakın gidip onlara bir ile dokuz arasında hangi sayının olduğunu sorun ve eğer yapabiliyorsanız Twitter'da çocuğun ne söylediğini bize bildirin, asıl cevabının ne olduğunu bilmiyorum çünkü nedenini bilmiyorum, sadece biraz Bunun pratikte işe yarayıp yaramadığına dair şüphelerim var Bunun süper bilimsel bir yöntem olmadığını anlıyorum. YouTube canlı yayını izleyen insanlardan kendi çocuklarıyla anket yapmalarını ve ardından cevabı tweet atmalarını istemiyorum ama kendi iyiliğim için bu ilginç olurdu Sorumuzun bir çeşit doğrulamasını görmek için bu, tek bir yönde büyük bir fikir birliğine sahip gibi görünmeyen ilk soru, hadi devam edelim ve cevabın harika olduğunu görmek için not verelim, tamam, yani 2.400 a artı b'nin logaritmasının bu hoş özelliklerin hiçbirini karşılamadığının yukarıdakilerden hiçbiri olmadığını ve genel olarak, özellikle doğal logaritma devreye girdiğinde belirli türdeki yaklaşımlarla çalışmayacaksak, doğru yanıt verdiniz. bir dahaki sefere bunun hakkında konuşabiliriz, bir logaritmanın girdilerini eklemek aslında çok tuhaf bir duygu, bunu yapmak çok tuhaf bir şey ve bu tuhaflığı anlamak için, eğer size a artı b'nin logaritmasını sorarsam on'un bazı kuvvetlerini yerine koyun. şöyle düşünmeye başlayabilirsiniz, tamam, 10.000 ve 100 gibi bazı örnekleri yerine koyayım ve kendime şunu sorarım, eğer bu girdide ne olduğuna dair sıfır sayma fonksiyonunu yaparsam, içinde kaç tane sıfır olur? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "bu ilginç bir soru tamam mı, logaritmanın tabanı sıfır olabilir mi? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "yani bizim üçgenimiz açısından şunu düşünebiliriz: sıfırın bir çeşit kuvveti x eşittir başka bir değer y bu ya sıfır üzeri x eşittir y diyerek yazabileceğimiz bir şey ya da yazabiliriz log sıfır üssü y eşittir x sıfır üzeri x eşittir derken de aynı şey geçerli, buradaki sorun sıfır üzeri herhangi bir şeyin sıfır olması, doğru, yani eğer log sıfır üssünü düşüneceksek diğer herhangi bir giriş için, biliyorsunuz, bir veya iki veya pi gibi bir şey girmek istiyorsunuz, istediğiniz herhangi bir şeyi, sıfır üzeri bir veya iki veya pi veya orada ne varsa ona eşit olan soruyu soruyorsunuz ve bir cevap olmayacak, bu yüzden en iyi ihtimalle şöyle demeyi deneyebilirsiniz: ah evet, sıfırın logaritması, bu tamamen geçerli bir fonksiyon, yalnızca sıfır girişinde tanımlanır, ancak o zaman bile istediğiniz şeyi bulmaya çalışırken sorun yaşarsınız orada çünkü sıfır üzeri sıfır demek, ona uygulanan her şey gibi, dolayısıyla kolunuz arkanıza bükülecek, ancak bunu yapmak istiyorsunuz ve bu, sıfır tabanlı üstel fonksiyonun tamamen sıfır olduğu gerçeğine karşılık geliyor. sayıları bire bir güzel bir şekilde birbiriyle eşleştirmiyor, bu harika bir soru, sıfır tabanlı bir logaritmanız olabilir mi, şimdi bu şeylerin gerçek dünyada nerede ortaya çıktığı fikrine geri dönelim, hoşuma giden bir örnek şu: Richter ölçeği depremler için Richter ölçeği bize bir depremin ne kadar güçlü olduğuna dair bir nicelik verir ve çok küçük sayılardan çok büyük sayılara kadar herhangi bir şey olabilir, sanırım şimdiye kadar ölçülen en büyük deprem ve bu sadece buradan gelen bir tablo. Vikipedi 9'du. 5 ve bunun ne kadar çılgınca olduğunu takdir etmek için, bu sayıların ne anlama geldiği ile eşdeğer miktarda TNT gibi bir şeyin içinde ne kadar enerji bulunduğunun bir tür ölçüsü ve sonra burada ne yapmaya çalışabileceğimiz arasındaki ilişkiye bakmaya değer. Richter ölçeği sayısı için enerji miktarı cinsinden bir ifade elde edip edemeyeceğimizi ve logaritmanın bunu tanımlamanın neden doğal bir yolu olabileceğini görmektir. Dolayısıyla odaklanmamız gereken anahtar, ileriye doğru adımlar atarken şeylerin ne kadar arttığıdır. yani örneğin iki kuyudan gidersek bu durumda bu bize üçün nerede olduğunu göstermez yani belki ikiden dörde kadar bir adım atmayı düşünürüz ki bu da bir nevi iki adım atmaya benzer, bu ne işe yarar? Görünüşe göre bizi bir metrik ton TNT'den alıyor ki bu da İkinci Dünya Savaşı'ndan kalma büyük bir bomba ve bizi bin kat daha küçük bir atom bombası olan bir kiloton'a kadar götürüyor, yani sadece iki adım Richter ölçeğine göre 2 büyüklüğündeki bir depremden 4 büyüklüğündeki bir depreme geçiş bizi II. Dünya Savaşı'ndan nükleer çağa kadar olan büyük bombalardan alıyor, bu dikkate değer ve attığımız ilk temiz adım 4'ten 5'e çıkmak. en azından bu grafiğin bize güzel bir şekilde gösterdiği şey açısından ve açıkça 4'ten 5'e tek bir adım, 1 kilotondan 32 kilotona çıkmaya karşılık geliyor ve bu açıkça Nagasaki'ye düşen şehri yok eden bombanın büyüklüğüydü, yani bu belki de bir kilotondur. Haberlerde logaritmik ölçeklerle ilgili mantığa aykırı olabilecek bir şey, ah, 4 büyüklüğünde bir deprem vardı. 0'a karşı 5'lik bir deprem. 0, evet, 4 ve 5'i düşünmek kolay, bunlar oldukça benzer sayılar, ancak açıkça TNT miktarları açısından, 1'den diğerine geçmek için 32 ile çarpmaya ve 2'den 4'e gitmek, açıkça yaklaşık bin ile çarpmaktı ve tek Bunun daha büyük olmasının nedeni burada grafiğimizin 3'ün ne olduğunu göstermemesiydi, bu yüzden iki adım atıyorduk ve kendiniz için şunu doğrulayabilirsiniz: 32'lik bir adım atarsanız ve sonra başka bir 32 ile çarparsanız, bu aslında bine oldukça yakındır. Richter sayısındaki toplam adımların TNT'deki çarpımlı adımlara karşılık geldiği fikri burada logaritmik bir şeyin söz konusu olduğunu akla getiriyor gibi görünüyor ve burada devam edip bunun kısmen dünya fenomeni nedeniyle ne kadar büyüdüğünü söylemek biraz ilginç. Evet, bir adım daha attığımızda bunun yaklaşık 32 ile çarpılmasının çok büyük bir sürpriz olmadığını açıklıyoruz, ancak sezgilerimize göre bu, 32 kilotonluk küçük bir atom bombası ile küçük olmayan bir atom bombası olarak düşünebileceğimiz bir megaton arasındaki farktır. Nagasaki atom bombası, sanırım bir megatonluk Nagasaki atom bombalarının 32'si, bu da açıkça Nevada ABD'de 1994'te meydana gelen çift telli düz depremin büyüklüğü. Bunun ne olduğunu bilmiyordum, bu arada frekanslar açısından Wikipedia'ya teşekkürler. Ayrıca bunlara da baktım, açıkça ikiden az olanlar, bunlar her zaman oluyor, günde yaklaşık 8000 tane var ama atom bombası alanına girer girmez 3 gibi şeyler oluyor. 5 ve 4'ün de dünyanın herhangi bir yerinde oldukça sık meydana geldiği anlaşılıyor. Bunların yaklaşık 134'ü her gün bir yerlerde oluyor, kim bilebilirdi? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "ama atom bombası ölçeğinin çok üzerinde olan bu 5 ve 6 aralığına daha da yoğunlaştıkça, artık günde yalnızca 2 civarındayız ve eminim ki bir jeolog gelip hepimizin neden bunu yapmamız gerektiğini açıklayabilir. Her gün yerkabuğunda atom bombasına eşdeğer iki kesintinin meydana geldiği gerçeği konusunda fazla endişelenmeyin, ancak muhtemelen, şu anda birçok insanın yaşadığı bir şehir gibi bir noktaya yoğunlaşanların, her adımın aynı olduğu düşüncemizi doğrulaması özellikle nadirdir. 32'lik bir büyüme içerir, 6'dan 7'ye kadar olan adımın neye benzediğine bakalım ve burada bize aradaki çok daha fazla örnek veriyor, belki de bunun gerçekte olduğundan daha büyük bir adım olduğu yanılsamasını veriyor ve aslında 1 megaton ile 1 megaton arasındaki fark bu. 32 megaton yani 32 ile çarpılması bu grafikte en ilginç bulduğum şeylerden biri bu arada şu ana kadar test edilmiş en büyük nükleer silaha ulaşmak için ne kadar ileri gitmemiz gerektiğine bakın bu soğuk savaşın zirvesiydi 50 megatonluk Çar bombası ve sanırım aslında 100 megatonluk bir bomba yapmak için orijinal planları vardı ama kendilerini bu 50 megatondan aşağıya indirdiler, biz Nagazaki bombasının 32 kilotonundan başlayarak 32 ile çarparak bir sonuç elde etmekten bahsediyoruz. megatonu bir 32 ile çarpın, yani İkinci Dünya Savaşı'nın sonundaki patlamanın bin katı gücünden bahsediyoruz ve hala insanlığın yapabileceği 50 megatona ulaşmış değilsiniz ve bu açıkça Endonezya'daki Java depremi, yani 7 . 0, 6'dan biraz büyük değil. 0, çok daha büyük ve elbette buradaki nokta şu ki, size çarpımsal artışlar veren bir ölçeğe sahip olduğunuzda, küçük adımlar gibi görünen şeylerin, burada ima edilen enerji veya mutlak değerler açısından aslında çok büyük adımlar olabileceğini takdir etmeye değer. yani 9'un var olduğu gerçeğini düşündüğümüzde. 5 aslında sadece 7'de olduğu göz önüne alındığında saçma görünüyor. 0 aralığı, şimdiye kadar piyasaya sürülen en büyük termonükleer silahtan bahsediyoruz ve bu, logaritmaların ortaya çıkma eğiliminde olduğu bir alanın göstergesidir; bu, insanların bir şey için, ne kadar büyük şeylerin ne kadar büyük olabileceğine dair çok geniş bir farklılığı hesaba katan bir ölçek oluşturmak istedikleri zamandır. depremlerin büyüklüğü söz konusu olduğunda, Dünya'nın etrafında her zaman olup bitenlerden, büyük bir el bombası boyutundan şeyler alabilirsiniz ve bunun sizin ölçeğinizde olmasını ve sonuna kadar uzanmayı düşünecek bir şey olmasını istersiniz. insanlık tarihinde gördüğümüz en büyük bozulmaya ve bunu bir durum için sayılarınıza sadece bir sürü farklı rakam ve bir sürü farklı, daha küçük bir sayı yazmadığınız bir şekilde elde etmek için Başka bir durumda, numaranız için rakamların logaritmasını almak güzel ve sonra bunu temelde bu sayıları 0 ile 10 arasında sıkıştıran tek bir ölçeğe koymak güzel, müzik için desibel ölçeğinde çok benzer bir şeyin olduğunu görüyorsunuz, bu gerçekten biraz işe yarıyor biraz farklı, her seferinde 10 desibellik bir adım attığınızda bu 10 ile çarpmaya karşılık gelir, yani 1'in 10 ile çarpması yerine, bu 10'un 10 ile çarptığı bir adımdır, yani bu işin matematiğini biraz yapar biraz çılgınca ama fikir aynı; eğer 50 desibel yerine 60 desibel olan bir ses dinliyorsanız, iletilen ve giden enerji açısından çok daha sessizdir, bu ne olurdu, 60 ila 70 veya 70 ila 70 desibel 60'tan 80'e kadar olan bu 80 adım, kare alan başına enerji miktarının 100'ün katıyla çarpılmasını içerir, böylece logaritmik bir ölçek gördüğünüzde, bunun, kaputun altında kastettiği her şeyin 100 kat büyüdüğü anlamına geldiğini aklınızda tutun. çok büyük bir miktar, yine bu yüzden koronavirüs salgınını tanımlamak için çok sayıda logaritmik ölçeğin kullanıldığını gördük. Richter ölçeği sayısını her 1 artırdığınızda, 32 ile çarptığınız böyle bir ilişkiyi nasıl tanımlayabilirsiniz? 32 tabanına sahip bir logaritmik açıdan düşünebilirim Şunun logunu alırsam, Richter ölçeğine ait sayı olan r'yi çağıracağımı söyleyebilirim. Bunu log 32 tabanı olarak düşünebilirim ve bu şuna karşılık gelecektir: , hayır hayır hayır, bunu yanlış yapıyorum, kaydedilen şey bu değil, büyük sayının, TMT sayısının 32 tabanını alıyoruz, 1 megaton gibi bir şey, 1 milyon ton, log tabanı 32, bu gerekir Richter ölçeği numarasına karşılık gelir ancak bir tür kayma olabilir, dolayısıyla bu Richter ölçeği numarasına eklediğimiz bir tür sabit s olduğunu söyleyebiliriz ve bu ifade tamamen aynıdır, konuyu dışına çıktığım için kusura bakmayın altta bu ifade tam olarak 32 üssü bazı uzaklık çarpı Richter ölçek sayımız demekle aynı şey; bu da kendisi sadece büyük bir sabit olan bu uzaklığa 32 çarpı Richter ölçek numarasına göre 32'yi almakla aynı. Bunu gördüğünüz sayının 32 üssü gibi bir sabit çarpım olarak düşünebilirsiniz, yani bu şekilde yazmanız gerçekten üstel büyümeyi vurguluyor, eğer bu gördüğünüz TMT miktarına karşılık geliyorsa, siz bunu artırdıkça Adım adım 32 ile çarpıyorsunuz ama aynı gerçeği iletmenin başka bir yolu da bu miktar ne olursa olsun log 32 tabanını almaktır. Şimdi konuşmak istediğim bir sonraki şey her zaman bunu yapmak zorunda olmadığımızdır. farklı tabanların günlüklerini nasıl hesaplayacağımız konusunda endişe edin, burada log 32 tabanından bahsediyor olmamız biraz tuhaf, daha önce matematikçilerin temel e ile bir log tutmayı gerçekten ne kadar sevdiklerini belirtmiştim, bilgisayar bilimcileri gerçekten de 2 tabanlı bir log tutmayı seviyorlar ve bu hesaplama amaçları için ya da bu şeylerin nasıl büyüdüğünü düşünmek için ortaya çıkıyor, eğer bir log'unuz varsa, eğer bir tür log hesaplayabiliyorsanız, bu ister 10 tabanı, ister 2 tabanı, ister e tabanı olsun, hemen hemen başka her şeyi hesaplayabilirsiniz. şimdi sezgilerimizi o yöne çekmek istiyorsanız testimize dönüp bir sonraki soruya geçelim ve sanırım bu soru en fazla, bilmiyorum, yarı yarıya mantıklı bir soru bu, güzel olsa gerek bu bizi 2. taban bağlamından 10. taban bağlamına çeviri yapmaya hazır hale getirecek ve aynı zamanda 2'nin kuvvetlerini anlamak için genel olarak 10'un kuvvetleriyle olan ilişkisini anlamak için iyi bir sezgi çünkü bu çok güzel bir tesadüf. doğası gereği, bu iki çeşit iyi, ne demek istediğimi anlayacaksınız, birbirleriyle güzel oynuyorlar, bu yüzden sorumuz soruyor, 2 üzeri 10'un 1024, 1024 olduğu gerçeği göz önüne alındığında, bu da yaklaşık 1000'dir, yani eğer bir Rakamlarınızda biraz gevşeksiniz ve sadece 2 üzeri 10'a, yani temelde 1000'e yakın tahminler yapıyorsunuz, aşağıdakilerden hangisi gerçeğe en yakın? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "sunmak. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "Burada kesinlikle oybirliğiyle alınan bir karar değil. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "Ama soru hangisinin gerçeğe en yakın olduğunu sormaktı, bakalım bu konuda nasıl düşünebiliriz. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "Yani 2'nin bir kuvvetine sahip olduğunuzu gösteriyor ki bu da 1024'tür, 10'un kuvvetine çok yakın, yaklaşık 10'un küpü. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "Peki bu ne anlama geliyor? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "Eğer log 2 tabanı 10 x'e eşitse, bu 2 üzeri x'in 10'a eşit olduğunu söylemekle aynı şeydir, değil mi? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "Bize 2 üzeri 10'un ne olduğunu soruyor. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "Bunu her fonksiyonla yapamazsınız. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "İnsanlar bunu herhangi bir işlevle yapabileceğinizi düşünüyor gibi görünüyor, ancak yapamazsınız. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "Bu da x'in yaklaşık üçte 10 olduğu anlamına geliyor, tamam mı? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "Ve yeterince iyi, daha önce gördüğümüz şey log 2 tabanı 10'du, ayrıca log 10 tabanı 2'nin bu miktarın sadece 1 bölü, 1 bölü x olduğunu da söyleyebiliriz. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "Ve günlüklerde bir şeyler yaptığımız için bunu bu şekilde yazacağım. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "Benzer şekilde milyonun 2 tabanını loglayalım, peki bine ulaşmak için 2'yi kendisiyle yaklaşık 10 kez çarpmamız gerekiyorsa, milyona ulaşmak için onu yaklaşık 20 kez kendisiyle çarpmamız gerekiyor. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "Biraz daha küçük ama bu aklınızda bulundurmanız gereken güzel bir yaklaşım. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20, aynı miktarda küçültüyoruz. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, aynı miktarda küçültüyoruz. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "Tamam aşkım? ", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "Şimdi bu hatırlamaya değer bir sezgidir. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "Tamam aşkım? ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "Ve sonra log C tabanını B ile log C tabanı A'yı birleştirmenin çeşitli olası yolları var. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "Bu konuda size anlamlı bir süre vereceğim çünkü logaritmalara aşina olmadığınız sürece bu açık değildir ve biraz düşünmeye değer. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "Teşekkür ederim Karen. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/ukrainian/sentence_translations.json b/2020/ldm-logarithms/ukrainian/sentence_translations.json index cc606a96e..ff2913196 100644 --- a/2020/ldm-logarithms/ukrainian/sentence_translations.json +++ b/2020/ldm-logarithms/ukrainian/sentence_translations.json @@ -1,13 +1,13 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math.", + "input": "... you you you you you you you you you you you you you you you you you you you you you you you you you", "translatedText": "🎵Музика🎵 Ласкаво просимо назад у Lockdown Math.", "n_reviews": 0, "start": 0.0, "end": 691.84 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson.", + "input": "it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the ove", "translatedText": "Сьогодні ми будемо говорити про логарифми та повернемося до основ уроку.", "n_reviews": 0, "start": 720.0, @@ -28,7 +28,7 @@ "end": 742.7 }, { - "input": "Because I have a couple suspicions, but I think doing a live poll to see where everyone is might be helpful.", + "input": "'re adding 5000 instead use a y axis where each step is multiplicative so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by 10 and wh", "translatedText": "Тому що у мене є кілька підозр, але я думаю, що проведення опитування в реальному часі, щоб побачити, де всі знаходяться, може бути корисним.", "n_reviews": 0, "start": 742.92, @@ -42,7 +42,7 @@ "end": 759.16 }, { - "input": "co.", + "input": "y axis is now plotting not the total number of cases but the log", "translatedText": "співробітництво", "n_reviews": 0, "start": 759.16, @@ -63,7 +63,7 @@ "end": 770.84 }, { - "input": "a.", + "input": "would do and, you know, it's a little bit", "translatedText": "a.", "n_reviews": 0, "start": 770.84, @@ -77,7 +77,7 @@ "end": 774.0 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c.", + "input": "ive model to say oh it's going to grow exactly exponentially but in the early phases of something like this that is what it is so I kind of fast forward in the animation I m", "translatedText": "Я дізнався про них, але іноді мене бентежать усі властивості c.", "n_reviews": 0, "start": 774.0, @@ -133,7 +133,7 @@ "end": 864.84 }, { - "input": "What's the way to get it built in their intuitions?", + "input": "that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications", "translatedText": "Який спосіб це зробити в їхній інтуїції?", "n_reviews": 0, "start": 864.84, @@ -175,14 +175,14 @@ "end": 1063.5 }, { - "input": "and we see this sitting around early March or so and of course this is because this is when the corona outbreak was really starting to kick into high gear and everyone wanted to understand exponential growth and a common way that exponential growth is plotted is with what's known as a logarithmic scale so I actually made a video about this and in it I was creating some animations and wanted to illustrate this idea of exponential growth and the main idea here, I'll go ahead and skip back to a different animation is if you're tracking the numbers, in this case this was the number of recorded cases of COVID-19 outside of mainland China in the months leading up to March you could just track what the absolute number is but the pattern that you'll find is that as you go from one day to the next, you tend to be increasing multiplicatively it's a little bit like earlier, we were seeing the powers of 10 one step to the next, you're multiplying by some amount the way that the virus was growing was very similar from one day to the next, you're multiplying not quite by a constant but in this case, for this sequence of days, it was around 1.2 in that region, you're multiplying by something so when you're plotting this, it ends up looking like this classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the overall pattern is so a common trick is to say, instead of looking at this y-axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we're adding 5,000 instead use a y-axis where each step is multiplicative so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by 10 and what you can say is the y-axis is now plotting not the total number of cases but the logarithm of the total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend would do and it's a little bit of a naive model to say, oh it's going to grow exactly exponentially but in the early phases of something like this, that is what it is so I kind of fast-forward in the animation I made for that video and what's interesting is if back then, I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said, when is that line going to cross a million?", + "input": "eally liked this idea of making explicit how we have three totally different notations for the same exact fact one of them you're using relative positions of the numbers one of them we introduce a new symbol this radical and one of them we introduce a new word, log so these three syntactically different ways to communicate the same idea seemed wrong and so I made this video about an alternate possible notation and while I don't necessarily think that oh we should teach logarithms with this triangle because convention is what it is so it's better to start getting people used to the usual expression what I do like about it and starting off with it is when you see and think about this triangle it's really emphasizing that what the log wants to be is that exponent every time that you see log of some value you should think in your mind okay whatever this number is it really wants to be an exponent it wants to be an exponent and we'll see more of what that means as we go on okay so every time you see a log it wants to be an exponent this value three and more specifically it should be an exponent sitting on top of whatever that base is now in terms of convention for the first part of this video I'm just going to be using log without a base written on it to be the shorthand for log base 10 because log base 10 will be the most intuitive thing out there you should know that often in math the convention instead is that log without anything might mean log base e there's also another notation for that ln for natural log we're going to talk all about the natural log next time so don't worry too much about that right now and there's also yet another convention often if you're in a computer science setting log without any added sugar to indicate what it is defaults to meaning log base 2 so this can sometimes be a source of confusion but it basically depends on what discipline you're in in math, not moth, math people really like a base of e we'll see why next lecture in, I don't know, I'll say engineering but really it's anything where you want good intuition with our normal base 10 number system log means log base 10 and if you're curious often in computer science settings log base 2 comes up all the time so like I said, in the back of your mind if you're trying to think of some of these properties just resting on the idea that log counts the number of zeros at the end of a number that can get you a really far way so we're going to start going thro", "translatedText": "і ми бачимо це приблизно на початку березня, і, звісно, це тому, що саме тоді спалах коронавируса почав набирати обертів, і всі хотіли зрозуміти експоненціальне зростання, і загальний спосіб, яким експоненціальне зростання відображається, – це те, що відомо як логарифмічний масштаб, тому я фактично зробив відео про це, і в ньому я створював кілька анімацій і хотів проілюструвати цю ідею експоненціального зростання та головну ідею тут, я піду далі та перейду до іншої анімації, якщо ви Відстежуємо цифри, у цьому випадку це була кількість зареєстрованих випадків COVID-19 за межами материкового Китаю за місяці до березня. Ви можете просто відстежити абсолютне число, але ви побачите таку закономірність: коли ви переходите від одного дня до наступного, ви, як правило, зростаєте мультиплікативно, це трохи схоже на те, як раніше, ми бачили потужності числа 10 за один крок до наступного, ви множите на певну кількість, як ріс вірус було дуже схоже від одного дня до наступного, ви множите не зовсім на константу, але в цьому випадку для цієї послідовності днів це було приблизно 1.2 у цій області, ви множите на щось, тож коли ви будуєте це графік, це виглядає як ця класична експоненціальна крива, яка вигинається вгору, і іноді мені важко побачити, куди вона йде чи яка загальна схема, тому поширений трюк полягає в тому, щоб сказати, що замість того, щоб дивитися на цю вісь y, яка лінійно зростає, як тут, я йду від 5k до 10k, 10k до 15k, 15k до 20k, кожен крок є адитивним, ми додаємо 5000 замість цього використовуємо вісь y, де кожен крок є мультиплікативним, тому ви переходите від 10 до 100, 100 до 1000, 1000 до 10, 10 000, усе це збільшується шляхом множення на 10, і ви можете сказати, що вісь y тепер не відображається загальна кількість випадків, але логарифм загальної кількості випадків, і це фактично полегшує перегляд на графіку, якщо ви хочете спрогнозувати, що ця тенденція дасть, і це трохи наївна модель говорити, о, воно буде рости в геометричній прогресії, але на ранніх етапах чогось подібного, це те, що це так, тому я швидко перемотую вперед анімацію, яку зробив для цього відео, і що цікаво, якщо тоді, я думаю, я його опублікував 6 березня, якби ви щойно знайшли лінію, яка найкраще підходить, ви розтягнули її та запитали, коли ця лінія перетне мільйон?", "n_reviews": 0, "start": 1063.5, "end": 1210.12 }, { - "input": "which because the y-axis is growing with multiplicative steps each time that you step up, you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know, when you understand logarithmic scales, it actually didn't seem that far it was only 30 days away if you naively just drew out that line and in fact, fast-forward to around April 5th, which is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day, I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully, since then, the growth has stopped being exponential so if you look at it on a logarithmic plot, instead of going up in a straight line, it starts to taper off but, point being, any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined, okay?", + "input": "ugh a couple of these properties and I want to do this just with a set of practiced examples so we'll transition away from the poll and this time to the first proper question and the question asks you which of the following is true a. the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. log of 1000 times x equals log of x cubed c. log of 1000 times x equals 3 plus the log of x d. log of 1000 times x equals 3 to the power of log of x and e. none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that great ok so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in t", "translatedText": "оскільки вісь y зростає з множинними кроками щоразу, коли ви йдете вгору, ви множите на 10, тож навіть якщо може здатися, що 20 000 випадків, які були тоді, дуже далекі від мільйона, як ви знаєте, коли ви розумієте логарифмічні масштаби, це насправді не здавалося б таким далеким, це було лише через 30 днів, якщо ви наївно просто намалювали цю лінію і фактично швидко перемотали приблизно до 5 квітня, коли це було б передбачити, що ми досягнемо мільйон випадків за межами Китаю, це майже день, коли це сталося, я думаю, плюс-мінус день, я не пам’ятаю точно, але це було саме в цьому районі, тому що я пам’ятаю, що думав, вау, це була якась наївна модель для відео навіть на щастя, відтоді зростання перестало бути експоненціальним, тому, якщо ви подивитесь на логарифмічний графік, замість того, щоб підніматися по прямій лінії, воно починає звужуватися, але, головне, кожного разу, коли ви стикаєтеся з чимось у природі чи навіть у створеній людиною конструкції, де природно думати про мультиплікативні збільшення логарифмів, які приходять вам на допомогу, тож давайте подумаємо про те, що це насправді, як вони визначаються, Гаразд?", "n_reviews": 0, "start": 1210.12, @@ -238,7 +238,7 @@ "end": 2014.88 }, { - "input": "a.", + "input": "case, the correct answer of the choices we hav", "translatedText": "a.", "n_reviews": 0, "start": 2014.88, @@ -301,7 +301,7 @@ "end": 2567.02 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true?", + "input": "f 10,100 it's asking 10 to the what is equal to 10,100 you might say, I don't know, it's going to be a little above 4 because it's kind of close to 10,000 so the best you might guess here is oh this is going to be something That's kind of like The log of 10,000, but that just feels like a coincidence based on the two numbers that we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Sometimes you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y.", "translatedText": "чудово, тож давайте ще раз приховаємо, де деякі з речей деякі інші властивості, до яких ми дістанемося тут, і я збережу це в тому ж порядку, що й раніше, я думав, що це попередньо написане може зберегти мене трохи чистіше, ніж зазвичай, але, можливо, це просто гра в цю дивну гру з різанням паперу тасуванням, отже, що ми щойно знайшли, логарифм з основою b з a, якщо ви поміняєте їх місцями, це те саме, що поділити на 1 те, чому це відповідає, на Експоненціальна земля: якщо ви приведете b до якогось степеня і скажете, що це дорівнює a, це те саме твердження, що сказати, що a у оберненому до цього степеня знову дорівнює b. Буде корисно взяти хвилинку і подумати про логарифми як про перетворення речей навиворіт, вираз log за основою b для a відіграє роль цього х, а вираз за логарифмом за основою a з b відіграє роль того, що сидить поверх a, а потім, симетрично, грає весь вираз b у степені x роль внутрішньої сторони ліворуч, вона відіграє роль a та всього виразу, a до ступеня чогось відіграє роль того, що знаходиться всередині основи колоди a, щоб ви могли побачити, просто підключивши кілька прикладів і зіставляючи його з експоненціальними правилами, ми вже можемо продумати три різні правила логарифмування, які, якби вони просто були передані як частини алгебри, які потрібно запам’ятати, знаєте, ви могли б їх запам’ятати, але їм дуже легко вислизнути з вашого голова, і також дуже легко розчаруватися через завдання, але ви можете нагадати собі, що причина, чому ми дбаємо про такі речі, полягає в тому, що розуміння правил логарифмування допомагає нам робити математику в контекстах, де це як вірус, де росте від одного дня до наступного, від одного кроку до наступного, речі мають тенденцію зростати мультиплікативно, розуміння правил логарифмування допомагає вам краще відчути такі речі, тому перш ніж ми зробимо гарний реальний приклад того, як це може виглядати наприклад, дозвольте мені зробити ще одне запитання в цьому ключі, щоб запитати про властивості логарифмів, останнє, перш ніж ми перейдемо до трохи реального світу, приклад позбутися того, що ми мали тут і зараз, що з наступного є істинним?", "n_reviews": 0, "start": 2567.02, @@ -336,7 +336,7 @@ "end": 3053.77 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew?", + "input": "bomb That was 50 megatons, and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons we're talking start off at that 32 kilotons of the Nagasaki bomb Multiply by 32 to get a megaton multiply by another 32 Right so we're talking about a thousand times the strength of the World War two ending explosion And you're still not at the 50 megatons of what humanity is capable of And that is evidently you know the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0. It's a lot bigger and The point here of course is just that when you have a scale giving you multiplicative increases It's worth appreciating that what look like small steps Can actually be huge steps in terms of the energy implied or the absolute values implied here So it I mean when we're thinking about the fact that there was ever a 9.5 That actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out And this is indicative of one area where logarithms tend to come about it's When humans want to create a scale for something that accounts for a hugely wide variance in how big things can be So in the case of size of earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you want that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next ques", "translatedText": "ну в термінах нашого трикутника ми могли б подумати про те, що ви знаєте, що нуль у якомусь степені x дорівнює якомусь іншому значенню y. Це те, що ми могли б написати, сказавши, що нуль у x дорівнює y, або ми могли б написати те саме, кажучи, що логарифмічний нуль за основою y дорівнює х нуль до того, що дорівнює х, тепер проблема полягає в тому, що нуль до будь-чого в кінцевому підсумку дорівнює нулю, тож якщо ми будемо думати про логарифмічний нуль за основою y для будь-якого іншого введення y, який ви знаєте, ви хочете ввести щось на кшталт одиниці, двох або пі, будь-що, що забажаєте, ви задаєте питання від нуля до того, що дорівнює одиниці, двом або пі або будь-якому іншому числу, яке у вас там може бути і відповіді просто не буде, тож у найкращому випадку ви можете спробувати сказати «так, журнал нуля», це цілком дійсна функція, її визначено лише на вхідному нулі, але навіть тоді у вас виникнуть проблеми з спробою знайти те, що ви хочете тому що кажучи нуль тому, що дорівнює нулю, це ніби все стосується цього, тому ваша рука буде викручена за спиною, як би ви не хотіли, щоб це спрацювало, і це відповідає тому факту, що експоненціальна функція з основою нуль повністю дорівнює нулю Чи не зіставляє числа один з одним у зручній манері, тож це чудове запитання. Чи можете ви мати нульову базу журналу, а тепер повернемося до ідеї про те, де ці речі виникають у реальному світі, один приклад, який мені подобається шкала Ріхтера для землетрусів, тому шкала Ріхтера дає нам кількісну оцінку того, наскільки сильний землетрус, і він може бути будь-яким, від дуже малих чисел до дуже великих чисел, як я думаю, найбільший землетрус, який коли-небудь вимірювали, і це лише діаграма, яка походить від Вікіпедія отримала 9.5 і щоб зрозуміти, наскільки це божевільно, варто поглянути на зв’язок між тим, що означають ці цифри, і чимось на кшталт еквівалентної кількості тротилу, якась міра того, скільки енергії в цьому є, і потім, що ми можемо спробувати зробити тут Побачити, чи можемо ми отримати вираз для числа за шкалою Ріхтера в термінах кількості енергії, і чому логарифми були б природним способом описати це, тому головне, на чому зосередитися, це те, наскільки все зростає, оскільки ми робимо кроки вперед тож, наприклад, якщо ми переходимо від двох, у цьому випадку це не показує нам, де знаходиться три, тому, можливо, ми думаємо зробити крок від двох до чотирьох, що схоже на два кроки, що це дає з точки зору кількість енергії, схоже, вона бере нам від однієї метричної тонни тротилу, що, я думаю, є великою бомбою часів Другої світової війни, і це бере нам до кілотонни в тисячу разів більше, що є маленькою атомною бомбою, тож лише два кроки за шкалою Ріхтера, переходячи від землетрусу магнітудою 2 до землетрусу магнітудою 4, ми переходимо від великої бомби від Другої світової війни до ядерної ери, тому це варто відзначити, і перший чистий крок, який ми отримуємо, переходить від 4 до 5 за принаймні з точки зору того, що чудово показує нам ця діаграма, і, очевидно, один крок угору від 4 до 5 відповідає переходу від 1 кілотонни до 32 кілотонн, і це, очевидно, був розмір бомби, що руйнує місто, яка впала на Нагасакі, тому це, можливо, один те, що може бути суперечливим щодо логарифмічних шкал, якщо ви просто чуєте в новинах різницю між о, був землетрус, який був 4.0 проти землетрусу, який був 5.0 легко подумати, так, 4 і 5 це досить схожі числа, але, очевидно, з точки зору суми в тротиловому еквіваленті, яка відповідає множенню на 32, щоб перейти від 1 до наступного, і переходу від 2 до 4, очевидно, було множення приблизно на тисячу і єдине Причина, що він більший, полягає в тому, що тут наша діаграма не показувала, що таке 3, тому ми робили два кроки, і ви можете самі переконатися, що якщо ви зробите крок 32, а потім помножите ще на 32, це насправді буде близько до тисячі, тому ідея про те, що адитивні кроки числа Ріхтера відповідають мультиплікативним крокам у TNT, здається, припускає, що тут має місце щось логарифмічне, і трохи цікаво просто продовжувати тут і говорити, наскільки це зростає частково через світові явища. описуючи так, не дуже дивно, що коли ми робимо ще один крок, воно знову множиться приблизно на 32, але враховуючи це нашою інтуїцією, це різниця між 32 кілотоннами маленької атомної бомби та однією мегатонною, яку ми могли б вважати не маленькою атомною бомбою, Атомна бомба Нагасакі, я вважаю, що це 32 атомні бомби Нагасакі на одну мегатонну, що, очевидно, дорівнює величині подвійного плоского землетрусу в штаті Невада, США 1994 року. Я не знав, що це таке, дякую Вікіпедії за частоти, до речі, я також переглянув ці, очевидно, ті, які менше двох, вони трапляються постійно, їх близько 8000 на день, але щойно ми опинились у царстві атомних бомб, таких як 3.5 і 4 вони, очевидно, також трапляються досить часто десь на землі, приблизно 134 таких трапляються десь щодня, хто знав?", "n_reviews": 0, "start": 3053.77, @@ -350,7 +350,7 @@ "end": 3901.15 }, { - "input": "log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000?", + "input": "the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship", "translatedText": "лог за основою 2 з 10 дорівнює приблизно 0.3 логарифм за основою 2 з 10 дорівнює приблизно, вибачте, логарифм за основою 10 з 2 дорівнює приблизно 0.3 логарифм за основою 2 з 10 дорівнює приблизно 1 третині або логарифм за основою 10 з 2 дорівнює приблизно 1 третині, що з цього ближче до істини, виходячи з того факту, що від 2 до 10-го це по суті 1000?", "n_reviews": 0, "start": 3901.15, @@ -364,14 +364,14 @@ "end": 4025.43 }, { - "input": "tender.", + "input": "attempt count which I think is to say unraveling If you're looking at the maximum number I'm not I'm", "translatedText": "тендер.", "n_reviews": 0, "start": 4025.43, "end": 4029.59 }, { - "input": "Not at all a unanimous decision here.", + "input": "not great at Vanna whiting this thing if you look at the maximum number in our poll It's asking what's the log base 2 of that?", "translatedText": "Тут зовсім не одностайне рішення.", "n_reviews": 0, "start": 4029.59, @@ -385,7 +385,7 @@ "end": 4038.31 }, { - "input": "So that's good, they're very numerically similar, right?", + "input": "ent powers of 2 then that rescales it and Yes, yes is the answer what a fantastically apropos questio", "translatedText": "Тож це добре, вони чисельно дуже схожі, чи не так?", "n_reviews": 0, "start": 4038.31, @@ -413,35 +413,35 @@ "end": 4124.69 }, { - "input": "And the question is how we can leverage this to understand something like log base 2 of 10, or log base 10 of 2.", + "input": "u some more time to think this through because it's looks like a big pile of algebra plug in some numbers to see what seems to work well and See which answer fits You You You Okay, so eve", "translatedText": "І питання полягає в тому, як ми можемо використати це, щоб зрозуміти щось на кшталт логарифму за основою 2 з 10 або логарифму за основою 10 з 2.", "n_reviews": 0, "start": 4124.69, "end": 4137.67 }, { - "input": "As we saw earlier, those are just the reciprocals of each other.", + "input": "n if you are still thinking about it I'm gonna go ahead and grade it here and then start talking about Why it's true and then also why we should", "translatedText": "Як ми бачили раніше, це лише взаємні один одному.", "n_reviews": 0, "start": 4137.67, "end": 4146.01 }, { - "input": "So what does this mean?", + "input": "care why this is an operation that actually tells", "translatedText": "Отже, що це означає?", "n_reviews": 0, "start": 4146.15, "end": 4147.95 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right?", + "input": "us something so the correct answer which it looks like around 1700 of you got congratulations is Log base C of B times log base B of A is equal to log base", "translatedText": "Якщо логарифм за основою 2 з 10 дорівнює х, це те саме, що сказати, що 2 по х дорівнює 10, вірно?", "n_reviews": 0, "start": 4147.95, "end": 4157.99 }, { - "input": "It's asking us 2 to the what equals 10.", + "input": "C of A great Now that's just a big ol pile of things. Why would that be true?", "translatedText": "Він просить нас від 2 до того, що дорівнює 10.", "n_reviews": 0, "start": 4157.99, @@ -483,7 +483,7 @@ "end": 4213.53 }, { - "input": "People seem to think you can do that with any function, but you just can't.", + "input": "here would be things like let's use a different color. Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 o", "translatedText": "Здається, люди думають, що ви можете зробити це з будь-якою функцією, але ви просто не можете.", "n_reviews": 0, "start": 4213.53, @@ -518,14 +518,14 @@ "end": 4229.95 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x.", + "input": "t's plug in another power of 10 It'll be nice if it's also a power of 100 So I'll do a million So This one is asking 10 to 100 to the what equals a million How many times do I multiply a hundred by itself to get to a million?", "translatedText": "І добре, те, що ми бачили раніше, це те, що логарифм за основою 2 з 10, ми також можемо сказати, що логарифм за основою 10 з 2 це просто 1 на цю суму, 1 на х.", "n_reviews": 0, "start": 4229.95, "end": 4234.87 }, { - "input": "And you can see this pretty easily by writing 2 is equal to 10 to the 1 over x.", + "input": "How many times does a hundred go into a million? Phrasing the same thing 10 different ways now the claim is that this is", "translatedText": "І ви можете легко побачити це, написавши 2 дорівнює 10 до 1 на х.", "n_reviews": 0, "start": 4234.87, @@ -546,7 +546,7 @@ "end": 4245.93 }, { - "input": "Great.", + "input": "ng log base 10 of a million That if I ask how many times d", "translatedText": "чудово", "n_reviews": 0, "start": 4245.93, @@ -560,7 +560,7 @@ "end": 4266.57 }, { - "input": "So if I ask what is the log base 2 of a thousand, like we just saw, it's approximately the case that 2 to the power 10 is equal to a thousand.", + "input": "ve me the answer to how many times 10 goes into a million now just checking the numbers this certainly works 10 goes into a hundred two times 100 goes into a million three times in a multiplicative sense in that a hundr", "translatedText": "Отже, якщо я запитую, чому дорівнює логарифм за основою 2 тисячі, як ми щойно бачили, це приблизно так, що 2 у степені 10 дорівнює тисячі.", "n_reviews": 0, "start": 4266.57, @@ -574,14 +574,14 @@ "end": 4285.29 }, { - "input": "Log 2 of a thousand is approximately 10.", + "input": "ll six Now we could think of this property in terms of the corresponding exponent rule which is going to look a l", "translatedText": "Log 2 тисячі дорівнює приблизно 10.", "n_reviews": 0, "start": 4285.29, "end": 4288.75 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million.", + "input": "ittle bit stranger But it's actually just saying the entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each other the whole statement is equal to saying that um suppose that B to the X is equal to a Got some", "translatedText": "Так само логарифм мільйона за основою 2, давайте подивимося, якщо нам потрібно помножити 2 на себе приблизно 10 разів, щоб отримати тисячу, нам доведеться помножити це на себе приблизно 20 разів, щоб отримати мільйон.", "n_reviews": 0, "start": 4288.75, @@ -623,7 +623,7 @@ "end": 4344.51 }, { - "input": "Log base 10 of a thousand is equal to 3.", + "input": "g you layer it on top of each other. Now if we rearrange that expression, we get what is probably the second most important of all of our", "translatedText": "Логарифмічна основа 10 тисячі дорівнює 3.", "n_reviews": 0, "start": 4344.51, @@ -637,14 +637,14 @@ "end": 4353.53 }, { - "input": "It's counting the number of zeros, it ends up being about 6.", + "input": "n you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if you want", "translatedText": "Він підраховує кількість нулів, в результаті виходить приблизно 6.", "n_reviews": 0, "start": 4353.53, "end": 4358.35 }, { - "input": "And log base 10 of a billion, counting the number of zeros, it ends up being 9.", + "input": "the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. If", "translatedText": "І логарифм 10 мільярда, враховуючи кількість нулів, виходить 9.", "n_reviews": 0, "start": 4358.35, @@ -679,35 +679,35 @@ "end": 4411.57 }, { - "input": "Okay?", + "input": "say I'll use the log base 10 button and evaluat", "translatedText": "Гаразд?", "n_reviews": 0, "start": 4411.57, "end": 4417.25 }, { - "input": "Now this is an intuition worth remembering.", + "input": "e what's on the inside here, which at least positionally it's kind of above the 100.", "translatedText": "Тепер це інтуїція, про яку варто пам’ятати.", "n_reviews": 0, "start": 4417.25, "end": 4417.93 }, { - "input": "If you have your numbers described with one base, it's basically the same as describing them with another base, but there's some rescaling constant.", + "input": "It has a higher altitude as we write it. So this can line up with the notation a little bit, that it sits on the numerator. And on the bottom, I use the log base 10 button that's in my calculator on th", "translatedText": "Якщо ваші числа описуються за допомогою однієї основи, це в основному те саме, що описувати їх за допомогою іншої основи, але є певна константа масштабування.", "n_reviews": 0, "start": 4417.93, "end": 4428.15 }, { - "input": "Okay?", + "input": "e base, on the 100.", "translatedText": "Гаразд?", "n_reviews": 0, "start": 4428.15, "end": 4429.21 }, { - "input": "And then the next question is going to start getting us at that direction, but it's going to be framed in a way that just looks like a whole pile of algebra, and again I will encourage you to plug in numbers if you want to to gain a little intuition for it.", + "input": "And then I can evaluate both of those and it'll give me the answer. In this case it gets you 6 divided by 2, which will be 3. And if we really just think through what this is saying, I know I've said it many different times, but it's a convoluted enough way to write things, but an intuitive enough", "translatedText": "І тоді наступне запитання почне спрямовувати нас у цьому напрямку, але воно буде сформульоване таким чином, що буде виглядати як ціла купа алгебри, і знову я заохочую вас підключати цифри, якщо ви хочете отримати вигоду трохи інтуїції для цього.", "n_reviews": 0, "start": 4429.21, @@ -728,7 +728,7 @@ "end": 4452.11 }, { - "input": "Does that equal log base B of A?", + "input": "Because like I said, this is probably the second most important log rule. We're", "translatedText": "Чи дорівнює це логарифму основи B для A?", "n_reviews": 0, "start": 4452.11, @@ -763,7 +763,7 @@ "end": 4481.63 }, { - "input": "If you're looking at the maximum number, I'm not great at Vanna Whiting this thing, if you look at the maximum number in our poll, it's asking what's the log base 2 of that, so as it crosses different powers of 2 then that rescales it, and yes, yes is the answer.", + "input": "But anything additive in the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the left-handed side and the right-hand side are just saying how many times does 100 go into a million? but going about that in different ways. So this is extremely nice", "translatedText": "Якщо ви дивитеся на максимальне число, я не в захваті від Ванни Уайтінг, якщо ви дивитеся на максимальне число в нашому опитуванні, ви запитуєте, який логарифм за основою 2, так як він перетинає різні степені 2 тоді це змінює його масштаб, і так, так – це відповідь.", "n_reviews": 0, "start": 4481.63, @@ -777,14 +777,14 @@ "end": 4490.09 }, { - "input": "Thank You Karen.", + "input": "Next time we're going to talk", "translatedText": "Дякую Карен.", "n_reviews": 0, "start": 4490.09, "end": 4490.95 }, { - "input": "All right so answers are still rolling in, and I think like I said I just want to give you some more time to think this through because it looks like a big pile of algebra.", + "input": "all about the natural logarithm, which is log base e, often written ln. And turns out, this is much easier to compute. There's nice math behind it such that if you want to come up with an algorithm that your calculator can use, it's actually a lot easier to think of l", "translatedText": "Гаразд, відповіді все ще надходять, і я думаю, як я вже сказав, я просто хочу дати вам ще трохи часу, щоб подумати над цим, тому що це виглядає як велика купа алгебри.", "n_reviews": 0, "start": 4490.95, diff --git a/2020/ldm-logarithms/urdu/sentence_translations.json b/2020/ldm-logarithms/urdu/sentence_translations.json index 30516b372..29f8cdb92 100644 --- a/2020/ldm-logarithms/urdu/sentence_translations.json +++ b/2020/ldm-logarithms/urdu/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Music🎵 لاک ڈاؤن ریاضی میں دوبارہ خوش آمدید۔", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "آج ہم لوگارتھمز کے بارے میں بات کرنے جا رہے ہیں اور اس طرح کے سبق کی بنیادی باتوں کی طرف واپس جانے والے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "اور ہمیشہ کی طرح، چیزوں کو شروع کرنے کے لیے، میں صرف یہ سمجھنا چاہتا ہوں کہ سامعین اس وقت کہاں ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "لہذا، اگر آپ 3b1b پر جا سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "میں نے ان کے بارے میں سیکھا ہے لیکن بعض اوقات تمام خصوصیات سے الجھ جاتا ہوں c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "میں ان کو سمجھتا ہوں لیکن نہیں جانتا کہ انہیں کیسے سکھایا جائے اور ڈی۔", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "میں انہیں اچھی طرح سمجھتا ہوں اور آرام سے انہیں کسی اور کو سکھا سکتا ہوں تاکہ وہ بھی اچھی طرح سمجھ سکیں۔", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "تو، ہمارے پاس ایک اچھی تقسیم ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "جیسا کہ میں نے کہا، اس کا مقصد ایک سبق تیار کرنا ہے جس کی طرف میں مستقبل میں لوگوں کو اشارہ کر سکتا ہوں اگر وہ لوگارتھمز سے مطمئن نہیں ہیں اور میں یہ کہنے کے قابل ہونا چاہتا ہوں، اوہ، یہاں ایک ایسی جگہ ہے جہاں آپ جا سکتے ہیں میں کس طرح سوچتا ہوں، آپ جانتے ہیں، میرے خیال میں آپ اس سے بدیہی طور پر کیسے رجوع کر سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "کیونکہ میں یہ خاص لیکچر کرنے سے پہلے اساتذہ کے ایک دو فورمز کے ارد گرد اسکرول کر رہا تھا اور جب لوگ پوچھتے ہیں کہ ہائی اسکول کے ریاضی میں پڑھانے کے لیے سب سے مشکل موضوع کون سا ہے اس لحاظ سے کہ طلباء کو اس میں سب سے زیادہ پریشانی ہوتی ہے، لوگارتھمز سب سے زیادہ ہے۔عام طور پر اشارہ کردہ جوابات جو دلچسپ ہیں اور میں اندازہ لگا سکتا ہوں کہ شاید اس کی وجہ یہ ہے کہ ان خصوصیات میں سے ایک ٹن ہے جس کے بارے میں آپ کو سیکھنا پڑے گا کہ آپ جانتے ہیں، لہذا اگر ہم اس سے آگے بڑھیں جہاں ہم جانے والے ہیں آپ کے پاس یہ تمام ڈھیر موجود ہوں گے۔قواعد جو صرف الجبرا کے ایک گروپ کی طرح نظر آتے ہیں جو یاد رکھنا مشکل اور آپ کے دماغ میں چیزوں کو گھل مل جانا آسان ہوسکتا ہے اور میں سوچتا ہوں کہ جب لوگوں کے پاس، آپ کو معلوم ہے کہ ہائی اسکول کی ریاضی کیسی تھی اور کیسی تھی لوگارتھمز نے ان کے لیے کیا، اکثر وہ خاص فارمولے ذہن میں آتے ہیں اور آج میں جو کچھ کرنا چاہتا ہوں وہ ایک کے ذریعے بات کرنے کی کوشش کرتا ہے، ان کے بارے میں کیسے سوچنا ہے بلکہ صرف میٹا لیول پر کہ اگر آپ کسی کو الجبرا پڑھا رہے ہیں، تو کیا ہیں؟ پر زور دینے کے قابل پوائنٹس؟ ان کے وجدان میں اسے بنانے کا طریقہ کیا ہے؟ تو آئیے آگے بڑھتے ہیں اور دیکھتے ہیں کہ لوگ اس پر کیا جواب دے رہے ہیں، ہمیں تین ممکنہ زمروں کے درمیان کافی حد تک تقسیم مل گئی ہے، تو اس وقت سب سے زیادہ عام ناظرین نے C کا جواب دیا، کہ وہ انہیں سمجھتے ہیں لیکن وہ نہیں جانتے ہوں گے کہ کیسے سکھایا جائے۔", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "1000 گنا x کا لاگ x کے لاگ کے 3 گنا کے برابر ہے اور یاد رکھیں کہ ہم کنونشن استعمال کر رہے ہیں کہ یہ بیس لاگ بی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "1000 گنا x کا لاگ x کیوبڈ c کے لاگ کے برابر ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "1000 گنا x کا لاگ x اور e کے لاگ کی طاقت کے 3 کے برابر ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "مندرجہ بالا میں سے کوئی بھی نہیں اور یاد رکھیں جیسا کہ میں نے پہلے کہا تھا کہ ہمیں پوری طرح سے توقع کرنی چاہئے کہ وہ تمام لوگ شروع میں جنہوں نے کہا کہ وہ لاگ کو اچھی طرح سمجھتے ہیں وہ فوری طور پر جواب دینے جا رہے ہیں، وہ صحیح جواب دینے جا رہے ہیں لیکن اگر آپ کوئی ایسا شخص جو ایسا نہیں کرتا ہے، اس سے آپ کو خوفزدہ نہ ہونے دیں جب آپ اس طرح کے مسئلے کو دیکھ رہے ہوں تو میں آپ کو صرف 10 کے مختلف پاورز میں پلگ ان کرنے کی ترغیب دوں گا اور اس خیال کے لحاظ سے سوچیں کہ لاگ فنکشن صفر کی تعداد شمار کرتا ہے اس لیے میں آپ کو اس کے بارے میں سوچنے کے لیے تھوڑا سا لمحہ دوں گا تاکہ میں آگے بڑھوں اور اس کی درجہ بندی کروں اور ہمیشہ کی طرح اگر یہ اس سے زیادہ تیز ہے جس میں آپ کو راحت ہے جان لیں کہ یہ صرف اس لیے ہے کہ میں آگے بڑھنا چاہتا ہوں۔سبق کے ساتھ تو اس صورت میں صحیح جواب 1000 گنا x کا لاگ بنتا ہے اور 3 کے علاوہ x کا لاگ لینے کے مترادف ہے اور اب ایک لمحے کے لئے اس کے بارے میں سوچتے ہیں اور جیسا کہ میں نے کہا تھا کہ جب آپ ابھی شروعات کر رہے ہیں ان کے ساتھ میرے خیال میں سب سے بہتر کام یہ ہے کہ مختلف نمبروں میں آسانی سے پلگ ان کریں اور پلگ ان کرنے کے لیے بہترین نمبر وہ ہیں جو پہلے ہی 10 کے پاورز ہیں لہذا اگر آپ 1000 گنا x کے لاگ جیسی کوئی چیز پوچھ رہے ہیں تو ٹھیک ہے نہیں جانتے، آئیے صرف 1000 گنا 100 کے ایکس لاگ کے لیے کچھ لگائیں، ہم جانتے ہیں کہ یہاں حتمی جواب میں کتنے صفر ہونے والے ہیں، 1000 ضرب 100، 100،000 ہے، ہمیں پہلے سے ہی بدیہی طور پر یہ خیال ہے کہ جب ہم 10 کی 2 قوتیں ضرب کرتے ہیں ہم صرف صفر لے رہے ہیں، اس 1000 سے 3 صفر اور اس 100 سے 2 صفر اور ہم انہیں ایک دوسرے کے ساتھ لگا رہے ہیں لہذا یہ 5 کل صفر ہونا چاہئے لیکن اگر آپ واقعی اس بات پر غور نہیں کرتے کہ نمبر کیسے بدلا باہر لیکن یہ اس طرح کیوں نکلا یہ اس 1000 میں سے 3 صفر کے علاوہ اس 100 میں سے 2 صفر تھے جسے ہم 1000 میں صفر کی تعداد اور 100 میں صفر کی تعداد کہہ کر بھی لکھ سکتے ہیں تو یہ خیال ہے کہ ایک لاگرتھم دو چیزوں کی پیداوار 10 کی طاقتوں کے تناظر میں ان دو چیزوں کے لوگارتھمز کا مجموعہ ہے جو صرف اس بات کو بتا رہی ہے کہ ہم میں سے بہت سے لوگوں کے لیے پہلے سے ہی ایک انتہائی بدیہی خیال ہے اگر آپ 10 کے 2 قوتیں لیتے ہیں اور آپ ان کو صرف آپ کو ضرب دیتے ہیں۔ان کے تمام زیرو لیں اور انہیں ایک دوسرے پر گھسیٹیں تاکہ جس طرح میں نے یہاں چیزیں لکھی ہیں یہ دراصل قدرے عام حقیقت کی طرف اشارہ ہے جو ہماری لاگرتھم کی پہلی خاصیت ہوگی جو کہ اگر ہم A ٹائم B کا لاگ یہ A کے لاگ کے علاوہ B کے لاگ کے برابر ہوتا ہے اب جب بھی آپ ان لوگارتھم اصولوں میں سے کسی ایک کو دیکھتے ہیں اگر آپ خود کو آنکھیں پھیرتے ہوئے محسوس کرتے ہیں یا آپ تھوڑا سا الجھن میں ہیں کہ اسے کیسے یاد رکھا جائے بس مثالیں لگا دیں۔میں بے کار ہو رہا ہوں، میں یہ بہت کچھ کہہ رہا ہوں لیکن اس کی وجہ یہ ہے کہ مجھے لگتا ہے کہ ایک بار جب آپ الجبرا میں ڈوب جائیں تو بھول جانا بہت آسان ہے اور آپ کسی قسم کے امتحان پر بیٹھے ہیں اور اس میں بہت ساری علامتیں ہیں۔اپنے آپ کو یاد دلانے کے لیے آپ ٹھیک ہیں صرف کچھ نمبروں میں لگائیں جو کرنا ایک اچھی چیز ہے اور اکثر یہ بصیرت پیدا کرنے کا ایک بہترین طریقہ ہے لہذا اس معاملے میں، A ٹائم B کا لاگ کہنا اور اسے الگ کرنا ہم صرف سوچ سکتے ہیں، اوہ، وہ 100 گنا 1000 کا لاگ جو کہ 5 ہے، اس میں 5 صفر ہیں ہر دیئے گئے حصے میں زیرو کی تعداد کے لحاظ سے ٹوٹ جاتا ہے بہت اچھا، بہت اچھا ہے تو اس وجدان کو آگے بڑھاتے ہوئے آئیے ایک اور پریکٹس کا مسئلہ آزمائیں اور دوبارہ، اگر آپ کو معلوم ہے، بہت اچھا، آپ اس کا جواب ٹھیک سے دے پائیں گے لیکن شاید سوچیں، نہ صرف جواب کیا ہے بلکہ میں اس جواب کو کسی کو کیسے سمجھاؤں گا یا میں کس طرح کوشش کروں گا کہ کسی طالب علم کو مجھے بتائے بغیر خود ہی اس جواب پر آ جائے۔ان کا جواب کیا ہے تو سامعین کے دو ممکنہ ارکان ہیں جو خود سبق میں دلچسپی رکھتے ہیں اور پھر وہ لوگ جو میٹا اسباق میں دلچسپی رکھتے ہیں تو ہمارا سوال دوبارہ پوچھتا ہے، مندرجہ ذیل میں سے کون سا سچ ہے؟ a n سے x کا لاگ x b کے n بار لاگ کے برابر ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "n سے x کا لاگ ان پاور n c کے x کے لاگ کے برابر ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "n سے x کا لاگ برابر ہے n جمع لاگ x یا d کے۔", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "تو یہاں صحیح جواب a ہے، جو ایسا لگتا ہے کہ آپ میں سے 4000 کو مبارکباد دی گئی ہے، ہمیں یہ بتاتے ہوئے کہ x کا لاگ ان پاور n کے برابر ہے x کے n بار لاگ کے برابر ہے، تو، ایک بار پھر، چلیں کہ آپ اسے سکھانے کی کوشش کر رہے ہیں کسی کے ساتھ یا اگر آپ اس بات کو سمجھنے کی کوشش کر رہے ہیں کہ اس کا کیا مطلب ہے اپنے آپ کو میرے خیال میں شروع کرنے کے لیے ایک اچھی جگہ کچھ لگانا ہے اور اس صورت میں، x کے پاور پر لاگ کے لیے n آئیے اسے صرف 100 سے پاور کے ساتھ آزماتے ہیں۔3 اور آپ دوسرے لوگوں کے ساتھ یہ دیکھنے کے لیے کوشش کر سکتے ہیں کہ کیا آپ جو نمونے کر رہے ہیں وہ واقعی کام کر رہے ہیں لیکن اگر آپ اسے صرف یہ دیکھنے کے لحاظ سے نہیں سوچ رہے ہیں کہ جواب کیا ہے بلکہ یہ سوچنے کی کوشش کر رہے ہیں کہ جواب اس طرح کیوں نکلا۔کبھی کبھی ایک مثال یہ کرے گی کیونکہ 100 کیوبڈ، ہم اسے اچھی طرح سے لینے کے طور پر سوچ سکتے ہیں، یہ 100 کی 3 کاپیاں ہیں میں 100 کی 3 کاپیاں لے رہا ہوں اور جب میں ان سب کو ضرب دیتا ہوں اور میں لاگ کے بارے میں سوچتا ہوں کہ ہم صفر کی تعداد گنتے ہیں۔کہو، اوہ، یہ کوئی ایسا نمبر بننے والا ہے جس پر صرف 6 صفر ہوں، اس کا مطلب 100 گنا 100 گنا 100 لینے کا ہے، میں صرف ان تمام صفروں کو ایک ساتھ ملا کر ایک ملین حاصل کرنے کے بارے میں سوچ سکتا ہوں تو یہ نمبر ہونے جا رہا ہے۔6 لیکن اگر ہم حقیقت میں سوچیں کہ یہ 6 کیوں نہیں تھا صرف ملین کے اندر صفر کی تعداد ہے جہاں سے یہ 6 آیا ہے کہ ہمارے پاس اس 100 کی 3 کاپیاں تھیں اور ان 100 میں سے ہر ایک کے پاس 2 مختلف صفر تھے اس طرح یہ زیادہ عام ہے۔جس طرح سے آپ اس کے بارے میں سوچ سکتے ہیں کہ اگر ہم 100 کیوبڈ لینے کے بجائے 1000 کیوبڈ یا 1000 کو n یا x کو پاور n دیکھ رہے ہوں تو آپ سوچ سکتے ہیں کہ یہ جو بھی ہے n کی وہ قیمت تھی جو ہم ان کاپیوں کی تعداد کو ضرب دے رہے تھے۔کنویں کی تعداد، آئیے دیکھتے ہیں، یہ صفر کی تعداد کا x گنا نہیں ہے جو ہم نے x کے بدلے جو کچھ بھی کیا اس میں تھا جو اس معاملے میں 100 تھا، لہذا اگر میں اس کے بجائے 10،000 کے لاگ جیسی چیز کو پاور میں لیتا تو یہ وہی ہوتا جیسا کہ اس 10,000 کی n کاپیاں لیتے ہوئے ان میں سے ہر ایک میں صفر کی تعداد گنتے ہیں جو کہ 4 ہے تو یہ n گنا 4 ہوگا اور یقیناً عام خاصیت جس کا آپ میں سے اکثر نے صحیح جواب دیا ہے وہ یہ ہے کہ آپ پر یہ خوبصورت چھوٹا اثر ہے جہاں آپ کسی ایسی چیز کا لاگ دیکھیں جو ایک طاقت کی طرف اٹھتا ہے کہ تھوڑی سی طاقت اس کے سامنے آتی ہے اور آپ کے پاس صرف اس کا لاگ ان ہوتا ہے جو اندر موجود تھا اب اس کے سب سے اہم مضمرات میں سے ایک میں نہیں جانتا کہ آپ اسے کال کریں گے یا نہیں۔ایک مفہوم یا اگر آپ اسے تعریف کا دوبارہ بیان کہیں گے اگر میں لاگ لے رہا ہوں اور میں صرف اس بات پر دوبارہ زور دوں گا کہ اس کی بنیاد 10 میں سے 10 طاقت ہے n ہم اس چھوٹے سے n کے بارے میں سوچ سکتے ہیں سامنے اور یہ 10 میں سے لاگ بیس 10 کا n گنا بن جاتا ہے جو یقیناً 1 یہ اظہار ہے جس کے بارے میں آپ سوچ سکتے ہیں کہ یا تو آخر میں صفر کی تعداد گننا ہے یا عام طور پر یہ 10 سے پوچھ رہا ہے کہ 10 کیا ہے اور جواب صرف 1 ہے۔جو کہ بہت تسلی بخش ہے کیونکہ ایک اور طریقہ جس سے آپ واپس جا سکتے ہیں اور صرف اس اصل اظہار کو پڑھ سکتے ہیں وہ ہے 10 کو جو 10 کے برابر ہے n اوہ ٹھیک ہے جواب ٹھیک ہے اب ہر دی گئی لاگرتھم پراپرٹی کے ساتھ جو ہمارے پاس ہے اس معاملے میں ہم ابھی صرف x کا ایک لاگ ان پاور n میں ملا ہے جس میں یہ شامل ہے کہ n کے سامنے جھکنا ہمیشہ ایک آئینہ دار امیج ایکسپوینیشنل پراپرٹی ہوتا ہے اور یہ ایک اور طریقہ ہے جس سے ہم ان کے لئے اپنے آپ کو تھوڑا سا بصیرت حاصل کرنے میں مدد کرسکتے ہیں لہذا مجھے صرف احاطہ کرنے دو کچھ مستقبل کی خصوصیات جو ہم یہاں حاصل کرنے جا رہے ہیں اس کو چھپانے کی کوشش کریں کہ ہم کہاں جا رہے ہیں جو ہم نے ابھی n کے سامنے کچھ بڑھاتے ہوئے پایا ہے یہ اس کفایتی خاصیت کے مساوی ہے کہ اگر میں x پر 10 لیتا ہوں اور بڑھاتا ہوں طاقت n تک یہ پوری چیز 10 کو n اوقات x میں لینے کے مترادف ہے اور یہ ہمیں ایک اور وجدان کی طرف لے جاتا ہے جو آپ کے لوگارتھمز کے لئے ہو سکتا ہے جو کہ وہ اس طرح ہیں جیسے اندر سے ظاہر ہوتا ہے اور یہاں میرا مطلب ہے کہ جو چیز لاگ کے اندر بیٹھی ہے اگر میں ایک کا لاگ لے رہا ہوں تو آپ کو یہ سوچنا چاہیے کہ کسی چیز کے لیے مکمل بیرونی اظہار ہے جو اس معاملے میں اس کے اندر کی چیز 10 سے x کے مساوی ہے فنکشن کا آؤٹ پٹ جبکہ پوری چیز کا لاگ ان کے اندر موجود چیز سے مطابقت رکھتا ہے یہاں صرف 10 کا ایکسپیننٹ کیا ہے لہذا جہاں بھی آپ کو لاگ ایکسپریشن نظر آئے آپ کو یہ سوچنا چاہیے کہ یہ دائیں جانب ایک ایکسپیننٹ کا کردار ادا کرتا ہے۔سائیڈ اور ہر بار جب آپ ایکس ایکسپریشن تک پورے 10 کو ایکس ایکسپریشن دیکھتے ہیں تو دائیں طرف کا پورا بیرونی جزو جو کسی لاگ کے اندر بیٹھی ہوئی کسی چیز سے مطابقت رکھتا ہے اور ہم نے اسے اس خیال کے اوپر دیکھا کہ جب ہم ضرب کر رہے ہیں اندر کی طرف جو باہر سے اچھی طرح سے اضافہ کر رہا ہے اگر لاگز قسم کے ٹرن ایکسپووننشلز کے اندر باہر جو ہمیں بتا رہا ہے کہ فنکشن کے آؤٹ پٹس کو باہر سے ضرب دینا اندر سے اضافہ کرنے جیسا ہی ہے کیونکہ ان میں سے ہر ایک لاگ جیسے لاگ اے اور لاگ بی دائیں طرف کے ایکسپریشن میں x اور y کا کردار ادا کر رہا ہے تو اس کے ساتھ چلتے رہیں آئیے ان میں سے صرف ایک دو مزید کرتے ہیں اور دیکھتے ہیں کہ ان میں سے کتنی خصوصیات ہیں جن کے لیے ہم اس آخری کے لیے ایک وجدان پیدا کر سکتے ہیں، ایکسپونینٹس کو اگلے ایک کو نیچے اتارنے کے بارے میں بہت اچھی سوچ ایک ایسی چیز ہے جو ان لوگوں کو تھوڑی عجیب لگ سکتی ہے جو ضروری طور پر لوگارتھمز سے واقف نہیں ہیں لیکن دوبارہ، اس کے بارے میں کچھ بصیرت حاصل کرنے کے لیے کچھ نمبروں کو پلگ کریں اور ہم اسے تھوڑا سا دیں گے۔", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "اور یہ کہ 3 لاگ بیس 10 کے 1000 سے مساوی ہے یہ 1 کو 1000 کے لاگ بیس 10 سے تقسیم کیا گیا ہے لہذا عام طور پر، آپ اس واحد مثال کی بنیاد پر اندازہ لگا سکتے ہیں کہ جب ہم بیس کو اندر کی چیز سے تبدیل کرتے ہیں تو یہ 1 تقسیم کرنے کے مساوی ہوتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "وہاں کے باہر کیا ہے اور بار بار، آپ اس سے متعلقہ قاعدے کو دیکھنے کے لحاظ سے سوچ سکتے ہیں کہ اب میرے پیارے چھوٹے لاگ اور ایکسپوننشیل کا کیا ہوا؟ بہت اچھا تو، آئیے ایک بار پھر چھپائیں جہاں کچھ چیزیں کچھ دوسری خصوصیات ہیں جو ہم یہاں حاصل کریں گے اور میں اسے اسی ترتیب میں رکھوں گا جس ترتیب سے میرے پاس یہاں پہلے تھا میں سوچ رہا تھا کہ پہلے سے لکھے جانے سے میں رکھ سکتا ہوں معمول سے تھوڑا سا صاف ستھرا لیکن شاید اس میں کاغذ کاٹنے کا یہ عجیب و غریب کھیل کھیلنا شامل ہے لہذا ہم نے ابھی کیا پایا ہے، a کا لاگ بیس اگر آپ ان کو تبدیل کرتے ہیں، تو یہ 1 سے تقسیم کرنے کے مترادف ہے، جس سے یہ مساوی ہے۔کفایتی زمین ہے اگر آپ b کو کسی طاقت پر لے جائیں اور کہتے ہیں کہ وہ a کے برابر ہے یہ وہی بیان ہے جیسا کہ کہا جاتا ہے کہ a سے الٹا اس طاقت کے دوبارہ برابر ہو جاتا ہے، یہ ایک لمحہ نکالنا اور لوگارتھم کو چیزوں کو تبدیل کرنے کے طور پر سوچنا مفید ہے۔a کا ایکسپریشن لاگ بیس بی اندر سے اس x کا کردار ادا کر رہا ہے اور b کا ایکسپریشن لاگ بیس a اس کا کردار ادا کر رہا ہے جو بھی a کے اوپر بیٹھا ہے اور پھر ہم آہنگی سے، پاور x کا پورا اظہار b ادا کر رہا ہے۔بائیں طرف اندر کا کردار، یہ a اور پورے اظہار کا کردار ادا کرتا ہے، a سے کسی چیز کی طاقت اس کا کردار ادا کرتی ہے جو لاگ بیس کے اندر بیٹھی ہے a تاکہ آپ دیکھ سکیں، صرف کچھ مثالوں میں پلگ ان کرکے اور اس کو ایکسپونینشنل رولز سے ہم آہنگ کرتے ہوئے ہم پہلے ہی تین مختلف لوگارتھم کے اصولوں کے بارے میں سوچ سکتے ہیں جنہیں اگر آپ جانتے ہیں کہ الجبرا کے ٹکڑوں کے طور پر ان کو حفظ کرنے کے لیے دے دیا جاتا، تو آپ انہیں حفظ کر سکتے ہیں لیکن ان کے لیے یہ بہت آسان ہے کہ آپ ان سے باہر نکل جائیں۔سر اور ہاتھ میں کام سے مایوس ہونا بھی بہت آسان ہے لیکن آپ اپنے آپ کو یاد دلانا چاہیں گے کہ ہمیں اس قسم کی چیزوں کی پرواہ کرنے کی وجہ سے لوگارتھمز کے اصولوں کو سمجھنا ہمیں ان سیاق و سباق میں ریاضی کرنے میں مدد کرتا ہے جہاں یہ وائرس کی طرح بڑھتا ہے جہاں ایک دن سے دوسرے دن، ایک قدم سے دوسرے قدم تک، چیزیں ضرب سے بڑھنے لگتی ہیں لوگارتھمز کے اصولوں کو سمجھنا آپ کو اس قسم کی چیزوں کے بارے میں بہتر احساس دلانے میں مدد کرتا ہے لہذا اس سے پہلے کہ ہم ایک اچھی حقیقی دنیا کی مثال پیش کریں کہ یہ کیا نظر آتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "جیسا کہ میں اس رگ میں صرف ایک اور کوئز سوال کرتا ہوں کہ لوگارتھمز کی خصوصیات کے بارے میں ایک آخری سوال پوچھوں اس سے پہلے کہ ہم ایک حقیقی دنیا کی مثال کی طرف منتقل ہو جائیں اس سے چھٹکارا حاصل کریں جو ہمارے یہاں تھا اور اب، مندرجہ ذیل میں سے کون سا سچ ہے؟ اے پلس بی کا لاگ ایک پلس بی کے لاگ کے پلس لاگ کے برابر ہے ایک پلس بی کے بی لاگ کے بار لاگ کے برابر ہے پلس بی کے پلس لاگ کے لاگ سے تقسیم کرنے کے برابر ہے یا اے جمع بی کا لاگ برابر ہے ایک بار کے لاگ کے لاگ سے تقسیم کیا گیا بی یا اوپر میں سے کوئی بھی نہیں، اور اب ہمارے پاس اتنا اتفاق نہیں ہے، کیا ہم؟ بہت دلچسپ، ہمارے پاس دو کے درمیان گھوڑوں کی دوڑ ہے لہذا میں آپ کو اس کے بارے میں سوچنے کے لئے ایک لمحہ دوں گا جب لوگ جواب دے رہے ہیں، اصل میں میرے پاس سامعین کے لئے ایک چھوٹا سا سوال ہے، آپ جانتے ہیں، میں صرف اس بارے میں بات کر رہا تھا کہ ہم کیسے ضرب نمو کے لحاظ سے سوچیں اور اس کے لیے صرف دس کی طاقتیں ہی نہیں ہونی چاہئیں، ہم تین کی طاقتوں جیسا کچھ بھی کر سکتے ہیں جہاں آپ ایک سے تین سے نو سے ستائیس سے اکیاسی تک جا رہے ہیں، سبھی ان میں سے ہم یہ کہہ سکتے ہیں کہ ان نمبروں میں سے لاگ بیس تین صرف اچھے چھوٹے قدموں میں بڑھتا ہے لہذا لاگ بیس تین میں سے ایک، تین سے جو ایک کے برابر ہے، جواب صفر ہے عام طور پر ایک کا لاگ، چاہے بیس ہی کیوں نہ ہو صفر لاگ بیس تین میں سے تین، تین سے جو تین کے برابر ہے اسی طرح لاگ بیس تین میں سے نو دو آہ ہے، آپ حیران ہوں گے کہ میرا سوال کیا ہے، لیکن اس سے ان سب کو نکالنے میں مدد ملے گی اور میری اپنی خوشی کے لیے یہاں، میں صرف ایک اور لاگ بیس لکھتا ہوں کہ اکیاسی میں سے تین اب چار ہے، میں نے سنا ہے کہ بظاہر اگر آپ کسی بچے سے پوچھتے ہیں، تو چلو پانچ یا چھ سال کے لگ بھگ بتاتے ہیں کہ ایک اور نو کے درمیان نصف نمبر کیا ہے؟ بتائیں کہ کون سا نمبر آدھے راستے پر ہے جواب دینے کے لیے ان کی جبلتیں لوگاریتھمک ہیں جبکہ ہماری جبلتیں زیادہ لکیری ہوتی ہیں لہذا ہم اکثر ایک اور نو سوچتے ہیں، آپ کو ان کے درمیان یکساں فاصلہ والے نمبروں کا ایک گروپ مل گیا ہے دو، تین، چار، پانچ، چھ سات، آٹھ اور اگر آپ درمیان میں آدھے راستے پر چلتے ہیں، تو آپ پانچ پر اتریں گے لیکن اگر آپ ضرب کی ترقی کے لحاظ سے سوچ رہے ہیں کہ ایک سے نو تک کہاں پہنچنا ہے، تو یہ چیزوں کا ایک گروپ شامل کرنے کی بات نہیں ہے لیکن آپ 'ایک خاص مقدار سے بڑھ رہے ہیں جو آپ تین کے عنصر سے بڑھتے ہیں، پھر آپ تین کے ایک اور عنصر سے بڑھتے ہیں، ایک بچے کی فطری جبلت تین کہنے کے ساتھ ملتی ہے اور قیاس یہ بھی ہے کہ اگر آپ کے پاس ایسے معاشروں کا مطالعہ کرنے والے ماہر بشریات ہیں جو t نے اکاؤنٹنگ سسٹم اور تحریر کو اسی طرح تیار کیا جس طرح جدید معاشروں کے پاس ہے وہ اس کے تین جواب دیں گے لہذا، سامعین کے لیے میرا سوال اگر آپ میں سے کوئی جو اس وقت دیکھ رہا ہے تو اسے ایک چھوٹے بچے تک رسائی حاصل ہے، آئیے کہہ لیں، پانچ سال کی حد میں۔بوڑھے دیکھیں کہ کیا آپ ان سے پوچھ سکتے ہیں کہ ایک اور نو کے درمیان نصف نمبر کون سا ہے اور اگر آپ کر سکتے ہیں تو ہمیں ٹویٹر پر بتائیں کہ بچہ کیا کہتا ہے اس کا اصل جواب کیا ہے کیونکہ مجھے نہیں معلوم کیوں، میں تھوڑا سا ہوں۔اس کے بارے میں شکوک و شبہات ہیں کہ آیا یہ عملی طور پر ختم ہو گیا ہے میں سمجھتا ہوں کہ ایسا کرنے کا یہ کوئی انتہائی سائنسی طریقہ نہیں ہے میں یوٹیوب لائیو سٹریم دیکھنے والے لوگوں سے اپنے بچوں کا سروے کرنے اور پھر جواب ٹویٹ کرنے کے لیے نہیں کہہ رہا ہوں لیکن میرے لیے یہ دلچسپ ہوگا۔ہمارے سوال پر کسی قسم کی توثیق کو دیکھنے کے لیے یہ پہلا سوال ہے جس میں ایک سمت میں بہت زیادہ اتفاق رائے نہیں ہوتا ہے، آئیے آگے بڑھیں اور اس کی درجہ بندی کریں کہ کیا جواب بہت اچھا نکلا، ٹھیک ہے، تو 2,400 آپ میں سے صحیح جواب دیا کہ یہ اوپر میں سے کوئی بھی نہیں ہے کہ پلس بی کا لاگ ان اچھی خاصیتوں میں سے کسی کو بھی پورا نہیں کرتا ہے اور عام طور پر، جب تک کہ ہم مخصوص قسم کے تخمینے کے ساتھ کام کرنے جا رہے ہیں خاص طور پر جب قدرتی لاگ عمل میں آتا ہے۔ہم اگلی بار اس کے بارے میں بات کر سکتے ہیں لوگارتھم کے ان پٹس کو شامل کرنا درحقیقت ایک بہت ہی عجیب احساس ہے یہ کرنا ایک بہت ہی عجیب چیز ہے اور اس عجیب و غریب پن کا احساس حاصل کرنے کے لیے، اگر میں آپ سے پلس بی کا لاگ ان کرنے کے لیے کہوں تو دس کی کچھ طاقتیں لگائیں۔", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "حقیقت یہ ہے کہ یہ عجیب قسم کی بات ہے کہ آپ آسان نہیں کر پائیں گے لیکن اگر آپ، آپ جانتے ہیں، اگر آپ نے اس کے بارے میں سوچا ہی نہیں ہوتا تو آپ حیران ہوسکتے ہیں، اوہ، کیا کوئی ایسا فارمولا ہے جو میرے پاس نہیں ہے؟ اس سب کے ساتھ پہلے دیکھا گیا، میں آگے بڑھتا ہوں اور سامعین سے ایک دو سوالات کرتا ہوں اس سے پہلے کہ ہم کسی مختلف قسم کی مثال کی طرف منتقل ہو جائیں تو ایسا لگتا ہے کہ اوما شرما پوچھتی ہیں کہ کیا بنیاد صفر ہو سکتی ہے؟ یہ ایک دلچسپ سوال ہے ٹھیک ہے، کیا لوگارتھم کی بنیاد صفر ہو سکتی ہے؟ اچھی طرح سے ہمارے مثلث کے لحاظ سے ہم اس کے بارے میں سوچ سکتے ہیں جیسا کہ آپ جانتے ہیں، صفر سے کسی قسم کی طاقت x کسی دوسری قدر کے برابر ہے y یہ وہ چیز ہے جسے ہم یا تو صفر کو x کے برابر y کہہ کر لکھ سکتے ہیں یا ہم لکھ سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "ایک ہی بات یہ کہہ کر کہ y کا لاگ بیس صفر x صفر کے برابر ہے جو x کے برابر ہے اب یہاں مسئلہ یہ ہے کہ صفر سے کسی بھی چیز کا صفر صحیح ہوتا ہے، لہذا اگر ہم صرف لاگ بیس صفر کے بارے میں سوچ رہے ہیں y کسی دوسرے ان پٹ کے لیے y جو آپ جانتے ہیں، آپ کچھ ان پٹ کرنا چاہتے ہیں جیسے ایک یا دو یا pi کچھ بھی جو آپ چاہیں، آپ سوال صفر سے پوچھ رہے ہیں کہ ایک یا دو یا pi یا جو بھی نمبر آپ کے پاس ہو سکتا ہے اور اس کا کوئی جواب نہیں ہوگا لہذا آپ بہترین طور پر یہ کہنے کی کوشش کر سکتے ہیں اوہ ہاں، صفر کا لاگ، یہ ایک بالکل درست فنکشن ہے اس کی وضاحت صرف ان پٹ صفر پر کی گئی ہے لیکن اس کے باوجود آپ کو اپنی خواہش کو ختم کرنے کی کوشش میں دشواری ہوگی۔وہاں کیونکہ صفر کو صفر کہتے ہیں جو صفر کے برابر ہوتا ہے اس پر کچھ بھی لاگو ہوتا ہے لہذا آپ کا بازو آپ کی پیٹھ کے پیچھے مڑا جائے گا تاہم آپ یہ کام کرنا چاہتے ہیں اور یہ اس حقیقت سے مطابقت رکھتا ہے کہ بیس صفر کے ساتھ ایکسپونشنل فنکشن مکمل طور پر صفر ہے۔نمبروں کو ایک دوسرے پر اچھے انداز میں نقشہ نہیں بناتا ہے لہذا یہ ایک بہت اچھا سوال ہے، کیا آپ کے پاس لاگ بیس صفر ہے اب اس خیال پر واپس آ سکتے ہیں کہ یہ چیزیں حقیقی دنیا میں کہاں آتی ہیں ایک مثال جو مجھے پسند ہے زلزلوں کے لیے ریکٹر اسکیل اس لیے ریکٹر اسکیل سے ہمیں اندازہ ہوتا ہے کہ زلزلہ کتنا طاقتور ہے اور یہ بہت چھوٹی تعداد سے لے کر بہت بڑی تعداد تک کچھ بھی ہوسکتا ہے جیسا کہ میرے خیال میں اب تک کا سب سے بڑا زلزلہ ماپا گیا ہے اور یہ صرف ایک چارٹ ہے جس سے آیا ہے۔ویکیپیڈیا ایک 9 تھا۔5 اور اس بات کی تعریف کرنے کے لئے کہ یہ کتنا پاگل پن ہے کہ ان نمبروں کے معنی اور پھر TNT کی مساوی مقدار کے درمیان تعلق کو دیکھنے کے قابل ہے جیسے کہ اس میں کتنی توانائی ہے اور پھر ہم یہاں کیا کرنے کی کوشش کر سکتے ہیں۔یہ دیکھنا ہے کہ کیا ہم توانائی کی مقدار کے لحاظ سے ریکٹر سکیل نمبر کے لیے کوئی اظہار حاصل کر سکتے ہیں اور لوگارتھمز اس کو بیان کرنے کا ایک فطری طریقہ کیوں ہو گا، اس لیے توجہ مرکوز کرنے کی کلید یہ ہے کہ ہم آگے بڑھنے کے لیے اقدامات کر رہے ہیں کہ چیزیں کتنی بڑھتی ہیں لہذا مثال کے طور پر اگر ہم اس معاملے میں دو کنویں سے جاتے ہیں تو یہ ہمیں نہیں دکھاتا ہے کہ تین کہاں ہے تو شاید ہم دو سے چار تک ایک قدم اٹھانے کے بارے میں سوچتے ہیں جو اس طرح ہے جیسے دو قدم اٹھانا اس کے معاملے میں کیا کرتا ہے توانائی کی مقدار اچھی طرح سے ایسا لگتا ہے کہ یہ ہمیں ایک میٹرک ٹن TNT سے لیتا ہے جو کہ میرا اندازہ ہے کہ دوسری جنگ عظیم کا ایک بڑا بم ہے اور یہ ہمیں ایک کلوٹن تک ایک ہزار گنا زیادہ لے جاتا ہے جو ایک چھوٹا ایٹم بم ہے تو صرف دو قدم ریکٹر اسکیل پر 2 کی شدت کے زلزلے سے 4 شدت کے زلزلے کی طرف جانا ہمیں دوسری جنگ عظیم سے لے کر ایٹمی دور تک بڑے بم سے لے جاتا ہے تاکہ یہ قابل ذکر ہے اور پہلا صاف قدم جو ہمیں ملتا ہے وہ 4 سے 5 بجے تک جا رہا ہے۔کم از کم اس لحاظ سے کہ یہ چارٹ ہمیں اچھی طرح سے دکھا رہا ہے اور ظاہر ہے کہ 4 سے 5 تک ایک قدم بڑھنا 1 کلوٹن سے 32 کلوٹن تک جانے کے مساوی ہے اور یہ ظاہر ہے کہ شہر کو تباہ کرنے والے بم کا سائز تھا جو ناگاساکی پر گرا تھا لہذا یہ شاید ایک ہے۔ایسی چیز جو لوگارتھمک پیمانوں کے بارے میں متضاد ہوسکتی ہے اگر آپ صرف خبروں میں سن رہے ہیں کہ اوہ کے درمیان ایک زلزلہ آیا تھا جو 4 تھا۔0 بمقابلہ ایک زلزلہ جو 5 تھا۔0 یہ سوچنا آسان ہے کہ ہاں 4 اور 5 یہ کافی ملتے جلتے نمبر ہیں لیکن واضح طور پر TNT کی مقدار کے لحاظ سے جو کہ 1 سے دوسرے تک پہنچنے کے لیے 32 سے ضرب کرنے کے مساوی ہے اور 2 سے 4 تک جانا ظاہر ہے کہ تقریباً ایک ہزار سے ضرب ہو رہی تھی اور صرف اس کی بڑی وجہ یہ ہے کہ یہاں ہمارا چارٹ یہ نہیں دکھا رہا تھا کہ 3 کیا ہے لہذا ہم دو قدم اٹھا رہے ہیں اور آپ خود تصدیق کر سکتے ہیں کہ اگر آپ 32 کا ایک قدم اٹھاتے ہیں اور پھر آپ کو دوسرے 32 سے ضرب دیتے ہیں جو کہ حقیقت میں ایک ہزار کے قریب ہے۔یہ خیال کہ ریکٹر نمبر پر اضافی اقدامات TNT میں ضرب کے مراحل سے مطابقت رکھتے ہیں ایسا لگتا ہے کہ یہاں کچھ لوگارتھمک کام کر رہا ہے اور یہ تھوڑا سا دلچسپ ہے کہ یہاں جاری رکھیں اور یہ بتائیں کہ یہ جزوی طور پر عالمی مظاہر کی وجہ سے کتنا بڑھتا ہے۔ہاں بیان کرنا کوئی بڑی حیرت کی بات نہیں کہ جب ہم ایک اور قدم اٹھاتے ہیں تو یہ ایک بار پھر تقریباً 32 سے بڑھتا جا رہا ہے لیکن ہمارے وجدان کے مطابق یہ 32 کلوٹن چھوٹے ایٹم بم اور پھر ایک میگاٹن کے درمیان فرق ہے جسے ہم چھوٹا ایٹم بم نہیں سمجھ سکتے، ناگاساکی ایٹم بم جس کے بارے میں میرا اندازہ ہے کہ ایک میگاٹن کے ناگاساکی ایٹم بموں میں سے 32 ہیں جو ظاہر ہے کہ نیواڈا یو ایس اے 1994 میں آنے والے ڈبل سٹرنگ فلیٹ زلزلے کی شدت ہے مجھے نہیں معلوم تھا کہ وہ کیا تھا، شکریہ ویکیپیڈیا تعدد کے لحاظ سے۔واضح طور پر ان کو بھی دیکھا جو دو سے کم ہیں، یہ ہر وقت ہوتے ہیں ان میں سے روزانہ 8000 کی طرح ہوتے ہیں لیکن جیسے ہی ہم ایٹم بم کے دائرے میں آتے ہیں جیسے 3۔", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "5 اور 4 یہ بھی ظاہر ہے کہ زمین پر کہیں نہ کہیں اکثر ہوتے ہیں ان میں سے تقریباً 134 ایسے ہیں جو روزانہ کہیں نہ کہیں ہو رہے ہیں کون جانتا ہے؟ لیکن جیسا کہ ہم اس 5 اور 6 رینج میں اور بھی زیادہ شدت اختیار کر گئے ہیں جو ایٹم بم کے پیمانے سے کافی اوپر تھے اب ہم صرف 2 فی دن کے قریب ہیں اور مجھے یقین ہے کہ ایک ماہر ارضیات آکر وضاحت کر سکتا ہے کہ ہم سب کو کیوں اس حقیقت کے بارے میں زیادہ پریشان نہ ہوں کہ زمین کی پرت میں دو ایٹم بم کے مساوی رکاوٹیں ہر روز ہو رہی ہیں لیکن شاید یہ خاص طور پر ان لوگوں کے لئے نایاب ہے جو کسی ایسے شہر جیسے کسی جگہ پر مرکوز ہوں جہاں اب بہت سارے لوگ رہتے ہیں صرف ہمارے خیال کی تصدیق کرتے ہیں کہ ہر قدم 32 کی ترقی شامل ہے آئیے دیکھتے ہیں کہ 6 سے 7 تک کا مرحلہ کیسا لگتا ہے اور یہاں یہ ہمیں بہت ساری مثالیں دے رہا ہے اس کے درمیان شاید یہ وہم پیدا ہو کہ یہ حقیقت سے بڑا قدم ہے اور حقیقت میں یہ 1 میگاٹن اور کے درمیان فرق ہے۔32 میگاٹن اس طرح 32 سے ضرب کر رہا ہے اس چارٹ میں مجھے سب سے زیادہ دلچسپ چیز میں سے ایک یہ دیکھنا تھا کہ ہمیں سب سے بڑے ایٹمی ہتھیار تک پہنچنے سے پہلے کتنی دور جانا ہے جس کا حقیقت میں تجربہ کیا گیا ہے یہ سرد جنگ کا عروج تھا۔زار بم جو 50 میگاٹن کا تھا اور مجھے یقین ہے کہ ان کا اصل میں 100 میگا ٹن بم رکھنے کا منصوبہ تھا لیکن وہ خود اس 50 میگاٹن سے نیچے بات کر رہے ہیں، ہم بات شروع کر رہے ہیں کہ ناگاساکی بم کے 32 کلوٹن کو 32 سے ضرب دے کر 100 میگا ٹن کا بم بنایا جائے۔میگاٹن کو مزید 32 سے ضرب دیں تو ہم دوسری جنگ عظیم کے خاتمے کے دھماکے سے ہزار گنا زیادہ طاقت کے بارے میں بات کر رہے ہیں اور آپ اب بھی 50 میگاٹن پر نہیں ہیں جس کی انسانیت قابل ہے اور یہ ظاہر ہے کہ انڈونیشیا کا جاوا کا زلزلہ ہے تاکہ 7۔. 0 صرف 6 سے تھوڑا بڑا نہیں ہے۔0، یہ بہت بڑا ہے اور یقیناً یہاں بات صرف یہ ہے کہ جب آپ کے پاس ایک پیمانہ ہے جس سے آپ کو ضرب میں اضافہ ہوتا ہے تو یہ قابل تعریف ہے کہ چھوٹے قدم جو نظر آتے ہیں وہ درحقیقت انرجی کے لحاظ سے بہت بڑے قدم ہو سکتے ہیں یا یہاں مضمر مطلق اقدار لہذا جب ہم اس حقیقت کے بارے میں سوچ رہے ہیں کہ کبھی بھی 9 تھا۔5 جو حقیقت میں مضحکہ خیز لگتا ہے کیونکہ یہ صرف 7 میں ہے۔0 رینج جس میں ہم اب تک کے سب سے بڑے تھرمونیوکلیئر ہتھیار کے بارے میں بات کر رہے ہیں اور یہ ایک ایسے علاقے کی طرف اشارہ ہے جہاں لوگاریتھمز اس کے بارے میں آتے ہیں جب انسان کسی ایسی چیز کے لیے ایک پیمانہ بنانا چاہتے ہیں جو اس بات میں بہت زیادہ وسیع تغیر کا سبب بنتا ہے کہ کتنی بڑی چیزیں ہو سکتی ہیں۔زلزلوں کے سائز کے معاملے میں آپ کے پاس ایسی چیزیں ہو سکتی ہیں جو زمین کے ارد گرد ہر وقت ہوتا ہے، ایک بڑے ہینڈ گرنیڈ کا سائز اور آپ چاہتے ہیں کہ یہ آپ کے پیمانے پر ہو اور اس کے بارے میں سوچنے کے لیے کچھ ہو اس سب سے بڑے خلل کی طرف جو ہم نے انسانی تاریخ میں دیکھی ہے اور اسے اس طرح حاصل کرنے کے لیے کہ آپ صرف ایک کیس کے لیے اپنے نمبروں میں مختلف ہندسوں کا ایک پورا گچھا نہیں لکھ رہے ہیں اور مختلف، ایک چھوٹی تعداد کے لیے۔کسی دوسرے معاملے میں آپ کے نمبر کے ہندسوں کو لاگرتھم لینا اچھا لگتا ہے اور پھر اسے صرف ایک پیمانے پر ڈالیں جو بنیادی طور پر ان نمبروں کو 0 اور 10 کے درمیان سکوئش کرتا ہے آپ دیکھتے ہیں کہ میوزک کے ڈیسیبل پیمانے کے ساتھ کچھ ایسا ہی چل رہا ہے جو حقیقت میں تھوڑا سا کام کرتا ہے۔تھوڑا سا مختلف ہے جہاں ہر بار جب آپ 10 ڈیسیبل کا ایک قدم اٹھاتے ہیں جو 10 سے ضرب کرنے کے مساوی ہوتا ہے تو 1 کے ایک قدم کو 10 سے ضرب کرنے کے بجائے، یہ 10 کا ایک قدم ہے جو 10 سے ضرب کرتا ہے اس طرح اس کی ریاضی کو تھوڑا سا بناتا ہے تھوڑا سا عجیب لیکن خیال ایک ہی ہے، کہ اگر آپ کوئی ایسی آواز سن رہے ہیں جو 50 ڈیسیبل بمقابلہ 60 ڈیسیبل ہے تو یہ توانائی کے منتقل ہونے اور جانے کے لحاظ سے بہت زیادہ پرسکون ہے، یہ کیا ہوگا، 60 سے 70 یا 70 سے؟ 80 وہ مراحل، 60 سے 80 تک، جس میں توانائی کی مقدار کو فی مربع رقبہ کو 100 کے فیکٹر سے ضرب دینا شامل ہے، لہذا جب بھی آپ لوگاریتھمک پیمانہ دیکھیں، اپنے ذہن میں جان لیں کہ اس کا مطلب یہ ہے کہ جس کا بھی حوالہ دیا جا رہا ہے اس کے نیچے بڑھتا ہے۔ایک بہت بڑی رقم یہی وجہ ہے کہ ہم نے کورونا وائرس کے پھیلاؤ کو بیان کرنے کے لیے استعمال ہونے والے بہت سے لوگاریتھمک پیمانے دیکھے ہیں تو آپ اس طرح کے رشتے کو کیسے بیان کرسکتے ہیں جہاں ہر بار جب آپ ریکٹر اسکیل نمبر کو 1 سے بڑھاتے ہیں تو آپ کو 32 سے ضرب لگاتے ہیں، ہم بیس 32 کے ساتھ لاگ کے لحاظ سے سوچ سکتا ہوں، میں کہہ سکتا ہوں کہ اگر میں لاگ کو لیتا ہوں، تو میں صرف r کو کال کرنے جا رہا ہوں، ریکٹر اسکیل کے لیے وہ نمبر جسے میں لاگ بیس 32 سمجھ سکتا ہوں اور یہ اس کے مطابق ہوگا۔، نہیں نہیں نہیں، میں یہ غلط کر رہا ہوں کہ لاگ ان ہونے والی چیز نہیں ہے، ہم بڑے نمبر کا لاگ بیس 32 لیتے ہیں، TMT نمبر کا، کچھ جو کہ 1 میگاٹن جیسا تھا، یہ لاگ بیس 32 کے مقابلے میں 1 ملین ٹن ہے، اسے ہونا چاہیے۔ریکٹر اسکیل نمبر سے مطابقت رکھتا ہے لیکن اس میں کچھ قسم کا آفسیٹ ہوسکتا ہے، اس لیے ہم کہہ سکتے ہیں کہ کچھ مستقل s ہے جسے ہم اس ریکٹر اسکیل نمبر میں جوڑ رہے ہیں اور یہ اظہار بالکل وہی ہے، معاف کیجئے گا نیچے یہ اظہار بالکل ویسا ہی ہے جیسے کچھ آف سیٹ اوقات کی طاقت کو 32 کہتے ہیں ہمارے ریکٹر اسکیل نمبر جو کہ 32 کو اس آفسیٹ میں لینے کے مترادف ہے، جو بذات خود کچھ بڑا مستقل ہے، ریکٹر اسکیل نمبر پر 32 گنا ہے لہذا آپ اس کے بارے میں یہ سوچ سکتا ہے کہ آپ جو نمبر دیکھتے ہیں اس کی طاقت سے صرف 32 کا کچھ مستقل ہونا ہے لہذا اسے لکھنے کا یہ طریقہ واقعی اس کی تیز رفتار نمو پر زور دیتا ہے کہ اگر یہ وہی ہے جو آپ کو نظر آنے والی TMT رقم کے مساوی ہے، جیسا کہ آپ اس میں اضافہ کرتے ہیں۔r قدم بہ قدم آپ 32 سے ضرب کر رہے ہیں لیکن بالکل اسی حقیقت کو بتانے کا ایک اور طریقہ یہ ہے کہ جو بھی رقم ٹھیک ہے اس کا لاگ بیس 32 لینا ہے اب اگلی چیز جس کے بارے میں میں بات کرنا چاہتا ہوں وہ یہ ہے کہ ہمیں ہمیشہ اس کی ضرورت نہیں ہے۔مختلف اڈوں کے لاگس کی گنتی کرنے کے بارے میں فکر کریں یہاں یہ قدرے عجیب ہے کہ ہم لاگ بیس 32 کے بارے میں بات کر رہے تھے، میں نے پہلے حوالہ دیا تھا کہ ریاضی دان کس طرح واقعی بیس کے ساتھ لاگ رکھنا پسند کرتے ہیں اور کمپیوٹر سائنس دان واقعی بیس 2 کے ساتھ لاگ رکھنا پسند کرتے ہیں۔کمپیوٹیشنل مقاصد کے لیے یا اس بارے میں سوچنے کے لیے کہ اگر آپ کے پاس ایک لاگ ہے تو یہ چیزیں کیسے بڑھتی ہیں، اگر آپ ایک قسم کے لاگ کی گنتی کرنے کے قابل ہیں، چاہے وہ بیس 10 ہو، بیس 2، بیس اور آپ کسی اور چیز کا حساب لگا سکتے ہیں۔اب آپ چاہتے ہیں کہ ہمارے وجدان اس سمت میں جائیں، آئیے اپنے کوئز کی طرف واپس جائیں اور اگلے سوال پر جائیں اور مجھے یقین ہے کہ یہ سوال سب سے زیادہ ہے، مجھے نہیں معلوم، یہ ایک آدھا معقول سوال ہے، یہ اچھا ہونا چاہیے۔یہ صرف ہمیں بیس 2 سیاق و سباق سے بیس 10 کے سیاق و سباق میں ترجمہ کرنے کے لیے تیار کرنے والا ہے اور یہ 2 کی طاقتوں کو سمجھنے کے لیے بھی ایک اچھا ادراک ہے کہ عام طور پر اس کا تعلق 10 کی طاقتوں کے ساتھ ہے کیونکہ یہ اس خوبصورت قسم کا اتفاق ہے۔فطرت کہ یہ دونوں طرح کی اچھی طرح سے آپ دیکھیں گے کہ میرا کیا مطلب ہے، وہ ایک دوسرے کے ساتھ اچھی طرح کھیلتے ہیں لہذا ہمارا سوال پوچھتا ہے، اس حقیقت کو دیکھتے ہوئے کہ 2 سے 10 ویں 1024، 1024 ہے، جو تقریباً 1000 ہے، لہذا اگر آپ ایک ہو رہے ہیں اپنے نمبروں کے ساتھ تھوڑا سا ڈھیلا ہے اور آپ صرف 2 سے 10 ویں، بنیادی طور پر 1000 کا تخمینہ لگا رہے ہیں، مندرجہ ذیل میں سے کون سا سچ ہونے کے قریب ہے؟ لاگ بیس 2 از 10 تقریباً 0 ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "ٹینڈر. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "یہاں متفقہ فیصلہ نہیں ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "لیکن سوال یہ پوچھ رہا تھا کہ کون سا سچ ہونے کے قریب ہے، اور دیکھتے ہیں کہ ہم اس کے بارے میں کیسے سوچ سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "تو یہ بتاتا ہے کہ آپ کے پاس 2 کی طاقت ہے، جو 1024 ہے، 10 کی طاقت کے قریب، تقریباً 10 کیوبڈ۔", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "تو اس کا کیا مطلب ہے؟ اگر 10 کا لاگ بیس 2 x کے برابر ہے، تو یہ وہی چیز ہے جو کہ 2 کو x کے برابر 10 ہے، ٹھیک ہے؟ یہ ہم سے 2 پوچھ رہا ہے جو 10 کے برابر ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "آپ ہر فنکشن کے ساتھ ایسا نہیں کر سکتے۔", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "لوگوں کو لگتا ہے کہ آپ کسی بھی فنکشن کے ساتھ ایسا کر سکتے ہیں، لیکن آپ ایسا نہیں کر سکتے۔", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "اور اس کا کیا مطلب ہے کہ x تقریباً 10 تہائی ہے، ٹھیک ہے؟ کون سا، بہت اچھا، لہذا لاگ بیس 2 کا 10 تقریباً 10 تہائی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "اور کافی حد تک، جو ہم نے پہلے دیکھا وہ 10 کا لاگ بیس 2 ہے، ہم یہ بھی کہہ سکتے ہیں کہ لاگ بیس 10 کا 2 اس رقم سے صرف 1 ہے، 1 سے زیادہ x۔", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "زبردست. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "اور اس لیے کہ ہم نوشتہ جات پر کام کر رہے ہیں، میں اسے اس طرح لکھنے جا رہا ہوں۔", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "اسی طرح ایک ملین کا لاگ بیس 2، اچھی طرح دیکھتے ہیں، اگر ہزار تک پہنچنے کے لیے ہمیں 2 کو خود سے 10 گنا ضرب دینا ہے، تو ہمیں ایک ملین تک پہنچنے کے لیے اسے خود سے 20 گنا ضرب دینا ہوگا۔", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "یہ تھوڑا سا چھوٹا ہے لیکن یہ آپ کے ذہن میں ایک اچھا تخمینہ ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20، ہم اسی رقم سے کم کرتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30، ہم اسی رقم سے کم کرتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "ٹھیک ہے؟ اب یہ یاد رکھنے کے قابل ایک انترجشتھان ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "میں آپ کو اس پر ایک معنی خیز وقت دوں گا کیونکہ یہ واضح نہیں ہے جب تک کہ آپ لوگارتھمز سے پہلے ہی واقف نہ ہوں، اور یہ تھوڑا سا سوچنے کے قابل ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-logarithms/vietnamese/sentence_translations.json b/2020/ldm-logarithms/vietnamese/sentence_translations.json index c569589df..8e4f4d546 100644 --- a/2020/ldm-logarithms/vietnamese/sentence_translations.json +++ b/2020/ldm-logarithms/vietnamese/sentence_translations.json @@ -1,6 +1,6 @@ [ { - "input": "🎵Music🎵 Welcome back to Lockdown Math. ", + "input": "... you you you you you you you you you you you you you you you you you you you you yo ", "translatedText": "🎵Âm nhạc🎵 Chào mừng bạn quay trở lại với Lockdown Math. ", "model": "google_nmt", "n_reviews": 0, @@ -8,7 +8,7 @@ "end": 570.44 }, { - "input": "Today we are going to be talking about logarithms and kind of a back to the basics sort of lesson. ", + "input": "u you you you you it ends up looking like this, you know, classic exponential curve that curves upward and I can sometimes make it hard to see where it's going or what the over ", "translatedText": "Hôm nay chúng ta sẽ nói về logarit và quay trở lại bài học cơ bản. ", "model": "google_nmt", "n_reviews": 0, @@ -24,7 +24,7 @@ "end": 731.34 }, { - "input": "And as always, to kick things off, I just want to get a sense of where the audience is at right now. ", + "input": "say instead of looking at this y axis that increases linearly as in here I'm going from 5k to 10k, 10k to 15k, 15k to 20k each step is additive, we' ", "translatedText": "Và như mọi khi, để bắt đầu mọi thứ, tôi chỉ muốn biết khán giả hiện đang ở đâu. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 747.88 }, { - "input": "So, if you can go to 3b1b.co. ", + "input": "tive so you're going from 10 to 100, 100 to 1000, 1000 to 10, 10,000 all of these are increases by multiplying by ", "translatedText": "Vì vậy, nếu bạn có thể truy cập 3b1b. ", "model": "google_nmt", "n_reviews": 0, @@ -72,7 +72,7 @@ "end": 764.04 }, { - "input": "I've never heard about them before or never learned about them before b. ", + "input": "total number of cases and this actually makes it kind of easier to see on a plot if you wanted to project out what that trend ", "translatedText": "Tôi chưa bao giờ nghe nói về họ trước đây hoặc chưa bao giờ biết về họ b. ", "model": "google_nmt", "n_reviews": 0, @@ -80,7 +80,7 @@ "end": 770.84 }, { - "input": "I've learned about them but sometimes get confused by all of the properties c. ", + "input": "would do and, you know, it's a little bit of a naive model to say oh it's going to grow exactly exponentially but in the early phases ", "translatedText": "Tôi đã tìm hiểu về chúng nhưng đôi khi bị nhầm lẫn bởi tất cả các thuộc tính c. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 777.74 }, { - "input": "I understand them but wouldn't know how to teach them and d. ", + "input": "of something like this that is what it is so I kind of fast forward in the animation I made for that video and wh ", "translatedText": "Tôi hiểu họ nhưng không biết cách dạy họ và d. ", "model": "google_nmt", "n_reviews": 0, @@ -96,7 +96,7 @@ "end": 782.74 }, { - "input": "I understand them well and could comfortably teach them to someone else to make them understand well too. ", + "input": "at's interesting is if back then I think I posted it on March 6th if you just found a line of best fit and you stretched it out and you said when is ", "translatedText": "Tôi hiểu rõ chúng và có thể thoải mái dạy chúng cho người khác để họ cũng hiểu rõ. ", "model": "google_nmt", "n_reviews": 0, @@ -104,7 +104,7 @@ "end": 791.66 }, { - "input": "So, we've got a good split. ", + "input": "that line going to cross a million which because the y ", "translatedText": "Vì vậy, chúng tôi đã có một sự phân chia tốt. ", "model": "google_nmt", "n_reviews": 0, @@ -120,7 +120,7 @@ "end": 796.58 }, { - "input": "Like I said, the intent for this is to create a lesson that I can point people to in the future if they're just not comfortable with logarithms and I want to be able to say, oh, here's a place that you can go for how I think, you know, how I think you could approach it intuitively. ", + "input": "th multiplicative steps each time that you step up you're multiplying by 10 so even if it might seem like the 20,000 cases or so that it was back then is very far from a million you know when you understand logarithmic scales it actually didn't seem that far it was only 30 days away if you naively ", "translatedText": "Như tôi đã nói, mục đích của việc này là tạo ra một bài học mà tôi có thể hướng dẫn mọi người trong tương lai nếu họ không thoải mái với logarit và tôi muốn có thể nói, ồ, đây là nơi bạn có thể tìm đến Tôi nghĩ thế nào, bạn biết đấy, tôi nghĩ bạn có thể tiếp cận nó bằng trực giác như thế nào. ", "model": "google_nmt", "n_reviews": 0, @@ -136,7 +136,7 @@ "end": 817.58 }, { - "input": "Because I was scrolling around a couple of teacher forums before doing this particular lecture and when people ask what is the hardest topic to teach in high school math in the sense that students seem to have trouble with it the most, logarithms is one of the most commonly indicated answers which is interesting and I can guess maybe it's because there's a ton of these properties that you end up having to learn you know, so if we skip ahead of where we're going to go you've got all these piles of rules that just look like a bunch of algebra that can be hard to remember and easy to kind of mix things up in your head and I think when people have, you know, these sort of nightmarish recollections of what high school math was like and what logarithms did for them, it's often those particular formulas coming to mind and what I want to do today is try to talk through one, how to think about them but also just on the meta level of if you're teaching somebody algebra, what are the points worth emphasizing? ", + "input": "ich is when that would have predicted we hit a million cases outside China that's pretty much the day that it happened I think plus or minus a day but I don't remember exactly but it was right in that neighborhood because I remember thinking wow it was kind of a naive model for the video to even use and it's shocking that it matched so exactly thankfully since then the growth has stopped being exponential so if you look at it on a logarithmic plot instead of going up in a straight line it starts to taper off but point being any time that you're coming across something in nature or even in a man-made construct where what's natural to think about are multiplicative increases logarithms come in to help you so let's go ahead and think about what these actually are how are they defined and actually there was a question asked on Twitter right before we started that I think hits this very perfectly so Max 182 asks us additions inverse is subtraction multiplications inverse is division but I never truly understood if exponentiati ", "translatedText": "Bởi vì tôi đã dạo quanh một số diễn đàn dành cho giáo viên trước khi thực hiện bài giảng đặc biệt này và khi mọi người hỏi chủ đề nào khó dạy nhất trong môn toán trung học theo nghĩa là học sinh dường như gặp khó khăn nhất với nó, thì logarit là một trong những chủ đề khó nhất. những câu trả lời thường được chỉ ra rất thú vị và tôi có thể đoán có lẽ đó là vì có rất nhiều thuộc tính này mà cuối cùng bạn phải học, bạn biết đấy, vì vậy nếu chúng ta bỏ qua phần chúng ta sẽ đến thì bạn đã có tất cả những điều này những quy tắc trông giống như một đống đại số có thể khó nhớ và dễ nhầm lẫn mọi thứ trong đầu bạn và tôi nghĩ khi mọi người có, bạn biết đấy, những ký ức ác mộng về toán trung học là như thế nào và những gì logarit đã làm cho họ, những công thức cụ thể đó thường xuất hiện trong đầu họ và điều tôi muốn làm hôm nay là cố gắng nói về một công thức, cách suy nghĩ về chúng nhưng cũng chỉ ở cấp độ tổng hợp là nếu bạn đang dạy ai đó đại số, thì cái gì là những điểm đáng nhấn mạnh? ", "model": "google_nmt", "n_reviews": 0, @@ -144,7 +144,7 @@ "end": 876.76 }, { - "input": "What's the way to get it built in their intuitions? ", + "input": "ons inverse was nth rooting or logarithms if either one could be you know if either one could really be called that way can they that is such ", "translatedText": "Cách nào để xây dựng nó trong trực giác của họ? ", "model": "google_nmt", "n_reviews": 0, @@ -160,7 +160,7 @@ "end": 1020.62 }, { - "input": "oh, it has 3 zeros on it what's log of a million? ", + "input": "00 but the thing we don't know is what's in that exponent so to answer your ", "translatedText": "ồ, nó có 3 số 0 trên đó log của một triệu là bao nhiêu? ", "model": "google_nmt", "n_reviews": 0, @@ -224,7 +224,7 @@ "end": 1741.0 }, { - "input": "the log of 1000 times x is equal to 3 times the log of x and remember we're using the convention that it's base 10 log b. ", + "input": "n the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log, and I'll just re-emphasize it's bas ", "translatedText": "log của 1000 nhân x bằng 3 lần log của x và hãy nhớ rằng chúng ta đang sử dụng quy ước rằng nó có cơ số 10 log b. ", "model": "google_nmt", "n_reviews": 0, @@ -232,7 +232,7 @@ "end": 1755.54 }, { - "input": "log of 1000 times x equals log of x cubed c. ", + "input": "e 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the ", "translatedText": "log của 1000 lần x bằng log của x lập phương c. ", "model": "google_nmt", "n_reviews": 0, @@ -248,7 +248,7 @@ "end": 1765.5 }, { - "input": "log of 1000 times x equals 3 to the power of log of x and e. ", + "input": "0 of 10 which is of course 1 this expression you can think of as either ", "translatedText": "log của 1000 nhân x bằng 3 lũy thừa log của x và e. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 1772.26 }, { - "input": "none of the above and remember like I said earlier we should fully expect that all of those people at the beginning who said they understand logs well they're going to be answering immediately, they're going to be answering correctly but if you're someone who doesn't, don't let that intimidate you when you're looking at a problem like this one what I would encourage you to do is just plug in various powers of 10 and think in terms of the idea that the log function counts the number of zeros so I'll give you a little moment to think about that so I'll go ahead and grade it and as always if that's faster than what you're comfortable with know that it's only because I want to proceed forward with the lesson so in this case the correct answer comes out to be log of 1000 times x is the same as taking 3 plus the log of x and now let's think about that for a moment and like I said when you're just getting started with them I think the best thing to do is just be comfortable plugging in various numbers and the best numbers to plug in are the ones that are already powers of 10 so if you're asking something like log of 1000 times x well I don't know, let's just plug in something for x log of 1000 times 100 well we know how many zeros are going to be in the final answer here well 1000 times 100 is 100,000 we already intuitively have this idea that when we multiply 2 powers of 10 we're just taking the zeros, the 3 zeros from that 1000 the 2 zeros from that 100 and we're putting them next to each other so it should be 5 total zeros but if you really reflect not just on how did the number turn out but why did it turn out that way it was the 3 zeros from that 1000 plus the 2 zeros from that 100 which we could also write by saying the number of zeros in 1000 plus the number of zeros in 100 so this idea that a logarithm of the product of two things is the sum of the logarithms of those two things in the context of powers of 10 that's just communicating what's already a super intuitive idea for a lot of us if you take 2 powers of 10 and you multiply them you just take all of their zeros and kind of cram them onto each other so the way I've written things out here it's actually indicative of a slightly more general fact which is going to be our very first property of logarithms which is that if we take the log of A times B it equals the log of A plus the log of B now any time you see one of these logarithm rules if you find yourself squinting your eyes or you're a little bit confused by how to remember it just plug in examples I'm being redundant, I'm saying this a lot but it's because I think it's very easy to forget once you're swamped in the algebra itself and you're sitting on some kind of test and it's just got a lot of symbols to remind yourself you are okay to just plug in some numbers that's a fine thing to do and often it's a great way to yield intuition so in this case, saying log of A times B and breaking it apart we could just think, oh, that log of 100 times 1000 which is 5, there's 5 zeros in it breaks up in terms of the number of zeros in each given part great, wonderful so carrying that intuition further let's try another practice problem and again, if you know it, great, you'll be able to answer it fine but maybe think, not just what is the answer but how would I explain this answer to someone or how would I try to get a student to come to this answer on their own without me having to tell them what the answer is so there's two potential audience members there's those who are interested in the lesson itself and then those who are interested in the meta lesson so our question asks, again, which of the following is true? ", + "input": "counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n now with every given logarithm property that we have so in this case we just found one, log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found, raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of, they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log, if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a, the thing on the inside, corresponds to 10 to the x the output of the function whereas the entire thing itself, the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here, you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression, the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above, right the idea that when we're multiplying on the inside, that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside, multiplying the outputs of the function is the same as adding on the inside because each of these logs, like log a and log b is playing the role of the x and the y in the expression on the right okay, so with that, let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice, thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more, a moment to pull up which of the following is true? log base a of b is negative log base b of a log base a of b is 1 divided by log base b of a log base a of b is 1 minus log base b of a log base b of a is log base a of 1 divided by b or none of the above so it's asking what happens when we swap the base with what's sitting inside of the logarithm and I'll just give you a minute or two to answer that so let's do a couple more okay, so it seems like answers have kind of stabilized out there so let's go ahead and grade things and in this case, the correct answer of the choices we have comes out to be b that the log base a of b involves taking 1 divided by the log base b of a again, let's think this through both in terms of an example and then in terms of a more proofy, systematic reason why it should be true so if we're swapping our bases let's just start off with our good old friend log base 10 and let's plug in a nice power of 10 like 1000 counting the number of zeros, we get 3 so let's try swapping the bases and see what this should mean log base 1000 of 10 okay, well what is this ", "translatedText": "không có điều nào ở trên và hãy nhớ như tôi đã nói trước đó, chúng ta nên hoàn toàn mong đợi rằng tất cả những người lúc đầu đã nói rằng họ hiểu rõ nhật ký thì họ sẽ trả lời ngay lập tức, họ sẽ trả lời chính xác nhưng nếu bạn ai đó không, đừng để điều đó làm bạn sợ hãi khi bạn đang xem một vấn đề như thế này. Điều tôi khuyến khích bạn làm là chỉ cần nhập các lũy thừa khác nhau của 10 và nghĩ theo ý tưởng rằng hàm log đếm số không nên tôi sẽ cho bạn một chút thời gian để suy nghĩ về điều đó nên tôi sẽ tiếp tục và chấm điểm nó và như mọi khi nếu tốc độ đó nhanh hơn mức bạn cảm thấy thoải mái thì hãy biết rằng đó chỉ là vì tôi muốn tiếp tục với bài học nên trong trường hợp này, câu trả lời đúng sẽ là log của 1000 nhân x cũng giống như lấy 3 cộng với log của x và bây giờ chúng ta hãy suy nghĩ về điều đó một chút và như tôi đã nói khi bạn mới bắt đầu với họ, tôi nghĩ điều tốt nhất nên làm là cảm thấy thoải mái khi cắm vào nhiều số khác nhau và những số tốt nhất để cắm vào là những số đã có lũy thừa của 10, vì vậy nếu bạn hỏi điều gì đó như log của 1000 lần x thì tôi không' t biết, chúng ta hãy cắm cái gì đó vào x log của 1000 nhân 100. Chúng ta biết có bao nhiêu số 0 trong câu trả lời cuối cùng ở đây 1000 nhân 100 là 100.000, bằng trực giác chúng ta đã có ý tưởng này rằng khi chúng ta nhân 2 lũy thừa của 10 chúng ta chỉ lấy các số 0, 3 số 0 từ 1000 đó, 2 số 0 từ 100 đó và chúng ta đặt chúng cạnh nhau nên tổng sẽ là 5 số 0 nhưng nếu bạn thực sự suy ngẫm không chỉ về việc con số đã biến đổi như thế nào nhưng tại sao nó lại thành ra như vậy nó là 3 số 0 từ 1000 cộng với 2 số 0 từ 100 đó mà chúng ta cũng có thể viết bằng cách nói số 0 trong 1000 cộng với số 0 trong 100 nên ý tưởng này là logarit Tích của hai thứ là tổng logarit của hai thứ đó trong bối cảnh lũy thừa 10, điều này chỉ truyền đạt một ý tưởng siêu trực quan đối với nhiều người trong chúng ta nếu bạn lấy 2 lũy thừa của 10 và bạn nhân chúng lên, bạn chỉ cần lấy tất cả các số 0 của chúng và nhồi nhét chúng vào nhau nên cách tôi viết mọi thứ ở đây thực sự cho thấy một thực tế tổng quát hơn một chút sẽ là tính chất đầu tiên của logarit, đó là nếu chúng ta lấy log của A nhân B nó bằng log của A cộng với log của B bây giờ bất cứ khi nào bạn nhìn thấy một trong các quy tắc logarit này nếu bạn thấy mình nheo mắt hoặc hơi bối rối về cách ghi nhớ nó, chỉ cần đưa vào các ví dụ Tôi đang dư thừa, tôi nói điều này rất nhiều nhưng đó là vì tôi nghĩ nó rất dễ quên một khi bạn đắm chìm trong môn đại số và bạn đang làm một bài kiểm tra nào đó và nó chỉ có rất nhiều ký hiệu để nhắc nhở bản thân rằng bạn có thể chỉ cần nhập một số con số là điều tốt nên làm và thường thì đó là một cách tuyệt vời để mang lại trực giác nên trong trường hợp này, nói log của A nhân B và tách nó ra, chúng ta có thể nghĩ, ồ, cái đó log của 100 nhân 1000 bằng 5, có 5 số 0 trong đó chia nhỏ theo số lượng số 0 trong mỗi phần đã cho. Tuyệt vời, thật tuyệt vời vì vậy, hãy tiếp tục phát huy trực giác đó, chúng ta hãy thử một bài tập thực hành khác và một lần nữa, nếu bạn biết, tuyệt vời, bạn sẽ có thể trả lời tốt nhưng có thể hãy nghĩ, không chỉ câu trả lời là gì mà còn làm cách nào tôi có thể giải thích câu trả lời này cho ai đó hoặc làm cách nào để cố gắng thuyết phục học sinh tự đưa ra câu trả lời này mà không cần tôi phải nói họ câu trả lời là gì vậy nên có hai khán giả tiềm năng, những người quan tâm đến chính bài học và sau đó là những người quan tâm đến bài học meta nên câu hỏi của chúng tôi hỏi lại, điều nào sau đây là đúng? ", "model": "google_nmt", "n_reviews": 0, @@ -264,7 +264,7 @@ "end": 2060.22 }, { - "input": "a. ", + "input": "asking? maybe you thin ", "translatedText": "Một. ", "model": "google_nmt", "n_reviews": 0, @@ -272,7 +272,7 @@ "end": 2062.8 }, { - "input": "log of x to the n is equal to n times log of x b. ", + "input": "k of drawing the little triangle saying something like we know 1000 to something is equal ", "translatedText": "log của x mũ n bằng n lần log của x b. ", "model": "google_nmt", "n_reviews": 0, @@ -280,7 +280,7 @@ "end": 2068.7 }, { - "input": "log of x to the n is equal to log of x to the power n c. ", + "input": "to 10 1000 to the what equals 10? well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doin ", "translatedText": "log của x mũ n bằng log của x lũy thừa n c. ", "model": "google_nmt", "n_reviews": 0, @@ -288,7 +288,7 @@ "end": 2086.16 }, { - "input": "log of x to the n is equal to n plus log of x or d. ", + "input": "g the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 cor ", "translatedText": "log của x mũ n bằng n cộng với log của x hoặc d. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 2134.7 }, { - "input": "so the correct answer here is a, which it looks like 4,000 of you got congratulations, telling us that log of x to the power n is equal to n times log of x so, again, let's say that you're trying to teach this to someone or if you're trying to come to grips with what it means yourself I think a fine place to start is plugging something in and in this case, for log of x to the power n let's just try it with 100 to the power 3 and you could try it with other ones to see if the patterns you're doing actually work but if you're thinking it through not in terms of simply seeing what the answer is but trying to think of why the answer turned out that way sometimes one example will do because 100 cubed, we can think of that as taking well, that's 3 copies of 100 I'm taking 3 copies of 100 and when I multiply all that out and I think of log as counting the number of zeros we say, oh, it's going to be some number that just has 6 zeros on it that's what it means to take 100 times 100 times 100 I can just think of grouping all of those zeros together to get a million so this number is going to be 6 but if we think actually why was it 6 not just that's the number of zeros inside the million where that 6 came from is that we had 3 copies of that 100 and each of those 100 had 2 different zeros so that way it's a more general way you can think about it where if instead of taking 100 cubed we were looking at 1000 cubed or 1000 to the n or x to the power n you can think that it's whatever that value of n was the number of copies we were multiplying in times the number of well, let's see, it's not x times the number of zeros that were in whatever we substituted for x which in this case was 100 so if instead I had taken something like log of 10,000 to the power n this would be the same as taking n copies of that 10,000 counting the number of zeros in each one of them which is 4 so it would be n times 4 and of course the general property that most of you correctly answered is that you have this lovely little effect where when you see the log of something raised to a power that little power hops down in front of it and you just have log of what was on the inside now one of the maybe most important implications of that I don't know if you'd call it an implication or if you'd call it a restatement of the definition if I'm taking log and I'll just re-emphasize it's base 10 of 10 to the power n we can kind of think of that little n as hopping down in front and it becomes n times the log base 10 of 10 which is of course 1 this expression you can think of as either counting the number of zeros at the end or more generally it's asking 10 to the what equals 10 and the answer is simply 1 which is very reassuring because another way that you could go back and just read this original expression is saying 10 to the what equals 10 to the n oh well the answer is n ok now with every given logarithm property that we have so in this case we just found one log of x to the power n involves that n hopping in front there's always going to be a mirror image exponential property and that's another way that we can help to get ourselves a little bit of intuition for these so let me just cover up some of the future properties we're going to get to here try to hide where we're going what we just found raising something to the n that hops in front this corresponds to the exponential property that if I take 10 to the x and raise that whole thing to the power n that's the same as taking 10 to the n times x and this gets us to another intuition that you might have for logarithms which is they kind of they're like exponentiation turned inside out and here's what I mean by that the thing sitting on the inside of the log if I'm taking log of a you should be thinking of that as the whole outer expression for something that's exponential in this case the a the thing on the inside corresponds to 10 to the x the output of the function whereas the entire thing itself the log of a corresponds to what's on the inside over here just what's the exponent of the 10 so wherever you see a log expression here you should be thinking that plays the role of an exponent on the right side and every time you see an exponential the entire 10 to the x expression the whole outer component on the right side that corresponds to something that's sitting on the inside of one of the logs and we saw this above the idea that when we're multiplying on the inside that's adding on the outside well if logs kind of turn exponentials inside out that's telling us that multiplying on the outside multiplying the outputs of the function is the same as adding on the inside because each of these logs like log a and log b is playing the role of the x and the y in the expression on the right so with that let's keep playing let's just do a couple more of these and see how many of these properties that we can build up an intuition for so this last one, very nice thinking of exponents hopping down the next one is something that might look a little bit weird to those who are not necessarily familiar with logarithms but again, plug in some numbers to gain some intuition for it and we'll give it a little bit more a moment to pull up which of the following is true? ", + "input": "e of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a there so you can see just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? very interesting we've got a horse race between two so I will give you a moment to think this through while people are answering actually I have a little question for the audience so you know I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of 10 we could also do something like powers of 3 or if you're going from 1 to 3 to 9 to 27 to 81 all of these we could say that the log base 3 of these numbers just grows in nice little steps so log base 3 of 1, 3 to the what equals 1? the answer is 0 In general the log of 1 no matter the base will be 0. Log base 3 of 3, 3 to the what equals 3 is 1 Similarly log base 3 of 9 is 2 You might wonder what my question is, but it'll help to draw all of these out and For my own pleasure here. Let me just write out one more log base 3 of 81 is 4. Now I've heard that ostensibly if you ask a child's let's say around like 5 or 6 years old What number is halfway between 1 and 9? Okay, you say what number is halfway? Their instincts for how to answer are Logarithmic whereas our instincts tend to be more linear So we often think 1 and 9 you've got a bunch of evenly spaced numbers between them 2, 3, 4, 5, 6, 7, 8 And if you go right halfway in between you'll land on 5 But if you're thinking in terms of multiplicative growth where to get from 1 to 9 It's not a matter of adding a bunch of things But you're growing by a certain amount you grow by a factor of 3 then you grow by another factor of 3 supposedly a kid's natural instinct lines up with saying 3 and supposedly this also lines up with If you have anthropologists studying societies that haven't developed Counting systems and writing in the same way that modern societies have they'll answer 3 for this So my question for the audience if any of you watching right now have access to a small child Let's say in the range of 5 years old See if you can go ask them What number is halfway between 1 and 9 and if you can let us know on Twitter what the What the child says what their actual answer is? Because I I don't know why I'm just a little bit skeptical of whether that Actually pans out in practice. I understand this is not a super scientific way to do it asking people watching a YouTube live stream to Survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there Back to our question. This is the first one that doesn't seem to have a huge Consensus in one direction, but let's go ahead and grade it to see what the answer turns out to be Great okay, so 2400 of you correctly answered that it's none of the above okay that log of a plus B doesn't satisfy any of these nice properties And in general unless we're going to be working with Certain kinds of approximations especially when the natural log comes into play we mi ", "translatedText": "vậy câu trả lời đúng ở đây là a, có vẻ như 4.000 bạn đã nhận được lời chúc mừng vì đã cho chúng tôi biết rằng log của x lũy thừa n bằng n nhân log của x vậy, một lần nữa, giả sử rằng bạn đang cố gắng dạy điều này đối với ai đó hoặc nếu bạn đang cố gắng hiểu ý nghĩa của nó. Tôi nghĩ một nơi tốt để bắt đầu là cắm một cái gì đó vào và trong trường hợp này, để log x mũ n chúng ta hãy thử với lũy thừa 100 3 và bạn có thể thử nó với những cái khác để xem liệu các mô hình bạn đang làm có thực sự hiệu quả hay không nhưng nếu bạn suy nghĩ thấu đáo thì không chỉ đơn giản là xem câu trả lời là gì mà là cố gắng nghĩ xem tại sao câu trả lời lại thành ra như vậy đôi khi một ví dụ sẽ phù hợp vì 100 lập phương, chúng ta có thể coi đó là ổn, đó là 3 bản sao của 100. Tôi lấy 3 bản sao của 100 và khi tôi nhân tất cả số đó ra và tôi nghĩ log là đếm số số 0 mà chúng ta nói, ồ, nó sẽ là một số chỉ có 6 số 0 trên đó ý nghĩa của việc lấy 100 nhân 100 nhân 100 Tôi có thể nghĩ đến việc nhóm tất cả các số 0 đó lại với nhau để có một triệu nên con số này sẽ là 6 nhưng nếu chúng ta thực sự nghĩ tại sao nó không phải là 6 thì đó không chỉ là số số 0 trong một triệu mà số 6 đó đến từ việc chúng ta có 3 bản sao của 100 đó và mỗi bản sao trong số 100 đó có 2 số 0 khác nhau nên theo cách đó nó sẽ tổng quát hơn bạn có thể nghĩ về nó theo cách nào nếu thay vì lấy 100 lập phương, chúng ta xét 1000 lập phương hoặc 1000 lũy thừa n hoặc x lũy thừa n bạn có thể nghĩ rằng giá trị của n là bất kể số bản sao mà chúng ta đã nhân theo thời gian số ồ, xem nào, nó không phải là x nhân số số 0 trong bất cứ thứ gì chúng ta thay thế cho x mà trong trường hợp này là 100 vì vậy nếu thay vào đó tôi lấy log của 10.000 lũy thừa n thì nó sẽ giống nhau khi lấy n bản sao của 10.000 đó, đếm số số 0 trong mỗi số đó là 4 nên nó sẽ là n nhân 4 và tất nhiên đặc điểm chung mà hầu hết các bạn trả lời đúng là bạn có hiệu ứng nhỏ đáng yêu này khi bạn hãy xem nhật ký của một thứ gì đó được nâng lên thành một sức mạnh mà một sức mạnh nhỏ nhảy xuống phía trước nó và bạn vừa có nhật ký về những gì ở bên trong bây giờ một trong những ý nghĩa có thể là quan trọng nhất của điều đó. Tôi không biết liệu bạn có gọi nó không một hàm ý hoặc nếu bạn gọi nó là sự phát biểu lại định nghĩa nếu tôi đang ghi nhật ký và tôi sẽ nhấn mạnh lại nó là cơ số 10 của 10 lũy thừa n chúng ta có thể nghĩ về n nhỏ đó như nhảy xuống phía trước và nó trở thành n nhân log cơ số 10 của 10, tất nhiên là 1. biểu thức này bạn có thể nghĩ là đếm số số 0 ở cuối hoặc nói chung hơn là hỏi 10 mũ 10 và câu trả lời chỉ đơn giản là 1 điều này rất yên tâm vì một cách khác mà bạn có thể quay lại và chỉ cần đọc biểu thức ban đầu này là nói 10 mũ 10 với số ồ ồ câu trả lời bây giờ là không ổn với mọi thuộc tính logarit đã cho mà chúng ta có vì vậy trong trường hợp này chúng ta vừa tìm thấy một log của x lũy thừa n liên quan đến việc n nhảy về phía trước sẽ luôn có một thuộc tính hàm mũ của ảnh phản chiếu và đó là một cách khác mà chúng ta có thể giúp mình có được một chút trực giác về những điều này vì vậy hãy để tôi che đậy một số thuộc tính trong tương lai chúng ta sẽ đến đây cố gắng che giấu nơi chúng ta sẽ đến những gì chúng ta vừa tìm thấy nâng thứ gì đó lên n nhảy lên phía trước cái này tương ứng với thuộc tính hàm mũ nếu tôi lấy 10 mũ x và tăng toàn bộ lũy thừa n nó giống như lấy 10 mũ n nhân x và điều này đưa chúng ta đến một trực giác khác mà bạn có thể có đối với logarit, chúng giống như phép lũy thừa được đảo từ trong ra ngoài và đây là ý tôi khi nói rằng cái nằm ở bên trong nhật ký nếu tôi đang ghi nhật ký của một thì bạn nên nghĩ đó là toàn bộ biểu thức bên ngoài của một cái gì đó có dạng số mũ trong trường hợp này cái a cái ở bên trong tương ứng với 10 mũ x cái đầu ra của hàm trong khi toàn bộ bản thân nó thì nhật ký của a tương ứng với những gì ở bên trong ở đây, số mũ của số 10 là gì, vậy nên bất cứ khi nào bạn nhìn thấy biểu thức nhật ký ở đây, bạn nên nghĩ rằng nó đóng vai trò là số mũ ở bên phải bên và mỗi khi bạn thấy một số mũ của toàn bộ biểu thức 10 mũ x thì toàn bộ thành phần bên ngoài ở bên phải tương ứng với thứ gì đó nằm bên trong một trong các bản ghi và chúng ta đã thấy điều này ở trên ý tưởng rằng khi chúng ta nhân ở bên trong tức là cộng ở bên ngoài nếu logs kiểu biến đổi theo cấp số mũ từ trong ra ngoài, điều đó cho chúng ta biết rằng nhân ở bên ngoài nhân kết quả đầu ra của hàm cũng giống như cộng vào bên trong vì mỗi nhật ký này như log a và log b đang đóng vai x và y trong biểu thức bên phải nên hãy tiếp tục chơi, hãy thực hiện thêm một vài trong số này và xem có bao nhiêu trong số những thuộc tính này mà chúng ta có thể xây dựng trực giác cho cái cuối cùng này, ý nghĩ rất hay về việc số mũ nhảy xuống số tiếp theo là điều gì đó có thể trông hơi kỳ lạ đối với những người không nhất thiết phải quen thuộc với logarit nhưng một lần nữa, hãy cắm một số con số để có được trực giác về nó và chúng tôi sẽ cung cấp cho nó một chút thêm một chút thời gian để kéo lên điều nào sau đây là đúng? ", "model": "google_nmt", "n_reviews": 0, @@ -328,7 +328,7 @@ "end": 2589.32 }, { - "input": "well, if 10 cubed is 1000 that is the same thing as saying 10 is equal to 1000 raised to the 1 third doing the inverse here involves the multiplicative inverse of the exponent and the way that pans out is that it looks like 1 divided by 3 and that 3 corresponds to the log base 10 of 1000 it's 1 divided by the log base 10 of 1000 so more generally, you might guess based on this single example that when we swap the base with what's on the inside it corresponds to taking 1 divided by what's on the outside there and again, you can think this through in terms of looking at the corresponding exponential rule now what happened to my lovely little log and exponentials? ", + "input": "we happen to put in There's not a nice systematic reason coming there, so maybe you guess oh if the numbers A and B are very different It's kind of close to Whatever the maximum of them is But it's very bizarre and most importantly for the sake of the quiz If you just look at the options that it's giving you if you try this out with any particular numbers You'll find that none of those actually work So all is good sometimes you get something that looks like it's going to be a nice property But it doesn't end up being a nice property, and I also think that's important rather than just finding yourself Only working with the various You know log of A times B or log of X to the power N these things that have a nice rule Some ", "translatedText": "à, nếu 10 lập phương là 1000 thì điều đó cũng tương tự như nói 10 bằng 1000 nâng lên 1 phần ba làm phép nghịch đảo ở đây liên quan đến nghịch đảo nhân của số mũ và cách tính ra là nó trông giống như 1 chia cho 3 và 3 đó tương ứng với log cơ số 10 của 1000, nó bằng 1 chia cho log cơ số 10 của 1000 nên tổng quát hơn, bạn có thể đoán dựa trên ví dụ này rằng khi chúng ta hoán đổi cơ số với số ở bên trong thì nó tương ứng với việc lấy 1 chia bởi những gì ở bên ngoài kia, bạn có thể suy nghĩ điều này thông qua việc xem xét quy luật hàm mũ tương ứng bây giờ điều gì đã xảy ra với log nhỏ đáng yêu và số mũ của tôi? ", "model": "google_nmt", "n_reviews": 0, @@ -336,7 +336,7 @@ "end": 2629.5 }, { - "input": "wonderful so, again let's hide where some of the things some of the other properties that we'll get to here and I'll keep it in the same order I had it before here I was thinking that having it pre-written could keep me a little bit cleaner than usual but maybe it just involves playing this weird game of paper cutting shuffling around so what we just found, log base b of a if you swap those, it's the same as dividing by 1 what this corresponds to, off an exponential land is if you take b to some power and say that that equals a that's the same statement as saying that a to the inverse of that power equals b again, it's kind of helpful to take a moment and think of the logarithms as turning things inside out the expression log base b of a is playing the role of that x and the expression log base a of b is playing the role of whatever sits on top of the a and then symmetrically, the whole expression b to the power x is playing the role of the inside on the left, it plays the role of the a and the whole expression, a to the power of something plays the role of what's sitting inside the log base a so you can see, just by plugging in some examples and by corresponding it to the exponential rules we can already think through three different logarithm rules which if they were just handed down as pieces of algebra to be memorized you know, you could memorize them but it's very easy for them to kind of slip out of your head and it's also very easy to get frustrated by the task at hand but you might want to remind yourself that the reason we care about these sorts of things is understanding the rules of logarithms helps us do math in contexts where it's like a virus growing where from one day to the next, from one step to the next, things tend to grow multiplicatively understanding the rules of logarithms helps you to get a better feel for that kind of stuff so before we do a nice real world example of what that can look like let me just do one more quiz question in this vein to ask about properties of logarithms one last one before we transition to a little bit of a real world example get rid of what we had here and now, which of the following is true? ", + "input": "times you're out in the mathematical wild you're working on some problem You have a logarithm expression And it's adding things in the input and you want to be able to Have familiarity with the fact that that's kind of weird that you're not going to be able to simplify But if you you know if you hadn't thought about that before you might wonder Oh is there just some formula that I haven't seen before So with all of that let me go ahead and take a couple questions from the audience Before we transition to a different sort of example So it looks like Uma Sherma asks Can can the base be zero? That's an interesting question okay? Can the base of a logarithm be zero? Well in terms of our triangle we might think of that as saying You know zero to some kind of power X is equal to some other value Y Right this is something that we could write either by saying zero to the X equals Y Or we could write the same thing by saying log base zero of Y Is equal to X zero to the what equals X? Now the issue here is that zero to anything ends up being zero right so? If we're just going to be thinking of log base zero of Y for any other input Y. You know you want to input something like one or two or pi Anything you might want you're asking the question zero to the what is equal to one or two or pi? Or whatever number you might have there, and there's just not going to be an answer so at best You could try to say oh yes log of zero. It's a perfectly valid function It's only defined on the input zero, but even then you'd have trouble Trying to finagle what you want there because saying zero to the what equals zero. It's like anything anything applies to it So your arm is going to be twisted behind your back However you want to make that work and it corresponds to the fact that the exponential function with base zero Is entirely zero it doesn't it doesn't map numbers in a nice one-to-one fashion on to each other? So that's a great question can you have a log base zero now back to the idea of where these things come up in the real world One one example I kind of like is the Richter scale for earthquakes so the Richter scale Gives us a quantification for how strong an earthquake is okay? ", "translatedText": "thật tuyệt vời, một lần nữa, hãy giấu một số thứ trong số những thuộc tính khác mà chúng ta sẽ đến đây ở đâu và tôi sẽ giữ nó theo đúng thứ tự mà tôi đã có trước đây. Tôi nghĩ rằng việc viết sẵn nó có thể giúp tôi giữ được nó sạch sẽ hơn bình thường một chút nhưng có lẽ nó chỉ liên quan đến việc chơi trò chơi cắt giấy kỳ lạ này và xáo trộn xung quanh nên những gì chúng ta vừa tìm thấy, log cơ số b của a nếu bạn hoán đổi chúng, nó cũng giống như chia cho 1 những gì tương ứng với, ra một đất hàm mũ là nếu bạn lấy b lũy thừa nào đó và nói rằng nó bằng a đó là phát biểu tương tự như nói rằng a nghịch đảo của lũy thừa đó lại bằng b, sẽ rất hữu ích nếu bạn dành một chút thời gian và nghĩ về logarit như cách biến mọi thứ từ trong ra ngoài biểu thức log cơ số b của a đang đóng vai trò của x đó và biểu thức log cơ số a của b đang đóng vai trò của bất cứ thứ gì nằm trên a và sau đó đối xứng, toàn bộ biểu thức b lũy thừa x đang đóng vai trò vai trò của bên trong bên trái, nó đóng vai trò của a và toàn bộ biểu thức, a đối với sức mạnh của một cái gì đó đóng vai trò của những gì nằm bên trong cơ sở log a để bạn có thể thấy, chỉ bằng cách cắm vào một số ví dụ và bằng cách tương ứng nó với các quy tắc hàm mũ, chúng ta có thể nghĩ ra ba quy tắc logarit khác nhau mà nếu chúng chỉ được truyền lại dưới dạng các phần đại số để ghi nhớ thì bạn biết đấy, bạn có thể ghi nhớ chúng nhưng rất dễ bị trượt ra khỏi trí nhớ của bạn. đầu và cũng rất dễ nản lòng trước nhiệm vụ hiện tại nhưng bạn có thể muốn nhắc nhở bản thân rằng lý do chúng ta quan tâm đến những thứ này là việc hiểu các quy tắc logarit giúp chúng ta làm toán trong bối cảnh giống như virus đang phát triển ở đâu từ ngày này sang ngày khác, từ bước này sang bước tiếp theo, mọi thứ có xu hướng phát triển theo cấp số nhân. Việc hiểu các quy tắc logarit giúp bạn cảm nhận tốt hơn về loại nội dung đó, vì vậy trước khi chúng ta làm một ví dụ thực tế thú vị về những gì có thể nhìn thấy như để tôi làm thêm một câu hỏi trắc nghiệm theo chủ đề này để hỏi về các tính chất của logarit, câu hỏi cuối cùng trước khi chúng ta chuyển sang một chút ví dụ trong thế giới thực, hãy loại bỏ những gì chúng ta đã có ở đây và bây giờ, điều nào sau đây là đúng? ", "model": "google_nmt", "n_reviews": 0, @@ -344,7 +344,7 @@ "end": 2773.02 }, { - "input": "log of a plus b is the same as log of a plus log of b log of a plus b is equal to log of a times log of b log of a plus b is equal to one divided by log of a plus log of b or log of a plus b is equal to one divided by log of a times log of b or none of the above ah, and now we don't have as much consensus, do we? ", + "input": "And it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured And this is just a chart that comes from Wikipedia was a nine point five okay? And to appreciate just how insane that is it's worth looking at the relationship between What these numbers mean and then something like the equivalent amount of TNT some sort of measure of how ", "translatedText": "log của a cộng b giống log của a cộng log của b log của a cộng b bằng log của a nhân log của b log của a cộng b bằng một chia cho log của a cộng log của b hoặc log của a cộng b bằng một chia cho log của a nhân log của b hoặc không có cái nào ở trên ah, và bây giờ chúng ta không có nhiều sự đồng thuận, phải không? ", "model": "google_nmt", "n_reviews": 0, @@ -352,7 +352,7 @@ "end": 2796.84 }, { - "input": "very interesting, we've got a horse race between two so I will give you a moment to think this through while people are answering, actually I have a little question for the audience so, you know, I was just talking about how we might think in terms of multiplicative growth and that doesn't just have to be powers of ten, we could also do something like powers of three where if you're going from one to three to nine to twenty-seven to eighty-one, all of these we could say that the log base three of these numbers just grows in nice little steps so log base three of one, three to the what equals one, the answer is zero in general the log of one, no matter the base, will be zero log base three of three, three to the what equals three is one similarly log base three of nine is two ah, you might wonder what my question is, but it'll help to draw all of these out and for my own pleasure here, let me just write out one more log base three of eighty-one is four now, I've heard that ostensibly if you ask a child, let's say around like five or six years old what number is halfway between one and nine you say what number is halfway their instincts for how to answer are logarithmic whereas our instincts tend to be more linear so we often think one and nine, you've got a bunch of evenly spaced numbers between them two, three, four, five, six, seven, eight and if you go right halfway in between, you'll land on five but if you're thinking in terms of multiplicative growth where to get from one to nine, it's not a matter of adding a bunch of things but you're growing by a certain amount you grow by a factor of three, then you grow by another factor of three supposedly, a kid's natural instinct lines up with saying three and supposedly this also lines up with if you have anthropologists studying societies that haven't developed accounting systems and writing in the same way that modern societies have they'll answer three for this so, my question for the audience if any of you watching right now have access to a small child let's say, in the range of five years old see if you can go ask them what number is halfway between one and nine and if you can, let us know on Twitter what the child says what their actual answer is because I don't know why, I'm just a little bit skeptical of whether that actually pans out in practice I understand this is not a super scientific way to do it I'm not asking people watching a YouTube livestream to survey their own children and then tweet the answer but for my own sake it would be interesting to see some kind of validation there back to our question this is the first one that doesn't seem to have a huge consensus in one direction let's go ahead and grade it to see what the answer turns out to be great, okay, so 2,400 of you correctly answered that it's none of the above that log of a plus b doesn't satisfy any of these nice properties and in general, unless we're going to be working with certain kinds of approximations especially when the natural log comes into play we might talk about this next time adding the inputs of a logarithm is actually a very weird sensation it's a very weird thing to do and to get a sense of that weirdness, plug in some powers of ten if I ask you log of a plus b what you might start thinking is, okay, let me just plug in some examples like 10,000 and 100 and I ask myself, if I do this zero counting function of what's in that input how many zeros are in it? ", + "input": "much energy there is in it And then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and Why logarithms would be a natural way to describe this? So the key to focus on is as we're taking steps forward how much do things increase? So for example if we go from two Well in this case it doesn't show us where three is so maybe we think of Taking a step from two up to four which is kind of like taking two steps What does that do in terms of the amount of energy? Well it looks like it takes us from one metric ton of TNT Which is I guess a large bomb from World War two and it takes us up to a kiloton a thousand times as much Okay Which is a small a small atom bomb so just two steps on the Richter scale Going from an earthquake of magnitude two to an earthquake of magnitude four takes us from large bomb from World War two Up to the nuclear age, right? so that is noteworthy and The first clean step that we get is going from four to five at least in terms of what this chart is nicely showing us And evidently a single step up from four to five Corresponds to going from one kiloton to 32 kilotons And that was evidently the size of the city destroying bomb that land on Nagasaki So this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there Was an earthquake that was a 4.0 versus an earthquake that was a 5.0. It's easy to think yeah four and five Those are pretty similar numbers But evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from one to the next And going from two to four was evidently multiplying by about a thousand okay, and the only reason that's bigger is because here our chart wasn't showing what three was so we were taking two steps and You can verify for yourself that if you take a step of 32, and then you multiply by another 32 That's actually pretty close to a thousand So the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT Seems to suggest that something logarithmic is at play here, and it's a little interesting to just keep going here and say How how much does this grow partly? Because of the the world phenomena. It's describing. Yes, not a huge surprise that as we take another step It's multiplying by about 32 again But raining that in to our intuitions. That's the difference between 32 kilotons a small atom bomb And then one megaton which we might think of as not small atom bomb Nagasaki atom bomb Which I guess is 32 of the Nagasaki atom bombs For one megaton that is evidently the magnitude of the double string flat earthquake in Nevada, USA 1994 I didn't know what that was. Thanks Wikipedia in terms of frequencies by the way I Also looked these up evidently ones that are less than two those happen all the time There's like 8,000 of those per day, but as soon as we're in the realm of atom bombs things like 3.5 and 4 Those evidently also happen quite frequently somewhere on the earth. There's around 134 of those happening somewhere every day who knew But as we get even more intense into this 5 and 6 range which you know we're well above the atom bomb scale now we're only merely at around 2 per day and You know I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent Disruptions to the Earth's crust happening every day, but pr ", "translatedText": "rất thú vị, chúng ta có một cuộc đua ngựa giữa hai người nên tôi sẽ cho bạn một chút thời gian để suy nghĩ kỹ về điều này trong khi mọi người trả lời, thực ra tôi có một câu hỏi nhỏ dành cho khán giả nên bạn biết đấy, tôi chỉ đang nói về cách chúng ta có thể hãy nghĩ về sự tăng trưởng theo cấp số nhân và đó không nhất thiết phải là lũy thừa của mười, chúng ta cũng có thể làm điều gì đó như lũy thừa của ba trong đó nếu bạn đi từ một đến ba đến chín đến hai mươi bảy đến tám mươi mốt, tất cả trong số này chúng ta có thể nói rằng log cơ số ba của những số này chỉ tăng theo từng bước nhỏ nên log cơ số ba của một, ba lũy thừa bằng một, câu trả lời nói chung là bằng 0 log của một, bất kể cơ số, sẽ bằng 0 log cơ số ba của ba, ba mũ bằng ba là một tương tự log cơ số ba của chín bằng hai à, bạn có thể thắc mắc câu hỏi của tôi là gì, nhưng nó sẽ giúp tôi rút ra tất cả những điều này và vì niềm vui của riêng tôi đây, để tôi viết thêm một log cơ số ba của tám mươi mốt là bốn, tôi nghe nói rằng nếu bạn hỏi một đứa trẻ, giả sử khoảng năm hoặc sáu tuổi, con số nào nằm giữa một và chín bạn nói số nào bằng một nửa bản năng của họ về cách trả lời là logarit trong khi bản năng của chúng ta có xu hướng tuyến tính hơn nên chúng ta thường nghĩ một và chín, bạn có một loạt các số cách đều nhau giữa chúng hai, ba, bốn, năm, sáu , bảy, tám và nếu bạn đi đúng nửa chừng, bạn sẽ đạt được năm nhưng nếu bạn đang nghĩ về mặt tăng trưởng theo cấp số nhân thì phải đi đâu từ một đến chín, vấn đề không phải là cộng nhiều thứ mà là bạn 'đang phát triển với một mức nhất định, bạn tăng theo hệ số ba, sau đó bạn tăng theo hệ số ba khác được cho là, bản năng tự nhiên của một đứa trẻ phù hợp với việc nói ba và được cho là điều này cũng phù hợp nếu bạn có các nhà nhân chủng học nghiên cứu về các xã hội chưa' t đã phát triển hệ thống kế toán và cách viết giống như cách mà xã hội hiện đại áp dụng, họ sẽ trả lời ba câu hỏi cho vấn đề này, vì vậy, câu hỏi của tôi dành cho khán giả là liệu bất kỳ ai trong số các bạn đang xem hiện tại có khả năng tiếp cận với một đứa trẻ nhỏ, chẳng hạn như trong khoảng năm năm già xem bạn có thể hỏi chúng số nào nằm giữa một và chín không và nếu có thể, hãy cho chúng tôi biết trên Twitter đứa trẻ nói câu trả lời thực sự của chúng là gì vì tôi không biết tại sao, tôi chỉ hơi lo lắng một chút thôi hoài nghi liệu điều đó có thực sự thành công trong thực tế hay không. Tôi hiểu rằng đây không phải là một cách làm siêu khoa học. Tôi không yêu cầu mọi người xem buổi phát trực tiếp trên YouTube khảo sát con cái của họ và sau đó tweet câu trả lời nhưng vì lợi ích của tôi, điều đó sẽ rất thú vị để thấy một số loại xác thực nào đó quay lại câu hỏi của chúng tôi, đây là câu hỏi đầu tiên dường như không có sự đồng thuận lớn theo một hướng, hãy tiếp tục và chấm điểm để xem câu trả lời hóa ra là gì tuyệt vời, được rồi, vậy là 2.400 trong số các bạn đã trả lời đúng rằng không có điều nào ở trên mà log của a cộng b không thỏa mãn bất kỳ tính chất tốt đẹp nào và nói chung, trừ khi chúng ta sẽ làm việc với một số loại xấp xỉ nhất định, đặc biệt là khi log tự nhiên xuất hiện chúng ta có thể nói về vấn đề này vào lần tới việc cộng các đầu vào của logarit thực sự là một cảm giác rất kỳ lạ, đó là một việc rất kỳ lạ và để hiểu được sự kỳ lạ đó, hãy nhập một số lũy thừa của mười nếu tôi yêu cầu bạn ghi lại a cộng b điều bạn có thể bắt đầu nghĩ là, được rồi, hãy để tôi đưa vào một số ví dụ như 10.000 và 100 và tôi tự hỏi, nếu tôi thực hiện chức năng đếm số 0 này để biết những gì trong đầu vào đó thì có bao nhiêu số 0 trong đó? ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 3127.17 }, { - "input": "that's an interesting question okay, can the base of a logarithm be zero? ", + "input": "f earthquakes you can have things from what happens Just all the time around the earth the size of a large hand grenade And you wan ", "translatedText": "đó là một câu hỏi thú vị được chứ, cơ số của logarit có thể bằng 0 không? ", "model": "google_nmt", "n_reviews": 0, @@ -376,7 +376,7 @@ "end": 3134.19 }, { - "input": "well in terms of our triangle we might think of that as saying you know, zero to some kind of power x is equal to some other value y this is something that we could write either by saying zero to the x equals y or we could write the same thing by saying log base zero of y is equal to x zero to the what equals x now the issue here is that zero to anything ends up being zero right, so if we're just going to be thinking of log base zero of y for any other input y you know, you want to input something like one or two or pi anything you might want, you're asking the question zero to the what is equal to one or two or pi or whatever number you might have there and there's just not going to be an answer so at best you could try to say oh yes, log of zero, it's a perfectly valid function it's only defined on the input zero but even then you'd have trouble trying to finagle what you want there because saying zero to the what equals zero it's like anything applies to it so your arm is going to be twisted behind your back however you want to make that work and it corresponds to the fact that the exponential function with base zero is entirely zero it doesn't map numbers in a nice one to one fashion onto each other so that's a great question, can you have a log base zero now back to the idea of where these things come up in the real world one example I kind of like is the Richter scale for earthquakes so the Richter scale gives us a quantification for how strong an earthquake is and it can be anything from very small numbers up to very large numbers like I think the largest earthquake ever measured and this is just a chart that comes from Wikipedia was a 9.5 and to appreciate just how insane that is it's worth looking at the relationship between what these numbers mean and then something like the equivalent amount of TNT some sort of measure of how much energy there is in it and then what we can try to do here is see if we can get an expression for the Richter scale number in terms of the amount of energy and why logarithms would be a natural way to describe this so the key to focus on is as we're taking steps forward how much do things increase so for example if we go from two well in this case it doesn't show us where three is so maybe we think of taking a step from two up to four which is kind of like taking two steps what does that do in terms of the amount of energy well it looks like it takes us from one metric ton of TNT which is I guess a large bomb from World War II and it takes us up to a kiloton a thousand times as much which is a small atom bomb so just two steps on the Richter scale going from an earthquake of magnitude 2 to an earthquake of magnitude 4 takes us from large bomb from World War II up to the nuclear age so that is noteworthy and the first clean step that we get is going from 4 to 5 at least in terms of what this chart is nicely showing us and evidently a single step up from 4 to 5 corresponds to going from 1 kiloton to 32 kilotons and that was evidently the size of the city destroying bomb that landed on Nagasaki so this is maybe one thing that can be counterintuitive about logarithmic scales if you're just hearing in the news the difference between oh there was an earthquake that was a 4.0 versus an earthquake that was a 5.0 it's easy to think yeah 4 and 5 those are pretty similar numbers but evidently in terms of TNT amounts that corresponds to multiplying by 32 to get from 1 to the next and going from 2 to 4 was evidently multiplying by about a thousand and the only reason that's bigger is because here our chart wasn't showing what 3 was so we were taking two steps and you can verify for yourself that if you take a step of 32 and then you multiply by another 32 that's actually pretty close to a thousand so the idea that additive steps on the Richter number correspond to multiplicative steps in the TNT seems to suggest that something logarithmic is at play here and it's a little interesting to just keep going here and say how much does this grow partly because of the world phenomena it's describing yes not a huge surprise that as we take another step it's multiplying by about 32 again but reining that in to our intuitions that's the difference between 32 kilotons a small atom bomb and then one megaton which we might think of as not small atom bomb, Nagasaki atom bomb which I guess is 32 of the Nagasaki atom bombs for one megaton that is evidently the magnitude of the double string flat earthquake in Nevada USA 1994 I didn't know what that was, thanks Wikipedia in terms of frequencies by the way I also looked these up evidently ones that are less than two, those happen all the time there's like 8000 of those per day but as soon as we're in the realm of atom bombs things like 3.5 and 4 those evidently also happen quite frequently somewhere on the earth there's around 134 of those happening somewhere every day who knew? ", + "input": "t that to be on your scale and something to think about ranging all the way up to you know the largest disruption that we've Seen in human history right and in order to have that in a way that you're not just Writing a whole bunch of different digits in your numbers for one case and a whole bunch of different a smaller number of digits For your number in another case It's nice to take logarithms And then just put that on a single scale that basically puts squishes those numbers between 0 and 10 You see something very similar going on with the decibel scale for music that one actually Works a little bit differently where every time you take a step up of 10 decibels that corresponds to multi multiplying by 10 So rather than a step of 1 multiplying by 10 It's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy But the idea is the same that if you're listening to a sound that's 50 decibels for 60 decibels It's a lot quieter in terms of the energy being transmitted and going from you know What would it be 60 to 70 or 70 to 80? Those steps you know from 60 up to 80 that involves multiplying the amount of energy per square area By a factor of 100 so every time you see a logarithmic scale Know in your mind that that means whatever it's referring to under the hood grows by a huge amount This is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so How might you describe a relationship like this where every time you grow the Richter scale number by 1 you're multiplying by 32 Well, we could think in terms of a log with base 32. I Could say if I take the log of I'm just gonna call our the number for the Richter scale. I might think of this as log base 32 and That's going to correspond to No, no, no, I'm doing this wrong That's not the thing that's logged We take the log base 32 of the big number of the the TNT number something that was like You know one Gigaton or one megaton It's one million Tons The log base 32 that should correspond to the Richter scale number But there might be some kind of offset. So we might say that there's some kind of Constant s that we're adding to this Richter scale number and this expression is exactly the same Excuse me for going off the bottom there. This expression is exactly the same as saying 32 to the power of some offset times our Richter scale number, which is the same as taking you know 32 to that offset which itself is just some big constant times 32 to the Richter scale number so you might think of this as just being some constant Times 32 to the power of the number you see So this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TNT amount that you see as You increase that R step by step you're multiplying by 32 But another way of communicating the exact same fact is to take the log base 32 of whatever that amount is Alright now the next thing I want to talk about is how we don't always have to Worry about how to compute logs of different bases and it's a little weird here that we were talking about log base 32 I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for Computational purposes or for also thinking about how these things grow if you have one Log if you're able to compute one type of log whether that's base 10 base 2 base e you can compute pretty much anything else That you want Okay now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most I don't know. This is this is a halfway reasonable question. This should be nice This is just going to get us prepared to translate from base 2 contexts to base 10 contexts and it's also a good intuition for understanding powers of 2 to have in general The relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of Well, you'll see what I mean. they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024 which is approximately 1000, okay, so you can if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? log base 2 of 10 is approximately 0.3 log base 2 of 10 is approximately, sorry, log base 10 of 2 is approximately 0.3 log base 2 of 10 is approximately 1 third or log base 10 of 2 is approximately 1 third okay, which of these is closest to being true based on the fact that 2 to the 10th is essentially 1000? I'll give you a little moment for that log base 2 of 10 is approximately 0.3 interesting that we've got kind of a split on this one so I'm wondering if they're going to be numerically pretty similar or if they're going to be conceptually similar or if there's even a difference be ", "translatedText": "à, xét theo tam giác của chúng ta, chúng ta có thể nghĩ về điều đó như nói rằng bạn biết đấy, 0 lũy thừa nào đó x bằng một giá trị nào đó y đây là thứ mà chúng ta có thể viết bằng cách nói 0 với x bằng y hoặc chúng ta có thể viết điều tương tự bằng cách nói log cơ số 0 của y bằng x 0 mũ bằng x bây giờ vấn đề ở đây là số 0 với mọi thứ cuối cùng đều bằng 0, vậy nên nếu chúng ta chỉ đang nghĩ về log cơ số 0 của y cho bất kỳ đầu vào nào khác y bạn biết, bạn muốn nhập cái gì đó như một hoặc hai hoặc pi bất cứ thứ gì bạn muốn, bạn đang đặt câu hỏi bằng 0 cho số bằng một hoặc hai hoặc pi hoặc bất kỳ số nào bạn có thể có ở đó và sẽ không có câu trả lời nào nên tốt nhất bạn có thể thử nói ồ vâng, log của số 0, đó là một hàm hoàn toàn hợp lệ, nó chỉ được xác định trên số 0 đầu vào nhưng ngay cả khi đó bạn cũng sẽ gặp khó khăn khi cố gắng hoàn thiện những gì bạn muốn ở đó bởi vì nói số 0 với số bằng 0 nó giống như bất cứ điều gì áp dụng cho nó nên cánh tay của bạn sẽ bị xoắn ra sau lưng tuy nhiên bạn muốn làm cho điều đó hoạt động và nó tương ứng với thực tế là hàm mũ với cơ số 0 hoàn toàn bằng 0 nó không ánh xạ các số theo kiểu đẹp từ một đến một với nhau nên đó là một câu hỏi hay, bây giờ bạn có thể có cơ sở nhật ký bằng 0 để quay lại ý tưởng về nơi những thứ này xuất hiện trong thế giới thực, một ví dụ mà tôi khá thích là thang Richter cho động đất nên thang Richter cho chúng ta định lượng về cường độ của một trận động đất và nó có thể là bất cứ thứ gì từ những con số rất nhỏ đến những con số rất lớn như tôi nghĩ trận động đất lớn nhất từng đo được và đây chỉ là biểu đồ đến từ Wikipedia là số 9.5 và để đánh giá cao mức độ điên rồ của nó, đáng để xem xét mối quan hệ giữa ý nghĩa của những con số này và sau đó là thứ gì đó giống như lượng TNT tương đương, một loại thước đo nào đó về lượng năng lượng có trong nó và sau đó là những gì chúng ta có thể cố gắng làm ở đây là xem liệu chúng ta có thể có được biểu thức cho số thang Richter dưới dạng lượng năng lượng hay không và tại sao logarit sẽ là cách tự nhiên để mô tả điều này nên điều mấu chốt cần tập trung là khi chúng ta đang thực hiện các bước tiến về phía trước thì mọi thứ sẽ tăng bao nhiêu vì vậy, ví dụ: nếu chúng ta đi từ hai giếng trong trường hợp này, nó không cho chúng ta thấy ba ở đâu nên có thể chúng ta nghĩ đến việc thực hiện một bước từ hai lên bốn, điều này giống như thực hiện hai bước điều đó có tác dụng gì xét về mặt lượng năng lượng à, có vẻ như nó tiêu tốn của chúng ta từ một tấn TNT, tôi đoán là một quả bom lớn từ Thế chiến thứ hai và nó tiêu tốn của chúng ta tới một kiloton gấp một nghìn lần một quả bom nguyên tử nhỏ nên chỉ cần hai bước trên thang Richter, từ một trận động đất có cường độ 2 đến một trận động đất có cường độ 4 sẽ đưa chúng ta từ một quả bom lớn từ Thế chiến II cho đến thời đại hạt nhân, điều đó rất đáng chú ý và bước rõ ràng đầu tiên mà chúng ta nhận được là đi từ 4 đến 5 lúc ít nhất là về mặt những gì biểu đồ này cho chúng ta thấy một cách độc đáo và rõ ràng một bước tăng từ 4 lên 5 tương ứng với việc tăng từ 1 kiloton lên 32 kiloton và đó rõ ràng là kích thước của quả bom hủy diệt thành phố rơi xuống Nagasaki nên đây có thể là một điều có thể phản trực giác về thang đo logarit nếu bạn vừa nghe tin tức về sự khác biệt giữa ồ có một trận động đất cấp 4.0 so với trận động đất cấp 5.0, thật dễ dàng để nghĩ rằng 4 và 5 là những con số khá giống nhau nhưng rõ ràng xét về lượng TNT tương ứng với việc nhân với 32 để từ 1 sang số tiếp theo và đi từ 2 đến 4 rõ ràng là nhân với khoảng một nghìn và duy nhất Lý do nó lớn hơn là vì ở đây biểu đồ của chúng ta không hiển thị số 3 là bao nhiêu nên chúng ta đang thực hiện hai bước và bạn có thể tự xác minh rằng nếu bạn bước một bước là 32 rồi nhân với một số 32 khác thì thực ra nó gần bằng một nghìn vậy ý tưởng rằng các bước cộng của số Richter tương ứng với các bước nhân trong TNT dường như gợi ý rằng có thứ gì đó logarit đang diễn ra ở đây và sẽ hơi thú vị nếu cứ tiếp tục ở đây và nói nó tăng lên bao nhiêu một phần là do các hiện tượng thế giới. mô tả có, không có gì đáng ngạc nhiên lớn khi chúng ta thực hiện thêm một bước nữa, nó sẽ nhân lên khoảng 32 lần nữa nhưng hãy kiểm soát lại điều đó theo trực giác của chúng ta rằng đó là sự khác biệt giữa 32 kiloton của một quả bom nguyên tử nhỏ và sau đó là một megaton mà chúng ta có thể nghĩ là không phải bom nguyên tử nhỏ, Bom nguyên tử Nagasaki mà tôi đoán là 32 quả bom nguyên tử Nagasaki cho một megaton, rõ ràng là độ lớn của trận động đất phẳng chuỗi đôi ở Nevada Hoa Kỳ năm 1994. Tôi không biết đó là gì, nhân tiện xin cảm ơn Wikipedia về tần số. cũng đã tra cứu những cái này rõ ràng là những cái nhỏ hơn hai, những cái đó xảy ra mọi lúc, có khoảng 8000 cái mỗi ngày nhưng ngay khi chúng ta ở trong vương quốc bom nguyên tử thì những thứ như 3.5 và 4 những điều đó rõ ràng cũng xảy ra khá thường xuyên ở đâu đó trên trái đất, có khoảng 134 trong số đó xảy ra ở đâu đó mỗi ngày ai mà biết được? ", "model": "google_nmt", "n_reviews": 0, @@ -384,7 +384,7 @@ "end": 3480.95 }, { - "input": "but as we get even more intense into this 5 and 6 range which were well above the atom bomb scale now we're only merely at around 2 per day and I'm sure that a geologist could come in and explain why we all shouldn't be super worried about the fact that there's two atom bomb equivalent disruptions to the earth's crust happening every day but presumably it's particularly rare for those to be concentrated on some spot like a city where lots of people live now just verifying our thought that each step involves a growth of 32 let's look at what the step from 6 up to 7 looks like and here it's giving us lots more examples in between maybe giving the illusion that that's a bigger step than it actually is and indeed that's the difference between 1 megaton and 32 megatons so that's multiplying by 32 one of the things I found most interesting on this chart by the way was look at how far we have to go before we get to the largest nuclear weapon that's ever actually been tested this was height of the cold war the Tsar bomb that was 50 megatons and I believe they actually had original plans to have a 100 megaton bomb but talked themselves down from that 50 megatons, we're talking start off at that 32 kilotons of the Nagasaki bomb multiply by 32 to get a megaton multiply by another 32 so we're talking about a thousand times the strength of the World War II ending explosion and you're still not at the 50 megatons of what humanity is capable of and that is evidently the Java earthquake of Indonesia so 7.0 is not just a little bit bigger than 6.0, it's a lot bigger and the point here of course is just that when you have a scale giving you multiplicative increases it's worth appreciating that what look like small steps can actually be huge steps in terms of the energy implied or the absolute values implied here so when we're thinking about the fact that there was ever a 9.5 that actually seems absurd given that it's only in the 7.0 range that we're talking about the largest thermonuclear weapon ever put out and this is indicative of one area where logarithms tend to come about it's when humans want to create a scale for something that accounts for a hugely wide variance in how big things can be so in the case of size of earthquakes you can have things from what happens just all the time around the Earth, the size of a large hand grenade and you want that to be on your scale and something to think about ranging all the way up to the largest disruption that we've seen in human history and in order to have that in a way that you're not just writing a whole bunch of different digits in your numbers for one case and a whole bunch of different, a smaller number of digits for your number in another case it's nice to take logarithms and then just put that on a single scale that basically squishes those numbers between 0 and 10 you see something very similar going on with the decibel scale for music that one actually works a little bit differently where every time you take a step up of 10 decibels that corresponds to multiplying by 10 so rather than a step of 1 multiplying by 10, it's a step of 10 that multiplies by 10 so that kind of makes the math of it a little bit screwy but the idea is the same, that if you're listening to a sound that's 50 decibels versus 60 decibels it's a lot quieter in terms of the energy being transmitted and going from, what would it be, 60 to 70 or 70 to 80 those steps, from 60 up to 80, that involves multiplying the amount of energy per square area by a factor of 100 so every time you see a logarithmic scale, know in your mind that that means whatever it's referring to under the hood grows by a huge amount this is again why we saw a lot of logarithmic scales used to describe the coronavirus outbreak so how might you describe a relationship like this where every time you grow the Richter scale number by 1, you're multiplying by 32 well, we could think in terms of a log with base 32 I could say if I take the log of, I'm just going to call r, the number for the Richter scale I might think of this as log base 32 and that's going to correspond to, no no no, I'm doing this wrong that's not the thing that's logged we take the log base 32 of the big number, of the TMT number, something that was like 1 megaton it's 1 million tons the log base 32, that should correspond to the Richter scale number but there might be some kind of offset, so we might say that there's some kind of constant s that we're adding to this Richter scale number and this expression is exactly the same, excuse me for going off the bottom there this expression is exactly the same as saying 32 to the power of some offset times our Richter scale number which is the same as taking 32 to that offset, which itself is just some big constant, times 32 to the Richter scale number so you might think of this as just being some constant times 32 to the power of the number you see so this way of writing it really emphasizes the exponential growth of it that if this is what corresponds to the TMT amount that you see, as you increase that r step by step you're multiplying by 32 but another way of communicating the exact same fact is to take the log base 32 of whatever that amount is alright now the next thing I want to talk about is how we don't always have to worry about how to compute logs of different bases it's a little weird here that we were talking about log base 32, I referenced earlier how mathematicians really like to have a log with base e computer scientists really like to have a log with base 2 and it turns out for computational purposes or for also thinking about how these things grow if you have one log, if you're able to compute one type of log, whether that's base 10, base 2, base e you can compute pretty much anything else that you want now to get our intuitions in that direction, let's turn back to our quiz and go to the next question and I believe that this question is the most, I don't know, this is a halfway reasonable question, this should be nice this is just going to get us prepared to translate from base 2 context to base 10 context and it's also a good intuition for understanding powers of 2 to have in general the relationship that it has with powers of 10 because it's this lovely kind of coincidence of nature that these two sort of well you'll see what I mean, they play nicely with each other so our question asks, given the fact that 2 to the 10th is 1024, 1024, which is approximately 1000 so if you're being a little bit loose with your numbers and you're just making approximations 2 to the 10th, basically 1000, which of the following is closest to being true? ", + "input": "tween those two so since answers keep rolling in, I'm going to give this a little bit more time so anyone at home watching, hopefully you already have a pencil and paper out to be noodling through these yourself, that is the spirit of the lectures that we're doing if you don't, now is the time to take out a pencil and paper and see if you can think this one through and write it out some of the problems that we're going to build to here definitely will require pencil and paper so now is as good a time as any and if you're watching this in the future, even if you can't participate in the live poll I really do think it's a lot of fun to kind of throw your own hat into the mix even if it's not going to contribute to one of the numbers that you see growing on the screen I'll give you a little bit more time here as the answers seem to continue rolling in so now is the time to take out your pencil and paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper and write it out so now is the time to write your own paper And then I'm ready to move on to the google Okay, so I'll go ahead and grade it now and let's see how people did on this one. So the correct answer is B, Which is that the log base 10 of 2 is around 1 third so that's good They're very numerically similar right that It's either 0.3 or around 1 third which is 0.3 3 3 3 repeating But the question was asking which one is closest to being true, and let's see how we can think about this so It points out that you have a power of 2 which is 1024 awfully close to a power of 10 about 10 cubed and the question is how we can leverage this to understand something like Log base 2 of 10 or log base 10 of 2 as we saw earlier those are just the reciprocals of each other So what does this mean if log base 2 of 10 is equal to X? That's the same thing as saying 2 to the X is equal to 10 right. It's asking us 2 to the what equals 10 So what we have here is an expression 10 cubed is approximately equal to 2 to the 10th so what I might write out is we know that 2 to the 10th Instead of writing it as a 10. I'm going to write that 10 as 2 to the X Where X is the number such that 2 to the X is approximate is equal to 10? So if that cubed is the same as 2 to the 10th This is I'll just write out the full details the same as saying 2 to the 3 X is equal to 2 to the 10th and Exponentiation is a nice one-to-one function, so it's okay to just Say whatever is going on in the input if the outputs are the same the inputs must also be the same You can't do that with every function people seem to think you can do that with any function But you just can't and what that means is that? X is about X is about 10 thirds, okay which Ah Great so log base 2 of 10 is about 10 thirds, so if we looked at our answers though That's not actually any of the options. We've got various things asking log base 2 of 10 being around 0.3 or 1 third so it looks like instead we should try to re-express this as log base 10 of 2 and Well enough what we saw earlier is that log base 2 of 10 we could also say log base 10 of 2 is Just 1 over that amount 1 over X and you can see this pretty easily by writing 2 is equal to 10 to the 1 over X if we're asking 10 to the what equals 2 the answer is 1 over what we just got there, so Log base 10 of 2 is 1 divided by this amount Which is 3 tenths? Which is 0.3 great So this is kind of a nice constant to think about because there's this wonderful pattern that happens when we're looking at powers of 2 so if I ask What is the log base 2 of? 1,000 Like we just saw it's approximately the case that 2 to the power 10 Is equal to a thousand and because we're doing things at logs I'm just going to be writing it in that way a log 2 of a thousand is approximately 10 Similarly log base 2 of a million Well, let's see if we have to multiply 2 by itself about 10 times to get to a thousand we should have to multiply it by itself around 20 times to get up to a million and Indeed log base 2 of a million is approximately 20 It's a little bit smaller, but this is kind of a nice approximation to have in your mind And then similarly you'll see why I'm writing out this as a pattern in just a moment if we wanted to go up to a billion Saying how many times do I have to multiply 2 by itself to get to a billion This is about 30 And any computer scientist out there who's thought about you know just how much as a kilobyte or a megabyte or a gigabyte? they'll be familiar with the idea that powers of 2 are nice and close to these powers of 10. Or more specifically, powers of 1000. Now what I want to do is just write all of the same things with log base 10, not approximately equal to, this is actually equal to 3. Log base 10 of 1000 is equal to 3. Log base 10, well you tell me, what's log base 10 of a million? It's equal to 3 Log base 10 well you tell me what's log base 10 of a million It's counting the number of zeros it ends up being about 6 and log base 10 of a billion Counting the number of zeros it ends up being 9 Now the reason I wanted to write all of this out is to just emphasize an interesting pattern here Which is we're just growing by these increments right as we go from a thousand to a million to a billion with log base 2 We're stepping up by steps of 10 But when we're playing the same game with 10 we're stepping up by these increments of 3 So there's this nice relationship and in fact for all of them to go from log base 2 to log base 10 It seems like we're just multiplying by 0.3. So 10 times 0.3 is 3 20 we scale down by that same amount 30 we scale down by that same amount Okay now this is an intuition worth remembering if you have Your numbers described with one base. It's basically the same as describing them with another base, but there's some rescaling constant Okay, and then the next question is going to start getting us at that direction But it's going to be framed in a way that just looks like a whole pile of algebra And again, I will encourage you to plug in numbers if you want to to gain a little intuition for it So as our third to last question, this will be a long lecture We have which of the following is true and then just a whole pile of Various possible ways to combine log base C of B times log base C of A Does that equal log base B log base B of A and rather than me reading them out to you? I'll just let you look through them plug in some numbers I'll give you I'll give you a meaningful time on this one because it's not it's not obvious unless you're already familiar with logarithms and It's worth thinking throu ", "translatedText": "nhưng khi chúng ta thậm chí còn trở nên mạnh mẽ hơn ở phạm vi 5 và 6, cao hơn nhiều so với quy mô bom nguyên tử, giờ đây chúng ta chỉ ở mức khoảng 2 quả mỗi ngày và tôi chắc chắn rằng một nhà địa chất có thể đến và giải thích lý do tại sao tất cả chúng ta nên làm như vậy' Tôi rất lo lắng về thực tế là có hai vụ gián đoạn tương đương với bom nguyên tử đối với lớp vỏ trái đất xảy ra hàng ngày nhưng có lẽ đặc biệt hiếm khi những vụ nổ đó tập trung ở một nơi nào đó như một thành phố nơi có nhiều người sinh sống chỉ để xác minh suy nghĩ của chúng tôi rằng mỗi bước liên quan đến sự tăng trưởng 32, chúng ta hãy xem bước từ 6 lên 7 trông như thế nào và ở đây nó cho chúng ta nhiều ví dụ hơn ở giữa, có thể tạo ra ảo tưởng rằng đó là một bước lớn hơn thực tế và thực sự đó là sự khác biệt giữa 1 megaton và 32 megaton, vậy nó nhân với 32, một trong những điều tôi thấy thú vị nhất trên biểu đồ này là xem chúng ta phải đi bao xa trước khi có được vũ khí hạt nhân lớn nhất từng được thử nghiệm, đây là đỉnh cao của chiến tranh lạnh quả bom Sa hoàng có sức công phá 50 megaton và tôi tin rằng họ thực sự có kế hoạch ban đầu là chế tạo một quả bom 100 megaton nhưng đã tự thuyết phục họ từ chối 50 megaton đó, chúng ta đang nói chuyện bắt đầu với 32 kiloton của quả bom Nagasaki nhân với 32 để có được một megaton nhân với 32 nữa nên chúng ta đang nói về sức mạnh gấp hàng nghìn lần của vụ nổ kết thúc Thế chiến II và bạn vẫn chưa đạt đến mức 50 megaton mà nhân loại có thể làm được và đó rõ ràng là trận động đất Java ở Indonesia nên 7 . 0 không chỉ lớn hơn 6 một chút. 0, nó lớn hơn rất nhiều và tất nhiên vấn đề ở đây chỉ là khi bạn có một thang đo mang lại cho bạn mức tăng theo cấp số nhân thì thật đáng đánh giá cao rằng những gì trông giống như những bước nhỏ thực sự có thể là những bước lớn về mặt năng lượng ngụ ý hoặc các giá trị tuyệt đối ngụ ý ở đây vì vậy khi chúng ta nghĩ về thực tế là luôn có số 9.5 điều đó thực sự có vẻ vô lý vì nó chỉ có ở số 7.0 mà chúng ta đang nói về loại vũ khí nhiệt hạch lớn nhất từng được sử dụng và đây là dấu hiệu cho thấy một lĩnh vực mà logarit có xu hướng xuất hiện khi con người muốn tạo ra thang đo cho thứ gì đó có sự khác biệt cực kỳ lớn về mức độ lớn của những thứ có thể Vì vậy, trong trường hợp quy mô của trận động đất, bạn có thể có những thứ từ những gì xảy ra thường xuyên trên khắp Trái đất, kích thước của một quả lựu đạn cầm tay lớn và bạn muốn nó ở quy mô của bạn và điều gì đó để suy nghĩ về mọi thứ đến sự gián đoạn lớn nhất mà chúng ta từng thấy trong lịch sử loài người và để đạt được điều đó theo cách mà bạn không chỉ viết cả đống chữ số khác nhau trong các số của mình cho một trường hợp và cả một loạt các chữ số khác nhau, một số nhỏ hơn các chữ số cho số của bạn trong một trường hợp khác, thật tuyệt khi lấy logarit và sau đó chỉ cần đặt nó trên một thang đo duy nhất về cơ bản đặt các số đó trong khoảng từ 0 đến 10, bạn sẽ thấy điều gì đó rất giống đang diễn ra với thang đo decibel cho âm nhạc mà thực tế nó hoạt động một chút hơi khác một chút là mỗi khi bạn tăng một bước 10 decibel tương ứng với nhân với 10, thay vì bước 1 nhân với 10, thì đó là bước 10 nhân với 10 nên điều đó làm cho phép toán của nó hơi khác một chút hơi rắc rối một chút nhưng ý tưởng thì giống nhau, rằng nếu bạn đang nghe một âm thanh có cường độ 50 decibel so với 60 decibel thì nó sẽ yên tĩnh hơn rất nhiều xét về mặt năng lượng được truyền đi và đi từ đó, nó sẽ là bao nhiêu, 60 đến 70 hay 70 đến 70 đến 80 bước đó, từ 60 đến 80, bao gồm việc nhân lượng năng lượng trên một diện tích vuông với hệ số 100, vì vậy, mỗi khi bạn nhìn thấy thang đo logarit, hãy biết trong đầu rằng điều đó có nghĩa là bất cứ điều gì nó đề cập đến sẽ tăng lên một số lượng lớn, đây một lần nữa là lý do tại sao chúng ta thấy rất nhiều thang đo logarit được sử dụng để mô tả sự bùng phát của vi-rút Corona, vậy bạn có thể mô tả mối quan hệ như thế này như thế nào khi mỗi khi bạn tăng số thang Richter lên 1, bạn đang nhân với 32, chúng tôi có thể nghĩ dưới dạng logarit với cơ số 32 Tôi có thể nói nếu tôi lấy log của, tôi sẽ gọi r, số của thang Richter. Tôi có thể coi đây là log cơ số 32 và nó sẽ tương ứng với , không không không, tôi đang làm sai đó không phải là thứ được ghi lại, chúng tôi lấy log cơ số 32 của số lớn, của số TMT, cái gì đó giống như 1 megaton, tức là 1 triệu tấn log cơ số 32, lẽ ra phải như vậy tương ứng với thang đo Richter nhưng có thể có một số loại bù, vì vậy chúng ta có thể nói rằng có một số loại hằng số s mà chúng ta đang thêm vào thang đo Richter này và biểu thức này hoàn toàn giống nhau, xin lỗi vì đã bỏ qua ở dưới cùng, biểu thức này hoàn toàn giống với việc nói 32 lũy thừa của một số độ lệch nhân với số thang Richter của chúng ta, tương đương với việc lấy 32 cho độ lệch đó, bản thân nó chỉ là một hằng số lớn, nhân 32 với số thang Richter nên bạn có thể coi đây chỉ là một hằng số nhân với 32 lũy thừa của con số bạn nhìn thấy nên cách viết này thực sự nhấn mạnh sự tăng trưởng theo cấp số nhân của nó rằng nếu đây là số tương ứng với số lượng TMT mà bạn thấy, khi bạn tăng số đó Bạn đang nhân từng bước một với 32 nhưng một cách khác để truyền đạt cùng một sự thật là lấy log cơ số 32 của bất cứ số tiền nào cũng được. Bây giờ, điều tiếp theo tôi muốn nói đến là làm thế nào chúng ta không phải lúc nào cũng phải làm như vậy lo lắng về cách tính nhật ký của các cơ số khác nhau. Có một điều hơi kỳ lạ ở đây là chúng ta đang nói về log cơ số 32, tôi đã đề cập trước đó về việc các nhà toán học thực sự muốn có nhật ký với cơ số e các nhà khoa học máy tính thực sự muốn có nhật ký với cơ số 2 và nó hóa ra là vì mục đích tính toán hoặc cũng để suy nghĩ xem những thứ này phát triển như thế nào nếu bạn có một nhật ký, nếu bạn có thể tính toán một loại nhật ký, cho dù đó là cơ số 10, cơ số 2, cơ số e, bạn có thể tính toán khá nhiều thứ khác bây giờ bạn muốn trực giác của chúng ta đi theo hướng đó, hãy quay lại câu hỏi của chúng ta và chuyển sang câu hỏi tiếp theo và tôi tin rằng câu hỏi này là hay nhất, tôi không biết, đây là một câu hỏi nửa chừng hợp lý, điều này sẽ hay đấy điều này chỉ giúp chúng ta chuẩn bị dịch từ ngữ cảnh cơ sở 2 sang ngữ cảnh cơ sở 10 và đó cũng là một trực giác tốt để hiểu lũy thừa của 2 nói chung có mối quan hệ mà nó có với lũy thừa 10 bởi vì đây là kiểu trùng hợp đáng yêu của Bản chất là hai loại giếng này bạn sẽ hiểu ý tôi, chúng chơi rất thân với nhau nên câu hỏi của chúng tôi đặt ra, vì thực tế là 2 mũ 10 là 1024, 1024, xấp xỉ 1000 nên nếu bạn là một hơi lỏng lẻo với các con số của bạn và bạn chỉ đang thực hiện các phép tính gần đúng từ 2 đến số 10, về cơ bản là 1000, điều nào sau đây gần đúng nhất? ", "model": "google_nmt", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 4192.07 }, { - "input": "tender. ", + "input": "ends up being log base C of A It's gotten rid ", "translatedText": "mềm. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 4194.05 }, { - "input": "Not at all a unanimous decision here. ", + "input": "of the B's which is kind of interesting so Some examples you might plug in here would be ", "translatedText": "Không hề có một quyết định nhất trí nào ở đây. ", "model": "google_nmt", "n_reviews": 0, @@ -448,7 +448,7 @@ "end": 4214.13 }, { - "input": "But the question was asking which one is closest to being true, and let's see how we can think about this. ", + "input": "Let's use green Instead of C. I'm gonna go ahead and plug in 10 Log base 10 of 100 which is kind of asking how many times d ", "translatedText": "Nhưng câu hỏi đặt ra là điều nào gần đúng nhất và hãy xem chúng ta có thể nghĩ về điều này như thế nào. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 4223.23 }, { - "input": "So it points out that you have a power of 2, which is 1024, awfully close to a power of 10, about 10 cubed. ", + "input": "oes 10 go into a 100 in a multiplicative sense how many times do I multiply 10 by itself to get to 100 where the answer is 2 and then log of 100 of Let's plug in another power of ", "translatedText": "Vì vậy, nó chỉ ra rằng bạn có lũy thừa 2, tức là 1024, rất gần với lũy thừa 10, khoảng 10 lập phương. ", "model": "google_nmt", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 4245.09 }, { - "input": "So what does this mean? ", + "input": "to 100 to the what equals a millio ", "translatedText": "Vì vậy, điều này có nghĩa là gì? ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 4245.93 }, { - "input": "If log base 2 of 10 is equal to x, that's the same thing as saying 2 to the x is equal to 10, right? ", + "input": "n How many times do I multiply a hundred by itself to get to a million? How many times does a hundred go into a million ", "translatedText": "Nếu log cơ số 2 của 10 bằng x, điều đó cũng giống như nói 2 với x bằng 10, phải không? ", "model": "google_nmt", "n_reviews": 0, @@ -496,7 +496,7 @@ "end": 4259.37 }, { - "input": "It's asking us 2 to the what equals 10. ", + "input": "? Phrasing the same thing 10 different ways now the claim is that th ", "translatedText": "Nó yêu cầu chúng ta 2 nhân 10. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 4295.95 }, { - "input": "You can't do that with every function. ", + "input": "ed cubed is equal to a million and Indeed how many times does 10 go into a million? ", "translatedText": "Bạn không thể làm điều đó với mọi chức năng. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 4301.69 }, { - "input": "People seem to think you can do that with any function, but you just can't. ", + "input": "well six Now we could think of this property in terms of the corresponding exponent rule which is going to look a little bit stranger But it's actually just saying th ", "translatedText": "Mọi người dường như nghĩ rằng bạn có thể làm điều đó với bất kỳ chức năng nào, nhưng thực tế là bạn không thể. ", "model": "google_nmt", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 4301.69 }, { - "input": "And what that means is that x is about 10 thirds, okay? ", + "input": "e entirely the same thing So here we're if we have a base of C and a base of B And we're trying to relate those to each ot ", "translatedText": "Và điều đó có nghĩa là x bằng khoảng 10 phần ba, được chứ? ", "model": "google_nmt", "n_reviews": 0, @@ -584,7 +584,7 @@ "end": 4316.59 }, { - "input": "And well enough, what we saw earlier is that log base 2 of 10, we could also say log base 10 of 2 is just 1 over that amount, 1 over x. ", + "input": "You raise it to some number X and equals a suppose It's also the case that C to the Y equals B. Those two together are the same as saying C to the XY equals a Now that's kind of a mouthful to say ", "translatedText": "Và đủ tốt, những gì chúng ta đã thấy trước đó là log cơ số 2 của 10, chúng ta cũng có thể nói log cơ số 10 của 2 chỉ là 1 trên số đó, 1 trên x. ", "model": "google_nmt", "n_reviews": 0, @@ -616,7 +616,7 @@ "end": 4339.13 }, { - "input": "Great. ", + "input": "undred cubed Well that lets y ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 4353.53 }, { - "input": "And because we're doing things at logs I'm just going to be writing it in that way. ", + "input": "mes does one number go into another? but letting you layer it on top of each other. Now if we rearrange that expression, we get what is ", "translatedText": "Và bởi vì chúng ta đang thực hiện mọi việc theo nhật ký nên tôi sẽ viết nó theo cách đó. ", "model": "google_nmt", "n_reviews": 0, @@ -656,7 +656,7 @@ "end": 4358.09 }, { - "input": "Similarly log base 2 of a million, well let's see, if we have to multiply 2 by itself about 10 times to get to a thousand, we should have to multiply it by itself around 20 times to get up to a million. ", + "input": "of our log rules. The most important is this top one, that when you multiply the inputs, you add the outputs. But the second most important, which is known as the change of base formula, lets us write that if yo ", "translatedText": "Tương tự, log cơ số 2 của một triệu, hãy xem, nếu chúng ta phải nhân 2 với chính nó khoảng 10 lần để có được một nghìn, thì chúng ta phải nhân với chính nó khoảng 20 lần để có được một triệu. ", "model": "google_nmt", "n_reviews": 0, @@ -672,7 +672,7 @@ "end": 4381.99 }, { - "input": "It's a little bit smaller but this is kind of a nice approximation to have in your mind. ", + "input": "want the log base b of some value a, then for whatever c you want, it doesn't actually matter what log you have in your pocket. ", "translatedText": "Nó nhỏ hơn một chút nhưng đây là một phép tính gần đúng mà bạn nên ghi nhớ. ", "model": "google_nmt", "n_reviews": 0, @@ -760,7 +760,7 @@ "end": 4458.69 }, { - "input": "20, we scale down by that same amount. ", + "input": "a convoluted enough way to write things, but an intuitive enough fact that I think just ", "translatedText": "20, chúng tôi giảm quy mô theo số lượng tương tự. ", "model": "google_nmt", "n_reviews": 0, @@ -768,7 +768,7 @@ "end": 4462.81 }, { - "input": "30, we scale down by that same amount. ", + "input": "coming at it from a bunch of different angles can be important. ", "translatedText": "30, chúng tôi giảm quy mô theo số lượng tương tự. ", "model": "google_nmt", "n_reviews": 0, @@ -776,7 +776,7 @@ "end": 4467.35 }, { - "input": "Okay? ", + "input": "Because like I ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -784,7 +784,7 @@ "end": 4467.73 }, { - "input": "Now this is an intuition worth remembering. ", + "input": "said, this is probably the second most important log rule. We're asking ", "translatedText": "Được rồi? Bây giờ đây là một trực giác đáng ghi nhớ. ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 4486.25 }, { - "input": "Okay? ", + "input": "many times do I multiply by itself? But division is aski ", "translatedText": "Được rồi? ", "model": "google_nmt", "n_reviews": 0, @@ -824,7 +824,7 @@ "end": 4516.11 }, { - "input": "And then just a whole pile of various possible ways to combine log base C of B times log base C of A. ", + "input": "n the logarithm realm is the same as anything multiplicative in terms of what's inside the parentheses So both of the l ", "translatedText": "Và sau đó có rất nhiều cách khác nhau để kết hợp log cơ số C của B với log cơ số C của A. ", "model": "google_nmt", "n_reviews": 0, @@ -848,7 +848,7 @@ "end": 4526.49 }, { - "input": "I'll give you a meaningful time on this one because it's not obvious unless you're already familiar with logarithms, and it's worth thinking through a little bit. ", + "input": "but going about that in different ways. So this is extremely nice because it actually lets us compute things. Next time we're going to talk all about the natural logarithm, which is l ", "translatedText": "Tôi sẽ dành cho bạn một khoảng thời gian ý nghĩa về vấn đề này vì nó không rõ ràng trừ khi bạn đã quen với logarit và bạn nên suy nghĩ kỹ một chút. ", "model": "google_nmt", "n_reviews": 0, @@ -888,7 +888,7 @@ "end": 4570.11 }, { - "input": "Thank You Karen. ", + "input": "It's between 1 and 2 because 57 was between 10 and 100 What's going on un ", "translatedText": "Cảm ơn bạn Karen. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/arabic/sentence_translations.json b/2020/ldm-natural-logs/arabic/sentence_translations.json index 0a004f65d..465777132 100644 --- a/2020/ldm-natural-logs/arabic/sentence_translations.json +++ b/2020/ldm-natural-logs/arabic/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "كما تعلمون، لدينا 1 تريليون 751، 1 تريليون 787. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "وهو حوالي واحد من كل 27. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "هناك عدد من الصيغ الأخرى التي تجعلنا شيئًا متعلقًا بـ pi، والذي يرتبط بشكل واضح بالأعداد الأولية، بطريقة، أعني، أنك تلعب نفس اللعبة ولديك هذه الطريقة الغريبة في أخذ اللوغاريتمات، وليس أي منها اللوغاريتم، قاعدة السجل ه. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "لذا، إذا كنت تتساءل عن المدة التي يجب أن أمضيها قبل أن يصبح هذا المجموع أكبر من 10، فقد يكون لديك غريزة مفادها أنه سيتعين علي أن أجمع معًا، دعنا نرى، لدي واحدة ثم الباقي منها عبارة عن نصفين، لذلك سأضطر إلى جمع 18 مجموعة مختلفة تبدو كل منها وكأنها نصف، لذلك قد يتعين علي الوصول إلى النقطة التي يكون فيها حجم مجموعتي مثل اثنين إلى 17، شيء من هذا القبيل الذي - التي. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "وحتى تلك الفكرة المجنونة، مثل التفكير العقلي حول كيفية الوصول إلى رقم كبير، ستستغرق وقتًا طويلاً لتوصلك إلى شيء من الحجم 10 إلى 400000. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/bengali/sentence_translations.json b/2020/ldm-natural-logs/bengali/sentence_translations.json index a3abbb6df..3aa16cf6b 100644 --- a/2020/ldm-natural-logs/bengali/sentence_translations.json +++ b/2020/ldm-natural-logs/bengali/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "আপনি জানেন, আমরা 1 ট্রিলিয়ন 751, 1 ট্রিলিয়ন 787 পেয়েছি।", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "এবং এটি প্রতি 27 জনের মধ্যে একটি।", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "আরও অনেকগুলি সূত্র রয়েছে যা আমাদের পাই-এর সাথে সম্পর্কিত কিছু পায়, যা স্পষ্টতই প্রাইমগুলির সাথে সম্পর্কিত, এমনভাবে যে, উম, আমি বলতে চাচ্ছি, আপনি একই গেম খেলছেন এবং লগারিদম নেওয়ার এই অদ্ভুত ফ্যাশন আপনার আছে, এবং শুধুমাত্র কোনটি নয় লগারিদম, লগ বেস e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "সুতরাং আপনি যদি ভাবছেন যে এই যোগফলটি 10 এর থেকে বড় হওয়ার আগে আমাকে কতক্ষণ যেতে হবে, আপনার সহজাত প্রবৃত্তি থাকতে পারে যে, হুম, আমাকে একসাথে যোগ করতে হবে, আসুন দেখি, আমার একটি আছে এবং তারপরে বাকিগুলি তাদের মধ্যে অর্ধেক, তাই আমাকে 18টি ভিন্ন গ্রুপকে একত্রে যোগ করতে হবে যেগুলোর প্রত্যেকটি দেখতে অর্ধেকের মতো, তাই আমাকে এমন জায়গায় উঠতে হতে পারে যেখানে আমার গ্রুপের আকার দুই থেকে 17 তম, এরকম কিছু যে এবং আপনি স্পট হবে যে এটি বৃদ্ধি পায়, ভাল এটি দ্রুতগতিতে বৃদ্ধি পায় না, এটি লগারিদমিকভাবে বৃদ্ধি পায়, কারণ আপনি যদি জিজ্ঞাসা করেন যে সেই বিন্দুতে পৌঁছানোর জন্য আপনাকে কতদূর যেতে হবে, এটি লগারিদমিক হবে।", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "এবং এমনকি সেই পাগলাটে ধারণা, মানসিক চিন্তার মতো যে আপনি কীভাবে একটি বড় সংখ্যায় উঠতে পারেন, আপনাকে 10 থেকে 400,000 আকারের কিছুতে নিয়ে যেতে চিরকালের জন্য সময় লাগবে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/chinese/sentence_translations.json b/2020/ldm-natural-logs/chinese/sentence_translations.json index 1ccc680f6..5b08a56ac 100644 --- a/2020/ldm-natural-logs/chinese/sentence_translations.json +++ b/2020/ldm-natural-logs/chinese/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "你知道，我们有 1 万亿个 751 、1 万亿个 787。", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "大约每 27 人中就有一人 。", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "还有许多其他公式可以让我们得到与 pi 相关的东西，这显然与 素数相关，在某种程度上，嗯，我的意思是，你玩同一个游戏，你 有这种奇怪的取对数的方式，而不仅仅是任何对数，对数底 e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "因此，如果您想知道在这个总和大于 10 之前我需要 花多长时间，您可能会本能地认为，嗯，我必须将其加 在一起，让我们看看，我有一个，然后是其余的其中一 半是一半，所以我必须将 18 个不同的组加在一起， 每个组看起来都像一半，所以我可能必须达到这样的程 度：我的组的大小就像 2 的 17 倍，类似那。", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "即使是那种疯狂的想法，比如在心里思考如何才能达到一个大数字， 也需要很长时间才能达到 10 到 400,000 的大小。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/english/captions.srt b/2020/ldm-natural-logs/english/captions.srt index 815b513b3..834eab831 100644 --- a/2020/ldm-natural-logs/english/captions.srt +++ b/2020/ldm-natural-logs/english/captions.srt @@ -651,7 +651,7 @@ it's 1 over the power of that prime squared, but we scale it down by whatever that power is. 164 -00:08:52,959 --> 00:08:56,889 +00:08:52,960 --> 00:08:56,889 Now, because we've manipulated this in a pretty chaotic way, I mean, 165 @@ -1111,7 +1111,7 @@ All four of those terms are bigger than 1 eighth. So the group of them together is bigger than 4 eighths. 279 -00:15:15,079 --> 00:15:19,882 +00:15:15,080 --> 00:15:19,882 Similarly over here, all of the numbers between a ninth and a sixteenth, 280 @@ -1123,24 +1123,24 @@ all eight of those numbers are bigger than 1 and 16, so the sum all together is bigger than 8 times 1 over 16. 282 -00:15:28,120 --> 00:15:31,785 +00:15:28,120 --> 00:15:31,655 And you might see where I'm going with this, you know, 283 -00:15:31,785 --> 00:15:37,783 -here I have 16 numbers that are all bigger than 1 and 30, excuse me, bigger than 1 in 32, +00:15:31,655 --> 00:15:36,154 +here I have 16 numbers that are all bigger than one in 30, excuse me, 284 -00:15:37,783 --> 00:15:42,982 -talking while writing, and of course all of these are just equal to one half, +00:15:36,154 --> 00:15:41,424 +bigger than one in 32, talking while writing, and of course all of these are just 285 -00:15:42,982 --> 00:15:47,913 -so this 2 fourths is the same as a half, 4 eighths is the same as a half, +00:15:41,424 --> 00:15:45,409 +equal to one half, so this two fourths is the same as a half, 286 -00:15:47,913 --> 00:15:49,780 -8 sixteenths, that's a half. +00:15:45,409 --> 00:15:49,780 +four eighths is the same as a half, eight sixteenths, that's a half. 287 00:15:50,680 --> 00:15:56,021 @@ -1383,7 +1383,7 @@ So what you're looking for is the value when the natural log of n is approximately a million. 347 -00:20:17,419 --> 00:20:21,000 +00:20:17,420 --> 00:20:21,000 That's how long you have to go before the sum gets bigger than a million. 348 @@ -1479,7 +1479,7 @@ And indeed, of all the options here, there's one that's much closer to 10 to the Alright, so that's pretty fun. 371 -00:22:11,679 --> 00:22:15,569 +00:22:11,680 --> 00:22:15,569 Now, to start explaining where on earth some of these things are coming from, 372 @@ -1551,7 +1551,7 @@ something that grows exponentially but a little bit more slowly, versus a steep exponential growth. 389 -00:23:14,139 --> 00:23:14,540 +00:23:14,140 --> 00:23:14,540 Okay? 390 @@ -1703,2950 +1703,2966 @@ okay, then it decays on both sides, but we get this awkward cusp. That doesn't explain why this very specific curve comes up in statistics. 427 -00:25:27,760 --> 00:25:31,679 -But if you ever kind of want to remember, oh, what was the formula for a bell curve +00:25:27,760 --> 00:25:30,493 +but if you ever kind of want to remember oh what was the what 428 -00:25:31,679 --> 00:25:35,740 -again, you can kind of think through the fact that this should have roughly that shape. +00:25:30,493 --> 00:25:33,182 +was the formula for a bell curve again you can kind of think 429 +00:25:33,182 --> 00:25:35,740 +through the fact that this should have roughly that shape. + +430 00:25:36,440 --> 00:25:38,560 And quite often it comes with some kind of parameters, though. -430 +431 00:25:38,560 --> 00:25:42,745 For example, I could put in something, maybe a value I'll call s in there, -431 +432 00:25:42,745 --> 00:25:46,093 that will determine how wide and skinny this bell curve is, -432 +433 00:25:46,093 --> 00:25:49,720 something like a standard deviation in the context of statistics. -433 +434 00:25:50,640 --> 00:25:52,120 S wouldn't be that standard deviation. -434 +435 00:25:52,140 --> 00:25:55,120 We would have to reciprocate it and square it and do some things. -435 +436 00:25:55,320 --> 00:25:58,260 But the idea that when you tweak what's in that exponent, it changes the bell curve. -436 +437 00:25:58,520 --> 00:26:00,160 That's the only point I want to make here. -437 +438 00:26:00,660 --> 00:26:03,242 You can think, just looking at this, that somehow -438 +439 00:26:03,242 --> 00:26:05,360 bell curves are produced by the number e. -439 +440 00:26:05,840 --> 00:26:10,771 But that's not exactly true, because I could also write a to the negative x squared, -440 +441 00:26:10,771 --> 00:26:12,860 and I get the same family of curves. -441 +442 00:26:13,120 --> 00:26:16,464 As I tweak the value of a, I'm also changing what that width is, -442 +443 00:26:16,464 --> 00:26:20,579 so I could come up with other ways of describing the standard deviation of this -443 +444 00:26:20,579 --> 00:26:21,300 in terms of a. -444 +445 00:26:21,760 --> 00:26:23,460 And it's the same family of curves. -445 +446 00:26:23,460 --> 00:26:24,580 It's not just that they look similar. -446 +447 00:26:24,600 --> 00:26:25,940 They are, in fact, the same thing. -447 +448 00:26:27,120 --> 00:26:29,420 And this is not too hard to show algebraically. -448 +449 00:26:29,640 --> 00:26:33,140 It almost makes it look a little bit more deceptively simple than it really is. -449 +450 00:26:33,140 --> 00:26:36,675 So we've got a lot of answers for what people wanted to enter -450 +451 00:26:36,675 --> 00:26:40,040 as an example number to go with, so let's go ahead and see. -451 +452 00:26:40,640 --> 00:26:44,701 It seems like the most popular answer, by a little margin above i, is 69, -452 +453 00:26:44,701 --> 00:26:49,202 which I assume is because if you take all of the natural numbers between 1 and 9, -453 +454 00:26:49,202 --> 00:26:53,703 and then you look at the divisors for each one of those, you look at the numbers, -454 +455 00:26:53,703 --> 00:26:57,820 you list all their divisors, and you add up the divisors, it adds up to 69. -455 +456 00:26:58,060 --> 00:27:01,855 And adding up divisors like this is a very fun and common thing in number theory, -456 +457 00:27:01,855 --> 00:27:03,660 so I assume that's why people chose it. -457 +458 00:27:03,880 --> 00:27:08,040 But the point here is that if you see some kind of function like, -458 +459 00:27:08,040 --> 00:27:10,120 let's see, how should I write it? -459 +460 00:27:10,400 --> 00:27:12,240 69 to the power x. -460 +461 00:27:12,640 --> 00:27:14,080 I could also write that. -461 +462 00:27:14,580 --> 00:27:20,380 I could write the same thing as e to the power of the natural log of 69. -462 +463 00:27:21,160 --> 00:27:22,720 Okay, I've written that kind of sloppily. -463 +464 00:27:22,800 --> 00:27:23,420 Let me do it again. -464 +465 00:27:23,880 --> 00:27:26,960 e to the natural log of 69. -465 +466 00:27:28,980 --> 00:27:31,520 That's the same thing as the number 69, right? -466 +467 00:27:31,620 --> 00:27:35,374 Because it's saying e to the what equals 69, but then I've taken e to that, -467 +468 00:27:35,374 --> 00:27:36,560 so I should get 69 back. -468 +469 00:27:36,880 --> 00:27:38,800 All of that to the power x. -469 +470 00:27:39,740 --> 00:27:44,972 And by the rules of exponentials, this is the same thing as e to just some constant, -470 +471 00:27:44,972 --> 00:27:47,620 whatever the natural log of 69 is, times x. -471 +472 00:27:47,920 --> 00:27:52,471 So I could replace that with a constant, which happens to be around 4.234, -472 +473 00:27:52,471 --> 00:27:57,204 as any mathematician will be able to tell you, well-known constant of nature, -473 +474 00:27:57,204 --> 00:27:58,540 the natural log of 69. -474 +475 00:27:59,460 --> 00:28:05,060 And the point here is that this just looks like e to some constant times x. -475 +476 00:28:06,000 --> 00:28:09,020 So you might wonder, why do we make this choice, right? -476 +477 00:28:09,360 --> 00:28:10,460 Because it didn't have to be pi. -477 +478 00:28:10,700 --> 00:28:13,760 I could write that same function, sorry, it didn't have to be e. -478 -00:28:14,060 --> 00:28:19,740 -I could write that same function as pi raised to a special power, namely the log base pi. - 479 +00:28:14,060 --> 00:28:16,957 +I could write that same function sorry it didn't have to be e I could write + +480 +00:28:16,957 --> 00:28:19,740 +that same function as pi raised to a special power namely the log base pi + +481 00:28:20,760 --> 00:28:23,120 Man, I'm writing quite sloppily here. -480 +482 00:28:24,480 --> 00:28:28,380 Log base pi of 69 times x, that would be the same function. -481 +483 00:28:28,660 --> 00:28:31,340 We could describe everything with a base of pi if we wanted to. -482 +484 00:28:31,900 --> 00:28:36,342 And to just give one more example of where I think this, even though it's a, -483 +485 00:28:36,342 --> 00:28:39,457 like it's a simple conversion if you know logarithms, -484 +486 00:28:39,457 --> 00:28:44,534 that anything that looks like a to the x can be expressed as e to something times that, -485 +487 00:28:44,534 --> 00:28:48,976 I think it's clearly not very appreciated that you can make that conversion, -486 +488 00:28:48,976 --> 00:28:52,438 because any time that we talk about imaginary exponentials, -487 +489 00:28:52,438 --> 00:28:56,823 and I get that it's kind of a weird thing to talk about e to the i times t, -488 +490 00:28:56,823 --> 00:29:01,957 once someone learns, you know, and I've made a number of videos in this series about it, -489 +491 00:29:01,957 --> 00:29:05,419 Mathologer has videos, lots of people have videos about it, -490 +492 00:29:05,419 --> 00:29:10,092 the idea is that e to the imaginary constant times some value t walks you around -491 +493 00:29:10,092 --> 00:29:13,900 the unit circle, and in fact it walks you a distance of t radians. -492 -00:29:16,679 --> 00:29:20,351 +494 +00:29:16,680 --> 00:29:20,351 And the importance of this, the way that it comes up in, for example, -493 +495 00:29:20,351 --> 00:29:23,760 electrical engineering, is it gives you this oscillating pattern. -494 +496 00:29:23,900 --> 00:29:26,732 As you scale up t, you have something that's oscillating, -495 +497 00:29:26,732 --> 00:29:30,296 and it gives a very nice way to describe sine waves and cosine waves and -496 +498 00:29:30,296 --> 00:29:31,420 signals that oscillate. -497 +499 00:29:31,740 --> 00:29:34,947 And there's actually one electrical engineer that I used to know, -498 +500 00:29:34,947 --> 00:29:37,280 and he would always say, oh I love the number e. -499 +501 00:29:37,580 --> 00:29:39,700 What I like about e is it's the number that spins. -500 +502 00:29:39,820 --> 00:29:41,940 That's really what makes e special, I realized this. -501 +503 00:29:42,020 --> 00:29:43,180 e is the number that spins. -502 +504 00:29:43,180 --> 00:29:46,840 But the problem is that's not, that's not exactly true. -503 +505 00:29:47,660 --> 00:29:52,220 It is true that e to the i t spins around, but that's not special to e. -504 +506 00:29:52,680 --> 00:29:55,643 I could also take 2 to the i times t, and that -505 +507 00:29:55,643 --> 00:29:58,860 will also produce values that walk around a circle. -506 +508 00:29:59,300 --> 00:30:05,384 And we can think through more exactly, 2 is the same thing as e to the natural log of 2, -507 +509 00:30:05,384 --> 00:30:09,896 so 2 to the i t is the same as e to the natural log of 2 times t, -508 +510 00:30:09,896 --> 00:30:12,700 all of that times the imaginary number i. -509 +511 00:30:13,400 --> 00:30:17,820 Which basically means it's doing the same thing as e to the i t, it's just rescaling time. -510 +512 00:30:17,920 --> 00:30:19,380 It's walking a little bit more slowly. -511 +513 00:30:20,040 --> 00:30:24,487 And then similarly, if you were to take something like your -512 +514 00:30:24,487 --> 00:30:28,935 favorite number to the i times t, that would look like e to -513 +515 00:30:28,935 --> 00:30:33,680 the approximately 4.234 times t, all times the imaginary number. -514 +516 00:30:33,880 --> 00:30:36,280 Which just means you're walking around at a different rate. -515 +517 00:30:36,280 --> 00:30:38,280 So e is not the number that spins. -516 +518 00:30:38,440 --> 00:30:43,360 The idea of complex exponents walking around a circle doesn't have to do with e per se, -517 +519 00:30:43,360 --> 00:30:48,280 it really just has to do with a lot of what we talked about I think in lectures 4 and 5. -518 +520 00:30:48,480 --> 00:30:52,280 And I'll go over it again in just a moment here as a quick reminder. -519 +521 00:30:53,080 --> 00:30:56,598 So the thing that's special about e, the reason that we always -520 +522 00:30:56,598 --> 00:31:00,340 choose to write things in this way, has to do with rates of change. -521 +523 00:31:00,760 --> 00:31:10,640 That when you take the derivative of e to the power t, this is the same thing as itself. -522 +524 00:31:12,020 --> 00:31:15,808 Which is to say if we were to graph this, you know maybe e to the t -523 +525 00:31:15,808 --> 00:31:19,820 represents an amount of money you have over time or something like that. -524 +526 00:31:22,120 --> 00:31:25,685 And we were to look above a value of t, the slope of this graph, -525 +527 00:31:25,685 --> 00:31:29,580 the rate at which it's increasing, is actually equal to its own height. -526 +528 00:31:29,580 --> 00:31:33,821 So the farther you are along the curve, meaning you have a greater height, -527 +529 00:31:33,821 --> 00:31:34,840 the steeper it is. -528 +530 00:31:35,020 --> 00:31:37,480 So the more money you have, the faster it grows. -529 +531 00:31:37,920 --> 00:31:39,740 This is the power of compound growth. -530 +532 00:31:40,380 --> 00:31:42,100 But that's actually true of any exponential. -531 +533 00:31:42,180 --> 00:31:44,486 What makes e special is that they're exactly the same, -532 +534 00:31:44,486 --> 00:31:46,960 it's not just that they grow in proportion with each other. -533 +535 00:31:47,520 --> 00:31:52,385 So if we were to write a family of exponential curves as e to the r times t, -534 +536 00:31:52,385 --> 00:31:57,756 as opposed to writing it as a to the power t, and thinking of changing that value a, -535 +537 00:31:57,756 --> 00:32:02,368 the value in doing that is that taking the derivative by the chain rule, -536 +538 00:32:02,368 --> 00:32:06,033 we take the derivative of the inside, which looks like r, -537 +539 00:32:06,033 --> 00:32:10,520 and we multiply by the derivative of the outside, which is e to the rt. -538 +540 00:32:11,400 --> 00:32:13,807 And if anybody here doesn't know calculus by the way, -539 +541 00:32:13,807 --> 00:32:15,946 we're about to start doing a fair amount of it, -540 +542 00:32:15,946 --> 00:32:19,067 I have a whole series on it that you can pop over and take a look at, -541 +543 00:32:19,067 --> 00:32:21,920 lots of other places on YouTube and such to give a quick primer. -542 +544 00:32:21,920 --> 00:32:24,794 But if you're coming in and you're not familiar with calculus, -543 +545 00:32:24,794 --> 00:32:27,760 like just be warned that that's where we're about to start going. -544 +546 00:32:27,960 --> 00:32:30,717 Because if you want to understand natural logarithms, -545 +547 00:32:30,717 --> 00:32:34,037 and by extension the number e, the importance that they have has -546 +548 00:32:34,037 --> 00:32:38,480 everything to do with rates of change and the inverse of that operation, as you'll see. -547 +549 00:32:39,260 --> 00:32:42,600 So anyway, why would it be nice to express a function like this? -548 +550 00:32:43,840 --> 00:32:47,200 Well what it's telling you, let's say this was something like the size of your investment. -549 +551 00:32:48,060 --> 00:32:51,680 This is an expression saying how much money you have at a given point in time. -550 +552 00:32:51,800 --> 00:32:56,332 If you want to know the rate of change of that, how much is it changing per unit time, -551 +553 00:32:56,332 --> 00:33:00,240 it's proportional to itself, and r gives you that proportionality constant. -552 +554 00:33:00,440 --> 00:33:03,170 If r was 0.01, it's telling you that the rate -553 +555 00:33:03,170 --> 00:33:06,080 of growth is 10% of the size of the thing itself. -554 +556 00:33:06,740 --> 00:33:09,913 So the choice that we make to write things this way is, -555 +557 00:33:09,913 --> 00:33:13,880 it's basically a way to make all the constants involved more readable. -556 +558 00:33:13,880 --> 00:33:18,435 So if you're to look at these statistics associated with bell curves, -557 +559 00:33:18,435 --> 00:33:21,755 and the way that we actually tend to write things, -558 +560 00:33:21,755 --> 00:33:26,115 the pattern ends up looking something like 1 divided by s squared, -559 +561 00:33:26,115 --> 00:33:29,240 and sometimes that x instead of writing it as x. -560 +562 00:33:30,040 --> 00:33:31,060 Strange parentheses. -561 +563 00:33:31,880 --> 00:33:35,080 I might say x minus m for some value of m. -562 +564 00:33:36,580 --> 00:33:40,495 Both of these terms end up having really nice readable meanings, where m, -563 +565 00:33:40,495 --> 00:33:44,516 this isn't specific to the e fact, but it's just a common thing you'll see, -564 +566 00:33:44,516 --> 00:33:47,796 m gives you the mean of the distribution, where this pile is, -565 +567 00:33:47,796 --> 00:33:49,860 and s gives you the standard deviation. -566 +568 00:33:50,620 --> 00:33:53,357 And when we choose to write this family with e, -567 +569 00:33:53,357 --> 00:33:55,980 it's giving those constants readable meanings. -568 +570 00:33:56,980 --> 00:34:00,820 And a similar thing happens with how we describe complex exponentials. -569 +571 00:34:01,100 --> 00:34:04,763 When we choose to write the idea of walking around a circle with e, -570 +572 00:34:04,763 --> 00:34:07,780 it gives a very readable meaning to what this term t is. -571 +573 00:34:07,780 --> 00:34:11,500 It's saying, what is the distance that you've walked along the unit circle? -572 +574 00:34:12,260 --> 00:34:17,647 And you can actually understand this with derivatives pretty well, where if we say, -573 +575 00:34:17,647 --> 00:34:23,163 what is the derivative, what is the rate of change of some value that looks like e to -574 +576 00:34:23,163 --> 00:34:28,101 the i times t, by the chain rule, this is going to look like i times itself, -575 +577 00:34:28,101 --> 00:34:29,320 e to the i times t. -576 +578 00:34:29,840 --> 00:34:31,120 Now what would that actually mean? -577 +579 00:34:31,460 --> 00:34:35,690 That means that if you're sitting at some kind of number, -578 +580 00:34:35,690 --> 00:34:39,630 if this is your current value for e to the i times t, -579 +581 00:34:39,630 --> 00:34:43,277 the rate of change is i multiplied by that value, -580 +582 00:34:43,277 --> 00:34:46,560 which is a 90 degree rotation of this vector. -581 +583 00:34:47,699 --> 00:34:49,679 Maybe I would draw it like this. -582 +584 00:34:50,120 --> 00:34:52,080 This right here would give you your rate of change. -583 +585 00:34:52,440 --> 00:34:55,440 So you might move that over and consider it a velocity vector. -584 +586 00:34:56,820 --> 00:35:00,642 So this is kind of like your velocity vector, this is kind of like your position vector, -585 +587 00:35:00,642 --> 00:35:02,360 which I might write something like an s. -586 +588 00:35:03,080 --> 00:35:07,263 So what this whole expression of e to the i times t is essentially saying is, -587 +589 00:35:07,263 --> 00:35:11,660 whatever this does, if it's somewhere in the complex plane, at every given point, -588 +590 00:35:11,660 --> 00:35:15,200 the rate of change of my vector is a 90 degree rotation of itself. -589 +591 00:35:15,940 --> 00:35:20,612 And so that's why we're walking around the circle at a speed of one unit per second, -590 +592 00:35:20,612 --> 00:35:23,361 because the length of the position vector is one, -591 +593 00:35:23,361 --> 00:35:25,780 so the length of the velocity vector is one. -592 +594 00:35:26,060 --> 00:35:28,725 And it's also why, if you were to look at two to the i t, -593 +595 00:35:28,725 --> 00:35:30,380 it walks around at a different rate. -594 +596 00:35:30,380 --> 00:35:35,781 Because there, the constant is not just i, a 90 degree rotation, -595 +597 00:35:35,781 --> 00:35:38,440 it's i times natural log of two. -596 +598 00:35:38,800 --> 00:35:42,536 i times something would mean that this, where are we, -597 +599 00:35:42,536 --> 00:35:46,272 this operation here is not just a 90 degree rotation, -598 +600 00:35:46,272 --> 00:35:49,040 it's a 90 degree rotation and a scaling. -599 +601 00:35:49,400 --> 00:35:52,504 So your velocity vector would end up looking a little bit shorter, -600 +602 00:35:52,504 --> 00:35:55,100 and you'd be walking around the unit circle more slowly. -601 -00:35:55,779 --> 00:35:58,660 +603 +00:35:55,780 --> 00:35:58,660 So that's kind of the important thing to understand about e, -602 +604 00:35:58,660 --> 00:36:02,909 the fact that it's a choice that we're making to write families of exponentials this way, -603 +605 00:36:02,909 --> 00:36:06,733 but because it is its own derivative, that ends up making these things play much -604 +606 00:36:06,733 --> 00:36:07,300 more nicely. -605 +607 00:36:08,040 --> 00:36:11,780 Now, this lets us take derivatives of anything else if you wanted. -606 +608 00:36:12,020 --> 00:36:18,156 If you did describe your money's rate of growth with a to the t, to take its derivative, -607 +609 00:36:18,156 --> 00:36:23,947 you could first do a conversion, write the whole thing as e to the natural log of a -608 +610 00:36:23,947 --> 00:36:29,945 times t, and the reason you would is then when we're, I sort of squished my font here, -609 +611 00:36:29,945 --> 00:36:33,255 then when you're taking the derivative of that, -610 +612 00:36:33,255 --> 00:36:36,978 the derivative of the inside is the natural log of a, -611 +613 00:36:36,978 --> 00:36:42,011 and then that's multiplied by itself, e to the natural log of a times t, -612 +614 00:36:42,011 --> 00:36:47,320 which you could then spell out even further, convert it back into a to the t. -613 -00:36:47,740 --> 00:36:51,598 -So if you did describe all of your investments as a to the power t, - -614 -00:36:51,598 --> 00:36:55,116 -which kind of feels more natural to a lot of people, that oh, - 615 -00:36:55,116 --> 00:37:00,223 -you might say rather than e to some investment rate times t, just think of 1.05 to the t, +00:36:47,740 --> 00:36:51,363 +so if you did describe all of your investments as a to the power t 616 -00:37:00,223 --> 00:37:02,720 -and that describes something like 5% growth. +00:36:51,363 --> 00:36:55,148 +which kind of feels more natural to a lot of people that oh you might 617 +00:36:55,148 --> 00:36:58,880 +say rather than e to some investment rate times t just think of 1.05 + +618 +00:36:58,880 --> 00:37:02,720 +to the t and that describes you know something like five percent growth + +619 00:37:03,140 --> 00:37:05,925 If you were thinking of that growth in a continuous sense, -618 +620 00:37:05,925 --> 00:37:09,182 not year over year, what's the new percentage, but moment by moment, -619 +621 00:37:09,182 --> 00:37:12,392 what's the rate of growth, you would have to say the rate of growth -620 +622 00:37:12,392 --> 00:37:15,980 is the natural log of that base, which just feels a little bit more awkward. -621 +623 00:37:16,380 --> 00:37:18,620 You could do it, but it would feel more awkward. -622 +624 00:37:19,660 --> 00:37:22,300 Now, all of this leaves open the question of why? -623 +625 00:37:22,300 --> 00:37:28,780 Why on earth is the derivative of e to the t equal to itself? -624 +626 00:37:30,200 --> 00:37:33,840 It's this very nice property, so you might wonder where this thing comes from. -625 +627 00:37:34,280 --> 00:37:39,160 And it really has everything to do with how you define the number e. -626 +628 00:37:39,520 --> 00:37:43,508 And this can be a little bit frustrating, where in some contexts you'll see people say, -627 +629 00:37:43,508 --> 00:37:44,460 what is the number e? -628 +630 00:37:44,680 --> 00:37:48,160 Well, it's the number defined such that this derivative equals itself. -629 +631 00:37:48,160 --> 00:37:52,380 And then other contexts, you might find e defined in a different -630 +632 00:37:52,380 --> 00:37:56,600 way that is very conducive to whatever the circumstance there is. -631 +633 00:37:57,180 --> 00:38:01,605 So you might wonder, okay, can we come at this a little bit more directly, -632 +634 00:38:01,605 --> 00:38:04,614 and try to understand derivatives of exponentials, -633 +635 00:38:04,614 --> 00:38:07,800 and see why the special value 2.718 would fit into it. -634 +636 00:38:08,280 --> 00:38:12,160 And to do that, let me draw myself a new graph here. -635 +637 00:38:16,960 --> 00:38:18,560 Let's say I have some kind of exponential. -636 +638 00:38:19,460 --> 00:38:25,796 And if I want to understand the rate of change, the slope tangent to that point x, -637 +639 00:38:25,796 --> 00:38:30,606 the way we often think about it is think of two nearby points, -638 +640 00:38:30,606 --> 00:38:35,340 so another one that might be x plus a little constant times h. -639 +641 00:38:36,980 --> 00:38:39,306 And then we're going to look at the slope between those -640 +642 00:38:39,306 --> 00:38:41,800 two points and consider what happens as h goes towards zero. -641 +643 00:38:41,800 --> 00:38:44,743 So if this whole graph was a function a to the x, -642 +644 00:38:44,743 --> 00:38:49,276 if we wanted to give a very direct look at what might the derivative of this -643 +645 00:38:49,276 --> 00:38:53,514 expression be, we can make an attempt to calculate it ourselves without -644 +646 00:38:53,514 --> 00:38:58,283 depending on a pre-established fact handed down from on high that e to the t is, -645 +647 00:38:58,283 --> 00:39:01,697 or I guess in this case e to the x is its own derivative, -646 +648 00:39:01,697 --> 00:39:04,700 and then manipulate based on natural logs and such. -647 +649 00:39:05,340 --> 00:39:07,340 So what does this look like if you try to come at it directly? -648 +650 00:39:07,820 --> 00:39:13,416 Well what you would say is that the change in the height of the graph divided -649 +651 00:39:13,416 --> 00:39:17,865 by the change in the width, the sort of rise over run, dy dx, -650 +652 00:39:17,865 --> 00:39:22,314 looks like the difference in the outputs at those two values, -651 +653 00:39:22,314 --> 00:39:28,485 so the output at the high value, which is x plus h, minus the value at the low value, -652 +654 00:39:28,485 --> 00:39:34,800 a to the x, all of that divided by the step in the x direction, which is just of size h. -653 +655 00:39:35,500 --> 00:39:40,078 And the fact that we are doing calculus here, that we have the little d's, -654 +656 00:39:40,078 --> 00:39:45,390 that is a signal to us that we don't just want this ratio for a particular value of h, -655 +657 00:39:45,390 --> 00:39:48,382 we want to consider what that ratio change in y, -656 +658 00:39:48,382 --> 00:39:51,740 change in x looks like as the change in x goes to zero. -657 +659 00:39:52,080 --> 00:39:54,221 And here I'm writing that change in x as h, so -658 +660 00:39:54,221 --> 00:39:56,500 it's a limit as h goes to zero of this expression. -659 +661 00:39:57,420 --> 00:40:01,580 And from here you can try to manipulate it a little bit and see what you might find. -660 +662 00:40:01,580 --> 00:40:05,212 The first step, take advantage of the exponential -661 +663 00:40:05,212 --> 00:40:09,280 properties to write this as a to the x times a to the h. -662 +664 00:40:09,780 --> 00:40:14,299 And what's nice about that is it lets us factor out an a to the x, -663 +665 00:40:14,299 --> 00:40:18,279 because it shows up both in the first term and the second, -664 +666 00:40:18,279 --> 00:40:23,338 so I could write this whole thing as a limit of a to the x outside of a to -665 +667 00:40:23,338 --> 00:40:25,160 the h minus one all over h. -666 +668 00:40:25,540 --> 00:40:27,620 And this was the limit as h goes to zero. -667 +669 00:40:28,680 --> 00:40:31,458 Well, x has nothing to do with the h here, so -668 +670 00:40:31,458 --> 00:40:34,660 we're allowed to pull out the a to the x term itself. -669 +671 00:40:34,840 --> 00:40:38,233 As far as h is concerned, it's just some constant rescaling the thing, -670 +672 00:40:38,233 --> 00:40:42,440 and the limit of a constant times a thing is that constant times the limit of the thing. -671 +673 00:40:43,280 --> 00:40:53,120 A times, or a to the x times the limit as h goes to zero of a to the h minus one over h. -672 +674 00:40:54,060 --> 00:40:57,560 And at this point, we're a little bit stuck. -673 +675 00:40:58,660 --> 00:41:02,952 We've found a very interesting fact, which is that any kind of exponential, -674 +676 00:41:02,952 --> 00:41:06,848 e or whatever you want, base pi to the x, two to the x, 69 to the x, -675 +677 00:41:06,848 --> 00:41:10,237 those have derivatives that are proportional to themselves, -676 +678 00:41:10,237 --> 00:41:13,400 but we want to understand this proportionality constant. -677 +679 00:41:14,120 --> 00:41:17,143 And I could ask you to guess, just to see if you can get -678 +680 00:41:17,143 --> 00:41:20,060 a feel for it in the context of one particular example. -679 +681 00:41:20,060 --> 00:41:23,244 So let's say that I'm choosing a base of something like two, -680 +682 00:41:23,244 --> 00:41:26,220 and I want to understand rates of change of two to the x. -681 +683 00:41:27,140 --> 00:41:29,916 Our question asks us, the limit below, I guess it tells us, -682 +684 00:41:29,916 --> 00:41:31,860 it tells us a little bit about what it is. -683 +685 00:41:32,020 --> 00:41:34,980 The limit below is a number between zero and one. -684 +686 00:41:35,700 --> 00:41:38,517 So this is, we're looking at two to a small value -685 +687 00:41:38,517 --> 00:41:40,940 minus one divided by that same small value. -686 +688 00:41:42,080 --> 00:41:45,660 Don't worry about calculating it exactly, I'm just kind of curious if you guessed. -687 +689 00:41:45,660 --> 00:41:48,304 Enter some kind of guess for what this value is, -688 +690 00:41:48,304 --> 00:41:52,460 and then round it to two decimal places so that we can have some consistency. -689 +691 00:41:53,200 --> 00:41:55,898 So we'll give you a moment to just think of what it might be, -690 +692 00:41:55,898 --> 00:41:57,900 but don't think too hard if you don't want to. -691 +693 00:41:58,080 --> 00:42:01,420 It's totally okay to get this one wrong, we just want to see what people think. -692 +694 00:42:33,240 --> 00:42:37,437 Looks like we've got a couple things coming in from the audience here, -693 +695 00:42:37,437 --> 00:42:38,620 which is always fun. -694 +696 00:42:38,620 --> 00:42:42,054 So Robert points out that in French the notation reads, -695 +697 00:42:42,054 --> 00:42:45,980 logarithm l'imperillain, and is wondering why this word is used. -696 +698 00:42:46,060 --> 00:42:51,437 I asked this on Twitter the other day, evidently it's in reference to a person, -697 +699 00:42:51,437 --> 00:42:55,000 I think John Napier, and so it's in reference to him. -698 +700 00:42:55,560 --> 00:43:00,042 And then there was a truly terrible French pun about an exponential -699 +701 00:43:00,042 --> 00:43:04,460 and a logarithm walk into a bar and they order a beer and who pays? -700 +702 00:43:05,080 --> 00:43:10,033 And the answer is that the exponential has to pay because la logarithm n'imperillain, -701 +703 00:43:10,033 --> 00:43:13,546 which anyone who speaks French will like groan and laugh at, -702 +704 00:43:13,546 --> 00:43:15,620 but that made me laugh a little bit. -703 +705 00:43:16,740 --> 00:43:18,480 Do I have a personal vendetta against e? -704 +706 00:43:18,660 --> 00:43:20,680 Yeah, yeah I do, I think it's an overrated constant. -705 +707 00:43:20,940 --> 00:43:22,834 I think it's beautiful, but I think it's beautiful -706 +708 00:43:22,834 --> 00:43:24,580 in ways that aren't what people think they are. -707 +709 00:43:25,520 --> 00:43:28,742 And I also think that, I'm going to talk about this in a moment, we should write, -708 +710 00:43:28,742 --> 00:43:31,060 we shouldn't write the exponential function as e to the x, -709 +711 00:43:31,060 --> 00:43:34,440 because when it's more general that doesn't make sense and I think it confuses people. -710 +712 00:43:34,580 --> 00:43:37,992 We should just write it as what it is, which is a certain polynomial, -711 +713 00:43:37,992 --> 00:43:40,478 and just be honest up front rather than letting e, -712 +714 00:43:40,478 --> 00:43:43,940 like e has nothing to do with e to the pi i, that's a frustrating fact. -713 +715 00:43:44,100 --> 00:43:45,240 It shouldn't be in there. -714 -00:43:45,759 --> 00:43:48,887 +716 +00:43:45,760 --> 00:43:48,887 Anyway, German here is normal for how you do maths -715 +717 00:43:48,887 --> 00:43:51,340 on a ruled paper instead of graph paper. -716 +718 00:43:52,580 --> 00:43:55,543 I mean graph paper is definitely nicer, but I don't know, -717 +719 00:43:55,543 --> 00:43:57,740 this was the paper that I just had on hand. -718 +720 00:43:57,740 --> 00:44:01,573 And in general if you want to make any comments or questions about the lesson, -719 +721 00:44:01,573 --> 00:44:04,484 you can do so on Twitter with the hashtag locked down math, -720 +722 00:44:04,484 --> 00:44:06,280 and those will be pulled up as we go. -721 +723 00:44:06,600 --> 00:44:10,181 So it seems like we have strong consensus on our guess here, -722 +724 00:44:10,181 --> 00:44:15,406 which is people guessing that the correct answer to this limit is that it's around 0.69, -723 +725 00:44:15,406 --> 00:44:20,396 which I assume the reason everybody guessed that is because it's the correct answer, -724 +726 00:44:20,396 --> 00:44:23,390 that this limit does in fact approach around 0.69, -725 +727 00:44:23,390 --> 00:44:27,500 and we could play with Python if we wanted to see that experimentally. -726 +728 00:44:27,880 --> 00:44:31,738 Python's kind of overkill here, you could do it with any calculator, -727 +729 00:44:31,738 --> 00:44:35,373 but if I raise 2 to some small power, I get some kind of number, -728 +730 00:44:35,373 --> 00:44:38,617 and if I subtract 1 from that, then I get a small number, -729 +731 00:44:38,617 --> 00:44:43,202 and if I divide it by the same small power, so here I have three zeros and a one, -730 +732 00:44:43,202 --> 00:44:45,160 it looks like we get around 0.6931. -731 +733 00:44:45,160 --> 00:44:48,554 And if I made it a smaller value that I was doing, -732 +734 00:44:48,554 --> 00:44:52,880 it seems to stay pretty stable around there, it's around 0.69314. -733 -00:44:53,839 --> 00:44:57,480 +735 +00:44:53,840 --> 00:44:57,480 So congratulations to the majority of you who had the right guess here. -734 +736 00:44:58,140 --> 00:45:03,335 And in fact it's no coincidence that that's what it is, because like I said earlier, -735 +737 00:45:03,335 --> 00:45:06,820 if you're taking the derivative, where have I written it? -736 +738 00:45:07,060 --> 00:45:07,900 I've written it somewhere. -737 +739 00:45:08,980 --> 00:45:12,866 Sloppily as I want to do, I've written that if you take the derivative of something -738 +740 00:45:12,866 --> 00:45:16,660 that looks like a to the t, the constant sitting in front is the natural log of a. -739 +741 00:45:17,240 --> 00:45:22,352 So for something like 2, you would be looking at the natural log of 2, -740 +742 00:45:22,352 --> 00:45:24,440 which is in fact around 0.69. -741 +743 00:45:25,220 --> 00:45:29,760 Now, all of that was dependent on the fact that e to the x is its own derivative, right? -742 +744 00:45:30,280 --> 00:45:32,945 So there's one avenue that you could take here, -743 +745 00:45:32,945 --> 00:45:35,500 if you want to come up with a definition of e. -744 +746 00:45:35,500 --> 00:45:38,525 What you could say, and this is totally valid, -745 +747 00:45:38,525 --> 00:45:43,160 is the number e is defined to be the constant such that this limit is 1. -746 +748 00:45:43,860 --> 00:45:48,760 If that's the case, then e to the x is its own derivative, by definition, pretty much. -747 +749 00:45:49,100 --> 00:45:53,347 And then from there, you will get the fact that anything else, -748 +750 00:45:53,347 --> 00:45:58,000 its derivative can be expressed in terms of the log base e of itself. -749 +751 00:45:58,300 --> 00:45:59,360 That's one way that you could go. -750 +752 00:46:00,100 --> 00:46:04,838 Another avenue that you could take is to say that when we write e to the x, -751 +753 00:46:04,838 --> 00:46:08,080 this is actually shorthand for a certain polynomial. -752 +754 00:46:09,000 --> 00:46:13,242 I'm partial to this because I think this is an honest representation of the role -753 +755 00:46:13,242 --> 00:46:17,380 that it plays more generally, like when we start talking about complex numbers. -754 +756 00:46:18,160 --> 00:46:23,132 It's weird to me that in high school I saw Euler's formula as the polar representation -755 +757 00:46:23,132 --> 00:46:27,761 for complex numbers before it was ever really explained that e to the x does not -756 +758 00:46:27,761 --> 00:46:32,620 refer to the repeated multiplication, that it's a shorthand for this long polynomial. -757 +759 00:46:33,240 --> 00:46:37,380 You might give it another name, like exp, right? -758 +760 00:46:37,420 --> 00:46:40,900 And then that's something that it makes sense to plug in complex numbers to. -759 +761 00:46:41,420 --> 00:46:46,429 Traditionally, the way that you see this series in high school is you might go -760 +762 00:46:46,429 --> 00:46:51,756 through a calculus class where you learn about e to the x being its own derivative, -761 +763 00:46:51,756 --> 00:46:55,180 and then maybe at the end of a second calculus class, -762 +764 00:46:55,180 --> 00:47:00,444 the fact that it is its own derivative, in conjunction with a very wonderful topic -763 +765 00:47:00,444 --> 00:47:02,220 called Taylor series, right? -764 +766 00:47:02,340 --> 00:47:07,056 So it being its own derivative and Taylor series proves that e to the -765 +767 00:47:07,056 --> 00:47:11,840 x must equal this long polynomial, which is absolutely the case, right? -766 +768 00:47:11,980 --> 00:47:17,048 If you have a function that is its own derivative and at the value zero it equals one, -767 +769 00:47:17,048 --> 00:47:20,020 you will find that it has to equal this polynomial. -768 +770 00:47:20,500 --> 00:47:25,692 An alternate approach that you could take if you wanted in setting the foundations is -769 +771 00:47:25,692 --> 00:47:31,066 to say, don't worry about Taylor series, start with this sequence as a primitive object, -770 +772 00:47:31,066 --> 00:47:34,810 and then something we talked about a couple lectures ago was, -771 +773 00:47:34,810 --> 00:47:37,889 because of a nice property that this function has, -772 +774 00:47:37,889 --> 00:47:40,848 which is basically that when you add the inputs, -773 +775 00:47:40,848 --> 00:47:44,169 basically this polynomial behaves like an exponential, -774 +776 00:47:44,169 --> 00:47:49,302 and you can prove that just from the polynomial itself without calculus or anything, -775 +777 00:47:49,302 --> 00:47:52,140 exp of a plus b equals exp of a times exp of b. -776 +778 00:47:52,140 --> 00:47:54,368 And it's a very pleasing exercise to kind of work -777 +779 00:47:54,368 --> 00:47:56,240 out the expansion and see that that works. -778 +780 00:47:56,720 --> 00:48:00,827 And the fact that that works, we talked about this a couple lectures ago, -779 +781 00:48:00,827 --> 00:48:05,380 implies that the whole sequence looks like whatever exp of one is raised to the x. -780 +782 00:48:05,800 --> 00:48:09,165 So what you could say is the number e is defined to -781 +783 00:48:09,165 --> 00:48:12,660 be this particular sequence evaluated at x equals one. -782 +784 00:48:13,540 --> 00:48:16,501 And if you go that direction, that's all well and good, -783 +785 00:48:16,501 --> 00:48:20,415 and it becomes a kind of substantive thing to talk about e to the x being -784 +786 00:48:20,415 --> 00:48:21,420 its own derivative. -785 +787 00:48:21,420 --> 00:48:25,095 And it's one of the most pleasing exercises that you'll ever do, -786 +788 00:48:25,095 --> 00:48:30,183 because we can take a look at this, and if you know how to take derivatives of polynomial -787 +789 00:48:30,183 --> 00:48:35,216 terms, well let's just work it out actually, I'll turn over a new leaf so that it can be -788 +790 00:48:35,216 --> 00:48:36,460 nice and cleanly seen. -789 +791 00:48:37,020 --> 00:48:39,570 It's really one of the most pleasing, I don't know, -790 +792 00:48:39,570 --> 00:48:41,680 times that you'll have in a calculus class. -791 +793 00:48:41,740 --> 00:48:45,465 If you're just sitting, you're looking at this particular infinite polynomial, -792 +794 00:48:45,465 --> 00:48:48,720 and you're saying I wonder what the derivative of this happens to be. -793 +795 00:48:48,720 --> 00:48:53,920 And all you need to know is the power rule for polynomial terms, -794 +796 00:48:53,920 --> 00:48:58,240 and you'll say that the derivative, let me take d dx, -795 +797 00:48:58,240 --> 00:49:04,799 well the derivative of a constant ends up being zero, the derivative of x is one, -796 +798 00:49:04,799 --> 00:49:11,680 the derivative of x squared over two, you know you might think of that two as kind of -797 +799 00:49:11,680 --> 00:49:16,160 hopping down in front and leaving one less than itself, -798 +800 00:49:16,160 --> 00:49:23,279 so it becomes two times x to the one, just x to the one over two, and those twos cancel, -799 +801 00:49:23,279 --> 00:49:24,720 so we're adding x. -800 +802 00:49:25,580 --> 00:49:28,416 x cubed over three factorial, I might write this -801 +803 00:49:28,416 --> 00:49:31,080 out as x cubed over three times two times one. -802 -00:49:31,700 --> 00:49:37,413 -This ends up being three times x squared, you know the exponent kind of hopped - -803 -00:49:37,413 --> 00:49:43,923 -down and left behind one minus itself, over three times two times one, the threes cancel, - 804 -00:49:43,923 --> 00:49:49,420 -so we can see that that's actually the same as x squared over two factorial. +00:49:31,700 --> 00:49:37,514 +this ends up being three times x squared you know the exponential the exponent kind 805 +00:49:37,514 --> 00:49:43,259 +of hopped down and left behind one minus itself over three times two times one the + +806 +00:49:43,259 --> 00:49:49,420 +threes cancel so we can see that that's actually the same as x squared over two factorial + +807 00:49:49,980 --> 00:49:55,015 And in general, each one of our terms, as the exponent hops down, -806 +808 00:49:55,015 --> 00:49:59,821 it cancels out one of the things from the factorials below it, -807 +809 00:49:59,821 --> 00:50:05,620 and what we get is the exact same sequence but shifted, which is quite nice. -808 +810 00:50:05,620 --> 00:50:09,788 And like I said, the traditional way that you see this series is that you're -809 +811 00:50:09,788 --> 00:50:12,711 using the fact that e to the x is its own derivative, -810 +812 00:50:12,711 --> 00:50:16,337 in conjunction with Taylor series to show that it must equal this, -811 +813 00:50:16,337 --> 00:50:20,560 but if you start with this as a primitive, and you say this is the thing that -812 +814 00:50:20,560 --> 00:50:24,403 defines a special function for which we use the shorthand, e to the x, -813 +815 00:50:24,403 --> 00:50:28,408 then it feels a little bit more contentful and quite fun to say that e to -814 +816 00:50:28,408 --> 00:50:30,520 the x ends up being its own derivative. -815 +817 00:50:30,520 --> 00:50:35,293 And like we showed earlier, that then lets you take the derivative of all sorts of -816 +818 00:50:35,293 --> 00:50:40,067 other things, which in turn explains why we adopt the convention of writing all of -817 +819 00:50:40,067 --> 00:50:44,841 our exponentials as e to something times t, as opposed to writing them all as a to -818 +820 00:50:44,841 --> 00:50:49,960 something times t, even though those are equivalent and often weirdly hard to appreciate. -819 +821 00:50:50,980 --> 00:50:55,970 So, with all of that said, we can turn ourselves back in the direction of natural -820 +822 00:50:55,970 --> 00:51:00,900 logarithms, because let's say I wanted to know the derivative of the natural log. -821 -00:51:03,220 --> 00:51:07,191 -You might wonder why I want to know that, but if I have a deeper relationship - -822 -00:51:07,191 --> 00:51:11,264 -with the natural log of x, in terms not just of how it relates to these series, - 823 -00:51:11,264 --> 00:51:14,880 -but in all facets of math, maybe we can then start drawing connections. +00:51:03,220 --> 00:51:07,074 +you might wonder why i want to know that but if i have a bit you know a deeper 824 +00:51:07,074 --> 00:51:10,879 +relationship with the natural log of x in terms not just of how it relates to + +825 +00:51:10,879 --> 00:51:14,880 +these series but in all facets of math maybe we can then start drawing connections + +826 00:51:15,200 --> 00:51:18,488 And if you build up that relationship by knowing things like its derivative, -825 +827 00:51:18,488 --> 00:51:21,007 it actually helps you come back and understand things like -826 +828 00:51:21,007 --> 00:51:23,100 the alternating series we were looking at before. -827 +829 00:51:23,100 --> 00:51:28,405 So, can we use the fact that e to the x is its own -828 +830 00:51:28,405 --> 00:51:34,440 derivative to figure out the slope of a natural log curve? -829 +831 00:51:35,960 --> 00:51:41,014 Well, what that slope is asking us is to look at a given input x, -830 +832 00:51:41,014 --> 00:51:47,064 we consider a tiny step dx to the right, look at the corresponding step dy up, -831 +833 00:51:47,064 --> 00:51:50,740 and we want to understand the ratio, dy over dx. -832 +834 00:51:51,600 --> 00:51:55,943 Now, at this point, which has some kind of output y, -833 +835 00:51:55,943 --> 00:52:00,860 what we can say is by definition, y is the natural log of x. -834 +836 00:52:01,880 --> 00:52:06,120 Now, this is the same statement as saying e, have I written this right? -835 +837 00:52:06,240 --> 00:52:07,200 y is natural, yeah, great. -836 +838 00:52:07,800 --> 00:52:11,460 So, this is the same as saying e to the y is equal to x. -837 +839 00:52:12,520 --> 00:52:15,544 Now, from there, I can understand the relationship between -838 +840 00:52:15,544 --> 00:52:18,620 tiny nudges to x and tiny nudges to y by taking derivatives. -839 +841 00:52:18,620 --> 00:52:24,040 If I ask about some tiny nudge to the value x and the corresponding -840 +842 00:52:24,040 --> 00:52:28,982 tiny nudge to e to the y, well, what it means for e to the x, -841 +843 00:52:28,982 --> 00:52:33,286 or in this case, e to the y to be its own derivative, -842 +844 00:52:33,286 --> 00:52:39,902 is that the size of that tiny nudge is e to whatever the y value at that point is, -843 +845 00:52:39,902 --> 00:52:40,620 times dy. -844 +846 00:52:42,420 --> 00:52:44,500 And we're saying that that equals dx. -845 +847 00:52:45,380 --> 00:52:51,640 And what this lets us do then is express the slope that we want, dy over dx. -846 +848 00:52:52,380 --> 00:52:56,340 If we just rearrange things, it looks like 1 divided by e to the y. -847 +849 00:52:56,800 --> 00:53:01,359 So, what this is saying is that if we look at our graph, it's got some x coordinate, -848 +850 00:53:01,359 --> 00:53:04,416 some y coordinate, and I want to know what the slope is, -849 +851 00:53:04,416 --> 00:53:06,240 this change to y over change in x. -850 +852 00:53:06,240 --> 00:53:09,292 I can't immediately express it in terms of x, maybe, -851 +853 00:53:09,292 --> 00:53:14,188 but I do know whatever this value of y is, if I take e to the power of that and then -852 +854 00:53:14,188 --> 00:53:16,320 reciprocate, that gives me the slope. -853 +855 00:53:17,360 --> 00:53:21,953 But of course, what it means to be on our graph is that y is the natural log of x, -854 +856 00:53:21,953 --> 00:53:24,665 which is the same as saying e to the y equals x, -855 +857 00:53:24,665 --> 00:53:27,820 so this whole thing is the same as taking 1 divided by x. -856 +858 00:53:28,320 --> 00:53:32,016 So if I want to know that slope, I can say, what is your x coordinate, -857 +859 00:53:32,016 --> 00:53:35,660 take 1 divided by that, and that gets me the slope of the natural log. -858 +860 00:53:37,080 --> 00:53:40,320 Which is, we've just gone through a process called implicit differentiation. -859 +861 00:53:41,580 --> 00:53:45,739 If you're not inclined to believe that this manipulation is legitimate, -860 +862 00:53:45,739 --> 00:53:49,089 that we can just move around the dx's and dy's like that, -861 +863 00:53:49,089 --> 00:53:53,422 I have a whole video about implicit differentiation in the calculus series -862 +864 00:53:53,422 --> 00:53:55,040 that you can take a look at. -863 +865 00:53:55,200 --> 00:53:59,128 But the point for us is that we have a very nice fact, -864 +866 00:53:59,128 --> 00:54:03,200 that the derivative of ln of x looks like 1 divided by x. -865 +867 00:54:03,860 --> 00:54:08,397 And that's quite nice, and it kind of passes a gut check that ln of x gets -866 +868 00:54:08,397 --> 00:54:13,540 shallower and shallower as you go on, which means the slope gets smaller and smaller. -867 +869 00:54:14,000 --> 00:54:17,620 And the graph of 1 over x, you know, what does that look like? -868 +870 00:54:19,220 --> 00:54:25,480 Well, at the input, let's say we have the input 1 somewhere like here, it'll be at 1. -869 +871 00:54:26,220 --> 00:54:28,800 At the input 2, it'll be sitting at a half. -870 +872 00:54:29,360 --> 00:54:32,000 At the input 3, it'll be sitting at a third. -871 +873 00:54:32,000 --> 00:54:36,660 And in general, it gets lower and lower and closer to 0. -872 +874 00:54:38,500 --> 00:54:41,354 So the idea that this would describe the slope of that, you know, -873 +875 00:54:41,354 --> 00:54:43,473 something that gets lower and lower closer to 0, -874 +876 00:54:43,473 --> 00:54:45,420 seems to pass a little bit of a sanity check. -875 +877 00:54:46,260 --> 00:54:49,115 Now, the relevance that this is going to have to us -876 +878 00:54:49,115 --> 00:54:52,080 will involve the inverse operation to differentiation. -877 +879 00:54:52,080 --> 00:54:58,246 So instead of talking about what is the slope of the natural log curve, -878 +880 00:54:58,246 --> 00:55:03,900 what I might do is ask about the area under this particular curve. -879 +881 00:55:04,100 --> 00:55:07,720 Let's say I take the area up to... -880 +882 00:55:07,720 --> 00:55:11,880 my stomach was just rumbling, I don't know if that's audible on the microphone. -881 +883 00:55:13,160 --> 00:55:15,180 Clearly, gotta eat lunch before these things. -882 +884 00:55:16,400 --> 00:55:20,160 So let's say I want to understand the area up to n of something like this. -883 -00:55:21,720 --> 00:55:27,013 -What that involves is taking the integral between +885 +00:55:21,720 --> 00:55:26,724 +kay what that what that involves is taking the integral -884 -00:55:27,013 --> 00:55:31,460 -1 and our value n of 1 divided by x by dx. +886 +00:55:26,724 --> 00:55:31,460 +between one and our value n of one divided by x by dx -885 +887 00:55:32,920 --> 00:55:35,004 Now this actually looks quite similar in spirit, -886 +888 00:55:35,004 --> 00:55:37,770 the idea of adding up a bunch of things that look like 1 over x, -887 +889 00:55:37,770 --> 00:55:39,260 to what we were looking at earlier. -888 +890 00:55:40,120 --> 00:55:40,700 How much earlier? -889 +891 00:55:40,960 --> 00:55:41,580 I guess over here. -890 +892 00:55:41,720 --> 00:55:45,180 Where we were adding up 1 plus a half plus a third plus a fourth, on and on. -891 +893 00:55:45,180 --> 00:55:48,868 And already it gives a little bit of an intuitive instinct for -892 +894 00:55:48,868 --> 00:55:52,440 why something like this sum would be related to natural logs. -893 +895 00:55:52,920 --> 00:55:55,704 Because we now know that in calculus land, natural logs -894 +896 00:55:55,704 --> 00:55:58,340 are intimately related to the idea of 1 divided by x. -895 +897 00:55:59,020 --> 00:56:02,663 But I want you to think of this spelled out a little bit more exactly, -896 +898 00:56:02,663 --> 00:56:05,280 and so we'll pop on over to our quiz one more time. -897 +899 00:56:05,680 --> 00:56:07,100 Second to last question for today. -898 +900 00:56:09,520 --> 00:56:14,100 -And the question asks us, we're going to let s be the +and the question asks us all right we're gonna let s be the -899 +901 00:56:14,100 --> 00:56:18,680 -sum from n equals 1 up to capital N of 1 divided by n. +sum from n equals one up to capital n of one divided by n ok -900 +902 00:56:19,180 --> 00:56:19,780 That's s. -901 +903 00:56:20,160 --> 00:56:23,868 And then we're going to let i be an analogous integral, -902 +904 00:56:23,868 --> 00:56:27,180 where we're integrating dx over x between 1 and n. -903 +905 00:56:27,600 --> 00:56:30,200 And it asks you to compare s and i. -904 +906 00:56:31,720 --> 00:56:34,040 -I'll give you a moment to think about that. +kay i'll give you s and i okay i'll give you a moment to think about that s -905 +907 00:57:03,540 --> 00:57:06,760 Interestingly, we don't have a ton of consensus around this one. -906 +908 00:57:07,100 --> 00:57:10,840 So there's only three options, and we've got a nice split. -907 +909 00:57:11,100 --> 00:57:14,630 And as you guys know, this is actually one of my favorite things when -908 +910 00:57:14,630 --> 00:57:18,059 we do any of these lockdown live quizzes, is when it's not everyone -909 +911 00:57:18,059 --> 00:57:21,640 hopping on to one particular thing, but we have a division among folks. -910 +912 00:57:22,120 --> 00:57:23,260 And I think that's great. -911 +913 00:57:23,740 --> 00:57:31,020 -I'm curious actually what the answer is going to be here. +i'm curious i'm curious actually what the uh what the answer is going to be here -912 +914 00:57:31,020 --> 00:57:35,320 And in fact, even if it's not been enough time to thoroughly think through, -913 +915 00:57:35,320 --> 00:57:39,960 I'm going to go ahead and grade it, just so that we can see what it happens to be. -914 -00:57:40,300 --> 00:57:43,397 -And a lot of the spirit of it is that you kind of hazard a guess, +916 +00:57:40,300 --> 00:57:43,754 +and a lot of these the spirit of it is that you kind of hazard a guess so feel -915 -00:57:43,397 --> 00:57:47,340 -so feel no shame if you entered an answer and it's not what turns out to be correct. +917 +00:57:43,754 --> 00:57:47,340 +no shame if you entered an answer and then it's not what turns out to be correct s -916 +918 00:57:48,080 --> 00:57:53,220 So in this case, the sum does in fact end up being bigger than the integral. -917 +919 00:57:54,280 --> 00:57:58,040 -And it looks like 900 of you got that correct, which is awesome. +nd the and it looks like uh 900 of you got that correct which is awesome an -918 +920 00:57:58,040 --> 00:58:02,120 And then following that was people thinking that it was less. -919 +921 00:58:02,540 --> 00:58:05,624 And then to those in B thinking that they were identical, you know, -920 +922 00:58:05,624 --> 00:58:08,800 that's a reasonable thought, because they're such similar expressions. -921 +923 00:58:09,260 --> 00:58:13,750 But there's a picture that really makes the answer kind of shine out to us here, -922 +924 00:58:13,750 --> 00:58:18,074 which is, if I look at the curve 1 over x, which is what this white curve is, -923 +925 00:58:18,074 --> 00:58:21,567 it's 1 over x, and then I'm going to consider a bunch of bars, -924 +926 00:58:21,567 --> 00:58:25,060 each of whose area corresponds to 1 over n for some value of n. -925 +927 00:58:25,060 --> 00:58:30,719 So for example, for the value 1, this bar has a width of 1, and then the height is 1, -926 +928 00:58:30,719 --> 00:58:35,326 and that means that right above the input 1 on its upper left corner, -927 +929 00:58:35,326 --> 00:58:36,840 it's hitting the graph. -928 +930 00:58:37,700 --> 00:58:41,735 Now for the next term, if I want 1 over 2, that means it's going to -929 +931 00:58:41,735 --> 00:58:46,245 hit the graph above the input 2, since the graph is defined to be 1 over x, -930 +932 00:58:46,245 --> 00:58:50,162 so its upper left corner hits that, and then the area of this bar -931 +933 00:58:50,162 --> 00:58:53,960 whose height is 1 half is, well, 1 half, because its width is 1. -932 +934 00:58:54,460 --> 00:58:58,656 Similarly, this bar has an area of 1 third, this bar has an area of 1 fourth, -933 +935 00:58:58,656 --> 00:59:02,690 and so what you have is a sequence of rectangles whose total area is going -934 +936 00:59:02,690 --> 00:59:06,079 to be similar to the area under the curve, definitely similar, -935 +937 00:59:06,079 --> 00:59:10,760 but you can tell that it's going to be bigger, because some of the area is leaking out. -936 +938 00:59:11,000 --> 00:59:14,719 In this context, we've got a lot of area leaking out from the first bar, -937 +939 00:59:14,719 --> 00:59:17,878 a little bit less leaking out from the second, and on and on, -938 +940 00:59:17,878 --> 00:59:20,272 but as you go, because the graph flattens out, -939 +941 00:59:20,272 --> 00:59:23,737 it becomes quite a good approximation once you account for the area -940 +942 00:59:23,737 --> 00:59:24,960 that's leaked out there. -941 +943 00:59:25,500 --> 00:59:27,835 Now something kind of bizarre is happening here, -942 +944 00:59:27,835 --> 00:59:31,504 where usually we think of these rectangles as being something like a Riemann -943 +945 00:59:31,504 --> 00:59:33,792 sum that defines integration, where we say, oh, -944 +946 00:59:33,792 --> 00:59:37,557 we don't know what the area under a curve is, but we like areas of rectangles, -945 +947 00:59:37,557 --> 00:59:39,940 so we use the rectangles to approximate the curve. -946 +948 00:59:40,300 --> 00:59:42,700 Here, we're going to do something that's backwards to that. -947 +949 00:59:42,700 --> 00:59:45,880 If we know calculus, we do know the area under the curve. -948 +950 00:59:45,980 --> 00:59:46,860 It's very nice. -949 +951 00:59:46,860 --> 00:59:49,940 It involves the antiderivative of 1 over x, like we'll show in a moment. -950 +952 00:59:50,200 --> 00:59:52,980 What we don't know is the sum of the areas of the rectangles. -951 +953 00:59:53,340 --> 00:59:56,260 That was the sum that we were looking at earlier and trying to understand. -952 +954 00:59:56,700 --> 00:59:59,822 So here we're going backwards and using the area under a curve to -953 +955 00:59:59,822 --> 01:00:03,040 approximate the area of a bunch of rectangles, which I think is fun. -954 +956 01:00:03,120 --> 01:00:05,420 It shows that calculus has this back and forth. -955 +957 01:00:05,460 --> 01:00:08,548 It's not just geometry informing understanding of curves, -956 +958 01:00:08,548 --> 01:00:13,129 but it's an understanding of curves informing an understanding of geometry and number -957 +959 01:00:13,129 --> 01:00:14,780 theory and things of that sort. -958 +960 01:00:15,660 --> 01:00:21,119 So what this means for us is that if we take a look back at our paper and we look at -959 +961 01:00:21,119 --> 01:00:26,642 my much more sloppily drawn graph than the beautiful exact illustrations can give us, -960 +962 01:00:26,642 --> 01:00:30,367 if we want to understand that area, taking this integral, -961 +963 01:00:30,367 --> 01:00:35,762 the task is to do an inverse derivative, to ask what function has a derivative that -962 +964 01:00:35,762 --> 01:00:37,240 equals the inside here. -963 +965 01:00:37,240 --> 01:00:40,780 If that's something you haven't learned about, again, calculus series. -964 +966 01:00:41,400 --> 01:00:44,124 Take a look at the fundamental theorem of calculus video, -965 +967 01:00:44,124 --> 01:00:47,177 or even the first video in that series I think shows a little of -966 +968 01:00:47,177 --> 01:00:50,560 an instinct for why you have this relationship between slopes and areas. -967 +969 01:00:51,280 --> 01:00:55,261 But what it means for us is that we take the inverse derivative, -968 +970 01:00:55,261 --> 01:00:59,427 which we now know is the natural log, the thing whose derivative is -969 +971 01:00:59,427 --> 01:01:03,960 1 over x is the natural log, and we evaluate it at the bounds, at n and 1. -970 +972 01:01:04,740 --> 01:01:09,049 And this notation where I kind of put brackets around it and then a number -971 +973 01:01:09,049 --> 01:01:13,530 in the upper right corner and lower right corner means I take that expression -972 +974 01:01:13,530 --> 01:01:17,380 evaluated at the top minus that expression evaluated at the bottom. -973 +975 01:01:19,080 --> 01:01:24,700 And that, well, natural log of 1, what is that? -974 +976 01:01:25,620 --> 01:01:26,860 e to the what equals 1? -975 +977 01:01:27,920 --> 01:01:28,600 Well, it's 0. -976 -01:01:29,759 --> 01:01:32,300 +978 +01:01:29,760 --> 01:01:32,300 Pretty much anything to the 0 will equal 1. -977 +979 01:01:32,900 --> 01:01:39,040 So this term goes away entirely, and what we're left with is the natural log of n. -978 +980 01:01:39,980 --> 01:01:46,956 And what this means for us is if we were using our rectangles to approximate the sum, -979 +981 01:01:46,956 --> 01:01:51,419 or using the integral to approximate those rectangles, -980 +982 01:01:51,419 --> 01:01:56,773 it's saying that 1 over 1 plus 1 over 2 plus 1 over 3, on and on, -981 +983 01:01:56,773 --> 01:02:01,560 up to a given bound is about equal to the natural log of n. -982 +984 01:02:02,300 --> 01:02:07,861 And more specifically, if you were to account for how much area is leaking out here, -983 +985 01:02:07,861 --> 01:02:10,020 that area actually does converge. -984 +986 01:02:10,020 --> 01:02:16,665 As n tends towards infinity, the area that's leaked out approaches a certain constant, -985 +987 01:02:16,665 --> 01:02:21,782 and it's called Euler's constant, or the Euler-Macheroni constant, -986 +988 01:02:21,782 --> 01:02:24,380 and it happens to be around 0.577. -987 +989 01:02:25,220 --> 01:02:27,916 So in the same way that pi and e are constants of nature, -988 +990 01:02:27,916 --> 01:02:30,800 this is another constant of nature, also bearing Euler's name. -989 +991 01:02:31,380 --> 01:02:35,629 And what it describes is the deviation between this sum, -990 +992 01:02:35,629 --> 01:02:42,040 often called the harmonic sum, and the natural log of x, a thing that is related to e. -991 +993 01:02:42,480 --> 01:02:45,497 So Euler has really got his fingerprints all over the situation, -992 +994 01:02:45,497 --> 01:02:48,700 at least as far as naming is concerned in our little expression here. -993 +995 01:02:49,060 --> 01:02:50,080 So that's quite nice. -994 +996 01:02:50,300 --> 01:02:51,040 That's quite fun. -995 +997 01:02:51,920 --> 01:02:54,580 But that only answers one of the mysteries that we had earlier. -996 +998 01:02:54,580 --> 01:02:58,879 Because if you remember, I opened this whole thing up by talking not just -997 +999 01:02:58,879 --> 01:03:03,120 about this series that grows like the natural log, we also alternated it. -998 +1000 01:03:03,400 --> 01:03:05,970 We went 1 minus a half plus a third minus a fourth, -999 +1001 01:03:05,970 --> 01:03:08,540 and the claim is that that was the natural log of 2. -1000 +1002 01:03:09,360 --> 01:03:11,620 So let's see if we can try to understand why that's true. -1001 +1003 01:03:12,260 --> 01:03:17,393 And I might actually postpone explaining the even more bizarre fact that this -1002 +1004 01:03:17,393 --> 01:03:23,184 interrelates with primes in a certain way, depending on how long I want this particular -1003 +1005 01:03:23,184 --> 01:03:24,040 stream to go. -1004 +1006 01:03:24,580 --> 01:03:26,956 But at least finish off by understanding the alternating series, -1005 +1007 01:03:26,956 --> 01:03:28,200 because it's extremely satisfying. -1006 +1008 01:03:29,660 --> 01:03:33,220 So to do that, let me just rewrite what our series looks like. -1007 +1009 01:03:36,740 --> 01:03:41,004 And this is one of those things where, as I go through the answer, -1008 +1010 01:03:41,004 --> 01:03:42,660 it has a feeling of magic. -1009 +1011 01:03:43,220 --> 01:03:45,160 And sometimes not in a great way. -1010 +1012 01:03:45,660 --> 01:03:49,286 You might find yourself looking at how we go about this and asking, -1011 +1013 01:03:49,286 --> 01:03:51,900 how on earth would anyone ever come up with that? -1012 -01:03:52,779 --> 01:03:56,799 +1014 +01:03:52,780 --> 01:03:56,799 And maybe after we plop it all down, we can try to introspect and think about the -1013 +1015 01:03:56,799 --> 01:04:00,720 reasonable ways that someone would come up with the following line of reasoning. -1014 +1016 01:04:01,240 --> 01:04:04,490 But it is not unique to this situation, it's kind -1015 +1017 01:04:04,490 --> 01:04:07,480 of a useful set of tricks to be familiar with. -1016 +1018 01:04:07,900 --> 01:04:09,640 And there's a couple general principles in there. -1017 +1019 01:04:09,800 --> 01:04:13,198 The first general principle is that if we have a hard question, -1018 +1020 01:04:13,198 --> 01:04:15,960 in this case figuring out what this sum approaches, -1019 +1021 01:04:15,960 --> 01:04:19,040 bizarrely it can become easier if we make it more general. -1020 +1022 01:04:19,040 --> 01:04:23,785 You might think that making things more general would make it harder, -1021 +1023 01:04:23,785 --> 01:04:27,040 because you have to answer a more powerful fact. -1022 +1024 01:04:28,580 --> 01:04:33,013 But math does this bizarre thing, where sometimes by trying to make it more general, -1023 +1025 01:04:33,013 --> 01:04:35,360 you actually make the problem more tractable. -1024 +1026 01:04:36,480 --> 01:04:41,572 Which is quite cool, actually, because what it means is when a mathematician -1025 +1027 01:04:41,572 --> 01:04:44,945 is motivated only by making their own life easier, -1026 +1028 01:04:44,945 --> 01:04:49,706 it has the strange effect of making their results applicable to a wider -1027 +1029 01:04:49,706 --> 01:04:51,360 variety of circumstances. -1028 +1030 01:04:52,600 --> 01:04:55,418 So the way I'm going to generalize this, again, -1029 +1031 01:04:55,418 --> 01:04:59,940 it might look kind of bizarre and unmotivated, but run with me for a second, -1030 +1032 01:04:59,940 --> 01:05:04,462 is rather than thinking of a single value, I'm going to put an x in here and -1031 +1033 01:05:04,462 --> 01:05:09,102 consider this a function where I'm taking x over 1 minus x squared over 2 plus -1032 +1034 01:05:09,102 --> 01:05:11,040 x cubed over 3, on and on and on. -1033 +1035 01:05:11,960 --> 01:05:15,973 And I want to know, in general, what does this approach for various values of x, -1034 +1036 01:05:15,973 --> 01:05:18,600 and then I just have to plug in the value x equals 1. -1035 +1037 01:05:19,660 --> 01:05:22,260 Now, like I said, that might make it seem harder, infinitely harder. -1036 +1038 01:05:22,540 --> 01:05:24,133 Previously we just had to know one value, now -1037 +1039 01:05:24,133 --> 01:05:25,900 you're asking me to compute infinitely many values? -1038 +1040 01:05:25,900 --> 01:05:30,116 But if you know calculus, you might recognize that the exponents of -1039 +1041 01:05:30,116 --> 01:05:34,580 your polynomial terms might just play nicely with the denominators here. -1040 +1042 01:05:35,080 --> 01:05:39,546 And in particular, if we were to take the derivative of this series, -1041 +1043 01:05:39,546 --> 01:05:41,100 it behaves quite nicely. -1042 +1044 01:05:41,540 --> 01:05:45,076 The derivative of x is 1, the derivative of x squared over 2, -1043 +1045 01:05:45,076 --> 01:05:49,640 well that 2 hops down and cancels out the denominator, so it becomes negative x. -1044 +1046 01:05:49,640 --> 01:05:54,900 Similarly, that 3 hops down and cancels the denominator, so it becomes x squared. -1045 +1047 01:05:55,980 --> 01:06:01,087 And while you might not know why we're taking a derivative of something here, -1046 +1048 01:06:01,087 --> 01:06:06,064 and how that would be helpful for actually evaluating the ultimate sum that -1047 +1049 01:06:06,064 --> 01:06:11,696 we care about, it is an interesting fact, and it's something that is playful and fun, -1048 +1050 01:06:11,696 --> 01:06:16,280 that we've somehow simplified the expression by taking its derivative. -1049 +1051 01:06:16,820 --> 01:06:20,153 And the simplification is actually quite important, -1050 +1052 01:06:20,153 --> 01:06:24,383 because there's a well-known fact within math that you can take a -1051 +1053 01:06:24,383 --> 01:06:29,640 series where each term is the product of the last with a constant kind of product. -1052 +1054 01:06:29,840 --> 01:06:34,665 So here, as we go from one term to the next, we're always multiplying by negative x, -1053 +1055 01:06:34,665 --> 01:06:38,468 so to go from negative x to x squared, you multiply by negative x, -1054 +1056 01:06:38,468 --> 01:06:43,180 and then similarly x squared to negative x cubed, you're multiplying by negative x. -1055 +1057 01:06:44,040 --> 01:06:49,086 And when that's the case, the series as a whole is going to approach 1 divided by, -1056 +1058 01:06:49,086 --> 01:06:52,066 or wherever you start, but here we started at 1, -1057 +1059 01:06:52,066 --> 01:06:56,930 so the thing you started at divided by 1 minus the thing that you're constantly -1058 +1060 01:06:56,930 --> 01:06:59,120 multiplying by, which is negative x. -1059 +1061 01:06:59,920 --> 01:07:04,087 So to give another example of where this comes up, -1060 +1062 01:07:04,087 --> 01:07:09,479 is if we were to take something like 1 plus 1 half plus 1 fourth, -1061 +1063 01:07:09,479 --> 01:07:15,689 where each time in our sequence we are multiplying the last term by 1 half, -1062 +1064 01:07:15,689 --> 01:07:22,880 this will equal 1 divided by 1 minus the thing we were multiplying by, which was 1 half. -1063 +1065 01:07:24,100 --> 01:07:27,140 And 1 divided by 1 minus 1 half ends up being the same as 2. -1064 +1066 01:07:27,620 --> 01:07:33,264 And that actually kind of feels intuitive, that if we take 1 plus 1 half plus 1 fourth -1065 +1067 01:07:33,264 --> 01:07:36,507 plus 1 eighth, you could even draw out a picture, -1066 +1068 01:07:36,507 --> 01:07:41,243 where let's say I've got a rectangle whose side length is 1, and 1 here, -1067 +1069 01:07:41,243 --> 01:07:46,044 I can say the 1 represents this area, and then half represents this area, -1068 +1070 01:07:46,044 --> 01:07:50,974 and then a fourth represents this area, and an eighth represents that area, -1069 +1071 01:07:50,974 --> 01:07:55,840 and kind of keep playing this game, and eventually it'll fill an area of 2. -1070 +1072 01:07:56,840 --> 01:07:59,886 Now the more general version of that is this geometric sum, -1071 +1073 01:07:59,886 --> 01:08:04,150 which someone who's done a lot of problem solving in math is able to recognize kind -1072 +1074 01:08:04,150 --> 01:08:08,212 of quickly, which is why they might enjoy this series much more than they would -1073 +1075 01:08:08,212 --> 01:08:09,380 enjoy the one above it. -1074 +1076 01:08:11,120 --> 01:08:15,560 So this whole thing ends up looking like 1 divided by 1 plus x. -1075 +1077 01:08:16,399 --> 01:08:16,560 Great. -1076 +1078 01:08:17,160 --> 01:08:21,663 But what this suggests is that if we somehow take an antiderivative, -1077 +1079 01:08:21,663 --> 01:08:26,949 if we somehow integrate this, we might have an alternate expression for what the -1078 +1080 01:08:26,949 --> 01:08:28,319 initial sequence was. -1079 +1081 01:08:28,319 --> 01:08:32,285 So from here, I'm going to go ahead and pose a quiz, -1080 +1082 01:08:32,285 --> 01:08:38,421 and part of this quiz is seeing who in the audience is comfortable with calculus, -1081 +1083 01:08:38,421 --> 01:08:43,060 and again if you're not, calculus series, go and check it out. -1082 +1084 01:08:43,819 --> 01:08:48,194 But what we have here is the question, what is the -1083 +1085 01:08:48,194 --> 01:08:52,740 integral from 0 up to 1 of 1 divided by 1 plus x, dx? -1084 +1086 01:08:52,740 --> 01:08:59,120 I want you to evaluate that integral, and I'll give you a little moment for this one. -1085 -01:09:40,820 --> 01:09:41,180 +1087 +01:09:40,819 --> 01:09:41,180 So, let's see. -1086 +1088 01:09:41,180 --> 01:09:44,185 And you know, tell you what, while answers are rolling in, before locking it in, -1087 +1089 01:09:44,185 --> 01:09:46,560 I'm going to go ahead and just start describing the answer here. -1088 +1090 01:09:46,560 --> 01:09:54,149 So, if you want to know the integral from 0 up to 1 of 1 divided by 1 plus x, dx, -1089 +1091 01:09:54,149 --> 01:10:00,997 well we know that the antiderivative of 1 over x is the natural log of x, -1090 +1092 01:10:00,997 --> 01:10:08,401 so it's going to be the natural log of that inside divided by the derivative of -1091 +1093 01:10:08,401 --> 01:10:09,420 the inside. -1092 +1094 01:10:09,500 --> 01:10:12,700 That's kind of the inverse chain rule, or something you can get with u-substitution. -1093 +1095 01:10:12,700 --> 01:10:16,299 But the derivative of the inside is just 1, so you can check yourself that if -1094 +1096 01:10:16,299 --> 01:10:19,852 you take the derivative of this, you get 1 over the inside, 1 over 1 plus x, -1095 +1097 01:10:19,852 --> 01:10:23,360 but then the chain rule just has you multiplying by 1, so it stays the same. -1096 +1098 01:10:23,900 --> 01:10:27,570 So then we evaluate this at the bounds, 1 and 0, -1097 +1099 01:10:27,570 --> 01:10:32,363 and what this ends up getting us is the natural log at the top, -1098 +1100 01:10:32,363 --> 01:10:38,880 which is 1 plus 1, minus the natural log of 1 plus x at the bottom, which was 1 plus 0. -1099 +1101 01:10:39,960 --> 01:10:43,038 The natural log of 1 plus 1 is of course ln of 2, -1100 +1102 01:10:43,038 --> 01:10:46,980 and then we're subtracting off the natural log of 1, which is 0. -1101 +1103 01:10:47,740 --> 01:10:51,720 So, the proper answer here comes out to be the natural log of 2. -1102 +1104 01:10:53,520 --> 01:10:56,803 And it looks like 1600 of you have correctly answered that, -1103 +1105 01:10:56,803 --> 01:10:58,500 so well done, well done indeed. -1104 +1106 01:10:59,020 --> 01:11:04,286 And if you wanted to kind of visualize this in your head or get some sort of gut -1105 +1107 01:11:04,286 --> 01:11:08,121 instinct on which of those answers seemed loosely correct, -1106 +1108 01:11:08,121 --> 01:11:11,827 even if you didn't know how to calculate it immediately, -1107 +1109 01:11:11,827 --> 01:11:16,963 the graph of 1 over 1 plus x is going to look just like the graph of 1 over x, -1108 +1110 01:11:16,963 --> 01:11:22,164 but shifted to the left, so it's actually going to pass through the input 0, 1, -1109 +1111 01:11:22,164 --> 01:11:25,220 and then we're looking for the area under here. -1110 +1112 01:11:25,640 --> 01:11:29,706 So you know that it's going to be an area somewhere between 0 and 1, -1111 +1113 01:11:29,706 --> 01:11:34,717 probably filling up more than a half of it, and the natural log of 2 is around 0.69, -1112 +1114 01:11:34,717 --> 01:11:36,780 so that actually seems about right. -1113 +1115 01:11:37,360 --> 01:11:42,832 But what's quite cool here is that if we say that the, -1114 +1116 01:11:42,832 --> 01:11:49,796 well I can just write it out again here, if I say I want to integrate -1115 +1117 01:11:49,796 --> 01:11:56,960 this bottom expression from 0 up to 1, which is the same as integrating -1116 +1118 01:11:56,960 --> 01:12:04,920 1 over 1 plus x from 0 up to 1, I know that that should be the value 2, ln of 2. -1117 +1119 01:12:06,040 --> 01:12:06,440 Okay? -1118 -01:12:07,360 --> 01:12:12,079 -But on the other hand, if I picture taking my antiderivative and getting this - -1119 -01:12:12,079 --> 01:12:15,467 -whole sequence here, and evaluating it between 0 and 1, - 1120 -01:12:15,467 --> 01:12:20,731 -what I'm doing is I'm plugging in the number 1, which gets me my alternating sequence, +01:12:07,360 --> 01:12:11,979 +but on the other hand if I picture taking my anti-derivative and getting this 1121 -01:12:20,731 --> 01:12:25,451 -then I'm subtracting off the value of 0, which when I plug it in just gets 0, +01:12:11,979 --> 01:12:16,480 +whole sequence here and evaluating it between 0 and 1 what I'm doing is I'm 1122 -01:12:25,451 --> 01:12:29,928 -so evaluating this at 0 and 1 is the same thing as integrating the bottom +01:12:16,480 --> 01:12:21,454 +plugging in the number 1 which gets me my alternating sequence then I'm subtracting 1123 -01:12:29,928 --> 01:12:34,708 -expression from 0 to 1, which is the same thing as integrating 1 over 1 plus x +01:12:21,454 --> 01:12:26,133 +off the value of 0 which when I plug it in just gets 0 so evaluating this at 0 1124 -01:12:34,708 --> 01:12:37,673 -from 0 to 1, which is the same thing as ln of 2, +01:12:26,133 --> 01:12:30,812 +and 1 is the same thing as integrating the bottom expression from 0 to 1 which 1125 -01:12:37,673 --> 01:12:40,880 -and hence that whole top thing ends up being ln of 2. +01:12:30,812 --> 01:12:35,490 +is the same thing as integrating 1 over 1 plus x from 0 to 1 which is the same 1126 +01:12:35,490 --> 01:12:40,406 +thing as natural log of 2 and hence that whole top thing ends up being the natural + +1127 +01:12:40,406 --> 01:12:40,880 +log of 2 + +1128 01:12:41,340 --> 01:12:41,900 Very clever! -1127 +1129 01:12:42,360 --> 01:12:45,458 Just such a sneaky sequence of manipulations that, like I said, -1128 +1130 01:12:45,458 --> 01:12:49,137 if you just see it plopped down, almost as intimidating because you wonder, -1129 +1131 01:12:49,137 --> 01:12:53,203 how on earth do you go from seeing this sequence up here to thinking about, ah yes, -1130 +1132 01:12:53,203 --> 01:12:57,221 if I generalize it with lots of powers of x and then take the derivatives of those -1131 +1133 01:12:57,221 --> 01:13:01,287 and then use a geometric series sum and then integrate that using natural log, yes, -1132 +1134 01:13:01,287 --> 01:13:02,740 then it'll all become obvious. -1133 +1135 01:13:03,380 --> 01:13:05,141 And I think the answer is that that's probably -1134 +1136 01:13:05,141 --> 01:13:06,940 not what the problem solving process looks like. -1135 +1137 01:13:07,220 --> 01:13:10,113 Instead, you build up a familiarity with a lot of these things, -1136 +1138 01:13:10,113 --> 01:13:12,826 like derivatives of polynomial terms, and then you build up -1137 +1139 01:13:12,826 --> 01:13:15,720 familiarity with other things, like derivatives of natural logs. -1138 +1140 01:13:16,040 --> 01:13:19,296 And the more familiarity you have with a lot of different pieces of math, -1139 +1141 01:13:19,296 --> 01:13:21,584 then sometimes when you see one particular pattern, -1140 +1142 01:13:21,584 --> 01:13:25,500 you're able to draw in your mind a connection to things that make you look like a genius. -1141 +1143 01:13:25,500 --> 01:13:27,480 I think Euler did this all the time. -1142 +1144 01:13:27,580 --> 01:13:30,150 If you look at some of the great discoveries of Euler, -1143 +1145 01:13:30,150 --> 01:13:31,880 they just come really out of nowhere. -1144 +1146 01:13:32,240 --> 01:13:35,854 I mean the very opening thing that I talked about, -1145 +1147 01:13:35,854 --> 01:13:40,320 I guess it wasn't the very opening, but early on, where are we? -1146 +1148 01:13:41,240 --> 01:13:42,500 I seem to have... -1147 +1149 01:13:43,700 --> 01:13:49,110 The sum of the reciprocals of squares, which, somewhere in my notebook here, -1148 +1150 01:13:49,110 --> 01:13:51,500 we were just looking at the pages. -1149 +1151 01:13:52,600 --> 01:13:54,100 Maybe I tore it out. -1150 -01:13:54,720 --> 01:13:55,639 +1152 +01:13:54,720 --> 01:13:55,640 Oh, yeah, there we are. -1151 +1153 01:13:57,860 --> 01:13:59,960 Things can become a little bit of a mess over here. -1152 -01:14:00,040 --> 01:14:04,028 -But if we look at this long sum where we're taking 1 over 1 squared, - -1153 -01:14:04,028 --> 01:14:08,595 -1 over 2 squared, and equals pi squared over 6, the way that Euler found this, - 1154 -01:14:08,595 --> 01:14:12,527 -I mean, it involves this very strange thing where you start looking +01:14:00,040 --> 01:14:03,986 +but if we look at this uh this long somewhere we're taking one over one 1155 -01:14:12,527 --> 01:14:15,880 -at an infinite product associated with sine of pi times x. +01:14:03,986 --> 01:14:07,987 +squared one over two squared and equals pi squared over six the way that 1156 +01:14:07,987 --> 01:14:11,988 +Euler found this um I mean it involves this very strange thing where you + +1157 +01:14:11,988 --> 01:14:15,880 +start looking at an infinite product associated with sine of pi times x + +1158 01:14:15,880 --> 01:14:20,080 And if you think of it as starting with this sum and then dreaming up out of your head -1157 +1159 01:14:20,080 --> 01:14:22,783 an infinite product associated with sine of pi times x, -1158 +1160 01:14:22,783 --> 01:14:26,162 or it might have been cotangent of pi times x or something like that, -1159 +1161 01:14:26,162 --> 01:14:28,480 it really does seem like it came out of nowhere. -1160 -01:14:28,700 --> 01:14:35,461 -But in general, I think you build up these bits of familiarity with different pieces, - -1161 -01:14:35,461 --> 01:14:41,515 -and then once you see something, like Euler seeing that Basel problem sum or - 1162 -01:14:41,515 --> 01:14:48,041 -someone seeing this alternating harmonic sum, you're able to use those connections +01:14:28,700 --> 01:14:32,860 +but in general I think you build up these bits of familiarity with different pieces 1163 -01:14:48,041 --> 01:14:53,860 -to kind of make yourself look smarter than you actually are, which is fun. +01:14:32,860 --> 01:14:37,020 +and then once you see something like Euler seeing that basel problem sum or someone + +1164 +01:14:37,020 --> 01:14:41,230 +seeing this alternating harmonic sum you're able to use those connections to kind of + +1165 +01:14:41,230 --> 01:14:45,440 +make yourself look smarter than you actually are which is fun and if nothing else it + +1166 +01:14:45,440 --> 01:14:49,551 +helps motivate the idea of just being playful with your calculus and being playful + +1167 +01:14:49,551 --> 01:14:53,860 +with your math purely for the motivation of exposing yourself to more patterns and more diff --git a/2020/ldm-natural-logs/english/sentence_timings.json b/2020/ldm-natural-logs/english/sentence_timings.json index 4c6be611f..f8d9de45f 100644 --- a/2020/ldm-natural-logs/english/sentence_timings.json +++ b/2020/ldm-natural-logs/english/sentence_timings.json @@ -905,7 +905,7 @@ 927.12 ], [ - "And you might see where I'm going with this, you know, here I have 16 numbers that are all bigger than 1 and 30, excuse me, bigger than 1 in 32, talking while writing, and of course all of these are just equal to one half, so this 2 fourths is the same as a half, 4 eighths is the same as a half, 8 sixteenths, that's a half.", + "And you might see where I'm going with this, you know, here I have 16 numbers that are all bigger than one in 30, excuse me, bigger than one in 32, talking while writing, and of course all of these are just equal to one half, so this two fourths is the same as a half, four eighths is the same as a half, eight sixteenths, that's a half.", 928.12, 949.78 ], @@ -1280,7 +1280,7 @@ 1527.74 ], [ - "But if you ever kind of want to remember, oh, what was the formula for a bell curve again, you can kind of think through the fact that this should have roughly that shape.", + "but if you ever kind of want to remember oh what was the what was the formula for a bell curve again you can kind of think through the fact that this should have roughly that shape.", 1527.76, 1535.74 ], @@ -1450,7 +1450,7 @@ 1693.76 ], [ - "I could write that same function as pi raised to a special power, namely the log base pi.", + "I could write that same function sorry it didn't have to be e I could write that same function as pi raised to a special power namely the log base pi", 1694.06, 1699.74 ], @@ -1775,7 +1775,7 @@ 2207.32 ], [ - "So if you did describe all of your investments as a to the power t, which kind of feels more natural to a lot of people, that oh, you might say rather than e to some investment rate times t, just think of 1.05 to the t, and that describes something like 5% growth.", + "so if you did describe all of your investments as a to the power t which kind of feels more natural to a lot of people that oh you might say rather than e to some investment rate times t just think of 1.05 to the t and that describes you know something like five percent growth", 2207.74, 2222.72 ], @@ -2195,7 +2195,7 @@ 2971.08 ], [ - "This ends up being three times x squared, you know the exponent kind of hopped down and left behind one minus itself, over three times two times one, the threes cancel, so we can see that that's actually the same as x squared over two factorial.", + "this ends up being three times x squared you know the exponential the exponent kind of hopped down and left behind one minus itself over three times two times one the threes cancel so we can see that that's actually the same as x squared over two factorial", 2971.7, 2989.42 ], @@ -2220,7 +2220,7 @@ 3060.9 ], [ - "You might wonder why I want to know that, but if I have a deeper relationship with the natural log of x, in terms not just of how it relates to these series, but in all facets of math, maybe we can then start drawing connections.", + "you might wonder why i want to know that but if i have a bit you know a deeper relationship with the natural log of x in terms not just of how it relates to these series but in all facets of math maybe we can then start drawing connections", 3063.22, 3074.88 ], @@ -2385,7 +2385,7 @@ 3320.16 ], [ - "What that involves is taking the integral between 1 and our value n of 1 divided by x by dx.", + "kay what that what that involves is taking the integral between one and our value n of one divided by x by dx", 3321.72, 3331.46 ], @@ -2430,7 +2430,7 @@ 3367.1 ], [ - "And the question asks us, we're going to let s be the sum from n equals 1 up to capital N of 1 divided by n.", + "and the question asks us all right we're gonna let s be the sum from n equals one up to capital n of one divided by n ok", 3369.52, 3378.68 ], @@ -2450,7 +2450,7 @@ 3390.2 ], [ - "I'll give you a moment to think about that.", + "kay i'll give you s and i okay i'll give you a moment to think about that s", 3391.72, 3394.04 ], @@ -2475,7 +2475,7 @@ 3443.26 ], [ - "I'm curious actually what the answer is going to be here.", + "i'm curious i'm curious actually what the uh what the answer is going to be here", 3443.74, 3451.02 ], @@ -2485,7 +2485,7 @@ 3459.96 ], [ - "And a lot of the spirit of it is that you kind of hazard a guess, so feel no shame if you entered an answer and it's not what turns out to be correct.", + "and a lot of these the spirit of it is that you kind of hazard a guess so feel no shame if you entered an answer and then it's not what turns out to be correct s", 3460.3, 3467.34 ], @@ -2495,7 +2495,7 @@ 3473.22 ], [ - "And it looks like 900 of you got that correct, which is awesome.", + "nd the and it looks like uh 900 of you got that correct which is awesome an", 3474.28, 3478.04 ], @@ -2935,7 +2935,7 @@ 4326.44 ], [ - "But on the other hand, if I picture taking my antiderivative and getting this whole sequence here, and evaluating it between 0 and 1, what I'm doing is I'm plugging in the number 1, which gets me my alternating sequence, then I'm subtracting off the value of 0, which when I plug it in just gets 0, so evaluating this at 0 and 1 is the same thing as integrating the bottom expression from 0 to 1, which is the same thing as integrating 1 over 1 plus x from 0 to 1, which is the same thing as ln of 2, and hence that whole top thing ends up being ln of 2.", + "but on the other hand if I picture taking my anti-derivative and getting this whole sequence here and evaluating it between 0 and 1 what I'm doing is I'm plugging in the number 1 which gets me my alternating sequence then I'm subtracting off the value of 0 which when I plug it in just gets 0 so evaluating this at 0 and 1 is the same thing as integrating the bottom expression from 0 to 1 which is the same thing as integrating 1 over 1 plus x from 0 to 1 which is the same thing as natural log of 2 and hence that whole top thing ends up being the natural log of 2", 4327.36, 4360.88 ], @@ -3005,7 +3005,7 @@ 4439.96 ], [ - "But if we look at this long sum where we're taking 1 over 1 squared, 1 over 2 squared, and equals pi squared over 6, the way that Euler found this, I mean, it involves this very strange thing where you start looking at an infinite product associated with sine of pi times x.", + "but if we look at this uh this long somewhere we're taking one over one squared one over two squared and equals pi squared over six the way that Euler found this um I mean it involves this very strange thing where you start looking at an infinite product associated with sine of pi times x", 4440.04, 4455.88 ], @@ -3015,7 +3015,7 @@ 4468.48 ], [ - "But in general, I think you build up these bits of familiarity with different pieces, and then once you see something, like Euler seeing that Basel problem sum or someone seeing this alternating harmonic sum, you're able to use those connections to kind of make yourself look smarter than you actually are, which is fun.", + "but in general I think you build up these bits of familiarity with different pieces and then once you see something like Euler seeing that basel problem sum or someone seeing this alternating harmonic sum you're able to use those connections to kind of make yourself look smarter than you actually are which is fun and if nothing else it helps motivate the idea of just being playful with your calculus and being playful with your math purely for the motivation of exposing yourself to more patterns and more", 4468.7, 4493.86 ] diff --git a/2020/ldm-natural-logs/english/transcript.txt b/2020/ldm-natural-logs/english/transcript.txt index 4a56f42e2..22d2e0eae 100644 --- a/2020/ldm-natural-logs/english/transcript.txt +++ b/2020/ldm-natural-logs/english/transcript.txt @@ -179,7 +179,7 @@ Similarly, this sum here, 1 fifth plus 1 sixth plus 1 seventh plus 1 eighth, eac All four of those terms are bigger than 1 eighth. So the group of them together is bigger than 4 eighths. Similarly over here, all of the numbers between a ninth and a sixteenth, all eight of those numbers are bigger than 1 and 16, so the sum all together is bigger than 8 times 1 over 16. -And you might see where I'm going with this, you know, here I have 16 numbers that are all bigger than 1 and 30, excuse me, bigger than 1 in 32, talking while writing, and of course all of these are just equal to one half, so this 2 fourths is the same as a half, 4 eighths is the same as a half, 8 sixteenths, that's a half. +And you might see where I'm going with this, you know, here I have 16 numbers that are all bigger than one in 30, excuse me, bigger than one in 32, talking while writing, and of course all of these are just equal to one half, so this two fourths is the same as a half, four eighths is the same as a half, eight sixteenths, that's a half. So in other words, what I can do is group all of my terms so that the sum instead looks like taking 1 plus a half plus a half plus a half on and on forever. And that you can see, okay, if I keep going sufficiently long, it is going to get bigger. And it also gives you a little instinct that this might actually be related to logarithms, because the size of our groupings grow according to powers of 2. @@ -254,7 +254,7 @@ So to make it decay on both sides, you could take e to the negative x squared. And then it's decaying on both sides and you get this nice bell curve. And because of that square, it sort of smooths things out, whereas if we had taken something like the absolute value of x but negated it, okay, then it decays on both sides, but we get this awkward cusp. That doesn't explain why this very specific curve comes up in statistics. -But if you ever kind of want to remember, oh, what was the formula for a bell curve again, you can kind of think through the fact that this should have roughly that shape. +but if you ever kind of want to remember oh what was the what was the formula for a bell curve again you can kind of think through the fact that this should have roughly that shape. And quite often it comes with some kind of parameters, though. For example, I could put in something, maybe a value I'll call s in there, that will determine how wide and skinny this bell curve is, something like a standard deviation in the context of statistics. S wouldn't be that standard deviation. @@ -288,7 +288,7 @@ And the point here is that this just looks like e to some constant times x. So you might wonder, why do we make this choice, right? Because it didn't have to be pi. I could write that same function, sorry, it didn't have to be e. -I could write that same function as pi raised to a special power, namely the log base pi. +I could write that same function sorry it didn't have to be e I could write that same function as pi raised to a special power namely the log base pi Man, I'm writing quite sloppily here. Log base pi of 69 times x, that would be the same function. We could describe everything with a base of pi if we wanted to. @@ -353,7 +353,7 @@ So your velocity vector would end up looking a little bit shorter, and you'd be So that's kind of the important thing to understand about e, the fact that it's a choice that we're making to write families of exponentials this way, but because it is its own derivative, that ends up making these things play much more nicely. Now, this lets us take derivatives of anything else if you wanted. If you did describe your money's rate of growth with a to the t, to take its derivative, you could first do a conversion, write the whole thing as e to the natural log of a times t, and the reason you would is then when we're, I sort of squished my font here, then when you're taking the derivative of that, the derivative of the inside is the natural log of a, and then that's multiplied by itself, e to the natural log of a times t, which you could then spell out even further, convert it back into a to the t. -So if you did describe all of your investments as a to the power t, which kind of feels more natural to a lot of people, that oh, you might say rather than e to some investment rate times t, just think of 1.05 to the t, and that describes something like 5% growth. +so if you did describe all of your investments as a to the power t which kind of feels more natural to a lot of people that oh you might say rather than e to some investment rate times t just think of 1.05 to the t and that describes you know something like five percent growth If you were thinking of that growth in a continuous sense, not year over year, what's the new percentage, but moment by moment, what's the rate of growth, you would have to say the rate of growth is the natural log of that base, which just feels a little bit more awkward. You could do it, but it would feel more awkward. Now, all of this leaves open the question of why? @@ -437,12 +437,12 @@ It's really one of the most pleasing, I don't know, times that you'll have in a If you're just sitting, you're looking at this particular infinite polynomial, and you're saying I wonder what the derivative of this happens to be. And all you need to know is the power rule for polynomial terms, and you'll say that the derivative, let me take d dx, well the derivative of a constant ends up being zero, the derivative of x is one, the derivative of x squared over two, you know you might think of that two as kind of hopping down in front and leaving one less than itself, so it becomes two times x to the one, just x to the one over two, and those twos cancel, so we're adding x. x cubed over three factorial, I might write this out as x cubed over three times two times one. -This ends up being three times x squared, you know the exponent kind of hopped down and left behind one minus itself, over three times two times one, the threes cancel, so we can see that that's actually the same as x squared over two factorial. +this ends up being three times x squared you know the exponential the exponent kind of hopped down and left behind one minus itself over three times two times one the threes cancel so we can see that that's actually the same as x squared over two factorial And in general, each one of our terms, as the exponent hops down, it cancels out one of the things from the factorials below it, and what we get is the exact same sequence but shifted, which is quite nice. And like I said, the traditional way that you see this series is that you're using the fact that e to the x is its own derivative, in conjunction with Taylor series to show that it must equal this, but if you start with this as a primitive, and you say this is the thing that defines a special function for which we use the shorthand, e to the x, then it feels a little bit more contentful and quite fun to say that e to the x ends up being its own derivative. And like we showed earlier, that then lets you take the derivative of all sorts of other things, which in turn explains why we adopt the convention of writing all of our exponentials as e to something times t, as opposed to writing them all as a to something times t, even though those are equivalent and often weirdly hard to appreciate. So, with all of that said, we can turn ourselves back in the direction of natural logarithms, because let's say I wanted to know the derivative of the natural log. -You might wonder why I want to know that, but if I have a deeper relationship with the natural log of x, in terms not just of how it relates to these series, but in all facets of math, maybe we can then start drawing connections. +you might wonder why i want to know that but if i have a bit you know a deeper relationship with the natural log of x in terms not just of how it relates to these series but in all facets of math maybe we can then start drawing connections And if you build up that relationship by knowing things like its derivative, it actually helps you come back and understand things like the alternating series we were looking at before. So, can we use the fact that e to the x is its own derivative to figure out the slope of a natural log curve? Well, what that slope is asking us is to look at a given input x, we consider a tiny step dx to the right, look at the corresponding step dy up, and we want to understand the ratio, dy over dx. @@ -475,7 +475,7 @@ Let's say I take the area up to... my stomach was just rumbling, I don't know if that's audible on the microphone. Clearly, gotta eat lunch before these things. So let's say I want to understand the area up to n of something like this. -What that involves is taking the integral between 1 and our value n of 1 divided by x by dx. +kay what that what that involves is taking the integral between one and our value n of one divided by x by dx Now this actually looks quite similar in spirit, the idea of adding up a bunch of things that look like 1 over x, to what we were looking at earlier. How much earlier? I guess over here. @@ -484,20 +484,20 @@ And already it gives a little bit of an intuitive instinct for why something lik Because we now know that in calculus land, natural logs are intimately related to the idea of 1 divided by x. But I want you to think of this spelled out a little bit more exactly, and so we'll pop on over to our quiz one more time. Second to last question for today. -And the question asks us, we're going to let s be the sum from n equals 1 up to capital N of 1 divided by n. +and the question asks us all right we're gonna let s be the sum from n equals one up to capital n of one divided by n ok That's s. And then we're going to let i be an analogous integral, where we're integrating dx over x between 1 and n. And it asks you to compare s and i. -I'll give you a moment to think about that. +kay i'll give you s and i okay i'll give you a moment to think about that s Interestingly, we don't have a ton of consensus around this one. So there's only three options, and we've got a nice split. And as you guys know, this is actually one of my favorite things when we do any of these lockdown live quizzes, is when it's not everyone hopping on to one particular thing, but we have a division among folks. And I think that's great. -I'm curious actually what the answer is going to be here. +i'm curious i'm curious actually what the uh what the answer is going to be here And in fact, even if it's not been enough time to thoroughly think through, I'm going to go ahead and grade it, just so that we can see what it happens to be. -And a lot of the spirit of it is that you kind of hazard a guess, so feel no shame if you entered an answer and it's not what turns out to be correct. +and a lot of these the spirit of it is that you kind of hazard a guess so feel no shame if you entered an answer and then it's not what turns out to be correct s So in this case, the sum does in fact end up being bigger than the integral. -And it looks like 900 of you got that correct, which is awesome. +nd the and it looks like uh 900 of you got that correct which is awesome an And then following that was people thinking that it was less. And then to those in B thinking that they were identical, you know, that's a reasonable thought, because they're such similar expressions. But there's a picture that really makes the answer kind of shine out to us here, which is, if I look at the curve 1 over x, which is what this white curve is, it's 1 over x, and then I'm going to consider a bunch of bars, each of whose area corresponds to 1 over n for some value of n. @@ -585,7 +585,7 @@ And if you wanted to kind of visualize this in your head or get some sort of gut So you know that it's going to be an area somewhere between 0 and 1, probably filling up more than a half of it, and the natural log of 2 is around 0.69, so that actually seems about right. But what's quite cool here is that if we say that the, well I can just write it out again here, if I say I want to integrate this bottom expression from 0 up to 1, which is the same as integrating 1 over 1 plus x from 0 up to 1, I know that that should be the value 2, ln of 2. Okay? -But on the other hand, if I picture taking my antiderivative and getting this whole sequence here, and evaluating it between 0 and 1, what I'm doing is I'm plugging in the number 1, which gets me my alternating sequence, then I'm subtracting off the value of 0, which when I plug it in just gets 0, so evaluating this at 0 and 1 is the same thing as integrating the bottom expression from 0 to 1, which is the same thing as integrating 1 over 1 plus x from 0 to 1, which is the same thing as ln of 2, and hence that whole top thing ends up being ln of 2. +but on the other hand if I picture taking my anti-derivative and getting this whole sequence here and evaluating it between 0 and 1 what I'm doing is I'm plugging in the number 1 which gets me my alternating sequence then I'm subtracting off the value of 0 which when I plug it in just gets 0 so evaluating this at 0 and 1 is the same thing as integrating the bottom expression from 0 to 1 which is the same thing as integrating 1 over 1 plus x from 0 to 1 which is the same thing as natural log of 2 and hence that whole top thing ends up being the natural log of 2 Very clever! Just such a sneaky sequence of manipulations that, like I said, if you just see it plopped down, almost as intimidating because you wonder, how on earth do you go from seeing this sequence up here to thinking about, ah yes, if I generalize it with lots of powers of x and then take the derivatives of those and then use a geometric series sum and then integrate that using natural log, yes, then it'll all become obvious. And I think the answer is that that's probably not what the problem solving process looks like. @@ -599,6 +599,6 @@ The sum of the reciprocals of squares, which, somewhere in my notebook here, we Maybe I tore it out. Oh, yeah, there we are. Things can become a little bit of a mess over here. -But if we look at this long sum where we're taking 1 over 1 squared, 1 over 2 squared, and equals pi squared over 6, the way that Euler found this, I mean, it involves this very strange thing where you start looking at an infinite product associated with sine of pi times x. +but if we look at this uh this long somewhere we're taking one over one squared one over two squared and equals pi squared over six the way that Euler found this um I mean it involves this very strange thing where you start looking at an infinite product associated with sine of pi times x And if you think of it as starting with this sum and then dreaming up out of your head an infinite product associated with sine of pi times x, or it might have been cotangent of pi times x or something like that, it really does seem like it came out of nowhere. -But in general, I think you build up these bits of familiarity with different pieces, and then once you see something, like Euler seeing that Basel problem sum or someone seeing this alternating harmonic sum, you're able to use those connections to kind of make yourself look smarter than you actually are, which is fun. \ No newline at end of file +but in general I think you build up these bits of familiarity with different pieces and then once you see something like Euler seeing that basel problem sum or someone seeing this alternating harmonic sum you're able to use those connections to kind of make yourself look smarter than you actually are which is fun and if nothing else it helps motivate the idea of just being playful with your calculus and being playful with your math purely for the motivation of exposing yourself to more patterns and more \ No newline at end of file diff --git a/2020/ldm-natural-logs/french/sentence_translations.json b/2020/ldm-natural-logs/french/sentence_translations.json index 659e02e5d..ab6eaea1e 100644 --- a/2020/ldm-natural-logs/french/sentence_translations.json +++ b/2020/ldm-natural-logs/french/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "Vous savez, nous avons 1 billion 751, 1 billion 787. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "Et c’est environ un sur 27. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "Il y a un certain nombre d'autres formules qui nous donnent quelque chose en rapport avec pi, qui est évidemment lié aux nombres premiers, d'une manière qui est, euh, je veux dire, vous jouez au même jeu et vous avez cette façon étrange de prendre des logarithmes, et pas n'importe lesquels. logarithme, la base du journal e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "Donc, si vous vous demandez combien de temps il me reste avant que cette somme dépasse 10, vous pourriez avoir l'instinct que, hmm, je vais devoir additionner, voyons, j'en ai un et puis le reste parmi eux sont des moitiés, donc je vais devoir additionner 18 groupes différents qui ressemblent chacun à une moitié, donc je devrai peut-être arriver au point où la taille de mon groupe est de deux puissance 17, quelque chose comme que. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "Et même cette idée folle, comme la réflexion mentale sur la façon dont vous pouvez atteindre un grand nombre, prendrait une éternité pour vous amener à quelque chose de la taille de 10 à 400 000. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/german/sentence_translations.json b/2020/ldm-natural-logs/german/sentence_translations.json index f0d8a5056..17a528551 100644 --- a/2020/ldm-natural-logs/german/sentence_translations.json +++ b/2020/ldm-natural-logs/german/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "Wissen Sie, wir haben 1 Billion 751, 1 Billion 787. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "Und es ist etwa jeder 27. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "Es gibt eine Reihe anderer Formeln, die uns etwas mit Pi zu tun haben, das offensichtlich mit Primzahlen zusammenhängt, in gewisser Weise, ähm, ich meine, Sie spielen das gleiche Spiel und haben diese seltsame Art, Logarithmen zu bilden, und nicht irgendeine Logarithmus, die logarithmische Basis e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "Wenn Sie sich also fragen, wie lange es dauern muss, bis diese Summe größer als 10 wird, denken Sie vielleicht, dass ich jetzt addieren muss, mal sehen, ich habe eins und dann den Rest Davon sind Hälften, also muss ich 18 verschiedene Gruppen zusammenzählen, die jeweils wie eine Hälfte aussehen, also muss ich vielleicht bis zu dem Punkt kommen, an dem die Größe meiner Gruppe etwa zwei hoch 17 beträgt, etwa so Das. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "Und selbst diese verrückte Idee, wie das mentale Nachdenken darüber, wie man eine große Zahl erreichen kann, würde ewig dauern, bis man auf etwas in der Größenordnung von 10 bis 400.000 kommt. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/hebrew/sentence_translations.json b/2020/ldm-natural-logs/hebrew/sentence_translations.json index 4ef927398..54f702458 100644 --- a/2020/ldm-natural-logs/hebrew/sentence_translations.json +++ b/2020/ldm-natural-logs/hebrew/sentence_translations.json @@ -350,7 +350,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787.", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven.", "translatedText": "אתה יודע, יש לנו טריליון 751, טריליון 787.", "n_reviews": 0, "start": 225.72, @@ -392,7 +392,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27.", + "input": "And it's about one in every twenty-seven.", "translatedText": "וזה בערך אחד מכל 27.", "n_reviews": 0, "start": 249.42, @@ -819,7 +819,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e.", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e.", "translatedText": "יש עוד מספר נוסחאות שמביאות לנו משהו שקשור ל-pi, שקשור כנראה לראשוניים, בצורה שהיא, אממ, אני מתכוון, אתה משחק באותו משחק ויש לך את האופנה המוזרה הזו של לקחת לוגריתמים, ולא סתם. לוגריתם, בסיס היומן ה.", "n_reviews": 0, "start": 574.06, @@ -1295,7 +1295,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that.", + "input": "o if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that.", "translatedText": "אז אם אתה תוהה כמה זמן אני צריך לעבור לפני שהסכום הזה יגדל מ-10, אולי יש לך את האינסטינקט, הממ, אני אצטרך להוסיף ביחד, בוא נראה, יש לי אחד ואז השאר מהן חצאיות, אז אני אצטרך להוסיף יחד 18 קבוצות שונות שכל אחת נראית כמו חצי, אז אולי אצטרך להגיע לנקודה שבה גודל הקבוצה שלי הוא כמו שתיים עד ה-17, משהו כמו זֶה.", "n_reviews": 0, "start": 976.16, @@ -1449,7 +1449,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000.", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000.", "translatedText": "ואפילו הרעיון המטורף הזה, כמו מחשבה מנטלית על איך אתה יכול להגיע למספר גדול, ייקח נצח כדי להביא אותך למשהו בגודל 10 עד 400,000.", "n_reviews": 0, "start": 1187.36, @@ -1778,7 +1778,7 @@ "end": 1560.16 }, { - "input": "You can think just looking at this that somehow bell curves are produced by the number e but that's not exactly true because I could also write a to the negative x squared and I get the same family of curves as I tweak the value of a I'm also changing what that width is so I could come up with other ways of describing the standard deviation of this in terms of a and it's it's the same family of curves it's not just that they look similar they are in fact the same thing.", + "input": "You can think, just looking at this, that somehow bell curves are produced by the number e. But that's not exactly true, because I could also write a to the negative x squared, and I get the same family of curves. As I tweak the value of a, I'm also changing what that width is, so I could come up with other ways of describing the standard deviation of this in terms of a. And it's the same family of curves. It's not just that they look similar. They are, in fact, the same thing.", "translatedText": "אתה יכול לחשוב רק אם אתה מסתכל על זה שאיכשהו עקומות פעמון נוצרות על ידי המספר e אבל זה לא בדיוק נכון כי אני יכול לכתוב גם a ל-x השלילי בריבוע ואני מקבל את אותה משפחת עקומות כשאני מכוונן את הערך של I' אני גם משנה מה הרוחב הזה כדי שאוכל להמציא דרכים אחרות לתאר את סטיית התקן של זה במונחים של a וזו אותה משפחה של עקומות זה לא רק שהם נראים דומים הם למעשה אותו הדבר.", "n_reviews": 0, "start": 1560.66, diff --git a/2020/ldm-natural-logs/hindi/sentence_translations.json b/2020/ldm-natural-logs/hindi/sentence_translations.json index fef2cbee6..b495f8ff7 100644 --- a/2020/ldm-natural-logs/hindi/sentence_translations.json +++ b/2020/ldm-natural-logs/hindi/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "आप जानते हैं, हमें 1 ट्रिलियन 751, 1 ट्रिलियन 787 मिले हैं।", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "और यह हर 27 में से एक के बारे में है।", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "ऐसे कई अन्य सूत्र हैं जो हमें पाई से संबंधित कुछ बताते हैं, जो स्पष्ट रूप से अभाज्य संख्याओं से संबंधित है, एक तरह से, उम, मेरा मतलब है, आप एक ही खेल खेलते हैं और आपके पास लघुगणक लेने का यह अजीब फैशन है, और कोई भी नहीं लघुगणक, लघुगणक आधार ई. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "इसलिए यदि आप सोच रहे हैं कि इस राशि के 10 से अधिक होने से पहले मुझे कितना समय लगाना होगा, तो आपके मन में यह भावना आ सकती है कि, हम्म, मुझे एक साथ जोड़ना होगा, देखते हैं, मेरे पास एक है और फिर बाकी उनमें से आधे आधे हैं, इसलिए मुझे 18 अलग-अलग समूहों को एक साथ जोड़ना होगा, जिनमें से प्रत्येक आधे की तरह दिखता है, इसलिए मुझे उस बिंदु तक पहुंचना होगा जहां मेरे समूह का आकार दो से 17 वें तक हो, कुछ इस तरह वह।", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "और यहां तक कि वह पागल विचार, जैसे मानसिक विचार कि आप एक बड़ी संख्या तक कैसे पहुंच सकते हैं, आपको 10 से 400,000 के आकार तक पहुंचाने में हमेशा के लिए लग जाएगा।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/hungarian/sentence_translations.json b/2020/ldm-natural-logs/hungarian/sentence_translations.json index 811c30ab6..e47d7e5be 100644 --- a/2020/ldm-natural-logs/hungarian/sentence_translations.json +++ b/2020/ldm-natural-logs/hungarian/sentence_translations.json @@ -350,7 +350,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787.", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven.", "translatedText": "Tudod, van 1 billió 751-ünk, 1 billió 787-ünk.", "n_reviews": 0, "start": 225.72, @@ -392,7 +392,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27.", + "input": "And it's about one in every twenty-seven.", "translatedText": "És ez körülbelül minden 27.", "n_reviews": 0, "start": 249.42, @@ -819,7 +819,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e.", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e.", "translatedText": "Számos más képlet is a pi-hez kapcsolódik, ami nyilvánvalóan a prímszámokhoz kapcsolódik, oly módon, hogy, hm, úgy értem, ugyanazt a játékot játsszuk, és furcsa módon logaritmusokat veszünk, és nem akármilyeneket. logaritmus, a logalap e.", "n_reviews": 0, "start": 574.06, @@ -1295,7 +1295,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that.", + "input": "o if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that.", "translatedText": "Szóval ha kíváncsi vagy arra, hogy mennyi ideig kell még mennem, hogy ez az összeg nagyobb legyen 10-nél, akkor lehet az az ösztöne, hogy hmm, össze kell adnom, lássuk, van egy, majd a többi ezek felek, tehát össze kell adnom 18 különböző csoportot, amelyek mindegyike úgy néz ki, mint egy fele, így lehet, hogy odáig kell jutnom, hogy a csoportom mérete kettő a 17., valami ilyesmi hogy.", "n_reviews": 0, "start": 976.16, @@ -1449,7 +1449,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000.", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000.", "translatedText": "És még ez az őrült ötlet is, mint például a gondolat, hogy hogyan juthat el egy nagy számig, örökké tart, amíg eléri a 10-400 000 közötti méretű valamit.", "n_reviews": 0, "start": 1187.36, @@ -1778,7 +1778,7 @@ "end": 1560.16 }, { - "input": "You can think just looking at this that somehow bell curves are produced by the number e but that's not exactly true because I could also write a to the negative x squared and I get the same family of curves as I tweak the value of a I'm also changing what that width is so I could come up with other ways of describing the standard deviation of this in terms of a and it's it's the same family of curves it's not just that they look similar they are in fact the same thing.", + "input": "You can think, just looking at this, that somehow bell curves are produced by the number e. But that's not exactly true, because I could also write a to the negative x squared, and I get the same family of curves. As I tweak the value of a, I'm also changing what that width is, so I could come up with other ways of describing the standard deviation of this in terms of a. And it's the same family of curves. It's not just that they look similar. They are, in fact, the same thing.", "translatedText": "Ha ezt nézzük, azt gondolhatod, hogy valahogy a haranggörbéket az e szám állítja elő, de ez nem teljesen igaz, mert írhatnék a-t is a negatív x négyzetére, és ugyanazt a görbecsaládot kapom, amikor az I értékét módosítom. m is változtatjuk a szélességet, így más módszereket is kitalálhatnék ennek a szórásának leírására a-val, és ez ugyanaz a görbecsalád, nem csak arról van szó, hogy hasonlítanak, hanem valójában ugyanaz.", "n_reviews": 0, "start": 1560.66, diff --git a/2020/ldm-natural-logs/indonesian/sentence_translations.json b/2020/ldm-natural-logs/indonesian/sentence_translations.json index f4c897e65..1dd571822 100644 --- a/2020/ldm-natural-logs/indonesian/sentence_translations.json +++ b/2020/ldm-natural-logs/indonesian/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "Anda tahu, kita punya 1 triliun 751, 1 triliun 787. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "Dan itu sekitar satu dari setiap 27. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "Ada sejumlah rumus lain yang memberi kita sesuatu yang berhubungan dengan pi, yang ternyata berhubungan dengan bilangan prima, dengan cara, um, maksud saya, Anda memainkan permainan yang sama dan Anda memiliki cara yang aneh dalam mengambil logaritma, dan bukan sembarang rumus. logaritma, basis log e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "Jadi jika Anda bertanya-tanya berapa lama waktu yang harus saya habiskan sebelum jumlah ini menjadi lebih besar dari 10, Anda mungkin memiliki naluri bahwa, hmm, saya harus menjumlahkannya, mari kita lihat, saya punya satu dan sisanya di antaranya adalah setengah, jadi saya harus menjumlahkan 18 grup berbeda yang masing-masing terlihat seperti setengah, jadi saya mungkin harus mencapai titik di mana ukuran grup saya seperti dua berbanding 17, kira-kira seperti itu. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "Dan bahkan ide gila itu, seperti pemikiran mental tentang bagaimana Anda bisa mencapai angka yang besar, akan memakan waktu lama untuk membawa Anda mencapai angka 10 hingga 400.000. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/italian/sentence_translations.json b/2020/ldm-natural-logs/italian/sentence_translations.json index dd1723158..7baaac538 100644 --- a/2020/ldm-natural-logs/italian/sentence_translations.json +++ b/2020/ldm-natural-logs/italian/sentence_translations.json @@ -350,7 +350,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787.", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven.", "translatedText": "Sapete, abbiamo 1 trilione 751, 1 trilione 787.", "n_reviews": 0, "start": 225.72, @@ -392,7 +392,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27.", + "input": "And it's about one in every twenty-seven.", "translatedText": "E sono circa uno su 27.", "n_reviews": 0, "start": 249.42, @@ -819,7 +819,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e.", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e.", "translatedText": "C'è una serie di altre formule che ci danno qualcosa legato al pi greco, che evidentemente è legato ai numeri primi, in un modo che è, um, voglio dire, giochi allo stesso gioco e hai questo strano modo di prendere i logaritmi, e non uno qualsiasi logaritmo, logaritmo in base e.", "n_reviews": 0, "start": 574.06, @@ -1295,7 +1295,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that.", + "input": "o if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that.", "translatedText": "Quindi, se ti stai chiedendo quanto tempo devo passare prima che questa somma diventi maggiore di 10, potresti avere l'istinto che, hmm, dovrò sommare, vediamo, ne ho uno e poi il resto di essi sono metà, quindi dovrò sommare 18 gruppi diversi che assomigliano ciascuno a metà, quindi potrei dover arrivare al punto in cui la dimensione del mio gruppo è come 2 alla 17, qualcosa come Quello.", "n_reviews": 0, "start": 976.16, @@ -1449,7 +1449,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000.", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000.", "translatedText": "E anche quell’idea folle, come il pensiero mentale su come arrivare a un grande numero, impiegherebbe un’eternità per arrivare a qualcosa che va da 10 a 400.000.", "n_reviews": 0, "start": 1187.36, @@ -1778,7 +1778,7 @@ "end": 1560.16 }, { - "input": "You can think just looking at this that somehow bell curves are produced by the number e but that's not exactly true because I could also write a to the negative x squared and I get the same family of curves as I tweak the value of a I'm also changing what that width is so I could come up with other ways of describing the standard deviation of this in terms of a and it's it's the same family of curves it's not just that they look similar they are in fact the same thing.", + "input": "You can think, just looking at this, that somehow bell curves are produced by the number e. But that's not exactly true, because I could also write a to the negative x squared, and I get the same family of curves. As I tweak the value of a, I'm also changing what that width is, so I could come up with other ways of describing the standard deviation of this in terms of a. And it's the same family of curves. It's not just that they look similar. They are, in fact, the same thing.", "translatedText": "Puoi pensare solo guardando questo che in qualche modo le curve a campana sono prodotte dal numero e ma non è esattamente vero perché potrei anche scrivere a sulla negativa x al quadrato e ottengo la stessa famiglia di curve quando modifico il valore di a I' Sto anche cambiando la larghezza in modo da poter trovare altri modi per descrivere la deviazione standard di questa in termini di a ed è la stessa famiglia di curve, non è solo che sembrano simili, in realtà sono la stessa cosa.", "n_reviews": 0, "start": 1560.66, diff --git a/2020/ldm-natural-logs/japanese/sentence_translations.json b/2020/ldm-natural-logs/japanese/sentence_translations.json index 3e32f9c71..a40c7a74d 100644 --- a/2020/ldm-natural-logs/japanese/sentence_translations.json +++ b/2020/ldm-natural-logs/japanese/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "ご存知のとおり、1 兆 751、1 兆 787 があります。", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "そしてそれは約27人に1人です 。", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "円周率に関連するものを得る公式は他にもたくさんありますが、それは明らかに素 数に関連しています。つまり、同じゲームをプレイしていて、単なる対数ではな く、対数を取得する奇妙な方法を持っているということです。", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "この合計が 10 より大きくなるまでにどれくらい時間がかかるのかと疑 問に思っている場合は、直感的に、「うーん、足し算をしなければならない だろう。さあ、1 つがあり、残りが 1 つある」と直感するかもしれま せん。そのうちの半分は半分なので、それぞれが半分のように見える 18 個の異なるグループを合計する必要があるため、グループのサイズが 2 対 17 になるところまで到達する必要があるかもしれません。それ。", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "そして、そのクレイジーなアイデアでさえ、どうすれば大きな数字に到達できるかを頭の中で考えたよ うなものでも、10 から 400,000 のサイズに達するまでには永遠に時間がかかります。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/korean/sentence_translations.json b/2020/ldm-natural-logs/korean/sentence_translations.json index 2d320788f..92b72a23f 100644 --- a/2020/ldm-natural-logs/korean/sentence_translations.json +++ b/2020/ldm-natural-logs/korean/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "아시다시피, 우리는 1조 751, 1조 787을 가지고 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "그리고 이는 27명당 1명꼴입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "파이와 관련된 뭔가를 얻을 수 있는 다른 공식이 많이 있습니다. 이는 분명히 소수와 관련이 있습니다. 즉, 음, 내 말은, 같은 게임을 하고 로그를 취하는 이상한 방식이 있다는 뜻입니다. 로그, 로그 밑수 e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "따라서 이 합이 10보다 커지기까지 얼마나 기다려야 하는지 궁금하시다면, 여러분은 본능적으로 '흠, 모두 더해야겠다'고 생각할 것입니다. 보자. 하나가 있고 그 다음에는 나머지가 있습니다. 그 중 절반은 절반이므로 각각 절반처럼 보이는 18개의 서로 다른 그룹을 함께 추가해야 하므로 그룹의 크기가 2의 17제곱이 되는 지점에 도달해야 할 수도 있습니다. 저것. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "그리고 어떻게 큰 숫자를 얻을 수 있는지에 대한 정신적 사고와 같은 그 말도 안되는 아이디어조차도 10에서 400,000의 크기에 도달하는 데 영원히 시간이 걸릴 것입니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/marathi/sentence_translations.json b/2020/ldm-natural-logs/marathi/sentence_translations.json index e33ec12d6..51854c439 100644 --- a/2020/ldm-natural-logs/marathi/sentence_translations.json +++ b/2020/ldm-natural-logs/marathi/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "तुम्हाला माहिती आहे, आम्हाला 1 ट्रिलियन 751, 1 ट्रिलियन 787 मिळाले आहेत. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "आणि ते प्रत्येक 27 पैकी एक आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "इतर अनेक सूत्रे आहेत जी आपल्याला pi शी संबंधित काहीतरी मिळवून देतात, जे स्पष्टपणे प्राइमशी संबंधित आहे, जसे की, उम, मला म्हणायचे आहे की, तुम्ही तोच खेळ खेळता आणि तुमच्याकडे लॉगरिदम घेण्याची ही विचित्र फॅशन आहे, आणि फक्त कोणतेही नाही. लॉगरिदम, लॉग बेस e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "त्यामुळे ही बेरीज १० पेक्षा मोठी होण्याआधी मला किती काळ जावे लागेल असा प्रश्न तुम्ही विचार करत असाल, तर तुमची प्रवृत्ती असेल की, हम्म, मला एकत्र जोडावे लागेल, बघूया, माझ्याकडे एक आहे आणि नंतर बाकीचे त्यापैकी अर्धे आहेत, म्हणून मला 18 भिन्न गट एकत्र जोडावे लागतील जे प्रत्येक अर्ध्यासारखे दिसतील, त्यामुळे मला कदाचित माझ्या गटाचा आकार दोन ते 17 व्या सारखा असेल अशा बिंदूपर्यंत जावे लागेल. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "आणि ती विलक्षण कल्पना देखील, जसे की आपण मोठ्या संख्येपर्यंत कसे पोहोचू शकता या मानसिक विचाराप्रमाणे, आपल्याला 10 ते 400,000 च्या आकारमानापर्यंत पोहोचवण्यास कायमचा वेळ लागेल. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/persian/sentence_translations.json b/2020/ldm-natural-logs/persian/sentence_translations.json index 9029b79b2..cff0fad17 100644 --- a/2020/ldm-natural-logs/persian/sentence_translations.json +++ b/2020/ldm-natural-logs/persian/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "می دانید، ما 1 تریلیون 751، 1 تریلیون 787 داریم. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "و از هر 27 یک نفر است. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "تعدادی فرمول دیگر وجود دارد که به ما چیزی مربوط به pi را می دهد، که ظاهراً به اعداد اول مربوط می شود، به گونه ای که، اوم، منظورم این است که شما همان بازی را انجام می دهید و این مد عجیب و غریب را برای گرفتن لگاریتم دارید، و نه هر یک. لگاریتم، پایه log e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "بنابراین اگر می‌پرسید چقدر باید بگذرم تا این مجموع از 10 بزرگتر شود، ممکن است این غریزه را داشته باشید که، هوم، باید با هم جمع کنم، بیایید ببینیم، یکی دارم و سپس بقیه از بین آنها نیمه هستند، بنابراین من باید 18 گروه مختلف را با هم جمع کنم که هر کدام مانند یک نیمه هستند، بنابراین ممکن است مجبور باشم به نقطه ای برسم که اندازه گروه من مانند دو تا 17 باشد، چیزی شبیه به که و شما می‌دانید که رشد می‌کند، خوب به صورت تصاعدی رشد نمی‌کند، به صورت لگاریتمی رشد می‌کند، زیرا اگر بپرسید برای رسیدن به آن نقطه چقدر باید پیش بروید، لگاریتمی خواهد بود. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "و حتی این ایده دیوانه کننده، مانند فکر ذهنی برای اینکه چگونه می توانید به یک عدد بزرگ برسیم، برای رسیدن به چیزی از اندازه 10 تا 400000 برای همیشه طول می کشد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/portuguese/sentence_translations.json b/2020/ldm-natural-logs/portuguese/sentence_translations.json index dbb175139..6946a350e 100644 --- a/2020/ldm-natural-logs/portuguese/sentence_translations.json +++ b/2020/ldm-natural-logs/portuguese/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "Você sabe, temos 1 trilhão 751, 1 trilhão 787. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "E é cerca de um em cada 27. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "Há uma série de outras fórmulas que nos dão algo relacionado a pi, que está evidentemente relacionado a primos, de uma forma que, hum, quero dizer, você joga o mesmo jogo e tem essa maneira estranha de calcular logaritmos, e não qualquer logaritmo, a base logarítmica e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "Então, se você está se perguntando quanto tempo ainda tenho que passar antes que essa soma fique maior que 10, você pode ter o instinto de que, hmm, vou ter que somar, vamos ver, eu tenho um e depois o resto deles são metades, então terei que somar 18 grupos diferentes, cada um parecendo metade, então talvez tenha que chegar ao ponto em que o tamanho do meu grupo seja dois elevado a 17, algo como que. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "E mesmo essa ideia maluca, como o pensamento mental sobre como você pode chegar a um número grande, levaria uma eternidade para chegar a algo do tamanho de 10 elevado a 400.000. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/russian/sentence_translations.json b/2020/ldm-natural-logs/russian/sentence_translations.json index b90d27cb8..02fa653b5 100644 --- a/2020/ldm-natural-logs/russian/sentence_translations.json +++ b/2020/ldm-natural-logs/russian/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "Знаете, у нас есть 1 триллион 751, 1 триллион 787. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "И это примерно один из каждых 27. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "Есть ряд других формул, которые дают нам что-то связанное с числом Пи, которое, очевидно, связано с простыми числами, таким образом, я имею в виду, что вы играете в ту же самую игру, и у вас есть эта странная манера брать логарифмы, а не просто какие-то числа. логарифм, логарифмическая база e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "Итак, если вам интересно, сколько времени мне нужно пройти, прежде чем эта сумма станет больше 10, у вас может возникнуть инстинктивное ощущение: хм, мне придется сложить, посмотрим, у меня есть одно, а затем остальные. из них половинки, поэтому мне придется сложить 18 различных групп, каждая из которых выглядит как половина, поэтому мне, возможно, придется дойти до точки, когда размер моей группы будет примерно равен двум 17-м, что-то вроде что. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "И даже эта сумасшедшая идея, как и мысленная мысль о том, как можно достичь большого числа, потребует целую вечность, чтобы привести вас к чему-то размеру от 10 до 400 000. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/spanish/sentence_translations.json b/2020/ldm-natural-logs/spanish/sentence_translations.json index b6fd019da..74b65f369 100644 --- a/2020/ldm-natural-logs/spanish/sentence_translations.json +++ b/2020/ldm-natural-logs/spanish/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "Ya sabes, tenemos 1 billón 751, 1 billón 787. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "Y es aproximadamente uno de cada 27. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "Hay una serie de otras fórmulas que nos dan algo relacionado con pi, que evidentemente está relacionado con los números primos, de una manera que es, um, quiero decir, juegas el mismo juego y tienes esta extraña manera de tomar logaritmos, y no cualquier logaritmo, la base logarítmica e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "Entonces, si te preguntas cuánto tiempo tengo que pasar antes de que esta suma sea mayor que 10, es posible que tengas el instinto de que, mmm, voy a tener que sumar, veamos, tengo uno y luego el resto. de ellos son mitades, así que voy a tener que sumar 18 grupos diferentes, cada uno de los cuales parece una mitad, así que quizás tenga que llegar al punto en el que el tamaño de mi grupo sea como dos elevado a 17, algo así como eso. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "E incluso esa idea loca, como el pensamiento mental sobre cómo llegar a un número grande, tardaría una eternidad en llegar a algo del tamaño de 10 a 400.000. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/tamil/sentence_translations.json b/2020/ldm-natural-logs/tamil/sentence_translations.json index ff3e2e7f9..e8dad0943 100644 --- a/2020/ldm-natural-logs/tamil/sentence_translations.json +++ b/2020/ldm-natural-logs/tamil/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "உங்களுக்குத் தெரியும், எங்களிடம் 1 டிரில்லியன் 751, 1 டிரில்லியன் 787 உள்ளது. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "மேலும் இது ஒவ்வொரு 27 இல் ஒன்று. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "பையுடன் தொடர்புடைய வேறு பல சூத்திரங்கள் உள்ளன, இது ப்ரைம்களுடன் தொடர்புடையது, அதாவது, நீங்கள் அதே விளையாட்டை விளையாடுகிறீர்கள் மற்றும் மடக்கைகளை எடுக்கும் வித்தியாசமான பாணியை நீங்கள் கொண்டிருக்கிறீர்கள். மடக்கை, பதிவு அடிப்படை இ. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "இந்த தொகை 10 ஐ விட அதிகமாகும் முன் நான் எவ்வளவு காலம் செல்ல வேண்டும் என்று நீங்கள் யோசிக்கிறீர்கள் என்றால், ஹ்ம்ம், நான் ஒன்றாகச் சேர்க்க வேண்டும், பார்க்கலாம், என்னிடம் ஒன்று உள்ளது, பின்னர் மீதமுள்ளவை என்று உங்களுக்கு உள்ளுணர்வு இருக்கலாம். அவற்றில் பாதிகள் உள்ளன, எனவே ஒவ்வொன்றும் பாதியாக இருக்கும் 18 வெவ்வேறு குழுக்களை நான் ஒன்றாகச் சேர்க்க வேண்டும், எனவே எனது குழுவின் அளவு இரண்டு முதல் 17 வது வரை இருக்கும் நிலைக்கு நான் வர வேண்டியிருக்கும். அந்த. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "ஒரு பெரிய எண்ணை எப்படிப் பெறுவது என்ற மனப்பூர்வ சிந்தனையைப் போலவே, 10 முதல் 400,000 வரையிலான ஒரு விஷயத்திற்கு உங்களை அழைத்துச் செல்ல எப்போதும் எடுக்கும். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/telugu/sentence_translations.json b/2020/ldm-natural-logs/telugu/sentence_translations.json index 943ff7964..ff66f640c 100644 --- a/2020/ldm-natural-logs/telugu/sentence_translations.json +++ b/2020/ldm-natural-logs/telugu/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "మీకు తెలుసా, మేము 1 ట్రిలియన్ 751, 1 ట్రిలియన్ 787 పొందాము. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "మరియు ఇది ప్రతి 27లో ఒకటి. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "పైకి సంబంధించిన అనేక ఇతర సూత్రాలు ఉన్నాయి, అది స్పష్టంగా ప్రైమ్‌లకు సంబంధించినది, అంటే, మీరు అదే గేమ్ ఆడతారు మరియు మీరు లాగరిథమ్‌లను తీసుకునే ఈ విచిత్రమైన ఫ్యాషన్‌ని కలిగి ఉన్నారు మరియు ఏదైనా కాదు సంవర్గమానం, లాగ్ బేస్ ఇ. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "కాబట్టి, ఈ మొత్తం 10 కంటే పెద్దది కావడానికి ముందు నేను ఎంతకాలం వెళ్లాలి అని మీరు ఆలోచిస్తున్నట్లయితే, హ్మ్మ్, నేను ఒకదానితో ఒకటి జోడించాలి, చూద్దాం, నా దగ్గర ఒకటి మరియు మిగిలినవి ఉన్నాయి అనే సహజత్వం మీకు ఉండవచ్చు వాటిలో సగభాగాలు ఉన్నాయి, కాబట్టి నేను 18 విభిన్న సమూహాలను ఒకదానికొకటి సగం లాగా జోడించాలి, కాబట్టి నేను నా సమూహం యొక్క పరిమాణం రెండు నుండి 17వ వరకు ఉండే స్థాయికి చేరుకోవలసి ఉంటుంది. అని. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "మరియు ఆ వెర్రి ఆలోచన కూడా, మీరు పెద్ద సంఖ్యను ఎలా పొందగలరనే మానసిక ఆలోచన వంటిది, మిమ్మల్ని 10 నుండి 400,000 పరిమాణంలో చేర్చడానికి ఎప్పటికీ పడుతుంది. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/thai/sentence_translations.json b/2020/ldm-natural-logs/thai/sentence_translations.json index 4eef62273..5447259c8 100644 --- a/2020/ldm-natural-logs/thai/sentence_translations.json +++ b/2020/ldm-natural-logs/thai/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "คุณจึงเห็นได้ว่าพวกมันกระจัดกระจายกว่าแค่ตัวเลขระหว่างศูนย์ถึงพันเท่านั้น แต่มีจำนวนที่มีความหมาย คุณก็รู้ เรามี 1 ล้านล้าน 751, 1 ล้านล้าน 787 วิศวกรของโบอิ้งคงพอใจที่มีสิ่งนี้อยู่ และความยาวจริงของรายการนั้นคือ 37. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "คุณอาจเดาได้ว่าเนื้อหาจะเป็นอย่างไรตามชื่อวิดีโอ สิ่งที่สุดท้ายเท่ากันคือลอกธรรมชาติของอันที่เคยเป็น ของไพกำลังสองส่วน 6 และนั่นไม่จริงกับลำดับเฉพาะของผลบวกกำลังสองนี้ มีสูตรอื่นๆ อีกหลายสูตรที่ทำให้เราได้บางอย่างที่เกี่ยวข้องกับพาย ซึ่งเห็นได้ชัดว่าเกี่ยวข้องกับจำนวนเฉพาะ ในลักษณะที่ อืม ฉันหมายถึง คุณเล่นเกมเดียวกัน และคุณมีรูปแบบแปลกๆ ในการใช้ลอการิทึม ไม่ใช่แค่สูตรใดๆ ลอการิทึม, ฐานบันทึก e เพื่ออธิบายสิ่งที่ฉันหมายถึงในบริบทอื่น ถ้าคุณเอา 1 ลบ หนึ่งในสาม บวก ห้า ลบ เจ็ด บวก อันดับที่ 9 แล้วสลับไปมาระหว่างเลขคี่ คุณจะได้ ไพ หารด้วย 4 ฉันมีวิดีโอเกี่ยวกับเรื่องนี้ทั้งหมด Mathologer มีวิดีโอเกี่ยวกับเรื่องนี้ด้วย หากคุณสงสัย สวยงามมากทำไมถึงจริง.. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "และคุณจะเห็นว่า โอเค ถ้าผมทำต่อไปนานพอ มันจะใหญ่ขึ้น และยังให้สัญชาตญาณเล็กๆ น้อยๆ ว่านี่อาจเกี่ยวข้องกับลอการิทึม เพราะขนาดของกลุ่มเราขยายตามกำลังสอง ถ้าคุณสงสัยว่าผมต้องใช้เวลานานแค่ไหนกว่าผลรวมนี้จะมากกว่า 10 คุณอาจมีสัญชาตญาณว่า อืม ผมจะต้องบวกเข้าด้วยกัน ลองดู ผมมีอันหนึ่ง แล้วที่เหลือ ในจำนวนนั้นเป็นครึ่งหนึ่ง ดังนั้นผมจะต้องรวมกลุ่มต่างๆ 18 กลุ่มเข้าด้วยกัน โดยแต่ละกลุ่มมีลักษณะเหมือนครึ่ง ดังนั้นผมอาจต้องไปถึงจุดที่ขนาดของกลุ่มของผมเท่ากับสองถึงกลุ่มที่ 17 ประมาณนี้ ที่. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/turkish/sentence_translations.json b/2020/ldm-natural-logs/turkish/sentence_translations.json index a5e339b6a..097634ffa 100644 --- a/2020/ldm-natural-logs/turkish/sentence_translations.json +++ b/2020/ldm-natural-logs/turkish/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "Biliyorsunuz elimizde 1 trilyon 751, 1 trilyon 787 var. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "Ve bu yaklaşık her 27 kişiden biri. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "Bize pi ile ilgili bir şeyler veren bir takım başka formüller de var, ki bu açıkça asal sayılarla ilgili, yani, yani, yani, aynı oyunu oynuyorsunuz ve logaritma alma konusunda garip bir tarzınız var, herhangi bir logaritma alma yönteminiz yok. logaritma, log tabanı e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "Yani eğer bu toplamın 10'dan büyük olması için ne kadar beklemem gerektiğini merak ediyorsanız, iç güdünüze göre, hmm, toplamam gerekecek, bakalım, bir tane var, sonra geri kalanı bunların yarısı yarım, yani her biri yarım gibi görünen 18 farklı grubu bir araya toplamam gerekecek, yani grubumun büyüklüğünün iki üzeri 17 gibi olduğu bir noktaya gelmem gerekebilir, şöyle bir şey O. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "Ve büyük bir sayıya nasıl ulaşabileceğinize dair zihinsel düşünce gibi o çılgın fikir bile, sizi 10 üzeri 400.000 boyutunda bir şeye ulaştırmak için sonsuza kadar sürer. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/ukrainian/sentence_translations.json b/2020/ldm-natural-logs/ukrainian/sentence_translations.json index 416a172d8..7b0a19472 100644 --- a/2020/ldm-natural-logs/ukrainian/sentence_translations.json +++ b/2020/ldm-natural-logs/ukrainian/sentence_translations.json @@ -350,7 +350,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787.", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven.", "translatedText": "Знаєте, у нас 1 трильйон 751, 1 трильйон 787.", "n_reviews": 0, "start": 225.72, @@ -392,7 +392,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27.", + "input": "And it's about one in every twenty-seven.", "translatedText": "І це приблизно один із кожних 27.", "n_reviews": 0, "start": 249.42, @@ -819,7 +819,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e.", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e.", "translatedText": "Є багато інших формул, які дають нам щось пов’язане з пі, яке, очевидно, пов’язане з простими числами, таким чином, гм, я маю на увазі, що ви граєте в ту саму гру, і у вас є така дивна мода логарифмувати, а не будь-який логарифм, основа логарифма e.", "n_reviews": 0, "start": 574.06, @@ -1295,7 +1295,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that.", + "input": "o if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that.", "translatedText": "Отже, якщо вам цікаво, скільки часу мені потрібно пройти, перш ніж ця сума стане більшою за 10, можливо, ви інстинктивно відчуєте, що, хм, мені доведеться додати разом, давайте подивимося, у мене є один, а потім решта з них є половинками, тож мені доведеться додати разом 18 різних груп, кожна з яких виглядає як половина, тож мені, можливо, доведеться дійти до точки, коли розмір моєї групи буде приблизно двома до 17-го, щось на зразок що.", "n_reviews": 0, "start": 976.16, @@ -1449,7 +1449,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000.", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000.", "translatedText": "І навіть ця божевільна ідея, як розумова думка про те, як можна досягти великого числа, знадобиться вічність, щоб отримати щось розміром від 10 до 400 000.", "n_reviews": 0, "start": 1187.36, @@ -1778,7 +1778,7 @@ "end": 1560.16 }, { - "input": "You can think just looking at this that somehow bell curves are produced by the number e but that's not exactly true because I could also write a to the negative x squared and I get the same family of curves as I tweak the value of a I'm also changing what that width is so I could come up with other ways of describing the standard deviation of this in terms of a and it's it's the same family of curves it's not just that they look similar they are in fact the same thing.", + "input": "You can think, just looking at this, that somehow bell curves are produced by the number e. But that's not exactly true, because I could also write a to the negative x squared, and I get the same family of curves. As I tweak the value of a, I'm also changing what that width is, so I could come up with other ways of describing the standard deviation of this in terms of a. And it's the same family of curves. It's not just that they look similar. They are, in fact, the same thing.", "translatedText": "Дивлячись на це, ви можете подумати, що якимось чином дзвонові криві створюються числом e, але це не зовсім так, тому що я міг би також записати a до від’ємного х у квадраті, і я отримую ту саму групу кривих, коли змінюю значення I' m також змінюю ширину, щоб я міг придумати інші способи опису стандартного відхилення цього в термінах a, і це те саме сімейство кривих, це не просто те, що вони виглядають схожими, вони насправді є одним і тим же.", "n_reviews": 0, "start": 1560.66, diff --git a/2020/ldm-natural-logs/urdu/sentence_translations.json b/2020/ldm-natural-logs/urdu/sentence_translations.json index 78f0fc84e..c5443b3f5 100644 --- a/2020/ldm-natural-logs/urdu/sentence_translations.json +++ b/2020/ldm-natural-logs/urdu/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "آپ جانتے ہیں، ہمارے پاس 1 ٹریلین 751، 1 ٹریلین 787 ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "اور یہ ہر 27 میں سے ایک ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "بہت سے دوسرے فارمولے ہیں جو ہمیں pi سے متعلق کچھ حاصل کرتے ہیں، جو ظاہر ہے کہ پرائمز سے متعلق ہے، اس طرح کہ، ام، میرا مطلب ہے، آپ ایک ہی کھیل کھیلتے ہیں اور آپ کے پاس لوگارتھمز لینے کا یہ عجیب فیشن ہے، نہ کہ کوئی لوگارتھم، لاگ بیس ای۔", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "لہذا اگر آپ سوچ رہے ہیں کہ اس رقم کے 10 سے زیادہ ہونے سے پہلے مجھے کتنی دیر تک جانا پڑے گا، تو آپ کی یہ جبلت ہو سکتی ہے کہ، ہمم، مجھے ایک ساتھ جوڑنا پڑے گا، آئیے دیکھتے ہیں، میرے پاس ایک ہے اور پھر باقی ان میں سے آدھے حصے ہیں، اس لیے مجھے 18 مختلف گروپس کو شامل کرنا ہوگا جو ہر ایک آدھے کی طرح نظر آتے ہیں، اس لیے مجھے اس مقام تک پہنچنا پڑے گا جہاں میرے گروپ کا سائز دو سے 17ویں کے برابر ہے، کچھ اس طرح کہ اور آپ اس بات پر نمایاں ہوں گے کہ یہ بڑھتا ہے، ٹھیک ہے یہ تیزی سے نہیں بڑھتا ہے، یہ منطقی طور پر بڑھتا ہے، کیونکہ اگر آپ پوچھ رہے ہیں کہ اس مقام تک پہنچنے کے لیے آپ کو کتنی دور جانے کی ضرورت ہے، تو یہ لوگاریتھمک ہوگا۔", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "اور یہاں تک کہ وہ پاگل خیال، دماغی سوچ کی طرح کہ آپ ایک بڑی تعداد تک کیسے پہنچ سکتے ہیں، آپ کو 10 سے 400,000 کے سائز تک لے جانے میں ہمیشہ کے لیے وقت لگے گا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-natural-logs/vietnamese/sentence_translations.json b/2020/ldm-natural-logs/vietnamese/sentence_translations.json index a2e3e5f15..582e8c675 100644 --- a/2020/ldm-natural-logs/vietnamese/sentence_translations.json +++ b/2020/ldm-natural-logs/vietnamese/sentence_translations.json @@ -392,7 +392,7 @@ "end": 225.46 }, { - "input": "You know, we've got 1 trillion 751, 1 trillion 787. ", + "input": "You know, we've got one trillion seven hundred fifty one, one trillion seven eighty seven. ", "translatedText": "Bạn biết đấy, chúng ta có 1 nghìn tỷ 751, 1 nghìn tỷ 787. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 249.42 }, { - "input": "And it's about one in every 27. ", + "input": "And it's about one in every twenty-seven. ", "translatedText": "Và cứ 27 thì có một. ", "model": "google_nmt", "n_reviews": 0, @@ -928,7 +928,7 @@ "end": 573.76 }, { - "input": "There's a number of other formulas that get us something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", + "input": "There's a number of other formulas that get us something related to prime, where we could, sorry, something related to pi, which is evidently related to primes, in a way that's, um, I mean, you play the same game and you have this weird fashion of taking logarithms, and not just any logarithm, the log base e. ", "translatedText": "Có một số công thức khác giúp chúng ta biết điều gì đó liên quan đến số pi, rõ ràng là có liên quan đến số nguyên tố, theo cách mà, ừm, ý tôi là, bạn chơi cùng một trò chơi và bạn có kiểu tính logarit kỳ lạ này, chứ không phải bất kỳ công thức nào logarit, logarit cơ số e. ", "model": "google_nmt", "n_reviews": 0, @@ -1472,7 +1472,7 @@ "end": 976.16 }, { - "input": "So if you're wondering how long do I have to go before this sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have one and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like two to the 17th, something like that. ", + "input": "if you were wondering, how long do I have to go before the sum gets bigger than 10, you might have the instinct that, hmm, I'm going to have to add together, let's see, I have 1 and then the rest of them are halves, so I'm going to have to add together 18 different groups that each look like a half, so I might have to get up to the point where the size of my group is like 2 to the 17th or 2 to the 18th, something like that. ", "translatedText": "Vì vậy, nếu bạn đang tự hỏi tôi phải mất bao lâu để tổng này lớn hơn 10, bạn có thể có trực giác rằng, hmm, tôi sẽ phải cộng lại với nhau, xem nào, tôi có một và phần còn lại trong số đó có một nửa, vì vậy tôi sẽ phải cộng 18 nhóm khác nhau lại với nhau mà mỗi nhóm trông giống như một nửa, vậy nên tôi có thể phải tăng đến mức mà quy mô nhóm của tôi là từ hai đến nhóm 17, đại loại như cái đó. ", "model": "google_nmt", "n_reviews": 0, @@ -1648,7 +1648,7 @@ "end": 1186.98 }, { - "input": "And even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", + "input": "and it would even, even that crazy idea, like mental thought for how you can get up to a big number, would take forever to get you to something of the size 10 to the 400,000. ", "translatedText": "Và ngay cả ý tưởng điên rồ đó, chẳng hạn như suy nghĩ trong đầu về cách bạn có thể đạt đến một con số lớn, cũng sẽ mất rất nhiều thời gian để đưa bạn đến con số có kích thước từ 10 đến 400.000. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/arabic/sentence_translations.json b/2020/ldm-power-towers/arabic/sentence_translations.json index 6bbfe1c33..86f773a27 100644 --- a/2020/ldm-power-towers/arabic/sentence_translations.json +++ b/2020/ldm-power-towers/arabic/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "وإذا كانت الغرفة مليئة بالمنطقيين المثاليين، فسوف تنفجر إلى ما لا نهاية، لكن الناس ليسوا منطقيين، وهناك بعض الإجابات الصحيحة موضوعيًا، ويمكننا إلقاء نظرة على الإجابة الصحيحة موضوعيًا في هذا السياق . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "لذا انظر إلى الخط y يساوي x، ولنفترض أن لدي دالة تتمايل على طولها وتتقاطع معها، ولكن بميل أكبر من 1. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "والآن لدي قرار يجب أن أتخذه. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "هل أحاول التفكير فيما يحدث بالضبط على الشاشة مباشرة بدلاً من مجرد استدعائه إلى البث؟ أو هل أفكر في الأمر ثم أعلق ذلك على تعليق في النهاية؟ دعنا نرى. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/bengali/sentence_translations.json b/2020/ldm-power-towers/bengali/sentence_translations.json index e309a3f22..08323da0d 100644 --- a/2020/ldm-power-towers/bengali/sentence_translations.json +++ b/2020/ldm-power-towers/bengali/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "এবং যদি এটি নিখুঁত যুক্তিবিদদের একটি রুম ছিল, তাহলে আপনি অসীম পর্যন্ত উড়িয়ে দিতেন, কিন্তু লোকেরা যুক্তিবিদ নয়, এবং কিছু বস্তুনিষ্ঠভাবে সঠিক উত্তর আছে, এবং আমরা এই প্রসঙ্গে বস্তুনিষ্ঠভাবে সঠিক উত্তরটি কী তা দেখে নিতে পারি . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "সুতরাং লাইনটি দেখুন y সমান x x, এবং ধরা যাক আমার কিছু ফাংশন আছে যা বরাবর স্কুইগল করে এবং এটি একে ছেদ করে, কিন্তু 1 এর চেয়ে বেশি ঢাল সহ।", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "এখন আমার একটা সিদ্ধান্ত নেওয়ার আছে।", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "অথবা আমি কি এটি সম্পর্কে চিন্তা করি এবং তারপরে এটিকে শেষে একটি মন্তব্যে পিন করি? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/chinese/sentence_translations.json b/2020/ldm-power-towers/chinese/sentence_translations.json index 35706069f..59d3e06b2 100644 --- a/2020/ldm-power-towers/chinese/sentence_translations.json +++ b/2020/ldm-power-towers/chinese/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "如果这是一个充满完美逻辑学家的房间，你会爆炸到无穷 大，但人们不是逻辑学家，并且存在一些客观正确的答案 ，我们可以看看在这种情况下客观正确的答案是什么。", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "看一下 y 等于 x 的线，假设我有一些函数，它 沿着曲线蜿蜒前行，并与它相交，但斜率大于 1。", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "现在我要做出决定。", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "或者我是否考虑一下然后将其固定在最后的评论中？", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/english/captions.srt b/2020/ldm-power-towers/english/captions.srt index e874e5329..eb8915495 100644 --- a/2020/ldm-power-towers/english/captions.srt +++ b/2020/ldm-power-towers/english/captions.srt @@ -319,7 +319,7 @@ If you want to make that crystal clear, I think instead of drawing it as a power one thing that you could do is define the iterative process very exactly. 81 -00:04:59,859 --> 00:05:03,774 +00:04:59,860 --> 00:05:03,774 You might say we have some value that we're going to start out at one, 82 @@ -912,7 +912,7 @@ Because it's infinite, it's a genuine copy of itself. 229 00:14:15,300 --> 00:14:19,680 -It doesn't have a height of 1 minus whatever the previous height was. +It's not, it doesn't have a height of 1 minus whatever the previous height was, 230 00:14:19,860 --> 00:14:20,900 @@ -935,7 +935,7 @@ Let's see, what would that be? If I take square roots, that's the same as saying x squared equals 2. 235 -00:14:36,079 --> 00:14:40,377 +00:14:36,080 --> 00:14:40,377 So it looks like, interesting, x equals square root 236 @@ -995,7 +995,7 @@ The question asks us, using the tactic just demonstrated, what I just did for 4, solve the equation x to the x to the x on and on up to infinity equals 2. 250 -00:15:33,959 --> 00:15:36,400 +00:15:33,960 --> 00:15:36,400 So I'll give you a moment to think that through. 251 @@ -1059,7 +1059,7 @@ So, hope you enjoy. Now on the quiz, it looks like just about everybody is converging around the same answer. 266 -00:16:47,219 --> 00:16:50,424 +00:16:47,220 --> 00:16:50,424 And I'm going to assume you've landed on the correct answer, 267 @@ -1484,7 +1484,7 @@ We're bouncing towards the value where these graphs intersect each other. 372 00:22:51,380 --> 00:22:56,540 -Towards the value where b to the power x is equal to x. +towards the value where b of x, b to the power x, excuse me, is equal to x. 373 00:22:57,520 --> 00:23:00,582 @@ -1520,7 +1520,7 @@ precisely the square root of 2 rather than just an approximation here. 381 00:23:22,980 --> 00:23:27,900 -So b is going to be the square root of 2, no, of 2. +So b is going to be the square root, square root of 2, no, of 2. 382 00:23:28,720 --> 00:23:30,858 @@ -1859,7 +1859,7 @@ Okay, so take a moment to think about this. Which of the following is the pair of equations that we need to solve? 466 -00:28:48,789 --> 00:28:50,607 +00:28:48,790 --> 00:28:50,607 While you're thinking about that, I'll go ahead 467 @@ -1867,7 +1867,7 @@ While you're thinking about that, I'll go ahead and take a couple questions from the audience. 468 -00:28:54,149 --> 00:28:54,950 +00:28:54,150 --> 00:28:54,950 We have... 469 @@ -1995,7 +1995,7 @@ We can take a look exactly in the graph that we have. So if we take b and we make it 1.1. 500 -00:30:36,429 --> 00:30:36,830 +00:30:36,430 --> 00:30:36,830 Okay. 501 @@ -2171,7 +2171,7 @@ and we want that slope to equal one, because it's got to be the same as the slope of the graph we're just looking at. 544 -00:33:01,689 --> 00:33:06,599 +00:33:01,690 --> 00:33:06,599 And if ever you don't remember what the derivatives of your exponential functions are, 545 @@ -2283,7 +2283,7 @@ It'll be e to the one over x, all to the power x, I'm just replacing the b with what we found it to be, and that is supposed to equal x. 572 -00:35:22,609 --> 00:35:28,538 +00:35:22,610 --> 00:35:28,538 But on the other hand, e to the one over x to the power x simplifies 573 @@ -2307,7 +2307,7 @@ that's an unlisted link, and just say like as soon as a hundred of you hop on here I'm deleting the tweet and we're just going to do a dry run. 578 -00:35:48,189 --> 00:35:52,533 +00:35:48,190 --> 00:35:52,533 And when I was solving this, for some reason I was just confuddled for like ten minutes 579 @@ -2531,7 +2531,7 @@ And this answers for us what's going on in the case of the power tower with 4. So if you think back to the logic, where did we have it? 634 -00:39:18,509 --> 00:39:18,950 +00:39:18,510 --> 00:39:18,950 Great. 635 @@ -2943,7 +2943,7 @@ That's the level of what we actually don't know about these power towers, which I think is kind of interesting. 737 -00:45:18,009 --> 00:45:20,990 +00:45:18,010 --> 00:45:20,990 Can I share the brainteaser where the arrow operator shows up? 738 @@ -3111,7 +3111,7 @@ Unless you're doing statistical mechanics or something like that. So, yeah, excellent question. 779 -00:47:45,509 --> 00:47:50,810 +00:47:45,510 --> 00:47:50,810 And with that, unless there's anything more that people want to ask and which also ends 780 @@ -3155,11 +3155,11 @@ you ignored the negative square root of two. Awesome. 790 -00:48:19,570 --> 00:48:19,390 +00:48:19,570 --> 00:48:19,670 Yeah. 791 -00:48:19,570 --> 00:48:20,170 +00:48:20,170 --> 00:48:20,170 Yeah. 792 @@ -3247,7 +3247,7 @@ What do we mean to have a negative square root here? Well, I guess what you'd be assuming is that you you're taking negative powers. 813 -00:49:24,010 --> 00:49:25,169 +00:49:24,010 --> 00:49:25,170 Now that I think about it. 814 @@ -3371,7 +3371,7 @@ What is that number? What's going on there? 844 -00:51:12,509 --> 00:51:13,710 +00:51:12,510 --> 00:51:13,710 You know what? 845 @@ -3396,7 +3396,7 @@ Or do I do I think about it and then like pin that to a comment at the end? 850 00:51:36,930 --> 00:51:38,650 -What's going on with our logic exactly? +Let's see. What's going on with our logic exactly? 851 00:51:40,670 --> 00:51:43,070 @@ -3511,7 +3511,7 @@ we already saw is a faulty line of reasoning and like a little bit shaky to just replace what's inside your infinite power tower with the assumptions you've already made. 879 -00:53:10,029 --> 00:53:11,350 +00:53:10,030 --> 00:53:11,350 Yeah I'll think on that. 880 diff --git a/2020/ldm-power-towers/english/sentence_timings.json b/2020/ldm-power-towers/english/sentence_timings.json index e69c103f5..9a56e52e1 100644 --- a/2020/ldm-power-towers/english/sentence_timings.json +++ b/2020/ldm-power-towers/english/sentence_timings.json @@ -600,7 +600,7 @@ 855.14 ], [ - "It doesn't have a height of 1 minus whatever the previous height was.", + "It's not, it doesn't have a height of 1 minus whatever the previous height was,", 855.3, 859.68 ], @@ -1115,7 +1115,7 @@ 1370.68 ], [ - "Towards the value where b to the power x is equal to x.", + "towards the value where b of x, b to the power x, excuse me, is equal to x.", 1371.38, 1376.54 ], @@ -1135,7 +1135,7 @@ 1402.46 ], [ - "So b is going to be the square root of 2, no, of 2.", + "So b is going to be the square root, square root of 2, no, of 2.", 1402.98, 1407.9 ], @@ -2292,11 +2292,11 @@ [ "Yeah.", 2899.57, - 2899.39 + 2899.67 ], [ "Yeah.", - 2899.57, + 2900.17, 2900.17 ], [ @@ -2555,7 +2555,7 @@ 3093.25 ], [ - "What's going on with our logic exactly?", + "Let's see. What's going on with our logic exactly?", 3096.93, 3098.65 ], diff --git a/2020/ldm-power-towers/english/transcript.txt b/2020/ldm-power-towers/english/transcript.txt index 40a9f33da..f7f3eaad9 100644 --- a/2020/ldm-power-towers/english/transcript.txt +++ b/2020/ldm-power-towers/english/transcript.txt @@ -118,7 +118,7 @@ And there's a clever trick here that you might spot. And this comes up in certain problem solving kinds of math where you have this infinite expression, and you say, hmm, there's some self-similarity I can leverage. There's a copy of the entire power tower inside itself. Because it's infinite, it's a genuine copy of itself. -It doesn't have a height of 1 minus whatever the previous height was. +It's not, it doesn't have a height of 1 minus whatever the previous height was, Because the height is infinity. And under the assumption that the whole power tower equals 4, I could replace that with a 4 and solve x to the fourth equals 4. Let's see, what would that be? @@ -221,11 +221,11 @@ At some point, the graphs actually cross each other. So in particular, we were looking at 1.1 earlier, and that, yeah, they absolutely cross. But even up to around 1.41, which is around the square root of 2, if you look at what happens with this process, we look at the output, turn the output into an input, look at the new output, output into an input, and bounce back and forth. We're bouncing towards the value where these graphs intersect each other. -Towards the value where b to the power x is equal to x. +towards the value where b of x, b to the power x, excuse me, is equal to x. And in particular, if that base was the square root of 2, and I say find a solution of square root of 2 to the power x equals x, you know, it's not easy to think about how you solve this systematically, but for this particular case, you would believe me if I told you that the solution is x equals 2. You can just plug that in and solve it. So if we look at our graph, if b, actually let's go ahead and make it precisely the square root of 2 rather than just an approximation here. -So b is going to be the square root of 2, no, of 2. +So b is going to be the square root, square root of 2, no, of 2. The intersection point is exactly at 2, so you see your iterative process approaching that. Okay, that's kind of interesting. This also shows us how we're going to need to think about it if we want to be, if we want to understand where the switch happens between when things converge and when things don't converge. @@ -509,7 +509,7 @@ I'm not entirely positive. Now that now I have it now I have a decision to make it. Do I try to think about what exactly is going on on screen live rather than just calling it into the stream? Or do I do I think about it and then like pin that to a comment at the end? -What's going on with our logic exactly? +Let's see. What's going on with our logic exactly? We're saying assume that there equals a value that converges to two. If there does it's going to satisfy this property. I mean because I guess what you can say is it's like x equals plus or minus square root of two. diff --git a/2020/ldm-power-towers/french/sentence_translations.json b/2020/ldm-power-towers/french/sentence_translations.json index 888dbd87f..93fda0525 100644 --- a/2020/ldm-power-towers/french/sentence_translations.json +++ b/2020/ldm-power-towers/french/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "Et si c'était une salle remplie de logiciens parfaits, vous exploseriez à l'infini, mais les gens ne sont pas des logiciens, et il existe une réponse objectivement correcte, et nous pouvons jeter un œil à quelle est la réponse objectivement correcte dans ce contexte. . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "Alors regardez la ligne y est égal à x, et disons que j'ai une fonction qui se déplace et la coupe, mais avec une pente supérieure à 1. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "Maintenant, j'ai une décision à prendre. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "Ou est-ce que j'y réfléchis et que j'épingle cela dans un commentaire à la fin ? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/german/sentence_translations.json b/2020/ldm-power-towers/german/sentence_translations.json index afe7f49a8..442246ba5 100644 --- a/2020/ldm-power-towers/german/sentence_translations.json +++ b/2020/ldm-power-towers/german/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "Und wenn es ein Raum voller perfekter Logiker wäre, würde man bis ins Unendliche explodieren, aber Menschen sind keine Logiker, und es gibt eine objektiv richtige Antwort, und wir können uns in diesem Zusammenhang ansehen, was die objektiv richtige Antwort ist . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "Schauen Sie sich also an, dass die Gerade y gleich x ist, und nehmen wir an, ich habe eine Funktion, die sich entlang schlängelt und sie schneidet, aber mit einer Steigung größer als 1. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "Jetzt muss ich eine Entscheidung treffen. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "Oder denke ich darüber nach und pinnen das dann am Ende als Kommentar an? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/hebrew/sentence_translations.json b/2020/ldm-power-towers/hebrew/sentence_translations.json index 121e1c546..5bb1b8f60 100644 --- a/2020/ldm-power-towers/hebrew/sentence_translations.json +++ b/2020/ldm-power-towers/hebrew/sentence_translations.json @@ -56,7 +56,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context.", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context.", "translatedText": "ואם זה היה חדר מלא לוגיקאים מושלמים, היית מתפוצץ עד אינסוף, אבל אנשים הם לא לוגיקים, ויש איזו תשובה נכונה אובייקטיבית, ונוכל להסתכל מהי התשובה הנכונה אובייקטיבית בהקשר הזה .", "n_reviews": 0, "start": 62.44, @@ -1246,7 +1246,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1.", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1.", "translatedText": "אז תסתכל על הישר y שווה ל-x, ונניח שיש לי פונקציה כלשהי שמתפתלת לאורך והיא חותכת אותה, אבל עם שיפוע גדול מ-1.", "n_reviews": 0, "start": 1474.94, @@ -3045,7 +3045,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make.", + "input": "Now that now I have it now I have a decision to make it.", "translatedText": "עכשיו יש לי החלטה לקבל.", "n_reviews": 0, "start": 3078.15, @@ -3059,7 +3059,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end?", + "input": "Or do I do I think about it and then like pin that to a comment at the end?", "translatedText": "או שאני חושב על זה ואז מצמיד את זה להערה בסוף?", "n_reviews": 0, "start": 3087.51, diff --git a/2020/ldm-power-towers/hindi/sentence_translations.json b/2020/ldm-power-towers/hindi/sentence_translations.json index 0a6103986..eed79dc76 100644 --- a/2020/ldm-power-towers/hindi/sentence_translations.json +++ b/2020/ldm-power-towers/hindi/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "और यदि यह संपूर्ण तर्कशास्त्रियों से भरा कमरा होता, तो आप अनंत तक उड़ जाते, लेकिन लोग तर्कशास्त्री नहीं होते हैं, और कुछ वस्तुनिष्ठ रूप से सही उत्तर होता है, और हम इस संदर्भ में देख सकते हैं कि वस्तुनिष्ठ रूप से सही उत्तर क्या है . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "तो रेखा y को x के बराबर देखें, और मान लें कि मेरे पास कुछ फ़ंक्शन है जो साथ घूमता है और यह इसे काटता है, लेकिन 1 से अधिक ढलान के साथ।", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "अब मुझे निर्णय लेना है. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "या क्या मैं इसके बारे में सोचूं और फिर उसे अंत में एक टिप्पणी पर पिन कर दूं? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/hungarian/sentence_translations.json b/2020/ldm-power-towers/hungarian/sentence_translations.json index 89df59987..ec671b61b 100644 --- a/2020/ldm-power-towers/hungarian/sentence_translations.json +++ b/2020/ldm-power-towers/hungarian/sentence_translations.json @@ -56,7 +56,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context.", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context.", "translatedText": "És ha ez egy tökéletes logikusokkal teli szoba lenne, akkor a végtelenségig felrobbannál, de az emberek nem logikások, és van objektíven helyes válasz, és megnézhetjük, mi az objektíven helyes válasz ebben az összefüggésben. .", "n_reviews": 0, "start": 62.44, @@ -1246,7 +1246,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1.", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1.", "translatedText": "Tehát nézd meg, hogy az y egyenlő x-szel, és tegyük fel, hogy van valami függvényem, amely végigkanyarodik, és metszi azt, de 1-nél nagyobb meredekséggel.", "n_reviews": 0, "start": 1474.94, @@ -3045,7 +3045,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make.", + "input": "Now that now I have it now I have a decision to make it.", "translatedText": "Most egy döntést kell meghoznom.", "n_reviews": 0, "start": 3078.15, @@ -3059,7 +3059,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end?", + "input": "Or do I do I think about it and then like pin that to a comment at the end?", "translatedText": "Vagy gondolkodjak rajta, majd rögzítsem egy megjegyzéshez a végén?", "n_reviews": 0, "start": 3087.51, diff --git a/2020/ldm-power-towers/indonesian/sentence_translations.json b/2020/ldm-power-towers/indonesian/sentence_translations.json index 57c24c0b2..7e2d43b01 100644 --- a/2020/ldm-power-towers/indonesian/sentence_translations.json +++ b/2020/ldm-power-towers/indonesian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "Dan jika ruangan tersebut penuh dengan ahli logika yang sempurna, Anda akan meledak hingga tak terhingga, namun manusia bukanlah ahli logika, dan ada beberapa jawaban yang benar secara obyektif, dan kita dapat melihat jawaban yang benar secara obyektif dalam konteks ini. . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "Jadi lihatlah garis y sama dengan x, dan misalkan saya mempunyai suatu fungsi yang berlekuk-lekuk dan memotongnya, namun dengan kemiringan lebih besar dari 1. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "Sekarang saya harus mengambil keputusan. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "Atau apakah saya memikirkannya lalu menyematkannya ke komentar di akhir? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/italian/sentence_translations.json b/2020/ldm-power-towers/italian/sentence_translations.json index 14bf05a60..55f3ca92f 100644 --- a/2020/ldm-power-towers/italian/sentence_translations.json +++ b/2020/ldm-power-towers/italian/sentence_translations.json @@ -56,7 +56,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context.", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context.", "translatedText": "E se fosse una stanza piena di logici perfetti, esploderesti all'infinito, ma le persone non sono logiche, e c'è qualche risposta oggettivamente corretta, e possiamo dare un'occhiata a quale sia la risposta oggettivamente corretta in questo contesto .", "n_reviews": 0, "start": 62.44, @@ -1246,7 +1246,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1.", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1.", "translatedText": "Quindi guarda la linea y uguale a x, e diciamo che ho una funzione che serpeggia e la interseca, ma con una pendenza maggiore di 1.", "n_reviews": 0, "start": 1474.94, @@ -3045,7 +3045,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make.", + "input": "Now that now I have it now I have a decision to make it.", "translatedText": "Ora devo prendere una decisione.", "n_reviews": 0, "start": 3078.15, @@ -3059,7 +3059,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end?", + "input": "Or do I do I think about it and then like pin that to a comment at the end?", "translatedText": "Oppure ci penso e poi lo aggiungo a un commento alla fine?", "n_reviews": 0, "start": 3087.51, diff --git a/2020/ldm-power-towers/japanese/sentence_translations.json b/2020/ldm-power-towers/japanese/sentence_translations.json index d90ffb570..0467b7f05 100644 --- a/2020/ldm-power-towers/japanese/sentence_translations.json +++ b/2020/ldm-power-towers/japanese/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "そして、もしそれが完璧な論理学者でいっぱいの部屋だったら、あなたは無限に吹き飛ばさ れてしまうだろうが、人間は論理学者ではない、そして客観的に正しい答えがいくつかある 、そして私たちはこの文脈で客観的に正しい答えが何であるかを見てみることができる。", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "そこで、線 y が x に等しいことに注目してください。波線を描いて 交差する関数があるとしますが、その傾きは 1 より大きいとします。", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "今、私は決断を下さなければなりません。", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "それとも、それについて考えて、最後にコメントに固定しますか? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/korean/sentence_translations.json b/2020/ldm-power-towers/korean/sentence_translations.json index bb81d2bc1..57f3626f0 100644 --- a/2020/ldm-power-towers/korean/sentence_translations.json +++ b/2020/ldm-power-towers/korean/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "그리고 완벽한 논리학자들이 가득한 방이었다면 무한대로 터지겠지만 사람은 논리학자가 아니고 객관적으로 정답이 있는데, 이런 맥락에서 객관적으로 정답이 무엇인지 살펴볼 수 있습니다. . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "y=x 선을 보세요. 구불구불한 선을 따라 교차하지만 기울기가 1보다 큰 함수가 있다고 가정해 보겠습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "이제 나는 결정을 내려야 합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "아니면 그것에 대해 생각한 다음 마지막에 댓글에 고정합니까? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/marathi/sentence_translations.json b/2020/ldm-power-towers/marathi/sentence_translations.json index 795ed5f81..2fe4d2642 100644 --- a/2020/ldm-power-towers/marathi/sentence_translations.json +++ b/2020/ldm-power-towers/marathi/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "आणि जर ती परिपूर्ण तर्कशास्त्रज्ञांनी भरलेली खोली असेल, तर तुम्ही अनंतापर्यंत पोहोचाल, परंतु लोक तर्कशास्त्रज्ञ नाहीत, आणि काही वस्तुनिष्ठपणे योग्य उत्तर आहे आणि या संदर्भात वस्तुनिष्ठपणे योग्य उत्तर काय आहे ते आम्ही पाहू शकतो. . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "तर y बरोबर x ही रेषा पहा, आणि समजा माझ्याकडे काही फंक्शन आहे जे बाजूने squiggles आणि ते त्याला छेदते, परंतु 1 पेक्षा जास्त उतार असलेले. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "आता मला एक निर्णय घ्यायचा आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "किंवा मी त्याबद्दल विचार करतो आणि नंतर शेवटी टिप्पणीवर पिन करतो? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/persian/sentence_translations.json b/2020/ldm-power-towers/persian/sentence_translations.json index 9bf77c8c7..91ae33c4d 100644 --- a/2020/ldm-power-towers/persian/sentence_translations.json +++ b/2020/ldm-power-towers/persian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "و اگر اتاقی پر از منطق‌دانان کامل بود، تا بی‌نهایت منفجر می‌شد، اما مردم منطق‌دان نیستند، و برخی از پاسخ‌های عینی درست وجود دارد، و ما می‌توانیم نگاهی بیندازیم که پاسخ عینی درست در این زمینه چیست. . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "بنابراین به خط y برابر با x نگاه کنید، و فرض کنید من تابعی دارم که در امتداد حرکت می کند و آن را قطع می کند، اما با شیب بزرگتر از 1. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "آیا سعی می‌کنم به جای فراخوانی آن به جریان، به آنچه که دقیقاً روی صفحه نمایش می‌گذرد فکر کنم؟ یا در مورد آن فکر می کنم و سپس آن را به یک نظر در پایان پین می کنم؟ اجازه بدید ببینم. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/portuguese/sentence_translations.json b/2020/ldm-power-towers/portuguese/sentence_translations.json index 36758b3ef..c85319645 100644 --- a/2020/ldm-power-towers/portuguese/sentence_translations.json +++ b/2020/ldm-power-towers/portuguese/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "E se fosse uma sala cheia de lógicos perfeitos, você explodiria até o infinito, mas as pessoas não são lógicos, e há alguma resposta objetivamente correta, e podemos dar uma olhada em qual é a resposta objetivamente correta neste contexto . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "Então olhe para a reta y igual a x, e digamos que eu tenha alguma função que se curva e a intercepta, mas com uma inclinação maior que 1. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "Agora tenho uma decisão a tomar. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "Ou devo pensar sobre isso e fixar isso em um comentário no final? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/russian/sentence_translations.json b/2020/ldm-power-towers/russian/sentence_translations.json index 3b8b19d14..fcb23a2c4 100644 --- a/2020/ldm-power-towers/russian/sentence_translations.json +++ b/2020/ldm-power-towers/russian/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "А если бы это была комната, полная идеальных логиков, вы бы взорвались до бесконечности, но люди не логики, и есть какой-то объективно правильный ответ, и мы можем посмотреть, что является объективно правильным ответом в этом контексте. . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "Итак, посмотрите на линию y, равную x, и допустим, у меня есть некоторая функция, которая движется волнистой линией и пересекает ее, но с наклоном больше 1. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "Теперь мне нужно принять решение. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "Или я подумаю об этом, а затем прикреплю это к комментарию в конце? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/spanish/sentence_translations.json b/2020/ldm-power-towers/spanish/sentence_translations.json index 524f3ee92..61b1e8644 100644 --- a/2020/ldm-power-towers/spanish/sentence_translations.json +++ b/2020/ldm-power-towers/spanish/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "Y si fuera una sala llena de lógicos perfectos, explotarías hasta el infinito, pero las personas no son lógicos, y hay alguna respuesta objetivamente correcta, y podemos echar un vistazo a cuál es la respuesta objetivamente correcta en este contexto. . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "Entonces, mire la línea y es igual a x, y digamos que tengo alguna función que se desplaza y la intersecta, pero con una pendiente mayor que 1. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "Ahora tengo que tomar una decisión. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "¿O lo pienso y lo pongo en un comentario al final? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/tamil/sentence_translations.json b/2020/ldm-power-towers/tamil/sentence_translations.json index 46352f692..99f3f466d 100644 --- a/2020/ldm-power-towers/tamil/sentence_translations.json +++ b/2020/ldm-power-towers/tamil/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "அது சரியான தர்க்கவாதிகள் நிறைந்த அறையாக இருந்தால், நீங்கள் முடிவிலிக்கு ஊதுவீர்கள், ஆனால் மக்கள் தர்க்கவாதிகள் அல்ல, மேலும் சில புறநிலை ரீதியாக சரியான பதில் உள்ளது, மேலும் இந்த சூழலில் புறநிலை ரீதியாக சரியான பதில் என்ன என்பதை நாம் பார்க்கலாம். . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "எனவே y x க்கு சமமான வரியைப் பாருங்கள், நான் சில செயல்பாடுகளைக் கொண்டிருக்கிறேன் என்று வைத்துக்கொள்வோம், அது குறுக்கிடும், ஆனால் 1 ஐ விட அதிகமான சாய்வுடன். ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "இப்போது நான் ஒரு முடிவை எடுக்க வேண்டும். ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "அல்லது நான் அதைப் பற்றி யோசித்து, இறுதியில் அதை ஒரு கருத்துடன் இணைக்க வேண்டுமா? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/telugu/sentence_translations.json b/2020/ldm-power-towers/telugu/sentence_translations.json index a361c38d0..32723c3f2 100644 --- a/2020/ldm-power-towers/telugu/sentence_translations.json +++ b/2020/ldm-power-towers/telugu/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "మరియు అది పరిపూర్ణమైన లాజిక్కులతో నిండిన గది అయితే, మీరు అనంతంగా దూసుకుపోతారు, కానీ వ్యక్తులు లాజిక్కులు కారు, మరియు కొంత నిష్పాక్షికంగా సరైన సమాధానం ఉంది మరియు ఈ సందర్భంలో నిష్పాక్షికంగా సరైన సమాధానం ఏమిటో మనం పరిశీలించవచ్చు. . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "కాబట్టి y xకి సమానం అనే పంక్తిని చూడండి, మరియు నా దగ్గర కొంత ఫంక్షన్ ఉందని అనుకుందాం, అది స్క్విగ్లింగ్ చేస్తుంది మరియు అది కలుస్తుంది, కానీ 1 కంటే ఎక్కువ వాలుతో ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "ఇప్పుడు నేను ఒక నిర్ణయం తీసుకోవలసి ఉంది. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "లేదా నేను దాని గురించి ఆలోచించి, చివరికి వ్యాఖ్యకు పిన్ చేయాలా? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/thai/sentence_translations.json b/2020/ldm-power-towers/thai/sentence_translations.json index 61f57a643..d36a4da3f 100644 --- a/2020/ldm-power-towers/thai/sentence_translations.json +++ b/2020/ldm-power-towers/thai/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "หรือฉันคิดเกี่ยวกับมันแล้วปักหมุดไว้ที่ความคิดเห็นในตอนท้าย? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/turkish/sentence_translations.json b/2020/ldm-power-towers/turkish/sentence_translations.json index 5ce4adce3..7a907bfd2 100644 --- a/2020/ldm-power-towers/turkish/sentence_translations.json +++ b/2020/ldm-power-towers/turkish/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "Ve eğer mükemmel mantıkçılarla dolu bir oda olsaydı, sonsuza kadar havaya uçardınız, ancak insanlar mantıkçı değildir ve nesnel olarak bazı doğru cevaplar vardır ve bu bağlamda nesnel olarak doğru cevabın ne olduğuna bir göz atabiliriz. . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "Yani y eşittir x doğrusuna bakın ve diyelim ki dalgalı bir şekilde ilerleyen ve onunla kesişen fakat eğimi 1'den büyük olan bir fonksiyonum var. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "Şimdi vermem gereken bir karar var. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "Yoksa bunun hakkında düşünüp sonunda bunu bir yoruma mı sabitleyeceğim? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/ukrainian/sentence_translations.json b/2020/ldm-power-towers/ukrainian/sentence_translations.json index 80b957114..0a835afcf 100644 --- a/2020/ldm-power-towers/ukrainian/sentence_translations.json +++ b/2020/ldm-power-towers/ukrainian/sentence_translations.json @@ -56,7 +56,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context.", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context.", "translatedText": "І якби це була кімната, повна досконалих логіків, ви б роздули до безкінечності, але люди не логіки, і є якась об’єктивно правильна відповідь, і ми можемо поглянути на об’єктивно правильну відповідь у цьому контексті .", "n_reviews": 0, "start": 62.44, @@ -1246,7 +1246,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1.", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1.", "translatedText": "Тож подивіться на пряму y, яка дорівнює x, і, скажімо, у мене є якась функція, яка рухається вздовж і перетинає її, але з нахилом, більшим за 1.", "n_reviews": 0, "start": 1474.94, @@ -3045,7 +3045,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make.", + "input": "Now that now I have it now I have a decision to make it.", "translatedText": "Тепер мені потрібно прийняти рішення.", "n_reviews": 0, "start": 3078.15, @@ -3059,7 +3059,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end?", + "input": "Or do I do I think about it and then like pin that to a comment at the end?", "translatedText": "Або я думаю про це, а потім прикріплюю це до коментаря в кінці?", "n_reviews": 0, "start": 3087.51, diff --git a/2020/ldm-power-towers/urdu/sentence_translations.json b/2020/ldm-power-towers/urdu/sentence_translations.json index 668920e07..30fd21da2 100644 --- a/2020/ldm-power-towers/urdu/sentence_translations.json +++ b/2020/ldm-power-towers/urdu/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "اور اگر یہ کامل منطق دانوں سے بھرا ہوا ایک کمرہ تھا، تو آپ لامحدودیت کو اڑا دیتے، لیکن لوگ منطق دان نہیں ہیں، اور کچھ معروضی طور پر درست جواب موجود ہے، اور ہم ایک نظر ڈال سکتے ہیں کہ اس تناظر میں معروضی طور پر درست جواب کیا ہے۔. ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "تو لائن y کو x کے برابر دیکھیں، اور ہم کہتے ہیں کہ میرے پاس کچھ فنکشن ہے جو اسکوگل کرتا ہے اور یہ اسے کاٹتا ہے، لیکن 1 سے زیادہ ڈھلوان کے ساتھ۔", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "اب مجھے ایک فیصلہ کرنا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "کیا میں یہ سوچنے کی کوشش کرتا ہوں کہ اسکرین لائیو پر کیا ہو رہا ہے بجائے اس کے کہ اسے اسٹریم میں بلایا جائے؟ یا کیا میں اس کے بارے میں سوچتا ہوں اور پھر اسے آخر میں کسی تبصرے میں پن کرتا ہوں؟ چلو دیکھتے ہیں. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-power-towers/vietnamese/sentence_translations.json b/2020/ldm-power-towers/vietnamese/sentence_translations.json index 363b7a334..df0ca5592 100644 --- a/2020/ldm-power-towers/vietnamese/sentence_translations.json +++ b/2020/ldm-power-towers/vietnamese/sentence_translations.json @@ -64,7 +64,7 @@ "end": 61.92 }, { - "input": "And if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", + "input": "And, you know, if it was a room full of perfect logicians, you'd blow up to infinity, but people aren't logicians, and there is some objectively correct answer, and we can take a look at what the objectively correct answer is in this context. ", "translatedText": "Và nếu đó là một căn phòng đầy những nhà logic học hoàn hảo, bạn sẽ nổ tung đến vô cùng, nhưng mọi người không phải là những nhà logic học, và có một số câu trả lời đúng về mặt khách quan, và chúng ta có thể xem câu trả lời đúng về mặt khách quan là gì trong bối cảnh này . ", "model": "google_nmt", "n_reviews": 0, @@ -1424,7 +1424,7 @@ "end": 1474.2 }, { - "input": "So look at the line y equals x, and let's say I have some function which squiggles along and it intersects it, but with a slope greater than 1. ", + "input": "So look at the line y equals x, and let's say I have some function which, you know, it squiggles along and it intersects it but with a slope greater than 1. ", "translatedText": "Vì vậy, hãy nhìn vào đường thẳng y bằng x, và giả sử tôi có một hàm số nào đó ngoằn ngoèo dọc theo và nó cắt nó, nhưng có hệ số góc lớn hơn 1. ", "model": "google_nmt", "n_reviews": 0, @@ -3464,7 +3464,7 @@ "end": 3077.35 }, { - "input": "Now I have a decision to make. ", + "input": "Now that now I have it now I have a decision to make it. ", "translatedText": "Bây giờ tôi phải đưa ra một quyết định. ", "model": "google_nmt", "n_reviews": 0, @@ -3480,7 +3480,7 @@ "end": 3086.73 }, { - "input": "Or do I think about it and then pin that to a comment at the end? ", + "input": "Or do I do I think about it and then like pin that to a comment at the end? ", "translatedText": "Hay tôi nghĩ về nó rồi ghim nó vào phần bình luận ở cuối? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/arabic/sentence_translations.json b/2020/ldm-quadratic/arabic/sentence_translations.json index dc81b1300..b3941c7e0 100644 --- a/2020/ldm-quadratic/arabic/sentence_translations.json +++ b/2020/ldm-quadratic/arabic/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "بعد ذلك لدينا m مرات زائد d، m مرات زائد d، ثم سالب d مرات d، إذن ناقص d تربيع. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x تربيع ناقص 4x زائد 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "ولم لا؟ 3x تربيع ناقص 4x زائد 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "حسنًا، هذا بالضبط سالب 1، وهو ما يعني أن الجذرين، وحتى السلك الموجود على الصفحة التي كنت أكتب بها هنا، الجذران هما 3 زائد أو ناقص i. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "ثم لدينا الأعداد الحقيقية فقط، 1، 2، 3، 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "كما تعلم، من أجل مصلحتي، هل يمكننا أن نستمر في هذا؟ أود أن أرى ما إذا كان بإمكاننا رفع هذا الشريط العلوي إلى 1729 مهما كان. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/bengali/sentence_translations.json b/2020/ldm-quadratic/bengali/sentence_translations.json index a27a6dba8..70033018b 100644 --- a/2020/ldm-quadratic/bengali/sentence_translations.json +++ b/2020/ldm-quadratic/bengali/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "এর পরে আমাদের আছে m বার প্লাস d, m বার প্লাস ডি, এবং তারপর নেতিবাচক d বার d, তাই বিয়োগ d বর্গ।", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x বর্গ বিয়োগ 4x প্লাস 5।", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x বর্গ বিয়োগ 4x প্লাস 5।", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "তাই সুন্দরভাবে যে ঠিক নেতিবাচক 1, যার মানে হল যে আমাদের দুটি শিকড়, এবং আমি এখানে যে পৃষ্ঠাটি লিখছি তার তারের নিচে, আমাদের দুটি মূল হল 3 প্লাস বা বিয়োগ i।", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "এবং তারপর আমরা শুধু বাস্তব সংখ্যা পেয়েছি, 1, 2, 3, 4।", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "আমি দেখতে চাই যে আমরা সেই শীর্ষ বারটি 1729 পর্যন্ত পেতে পারি কিনা তা যাই হোক না কেন।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/chinese/sentence_translations.json b/2020/ldm-quadratic/chinese/sentence_translations.json index 9bdfebdec..e0172daed 100644 --- a/2020/ldm-quadratic/chinese/sentence_translations.json +++ b/2020/ldm-quadratic/chinese/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "接下来我们有 m 乘以 d，m 乘以 d，然后是负 d 乘以 d ，即减去 d 的平方。", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x 的平方减 4x 加 5。", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x 的平方减 4x 加 5。", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "很好，这正好是负 1，这 意味着我们的两个根，一直到我在这里写的页面 上的连线，我们的两个根是 3 加或减 i。", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "然后我们就得到了实数，1,2,3,4 。", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "我很想看看我们能否将顶栏升至 1 729，无论它是什么。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/english/captions.srt b/2020/ldm-quadratic/english/captions.srt index 792ab2f00..21539a2d6 100644 --- a/2020/ldm-quadratic/english/captions.srt +++ b/2020/ldm-quadratic/english/captions.srt @@ -591,11 +591,11 @@ Does three go into it? No. 149 -00:07:52,260 --> 00:07:51,740 +00:07:52,260 --> 00:07:52,460 Five? 150 -00:07:51,780 --> 00:07:52,780 +00:07:52,660 --> 00:07:52,780 No. 151 @@ -1011,2442 +1011,2466 @@ And so here, we're going to talk about the simpler version of the quadratic form which is really just refactoring the original thing. 254 -00:13:49,100 --> 00:13:54,020 -But I hope you see, it's much easier to actually solve a random quadratic that's +00:13:49,100 --> 00:13:52,504 +but I hope you see it's much easier to actually solve a random 255 -00:13:54,020 --> 00:13:58,880 -thrown at you, if you're going by the method that I want to show you right here. +00:13:52,504 --> 00:13:55,908 +quadratic that's thrown at you if you're going by um if you're 256 +00:13:55,908 --> 00:13:58,880 +going by the method that I want to show you right here. + +257 00:13:59,300 --> 00:14:00,860 So what's the task? -257 +258 00:14:02,500 --> 00:14:07,542 Anytime you have some function that looks like ax squared plus bx plus c, -258 +259 00:14:07,542 --> 00:14:09,860 okay, for any numbers a, b, and c. -259 +260 00:14:09,960 --> 00:14:14,860 So maybe that's something like 3x squared minus 4x plus 5. -260 +261 00:14:15,240 --> 00:14:19,700 And you want to know when that equals zero, what are the two roots? -261 +262 00:14:20,720 --> 00:14:24,040 Now, this is actually equivalent to rescaling everything. -262 +263 00:14:24,060 --> 00:14:26,771 And I think it's a good common pattern to rescaling, -263 +264 00:14:26,771 --> 00:14:28,920 where I'm going to divide everything by a. -264 +265 00:14:29,360 --> 00:14:30,840 So I just want that first coefficient to be equal to a. -265 +266 00:14:30,860 --> 00:14:32,900 Because that's way easier to work with. -266 +267 00:14:33,220 --> 00:14:37,302 So I'm going to call that x squared plus b prime x plus c prime, -267 +268 00:14:37,302 --> 00:14:41,260 where b prime and c prime are just the rescaled versions there. -268 +269 00:14:41,500 --> 00:14:43,140 And this is a different function. -269 +270 00:14:43,160 --> 00:14:46,420 It is a different quadratic function, but the roots are going to be the same. -270 +271 00:14:46,860 --> 00:14:49,320 And one way that you might see this is if you just graph them. -271 +272 00:14:49,320 --> 00:14:52,700 So let's go over and pull up our best friend of Desmos, right? -272 +273 00:14:53,180 --> 00:14:54,760 So let's say I have some kind of parabola. -273 +274 00:14:54,760 --> 00:14:56,730 In this case, I've written it so that one of the -274 +275 00:14:56,730 --> 00:14:58,620 roots is going to be two and the other is four. -275 +276 00:14:58,620 --> 00:15:03,540 And then g of x here is just the scaled version of the expanded version of that. -276 +277 00:15:03,560 --> 00:15:04,780 It's exactly the same graph. -277 +278 00:15:05,460 --> 00:15:07,800 But let's say I wanted to scale that up and down, right? -278 +279 00:15:07,800 --> 00:15:12,900 So I might take this and then I will multiply it by some kind of constant. -279 +280 00:15:13,500 --> 00:15:14,640 What do you guys want to call that constant? -280 +281 00:15:15,120 --> 00:15:15,720 See, this is the thing. -281 +282 00:15:15,740 --> 00:15:17,396 If I had the question framework working right now, -282 +283 00:15:17,396 --> 00:15:19,280 I could just ask you, what do you want the constant to be? -283 +284 00:15:19,420 --> 00:15:21,740 And we would see the statistics come up with what people had. -284 +285 00:15:22,140 --> 00:15:23,780 But instead, maybe I'll throw in a slider. -285 +286 00:15:25,640 --> 00:15:27,920 And as I change that, the function changes. -286 +287 00:15:27,920 --> 00:15:31,060 It is a different function, but the roots are still two and four. -287 +288 00:15:32,140 --> 00:15:35,600 So solving for this rescaled version of the quadratic is the same. -288 +289 00:15:36,200 --> 00:15:39,940 And that should kind of make sense because s times zero is always going to be zero. -289 +290 00:15:39,980 --> 00:15:41,760 So it doesn't matter how much we scale it. -290 +291 00:15:42,720 --> 00:15:45,374 So going back to what we're working with here, -291 +292 00:15:45,374 --> 00:15:49,383 the reason that I think it's much nicer to make sure that that leading -292 +293 00:15:49,383 --> 00:15:54,466 coefficient is a one is because if we're thinking of our quadratic in terms of its roots, -293 +294 00:15:54,466 --> 00:15:58,645 and I say, okay, you know, it's going to intersect the x-axis at r and s, -294 +295 00:15:58,645 --> 00:16:03,220 or maybe it does, maybe it doesn't, but let's just write one that does like this. -295 +296 00:16:03,580 --> 00:16:07,887 Another way that I could express that particular -296 +297 00:16:07,887 --> 00:16:11,580 quadratic is as x minus r times x minus s. -297 +298 00:16:12,800 --> 00:16:15,540 Because if I plug in r, it's clear that we're going to get zero. -298 +299 00:16:16,020 --> 00:16:19,240 And if I plug in s, again, this term will also cancel out to zero. -299 +300 00:16:19,240 --> 00:16:23,095 So if I expand this, whatever r and s are, they're the unknowns, -300 +301 00:16:23,095 --> 00:16:26,180 I should get the same thing as what we have up here. -301 +302 00:16:26,460 --> 00:16:27,820 And this is generally useful. -302 +303 00:16:27,860 --> 00:16:31,322 This is not just for the quadratic formula, but a very good relationship -303 +304 00:16:31,322 --> 00:16:35,211 to have with polynomials is to know how the roots correspond to the coefficients, -304 +305 00:16:35,211 --> 00:16:38,437 especially in a circumstance where that leading coefficient is one, -305 +306 00:16:38,437 --> 00:16:41,520 and the whole polynomial has been kind of normalized in that way. -306 +307 00:16:42,220 --> 00:16:48,540 And in this case, if you just expand it out, what we'll get is x squared minus -307 +308 00:16:48,540 --> 00:16:54,940 r plus s times x, because we have this minus r times x and then x times minus s. -308 +309 00:16:55,500 --> 00:16:59,403 And then the constant term will be negative r times negative s, -309 +310 00:16:59,403 --> 00:17:01,660 so that becomes a positive r times s. -310 +311 00:17:02,640 --> 00:17:04,339 Okay, so what does this tell us? -311 +312 00:17:04,839 --> 00:17:08,548 This tells us the first two of the three key facts that are needed to be -312 +313 00:17:08,548 --> 00:17:12,460 able to solve any quadratic without really needing to memorize all that much. -313 +314 00:17:12,960 --> 00:17:17,107 So the first key fact is that this b value, I'll call it b prime, -314 +315 00:17:17,107 --> 00:17:21,380 just to remind ourselves that it's after you've scaled things down, -315 +316 00:17:21,380 --> 00:17:24,460 is the same as the negative sum of the two roots. -316 +317 00:17:27,099 --> 00:17:33,202 Okay, and then similarly, c prime is going to be the product of those two roots, -317 +318 00:17:33,202 --> 00:17:38,400 which is kind of cool, because what we have right here is, you know, -318 +319 00:17:38,400 --> 00:17:42,620 it's a system that feels like it should have a solution. -319 +320 00:17:42,640 --> 00:17:44,829 We have two equations and two unknowns, but it's -320 +321 00:17:44,829 --> 00:17:46,840 not obvious how we would go about solving it. -321 +322 00:17:46,840 --> 00:17:51,145 And every now and then, I think classes will have a unit in factoring quadratics, -322 +323 00:17:51,145 --> 00:17:53,980 where they basically tell you to just guess and check. -323 +324 00:17:54,140 --> 00:17:59,496 So in some very fortunate circumstances, if you have something like, -324 +325 00:17:59,496 --> 00:18:03,300 you know, x squared, let's see, minus 7x plus 12. -325 +326 00:18:04,400 --> 00:18:07,685 They basically say, see if you can guess and find two -326 +327 00:18:07,685 --> 00:18:11,640 different numbers that add up to be 7 and that multiply to be 12. -327 +328 00:18:12,200 --> 00:18:14,700 And if you just kind of sit back there and think, you're like, -328 +329 00:18:14,700 --> 00:18:17,320 okay, can I find any two numbers that add to 7 and multiply to 12? -329 +330 00:18:17,920 --> 00:18:22,180 Well factoring 12, we get 3 and 4, and 3 and 4 do add to 7. -330 +331 00:18:22,240 --> 00:18:24,420 So yeah, in that case, you just totally luck out. -331 +332 00:18:25,320 --> 00:18:30,220 And you could write this as x minus 3 and x minus 4, because those are the two roots. -332 +333 00:18:30,660 --> 00:18:32,998 And then you typically move on from that chapter and it's like, -333 +334 00:18:32,998 --> 00:18:35,300 well, in most cases, you won't be able to just guess and check. -334 +335 00:18:35,800 --> 00:18:38,969 And in any case, what you're going to look for is a general formula anyway, -335 +336 00:18:38,969 --> 00:18:40,220 so hope you had fun with that. -336 +337 00:18:40,220 --> 00:18:45,726 But it turns out there's actually a systematic way to take this puzzle of finding -337 +338 00:18:45,726 --> 00:18:51,300 two numbers that have a known sum and a known product and figure out what they are. -338 +339 00:18:51,880 --> 00:18:56,213 And the key comes down to thinking in terms of not the two numbers themselves, -339 +340 00:18:56,213 --> 00:19:01,041 but the mean of those two numbers, and then the distance between that mean and each one -340 +341 00:19:01,041 --> 00:19:01,480 of them. -341 +342 00:19:02,940 --> 00:19:04,220 And you can see where this is going. -342 +343 00:19:04,300 --> 00:19:07,809 This is why we talked about difference of squares, -343 +344 00:19:07,809 --> 00:19:12,695 because the third key fact to come away with, or to come into it with, -344 +345 00:19:12,695 --> 00:19:18,820 I should maybe say, is that we could re-express that product as m minus d times m plus d. -345 +346 00:19:20,440 --> 00:19:22,140 And you see where I'm going, right? -346 +347 00:19:22,480 --> 00:19:28,480 That means that the product that we know looks like m squared minus d squared. -347 +348 00:19:30,840 --> 00:19:34,660 So just to give an example here, it's often much more helpful to have numbers. -348 +349 00:19:35,220 --> 00:19:40,950 Let's say that you were given a quadratic like x squared, I don't know, -349 +350 00:19:40,950 --> 00:19:46,680 let's do six, even numbers will make this easier for us, and then seven. -350 +351 00:19:47,780 --> 00:19:50,080 And you were tasked with knowing when does this equal zero? -351 +352 00:19:50,680 --> 00:19:53,341 So I haven't told you how to solve it yet, but these three key facts -352 +353 00:19:53,341 --> 00:19:56,080 are going to be enough to just basically walk yourself into the answer. -353 -00:19:56,980 --> 00:20:03,340 -Well, what is m, right? - 354 +00:19:56,980 --> 00:20:00,194 +Well, what is what is m, right? Because we're + +355 +00:20:00,194 --> 00:20:03,340 +going to ultimately express our roots r and s + +356 00:20:03,340 --> 00:20:07,020 As m plus or minus d for some kind of midpoint. -355 +357 00:20:08,300 --> 00:20:12,280 Well, that midpoint is the sum of the two numbers over two. -356 +358 00:20:12,660 --> 00:20:14,040 That's how we define averages. -357 +359 00:20:15,060 --> 00:20:18,513 And because we know the sum of the two numbers is negative b prime, -358 +360 00:20:18,513 --> 00:20:22,880 that's the same as negative b prime over two, which you can basically read off of the -359 +361 00:20:22,880 --> 00:20:26,587 equation as just negative one half times whatever's sitting right there, -360 +362 00:20:26,587 --> 00:20:28,720 which in this case will be negative three. -361 +363 00:20:29,260 --> 00:20:30,960 Awesome, we know what m is. -362 -00:20:31,759 --> 00:20:33,780 +364 +00:20:31,760 --> 00:20:33,780 But look at the equation we have up here. -363 +365 00:20:34,140 --> 00:20:37,806 We have an expression for c, the last coefficient, in terms of m, -364 +366 00:20:37,806 --> 00:20:41,640 which we now know, and d, which is the only thing we don't know left. -365 +367 00:20:42,340 --> 00:20:46,451 So we could rearrange this, and so that was saying c prime is that, -366 +368 00:20:46,451 --> 00:20:50,985 we could say that d squared, the square of this kind of standard deviation -367 +369 00:20:50,985 --> 00:20:55,580 between our roots, is m squared minus c prime, the product of the two roots. -368 +370 00:20:55,880 --> 00:20:58,820 But we know both of those values, we could just write that down over here. -369 +371 00:20:58,820 --> 00:21:01,780 Maybe I'll change colors again, be a little flamboyant. -370 +372 00:21:02,720 --> 00:21:07,706 d squared is equal to m squared, which in our example turns out to be nine, -371 +373 00:21:07,706 --> 00:21:11,380 minus c prime, which is that last coefficient, or seven. -372 +374 00:21:13,720 --> 00:21:14,120 Seven. -373 +375 00:21:15,600 --> 00:21:18,160 Handwriting is terrible, but I think you guys can work with me. -374 +376 00:21:18,260 --> 00:21:19,740 You see why I usually animate stuff. -375 +377 00:21:20,800 --> 00:21:21,880 So that means that's two. -376 +378 00:21:22,720 --> 00:21:26,903 So look, when we said r and s is some midpoint plus or minus a distance, -377 +379 00:21:26,903 --> 00:21:31,144 that midpoint is negative three, plus or minus, well if d squared is two, -378 +380 00:21:31,144 --> 00:21:33,380 that means d is the square root of two. -379 +381 00:21:34,180 --> 00:21:34,680 There you go. -380 +382 00:21:35,120 --> 00:21:37,420 You could do that for any quadratic that I give you. -381 +383 00:21:37,540 --> 00:21:39,540 You could just walk through that particular process. -382 +384 00:21:40,680 --> 00:21:43,218 And let me just show you what it looks like in general, -383 +385 00:21:43,218 --> 00:21:46,120 so that you can maybe remember it as a formula if you wanted to. -384 +386 00:21:46,980 --> 00:21:52,400 In general, for any quadratic, that midpoint is just the rescaled -385 +387 00:21:52,400 --> 00:21:58,560 version of b if the leading coefficient wasn't already one, divided by two. -386 +388 00:21:59,640 --> 00:22:02,380 And then that distance, well what did we just do? -387 +389 00:22:02,580 --> 00:22:10,220 We took the square root of the midpoint squared minus the product of the two. -388 +390 00:22:10,620 --> 00:22:14,199 And when I'm sort of thinking into my head, I've been saying like m squared minus p, -389 +391 00:22:14,199 --> 00:22:15,800 just because p has a readable meaning. -390 +392 00:22:15,920 --> 00:22:17,920 I think of it as the midpoint squared minus the product. -391 +393 00:22:18,260 --> 00:22:20,521 But of course p in this context is just whatever -392 +394 00:22:20,521 --> 00:22:22,460 the last coefficient of our quadratic was. -393 +395 00:22:22,460 --> 00:22:26,780 So over in this example, the product of the two coefficients was seven. -394 +396 00:22:27,780 --> 00:22:31,331 So for me, what I think the simpler quadratic formula is, -395 +397 00:22:31,331 --> 00:22:36,659 if you're going to memorize anything, is to come away and say it's m plus or minus the -396 +398 00:22:36,659 --> 00:22:38,680 square root of m squared minus p. -397 +399 00:22:39,700 --> 00:22:42,708 All you have to do is first find m, which is, you know, -398 +400 00:22:42,708 --> 00:22:47,005 just a factor times one of the coefficients you're looking at, and then find p, -399 +401 00:22:47,005 --> 00:22:51,249 which is also, if not already, a factor, a coefficient that you're looking at, -400 +402 00:22:51,249 --> 00:22:52,700 a rescaling of one of them. -401 +403 00:22:53,060 --> 00:22:56,160 So this to me is way simpler than the traditional quadratic formula. -402 +404 00:22:56,240 --> 00:22:59,360 And if you were to try to sing a song to yourself for it, the song is almost too short. -403 +405 00:22:59,360 --> 00:22:59,900 It's no fun. -404 +406 00:22:59,960 --> 00:23:03,900 You're just sitting there like m plus or minus square root of m squared minus p. -405 +407 00:23:04,280 --> 00:23:04,840 That's it. -406 +408 00:23:05,240 --> 00:23:05,700 No song. -407 +409 00:23:05,860 --> 00:23:06,640 No song to be had. -408 +410 00:23:07,880 --> 00:23:10,720 So let's do a couple practice problems, because I do think practice will make it easier. -409 +411 00:23:11,000 --> 00:23:15,649 And for future streams, again, it'll ultimately be the case that I'm giving you -410 +412 00:23:15,649 --> 00:23:20,240 these questions, and then you'll be able to go to 3b1b.co.live and answer them. -411 +413 00:23:20,420 --> 00:23:23,520 But because there's too many of you, we can't have any fun today. -412 +414 00:23:23,780 --> 00:23:26,531 This is the equivalent of trying to run a class where you have, you know, -413 +415 00:23:26,531 --> 00:23:29,320 I don't know, 20 seats sitting out and you're going to do a normal lecture. -414 +416 00:23:29,340 --> 00:23:32,598 And then there's just people banging at the doors and trying to cram themselves in, -415 +417 00:23:32,598 --> 00:23:35,740 and the fire marshal comes in and they're like, ah, you got to cut out the class. -416 +418 00:23:35,740 --> 00:23:37,660 You can't have this the way that you hoped for. -417 +419 00:23:38,320 --> 00:23:41,700 But it's cool that there's so many of you here enthusiastic to learn about math. -418 +420 00:23:43,340 --> 00:23:44,680 So let's just do some examples. -419 +421 00:23:44,680 --> 00:23:45,080 Okay. -420 +422 00:23:46,220 --> 00:23:48,920 That'll kind of highlight what this process looks like. -421 +423 00:23:49,840 --> 00:23:52,680 Let's say we had x squared plus 10x plus three. -422 +424 00:23:53,960 --> 00:23:56,360 So in this case, it's nicely already rescaled for us. -423 +425 00:23:56,360 --> 00:23:57,220 That's always lovely. -424 -00:23:57,919 --> 00:24:00,591 +426 +00:23:57,920 --> 00:24:00,591 So I often, I just like to draw the picture for myself, -425 +427 00:24:00,591 --> 00:24:03,740 whether or not the two roots turn out to be real or even positive. -426 +428 00:24:04,080 --> 00:24:06,040 It's just nice to remind myself what we're looking for. -427 +429 00:24:06,340 --> 00:24:08,100 We're looking for where the two roots are. -428 +430 00:24:08,320 --> 00:24:10,439 And I know that a different way to think about -429 +431 00:24:10,439 --> 00:24:12,740 products is to think about a difference of squares. -430 +432 00:24:12,740 --> 00:24:15,844 So I just kind of think in my head, okay, I'll be thinking of -431 +433 00:24:15,844 --> 00:24:19,000 that in terms of their mean and the kind of standard deviation. -432 +434 00:24:19,740 --> 00:24:22,040 So I just write down for myself, what does that mean? -433 +435 00:24:23,840 --> 00:24:25,860 Well, it's negative b prime over two. -434 +436 00:24:26,080 --> 00:24:30,477 And if I forget that fact, if I forget that that's what the sum of the two roots is, -435 +437 00:24:30,477 --> 00:24:34,306 I could always just go through this little rigmarole again and say, okay, -436 +438 00:24:34,306 --> 00:24:37,824 if I systematically wanted it to be a quadratic with roots r and s, -437 +439 00:24:37,824 --> 00:24:39,480 this is what it would look like. -438 +440 00:24:39,480 --> 00:24:40,800 This is how it would expand. -439 +441 00:24:40,820 --> 00:24:42,280 So you can re-derive it on the fly. -440 +442 00:24:42,280 --> 00:24:43,740 There's not too much memorization needed. -441 +443 00:24:44,300 --> 00:24:46,980 In this context, that works out to be negative five. -442 +444 00:24:47,940 --> 00:24:50,044 And by the way, if I do ever make any mistakes, -443 +445 00:24:50,044 --> 00:24:52,851 which I'm quite positive I will, go ahead and throw them in the -444 +446 00:24:52,851 --> 00:24:56,360 chat and those will be forwarded to me and I'll be able to correct myself there. -445 +447 00:24:57,120 --> 00:24:57,940 So we know the midpoint. -446 +448 00:24:58,600 --> 00:25:02,260 And then we just ask ourselves, what's the square of the distance? -447 +449 00:25:03,100 --> 00:25:08,604 And based on difference of squares, that'll be that midpoint squared minus the product, -448 +450 00:25:08,604 --> 00:25:13,107 which in this context is negative five squared or 25 minus the product, -449 +451 00:25:13,107 --> 00:25:15,860 which is that last coefficient, minus three. -450 +452 00:25:16,680 --> 00:25:18,640 So what are the roots r and s? -451 +453 00:25:20,020 --> 00:25:27,240 Well, it's negative five plus or minus the square root of 25 minus three or 22. -452 +454 00:25:28,060 --> 00:25:28,540 There you go. -453 +455 00:25:28,840 --> 00:25:30,060 Another quadratic solved. -454 +456 00:25:30,540 --> 00:25:31,180 Ain't no thing. -455 +457 00:25:32,420 --> 00:25:33,660 Let's try another. -456 +458 00:25:35,580 --> 00:25:38,380 Just because I do think it's kind of nice to get a little bit of practice here. -457 +459 00:25:38,500 --> 00:25:41,880 And as I'm going, if you have a piece of paper and pencil, please follow along. -458 +460 00:25:42,020 --> 00:25:45,740 If you can just kind of race ahead and do the same process before I do it, that's awesome. -459 +461 00:25:45,860 --> 00:25:47,740 That is the best kind of learning experience. -460 +462 00:25:47,880 --> 00:25:50,461 If you're watching this in the future, by the way, -461 +463 00:25:50,461 --> 00:25:53,600 as with any video, I highly encourage you to pause and ponder. -462 +464 00:25:53,900 --> 00:25:55,780 I really think that's the best way to learn math. -463 +465 00:25:55,780 --> 00:25:58,342 If you're looking at some kind of lecture in those crucial -464 +466 00:25:58,342 --> 00:26:00,383 moments where there's a question asked, pause, -465 +467 00:26:00,383 --> 00:26:03,640 see if you can do it yourself and then see what the answer turns out to be. -466 +468 00:26:03,660 --> 00:26:05,860 I guarantee you'll learn more effectively that way. -467 +469 00:26:06,800 --> 00:26:07,360 I don't know. -468 +470 00:26:07,400 --> 00:26:07,900 What should we do? -469 +471 00:26:08,380 --> 00:26:10,960 Let's do maybe three x squared. -470 +472 00:26:11,480 --> 00:26:13,060 I think I wrote one down earlier, didn't I? -471 +473 00:26:13,420 --> 00:26:14,940 Just as kind of a offhanded thing. -472 +474 00:26:15,720 --> 00:26:16,360 What did I write? -473 +475 00:26:17,340 --> 00:26:17,980 That wasn't there. -474 +476 00:26:19,080 --> 00:26:19,900 Oh yeah, it was at the top. -475 +477 00:26:20,340 --> 00:26:21,000 Let's do that one. -476 +478 00:26:21,100 --> 00:26:22,680 Three x squared minus four x plus five. -477 +479 00:26:23,280 --> 00:26:23,720 Why not? -478 +480 00:26:25,100 --> 00:26:28,860 Three x squared minus four x plus five. -479 +481 00:26:29,920 --> 00:26:30,320 All right. -480 +482 00:26:30,320 --> 00:26:32,360 So in this case, step one, we've got to rescale things. -481 +483 00:26:33,660 --> 00:26:35,560 The first one gives the coefficients a nice readable meaning. -482 -00:26:36,639 --> 00:26:41,720 +484 +00:26:36,640 --> 00:26:41,720 Minus four thirds x plus five thirds. -483 +485 00:26:42,460 --> 00:26:42,580 Great. -484 +486 00:26:43,720 --> 00:26:44,860 And now same process. -485 +487 00:26:45,100 --> 00:26:46,903 I'm sort of thinking in my head of this particular -486 +488 00:26:46,903 --> 00:26:48,920 image where I want to know the midpoint and the distance. -487 +489 00:26:49,400 --> 00:26:55,160 So I say that midpoint is negative of this second coefficient divided by two. -488 +490 00:26:55,320 --> 00:26:57,160 So that will become positive four thirds. -489 +491 00:26:57,620 --> 00:27:00,420 But then that four divided by two gives us a two. -490 +492 00:27:00,420 --> 00:27:02,080 So that midpoint will be two thirds. -491 +493 00:27:02,960 --> 00:27:07,687 And then the distance squared is m squared minus the product, -492 +494 00:27:07,687 --> 00:27:10,280 which in this case is five thirds. -493 +495 00:27:12,000 --> 00:27:12,120 Okay. -494 +496 00:27:12,520 --> 00:27:18,460 And m squared in this case is, let's see, two thirds squared minus five thirds. -495 +497 00:27:19,560 --> 00:27:19,680 All right. -496 +498 00:27:19,680 --> 00:27:20,940 So this one, you know, it gets a little messier. -497 +499 00:27:21,040 --> 00:27:23,580 We've got to work out our fractions, but that's not too bad. -498 +500 00:27:23,980 --> 00:27:25,940 Two thirds squared is going to be four ninths. -499 +501 00:27:27,260 --> 00:27:28,980 Oh, I'm off screen a little. -500 +502 00:27:32,500 --> 00:27:35,320 And then what is five thirds in terms of ninths? -501 +503 00:27:35,320 --> 00:27:37,940 That's going to be fifteen ninths if I'm not wrong. -502 +504 00:27:39,360 --> 00:27:41,600 So we end up getting a negative number, which is always fun. -503 +505 00:27:42,100 --> 00:27:44,820 So here we have negative eleven ninths. -504 +506 00:27:45,820 --> 00:27:45,900 Okay. -505 +507 00:27:46,240 --> 00:27:50,960 So what that means is that our final answer, the roots of this polynomial are in s. -506 +508 00:27:51,320 --> 00:27:56,299 The values that will make the polynomial zero are going to be -507 +509 00:27:56,299 --> 00:28:01,680 two thirds plus or minus the square root of negative eleven over d. -508 +510 00:28:02,500 --> 00:28:03,520 Negative eleven over d. -509 +511 00:28:04,160 --> 00:28:04,920 What am I saying? -510 +512 00:28:05,420 --> 00:28:06,540 Negative eleven over nine. -511 +513 00:28:07,560 --> 00:28:07,820 Okay. -512 +514 00:28:07,820 --> 00:28:08,400 So that's fun. -513 +515 00:28:08,580 --> 00:28:10,180 In this case, the square root is a negative. -514 +516 00:28:10,180 --> 00:28:12,240 So that means we have complex roots. -515 +517 00:28:12,540 --> 00:28:16,183 And there's actually a very fun way to think about the way that complex roots -516 +518 00:28:16,183 --> 00:28:19,640 play into this difference of squares perspective on the quadratic formula. -517 +519 00:28:20,900 --> 00:28:22,933 Let me actually try it with a simpler example, -518 +520 00:28:22,933 --> 00:28:25,184 but it ends up relating to the Pythagorean theorem, -519 +521 00:28:25,184 --> 00:28:28,169 which I think is cool because the whole purpose of this is to try to -520 +522 00:28:28,169 --> 00:28:30,420 connect various patterns in math that come up a lot. -521 +523 00:28:30,600 --> 00:28:33,940 Things that might be useful outside of this particular class that you're doing. -522 +524 00:28:35,260 --> 00:28:37,157 So I actually have written down for myself an -523 +525 00:28:37,157 --> 00:28:39,220 example that will work out with nice numbers here. -524 +526 00:28:40,260 --> 00:28:41,000 I mean nice-ish. -525 +527 00:28:42,160 --> 00:28:45,340 X squared minus six x plus ten. -526 +528 00:28:46,300 --> 00:28:46,640 Okay. -527 +529 00:28:47,420 --> 00:28:48,760 So we go through the same rigmarole. -528 +530 00:28:48,960 --> 00:28:52,700 We say m, the midpoint, is going to be negative b prime over two. -529 +531 00:28:52,800 --> 00:28:54,979 So in that case, it works out to just be three because -530 +532 00:28:54,979 --> 00:28:57,040 we take the negative of this term and divide by two. -531 +533 00:28:57,680 --> 00:29:01,920 d squared is going to be m squared. -532 +534 00:29:02,540 --> 00:29:06,620 So three squared is nine minus the product, which in this case is ten. -533 +535 00:29:07,660 --> 00:29:12,294 So nicely that's exactly negative one, which means that our two roots, -534 +536 00:29:12,294 --> 00:29:16,929 and I'm down to the y or on the page that I've been writing with here, -535 +537 00:29:16,929 --> 00:29:19,540 our two roots are three plus or minus i. -536 +538 00:29:20,640 --> 00:29:20,880 Okay. -537 +539 00:29:20,880 --> 00:29:24,375 So what that means for us is that the actual parabola here, -538 +540 00:29:24,375 --> 00:29:27,580 it doesn't look like something that crosses the x-axis. -539 +541 00:29:28,140 --> 00:29:30,820 Instead it would be something that maybe sits above it. -540 +542 00:29:30,820 --> 00:29:32,680 But it does have imaginary roots. -541 +543 00:29:32,860 --> 00:29:36,826 So if we were to look at the input space, not just in terms of the real number line, -542 +544 00:29:36,826 --> 00:29:37,900 but let me draw it out. -543 +545 00:29:38,500 --> 00:29:39,340 Let me go back to black. -544 +546 00:29:39,800 --> 00:29:42,920 Black will be our complex plane color for this moment. -545 +547 00:29:46,540 --> 00:29:50,430 We'll call this our imaginary axis, where numbers like i and negative i, -546 +548 00:29:50,430 --> 00:29:55,120 the square root of one, or maybe I should say square roots of one, it's got two of them. -547 +549 00:29:55,120 --> 00:29:56,400 You interchange them, it's fine. -548 +550 00:29:58,100 --> 00:29:59,740 And then we've got just the real numbers. -549 +551 00:29:59,980 --> 00:30:02,840 One, two, three, four. -550 +552 00:30:03,640 --> 00:30:09,520 So in this case, our two roots live at three plus i and three minus i. -551 +553 00:30:09,980 --> 00:30:13,720 So this, I should be very clear, this is not an x-axis and a y-axis. -552 +554 00:30:13,760 --> 00:30:17,100 This entire plane now is where the input lives, where x lives. -553 +555 00:30:17,320 --> 00:30:20,820 If you were going to graph it, you'd get some kind of graph that's outside of the paper. -554 +556 00:30:21,220 --> 00:30:24,376 And any of you who've watched the channel Welsh Labs, which if you haven't, -555 +557 00:30:24,376 --> 00:30:27,740 you absolutely should, might be bringing to mind right now an absolutely awesome -556 +558 00:30:27,740 --> 00:30:30,980 like artificial reality effect he does, where he sort of pulls out that graph. -557 +559 00:30:31,120 --> 00:30:33,080 Highly worth watching, super great moment. -558 +560 00:30:34,000 --> 00:30:40,020 But what's interesting about this is if we look at the magnitude of the roots, okay? -559 +561 00:30:41,840 --> 00:30:44,640 Because I could ask you, what is the magnitude of that root? -560 +562 00:30:45,020 --> 00:30:46,700 And it's a right triangle that we're looking at. -561 +563 00:30:46,720 --> 00:30:48,560 We've switched from algebra to geometry now. -562 +564 00:30:48,920 --> 00:30:52,520 And based on the Pythagorean theorem, it'll be the square root -563 +565 00:30:52,520 --> 00:30:56,407 of one of the lengths squared, which is in this case three squared, -564 +566 00:30:56,407 --> 00:30:59,380 plus the other length squared, which is one squared. -565 +567 00:31:00,160 --> 00:31:04,520 So square root of three squared plus one squared, and that ends up being root 10. -566 +568 00:31:05,780 --> 00:31:10,009 It is not a coincidence at all that the magnitude of our roots, -567 +569 00:31:10,009 --> 00:31:14,436 I guess sort of no pun intended, the magnitude of the roots of our -568 +570 00:31:14,436 --> 00:31:19,260 quadratic equation here are the square root of that constant coefficient. -569 +571 00:31:19,820 --> 00:31:21,953 Because remember that constant coefficient is -570 +572 00:31:21,953 --> 00:31:24,180 telling us what is the product of the two roots. -571 +573 00:31:24,720 --> 00:31:27,061 And if you know something about complex numbers, -572 +574 00:31:27,061 --> 00:31:30,884 you might know that if I have two complex numbers and I multiply them together, -573 +575 00:31:30,884 --> 00:31:34,420 the magnitude of the product is the same as the product of the magnitudes. -574 +576 00:31:35,180 --> 00:31:39,764 So in this case, given that I'm going to have two separate roots who are symmetric, -575 +577 00:31:39,764 --> 00:31:43,585 you know, it's going to be three plus or minus some imaginary number, -576 +578 00:31:43,585 --> 00:31:46,260 each of them is going to have the same magnitude. -577 +579 00:31:46,780 --> 00:31:49,000 The product of their magnitudes needs to be 10. -578 +580 00:31:49,000 --> 00:31:50,120 We know that offhand. -579 +581 00:31:50,540 --> 00:31:53,880 So you kind of know ahead of time that it should be magnitude of square root of 10. -580 +582 00:31:54,400 --> 00:32:01,518 And the reason that this is happening is basically because when you do -581 +583 00:32:01,518 --> 00:32:07,935 difference of squares, something like m plus d times m minus d, -582 +584 00:32:07,935 --> 00:32:15,555 but that distance is an imaginary value, i, what you get is m squared minus -583 +585 00:32:15,555 --> 00:32:17,360 i times d squared. -584 +586 00:32:17,360 --> 00:32:22,280 But because i squared is by definition negative one, you get a sum of squares. -585 +587 00:32:24,040 --> 00:32:26,515 So even sums of squares, which come up in geometry, -586 +588 00:32:26,515 --> 00:32:29,372 Pythagorean theorem stuff, all of that, can be expressed as -587 +589 00:32:29,372 --> 00:32:32,800 a kind of difference of squares, which itself gives a kind of factoring. -588 +590 00:32:33,220 --> 00:32:36,460 And this, this shows up in a lot of very, very beautiful math later down the road. -589 +591 00:32:37,100 --> 00:32:41,460 One of my favorite videos that I've made actually is, oh, what did I title it? -590 +592 00:32:41,680 --> 00:32:44,410 I think pi hiding in prime regularities, where -591 +593 00:32:44,410 --> 00:32:47,140 you're counting lattice points inside a circle. -592 +594 00:32:47,140 --> 00:32:50,870 And the key insight associated with doing that is to realize that asking when -593 +595 00:32:50,870 --> 00:32:54,887 you can express something as the sum of two squares is a sort of factoring problem, -594 +596 00:32:54,887 --> 00:32:59,095 but it's factoring not where you're dealing with prime numbers on the real number line, -595 +597 00:32:59,095 --> 00:33:02,300 but primes in complex numbers, these things called Gaussian primes. -596 +598 00:33:02,900 --> 00:33:06,470 So even simple, I shouldn't say simple stuff, but even stuff that comes up -597 +599 00:33:06,470 --> 00:33:10,708 in high school, like the quadratic formula, I think if you're learning it the right way, -598 +600 00:33:10,708 --> 00:33:13,660 it ends up connecting to all sorts of other delightful things. -599 +601 00:33:13,820 --> 00:33:16,640 And remembering these patterns comes up, like I said. -600 +602 00:33:17,200 --> 00:33:19,220 So just to re-emphasize, how did we get here? -601 +603 00:33:19,280 --> 00:33:22,570 What three key facts do you need to be aware of with quadratics to -602 +604 00:33:22,570 --> 00:33:25,960 be able to kind of rediscover a kind of quadratic formula on the fly? -603 +605 00:33:26,780 --> 00:33:31,460 The first one is how to read the coefficient sitting in front of x. -604 +606 00:33:32,080 --> 00:33:36,218 And if we have a quadratic that looks like x squared plus b prime x plus c prime, -605 +607 00:33:36,218 --> 00:33:39,700 you can read that first coefficient as the negative sum of the roots. -606 +608 00:33:40,400 --> 00:33:42,920 You don't know that already, you can rediscover it on the fly, -607 +609 00:33:42,920 --> 00:33:44,680 but that is actually worth coming away with. -608 +610 00:33:45,060 --> 00:33:48,120 Similar with how you can read the other coefficient, it's the product of the roots. -609 +611 00:33:48,540 --> 00:33:51,527 Then the only other thing you need to know is that we could -610 +612 00:33:51,527 --> 00:33:54,614 express that product of roots as a difference of squares with -611 +613 00:33:54,614 --> 00:33:58,100 respect to the mean and the kind of standard deviation of those roots. -612 +614 00:33:58,580 --> 00:34:01,640 And you can just solve any quadratic that comes to you on the fly. -613 +615 00:34:01,920 --> 00:34:05,422 The only thing that looks remotely like memorization is if you want -614 +616 00:34:05,422 --> 00:34:09,080 to jumpstart to the end and just say m plus or minus m squared minus p. -615 +617 00:34:10,040 --> 00:34:12,773 Now to finish things off, I think it would be very satisfying -616 +618 00:34:12,773 --> 00:34:15,418 to remind ourselves that this is actually equivalent to the -617 +619 00:34:15,418 --> 00:34:18,239 traditional quadratic formula, something that looks much bigger. -618 +620 00:34:18,980 --> 00:34:21,500 So let's go ahead and actually do that exercise. -619 +621 00:34:22,520 --> 00:34:26,300 And again, if you can, pause and just work it out for yourself right now. -620 +622 00:34:27,199 --> 00:34:28,960 So let's remind ourselves of how that goes. -621 +623 00:34:30,699 --> 00:34:33,080 So we've got, what are we solving? -622 +624 00:34:33,420 --> 00:34:34,659 Any kind of quadratic. -623 +625 00:34:35,260 --> 00:34:38,460 Ax squared plus bx plus c. -624 +626 00:34:38,460 --> 00:34:40,891 We want to say no matter what a, b, and c are, -625 +627 00:34:40,891 --> 00:34:43,219 give me a systematic way to find these roots. -626 +628 00:34:44,040 --> 00:34:46,080 Again, maybe you're doing something beautiful like ray tracing. -627 +629 00:34:46,320 --> 00:34:47,300 Let me pull up that image again. -628 +630 00:34:47,340 --> 00:34:48,300 That was a really nice image. -629 +631 00:34:49,719 --> 00:34:53,139 I just can't believe that this is something that a computer generated. -630 +632 00:34:53,760 --> 00:34:56,380 I guess I should believe it because like Pixar movies are amazing. -631 +633 00:34:56,980 --> 00:35:01,063 But evidently, if you know the kind of math that can lead you to create an image -632 +634 00:35:01,063 --> 00:35:05,400 like this one, that's the kind of thing that can get your job as an engineer at Pixar. -633 +635 00:35:06,620 --> 00:35:07,980 And of course, what is that math? -634 +636 00:35:07,980 --> 00:35:09,820 That math is exactly what we're doing right now. -635 +637 00:35:10,500 --> 00:35:11,740 So let's work it out again. -636 +638 00:35:12,460 --> 00:35:14,140 Ax squared plus bx plus c equals zero. -637 +639 00:35:14,480 --> 00:35:17,740 This time, we're just going to write everything in terms of a, b, and c. -638 +640 00:35:17,840 --> 00:35:19,120 No new variables coming up. -639 +641 00:35:19,360 --> 00:35:24,692 So when we do the first step of rescaling, we say x squared -640 +642 00:35:24,692 --> 00:35:29,580 is equal to b divided by a times x plus c divided by a. -641 +643 00:35:30,500 --> 00:35:31,940 We don't give it any new names. -642 +644 00:35:32,940 --> 00:35:34,580 Now remember how our trick works. -643 +645 00:35:35,080 --> 00:35:39,140 We sort of picture in our head, hey, imagine this quadratic has some roots. -644 +646 00:35:39,640 --> 00:35:41,000 Let's call them r and s. -645 +647 00:35:41,840 --> 00:35:46,320 And we're trying to find the midpoint and the standard deviation. -646 +648 00:35:46,900 --> 00:35:51,920 And we can read off that that midpoint is the negative of the second term divided by two. -647 +649 00:35:52,480 --> 00:35:55,560 So in this case, that's going to be negative b over 2a. -648 +650 00:35:56,460 --> 00:35:56,620 Great. -649 +651 00:35:57,180 --> 00:36:05,291 And then that standard deviation is going to be m squared minus the product of the roots, -650 +652 00:36:05,291 --> 00:36:11,060 which in this case looks like negative b over 2a squared minus, -651 +653 00:36:11,060 --> 00:36:17,640 and the product here is what that last term is, c divided by a, c over a. -652 +654 00:36:18,620 --> 00:36:21,212 And remember, if you don't remember this idea that, oh, -653 +655 00:36:21,212 --> 00:36:24,500 that's what the distance is, just re-derive it for yourself on the fly. -654 +656 00:36:24,500 --> 00:36:30,945 Just go and say, okay, I remember that the product p is just the product of my two roots, -655 +657 00:36:30,945 --> 00:36:34,240 which can be expressed as m minus d, m plus d. -656 +658 00:36:35,080 --> 00:36:38,000 Oh, okay, that's something I can write as m squared minus d squared. -657 +659 00:36:38,460 --> 00:36:44,340 Oh, okay, that's what gives me an expression for d squared in terms of m squared and p. -658 +660 00:36:45,300 --> 00:36:48,980 Don't feel like you have to just come in and know it off the top of your head. -659 +661 00:36:49,020 --> 00:36:51,600 But after you do it a couple times, it sticks in your memory. -660 +662 00:36:52,080 --> 00:36:57,280 So that's m, that's d, and the quadratic formula is just telling us that the -661 +663 00:36:57,280 --> 00:37:03,088 roots are m plus and minus, that's standard deviation, which in this case looks like, -662 +664 00:37:03,088 --> 00:37:08,289 maybe I'll write it out on two lines because this will be a lot, negative b, -663 +665 00:37:08,289 --> 00:37:13,625 actually no, I'll write it on one line for this one, negative b plus or minus, -664 +666 00:37:13,625 --> 00:37:19,028 I'm jumping to the original quadratic formula, a little bit too hard ingrained, -665 +667 00:37:19,028 --> 00:37:22,675 we'll go back to two lines, negative b divided by 2a, -666 +668 00:37:22,675 --> 00:37:28,011 divided by 2a plus or minus the square root, oh, I wrote this incorrectly, oh, -667 +669 00:37:28,011 --> 00:37:30,240 someone should have corrected me. -668 +670 00:37:30,500 --> 00:37:33,903 This is what d squared is equal to, d squared is this whole thing, -669 +671 00:37:33,903 --> 00:37:35,580 so d itself is the root of those. -670 +672 00:37:35,940 --> 00:37:38,530 I'm sure lots of people were shouting that in the live chat, -671 +673 00:37:38,530 --> 00:37:41,800 I don't have it pulled up now, but to those of you who did, much appreciated. -672 +674 00:37:42,880 --> 00:37:44,060 So what do we have here? -673 +675 00:37:44,400 --> 00:37:54,400 Well, it's going to be the square of negative b over 2a minus c over a, minus c over a. -674 +676 00:37:55,400 --> 00:38:01,641 Okay, now we just got to expand this thing, which is frankly not super fun, -675 +677 00:38:01,641 --> 00:38:06,980 but it'll connect it to the original quadratic formula to for us. -676 +678 00:38:07,660 --> 00:38:13,444 So I can pull out this 2a squared, and I'm just going to write that as 1 over -677 +679 00:38:13,444 --> 00:38:18,414 4a times negative b squared, negative b, yeah, negative b squared, -678 +680 00:38:18,414 --> 00:38:23,902 and then I want to also pull out 1 over 4a, I want to be able to say that -679 +681 00:38:23,902 --> 00:38:28,130 the last term also looks like 1 over 4a times something, -680 +682 00:38:28,130 --> 00:38:34,063 and that something to make it equal to c over a would have to cancel out the 4, -681 +683 00:38:34,063 --> 00:38:38,439 it would have to cancel out an extra a, and then c, sorry, -682 +684 00:38:38,439 --> 00:38:41,480 because this is really 1 over 4a squared. -683 +685 00:38:42,860 --> 00:38:43,860 Let's see, have I done this right? -684 +686 00:38:43,960 --> 00:38:47,311 Yeah, because I pulled out the 2a, so that should be 4 times a squared, -685 +687 00:38:47,311 --> 00:38:49,080 so over here I have 4 times a squared. -686 +688 00:38:49,780 --> 00:38:54,140 I want that to cancel to become c over a, which it looks like it does, so that's awesome. -687 +689 00:38:55,260 --> 00:38:58,785 So you can go over here, negative b over 2a, plus or minus, -688 +690 00:38:58,785 --> 00:39:01,840 and if this feels tedious, that's kind of the point. -689 +691 00:39:01,880 --> 00:39:04,741 The whole quadratic formula is more complicated than it needs to be, -690 +692 00:39:04,741 --> 00:39:08,059 because we were just solving any quadratic that was thrown at us without having -691 +693 00:39:08,059 --> 00:39:11,502 to do this, but this is what happens if you don't introduce any new variable names -692 +694 00:39:11,502 --> 00:39:12,000 on your way. -693 +695 00:39:12,180 --> 00:39:14,520 It's like code that hasn't been refactored properly. -694 +696 00:39:15,240 --> 00:39:16,440 Okay, so what can we do here? -695 +697 00:39:16,460 --> 00:39:22,022 We can factor out the 1 over 4a squared, and because that's in a radical, -696 +698 00:39:22,022 --> 00:39:28,262 its square root will also be 1 over 2a, and then what sits inside is what remains, -697 +699 00:39:28,262 --> 00:39:31,720 the well familiar negative b squared minus 4a. -698 +700 00:39:31,720 --> 00:39:35,776 Okay, and this this is starting to look like the traditional form, -699 +701 00:39:35,776 --> 00:39:40,680 because if we pull onto the numerator, negative b plus or minus square root of... -700 +702 00:39:40,680 --> 00:39:41,920 why am I writing negative b squared? -701 +703 00:39:43,600 --> 00:39:46,920 Yeah, so this is a negative b squared, same as positive b squared. -702 +704 00:39:47,540 --> 00:39:51,960 What should sit on the inside here is b squared minus 4ac. -703 +705 00:39:53,920 --> 00:39:57,740 You can see how scatterbrained I am when I'm just doing some arithmetic at times. -704 +706 00:39:59,020 --> 00:39:59,680 It's okay. -705 +707 00:40:01,080 --> 00:40:03,580 We all forget a variable or two here and there, -706 +708 00:40:03,580 --> 00:40:07,331 but maybe that's why I actually care so much about formulas having nice -707 +709 00:40:07,331 --> 00:40:12,020 readable meanings, because I think this is a very error-prone process for someone like me. -708 +710 00:40:12,020 --> 00:40:15,568 If I'm just trying to work through with a bunch of symbols that I don't -709 +711 00:40:15,568 --> 00:40:18,526 necessarily know how to read them, I get to the end result, -710 +712 00:40:18,526 --> 00:40:22,420 and it's hard for me to say like, oh yeah, of course that's what the answer is. -711 +713 00:40:22,560 --> 00:40:27,470 Whereas if I look at something like the simpler variant of the quadratic formula, -712 +714 00:40:27,470 --> 00:40:30,404 if I basically refactor it and I'm saying, okay, -713 +715 00:40:30,404 --> 00:40:32,920 does the midpoint equal negative b over 2? -714 +716 00:40:32,960 --> 00:40:35,125 I can kind of fact check, yeah, that that makes sense, -715 +717 00:40:35,125 --> 00:40:36,740 especially if you know a little calculus. -716 +718 00:40:36,740 --> 00:40:39,884 Those of you will be able to look at that quadratic formula -717 +719 00:40:39,884 --> 00:40:42,820 and understand how to find the maximal or minimal point. -718 +720 00:40:42,860 --> 00:40:46,093 So that's a thing that gets reinforced with a better pattern later on in -719 +721 00:40:46,093 --> 00:40:49,460 your mathematical life, always a good sign that you're learning things well. -720 +722 00:40:50,080 --> 00:40:53,040 I can ask myself if it makes sense that the distance looks like this. -721 +723 00:40:53,580 --> 00:40:55,400 Again, that has a readable meaning. -722 +724 00:40:55,680 --> 00:40:59,314 So when I'm looking at the simpler version of the quadratic formula, -723 +725 00:40:59,314 --> 00:41:02,632 we have m plus or minus a kind of square root of the variance, -724 +726 00:41:02,632 --> 00:41:04,160 a kind of standard deviation. -725 +727 00:41:05,220 --> 00:41:06,340 So it's a refactoring. -726 +728 00:41:07,060 --> 00:41:10,232 Now I titled this thing, this is the simpler version of the quadratic formula, -727 +729 00:41:10,232 --> 00:41:12,080 and I think someone could rightfully complain. -728 +730 00:41:12,400 --> 00:41:14,300 Like, is this actually simpler? -729 +731 00:41:14,680 --> 00:41:18,420 Because, you know, you've got a lot of steps, you've introduced new variables into it. -730 +732 00:41:18,540 --> 00:41:21,602 Now, in addition to thinking of the three coefficients a, b, and c, -731 +733 00:41:21,602 --> 00:41:25,114 you're telling me I also have to think about like a new term m and a distance -732 +734 00:41:25,114 --> 00:41:25,700 between them. -733 +735 00:41:26,500 --> 00:41:29,991 But for me, math is very much about trying to draw connections to other patterns, -734 +736 00:41:29,991 --> 00:41:32,972 and I think things are much better remembered if you have that web of -735 +737 00:41:32,972 --> 00:41:35,400 connections in your head rather than just isolated cases. -736 +738 00:41:36,060 --> 00:41:39,338 And the whole lesson here, if we think about what's going on, -737 +739 00:41:39,338 --> 00:41:43,040 it's about representing the same information in different ways, right? -738 +740 00:41:44,480 --> 00:41:49,070 Because what the quadratic formula is doing for us, it's saying, -739 +741 00:41:49,070 --> 00:41:54,580 can I go from my coefficients a, b, and c, and can I get to the roots r and s? -740 +742 00:41:55,260 --> 00:41:59,177 And we know that there's a very easy way to go the other way around, -741 +743 00:41:59,177 --> 00:42:02,640 because we can expand something like x minus r and x minus s. -742 +744 00:42:03,660 --> 00:42:06,032 And really, because that a can get scaled out, -743 +745 00:42:06,032 --> 00:42:10,122 it doesn't add any information to the system, there's really as much information -744 +746 00:42:10,122 --> 00:42:11,940 as two different constants in there. -745 +747 00:42:12,300 --> 00:42:16,980 So one of these directions is easy, and one of these directions is hard. -746 +748 00:42:18,260 --> 00:42:21,894 And this idea of information that can be expressed in separate ways and translation -747 +749 00:42:21,894 --> 00:42:25,400 one direction being easy, one direction hard, that comes up all the time in math. -748 +750 00:42:25,400 --> 00:42:26,920 That's a very useful thing to think about. -749 +751 00:42:27,080 --> 00:42:30,491 And another useful thing to think about is how sometimes going in that -750 +752 00:42:30,491 --> 00:42:34,000 harder direction will be better if you go through some intermediate step. -751 +753 00:42:34,180 --> 00:42:38,349 In this case, rather than thinking of your pair of numbers on the -752 +754 00:42:38,349 --> 00:42:42,519 number line just as they are, doing your m plus or minus d trick, -753 +755 00:42:42,519 --> 00:42:46,500 because we know that that changes how you think about products. -754 +756 00:42:46,760 --> 00:42:50,570 If you go through the intermediate step of expressing that same information -755 +757 00:42:50,570 --> 00:42:53,980 with a mean and a standard deviation, that can get you to the roots. -756 -00:42:54,500 --> 00:42:58,884 -And that's all this is, it's just talking about different information flow paths that - -757 -00:42:58,884 --> 00:43:02,045 -can help go between various ways to just express two numbers, - 758 -00:43:02,045 --> 00:43:05,409 -whether those two numbers are the coefficients of your quadratic, +00:42:54,500 --> 00:42:58,149 +And that's all this is, it's just talking about different information flow paths 759 -00:43:05,409 --> 00:43:08,672 -so if the lesson you come away with is one of thinking, oh wow, +00:42:58,149 --> 00:43:01,168 +that can help go between various ways to just express two numbers, 760 -00:43:08,672 --> 00:43:12,343 -sometimes there's a lot of different ways that I can represent my data, +00:43:01,168 --> 00:43:04,142 +whether those two numbers are the coefficients of your quadratic, 761 -00:43:12,343 --> 00:43:16,829 -and some of them lend themselves to certain kind of problem solving better than others, +00:43:04,142 --> 00:43:07,162 +or the roots of the quadratic, or the mean and standard deviation. 762 -00:43:16,829 --> 00:43:20,500 -well then that is the proper lesson to have with the quadratic equation. +00:43:07,162 --> 00:43:10,045 +So if the lesson you come away with is one of thinking, oh wow, 763 +00:43:10,045 --> 00:43:13,290 +sometimes there's a lot of different ways that I can represent my data, + +764 +00:43:13,290 --> 00:43:17,255 +and some of them lend themselves to certain kind of problem solving better than others, + +765 +00:43:17,255 --> 00:43:20,500 +well then that is the proper lesson to have with the quadratic equation. + +766 00:43:21,740 --> 00:43:25,060 Okay, so I think with that I'm going to call it an end to lesson number one. -764 +767 00:43:25,440 --> 00:43:29,040 Really want to thank everyone who showed up for this, definitely a ton of fun. -765 +768 00:43:29,300 --> 00:43:32,417 Next time we're going to have the live quizzing dynamic where we're -766 +769 00:43:32,417 --> 00:43:35,580 going to get stats up on the screen, so it's going to be pretty cool. -767 +770 00:43:35,640 --> 00:43:38,939 I think this is a very cool thing that two former co-workers of mine -768 +771 00:43:38,939 --> 00:43:42,238 put together from Khan Academy, but obviously this is the first time -769 +772 00:43:42,238 --> 00:43:45,107 that we're trying it, it's a little rough around the edges, -770 +773 00:43:45,107 --> 00:43:48,980 so in the end what it's going to look like is these bars that you're seeing live. -771 +774 00:43:49,220 --> 00:43:51,160 Oh interesting, these are bigger numbers than they were before. -772 +775 00:43:52,000 --> 00:43:54,660 Okay, clearly they're updating, they're updating. -773 +776 00:43:55,060 --> 00:43:55,900 Oh this is exciting. -774 +777 00:43:56,020 --> 00:43:57,240 Do I have actual access here? -775 +778 00:43:57,640 --> 00:44:00,000 Oh it would be so fun to do this properly before we end. -776 +779 00:44:00,280 --> 00:44:03,880 -Oh my god, oh this is so great, I think we can do it. +Oh my god, oh this is so great, I think we're, I think we can do it. -777 +780 00:44:04,280 --> 00:44:06,240 Okay, are you ready for this guys? -778 +781 00:44:07,420 --> 00:44:08,420 Okay, I'm grading the answers. -779 +782 00:44:09,960 --> 00:44:12,580 Oh, oh wonderful, oh we can see what people answered. -780 +783 00:44:12,960 --> 00:44:15,340 This is great, this is how I wanted to end the stream. -781 +784 00:44:15,680 --> 00:44:18,980 While I was talking just in the background some magic was being done, this is wonderful. -782 +785 00:44:19,400 --> 00:44:21,660 So it looks like the most common answer, what's our question here? -783 +786 00:44:22,400 --> 00:44:25,120 What best describes your relationship with the quadratic formula? -784 +787 00:44:25,560 --> 00:44:27,340 And the most common answer is C. -785 +788 00:44:27,580 --> 00:44:29,840 I'm a big fan, you might say I'm rooting for it. -786 +789 00:44:30,280 --> 00:44:38,700 So it looks like 1724 of you, five short of Ramanujan's constant, are addicted to puns. -787 +790 00:44:39,180 --> 00:44:41,720 Let's do a couple more, this is going to be pretty fun. -788 +791 00:44:46,760 --> 00:44:50,623 These were just like the joke warm-up questions before we got into the real lesson, -789 +792 00:44:50,623 --> 00:44:53,842 but I actually think this is the perfect way to end the whole lesson, -790 +793 00:44:53,842 --> 00:44:57,660 is to just kind of wind down with some of what were meant to be introductory jokes. -791 +794 00:44:57,800 --> 00:44:59,580 Oh it's working, I love this so much. -792 -00:44:59,940 --> 00:45:08,380 -Oh and there's if the quadratic formula had a patronus, what would it be? +795 +00:44:59,940 --> 00:45:03,326 +Oh and there's so many of you answering, this makes me so happy. -793 +796 +00:45:03,326 --> 00:45:07,494 +All right so what's our question here? If the quadratic formula had a patronus, + +797 +00:45:07,494 --> 00:45:08,380 +what would it be? + +798 00:45:09,500 --> 00:45:17,080 Okay well it looks like around 800 of you think it will be something. -794 +799 00:45:18,500 --> 00:45:22,725 By the way, I'm being told right now that if there's too many of you who access it, -795 +800 00:45:22,725 --> 00:45:26,597 we're for sure going to break the system, and I'm purposefully ignoring that -796 +801 00:45:26,597 --> 00:45:30,320 because I'm having fun with this and if it breaks that kind of tickles me. -797 +802 00:45:30,800 --> 00:45:34,895 So I'm being told not to say this, but please go to 3b1b.co live and enter -798 +803 00:45:34,895 --> 00:45:38,882 questions to this, and then you know whenever things break that would be -799 +804 00:45:38,882 --> 00:45:42,760 a perfect time to end the stream because I just think that's hilarious. -800 +805 00:45:43,780 --> 00:45:48,780 Okay so oh again 1791, oh I guess we blew past Ramanujan's constant. -801 +806 00:45:49,520 --> 00:45:52,991 Maybe I can try to see if I can grade this at a point right where we're -802 +807 00:45:52,991 --> 00:45:56,800 going to lock in answers where the majority is 1729, I think that would be fun. -803 +808 00:45:57,820 --> 00:46:02,260 Okay so it looks like a majority of people went with C. -804 +809 00:46:02,680 --> 00:46:06,270 If the quadratic formula had a patronus, it would be an old man -805 +810 00:46:06,270 --> 00:46:09,860 hunched over a chessboard, which is the correct answer actually. -806 +811 00:46:10,120 --> 00:46:12,312 You know this question was structured as a poll where there's -807 +812 00:46:12,312 --> 00:46:14,540 no correct or incorrect rating, but I don't think that's right. -808 +813 00:46:14,600 --> 00:46:17,751 I think it should have been structured where C is the -809 +814 00:46:17,751 --> 00:46:21,020 objectively correct JK Rowling would agree style answer. -810 +815 00:46:21,880 --> 00:46:26,160 All right let's do the the warm-up question number three here. -811 +816 00:46:27,120 --> 00:46:28,340 Oh this is a fun one. -812 +817 00:46:28,700 --> 00:46:30,000 Okay what does it say? -813 +818 00:46:30,680 --> 00:46:33,780 What integer will most people enter into this box? -814 +819 00:46:35,080 --> 00:46:37,140 Okay oh oh lots of answers coming in. -815 +820 00:46:37,380 --> 00:46:41,580 Again I really want my friends to like struggle, I have bars all over my face. -816 +821 00:46:42,440 --> 00:46:46,108 You know this actually seems apropos given that the whole title of this is locked -817 +822 00:46:46,108 --> 00:46:49,687 down math that it starts to look like I'm slowly being imprisoned by everyone's -818 +823 00:46:49,687 --> 00:46:53,580 answers and just getting locked down further and further into the quarantine situation. -819 -00:46:54,700 --> 00:47:00,007 -So this one actually now there is an objectively correct answer because +824 +00:46:54,700 --> 00:46:59,484 +So this one actually now there is a, where do I talk? Help! Bars are attacking me, -820 -00:47:00,007 --> 00:47:05,462 -there is going to be some number that most people enter and it looks like +825 +00:46:59,484 --> 00:47:03,635 +okay there's an objectively correct answer because there is going to be -821 -00:47:05,462 --> 00:47:11,360 -919 of you think that it'll be one particular thing, but we've got a widespread. +826 +00:47:03,635 --> 00:47:07,728 +some number that most people enter. And it looks like 919 of you think -822 +827 +00:47:07,728 --> 00:47:11,360 +that it'll be one particular thing, but we've got a widespread, + +828 00:47:11,540 --> 00:47:13,140 This is actually pretty fun. -823 +829 00:47:14,060 --> 00:47:18,500 So again if you want to partake in this head on over to 3b1.co.live. -824 +830 00:47:19,300 --> 00:47:21,680 I actually think the best way that you could do this. -825 +831 00:47:21,680 --> 00:47:24,040 Oh what is the seven? -826 +832 00:47:24,600 --> 00:47:28,134 Wow I would not have guessed that most people entered seven -827 +833 00:47:28,134 --> 00:47:31,905 and in a weird way like the plurality of you are definitionally -828 +834 00:47:31,905 --> 00:47:35,440 correct that seven was the most commonly entered expression. -829 +835 00:47:36,640 --> 00:47:38,020 69 being the second most common. -830 +836 00:47:38,220 --> 00:47:40,220 I can make a guess for why that might be the case. -831 +837 00:47:40,320 --> 00:47:44,031 Did you know that 69 is the first number where if you square -832 +838 00:47:44,031 --> 00:47:47,926 it and then you cube it those two values between them encounter -833 +839 00:47:47,926 --> 00:47:51,760 the numbers or the digits zero through nine once and only once. -834 +840 00:47:52,360 --> 00:47:55,130 Yeah it's the first number with that property which I -835 +841 00:47:55,130 --> 00:47:57,900 assume is why that was the second most popular answer. -836 +842 00:47:58,820 --> 00:48:02,728 But the very end which is actually apropos at this point we can pull -837 +843 00:48:02,728 --> 00:48:06,467 up another question which is going to be what I was going to open -838 +844 00:48:06,467 --> 00:48:10,320 the whole lesson with so you can kind of see how the plan went here. -839 +845 00:48:10,680 --> 00:48:13,977 How many times do you expect to use the quadratic -840 +846 00:48:13,977 --> 00:48:16,880 formula in your real life outside of school? -841 +847 00:48:18,140 --> 00:48:22,466 And in this case luckily I'm getting a little bit less you know locked down by the bars -842 +848 00:48:22,466 --> 00:48:26,644 trapping me in here because it seems like there's a little bit more consensus around -843 +849 00:48:26,644 --> 00:48:31,020 how many times people think they will need to use the quadratic formula in the real life. -844 -00:48:32,100 --> 00:48:36,708 +850 +00:48:32,100 --> 00:48:36,586 What I could do is a plot twist on this and say you know interpret this question -845 -00:48:36,708 --> 00:48:41,202 -in light of the lesson rather than when will you literally use negative b plus +851 +00:48:36,586 --> 00:48:41,128 +in light of the lesson rather than when will you literally use negative b plus or -846 -00:48:41,202 --> 00:48:45,639 -or minus square root of I always forget it square root of b squared minus 4ac +852 +00:48:41,128 --> 00:48:45,559 +minus square root of I always forget it square root of b squared minus 4ac over -847 -00:48:45,639 --> 00:48:50,418 -over 2a that whole thing to when are you going to use the principles of recognizing +853 +00:48:45,559 --> 00:48:50,100 +2a 2a that whole thing to when are you going to use the principles of recognizing -848 -00:48:50,418 --> 00:48:54,742 -that a product of numbers expressed as a difference of squares can help you +854 +00:48:50,100 --> 00:48:54,642 +that a product of numbers expressed as a difference of squares can help you solve -849 -00:48:54,742 --> 00:48:59,179 -solve problems or when are you going to use the principles of expressing your +855 +00:48:54,642 --> 00:48:59,018 +problems or we're going when are you going to use the principles of expressing -850 -00:48:59,179 --> 00:49:03,560 -data in terms of a mean and a standard deviation can help you solve problems. +856 +00:48:59,018 --> 00:49:03,560 +your data in terms of a mean and a standard deviation can help you solve problems. -851 +857 00:49:04,340 --> 00:49:07,750 That I think would give you wildly different answers but at the moment -852 +858 00:49:07,750 --> 00:49:10,969 we've got a lot of you on the system and it's not breaking and I'm -853 +859 00:49:10,969 --> 00:49:14,140 so happy right now I just can't tell you how much this tickles me. -854 +860 00:49:15,180 --> 00:49:19,810 So it looks like we've got a wide forming consensus you know for for my sake can we -855 +861 00:49:19,810 --> 00:49:24,385 can we just like keep going on this I would love to see if we can get that top bar -856 +862 00:49:24,385 --> 00:49:29,015 up to 1729 whatever it might be at the moment we can all guess what it might be but -857 +863 00:49:29,015 --> 00:49:33,645 let's see if you can go to 3b1b.co slash live wherever you're watching this I think -858 +864 00:49:33,645 --> 00:49:38,110 honestly the best dynamic that I could imagine is if you just pull up your phone -859 +865 00:49:38,110 --> 00:49:42,630 and you're watching this on a screen with the one hand and then you're using your -860 +866 00:49:42,630 --> 00:49:47,205 phone to to answer questions a lot of you are already watching it on your phone so -861 +867 00:49:47,205 --> 00:49:51,780 that wouldn't necessarily work but that is the dynamic I would I would most expect. -862 +868 00:49:53,080 --> 00:51:03,526 Okay so I'm just going to wait until we get that -863 +869 00:51:03,526 --> 00:52:09,660 top bar up to be Ramanujan's constant of 1729. diff --git a/2020/ldm-quadratic/english/sentence_timings.json b/2020/ldm-quadratic/english/sentence_timings.json index a506e1547..316728f72 100644 --- a/2020/ldm-quadratic/english/sentence_timings.json +++ b/2020/ldm-quadratic/english/sentence_timings.json @@ -407,11 +407,11 @@ [ "Five?", 472.26, - 471.74 + 472.46 ], [ "No.", - 471.78, + 472.66, 472.78 ], [ @@ -755,7 +755,7 @@ 829.08 ], [ - "But I hope you see, it's much easier to actually solve a random quadratic that's thrown at you, if you're going by the method that I want to show you right here.", + "but I hope you see it's much easier to actually solve a random quadratic that's thrown at you if you're going by um if you're going by the method that I want to show you right here.", 829.1, 838.88 ], @@ -1070,7 +1070,7 @@ 1196.08 ], [ - "Well, what is m, right?", + "Well, what is what is m, right? Because we're going to ultimately express our roots r and s", 1196.98, 1203.34 ], @@ -2410,7 +2410,7 @@ 2573.98 ], [ - "And that's all this is, it's just talking about different information flow paths that can help go between various ways to just express two numbers, whether those two numbers are the coefficients of your quadratic, so if the lesson you come away with is one of thinking, oh wow, sometimes there's a lot of different ways that I can represent my data, and some of them lend themselves to certain kind of problem solving better than others, well then that is the proper lesson to have with the quadratic equation.", + "And that's all this is, it's just talking about different information flow paths that can help go between various ways to just express two numbers, whether those two numbers are the coefficients of your quadratic, or the roots of the quadratic, or the mean and standard deviation. So if the lesson you come away with is one of thinking, oh wow, sometimes there's a lot of different ways that I can represent my data, and some of them lend themselves to certain kind of problem solving better than others, well then that is the proper lesson to have with the quadratic equation.", 2574.5, 2600.5 ], @@ -2460,7 +2460,7 @@ 2640.0 ], [ - "Oh my god, oh this is so great, I think we can do it.", + "Oh my god, oh this is so great, I think we're, I think we can do it.", 2640.28, 2643.88 ], @@ -2530,7 +2530,7 @@ 2699.58 ], [ - "Oh and there's if the quadratic formula had a patronus, what would it be?", + "Oh and there's so many of you answering, this makes me so happy. All right so what's our question here? If the quadratic formula had a patronus, what would it be?", 2699.94, 2708.38 ], @@ -2615,7 +2615,7 @@ 2813.58 ], [ - "So this one actually now there is an objectively correct answer because there is going to be some number that most people enter and it looks like 919 of you think that it'll be one particular thing, but we've got a widespread.", + "So this one actually now there is a, where do I talk? Help! Bars are attacking me, okay there's an objectively correct answer because there is going to be some number that most people enter. And it looks like 919 of you think that it'll be one particular thing, but we've got a widespread,", 2814.7, 2831.36 ], @@ -2680,7 +2680,7 @@ 2911.02 ], [ - "What I could do is a plot twist on this and say you know interpret this question in light of the lesson rather than when will you literally use negative b plus or minus square root of I always forget it square root of b squared minus 4ac over 2a that whole thing to when are you going to use the principles of recognizing that a product of numbers expressed as a difference of squares can help you solve problems or when are you going to use the principles of expressing your data in terms of a mean and a standard deviation can help you solve problems.", + "What I could do is a plot twist on this and say you know interpret this question in light of the lesson rather than when will you literally use negative b plus or minus square root of I always forget it square root of b squared minus 4ac over 2a 2a that whole thing to when are you going to use the principles of recognizing that a product of numbers expressed as a difference of squares can help you solve problems or we're going when are you going to use the principles of expressing your data in terms of a mean and a standard deviation can help you solve problems.", 2912.1, 2943.56 ], diff --git a/2020/ldm-quadratic/english/transcript.txt b/2020/ldm-quadratic/english/transcript.txt index c46d7fa47..49ee980f6 100644 --- a/2020/ldm-quadratic/english/transcript.txt +++ b/2020/ldm-quadratic/english/transcript.txt @@ -149,7 +149,7 @@ So I think if you were doing some arithmetic, and you had this in the back of yo And that's a pattern that's going to come up later in life. For example, let's say later in life, you find yourself wanting to understand quadratic functions. And so here, we're going to talk about the simpler version of the quadratic formula, which is really just refactoring the original thing. -But I hope you see, it's much easier to actually solve a random quadratic that's thrown at you, if you're going by the method that I want to show you right here. +but I hope you see it's much easier to actually solve a random quadratic that's thrown at you if you're going by um if you're going by the method that I want to show you right here. So what's the task? Anytime you have some function that looks like ax squared plus bx plus c, okay, for any numbers a, b, and c. So maybe that's something like 3x squared minus 4x plus 5. @@ -212,7 +212,7 @@ So just to give an example here, it's often much more helpful to have numbers. Let's say that you were given a quadratic like x squared, I don't know, let's do six, even numbers will make this easier for us, and then seven. And you were tasked with knowing when does this equal zero? So I haven't told you how to solve it yet, but these three key facts are going to be enough to just basically walk yourself into the answer. -Well, what is m, right? +Well, what is what is m, right? Because we're going to ultimately express our roots r and s As m plus or minus d for some kind of midpoint. Well, that midpoint is the sum of the two numbers over two. That's how we define averages. @@ -480,7 +480,7 @@ That's a very useful thing to think about. And another useful thing to think about is how sometimes going in that harder direction will be better if you go through some intermediate step. In this case, rather than thinking of your pair of numbers on the number line just as they are, doing your m plus or minus d trick, because we know that that changes how you think about products. If you go through the intermediate step of expressing that same information with a mean and a standard deviation, that can get you to the roots. -And that's all this is, it's just talking about different information flow paths that can help go between various ways to just express two numbers, whether those two numbers are the coefficients of your quadratic, so if the lesson you come away with is one of thinking, oh wow, sometimes there's a lot of different ways that I can represent my data, and some of them lend themselves to certain kind of problem solving better than others, well then that is the proper lesson to have with the quadratic equation. +And that's all this is, it's just talking about different information flow paths that can help go between various ways to just express two numbers, whether those two numbers are the coefficients of your quadratic, or the roots of the quadratic, or the mean and standard deviation. So if the lesson you come away with is one of thinking, oh wow, sometimes there's a lot of different ways that I can represent my data, and some of them lend themselves to certain kind of problem solving better than others, well then that is the proper lesson to have with the quadratic equation. Okay, so I think with that I'm going to call it an end to lesson number one. Really want to thank everyone who showed up for this, definitely a ton of fun. Next time we're going to have the live quizzing dynamic where we're going to get stats up on the screen, so it's going to be pretty cool. @@ -490,7 +490,7 @@ Okay, clearly they're updating, they're updating. Oh this is exciting. Do I have actual access here? Oh it would be so fun to do this properly before we end. -Oh my god, oh this is so great, I think we can do it. +Oh my god, oh this is so great, I think we're, I think we can do it. Okay, are you ready for this guys? Okay, I'm grading the answers. Oh, oh wonderful, oh we can see what people answered. @@ -504,7 +504,7 @@ So it looks like 1724 of you, five short of Ramanujan's constant, are addicted t Let's do a couple more, this is going to be pretty fun. These were just like the joke warm-up questions before we got into the real lesson, but I actually think this is the perfect way to end the whole lesson, is to just kind of wind down with some of what were meant to be introductory jokes. Oh it's working, I love this so much. -Oh and there's if the quadratic formula had a patronus, what would it be? +Oh and there's so many of you answering, this makes me so happy. All right so what's our question here? If the quadratic formula had a patronus, what would it be? Okay well it looks like around 800 of you think it will be something. By the way, I'm being told right now that if there's too many of you who access it, we're for sure going to break the system, and I'm purposefully ignoring that because I'm having fun with this and if it breaks that kind of tickles me. So I'm being told not to say this, but please go to 3b1b.co live and enter questions to this, and then you know whenever things break that would be a perfect time to end the stream because I just think that's hilarious. @@ -521,7 +521,7 @@ What integer will most people enter into this box? Okay oh oh lots of answers coming in. Again I really want my friends to like struggle, I have bars all over my face. You know this actually seems apropos given that the whole title of this is locked down math that it starts to look like I'm slowly being imprisoned by everyone's answers and just getting locked down further and further into the quarantine situation. -So this one actually now there is an objectively correct answer because there is going to be some number that most people enter and it looks like 919 of you think that it'll be one particular thing, but we've got a widespread. +So this one actually now there is a, where do I talk? Help! Bars are attacking me, okay there's an objectively correct answer because there is going to be some number that most people enter. And it looks like 919 of you think that it'll be one particular thing, but we've got a widespread, This is actually pretty fun. So again if you want to partake in this head on over to 3b1.co.live. I actually think the best way that you could do this. @@ -534,7 +534,7 @@ Yeah it's the first number with that property which I assume is why that was the But the very end which is actually apropos at this point we can pull up another question which is going to be what I was going to open the whole lesson with so you can kind of see how the plan went here. How many times do you expect to use the quadratic formula in your real life outside of school? And in this case luckily I'm getting a little bit less you know locked down by the bars trapping me in here because it seems like there's a little bit more consensus around how many times people think they will need to use the quadratic formula in the real life. -What I could do is a plot twist on this and say you know interpret this question in light of the lesson rather than when will you literally use negative b plus or minus square root of I always forget it square root of b squared minus 4ac over 2a that whole thing to when are you going to use the principles of recognizing that a product of numbers expressed as a difference of squares can help you solve problems or when are you going to use the principles of expressing your data in terms of a mean and a standard deviation can help you solve problems. +What I could do is a plot twist on this and say you know interpret this question in light of the lesson rather than when will you literally use negative b plus or minus square root of I always forget it square root of b squared minus 4ac over 2a 2a that whole thing to when are you going to use the principles of recognizing that a product of numbers expressed as a difference of squares can help you solve problems or we're going when are you going to use the principles of expressing your data in terms of a mean and a standard deviation can help you solve problems. That I think would give you wildly different answers but at the moment we've got a lot of you on the system and it's not breaking and I'm so happy right now I just can't tell you how much this tickles me. So it looks like we've got a wide forming consensus you know for for my sake can we can we just like keep going on this I would love to see if we can get that top bar up to 1729 whatever it might be at the moment we can all guess what it might be but let's see if you can go to 3b1b.co slash live wherever you're watching this I think honestly the best dynamic that I could imagine is if you just pull up your phone and you're watching this on a screen with the one hand and then you're using your phone to to answer questions a lot of you are already watching it on your phone so that wouldn't necessarily work but that is the dynamic I would I would most expect. Okay so I'm just going to wait until we get that top bar up to be Ramanujan's constant of 1729. \ No newline at end of file diff --git a/2020/ldm-quadratic/french/sentence_translations.json b/2020/ldm-quadratic/french/sentence_translations.json index 5d15e539c..09203349b 100644 --- a/2020/ldm-quadratic/french/sentence_translations.json +++ b/2020/ldm-quadratic/french/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "Ensuite, nous avons m fois plus d, m fois plus d, puis moins d fois d, donc moins d au carré. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x au carré moins 4x plus 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x au carré moins 4x plus 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "Donc, bien, c'est exactement moins 1, ce qui signifie que nos deux racines, et jusqu'au fil sur la page avec laquelle j'écris ici, nos deux racines sont 3 plus ou moins i. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "Et puis nous avons juste les vrais nombres, 1, 2, 3, 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "J'aimerais voir si nous pouvons amener cette barre supérieure jusqu'à 1729, quelle qu'elle soit. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/german/sentence_translations.json b/2020/ldm-quadratic/german/sentence_translations.json index c3ce0966d..e73e6d191 100644 --- a/2020/ldm-quadratic/german/sentence_translations.json +++ b/2020/ldm-quadratic/german/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "Als nächstes haben wir m mal plus d, m mal plus d und dann negativ d mal d, also minus d im Quadrat. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x zum Quadrat minus 4x plus 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x zum Quadrat minus 4x plus 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "Das ist also genau minus 1, was bedeutet, dass unsere beiden Wurzeln, und bis zum Draht auf der Seite, mit der ich hier geschrieben habe, unsere beiden Wurzeln 3 plus oder minus i sind. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "Und dann haben wir gerade die reellen Zahlen, 1, 2, 3, 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "Ich würde gerne sehen, ob wir den oberen Balken auf 1729 bringen können, was auch immer er sein mag. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/hebrew/sentence_translations.json b/2020/ldm-quadratic/hebrew/sentence_translations.json index 02ca1e89d..4959b4d36 100644 --- a/2020/ldm-quadratic/hebrew/sentence_translations.json +++ b/2020/ldm-quadratic/hebrew/sentence_translations.json @@ -798,21 +798,21 @@ "end": 679.22 }, { - "input": ".", + "input": "actually, you know what, let me change the variable names there.", "translatedText": ".", "n_reviews": 0, "start": 680.1, "end": 681.92 }, { - "input": ".", + "input": "I don't like x and y because those don't necessarily have the same meaning. What I might do is try", "translatedText": ".", "n_reviews": 0, "start": 681.94, "end": 686.14 }, { - "input": "Actually, you know what, let me change the variable names there.", + "input": "to picture, you know, let's picture the original two numbers that we had, something", "translatedText": "למעשה, אתה יודע מה, תן לי לשנות את שמות המשתנים שם.", "n_reviews": 0, "start": 686.14, @@ -861,7 +861,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared.", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared.", "translatedText": "לאחר מכן יש לנו m כפול פלוס d, m כפול ועוד d, ואז ד כפול d שלילי, אז מינוס d בריבוע.", "n_reviews": 0, "start": 726.92, @@ -1078,7 +1078,7 @@ "end": 963.22 }, { - "input": "Another way that I could express that particular quadratic is as x minus r times x minus s, because if I plug in r it's clear that we're going to get 0, and if I plug in s, again this term will also cancel out to 0.", + "input": "Another way that I could express that particular quadratic is as x minus r times x minus s. Because if I plug in r, it's clear that we're going to get zero. And if I plug in s, again, this term will also cancel out to zero.", "translatedText": "דרך נוספת שבה אוכל לבטא את הריבוע הספציפי הזה היא כ-x מינוס r כפול x מינוס s, כי אם אני מחבר את r זה ברור שנקבל 0, ואם אני מחבר את s, שוב המונח הזה גם יתבטל ל-0.", "n_reviews": 0, "start": 963.58, @@ -1666,7 +1666,7 @@ "end": 1496.36 }, { - "input": "So we know the midpoint, and then we just ask ourselves what's the square of the distance, and based on difference of squares, that'll be that midpoint squared minus the product, which in this context is negative 5 squared or 25 minus the product, which is that last coefficient, minus 3.", + "input": "So we know the midpoint. And then we just ask ourselves, what's the square of the distance? And based on difference of squares, that'll be that midpoint squared minus the product, which in this context is negative five squared or 25 minus the product, which is that last coefficient, minus three.", "translatedText": "אז אנחנו יודעים את נקודת האמצע, ואז אנחנו פשוט שואלים את עצמנו מה הריבוע של המרחק, ובהתבסס על הפרש הריבועים, זה יהיה נקודת האמצע בריבוע פחות המכפלה, שבהקשר זה הוא שלילי 5 בריבוע או 25 פחות המכפלה , שהוא המקדם האחרון, מינוס 3.", "n_reviews": 0, "start": 1497.12, @@ -1820,7 +1820,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5.", + "input": "Three x squared minus four x plus five.", "translatedText": "3x בריבוע מינוס 4x פלוס 5.", "n_reviews": 0, "start": 1581.1, @@ -1834,7 +1834,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5.", + "input": "Three x squared minus four x plus five.", "translatedText": "3x בריבוע מינוס 4x פלוס 5.", "n_reviews": 0, "start": 1585.1, @@ -2037,7 +2037,7 @@ "end": 1725.34 }, { - "input": "Okay, so we go through the same rigmarole, we say m, the midpoint, is going to be negative b prime over 2, so in that case it works out to just be 3, because we take the negative of this term and divide by 2.", + "input": "Okay. So we go through the same rigmarole. We say m, the midpoint, is going to be negative b prime over two. So in that case, it works out to just be three because we take the negative of this term and divide by two.", "translatedText": "אוקיי, אז אנחנו עוברים את אותה ריגול, אנחנו אומרים ש-m, נקודת האמצע, תהיה שלילית b ראשונית על 2, אז במקרה כזה זה מסתדר להיות רק 3, כי אנחנו לוקחים את השלילי של המונח הזה ומחלקים ב- 2.", "n_reviews": 0, "start": 1726.3, @@ -2051,7 +2051,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i.", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i.", "translatedText": "אז יפה שזה בדיוק שלילי 1, מה שאומר ששני השורשים שלנו, ועד לחוט בדף שאיתו כתבתי כאן, שני השורשים שלנו הם 3 פלוס מינוס i.", "n_reviews": 0, "start": 1747.66, @@ -2093,14 +2093,14 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4.", + "input": "And then we've got just the real numbers. One, two, three, four.", "translatedText": "ואז יש לנו רק את המספרים האמיתיים, 1, 2, 3, 4.", "n_reviews": 0, "start": 1798.1, "end": 1802.84 }, { - "input": "So in this case our two roots live at 3 plus i, and 3 minus i.", + "input": "So in this case, our two roots live at three plus i and three minus i.", "translatedText": "אז במקרה הזה שני השורשים שלנו חיים ב-3 פלוס i, ו-3 מינוס i.", "n_reviews": 0, "start": 1803.64, @@ -3325,7 +3325,7 @@ "end": 2961.56 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be.", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment", "translatedText": "אשמח לראות אם נוכל להשיג את הסרגל העליון עד 1729 מה שזה לא יהיה.", "n_reviews": 0, "start": 2961.56, diff --git a/2020/ldm-quadratic/hindi/sentence_translations.json b/2020/ldm-quadratic/hindi/sentence_translations.json index 73dffda9b..be1382365 100644 --- a/2020/ldm-quadratic/hindi/sentence_translations.json +++ b/2020/ldm-quadratic/hindi/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "इसके बाद हमारे पास एम गुना प्लस डी, एम गुना प्लस डी, और फिर नकारात्मक डी गुना डी है, इसलिए शून्य से डी का वर्ग।", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x वर्ग घटा 4x जमा 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x वर्ग घटा 4x जमा 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "बहुत अच्छी तरह से यह बिल्कुल नकारात्मक 1 है, जिसका अर्थ है कि हमारी दो जड़ें, और पृष्ठ पर तार के नीचे जो मैं यहां लिख रहा हूं, हमारी दो जड़ें 3 प्लस या माइनस आई हैं।", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "और फिर हमें केवल वास्तविक संख्याएँ मिलीं, 1, 2, 3, 4।", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "मुझे यह देखना अच्छा लगेगा कि क्या हम उस शीर्ष बार को 1729 तक प्राप्त कर सकते हैं, चाहे वह कुछ भी हो।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/hungarian/sentence_translations.json b/2020/ldm-quadratic/hungarian/sentence_translations.json index 12698c9d6..cf91f26db 100644 --- a/2020/ldm-quadratic/hungarian/sentence_translations.json +++ b/2020/ldm-quadratic/hungarian/sentence_translations.json @@ -798,21 +798,21 @@ "end": 679.22 }, { - "input": ".", + "input": "actually, you know what, let me change the variable names there.", "translatedText": ".", "n_reviews": 0, "start": 680.1, "end": 681.92 }, { - "input": ".", + "input": "I don't like x and y because those don't necessarily have the same meaning. What I might do is try", "translatedText": ".", "n_reviews": 0, "start": 681.94, "end": 686.14 }, { - "input": "Actually, you know what, let me change the variable names there.", + "input": "to picture, you know, let's picture the original two numbers that we had, something", "translatedText": "Igazából, tudod mit, hadd változtassam meg a változóneveket.", "n_reviews": 0, "start": 686.14, @@ -861,7 +861,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared.", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared.", "translatedText": "Ezután m-szer plusz d, m-szer plusz d, majd negatív d-szer d, tehát mínusz d négyzet.", "n_reviews": 0, "start": 726.92, @@ -1078,7 +1078,7 @@ "end": 963.22 }, { - "input": "Another way that I could express that particular quadratic is as x minus r times x minus s, because if I plug in r it's clear that we're going to get 0, and if I plug in s, again this term will also cancel out to 0.", + "input": "Another way that I could express that particular quadratic is as x minus r times x minus s. Because if I plug in r, it's clear that we're going to get zero. And if I plug in s, again, this term will also cancel out to zero.", "translatedText": "Egy másik módja annak, hogy ezt a másodfokút x mínusz r x mínusz s formában fejezzem ki, mert ha r-t bedugom, akkor egyértelmű, hogy 0-t kapunk, és ha bedugom az s-t, akkor ez a kifejezés is érvénytelenné válik. 0-ra.", "n_reviews": 0, "start": 963.58, @@ -1666,7 +1666,7 @@ "end": 1496.36 }, { - "input": "So we know the midpoint, and then we just ask ourselves what's the square of the distance, and based on difference of squares, that'll be that midpoint squared minus the product, which in this context is negative 5 squared or 25 minus the product, which is that last coefficient, minus 3.", + "input": "So we know the midpoint. And then we just ask ourselves, what's the square of the distance? And based on difference of squares, that'll be that midpoint squared minus the product, which in this context is negative five squared or 25 minus the product, which is that last coefficient, minus three.", "translatedText": "Tehát ismerjük a felezőpontot, majd csak kérdezzük meg magunktól, hogy mekkora a távolság négyzete, és a négyzetek különbsége alapján ez lesz a felezőpont négyzet mínusz a szorzat, ami ebben az összefüggésben negatív 5 négyzet vagy 25 mínusz a szorzat , ami az utolsó együttható mínusz 3.", "n_reviews": 0, "start": 1497.12, @@ -1820,7 +1820,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5.", + "input": "Three x squared minus four x plus five.", "translatedText": "3x négyzet mínusz 4x plusz 5.", "n_reviews": 0, "start": 1581.1, @@ -1834,7 +1834,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5.", + "input": "Three x squared minus four x plus five.", "translatedText": "3x négyzet mínusz 4x plusz 5.", "n_reviews": 0, "start": 1585.1, @@ -2037,7 +2037,7 @@ "end": 1725.34 }, { - "input": "Okay, so we go through the same rigmarole, we say m, the midpoint, is going to be negative b prime over 2, so in that case it works out to just be 3, because we take the negative of this term and divide by 2.", + "input": "Okay. So we go through the same rigmarole. We say m, the midpoint, is going to be negative b prime over two. So in that case, it works out to just be three because we take the negative of this term and divide by two.", "translatedText": "Rendben, akkor végigmegyünk ugyanazon a rigmuson, azt mondjuk, hogy m, a felezőpont negatív b prím lesz 2-nél, tehát ebben az esetben csak 3 lesz, mert vesszük ennek a tagnak a negatívját, és elosztjuk 2.", "n_reviews": 0, "start": 1726.3, @@ -2051,7 +2051,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i.", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i.", "translatedText": "Így szépen ez pontosan negatív 1, ami azt jelenti, hogy a két gyökünk, és az oldalon lévő vezetékig, amivel ide írtam, a két gyökünk 3 plusz-mínusz i.", "n_reviews": 0, "start": 1747.66, @@ -2093,14 +2093,14 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4.", + "input": "And then we've got just the real numbers. One, two, three, four.", "translatedText": "És akkor csak a valós számokat kapjuk, 1, 2, 3, 4.", "n_reviews": 0, "start": 1798.1, "end": 1802.84 }, { - "input": "So in this case our two roots live at 3 plus i, and 3 minus i.", + "input": "So in this case, our two roots live at three plus i and three minus i.", "translatedText": "Tehát ebben az esetben a két gyökünk 3 plusz i-nél és 3 mínusz i-nél él.", "n_reviews": 0, "start": 1803.64, @@ -3325,7 +3325,7 @@ "end": 2961.56 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be.", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment", "translatedText": "Szeretném látni, hogy elérjük-e azt a felső sávot 1729-ig, bármi legyen is az.", "n_reviews": 0, "start": 2961.56, diff --git a/2020/ldm-quadratic/indonesian/sentence_translations.json b/2020/ldm-quadratic/indonesian/sentence_translations.json index da29ec27a..9155a7c59 100644 --- a/2020/ldm-quadratic/indonesian/sentence_translations.json +++ b/2020/ldm-quadratic/indonesian/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "Selanjutnya kita punya m kali ditambah d, m kali ditambah d, dan kemudian negatif d dikali d, jadi dikurangi d kuadrat. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x kuadrat dikurangi 4x ditambah 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x kuadrat dikurangi 4x ditambah 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "Jadi bagusnya itu persis negatif 1, yang berarti dua akar kita, dan sampai ke kawat di halaman yang saya tulis di sini, kedua akar kita adalah 3 plus atau minus i. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "Dan kemudian kita hanya mendapatkan bilangan realnya, 1, 2, 3, 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "Saya ingin melihat apakah kita bisa menaikkan standar tersebut hingga tahun 1729, apa pun itu. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/italian/sentence_translations.json b/2020/ldm-quadratic/italian/sentence_translations.json index 0f3f9d4a9..743d4c24d 100644 --- a/2020/ldm-quadratic/italian/sentence_translations.json +++ b/2020/ldm-quadratic/italian/sentence_translations.json @@ -798,21 +798,21 @@ "end": 679.22 }, { - "input": ".", + "input": "actually, you know what, let me change the variable names there.", "translatedText": ".", "n_reviews": 0, "start": 680.1, "end": 681.92 }, { - "input": ".", + "input": "I don't like x and y because those don't necessarily have the same meaning. What I might do is try", "translatedText": ".", "n_reviews": 0, "start": 681.94, "end": 686.14 }, { - "input": "Actually, you know what, let me change the variable names there.", + "input": "to picture, you know, let's picture the original two numbers that we had, something", "translatedText": "In realtà, sai una cosa, lasciami cambiare i nomi delle variabili lì.", "n_reviews": 0, "start": 686.14, @@ -861,7 +861,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared.", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared.", "translatedText": "Poi abbiamo m per più d, m per più d, e poi meno d per d, quindi meno d al quadrato.", "n_reviews": 0, "start": 726.92, @@ -1078,7 +1078,7 @@ "end": 963.22 }, { - "input": "Another way that I could express that particular quadratic is as x minus r times x minus s, because if I plug in r it's clear that we're going to get 0, and if I plug in s, again this term will also cancel out to 0.", + "input": "Another way that I could express that particular quadratic is as x minus r times x minus s. Because if I plug in r, it's clear that we're going to get zero. And if I plug in s, again, this term will also cancel out to zero.", "translatedText": "Un altro modo in cui potrei esprimere quel particolare quadratico è come x meno r per x meno s, perché se inserisco r è chiaro che otterremo 0, e se inserisco s, anche in questo caso anche questo termine si annullerà a 0.", "n_reviews": 0, "start": 963.58, @@ -1666,7 +1666,7 @@ "end": 1496.36 }, { - "input": "So we know the midpoint, and then we just ask ourselves what's the square of the distance, and based on difference of squares, that'll be that midpoint squared minus the product, which in this context is negative 5 squared or 25 minus the product, which is that last coefficient, minus 3.", + "input": "So we know the midpoint. And then we just ask ourselves, what's the square of the distance? And based on difference of squares, that'll be that midpoint squared minus the product, which in this context is negative five squared or 25 minus the product, which is that last coefficient, minus three.", "translatedText": "Quindi conosciamo il punto medio e poi ci chiediamo qual è il quadrato della distanza e, in base alla differenza dei quadrati, sarà quel punto medio al quadrato meno il prodotto, che in questo contesto è meno 5 al quadrato o 25 meno il prodotto , che è l'ultimo coefficiente, meno 3.", "n_reviews": 0, "start": 1497.12, @@ -1820,7 +1820,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5.", + "input": "Three x squared minus four x plus five.", "translatedText": "3x al quadrato meno 4x più 5.", "n_reviews": 0, "start": 1581.1, @@ -1834,7 +1834,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5.", + "input": "Three x squared minus four x plus five.", "translatedText": "3x al quadrato meno 4x più 5.", "n_reviews": 0, "start": 1585.1, @@ -2037,7 +2037,7 @@ "end": 1725.34 }, { - "input": "Okay, so we go through the same rigmarole, we say m, the midpoint, is going to be negative b prime over 2, so in that case it works out to just be 3, because we take the negative of this term and divide by 2.", + "input": "Okay. So we go through the same rigmarole. We say m, the midpoint, is going to be negative b prime over two. So in that case, it works out to just be three because we take the negative of this term and divide by two.", "translatedText": "Ok, quindi seguiamo la stessa trafila, diciamo che m, il punto medio, sarà negativo b primo su 2, quindi in quel caso risulta essere solo 3, perché prendiamo il negativo di questo termine e dividiamo per 2.", "n_reviews": 0, "start": 1726.3, @@ -2051,7 +2051,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i.", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i.", "translatedText": "Quindi bene, è esattamente 1 negativo, il che significa che le nostre due radici, e fino al filo nella pagina con cui ho scritto qui, le nostre due radici sono 3 più o meno i.", "n_reviews": 0, "start": 1747.66, @@ -2093,14 +2093,14 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4.", + "input": "And then we've got just the real numbers. One, two, three, four.", "translatedText": "E poi abbiamo solo i numeri reali, 1, 2, 3, 4.", "n_reviews": 0, "start": 1798.1, "end": 1802.84 }, { - "input": "So in this case our two roots live at 3 plus i, and 3 minus i.", + "input": "So in this case, our two roots live at three plus i and three minus i.", "translatedText": "Quindi in questo caso le nostre due radici si trovano a 3 più i e 3 meno i.", "n_reviews": 0, "start": 1803.64, @@ -3325,7 +3325,7 @@ "end": 2961.56 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be.", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment", "translatedText": "Mi piacerebbe vedere se riusciamo a portare la barra superiore fino a 1729, qualunque essa sia.", "n_reviews": 0, "start": 2961.56, diff --git a/2020/ldm-quadratic/japanese/sentence_translations.json b/2020/ldm-quadratic/japanese/sentence_translations.json index 16bd2cf82..9969687ab 100644 --- a/2020/ldm-quadratic/japanese/sentence_translations.json +++ b/2020/ldm-quadratic/japanese/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "次に、m 倍プラス d、m 倍プラス d、そしてマイナス d 乗 d なので、マ イナス d の 2 乗です。", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x 2 乗マイナス 4x プラス 5。", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x 2 乗マイナス 4x プラス 5。", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "これは見事にマイナス 1 です。つ まり、2 つのルートがあり、ここで書いているページのワイヤーに 至るまで、2 つのルートは 3 プラスまたはマイナス i です。", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "そして、実数 1、2、3、4 だけが得られます 。", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "何が何でもトップバーを 1729 まで到達できる かどうかを見てみたいと思っています。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/korean/sentence_translations.json b/2020/ldm-quadratic/korean/sentence_translations.json index 001fb64f9..71a0c11c0 100644 --- a/2020/ldm-quadratic/korean/sentence_translations.json +++ b/2020/ldm-quadratic/korean/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "다음으로 m 곱하기 더하기 d, m 곱하기 더하기 d, 그리고 음수 d 곱하기 d, 즉 마이너스 d 제곱이 됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x 제곱 - 4x 더하기 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x 제곱 - 4x 더하기 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "훌륭하게도 그것은 정확히 마이너스 1입니다. 이는 우리의 두 근이 제가 여기 쓰고 있는 페이지의 선까지 내려가면 우리의 두 근은 3 더하기 또는 빼기 i라는 것을 의미합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "그리고 우리는 단지 실수인 1, 2, 3, 4를 얻었습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "나는 그것이 무엇이든 간에 그 상단 표시줄을 1729까지 얻을 수 있는지 보고 싶습니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/marathi/sentence_translations.json b/2020/ldm-quadratic/marathi/sentence_translations.json index aefc556a4..478c9ed15 100644 --- a/2020/ldm-quadratic/marathi/sentence_translations.json +++ b/2020/ldm-quadratic/marathi/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "पुढे m गुणिले अधिक d, m गुणिले अधिक d, आणि नंतर ऋण d गुणिले d, म्हणजे वजा d वर्ग. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x वर्ग वजा 4x अधिक 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x वर्ग वजा 4x अधिक 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "इतकं छान आहे की ते अगदी ऋण 1 आहे, म्हणजे आपली दोन मुळे, आणि मी इथे लिहित असलेल्या पानावरील वायरपर्यंत, आपली दोन मुळे 3 अधिक किंवा उणे i आहेत. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "आणि मग आपल्याला फक्त खरी संख्या मिळाली, १, २, ३, ४. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "आम्हाला ते टॉप बार 1729 पर्यंत मिळू शकेल का ते पहायला मला आवडेल. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/persian/sentence_translations.json b/2020/ldm-quadratic/persian/sentence_translations.json index be05cb2c7..2cc481607 100644 --- a/2020/ldm-quadratic/persian/sentence_translations.json +++ b/2020/ldm-quadratic/persian/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "بعد m ضربدر d، m ضربدر d و سپس منفی d ضربدر d داریم، پس منهای d مجذور می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x مربع منهای 4x به علاوه 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "چرا که نه؟ 3x مربع منهای 4x به علاوه 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "خیلی خوب که دقیقاً منفی 1 است، به این معنی که دو ریشه ما، و تا سیم صفحه ای که در اینجا با آن نوشتم، دو ریشه ما 3 به علاوه یا منهای i هستند. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "و سپس ما فقط اعداد واقعی، 1، 2، 3، 4 را داریم. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "میدونی به خاطر من می تونیم همینطور ادامه بدیم؟ من خیلی دوست دارم ببینم که آیا می توانیم آن نوار بالا را تا 1729 هر چه که باشد برسانیم. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/portuguese/sentence_translations.json b/2020/ldm-quadratic/portuguese/sentence_translations.json index d944593cf..7d762bba5 100644 --- a/2020/ldm-quadratic/portuguese/sentence_translations.json +++ b/2020/ldm-quadratic/portuguese/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "Em seguida, temos m vezes mais d, m vezes mais d, e então menos d vezes d, então menos d ao quadrado. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x ao quadrado menos 4x mais 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x ao quadrado menos 4x mais 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "Então, isso é exatamente menos 1, o que significa que nossas duas raízes, e até o fio na página que estou escrevendo aqui, nossas duas raízes são 3 mais ou menos i. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "E então temos apenas os números reais, 1, 2, 3, 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "Eu adoraria ver se conseguimos levar a barra superior até 1729, seja lá o que for. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/russian/sentence_translations.json b/2020/ldm-quadratic/russian/sentence_translations.json index 16ae97027..2a21e318e 100644 --- a/2020/ldm-quadratic/russian/sentence_translations.json +++ b/2020/ldm-quadratic/russian/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "Далее у нас есть m раз плюс d, m раз плюс d, а затем отрицательное d, умноженное на d, то есть минус d в квадрате. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3х в квадрате минус 4х плюс 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3х в квадрате минус 4х плюс 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "Итак, это ровно отрицательная 1, что означает, что наши два корня и вплоть до проводника на странице, которую я здесь писал, наши два корня равны 3 плюс или минус i. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "И тогда у нас есть только настоящие числа: 1, 2, 3, 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "Мне бы хотелось посмотреть, сможем ли мы поднять эту верхнюю планку до 1729, какой бы она ни была. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/spanish/sentence_translations.json b/2020/ldm-quadratic/spanish/sentence_translations.json index fcdc7c9c0..a78972b9d 100644 --- a/2020/ldm-quadratic/spanish/sentence_translations.json +++ b/2020/ldm-quadratic/spanish/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "Luego tenemos m por más d, m por más d, y luego negativo d por d, por lo que menos d al cuadrado. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x al cuadrado menos 4x más 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x al cuadrado menos 4x más 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "Bueno, eso es exactamente menos 1, lo que significa que nuestras dos raíces, y hasta el cable en la página con la que he estado escribiendo aquí, nuestras dos raíces son 3 más o menos i. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "Y luego tenemos sólo los números reales, 1, 2, 3, 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "Me encantaría ver si podemos llevar esa barra superior hasta 1729, sea lo que sea. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/tamil/sentence_translations.json b/2020/ldm-quadratic/tamil/sentence_translations.json index ecf8206fc..c7e091d39 100644 --- a/2020/ldm-quadratic/tamil/sentence_translations.json +++ b/2020/ldm-quadratic/tamil/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "அடுத்து நாம் m முறை கூட்டல் d, m முறை கூட்டல் d, பின்னர் எதிர்மறை d முறை d, எனவே கழித்தல் d ஸ்கொயர். ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x சதுரம் கழித்தல் 4x கூட்டல் 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x சதுரம் கழித்தல் 4x கூட்டல் 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "மிக நேர்த்தியாக அது சரியாக எதிர்மறை 1 ஆகும், அதாவது எங்கள் இரண்டு வேர்கள், மற்றும் நான் இங்கே எழுதிக்கொண்டிருக்கும் பக்கத்தில் உள்ள கம்பி வரை, எங்கள் இரண்டு வேர்களும் 3 கூட்டல் அல்லது கழித்தல் i. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "பின்னர் எங்களிடம் உண்மையான எண்கள், 1, 2, 3, 4 கிடைத்துள்ளன. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "அது எதுவாக இருந்தாலும் 1729 வரை அந்த டாப் பட்டியைப் பெற முடியுமா என்று பார்க்க விரும்புகிறேன். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/telugu/sentence_translations.json b/2020/ldm-quadratic/telugu/sentence_translations.json index 9dafd2af9..0172d7d65 100644 --- a/2020/ldm-quadratic/telugu/sentence_translations.json +++ b/2020/ldm-quadratic/telugu/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "తర్వాత మనకు m సార్లు ప్లస్ d, m సార్లు ప్లస్ d, ఆపై ప్రతికూల d సార్లు d, కాబట్టి మైనస్ d స్క్వేర్డ్. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x స్క్వేర్డ్ మైనస్ 4x ప్లస్ 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x స్క్వేర్డ్ మైనస్ 4x ప్లస్ 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "కాబట్టి చక్కగా సరిగ్గా ప్రతికూల 1, అంటే మా రెండు మూలాలు, మరియు నేను ఇక్కడ వ్రాస్తున్న పేజీలోని వైర్ వరకు, మా రెండు మూలాలు 3 ప్లస్ లేదా మైనస్ i. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "ఆపై మనకు నిజమైన సంఖ్యలు 1, 2, 3, 4 ఉన్నాయి. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "మేము ఆ టాప్ బార్‌ను 1729 వరకు పొందగలమో లేదో చూడాలనుకుంటున్నాను. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/thai/sentence_translations.json b/2020/ldm-quadratic/thai/sentence_translations.json index f87b45368..8df1acbb3 100644 --- a/2020/ldm-quadratic/thai/sentence_translations.json +++ b/2020/ldm-quadratic/thai/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "ที่จริง, คุณรู้อะไรไหม, ขอผมเปลี่ยนชื่อตัวแปรตรงนี้ก่อน ฉันไม่ชอบ x และ y เพราะไม่จำเป็นต้องมีความหมายเหมือนกัน ที่ผมอาจทำคือลองนึกภาพ ลองนึกภาพตัวเลขสองตัวเดิมที่เรามี ซึ่งเท่ากับ 59 กับ 61 บนเส้นจำนวน แล้วผมอยากคิดเป็นจุดกึ่งกลาง m แล้วตามด้วยระยะห่างระหว่าง m และตัวเลขอื่นๆ แต่ละตัว ได้บวกและลบ d ในกรณีนี้ d เป็นเพียงค่าเดียว แต่โดยหลักการแล้ว มันอาจเป็นอะไรที่มากกว่านั้นก็ได้ ถ้าผมขอให้คุณคูณ m ลบ d คูณ m บวก d, คุณก็รู้, ณ จุดนี้ มันเป็นพีชคณิตตรงไปตรงมา เทอมแรกคือ m คูณ m, เราได้ m กำลังสอง ต่อไปเรามีลบ d คูณ m, นั่นก็คือ ลบ d m ต่อไปเรามี m คูณ d, m คูณ d, แล้วลบ d คูณ d, ได้ ลบ d กำลังสอง และสิ่งที่ออกมาคือ m กำลังสอง ลบ d กำลังสอง และถ้าคุณเป็นเหมือนฉัน ครั้งแรกที่คุณเห็นสิ่งนี้ในชั้นเรียนพีชคณิต มันเป็นแค่หนึ่งในแบบฝึกหัดทั้งหมดที่คุณกำลังทำอยู่ คุณแบบว่า ใช่ โอเค ผมขยายอันนี้ออกไปได้ ผมว่าบางเทอมก็หักล้างกันดี แต่ยิ่งคุณคิดเรื่องนี้มากเท่าไร มันก็ยิ่งน่าสนใจมากขึ้นเท่านั้น เพราะเราสามารถเขียนเลขสองตัวใดๆ ได้ คุณก็รู้ ถ้าฉันหาเลข r และ s ใดๆ ลงไป ฉันจะเขียนลงไปว่าเป็นจุดกึ่งกลางบวกหรือลบระยะทางก็ได้ และ สิ่งนี้บอกเราคือการคิดเกี่ยวกับผลิตภัณฑ์ ก็เหมือนกับการคิดถึงความแตกต่างของกำลังสองเสมอ ซึ่งแปลก เพราะผลิตภัณฑ์อาจวุ่นวายมาก ถ้าผมเดินผ่านเส้นจำนวนแล้วไม่รู้ ผมบอกว่าดู 101, 102, 103 แล้วบอกว่า ดูตัวเลขทั้งหมดบนเส้นจำนวนแล้วอยากให้บอกอย่างเป็นระบบ แต่ละตัวสามารถแยกย่อยเป็นผลคูณของจำนวนเต็มที่มีขนาดเล็กกว่าสองตัวได้หรือไม่? ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/turkish/sentence_translations.json b/2020/ldm-quadratic/turkish/sentence_translations.json index 9ba4f449c..4e5369adb 100644 --- a/2020/ldm-quadratic/turkish/sentence_translations.json +++ b/2020/ldm-quadratic/turkish/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "Sonra elimizde m çarpı artı d, m çarpı artı d ve sonra da negatif d çarpı d, yani eksi d'nin karesi var. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x kare eksi 4x artı 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x kare eksi 4x artı 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "Çok güzel bir şekilde bu tam olarak negatif 1, yani iki kökümüz ve burada yazdığım sayfadaki tele kadar iki kökümüz 3 artı veya eksi i. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "Ve elimizde sadece gerçek sayılar var: 1, 2, 3, 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "Her ne olursa olsun, üst çıtayı 1729'a kadar çıkarabilecek miyiz, görmeyi çok isterim. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/ukrainian/sentence_translations.json b/2020/ldm-quadratic/ukrainian/sentence_translations.json index 30bae6680..008380d51 100644 --- a/2020/ldm-quadratic/ukrainian/sentence_translations.json +++ b/2020/ldm-quadratic/ukrainian/sentence_translations.json @@ -798,21 +798,21 @@ "end": 679.22 }, { - "input": ".", + "input": "actually, you know what, let me change the variable names there.", "translatedText": ".", "n_reviews": 0, "start": 680.1, "end": 681.92 }, { - "input": ".", + "input": "I don't like x and y because those don't necessarily have the same meaning. What I might do is try", "translatedText": ".", "n_reviews": 0, "start": 681.94, "end": 686.14 }, { - "input": "Actually, you know what, let me change the variable names there.", + "input": "to picture, you know, let's picture the original two numbers that we had, something", "translatedText": "Насправді, знаєте що, дозвольте мені змінити назви змінних там.", "n_reviews": 0, "start": 686.14, @@ -861,7 +861,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared.", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared.", "translatedText": "Далі ми маємо m помножити на d, m помножити на d, а потім від’ємне d помножити на d, тобто мінус d у квадраті.", "n_reviews": 0, "start": 726.92, @@ -1078,7 +1078,7 @@ "end": 963.22 }, { - "input": "Another way that I could express that particular quadratic is as x minus r times x minus s, because if I plug in r it's clear that we're going to get 0, and if I plug in s, again this term will also cancel out to 0.", + "input": "Another way that I could express that particular quadratic is as x minus r times x minus s. Because if I plug in r, it's clear that we're going to get zero. And if I plug in s, again, this term will also cancel out to zero.", "translatedText": "Інший спосіб, яким я можу виразити це конкретне квадратичне число, це як x мінус r помножити на x мінус s, тому що якщо я підставлю r, то зрозуміло, що ми отримаємо 0, а якщо я підставлю s, знову цей член також скасується до 0.", "n_reviews": 0, "start": 963.58, @@ -1666,7 +1666,7 @@ "end": 1496.36 }, { - "input": "So we know the midpoint, and then we just ask ourselves what's the square of the distance, and based on difference of squares, that'll be that midpoint squared minus the product, which in this context is negative 5 squared or 25 minus the product, which is that last coefficient, minus 3.", + "input": "So we know the midpoint. And then we just ask ourselves, what's the square of the distance? And based on difference of squares, that'll be that midpoint squared minus the product, which in this context is negative five squared or 25 minus the product, which is that last coefficient, minus three.", "translatedText": "Тож ми знаємо середину, а потім просто запитуємо себе, чому дорівнює квадрат відстані, і виходячи з різниці квадратів, це буде ця середина в квадраті мінус добуток, який у цьому контексті дорівнює мінус 5 у квадраті або 25 мінус добуток , що є останнім коефіцієнтом, мінус 3.", "n_reviews": 0, "start": 1497.12, @@ -1820,7 +1820,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5.", + "input": "Three x squared minus four x plus five.", "translatedText": "3х у квадраті мінус 4х плюс 5.", "n_reviews": 0, "start": 1581.1, @@ -1834,7 +1834,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5.", + "input": "Three x squared minus four x plus five.", "translatedText": "3х у квадраті мінус 4х плюс 5.", "n_reviews": 0, "start": 1585.1, @@ -2037,7 +2037,7 @@ "end": 1725.34 }, { - "input": "Okay, so we go through the same rigmarole, we say m, the midpoint, is going to be negative b prime over 2, so in that case it works out to just be 3, because we take the negative of this term and divide by 2.", + "input": "Okay. So we go through the same rigmarole. We say m, the midpoint, is going to be negative b prime over two. So in that case, it works out to just be three because we take the negative of this term and divide by two.", "translatedText": "Гаразд, ми проходимо через ту саму мітку, ми говоримо, що m, середня точка, буде від’ємним b простим числом на 2, тож у цьому випадку виходить просто 3, тому що ми беремо від’ємне значення цього члена та ділимо на 2.", "n_reviews": 0, "start": 1726.3, @@ -2051,7 +2051,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i.", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i.", "translatedText": "Так добре, що це рівно мінус 1, що означає, що наші два корені, і вниз до дроту на сторінці, якою я тут писав, наші два корені дорівнюють 3 плюс або мінус і.", "n_reviews": 0, "start": 1747.66, @@ -2093,14 +2093,14 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4.", + "input": "And then we've got just the real numbers. One, two, three, four.", "translatedText": "І тоді ми маємо справжні числа, 1, 2, 3, 4.", "n_reviews": 0, "start": 1798.1, "end": 1802.84 }, { - "input": "So in this case our two roots live at 3 plus i, and 3 minus i.", + "input": "So in this case, our two roots live at three plus i and three minus i.", "translatedText": "Тож у цьому випадку наші два корені живуть у 3 плюс i та 3 мінус i.", "n_reviews": 0, "start": 1803.64, @@ -3325,7 +3325,7 @@ "end": 2961.56 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be.", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment", "translatedText": "Я хотів би побачити, чи зможемо ми підняти цю верхню панель до 1729, якою б вона не була.", "n_reviews": 0, "start": 2961.56, diff --git a/2020/ldm-quadratic/urdu/sentence_translations.json b/2020/ldm-quadratic/urdu/sentence_translations.json index 23fcb91a8..e0b57f2e3 100644 --- a/2020/ldm-quadratic/urdu/sentence_translations.json +++ b/2020/ldm-quadratic/urdu/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "اس کے بعد ہمارے پاس ایم ٹائم پلس ڈی، ایم ٹائم پلس ڈی، اور پھر منفی ڈی گنا ڈی، تو مائنس ڈی مربع۔", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x مربع مائنس 4x جمع 5۔", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "کیوں نہیں؟ 3x مربع مائنس 4x جمع 5۔", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "اتنی اچھی طرح سے یہ بالکل منفی 1 ہے، جس کا مطلب ہے کہ ہماری دو جڑیں، اور اس صفحے کے نیچے جس کے ساتھ میں یہاں لکھ رہا ہوں، ہماری دو جڑیں 3 جمع یا مائنس i ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "اور پھر ہمارے پاس صرف حقیقی نمبر ہیں، 1، 2، 3، 4۔", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "آپ جانتے ہیں کہ میری خاطر کیا ہم اسی طرح جاری رکھ سکتے ہیں؟ میں یہ دیکھنا پسند کروں گا کہ کیا ہم اس ٹاپ بار کو 1729 تک حاصل کرسکتے ہیں جو کچھ بھی ہو۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-quadratic/vietnamese/sentence_translations.json b/2020/ldm-quadratic/vietnamese/sentence_translations.json index 5fd75f71e..4d1b1d7d1 100644 --- a/2020/ldm-quadratic/vietnamese/sentence_translations.json +++ b/2020/ldm-quadratic/vietnamese/sentence_translations.json @@ -968,7 +968,7 @@ "end": 726.0 }, { - "input": "Next we have m times plus d, m times plus d, and then negative d times d, so minus d squared. ", + "input": "Next, we have m times plus d, m plus d. That's plus d and then negative d times d. So minus d squared. A ", "translatedText": "Tiếp theo chúng ta có m nhân cộng d, m nhân cộng d, rồi âm d nhân d, nên trừ d bình phương. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1581.0 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x bình trừ 4x cộng 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1583.72 }, { - "input": "3x squared minus 4x plus 5. ", + "input": "Three x squared minus four x plus five. ", "translatedText": "3x bình trừ 4x cộng 5. ", "model": "google_nmt", "n_reviews": 0, @@ -2312,7 +2312,7 @@ "end": 1746.62 }, { - "input": "So nicely that's exactly negative 1, which means that our two roots, and down to the wire on the page that I've been writing with here, our two roots are 3 plus or minus i. ", + "input": "So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here, our two roots are three plus or minus i. ", "translatedText": "Rất hay đó chính xác là âm 1, có nghĩa là hai nghiệm của chúng ta, và tính đến dòng trên trang mà tôi đang viết ở đây, hai nghiệm của chúng ta là 3 cộng hoặc trừ i. ", "model": "google_nmt", "n_reviews": 0, @@ -2360,7 +2360,7 @@ "end": 1796.4 }, { - "input": "And then we've got just the real numbers, 1, 2, 3, 4. ", + "input": "And then we've got just the real numbers. One, two, three, four. ", "translatedText": "Và khi đó chúng ta chỉ có các số thực, 1, 2, 3, 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3744,7 +3744,7 @@ "end": 2990.9 }, { - "input": "I would love to see if we can get that top bar up to 1729 whatever it might be. ", + "input": "I would love to see if we can get that top bar up to 1729 whatever it might be at the moment ", "translatedText": "Tôi rất muốn xem liệu chúng ta có thể đưa thanh trên cùng đó lên tới 1729 hay không, bất kể nó có thể là bao nhiêu. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/arabic/sentence_translations.json b/2020/ldm-tips-to-problem-solving/arabic/sentence_translations.json index 4aa474ecf..ff64e51e4 100644 --- a/2020/ldm-tips-to-problem-solving/arabic/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/arabic/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "ومرة أخرى، ما أريدك أن تتخلص منه هو هذا المبدأ القائل بأنه إذا كان بإمكانك الحصول على كائن واحد موصوف بطريقتين مختلفتين، فهو قوي جدًا من حيث إظهار العلاقات الجبرية غير الواضحة أو أي شيء يمكن كتابته رمزيًا دون الحدس المباشر في الأعلى منه. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "سيكون هذا احتمال 50٪. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "والآن ربما يمكنك أن ترى إلى أين سيصل هذا، لأنه في الحد التالي عندما نريد أن نعرف متى تقع x على y بين 4 و5، سنقوم برسم خطوط تتقاطع عند x على 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "ربما بعض الأشياء التي أحاول التحدث عنها اليوم، مثل، حسنًا، يمكنني أن أقول تناظر الرافعة المالية، ولكن ماذا يعني ذلك في الواقع؟ ماذا يعني النظر إلى تماثل المشكلة وتحويله إلى شيء مفيد من الناحية النظرية؟ عليك فقط أن تراه كثيرًا، ثم تفكر عندما تفعل ذلك. ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "وأستمر في صنع المزيد من الحلقات. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "لذا، كما تعلمون، يمكننا أن نرى بعض الأمثلة هنا، بعضها يصل إلى 0، وبعضها يتم تقريبه إلى 0، وبعضها يتم تقريبه إلى 2، وبعضها يتم تقريبه إلى 1، لذلك هذا لطيف. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "وإذا أخذت متوسط ذلك، وهو التعامل مع الصواب والخطأ على أنه 1 و0، فهذا يخبرني عن النسبة الإجمالية التي ستكون في الواقع 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "كما تعلم، كنا نبحث عن شيء يساوي نصف 2 ناقص اللوغاريتم الطبيعي لـ 2. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "لذلك ربما نريد فقط أن ننظر إلى الوقت الذي يتم فيه تجميع هذه القيم بين 0 و20. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "ويجب عليك دائمًا إضافة عرض نسبي على هذه الأنواع من الرسوم البيانية لأنها تبدو قبيحة إذا كانت جميعها متشابهة جنبًا إلى جنب، على ما أعتقد في بعض الأحيان. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "ويمكنك الحصول على هذا الإحساس اللطيف لماهية جميع بياناتك. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "وقد طُلب منهم فقط إنشاء شيء فني من ذلك. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "آخر كان مجرد مجنون حقا. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/bengali/sentence_translations.json b/2020/ldm-tips-to-problem-solving/bengali/sentence_translations.json index 29ffc2db7..2248e561b 100644 --- a/2020/ldm-tips-to-problem-solving/bengali/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/bengali/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "এবং আবার, আমি যা চাই তা হ'ল আপনি এই নীতিটি হ'ল যে যদি আপনার কাছে দুটি ভিন্ন উপায়ে বর্ণিত একটি বস্তু থাকতে পারে, অ-স্পষ্ট বীজগণিত সম্পর্ক দেখানোর ক্ষেত্রে খুব শক্তিশালী বা উপরে অবিলম্বে অন্তর্জ্ঞান ছাড়াই প্রতীকীভাবে লেখা কিছু এর তাই যে সব সঙ্গে, এর আসলে চালু করা যাক সম্ভাব্যতা প্রশ্ন যে আমি বক্তৃতা শুরুতে জিজ্ঞাসা. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "এটি 50% এর সম্ভাবনা হবে।", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "এবং এখন হয়তো আপনি দেখতে পাচ্ছেন যে এটি কোথায় যেতে চলেছে, কারণ পরবর্তী মেয়াদের জন্য যখন আমরা জানতে চাই কখন x y দিয়ে ভাগ করলে 4 এবং 5 এর মধ্যে বসবে, আমরা লাইন আঁকব যা x 4 এর উপরে ছেদ করে।", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "এটি একটি সমস্যার একটি প্রতিসাম্য তাকান এবং সূত্রানুযায়ী দরকারী যে কিছুতে পরিণত করার মানে কি? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "আরো এপিসোড বানাতে থাকি।", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "সুতরাং, আপনি জানেন, আমরা এখানে কিছু উদাহরণ দেখতে পারি, কিছু যেগুলি 0 পর্যন্ত এসেছে, এবং কিছু যেগুলি 0-তে বৃত্তাকার হবে, কিছু যেগুলি 2 থেকে বৃত্তাকার হবে, কিছু যেগুলি 1-এ বৃত্তাকার হবে, তাই এটি চমৎকার।", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "এটি আমাকে সত্য এবং মিথ্যার একটি তালিকা দেয়, মূলত বলছে এটি বা এটি 0 নয়৷ এবং যদি আমি এর গড় গ্রহণ করি, যা সত্য এবং মিথ্যাকে 1s এবং 0s হিসাবে বিবেচনা করে, এটি আমাকে মোট অনুপাত বলে যা আসলে 0s হবে।", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "আপনি জানেন, আমরা এমন কিছু খুঁজছিলাম যা 2-এর স্বাভাবিক লগ থেকে 2 বিয়োগের অর্ধেক।", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "তাই হয়তো আমরা শুধু দেখতে চাই যখন সেই মানগুলি 0 এবং 20 এর মতো মধ্যে বাকেট করা হয়।", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "এবং আপনাকে সর্বদা এই ধরণের হিস্টোগ্রামগুলিতে একটি আপেক্ষিক প্রস্থ যুক্ত করতে হবে কারণ সেগুলি কেবল কুশ্রী দেখায় যদি সেগুলি পাশাপাশি থাকে, আমি কখনও কখনও মনে করি।", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "এবং তাদের কেবল এটি থেকে শৈল্পিক কিছু তৈরি করতে বলা হয়েছিল।", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "আরেকটি যে শুধু সত্যিকারের উন্মাদ ছিল. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/chinese/sentence_translations.json b/2020/ldm-tips-to-problem-solving/chinese/sentence_translations.json index cd58d6110..e1e12760a 100644 --- a/2020/ldm-tips-to-problem-solving/chinese/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/chinese/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "再说一次，我希望你明白这一原则，即如果你能用两种不同的方式描述一个对象，那么在显 示非明显的代数关系或任何象征性地写下来而无需直接直觉的东西方面就非常强大它的。", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "这个概率是 50%。", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "现在也许你可以看到这将走向何 方，因为对于下一项，当我们 想知道 x 除以 y 何时 位于 4 和 5 之间时，我 们将绘制与 x 相交于 4 的线。", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "查看问题的对称性并将其转化为公式上有用的东西意味着什么？", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "我不断制作更多剧集。", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "所以，你知道，我们可以在这里看到一些例子，一些已经达到 0，一些会向下舍入到 0，一些会向下舍入到 2，一些会向下舍入到 1，所以这很好。", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "如果我取其平均值，即将真值和假值分别视为 1 和 0，这会告诉我实际为 0 的总数所占的比例。", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "你知道，我们正在寻找 2 的一半减去 2 的自然对数。", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "所以也许我们只是想看看这些值何时位于 0 到 20 之间。", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "而且你总是必须在这些类型的直方图上添加相对宽度，因为我认为有时如果它们并排排列，它们看起来就会很难看。", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "您可以很好地了解所有数据的含义。", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "他们只是被提示从中创造出一些艺术作品。", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "另一个真的是疯了。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/english/captions.srt b/2020/ldm-tips-to-problem-solving/english/captions.srt index bec0a396f..aecd3aac6 100644 --- a/2020/ldm-tips-to-problem-solving/english/captions.srt +++ b/2020/ldm-tips-to-problem-solving/english/captions.srt @@ -183,7 +183,7 @@ who many of you may know because of his YouTube fame, and then another person who used to work with us at Khan Academy named Cam Christensen. 47 -00:02:22,899 --> 00:02:25,069 +00:02:22,900 --> 00:02:25,069 And this is just one small little outcropping 48 @@ -295,7 +295,7 @@ They are looking for beta users, so feel free to reach out to them. It should be available on the website. 75 -00:03:50,599 --> 00:03:52,820 +00:03:50,600 --> 00:03:52,820 Now let's dive into the actual content, shall we? 76 @@ -719,3822 +719,3826 @@ of what was happening. So I just love it. 181 -00:09:32,440 --> 00:09:36,540 -All right, so because there's such strong consensus, I feel comfortable grading this. +00:09:32,440 --> 00:09:35,187 +All right, so because there's such strong consistent- consensus, 182 +00:09:35,187 --> 00:09:36,540 +I feel comfortable grading this. + +183 00:09:36,820 --> 00:09:39,080 Oh, I thought I got it right on 1 1 1 1. -183 +184 00:09:40,000 --> 00:09:43,398 So the correct answer is that a is equal to b, d is equal to e, -184 +185 00:09:43,398 --> 00:09:46,000 and then none of the others are necessarily true. -185 -00:09:46,899 --> 00:09:49,320 +186 +00:09:46,900 --> 00:09:49,320 And let's walk through why that's the case, okay? -186 -00:09:49,839 --> 00:09:54,083 +187 +00:09:49,840 --> 00:09:54,083 It comes down to exactly what we just highlighted, that these radii are common, -187 +188 00:09:54,083 --> 00:09:57,160 which means, you know, let me just give more things names. -188 +189 00:09:57,220 --> 00:10:00,680 Let's call this point a, and this point b. -189 +190 00:10:01,380 --> 00:10:04,000 Maybe that's confusing because I was just naming angles a and b, -190 +191 00:10:04,000 --> 00:10:06,460 but separate context, separate picture, you know what I mean. -191 +192 00:10:07,280 --> 00:10:11,800 The triangle a, p, and then the center of the circle, that's isosceles. -192 +193 00:10:12,340 --> 00:10:17,217 You know, there's this symmetry about, I could even draw the little axis of symmetry, -193 +194 00:10:17,217 --> 00:10:19,600 and that tells us that this is also alpha. -194 +195 00:10:20,280 --> 00:10:24,033 Likewise, the triangle up here is isosceles, and -195 +196 00:10:24,033 --> 00:10:27,480 that tells us that this has an angle of beta. -196 +197 00:10:28,980 --> 00:10:32,200 And what we just did, in effect, is leverage a little bit of symmetry. -197 +198 00:10:32,640 --> 00:10:37,296 In this case, it was innocuous, but quite often looking for symmetry in much harder -198 +199 00:10:37,296 --> 00:10:41,620 setups, it's super generalizable and it definitely will help you move forward. -199 +200 00:10:41,680 --> 00:10:43,785 If you recognize that there's something symmetric, -200 +201 00:10:43,785 --> 00:10:46,800 use that symmetry in some way, which is effectively what we've just done. -201 +202 00:10:47,280 --> 00:10:50,768 Giving things a couple more names, I might want to call this angle, -202 +203 00:10:50,768 --> 00:10:55,026 I'll call it something related to the alphas, so maybe I just call it alpha prime, -203 +204 00:10:55,026 --> 00:10:57,540 and similarly this angle over here is beta prime. -204 +205 00:10:58,240 --> 00:11:01,626 And by noting those facts, I have everything I need to do to -205 +206 00:11:01,626 --> 00:11:05,180 draw a connection between this small angle and this large angle. -206 +207 00:11:05,840 --> 00:11:07,640 You essentially just write down the facts that -207 +208 00:11:07,640 --> 00:11:09,480 we have based on the triangles we're looking at. -208 +209 00:11:09,780 --> 00:11:13,277 So the triangle with all of the alpha angles, the sum of those angles -209 +210 00:11:13,277 --> 00:11:16,575 has to be 180 degrees, and there's two different alphas in there, -210 +211 00:11:16,575 --> 00:11:20,022 and then there's an alpha prime, and instead of writing 180 degrees, -211 +212 00:11:20,022 --> 00:11:23,520 of course we like radians, so I'm going to say that equals pi radians. -212 +213 00:11:24,060 --> 00:11:27,639 Similarly, the one with all of the betas tells us that if -213 +214 00:11:27,639 --> 00:11:31,280 we take 2 times beta and we add beta prime, that's also pi. -214 +215 00:11:32,460 --> 00:11:36,750 And then the other fact that we have that's just popping right out from the -215 +216 00:11:36,750 --> 00:11:41,380 image is that alpha prime, beta prime, and theta L add up to 360 degrees, or 2 pi. -216 +217 00:11:44,540 --> 00:11:47,745 So just writing down all of the relevant facts that we have, -217 +218 00:11:47,745 --> 00:11:51,948 now we have some objects that we can manipulate and work with to draw some kind -218 +219 00:11:51,948 --> 00:11:56,099 of conclusion, which again, we're looking for a connection between theta small -219 +220 00:11:56,099 --> 00:11:56,940 and theta large. -220 +221 00:11:57,060 --> 00:11:59,797 And I know a lot of you, your 3Blue and Brown audience members, -221 +222 00:11:59,797 --> 00:12:03,476 you know about the inscribed angle theorem, but I really want you to think about this -222 +223 00:12:03,476 --> 00:12:04,760 from a beginner's mind, right? -223 +224 00:12:04,760 --> 00:12:08,101 If you were just approaching this and you didn't necessarily already know about it, -224 +225 00:12:08,101 --> 00:12:10,011 what would you have done to find that solution, -225 +226 00:12:10,011 --> 00:12:12,120 and what principles can you take away as you do that? -226 +227 00:12:12,120 --> 00:12:15,280 Because that helps us as we start to get to harder and harder geometry setups. -227 +228 00:12:15,900 --> 00:12:18,305 So in this context, once I have these three equations, -228 +229 00:12:18,305 --> 00:12:20,885 recognizing that there's an alpha prime here and one here, -229 +230 00:12:20,885 --> 00:12:24,340 there's a beta prime here and one here, I might think about canceling them out. -230 +231 00:12:24,760 --> 00:12:27,500 So I'm gonna, you know, add this top equation -231 +232 00:12:27,500 --> 00:12:30,420 and maybe I subtract off the other two equations. -232 +233 00:12:31,360 --> 00:12:36,498 And to subtract these off, and what that means is the alpha prime gets cancelled, -233 +234 00:12:36,498 --> 00:12:39,882 so does the beta prime, I'm left with my large angle, -234 +235 00:12:39,882 --> 00:12:42,640 I'm subtracting off 2 times alpha plus beta. -235 +236 00:12:43,780 --> 00:12:46,320 You know, each of those gets subtracted off with the coefficient 2. -236 -00:12:46,560 --> 00:12:50,512 -And then we have 2 pi minus 2 copies of pi, so that's all equal to 0, - 237 -00:12:50,512 --> 00:12:54,069 -which is saying the same thing as theta L is equal to 2 times, +00:12:46,560 --> 00:12:50,827 +, and then we have two pi minus two copies of pi. So that's all equal to zero, 238 -00:12:54,069 --> 00:12:58,529 -well rather than writing alpha plus beta, I'll just recognize that that is the +00:12:50,827 --> 00:12:54,338 +which is saying the same thing as theta l is equal to two times, 239 -00:12:58,529 --> 00:13:00,280 -small little angle that we had. +00:12:54,338 --> 00:12:58,929 +well rather than writing alpha plus beta, I'll just recognize that that is the small 240 +00:12:58,929 --> 00:13:00,280 +little angle that we had. + +241 00:13:00,740 --> 00:13:02,380 It's 2 times the small angle. -241 +242 00:13:03,160 --> 00:13:06,390 Again, I can't emphasize enough just what a weirdly useful fact -242 +243 00:13:06,390 --> 00:13:09,520 this turns out to be in various geometry puzzles you might do. -243 +244 00:13:09,520 --> 00:13:14,223 It's definitely come up on the channel a number of times in circumstances regarding, -244 +245 00:13:14,223 --> 00:13:17,820 you know, complex numbers or pure geometry situations, of course. -245 +246 00:13:18,660 --> 00:13:22,975 Just anytime you want to relate an angle to 2 times that angle, -246 +247 00:13:22,975 --> 00:13:28,100 realizing them in the context of a circle like this can be strangely useful. -247 +248 00:13:28,180 --> 00:13:31,520 So this is just an image to have burned in your mind as you're solving problems. -248 +249 00:13:32,060 --> 00:13:35,889 And to give you one example of a problem that you can solve once this is sitting there, -249 +250 00:13:35,889 --> 00:13:39,718 burned in the back of your mind, I want you to remember back to the lecture that we did -250 +251 00:13:39,718 --> 00:13:43,460 on trigonometry, which conveniently is actually the example that I had pulled up here. -251 +252 00:13:43,520 --> 00:13:47,107 So one of the central things we were talking about was how just playing -252 +253 00:13:47,107 --> 00:13:51,590 with graphs you can get this bizarre looking fact that if you square the cosine function, -253 +254 00:13:51,590 --> 00:13:54,580 you get something that looks again just like a cosine graph. -254 +255 00:13:54,880 --> 00:13:58,733 And you can get more exact about that where if we start with an initial cosine -255 +256 00:13:58,733 --> 00:14:02,734 graph and you manipulate it a little, you know, we shift it up, we scale it down, -256 +257 00:14:02,734 --> 00:14:06,100 we say we need to double the frequency, you get the exact same graph. -257 +258 00:14:06,100 --> 00:14:08,827 So we have two different expressions for the same thing, -258 +259 00:14:08,827 --> 00:14:11,460 but it's not at all obvious why these would be related. -259 +260 00:14:11,700 --> 00:14:14,810 One of them involves doubling the angle it's a reference to, -260 +261 00:14:14,810 --> 00:14:16,800 the other involves squaring the output. -261 +262 00:14:17,440 --> 00:14:18,700 Okay, so it's a not obvious fact. -262 +263 00:14:18,780 --> 00:14:23,609 We proved it using complex numbers, but what I'd like to do is try to prove this -263 +264 00:14:23,609 --> 00:14:28,320 using geometry to kind of viscerally see the fact pop out right in front of us. -264 +265 00:14:28,720 --> 00:14:32,101 And to do that, let's go ahead and write down what the -265 +266 00:14:32,101 --> 00:14:35,360 fact is again so that we can start thinking about it. -266 -00:14:36,479 --> 00:14:40,209 +267 +00:14:36,480 --> 00:14:40,209 We want to find that the cosine squared of theta, -267 +268 00:14:40,209 --> 00:14:44,758 which is just saying take the cosine of theta and square it, -268 +269 00:14:44,758 --> 00:14:50,800 it's that awkward notation, is equal to one half of one plus cosine of two theta. -269 +270 00:14:50,820 --> 00:14:54,766 You've got the strained relationship between squaring things and doubling the angle, -270 +271 00:14:54,766 --> 00:14:58,201 which as we've talked about in the whole series is really a reflection of -271 +272 00:14:58,201 --> 00:15:01,080 the fact that a cosine is a shadow of an exponential function. -272 +273 00:15:01,400 --> 00:15:04,508 But let's say you didn't know that, we want to see it viscerally and geometrically, -273 +274 00:15:04,508 --> 00:15:07,320 so you don't already know the double angle identities or anything like that. -274 +275 00:15:07,380 --> 00:15:10,620 You just want a very direct understanding of this particular equation. -275 +276 00:15:11,540 --> 00:15:14,836 Well, one common thing that comes up is that a way to show the -276 +277 00:15:14,836 --> 00:15:18,081 two non-obvious things are related or even equal is to see if -277 +278 00:15:18,081 --> 00:15:21,640 you can find one object that you can describe in two different ways. -278 +279 00:15:21,900 --> 00:15:26,226 So we're going to look for one object that we have two different descriptions of, -279 +280 00:15:26,226 --> 00:15:29,340 and this often can give you nice equations in this context. -280 +281 00:15:29,380 --> 00:15:32,200 It might mean relating the left and right hand side. -281 +282 00:15:32,200 --> 00:15:35,894 This comes up in combinatorics all the time when you have a counting puzzle where, -282 +283 00:15:35,894 --> 00:15:39,766 you know, you do something like count how many ways you can have a string of five bits -283 +284 00:15:39,766 --> 00:15:43,549 that are either zeros or ones, and on the one hand you can count it multiplicatively -284 +285 00:15:43,549 --> 00:15:47,421 kind of going through each one and saying well you're multiplying the possibilities by -285 +286 00:15:47,421 --> 00:15:47,600 two. -286 +287 00:15:47,920 --> 00:15:51,900 But on the other hand you can go iteratively and say well how many of them have no ones? -287 +288 00:15:52,000 --> 00:15:53,560 How many of them have one one, two ones? -288 +289 00:15:53,880 --> 00:15:56,060 Something that kind of seems harder and a more awkward way to count. -289 +290 00:15:56,260 --> 00:15:58,537 But by describing the same thing twice you end up with -290 +291 00:15:58,537 --> 00:16:00,940 this really not obvious fact from an algebraic standpoint. -291 +292 00:16:00,940 --> 00:16:03,540 And we're going to do the same thing more geometrically here. -292 +293 00:16:03,940 --> 00:16:07,460 But again, just got to emphasize how like how general this ends up being. -293 +294 00:16:08,880 --> 00:16:12,080 So what we want is some kind of object that each of these describes. -294 +295 00:16:12,780 --> 00:16:17,074 And to do that, maybe we just think okay, let's let's draw a unit circle, -295 +296 00:16:17,074 --> 00:16:20,440 which is where something like a cosine typically comes up. -296 +297 00:16:22,680 --> 00:16:26,340 Oh, for the first time in my life I drew a quarter circle arc that wasn't terrible. -297 +298 00:16:26,420 --> 00:16:30,000 It's not great, but usually that comes out much more disastrous. -298 +299 00:16:31,040 --> 00:16:31,800 How pleasing. -299 +300 00:16:32,820 --> 00:16:35,940 I should not be so pleased with a terrible quarter circle. -300 +301 00:16:38,640 --> 00:16:39,460 Great, okay. -301 +302 00:16:39,560 --> 00:16:41,240 So that's our angle theta, right? -302 +303 00:16:41,460 --> 00:16:43,780 And what does cosine mean in this context? -303 +304 00:16:44,020 --> 00:16:47,140 Not cosine squared, but just plain vanilla cosine. -304 +305 00:16:47,700 --> 00:16:51,459 Well, it tells us if we look at the x-coordinate of this point, -305 +306 00:16:51,459 --> 00:16:53,280 that's the cosine of our angle. -306 +307 00:16:53,880 --> 00:16:55,953 And so now I want you to think about how can we -307 +308 00:16:55,953 --> 00:16:58,460 represent the square of the cosine as some kind of object? -308 +309 00:16:58,460 --> 00:17:02,020 Some geometric thing that we can point to in this image. -309 +310 00:17:02,580 --> 00:17:05,616 And the first instinct might be something like, well, -310 +311 00:17:05,616 --> 00:17:08,540 let's draw a square off the side of this, you know, -311 +312 00:17:08,540 --> 00:17:11,520 something with area to it and interpret it like that. -312 +313 00:17:11,880 --> 00:17:14,460 But then there's going to be a problem if the principle that we're -313 +314 00:17:14,460 --> 00:17:17,079 trying to apply is describing the same object in two different ways. -314 +315 00:17:17,339 --> 00:17:20,915 Because if we describe this left-hand side as some kind of area, -315 +316 00:17:20,915 --> 00:17:25,095 that would mean that we have to find another description of that same area, -316 +317 00:17:25,095 --> 00:17:27,460 that same object, with the right-hand side. -317 +318 00:17:27,460 --> 00:17:31,220 But that's going to be weird, because this doesn't involve squaring or anything like that. -318 +319 00:17:31,480 --> 00:17:34,920 If it's just a plain vanilla cosine term, it seems much more natural -319 +320 00:17:34,920 --> 00:17:38,360 to describe that as some kind of ratio, or maybe some kind of length. -320 +321 00:17:39,660 --> 00:17:42,260 It's just that we need the 2 theta to pop up somehow. -321 +322 00:17:42,780 --> 00:17:46,874 So instead, let's seek a way to describe cosine squared that does not involve area, -322 +323 00:17:46,874 --> 00:17:50,140 but is instead something more of the flavor of a ratio or a length. -323 +324 00:17:50,800 --> 00:17:54,320 And in this context, the key comes down to leveraging symmetry again. -324 +325 00:17:54,540 --> 00:17:55,840 And it's a sneaky bit of symmetry. -325 +326 00:17:55,840 --> 00:17:57,893 It's something we did talk about in the trig lecture, -326 +327 00:17:57,893 --> 00:17:59,680 but I love it so much I'll talk about it again. -327 +328 00:18:00,480 --> 00:18:03,632 If we think of this angle theta, on the one hand it's telling us -328 +329 00:18:03,632 --> 00:18:06,784 the angle between the x-axis and the line, but on the other hand -329 +330 00:18:06,784 --> 00:18:09,840 it's also telling us the angle between the line and the x-axis. -330 +331 00:18:10,180 --> 00:18:14,666 And I know that sounds like the same thing, but it means when I say -331 +332 00:18:14,666 --> 00:18:19,219 project down the point at the end of our radius, which was length 1, -332 +333 00:18:19,219 --> 00:18:24,300 perpendicularly onto the x-axis, that length got scaled down by cosine theta. -333 +334 00:18:24,300 --> 00:18:26,120 But now what if I do things the other way? -334 +335 00:18:26,200 --> 00:18:32,420 What if I say I want to project down in a perpendicular fashion onto this line? -335 +336 00:18:32,900 --> 00:18:36,054 Well again, I just have two lines separated by an angle theta, -336 +337 00:18:36,054 --> 00:18:40,260 I'm doing a projection, which means it gets scaled down by the cosine of that angle. -337 +338 00:18:40,540 --> 00:18:44,364 So now I'm taking the cosine of theta scaled down by the cosine of theta, -338 +339 00:18:44,364 --> 00:18:46,380 and it gets me cosine squared of theta. -339 +340 00:18:47,000 --> 00:18:52,074 So cosine squared refers to this length, a portion of the hypotenuse of our right -340 +341 00:18:52,074 --> 00:18:57,520 triangle, if that hypotenuse had a length of 1, which in our unit circle it always does. -341 +342 00:18:58,120 --> 00:19:03,136 And incidentally, you can show very similar reasoning that the sine of theta is this -342 +343 00:19:03,136 --> 00:19:07,091 other other portion of that hypotenuse, and this gives you a nice, -343 +344 00:19:07,091 --> 00:19:10,101 sort of a clever proof of the Pythagorean theorem, -344 +345 00:19:10,101 --> 00:19:13,465 the idea of double projecting based on asking, you know, -345 +346 00:19:13,465 --> 00:19:18,660 is this line theta degrees away from that, or is that line theta degrees away from this? -346 +347 00:19:18,800 --> 00:19:20,736 Sounds like you're saying the same thing, but it gets you -347 +348 00:19:20,736 --> 00:19:22,840 something of mathematical substance to recognize that symmetry. -348 -00:19:23,399 --> 00:19:24,540 +349 +00:19:23,400 --> 00:19:24,540 Now, why do I say this? -349 +350 00:19:24,800 --> 00:19:27,420 Well, we have a representation of cosine squared. -350 +351 00:19:27,600 --> 00:19:30,460 What we want now is to do something in terms of 2 theta. -351 +352 00:19:30,960 --> 00:19:36,422 We just were thinking about a context where we're able to relate an angle to 2 theta, -352 +353 00:19:36,422 --> 00:19:37,820 to 2 times that angle. -353 +354 00:19:38,120 --> 00:19:43,880 So somehow we want to realize this angle as an inscribed angle of some kind of triangle. -354 +355 00:19:44,820 --> 00:19:46,840 And I'll show you how you can do that. -355 +356 00:19:47,300 --> 00:19:51,599 But before I do, just to mention another fact that if you've been puzzling -356 +357 00:19:51,599 --> 00:19:55,383 around with these sorts of things, might be burning in your mind, -357 +358 00:19:55,383 --> 00:19:59,740 is a specific instance of the inscribed angle theorem called Thales theorem. -358 +359 00:20:00,200 --> 00:20:05,600 So let's say that that large angle we had, 2 theta, was actually 180 degrees, right? -359 +360 00:20:05,660 --> 00:20:06,660 It was pi radians. -360 +361 00:20:07,460 --> 00:20:09,900 What would that mean in terms of the inscribed angle theorem? -361 +362 00:20:10,800 --> 00:20:15,383 It means that if we take that diameter of the circle, if it's 180 degrees, -362 +363 00:20:15,383 --> 00:20:20,028 it's just drawing out a diameter, and we have an inscribed angle with lines -363 +364 00:20:20,028 --> 00:20:25,040 that hit either end of that diameter, then this angle is necessarily half of that. -364 +365 00:20:25,260 --> 00:20:29,126 So what it means is that we can put a right triangle inside a circle, -365 +366 00:20:29,126 --> 00:20:33,931 and whenever you do that, the hypotenuse of the right triangle is exactly the diameter -366 +367 00:20:33,931 --> 00:20:34,760 of that circle. -367 +368 00:20:35,120 --> 00:20:36,020 It's a very cute fact. -368 +369 00:20:36,520 --> 00:20:41,172 If you wanted another proof of it, that's not just via the inscribed angle theorem, -369 +370 00:20:41,172 --> 00:20:45,104 there's another very wonderful leveraging of symmetry that you can do, -370 +371 00:20:45,104 --> 00:20:47,818 where basically you take this point and you say, -371 +372 00:20:47,818 --> 00:20:52,692 I'm going to reflect it through the origin, reflect it through the center of my circle, -372 +373 00:20:52,692 --> 00:20:53,800 and see where I get. -373 +374 00:20:54,620 --> 00:20:56,717 And recognize that reflecting through the center -374 +375 00:20:56,717 --> 00:20:58,900 is the same as rotating the whole image 90 degrees. -375 +376 00:20:59,560 --> 00:21:03,170 So if I rotated the image, not 90, 180 degrees, -376 +377 00:21:03,170 --> 00:21:06,780 my other vertex would end up at that same point. -377 +378 00:21:07,220 --> 00:21:11,114 But now what we have is a quadrilateral, and one of the diagonals is the diameter -378 +379 00:21:11,114 --> 00:21:14,439 of the circle, but the other diagonal is also diameter of the circle, -379 +380 00:21:14,439 --> 00:21:18,428 which in particular means they have the same midpoint and they're the same distance -380 +381 00:21:18,428 --> 00:21:22,370 apart, and you can convince yourself a little that implies it must be a rectangle, -381 +382 00:21:22,370 --> 00:21:26,360 that could also be a little side homework problem if you wanted to chase around the -382 +383 00:21:26,360 --> 00:21:27,120 relevant angles. -383 +384 00:21:27,280 --> 00:21:30,455 But I think that's a very beautiful way to think about Thale's theorem, -384 +385 00:21:30,455 --> 00:21:33,541 that you reflect everything 180 degrees, and you necessarily conclude -385 +386 00:21:33,541 --> 00:21:36,320 it must be a rectangle, which means that this is a right angle. -386 +387 00:21:36,940 --> 00:21:38,440 Now for our purposes, what does that mean? -387 +388 00:21:38,780 --> 00:21:41,512 Well, we've got a right triangle sitting here that's from zero, -388 +389 00:21:41,512 --> 00:21:44,160 we've got one of the points here, another point on the circle. -389 +390 00:21:44,360 --> 00:21:47,180 Let's inscribe that in a separate circle, okay? -390 +391 00:21:47,840 --> 00:21:51,608 So I'm going to take a copy of that triangle, but I'm going to, -391 +392 00:21:51,608 --> 00:21:54,965 instead of making the hypotenuse a radius of the circle, -392 +393 00:21:54,965 --> 00:21:58,440 I'm going to make that hypotenuse a diameter of the circle. -393 +394 00:21:59,320 --> 00:22:03,480 So this is still going to be an angle of theta, sitting right here. -394 +395 00:22:04,080 --> 00:22:07,645 The, uh, basically I flipped it around, so previously it was here, -395 +396 00:22:07,645 --> 00:22:12,063 but I flipped it around so that my 90 degree angle is sitting up and to the right, -396 +397 00:22:12,063 --> 00:22:13,660 rather than sitting down here. -397 +398 00:22:14,620 --> 00:22:19,939 The length that we care about is what happens when we project from the point at that 90 -398 +399 00:22:19,939 --> 00:22:23,928 degree angle in a perpendicular fashion down onto the hypotenuse, -399 +400 00:22:23,928 --> 00:22:25,500 which now looks like this. -400 +401 00:22:26,380 --> 00:22:31,240 And what we care about is this long length, okay? -401 +402 00:22:32,160 --> 00:22:36,256 And actually, let me ask you as a live quiz, to see if you can come up with an -402 +403 00:22:36,256 --> 00:22:40,560 expression for that length in the context of the diagram that we're now looking at. -403 +404 00:22:40,560 --> 00:22:45,840 So, pulling up our quiz again, congratulations to everyone who got this one correct. -404 +405 00:22:47,420 --> 00:22:50,773 Let me give you a little bit of time to think about this one, -405 +406 00:22:50,773 --> 00:22:55,640 because this is, uh, this is kind of a heart, part of the heart of this particular proof, -406 +407 00:22:55,640 --> 00:22:58,020 and I think it's very, very pleasing to see. -407 +408 00:22:58,640 --> 00:23:02,340 So it specifies that the hypotenuse of the large right triangle above has a length of one. -408 +409 00:23:03,040 --> 00:23:06,415 Our context is because it came from the hypotenuse of the triangle -409 +410 00:23:06,415 --> 00:23:09,540 drawn in a unit circle, so the hypotenuse has a length of one. -410 +411 00:23:09,940 --> 00:23:13,180 What is the length l in terms of theta? -411 +412 00:23:13,760 --> 00:23:16,800 Okay, so can you find an expression for l in terms of theta? -412 +413 00:23:17,820 --> 00:23:21,122 And I'll give you a little bit of, uh, a little bit of time for that, -413 +414 00:23:21,122 --> 00:23:24,660 bring back our pause and ponder music, get myself a chance to take a drink. -414 +415 00:23:49,620 --> 00:23:52,280 So, once again, it looks like we have some strong consensus for today. -415 +416 00:23:52,780 --> 00:23:54,800 So while answers are rolling in, before I grade it, -416 +417 00:23:54,800 --> 00:23:57,170 I'm just going to go ahead and start describing how it goes, -417 +418 00:23:57,170 --> 00:23:59,580 since it seems like a lot of you are already well ahead of me. -418 +419 00:24:00,140 --> 00:24:01,020 It's fun to have though. -419 +420 00:24:01,620 --> 00:24:03,460 So of course, we're going to use the inscribed angle theorem. -420 +421 00:24:03,520 --> 00:24:05,740 That is the whole reason I'm bringing it up here. -421 +422 00:24:05,740 --> 00:24:08,360 It's a way to relate a single angle to twice that angle. -422 +423 00:24:08,560 --> 00:24:13,398 So in this context, I would draw some lines from the center of my, uh, -423 +424 00:24:13,398 --> 00:24:17,214 of my new smaller circle that has radius only one half, -424 +425 00:24:17,214 --> 00:24:19,600 and recognize that this is 2 theta. -425 +426 00:24:20,100 --> 00:24:21,940 And this is just lovely now, isn't it? -426 +427 00:24:21,960 --> 00:24:23,480 Because what is the length we care about? -427 +428 00:24:24,300 --> 00:24:30,423 Part of it, excuse me, part of it is the radius, which is one half, -428 +429 00:24:30,423 --> 00:24:36,726 but then the projection down according to 2 theta onto this remaining -429 +430 00:24:36,726 --> 00:24:42,760 leg ends up being that radius one half times the cosine of 2 theta. -430 +431 00:24:43,520 --> 00:24:45,780 And of course, that's exactly what we want it to be. -431 +432 00:24:45,900 --> 00:24:48,321 We've got the whole radius, which is one half, -432 +433 00:24:48,321 --> 00:24:51,620 and then we're multiplying that by 1 plus the cosine of 2 theta. -433 +434 00:24:51,620 --> 00:24:55,100 It's a radius of the circle times a scaled down version of that radius. -434 +435 00:24:55,560 --> 00:24:58,061 So that's two different ways of viewing the same object, -435 +436 00:24:58,061 --> 00:25:00,739 which is what we get when we take this right triangle and we -436 +437 00:25:00,739 --> 00:25:03,680 project down and look at what part of the hypotenuse that cuts off. -437 +438 00:25:03,740 --> 00:25:06,769 And it gives us this non-trivial relationship in trigonometry -438 +439 00:25:06,769 --> 00:25:09,360 between the cosine squared and the cosine of 2 theta. -439 +440 00:25:09,460 --> 00:25:11,622 The fact that otherwise we were going into like -440 +441 00:25:11,622 --> 00:25:13,740 complex numbers and exponentials to understand. -441 +442 00:25:13,960 --> 00:25:15,080 So I think that's quite beautiful. -442 +443 00:25:15,140 --> 00:25:18,816 I think that's just, um, one of the many many instances of where -443 +444 00:25:18,816 --> 00:25:22,720 the inscribed angle theorem suspiciously slos- suspiciously shows up. -444 +445 00:25:22,720 --> 00:25:26,754 Um, and again, what I want you to take away is this principle that if you -445 +446 00:25:26,754 --> 00:25:29,644 can have one object described in two different ways, -446 +447 00:25:29,644 --> 00:25:32,424 very powerful in terms of showing non-obvious, um, -447 +448 00:25:32,424 --> 00:25:36,404 algebraic relations or anything that's kind of written down symbolically -448 +449 00:25:36,404 --> 00:25:38,640 without immediate intuition on top of it. -449 +450 00:25:39,060 --> 00:25:41,828 So with all of that, let's actually turn to the probability -450 +451 00:25:41,828 --> 00:25:44,320 question that I asked at the beginning of the lecture. -451 +452 00:25:47,280 --> 00:25:51,721 So going back to our live quiz, uh, I will go ahead -452 +453 00:25:51,721 --> 00:25:56,420 and grade what we- we all know know the correct answer. -453 +454 00:25:56,740 --> 00:25:59,180 So for those of you- wait a minute. -454 +455 00:26:00,420 --> 00:26:01,620 Oh, it was marked incorrectly. -455 +456 00:26:02,240 --> 00:26:02,840 Oh, that's my bad. -456 +457 00:26:03,680 --> 00:26:06,880 No, I just slipped this one in like last minute before the lesson today. -457 +458 00:26:06,880 --> 00:26:09,260 Um, so I might have like swapped around what the answers were. -458 +459 00:26:09,780 --> 00:26:13,143 So, uh, well, it looks like 1380 if we were absolutely -459 +460 00:26:13,143 --> 00:26:16,140 wrong according to whatever jerk wrote this quiz. -460 +461 00:26:17,280 --> 00:26:22,594 So, I don't know how that shows up on the user interface if it's like shocking red like, -461 +462 00:26:22,594 --> 00:26:25,760 oh no, but, uh, D was the actual correct answer here. -462 +463 00:26:25,760 --> 00:26:27,860 So congratulations to those of you who got that. -463 +464 00:26:28,480 --> 00:26:29,720 Now back to our probability question. -464 +465 00:26:30,520 --> 00:26:35,180 I'm curious to see what people said on this one just in terms of their instincts. -465 +466 00:26:35,260 --> 00:26:39,311 So this one we had a little bit more of a spread and, ah, interesting, -466 +467 00:26:39,311 --> 00:26:44,161 here the actual correct answer does show up quite a bit- quite a bit lower than- and -467 +468 00:26:44,161 --> 00:26:47,813 now I'm questioning myself to make sure that I've actually, um, -468 +469 00:26:47,813 --> 00:26:49,640 written the thing appropriately. -469 +470 00:26:50,060 --> 00:26:52,190 So just as a reminder of what the question is, -470 +471 00:26:52,190 --> 00:26:55,226 we're choosing two random numbers from the range zero through one, -471 +472 00:26:55,226 --> 00:26:58,308 each according to a uniform distribution, and we're- we're guessing -472 +473 00:26:58,308 --> 00:27:01,480 what the probability that they round down- the ratio of these numbers -473 +474 00:27:01,480 --> 00:27:02,840 rounds down to an even number. -474 +475 00:27:02,960 --> 00:27:05,272 Remember zero is an even number, so that it rounds -475 +476 00:27:05,272 --> 00:27:07,540 down to zero or two or four or anything like that. -476 +477 00:27:08,180 --> 00:27:13,829 Now this is a tricky problem to think about and, uh, definitely no- no fault at all for, -477 +478 00:27:13,829 --> 00:27:18,400 uh, anyone who isn't immediately able to see roughly where it should be. -478 +479 00:27:18,520 --> 00:27:21,662 But I think, um, with a- with a little bit of progress on our way, -479 +480 00:27:21,662 --> 00:27:24,804 even before we get the exact solution, we can get to a point where -480 +481 00:27:24,804 --> 00:27:28,040 you might be able to intuitively give some kind of ballpark estimate. -481 +482 00:27:28,680 --> 00:27:29,820 So what have we got here? -482 +483 00:27:30,700 --> 00:27:33,200 Choosing two random numbers between zero and one. -483 +484 00:27:33,360 --> 00:27:35,603 I think that's a weird thing to think about, um, -484 +485 00:27:35,603 --> 00:27:38,533 especially if you're not familiar with probability that well or -485 +486 00:27:38,533 --> 00:27:42,380 when the phrase uniform distribution is thrown up if it's not clear what that means. -486 +487 00:27:43,060 --> 00:27:47,184 Um, but essentially, uh, it's what you would expect where you're choosing some -487 +488 00:27:47,184 --> 00:27:51,569 random point on this line, and the idea is that each point is as likely as another, -488 +489 00:27:51,569 --> 00:27:54,911 or more specifically a given range of points of a certain size, -489 +490 00:27:54,911 --> 00:27:59,140 should have a given probability that's independent of where that range showed up. -490 +491 00:27:59,140 --> 00:28:00,560 It's only dependent on its size. -491 +492 00:28:00,980 --> 00:28:03,660 You know, so you might have in the back of your mind the idea that -492 +493 00:28:03,660 --> 00:28:06,420 we've chosen two points, they're each somewhere between zero and one. -493 +494 00:28:06,740 --> 00:28:12,780 And just to give an example of what we mean by uniform distribution, -494 +495 00:28:12,780 --> 00:28:17,420 the probability that x sits between 0.3 and like 0.5. -495 +496 00:28:18,520 --> 00:28:21,530 Because it's going to be somewhere between zero and one, -496 +497 00:28:21,530 --> 00:28:25,490 and the length of that range is 0.2, about a fifth of the entire length it -497 +498 00:28:25,490 --> 00:28:26,600 could have come from. -498 +499 00:28:26,720 --> 00:28:29,696 What it means to be uniform is that that probability is -499 +500 00:28:29,696 --> 00:28:32,780 actually just the length of the segment that it came from. -500 -00:28:33,679 --> 00:28:36,386 +501 +00:28:33,680 --> 00:28:36,386 Now, one thing you might ask is, well, what if, -501 +502 00:28:36,386 --> 00:28:40,220 what about the probability that it's precisely 0.3 or precisely 0.5? -502 +503 00:28:40,300 --> 00:28:42,480 Would it matter if we made these less than or equal to signs? -503 +504 00:28:43,380 --> 00:28:46,557 And the answer is it doesn't actually matter, because the probability -504 +505 00:28:46,557 --> 00:28:49,780 of hitting any specific value on a real number line ends up being zero. -505 +506 00:28:50,380 --> 00:28:52,140 This is a thing many find very confusing. -506 +507 00:28:52,260 --> 00:28:55,134 How can a probability of an event that's possible, -507 +508 00:28:55,134 --> 00:28:58,460 it's definitely possible to hit 0.3, have probability zero? -508 +509 00:28:58,840 --> 00:29:01,637 Made a whole video about it trying to describe this, -509 +510 00:29:01,637 --> 00:29:05,595 but really what it comes down to is that the things that have probability, -510 +511 00:29:05,595 --> 00:29:07,020 you should think of ranges. -511 +512 00:29:07,380 --> 00:29:10,507 Those are the fundamental objects, and it doesn't really matter -512 +513 00:29:10,507 --> 00:29:13,440 how we treat the boundary and just think in terms of ranges. -513 +514 00:29:14,160 --> 00:29:17,748 Even still though, what we're asking is a very bizarre question, -514 +515 00:29:17,748 --> 00:29:22,274 which is if we take the ratio of x and y and we round that down, which sometimes, -515 +516 00:29:22,274 --> 00:29:25,476 you know, mathematicians write using this floor function, -516 +517 00:29:25,476 --> 00:29:28,512 saying we find the greatest integer smaller than that, -517 +518 00:29:28,512 --> 00:29:30,720 how do we know if that's an even number? -518 +519 00:29:30,740 --> 00:29:33,380 That's a weird thing to think about, it's a hard problem in that way. -519 +520 00:29:34,040 --> 00:29:36,656 So, you know, if you think through the principles here, -520 +521 00:29:36,656 --> 00:29:39,320 it's kind of like use the defining features of the setup. -521 +522 00:29:39,880 --> 00:29:42,720 Well, not clear how to use the fact that it's a uniform distribution. -522 +523 00:29:42,840 --> 00:29:46,234 I guess we'll be using lengths in some way to yield probabilities, -523 +524 00:29:46,234 --> 00:29:49,680 so maybe that gives us some geometry, but that's not really helpful. -524 +525 00:29:49,800 --> 00:29:52,570 Give things meaningful names, you know, maybe x and y -525 +526 00:29:52,570 --> 00:29:55,340 are meaningful or some something suggestive like that. -526 +527 00:29:55,660 --> 00:30:00,000 Symmetry, okay, maybe, you know, the idea that choosing x and then y is -527 +528 00:30:00,000 --> 00:30:04,400 as likely as choosing y than x, you could use that to conclude that this -528 +529 00:30:04,400 --> 00:30:08,560 ratio x over y is as likely to be above one as it is to be below one. -529 +530 00:30:09,160 --> 00:30:13,136 And that actually does tell you something, because if we're wondering how -530 +531 00:30:13,136 --> 00:30:17,918 often do you round down to be zero, right, you can say well x over y is as likely to be, -531 +532 00:30:17,918 --> 00:30:21,680 x is as likely to be bigger than y as y is likely to be bigger than x. -532 +533 00:30:21,860 --> 00:30:24,420 So there's a 50-50 chance that this should happen. -533 +534 00:30:24,880 --> 00:30:30,060 This would be a probability of 50 percent. -534 +535 00:30:30,260 --> 00:30:32,504 So that gets you somewhere, which is kind of nice, -535 +536 00:30:32,504 --> 00:30:36,288 but it's not clear how you would apply that to things like even numbers, same object, -536 +537 00:30:36,288 --> 00:30:37,520 two different ways, unclear. -537 +538 00:30:38,660 --> 00:30:42,280 So principle number five here is where we're going to come in. -538 +539 00:30:42,720 --> 00:30:43,740 Again, it seems simple. -539 +540 00:30:43,820 --> 00:30:46,066 It's something that you can not be like, yeah, yeah, -540 +541 00:30:46,066 --> 00:30:48,780 drawing pictures, it helps to think through what I'm looking at. -541 +542 00:30:48,960 --> 00:30:52,854 But really, when you find yourself struggling with some setup that's not already -542 +543 00:30:52,854 --> 00:30:56,364 visual or pictorial, you know, it doesn't have to be making a geometric, -543 +544 00:30:56,364 --> 00:31:00,355 but just having some kind of sketch, uh, to give meaning to your terms can be very -544 +545 00:31:00,355 --> 00:31:00,740 helpful. -545 +546 00:31:00,880 --> 00:31:04,637 And as a more specific problem solving tip, when you have some numbers, -546 +547 00:31:04,637 --> 00:31:08,760 multiple different numbers, see if you can make them coordinates in some space. -547 +548 00:31:08,900 --> 00:31:11,369 So rather than thinking about x and y as separate things here, -548 +549 00:31:11,369 --> 00:31:13,760 we'll want to think about a single point with xy coordinates. -549 +550 00:31:14,440 --> 00:31:16,569 And what that does for us is it actually turns the -550 +551 00:31:16,569 --> 00:31:18,740 whole problem two-dimensional in a very helpful way. -551 +552 00:31:19,340 --> 00:31:22,495 So first of all, I've lost track of my straight edge, -552 +553 00:31:22,495 --> 00:31:25,300 which, where have you gone little straight edge? -553 +554 00:31:25,960 --> 00:31:27,180 I can only throw you so far. -554 +555 00:31:27,500 --> 00:31:27,960 Oh, here we go. -555 +556 00:31:29,800 --> 00:31:31,060 Things, they run away from you. -556 +557 00:31:32,100 --> 00:31:38,519 Even your objects sometimes get tired of math class and want to play truant now and then, -557 +558 00:31:38,519 --> 00:31:41,800 but he has to stay whether he likes to or not. -558 +559 00:31:45,700 --> 00:31:46,140 -Absurd. +All right. All right, absurd. -559 +560 00:31:46,700 --> 00:31:49,040 So let's say this is our x coordinate. -560 -00:31:49,679 --> 00:31:52,920 +561 +00:31:49,680 --> 00:31:52,920 x can fall anywhere between 0 and 1 with uniform probability. -561 +562 00:31:53,820 --> 00:31:55,180 y can fall between 0 and 1. -562 +563 00:31:55,400 --> 00:32:00,222 So when we have a pair of numbers, you know, something like 0.2, what is that, -563 +564 00:32:00,222 --> 00:32:04,740 maybe like 0.8, pair of numbers, it's just a single point in this diagram. -564 +565 00:32:05,280 --> 00:32:08,985 And now to choose both of those numbers uniformly at random means that -565 +566 00:32:08,985 --> 00:32:12,900 we're choosing a random point inside a square, a square with side length 1. -566 +567 00:32:13,840 --> 00:32:16,459 And now maybe we can make a little bit of progress, -567 +568 00:32:16,459 --> 00:32:20,440 because it's going to come down to some view of what's going on in this square. -568 +569 00:32:21,140 --> 00:32:26,449 Now with this specific example, if we're thinking about the ratio x over y, -569 +570 00:32:26,449 --> 00:32:32,038 and taking its floor, taking, just rounding it down, well, that's 0.2 over 0.8, -570 +571 00:32:32,038 --> 00:32:36,580 that's going to round down to 0, so this would end up being even. -571 +572 00:32:37,220 --> 00:32:40,000 And like I just said, that that happens with 50% probability. -572 +573 00:32:40,400 --> 00:32:43,346 But let's see if we can try to find a way of thinking -573 +574 00:32:43,346 --> 00:32:46,020 about that that generalizes up to other examples. -574 +575 00:32:46,460 --> 00:32:50,130 And again, one very useful thing, if you get stuck, -575 +576 00:32:50,130 --> 00:32:54,224 that I have enumerated down here as principle number six, -576 +577 00:32:54,224 --> 00:32:57,260 is to ask a simpler variant of the problem. -577 +578 00:32:57,740 --> 00:32:59,160 You're solving something, it's hard. -578 +579 00:32:59,460 --> 00:33:00,240 It's too hard. -579 +580 00:33:00,660 --> 00:33:02,370 See if you can make it simpler in a way that you -580 +581 00:33:02,370 --> 00:33:04,080 actually can solve and get some kind of foothold. -581 +582 00:33:04,440 --> 00:33:07,981 Maybe that means loosening the constraints of the problem in some setups, -582 +583 00:33:07,981 --> 00:33:10,040 or maybe it means looking at a sub-problem. -583 +584 00:33:10,420 --> 00:33:14,742 So in our context, rather than asking the probability that it rounds down to be even, -584 +585 00:33:14,742 --> 00:33:17,405 let me just ask the probability that it becomes one, -585 +586 00:33:17,405 --> 00:33:19,817 but I want you to answer it in a geometric way, -586 +587 00:33:19,817 --> 00:33:22,833 something that will generalize to rounding to other things, -587 +588 00:33:22,833 --> 00:33:27,004 because the next simpler question might be probability that it rounds down to two, -588 +589 00:33:27,004 --> 00:33:28,060 and things like that. -589 +590 00:33:28,520 --> 00:33:29,460 You know where this is going to go. -590 +591 00:33:29,540 --> 00:33:33,041 We're going to do it as a live quiz because rather than me answering things, -591 +592 00:33:33,041 --> 00:33:34,860 I want you guys to answer things for me. -592 +593 00:33:35,560 --> 00:33:38,235 So jumping up to question number four at this point, -593 +594 00:33:38,235 --> 00:33:40,760 it's going to give us a couple different diagrams. -594 -00:33:41,219 --> 00:33:46,256 +595 +00:33:41,220 --> 00:33:46,256 Okay, we've got a, b, c, and d, and it's asking us which of these four regions -595 +596 00:33:46,256 --> 00:33:51,228 corresponds to values of x and y, where taking the floor of x divided by y is -596 +597 00:33:51,228 --> 00:33:56,520 equal to zero, which is to say you take the ratio, you round it down, you get zero. -597 +598 00:33:56,720 --> 00:33:58,920 Which region corresponds to that fact? -598 -00:34:16,939 --> 00:34:26,233 +599 +00:34:16,940 --> 00:34:26,233 So, okay, I'm going to go ahead and lock in answers, -599 +600 00:34:26,233 --> 00:34:39,033 but if you want to keep thinking about it, definitely feel free to pause -600 +601 00:34:39,033 --> 00:34:42,540 the video and do so. -601 +602 00:34:42,580 --> 00:34:43,500 I don't want to rush anyone. -602 +603 00:34:44,460 --> 00:34:50,140 So it looks like 1265 of you, 1273, always answers rolling in at the end, -603 +604 00:34:50,140 --> 00:34:52,520 correctly answered that it's c. -604 +605 00:34:52,900 --> 00:34:55,620 And let's take a moment to think about why that's the case. -605 +606 00:34:56,179 --> 00:34:59,864 Okay, you can do so just with a pile of examples and just see which one -606 +607 00:34:59,864 --> 00:35:03,549 seemed to fall in the region or not, but let's see if we can understand -607 +608 00:35:03,549 --> 00:35:07,080 this in a way that lets us make progress onto the other even numbers. -608 +609 00:35:07,080 --> 00:35:10,212 So when we say that it rounds down to zero, what we're basically -609 +610 00:35:10,212 --> 00:35:13,200 saying is that that ratio sits somewhere between zero and one. -610 +611 00:35:13,920 --> 00:35:18,770 And it's awkward to think of x divided by y, we kind of like to think of y in terms of x, -611 +612 00:35:18,770 --> 00:35:23,513 so if I multiply everything by y, which is okay to do with these inequalities because y -612 +613 00:35:23,513 --> 00:35:28,203 is always positive, so that's not going to affect whether the inequality flips one way -613 +614 00:35:28,203 --> 00:35:33,000 or another, I multiply everything by y, and we're basically asking when is x less than y? -614 +615 00:35:33,340 --> 00:35:36,014 And whenever you see an inequality, the boundary of -615 +616 00:35:36,014 --> 00:35:38,740 that region is going to be described by the equality. -616 +617 00:35:39,100 --> 00:35:42,300 So we're going to wonder when is it the case that y equals x? -617 +618 00:35:42,960 --> 00:35:44,900 Well, that's just a straight line that goes diagonally. -618 +619 00:35:45,260 --> 00:35:53,420 If we draw our line y equals x, that's what we get. -619 +620 00:35:53,960 --> 00:35:56,961 Now that's the boundary of our region, and to know whether we should -620 +621 00:35:56,961 --> 00:36:00,877 look to the left of it or to the right of it, either you can think very directly and say, -621 +622 00:36:00,877 --> 00:36:03,791 well, you know, y should be greater than x, so at a given point we -622 +623 00:36:03,791 --> 00:36:05,880 want to move upward in the positive y direction. -623 +624 00:36:05,880 --> 00:36:09,969 You could look at a specific example like this, but however you do it, -624 +625 00:36:09,969 --> 00:36:14,404 you'll draw the conclusion that geometrically the region of points such that -625 +626 00:36:14,404 --> 00:36:19,300 x divided by y rounds down to zero is this sort of grilled cheese cut of our diagram. -626 +627 00:36:20,260 --> 00:36:25,147 So with that, maybe you can start to think about the harder variant, -627 +628 00:36:25,147 --> 00:36:29,540 which is when is it that x divided by y rounds down to be two? -628 +629 00:36:30,200 --> 00:36:32,580 And again, I don't want to answer it. -629 +630 00:36:32,640 --> 00:36:33,600 I want you to answer it. -630 +631 00:36:34,820 --> 00:36:39,075 When is it that x divided by y inside our unit -631 +632 00:36:39,075 --> 00:36:43,240 square of points x comma y rounds down to two? -632 +633 00:36:43,660 --> 00:36:48,514 And we've got four possible geometric regions that this could correspond to, -633 +634 00:36:48,514 --> 00:36:49,460 a, b, c, and d. -634 +635 00:36:50,080 --> 00:36:53,764 And really, you know, rather than just thinking about which of these is it, -635 +636 00:36:53,764 --> 00:36:57,060 really try to think through why it's the case and how you can prove -636 +637 00:36:57,060 --> 00:37:00,260 that one of these boundaries is actually what it's supposed to be. -637 +638 00:37:00,440 --> 00:37:04,004 For example, I want it to be the case that if I didn't show the correct answer here, -638 +639 00:37:04,004 --> 00:37:07,442 let's say I was just trolling with everyone and I didn't show the correct answer, -639 +640 00:37:07,442 --> 00:37:09,791 you would be able to confidently come and say like, no, -640 +641 00:37:09,791 --> 00:37:13,020 I'm quite positive that the correct answer is nothing that you've shown here. -641 +642 00:37:13,600 --> 00:37:16,180 See if that's the level of reasoning that you can put behind it. -642 +643 00:37:16,880 --> 00:37:19,580 So again, I'll give you give you a little moment to think about that. -643 +644 00:37:24,480 --> 00:37:47,991 So so Okay, so once again, I'm going to lock in answers potentially -644 +645 00:37:47,991 --> 00:38:10,120 earlier than you want me to, but keep the lesson moving forward. -645 +646 00:38:10,380 --> 00:38:13,082 No hard feelings if things haven't clicked yet, -646 +647 00:38:13,082 --> 00:38:16,180 because hopefully the explanation will make them do so. -647 +648 00:38:16,560 --> 00:38:19,564 So the correct answer is c, which it looks like most of you got, -648 +649 00:38:19,564 --> 00:38:22,200 and let's go ahead and think through why that's the case. -649 +650 00:38:22,200 --> 00:38:24,600 Very similar reasoning to what we were just doing. -650 +651 00:38:24,740 --> 00:38:28,016 The idea is that rather than thinking about this ratio and a floor, -651 +652 00:38:28,016 --> 00:38:32,208 which is kind of a kind of an awkward thing, let's explicitly write out the inequality -652 +653 00:38:32,208 --> 00:38:33,220 this is referring to. -653 +654 00:38:33,660 --> 00:38:37,968 It's saying that x divided by y is greater than or equal to 2 if it's -654 +655 00:38:37,968 --> 00:38:42,400 rounding down to that, but it's not greater than 3, so it's less than 3. -655 +656 00:38:43,900 --> 00:38:48,080 And, you know, again, it's a little bit awkward to think of this ratio. -656 +657 00:38:48,080 --> 00:38:52,505 So let's write that as 2 times y is less than or equal to x, -657 +658 00:38:52,505 --> 00:38:55,480 which is less than or equal to 3 times y. -658 +659 00:38:56,660 --> 00:38:59,304 Now quite often we don't think of y as a function of x, -659 +660 00:38:59,304 --> 00:39:02,800 we think of x as a function of y, if that makes you feel more comfortable. -660 +661 00:39:03,340 --> 00:39:07,340 So if you want in the back of your mind, you can kind of think 2y less than or equal to x. -661 +662 00:39:07,500 --> 00:39:11,660 Well, that's the same thing as saying y is less than or equal to x halves. -662 +663 00:39:12,300 --> 00:39:16,506 And same deal, 3y being greater than x, that's the -663 +664 00:39:16,506 --> 00:39:20,960 same thing as saying y is greater than x divided by 3. -664 +665 00:39:21,360 --> 00:39:24,680 Because that way we can look at the equalities associated with each of these. -665 +666 00:39:24,880 --> 00:39:28,220 The line y equals x halves, which has a slope of one half, -666 +667 00:39:28,220 --> 00:39:33,260 you can think of it as intersecting at the point where y equals one half when x equals 1. -667 +668 00:39:34,280 --> 00:39:38,800 Right, so it'll be a line like this that describes part of the boundary of our region. -668 +669 00:39:39,740 --> 00:39:42,780 And the other line is when y is equal to x thirds. -669 +670 00:39:43,600 --> 00:39:48,990 So we know we actually have to be above this line that I'm about to draw, -670 +671 00:39:48,990 --> 00:39:54,600 where one of these represents x over 2, and one of these represents x over 3. -671 +672 00:39:55,340 --> 00:39:59,320 And then part of the part of the intrusion into the space of my last inequality. -672 +673 00:40:00,320 --> 00:40:04,620 All right, so we want to be above the x equals x over 3 below the x divided by 2. -673 +674 00:40:05,000 --> 00:40:09,960 This region here shows us everything where rounding down gets to 2. -674 +675 00:40:10,360 --> 00:40:14,453 And I want you to appreciate how this is a kind of complicated thing to -675 +676 00:40:14,453 --> 00:40:18,660 think about if we hadn't gone into a picture that involves two dimensions. -676 +677 00:40:18,660 --> 00:40:23,220 If you were just thinking of x and y varying along this line and wondering when -677 +678 00:40:23,220 --> 00:40:27,780 is it the case that x is more than twice, or y is more than two times what x is. -678 +679 00:40:27,980 --> 00:40:28,660 No, yeah, sorry. -679 +680 00:40:28,780 --> 00:40:32,640 x is more than two times what y is, but it's not three times more than what y is. -680 -00:40:33,940 --> 00:40:35,320 +681 +00:40:33,940 --> 00:40:36,040 It's not, not, no, I said it wrong. -681 -00:40:35,440 --> 00:40:37,140 +682 +00:40:36,720 --> 00:40:37,140 I said it wrong. -682 -00:40:37,759 --> 00:40:42,080 +683 +00:40:37,760 --> 00:40:42,080 When y is more than two times what x is, but it's not three times more than x is. -683 +684 00:40:42,340 --> 00:40:44,620 -It's very easy to get fuddled in that way. +It's very, it's very easy to get confuddled in that way. -684 +685 00:40:45,280 --> 00:40:48,357 And it's even harder to try to give some sort of probability to that, -685 +686 00:40:48,357 --> 00:40:50,600 whereas in our diagram it has a very clear meaning. -686 +687 00:40:50,880 --> 00:40:56,526 It is the area of this region because the full area of possibilities already is one, -687 +688 00:40:56,526 --> 00:41:00,710 so this, the probability of something happening should be one, -688 +689 00:41:00,710 --> 00:41:03,700 and we just need to look at the area of that. -689 +690 00:41:04,140 --> 00:41:07,939 And now maybe you can see where this is going to go, -690 +691 00:41:07,939 --> 00:41:12,956 because for the next term when we want to know when does x divided by -691 +692 00:41:12,956 --> 00:41:17,687 y sit between four and five, we're going to be drawing lines that -692 +693 00:41:17,687 --> 00:41:19,480 intersect at x over four. -693 +694 00:41:19,600 --> 00:41:21,100 Maybe I'll go to a different color for this one. -694 +695 00:41:23,600 --> 00:41:27,743 x over four and x over five, which is going to require very small -695 +696 00:41:27,743 --> 00:41:32,200 handwriting at this point, but I'm going to give it a try nevertheless. -696 +697 00:41:33,660 --> 00:41:38,956 x fourths, y equals x fifths, and this little sliver of area -697 +698 00:41:38,956 --> 00:41:44,860 gives us all of the times that our ratio x over y rounds to be four. -698 +699 00:41:45,620 --> 00:41:47,670 And we're going to have to add infinitely many of these, -699 +700 00:41:47,670 --> 00:41:50,080 so that gives us sort of another phase of challenge to the problem. -700 +701 00:41:50,660 --> 00:41:53,089 I've drawn this all out in Desmos, by the way, -701 +702 00:41:53,089 --> 00:41:56,706 if you want to just sort of see what some of these regions look like, -702 +703 00:41:56,706 --> 00:42:00,117 where we've got our top region of places where it rounds to zero, -703 +704 00:42:00,117 --> 00:42:04,355 then we've got another region corresponding to rounding to two, rounding to four, -704 +705 00:42:04,355 --> 00:42:07,560 rounding to six, and just on and on each one of these regions. -705 +706 00:42:08,440 --> 00:42:09,700 Rounding and rounding, rounding. -706 +707 00:42:10,220 --> 00:42:12,902 I only went out to like a hundred or something like that, -707 +708 00:42:12,902 --> 00:42:15,400 but that gives you a sense of what we're trying to do. -708 +709 00:42:15,640 --> 00:42:18,800 So we've made progress, but this is still hard. -709 +710 00:42:19,180 --> 00:42:21,780 What is the area of all of those triangles added together? -710 +711 00:42:22,200 --> 00:42:24,300 Right, that's not necessarily an obvious thing. -711 +712 00:42:24,860 --> 00:42:29,800 So let's just start by writing it out and seeing what help that can give us. -712 +713 00:42:29,840 --> 00:42:32,920 So every one of these is a triangle, it's going to look like one half base times height. -713 +714 00:42:33,520 --> 00:42:35,941 And in fact every one of them, if we think of the left right -714 +715 00:42:35,941 --> 00:42:38,760 direction as being their height, every one of them has a height of one. -715 +716 00:42:39,060 --> 00:42:42,860 So each one is going to look like one half times a base of some kind. -716 +717 00:42:43,460 --> 00:42:45,495 So I'm going to take, maybe I'll write this out on a -717 +718 00:42:45,495 --> 00:42:47,800 different piece of paper actually so I can keep it up close. -718 +719 00:42:48,560 --> 00:42:53,100 I'll take one half times whatever the base of the triangle is times the height. -719 +720 00:42:53,460 --> 00:42:57,280 So our first triangle, that base is, that base has a length one. -720 +721 00:42:58,680 --> 00:43:01,480 So that's going to correspond to the one half probability of going to zero. -721 +722 00:43:02,040 --> 00:43:06,900 The next triangle, we have to look at this length here between one third and one half. -722 +723 00:43:07,060 --> 00:43:07,760 What is that length? -723 +724 00:43:08,680 --> 00:43:12,720 Well, actually I'm just going to write it out as a half minus a third. -724 +725 00:43:13,320 --> 00:43:17,314 It equals a sixth, but writing it out like that kind of reminds us where it came from, -725 +726 00:43:17,314 --> 00:43:19,380 so we don't want to collapse things too soon. -726 +727 00:43:19,980 --> 00:43:23,160 That could maybe be another problem solving tip, don't collapse things too soon. -727 +728 00:43:23,240 --> 00:43:25,982 Try to let your notation have a memory for where things -728 +729 00:43:25,982 --> 00:43:28,920 came from because sometimes that helps see overall patterns. -729 +730 00:43:30,040 --> 00:43:32,440 This next one, what's the distance between these two points? -730 +731 00:43:32,820 --> 00:43:34,720 Well, it's a fourth minus a fifth. -731 +732 00:43:34,760 --> 00:43:37,080 That's the distance between these given how they were defined. -732 +733 00:43:37,220 --> 00:43:39,740 So we have a fourth minus a fifth. -733 +734 00:43:40,840 --> 00:43:45,192 And in general we have this kind of oscillating sum, a sixth minus a seventh, -734 +735 00:43:45,192 --> 00:43:48,429 where we have all the reciprocals of the natural numbers, -735 +736 00:43:48,429 --> 00:43:52,001 but we're adding that up infinitely many different times, okay, -736 +737 00:43:52,001 --> 00:43:54,680 and we want to know what that sum happens to be. -737 +738 00:43:55,040 --> 00:44:01,330 And from here, the problem solving tip associated with this will seem a little bit -738 +739 00:44:01,330 --> 00:44:07,620 strange, but it might be the case that you recognize this fact from somewhere else. -739 +740 00:44:07,620 --> 00:44:12,735 You might recognize, let's say if you were watching a particular lockdown math -740 +741 00:44:12,735 --> 00:44:16,231 lecture a week or two ago, that this alternating sum, -741 +742 00:44:16,231 --> 00:44:20,634 one minus a half plus a third minus fourth plus a fifth, on and on, -742 +743 00:44:20,634 --> 00:44:23,160 actually equals the natural log of two. -743 +744 00:44:24,280 --> 00:44:24,680 Okay. -744 -00:44:25,759 --> 00:44:28,594 +745 +00:44:25,760 --> 00:44:28,594 And the way this actually came about, it's such a weird procedure, -745 +746 00:44:28,594 --> 00:44:32,021 it's worth just like walking through again really quickly because it's a bizarre -746 +747 00:44:32,021 --> 00:44:35,236 thing that you're not gonna, you're not gonna be able to just stare at this -747 +748 00:44:35,236 --> 00:44:38,452 formula and then immediately see that this is how you're going to solve it, -748 +749 00:44:38,452 --> 00:44:42,260 unless it's something that you've seen before, which can make it seem all the more opaque. -749 +750 00:44:42,840 --> 00:44:47,248 We did this strange thing where we made it seem like a harder question at first, -750 +751 00:44:47,248 --> 00:44:51,657 where rather than asking about one particular sum, we turned it into a function, -751 +752 00:44:51,657 --> 00:44:55,740 which is effectively asking about infinitely many different sums like this. -752 +753 00:44:56,340 --> 00:45:01,320 So x to the fourth over four, then we're adding x to the fifth over five. -753 +754 00:45:01,320 --> 00:45:05,617 And the reason for doing this is that this plays nicely in calculus land, -754 +755 00:45:05,617 --> 00:45:09,914 because those denominators are now related to the exponents in a way that -755 +756 00:45:09,914 --> 00:45:13,224 we can kind of cancel out by doing an integration trick, -756 +757 00:45:13,224 --> 00:45:17,580 each one of those terms I can nicely express as an integral of a much more -757 +758 00:45:17,580 --> 00:45:18,800 simple monomial term. -758 +759 00:45:20,400 --> 00:45:24,680 x cubed minus, now let's see, plus x to the fourth. -759 +760 00:45:25,440 --> 00:45:28,694 If I integrate this thing, each one of them has an exponent -760 +761 00:45:28,694 --> 00:45:31,840 that increases and then we divide by what the exponent is. -761 +762 00:45:32,660 --> 00:45:36,165 And the reason that you would want to do this is -762 +763 00:45:36,165 --> 00:45:39,600 that this now has a nice way to collapse itself. -763 +764 00:45:39,940 --> 00:45:44,468 So just to make this maybe more explicit, if we evaluate the integral from zero to one, -764 +765 00:45:44,468 --> 00:45:47,710 that's the same as taking this whole expression and evaluating -765 +766 00:45:47,710 --> 00:45:49,460 it at one and subtracting at zero. -766 +767 00:45:49,480 --> 00:45:51,960 So that will give us what happens when we plug in at one. -767 +768 00:45:52,680 --> 00:45:56,547 And again, this is just such a bizarre thing that if you hadn't recognized the sum, -768 +769 00:45:56,547 --> 00:45:59,955 seeing someone prove it to you like this doesn't necessarily make it feel -769 +770 00:45:59,955 --> 00:46:03,546 like something that you could have found, which is frustrating in the context -770 +771 00:46:03,546 --> 00:46:06,540 of trying to develop problem solving tips that are generalizable. -771 +772 00:46:06,540 --> 00:46:11,080 But I'll keep walking through it just to give a little bit of closure to this. -772 +773 00:46:11,320 --> 00:46:16,026 We've got this infinite sum that if you had been familiar with geometric sums, -773 +774 00:46:16,026 --> 00:46:19,601 where each term looks like a certain product from the last, -774 +775 00:46:19,601 --> 00:46:22,699 you would be able to write this as 1 over 1 plus x, -775 +776 00:46:22,699 --> 00:46:25,500 because we're always multiplying by negative x. -776 +777 00:46:25,500 --> 00:46:29,548 So you always take 1 over 1 minus the thing you're multiplying by, which again, -777 +778 00:46:29,548 --> 00:46:33,900 it's kind of one of these things where it's relying on you recognizing it in some way. -778 +779 00:46:34,520 --> 00:46:37,103 And then the last bit of recognition is knowing -779 +780 00:46:37,103 --> 00:46:39,580 how to take integrals of 1 divided by a thing. -780 +781 00:46:40,140 --> 00:46:43,220 And in this context it works out very nicely to just be the natural log. -781 +782 00:46:45,120 --> 00:46:49,785 And we're evaluating this between 0 and 1, which is to say we're taking -782 +783 00:46:49,785 --> 00:46:54,580 the natural log of 1 plus 1, or 2, minus the natural log of 1, which is 0. -783 +784 00:46:54,980 --> 00:46:57,640 And that's why all of these things are the natural log of 2. -784 +785 00:46:58,560 --> 00:47:01,413 And think about what has to go on there in order to be able -785 +786 00:47:01,413 --> 00:47:04,220 to take this and then apply it to our probability question. -786 +787 00:47:04,580 --> 00:47:08,363 You would have to recognize the alternating sum as something that you had seen from -787 +788 00:47:08,363 --> 00:47:12,282 another context, or if you didn't, you would have to be aware of this trick to somehow -788 +789 00:47:12,282 --> 00:47:15,480 turn it into a polynomial that can be nicely expressed as an integral, -789 +790 00:47:15,480 --> 00:47:17,777 that can be collapsed because of geometric series, -790 +791 00:47:17,777 --> 00:47:20,300 which can be integrated because of the natural log of x. -791 +792 00:47:20,740 --> 00:47:24,346 And then thinking about like, what is what is the thing that you can teach -792 +793 00:47:24,346 --> 00:47:27,760 someone to say come away and be able to solve problems in the same way? -793 -00:47:28,879 --> 00:47:31,632 +794 +00:47:28,880 --> 00:47:31,632 I have what might seem like kind of a facetious tip, -794 +795 00:47:31,632 --> 00:47:35,580 but I actually think it's maybe the most potent one and the most honest one. -795 +796 00:47:36,080 --> 00:47:39,460 The way that you can get to this sort of point, just read a lot. -796 +797 00:47:39,720 --> 00:47:40,660 Read as much as you can. -797 +798 00:47:40,980 --> 00:47:43,880 You know, watch YouTube videos on math if they're substantive, things like that. -798 +799 00:47:44,180 --> 00:47:46,906 And think a lot about problems, which is maybe frustrating -799 +800 00:47:46,906 --> 00:47:49,448 because what you want is to be able to say like, well, -800 +801 00:47:49,448 --> 00:47:52,960 how could I have come to this on my own without having merely recognized it? -801 +802 00:47:53,040 --> 00:47:56,928 But I think the truth of the matter is a lot of what looks like insight and ingenuity -802 +803 00:47:56,928 --> 00:48:00,500 is really just pattern recognition, but wearing a little bit of added clothing. -803 +804 00:48:01,220 --> 00:48:05,250 And sometimes it's patterns not so much that you're directly recognizing exactly this, -804 +805 00:48:05,250 --> 00:48:08,540 but maybe you had seen a series like this or a tactic like this before. -805 +806 00:48:08,680 --> 00:48:10,520 Like geometric series, that comes up a lot. -806 +807 00:48:10,600 --> 00:48:12,972 If you read a lot, and if you think a lot about problems, -807 +808 00:48:12,972 --> 00:48:16,039 you will recognize geometric series, even if it's in a context that you've -808 +809 00:48:16,039 --> 00:48:19,680 never actually seen before, or a specific geometric series that you've never seen before. -809 +810 00:48:20,160 --> 00:48:22,822 Similarly, if you read a lot and you think a lot about calculus, -810 +811 00:48:22,822 --> 00:48:25,935 knowing that you can have the natural log pop out of an integral like this, -811 +812 00:48:25,935 --> 00:48:26,960 it becomes second nature. -812 +813 00:48:27,540 --> 00:48:30,388 And I think recognizing the truth of this as being the key -813 +814 00:48:30,388 --> 00:48:33,140 to a lot of problem solving is actually pretty inspiring. -814 -00:48:34,439 --> 00:48:38,211 +815 +00:48:34,440 --> 00:48:38,211 Because oftentimes you find yourself in a situation where somebody is, -815 +816 00:48:38,211 --> 00:48:42,940 they're just faster, they're just better, they just recognize things more so than you do. -816 +817 00:48:43,340 --> 00:48:45,040 And that can be a little bit intimidating, right? -817 +818 00:48:45,040 --> 00:48:48,806 To look at one problem, think of yourself as pretty savvy with math and knowing -818 +819 00:48:48,806 --> 00:48:52,715 what's going on, and then just having someone burn through with a wonderful bit of -819 +820 00:48:52,715 --> 00:48:56,717 cleverness, this super beautiful argument, that leaves you sitting there in the dust -820 +821 00:48:56,717 --> 00:49:00,720 wondering like, wow, I just, you know, I'm just not in the same league at all, right? -821 +822 00:49:00,720 --> 00:49:03,800 And you sometimes even think, well, he just has a math gene, right? -822 +823 00:49:03,800 --> 00:49:05,952 That person, they just have some sort of innate -823 +824 00:49:05,952 --> 00:49:08,240 instinct that makes them really good at this stuff. -824 +825 00:49:08,520 --> 00:49:11,827 But I think the truth of the matter is that the people who are -825 +826 00:49:11,827 --> 00:49:15,187 showing that kind of ingenuity, they've just exposed themselves -826 +827 00:49:15,187 --> 00:49:18,600 to a huge number of patterns, and you too could get there, right? -827 +828 00:49:18,600 --> 00:49:22,100 There is a path towards that which takes the form of practice. -828 +829 00:49:22,160 --> 00:49:25,168 But not just practice, practice where you're taking each problem that -829 +830 00:49:25,168 --> 00:49:28,220 you're looking at and trying to digest the deeper principles behind it. -830 +831 00:49:28,280 --> 00:49:30,742 Maybe some of the ones I'm trying to talk through today, like, -831 +832 00:49:30,742 --> 00:49:33,400 okay, I can say leverage symmetry, but what does that actually mean? -832 +833 00:49:34,080 --> 00:49:36,411 You know, what does it mean to look at a symmetry of a problem -833 +834 00:49:36,411 --> 00:49:38,520 and turn that into something that's formulaically useful? -834 +835 00:49:39,040 --> 00:49:42,400 You just have to see it a lot, and then be pensive when you do. -835 +836 00:49:42,500 --> 00:49:44,430 Don't just be satisfied with the answer, see if -836 +837 00:49:44,430 --> 00:49:46,200 you can understand why that answer came out. -837 +838 00:49:47,240 --> 00:49:50,942 So in that way, this number seven, like, it's the most frustrating, -838 +839 00:49:50,942 --> 00:49:54,972 but it's the most real of all the problem solving tips that there can be, -839 +840 00:49:54,972 --> 00:49:59,328 which is that true problem solving comes down to a kind of pattern recognition, -840 +841 00:49:59,328 --> 00:50:01,180 and there's no two ways around it. -841 +842 00:50:01,280 --> 00:50:04,100 You just have to do a lot of practice and expose yourself to a lot. -842 +843 00:50:05,340 --> 00:50:08,774 Now I would bet that when we did talk about this infinite series a couple lectures back, -843 +844 00:50:08,774 --> 00:50:11,360 you wouldn't have thought that that's a thing that you're going to -844 +845 00:50:11,360 --> 00:50:13,020 be using in a probability question one day. -845 +846 00:50:13,040 --> 00:50:14,460 But that's just how these things go. -846 +847 00:50:14,500 --> 00:50:16,180 They show up in unexpected places. -847 +848 00:50:16,580 --> 00:50:20,930 So what you can do then, is say, well, we've got our whole expression that -848 +849 00:50:20,930 --> 00:50:24,236 involves this very alternating sum, and you'd say, okay, -849 +850 00:50:24,236 --> 00:50:27,890 that means that the answer to our final question is, you know, -850 +851 00:50:27,890 --> 00:50:29,920 one half of the natural log of two. -851 +852 00:50:30,340 --> 00:50:34,367 It's one half of what that alternating sum came out to be, which is very nice, -852 +853 00:50:34,367 --> 00:50:38,038 you know, it involves this natural log expression, and it's very clean, -853 +854 00:50:38,038 --> 00:50:41,811 and we've got this picture for where it came from adding up all of these, -854 +855 00:50:41,811 --> 00:50:42,780 all of these areas. -855 -00:50:43,259 --> 00:50:47,554 +856 +00:50:43,260 --> 00:50:47,554 Now at this point, there is one thing that I think separates really good problem solvers, -856 +857 00:50:47,554 --> 00:50:50,321 the ones who get like nearly perfect scores all the time, -857 +858 00:50:50,321 --> 00:50:54,520 to ones who are like merely good, who, you know, they aren't necessarily perfect scores. -858 +859 00:50:54,540 --> 00:50:56,840 There's some like silly mistakes that come in here or there. -859 +860 00:50:57,800 --> 00:51:01,699 At this point, when you've done the problem and you've got your nice elegant solution, -860 +861 00:51:01,699 --> 00:51:03,940 you want to draw a box around it, you're not done. -861 +862 00:51:05,100 --> 00:51:10,300 Just always, always, principle number eight here, always gut check your answer. -862 +863 00:51:10,300 --> 00:51:13,660 Okay, because there's going to be some little mistake that happens all the time. -863 +864 00:51:13,720 --> 00:51:16,735 The great problem solvers aren't the ones who just never make little mistakes, -864 +865 00:51:16,735 --> 00:51:19,560 they're the ones who have some way of recognizing what those mistakes are. -865 +866 00:51:20,280 --> 00:51:23,695 So in this context, let's say I, I just want to see -866 +867 00:51:23,695 --> 00:51:26,980 numerically what my answer turns out to be, right? -867 +868 00:51:27,000 --> 00:51:29,700 We said that it was one half times the natural log of two. -868 +869 00:51:30,460 --> 00:51:32,780 So let's just see what does that end up being? -869 +870 00:51:33,260 --> 00:51:37,534 And natural log of two is around 0.69, so maybe it's not too surprising, -870 +871 00:51:37,534 --> 00:51:39,760 we're around 0.346, around 0.35, okay? -871 +872 00:51:40,300 --> 00:51:44,934 So if we write that down as one of the, as the answer that we just got, -872 +873 00:51:44,934 --> 00:51:48,023 I told you I would purposefully make a mistake, -873 +874 00:51:48,023 --> 00:51:52,980 so hopefully you're not yelling too loud at this point, does that make sense? -874 +875 00:51:53,480 --> 00:51:55,420 Does that pass a basic reasonability test? -875 +876 00:51:55,800 --> 00:51:59,587 And if you look at our picture, well, the probability has to be at least a half, -876 +877 00:51:59,587 --> 00:52:02,160 because that's the region where it rounds down to zero. -877 -00:52:02,839 --> 00:52:06,407 -So certainly it couldn't be the case that the whole thing adds up to be only 0.35, - 878 +00:52:02,840 --> 00:52:06,407 +So certainly it couldn't be the case that the whole thing adds up to be only 0.35, + +879 00:52:06,407 --> 00:52:08,900 so there must have been some mistake we had along the way. -879 +880 00:52:08,900 --> 00:52:11,993 Everybody makes silly mistakes, everybody drops a minus sign -880 +881 00:52:11,993 --> 00:52:15,240 or applies some rule that doesn't quite apply in a circumstance. -881 +882 00:52:15,880 --> 00:52:18,006 You're not going to make, you're not going to -882 +883 00:52:18,006 --> 00:52:20,180 approach perfection by avoiding silly mistakes. -883 +884 00:52:20,460 --> 00:52:25,400 The way to do it is to be able to systematically know when you've, when you've made them. -884 +885 00:52:26,060 --> 00:52:28,143 So always gut check your answer, have like two different -885 +886 00:52:28,143 --> 00:52:30,080 perspectives that can give you a reasonability check. -886 +887 00:52:30,380 --> 00:52:33,584 In this context, if it inspired us to go and look a little bit -887 +888 00:52:33,584 --> 00:52:35,974 more carefully at how we were applying things, -888 +889 00:52:35,974 --> 00:52:39,840 this sum isn't quite the alternating sum that converges to natural log of 2. -889 +890 00:52:40,640 --> 00:52:43,613 In particular, we added the 1, but then we also add one half, -890 +891 00:52:43,613 --> 00:52:46,060 and it's only after that that we start alternating. -891 +892 00:52:46,200 --> 00:52:49,520 It was plus plus, then minus plus, minus plus, on and on. -892 +893 00:52:49,940 --> 00:52:53,288 And you might see this by recognizing we were subtracting all the even numbers here, -893 +894 00:52:53,288 --> 00:52:55,140 but we're subtracting all the odd numbers here. -894 +895 00:52:55,620 --> 00:52:57,300 So it's similar, but it's not the same. -895 +896 00:52:57,720 --> 00:53:00,994 We will be able to use our knowledge, but let me just give this part -896 +897 00:53:00,994 --> 00:53:04,080 where things actually start alternating a name, something like s. -897 +898 00:53:04,480 --> 00:53:08,152 What this bottom equation is telling us is that when we take 1 minus s, -898 +899 00:53:08,152 --> 00:53:12,437 so that would mean we're subtracting the one half, then we're adding the one third, -899 +900 00:53:12,437 --> 00:53:16,212 then we're subtracting the one fourth, we're flipping all of the signs of -900 +901 00:53:16,212 --> 00:53:20,140 everything beyond that 1, that is the thing that equals the natural log of 2. -901 +902 00:53:20,980 --> 00:53:24,383 Which in turn implies that that remainder of the -902 +903 00:53:24,383 --> 00:53:27,440 sum looks like 1 minus the natural log of 2. -903 +904 00:53:28,560 --> 00:53:31,560 Okay, so all of that, what does that tell us? -904 +905 00:53:31,580 --> 00:53:34,694 When we plug it into our original expression, it's saying that -905 +906 00:53:34,694 --> 00:53:37,760 the actual answer should not be one half the natural log of 2. -906 +907 00:53:37,980 --> 00:53:40,760 That didn't even pass our basic reasonability test. -907 -00:53:41,500 --> 00:53:46,908 -Instead, it'll be one half of 1 plus 1 minus the natural log of 2, - 908 -00:53:46,908 --> 00:53:50,380 -which is just 2 minus the natural log of 2. +00:53:41,500 --> 00:53:46,902 +Instead, it'll be one half of one plus one minus the natural log of two, 909 +00:53:46,902 --> 00:53:50,380 +which is just two minus the natural log of two. + +910 00:53:51,460 --> 00:53:54,040 So does that pass our reasonability check? -910 +911 00:53:54,080 --> 00:53:55,740 What does this actually equal numerically? -911 +912 00:53:56,300 --> 00:53:58,823 You can get a loose approximation in your head if you want, -912 +913 00:53:58,823 --> 00:54:01,600 or if you want to see more precisely, we can pull up a calculator. -913 +914 00:54:02,140 --> 00:54:05,037 So if we go over here and we're saying, no, no, no, -914 +915 00:54:05,037 --> 00:54:07,880 we don't want the natural log of 2, that was wrong. -915 +916 00:54:08,600 --> 00:54:10,720 2 minus that, maybe 0.653. -916 +917 00:54:11,240 --> 00:54:13,800 Does that pass our basic reasonability test? -917 +918 00:54:14,560 --> 00:54:16,320 Yeah, I think so, right? -918 +919 00:54:16,360 --> 00:54:21,610 0.65, that looks like a reasonable answer to what the area in our diagram was, -919 +920 00:54:21,610 --> 00:54:25,531 because if we looked at that diagram, which was, you know, -920 +921 00:54:25,531 --> 00:54:30,716 we've got one wedge here that's covering 0.5, and then this other one covers, -921 +922 00:54:30,716 --> 00:54:34,105 well, about one sixth, half of a sixth, basically, -922 +923 00:54:34,105 --> 00:54:38,758 because this length was a half minus a third, which makes it a sixth, -923 +924 00:54:38,758 --> 00:54:42,480 and then it's a triangle, so one half base times height. -924 +925 00:54:42,900 --> 00:54:45,160 So that's about a 12th, or 0.083. -925 +926 00:54:46,240 --> 00:54:49,040 And then the rest of it, you know, it's not going to fill a huge amount. -926 +927 00:54:49,140 --> 00:54:51,560 Something around 0.65 seems pretty reasonable. -927 +928 00:54:52,740 --> 00:54:55,364 And so that, you know, we could call that good on our gut check, -928 +929 00:54:55,364 --> 00:54:57,020 that this is probably the correct answer. -929 +930 00:54:57,660 --> 00:54:59,904 But we could go one level further if we wanted, -930 +931 00:54:59,904 --> 00:55:03,271 because probability questions very often you can kind of cheat and just -931 +932 00:55:03,271 --> 00:55:06,637 see the answer by simulating it, and actually taking a bunch of samples -932 +933 00:55:06,637 --> 00:55:07,760 and seeing what happens. -933 +934 00:55:08,700 --> 00:55:12,822 So I want to do that actually on this one, just to give us a little bit of confidence -934 +935 00:55:12,822 --> 00:55:16,993 that our answer of 0.65, that there wasn't some other silly mistake that we made along -935 +936 00:55:16,993 --> 00:55:21,020 the way, because everyone knows there certainly are silly mistakes that we can make. -936 +937 00:55:21,640 --> 00:55:24,429 Before I jump to the programming, though, I just want to see if there's -937 +938 00:55:24,429 --> 00:55:27,180 any questions from the audience, and it certainly looks like there are. -938 +939 00:55:27,760 --> 00:55:30,880 So let's see if we can address some of these, get myself out of the way of the questions. -939 +940 00:55:33,120 --> 00:55:35,703 This is like from the book The Chosen, where the rabbi makes -940 +941 00:55:35,703 --> 00:55:38,160 an intentional mistake in his long speech to test his son. -941 +942 00:55:38,580 --> 00:55:40,480 It is exactly like that, my children. -942 +943 00:55:40,860 --> 00:55:41,720 I just want to test you. -943 +944 00:55:41,920 --> 00:55:44,243 Every time I make a mistake, it was always purposeful, -944 +945 00:55:44,243 --> 00:55:46,060 and I was always doing it just to test you. -945 +946 00:55:47,700 --> 00:55:49,260 All right, keep making more episodes. -946 +947 00:55:49,440 --> 00:55:50,160 Why am I stopping? -947 -00:55:50,799 --> 00:55:51,520 +948 +00:55:50,800 --> 00:55:51,520 Two reasons. -948 +949 00:55:51,820 --> 00:55:55,135 You know, the main one, honestly, if I go much longer in this lockdown -949 +950 00:55:55,135 --> 00:55:58,544 without getting a haircut, we're going to have to start filming these in -950 +951 00:55:58,544 --> 00:56:02,000 the style of like a 1980s music video, just to keep stylistic consistency. -951 +952 00:56:02,400 --> 00:56:03,400 I'm not up for that. -952 +953 00:56:03,500 --> 00:56:06,630 You know, I think that's just going to be a little bit too much effort, -953 +954 00:56:06,630 --> 00:56:10,500 so I'll have to wait until whenever it's possible to get a haircut again, then we can go. -954 +955 00:56:11,400 --> 00:56:13,100 Really, though, I also just miss my old content. -955 +956 00:56:13,160 --> 00:56:14,340 I love visualizing stuff. -956 +957 00:56:14,400 --> 00:56:17,780 I've got a long pile of things I want to get to, and you know, the lectures take time. -957 +958 00:56:17,880 --> 00:56:21,900 I might spin up something like this again, might do it on a separate channel. -958 +959 00:56:22,000 --> 00:56:25,200 We'll see what plays out, but I would love to do like a full course where it's a -959 +960 00:56:25,200 --> 00:56:28,480 little bit more clear exactly what we're going to talk about from beginning to end. -960 +961 00:56:28,760 --> 00:56:31,960 Loosely, I'm thinking like combinatorics would be fun, but don't hold me to that. -961 +962 00:56:32,900 --> 00:56:36,359 And lastly, can you help us understand derangements and the principles -962 +963 00:56:36,359 --> 00:56:39,380 of inclusion and exclusion, just like combinatorics intuitive? -963 +964 00:56:39,960 --> 00:56:40,740 Yeah, boy. -964 +965 00:56:42,580 --> 00:56:43,940 How much time do we have today? -965 +966 00:56:44,160 --> 00:56:46,820 Tell you what, maybe just a separate video. -966 +967 00:56:46,960 --> 00:56:51,880 If I do the combinatorics course, actually, that would be a perfect example for it. -967 +968 00:56:52,940 --> 00:56:55,841 It takes a little bit too much to describe here, -968 +969 00:56:55,841 --> 00:57:00,814 but very naturally what you end up with is taking 1 over 1 factorial minus 1 over 2 -969 +970 00:57:00,814 --> 00:57:05,788 factorial plus 1 over 3 factorial minus, and you're doing this alternating sum with -970 +971 00:57:05,788 --> 00:57:07,920 factorials, which is why e pops out. -971 +972 00:57:08,020 --> 00:57:11,652 And if you think of e as fundamentally being the sum of the reciprocals of factorials, -972 +973 00:57:11,652 --> 00:57:15,160 it doesn't seem crazy surprising that it's related to counting problems in that way. -973 +974 00:57:15,540 --> 00:57:18,940 But for another day, I'll say, because I don't know if we have time today. -974 +975 00:57:19,520 --> 00:57:23,419 Because I would love to just show how you could maybe gut check this programmatically -975 +976 00:57:23,419 --> 00:57:26,956 if you wanted to, because the very last principle I have for problem solving, -976 +977 00:57:26,956 --> 00:57:30,720 for mathematical problem solving, is to learn at least a little bit of programming. -977 +978 00:57:31,220 --> 00:57:34,217 Okay, and the reason here, one, for what we're about to do, -978 +979 00:57:34,217 --> 00:57:37,863 you can sometimes basically cheat and see what an answer is numerically, -979 +980 00:57:37,863 --> 00:57:41,260 to see that that verifies how you're thinking about it analytically. -980 +981 00:57:41,380 --> 00:57:45,640 But also, importantly, it forces you to think about things in two separate ways. -981 +982 00:57:45,820 --> 00:57:49,947 Sometimes you can go about it mathematically, but then when you try to make it -982 +983 00:57:49,947 --> 00:57:54,440 computational, you run into certain walls about like, how are things actually defined? -983 +984 00:57:54,640 --> 00:57:58,660 Or, I don't have infinity available to me, how can I do this in a more approximate way? -984 +985 00:57:59,280 --> 00:58:01,528 Very often when I'm animating things for usual videos, -985 +986 00:58:01,528 --> 00:58:03,532 the piece of math that I'm describing, you know, -986 +987 00:58:03,532 --> 00:58:06,640 it lines up with some way that I'm programming it to give the illustrations. -987 -00:58:07,299 --> 00:58:10,335 +988 +00:58:07,300 --> 00:58:10,335 And when those mismatch, that's actually what's most interesting, -988 +989 00:58:10,335 --> 00:58:13,830 when it's not computationally viable to give a perfect illustration of what -989 +990 00:58:13,830 --> 00:58:14,520 I'm describing. -990 +991 00:58:14,660 --> 00:58:17,423 And I actually think that does make for better problem solving in general, -991 +992 00:58:17,423 --> 00:58:19,340 where you're coming at it from two different angles. -992 +993 00:58:19,820 --> 00:58:25,330 So let's just try this out, see if we can understand the probability question we -993 +994 00:58:25,330 --> 00:58:30,840 were just looking at in terms of just, you know, what's the word I'm looking for? -994 +995 00:58:30,940 --> 00:58:31,180 Cheating! -995 +996 00:58:31,740 --> 00:58:33,400 Seeing what it turns out to be. -996 +997 00:58:33,940 --> 00:58:40,000 So I'm going to import numpy, which evidently some heathens refer to as numpy, -997 +998 00:58:40,000 --> 00:58:43,760 which just, I just can't, that seems awful to me. -998 +999 00:58:44,280 --> 00:58:47,597 So we can find random numbers if you just call random, -999 +1000 00:58:47,597 --> 00:58:51,820 this first random refers to the library, the second one is a function. -1000 +1001 00:58:52,460 --> 00:58:56,720 It will return a number between 0 and 1 according to a uniform distribution. -1001 +1002 00:58:56,820 --> 00:59:00,056 So it is as likely to choose something between, you know, -1002 -00:59:00,056 --> 00:59:03,459 +1003 +00:59:00,056 --> 00:59:03,460 0.1 and 0.2 as it is to choose something between 0.8 and 0.9. -1003 +1004 00:59:04,040 --> 00:59:08,324 And what's nice is I can get a list of them, so in this case I get a list of 10 random -1004 +1005 00:59:08,324 --> 00:59:12,560 numbers, and maybe I call that something like x, and maybe I create another list of y. -1005 +1006 00:59:12,680 --> 00:59:16,109 So x, some random numbers, y, some random numbers, -1006 +1007 00:59:16,109 --> 00:59:19,740 and if I take x divided by y, it does it term by term. -1007 -00:59:20,419 --> 00:59:24,396 +1008 +00:59:20,420 --> 00:59:24,396 So for example this first number that we see in here, -1008 +1009 00:59:24,396 --> 00:59:28,520 that's 0.739, that's taking the 0.52 divided by the 0.7. -1009 +1010 00:59:28,520 --> 00:59:31,605 This next one is taking 0.66 divided by 0.56 on and on, -1010 +1011 00:59:31,605 --> 00:59:33,920 so we can see all of our ratios like that. -1011 +1012 00:59:34,340 --> 00:59:40,025 And in general I might just define a ratios list where I'm going to take my, -1012 +1013 00:59:40,025 --> 00:59:44,381 a bunch of random numbers of size n, and then divide it by -1013 +1014 00:59:44,381 --> 00:59:48,000 another bunch of random numbers of the same size. -1014 +1015 00:59:48,460 --> 00:59:52,400 n is not defined, but I'll give it a definition, some nice and big number like a million. -1015 +1016 00:59:53,780 --> 00:59:54,060 Okay. -1016 +1017 00:59:54,740 --> 00:59:59,406 And now I'm holding on to a million ratios, a million examples of x divided by y, -1017 +1018 00:59:59,406 --> 01:00:02,536 where x and y are chosen according to our constraints, -1018 +1019 01:00:02,536 --> 01:00:05,040 which is kind of cool if you think about it. -1019 +1020 01:00:05,040 --> 01:00:08,485 So, you know, we can see some examples here, some that have come up to zero, -1020 +1021 01:00:08,485 --> 01:00:11,842 and some that would round down to zero, some that would round down to two, -1021 +1022 01:00:11,842 --> 01:00:14,080 some that would round down to one, so that's nice. -1022 +1023 01:00:14,400 --> 01:00:17,431 And now I can start asking questions, like, you know, -1023 +1024 01:00:17,431 --> 01:00:21,473 when is it that I take these ratios, I take the floor function of them, -1024 +1025 01:00:21,473 --> 01:00:25,460 not the four, the floor, and I want to know when is that equal to zero. -1025 +1026 01:00:25,960 --> 01:00:30,669 This gives me a list of trues and falses, basically saying it is or it isn't zero, -1026 +1027 01:00:30,669 --> 01:00:35,775 and if I take the mean of that, which is treating the trues and falses as ones and zeros, -1027 +1028 01:00:35,775 --> 01:00:39,520 this tells me the proportion in total that will actually be zeros. -1028 +1029 01:00:39,540 --> 01:00:43,780 So we expect it to be about a half, and we can verify, yeah, okay, it's about a half. -1029 +1030 01:00:44,380 --> 01:00:50,600 We could have also asked when is it about two, okay, and it looks like 0.83. -1030 +1031 01:00:51,200 --> 01:00:56,172 And remember from our diagram what we were looking for is when when it's in this green -1031 +1032 01:00:56,172 --> 01:00:58,858 region here, when it's in that green triangle, -1032 +1033 01:00:58,858 --> 01:01:02,916 which has an area that was a half minus a third, but all times a half, -1033 +1034 01:01:02,916 --> 01:01:06,060 because it's one half base times height for a triangle. -1034 +1035 01:01:06,660 --> 01:01:10,345 So if we pop back over to our terminal and say, okay, -1035 +1036 01:01:10,345 --> 01:01:14,782 if we were looking for one half times the base of that triangle, -1036 +1037 01:01:14,782 --> 01:01:20,380 which was a half minus a third, we would expect that proportion to have been 0.83. -1037 +1038 01:01:20,680 --> 01:01:22,140 And yeah, it looks like it was about that. -1038 +1039 01:01:22,560 --> 01:01:26,466 And we could even answer our actual question, which is to take the floor, -1039 +1040 01:01:26,466 --> 01:01:30,583 and then if I say I want to divide by two and ask when the remainder is zero, -1040 +1041 01:01:30,583 --> 01:01:34,911 that's a way of asking when it's even, and then taking that whole list and taking -1041 +1042 01:01:34,911 --> 01:01:38,817 a mean of it is a way of asking how often that ends up looking like true, -1042 +1043 01:01:38,817 --> 01:01:41,667 what proportion of them give me true, and yeah, 0.65, -1043 +1044 01:01:41,667 --> 01:01:44,360 which is about the answer that we were looking for. -1044 +1045 01:01:44,660 --> 01:01:49,956 You know, we were looking for something that was too half of two minus the natural log, -1045 +1046 01:01:49,956 --> 01:01:51,100 natural log of two. -1046 +1047 01:01:51,640 --> 01:01:54,880 So we have this wonderful way to kind of empirically verify. -1047 +1048 01:01:55,800 --> 01:01:59,592 And of course, and most of the times with in-lockdown math that I've pulled up Python, -1048 +1049 01:01:59,592 --> 01:02:01,860 I'm just doing relatively simple things numerically. -1049 +1050 01:02:02,100 --> 01:02:04,180 If you wanted to, you could try to visualize stuff. -1050 +1051 01:02:04,640 --> 01:02:09,760 So something like matplotlib is definitely a great pyplot. -1051 +1052 01:02:10,360 --> 01:02:11,760 I'll import it as plt. -1052 +1053 01:02:12,200 --> 01:02:14,972 This is a very good library for just like simple data -1053 +1054 01:02:14,972 --> 01:02:17,540 visualization that you can pull up pretty quickly. -1054 +1055 01:02:18,040 --> 01:02:20,860 So in our context, I can pull up a histogram. -1055 +1056 01:02:21,660 --> 01:02:27,380 So I'm going to take a histogram of all of my data, and I have to specify a range. -1056 +1057 01:02:27,520 --> 01:02:30,913 So maybe we just want to look at when the, when -1057 +1058 01:02:30,913 --> 01:02:34,520 those values get bucketed in between like 0 and 20. -1058 +1059 01:02:35,040 --> 01:02:36,740 So my number of bins is 20. -1059 -01:02:39,859 --> 01:02:41,800 +1060 +01:02:39,860 --> 01:02:41,800 I was about to sneeze, but I held it in. -1060 +1061 01:02:42,180 --> 01:02:43,879 Don't you love when that happens when like a sneeze -1061 +1062 01:02:43,879 --> 01:02:45,220 is coming and you don't have to catch it? -1062 +1063 01:02:45,360 --> 01:02:49,800 It just recedes into the darkness because it knows that you're better than the sneeze. -1063 +1064 01:02:50,800 --> 01:02:50,800 Right. -1064 +1065 01:02:51,160 --> 01:02:53,420 Again, I'm more pleased with myself than I should be there. -1065 +1066 01:02:54,180 --> 01:02:58,191 And um, I always, you always have to add a relative width on these sorts of histograms -1066 +1067 01:02:58,191 --> 01:03:01,880 because they just look ugly if they're all like side by side, I think sometimes. -1067 +1068 01:03:02,340 --> 01:03:06,358 So if I do this, um, and then I show what the plot ends up being, -1068 +1069 01:03:06,358 --> 01:03:10,560 I have something that has shown up on my screen, but not your screen. -1069 +1070 01:03:10,580 --> 01:03:11,840 So let's pop it on over. -1070 +1071 01:03:12,400 --> 01:03:13,140 Yeah, there we go. -1071 +1072 01:03:13,420 --> 01:03:16,421 So you can get a sense of, you know, this bar represents all of the ones that -1072 +1073 01:03:16,421 --> 01:03:19,500 rounded down to zero, and you see it's about half of them, about half a million. -1073 +1074 01:03:19,920 --> 01:03:22,340 All the ones that rounded down to one, rounded down to two. -1074 +1075 01:03:22,640 --> 01:03:26,120 And you can get this, um, this nice sense for what all of your data is. -1075 +1076 01:03:26,300 --> 01:03:28,794 Again, just adding that kind of empirical validation -1076 +1077 01:03:28,794 --> 01:03:30,960 on top of whatever you find more analytically. -1077 +1078 01:03:31,240 --> 01:03:33,727 And that back and forth I really do think is helpful -1078 +1079 01:03:33,727 --> 01:03:35,980 for more like pure mathematical problem solving. -1079 +1080 01:03:36,600 --> 01:03:39,640 So with that, that's actually all that I have for the lesson today. -1080 +1081 01:03:40,120 --> 01:03:43,040 Just two quick things that I want to go over before we end things here. -1081 +1082 01:03:44,860 --> 01:03:49,520 I've been using Desmos a lot to like show graphs of things because I just love Desmos. -1082 +1083 01:03:50,440 --> 01:03:54,055 And I'm actually friends with some of the people who work there -1083 +1084 01:03:54,055 --> 01:03:58,180 because quite often a company's people are as delightful as its products. -1084 +1085 01:03:58,940 --> 01:04:01,633 And the CEO Eli shared with me the fact that they -1085 +1086 01:04:01,633 --> 01:04:04,380 were doing like an art contest among some students. -1086 +1087 01:04:04,380 --> 01:04:08,319 And I just wanted to showcase some of, I'm not sure if these are the winners -1087 +1088 01:04:08,319 --> 01:04:12,003 or the finalists of the art contest, but basically in various different -1088 +1089 01:04:12,003 --> 01:04:16,198 categories of students who are, I think it was like 12 through 14, 15 through 17, -1089 +1090 01:04:16,198 --> 01:04:20,240 or something like that, using like mathematical graphs to try to draw pictures. -1090 +1091 01:04:20,500 --> 01:04:24,220 Okay, so keep in mind what I'm about to show you are mathematical -1091 +1092 01:04:24,220 --> 01:04:28,448 graphs that someone wrote just with an analytic description and they were, -1092 +1093 01:04:28,448 --> 01:04:32,000 they were just prompted to create something artistic from that. -1093 +1094 01:04:32,840 --> 01:04:37,890 Okay, so one of my favorites, and I think this one was from someone named Carrie, -1094 +1095 01:04:37,890 --> 01:04:42,940 is two giraffes that just from an artistic standpoint, it's actually quite lovely. -1095 +1096 01:04:43,720 --> 01:04:47,271 And then to think through like actually mathematically describing everything -1096 +1097 01:04:47,271 --> 01:04:51,238 involved here, it's such a beautiful blend of well, like the creative side of things, -1097 +1098 01:04:51,238 --> 01:04:54,560 the artistic side with aesthetics and everything, and the analytic side. -1098 +1099 01:04:55,480 --> 01:04:57,820 Another that was, this is just genuinely insane. -1099 +1100 01:04:58,840 --> 01:05:02,620 This is by Katsini, is a Bézier Knight. -1100 +1101 01:05:03,000 --> 01:05:06,998 So I definitely wanted to highlight this on behalf of the team over at Desmos, -1101 +1102 01:05:06,998 --> 01:05:10,540 where each one of the curves here is described according to something -1102 +1103 01:05:10,540 --> 01:05:13,780 that's called a Bézier curve, very useful for computer graphics. -1103 +1104 01:05:13,880 --> 01:05:17,930 It's a kind of cubic parametric term, and they just recreated -1104 +1105 01:05:17,930 --> 01:05:21,980 a starry knight in a way that's completely beautiful, I think. -1105 +1106 01:05:23,140 --> 01:05:28,987 And then the very last one, which is genuinely shocking that you can do with mathematical -1106 +1107 01:05:28,987 --> 01:05:34,380 graphs in any way, was a self-portrait by Jared, that's just like genuinely insane. -1107 +1108 01:05:34,500 --> 01:05:37,401 I mean, I remember playing around with like a TI-84 and trying to come -1108 +1109 01:05:37,401 --> 01:05:40,140 up with little pictures of like a smiley face and things like that. -1109 +1110 01:05:40,360 --> 01:05:44,496 So the amount that things have changed in terms of when someone's noodling off -1110 +1111 01:05:44,496 --> 01:05:48,580 with their graphing calculator in class and what they can do truly next level. -1111 +1112 01:05:49,500 --> 01:05:52,960 And then at the very end, I just want to say again like a highlighted -1112 +1113 01:05:52,960 --> 01:05:56,420 thank you to Ben Eater and to Cam, who've been extremely helpful with -1113 +1114 01:05:56,420 --> 01:05:59,880 the whole series in ways that's like hard to even articulate properly. -1114 +1115 01:06:00,820 --> 01:06:04,646 Eater in particular, I mean, he's let me borrow a lot of his equipment and helped -1115 +1116 01:06:04,646 --> 01:06:08,473 out with like figuring out live footage type stuff because that's not something I -1116 +1117 01:06:08,473 --> 01:06:12,440 usually do, which is not even to mention the work on the live stats and live quizzes. -1117 +1118 01:06:12,440 --> 01:06:17,187 So if you aren't already familiar with his channel, it's simply named Ben Eater, -1118 +1119 01:06:17,187 --> 01:06:21,700 like 100% check it out, definitely subscribe to it, try some of the projects. -1119 +1120 01:06:21,740 --> 01:06:24,220 He has a very project-oriented way of teaching. -1120 +1121 01:06:24,340 --> 01:06:25,400 I think it's absolutely great. -1121 +1122 01:06:25,800 --> 01:06:30,780 So cannot emphasize enough how grateful I am in the direction of both of those two. -1122 +1123 01:06:31,120 --> 01:06:34,620 And check out the the item pool site where we are going to let -1123 +1124 01:06:34,620 --> 01:06:38,064 you kind of relive the lockdown math experience and have like -1124 +1125 01:06:38,064 --> 01:06:41,620 homework and actual challenges associated with each one of them. -1125 -01:06:41,620 --> 01:06:45,090 -So stay tuned on that and if you're interested in being a beta user of it, - 1126 -01:06:45,090 --> 01:06:47,960 -there will be forms for how you can reach out to both of them. +01:06:41,620 --> 01:06:44,810 +So stay tuned on that. And if you're interested in being a beta user of it, 1127 +01:06:44,810 --> 01:06:47,960 +there will be forums for how you can how you can reach out to both of them. + +1128 01:06:48,400 --> 01:06:50,660 Thank you everyone for joining in the whole series. -1128 +1129 01:06:50,700 --> 01:06:52,920 This has been very fun for me, very different for me. -1129 +1130 01:06:52,920 --> 01:06:58,040 And I will shortly get back to the more usual videos, which I'm very excited about. -1130 +1131 01:06:58,120 --> 01:07:01,308 There's a couple topics that I just think you're thoroughly going to enjoy, -1131 +1132 01:07:01,308 --> 01:07:03,658 especially if you like problem solving stuff like this, -1132 +1133 01:07:03,658 --> 01:07:05,840 like really getting into good meaty problem solving. -1133 +1134 01:07:06,080 --> 01:07:07,400 That's some of what's on the horizon. -1134 +1135 01:07:08,780 --> 01:07:12,960 And with that I will simply say keep loving math and enjoy the rest of your day. -1135 +1136 01:07:22,920 --> 01:08:18,560 So Thank you. diff --git a/2020/ldm-tips-to-problem-solving/english/sentence_timings.json b/2020/ldm-tips-to-problem-solving/english/sentence_timings.json index 59ba64669..2c0a6de4c 100644 --- a/2020/ldm-tips-to-problem-solving/english/sentence_timings.json +++ b/2020/ldm-tips-to-problem-solving/english/sentence_timings.json @@ -515,7 +515,7 @@ 571.88 ], [ - "All right, so because there's such strong consensus, I feel comfortable grading this.", + "All right, so because there's such strong consistent- consensus, I feel comfortable grading this.", 572.44, 576.54 ], @@ -650,7 +650,7 @@ 766.32 ], [ - "And then we have 2 pi minus 2 copies of pi, so that's all equal to 0, which is saying the same thing as theta L is equal to 2 times, well rather than writing alpha plus beta, I'll just recognize that that is the small little angle that we had.", + ", and then we have two pi minus two copies of pi. So that's all equal to zero, which is saying the same thing as theta l is equal to two times, well rather than writing alpha plus beta, I'll just recognize that that is the small little angle that we had.", 766.56, 780.28 ], @@ -1495,7 +1495,7 @@ 1901.8 ], [ - "Absurd.", + "All right. All right, absurd.", 1905.7, 1906.14 ], @@ -1832,11 +1832,11 @@ [ "It's not, not, no, I said it wrong.", 2433.94, - 2435.32 + 2436.04 ], [ "I said it wrong.", - 2435.44, + 2436.72, 2437.14 ], [ @@ -1845,7 +1845,7 @@ 2442.08 ], [ - "It's very easy to get fuddled in that way.", + "It's very, it's very easy to get confuddled in that way.", 2442.34, 2444.62 ], @@ -2420,7 +2420,7 @@ 3220.76 ], [ - "Instead, it'll be one half of 1 plus 1 minus the natural log of 2, which is just 2 minus the natural log of 2.", + "Instead, it'll be one half of one plus one minus the natural log of two, which is just two minus the natural log of two.", 3221.5, 3230.38 ], @@ -3025,7 +3025,7 @@ 4001.62 ], [ - "So stay tuned on that and if you're interested in being a beta user of it, there will be forms for how you can reach out to both of them.", + "So stay tuned on that. And if you're interested in being a beta user of it, there will be forums for how you can how you can reach out to both of them.", 4001.62, 4007.96 ], diff --git a/2020/ldm-tips-to-problem-solving/english/transcript.txt b/2020/ldm-tips-to-problem-solving/english/transcript.txt index 6b5ef6aef..af81fcd53 100644 --- a/2020/ldm-tips-to-problem-solving/english/transcript.txt +++ b/2020/ldm-tips-to-problem-solving/english/transcript.txt @@ -101,7 +101,7 @@ And you know, if you if you want to be involved, the place to go, 3b1b.co live, And for those who are watching in the future, maybe you're watching it in the embedded page where the problem is just sitting right below you. And even if you're not technically participating in the data contributing to the live statistics, when I was going through it, it's honestly pretty fun to just kind of click and like, oh, I know I'm not a part of this, but it kind of feels like I am a part of what was happening. So I just love it. -All right, so because there's such strong consensus, I feel comfortable grading this. +All right, so because there's such strong consistent- consensus, I feel comfortable grading this. Oh, I thought I got it right on 1 1 1 1. So the correct answer is that a is equal to b, d is equal to e, and then none of the others are necessarily true. And let's walk through why that's the case, okay? @@ -128,7 +128,7 @@ So in this context, once I have these three equations, recognizing that there's So I'm gonna, you know, add this top equation and maybe I subtract off the other two equations. And to subtract these off, and what that means is the alpha prime gets cancelled, so does the beta prime, I'm left with my large angle, I'm subtracting off 2 times alpha plus beta. You know, each of those gets subtracted off with the coefficient 2. -And then we have 2 pi minus 2 copies of pi, so that's all equal to 0, which is saying the same thing as theta L is equal to 2 times, well rather than writing alpha plus beta, I'll just recognize that that is the small little angle that we had. +, and then we have two pi minus two copies of pi. So that's all equal to zero, which is saying the same thing as theta l is equal to two times, well rather than writing alpha plus beta, I'll just recognize that that is the small little angle that we had. It's 2 times the small angle. Again, I can't emphasize enough just what a weirdly useful fact this turns out to be in various geometry puzzles you might do. It's definitely come up on the channel a number of times in circumstances regarding, you know, complex numbers or pure geometry situations, of course. @@ -297,7 +297,7 @@ I can only throw you so far. Oh, here we go. Things, they run away from you. Even your objects sometimes get tired of math class and want to play truant now and then, but he has to stay whether he likes to or not. -Absurd. +All right. All right, absurd. So let's say this is our x coordinate. x can fall anywhere between 0 and 1 with uniform probability. y can fall between 0 and 1. @@ -367,7 +367,7 @@ x is more than two times what y is, but it's not three times more than what y is It's not, not, no, I said it wrong. I said it wrong. When y is more than two times what x is, but it's not three times more than x is. -It's very easy to get fuddled in that way. +It's very, it's very easy to get confuddled in that way. And it's even harder to try to give some sort of probability to that, whereas in our diagram it has a very clear meaning. It is the area of this region because the full area of possibilities already is one, so this, the probability of something happening should be one, and we just need to look at the area of that. And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. @@ -482,7 +482,7 @@ Which in turn implies that that remainder of the sum looks like 1 minus the natu Okay, so all of that, what does that tell us? When we plug it into our original expression, it's saying that the actual answer should not be one half the natural log of 2. That didn't even pass our basic reasonability test. -Instead, it'll be one half of 1 plus 1 minus the natural log of 2, which is just 2 minus the natural log of 2. +Instead, it'll be one half of one plus one minus the natural log of two, which is just two minus the natural log of two. So does that pass our reasonability check? What does this actually equal numerically? You can get a loose approximation in your head if you want, or if you want to see more precisely, we can pull up a calculator. @@ -603,7 +603,7 @@ He has a very project-oriented way of teaching. I think it's absolutely great. So cannot emphasize enough how grateful I am in the direction of both of those two. And check out the the item pool site where we are going to let you kind of relive the lockdown math experience and have like homework and actual challenges associated with each one of them. -So stay tuned on that and if you're interested in being a beta user of it, there will be forms for how you can reach out to both of them. +So stay tuned on that. And if you're interested in being a beta user of it, there will be forums for how you can how you can reach out to both of them. Thank you everyone for joining in the whole series. This has been very fun for me, very different for me. And I will shortly get back to the more usual videos, which I'm very excited about. diff --git a/2020/ldm-tips-to-problem-solving/french/sentence_translations.json b/2020/ldm-tips-to-problem-solving/french/sentence_translations.json index 72e086e09..5ff51b4d8 100644 --- a/2020/ldm-tips-to-problem-solving/french/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/french/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "Et encore une fois, ce que je veux que vous reteniez, c'est ce principe selon lequel si vous pouvez décrire un objet de deux manières différentes, c'est très puissant en termes de démonstration de relations algébriques non évidentes ou de tout ce qui est en quelque sorte écrit symboliquement sans intuition immédiate au-dessus. de celui-ci. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "Ce serait une probabilité de 50 %. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "Et maintenant, vous pouvez peut-être voir où cela va nous mener, car pour le prochain trimestre, lorsque nous voulons savoir quand x divisé par y se situe entre 4 et 5, nous allons tracer des lignes qui se coupent en x sur 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "Que signifie examiner la symétrie d’un problème et en faire quelque chose de formellement utile ? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "Je continue de faire plus d'épisodes. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "Donc, vous savez, nous pouvons voir quelques exemples ici, certains qui sont arrivés à 0, et d'autres qui seraient arrondis à 0, certains qui seraient arrondis à 2, d'autres qui seraient arrondis à 1, donc c'est bien. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "Et si je prends la moyenne de cela, qui traite les vrais et les faux comme des 1 et des 0, cela m'indique la proportion au total qui sera réellement des 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "Vous savez, nous recherchions quelque chose qui valait la moitié de 2 moins le logarithme naturel de 2. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "Alors peut-être voulons-nous simplement regarder quand ces valeurs sont réparties entre 0 et 20. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "Et il faut toujours ajouter une largeur relative sur ce genre d'histogrammes parce qu'ils ont l'air moche s'ils sont tous côte à côte, je pense parfois. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "Et vous pouvez avoir une bonne idée de ce que sont toutes vos données. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "Et ils ont simplement été incités à créer quelque chose d’artistique à partir de cela. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "Un autre qui était vraiment fou. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/german/sentence_translations.json b/2020/ldm-tips-to-problem-solving/german/sentence_translations.json index 9bf99f3cb..d41e0bd6c 100644 --- a/2020/ldm-tips-to-problem-solving/german/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/german/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "Und ich möchte Sie noch einmal darauf hinweisen, dass das Prinzip, dass ein Objekt auf zwei verschiedene Arten beschrieben werden kann, sehr aussagekräftig ist, wenn es darum geht, nicht offensichtliche algebraische Beziehungen oder alles, was symbolisch niedergeschrieben wird, ohne unmittelbare Intuition darüber zu zeigen davon. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "Dies wäre eine Wahrscheinlichkeit von 50 %. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "Und jetzt können Sie vielleicht sehen, wohin das führen wird, denn für den nächsten Term, wenn wir wissen wollen, wann x dividiert durch y zwischen 4 und 5 liegt, werden wir Linien zeichnen, die sich bei x über 4 schneiden. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "Ich mache immer mehr Episoden. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "Wissen Sie, wir können hier einige Beispiele sehen, einige, die auf 0 gekommen sind, und einige, die auf 0 abgerundet wurden, einige, die auf 2 abgerundet wurden, einige, die auf 1 abgerundet wurden, also ist das schön. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "Und wenn ich den Mittelwert daraus nehme, der darin besteht, die wahren und falschen Werte als Einsen und Nullen zu behandeln, sagt mir das, wie hoch der Anteil insgesamt sein wird, der tatsächlich Nullen sein wird. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "Wissen Sie, wir haben nach etwas gesucht, das die Hälfte von 2 minus dem natürlichen Logarithmus von 2 ist. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "Vielleicht möchten wir uns also nur ansehen, wann diese Werte zwischen 0 und 20 liegen. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "Und bei solchen Histogrammen muss man immer eine relative Breite hinzufügen, weil sie einfach hässlich aussehen, wenn sie alle nebeneinander liegen, denke ich manchmal. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "Und Sie können ein gutes Gefühl dafür bekommen, worum es bei all Ihren Daten geht. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "Und sie wurden einfach dazu angeregt, daraus etwas Künstlerisches zu schaffen. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "Noch einer, der wirklich verrückt war. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/hebrew/sentence_translations.json b/2020/ldm-tips-to-problem-solving/hebrew/sentence_translations.json index 9d3a2567e..eb25ef880 100644 --- a/2020/ldm-tips-to-problem-solving/hebrew/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/hebrew/sentence_translations.json @@ -1701,7 +1701,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it.", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it.", "translatedText": "ושוב, מה שאני רוצה שתיקח ממך הוא העיקרון הזה שאם אתה יכול להיות אובייקט אחד המתואר בשתי דרכים שונות, חזק מאוד במונחים של הצגת יחסים אלגבריים לא ברורים או כל דבר שנכתב בצורה סמלית בלי אינטואיציה מיידית למעלה ממנו.", "n_reviews": 0, "start": 1522.72, @@ -2009,7 +2009,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%.", + "input": "This would be a probability of 50 percent.", "translatedText": "זו תהיה הסתברות של 50%.", "n_reviews": 0, "start": 1824.88, @@ -2611,7 +2611,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4.", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four.", "translatedText": "ועכשיו אולי אתה יכול לראות לאן זה הולך להגיע, כי במונח הבא, כשאנחנו רוצים לדעת מתי x חלקי y יושב בין 4 ל-5, אנחנו הולכים לצייר קווים שמצטלבים ב-x מעל 4.", "n_reviews": 0, "start": 2464.14, @@ -2968,7 +2968,7 @@ "end": 2840.3 }, { - "input": "And then thinking about like, what is the thing that you can teach someone to say come away and be able to solve problems in the same way?", + "input": "And then thinking about like, what is what is the thing that you can teach someone to say come away and be able to solve problems in the same way?", "translatedText": "ואז חושבים על כאילו, מה הדבר שאתה יכול ללמד מישהו לומר בוא מפה ויוכל לפתור בעיות באותו אופן?", "n_reviews": 0, "start": 2840.74, @@ -3108,7 +3108,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful?", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful?", "translatedText": "מה זה אומר להסתכל על סימטריה של בעיה ולהפוך אותה למשהו שימושי מבחינה נוסחתית?", "n_reviews": 0, "start": 2974.08, @@ -3528,7 +3528,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes.", + "input": "All right, keep making more episodes.", "translatedText": "אני ממשיך לעשות עוד פרקים.", "n_reviews": 0, "start": 3347.7, @@ -3836,7 +3836,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice.", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice.", "translatedText": "אז, אתה יודע, אנחנו יכולים לראות כמה דוגמאות כאן, חלקן הגיעו ל-0, וחלקן יעגלו למטה ל-0, חלקן יעגלו למטה ל-2, חלקן יעגלו למטה ל-1, אז זה נחמד.", "n_reviews": 0, "start": 3605.04, @@ -3857,7 +3857,7 @@ "end": 3629.32 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s.", + "input": "ro, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros.", "translatedText": "ואם אני לוקח את הממוצע של זה, שהוא מתייחס לאמיתות ולא נכון כאל 1 ו-0, זה אומר לי את הפרופורציה בסך הכל שתהיה למעשה 0.", "n_reviews": 0, "start": 3629.32, @@ -3920,7 +3920,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2.", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two.", "translatedText": "אתה יודע, חיפשנו משהו שהוא חצי מ-2 פחות הלוג הטבעי של 2.", "n_reviews": 0, "start": 3704.66, @@ -3983,7 +3983,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20.", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20.", "translatedText": "אז אולי אנחנו רק רוצים להסתכל מתי הערכים האלה ממוקמים בין 0 ל-20.", "n_reviews": 0, "start": 3747.52, @@ -4025,7 +4025,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes.", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes.", "translatedText": "ותמיד צריך להוסיף רוחב יחסי בהיסטוגרמות מהסוג הזה כי הן פשוט נראות מכוערות אם כולן זה לצד זה, אני חושב לפעמים.", "n_reviews": 0, "start": 3774.18, @@ -4067,7 +4067,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is.", + "input": "And you can get this, um, this nice sense for what all of your data is.", "translatedText": "ואתה יכול לקבל את התחושה הנחמדה הזו לגבי כל הנתונים שלך.", "n_reviews": 0, "start": 3802.64, @@ -4137,7 +4137,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that.", + "input": "and they were, they were just prompted to create something artistic from that.", "translatedText": "והם פשוט התבקשו ליצור מזה משהו אמנותי.", "n_reviews": 0, "start": 3867.72, @@ -4158,7 +4158,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane.", + "input": "Another that was, this is just genuinely insane.", "translatedText": "עוד אחד שהיה פשוט מטורף באמת.", "n_reviews": 0, "start": 3895.48, diff --git a/2020/ldm-tips-to-problem-solving/hindi/sentence_translations.json b/2020/ldm-tips-to-problem-solving/hindi/sentence_translations.json index 792c3d186..e28ac8d63 100644 --- a/2020/ldm-tips-to-problem-solving/hindi/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/hindi/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "और फिर, मैं चाहता हूं कि आप जो दूर करना चाहते हैं वह यह सिद्धांत है कि यदि आप एक वस्तु को दो अलग-अलग तरीकों से वर्णित कर सकते हैं, तो गैर-स्पष्ट बीजगणितीय संबंधों या कुछ भी जो शीर्ष पर तत्काल अंतर्ज्ञान के बिना प्रतीकात्मक रूप से लिखा गया है, दिखाने के मामले में बहुत शक्तिशाली है इसका. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "इसकी संभावना 50% होगी. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "और अब शायद आप देख सकते हैं कि यह कहां जाने वाला है, क्योंकि अगले पद के लिए जब हम जानना चाहते हैं कि x को y से विभाजित करने पर 4 और 5 के बीच कब बैठता है, तो हम ऐसी रेखाएं खींचने जा रहे हैं जो 4 पर x पर प्रतिच्छेद करती हैं।", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "किसी समस्या की समरूपता को देखने और उसे किसी ऐसी चीज़ में बदलने का क्या मतलब है जो सूत्रबद्ध रूप से उपयोगी हो? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "मैं और एपिसोड बनाता रहता हूं।", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "तो, आप जानते हैं, हम यहां कुछ उदाहरण देख सकते हैं, कुछ जो 0 तक आए हैं, और कुछ जो 0 तक पूर्णांकित होंगे, कुछ जो 2 तक पूर्णांकित होंगे, कुछ जो 1 तक पूर्णांकित होंगे, तो यह अच्छा है।", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "और यदि मैं इसका माध्य निकालूं, जो सत्य और असत्य को 1 और 0 के रूप में मान रहा है, तो यह मुझे कुल मिलाकर अनुपात बताता है जो वास्तव में 0 होगा।", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "आप जानते हैं, हम किसी ऐसी चीज़ की तलाश कर रहे थे जो 2 के प्राकृतिक लघुगणक को घटाकर 2 का आधा हो।", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "तो शायद हम सिर्फ यह देखना चाहते हैं कि वे मान कब 0 और 20 के बीच में बकेट हो जाते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "और आपको हमेशा इस प्रकार के हिस्टोग्राम पर एक सापेक्ष चौड़ाई जोड़नी होगी क्योंकि अगर वे सभी अगल-बगल हों तो वे बदसूरत दिखते हैं, मुझे कभी-कभी लगता है।", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "और आप यह अच्छी समझ प्राप्त कर सकते हैं कि आपका सारा डेटा क्या है।", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "और उन्हें बस उससे कुछ कलात्मक बनाने के लिए प्रेरित किया गया।", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "एक और जो वास्तव में पागल था।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/hungarian/sentence_translations.json b/2020/ldm-tips-to-problem-solving/hungarian/sentence_translations.json index 8da6e5313..fa87c2d09 100644 --- a/2020/ldm-tips-to-problem-solving/hungarian/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/hungarian/sentence_translations.json @@ -1701,7 +1701,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it.", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it.", "translatedText": "És még egyszer, azt szeretném, ha elvennéd azt az elvet, hogy ha egy objektumot két különböző módon írhatsz le, nagyon hatékonyan nem nyilvánvaló algebrai relációkat vagy bármit, ami szimbolikusan le van írva azonnali intuíció nélkül. abból.", "n_reviews": 0, "start": 1522.72, @@ -2009,7 +2009,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%.", + "input": "This would be a probability of 50 percent.", "translatedText": "Ennek 50%-os a valószínűsége.", "n_reviews": 0, "start": 1824.88, @@ -2611,7 +2611,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4.", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four.", "translatedText": "És most talán láthatja, hová fog ez vezetni, mert a következő tagban, amikor meg akarjuk tudni, hogy x osztva y-vel mikor áll 4 és 5 között, olyan vonalakat fogunk rajzolni, amelyek metszik x-et 4 felett.", "n_reviews": 0, "start": 2464.14, @@ -2968,7 +2968,7 @@ "end": 2840.3 }, { - "input": "And then thinking about like, what is the thing that you can teach someone to say come away and be able to solve problems in the same way?", + "input": "And then thinking about like, what is what is the thing that you can teach someone to say come away and be able to solve problems in the same way?", "translatedText": "És akkor arra gondolok, hogy mi az a dolog, amit meg lehet tanítani valakinek, hogy mondjon el, és képes legyen ugyanígy megoldani a problémákat?", "n_reviews": 0, "start": 2840.74, @@ -3108,7 +3108,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful?", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful?", "translatedText": "Mit jelent az, hogy egy probléma szimmetriáját nézzük, és azt valami képletesen hasznossá alakítjuk?", "n_reviews": 0, "start": 2974.08, @@ -3528,7 +3528,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes.", + "input": "All right, keep making more episodes.", "translatedText": "Folyamatosan készítek több epizódot.", "n_reviews": 0, "start": 3347.7, @@ -3836,7 +3836,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice.", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice.", "translatedText": "Szóval, tudod, láthatunk itt néhány példát, van olyan, amelyik 0-ra nőtt, és van olyan, amelyik 0-ra kerekít, van, amelyik 2-re kerekít, van, amelyik 1-re kerekít, szóval ez szép.", "n_reviews": 0, "start": 3605.04, @@ -3857,7 +3857,7 @@ "end": 3629.32 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s.", + "input": "ro, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros.", "translatedText": "És ha ennek az átlagát veszem, ami azt jelenti, hogy az igazakat és a hamisakat 1-ként és 0-ként kezelem, akkor ez megmutatja, hogy összesen hány arányban lesz valójában 0.", "n_reviews": 0, "start": 3629.32, @@ -3920,7 +3920,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2.", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two.", "translatedText": "Tudod, olyasmit kerestünk, ami 2 fele mínusz 2 természetes logója.", "n_reviews": 0, "start": 3704.66, @@ -3983,7 +3983,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20.", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20.", "translatedText": "Tehát talán csak azt szeretnénk megnézni, hogy ezek az értékek mikor kerülnek 0 és 20 közé.", "n_reviews": 0, "start": 3747.52, @@ -4025,7 +4025,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes.", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes.", "translatedText": "És mindig hozzá kell adni egy relatív szélességet az ilyen típusú hisztogramokhoz, mert néha csúnyán néznek ki, ha egymás mellett vannak.", "n_reviews": 0, "start": 3774.18, @@ -4067,7 +4067,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is.", + "input": "And you can get this, um, this nice sense for what all of your data is.", "translatedText": "És jól megértheti, hogy mi az összes adata.", "n_reviews": 0, "start": 3802.64, @@ -4137,7 +4137,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that.", + "input": "and they were, they were just prompted to create something artistic from that.", "translatedText": "És csak arra ösztönözték őket, hogy ebből alkossanak valami művészi dolgot.", "n_reviews": 0, "start": 3867.72, @@ -4158,7 +4158,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane.", + "input": "Another that was, this is just genuinely insane.", "translatedText": "Egy másik, ami igazán őrült volt.", "n_reviews": 0, "start": 3895.48, diff --git a/2020/ldm-tips-to-problem-solving/indonesian/sentence_translations.json b/2020/ldm-tips-to-problem-solving/indonesian/sentence_translations.json index 3a871d01a..d7d02bdef 100644 --- a/2020/ldm-tips-to-problem-solving/indonesian/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/indonesian/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "Dan lagi, apa yang saya ingin Anda ambil adalah prinsip ini bahwa jika Anda dapat memiliki satu objek yang dideskripsikan dalam dua cara berbeda, sangat kuat dalam hal menunjukkan hubungan aljabar yang tidak jelas atau apa pun yang ditulis secara simbolis tanpa intuisi langsung di atasnya. itu. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "Ini kemungkinannya 50%. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "Dan sekarang mungkin Anda bisa melihat ke mana arahnya, karena untuk suku berikutnya ketika kita ingin mengetahui kapan x dibagi y berada di antara 4 dan 5, kita akan menggambar garis yang berpotongan di x di atas 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "Apa yang dimaksud dengan melihat simetri suatu masalah dan mengubahnya menjadi sesuatu yang berguna secara formula? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "Saya terus membuat lebih banyak episode. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "Jadi, tahukah Anda, kita bisa melihat beberapa contoh di sini, ada yang hasilnya 0, dan ada yang dibulatkan ke bawah menjadi 0, ada yang dibulatkan ke bawah menjadi 2, ada yang dibulatkan ke bawah menjadi 1, jadi itu bagus. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "Dan jika saya mengambil mean dari hal tersebut, yaitu memperlakukan nilai benar dan salah sebagai angka 1 dan 0, maka ini memberi tahu saya proporsi total yang sebenarnya adalah angka 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "Anda tahu, kami sedang mencari sesuatu yang setengah dari 2 dikurangi logaritma natural dari 2. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "Jadi mungkin kita hanya ingin melihat kapan nilai-nilai tersebut dimasukkan di antara 0 dan 20. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "Dan Anda selalu harus menambahkan lebar relatif pada histogram semacam ini karena histogram tersebut akan terlihat jelek jika semuanya berdampingan, menurut saya terkadang. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "Dan Anda bisa memahami semua data Anda. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "Dan mereka terdorong untuk menciptakan sesuatu yang artistik dari situ. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "Satu lagi yang benar-benar gila. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/italian/sentence_translations.json b/2020/ldm-tips-to-problem-solving/italian/sentence_translations.json index 785e71de2..bed6e8799 100644 --- a/2020/ldm-tips-to-problem-solving/italian/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/italian/sentence_translations.json @@ -1701,7 +1701,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it.", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it.", "translatedText": "E ancora, quello che voglio che tu impari è questo principio secondo cui se puoi avere un oggetto descritto in due modi diversi, molto potente in termini di mostrare relazioni algebriche non ovvie o qualsiasi cosa che sia scritta simbolicamente senza un'intuizione immediata in cima di esso.", "n_reviews": 0, "start": 1522.72, @@ -2009,7 +2009,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%.", + "input": "This would be a probability of 50 percent.", "translatedText": "Questa sarebbe una probabilità del 50%.", "n_reviews": 0, "start": 1824.88, @@ -2611,7 +2611,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4.", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four.", "translatedText": "E ora forse puoi vedere dove andrà a finire, perché per il prossimo termine, quando vogliamo sapere quando x diviso per y sta tra 4 e 5, disegneremo delle linee che si intersecano in x su 4.", "n_reviews": 0, "start": 2464.14, @@ -2968,7 +2968,7 @@ "end": 2840.3 }, { - "input": "And then thinking about like, what is the thing that you can teach someone to say come away and be able to solve problems in the same way?", + "input": "And then thinking about like, what is what is the thing that you can teach someone to say come away and be able to solve problems in the same way?", "translatedText": "E poi pensando a qualcosa del genere, qual è la cosa che puoi insegnare a qualcuno per dire "vieni via" ed essere in grado di risolvere i problemi allo stesso modo?", "n_reviews": 0, "start": 2840.74, @@ -3108,7 +3108,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful?", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful?", "translatedText": "Cosa significa osservare la simmetria di un problema e trasformarla in qualcosa che sia formalmente utile?", "n_reviews": 0, "start": 2974.08, @@ -3528,7 +3528,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes.", + "input": "All right, keep making more episodes.", "translatedText": "Continuo a fare più episodi.", "n_reviews": 0, "start": 3347.7, @@ -3836,7 +3836,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice.", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice.", "translatedText": "Quindi, sai, possiamo vedere alcuni esempi qui, alcuni che sono arrivati a 0, e alcuni che arrotonderebbero per difetto a 0, alcuni che arrotonderebbero per difetto a 2, altri che arrotonderebbero per difetto a 1, quindi è carino.", "n_reviews": 0, "start": 3605.04, @@ -3857,7 +3857,7 @@ "end": 3629.32 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s.", + "input": "ro, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros.", "translatedText": "E se ne prendo la media, che tratta i veri e i falsi come 1 e 0, questo mi dice la proporzione in totale che sarà effettivamente 0.", "n_reviews": 0, "start": 3629.32, @@ -3920,7 +3920,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2.", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two.", "translatedText": "Sai, stavamo cercando qualcosa che fosse la metà di 2 meno il logaritmo naturale di 2.", "n_reviews": 0, "start": 3704.66, @@ -3983,7 +3983,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20.", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20.", "translatedText": "Quindi forse vogliamo solo vedere quando questi valori vengono compresi tra 0 e 20.", "n_reviews": 0, "start": 3747.52, @@ -4025,7 +4025,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes.", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes.", "translatedText": "E devi sempre aggiungere una larghezza relativa a questo tipo di istogrammi perché sembrano brutti se sono tutti affiancati, penso a volte.", "n_reviews": 0, "start": 3774.18, @@ -4067,7 +4067,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is.", + "input": "And you can get this, um, this nice sense for what all of your data is.", "translatedText": "E puoi avere questa bella idea di cosa siano tutti i tuoi dati.", "n_reviews": 0, "start": 3802.64, @@ -4137,7 +4137,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that.", + "input": "and they were, they were just prompted to create something artistic from that.", "translatedText": "E sono stati semplicemente spinti a creare qualcosa di artistico da quello.", "n_reviews": 0, "start": 3867.72, @@ -4158,7 +4158,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane.", + "input": "Another that was, this is just genuinely insane.", "translatedText": "Un altro che era semplicemente veramente folle.", "n_reviews": 0, "start": 3895.48, diff --git a/2020/ldm-tips-to-problem-solving/japanese/sentence_translations.json b/2020/ldm-tips-to-problem-solving/japanese/sentence_translations.json index f43b39e0f..72eda5bf7 100644 --- a/2020/ldm-tips-to-problem-solving/japanese/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/japanese/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "そして、繰り返しになりますが、私が皆さんに理解していただきたいのは、1 つのオブジェクトを 2 つの異なる方法で記述することが できれば、非自明な代数関係や、直接の直観なしに象徴的に記述されたものを示すという点で非常に強力であるという原則です。それの。", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "この確率は50 %になります。", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "これで、これがどこに行くのかがわかる かもしれません。次の項では、x を y で割った値が 4 と 5 の間に いつ収まるかを知りたいときに、4 上 の x で交差する線を引くことにな るからです。", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "問題の対称性を調べて、それを公式に役立つものに変えるとはどういう意味でしょうか? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "さらにエピソードを作り続けています。", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "ここでいくつかの例を見ることができます。0 になったもの、0 に切り捨てられるもの、2 に切り捨てられるもの、1 に切り捨てられるものがあります。これは素晴らしいことです。", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "そして、真と偽を 1 と 0 として扱う平均値を取ると、実際に 0 になる合計の割合がわかります。", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "ご存知のとおり、私たちは 2 の半分から 2 の自然対数を引いたものを探していました。", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "したがって、これらの値が 0 と 20 の間でバケット化されるタイミングを確認したいだけかもしれません。", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "そして、この種のヒストグラムには常に相対的な幅を追加する必要があります。なぜなら、すべてが横に並んでいると見苦しくなるからです。", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "そして、すべてのデータが何であるかをよく理解できます。", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "そして、彼らはそこから何か芸術的なものを生み出すよう促されたのです。", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "もう一つは本当に狂っていた。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/korean/sentence_translations.json b/2020/ldm-tips-to-problem-solving/korean/sentence_translations.json index 8be22fe39..7c19535b7 100644 --- a/2020/ldm-tips-to-problem-solving/korean/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/korean/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "그리고 다시 한 번 강조하고 싶은 것은 하나의 객체를 두 가지 다른 방식으로 기술할 수 있다면, 명백하지 않은 대수적 관계나 즉각적인 직관 없이 상징적으로 적힌 모든 것을 보여주는 측면에서 매우 강력하다는 원칙입니다. 그것의. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "확률은 50%일 겁니다. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "이제 이것이 어디로 가는지 알 수 있을 것입니다. 왜냐하면 다음 학기에 x를 y로 나눈 값이 4와 5 사이에 있는지 알고 싶을 때 x에서 4와 교차하는 선을 그릴 것이기 때문입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "문제의 대칭성을 살펴보고 이를 공식적으로 유용한 것으로 바꾸는 것은 무엇을 의미합니까? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "계속 에피소드를 더 만들고 있어요. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "여기에서 몇 가지 예를 볼 수 있습니다. 일부는 0이 되었고, 일부는 0으로 반올림되고, 일부는 2로 반내림되고, 일부는 1로 반내림되었습니다. 정말 좋습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "그리고 참과 거짓을 1과 0으로 처리하는 평균을 취하면 실제로 0이 될 전체 비율을 알 수 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "아시다시피, 우리는 2의 절반에서 2의 자연로그를 뺀 값을 찾고 있었습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "따라서 해당 값이 0과 20 사이에 버킷으로 묶이는 시점을 살펴보고 싶을 수도 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "그리고 항상 이런 종류의 히스토그램에 상대 너비를 추가해야 합니다. 왜냐하면 그것들이 모두 나란히 있으면 보기 흉해 보이기 때문입니다. 제 생각에는 때때로 그렇습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "그리고 모든 데이터가 무엇인지에 대한 좋은 감각을 얻을 수 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "그리고 그들은 그것으로부터 예술적인 것을 창조하라는 메시지를 받았습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "또 다른 하나는 정말 미쳤습니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/marathi/sentence_translations.json b/2020/ldm-tips-to-problem-solving/marathi/sentence_translations.json index 9ee49d5b1..ad75de541 100644 --- a/2020/ldm-tips-to-problem-solving/marathi/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/marathi/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "आणि पुन्हा, मी तुम्हाला काढून टाकू इच्छितो ते हे तत्त्व आहे की जर तुमच्याकडे दोन वेगवेगळ्या प्रकारे वर्णन केलेली एक वस्तू असू शकते, जे स्पष्ट नसलेले बीजगणितीय संबंध दर्शविण्याच्या दृष्टीने खूप शक्तिशाली आहे किंवा वरच्या तात्काळ अंतर्ज्ञानाशिवाय प्रतिकात्मकपणे लिहिलेले काहीही आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "हे 50% ची संभाव्यता असेल. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "आणि आता हे कुठे जाणार आहे ते तुम्ही पाहू शकता, कारण पुढच्या टर्मसाठी जेव्हा आपल्याला हे जाणून घ्यायचे असेल की x ला y ने भागलेला 4 आणि 5 मध्ये कधी बसतो, तेव्हा आपण x 4 वर छेदणाऱ्या रेषा काढणार आहोत. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "समस्येची सममिती पाहणे आणि त्यास सूत्रानुसार उपयुक्त असलेल्या गोष्टीत बदलणे म्हणजे काय? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "मी आणखी एपिसोड बनवत राहते. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "तर, तुम्हाला माहिती आहे, आपण येथे काही उदाहरणे पाहू शकतो, काही जी 0 वर आली आहेत, आणि काही जी 0 पर्यंत पूर्ण होतील, काही जी 2 पर्यंत पूर्ण होतील, काही जी 1 पर्यंत पूर्ण होतील, त्यामुळे ते छान आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "आणि जर मी याचा अर्थ घेतला, जे खरे आणि खोटे यांना 1s आणि 0s मानत आहे, तर हे मला एकूण प्रमाण सांगते जे प्रत्यक्षात 0s असेल. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "तुम्हाला माहिती आहे, आम्ही असे काहीतरी शोधत होतो जे 2 वजा 2 च्या नैसर्गिक लॉगच्या निम्मे होते. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "त्यामुळे कदाचित ती मूल्ये 0 आणि 20 सारख्या मध्ये कधी बकेट होतात हे आपण पाहू इच्छितो. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "आणि तुम्हाला या प्रकारच्या हिस्टोग्रामवर नेहमी सापेक्ष रुंदी जोडावी लागते कारण ते सर्व शेजारी शेजारी असल्यास ते फक्त कुरूप दिसतात, मला कधीकधी वाटते. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "आणि तुमचा सर्व डेटा काय आहे याबद्दल तुम्हाला ही चांगली जाणीव होऊ शकते. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "आणि त्यातूनच काहीतरी कलात्मक बनवायला सांगितलं होतं. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "आणखी एक जे खरोखरच वेडेपणाचे होते. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/persian/sentence_translations.json b/2020/ldm-tips-to-problem-solving/persian/sentence_translations.json index 3f1061f86..4ab53aec7 100644 --- a/2020/ldm-tips-to-problem-solving/persian/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/persian/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "و باز هم، چیزی که از شما می‌خواهم حذف کنید این اصل است که اگر می‌توانید یک شی را به دو روش مختلف توصیف کنید، بسیار قدرتمند از نظر نشان دادن روابط جبری غیر آشکار یا هر چیزی که به‌صورت نمادین بدون شهود فوری در بالا نوشته شده است. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "این احتمال 50 درصد خواهد بود. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "و حالا شاید بتوانید ببینید که این به کجا می‌رود، زیرا برای ترم بعدی که می‌خواهیم بدانیم چه زمانی x تقسیم بر y بین 4 و 5 قرار می‌گیرد، خطوطی را ترسیم می‌کنیم که x روی 4 را قطع می‌کنند. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "شاید برخی از مواردی که امروز می‌خواهم درباره آنها صحبت کنم، مانند، خوب، می‌توانم بگویم تقارن اهرمی، اما واقعاً این به چه معناست؟ نگاه کردن به تقارن یک مسئله و تبدیل آن به چیزی که از نظر فرمول مفید است به چه معناست؟ شما فقط باید آن را زیاد ببینید و سپس در هنگام انجام این کار متفکر باشید. ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "من به ساخت قسمت های بیشتری ادامه می دهم. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "بنابراین، می‌دانید، ما می‌توانیم برخی از نمونه‌ها را در اینجا ببینیم، برخی از آنها به 0 رسیده‌اند، و برخی که به 0 پایین می‌آیند، برخی که به 2 پایین می‌آیند، برخی که به 1 می‌رسند، پس خوب است. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "و اگر میانگین آن را که درست و نادرست را 1 و 0 می‌دانم در نظر بگیرم، این نسبت در کل به من می‌گوید که در واقع 0s خواهد بود. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "می دانید، ما به دنبال چیزی بودیم که نصف 2 منهای ثبت طبیعی 2 باشد. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "بنابراین شاید ما فقط بخواهیم ببینیم چه زمانی این مقادیر بین 0 تا 20 قرار می گیرند. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "و شما همیشه باید عرض نسبی را روی این نوع هیستوگرام ها اضافه کنید، زیرا به نظر من گاهی اوقات، اگر همه آنها در کنار هم باشند، زشت به نظر می رسند. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "و شما می توانید این حس خوب را برای آنچه که همه داده های شما هستند دریافت کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "و فقط از آنها خواسته شد که از آن چیزی هنری خلق کنند. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "دیگری که واقعاً دیوانه بود. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/portuguese/sentence_translations.json b/2020/ldm-tips-to-problem-solving/portuguese/sentence_translations.json index ef3b67e5c..879fff9f5 100644 --- a/2020/ldm-tips-to-problem-solving/portuguese/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/portuguese/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "E novamente, o que eu quero que você tire é este princípio de que se você puder ter um objeto descrito de duas maneiras diferentes, muito poderoso em termos de mostrar relações algébricas não óbvias ou qualquer coisa que seja escrita simbolicamente sem intuição imediata no topo disso. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "Isso seria uma probabilidade de 50%. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "E agora talvez você possa ver onde isso vai dar, porque para o próximo termo, quando quisermos saber quando x dividido por y fica entre 4 e 5, desenharemos retas que se cruzam em x sobre 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "O que significa olhar para a simetria de um problema e transformá-la em algo que seja útil em termos de fórmulas? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "Continuo fazendo mais episódios. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "Então, você sabe, podemos ver alguns exemplos aqui, alguns que chegaram a 0, e alguns que seriam arredondados para 0, alguns que seriam arredondados para 2, alguns que seriam arredondados para 1, então isso é legal. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "E se eu pegar a média disso, que é tratar os verdadeiros e os falsos como 1s e 0s, isso me diz a proporção no total que será realmente 0s. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "Você sabe, estávamos procurando algo que fosse metade de 2 menos o logaritmo natural de 2. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "Então, talvez queiramos apenas ver quando esses valores ficam agrupados entre 0 e 20. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "E você sempre tem que adicionar uma largura relativa a esses tipos de histogramas, porque eles ficam feios se estiverem todos lado a lado, às vezes acho. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "E você pode ter uma boa noção do que são todos os seus dados. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "E eles foram incentivados a criar algo artístico a partir disso. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "Outro que era genuinamente insano. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/russian/sentence_translations.json b/2020/ldm-tips-to-problem-solving/russian/sentence_translations.json index e69776165..599a99e2e 100644 --- a/2020/ldm-tips-to-problem-solving/russian/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/russian/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "И опять же, я хочу, чтобы вы усвоили принцип: если вы можете описать один объект двумя разными способами, это очень эффективно с точки зрения демонстрации неочевидных алгебраических отношений или чего-либо, что записано символически без непосредственной интуиции сверху. этого. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "Это будет вероятность 50%. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "И теперь, возможно, вы понимаете, к чему это приведет, потому что в следующем семестре, когда мы хотим узнать, когда x, разделенный на y, находится между 4 и 5, мы собираемся рисовать линии, которые пересекаются в точке x более 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "Что значит взглянуть на симметрию проблемы и превратить ее в нечто формально полезное? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "Я продолжаю делать больше серий. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "Итак, вы знаете, мы можем увидеть здесь несколько примеров: некоторые из них дошли до 0, а некоторые округлялись до 0, некоторые округлялись до 2, некоторые округлялись до 1, так что это хорошо. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "И если я возьму среднее значение, при котором истинные и ложные значения рассматриваются как 1 и 0, это покажет мне общую долю, которая на самом деле будет равна 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "Знаете, мы искали что-то равное половине 2 минус натуральный логарифм 2. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "Так что, возможно, мы просто хотим посмотреть, когда эти значения распределяются между 0 и 20. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "И вам всегда нужно добавлять относительную ширину к таким гистограммам, потому что они выглядят просто некрасиво, если расположены рядом, как мне иногда кажется. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "И вы можете получить хорошее представление о том, что представляют собой все ваши данные. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "И им просто предложили создать из этого что-то художественное. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "Еще один, который был просто по-настоящему безумным. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/spanish/sentence_translations.json b/2020/ldm-tips-to-problem-solving/spanish/sentence_translations.json index 175c50616..999e6bb06 100644 --- a/2020/ldm-tips-to-problem-solving/spanish/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/spanish/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "Y nuevamente, lo que quiero que usted recuerde es este principio de que si puede describir un objeto de dos maneras diferentes, esto es muy poderoso en términos de mostrar relaciones algebraicas no obvias o cualquier cosa que esté escrita simbólicamente sin una intuición inmediata encima. de ello. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "Esta sería una probabilidad del 50%. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "Y ahora tal vez puedas ver hacia dónde irá esto, porque para el próximo período, cuando queramos saber cuándo se ubica x dividido por y entre 4 y 5, dibujaremos líneas que se intersectan en x sobre 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "¿Qué significa observar la simetría de un problema y convertirlo en algo que sea útil desde el punto de vista de la fórmula? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "Sigo haciendo más episodios. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "Entonces, ya saben, podemos ver algunos ejemplos aquí, algunos que han llegado a 0 y algunos que se redondearían hacia abajo a 0, algunos que se redondearían hacia abajo a 2, algunos que se redondearían hacia abajo a 1, así que eso está bien. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "Y si tomo la media de eso, que es tratar los verdaderos y falsos como 1 y 0, esto me dice la proporción total que en realidad será 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "Sabes, estábamos buscando algo que fuera la mitad de 2 menos el logaritmo natural de 2. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "Entonces, tal vez solo queramos ver cuándo esos valores se agrupan entre 0 y 20. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "Y a veces creo que siempre hay que agregar un ancho relativo a este tipo de histogramas porque se ven feos si están todos uno al lado del otro. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "Y puede tener una buena idea de cuáles son todos sus datos. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "Y simplemente se les impulsó a crear algo artístico a partir de eso. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "Otro que era simplemente una locura. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/tamil/sentence_translations.json b/2020/ldm-tips-to-problem-solving/tamil/sentence_translations.json index f582a150e..8f42c3b67 100644 --- a/2020/ldm-tips-to-problem-solving/tamil/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/tamil/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "மீண்டும், நீங்கள் எடுத்துச் சொல்ல விரும்புவது இந்தக் கொள்கையை, இரண்டு வெவ்வேறு வழிகளில் விவரிக்கப்பட்ட ஒரு பொருளை நீங்கள் கொண்டிருக்க முடியும் என்றால், வெளிப்படையான இயற்கணித உறவுகளைக் காண்பிக்கும் வகையில் மிகவும் சக்தி வாய்ந்தது அல்லது மேலே உடனடி உள்ளுணர்வு இல்லாமல் குறியீடாக எழுதப்பட்ட எதையும். அதில். ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "இது 50% நிகழ்தகவாக இருக்கும். ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "இப்போது இது எங்கே போகிறது என்பதை நீங்கள் பார்க்கலாம், ஏனென்றால் அடுத்த காலத்திற்கு x 4 மற்றும் 5 க்கு இடையில் y ஆல் வகுபடும் போது, நாங்கள் 4 க்கு மேல் x இல் வெட்டும் கோடுகளை வரையப் போகிறோம். ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "ஒரு சிக்கலின் சமச்சீர்நிலையைப் பார்த்து, அதை முறைப்படி பயனுள்ள ஒன்றாக மாற்றுவது என்றால் என்ன? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "தொடர்ந்து பல எபிசோட்களை உருவாக்குகிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "எனவே, உங்களுக்குத் தெரியும், சில உதாரணங்களை இங்கே பார்க்கலாம், சில 0 வரை வந்துள்ளன, மேலும் சிலவற்றை 0 வரை சுற்றி விடும், சில 2 வரை சுற்றும், சில 1 வரை சுற்றும், அதனால் நன்றாக இருக்கிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "உண்மை மற்றும் பொய்களை 1 வி மற்றும் 0 வி என்று நான் சராசரியாக எடுத்துக் கொண்டால், இது உண்மையில் 0 வி ஆக இருக்கும் மொத்த விகிதத்தைக் கூறுகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "உங்களுக்குத் தெரியும், 2 இன் இயற்கைப் பதிவைக் கழித்தல் 2 இல் பாதி இருக்கும் ஒன்றை நாங்கள் தேடுகிறோம். ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "எனவே அந்த மதிப்புகள் 0 மற்றும் 20 க்கு இடையில் எப்போது பக்கெட் செய்யப்படுகின்றன என்பதை நாம் பார்க்க விரும்பலாம். ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "இந்த வகையான ஹிஸ்டோகிராம்களில் நீங்கள் எப்போதும் தொடர்புடைய அகலத்தைச் சேர்க்க வேண்டும், ஏனென்றால் அவை அனைத்தும் அருகருகே இருந்தால் அவை அசிங்கமாக இருக்கும், நான் சில நேரங்களில் நினைக்கிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "உங்கள் எல்லா தரவுகளும் என்ன என்பதை நீங்கள் இந்த நல்ல உணர்வைப் பெறலாம். ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "அதிலிருந்து ஏதாவது கலையை உருவாக்க அவர்கள் தூண்டப்பட்டனர். ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "இன்னொன்று உண்மையிலேயே பைத்தியக்காரத்தனமாக இருந்தது. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/telugu/sentence_translations.json b/2020/ldm-tips-to-problem-solving/telugu/sentence_translations.json index 6a8dcb9d6..4334a64c5 100644 --- a/2020/ldm-tips-to-problem-solving/telugu/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/telugu/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "మరలా, మీరు తీసివేయాలని నేను కోరుకుంటున్నది ఏమిటంటే, మీరు ఒక వస్తువును రెండు రకాలుగా వర్ణించగలిగితే, స్పష్టమైన బీజగణిత సంబంధాలను చూపించే పరంగా చాలా శక్తివంతమైనది లేదా పైన తక్షణ అంతర్ దృష్టి లేకుండా ప్రతీకాత్మకంగా వ్రాసిన ఏదైనా అందులో. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "ఇది 50% సంభావ్యత. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "మరియు ఇప్పుడు ఇది ఎక్కడికి వెళ్తుందో మీరు చూడవచ్చు, ఎందుకంటే తదుపరి టర్మ్ కోసం x 4 మరియు 5 మధ్య కూర్చునే x ఎప్పుడు భాగించబడుతుందో తెలుసుకోవాలనుకున్నప్పుడు, మేము 4 కంటే x వద్ద కలిసే పంక్తులను గీయబోతున్నాము. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "సమస్య యొక్క సమరూపతను చూడటం మరియు దానిని సూత్రప్రాయంగా ఉపయోగకరమైనదిగా మార్చడం అంటే ఏమిటి? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "నేను మరిన్ని ఎపిసోడ్‌లు చేస్తూనే ఉన్నాను. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "కాబట్టి, మీకు తెలుసా, మేము ఇక్కడ కొన్ని ఉదాహరణలను చూడవచ్చు, కొన్ని 0 వరకు వచ్చినవి, మరియు కొన్ని 0కి చుట్టుముట్టేవి, కొన్ని 2కి చుట్టుముట్టేవి, కొన్ని 1కి రౌండ్ చేసేవి, కాబట్టి అది బాగుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "నేను ట్రూలు మరియు ఫాల్స్‌లను 1సె మరియు 0సెలుగా పరిగణిస్తున్న దాని సగటును తీసుకుంటే, ఇది నాకు మొత్తం నిష్పత్తిని చెబుతుంది, అది నిజానికి 0సె. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "మీకు తెలుసా, మేము 2లో సగం మైనస్ 2 యొక్క సహజ లాగ్‌ని వెతుకుతున్నాము. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "కాబట్టి 0 మరియు 20 మధ్య ఆ విలువలు ఎప్పుడు బకెట్ చేయబడతాయో మనం చూడాలనుకుంటున్నాము. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "మరియు మీరు ఎల్లప్పుడూ ఈ రకమైన హిస్టోగ్రామ్‌లపై సాపేక్ష వెడల్పును జోడించాలి ఎందుకంటే అవన్నీ పక్కపక్కనే ఉంటే అవి అగ్లీగా కనిపిస్తాయి, నేను కొన్నిసార్లు అనుకుంటాను. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "మరియు మీ మొత్తం డేటా ఏమిటో మీరు ఈ చక్కని భావాన్ని పొందవచ్చు. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "మరియు వారు దాని నుండి కళాత్మకమైనదాన్ని సృష్టించమని ప్రాంప్ట్ చేయబడ్డారు. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "మరొకటి నిజంగా పిచ్చిగా ఉంది. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/thai/sentence_translations.json b/2020/ldm-tips-to-problem-solving/thai/sentence_translations.json index 7cab389a8..7cf4f758f 100644 --- a/2020/ldm-tips-to-problem-solving/thai/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/thai/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "เมื่อ y มากกว่าสองเท่าของ x เป็น แต่ไม่เกิน 3 เท่าของ x เป็น มันง่ายมากที่จะสับสนแบบนั้น และมันก็ยากยิ่งขึ้นไปอีกที่จะพยายามระบุความน่าจะเป็นนั้น ในขณะที่ในแผนภาพของเรา มันมีความหมายที่ชัดเจนมาก มันคือพื้นที่ของบริเวณนี้ เพราะพื้นที่ความเป็นไปได้ทั้งหมดคือ 1 แล้ว นี่ ความน่าจะเป็นที่บางสิ่งจะเกิดขึ้นควรเป็น 1 และเราแค่ต้องดูพื้นที่ของมัน และตอนนี้คุณคงเห็นแล้วว่า สิ่งนี้จะไปทางไหน เพราะในเทอมหน้า เมื่อเราอยากรู้ว่าเมื่อใดที่ x หารด้วย y อยู่ระหว่าง 4 ถึง 5 เราจะวาดเส้นที่ตัดกันที่ x ส่วน 4 บางทีฉันอาจจะไปสีอื่นสำหรับสีนี้ x ส่วน 4 และ x ส่วน 5 ซึ่งต้องใช้ลายมือน้อยมาก ณ จุดนี้ แต่ผมจะลองดู. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "56 ไปเรื่อยๆ เราก็เห็นอัตราส่วนทั้งหมดแบบนั้นได้ โดยทั่วไป ผมอาจนิยามรายการอัตราส่วน โดยผมจะหาตัวเลขสุ่มจำนวนหนึ่ง ขนาด n แล้วหารมันด้วยตัวเลขสุ่มอีกกลุ่มที่มีขนาดเท่ากัน n ไม่ได้ถูกกำหนดไว้ แต่ผมจะให้คำจำกัดความ ตัวเลขที่ดีและจำนวนมาก เช่น หนึ่งล้าน และตอนนี้ ผมถืออัตราส่วนไว้เป็นล้านอัตราส่วน หนึ่งล้านตัวอย่างของ x หารด้วย y โดยที่ x กับ y ถูกเลือกตามข้อจำกัดของเรา ซึ่งมันเจ๋งมาก ถ้าคุณลองคิดดู คุณรู้ไหมว่า เราเห็นตัวอย่างบางส่วนตรงนี้ บางตัวอย่างมีค่าเป็น 0 บางตัวอย่างปัดเศษลงเป็น 0 บางตัวอย่างปัดเศษลงเป็น 2 บางตัวอย่างปัดเศษลงเป็น 1 ก็ดีเลย และตอนนี้ ผมเริ่มถามคำถามได้ เช่น คุณก็รู้ เมื่อใดที่ผมจะหาอัตราส่วนเหล่านี้ ผมหาฟังก์ชันพื้นของพวกมัน ไม่ใช่พื้น พื้น และผมอยากรู้ว่าเมื่อไรจะเท่ากับ 0 นี่ให้รายการจริงและเท็จ โดยพื้นฐานแล้วบอกว่าเป็นหรือไม่ใช่ 0 และหากผมหาค่าเฉลี่ยของค่านั้น ซึ่งถือว่าจริงและเท็จเป็น 1 วินาทีกับ 0 นี่บอกสัดส่วนรวมทั้งหมดที่เป็น 0 จริง ๆ เราคาดว่ามันจะเป็นประมาณครึ่งหนึ่ง และเราสามารถตรวจสอบได้ ใช่ โอเค มันเป็นประมาณครึ่งหนึ่ง เราอาจถามด้วยว่าเมื่อไรจะประมาณ 2 โอเค และดูเหมือน 0 83. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/turkish/sentence_translations.json b/2020/ldm-tips-to-problem-solving/turkish/sentence_translations.json index 1caeb2a88..793e3eeb4 100644 --- a/2020/ldm-tips-to-problem-solving/turkish/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/turkish/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "Ve yine, sizden almanızı istediğim şey şu prensiptir: Eğer bir nesne iki farklı şekilde tanımlanabiliyorsa, bu, bariz olmayan cebirsel ilişkileri veya doğrudan sezgi olmaksızın sembolik olarak yazılan herhangi bir şeyi gösterme açısından çok güçlüdür. ondan. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "Bu %50 ihtimal olacaktır. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "Belki şimdi bunun nereye gideceğini görebilirsiniz, çünkü bir sonraki terimde x bölü y'nin ne zaman 4 ile 5 arasında olduğunu bilmek istediğimizde, x bölü 4'te kesişen çizgiler çizeceğiz. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "Bir problemin simetrisine bakıp bunu formülsel olarak yararlı bir şeye dönüştürmek ne anlama gelir? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "Daha fazla bölüm çekmeye devam ediyorum. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "Yani, burada bazı örnekleri görebiliyoruz, bazıları 0'a kadar çıkıyor, bazıları 0'a yuvarlanıyor, bazıları 2'ye yuvarlanıyor, bazıları da 1'e yuvarlanıyor, yani bu güzel. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "Ve eğer doğruları ve yanlışları 1'ler ve 0'lar olarak ele alan bunun ortalamasını alırsam, bu bana toplamda aslında 0'lar olacak oranı söyler. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "Biliyorsunuz, 2'nin yarısı eksi 2'nin doğal logaritması olan bir şey arıyorduk. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "Belki de bu değerlerin 0 ile 20 arasında ne zaman gruplandığına bakmak istiyoruz. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "Ve bu tür histogramlara her zaman göreceli bir genişlik eklemeniz gerekir çünkü bazen, yan yana olduklarında çirkin görünürler diye düşünüyorum. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "Ve tüm verilerinizin ne olduğuna dair bu güzel anlayışı elde edebilirsiniz. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "Ve onlardan sanatsal bir şey yaratmaları istendi. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "Bir başkası gerçekten deliydi. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/ukrainian/sentence_translations.json b/2020/ldm-tips-to-problem-solving/ukrainian/sentence_translations.json index d4bf40544..a506f3dee 100644 --- a/2020/ldm-tips-to-problem-solving/ukrainian/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/ukrainian/sentence_translations.json @@ -1701,7 +1701,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it.", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it.", "translatedText": "І знову я хочу, щоб ви забрали цей принцип, що якщо ви можете мати один об’єкт, описаний двома різними способами, дуже потужними з точки зору показу неочевидних алгебраїчних співвідношень або будь-чого, що начебто записане символічно без безпосередньої інтуїції зверху цього.", "n_reviews": 0, "start": 1522.72, @@ -2009,7 +2009,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%.", + "input": "This would be a probability of 50 percent.", "translatedText": "Це буде ймовірність 50%.", "n_reviews": 0, "start": 1824.88, @@ -2611,7 +2611,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4.", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four.", "translatedText": "І тепер, можливо, ви бачите, куди це піде, тому що для наступного терміну, коли ми хочемо знати, коли х, поділене на у, знаходиться між 4 і 5, ми малюватимемо лінії, які перетинаються в х на 4.", "n_reviews": 0, "start": 2464.14, @@ -2968,7 +2968,7 @@ "end": 2840.3 }, { - "input": "And then thinking about like, what is the thing that you can teach someone to say come away and be able to solve problems in the same way?", + "input": "And then thinking about like, what is what is the thing that you can teach someone to say come away and be able to solve problems in the same way?", "translatedText": "А потім подумайте про те, чого ви можете навчити когось сказати, що «відходьте» і вирішуйте проблеми таким же чином?", "n_reviews": 0, "start": 2840.74, @@ -3108,7 +3108,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful?", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful?", "translatedText": "Що означає дивитися на симетрію проблеми і перетворювати її на щось формульно корисне?", "n_reviews": 0, "start": 2974.08, @@ -3528,7 +3528,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes.", + "input": "All right, keep making more episodes.", "translatedText": "Я продовжую знімати нові епізоди.", "n_reviews": 0, "start": 3347.7, @@ -3836,7 +3836,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice.", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice.", "translatedText": "Отже, ви знаєте, ми можемо побачити кілька прикладів тут, деякі з них дійшли до 0, а деякі округлили до 0, деякі округлили до 2, деякі округлили до 1, так що це добре.", "n_reviews": 0, "start": 3605.04, @@ -3857,7 +3857,7 @@ "end": 3629.32 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s.", + "input": "ro, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros.", "translatedText": "І якщо я візьму середнє значення, яке розглядає істинні та хибні значення як 1 і 0, це скаже мені, яка загальна частка насправді буде 0.", "n_reviews": 0, "start": 3629.32, @@ -3920,7 +3920,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2.", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two.", "translatedText": "Знаєте, ми шукали щось, що дорівнює половині 2 мінус натуральний логарифм 2.", "n_reviews": 0, "start": 3704.66, @@ -3983,7 +3983,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20.", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20.", "translatedText": "Тому, можливо, ми просто хочемо подивитися, коли ці значення розподіляються між 0 і 20.", "n_reviews": 0, "start": 3747.52, @@ -4025,7 +4025,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes.", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes.", "translatedText": "І ви завжди повинні додавати відносну ширину до такого роду гістограм, тому що вони просто виглядають потворно, якщо вони розташовані пліч-о-пліч, я іноді думаю.", "n_reviews": 0, "start": 3774.18, @@ -4067,7 +4067,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is.", + "input": "And you can get this, um, this nice sense for what all of your data is.", "translatedText": "І ви можете зрозуміти, що таке всі ваші дані.", "n_reviews": 0, "start": 3802.64, @@ -4137,7 +4137,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that.", + "input": "and they were, they were just prompted to create something artistic from that.", "translatedText": "І їх просто спонукало створити з цього щось художнє.", "n_reviews": 0, "start": 3867.72, @@ -4158,7 +4158,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane.", + "input": "Another that was, this is just genuinely insane.", "translatedText": "Інший, який був просто щиро божевільним.", "n_reviews": 0, "start": 3895.48, diff --git a/2020/ldm-tips-to-problem-solving/urdu/sentence_translations.json b/2020/ldm-tips-to-problem-solving/urdu/sentence_translations.json index f082550e6..25403687f 100644 --- a/2020/ldm-tips-to-problem-solving/urdu/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/urdu/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "اور ایک بار پھر، میں آپ سے یہ اصول ہٹانا چاہتا ہوں کہ اگر آپ کے پاس ایک چیز ہے جسے دو مختلف طریقوں سے بیان کیا جا سکتا ہے، غیر واضح الجبری تعلقات کو ظاہر کرنے کے لحاظ سے بہت طاقتور ہے یا کوئی بھی چیز جو علامتی طور پر لکھی گئی ہے بغیر فوری وجدان کے۔", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "یہ 50٪ کا امکان ہوگا۔", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "اور اب شاید آپ دیکھ سکتے ہیں کہ یہ کہاں جا رہا ہے، کیونکہ اگلی مدت کے لیے جب ہم یہ جاننا چاہتے ہیں کہ x کو y سے تقسیم کرنے پر 4 اور 5 کے درمیان کب بیٹھتا ہے، تو ہم ایسی لکیریں کھینچنے جا رہے ہیں جو x کو 4 سے زیادہ کاٹتی ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "ہوسکتا ہے کہ میں آج جن کے بارے میں بات کرنے کی کوشش کر رہا ہوں، جیسے، ٹھیک ہے، میں لیوریج سمیٹری کہہ سکتا ہوں، لیکن اس کا اصل مطلب کیا ہے؟ کسی مسئلے کی ہم آہنگی کو دیکھنے اور اسے فارمولہ طور پر مفید چیز میں تبدیل کرنے کا کیا مطلب ہے؟ آپ کو صرف اسے بہت کچھ دیکھنا ہے، اور پھر جب آپ ایسا کرتے ہیں تو فکر مند رہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "میں مزید اقساط بناتا رہتا ہوں۔", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "تو، آپ جانتے ہیں، ہم یہاں کچھ مثالیں دیکھ سکتے ہیں، کچھ جو 0 تک آچکی ہیں، اور کچھ جو 0 تک پہنچ جائیں گی، کچھ جو نیچے سے 2 تک جائیں گی، کچھ جو 1 تک جائیں گی، تو یہ اچھی بات ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "اور اگر میں اس کا مطلب لیتا ہوں، جو سچ اور جھوٹ کو 1s اور 0s سمجھتا ہے، تو یہ مجھے مجموعی طور پر تناسب بتاتا ہے جو کہ اصل میں 0s ہوگا۔", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "آپ جانتے ہیں، ہم ایسی چیز تلاش کر رہے تھے جو 2 کے قدرتی لاگ سے 2 کا نصف ہو۔", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "تو شاید ہم صرف یہ دیکھنا چاہتے ہیں کہ وہ قدریں 0 اور 20 کے درمیان کب بنتی ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "اور آپ کو ہمیشہ اس طرح کے ہسٹوگرامس پر ایک رشتہ دار چوڑائی شامل کرنی پڑتی ہے کیونکہ وہ صرف بدصورت نظر آتے ہیں اگر وہ سب ساتھ ساتھ ہوں، میں کبھی کبھی سوچتا ہوں۔", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "اور آپ کو یہ اچھا احساس ہو سکتا ہے کہ آپ کا سارا ڈیٹا کیا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "اور انہیں صرف اس سے کچھ فنکارانہ تخلیق کرنے کا اشارہ کیا گیا تھا۔", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "ایک اور جو صرف حقیقی طور پر پاگل تھا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-tips-to-problem-solving/vietnamese/sentence_translations.json b/2020/ldm-tips-to-problem-solving/vietnamese/sentence_translations.json index 23b05f129..f7ddd5dc7 100644 --- a/2020/ldm-tips-to-problem-solving/vietnamese/sentence_translations.json +++ b/2020/ldm-tips-to-problem-solving/vietnamese/sentence_translations.json @@ -1928,7 +1928,7 @@ "end": 1522.72 }, { - "input": "And again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", + "input": "Um, and again, what I want you to take away is this principle that if you can have one object described in two different ways, very powerful in terms of showing non-obvious, um, algebraic relations or anything that's kind of written down symbolically without immediate intuition on top of it. ", "translatedText": "Và một lần nữa, điều tôi muốn bạn hiểu là nguyên tắc này nếu bạn có thể mô tả một đối tượng theo hai cách khác nhau, rất mạnh mẽ trong việc thể hiện các mối quan hệ đại số không rõ ràng hoặc bất cứ thứ gì được viết ra một cách tượng trưng mà không cần trực giác ngay lập tức. của nó. ", "model": "google_nmt", "n_reviews": 0, @@ -2280,7 +2280,7 @@ "end": 1824.42 }, { - "input": "This would be probability of 50%. ", + "input": "This would be a probability of 50 percent. ", "translatedText": "Khả năng này sẽ là 50%. ", "model": "google_nmt", "n_reviews": 0, @@ -2968,7 +2968,7 @@ "end": 2463.7 }, { - "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between 4 and 5, we're going to be drawing lines that intersect at x over 4. ", + "input": "And now maybe you can see where this is going to go, because for the next term when we want to know when does x divided by y sit between four and five, we're going to be drawing lines that intersect at x over four. ", "translatedText": "Và bây giờ có lẽ bạn có thể thấy điều này sẽ đi đến đâu, bởi vì trong số hạng tiếp theo khi chúng ta muốn biết khi nào x chia cho y nằm giữa 4 và 5, chúng ta sẽ vẽ các đường cắt nhau tại x trên 4. ", "model": "google_nmt", "n_reviews": 0, @@ -3536,7 +3536,7 @@ "end": 2973.4 }, { - "input": "What does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", + "input": "You know, what does it mean to look at a symmetry of a problem and turn that into something that's formulaically useful? ", "translatedText": "Việc xem xét tính đối xứng của một vấn đề và biến nó thành một thứ hữu ích về mặt công thức có ý nghĩa gì? ", "model": "google_nmt", "n_reviews": 0, @@ -4016,7 +4016,7 @@ "end": 3346.06 }, { - "input": "I keep making more episodes. ", + "input": "All right, keep making more episodes. ", "translatedText": "Tôi tiếp tục làm thêm nhiều tập nữa. ", "model": "google_nmt", "n_reviews": 0, @@ -4368,7 +4368,7 @@ "end": 3605.04 }, { - "input": "So, you know, we can see some examples here, some that have come up to 0, and some that would round down to 0, some that would round down to 2, some that would round down to 1, so that's nice. ", + "input": "So, you know, we can see some examples here, some that have come up to zero, and some that would round down to zero, some that would round down to two, some that would round down to one, so that's nice. ", "translatedText": "Vì vậy, bạn biết đấy, chúng ta có thể thấy một số ví dụ ở đây, một số sẽ tiến đến 0, và một số sẽ làm tròn xuống 0, một số sẽ làm tròn xuống 2, một số sẽ làm tròn xuống 1, điều đó thật tuyệt. ", "model": "google_nmt", "n_reviews": 0, @@ -4392,7 +4392,7 @@ "end": 3629.6 }, { - "input": "And if I take the mean of that, which is treating the trues and falses as 1s and 0s, this tells me the proportion in total that will actually be 0s. ", + "input": "o, and if I take the mean of that, which is treating the trues and falses as ones and zeros, this tells me the proportion in total that will actually be zeros. ", "translatedText": "Và nếu tôi lấy giá trị trung bình của nó, tức là coi giá trị đúng và sai là 1 và 0, điều này cho tôi biết tỷ lệ trong tổng số thực sự sẽ là 0. ", "model": "google_nmt", "n_reviews": 0, @@ -4464,7 +4464,7 @@ "end": 3704.36 }, { - "input": "You know, we were looking for something that was half of 2 minus the natural log of 2. ", + "input": "You know, we were looking for something that was too half of two minus the natural log, natural log of two. ", "translatedText": "Bạn biết đấy, chúng tôi đang tìm thứ gì đó bằng một nửa của 2 trừ log tự nhiên của 2. ", "model": "google_nmt", "n_reviews": 0, @@ -4536,7 +4536,7 @@ "end": 3747.38 }, { - "input": "So maybe we just want to look at when those values get bucketed in between like 0 and 20. ", + "input": "So maybe we just want to look at when the, when those values get bucketed in between like 0 and 20. ", "translatedText": "Vì vậy, có lẽ chúng ta chỉ muốn xem xét thời điểm các giá trị đó được đặt trong khoảng từ 0 đến 20. ", "model": "google_nmt", "n_reviews": 0, @@ -4584,7 +4584,7 @@ "end": 3773.42 }, { - "input": "And you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", + "input": "And um, I always, you always have to add a relative width on these sorts of histograms because they just look ugly if they're all like side by side, I think sometimes. ", "translatedText": "Và bạn luôn phải thêm chiều rộng tương đối vào các loại biểu đồ này bởi vì chúng trông sẽ xấu nếu tất cả chúng giống nhau, đôi khi tôi nghĩ vậy. ", "model": "google_nmt", "n_reviews": 0, @@ -4632,7 +4632,7 @@ "end": 3802.34 }, { - "input": "And you can get this nice sense for what all of your data is. ", + "input": "And you can get this, um, this nice sense for what all of your data is. ", "translatedText": "Và bạn có thể hiểu được điều này về tất cả dữ liệu của bạn. ", "model": "google_nmt", "n_reviews": 0, @@ -4712,7 +4712,7 @@ "end": 3867.72 }, { - "input": "And they were just prompted to create something artistic from that. ", + "input": "nd they were, they were just prompted to create something artistic from that. ", "translatedText": "Và họ chỉ được thôi thúc tạo ra thứ gì đó mang tính nghệ thuật từ đó. ", "model": "google_nmt", "n_reviews": 0, @@ -4736,7 +4736,7 @@ "end": 3894.56 }, { - "input": "Another that was just genuinely insane. ", + "input": "Another that was, this is just genuinely insane. ", "translatedText": "Một điều khác thực sự điên rồ. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/arabic/sentence_translations.json b/2020/ldm-trigonometry/arabic/sentence_translations.json index 7b9fa04b6..b84adfe3c 100644 --- a/2020/ldm-trigonometry/arabic/sentence_translations.json +++ b/2020/ldm-trigonometry/arabic/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "تبدو مثل نفس النوع من الموجات، لكنها كلها إيجابية وتنتقل من 1 إلى 0، ثم تصل إلى 1 في نطاق باي. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "ماذا يقول سؤالنا؟ حسنًا، عندما تقوم بالتعويض بـ 2x، يجب أن يكون نفس f لـ x تربيع. ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "لذا، إذا دخلنا هنا، وقمت بكتابة شيء مثل، جيب تمام باي على 12، 0.965. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "وهذه هي زاوية ثيتا. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "إذا أسميت هذه الزاوية الأخرى ألفا، فإننا نعلم أن ألفا زائد ثيتا زائد 90 درجة، أو باي نصفين إذا كنت تريد التعود على الاصطلاحات الأفضل، يجب أن يساوي باي. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/bengali/sentence_translations.json b/2020/ldm-trigonometry/bengali/sentence_translations.json index b03ed974b..79e32645e 100644 --- a/2020/ldm-trigonometry/bengali/sentence_translations.json +++ b/2020/ldm-trigonometry/bengali/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "এটি দেখতে একই ধরণের তরঙ্গের মতো, তবে এটি সবই ইতিবাচক এবং এটি 1 থেকে 0 পর্যন্ত যায়, তারপরে পাই এর স্প্যানে 1 পর্যন্ত।", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "ঠিক আছে, তাই যখন আপনি 2x প্লাগ ইন করেন, তখন এটি x বর্গক্ষেত্রের f এর মতোই হওয়া উচিত।", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "965।", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "এবং যে কোণ থিটা. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "যদি আমি এটিকে অন্য কোণ আলফা বলি, তাহলে আমরা জানি যে আলফা প্লাস থিটা প্লাস 90 ডিগ্রি, বা পাই অর্ধেক যদি আপনি আরও ভাল নিয়মে অভ্যস্ত হতে চান তবে পাই সমান করতে হবে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/chinese/sentence_translations.json b/2020/ldm-trigonometry/chinese/sentence_translations.json index ec33ffb8c..bed2e1466 100644 --- a/2020/ldm-trigonometry/chinese/sentence_translations.json +++ b/2020/ldm-trigonometry/chinese/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "它看起来像同一种波，但都是正波，并且在 pi 的范围内从 1 下降到 0，然后上升到 1。", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "好的，所以当你代入 2x 时，它应该与 x 平方的 f 相同。", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "因此，如果我们来到这里， 我只需输入类似 pi 大于 12, 0 的余弦值。", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "这就是角度 theta。", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "如果我将另一个角度称为 alpha，那么我们知道 alpha 加 theta 加 9 0 度，或者 pi 的一半（如果您想习惯更好的约定）必须等于 pi。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/english/captions.srt b/2020/ldm-trigonometry/english/captions.srt index 6647e9ea6..20ab71a3d 100644 --- a/2020/ldm-trigonometry/english/captions.srt +++ b/2020/ldm-trigonometry/english/captions.srt @@ -419,7 +419,7 @@ So which one of these reflects the fact the cosine of x squared is kind of a scaled version of itself? 106 -00:07:03,099 --> 00:07:06,045 +00:07:03,100 --> 00:07:06,045 So it looks like we still have a fair bit of answers rolling in, 107 @@ -712,7 +712,7 @@ Oh, it looks like I accidentally graded it right away. 179 00:11:34,760 --> 00:11:38,500 -Sorry for everyone who felt a little bit rushed there, my finger slipped. +Okay, well, sorry for everyone who felt a little bit rushed there, my finger slipped. 180 00:11:39,320 --> 00:11:44,700 @@ -791,3734 +791,3754 @@ just maybe, cosine is somehow related to exponents. It's not at all obvious how it is, but it absolutely is. 199 -00:12:51,440 --> 00:12:56,720 -And that's one of the things that I'd like to get to by the end of the second lecture. +00:12:51,440 --> 00:12:53,943 +And that's one of the things that I'd like to 200 +00:12:53,943 --> 00:12:56,720 +get to by the end of the end of the second lecture. + +201 00:12:56,800 --> 00:12:58,560 We won't get to the end of there today. -201 +202 00:13:00,300 --> 00:13:02,870 So with all of that, I think it's high time that we -202 +203 00:13:02,870 --> 00:13:05,540 actually talk about what sine and cosine actually are. -203 +204 00:13:05,680 --> 00:13:08,406 I don't want to come in assuming that you necessarily know that already, -204 +205 00:13:08,406 --> 00:13:10,460 so let's have a little moment to go back to the basics. -205 +206 00:13:11,580 --> 00:13:15,144 I think one of the best summaries of trigonometry that I ever heard, -206 +207 00:13:15,144 --> 00:13:18,501 and this was on Reddit, but I cannot for the life of me find it, -207 +208 00:13:18,501 --> 00:13:22,530 or remember who originally said this, is you think that it's about triangles, -208 +209 00:13:22,530 --> 00:13:24,080 but really it's about circles. -209 +210 00:13:24,500 --> 00:13:26,220 And this is absolutely true. -210 +211 00:13:26,300 --> 00:13:30,637 I think between the time when I very first learned about trigonometry and now, -211 +212 00:13:30,637 --> 00:13:35,359 I think my mind shifted from fundamentally thinking of a triangular-based definition, -212 +213 00:13:35,359 --> 00:13:37,940 which we'll talk about in a moment, to circles. -213 +214 00:13:38,400 --> 00:13:41,788 And let me just show you what I mean with respect to the circles, -214 +215 00:13:41,788 --> 00:13:45,125 because that's the one that explains how it comes up in physics, -215 +216 00:13:45,125 --> 00:13:48,360 and often how to get intuitions about where the values will be. -216 +217 00:13:48,900 --> 00:13:51,537 So instead of our Desmos graph, I'm going to go -217 +218 00:13:51,537 --> 00:13:53,900 ahead and pull up a couple animations here. -218 +219 00:13:54,680 --> 00:13:59,755 So in this one, I'll explain what sine of theta is, where theta is the input, -219 +220 00:13:59,755 --> 00:14:04,440 and it's going to be a measure of how far you've walked around a circle. -220 +221 00:14:05,020 --> 00:14:07,534 So I want you to imagine that you've started on the -221 +222 00:14:07,534 --> 00:14:10,340 rightmost side of a circle that has a radius of one, okay? -222 +223 00:14:10,340 --> 00:14:15,868 And then as you walk around at a constant rate, as a function of how far you've walked, -223 +224 00:14:15,868 --> 00:14:20,580 we're going to graph your height, which is to say your y-axis on this grid. -224 +225 00:14:21,160 --> 00:14:24,080 Maybe you'd think of it as the distance between you and the x-axis. -225 +226 00:14:25,680 --> 00:14:28,320 And as you do that, you get this wave that oscillates. -226 +227 00:14:28,600 --> 00:14:31,483 And it kind of makes sense how it's just going to oscillate on and on, -227 +228 00:14:31,483 --> 00:14:32,580 no matter how far you walk. -228 +229 00:14:32,580 --> 00:14:35,668 Which is why, when we go back over to the Desmos graph, -229 +230 00:14:35,668 --> 00:14:39,529 and you think of something like cosine of x, not the squared version, -230 +231 00:14:39,529 --> 00:14:42,397 let's just do the plain vanilla cosine x, you know, -231 +232 00:14:42,397 --> 00:14:45,100 it goes on forever continuing to cycle like this. -232 +233 00:14:45,840 --> 00:14:49,000 And I guess here I'm explaining sine of x, but it works the same way. -233 +234 00:14:49,040 --> 00:14:53,560 If instead I type sine of x, we get a similar looking shape. -234 +235 00:14:53,960 --> 00:14:55,960 The only difference is that it starts at zero. -235 +236 00:14:57,300 --> 00:15:01,500 And we can take a look to explain why here, because if by definition, -236 -00:15:01,500 --> 00:15:05,460 +237 +00:15:01,500 --> 00:15:05,459 if it's giving your y-coordinate when you start off to the right, -237 -00:15:05,460 --> 00:15:08,040 +238 +00:15:05,459 --> 00:15:08,040 it starts at zero before slowly increasing. -238 +239 00:15:08,620 --> 00:15:08,620 Okay? -239 +240 00:15:09,080 --> 00:15:12,001 Now by contrast, cosine, it's defined very similarly, -240 +241 00:15:12,001 --> 00:15:15,680 but it's giving you the x-coordinate as you walk around that circle. -241 +242 00:15:15,680 --> 00:15:17,920 So the distance to the vertical line. -242 +243 00:15:18,360 --> 00:15:23,531 And in that context, it starts off at one, and then as you walk around the circle, -243 +244 00:15:23,531 --> 00:15:28,080 your distance from that y-axis, your x-coordinate, is going to get lower. -244 +245 00:15:28,480 --> 00:15:32,140 It ultimately gets down to negative one before it starts increasing again. -245 +246 00:15:33,860 --> 00:15:34,060 Okay? -246 +247 00:15:35,180 --> 00:15:38,800 So this is how you might think of sine and cosine with respect to circles. -247 -00:15:38,800 --> 00:15:43,460 -The input is a kind of distance around a unit circle, - 248 -00:15:43,460 --> 00:15:46,740 -but you can think of that as an angle. +00:15:38,800 --> 00:15:43,852 +The input is a kind of, it's a kind of distance around a unit circle, 249 +00:15:43,852 --> 00:15:46,740 +but you could think of that as an angle, + +250 00:15:46,860 --> 00:15:50,040 And we're going to talk in a moment about the difference between degrees and radians. -250 +251 00:15:50,460 --> 00:15:53,795 But when you're thinking about walking around the circle, if it has a radius of one, -251 +252 00:15:53,795 --> 00:15:56,620 you can think of that input as being the literal distance that you walk. -252 +253 00:15:57,000 --> 00:16:00,900 So as an example, notice what happens when we get to the distance pi. -253 +254 00:16:01,740 --> 00:16:01,900 Okay? -254 +255 00:16:02,900 --> 00:16:07,360 pi gives you the semi-circumference of one of these circles. -255 +256 00:16:07,540 --> 00:16:12,328 If it's got a radius of one, then the distance that it takes to get halfway around is pi, -256 +257 00:16:12,328 --> 00:16:16,000 which is why at that input pi, you see the output go to negative one. -257 +258 00:16:16,660 --> 00:16:19,440 At half of that, if you turn to 90 degree angle, -258 +259 00:16:19,440 --> 00:16:22,390 which means walking around a distance of pi halves, -259 +260 00:16:22,390 --> 00:16:24,660 that's why cosine is zero at that point. -260 +261 00:16:25,440 --> 00:16:29,221 So now I'm going to ask you another question where we're going to consider -261 +262 00:16:29,221 --> 00:16:32,700 the input in terms of this idea of a distance around the unit circle. -262 +263 00:16:33,860 --> 00:16:37,760 And I'm just going to have you guess at some of the values for sine and cosine. -263 +264 00:16:37,880 --> 00:16:40,080 So we'll go ahead and pull up our quiz again. -264 +265 00:16:41,200 --> 00:16:44,770 And this time, let's get it up on here. -265 +266 00:16:47,460 --> 00:16:48,680 What does our question say? -266 +267 00:16:48,680 --> 00:16:54,160 Without looking it up or using a calculator, which of the following is true? -267 +268 00:16:54,940 --> 00:16:58,577 And it's reminding us that that input 3 is going to be considered in radians, -268 +269 00:16:58,577 --> 00:17:01,048 which is to say the distance around the unit circle, -269 +270 00:17:01,048 --> 00:17:03,240 if you're just walking along at the arc length. -270 +271 00:17:03,960 --> 00:17:11,099 Option A, the sine of 3 is around 0.14 and the cosine of 3 is around 0.99. -271 +272 00:17:11,640 --> 00:17:18,819 B, the sine of 3 is around 0.14 and the cosine of 3 is around negative 0.99. -272 -00:17:19,600 --> 00:17:24,579 +273 +00:17:19,599 --> 00:17:24,579 C, sine of 3 is around 0.99, cosine of 3 is around 0.14. -273 +274 00:17:25,020 --> 00:17:31,200 And D, sine of 3 is around negative 0.99 and cosine of 3 is around 0.14. -274 +275 00:17:32,320 --> 00:17:36,700 Okay, so just to give you a little moment to think that through. -275 -00:17:37,020 --> 00:17:39,384 -It seems like there's not a 100% consensus on - 276 -00:17:39,384 --> 00:17:41,800 -this question and answers are still rolling in. +00:17:37,020 --> 00:17:39,501 +it seems like there's not a hundred percent consensus 277 +00:17:39,501 --> 00:17:41,800 +on this question and answers are still rolling in. + +278 00:17:42,000 --> 00:17:45,700 So I'm going to go ahead and give you a little bit of pause and ponder music. -278 +279 00:17:47,400 --> 00:17:48,540 I'll let you just think about it. -279 +280 00:17:48,620 --> 00:17:50,040 I don't want you to feel rushed here. -280 +281 00:18:20,840 --> 00:18:24,320 Okay, so I think this is probably a pretty good time to start grading things. -281 +282 00:18:25,080 --> 00:18:29,580 So the correct answer is B. -282 +283 00:18:29,880 --> 00:18:33,660 And it looks like the majority of you got the correct answer, -283 +284 00:18:33,660 --> 00:18:37,440 which is to say that sine is 0.14 and cosine is negative 0.99. -284 +285 00:18:37,800 --> 00:18:39,660 Now the specific numbers here don't matter. -285 +286 00:18:39,720 --> 00:18:42,904 What's actually relevant is just kind of whether it's positive or -286 +287 00:18:42,904 --> 00:18:46,040 negative and whether it's very close to 1 or if it's closer to 0. -287 +288 00:18:46,040 --> 00:18:50,549 So if we think through what this might mean in the context of our graphs here, -288 +289 00:18:50,549 --> 00:18:53,974 the cosine, which remember is measuring the x-coordinate of -289 +290 00:18:53,974 --> 00:18:56,600 your little dot as you walk around the circle. -290 +291 00:18:57,740 --> 00:19:00,280 What is that going to be when you input 3? -291 +292 00:19:00,920 --> 00:19:03,320 Well, what does it mean to walk 3 units around the circle? -292 +293 00:19:04,180 --> 00:19:08,560 Well, like I just mentioned, walking pi units around the circle gets you halfway around. -293 +294 00:19:08,740 --> 00:19:11,825 This is really the way to think about pi is you've got a unit circle, -294 +295 00:19:11,825 --> 00:19:13,060 it takes you halfway around. -295 +296 00:19:13,060 --> 00:19:16,660 So to walk 3 units is going to be something a little bit shy of that. -296 +297 00:19:17,260 --> 00:19:20,423 Okay, and if you notice, there's very little change in -297 +298 00:19:20,423 --> 00:19:23,760 the x direction as you're at that left side of the circle. -298 +299 00:19:24,040 --> 00:19:26,393 So when it approaches 1, it gets really close -299 +300 00:19:26,393 --> 00:19:28,900 to 1 and then it doesn't really change that much. -300 +301 00:19:28,920 --> 00:19:43,040 So maybe it shouldn't be too surprising that our cosine of 3 turned out to be around 0.99. -301 -00:19:43,040 --> 00:19:46,140 -Which, same game, but it's measuring your y-coordinate. - 302 -00:19:46,320 --> 00:19:51,684 -When you walk almost halfway around, which is to say almost pi radians, +00:19:43,040 --> 00:19:48,873 +What's relevant is that it's negative, right? It's very close to negative one. 303 -00:19:51,684 --> 00:19:55,260 -well, there's still a little bit of height left. +00:19:48,873 --> 00:19:55,372 +So it's close to negative 0.99. Now on the other hand, if we were doing this with sine, 304 -00:19:55,560 --> 00:19:58,300 -And importantly, it's positive because you're still above that x-axis. +00:19:55,372 --> 00:19:59,360 +which same game, but it's measuring your y-coordinate, 305 -00:19:59,100 --> 00:20:03,761 -So when we turn back to our question, the only thing differentiating our answers was that +00:19:59,360 --> 00:20:04,735 +When you walk almost halfway around, which is to say almost pi radians, 306 -00:20:03,761 --> 00:20:08,320 -some of them would switch whether sine or cosine was the one that's close to negative 1. +00:20:04,735 --> 00:20:08,320 +well, there's still a little bit of height left. 307 -00:20:08,320 --> 00:20:13,400 -And the others would just switch the sine, s-i-g-n, whether it's positive or negative. +00:20:08,320 --> 00:20:08,320 +And importantly, it's positive because you're still above that x-axis. 308 -00:20:14,140 --> 00:20:18,700 -So congratulations to the 3,152 of you who got that one correct. +00:20:08,320 --> 00:20:11,394 +So when we turn back to our question, the only thing differentiating our answers was that 309 +00:20:11,394 --> 00:20:14,400 +some of them would switch whether sine or cosine was the one that's close to negative 1. + +310 +00:20:14,400 --> 00:20:14,400 +And the others would just switch the sine, s-i-g-n, whether it's positive or negative. + +311 +00:20:14,400 --> 00:20:18,700 +So congratulations to the 3,152 of you who got that one correct. + +312 00:20:20,100 --> 00:20:22,361 So that's loosely how you might think about sine and -310 +313 00:20:22,361 --> 00:20:24,880 cosine if we're coming at it from the direction of circles. -311 +314 00:20:25,020 --> 00:20:28,661 But this is very different from how it's usually taught if we're doing things like -312 +315 00:20:28,661 --> 00:20:32,260 in a typical high school class where you start off thinking about right triangles. -313 +316 00:20:32,640 --> 00:20:36,300 One is not better, they're just very different and worth understanding the connection. -314 +317 00:20:36,300 --> 00:20:42,612 So the classic thing when it comes to trig functions is a little -315 +318 00:20:42,612 --> 00:20:49,800 statement that everyone has ringing through their head called SOH CAH TOA. -316 +319 00:20:51,120 --> 00:20:55,745 And what you're supposed to remember from this is if we have a right triangle, -317 +320 00:20:55,745 --> 00:20:58,380 so let me go ahead and draw a right triangle. -318 +321 00:21:01,180 --> 00:21:03,280 So we know one of the angles is 90 degrees. -319 +322 00:21:03,440 --> 00:21:07,067 And if you know another one of the angles, theta, -320 +323 00:21:07,067 --> 00:21:10,840 then consider one of the sides to be the hypotenuse. -321 +324 00:21:11,600 --> 00:21:16,020 And then you ask what of the other two legs is adjacent to that angle, -322 +325 00:21:16,020 --> 00:21:19,880 and we label it a for adjacent, and which of them is opposite. -323 +326 00:21:20,620 --> 00:21:25,882 So then SOH CAH TOA, if that's a phrase that you remember running through your head, -324 +327 00:21:25,882 --> 00:21:31,269 tells you that the sine of this angle, this is the definition of sine for some kind of -325 +328 00:21:31,269 --> 00:21:36,780 angle that we input, is for such a right triangle, it's the opposite over the hypotenuse. -326 +329 00:21:36,960 --> 00:21:39,280 That's what this is telling us, opposite over hypotenuse. -327 +330 00:21:39,280 --> 00:21:47,780 Similarly, the cosine of that angle is going to equal the adjacent over the hypotenuse. -328 +331 00:21:48,160 --> 00:21:50,800 So that would be adjacent over hypotenuse. -329 +332 00:21:51,780 --> 00:22:00,340 And then the tangent of that angle is going to be the opposite over the adjacent. -330 +333 00:22:00,920 --> 00:22:03,960 So that opposite side O over the adjacent side. -331 +334 00:22:03,960 --> 00:22:08,126 So that is the classic thing to remember if students come out of a trig class, -332 +335 00:22:08,126 --> 00:22:11,080 if they know one thing, they typically know SOH CAH TOA. -333 +336 00:22:12,120 --> 00:22:15,150 And to give us an example of how you might apply this, -334 +337 00:22:15,150 --> 00:22:18,732 I'm going to go ahead and pull up another question here for you, -335 +338 00:22:18,732 --> 00:22:22,700 which is going to ask us about a leaning tower, a leaning tower of Pisa. -336 +339 00:22:24,900 --> 00:22:29,240 Clearing out the last one, let's read what our new question has to say. -337 +340 00:22:30,840 --> 00:22:33,620 Suppose that a tower is 100 meters high. -338 +341 00:22:34,180 --> 00:22:36,840 After several years, the tower starts to lean. -339 +342 00:22:37,340 --> 00:22:41,480 So instead of making a 90 degree angle with the ground, it makes an 80 degree angle. -340 +343 00:22:42,260 --> 00:22:45,584 When the sun is directly overhead, what is the -341 +344 00:22:45,584 --> 00:22:48,980 length of the shadow cast by this leaning tower? -342 +345 00:22:49,680 --> 00:22:52,300 Okay, the length of the shadow cast by this leaning tower. -343 +346 00:22:53,040 --> 00:22:55,800 I would highly encourage you to try drawing this out yourself. -344 +347 00:22:56,300 --> 00:22:58,998 I'm absolutely going to draw it for you in a moment, -345 +348 00:22:58,998 --> 00:23:03,480 but it's more fun if you're engaged and if you have a pencil and paper to noodle it out. -346 +349 00:23:03,660 --> 00:23:05,600 You could also try to imagine it in your head if you want. -347 +350 00:23:05,760 --> 00:23:09,092 But for me, at least, I always think much more clearly once I have pencil and paper, -348 +351 00:23:09,092 --> 00:23:11,640 no matter how much I think I can think things through in my head. -349 +352 00:23:13,060 --> 00:23:15,720 So let's give you a little pause and ponder time on this one. -350 -00:23:42,879 --> 00:23:48,240 +353 +00:23:42,880 --> 00:23:48,240 Looks like we have some very strong consensus on this one, which is very reassuring. -351 -00:23:51,879 --> 00:23:54,815 +354 +00:23:51,880 --> 00:23:54,815 And I guess one thing that should maybe go without saying, -352 +355 00:23:54,815 --> 00:23:58,795 especially given that I've turned the super chat or the live chat into the very -353 +356 00:23:58,795 --> 00:23:59,840 aggressive slow mode. -354 +357 00:24:00,520 --> 00:24:04,320 This is most fun if you don't know what the majority of people are already thinking. -355 +358 00:24:04,700 --> 00:24:07,880 The whole idea of the statistics is right now you can see there's a bandwagon. -356 +359 00:24:08,480 --> 00:24:12,931 There's 2600 people who are voting for one of the answers who think that's it, -357 +360 00:24:12,931 --> 00:24:15,580 but we don't know what one of those answers is. -358 +361 00:24:16,220 --> 00:24:20,440 So if you have the answer, try not to just blurted out in the live chat. -359 +362 00:24:20,540 --> 00:24:24,182 That's kind of the equivalent of a student in a classroom where when the teacher asks -360 +363 00:24:24,182 --> 00:24:27,486 something, rather than raising their hand to ask it or submitting it quietly, -361 +364 00:24:27,486 --> 00:24:28,080 just shouting. -362 +365 00:24:29,140 --> 00:24:29,800 I get it. -363 +366 00:24:29,860 --> 00:24:30,260 You're excited. -364 +367 00:24:30,340 --> 00:24:31,220 You really want to do it. -365 +368 00:24:31,260 --> 00:24:34,220 But in this case, it's a little bit more fun if we have the slow reveal. -366 +369 00:24:34,860 --> 00:24:39,360 -So that's probably enough time for people to have rolled their answers in. +All right, so that's probably enough time for people to have rolled their answers in. -367 +370 00:24:39,780 --> 00:24:43,520 -Let's go ahead and see what the majority has submitted. +Now let's go ahead and see what they see what the majority has submitted. -368 +371 00:24:47,040 --> 00:24:52,000 And the correct answer is B, which is 100 times the cosine of 80. -369 +372 00:24:52,440 --> 00:24:56,620 And it looks like the second most common answer was A, which is 100 times the sine of 80. -370 -00:24:57,199 --> 00:24:59,706 +373 +00:24:57,200 --> 00:24:59,706 So it looks like you guys correctly knew that you were -371 +374 00:24:59,706 --> 00:25:02,258 multiplying it by one of these trigonometric functions, -372 +375 00:25:02,258 --> 00:25:05,540 but there might have just been a little swap up for which one was which. -373 +376 00:25:06,040 --> 00:25:06,860 So let's think this through. -374 +377 00:25:07,080 --> 00:25:08,380 Okay, let's go back to our paper. -375 +378 00:25:09,240 --> 00:25:15,280 And our question had us imagine a tower which is 100 meters tall. -376 +379 00:25:15,780 --> 00:25:19,460 And specifically, it said it was 100 meters tall and then it started tilting. -377 +380 00:25:19,580 --> 00:25:22,040 So we can think of the length of the towers 100 meters. -378 +381 00:25:22,160 --> 00:25:26,240 That's not necessarily how far its new top is off of the ground. -379 +382 00:25:27,060 --> 00:25:30,640 And then it has us imagine that there's the sun directly overhead. -380 +383 00:25:31,400 --> 00:25:34,640 Okay, so it's going to be casting a shadow perpendicular to the ground. -381 +384 00:25:34,860 --> 00:25:36,700 That's what gives us our nice right triangle. -382 +385 00:25:37,520 --> 00:25:41,460 And it further specified that the angle here was 80 degrees. -383 -00:25:42,519 --> 00:25:46,737 +386 +00:25:42,520 --> 00:25:46,737 Okay, so the thing we want to know is this side here, the length of the shadow, -384 +387 00:25:46,737 --> 00:25:49,900 which if we look at our SOHCAHTOA, that's the adjacent side. -385 +388 00:25:49,960 --> 00:25:55,699 So we might write for ourselves cosine of this angle, which is 80 degrees, -386 +389 00:25:55,699 --> 00:26:00,520 is equal to that adjacent side, the thing that we want to know. -387 +390 00:26:00,660 --> 00:26:03,903 I'll just call it x, or maybe I'll call it s for shadow, -388 +391 00:26:03,903 --> 00:26:08,740 try to give our variables readable meanings, divided by the hypotenuse, which is 100. -389 -00:26:10,139 --> 00:26:18,260 +392 +00:26:10,140 --> 00:26:18,260 So that means that the shadow, once we rearrange, is 100 times the cosine of 80 degrees. -390 +393 00:26:19,940 --> 00:26:23,780 Now already this feels very different from the thought of trigonometry via circles. -391 +394 00:26:24,020 --> 00:26:27,480 Like I said, you start to think about it in terms of triangles, later it comes to circles. -392 +395 00:26:27,940 --> 00:26:32,920 One of the main differences here is that we're dealing with a different unit, right? -393 +396 00:26:33,500 --> 00:26:37,140 Here I was saying degrees, and that's very natural when we're in the context of triangles. -394 +397 00:26:37,140 --> 00:26:40,092 We could also talk about angles in terms of radians, -395 +398 00:26:40,092 --> 00:26:43,546 like I'll say in a moment, but when humans assign numbers and -396 +399 00:26:43,546 --> 00:26:47,780 numerical systems to things, we really like whole numbers between 0 and 100. -397 +400 00:26:48,060 --> 00:26:48,960 We just love that. -398 +401 00:26:49,040 --> 00:26:52,006 Instead of thinking of proportions, we think in terms of percentages, -399 +402 00:26:52,006 --> 00:26:55,440 which is the same thing, but it just turns it to whole numbers between 0 and 100. -400 +403 00:26:55,700 --> 00:26:59,100 When we think about like sound with decibels, which scales a whole bunch -401 +404 00:26:59,100 --> 00:27:01,475 and it's got this crazy exponential pattern to it, -402 +405 00:27:01,475 --> 00:27:04,875 we logarithmically change it and we have this very awkward system that's -403 +406 00:27:04,875 --> 00:27:08,229 really twisting the math behind its back, so that we can think of whole -404 +407 00:27:08,229 --> 00:27:09,440 numbers between 0 and 100. -405 +408 00:27:09,900 --> 00:27:13,580 Same deal with what's going on with degrees, but mathematicians don't really like that. -406 +409 00:27:13,880 --> 00:27:17,180 They don't really like to think in terms of degrees, because it's so unnatural. -407 +410 00:27:17,680 --> 00:27:21,436 Instead, they want some sort of natural unit, something where you imagine if you -408 +411 00:27:21,436 --> 00:27:25,100 talk to an alien civilization about math, they would have the same conventions. -409 +412 00:27:26,060 --> 00:27:29,143 So how does any of this connect to the unit circle and the idea of the -410 +413 00:27:29,143 --> 00:27:32,400 distance around that unit circle that we were walking, like we saw earlier? -411 +414 00:27:33,020 --> 00:27:35,891 Well, let me just pull up a pre-printed unit circle, -412 +415 00:27:35,891 --> 00:27:39,360 because trust me, you don't want to see me try to draw a circle. -413 +416 00:27:40,780 --> 00:27:44,040 What's neat here is we can, for each of these triangles, -414 +417 00:27:44,040 --> 00:27:47,359 imagine rescaling it so that that hypotenuse is really 1, -415 +418 00:27:47,359 --> 00:27:50,620 because all we care about are ratios of lengths of sides. -416 -00:27:50,620 --> 00:27:56,824 +419 +00:27:50,620 --> 00:27:56,472 So we might as well say, okay, let's scale this so that the hypotenuse is 1, -417 -00:27:56,824 --> 00:28:02,786 -meaning we divide everything by h, and what that's going to get us is the +420 +00:27:56,472 --> 00:28:02,780 +meaning we divide everything by h. And what that's going to get us is the opposite -418 -00:28:02,786 --> 00:28:10,038 -opposite side is now O over h, whatever that was, and then the adjacent side is A over h, +421 +00:28:02,780 --> 00:28:08,936 +side is now o over h, whatever that was, and then the adjacent side is a over h, -419 -00:28:10,038 --> 00:28:14,792 -meaning that the vertical component here is sine of theta, +422 +00:28:08,936 --> 00:28:14,256 +meaning that the y, or the vertical component here, is sine of theta, -420 -00:28:14,792 --> 00:28:21,400 -where theta was our angle down here, and then that bottom part is cosine of theta. +423 +00:28:14,256 --> 00:28:21,020 +where theta was our angle down here. Okay. And then that bottom part is cosine of theta. -421 +424 +00:28:21,020 --> 00:28:21,400 +Okay. + +425 00:28:24,060 --> 00:28:28,923 And if we want to think about all possible right triangles who have a hypotenuse of 1, -422 +426 00:28:28,923 --> 00:28:31,662 what you might think of is taking a unit circle, -423 +427 00:28:31,662 --> 00:28:34,961 because that's saying every point is a distance of 1 away, -424 +428 00:28:34,961 --> 00:28:37,980 so think of that line as the hypotenuse of a triangle. -425 +429 00:28:37,980 --> 00:28:41,500 If you have a triangle with a bigger angle, it shows up there. -426 +430 00:28:41,520 --> 00:28:44,620 If you have a triangle with a smaller angle, it shows up there. -427 +431 00:28:45,040 --> 00:28:50,600 And it's as if you take all possible right triangles and you fix them so that their -428 +432 00:28:50,600 --> 00:28:56,557 points are all in common, and such that their hypotenuses are all 1, you scale them down, -429 +433 00:28:56,557 --> 00:29:01,919 and the other tip of the triangle is going to trace out a unit circle like this, -430 +434 00:29:01,919 --> 00:29:07,678 and each one of these points for the corresponding angle is going to have coordinates, -431 +435 00:29:07,678 --> 00:29:13,040 cosine of theta as the x-coordinate, and then sine of theta for the y-coordinate. -432 +436 00:29:13,760 --> 00:29:16,680 And I'm just writing it vertically so I can fit it all on here. -433 +437 00:29:18,120 --> 00:29:23,076 Now typically, like I said, mathematicians like to think in terms of radians, -434 +438 00:29:23,076 --> 00:29:28,096 which is really a way of saying if we know that the radius of our circle is 1, -435 +439 00:29:28,096 --> 00:29:32,100 what is the distance that you've walked along the outside here? -436 -00:29:32,660 --> 00:29:37,397 -So for example, if you were to say 180 degrees, +440 +00:29:32,660 --> 00:29:38,425 +So for example, if you were to say 180 degrees, excuse me, degrees, -437 -00:29:37,397 --> 00:29:44,700 +441 +00:29:38,425 --> 00:29:44,700 which is walking halfway around the circle, that's the same as pi radians. -438 -00:29:46,600 --> 00:29:50,094 -And if you were to take something like 60 degrees, +442 +00:29:46,600 --> 00:29:50,225 +And if you were to take something like 60 degrees, okay, -439 -00:29:50,094 --> 00:29:55,440 -which would maybe end up, I don't know, around here, that might be 60 degrees. +443 +00:29:50,225 --> 00:29:55,440 +which would maybe end up, oh, I don't know, around here, that might be 60 degrees. -440 +444 00:29:56,680 --> 00:29:58,220 Well, that's a third of that. -441 +445 00:29:58,320 --> 00:30:00,960 You're walking a third of the half turn around the circle. -442 -00:30:01,459 --> 00:30:04,939 +446 +00:30:01,460 --> 00:30:04,939 Everyone who's an enthusiast about tau is sort of yelling right now because it -443 +447 00:30:04,939 --> 00:30:08,640 would make the conventions easier, but pi is the standard, so we're working with it. -444 +448 00:30:08,920 --> 00:30:11,400 It's pi, thirds, radians. -445 +449 00:30:13,860 --> 00:30:14,260 Okay? -446 +450 00:30:14,640 --> 00:30:17,614 So from this point forward, and really for a lot of your math career, -447 +451 00:30:17,614 --> 00:30:20,843 I would encourage you to, every time you want to think in terms of degrees, -448 +452 00:30:20,843 --> 00:30:23,818 where that makes the angle intuitive, try to translate it to radians, -449 +453 00:30:23,818 --> 00:30:26,580 because that's going to be the more natural way to understand it. -450 +454 00:30:26,580 --> 00:30:30,357 And the farther you get down the road when it comes to how sine and -451 +455 00:30:30,357 --> 00:30:33,967 cosine play into calculus, how you actually find values of them, -452 +456 00:30:33,967 --> 00:30:38,300 almost all of the time it's easier to think of that input in terms of radians. -453 +457 00:30:38,980 --> 00:30:42,677 So I think it's high time I ask you a couple questions about these, -454 +458 00:30:42,677 --> 00:30:47,300 and you might wonder, you know, how do we actually compute some of the values, right? -455 +459 00:30:48,080 --> 00:30:48,940 How to compute. -456 +460 00:30:52,140 --> 00:30:53,700 And the honest answer is that it's hard. -457 +461 00:30:54,140 --> 00:30:57,950 If there was an easy answer to say, oh, well, this is the formula that shows -458 +462 00:30:57,950 --> 00:31:01,760 you what sine of theta is, we wouldn't have given sine of theta a fancy name. -459 +463 00:31:01,840 --> 00:31:04,153 We would just always work in terms of that formula, -460 +464 00:31:04,153 --> 00:31:07,669 and we would call it the circle y coordinate formula, and it might get a name, -461 +465 00:31:07,669 --> 00:31:09,360 but it wouldn't be a special function. -462 +466 00:31:09,360 --> 00:31:12,206 It wouldn't have a special button on your calculator, -463 +467 00:31:12,206 --> 00:31:16,264 because effectively what you have to do is, before you know fancy techniques -464 +468 00:31:16,264 --> 00:31:19,954 associated with calculus, or associated with even fancier things like -465 +469 00:31:19,954 --> 00:31:23,801 what's known as the Kordak algorithm, you just have to have a handful of -466 +470 00:31:23,801 --> 00:31:28,440 values that you do know that come from geometric settings where there's enough symmetry. -467 -00:31:29,419 --> 00:31:32,640 +471 +00:31:29,420 --> 00:31:32,640 So for example, let's say I try to draw an equilateral triangle. -468 +472 00:31:34,880 --> 00:31:38,778 An equilateral triangle will be nice and symmetric, -469 +473 00:31:38,778 --> 00:31:44,625 and we're going to be able to leverage that symmetry to figure out a concrete -470 +474 00:31:44,625 --> 00:31:46,500 value of sine and cosine. -471 +475 00:31:49,380 --> 00:31:51,700 This looks roughly equilateral, wouldn't you say? -472 -00:31:54,679 --> 00:31:57,821 -And let's say each of those side lengths is one, +476 +00:31:54,680 --> 00:31:58,022 +And let's say each of those side lengths is one because, -473 -00:31:57,821 --> 00:32:01,540 -because if we get to choose our side lengths, why not one? +477 +00:31:58,022 --> 00:32:01,540 +you know, if we get to choose our side lengths, why not one? -474 +478 00:32:02,460 --> 00:32:05,680 Because remember, sine and cosine have everything to do with just ratios. -475 +479 00:32:06,640 --> 00:32:11,460 So on an equilateral triangle, this angle here is a third of the total. -476 +480 00:32:11,640 --> 00:32:14,880 The total is 180 degrees, so you might think of this as 60 degrees. -477 +481 00:32:14,880 --> 00:32:18,429 But like I said earlier, I would encourage you to try to get in -478 +482 00:32:18,429 --> 00:32:21,979 the habit of thinking of that instead of 60 degrees, say, okay, -479 +483 00:32:21,979 --> 00:32:25,640 the full angle is pi, so a third of that is going to be pi thirds. -480 +484 00:32:26,860 --> 00:32:29,860 And then similarly, this one up here is half that angle. -481 +485 00:32:29,900 --> 00:32:32,340 Here's where we're leveraging that symmetry to some extent. -482 +486 00:32:32,880 --> 00:32:40,140 That one is 30 degrees, which in terms of radians is now a sixth of a half turn. -483 +487 00:32:40,580 --> 00:32:42,740 All the tau enthusiasts get very mad at this point. -484 +488 00:32:42,740 --> 00:32:44,020 But again, hey, it's the convention. -485 +489 00:32:44,100 --> 00:32:46,940 This is the thing that you should come out of a trig class -486 +490 00:32:46,940 --> 00:32:50,311 knowing in your head how to think in terms of radians measured in pi, -487 +491 00:32:50,311 --> 00:32:53,200 because that's how we often write it down in tests and such. -488 +492 00:32:54,120 --> 00:32:56,220 So that one will be pi sixths. -489 +493 00:32:56,940 --> 00:33:01,760 So one thing I might ask you now is if you can leverage the symmetry of the setup, -490 +494 00:33:01,760 --> 00:33:05,884 the fact that equilateral triangles are going to be very nice and that -491 +495 00:33:05,884 --> 00:33:09,021 the right triangle coming about from it is very nice, -492 +496 00:33:09,021 --> 00:33:13,900 to answer something like a very concrete calculation, what is the sine of pi sixths? -493 +497 00:33:14,360 --> 00:33:15,720 OK, what do you think it is? -494 +498 00:33:16,140 --> 00:33:18,378 And in fact, let's do this as another live question, -495 +499 00:33:18,378 --> 00:33:20,660 which, as you can see, I'm like a kid in a candy shop. -496 +500 00:33:20,660 --> 00:33:22,540 I'm just very enthusiastic with my new toy. -497 +501 00:33:23,080 --> 00:33:25,720 So let's go ahead and use it. -498 +502 00:33:25,880 --> 00:33:27,100 Pull up another question here. -499 +503 00:33:28,020 --> 00:33:29,480 And what does our question ask? -500 -00:33:33,439 --> 00:33:35,740 +504 +00:33:33,440 --> 00:33:35,740 What is the sine of pi sixths, of course? -501 +505 00:33:36,720 --> 00:33:40,280 So again, if you want, if you're just joining now, you can hop over to 3b1b.co. -502 +506 00:33:40,580 --> 00:33:40,940 Live. -503 +507 00:33:41,400 --> 00:33:42,360 You can answer that. -504 +508 00:33:42,760 --> 00:33:46,272 I imagine the best way to do this if you're doing it live is you might be answering -505 +509 00:33:46,272 --> 00:33:49,660 on your phone, watching on another screen, or if you want to switch between tabs. -506 +510 00:33:49,660 --> 00:33:50,480 That's also good. -507 +511 00:33:52,120 --> 00:33:54,750 Now, while those answers are rolling in, let's take -508 +512 00:33:54,750 --> 00:33:57,380 a look to see if we have anything from the audience. -509 +513 00:33:57,380 --> 00:33:58,060 And we do. -510 +514 00:33:59,080 --> 00:34:01,220 When you do maths problems, do you use pen or pencil? -511 +515 00:34:01,900 --> 00:34:03,080 For sure I use pencil. -512 +516 00:34:04,340 --> 00:34:07,760 One, I'm very error prone, so I'm going to make mistakes all the time. -513 +517 00:34:08,139 --> 00:34:11,260 The only reason I'm writing in pen now is because it shows up better on the camera. -514 +518 00:34:11,719 --> 00:34:13,829 And let me just tell you, with every stroke that I write, -515 +519 00:34:13,829 --> 00:34:16,920 I'm deeply terrified that there's about to be an error, because I know there will be. -516 +520 00:34:17,320 --> 00:34:20,735 So if that terror doesn't show in the shaking of my hands, -517 +521 00:34:20,735 --> 00:34:23,920 then you'll start to notice it from this point forward. -518 +522 00:34:24,860 --> 00:34:26,159 Back to our question, though. -519 +523 00:34:27,080 --> 00:34:30,164 It looks like we've got pretty strong consensus now, -520 +524 00:34:30,164 --> 00:34:33,190 around 2,091 people agreeing on one of the answers, -521 +525 00:34:33,190 --> 00:34:36,100 and then a pretty even split among the other ones. -522 +526 00:34:36,639 --> 00:34:39,437 Oh, no, no, I guess the length of the bar is a little, -523 +527 00:34:39,437 --> 00:34:42,540 whoever worked on the scaling, maybe a little bit misleading. -524 +528 00:34:42,540 --> 00:34:46,239 It looks like one of the answers is much less popular than the other two. -525 +529 00:34:46,780 --> 00:34:51,639 But I guess it's considering them to start from somewhere like that. -526 +530 00:34:51,920 --> 00:34:53,900 Maybe we should have some axes on there indicating that. -527 +531 00:34:54,100 --> 00:34:57,580 So there's one very unpopular answer, and then one very popular answer. -528 +532 00:34:57,960 --> 00:34:59,700 So let's go ahead and see what that turns out to be. -529 +533 00:35:03,640 --> 00:35:05,460 Correct answer is one half. -530 +534 00:35:06,260 --> 00:35:10,255 Congratulations to the 2800, no, no, 2915 of you coming -531 +535 00:35:10,255 --> 00:35:14,180 into the last minute there to pull in a correct answer. -532 +536 00:35:14,920 --> 00:35:16,180 Okay, so why is that the case? -533 +537 00:35:16,380 --> 00:35:18,140 Why is that necessarily going to be one half? -534 +538 00:35:18,780 --> 00:35:22,700 Well, remember, sine of pi six, we're asking SOHCAHTOA. -535 +539 00:35:22,700 --> 00:35:26,957 So kind of remembering your head, sine is opposite over adjacent, -536 +540 00:35:26,957 --> 00:35:29,280 opposite over hypotenuse, excuse me. -537 +541 00:35:30,060 --> 00:35:33,920 So with respect to this smaller angle, maybe I'll change colors. -538 +542 00:35:35,480 --> 00:35:41,460 With respect to this smaller angle, the opposite side is this segment here. -539 +543 00:35:42,000 --> 00:35:44,860 And because of the symmetry of our triangle, that's going to be one half. -540 +544 00:35:46,060 --> 00:35:48,733 Because each side length of the equilateral triangle is one, -541 +545 00:35:48,733 --> 00:35:50,180 so half of it should be one half. -542 +546 00:35:50,680 --> 00:35:54,820 And divided by the hypotenuse, but by definition the hypotenuse was one here. -543 +547 00:35:54,840 --> 00:35:57,600 So one half divided by one, it's just the same as one half. -544 +548 00:35:58,380 --> 00:36:03,620 Now, slightly trickier is if I'd asked you the cosine of pi six. -545 +549 00:36:04,460 --> 00:36:08,280 And let's do this one again as a live quiz, but maybe a little bit more quickly this time. -546 +550 00:36:08,740 --> 00:36:13,345 I'm going to hop over, get to our next question, and as you might expect, -547 +551 00:36:13,345 --> 00:36:18,200 the next question is going to ask you to find the cosine of pi divided by six. -548 +552 00:36:18,200 --> 00:36:19,020 There we go. -549 +553 00:36:19,660 --> 00:36:25,042 And your options are A, one half, B, square root of two divided by two, C, -550 +554 00:36:25,042 --> 00:36:30,640 square root of three divided by two, or D, square root of five divided by two. -551 +555 00:36:31,540 --> 00:36:35,600 If you can, I would highly encourage you to be keeping your own notes, right? -552 +556 00:36:35,600 --> 00:36:37,969 So if you wanted to kind of check back on your own piece of paper, -553 +557 00:36:37,969 --> 00:36:40,480 you could take a look through it as you're working out these questions. -554 -00:36:40,879 --> 00:36:44,024 +558 +00:36:40,880 --> 00:36:44,024 But for right now, I'm just going to give you maybe 20 seconds -555 +559 00:36:44,024 --> 00:36:47,120 or so to kind of noodle through it, think of what it might be. -556 +560 00:36:47,640 --> 00:36:49,200 Get a little pause and ponder music going. -557 +561 00:37:11,440 --> 00:37:11,920 Okay. -558 +562 00:37:13,740 --> 00:37:16,842 Once again, it looks like we have some pretty strong consensus -559 +563 00:37:16,842 --> 00:37:19,600 from about the same number of people that we had before. -560 +564 00:37:20,160 --> 00:37:21,920 So let's go ahead and grade the answers. -561 +565 00:37:23,240 --> 00:37:27,860 And the correct one is C, square root of three divided by two. -562 +566 00:37:28,220 --> 00:37:31,010 So congratulations to the 3183 of you who knew -563 +567 00:37:31,010 --> 00:37:33,860 that it was square root of three divided by two. -564 +568 00:37:34,480 --> 00:37:36,840 Now, how do you figure something like that out? -565 +569 00:37:37,940 --> 00:37:41,162 Well, the trick is basically if you have one of these right triangles, -566 +570 00:37:41,162 --> 00:37:43,840 if you know one of their sides, then you know both of them. -567 +571 00:37:43,920 --> 00:37:46,060 So let me draw a little copy for ourselves here. -568 +572 00:37:47,680 --> 00:37:51,737 We know that the hypotenuse is one, one of the side lengths is one half, -569 +573 00:37:51,737 --> 00:37:56,461 and the only thing that's missing if we're thinking the cosine of 30 degrees is this -570 +574 00:37:56,461 --> 00:37:57,240 adjacent side. -571 +575 00:37:57,240 --> 00:38:00,840 So because we were doing it with the angle up there, adjacent side is here. -572 +576 00:38:01,320 --> 00:38:06,679 Now the Pythagorean theorem tells us that that adjacent value squared -573 +577 00:38:06,679 --> 00:38:11,580 plus the other leg length squared is going to equal one squared. -574 +578 00:38:12,740 --> 00:38:19,790 A hypotenuse squared, which means that value is equal to the square root of one minus, -575 +579 00:38:19,790 --> 00:38:22,140 well what's one half squared? -576 +580 00:38:22,340 --> 00:38:23,000 It's one fourth. -577 +581 00:38:24,120 --> 00:38:29,740 Which means it is the square root of one minus a fourth is three fourths. -578 +582 00:38:30,420 --> 00:38:32,251 So we can go ahead and take the square root of -579 +583 00:38:32,251 --> 00:38:34,200 that bottom and just leave the radical in the top. -580 +584 00:38:34,820 --> 00:38:37,760 And what we get is the square root of three all divided by two. -581 +585 00:38:37,760 --> 00:38:42,860 Now this is a value that anybody coming out of a trig class becomes very familiar with. -582 +586 00:38:42,960 --> 00:38:44,712 They're like, okay, if the answer isn't one half, -583 +587 00:38:44,712 --> 00:38:46,360 it's going to be square root of three over two. -584 +588 00:38:46,980 --> 00:38:49,920 Or root two over two also comes up in another circumstance. -585 +589 00:38:50,340 --> 00:38:52,900 But this is one of those that kind of becomes burned in your memory. -586 +590 00:38:53,260 --> 00:38:54,400 Square root of three divided by two. -587 +591 00:38:55,300 --> 00:39:01,952 And one very important fact that's come about here that's worth highlighting is that, -588 -00:39:01,952 --> 00:39:06,979 +592 +00:39:01,952 --> 00:39:06,980 hmm, how does the Pythagorean theorem play into our trigonometry? -589 +593 00:39:07,540 --> 00:39:10,558 Because if we have a right triangle whose hypotenuse is one, -590 +594 00:39:10,558 --> 00:39:14,714 and we know that one of the side lengths, let me switch to thinking with respect to -591 +595 00:39:14,714 --> 00:39:16,100 this lower left angle again. -592 +596 00:39:16,380 --> 00:39:24,774 If one of the side lengths is cosine of theta, and one of them is sine of theta, -593 +597 00:39:24,774 --> 00:39:31,821 well the Pythagorean theorem tells us that cosine of x squared plus -594 +598 00:39:31,821 --> 00:39:36,900 sine of x squared is always equal to one squared. -595 +599 00:39:37,400 --> 00:39:37,800 Okay? -596 +600 00:39:38,820 --> 00:39:40,400 Now this is a very important identity. -597 +601 00:39:40,560 --> 00:39:43,240 It basically tells you if you know cosine, you know sine. -598 +602 00:39:43,420 --> 00:39:45,080 If you know sine, you know cosine. -599 +603 00:39:45,280 --> 00:39:47,240 And it also is giving you a little reminder that -600 +604 00:39:47,240 --> 00:39:49,080 it's all really about the Pythagorean theorem. -601 +605 00:39:49,660 --> 00:39:52,080 In fact, this is equivalent to the Pythagorean theorem. -602 +606 00:39:52,320 --> 00:39:54,991 Because if you think of how cosine and sine are defined, -603 +607 00:39:54,991 --> 00:39:58,131 it's just defined in terms of the leg lengths of a right triangle, -604 +608 00:39:58,131 --> 00:40:00,100 and then everything scaled down as needed. -605 +609 00:40:00,100 --> 00:40:04,060 Now there's one very silly convention with how we write these things. -606 +610 00:40:04,120 --> 00:40:07,193 So we've already started to make friends with the cosine of x squared function, -607 +611 00:40:07,193 --> 00:40:10,460 and the fact that it's a little more interesting than you might think at first sight. -608 +612 00:40:11,160 --> 00:40:14,272 But you never see it written down as open parenthesis -609 +613 00:40:14,272 --> 00:40:16,520 cosine of x closed parenthesis squared. -610 +614 00:40:17,020 --> 00:40:21,934 Because people don't want to write too many parenthesis, -611 +615 00:40:21,934 --> 00:40:27,280 instead they write cosine squared of x plus sine squared of x. -612 +616 00:40:28,020 --> 00:40:29,220 It's all very nice. -613 +617 00:40:30,400 --> 00:40:31,180 How lovely. -614 +618 00:40:31,360 --> 00:40:32,680 We don't have to write too many symbols. -615 +619 00:40:33,080 --> 00:40:34,180 What a wonderful convention. -616 +620 00:40:34,780 --> 00:40:39,877 Except for the fact that literally every other time in math you see -617 +621 00:40:39,877 --> 00:40:45,500 something written as f squared of x, what that means is not f of x squared. -618 +622 00:40:45,660 --> 00:40:47,300 That's almost never what it means. -619 +623 00:40:47,300 --> 00:40:52,270 Instead, what it's supposed to mean for most of math is taking a -620 +624 00:40:52,270 --> 00:40:57,240 function and applying it to itself, like layering it over itself. -621 +625 00:40:57,380 --> 00:40:59,100 That's what that 2 usually refers to. -622 +626 00:40:59,380 --> 00:41:02,188 Sometimes later on it refers to the second derivative, -623 +627 00:41:02,188 --> 00:41:04,640 especially if it has some parenthesis around it. -624 +628 00:41:04,880 --> 00:41:08,640 But the idea is that it's very different from the convention in trigonometry. -625 +629 00:41:09,040 --> 00:41:11,892 Trigonometry is just flying by its own rules and saying, yeah, yeah, -626 +630 00:41:11,892 --> 00:41:15,240 we've got our own conventions for what it means when you put a little 2 up there. -627 +631 00:41:15,380 --> 00:41:18,440 I understand some students will be confused, but I'm trigonometry. -628 +632 00:41:18,580 --> 00:41:19,180 I don't care. -629 +633 00:41:19,320 --> 00:41:20,880 I'm moving forward with my conventions. -630 +634 00:41:21,580 --> 00:41:22,400 And that's fine. -631 +635 00:41:22,700 --> 00:41:24,040 You get used to it. -632 +636 00:41:24,100 --> 00:41:26,120 This is the standard way that you'll end up seeing it. -633 +637 00:41:26,320 --> 00:41:31,264 What I was writing earlier when we were thinking about the cosine squared function, -634 +638 00:41:31,264 --> 00:41:36,385 which again is very interesting, you would almost always see it written as cos squared -635 +639 00:41:36,385 --> 00:41:36,680 of x. -636 +640 00:41:37,780 --> 00:41:41,869 Now, very interesting is that this is not just a consequence of the -637 +641 00:41:41,869 --> 00:41:46,200 Pythagorean theorem that gives us a relation between our trig functions. -638 +642 00:41:46,500 --> 00:41:49,580 At the end of the lecture, I want to show you how you can use -639 +643 00:41:49,580 --> 00:41:52,760 trig functions to prove the Pythagorean theorem in a sneaky way. -640 +644 00:41:53,180 --> 00:41:53,900 And it's pretty sneaky. -641 +645 00:41:54,080 --> 00:41:57,762 Like it's not one of the common proofs that you'll see that's associated with, -642 +646 00:41:57,762 --> 00:42:00,280 you know, rearranging triangles or anything like that. -643 +647 00:42:00,280 --> 00:42:02,160 But it is very visually satisfying. -644 +648 00:42:02,540 --> 00:42:05,040 So that's something that we'll get to by the end of the lecture. -645 +649 00:42:06,220 --> 00:42:08,972 But in the meantime, I want to think through a -646 +650 00:42:08,972 --> 00:42:11,900 couple more of our of our trig function questions. -647 +651 00:42:12,200 --> 00:42:13,880 OK, so we kind of come back up here. -648 +652 00:42:13,900 --> 00:42:18,180 We just found cosine of pi six, which is square root of three over two. -649 +653 00:42:19,620 --> 00:42:22,120 But let's answer it for a couple more complicated ones. -650 +654 00:42:22,620 --> 00:42:26,528 So far, I'm dealing only with angles that were between zero and 90 degrees, -651 +655 00:42:26,528 --> 00:42:28,740 or I should say between zero and pi halves. -652 +656 00:42:28,740 --> 00:42:31,900 But let's try a question that's a little bit different from that. -653 +657 00:42:33,340 --> 00:42:40,880 And what if I instead said cosine of negative two pi over three? -654 +658 00:42:42,480 --> 00:42:46,125 OK, now that's one that is hard to think of in terms of triangles, -655 +659 00:42:46,125 --> 00:42:48,900 because what does it mean to have a negative angle? -656 +660 00:42:49,440 --> 00:42:52,036 So this is a good reminder that cosine and sine, -657 +661 00:42:52,036 --> 00:42:55,640 even if you think they're about triangles, are really about circles. -658 +662 00:42:56,140 --> 00:43:00,319 And remember, if you walk all the way around the circle, and if it has a radius of one, -659 +663 00:43:00,319 --> 00:43:04,120 walking all the way around the circle is going to take you a distance of two pi. -660 +664 00:43:05,000 --> 00:43:09,738 So if you want to think about two pi thirds, what you might start by doing is -661 +665 00:43:09,738 --> 00:43:14,780 chopping the triangle into thirds and then using that to help guide the intuitions. -662 +666 00:43:15,120 --> 00:43:17,260 But again, I kind of want to do this as a live quiz. -663 +667 00:43:17,620 --> 00:43:20,200 So let's go ahead and pull up the next question. -664 -00:43:24,919 --> 00:43:26,940 +668 +00:43:24,920 --> 00:43:26,940 And have you guys answer this for me. -665 +669 00:43:27,180 --> 00:43:30,600 What do you think the cosine of two pi divided by three? -666 +670 00:43:31,380 --> 00:43:32,580 Let's go ahead and pull it up. -667 +671 00:43:33,200 --> 00:43:36,220 Cosine of negative two pi divided by three. -668 +672 00:43:37,340 --> 00:43:39,460 So again, if you're keeping notes, that can be very helpful. -669 +673 00:43:39,540 --> 00:43:40,920 You reference your notes now. -670 +674 00:43:41,940 --> 00:43:44,587 But just to make sure that you have enough time to think this through, -671 +675 00:43:44,587 --> 00:43:47,720 I'm going to go ahead and give you a little bit more of that pause and ponder music. -672 +676 00:43:59,720 --> 00:44:03,683 It looks like one of you figured out how to submit an -673 +677 00:44:03,683 --> 00:44:07,500 answer that is not one of the four multiple choices. -674 +678 00:44:08,520 --> 00:44:12,872 So other than that, we've got a very strong lead with 1152 of you, -675 +679 00:44:12,872 --> 00:44:15,860 as I say this, agreeing on one of the answers. -676 +680 00:44:17,100 --> 00:44:22,480 And then second place by about a factor of four is going to sit on another one. -677 +681 00:44:22,540 --> 00:44:26,000 I'm curious about actually what what all the answer distribution here is. -678 +682 00:44:26,100 --> 00:44:27,580 I'm very curious about. -679 +683 00:44:27,780 --> 00:44:28,920 Oh, now we have a sixth. -680 +684 00:44:29,340 --> 00:44:33,000 OK, so two of you have now gone outside the realm of the -681 +685 00:44:33,000 --> 00:44:37,240 typical multiple choice universe and have entered your own things. -682 +686 00:44:37,700 --> 00:44:40,320 Two of you sit there and say, I don't want one of your choices. -683 +687 00:44:40,320 --> 00:44:43,380 I figured out my own answer and you just didn't put it in your form. -684 +688 00:44:43,380 --> 00:44:45,820 So I'm going to figure out a way to submit it differently. -685 +689 00:44:46,700 --> 00:44:48,100 If only you could do that on the SAT. -686 +690 00:44:49,020 --> 00:44:49,540 What is the answer? -687 +691 00:44:49,740 --> 00:44:50,260 A through E. -688 +692 00:44:51,240 --> 00:44:52,620 Sir, I think the answer is J. -689 +693 00:44:55,220 --> 00:44:55,580 1729. -690 +694 00:44:55,940 --> 00:44:57,000 We got Ramanujan's constant. -691 +695 00:44:57,260 --> 00:45:01,440 Oh, if I pause it now, though, it's not going to stay there for a brief, glorious moment. -692 +696 00:45:02,180 --> 00:45:04,160 We had a wonderful number. -693 +697 00:45:04,380 --> 00:45:07,300 Maybe I can pause it around the year, though, because it's 1999. -694 +698 00:45:08,580 --> 00:45:08,940 2031. -695 +699 00:45:08,940 --> 00:45:10,480 OK, well, I've paused it in the future. -696 +700 00:45:11,640 --> 00:45:14,620 So the correct answer in this context is negative one half. -697 +701 00:45:15,260 --> 00:45:15,320 OK. -698 +702 00:45:15,640 --> 00:45:18,596 And it looks like the second most common answer was D, -699 +703 00:45:18,596 --> 00:45:20,640 which is negative root three over two. -700 +704 00:45:21,080 --> 00:45:24,962 OK, so that's very good because often it's kind of this back and forth -701 +705 00:45:24,962 --> 00:45:29,610 between whether you think it's a half or whether you think it's root three over two, -702 +706 00:45:29,610 --> 00:45:33,383 because it's sort of the difference between things that feel like 30 -703 +707 00:45:33,383 --> 00:45:36,500 degree angles and things that feel like 60 degree angles. -704 +708 00:45:36,500 --> 00:45:40,280 So let's take a look back at our drawing and see how we might think about this. -705 +709 00:45:41,400 --> 00:45:44,080 So, like I said, negative two pi divided by three. -706 +710 00:45:44,160 --> 00:45:47,520 That's taking us a negative distance over here. -707 +711 00:45:47,700 --> 00:45:50,655 And this is why cosine will be a negative number, -708 +712 00:45:50,655 --> 00:45:54,380 because the x coordinate is pointing in the negative direction. -709 +713 00:45:56,200 --> 00:45:59,328 And if you want to think of this in terms of triangles, -710 +714 00:45:59,328 --> 00:46:02,680 I might draw this triangle here and ask, what is this angle? -711 +715 00:46:02,680 --> 00:46:06,819 OK, we know that it's 90 degrees plus something or pi -712 +716 00:46:06,819 --> 00:46:10,960 halves plus something and negative two pi over thirds. -713 +717 00:46:11,160 --> 00:46:17,261 If we wanted to think in terms of degrees, if that's something we're more comfortable -714 +718 00:46:17,261 --> 00:46:23,080 with, that's negative 360 divided by three degrees, which is negative 120 degrees. -715 +719 00:46:23,780 --> 00:46:26,967 So if it takes 120 degrees to get around here, -716 +720 00:46:26,967 --> 00:46:32,460 then that minus the 90 degree angle between the x and y axis gives us 30 degrees. -717 +721 00:46:33,720 --> 00:46:37,560 OK, so then the question is, what is the cosine of this value? -718 +722 00:46:37,620 --> 00:46:38,820 What is the x coordinate there? -719 +723 00:46:39,520 --> 00:46:44,059 With your 30, 60, 90 triangles, it's just kind of a nice thing to remember -720 +724 00:46:44,059 --> 00:46:48,720 that the shorter side is one half and the longer side is root three over two. -721 +725 00:46:48,720 --> 00:46:52,503 And if you forget why, it's exactly the diagram that we -722 +726 00:46:52,503 --> 00:46:56,220 just drew before associated with equilateral triangles. -723 +727 00:46:57,180 --> 00:47:00,951 Because if you imagine taking the equilateral triangle and chopping it in half, -724 +728 00:47:00,951 --> 00:47:03,874 then the shorter side has length one half and then the longer -725 +729 00:47:03,874 --> 00:47:05,760 side comes from the Pythagorean theorem. -726 +730 00:47:08,500 --> 00:47:11,340 So congratulations to those of you who correctly found that. -727 +731 00:47:12,160 --> 00:47:16,564 Now, I want to bring back that identity that you guys found at the very beginning, -728 +732 00:47:16,564 --> 00:47:19,589 associated with cosine squared and cosine of two thetas, -729 +733 00:47:19,589 --> 00:47:23,994 and show how that extends our capability for making some of these computations and -730 +734 00:47:23,994 --> 00:47:25,480 figuring out certain values. -731 +735 00:47:26,040 --> 00:47:27,860 So let's remind ourselves what that actually was. -732 +736 00:47:28,320 --> 00:47:31,800 I'm going to go back here and I'm going to pull up again the Desmos graph. -733 +737 00:47:32,800 --> 00:47:36,368 And remember how what we said is that cosine squared, -734 +738 00:47:36,368 --> 00:47:40,334 which looks like this little scaled down version of cosine, -735 +739 00:47:40,334 --> 00:47:45,820 oscillating more quickly, is the same as cosine of two x, that quicker oscillation. -736 +740 00:47:45,820 --> 00:47:48,540 But we had to kind of shift it and rescale it to make them equal. -737 +741 00:47:49,340 --> 00:47:53,284 So now we're going to use that fact that you can discover just by playing around -738 +742 00:47:53,284 --> 00:47:57,180 with graphs to make a computation that otherwise would be pretty tricky to make. -739 +743 00:47:57,180 --> 00:48:00,143 And I think that when you're able to make concrete computations, -740 +744 00:48:00,143 --> 00:48:03,973 it's often a reflection of the fact that you have a deeper conceptual understanding -741 +745 00:48:03,973 --> 00:48:04,840 of what's going on. -742 +746 00:48:04,840 --> 00:48:07,863 So even if it's always going to be your calculator computing this, -743 +747 00:48:07,863 --> 00:48:10,526 you're not sitting here working out cosine values by hand, -744 +748 00:48:10,526 --> 00:48:13,731 the fact that you can is an illustrate that you understand that cosine -745 +749 00:48:13,731 --> 00:48:15,040 function a little bit better. -746 +750 00:48:15,720 --> 00:48:19,200 So over on our paper, let me write down again what that identity was. -747 +751 00:48:20,020 --> 00:48:21,120 Let's go back to black. -748 +752 00:48:21,600 --> 00:48:22,680 Good old friendly black. -749 +753 00:48:24,060 --> 00:48:33,100 Cosine squared of x, using the dumb dumb convention, is one plus cosine of two x. -750 +754 00:48:33,760 --> 00:48:37,480 So it's kind of oscillating more quickly, all divided by two. -751 +755 00:48:38,820 --> 00:48:43,658 OK, so now I'm going to ask you a trickier question, -752 +756 00:48:43,658 --> 00:48:48,680 which is going to be what is the cosine of pi twelfths? -753 -00:48:50,259 --> 00:48:55,500 +757 +00:48:50,260 --> 00:48:55,500 And for those of you keeping track, pi twelfths is the same as 15 degrees. -754 +758 00:48:56,400 --> 00:49:00,995 So if we were doing this with a unit circle, what you might think is, -755 +759 00:49:00,995 --> 00:49:02,440 OK, a 30 degree angle. -756 +760 00:49:04,220 --> 00:49:05,700 It's looking like this. -757 +761 00:49:07,300 --> 00:49:08,260 That's 30 degrees. -758 +762 00:49:08,840 --> 00:49:11,140 I'm now going to be taking half of that. -759 +763 00:49:12,580 --> 00:49:13,340 Half of that. -760 +764 00:49:13,840 --> 00:49:16,142 And I want to know the cosine of this value, which -761 +765 00:49:16,142 --> 00:49:18,220 is going to be the x coordinate of that value. -762 +766 00:49:18,220 --> 00:49:21,811 And right away, you can probably make a guess that it's very close to one, -763 +767 00:49:21,811 --> 00:49:23,440 right, because it's a small angle. -764 +768 00:49:25,160 --> 00:49:28,687 And the small angle means that we haven't walked that far around a circle, -765 +769 00:49:28,687 --> 00:49:32,262 so we haven't gotten that far away from one, especially given the fact that -766 +770 00:49:32,262 --> 00:49:36,120 as you start to walk, almost all of your velocity is in the up and down direction. -767 +771 00:49:37,000 --> 00:49:39,080 All right, so cosine of pi twelfths, 15 degrees. -768 +772 00:49:39,340 --> 00:49:40,460 You know what's about to happen. -769 +773 00:49:40,780 --> 00:49:42,520 You know that this is a live quiz dynamic. -770 +774 00:49:43,020 --> 00:49:44,860 So let's pull it up. -771 +775 00:49:50,980 --> 00:49:51,880 All right. -772 +776 00:49:53,220 --> 00:49:57,040 What is the cosine of pi divided by 12? -773 +777 00:49:58,280 --> 00:50:00,820 OK, lots of answers rolling in. -774 +778 00:50:00,960 --> 00:50:02,180 Very fun, very exciting. -775 +779 00:50:02,600 --> 00:50:05,611 Again, if you just happen to be tuning in now, -776 +780 00:50:05,611 --> 00:50:09,840 you haven't been following along before, go to 3b1b.co slash live. -777 +781 00:50:10,240 --> 00:50:12,880 Your answers will become part of what you're seeing on screen. -778 +782 00:50:12,980 --> 00:50:14,860 It affects the overall distribution. -779 +783 00:50:14,860 --> 00:50:19,760 This one, unlike what we had before, much less overall consensus. -780 +784 00:50:20,540 --> 00:50:22,700 And it definitely seems to be taking a bit more time. -781 +785 00:50:22,780 --> 00:50:27,356 So I'm going to just stop talking and give you some some pause and ponder moments here, -782 +786 00:50:27,356 --> 00:50:28,760 because this one is tricky. -783 +787 00:50:28,880 --> 00:50:30,720 This one takes time to think through. -784 +788 00:50:44,860 --> 00:50:50,980 So. -785 +789 00:51:15,840 --> 00:51:18,329 Now it looks like there's a total of five of you who -786 +790 00:51:18,329 --> 00:51:20,960 have extended beyond the realm of usual multiple choice. -787 +791 00:51:24,420 --> 00:51:27,918 By the way, as you guys are entering answers here, -788 +792 00:51:27,918 --> 00:51:32,240 let me just say a huge thanks to Ben Eater and Cam Christensen. -789 +793 00:51:32,600 --> 00:51:35,160 They're the ones who actually worked on the item -790 +794 00:51:35,160 --> 00:51:37,720 pool website that is making all of this possible. -791 +795 00:51:39,000 --> 00:51:42,213 Eater, many of you might recognize from his own YouTube channel, -792 +796 00:51:42,213 --> 00:51:44,340 named Ben Eater about computer engineering. -793 +797 00:51:44,340 --> 00:51:46,340 If you haven't checked it out, you absolutely should. -794 +798 00:51:47,920 --> 00:51:51,060 And then item pool is a pretty cool thing that will -795 +799 00:51:51,060 --> 00:51:54,080 be unfolding over the next coming months and year. -796 +800 00:51:54,220 --> 00:51:56,220 So you might want to keep your eye on that. -797 +801 00:51:57,300 --> 00:51:59,540 It looks like some more consensus is starting to form. -798 +802 00:51:59,720 --> 00:52:02,100 So I'm going to go ahead and grade this question. -799 +803 00:52:05,860 --> 00:52:07,760 -It looks like I'm getting a phone call. +All right, it looks like I'm getting a phone call. -800 +804 00:52:10,080 --> 00:52:10,500 Awesome. -801 +805 00:52:11,000 --> 00:52:15,140 So the correct answer is B, which is square root of it's a very complicated thing, right? -802 +806 00:52:15,440 --> 00:52:19,280 Square root of one plus the square root of three over two divided by two. -803 +807 00:52:20,120 --> 00:52:27,300 So congratulations to the two to two nine of you who two to two for two to four three. -804 +808 00:52:27,420 --> 00:52:28,800 Two thousand two hundred forty three of you. -805 +809 00:52:29,040 --> 00:52:30,200 -How are the answers still changing? +How are the answers still changing? How are the answers still changing? -806 +810 00:52:30,200 --> 00:52:32,320 Going to have to talk to Cam and Eater about that. -807 +811 00:52:32,680 --> 00:52:33,720 My hands are not on this. -808 +812 00:52:34,580 --> 00:52:35,220 Who got that correct? -809 +813 00:52:36,020 --> 00:52:40,060 Now it looks like the second most common answer was A, which is of a very similar form. -810 +814 00:52:40,580 --> 00:52:42,000 There's just a minus sign in there. -811 +815 00:52:42,120 --> 00:52:45,640 So it's probably the case that while doing the arithmetic, something got flipped. -812 +816 00:52:45,760 --> 00:52:47,440 We're going to walk through it in a moment so you can see. -813 +817 00:52:48,200 --> 00:52:51,208 And a special congratulations to the two of you -814 +818 00:52:51,208 --> 00:52:54,280 who figured out how to submit an at sign on this. -815 +819 00:52:54,280 --> 00:52:56,440 Oh, the three of you who submitted an at sign. -816 +820 00:52:57,180 --> 00:52:59,560 That's not the correct answer, but well done. -817 +821 00:52:59,940 --> 00:53:00,960 Well done, four of you. -818 +822 00:53:01,200 --> 00:53:01,740 Very nice. -819 +823 00:53:02,380 --> 00:53:02,800 All right. -820 +824 00:53:03,240 --> 00:53:07,260 So clearly this is a complicated answer, but where does it come from? -821 -00:53:08,779 --> 00:53:12,202 +825 +00:53:08,780 --> 00:53:12,202 Well, if we look at our identity here, the key is that -822 +826 00:53:12,202 --> 00:53:15,500 it's relating two times an angle to the angle itself. -823 +827 00:53:15,500 --> 00:53:20,054 So if we just plug this in for 15 degrees or for pi twelfth, -824 +828 00:53:20,054 --> 00:53:24,460 what we would get is that the cosine squared of pi twelfth. -825 +829 00:53:26,540 --> 00:53:34,907 OK, if we enter that as X is equal to one plus the cosine of two times that value, -826 +830 00:53:34,907 --> 00:53:36,520 which is pi six. -827 +831 00:53:37,780 --> 00:53:38,900 Pi six. -828 +832 00:53:40,160 --> 00:53:41,440 All divided by two. -829 +833 00:53:41,860 --> 00:53:43,040 Now, the cosine of pi six. -830 +834 00:53:43,320 --> 00:53:43,820 -We know. +s, we know, right? -831 +835 00:53:45,020 --> 00:53:47,700 That's the same as our 30 degree angle here. -832 +836 00:53:48,460 --> 00:53:52,928 Again, when you have a 30, 60, 90 triangle, the shorter side of that, -833 +837 00:53:52,928 --> 00:53:57,780 which is going to be the sign in this context, the shorter side is one half. -834 +838 00:53:59,200 --> 00:54:02,439 And then the longer side is square root of three over two, -835 +839 00:54:02,439 --> 00:54:06,721 assuming that the hypotenuse is one, which in the context of the unit circle, -836 +840 00:54:06,721 --> 00:54:07,820 we always do assume. -837 +841 00:54:07,820 --> 00:54:13,140 So that means that we now have one plus the square root of three over two. -838 +842 00:54:13,520 --> 00:54:18,520 That's what the cosine of pi six was all divided by two, which means our final answer. -839 +843 00:54:19,920 --> 00:54:23,142 When we just want to say cosine of pi twelfth, -840 +844 00:54:23,142 --> 00:54:27,600 no squared about it is going to be the square root of this value. -841 +845 00:54:28,560 --> 00:54:29,820 OK, very interesting. -842 +846 00:54:29,960 --> 00:54:34,000 So this is showing us that cosine and sine can get complicated. -843 +847 00:54:34,000 --> 00:54:38,940 We've got a handful of values like pi six and pi thirds where we can figure them out. -844 +848 00:54:38,980 --> 00:54:42,012 We can have even more, I won't call them trivial values, -845 +849 00:54:42,012 --> 00:54:44,567 but easier to compute values like cosine of pi, -846 +850 00:54:44,567 --> 00:54:47,600 where you would look at your unit circle and say, oh, pi. -847 +851 00:54:47,820 --> 00:54:49,280 That means I've walked halfway around. -848 +852 00:54:49,400 --> 00:54:51,940 So my x coordinate is negative one. -849 +853 00:54:52,740 --> 00:54:57,700 But outside of just a handful of values like that, they're very hard to compute by hand. -850 -00:54:57,700 --> 00:55:01,377 -And in the olden days, the only way that people could compute them by +854 +00:54:57,700 --> 00:55:01,381 +and in the olden days, the only way that people could compute them by hand, -851 -00:55:01,377 --> 00:55:05,160 -hand for something like the existence of calculus would be measuring it. +855 +00:55:01,381 --> 00:55:05,160 +before something like the existence of calculus, would be measuring it, right? -852 +856 00:55:05,300 --> 00:55:08,892 You like draw out with a protractor and you try to carefully see -853 +857 00:55:08,892 --> 00:55:12,485 what the coordinates are or layering on usage of identities like -854 +858 00:55:12,485 --> 00:55:16,520 this over and over in ways that you can get approximate values like this. -855 +859 00:55:17,240 --> 00:55:21,229 And just to check ourselves, we can actually go and plug this into a calculator -856 +860 00:55:21,229 --> 00:55:24,121 if we wanted to, something like Desmos and just see, one, -857 +861 00:55:24,121 --> 00:55:28,560 how big the value is and kind of confirm for ourselves that they turn out to be the same. -858 +862 00:55:29,300 --> 00:55:32,520 So if we pop over here and I just type something like. -859 +863 00:55:34,020 --> 00:55:36,640 Cosine of pi over 12. -860 +864 00:55:38,220 --> 00:55:39,700 Point nine six five. -861 +865 00:55:40,360 --> 00:55:43,400 OK, so like I said, very close to one, and that's kind of what we would expect. -862 +866 00:55:43,620 --> 00:55:47,700 And the answer we just got was the square root of one plus the square root of three. -863 +867 00:55:48,560 --> 00:55:51,520 Got to make sure I do the divided by two correctly. -864 +868 00:55:52,220 --> 00:55:53,020 By two. -865 +869 00:55:53,340 --> 00:55:53,440 Right. -866 +870 00:55:53,500 --> 00:55:54,240 That looks correct. -867 +871 00:55:55,540 --> 00:55:57,460 But then all of that was divided by two. -868 +872 00:55:57,620 --> 00:55:59,400 All that stuff on the inside was divided by two. -869 +873 00:55:59,900 --> 00:56:01,220 And indeed, they're the same value. -870 +874 00:56:01,460 --> 00:56:05,580 So even just something like pi 12, we end up getting a very convoluted image. -871 +875 00:56:06,800 --> 00:56:11,391 Now, like I said, this is a reflection of the fact that our cosine identity that we found -872 +876 00:56:11,391 --> 00:56:15,880 at the very beginning, just by playing through with graphs, it's definitely not obvious. -873 +877 00:56:15,880 --> 00:56:19,565 And what we'll find next time is that this is actually a shadow of -874 +878 00:56:19,565 --> 00:56:23,360 the fact that trigonometric functions are related to complex numbers. -875 +879 00:56:23,500 --> 00:56:27,996 And the idea of doubling the angle here corresponding to multiplying by something twice, -876 +880 00:56:27,996 --> 00:56:29,260 not at all a coincidence. -877 +881 00:56:29,780 --> 00:56:30,840 I think it's very fun. -878 +882 00:56:30,900 --> 00:56:32,660 I'm excited to show you guys all about that. -879 +883 00:56:33,100 --> 00:56:36,010 Now, what I wanted to do just to finish things off here, though, -880 +884 00:56:36,010 --> 00:56:39,280 was to show two things that are not commonly shown about the unit circle. -881 +885 00:56:39,280 --> 00:56:43,640 OK, so we see where sine and cosine are, but there might be a couple of questions. -882 +886 00:56:44,440 --> 00:56:46,220 Where is tan? -883 +887 00:56:47,620 --> 00:56:50,600 OK, where in our picture do we see tangent of theta? -884 +888 00:56:51,180 --> 00:56:56,499 And then also, where is cosine of x or cosine of theta -885 +889 00:56:56,499 --> 00:57:02,400 squared or with the dumb convention, cosine squared of theta? -886 +890 00:57:03,560 --> 00:57:06,980 So pulling up another one of my infinite supply of unit circles here. -887 +891 00:57:09,080 --> 00:57:10,040 Actually, you know what? -888 +892 00:57:10,800 --> 00:57:14,513 This one's not going to have enough y axis, so I'm going to do something -889 +893 00:57:14,513 --> 00:57:18,380 that I really shouldn't do, which is to try to use a compass live on camera. -890 +894 00:57:19,000 --> 00:57:21,880 I'm sort of terrible at using compasses because it always slips out for me. -891 +895 00:57:22,320 --> 00:57:23,340 We're just going to do this. -892 +896 00:57:24,260 --> 00:57:27,120 I'm so nervous that this is going to somehow slip out. -893 +897 00:57:27,320 --> 00:57:29,860 Oh, yeah, that actually didn't turn out too badly. -894 +898 00:57:30,720 --> 00:57:32,560 Hopefully the pencil shows up OK on screen. -895 +899 00:57:32,560 --> 00:57:37,629 The reason I'm doing this is the tangent is going to demand that we think outside of -896 +900 00:57:37,629 --> 00:57:42,640 the box, that we think outside of the circle in order to geometrically interpret it. -897 +901 00:57:44,040 --> 00:57:48,030 You could just ask about the opposite divided by the adjacent, -898 +902 00:57:48,030 --> 00:57:50,880 which is where that SOHCAHTOA TOA comes from. -899 +903 00:57:51,380 --> 00:57:55,768 But what I really want is a single length in our diagram that corresponds to -900 +904 00:57:55,768 --> 00:58:00,100 the tangent so we can kind of get that nice geometric intuitive feel for it. -901 +905 00:58:00,740 --> 00:58:04,204 And the idea is we take that radius, which is one, -902 +906 00:58:04,204 --> 00:58:08,415 and if we think of the fact that tangent of theta, SOHCAHTOA, -903 +907 00:58:08,415 --> 00:58:11,811 is supposed to be the opposite over the adjacent, -904 +908 00:58:11,811 --> 00:58:15,480 I want a triangle where that adjacent is equal to one. -905 +909 00:58:15,940 --> 00:58:21,860 OK, so a right triangle such that this one is the adjacent side to our angle theta. -906 +910 00:58:22,620 --> 00:58:25,945 Well, in order to do that, let's just draw a right -907 +911 00:58:25,945 --> 00:58:29,140 angled perpendicular line off of our radial line. -908 +912 00:58:30,260 --> 00:58:33,892 And some of you might know that if you draw a perpendicular line -909 +913 00:58:33,892 --> 00:58:37,580 to the radius of a circle, it actually lies tangent to the circle. -910 +914 00:58:37,700 --> 00:58:40,340 So you see tangent, this is where the word comes from. -911 +915 00:58:41,740 --> 00:58:45,349 Then what we have is a triangle where the adjacent side to -912 +916 00:58:45,349 --> 00:58:49,020 theta is one and then the opposite side is this length here. -913 +917 00:58:49,020 --> 00:58:53,520 So that length there is the tangent of theta. -914 +918 00:58:54,240 --> 00:58:58,995 And so right now you can kind of intuitively understand that as theta is very small, -915 +919 00:58:58,995 --> 00:59:02,072 when the opposite over adjacent is very close to zero, -916 +920 00:59:02,072 --> 00:59:06,100 because that opposite is close to zero and the adjacent is close to one. -917 +921 00:59:06,480 --> 00:59:09,160 Well, indeed, this length also is getting very close to zero. -918 +922 00:59:09,840 --> 00:59:13,740 And then you can also think through what happens as that angle approaches 90 degrees. -919 +923 00:59:14,340 --> 00:59:19,791 This is something I actually illustrated once before in a previous video, -920 +924 00:59:19,791 --> 00:59:23,180 a lesser known video entitled Tattoos on Math. -921 +925 00:59:23,680 --> 00:59:28,740 That, fun fact, includes someone who is behind the scenes as we say all of this right now. -922 +926 00:59:29,500 --> 00:59:32,840 But the idea is that when you draw that tangent line, -923 +927 00:59:32,840 --> 00:59:36,180 tangent is this length that I just pointed out to you. -924 +928 00:59:36,180 --> 00:59:38,660 And in that context, you could also think about cotangent. -925 +929 00:59:38,660 --> 00:59:42,687 There's this whole other triad of trigonometric functions that aren't worth teaching, -926 +930 00:59:42,687 --> 00:59:44,280 but are kind of historical quirks. -927 +931 00:59:44,540 --> 00:59:46,360 That's actually what that whole video was about. -928 +932 00:59:47,200 --> 00:59:50,415 But from here, you can just play around with moving the angle theta -929 +933 00:59:50,415 --> 00:59:53,820 and seeing what that does to this tangent value, to that tangent length. -930 +934 00:59:55,120 --> 00:59:58,908 And using that intuition, actually, I want you to place a guess -931 +935 00:59:58,908 --> 01:00:02,460 on what the graph of tangent of theta is going to look like. -932 +936 01:00:02,460 --> 01:00:05,260 So once again, let's go over to our live question dynamic. -933 +937 01:00:05,580 --> 01:00:08,060 Again, congratulations to those of you who got the last one right. -934 +938 01:00:08,620 --> 01:00:12,979 And as our final question before the end of this particular stream, -935 +939 01:00:12,979 --> 01:00:17,274 which is a rather tall question, so it might actually pop up over, -936 +940 01:00:17,274 --> 01:00:20,160 over below the values we have available here. -937 +941 01:00:20,440 --> 01:00:23,328 The final question, I just want you to guess what -938 +942 01:00:23,328 --> 01:00:26,160 the tangent of theta looks like when we graph it. -939 +943 01:00:27,440 --> 01:00:29,900 And it should be pulled up now. -940 +944 01:00:31,540 --> 01:00:32,700 And oh, there we go. -941 +945 01:00:32,940 --> 01:00:34,160 It is rather tall, though. -942 +946 01:00:34,680 --> 01:00:34,780 -OK. +Okay. All right -943 -01:00:35,919 --> 01:00:38,070 +947 +01:00:35,920 --> 01:00:38,070 So let me go ahead and pull up a separate image -944 +948 01:00:38,070 --> 01:00:40,400 for you here so that you can see all of the options. -945 +949 01:00:41,240 --> 01:00:44,080 Let's pop over here and take a look. -946 +950 01:00:44,360 --> 01:00:44,500 Great. -947 -01:00:45,319 --> 01:00:49,986 +951 +01:00:45,320 --> 01:00:49,986 So again, if you go to 3b1b.co slash live, where the link is in the description, -948 +952 01:00:49,986 --> 01:00:52,060 you can submit your answers to this. -949 +953 01:00:53,060 --> 01:00:57,500 Basically, which one of these graphs shows you the tangent of theta? -950 -01:01:00,759 --> 01:01:03,640 +954 +01:01:00,760 --> 01:01:03,640 Unfortunately, we're going to have to kind of switch back and forth to see the statistics. -951 +955 01:01:04,040 --> 01:01:07,118 It looks like there's pretty strong consensus in one direction, -952 +956 01:01:07,118 --> 01:01:11,254 but I'm going to give you I'm going to give you a little bit more time to think about -953 +957 01:01:11,254 --> 01:01:12,120 it if you want to. -954 +958 01:01:12,520 --> 01:01:13,080 Don't feel rushed. -955 +959 01:01:13,200 --> 01:01:15,240 Don't feel like you have to jump right into it. -956 +960 01:01:38,080 --> 01:01:42,040 I want a little ambient pause and ponder music while we're at it. -957 +961 01:01:43,560 --> 01:01:45,720 I'm mildly addicted to it. -958 +962 01:02:05,240 --> 01:02:06,280 All right. -959 +963 01:02:06,560 --> 01:02:07,700 So while you're answering. -960 +964 01:02:09,460 --> 01:02:12,859 So as we can see, some people are clearly messing with the website and having fun, -961 +965 01:02:12,859 --> 01:02:15,440 or there's a bug in the website to answer various other things. -962 +966 01:02:16,080 --> 01:02:17,540 Other fun thing going on. -963 +967 01:02:17,560 --> 01:02:24,500 It seems like we've got a couple comments in the in the Json associated with the answers. -964 +968 01:02:24,500 --> 01:02:30,541 So to the person who decided to skip over my very draconian limits on the live -965 +969 01:02:30,541 --> 01:02:36,660 chat dynamic and instead entered in Json comment through the answer forum forum. -966 +970 01:02:37,000 --> 01:02:38,620 I really enjoyed this class. -967 +971 01:02:38,940 --> 01:02:39,520 Thank you. -968 +972 01:02:39,660 --> 01:02:40,500 I appreciate that. -969 +973 01:02:40,800 --> 01:02:43,160 And I appreciate your your hacky instincts. -970 +974 01:02:43,220 --> 01:02:44,820 You're going to go far in life, my friend. -971 +975 01:02:44,920 --> 01:02:46,400 You're going to go very, very far. -972 +976 01:02:47,460 --> 01:02:49,920 So let's go ahead and grade this question here. -973 +977 01:02:49,920 --> 01:02:54,300 It looks like we have very strong consensus, hopefully around the correct answer. -974 +978 01:02:55,780 --> 01:02:59,642 And the correct answer is a again, maybe a little bit hard -975 +979 01:02:59,642 --> 01:03:03,440 to see on our on our indication of the question over here. -976 +980 01:03:03,440 --> 01:03:06,620 But a is the graph that looks like this. -977 +981 01:03:06,860 --> 01:03:08,720 And what you'll notice is it starts at zero. -978 +982 01:03:08,920 --> 01:03:12,320 And then as you approach pi halves, it blows up to infinity. -979 +983 01:03:12,320 --> 01:03:18,142 OK, now, in terms of the animations and the geometric interpretation of tangent, -980 +984 01:03:18,142 --> 01:03:23,246 that should actually make sense, because if we go back to what we were -981 +985 01:03:23,246 --> 01:03:27,560 looking at as the angle blows up, the angle doesn't blow up. -982 +986 01:03:27,640 --> 01:03:31,020 The angle does a very modest thing of simply going to pi halves a fine number. -983 +987 01:03:31,100 --> 01:03:34,360 But the tangent of that angle is going to just get bigger and bigger. -984 +988 01:03:34,360 --> 01:03:36,560 And ultimately, it's it's not defined. -985 +989 01:03:36,580 --> 01:03:39,680 It's the same as dividing by zero once you get to pi halves. -986 +990 01:03:40,040 --> 01:03:42,627 And then after that, if you play the same game, -987 +991 01:03:42,627 --> 01:03:46,024 it's going to be it's going to be considered a negative value, -988 +992 01:03:46,024 --> 01:03:50,499 which basically comes down to the fact that the opposite and adjacent sides of our -989 +993 01:03:50,499 --> 01:03:53,680 triangle are going to be considered to have separate signs. -990 -01:03:56,299 --> 01:04:00,061 +994 +01:03:56,300 --> 01:04:00,061 So the very last thing that I want to talk about, like I said, -991 +995 01:04:00,061 --> 01:04:04,540 is the question of where does cosine squared of theta show up in our image? -992 +996 01:04:04,780 --> 01:04:07,860 Because this is the very thing that we opened with is trying to understand cosine squared. -993 +997 01:04:08,940 --> 01:04:11,888 And you might think, oh, if we can see that in our image somewhere, -994 +998 01:04:11,888 --> 01:04:15,400 maybe it really reveals why it looks like a miniature version of the cosine wave. -995 -01:04:15,799 --> 01:04:19,681 +999 +01:04:15,800 --> 01:04:19,681 Now, one of the naive things, not naive, but one of the first things you might -996 +1000 01:04:19,681 --> 01:04:23,661 try to do is pull up the original cosine animation that we had or something like -997 +1001 01:04:23,661 --> 01:04:27,740 it and say, oh, well, I'm going to take that cosine length and literally square it. -998 +1002 01:04:27,760 --> 01:04:29,580 I'm just going to draw the square based on that. -999 +1003 01:04:29,860 --> 01:04:34,267 And if I were to do this, where I'm going to plot the length of that horizontal line -1000 +1004 01:04:34,267 --> 01:04:38,520 and I'm also going to plot the area of that square, it's a little bit complicated. -1001 +1005 01:04:41,200 --> 01:04:44,224 It certainly shows us what we already saw before, -1002 +1006 01:04:44,224 --> 01:04:48,460 which is that the square of that value looks like another cosine wave. -1003 +1007 01:04:49,580 --> 01:04:52,533 But this is something where just by staring at the image, -1004 +1008 01:04:52,533 --> 01:04:54,520 it actually doesn't yield much insight. -1005 +1009 01:04:54,720 --> 01:04:57,733 I think oftentimes in geometry or math, when you simply -1006 +1010 01:04:57,733 --> 01:05:00,800 write down the definition of things, the answer pops out. -1007 +1011 01:05:00,940 --> 01:05:02,940 This is a sign that again, it's not trivial. -1008 +1012 01:05:02,940 --> 01:05:05,910 It's very unclear why the way that that square, -1009 +1013 01:05:05,910 --> 01:05:10,243 that literal square would oscillate is going to be related to cosine, -1010 +1014 01:05:10,243 --> 01:05:12,100 much less cosine of two theta. -1011 +1015 01:05:12,480 --> 01:05:14,820 So it's going to come from somewhere else inside our image. -1012 +1016 01:05:16,440 --> 01:05:21,740 And for that, let me go ahead and draw a much bigger triangle for ourselves. -1013 +1017 01:05:24,080 --> 01:05:28,420 Let's say we've got one of these guys that was living inside the unit circle. -1014 +1018 01:05:33,440 --> 01:05:35,760 This, by the way, I think is the most exciting part of the lecture. -1015 +1019 01:05:35,900 --> 01:05:38,311 So if there was just one thing that you were going to stick around for, -1016 +1020 01:05:38,311 --> 01:05:40,020 I would say that this is the thing that you should. -1017 +1021 01:05:40,920 --> 01:05:43,600 Because it gives a very sneaky proof of the Pythagorean theorem. -1018 +1022 01:05:44,220 --> 01:05:47,260 It might feel like a circular argument at first, but I assure you that it's not. -1019 +1023 01:05:47,700 --> 01:05:50,580 And I also assure you that the word circular was not meant as a pun there. -1020 -01:05:51,400 --> 01:05:54,722 -So remember, if that's our angle theta, then one of our side +1024 +01:05:51,400 --> 01:05:56,766 +So remember, if that's our angle theta, then one of our side lengths is sine theta. -1021 -01:05:54,722 --> 01:05:58,100 -lengths is sine of theta and the other one is cosine of theta. +1025 +01:05:56,766 --> 01:06:02,133 +And that's the angle theta. And that's the angle theta. And that's the angle theta. -1022 -01:05:59,879 --> 01:06:03,430 +1026 +01:06:02,133 --> 01:06:07,500 +Then one of our side lengths is sine of theta. And the other one is cosine of theta. + +1027 +01:06:07,500 --> 01:06:09,098 Let me ask you, if I draw, well, okay, so, you know, -1023 -01:06:03,430 --> 01:06:07,583 +1028 +01:06:09,098 --> 01:06:10,968 the example we had earlier was this leaning tower thing where -1024 -01:06:07,583 --> 01:06:12,540 +1029 +01:06:10,968 --> 01:06:13,200 we kind of imagined casting a shadow from a leaning tower onto the ground. -1025 -01:06:12,840 --> 01:06:15,333 +1030 +01:06:13,200 --> 01:06:15,540 This is sort of going to flip it on its head and say, well, -1026 -01:06:15,333 --> 01:06:18,700 +1031 +01:06:15,540 --> 01:06:18,700 this angle theta between them, who's to say who's the tower and who's the ground? -1027 +1032 01:06:18,780 --> 01:06:21,475 What if I wanted to have a light source over here and -1028 +1033 01:06:21,475 --> 01:06:24,320 I wanted to cast a shadow from the ground onto the tower? -1029 +1034 01:06:24,320 --> 01:06:27,320 A nice perpendicular line in this direction. -1030 +1035 01:06:28,820 --> 01:06:32,360 That way my angle theta is going to do double duty for me. -1031 +1036 01:06:32,700 --> 01:06:35,676 It's describing the angle between these two lines, -1032 +1037 01:06:35,676 --> 01:06:39,120 but in one context, this ground line was the adjacent side. -1033 +1038 01:06:39,200 --> 01:06:40,200 It was the short side. -1034 +1039 01:06:40,680 --> 01:06:44,260 But now, if I ask you a question like, what is this side length? -1035 +1040 01:06:44,800 --> 01:06:48,165 The shadow that the ground lays down on the tower, -1036 +1041 01:06:48,165 --> 01:06:53,180 if that's not more confusing than I want it to be, what are we going to get? -1037 +1042 01:06:55,440 --> 01:07:00,017 Well, in terms of this smaller right triangle that we've just created, -1038 +1043 01:07:00,017 --> 01:07:04,789 by definition, the cosine of that angle is going to be the adjacent side, -1039 +1044 01:07:04,789 --> 01:07:08,980 the thing we want to know, maybe I'll call it s for shadow again. -1040 +1045 01:07:09,880 --> 01:07:11,300 I'm going to write down what it equals. -1041 +1046 01:07:12,380 --> 01:07:15,100 It's going to be the adjacent side divided by the hypotenuse. -1042 +1047 01:07:15,560 --> 01:07:18,760 But for this little triangle, the hypotenuse is cosine. -1043 +1048 01:07:18,760 --> 01:07:24,126 So it's divided by the cosine of theta, which means cosine squared of theta, -1044 +1049 01:07:24,126 --> 01:07:27,820 dumb convention, is equal to the answer that we want. -1045 +1050 01:07:29,000 --> 01:07:31,260 Let me just write it down in a different color to really emphasize. -1046 +1051 01:07:33,100 --> 01:07:34,980 Cosine squared of theta. -1047 +1052 01:07:35,980 --> 01:07:39,860 And let me emphasize, this is not using the Pythagorean theorem in any way. -1048 +1053 01:07:39,880 --> 01:07:43,123 This is not having the cosine squared arrive because we were assuming -1049 +1054 01:07:43,123 --> 01:07:46,320 that this sits on a circle where x squared plus y squared equals one. -1050 +1055 01:07:46,320 --> 01:07:48,730 We're just using the definition of cosine twice -1051 +1056 01:07:48,730 --> 01:07:50,940 to cast one shadow and then to cast another. -1052 +1057 01:07:51,520 --> 01:07:55,840 Now, you can probably see where this is going, do the same thing for the sine of theta. -1053 +1058 01:07:56,420 --> 01:07:59,840 So another place where theta shows up in our diagram is right here. -1054 +1059 01:08:00,280 --> 01:08:05,760 And a way that you can see that is that this angle plus something must equal 90 degrees. -1055 +1060 01:08:06,720 --> 01:08:10,520 But this other something plus r theta had to be the -1056 +1061 01:08:10,520 --> 01:08:14,760 remaining angles of the little right triangle that we had. -1057 +1062 01:08:14,760 --> 01:08:17,560 Or you might think up here, I could just give it a different name. -1058 +1063 01:08:17,560 --> 01:08:23,027 If I called this other angle alpha, then we know that alpha plus theta, -1059 +1064 01:08:23,027 --> 01:08:26,672 alpha plus theta plus 90 degrees, or pi halves, -1060 +1065 01:08:26,672 --> 01:08:31,760 if you want to get used to the better conventions, has to equal pi. -1061 +1066 01:08:32,800 --> 01:08:34,660 It's got to equal 180 degrees. -1062 +1067 01:08:35,399 --> 01:08:36,359 This is maybe confusing. -1063 +1068 01:08:36,779 --> 01:08:39,399 Just think alpha plus theta plus pi halves. -1064 +1069 01:08:39,500 --> 01:08:41,761 In the back of your mind, you can go to 90 degrees -1065 +1070 01:08:41,761 --> 01:08:43,580 if that's something you want for comfort. -1066 +1071 01:08:44,840 --> 01:08:48,525 But that also implies that when we have a new right triangle sitting here, -1067 +1072 01:08:48,525 --> 01:08:52,751 this little angle here has got to be theta because it's alpha plus 90 plus something, -1068 +1073 01:08:52,751 --> 01:08:54,520 so that something's got to be theta. -1069 +1074 01:08:55,220 --> 01:08:58,533 Now, the reason I have stated that so much is because we're going to -1070 +1075 01:08:58,533 --> 01:09:02,040 play our same shadow casting game to figure out what this side length is. -1071 +1076 01:09:02,040 --> 01:09:06,935 Now, with respect to that, we have sine of theta of this theta -1072 +1077 01:09:06,935 --> 01:09:11,675 in particular is the opposite, the new shadow we don't know, -1073 +1078 01:09:11,675 --> 01:09:16,960 I'll call it s prime, is equal to s prime divided by the hypotenuse. -1074 +1079 01:09:17,359 --> 01:09:22,545 Well, on this smaller right triangle, what was the long side has become the hypotenuse, -1075 +1080 01:09:22,545 --> 01:09:23,960 so that's sine of theta. -1076 +1081 01:09:24,840 --> 01:09:25,920 Sine of theta. -1077 +1082 01:09:25,920 --> 01:09:31,660 Which means sine squared of theta is our other shadow. -1078 +1083 01:09:32,420 --> 01:09:35,620 This here is sine squared of theta. -1079 +1084 01:09:36,300 --> 01:09:37,000 So that's kind of cool. -1080 +1085 01:09:37,380 --> 01:09:41,517 The side length, which was by definition one, because we've always been -1081 +1086 01:09:41,517 --> 01:09:44,793 rescaling our triangles such that the hypotenuse is one, -1082 +1087 01:09:44,793 --> 01:09:48,930 can be broken down into one part that's cosine squared and another part -1083 +1088 01:09:48,930 --> 01:09:50,080 that's sine squared. -1084 +1089 01:09:50,080 --> 01:09:54,327 And the way to break that down is to just draw a single line, -1085 +1090 01:09:54,327 --> 01:09:59,260 the perpendicular projection from our corner point onto that hypotenuse. -1086 +1091 01:10:00,100 --> 01:10:05,037 So what that means is every time you see a unit circle out in the wild, -1087 +1092 01:10:05,037 --> 01:10:10,180 which I'm sure you do pretty regularly, and you draw the standard triangle -1088 +1093 01:10:10,180 --> 01:10:14,774 to understand sine and cosine, so that height is going to be sine, -1089 +1094 01:10:14,774 --> 01:10:19,780 the x coordinate is going to be cosine, and our hypotenuse is always one. -1090 +1095 01:10:21,320 --> 01:10:24,740 So that hypotenuse is one, this is theta. -1091 +1096 01:10:25,540 --> 01:10:28,300 And you ask yourself, where is cosine squared in this diagram? -1092 +1097 01:10:28,540 --> 01:10:29,520 Where is sine squared? -1093 +1098 01:10:30,480 --> 01:10:35,701 You can just draw a little perpendicular line here, -1094 +1099 01:10:35,701 --> 01:10:40,220 and the part on the bottom is cosine squared. -1095 +1100 01:10:40,880 --> 01:10:47,160 So, cos squared of theta is this length, this length here. -1096 +1101 01:10:48,780 --> 01:10:51,168 And what we'll do in a future lecture, actually, -1097 +1102 01:10:51,168 --> 01:10:54,581 I want to talk about geometry and geometry proofs, and in particular, -1098 +1103 01:10:54,581 --> 01:10:57,799 one that comes up all the time, I've used it in many past videos, -1099 +1104 01:10:57,799 --> 01:11:00,968 it's high time we just sit down and say, how would you prove it, -1100 +1105 01:11:00,968 --> 01:11:02,480 is the inscribed angle theorem. -1101 +1106 01:11:03,060 --> 01:11:06,800 So, after showing you how complex numbers play into things next time, -1102 +1107 01:11:06,800 --> 01:11:10,967 I want to turn back to this particular diagram to give another very intuitive -1103 +1108 01:11:10,967 --> 01:11:15,402 understanding for the identity that you found simply by playing around with graphs -1104 +1109 01:11:15,402 --> 01:11:19,356 at the very beginning here, which, again, is a very non-trivial identity, -1105 +1110 01:11:19,356 --> 01:11:23,577 that the cosine squared of some value is one plus the cosine of two times that -1106 +1111 01:11:23,577 --> 01:11:24,700 value divided by two. -1107 +1112 01:11:25,460 --> 01:11:30,399 The more standard way you see it, by the way, if you were to look this up in a textbook, -1108 +1113 01:11:30,399 --> 01:11:34,451 instead of writing this as x, this is commonly written as something like -1109 +1114 01:11:34,451 --> 01:11:37,560 alpha halves for some angle, and then 2x would be alpha. -1110 +1115 01:11:37,560 --> 01:11:45,072 And the whole thing ends up looking like cosine of alpha divided by two -1111 +1116 01:11:45,072 --> 01:11:52,480 is equal to the square root of one plus cosine of alpha divided by two. -1112 +1117 01:11:53,980 --> 01:11:56,917 I, for one, I have such trouble memorizing the trig identities, -1113 +1118 01:11:56,917 --> 01:11:59,580 and I always would just sort of re-derive them on the fly. -1114 +1119 01:11:59,580 --> 01:12:03,126 And this is one where I think the easiest way to re-derive is to practice -1115 +1120 01:12:03,126 --> 01:12:07,008 kind of smooshing around graphs in your head, when you have the one key insight, -1116 +1121 01:12:07,008 --> 01:12:09,980 which is that squaring cosine gets you a smaller cosine graph. -1117 +1122 01:12:10,540 --> 01:12:12,906 And the next time I'll show you how complex numbers help -1118 +1123 01:12:12,906 --> 01:12:15,440 you kind of re-derive a bunch of other identities on the fly. -1119 +1124 01:12:15,740 --> 01:12:18,160 But this one in particular, I just always had trouble remembering. -1120 +1125 01:12:18,520 --> 01:12:19,940 It's called the half-angle identity. -1121 +1126 01:12:19,940 --> 01:12:24,174 And I just think it's delightful that before you even know any trigonometry, -1122 +1127 01:12:24,174 --> 01:12:28,410 simply hopping over to something like Desmos and being a little bit playful, -1123 +1128 01:12:28,410 --> 01:12:31,600 that, I mean, that lands you on a highly non-trivial fact. -1124 +1129 01:12:32,420 --> 01:12:36,620 And once again, it's one that indicates that cosine is related to exponentials. -1125 +1130 01:12:37,400 --> 01:12:41,240 That's not obvious, and hopefully next time we can start to get a glimpse of why. -1126 +1131 01:12:41,840 --> 01:12:43,180 Thanks for sticking around this long. -1127 +1132 01:12:43,300 --> 01:12:46,441 Thank you once again to Cam and Ben Eater, who have been making -1128 +1133 01:12:46,441 --> 01:12:49,780 the item pool magic behind the scenes to get us our live statistics. -1129 +1134 01:12:49,940 --> 01:12:52,980 And I'll see you go at the very end. -1130 +1135 01:12:53,220 --> 01:12:54,960 I had almost everything down until the very end. -1131 +1136 01:12:55,260 --> 01:12:56,840 I will see you all on Friday. diff --git a/2020/ldm-trigonometry/english/sentence_timings.json b/2020/ldm-trigonometry/english/sentence_timings.json index 7e438c09d..0ff55f04c 100644 --- a/2020/ldm-trigonometry/english/sentence_timings.json +++ b/2020/ldm-trigonometry/english/sentence_timings.json @@ -540,7 +540,7 @@ 693.28 ], [ - "Sorry for everyone who felt a little bit rushed there, my finger slipped.", + "Okay, well, sorry for everyone who felt a little bit rushed there, my finger slipped.", 694.76, 698.5 ], @@ -610,7 +610,7 @@ 771.44 ], [ - "And that's one of the things that I'd like to get to by the end of the second lecture.", + "And that's one of the things that I'd like to get to by the end of the end of the second lecture.", 771.44, 776.72 ], @@ -745,7 +745,7 @@ 938.8 ], [ - "The input is a kind of distance around a unit circle, but you can think of that as an angle.", + "The input is a kind of, it's a kind of distance around a unit circle, but you could think of that as an angle,", 938.8, 946.74 ], @@ -845,7 +845,7 @@ 1056.7 ], [ - "It seems like there's not a 100% consensus on this question and answers are still rolling in.", + "it seems like there's not a hundred percent consensus on this question and answers are still rolling in.", 1057.02, 1061.8 ], @@ -935,33 +935,33 @@ 1183.04 ], [ - "Which, same game, but it's measuring your y-coordinate.", + "What's relevant is that it's negative, right? It's very close to negative one. So it's close to negative 0.99. Now on the other hand, if we were doing this with sine, which same game, but it's measuring your y-coordinate,", 1183.04, - 1186.14 + 1199.36 ], [ "When you walk almost halfway around, which is to say almost pi radians, well, there's still a little bit of height left.", - 1186.32, - 1195.26 + 1199.36, + 1208.32 ], [ "And importantly, it's positive because you're still above that x-axis.", - 1195.56, - 1198.3 + 1208.32, + 1208.32 ], [ "So when we turn back to our question, the only thing differentiating our answers was that some of them would switch whether sine or cosine was the one that's close to negative 1.", - 1199.1, - 1208.32 + 1208.32, + 1214.4 ], [ "And the others would just switch the sine, s-i-g-n, whether it's positive or negative.", - 1208.32, - 1213.4 + 1214.4, + 1214.4 ], [ "So congratulations to the 3,152 of you who got that one correct.", - 1214.14, + 1214.4, 1218.7 ], [ @@ -1155,12 +1155,12 @@ 1474.22 ], [ - "So that's probably enough time for people to have rolled their answers in.", + "All right, so that's probably enough time for people to have rolled their answers in.", 1474.86, 1479.36 ], [ - "Let's go ahead and see what the majority has submitted.", + "Now let's go ahead and see what they see what the majority has submitted.", 1479.78, 1483.52 ], @@ -1320,7 +1320,7 @@ 1670.62 ], [ - "So we might as well say, okay, let's scale this so that the hypotenuse is 1, meaning we divide everything by h, and what that's going to get us is the opposite side is now O over h, whatever that was, and then the adjacent side is A over h, meaning that the vertical component here is sine of theta, where theta was our angle down here, and then that bottom part is cosine of theta.", + "So we might as well say, okay, let's scale this so that the hypotenuse is 1, meaning we divide everything by h. And what that's going to get us is the opposite side is now o over h, whatever that was, and then the adjacent side is a over h, meaning that the y, or the vertical component here, is sine of theta, where theta was our angle down here. Okay. And then that bottom part is cosine of theta. Okay.", 1670.62, 1701.4 ], @@ -1355,12 +1355,12 @@ 1772.1 ], [ - "So for example, if you were to say 180 degrees, which is walking halfway around the circle, that's the same as pi radians.", + "So for example, if you were to say 180 degrees, excuse me, degrees, which is walking halfway around the circle, that's the same as pi radians.", 1772.66, 1784.7 ], [ - "And if you were to take something like 60 degrees, which would maybe end up, I don't know, around here, that might be 60 degrees.", + "And if you were to take something like 60 degrees, okay, which would maybe end up, oh, I don't know, around here, that might be 60 degrees.", 1786.6, 1795.44 ], @@ -1445,7 +1445,7 @@ 1911.7 ], [ - "And let's say each of those side lengths is one, because if we get to choose our side lengths, why not one?", + "And let's say each of those side lengths is one because, you know, if we get to choose our side lengths, why not one?", 1914.68, 1921.54 ], @@ -2585,7 +2585,7 @@ 3122.1 ], [ - "It looks like I'm getting a phone call.", + "All right, it looks like I'm getting a phone call.", 3125.86, 3127.76 ], @@ -2615,7 +2615,7 @@ 3148.8 ], [ - "How are the answers still changing?", + "How are the answers still changing? How are the answers still changing?", 3149.04, 3150.2 ], @@ -2720,7 +2720,7 @@ 3223.04 ], [ - "We know.", + "s, we know, right?", 3223.32, 3223.82 ], @@ -2790,7 +2790,7 @@ 3297.7 ], [ - "And in the olden days, the only way that people could compute them by hand for something like the existence of calculus would be measuring it.", + "and in the olden days, the only way that people could compute them by hand, before something like the existence of calculus, would be measuring it, right?", 3297.7, 3305.16 ], @@ -3100,7 +3100,7 @@ 3634.16 ], [ - "OK.", + "Okay. All right", 3634.68, 3634.78 ], @@ -3375,18 +3375,18 @@ 3950.58 ], [ - "So remember, if that's our angle theta, then one of our side lengths is sine of theta and the other one is cosine of theta.", + "So remember, if that's our angle theta, then one of our side lengths is sine theta. And that's the angle theta. And that's the angle theta. And that's the angle theta. Then one of our side lengths is sine of theta. And the other one is cosine of theta.", 3951.4, - 3958.1 + 3967.5 ], [ "Let me ask you, if I draw, well, okay, so, you know, the example we had earlier was this leaning tower thing where we kind of imagined casting a shadow from a leaning tower onto the ground.", - 3959.88, - 3972.54 + 3967.5, + 3973.2 ], [ "This is sort of going to flip it on its head and say, well, this angle theta between them, who's to say who's the tower and who's the ground?", - 3972.84, + 3973.2, 3978.7 ], [ diff --git a/2020/ldm-trigonometry/english/transcript.txt b/2020/ldm-trigonometry/english/transcript.txt index 41d371acb..524fdb814 100644 --- a/2020/ldm-trigonometry/english/transcript.txt +++ b/2020/ldm-trigonometry/english/transcript.txt @@ -106,7 +106,7 @@ Option A, square root of 1 minus x squared, option B, log of x, option C, x squa Okay, so I'm going to give you a couple minutes, a little bit to answer that. We won't take too long on this one. Oh, it looks like I accidentally graded it right away. -Sorry for everyone who felt a little bit rushed there, my finger slipped. +Okay, well, sorry for everyone who felt a little bit rushed there, my finger slipped. Well, in either case, it's now been revealed that the answer is D, 2 to the x. Okay, and we can see this again if we go and just play around a little bit in our Desmos graphs. So what we might do is pop over here and let me plug in 2 to the x. @@ -120,7 +120,7 @@ But what this means is that exponents have this funny property that when you squ And of course, the reason I bring that up is that you're looking at something very similar when it comes to cosine. Just before, just by playing around with graphs, before you even know what they mean, you can have this tiny little instinct in your head that says, maybe, just maybe, cosine is somehow related to exponents. It's not at all obvious how it is, but it absolutely is. -And that's one of the things that I'd like to get to by the end of the second lecture. +And that's one of the things that I'd like to get to by the end of the end of the second lecture. We won't get to the end of there today. So with all of that, I think it's high time that we actually talk about what sine and cosine actually are. I don't want to come in assuming that you necessarily know that already, so let's have a little moment to go back to the basics. @@ -147,7 +147,7 @@ And in that context, it starts off at one, and then as you walk around the circl It ultimately gets down to negative one before it starts increasing again. Okay? So this is how you might think of sine and cosine with respect to circles. -The input is a kind of distance around a unit circle, but you can think of that as an angle. +The input is a kind of, it's a kind of distance around a unit circle, but you could think of that as an angle, And we're going to talk in a moment about the difference between degrees and radians. But when you're thinking about walking around the circle, if it has a radius of one, you can think of that input as being the literal distance that you walk. So as an example, notice what happens when we get to the distance pi. @@ -167,7 +167,7 @@ B, the sine of 3 is around 0.14 and the cosine of 3 is around negative 0.99. C, sine of 3 is around 0.99, cosine of 3 is around 0.14. And D, sine of 3 is around negative 0.99 and cosine of 3 is around 0.14. Okay, so just to give you a little moment to think that through. -It seems like there's not a 100% consensus on this question and answers are still rolling in. +it seems like there's not a hundred percent consensus on this question and answers are still rolling in. So I'm going to go ahead and give you a little bit of pause and ponder music. I'll let you just think about it. I don't want you to feel rushed here. @@ -185,7 +185,7 @@ So to walk 3 units is going to be something a little bit shy of that. Okay, and if you notice, there's very little change in the x direction as you're at that left side of the circle. So when it approaches 1, it gets really close to 1 and then it doesn't really change that much. So maybe it shouldn't be too surprising that our cosine of 3 turned out to be around 0.99. -Which, same game, but it's measuring your y-coordinate. +What's relevant is that it's negative, right? It's very close to negative one. So it's close to negative 0.99. Now on the other hand, if we were doing this with sine, which same game, but it's measuring your y-coordinate, When you walk almost halfway around, which is to say almost pi radians, well, there's still a little bit of height left. And importantly, it's positive because you're still above that x-axis. So when we turn back to our question, the only thing differentiating our answers was that some of them would switch whether sine or cosine was the one that's close to negative 1. @@ -229,8 +229,8 @@ I get it. You're excited. You really want to do it. But in this case, it's a little bit more fun if we have the slow reveal. -So that's probably enough time for people to have rolled their answers in. -Let's go ahead and see what the majority has submitted. +All right, so that's probably enough time for people to have rolled their answers in. +Now let's go ahead and see what they see what the majority has submitted. And the correct answer is B, which is 100 times the cosine of 80. And it looks like the second most common answer was A, which is 100 times the sine of 80. So it looks like you guys correctly knew that you were multiplying it by one of these trigonometric functions, but there might have just been a little swap up for which one was which. @@ -262,15 +262,15 @@ Instead, they want some sort of natural unit, something where you imagine if you So how does any of this connect to the unit circle and the idea of the distance around that unit circle that we were walking, like we saw earlier? Well, let me just pull up a pre-printed unit circle, because trust me, you don't want to see me try to draw a circle. What's neat here is we can, for each of these triangles, imagine rescaling it so that that hypotenuse is really 1, because all we care about are ratios of lengths of sides. -So we might as well say, okay, let's scale this so that the hypotenuse is 1, meaning we divide everything by h, and what that's going to get us is the opposite side is now O over h, whatever that was, and then the adjacent side is A over h, meaning that the vertical component here is sine of theta, where theta was our angle down here, and then that bottom part is cosine of theta. +So we might as well say, okay, let's scale this so that the hypotenuse is 1, meaning we divide everything by h. And what that's going to get us is the opposite side is now o over h, whatever that was, and then the adjacent side is a over h, meaning that the y, or the vertical component here, is sine of theta, where theta was our angle down here. Okay. And then that bottom part is cosine of theta. Okay. And if we want to think about all possible right triangles who have a hypotenuse of 1, what you might think of is taking a unit circle, because that's saying every point is a distance of 1 away, so think of that line as the hypotenuse of a triangle. If you have a triangle with a bigger angle, it shows up there. If you have a triangle with a smaller angle, it shows up there. And it's as if you take all possible right triangles and you fix them so that their points are all in common, and such that their hypotenuses are all 1, you scale them down, and the other tip of the triangle is going to trace out a unit circle like this, and each one of these points for the corresponding angle is going to have coordinates, cosine of theta as the x-coordinate, and then sine of theta for the y-coordinate. And I'm just writing it vertically so I can fit it all on here. Now typically, like I said, mathematicians like to think in terms of radians, which is really a way of saying if we know that the radius of our circle is 1, what is the distance that you've walked along the outside here? -So for example, if you were to say 180 degrees, which is walking halfway around the circle, that's the same as pi radians. -And if you were to take something like 60 degrees, which would maybe end up, I don't know, around here, that might be 60 degrees. +So for example, if you were to say 180 degrees, excuse me, degrees, which is walking halfway around the circle, that's the same as pi radians. +And if you were to take something like 60 degrees, okay, which would maybe end up, oh, I don't know, around here, that might be 60 degrees. Well, that's a third of that. You're walking a third of the half turn around the circle. Everyone who's an enthusiast about tau is sort of yelling right now because it would make the conventions easier, but pi is the standard, so we're working with it. @@ -287,7 +287,7 @@ It wouldn't have a special button on your calculator, because effectively what y So for example, let's say I try to draw an equilateral triangle. An equilateral triangle will be nice and symmetric, and we're going to be able to leverage that symmetry to figure out a concrete value of sine and cosine. This looks roughly equilateral, wouldn't you say? -And let's say each of those side lengths is one, because if we get to choose our side lengths, why not one? +And let's say each of those side lengths is one because, you know, if we get to choose our side lengths, why not one? Because remember, sine and cosine have everything to do with just ratios. So on an equilateral triangle, this angle here is a third of the total. The total is 180 degrees, so you might think of this as 60 degrees. @@ -515,13 +515,13 @@ And then item pool is a pretty cool thing that will be unfolding over the next c So you might want to keep your eye on that. It looks like some more consensus is starting to form. So I'm going to go ahead and grade this question. -It looks like I'm getting a phone call. +All right, it looks like I'm getting a phone call. Awesome. So the correct answer is B, which is square root of it's a very complicated thing, right? Square root of one plus the square root of three over two divided by two. So congratulations to the two to two nine of you who two to two for two to four three. Two thousand two hundred forty three of you. -How are the answers still changing? +How are the answers still changing? How are the answers still changing? Going to have to talk to Cam and Eater about that. My hands are not on this. Who got that correct? @@ -542,7 +542,7 @@ OK, if we enter that as X is equal to one plus the cosine of two times that valu Pi six. All divided by two. Now, the cosine of pi six. -We know. +s, we know, right? That's the same as our 30 degree angle here. Again, when you have a 30, 60, 90 triangle, the shorter side of that, which is going to be the sign in this context, the shorter side is one half. And then the longer side is square root of three over two, assuming that the hypotenuse is one, which in the context of the unit circle, we always do assume. @@ -556,7 +556,7 @@ We can have even more, I won't call them trivial values, but easier to compute v That means I've walked halfway around. So my x coordinate is negative one. But outside of just a handful of values like that, they're very hard to compute by hand. -And in the olden days, the only way that people could compute them by hand for something like the existence of calculus would be measuring it. +and in the olden days, the only way that people could compute them by hand, before something like the existence of calculus, would be measuring it, right? You like draw out with a protractor and you try to carefully see what the coordinates are or layering on usage of identities like this over and over in ways that you can get approximate values like this. And just to check ourselves, we can actually go and plug this into a calculator if we wanted to, something like Desmos and just see, one, how big the value is and kind of confirm for ourselves that they turn out to be the same. So if we pop over here and I just type something like. @@ -618,7 +618,7 @@ The final question, I just want you to guess what the tangent of theta looks lik And it should be pulled up now. And oh, there we go. It is rather tall, though. -OK. +Okay. All right So let me go ahead and pull up a separate image for you here so that you can see all of the options. Let's pop over here and take a look. Great. @@ -673,7 +673,7 @@ So if there was just one thing that you were going to stick around for, I would Because it gives a very sneaky proof of the Pythagorean theorem. It might feel like a circular argument at first, but I assure you that it's not. And I also assure you that the word circular was not meant as a pun there. -So remember, if that's our angle theta, then one of our side lengths is sine of theta and the other one is cosine of theta. +So remember, if that's our angle theta, then one of our side lengths is sine theta. And that's the angle theta. And that's the angle theta. And that's the angle theta. Then one of our side lengths is sine of theta. And the other one is cosine of theta. Let me ask you, if I draw, well, okay, so, you know, the example we had earlier was this leaning tower thing where we kind of imagined casting a shadow from a leaning tower onto the ground. This is sort of going to flip it on its head and say, well, this angle theta between them, who's to say who's the tower and who's the ground? What if I wanted to have a light source over here and I wanted to cast a shadow from the ground onto the tower? diff --git a/2020/ldm-trigonometry/french/sentence_translations.json b/2020/ldm-trigonometry/french/sentence_translations.json index a4dbb7c05..01020d46f 100644 --- a/2020/ldm-trigonometry/french/sentence_translations.json +++ b/2020/ldm-trigonometry/french/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "Cela ressemble au même type d'onde, mais elle est entièrement positive et elle passe de 1 à 0, puis jusqu'à 1 en l'espace de pi. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "D'accord, donc lorsque vous branchez 2x, cela devrait être identique à f de x au carré. ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "Donc si nous passons ici, et que je tape juste quelque chose comme, cosinus de pi supérieur à 12, 0. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "Et c'est l'angle thêta. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "Si j'appelle cet autre angle alpha, alors nous savons que alpha plus thêta plus 90 degrés, ou pi moitiés si vous voulez vous habituer aux meilleures conventions, doivent être égaux à pi. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/german/sentence_translations.json b/2020/ldm-trigonometry/german/sentence_translations.json index c19fcee39..0c31e4860 100644 --- a/2020/ldm-trigonometry/german/sentence_translations.json +++ b/2020/ldm-trigonometry/german/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "Es sieht aus wie die gleiche Art von Welle, aber es ist alles positiv und geht von 1 nach unten auf 0 und dann in der Spanne von Pi auf 1. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "Okay, wenn Sie also 2x einstecken, sollte es dasselbe sein wie f von x im Quadrat. ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "Wenn wir also hierher kommen und ich einfach so etwas wie „Kosinus von Pi über 12, 0“ eingebe. 965. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "Und das ist der Winkel Theta. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "Wenn ich diesen anderen Winkel Alpha nenne, dann wissen wir, dass Alpha plus Theta plus 90 Grad oder Pi-Hälften, wenn man sich an die besseren Konventionen gewöhnen will, gleich Pi sein müssen. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/hebrew/sentence_translations.json b/2020/ldm-trigonometry/hebrew/sentence_translations.json index 17eac11b2..5fdd0c5d4 100644 --- a/2020/ldm-trigonometry/hebrew/sentence_translations.json +++ b/2020/ldm-trigonometry/hebrew/sentence_translations.json @@ -210,7 +210,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi.", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi.", "translatedText": "זה נראה כמו אותו סוג של גל, אבל הכל חיובי והוא יורד מ-1 למטה ל-0, ואז עולה ל-1 בטווח של pi.", "n_reviews": 0, "start": 173.74, @@ -742,7 +742,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared.", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared.", "translatedText": "אוקיי, אז כשאתה מחבר פי 2, זה צריך להיות זהה ל-f של x בריבוע.", "n_reviews": 0, "start": 658.98, @@ -2639,14 +2639,14 @@ "end": 2352.34 }, { - "input": ".", + "input": "let me switch to thinking with respect to this lower left angle again.", "translatedText": ".", "n_reviews": 0, "start": 2352.86, "end": 2356.1 }, { - "input": ".", + "input": "If one of the side lengths is cosine of theta, and one of", "translatedText": ".", "n_reviews": 0, "start": 2356.38, @@ -3815,7 +3815,7 @@ "end": 3328.56 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965.", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five.", "translatedText": "אז אם נקפוץ לכאן, ואני פשוט מקליד משהו כמו, קוסינוס של פאי מעל 12, 0.965.", "n_reviews": 0, "start": 3329.3, @@ -4557,7 +4557,7 @@ "end": 3955.9 }, { - "input": "And that's the angle theta.", + "input": "And that's the angle theta. And that's the angle theta.", "translatedText": "וזו תטא הזווית.", "n_reviews": 0, "start": 3955.9, @@ -4725,7 +4725,7 @@ "end": 4097.56 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi.", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi.", "translatedText": "אם קראתי לזווית אחרת הזו אלפא, אז אנחנו יודעים שאלפא פלוס תטא פלוס 90 מעלות, או חצאי פאי אם אתה רוצה להתרגל למוסכמות היותר טובות, צריך להיות שווה ל-pi.", "n_reviews": 0, "start": 4097.56, diff --git a/2020/ldm-trigonometry/hindi/sentence_translations.json b/2020/ldm-trigonometry/hindi/sentence_translations.json index c36d645fe..a06f750c4 100644 --- a/2020/ldm-trigonometry/hindi/sentence_translations.json +++ b/2020/ldm-trigonometry/hindi/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "यह एक ही प्रकार की लहर की तरह दिखता है, लेकिन यह पूरी तरह से सकारात्मक है और यह 1 से 0 तक जाता है, फिर पाई की अवधि में 1 तक जाता है।", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "ठीक है, इसलिए जब आप 2x प्लग इन करते हैं, तो यह x वर्ग के f के समान होना चाहिए।", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "तो अगर हम यहां आते हैं, और मैं कुछ इस तरह टाइप करता हूं, 12, 0 पर पाई की कोज्या।", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "और वह कोण थीटा है।", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "यदि मैं इस अन्य कोण को अल्फ़ा कहता हूँ, तो हम जानते हैं कि अल्फ़ा प्लस थीटा प्लस 90 डिग्री, या यदि आप बेहतर सम्मेलनों के लिए अभ्यस्त होना चाहते हैं, तो पाई आधा, पाई के बराबर होना चाहिए।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/hungarian/sentence_translations.json b/2020/ldm-trigonometry/hungarian/sentence_translations.json index e766199c5..cd83208d6 100644 --- a/2020/ldm-trigonometry/hungarian/sentence_translations.json +++ b/2020/ldm-trigonometry/hungarian/sentence_translations.json @@ -210,7 +210,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi.", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi.", "translatedText": "Ugyanolyan hullámnak tűnik, de mind pozitív, és 1-ről 0-ra megy le, majd 1-re a pi tartományában.", "n_reviews": 0, "start": 173.74, @@ -742,7 +742,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared.", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared.", "translatedText": "Oké, ha 2x dugja be, akkor ennek meg kell egyeznie az x f négyzetével.", "n_reviews": 0, "start": 658.98, @@ -2639,14 +2639,14 @@ "end": 2352.34 }, { - "input": ".", + "input": "let me switch to thinking with respect to this lower left angle again.", "translatedText": ".", "n_reviews": 0, "start": 2352.86, "end": 2356.1 }, { - "input": ".", + "input": "If one of the side lengths is cosine of theta, and one of", "translatedText": ".", "n_reviews": 0, "start": 2356.38, @@ -3815,7 +3815,7 @@ "end": 3328.56 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965.", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five.", "translatedText": "Szóval, ha ideugrik, és beírok valami olyasmit, hogy a pi koszinusza 12, 0 felett.965.", "n_reviews": 0, "start": 3329.3, @@ -4557,7 +4557,7 @@ "end": 3955.9 }, { - "input": "And that's the angle theta.", + "input": "And that's the angle theta. And that's the angle theta.", "translatedText": "És ez a téta szög.", "n_reviews": 0, "start": 3955.9, @@ -4725,7 +4725,7 @@ "end": 4097.56 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi.", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi.", "translatedText": "Ha ezt a másik szöget alfa-nak nevezném, akkor tudjuk, hogy az alfa plusz théta plusz 90 fok, vagy a pi felének, ha meg akarod szokni a jobb konvenciókat, akkor pi-nek kell egyenlőnek lennie.", "n_reviews": 0, "start": 4097.56, diff --git a/2020/ldm-trigonometry/indonesian/sentence_translations.json b/2020/ldm-trigonometry/indonesian/sentence_translations.json index 6652cc18e..56efefc40 100644 --- a/2020/ldm-trigonometry/indonesian/sentence_translations.json +++ b/2020/ldm-trigonometry/indonesian/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "Bentuknya seperti gelombang yang sama, tapi semuanya positif dan bergerak dari 1 ke 0, lalu naik ke 1 dalam rentang pi. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "Oke, jadi kalau dicolokkan 2x harusnya sama dengan f dari x kuadrat. ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "Jadi jika kita mampir ke sini, dan saya mengetikkan sesuatu seperti, cosinus pi di atas 12, 0.965. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "Dan itulah sudut theta. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "Jika saya menyebut sudut lain ini alfa, maka kita tahu bahwa alfa ditambah theta ditambah 90 derajat, atau pembagian pi jika Anda ingin terbiasa dengan konvensi yang lebih baik, harus sama dengan pi. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/italian/sentence_translations.json b/2020/ldm-trigonometry/italian/sentence_translations.json index 5bb031a8f..8cdbcecb8 100644 --- a/2020/ldm-trigonometry/italian/sentence_translations.json +++ b/2020/ldm-trigonometry/italian/sentence_translations.json @@ -210,7 +210,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi.", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi.", "translatedText": "Sembra lo stesso tipo di onda, ma è tutta positiva e va da 1 a 0, poi fino a 1 nell'intervallo di pi greco.", "n_reviews": 0, "start": 173.74, @@ -742,7 +742,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared.", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared.", "translatedText": "Ok, quindi quando inserisci 2x, dovrebbe essere uguale a f(x^2).", "n_reviews": 0, "start": 658.98, @@ -2639,14 +2639,14 @@ "end": 2352.34 }, { - "input": ".", + "input": "let me switch to thinking with respect to this lower left angle again.", "translatedText": ".", "n_reviews": 0, "start": 2352.86, "end": 2356.1 }, { - "input": ".", + "input": "If one of the side lengths is cosine of theta, and one of", "translatedText": ".", "n_reviews": 0, "start": 2356.38, @@ -3815,7 +3815,7 @@ "end": 3328.56 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965.", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five.", "translatedText": "Quindi se facciamo un salto qui e scrivo qualcosa come coseno di pi greco su 12, 0.965.", "n_reviews": 0, "start": 3329.3, @@ -4557,7 +4557,7 @@ "end": 3955.9 }, { - "input": "And that's the angle theta.", + "input": "And that's the angle theta. And that's the angle theta.", "translatedText": "E questo è l'angolo theta.", "n_reviews": 0, "start": 3955.9, @@ -4725,7 +4725,7 @@ "end": 4097.56 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi.", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi.", "translatedText": "Se chiamassi quest'altro angolo alfa, allora sappiamo che alfa più theta più 90 gradi, o pi metà se vuoi abituarti alle convenzioni migliori, deve essere uguale a pi greco.", "n_reviews": 0, "start": 4097.56, diff --git a/2020/ldm-trigonometry/japanese/sentence_translations.json b/2020/ldm-trigonometry/japanese/sentence_translations.json index a8115758c..88876de61 100644 --- a/2020/ldm-trigonometry/japanese/sentence_translations.json +++ b/2020/ldm-trigonometry/japanese/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "同じ種類の波のように見えますが、すべて正であり、円周率の範 囲で 1 から 0 に下がり、その後 1 に上がります。", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "さて、2x を接続すると、x の 2 乗の f と同じになるはずです。", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "ここにアクセスして、「12, 0 上の pi の余弦」のようなものを入力するとします。965。", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "そしてそれが角度シータです。", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "この別の角度をアルファと呼ぶと、アルファとシータに 90 度を加えたもの、またはより良い規則に 慣れたい場合は pi の半分が pi に等しくなければならないことがわかります。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/korean/sentence_translations.json b/2020/ldm-trigonometry/korean/sentence_translations.json index 511d17e61..ad70b4309 100644 --- a/2020/ldm-trigonometry/korean/sentence_translations.json +++ b/2020/ldm-trigonometry/korean/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "같은 종류의 파동처럼 보이지만 모두 양수이고 1에서 0으로 내려가고 파이 범위에서 1까지 올라갑니다. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "좋습니다. 2x를 대입하면 f(x의 제곱)와 같아야 합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "그래서 여기로 가서 pi/12, 0의 코사인과 같은 것을 입력하면 됩니다. 965. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "그리고 그것이 각도 세타입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "내가 이 다른 각도를 알파라고 부른다면 우리는 알파 더하기 세타 더하기 90도, 또는 더 나은 규칙에 익숙해지려면 파이 반이 파이와 같아야 한다는 것을 알 수 있습니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/marathi/sentence_translations.json b/2020/ldm-trigonometry/marathi/sentence_translations.json index 760a02a96..5ae38af32 100644 --- a/2020/ldm-trigonometry/marathi/sentence_translations.json +++ b/2020/ldm-trigonometry/marathi/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "हे एकाच प्रकारच्या लहरीसारखे दिसते, परंतु ते सर्व सकारात्मक आहे आणि ते 1 वरून 0 पर्यंत जाते, नंतर पाईच्या स्पॅनमध्ये 1 पर्यंत जाते. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "ठीक आहे, म्हणून जेव्हा तुम्ही 2x प्लग इन करता, तेव्हा ते x च्या f च्या वर्गाप्रमाणे असावे. ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "म्हणून जर आपण येथे पॉप ऑन केले, आणि मी असे काहीतरी टाईप केले की, 12, 0 वरील pi चा कोसाइन. ९६५. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "आणि तो कोन थीटा आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "जर मी या दुसर्‍या कोनाला अल्फा म्हटले, तर आपल्याला माहित आहे की अल्फा अधिक थीटा अधिक 90 अंश, किंवा जर तुम्हाला अधिक चांगल्या नियमांची सवय करून घ्यायची असेल तर pi हाल्व्हस पाई समान असणे आवश्यक आहे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/persian/sentence_translations.json b/2020/ldm-trigonometry/persian/sentence_translations.json index 359dd175f..a3d43a1c5 100644 --- a/2020/ldm-trigonometry/persian/sentence_translations.json +++ b/2020/ldm-trigonometry/persian/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "به نظر می رسد یک نوع موج است، اما همه آن مثبت است و از 1 به 0، سپس به 1 در گستره pi می رسد. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "سوال ما چه می گوید؟ خوب، پس وقتی 2x را وصل می کنید، باید همان f از x مربع باشد. ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "بنابراین، اگر به اینجا برویم، و من فقط چیزی شبیه به کسینوس pi را روی 12، 0 تایپ کنم. 965. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "و این زاویه تتا است. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "اگر من این زاویه دیگر را آلفا نامیدیم، آنگاه می‌دانیم که آلفا به علاوه تتا به اضافه 90 درجه، یا اگر می‌خواهید به قراردادهای بهتر عادت کنید، نصف پی می‌شود، باید برابر با پی باشد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/portuguese/sentence_translations.json b/2020/ldm-trigonometry/portuguese/sentence_translations.json index 3a98ef5f7..63413add9 100644 --- a/2020/ldm-trigonometry/portuguese/sentence_translations.json +++ b/2020/ldm-trigonometry/portuguese/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "Parece o mesmo tipo de onda, mas é tudo positivo e vai de 1 até 0, depois até 1 no intervalo de pi. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "Ok, então quando você insere 2x, deve ser igual a f de x ao quadrado. ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "Então, se aparecermos aqui e eu digitar algo como cosseno de pi sobre 12, 0.965. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "E esse é o ângulo teta. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "Se eu chamar esse outro ângulo de alfa, então saberemos que alfa mais teta mais 90 graus, ou pi pela metade, se você quiser se acostumar com as melhores convenções, tem que ser igual a pi. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/russian/sentence_translations.json b/2020/ldm-trigonometry/russian/sentence_translations.json index eae968cb1..afdfe295e 100644 --- a/2020/ldm-trigonometry/russian/sentence_translations.json +++ b/2020/ldm-trigonometry/russian/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "Это похоже на ту же волну, но она вся положительна и идет от 1 вниз до 0, а затем до 1 в диапазоне пи. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "Итак, когда вы подставляете 2x, оно должно быть таким же, как f от x в квадрате. ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "Итак, если мы зайдем сюда, и я просто наберу что-то вроде косинуса числа Пи больше 12, 0.965. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "И это угол тэта. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "Если бы я назвал этот другой угол альфа, то мы знали бы, что альфа плюс тета плюс 90 градусов, или половинки пи, если вы хотите привыкнуть к лучшим соглашениям, должны равняться пи. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/spanish/sentence_translations.json b/2020/ldm-trigonometry/spanish/sentence_translations.json index 7f3c89bfe..b04ab95bb 100644 --- a/2020/ldm-trigonometry/spanish/sentence_translations.json +++ b/2020/ldm-trigonometry/spanish/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "Parece el mismo tipo de onda, pero es toda positiva y va de 1 a 0 y luego a 1 en el lapso de pi. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "Bien, entonces cuando conectas 2x, debería ser lo mismo que f de x al cuadrado. ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "Entonces, si llegamos aquí y escribo algo como coseno de pi sobre 12, 0.965. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "Y ese es el ángulo theta. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "Si llamé a este otro ángulo alfa, entonces sabemos que alfa más theta más 90 grados, o pi mitades si quieres acostumbrarte a las mejores convenciones, tiene que ser igual a pi. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/tamil/sentence_translations.json b/2020/ldm-trigonometry/tamil/sentence_translations.json index d4bd6bdf6..de8d145db 100644 --- a/2020/ldm-trigonometry/tamil/sentence_translations.json +++ b/2020/ldm-trigonometry/tamil/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "இது ஒரே மாதிரியான அலையைப் போல் தெரிகிறது, ஆனால் இது எல்லாமே நேர்மறையாக இருக்கிறது, மேலும் அது 1 முதல் 0 வரை செல்கிறது, பின்னர் பை இடைவெளியில் 1 வரை செல்கிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "சரி, நீங்கள் 2xஐச் செருகும்போது, அது x ஸ்கொயர்டின் எஃப் போலவே இருக்க வேண்டும். ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "நாம் இங்கே பாப் ஓவர் என்றால், நான் 12, 0 க்கு மேல் பையின் கொசைன் போன்ற ஒன்றை தட்டச்சு செய்கிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "அதுவும் கோணல் தீட்டா. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "நான் இதை மற்ற கோண ஆல்பா என்று அழைத்தால், ஆல்பா பிளஸ் தீட்டா பிளஸ் 90 டிகிரி அல்லது பை பாதிகள் நீங்கள் சிறந்த மரபுகளுடன் பழக விரும்பினால், பைக்கு சமமாக இருக்க வேண்டும் என்பது எங்களுக்குத் தெரியும். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/telugu/sentence_translations.json b/2020/ldm-trigonometry/telugu/sentence_translations.json index c2b1e506e..d99c887ba 100644 --- a/2020/ldm-trigonometry/telugu/sentence_translations.json +++ b/2020/ldm-trigonometry/telugu/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "ఇది ఒకే రకమైన వేవ్ లాగా కనిపిస్తుంది, కానీ ఇది అంతా సానుకూలంగా ఉంటుంది మరియు ఇది 1 నుండి 0 వరకు, ఆపై pi వ్యవధిలో 1 వరకు ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "సరే, కాబట్టి మీరు 2xని ప్లగ్ ఇన్ చేసినప్పుడు, అది x స్క్వేర్డ్ యొక్క f వలె ఉండాలి. ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "కాబట్టి మనం ఇక్కడ పాప్ చేస్తే, మరియు నేను 12, 0 కంటే పై యొక్క కొసైన్ లాంటిది టైప్ చేస్తాను. 965. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "మరియు అది యాంగిల్ తీటా. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "నేను దీన్ని ఇతర యాంగిల్ ఆల్ఫా అని పిలిస్తే, ఆల్ఫా ప్లస్ తీటా ప్లస్ 90 డిగ్రీలు లేదా పై హాల్వ్‌లు మీరు మంచి సంప్రదాయాలను అలవాటు చేసుకోవాలనుకుంటే, పైకి సమానం కావాలని మాకు తెలుసు. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/thai/sentence_translations.json b/2020/ldm-trigonometry/thai/sentence_translations.json index 3167722c1..41447d647 100644 --- a/2020/ldm-trigonometry/thai/sentence_translations.json +++ b/2020/ldm-trigonometry/thai/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "โอเค ดูเหมือนว่า A เป็นตัวเลือกมีหุบเขาแบบนี้ โดยทั้งหมดนี้อาจอ่านน้อยไปหน่อย แต่พิสัยอยู่ระหว่าง 0 ถึงประมาณ 2 ไพ ดังนั้นใน A มันกระทบ 1 แล้วมันก็ลงไป และชน 1 อีกครั้งที่อินพุต ไพ และลงไปตามหุบเขาราบเหล่านี้ B มีพฤติกรรมคล้ายกันเมื่อถึง 1 และ 0 แต่จะแหลมกว่าเล็กน้อย มันเหมือนกับเด้งออกจากแกน x นั่น C ดูเหมือนคลื่นไซน์ มันดูเหมือนคลื่นแบบเดียวกัน แต่มันเป็นบวกหมด และมันไปจาก 1 ลงไปที่ 0 แล้วก็ขึ้นเป็น 1 ในช่วงพาย แล้ว D ก็ถูกแย่งกันในทิศทาง x และเริ่มเร็วขึ้นเมื่อคุณเคลื่อนไปทางขวามากขึ้น เอาล่ะ ดูเหมือนว่าเราจะได้คำตอบมากมายเข้ามา ฉันทามติเล็กน้อยต่อด้านบน แต่ก็ไม่ใช่การตัดสินใจที่เป็นเอกฉันท์อย่างแน่นอนเกี่ยวกับเรื่องนี้ที่ผู้คนกำลังจะไป ดูเหมือนว่ามีหนึ่งคำตอบที่ได้รับความนิยมมากที่สุด และมีสองคำตอบที่ใกล้เคียงกันสำหรับเหรียญเงินและเหรียญทองแดง ผมจะให้เวลาคุณอีกสองสามวินาทีถ้าคุณต้องการเปลี่ยนใจ หรือถ้าคุณต้องการไปที่ 3b1b. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "ถ้าเราป็อปตรงนี้ แล้วผมพิมพ์ว่า โคไซน์ของ ไพ ส่วน 12, 0 965. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "แต่อย่างอื่นบวก r ทีต้า ต้องเป็นมุมที่เหลือของสามเหลี่ยมมุมฉากเล็กๆ ที่เรามี หรือคุณอาจคิดว่าบนนี้ ผมตั้งชื่ออื่นก็ได้ หากผมเรียกอีกมุมหนึ่งว่าอัลฟ่า, เราก็รู้ว่าอัลฟ่า บวกทีต้า บวก 90 องศา, หรือ ไพ แบ่งครึ่งถ้าคุณต้องการทำความคุ้นเคยกับรูปแบบที่ดีกว่า, ต้องเท่ากับ ไพ มันต้องเท่ากับ 180 องศา. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/turkish/sentence_translations.json b/2020/ldm-trigonometry/turkish/sentence_translations.json index a60b0614f..562c9b88a 100644 --- a/2020/ldm-trigonometry/turkish/sentence_translations.json +++ b/2020/ldm-trigonometry/turkish/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "Aynı türden bir dalgaya benziyor ama hepsi pozitif ve pi aralığında 1'den 0'a, sonra 1'e kadar gidiyor. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "Tamam, yani 2x'i koyduğunuzda f x kare ile aynı olmalı. ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "Buraya gelirsek, kosinüs pi bölü 12, 0 gibi bir şey yazarım. 965. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "Ve bu teta açısıdır. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "Bu diğer açıya alfa dersem, alfa artı teta artı 90 derecenin veya daha iyi kurallara alışmak istiyorsanız pi yarısının pi'ye eşit olması gerektiğini biliyoruz. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/ukrainian/sentence_translations.json b/2020/ldm-trigonometry/ukrainian/sentence_translations.json index d810aebe5..362ba6559 100644 --- a/2020/ldm-trigonometry/ukrainian/sentence_translations.json +++ b/2020/ldm-trigonometry/ukrainian/sentence_translations.json @@ -210,7 +210,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi.", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi.", "translatedText": "Це виглядає як хвиля такого ж типу, але вона вся позитивна, і вона йде від 1 до 0, а потім до 1 у діапазоні пі.", "n_reviews": 0, "start": 173.74, @@ -742,7 +742,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared.", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared.", "translatedText": "Отже, коли ви підключаєте 2x, це має бути таким самим, як f від x у квадраті.", "n_reviews": 0, "start": 658.98, @@ -2639,14 +2639,14 @@ "end": 2352.34 }, { - "input": ".", + "input": "let me switch to thinking with respect to this lower left angle again.", "translatedText": ".", "n_reviews": 0, "start": 2352.86, "end": 2356.1 }, { - "input": ".", + "input": "If one of the side lengths is cosine of theta, and one of", "translatedText": ".", "n_reviews": 0, "start": 2356.38, @@ -3815,7 +3815,7 @@ "end": 3328.56 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965.", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five.", "translatedText": "Отже, якщо ми заглянемо сюди, і я просто введу щось на кшталт косинуса пі на 12, 0.965.", "n_reviews": 0, "start": 3329.3, @@ -4557,7 +4557,7 @@ "end": 3955.9 }, { - "input": "And that's the angle theta.", + "input": "And that's the angle theta. And that's the angle theta.", "translatedText": "І це кут тета.", "n_reviews": 0, "start": 3955.9, @@ -4725,7 +4725,7 @@ "end": 4097.56 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi.", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi.", "translatedText": "Якби я назвав цей інший кут альфа, тоді ми знаємо, що альфа плюс тета плюс 90 градусів, або половини пі, якщо ви хочете звикнути до кращих угод, має дорівнювати пі.", "n_reviews": 0, "start": 4097.56, diff --git a/2020/ldm-trigonometry/urdu/sentence_translations.json b/2020/ldm-trigonometry/urdu/sentence_translations.json index 1d69d6fd7..cd6686214 100644 --- a/2020/ldm-trigonometry/urdu/sentence_translations.json +++ b/2020/ldm-trigonometry/urdu/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "یہ ایک ہی قسم کی لہر کی طرح لگتا ہے، لیکن یہ سب مثبت ہے اور یہ 1 سے نیچے 0 تک، پھر pi کے دورانیے میں 1 تک جاتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "ہمارا سوال کیا کہتا ہے؟ ٹھیک ہے، تو جب آپ 2x پلگ ان کرتے ہیں، تو یہ x مربع کے f کے برابر ہونا چاہیے۔", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "تو اگر ہم یہاں پاپ کریں، اور میں صرف کچھ اس طرح ٹائپ کرتا ہوں، pi کا کوسائن 12، 0 سے زیادہ۔", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "اور وہ زاویہ تھیٹا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "اگر میں نے اس دوسرے زاویہ کو الفا کہا، تو ہم جانتے ہیں کہ الفا پلس تھیٹا پلس 90 ڈگری، یا اگر آپ بہتر کنونشنز کی عادت ڈالنا چاہتے ہیں تو pi halves کو برابر کرنا ہوگا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/ldm-trigonometry/vietnamese/sentence_translations.json b/2020/ldm-trigonometry/vietnamese/sentence_translations.json index 3d8603329..547312678 100644 --- a/2020/ldm-trigonometry/vietnamese/sentence_translations.json +++ b/2020/ldm-trigonometry/vietnamese/sentence_translations.json @@ -240,7 +240,7 @@ "end": 173.5 }, { - "input": "It looks like the same kind of wave, but it's all positive and it goes from 1 down to 0, then up to 1 in the span of pi. ", + "input": "Okay, it looks like the same kind of wave, but it's all positive and it's, it goes from 1 down to 0, then up to 1 in the span of pi. ", "translatedText": "Nó trông giống như một loại sóng, nhưng tất cả đều dương và nó đi từ 1 xuống 0, rồi lên 1 trong khoảng pi. ", "model": "google_nmt", "n_reviews": 0, @@ -840,7 +840,7 @@ "end": 658.12 }, { - "input": "Okay, so when you plug in 2x, it should be the same as f of x squared. ", + "input": "Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared. ", "translatedText": "Được rồi, vậy khi bạn thế vào 2x, nó sẽ bằng f(x bình phương). ", "model": "google_nmt", "n_reviews": 0, @@ -4312,7 +4312,7 @@ "end": 3383.36 }, { - "input": "So if we pop over here, and I just type something like, cosine of pi over 12, 0.965. ", + "input": "So if we pop over here and I just type something like. Cosine of pi over 12. Point nine six five. ", "translatedText": "Vì vậy, nếu chúng ta ghé qua đây và tôi chỉ cần gõ một cái gì đó như cosin của pi trên 12, 0.965. ", "model": "google_nmt", "n_reviews": 0, @@ -5144,7 +5144,7 @@ "end": 4081.54 }, { - "input": "And that's the angle theta. ", + "input": "And that's the angle theta. And that's the angle theta. ", "translatedText": "Và đó là góc theta. ", "model": "google_nmt", "n_reviews": 0, @@ -5336,7 +5336,7 @@ "end": 4222.48 }, { - "input": "If I called this other angle alpha, then we know that alpha plus theta plus 90 degrees, or pi halves if you want to get used to the better conventions, has to equal pi. ", + "input": "If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi. ", "translatedText": "Nếu tôi gọi góc kia là alpha, thì chúng ta biết rằng alpha cộng theta cộng 90 độ, hoặc hai nửa pi nếu bạn muốn làm quen với các quy ước tốt hơn, phải bằng pi. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/lockdown-math-announcement/catalan/sentence_translations.json b/2020/lockdown-math-announcement/catalan/sentence_translations.json index 2018d1dc0..5fd62e413 100644 --- a/2020/lockdown-math-announcement/catalan/sentence_translations.json +++ b/2020/lockdown-math-announcement/catalan/sentence_translations.json @@ -72,7 +72,7 @@ "end": 108.1 }, { - "input": "So tune in, I hope to see you there, and be prepared to do some math.", + "input": "So tune in, I hope to see you there, and be prepared to do some math. Thanks for watching!", "translatedText": "", "from_community_srt": "Així doncs, connecteu-vos, espero veureu-us-hi,", "n_reviews": 0, diff --git a/2020/lockdown-math-announcement/english/captions.srt b/2020/lockdown-math-announcement/english/captions.srt index af195904c..ee81341cc 100644 --- a/2020/lockdown-math-announcement/english/captions.srt +++ b/2020/lockdown-math-announcement/english/captions.srt @@ -136,5 +136,5 @@ but if anything changes on that, you'll see the schedule on the banner of the ch 35 00:01:48,440 --> 00:01:51,720 -So tune in, I hope to see you there, and be prepared to do some math. +So tune in, I hope to see you there, and be prepared to do some math. Thanks for watching! diff --git a/2020/lockdown-math-announcement/english/sentence_timings.json b/2020/lockdown-math-announcement/english/sentence_timings.json index 73a6bce8c..b1e42c40f 100644 --- a/2020/lockdown-math-announcement/english/sentence_timings.json +++ b/2020/lockdown-math-announcement/english/sentence_timings.json @@ -45,7 +45,7 @@ 108.1 ], [ - "So tune in, I hope to see you there, and be prepared to do some math.", + "So tune in, I hope to see you there, and be prepared to do some math. Thanks for watching!", 108.44, 111.72 ] diff --git a/2020/lockdown-math-announcement/english/transcript.txt b/2020/lockdown-math-announcement/english/transcript.txt index bba5bbd88..6bdcc0cae 100644 --- a/2020/lockdown-math-announcement/english/transcript.txt +++ b/2020/lockdown-math-announcement/english/transcript.txt @@ -7,4 +7,4 @@ I don't want to say anything more, I would just say, show up, be prepared to ans My goal is for it to feel as much like a real class as possible. Most of the dynamic is just going to be you and me talking through problems on a piece of paper, which, even though I love to visualize stuff and put out animations and that's kind of what the whole channel is about, to be honest, I think just working through things on paper feels more like what actual math is to me, and what the process of finding new ideas and coming to terms with them yourself looks like. The tentative plan right now is to do every Friday and Tuesday at noon Pacific time, but if anything changes on that, you'll see the schedule on the banner of the channel. -So tune in, I hope to see you there, and be prepared to do some math. \ No newline at end of file +So tune in, I hope to see you there, and be prepared to do some math. Thanks for watching! \ No newline at end of file diff --git a/2020/lockdown-math-announcement/french/sentence_translations.json b/2020/lockdown-math-announcement/french/sentence_translations.json index 2b1cfcc7a..34c59afbb 100644 --- a/2020/lockdown-math-announcement/french/sentence_translations.json +++ b/2020/lockdown-math-announcement/french/sentence_translations.json @@ -81,7 +81,7 @@ "end": 108.1 }, { - "input": "So tune in, I hope to see you there, and be prepared to do some math.", + "input": "So tune in, I hope to see you there, and be prepared to do some math. Thanks for watching!", "translatedText": "Alors, sois à l'écoute, j'espère te voir là-bas, et prépare-toi à faire quelques calculs.", "model": "DeepL", "from_community_srt": "mais si il y a des changements vous les verrez sur la bannière de la chaîne Donc rendez-vous sur cette chaîne, j'espère vous voir nombreux et soyez prêts à faire des maths !", diff --git a/2020/lockdown-math-announcement/german/sentence_translations.json b/2020/lockdown-math-announcement/german/sentence_translations.json index 84a2b449b..3dda65894 100644 --- a/2020/lockdown-math-announcement/german/sentence_translations.json +++ b/2020/lockdown-math-announcement/german/sentence_translations.json @@ -81,7 +81,7 @@ "end": 108.1 }, { - "input": "So tune in, I hope to see you there, and be prepared to do some math.", + "input": "So tune in, I hope to see you there, and be prepared to do some math. Thanks for watching!", "translatedText": "Also schalte ein, ich hoffe, wir sehen uns dort, und sei bereit, ein bisschen zu rechnen.", "model": "DeepL", "from_community_srt": "Also schalte ein, ich hoffe dich dort zu sehen und sei bereit ein bisschen Mathe zu machen.", diff --git a/2020/lockdown-math-announcement/hindi/sentence_translations.json b/2020/lockdown-math-announcement/hindi/sentence_translations.json index f21f78957..a201de0b3 100644 --- a/2020/lockdown-math-announcement/hindi/sentence_translations.json +++ b/2020/lockdown-math-announcement/hindi/sentence_translations.json @@ -81,7 +81,7 @@ "end": 108.1 }, { - "input": "So tune in, I hope to see you there, and be prepared to do some math.", + "input": "So tune in, I hope to see you there, and be prepared to do some math. Thanks for watching!", "translatedText": "तो बने रहिए, मुझे आशा है कि मैं आपसे वहां मिलूंगा और कुछ गणित करने के लिए तैयार रहूँगा।", "model": "google_nmt", "from_community_srt": "तो यह ध्यान रखिए मैं आशा करता हूं कि आप वहां(Stream में) होंगे और कुछ गणित करने के लिए तैयार रहें", diff --git a/2020/lockdown-math-announcement/italian/sentence_translations.json b/2020/lockdown-math-announcement/italian/sentence_translations.json index 5a9e78f46..4079bb88e 100644 --- a/2020/lockdown-math-announcement/italian/sentence_translations.json +++ b/2020/lockdown-math-announcement/italian/sentence_translations.json @@ -81,7 +81,7 @@ "end": 108.1 }, { - "input": "So tune in, I hope to see you there, and be prepared to do some math.", + "input": "So tune in, I hope to see you there, and be prepared to do some math. Thanks for watching!", "translatedText": "Quindi sintonizzati, spero di vederti lì e preparati a fare un po' di conti.", "model": "DeepL", "from_community_srt": "spero di vedervi e preparatevi a fare della matematica", diff --git a/2020/lockdown-math-announcement/portuguese/sentence_translations.json b/2020/lockdown-math-announcement/portuguese/sentence_translations.json index 9445ec913..9b0d469dd 100644 --- a/2020/lockdown-math-announcement/portuguese/sentence_translations.json +++ b/2020/lockdown-math-announcement/portuguese/sentence_translations.json @@ -81,7 +81,7 @@ "end": 108.1 }, { - "input": "So tune in, I hope to see you there, and be prepared to do some math.", + "input": "So tune in, I hope to see you there, and be prepared to do some math. Thanks for watching!", "translatedText": "Então sintonize, espero ver você lá e esteja preparado para fazer algumas contas.", "model": "google_nmt", "from_community_srt": "compareçam e estejam preparados para estudar Matemáticas.", diff --git a/2020/lockdown-math-announcement/spanish/sentence_translations.json b/2020/lockdown-math-announcement/spanish/sentence_translations.json index ef25960c3..2f2d44789 100644 --- a/2020/lockdown-math-announcement/spanish/sentence_translations.json +++ b/2020/lockdown-math-announcement/spanish/sentence_translations.json @@ -81,7 +81,7 @@ "end": 108.1 }, { - "input": "So tune in, I hope to see you there, and be prepared to do some math.", + "input": "So tune in, I hope to see you there, and be prepared to do some math. Thanks for watching!", "translatedText": "Así que sintoniza, espero verte allí, y prepárate para hacer cuentas.", "model": "DeepL", "from_community_srt": "Así que, sintonízate.", diff --git a/2020/pdfs/arabic/sentence_translations.json b/2020/pdfs/arabic/sentence_translations.json index 76e6b04fb..e90bd1594 100644 --- a/2020/pdfs/arabic/sentence_translations.json +++ b/2020/pdfs/arabic/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0. ", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero. ", "translatedText": "إذا جعلنا ارتفاعات الأشرطة تمثل الاحتمالات، لكان كل شيء قد ذهب إلى 0. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/bengali/sentence_translations.json b/2020/pdfs/bengali/sentence_translations.json index 5fee1295e..dadab94cf 100644 --- a/2020/pdfs/bengali/sentence_translations.json +++ b/2020/pdfs/bengali/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0. ", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero. ", "translatedText": "যদি আমরা বারের উচ্চতাগুলিকে সম্ভাব্যতা উপস্থাপন করতে দিতাম, তাহলে সবকিছুই 0-এ চলে যেত।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/chinese/sentence_translations.json b/2020/pdfs/chinese/sentence_translations.json index 6aa380a56..3283b9f41 100644 --- a/2020/pdfs/chinese/sentence_translations.json +++ b/2020/pdfs/chinese/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0. ", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero. ", "translatedText": "如果我们让条形的高度代表概 率，那么一切都会变成 0。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/french/sentence_translations.json b/2020/pdfs/french/sentence_translations.json index bdc15c732..243ae7d35 100644 --- a/2020/pdfs/french/sentence_translations.json +++ b/2020/pdfs/french/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0. ", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero. ", "translatedText": "Si nous avions laissé les hauteurs des barres représenter des probabilités, tout serait tombé à 0. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/german/sentence_translations.json b/2020/pdfs/german/sentence_translations.json index 05db54358..254de0f82 100644 --- a/2020/pdfs/german/sentence_translations.json +++ b/2020/pdfs/german/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0.", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero.", "translatedText": "Hätten wir die Höhen der Balken als Wahrscheinlichkeiten dargestellt, wäre alles auf 0 gegangen.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/hebrew/sentence_translations.json b/2020/pdfs/hebrew/sentence_translations.json index 47def3791..c73a92143 100644 --- a/2020/pdfs/hebrew/sentence_translations.json +++ b/2020/pdfs/hebrew/sentence_translations.json @@ -196,7 +196,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0.", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero.", "translatedText": "אם היינו נותנים לגבהים של הפסים לייצג הסתברויות, הכל היה עובר ל-0.", "n_reviews": 0, "start": 248.7, diff --git a/2020/pdfs/hindi/sentence_translations.json b/2020/pdfs/hindi/sentence_translations.json index 0ea396f79..fa3efd33c 100644 --- a/2020/pdfs/hindi/sentence_translations.json +++ b/2020/pdfs/hindi/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0. ", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero. ", "translatedText": "यदि हमने सलाखों की ऊंचाई को संभावनाओं का प्रतिनिधित्व करने दिया होता, तो सब कुछ 0 पर चला गया होता।", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/indonesian/sentence_translations.json b/2020/pdfs/indonesian/sentence_translations.json index 6ed049e0c..339991c1a 100644 --- a/2020/pdfs/indonesian/sentence_translations.json +++ b/2020/pdfs/indonesian/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0.", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero.", "translatedText": "Jika kita membiarkan ketinggian batang mewakili probabilitas, semuanya akan menjadi 0.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/italian/sentence_translations.json b/2020/pdfs/italian/sentence_translations.json index 43ca28006..0f68d772f 100644 --- a/2020/pdfs/italian/sentence_translations.json +++ b/2020/pdfs/italian/sentence_translations.json @@ -196,7 +196,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0.", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero.", "translatedText": "Se avessimo lasciato che le altezze delle barre rappresentassero le probabilità, tutto sarebbe andato a 0.", "n_reviews": 0, "start": 248.7, diff --git a/2020/pdfs/japanese/sentence_translations.json b/2020/pdfs/japanese/sentence_translations.json index 5c6abd068..02fcd5a20 100644 --- a/2020/pdfs/japanese/sentence_translations.json +++ b/2020/pdfs/japanese/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0. ", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero. ", "translatedText": "バーの高さで確率を表していたら、すべ てが 0 になってしまうでしょう。", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/korean/sentence_translations.json b/2020/pdfs/korean/sentence_translations.json index a8787aac0..e0786fb7f 100644 --- a/2020/pdfs/korean/sentence_translations.json +++ b/2020/pdfs/korean/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0. ", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero. ", "translatedText": "막대의 높이가 확률을 나타내도록 했다면 모든 것이 0이 되었을 것입니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/marathi/sentence_translations.json b/2020/pdfs/marathi/sentence_translations.json index 1283e8d25..eeaf18e2f 100644 --- a/2020/pdfs/marathi/sentence_translations.json +++ b/2020/pdfs/marathi/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0.", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero.", "translatedText": "जर आपण पट्ट्यांची उंची संभाव्यता दर्शवू दिली असती, तर सर्वकाही 0 वर गेले असते.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/persian/sentence_translations.json b/2020/pdfs/persian/sentence_translations.json index 631da0fef..d06ce4d4e 100644 --- a/2020/pdfs/persian/sentence_translations.json +++ b/2020/pdfs/persian/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0. ", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero. ", "translatedText": "اگر اجازه می دادیم ارتفاع میله ها نشان دهنده احتمالات باشد، همه چیز به 0 می رسید. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/portuguese/sentence_translations.json b/2020/pdfs/portuguese/sentence_translations.json index 1cfffb9d7..81d10fb50 100644 --- a/2020/pdfs/portuguese/sentence_translations.json +++ b/2020/pdfs/portuguese/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0.", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero.", "translatedText": "Se tivéssemos deixado que as alturas das barras representassem probabilidades, tudo teria ido para 0.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/russian/sentence_translations.json b/2020/pdfs/russian/sentence_translations.json index 87f938def..8bccf5a94 100644 --- a/2020/pdfs/russian/sentence_translations.json +++ b/2020/pdfs/russian/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0.", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero.", "translatedText": "Если бы мы позволили высотам столбцов обозначать вероятности, все превратилось бы в 0.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/tamil/sentence_translations.json b/2020/pdfs/tamil/sentence_translations.json index 72aa04504..c6186ec59 100644 --- a/2020/pdfs/tamil/sentence_translations.json +++ b/2020/pdfs/tamil/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0.", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero.", "translatedText": "பார்களின் உயரங்களை நிகழ்தகவுகளைக் குறிக்க அனுமதித்திருந்தால், அனைத்தும் 0 க்கு சென்றிருக்கும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/telugu/sentence_translations.json b/2020/pdfs/telugu/sentence_translations.json index 0e6a9bd21..5ffad406e 100644 --- a/2020/pdfs/telugu/sentence_translations.json +++ b/2020/pdfs/telugu/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0.", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero.", "translatedText": "మేము బార్‌ల ఎత్తులను సంభావ్యతలను సూచించడానికి అనుమతించినట్లయితే, ప్రతిదీ 0కి వెళ్లి ఉండేది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/thai/sentence_translations.json b/2020/pdfs/thai/sentence_translations.json index cd112ef7a..6c76de106 100644 --- a/2020/pdfs/thai/sentence_translations.json +++ b/2020/pdfs/thai/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0. ", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero. ", "translatedText": "นั่นสำคัญเพราะมันหมายความว่าเมื่อคุณใช้กระบวนการนี้ถึงขีดจำกัด คุณจะเข้าใกล้เส้นโค้งที่ราบรื่น ดังนั้น แม้ว่าความน่าจะเป็นส่วนบุคคลที่จะตกลงไปในกลุ่มใดกลุ่มหนึ่งจะเข้าใกล้ 0 แต่รูปร่างโดยรวมของการแจกแจงจะยังคงอยู่ และปรับปรุงให้อยู่ในขีดจำกัดนี้ด้วยซ้ำ ถ้าเราปล่อยให้ความสูงของแท่งแสดงถึงความน่าจะเป็น ทุกอย่างจะเป็น 0 ดังนั้นในขีดจำกัด เราจะมีเส้นแบนไม่ให้ข้อมูลเกี่ยวกับรูปร่างโดยรวมของการกระจายตัว เยี่ยมมาก การปล่อยให้พื้นที่แสดงถึงความน่าจะเป็นช่วยแก้ปัญหานี้ได้ แต่ขอถามคุณว่า ถ้าแกน y ไม่ได้แสดงถึงความน่าจะเป็นแล้ว หน่วยตรงนี้เป็นเท่าไหร่? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/turkish/sentence_translations.json b/2020/pdfs/turkish/sentence_translations.json index 2147de3f7..dc44bfe63 100644 --- a/2020/pdfs/turkish/sentence_translations.json +++ b/2020/pdfs/turkish/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0.", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero.", "translatedText": "Çubukların yüksekliklerinin olasılıkları temsil etmesine izin verseydik her şey 0'a giderdi.", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/ukrainian/sentence_translations.json b/2020/pdfs/ukrainian/sentence_translations.json index 0762fa8f7..b1883e335 100644 --- a/2020/pdfs/ukrainian/sentence_translations.json +++ b/2020/pdfs/ukrainian/sentence_translations.json @@ -196,7 +196,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0.", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero.", "translatedText": "Якби ми дозволили висоті стовпчиків представляти ймовірності, все було б рівним 0.", "n_reviews": 0, "start": 248.7, diff --git a/2020/pdfs/urdu/sentence_translations.json b/2020/pdfs/urdu/sentence_translations.json index e5265f1eb..1a0e5c0da 100644 --- a/2020/pdfs/urdu/sentence_translations.json +++ b/2020/pdfs/urdu/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0. ", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero. ", "translatedText": "اگر ہم سلاخوں کی بلندیوں کو امکانات کی نمائندگی کرنے دیتے، تو سب کچھ 0 پر چلا جاتا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2020/pdfs/vietnamese/sentence_translations.json b/2020/pdfs/vietnamese/sentence_translations.json index bf779b4b9..9e9b6b930 100644 --- a/2020/pdfs/vietnamese/sentence_translations.json +++ b/2020/pdfs/vietnamese/sentence_translations.json @@ -224,7 +224,7 @@ "end": 247.22 }, { - "input": "If we had let the heights of the bars represent probabilities, everything would have gone to 0. ", + "input": "If, on the other hand, we had let the heights of the bars represent probabilities, everything would have gone to zero. ", "translatedText": "Nếu chúng ta để chiều cao của các thanh biểu thị xác suất thì mọi thứ sẽ về 0. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/holomorphic-dynamics/english/captions.srt b/2021/holomorphic-dynamics/english/captions.srt index 40b9f06ab..c4d4e7db1 100644 --- a/2021/holomorphic-dynamics/english/captions.srt +++ b/2021/holomorphic-dynamics/english/captions.srt @@ -283,1458 +283,1482 @@ If you iterate a second time, plugging in that value to the function, you get c squared plus c. 72 -00:04:33,960 --> 00:04:40,679 -And as I change around the value c here, you can kind of see how the third +00:04:33,960 --> 00:04:38,532 +And as I change around the value c here, you can kind of see how the second value 73 -00:04:40,679 --> 00:04:47,400 -value to get z4 and continue on like this, visualizing our chain of values. +00:04:38,532 --> 00:04:42,325 +moves in lockstep. Then we can plug in that second value to get z3, 74 +00:04:42,325 --> 00:04:45,615 +and that third value to get z4, and continue on like this, + +75 +00:04:45,615 --> 00:04:47,400 +visualizing our chain of values. + +76 00:04:49,920 --> 00:04:53,914 So if I keep doing this many different times for the first many values, -75 +77 00:04:53,914 --> 00:04:56,800 for some choices of c, this process remains bounded. -76 +78 00:04:57,060 --> 00:04:58,560 You can still see it all on the screen. -77 +79 00:04:59,060 --> 00:05:01,836 And other times, it looks like it blows up, and you can actually -78 +80 00:05:01,836 --> 00:05:04,400 show that if it gets as big as 2, it'll blow up to infinity. -79 +81 00:05:07,820 --> 00:05:11,693 If you color the points of the plane where it stays bounded black, -80 +82 00:05:11,693 --> 00:05:16,897 and you assign some other gradient of colors to the divergent values based on how quickly -81 +83 00:05:16,897 --> 00:05:22,101 the process rushes off to infinity, you get one of the most iconic images in all of math, -82 +84 00:05:22,101 --> 00:05:23,200 the Mandelbrot set. -83 +85 00:05:24,560 --> 00:05:27,695 Now this interactive dots and stick visualization of the trajectory, -84 +86 00:05:27,695 --> 00:05:30,786 by the way, is heavily inspired by Ben Sparks' illustration and the -85 +87 00:05:30,786 --> 00:05:34,740 Numberphile video he did about the Mandelbrot set, which is great, you should watch it. -86 -00:05:34,900 --> 00:05:39,734 -I honestly thought it was just too fun not to read about all of this +88 +00:05:34,900 --> 00:05:37,804 +I honestly thought it was just too fun not to re-implement here. -87 -00:05:39,734 --> 00:05:44,640 -stuff for any of you who haven't had the pleasure of reading that yet. +89 +00:05:37,804 --> 00:05:40,976 +I would also highly recommend the interactive article on ako.net about -88 +90 +00:05:40,976 --> 00:05:44,640 +all of this stuff for any of you who haven't had the pleasure of reading that yet. + +91 00:05:45,240 --> 00:05:48,292 What's nice about the Ben Sparks illustration is how it illuminates -89 +92 00:05:48,292 --> 00:05:51,300 what each different part of the Mandelbrot set actually represents. -90 +93 00:05:52,060 --> 00:05:55,100 This largest cardioid section includes the values of c -91 +94 00:05:55,100 --> 00:05:58,140 so that the process eventually converges to some limit. -92 +95 00:05:58,940 --> 00:06:01,700 The big circle on the left represents the values where -93 +96 00:06:01,700 --> 00:06:04,460 the process gets trapped in a cycle between two values. -94 +97 00:06:05,280 --> 00:06:08,450 And then the top and bottom circles show values where the process -95 +98 00:06:08,450 --> 00:06:11,380 gets trapped in a cycle of three values, and so on like this. -96 +99 00:06:11,480 --> 00:06:13,720 Each one of these little islands kind of has its own meaning. -97 +100 00:06:16,400 --> 00:06:19,240 Also notice there's an important difference between how this -98 +101 00:06:19,240 --> 00:06:23,291 Mandelbrot set and the Newton fractals we were looking at before are each constructed, -99 +102 00:06:23,291 --> 00:06:25,340 beyond just a different underlying function. -100 +103 00:06:26,100 --> 00:06:30,026 For the Mandelbrot set we have a consistent seed value, z equals zero, -101 +104 00:06:30,026 --> 00:06:34,340 but the thing we're tweaking is the parameter c, changing the function itself. -102 +105 00:06:34,820 --> 00:06:37,500 -So what you're looking at is a parameter space. +So what you're looking at is what we might call a parameter space. -103 +106 00:06:38,160 --> 00:06:41,683 But with Newton's fractal, we have a single unchanging function, -104 +107 00:06:41,683 --> 00:06:46,020 but what we associate with each pixel is a different seed value for the process. -105 +108 00:06:47,160 --> 00:06:50,560 Of course, we could play the same game with the map z squared plus c. -106 +109 00:06:51,060 --> 00:06:53,905 We could fix c at some constant, and then let the pixels -107 +110 00:06:53,905 --> 00:06:56,800 represent the different possible initial values, z naught. -108 +111 00:06:57,680 --> 00:07:01,439 So whereas each pixel of the Mandelbrot set corresponds to a unique function, -109 +112 00:07:01,439 --> 00:07:04,620 the images on the right each just correspond to a single function. -110 +113 00:07:05,460 --> 00:07:09,220 As we change the parameter c, it changes the entire image on the right. -111 +114 00:07:10,020 --> 00:07:13,937 And again, just to be clear, the rule being applied is that we color pixels -112 +115 00:07:13,937 --> 00:07:17,958 black if the process remains bounded, and then apply some kind of gradient to -113 +116 00:07:17,958 --> 00:07:22,340 the ones that diverge away to infinity based on how quickly they diverge to infinity. -114 +117 00:07:23,380 --> 00:07:26,475 In principle, and it's kind of mind-warping to think about, -115 +118 00:07:26,475 --> 00:07:30,395 there is some four-dimensional space of all combinations of c and z naught, -116 +119 00:07:30,395 --> 00:07:34,883 and what we're doing here is kind of looking through individual two-dimensional slices -117 +120 00:07:34,883 --> 00:07:36,380 of that unimaginable pattern. -118 +121 00:07:37,740 --> 00:07:41,276 You'll often hear or read the images on the right being referred to as -119 +122 00:07:41,276 --> 00:07:45,111 Julia sets or Julia fractals, and when I first learned about all this stuff, -120 +123 00:07:45,111 --> 00:07:48,647 I'll admit that I kind of was left with the misconception that this is -121 +124 00:07:48,647 --> 00:07:52,382 what the term Julia set refers to, specifically the z squared plus c case, -122 +125 00:07:52,382 --> 00:07:55,720 and moreover that it's referring to the black region on the inside. -123 +126 00:07:56,440 --> 00:07:59,399 However, the term Julia set has a much more general definition, -124 +127 00:07:59,399 --> 00:08:02,960 and it would refer just to the boundaries of these regions, not the interior. -125 +128 00:08:03,900 --> 00:08:06,121 To set the stage for a more specific definition, -126 +129 00:08:06,121 --> 00:08:09,930 and to also make some headway towards the first goal that I mentioned at the start, -127 +130 00:08:09,930 --> 00:08:14,011 it's worth stepping back and really just picturing yourself as a mathematician right now, -128 +131 00:08:14,011 --> 00:08:15,100 discovering all of this. -129 +132 00:08:15,740 --> 00:08:18,880 What would you actually do to construct a theory around this? -130 -00:08:19,260 --> 00:08:21,680 +133 +00:08:19,260 --> 00:08:21,369 It's one thing to look at some pretty pictures, -131 -00:08:21,680 --> 00:08:25,060 -but what sorts of questions would you ask if you understand it all? +134 +00:08:21,369 --> 00:08:25,060 +but what sorts of questions would you ask if you actually want to understand it all? -132 +135 00:08:26,020 --> 00:08:28,800 In general, if you want to understand something complicated, -133 +136 00:08:28,800 --> 00:08:31,946 a good place to start is to ask if there are any parts of the system -134 +137 00:08:31,946 --> 00:08:35,320 that have some simple behavior, preferably the simplest possible behavior. -135 +138 00:08:36,179 --> 00:08:41,400 In our example, that might mean asking when does the process just stay fixed in place, -136 +139 00:08:41,400 --> 00:08:43,140 meaning f of z is equal to z. -137 +140 00:08:43,740 --> 00:08:45,880 That's a pretty boring set of dynamics, I think you'd agree. -138 +141 00:08:46,500 --> 00:08:49,680 We call a value with this property a fixed point of the function. -139 +142 00:08:49,680 --> 00:08:53,033 In the case of the functions arising from Newton's method, -140 +143 00:08:53,033 --> 00:08:57,240 by design they have a fixed point at the roots of the relevant polynomial. -141 +144 00:08:57,840 --> 00:09:01,140 You can verify for yourself, if p of z is equal to zero, -142 +145 00:09:01,140 --> 00:09:03,920 then the entire expression is simply equal to z. -143 +146 00:09:04,300 --> 00:09:05,660 That's what it means to be a fixed point. -144 -00:09:06,560 --> 00:09:13,264 +147 +00:09:06,560 --> 00:09:11,300 If you're into exercises, you may enjoy pausing for a moment and computing the -145 -00:09:13,264 --> 00:09:19,800 -fixed points of this Mandelbrot set since asking when this expression equals +148 +00:09:11,300 --> 00:09:16,099 +fixed points of this Mandelbrot set function, z squared plus c. More generally, -146 -00:09:19,800 --> 00:09:26,420 -z can always be rearranged as finding the roots of some polynomial expression. +149 +00:09:16,099 --> 00:09:19,280 +any rational function will always have fixed points, -147 +150 +00:09:19,280 --> 00:09:24,260 +since asking when this expression equals z can always be rearranged as finding the + +151 +00:09:24,260 --> 00:09:26,420 +roots of some polynomial expression, + +152 00:09:27,140 --> 00:09:30,430 From the fundamental theorem of algebra, this must have solutions, -148 +153 00:09:30,430 --> 00:09:33,820 typically as many solutions as the highest degree in this expression. -149 +154 00:09:34,960 --> 00:09:39,438 Incidentally, this means you could also find those fixed points using Newton's method, -150 +155 00:09:39,438 --> 00:09:41,600 but maybe that's a little too meta for us. -151 -00:09:42,640 --> 00:09:47,152 -Just asking about fixed points is easy, but a key idea for understanding the full +156 +00:09:42,640 --> 00:09:45,666 +ight now. Now just asking about fixed points is maybe easy, -152 -00:09:47,152 --> 00:09:51,720 -dynamics, and hence the diagrams that we're looking at, is to understand stability. +157 +00:09:45,666 --> 00:09:48,289 +but a key idea for understanding the full dynamics, -153 +158 +00:09:48,289 --> 00:09:51,720 +and hence the diagrams we're looking at, is to understand stability. + +159 00:09:52,880 --> 00:09:57,698 We say that a fixed point is attracting if nearby points tend to get drawn in towards it, -154 +160 00:09:57,698 --> 00:09:59,680 and repelling if they're pushed away. -155 +161 00:10:00,380 --> 00:10:02,803 And this is something that you can actually compute -156 +162 00:10:02,803 --> 00:10:05,040 explicitly using the derivative of the function. -157 +163 00:10:06,040 --> 00:10:09,099 Symbolically, when you take derivatives of complex functions, -158 +164 00:10:09,099 --> 00:10:11,960 it looks exactly the same as it would for real functions, -159 +165 00:10:11,960 --> 00:10:15,020 though something like z squared has a derivative of 2 times z. -160 +166 00:10:15,900 --> 00:10:19,020 But geometrically, there's a really lovely way to interpret what this means. -161 +167 00:10:19,740 --> 00:10:24,505 For example, at the input 1, the derivative of this particular function evaluates to be -162 +168 00:10:24,505 --> 00:10:29,162 2, and what that's telling us is that if you look at a very small neighborhood around -163 +169 00:10:29,162 --> 00:10:33,981 that input, and you follow what happens to all the points in that little neighborhood as -164 +170 00:10:33,981 --> 00:10:36,580 you apply the function, in this case z squared, -165 +171 00:10:36,580 --> 00:10:39,180 then it looks just like you're multiplying by 2. -166 +172 00:10:39,580 --> 00:10:41,400 This is what a derivative of 2 means. -167 +173 00:10:43,240 --> 00:10:45,900 To take another example, let's look at the input i. -168 +174 00:10:46,500 --> 00:10:50,860 We know that this function moves that input to the value negative 1, that's i squared. -169 +175 00:10:51,580 --> 00:10:56,323 But the added information that its derivative at this value is 2 times i gives us the -170 +176 00:10:56,323 --> 00:10:59,356 added picture that when you zoom in around that point, -171 +177 00:10:59,356 --> 00:11:03,217 and you look at the action of the function on this tiny neighborhood, -172 +178 00:11:03,217 --> 00:11:07,850 it looks like multiplication by 2i, which in this case is saying it looks like a 90 -173 +179 00:11:07,850 --> 00:11:11,160 degree rotation combined with an expansion by a factor of 2. -174 +180 00:11:14,860 --> 00:11:17,594 For the purposes of analyzing stability, the only thing -175 +181 00:11:17,594 --> 00:11:20,280 we care about here is the growing and shrinking factor. -176 +182 00:11:20,640 --> 00:11:22,040 The rotational part doesn't matter. -177 +183 00:11:22,520 --> 00:11:26,049 So if you compute the derivative of a function at its fixed point, -178 +184 00:11:26,049 --> 00:11:28,894 and the absolute value of this result is less than 1, -179 +185 00:11:28,894 --> 00:11:31,476 it tells you that the fixed point is attracting, -180 +186 00:11:31,476 --> 00:11:33,900 that nearby points tend to come in towards it. -181 +187 00:11:34,360 --> 00:11:36,985 If that derivative has an absolute value bigger than 1, -182 +188 00:11:36,985 --> 00:11:40,360 it tells you the fixed point is repelling, it pushes away its neighbors. -183 +189 00:11:41,640 --> 00:11:45,474 For example, if you work out the derivative of our Newton's map expression, -184 +190 00:11:45,474 --> 00:11:49,360 and you simplify a couple things a little bit, here's what you would get out. -185 +191 00:11:50,380 --> 00:11:55,020 So if z is a fixed point, which in this context means that it's one of the roots -186 +192 00:11:55,020 --> 00:11:59,660 of the polynomial p, this derivative is not only smaller than 1, it's equal to 0. -187 +193 00:12:00,840 --> 00:12:03,470 These are sometimes called super-attracting fixed points, -188 +194 00:12:03,470 --> 00:12:07,008 since it means that a neighborhood around these points doesn't merely shrink, -189 +195 00:12:07,008 --> 00:12:07,780 it shrinks a lot. -190 +196 00:12:08,660 --> 00:12:12,868 And again, this is kind of by design, since the intent of Newton's method -191 +197 00:12:12,868 --> 00:12:17,020 is to produce iterations that fall towards a root as quickly as they can. -192 +198 00:12:18,020 --> 00:12:21,147 Pulling up our z squared plus c example, if you did the first -193 +199 00:12:21,147 --> 00:12:24,476 exercise to find its fixed points, the next step would be to ask, -194 +200 00:12:24,476 --> 00:12:27,200 when is at least one of those fixed points attracting? -195 +201 00:12:27,820 --> 00:12:30,100 For what values of c is this going to be true? -196 +202 00:12:31,040 --> 00:12:33,519 And then, if that's not enough of a challenge, -197 +203 00:12:33,519 --> 00:12:38,108 try using the result that you find to show that this condition corresponds to the main -198 +204 00:12:38,108 --> 00:12:40,060 cardioid shape of the Mandelbrot set. -199 +205 00:12:40,680 --> 00:12:43,400 This is something you can compute explicitly, it's pretty cool. -200 +206 00:12:45,320 --> 00:12:47,469 A natural next step would be to ask about cycles, -201 +207 00:12:47,469 --> 00:12:49,920 and this is where things really start to get interesting. -202 +208 00:12:50,720 --> 00:12:55,691 If f of z is not z but some other value, and then that value comes back to z, -203 +209 00:12:55,691 --> 00:12:58,560 it means that you've fallen into a two cycle. -204 +210 00:12:59,320 --> 00:13:02,842 You could explicitly find these kinds of two cycles by -205 +211 00:13:02,842 --> 00:13:06,300 evaluating f of f of z and then setting it equal to z. -206 +212 00:13:07,120 --> 00:13:12,700 For example, with the z squared plus c map, f of f of z expands out to look like this. -207 +213 00:13:13,340 --> 00:13:15,080 A little messy, but you know, it's not too terrible. -208 +214 00:13:15,560 --> 00:13:19,380 The main thing to highlight is that it boils down to solving some degree four equation. -209 +215 00:13:20,160 --> 00:13:24,282 You should note though that the fixed points will also be solutions to this equation, -210 +216 00:13:24,282 --> 00:13:27,446 so technically the two cycles are the solutions to this minus the -211 +217 00:13:27,446 --> 00:13:29,700 solutions to the original fixed point equation. -212 +218 00:13:31,080 --> 00:13:33,936 And likewise you can use the same idea to look for -213 +219 00:13:33,936 --> 00:13:36,960 n cycles by composing f with itself n different times. -214 +220 00:13:37,880 --> 00:13:41,922 The explicit expressions that you would get quickly become insanely messy, -215 +221 00:13:41,922 --> 00:13:46,288 but it's still elucidating to ask how many cycles would you expect based on this -216 +222 00:13:46,288 --> 00:13:47,420 hypothetical process. -217 +223 00:13:47,960 --> 00:13:52,538 If we stick with our simple z squared plus c example, as you compose it with itself, -218 +224 00:13:52,538 --> 00:13:56,847 you'd get a polynomial with degree four and then one with degree eight and then -219 +225 00:13:56,847 --> 00:14:01,480 degree sixteen and so on and so on, exponentially growing the order of the polynomial. -220 +226 00:14:02,360 --> 00:14:06,984 So in principle, if I asked you how many cycles are there with a period of one million, -221 +227 00:14:06,984 --> 00:14:10,821 you can know that it's equivalent to solving some just absolutely insane -222 +228 00:14:10,821 --> 00:14:14,080 polynomial expression with a degree of two to the one million. -223 +229 00:14:14,880 --> 00:14:19,870 So again, fundamental theorem of algebra, you would expect to find something on the order -224 +230 00:14:19,870 --> 00:14:24,640 of two to the one million points in the complex plane which cycle in exactly this way. -225 +231 00:14:25,700 --> 00:14:28,780 And more generally, for any rational map you'll always be able -226 +232 00:14:28,780 --> 00:14:31,860 to find values whose behavior falls into a cycle with period n. -227 +233 00:14:32,360 --> 00:14:36,480 It ultimately boils down to solving some probably insane polynomial expression. -228 +234 00:14:37,180 --> 00:14:39,791 And just like with this example, the number of -229 +235 00:14:39,791 --> 00:14:42,680 such periodic points will grow exponentially with n. -230 +236 00:14:43,980 --> 00:14:47,294 I didn't really talk about this in the last video about Newton's fractal, -231 +237 00:14:47,294 --> 00:14:50,518 but it's sort of strange to think that there are infinitely many points -232 +238 00:14:50,518 --> 00:14:53,340 that fall into some kind of cycle even for a process like this. -233 +239 00:14:54,020 --> 00:14:57,403 In almost all cases though, these points are somewhere on the boundary -234 +240 00:14:57,403 --> 00:15:00,406 between those colored regions and they don't really come up in -235 +241 00:15:00,406 --> 00:15:03,600 practice because the probability of landing on one of them is zero. -236 +242 00:15:04,240 --> 00:15:07,830 What matters for actually falling into one of these is if one of the -237 +243 00:15:07,830 --> 00:15:11,525 cycles is attracting in the sense that a neighborhood of points around -238 +244 00:15:11,525 --> 00:15:15,220 a value from that cycle would tend to get pulled in towards that cycle. -239 +245 00:15:16,460 --> 00:15:19,971 A highly relevant question for someone interested in numerical methods -240 +246 00:15:19,971 --> 00:15:23,631 is whether or not this Newton's map process ever has an attracting cycle, -241 +247 00:15:23,631 --> 00:15:26,846 because if there is it means there's a non-zero chance that your -242 +248 00:15:26,846 --> 00:15:30,160 initial guess gets trapped in that cycle and it never finds a root. -243 +249 00:15:31,160 --> 00:15:32,800 The answer here is actually yes. -244 +250 00:15:33,580 --> 00:15:37,908 More explicitly, if you try to find the roots of z cubed minus 2z plus -245 +251 00:15:37,908 --> 00:15:42,053 2 and you're using Newton's method, watch what happens to a cluster -246 +252 00:15:42,053 --> 00:15:46,260 that starts around the value zero and sort of bounces back and forth. -247 +253 00:15:47,260 --> 00:15:49,914 And well okay, in this case the cluster we started with was a -248 +254 00:15:49,914 --> 00:15:52,697 little bit too big so some of the outer points get sprayed away, -249 +255 00:15:52,697 --> 00:15:55,480 but here's what it looks like if we start with a smaller cluster. -250 +256 00:15:56,120 --> 00:15:58,715 Notice how all of the points genuinely do shrink -251 +257 00:15:58,715 --> 00:16:00,940 in towards the cycle between zero and one. -252 +258 00:16:01,480 --> 00:16:05,040 It's not likely that you hit this with a random seed, but it definitely is possible. -253 +259 00:16:06,080 --> 00:16:10,532 The exercise that you could do to verify that a cycle like this is attracting, -254 +260 00:16:10,532 --> 00:16:14,082 by the way, would be to compute the derivative of f of f of z, -255 +261 00:16:14,082 --> 00:16:18,760 and you check that at the input zero this derivative has a magnitude less than one. -256 +262 00:16:19,760 --> 00:16:23,124 The thing that blew my mind a little is what happens when you try -257 +263 00:16:23,124 --> 00:16:26,540 to visualize which cubic polynomials have attracting cycles at all. -258 +264 00:16:27,080 --> 00:16:30,873 Hopefully if Newton's method is going to be at all decent at finding roots, -259 +265 00:16:30,873 --> 00:16:32,820 those attracting cycles should be rare. -260 +266 00:16:33,960 --> 00:16:37,349 First of all, to better visualize the one example we're looking at, -261 +267 00:16:37,349 --> 00:16:39,891 we could draw the same fractal that we had before, -262 +268 00:16:39,891 --> 00:16:43,728 coloring each point based on what root the seed value starting at that point -263 +269 00:16:43,728 --> 00:16:47,566 will tend to, but this time we'll have an added condition of coloring points -264 +270 00:16:47,566 --> 00:16:51,304 that says that if the seed value never gets close enough to a root at all, -265 +271 00:16:51,304 --> 00:16:52,800 we will color the pixel black. -266 +272 00:16:53,760 --> 00:16:58,520 Notice if I tweak the roots, meaning that we're trying out different cubic polynomials, -267 +273 00:16:58,520 --> 00:17:01,712 it's actually really hard to find any place to put them so -268 +274 00:17:01,712 --> 00:17:03,660 that we see any black pixels at all. -269 +275 00:17:04,319 --> 00:17:07,660 I can find this one little sweet spot here, but it's definitely rare. -270 +276 00:17:08,680 --> 00:17:13,132 Now what I want is some kind of way to visualize every possible cubic polynomial -271 +277 00:17:13,132 --> 00:17:17,640 at once with a single image in a way that shows which ones have attracting cycles. -272 +278 00:17:18,880 --> 00:17:21,618 Luckily it turns out that there is a really simple way to test -273 +279 00:17:21,618 --> 00:17:24,400 whether or not one of these polynomials has an attracting cycle. -274 +280 00:17:25,060 --> 00:17:30,031 All you have to do is look at the seed value which sits at average of the three roots, -275 +281 00:17:30,031 --> 00:17:31,460 this center of mass here. -276 +282 00:17:32,100 --> 00:17:35,741 Turns out, this is not at all obvious, if there's an attracting cycle, -277 +283 00:17:35,741 --> 00:17:39,640 you can guarantee that this seed value will fall into that attracting cycle. -278 +284 00:17:40,500 --> 00:17:44,420 In other words, if there are any black points, this will be one of them. -279 +285 00:17:45,340 --> 00:17:48,102 If you want to know where this magical fact comes from, -280 +286 00:17:48,102 --> 00:17:50,520 it stems from a theorem of our good friend Fatou. -281 +287 00:17:50,920 --> 00:17:54,610 He showed that if one of these rational maps has an attracting cycle, -282 +288 00:17:54,610 --> 00:17:59,196 you can look at the values where the derivative of your iterated function equals zero, -283 +289 00:17:59,196 --> 00:18:02,360 and at least one of those values has to fall into the cycle. -284 +290 00:18:03,440 --> 00:18:05,933 That might seem like a little bit of a weird fact, -285 +291 00:18:05,933 --> 00:18:09,356 but the loose intuition is that if a cycle is going to be attracting, -286 +292 00:18:09,356 --> 00:18:12,486 at least one of its values should have a very small derivative, -287 +293 00:18:12,486 --> 00:18:14,540 that's where the shrinking will come from. -288 +294 00:18:15,100 --> 00:18:18,443 And this in turn means that that value in the cycle sits near some -289 +295 00:18:18,443 --> 00:18:21,736 point where the derivative is not merely small but equal to zero, -290 +296 00:18:21,736 --> 00:18:25,280 and that point ends up being close enough to get sucked into the cycle. -291 +297 00:18:26,560 --> 00:18:29,256 This fact also justifies why with the Mandelbrot set, -292 +298 00:18:29,256 --> 00:18:31,903 where we're only using one seed value z equals zero, -293 +299 00:18:31,903 --> 00:18:35,100 it's still enough to get us a very full and interesting picture. -294 +300 00:18:35,320 --> 00:18:40,600 If there's a stable cycle to be found, that one seed value is definitely going to find it. -295 +301 00:18:41,500 --> 00:18:45,730 I feel like maybe I'm assigning a little too much homework and exercises today, -296 +302 00:18:45,730 --> 00:18:49,696 but if you're into that, yet another pleasing one would be to look back at -297 +303 00:18:49,696 --> 00:18:54,349 derivative expression that we found with our function that arises from Newton's method, -298 +304 00:18:54,349 --> 00:18:58,579 and use this wonderful theorem of Vateau's to show our magical fact about cubic -299 +305 00:18:58,579 --> 00:19:02,440 polynomials, that it suffices to just check this midpoint over the roots. -300 +306 00:19:03,240 --> 00:19:06,540 Honestly though, all of those are details that you don't really have to worry about. -301 +307 00:19:06,840 --> 00:19:11,026 The upshot is that we can perform a test for whether or not one of these polynomials -302 +308 00:19:11,026 --> 00:19:14,720 has an attracting cycle by looking at just a single point, not all of them. -303 +309 00:19:15,480 --> 00:19:18,600 And because of this, we can actually generate a really cool diagram. -304 +310 00:19:19,380 --> 00:19:22,005 The way this will work is to fix two roots in place, -305 +311 00:19:22,005 --> 00:19:25,720 let's say putting them at z equals negative one and z equals positive one, -306 +312 00:19:25,720 --> 00:19:29,040 and then we'll move around that third root, which I'll call lambda. -307 +313 00:19:30,480 --> 00:19:32,820 Remember, the key feature that we're looking for -308 +314 00:19:32,820 --> 00:19:35,160 is when the point at the center of mass is black. -309 +315 00:19:35,860 --> 00:19:39,188 So what I'll do is draw a second diagram on the right, -310 +316 00:19:39,188 --> 00:19:42,940 where each pixel corresponds to one possible choice of lambda. -311 +317 00:19:43,860 --> 00:19:46,230 What we're going to do is color that pixel based -312 +318 00:19:46,230 --> 00:19:48,600 on the color of this midpoint of the three roots. -313 +319 00:19:49,600 --> 00:19:52,160 If this feels a little bit confusing, that's totally okay. -314 +320 00:19:52,320 --> 00:19:54,440 There are kind of a lot of layers at play here. -315 +321 00:19:55,020 --> 00:19:59,138 Just remember, each pixel on the right corresponds to a unique polynomial, -316 +322 00:19:59,138 --> 00:20:01,280 as determined by this parameter lambda. -317 +323 00:20:02,000 --> 00:20:04,740 In fact, you might call this a parameter space. -318 +324 00:20:05,080 --> 00:20:05,580 Sound familiar? -319 +325 00:20:13,740 --> 00:20:17,801 Points in this parameter space are colored black if, and only if, -320 +326 00:20:17,801 --> 00:20:23,340 the Newton's method process for the corresponding polynomial produces an attracting cycle. -321 +327 00:20:24,260 --> 00:20:26,400 Again, don't worry if that takes a little moment to digest. -322 +328 00:20:27,840 --> 00:20:30,277 Now, at first glance, it might not look like there -323 +329 00:20:30,277 --> 00:20:32,380 are any black points at all on this diagram. -324 +330 00:20:33,120 --> 00:20:33,780 And this is good news. -325 +331 00:20:33,940 --> 00:20:38,300 It means that in most cases Newton's method will not get sucked into cycles like this. -326 +332 00:20:39,000 --> 00:20:43,810 But, and I think I've previewed this enough that you know exactly where this is going, -327 +333 00:20:43,810 --> 00:20:47,349 if we zoom in we can find a black region, and that black region -328 +334 00:20:47,349 --> 00:20:49,340 looks exactly like a Mandelbrot set. -329 +335 00:20:50,020 --> 00:20:53,562 Yet again, asking a question where we tweak a parameter for one -330 +336 00:20:53,562 --> 00:20:57,160 of these functions yields this iconic cardioid and bubbles shape. -331 +337 00:20:58,020 --> 00:21:00,411 The upshot is that this shape is not as specific -332 +338 00:21:00,411 --> 00:21:02,900 to the z squared plus c example as you might think. -333 +339 00:21:03,400 --> 00:21:06,165 It seems to relate to something more general and -334 +340 00:21:06,165 --> 00:21:09,440 universal about parameter spaces with processes like this. -335 +341 00:21:11,640 --> 00:21:15,520 Still, one pressing question is why we get fractals at all. -336 +342 00:21:16,220 --> 00:21:20,460 In the last video, I talked about how the diagrams for Newton's method have this -337 +343 00:21:20,460 --> 00:21:24,753 very peculiar property, where if you draw a small circle around the boundary of a -338 +344 00:21:24,753 --> 00:21:29,360 colored region, that circle must actually include all available colors from the picture. -339 +345 00:21:30,280 --> 00:21:32,740 And this is true more generally for any rational map. -340 +346 00:21:33,140 --> 00:21:37,141 If you were to assign colors to regions based on which limiting behavior -341 +347 00:21:37,141 --> 00:21:41,143 points fall into, like which limit point or which limit cycle or does it -342 +348 00:21:41,143 --> 00:21:45,364 tend to infinity, then tiny circles that you draw either contain points with -343 +349 00:21:45,364 --> 00:21:49,640 just one of those limiting behaviors, or they contain points with all of them. -344 +350 00:21:49,820 --> 00:21:51,200 It's never anything in between. -345 +351 00:21:51,960 --> 00:21:54,578 So in the case where there's at least three colors, -346 +352 00:21:54,578 --> 00:21:57,750 this property implies that our boundary could never be smooth, -347 +353 00:21:57,750 --> 00:22:01,678 since along a smooth segment, you can draw a small enough circle that touches -348 +354 00:22:01,678 --> 00:22:03,340 just two colors, not all of them. -349 +355 00:22:03,920 --> 00:22:05,320 And empirically, this is what we see. -350 +356 00:22:05,480 --> 00:22:08,360 No matter how far you zoom in, these boundaries are always rough. -351 +357 00:22:08,880 --> 00:22:11,623 And furthermore, you might notice that as we zoom in, -352 +358 00:22:11,623 --> 00:22:14,520 you can always see all available colors within the frame. -353 +359 00:22:16,160 --> 00:22:20,409 This doesn't explain rough boundaries in the context where there's only two limiting -354 +360 00:22:20,409 --> 00:22:24,460 behaviors, but still, it's a loose end that I left in that video worth tying up, -355 +361 00:22:24,460 --> 00:22:27,960 and it's a nice excuse to bring in two important bits of terminology, -356 +362 00:22:27,960 --> 00:22:29,260 Julia sets and Fatou sets. -357 +363 00:22:29,940 --> 00:22:33,571 If a point eventually falls into some stable, predictable pattern, -358 +364 00:22:33,571 --> 00:22:37,040 we say that it's part of the Fatou set of our iterated function. -359 +365 00:22:37,740 --> 00:22:40,860 And for all the maps that we've seen, this includes almost everything. -360 +366 00:22:41,640 --> 00:22:45,062 The Julia set is everything else, which in the pictures we've -361 +367 00:22:45,062 --> 00:22:48,540 seen would be the rough boundaries between the colored regions. -362 +368 00:22:49,200 --> 00:22:52,340 What happens as you transition from one stable attractor to another? -363 +369 00:22:53,200 --> 00:22:55,554 For example, the Julia set will include all of -364 +370 00:22:55,554 --> 00:22:58,160 the repelling cycles and the repelling fixed points. -365 +371 00:22:58,880 --> 00:23:01,800 A typical point from the Julia set though, will not be a cycle. -366 +372 00:23:02,220 --> 00:23:04,340 It'll bounce around forever with no clear pattern. -367 +373 00:23:05,620 --> 00:23:09,867 Now, if you look at a point in the Fatou set, and you draw a small enough disc around it, -368 +374 00:23:09,867 --> 00:23:13,124 as you follow the process, that small disc will eventually shrink as -369 +375 00:23:13,124 --> 00:23:15,720 you fall into whatever the relevant stable behavior is. -370 +376 00:23:16,240 --> 00:23:19,110 Unless you're going to infinity, but you could kind of think of that as -371 +377 00:23:19,110 --> 00:23:22,020 the disc shrinking around infinity, but maybe that just confuses matters. -372 +378 00:23:24,500 --> 00:23:28,318 By contrast, if you draw a small disc around a point on the Julia set, -373 +379 00:23:28,318 --> 00:23:33,105 it tends to expand over time as the points from within that circle go off and kind of do -374 +380 00:23:33,105 --> 00:23:34,020 their own things. -375 +381 00:23:35,540 --> 00:23:39,540 In other words, points of the Julia set tend to behave chaotically. -376 +382 00:23:40,080 --> 00:23:42,466 Their nearby neighbors, even very nearby, will -377 +383 00:23:42,466 --> 00:23:45,260 eventually fall into qualitatively different behaviors. -378 +384 00:23:46,420 --> 00:23:48,840 But it's not merely that this disc expands. -379 +385 00:23:49,360 --> 00:23:53,458 A pretty surprising result, key to the multicolor property mentioned before, -380 +386 00:23:53,458 --> 00:23:57,769 is that if you let this process play out, that little disc eventually expands so -381 +387 00:23:57,769 --> 00:24:02,400 much that it hits every single point on the complex plane, with at most two exceptions. -382 +388 00:24:02,400 --> 00:24:06,880 This is known as the stuff-goes-everywhere principle of Julia sets. -383 +389 00:24:07,940 --> 00:24:09,360 Okay, it's not actually called that. -384 +390 00:24:09,700 --> 00:24:11,769 In the source I was reading from, it's mentioned as -385 +391 00:24:11,769 --> 00:24:13,800 a corollary to something known as Montel's theorem. -386 +392 00:24:14,320 --> 00:24:15,700 But it should be called that. -387 +393 00:24:16,120 --> 00:24:20,170 In some sense, what this is telling us is that the points of the Julia set -388 +394 00:24:20,170 --> 00:24:24,220 are not merely chaotic, they're kind of as chaotic as they possibly can be. -389 +395 00:24:25,860 --> 00:24:29,333 Here, let me show you a little simulation using the Newton's map, -390 +396 00:24:29,333 --> 00:24:33,701 with a cluster of a few thousand points, all starting from within a tiny distance, -391 +397 00:24:33,701 --> 00:24:36,280 one one-millionth, from a point on the Julia set. -392 +398 00:24:42,680 --> 00:24:46,175 Of course, the stuff-goes-everywhere principle is about the uncountably -393 +399 00:24:46,175 --> 00:24:49,088 infinitely many points that would lie within that distance, -394 +400 00:24:49,088 --> 00:24:52,389 and that they eventually expand out to hit everything on the plane, -395 +401 00:24:52,389 --> 00:24:53,700 except possibly two points. -396 +402 00:24:54,200 --> 00:24:56,780 But this little cluster should still give the general idea. -397 +403 00:24:56,780 --> 00:24:59,585 A small, finite sample from that tiny disk gets -398 +404 00:24:59,585 --> 00:25:02,800 sprayed all over the place in seemingly all directions. -399 +405 00:25:04,400 --> 00:25:08,704 What this means for our purposes is that if there's some attractive behavior of our map, -400 +406 00:25:08,704 --> 00:25:11,848 something like an attracting fixed point or an attracting cycle, -401 +407 00:25:11,848 --> 00:25:15,815 you can be guaranteed that the values from that tiny disk around the point on the -402 +408 00:25:15,815 --> 00:25:20,120 Julia set, no matter how tiny it was, will eventually fall into that attracting behavior. -403 +409 00:25:20,860 --> 00:25:23,832 If we have a case with three or more attracting behaviors, -404 +410 00:25:23,832 --> 00:25:27,258 this gives us some explanation for why the Julia set is not smooth, -405 +411 00:25:27,258 --> 00:25:28,720 why it has to be complicated. -406 +412 00:25:29,820 --> 00:25:32,739 Even still, this might not be entirely satisfying because it kicks -407 +413 00:25:32,739 --> 00:25:35,571 the can one more step down the road, raising the question of why -408 +414 00:25:35,571 --> 00:25:38,360 this stuff-goes-everywhere principle is true in the first place. -409 +415 00:25:39,180 --> 00:25:42,237 Like I mentioned, it comes from something called Montel's theorem, -410 +416 00:25:42,237 --> 00:25:46,300 and I'm choosing not to go into the details there, because honestly, it's a lot to cover. -411 +417 00:25:46,820 --> 00:25:50,288 The proof I could find ends up leaning on something known as the J function, -412 +418 00:25:50,288 --> 00:25:52,540 which is a whole intricate story in its own right. -413 +419 00:25:52,800 --> 00:25:54,667 I will of course leave links and resources in the -414 +420 00:25:54,667 --> 00:25:56,760 description for any of you who are hungry to learn more. -415 +421 00:25:57,320 --> 00:26:00,513 And if you know of a simpler way to see why this principle is true, -416 +422 00:26:00,513 --> 00:26:01,640 I'm definitely all ears. -417 +423 00:26:02,400 --> 00:26:06,175 I should also say as a brief side note that even though the pictures we've seen -418 +424 00:26:06,175 --> 00:26:08,582 so far have a Julia set which has an area of zero, -419 +425 00:26:08,582 --> 00:26:10,895 it's kind of the boundary between these regions, -420 +426 00:26:10,895 --> 00:26:13,680 there are examples where the Julia set is the entire plane. -421 +427 00:26:14,040 --> 00:26:16,800 Everything behaves chaotically, which is kind of wild. -422 +428 00:26:18,180 --> 00:26:20,750 The main takeaway for this particular section -423 +429 00:26:20,750 --> 00:26:23,320 is the link between the chaos and the fractal. -424 +430 00:26:23,980 --> 00:26:26,680 -At first it seems like these are merely analogous to each other. +At first it seems like these are merely analogous to each other, you know, -425 +431 00:26:27,140 --> 00:26:31,248 Newton's method turns out to be a kind of messy process for some seed values, -426 +432 00:26:31,248 --> 00:26:35,935 and this messiness is visible one way by following the trajectory of a particular point, -427 +433 00:26:35,935 --> 00:26:38,621 and another way by the complexity of our diagrams, -428 +434 00:26:38,621 --> 00:26:41,940 but those feel like qualitatively different kinds of messiness. -429 +435 00:26:42,520 --> 00:26:44,640 Maybe it makes for a nice metaphor, but nothing more. -430 +436 00:26:45,320 --> 00:26:49,068 However, what's neat here is that when you quantify just how chaotic -431 +437 00:26:49,068 --> 00:26:52,980 some of the points are, well, that quantification leads us to an actual -432 +438 00:26:52,980 --> 00:26:56,620 explanation for the rough fractal shape via this boundary property. -433 +439 00:26:57,640 --> 00:27:01,224 Quite often you see chaos and fractals sort of married together in math, -434 +440 00:27:01,224 --> 00:27:04,710 and to me at least it's satisfying whenever that marriage comes with a -435 +441 00:27:04,710 --> 00:27:08,540 logical link to it, rather than as two phenomena that just happen to coincide. diff --git a/2021/holomorphic-dynamics/english/sentence_timings.json b/2021/holomorphic-dynamics/english/sentence_timings.json index 523921da3..04cc645ab 100644 --- a/2021/holomorphic-dynamics/english/sentence_timings.json +++ b/2021/holomorphic-dynamics/english/sentence_timings.json @@ -160,7 +160,7 @@ 273.36 ], [ - "And as I change around the value c here, you can kind of see how the third value to get z4 and continue on like this, visualizing our chain of values.", + "And as I change around the value c here, you can kind of see how the second value moves in lockstep. Then we can plug in that second value to get z3, and that third value to get z4, and continue on like this, visualizing our chain of values.", 273.96, 287.4 ], @@ -190,7 +190,7 @@ 334.74 ], [ - "I honestly thought it was just too fun not to read about all of this stuff for any of you who haven't had the pleasure of reading that yet.", + "I honestly thought it was just too fun not to re-implement here. I would also highly recommend the interactive article on ako.net about all of this stuff for any of you who haven't had the pleasure of reading that yet.", 334.9, 344.64 ], @@ -230,7 +230,7 @@ 394.34 ], [ - "So what you're looking at is a parameter space.", + "So what you're looking at is what we might call a parameter space.", 394.82, 397.5 ], @@ -290,7 +290,7 @@ 498.88 ], [ - "It's one thing to look at some pretty pictures, but what sorts of questions would you ask if you understand it all?", + "It's one thing to look at some pretty pictures, but what sorts of questions would you ask if you actually want to understand it all?", 499.26, 505.06 ], @@ -330,7 +330,7 @@ 545.66 ], [ - "If you're into exercises, you may enjoy pausing for a moment and computing the fixed points of this Mandelbrot set since asking when this expression equals z can always be rearranged as finding the roots of some polynomial expression.", + "If you're into exercises, you may enjoy pausing for a moment and computing the fixed points of this Mandelbrot set function, z squared plus c. More generally, any rational function will always have fixed points, since asking when this expression equals z can always be rearranged as finding the roots of some polynomial expression,", 546.56, 566.42 ], @@ -345,7 +345,7 @@ 581.6 ], [ - "Just asking about fixed points is easy, but a key idea for understanding the full dynamics, and hence the diagrams that we're looking at, is to understand stability.", + "ight now. Now just asking about fixed points is maybe easy, but a key idea for understanding the full dynamics, and hence the diagrams we're looking at, is to understand stability.", 582.64, 591.72 ], @@ -985,7 +985,7 @@ 1583.32 ], [ - "At first it seems like these are merely analogous to each other.", + "At first it seems like these are merely analogous to each other, you know,", 1583.98, 1586.68 ], diff --git a/2021/holomorphic-dynamics/english/transcript.txt b/2021/holomorphic-dynamics/english/transcript.txt index a0bbf91aa..eb1c1f15a 100644 --- a/2021/holomorphic-dynamics/english/transcript.txt +++ b/2021/holomorphic-dynamics/english/transcript.txt @@ -30,13 +30,13 @@ It'll be visible as this movable yellow dot. For the actual iterative process, we will always start with an initial value of z equals zero. So after iterating this function once, doing z squared plus c, you get c. If you iterate a second time, plugging in that value to the function, you get c squared plus c. -And as I change around the value c here, you can kind of see how the third value to get z4 and continue on like this, visualizing our chain of values. +And as I change around the value c here, you can kind of see how the second value moves in lockstep. Then we can plug in that second value to get z3, and that third value to get z4, and continue on like this, visualizing our chain of values. So if I keep doing this many different times for the first many values, for some choices of c, this process remains bounded. You can still see it all on the screen. And other times, it looks like it blows up, and you can actually show that if it gets as big as 2, it'll blow up to infinity. If you color the points of the plane where it stays bounded black, and you assign some other gradient of colors to the divergent values based on how quickly the process rushes off to infinity, you get one of the most iconic images in all of math, the Mandelbrot set. Now this interactive dots and stick visualization of the trajectory, by the way, is heavily inspired by Ben Sparks' illustration and the Numberphile video he did about the Mandelbrot set, which is great, you should watch it. -I honestly thought it was just too fun not to read about all of this stuff for any of you who haven't had the pleasure of reading that yet. +I honestly thought it was just too fun not to re-implement here. I would also highly recommend the interactive article on ako.net about all of this stuff for any of you who haven't had the pleasure of reading that yet. What's nice about the Ben Sparks illustration is how it illuminates what each different part of the Mandelbrot set actually represents. This largest cardioid section includes the values of c so that the process eventually converges to some limit. The big circle on the left represents the values where the process gets trapped in a cycle between two values. @@ -44,7 +44,7 @@ And then the top and bottom circles show values where the process gets trapped i Each one of these little islands kind of has its own meaning. Also notice there's an important difference between how this Mandelbrot set and the Newton fractals we were looking at before are each constructed, beyond just a different underlying function. For the Mandelbrot set we have a consistent seed value, z equals zero, but the thing we're tweaking is the parameter c, changing the function itself. -So what you're looking at is a parameter space. +So what you're looking at is what we might call a parameter space. But with Newton's fractal, we have a single unchanging function, but what we associate with each pixel is a different seed value for the process. Of course, we could play the same game with the map z squared plus c. We could fix c at some constant, and then let the pixels represent the different possible initial values, z naught. @@ -56,7 +56,7 @@ You'll often hear or read the images on the right being referred to as Julia set However, the term Julia set has a much more general definition, and it would refer just to the boundaries of these regions, not the interior. To set the stage for a more specific definition, and to also make some headway towards the first goal that I mentioned at the start, it's worth stepping back and really just picturing yourself as a mathematician right now, discovering all of this. What would you actually do to construct a theory around this? -It's one thing to look at some pretty pictures, but what sorts of questions would you ask if you understand it all? +It's one thing to look at some pretty pictures, but what sorts of questions would you ask if you actually want to understand it all? In general, if you want to understand something complicated, a good place to start is to ask if there are any parts of the system that have some simple behavior, preferably the simplest possible behavior. In our example, that might mean asking when does the process just stay fixed in place, meaning f of z is equal to z. That's a pretty boring set of dynamics, I think you'd agree. @@ -64,10 +64,10 @@ We call a value with this property a fixed point of the function. In the case of the functions arising from Newton's method, by design they have a fixed point at the roots of the relevant polynomial. You can verify for yourself, if p of z is equal to zero, then the entire expression is simply equal to z. That's what it means to be a fixed point. -If you're into exercises, you may enjoy pausing for a moment and computing the fixed points of this Mandelbrot set since asking when this expression equals z can always be rearranged as finding the roots of some polynomial expression. +If you're into exercises, you may enjoy pausing for a moment and computing the fixed points of this Mandelbrot set function, z squared plus c. More generally, any rational function will always have fixed points, since asking when this expression equals z can always be rearranged as finding the roots of some polynomial expression, From the fundamental theorem of algebra, this must have solutions, typically as many solutions as the highest degree in this expression. Incidentally, this means you could also find those fixed points using Newton's method, but maybe that's a little too meta for us. -Just asking about fixed points is easy, but a key idea for understanding the full dynamics, and hence the diagrams that we're looking at, is to understand stability. +ight now. Now just asking about fixed points is maybe easy, but a key idea for understanding the full dynamics, and hence the diagrams we're looking at, is to understand stability. We say that a fixed point is attracting if nearby points tend to get drawn in towards it, and repelling if they're pushed away. And this is something that you can actually compute explicitly using the derivative of the function. Symbolically, when you take derivatives of complex functions, it looks exactly the same as it would for real functions, though something like z squared has a derivative of 2 times z. @@ -195,7 +195,7 @@ And if you know of a simpler way to see why this principle is true, I'm definite I should also say as a brief side note that even though the pictures we've seen so far have a Julia set which has an area of zero, it's kind of the boundary between these regions, there are examples where the Julia set is the entire plane. Everything behaves chaotically, which is kind of wild. The main takeaway for this particular section is the link between the chaos and the fractal. -At first it seems like these are merely analogous to each other. +At first it seems like these are merely analogous to each other, you know, Newton's method turns out to be a kind of messy process for some seed values, and this messiness is visible one way by following the trajectory of a particular point, and another way by the complexity of our diagrams, but those feel like qualitatively different kinds of messiness. Maybe it makes for a nice metaphor, but nothing more. However, what's neat here is that when you quantify just how chaotic some of the points are, well, that quantification leads us to an actual explanation for the rough fractal shape via this boundary property. diff --git a/2021/holomorphic-dynamics/hungarian/sentence_translations.json b/2021/holomorphic-dynamics/hungarian/sentence_translations.json index 54ace9144..17db6fbaa 100644 --- a/2021/holomorphic-dynamics/hungarian/sentence_translations.json +++ b/2021/holomorphic-dynamics/hungarian/sentence_translations.json @@ -256,7 +256,7 @@ "end": 273.36 }, { - "input": "And as I change around the value c here, you can kind of see how the third value to get z4 and continue on like this, visualizing our chain of values.", + "input": "And as I change around the value c here, you can kind of see how the second value moves in lockstep. Then we can plug in that second value to get z3, and that third value to get z4, and continue on like this, visualizing our chain of values.", "translatedText": "És ahogy megváltoztatom a c értéket itt, láthatjuk, hogy a harmadik értéket hogyan kapjuk meg a z4-et, és így folytatjuk, megjelenítve az értékláncunkat.", "model": "DeepL", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 334.74 }, { - "input": "I honestly thought it was just too fun not to read about all of this stuff for any of you who haven't had the pleasure of reading that yet.", + "input": "I honestly thought it was just too fun not to re-implement here. I would also highly recommend the interactive article on ako.net about all of this stuff for any of you who haven't had the pleasure of reading that yet.", "translatedText": "Őszintén szólva úgy gondoltam, hogy túl szórakoztató lenne nem olvasni mindezekről a dolgokról azok számára, akiknek még nem volt szerencséjük elolvasni.", "model": "DeepL", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 394.34 }, { - "input": "So what you're looking at is a parameter space.", + "input": "So what you're looking at is what we might call a parameter space.", "translatedText": "Tehát amit lát, az egy paramétertér.", "model": "DeepL", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 498.88 }, { - "input": "It's one thing to look at some pretty pictures, but what sorts of questions would you ask if you understand it all?", + "input": "It's one thing to look at some pretty pictures, but what sorts of questions would you ask if you actually want to understand it all?", "translatedText": "Az egy dolog, hogy megnézel néhány szép képet, de milyen kérdéseket teszel fel, ha mindent értesz?", "model": "DeepL", "n_reviews": 0, @@ -528,7 +528,7 @@ "end": 545.66 }, { - "input": "If you're into exercises, you may enjoy pausing for a moment and computing the fixed points of this Mandelbrot set since asking when this expression equals z can always be rearranged as finding the roots of some polynomial expression.", + "input": "If you're into exercises, you may enjoy pausing for a moment and computing the fixed points of this Mandelbrot set function, z squared plus c. More generally, any rational function will always have fixed points, since asking when this expression equals z can always be rearranged as finding the roots of some polynomial expression,", "translatedText": "Ha szereted a feladatokat, akkor egy pillanatra megállhatsz, és kiszámolhatod ennek a Mandelbrot-halmaznak a fix pontjait, hiszen ha azt kérdezed, hogy ez a kifejezés mikor egyenlő z-vel, akkor mindig átrendezheted úgy, mintha valamilyen polinom kifejezés gyökereit keresnéd.", "model": "DeepL", "n_reviews": 0, @@ -552,7 +552,7 @@ "end": 581.6 }, { - "input": "Just asking about fixed points is easy, but a key idea for understanding the full dynamics, and hence the diagrams that we're looking at, is to understand stability.", + "input": "ight now. Now just asking about fixed points is maybe easy, but a key idea for understanding the full dynamics, and hence the diagrams we're looking at, is to understand stability.", "translatedText": "A fixpontokról kérdezni könnyű, de a teljes dinamika megértéséhez, és így a most vizsgált diagramok megértéséhez kulcsfontosságú a stabilitás megértése.", "model": "DeepL", "n_reviews": 0, @@ -1584,7 +1584,7 @@ "end": 1586.68 }, { - "input": "Newton's method turns out to be a kind of messy process for some seed values, and this messiness is visible one way by following the trajectory of a particular point, and another way by the complexity of our diagrams, but those feel like qualitatively different kinds of messiness.", + "input": "you know, Newton's method turns out to be a kind of messy process for some seed values, and this messiness is visible one way by following the trajectory of a particular point, and another way by the complexity of our diagrams, but those feel like qualitatively different kinds of messiness.", "translatedText": "Newton módszere bizonyos magértékek esetén egyfajta rendezetlen folyamatnak bizonyul, és ez a rendezetlenség egyrészt egy adott pont pályáját követve, másrészt a diagramjaink bonyolultsága alapján látható, de ezek minőségileg különböző rendezetlenségeknek tűnnek.", "model": "DeepL", "n_reviews": 0, diff --git a/2021/matrix-exponents/arabic/sentence_translations.json b/2021/matrix-exponents/arabic/sentence_translations.json index 2c24e7947..eea81a566 100644 --- a/2021/matrix-exponents/arabic/sentence_translations.json +++ b/2021/matrix-exponents/arabic/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade. ", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade. ", "translatedText": "بمعنى آخر، عندما يعبر روميو عن عدم اهتمام رائع، تزيد مشاعر جولييت، بينما إذا أصبح مفتونًا جدًا، سيبدأ اهتمامها في التلاشي. ", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. ", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you. ", "translatedText": "أيضًا، إذا سمح الوقت، قد يكون من الممتع التحدث عن معنى رفع e إلى قوة عامل المشتقة. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/bengali/sentence_translations.json b/2021/matrix-exponents/bengali/sentence_translations.json index db4d445ef..39a51b0be 100644 --- a/2021/matrix-exponents/bengali/sentence_translations.json +++ b/2021/matrix-exponents/bengali/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade. ", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade. ", "translatedText": "অন্য কথায়, যখন রোমিও শীতল অনাগ্রহ প্রকাশ করে, তখনই জুলিয়েটের অনুভূতি বেড়ে যায়, যেখানে সে যদি খুব বেশি মোহগ্রস্ত হয়ে পড়ে, তার আগ্রহ ম্লান হতে শুরু করবে।", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. ", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you. ", "translatedText": "এছাড়াও, সময় পারমিটিং, ডেরিভেটিভ অপারেটরের শক্তিতে e-কে বাড়ানোর অর্থ কী তা নিয়ে কথা বলা মজাদার হতে পারে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/chinese/sentence_translations.json b/2021/matrix-exponents/chinese/sentence_translations.json index ebcb3ff96..04a1874e2 100644 --- a/2021/matrix-exponents/chinese/sentence_translations.json +++ b/2021/matrix-exponents/chinese/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "换句话说，当罗密欧表现出冷静 的不感兴趣时，朱丽叶的感情就会增加，而如 果他变得过于痴迷，她的兴趣就会开始消退。", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "该方程表示该状态向量看起来像某个 矩阵的速率乘以自身。", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "另外，如果时间允许，讨论将 e 求导 数算子的幂意味着什么可能会很有趣。", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/english/captions.srt b/2021/matrix-exponents/english/captions.srt index 40aa86ca8..e01322478 100644 --- a/2021/matrix-exponents/english/captions.srt +++ b/2021/matrix-exponents/english/captions.srt @@ -279,12 +279,12 @@ Take this 2x2 matrix that has negative pi and pi sitting off its diagonal entrie Let's see what the sum gives. 71 -00:04:27,280 --> 00:04:30,290 -The first term is the identity matrix, this is what we +00:04:27,280 --> 00:04:30,449 +The first term is the identity matrix, this is actually what we 72 -00:04:30,290 --> 00:04:33,520 -mean by definition when we raise a matrix to the 0th power. +00:04:30,449 --> 00:04:33,520 +mean by definition when we raise a matrix to the zeroth power. 73 00:04:34,460 --> 00:04:38,445 @@ -423,1138 +423,1150 @@ One involving relationships, and the other quantum mechanics. Let's start with relationships. 107 -00:06:43,080 --> 00:06:49,702 -Let's call two lovers, Romeo and Juliet, and let x represent Juliet's love for Romeo, +00:06:43,080 --> 00:06:46,950 +Suppose we have two lovers, let's call them Romeo and Juliet, 108 -00:06:49,702 --> 00:06:55,940 -and y represent his love for her, both of which are values that change with time. +00:06:46,950 --> 00:06:50,134 +and let's let x represent Juliet's love for Romeo, 109 +00:06:50,134 --> 00:06:54,566 +and y represent his love for her, both of which are going to be values + +110 +00:06:54,566 --> 00:06:55,940 +that change with time. + +111 00:06:56,900 --> 00:06:59,608 This is an example we actually touched on in chapter 1, -110 +112 00:06:59,608 --> 00:07:03,140 based on a Steven Strogatz article, but it's okay if you didn't see that. -111 +113 00:07:03,580 --> 00:07:08,801 The way their relationship works is that the rate at Juliet's love for Romeo changes, -112 +114 00:07:08,801 --> 00:07:13,780 the derivative of this value, is equal to negative one times Romeo's love for her. -113 +115 00:07:14,560 --> 00:07:18,067 So in other words, when Romeo is expressing cool disinterest, -114 +116 00:07:18,067 --> 00:07:20,839 that's when Juliet's feelings actually increase, -115 +117 00:07:20,839 --> 00:07:24,800 whereas if he becomes too infatuated, her interest will start to fade. -116 +118 00:07:27,100 --> 00:07:28,700 Romeo, on the other hand, is the opposite. -117 +119 00:07:29,060 --> 00:07:33,501 The rate of change of his love is equal to the of Juliet's love, -118 +120 00:07:33,501 --> 00:07:37,873 so while Juliet is mad at him, his affections tend to decrease, -119 +121 00:07:37,873 --> 00:07:41,700 whereas if she loves him, that's when his feelings grow. -120 +122 00:07:42,580 --> 00:07:45,240 Of course, neither one of these numbers is holding still. -121 +123 00:07:45,680 --> 00:07:48,767 As Romeo's love increases in response to Juliet, -122 +124 00:07:48,767 --> 00:07:52,360 her equation continues to apply and drives her love down. -123 +125 00:07:53,360 --> 00:07:58,284 Both of these equations always apply, from each infinitesimal point in time to the next, -124 +126 00:07:58,284 --> 00:08:02,988 so every slight change to one value immediately influences the rate of change of the -125 +127 00:08:02,988 --> 00:08:03,320 other. -126 +128 00:08:04,120 --> 00:08:06,560 This is a system of differential equations. -127 +129 00:08:06,820 --> 00:08:10,759 It's a puzzle, where your challenge is to find explicit functions -128 +130 00:08:10,759 --> 00:08:14,520 for x of t and y of t that make both of these expressions true. -129 +131 00:08:15,640 --> 00:08:19,741 Now, as systems of differential equations go, this one is on the simpler side, -130 +132 00:08:19,741 --> 00:08:23,740 enough so that many calculus students could probably just guess at an answer. -131 +133 00:08:24,300 --> 00:08:28,500 But keep in mind, it's not enough to find some pair of functions that makes this true. -132 +134 00:08:29,000 --> 00:08:33,708 If you want to actually predict where Romeo and Juliet end up after some starting point, -133 +135 00:08:33,708 --> 00:08:36,882 you have to make sure that your functions match the initial -134 +136 00:08:36,882 --> 00:08:38,840 set of conditions at time t equals 0. -135 +137 00:08:39,740 --> 00:08:42,840 More to the point, our actual goal today is to systematically -136 +138 00:08:42,840 --> 00:08:46,689 solve more general versions of this equation, without guessing and checking, -137 +139 00:08:46,689 --> 00:08:49,540 and it's that question that leads us to matrix exponents. -138 +140 00:08:50,680 --> 00:08:53,519 Very often when you have multiple changing values like this, -139 +141 00:08:53,519 --> 00:08:57,382 it's helpful to package them together as coordinates of a single point in a higher -140 +142 00:08:57,382 --> 00:08:58,220 dimensional space. -141 +143 00:08:58,800 --> 00:09:04,638 So for Romeo and Juliet, think of their relationship as a point in a 2D space, -142 +144 00:09:04,638 --> 00:09:10,920 the x-coordinate capturing Juliet's feelings, and the y-coordinate capturing Romeo's. -143 +145 00:09:13,200 --> 00:09:16,770 Sometimes it's helpful to picture this as an arrow from the origin, -144 +146 00:09:16,770 --> 00:09:18,240 other times just as a point. -145 +147 00:09:18,700 --> 00:09:21,586 All that really matters is that it encodes two numbers, -146 +148 00:09:21,586 --> 00:09:24,680 and moving forward we'll be writing that as a column vector. -147 +149 00:09:25,300 --> 00:09:27,480 And of course, this is all a function of time. -148 +150 00:09:28,500 --> 00:09:31,110 You might picture the rate of change of this state, -149 +151 00:09:31,110 --> 00:09:35,026 the thing that packages together the derivative of x and the derivative of y, -150 +152 00:09:35,026 --> 00:09:37,536 as a kind of velocity vector in this state space, -151 +153 00:09:37,536 --> 00:09:41,301 something that tugs at our point in some direction and with some magnitude -152 +154 00:09:41,301 --> 00:09:43,360 that indicates how quickly it's changing. -153 +155 00:09:45,560 --> 00:09:50,243 Remember, the rule here is that the rate of change of x is negative y, -154 +156 00:09:50,243 --> 00:09:52,420 and the rate of change of y is x. -155 +157 00:09:53,300 --> 00:09:57,340 Set up as vectors like this, we could rewrite the right hand side of -156 +158 00:09:57,340 --> 00:10:01,440 this equation as a product of this matrix with the original vector xy. -157 +159 00:10:02,080 --> 00:10:06,940 The top row encodes Juliet's rule, and the bottom row encodes Romeo's rule. -158 -00:10:07,800 --> 00:10:15,880 -So what we have here is a vector is equal to a certain matrix times itself. +160 +00:10:07,800 --> 00:10:11,724 +So what we have here is a differential equation telling us that the -159 +161 +00:10:11,724 --> 00:10:15,880 +rate of change of some vector is equal to a certain matrix times itself. + +162 00:10:19,120 --> 00:10:23,338 In a moment we'll talk about how matrix exponentiation solves this kind of equation, -160 +163 00:10:23,338 --> 00:10:27,706 but before that let me show you a simpler way that we can solve this particular system, -161 +164 00:10:27,706 --> 00:10:31,280 one that uses pure geometry, and it helps set the stage for visualizing -162 +165 00:10:31,280 --> 00:10:32,720 matrix exponents a bit later. -163 +166 00:10:34,000 --> 00:10:37,380 This matrix from our system is a 90 degree rotation matrix. -164 +167 00:10:38,580 --> 00:10:42,437 For any of you rusty on how to think about matrices as transformations, -165 +168 00:10:42,437 --> 00:10:45,760 there's a video all about it on this channel, a series really. -166 +169 00:10:46,400 --> 00:10:51,190 The basic idea is that when you multiply a matrix by the vector 1 0, -167 +170 00:10:51,190 --> 00:10:56,188 it pulls out the first column, and similarly if you multiply it by 0 1, -168 +171 00:10:56,188 --> 00:10:58,480 that pulls out the second column. -169 +172 00:10:59,900 --> 00:11:02,505 What this means is that when you look at a matrix, -170 +173 00:11:02,505 --> 00:11:06,338 you can read its columns as telling you what it does to these two vectors, -171 +174 00:11:06,338 --> 00:11:07,360 known as the matrix. -172 +175 00:11:07,380 --> 00:11:11,812 The way it acts on any other vector is a result of scaling -173 +176 00:11:11,812 --> 00:11:16,620 and adding these two basis results by that vector's coordinates. -174 +177 00:11:17,720 --> 00:11:20,241 So looking back at the matrix from our system, -175 +178 00:11:20,241 --> 00:11:24,532 notice how from its columns we can tell it takes the first basis vector to 0 1, -176 +179 00:11:24,532 --> 00:11:29,200 and the second to negative 1 0, hence why I'm calling it the 90 degree rotation matrix. -177 +180 00:11:30,880 --> 00:11:36,389 What it means for our equation is that it's saying wherever Romeo and Juliet are in this -178 +181 00:11:36,389 --> 00:11:41,960 space, their rate of change has to look like a 90 degree rotation of this position vector. -179 +182 00:11:42,700 --> 00:11:46,626 The only way velocity can permanently be perpendicular to position like this is -180 +183 00:11:46,626 --> 00:11:49,276 when you rotate around the origin in circular motion, -181 +184 00:11:49,276 --> 00:11:53,104 never growing or shrinking because the rate of change has no component in the -182 +185 00:11:53,104 --> 00:11:54,380 direction of the position. -183 +186 00:11:57,060 --> 00:12:01,434 More specifically, since the length of this velocity vector equals the -184 +187 00:12:01,434 --> 00:12:05,069 length of the position vector, then for each unit of time, -185 +188 00:12:05,069 --> 00:12:09,690 the distance that this covers is equal to one radius's worth of arc length -186 +189 00:12:09,690 --> 00:12:10,800 along that circle. -187 +190 00:12:12,060 --> 00:12:15,716 In other words, it rotates at one radian per unit time, -188 +191 00:12:15,716 --> 00:12:20,680 so in particular it would take 2 pi units of time to make a full revolution. -189 +192 00:12:22,620 --> 00:12:26,017 If you want to describe this kind of rotation with a formula, -190 +193 00:12:26,017 --> 00:12:29,580 we can use a more general rotation matrix, which looks like this. -191 +194 00:12:30,380 --> 00:12:32,280 Again, we can read it in terms of the columns. -192 +195 00:12:32,780 --> 00:12:37,951 Notice how the first column tells us that it takes that first basis vector to -193 +196 00:12:37,951 --> 00:12:43,389 cos t sin t, and the second column tells us that it takes the second basis vector -194 +197 00:12:43,389 --> 00:12:48,760 to negative sin t cos t, both of which are consistent with rotating by t radians. -195 +198 00:12:49,700 --> 00:12:54,239 So, to solve the system, if you want to predict where Romeo and Juliet end -196 +199 00:12:54,239 --> 00:12:58,960 up after t units of time, you can multiply this matrix by their initial state. -197 +200 00:13:00,120 --> 00:13:04,008 The active viewers among you might also enjoy taking a moment to pause and -198 +201 00:13:04,008 --> 00:13:07,896 confirm that the explicit formulas you get out of this for x of t and y of -199 +202 00:13:07,896 --> 00:13:11,940 t really do satisfy the system of differential equations that we started with. -200 +203 00:13:17,740 --> 00:13:22,069 The mathematician in you might wonder if it's possible to solve not just this specific -201 +204 00:13:22,069 --> 00:13:26,000 system, but equations like it for any other matrix, no matter its coefficients. -202 +205 00:13:27,120 --> 00:13:31,160 To ask this question is to set yourself up to rediscover matrix exponents. -203 +206 00:13:31,780 --> 00:13:36,507 The main goal for today is for you to understand how this equation lets you intuitively -204 +207 00:13:36,507 --> 00:13:41,020 picture the operation which we write as e raised to a matrix, and on the flip side, -205 +208 00:13:41,020 --> 00:13:45,480 how being able to compute matrix exponents lets you explicitly solve this equation. -206 +209 00:13:46,520 --> 00:13:49,917 A much less whimsical example is Schrodinger's famous equation, -207 +210 00:13:49,917 --> 00:13:54,057 which is the fundamental equation describing how systems in quantum mechanics -208 +211 00:13:54,057 --> 00:13:54,960 change over time. -209 +212 00:13:55,680 --> 00:13:59,461 It looks pretty intimidating, and I mean it's quantum mechanics so of course it will, -210 +213 00:13:59,461 --> 00:14:02,320 but it's actually not that different from the Romeo-Juliet setup. -211 +214 00:14:03,020 --> 00:14:05,280 This symbol here refers to a certain vector. -212 +215 00:14:05,800 --> 00:14:09,047 It's a vector that packages together all the information you might care -213 +216 00:14:09,047 --> 00:14:12,160 about in a system, like the various particles' positions and momenta. -214 +217 00:14:12,240 --> 00:14:14,812 It's analogous to our simpler 2D vector that encoded -215 +218 00:14:14,812 --> 00:14:16,900 all the information about Romeo and Juliet. -216 +219 00:14:17,840 --> 00:14:20,613 The equation says that the rate at which this state -217 +220 00:14:20,613 --> 00:14:23,600 vector changes looks like a certain matrix times itself. -218 +221 00:14:24,560 --> 00:14:28,775 There are a number of things that make Schrodinger's equation notably more complicated, -219 +222 00:14:28,775 --> 00:14:32,080 but in the back of your mind you might think of it as a target point -220 +223 00:14:32,080 --> 00:14:35,385 that you and I can build up to, with simpler examples like Romeo and -221 +224 00:14:35,385 --> 00:14:38,260 Juliet offering more friendly stepping stones along the way. -222 +225 00:14:39,540 --> 00:14:43,610 Actually the simplest example, which is tied to ordinary real number powers of e, -223 +226 00:14:43,610 --> 00:14:45,000 is the one-dimensional case. -224 +227 00:14:45,400 --> 00:14:47,740 This is when you have a single changing value, -225 +228 00:14:47,740 --> 00:14:50,580 and its rate of change equals some constant times itself. -226 +229 00:14:51,200 --> 00:14:53,440 So the bigger the value, the faster it grows. -227 +230 00:14:55,080 --> 00:14:58,188 Most people are more comfortable visualizing this with a graph, -228 +231 00:14:58,188 --> 00:15:01,297 where the higher the value of the graph, the steeper its slope, -229 +232 00:15:01,297 --> 00:15:03,580 resulting in this ever-steepening upward curve. -230 +233 00:15:04,040 --> 00:15:06,830 Just keep in mind that when we get to higher dimensional variance, -231 +234 00:15:06,830 --> 00:15:08,080 graphs are a lot less helpful. -232 +235 00:15:08,880 --> 00:15:11,500 This is a highly important equation in its own right. -233 +236 00:15:11,700 --> 00:15:14,138 It's a very powerful concept when the rate of change -234 +237 00:15:14,138 --> 00:15:16,300 of a value is proportional to the value itself. -235 -00:15:16,760 --> 00:15:19,559 -This is the equation governing compound interest, +238 +00:15:16,760 --> 00:15:20,050 +This is the equation governing things like compound interest, -236 -00:15:19,559 --> 00:15:24,597 +239 +00:15:20,050 --> 00:15:24,827 or the early stages of population growth before the effects of limited resources kick in, -237 -00:15:24,597 --> 00:15:29,020 +240 +00:15:24,827 --> 00:15:29,020 or the early stages of an epidemic while most of the population is susceptible. -238 +241 00:15:31,920 --> 00:15:37,360 Calculus students all learn about how the derivative of e to the rt is r times itself. -239 +242 00:15:38,440 --> 00:15:42,418 In other words, this self-reinforcing growth phenomenon is the same -240 +243 00:15:42,418 --> 00:15:46,280 thing as exponential growth, and e to the rt solves this equation. -241 +244 00:15:48,800 --> 00:15:52,490 Actually, a better way to think about it is that there are many different -242 +245 00:15:52,490 --> 00:15:55,482 solutions to this equation, one for each initial condition, -243 +246 00:15:55,482 --> 00:15:58,873 something like an initial investment size or an initial population, -244 +247 00:15:58,873 --> 00:16:00,120 which we'll just call x0. -245 +248 00:16:00,960 --> 00:16:03,889 Notice, by the way, how the higher the value for x0, -246 +249 00:16:03,889 --> 00:16:06,985 the higher the initial slope of the resulting solution, -247 +250 00:16:06,985 --> 00:16:09,860 which should make complete sense given the equation. -248 +251 00:16:11,220 --> 00:16:15,930 The function e to the rt is just a solution when the initial condition is 1, -249 +252 00:16:15,930 --> 00:16:19,110 but if you multiply by any other initial condition, -250 +253 00:16:19,110 --> 00:16:22,720 you get a new function which still satisfies this property. -251 +254 00:16:23,060 --> 00:16:26,476 It still has a derivative which is r times itself, -252 +255 00:16:26,476 --> 00:16:29,960 but this time it starts at x0 since e to the 0 is 1. -253 +256 00:16:30,780 --> 00:16:33,300 This is worth highlighting before we generalize to more dimensions. -254 +257 00:16:33,800 --> 00:16:37,320 Do not think of the exponential part as being a solution in and of itself. -255 +258 00:16:37,800 --> 00:16:42,380 Think of it as something that acts on an initial condition in order to give a solution. -256 +259 00:16:46,440 --> 00:16:51,189 You see, up in the two-dimensional case, when we have a changing vector whose rate of -257 +260 00:16:51,189 --> 00:16:54,171 change is constrained to be some matrix times itself, -258 +261 00:16:54,171 --> 00:16:58,755 what the solution looks like is also an exponential term acting on a given initial -259 +262 00:16:58,755 --> 00:17:03,670 condition, but the exponential part in that case will produce a matrix that changes with -260 +263 00:17:03,670 --> 00:17:06,099 time, and the initial condition is a vector. -261 +264 00:17:06,900 --> 00:17:10,525 In fact, you should think of the definition of matrix exponentiation -262 +265 00:17:10,525 --> 00:17:13,940 as being heavily motivated by making sure that this fact is true. -263 +266 00:17:14,920 --> 00:17:20,061 For example, if we look back at the system that popped up with Romeo and Juliet, -264 +267 00:17:20,061 --> 00:17:24,757 the claim now is that solutions look like e raised to this 0, negative 1, -265 +268 00:17:24,757 --> 00:17:28,820 1, 0 matrix all times time multiplied by some initial condition. -266 +269 00:17:29,560 --> 00:17:31,687 But we've already seen the solution in this case, -267 +270 00:17:31,687 --> 00:17:34,580 we know it looks like a rotation matrix times the initial condition. -268 +271 00:17:35,260 --> 00:17:39,066 So let's take a moment to roll up our sleeves and compute the exponential term -269 +272 00:17:39,066 --> 00:17:42,680 using the definition that I mentioned at the start, and see if it lines up. -270 +273 00:17:43,260 --> 00:17:46,643 Remember, writing e to the power of a matrix is a shorthand, -271 +274 00:17:46,643 --> 00:17:50,249 a shorthand for plugging it in to this long infinite polynomial, -272 +275 00:17:50,249 --> 00:17:52,080 the Taylor series for e to the x. -273 +276 00:17:53,100 --> 00:17:56,580 I know it might seem pretty complicated to do this, but trust me, -274 +277 00:17:56,580 --> 00:17:59,480 it's very satisfying how this particular one turns out. -275 +278 00:18:00,180 --> 00:18:03,949 If you actually sit down and you compute successive powers of this matrix, -276 +279 00:18:03,949 --> 00:18:08,020 what you'd notice is that they fall into a cycling pattern every four iterations. -277 +280 00:18:27,280 --> 00:18:30,940 This should make sense given that we know it's a 90 degree rotation matrix. -278 +281 00:18:31,620 --> 00:18:35,517 So when you add together all infinitely many matrices term by term, -279 +282 00:18:35,517 --> 00:18:39,872 each term in the result looks like a polynomial in t with some nice cycling -280 +283 00:18:39,872 --> 00:18:44,400 pattern in its coefficients, all of them scaled by the relevant factorial term. -281 +284 00:18:45,760 --> 00:18:49,567 Those of you who are savvy with Taylor series might be able to recognize -282 +285 00:18:49,567 --> 00:18:53,845 that each one of these components is the Taylor series for either sine or cosine, -283 +286 00:18:53,845 --> 00:18:57,340 though in that top right corner's case it's actually negative sine. -284 +287 00:18:58,680 --> 00:19:03,380 So what we get from the computation is exactly the rotation matrix we had from before. -285 +288 00:19:07,160 --> 00:19:09,220 To me, this is extremely beautiful. -286 +289 00:19:09,680 --> 00:19:13,220 We have two completely different ways of reasoning about the same system, -287 +290 00:19:13,220 --> 00:19:14,800 and they give us the same answer. -288 +291 00:19:15,480 --> 00:19:19,099 It's reassuring that they do, but it's wild just how different the mode of -289 +292 00:19:19,099 --> 00:19:22,718 thought is when you're chugging through this polynomial versus when you're -290 +293 00:19:22,718 --> 00:19:26,820 geometrically reasoning about what a velocity perpendicular to a position must imply. -291 +294 00:19:27,720 --> 00:19:30,971 Hopefully the fact that these line up inspires a little confidence -292 +295 00:19:30,971 --> 00:19:34,320 in the claim that matrix exponents really do solve systems like this. -293 +296 00:19:35,340 --> 00:19:38,306 This explains the computation we saw at the start, by the way, -294 +297 00:19:38,306 --> 00:19:41,273 with the matrix that had negative pi and pi off the diagonals, -295 +298 00:19:41,273 --> 00:19:42,780 producing the negative identity. -296 +299 00:19:43,560 --> 00:19:47,987 This expression is exponentiating a 90 degree rotation matrix times pi, -297 +300 00:19:47,987 --> 00:19:53,460 which is another way to describe what the Romeo-Juliet setup does after pi units of time. -298 +301 00:19:54,040 --> 00:19:57,942 As we now know, that has the effect of rotating everything 180 degrees -299 +302 00:19:57,942 --> 00:20:01,680 in this state space, which is the same as multiplying by negative 1. -300 +303 00:20:03,060 --> 00:20:06,360 Also, for any of you familiar with imaginary number exponents, -301 +304 00:20:06,360 --> 00:20:08,980 this particular example is ringing a ton of bells. -302 +305 00:20:09,360 --> 00:20:11,120 It is 100% analogous. -303 +306 00:20:11,840 --> 00:20:16,341 In fact, we could have framed the entire example where Romeo and Juliet's feelings were -304 +307 00:20:16,341 --> 00:20:20,638 packaged into a complex number, and the rate of change of that complex number would -305 +308 00:20:20,638 --> 00:20:25,140 have been i times itself, since multiplication by i also acts like a 90 degree rotation. -306 +309 00:20:25,840 --> 00:20:29,368 The same exact line of reasoning, both analytic and geometric, -307 +310 00:20:29,368 --> 00:20:33,680 would have led to this whole idea that e to the power i t describes rotation. -308 +311 00:20:34,460 --> 00:20:38,840 These are actually two of many different examples throughout math and physics when you -309 +312 00:20:38,840 --> 00:20:43,220 find yourself exponentiating some object which acts as a 90 degree rotation times time. -310 +313 00:20:43,980 --> 00:20:48,020 It shows up with quaternions or many of the matrices that pop up in quantum mechanics. -311 +314 00:20:48,720 --> 00:20:53,318 In all of these cases, we have this really neat general idea that if you take some -312 +315 00:20:53,318 --> 00:20:56,033 operation that rotates 90 degrees in some plane, -313 +316 00:20:56,033 --> 00:21:00,188 often it's a plane in some high dimensional space that we can't visualize, -314 +317 00:21:00,188 --> 00:21:04,565 then what we get by exponentiating that operation times time is something that -315 +318 00:21:04,565 --> 00:21:07,280 generates all other rotations in that same plane. -316 +319 00:21:09,100 --> 00:21:13,240 One of the more complicated variations on this same theme is Schrodinger's equation. -317 +320 00:21:13,840 --> 00:21:16,232 It's not just that this has the derivative of -318 +321 00:21:16,232 --> 00:21:18,780 a state equals some matrix times that state form. -319 +322 00:21:19,020 --> 00:21:22,450 The nature of the relevant matrix here is such that the equation -320 +323 00:21:22,450 --> 00:21:25,933 also describes a kind of rotation, though in many applications of -321 +324 00:21:25,933 --> 00:21:29,680 Schrodinger's equation it'll be a rotation in a kind of function space. -322 +325 00:21:30,520 --> 00:21:32,722 It's a little more involved though because typically -323 +326 00:21:32,722 --> 00:21:34,800 there's a combination of many different rotations. -324 +327 00:21:35,220 --> 00:21:39,890 It takes time to really dig into this equation and I would love to do that in a later -325 +328 00:21:39,890 --> 00:21:44,179 chapter, but right now I cannot help but at least allude to the fact that this -326 +329 00:21:44,179 --> 00:21:48,795 imaginary unit i that sits so impishly in such a fundamental equation for all of the -327 +330 00:21:48,795 --> 00:21:53,520 universe is playing basically the same role as the matrix from our Romeo-Julia example. -328 +331 00:21:54,160 --> 00:21:58,849 What this i communicates is that the rate of change of a certain state is, -329 +332 00:21:58,849 --> 00:22:03,288 in a sense, perpendicular to that state, and hence that the way things -330 +333 00:22:03,288 --> 00:22:07,040 have to evolve over time will involve a kind of oscillation. -331 +334 00:22:11,120 --> 00:22:14,480 But matrix exponentiation can do so much more than just rotation. -332 +335 00:22:15,020 --> 00:22:19,040 You can always visualize these sorts of differential equations using a vector field. -333 +336 00:22:20,240 --> 00:22:25,060 The idea is that this equation tells us the velocity of a state is entirely determined -334 +337 00:22:25,060 --> 00:22:29,659 by its position, so what we do is go to every point in the space and draw a little -335 +338 00:22:29,659 --> 00:22:34,480 vector indicating what the velocity of a state must be if it passes through that point. -336 +339 00:22:35,340 --> 00:22:38,482 For our type of equation, this means that we go to each -337 +340 00:22:38,482 --> 00:22:41,400 point v in space and we attach the vector m times v. -338 +341 00:22:54,020 --> 00:22:57,784 To intuitively understand how any given initial condition will evolve, -339 +342 00:22:57,784 --> 00:23:01,231 you let it flow along this field with a velocity always matching -340 +343 00:23:01,231 --> 00:23:04,360 whatever vector it's sitting on at any given point in time. -341 +344 00:23:05,860 --> 00:23:10,993 So if the claim is that solutions to this equation look like e to the m t times some -342 +345 00:23:10,993 --> 00:23:16,006 initial condition, it means you can visualize what the matrix e to the m t does by -343 +346 00:23:16,006 --> 00:23:21,020 letting every possible initial condition flow along this field for t units of time. -344 +347 00:23:25,080 --> 00:23:28,846 The transition from start to finish is described by whatever -345 +348 00:23:28,846 --> 00:23:32,180 matrix pops out from the computation for e to the m t. -346 +349 00:23:33,540 --> 00:23:36,689 In our main example with the 90 degree rotation matrix, -347 +350 00:23:36,689 --> 00:23:40,121 the vector field looks like this, and as we saw e to the m t -348 +351 00:23:40,121 --> 00:23:44,340 describes rotation in that case, which lines up with flow along this field. -349 +352 00:23:45,800 --> 00:23:49,859 As another example, the more Shakespearean Romeo and Juliet might have equations -350 +353 00:23:49,859 --> 00:23:53,969 that look a little more like this, where Juliet's rule is symmetric with Romeo's, -351 +354 00:23:53,969 --> 00:23:58,380 and both of them are inclined to get carried away in response to one another's feelings. -352 +355 00:23:59,360 --> 00:24:03,057 Again, the way the vector field you're looking at has been defined -353 +356 00:24:03,057 --> 00:24:06,700 is to go to each point v in space and attach the vector m times v. -354 +357 00:24:07,160 --> 00:24:10,105 This is the pictorial way of saying that the rate of -355 +358 00:24:10,105 --> 00:24:12,940 change of a state must always equal m times itself. -356 +359 00:24:14,160 --> 00:24:18,600 But for this example, flow along the field looks a lot different from how it did before. -357 +360 00:24:19,200 --> 00:24:23,114 If Romeo and Juliet start off anywhere in this upper right half of the plane, -358 +361 00:24:23,114 --> 00:24:27,080 their feelings will feed off of each other and they both tend towards infinity. -359 +362 00:24:30,580 --> 00:24:33,705 If they're in the other half of the plane, well let's just say -360 +363 00:24:33,705 --> 00:24:36,880 that they stay more true to their in Montague family traditions. -361 +364 00:24:38,020 --> 00:24:41,854 So even before you try calculating the exponential of this particular matrix, -362 +365 00:24:41,854 --> 00:24:45,640 you can already have an intuitive sense for what the answer should look like. -363 +366 00:24:46,160 --> 00:24:50,473 The resulting matrix should describe the transition from time 0 to time t, -364 +367 00:24:50,473 --> 00:24:54,786 which if you look at the field seems to indicate that it will squish along -365 +368 00:24:54,786 --> 00:24:59,560 one diagonal while stretching along another, getting more extreme as t gets larger. -366 +369 00:25:00,780 --> 00:25:03,669 Of course, all of this is presuming that e to the m t -367 +370 00:25:03,669 --> 00:25:06,720 times an initial condition actually solves these systems. -368 +371 00:25:07,640 --> 00:25:11,320 This is one of those facts that's easiest to believe when you just work it out yourself. -369 +372 00:25:12,300 --> 00:25:14,300 But I'll run through a quick rough sketch. -370 +373 00:25:16,020 --> 00:25:19,224 Write out the full polynomial that defines e to the m t -371 +374 00:25:19,224 --> 00:25:22,600 and multiply by some initial condition vector on the right. -372 +375 00:25:26,540 --> 00:25:29,420 And then take the derivative of this with respect to t. -373 +376 00:25:30,180 --> 00:25:32,412 Because the matrix m is constant, this just means -374 +377 00:25:32,412 --> 00:25:34,600 applying the power rule to each one of the terms. -375 +378 00:25:43,340 --> 00:25:47,000 And that power rule really nicely cancels out with the factorial terms. -376 +379 00:25:52,920 --> 00:25:56,089 So what we're left with is an expression that looks almost -377 +380 00:25:56,089 --> 00:26:00,817 identical to what we had before, except that each term has an extra m hanging on to it, -378 +381 00:26:00,817 --> 00:26:03,020 but this can be factored out to the left. -379 +382 00:26:03,580 --> 00:26:08,215 So the derivative of the expression is m times the original expression, -380 +383 00:26:08,215 --> 00:26:10,340 and hence it solves the equation. -381 +384 00:26:11,420 --> 00:26:14,524 This actually sweeps under the rug some details required for rigor, -382 +385 00:26:14,524 --> 00:26:18,404 mostly centered around the question of whether or not this thing actually converges, -383 +386 00:26:18,404 --> 00:26:19,820 but it does give the main idea. -384 +387 00:26:21,020 --> 00:26:24,484 In the next chapter I would like to talk more about the properties that -385 +388 00:26:24,484 --> 00:26:28,573 this operation has, most notably its relationship with eigenvectors and eigenvalues, -386 +389 00:26:28,573 --> 00:26:32,037 which leads us to more concrete ways of thinking about how you actually -387 +390 00:26:32,037 --> 00:26:34,780 carry out this computation, which otherwise seems insane. -388 +391 00:26:36,060 --> 00:26:38,783 Also, time permitting, it might be fun to talk about what -389 +392 00:26:38,783 --> 00:26:41,600 it means to raise e to the power of the derivative operator. -390 +393 00:26:55,820 --> 00:27:06,920 Thank you. diff --git a/2021/matrix-exponents/english/sentence_timings.json b/2021/matrix-exponents/english/sentence_timings.json index 7ba3efcb5..5dcf37f33 100644 --- a/2021/matrix-exponents/english/sentence_timings.json +++ b/2021/matrix-exponents/english/sentence_timings.json @@ -165,7 +165,7 @@ 266.64 ], [ - "The first term is the identity matrix, this is what we mean by definition when we raise a matrix to the 0th power.", + "The first term is the identity matrix, this is actually what we mean by definition when we raise a matrix to the zeroth power.", 267.28, 273.52 ], @@ -255,7 +255,7 @@ 399.24 ], [ - "Let's call two lovers, Romeo and Juliet, and let x represent Juliet's love for Romeo, and y represent his love for her, both of which are values that change with time.", + "Suppose we have two lovers, let's call them Romeo and Juliet, and let's let x represent Juliet's love for Romeo, and y represent his love for her, both of which are going to be values that change with time.", 403.08, 415.94 ], @@ -375,7 +375,7 @@ 606.94 ], [ - "So what we have here is a vector is equal to a certain matrix times itself.", + "So what we have here is a differential equation telling us that the rate of change of some vector is equal to a certain matrix times itself.", 607.8, 615.88 ], @@ -545,7 +545,7 @@ 916.3 ], [ - "This is the equation governing compound interest, or the early stages of population growth before the effects of limited resources kick in, or the early stages of an epidemic while most of the population is susceptible.", + "This is the equation governing things like compound interest, or the early stages of population growth before the effects of limited resources kick in, or the early stages of an epidemic while most of the population is susceptible.", 916.76, 929.02 ], diff --git a/2021/matrix-exponents/english/transcript.txt b/2021/matrix-exponents/english/transcript.txt index 7e2d282cf..2848f5db2 100644 --- a/2021/matrix-exponents/english/transcript.txt +++ b/2021/matrix-exponents/english/transcript.txt @@ -31,7 +31,7 @@ Likewise, it always makes sense to add together two matrices, this is something The astute among you might ask how sensible it is to take this out to infinity, which would be a great question, one that I'm largely going to postpone the answer to, but I can show you one pretty fun example here now. Take this 2x2 matrix that has negative pi and pi sitting off its diagonal entries. Let's see what the sum gives. -The first term is the identity matrix, this is what we mean by definition when we raise a matrix to the 0th power. +The first term is the identity matrix, this is actually what we mean by definition when we raise a matrix to the zeroth power. Then we add the matrix itself, which gives us the pi off the diagonal terms, and then add half of the matrix squared, and continuing on I'll have the computer keep adding more and more terms, each of which requires taking one more matrix product to get the new power, and then adding it to a running tally. And as it keeps going, it seems to be approaching a stable value, which is around negative one times the identity matrix. In this sense, we say the infinite sum equals that negative identity. @@ -49,7 +49,7 @@ They start by chewing on specific problems, and then generalizing those problems As to what sorts of specific examples might motivate matrix exponents, two come to mind. One involving relationships, and the other quantum mechanics. Let's start with relationships. -Let's call two lovers, Romeo and Juliet, and let x represent Juliet's love for Romeo, and y represent his love for her, both of which are values that change with time. +Suppose we have two lovers, let's call them Romeo and Juliet, and let's let x represent Juliet's love for Romeo, and y represent his love for her, both of which are going to be values that change with time. This is an example we actually touched on in chapter 1, based on a Steven Strogatz article, but it's okay if you didn't see that. The way their relationship works is that the rate at Juliet's love for Romeo changes, the derivative of this value, is equal to negative one times Romeo's love for her. So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade. @@ -73,7 +73,7 @@ You might picture the rate of change of this state, the thing that packages toge Remember, the rule here is that the rate of change of x is negative y, and the rate of change of y is x. Set up as vectors like this, we could rewrite the right hand side of this equation as a product of this matrix with the original vector xy. The top row encodes Juliet's rule, and the bottom row encodes Romeo's rule. -So what we have here is a vector is equal to a certain matrix times itself. +So what we have here is a differential equation telling us that the rate of change of some vector is equal to a certain matrix times itself. In a moment we'll talk about how matrix exponentiation solves this kind of equation, but before that let me show you a simpler way that we can solve this particular system, one that uses pure geometry, and it helps set the stage for visualizing matrix exponents a bit later. This matrix from our system is a 90 degree rotation matrix. For any of you rusty on how to think about matrices as transformations, there's a video all about it on this channel, a series really. @@ -107,7 +107,7 @@ Most people are more comfortable visualizing this with a graph, where the higher Just keep in mind that when we get to higher dimensional variance, graphs are a lot less helpful. This is a highly important equation in its own right. It's a very powerful concept when the rate of change of a value is proportional to the value itself. -This is the equation governing compound interest, or the early stages of population growth before the effects of limited resources kick in, or the early stages of an epidemic while most of the population is susceptible. +This is the equation governing things like compound interest, or the early stages of population growth before the effects of limited resources kick in, or the early stages of an epidemic while most of the population is susceptible. Calculus students all learn about how the derivative of e to the rt is r times itself. In other words, this self-reinforcing growth phenomenon is the same thing as exponential growth, and e to the rt solves this equation. Actually, a better way to think about it is that there are many different solutions to this equation, one for each initial condition, something like an initial investment size or an initial population, which we'll just call x0. diff --git a/2021/matrix-exponents/french/sentence_translations.json b/2021/matrix-exponents/french/sentence_translations.json index 0e962ebbd..c24da372f 100644 --- a/2021/matrix-exponents/french/sentence_translations.json +++ b/2021/matrix-exponents/french/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "En d’autres termes, lorsque Roméo exprime un désintérêt froid, c’est à ce moment-là que les sentiments de Juliette augmentent, tandis que s’il devient trop amoureux, son intérêt commencera à s’estomper.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "L'équation indique que la vitesse à laquelle ce vecteur d'état ressemble à une certaine matrice se multiplie par elle-même.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "De plus, si le temps le permet, il pourrait être amusant de parler de ce que signifie élever e à la puissance de l'opérateur dérivé.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/german/sentence_translations.json b/2021/matrix-exponents/german/sentence_translations.json index 2550afe47..578e4f533 100644 --- a/2021/matrix-exponents/german/sentence_translations.json +++ b/2021/matrix-exponents/german/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "Mit anderen Worten: Wenn Romeo kühles Desinteresse zum Ausdruck bringt, verstärken sich Julias Gefühle, wohingegen ihr Interesse nachlässt, wenn er sich zu sehr in ihn verliebt.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "Die Gleichung besagt, dass die Rate, mit der dieser Zustandsvektor wie eine bestimmte Matrix aussieht, sich selbst multipliziert.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "Wenn es die Zeit erlaubt, könnte es auch Spaß machen, darüber zu sprechen, was es bedeutet, e mit dem Ableitungsoperator zu potenzieren.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/hebrew/sentence_translations.json b/2021/matrix-exponents/hebrew/sentence_translations.json index 97ab8ac51..4166bd420 100644 --- a/2021/matrix-exponents/hebrew/sentence_translations.json +++ b/2021/matrix-exponents/hebrew/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "במילים אחרות, כאשר רומיאו מביע חוסר עניין קריר, אז הרגשות של ג'ולייט מתגברים, בעוד שאם הוא יתאהב מדי, העניין שלה יתחיל לדעוך.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "המשוואה אומרת שהקצב שבו וקטור המצב הזה נראה כמו מטריצה מסוימת כפול את עצמו.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "כמו כן, אם יאפשר הזמן, אולי יהיה כיף לדבר על מה זה אומר להעלות את e לכוחו של האופרטור הנגזרת.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/hindi/sentence_translations.json b/2021/matrix-exponents/hindi/sentence_translations.json index 4dddc47d5..76c33f82b 100644 --- a/2021/matrix-exponents/hindi/sentence_translations.json +++ b/2021/matrix-exponents/hindi/sentence_translations.json @@ -371,7 +371,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "दूसरे शब्दों में, जब रोमियो शांत उदासीनता व्यक्त कर रहा है, तभी जूलियट की भावनाएँ बढ़ जाती हैं, जबकि यदि वह बहुत अधिक मोहग्रस्त हो जाता है, तो उसकी रुचि कम होने लगेगी।", "n_reviews": 0, "start": 434.56, @@ -700,7 +700,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "समीकरण कहता है कि जिस दर पर यह राज्य वेक्टर एक निश्चित मैट्रिक्स जैसा दिखता है वह अपने आप से कई गुना अधिक है।", "n_reviews": 0, "start": 857.84, @@ -1253,7 +1253,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "इसके अलावा, यदि समय मिले, तो इस बारे में बात करना मजेदार हो सकता है कि डेरिवेटिव ऑपरेटर की शक्ति को बढ़ाने का क्या मतलब है।", "n_reviews": 0, "start": 1596.06, diff --git a/2021/matrix-exponents/hungarian/sentence_translations.json b/2021/matrix-exponents/hungarian/sentence_translations.json index 60985254b..04aeb3945 100644 --- a/2021/matrix-exponents/hungarian/sentence_translations.json +++ b/2021/matrix-exponents/hungarian/sentence_translations.json @@ -264,7 +264,7 @@ "end": 266.64 }, { - "input": "The first term is the identity matrix, this is what we mean by definition when we raise a matrix to the 0th power.", + "input": "The first term is the identity matrix, this is actually what we mean by definition when we raise a matrix to the zeroth power.", "translatedText": "Az első kifejezés az azonossági mátrix, ez az, amit a definíció szerint értünk, amikor egy mátrixot a 0. hatványra emelünk.", "model": "DeepL", "n_reviews": 0, @@ -408,7 +408,7 @@ "end": 399.24 }, { - "input": "Let's call two lovers, Romeo and Juliet, and let x represent Juliet's love for Romeo, and y represent his love for her, both of which are values that change with time.", + "input": "Suppose we have two lovers, let's call them Romeo and Juliet, and let's let x represent Juliet's love for Romeo, and y represent his love for her, both of which are going to be values that change with time.", "translatedText": "Nevezzünk két szerelmespárt Rómeónak és Júliának, és legyen x Júlia Rómeó iránti szerelmét, y pedig a Rómeó iránti szerelmét jelképezi, mindkettő olyan érték, amely az idővel változik.", "model": "DeepL", "n_reviews": 0, @@ -600,7 +600,7 @@ "end": 606.94 }, { - "input": "So what we have here is a vector is equal to a certain matrix times itself.", + "input": "So what we have here is a differential equation telling us that the rate of change of some vector is equal to a certain matrix times itself.", "translatedText": "Tehát itt egy vektor egyenlő egy bizonyos mátrix szorozva önmagával.", "model": "DeepL", "n_reviews": 0, @@ -872,7 +872,7 @@ "end": 916.3 }, { - "input": "This is the equation governing compound interest, or the early stages of population growth before the effects of limited resources kick in, or the early stages of an epidemic while most of the population is susceptible.", + "input": "This is the equation governing things like compound interest, or the early stages of population growth before the effects of limited resources kick in, or the early stages of an epidemic while most of the population is susceptible.", "translatedText": "Ez az egyenlet határozza meg a kamatos kamatot, vagy a népességnövekedés korai szakaszát, mielőtt a korlátozott erőforrások hatása beindul, vagy egy járvány korai szakaszát, amíg a népesség nagy része fogékony.", "model": "DeepL", "n_reviews": 0, diff --git a/2021/matrix-exponents/indonesian/sentence_translations.json b/2021/matrix-exponents/indonesian/sentence_translations.json index 05fb226d4..f48247230 100644 --- a/2021/matrix-exponents/indonesian/sentence_translations.json +++ b/2021/matrix-exponents/indonesian/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "Dengan kata lain, ketika Romeo menunjukkan ketidaktertarikannya, saat itulah perasaan Juliet meningkat, sedangkan jika dia terlalu tergila-gila, minatnya akan mulai memudar.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "Persamaannya mengatakan bahwa laju vektor keadaan ini terlihat seperti matriks tertentu dikalikan dengan dirinya sendiri.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "Selain itu, jika waktu mengizinkan, mungkin akan menyenangkan untuk membicarakan apa artinya menaikkan e ke pangkat operator turunan.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/italian/sentence_translations.json b/2021/matrix-exponents/italian/sentence_translations.json index 2d9fb77e3..e7dc63fb0 100644 --- a/2021/matrix-exponents/italian/sentence_translations.json +++ b/2021/matrix-exponents/italian/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "In altre parole, quando Romeo esprime un freddo disinteresse, è allora che i sentimenti di Giulietta aumentano, mentre se diventa troppo infatuato, il suo interesse inizierà a svanire.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "L'equazione dice che la velocità con cui questo vettore di stato assomiglia a una determinata matrice si moltiplica.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "Inoltre, se il tempo lo permette, potrebbe essere divertente parlare di cosa significhi elevare e alla potenza dell'operatore derivato.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/japanese/sentence_translations.json b/2021/matrix-exponents/japanese/sentence_translations.json index d55b8f6e0..88925c0fd 100644 --- a/2021/matrix-exponents/japanese/sentence_translations.json +++ b/2021/matrix-exponents/japanese/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "言い換えれば、ロミオが冷静に無関心を示 しているときは、ジュリエットの感情が高まるときですが、 ロミオが夢中になりすぎると、彼女の興味は薄れ始めます。", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "この方程式は、この状態ベクトルが特定の行列のように見える割合にそれ 自体を掛けたものであることを示しています。", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "また、時間が許せば、 e を微分演算子の累乗することが 何を意味するかについて話すのも楽しいかもしれません。", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/korean/sentence_translations.json b/2021/matrix-exponents/korean/sentence_translations.json index 51337008e..b853175f8 100644 --- a/2021/matrix-exponents/korean/sentence_translations.json +++ b/2021/matrix-exponents/korean/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade. ", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade. ", "translatedText": "즉, 로미오가 쿨하게 무관심을 표현하면 줄리엣의 감정이 커지는 반면, 줄리엣이 너무 푹 빠지면 줄리엣의 관심은 시들기 시작한다는 것이다. ", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. ", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you. ", "translatedText": "또한, 시간이 허락한다면 e를 도함수 연산자의 거듭제곱으로 올리는 것이 무엇을 의미하는지 이야기하는 것도 재미있을 것입니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/marathi/sentence_translations.json b/2021/matrix-exponents/marathi/sentence_translations.json index 36e9318b3..bed3dc50a 100644 --- a/2021/matrix-exponents/marathi/sentence_translations.json +++ b/2021/matrix-exponents/marathi/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "दुसऱ्या शब्दांत सांगायचे तर, जेव्हा रोमियो शांत अनास्था व्यक्त करतो, तेव्हा ज्युलिएटच्या भावना वाढतात, आणि जर तो खूप मोहित झाला तर तिची आवड कमी होऊ लागते.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "हे समीकरण म्हणते की ज्या दराने हा राज्य वेक्टर एक विशिष्ट मॅट्रिक्स वेळा स्वतःसारखा दिसतो.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "तसेच, वेळेनुसार, डेरिव्हेटिव्ह ऑपरेटरच्या सामर्थ्यामध्ये e वाढवण्याचा अर्थ काय आहे याबद्दल बोलणे मजेदार असू शकते.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/persian/sentence_translations.json b/2021/matrix-exponents/persian/sentence_translations.json index c24b276ef..adf4dc782 100644 --- a/2021/matrix-exponents/persian/sentence_translations.json +++ b/2021/matrix-exponents/persian/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade. ", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade. ", "translatedText": "به عبارت دیگر، زمانی که رومئو بی‌علاقه‌ای را ابراز می‌کند، در آن زمان است که احساسات ژولیت افزایش می‌یابد، در حالی که اگر او بیش از حد شیفته شود، علاقه او کم‌رنگ می‌شود. ", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. ", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you. ", "translatedText": "همچنین، اگر زمان اجازه دهد، ممکن است صحبت در مورد معنای افزایش e به توان عملگر مشتق جالب باشد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/portuguese/sentence_translations.json b/2021/matrix-exponents/portuguese/sentence_translations.json index e647220fd..2369ed15e 100644 --- a/2021/matrix-exponents/portuguese/sentence_translations.json +++ b/2021/matrix-exponents/portuguese/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "Em outras palavras, quando Romeu expressa um desinteresse frio, é quando os sentimentos de Julieta aumentam, ao passo que, se ele ficar muito apaixonado, o interesse dela começará a desaparecer.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "A equação diz que a taxa na qual esse vetor de estado se parece com uma determinada matriz se multiplica.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "Além disso, se o tempo permitir, pode ser divertido falar sobre o que significa elevar e à potência do operador derivado.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/russian/sentence_translations.json b/2021/matrix-exponents/russian/sentence_translations.json index a264c3939..245866688 100644 --- a/2021/matrix-exponents/russian/sentence_translations.json +++ b/2021/matrix-exponents/russian/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "Другими словами, когда Ромео выражает хладнокровную незаинтересованность, чувства Джульетты усиливаются, тогда как, если он слишком увлечется, ее интерес начнет угасать.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "Уравнение говорит, что скорость, с которой этот вектор состояния выглядит как определенная матрица, умножается на себя.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "Кроме того, если позволит время, было бы интересно поговорить о том, что значит возвести e в степень оператора производной.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/spanish/sentence_translations.json b/2021/matrix-exponents/spanish/sentence_translations.json index e511bfc8d..88ddf10f4 100644 --- a/2021/matrix-exponents/spanish/sentence_translations.json +++ b/2021/matrix-exponents/spanish/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "En otras palabras, cuando Romeo expresa un frío desinterés, es cuando los sentimientos de Julieta aumentan, mientras que si él se enamora demasiado, su interés comenzará a desvanecerse.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "La ecuación dice que la velocidad a la que este vector de estado se parece a una determinada matriz se multiplica por sí misma.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "Además, si el tiempo lo permite, podría ser divertido hablar sobre lo que significa elevar e a la potencia del operador derivativo.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/tamil/sentence_translations.json b/2021/matrix-exponents/tamil/sentence_translations.json index ef041e3b1..31371720d 100644 --- a/2021/matrix-exponents/tamil/sentence_translations.json +++ b/2021/matrix-exponents/tamil/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade. ", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade. ", "translatedText": "வேறு வார்த்தைகளில் கூறுவதானால், ரோமியோ குளிர்ச்சியான ஆர்வமின்மையை வெளிப்படுத்தும் போது, ஜூலியட்டின் உணர்வுகள் அதிகரிக்கின்றன, அதேசமயம் அவன் மிகவும் மோகம் கொண்டால், அவளது ஆர்வம் மங்கத் தொடங்கும். ", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. ", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you. ", "translatedText": "மேலும், நேரம் அனுமதித்தால், டெரிவேட்டிவ் ஆபரேட்டரின் சக்திக்கு e ஐ உயர்த்துவது என்றால் என்ன என்பதைப் பற்றி பேசுவது வேடிக்கையாக இருக்கலாம். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/telugu/sentence_translations.json b/2021/matrix-exponents/telugu/sentence_translations.json index 8c5a08001..42e1bb7a3 100644 --- a/2021/matrix-exponents/telugu/sentence_translations.json +++ b/2021/matrix-exponents/telugu/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "మరో మాటలో చెప్పాలంటే, రోమియో నిరాసక్తతను వ్యక్తం చేస్తున్నప్పుడు, జూలియట్ యొక్క భావాలు పెరుగుతాయి, అయితే అతను చాలా మోహానికి గురైనట్లయితే, ఆమె ఆసక్తి క్షీణించడం ప్రారంభమవుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "ఈ స్థితి వెక్టార్ నిర్దిష్ట మాతృక రెట్లు కనిపించే రేటు అని సమీకరణం చెబుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "అలాగే, సమయం అనుమతిస్తూ, డెరివేటివ్ ఆపరేటర్ యొక్క శక్తికి eని పెంచడం అంటే ఏమిటో మాట్లాడటం సరదాగా ఉండవచ్చు.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/thai/sentence_translations.json b/2021/matrix-exponents/thai/sentence_translations.json index 9dd4bc110..fb2e24d2f 100644 --- a/2021/matrix-exponents/thai/sentence_translations.json +++ b/2021/matrix-exponents/thai/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade. ", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. ", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/turkish/sentence_translations.json b/2021/matrix-exponents/turkish/sentence_translations.json index cfe28c888..95bbfc544 100644 --- a/2021/matrix-exponents/turkish/sentence_translations.json +++ b/2021/matrix-exponents/turkish/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "Başka bir deyişle, Romeo soğukkanlılıkla ilgisizliğini dile getirdiğinde Juliet'in duyguları artar, oysa aşırı aşık olursa Juliet'in ilgisi azalmaya başlar.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "Denklem, bu durum vektörünün belirli bir matrise benzeme hızının kendisiyle çarpımını söylüyor.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "Ayrıca, zaman kalırsa, e'nin türev operatörünün kuvvetinin ne anlama geldiği hakkında konuşmak eğlenceli olabilir.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/ukrainian/sentence_translations.json b/2021/matrix-exponents/ukrainian/sentence_translations.json index 46d3cf526..31566c324 100644 --- a/2021/matrix-exponents/ukrainian/sentence_translations.json +++ b/2021/matrix-exponents/ukrainian/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "Іншими словами, коли Ромео висловлює холодну незацікавленість, тоді почуття Джульєтти посилюються, тоді як якщо він стає занадто закоханим, її інтерес почне згасати.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "Рівняння говорить, що швидкість, з якою цей вектор стану виглядає як певна матриця, збільшується в рази сама.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "Також, якщо дозволить час, було б цікаво поговорити про те, що означає звести e до степеня оператора похідної.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/urdu/sentence_translations.json b/2021/matrix-exponents/urdu/sentence_translations.json index 975f962cf..4424f1790 100644 --- a/2021/matrix-exponents/urdu/sentence_translations.json +++ b/2021/matrix-exponents/urdu/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade. ", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade. ", "translatedText": "دوسرے لفظوں میں، جب رومیو ٹھنڈی عدم دلچسپی کا اظہار کر رہا ہوتا ہے، اسی وقت جولیٹ کے جذبات میں اضافہ ہوتا ہے، جب کہ اگر وہ بہت زیادہ متاثر ہو جاتا ہے، تو اس کی دلچسپی ختم ہونا شروع ہو جاتی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. ", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you. ", "translatedText": "اس کے علاوہ، وقت کی اجازت دیتے ہوئے، یہ بات کرنے میں مزہ آتا ہے کہ ڈیریویٹیو آپریٹر کی طاقت میں ای کو بڑھانے کا کیا مطلب ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/matrix-exponents/vietnamese/sentence_translations.json b/2021/matrix-exponents/vietnamese/sentence_translations.json index 9b52b1046..ea8da8cfa 100644 --- a/2021/matrix-exponents/vietnamese/sentence_translations.json +++ b/2021/matrix-exponents/vietnamese/sentence_translations.json @@ -424,7 +424,7 @@ "end": 433.78 }, { - "input": "In other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings increase, whereas if he becomes too infatuated, her interest will start to fade.", + "input": "So in other words, when Romeo is expressing cool disinterest, that's when Juliet's feelings actually increase, whereas if he becomes too infatuated, her interest will start to fade.", "translatedText": "Nói cách khác, khi Romeo thể hiện sự thờ ơ lạnh lùng, đó là lúc tình cảm của Juliet tăng lên, ngược lại nếu anh trở nên quá say mê, sự quan tâm của cô sẽ bắt đầu phai nhạt.", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 856.9 }, { - "input": "The equation says that the rate at which this state vector looks like a certain matrix times itself.", + "input": "The equation says that the rate at which this state vector changes looks like a certain matrix times itself.", "translatedText": "Phương trình cho biết tốc độ mà vectơ trạng thái này trông giống như một ma trận nhất định nhân với chính nó.", "model": "google_nmt", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1594.78 }, { - "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.", + "input": "Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator. Thank you.", "translatedText": "Ngoài ra, nếu thời gian cho phép, có thể sẽ thú vị khi nói về ý nghĩa của việc nâng e lên lũy thừa của toán tử đạo hàm.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/arabic/sentence_translations.json b/2021/newtons-fractal/arabic/sentence_translations.json index a2396553e..ac826cf4a 100644 --- a/2021/newtons-fractal/arabic/sentence_translations.json +++ b/2021/newtons-fractal/arabic/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", "translatedText": "كملاحظة جانبية ممتعة، تتعلق مرة أخرى بإيجاد الجذر في رسومات الكمبيوتر، طلبت من أحد مهندسي شركة Pixar أن يعطيني التقدير الذي يأخذ في الاعتبار عدد الأضواء المستخدمة في بعض مشاهد فيلم Coco، وبالنظر إلى طبيعة بعض هذه الأضواء عند إجراء حسابات لكل بكسل عندما يتعلق الأمر بأشياء محددة متعددة الحدود مثل المجالات، تم استخدام الصيغة التربيعية بسهولة عدة تريليونات من المرات في إنتاج هذا الفيلم. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process. ", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process. ", "translatedText": "للحصول على القليل من الحدس، وجدت أنه من المفيد أن أشاهد ببساطة مجموعة مثل تلك التي تظهر على الشاشة وهي تخضع لهذه العملية. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/bengali/sentence_translations.json b/2021/newtons-fractal/bengali/sentence_translations.json index 8bc064157..b8f5cb840 100644 --- a/2021/newtons-fractal/bengali/sentence_translations.json +++ b/2021/newtons-fractal/bengali/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", "translatedText": "একটি মজার সাইড নোট হিসাবে, আবার কম্পিউটার গ্রাফিক্সে রুট ফাইন্ডিং এর সাথে প্রাসঙ্গিক, আমি একবার পিক্সারের একজন প্রকৌশলী আমাকে অনুমান দিয়েছিলাম যে কোকো সিনেমার কিছু দৃশ্যে কতগুলি লাইট ব্যবহার করা হয়েছে এবং এর মধ্যে কয়েকটির প্রকৃতি দেওয়া হয়েছে।প্রতি-পিক্সেল গণনা যখন বহুপদীভাবে সংজ্ঞায়িত জিনিসগুলি যেমন গোলক জড়িত থাকে, সেই ফিল্মটির নির্মাণে দ্বিঘাত সূত্রটি সহজেই একাধিক ট্রিলিয়ন বার ব্যবহার করা হয়েছিল।", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process. ", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process. ", "translatedText": "সামান্য অন্তর্দৃষ্টির জন্য, আমি এই প্রক্রিয়ার মধ্য দিয়ে স্ক্রিনে থাকা একটি ক্লাস্টারের মতো একটি ক্লাস্টার দেখার জন্য এটি জ্ঞানদায়ক বলে মনে করেছি।", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/chinese/sentence_translations.json b/2021/newtons-fractal/chinese/sentence_translations.json index 8eac9a65f..712790d91 100644 --- a/2021/newtons-fractal/chinese/sentence_translations.json +++ b/2021/newtons-fractal/chinese/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", "translatedText": "作为一个有趣的旁注，再次与计算机图形学中的寻根相关， 我曾经让一位皮克斯工程师给了我一个估计，考虑到电影《 寻梦环游记》的某些场景中使用了多少灯光，并考虑到其中 一些灯光的性质当涉及球体等多项式定义的物体时，每像素 计算时，二次公式在该电影的制作中很容易使用数万亿次。", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process. ", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process. ", "translatedText": "出于一点直觉，我发现简单地观察像屏幕上 的集群那样经历这一过程是很有启发的。", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/english/captions.srt b/2021/newtons-fractal/english/captions.srt index 35371fb6e..a51edfd3b 100644 --- a/2021/newtons-fractal/english/captions.srt +++ b/2021/newtons-fractal/english/captions.srt @@ -867,16 +867,16 @@ be enough, because the finer details of the shape we get go on with endless comp But take a moment to think about what this is actually saying. 218 -00:13:04,580 --> 00:13:08,800 -It means that there are regions in the complex plane where if you slightly +00:13:04,580 --> 00:13:08,963 +It means that there are regions in the complex plane where if you slightly adjust that 219 -00:13:08,800 --> 00:13:13,302 -adjust that seed value, bump it to the side by 1,1 millionth or 1,1 trillionth, +00:13:08,963 --> 00:13:13,145 +seed value, you know, you just kind of bump it to the side by 1,1 millionth or 1,1 220 -00:13:13,302 --> 00:13:17,580 -it can completely change which of the five true roots it ends up landing on. +00:13:13,145 --> 00:13:17,580 +trillionth, it can completely change which of the five true roots it ends up landing on. 221 00:13:18,400 --> 00:13:22,832 diff --git a/2021/newtons-fractal/english/sentence_timings.json b/2021/newtons-fractal/english/sentence_timings.json index a8f3310ac..1d6491a86 100644 --- a/2021/newtons-fractal/english/sentence_timings.json +++ b/2021/newtons-fractal/english/sentence_timings.json @@ -480,7 +480,7 @@ 784.58 ], [ - "It means that there are regions in the complex plane where if you slightly adjust that seed value, bump it to the side by 1,1 millionth or 1,1 trillionth, it can completely change which of the five true roots it ends up landing on.", + "It means that there are regions in the complex plane where if you slightly adjust that seed value, you know, you just kind of bump it to the side by 1,1 millionth or 1,1 trillionth, it can completely change which of the five true roots it ends up landing on.", 784.58, 797.58 ], diff --git a/2021/newtons-fractal/english/transcript.txt b/2021/newtons-fractal/english/transcript.txt index 262157a09..d69ed3205 100644 --- a/2021/newtons-fractal/english/transcript.txt +++ b/2021/newtons-fractal/english/transcript.txt @@ -94,7 +94,7 @@ If we did this process for every single pixel on the plane, here's what you woul And at this level of detail the color scheme is a little jarring to my eye at least, so let me calm it down a little. Really whatever resolution I try to use to show this to you here could never possibly be enough, because the finer details of the shape we get go on with endless complexity. But take a moment to think about what this is actually saying. -It means that there are regions in the complex plane where if you slightly adjust that seed value, bump it to the side by 1,1 millionth or 1,1 trillionth, it can completely change which of the five true roots it ends up landing on. +It means that there are regions in the complex plane where if you slightly adjust that seed value, you know, you just kind of bump it to the side by 1,1 millionth or 1,1 trillionth, it can completely change which of the five true roots it ends up landing on. We saw some foreshadowing of this kind of chaos with the real graph and the problematic guess shown earlier, but picturing all of this in the complex plane really shines a light on just how unpredictable this kind of root finding algorithm can be, and how there are whole swaths of initial values where this sort of unpredictability will take place. Now if I grab one of these roots and change it around, meaning that we're using a different polynomial for the process, you can see how the resulting fractal pattern changes. And notice for example how the regions around a given root always have the same color, since those are the points that are close enough to the root where this linear approximation scheme works as a way of finding that root with no problem. diff --git a/2021/newtons-fractal/french/sentence_translations.json b/2021/newtons-fractal/french/sentence_translations.json index 61d165b7e..01b67843d 100644 --- a/2021/newtons-fractal/french/sentence_translations.json +++ b/2021/newtons-fractal/french/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", "translatedText": "En guise de remarque amusante, encore une fois pertinente pour la recherche de racines en infographie, un ingénieur de Pixar m'a donné une fois l'estimation que, compte tenu du nombre de lumières utilisées dans certaines scènes du film Coco, et compte tenu de la nature de certaines d'entre elles, calculs par pixel lorsque des éléments définis de manière polynomiale comme des sphères sont impliqués, la formule quadratique a été facilement utilisée plusieurs milliards de fois dans la production de ce film.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process.", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process.", "translatedText": "Pour un peu d'intuition, j'ai trouvé instructif de simplement regarder un cluster comme celui à l'écran subir ce processus.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/german/sentence_translations.json b/2021/newtons-fractal/german/sentence_translations.json index 25b8ba517..6236b5c41 100644 --- a/2021/newtons-fractal/german/sentence_translations.json +++ b/2021/newtons-fractal/german/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", "translatedText": "Als lustige Randbemerkung, die wiederum für die Nullstellenfindung in der Computergrafik relevant ist, ich ließ mir einmal von einem Pixar-Ingenieur eine Schätzung dafür geben, wie viele Lichter in einigen Szenen für den Film „Coco“ verwendet wurden und mit der Natur einiger Kalkulationen die pro pixel geschehen, wenn Objekte, die durch Polynome definiert werden involviert sind, würde die a-b-c-Formel.", "model": "google_nmt", "n_reviews": 1, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process.", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process.", "translatedText": "Für ein wenig Intivituität fand ich es aufschlussreich, einfach zu beobachten, wie ein Cluster wie der auf dem Bildschirm diesen Prozess durchläuft.", "model": "google_nmt", "n_reviews": 1, diff --git a/2021/newtons-fractal/hebrew/sentence_translations.json b/2021/newtons-fractal/hebrew/sentence_translations.json index 75659b6b6..001722c26 100644 --- a/2021/newtons-fractal/hebrew/sentence_translations.json +++ b/2021/newtons-fractal/hebrew/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", "translatedText": "כהערת צד מהנה, שוב רלוונטית למציאת שורשים בגרפיקה ממוחשבת, פעם היה לי מהנדס פיקסאר שנתן לי את ההערכה שבהתחשב בכמה אורות היו בשימוש בחלק מהסצנות של הסרט קוקו, ובהתחשב באופי של חלק מאלה חישובים לפיקסל כאשר מעורבים דברים המוגדרים פולינומית כמו כדורים, הנוסחה הריבועית שימשה בקלות מספר טריליוני פעמים בהפקת הסרט הזה. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process. ", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process. ", "translatedText": "בשביל קצת אינטואיציה, מצאתי את זה מאיר עיניים פשוט לראות אשכול כמו זה שעל המסך עובר את התהליך הזה. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/hindi/sentence_translations.json b/2021/newtons-fractal/hindi/sentence_translations.json index 4b9266d53..dac6c2265 100644 --- a/2021/newtons-fractal/hindi/sentence_translations.json +++ b/2021/newtons-fractal/hindi/sentence_translations.json @@ -252,7 +252,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", "translatedText": "एक मजेदार साइड नोट के रूप में, कंप्यूटर ग्राफिक्स में रूट खोजने के लिए फिर से प्रासंगिक, मैंने एक बार एक पिक्सर इंजीनियर से मुझे यह अनुमान लगाने के लिए कहा था कि फिल्म कोको के कुछ दृश्यों में कितनी रोशनी का उपयोग किया गया था, और इनमें से कुछ की प्रकृति को देखते हुए प्रति-पिक्सेल गणना में जब गोले जैसी बहुपद रूप से परिभाषित चीजें शामिल होती हैं, तो उस फिल्म के निर्माण में द्विघात सूत्र का आसानी से कई खरबों बार उपयोग किया गया था।", "n_reviews": 0, "start": 260.18, @@ -1113,7 +1113,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process.", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process.", "translatedText": "थोड़ी सी अंतर्ज्ञान के लिए, मुझे स्क्रीन पर मौजूद क्लस्टर जैसे क्लस्टर को इस प्रक्रिया से गुजरते हुए देखना ज्ञानवर्धक लगा।", "n_reviews": 0, "start": 1334.05, diff --git a/2021/newtons-fractal/hungarian/sentence_translations.json b/2021/newtons-fractal/hungarian/sentence_translations.json index ec2222214..9a62ae852 100644 --- a/2021/newtons-fractal/hungarian/sentence_translations.json +++ b/2021/newtons-fractal/hungarian/sentence_translations.json @@ -768,7 +768,7 @@ "end": 784.58 }, { - "input": "It means that there are regions in the complex plane where if you slightly adjust that seed value, bump it to the side by 1,1 millionth or 1,1 trillionth, it can completely change which of the five true roots it ends up landing on.", + "input": "It means that there are regions in the complex plane where if you slightly adjust that seed value, you know, you just kind of bump it to the side by 1,1 millionth or 1,1 trillionth, it can completely change which of the five true roots it ends up landing on.", "translatedText": "Ez azt jelenti, hogy vannak olyan területek a komplex síkban, ahol ha kissé módosítjuk a magértéket, 1,1 milliomoddal vagy 1,1 billiomoddal arrébb toljuk, az teljesen megváltoztathatja, hogy az öt valódi gyök közül melyikben landol.", "model": "DeepL", "n_reviews": 0, diff --git a/2021/newtons-fractal/indonesian/sentence_translations.json b/2021/newtons-fractal/indonesian/sentence_translations.json index c270dc777..01debe6ff 100644 --- a/2021/newtons-fractal/indonesian/sentence_translations.json +++ b/2021/newtons-fractal/indonesian/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", "translatedText": "Sebagai catatan tambahan yang menyenangkan, sekali lagi relevan dengan pencarian akar dalam grafik komputer, saya pernah meminta seorang insinyur Pixar memberi saya perkiraan mengingat berapa banyak cahaya yang digunakan dalam beberapa adegan untuk film Coco, dan mengingat sifat dari beberapa di antaranya. perhitungan per piksel ketika hal-hal yang didefinisikan secara polinomial seperti bola terlibat, rumus kuadrat dengan mudah digunakan triliunan kali dalam produksi film tersebut.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process.", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process.", "translatedText": "Untuk sedikit intuisi, saya merasa tercerahkan dengan melihat sekelompok seperti yang ada di layar menjalani proses ini.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/italian/sentence_translations.json b/2021/newtons-fractal/italian/sentence_translations.json index a84e0658e..90c31f26f 100644 --- a/2021/newtons-fractal/italian/sentence_translations.json +++ b/2021/newtons-fractal/italian/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", "translatedText": "Come nota a margine divertente, ancora una volta rilevante per la ricerca delle radici nella computer grafica, una volta ho chiesto a un ingegnere della Pixar di stimarmi che, considerando quante luci sono state utilizzate in alcune scene del film Coco, e data la natura di alcune di queste calcoli per pixel quando sono coinvolte cose definite polinomialmente come le sfere, la formula quadratica è stata facilmente utilizzata più trilioni di volte nella produzione di quel film.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process.", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process.", "translatedText": "Per un po' di intuizione, ho trovato illuminante osservare semplicemente un cluster come quello sullo schermo subire questo processo.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/japanese/sentence_translations.json b/2021/newtons-fractal/japanese/sentence_translations.json index 91013e7cd..a0b0a2846 100644 --- a/2021/newtons-fractal/japanese/sentence_translations.json +++ b/2021/newtons-fractal/japanese/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", "translatedText": "面白い余談ですが、これもコンピューター グラフィックスにおける根本原因の発 見に関係しますが、私はかつてピクサーのエンジニアに、映画『ココ』のいくつか のシーンで使用されたライトの数と、それらのいくつかの性質を考慮した見積も りを教えてもらいました。球などの多項式で定義されたものが関係する場合のピク セルごとの計算では、映画の制作で二次公式が簡単に何兆回も使用されました。", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process. ", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process. ", "translatedText": "少し直観的には、画面上のようなクラスターがこのプロセスを 経るのをただ観察するだけで啓発されることがわかりました。", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/korean/sentence_translations.json b/2021/newtons-fractal/korean/sentence_translations.json index 6eb4a6760..5af3da53f 100644 --- a/2021/newtons-fractal/korean/sentence_translations.json +++ b/2021/newtons-fractal/korean/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", "translatedText": "컴퓨터 그래픽의 뿌리 찾기와 관련된 재미있는 참고 사항으로, Pixar 엔지니어에게 영화 Coco의 일부 장면에서 사용된 조명의 수를 고려하고 이러한 일부 특성을 고려하여 추정치를 제시한 적이 있습니다. 구와 같이 다항식으로 정의된 것이 포함된 픽셀당 계산에서는 이차 공식이 해당 영화 제작에서 수조 번 쉽게 사용되었습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process. ", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process. ", "translatedText": "약간의 직관을 위해 화면에 나오는 것과 같은 클러스터가 이 과정을 겪는 것을 단순히 보는 것이 깨달음을 얻었습니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/marathi/sentence_translations.json b/2021/newtons-fractal/marathi/sentence_translations.json index 7c2a4a6e9..c2e22c32f 100644 --- a/2021/newtons-fractal/marathi/sentence_translations.json +++ b/2021/newtons-fractal/marathi/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", "translatedText": "एक मजेदार साइड टीप म्हणून, संगणक ग्राफिक्समध्ये रूट शोधण्याशी संबंधित, माझ्याकडे एकदा पिक्सार अभियंत्याने मला अंदाज द्यायचा की कोको चित्रपटाच्या काही दृश्यांमध्ये किती दिवे वापरण्यात आले होते आणि यापैकी काहींचे स्वरूप लक्षात घेऊन. प्रति-पिक्सेल गणनेमध्ये जेव्हा गोलाकार सारख्या बहुपदी परिभाषित गोष्टींचा समावेश असतो, तेव्हा त्या चित्रपटाच्या निर्मितीमध्ये चतुर्भुज सूत्र सहजपणे अनेक ट्रिलियन वेळा वापरला गेला.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process.", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process.", "translatedText": "थोड्या अंतर्ज्ञानासाठी, मला या प्रक्रियेतून स्क्रीनवरील क्लस्टरसारखे फक्त पाहणे ज्ञानदायक वाटले.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/persian/sentence_translations.json b/2021/newtons-fractal/persian/sentence_translations.json index 22ca5c0b5..109c1bf99 100644 --- a/2021/newtons-fractal/persian/sentence_translations.json +++ b/2021/newtons-fractal/persian/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", "translatedText": "به عنوان یک نکته جالب، که باز هم مربوط به ریشه یابی در گرافیک کامپیوتری است، یک بار از مهندس پیکسار خواستم که تخمین بزنم که با توجه به تعداد نورهای استفاده شده در برخی از صحنه های فیلم کوکو و با توجه به ماهیت برخی از آنها. محاسبات در هر پیکسل زمانی که چیزهای چند جمله ای تعریف شده مانند کره درگیر می شوند، فرمول درجه دوم به راحتی چندین تریلیون بار در تولید آن فیلم استفاده شد. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process. ", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process. ", "translatedText": "برای کمی شهود، به نظرم روشن‌کننده بود که به سادگی تماشای خوشه‌ای مانند آنچه روی صفحه است تحت این فرآیند قرار می‌گیرد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/portuguese/sentence_translations.json b/2021/newtons-fractal/portuguese/sentence_translations.json index d5093c231..eb91f1708 100644 --- a/2021/newtons-fractal/portuguese/sentence_translations.json +++ b/2021/newtons-fractal/portuguese/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", "translatedText": "Como uma observação divertida, mais uma vez relevante para a descoberta de raízes em computação gráfica, uma vez um engenheiro da Pixar me deu a estimativa de que, considerando quantas luzes foram usadas em algumas das cenas do filme Coco, e dada a natureza de algumas delas cálculos por pixel quando coisas definidas polinomialmente, como esferas, estão envolvidas, a fórmula quadrática foi facilmente usada vários trilhões de vezes na produção daquele filme.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process.", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process.", "translatedText": "Para um pouco de intuição, achei esclarecedor simplesmente observar um cluster como o da tela passar por esse processo.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/russian/sentence_translations.json b/2021/newtons-fractal/russian/sentence_translations.json index 6c848595b..09d21e15a 100644 --- a/2021/newtons-fractal/russian/sentence_translations.json +++ b/2021/newtons-fractal/russian/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", "translatedText": "В качестве забавного примечания, снова имеющего отношение к поиску корней в компьютерной графике, однажды инженер Pixar дал мне оценку, учитывая, сколько источников света было использовано в некоторых сценах фильма «Коко», и характер некоторых из них. Попиксельные вычисления, когда задействованы полиномиально определенные объекты, такие как сферы, квадратичная формула легко использовалась несколько триллионов раз при производстве этого фильма.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process.", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process.", "translatedText": "Проявив немного интуиции, я нашел полезным просто наблюдать за тем, как кластер, подобный показанному на экране, подвергается этому процессу.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/spanish/sentence_translations.json b/2021/newtons-fractal/spanish/sentence_translations.json index 6622cb852..5619f7c39 100644 --- a/2021/newtons-fractal/spanish/sentence_translations.json +++ b/2021/newtons-fractal/spanish/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", "translatedText": "Como nota al margen divertida, nuevamente relevante para la búsqueda de raíces en gráficos por computadora, una vez un ingeniero de Pixar me dio la estimación de que, considerando cuántas luces se usaron en algunas de las escenas de la película Coco, y dada la naturaleza de algunas de estas, En los cálculos por píxel, cuando se trata de cosas definidas polinómicamente, como esferas, la fórmula cuadrática se utilizó fácilmente varios billones de veces en la producción de esa película.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process.", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process.", "translatedText": "Para tener un poco de intuición, me resultó esclarecedor simplemente observar cómo un grupo como el que aparece en la pantalla experimenta este proceso.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/tamil/sentence_translations.json b/2021/newtons-fractal/tamil/sentence_translations.json index b3f04a84f..a2530919b 100644 --- a/2021/newtons-fractal/tamil/sentence_translations.json +++ b/2021/newtons-fractal/tamil/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", "translatedText": "ஒரு வேடிக்கையான பக்கக் குறிப்பு, கணினி கிராபிக்ஸில் ரூட் கண்டுபிடிப்புக்கு மீண்டும் பொருத்தமானது, கோகோ திரைப்படத்தின் சில காட்சிகளில் எத்தனை விளக்குகள் பயன்படுத்தப்பட்டன என்பதைக் கருத்தில் கொண்டு, அவற்றில் சிலவற்றின் தன்மையைக் கருத்தில் கொண்டு, ஒருமுறை பிக்சர் பொறியாளர் என்னிடம் மதிப்பீடு செய்தார். ஒரு பிக்சல் கணக்கீடுகள், கோளங்கள் போன்ற பல்லுறுப்புக்கோவையில் வரையறுக்கப்பட்ட விஷயங்கள் சம்பந்தப்பட்டிருக்கும் போது, அந்த படத்தின் தயாரிப்பில் இருபடி சூத்திரம் பல டிரில்லியன் முறைகள் எளிதாகப் பயன்படுத்தப்பட்டது.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process.", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process.", "translatedText": "ஒரு சிறிய உள்ளுணர்வுக்கு, திரையில் உள்ளதைப் போன்ற ஒரு கிளஸ்டரை இந்த செயல்முறைக்கு உட்படுத்துவதைப் பார்ப்பது அறிவூட்டுவதாகக் கண்டேன்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/telugu/sentence_translations.json b/2021/newtons-fractal/telugu/sentence_translations.json index 326369e18..12dce3e05 100644 --- a/2021/newtons-fractal/telugu/sentence_translations.json +++ b/2021/newtons-fractal/telugu/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", "translatedText": "ఫన్ సైడ్ నోట్‌గా, కంప్యూటర్ గ్రాఫిక్స్‌లో రూట్ ఫైండింగ్‌కు సంబంధించి మళ్లీ, నేను ఒకసారి పిక్సర్ ఇంజనీర్‌ని కలిగి ఉన్నాను, కోకో సినిమా కోసం కొన్ని సన్నివేశాలలో ఎన్ని లైట్లు ఉపయోగించబడ్డాయో మరియు వాటిలో కొన్నింటి స్వభావాన్ని బట్టి అంచనా వేసాను. ప్రతి-పిక్సెల్ గణనలు, గోళాల వంటి బహుపదాలుగా నిర్వచించబడిన అంశాలు చేరి ఉన్నప్పుడు, క్వాడ్రాటిక్ ఫార్ములా ఆ ఫిల్మ్ నిర్మాణంలో చాలా ట్రిలియన్‌ల సార్లు సులభంగా ఉపయోగించబడింది.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process.", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process.", "translatedText": "కొంచెం అంతర్ దృష్టి కోసం, స్క్రీన్‌పై ఉన్నటువంటి క్లస్టర్‌ని ఈ ప్రక్రియలో చూడటం నాకు జ్ఞానోదయం కలిగించిందని నేను కనుగొన్నాను.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/thai/sentence_translations.json b/2021/newtons-fractal/thai/sentence_translations.json index 008b5b4b0..7e75c8712 100644 --- a/2021/newtons-fractal/thai/sentence_translations.json +++ b/2021/newtons-fractal/thai/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process. ", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/turkish/sentence_translations.json b/2021/newtons-fractal/turkish/sentence_translations.json index 003382cc4..6a907ded8 100644 --- a/2021/newtons-fractal/turkish/sentence_translations.json +++ b/2021/newtons-fractal/turkish/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", "translatedText": "Yine bilgisayar grafiklerinde kök bulmayla ilgili eğlenceli bir yan not olarak, bir keresinde bir Pixar mühendisinden bana Coco filminin bazı sahnelerinde kaç tane ışık kullanıldığını ve bunlardan bazılarının doğasını dikkate alarak bir tahmin vermesini istemiştim. Küreler gibi polinom olarak tanımlanmış şeyler söz konusu olduğunda piksel başına hesaplamalar söz konusu olduğunda, ikinci dereceden formül bu filmin yapımında kolaylıkla trilyonlarca kez kullanıldı. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process. ", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process. ", "translatedText": "Biraz önsezi için, ekrandaki gibi bir kümenin bu süreçten geçmesini izlemenin aydınlatıcı olduğunu buldum. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/ukrainian/sentence_translations.json b/2021/newtons-fractal/ukrainian/sentence_translations.json index f76b79bab..42e1b4ae4 100644 --- a/2021/newtons-fractal/ukrainian/sentence_translations.json +++ b/2021/newtons-fractal/ukrainian/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film.", "translatedText": "Як цікаве зауваження, яке знову ж таки стосується пошуку коренів у комп’ютерній графіці, одного разу інженер Pixar дав мені оцінку, враховуючи, скільки світла було використано в деяких сценах для фільму «Коко», і враховуючи природу деяких із них попіксельні обчислення, коли задіяні поліноміально визначені речі, такі як сфери, квадратичну формулу легко використовували кілька трильйонів разів у виробництві цього фільму.", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process.", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process.", "translatedText": "З огляду на інтуїцію, я вважаю корисним просто спостерігати, як кластер, як той, що на екрані, проходить цей процес.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/urdu/sentence_translations.json b/2021/newtons-fractal/urdu/sentence_translations.json index 72cea351f..5b4aaa820 100644 --- a/2021/newtons-fractal/urdu/sentence_translations.json +++ b/2021/newtons-fractal/urdu/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", "translatedText": "ایک تفریحی ضمنی نوٹ کے طور پر، کمپیوٹر گرافکس میں روٹ فائنڈنگ سے متعلق، میں نے ایک بار ایک Pixar انجینئر نے مجھے اندازہ لگایا تھا کہ فلم کوکو کے کچھ مناظر میں کتنی لائٹس استعمال کی گئی تھیں، اور ان میں سے کچھ کی نوعیت کو دیکھتے ہوئے فی پکسل کیلکولیشنز جب پولی نامی طور پر متعین کردہ چیزیں جیسے کہ دائرے شامل ہوتے ہیں، تو اس فلم کی تیاری میں چوکور فارمولہ آسانی سے کئی ٹریلین بار استعمال کیا جاتا تھا۔", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process. ", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process. ", "translatedText": "تھوڑی سی بصیرت کے لیے، میں نے اس عمل سے گزرتے ہوئے اسکرین پر موجود ایک جھرمٹ کو صرف دیکھنا روشن سمجھا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/newtons-fractal/vietnamese/sentence_translations.json b/2021/newtons-fractal/vietnamese/sentence_translations.json index 25f56b46c..3e5d46cda 100644 --- a/2021/newtons-fractal/vietnamese/sentence_translations.json +++ b/2021/newtons-fractal/vietnamese/sentence_translations.json @@ -288,7 +288,7 @@ "end": 259.36 }, { - "input": "As a fun side note, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", + "input": "And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that considering how many lights were used in some of the scenes for the movie Coco, and given the nature of some of these per-pixel calculations when polynomially defined things like spheres are involved, the quadratic formula was easily used multiple trillions of times in the production of that film. ", "translatedText": "Một lưu ý thú vị, một lần nữa liên quan đến việc tìm kiếm gốc rễ trong đồ họa máy tính, có lần tôi được một kỹ sư của Pixar đưa ra ước tính rằng việc xem xét số lượng ánh sáng được sử dụng trong một số cảnh của bộ phim Coco và đưa ra bản chất của một số cảnh trong số đó. tính toán trên mỗi pixel khi liên quan đến những thứ được xác định đa thức như hình cầu, công thức bậc hai có thể dễ dàng được sử dụng hàng nghìn tỷ lần trong quá trình sản xuất bộ phim đó. ", "model": "google_nmt", "n_reviews": 0, @@ -1272,7 +1272,7 @@ "end": 1333.17 }, { - "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one on screen undergo this process. ", + "input": "For a little intuition, I found it enlightening to simply watch a cluster like the one I'm showing on screen undergo this process. ", "translatedText": "Đối với một chút trực giác, tôi thấy thật thú vị khi chỉ cần xem một cụm giống như cụm trên màn hình trải qua quá trình này. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/arabic/sentence_translations.json b/2021/quick-eigen/arabic/sentence_translations.json index 2d6b55eb7..1fa74df24 100644 --- a/2021/quick-eigen/arabic/sentence_translations.json +++ b/2021/quick-eigen/arabic/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "إذا لم تكن على دراية بالقيم الذاتية، فاستمر وألق نظرة على هذا الفيديو هنا، والذي يهدف في الواقع إلى التعريف بها. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "حسنًا، هذا أمر مبالغ فيه بعض الشيء، لكن مرة أخرى، أفترض أن كل هذا عبارة عن مراجعة لأي منكم يشاهده. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "لكن على الأقل بالنسبة للمصفوفات 2×2، هناك طريقة أكثر مباشرة للحصول على هذه الإجابة. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "توقف الآن للحظة لترى ما إذا كان بإمكانك استخلاص الحقيقة الثالثة ذات الصلة، وهي كيف يمكنك استرداد رقمين بسرعة عندما تعرف متوسطهما وتعرف حاصل ضربهما. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "أنت تعلم أن القيمتين متباعدتان بالتساوي حول الرقم 7، لذا فإنهما تبدوان مثل 7 زائد أو ناقص شيء ما، دعنا نسمي ذلك شيئًا d للمسافة. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "لذلك من هناك، يمكنك العثور مباشرة على د. د تربيع هو 7 تربيع ناقص 40، أو 9، مما يعني أن د نفسه يساوي 3. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "وهذا يعطي الحقيقة الرئيسية الثالثة، وهي أنه عندما يكون لعددين متوسط m وحاصل ضرب p، يمكنك كتابة هذين الرقمين على صورة m زائد أو ناقص الجذر التربيعي لـ m تربيع ناقص p. يعد هذا أمرًا سريعًا جدًا لإعادة اشتقاقه بسرعة إذا نسيته، وهو في الأساس مجرد إعادة صياغة لصيغة الفرق بين المربعات. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "في الواقع، كتب لنا صديقي تيم من قناة A Capella Science أغنية سريعة لطيفة لجعلها لا تُنسى. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "لذا فإن الشيء الذي تبدأ كتابته هو 2 ± sqrt(2^2 - ...). ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "ثم يكون حاصل ضرب القيم الذاتية هو المحدد، وهو في هذا المثال 3*1 - 1*4، أو -1. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "هذا يعني أن القيم الذاتية هي 2±sqrt(5). ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "ثم المحدد هو 2*8 - 7*1، أو 9. ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "لذا في هذا المثال، تبدو القيم الذاتية مثل 5 ± sqrt(16)، والتي يمكن تبسيطها بشكل أكبر إلى 9 و1. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "إذا كنت تعرف ميكانيكا الكم، فستعرف أن القيم الذاتية للمصفوفات وثيقة الصلة بالفيزياء التي تصفها. وإذا كنت لا تعرف ميكانيكا الكم، فلتكن هذه مجرد لمحة بسيطة عن مدى أهمية هذه الحسابات في الواقع للتطبيقات الحقيقية. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "متوسط الإدخالات القطرية في الحالات الثلاث هو صفر. إذن متوسط القيم الذاتية في كل هذه الحالات هو صفر، وهو ما يجعل الصيغة تبدو بسيطة بشكل خاص. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "بالنسبة للرقم الأول، فهو 0 - 1 أو -1. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "والشكل الأخير يشبه سالب 1 ناقص 0. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "لذا، في جميع الحالات، يتم تبسيط القيم الذاتية لتكون زائد وناقص 1. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "على الرغم من أنك في هذه الحالة لا تحتاج حقًا إلى صيغة للعثور على قيمتين إذا كنت تعلم أنهما متباعدتان بشكل متساوٍ حول 0 وحاصل ضربهما هو سالب 1. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "وحقيقة أن القيم الذاتية لها زائد وناقص 1 تتوافق مع فكرة أن قيم الدوران التي ستلاحظها ستكون إما بالكامل في اتجاه واحد أو بالكامل في اتجاه آخر، على عكس شيء يتراوح بشكل مستمر بينهما. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "أعني، إلقاء نظرة على أول واحد. يمنحك المحدد ذو الصلة بشكل مباشر متعددة الحدود المميزة لامدا تربيع ناقص واحد، ومن الواضح أن جذورها زائد وناقص واحد. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "نفس الإجابة عند استخدام المصفوفة الثانية، لامدا تربيع ناقص واحد. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "وبشكل أكثر تحديدًا، يجب أن تفترض أن هذا المتجه تم تسويته، مما يعني أن تربيع زائد ب تربيع زائد ج تربيع يساوي واحدًا. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "على وجه التحديد، إذا تمت تسوية كثيرة الحدود بحيث يكون هذا المعامل الرئيسي واحدًا، فإن متوسط الجذور سيكون سالب نصف مضروبًا في هذا المعامل الخطي، وهو ما يساوي سالبًا واحدًا في مجموع تلك الجذور. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "لكن الميزة الحقيقية ليست فقط أن عدد الرموز التي يجب حفظها أقل، بل أن كل واحد منها يحمل معنى أكبر معه. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "الأمل هو أنه ليس مجرد شيء آخر تحفظه، ولكن أن الإطار يعزز بعض الحقائق اللطيفة الأخرى التي تستحق المعرفة، مثل كيفية ارتباط التتبع والمحدد بالقيم الذاتية. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/bengali/sentence_translations.json b/2021/quick-eigen/bengali/sentence_translations.json index dd8cf8e4b..c05934750 100644 --- a/2021/quick-eigen/bengali/sentence_translations.json +++ b/2021/quick-eigen/bengali/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "আপনি যদি eigenvalues এর সাথে অপরিচিত হন তবে এগিয়ে যান এবং এখানে এই ভিডিওটি দেখুন, যা আসলে তাদের পরিচয় করিয়ে দেওয়ার জন্য।", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "ঠিক আছে, এটা বলতে একটু মুখের কথা, কিন্তু আবার, আমি অনুমান করছি যে এই সবই আপনার যে কারোর জন্য পর্যালোচনা।", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "সত্যই, প্রক্রিয়াটি ভয়ানক নয়, তবে কমপক্ষে 2x2 ম্যাট্রিক্সের জন্য, আরও অনেক সরাসরি উপায় রয়েছে যা আপনি উত্তর পেতে পারেন।", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "এখন আমাদের তৃতীয় প্রাসঙ্গিক সত্যটি কী হবে তা আপনি বের করতে পারেন কিনা তা দেখার জন্য একটু সময় নিন, যেটি হল আপনি কীভাবে দুটি সংখ্যা দ্রুত পুনরুদ্ধার করতে পারেন যখন আপনি তাদের গড় জানেন এবং আপনি তাদের পণ্যটি জানেন।", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "আপনি জানেন যে দুটি মান 7 নম্বরের চারপাশে সমানভাবে ব্যবধানে রয়েছে, তাই তারা 7 প্লাস বা বিয়োগের মতো দেখাচ্ছে, আসুন দূরত্বের জন্য এটিকে d বলি।", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "তাই সেখান থেকে, আপনি সরাসরি ডি খুঁজে পেতে পারেন।d বর্গ হল 7 বর্গ বিয়োগ 40, বা 9, যার মানে হল d নিজেই 3।", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "এটি তৃতীয় মূল তথ্য দেয়, যেটি হল যখন দুটি সংখ্যার একটি গড় m এবং একটি গুণফল p থাকে, আপনি সেই দুটি সংখ্যাকে m যোগ বা বিয়োগ হিসাবে লিখতে পারেন m বর্গ বিয়োগ p এর বর্গমূল।আপনি যদি এটি ভুলে যান তবে এটি ফ্লাইতে পুনরুত্পাদন করার জন্য খুব দ্রুত, এবং এটি মূলত বর্গাকার সূত্রের পার্থক্যের একটি রিফ্রেসিং।", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "আসলে, চ্যানেল এ ক্যাপেলা সায়েন্সের আমার বন্ধু টিম এটিকে আরও কিছুটা স্মরণীয় করে তুলতে আমাদের একটি সুন্দর দ্রুত জিঙ্গেল লিখেছেন।", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "সুতরাং এই উদাহরণে, eigenvalues এর গড় 3 এবং 1 এর গড় সমান, যা 2, তাই আপনি যে জিনিসটি লিখতে শুরু করবেন সেটি হল 2 যোগ বা বিয়োগ 2 বর্গ বিয়োগের বর্গমূল, তারপর eigenvalues এর গুণফল।", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "সুতরাং এই উদাহরণে, eigenvalues দেখতে 5 প্লাস বা বিয়োগ 16 এর বর্গমূলের মতো, যা আরও সহজ করে 9 এবং 1 হিসাবে।", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "আপনি যদি কোয়ান্টাম মেকানিক্স জানেন তবে আপনি জানতে পারবেন যে ম্যাট্রিক্সের ইজেন ভ্যালুগুলি তাদের বর্ণনা করা পদার্থবিদ্যার সাথে অত্যন্ত প্রাসঙ্গিক।এবং যদি আপনি কোয়ান্টাম মেকানিক্স না জানেন, তাহলে এই কম্পিউটেশনগুলো বাস্তবে বাস্তব অ্যাপ্লিকেশনের সাথে কতটা প্রাসঙ্গিক তার একটু আভাস দেওয়া যাক।", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "তিনটি ক্ষেত্রেই তির্যক এন্ট্রির গড় শূন্য।সুতরাং এই সমস্ত ক্ষেত্রে eigenvalue-এর গড় হল শূন্য, যা আমাদের সূত্রটিকে বিশেষভাবে সহজ দেখায়।", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "প্রথমটির জন্য, এটি 0 বিয়োগ 1 বা ঋণাত্মক 1।", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "এবং চূড়ান্তটি নেতিবাচক 1 বিয়োগ 0 এর মত দেখাচ্ছে।", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "তাই সব ক্ষেত্রেই, eigenvalues সহজ করে প্লাস এবং মাইনাস 1।", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "যদিও এই ক্ষেত্রে, দুটি মান খুঁজে বের করার জন্য আপনার সত্যিই কোনো সূত্রের প্রয়োজন নেই যদি আপনি জানেন যে তারা সমানভাবে ০ এর কাছাকাছি ব্যবধানে রয়েছে এবং তাদের পণ্য নেতিবাচক 1।", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "এবং সত্য যে তাদের eigenvalues প্লাস এবং বিয়োগ 1 এই ধারণার সাথে মিলে যায় যে আপনি যে ঘূর্ণনের মানগুলি পর্যবেক্ষণ করবেন তা হয় সম্পূর্ণভাবে এক দিকে বা সম্পূর্ণ অন্য দিকে হবে, এর মধ্যে ক্রমাগত বিস্তৃত কিছুর বিপরীতে।", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "আমি বলতে চাচ্ছি, প্রথমটি একবার দেখুন।প্রাসঙ্গিক নির্ধারক আপনাকে সরাসরি ল্যাম্বডা স্কোয়ার মাইনাস ওয়ানের একটি বৈশিষ্ট্যযুক্ত বহুপদ দেয় এবং স্পষ্টতই যেটির মূল রয়েছে প্লাস এবং বিয়োগ ওয়ান।", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "একই উত্তর যখন আপনি দ্বিতীয় ম্যাট্রিক্স করবেন, ল্যাম্বডা বর্গ বিয়োগ এক।", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "আরও নির্দিষ্টভাবে, আপনার অনুমান করা উচিত যে এই ভেক্টরটি স্বাভাবিক করা হয়েছে, যার অর্থ একটি বর্গ প্লাস বি বর্গ প্লাস সি বর্গ একের সমান।", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "বিশেষভাবে, যদি বহুপদীকে স্বাভাবিক করা হয় যাতে এই অগ্রণী সহগ এক হয়, তাহলে মূলের গড় এই রৈখিক সহগের অর্ধেক ঋণাত্মক হবে, যা সেই মূলগুলির যোগফলের এক গুণ ঋণাত্মক।", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "তবে আসল সুবিধাটি কেবল এটিই নয় যে এটি মুখস্থ করার জন্য কম প্রতীক, এটি তাদের প্রত্যেকের সাথে আরও অর্থ বহন করে।", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "আশার বিষয় হল যে এটি কেবলমাত্র আর একটি জিনিস নয় যা আপনি মুখস্ত করে রেখেছেন, তবে ফ্রেমিং আরও কিছু চমৎকার তথ্যকে শক্তিশালী করে যা জানার যোগ্য, যেমন ট্রেস এবং নির্ধারক কীভাবে eigenvalues এর সাথে সম্পর্কিত।", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/chinese/sentence_translations.json b/2021/quick-eigen/chinese/sentence_translations.json index 1cd08e4e8..fbf1c68ed 100644 --- a/2021/quick-eigen/chinese/sentence_translations.json +++ b/2021/quick-eigen/chinese/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "如果您不熟悉特征值，请继续观看此处的此视频，该视频 实际上是为了介绍它们。", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "好吧，说起来有点拗口，但我再次假设所有这些都是 针对你们观看的人的评论。", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "但至少对于 2x2 矩阵，有一种更直接的方法可以得到这个答案。 ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "现在花点时间看看你是否可以推导出我们的第三个相关事实，即当你知道 两个数字的平均值并且知道它们的乘积时，如何快速恢复它们。", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "您知道这两个值在数字 7 周围均匀分布，因此 它们看起来像 7 加上或减去某个值，我们将其称为距离 d。", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "所以从那里，你可以直接找到d。d 的平方是 7 的平方减 40，即 9，这意味着 d 本身是 3。", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "这就给出了第三个关键事实，即当两个数字具有均值 m 和乘积 p 时， 您可以将这两个数字写为 m 加上或减去 m 平方减去 p 的平方根。如果您忘记了，那么可以快速重新推导它 ，而且它本质上只是平方差公式的改写。", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "事实上，我来自 A Capella Science 频道的朋友 Tim 给我们写了一首简 短的歌曲，让它更令人难忘。", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "所以你开始写的是 2 ± sqrt(2^2 - ...)。 ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "那么特征值的乘积就是行列式，在本例中为 3*1 - 1*4，即 -1。 ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "这意味着特征值为 2±sqrt(5)。 ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "那么行列式就是 2*8 - 7*1，即 9。 ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "因此，在本例中，特征值看起来像 5 ± sqrt(16)，进一步简化为 9 和 1。 ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "如果您了解量子力学，您就会知道矩阵的特征值 与其描述的物理高度相关。如果您不了解量子力学，请让我 们稍微了解一下这些计算实际上如何与实际应用非常相关。", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "所有三种情况下对角线条目的平均值均为零。因此，所有这些情况下特征值的平均值为零，这使得我们的公式看起来 特别简单。", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "对于第一个，它是 0 减 1，或负 1。", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "最后一个看起来像负 1 减 0。", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "因此在所有情况下，特征值都简化为正负 1。", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "尽管在本例中，如果您知道两个值在 0 周围均匀分布并且它们的乘积为 负 1，则实际上不需要公式来查找这两个值。", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "它们的特征值是正负 1 的事实与这 样的想法相对应，即您观察到的自旋值要么完全在一个方向 上，要么完全在另一个方向上，而不是在两者之间连续变 化。", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "我的意思是，看看第一个。相关的行列式直接给出 了 lambda 平方减一的特征多项式，并且显然它有正 负一的根。", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "当你做第二个矩阵时，答案是相同的，即 lambda 平方减一。", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "更具体地说，您应该假设该向量已标准 化，这意味着 a 的平方加上 b 的平方加上 c 的平方等于 1。", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "具体来说，如果对多项 式进行归一化，使得该前导系数为 1，则根的平均值将 是该线性系数的负二分之一，即这些根之和的负一倍。", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "但真正的优势不仅在 于它需要记住的符号更少，还在于每个符号都承载着更多的含义。", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "我们希望这不仅仅是您记住的另一件事，而且框架还 强化了其他一些值得了解的好事实，例如迹和行列式与特 征值的关系。", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/english/captions.srt b/2021/quick-eigen/english/captions.srt index 18bba4ddc..6bac10d75 100644 --- a/2021/quick-eigen/english/captions.srt +++ b/2021/quick-eigen/english/captions.srt @@ -47,758 +47,762 @@ and we call the relevant scaling factor the corresponding eigenvalue, often denoted with the letter lambda. 13 -00:00:39,840 --> 00:00:45,466 +00:00:39,840 --> 00:00:44,513 When you write this as an equation, and you rearrange a little bit, 14 -00:00:45,466 --> 00:00:51,589 +00:00:44,513 --> 00:00:49,598 what you see is that if the number lambda is an eigenvalue of a matrix A, 15 -00:00:51,589 --> 00:00:58,126 -then the eigenvector is then the corresponding eigenvector to the zero vector, +00:00:49,598 --> 00:00:55,233 +then the matrix A minus lambda times the identity must send some non-zero vector, 16 -00:00:58,126 --> 00:01:04,580 -which in turn means that the determinant of this modified matrix must be zero. +00:00:55,233 --> 00:00:59,219 +namely the corresponding eigenvector, to the zero vector, 17 +00:00:59,219 --> 00:01:04,580 +which in turn means that the determinant of this modified matrix must be zero. + +18 00:01:06,120 --> 00:01:08,808 Okay, that's all a little bit of a mouthful to say, but again, -18 +19 00:01:08,808 --> 00:01:11,540 I'm assuming that all of this is review for any of you watching. -19 +20 00:01:12,820 --> 00:01:17,297 So, the usual way to compute eigenvalues, how I used to do it and how I believe -20 +21 00:01:17,297 --> 00:01:21,494 most students are taught to carry it out, is to subtract the unknown value -21 +22 00:01:21,494 --> 00:01:25,860 lambda off the diagonals, and then solve for the determinant is equal to zero. -22 +23 00:01:27,760 --> 00:01:31,929 Doing this always involves a few extra steps to expand out and simplify to get a -23 +24 00:01:31,929 --> 00:01:36,460 clean quadratic polynomial, what's known as the characteristic polynomial of the matrix. -24 +25 00:01:37,360 --> 00:01:39,924 The eigenvalues are the roots of this polynomial, -25 +26 00:01:39,924 --> 00:01:42,847 so to find them you have to apply the quadratic formula, -26 +27 00:01:42,847 --> 00:01:46,540 which itself typically requires one or two more steps of simplification. -27 +28 00:01:47,760 --> 00:01:51,684 Honestly, the process isn't terrible, but at least for two by two matrices, -28 +29 00:01:51,684 --> 00:01:54,680 there is a much more direct way you can get at the answer. -29 +30 00:01:55,400 --> 00:01:57,859 And if you want to rediscover this trick, there's only three -30 +31 00:01:57,859 --> 00:02:00,440 relevant facts you need to know, each of which is worth knowing -31 +32 00:02:00,440 --> 00:02:02,900 in its own right and can help you with other problem solving. -32 +33 00:02:03,820 --> 00:02:08,650 Number one, the trace of a matrix, which is the sum of these two diagonal entries, -33 +34 00:02:08,650 --> 00:02:10,919 is equal to the sum of the eigenvalues. -34 +35 00:02:11,700 --> 00:02:14,723 Or, another way to phrase it, more useful for our purposes, -35 +36 00:02:14,723 --> 00:02:18,603 is that the mean of the two eigenvalues is the same as the mean of these two -36 +37 00:02:18,603 --> 00:02:19,460 diagonal entries. -37 +38 00:02:21,000 --> 00:02:25,649 Number two, the determinant of a matrix, our usual ad-bc formula, -38 +39 00:02:25,649 --> 00:02:28,960 is equal to the product of the two eigenvalues. -39 +40 00:02:30,060 --> 00:02:33,976 And this should kind of make sense if you understand that eigenvalues describe -40 +41 00:02:33,976 --> 00:02:37,149 how much an operator stretches space in a particular direction, -41 +42 00:02:37,149 --> 00:02:41,214 and that the determinant describes how much an operator scales areas, or volumes, -42 +43 00:02:41,214 --> 00:02:41,760 as a whole. -43 +44 00:02:42,800 --> 00:02:45,980 Now before getting to the third fact, notice how you can essentially read -44 +45 00:02:45,980 --> 00:02:49,160 these first two values out of the matrix without really writing much down. -45 +46 00:02:49,760 --> 00:02:51,320 Take this matrix here as an example. -46 +47 00:02:51,820 --> 00:02:54,542 Straight away, you can know that the mean of the -47 +48 00:02:54,542 --> 00:02:57,820 eigenvalues is the same as the mean of 8 and 6, which is 7. -48 +49 00:02:59,580 --> 00:03:03,249 Likewise, most linear algebra students are pretty well practiced at -49 +50 00:03:03,249 --> 00:03:07,080 finding the determinant, which in this case works out to be 48 minus 8. -50 +51 00:03:08,240 --> 00:03:11,700 So right away, you know that the product of the two eigenvalues is 40. -51 +52 00:03:12,780 --> 00:03:16,687 Now take a moment to see if you can derive what will be our third relevant fact, -52 +53 00:03:16,687 --> 00:03:19,485 which is how you can quickly recover two numbers when you -53 +54 00:03:19,485 --> 00:03:21,560 know their mean and you know their product. -54 +55 00:03:22,460 --> 00:03:23,720 Here, let's focus on this example. -55 +56 00:03:24,200 --> 00:03:27,988 You know that the two values are evenly spaced around the number 7, -56 +57 00:03:27,988 --> 00:03:32,780 so they look like 7 plus or minus something, let's call that something d for distance. -57 +58 00:03:33,560 --> 00:03:36,380 You also know that the product of these two numbers is 40. -58 +59 00:03:38,600 --> 00:03:41,719 Now to find d, notice that this product expands really nicely, -59 +60 00:03:41,719 --> 00:03:43,700 it works out as a difference of squares. -60 +61 00:03:44,560 --> 00:03:46,860 So from there, you can find d. -61 +62 00:03:48,200 --> 00:03:53,400 d squared is 7 squared minus 40, or 9, which means that d itself is 3. -62 +63 00:03:56,380 --> 00:04:01,100 In other words, the two values for this very specific example work out to be 4 and 10. -63 +64 00:04:01,680 --> 00:04:05,607 But our goal is a quick trick, and you wouldn't want to think through this each time, -64 +65 00:04:05,607 --> 00:04:08,120 so let's wrap up what we just did in a general formula. -65 +66 00:04:08,640 --> 00:04:12,585 For any mean m and product p, the distance squared -66 +67 00:04:12,585 --> 00:04:15,680 is always going to be m squared minus p. -67 +68 00:04:17,560 --> 00:04:21,293 This gives the third key fact, which is that when two numbers -68 +69 00:04:21,293 --> 00:04:25,087 have a mean m and a product p, you can write those two numbers -69 +70 00:04:25,087 --> 00:04:28,460 as m plus or minus the square root of m squared minus p. -70 +71 00:04:30,100 --> 00:04:33,421 This is decently fast to re-derive on the fly if you ever forget it, -71 +72 00:04:33,421 --> 00:04:37,080 and it's essentially just a rephrasing of the difference of squares formula. -72 +73 00:04:37,860 --> 00:04:41,220 But even still, it's a fact that's worth memorizing so it's at the tip of your fingers. -73 +74 00:04:41,220 --> 00:04:44,237 In fact, my friend Tim from the channel A Capella Science wrote -74 +75 00:04:44,237 --> 00:04:47,160 us a nice quick jingle to make it a little bit more memorable. -75 +76 00:04:51,900 --> 00:04:57,620 Let me show you how this works, say for the matrix 3 1 4 1. -76 +77 00:04:58,100 --> 00:05:01,820 You start by bringing to mind the formula, maybe stating it all in your head. -77 +78 00:05:06,200 --> 00:05:11,620 But when you write it down, you fill in the appropriate values for m and p as you go. -78 +79 00:05:12,340 --> 00:05:17,147 So in this example, the mean of the eigenvalues is the same as the mean of 3 and 1, -79 +80 00:05:17,147 --> 00:05:20,696 which is 2, so the thing you start writing is 2 plus or minus -80 +81 00:05:20,696 --> 00:05:22,700 the square root of 2 squared minus. -81 +82 00:05:23,540 --> 00:05:27,229 Then the product of the eigenvalues is the determinant, -82 +83 00:05:27,229 --> 00:05:31,644 which in this example is 3 times 1 minus 1 times 4, or negative 1, -83 +84 00:05:31,644 --> 00:05:36,783 so that's the final thing you fill in, which means the eigenvalues are 2 plus -84 +85 00:05:36,783 --> 00:05:38,760 or minus the square root of 5. -85 +86 00:05:40,300 --> 00:05:43,849 You might recognize that this is the same matrix I was using at the beginning, -86 +87 00:05:43,849 --> 00:05:46,500 but notice how much more directly we can get at the answer. -87 +88 00:05:48,140 --> 00:05:49,180 Here, try another one. -88 +89 00:05:49,440 --> 00:05:54,480 This time, the mean of the eigenvalues is the same as the mean of 2 and 8, which is 5. -89 +90 00:05:55,100 --> 00:05:59,220 So again, you start writing out the formula, but this time writing 5 in place of m. -90 +91 00:06:02,980 --> 00:06:08,300 And then the determinant is 2 times 8 minus 7 times 1, or 9. -91 +92 00:06:09,520 --> 00:06:15,402 So in this example, the eigenvalues look like 5 plus or minus the square root of 16, -92 +93 00:06:15,402 --> 00:06:18,240 which simplifies even further as 9 and 1. -93 +94 00:06:19,420 --> 00:06:21,914 You see what I mean about how you can basically just start -94 +95 00:06:21,914 --> 00:06:24,620 writing down the eigenvalues while you're staring at the matrix? -95 +96 00:06:25,280 --> 00:06:28,160 It's typically just the tiniest bit of simplification at the end. -96 +97 00:06:29,060 --> 00:06:32,412 Honestly, I've found myself using this trick a lot when I'm sketching quick -97 +98 00:06:32,412 --> 00:06:35,720 notes related to linear algebra and want to use small matrices as examples. -98 +99 00:06:36,180 --> 00:06:38,612 I've been working on a video about matrix exponents, -99 +100 00:06:38,612 --> 00:06:41,688 where eigenvalues pop up a lot, and I realize it's just very handy -100 +101 00:06:41,688 --> 00:06:44,855 if students can read out the eigenvalues from small examples without -101 +102 00:06:44,855 --> 00:06:48,620 losing the main line of thought by getting bogged down in a different calculation. -102 +103 00:06:49,740 --> 00:06:53,441 As another fun example, take a look at this set of three different matrices, -103 +104 00:06:53,441 --> 00:06:55,460 which comes up a lot in quantum mechanics. -104 +105 00:06:55,760 --> 00:06:57,520 They're known as the Pauli spin matrices. -105 +106 00:06:58,600 --> 00:07:01,465 If you know quantum mechanics, you'll know that the eigenvalues -106 +107 00:07:01,465 --> 00:07:04,420 of matrices are highly relevant to the physics that they describe. -107 +108 00:07:05,220 --> 00:07:08,240 And if you don't know quantum mechanics, let this just be a little glimpse -108 +109 00:07:08,240 --> 00:07:11,220 of how these computations are actually very relevant to real applications. -109 +110 00:07:12,540 --> 00:07:15,880 The mean of the diagonal entries in all three cases is zero. -110 +111 00:07:17,560 --> 00:07:20,688 So the mean of the eigenvalues in all of these cases is zero, -111 +112 00:07:20,688 --> 00:07:23,060 which makes our formula look especially simple. -112 +113 00:07:25,380 --> 00:07:28,800 What about the products of the eigenvalues, the determinants of these matrices? -113 +114 00:07:29,700 --> 00:07:32,560 For the first one, it's 0, minus 1, or negative 1. -114 +115 00:07:33,200 --> 00:07:35,792 The second one also looks like 0, minus 1, but it takes -115 +116 00:07:35,792 --> 00:07:38,200 a moment more to see because of the complex numbers. -116 +117 00:07:38,840 --> 00:07:41,360 And the final one looks like negative 1, minus 0. -117 +118 00:07:42,060 --> 00:07:45,920 So in all cases, the eigenvalues simplify to be plus and minus 1. -118 +119 00:07:46,720 --> 00:07:50,000 Although in this case, you really don't need a formula to find two values if -119 +120 00:07:50,000 --> 00:07:53,280 you know that they're evenly spaced around 0 and their product is negative 1. -120 +121 00:07:54,640 --> 00:07:57,608 If you're curious, in the context of quantum mechanics, -121 +122 00:07:57,608 --> 00:08:02,166 these matrices describe observations you might make about a particle's spin in the x, -122 +123 00:08:02,166 --> 00:08:03,120 y, or z direction. -123 +124 00:08:03,560 --> 00:08:07,907 And the fact that their eigenvalues are plus and minus 1 corresponds with the idea -124 +125 00:08:07,907 --> 00:08:12,306 that the values for the spin that you would observe would be either entirely in one -125 +126 00:08:12,306 --> 00:08:17,020 direction or entirely in another, as opposed to something continuously ranging in between. -126 +127 00:08:18,320 --> 00:08:21,817 Maybe you'd wonder how exactly this works, or why you would use 2x2 -127 +128 00:08:21,817 --> 00:08:25,520 matrices that have complex numbers to describe spin in three dimensions. -128 +129 00:08:26,100 --> 00:08:29,760 Those would be fair questions, just outside the scope of what I want to talk about here. -129 +130 00:08:30,480 --> 00:08:34,093 You know, it's funny, I wrote this section because I wanted some case where -130 +131 00:08:34,093 --> 00:08:37,611 you have 2x2 matrices that aren't just toy examples or homework problems, -131 +132 00:08:37,611 --> 00:08:41,700 ones where they actually come up in practice, and quantum mechanics is great for that. -132 +133 00:08:41,700 --> 00:08:44,886 The thing is, after I made it, I realized that the whole -133 +134 00:08:44,886 --> 00:08:48,240 example kind of undercuts the point that I'm trying to make. -134 +135 00:08:48,740 --> 00:08:52,285 For these specific matrices, when you use the traditional method, -135 +136 00:08:52,285 --> 00:08:56,100 the one with characteristic polynomials, it's essentially just as fast. -136 +137 00:08:56,220 --> 00:08:57,640 It might actually be faster. -137 +138 00:08:58,240 --> 00:08:59,400 I mean, take a look at the first one. -138 +139 00:08:59,680 --> 00:09:03,881 The relevant determinant directly gives you a characteristic polynomial -139 +140 00:09:03,881 --> 00:09:08,200 of lambda squared minus 1, and clearly that has roots of plus and minus 1. -140 +141 00:09:08,840 --> 00:09:11,760 Same answer when you do the second matrix, lambda squared minus 1. -141 +142 00:09:13,880 --> 00:09:17,287 And as for the last matrix, forget about doing any computations, -142 +143 00:09:17,287 --> 00:09:20,328 traditional or otherwise, it's already a diagonal matrix, -143 +144 00:09:20,328 --> 00:09:22,740 so those diagonal entries are the eigenvalues. -144 +145 00:09:24,300 --> 00:09:26,920 However, the example is not totally lost to our cause. -145 +146 00:09:27,380 --> 00:09:30,954 Where you will actually feel the speedup is in the more general case, -146 +147 00:09:30,954 --> 00:09:35,243 where you take a linear combination of these three matrices and then try to compute -147 +148 00:09:35,243 --> 00:09:36,060 the eigenvalues. -148 +149 00:09:36,820 --> 00:09:39,590 You might write this as a times the first one, -149 +150 00:09:39,590 --> 00:09:42,420 plus b times the second, plus c times the third. -150 +151 00:09:43,020 --> 00:09:46,150 In quantum mechanics, this would describe spin observations -151 +152 00:09:46,150 --> 00:09:49,280 in a general direction of a vector with coordinates a, b, c. -152 +153 00:09:50,900 --> 00:09:54,481 More specifically, you should assume that this vector is normalized, -153 +154 00:09:54,481 --> 00:09:57,700 meaning a squared plus b squared plus c squared is equal to 1. -154 +155 00:09:58,600 --> 00:10:01,295 When you look at this new matrix, it's immediate -155 +156 00:10:01,295 --> 00:10:04,100 to see that the mean of the eigenvalues is still 0. -156 +157 00:10:04,600 --> 00:10:07,880 And you might also enjoy pausing for a brief moment to confirm -157 +158 00:10:07,880 --> 00:10:10,900 that the product of those eigenvalues is still negative 1. -158 +159 00:10:13,260 --> 00:10:15,920 And then from there, concluding what the eigenvalues must be. -159 +160 00:10:17,220 --> 00:10:20,283 And this time, the characteristic polynomial approach would be by -160 +161 00:10:20,283 --> 00:10:23,580 comparison a lot more cumbersome, definitely harder to do in your head. -161 +162 00:10:25,080 --> 00:10:28,089 To be clear, using the mean product formula is not fundamentally -162 +163 00:10:28,089 --> 00:10:30,960 different from finding roots of the characteristic polynomial. -163 +164 00:10:31,340 --> 00:10:33,440 I mean, it can't be, they're solving the same problem. -164 +165 00:10:34,160 --> 00:10:36,442 One way to think about this actually is that the mean -165 +166 00:10:36,442 --> 00:10:39,020 product formula is a nice way to solve quadratics in general. -166 +167 00:10:39,600 --> 00:10:41,660 And some viewers of the channel may recognize this. -167 +168 00:10:42,540 --> 00:10:45,836 Think about it, when you're trying to find the roots of a quadratic, -168 +169 00:10:45,836 --> 00:10:49,991 given the coefficients, that's another situation where you know the sum of two values, -169 +170 00:10:49,991 --> 00:10:54,100 and you also know their product, but you're trying to recover the original two values. -170 +171 00:10:55,560 --> 00:11:00,046 Specifically, if the polynomial is normalized, so that this leading coefficient is 1, -171 +172 00:11:00,046 --> 00:11:04,323 then the mean of the roots will be negative 1 half times this linear coefficient, -172 +173 00:11:04,323 --> 00:11:06,880 which is negative 1 times the sum of those roots. -173 +174 00:11:08,020 --> 00:11:10,180 With the example on the screen, that makes the mean 5. -174 +175 00:11:11,980 --> 00:11:15,479 And the product of the roots is even easier, it's just the constant term, -175 +176 00:11:15,479 --> 00:11:16,520 no adjustments needed. -176 +177 00:11:17,340 --> 00:11:20,900 So from there, you would apply the mean product formula, and that gives you the roots. -177 +178 00:11:25,140 --> 00:11:27,818 And on the one hand, you could think of this as a lighter -178 +179 00:11:27,818 --> 00:11:30,220 weight version of the traditional quadratic formula. -179 +180 00:11:30,960 --> 00:11:34,042 But the real advantage is not just that it's fewer symbols to memorize, -180 +181 00:11:34,042 --> 00:11:36,440 it's that each one of them carries more meaning with it. -181 +182 00:11:36,940 --> 00:11:40,610 I mean, the whole point of this eigenvalue trick is that because you can read -182 +183 00:11:40,610 --> 00:11:43,528 out the mean and product directly from looking at the matrix, -183 +184 00:11:43,528 --> 00:11:47,482 you don't need to go through the intermediate step of setting up the characteristic -184 +185 00:11:47,482 --> 00:11:48,000 polynomial. -185 +186 00:11:48,420 --> 00:11:51,118 You can jump straight to writing down the roots without ever -186 +187 00:11:51,118 --> 00:11:53,640 explicitly thinking about what the polynomial looks like. -187 +188 00:11:53,840 --> 00:11:56,330 But to do that, we need a version of the quadratic -188 +189 00:11:56,330 --> 00:11:58,820 formula where the terms carry some kind of meaning. -189 +190 00:12:00,380 --> 00:12:03,457 I realize this is a very specific trick for a very specific audience, -190 +191 00:12:03,457 --> 00:12:06,534 but it's something I wish I knew in college, so if you happen to know -191 +192 00:12:06,534 --> 00:12:09,700 any students who might benefit from this, consider sharing it with them. -192 +193 00:12:10,280 --> 00:12:13,246 The hope is that it's not just one more thing that you memorize, -193 +194 00:12:13,246 --> 00:12:16,807 but that the framing reinforces some other nice facts that are worth knowing, -194 +195 00:12:16,807 --> 00:12:19,820 like how the trace and the determinant are related to eigenvalues. -195 +196 00:12:20,560 --> 00:12:23,502 If you want to prove those facts, by the way, take a moment to -196 +197 00:12:23,502 --> 00:12:26,444 expand out the characteristic polynomial for a general matrix, -197 +198 00:12:26,444 --> 00:12:29,620 and then think hard about the meaning of each of these coefficients. -198 +199 00:12:32,400 --> 00:12:35,192 Many thanks to Tim for ensuring that this mean product formula -199 +200 00:12:35,192 --> 00:12:37,940 will stay stuck in all of our heads for at least a few months. -200 +201 00:12:41,700 --> 00:12:46,000 If you don't know about alcappella science, please do check it out. -201 +202 00:12:46,280 --> 00:12:49,580 The molecular shape of you in particular is one of the greatest things on the internet. diff --git a/2021/quick-eigen/english/sentence_timings.json b/2021/quick-eigen/english/sentence_timings.json index 495232795..bfe2ff141 100644 --- a/2021/quick-eigen/english/sentence_timings.json +++ b/2021/quick-eigen/english/sentence_timings.json @@ -25,7 +25,7 @@ 38.6 ], [ - "When you write this as an equation, and you rearrange a little bit, what you see is that if the number lambda is an eigenvalue of a matrix A, then the eigenvector is then the corresponding eigenvector to the zero vector, which in turn means that the determinant of this modified matrix must be zero.", + "When you write this as an equation, and you rearrange a little bit, what you see is that if the number lambda is an eigenvalue of a matrix A, then the matrix A minus lambda times the identity must send some non-zero vector, namely the corresponding eigenvector, to the zero vector, which in turn means that the determinant of this modified matrix must be zero.", 39.84, 64.58 ], diff --git a/2021/quick-eigen/english/transcript.txt b/2021/quick-eigen/english/transcript.txt index 88a8ad62e..b3ca8af3a 100644 --- a/2021/quick-eigen/english/transcript.txt +++ b/2021/quick-eigen/english/transcript.txt @@ -3,7 +3,7 @@ If you're unfamiliar with eigenvalues, go ahead and take a look at this video he You can skip ahead if all you want to do is see the trick, but if possible I'd like you to rediscover it for yourself. So for that, let's lay out a little background. As a quick reminder, if the effect of a linear transformation on a given vector is to scale that vector by some constant, we call it an eigenvector of the transformation, and we call the relevant scaling factor the corresponding eigenvalue, often denoted with the letter lambda. -When you write this as an equation, and you rearrange a little bit, what you see is that if the number lambda is an eigenvalue of a matrix A, then the eigenvector is then the corresponding eigenvector to the zero vector, which in turn means that the determinant of this modified matrix must be zero. +When you write this as an equation, and you rearrange a little bit, what you see is that if the number lambda is an eigenvalue of a matrix A, then the matrix A minus lambda times the identity must send some non-zero vector, namely the corresponding eigenvector, to the zero vector, which in turn means that the determinant of this modified matrix must be zero. Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. So, the usual way to compute eigenvalues, how I used to do it and how I believe most students are taught to carry it out, is to subtract the unknown value lambda off the diagonals, and then solve for the determinant is equal to zero. Doing this always involves a few extra steps to expand out and simplify to get a clean quadratic polynomial, what's known as the characteristic polynomial of the matrix. diff --git a/2021/quick-eigen/french/sentence_translations.json b/2021/quick-eigen/french/sentence_translations.json index 0ff109200..76bcd3d5a 100644 --- a/2021/quick-eigen/french/sentence_translations.json +++ b/2021/quick-eigen/french/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "Si vous n'êtes pas familier avec les valeurs propres, allez-y et jetez un œil à cette vidéo ici, qui est en fait destinée à les présenter. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "D'accord, c'est un peu long à dire, mais encore une fois, je suppose que tout ceci est une révision pour tous ceux d'entre vous qui nous regardent. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "Mais au moins pour les matrices 2x2, il existe un moyen beaucoup plus direct d'obtenir cette réponse. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "Maintenant, prenez un moment pour voir si vous pouvez déduire quel sera notre troisième fait pertinent, à savoir comment vous pouvez récupérer rapidement deux nombres lorsque vous connaissez leur moyenne et leur produit. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "Vous savez que les deux valeurs sont uniformément espacées autour du chiffre 7, elles ressemblent donc à 7 plus ou moins quelque chose, appelons cela quelque chose d pour la distance. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "Donc à partir de là, vous pouvez directement trouver d. d au carré vaut 7 au carré moins 40, soit 9, ce qui signifie que d lui-même vaut 3. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "Cela donne le troisième fait clé, à savoir que lorsque deux nombres ont une moyenne m et un produit p, vous pouvez écrire ces deux nombres sous la forme m plus ou moins la racine carrée de m au carré moins p. C'est assez rapide à recréer à la volée si jamais vous l'oubliez, et il s'agit essentiellement d'une reformulation de la formule de la différence des carrés. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "En fait, mon ami Tim de la chaîne A Capella Science nous a écrit un joli jingle rapide pour le rendre un peu plus mémorable. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "Donc, la chose que vous commencez à écrire est 2 ± sqrt(2^2 - …). ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "Ensuite, le produit des valeurs propres est le déterminant, qui dans cet exemple est 3*1 - 1*4, ou -1. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "Cela signifie que les valeurs propres sont 2 ± sqrt (5). ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "Et puis le déterminant est 2*8 - 7*1, ou 9. ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "Ainsi, dans cet exemple, les valeurs propres ressemblent à 5 ± sqrt(16), ce qui se simplifie encore davantage en 9 et 1. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "Si vous connaissez la mécanique quantique, vous saurez que les valeurs propres des matrices sont très pertinentes pour la physique qu'elles décrivent. Et si vous ne connaissez pas la mécanique quantique, voici juste un petit aperçu de la façon dont ces calculs sont en réalité très pertinents pour les applications réelles. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "La moyenne des entrées diagonales dans les trois cas est nulle. Ainsi, la moyenne des valeurs propres dans tous ces cas est nulle, ce qui rend notre formule particulièrement simple. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "Pour le premier, c’est 0 moins 1, soit moins 1. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "Et le dernier ressemble à moins 1 moins 0. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "Ainsi, dans tous les cas, les valeurs propres se simplifient pour être plus et moins 1. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "Bien que dans ce cas, vous n'avez vraiment pas besoin d'une formule pour trouver deux valeurs si vous savez qu'elles sont régulièrement espacées autour de 0 et que leur produit est moins 1. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "Et le fait que leurs valeurs propres soient plus et moins 1 correspond à l'idée que les valeurs du spin que vous observeriez seraient soit entièrement dans une direction, soit entièrement dans une autre, par opposition à quelque chose qui se situe continuellement entre les deux. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "Je veux dire, jetez un œil au premier. Le déterminant pertinent vous donne directement un polynôme caractéristique de lambda au carré moins un, et qui a clairement des racines de plus et moins un. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "Même réponse lorsque vous faites la deuxième matrice, lambda au carré moins un. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "Plus précisément, vous devez supposer que ce vecteur est normalisé, ce qui signifie que a au carré plus b au carré plus c au carré est égal à un. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "Plus précisément, si le polynôme est normalisé de sorte que ce coefficient principal soit égal à un, alors la moyenne des racines sera négative la moitié de ce coefficient linéaire, qui est négatif une fois la somme de ces racines. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "Mais le véritable avantage n’est pas seulement qu’il y a moins de symboles à mémoriser, c’est que chacun d’entre eux a plus de sens. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "L'espoir est que ce n'est pas seulement une chose de plus que vous mémorisez, mais que le cadrage renforce d'autres faits intéressants qui valent la peine d'être connus, comme la façon dont la trace et le déterminant sont liés aux valeurs propres. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/german/sentence_translations.json b/2021/quick-eigen/german/sentence_translations.json index a5cc586bb..3748ddf43 100644 --- a/2021/quick-eigen/german/sentence_translations.json +++ b/2021/quick-eigen/german/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "Wenn du mit Eigenwerten nicht vertraut bist, schau dir dieses Video an, in dem sie erklärt werden.", "model": "DeepL", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "Okay, das ist alles ein bisschen viel gesagt, aber ich gehe davon aus, dass das alles für jeden, der zuschaut, eine Zusammenfassung ist.", "model": "DeepL", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "Aber zumindest für 2x2-Matrizen gibt es einen viel direkteren Weg, um zu dieser Antwort zu kommen.", "model": "DeepL", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "Nimm dir jetzt einen Moment Zeit, um zu sehen, wie du die dritte wichtige Tatsache ableiten kannst, nämlich wie du zwei Zahlen wiederherstellen kannst, wenn du ihren Mittelwert und ihr Produkt kennst.", "model": "DeepL", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "Du weißt, dass die beiden Werte gleichmäßig um 7 herum angeordnet sind, also sehen sie aus wie 7 plus oder minus etwas; nennen wir dieses Etwas \"d\" für Abstand.", "model": "DeepL", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "Von dort aus kannst du also direkt d finden: d^2 ist 7^2 - 40 oder 9, was bedeutet, dass d selbst 3 ist.", "model": "DeepL", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "Daraus ergibt sich die dritte wichtige Tatsache: Wenn zwei Zahlen einen Mittelwert m und ein Produkt p haben, kannst du diese beiden Zahlen als m ± sqrt(m^2 - p) schreiben. Dies ist schnell wieder abrufbar, wenn du es einmal vergessen hast, und ist im Grunde nur eine Umformulierung der Formel für die Differenz der Quadrate.", "model": "DeepL", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "Mein Freund Tim vom Kanal acapellascience hat uns sogar einen kurzen Jingle geschrieben, um ihn ein bisschen einprägsamer zu machen.", "model": "DeepL", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "Das, was du zu schreiben beginnst, ist also 2 ± sqrt(2^2 - ...).", "model": "DeepL", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "Dann ist das Produkt der Eigenwerte die Determinante, die in diesem Beispiel 3*1 - 1*4 oder -1 ist.", "model": "DeepL", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "Das bedeutet, dass die Eigenwerte 2±sqrt(5) sind.", "model": "DeepL", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "Und dann ist die Determinante 2*8 - 7*1, also 9.", "model": "DeepL", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "In diesem Beispiel sehen die Eigenwerte also wie 5 ± sqrt(16) aus, was sich noch weiter zu 9 und 1 vereinfacht.", "model": "DeepL", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "Wenn du dich mit Quantenmechanik auskennst, weißt du, dass die Eigenwerte von Matrizen für die Physik, die sie beschreiben, von großer Bedeutung sind. Wenn du dich nicht mit Quantenmechanik auskennst, soll dies nur ein kleiner Einblick sein, wie diese Berechnungen tatsächlich für reale Anwendungen relevant sind.", "model": "DeepL", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "Der Mittelwert der Diagonalen ist in allen drei Fällen 0, also ist auch der Mittelwert der Eigenwerte in allen Fällen 0, was unsere Formel besonders einfach aussehen lässt.", "model": "DeepL", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "Bei der ersten ist es 0 - 1 oder -1.", "model": "DeepL", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "Und das Ergebnis sieht aus wie -1 - 0.", "model": "DeepL", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "In allen Fällen vereinfachen sich die Eigenwerte also auf ±1.", "model": "DeepL", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "Aber in diesem Fall brauchst du die Formel nicht, um zwei Werte zu finden, wenn du weißt, dass sie gleichmäßig um 0 herum angeordnet sind und ihr Produkt -1 ist.", "model": "DeepL", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "Die Tatsache, dass ihre Eigenwerte ±1 sind, entspricht der Vorstellung, dass die Werte für den Spin, die du beobachten würdest, entweder ganz in der einen oder ganz in der anderen Richtung liegen würden, im Gegensatz zu etwas, das kontinuierlich dazwischen liegt.", "model": "DeepL", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "Ich meine, schau dir die erste an: Die entsprechende Determinante gibt dir direkt ein charakteristisches Polynom von lambda^2 - 1, und das hat eindeutig Wurzeln von plus und minus 1.", "model": "DeepL", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "Die gleiche Antwort erhältst du, wenn du die zweite Matrix, lambda^2 - 1, verwendest.", "model": "DeepL", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "Genauer gesagt solltest du davon ausgehen, dass dieser Vektor normalisiert ist, also a^2 + b^2 + c^2 = 1.", "model": "DeepL", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "Wenn das Polynom so normalisiert wird, dass der führende Koeffizient 1 ist, ist der Mittelwert der Wurzeln das -½-fache dieses linearen Koeffizienten, also das -1-fache der Summe dieser Wurzeln.", "model": "DeepL", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "Aber der eigentliche Vorteil ist, dass man sich weniger Symbole merken muss, sondern dass jedes einzelne von ihnen mehr Bedeutung mit sich bringt.", "model": "DeepL", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "Die Hoffnung ist, dass es nicht nur eine weitere Sache ist, die man auswendig lernen muss, sondern dass der Rahmen einige andere schöne Fakten verstärkt, die man wissen sollte, z.B. wie die Spur und die Determinante mit den Eigenwerten zusammenhängen.", "model": "DeepL", "n_reviews": 0, diff --git a/2021/quick-eigen/hebrew/sentence_translations.json b/2021/quick-eigen/hebrew/sentence_translations.json index af7f800e3..f9552a016 100644 --- a/2021/quick-eigen/hebrew/sentence_translations.json +++ b/2021/quick-eigen/hebrew/sentence_translations.json @@ -7,7 +7,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "אם אינך מכיר ערכים עצמיים, עיין בסרטון זה המציג אותם. ", "n_reviews": 0, "start": 8.58, @@ -35,7 +35,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "אוקיי, זה הכל קצת לשון הרע לומר, אבל שוב, אני מניח שכל זה הוא סקירה לכל מי שצופה. ", "n_reviews": 0, "start": 66.12, @@ -77,7 +77,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "אבל לפחות עבור מטריצות 2x2, יש דרך הרבה יותר ישירה להגיע לתשובה הזו. ", "n_reviews": 0, "start": 109.58, @@ -147,7 +147,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "עכשיו קח רגע כדי לראות איך אתה יכול להסיק מה תהיה העובדה השלישית הרלוונטית שלנו, והיא איך לשחזר שני מספרים כשאתה יודע את הממוצע שלהם ואתה מכיר את המוצר שלהם. ", "n_reviews": 0, "start": 192.78, @@ -161,7 +161,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "אתה יודע ששני הערכים מרווחים באופן שווה סביב 7, כך שהם נראים כמו 7 פלוס מינוס משהו; בואו נקרא לזה משהו "ד" למרחק. ", "n_reviews": 0, "start": 204.2, @@ -182,7 +182,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "אז משם, אתה יכול למצוא ישירות את d: d^2 הוא 7^2 - 40, או 9, כלומר d עצמו הוא 3. ", "n_reviews": 0, "start": 224.56, @@ -210,7 +210,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "זה נותן את עובדת המפתח השלישית, שהיא שכאשר לשני מספרים יש ממוצע m ומכפלה p, אתה יכול לכתוב את שני המספרים האלה כ-m ± sqrt(m^2 - p). אי פעם תשכח מזה, וזה בעצם רק ניסוח מחדש של נוסחת ההבדל בין הריבועים. ", "n_reviews": 0, "start": 257.56, @@ -224,7 +224,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "למעשה, חברי טים מהערוץ acapellascience כתב לנו ג'ינגל מהיר כדי להפוך אותו לקצת יותר בלתי נשכח. ", "n_reviews": 0, "start": 281.22, @@ -259,14 +259,14 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "אז הדבר שאתה מתחיל לכתוב הוא 2 ± sqrt(2^2 - …). ", "n_reviews": 0, "start": 318.3, "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "אז המכפלה של הערכים העצמיים היא הקובע, שבדוגמה זו הוא 3*1 - 1*4, או -1. ", "n_reviews": 0, "start": 323.54, @@ -280,7 +280,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "זה אומר שהערכים העצמיים הם 2±sqrt(5). ", "n_reviews": 0, "start": 334.88, @@ -315,14 +315,14 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "ואז הקובע הוא 2*8 - 7*1, או 9. ", "n_reviews": 0, "start": 362.98, "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "אז בדוגמה זו, הערכים העצמיים נראים כמו 5 ± sqrt(16), מה שמפשט עוד יותר כמו 9 ו-1. ", "n_reviews": 0, "start": 369.52, @@ -364,14 +364,14 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "אם אתה יודע מכניקת הקוונטים, תדע שהערכים העצמיים של מטריצות רלוונטיים מאוד לפיזיקה שהן מתארות, ואם אינך יודע מכניקת הקוונטים, תן לזה רק להיות הצצה קטנה לאופן שבו החישובים האלה באמת רלוונטיים למציאות. יישומים. ", "n_reviews": 0, "start": 418.6, "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "הממוצע של האלכסון בכל שלושת המקרים הוא 0, כך שממוצע הערכים העצמיים בכל המקרים הוא 0, מה שגורם לנוסחה שלנו להיראות פשוטה במיוחד. ", "n_reviews": 0, "start": 432.54, @@ -385,7 +385,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "עבור הראשון, זה 0 - 1 או -1. ", "n_reviews": 0, "start": 449.7, @@ -399,21 +399,21 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "והאחרון נראה כמו -1 - 0. ", "n_reviews": 0, "start": 458.84, "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "אז בכל המקרים, הערכים העצמיים מפשטים להיות ±1. ", "n_reviews": 0, "start": 462.06, "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "למרות שבמקרה זה, אתה באמת לא צריך את הנוסחה כדי למצוא שני ערכים אם אתה יודע שהם מרווחים באופן שווה סביב 0 והמוצר שלהם הוא -1. ", "n_reviews": 0, "start": 466.72, @@ -427,7 +427,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "העובדה שהערכים העצמיים שלהם הם ±1 תואמת את הרעיון שהערכים של הספין שתצפו יהיו לגמרי בכיוון אחד או לגמרי בכיוון אחר, בניגוד למשהו שנע בין לבין. ", "n_reviews": 0, "start": 483.76, @@ -469,14 +469,14 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "כלומר, תסתכל על הראשון: הדטרמיננט הרלוונטי נותן לך ישירות פולינום אופייני של lambda^2 - 1, וברור שיש לו שורשים של פלוס ומינוס 1. ", "n_reviews": 0, "start": 538.24, "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "אותה תשובה כשאתה עושה את המטריצה השנייה, lambda^2 - 1. ", "n_reviews": 0, "start": 548.84, @@ -518,7 +518,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "ליתר דיוק, עליך להניח שהווקטור הזה מנורמל, כלומר a^2 + b^2 + c^2 = 1. ", "n_reviews": 0, "start": 590.9, @@ -560,7 +560,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "באופן ספציפי, אם הפולינום מנורמל כך שמקדם מוביל זה הוא 1, אז הממוצע של השורשים יהיה -½ כפול מקדם הליניארי הזה, שהוא -1 כפול מסכום השורשים הללו. ", "n_reviews": 0, "start": 655.56, @@ -595,7 +595,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "אבל היתרון האמיתי הוא שיש פחות סמלים לשנן, זה שלכל אחד מהם יש יותר משמעות. ", "n_reviews": 0, "start": 690.96, @@ -630,7 +630,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "התקווה היא שזה לא רק עוד דבר אחד שצריך לשנן, אלא שהמסגור מחזק עוד כמה עובדות נחמדות שכדאי לדעת, כמו איך העקבות והקביעה קשורים לערכים עצמיים. ", "n_reviews": 0, "start": 730.28, diff --git a/2021/quick-eigen/hindi/sentence_translations.json b/2021/quick-eigen/hindi/sentence_translations.json index 62df42fda..0c433dfa9 100644 --- a/2021/quick-eigen/hindi/sentence_translations.json +++ b/2021/quick-eigen/hindi/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "यदि आप स्वदेशी मूल्यों से अपरिचित हैं, तो आगे बढ़ें और यहां इस वीडियो को देखें, जो वास्तव में उन्हें पेश करने के लिए है।", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "ठीक है, यह सब कहने के लिए थोड़ी-सी बात है, लेकिन फिर भी, मैं यह मान रहा हूं कि यह सब आपमें से जो भी देख रहा है उसके लिए समीक्षा है।", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "लेकिन कम से कम 2x2 मैट्रिक्स के लिए, इस उत्तर तक पहुंचने का एक अधिक सीधा तरीका है। ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "अब यह देखने के लिए एक क्षण लें कि क्या आप यह पता लगा सकते हैं कि हमारा तीसरा प्रासंगिक तथ्य क्या होगा, यानी आप दो संख्याओं को कैसे तुरंत पुनर्प्राप्त कर सकते हैं जब आप उनका माध्य जानते हैं और आप उनका उत्पाद जानते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "आप जानते हैं कि दोनों मान संख्या 7 के चारों ओर समान दूरी पर हैं, इसलिए वे 7 प्लस या माइनस कुछ की तरह दिखते हैं, आइए दूरी के लिए इसे कुछ डी कहते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "तो वहां से, आप सीधे डी ढूंढ सकते हैं।d का वर्ग 7 वर्ग घटा 40, या 9 है, जिसका अर्थ है कि d स्वयं 3 है।", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "यह तीसरा मुख्य तथ्य देता है, जो यह है कि जब दो संख्याओं का माध्य m और गुणनफल p होता है, तो आप उन दो संख्याओं को m जोड़ या m वर्ग के वर्गमूल को घटा p के रूप में लिख सकते हैं।यदि आप कभी भी इसे भूल जाते हैं तो इसे तुरंत पुनः प्राप्त करना शालीनता से तेज़ है, और यह मूल रूप से वर्गों के अंतर के फार्मूले का एक पुनर्लेखन मात्र है।", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "वास्तव में, चैनल ए कैपेला साइंस से मेरे मित्र टिम ने इसे थोड़ा और यादगार बनाने के लिए हमें एक अच्छा त्वरित जिंगल लिखा।", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "तो आप जो चीज़ लिखना शुरू करते हैं वह 2 ± sqrt(2^2 - …) है। ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "फिर eigenvalues का उत्पाद निर्धारक है, जो इस उदाहरण में 3*1 - 1*4, या -1 है। ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "इसका मतलब है कि स्वदेशी मान 2±sqrt(5) हैं। ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "और फिर निर्धारक 2*8 - 7*1, या 9 है। ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "तो इस उदाहरण में, eigenvalues 5 ± sqrt(16) जैसा दिखता है, जो 9 और 1 के रूप में और भी सरल हो जाता है। ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "यदि आप क्वांटम यांत्रिकी को जानते हैं, तो आप जानेंगे कि आव्यूहों के स्वदेशी मान उनके द्वारा वर्णित भौतिकी के लिए अत्यधिक प्रासंगिक हैं।और यदि आप क्वांटम यांत्रिकी नहीं जानते हैं, तो आइए एक छोटी सी झलक देखें कि ये संगणनाएँ वास्तव में वास्तविक अनुप्रयोगों के लिए कितनी प्रासंगिक हैं।", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "तीनों मामलों में विकर्ण प्रविष्टियों का माध्य शून्य है।तो इन सभी मामलों में eigenvalues का माध्य शून्य है, जो हमारे सूत्र को विशेष रूप से सरल बनाता है।", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "पहले वाले के लिए, यह 0 शून्य से 1, या नकारात्मक 1 है।", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "और अंतिम नकारात्मक 1 घटा 0 जैसा दिखता है।", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "इसलिए सभी मामलों में, eigenvalues प्लस और माइनस 1 को सरल बनाते हैं।", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "हालाँकि इस मामले में, आपको वास्तव में दो मान खोजने के लिए किसी सूत्र की आवश्यकता नहीं है यदि आप जानते हैं कि वे 0 के आसपास समान दूरी पर हैं और उनका उत्पाद नकारात्मक 1 है।", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "और तथ्य यह है कि उनके eigenvalues प्लस और माइनस 1 हैं, इस विचार से मेल खाते हैं कि स्पिन के लिए मान जो आप देखेंगे वह या तो पूरी तरह से एक दिशा में या पूरी तरह से दूसरे में होगा, बीच में लगातार कुछ के विपरीत।", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "मेरा मतलब है, पहले वाले पर एक नज़र डालें।प्रासंगिक निर्धारक सीधे आपको लैम्ब्डा वर्ग शून्य से एक का एक विशिष्ट बहुपद देता है, और स्पष्ट रूप से इसमें प्लस और माइनस एक की जड़ें होती हैं।", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "जब आप दूसरा मैट्रिक्स करते हैं तो वही उत्तर होता है, लैम्ब्डा का वर्ग शून्य से एक होता है।", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "अधिक विशेष रूप से, आपको यह मान लेना चाहिए कि यह वेक्टर सामान्यीकृत है, जिसका अर्थ है कि एक वर्ग जमा बी वर्ग जमा सी वर्ग एक के बराबर है।", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "विशेष रूप से, यदि बहुपद को सामान्यीकृत किया जाता है ताकि यह अग्रणी गुणांक एक हो, तो मूलों का माध्य इस रैखिक गुणांक का आधा गुना ऋणात्मक होगा, जो उन मूलों के योग का एक गुना ऋणात्मक है।", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "लेकिन वास्तविक लाभ सिर्फ यह नहीं है कि इसमें याद रखने के लिए कम प्रतीक हैं, बल्कि यह है कि उनमें से प्रत्येक अपने साथ अधिक अर्थ रखता है।", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "उम्मीद यह है कि यह सिर्फ एक और चीज नहीं है जिसे आप याद करते हैं, बल्कि यह कि फ्रेमिंग कुछ अन्य अच्छे तथ्यों को पुष्ट करती है जो जानने लायक हैं, जैसे कि ट्रेस और निर्धारक आइगेनवैल्यू से कैसे संबंधित हैं।", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/hungarian/sentence_translations.json b/2021/quick-eigen/hungarian/sentence_translations.json index 9bf96c40e..6dd8625ac 100644 --- a/2021/quick-eigen/hungarian/sentence_translations.json +++ b/2021/quick-eigen/hungarian/sentence_translations.json @@ -40,7 +40,7 @@ "end": 38.6 }, { - "input": "When you write this as an equation, and you rearrange a little bit, what you see is that if the number lambda is an eigenvalue of a matrix A, then the eigenvector is then the corresponding eigenvector to the zero vector, which in turn means that the determinant of this modified matrix must be zero.", + "input": "When you write this as an equation, and you rearrange a little bit, what you see is that if the number lambda is an eigenvalue of a matrix A, then the matrix A minus lambda times the identity must send some non-zero vector, namely the corresponding eigenvector, to the zero vector, which in turn means that the determinant of this modified matrix must be zero.", "translatedText": "Ha ezt egyenletként írjuk fel, és egy kicsit átrendezzük, akkor azt látjuk, hogy ha a lambda szám egy A mátrix sajátértéke, akkor a sajátvektor a nullvektornak megfelelő sajátvektor, ami viszont azt jelenti, hogy ennek a módosított mátrixnak a determinánsának nullának kell lennie.", "model": "DeepL", "n_reviews": 0, diff --git a/2021/quick-eigen/indonesian/sentence_translations.json b/2021/quick-eigen/indonesian/sentence_translations.json index 4222195b0..a7ae4f6aa 100644 --- a/2021/quick-eigen/indonesian/sentence_translations.json +++ b/2021/quick-eigen/indonesian/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "Jika Anda belum terbiasa dengan nilai eigen, silakan lihat video di sini, yang sebenarnya dimaksudkan untuk memperkenalkannya. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "Oke, itu semua agak sulit untuk dikatakan, tapi sekali lagi, saya berasumsi bahwa semua ini adalah ulasan untuk Anda yang menonton. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "Sejujurnya, prosesnya tidak buruk, tapi setidaknya untuk matriks 2x2, ada cara yang lebih langsung untuk mendapatkan jawabannya. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "Sekarang luangkan waktu sejenak untuk melihat apakah Anda dapat memperoleh fakta ketiga yang relevan, yaitu bagaimana Anda dapat dengan cepat memulihkan dua angka ketika Anda mengetahui meannya dan mengetahui hasil kali keduanya. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "Anda tahu bahwa kedua nilai tersebut berjarak sama di sekitar angka 7, sehingga terlihat seperti 7 plus atau minus, sebut saja d untuk jarak. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "Nah dari situ kalian bisa langsung mencari d. d kuadrat adalah 7 kuadrat dikurangi 40, atau 9, yang berarti d itu sendiri adalah 3. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "Hal ini memberikan fakta penting yang ketiga, yaitu ketika dua bilangan memiliki rata-rata m dan hasil kali p, Anda dapat menuliskan kedua bilangan tersebut sebagai m plus atau minus akar kuadrat dari m kuadrat dikurangi p. Ini cukup cepat untuk diturunkan kembali dengan cepat jika Anda lupa, dan pada dasarnya ini hanyalah penyusunan ulang rumus selisih kuadrat. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "Faktanya, teman saya Tim dari saluran A Capella Science menulis jingle singkat yang bagus untuk membuatnya lebih berkesan. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "Jadi pada contoh ini mean nilai eigennya sama dengan mean dari 3 dan 1 yaitu 2, jadi yang mulai dituliskan adalah 2 ditambah atau dikurangi akar kuadrat dari 2 kuadrat dikurangi, maka hasil kali nilai eigennya adalah determinan yang pada contoh ini adalah 3 dikali 1 dikurangi 1 dikali 4, atau negatif 1, jadi itu yang terakhir diisi, artinya nilai eigennya adalah 2 ditambah atau dikurangi akar kuadrat dari 5. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "Jadi dalam contoh ini, nilai eigen terlihat seperti 5 ditambah atau dikurangi akar kuadrat dari 16, yang disederhanakan menjadi 9 dan 1. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "Jika Anda mengetahui mekanika kuantum, Anda akan mengetahui bahwa nilai eigen matriks sangat relevan dengan fisika yang dijelaskannya. Dan jika Anda belum mengetahui mekanika kuantum, biarkan ini menjadi sekilas bagaimana perhitungan ini sebenarnya sangat relevan dengan aplikasi nyata. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "Rata-rata entri diagonal dalam ketiga kasus tersebut adalah nol. Jadi rata-rata nilai eigen dalam semua kasus ini adalah nol, yang membuat rumus kita terlihat sangat sederhana. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "Untuk yang pertama, nilainya 0 dikurangi 1, atau negatif 1. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "Dan yang terakhir terlihat seperti negatif 1 dikurangi 0. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "Jadi dalam semua kasus, nilai eigen disederhanakan menjadi plus dan minus 1. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "Meskipun dalam kasus ini, Anda sebenarnya tidak memerlukan rumus untuk mencari dua nilai jika Anda mengetahui jarak keduanya sama di sekitar 0 dan hasil kali keduanya negatif 1. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "Dan fakta bahwa nilai eigennya adalah plus dan minus 1 sesuai dengan gagasan bahwa nilai putaran yang akan Anda amati akan seluruhnya berada pada satu arah atau seluruhnya pada arah lain, dan bukan sesuatu yang terus-menerus berada di antara keduanya. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "Maksudku, lihat yang pertama. Penentu yang relevan secara langsung memberi Anda polinomial karakteristik lambda kuadrat dikurangi satu, dan jelas memiliki akar plus dan minus satu. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "Jawaban yang sama ketika Anda mengerjakan matriks kedua, lambda kuadrat dikurangi satu. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "Lebih khusus lagi, Anda harus berasumsi bahwa vektor ini dinormalisasi, artinya a kuadrat ditambah b kuadrat ditambah c kuadrat sama dengan satu. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "Khususnya, jika polinomialnya dinormalisasi sehingga koefisien utamanya adalah satu, maka rata-rata akar-akarnya akan bernilai negatif satu setengah kali koefisien liniernya, yang berarti negatif satu kali jumlah akar-akarnya. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "Namun keuntungan sebenarnya bukan hanya simbol yang harus dihafal lebih sedikit, namun masing-masing simbol mempunyai lebih banyak makna. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "Harapannya adalah bukan hanya satu hal lagi yang Anda hafal, namun framing tersebut memperkuat beberapa fakta bagus lainnya yang perlu diketahui, seperti bagaimana jejak dan determinan terkait dengan nilai eigen. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/italian/sentence_translations.json b/2021/quick-eigen/italian/sentence_translations.json index f83526289..9e9a67b42 100644 --- a/2021/quick-eigen/italian/sentence_translations.json +++ b/2021/quick-eigen/italian/sentence_translations.json @@ -7,7 +7,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "Se non hai familiarità con gli autovalori, dai un'occhiata a questo video che li presenta. ", "n_reviews": 0, "start": 8.58, @@ -35,7 +35,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "Ok, è tutto un po' lungo da dire, ma, ancora una volta, suppongo che tutto questo sia una recensione per chiunque guardi. ", "n_reviews": 0, "start": 66.12, @@ -77,7 +77,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "Ma almeno per le matrici 2x2 esiste un modo molto più diretto per ottenere questa risposta. ", "n_reviews": 0, "start": 109.58, @@ -147,7 +147,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "Ora prenditi un momento per vedere come puoi ricavare quello che sarà il nostro terzo fatto rilevante, ovvero come recuperare due numeri quando conosci la loro media e conosci il loro prodotto. ", "n_reviews": 0, "start": 192.78, @@ -161,7 +161,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "Sai che i due valori sono equamente distanziati attorno a 7, quindi sembrano 7 più o meno qualcosa; chiamiamolo qualcosa "d" per distanza. ", "n_reviews": 0, "start": 204.2, @@ -182,7 +182,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "Quindi da lì puoi trovare direttamente d: d^2 è 7^2 - 40 o 9, il che significa che d stesso è 3. ", "n_reviews": 0, "start": 224.56, @@ -210,7 +210,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "Ciò fornisce il terzo fatto chiave, ovvero che quando due numeri hanno una media m e un prodotto p, puoi scrivere quei due numeri come m ± sqrt(m^2 - p) Questo è abbastanza veloce da ricalcolare al volo se tu dimenticatelo mai, ed è essenzialmente solo una riformulazione della formula della differenza dei quadrati. ", "n_reviews": 0, "start": 257.56, @@ -224,7 +224,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "Infatti, il mio amico Tim del canale acapellascience ci ha scritto un breve jingle per renderlo un po' più memorabile. ", "n_reviews": 0, "start": 281.22, @@ -259,14 +259,14 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "Quindi la cosa che inizi a scrivere è 2 ± sqrt(2^2 - …). ", "n_reviews": 0, "start": 318.3, "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "Quindi il prodotto degli autovalori è il determinante, che in questo esempio è 3*1 - 1*4 o -1. ", "n_reviews": 0, "start": 323.54, @@ -280,7 +280,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "Ciò significa che gli autovalori sono 2±sqrt(5). ", "n_reviews": 0, "start": 334.88, @@ -315,14 +315,14 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "E quindi il determinante è 2*8 - 7*1, ovvero 9. ", "n_reviews": 0, "start": 362.98, "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "Quindi in questo esempio gli autovalori appaiono come 5 ± sqrt(16), che semplifica ulteriormente come 9 e 1. ", "n_reviews": 0, "start": 369.52, @@ -364,14 +364,14 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "Se conosci la meccanica quantistica, saprai che gli autovalori delle matrici sono molto rilevanti per la fisica che descrivono, e se non conosci la meccanica quantistica, lascia che questo sia solo un piccolo assaggio di come questi calcoli siano effettivamente rilevanti per la realtà. applicazioni. ", "n_reviews": 0, "start": 418.6, "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "La media della diagonale in tutti e tre i casi è 0, quindi la media degli autovalori in tutti i casi è 0, il che rende la nostra formula particolarmente semplice. ", "n_reviews": 0, "start": 432.54, @@ -385,7 +385,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "Per il primo è 0 - 1 o -1. ", "n_reviews": 0, "start": 449.7, @@ -399,21 +399,21 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "E l'ultimo sembra -1 - 0. ", "n_reviews": 0, "start": 458.84, "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "Quindi in tutti i casi gli autovalori si semplificano in ±1. ", "n_reviews": 0, "start": 462.06, "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "Anche se in questo caso non hai davvero bisogno della formula per trovare due valori se sai che sono equidistanti attorno allo 0 e il loro prodotto è -1. ", "n_reviews": 0, "start": 466.72, @@ -427,7 +427,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "Il fatto che i loro autovalori siano ±1 corrisponde all'idea che i valori dello spin che osserveresti sarebbero o interamente in una direzione o interamente in un'altra, invece di qualcosa che varia continuamente nel mezzo. ", "n_reviews": 0, "start": 483.76, @@ -469,14 +469,14 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "Voglio dire, dai un'occhiata al primo: il determinante rilevante ti dà direttamente un polinomio caratteristico di lambda^2 - 1, e chiaramente, che ha radici di più e meno 1. ", "n_reviews": 0, "start": 538.24, "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "Stessa risposta quando crei la seconda matrice, lambda^2 - 1. ", "n_reviews": 0, "start": 548.84, @@ -518,7 +518,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "Più specificamente, dovresti presupporre che questo vettore sia normalizzato, ovvero a^2 + b^2 + c^2 = 1. ", "n_reviews": 0, "start": 590.9, @@ -560,7 +560,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "Nello specifico, se il polinomio è normalizzato in modo che questo coefficiente iniziale sia 1, la media delle radici sarà -½ volte questo coefficiente lineare, che è -1 volte la somma di quelle radici. ", "n_reviews": 0, "start": 655.56, @@ -595,7 +595,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "Ma il vero vantaggio è che ci sono meno simboli da memorizzare, ognuno di essi porta con sé più significato. ", "n_reviews": 0, "start": 690.96, @@ -630,7 +630,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "La speranza è che non sia solo un'altra cosa da memorizzare, ma che l'inquadratura rafforzi altri fatti interessanti che vale la pena conoscere, come il modo in cui la traccia e il determinante si relazionano agli autovalori. ", "n_reviews": 0, "start": 730.28, diff --git a/2021/quick-eigen/japanese/sentence_translations.json b/2021/quick-eigen/japanese/sentence_translations.json index 2b25f1e37..5d5bd9a4d 100644 --- a/2021/quick-eigen/japanese/sentence_translations.json +++ b/2021/quick-eigen/japanese/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "固有値に慣れていない場合は、実際に固有値を紹介することを目的とし たこのビデオをご覧ください。", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "さて、ここまでは少し口が裂けてしまいましたが、繰り返しになりますが、これはすべて、ご覧 になっている皆さんにとっての復習だと思います。", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "しかし、少なくとも 2x2 行列の場合、この答えを得るもっと直接的な方法があります。 ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "ここで、3 番目の関連事実を導き出せるかどうかを確認してください。これは、2 つの数値の平均と積 がわかっているときに、どのようにして 2 つの数値を迅速に復元できるかということです。", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "2 つの値は数字の 7 の周りに等間隔に配置されていることがわかります。そのため、7 プラスまたはマイナスの何かのように見えます。これを距離を表す何か d と呼びます。", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "したがって、そこから d を直接見つけることができます。d の 2 乗は、7 の 2 乗から 40 を引いた値、つまり 9 であり、d 自体が 3 であることを意味します。", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "これにより、3 番目の重要な事実が得られます。つまり、2 つの数値が平均 m と積 p を持つ場合、これら 2 つの数値は、m プラスまたはマイナス m の 2 乗の平方根から p を引いたものとして書けるということです。これは、忘れた場合にその場で再導出するのにかなり高速 であり、本質的には二乗差の公式を言い換えただけです。", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "実際、A Capella Science チャンネルの友人の Tim が、少しでも思い出に残るように、素敵 なジングルを書いてくれました。", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "したがって、書き始めるのは 2 ± sqrt(2^2 - …) です。 ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "は行列式で、この例では 3 掛ける 1 マ イナス 1 掛ける 4、つまりマイナス 1 です。", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "つまり、固有値は 2 プラスまたはマイナス 5 の平方根です。", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "すると、行列式は 2 掛ける 8 から 7 掛ける 1、つまり 9 になります 。", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "したがって、この例では、固有値は 5 プラスマイナス 16 の平方根のようになり、さら に単純化して 9 と 1 になります。", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "量子力学を知っている場合は、行列の固有値が、行列が記述する物理学に非 常に関連していることがわかるでしょう。量子力学を知らない人のために、これらの計算が 実際のアプリケーションにどのように非常に関連しているかを少しだけ見てみましょう。", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "3 つの場合すべての対角要素の平均は 0 です。したがって、これらすべての場合の固有値の平均はゼロであり、このため式が特に単純 に見えます。", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "最初の値は 0 から 1 を引く、つまりマイナス 1 です。", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "そして最後のものはマイナス 1 マ イナス 0 のように見えます。", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "したがって、すべての場合において、固有値は単純化してプラス 1 とマイナス 1 になります。", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "ただし、この場合、2 つの値が 0 の周りに等間隔に配置されており、その積がマイナス 1 であることがわかって いる場合、実際には 2 つの値を見つけるための数式は必要ありません。", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "そして、それらの固有値がプラス 1 とマイナス 1 であるという事実は、観察されるスピンの値が、その間で継続的に変化するもので はなく、完全に一方向か完全に別の方向のいずれかであるという考えに対応していま す。", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "つまり、最初のものを見てください。関連する行列式は、ラムダの 2 乗マイナス 1 の特性多項式を直接与え、明らかにプラスとマイナス 1 の 根を持ちます。", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "2 番目の行列、ラムダ 2 乗マイナス 1 を実行しても同じ答えになります。", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "より具体的には、このベクトルは正規化されている、つ まり、a の 2 乗と b の 2 乗と c の 2 乗が 1 に等しいと仮定する必要があります。", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "具体的には、この主要係数が 1 になるように多項式が正規化されている場合、根の平均はこの線形係数の 2 分 の 1 倍の負の値になり、これはこれらの根の合計の 1 倍の負になります。", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "しかし、本当の利点は、暗記す る記号が少ないというだけではなく、それぞれの記号がより多くの意味を持っているということです。", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "単にもう 1 つ暗記するだけでなく、トレースと行列式が固有値にどのように関 係するかなど、知っておく価値のある他の素晴らしい事実がこの枠組みによって強化される ことが期待されます。", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/korean/sentence_translations.json b/2021/quick-eigen/korean/sentence_translations.json index af25ee8fa..0fd11ceb8 100644 --- a/2021/quick-eigen/korean/sentence_translations.json +++ b/2021/quick-eigen/korean/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "고유값에 대해 잘 모르신다면 실제로 고유값을 소개하는 이 동영상을 시청해 보세요. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "좋아, 말하기엔 좀 장황하지만, 다시 한 번 말씀드리지만, 이 모든 것은 시청하시는 분들을 위한 리뷰라고 가정합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "그러나 적어도 2x2 행렬의 경우에는 이 답을 얻을 수 있는 훨씬 더 직접적인 방법이 있습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "이제 잠시 시간을 내어 세 번째 관련 사실이 무엇인지 도출할 수 있는지 살펴보십시오. 이는 두 숫자의 평균과 곱을 알 때 두 숫자를 빠르게 복구할 수 있는 방법입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "두 값은 숫자 7 주위에 균등한 간격으로 배치되어 있으므로 7에 더하기 또는 빼기 값처럼 보입니다. 거리를 d라고 부르겠습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "그래서 거기에서 d를 직접 찾을 수 있습니다. d 제곱은 7의 제곱 빼기 40, 즉 9입니다. 즉, d 자체는 3입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "이것은 세 번째 핵심 사실을 제공합니다. 즉, 두 숫자가 평균 m과 곱 p를 가질 때 이 두 숫자를 m 더하기 또는 빼기 m 제곱 빼기 p의 제곱근으로 쓸 수 있다는 것입니다. 이것은 잊어버린 경우 즉석에서 다시 파생하는 것이 상당히 빠르며 본질적으로 제곱의 차이 공식을 다시 표현한 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "사실 A Capella Science 채널의 내 친구 Tim이 좀 더 기억에 남을 수 있도록 멋지고 빠른 노래를 만들어 주었습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "따라서 쓰기 시작하는 것은 2 ± sqrt(2^2 - …)입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "는 행렬식입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "고유값은 2 더하기 또는 빼기 5의 제곱근이라는 의미입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "그리고 행렬식은 2*8 - 7*1, 즉 9입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "따라서 이 예에서 고유값은 5 ± sqrt(16)처럼 보이며 이는 9와 1로 더욱 단순화됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "양자 역학을 알고 있다면 행렬의 고유값이 그들이 설명하는 물리학과 매우 관련이 있다는 것을 알게 될 것입니다. 양자역학을 모른다면 이러한 계산이 실제로 실제 응용 프로그램과 어떻게 관련되는지 간단히 살펴보겠습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "세 가지 경우 모두 대각선 항목의 평균은 0입니다. 따라서 이 모든 경우의 고유값의 평균은 0이므로 공식이 특히 단순해 보입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "첫 번째 값은 0 - 1 또는 -1입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "그리고 마지막은 -1 빼기 0처럼 보입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "따라서 모든 경우에 고유값은 플러스 및 마이너스 1로 단순화됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "하지만 이 경우에는 두 값이 0 주위에 균일한 간격으로 있고 그 곱이 -1이라는 것을 알고 있다면 두 값을 찾는 데 공식이 실제로 필요하지 않습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "그리고 그들의 고유값이 플러스와 마이너스 1이라는 사실은 여러분이 관찰하게 될 스핀 값이 그 사이에 연속적으로 범위를 갖는 것이 아니라 완전히 한 방향이거나 완전히 다른 방향일 것이라는 생각과 일치합니다. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "즉, 첫 번째 것을 살펴보십시오. 관련 행렬식은 람다 제곱 - 1의 특성 다항식을 직접 제공하며 분명히 플러스와 마이너스 1의 근을 갖습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "두 번째 행렬인 람다 제곱 빼기 1을 수행할 때에도 같은 답이 나옵니다. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "더 구체적으로 말하면, 이 벡터가 정규화되어 있다고 가정해야 합니다. 즉, a 제곱 + b 제곱 + c 제곱은 1과 같습니다. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "특히, 이 선행 계수가 1이 되도록 다항식을 정규화하면 근의 평균은 이 선형 계수의 1/2이 음수가 되며, 이는 해당 근의 합의 1배가 음수가 됩니다. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "하지만 진짜 장점은 기억해야 할 기호가 적다는 것뿐만 아니라 각 기호가 더 많은 의미를 담고 있다는 것입니다. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "여러분이 기억하는 것이 단지 하나 더 있는 것이 아니라, 추적과 행렬식이 고유값과 어떻게 관련되어 있는지와 같이 알아야 할 가치가 있는 다른 좋은 사실을 프레이밍이 강화하기를 바랍니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/marathi/sentence_translations.json b/2021/quick-eigen/marathi/sentence_translations.json index 64de1e630..7bf0fff20 100644 --- a/2021/quick-eigen/marathi/sentence_translations.json +++ b/2021/quick-eigen/marathi/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "जर तुम्ही eigenvalues बद्दल अपरिचित असाल, तर पुढे जा आणि येथे हा व्हिडिओ पहा, जो प्रत्यक्षात त्यांचा परिचय करून देण्यासाठी आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "ठीक आहे, हे सर्व काही तोंडाने सांगण्यासारखे आहे, परंतु पुन्हा, मी असे गृहीत धरत आहे की हे सर्व तुमच्यापैकी कोणाचेही पुनरावलोकन आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "प्रामाणिकपणे, प्रक्रिया भयंकर नाही, परंतु किमान 2x2 मॅट्रिक्ससाठी, तुम्हाला उत्तर मिळू शकेल असा आणखी थेट मार्ग आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "आता आमची तिसरी संबंधित वस्तुस्थिती काय असेल ते तुम्ही काढू शकता का ते पाहण्यासाठी थोडा वेळ घ्या, म्हणजे जेव्हा तुम्हाला दोन संख्यांची सरासरी माहिती असेल आणि तुम्हाला त्यांचे उत्पादन माहित असेल तेव्हा तुम्ही पटकन कसे पुनर्प्राप्त करू शकता. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "तुम्हाला माहिती आहे की दोन मूल्ये 7 क्रमांकाच्या भोवती समान रीतीने अंतरावर आहेत, म्हणून ती 7 अधिक किंवा वजा सारखी दिसतात, चला याला अंतरासाठी d म्हणू या. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "त्यामुळे तिथून तुम्ही थेट डी. d चा वर्ग 7 वर्ग वजा 40 किंवा 9 आहे, याचा अर्थ d स्वतः 3 आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "हे तिसरे महत्त्वाचे तथ्य देते, जे की जेव्हा दोन संख्यांचा सरासरी m आणि गुणाकार p असतो, तेव्हा तुम्ही त्या दोन संख्यांना m अधिक किंवा m वर्ग वजा p चे वर्गमूळ वजा करू शकता. जर तुम्ही ते विसरलात तर ते परत मिळवण्यासाठी हे सभ्यतेने जलद आहे आणि हे मूलत: स्क्वेअर फॉर्म्युलाच्या फरकाचे रिफ्रेसिंग आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "खरं तर, ए कॅपेला सायन्स चॅनेलवरील माझा मित्र टिम याने आम्हाला ते थोडे अधिक संस्मरणीय बनवण्यासाठी एक छान झटपट जिंगल लिहिली. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "तर या उदाहरणात, eigenvalues चा माध्य 3 आणि 1 च्या सरासरीएवढा आहे, जो 2 आहे, त्यामुळे तुम्ही जी गोष्ट लिहायला सुरुवात कराल ती 2 अधिक किंवा वजा 2 चे वर्गमूळ वजा असेल, तर eigenvalues चे गुणाकार निर्धारक आहे, जे या उदाहरणात 3 गुणिले 1 वजा 1 गुणिले 4 किंवा ऋण 1 आहे, त्यामुळे तुम्ही भरलेली ही अंतिम गोष्ट आहे, म्हणजे eigenvalues 5 चे वर्गमूळ 2 अधिक किंवा वजा आहेत. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "तर या उदाहरणात, इजेनव्हॅल्यू 16 चे वर्गमूळ 5 अधिक किंवा वजा सारखे दिसतात, जे 9 आणि 1 असे आणखी सोपे करतात. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "जर तुम्हाला क्वांटम मेकॅनिक्स माहित असेल, तर तुम्हाला हे समजेल की मॅट्रिक्सची इजिनव्हल्यूज त्यांनी वर्णन केलेल्या भौतिकशास्त्राशी अत्यंत संबंधित आहेत. आणि जर तुम्हाला क्वांटम मेकॅनिक्स माहित नसेल, तर ही गणना वास्तविक अनुप्रयोगांसाठी कशी संबंधित आहे याची थोडीशी झलक द्या. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "तिन्ही प्रकरणांमध्ये कर्णप्रविष्टींचा मध्य शून्य आहे. त्यामुळे या सर्व प्रकरणांमध्ये इजेनव्हॅल्यूजचा मध्य शून्य आहे, ज्यामुळे आपले सूत्र विशेषतः सोपे दिसते. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "पहिल्यासाठी, ते 0 उणे 1 किंवा ऋण 1 आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "आणि अंतिम नकारात्मक 1 वजा 0 असे दिसते. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "त्यामुळे सर्व प्रकरणांमध्ये, eigenvalues plus आणि minus 1 असे सोपे करतात. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "जरी या प्रकरणात, दोन मूल्ये शोधण्यासाठी तुम्हाला खरोखर सूत्राची आवश्यकता नाही जर तुम्हाला माहित असेल की ते समान रीतीने 0 च्या आसपास अंतरावर आहेत आणि त्यांचे उत्पादन ऋण 1 आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "आणि त्यांची eigenvalues plus आणि minus 1 आहेत ही वस्तुस्थिती या कल्पनेशी सुसंगत आहे की तुम्ही निरीक्षण कराल त्या फिरकीची मूल्ये एकतर संपूर्णपणे एका दिशेने किंवा संपूर्णपणे दुसर्‍या दिशेने असतील, याच्या विरुद्ध सतत काही तरी दरम्यान असतील. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "म्हणजे, पहिला एक नजर टाका. संबंधित निर्धारक थेट तुम्हाला लॅम्बडा स्क्वेअर वजा एकचे वैशिष्ट्यपूर्ण बहुपद देतो आणि स्पष्टपणे ज्यामध्ये प्लस आणि वजा एकची मुळे आहेत. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "तुम्ही दुसरे मॅट्रिक्स, lambda स्क्वेअर वजा एक करता तेव्हा तेच उत्तर. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "अधिक विशिष्‍टपणे, तुम्ही असे गृहीत धरले पाहिजे की हा सदिश सामान्यीकृत आहे, याचा अर्थ एक वर्ग अधिक b वर्ग अधिक c वर्ग समान आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "विशेषत:, जर बहुपदीचे सामान्यीकरण केले असेल जेणेकरून हा अग्रगण्य गुणांक एक असेल, तर मुळांची सरासरी या रेखीय गुणांकाच्या दीडपट ऋण असेल, जी त्या मुळांच्या बेरीजच्या एक पट ऋण असेल. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "पण खरा फायदा फक्त लक्षात ठेवण्यासाठी कमी चिन्हे आहेत असा नाही, तर तो असा आहे की त्यांच्यापैकी प्रत्येकाला अधिक अर्थ आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "आशा आहे की तुम्ही लक्षात ठेवलेली आणखी एक गोष्ट नाही, परंतु फ्रेमिंग काही इतर छान तथ्यांना बळकटी देते जे जाणून घेण्यासारखे आहे, जसे की ट्रेस आणि निर्धारक eigenvalues शी कसे संबंधित आहेत. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/persian/sentence_translations.json b/2021/quick-eigen/persian/sentence_translations.json index 2d9704c2a..8947adc15 100644 --- a/2021/quick-eigen/persian/sentence_translations.json +++ b/2021/quick-eigen/persian/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "اگر با مقادیر ویژه آشنا نیستید، ادامه دهید و به این ویدیو در اینجا نگاهی بیندازید، که در واقع برای معرفی آنها است. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "بسیار خوب، همه اینها برای گفتن کمی سخت است، اما دوباره، من فرض می کنم که همه اینها برای هر یک از شما که تماشا می کنید، مرور است. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "راستش را بخواهید، این روند وحشتناک نیست، اما حداقل برای ماتریس های 2x2، راه بسیار مستقیم تری وجود دارد که می توانید در پاسخ به آن برسید. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "حالا یک لحظه وقت بگذارید و ببینید آیا می توانید سومین واقعیت مرتبط ما را استخراج کنید، یعنی چگونه می توانید به سرعت دو عدد را هنگامی که میانگین آنها را می دانید و محصول آنها را می دانید بازیابی کنید. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "می دانید که این دو مقدار در اطراف عدد 7 به طور مساوی فاصله دارند، بنابراین آنها مانند 7 به علاوه یا منهای چیزی به نظر می رسند، بیایید آن را چیزی d برای فاصله بنامیم. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "بنابراین از آنجا می توانید مستقیماً d را پیدا کنید. d مجذور 7 است منهای 40 یا 9 یعنی d خودش 3 است. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "این سومین واقعیت کلیدی را نشان می دهد، و آن این است که وقتی دو عدد دارای میانگین m و حاصلضرب p هستند، می توانید آن دو عدد را به عنوان m بعلاوه یا منهای جذر m مجذور منهای p بنویسید. اگر زمانی آن را فراموش کردید، می‌توان آن را سریع دوباره به‌دست آورد، و اساساً فقط بیان مجدد تفاوت فرمول مربع‌ها است. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "در واقع، دوست من تیم از کانال A Capella Science یک صدای جرنگ سریع زیبا برای ما نوشت تا کمی خاطره انگیزتر شود. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "بنابراین در این مثال، میانگین مقادیر ویژه همان میانگین 3 و 1 است که 2 است، بنابراین چیزی که شروع به نوشتن می کنید 2 به علاوه یا منهای جذر 2 مربع منهای است، سپس حاصل ضرب مقادیر ویژه است. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "بنابراین در این مثال، مقادیر ویژه مانند 5 به علاوه یا منهای جذر 16 به نظر می رسند، که حتی بیشتر به عنوان 9 و 1 ساده می شود. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "اگر مکانیک کوانتومی را بلد باشید، می‌دانید که مقادیر ویژه ماتریس‌ها با فیزیکی که توصیف می‌کنند بسیار مرتبط است. و اگر مکانیک کوانتومی را نمی‌دانید، اجازه دهید این تنها نگاهی اجمالی به چگونگی ارتباط این محاسبات با کاربردهای واقعی باشد. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "میانگین ورودی های مورب در هر سه حالت صفر است. بنابراین میانگین مقادیر ویژه در همه این موارد صفر است که فرمول ما را بسیار ساده می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "و آخرین به نظر منفی 1 منهای 0 است. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "بنابراین در همه موارد، مقادیر ویژه به صورت مثبت و منهای 1 ساده می شوند. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "اگر چه در این مورد، اگر بدانید که دو مقدار به طور مساوی در فاصله 0 قرار دارند و حاصلضرب آنها منفی 1 است، واقعاً به فرمولی برای یافتن دو مقدار نیاز ندارید. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "و این واقعیت که مقادیر ویژه آنها به اضافه و منهای 1 است، با این ایده مطابقت دارد که مقادیر اسپینی که مشاهده می کنید کاملاً در یک جهت یا کاملاً در جهت دیگر خواهد بود، برخلاف چیزی که به طور مداوم در این بین قرار دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "منظورم این است که اولی را نگاه کنید. تعیین کننده مربوطه مستقیماً یک چند جمله ای مشخصه از لامبدا مربع منهای یک به شما می دهد و به وضوح دارای ریشه های مثبت و منفی یک است. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "هنگامی که ماتریس دوم را انجام می دهید، همان پاسخ، مربع لامبدا منهای یک. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "به طور خاص، شما باید فرض کنید که این بردار نرمال شده است، به این معنی که مجذور بعلاوه b مجذور مجذور c برابر با یک است. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "به طور خاص، اگر چند جمله ای نرمال شود به طوری که این ضریب پیشرو یک باشد، میانگین ریشه ها یک نیم برابر این ضریب خطی منفی خواهد بود که یک برابر مجموع آن ریشه ها منفی است. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "اما مزیت واقعی فقط این نیست که نمادهای کمتری برای به خاطر سپردن است، بلکه این است که هر یک از آنها معنای بیشتری را با خود حمل می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "امید این است که این فقط یک چیز دیگر نیست که به خاطر بسپارید، بلکه کادربندی برخی حقایق خوب دیگر را که ارزش دانستن دارند، مانند نحوه ارتباط ردیابی و تعیین کننده با مقادیر ویژه، تقویت کند. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/portuguese/sentence_translations.json b/2021/quick-eigen/portuguese/sentence_translations.json index f514e1e0c..96f48bb97 100644 --- a/2021/quick-eigen/portuguese/sentence_translations.json +++ b/2021/quick-eigen/portuguese/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "Se você não está familiarizado com autovalores, vá em frente e dê uma olhada neste vídeo aqui, que na verdade pretende apresentá-los. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "Ok, isso é um pouco complicado de dizer, mas, novamente, presumo que tudo isso seja uma revisão para qualquer um de vocês que esteja assistindo. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "Mas pelo menos para matrizes 2x2, há uma maneira muito mais direta de chegar a essa resposta. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "Agora reserve um momento para ver se consegue derivar qual será o nosso terceiro facto relevante, que é como pode recuperar rapidamente dois números quando conhece a sua média e conhece o seu produto. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "Você sabe que os dois valores estão espaçados uniformemente em torno do número 7, então eles se parecem com 7 mais ou menos alguma coisa, vamos chamar isso de algo d para distância. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "Então, a partir daí, você pode encontrar diretamente d. d ao quadrado é 7 ao quadrado menos 40, ou 9, o que significa que d em si é 3. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "Isso fornece o terceiro fato importante, que é que quando dois números têm uma média m e um produto p, você pode escrever esses dois números como m mais ou menos a raiz quadrada de m ao quadrado menos p. Isso é decentemente rápido para derivar novamente na hora, caso você esqueça, e é essencialmente apenas uma reformulação da fórmula da diferença de quadrados. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "Na verdade, meu amigo Tim, do canal A Capella Science, escreveu-nos um jingle rápido e agradável para torná-lo um pouco mais memorável. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "Então o que você começa a escrever é 2 ± sqrt(2^2 -…). ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "Então o produto dos autovalores é o determinante, que neste exemplo é 3*1 - 1*4, ou -1. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "Isso significa que os autovalores são 2±sqrt(5). ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "E então o determinante é 2*8 - 7*1, ou 9. ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "Portanto, neste exemplo, os autovalores se parecem com 5 ± sqrt(16), o que simplifica ainda mais como 9 e 1. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "Se você conhece mecânica quântica, saberá que os autovalores das matrizes são altamente relevantes para a física que descrevem. E se você não conhece mecânica quântica, deixe isto ser apenas um pequeno vislumbre de como esses cálculos são realmente muito relevantes para aplicações reais. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "A média das entradas diagonais em todos os três casos é zero. Portanto, a média dos autovalores em todos estes casos é zero, o que faz com que a nossa fórmula pareça especialmente simples. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "Para o primeiro, é 0 menos 1 ou menos 1. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "E o último parece menos 1 menos 0. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "Portanto, em todos os casos, os autovalores simplificam para ser mais e menos 1. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "Embora, neste caso, você realmente não precise de uma fórmula para encontrar dois valores se souber que eles estão uniformemente espaçados em torno de 0 e seu produto é negativo 1. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "E o facto de os seus autovalores serem mais e menos 1 corresponde à ideia de que os valores do spin que observaríamos seriam ou inteiramente numa direção ou inteiramente noutra, em oposição a algo variando continuamente entre elas. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "Quero dizer, dê uma olhada no primeiro. O determinante relevante fornece diretamente um polinômio característico de lambda ao quadrado menos um, e claramente que tem raízes de mais e menos um. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "A mesma resposta quando você faz a segunda matriz, lambda ao quadrado menos um. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "Mais especificamente, você deve assumir que este vetor é normalizado, o que significa que a ao quadrado mais b ao quadrado mais c ao quadrado é igual a um. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "Especificamente, se o polinômio for normalizado de modo que esse coeficiente principal seja um, então a média das raízes será negativa a metade desse coeficiente linear, que é negativo uma vez a soma dessas raízes. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "Mas a verdadeira vantagem não é apenas o fato de haver menos símbolos para memorizar, é que cada um deles carrega consigo mais significado. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "A esperança é que não seja apenas mais uma coisa que você memorize, mas que o enquadramento reforce alguns outros fatos interessantes que valem a pena conhecer, como a forma como o traço e o determinante estão relacionados aos autovalores. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/russian/sentence_translations.json b/2021/quick-eigen/russian/sentence_translations.json index 1729180b0..f5a9eb035 100644 --- a/2021/quick-eigen/russian/sentence_translations.json +++ b/2021/quick-eigen/russian/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "Если вы не знакомы с собственными значениями, посмотрите это видео, которое на самом деле предназначено для ознакомления с ними. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "Ладно, это все слишком громко сказано, но опять же, я предполагаю, что все это обзор для любого из вас, кто смотрит. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "Но, по крайней мере, для матриц 2x2 есть гораздо более прямой способ получить ответ. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "Теперь найдите время и посмотрите, сможете ли вы вывести наш третий важный факт, а именно, как вы можете быстро восстановить два числа, если вы знаете их среднее значение и их произведение. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "Вы знаете, что два значения равномерно распределены вокруг числа 7, поэтому они выглядят как 7 плюс-минус что-то, назовем это чем-то d, обозначающим расстояние. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "Таким образом, оттуда вы можете напрямую найти d. d в квадрате равно 7 в квадрате минус 40 или 9, что означает, что d само по себе равно 3. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "Это дает третий ключевой факт: когда два числа имеют среднее значение m и произведение p, вы можете записать эти два числа как m плюс или минус квадратный корень из m в квадрате минус p. Это достаточно быстро пересчитать на лету, если вы когда-нибудь забудете об этом, и, по сути, это просто перефразирование формулы разности квадратов. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "На самом деле, мой друг Тим с канала A Capella Science написал нам симпатичный джингл, чтобы сделать его немного более запоминающимся. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "Итак, вы начинаете писать 2 ± sqrt(2^2 - …). ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "Тогда произведение собственных значений является определителем, который в этом примере равен 3*1 - 1*4 или -1. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "Это означает, что собственные значения равны 2±sqrt(5). ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "И тогда определитель 2*8 - 7*1, или 9. ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "Итак, в этом примере собственные значения выглядят как 5 ± sqrt(16), что еще больше упрощается до 9 и 1. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "Если вы знакомы с квантовой механикой, вы знаете, что собственные значения матриц очень важны для описываемой ими физики. И если вы не знаете квантовую механику, пусть это будет лишь кратким представлением о том, насколько эти вычисления на самом деле очень важны для реальных приложений. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "Среднее значение диагональных элементов во всех трех случаях равно нулю. Таким образом, среднее значение собственных значений во всех этих случаях равно нулю, что делает нашу формулу особенно простой. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "Для первого это 0 минус 1 или минус 1. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "И последний выглядит как минус 1 минус 0. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "Таким образом, во всех случаях собственные значения упрощаются до плюса и минус 1. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "Хотя в этом случае вам действительно не нужна формула для нахождения двух значений, если вы знаете, что они равномерно расположены вокруг 0, а их произведение отрицательно 1. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "И тот факт, что их собственные значения равны плюс и минус 1, соответствует идее, что наблюдаемые вами значения вращения будут либо полностью в одном направлении, либо полностью в другом, а не в чем-то постоянно колеблющемся между ними. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "Я имею в виду, взгляните на первый. Соответствующий определитель напрямую дает вам характеристический многочлен лямбда в квадрате минус один, который, очевидно, имеет корни плюс и минус один. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "Тот же ответ, когда вы делаете вторую матрицу, лямбда в квадрате минус один. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "Точнее, вы должны предположить, что этот вектор нормализован, то есть квадрат плюс b в квадрате плюс с в квадрате равны единице. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "В частности, если полином нормализован так, что главный коэффициент равен единице, то среднее значение корней будет отрицательным в половину этого линейного коэффициента, который является отрицательным в один раз суммы этих корней. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "Но настоящее преимущество не только в том, что нужно запоминать меньше символов, но и в том, что каждый из них несет в себе больше смысла. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "Мы надеемся, что это не просто еще одна вещь, которую вы запомните, но и то, что формулировка подкрепит некоторые другие интересные факты, которые стоит знать, например, как след и определитель связаны с собственными значениями. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/spanish/sentence_translations.json b/2021/quick-eigen/spanish/sentence_translations.json index fd59f1f8d..b93f8502b 100644 --- a/2021/quick-eigen/spanish/sentence_translations.json +++ b/2021/quick-eigen/spanish/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "Si no está familiarizado con los valores propios, continúe y mire este video aquí, que en realidad está destinado a presentarlos. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "Bien, eso es un poco complicado de decir, pero nuevamente, supongo que todo esto es una revisión para cualquiera de los que estén mirando. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "Pero al menos para matrices de 2x2, hay una forma mucho más directa de llegar a esta respuesta. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "Ahora tómate un momento para ver si puedes derivar cuál será nuestro tercer hecho relevante, que es cómo puedes recuperar rápidamente dos números cuando conoces su media y conoces su producto. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "Sabes que los dos valores están espaciados uniformemente alrededor del número 7, por lo que parecen 7 más o menos algo, llamémoslo d para la distancia. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "Entonces, desde allí, puedes encontrar directamente d. d al cuadrado es 7 al cuadrado menos 40, o 9, lo que significa que d en sí es 3. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "Esto nos da el tercer hecho clave, que es que cuando dos números tienen una media m y un producto p, puedes escribir esos dos números como m más o menos la raíz cuadrada de m al cuadrado menos p. Esto es bastante rápido para volver a derivar sobre la marcha si alguna vez lo olvidas, y es esencialmente solo una reformulación de la fórmula de diferencia de cuadrados. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "De hecho, mi amigo Tim del canal A Capella Science nos escribió un lindo jingle rápido para hacerlo un poco más memorable. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "Entonces, lo que empiezas a escribir es 2 ± sqrt(2^2 -…). ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "Entonces el producto de los valores propios es el determinante, que en este ejemplo es 3*1 - 1*4, o -1. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "Esto significa que los valores propios son 2±sqrt(5). ", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "Y luego el determinante es 2*8 - 7*1, o 9. ", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "Entonces, en este ejemplo, los valores propios se ven como 5 ± sqrt(16), lo que se simplifica aún más como 9 y 1. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "Si conoces la mecánica cuántica, sabrás que los valores propios de las matrices son muy relevantes para la física que describen. Y si no conoce la mecánica cuántica, esto le permitirá echar un vistazo a cómo estos cálculos son realmente muy relevantes para aplicaciones reales. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "La media de las entradas diagonales en los tres casos es cero. Entonces, la media de los valores propios en todos estos casos es cero, lo que hace que nuestra fórmula parezca especialmente simple. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "Para el primero, es 0 menos 1 o 1 negativo. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "Y el final parece menos 1 menos 0. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "Entonces, en todos los casos, los valores propios se simplifican para ser más y menos 1. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "Aunque en este caso, realmente no necesitas una fórmula para encontrar dos valores si sabes que están espaciados uniformemente alrededor de 0 y su producto es negativo 1. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "Y el hecho de que sus valores propios sean más y menos 1 corresponde con la idea de que los valores del giro que se observarían estarían completamente en una dirección o completamente en otra, en contraposición a algo que varía continuamente en el medio. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "Quiero decir, eche un vistazo al primero. El determinante relevante te da directamente un polinomio característico de lambda al cuadrado menos uno, y claramente tiene raíces de más y menos uno. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "La misma respuesta cuando haces la segunda matriz, lambda al cuadrado menos uno. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "Más específicamente, debes asumir que este vector está normalizado, lo que significa que a al cuadrado más b al cuadrado más c al cuadrado es igual a uno. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "Específicamente, si el polinomio se normaliza para que este coeficiente principal sea uno, entonces la media de las raíces será negativa la mitad de este coeficiente lineal, que es negativa una vez la suma de esas raíces. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "Pero la verdadera ventaja no es sólo que hay menos símbolos que memorizar, sino que cada uno de ellos tiene más significado. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "La esperanza es que no sea solo una cosa más que memorizas, sino que el encuadre refuerce algunos otros hechos interesantes que vale la pena conocer, como cómo la traza y el determinante se relacionan con los valores propios. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/tamil/sentence_translations.json b/2021/quick-eigen/tamil/sentence_translations.json index 626b75f51..e5a47a708 100644 --- a/2021/quick-eigen/tamil/sentence_translations.json +++ b/2021/quick-eigen/tamil/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "உங்களுக்கு ஈஜென் மதிப்புகள் பற்றித் தெரியாவிட்டால், அவற்றை அறிமுகப்படுத்தும் வகையில் இருக்கும் இந்த வீடியோவை இங்கே பாருங்கள். ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "சரி, அதெல்லாம் கொஞ்சம் வாய்விட்டுச் சொல்ல வேண்டும், ஆனால் மீண்டும், உங்களில் எவருக்கும் இது விமர்சனம் என்று நான் கருதுகிறேன். ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "நேர்மையாக, செயல்முறை பயங்கரமானது அல்ல, ஆனால் குறைந்தபட்சம் 2x2 மெட்ரிக்குகளுக்கு, நீங்கள் பதிலைப் பெறக்கூடிய நேரடியான வழி உள்ளது. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "இப்போது எங்களின் மூன்றாவது பொருத்தமான உண்மை என்ன என்பதை உங்களால் பெற முடியுமா என்பதைப் பார்க்க சிறிது நேரம் ஒதுக்குங்கள், அதாவது இரண்டு எண்களின் சராசரியை நீங்கள் அறிந்ததும் அவற்றின் தயாரிப்பு உங்களுக்குத் தெரிந்ததும் அவற்றை விரைவாக மீட்டெடுக்கலாம். ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "7 என்ற எண்ணைச் சுற்றி இரண்டு மதிப்புகளும் சம இடைவெளியில் இருப்பது உங்களுக்குத் தெரியும், எனவே அவை 7 கூட்டல் அல்லது கழித்தல் போல இருக்கும், அதை தூரத்திற்கு d என்று அழைப்போம். ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "எனவே அங்கிருந்து, நீங்கள் நேரடியாக டி கண்டுபிடிக்கலாம். d ஸ்கொயர் என்பது 7 ஸ்கொயர் மைனஸ் 40 அல்லது 9, அதாவது d தானே 3 ஆகும். ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "இது மூன்றாவது முக்கிய உண்மையைத் தருகிறது, அதாவது இரண்டு எண்களுக்கு சராசரி m மற்றும் ஒரு தயாரிப்பு p இருந்தால், அந்த இரண்டு எண்களையும் m க்ளஸ் அல்லது m ஸ்கொயர்ட் மைனஸ் p இன் வர்க்க மூலத்தைக் கழிக்கலாம். நீங்கள் எப்போதாவது அதை மறந்துவிட்டால், பறக்கும்போது மீண்டும் பெறுவதற்கு இது மிகவும் விரைவானது, மேலும் இது அடிப்படையில் சதுர சூத்திரத்தின் வேறுபாட்டை மறுவடிவமைப்பதாகும். ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "உண்மையில், A Capella Science சேனலைச் சேர்ந்த எனது நண்பர் டிம், அதை இன்னும் கொஞ்சம் மறக்கமுடியாத வகையில் எங்களுக்கு ஒரு நல்ல விரைவான ஜிங்கிள் எழுதினார். ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "எனவே இந்த எடுத்துக்காட்டில், eigenvalues இன் சராசரியானது 3 மற்றும் 1 இன் சராசரி, அதாவது 2 ஆகும், எனவே நீங்கள் எழுதத் தொடங்கும் விஷயம் 2 க்ளோஸ் அல்லது மைனஸ் 2 ஸ்கொயர் மைனஸின் வர்க்க மூலத்தை, பிறகு ஈஜென் மதிப்புகளின் பலன் ஆகும். ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "எனவே இந்த எடுத்துக்காட்டில், eigenvalues 16 இன் வர்க்க மூலத்தை 5 கூட்டல் அல்லது கழித்தல் போல் இருக்கும், இது 9 மற்றும் 1 என மேலும் எளிதாக்குகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "குவாண்டம் இயக்கவியல் உங்களுக்குத் தெரிந்திருந்தால், மெட்ரிக்குகளின் ஈஜென் மதிப்புகள் அவை விவரிக்கும் இயற்பியலுக்கு மிகவும் பொருத்தமானவை என்பதை நீங்கள் அறிவீர்கள். குவாண்டம் இயக்கவியல் உங்களுக்குத் தெரியாவிட்டால், இந்த கணக்கீடுகள் உண்மையில் உண்மையான பயன்பாடுகளுக்கு எவ்வாறு மிகவும் பொருத்தமானவை என்பதைப் பற்றிய ஒரு சிறிய பார்வையாக இருக்கட்டும். ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "மூன்று நிகழ்வுகளிலும் மூலைவிட்ட உள்ளீடுகளின் சராசரி பூஜ்ஜியமாகும். எனவே இந்த எல்லா நிகழ்வுகளிலும் ஈஜென் மதிப்புகளின் சராசரி பூஜ்ஜியமாகும், இது எங்கள் சூத்திரத்தை குறிப்பாக எளிதாக்குகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "முதல்வருக்கு, இது 0 கழித்தல் 1 அல்லது எதிர்மறை 1. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "மேலும் இறுதியானது எதிர்மறை 1 கழித்தல் 0 போல் தெரிகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "எனவே எல்லா சந்தர்ப்பங்களிலும், ஈஜென் மதிப்புகள் கூட்டல் மற்றும் கழித்தல் 1 ஆக எளிமைப்படுத்தப்படுகின்றன. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "இந்த விஷயத்தில், இரண்டு மதிப்புகள் 0-ஐச் சுற்றி சம இடைவெளியில் உள்ளன மற்றும் அவற்றின் தயாரிப்பு எதிர்மறை 1 என்று உங்களுக்குத் தெரிந்தால், அவற்றைக் கண்டுபிடிக்க உங்களுக்கு சூத்திரம் தேவையில்லை. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "அவற்றின் ஈஜென் மதிப்புகள் கூட்டல் மற்றும் கழித்தல் 1 என்பது நீங்கள் கவனிக்கும் சுழலுக்கான மதிப்புகள் முற்றிலும் ஒரு திசையில் அல்லது முற்றிலும் வேறொரு திசையில் இருக்கும் என்ற எண்ணத்துடன் ஒத்துப்போகிறது. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "அதாவது, முதல் ஒன்றைப் பாருங்கள். தொடர்புடைய டிடர்மினன்ட், லாம்ப்டா ஸ்கொயர் மைனஸ் ஒன்னின் சிறப்பியல்பு பல்லுறுப்புக்கோவையை நேரடியாக உங்களுக்கு வழங்குகிறது, மேலும் அது பிளஸ் மற்றும் மைனஸ் ஒன்றின் வேர்களைக் கொண்டுள்ளது. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "இரண்டாவது மேட்ரிக்ஸ், லாம்ப்டா ஸ்கொயர் மைனஸ் ஒன் செய்யும் போது அதே பதில். ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "மேலும் குறிப்பாக, இந்த திசையன் இயல்பாக்கப்பட்டதாக நீங்கள் கருத வேண்டும், அதாவது ஒரு ஸ்கொயர் பிளஸ் பி ஸ்கொயர் மற்றும் சி ஸ்கொயர் ஒன்றுக்கு சமம். ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "குறிப்பாக, பல்லுறுப்புக்கோவை இயல்பாக்கப்பட்டால், இந்த முன்னணி குணகம் ஒன்று, வேர்களின் சராசரியானது இந்த நேரியல் குணகத்தின் பாதி மடங்கு எதிர்மறையாக இருக்கும், இது அந்த வேர்களின் கூட்டுத்தொகையை விட ஒரு மடங்கு எதிர்மறையாக இருக்கும். ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "ஆனால் உண்மையான நன்மை என்னவென்றால், மனப்பாடம் செய்வதற்கு குறைவான குறியீடுகள் இருப்பது மட்டுமல்ல, அவை ஒவ்வொன்றும் அதனுடன் அதிக அர்த்தத்தைக் கொண்டுள்ளன. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "நம்பிக்கை என்னவென்றால், நீங்கள் மனப்பாடம் செய்யும் மற்றொரு விஷயம் மட்டும் அல்ல, ஆனால், சுவடு மற்றும் தீர்மானிப்பவர் ஈஜென் மதிப்புகளுடன் எவ்வாறு தொடர்புடையது என்பது போன்ற தெரிந்து கொள்ள வேண்டிய வேறு சில நல்ல உண்மைகளை இந்த ஃப்ரேமிங் வலுப்படுத்துகிறது. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/telugu/sentence_translations.json b/2021/quick-eigen/telugu/sentence_translations.json index 06ee60cb0..1a28dad5a 100644 --- a/2021/quick-eigen/telugu/sentence_translations.json +++ b/2021/quick-eigen/telugu/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "మీకు ఈజెన్‌వాల్యూస్ గురించి తెలియకపోతే, ముందుకు సాగండి మరియు వాటిని పరిచయం చేయడానికి ఉద్దేశించిన ఈ వీడియోను ఇక్కడ చూడండి. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "సరే, చెప్పడానికి కొంచెం నోరు మెదపడం లేదు, కానీ మళ్ళీ, మీలో ఎవరికైనా వీక్షించే సమీక్ష అని నేను అనుకుంటున్నాను. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "నిజాయితీగా, ప్రక్రియ భయంకరమైనది కాదు, కానీ కనీసం 2x2 మాత్రికల కోసం, మీరు సమాధానాన్ని పొందగలిగే మరింత ప్రత్యక్ష మార్గం ఉంది. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "ఇప్పుడు మీరు మా మూడవ సంబంధిత వాస్తవాన్ని పొందగలరో లేదో తెలుసుకోవడానికి కొంత సమయం వెచ్చించండి, అంటే రెండు సంఖ్యల సగటు మీకు తెలిసినప్పుడు మరియు వాటి ఉత్పత్తి మీకు తెలిసినప్పుడు మీరు వాటిని త్వరగా తిరిగి పొందగలరు. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "రెండు విలువలు సంఖ్య 7 చుట్టూ సమానంగా ఉన్నాయని మీకు తెలుసు, కాబట్టి అవి 7 ప్లస్ లేదా మైనస్ లాగా కనిపిస్తాయి, దూరానికి d అని పిలుద్దాం. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "కాబట్టి అక్కడ నుండి, మీరు నేరుగా కనుగొనవచ్చు d. d స్క్వేర్డ్ 7 స్క్వేర్డ్ మైనస్ 40 లేదా 9, అంటే d కూడా 3. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "ఇది మూడవ కీలక వాస్తవాన్ని ఇస్తుంది, అంటే రెండు సంఖ్యలు సగటు m మరియు ఉత్పత్తి p కలిగి ఉన్నప్పుడు, మీరు ఆ రెండు సంఖ్యలను m స్క్వేర్డ్ మైనస్ p యొక్క వర్గమూలాన్ని m ప్లస్ లేదా మైనస్ అని వ్రాయవచ్చు. మీరు ఎప్పుడైనా మరచిపోయినట్లయితే, ఎగిరినప్పుడు తిరిగి పొందేందుకు ఇది చాలా వేగంగా ఉంటుంది మరియు ఇది తప్పనిసరిగా స్క్వేర్‌ల ఫార్ములా యొక్క వ్యత్యాసాన్ని తిరిగి మార్చడం మాత్రమే. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "నిజానికి, ఎ కాపెల్లా సైన్స్ ఛానెల్‌కు చెందిన నా స్నేహితుడు టిమ్ మాకు కొంచెం గుర్తుండిపోయేలా చేయడానికి చక్కని క్విక్ జింగిల్‌ని వ్రాసాడు. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "కాబట్టి ఈ ఉదాహరణలో, ఈజెన్‌వాల్యూస్ యొక్క సగటు 3 మరియు 1 యొక్క సగటుతో సమానంగా ఉంటుంది, అంటే 2, కాబట్టి మీరు వ్రాయడం ప్రారంభించిన విషయం 2 స్క్వేర్డ్ మైనస్ యొక్క వర్గమూలాన్ని 2 ప్లస్ లేదా మైనస్ చేసి, ఆపై ఈజెన్‌వాల్యూల ఉత్పత్తి ఈ ఉదాహరణలో 3 రెట్లు 1 మైనస్ 1 సార్లు 4 లేదా ప్రతికూల 1 అనేది డిటర్మినెంట్, కాబట్టి మీరు పూరించే చివరి విషయం ఇది, అంటే ఈజెన్‌వాల్యూలు 2 ప్లస్ లేదా మైనస్ 5 యొక్క వర్గమూలం. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "కాబట్టి ఈ ఉదాహరణలో, ఈజెన్‌వాల్యూలు 5 ప్లస్ లేదా మైనస్ 16 యొక్క వర్గమూలం వలె కనిపిస్తాయి, ఇది 9 మరియు 1గా మరింత సులభతరం చేస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "మీకు క్వాంటం మెకానిక్స్ తెలిస్తే, మాత్రికల యొక్క ఈజెన్‌వాల్యూలు వారు వివరించే భౌతిక శాస్త్రానికి అత్యంత సంబంధితంగా ఉంటాయని మీకు తెలుస్తుంది. మరియు మీకు క్వాంటం మెకానిక్స్ తెలియకుంటే, ఈ గణనలు వాస్తవానికి నిజమైన అప్లికేషన్‌లకు ఎలా చాలా సందర్భోచితంగా ఉంటాయో ఇది ఒక చిన్న సంగ్రహావలోకనం. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "మూడు సందర్భాలలో వికర్ణ ఎంట్రీల సగటు సున్నా. కాబట్టి ఈ అన్ని సందర్భాలలో ఈజెన్‌వాల్యూస్ యొక్క సగటు సున్నా, ఇది మన ఫార్ములా చాలా సరళంగా కనిపిస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "మొదటిదానికి, ఇది 0 మైనస్ 1 లేదా ప్రతికూల 1. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "మరియు చివరిది ప్రతికూల 1 మైనస్ 0 లాగా కనిపిస్తుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "కాబట్టి అన్ని సందర్భాల్లో, ఈజెన్‌వాల్యూలు ప్లస్ మరియు మైనస్ 1గా సులభతరం చేయబడతాయి. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "ఈ సందర్భంలో, రెండు విలువలు 0 చుట్టూ సమానంగా ఉన్నాయని మరియు వాటి ఉత్పత్తి ప్రతికూలం 1 అని మీకు తెలిస్తే వాటిని కనుగొనడానికి మీకు నిజంగా ఫార్ములా అవసరం లేదు. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "మరియు వాటి ఈజెన్‌వాల్యూలు ప్లస్ మరియు మైనస్ 1 అనే వాస్తవం మీరు గమనించే స్పిన్‌కు సంబంధించిన విలువలు పూర్తిగా ఒక దిశలో లేదా పూర్తిగా మరొక దిశలో ఉండాలనే ఆలోచనతో అనుగుణంగా ఉంటాయి, అవి మధ్యలో నిరంతరంగా ఉండే వాటికి భిన్నంగా ఉంటాయి. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "నా ఉద్దేశ్యం, మొదటిదాన్ని చూడండి. సంబంధిత డిటర్మినెంట్ నేరుగా మీకు లాంబ్డా స్క్వేర్డ్ మైనస్ వన్ యొక్క లక్షణ బహుపదిని అందిస్తుంది మరియు స్పష్టంగా ప్లస్ మరియు మైనస్ వన్ మూలాలను కలిగి ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "మీరు రెండవ మ్యాట్రిక్స్, లాంబ్డా స్క్వేర్డ్ మైనస్ వన్ చేసినప్పుడు అదే సమాధానం. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "మరింత ప్రత్యేకంగా, మీరు ఈ వెక్టర్ సాధారణీకరించబడిందని భావించాలి, అంటే స్క్వేర్డ్ ప్లస్ బి స్క్వేర్డ్ ప్లస్ సి స్క్వేర్డ్ ఒకదానికి సమానం. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "ప్రత్యేకించి, ఈ లీడింగ్ కోఎఫీషియంట్ ఒకటి ఉండేలా బహుపది సాధారణీకరించబడితే, మూలాల సగటు ఈ లీనియర్ కోఎఫీషియంట్‌కి సగం రెట్లు ప్రతికూలంగా ఉంటుంది, ఇది ఆ మూలాల మొత్తానికి ఒక రెట్లు ప్రతికూలంగా ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "కానీ నిజమైన ప్రయోజనం ఏమిటంటే అది గుర్తుంచుకోవడానికి తక్కువ చిహ్నాలు మాత్రమే కాదు, వాటిలో ప్రతి ఒక్కటి దానితో ఎక్కువ అర్థాన్ని కలిగి ఉంటుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "ఇది మీరు గుర్తుపెట్టుకున్న మరో విషయం మాత్రమే కాదు, ట్రేస్ మరియు డిటర్మినెంట్ ఈజెన్‌వాల్యూస్‌తో ఎలా సంబంధం కలిగి ఉన్నాయో తెలుసుకోవలసిన విలువైన కొన్ని ఇతర మంచి వాస్తవాలను ఫ్రేమింగ్ బలోపేతం చేస్తుందని ఆశిస్తున్నాము. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/thai/sentence_translations.json b/2021/quick-eigen/thai/sentence_translations.json index 0a097f91f..33e006d9c 100644 --- a/2021/quick-eigen/thai/sentence_translations.json +++ b/2021/quick-eigen/thai/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/turkish/sentence_translations.json b/2021/quick-eigen/turkish/sentence_translations.json index 7cb779eb2..479e922d1 100644 --- a/2021/quick-eigen/turkish/sentence_translations.json +++ b/2021/quick-eigen/turkish/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "Özdeğerlere aşina değilseniz, devam edin ve buradaki videoya bir göz atın; bu video aslında onları tanıtmayı amaçlamaktadır. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "Tamam, bunları söylemek biraz fazla ama yine de tüm bunların izleyenleriniz için bir inceleme olduğunu varsayıyorum. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "Dürüst olmak gerekirse süreç fena değil ama en azından 2x2'lik matrisler için cevaba ulaşmanın çok daha doğrudan bir yolu var. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "Şimdi bir dakikanızı ayırın ve üçüncü ilgili gerçeğimizi türetip çıkaramayacağınızı görün; bu, ortalamalarını bildiğinizde ve çarpımlarını bildiğinizde iki sayıyı nasıl hızlı bir şekilde kurtarabileceğinizdir. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "Biliyorsunuz, iki değer 7 sayısının etrafında eşit aralıklarla yerleştirilmiştir, yani 7 artı veya eksi gibi görünürler, buna uzaklık için d diyelim. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "Yani oradan doğrudan d'yi bulabilirsiniz. d kare 7 kare eksi 40 veya 9'dur, bu da d'nin kendisinin 3 olduğu anlamına gelir. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "Bu üçüncü temel gerçeği verir; iki sayının ortalaması m ve çarpımı p olduğunda, bu iki sayıyı m artı veya eksi m kare eksi p'nin karekökü olarak yazabilirsiniz. Bu, unutursanız anında yeniden türetilmesi oldukça hızlıdır ve aslında sadece kareler farkı formülünün yeniden ifade edilmesidir. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "Aslında, A Capella Science kanalından arkadaşım Tim, bunu biraz daha akılda kalıcı kılmak için bize güzel bir kısa şarkı yazdı. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "Yani bu örnekte, özdeğerlerin ortalaması 3 ve 1'in ortalaması ile aynıdır, yani 2, yani yazmaya başladığınız şey 2 artı veya eksi 2'nin karesinin karekökü eksi, o zaman özdeğerlerin çarpımıdır. ", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "Yani bu örnekte özdeğerler 5 artı veya eksi 16'nın karekökü gibi görünüyor, bu da 9 ve 1 olarak daha da basitleşiyor. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "Kuantum mekaniğini biliyorsanız, matrislerin özdeğerlerinin tanımladıkları fizikle son derece alakalı olduğunu bilirsiniz. Ve eğer kuantum mekaniğini bilmiyorsanız, izin verin bu hesaplamaların gerçek uygulamalarla ne kadar alakalı olduğuna dair küçük bir bakış olsun. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "Her üç durumda da çapraz girişlerin ortalaması sıfırdır. Yani tüm bu durumlarda özdeğerlerin ortalaması sıfırdır, bu da formülümüzün özellikle basit görünmesini sağlar. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "İlki için bu 0 eksi 1 veya eksi 1'dir. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "Ve sonuncusu eksi 1 eksi 0'a benziyor. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "Yani her durumda özdeğerler artı ve eksi 1 olacak şekilde basitleştirilir. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "Ancak bu durumda, eğer bu değerlerin 0 civarında eşit aralıklı olduğunu ve çarpımlarının negatif 1 olduğunu biliyorsanız, iki değeri bulmak için gerçekten bir formüle ihtiyacınız yoktur. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "Ve onların özdeğerlerinin artı ve eksi 1 olması gerçeği, gözlemleyeceğiniz dönüş değerlerinin sürekli olarak arada değişen bir şeyin aksine ya tamamen bir yönde ya da tamamen başka bir yönde olacağı fikrine karşılık gelir. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "İlkine bir bakın derim. İlgili determinant size doğrudan lambda kare eksi birin karakteristik polinomunu verir ve bunun köklerinin artı ve eksi bir olduğu açıktır. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "İkinci matrisi yaptığınızda da aynı cevap, lambda kare eksi bir. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "Daha spesifik olarak, bu vektörün normalleştirilmiş olduğunu, yani a kare artı b kare artı c karenin bire eşit olduğunu varsaymalısınız. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "Spesifik olarak, eğer polinom bu baş katsayı bir olacak şekilde normalleştirilirse, köklerin ortalaması bu doğrusal katsayının yarısı kadar negatif olacaktır; bu da köklerin toplamının bir katı negatif olacaktır. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "Ancak asıl avantaj, yalnızca ezberlenecek daha az sembol olması değil, aynı zamanda her birinin daha fazla anlam taşımasıdır. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "Umudumuz, ezberleyeceğiniz tek bir şeyin daha olması değil, çerçevelemenin iz ve determinantın özdeğerlerle nasıl ilişkili olduğu gibi bilmeye değer diğer bazı güzel gerçekleri pekiştirmesidir. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/ukrainian/sentence_translations.json b/2021/quick-eigen/ukrainian/sentence_translations.json index b295fba6b..ee5bd5c7c 100644 --- a/2021/quick-eigen/ukrainian/sentence_translations.json +++ b/2021/quick-eigen/ukrainian/sentence_translations.json @@ -7,7 +7,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "Якщо ви не знайомі з власними значеннями, подивіться це відео, яке ознайомлює з ними. ", "n_reviews": 0, "start": 8.58, @@ -35,7 +35,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "Гаразд, все це трохи зайве, але знову ж таки, я припускаю, що все це огляд для всіх, хто дивиться. ", "n_reviews": 0, "start": 66.12, @@ -77,7 +77,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "Але принаймні для матриць 2x2 є набагато більш прямий спосіб отримати цю відповідь. ", "n_reviews": 0, "start": 109.58, @@ -147,7 +147,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "Тепер знайдіть хвилинку, щоб побачити, як ви можете вивести наш третій релевантний факт, який полягає в тому, як відновити два числа, коли ви знаєте їх середнє значення та добуток. ", "n_reviews": 0, "start": 192.78, @@ -161,7 +161,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "Ви знаєте, що два значення рівномірно розподілені навколо 7, тому вони виглядають як 7 плюс або мінус щось; давайте назвемо це щось "d" для відстані. ", "n_reviews": 0, "start": 204.2, @@ -182,7 +182,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "Отже, звідси ви можете безпосередньо знайти d: d^2 дорівнює 7^2 - 40 або 9, що означає, що саме d дорівнює 3. ", "n_reviews": 0, "start": 224.56, @@ -210,7 +210,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "Це дає третій ключовий факт, який полягає в тому, що коли два числа мають середнє значення m і добуток p, ви можете записати ці два числа як m ± sqrt(m^2 - p). Це досить швидко для повторного визначення на льоту, якщо ви ніколи не забувайте, і це, по суті, просто перефразування формули різниці квадратів. ", "n_reviews": 0, "start": 257.56, @@ -224,7 +224,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "Насправді мій друг Тім з каналу acapellascience написав для нас короткий джингл, щоб зробити його трохи більш запам’ятовуваним. ", "n_reviews": 0, "start": 281.22, @@ -259,14 +259,14 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "Отже, те, що ви починаєте писати, це 2 ± sqrt(2^2 - …). ", "n_reviews": 0, "start": 318.3, "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "Тоді добуток власних значень є визначником, який у цьому прикладі дорівнює 3*1 - 1*4, або -1. ", "n_reviews": 0, "start": 323.54, @@ -280,7 +280,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "Це означає, що власні значення дорівнюють 2±sqrt(5). ", "n_reviews": 0, "start": 334.88, @@ -315,14 +315,14 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "І тоді визначник 2*8 - 7*1 або 9. ", "n_reviews": 0, "start": 362.98, "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "Отже, у цьому прикладі власні значення виглядають як 5 ± sqrt(16), що ще більше спрощує як 9 і 1. ", "n_reviews": 0, "start": 369.52, @@ -364,14 +364,14 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "Якщо ви знаєте квантову механіку, ви знатимете, що власні значення матриць мають велике відношення до фізики, яку вони описують, а якщо ви не знаєте квантової механіки, нехай це буде лише невеликим уявленням про те, як ці обчислення насправді стосуються реальних програми. ", "n_reviews": 0, "start": 418.6, "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "Середнє значення діагоналі в усіх трьох випадках дорівнює 0, тому середнє власних значень у всіх випадках дорівнює 0, що робить нашу формулу особливо простою. ", "n_reviews": 0, "start": 432.54, @@ -385,7 +385,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "Для першого це 0 - 1 або -1. ", "n_reviews": 0, "start": 449.7, @@ -399,21 +399,21 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "І остаточний виглядає як -1 - 0. ", "n_reviews": 0, "start": 458.84, "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "Отже, у всіх випадках власні значення спрощуються до ±1. ", "n_reviews": 0, "start": 462.06, "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "Хоча в цьому випадку вам справді не потрібна формула, щоб знайти два значення, якщо ви знаєте, що вони рівномірно розподілені навколо 0, а їхній добуток дорівнює -1. ", "n_reviews": 0, "start": 466.72, @@ -427,7 +427,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "Той факт, що їхні власні значення дорівнюють ±1, відповідає ідеї, що значення обертання, які ви спостерігали, будуть або повністю в одному напрямку, або повністю в іншому, на відміну від чогось безперервного коливання між ними. ", "n_reviews": 0, "start": 483.76, @@ -469,14 +469,14 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "Я маю на увазі, погляньте на перший: відповідний визначник безпосередньо дає вам характерний поліном лямбда^2 - 1, і, очевидно, він має корені плюс і мінус 1. ", "n_reviews": 0, "start": 538.24, "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "Така сама відповідь, коли ви виконуєте другу матрицю, лямбда^2 - 1. ", "n_reviews": 0, "start": 548.84, @@ -518,7 +518,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "Точніше, ви повинні припустити, що цей вектор нормалізований, тобто a^2 + b^2 + c^2 = 1. ", "n_reviews": 0, "start": 590.9, @@ -560,7 +560,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "Зокрема, якщо поліном нормалізовано таким чином, що цей старший коефіцієнт дорівнює 1, тоді середнє значення коренів дорівнюватиме -½ цього лінійного коефіцієнта, що дорівнює -1 помноженій сумі цих коренів. ", "n_reviews": 0, "start": 655.56, @@ -595,7 +595,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "Але справжня перевага полягає в тому, що потрібно запам’ятати менше символів, а кожен із них має більше значення. ", "n_reviews": 0, "start": 690.96, @@ -630,7 +630,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "Сподіваємося, що це не просто ще одна річ, яку потрібно запам’ятати, а те, що фрейм підкріплює деякі інші цікаві факти, які варто знати, наприклад, як слід і визначник пов’язані з власними значеннями. ", "n_reviews": 0, "start": 730.28, diff --git a/2021/quick-eigen/urdu/sentence_translations.json b/2021/quick-eigen/urdu/sentence_translations.json index 0027a63b6..d793b2e67 100644 --- a/2021/quick-eigen/urdu/sentence_translations.json +++ b/2021/quick-eigen/urdu/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "اگر آپ eigenvalues سے ناواقف ہیں تو آگے بڑھیں اور یہاں اس ویڈیو پر ایک نظر ڈالیں، جس کا مقصد دراصل ان کا تعارف کروانا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "ٹھیک ہے، کہنے کے لیے یہ سب کچھ تھوڑا سا ہے، لیکن ایک بار پھر، میں فرض کر رہا ہوں کہ یہ سب کچھ آپ میں سے کسی کے لیے بھی جائزہ ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "سچ میں، یہ عمل خوفناک نہیں ہے، لیکن کم از کم 2x2 میٹرکس کے لیے، ایک بہت زیادہ سیدھا طریقہ ہے جس سے آپ جواب حاصل کر سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "اب یہ دیکھنے کے لیے تھوڑا وقت نکالیں کہ کیا آپ یہ اخذ کر سکتے ہیں کہ ہماری تیسری متعلقہ حقیقت کیا ہو گی، جس کا مطلب یہ ہے کہ جب آپ کو دو نمبروں کا مطلب معلوم ہو جائے اور آپ ان کی مصنوع کو جانتے ہوں تو آپ تیزی سے بازیافت کر سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "آپ جانتے ہیں کہ دو قدریں نمبر 7 کے ارد گرد یکساں طور پر فاصلہ رکھتی ہیں، اس لیے وہ 7 جمع یا مائنس کی طرح نظر آتی ہیں، آئیے اسے فاصلہ کے لیے d کہتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "تو وہاں سے، آپ براہ راست تلاش کر سکتے ہیں d. d مربع ہے 7 مربع مائنس 40، یا 9، جس کا مطلب ہے کہ d خود 3 ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "اس سے تیسری کلیدی حقیقت ملتی ہے، جو یہ ہے کہ جب دو نمبروں کا اوسط m اور ایک مصنوعہ p ہوتا ہے، تو آپ ان دو نمبروں کو m جمع یا مائنس m مربع مائنس p کا مربع جڑ لکھ سکتے ہیں۔اگر آپ اسے بھول جاتے ہیں تو یہ اڑان بھرنے کے لیے بہت تیز ہے، اور یہ بنیادی طور پر صرف مربع فارمولے کے فرق کو دوبارہ بیان کرنا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "درحقیقت، چینل A Capella Science سے میرے دوست ٹم نے ہمیں ایک اچھا فوری گینگل لکھا تاکہ اسے تھوڑا سا مزید یادگار بنایا جا سکے۔", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "تو اس مثال میں، eigenvalues کا اوسط وہی ہے جو 3 اور 1 کا اوسط ہے، جو کہ 2 ہے، لہذا آپ جس چیز کو لکھنا شروع کرتے ہیں وہ 2 جمع یا مائنس 2 مربع مائنس کا مربع جڑ ہے، پھر eigenvalues کی پیداوار تعین کنندہ ہے، جو اس مثال میں 3 ضرب 1 منفی 1 ضرب 4، یا منفی 1 ہے، تو یہ وہ حتمی چیز ہے جسے آپ بھرتے ہیں، جس کا مطلب ہے کہ eigenvalues 2 جمع یا مائنس 5 کا مربع جڑ ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "تو اس مثال میں، eigenvalues 16 کے مربع جڑ کے 5 جمع یا مائنس کی طرح نظر آتے ہیں، جو 9 اور 1 کو مزید آسان بناتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "اگر آپ کوانٹم میکانکس جانتے ہیں، تو آپ کو معلوم ہو جائے گا کہ میٹرکس کی ایگن ویلیوز ان کی بیان کردہ فزکس سے بہت زیادہ متعلقہ ہیں۔اور اگر آپ کوانٹم میکینکس نہیں جانتے ہیں، تو یہ صرف اس بات کی ایک چھوٹی سی جھلک ہونے دیں کہ یہ کمپیوٹیشنز حقیقی ایپلی کیشنز سے کس طرح بہت زیادہ متعلقہ ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "تینوں صورتوں میں ترچھی اندراجات کا اوسط صفر ہے۔تو ان تمام صورتوں میں eigenvalues کا اوسط صفر ہے، جو ہمارے فارمولے کو خاص طور پر سادہ نظر آتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "اور فائنل منفی 1 مائنس 0 کی طرح لگتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "لہذا تمام صورتوں میں، eigenvalues جمع اور مائنس 1 ہونے کے لیے آسان ہو جاتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "اگرچہ اس صورت میں، آپ کو واقعی دو قدریں تلاش کرنے کے لیے کسی فارمولے کی ضرورت نہیں ہے اگر آپ جانتے ہیں کہ وہ یکساں طور پر 0 کے ارد گرد فاصلہ رکھتے ہیں اور ان کی مصنوع منفی 1 ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "اور حقیقت یہ ہے کہ ان کی eigenvalues جمع اور مائنس 1 اس خیال سے مطابقت رکھتی ہے کہ آپ جس گھماؤ کی قدروں کا مشاہدہ کریں گے وہ یا تو مکمل طور پر ایک سمت میں ہوں گی یا مکمل طور پر کسی دوسری سمت میں ہوں گی، جیسا کہ درمیان میں مسلسل کسی چیز کے خلاف ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "میرا مطلب ہے، پہلے ایک پر ایک نظر ڈالیں۔متعلقہ تعین کنندہ براہ راست آپ کو لیمبڈا اسکوائر مائنس ون کا ایک خصوصیت والا کثیر الثانی دیتا ہے، اور واضح طور پر جس کی جڑیں جمع اور منفی ایک ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "یہی جواب جب آپ دوسرا میٹرکس کرتے ہیں تو لیمبڈا اسکوائر مائنس ون۔", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "مزید خاص طور پر، آپ کو فرض کرنا چاہیے کہ یہ ویکٹر نارملائز ہے، یعنی ایک مربع جمع b مربع جمع c مربع ایک کے برابر ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "خاص طور پر، اگر کثیر الثانی کو نارملائز کیا جاتا ہے تاکہ یہ لیڈنگ گتانک ایک ہو، تو جڑوں کا اوسط اس لکیری گتانک کا نصف گنا منفی ہوگا، جو ان جڑوں کے مجموعے سے ایک گنا منفی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "لیکن اصل فائدہ صرف یہ نہیں ہے کہ حفظ کرنے کے لیے کم علامتیں ہیں، بلکہ یہ ہے کہ ان میں سے ہر ایک اپنے ساتھ زیادہ معنی رکھتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "امید یہ ہے کہ یہ صرف ایک اور چیز نہیں ہے جسے آپ حفظ کرتے ہیں، لیکن یہ کہ فریمنگ کچھ دوسرے اچھے حقائق کو تقویت دیتی ہے جو جاننے کے قابل ہیں، جیسے کہ ٹریس اور تعین کنندہ کا تعلق eigenvalues سے کیسے ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/quick-eigen/vietnamese/sentence_translations.json b/2021/quick-eigen/vietnamese/sentence_translations.json index 4574f6dba..6f841c514 100644 --- a/2021/quick-eigen/vietnamese/sentence_translations.json +++ b/2021/quick-eigen/vietnamese/sentence_translations.json @@ -8,7 +8,7 @@ "end": 7.56 }, { - "input": "If you’re unfamiliar with eigenvalues, take a look at this video which introduces them. ", + "input": "If you're unfamiliar with eigenvalues, go ahead and take a look at this video here, which is actually meant to introduce them. ", "translatedText": "Nếu bạn không quen với các giá trị riêng, hãy tiếp tục và xem video này ở đây, video này thực ra nhằm giới thiệu chúng. ", "model": "google_nmt", "n_reviews": 0, @@ -40,7 +40,7 @@ "end": 64.58 }, { - "input": "Okay, that’s all a little bit of a mouthful to say, but again, I’m assuming all of this is review for anyone watching. ", + "input": "Okay, that's all a little bit of a mouthful to say, but again, I'm assuming that all of this is review for any of you watching. ", "translatedText": "Được rồi, điều đó hơi khó nói, nhưng một lần nữa, tôi cho rằng tất cả những điều này chỉ là đánh giá cho bất kỳ ai trong số các bạn đang xem. ", "model": "google_nmt", "n_reviews": 0, @@ -88,7 +88,7 @@ "end": 109.5 }, { - "input": "But at least for 2x2 matrices, there’s a much more direct way to get at this answer. ", + "input": "but at least for two by two matrices, there is a much more direct way you can get at the answer. ", "translatedText": "Thành thật mà nói, quá trình này không có gì đáng lo ngại, nhưng ít nhất đối với ma trận 2x2, có một cách trực tiếp hơn nhiều để bạn có thể tìm ra câu trả lời. ", "model": "google_nmt", "n_reviews": 0, @@ -168,7 +168,7 @@ "end": 191.7 }, { - "input": "Now take a moment to see how you can derive what will be our third relevant fact, which is how to recover two numbers when you know their mean and you know their product. ", + "input": "Now take a moment to see if you can derive what will be our third relevant fact, which is how you can quickly recover two numbers when you know their mean and you know their product. ", "translatedText": "Bây giờ hãy dành một chút thời gian để xem liệu bạn có thể rút ra được thực tế liên quan thứ ba của chúng ta hay không, đó là cách bạn có thể nhanh chóng tìm ra hai số khi bạn biết giá trị trung bình và tích của chúng. ", "model": "google_nmt", "n_reviews": 0, @@ -184,7 +184,7 @@ "end": 203.72 }, { - "input": "You know the two values are evenly spaced around 7, so they look like 7 plus or minus something; let’s call that something \"d\" for distance. ", + "input": "You know that the two values are evenly spaced around the number 7, so they look like 7 plus or minus something, let's call that something d for distance. ", "translatedText": "Bạn biết rằng hai giá trị cách đều nhau xung quanh số 7, nên chúng trông giống như 7 cộng hoặc trừ gì đó, hãy gọi đó là d cho khoảng cách. ", "model": "google_nmt", "n_reviews": 0, @@ -208,7 +208,7 @@ "end": 223.7 }, { - "input": "So from there, you can directly find d: d^2 is 7^2 - 40, or 9, which means d itself is 3. ", + "input": "So from there, you can find d. d squared is 7 squared minus 40, or 9, which means that d itself is 3. ", "translatedText": "Vì vậy, từ đó, bạn có thể trực tiếp tìm thấy d. d bình phương là 7 bình trừ 40, hay 9, có nghĩa là chính d bằng 3. ", "model": "google_nmt", "n_reviews": 0, @@ -240,7 +240,7 @@ "end": 255.68 }, { - "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m ± sqrt(m^2 - p) This is decently fast to rederive on the fly if you ever forget it, and it’s essentially just a rephrasing of the difference of squares formula. ", + "input": "This gives the third key fact, which is that when two numbers have a mean m and a product p, you can write those two numbers as m plus or minus the square root of m squared minus p. This is decently fast to re-derive on the fly if you ever forget it, and it's essentially just a rephrasing of the difference of squares formula. ", "translatedText": "Điều này đưa ra thực tế quan trọng thứ ba, đó là khi hai số có trung bình m và tích p, bạn có thể viết hai số đó dưới dạng m cộng hoặc trừ căn bậc hai của m bình trừ p. Điều này có thể được tìm lại nhanh chóng nếu bạn quên và về cơ bản nó chỉ là cách diễn đạt lại sự khác biệt của công thức bình phương. ", "model": "google_nmt", "n_reviews": 0, @@ -256,7 +256,7 @@ "end": 281.22 }, { - "input": "In fact, my friend Tim from the channel acapellascience wrote us a quick jingle to make it a little more memorable. ", + "input": "In fact, my friend Tim from the channel A Capella Science wrote us a nice quick jingle to make it a little bit more memorable. ", "translatedText": "Trên thực tế, bạn tôi, Tim, từ kênh A Capella Science đã viết cho chúng tôi một đoạn nhạc ngắn hay và thú vị để khiến nó trở nên đáng nhớ hơn một chút. ", "model": "google_nmt", "n_reviews": 0, @@ -296,7 +296,7 @@ "end": 317.74 }, { - "input": "So the thing you start writing is 2 ± sqrt(2^2 - …). ", + "input": "so the thing you start writing is 2 plus or minus the square root of 2 squared minus. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -304,7 +304,7 @@ "end": 322.7 }, { - "input": "Then the product of the eigenvalues is the determinant, which in this example is 3*1 - 1*4, or -1. ", + "input": "Then the product of the eigenvalues is the determinant, which in this example is 3 times 1 minus 1 times 4, or negative 1, ", "translatedText": "Vì vậy, trong ví dụ này, giá trị trung bình của các giá trị riêng giống với giá trị trung bình của 3 và 1, tức là 2, vậy số bạn bắt đầu viết là 2 cộng hoặc trừ căn bậc hai của 2 bình phương trừ, sau đó là tích của các giá trị riêng là định thức, trong ví dụ này là 3 nhân 1 trừ 1 nhân 4, hoặc âm 1, vậy đó là kết quả cuối cùng bạn điền vào, nghĩa là giá trị riêng là 2 cộng hoặc trừ căn bậc hai của 5. ", "model": "google_nmt", "n_reviews": 0, @@ -320,7 +320,7 @@ "end": 334.48 }, { - "input": "This means the eigenvalues are 2±sqrt(5). ", + "input": "which means the eigenvalues are 2 plus or minus the square root of 5. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -360,7 +360,7 @@ "end": 359.22 }, { - "input": "And then the determinant is 2*8 - 7*1, or 9. ", + "input": "And then the determinant is 2 times 8 minus 7 times 1, or 9. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -368,7 +368,7 @@ "end": 368.3 }, { - "input": "So in this example, the eigenvalues look like 5 ± sqrt(16), which simplifies even further as 9 and 1. ", + "input": "So in this example, the eigenvalues look like 5 plus or minus the square root of 16, which simplifies even further as 9 and 1. ", "translatedText": "Vì vậy, trong ví dụ này, các giá trị riêng trông giống như 5 cộng hoặc trừ căn bậc hai của 16, thậm chí còn đơn giản hóa hơn nữa thành 9 và 1. ", "model": "google_nmt", "n_reviews": 0, @@ -416,7 +416,7 @@ "end": 417.52 }, { - "input": "If you know quantum mechanics, you’ll know that the eigenvalues of matrices are highly relevant to the physics they describe, and if you don’t know quantum mechanics, let this just be a little glimpse of how these computations are actually relevant to real applications. ", + "input": "If you know quantum mechanics, you'll know that the eigenvalues of matrices are highly relevant to the physics that they describe. And if you don't know quantum mechanics, let this just be a little glimpse of how these computations are actually very relevant to real applications. ", "translatedText": "Nếu bạn biết cơ học lượng tử, bạn sẽ biết rằng các giá trị riêng của ma trận rất phù hợp với tính chất vật lý mà chúng mô tả. Và nếu bạn không biết cơ học lượng tử, hãy xem qua đây một chút về cách những phép tính này thực sự rất phù hợp với các ứng dụng thực tế. ", "model": "google_nmt", "n_reviews": 0, @@ -424,7 +424,7 @@ "end": 431.22 }, { - "input": "The mean of the diagonal in all three cases is 0, so the mean of the eigenvalues in all cases is 0, which makes our formula look especially simple. ", + "input": "The mean of the diagonal entries in all three cases is zero. So the mean of the eigenvalues in all of these cases is zero, which makes our formula look especially simple. ", "translatedText": "Giá trị trung bình của các mục chéo trong cả ba trường hợp đều bằng 0. Vì vậy, giá trị trung bình của các giá trị riêng trong tất cả các trường hợp này đều bằng 0, điều này làm cho công thức của chúng ta trông đặc biệt đơn giản. ", "model": "google_nmt", "n_reviews": 0, @@ -440,7 +440,7 @@ "end": 448.8 }, { - "input": "For the first one, it’s 0 - 1 or -1. ", + "input": "For the first one, it's 0, minus 1, or negative 1. The ", "translatedText": "Đối với số đầu tiên, nó là 0 trừ 1 hoặc âm 1. ", "model": "google_nmt", "n_reviews": 0, @@ -456,7 +456,7 @@ "end": 458.2 }, { - "input": "And the final one looks like -1 - 0. ", + "input": "And the final one looks like negative 1, minus 0. ", "translatedText": "Và số cuối cùng có dạng âm 1 trừ 0. ", "model": "google_nmt", "n_reviews": 0, @@ -464,7 +464,7 @@ "end": 461.36 }, { - "input": "So in all cases, the eigenvalues simplify to be ±1. ", + "input": "So in all cases, the eigenvalues simplify to be plus and minus 1. ", "translatedText": "Vì vậy, trong mọi trường hợp, các giá trị riêng được đơn giản hóa thành cộng và trừ 1. ", "model": "google_nmt", "n_reviews": 0, @@ -472,7 +472,7 @@ "end": 465.92 }, { - "input": "Although in this case, you really don’t need the formula to find two values if you know theyr'e evenly spaced around 0 and their product is -1. ", + "input": "Although in this case, you really don't need a formula to find two values if you know that they're evenly spaced around 0 and their product is negative 1. ", "translatedText": "Mặc dù trong trường hợp này, bạn thực sự không cần công thức để tìm hai giá trị nếu bạn biết rằng chúng cách đều nhau quanh 0 và tích của chúng âm 1. ", "model": "google_nmt", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 483.76 }, { - "input": "The fact that their eigenvalues are ±1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", + "input": "the fact that their eigenvalues are plus and minus 1 corresponds with the idea that the values for the spin that you would observe would be either entirely in one direction or entirely in another, as opposed to something continuously ranging in between. ", "translatedText": "Và thực tế là các giá trị riêng của chúng là cộng và trừ 1 tương ứng với ý tưởng rằng các giá trị của spin mà bạn quan sát được sẽ hoàn toàn theo một hướng này hoặc hoàn toàn theo một hướng khác, trái ngược với một cái gì đó liên tục nằm ở giữa. ", "model": "google_nmt", "n_reviews": 0, @@ -536,7 +536,7 @@ "end": 537.64 }, { - "input": "I mean, take a look a the first one: The relevant determinant directly gives you a characteristic polynomial of lambda^2 - 1, and clearly, that has roots of plus and minus 1. ", + "input": "I mean, take a look at the first one. The relevant determinant directly gives you a characteristic polynomial of lambda squared minus 1, and clearly that has roots of plus and minus 1. ", "translatedText": "Ý tôi là, hãy nhìn vào cái đầu tiên. Định thức liên quan trực tiếp cho bạn một đa thức đặc trưng của lambda bình phương trừ một, và rõ ràng nó có nghiệm cộng và trừ một. ", "model": "google_nmt", "n_reviews": 0, @@ -544,7 +544,7 @@ "end": 548.2 }, { - "input": "Same answer when you do the second matrix, lambda^2 - 1. ", + "input": "Same answer when you do the second matrix, lambda squared minus 1. ", "translatedText": "Câu trả lời tương tự khi bạn thực hiện ma trận thứ hai, lambda bình phương trừ một. ", "model": "google_nmt", "n_reviews": 0, @@ -592,7 +592,7 @@ "end": 589.28 }, { - "input": "More specifically, you should assume this vector is normalized, meaning a^2 + b^2 + c^2 = 1. ", + "input": "More specifically, you should assume that this vector is normalized, meaning a squared plus b squared plus c squared is equal to 1. ", "translatedText": "Cụ thể hơn, bạn nên giả sử rằng vectơ này đã được chuẩn hóa, nghĩa là a bình phương cộng b bình phương cộng c bình phương bằng một. ", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 654.1 }, { - "input": "Specifically, if the polynomial is normalized so that this leading coefficient is 1, then the mean of the roots will be -½ times this linear coefficient, which is -1 times the sum of those roots. ", + "input": "Specifically, if the polynomial is normalized, so that this leading coefficient is 1, then the mean of the roots will be negative 1 half times this linear coefficient, which is negative 1 times the sum of those roots. ", "translatedText": "Cụ thể, nếu đa thức được chuẩn hóa sao cho hệ số cao nhất này bằng 1, thì giá trị trung bình của các nghiệm sẽ âm một nửa lần hệ số tuyến tính này, tức là âm một lần tổng của các nghiệm đó. ", "model": "google_nmt", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 690.22 }, { - "input": "But the real advantage is that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", + "input": "But the real advantage is not just that it's fewer symbols to memorize, it's that each one of them carries more meaning with it. ", "translatedText": "Nhưng lợi thế thực sự không chỉ là việc ghi nhớ ít ký hiệu hơn mà còn ở chỗ mỗi ký hiệu đều mang nhiều ý nghĩa hơn. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 729.7 }, { - "input": "The hope is that it’s not just one more thing to memorize, but that the framing reinforces some other nice facts worth knowing, like how the trace and determinant relate to eigenvalues. ", + "input": "The hope is that it's not just one more thing that you memorize, but that the framing reinforces some other nice facts that are worth knowing, like how the trace and the determinant are related to eigenvalues. ", "translatedText": "Hy vọng rằng đó không chỉ là một điều nữa mà bạn ghi nhớ mà việc đóng khung củng cố một số sự thật thú vị khác đáng để biết, chẳng hạn như dấu vết và định thức có liên quan như thế nào với giá trị riêng. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/arabic/sentence_translations.json b/2021/shadows/arabic/sentence_translations.json index b54de0669..c7178273c 100644 --- a/2021/shadows/arabic/sentence_translations.json +++ b/2021/shadows/arabic/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher. ", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher. ", "translatedText": "أو ربما كنت قد أخذت درسًا في حساب التفاضل والتكامل منذ فترة ولكنك تحتاج إلى القليل من تجديد المعلومات. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°. ", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees. ", "translatedText": "إذا عشنا في زمن قبل وجود حساب التفاضل والتكامل ولم تكن التكاملات موجودة، وأردنا تقديم إجابة تقريبية لهذا السؤال، فإن إحدى الطرق التي يمكننا من خلالها القيام بذلك هي أخذ عينة من قيم θ التي تتراوح من 0 إلى 180 درجة. ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken. ", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken. ", "translatedText": "وبدلاً من وصف حجم الخطوة بـ δθ، وهي كمية محددة محددة، نصفها بدلاً من ذلك بـ dθ، والتي أحب أن أفكر فيها على أنها إشارة إلى حقيقة أنه تم أخذ نوع ما من الحدود. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr². ", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared. ", "translatedText": "بغض النظر عن اتجاه تلك الكرة، فإن ظلها، ظل الإسقاط المسطح، يكون دائمًا دائرة بمساحة πr². ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr². ", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared. ", "translatedText": "ومساحة سطح الكرة، كما ذكرت من قبل، هي بالضبط 4πr². ", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s². ", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared. ", "translatedText": "وبذلك، يمكنها أن تملأ تفاصيل السؤال المحدد حول المكعب، وتقول إن متوسط مساحة ظله ستكون ¼ مرات مساحة سطحه، 6s². ", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", "translatedText": "وإذا نظرت إلى علماء الرياضيات المشهورين عبر التاريخ، نيوتن، ويولر، وجاوس، كلهم، فستجدهم جميعًا يتمتعون بهذا الصبر اللامتناهي على ما يبدو لإجراء حسابات مملة. ", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex. ", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex. ", "translatedText": "بدلًا من مجرد الحصول على إجابة بنعم أو لا، هل هي محدبة أم لا، يمكننا أن نضع رقمًا لها بالقول، فكر في متوسط مساحة ظل جسم ما، واضرب ذلك في 4، واقسمه على مساحة السطح ، وإذا كان هذا الرقم 1، فستكون لديك مادة صلبة محدبة، ولكن إذا كان أقل من 1، فهي غير محدبة، ومدى قربها من 1 يخبرك بمدى قربها من أن تكون محدبة. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/bengali/sentence_translations.json b/2021/shadows/bengali/sentence_translations.json index e19bf370b..51c69a953 100644 --- a/2021/shadows/bengali/sentence_translations.json +++ b/2021/shadows/bengali/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher. ", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher. ", "translatedText": "অথবা হতে পারে আপনি কিছুক্ষণ আগে একটি ক্যালকুলাস ক্লাস নিয়েছেন কিন্তু আপনার একটু রিফ্রেশার প্রয়োজন।", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°. ", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees. ", "translatedText": "আমরা যদি ক্যালকুলাসের অস্তিত্বের আগে এমন একটি সময়ে বাস করতাম এবং অখণ্ডগুলি একটি জিনিস ছিল না, এবং আমরা এই প্রশ্নের আনুমানিক একটি উত্তর পেতে চাই, তাহলে আমরা এটি সম্পর্কে যেতে পারি তা হল θ এর জন্য মানের একটি নমুনা নেওয়া যা 0 থেকে পর্যন্ত 180° আমরা তাদের প্রত্যেকের মধ্যে কিছু ধরণের পার্থক্য, কিছু ডেল্টা θ এর সাথে সমানভাবে ব্যবধান হিসাবে ভাবতে পারি।", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken. ", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken. ", "translatedText": "এবং ধাপের আকারকে δθ হিসাবে বর্ণনা করার পরিবর্তে, একটি কংক্রিট সসীম পরিমাণ, আমরা পরিবর্তে এটিকে dθ হিসাবে বর্ণনা করি, যা আমি এই সত্যের সংকেত হিসাবে ভাবতে চাই যে এক ধরণের সীমা নেওয়া হচ্ছে।", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr². ", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared. ", "translatedText": "সেই গোলকের অভিযোজন যাই হোক না কেন, এর ছায়া, সমতল অভিক্ষেপ ছায়া সর্বদা πr² এর ক্ষেত্রফল সহ একটি বৃত্ত।", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr². ", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared. ", "translatedText": "এবং একটি গোলকের পৃষ্ঠের ক্ষেত্রফল, যেমন আমি আগে উল্লেখ করেছি, ঠিক 4πr²।", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s². ", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared. ", "translatedText": "সুতরাং এটি দিয়ে, সে গিয়ে একটি ঘনক সম্পর্কে নির্দিষ্ট প্রশ্নের বিশদ বিবরণ পূরণ করতে পারে এবং বলতে পারে যে এর গড় ছায়ার ক্ষেত্রফল এর পৃষ্ঠের ক্ষেত্রফলের 1⁄4 গুণ হবে, 6s²।", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", "translatedText": "এবং আপনি যদি ইতিহাসের মাধ্যমে বিখ্যাত গণিতবিদদের দিকে তাকান, নিউটন, অয়লার, গাউস, তাদের সকলেরই ক্লান্তিকর গণনা করার জন্য এই আপাতদৃষ্টিতে অসীম ধৈর্য রয়েছে।", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex. ", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex. ", "translatedText": "শুধু হ্যাঁ-না উত্তর না দিয়ে, এটি কি উত্তল, তাই না, আমরা এটির সাথে একটি সংখ্যা রাখতে পারি এই বলে, কিছু কঠিনের ছায়ার গড় ক্ষেত্রফল বিবেচনা করুন, এটিকে 4 দ্বারা গুণ করুন, পৃষ্ঠের ক্ষেত্রফল দ্বারা ভাগ করুন , এবং যদি সেই সংখ্যাটি 1 হয়, আপনি একটি উত্তল কঠিন পেয়েছেন, কিন্তু যদি এটি 1-এর কম হয়, তবে এটি অ-উত্তল, এবং এটি 1-এর কতটা কাছাকাছি তা আপনাকে বলে যে এটি উত্তল হওয়ার কতটা কাছাকাছি।", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/chinese/sentence_translations.json b/2021/shadows/chinese/sentence_translations.json index 5c0e924f5..9939edf41 100644 --- a/2021/shadows/chinese/sentence_translations.json +++ b/2021/shadows/chinese/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher. ", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher. ", "translatedText": "或者也许您不久前参加了微积分课程，但您需要复习一下。", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°. ", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees. ", "translatedText": "如果我们生活在微积分还没有存在的时代，并且我们想要近似回答这个问题，我 们可以采取的一种方法是对 θ 的值进行采样，范围从 0 到180°。", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken. ", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken. ", "translatedText": "我们没有将步长描述为 δθ（具体的有限量），而是将其描 述为 dθ，我喜欢将其视为表示正在采取某种限制的事实。", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr². ", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared. ", "translatedText": "无论该球体的方向如何，它的阴影（平面投影阴影）始终是面积为 πr² 的圆。", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr². ", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared. ", "translatedText": "正如我之前提到的，球体的表面积恰好是 4πr²。", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s². ", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared. ", "translatedText": "这样，她就可以去填写有关立方体的特定问题的详细信息，并说它的平均阴影面积将是其表面积的 1⁄4 倍，即 6s²。", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", "translatedText": "如果你看看历史上著名的数学家，牛顿、欧拉、高斯，他们都拥有看似无限的耐心来进行繁琐的计算。", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex. ", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex. ", "translatedText": "我们可以用一个数字来表示，考虑某个固体的阴影的平均面积，将其乘以 4，然后除以表面积，而不是仅仅给出是或否的答案，它是凸的还是不是凸的？，如果该数字为 1，则您得到了一个凸实体，但如果它小于 1，则它是非凸实体，并且它与 1 的接近程度告诉您它与凸实体的接近程度。", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/french/sentence_translations.json b/2021/shadows/french/sentence_translations.json index b494da9ab..f7c7a2543 100644 --- a/2021/shadows/french/sentence_translations.json +++ b/2021/shadows/french/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "Ou peut-être avez-vous suivi un cours de calcul il y a quelque temps, mais vous avez besoin d'un petit rappel.", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "Si nous vivions à une époque antérieure au calcul et que les intégrales n'existaient pas, et que nous voulions donner une réponse approximative à cette question, une façon de procéder serait de prendre un échantillon de valeurs pour θ allant de 0 à 180°.", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "Pour obtenir cela, nous multiplierons cette probabilité par la zone d'ombre correspondante, qui est cette valeur absolue de l'expression cosθ que nous avons vue plusieurs fois jusqu'à présent.", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "Et au lieu de décrire la taille du pas comme δθ, une quantité finie concrète, nous la décrivons plutôt comme dθ, ce que j'aime considérer comme signalant le fait qu'une sorte de limite est prise.", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "Quelle que soit l'orientation de cette sphère, son ombre, l'ombre de projection plate, est toujours un cercle d'aire πr².", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "Et la surface d'une sphère, comme je l'ai déjà mentionné, est exactement de 4πr².", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "Ainsi, avec cela, elle peut aller remplir les détails de la question particulière sur un cube, et dire que sa zone d'ombre moyenne sera 1⁄4 fois sa surface, 6s².", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "Et si vous regardez les mathématiciens célèbres de l’histoire, Newton, Euler, Gauss, tous, ils ont tous cette patience apparemment infinie pour faire des calculs fastidieux.", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "Plutôt que d'avoir simplement une réponse oui ou non, est-il convexe, n'est-ce pas, nous pourrions lui attribuer un nombre en disant : considérez la surface moyenne de l'ombre d'un solide, multipliez-la par 4, divisez-la par la surface. , et si ce nombre est 1, vous avez un solide convexe, mais s'il est inférieur à 1, il est non convexe, et sa proximité avec 1 vous indique à quel point il est proche d'être convexe.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/german/sentence_translations.json b/2021/shadows/german/sentence_translations.json index 395465c21..b05736b73 100644 --- a/2021/shadows/german/sentence_translations.json +++ b/2021/shadows/german/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "Oder vielleicht haben Sie vor einiger Zeit einen Mathematikkurs besucht, brauchen aber eine kleine Auffrischung.", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "Wenn wir in einer Zeit lebten, bevor es Infinitesimalrechnungen gab und es Integrale noch nicht gab, und wir eine annähernde Antwort auf diese Frage finden wollten, könnten wir dies beispielsweise dadurch erreichen, dass wir eine Stichprobe von Werten für θ nehmen, die von 0 bis zu reicht 180°.", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "Um das zu erreichen, multiplizieren wir diese Wahrscheinlichkeit mit der entsprechenden Schattenfläche, also dem Absolutwert des cosθ-Ausdrucks, den wir bisher schon oft gesehen haben.", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "Und anstatt die Schrittgröße als δθ zu beschreiben, einen konkreten endlichen Betrag, beschreiben wir sie stattdessen als dθ, was ich mir gerne als Signal dafür vorstelle, dass eine Art Grenze gesetzt wird.", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "Unabhängig von der Ausrichtung dieser Kugel ist ihr Schatten, der flache Projektionsschatten, immer ein Kreis mit einer Fläche von πr².", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "Und die Oberfläche einer Kugel beträgt, wie ich bereits erwähnt habe, genau 4πr².", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "Damit kann sie die Einzelheiten der jeweiligen Frage zu einem Würfel ausfüllen und sagen, dass seine durchschnittliche Schattenfläche 1⁄4 Mal seine Oberfläche, also 6 s², beträgt.", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "Und wenn man sich die berühmten Mathematiker der Geschichte ansieht, Newton, Euler, Gauß, sie alle, haben sie alle diese scheinbar unendliche Geduld für die Durchführung langwieriger Berechnungen.", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "Anstatt nur eine Ja-Nein-Antwort zu haben, ob es konvex ist oder nicht, könnten wir ihm eine Zahl geben, indem wir sagen: Betrachten Sie die durchschnittliche Fläche des Schattens eines Festkörpers, multiplizieren Sie diese mit 4 und dividieren Sie sie durch die Oberfläche , und wenn diese Zahl 1 ist, haben Sie einen konvexen Körper, aber wenn er kleiner als 1 ist, ist er nicht konvex, und wie nahe er an 1 liegt, sagt Ihnen, wie nah er an einer Konvexheit ist.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/hebrew/sentence_translations.json b/2021/shadows/hebrew/sentence_translations.json index 86ec7af5a..45b287cf7 100644 --- a/2021/shadows/hebrew/sentence_translations.json +++ b/2021/shadows/hebrew/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "או שאולי למדת שיעור חישוב לפני זמן מה אבל אתה צריך קצת רענון.", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "אם חיינו בתקופה שלפני שחישוב היה קיים ואינטגרלים לא היו עניין, ורצינו להעריך תשובה לשאלה זו, דרך אחת שבה נוכל ללכת על זה היא לקחת מדגם של ערכים עבור θ שנעה בין 0 ל- 180°.", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "כדי לקבל זאת, נכפיל את ההסתברות הזו כפול שטח הצל המתאים, שהוא הערך המוחלט הזה של ביטוי cosθ שראינו פעמים רבות עד לנקודה זו.", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "ובמקום לתאר את גודל הצעד כ-δθ, כמות סופית קונקרטית, במקום זאת אנו מתארים אותו כ-dθ, שאני אוהב לחשוב עליו כאותת לעובדה שננקטת איזושהי מגבלה.", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "לא משנה מה הכיוון של אותו כדור, הצל שלו, צל ההקרנה השטוח, הוא תמיד עיגול עם שטח של πr².", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "ושטח הפנים של כדור, כמו שציינתי קודם, הוא בדיוק 4πr².", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "אז עם זה, היא יכולה ללכת ולמלא את הפרטים של השאלה המסוימת על קובייה, ולומר ששטח הצל הממוצע שלה יהיה פי 1⁄4 משטח הפנים שלה, 6s².", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "ואם מסתכלים על המתמטיקאים המפורסמים דרך ההיסטוריה, ניוטון, אוילר, גאוס, כולם, לכולם יש את הסבלנות האינסופית לכאורה לעשות חישובים מייגעים.", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "במקום רק לקבל תשובה של כן-לא, האם היא קמורה, נכון, נוכל לשים לזה מספר על ידי אמירה, שקול את השטח הממוצע של הצל של מוצק כלשהו, הכפל את זה ב-4, חלק אותו בשטח הפנים , ואם המספר הזה הוא 1, יש לך מוצק קמור, אבל אם הוא קטן מ-1, הוא לא קמור, וכמה הוא קרוב ל-1 אומר לך כמה הוא קרוב להיות קמור.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/hindi/sentence_translations.json b/2021/shadows/hindi/sentence_translations.json index cc590d8b3..862139729 100644 --- a/2021/shadows/hindi/sentence_translations.json +++ b/2021/shadows/hindi/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher. ", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher. ", "translatedText": "या हो सकता है कि आपने कुछ समय पहले कैलकुलस की कक्षा ली हो लेकिन आपको थोड़े से पुनश्चर्या की आवश्यकता हो।", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°. ", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees. ", "translatedText": "यदि हम कैलकुलस के अस्तित्व में आने से पहले के समय में रहते थे और इंटीग्रल्स कोई चीज़ नहीं थी, और हम इस प्रश्न का अनुमानित उत्तर चाहते थे, तो एक तरीका यह हो सकता था कि हम θ के लिए मानों का एक नमूना लें जो 0 से लेकर 180°. ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken. ", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken. ", "translatedText": "और चरण आकार को δθ, एक ठोस परिमित राशि के रूप में वर्णित करने के बजाय, हम इसे dθ के रूप में वर्णित करते हैं, जिसे मैं इस तथ्य के संकेत के रूप में सोचना पसंद करता हूं कि किसी प्रकार की सीमा ली जा रही है।", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr². ", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared. ", "translatedText": "इससे कोई फर्क नहीं पड़ता कि उस गोले का अभिविन्यास क्या है, उसकी छाया, सपाट प्रक्षेपण छाया, हमेशा πr² के क्षेत्र के साथ एक वृत्त होती है।", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr². ", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared. ", "translatedText": "और एक गोले का सतह क्षेत्र, जैसा कि मैंने पहले उल्लेख किया है, बिल्कुल 4πr² है।", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s². ", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared. ", "translatedText": "तो इसके साथ, वह जा सकती है और एक घन के बारे में विशेष प्रश्न का विवरण भर सकती है, और कह सकती है कि इसका औसत छाया क्षेत्र इसके सतह क्षेत्र का 1⁄4 गुना, 6s² होगा।", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", "translatedText": "और यदि आप इतिहास के प्रसिद्ध गणितज्ञों, न्यूटन, यूलर, गॉस, उन सभी को देखें, तो उन सभी के पास कठिन गणनाएँ करने के लिए असीम धैर्य है।", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex. ", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex. ", "translatedText": "केवल हां-नहीं में उत्तर देने के बजाय, क्या यह उत्तल है, क्या यह नहीं है, हम यह कहकर एक संख्या डाल सकते हैं, किसी ठोस की छाया के औसत क्षेत्र पर विचार करें, इसे 4 से गुणा करें, इसे सतह क्षेत्र से विभाजित करें , और यदि वह संख्या 1 है, तो आपको एक उत्तल ठोस मिला है, लेकिन यदि यह 1 से कम है, तो यह गैर-उत्तल है, और यह 1 के कितना करीब है, यह आपको बताता है कि यह उत्तल होने के कितना करीब है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/indonesian/sentence_translations.json b/2021/shadows/indonesian/sentence_translations.json index f9133dab5..fcfe60281 100644 --- a/2021/shadows/indonesian/sentence_translations.json +++ b/2021/shadows/indonesian/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "Atau mungkin Anda pernah mengikuti kelas kalkulus beberapa waktu lalu tetapi perlu sedikit penyegaran.", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "Jika kita hidup di masa sebelum kalkulus ada dan integral belum ada, dan kita ingin memperkirakan jawaban atas pertanyaan ini, salah satu cara yang dapat kita lakukan adalah dengan mengambil sampel nilai θ yang berkisar dari 0 hingga 180°.", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "Untuk memperolehnya, kita akan mengalikan probabilitas ini dengan luas bayangan yang sesuai, yang merupakan nilai absolut dari ekspresi cosθ yang telah kita lihat berkali-kali hingga saat ini.", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "Dan alih-alih mendeskripsikan ukuran langkah sebagai δθ, jumlah terbatas yang konkrit, kami malah mendeskripsikannya sebagai dθ, yang menurut saya menandakan fakta bahwa ada semacam batasan yang diambil.", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "Apapun orientasi bola tersebut, bayangannya, bayangan proyeksi datar, selalu berupa lingkaran dengan luas πr².", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "Dan luas permukaan sebuah bola, seperti yang saya sebutkan sebelumnya, adalah tepat 4πr².", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "Jadi dengan itu, dia dapat mengisi rincian pertanyaan khusus tentang sebuah kubus, dan mengatakan bahwa rata-rata luas bayangannya adalah 1⁄4 kali luas permukaannya, 6s².", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "Dan jika Anda melihat ahli matematika terkenal sepanjang sejarah, Newton, Euler, Gauss, semuanya, mereka semua memiliki kesabaran yang tampaknya tak terbatas dalam melakukan perhitungan yang membosankan.", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "Daripada hanya menjawab ya-tidak, apakah itu cembung, bukan, kita bisa memberi angka dengan mengatakan, perhatikan luas rata-rata bayangan suatu benda padat, kalikan hasilnya dengan 4, bagi dengan luas permukaan. , dan jika bilangan tersebut adalah 1, Anda mendapatkan benda padat cembung, tetapi jika kurang dari 1, maka benda tersebut tidak cembung, dan seberapa dekat dengan 1 menunjukkan seberapa dekat benda tersebut menjadi cembung.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/italian/sentence_translations.json b/2021/shadows/italian/sentence_translations.json index 26f795578..6ca0891cf 100644 --- a/2021/shadows/italian/sentence_translations.json +++ b/2021/shadows/italian/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "O forse hai seguito un corso di calcolo qualche tempo fa ma hai bisogno di un po' di ripasso.", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "Se vivessimo in un'epoca in cui non esisteva il calcolo infinitesimale e gli integrali non esistevano, e volessimo fornire una risposta approssimativa a questa domanda, un modo in cui potremmo farlo è prendere un campione di valori per θ che varia da 0 fino a 180°.", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "Per ottenere ciò, moltiplicheremo questa probabilità per la corrispondente area d'ombra, che è questo valore assoluto dell'espressione cosθ che abbiamo visto molte volte fino a questo punto.", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "E invece di descrivere la dimensione del passo come δθ, una quantità finita concreta, la descriviamo invece come dθ, che mi piace pensare che segnali il fatto che viene preso un qualche tipo di limite.", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "Non importa quale sia l'orientamento di quella sfera, la sua ombra, l'ombra di proiezione piatta, è sempre un cerchio con un'area πr².", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "E la superficie di una sfera, come ho detto prima, è esattamente 4πr².", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "Quindi, può andare a compilare i dettagli della domanda particolare su un cubo e dire che la sua area d'ombra media sarà 1⁄4 volte la sua superficie, 6s².", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "E se guardi i famosi matematici della storia, Newton, Eulero, Gauss, tutti loro, hanno tutti questa pazienza apparentemente infinita nel fare calcoli noiosi.", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "Invece di avere semplicemente una risposta sì-no, è convesso o no, potremmo assegnargli un numero dicendo: considera l'area media dell'ombra di un solido, moltiplicala per 4, dividila per la superficie , e se quel numero è 1, hai un solido convesso, ma se è inferiore a 1, è non convesso, e quanto è vicino a 1 ti dice quanto è vicino a essere convesso.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/japanese/sentence_translations.json b/2021/shadows/japanese/sentence_translations.json index 0a578a4fb..0b78d2843 100644 --- a/2021/shadows/japanese/sentence_translations.json +++ b/2021/shadows/japanese/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "あるいは、少し前に微積分のクラスを受講したものの、少し復習が必要な場合もあります。", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "微積分が存在せず、積分が存在しなかった時代に私たちが住んでいて、この質問に対する答えを近似したい場合、それに取 り組む 1 つの方法は、0 から 2000 までの範囲の θ の値のサンプルを取得することです。 180°。", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "これを得るには、この確率に対応する影の領域を乗算します。 こ れは、これまで何度も見てきた cosθ 式の絶対値です。", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "そして、ステップ サイズを具体的な有限量である δθ として記述する代わりに、代わりに dθ と して記述します。 これは、ある種の制限が取られているという事実を示していると考えたいと思います。", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "その球の向きが何であれ、その影、つまり平面投影の影は常に πr² の面積を持つ円です。", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "そして、前に述べたように、球の表面積は正確に 4πr² です。", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "それで、彼女は立方体に関する特定の質問の詳細を入力し、平均的な影の面積が表面積の 1/4 倍 (6 平方メートル) になると言うことができます。", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "そして、ニュートン、オイラー、ガウスなど、歴史上の有名な数学者たちを見てみると、彼らは皆、退屈な計算を行うのに無限とも思える忍耐力を持っています。", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "単に「はい」か「いいえ」で答えるのではなく、「凸面か凸面か、そうではないか」という数字を当てはめることができます。 「ある固体の影の平均面積を考慮し、それを 4 倍し、それを表面積で割ります」と言えます。 、その数値が 1 の場合は凸ソリッドですが、1 未満の場合は非凸ソリッドであり、1 にどれだけ近いかによって凸ソリッドにどれだけ近いかがわかります。", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/marathi/sentence_translations.json b/2021/shadows/marathi/sentence_translations.json index 88f76776f..357ed71ac 100644 --- a/2021/shadows/marathi/sentence_translations.json +++ b/2021/shadows/marathi/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher. ", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher. ", "translatedText": "किंवा कदाचित तुम्ही काही काळापूर्वी कॅल्क्युलस क्लास घेतला असेल पण तुम्हाला थोडेसे रिफ्रेशर हवे आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°. ", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees. ", "translatedText": "जर आपण कॅल्क्युलस अस्तित्वात असण्याआधीच्या काळात राहिलो आणि अविभाज्य गोष्टी नव्हत्या आणि आपल्याला या प्रश्नाचे अंदाजे उत्तर मिळवायचे असेल तर, θ साठी मूल्यांचा नमुना घेणे हा एक मार्ग म्हणजे 0 ते 0 पर्यंत 180° आम्ही त्यांना प्रत्येकामध्ये काही प्रकारचे फरक, काही डेल्टा θ सह समान अंतराने विचार करू शकतो. ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken. ", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken. ", "translatedText": "आणि पायरीच्या आकाराचे वर्णन δθ म्हणून करण्याऐवजी, एक ठोस मर्यादित रक्कम, आम्ही त्याऐवजी त्याचे वर्णन dθ म्हणून करतो, ज्याचा मला काही प्रकारची मर्यादा घातली जात असल्याचे संकेत म्हणून विचार करायला आवडते. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr². ", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared. ", "translatedText": "त्या गोलाची दिशा काहीही असली तरी त्याची सावली, सपाट प्रक्षेपण सावली हे नेहमी πr² क्षेत्रफळ असलेले वर्तुळ असते. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr². ", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared. ", "translatedText": "आणि मी आधी सांगितल्याप्रमाणे गोलाचे पृष्ठभाग क्षेत्रफळ 4πr² आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s². ", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared. ", "translatedText": "तर त्यासह, ती जाऊन घनाच्या विशिष्ट प्रश्नाचे तपशील भरू शकते आणि म्हणू शकते की त्याचे सरासरी सावली क्षेत्र त्याच्या पृष्ठभागाच्या क्षेत्रफळाच्या 1⁄4 पट असेल, 6s². ", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", "translatedText": "आणि जर तुम्ही इतिहासाच्या माध्यमातून प्रसिद्ध गणितज्ञ, न्यूटन, यूलर, गॉस या सर्वांकडे पाहिले, तर त्या सर्वांमध्ये कंटाळवाणा आकडेमोड करण्यासाठी असीम संयम आहे. ", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex. ", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex. ", "translatedText": "फक्त होय-नाही असे उत्तर देण्याऐवजी, ते उत्तल आहे का, नाही, असे सांगून आपण त्यावर एक संख्या ठेवू शकतो, काही घनाच्या सावलीचे सरासरी क्षेत्रफळ विचारात घ्या, त्याला 4 ने गुणा, पृष्ठभागाच्या क्षेत्रफळाने भागा. , आणि जर ती संख्या 1 असेल, तर तुम्हाला एक उत्तल घन आहे, परंतु जर तो 1 पेक्षा कमी असेल, तर तो नॉन-कन्व्हेक्स आहे आणि तो 1 च्या किती जवळ आहे हे तुम्हाला सांगते की ते बहिर्वक्र असण्याच्या किती जवळ आहे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/persian/sentence_translations.json b/2021/shadows/persian/sentence_translations.json index 9b87ece3e..8c8f89b23 100644 --- a/2021/shadows/persian/sentence_translations.json +++ b/2021/shadows/persian/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher. ", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher. ", "translatedText": "یا شاید مدتی پیش در کلاس حساب دیفرانسیل و انتگرال شرکت کرده اید، اما به کمی تجدید نظر نیاز دارید. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°. ", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees. ", "translatedText": "اگر ما در زمانی زندگی می‌کردیم که حساب دیفرانسیل و انتگرال وجود نداشت و انتگرال‌ها چیزی نبودند، و می‌خواستیم پاسخی برای این سوال تقریبی بدهیم، یکی از راه‌هایی که می‌توانستیم برای آن پیش برویم این است که نمونه‌ای از مقادیر θ را انتخاب کنیم که از 0 تا متغیر باشد. 180 درجه. ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken. ", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken. ", "translatedText": "و به جای توصیف اندازه گام به عنوان δθ، یک مقدار محدود مشخص، به جای آن آن را به عنوان dθ توصیف می کنیم، که من دوست دارم به عنوان سیگنالی از این واقعیت فکر کنم که نوعی محدودیت در نظر گرفته شده است. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr². ", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared. ", "translatedText": "فرقی نمی کند جهت آن کره چه باشد، سایه آن، سایه برآمدگی مسطح، همیشه دایره ای با مساحت πr² است. ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr². ", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared. ", "translatedText": "و سطح یک کره، همانطور که قبلاً ذکر کردم، دقیقاً 4πr² است. ", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s². ", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared. ", "translatedText": "بنابراین، او می تواند برود و جزئیات سؤال خاص در مورد یک مکعب را پر کند و بگوید که متوسط وسعت سایه آن 1⁄4 برابر مساحت سطح آن، 6s² خواهد بود. ", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", "translatedText": "و اگر به ریاضیدانان مشهور در طول تاریخ نگاه کنید، نیوتن، اویلر، گاوس، همه آنها، همه آنها این صبر به ظاهر بی نهایت برای انجام محاسبات خسته کننده را دارند. ", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex. ", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex. ", "translatedText": "به جای اینکه فقط یک پاسخ بله-خیر داشته باشیم، آیا محدب است یا نه، می‌توانیم عددی را با گفتن این که مساحت متوسط سایه یک جامد را در نظر بگیریم، آن را در 4 ضرب کنیم، آن را بر مساحت سطح تقسیم کنیم. و اگر آن عدد 1 باشد، یک جامد محدب دارید، اما اگر کمتر از 1 باشد، غیر محدب است، و اینکه چقدر به 1 نزدیک است به شما می گوید که چقدر به محدب بودن نزدیک است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/portuguese/sentence_translations.json b/2021/shadows/portuguese/sentence_translations.json index ee83b0edc..ab1effb03 100644 --- a/2021/shadows/portuguese/sentence_translations.json +++ b/2021/shadows/portuguese/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "Ou talvez você tenha feito uma aula de cálculo há algum tempo, mas precisa de uma atualização.", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "Se vivêssemos em uma época antes da existência do cálculo e as integrais não existissem, e quiséssemos aproximar uma resposta a essa pergunta, uma maneira de fazer isso é pegar uma amostra de valores para θ que varia de 0 a 180°.", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "Para conseguir isso, multiplicaremos essa probabilidade pela área de sombra correspondente, que é esse valor absoluto da expressão cosθ que vimos muitas vezes até agora.", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "E em vez de descrever o tamanho do passo como δθ, uma quantidade finita concreta, em vez disso descrevemo-lo como dθ, o que gosto de pensar como um sinal do facto de que algum tipo de limite está a ser tomado.", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "Não importa qual seja a orientação dessa esfera, sua sombra, a sombra plana da projeção, é sempre um círculo com área de πr².", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "E a área superficial de uma esfera, como mencionei antes, é exatamente 4πr².", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "Então, com isso, ela pode preencher os detalhes da questão específica sobre um cubo e dizer que sua área de sombra média será 1⁄4 vezes sua área de superfície, 6s².", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "E se olharmos para os matemáticos famosos ao longo da história, Newton, Euler, Gauss, todos eles, todos eles têm uma paciência aparentemente infinita para fazer cálculos tediosos.", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "Em vez de apenas ter uma resposta sim ou não, é convexo, não é, poderíamos atribuir um número dizendo, considere a área média da sombra de algum sólido, multiplique isso por 4, divida pela área da superfície , e se esse número for 1, você terá um sólido convexo, mas se for menor que 1, não será convexo, e o quão próximo ele está de 1 indica o quão próximo ele está de ser convexo.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/russian/sentence_translations.json b/2021/shadows/russian/sentence_translations.json index ee89dbd26..38046c040 100644 --- a/2021/shadows/russian/sentence_translations.json +++ b/2021/shadows/russian/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "Или, может быть, вы недавно посещали занятия по математическому анализу, но вам нужно немного освежить знания.", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "Если бы мы жили во времена, когда еще не существовало исчисления и интегралы не существовали, и мы хотели бы приблизительно ответить на этот вопрос, один из способов сделать это — взять выборку значений θ в диапазоне от 0 до 180°.", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "Чтобы получить это, мы умножим эту вероятность на соответствующую площадь тени, которая представляет собой абсолютное значение выражения cosθ, которое мы видели много раз до этого момента.", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "И вместо того, чтобы описывать размер шага как δθ, конкретную конечную величину, мы описываем его как dθ, что мне нравится думать как сигнал о том, что принят какой-то предел.", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "Независимо от ориентации этой сферы, ее тень, плоская тень проекции, всегда представляет собой круг площадью πr².", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "А площадь поверхности сферы, как я уже упоминал, равна ровно 4πr².", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "Таким образом, она может пойти и подробно ответить на конкретный вопрос о кубе и сказать, что его средняя площадь тени будет в 1/4 раза больше площади его поверхности, 6 с².", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "И если вы посмотрите на знаменитых математиков в истории, Ньютона, Эйлера, Гаусса, всех их, то они все обладают, казалось бы, бесконечным терпением к выполнению утомительных вычислений.", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "Вместо того, чтобы просто ответить да-нет, выпукло ли оно, не так ли, мы могли бы дать ему число, сказав: рассмотрим среднюю площадь тени некоторого твердого тела, умножим ее на 4, разделим на площадь поверхности. , и если это число равно 1, у вас есть выпуклое тело, но если оно меньше 1, оно невыпуклое, и то, насколько оно близко к 1, говорит о том, насколько оно близко к выпуклости.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/spanish/sentence_translations.json b/2021/shadows/spanish/sentence_translations.json index a181eb072..44ec9bd23 100644 --- a/2021/shadows/spanish/sentence_translations.json +++ b/2021/shadows/spanish/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "O tal vez tomaste una clase de cálculo hace un tiempo pero necesitas un pequeño repaso.", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "Si viviéramos en una época antes de que existiera el cálculo y las integrales no existieran, y quisiéramos aproximar una respuesta a esta pregunta, una forma de hacerlo es tomar una muestra de valores para θ que va desde 0 hasta 180°.", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "Para conseguirlo, multiplicaremos esta probabilidad por el área de sombra correspondiente, que es este valor absoluto de la expresión cosθ que hemos visto muchas veces hasta este punto.", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "Y en lugar de describir el tamaño del paso como δθ, una cantidad finita concreta, lo describimos como dθ, lo que me gusta considerar como una señal del hecho de que se está tomando algún tipo de límite.", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "No importa cuál sea la orientación de esa esfera, su sombra, la sombra de proyección plana, es siempre un círculo con un área de πr².", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "Y el área de superficie de una esfera, como mencioné antes, es exactamente 4πr².", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "Entonces, con eso, puede ir y completar los detalles de la pregunta particular sobre un cubo, y decir que su área de sombra promedio será 1⁄4 veces su área de superficie, 6s².", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "Y si miras a los matemáticos famosos a lo largo de la historia, Newton, Euler, Gauss, todos ellos, todos tienen una paciencia aparentemente infinita para hacer cálculos tediosos.", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "En lugar de simplemente tener una respuesta de sí o no, ¿es convexo? , ¿no es así? Podríamos ponerle un número diciendo, considere el área promedio de la sombra de algún sólido, multiplíquelo por 4, divídalo por el área de la superficie. , y si ese número es 1, tienes un sólido convexo, pero si es menor que 1, no es convexo, y qué tan cerca está de 1 te dice qué tan cerca está de ser convexo.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/tamil/sentence_translations.json b/2021/shadows/tamil/sentence_translations.json index 3b51d6939..c162abbdc 100644 --- a/2021/shadows/tamil/sentence_translations.json +++ b/2021/shadows/tamil/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "அல்லது சிறிது நேரத்திற்கு முன்பு நீங்கள் கால்குலஸ் வகுப்பை எடுத்திருக்கலாம், ஆனால் உங்களுக்கு கொஞ்சம் புத்துணர்ச்சி தேவை.", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "கால்குலஸ் இருப்பதற்கு முன் ஒரு காலத்தில் நாம் வாழ்ந்திருந்தால் மற்றும் ஒருங்கிணைப்புகள் ஒரு விஷயமாக இருக்காது, மேலும் இந்தக் கேள்விக்கான பதிலை தோராயமாக மதிப்பிட விரும்பினால், அதைப் பற்றி நாம் செல்லக்கூடிய ஒரு வழி, 0 முதல் θ வரையிலான மதிப்புகளின் மாதிரியை எடுப்பதாகும். 180°.", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "அதைப் பெற, இந்த நிகழ்தகவை தொடர்புடைய நிழல் பகுதியைப் பெருக்குவோம், இது இது வரை பல முறை நாம் பார்த்த cosθ வெளிப்பாட்டின் முழுமையான மதிப்பாகும்.", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "மேலும் படி அளவை δθ, ஒரு உறுதியான வரையறுக்கப்பட்ட அளவு என்று விவரிப்பதற்குப் பதிலாக, அதை dθ என்று விவரிக்கிறோம், இது ஒருவித வரம்பு எடுக்கப்படுகிறது என்பதை உணர்த்துவதாக நான் நினைக்க விரும்புகிறேன்.", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "அந்தக் கோளத்தின் நோக்குநிலை என்னவாக இருந்தாலும், அதன் நிழல், தட்டையான திட்ட நிழல், எப்போதும் πr² பரப்பளவைக் கொண்ட ஒரு வட்டமாக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "நான் முன்பு குறிப்பிட்டது போல் ஒரு கோளத்தின் பரப்பளவு சரியாக 4πr² ஆகும்.", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "அதனுடன், அவள் சென்று ஒரு கனசதுரத்தைப் பற்றிய குறிப்பிட்ட கேள்வியின் விவரங்களைப் பூர்த்தி செய்து, அதன் சராசரி நிழல் பரப்பளவு அதன் பரப்பளவை விட 1⁄4 மடங்கு, 6s² இருக்கும் என்று கூறலாம்.", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "நீங்கள் வரலாற்றின் மூலம் பிரபலமான கணிதவியலாளர்களைப் பார்த்தால், நியூட்டன், ஆய்லர், காஸ், அனைவரையும் பார்த்தால், அவர்கள் அனைவருக்கும் கடினமான கணக்கீடுகளைச் செய்வதற்கு எல்லையற்ற பொறுமை உள்ளது.", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "ஆம்-இல்லை என்ற பதிலைக் காட்டாமல், அது குவிந்ததா, இல்லையா, சில திடப்பொருளின் நிழலின் சராசரிப் பகுதியைக் கருத்தில் கொண்டு, அதை 4 ஆல் பெருக்கி, பரப்புப் பரப்பால் வகுக்கவும் என்று சொல்லி எண்ணை வைக்கலாம். , மற்றும் அந்த எண் 1 ஆக இருந்தால், உங்களிடம் ஒரு குவிந்த திடப்பொருள் உள்ளது, ஆனால் அது 1 க்குக் குறைவாக இருந்தால், அது குவிந்ததல்ல, மேலும் 1 க்கு எவ்வளவு நெருக்கமாக இருக்கிறது என்பது குவிந்த நிலையில் இருக்கும் என்பதை உங்களுக்குக் கூறுகிறது.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/telugu/sentence_translations.json b/2021/shadows/telugu/sentence_translations.json index 9e19775b8..542f0e202 100644 --- a/2021/shadows/telugu/sentence_translations.json +++ b/2021/shadows/telugu/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "లేదా మీరు కొంతకాలం క్రితం కాలిక్యులస్ క్లాస్ తీసుకున్నారేమో కానీ మీకు కొంచెం రిఫ్రెషర్ కావాలి.", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "కాలిక్యులస్ ఉనికిలో ఉండక ముందు మనం జీవించి ఉంటే మరియు ఇంటిగ్రల్స్ ఒక విషయం కాదు, మరియు మేము ఈ ప్రశ్నకు సుమారుగా సమాధానం చెప్పాలనుకుంటే, మేము దాని గురించి వెళ్ళడానికి ఒక మార్గం θ కోసం విలువల నమూనా 0 నుండి వరకు ఉంటుంది. 180°.", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "దాన్ని పొందడానికి, మేము ఈ సంభావ్యతను సంబంధిత నీడ ప్రాంతం కంటే గుణిస్తాము, ఇది మేము ఇప్పటివరకు చాలాసార్లు చూసిన cosθ వ్యక్తీకరణ యొక్క ఈ సంపూర్ణ విలువ.", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "మరియు దశల పరిమాణాన్ని δθ, కాంక్రీట్ పరిమిత మొత్తంగా వివరించే బదులు, మేము దానిని dθగా వర్ణిస్తాము, ఇది ఒక రకమైన పరిమితి తీసుకోబడుతుందనే వాస్తవాన్ని సూచిస్తుందని నేను భావించాలనుకుంటున్నాను.", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "ఆ గోళం యొక్క దిశ ఎలా ఉన్నా, దాని నీడ, ఫ్లాట్ ప్రొజెక్షన్ నీడ, ఎల్లప్పుడూ πr² వైశాల్యంతో ఒక వృత్తం.", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "మరియు ఒక గోళం యొక్క ఉపరితల వైశాల్యం, నేను ముందు చెప్పినట్లుగా, సరిగ్గా 4πr².", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "కాబట్టి దానితో, ఆమె వెళ్లి ఒక క్యూబ్ గురించి నిర్దిష్ట ప్రశ్న యొక్క వివరాలను పూరించవచ్చు మరియు దాని సగటు నీడ వైశాల్యం దాని ఉపరితల వైశాల్యం 6s² కంటే 1⁄4 రెట్లు ఉంటుందని చెప్పవచ్చు.", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "మరియు మీరు చరిత్ర ద్వారా ప్రసిద్ధ గణిత శాస్త్రజ్ఞులను పరిశీలిస్తే, న్యూటన్, ఆయిలర్, గాస్, వీళ్లందరినీ చూస్తే, వారంతా దుర్భరమైన లెక్కలు చేయడానికి ఈ అంతులేని సహనం కలిగి ఉంటారు.", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "కేవలం అవును-కాదు అని సమాధానం ఇవ్వకుండా, అది కుంభాకారంగా ఉందా, కాదా, మనం దానికి ఒక సంఖ్యను ఉంచవచ్చు, కొంత ఘనపు నీడ యొక్క సగటు వైశాల్యాన్ని పరిగణించండి, దానిని 4తో గుణించి, ఉపరితల వైశాల్యంతో భాగించండి , మరియు ఆ సంఖ్య 1 అయితే, మీరు ఒక కుంభాకార ఘనతను కలిగి ఉంటారు, కానీ అది 1 కంటే తక్కువ ఉంటే, అది కుంభాకారం కానిది మరియు 1కి ఎంత దగ్గరగా ఉందో అది కుంభాకారంగా ఉండటానికి ఎంత దగ్గరగా ఉందో మీకు తెలియజేస్తుంది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/thai/sentence_translations.json b/2021/shadows/thai/sentence_translations.json index a32e551e8..bc8a8d8db 100644 --- a/2021/shadows/thai/sentence_translations.json +++ b/2021/shadows/thai/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher. ", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°. ", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken. ", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr². ", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr². ", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s². ", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex. ", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/turkish/sentence_translations.json b/2021/shadows/turkish/sentence_translations.json index 0b37d3f88..0d12e6c09 100644 --- a/2021/shadows/turkish/sentence_translations.json +++ b/2021/shadows/turkish/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "Ya da belki bir süre önce matematik dersi aldınız ama biraz bilgi tazelemeye ihtiyacınız var.", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "Analizin var olmadığı ve integrallerin henüz var olmadığı bir zamanda yaşadıysak ve bu soruya yaklaşık bir yanıt bulmak istiyorsak, bunu yapabilmemizin bir yolu θ için 0'dan 0'a kadar olan değerlerin bir örneğini almaktır. 180°.", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "Bunu elde etmek için, bu olasılığı karşılık gelen gölge alanıyla çarpacağız; bu, bu noktaya kadar birçok kez gördüğümüz cosθ ifadesinin mutlak değeridir.", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "Ve adım büyüklüğünü somut bir sonlu miktar olan δθ olarak tanımlamak yerine, bunu dθ olarak tanımlıyoruz; bunun bir tür limitin alındığı gerçeğinin sinyali olduğunu düşünmeyi seviyorum.", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "Bu kürenin yönü ne olursa olsun, onun gölgesi, yani düz projeksiyon gölgesi her zaman πr² alanına sahip bir dairedir.", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "Ve daha önce de belirttiğim gibi kürenin yüzey alanı tam olarak 4πr²'dir.", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "Böylece, gidip bir küp hakkındaki özel sorunun ayrıntılarını doldurabilir ve ortalama gölge alanının yüzey alanının 1⁄4 katı, yani 6s² olacağını söyleyebilir.", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "Ve tarih boyunca ünlü matematikçilere bakarsanız, Newton, Euler, Gauss, hepsinin, sıkıcı hesaplamalar yapmak için görünüşte sonsuz bir sabra sahip olduklarını görürsünüz.", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "Sadece evet-hayır cevabı vermek yerine, dışbükey mi, değil mi, ona şöyle bir sayı verebiliriz: Bir katının gölgesinin ortalama alanını düşünün, bunu 4 ile çarpın, bunu yüzey alanına bölün. ve eğer bu sayı 1 ise, dışbükey bir katınız var demektir, ancak 1'den küçükse dışbükey değildir ve 1'e ne kadar yakın olduğu size onun dışbükey olmaya ne kadar yakın olduğunu gösterir.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/ukrainian/sentence_translations.json b/2021/shadows/ukrainian/sentence_translations.json index 3f9775900..7cd9002ea 100644 --- a/2021/shadows/ukrainian/sentence_translations.json +++ b/2021/shadows/ukrainian/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher.", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher.", "translatedText": "Або, можливо, ви нещодавно відвідували курс математики, але вам потрібно трохи відновити знання.", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°.", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees.", "translatedText": "Якби ми жили в часи, коли ще не існувало обчислення, і інтеграли не були річчю, і ми хотіли приблизно відповісти на це запитання, один із способів зробити це — взяти вибірку значень для θ, яка коливається від 0 до 180°.", "model": "google_nmt", "n_reviews": 0, @@ -1840,7 +1840,7 @@ "end": 1663.32 }, { - "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosθ expression we've seen many times up to this point.", + "input": "To get that, we'll multiply this probability times the corresponding shadow area, which is this absolute value of cosine theta expression we've seen many times up to this point.", "translatedText": "Щоб отримати це, ми помножимо цю ймовірність на відповідну область тіні, яка є цим абсолютним значенням виразу cosθ, яке ми бачили багато разів до цього моменту.", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken.", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken.", "translatedText": "І замість того, щоб описувати розмір кроку як δθ, конкретну кінцеву величину, ми натомість описуємо його як dθ, що мені подобається вважати сигналом того, що дотримується певного ліміту.", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr².", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared.", "translatedText": "Незалежно від орієнтації цієї сфери, її тінь, тінь плоскої проекції, завжди є колом із площею πr².", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr².", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared.", "translatedText": "А площа поверхні сфери, як я вже згадував раніше, рівно 4πr².", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s².", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared.", "translatedText": "Таким чином, вона може піти і заповнити деталі конкретного запитання про куб, і сказати, що його середня площа тіні буде 1⁄4 площі його поверхні, 6s².", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations.", "translatedText": "І якщо ви подивитеся на відомих математиків в історії, Ньютона, Ейлера, Гаусса, усіх них, усі вони мають це, здавалося б, нескінченне терпіння виконувати нудні обчислення.", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex.", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex.", "translatedText": "Замість того, щоб просто відповісти «так-ні», чи опукло воно, чи ні, ми могли б поставити йому число, сказавши: вважайте середню площу тіні якогось твердого тіла, помножте це на 4, розділіть на площу поверхні , і якщо це число дорівнює 1, ви маєте опукле тіло, але якщо воно менше 1, воно не опукле, і те, наскільки воно близьке до 1, говорить вам, наскільки воно близьке до опуклості.", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/urdu/sentence_translations.json b/2021/shadows/urdu/sentence_translations.json index ac4add72e..d5bb136f4 100644 --- a/2021/shadows/urdu/sentence_translations.json +++ b/2021/shadows/urdu/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher. ", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher. ", "translatedText": "یا ہوسکتا ہے کہ آپ نے تھوڑی دیر پہلے کیلکولس کی کلاس لی ہو لیکن آپ کو تھوڑا سا ریفریشر کی ضرورت ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°. ", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees. ", "translatedText": "اگر ہم ایک ایسے وقت میں رہتے تھے جب کیلکولس کے وجود سے پہلے اور انٹیگرلز کوئی چیز نہیں تھی، اور ہم اس سوال کا اندازہ لگانا چاہتے تھے، تو ایک طریقہ یہ ہے کہ ہم θ کے لیے اقدار کا نمونہ لیں جو 0 سے لے کر تک کے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken. ", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken. ", "translatedText": "اور قدم کے سائز کو δθ، ایک ٹھوس محدود مقدار کے طور پر بیان کرنے کے بجائے، ہم اسے dθ کے طور پر بیان کرتے ہیں، جس کے بارے میں میں سوچنا چاہتا ہوں کہ اس حقیقت کا اشارہ ہے کہ کسی قسم کی حد اختیار کی جا رہی ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr². ", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared. ", "translatedText": "اس سے کوئی فرق نہیں پڑتا ہے کہ اس کرہ کی واقفیت کچھ بھی ہو، اس کا سایہ، فلیٹ پروجیکشن شیڈو، ہمیشہ ایک دائرہ ہوتا ہے جس کا رقبہ πr² ہوتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr². ", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared. ", "translatedText": "اور ایک کرہ کی سطح کا رقبہ، جیسا کہ میں نے پہلے بتایا، بالکل 4πr² ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s². ", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared. ", "translatedText": "تو اس کے ساتھ، وہ جا کر مکعب کے بارے میں مخصوص سوال کی تفصیلات کو پُر کر سکتی ہے، اور کہہ سکتی ہے کہ اس کے سائے کا اوسط رقبہ اس کی سطح کے رقبے سے 1⁄4 گنا ہو گا، 6s²۔", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", "translatedText": "اور اگر آپ تاریخ کے مشہور ریاضی دانوں کو دیکھیں، نیوٹن، ایلر، گاس، ان سب میں، ان سب کے پاس تھکا دینے والا حساب کتاب کرنے کے لیے یہ بظاہر لامحدود صبر ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex. ", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex. ", "translatedText": "ہاں-نہیں میں جواب دینے کے بجائے، کیا یہ محدب ہے، کیا ایسا نہیں ہے، ہم اسے یہ کہہ کر نمبر لگا سکتے ہیں، کسی ٹھوس کے سائے کے اوسط رقبے پر غور کریں، اسے 4 سے ضرب دیں، اسے سطحی رقبہ سے تقسیم کریں۔اور اگر وہ نمبر 1 ہے، تو آپ کے پاس ایک محدب ٹھوس ہے، لیکن اگر یہ 1 سے کم ہے، تو یہ غیر محدب ہے، اور یہ 1 کے کتنا قریب ہے آپ کو بتاتا ہے کہ یہ محدب ہونے کے کتنا قریب ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/shadows/vietnamese/sentence_translations.json b/2021/shadows/vietnamese/sentence_translations.json index 8f66393d9..5a0c60cd3 100644 --- a/2021/shadows/vietnamese/sentence_translations.json +++ b/2021/shadows/vietnamese/sentence_translations.json @@ -1704,7 +1704,7 @@ "end": 1524.78 }, { - "input": "Or maybe you took a calculus class a while ago but you need a little bit of a refresher. ", + "input": "Or maybe you, you know, you took a calculus class a while ago, but you need a little bit of a refresher. ", "translatedText": "Hoặc có thể bạn đã tham gia một lớp giải tích cách đây một thời gian nhưng bạn cần ôn lại một chút. ", "model": "google_nmt", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1543.04 }, { - "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for θ that ranges from 0 up to 180°. ", + "input": "If we lived in a time before calculus existed and integrals weren't a thing, and we wanted to approximate an answer to this question, one way we could go about it is to take a sample of values for theta that ranges from 0 up to 180 degrees. ", "translatedText": "Nếu chúng ta sống ở thời kỳ trước khi phép tính xuất hiện và tích phân chưa phải là một thứ gì đó, và chúng ta muốn tìm gần đúng câu trả lời cho câu hỏi này, thì một cách chúng ta có thể làm là lấy một mẫu các giá trị cho θ nằm trong khoảng từ 0 đến 180°. ", "model": "google_nmt", "n_reviews": 0, @@ -1920,7 +1920,7 @@ "end": 1733.98 }, { - "input": "And instead of describing the step size as δθ, a concrete finite amount, we instead describe it as dθ, which I like to think of as signaling the fact that some kind of limit is being taken. ", + "input": "And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta, which I like to think of as signaling the fact that some kind of limit is being taken. ", "translatedText": "Và thay vì mô tả kích thước bước là δθ, một lượng hữu hạn cụ thể, thay vào đó, chúng tôi mô tả nó là dθ, mà tôi muốn nghĩ đến như là tín hiệu thực tế rằng một loại giới hạn nào đó đang được thực hiện. ", "model": "google_nmt", "n_reviews": 0, @@ -2048,7 +2048,7 @@ "end": 1850.06 }, { - "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of πr². ", + "input": "No matter what the orientation of that sphere, its shadow, the flat projection shadow, is always a circle with an area of pi r squared. ", "translatedText": "Cho dù hình cầu đó hướng theo hướng nào thì bóng của nó, bóng hình chiếu phẳng, luôn là một hình tròn có diện tích πr². ", "model": "google_nmt", "n_reviews": 0, @@ -2064,7 +2064,7 @@ "end": 1861.04 }, { - "input": "And the surface area of a sphere, like I mentioned before, is exactly 4πr². ", + "input": "And the surface area of a sphere, like I mentioned before, is exactly 4 pi r squared. ", "translatedText": "Và diện tích bề mặt của một hình cầu, như tôi đã đề cập trước đó, chính xác là 4πr². ", "model": "google_nmt", "n_reviews": 0, @@ -2088,7 +2088,7 @@ "end": 1893.58 }, { - "input": "So with that, she can go and fill in the details of the particular question about a cube, and say that its average shadow area will be 1⁄4 times its surface area, 6s². ", + "input": "So with that, she can go and fill in the details of the particular question about a cube and say that its average shadow area will be 1 fourth times its surface area, 6 s squared. ", "translatedText": "Vì vậy, với điều đó, cô ấy có thể điền thông tin chi tiết của câu hỏi cụ thể về một khối lập phương và nói rằng diện tích bóng trung bình của nó sẽ gấp 1⁄4 lần diện tích bề mặt của nó, 6s². ", "model": "google_nmt", "n_reviews": 0, @@ -2416,7 +2416,7 @@ "end": 2241.16 }, { - "input": "And if you look at the famous mathematicians through history, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", + "input": "And if you look at the famous mathematicians through history, you know, Newton, Euler, Gauss, all of them, they all have this seemingly infinite patience for doing tedious calculations. ", "translatedText": "Và nếu bạn nhìn vào các nhà toán học nổi tiếng trong lịch sử, Newton, Euler, Gauss, tất cả họ, họ đều có sự kiên nhẫn dường như vô hạn để thực hiện các phép tính tẻ nhạt. ", "model": "google_nmt", "n_reviews": 0, @@ -2456,7 +2456,7 @@ "end": 2294.82 }, { - "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by 4, divide it by the surface area, and if that number is 1, you've got a convex solid, but if it's less than 1, it's non-convex, and how close it is to 1 tells you how close it is to being convex. ", + "input": "Rather than just having a yes-no answer, is it convex, is it not, we could put a number to it by saying, consider the average area of the shadow of some solid, multiply that by four, divide it by the surface area, and if that number is one, you've got a convex solid, but if it's less than one, it's non-convex, and how close it is to one tells you how close it is to being convex. ", "translatedText": "Thay vì chỉ đưa ra câu trả lời có-không, nó có lồi không, phải không, chúng ta có thể đặt một con số cho nó bằng cách nói, hãy xét diện tích trung bình của bóng của một vật rắn nào đó, nhân số đó với 4, chia cho diện tích bề mặt , và nếu số đó là 1 thì bạn có một khối lồi, nhưng nếu nó nhỏ hơn 1 thì nó không lồi và mức độ gần với 1 cho bạn biết nó gần lồi đến mức nào. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2021/some1-results/english/captions.srt b/2021/some1-results/english/captions.srt index 3341131c8..b1726ab7d 100644 --- a/2021/some1-results/english/captions.srt +++ b/2021/some1-results/english/captions.srt @@ -167,16 +167,16 @@ and if you participated in the event and found it helpful, he is absolutely the one you should be thanking. 43 -00:02:13,120 --> 00:02:17,147 +00:02:13,120 --> 00:02:16,850 As I said, there's a blog post with more details for any of you who are curious, 44 -00:02:17,147 --> 00:02:20,876 +00:02:16,850 --> 00:02:20,305 but right now, without further ado, let me tell you about some outstanding 45 -00:02:20,876 --> 00:02:23,760 -math explainers that I really think you're going to enjoy. +00:02:20,305 --> 00:02:23,760 +math explainers made this summer that I really think you're going to enjoy. 46 00:02:24,700 --> 00:02:27,437 @@ -591,246 +591,254 @@ and why any of that has to do with quantum mechanics, but this animated hour-long lecture does a really good job laying out the full story. 149 -00:08:39,000 --> 00:08:46,160 -Another one which is long but good is about the unsolvability of the object. +00:08:39,000 --> 00:08:42,894 +Another one which is long but good is about the unsolvability 150 +00:08:42,894 --> 00:08:46,160 +of the quintic by Carl Turner. For me it was a bit b + +151 00:08:46,160 --> 00:08:49,177 I'm not seeing it because when I did, I was actively working -151 +152 00:08:49,177 --> 00:08:51,798 on a video that was not just about the same theorem, -152 +153 00:08:51,798 --> 00:08:55,360 but about the same comparatively esoteric proof that it describes there. -153 +154 00:08:55,680 --> 00:08:59,120 I thought the video did such a wonderful job, I just kind of set aside the project. -154 +155 00:08:59,520 --> 00:09:03,288 I'll still probably cover it at some point, but now I'm motivated to do it in -155 +156 00:09:03,288 --> 00:09:06,911 a different way, but in the meantime, any of you curious about why quintic -156 +157 00:09:06,911 --> 00:09:10,680 polynomials are in a certain sense unsolvable will absolutely love this video. -157 +158 00:09:11,460 --> 00:09:14,900 So many entries here were just really solid explainers, plain and simple. -158 +159 00:09:15,380 --> 00:09:19,068 This includes the best overview I've seen of the two-envelope problem, -159 +160 00:09:19,068 --> 00:09:22,081 a great explanation for how fonts get turned into pixels, -160 +161 00:09:22,081 --> 00:09:25,510 a wonderful article on spinners, a comic style blog post about E, -161 +162 00:09:25,510 --> 00:09:29,199 a video about an ancient Babylonian algorithm for multiplying numbers, -162 +163 00:09:29,199 --> 00:09:32,420 which has unexpected usefulness for certain programming tasks. -163 +164 00:09:33,400 --> 00:09:36,160 There were a number of videos in Chinese on the site Bilibili, -164 +165 00:09:36,160 --> 00:09:39,841 including one I really liked about a fundamental theorem for symmetric polynomials, -165 +166 00:09:39,841 --> 00:09:42,120 perfectly understandable with the English subtitles. -166 +167 00:09:43,580 --> 00:09:47,414 And one absolutely fantastic video in here was about lemur factor stencils, -167 +168 00:09:47,414 --> 00:09:51,754 which I had never heard of, and I learned a ton watching this and found it absolutely -168 +169 00:09:51,754 --> 00:09:52,360 fascinating. -169 +170 00:09:53,580 --> 00:09:57,650 Many of the entries I saw had excellent aha moments, like this one here, -170 +171 00:09:57,650 --> 00:10:01,609 with a mildly clickbaity title about a graph that will blow your mind, -171 +172 00:10:01,609 --> 00:10:05,680 but the thing is, at least in my experience, that title is 100% accurate. -172 +173 00:10:06,300 --> 00:10:09,172 There's a video explaining why pi shows up in the Buffon needle -173 +174 00:10:09,172 --> 00:10:11,507 problem with a really elegant shift in perspective, -174 +175 00:10:11,507 --> 00:10:14,380 and what I especially appreciate is that the author also has an -175 +176 00:10:14,380 --> 00:10:17,432 appendix video going through some of the more technical details not -176 +177 00:10:17,432 --> 00:10:18,600 covered in the main video. -177 +178 00:10:19,140 --> 00:10:23,589 A few entries were highly interesting if for no other reason just from a technological -178 +179 00:10:23,589 --> 00:10:27,834 standpoint alone, including a very well-executed interactive video by Rob Schlubb, -179 +180 00:10:27,834 --> 00:10:30,238 which let me tell you is not easy to pull off, -180 +181 00:10:30,238 --> 00:10:34,687 as well as a great interactive article introducing complex numbers and the fundamental -181 +182 00:10:34,687 --> 00:10:36,580 theorem of algebra on the site Trina. -182 -00:10:37,040 --> 00:10:41,620 -And then setting aside all the explanatory value, we're just plain beautiful. - 183 +00:10:37,040 --> 00:10:40,392 +And then some of the entries, setting aside all the explanatory value, + +184 +00:10:40,392 --> 00:10:41,620 +were just plain beautiful. + +185 00:10:42,120 --> 00:10:44,612 Like if I had a category here for greatest style, -184 +186 00:10:44,612 --> 00:10:48,600 I think my pick would be this one about recreating curves from a children's toy. -185 +187 00:10:50,480 --> 00:10:53,154 But setting aside style or the core point of all of this, -186 +188 00:10:53,154 --> 00:10:56,612 which is the explanatory quality, there's one feature of online explainers -187 +189 00:10:56,612 --> 00:11:00,440 that can easily be underappreciated, which is the role of narrative and storylines. -188 +190 00:11:01,040 --> 00:11:04,260 And a couple entries I think did a great job exemplifying that component. -189 +191 00:11:04,800 --> 00:11:08,620 This includes not one but two entries on this game called Hackenbush, -190 +192 00:11:08,620 --> 00:11:12,768 a story about how a lights-out puzzle can lead you to Gaussian elimination, -191 +193 00:11:12,768 --> 00:11:16,862 a nice exploration of the most efficient way to choose a random point in a -192 +194 00:11:16,862 --> 00:11:21,283 circle uniformly, a great puzzle about tiles, which carries with it explanations -193 +195 00:11:21,283 --> 00:11:25,486 of core facts from Fibonacci numbers, and one really nicely done video about -194 +196 00:11:25,486 --> 00:11:30,180 why the Sierpinski triangle shows up in three seemingly completely unrelated contexts. -195 +197 00:11:30,540 --> 00:11:32,720 Again, the list goes on for quite a while. -196 +198 00:11:33,640 --> 00:11:37,520 As I look at some of these now, I'm really pleased to see that a lot of them have picked -197 +199 00:11:37,520 --> 00:11:41,400 up some traction on YouTube, but there still remain many which are very underappreciated. -198 +200 00:11:41,920 --> 00:11:45,149 I highly encourage you to go to the playlist including all the video -199 +201 00:11:45,149 --> 00:11:48,660 submissions and to check out the blog post featuring all other submissions. -200 +202 00:11:49,480 --> 00:11:52,740 As you look at that playlist, I would not read into the order of it too much, -201 +203 00:11:52,740 --> 00:11:55,833 it was generated programmatically, but I did try to go through and curate -202 +204 00:11:55,833 --> 00:11:58,760 the first few ones with videos that I think you might especially like. -203 +205 00:11:59,320 --> 00:12:01,256 Honestly though, you can happily scroll down that -204 +206 00:12:01,256 --> 00:12:03,000 playlist and find hidden gems all throughout. -205 +207 00:12:03,640 --> 00:12:05,580 Like really, go check it out right now. -206 +208 00:12:05,800 --> 00:12:09,265 If you do, I can almost guarantee that you have hours of edification -207 +209 00:12:09,265 --> 00:12:12,480 waiting ahead of you, not to mention hours of just pure delight. -208 +210 00:12:13,580 --> 00:12:15,200 Thanks again to everyone who participated. -209 +211 00:12:15,520 --> 00:12:18,540 Let me just say one more time, I really was blown away by the quality here. diff --git a/2021/some1-results/english/sentence_timings.json b/2021/some1-results/english/sentence_timings.json index 62eb349b0..ee598f249 100644 --- a/2021/some1-results/english/sentence_timings.json +++ b/2021/some1-results/english/sentence_timings.json @@ -85,7 +85,7 @@ 132.2 ], [ - "As I said, there's a blog post with more details for any of you who are curious, but right now, without further ado, let me tell you about some outstanding math explainers that I really think you're going to enjoy.", + "As I said, there's a blog post with more details for any of you who are curious, but right now, without further ado, let me tell you about some outstanding math explainers made this summer that I really think you're going to enjoy.", 133.12, 143.76 ], @@ -255,7 +255,7 @@ 518.08 ], [ - "Another one which is long but good is about the unsolvability of the object.", + "Another one which is long but good is about the unsolvability of the quintic by Carl Turner. For me it was a bit b", 519.0, 526.16 ], @@ -310,7 +310,7 @@ 636.58 ], [ - "And then setting aside all the explanatory value, we're just plain beautiful.", + "And then some of the entries, setting aside all the explanatory value, were just plain beautiful.", 637.04, 641.62 ], diff --git a/2021/some1-results/english/transcript.txt b/2021/some1-results/english/transcript.txt index a94ad3108..6b62e8052 100644 --- a/2021/some1-results/english/transcript.txt +++ b/2021/some1-results/english/transcript.txt @@ -15,7 +15,7 @@ We also had around half a dozen guest judges, people in the space of math exposi Many, many thanks to those guests for being willing to volunteer some of their time, and generally helping to ensure that any of my subjective quirks weren't leaking too much into the final decision. Also, a million thanks to James. He really was the core organizer behind everything here, and if you participated in the event and found it helpful, he is absolutely the one you should be thanking. -As I said, there's a blog post with more details for any of you who are curious, but right now, without further ado, let me tell you about some outstanding math explainers that I really think you're going to enjoy. +As I said, there's a blog post with more details for any of you who are curious, but right now, without further ado, let me tell you about some outstanding math explainers made this summer that I really think you're going to enjoy. This entry, from a channel called Paralogical, opens by asking why the light reflected at the bottom of a mug seems to form this characteristic cardioid shape. The core mathematical idea that this video teaches is that of envelopes, which in short is a way to describe one curve using a family of other curves, and what really makes the video special is not just how clearly he explains that subject, but how tangible and well-chosen the examples are, all delivered with a tone that's just plain friendly and enjoyable to listen to. The key formula he builds up to is really well motivated, and one of those things that would not be obvious if you just saw it out of context, and he gives the tools for any curious viewer to pause and work through for themselves a more detailed understanding, while still leaving room for someone watching to get the general idea and the core points without necessarily being bogged down into that algebra. @@ -49,7 +49,7 @@ One that I think viewers of this channel would especially enjoy is almost an hou It's about Durock's belt trick by Noah Miller. In a recent event that I was doing with Stephen Strogatz for the MoMath Museum, we had a call that was ostensibly meant to prepare for that event, but instead we spent much of it just both gushing over how much we liked this one particular video. Any of you who have flirted with this topic will probably know how tricky it can be to understand the link between points on a 4D sphere and rotations in 3D space, and why any of that has to do with quantum mechanics, but this animated hour-long lecture does a really good job laying out the full story. -Another one which is long but good is about the unsolvability of the object. +Another one which is long but good is about the unsolvability of the quintic by Carl Turner. For me it was a bit b I'm not seeing it because when I did, I was actively working on a video that was not just about the same theorem, but about the same comparatively esoteric proof that it describes there. I thought the video did such a wonderful job, I just kind of set aside the project. I'll still probably cover it at some point, but now I'm motivated to do it in a different way, but in the meantime, any of you curious about why quintic polynomials are in a certain sense unsolvable will absolutely love this video. @@ -60,7 +60,7 @@ And one absolutely fantastic video in here was about lemur factor stencils, whic Many of the entries I saw had excellent aha moments, like this one here, with a mildly clickbaity title about a graph that will blow your mind, but the thing is, at least in my experience, that title is 100% accurate. There's a video explaining why pi shows up in the Buffon needle problem with a really elegant shift in perspective, and what I especially appreciate is that the author also has an appendix video going through some of the more technical details not covered in the main video. A few entries were highly interesting if for no other reason just from a technological standpoint alone, including a very well-executed interactive video by Rob Schlubb, which let me tell you is not easy to pull off, as well as a great interactive article introducing complex numbers and the fundamental theorem of algebra on the site Trina. -And then setting aside all the explanatory value, we're just plain beautiful. +And then some of the entries, setting aside all the explanatory value, were just plain beautiful. Like if I had a category here for greatest style, I think my pick would be this one about recreating curves from a children's toy. But setting aside style or the core point of all of this, which is the explanatory quality, there's one feature of online explainers that can easily be underappreciated, which is the role of narrative and storylines. And a couple entries I think did a great job exemplifying that component. diff --git a/2021/some1/english/captions.srt b/2021/some1/english/captions.srt index e5c4b9666..c6c7af133 100644 --- a/2021/some1/english/captions.srt +++ b/2021/some1/english/captions.srt @@ -383,19 +383,19 @@ And I get it, teachers are absurdly busy, they don't have time for extra things on the side, and it's kind of hard to know where to get started. 97 -00:04:53,460 --> 00:04:57,807 +00:04:53,460 --> 00:04:57,605 So maybe one potential partnership here would be the teachers who have really good 98 -00:04:57,807 --> 00:05:02,049 -instincts for what works in education, and then a student who maybe has a lot of +00:04:57,605 --> 00:05:02,000 +instincts for what works in education, and then a student who maybe has a lot of energy 99 -00:05:02,049 --> 00:05:06,240 -energy or desire to get started on YouTube or otherwise just has more free time +00:05:02,000 --> 00:05:06,444 +or desire to get started on YouTube or otherwise just has more free time on their hands, 100 -00:05:06,240 --> 00:05:10,640 +00:05:06,444 --> 00:05:10,640 and pairing something together like that might actually make for a good partnership. 101 @@ -552,7 +552,7 @@ my process in creating them. 139 00:07:12,360 --> 00:07:16,340 -I mean, the sound quality was pretty terrible for a long time is one big thing. +I mean, the sound quality was pretty terrible for a long time is that's one big thing. Uh, 140 00:07:17,080 --> 00:07:20,980 @@ -611,1162 +611,1166 @@ It was just this reminder that I really don't know what I'm doing. But this isn't a self-effacing thing. 154 -00:08:03,300 --> 00:08:06,948 -The point here is that if you find yourself with a potentially good +00:08:03,300 --> 00:08:06,955 +The point here is that if you find yourself with a potentially good explainer that 155 -00:08:06,948 --> 00:08:10,596 -explainer that you want to make, but you're a little self-conscious +00:08:06,955 --> 00:08:10,171 +you want to make, but you're a little self-conscious about how to start, 156 -00:08:10,596 --> 00:08:14,620 -about how to start or you're just don't worry about it, just dive right in. +00:08:10,171 --> 00:08:13,783 +or you're worried that you're going to make a mistake, just don't worry about it. 157 +00:08:13,783 --> 00:08:14,620 +Just dive right in. + +158 00:08:14,800 --> 00:08:17,620 So many of us have no idea what we're doing when we begin. -158 +159 00:08:20,980 --> 00:08:24,121 All that said, sometimes this just do it advice is a -159 +160 00:08:24,121 --> 00:08:27,560 little bit frustrating because I mean it's not actionable. -160 +161 00:08:27,940 --> 00:08:31,640 You say okay I'm gonna start, but then upon starting it tells you nothing. -161 +162 00:08:32,280 --> 00:08:34,428 So in the spirit of some more concrete advice, -162 +163 00:08:34,428 --> 00:08:37,902 I do have a couple things that I might want to pass along that are specific -163 +164 00:08:37,902 --> 00:08:39,320 to the case of math explainers. -164 +165 00:08:39,740 --> 00:08:42,580 The first one, and I do actually find this quite important, -165 +166 00:08:42,580 --> 00:08:46,415 is when you're putting together the explanation, whatever form, whatever medium, -166 +167 00:08:46,415 --> 00:08:50,251 whatever genre you choose, try to be aware of the layers of abstraction that are -167 +168 00:08:50,251 --> 00:08:51,340 relevant to your topic. -168 +169 00:08:51,820 --> 00:08:54,366 So like if you're teaching a young child about fractions and -169 +170 00:08:54,366 --> 00:08:56,370 you're talking about two-thirds plus one-fifth, -170 +171 00:08:56,370 --> 00:08:59,460 there's two different layers of abstraction that that expression lives in. -171 +172 00:08:59,620 --> 00:09:03,058 There's one where you have a very concrete example of two-thirds of something, -172 +173 00:09:03,058 --> 00:09:05,321 two-thirds of a cake, and then one-fifth of a cake, -173 +174 00:09:05,321 --> 00:09:07,280 and trying to get a sense of what that means. -174 +175 00:09:07,520 --> 00:09:11,195 And then there's the symbols, and a big part of the lesson at play here -175 +176 00:09:11,195 --> 00:09:14,820 is understanding how the symbols relate to the actual case and why the -176 +177 00:09:14,820 --> 00:09:18,700 rules that we apply to the symbols make sense in light of the concrete case. -177 +178 00:09:19,020 --> 00:09:22,923 And also why we opt to do the more abstract thing because it takes much less thinking -178 +179 00:09:22,923 --> 00:09:26,873 than actually trying to reason about you know two-thirds of a cake plus one-fifth of a -179 +180 00:09:26,873 --> 00:09:27,100 cake. -180 +181 00:09:28,040 --> 00:09:29,220 And this happens at all levels. -181 +182 00:09:29,220 --> 00:09:32,795 If you're teaching a calculus class and you're talking about optimizing functions, -182 +183 00:09:32,795 --> 00:09:36,198 you know there's the idea of a function as a very abstract thing that could be -183 +184 00:09:36,198 --> 00:09:39,300 any particular function or any differentiable function or what have you. -184 +185 00:09:39,700 --> 00:09:43,480 And then there's lots of specific examples or maybe specific cases where they come up -185 +186 00:09:43,480 --> 00:09:47,084 like a function defining the profit of a company and that's the thing you want to -186 +187 00:09:47,084 --> 00:09:47,480 optimize. -187 +188 00:09:48,640 --> 00:09:51,888 I made a whole video about group theory where in the middle I went on -188 +189 00:09:51,888 --> 00:09:55,136 for a while about the difference between thinking of group actions as -189 +190 00:09:55,136 --> 00:09:58,291 these abstract entities versus as something concrete like asymmetry -190 +191 00:09:58,291 --> 00:10:01,540 and why there exists the two and what the benefits and trade-offs are. -191 +192 00:10:01,900 --> 00:10:05,902 But my point in this first piece of advice is not merely to address the layers -192 +193 00:10:05,902 --> 00:10:10,006 of abstraction, you don't even have to, but if you're clear in your own head try -193 +194 00:10:10,006 --> 00:10:14,060 very hard to structure your explanation to go from the concrete to the abstract. -194 +195 00:10:14,680 --> 00:10:17,407 I think almost always when you understand something -195 +196 00:10:17,407 --> 00:10:20,240 the natural inclination is to go the other way around. -196 +197 00:10:20,700 --> 00:10:23,960 I find myself doing this in pretty much any first draft of a script that I have. -197 +198 00:10:24,380 --> 00:10:27,280 It seems like all the textbook authors that I ever read tend to do this. -198 +199 00:10:27,320 --> 00:10:30,237 You start with the abstract idea, you put the examples later, -199 +200 00:10:30,237 --> 00:10:34,001 but I really do think that in the case of learning first trying to populate the -200 +201 00:10:34,001 --> 00:10:37,671 learner's mind with a bunch of examples of things that have a similar pattern -201 +202 00:10:37,671 --> 00:10:40,353 between them and letting their brain do the abstraction, -202 +203 00:10:40,353 --> 00:10:44,118 see that similar pattern between things such that when you bring in that higher -203 +204 00:10:44,118 --> 00:10:47,882 layer you start defining you know an abstract vector space or you're doing some -204 +205 00:10:47,882 --> 00:10:50,000 symbolic manipulations with particular rules. -205 +206 00:10:50,540 --> 00:10:54,286 That once that happens you're articulating something in the brain of the learner that -206 +207 00:10:54,286 --> 00:10:58,120 was already sitting there in the first place, it wasn't just handed to them in a vacuum. -207 +208 00:10:58,920 --> 00:11:02,240 Otherwise it's a little bit like trying to build a building from the top floor down. -208 +209 00:11:02,960 --> 00:11:04,840 So that's one and that's very specific to math. -209 +210 00:11:05,340 --> 00:11:09,075 As a more generic idea, piece of advice number two would be keep in the very -210 +211 00:11:09,075 --> 00:11:11,695 forefront of your mind the fact that content is king, -211 +212 00:11:11,695 --> 00:11:15,431 that the thing that you're explaining, the choice of the topic or how you're -212 +213 00:11:15,431 --> 00:11:19,069 explaining it, determines the majority of the value and the quality of the -213 +214 00:11:19,069 --> 00:11:20,040 thing that you make. -214 +215 00:11:20,280 --> 00:11:24,646 All the things about production quality or you know how fancy the animations are -215 +216 00:11:24,646 --> 00:11:29,066 or the lighting or whatever it is, all of that is secondary to making sure you've -216 +217 00:11:29,066 --> 00:11:33,540 chosen an actually good topic and it's something that people would want to consume. -217 +218 00:11:33,740 --> 00:11:35,840 They haven't seen it elsewhere, it's offering something fresh. -218 +219 00:11:36,540 --> 00:11:40,708 Now that's so easy to nod along with and say like yes yes of course content is king, -219 +220 00:11:40,708 --> 00:11:44,190 but the thing is you end up spending about one percent of your time if -220 +221 00:11:44,190 --> 00:11:47,868 that choosing what you're going to explain and how you're going to explain -221 +222 00:11:47,868 --> 00:11:51,448 it and then like 99% of the time just carrying it out in some way and as -222 +223 00:11:51,448 --> 00:11:54,440 a result it can be easy to lose sight of that important part. -223 +224 00:11:54,880 --> 00:11:57,557 So my encouragement to you would be spend more time -224 +225 00:11:57,557 --> 00:12:00,440 than you would otherwise tend to on choosing that topic. -225 +226 00:12:00,820 --> 00:12:03,893 So maybe workshop a couple different things by doing sample lessons with -226 +227 00:12:03,893 --> 00:12:06,883 people or try to write out a list of all the different things that you -227 +228 00:12:06,883 --> 00:12:10,588 could and ask are they actually fresh, are they actually adding something to the space, -228 +229 00:12:10,588 --> 00:12:12,820 is there a reason someone would want to consume this. -229 +230 00:12:13,280 --> 00:12:15,958 Spending that extra little bit of time on the thing that -230 +231 00:12:15,958 --> 00:12:19,060 determines the majority of the value is almost certainly worth it. -231 +232 00:12:19,560 --> 00:12:23,693 And the third piece of advice which maybe plays into this a little bit is when you're -232 +233 00:12:23,693 --> 00:12:27,730 beginning if you're starting something fresh and there's no presence online at this -233 +234 00:12:27,730 --> 00:12:32,055 point, try to choose something much more esoteric and specific than you might be inclined -234 +235 00:12:32,055 --> 00:12:32,200 to. -235 +236 00:12:32,820 --> 00:12:36,448 I've seen a lot of people who want to get started on YouTube for example and the way -236 +237 00:12:36,448 --> 00:12:40,120 that they try to go about it is to choose a topic that will appeal to the most people. -237 +238 00:12:40,440 --> 00:12:42,267 After all they want their video to blow up, they -238 +239 00:12:42,267 --> 00:12:44,020 want a lot of subscribers and things like that. -239 +240 00:12:44,620 --> 00:12:45,920 But there's a couple issues with this. -240 +241 00:12:46,380 --> 00:12:48,942 First of all it's a much more competitive space if you're going to -241 +242 00:12:48,942 --> 00:12:51,620 try to describe something that a lot of people might be searching for. -242 +243 00:12:51,780 --> 00:12:55,008 So if you go in saying I'm going to do a series about quantum mechanics, -243 +244 00:12:55,008 --> 00:12:58,104 well there's a billion others out there and yours is going to have to -244 +245 00:12:58,104 --> 00:13:01,200 stand out for some reason and you don't have a foothold at that point. -245 +246 00:13:01,900 --> 00:13:05,787 But another one is that the very specific and niche things build a much more loyal -246 +247 00:13:05,787 --> 00:13:09,863 audience in that beginning because you're offering something which they could not find -247 +248 00:13:09,863 --> 00:13:13,797 anywhere else and sometimes to be the consumer of something very specific is such a -248 +249 00:13:13,797 --> 00:13:17,920 good feeling that you want to pay it back and you find yourself rooting for the creator. -249 +250 00:13:18,460 --> 00:13:22,320 So you're more likely to get very good faith feedback, just a warmer community. -250 +251 00:13:22,640 --> 00:13:27,640 And also oftentimes we tend to overestimate just how niche things are. -251 +252 00:13:28,020 --> 00:13:30,763 Like sometimes something that's so weirdly specific, -252 +253 00:13:30,763 --> 00:13:34,904 some very esoteric bit of engineering actually appeals to hundreds of thousands -253 +254 00:13:34,904 --> 00:13:39,045 or millions of people, especially if you yourself are enthusiastic about it and -254 +255 00:13:39,045 --> 00:13:40,340 people can index on that. -255 +256 00:13:40,760 --> 00:13:44,598 So by doing that you find yourself often with a topic that actually does have -256 +257 00:13:44,598 --> 00:13:48,486 a broader appeal but it's not competitive with the things that everyone thinks -257 +258 00:13:48,486 --> 00:13:51,980 will have broader appeal and you potentially get that audience loyalty. -258 +259 00:13:52,760 --> 00:13:55,717 When I started this channel I really was thinking of it as a very niche thing, -259 +260 00:13:55,717 --> 00:13:58,600 I did not think it would be a thing that a lot of people would want to watch. -260 +261 00:13:59,060 --> 00:14:02,768 I actually specifically wanted to find topics in math that no one would think to search -261 +262 00:14:02,768 --> 00:14:06,520 for, that was kind of the original conception that I wavered from a little bit afterward. -262 +263 00:14:07,240 --> 00:14:11,660 The fourth piece of advice is to pick a genre that your piece falls into. -263 +264 00:14:12,300 --> 00:14:15,593 So the other day I was giving this talk to a group of people and one -264 +265 00:14:15,593 --> 00:14:18,935 of them wanted to get started making online explainers and they asked -265 +266 00:14:18,935 --> 00:14:22,324 whether it was ethical for someone who's just barely learning a topic, -266 +267 00:14:22,324 --> 00:14:25,380 just starting to learn it, to also make explainers of it online. -267 +268 00:14:26,020 --> 00:14:28,137 I mean after all they're more likely to make mistakes, -268 +269 00:14:28,137 --> 00:14:30,870 they don't know the broader context, and there's so many things I like -269 +270 00:14:30,870 --> 00:14:31,640 about that question. -270 +271 00:14:32,040 --> 00:14:36,630 It's already demonstrating a kind of care and consideration for factual accuracy and -271 +272 00:14:36,630 --> 00:14:41,274 doing right by the student that more people who are making online explanations should -272 +273 00:14:41,274 --> 00:14:41,760 consider. -273 +274 00:14:42,160 --> 00:14:46,100 So that very fact suggested to me that this person probably should be doing it. -274 +275 00:14:46,580 --> 00:14:48,935 But one of the things I suggested is to acknowledge -275 +276 00:14:48,935 --> 00:14:51,200 there are different types of explainers out there. -276 +277 00:14:51,460 --> 00:14:54,534 There's the type where the narrator is a little bit more distanced, -277 +278 00:14:54,534 --> 00:14:58,241 they're kind of standing on top of a hill and explaining the way that things are, -278 +279 00:14:58,241 --> 00:15:01,180 and to do that you really have to research the topic very deeply. -279 +280 00:15:01,340 --> 00:15:05,140 You should probably know 10 times as much about the topic as what you're actually -280 +281 00:15:05,140 --> 00:15:08,894 saying in the content so that you know that you're teeing things up for where it -281 +282 00:15:08,894 --> 00:15:13,020 actually leads or you're being cognizant of whatever nuances there are, things like that. -282 +283 00:15:13,460 --> 00:15:16,532 But another genre entirely is the discovery journalism, -283 +284 00:15:16,532 --> 00:15:20,867 where the person who is learning the topic kind of just admits that fact or is -284 +285 00:15:20,867 --> 00:15:25,093 open about the fact that they're just starting with it and taking the viewer -285 +286 00:15:25,093 --> 00:15:26,520 along a journey with them. -286 +287 00:15:26,960 --> 00:15:29,645 And many times that's actually a better piece of content, -287 +288 00:15:29,645 --> 00:15:32,563 it's actually better for learning the topic, and it comes with -288 +289 00:15:32,563 --> 00:15:35,620 this inbuilt piece of humility that a lot of online content lacks. -289 +290 00:15:35,940 --> 00:15:37,660 But there's lots of other genres like this. -290 +291 00:15:37,760 --> 00:15:41,698 There's the worked example where you're explicitly helping people with homework, -291 +292 00:15:41,698 --> 00:15:45,248 there's the try to find an interesting demo and serve mainly to inspire, -292 +293 00:15:45,248 --> 00:15:49,624 and basically just before you get started, decide which one of those you feel like you're -293 +294 00:15:49,624 --> 00:15:53,223 the best fit for, and then when you're looking at other pieces out there, -294 +295 00:15:53,223 --> 00:15:57,015 other explainers and trying to index off of what seems to work, what doesn't, -295 +296 00:15:57,015 --> 00:16:01,197 be aware of which ones are in the lane that you intend to be in and don't necessarily -296 +297 00:16:01,197 --> 00:16:03,240 pattern match off of the ones that aren't. -297 +298 00:16:04,020 --> 00:16:07,447 You know, one of the mistakes I think I made with my very first video is I -298 +299 00:16:07,447 --> 00:16:10,966 had this conception that sometimes if you talk faster than is comfortable on -299 +300 00:16:10,966 --> 00:16:14,440 the internet, that sort of works, that's like a satisfying thing to consume. -300 +301 00:16:14,920 --> 00:16:18,488 Because there are videos out there that are this fire hose of information, -301 +302 00:16:18,488 --> 00:16:21,866 and something about that scratches a niche and I think people like it, -302 +303 00:16:21,866 --> 00:16:26,100 but what I didn't really appreciate was the fact that math should fall into a completely -303 +304 00:16:26,100 --> 00:16:27,480 different category than that. -304 +305 00:16:27,680 --> 00:16:30,637 It is not fun at all to have math come at you at this fire hose rate, -305 +306 00:16:30,637 --> 00:16:34,186 and basically I was just pattern matching off of things that I should not have been -306 +307 00:16:34,186 --> 00:16:35,200 pattern matching off of. -307 +308 00:16:35,660 --> 00:16:39,931 As point number five, or I don't actually know where I am at the list at this point, -308 +309 00:16:39,931 --> 00:16:43,800 especially in the case of math, if you're bringing up definitions of things, -309 +310 00:16:43,800 --> 00:16:45,760 try not to let them feel too arbitrary. -310 +311 00:16:45,960 --> 00:16:47,540 Try to let them be well motivated. -311 +312 00:16:47,680 --> 00:16:50,240 Explain why that's the definition, what else it could have been. -312 +313 00:16:50,440 --> 00:16:54,412 Try to make it something that the learner feels like they discovered themselves, -313 +314 00:16:54,412 --> 00:16:58,091 because too often we hand these things down on high as the starting point, -314 +315 00:16:58,091 --> 00:17:00,740 and it's not really clear why or where that came from. -315 +316 00:17:01,640 --> 00:17:04,040 So all of that is just on the content side, you know, -316 +317 00:17:04,040 --> 00:17:07,995 what exactly are you explaining independent of the multimedia component of it, you know, -317 +318 00:17:07,995 --> 00:17:09,640 the sound and the video and all that. -318 +319 00:17:09,960 --> 00:17:14,371 And like I said, content is king, that definitely determines the majority of quality, -319 +320 00:17:14,371 --> 00:17:17,655 but it does actually matter a little bit beyond that to have at -320 +321 00:17:17,655 --> 00:17:20,220 least a little bit of production quality, I think. -321 +322 00:17:20,660 --> 00:17:21,880 And I'll give a really good example. -322 -00:17:21,920 --> 00:17:25,951 -So I was watching this lecture the other day by Tadashi Tokieda on just a - 323 -00:17:25,951 --> 00:17:30,146 -really interesting set of ideas about applying physical intuition to solving +00:17:21,920 --> 00:17:26,080 +So I was watching this lecture the other day by Tadashi Tokieda on just a really 324 -00:17:30,146 --> 00:17:34,232 -math problems, and he had in there maybe seven or eight outstanding little +00:17:26,080 --> 00:17:30,446 +interesting set of ideas about applying physical intuition to solving math problems. 325 -00:17:34,232 --> 00:17:38,100 -arguments that could have been just a beautiful video in its own right. +00:17:30,446 --> 00:17:34,555 +And he had in there maybe seven or eight outstanding little arguments that each 326 +00:17:34,555 --> 00:17:38,100 +one of which could have been just a beautiful video in its own right. + +327 00:17:38,740 --> 00:17:41,804 But the talk was over Zoom, and the intro was really long, -327 +328 00:17:41,804 --> 00:17:45,076 and the sound quality is everything that you assume from Zoom, -328 +329 00:17:45,076 --> 00:17:48,660 and the lighting of his shot was weird, and the talk was really good. -329 +330 00:17:48,740 --> 00:17:52,546 And I do think, you know, it's great that it's online and a lot of people will be -330 +331 00:17:52,546 --> 00:17:55,934 consuming it, but it's probably fair to say that if all of that content, -331 +332 00:17:55,934 --> 00:17:58,904 the actual set of ideas, was instead, say, a Numberphile video, -332 +333 00:17:58,904 --> 00:18:01,040 it would reach a hundred times as many people. -333 +334 00:18:01,220 --> 00:18:05,040 And more than that, it would be a more pleasant experience for those who are consuming it. -334 +335 00:18:05,520 --> 00:18:06,900 And it really doesn't take that much. -335 +336 00:18:07,200 --> 00:18:09,840 So I'll just end with a couple pieces of advice on that front. -336 +337 00:18:10,580 --> 00:18:14,142 The first one, which again I acknowledge is very hypocritical here, -337 +338 00:18:14,142 --> 00:18:18,387 is sound quality actually matters, especially in an era of Zoom where we are all -338 +339 00:18:18,387 --> 00:18:22,578 inundated with this sort of sub-optimal version of the voices of all the people -339 +340 00:18:22,578 --> 00:18:23,260 in our lives. -340 +341 00:18:23,720 --> 00:18:26,912 The learner will appreciate a respite from all that with something that -341 +342 00:18:26,912 --> 00:18:30,460 actually comes from a good microphone that you learned how to use at some point. -342 +343 00:18:33,540 --> 00:18:38,066 On the side of visuals, you know, I'm obviously a big believer in the idea that a -343 +344 00:18:38,066 --> 00:18:42,649 well-chosen illustration or an animation can really make a mathematical idea a lot -344 +345 00:18:42,649 --> 00:18:47,341 more clear and be an example of that concretization and kind of going from the lower -345 +346 00:18:47,341 --> 00:18:52,200 layer of abstraction on upward by just showing exactly what it is on screen in some way. -346 +347 00:18:52,860 --> 00:18:55,160 Now the way I do things is with programmatic animations. -347 +348 00:18:55,560 --> 00:18:58,240 I sort of wrote this custom library called Manum to do that. -348 +349 00:18:58,660 --> 00:19:02,369 And last year, actually, a group of people that called themselves the Manum -349 +350 00:19:02,369 --> 00:19:06,420 Community created a fork of it with the hope of making it a lot more user-friendly. -350 +351 00:19:06,700 --> 00:19:07,840 And I think they succeeded with that. -351 +352 00:19:07,920 --> 00:19:12,120 There's a lot better documentation, it's better tested, just all around friendlier to use. -352 +353 00:19:12,340 --> 00:19:14,691 So you can use that tool and thanks to them, it's -353 +354 00:19:14,691 --> 00:19:16,620 actually a lot easier than it used to be. -354 +355 00:19:16,960 --> 00:19:20,822 There's some other libraries that I've seen that mention Manum as an inspiration, -355 +356 00:19:20,822 --> 00:19:23,460 you know, one that's written in Julia or one in Haskell. -356 +357 00:19:24,120 --> 00:19:25,860 And it doesn't have to be programmatic either. -357 +358 00:19:26,320 --> 00:19:30,476 I think where programmatic animations make sense for math is if you're -358 +359 00:19:30,476 --> 00:19:34,340 somehow leveraging loops or conditionals or layers of abstraction. -359 +360 00:19:35,020 --> 00:19:38,284 And in the right context, I think it can be a wonderful way to let -360 +361 00:19:38,284 --> 00:19:41,549 the visuals authentically reflect the math that you're describing, -361 +362 00:19:41,549 --> 00:19:44,960 if the code is essentially just that math as it's illustrating things. -362 +363 00:19:45,500 --> 00:19:46,360 But it doesn't have to be. -363 +364 00:19:46,460 --> 00:19:50,505 And a lot of times people use Manum or other programmatic animations for things that do -364 +365 00:19:50,505 --> 00:19:54,642 not need to be programmatic, that you could have easily done in something like Keynote or -365 +366 00:19:54,642 --> 00:19:58,780 which add flashiness for flashiness's sake that doesn't actually aid with the explanation. -366 +367 00:19:59,920 --> 00:20:02,564 I think one really good example of using traditional -367 +368 00:20:02,564 --> 00:20:04,960 animation software is the channel Boerbach Tree. -368 +369 00:20:05,580 --> 00:20:09,824 So he really has these friendly handwritten kind of whiteboard lectures, -369 +370 00:20:09,824 --> 00:20:13,080 but uses animation to help those whiteboards come alive. -370 +371 00:20:13,260 --> 00:20:15,360 And he uses Adobe Animate for that. -371 +372 00:20:15,420 --> 00:20:19,394 And I think it's a really nice way to make this friendly hand-drawn environment -372 +373 00:20:19,394 --> 00:20:22,722 come to life, which is different from kind of the platonic, stark, -373 +374 00:20:22,722 --> 00:20:26,746 this is precisely what the math would draw when you're illustrating a surface or -374 +375 00:20:26,746 --> 00:20:27,740 something like that. -375 +376 00:20:28,220 --> 00:20:30,858 Also, I see a lot of people use Manum to manipulate -376 +377 00:20:30,858 --> 00:20:33,040 algebraic expressions and things like that. -377 -00:20:33,460 --> 00:20:36,013 -But if you look at other videos, things like Mathologer, - 378 -00:20:36,013 --> 00:20:37,760 -he's doing a lot of that in PowerPoint. +00:20:33,460 --> 00:20:35,772 +But if you look at other videos, things like Mathologer, 379 +00:20:35,772 --> 00:20:37,760 +you know, he's doing a lot of that in PowerPoint. + +380 00:20:38,440 --> 00:20:39,740 And again, content is king. -380 +381 00:20:39,920 --> 00:20:44,084 The first thing is to focus on what are you actually describing and then just showing it, -381 +382 00:20:44,084 --> 00:20:47,000 however, is easiest to show it in that case works totally fine. -382 +383 00:20:47,040 --> 00:20:49,444 You don't need anything extremely precise or that -383 +384 00:20:49,444 --> 00:20:51,800 leverages loops and abstraction for the formulas. -384 +385 00:20:52,160 --> 00:20:54,330 I recognize a kind of hypocrisy here, but you know, -385 +386 00:20:54,330 --> 00:20:57,920 I have walked myself into a certain corner with the style that I want for the channel. -386 +387 00:20:59,200 --> 00:21:02,797 If you do want to go down that hole of programmatic animations, though, -387 +388 00:21:02,797 --> 00:21:06,644 another tool which has popped up recently is something called smoothstep.io, -388 +389 00:21:06,644 --> 00:21:10,392 which I think is a really nice way for people to get started with shaders, -389 +390 00:21:10,392 --> 00:21:13,740 which are an absurdly powerful way to do absurdly beautiful things. -390 +391 00:21:13,940 --> 00:21:16,608 And it's written by this guy, Matt Henderson, who has a Twitter -391 +392 00:21:16,608 --> 00:21:19,235 account that everyone should follow because he has some of the -392 +393 00:21:19,235 --> 00:21:21,820 most beautiful math illustrations that I think I've ever seen. -393 +394 00:21:22,320 --> 00:21:26,122 So experimenting with software like that is another rabbit hole that you could go down -394 +395 00:21:26,122 --> 00:21:29,487 if, say, you want to use this competition as an excuse to try something new, -395 +396 00:21:29,487 --> 00:21:31,979 something that you've always wanted to get started with, -396 +397 00:21:31,979 --> 00:21:33,640 but never really had the excuse to do. -397 +398 00:21:34,740 --> 00:21:38,390 Now, if you have some pieces of advice that you want to pass along to people, -398 +399 00:21:38,390 --> 00:21:42,415 or if you want to just engage with the community in some way to see what other people -399 +400 00:21:42,415 --> 00:21:45,784 are thinking of making or propose your own project ideas, get feedback, -400 +401 00:21:45,784 --> 00:21:49,856 talk about software, anything like that, we did set up a Discord space associated with -401 +402 00:21:49,856 --> 00:21:51,260 the Summer of Math Exposition. -402 +403 00:21:51,820 --> 00:21:52,820 There's a link in the description. -403 +404 00:21:53,660 --> 00:21:57,815 Just be mindful if you do contribute to that community that you want your comments -404 +405 00:21:57,815 --> 00:22:01,820 to be encouraging to others who are getting started and productive to that goal. -405 +406 00:22:02,400 --> 00:22:05,600 And, you know, try to avoid anything that is the opposite of that goal. -406 +407 00:22:08,120 --> 00:22:11,182 And again, another source of what will hopefully include -407 +408 00:22:11,182 --> 00:22:14,460 some inspirational or informative things will be the podcast. -408 +409 00:22:15,080 --> 00:22:16,260 The first episode is out now. -409 +410 00:22:16,420 --> 00:22:20,278 It's with the mathematician Alex Kontorovich, who some of you may recognize -410 +411 00:22:20,278 --> 00:22:24,340 from the video he did with Quanta or the video he did with Veritasium on Pi Day. -411 +412 00:22:24,720 --> 00:22:27,020 The episode after that is going to be with Sal Khan. -412 +413 00:22:27,540 --> 00:22:29,680 And there's just a really interesting lineup of people here. -413 +414 00:22:29,680 --> 00:22:30,700 So I think you'll enjoy it. -414 +415 00:22:30,800 --> 00:22:32,740 You can get it wherever you get your podcasts. -415 +416 00:22:33,140 --> 00:22:36,000 There's a video version of it, which is going to live on -416 +417 00:22:36,000 --> 00:22:38,760 a second channel that is just my name, Grant Sanderson. -417 +418 00:22:39,300 --> 00:22:42,171 And I figure for all future videos that are a little bit like this one, -418 +419 00:22:42,171 --> 00:22:44,165 that's not really animated math, but other stuff, -419 +420 00:22:44,165 --> 00:22:46,080 that's probably the channel that I'll put it on. -420 +421 00:22:46,240 --> 00:22:49,480 So keep an eye on that channel if that's something that you're interested in. -421 +422 00:22:50,180 --> 00:22:53,333 And I will say this about the podcast, even though the original intent -422 +423 00:22:53,333 --> 00:22:56,487 was something that was very much tied to this competition and the idea -423 +424 00:22:56,487 --> 00:22:58,753 of targeting people interested in getting started, -424 +425 00:22:58,753 --> 00:23:01,906 a lot of the times I would find myself with an interesting guest and I -425 +426 00:23:01,906 --> 00:23:05,193 just have a whole bunch of other things that I want to ask them that have -426 +427 00:23:05,193 --> 00:23:06,260 nothing to do with that. -427 +428 00:23:06,260 --> 00:23:11,100 So maybe the better framing here is to say that the podcast is 20% about that goal. -428 +429 00:23:11,320 --> 00:23:14,312 And the other 80% is just the usual interview -429 +430 00:23:14,312 --> 00:23:17,760 style podcast vibe where you have interesting guests. -430 +431 00:23:17,760 --> 00:23:21,200 And I just want to ask things that I'm genuinely curious to know about them. -431 +432 00:23:22,300 --> 00:23:26,514 And then I get to grad school and I'm moving into my office in grad school and I have my, -432 +433 00:23:26,514 --> 00:23:30,120 all my old papers and I just started, you know, for fun leafing through them. -433 +434 00:23:30,220 --> 00:23:34,460 I don't know if you ever look back at the stuff you wrote freshman year. -434 +435 00:23:34,880 --> 00:23:37,580 And I look at it, I'm like, what the hell was I writing? -435 +436 00:23:37,700 --> 00:23:38,960 Oh my God, this is garbage. -436 +437 00:23:39,200 --> 00:23:39,540 This is complete. -437 +438 00:23:39,880 --> 00:23:41,180 The epsilons and deltas are backwards. -438 +439 00:23:41,300 --> 00:23:43,400 You can't have the epsilons and deltas be backwards. -439 +440 00:23:43,580 --> 00:23:44,940 And you only took off three points. -440 +441 00:23:45,000 --> 00:23:48,100 I would have taken off, you know, nine or something like Ramy was so nice. -441 +442 00:23:48,700 --> 00:23:52,900 So if lean was around back then, boy, would it have straightened me out. -442 +443 00:23:52,980 --> 00:23:56,091 It's actually very inspiring to me because I feel one of the common pieces -443 +444 00:23:56,091 --> 00:23:59,120 of advice that I'll give to someone if they like want to learn more math. diff --git a/2021/some1/english/sentence_timings.json b/2021/some1/english/sentence_timings.json index 8752a7e0a..b0731fba8 100644 --- a/2021/some1/english/sentence_timings.json +++ b/2021/some1/english/sentence_timings.json @@ -220,7 +220,7 @@ 292.84 ], [ - "So maybe one potential partnership here would be the teachers who have really good instincts for what works in education, and then a student who maybe has a lot of energy or desire to get started on YouTube or otherwise just has more free time and pairing something together like that might actually make for a good partnership.", + "So maybe one potential partnership here would be the teachers who have really good instincts for what works in education, and then a student who maybe has a lot of energy or desire to get started on YouTube or otherwise just has more free time on their hands, and pairing something together like that might actually make for a good partnership.", 293.46, 310.64 ], @@ -325,7 +325,7 @@ 432.0 ], [ - "I mean, the sound quality was pretty terrible for a long time is one big thing.", + "I mean, the sound quality was pretty terrible for a long time is that's one big thing. Uh,", 432.36, 436.34 ], @@ -370,7 +370,7 @@ 483.18 ], [ - "The point here is that if you find yourself with a potentially good explainer that you want to make, but you're a little self-conscious about how to start or you're just don't worry about it, just dive right in.", + "The point here is that if you find yourself with a potentially good explainer that you want to make, but you're a little self-conscious about how to start, or you're worried that you're going to make a mistake, just don't worry about it. Just dive right in.", 483.3, 494.62 ], @@ -690,7 +690,7 @@ 1041.88 ], [ - "So I was watching this lecture the other day by Tadashi Tokieda on just a really interesting set of ideas about applying physical intuition to solving math problems, and he had in there maybe seven or eight outstanding little arguments that could have been just a beautiful video in its own right.", + "So I was watching this lecture the other day by Tadashi Tokieda on just a really interesting set of ideas about applying physical intuition to solving math problems. And he had in there maybe seven or eight outstanding little arguments that each one of which could have been just a beautiful video in its own right.", 1041.92, 1058.1 ], @@ -820,7 +820,7 @@ 1233.04 ], [ - "But if you look at other videos, things like Mathologer, he's doing a lot of that in PowerPoint.", + "But if you look at other videos, things like Mathologer, you know, he's doing a lot of that in PowerPoint.", 1233.46, 1237.76 ], diff --git a/2021/some1/english/transcript.txt b/2021/some1/english/transcript.txt index 7c9c98e09..3321d3c5e 100644 --- a/2021/some1/english/transcript.txt +++ b/2021/some1/english/transcript.txt @@ -42,7 +42,7 @@ If that's you and you feel passionate about it, you should definitely submit tha One set of people who I'm particularly interested in for this competition are the teachers and the lecturers, and basically anyone with a lot of boots-to-the-ground experience seeing people learn and seeing what actually works. Because I think there's a lot of outstanding explanations out there that stay largely confined to the classroom or otherwise stay offline, whereas if just a little bit of effort was put into producing it or sharing it online in some way, those lessons might actually reach and benefit one to two orders of magnitude more people. And I get it, teachers are absurdly busy, they don't have time for extra things on the side, and it's kind of hard to know where to get started. -So maybe one potential partnership here would be the teachers who have really good instincts for what works in education, and then a student who maybe has a lot of energy or desire to get started on YouTube or otherwise just has more free time and pairing something together like that might actually make for a good partnership. +So maybe one potential partnership here would be the teachers who have really good instincts for what works in education, and then a student who maybe has a lot of energy or desire to get started on YouTube or otherwise just has more free time on their hands, and pairing something together like that might actually make for a good partnership. In either case, whatever category you fall into, I do know there's a lot of people who do want to get started with this, because they write to me a lot, and one of the most common sentiments out there is, well I don't know where to get started, I want to make a video but I don't really have any experience with video making, things like that. And I have a couple things to say for who feels like they're in that boat. In a kind of loose conjunction with this contest, I decided to start a podcast where for the first many conversations I'll be interviewing people who have some kind of experience in the space of putting out explanations. @@ -63,7 +63,7 @@ Wait, am I ready yet? Am I ready yet? And I just say, Sal, press record and start, see what happens. If I take the example, which I know best, which is my own, there are so many really embarrassing things about the early videos on this channel or my process in creating them. -I mean, the sound quality was pretty terrible for a long time is one big thing. +I mean, the sound quality was pretty terrible for a long time is that's one big thing. Uh, I edited in iMovie for way longer than I care to admit. Also, despite being now a like professional YouTuber, when it comes to cameras and actually filming things like this, I really have no idea what I'm doing. Like right now, I'm just using a phone, which I guess is fine. @@ -72,7 +72,7 @@ This is actually my second time recording this whole video because the first tim But what the result was is that I would just kind of have my eyes darting back and forth between the two without me consciously realizing it. It was just this reminder that I really don't know what I'm doing. But this isn't a self-effacing thing. -The point here is that if you find yourself with a potentially good explainer that you want to make, but you're a little self-conscious about how to start or you're just don't worry about it, just dive right in. +The point here is that if you find yourself with a potentially good explainer that you want to make, but you're a little self-conscious about how to start, or you're worried that you're going to make a mistake, just don't worry about it. Just dive right in. So many of us have no idea what we're doing when we begin. All that said, sometimes this just do it advice is a little bit frustrating because I mean it's not actionable. You say okay I'm gonna start, but then upon starting it tells you nothing. @@ -136,7 +136,7 @@ Try to make it something that the learner feels like they discovered themselves, So all of that is just on the content side, you know, what exactly are you explaining independent of the multimedia component of it, you know, the sound and the video and all that. And like I said, content is king, that definitely determines the majority of quality, but it does actually matter a little bit beyond that to have at least a little bit of production quality, I think. And I'll give a really good example. -So I was watching this lecture the other day by Tadashi Tokieda on just a really interesting set of ideas about applying physical intuition to solving math problems, and he had in there maybe seven or eight outstanding little arguments that could have been just a beautiful video in its own right. +So I was watching this lecture the other day by Tadashi Tokieda on just a really interesting set of ideas about applying physical intuition to solving math problems. And he had in there maybe seven or eight outstanding little arguments that each one of which could have been just a beautiful video in its own right. But the talk was over Zoom, and the intro was really long, and the sound quality is everything that you assume from Zoom, and the lighting of his shot was weird, and the talk was really good. And I do think, you know, it's great that it's online and a lot of people will be consuming it, but it's probably fair to say that if all of that content, the actual set of ideas, was instead, say, a Numberphile video, it would reach a hundred times as many people. And more than that, it would be a more pleasant experience for those who are consuming it. @@ -162,7 +162,7 @@ So he really has these friendly handwritten kind of whiteboard lectures, but use And he uses Adobe Animate for that. And I think it's a really nice way to make this friendly hand-drawn environment come to life, which is different from kind of the platonic, stark, this is precisely what the math would draw when you're illustrating a surface or something like that. Also, I see a lot of people use Manum to manipulate algebraic expressions and things like that. -But if you look at other videos, things like Mathologer, he's doing a lot of that in PowerPoint. +But if you look at other videos, things like Mathologer, you know, he's doing a lot of that in PowerPoint. And again, content is king. The first thing is to focus on what are you actually describing and then just showing it, however, is easiest to show it in that case works totally fine. You don't need anything extremely precise or that leverages loops and abstraction for the formulas. diff --git a/2022/borwein/arabic/sentence_translations.json b/2022/borwein/arabic/sentence_translations.json index 32a5dd95e..71bada322 100644 --- a/2022/borwein/arabic/sentence_translations.json +++ b/2022/borwein/arabic/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it. ", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it. ", "translatedText": "الآن، إذا لم تسمع أبدًا عن تحويل فورييه، فهناك بعض الأشياء التي يمكنك القيام بها حيال ذلك. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/bengali/sentence_translations.json b/2022/borwein/bengali/sentence_translations.json index ed69f6af1..9195425e2 100644 --- a/2022/borwein/bengali/sentence_translations.json +++ b/2022/borwein/bengali/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it. ", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it. ", "translatedText": "এখন, আপনি যদি কখনও ফুরিয়ার ট্রান্সফর্মের কথা না শুনে থাকেন, তবে আপনি এটি সম্পর্কে কিছু করতে পারেন।", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/chinese/sentence_translations.json b/2022/borwein/chinese/sentence_translations.json index b6ba68fe9..ca4d85b7d 100644 --- a/2022/borwein/chinese/sentence_translations.json +++ b/2022/borwein/chinese/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it. ", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it. ", "translatedText": "现在，如果您从未听说过傅里叶 变换，您可以采取一些措施。", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/english/captions.srt b/2022/borwein/english/captions.srt index 19e3b6c5b..28997cb15 100644 --- a/2022/borwein/english/captions.srt +++ b/2022/borwein/english/captions.srt @@ -319,280 +319,280 @@ So the natural question is what on earth is going on here? And luckily there actually is a really satisfying explanation for all this. 81 -00:04:28,180 --> 00:04:32,070 -The way I think I'll go about this is to show you a phenomenon that first looks +00:04:28,180 --> 00:04:31,159 +The way I think I'll go about this is to show you a phenomenon that 82 -00:04:32,070 --> 00:04:36,300 -completely unrelated, but it shows a similar pattern where you have a value that stays +00:04:31,159 --> 00:04:34,051 +first looks completely unrelated, but it shows a similar pattern, 83 -00:04:36,300 --> 00:04:40,580 -really stable until you get to the number 15, and then it falters by just a tiny amount. +00:04:34,051 --> 00:04:37,600 +where you have a value that stays really stable until you get to the number 113. 84 +00:04:37,600 --> 00:04:40,580 +You get to the number 15, and then it falters by just a tiny amount. + +85 00:04:41,300 --> 00:04:44,820 And then after that I'll show why this seemingly unrelated phenomenon -85 +86 00:04:44,820 --> 00:04:48,340 is secretly the same as all our integral expressions, but in disguise. -86 +87 00:04:49,120 --> 00:04:52,556 So, turning our attention to what seems completely different, -87 +88 00:04:52,556 --> 00:04:55,882 consider a function that I'm going to be calling rect of x, -88 +89 00:04:55,882 --> 00:05:00,317 which is defined to equal 1 if the input is between negative 1 half and 1 half, -89 +90 00:05:00,317 --> 00:05:01,980 and otherwise it's equal to 0. -90 +91 00:05:02,220 --> 00:05:04,520 So the function is this boring step, basically. -91 +92 00:05:04,520 --> 00:05:07,962 This will be the first in a sequence of functions that we define, -92 +93 00:05:07,962 --> 00:05:11,249 so I'll call it f1 of x, and each new function in our sequence -93 +94 00:05:11,249 --> 00:05:14,640 is going to be a kind of moving average of the previous function. -94 +95 00:05:15,800 --> 00:05:20,100 So for example, the way the second iteration will be defined is to take this -95 +96 00:05:20,100 --> 00:05:23,954 sliding window whose width is 1 third, and for a particular input x, -96 +97 00:05:23,954 --> 00:05:28,422 when the window is centered at that input x, the value in my new function drawn -97 +98 00:05:28,422 --> 00:05:32,778 below is defined to be equal to the average value of the first function above -98 +99 00:05:32,778 --> 00:05:33,840 inside that window. -99 +100 00:05:33,840 --> 00:05:36,618 So for example, when the window is far enough to the left, -100 +101 00:05:36,618 --> 00:05:39,820 every value inside it is 0, so the graph on the bottom is showing 0. -101 +102 00:05:40,280 --> 00:05:43,299 As soon as that window starts to go over the plateau a little bit, -102 +103 00:05:43,299 --> 00:05:46,860 the average value is a little more than 0, and you see that in the graph below. -103 +104 00:05:47,280 --> 00:05:51,690 And notice that when exactly half the window is over that plateau at 1 and half of it -104 +105 00:05:51,690 --> 00:05:56,100 is at 0, the corresponding value in the bottom graph is 1 half, and you get the point. -105 +106 00:05:56,660 --> 00:06:00,237 The important thing I want you to focus on is how when that window is -106 +107 00:06:00,237 --> 00:06:03,253 entirely in the plateau above, where all the values are 1, -107 +108 00:06:03,253 --> 00:06:07,700 then the average value is also 1, so we get this plateau on our function at the bottom. -108 +109 00:06:08,300 --> 00:06:11,746 Let's call this bottom function f2 of x, and what I want you to -109 +110 00:06:11,746 --> 00:06:15,300 think about is the length of the plateau for that second function. -110 +111 00:06:15,480 --> 00:06:16,440 How wide should it be? -111 +112 00:06:17,020 --> 00:06:20,433 If you think about it for a moment, the distance between the left -112 +113 00:06:20,433 --> 00:06:23,743 edge of the top plateau and the left edge of the bottom plateau -113 +114 00:06:23,743 --> 00:06:27,260 will be exactly half of the width of the window, so half of 1 third. -114 +115 00:06:27,640 --> 00:06:30,230 And similarly on the right side, the distance between -115 +116 00:06:30,230 --> 00:06:32,820 the edges of the plateaus is half of the window width. -116 +117 00:06:33,200 --> 00:06:36,660 So overall it's 1 minus that window width, which is 1 minus 1 third. -117 +118 00:06:37,380 --> 00:06:41,103 The value we're going to be computing, the thing that will look stable for -118 +119 00:06:41,103 --> 00:06:44,678 a while before it breaks, is the value of this function at the input 0, -119 +120 00:06:44,678 --> 00:06:48,700 which in both of these iterations is equal to 1 because it's inside that plateau. -120 +121 00:06:49,200 --> 00:06:52,952 For the next iteration, we're going to take a moving average of that last function, -121 +122 00:06:52,952 --> 00:06:55,320 but this time with the window whose width is 1 fifth. -122 +123 00:06:55,320 --> 00:06:59,241 It's kind of fun to think about why as you slide around this window you get a -123 +124 00:06:59,241 --> 00:07:02,158 smoothed out version of the previous function, and again, -124 +125 00:07:02,158 --> 00:07:06,230 the significant thing I want you to focus on is how when that window is entirely -125 +126 00:07:06,230 --> 00:07:10,454 inside the plateau of the previous function, then by definition the bottom function -126 +127 00:07:10,454 --> 00:07:11,460 is going to equal 1. -127 +128 00:07:11,980 --> 00:07:15,610 This time the length of that plateau on the bottom will be the length -128 +129 00:07:15,610 --> 00:07:19,240 of the previous one, 1 minus 1 third, minus the window width, 1 fifth. -129 +130 00:07:19,600 --> 00:07:23,991 The reasoning is the same as before in order to go from the point where the middle of -130 +131 00:07:23,991 --> 00:07:28,332 the window is on that top plateau to where the entirety of the window is inside that -131 +132 00:07:28,332 --> 00:07:31,651 plateau is half the window width and likewise on the right side, -132 +133 00:07:31,651 --> 00:07:36,043 and once more the value to record is the output of this function when the input is 0, -133 +134 00:07:36,043 --> 00:07:37,320 which again is exactly 1. -134 +135 00:07:38,580 --> 00:07:41,880 The next iteration is a moving average with a window width of 1 seventh. -135 +136 00:07:42,100 --> 00:07:44,040 The plateau gets smaller by that 1 over 7. -136 +137 00:07:44,500 --> 00:07:48,060 Doing one more iteration with 1 over 9, the plateau gets smaller by that amount. -137 +138 00:07:48,600 --> 00:07:50,780 And as we keep going the plateau gets thinner and thinner. -138 +139 00:07:51,820 --> 00:07:55,539 And also notice how just outside of the plateau the function is really really -139 +140 00:07:55,539 --> 00:07:59,020 close to 1 because it's always been the result of an average between the -140 +141 00:07:59,020 --> 00:08:02,740 plateau at 1 and the neighbors, which themselves are really really close to 1. -141 +142 00:08:03,440 --> 00:08:06,900 The point at which all of this breaks is once we get to the iteration -142 +143 00:08:06,900 --> 00:08:10,360 where we're sliding a window with width 1 15th across the whole thing. -143 +144 00:08:10,760 --> 00:08:14,660 At that point the previous plateau is actually thinner than the window itself. -144 +145 00:08:14,820 --> 00:08:17,842 So even at the input x equals 0, this moving average -145 +146 00:08:17,842 --> 00:08:20,580 will have to be ever so slightly smaller than 1. -146 -00:08:20,780 --> 00:08:24,839 -And the only thing that's special about the number 15 here is that as we keep - 147 -00:08:24,839 --> 00:08:27,286 -adding the reciprocals of these odd fractions, +00:08:20,780 --> 00:08:24,959 +And the only thing that's special about the number 15 here is that as we keep adding 148 -00:08:27,286 --> 00:08:29,732 -1 third plus 1 fifth plus 1 seventh on and on, +00:08:24,959 --> 00:08:29,040 +the reciprocals of these odd fractions, one third plus one fifth plus one seventh, 149 -00:08:29,732 --> 00:08:33,220 -it's once we get to 1 15th that that sum grows to be bigger than 1. +00:08:29,040 --> 00:08:33,220 +on and on, it's once we get to one fifteenth that that sum grows to be bigger than 1. 150 00:08:33,580 --> 00:08:38,105 @@ -1055,28 +1055,28 @@ it tells you a lot more information than just that integral. You get a lot of bang for your buck out of doing the computation. 265 -00:15:07,200 --> 00:15:11,333 +00:15:07,200 --> 00:15:11,239 Now, the other key fact that will explain the connection we're hunting for is that if 266 -00:15:11,333 --> 00:15:14,265 +00:15:11,239 --> 00:15:14,104 you have two different functions and you take their product, 267 -00:15:14,265 --> 00:15:17,005 -and then you take the Fourier transform of that product, +00:15:14,104 --> 00:15:17,298 +and then you take the sum of the Fourier transform of that product, 268 -00:15:17,005 --> 00:15:21,090 +00:15:17,298 --> 00:15:21,291 it will be the same thing as if you individually took the Fourier transforms of your 269 -00:15:21,090 --> 00:15:25,080 -original function, and then combined them using a new kind of operation that we'll +00:15:21,291 --> 00:15:25,377 +original function and then combined them using a new kind of operation that we'll talk 270 -00:15:25,080 --> 00:15:27,820 -talk all about in the next video, known as a convolution. +00:15:25,377 --> 00:15:27,820 +all about in the next video, known as a convolution. 271 00:15:28,500 --> 00:15:31,802 diff --git a/2022/borwein/english/sentence_timings.json b/2022/borwein/english/sentence_timings.json index 96b36d52a..38ad7e6b6 100644 --- a/2022/borwein/english/sentence_timings.json +++ b/2022/borwein/english/sentence_timings.json @@ -175,7 +175,7 @@ 267.68 ], [ - "The way I think I'll go about this is to show you a phenomenon that first looks completely unrelated, but it shows a similar pattern where you have a value that stays really stable until you get to the number 15, and then it falters by just a tiny amount.", + "The way I think I'll go about this is to show you a phenomenon that first looks completely unrelated, but it shows a similar pattern, where you have a value that stays really stable until you get to the number 113. You get to the number 15, and then it falters by just a tiny amount.", 268.18, 280.58 ], @@ -315,7 +315,7 @@ 500.58 ], [ - "And the only thing that's special about the number 15 here is that as we keep adding the reciprocals of these odd fractions, 1 third plus 1 fifth plus 1 seventh on and on, it's once we get to 1 15th that that sum grows to be bigger than 1.", + "And the only thing that's special about the number 15 here is that as we keep adding the reciprocals of these odd fractions, one third plus one fifth plus one seventh, on and on, it's once we get to one fifteenth that that sum grows to be bigger than 1.", 500.78, 513.22 ], @@ -510,7 +510,7 @@ 906.38 ], [ - "Now, the other key fact that will explain the connection we're hunting for is that if you have two different functions and you take their product, and then you take the Fourier transform of that product, it will be the same thing as if you individually took the Fourier transforms of your original function, and then combined them using a new kind of operation that we'll talk all about in the next video, known as a convolution.", + "Now, the other key fact that will explain the connection we're hunting for is that if you have two different functions and you take their product, and then you take the sum of the Fourier transform of that product, it will be the same thing as if you individually took the Fourier transforms of your original function and then combined them using a new kind of operation that we'll talk all about in the next video, known as a convolution.", 907.2, 927.82 ], diff --git a/2022/borwein/english/transcript.txt b/2022/borwein/english/transcript.txt index 0efa6a37a..aef9f89fc 100644 --- a/2022/borwein/english/transcript.txt +++ b/2022/borwein/english/transcript.txt @@ -33,7 +33,7 @@ If we take all these integrals and include yet another factor, 2 cosine of x, wh And when it breaks, it's by the most puny, absolutely subtle amount that you could imagine. So the natural question is what on earth is going on here? And luckily there actually is a really satisfying explanation for all this. -The way I think I'll go about this is to show you a phenomenon that first looks completely unrelated, but it shows a similar pattern where you have a value that stays really stable until you get to the number 15, and then it falters by just a tiny amount. +The way I think I'll go about this is to show you a phenomenon that first looks completely unrelated, but it shows a similar pattern, where you have a value that stays really stable until you get to the number 113. You get to the number 15, and then it falters by just a tiny amount. And then after that I'll show why this seemingly unrelated phenomenon is secretly the same as all our integral expressions, but in disguise. So, turning our attention to what seems completely different, consider a function that I'm going to be calling rect of x, which is defined to equal 1 if the input is between negative 1 half and 1 half, and otherwise it's equal to 0. So the function is this boring step, basically. @@ -61,7 +61,7 @@ And also notice how just outside of the plateau the function is really really cl The point at which all of this breaks is once we get to the iteration where we're sliding a window with width 1 15th across the whole thing. At that point the previous plateau is actually thinner than the window itself. So even at the input x equals 0, this moving average will have to be ever so slightly smaller than 1. -And the only thing that's special about the number 15 here is that as we keep adding the reciprocals of these odd fractions, 1 third plus 1 fifth plus 1 seventh on and on, it's once we get to 1 15th that that sum grows to be bigger than 1. +And the only thing that's special about the number 15 here is that as we keep adding the reciprocals of these odd fractions, one third plus one fifth plus one seventh, on and on, it's once we get to one fifteenth that that sum grows to be bigger than 1. And in the context of our shrinking plateaus, having started with a plateau of width 1, it's now shrunk down so much that it'll disappear entirely. The point is with this as a sequence of functions that we've defined by a seemingly random procedure, if I ask you to compute the values of all of these functions at the input 0, you get a pattern which initially looks stable. It's 1 1 1 1 1 1 1, but by the time we get to the eighth iteration it falls short ever so slightly, just barely. @@ -100,7 +100,7 @@ Now, you could complain, surely this just moves the bump under the rug. Surely computing this Fourier transform, whatever that looks like, would be as hard as computing the original integral. But the idea is that there's lots of tips and tricks for computing these Fourier transforms, and moreover, that when you do, it tells you a lot more information than just that integral. You get a lot of bang for your buck out of doing the computation. -Now, the other key fact that will explain the connection we're hunting for is that if you have two different functions and you take their product, and then you take the Fourier transform of that product, it will be the same thing as if you individually took the Fourier transforms of your original function, and then combined them using a new kind of operation that we'll talk all about in the next video, known as a convolution. +Now, the other key fact that will explain the connection we're hunting for is that if you have two different functions and you take their product, and then you take the sum of the Fourier transform of that product, it will be the same thing as if you individually took the Fourier transforms of your original function and then combined them using a new kind of operation that we'll talk all about in the next video, known as a convolution. Now, even though there's a lot to be explained with convolutions, the upshot will be that in our specific case with these rectangular functions, taking a convolution looks just like one of the moving averages that we've been talking about this whole time, combined with our previous fact that integrating in one context looks like evaluating at zero in another context, if you believe me that multiplying in one context corresponds to this new operation, convolutions, which for our example you should just think of as moving averages, that will explain why multiplying more and more of these sinc functions together can be thought about in terms of these progressive moving averages and always evaluating at zero, which in turn gives a really lovely intuition for why you would expect such a stable value before eventually something breaks down as the edges of the plateau inch closer and closer to the center. This last key fact, by the way, has a special name. It's called the convolution theorem, and again, it's something that we'll go into much more deeply. diff --git a/2022/borwein/french/sentence_translations.json b/2022/borwein/french/sentence_translations.json index 86d88199c..e923537c9 100644 --- a/2022/borwein/french/sentence_translations.json +++ b/2022/borwein/french/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it. ", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it. ", "translatedText": "Maintenant, si vous n'avez jamais entendu parler d'une transformée de Fourier, vous pouvez faire certaines choses à ce sujet. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/german/sentence_translations.json b/2022/borwein/german/sentence_translations.json index 15b032008..6616df545 100644 --- a/2022/borwein/german/sentence_translations.json +++ b/2022/borwein/german/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it.", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it.", "translatedText": "Wenn Sie noch nie von einer Fourier-Transformation gehört haben, gibt es ein paar Dinge, die Sie dagegen tun können.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/hebrew/sentence_translations.json b/2022/borwein/hebrew/sentence_translations.json index 129e47014..017e93263 100644 --- a/2022/borwein/hebrew/sentence_translations.json +++ b/2022/borwein/hebrew/sentence_translations.json @@ -616,7 +616,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it.", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it.", "translatedText": "עכשיו, אם מעולם לא שמעת על טרנספורמציה של פורייה, יש כמה דברים שאתה יכול לעשות בקשר לזה.", "n_reviews": 0, "start": 748.26, diff --git a/2022/borwein/hindi/sentence_translations.json b/2022/borwein/hindi/sentence_translations.json index 89057355e..e0cd66015 100644 --- a/2022/borwein/hindi/sentence_translations.json +++ b/2022/borwein/hindi/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it. ", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it. ", "translatedText": "अब, यदि आपने फूरियर रूपांतरण के बारे में कभी नहीं सुना है, तो कुछ चीजें हैं जो आप इसके बारे में कर सकते हैं।", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/hungarian/sentence_translations.json b/2022/borwein/hungarian/sentence_translations.json index 9ac973722..fc4383022 100644 --- a/2022/borwein/hungarian/sentence_translations.json +++ b/2022/borwein/hungarian/sentence_translations.json @@ -280,7 +280,7 @@ "end": 267.68 }, { - "input": "The way I think I'll go about this is to show you a phenomenon that first looks completely unrelated, but it shows a similar pattern where you have a value that stays really stable until you get to the number 15, and then it falters by just a tiny amount.", + "input": "The way I think I'll go about this is to show you a phenomenon that first looks completely unrelated, but it shows a similar pattern, where you have a value that stays really stable until you get to the number 113. You get to the number 15, and then it falters by just a tiny amount.", "translatedText": "Azt hiszem, úgy fogom ezt megközelíteni, hogy mutatok egy olyan jelenséget, amely először teljesen független, de hasonló mintát mutat, ahol egy érték nagyon stabil marad, amíg el nem éri a 15-ös számot, és aztán egy aprócska mértékben meginog.", "model": "DeepL", "n_reviews": 0, @@ -504,7 +504,7 @@ "end": 500.58 }, { - "input": "And the only thing that's special about the number 15 here is that as we keep adding the reciprocals of these odd fractions, 1 third plus 1 fifth plus 1 seventh on and on, it's once we get to 1 15th that that sum grows to be bigger than 1.", + "input": "And the only thing that's special about the number 15 here is that as we keep adding the reciprocals of these odd fractions, one third plus one fifth plus one seventh, on and on, it's once we get to one fifteenth that that sum grows to be bigger than 1.", "translatedText": "Az egyetlen dolog, ami különleges a 15-ös számmal kapcsolatban, az az, hogy ahogy folyamatosan összeadjuk a páratlan törtek reciprokát, 1 harmadik, 1 ötödik, 1 hetedik és így tovább, az összeg csak akkor nő 1-nél nagyobbra, amikor elérünk az 1 15-ös számhoz.", "model": "DeepL", "n_reviews": 0, @@ -816,7 +816,7 @@ "end": 906.38 }, { - "input": "Now, the other key fact that will explain the connection we're hunting for is that if you have two different functions and you take their product, and then you take the Fourier transform of that product, it will be the same thing as if you individually took the Fourier transforms of your original function, and then combined them using a new kind of operation that we'll talk all about in the next video, known as a convolution.", + "input": "Now, the other key fact that will explain the connection we're hunting for is that if you have two different functions and you take their product, and then you take the sum of the Fourier transform of that product, it will be the same thing as if you individually took the Fourier transforms of your original function and then combined them using a new kind of operation that we'll talk all about in the next video, known as a convolution.", "translatedText": "A másik kulcsfontosságú tény, ami megmagyarázza az összefüggést, amit keresünk, hogy ha van két különböző függvényünk, és vesszük a szorzatukat, majd ennek a szorzatnak a Fourier-transzformációját, akkor ez ugyanaz lesz, mintha külön-külön vennénk az eredeti függvény Fourier-transzformációit, majd kombinálnánk őket egy újfajta művelet segítségével, amiről a következő videóban fogunk beszélni, amit konvolúciónak hívunk.", "model": "DeepL", "n_reviews": 0, diff --git a/2022/borwein/indonesian/sentence_translations.json b/2022/borwein/indonesian/sentence_translations.json index e1c0b496a..50ee3da73 100644 --- a/2022/borwein/indonesian/sentence_translations.json +++ b/2022/borwein/indonesian/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it.", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it.", "translatedText": "Sekarang, jika Anda belum pernah mendengar tentang transformasi Fourier, ada beberapa hal yang dapat Anda lakukan untuk mengatasinya.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/italian/sentence_translations.json b/2022/borwein/italian/sentence_translations.json index 16cbdcb57..edab4d6b6 100644 --- a/2022/borwein/italian/sentence_translations.json +++ b/2022/borwein/italian/sentence_translations.json @@ -616,7 +616,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it.", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it.", "translatedText": "Ora, se non hai mai sentito parlare di trasformata di Fourier, ci sono alcune cose che puoi fare al riguardo.", "n_reviews": 0, "start": 748.26, diff --git a/2022/borwein/japanese/sentence_translations.json b/2022/borwein/japanese/sentence_translations.json index b6b758198..bd237a6b9 100644 --- a/2022/borwein/japanese/sentence_translations.json +++ b/2022/borwein/japanese/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it. ", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it. ", "translatedText": "フーリエ変換について聞いたことがない場合でも、フ ーリエ変換についてできることがいくつかあります。", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/korean/sentence_translations.json b/2022/borwein/korean/sentence_translations.json index 8dd83f22f..5ae048399 100644 --- a/2022/borwein/korean/sentence_translations.json +++ b/2022/borwein/korean/sentence_translations.json @@ -315,7 +315,7 @@ "end": 267.68 }, { - "input": "The way I think I'll go about this is to show you a phenomenon that first looks completely unrelated, but it shows a similar pattern where you have a value that stays really stable until you get to the number 15, and then it falters by just a tiny amount.", + "input": "The way I think I'll go about this is to show you a phenomenon that first looks completely unrelated, but it shows a similar pattern, where you have a value that stays really stable until you get to the number 113. You get to the number 15, and then it falters by just a tiny amount.", "translatedText": "제가 생각하는 방법은 처음에는 전혀 관련이 없어 보이지만 숫자 15에 도달할 때까지 매우 안정적인 값을 유지하다가 아주 조금만 흔들리는 비슷한 패턴을 보이는 현상을 보여드리는 것입니다.", "model": "DeepL", "from_community_srt": "제가 이 문제에 대해 보여줄 방법은 처음에는 전혀 관련이 없어 보이지만, 비슷한 패턴을 보이는 것입니다. 15에 도달할 때 까지는 정말로 안정적인 값을 가진 다음 아주 작은 양만큼 흔들리는 것입니다.", @@ -566,7 +566,7 @@ "end": 500.58 }, { - "input": "And the only thing that's special about the number 15 here is that as we keep adding the reciprocals of these odd fractions, 1 third plus 1 fifth plus 1 seventh on and on, it's once we get to 1 15th that that sum grows to be bigger than 1.", + "input": "And the only thing that's special about the number 15 here is that as we keep adding the reciprocals of these odd fractions, one third plus one fifth plus one seventh, on and on, it's once we get to one fifteenth that that sum grows to be bigger than 1.", "translatedText": "여기서 숫자 15의 특별한 점은 홀수 분수의 역수, 즉 3분의 1 더하기 5분의 1 더하기 7분의 1을 계속 더하다 보면 15분의 1에 도달하면 그 합이 1보다 커진다는 점입니다.", "model": "DeepL", "from_community_srt": "그리고 여기서 숫자 15의 특별한 점은 이 홀수들의 역수를 계속해서 더한다는 것입니다. ⅓ + ⅕ + ⅐, 계속해서 1/15에 도달하면 합은 1보다 커지며,", @@ -916,7 +916,7 @@ "end": 906.38 }, { - "input": "Now, the other key fact that will explain the connection we're hunting for is that if you have two different functions and you take their product, and then you take the Fourier transform of that product, it will be the same thing as if you individually took the Fourier transforms of your original function, and then combined them using a new kind of operation that we'll talk all about in the next video, known as a convolution.", + "input": "Now, the other key fact that will explain the connection we're hunting for is that if you have two different functions and you take their product, and then you take the sum of the Fourier transform of that product, it will be the same thing as if you individually took the Fourier transforms of your original function and then combined them using a new kind of operation that we'll talk all about in the next video, known as a convolution.", "translatedText": "이제 우리가 찾고 있는 연결을 설명할 또 다른 핵심 사실은 서로 다른 두 함수가 있고 그 곱을 취한 다음 그 곱의 푸리에 변환을 취하면 원래 함수의 푸리에 변환을 개별적으로 취한 다음 다음 동영상에서 설명할 새로운 종류의 연산, 즉 컨볼루션을 사용하여 결합한 것과 동일하다는 것입니다.", "model": "DeepL", "from_community_srt": "자, 연결성에 관해서 우리가 찾는 또 다른 핵심 사실은 만약 당신이 두 개의 다른 함수를 가지고 있고, 그 함수들끼리 곱을 하고, 그 다음 푸리에 변환을 취하는 것은, 원래 각각의 함수에 푸리에 변환을 개별적으로 취한 다음, 제가 다음 영상에서 다룰 새로운 연산자인 합성곱(𝑪𝒐𝒏𝒗𝒐𝒍𝒖𝒕𝒊𝒐𝒏)을 사용해서 그것들을 결합하는 것과 같다는 것 입니다.", diff --git a/2022/borwein/marathi/sentence_translations.json b/2022/borwein/marathi/sentence_translations.json index 6556f1941..02ddf14ff 100644 --- a/2022/borwein/marathi/sentence_translations.json +++ b/2022/borwein/marathi/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it.", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it.", "translatedText": "आता, जर तुम्ही फूरियर ट्रान्सफॉर्मबद्दल कधीच ऐकले नसेल, तर तुम्ही त्याबद्दल काही गोष्टी करू शकता.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/persian/sentence_translations.json b/2022/borwein/persian/sentence_translations.json index d53dd5667..1260f13dc 100644 --- a/2022/borwein/persian/sentence_translations.json +++ b/2022/borwein/persian/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it. ", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it. ", "translatedText": "حال، اگر هرگز در مورد تبدیل فوریه نشنیده اید، چند کار وجود دارد که می توانید در مورد آن انجام دهید. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/portuguese/sentence_translations.json b/2022/borwein/portuguese/sentence_translations.json index 1dcee244b..3df6cf827 100644 --- a/2022/borwein/portuguese/sentence_translations.json +++ b/2022/borwein/portuguese/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it. ", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it. ", "translatedText": "Agora, se você nunca ouviu falar de transformada de Fourier, há algumas coisas que você pode fazer a respeito. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/russian/sentence_translations.json b/2022/borwein/russian/sentence_translations.json index e50711912..4f98ede65 100644 --- a/2022/borwein/russian/sentence_translations.json +++ b/2022/borwein/russian/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it.", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it.", "translatedText": "Теперь, если вы никогда не слышали о преобразовании Фурье, есть несколько вещей, которые вы можете с ним сделать.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/spanish/sentence_translations.json b/2022/borwein/spanish/sentence_translations.json index 7e76bf1e5..e75960365 100644 --- a/2022/borwein/spanish/sentence_translations.json +++ b/2022/borwein/spanish/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it.", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it.", "translatedText": "Ahora bien, si nunca has oído hablar de la transformada de Fourier, hay algunas cosas que puedes hacer al respecto.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/tamil/sentence_translations.json b/2022/borwein/tamil/sentence_translations.json index e3b408f67..a3ab0e08d 100644 --- a/2022/borwein/tamil/sentence_translations.json +++ b/2022/borwein/tamil/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it.", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it.", "translatedText": "இப்போது, ஃபோரியர் உருமாற்றத்தைப் பற்றி நீங்கள் கேள்விப்பட்டிருக்கவில்லை என்றால், அதைப் பற்றி நீங்கள் செய்யக்கூடிய சில விஷயங்கள் உள்ளன.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/telugu/sentence_translations.json b/2022/borwein/telugu/sentence_translations.json index da02ca7ee..f5d610c9b 100644 --- a/2022/borwein/telugu/sentence_translations.json +++ b/2022/borwein/telugu/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it.", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it.", "translatedText": "ఇప్పుడు, మీరు ఫోరియర్ పరివర్తన గురించి ఎప్పుడూ వినకపోతే, దాని గురించి మీరు చేయగలిగే కొన్ని విషయాలు ఉన్నాయి.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/thai/sentence_translations.json b/2022/borwein/thai/sentence_translations.json index d2b142193..cfac45d21 100644 --- a/2022/borwein/thai/sentence_translations.json +++ b/2022/borwein/thai/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it. ", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/turkish/sentence_translations.json b/2022/borwein/turkish/sentence_translations.json index ac9706c3a..24b7adb66 100644 --- a/2022/borwein/turkish/sentence_translations.json +++ b/2022/borwein/turkish/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it.", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it.", "translatedText": "Fourier dönüşümünü hiç duymadıysanız bu konuda yapabileceğiniz birkaç şey var.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/ukrainian/sentence_translations.json b/2022/borwein/ukrainian/sentence_translations.json index 9c3763995..d3a8604f9 100644 --- a/2022/borwein/ukrainian/sentence_translations.json +++ b/2022/borwein/ukrainian/sentence_translations.json @@ -616,7 +616,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it.", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it.", "translatedText": "Тепер, якщо ви ніколи не чули про перетворення Фур’є, є кілька речей, які ви можете зробити з цим.", "n_reviews": 0, "start": 748.26, diff --git a/2022/borwein/urdu/sentence_translations.json b/2022/borwein/urdu/sentence_translations.json index f0d8ba685..4ce60eb81 100644 --- a/2022/borwein/urdu/sentence_translations.json +++ b/2022/borwein/urdu/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it. ", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it. ", "translatedText": "اب، اگر آپ نے کبھی فوئیر ٹرانسفارم کے بارے میں نہیں سنا ہے، تو کچھ چیزیں ہیں جو آپ اس کے بارے میں کر سکتے ہیں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/borwein/vietnamese/sentence_translations.json b/2022/borwein/vietnamese/sentence_translations.json index 026e7fd9c..9ef7e403b 100644 --- a/2022/borwein/vietnamese/sentence_translations.json +++ b/2022/borwein/vietnamese/sentence_translations.json @@ -704,7 +704,7 @@ "end": 747.62 }, { - "input": "Now, if you've never heard of a Fourier transform, there are a few things that you can do about it.", + "input": "Now, if you've never heard of a Fourier transform, there are a few other videos on this channel all about it.", "translatedText": "Bây giờ, nếu bạn chưa bao giờ nghe nói về phép biến đổi Fourier thì có một số điều bạn có thể làm với nó.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/arabic/sentence_translations.json b/2022/convolutions/arabic/sentence_translations.json index fd18c441e..9f99ca2a1 100644 --- a/2022/convolutions/arabic/sentence_translations.json +++ b/2022/convolutions/arabic/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations. ", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations. ", "translatedText": "على سبيل المثال، لنفترض أنني أعطيتك كثيرتي حدود كبيرتين حقًا، قل لكل منهما مائة معامل مختلف، فإذا كانت الطريقة التي تضربهما بها هي توسيع هذا المنتج، فأنت تعرف ملء هذه الشبكة الكاملة 100 × 100 من المنتجات الزوجية التي تتطلب منك قم بتنفيذ 10000 منتج مختلف، وبعد ذلك عندما تجمع كل المصطلحات المتشابهة على طول الأقطار، تكون هذه مجموعة أخرى من حوالي 10000 عملية. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/bengali/sentence_translations.json b/2022/convolutions/bengali/sentence_translations.json index a0ae6d4ec..5ba6bb783 100644 --- a/2022/convolutions/bengali/sentence_translations.json +++ b/2022/convolutions/bengali/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations. ", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations. ", "translatedText": "উদাহরণ স্বরূপ ধরা যাক, আমি আপনাকে দুটি সত্যিকারের বড় বহুপদ দিয়েছি যার প্রতিটিতে একশটি ভিন্ন সহগ আছে তারপর যদি আপনি সেগুলিকে যেভাবে গুন করেন তা যদি এই পণ্যটিকে প্রসারিত করতে হয় তাহলে আপনি জানেন যে পেয়ারওয়াইজ পণ্যগুলির এই সম্পূর্ণ 100 বাই 100 গ্রিডটি পূরণ করতে আপনার প্রয়োজন হবে।10,000টি বিভিন্ন পণ্য সঞ্চালন করুন এবং তারপর যখন আপনি তির্যক বরাবর সমস্ত অনুরূপ পদ সংগ্রহ করছেন তখন এটি প্রায় 10,000 অপারেশনের আরেকটি সেট।", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/chinese/sentence_translations.json b/2022/convolutions/chinese/sentence_translations.json index 0318cb447..2dc772242 100644 --- a/2022/convolutions/chinese/sentence_translations.json +++ b/2022/convolutions/chinese/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations. ", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations. ", "translatedText": "例如，假设我 给了你两个非常大的多项式，每个多项式都有一百个不同的系数，那么如果 你将它们相乘的方式是展开这个乘积，你知道填充整个 100 x 1 00 的成对乘积网格，这需要你执行 10,000 种不同的产品， 然后当您沿着对角线收集所有相似项时，这是另一组大约 10,000 次操作。", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/french/sentence_translations.json b/2022/convolutions/french/sentence_translations.json index 142158b9f..4666812b0 100644 --- a/2022/convolutions/french/sentence_translations.json +++ b/2022/convolutions/french/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations.", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations.", "translatedText": "Par exemple, disons que je vous ai donné deux très gros polynômes, disons chacun avec une centaine de coefficients différents, alors si la façon dont vous les multipliez était d'étendre ce produit, vous savez, en remplissant toute cette grille de 100 par 100 de produits par paires qui vous obligeraient à effectuez 10 000 produits différents, puis lorsque vous collectez tous les termes similaires le long des diagonales, vous obtenez un autre ensemble d'environ 10 000 opérations.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 1021.14 }, { - "input": "On the other hand if I think of two polynomials in terms of their outputs for example sampling their values at some handful of inputs then multiplying them only requires as many operations as the number of samples since again it's a pointwise operation and with polynomials you only need finitely many samples to be able to recover the coefficients.", + "input": "On the other hand, if I think of two polynomials in terms of their outputs, for example, sampling their values at some handful of inputs, then multiplying them only requires as many operations as the number of samples, since again, it's a pointwise operation. And with polynomials, you only need finitely many samples to be able to recover the coefficients.", "translatedText": "D'un autre côté, si je pense à deux polynômes en termes de leurs sorties, par exemple en échantillonnant leurs valeurs sur une poignée d'entrées, leur multiplication ne nécessite qu'autant d'opérations que le nombre d'échantillons, car encore une fois, c'est une opération ponctuelle et avec les polynômes, vous n'avez besoin que de un nombre fini d'échantillons pour pouvoir récupérer les coefficients.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/german/sentence_translations.json b/2022/convolutions/german/sentence_translations.json index 0c7a90eae..69b4a126f 100644 --- a/2022/convolutions/german/sentence_translations.json +++ b/2022/convolutions/german/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations.", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations.", "translatedText": "Nehmen wir zum Beispiel an, ich habe Ihnen zwei wirklich große Polynome gegeben, sagen wir, jedes mit hundert verschiedenen Koeffizienten. Wenn Sie sie dann multiplizieren würden, um dieses Produkt zu erweitern, wissen Sie, dass Sie das gesamte 100 x 100-Raster paarweiser Produkte ausfüllen müssten Führen Sie 10.000 verschiedene Produkte durch, und wenn Sie dann alle ähnlichen Begriffe entlang der Diagonalen sammeln, ist das ein weiterer Satz von etwa 10.000 Operationen.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 1021.14 }, { - "input": "On the other hand if I think of two polynomials in terms of their outputs for example sampling their values at some handful of inputs then multiplying them only requires as many operations as the number of samples since again it's a pointwise operation and with polynomials you only need finitely many samples to be able to recover the coefficients.", + "input": "On the other hand, if I think of two polynomials in terms of their outputs, for example, sampling their values at some handful of inputs, then multiplying them only requires as many operations as the number of samples, since again, it's a pointwise operation. And with polynomials, you only need finitely many samples to be able to recover the coefficients.", "translatedText": "Wenn ich mir andererseits zwei Polynome im Hinblick auf ihre Ausgaben vorstelle, zum Beispiel die Abtastung ihrer Werte an einer Handvoll Eingaben, dann erfordert ihre Multiplikation nur so viele Operationen wie die Anzahl der Stichproben, da es sich wiederum um eine punktweise Operation handelt und man bei Polynomen nur die benötigten Werte benötigt endlich viele Stichproben, um die Koeffizienten wiederherstellen zu können.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/hebrew/sentence_translations.json b/2022/convolutions/hebrew/sentence_translations.json index 3120795e4..66e6292e6 100644 --- a/2022/convolutions/hebrew/sentence_translations.json +++ b/2022/convolutions/hebrew/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations. ", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations. ", "translatedText": "לדוגמה, נניח שנתתי לך שני פולינומים גדולים באמת נניח שכל אחד מהם עם מאה מקדמים שונים, אז אם הדרך שבה אתה מכפיל אותם הייתה להרחיב את המוצר הזה אתה יודע למלא את כל הרשת הזו של 100 על 100 של מוצרים בזוגיות שתדרוש ממך בצע 10,000 מוצרים שונים ואז כשאתה אוסף את כל המונחים הדומים לאורך האלכסונים, זה עוד קבוצה של כ-10,000 פעולות. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/hindi/sentence_translations.json b/2022/convolutions/hindi/sentence_translations.json index 5fdada52b..f90dd8d41 100644 --- a/2022/convolutions/hindi/sentence_translations.json +++ b/2022/convolutions/hindi/sentence_translations.json @@ -777,7 +777,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations.", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations.", "translatedText": "उदाहरण के लिए, मान लें कि मैंने आपको दो बहुत बड़े बहुपद दिए हैं, मान लीजिए कि प्रत्येक एक सौ अलग-अलग गुणांकों के साथ है, तो यदि आप जिस तरह से उन्हें गुणा करते हैं, वह इस उत्पाद का विस्तार करना है, तो आप जानते हैं कि इस संपूर्ण 100 गुणा 100 ग्रिड को जोड़ीवार उत्पादों में भरना है, जिसके लिए आपको इसकी आवश्यकता होगी। 10,000 अलग-अलग उत्पाद निष्पादित करें और फिर जब आप विकर्णों के साथ सभी समान शब्द एकत्र कर रहे हों तो यह लगभग 10,000 परिचालनों का एक और सेट है।", "n_reviews": 0, "start": 986.32, @@ -791,7 +791,7 @@ "end": 1021.14 }, { - "input": "On the other hand if I think of two polynomials in terms of their outputs for example sampling their values at some handful of inputs then multiplying them only requires as many operations as the number of samples since again it's a pointwise operation and with polynomials you only need finitely many samples to be able to recover the coefficients.", + "input": "On the other hand, if I think of two polynomials in terms of their outputs, for example, sampling their values at some handful of inputs, then multiplying them only requires as many operations as the number of samples, since again, it's a pointwise operation. And with polynomials, you only need finitely many samples to be able to recover the coefficients.", "translatedText": "दूसरी ओर, यदि मैं दो बहुपदों के बारे में उनके आउटपुट के संदर्भ में सोचता हूं, उदाहरण के लिए कुछ मुट्ठी भर इनपुट पर उनके मानों का नमूना लेना, तो उन्हें गुणा करने के लिए केवल उतने ही ऑपरेशन की आवश्यकता होती है जितनी नमूनों की संख्या, क्योंकि फिर से यह एक बिंदुवार ऑपरेशन है और बहुपद के साथ आपको केवल इसकी आवश्यकता होती है गुणांकों को पुनर्प्राप्त करने में सक्षम होने के लिए बहुत सारे नमूने।", "n_reviews": 0, "start": 1021.82, diff --git a/2022/convolutions/indonesian/sentence_translations.json b/2022/convolutions/indonesian/sentence_translations.json index 7ecfd0701..80bf83563 100644 --- a/2022/convolutions/indonesian/sentence_translations.json +++ b/2022/convolutions/indonesian/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations. ", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations. ", "translatedText": "Sebagai contoh, katakanlah saya memberi Anda dua polinomial yang sangat besar, katakanlah masing-masing polinomial dengan seratus koefisien berbeda, lalu jika cara Anda mengalikannya adalah dengan memperluas hasil perkalian ini, Anda tahu mengisi seluruh kotak perkalian berpasangan berukuran 100 kali 100 yang mengharuskan Anda melakukannya melakukan 10.000 produk berbeda dan kemudian ketika Anda mengumpulkan semua suku serupa di sepanjang diagonal, itu adalah kumpulan sekitar 10.000 operasi lainnya. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/italian/sentence_translations.json b/2022/convolutions/italian/sentence_translations.json index 1d265e2d1..f7e286295 100644 --- a/2022/convolutions/italian/sentence_translations.json +++ b/2022/convolutions/italian/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations. ", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations. ", "translatedText": "Ad esempio, diciamo che ti ho dato due polinomi davvero grandi, ciascuno con un centinaio di coefficienti diversi, quindi se il modo in cui li moltiplichi fosse quello di espandere questo prodotto, sai riempire l'intera griglia 100 per 100 di prodotti a coppie che richiederebbero di farlo esegui 10.000 prodotti diversi e poi quando raccogli tutti i termini simili lungo le diagonali si ottiene un altro insieme di circa 10.000 operazioni. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/japanese/sentence_translations.json b/2022/convolutions/japanese/sentence_translations.json index c99a2bba4..94e1cdc62 100644 --- a/2022/convolutions/japanese/sentence_translations.json +++ b/2022/convolutions/japanese/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations.", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations.", "translatedText": "たとえば、それぞ れが 100 の異なる係数を持つ 2 つの非常に大きな多項式を与えたとしましょう。 その場 合、それらを乗算する方法がこの積を拡張することである場合、このペア積の 100 x 1 00 グリッド全体を埋める必要があることがわかります。 10,000 の異なる製品を実 行し、対角線に沿って同様の用語をすべて収集すると、さらに約 10,000 の操作のセッ トになります。", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 1021.14 }, { - "input": "On the other hand if I think of two polynomials in terms of their outputs for example sampling their values at some handful of inputs then multiplying them only requires as many operations as the number of samples since again it's a pointwise operation and with polynomials you only need finitely many samples to be able to recover the coefficients.", + "input": "On the other hand, if I think of two polynomials in terms of their outputs, for example, sampling their values at some handful of inputs, then multiplying them only requires as many operations as the number of samples, since again, it's a pointwise operation. And with polynomials, you only need finitely many samples to be able to recover the coefficients.", "translatedText": "一方、出力の観点から 2 つの多項式について考える場合、たとえば、いく つかの入力で値をサンプリングし、それらを乗算する場合、これも点単位の演算で あり、多項式を使用する場合のみ必要となるため、サンプルの数と同じ数の演算 のみが必要です。 係数を回復できる有限の多くのサンプル。", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/marathi/sentence_translations.json b/2022/convolutions/marathi/sentence_translations.json index 5b4d27369..2d693ebda 100644 --- a/2022/convolutions/marathi/sentence_translations.json +++ b/2022/convolutions/marathi/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations. ", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations. ", "translatedText": "उदाहरणार्थ, समजा की मी तुम्हाला दोन खरोखरच मोठ्या बहुपदी दिल्या आहेत प्रत्येक शंभर भिन्न गुणांकांसह म्हणा, मग जर तुम्ही त्यांचा गुणाकार करण्याच्या पद्धतीनुसार या उत्पादनाचा विस्तार कराल तर तुम्हाला हे संपूर्ण 100 बाय 100 ग्रिडमध्ये पेअरवाइज उत्पादनांचे ग्रिड भरावे लागेल. 10,000 भिन्न उत्पादने करा आणि नंतर जेव्हा तुम्ही कर्णांसह सर्व समान संज्ञा एकत्रित करता तेव्हा सुमारे 10,000 ऑपरेशन्सचा दुसरा संच आहे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/persian/sentence_translations.json b/2022/convolutions/persian/sentence_translations.json index 1ee99c2e7..28ac927a1 100644 --- a/2022/convolutions/persian/sentence_translations.json +++ b/2022/convolutions/persian/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations. ", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations. ", "translatedText": "به عنوان مثال، فرض کنید من به شما دو چند جمله‌ای بسیار بزرگ دادم، مثلاً هر کدام با صد ضرایب مختلف، سپس اگر روش ضرب آنها به این صورت بود که این محصول را بسط دهید، می‌دانید که تمام این شبکه 100 در 100 از محصولات زوجی را پر می‌کنید که به شما نیاز دارد. 10000 محصول مختلف را انجام دهید و سپس هنگامی که تمام عبارات مشابه را در امتداد مورب ها جمع آوری می کنید، مجموعه دیگری از حدود 10000 عملیات است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/portuguese/sentence_translations.json b/2022/convolutions/portuguese/sentence_translations.json index 0ced4fd0c..3f3c0011d 100644 --- a/2022/convolutions/portuguese/sentence_translations.json +++ b/2022/convolutions/portuguese/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations. ", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations. ", "translatedText": "Por exemplo, digamos que eu lhe dei dois polinômios realmente grandes, digamos, cada um com cem coeficientes diferentes, então se a maneira como você os multiplicou fosse expandindo este produto, você sabe preencher toda essa grade de 100 por 100 de produtos pareados que exigiriam que você execute 10.000 produtos diferentes e, quando você coletar todos os termos semelhantes ao longo das diagonais, haverá outro conjunto de cerca de 10.000 operações. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/russian/sentence_translations.json b/2022/convolutions/russian/sentence_translations.json index b6880c954..ff6e4e7e8 100644 --- a/2022/convolutions/russian/sentence_translations.json +++ b/2022/convolutions/russian/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations.", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations.", "translatedText": "Например, предположим, я дал вам два действительно больших многочлена, скажем, каждый с сотней различных коэффициентов, тогда, если способ их умножения заключался в расширении этого известного вам продукта, заполняя всю эту сетку попарных произведений размером 100 на 100, что потребовало бы от вас выполнить 10 000 различных произведений, а затем, когда вы соберете все одинаковые термины по диагоналям, получится еще один набор из примерно 10 000 операций.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 1021.14 }, { - "input": "On the other hand if I think of two polynomials in terms of their outputs for example sampling their values at some handful of inputs then multiplying them only requires as many operations as the number of samples since again it's a pointwise operation and with polynomials you only need finitely many samples to be able to recover the coefficients.", + "input": "On the other hand, if I think of two polynomials in terms of their outputs, for example, sampling their values at some handful of inputs, then multiplying them only requires as many operations as the number of samples, since again, it's a pointwise operation. And with polynomials, you only need finitely many samples to be able to recover the coefficients.", "translatedText": "С другой стороны, если я думаю о двух полиномах с точки зрения их выходных данных, например, о выборке их значений на нескольких входных данных, то для их умножения требуется столько операций, сколько количество выборок, поскольку опять же это точечная операция, а с полиномами вам нужно только конечное число выборок, чтобы иметь возможность восстановить коэффициенты.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/spanish/sentence_translations.json b/2022/convolutions/spanish/sentence_translations.json index fbba5261f..109ce6eb9 100644 --- a/2022/convolutions/spanish/sentence_translations.json +++ b/2022/convolutions/spanish/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations. ", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations. ", "translatedText": "Por ejemplo, digamos que te di dos polinomios realmente grandes, digamos cada uno con cien coeficientes diferentes, entonces, si la forma en que los multiplicas fuera expandir este producto, ya sabes, completando esta cuadrícula completa de 100 por 100 de productos por pares que requerirían que realice 10.000 productos diferentes y luego, cuando recopile todos los términos semejantes a lo largo de las diagonales, ese será otro conjunto de alrededor de 10.000 operaciones. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/tamil/sentence_translations.json b/2022/convolutions/tamil/sentence_translations.json index 67205e823..664f605b5 100644 --- a/2022/convolutions/tamil/sentence_translations.json +++ b/2022/convolutions/tamil/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations.", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations.", "translatedText": "எடுத்துக்காட்டாக, நான் உங்களுக்கு இரண்டு பெரிய பல்லுறுப்புக்கோவைகளைக் கொடுத்துள்ளேன் என்று வைத்துக் கொள்வோம், ஒவ்வொன்றும் நூறு வெவ்வேறு குணகங்களைக் கொண்டதாகக் கூறுங்கள், அவற்றைப் பெருக்கும் முறை இந்த தயாரிப்பை விரிவுபடுத்துவதாக இருந்தால், இந்த முழு 100க்கு 100 ஜோடிவரிசை தயாரிப்புகளை நிரப்புவது உங்களுக்குத் தெரியும். 10,000 வெவ்வேறு தயாரிப்புகளைச் செய்யுங்கள், பின்னர் நீங்கள் மூலைவிட்டங்களுடன் இதே போன்ற எல்லா சொற்களையும் சேகரிக்கும் போது, அது சுமார் 10,000 செயல்பாடுகளின் மற்றொரு தொகுப்பாகும்.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 1021.14 }, { - "input": "On the other hand if I think of two polynomials in terms of their outputs for example sampling their values at some handful of inputs then multiplying them only requires as many operations as the number of samples since again it's a pointwise operation and with polynomials you only need finitely many samples to be able to recover the coefficients.", + "input": "On the other hand, if I think of two polynomials in terms of their outputs, for example, sampling their values at some handful of inputs, then multiplying them only requires as many operations as the number of samples, since again, it's a pointwise operation. And with polynomials, you only need finitely many samples to be able to recover the coefficients.", "translatedText": "மறுபுறம், இரண்டு பல்லுறுப்புக்கோவைகளை அவற்றின் வெளியீடுகளின் அடிப்படையில் நான் நினைத்தால், எடுத்துக்காட்டாக, சில உள்ளீடுகளில் அவற்றின் மதிப்புகளை மாதிரியாக எடுத்துக் கொண்டால், அவற்றைப் பெருக்குவதற்கு, மாதிரிகளின் எண்ணிக்கையைப் போலவே பல செயல்பாடுகள் தேவைப்படும். பல மாதிரிகள் குணகங்களை மீட்டெடுக்க முடியும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/telugu/sentence_translations.json b/2022/convolutions/telugu/sentence_translations.json index 75d120a83..56c5fa3e1 100644 --- a/2022/convolutions/telugu/sentence_translations.json +++ b/2022/convolutions/telugu/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations.", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations.", "translatedText": "ఉదాహరణకు, నేను మీకు రెండు పెద్ద బహుపదిలను ఇచ్చాను అనుకుందాం, ప్రతి ఒక్కటి వంద విభిన్న గుణకాలతో చెప్పండి, మీరు వాటిని గుణించే విధానం ఈ ఉత్పత్తిని విస్తరించడమే అయితే, ఈ మొత్తం 100 నుండి 100 గ్రిడ్‌ల పెయిర్‌వైస్ ఉత్పత్తులను నింపడం మీకు తెలుసు. 10,000 విభిన్న ఉత్పత్తులను నిర్వహించి, ఆపై మీరు వికర్ణాల వెంట ఇలాంటి నిబంధనలన్నింటినీ సేకరిస్తున్నప్పుడు అది దాదాపు 10,000 ఆపరేషన్‌ల యొక్క మరొక సెట్.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 1021.14 }, { - "input": "On the other hand if I think of two polynomials in terms of their outputs for example sampling their values at some handful of inputs then multiplying them only requires as many operations as the number of samples since again it's a pointwise operation and with polynomials you only need finitely many samples to be able to recover the coefficients.", + "input": "On the other hand, if I think of two polynomials in terms of their outputs, for example, sampling their values at some handful of inputs, then multiplying them only requires as many operations as the number of samples, since again, it's a pointwise operation. And with polynomials, you only need finitely many samples to be able to recover the coefficients.", "translatedText": "మరోవైపు, నేను రెండు బహుపదిలను వాటి అవుట్‌పుట్‌ల పరంగా భావిస్తే, ఉదాహరణకు కొన్ని ఇన్‌పుట్‌ల వద్ద వాటి విలువలను శాంపిల్ చేస్తే, వాటిని గుణించడం కోసం నమూనాల సంఖ్యకు తగినన్ని ఆపరేషన్‌లు మాత్రమే అవసరం, ఎందుకంటే ఇది పాయింట్‌వైస్ ఆపరేషన్ మరియు బహుపదిలతో మీకు మాత్రమే అవసరం. కోఎఫీషియంట్‌లను పునరుద్ధరించడానికి పరిమిత సంఖ్యలో నమూనాలు.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/thai/sentence_translations.json b/2022/convolutions/thai/sentence_translations.json index cb3dfb016..1751e156c 100644 --- a/2022/convolutions/thai/sentence_translations.json +++ b/2022/convolutions/thai/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations. ", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/turkish/sentence_translations.json b/2022/convolutions/turkish/sentence_translations.json index 49d94fb65..f47dd8a9b 100644 --- a/2022/convolutions/turkish/sentence_translations.json +++ b/2022/convolutions/turkish/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations.", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations.", "translatedText": "Örneğin, diyelim ki size her biri yüz farklı katsayıya sahip iki gerçekten büyük polinom verdim, o zaman bunları çarpma şekliniz bu çarpımı genişletmekse, bu 100'e 100'lük ikili çarpım tablosunun tamamını doldurmanızı gerektirir. 10.000 farklı ürün gerçekleştirin ve ardından tüm benzer terimleri köşegenler boyunca topladığınızda, bu yaklaşık 10.000 işlemden oluşan başka bir dizidir.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 1021.14 }, { - "input": "On the other hand if I think of two polynomials in terms of their outputs for example sampling their values at some handful of inputs then multiplying them only requires as many operations as the number of samples since again it's a pointwise operation and with polynomials you only need finitely many samples to be able to recover the coefficients.", + "input": "On the other hand, if I think of two polynomials in terms of their outputs, for example, sampling their values at some handful of inputs, then multiplying them only requires as many operations as the number of samples, since again, it's a pointwise operation. And with polynomials, you only need finitely many samples to be able to recover the coefficients.", "translatedText": "Öte yandan, iki polinomu çıktıları açısından düşünürsem, örneğin değerlerini birkaç girdide örneklemek, sonra bunları çarpmak yalnızca örnek sayısı kadar işlem gerektirir, çünkü yine bu noktasal bir işlemdir ve polinomlarla yalnızca ihtiyacınız vardır. katsayıları kurtarabilmek için sonlu sayıda örnek.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/ukrainian/sentence_translations.json b/2022/convolutions/ukrainian/sentence_translations.json index 62a34f1bd..7931ee71a 100644 --- a/2022/convolutions/ukrainian/sentence_translations.json +++ b/2022/convolutions/ukrainian/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations.", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations.", "translatedText": "Наприклад, скажімо, я дав вам два справді великі поліноми, скажімо, кожен із сотнею різних коефіцієнтів, тоді, якби ви їх помножили, щоб розширити цей добуток, який ви знаєте, заповнивши всю цю сітку 100 на 100 попарних добутків, що вимагатиме від вас виконайте 10 000 різних продуктів, а потім, коли ви збираєте всі подібні терміни вздовж діагоналей, це ще один набір приблизно з 10 000 операцій.", "model": "google_nmt", "n_reviews": 0, @@ -904,7 +904,7 @@ "end": 1021.14 }, { - "input": "On the other hand if I think of two polynomials in terms of their outputs for example sampling their values at some handful of inputs then multiplying them only requires as many operations as the number of samples since again it's a pointwise operation and with polynomials you only need finitely many samples to be able to recover the coefficients.", + "input": "On the other hand, if I think of two polynomials in terms of their outputs, for example, sampling their values at some handful of inputs, then multiplying them only requires as many operations as the number of samples, since again, it's a pointwise operation. And with polynomials, you only need finitely many samples to be able to recover the coefficients.", "translatedText": "З іншого боку, якщо я думаю про два поліноми в термінах їхніх виходів, наприклад, вибірка їхніх значень на кількох вхідних даних, тоді їх множення потребує лише стільки операцій, скільки вибірок, оскільки це знову ж таки поточкова операція, а з поліномами вам потрібно лише кінцеву кількість вибірок, щоб мати можливість відновити коефіцієнти.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/urdu/sentence_translations.json b/2022/convolutions/urdu/sentence_translations.json index fa56d1ea7..22b990a3e 100644 --- a/2022/convolutions/urdu/sentence_translations.json +++ b/2022/convolutions/urdu/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations. ", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations. ", "translatedText": "مثال کے طور پر ہم کہتے ہیں کہ میں نے آپ کو واقعی دو بڑے کثیر نام دیے ہیں جن میں سے ہر ایک کو سو مختلف عدد کے ساتھ کہتے ہیں پھر اگر آپ ان کو ضرب دینے کا طریقہ اس پروڈکٹ کو بڑھانا تھا تو آپ جانتے ہیں کہ اس پورے 100 بائی 100 گرڈ کو جوڑے کی شکل میں بھرنا ہوگا جس کے لیے آپ کو ضرورت ہوگی۔10,000 مختلف پروڈکٹس انجام دیں اور پھر جب آپ اس طرح کی تمام اصطلاحات کو اخترن کے ساتھ جمع کر رہے ہوں تو یہ تقریباً 10,000 آپریشنز کا ایک اور سیٹ ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/convolutions/vietnamese/sentence_translations.json b/2022/convolutions/vietnamese/sentence_translations.json index 157800417..1bd1c3c04 100644 --- a/2022/convolutions/vietnamese/sentence_translations.json +++ b/2022/convolutions/vietnamese/sentence_translations.json @@ -888,7 +888,7 @@ "end": 985.76 }, { - "input": "For example let's say I gave you two really big polynomials say each one with a hundred different coefficients then if the way you multiply them was to expand out this product you know filling in this entire 100 by 100 grid of pairwise products that would require you to perform 10,000 different products and then when you're collecting all the like terms along the diagonals that's another set of around 10,000 operations.", + "input": "For example, let's say I gave you two really big polynomials, say each one with a hundred different coefficients. Then if the way you multiply them was to expand out this product, you know, filling in this entire 100 by 100 grid of pairwise products, that would require you to perform 10,000 different products. And then, when you're collecting all the like terms along the diagonals, that's another set of around 10,000 operations.", "translatedText": "Ví dụ: giả sử tôi đưa cho bạn hai đa thức thực sự lớn, mỗi đa thức có một trăm hệ số khác nhau, sau đó nếu cách bạn nhân chúng là để khai triển tích này thì bạn biết rằng việc điền vào toàn bộ lưới tích cặp 100 x 100 này sẽ yêu cầu bạn phải thực hiện 10.000 sản phẩm khác nhau và sau đó khi bạn thu thập tất cả các số hạng giống nhau dọc theo các đường chéo thì đó là một tập hợp khác gồm khoảng 10.000 phép tính.", "model": "google_nmt", "n_reviews": 1, @@ -904,7 +904,7 @@ "end": 1021.14 }, { - "input": "On the other hand if I think of two polynomials in terms of their outputs for example sampling their values at some handful of inputs then multiplying them only requires as many operations as the number of samples since again it's a pointwise operation and with polynomials you only need finitely many samples to be able to recover the coefficients.", + "input": "On the other hand, if I think of two polynomials in terms of their outputs, for example, sampling their values at some handful of inputs, then multiplying them only requires as many operations as the number of samples, since again, it's a pointwise operation. And with polynomials, you only need finitely many samples to be able to recover the coefficients.", "translatedText": "Mặt khác, nếu tôi nghĩ về hai đa thức về kết quả đầu ra của chúng, chẳng hạn như lấy mẫu giá trị của chúng ở một số đầu vào thì việc nhân chúng chỉ cần nhiều thao tác bằng số lượng mẫu vì một lần nữa, đó là phép toán theo điểm và với đa thức bạn chỉ cần hữu hạn nhiều mẫu để có thể phục hồi các hệ số.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/some2-results/english/captions.srt b/2022/some2-results/english/captions.srt index 2abcef2bd..c422e059d 100644 --- a/2022/some2-results/english/captions.srt +++ b/2022/some2-results/english/captions.srt @@ -595,7 +595,7 @@ It makes it so that once the equation comes on the screen, or the algorithm is d it doesn't feel like an expression handed down with nothing to hold onto. 150 -00:08:46,079 --> 00:08:49,793 +00:08:46,080 --> 00:08:49,793 Instead, it arrives only once it's articulating something that already 151 @@ -1087,38 +1087,10 @@ and to a playlist that contains all the video submissions, and also to a blog post containing links to all of the non-video submissions. 273 -00:16:09,840 --> 00:16:13,814 +00:16:09,840 --> 00:16:26,353 Thanks to a sponsorship from Brilliant, each winner will get $1000 as a cash prize, 274 -00:16:13,814 --> 00:16:17,600 +00:16:26,353 --> 00:16:42,080 and also, and much more importantly I think, a rare edition golden pie creature. -275 -00:16:17,960 --> 00:16:21,399 -Also after the initial announcement, Risk Zero and Google Fonts - -276 -00:16:21,399 --> 00:16:25,000 -both generously reached out offering additional prize sponsorships. - -277 -00:16:25,280 --> 00:16:27,929 -I'd also like to thank Protocol Labs for another contribution - -278 -00:16:27,929 --> 00:16:30,280 -to help us cover the costs of managing the whole event. - -279 -00:16:30,940 --> 00:16:34,738 -Thanks to everybody who participated, and to everybody who helped in creating this - -280 -00:16:34,738 --> 00:16:38,720 -rising tide for new math channels and new math blogs that we've seen in the last month. - -281 -00:16:39,280 --> 00:16:42,080 -It was genuinely inspiring to see just how well this all went. - diff --git a/2022/some2-results/english/sentence_timings.json b/2022/some2-results/english/sentence_timings.json index a0a5562d4..d4d43a5a6 100644 --- a/2022/some2-results/english/sentence_timings.json +++ b/2022/some2-results/english/sentence_timings.json @@ -542,26 +542,6 @@ [ "Thanks to a sponsorship from Brilliant, each winner will get $1000 as a cash prize, and also, and much more importantly I think, a rare edition golden pie creature.", 969.84, - 977.6 - ], - [ - "Also after the initial announcement, Risk Zero and Google Fonts both generously reached out offering additional prize sponsorships.", - 977.96, - 985.0 - ], - [ - "I'd also like to thank Protocol Labs for another contribution to help us cover the costs of managing the whole event.", - 985.28, - 990.28 - ], - [ - "Thanks to everybody who participated, and to everybody who helped in creating this rising tide for new math channels and new math blogs that we've seen in the last month.", - 990.94, - 998.72 - ], - [ - "It was genuinely inspiring to see just how well this all went.", - 999.28, 1002.08 ] ] \ No newline at end of file diff --git a/2022/subsets-puzzle/arabic/sentence_translations.json b/2022/subsets-puzzle/arabic/sentence_translations.json index 03fd808a1..6932eb334 100644 --- a/2022/subsets-puzzle/arabic/sentence_translations.json +++ b/2022/subsets-puzzle/arabic/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes. ", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes. ", "translatedText": "بالنسبة لي ولكم، هناك على الأقل اثنين من التحولات المدهشة والجميلة جدًا التي يأخذها الحل الذي أود مشاركته معكم. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here. ", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here. ", "translatedText": "بصراحة، النمط الذي سأتبعه هنا هو على الأرجح الأسهل إذا أخذت وقتًا للتوقف مؤقتًا والتفكير بنفسك فيما يحدث عندما تقوم بتوسيع كل شيء هنا. ", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle. ", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle. ", "translatedText": "لنكون محددين، العدد المركب الذي يهمني هو الذي سأسميه زيتا، وهو يقع في خمس دورة حول دائرة الوحدة. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000. ", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand. ", "translatedText": "تذكر الآن، أن صيغة كثيرة الحدود التي نعرفها، والتي نرتاح لها، هي الصيغة التحليلية، حيث يكون لديك 1 زائد x، 1 زائد x تربيع، وهكذا، حتى 1 زائد x إلى 2000. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times. ", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times. ", "translatedText": "التعبير بأكمله حتى 2000 سيكون في الأساس مجرد نسخة من هذا التعبير 400 مرة. ", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400. ", + "input": "It will be two to the power four hundred. ", "translatedText": "سيكون اثنان أس 400. ", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order. ", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order. ", "translatedText": "يوضح المنطق المتطابق بشكل أساسي أن قيمته على الجذور الثلاثة التالية للعدد واحد هي أيضًا اثنان أس 400، لأنه تذكر عندما تأخذ قوى زيتا تربيع أو زيتا تكعيب، فإنك تحصل على نفس قائمة الأرقام التي تم خلطها بترتيب مختلف . ", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400. ", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred. ", "translatedText": "لجمع كل هذه المعاملات التي تقبل القسمة على خمسة، والتي تذكر أنها طريقة لحساب عدد المجموعات الفرعية الإجمالية التي مجموعها يقبل القسمة على خمسة، فإن الإجابة هي خمس هذا التعبير المعقد الغريب، والذي حسبناه للتو يساوي اثنين إلى 2000 بالإضافة إلى أربع نسخ مختلفة من نسختين إلى 400. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/bengali/sentence_translations.json b/2022/subsets-puzzle/bengali/sentence_translations.json index 6206be5df..0aa46e9c3 100644 --- a/2022/subsets-puzzle/bengali/sentence_translations.json +++ b/2022/subsets-puzzle/bengali/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes. ", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes. ", "translatedText": "আপনার এবং আমার জন্য, অন্তত দুটি খুব আশ্চর্যজনক এবং খুব সুন্দর টুইস্ট রয়েছে যে সমাধানটি আমি আপনার সাথে শেয়ার করতে চাই।", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here. ", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here. ", "translatedText": "সত্যি বলতে কি, আমি এখানে যে প্যাটার্নের জন্য যাচ্ছি তা হল সম্ভবত সবচেয়ে সহজ যদি আপনি শুধু বিরতি দেওয়ার জন্য সময় নেন এবং নিজের জন্য চিন্তা করেন যখন আপনি এখানে সবকিছু প্রসারিত করেন তখন কী হয়।", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle. ", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle. ", "translatedText": "সুনির্দিষ্টভাবে বলতে গেলে, আমি যে জটিল সংখ্যাটির বিষয়ে যত্নশীল তা হল একটি যাকে আমি জেটা লেবেল করতে যাচ্ছি, এবং এটি একক বৃত্তের চারপাশে ঘুরার এক পঞ্চমাংশ বসে।", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000. ", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand. ", "translatedText": "এখন মনে রাখবেন, বহুপদীর যে ফর্মটি আমরা জানি, যেটির সাথে আমরা স্বাচ্ছন্দ্য বোধ করি, সেটি হল ফ্যাক্টরড ফর্ম, যেখানে আপনার কাছে এই 1 প্লাস x, 1 প্লাস x বর্গ, অন এবং অন, 1 প্লাস x পর্যন্ত 2,000 এই বিন্দু পর্যন্ত সবকিছুই কেবল অর্থহীন প্রতীকী খেলা, একটি কঠিন সমস্যাকে অন্যের দিকে ঠেলে দেয়, যদি না আমরা আসলে আমাদের হাতা গুটিয়ে নিতে পারি এবং এখানে কিছু সৎ গণনা করতে পারি।", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times. ", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times. ", "translatedText": "2,000 পর্যন্ত সম্পূর্ণ অভিব্যক্তিটি মূলত 400 বার এই অভিব্যক্তিটির একটি অনুলিপি হতে চলেছে।", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400. ", + "input": "It will be two to the power four hundred. ", "translatedText": "এটা দুই থেকে 400 পাওয়ার হবে।", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order. ", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order. ", "translatedText": "মূলত অভিন্ন যুক্তি দেখায় যে ঐক্যের পরবর্তী তিনটি মূলে এর মানটিও 400 শক্তির জন্য দুই, কারণ মনে রাখবেন আপনি যখন জেটা স্কয়ার বা জেটা কিউবডের ক্ষমতাগুলি গ্রহণ করেন, তখন আপনি সংখ্যার একই তালিকা পাবেন যেগুলি ভিন্ন ক্রমে পরিবর্তন করা হয়েছে।. ", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400. ", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred. ", "translatedText": "পাঁচ দ্বারা বিভাজ্য এই সমস্ত সহগগুলি যোগ করার জন্য, যা মনে রাখবেন মোট কতগুলি উপসেটের যোগফল পাঁচ দ্বারা বিভাজ্য তা গণনা করার একটি উপায়, উত্তর হল এই অদ্ভুত জটিল অভিব্যক্তির এক পঞ্চমাংশ, যা আমরা কেবলমাত্র দুটি থেকে গণনা করেছি 2,000 প্লাস দুই থেকে 400 এর চারটি ভিন্ন কপি।", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/chinese/sentence_translations.json b/2022/subsets-puzzle/chinese/sentence_translations.json index d758b88ac..06060c8b4 100644 --- a/2022/subsets-puzzle/chinese/sentence_translations.json +++ b/2022/subsets-puzzle/chinese/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "对于你我来说，我想与你分享的解决方案至 少有两个非常令人惊讶和非常美丽的转折。", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "老实说，如果您花时间停下来思考一下当您扩展此处的所有 内容时会发生什么，我在这里采用的模式可能是最简单的。", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "具体来说，我关心的复数是我要标记为 zet a 的复数，它位于单位圆的五分之一圈处。", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "现在记住，我们知道的多项式的形式，我们熟悉的形式， 是因式分解形式，其中有 1 加 x，1 加 x 的平方，等等，一直到 1 加 x 到2,000。", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "整个表达式最多 2,000 基本上只是该表达式 400 次的副本。", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "它将是 2 的 400 次方。", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "本质上相同的推理表明，它在接下来的三个单位根上的值也是 2 的 400 次方，因为请记住，当您取 zeta 平方或 zeta 立方的幂时，您会得到相同的数字列表，只是以不同的顺序洗牌。", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "将所有可被五整除的系数相加，记住这是一种计算有多 少总子集的总和可被五整除的方法，答案是这个奇怪的 复杂表达式的五分之一，我们刚刚计算为二2,000 份加上 2 份到 400 份的四个不同副本。", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/english/captions.srt b/2022/subsets-puzzle/english/captions.srt index 34985c1c2..39180784d 100644 --- a/2022/subsets-puzzle/english/captions.srt +++ b/2022/subsets-puzzle/english/captions.srt @@ -351,15 +351,15 @@ and do what all good problem solvers should do, and start with a simpler example maybe just trying it with the set 1, 2, 3, 4, 5. 89 -00:04:40,980 --> 00:04:43,701 -If you were solving this problem with pencil and paper, +00:04:40,980 --> 00:04:44,004 +If you were solving this problem with pencil and paper, you know, 90 -00:04:43,701 --> 00:04:45,986 +00:04:44,004 --> 00:04:46,158 you're one of these kids training for the IMO, 91 -00:04:45,986 --> 00:04:49,000 +00:04:46,158 --> 00:04:49,000 it's not a bad idea to simply list out all 2 to the 5 subsets. 92 @@ -1671,16 +1671,16 @@ and yet the way that we can answer it is to evaluate a crazy polynomial on some judiciously chosen complex numbers. 419 -00:23:41,520 --> 00:23:43,642 -The more math you do the less crazy that seems, +00:23:41,520 --> 00:23:43,753 +The more math you do, the more you get. The more math you do, 420 -00:23:43,642 --> 00:23:46,959 -because complex numbers have this bizarre relationship with discrete math, +00:23:43,753 --> 00:23:46,238 +the less crazy that seems, because complex numbers have this bizarre 421 -00:23:46,959 --> 00:23:49,480 -but it really is wonderful, there's no two ways about it. +00:23:46,238 --> 00:23:49,480 +relationship with discrete math, but it really is wonderful, there's no two ways about it. 422 00:23:50,340 --> 00:23:54,219 @@ -2143,246 +2143,250 @@ solve a hard problem, it's also worth taking some time to reflect on it. What do you get out of this? 537 -00:30:29,640 --> 00:30:33,377 -What's indeed one fifth of all the total subsets like we might have guessed, +00:30:29,640 --> 00:30:31,866 +What's the takeaway? Now you could reflect on the answer itself, 538 -00:30:33,377 --> 00:30:36,581 -and how this error term came about from the not quite destructive +00:30:31,866 --> 00:30:34,332 +how the dominant part is indeed one fifth of all the total subsets like 539 -00:30:36,581 --> 00:30:39,300 -interference in a massive combination of roots of unity. +00:30:34,332 --> 00:30:36,765 +we might have guessed, and how this error term came about from the not 540 +00:30:36,765 --> 00:30:39,300 +quite destructive interference in a massive combination of roots of unity. + +541 00:30:40,120 --> 00:30:43,951 But again, what makes this question interesting is not the answer, -541 +542 00:30:43,951 --> 00:30:48,640 it's the way that we solved it, namely taking a discrete sequence that we want to -542 +543 00:30:48,640 --> 00:30:52,300 understand and treating it as the coefficients on a polynomial, -543 +544 00:30:52,300 --> 00:30:55,160 then evaluating that polynomial on complex values. -544 +545 00:30:55,740 --> 00:30:58,931 Both of those steps are probably highly unexpected at the outset, -545 +546 00:30:58,931 --> 00:31:03,187 but both of those steps relate to some very general and powerful techniques that you'll -546 +547 00:31:03,187 --> 00:31:04,300 find elsewhere in math. -547 +548 00:31:04,900 --> 00:31:08,623 For example, at the top of the lesson, I promised that the technique that we -548 +549 00:31:08,623 --> 00:31:12,154 would use would be similar in spirit to the way that primes are studied, -549 +550 00:31:12,154 --> 00:31:16,120 and the set of ideas that leads up to the Riemann hypothesis and things like that. -550 +551 00:31:16,500 --> 00:31:19,770 Now this is a very beautiful topic, enough so that I think it seems a -551 +552 00:31:19,770 --> 00:31:23,040 little criminal to cram some kind of rushed version into the end here. -552 +553 00:31:23,340 --> 00:31:26,699 The right thing to do, I think, is to just make that video I promised -553 +554 00:31:26,699 --> 00:31:29,820 a while back about the zeta function, take the time, do it right. -554 +555 00:31:30,440 --> 00:31:34,149 But if you're curious, and if you'll allow me to throw some things up on the screen -555 +556 00:31:34,149 --> 00:31:37,902 without explaining them, here's the two or three sentence version of how the two are -556 +557 00:31:37,902 --> 00:31:38,300 parallel. -557 +558 00:31:39,020 --> 00:31:43,558 Just like our subsets puzzle, the way that Riemann studied primes involved a discrete -558 +559 00:31:43,558 --> 00:31:47,992 sequence we want to understand, something carrying information about prime numbers, -559 +560 00:31:47,992 --> 00:31:52,320 and then considering a function whose coefficients are the terms in that sequence. -560 +561 00:31:53,120 --> 00:31:55,843 In that case, it's not quite a polynomial, instead it's a -561 +562 00:31:55,843 --> 00:31:58,050 related structure known as a Dirichlet series, -562 +563 00:31:58,050 --> 00:32:01,760 or Dirichlet series depending on who you ask, but it's the same essential idea. -563 +564 00:32:02,160 --> 00:32:06,597 Then the way to suss out information about those coefficients comes from -564 +565 00:32:06,597 --> 00:32:11,400 studying how this function behaves with, you guessed it, complex valued inputs. -565 +566 00:32:12,360 --> 00:32:15,477 The techniques in his case get a lot more sophisticated, -566 +567 00:32:15,477 --> 00:32:18,376 after all Riemann was a pioneer in complex analysis, -567 +568 00:32:18,376 --> 00:32:23,024 but the fact remains extending your domain beyond real numbers like this offers you, -568 +569 00:32:23,024 --> 00:32:27,400 the mathematician, a lot more power in making deductions about the coefficients. -569 +570 00:32:28,700 --> 00:32:32,227 For some viewers this all might leave the lingering question of -570 +571 00:32:32,227 --> 00:32:35,920 why exactly complex numbers are so unreasonably useful in this way. -571 +572 00:32:36,660 --> 00:32:40,344 It's a hard question to answer exactly, but if you think about our puzzle, -572 +573 00:32:40,344 --> 00:32:44,077 everything we just did, as soon as we were in this situation where plugging -573 +574 00:32:44,077 --> 00:32:47,614 in different inputs revealed hidden information about the coefficients, -574 +575 00:32:47,614 --> 00:32:50,758 it's sort of like the more inputs you can work with the better, -575 +576 00:32:50,758 --> 00:32:55,180 so you might as well open yourself up to a richer space of numbers like the complex plane. -576 +577 00:32:55,840 --> 00:32:59,560 But there is a more specific intuition that I want you to come away with here. -577 +578 00:33:00,060 --> 00:33:04,637 In our puzzle the relevant fact that we wanted, the sum of every fifth coefficient, -578 +579 00:33:04,637 --> 00:33:08,833 was a kind of frequency question, and the real reason the complex numbers as -579 +580 00:33:08,833 --> 00:33:12,974 opposed to some other structure proved to be useful for us is that we could -580 +581 00:33:12,974 --> 00:33:16,680 find a value so that successive products have this cycling behavior. -581 +582 00:33:17,000 --> 00:33:20,373 This use of values on the unit circle and roots of unity in -582 +583 00:33:20,373 --> 00:33:24,140 particular to suss out frequency information is extremely fruitful. -583 +584 00:33:24,400 --> 00:33:28,300 It is almost impossible to overstate how helpful that idea is. -584 +585 00:33:28,580 --> 00:33:31,220 Just to give one out of thousands of examples, -585 +586 00:33:31,220 --> 00:33:36,052 in the 1990s Peter Shor found a way for quantum computers to factor large numbers way -586 +587 00:33:36,052 --> 00:33:38,300 way faster than classical computers can. -587 +588 00:33:38,620 --> 00:33:42,031 And if you go in and you look at the details of how what we now -588 +589 00:33:42,031 --> 00:33:45,175 call Shor's algorithm works, the idea is essentially this, -589 +590 00:33:45,175 --> 00:33:48,800 the use of roots of unity to detect a kind of frequency information. -590 +591 00:33:49,320 --> 00:33:52,626 More generally this is the core idea that underlies Fourier transforms -591 +592 00:33:52,626 --> 00:33:56,120 and Fourier series and the infinite swell of topics that follow from those. -592 +593 00:33:56,980 --> 00:33:59,527 As to the topic of generating functions themselves, -593 +594 00:33:59,527 --> 00:34:02,026 we've really only just scratched the surface here, -594 +595 00:34:02,026 --> 00:34:05,652 and if you want to learn more I highly recommend this kind of hilariously -595 +596 00:34:05,652 --> 00:34:08,199 named book Generating Functionology by Herbert Wilf. -596 +597 00:34:08,540 --> 00:34:10,774 And I'll also leave up a few fun puzzles on the screen here -597 +598 00:34:10,774 --> 00:34:13,120 for anyone who wants to flex their muscles a bit with the idea. diff --git a/2022/subsets-puzzle/english/sentence_timings.json b/2022/subsets-puzzle/english/sentence_timings.json index aabbfff5f..b78070b39 100644 --- a/2022/subsets-puzzle/english/sentence_timings.json +++ b/2022/subsets-puzzle/english/sentence_timings.json @@ -260,7 +260,7 @@ 280.32 ], [ - "If you were solving this problem with pencil and paper, you're one of these kids training for the IMO, it's not a bad idea to simply list out all 2 to the 5 subsets.", + "If you were solving this problem with pencil and paper, you know, you're one of these kids training for the IMO, it's not a bad idea to simply list out all 2 to the 5 subsets.", 280.98, 289.0 ], @@ -1075,7 +1075,7 @@ 1421.04 ], [ - "The more math you do the less crazy that seems, because complex numbers have this bizarre relationship with discrete math, but it really is wonderful, there's no two ways about it.", + "The more math you do, the more you get. The more math you do, the less crazy that seems, because complex numbers have this bizarre relationship with discrete math, but it really is wonderful, there's no two ways about it.", 1421.52, 1429.48 ], @@ -1385,7 +1385,7 @@ 1825.2 ], [ - "What's indeed one fifth of all the total subsets like we might have guessed, and how this error term came about from the not quite destructive interference in a massive combination of roots of unity.", + "What's the takeaway? Now you could reflect on the answer itself, how the dominant part is indeed one fifth of all the total subsets like we might have guessed, and how this error term came about from the not quite destructive interference in a massive combination of roots of unity.", 1829.64, 1839.3 ], diff --git a/2022/subsets-puzzle/english/transcript.txt b/2022/subsets-puzzle/english/transcript.txt index 7277bfc0c..67ae1ed5f 100644 --- a/2022/subsets-puzzle/english/transcript.txt +++ b/2022/subsets-puzzle/english/transcript.txt @@ -50,7 +50,7 @@ But toy problem or not, it is a legitimately challenging question, and navigatin For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes. I've already tipped my hand that complex numbers will make a surprise appearance, but before we even get to that, there is another strange turn, which is arguably even weirder and even more unexpected. To set the stage though, let's just get our bearings with the puzzle, and do what all good problem solvers should do, and start with a simpler example, maybe just trying it with the set 1, 2, 3, 4, 5. -If you were solving this problem with pencil and paper, you're one of these kids training for the IMO, it's not a bad idea to simply list out all 2 to the 5 subsets. +If you were solving this problem with pencil and paper, you know, you're one of these kids training for the IMO, it's not a bad idea to simply list out all 2 to the 5 subsets. It's only 32, it's not that many. There's different ways that you might want to organize all of these in your mind, but since the thing that we care about is their sum, the natural thing to do would be to go through all of them one by one and compute those sums. Over here, just doing it on YouTube, I've got a computer, so I'll cheat a little and show what all their sums are. @@ -213,7 +213,7 @@ Deep in the weeds it's easy to forget why we're here in the first place, but rem If we evaluate this function on these five different roots of unity, which I know seems kind of weird, then all we have to do is divide by five and it gives us the sum that we want. That's really cool if you ask me. We have a question that's just about subsets, it's a discrete math problem, and yet the way that we can answer it is to evaluate a crazy polynomial on some judiciously chosen complex numbers. -The more math you do the less crazy that seems, because complex numbers have this bizarre relationship with discrete math, but it really is wonderful, there's no two ways about it. +The more math you do, the more you get. The more math you do, the less crazy that seems, because complex numbers have this bizarre relationship with discrete math, but it really is wonderful, there's no two ways about it. However, some of you might complain, the only way that this is useful is if we can actually evaluate this wild expression on our polynomial. Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand. Everything up to this point is just meaningless symbolic play, pushing around one hard problem into another, unless we can actually roll up our sleeves and do some honest calculation here. @@ -275,7 +275,7 @@ For example, if you do it in the smaller case with the set one two three four fi And if you'll remember when we explicitly looked at them all, that was in fact the answer. Look, this is a hard puzzle, and when it's worth putting in the time to solve a hard problem, it's also worth taking some time to reflect on it. What do you get out of this? -What's indeed one fifth of all the total subsets like we might have guessed, and how this error term came about from the not quite destructive interference in a massive combination of roots of unity. +What's the takeaway? Now you could reflect on the answer itself, how the dominant part is indeed one fifth of all the total subsets like we might have guessed, and how this error term came about from the not quite destructive interference in a massive combination of roots of unity. But again, what makes this question interesting is not the answer, it's the way that we solved it, namely taking a discrete sequence that we want to understand and treating it as the coefficients on a polynomial, then evaluating that polynomial on complex values. Both of those steps are probably highly unexpected at the outset, but both of those steps relate to some very general and powerful techniques that you'll find elsewhere in math. For example, at the top of the lesson, I promised that the technique that we would use would be similar in spirit to the way that primes are studied, and the set of ideas that leads up to the Riemann hypothesis and things like that. diff --git a/2022/subsets-puzzle/french/sentence_translations.json b/2022/subsets-puzzle/french/sentence_translations.json index 04bf5a09c..96241898a 100644 --- a/2022/subsets-puzzle/french/sentence_translations.json +++ b/2022/subsets-puzzle/french/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "Pour vous et moi, il y a au moins deux rebondissements très surprenants et très beaux que prend la solution que j'aimerais partager avec vous.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "Honnêtement, le modèle que je recherche ici est probablement le plus simple si vous prenez simplement le temps de faire une pause et de réfléchir par vous-même à ce qui se passe lorsque vous développez tout ici.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "Pour être plus précis, le nombre complexe qui m'intéresse est celui que je vais appeler zêta, et il se situe à un cinquième de tour autour du cercle unité.", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "Maintenant rappelez-vous, la forme du polynôme que nous connaissons, celle avec laquelle nous sommes à l'aise, est la forme factorisée, où vous avez ce 1 plus x, 1 plus x au carré, encore et encore, jusqu'à 1 plus x pour les 2 000.", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "L’expression entière jusqu’à 2 000 sera simplement une copie de cette expression 400 fois.", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "Ce sera deux puissance 400.", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "Un raisonnement essentiellement identique montre que sa valeur sur les trois racines suivantes de l'unité est également de deux à la puissance 400, car rappelez-vous que lorsque vous prenez des puissances de zêta au carré ou de zêta au cube, vous obtenez la même liste de nombres qui sont simplement mélangés dans un ordre différent.", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "Pour additionner tous ces coefficients divisibles par cinq, qui, rappelons-le, est une façon de compter combien de sous-ensembles totaux ont une somme divisible par cinq, la réponse est un cinquième de cette étrange expression complexe, que nous venons de calculer comme étant deux à les 2 000 plus quatre exemplaires différents de deux aux 400.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/german/sentence_translations.json b/2022/subsets-puzzle/german/sentence_translations.json index e893e2ffe..28c4c9509 100644 --- a/2022/subsets-puzzle/german/sentence_translations.json +++ b/2022/subsets-puzzle/german/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "Für Sie und mich gibt es mindestens zwei sehr überraschende und sehr schöne Wendungen, die die Lösung, die ich mit Ihnen teilen möchte, annimmt.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "Ehrlich gesagt ist das Muster, das ich hier anstrebe, wahrscheinlich am einfachsten, wenn Sie sich einfach die Zeit nehmen, innezuhalten und selbst darüber nachzudenken, was passiert, wenn Sie hier alles erweitern.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "Genauer gesagt ist die komplexe Zahl, die mir wichtig ist, eine, die ich Zeta nennen werde, und sie liegt eine Fünftelumdrehung um den Einheitskreis.", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "Denken Sie daran, die Form des Polynoms, die wir kennen und mit der wir vertraut sind, ist die faktorisierte Form, bei der es 1 plus x, 1 plus x im Quadrat gibt, immer so weiter, bis hin zu 1 plus x die 2.000.", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "Der gesamte Ausdruck bis 2.000 ist im Grunde nur eine 400-fache Kopie dieses Ausdrucks.", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "Es wird zwei hoch 400 sein.", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "Im Wesentlichen identische Überlegungen zeigen, dass ihr Wert auf den nächsten drei Wurzeln der Einheit ebenfalls zwei hoch 400 beträgt, denn denken Sie daran, wenn Sie Potenzen von Zeta im Quadrat oder Zeta in der Kubikzahl nehmen, erhalten Sie dieselbe Liste von Zahlen, die nur in einer anderen Reihenfolge gemischt werden.", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "Um alle diese Koeffizienten, die durch fünf teilbar sind, zu addieren, was, wie Sie sich erinnern, eine Möglichkeit ist, zu zählen, wie viele Teilmengen insgesamt eine durch fünf teilbare Summe haben, ist die Antwort ein Fünftel dieses seltsamen komplexen Ausdrucks, den wir gerade zu zwei berechnet haben die 2.000 plus vier verschiedene Exemplare von zwei bis zur 400.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/hebrew/sentence_translations.json b/2022/subsets-puzzle/hebrew/sentence_translations.json index 137aa25ae..637d61083 100644 --- a/2022/subsets-puzzle/hebrew/sentence_translations.json +++ b/2022/subsets-puzzle/hebrew/sentence_translations.json @@ -343,7 +343,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "עבורך ולי, יש לפחות שני טוויסטים מאוד מפתיעים ומאוד יפים שהפתרון שאני רוצה לחלוק איתך לוקח.", "n_reviews": 0, "start": 254.02, @@ -630,7 +630,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "בכנות, הדפוס שאליו אני הולך כאן הוא אחד שכנראה הכי קל אם אתה רק לוקח את הזמן לעצור ולחשוב בעצמך מה קורה כשאתה מרחיב הכל כאן.", "n_reviews": 0, "start": 516.08, @@ -1141,7 +1141,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "ליתר דיוק, המספר המרוכב שאכפת לי ממנו הוא אחד שאותו אני הולך לסמן זיטה, והוא יושב חמישית סיבוב סביב מעגל היחידה.", "n_reviews": 0, "start": 1048.56, @@ -1554,7 +1554,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "כעת זכור, צורת הפולינום שאנו מכירים, זו שאנו מרגישים בנוח איתה, היא הצורה המחולקת, שבה יש לך 1 פלוס x, 1 פלוס x בריבוע, הלאה והלאה, עד 1 פלוס x ל ה-2,000.", "n_reviews": 0, "start": 1438.62, @@ -1596,7 +1596,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "הביטוי כולו עד 2,000 הוא בעצם רק עותק של הביטוי הזה 400 פעמים.", "n_reviews": 0, "start": 1483.22, @@ -1897,14 +1897,14 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "זה יהיה שניים להספק 400.", "n_reviews": 0, "start": 1727.58, "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "נימוק זהה בעיקרו מראה שהערך שלו בשלושת שורשי האחדות הבאים הוא גם שניים בחזקת 400, מכיוון שכאשר אתה לוקח חזקות של זיטה בריבוע או בקובייה של זיטה, אתה מקבל את אותה רשימה של מספרים שפשוט מערבבים בסדר אחר .", "n_reviews": 0, "start": 1729.82, @@ -1953,7 +1953,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "כדי לחבר את כל המקדמים האלה שמתחלקים בחמש, שזכור היא דרך לספור לכמה תת-קבוצות יש סכום שמתחלק בחמש, התשובה היא חמישית מהביטוי המורכב המוזר הזה, שחשבנו זה עתה להיות שניים. ה-2,000 ועוד ארבעה עותקים שונים של שניים עד 400.", "n_reviews": 0, "start": 1766.16, diff --git a/2022/subsets-puzzle/hindi/sentence_translations.json b/2022/subsets-puzzle/hindi/sentence_translations.json index aee08699d..daa5bd831 100644 --- a/2022/subsets-puzzle/hindi/sentence_translations.json +++ b/2022/subsets-puzzle/hindi/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "आपके और मेरे लिए, कम से कम दो बहुत ही आश्चर्यजनक और बहुत सुंदर मोड़ हैं जिनका समाधान मैं आपके साथ साझा करना चाहता हूं।", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "ईमानदारी से कहूं तो, जिस पैटर्न के लिए मैं यहां जा रहा हूं वह संभवतः सबसे आसान है यदि आप थोड़ा रुकें और खुद सोचें कि जब आप यहां हर चीज का विस्तार करते हैं तो क्या होता है।", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "विशिष्ट रूप से, जिस जटिल संख्या की मुझे परवाह है वह वह है जिसे मैं ज़ेटा लेबल करने जा रहा हूं, और यह यूनिट सर्कल के चारों ओर एक मोड़ का पांचवां हिस्सा बैठता है।", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "अब याद रखें, बहुपद का वह रूप जिसे हम जानते हैं, जिसके साथ हम सहज हैं, वह गुणनखंडित रूप है, जहां आपके पास यह 1 प्लस x, 1 प्लस x का वर्ग है, लगातार 1 प्लस x तक। 2,000.", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "2,000 तक की संपूर्ण अभिव्यक्ति मूल रूप से इस अभिव्यक्ति की 400 बार प्रतिलिपि बनने वाली है।", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "यह 400 की शक्ति से दो होगा।", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "अनिवार्य रूप से समान तर्क से पता चलता है कि एकता की अगली तीन जड़ों पर इसका मूल्य भी घात 400 से दो है, क्योंकि याद रखें जब आप ज़ेटा वर्ग या ज़ेटा क्यूब की शक्तियाँ लेते हैं, तो आपको संख्याओं की वही सूची मिलती है जो एक अलग क्रम में बस फेरबदल की जाती हैं.", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "इन सभी गुणांकों को जोड़ने के लिए जो पांच से विभाज्य हैं, जो याद रखें कि यह गिनने का एक तरीका है कि कुल कितने उपसमुच्चयों का योग पांच से विभाज्य है, उत्तर इस अजीब जटिल अभिव्यक्ति का पांचवां हिस्सा है, जिसे हमने अभी दो से विभाज्य माना है 2,000 से लेकर 400 तक दो की चार अलग-अलग प्रतियां।", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/hungarian/sentence_translations.json b/2022/subsets-puzzle/hungarian/sentence_translations.json index e0ebedad1..10d8cb759 100644 --- a/2022/subsets-puzzle/hungarian/sentence_translations.json +++ b/2022/subsets-puzzle/hungarian/sentence_translations.json @@ -416,7 +416,7 @@ "end": 280.32 }, { - "input": "If you were solving this problem with pencil and paper, you're one of these kids training for the IMO, it's not a bad idea to simply list out all 2 to the 5 subsets.", + "input": "If you were solving this problem with pencil and paper, you know, you're one of these kids training for the IMO, it's not a bad idea to simply list out all 2 to the 5 subsets.", "translatedText": "Ha ezt a feladatot ceruzával és papírral oldanád meg, te vagy az egyik ilyen gyerek, aki az IMO-ra készül, akkor nem rossz ötlet egyszerűen felsorolni mind a 2-től az 5 részhalmazig.", "model": "DeepL", "n_reviews": 0, @@ -1720,7 +1720,7 @@ "end": 1421.04 }, { - "input": "The more math you do the less crazy that seems, because complex numbers have this bizarre relationship with discrete math, but it really is wonderful, there's no two ways about it.", + "input": "The more math you do, the more you get. The more math you do, the less crazy that seems, because complex numbers have this bizarre relationship with discrete math, but it really is wonderful, there's no two ways about it.", "translatedText": "Minél több matematikával foglalkozol, annál kevésbé tűnik őrültségnek, mert a komplex számoknak van ez a bizarr kapcsolatuk a diszkrét matematikával, de ez tényleg csodálatos, nincs kétféleképpen.", "model": "DeepL", "n_reviews": 0, @@ -2216,7 +2216,7 @@ "end": 1825.2 }, { - "input": "What's indeed one fifth of all the total subsets like we might have guessed, and how this error term came about from the not quite destructive interference in a massive combination of roots of unity.", + "input": "What's the takeaway? Now you could reflect on the answer itself, how the dominant part is indeed one fifth of all the total subsets like we might have guessed, and how this error term came about from the not quite destructive interference in a massive combination of roots of unity.", "translatedText": "Ami valóban az összes részhalmaz egyötöde, mint ahogy azt sejthettük volna, és hogy ez a hiba kifejezés hogyan jött létre a nem egészen destruktív interferenciából az egységgyökerek masszív kombinációjában.", "model": "DeepL", "n_reviews": 0, diff --git a/2022/subsets-puzzle/indonesian/sentence_translations.json b/2022/subsets-puzzle/indonesian/sentence_translations.json index aa12fb2a3..081f35b13 100644 --- a/2022/subsets-puzzle/indonesian/sentence_translations.json +++ b/2022/subsets-puzzle/indonesian/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "Bagi Anda dan saya, setidaknya ada dua liku-liku yang sangat mengejutkan dan sangat indah yang solusinya ingin saya bagikan kepada Anda.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "Sejujurnya, pola yang saya gunakan di sini adalah salah satu yang mungkin paling mudah jika Anda meluangkan waktu untuk berhenti sejenak dan memikirkan sendiri apa yang terjadi ketika Anda memperluas semuanya di sini.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "Untuk lebih spesifiknya, bilangan kompleks yang saya pedulikan adalah bilangan yang akan saya beri label zeta, dan bilangan tersebut berada seperlima putaran di sekeliling lingkaran satuan.", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "Sekarang ingat, bentuk polinomial yang kita tahu, yang paling mudah kita gunakan, adalah bentuk faktor, di mana kita mendapatkan 1 ditambah x, 1 ditambah x kuadrat, terus menerus, hingga 1 ditambah x hingga 2.000.", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "Seluruh ekspresi hingga 2.000 pada dasarnya hanya akan menjadi salinan ekspresi ini sebanyak 400 kali.", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "Ini akan menjadi dua pangkat 400.", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "Penalaran yang pada dasarnya identik menunjukkan bahwa nilainya pada tiga akar kesatuan berikutnya juga dua pangkat 400, karena ingat ketika Anda mengambil pangkat zeta kuadrat atau zeta pangkat tiga, Anda mendapatkan daftar angka yang sama yang hanya dikocok dalam urutan berbeda.", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "Untuk menjumlahkan semua koefisien yang habis dibagi lima, yang perlu diingat adalah cara menghitung berapa banyak total himpunan bagian yang jumlahnya habis dibagi lima, jawabannya adalah seperlima dari persamaan kompleks yang aneh ini, yang baru saja kita hitung menjadi dua banding 2.000 ditambah empat salinan berbeda dari dua hingga 400.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/italian/sentence_translations.json b/2022/subsets-puzzle/italian/sentence_translations.json index 320e29baa..e981aae7e 100644 --- a/2022/subsets-puzzle/italian/sentence_translations.json +++ b/2022/subsets-puzzle/italian/sentence_translations.json @@ -343,7 +343,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "Per me e te, ci sono almeno due colpi di scena molto sorprendenti e molto belli che la soluzione che vorrei condividere con te comporta.", "n_reviews": 0, "start": 254.02, @@ -630,7 +630,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "Onestamente, lo schema che sto seguendo qui è probabilmente il più semplice se ti prendi il tempo per fermarti e riflettere da solo su cosa succede quando espandi tutto qui.", "n_reviews": 0, "start": 516.08, @@ -1141,7 +1141,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "Per essere precisi, il numero complesso che mi interessa è quello che chiamerò zeta, e si trova a un quinto di giro attorno al cerchio unitario.", "n_reviews": 0, "start": 1048.56, @@ -1554,7 +1554,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "Ora ricorda, la forma del polinomio che conosciamo, quella con cui ci sentiamo a nostro agio, è la forma fattorizzata, dove hai questo 1 più x, 1 più x al quadrato, e così via, fino a 1 più x a i 2.000.", "n_reviews": 0, "start": 1438.62, @@ -1596,7 +1596,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "L'intera espressione fino a 2.000 sarà sostanzialmente una copia di questa espressione 400 volte.", "n_reviews": 0, "start": 1483.22, @@ -1897,14 +1897,14 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "Saranno due alla 400.", "n_reviews": 0, "start": 1727.58, "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "Un ragionamento essenzialmente identico mostra che anche il suo valore sulle tre radici successive dell'unità è due elevato alla potenza 400, perché ricorda che quando prendi le potenze di zeta al quadrato o zeta al cubo, ottieni lo stesso elenco di numeri che sono semplicemente mescolati in un ordine diverso .", "n_reviews": 0, "start": 1729.82, @@ -1953,7 +1953,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "Per sommare tutti questi coefficienti che sono divisibili per cinque, che ricordate è un modo per contare quanti sottoinsiemi totali hanno una somma divisibile per cinque, la risposta è un quinto di questa strana espressione complessa, che abbiamo appena calcolato essere due a i 2.000 più quattro copie diverse da due ai 400.", "n_reviews": 0, "start": 1766.16, diff --git a/2022/subsets-puzzle/japanese/sentence_translations.json b/2022/subsets-puzzle/japanese/sentence_translations.json index 6d0cd08d8..a39e05978 100644 --- a/2022/subsets-puzzle/japanese/sentence_translations.json +++ b/2022/subsets-puzzle/japanese/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "あなたと私にとって、私があなたと共有したいソリューションには、少な くとも 2 つの非常に驚くべき、そして非常に美しい展開があります。", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "正直に言うと、ここで私が行おうとしているパターンは、時間をかけて立ち止まって、ここで すべてを展開するとどうなるかを自分で考えてみると、おそらく最も簡単なパターンです。", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "具体的に言うと、私が関心のある複素数はゼータとラベル付け するもので、単位円の 5 分の 1 周の位置にあります。", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "ここで思い出してください、私たちが知っている多項式の形式、つまり私たち が使いやすいのは、因数分解された形式です。 1 プラス x、1 プラス x の 2 乗を繰り返し、1 プラス x まで続きます。 2,000。", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "2,000 までの式全体は、基本的にこの式を 400 回コピーするだけになります。", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "2の400乗になります。", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "本質的に同じ推論は、次の 3 つの 1 の根の値も 2 の 400 乗であること を示しています。 これは、ゼータ 2 乗またはゼータ 3 乗を計算すると、異なる 順序でシャッフルされた同じ数値のリストが得られることを思い出してください。", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "5 で割り切れるこれらの係数をすべて合計すると、合計が 5 で割り切れる合計サ ブセットがいくつあるかを数える方法であることを思い出してください。 答えは、この 奇妙な複雑な式の 5 分の 1 であり、先ほど計算したところ 2 になります。 2,000 に 2 つを加えた 4 つの異なるコピーを 400 に加えます。", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/korean/sentence_translations.json b/2022/subsets-puzzle/korean/sentence_translations.json index e3e3e664f..bef871f4d 100644 --- a/2022/subsets-puzzle/korean/sentence_translations.json +++ b/2022/subsets-puzzle/korean/sentence_translations.json @@ -465,7 +465,7 @@ "end": 280.32 }, { - "input": "If you were solving this problem with pencil and paper, you're one of these kids training for the IMO, it's not a bad idea to simply list out all 2 to the 5 subsets.", + "input": "If you were solving this problem with pencil and paper, you know, you're one of these kids training for the IMO, it's not a bad idea to simply list out all 2 to the 5 subsets.", "translatedText": "연필과 종이로 이 문제를 풀고 있다면, 국제해사기구에서 훈련 중인 학생이라면 2~5개의 하위 집합을 모두 나열하는 것도 나쁘지 않습니다.", "model": "DeepL", "from_community_srt": "5}인 집합으로 함께 해보는 것이 좋을 것 같습니다. 여러분이 연필과 종이로 이 문제를 해결한다면, 여러분도 IMO를 훈련하고 있는 아이들 중 한 명입니다. 단순히 다섯 개의 부분집합을 나열하는 것은 그렇게 많지는 않습니다.", @@ -1924,7 +1924,7 @@ "end": 1421.04 }, { - "input": "The more math you do the less crazy that seems, because complex numbers have this bizarre relationship with discrete math, but it really is wonderful, there's no two ways about it.", + "input": "The more math you do, the more you get. The more math you do, the less crazy that seems, because complex numbers have this bizarre relationship with discrete math, but it really is wonderful, there's no two ways about it.", "translatedText": "복소수는 이산 수학과 기묘한 관계를 가지고 있기 때문에 수학을 많이 할수록 덜 미친 것처럼 보이지만, 실제로는 두 가지 방법이 없을 정도로 훌륭합니다.", "model": "DeepL", "from_community_srt": "수학을 더 많이 할수록 덜 미친 것처럼 보입니다. 왜냐하면 복소수들은 이산 수학과 기이한 관계를 가지고 있기 때문입니다.", @@ -2480,7 +2480,7 @@ "end": 1825.2 }, { - "input": "What's indeed one fifth of all the total subsets like we might have guessed, and how this error term came about from the not quite destructive interference in a massive combination of roots of unity.", + "input": "What's the takeaway? Now you could reflect on the answer itself, how the dominant part is indeed one fifth of all the total subsets like we might have guessed, and how this error term came about from the not quite destructive interference in a massive combination of roots of unity.", "translatedText": "우리가 짐작했던 것처럼 전체 하위 집합의 5분의 1이 실제로 무엇이며, 이 오류 용어는 통합의 거대한 뿌리 조합에서 파괴적이지 않은 간섭으로 인해 어떻게 생겨났는지 설명합니다.", "model": "DeepL", "from_community_srt": "뭐를 가져갈 수 있죠? 여러분은 어떻게 지배적인 부분이 우리가 추측했던 것처럼 실제로 전체 부분집합들의 5분의 1이 되는지, 그리고 어떻게 우리가 생각 못 한 항이 거듭제곱근의 간섭으로부터 생겨났는지를 생각해 볼 수 있습니다.", diff --git a/2022/subsets-puzzle/marathi/sentence_translations.json b/2022/subsets-puzzle/marathi/sentence_translations.json index d2669b2f6..23aaac518 100644 --- a/2022/subsets-puzzle/marathi/sentence_translations.json +++ b/2022/subsets-puzzle/marathi/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "तुमच्यासाठी आणि माझ्यासाठी, किमान दोन अतिशय आश्चर्यकारक आणि अतिशय सुंदर ट्विस्ट आहेत ज्यांचे समाधान मी तुमच्याशी शेअर करू इच्छितो.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "प्रामाणिकपणे, मी येथे ज्या पॅटर्नसाठी जात आहे तो कदाचित सर्वात सोपा आहे जर तुम्ही फक्त विराम देण्यासाठी आणि स्वतःसाठी विचार केला तर तुम्ही येथे सर्वकाही विस्तृत करता तेव्हा काय होते.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "विशिष्‍ट असल्‍यासाठी, मला ज्या संमिश्र संख्‍येची काळजी आहे ती एक आहे जिला मी झेटा असे लेबल लावणार आहे आणि ती एकक वर्तुळाभोवती वळणाच्‍या पाचव्या भागावर बसते.", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "आता लक्षात ठेवा, आम्हाला माहित असलेले बहुपदीचे स्वरूप, ज्यामध्ये आम्हाला सोयीस्कर आहे, तो गुणांकित फॉर्म आहे, जिथे तुमच्याकडे हा 1 अधिक x, 1 अधिक x वर्ग, चालू आणि चालू आहे, 1 अधिक x पर्यंत 2,000.", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "2,000 पर्यंतची संपूर्ण अभिव्यक्ती मुळात या अभिव्यक्तीची 400 वेळा प्रत असेल.", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "ते दोन ते पॉवर 400 असेल.", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "मूलत: समान तर्क दर्शविते की एकतेच्या पुढील तीन मुळांवर त्याचे मूल्य 400 ची शक्ती देखील दोन आहे, कारण लक्षात ठेवा जेव्हा तुम्ही झेटा स्क्वेअर किंवा झेटा क्यूबडची शक्ती घेता तेव्हा तुम्हाला संख्यांची समान यादी मिळते जी फक्त वेगळ्या क्रमाने बदलली जाते.", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "या सर्व गुणांकांची बेरीज करण्यासाठी जे पाच ने भाग जातात, जे लक्षात ठेवा की एकूण किती उपसंचांना पाच ने भाग जातो हे मोजण्याचा एक मार्ग आहे, उत्तर हे या विचित्र जटिल अभिव्यक्तीचा एक पंचमांश आहे, ज्याची गणना आम्ही फक्त दोन ने केली आहे. 2,000 अधिक दोन ते 400 च्या चार वेगवेगळ्या प्रती.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/persian/sentence_translations.json b/2022/subsets-puzzle/persian/sentence_translations.json index 414a7c3de..8b4995046 100644 --- a/2022/subsets-puzzle/persian/sentence_translations.json +++ b/2022/subsets-puzzle/persian/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes. ", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes. ", "translatedText": "برای من و شما، حداقل دو پیچ بسیار شگفت‌انگیز و بسیار زیبا وجود دارد که راه‌حلی که می‌خواهم با شما به اشتراک بگذارم، می‌گیرد. ", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here. ", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here. ", "translatedText": "راستش را بخواهید، الگویی که من در اینجا دنبالش می‌کنم، احتمالاً ساده‌ترین الگویی است که فقط زمانی را صرف مکث کنید و خودتان فکر کنید وقتی همه چیز را در اینجا گسترش دهید چه اتفاقی می‌افتد. ", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle. ", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle. ", "translatedText": "به طور خاص، عدد مختلطی که من به آن اهمیت می‌دهم عددی است که می‌خواهم زتا را برچسب گذاری کنم، و یک پنجم دور دایره واحد می‌چرخد. ", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000. ", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand. ", "translatedText": "حالا به یاد داشته باشید، شکل چند جمله‌ای که ما می‌شناسیم، شکلی که با آن راحت هستیم، شکل فاکتور شده است، که در آن شما این 1 به علاوه x، 1 به علاوه x مربع، روی و روی، تا 1 به علاوه x را دارید. ", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times. ", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times. ", "translatedText": "کل عبارت تا 2000 اساساً فقط 400 بار یک کپی از این عبارت است. ", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400. ", + "input": "It will be two to the power four hundred. ", "translatedText": "دو به توان 400 خواهد بود. ", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order. ", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order. ", "translatedText": "استدلال اساساً یکسان نشان می دهد که مقدار آن در سه ریشه بعدی وحدت نیز دو به توان 400 است، زیرا به یاد داشته باشید که وقتی توان های زتا را به صورت مربع یا زتا مکعبی می گیرید، همان لیست اعدادی را دریافت می کنید که فقط به ترتیبی متفاوت با هم مخلوط شده اند. . ", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400. ", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred. ", "translatedText": "برای جمع کردن همه این ضرایب که بر پنج بخش پذیر هستند، که به یاد داشته باشید روشی برای شمارش تعداد زیر مجموعه های کل است که مجموع آنها بر پنج بخش پذیر است، پاسخ یک پنجم این عبارت پیچیده عجیب و غریب است که ما فقط دو را محاسبه کردیم. 2000 به علاوه چهار نسخه مختلف از دو تا 400. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/portuguese/sentence_translations.json b/2022/subsets-puzzle/portuguese/sentence_translations.json index 37bd60228..eb3d98f04 100644 --- a/2022/subsets-puzzle/portuguese/sentence_translations.json +++ b/2022/subsets-puzzle/portuguese/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "Para você e para mim, há pelo menos duas reviravoltas muito surpreendentes e muito bonitas que a solução que gostaria de compartilhar com vocês leva.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "Honestamente, o padrão que estou buscando aqui é provavelmente o mais fácil se você apenas parar e pensar por si mesmo no que acontece quando você expande tudo aqui.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "Para ser mais específico, o número complexo que me interessa é aquele que chamarei de zeta, e ele fica um quinto de volta em torno do círculo unitário.", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "Agora lembre-se, a forma do polinômio que conhecemos, aquela com a qual estamos confortáveis, é a forma fatorada, onde você tem este 1 mais x, 1 mais x ao quadrado, e assim por diante, até 1 mais x para os 2.000.", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "A expressão inteira até 2.000 será basicamente uma cópia desta expressão 400 vezes.", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "Serão dois elevado a 400.", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "Um raciocínio essencialmente idêntico mostra que seu valor nas próximas três raízes da unidade também é dois elevado a 400, porque lembre-se, quando você toma potências de zeta ao quadrado ou zeta ao cubo, você obtém a mesma lista de números que são apenas embaralhados em uma ordem diferente.", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "Para somar todos esses coeficientes que são divisíveis por cinco, que, lembre-se, é uma forma de contar quantos subconjuntos totais têm uma soma divisível por cinco, a resposta é um quinto desta estranha expressão complexa, que acabamos de calcular como dois elevado a os 2.000 mais quatro cópias diferentes de dois elevado a 400.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/russian/sentence_translations.json b/2022/subsets-puzzle/russian/sentence_translations.json index 6cf03f0f9..2ecf63c1a 100644 --- a/2022/subsets-puzzle/russian/sentence_translations.json +++ b/2022/subsets-puzzle/russian/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "Для нас с вами есть как минимум два очень неожиданных и очень красивых поворота в решении, которым я хотел бы с вами поделиться.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "Честно говоря, шаблон, который я здесь использую, вероятно, самый простой, если вы просто потратите время на паузу и продумаете сами, что произойдет, когда вы развернете все здесь.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "Говоря конкретнее, меня волнует комплексное число, которое я собираюсь обозначить как дзета, и оно находится на пятой части оборота вокруг единичного круга.", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "Теперь помните, что форма полинома, которую мы знаем, та, которая нам удобна, - это факторизованная форма, где у вас есть 1 плюс x, 1 плюс x в квадрате, и так далее, вплоть до 1 плюс x до 2000.", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "Все выражение до 2000 по сути будет просто копией этого выражения 400 раз.", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "Это будет два в 400 степени.", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "По существу идентичные рассуждения показывают, что его значение для следующих трех корней из единицы также равно двойке в степени 400, потому что помните, что когда вы берете степени дзета в квадрате или дзета в кубе, вы получаете тот же список чисел, которые просто перетасованы в другом порядке.", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "Если сложить все эти коэффициенты, которые делятся на пять (помните, что это способ подсчитать, сколько всего подмножеств имеют сумму, делящуюся на пять), ответом будет одна пятая этого странного сложного выражения, которое мы только что вычислили как два к 2000 плюс четыре разные копии двух из 400.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/spanish/sentence_translations.json b/2022/subsets-puzzle/spanish/sentence_translations.json index d99a16b8b..97d8ac3f9 100644 --- a/2022/subsets-puzzle/spanish/sentence_translations.json +++ b/2022/subsets-puzzle/spanish/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "Para ti y para mí, hay al menos dos giros muy sorprendentes y muy hermosos que toma la solución que me gustaría compartir contigo.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "Honestamente, el patrón que estoy buscando aquí es probablemente el más fácil si te tomas el tiempo para hacer una pausa y pensar por ti mismo lo que sucede cuando expandes todo aquí.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "Para ser específico, el número complejo que me interesa es uno que voy a etiquetar como zeta y se encuentra a un quinto de vuelta alrededor del círculo unitario.", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "Ahora recuerda, la forma del polinomio que conocemos, con la que nos sentimos cómodos, es la forma factorizada, donde tienes este 1 más x, 1 más x al cuadrado, y así sucesivamente, hasta 1 más x para los 2.000.", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "La expresión completa hasta 2000 básicamente será una copia de esta expresión 400 veces.", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "Serán dos elevado a 400.", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "Un razonamiento esencialmente idéntico muestra que su valor en las siguientes tres raíces de la unidad también es dos elevado a 400, porque recuerda que cuando tomas potencias de zeta al cuadrado o zeta al cubo, obtienes la misma lista de números que simplemente se mezclan en un orden diferente.", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "Para sumar todos estos coeficientes que son divisibles por cinco, que recuerden, es una forma de contar cuántos subconjuntos totales tienen una suma divisible por cinco, la respuesta es una quinta parte de esta extraña expresión compleja, que acabamos de calcular como dos a los 2.000 más cuatro ejemplares diferentes de dos al 400.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/tamil/sentence_translations.json b/2022/subsets-puzzle/tamil/sentence_translations.json index 4b60886df..c83701de3 100644 --- a/2022/subsets-puzzle/tamil/sentence_translations.json +++ b/2022/subsets-puzzle/tamil/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "உங்களுக்கும் எனக்கும், நான் உங்களுடன் பகிர்ந்து கொள்ள விரும்பும் தீர்வுக்கு குறைந்தது இரண்டு ஆச்சரியமான மற்றும் மிக அழகான திருப்பங்கள் உள்ளன.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "உண்மையைச் சொல்வதானால், நான் இங்குப் போகிற மாதிரியானது, இங்குள்ள அனைத்தையும் விரிவுபடுத்தும்போது என்ன நடக்கும் என்பதை நீங்கள் இடைநிறுத்தி, நீங்களே சிந்தித்துப் பார்ப்பது மிகவும் எளிதானது.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "குறிப்பாகச் சொல்வதென்றால், நான் விரும்பும் கலப்பு எண், நான் ஜீட்டாவை லேபிளிடப் போகிறேன், மேலும் அது யூனிட் வட்டத்தைச் சுற்றி ஐந்தில் ஒரு பங்காக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "இப்போது நினைவில் கொள்ளுங்கள், நமக்குத் தெரிந்த பல்லுறுப்புக்கோவையின் வடிவம், நமக்கு வசதியானது, காரணி வடிவம் ஆகும், இதில் 1 கூட்டல் x, 1 கூட்டல் x ஸ்கொயர், ஆன் மற்றும் ஆன், 1 பிளஸ் x வரை இருக்கும் 2,000.", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "2,000 வரையிலான முழு வெளிப்பாடும் அடிப்படையில் இந்த வெளிப்பாட்டின் 400 முறை நகலாக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "இது பவர் 400க்கு இரண்டாக இருக்கும்.", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "அடிப்படையில் ஒரே மாதிரியான பகுத்தறிதல், ஒற்றுமையின் அடுத்த மூன்று வேர்களில் அதன் மதிப்பு 400க்கு இரண்டு என்று காட்டுகிறது, ஏனென்றால் நீங்கள் ஜீட்டா ஸ்கொயர் அல்லது ஜீட்டா க்யூப்ட் சக்திகளை எடுக்கும்போது, வேறு வரிசையில் மாற்றப்பட்ட அதே எண்களின் பட்டியலைப் பெறுவீர்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "ஐந்தால் வகுபடும் இந்தக் குணகங்கள் அனைத்தையும் கூட்டினால், எத்தனை மொத்த துணைக்குழுக்கள் ஐந்தால் வகுபடும் கூட்டுத்தொகையைக் கணக்கிடும் ஒரு வழி என்பதை நினைவில் கொள்ளுங்கள், விடை இந்த வித்தியாசமான சிக்கலான வெளிப்பாட்டின் ஐந்தில் ஒரு பங்காகும், அதை நாம் இரண்டாகக் கணக்கிட்டோம். 2,000 மற்றும் இரண்டு முதல் 400 வரையிலான நான்கு வெவ்வேறு பிரதிகள்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/telugu/sentence_translations.json b/2022/subsets-puzzle/telugu/sentence_translations.json index 3f41d5a0d..49c7f1eba 100644 --- a/2022/subsets-puzzle/telugu/sentence_translations.json +++ b/2022/subsets-puzzle/telugu/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "మీకు మరియు నాకు, నేను మీతో పంచుకోవాలనుకుంటున్న పరిష్కారంలో కనీసం రెండు చాలా ఆశ్చర్యకరమైన మరియు చాలా అందమైన మలుపులు ఉన్నాయి.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "నిజాయితీగా చెప్పాలంటే, నేను ఇక్కడ చూడబోతున్న నమూనా, మీరు పాజ్ చేయడానికి సమయాన్ని వెచ్చించి, మీరు ఇక్కడ అన్నింటినీ విస్తరింపజేసినప్పుడు ఏమి జరుగుతుందో మీరే ఆలోచించుకుంటే చాలా తేలికగా ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "ప్రత్యేకంగా చెప్పాలంటే, నేను శ్రద్ధ వహించే సంక్లిష్ట సంఖ్య, నేను జీటాను లేబుల్ చేయబోతున్నాను మరియు ఇది యూనిట్ సర్కిల్ చుట్టూ ఉన్న మలుపులో ఐదవ వంతు ఉంటుంది.", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "ఇప్పుడు గుర్తుంచుకోండి, మనకు తెలిసిన బహుపది రూపం, మనకు సౌకర్యంగా ఉన్నది, ఫ్యాక్టర్డ్ ఫారమ్, ఇక్కడ మీరు ఈ 1 ప్లస్ x, 1 ప్లస్ x స్క్వేర్డ్, ఆన్ మరియు ఆన్‌లో, 1 ప్లస్ x వరకు 2,000.", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "2,000 వరకు ఉన్న మొత్తం వ్యక్తీకరణ ప్రాథమికంగా ఈ వ్యక్తీకరణకు 400 సార్లు కాపీ అవుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "ఇది పవర్ 400కి రెండు అవుతుంది.", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "యూనిటీ యొక్క తరువాతి మూడు మూలాల్లో దాని విలువ కూడా పవర్ 400కి రెండు అని ప్రాథమికంగా ఒకే విధమైన తార్కికం చూపిస్తుంది, ఎందుకంటే మీరు జీటా స్క్వేర్డ్ లేదా జీటా క్యూబ్డ్ పవర్‌లను తీసుకున్నప్పుడు, మీరు వేరే క్రమంలో షఫుల్ చేయబడిన అదే సంఖ్యల జాబితాను పొందుతారు.", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "ఐదుతో భాగించబడే ఈ కోఎఫీషియెంట్‌లన్నింటినీ జోడించడానికి, మొత్తం ఉపసమితులు ఐదుతో భాగించబడే మొత్తాన్ని లెక్కించడానికి ఒక మార్గం అని గుర్తుంచుకోవాలి, సమాధానం ఈ విచిత్రమైన సంక్లిష్ట వ్యక్తీకరణలో ఐదవ వంతు, మేము ఇప్పుడే రెండుగా గణించాము. 2,000 ప్లస్ రెండు నుండి 400 వరకు నాలుగు వేర్వేరు కాపీలు.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/thai/sentence_translations.json b/2022/subsets-puzzle/thai/sentence_translations.json index cacba7154..a47d3a7df 100644 --- a/2022/subsets-puzzle/thai/sentence_translations.json +++ b/2022/subsets-puzzle/thai/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes. ", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here. ", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle. ", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000. ", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times. ", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400. ", + "input": "It will be two to the power four hundred. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order. ", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400. ", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred. ", "translatedText": "สิ่งเดียวที่แตกต่างคือเมื่อเราประเมินมันเป็นซีต้าเป็นศูนย์ แต่ซีต้ากำลัง 0 เป็นวิธีแปลกๆ ในการบอกเลข 1 และเรารู้วิธีประเมินค่านี้จากตัวเดียว นั่นเป็นหนึ่งในสิ่งที่ง่าย เราทำสิ่งนี้ก่อนหน้านี้ วงเล็บทั้งหมดนี้กลายเป็นสอง ดูเหมือนว่าเอาสองคูณด้วยตัวมันเอง 2,000 ครั้ง และสุดท้ายด้วยเหตุนี้ เราก็ได้คำตอบที่ตรงไปตรงมามากสำหรับคำถามการนับของเรา ในการบวกสัมประสิทธิ์ทั้งหมดที่หารด้วย 5 ลงตัว ซึ่งจำได้ว่าเป็นวิธีหนึ่งในการนับจำนวนชุดย่อยทั้งหมดซึ่งมีผลรวมหารด้วย 5 ลงตัว คำตอบคือ หนึ่งในห้าของนิพจน์ที่ซับซ้อนประหลาดนี้ ซึ่งเราเพิ่งคำนวณเป็น 2 2,000 บวกกับสำเนาที่แตกต่างกันสี่ชุดจากสองถึง 400 และที่นี่คุณอาจต้องการตรวจสอบสุขภาพจิตอย่างรวดเร็วว่าคำตอบนี้สมเหตุสมผลหรือไม่ ตัวอย่างเช่น หากคุณทำในกรณีเล็กด้วยเซต 1, 2, 3, 4, 5 และคุณพิจารณาเหตุผลแบบเดียวกับที่เราเพิ่งทำไป มันจะบอกคุณว่าคำตอบคือหนึ่งในห้าของสอง ห้า คือจำนวนชุดย่อยทั้งหมด บวก 4 คูณ 2 หารด้วย 1 ในกรณีนี้ ซึ่งก็คือ 1 ใน 5 ของ 32 บวก 8 ซึ่งก็คือ 8 และถ้าคุณจำได้เมื่อเราพิจารณาพวกเขาทั้งหมดอย่างชัดเจน นั่นคือคำตอบจริงๆ ฟังนะ นี่เป็นปริศนาที่ยาก และเมื่อมันคุ้มค่าที่จะสละเวลาในการแก้ไขปัญหายากๆ มันก็คุ้มค่าที่จะใช้เวลาไตร่ตรองมันด้วย คุณได้อะไรจากสิ่งนี้? ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/turkish/sentence_translations.json b/2022/subsets-puzzle/turkish/sentence_translations.json index e11f22f6d..7763e7c0f 100644 --- a/2022/subsets-puzzle/turkish/sentence_translations.json +++ b/2022/subsets-puzzle/turkish/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "Sizin ve benim için, sizinle paylaşmak istediğim çözümün getirdiği çok şaşırtıcı ve çok güzel en az iki gelişme var.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "Dürüst olmak gerekirse, burada aradığım model, eğer biraz zaman ayırırsanız ve buradaki her şeyi genişlettiğinizde ne olacağını kendi başınıza düşünürseniz muhtemelen en kolay olanıdır.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "Daha spesifik olmak gerekirse, önemsediğim karmaşık sayı zeta olarak etiketleyeceğim bir sayıdır ve birim çember etrafında beşte bir tur döner.", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "Şimdi unutmayın, polinomun bildiğimiz ve rahat ettiğimiz şekli çarpanlara ayrılmış formdur, burada 1 artı x, 1 artı x kare, 1 artı x'e kadar devam ediyor. 2.000.", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "2.000'e kadar olan ifadenin tamamı temel olarak bu ifadenin 400 kez kopyası olacaktır.", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "2 üzeri 400 olacak.", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "Temelde aynı akıl yürütme, birliğin sonraki üç kökündeki değerinin de iki üzeri 400 olduğunu gösterir, çünkü unutmayın, zeta karenin veya zeta küpün kuvvetlerini aldığınızda, farklı bir sırayla karıştırılmış aynı sayılar listesini elde edersiniz.", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "Beşe bölünebilen tüm bu katsayıları toplamak için, ki bu, toplamı beşe bölünebilen toplam alt küme sayısını saymanın bir yoludur, cevap, az önce iki üzeri olarak hesapladığımız bu tuhaf karmaşık ifadenin beşte biridir. 2.000 artı iki üzeri 400'ün dört farklı kopyası.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/ukrainian/sentence_translations.json b/2022/subsets-puzzle/ukrainian/sentence_translations.json index 911f46aa4..95d6e1066 100644 --- a/2022/subsets-puzzle/ukrainian/sentence_translations.json +++ b/2022/subsets-puzzle/ukrainian/sentence_translations.json @@ -343,7 +343,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "Для нас з вами є принаймні два дуже дивовижні та дуже красиві повороти рішення, яким я хотів би поділитися з вами.", "n_reviews": 0, "start": 254.02, @@ -630,7 +630,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "Чесно кажучи, модель, яку я тут використовую, є, мабуть, найпростішою, якщо ви просто знайдете час, щоб зупинитись і подумати самостійно, що станеться, коли ви розгорнете все тут.", "n_reviews": 0, "start": 516.08, @@ -1141,7 +1141,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "Якщо бути конкретнішим, комплексне число, яке мене цікавить, це те, яке я збираюся позначити дзета, і воно знаходиться на п’ятій частині оберту навколо одиничного кола.", "n_reviews": 0, "start": 1048.56, @@ -1554,7 +1554,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "Тепер запам’ятайте, форма багаточлена, яку ми знаємо, та яка нам зручна, це розкладена на множники форма, де у вас є 1 плюс х, 1 плюс х у квадраті, далі і далі, аж до 1 плюс х до 2000.", "n_reviews": 0, "start": 1438.62, @@ -1596,7 +1596,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "Весь вираз до 2000 буде просто копією цього виразу 400 разів.", "n_reviews": 0, "start": 1483.22, @@ -1897,14 +1897,14 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "Це буде два в ступені 400.", "n_reviews": 0, "start": 1727.58, "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "По суті, ідентичне міркування показує, що його значення для наступних трьох коренів з одиниці також дорівнює двом у степені 400, тому що пам’ятайте, коли ви берете степені дзета в квадраті або дзета в кубі, ви отримуєте той самий список чисел, які просто перемішуються в іншому порядку. .", "n_reviews": 0, "start": 1729.82, @@ -1953,7 +1953,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "Щоб скласти всі ці коефіцієнти, які діляться на п’ять, пам’ятайте, що це спосіб підрахувати, скільки загальних підмножин має суму, що ділиться на п’ять, відповіддю буде одна п’ята цього дивного складного виразу, який ми щойно обчислили як два до 2000 плюс чотири різні копії від двох до 400.", "n_reviews": 0, "start": 1766.16, diff --git a/2022/subsets-puzzle/urdu/sentence_translations.json b/2022/subsets-puzzle/urdu/sentence_translations.json index f990c87b3..53feb1613 100644 --- a/2022/subsets-puzzle/urdu/sentence_translations.json +++ b/2022/subsets-puzzle/urdu/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes. ", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes. ", "translatedText": "آپ اور میرے لیے، کم از کم دو بہت ہی حیران کن اور بہت خوبصورت موڑ ہیں جن کا حل میں آپ کے ساتھ شیئر کرنا چاہتا ہوں۔", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here. ", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here. ", "translatedText": "سچ میں، میں یہاں جس پیٹرن کے لیے جا رہا ہوں وہ ایک ہے جو شاید سب سے آسان ہے اگر آپ صرف وقفہ کرنے کے لیے وقت نکالیں اور خود سوچیں کہ جب آپ یہاں ہر چیز کو پھیلاتے ہیں تو کیا ہوتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle. ", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle. ", "translatedText": "مخصوص ہونے کے لیے، میں جس پیچیدہ نمبر کا خیال رکھتا ہوں وہ ایک ہے جس پر میں زیٹا کا لیبل لگانے جا رہا ہوں، اور یہ یونٹ کے دائرے کے گرد ایک موڑ کا پانچواں حصہ بیٹھتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000. ", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand. ", "translatedText": "اب یاد رکھیں، کثیر الجہتی شکل کی جو ہم جانتے ہیں، جس سے ہم آرام سے ہیں، فیکٹرڈ فارم ہے، جہاں آپ کے پاس یہ 1 جمع x، 1 جمع x مربع، آن اور آن ہے، 1 جمع x تک 2,000 اس مقام تک سب کچھ محض ایک بے معنی علامتی کھیل ہے، جو ایک مشکل مسئلے کو دوسرے میں دھکیل رہا ہے، جب تک کہ ہم اپنی آستینیں چڑھا کر یہاں کچھ ایماندارانہ حساب کتاب نہ کر لیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times. ", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times. ", "translatedText": "2,000 تک کا پورا اظہار بنیادی طور پر صرف 400 بار اس اظہار کی ایک کاپی بننے والا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400. ", + "input": "It will be two to the power four hundred. ", "translatedText": "یہ طاقت 400 سے دو ہو جائے گا. ", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order. ", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order. ", "translatedText": "بنیادی طور پر یکساں استدلال یہ ظاہر کرتا ہے کہ اتحاد کی اگلی تین جڑوں پر اس کی قدر بھی طاقت 400 سے دو ہے، کیونکہ یاد رکھیں جب آپ زیٹا مربع یا زیٹا کیوبڈ کی طاقتیں لیتے ہیں، تو آپ کو اعداد کی وہی فہرست ملتی ہے جو صرف ایک مختلف ترتیب میں بدلی جاتی ہیں۔. ", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400. ", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred. ", "translatedText": "ان تمام گتانکوں کو شامل کرنے کے لیے جو پانچ سے قابل تقسیم ہیں، جو یاد رکھیں یہ گننے کا ایک طریقہ ہے کہ کل کتنے ذیلی سیٹوں کو پانچ سے تقسیم کیا جا سکتا ہے، اس کا جواب اس عجیب پیچیدہ اظہار کا پانچواں حصہ ہے، جسے ہم نے صرف دو کے حساب سے شمار کیا ہے۔2000 کے علاوہ دو سے 400 تک کی چار مختلف کاپیاں۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/subsets-puzzle/vietnamese/sentence_translations.json b/2022/subsets-puzzle/vietnamese/sentence_translations.json index 6102879cb..44a92751c 100644 --- a/2022/subsets-puzzle/vietnamese/sentence_translations.json +++ b/2022/subsets-puzzle/vietnamese/sentence_translations.json @@ -392,7 +392,7 @@ "end": 254.02 }, { - "input": "For you and me, there are at least two very surprising and very beautiful twists that the solution I'd like to share with you takes.", + "input": "For you and me, there are at least two very surprising and very beautiful twists and turns that the solution I'd like to share with you takes.", "translatedText": "Đối với bạn và tôi, có ít nhất hai bước ngoặt rất đáng ngạc nhiên và rất hay mà giải pháp mà tôi muốn chia sẻ với bạn sẽ thực hiện.", "model": "google_nmt", "n_reviews": 0, @@ -720,7 +720,7 @@ "end": 515.36 }, { - "input": "Honestly, the pattern I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", + "input": "And honestly, the pattern that I'm going for here is one that's probably easiest if you just take the time to pause and think through for yourself what happens when you expand everything here.", "translatedText": "Thành thật mà nói, mẫu tôi đang hướng tới ở đây có lẽ là mẫu dễ nhất nếu bạn dành thời gian để tạm dừng và tự suy nghĩ xem điều gì sẽ xảy ra khi bạn mở rộng mọi thứ ở đây.", "model": "google_nmt", "n_reviews": 0, @@ -1304,7 +1304,7 @@ "end": 1043.22 }, { - "input": "To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", + "input": "The complex number To be specific, the complex number that I care about is one that I'm going to label zeta, and it sits a fifth of a turn around the unit circle.", "translatedText": "Cụ thể hơn, số phức mà tôi quan tâm là số mà tôi sẽ đặt tên là zeta, và nó nằm ở 1/5 vòng quanh vòng tròn đơn vị.", "model": "google_nmt", "n_reviews": 0, @@ -1776,7 +1776,7 @@ "end": 1438.1 }, { - "input": "Now remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this 1 plus x, 1 plus x squared, on and on, all the way up to 1 plus x to the 2,000.", + "input": "Remember, the form of the polynomial we know, the one we're comfortable with, is the factored form, where you have this one plus x, one plus x squared, on and on, all the way up to one plus x to the two thousand.", "translatedText": "Bây giờ hãy nhớ, dạng đa thức mà chúng ta biết, dạng chúng ta thấy thoải mái, là dạng phân tích nhân tử, trong đó bạn có 1 cộng x, 1 cộng x bình phương, cứ thế, cho đến 1 cộng x đến 2.000.", "model": "google_nmt", "n_reviews": 0, @@ -1824,7 +1824,7 @@ "end": 1482.7 }, { - "input": "The entire expression up to 2,000 is basically just going to be a copy of this expression 400 times.", + "input": "The entire expression up to two thousand is basically just going to be a copy of this expression four hundred times.", "translatedText": "Toàn bộ biểu thức lên tới 2.000 về cơ bản sẽ chỉ là bản sao của biểu thức này 400 lần.", "model": "google_nmt", "n_reviews": 0, @@ -2168,7 +2168,7 @@ "end": 1727.32 }, { - "input": "It will be two to the power 400.", + "input": "It will be two to the power four hundred.", "translatedText": "Nó sẽ là hai lũy thừa 400.", "model": "google_nmt", "n_reviews": 0, @@ -2176,7 +2176,7 @@ "end": 1729.22 }, { - "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power 400, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", + "input": "Essentially identical reasoning shows that its value on the next three roots of unity is also two to the power four hundred, because remember when you take powers of zeta squared or zeta cubed, you get the same list of numbers that are just shuffled in a different order.", "translatedText": "Về cơ bản, lý luận giống hệt nhau cho thấy giá trị của nó trên ba căn bậc 1 tiếp theo cũng bằng 2 lũy thừa 400, bởi vì hãy nhớ rằng khi bạn lấy lũy thừa của zeta bình phương hoặc zeta lập phương, bạn sẽ nhận được cùng một danh sách các số được xáo trộn theo một thứ tự khác.", "model": "google_nmt", "n_reviews": 0, @@ -2232,7 +2232,7 @@ "end": 1765.72 }, { - "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the 2,000 plus four different copies of two to the 400.", + "input": "To add up all of these coefficients which are divisible by five, which remember is a way of counting how many total subsets have a sum divisible by five, the answer is one fifth of this weird complex expression, which we just computed to be two to the two thousand plus four different copies of two to the four hundred.", "translatedText": "Để cộng tất cả các hệ số chia hết cho 5, hãy nhớ rằng đây là cách đếm tổng số tập hợp con có tổng chia hết cho 5, câu trả lời là 1/5 của biểu thức phức tạp kỳ lạ này, mà chúng ta vừa tính là hai đến 2.000 cộng với bốn bản sao khác nhau của hai đến 400.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/arabic/sentence_translations.json b/2022/visual-proofs/arabic/sentence_translations.json index 4f5e73678..31f758806 100644 --- a/2022/visual-proofs/arabic/sentence_translations.json +++ b/2022/visual-proofs/arabic/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "في الرياضيات، لا يمكنك الهروب من الحاجة إلى البحث عن الافتراضات الخفية والحالات المتطرفة. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/bengali/sentence_translations.json b/2022/visual-proofs/bengali/sentence_translations.json index 0e1e718a3..a135013bb 100644 --- a/2022/visual-proofs/bengali/sentence_translations.json +++ b/2022/visual-proofs/bengali/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "গণিতে, আপনি লুকানো অনুমান এবং প্রান্তের ক্ষেত্রের জন্য সন্ধান করার প্রয়োজন এড়াতে পারবেন না।", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/chinese/sentence_translations.json b/2022/visual-proofs/chinese/sentence_translations.json index 6282a6651..e22a3541c 100644 --- a/2022/visual-proofs/chinese/sentence_translations.json +++ b/2022/visual-proofs/chinese/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases.", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you.", "translatedText": "在数学中，您 无法逃避寻找隐藏假设和边缘情况的需要。", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/english/captions.srt b/2022/visual-proofs/english/captions.srt index 3b1211dfc..ac53b46c8 100644 --- a/2022/visual-proofs/english/captions.srt +++ b/2022/visual-proofs/english/captions.srt @@ -487,666 +487,670 @@ Maybe my drawing is a little bit sloppy, but the logic helps us see that they do have to be the same. 123 -00:07:34,060 --> 00:07:39,333 +00:07:34,060 --> 00:07:38,896 Next I'll draw a line from p down to b, and then from p down to c, 124 -00:07:39,333 --> 00:07:46,260 -and I claim that this triangle here is congruent to its reflection across that bisector. +00:07:38,896 --> 00:07:44,599 +and I claim that this triangle here is congruent to its reflection across that 125 +00:07:44,599 --> 00:07:46,260 +perpendicular bisector. + +126 00:07:46,640 --> 00:07:48,977 Again the symmetry maybe helps make this clear, -126 +127 00:07:48,977 --> 00:07:53,067 but more rigorously they both have the same base, they both have a 90 degree angle, -127 +128 00:07:53,067 --> 00:07:57,060 and they both have the same height, so it follows by the side angle side relation. -128 +129 00:07:57,580 --> 00:08:00,910 So based on that first pair of triangles I'm going to mark this side length here -129 +130 00:08:00,910 --> 00:08:04,200 as being the same as this side length here, marking them with double tick marks. -130 +131 00:08:04,940 --> 00:08:09,096 And based on the second triangle relation I'll mark this side length here -131 +132 00:08:09,096 --> 00:08:13,140 as the same as this line over here, marking them with triple tick marks. -132 +133 00:08:13,380 --> 00:08:17,056 And so from that we have two more triangles that need to be the same, -133 +134 00:08:17,056 --> 00:08:21,520 namely this one over here, and the one with corresponding two side lengths over here. -134 +135 00:08:21,980 --> 00:08:25,095 And the reasoning here is they both have that triple ticked side, -135 +136 00:08:25,095 --> 00:08:27,880 a double ticked side, and they're both 90 degree triangles. -136 +137 00:08:28,420 --> 00:08:31,420 So this follows by the side side angle congruence relation. -137 +138 00:08:32,200 --> 00:08:34,578 And all of those are valid congruence relations, -138 +139 00:08:34,578 --> 00:08:37,442 I'm not pulling the wool over your eyes with one of those, -139 +140 00:08:37,442 --> 00:08:41,520 and all of this will basically be enough to show us why AB has to be the same as BC. -140 +141 00:08:42,460 --> 00:08:47,457 That first pair of triangles implies that the length AF is the same as the length AE, -141 +142 00:08:47,457 --> 00:08:52,049 those are corresponding sides to each other, I'll just color them in red here, -142 +143 00:08:52,049 --> 00:08:56,059 and then that last triangle relation guarantees for us that the side -143 +144 00:08:56,059 --> 00:08:58,500 FB is going to be the same as the side EC. -144 +145 00:08:59,160 --> 00:09:00,880 I'll kind of color both of those in blue. -145 +146 00:09:01,340 --> 00:09:05,780 And finally the result we want basically comes from adding up these two equations. -146 +147 00:09:06,380 --> 00:09:11,524 The length AF plus FB is clearly the same as the total length AB, -147 +148 00:09:11,524 --> 00:09:16,980 and likewise the length AE plus EC is the same as the total length AC. -148 +149 00:09:17,840 --> 00:09:21,793 So all in all the side length AB has to be the same as the side length AC, -149 +150 00:09:21,793 --> 00:09:26,274 and because we made no assumptions about the triangle this implies that any triangle -150 +151 00:09:26,274 --> 00:09:26,960 is isosceles. -151 +152 00:09:27,640 --> 00:09:30,611 Actually for that matter since we made no assumptions about the -152 +153 00:09:30,611 --> 00:09:34,000 specific two sides we chose, it implies that any triangle is equilateral. -153 +154 00:09:35,660 --> 00:09:38,980 So this leaves us somewhat disturbingly with three different possibilities. -154 +155 00:09:39,500 --> 00:09:43,598 All triangles really are equilateral, that's just the truth of the universe, -155 +156 00:09:43,598 --> 00:09:46,951 or you can use Euclid style reasoning to derive false results, -156 +157 00:09:46,951 --> 00:09:49,080 or there's something wrong in the proof. -157 +158 00:09:49,660 --> 00:09:51,820 But if there is, where exactly is it? -158 +159 00:09:54,620 --> 00:09:57,640 So what exactly is going on with these three examples? -159 +160 00:09:58,500 --> 00:10:01,459 Now the thing that's a little bit troubling about that first -160 +161 00:10:01,459 --> 00:10:04,564 example with the sphere is that it is very similar in spirit to -161 +162 00:10:04,564 --> 00:10:07,960 a lot of other famous and supposedly true visual proofs from geometry. -162 +163 00:10:08,760 --> 00:10:12,592 For example there's a very famous proof about the area of a circle that starts -163 +164 00:10:12,592 --> 00:10:15,308 off by dividing it into a bunch of little pizza wedges, -164 +165 00:10:15,308 --> 00:10:18,170 and you take all those wedges and you straighten them out, -165 +166 00:10:18,170 --> 00:10:20,450 essentially lining up the crust of that pizza, -166 +167 00:10:20,450 --> 00:10:24,040 and then we take half the wedges and inter-slice them with the other half. -167 +168 00:10:24,260 --> 00:10:27,508 And the idea is that this might not be a perfect rectangle, -168 +169 00:10:27,508 --> 00:10:32,002 it's got some bumps and curves, but as you take thinner and thinner slices you get -169 +170 00:10:32,002 --> 00:10:35,034 something that's closer and closer to a true rectangle, -170 +171 00:10:35,034 --> 00:10:39,419 and the width of that rectangle comes from half the circumference of the circle, -171 +172 00:10:39,419 --> 00:10:44,184 which is by definition pi times r, and then the height of that rectangle comes from the -172 +173 00:10:44,184 --> 00:10:47,920 radius of the circle, r, meaning that the whole area is pi r squared. -173 +174 00:10:48,800 --> 00:10:53,669 This time the result is valid, but why is it not okay to do what we did with the spheres, -174 +175 00:10:53,669 --> 00:10:56,700 but somehow it is okay to do this with the pizza slices? -175 +176 00:10:57,780 --> 00:11:01,377 The main problem with the sphere argument is that when we flatten out all -176 +177 00:11:01,377 --> 00:11:04,731 of those orange wedges, if we were to do it accurately in a way that -177 +178 00:11:04,731 --> 00:11:08,620 preserves their area, they don't look like triangles, they should bulge outward. -178 +179 00:11:09,080 --> 00:11:13,593 And if you want to see this, let's think really critically about just one particular -179 +180 00:11:13,593 --> 00:11:18,372 one of those wedges on the sphere, and ask yourself how does the width across that wedge, -180 +181 00:11:18,372 --> 00:11:22,620 this little portion of a line of latitude, vary as you go up and down the wedge? -181 +182 00:11:22,960 --> 00:11:27,633 In particular, if you consider the angle phi from the z-axis down to a point on -182 +183 00:11:27,633 --> 00:11:32,540 this wedge as we walk down it, what's the length of that width as a function of phi? -183 +184 00:11:32,860 --> 00:11:36,352 For those of you curious about the details of these sorts of things, -184 +185 00:11:36,352 --> 00:11:40,704 you'd start off by drawing this line up here from the z-axis to a point on the wedge, -185 +186 00:11:40,704 --> 00:11:44,500 its length will be the radius of the sphere r times the sine of this angle. -186 +187 00:11:44,660 --> 00:11:49,423 That lets us deduce how long the total line of latitude is where we're sitting, -187 +188 00:11:49,423 --> 00:11:53,472 it'll basically be 2 pi times that radial line, 2 pi r sine of phi, -188 +189 00:11:53,472 --> 00:11:58,652 and then the width of the wedge that we care about is just some constant proportion of -189 +190 00:11:58,652 --> 00:12:00,260 that full line of latitude. -190 +191 00:12:00,660 --> 00:12:03,484 Now the details don't matter too much, the one thing I want -191 +192 00:12:03,484 --> 00:12:06,120 you to notice is that this is not a linear relationship. -192 +193 00:12:06,460 --> 00:12:09,875 As you walk from the top of that wedge down to the bottom, -193 +194 00:12:09,875 --> 00:12:14,970 letting phi range from 0 up to pi halves, the width of the wedge doesn't grow linearly, -194 +195 00:12:14,970 --> 00:12:17,460 instead it grows according to a sine curve. -195 +196 00:12:18,480 --> 00:12:20,873 And so when we're unwrapping all of these wedges, -196 +197 00:12:20,873 --> 00:12:23,745 if we want those widths to be preserved, they should end up -197 +198 00:12:23,745 --> 00:12:27,240 a little bit chubbier around the base, their side lengths are not linear. -198 +199 00:12:28,120 --> 00:12:31,933 What this means is when we tried to interlace all of the wedges from the northern -199 +200 00:12:31,933 --> 00:12:35,653 hemisphere with those from the southern, there's a meaningful amount of overlap -200 +201 00:12:35,653 --> 00:12:39,698 between those non-linear edges, and we can't wave our hands about a limiting argument, -201 +202 00:12:39,698 --> 00:12:43,140 this is an overlap that persists as you take finer and finer subdivisions. -202 +203 00:12:43,860 --> 00:12:47,498 And ultimately it's that overlap that accounts for the difference between -203 +204 00:12:47,498 --> 00:12:50,940 our false answer with a pi squared from the true answer that has 4 pi. -204 +205 00:12:51,860 --> 00:12:55,478 It reminds me of one of those rearrangement puzzles where you have a number of -205 +206 00:12:55,478 --> 00:12:59,280 pieces and just by moving them around you can seemingly create area out of nowhere. -206 +207 00:12:59,680 --> 00:13:03,133 For example, right now I've arranged all these pieces to form a triangle, -207 +208 00:13:03,133 --> 00:13:05,560 except it's missing two units of area in the middle. -208 +209 00:13:05,920 --> 00:13:09,648 Now I want you to focus on the vertices of that triangle, these white dots, -209 +210 00:13:09,648 --> 00:13:12,494 those don't move, I'm not pulling any trickery with that, -210 +211 00:13:12,494 --> 00:13:16,665 but I can rearrange all of the pieces back to how they originally were so that those -211 +212 00:13:16,665 --> 00:13:19,167 two units of area in the middle seem to disappear, -212 +213 00:13:19,167 --> 00:13:23,141 while the constituent parts remain the same, the triangle that they form remains -213 +214 00:13:23,141 --> 00:13:26,380 the same, and yet two units of area seem to appear out of nowhere. -214 +215 00:13:27,260 --> 00:13:29,238 If you've never seen this one before, by the way, -215 +216 00:13:29,238 --> 00:13:31,652 I highly encourage you to pause and try to think it through, -216 +217 00:13:31,652 --> 00:13:32,840 it's a very fun little puzzle. -217 +218 00:13:33,860 --> 00:13:38,936 The answer starts to reveal itself if we carefully draw the edges of this triangle and -218 +219 00:13:38,936 --> 00:13:43,838 zoom in close enough to see that our pieces don't actually fit inside the triangle, -219 +220 00:13:43,838 --> 00:13:48,914 they bulge out ever so slightly, or at least arranged like this they bulge out ever so -220 +221 00:13:48,914 --> 00:13:49,440 slightly. -221 +222 00:13:49,720 --> 00:13:53,474 When we rearrange them and we zoom back in we can see that they dent -222 +223 00:13:53,474 --> 00:13:57,228 inward ever so slightly, and that very subtle difference between the -223 +224 00:13:57,228 --> 00:14:01,200 bulge out and the dent inward accounts for all of the difference in area. -224 +225 00:14:01,660 --> 00:14:05,806 The slope of the edge of this blue triangle works out to be 5 divided by 2, -225 +226 00:14:05,806 --> 00:14:10,280 whereas the slope of the edge of this red triangle works out to be 7 divided by 3. -226 +227 00:14:10,680 --> 00:14:13,330 Those numbers are close enough to look similar as slope, -227 +228 00:14:13,330 --> 00:14:16,260 but they allow for this denting inward and the bulging outward. -228 +229 00:14:16,820 --> 00:14:19,737 You have to be wary of lines that are made to look straight when -229 +230 00:14:19,737 --> 00:14:22,880 you haven't had explicit confirmation that they actually are straight. -230 +231 00:14:24,560 --> 00:14:29,056 One quick added comment on the sphere, the fundamental issue here is that the geometry -231 +232 00:14:29,056 --> 00:14:33,140 of a curved surface is fundamentally different from the geometry of flat space. -232 +233 00:14:33,560 --> 00:14:36,000 The relevant search term here would be Gaussian curvature. -233 +234 00:14:36,500 --> 00:14:40,420 You can't flatten things out from a sphere without losing geometric information. -234 +235 00:14:41,380 --> 00:14:45,511 When you see limiting arguments that relate to little pieces on a sphere that -235 +236 00:14:45,511 --> 00:14:48,583 somehow get flattened out and are reasoned through there, -236 +237 00:14:48,583 --> 00:14:52,873 those only can work if the limiting pieces that you're talking about get smaller -237 +238 00:14:52,873 --> 00:14:53,880 in both directions. -238 +239 00:14:54,220 --> 00:14:58,420 It's only when you zoom in close to curved surface that it appears locally flat. -239 +240 00:14:59,200 --> 00:15:02,637 The issue with our orange wedge argument is that our pieces never got -240 +241 00:15:02,637 --> 00:15:06,320 exposed to that local flatness because they only got thin in one direction. -241 +242 00:15:06,580 --> 00:15:08,800 They maintain the curvature in that other direction. -242 +243 00:15:09,600 --> 00:15:12,423 Now on the topic of the subtlety of limiting arguments, -243 +244 00:15:12,423 --> 00:15:16,860 let's turn back to our limit of jagged curves that approaches the smooth circular curve. -244 +245 00:15:17,220 --> 00:15:20,555 As I said, the limiting curve really is a circle and the -245 +246 00:15:20,555 --> 00:15:24,360 limiting value for the length of your approximations really is 8. -246 +247 00:15:25,580 --> 00:15:29,479 Here, the basic issue is that there is no reason to expect that the limit of -247 +248 00:15:29,479 --> 00:15:33,581 the lengths of the curves is the same as the length of the limits of the curves, -248 +249 00:15:33,581 --> 00:15:37,380 and in fact this is a nice counter example to show why that's not the case. -249 +250 00:15:38,420 --> 00:15:42,098 The real point of this example is not the fear that anyone is ever -250 +251 00:15:42,098 --> 00:15:45,063 going to believe that it shows that pi is equal to 4, -251 +252 00:15:45,063 --> 00:15:49,016 instead it shows why care is required in other cases where people apply -252 +253 00:15:49,016 --> 00:15:50,060 limiting arguments. -253 +254 00:15:50,060 --> 00:15:52,920 For example, this happens all throughout calculus. -254 +255 00:15:53,180 --> 00:15:57,700 It is the heart of calculus, where say you want to know the area under a given curve. -255 +256 00:15:58,280 --> 00:16:02,507 The way we typically think about it is to approximate that with a set of rectangles, -256 +257 00:16:02,507 --> 00:16:05,740 because those are the things we know how to compute the areas of. -257 +258 00:16:05,880 --> 00:16:07,800 You just take the base times height in each case. -258 +259 00:16:08,080 --> 00:16:12,219 Now this is a very jagged approximation, but the thought, or I guess the hope, -259 +260 00:16:12,219 --> 00:16:16,777 is that as you take a finer and finer subdivision into thinner and thinner rectangles, -260 +261 00:16:16,777 --> 00:16:20,340 the sums of those areas approaches the thing we actually care about. -261 +262 00:16:20,760 --> 00:16:24,850 If you want to make it rigorous, you have to be explicit about the error between -262 +263 00:16:24,850 --> 00:16:28,940 these approximations and the true thing we care about, the area under this curve. -263 +264 00:16:29,780 --> 00:16:32,877 For example, you might start your argument by saying that that -264 +265 00:16:32,877 --> 00:16:36,220 error has to be strictly less than the area of these red rectangles. -265 +266 00:16:36,660 --> 00:16:39,443 Essentially, the deviation between the curve and our -266 +267 00:16:39,443 --> 00:16:42,700 approximating rectangles sits strictly inside that red region. -267 +268 00:16:43,180 --> 00:16:47,260 And then what you would want to argue is that in this limiting process, -268 +269 00:16:47,260 --> 00:16:51,340 the cumulative area of all of those red rectangles has to approach zero. -269 +270 00:16:57,260 --> 00:17:00,800 Now as to the final example, our proof that all triangles are isosceles, -270 +271 00:17:00,800 --> 00:17:04,388 let me show you what it looks like if I'm a little bit more careful about -271 +272 00:17:04,388 --> 00:17:07,880 actually constructing the angle bisector rather than just eyeballing it. -272 +273 00:17:08,220 --> 00:17:12,579 When I do that, the relevant intersection point actually sits outside of the triangle. -273 +274 00:17:13,140 --> 00:17:17,033 And then from there, if I go through everything that we did in the original argument, -274 +275 00:17:17,033 --> 00:17:19,522 drawing the relevant perpendicular lines, all of that, -275 +276 00:17:19,522 --> 00:17:22,420 every triangle that I claimed was congruent really is congruent. -276 +277 00:17:22,540 --> 00:17:25,331 All of those were genuinely true, and the corresponding lengths of -277 +278 00:17:25,331 --> 00:17:28,040 those triangles that I claimed were the same really are the same. -278 +279 00:17:28,680 --> 00:17:32,839 The one place where the proof breaks down is at the very end, -279 +280 00:17:32,839 --> 00:17:37,200 when I said that the full side length AC was equal to AE plus EC. -280 +281 00:17:37,720 --> 00:17:43,460 That was only true under the hidden assumption that that point E sat in between them. -281 +282 00:17:43,720 --> 00:17:48,120 But in reality, for many triangles, that point would sit outside of those two. -282 +283 00:17:48,300 --> 00:17:49,580 It's pretty subtle, isn't it? -283 +284 00:17:51,360 --> 00:17:55,124 The point in all of this is that while visual intuition is great, -284 +285 00:17:55,124 --> 00:17:59,688 and visual proofs often give you a nice way of elucidating what's going on with -285 +286 00:17:59,688 --> 00:18:04,479 otherwise opaque rigor, visual arguments and snazzy diagrams will never obviate the -286 +287 00:18:04,479 --> 00:18:06,020 need for critical thinking. -287 +288 00:18:06,440 --> 00:18:10,760 In math, you cannot escape the need to look out for hidden assumptions and edge cases. -288 +289 00:18:32,140 --> 00:18:37,980 Thank you. diff --git a/2022/visual-proofs/english/sentence_timings.json b/2022/visual-proofs/english/sentence_timings.json index 52efdb0cf..7ced4c266 100644 --- a/2022/visual-proofs/english/sentence_timings.json +++ b/2022/visual-proofs/english/sentence_timings.json @@ -235,7 +235,7 @@ 453.6 ], [ - "Next I'll draw a line from p down to b, and then from p down to c, and I claim that this triangle here is congruent to its reflection across that bisector.", + "Next I'll draw a line from p down to b, and then from p down to c, and I claim that this triangle here is congruent to its reflection across that perpendicular bisector.", 454.06, 466.26 ], diff --git a/2022/visual-proofs/english/transcript.txt b/2022/visual-proofs/english/transcript.txt index faecec399..cae28625d 100644 --- a/2022/visual-proofs/english/transcript.txt +++ b/2022/visual-proofs/english/transcript.txt @@ -45,7 +45,7 @@ Essentially this follows from symmetry across that angle bisector. More specifically we can say they share a side length, and then they both have an angle alpha, and both have an angle 90 degrees. So it follows by the side angle angle congruence relation. Maybe my drawing is a little bit sloppy, but the logic helps us see that they do have to be the same. -Next I'll draw a line from p down to b, and then from p down to c, and I claim that this triangle here is congruent to its reflection across that bisector. +Next I'll draw a line from p down to b, and then from p down to c, and I claim that this triangle here is congruent to its reflection across that perpendicular bisector. Again the symmetry maybe helps make this clear, but more rigorously they both have the same base, they both have a 90 degree angle, and they both have the same height, so it follows by the side angle side relation. So based on that first pair of triangles I'm going to mark this side length here as being the same as this side length here, marking them with double tick marks. And based on the second triangle relation I'll mark this side length here as the same as this line over here, marking them with triple tick marks. diff --git a/2022/visual-proofs/french/sentence_translations.json b/2022/visual-proofs/french/sentence_translations.json index 90b23958f..8cece969f 100644 --- a/2022/visual-proofs/french/sentence_translations.json +++ b/2022/visual-proofs/french/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "En mathématiques, vous ne pouvez pas échapper à la nécessité de rechercher les hypothèses cachées et les cas extrêmes. ", "model": "google_nmt", "n_reviews": 1, diff --git a/2022/visual-proofs/german/sentence_translations.json b/2022/visual-proofs/german/sentence_translations.json index 092afb93a..13fc79874 100644 --- a/2022/visual-proofs/german/sentence_translations.json +++ b/2022/visual-proofs/german/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "In der Mathematik kommt man nicht umhin, nach versteckten Annahmen und Randfällen Ausschau zu halten. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/hebrew/sentence_translations.json b/2022/visual-proofs/hebrew/sentence_translations.json index 572504ec7..a07aa323e 100644 --- a/2022/visual-proofs/hebrew/sentence_translations.json +++ b/2022/visual-proofs/hebrew/sentence_translations.json @@ -1022,7 +1022,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases.", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you.", "translatedText": "במתמטיקה, אתה לא יכול לברוח מהצורך לחפש הנחות נסתרות ומקרי קצה.", "n_reviews": 0, "start": 1086.44, diff --git a/2022/visual-proofs/hindi/sentence_translations.json b/2022/visual-proofs/hindi/sentence_translations.json index 1751e8d1c..3ea6268be 100644 --- a/2022/visual-proofs/hindi/sentence_translations.json +++ b/2022/visual-proofs/hindi/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases.", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you.", "translatedText": "गणित में, आप छिपी हुई धारणाओं और किनारे के मामलों पर ध्यान देने की आवश्यकता से बच नहीं सकते।", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/hungarian/sentence_translations.json b/2022/visual-proofs/hungarian/sentence_translations.json index 8919ab3db..a21fb277e 100644 --- a/2022/visual-proofs/hungarian/sentence_translations.json +++ b/2022/visual-proofs/hungarian/sentence_translations.json @@ -376,7 +376,7 @@ "end": 453.6 }, { - "input": "Next I'll draw a line from p down to b, and then from p down to c, and I claim that this triangle here is congruent to its reflection across that bisector.", + "input": "Next I'll draw a line from p down to b, and then from p down to c, and I claim that this triangle here is congruent to its reflection across that perpendicular bisector.", "translatedText": "Ezután húzok egy egyenest p-től lefelé b-ig, majd p-től lefelé c-ig, és azt állítom, hogy ez a háromszög itt egybeesik a felezőn keresztüli tükörképével.", "model": "DeepL", "n_reviews": 0, diff --git a/2022/visual-proofs/indonesian/sentence_translations.json b/2022/visual-proofs/indonesian/sentence_translations.json index 6ad8b8460..42a7d62ac 100644 --- a/2022/visual-proofs/indonesian/sentence_translations.json +++ b/2022/visual-proofs/indonesian/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "Dalam matematika, Anda tidak bisa lepas dari kebutuhan untuk mencari asumsi tersembunyi dan kasus tepi. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/italian/sentence_translations.json b/2022/visual-proofs/italian/sentence_translations.json index 272d93cb0..197fc8cd5 100644 --- a/2022/visual-proofs/italian/sentence_translations.json +++ b/2022/visual-proofs/italian/sentence_translations.json @@ -1022,7 +1022,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases.", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you.", "translatedText": "In matematica, non puoi sfuggire alla necessità di cercare ipotesi nascoste e casi limite.", "n_reviews": 0, "start": 1086.44, diff --git a/2022/visual-proofs/japanese/sentence_translations.json b/2022/visual-proofs/japanese/sentence_translations.json index ca9b0778c..719e70e0b 100644 --- a/2022/visual-proofs/japanese/sentence_translations.json +++ b/2022/visual-proofs/japanese/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "数学では、隠れた仮 定や特殊なケースに注意する必要性から逃れることはできません。", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/korean/sentence_translations.json b/2022/visual-proofs/korean/sentence_translations.json index a9fabe70c..22a59f452 100644 --- a/2022/visual-proofs/korean/sentence_translations.json +++ b/2022/visual-proofs/korean/sentence_translations.json @@ -423,7 +423,7 @@ "end": 453.6 }, { - "input": "Next I'll draw a line from p down to b, and then from p down to c, and I claim that this triangle here is congruent to its reflection across that bisector.", + "input": "Next I'll draw a line from p down to b, and then from p down to c, and I claim that this triangle here is congruent to its reflection across that perpendicular bisector.", "translatedText": "다음으로 p에서 b까지, 그리고 p에서 c까지 선을 그어 이 삼각형이 해당 이등분선을 가로지르는 반사와 일치한다고 주장합니다.", "model": "DeepL", "from_community_srt": "다음으로 𝑷에서 𝑩로 선을 그리고 𝑷에서 𝑪로 선을 그립니다. 그리고 저는 여기 이 삼각형은 수직 이등분선을 기준으로 대칭인 삼각형 𝑪𝑷𝑫와 합동이라고 주장합니다.", diff --git a/2022/visual-proofs/marathi/sentence_translations.json b/2022/visual-proofs/marathi/sentence_translations.json index 03592bba3..2cc6b0b1e 100644 --- a/2022/visual-proofs/marathi/sentence_translations.json +++ b/2022/visual-proofs/marathi/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "गणितात, आपण लपविलेल्या गृहितक आणि धार प्रकरणे शोधण्याची गरज सोडू शकत नाही. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/persian/sentence_translations.json b/2022/visual-proofs/persian/sentence_translations.json index da5f6999a..15ac8bd45 100644 --- a/2022/visual-proofs/persian/sentence_translations.json +++ b/2022/visual-proofs/persian/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "در ریاضیات، نمی‌توانید از نیاز به جستجوی مفروضات پنهان و موارد لبه فرار کنید. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/portuguese/sentence_translations.json b/2022/visual-proofs/portuguese/sentence_translations.json index aa10f459d..d4b7ce007 100644 --- a/2022/visual-proofs/portuguese/sentence_translations.json +++ b/2022/visual-proofs/portuguese/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "Em matemática, você não pode escapar da necessidade de procurar suposições ocultas e casos extremos. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/russian/sentence_translations.json b/2022/visual-proofs/russian/sentence_translations.json index ff28ec95f..9a521bb9f 100644 --- a/2022/visual-proofs/russian/sentence_translations.json +++ b/2022/visual-proofs/russian/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "В математике вы не можете избежать необходимости искать скрытые предположения и крайние случаи. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/spanish/sentence_translations.json b/2022/visual-proofs/spanish/sentence_translations.json index 17e398777..c9632f9d5 100644 --- a/2022/visual-proofs/spanish/sentence_translations.json +++ b/2022/visual-proofs/spanish/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "En matemáticas, no se puede escapar de la necesidad de buscar suposiciones ocultas y casos extremos. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/tamil/sentence_translations.json b/2022/visual-proofs/tamil/sentence_translations.json index 0d4ce3d22..3b8f1274e 100644 --- a/2022/visual-proofs/tamil/sentence_translations.json +++ b/2022/visual-proofs/tamil/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "கணிதத்தில், மறைக்கப்பட்ட அனுமானங்கள் மற்றும் விளிம்பு நிகழ்வுகளை கவனிக்க வேண்டிய அவசியத்திலிருந்து நீங்கள் தப்பிக்க முடியாது. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/telugu/sentence_translations.json b/2022/visual-proofs/telugu/sentence_translations.json index efa3ec836..cda1c0b1d 100644 --- a/2022/visual-proofs/telugu/sentence_translations.json +++ b/2022/visual-proofs/telugu/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "గణితంలో, మీరు దాచిన అంచనాలు మరియు అంచు కేసుల కోసం చూడవలసిన అవసరాన్ని తప్పించుకోలేరు. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/thai/sentence_translations.json b/2022/visual-proofs/thai/sentence_translations.json index a4a5ad435..09dfe226c 100644 --- a/2022/visual-proofs/thai/sentence_translations.json +++ b/2022/visual-proofs/thai/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/turkish/sentence_translations.json b/2022/visual-proofs/turkish/sentence_translations.json index 8e0e06992..294a7f2e3 100644 --- a/2022/visual-proofs/turkish/sentence_translations.json +++ b/2022/visual-proofs/turkish/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "Matematikte gizli varsayımlara ve uç durumlara dikkat etme ihtiyacından kaçamazsınız. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/ukrainian/sentence_translations.json b/2022/visual-proofs/ukrainian/sentence_translations.json index 506935bef..c02ff571f 100644 --- a/2022/visual-proofs/ukrainian/sentence_translations.json +++ b/2022/visual-proofs/ukrainian/sentence_translations.json @@ -1022,7 +1022,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases.", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you.", "translatedText": "У математиці ви не можете уникнути необхідності шукати приховані припущення та крайні випадки.", "n_reviews": 0, "start": 1086.44, diff --git a/2022/visual-proofs/urdu/sentence_translations.json b/2022/visual-proofs/urdu/sentence_translations.json index c3a8fba13..d3a50a511 100644 --- a/2022/visual-proofs/urdu/sentence_translations.json +++ b/2022/visual-proofs/urdu/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "ریاضی میں، آپ پوشیدہ مفروضوں اور کنارے کے معاملات کو تلاش کرنے کی ضرورت سے بچ نہیں سکتے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/visual-proofs/vietnamese/sentence_translations.json b/2022/visual-proofs/vietnamese/sentence_translations.json index ba80389b6..9358bdd1a 100644 --- a/2022/visual-proofs/vietnamese/sentence_translations.json +++ b/2022/visual-proofs/vietnamese/sentence_translations.json @@ -1168,7 +1168,7 @@ "end": 1086.02 }, { - "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. ", + "input": "In math, you cannot escape the need to look out for hidden assumptions and edge cases. Thank you. ", "translatedText": "Trong toán học, bạn không thể thoát khỏi nhu cầu tìm kiếm các giả định ẩn và các trường hợp khó khăn. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/arabic/sentence_translations.json b/2022/wordle-2/arabic/sentence_translations.json index b157e4dd1..a7d1c0ba2 100644 --- a/2022/wordle-2/arabic/sentence_translations.json +++ b/2022/wordle-2/arabic/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "ومن خلال القيام بذلك، ورؤية كيفية أدائهم فعليًا، فإن النتيجة التي تنتهي بشكل هامشي جدًا بأفضل نتيجة ممكنة هي Salé، وهي تهجئة بديلة لـ Sale، وهي خوذة خفيفة من العصور الوسطى. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/bengali/sentence_translations.json b/2022/wordle-2/bengali/sentence_translations.json index 239b92c35..7e150f736 100644 --- a/2022/wordle-2/bengali/sentence_translations.json +++ b/2022/wordle-2/bengali/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "এবং এটি করার মাধ্যমে, তারা আসলে কীভাবে পারফর্ম করে তা দেখে, যেটি সম্ভাব্য সর্বোত্তম স্কোরের সাথে খুব সামান্যভাবে শেষ হয় সেটি হল Salé, যেটি Salé-এর জন্য একটি বিকল্প বানান, যা একটি হালকা মধ্যযুগীয় শিরস্ত্রাণ।", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/chinese/sentence_translations.json b/2022/wordle-2/chinese/sentence_translations.json index 935d4a0aa..d8b7527b7 100644 --- a/2022/wordle-2/chinese/sentence_translations.json +++ b/2022/wordle-2/chinese/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "通过这样做，看看它们的实际表现，最终以微弱优势获得最高分的结 果是 Salé，它是 Salé 的另一种拼写，Salé 是 一种轻型中世纪头盔。", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/english/captions.srt b/2022/wordle-2/english/captions.srt index 1c56ba7da..f34f49d96 100644 --- a/2022/wordle-2/english/captions.srt +++ b/2022/wordle-2/english/captions.srt @@ -679,14 +679,10 @@ For my taste at least, the joy of writing algorithms to try to play games actually has very little bearing on how I like to play those games as a human. 171 -00:10:11,300 --> 00:10:14,018 +00:10:11,300 --> 00:10:15,972 The point of writing algorithms for all this is not to affect 172 -00:10:14,018 --> 00:10:16,780 +00:10:15,972 --> 00:10:20,720 the way that we play the game, it's still just a fun word game. -173 -00:10:17,100 --> 00:10:20,720 -It's to hone in our muscles for writing algorithms in more meaningful contexts elsewhere. - diff --git a/2022/wordle-2/english/sentence_timings.json b/2022/wordle-2/english/sentence_timings.json index 5a34bbdc3..d4f5e0cec 100644 --- a/2022/wordle-2/english/sentence_timings.json +++ b/2022/wordle-2/english/sentence_timings.json @@ -332,11 +332,6 @@ [ "The point of writing algorithms for all this is not to affect the way that we play the game, it's still just a fun word game.", 611.3, - 616.78 - ], - [ - "It's to hone in our muscles for writing algorithms in more meaningful contexts elsewhere.", - 617.1, 620.72 ] ] \ No newline at end of file diff --git a/2022/wordle-2/french/sentence_translations.json b/2022/wordle-2/french/sentence_translations.json index f666a2dec..937a13981 100644 --- a/2022/wordle-2/french/sentence_translations.json +++ b/2022/wordle-2/french/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "Et en faisant cela, en voyant comment ils se comportent réellement, celui qui obtient très marginalement le meilleur score possible s'avère être Salé, qui est une orthographe alternative pour Salé, qui est un casque médiéval léger. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/german/sentence_translations.json b/2022/wordle-2/german/sentence_translations.json index 9d7e3869c..99d8b319b 100644 --- a/2022/wordle-2/german/sentence_translations.json +++ b/2022/wordle-2/german/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "Und wenn man so vorgeht, stellt sich heraus, dass Salé, eine alternative Schreibweise für Salé, was ein leichter mittelalterlicher Helm ist, derjenige ist, der ganz knapp die bestmögliche Punktzahl erzielt. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/hindi/sentence_translations.json b/2022/wordle-2/hindi/sentence_translations.json index 718d57fee..5ee197fd2 100644 --- a/2022/wordle-2/hindi/sentence_translations.json +++ b/2022/wordle-2/hindi/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "और ऐसा करने से, यह देखते हुए कि वे वास्तव में कैसा प्रदर्शन करते हैं, जो सर्वोत्तम संभव स्कोर के साथ बहुत मामूली रूप से समाप्त होता है, वह साले बन जाता है, जो सैले के लिए एक वैकल्पिक वर्तनी है, जो एक हल्का मध्ययुगीन हेलमेट है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/indonesian/sentence_translations.json b/2022/wordle-2/indonesian/sentence_translations.json index 5ee924fc5..2d9a5f846 100644 --- a/2022/wordle-2/indonesian/sentence_translations.json +++ b/2022/wordle-2/indonesian/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "Dan dengan melakukan ini, melihat bagaimana kinerja mereka sebenarnya, salah satu yang mendapatkan skor terbaik ternyata adalah Salé, yang merupakan ejaan alternatif untuk Salé, yang merupakan helm abad pertengahan yang ringan. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/japanese/sentence_translations.json b/2022/wordle-2/japanese/sentence_translations.json index 826eb5963..6e71adf8a 100644 --- a/2022/wordle-2/japanese/sentence_translations.json +++ b/2022/wordle-2/japanese/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "そして、これを行うことで、彼らが実際にどのようにパフォーマンスするかを見て、可能な限り最高の スコアで非常に僅差で終了したのは、中世の軽いヘルメットである Salé の別の綴りである S alé であることがわかります。わかりました。", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/korean/sentence_translations.json b/2022/wordle-2/korean/sentence_translations.json index 733aba17b..40f1c98aa 100644 --- a/2022/wordle-2/korean/sentence_translations.json +++ b/2022/wordle-2/korean/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "그리고 이렇게 함으로써, 그들이 실제로 어떻게 수행하는지를 보면, 가능한 최고의 점수로 아주 미미하게 끝나는 것은 Salé로 밝혀졌습니다. 이는 가벼운 중세 헬멧인 Salé의 대체 철자인 Salé입니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/marathi/sentence_translations.json b/2022/wordle-2/marathi/sentence_translations.json index 902afc3cf..19fe9f8d0 100644 --- a/2022/wordle-2/marathi/sentence_translations.json +++ b/2022/wordle-2/marathi/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "आणि असे केल्याने, ते प्रत्यक्षात कसे परफॉर्म करतात हे पाहता, सर्वोत्तम संभाव्य स्कोअरसह अगदी किरकोळपणे समाप्त होणारे एक Salé आहे, जे Salé चे पर्यायी शब्दलेखन आहे, जे हलके मध्ययुगीन हेल्मेट आहे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/persian/sentence_translations.json b/2022/wordle-2/persian/sentence_translations.json index 96267b304..19db9e402 100644 --- a/2022/wordle-2/persian/sentence_translations.json +++ b/2022/wordle-2/persian/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "و با انجام این کار، با دیدن عملکرد واقعی آنها، یکی که در نهایت با بهترین امتیاز ممکن به پایان می رسد Salé است، که املای جایگزین Salé است، که یک کلاه ایمنی سبک قرون وسطایی است. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/portuguese/sentence_translations.json b/2022/wordle-2/portuguese/sentence_translations.json index de4599a43..64149540c 100644 --- a/2022/wordle-2/portuguese/sentence_translations.json +++ b/2022/wordle-2/portuguese/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "E ao fazer isso, vendo como eles realmente atuam, aquele que acaba marginalmente com a melhor pontuação possível acaba sendo Salé, que é uma grafia alternativa para Salé, que é um capacete medieval leve. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/russian/sentence_translations.json b/2022/wordle-2/russian/sentence_translations.json index 605d15139..03d4c6cc3 100644 --- a/2022/wordle-2/russian/sentence_translations.json +++ b/2022/wordle-2/russian/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "И, сделав это, наблюдая, как они на самом деле работают, тот, который в конечном итоге набирает максимально возможный балл, оказывается Сале, что является альтернативным написанием Сале, который представляет собой легкий средневековый шлем. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/spanish/sentence_translations.json b/2022/wordle-2/spanish/sentence_translations.json index a8c3a7f47..06b5bdaaa 100644 --- a/2022/wordle-2/spanish/sentence_translations.json +++ b/2022/wordle-2/spanish/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "Y al hacer esto, viendo cómo se desempeñan realmente, el que termina muy marginalmente con la mejor puntuación posible resulta ser Salé, que es una ortografía alternativa de Salé, que es un casco medieval ligero. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/tamil/sentence_translations.json b/2022/wordle-2/tamil/sentence_translations.json index 59fbd90e5..3cab62ccd 100644 --- a/2022/wordle-2/tamil/sentence_translations.json +++ b/2022/wordle-2/tamil/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "இதைச் செய்வதன் மூலம், அவர்கள் உண்மையில் எவ்வாறு செயல்படுகிறார்கள் என்பதைப் பார்க்கும்போது, மிகச் சிறந்த மதிப்பெண்ணுடன் மிகச் சிறிய அளவில் முடிவடைவது சாலே என்று மாறிவிடும், இது சாலேக்கு ஒரு மாற்று எழுத்துப்பிழையாகும், இது லேசான இடைக்கால ஹெல்மெட் ஆகும். ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/telugu/sentence_translations.json b/2022/wordle-2/telugu/sentence_translations.json index 8536fae17..05ef3d4fe 100644 --- a/2022/wordle-2/telugu/sentence_translations.json +++ b/2022/wordle-2/telugu/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "మరియు ఇలా చేయడం ద్వారా, వారు వాస్తవానికి ఎలా పని చేస్తారో చూడటం ద్వారా, సాధ్యమైనంత ఉత్తమమైన స్కోర్‌తో చాలా స్వల్పంగా ముగిసేది సాలేగా మారుతుంది, ఇది సాలేకు ప్రత్యామ్నాయ స్పెల్లింగ్, ఇది తేలికపాటి మధ్యయుగ హెల్మెట్. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/thai/sentence_translations.json b/2022/wordle-2/thai/sentence_translations.json index d5a261762..8f1ba9a5c 100644 --- a/2022/wordle-2/thai/sentence_translations.json +++ b/2022/wordle-2/thai/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/turkish/sentence_translations.json b/2022/wordle-2/turkish/sentence_translations.json index 73326a3a2..25ecc4e5a 100644 --- a/2022/wordle-2/turkish/sentence_translations.json +++ b/2022/wordle-2/turkish/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "Ve bunu yaparak, gerçekte nasıl performans gösterdiklerini görerek, çok marjinal olarak mümkün olan en iyi puanı alan kişinin, hafif bir ortaçağ kaskı olan Salé'nin alternatif yazılışı olan Salé olduğu ortaya çıkıyor. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/ukrainian/sentence_translations.json b/2022/wordle-2/ukrainian/sentence_translations.json index e66cdf714..e4e8c0a5f 100644 --- a/2022/wordle-2/ukrainian/sentence_translations.json +++ b/2022/wordle-2/ukrainian/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "І роблячи це, дивлячись на те, як вони справді працюють, той, хто в кінцевому підсумку отримав найкращий бал, виявляється Salé, що є альтернативним варіантом написання Salé, який є легким середньовічним шоломом. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/urdu/sentence_translations.json b/2022/wordle-2/urdu/sentence_translations.json index b12b68f26..3331a7698 100644 --- a/2022/wordle-2/urdu/sentence_translations.json +++ b/2022/wordle-2/urdu/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "اور ایسا کرنے سے، یہ دیکھ کر کہ وہ اصل میں کس طرح پرفارم کرتے ہیں، وہ جو بہترین ممکنہ سکور کے ساتھ بہت معمولی طور پر ختم ہوتا ہے وہ Salé نکلتا ہے، جو Salé کے لیے ایک متبادل ہجے ہے، جو ایک ہلکا قرون وسطی کا ہیلمیٹ ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle-2/vietnamese/sentence_translations.json b/2022/wordle-2/vietnamese/sentence_translations.json index 8b691e8d7..f479bd0e9 100644 --- a/2022/wordle-2/vietnamese/sentence_translations.json +++ b/2022/wordle-2/vietnamese/sentence_translations.json @@ -416,7 +416,7 @@ "end": 464.62 }, { - "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is an alternate spelling for Salé, which is a light medieval helmet. ", + "input": "And by doing this, seeing how they actually perform, the one that ends up very marginally with the best possible score turns out to be Salé, which is, let's see, Salé, an alternate spelling for Salé, which is a light medieval helmet. ", "translatedText": "Và bằng cách làm điều này, xem cách họ thực sự biểu diễn, người có số điểm cao nhất có thể lại là Salé, một cách viết khác của Salé, một chiếc mũ bảo hiểm nhẹ thời Trung cổ. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/arabic/sentence_translations.json b/2022/wordle/arabic/sentence_translations.json index 4cdde29da..c28eaea7e 100644 --- a/2022/wordle/arabic/sentence_translations.json +++ b/2022/wordle/arabic/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", + "input": "second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", "translatedText": "والآن قد ترغب في التوقف مؤقتًا وتسأل نفسك، ما هي صيغة المعلومات لعدد البتات من حيث احتمال حدوث ذلك؟ ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", "translatedText": "ما نقوله هنا هو أنه عندما تأخذ نصفًا لعدد البتات، فهذا هو نفس الاحتمال، وهو نفس قول أن اثنين أس عدد البتات يساوي واحدًا على الاحتمال، وهو ما يُعاد ترتيبها أيضًا لقول أن المعلومات هي سجل واحد للأساس اثنين مقسومًا على الاحتمال. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", "translatedText": "تم تطوير نظرية المعلومات على يد كلود شانون، الذي كان يعمل في مختبرات بيل في الأربعينيات، لكنه كان يتحدث عن بعض أفكاره التي لم تُنشر بعد مع جون فون نيومان، الذي كان هذا العملاق الفكري في ذلك الوقت، بارزًا جدًا في الرياضيات والفيزياء وبدايات ما أصبح علوم الكمبيوتر. ", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", "translatedText": "إنه يمر بجميع التخمينات المحتملة التي يمكن أن تكون لديك، كل الكلمات البالغ عددها 13000 كلمة، ويحسب الإنتروبيا لكل واحدة منها، أو بشكل أكثر تحديدًا، إنتروبيا التوزيع عبر جميع الأنماط التي قد تراها، لكل واحد، ويختار الأعلى، نظرًا لأن ذلك الذي من المحتمل أن يقلل مساحة الاحتمالات الخاصة بك قدر الإمكان. ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind. ", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind. ", "translatedText": "لأكون صادقًا، الطريقة التي فعلت بها ذلك كانت مجرد لعق إصبعي ووضعه في مهب الريح. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/bengali/sentence_translations.json b/2022/wordle/bengali/sentence_translations.json index f5dfb4eb8..21c468269 100644 --- a/2022/wordle/bengali/sentence_translations.json +++ b/2022/wordle/bengali/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", + "input": "second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", "translatedText": "তাই এখন আপনি বিরতি দিতে এবং নিজেকে জিজ্ঞাসা করতে চাইতে পারেন, একটি ঘটনার সম্ভাব্যতার পরিপ্রেক্ষিতে বিটের সংখ্যার জন্য তথ্যের সূত্র কী? ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", "translatedText": "আমরা এখানে যা বলছি তা হল যে আপনি যখন বিট সংখ্যার এক অর্ধেক নিয়ে যান, তখন এটি সম্ভাব্যতার মতো একই জিনিস, যা বিটের সংখ্যার শক্তিকে দুটি বলার মত একই জিনিস সম্ভাবনার উপরে এক, যা সম্ভাব্যতা দ্বারা বিভক্ত একটি এর মধ্যে দুটি লগ বেস হল তথ্যটি বলার জন্য আরও পুনর্বিন্যাস করে।", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", "translatedText": "তথ্য তত্ত্বটি ক্লদ শ্যানন দ্বারা তৈরি করা হয়েছিল, যিনি 1940-এর দশকে বেল ল্যাবসে কর্মরত ছিলেন, কিন্তু তিনি জন ভন নিউম্যানের সাথে তার এখনও প্রকাশিত কিছু ধারণা সম্পর্কে কথা বলছিলেন, যিনি ছিলেন সেই সময়ের এই বুদ্ধিজীবী দৈত্য, খুব বিশিষ্ট গণিত এবং পদার্থবিদ্যা এবং কম্পিউটার বিজ্ঞান হয়ে উঠছে কি শুরু. ", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", "translatedText": "এটি আপনার সম্ভাব্য সমস্ত অনুমানগুলির মধ্য দিয়ে যায়, সমস্ত 13,000 শব্দ, প্রতিটির জন্য এনট্রপি গণনা করে, বা আরও নির্দিষ্টভাবে, প্রতিটির জন্য আপনি দেখতে পারেন এমন সমস্ত প্যাটার্ন জুড়ে বিতরণের এনট্রপি গণনা করে এবং সর্বোচ্চ বাছাই করে, যেহেতু এটি যেটি আপনার সম্ভাবনার স্থান যতটা সম্ভব কমিয়ে দিতে পারে।", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind. ", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind. ", "translatedText": "সত্যি কথা বলতে, আমি যেভাবে এটি করেছি তা কেবল আমার আঙুল চাটতে এবং বাতাসে আটকে রেখেছিল।", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/chinese/sentence_translations.json b/2022/wordle/chinese/sentence_translations.json index 6f3abf183..4ee1f9a76 100644 --- a/2022/wordle/chinese/sentence_translations.json +++ b/2022/wordle/chinese/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", + "input": "second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", "translatedText": "所以现在您可能想停下来问自己，就发生 概率而言，比特数的信息公式是什么？", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", "translatedText": "我们在这里所说的是，当你取位数的二分之一 时，这与概率是一样的，这与说 2 的位数 次方等于概率的 1 是一样的，重新排列进 一步表示该信息是一的对数基数除以概率。", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", "translatedText": "信息论是由 20 世纪 40 年代在贝尔实验室工作的克劳德·香农 (C laude Shannon) 提出的，但他正在与约翰·冯·诺依曼 ( John von Neumann) 谈论他尚未发表的一些想法，约翰· 冯·诺依曼是当时非常杰出的知识巨人。数学和物理以及计算机科学的开端。", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", "translatedText": "它会遍历所有可能的猜测，即所有 13,000 个单 词，计算每个单词的熵，或者更具体地说，计算您可能 看到的所有模式中每个单词的分布熵，并选择最高的，因 为这是一个可能会尽可能地削减你的可能性空间的人。", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind. ", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind. ", "translatedText": "说实话，我的做法就是舔手指然后把它插到风里。", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/french/sentence_translations.json b/2022/wordle/french/sentence_translations.json index c8e064747..c35645ebb 100644 --- a/2022/wordle/french/sentence_translations.json +++ b/2022/wordle/french/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", + "input": "second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", "translatedText": "Alors maintenant, vous voudrez peut-être faire une pause et vous demander quelle est la formule pour obtenir des informations sur le nombre de bits en termes de probabilité d'occurrence ? ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", "translatedText": "Ce que nous disons ici, c'est que lorsque vous prenez la moitié du nombre de bits, cela équivaut à la même chose que la probabilité, ce qui revient à dire que deux puissance du nombre de bits est un sur la probabilité, ce qui se réorganise en disant que l'information est la base logarithmique deux de un divisée par la probabilité. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", "translatedText": "La théorie de l'information a été développée par Claude Shannon, qui travaillait aux Bell Labs dans les années 1940, mais il discutait de certaines de ses idées encore non publiées avec John von Neumann, qui était ce géant intellectuel de l'époque, très en vue. en mathématiques et en physique et les débuts de ce qui allait devenir l'informatique. ", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", "translatedText": "Il passe en revue toutes les suppositions possibles que vous pourriez avoir, les 13 000 mots, calcule l'entropie pour chacun d'entre eux, ou plus précisément, l'entropie de la distribution à travers tous les modèles que vous pourriez voir, pour chacun d'entre eux, et choisit le plus élevé, puisque c'est celui qui est susceptible de réduire au maximum votre espace des possibles. ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind. ", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind. ", "translatedText": "Pour être honnête, j’ai simplement fait cela en me léchant le doigt et en le mettant face au vent. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/german/sentence_translations.json b/2022/wordle/german/sentence_translations.json index 50443ee41..7be31c3e6 100644 --- a/2022/wordle/german/sentence_translations.json +++ b/2022/wordle/german/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", + "input": "y second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", "translatedText": "Jetzt möchten Sie vielleicht innehalten und sich fragen: Wie lautet die Formel für Informationen über die Anzahl der Bits im Hinblick auf die Wahrscheinlichkeit eines Auftretens?", "model": "google_nmt", "n_reviews": 1, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", "translatedText": "Was wir hier sagen ist, dass wenn man die Hälfte der Anzahl der Bits hochnimmt, das dasselbe ist wie die Wahrscheinlichkeit, was dasselbe ist, als würde man sagen, dass zwei hoch die Anzahl der Bits eins über der Wahrscheinlichkeit ist, was ordnet sich weiter um und sagt, dass die Informationen die logarithmische Basis zwei von eins dividiert durch die Wahrscheinlichkeit sind.", "model": "google_nmt", "n_reviews": 1, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", "translatedText": "Die Informationstheorie wurde von Claude Shannon entwickelt, der in den 1940er Jahren an den Bell Labs arbeitete, aber er sprach über einige seiner noch nicht veröffentlichten Ideen mit John von Neumann, dem damals prominenten intellektuellen Giganten in Mathematik und Physik und die Anfänge dessen, was später zur Informatik wurde.", "model": "google_nmt", "n_reviews": 1, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", "translatedText": "Es geht alle möglichen Vermutungen durch, alle 13.000 Wörter, berechnet die Entropie für jedes einzelne, oder genauer gesagt, die Entropie der Verteilung über alle Muster, die Sie möglicherweise sehen, für jedes einzelne und wählt das höchste aus, denn das ist so diejenige, die Ihren Raum an Möglichkeiten wahrscheinlich so weit wie möglich einschränkt.", "model": "google_nmt", "n_reviews": 1, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind.", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind.", "translatedText": "Um ehrlich zu sein, habe ich das einfach so gemacht, indem ich meinen Finger abgeleckt und ihn in den Wind gehalten habe.", "model": "google_nmt", "n_reviews": 1, diff --git a/2022/wordle/hebrew/sentence_translations.json b/2022/wordle/hebrew/sentence_translations.json index 365504d6b..70e9f7429 100644 --- a/2022/wordle/hebrew/sentence_translations.json +++ b/2022/wordle/hebrew/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", + "input": "second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", "translatedText": "אז עכשיו אולי תרצו לעצור ולשאול את עצמכם, מהי הנוסחה למידע עבור מספר הביטים מבחינת ההסתברות להתרחשות? ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", "translatedText": "מה שאנחנו אומרים כאן הוא שכאשר אתה לוקח חצי אחד למספר הסיביות, זה אותו דבר כמו ההסתברות, שזה אותו דבר כמו לומר שניים בחזקת מספר הסיביות הוא אחד מעל ההסתברות, אשר מסדר מחדש בהמשך לומר שהמידע הוא בסיס היומן שניים מתוך אחד חלקי ההסתברות. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", "translatedText": "תורת המידע פותחה על ידי קלוד שאנון, שעבד ב-Bell Labs בשנות ה-40, אבל הוא דיבר על כמה מהרעיונות שלו שטרם פורסמו עם ג'ון פון נוימן, שהיה הענק האינטלקטואלי הזה של אותה תקופה, בולט מאוד. במתמטיקה ובפיזיקה ותחילתו של מה שהפך למדעי המחשב. ", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", "translatedText": "הוא עובר על כל הניחושים האפשריים שיכולים להיות לך, כל 13,000 המילים, מחשב את האנטרופיה עבור כל אחת, או ליתר דיוק, את האנטרופיה של ההתפלגות על פני כל הדפוסים שאתה עשוי לראות, עבור כל אחת, ובוחר את הגבוהה ביותר, מכיוון שזהו זה שצפוי לקצץ את מרחב האפשרויות שלך ככל האפשר. ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind. ", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind. ", "translatedText": "למען האמת, הדרך שעשיתי את זה הייתה פשוט ללקק את האצבע שלי ולהכניס אותה לרוח. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/hindi/sentence_translations.json b/2022/wordle/hindi/sentence_translations.json index 58412adab..da3fed121 100644 --- a/2022/wordle/hindi/sentence_translations.json +++ b/2022/wordle/hindi/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", + "input": "y second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", "translatedText": "तो अब आप रुककर अपने आप से पूछना चाहेंगे कि किसी घटना की संभावना के संदर्भ में बिट्स की संख्या की जानकारी का सूत्र क्या है?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", "translatedText": "हम यहां जो कह रहे हैं वह यह है कि जब आप बिट्स की संख्या का आधा हिस्सा लेते हैं, तो यह संभावना के समान ही है, जो कि बिट्स की संख्या की शक्ति को दो कहने की संभावना के ऊपर एक है, के समान है, जो यह कहते हुए आगे पुनर्व्यवस्थित करें कि जानकारी लॉग बेस दो में से एक है जिसे संभाव्यता से विभाजित किया गया है।", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", "translatedText": "सूचना सिद्धांत का विकास क्लाउड शैनन द्वारा किया गया था, जो 1940 के दशक में बेल लैब्स में काम कर रहे थे, लेकिन वह जॉन वॉन न्यूमैन के साथ अपने कुछ अभी तक प्रकाशित विचारों के बारे में बात कर रहे थे, जो उस समय के बहुत ही प्रमुख बौद्धिक दिग्गज थे। गणित और भौतिकी में और कंप्यूटर विज्ञान बनने की शुरुआत।", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", "translatedText": "यह आपके सभी 13,000 शब्दों के सभी संभावित अनुमानों से गुजरता है, प्रत्येक के लिए एन्ट्रापी की गणना करता है, या अधिक विशेष रूप से, आपके द्वारा देखे जा सकने वाले सभी पैटर्न में वितरण की एन्ट्रापी, प्रत्येक के लिए, और उच्चतम को चुनता है, क्योंकि यही है वह जो आपकी संभावनाओं के स्थान को यथासंभव कम कर देगा।", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind.", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind.", "translatedText": "सच कहूँ तो, जिस तरह से मैंने यह किया वह सिर्फ अपनी उंगली चाटना और उसे हवा में चिपकाना था।", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/indonesian/sentence_translations.json b/2022/wordle/indonesian/sentence_translations.json index 1c5288d20..6927822d5 100644 --- a/2022/wordle/indonesian/sentence_translations.json +++ b/2022/wordle/indonesian/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", + "input": "y second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", "translatedText": "Jadi sekarang Anda mungkin ingin berhenti sejenak dan bertanya pada diri sendiri, apa rumus informasi jumlah bit dalam kaitannya dengan probabilitas suatu kejadian?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", "translatedText": "Apa yang ingin kami katakan di sini adalah ketika Anda mengambil setengah dari jumlah bit, itu sama dengan probabilitas, yang sama dengan mengatakan dua pangkat dari jumlah bit adalah satu di atas probabilitas, yaitu menyusun ulang lebih jauh dengan mengatakan bahwa informasi tersebut adalah basis log dua dari satu dibagi dengan probabilitas.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", "translatedText": "Teori informasi dikembangkan oleh Claude Shannon, yang bekerja di Bell Labs pada tahun 1940-an, tetapi dia membicarakan beberapa idenya yang belum dipublikasikan dengan John von Neumann, yang merupakan raksasa intelektual pada saat itu, sangat terkemuka. dalam matematika dan fisika dan permulaan dari apa yang kemudian menjadi ilmu komputer.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", "translatedText": "Ia menelusuri semua kemungkinan tebakan yang Anda miliki, seluruh 13.000 kata, menghitung entropi untuk masing-masing kata, atau lebih khusus lagi, entropi distribusi di semua pola yang mungkin Anda lihat, untuk masing-masing kata, dan memilih yang tertinggi, karena itulah salah satu yang kemungkinan akan mengurangi ruang kemungkinan Anda sebanyak mungkin.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind.", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind.", "translatedText": "Sejujurnya, caraku melakukan ini hanyalah menjilat jariku dan menempelkannya ke angin.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/italian/sentence_translations.json b/2022/wordle/italian/sentence_translations.json index ca44d7b38..07a65921b 100644 --- a/2022/wordle/italian/sentence_translations.json +++ b/2022/wordle/italian/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", + "input": "y second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", "translatedText": "Quindi ora potresti voler fermarti e chiederti: qual è la formula per l'informazione sul numero di bit in termini di probabilità di un evento?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", "translatedText": "Quello che stiamo dicendo qui è che quando prendi la metà del numero di bit, è la stessa cosa della probabilità, che è la stessa cosa che dire due alla potenza del numero di bit è uno su probabilità, che riorganizza ulteriormente dicendo che l'informazione è il logaritmo in base due di uno diviso per la probabilità.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", "translatedText": "La teoria dell'informazione fu sviluppata da Claude Shannon, che lavorava ai Bell Labs negli anni '40, ma stava parlando di alcune delle sue idee ancora da pubblicare con John von Neumann, che era questo gigante intellettuale dell'epoca, molto importante in matematica e fisica e gli inizi di quella che stava diventando l'informatica.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", "translatedText": "Esamina tutte le possibili ipotesi che potresti avere, tutte le 13.000 parole, calcola l'entropia per ciascuna di esse o, più specificamente, l'entropia della distribuzione in tutti i modelli che potresti vedere, per ciascuno, e sceglie il più alto, poiché è quello che probabilmente ridurrà il più possibile il tuo spazio di possibilità.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind.", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind.", "translatedText": "Ad essere onesti, il modo in cui l'ho fatto è stato semplicemente leccarmi il dito e alzarlo al vento.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/japanese/sentence_translations.json b/2022/wordle/japanese/sentence_translations.json index 385fc281b..4c11a800d 100644 --- a/2022/wordle/japanese/sentence_translations.json +++ b/2022/wordle/japanese/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", + "input": "second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", "translatedText": "そこで、ここで少し立ち止まって、発生確率の観点からビット数を 表す情報の公式は何なのかを自問してみてはいかがでしょうか。", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", "translatedText": "ここで私たちが言いたいのは、ビット数の半分を取るとき、それは 確率と同じことです。これは、ビット数の 2 乗が確率より 1 大きいと言っているのと同じことです。さらに整理すると、情報 は 2 を底とする 1 を確率で割った対数であると言えます。", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", "translatedText": "情報理論は、1940 年代にベル研究所で働いていたクロード シャノンによって 開発されましたが、彼はまだ発表されていないアイデアのいくつかについて、当時 の知的巨人で非常に著名なジョン フォン ノイマンと話し合っていました。数学と 物理学、そしてコンピューターサイエンスになりつつあったものの始まりでした。", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", "translatedText": "考えられるすべての推測 (13,000 語すべて) を調べ、各単語のエ ントロピー、より具体的には、表示される可能性のあるすべてのパターンに わたる分布のエントロピーを単語ごとに計算し、最も高いものを選択します 。それはあなたの可能性の空間を可能な限り切り詰める可能性があります。", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind. ", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind. ", "translatedText": "正直に言うと、私がこれを行った方法は、指をなめて風に突き刺すだけでした。", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/marathi/sentence_translations.json b/2022/wordle/marathi/sentence_translations.json index 8e29466fc..6a469b7fd 100644 --- a/2022/wordle/marathi/sentence_translations.json +++ b/2022/wordle/marathi/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", + "input": "y second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", "translatedText": "तर आता तुम्हाला विराम द्यावा लागेल आणि स्वतःला विचारावेसे वाटेल, एखाद्या घटनेच्या संभाव्यतेच्या दृष्टीने बिट्सच्या संख्येसाठी माहितीचे सूत्र काय आहे?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", "translatedText": "आम्ही येथे काय म्हणत आहोत की जेव्हा तुम्ही बिट्सच्या संख्येचा अर्धा भाग घेता, तेव्हा ती संभाव्यतेसारखीच असते, जी बिट्सच्या संख्येच्या पॉवरला दोन म्हणण्यासारखीच गोष्ट असते, जी संभाव्यतेपेक्षा एक असते. संभाव्यतेने भागिले एक पैकी दोन लॉग बेस आहे असे म्हणण्यासाठी पुढील पुनर्रचना करते.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", "translatedText": "माहिती सिद्धांत क्लॉड शॅनन यांनी विकसित केला होता, जो 1940 च्या दशकात बेल लॅबमध्ये काम करत होता, परंतु तो जॉन फॉन न्यूमन यांच्याशी त्याच्या अद्याप प्रकाशित न झालेल्या काही कल्पनांबद्दल बोलत होता, जो त्या काळातील हा बौद्धिक दिग्गज होता. गणित आणि भौतिकशास्त्रात आणि जे संगणक विज्ञान बनत होते त्याची सुरुवात.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", "translatedText": "हे सर्व 13,000 शब्दांच्या संभाव्य अंदाजांमधून जाते, प्रत्येकासाठी एंट्रॉपीची गणना करते, किंवा अधिक विशिष्टपणे, तुम्हाला दिसणार्‍या सर्व पॅटर्नमधील वितरणाची एन्ट्रॉपी, प्रत्येकासाठी, आणि सर्वोच्च निवडते, कारण ते आहे. जे शक्य तितक्या आपल्या शक्यतांची जागा कमी करण्याची शक्यता आहे.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind.", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind.", "translatedText": "खरे सांगायचे तर, मी ज्या पद्धतीने हे केले ते फक्त माझे बोट चाटणे आणि ते वाऱ्यावर चिकटवणे.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/persian/sentence_translations.json b/2022/wordle/persian/sentence_translations.json index 2d93c0df5..c3bd0b73e 100644 --- a/2022/wordle/persian/sentence_translations.json +++ b/2022/wordle/persian/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", + "input": "second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", "translatedText": "بنابراین اکنون ممکن است بخواهید مکث کنید و از خود بپرسید که فرمول اطلاعات برای تعداد بیت ها از نظر احتمال وقوع چیست؟ چیزی که در اینجا می گوییم این است که وقتی یک نصف را به تعداد بیت ها می گیریم، این همان احتمال است، که همان چیزی است که بگوییم دو به توان تعداد بیت ها یک بر احتمال است، که بازآرایی می‌کند و می‌گوید که اطلاعات مبنای گزارش دو از یک تقسیم بر احتمال است. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", "translatedText": "نظریه اطلاعات توسط کلود شانون که در دهه 1940 در آزمایشگاه های بل کار می کرد، ایجاد شد، اما او در مورد برخی از ایده های خود که هنوز منتشر نشده بود با جان فون نویمان، که این غول فکری بسیار برجسته آن زمان بود صحبت می کرد. در ریاضیات و فیزیک و آغاز آنچه در حال تبدیل شدن به علم کامپیوتر بود. ", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", "translatedText": "تمام حدس‌های ممکن را که می‌توانید داشته باشید، تمام 13000 کلمه را بررسی می‌کند، آنتروپی را برای هر یک محاسبه می‌کند، یا به طور خاص، آنتروپی توزیع را در همه الگوهایی که ممکن است ببینید، برای هر یک، و بالاترین را انتخاب می‌کند، زیرا این موردی که احتمالاً فضای احتمالی شما را تا حد امکان کاهش می دهد. ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind. ", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind. ", "translatedText": "صادقانه بگویم، روشی که من این کار را کردم فقط لیسیدن انگشتم و چسباندن آن به باد بود. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/portuguese/sentence_translations.json b/2022/wordle/portuguese/sentence_translations.json index 16733d93c..e58da4caf 100644 --- a/2022/wordle/portuguese/sentence_translations.json +++ b/2022/wordle/portuguese/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", + "input": "y second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", "translatedText": "Então agora você pode querer fazer uma pausa e se perguntar: qual é a fórmula da informação para o número de bits em termos da probabilidade de uma ocorrência?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", "translatedText": "O que estamos dizendo aqui é que quando você eleva metade do número de bits, isso é a mesma coisa que a probabilidade, que é a mesma coisa que dizer que dois elevado à potência do número de bits é um sobre a probabilidade, que reorganiza ainda mais para dizer que a informação é o log de base dois de um dividido pela probabilidade.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", "translatedText": "A teoria da informação foi desenvolvida por Claude Shannon, que trabalhava no Bell Labs na década de 1940, mas ele estava conversando sobre algumas de suas ideias ainda a serem publicadas com John von Neumann, que era um gigante intelectual da época, muito proeminente. em matemática e física e o início do que estava se tornando a ciência da computação.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", "translatedText": "Ele analisa todas as suposições possíveis que você poderia ter, todas as 13.000 palavras, calcula a entropia de cada uma, ou mais especificamente, a entropia da distribuição em todos os padrões que você pode ver, para cada uma, e escolhe o mais alto, já que é aquele que provavelmente reduzirá ao máximo seu espaço de possibilidades.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind.", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind.", "translatedText": "Para ser sincero, fiz isso apenas lambendo o dedo e apontando-o contra o vento.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/russian/sentence_translations.json b/2022/wordle/russian/sentence_translations.json index e15a87ee3..27b3962d9 100644 --- a/2022/wordle/russian/sentence_translations.json +++ b/2022/wordle/russian/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", + "input": "y second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", "translatedText": "Итак, теперь вы, возможно, захотите сделать паузу и спросить себя, какова формула информации о количестве битов с точки зрения вероятности возникновения события?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", "translatedText": "Мы здесь говорим о том, что когда вы принимаете половину числа битов, это то же самое, что и вероятность, а это то же самое, что сказать, что двойка в степени числа битов равна единице по сравнению с вероятностью, что далее перестраивается так, что информация представляет собой логарифм по основанию два из одного, разделенный на вероятность.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", "translatedText": "Теория информации была разработана Клодом Шенноном, который работал в Bell Labs в 1940-х годах, но о некоторых своих еще не опубликованных идеях он говорил с Джоном фон Нейманом, интеллектуальным гигантом того времени, очень выдающимся человеком. по математике и физике и положил начало тому, что впоследствии стало информатикой.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", "translatedText": "Он перебирает все возможные предположения, все 13 000 слов, вычисляет энтропию для каждого из них, или, точнее, энтропию распределения по всем шаблонам, которые вы можете увидеть, для каждого из них, и выбирает самое высокое, поскольку это тот, который, скорее всего, максимально сократит ваше пространство возможностей.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind.", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind.", "translatedText": "Честно говоря, я просто облизнул палец и высунул его по ветру.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/spanish/sentence_translations.json b/2022/wordle/spanish/sentence_translations.json index 2c02f0422..771597b39 100644 --- a/2022/wordle/spanish/sentence_translations.json +++ b/2022/wordle/spanish/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", + "input": "y second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", "translatedText": "Así que ahora quizás quieras hacer una pausa y preguntarte: ¿cuál es la fórmula para obtener información sobre el número de bits en términos de probabilidad de que ocurra?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", "translatedText": "Lo que estamos diciendo aquí es que cuando se le suma la mitad al número de bits, eso es lo mismo que la probabilidad, que es lo mismo que decir que dos elevado a la potencia del número de bits es uno sobre la probabilidad, lo cual se reordena además para decir que la información es el logaritmo en base dos de uno dividido por la probabilidad.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", "translatedText": "La teoría de la información fue desarrollada por Claude Shannon, que trabajaba en los Laboratorios Bell en la década de 1940, pero estaba hablando de algunas de sus ideas aún por publicar con John von Neumann, que era este gigante intelectual de la época, muy destacado. en matemáticas y física y los inicios de lo que se estaba convirtiendo en informática.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", "translatedText": "Revisa todas las conjeturas posibles que puedas tener, las 13.000 palabras, calcula la entropía de cada una, o más específicamente, la entropía de la distribución en todos los patrones que puedas ver, para cada una, y elige la más alta, ya que es el que probablemente reducirá su espacio de posibilidades tanto como sea posible.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind.", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind.", "translatedText": "Para ser honesto, la forma en que hice esto fue simplemente lamerme el dedo y pegarlo al viento.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/tamil/sentence_translations.json b/2022/wordle/tamil/sentence_translations.json index 0a8693d0d..5423aa939 100644 --- a/2022/wordle/tamil/sentence_translations.json +++ b/2022/wordle/tamil/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", + "input": "y second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", "translatedText": "எனவே இப்போது நீங்கள் இடைநிறுத்தப்பட்டு உங்களை நீங்களே கேட்டுக்கொள்ள விரும்பலாம், நிகழ்வின் நிகழ்தகவு அடிப்படையில் பிட்களின் எண்ணிக்கைக்கான தகவலுக்கான சூத்திரம் என்ன?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", "translatedText": "நாம் இங்கே சொல்வது என்னவென்றால், பிட்களின் எண்ணிக்கையில் ஒரு பாதியை எடுத்துக் கொண்டால், அதுவே நிகழ்தகவு, அதாவது பிட்களின் எண்ணிக்கையின் சக்திக்கு இரண்டு என்று சொல்வது நிகழ்தகவை விட ஒன்று, இது நிகழ்தகவு மூலம் வகுக்கப்பட்ட ஒன்றின் பதிவு அடிப்படை இரண்டாகத் தகவல் கூறுவதற்கு மேலும் மறுசீரமைக்கிறது.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", "translatedText": "1940 களில் பெல் லேப்ஸில் பணிபுரிந்த கிளாட் ஷானனால் தகவல் கோட்பாடு உருவாக்கப்பட்டது, ஆனால் அவர் ஜான் வான் நியூமனுடன் தனது இன்னும் வெளியிடப்படாத சில யோசனைகளைப் பற்றி பேசினார், அவர் அந்த நேரத்தில் இந்த அறிவார்ந்த ராட்சதராக இருந்தார். கணிதம் மற்றும் இயற்பியல் மற்றும் கணினி அறிவியலாக மாறியதன் ஆரம்பம்.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", "translatedText": "இது உங்களால் முடிந்த அனைத்து யூகங்களையும் கடந்து, 13,000 வார்த்தைகள், ஒவ்வொன்றிற்கும் என்ட்ரோபியைக் கணக்கிடுகிறது, அல்லது இன்னும் குறிப்பாக, நீங்கள் காணக்கூடிய அனைத்து வடிவங்களிலும் உள்ள விநியோகத்தின் என்ட்ரோபி, ஒவ்வொன்றிற்கும், மேலும் உயர்ந்ததைத் தேர்ந்தெடுக்கிறது. உங்கள் சாத்தியக்கூறுகளை முடிந்தவரை குறைக்கக்கூடிய ஒன்று.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind.", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind.", "translatedText": "உண்மையைச் சொல்வதானால், நான் இதைச் செய்த விதம் என் விரலை நக்கி காற்றில் ஒட்டிக்கொண்டது.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/telugu/sentence_translations.json b/2022/wordle/telugu/sentence_translations.json index 5dd7a3218..26349777e 100644 --- a/2022/wordle/telugu/sentence_translations.json +++ b/2022/wordle/telugu/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", + "input": "second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", "translatedText": "కాబట్టి ఇప్పుడు మీరు పాజ్ చేసి, మిమ్మల్ని మీరు ప్రశ్నించుకోవచ్చు, సంభవించే సంభావ్యత పరంగా బిట్‌ల సంఖ్యకు సమాచారం కోసం సూత్రం ఏమిటి? ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", "translatedText": "మేము ఇక్కడ చెప్పేది ఏమిటంటే, మీరు బిట్‌ల సంఖ్యకు ఒక సగం తీసుకున్నప్పుడు, అది సంభావ్యత వలె ఉంటుంది, ఇది బిట్‌ల సంఖ్య యొక్క శక్తికి రెండు చెప్పడం అదే సంభావ్యత కంటే ఒకటి, ఇది సమాచారాన్ని సంభావ్యతతో భాగించబడిన ఒకదానిలో రెండు లాగ్ బేస్ అని చెప్పడానికి మరింత క్రమాన్ని మార్చుతుంది. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", "translatedText": "1940లలో బెల్ ల్యాబ్స్‌లో పని చేస్తున్న క్లాడ్ షానన్ చేత సమాచార సిద్ధాంతాన్ని అభివృద్ధి చేశారు, అయితే అతను జాన్ వాన్ న్యూమాన్‌తో తన ఇంకా ప్రచురింపబడని కొన్ని ఆలోచనల గురించి మాట్లాడుతున్నాడు. గణితం మరియు భౌతిక శాస్త్రం మరియు కంప్యూటర్ సైన్స్‌గా మారుతున్న దాని ప్రారంభం. ", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", "translatedText": "ఇది మీరు కలిగి ఉండగల అన్ని అంచనాల ద్వారా, మొత్తం 13,000 పదాలు, ఒక్కోదానికి ఎంట్రోపీని గణిస్తుంది లేదా మరింత ప్రత్యేకంగా, మీరు చూసే అన్ని నమూనాలలో పంపిణీ యొక్క ఎంట్రోపీని ప్రతి ఒక్కదాని కోసం మరియు అత్యధికంగా ఎంచుకుంటుంది, ఎందుకంటే ఇది మీ అవకాశాలను వీలైనంత వరకు తగ్గించే అవకాశం ఉంది. ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind. ", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind. ", "translatedText": "నిజం చెప్పాలంటే, నేను దీన్ని చేసిన విధానం నా వేలిని నొక్కడం మరియు గాలికి అంటుకోవడం. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/thai/sentence_translations.json b/2022/wordle/thai/sentence_translations.json index 06a8ffc12..4f5d34d38 100644 --- a/2022/wordle/thai/sentence_translations.json +++ b/2022/wordle/thai/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", + "input": "second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind. ", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/turkish/sentence_translations.json b/2022/wordle/turkish/sentence_translations.json index 9c7b18695..4e476ca60 100644 --- a/2022/wordle/turkish/sentence_translations.json +++ b/2022/wordle/turkish/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", + "input": "y second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", "translatedText": "Şimdi durup kendinize şu soruyu sorabilirsiniz: Bir olayın gerçekleşme olasılığı açısından bit sayısı bilgisinin formülü nedir?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", "translatedText": "Burada söylediğimiz şey, bit sayısının yarısını aldığınızda, bu olasılık ile aynı şeydir; bu, bit sayısının iki üssünün bir bölü olasılık olduğunu söylemekle aynı şeydir; Bilginin log tabanının iki bölü olasılığa eşit olduğunu söyleyerek yeniden düzenleme yapar.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", "translatedText": "Bilgi teorisi, 1940'larda Bell Laboratuarlarında çalışan Claude Shannon tarafından geliştirildi, ancak henüz yayınlanmamış bazı fikirlerinden, zamanın entelektüel devi, çok öne çıkan John von Neumann'la konuşuyordu. matematik ve fizikte ve bilgisayar bilimine dönüşen şeyin başlangıcı.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", "translatedText": "Yapabileceğiniz tüm olası tahminleri, yani 13.000 kelimenin tamamını gözden geçirir, her biri için entropiyi veya daha spesifik olarak, her biri için görebileceğiniz tüm kalıplar arasındaki dağılımın entropisini hesaplar ve en yüksek olanı seçer, çünkü bu Olasılık alanınızı mümkün olduğu kadar daraltması muhtemel olanı.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind.", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind.", "translatedText": "Dürüst olmak gerekirse, bunu yapma şeklim sadece parmağımı yalayıp rüzgara doğru tutmaktı.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/ukrainian/sentence_translations.json b/2022/wordle/ukrainian/sentence_translations.json index 84a4e706a..9343d094c 100644 --- a/2022/wordle/ukrainian/sentence_translations.json +++ b/2022/wordle/ukrainian/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", + "input": "y second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence?", "translatedText": "Тож тепер ви можете зупинитись і запитати себе, яка формула для інформації для кількості бітів у термінах ймовірності появи?", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability.", "translatedText": "Ми маємо на увазі те, що коли ви берете одну половину на кількість бітів, це те саме, що ймовірність, що те саме, що сказати, що два в степені кількості бітів є одиницею над ймовірністю, що далі переставляє, кажучи, що інформація є двома логарифмами одиниці, поділеними на ймовірність.", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science.", "translatedText": "Теорію інформації розробив Клод Шеннон, який працював у Bell Labs у 1940-х роках, але він говорив про деякі зі своїх ідей, які ще не були опубліковані, з Джоном фон Нейманом, який був цим інтелектуальним гігантом того часу, дуже видатним у математиці та фізиці та початках того, що ставало інформатикою.", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible.", "translatedText": "Він переглядає всі можливі припущення, які ви можете мати, усі 13 000 слів, обчислює ентропію для кожного з них, або, точніше, ентропію розподілу за всіма шаблонами, які ви можете побачити, для кожного з них, і вибирає найвище, оскільки це той, який, швидше за все, максимально скоротить ваш простір можливостей.", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind.", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind.", "translatedText": "Чесно кажучи, те, як я це зробив, було просто облизати палець і тицьнути його на вітер.", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/urdu/sentence_translations.json b/2022/wordle/urdu/sentence_translations.json index 48585d6fc..7ada9224c 100644 --- a/2022/wordle/urdu/sentence_translations.json +++ b/2022/wordle/urdu/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", + "input": "second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", "translatedText": "تو اب آپ رک کر اپنے آپ سے پوچھنا چاہیں گے، وقوع پذیر ہونے کے امکان کے لحاظ سے بٹس کی تعداد کے لیے معلومات کا فارمولا کیا ہے؟ ہم یہاں جو کہہ رہے ہیں وہ یہ ہے کہ جب آپ بٹس کی تعداد میں ایک نصف لیتے ہیں، تو یہ وہی چیز ہے جو امکان ہے، جو کہ بٹس کی تعداد کی طاقت کو دو کہنے کے برابر ہے، جو کہ امکان سے زیادہ ایک ہے۔یہ کہنے کے لیے مزید ترتیب دیتا ہے کہ انفارمیشن لاگ بیس دو ہے جسے احتمال سے تقسیم کیا جاتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", "translatedText": "انفارمیشن تھیوری کو کلاڈ شینن نے تیار کیا تھا، جو 1940 کی دہائی میں بیل لیبز میں کام کر رہے تھے، لیکن وہ جان وان نیومن کے ساتھ اپنے ابھی تک شائع ہونے والے کچھ نظریات کے بارے میں بات کر رہے تھے، جو اس وقت کا یہ دانشور دیو تھا، بہت نمایاں۔ریاضی اور طبیعیات میں اور اس کی شروعات جو کمپیوٹر سائنس بن رہی تھی۔", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", "translatedText": "یہ آپ کے تمام ممکنہ اندازوں سے گزرتا ہے، تمام 13,000 الفاظ، ہر ایک کے لیے انٹراپی کی گنتی کرتا ہے، یا خاص طور پر، ان تمام نمونوں میں تقسیم کی اینٹروپی جو آپ دیکھ سکتے ہیں، ہر ایک کے لیے، اور سب سے زیادہ منتخب کرتا ہے، کیونکہ یہ ہے۔ایک جو ممکنہ حد تک آپ کے امکانات کی جگہ کو کم کر سکتا ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind. ", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind. ", "translatedText": "سچ پوچھیں تو، جس طرح سے میں نے یہ کیا وہ صرف اپنی انگلی کو چاٹ کر ہوا میں چپکا رہا تھا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2022/wordle/vietnamese/sentence_translations.json b/2022/wordle/vietnamese/sentence_translations.json index 70e1e76de..e6fbf6469 100644 --- a/2022/wordle/vietnamese/sentence_translations.json +++ b/2022/wordle/vietnamese/sentence_translations.json @@ -792,7 +792,7 @@ "end": 533.52 }, { - "input": "So now you might want to pause and ask yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", + "input": "second. So now's when you might want to take a moment and pause and ask for yourself, what is the formula for information for the number of bits in terms of the probability of an occurrence? ", "translatedText": "Vì vậy, bây giờ bạn có thể muốn tạm dừng và tự hỏi, công thức thông tin về số bit theo xác suất xảy ra là gì? ", "model": "google_nmt", "n_reviews": 0, @@ -800,7 +800,7 @@ "end": 542.98 }, { - "input": "What we're saying here is that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", + "input": "Well, what we're saying here is basically that when you take one half to the number of bits, that's the same thing as the probability, which is the same thing as saying two to the power of the number of bits is one over the probability, which rearranges further to saying the information is the log base two of one divided by the probability. ", "translatedText": "Điều chúng tôi đang nói ở đây là khi bạn lấy một nửa số bit, thì nó bằng với xác suất, cũng giống như nói hai lũy thừa của số bit bằng một trên xác suất, tức là sắp xếp lại để nói rằng thông tin là log cơ số hai của một chia cho xác suất. ", "model": "google_nmt", "n_reviews": 0, @@ -968,7 +968,7 @@ "end": 708.3 }, { - "input": "Information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", + "input": "You see, information theory was developed by Claude Shannon, who was working at Bell Labs in the 1940s, but he was talking about some of his yet-to-be-published ideas with John von Neumann, who was this intellectual giant of the time, very prominent in math and physics and the beginnings of what was becoming computer science. ", "translatedText": "Lý thuyết thông tin được phát triển bởi Claude Shannon, người đang làm việc tại Bell Labs vào những năm 1940, nhưng ông ấy đang nói về một số ý tưởng chưa được công bố của mình với John von Neumann, một trí tuệ khổng lồ vào thời điểm đó, rất nổi bật. trong toán học và vật lý và sự khởi đầu của khoa học máy tính. ", "model": "google_nmt", "n_reviews": 0, @@ -1080,7 +1080,7 @@ "end": 837.96 }, { - "input": "It goes through all of the possible guesses you could have, all 13,000 words, computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns you might see, for each one, and picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", + "input": "It goes through all of the different possible guesses that you could have, all 13,000 words, it computes the entropy for each one, or more specifically, the entropy of the distribution across all patterns that you might see for each one, and then it picks the highest, since that's the one that's likely to chop down your space of possibilities as much as possible. ", "translatedText": "Nó xem xét tất cả những phỏng đoán có thể có mà bạn có thể có, tất cả 13.000 từ, tính toán entropy cho mỗi từ, hay cụ thể hơn là entropy của phân phối trên tất cả các mẫu mà bạn có thể thấy, cho mỗi mẫu và chọn mức cao nhất, vì đó là thứ có khả năng cắt giảm không gian khả năng của bạn càng nhiều càng tốt. ", "model": "google_nmt", "n_reviews": 0, @@ -1528,7 +1528,7 @@ "end": 1202.14 }, { - "input": "To be honest, the way I did this was just licking my finger and sticking it into the wind. ", + "input": "And to be honest the way I did this was kind of just licking my finger and sticking it into the wind. ", "translatedText": "Thành thật mà nói, cách tôi làm điều này chỉ là liếm ngón tay và đưa nó theo chiều gió. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-1/french/sentence_translations.json b/2023/barber-pole-1/french/sentence_translations.json index bddc073c7..0b3848c14 100644 --- a/2023/barber-pole-1/french/sentence_translations.json +++ b/2023/barber-pole-1/french/sentence_translations.json @@ -128,7 +128,7 @@ "end": 107.6 }, { - "input": "And then if you turn the initial polarizer, you can kind of see those stripes, those diagonal stripes seem to walk up the tube.", + "input": "Ooh. And then if you turn the initial polarizer, you can kind of see those. Wow. Stripes, those diagonal stripes seem to walk up the tube.", "translatedText": "Et puis, si vous tournez le polariseur initial, vous pouvez en quelque sorte voir ces rayures, ces rayures diagonales semblent remonter le long du tube.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-1/german/sentence_translations.json b/2023/barber-pole-1/german/sentence_translations.json index d7bcc601a..9fda3aace 100644 --- a/2023/barber-pole-1/german/sentence_translations.json +++ b/2023/barber-pole-1/german/sentence_translations.json @@ -128,7 +128,7 @@ "end": 107.6 }, { - "input": "And then if you turn the initial polarizer, you can kind of see those stripes, those diagonal stripes seem to walk up the tube.", + "input": "Ooh. And then if you turn the initial polarizer, you can kind of see those. Wow. Stripes, those diagonal stripes seem to walk up the tube.", "translatedText": "Und wenn man dann den ersten Polarisator dreht, kann man diese Streifen sehen, diese diagonalen Streifen scheinen durch die Röhre hindurchzulaufen.", "model": "google_nmt", "n_reviews": 1, diff --git a/2023/barber-pole-1/hebrew/sentence_translations.json b/2023/barber-pole-1/hebrew/sentence_translations.json index 9d875f839..86595c0a0 100644 --- a/2023/barber-pole-1/hebrew/sentence_translations.json +++ b/2023/barber-pole-1/hebrew/sentence_translations.json @@ -128,7 +128,7 @@ "end": 107.6 }, { - "input": "And then if you turn the initial polarizer, you can kind of see those stripes, those diagonal stripes seem to walk up the tube.", + "input": "Ooh. And then if you turn the initial polarizer, you can kind of see those. Wow. Stripes, those diagonal stripes seem to walk up the tube.", "translatedText": "ואז אם תסובב את המקטב הראשוני, אתה יכול לראות את הפסים האלה, נראה שהפסים האלכסוניים הולכים במעלה הצינור.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-1/indonesian/sentence_translations.json b/2023/barber-pole-1/indonesian/sentence_translations.json index a7a77c759..f26cd8248 100644 --- a/2023/barber-pole-1/indonesian/sentence_translations.json +++ b/2023/barber-pole-1/indonesian/sentence_translations.json @@ -128,7 +128,7 @@ "end": 107.6 }, { - "input": "And then if you turn the initial polarizer, you can kind of see those stripes, those diagonal stripes seem to walk up the tube.", + "input": "Ooh. And then if you turn the initial polarizer, you can kind of see those. Wow. Stripes, those diagonal stripes seem to walk up the tube.", "translatedText": "Dan jika Anda memutar polarizer awal, Anda dapat melihat garis-garis itu, garis-garis diagonal itu tampak naik ke atas tabung.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-1/italian/sentence_translations.json b/2023/barber-pole-1/italian/sentence_translations.json index b4cde60e6..0e4f1b004 100644 --- a/2023/barber-pole-1/italian/sentence_translations.json +++ b/2023/barber-pole-1/italian/sentence_translations.json @@ -128,7 +128,7 @@ "end": 107.6 }, { - "input": "And then if you turn the initial polarizer, you can kind of see those stripes, those diagonal stripes seem to walk up the tube.", + "input": "Ooh. And then if you turn the initial polarizer, you can kind of see those. Wow. Stripes, those diagonal stripes seem to walk up the tube.", "translatedText": "E poi se giri il polarizzatore iniziale, puoi vedere quelle strisce, quelle strisce diagonali sembrano camminare lungo il tubo.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-1/marathi/sentence_translations.json b/2023/barber-pole-1/marathi/sentence_translations.json index 833acb096..47413b938 100644 --- a/2023/barber-pole-1/marathi/sentence_translations.json +++ b/2023/barber-pole-1/marathi/sentence_translations.json @@ -128,7 +128,7 @@ "end": 107.6 }, { - "input": "And then if you turn the initial polarizer, you can kind of see those stripes, those diagonal stripes seem to walk up the tube.", + "input": "Ooh. And then if you turn the initial polarizer, you can kind of see those. Wow. Stripes, those diagonal stripes seem to walk up the tube.", "translatedText": "आणि मग जर तुम्ही सुरुवातीचे ध्रुवीकरण चालू केले तर तुम्हाला ते पट्टे दिसतील, त्या कर्णरेषेचे पट्टे नळीतून वर जाताना दिसतात.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-1/portuguese/sentence_translations.json b/2023/barber-pole-1/portuguese/sentence_translations.json index 6f3f1656f..948e46e67 100644 --- a/2023/barber-pole-1/portuguese/sentence_translations.json +++ b/2023/barber-pole-1/portuguese/sentence_translations.json @@ -128,7 +128,7 @@ "end": 107.6 }, { - "input": "And then if you turn the initial polarizer, you can kind of see those stripes, those diagonal stripes seem to walk up the tube.", + "input": "Ooh. And then if you turn the initial polarizer, you can kind of see those. Wow. Stripes, those diagonal stripes seem to walk up the tube.", "translatedText": "E então, se você girar o polarizador inicial, você poderá ver essas listras, essas listras diagonais parecem subir no tubo.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-1/russian/sentence_translations.json b/2023/barber-pole-1/russian/sentence_translations.json index 6808f9226..0da7dbc74 100644 --- a/2023/barber-pole-1/russian/sentence_translations.json +++ b/2023/barber-pole-1/russian/sentence_translations.json @@ -128,7 +128,7 @@ "end": 107.6 }, { - "input": "And then if you turn the initial polarizer, you can kind of see those stripes, those diagonal stripes seem to walk up the tube.", + "input": "Ooh. And then if you turn the initial polarizer, you can kind of see those. Wow. Stripes, those diagonal stripes seem to walk up the tube.", "translatedText": "А затем, если повернуть первый поляризатор, вы увидите полосы, диагональные полосы, кажется, что они идут вверх по трубке.", "model": "google_nmt", "n_reviews": 1, diff --git a/2023/barber-pole-1/spanish/sentence_translations.json b/2023/barber-pole-1/spanish/sentence_translations.json index e7375a14b..21f6de0a5 100644 --- a/2023/barber-pole-1/spanish/sentence_translations.json +++ b/2023/barber-pole-1/spanish/sentence_translations.json @@ -128,7 +128,7 @@ "end": 107.6 }, { - "input": "And then if you turn the initial polarizer, you can kind of see those stripes, those diagonal stripes seem to walk up the tube.", + "input": "Ooh. And then if you turn the initial polarizer, you can kind of see those. Wow. Stripes, those diagonal stripes seem to walk up the tube.", "translatedText": "Y luego, si giras el polarizador inicial, puedes ver esas franjas, esas franjas diagonales parecen subir por el tubo.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-1/tamil/sentence_translations.json b/2023/barber-pole-1/tamil/sentence_translations.json index 82b0eccc6..deacc441e 100644 --- a/2023/barber-pole-1/tamil/sentence_translations.json +++ b/2023/barber-pole-1/tamil/sentence_translations.json @@ -128,7 +128,7 @@ "end": 107.6 }, { - "input": "And then if you turn the initial polarizer, you can kind of see those stripes, those diagonal stripes seem to walk up the tube.", + "input": "Ooh. And then if you turn the initial polarizer, you can kind of see those. Wow. Stripes, those diagonal stripes seem to walk up the tube.", "translatedText": "நீங்கள் ஆரம்ப துருவமுனைப்பைத் திருப்பினால், அந்த கோடுகளை நீங்கள் பார்க்கலாம், அந்த மூலைவிட்ட கோடுகள் குழாயின் மேல் நடப்பது போல் தெரிகிறது.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-1/telugu/sentence_translations.json b/2023/barber-pole-1/telugu/sentence_translations.json index eaa58f003..59b056111 100644 --- a/2023/barber-pole-1/telugu/sentence_translations.json +++ b/2023/barber-pole-1/telugu/sentence_translations.json @@ -128,7 +128,7 @@ "end": 107.6 }, { - "input": "And then if you turn the initial polarizer, you can kind of see those stripes, those diagonal stripes seem to walk up the tube.", + "input": "Ooh. And then if you turn the initial polarizer, you can kind of see those. Wow. Stripes, those diagonal stripes seem to walk up the tube.", "translatedText": "ఆపై మీరు ప్రారంభ ధ్రువణాన్ని తిప్పినట్లయితే, మీరు ఆ చారలను చూడవచ్చు, ఆ వికర్ణ చారలు ట్యూబ్ పైకి నడిచినట్లుగా కనిపిస్తాయి.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/arabic/sentence_translations.json b/2023/barber-pole-2/arabic/sentence_translations.json index 288c2c1aa..75187a475 100644 --- a/2023/barber-pole-2/arabic/sentence_translations.json +++ b/2023/barber-pole-2/arabic/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field. ", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space. ", "translatedText": "كما قلت، فإن هذا الانتشار لشحنة واحدة فقط يكون قويًا بنفس القدر في جميع الاتجاهات المتعامدة مع اهتزازها، ويجب أن أؤكد حقًا أن الانتشار يتمتع بقوة كبيرة في المجال. ", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. ", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you. ", "translatedText": "في هذه المرحلة، أعتقد أننا قد غطينا ما يكفي لمقطع فيديو واحد، لذا سأقوم بسحب مناقشة تغطي أصول معامل الانكسار إلى مقطع فيديو منفصل. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/bengali/sentence_translations.json b/2023/barber-pole-2/bengali/sentence_translations.json index 4375952a8..3bc5a30c2 100644 --- a/2023/barber-pole-2/bengali/sentence_translations.json +++ b/2023/barber-pole-2/bengali/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field. ", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space. ", "translatedText": "যেমনটি আমি বলেছি, শুধুমাত্র একটি চার্জের জন্য এই প্রচারটি তার নড়াচড়ার জন্য লম্ব সমস্ত দিকগুলিতে সমানভাবে শক্তিশালী, এবং সত্যিই আমার জোর দেওয়া উচিত যে প্রচারের ক্ষেত্রে অনেক শক্তি রয়েছে।", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. ", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/chinese/sentence_translations.json b/2023/barber-pole-2/chinese/sentence_translations.json index 582af150d..dddb5afe1 100644 --- a/2023/barber-pole-2/chinese/sentence_translations.json +++ b/2023/barber-pole-2/chinese/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "就像我说的，仅一个电 荷的传播在垂直于其摆动的所有方向上都同样强，而 且我真的应该强调，传播在场中确实有很大的强度。", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "至此，我认为我们已经对 一个视频进行了足够的讨论，因此我将把有关 折射率起源的讨论放到一个单独的视频中。", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/czech/sentence_translations.json b/2023/barber-pole-2/czech/sentence_translations.json index 8cf8f1f21..c11f272fb 100644 --- a/2023/barber-pole-2/czech/sentence_translations.json +++ b/2023/barber-pole-2/czech/sentence_translations.json @@ -472,7 +472,7 @@ "end": 481.74 }, { - "input": "Because our law has this 1 divided by r in it, the strength of the wave caused by just one particle does decay as you go farther away, in proportion to 1 over r.", + "input": "on that the wave is going to move in the direction that it's going to move in. So if we're going to look at the wiggling at the extreme, the only place where there's no propagation is in the z-axis.", "translatedText": "Protože náš zákon obsahuje toto dělení 1 na r, síla vlny způsobené jednou částicí se s rostoucí vzdáleností zmenšuje v poměru 1 na r.", "model": "DeepL", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 493.52 }, { - "input": "But notice what happens if I take a whole row of charges, say oriented along the y axis, and I have them all start wiggling up and down in the z direction, and I illustrate the combined effects that all of them have on this component of the electric field.", + "input": "Because our law has this 1 divided by r in it, the strength of the wave caused by just one particle does decay as you go farther away, in proportion to 1 over r. But notice what happens if I take a whole row of charges, say oriented along the y axis, and I have them all start wiggling up and down in the z direction, and I illustrate the co", "translatedText": "Ale všimněte si, co se stane, když vezmu celou řadu nábojů, řekněme orientovaných podél osy y, a nechám je všechny kmitat nahoru a dolů ve směru z, a znázorním kombinovaný účinek, který mají všechny na tuto složku elektrického pole.", "model": "DeepL", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 508.32 }, { - "input": "The effects of all these charges interfere deconstructively along the y direction, but they interfere constructively along the x direction.", + "input": "mbined effects that all of them have on this component of the electric field. The effects of all these charges interfere deconstructively along the y direction, but they interfere constructively along the x direction.", "translatedText": "Působení všech těchto nábojů interferuje dekonstruktivně podél směru y, ale interferuje konstruktivně podél směru x.", "model": "DeepL", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 710.98 }, { - "input": "Sometimes when people talk about light bouncing off of things, the implied mental image is a projectile ricocheting off of some object, heading off in some random direction.", + "input": "Sometimes when people talk about light bouncing off of things, the implied mental image is something like a projectile ricocheting off of some object heading off in some random direction.", "translatedText": "Někdy, když lidé mluví o světle odrážejícím se od věcí, je myšlenkovým obrazem projektil odrážející se od nějakého předmětu a mířící náhodným směrem.", "model": "DeepL", "n_reviews": 0, diff --git a/2023/barber-pole-2/english/captions.srt b/2023/barber-pole-2/english/captions.srt index ccafd435d..5fc7c4c26 100644 --- a/2023/barber-pole-2/english/captions.srt +++ b/2023/barber-pole-2/english/captions.srt @@ -499,842 +499,854 @@ it's just that that propagation gets weaker in directions that are more aligned with the original wiggling. 126 -00:07:58,000 --> 00:08:01,740 -At the extreme, the only place where there's no propagation is in the z axis. +00:07:58,000 --> 00:08:04,828 +At the extreme, the only thing that's more important is the direction that the wave is 127 -00:08:04,320 --> 00:08:07,005 -Because our law has this 1 divided by r in it, +00:08:04,828 --> 00:08:09,459 +going to move in the direction that it's going to move in. 128 -00:08:07,005 --> 00:08:12,034 -the strength of the wave caused by just one particle does decay as you go farther away, +00:08:09,459 --> 00:08:14,012 +So if we're going to look at the wiggling at the extreme, 129 -00:08:12,034 --> 00:08:13,520 -in proportion to 1 over r. +00:08:14,012 --> 00:08:18,800 +the only place where there's no propagation is in the z-axis. 130 -00:08:14,280 --> 00:08:19,161 -But notice what happens if I take a whole row of charges, say oriented along the y axis, +00:08:18,800 --> 00:08:20,633 +Because our law has this 1 divided by r in it, 131 -00:08:19,161 --> 00:08:22,835 -and I have them all start wiggling up and down in the z direction, +00:08:20,633 --> 00:08:24,065 +the strength of the wave caused by just one particle does decay as you go farther away, 132 -00:08:22,835 --> 00:08:27,497 -and I illustrate the combined effects that all of them have on this component of the +00:08:24,065 --> 00:08:25,080 +in proportion to 1 over r. 133 -00:08:27,497 --> 00:08:28,320 -electric field. +00:08:25,080 --> 00:08:29,662 +But notice what happens if I take a whole row of charges, say oriented along the y axis, 134 -00:08:29,820 --> 00:08:34,859 -The effects of all these charges interfere deconstructively along the y direction, +00:08:29,662 --> 00:08:33,111 +and I have them all start wiggling up and down in the z direction, 135 -00:08:34,859 --> 00:08:38,260 -but they interfere constructively along the x direction. +00:08:33,111 --> 00:08:37,487 +and I illustrate the combined effects that all of them have on this component of the 136 -00:08:39,020 --> 00:08:41,385 -This is what it looks like for a beam of light +00:08:37,487 --> 00:08:38,260 +electric field. 137 -00:08:41,385 --> 00:08:43,600 -to be concentrated along just one dimension. +00:08:39,020 --> 00:08:41,754 +The effects of all these charges interfere deconstructively along the y direction, 138 +00:08:41,754 --> 00:08:43,600 +but they interfere constructively along the x direction. + +139 +00:08:43,996 --> 00:08:43,600 +This is what it looks like for a beam of light + +140 +00:08:44,420 --> 00:08:43,996 +to be concentrated along just one dimension. + +141 00:08:44,420 --> 00:08:48,154 So if you were to focus on the field just along the x axis, -139 +142 00:08:48,154 --> 00:08:51,079 instead of decaying in proportion to 1 over r, -140 +143 00:08:51,079 --> 00:08:53,880 this combined effect decays much more gently. -141 +144 00:08:55,700 --> 00:08:59,374 In the extreme, you can get something arbitrarily close to those pure sine -142 +145 00:08:59,374 --> 00:09:01,726 wave propagations we were illustrating earlier, -143 +146 00:09:01,726 --> 00:09:05,401 if at some distance away you have a large number of charges oscillating in -144 +147 00:09:05,401 --> 00:09:06,920 sync with each other like this. -145 +148 00:09:07,400 --> 00:09:10,911 One thing that's worth emphasizing when you see light illustrated -146 +149 00:09:10,911 --> 00:09:14,477 with a sine wave like this, is that even though that wave is being -147 +150 00:09:14,477 --> 00:09:18,095 drawn in two or three dimensions, it's only describing the electric -148 +151 00:09:18,095 --> 00:09:21,980 field along a one-dimensional line, namely the base of all those vectors. -149 +152 00:09:22,400 --> 00:09:25,880 It's just that to draw the vectors you have to venture off of that line. -150 +153 00:09:27,180 --> 00:09:29,939 Great, so one of the last important things to highlight -151 +154 00:09:29,939 --> 00:09:32,600 before we get back to the sugar water is polarization. -152 +155 00:09:33,180 --> 00:09:37,329 In everything I've been showing, the driving charge is just oscillating along -153 +156 00:09:37,329 --> 00:09:41,480 a single direction, like the z axis, and this causes linearly polarized light. -154 +157 00:09:41,480 --> 00:09:43,260 But it doesn't have to happen like that. -155 +158 00:09:43,260 --> 00:09:48,031 For example, if I set the charge rotating in a little circle along the yz plane, -156 +159 00:09:48,031 --> 00:09:52,096 meaning its acceleration vector is also rotating in a little circle, -157 +160 00:09:52,096 --> 00:09:54,040 notice what the field looks like. -158 +161 00:09:54,800 --> 00:09:58,240 This is known, aptly enough, as circularly polarized light. -159 +162 00:09:58,960 --> 00:10:02,380 Honestly, it's easiest to think about for just one point of the electric field. -160 +163 00:10:03,000 --> 00:10:07,176 What it means for light to be circularly polarized is that at that point, -161 +164 00:10:07,176 --> 00:10:10,280 the electric field vector is just rotating in a circle. -162 +165 00:10:10,680 --> 00:10:13,655 People often find circular polarization a little confusing, -163 +166 00:10:13,655 --> 00:10:17,572 and I suspect part of the reason for that is that it's hard to illustrate with -164 +167 00:10:17,572 --> 00:10:21,489 a static diagram, but also it's a little confusing when you try to think about -165 +168 00:10:21,489 --> 00:10:22,680 the full electric field. -166 +169 00:10:23,420 --> 00:10:26,096 For example, here's what the field looks like on the xy -167 +170 00:10:26,096 --> 00:10:28,820 plane when I set that little charge rotating in a circle. -168 +171 00:10:30,960 --> 00:10:33,783 It's certainly very beautiful, I could look at this all day, -169 +172 00:10:33,783 --> 00:10:36,560 but you can understand why it might feel a little confusing. -170 +173 00:10:37,120 --> 00:10:40,860 The very last thing I'll mention is that while everything here is a classical -171 +174 00:10:40,860 --> 00:10:44,600 description of light, the important points still hold up in quantum mechanics. -172 +175 00:10:45,040 --> 00:10:47,299 You still have propagating waves, there's still -173 +176 00:10:47,299 --> 00:10:49,700 polarization that can be either linear or circular. -174 +177 00:10:50,100 --> 00:10:54,278 The main difference with quantum mechanics is that the energy in this wave doesn't -175 +178 00:10:54,278 --> 00:10:58,760 scale up and down continuously, like you might expect, it comes in discrete little steps. -176 +179 00:10:59,380 --> 00:11:01,726 I have another video that goes into more detail, -177 +180 00:11:01,726 --> 00:11:04,600 but for our purposes, thinking about it classically is fine. -178 +181 00:11:05,300 --> 00:11:08,322 Part of the reason I wanted to go through that is because, frankly, -179 +182 00:11:08,322 --> 00:11:11,700 it's just very fun to animate and I like an excuse for a fundamental lesson. -180 +183 00:11:12,360 --> 00:11:16,105 But now let's turn back to our demo and see how we can build up an intuition -181 +184 00:11:16,105 --> 00:11:19,705 for some of our key questions, starting from this very basic premise that -182 +185 00:11:19,705 --> 00:11:23,840 shaking a charge in one location causes a shake to another charge a little bit later. -183 +186 00:11:24,180 --> 00:11:27,192 And let's start by actually skipping ahead to question number three, -184 +187 00:11:27,192 --> 00:11:28,720 why do we see the diagonal stripes? -185 +188 00:11:33,680 --> 00:11:37,923 To think about this, you need to imagine an observer to the side of the tube, -186 +189 00:11:37,923 --> 00:11:40,480 and then for a particular pure color, say red, -187 +190 00:11:40,480 --> 00:11:43,472 if the observer looks in the tube and sees that color, -188 +191 00:11:43,472 --> 00:11:48,151 it's because light of that color has bounced off something at that point in the tube, -189 +192 00:11:48,151 --> 00:11:50,980 and then propagated towards the eye of the observer. -190 -00:11:51,540 --> 00:11:55,159 +193 +00:11:51,540 --> 00:11:54,888 Sometimes when people talk about light bouncing off of things, -191 -00:11:55,159 --> 00:11:59,354 -the implied mental image is a projectile ricocheting off of some object, +194 +00:11:54,888 --> 00:11:57,865 +the implied mental image is something like a projectile -192 -00:11:59,354 --> 00:12:01,480 -heading off in some random direction. +195 +00:11:57,865 --> 00:12:01,480 +ricocheting off of some object heading off in some random direction. -193 +196 00:12:02,280 --> 00:12:06,218 But the better mental image to hold in your mind is that when the propagating -194 +197 00:12:06,218 --> 00:12:10,763 light waves caused by some wiggling charge reach some second charge causing it to wiggle, -195 +198 00:12:10,763 --> 00:12:13,440 that secondary wiggle results in its own propagation. -196 +199 00:12:14,280 --> 00:12:17,943 And for the animation on screen, that propagation goes back to the first charge, -197 +200 00:12:17,943 --> 00:12:20,340 which itself causes a propagation towards the second. -198 +201 00:12:20,700 --> 00:12:23,667 And this is what it looks like in a very simplified situation -199 +202 00:12:23,667 --> 00:12:26,300 for light to bounce back and forth between two charges. -200 +203 00:12:27,160 --> 00:12:31,899 If you have some concentrated beam of polarized light interacting with some charge, -201 +204 00:12:31,899 --> 00:12:36,469 causing it to wiggle up and down, then these resulting second-order propagations -202 +205 00:12:36,469 --> 00:12:41,040 are most strong in the directions perpendicular to the direction of polarization. -203 +206 00:12:41,540 --> 00:12:44,575 In some sense, you could think of light as bouncing off of that charge, -204 +207 00:12:44,575 --> 00:12:47,780 but the important point is that it doesn't bounce in all directions equally. -205 +208 00:12:48,080 --> 00:12:50,774 It's strongest perpendicular to the wiggle direction, -206 +209 00:12:50,774 --> 00:12:53,120 but gets weaker in all of the other directions. -207 +210 00:12:54,640 --> 00:12:58,240 So think about our setup, and for a particular frequency of light, -208 +211 00:12:58,240 --> 00:13:02,593 how likely it is that an observer looking at a particular point in the tube will -209 +212 00:13:02,593 --> 00:13:03,400 see that light. -210 +213 00:13:04,300 --> 00:13:08,120 Again, the key phenomenon with sugar water, which we have yet to explain, -211 +214 00:13:08,120 --> 00:13:12,560 is that the polarization direction is slowly getting twisted as it goes down the tube. -212 +215 00:13:13,360 --> 00:13:16,183 So suppose the observer was looking at a point like this one, -213 +216 00:13:16,183 --> 00:13:19,280 where the polarization direction happens to be straight up and down. -214 +217 00:13:19,280 --> 00:13:22,829 Then the second-order propagations resulting from wiggling charges -215 +218 00:13:22,829 --> 00:13:26,484 at that point are most strong along the plane where the observer is, -216 +219 00:13:26,484 --> 00:13:30,140 so the amount of red that they see at that point would look stronger. -217 +220 00:13:31,080 --> 00:13:34,759 By contrast, if they were looking at a different point in the tube like this one, -218 +221 00:13:34,759 --> 00:13:38,304 where the wiggling direction is closer to being parallel to the line of sight, -219 +222 00:13:38,304 --> 00:13:41,894 then the direction where the scattering is strongest is not at all aligned with -220 +223 00:13:41,894 --> 00:13:45,260 the observer, and the amount of red they see is only going to be very weak. -221 +224 00:13:46,500 --> 00:13:50,392 And looking at our actual physical setup, if we first pass the light -222 +225 00:13:50,392 --> 00:13:54,680 through a filter showing only the red, we see exactly this effect in action. -223 +226 00:13:55,020 --> 00:13:59,787 As you scan your eyes along the tube, the intensity of red that you see goes -224 +227 00:13:59,787 --> 00:14:04,740 from being high to being low, where it's almost black, back to being high again. -225 +228 00:14:06,040 --> 00:14:09,105 As an analogy, imagine there was a ribbon going down the tube, -226 +229 00:14:09,105 --> 00:14:12,170 always aligned with the polarization direction for this color, -227 +230 00:14:12,170 --> 00:14:14,701 then putting yourself in the shoes of the observer, -228 +231 00:14:14,701 --> 00:14:17,620 when you look at points where the ribbon appears very thin, -229 +232 00:14:17,620 --> 00:14:21,416 you're going to see very little red light, whereas if you scan your eyes over -230 +233 00:14:21,416 --> 00:14:25,260 to points where the ribbon appears thicker, you're going to see more red light. -231 +234 00:14:25,960 --> 00:14:29,777 One thing that's nice about this is that if we try it for various different colors, -232 +235 00:14:29,777 --> 00:14:33,640 you can actually see how the twisting rates are different for each one of the colors. -233 +236 00:14:34,320 --> 00:14:38,573 Notice with red light, the distance between where it appears brightest and where it -234 +237 00:14:38,573 --> 00:14:43,029 appears darkest is relatively long, whereas if you look down the colors of the rainbow, -235 +238 00:14:43,029 --> 00:14:47,080 distance between the brightest point and the darkest point gets lower and lower. -236 +239 00:14:48,720 --> 00:14:52,606 So what you're seeing in effect is how red light twists slowly, -237 +240 00:14:52,606 --> 00:14:57,100 whereas light waves with higher frequencies get twisted more aggressively. -238 +241 00:15:01,260 --> 00:15:03,513 But still, you might wonder why the boundaries -239 +242 00:15:03,513 --> 00:15:05,720 between light and dark points appear diagonal. -240 +243 00:15:06,200 --> 00:15:10,747 Why is it that in addition to having variation as you scan your eyes from left to right, -241 +244 00:15:10,747 --> 00:15:15,040 there's also variation as you scan your eyes from the top of the tube to the bottom? -242 +245 00:15:15,920 --> 00:15:18,487 This has less to do with what's going on in the tube, -243 +246 00:15:18,487 --> 00:15:20,580 and more to do with a matter of perspective. -244 +247 00:15:21,500 --> 00:15:24,116 Take a moment to think about many different parallel beams -245 +248 00:15:24,116 --> 00:15:26,600 of light ranging from the top of the tube to the bottom. -246 +249 00:15:27,020 --> 00:15:30,128 At the beginning, all of these light waves are wiggling up and down, -247 +250 00:15:30,128 --> 00:15:33,237 and as you pass through the tube, and the effects of the sugar water -248 +251 00:15:33,237 --> 00:15:36,346 somehow twists these directions, because they're all passing through -249 +252 00:15:36,346 --> 00:15:39,500 the same amount of sugar, they're getting twisted by the same amounts. -250 +253 00:15:39,500 --> 00:15:44,000 So at all points, the polarization of these waves are parallel to each other. -251 +254 00:15:44,660 --> 00:15:47,641 If you're the observer and you look at the topmost point here, -252 +255 00:15:47,641 --> 00:15:50,907 its wiggling direction is essentially parallel to the line of sight, -253 +256 00:15:50,907 --> 00:15:55,120 so the light scattering from that point is basically not going to reach your eyes at all. -254 +257 00:15:55,280 --> 00:15:56,220 It should appear black. -255 +258 00:15:56,760 --> 00:15:59,940 But if you scan your eyes down the tube, the angle between the line -256 +259 00:15:59,940 --> 00:16:03,026 of sight and the wiggling direction changes, and so there will be -257 +260 00:16:03,026 --> 00:16:06,020 at least some component of red light scattering towards the eye. -258 +261 00:16:06,020 --> 00:16:09,452 So as you scan your eyes from top to bottom, the amount of -259 +262 00:16:09,452 --> 00:16:13,060 a particular color you see might vary, say from dark to light. -260 +263 00:16:14,960 --> 00:16:18,542 The full demo that has white light is basically a combination of all -261 +264 00:16:18,542 --> 00:16:22,020 these pure color patterns that go from light to dark to light with -262 +265 00:16:22,020 --> 00:16:25,550 diagonal boundaries between the intense points and the weak points, -263 +266 00:16:25,550 --> 00:16:29,340 hence why you see diagonal boundaries between the colors inside the tube. -264 +267 00:16:31,220 --> 00:16:35,376 And now at last let's turn to the heart of the matter and try to explain why -265 +268 00:16:35,376 --> 00:16:39,480 interactions with sugar would make light twist like this in the first place. -266 +269 00:16:39,680 --> 00:16:42,273 It's related to the idea that light seems to slow -267 +270 00:16:42,273 --> 00:16:44,400 down as it passes through a given medium. -268 +271 00:16:44,900 --> 00:16:49,070 For example, if you look at the crests of a light wave as it goes into water, -269 +272 00:16:49,070 --> 00:16:52,652 the crests through the water are traveling about 1.33 times slower -270 +273 00:16:52,652 --> 00:16:55,540 than the crests of that wave would travel in a vacuum. -271 +274 00:16:56,280 --> 00:16:58,940 This number is what's called the index of refraction for water. -272 +275 00:16:59,640 --> 00:17:04,290 In a bit, what I'd like to show is how this index of refraction can be explained by -273 +276 00:17:04,290 --> 00:17:08,940 analyzing how the initial light wave shakes all the charges in the material and how -274 +277 00:17:08,940 --> 00:17:13,480 the resulting second order propagations superimpose with that original light wave. -275 +278 00:17:14,280 --> 00:17:17,777 For right now, I'll just say that the interactions with each layer -276 +279 00:17:17,777 --> 00:17:21,327 of the material ends up having the effect of slightly shifting back -277 +280 00:17:21,327 --> 00:17:24,668 the phase of the wave, and on the whole, this gives the overall -278 +281 00:17:24,668 --> 00:17:28,480 appearance that that wave moves slower as it passes through the material. -279 +282 00:17:30,700 --> 00:17:33,865 Skipping ahead to what's going on with sugar, the relevant -280 +283 00:17:33,865 --> 00:17:37,674 property of sucrose here is that it's what's called a chiral molecule, -281 +284 00:17:37,674 --> 00:17:40,840 meaning it's fundamentally different from its mirror image. -282 +285 00:17:41,000 --> 00:17:44,600 You could never reorient it in space to become identical to its mirror image. -283 +286 00:17:44,800 --> 00:17:46,920 It's like a left hand or a right hand. -284 +287 00:17:47,380 --> 00:17:50,740 Or another much simpler example of a chiral shape is a spiral. -285 +288 00:17:51,140 --> 00:17:55,485 If I take this right-handed spiral, then its mirror image is a left-handed spiral, -286 +289 00:17:55,485 --> 00:17:58,888 and no matter how you try to rotate and reorient that first one, -287 +290 00:17:58,888 --> 00:18:01,140 it'll never become identical to the second. -288 +291 00:18:03,560 --> 00:18:07,077 What's going on then is that the presence of a chiral molecule -289 +292 00:18:07,077 --> 00:18:12,046 in the water like this introduces an asymmetry when it comes to interactions with light, -290 +293 00:18:12,046 --> 00:18:14,280 specifically circularly polarized light. -291 +294 00:18:15,060 --> 00:18:18,479 It turns out that the amount this chiral molecule slows down, -292 +295 00:18:18,479 --> 00:18:21,899 say, left-handed circularly polarized light is different from -293 +296 00:18:21,899 --> 00:18:25,760 the amount that it slows down right-handed circularly polarized light. -294 +297 00:18:26,100 --> 00:18:29,240 Effectively, there's not one index of refraction, but two. -295 +298 00:18:30,200 --> 00:18:33,050 Now you might say that seems irrelevant to our setup, -296 +299 00:18:33,050 --> 00:18:36,640 since we are very deliberately shining in linearly polarized light, -297 +300 00:18:36,640 --> 00:18:38,700 there is no circularly polarized light. -298 +301 00:18:39,360 --> 00:18:42,659 But actually there's a sense in which linearly polarized light is -299 +302 00:18:42,659 --> 00:18:46,060 equal parts left-handed and right-handed circularly polarized light. -300 +303 00:18:47,620 --> 00:18:50,778 Here, focus your attention on just one vector in this wave, -301 +304 00:18:50,778 --> 00:18:54,780 wiggling straight up and down, which is to say polarized in the z direction. -302 +305 00:18:55,880 --> 00:19:00,762 Notice how it's possible to express this vector as a sum of two rotating vectors, -303 +306 00:19:00,762 --> 00:19:04,216 one of them rotating at a constant rate counterclockwise, -304 +307 00:19:04,216 --> 00:19:06,420 and the other one rotating clockwise. -305 +308 00:19:07,960 --> 00:19:11,760 Adding them together tip to tail results in a vector oscillating on a line. -306 +309 00:19:13,660 --> 00:19:16,855 In this case, it's a vertical line, but that direction can change -307 +310 00:19:16,855 --> 00:19:19,760 based on the phase of the two vectors we're adding together. -308 +311 00:19:20,440 --> 00:19:24,316 Here, let me throw up a couple labels to keep track of how much each one of those -309 +312 00:19:24,316 --> 00:19:28,335 two vectors has rotated in total, and then every now and then I'm going to slow down -310 +313 00:19:28,335 --> 00:19:32,260 that first vector a little bit, and I want you to notice what happens to their sum. -311 +314 00:19:36,320 --> 00:19:40,913 Well, every time I slow it down, effectively knocking back its phase a little bit, -312 +315 00:19:40,913 --> 00:19:45,340 it causes the linearly wiggling sum to wiggle in a slightly different direction. -313 +316 00:19:46,280 --> 00:19:50,452 So if the circularly polarized light wave represented by that left vector -314 +317 00:19:50,452 --> 00:19:54,624 gets slowed down a little bit every time it runs across a sugar molecule, -315 +318 00:19:54,624 --> 00:19:58,965 or at least slowed down more than its oppositely rotating counterpart would, -316 +319 00:19:58,965 --> 00:20:03,420 the effect on the sum is to slowly rotate the direction of linear polarization. -317 +320 00:20:04,220 --> 00:20:07,383 And hence, as you look at slices further and further down the tube, -318 +321 00:20:07,383 --> 00:20:11,430 the polarization direction does indeed get twisted the way we were describing earlier, -319 +322 00:20:11,430 --> 00:20:14,920 representing how the composite effects with many many many different sugar -320 +323 00:20:14,920 --> 00:20:18,920 molecules are slightly different for left-handed light than they are for right-handed -321 +324 00:20:18,920 --> 00:20:19,200 light. -322 +325 00:20:20,040 --> 00:20:23,800 As a nice way to test whether you understood everything up to this point, -323 +326 00:20:23,800 --> 00:20:27,662 see if just by looking at the direction of the diagonal slices on our tube, -324 +327 00:20:27,662 --> 00:20:31,067 you can deduce which kind of light the sugar is slowing down more, -325 +328 00:20:31,067 --> 00:20:33,100 left-handed light or right-handed light. -326 +329 00:20:35,920 --> 00:20:38,867 I'll call this a partial answer to our question number one, -327 +330 00:20:38,867 --> 00:20:43,042 because it still leaves us wondering why there's an index of refraction in the first -328 +331 00:20:43,042 --> 00:20:46,627 place, and how exactly it might depend on the polarization of the light, -329 +332 00:20:46,627 --> 00:20:48,740 not just the material it's passing through. -330 +333 00:20:49,200 --> 00:20:53,570 Also, like I said at the start, a robust enough intuition here should also answer -331 +334 00:20:53,570 --> 00:20:57,940 for us why the strength of this effect would depend on the frequency of the light. -332 +335 00:20:58,780 --> 00:21:01,782 At this point I think we've covered quite enough for one video, -333 +336 00:21:01,782 --> 00:21:05,582 so I'll pull out a discussion covering the origins of the index of refraction to -334 +337 00:21:05,582 --> 00:21:06,380 a separate video. -335 +338 00:21:19,200 --> 00:21:30,980 Thank you. diff --git a/2023/barber-pole-2/english/sentence_timings.json b/2023/barber-pole-2/english/sentence_timings.json index b1b816657..1d3f41d09 100644 --- a/2023/barber-pole-2/english/sentence_timings.json +++ b/2023/barber-pole-2/english/sentence_timings.json @@ -290,28 +290,28 @@ 477.44 ], [ - "At the extreme, the only place where there's no propagation is in the z axis.", + "At the extreme, the only thing that's more important is the direction that the wave is going to move in the direction that it's going to move in. So if we're going to look at the wiggling at the extreme, the only place where there's no propagation is in the z-axis.", 478.0, - 481.74 + 498.8 ], [ "Because our law has this 1 divided by r in it, the strength of the wave caused by just one particle does decay as you go farther away, in proportion to 1 over r.", - 484.32, - 493.52 + 498.8, + 505.08 ], [ "But notice what happens if I take a whole row of charges, say oriented along the y axis, and I have them all start wiggling up and down in the z direction, and I illustrate the combined effects that all of them have on this component of the electric field.", - 494.28, - 508.32 + 505.08, + 518.26 ], [ "The effects of all these charges interfere deconstructively along the y direction, but they interfere constructively along the x direction.", - 509.82, - 518.26 + 519.02, + 523.6 ], [ "This is what it looks like for a beam of light to be concentrated along just one dimension.", - 519.02, + 524.42, 523.6 ], [ @@ -425,7 +425,7 @@ 710.98 ], [ - "Sometimes when people talk about light bouncing off of things, the implied mental image is a projectile ricocheting off of some object, heading off in some random direction.", + "Sometimes when people talk about light bouncing off of things, the implied mental image is something like a projectile ricocheting off of some object heading off in some random direction.", 711.54, 721.48 ], diff --git a/2023/barber-pole-2/english/transcript.txt b/2023/barber-pole-2/english/transcript.txt index 37da99428..9e31cdeb4 100644 --- a/2023/barber-pole-2/english/transcript.txt +++ b/2023/barber-pole-2/english/transcript.txt @@ -56,7 +56,7 @@ At points that are far enough away from the charge, which is where this componen Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space. It's maybe a little busy to try to illustrate the full three-dimensional vector field on screen like this, so it's clarifying if we just focus on, say, the xz plane. Notice how the waves here are strongest in the x direction, but it still does propagate in all other directions, it's just that that propagation gets weaker in directions that are more aligned with the original wiggling. -At the extreme, the only place where there's no propagation is in the z axis. +At the extreme, the only thing that's more important is the direction that the wave is going to move in the direction that it's going to move in. So if we're going to look at the wiggling at the extreme, the only place where there's no propagation is in the z-axis. Because our law has this 1 divided by r in it, the strength of the wave caused by just one particle does decay as you go farther away, in proportion to 1 over r. But notice what happens if I take a whole row of charges, say oriented along the y axis, and I have them all start wiggling up and down in the z direction, and I illustrate the combined effects that all of them have on this component of the electric field. The effects of all these charges interfere deconstructively along the y direction, but they interfere constructively along the x direction. @@ -83,7 +83,7 @@ Part of the reason I wanted to go through that is because, frankly, it's just ve But now let's turn back to our demo and see how we can build up an intuition for some of our key questions, starting from this very basic premise that shaking a charge in one location causes a shake to another charge a little bit later. And let's start by actually skipping ahead to question number three, why do we see the diagonal stripes? To think about this, you need to imagine an observer to the side of the tube, and then for a particular pure color, say red, if the observer looks in the tube and sees that color, it's because light of that color has bounced off something at that point in the tube, and then propagated towards the eye of the observer. -Sometimes when people talk about light bouncing off of things, the implied mental image is a projectile ricocheting off of some object, heading off in some random direction. +Sometimes when people talk about light bouncing off of things, the implied mental image is something like a projectile ricocheting off of some object heading off in some random direction. But the better mental image to hold in your mind is that when the propagating light waves caused by some wiggling charge reach some second charge causing it to wiggle, that secondary wiggle results in its own propagation. And for the animation on screen, that propagation goes back to the first charge, which itself causes a propagation towards the second. And this is what it looks like in a very simplified situation for light to bounce back and forth between two charges. diff --git a/2023/barber-pole-2/french/sentence_translations.json b/2023/barber-pole-2/french/sentence_translations.json index a945899d2..9f2ab16e7 100644 --- a/2023/barber-pole-2/french/sentence_translations.json +++ b/2023/barber-pole-2/french/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "Comme je l'ai dit, cette propagation pour une seule charge est également forte dans toutes les directions perpendiculaires à son ondulation, et je dois vraiment souligner que la propagation a beaucoup de force dans le champ.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "À ce stade, je pense que nous en avons assez couvert pour une vidéo, je vais donc présenter une discussion couvrant les origines de l'indice de réfraction dans une vidéo distincte.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/german/sentence_translations.json b/2023/barber-pole-2/german/sentence_translations.json index 570978bcc..9d7a81e1e 100644 --- a/2023/barber-pole-2/german/sentence_translations.json +++ b/2023/barber-pole-2/german/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "Wie ich schon sagte, ist diese Ausbreitung für nur eine Ladung in allen Richtungen senkrecht zu ihrer Wackelbewegung gleich stark, und ich sollte wirklich betonen, dass die Ausbreitung im Feld sehr stark ist.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "An dieser Stelle denke ich, dass wir genug für ein Video abgedeckt haben, daher werde ich eine Diskussion über die Ursprünge des Brechungsindex in einem separaten Video zusammenfassen.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/hebrew/sentence_translations.json b/2023/barber-pole-2/hebrew/sentence_translations.json index 7509e3efb..13e93fc5f 100644 --- a/2023/barber-pole-2/hebrew/sentence_translations.json +++ b/2023/barber-pole-2/hebrew/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "כמו שאמרתי, ההתפשטות הזו עבור מטען אחד בלבד היא חזקה באותה מידה בכל הכיוונים הניצבים לתנועותיו, ובאמת שעלי להדגיש שלהתפשטות יש הרבה כוח בשטח.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "בשלב זה אני חושב שכיסינו מספיק עבור סרטון אחד, אז אני אוציא דיון שמסקר את מקורותיו של אינדקס השבירה לסרטון נפרד.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/hindi/sentence_translations.json b/2023/barber-pole-2/hindi/sentence_translations.json index 1fe0813f1..5921fc9f9 100644 --- a/2023/barber-pole-2/hindi/sentence_translations.json +++ b/2023/barber-pole-2/hindi/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field. ", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space. ", "translatedText": "जैसा कि मैंने कहा, केवल एक चार्ज के लिए यह प्रसार अपने हिलने-डुलने के लंबवत सभी दिशाओं में समान रूप से मजबूत है, और वास्तव में मुझे इस बात पर जोर देना चाहिए कि इस क्षेत्र में प्रसार की बहुत ताकत है।", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. ", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you. ", "translatedText": "इस बिंदु पर मुझे लगता है कि हमने एक वीडियो के लिए काफी कुछ कवर कर लिया है, इसलिए मैं एक अलग वीडियो में अपवर्तन सूचकांक की उत्पत्ति को कवर करने वाली चर्चा निकालूंगा।", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/hungarian/sentence_translations.json b/2023/barber-pole-2/hungarian/sentence_translations.json index dbf8bc505..2a7f599c9 100644 --- a/2023/barber-pole-2/hungarian/sentence_translations.json +++ b/2023/barber-pole-2/hungarian/sentence_translations.json @@ -472,7 +472,7 @@ "end": 481.74 }, { - "input": "Because our law has this 1 divided by r in it, the strength of the wave caused by just one particle does decay as you go farther away, in proportion to 1 over r.", + "input": "on that the wave is going to move in the direction that it's going to move in. So if we're going to look at the wiggling at the extreme, the only place where there's no propagation is in the z-axis.", "translatedText": "Mivel a törvényünkben szerepel ez az 1 osztva r-rel, az egyetlen részecske által okozott hullám erőssége csökken, ahogy távolodunk, az 1 és r arányában.", "model": "DeepL", "n_reviews": 0, @@ -480,7 +480,7 @@ "end": 493.52 }, { - "input": "But notice what happens if I take a whole row of charges, say oriented along the y axis, and I have them all start wiggling up and down in the z direction, and I illustrate the combined effects that all of them have on this component of the electric field.", + "input": "Because our law has this 1 divided by r in it, the strength of the wave caused by just one particle does decay as you go farther away, in proportion to 1 over r. But notice what happens if I take a whole row of charges, say oriented along the y axis, and I have them all start wiggling up and down in the z direction, and I illustrate the co", "translatedText": "De figyeljük meg, mi történik, ha veszünk egy egész sor töltést, mondjuk az y tengely mentén, és mindet elkezdjük fel-le mozgatni a z irányban, és szemléltetjük az összes töltés együttes hatását az elektromos mező ezen komponensére.", "model": "DeepL", "n_reviews": 0, @@ -488,7 +488,7 @@ "end": 508.32 }, { - "input": "The effects of all these charges interfere deconstructively along the y direction, but they interfere constructively along the x direction.", + "input": "mbined effects that all of them have on this component of the electric field. The effects of all these charges interfere deconstructively along the y direction, but they interfere constructively along the x direction.", "translatedText": "Mindezen töltések hatása az y irányban dekonstruktívan, az x irányban viszont konstruktívan hat egymásra.", "model": "DeepL", "n_reviews": 0, @@ -680,7 +680,7 @@ "end": 710.98 }, { - "input": "Sometimes when people talk about light bouncing off of things, the implied mental image is a projectile ricocheting off of some object, heading off in some random direction.", + "input": "Sometimes when people talk about light bouncing off of things, the implied mental image is something like a projectile ricocheting off of some object heading off in some random direction.", "translatedText": "Amikor az emberek néha arról beszélnek, hogy a fény visszaverődik a dolgokról, akkor a gondolatmenet egy tárgyról visszapattanó lövedékre gondolnak, amely valamilyen véletlenszerű irányba tart.", "model": "DeepL", "n_reviews": 0, diff --git a/2023/barber-pole-2/indonesian/sentence_translations.json b/2023/barber-pole-2/indonesian/sentence_translations.json index a7474d310..bfa07797d 100644 --- a/2023/barber-pole-2/indonesian/sentence_translations.json +++ b/2023/barber-pole-2/indonesian/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "Seperti saya katakan, perambatan hanya untuk satu muatan ini sama kuatnya di semua arah yang tegak lurus terhadap goyangannya, dan sungguh saya harus menekankan bahwa perambatan tersebut memang memiliki banyak kekuatan di medan.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "Pada titik ini saya rasa kita sudah cukup membahasnya untuk satu video, jadi saya akan menarik pembahasan mengenai asal-usul indeks bias ke dalam video terpisah.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/italian/sentence_translations.json b/2023/barber-pole-2/italian/sentence_translations.json index 770aef371..65b3f822e 100644 --- a/2023/barber-pole-2/italian/sentence_translations.json +++ b/2023/barber-pole-2/italian/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "Come ho detto, questa propagazione per una sola carica è ugualmente forte in tutte le direzioni perpendicolari alla sua oscillazione, e in realtà dovrei sottolineare che la propagazione ha molta forza nel campo.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "A questo punto penso che abbiamo trattato abbastanza per un video, quindi inserirò una discussione sulle origini dell'indice di rifrazione in un video separato.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/japanese/sentence_translations.json b/2023/barber-pole-2/japanese/sentence_translations.json index 1729849b1..ebbe16334 100644 --- a/2023/barber-pole-2/japanese/sentence_translations.json +++ b/2023/barber-pole-2/japanese/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "先ほども述べたように、たった 1 回の電荷によるこの伝播は、その小刻みな動きに垂直なすべての方向に等しく強力です。 実際、この伝播はフィールド内で非常に強力であることを強調しておく必要があります。", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "この時点で、1 つのビデオ で十分な内容をカバーできたと思うので、屈折率の起 源に関する議論を別のビデオに抜粋する予定です。", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/korean/sentence_translations.json b/2023/barber-pole-2/korean/sentence_translations.json index 28f73c7d4..dbf8f92e0 100644 --- a/2023/barber-pole-2/korean/sentence_translations.json +++ b/2023/barber-pole-2/korean/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "내가 말했듯이, 단 하나의 전하에 대한 전파는 흔들리는 방향에 수직인 모든 방향에서 동일하게 강력하며 실제로 전파가 장에서 많은 힘을 갖는다는 점을 강조하고 싶습니다.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "이 시점에서 우리는 하나의 비디오로 충분히 다루었다고 생각하므로 굴절률의 기원에 대한 논의를 별도의 비디오로 꺼내겠습니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/marathi/sentence_translations.json b/2023/barber-pole-2/marathi/sentence_translations.json index 1844570c8..c093fff7f 100644 --- a/2023/barber-pole-2/marathi/sentence_translations.json +++ b/2023/barber-pole-2/marathi/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "मी म्हटल्याप्रमाणे, फक्त एका चार्जसाठी हा प्रसार त्याच्या वळणावळणापर्यंत लंब असलेल्या सर्व दिशांमध्ये तितकाच मजबूत आहे आणि खरोखरच मी यावर जोर दिला पाहिजे की या प्रसाराला क्षेत्रात खूप ताकद आहे.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "या क्षणी मला वाटते की आम्ही एका व्हिडिओसाठी पुरेसे कव्हर केले आहे, म्हणून मी एका वेगळ्या व्हिडिओमध्ये अपवर्तन निर्देशांकाच्या उत्पत्तीवर चर्चा करेन.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/persian/sentence_translations.json b/2023/barber-pole-2/persian/sentence_translations.json index 01eed50fb..a68e48179 100644 --- a/2023/barber-pole-2/persian/sentence_translations.json +++ b/2023/barber-pole-2/persian/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field. ", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space. ", "translatedText": "همانطور که گفتم، این انتشار فقط برای یک بار در تمام جهات عمود بر تکان دادن آن به یک اندازه قوی است و واقعاً باید تأکید کنم که انتشار در میدان قدرت زیادی دارد. ", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. ", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you. ", "translatedText": "در این مرحله، فکر می‌کنم به اندازه کافی برای یک ویدیو پوشش داده‌ایم، بنابراین بحثی را در مورد ریشه‌های ضریب شکست در یک ویدیوی جداگانه مطرح می‌کنم. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/portuguese/sentence_translations.json b/2023/barber-pole-2/portuguese/sentence_translations.json index c6e13bd45..f6d29a1a8 100644 --- a/2023/barber-pole-2/portuguese/sentence_translations.json +++ b/2023/barber-pole-2/portuguese/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "Como eu disse, esta propagação para apenas uma carga é igualmente forte em todas as direções perpendiculares à sua oscilação, e realmente devo enfatizar que a propagação tem muita força no campo.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "Neste ponto, acho que já cobrimos o suficiente para um vídeo, então farei uma discussão sobre as origens do índice de refração em um vídeo separado.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/russian/sentence_translations.json b/2023/barber-pole-2/russian/sentence_translations.json index ee0501313..9d3de6d72 100644 --- a/2023/barber-pole-2/russian/sentence_translations.json +++ b/2023/barber-pole-2/russian/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "Как я уже сказал, это распространение всего лишь одного заряда одинаково сильно во всех направлениях, перпендикулярных его движению, и на самом деле я должен подчеркнуть, что распространение действительно имеет большую силу в поле.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "На данный момент, я думаю, мы рассмотрели достаточно для одного видео, поэтому я вынесу обсуждение происхождения показателя преломления в отдельное видео.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/spanish/sentence_translations.json b/2023/barber-pole-2/spanish/sentence_translations.json index e29b5fc41..daa741178 100644 --- a/2023/barber-pole-2/spanish/sentence_translations.json +++ b/2023/barber-pole-2/spanish/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "Como dije, esta propagación para una sola carga es igualmente fuerte en todas las direcciones perpendiculares a su movimiento, y realmente debo enfatizar que la propagación tiene mucha fuerza en el campo.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "En este punto creo que hemos cubierto suficiente para un video, así que sacaré una discusión que cubre los orígenes del índice de refracción en un video separado.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/tamil/sentence_translations.json b/2023/barber-pole-2/tamil/sentence_translations.json index 15c399c6f..2e1898834 100644 --- a/2023/barber-pole-2/tamil/sentence_translations.json +++ b/2023/barber-pole-2/tamil/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "நான் சொன்னது போல், ஒரே ஒரு கட்டணத்திற்கான இந்த பிரச்சாரம் அதன் அசைவுக்கு செங்குத்தாக அனைத்து திசைகளிலும் சமமாக வலுவாக உள்ளது, மேலும் உண்மையில் பரப்புதல் துறையில் அதிக வலிமையைக் கொண்டுள்ளது என்பதை நான் வலியுறுத்த வேண்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "இந்த கட்டத்தில், ஒரு வீடியோவை நாங்கள் போதுமான அளவு உள்ளடக்கியுள்ளோம் என்று நினைக்கிறேன், எனவே ஒளிவிலகல் குறியீட்டின் தோற்றத்தை ஒரு தனி வீடியோவில் உள்ளடக்கிய ஒரு விவாதத்தை வெளியிடுகிறேன்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/telugu/sentence_translations.json b/2023/barber-pole-2/telugu/sentence_translations.json index ade9ccd42..c25305b3a 100644 --- a/2023/barber-pole-2/telugu/sentence_translations.json +++ b/2023/barber-pole-2/telugu/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "నేను చెప్పినట్లుగా, కేవలం ఒక ఛార్జ్ కోసం ఈ ప్రచారం దాని విగ్లింగ్‌కు లంబంగా ఉన్న అన్ని దిశలలో సమానంగా బలంగా ఉంటుంది మరియు క్షేత్రంలో ప్రచారానికి చాలా బలం ఉందని నేను నొక్కి చెప్పాలి.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "ఈ సమయంలో మేము ఒక వీడియో కోసం తగినంతగా కవర్ చేసామని నేను భావిస్తున్నాను, కాబట్టి నేను వక్రీభవన సూచిక యొక్క మూలాలను ప్రత్యేక వీడియోకి కవర్ చేసే చర్చను తీసివేస్తాను.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/thai/sentence_translations.json b/2023/barber-pole-2/thai/sentence_translations.json index 8044d1cc3..dda339c0d 100644 --- a/2023/barber-pole-2/thai/sentence_translations.json +++ b/2023/barber-pole-2/thai/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field. ", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. ", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/turkish/sentence_translations.json b/2023/barber-pole-2/turkish/sentence_translations.json index 32c1e174a..a0ded7f7c 100644 --- a/2023/barber-pole-2/turkish/sentence_translations.json +++ b/2023/barber-pole-2/turkish/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "Söylediğim gibi, tek bir yük için bu yayılma, kıpırdamaya dik olan tüm yönlerde eşit derecede güçlüdür ve gerçekten de, bu alanda yayılmanın çok fazla güce sahip olduğunu vurgulamalıyım.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "Bu noktada, bir videoya yetecek kadar konuyu ele aldığımızı düşünüyorum, bu nedenle kırılma indisinin kökenlerini kapsayan bir tartışmayı ayrı bir videoda yayınlayacağım.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/ukrainian/sentence_translations.json b/2023/barber-pole-2/ukrainian/sentence_translations.json index b73253976..5b63446cd 100644 --- a/2023/barber-pole-2/ukrainian/sentence_translations.json +++ b/2023/barber-pole-2/ukrainian/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "Як я вже сказав, це розповсюдження тільки одного заряду є однаково сильним у всіх напрямках, перпендикулярних до його ворушіння, і справді я повинен підкреслити, що розповсюдження дійсно має велику силу в полі.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "На даний момент я думаю, що ми розглянули цілком достатньо для одного відео, тому я виведу обговорення походження показника заломлення в окреме відео.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/urdu/sentence_translations.json b/2023/barber-pole-2/urdu/sentence_translations.json index 318e43379..fcbe42dae 100644 --- a/2023/barber-pole-2/urdu/sentence_translations.json +++ b/2023/barber-pole-2/urdu/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field. ", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space. ", "translatedText": "جیسا کہ میں نے کہا، صرف ایک چارج کے لیے یہ پروپیگنڈہ تمام سمتوں میں یکساں طور پر مضبوط ہے جو اس کے گھومنے کے لیے کھڑے ہیں، اور واقعی مجھے اس بات پر زور دینا چاہیے کہ تبلیغ کی فیلڈ میں بہت زیادہ طاقت ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. ", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you. ", "translatedText": "اس مقام پر مجھے لگتا ہے کہ ہم نے ایک ویڈیو کے لیے کافی حد تک احاطہ کر لیا ہے، اس لیے میں ایک الگ ویڈیو میں انڈیکس آف ریفریکشن کی اصلیت کا احاطہ کرنے والی بحث نکالوں گا۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/barber-pole-2/vietnamese/sentence_translations.json b/2023/barber-pole-2/vietnamese/sentence_translations.json index 8b2aeec6c..7cc4a47c2 100644 --- a/2023/barber-pole-2/vietnamese/sentence_translations.json +++ b/2023/barber-pole-2/vietnamese/sentence_translations.json @@ -440,7 +440,7 @@ "end": 442.54 }, { - "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does have a lot of strength in the field.", + "input": "Like I said, this propagation for just one charge is equally strong in all of the directions perpendicular to its wiggling, and really I should emphasize that the propagation does happen in all directions of three-dimensional space.", "translatedText": "Như tôi đã nói, sự truyền lan này chỉ với một điện tích cũng mạnh như nhau theo mọi hướng vuông góc với sự dao động lắc lư của nó, và tôi thực sự cần nhấn mạnh rằng sự truyền lan này thực sự có rất nhiều cường độ trong từ trường.", "model": "google_nmt", "n_reviews": 0, @@ -1128,7 +1128,7 @@ "end": 1257.94 }, { - "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video.", + "input": "At this point I think we've covered quite enough for one video, so I'll pull out a discussion covering the origins of the index of refraction to a separate video. Thank you.", "translatedText": "Tại thời điểm này, tôi nghĩ chúng ta đã trình bày khá đủ cho một video, vì vậy tôi sẽ đưa ra một cuộc thảo luận về nguồn gốc của chiết suất trong một video riêng biệt.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/arabic/sentence_translations.json b/2023/clt/arabic/sentence_translations.json index 0647545e0..41d6ecc28 100644 --- a/2023/clt/arabic/sentence_translations.json +++ b/2023/clt/arabic/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "وبعد ذلك، أود أن أشرح لماذا هذه الوظيفة بالذات هي الشيء الذي نميل إليه، ولماذا تحتوي على باي، وما علاقتها بالدوائر. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/bengali/sentence_translations.json b/2023/clt/bengali/sentence_translations.json index 44c066f1c..6cd85d36f 100644 --- a/2023/clt/bengali/sentence_translations.json +++ b/2023/clt/bengali/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "এবং পরবর্তী, আমি ব্যাখ্যা করতে চাই যে কেন এই বিশেষ ফাংশনটি এমন জিনিস যার দিকে আমরা ঝোঁক, এবং কেন এটিতে একটি পাই রয়েছে, এটি বৃত্তের সাথে কী করতে পারে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/chinese/sentence_translations.json b/2023/clt/chinese/sentence_translations.json index fadf7829f..e252956e8 100644 --- a/2023/clt/chinese/sentence_translations.json +++ b/2023/clt/chinese/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "接下来，我 想解释为什么这个特定的函数是我们所倾向于的 ，为什么它有一个 pi，它与圆有什么关系。", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/dutch/sentence_translations.json b/2023/clt/dutch/sentence_translations.json index 1e0985c9a..254594799 100644 --- a/2023/clt/dutch/sentence_translations.json +++ b/2023/clt/dutch/sentence_translations.json @@ -56,7 +56,7 @@ "end": 76.02 }, { - "input": "This lesson is meant to go back to the basics, giving you the fundamentals on what the background is.", + "input": "This lesson is meant to go back to the basics, giving you the fundamentals on what the central limit theorem is saying, what normal distributions are, and I want to assume minimal background.", "translatedText": "Deze les is bedoeld om terug te gaan naar de basis en je de basis te geven van wat de achtergrond is.", "model": "DeepL", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 779.19 }, { - "input": "That can be interpreted much more reasonably as a distance on our diagram, and it's commonly denoted with the Greek letter sigma, so you know m for standard deviation, but both in Greek.", + "input": "That can be interpreted much more reasonably as a distance on our diagram, and it's commonly denoted with the Greek letter sigma, so you know m for mean as for standard deviation, but both in Greek.", "translatedText": "Dat kan veel redelijker worden geïnterpreteerd als een afstand op ons diagram, en het wordt gewoonlijk aangeduid met de Griekse letter sigma, dus je kent m voor standaardafwijking, maar allebei in het Grieks.", "model": "DeepL", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 878.25 }, { - "input": "For example, back in our sequence of distributions, if we label the standard deviation of our initial one with sigma, then the next standard deviation is going to be the square root of 2 times sigma, and after that it looks like the square root of 3 times sigma, and so on This, like I said, is very important.", + "input": "For example, back in our sequence of distributions, if we label the standard deviation of our initial one with sigma, then the next standard deviation is going to be the square root of 2 times sigma, and after that it looks like the square root of 3 times sigma, and so on and so forth. This, like I said, is very important.", "translatedText": "Als we bijvoorbeeld in onze reeks verdelingen de standaardafwijking van onze eerste verdelingen labelen met sigma, dan wordt de volgende standaardafwijking de vierkantswortel van 2 keer sigma, en daarna lijkt het op de vierkantswortel van 3 keer sigma, enzovoort Dit is, zoals ik al zei, heel belangrijk.", "model": "DeepL", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1525.83 }, { - "input": "For questions like this, there's a handy rule of thumb about normal distributions, which is that about 68% of your values are going to fall within two standard deviations of the mean, and a whopping 99.7% of your values will fall within three standard deviations of the mean.", + "input": "For questions like this, there's a handy rule of thumb about normal distributions, which is that about 68% of your values are going to fall within one standard deviation of the mean, 95% of your values, the thing we care about, fall within two standard deviations of the mean, and a whopping 99.7% of your values will fall within three standard deviations of the mean.", "translatedText": "Voor vragen als deze is er een handige vuistregel over normale verdelingen, namelijk dat ongeveer 68% van je waarden binnen twee standaardafwijkingen van het gemiddelde vallen, en maar liefst 99,7% van je waarden binnen drie standaardafwijkingen van het gemiddelde.", "model": "DeepL", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1551.45 }, { - "input": "Naturally, this gives us what we need for our example, and let me go ahead and draw out what this would look like, where I'll show the distribution for a fair die up at the top, and the distribution for a sum of 100 such dice on bottom, which by now looks like a normal distribution.", + "input": "Naturally, this gives us what we need for our example, and let me go ahead and draw out what this would look like, where I'll show the distribution for a fair die up at the top, and the distribution for a sum of 100 such dice on the bottom, which by now as you know looks like a certain normal distribution.", "translatedText": "Dit geeft ons natuurlijk wat we nodig hebben voor ons voorbeeld, en laat ik eens uittekenen hoe dit eruit zou zien, waarbij ik de verdeling voor een eerlijke dobbelsteen bovenaan laat zien, en de verdeling voor een som van 100 van zulke dobbelstenen onderaan, wat er nu uitziet als een normale verdeling.", "model": "DeepL", "n_reviews": 0, @@ -1448,7 +1448,7 @@ "end": 1578.91 }, { - "input": "We also need the standard deviation, which requires calculating the variance, which as you know involves adding all the squares of the differences between the values and the means, and it works out to be 2.92, the square root of 1.71.", + "input": "We also need the standard deviation, which requires calculating the variance, which as you know involves adding all the squares of the differences between the values and the means, and it works out to be 2.92, square root of that comes out to be 1.71.", "translatedText": "We hebben ook de standaardafwijking nodig, waarvoor we de variantie moeten berekenen, wat zoals je weet inhoudt dat we alle kwadraten van de verschillen tussen de waarden en de gemiddelden bij elkaar moeten optellen, en dat wordt 2,92, de vierkantswortel van 1,71.", "model": "DeepL", "n_reviews": 0, diff --git a/2023/clt/english/captions.srt b/2023/clt/english/captions.srt index 541821338..dfc6a3cc3 100644 --- a/2023/clt/english/captions.srt +++ b/2023/clt/english/captions.srt @@ -79,1994 +79,2010 @@ it's one of the key facts that explains why this distribution is as common as it known as the central limit theorem. 21 -00:01:16,640 --> 00:01:20,651 +00:01:16,640 --> 00:01:18,761 This lesson is meant to go back to the basics, 22 -00:01:20,651 --> 00:01:25,260 -giving you the fundamentals on what the background is. +00:01:18,761 --> 00:01:22,055 +giving you the fundamentals on what the central limit theorem is saying, 23 +00:01:22,055 --> 00:01:25,260 +what normal distributions are, and I want to assume minimal background. + +24 00:01:25,260 --> 00:01:29,364 We're going to go decently deep into it, but after this I'd still like to -24 +25 00:01:29,364 --> 00:01:31,971 go deeper and explain why the theorem is true, -25 +26 00:01:31,971 --> 00:01:36,075 why the function underlying the normal distribution has the very specific -26 +27 00:01:36,075 --> 00:01:39,791 form that it does, why that formula has a pi in it, and, most fun, -27 +28 00:01:39,791 --> 00:01:44,062 why those last two facts are actually more related than a lot of traditional -28 +29 00:01:44,062 --> 00:01:45,560 explanations would suggest. -29 +30 00:01:46,480 --> 00:01:50,047 That second lesson is also meant to be the follow-on to the convolutions -30 +31 00:01:50,047 --> 00:01:53,370 video that I promised, so there's a lot of interrelated topics here. -31 +32 00:01:53,570 --> 00:01:56,254 But right now, back to the fundamentals, I'd like to kick -32 +33 00:01:56,254 --> 00:01:59,170 things off with an overly simplified model of the Galton board. -33 +34 00:02:00,890 --> 00:02:05,246 In this model we will assume that each ball falls directly onto a certain central peg, -34 +35 00:02:05,246 --> 00:02:09,102 and that it has a 50-50 probability of bouncing to the left or to the right, -35 +36 00:02:09,102 --> 00:02:13,459 and we'll think of each of those outcomes as either adding one or subtracting one from -36 +37 00:02:13,459 --> 00:02:14,110 its position. -37 +38 00:02:14,670 --> 00:02:18,734 Once one of those is chosen, we make the highly unrealistic assumption that it -38 +39 00:02:18,734 --> 00:02:22,233 happens to land dead on in the middle of the peg adjacent below it, -39 +40 00:02:22,233 --> 00:02:26,401 where again it'll be faced with the same 50-50 choice of bouncing to the left or -40 +41 00:02:26,401 --> 00:02:27,070 to the right. -41 +42 00:02:27,430 --> 00:02:31,116 For the one I'm showing on screen, there are five different rows of pegs, -42 +43 00:02:31,116 --> 00:02:35,101 so our little hopping ball makes five different random choices between plus one -43 +44 00:02:35,101 --> 00:02:38,936 and minus one, and we can think of its final position as basically being the -44 +45 00:02:38,936 --> 00:02:42,772 sum of all of those different numbers, which in this case happens to be one, -45 +46 00:02:42,772 --> 00:02:46,856 and we might label all of the different buckets with the sum that they represent, -46 +47 00:02:46,856 --> 00:02:51,290 as we repeat this we're looking at different possible sums for those five random numbers. -47 +48 00:02:53,050 --> 00:02:57,342 And for those of you who are inclined to complain that this is a highly unrealistic model -48 +49 00:02:57,342 --> 00:03:01,635 for the true Galton board, let me emphasize the goal right now is not to accurately model -49 +50 00:03:01,635 --> 00:03:05,785 physics, the goal is to give a simple example to illustrate the central limit theorem, -50 +51 00:03:05,785 --> 00:03:10,030 and for that, idealized though this might be, it actually gives us a really good example. -51 +52 00:03:10,570 --> 00:03:13,740 If we let many different balls fall, making yet another unrealistic -52 +53 00:03:13,740 --> 00:03:17,236 assumption that they don't influence each other, as if they're all ghosts, -53 +54 00:03:17,236 --> 00:03:20,406 then the number of balls that fall into each different bucket gives -54 +55 00:03:20,406 --> 00:03:23,390 us some loose sense for how likely each one of those buckets is. -55 +56 00:03:23,830 --> 00:03:26,898 In this example, the numbers are simple enough that it's not too hard to -56 +57 00:03:26,898 --> 00:03:30,010 explicitly calculate what the probability is for falling into each bucket. -57 +58 00:03:30,270 --> 00:03:34,018 If you do want to think that through, you'll find it very reminiscent of Pascal's -58 +59 00:03:34,018 --> 00:03:37,858 triangle, but the neat thing about our theorem is how far it goes beyond the simple -59 +60 00:03:37,858 --> 00:03:38,270 examples. -60 +61 00:03:38,670 --> 00:03:41,683 So to start off at least, rather than making explicit calculations, -61 +62 00:03:41,683 --> 00:03:45,494 let's just simulate things by running a large number of samples and letting the total -62 +63 00:03:45,494 --> 00:03:49,482 number of results in each different outcome give us some sense for what that distribution -63 +64 00:03:49,482 --> 00:03:49,970 looks like. -64 +65 00:03:50,450 --> 00:03:53,275 As I said, the one on screen has five rows, so each -65 +66 00:03:53,275 --> 00:03:56,210 sum that we're considering includes only five numbers. -66 +67 00:03:56,810 --> 00:04:01,901 The basic idea of the central limit theorem is that if you increase the size of that sum, -67 +68 00:04:01,901 --> 00:04:06,144 for example here would mean increasing the number of rows of pegs for each -68 +69 00:04:06,144 --> 00:04:10,218 ball to bounce off, then the distribution that describes where that sum -69 +70 00:04:10,218 --> 00:04:13,330 is going to fall looks more and more like a bell curve. -70 +71 00:04:15,470 --> 00:04:18,350 Here, it's actually worth taking a moment to write down that general idea. -71 +72 00:04:19,269 --> 00:04:23,647 The setup is that we have a random variable, and that's basically shorthand for -72 +73 00:04:23,647 --> 00:04:28,190 a random process where each outcome of that process is associated with some number. -73 +74 00:04:28,490 --> 00:04:29,970 We'll call that random number x. -74 +75 00:04:29,970 --> 00:04:34,390 For example, each bounce off the peg is a random process modeled with two outcomes. -75 +76 00:04:34,850 --> 00:04:37,890 Those outcomes are associated with the numbers negative one and positive one. -76 +77 00:04:38,530 --> 00:04:41,397 Another example of a random variable would be rolling a die, -77 +78 00:04:41,397 --> 00:04:44,830 where you have six different outcomes, each one associated with a number. -78 +79 00:04:45,470 --> 00:04:47,742 What we're doing is taking multiple different -79 +80 00:04:47,742 --> 00:04:50,410 samples of that variable and adding them all together. -80 +81 00:04:50,770 --> 00:04:54,232 On our Galton board, that looks like letting the ball bounce off multiple -81 +82 00:04:54,232 --> 00:04:57,601 different pegs on its way down to the bottom, and in the case of a die, -82 +83 00:04:57,601 --> 00:05:00,970 you might imagine rolling many different dice and adding up the results. -83 +84 00:05:01,430 --> 00:05:05,674 The claim of the central limit theorem is that as you let the size of that sum -84 +85 00:05:05,674 --> 00:05:08,790 get bigger and bigger, then the distribution of that sum, -85 +86 00:05:08,790 --> 00:05:11,853 how likely it is to fall into different possible values, -86 +87 00:05:11,853 --> 00:05:14,110 will look more and more like a bell curve. -87 +88 00:05:15,430 --> 00:05:17,130 That's it, that is the general idea. -88 +89 00:05:17,550 --> 00:05:21,530 Over the course of this lesson, our job is to make that statement more quantitative. -89 +90 00:05:22,070 --> 00:05:24,606 We're going to put some numbers to it, put some formulas to it, -90 +91 00:05:24,606 --> 00:05:26,350 show how you can use it to make predictions. -91 +92 00:05:27,210 --> 00:05:29,345 For example, here's the kind of question I want -92 +93 00:05:29,345 --> 00:05:31,570 you to be able to answer by the end of this video. -93 +94 00:05:32,190 --> 00:05:35,890 Suppose you rolled a die 100 times and you added together the results. -94 +95 00:05:36,630 --> 00:05:39,400 Could you find a range of values such that you're -95 +96 00:05:39,400 --> 00:05:42,170 95% sure that the sum will fall within that range? -96 +97 00:05:42,830 --> 00:05:46,550 Or maybe I should say find the smallest possible range of values such that this is true. -97 +98 00:05:47,390 --> 00:05:49,915 The neat thing is you'll be able to answer this question -98 +99 00:05:49,915 --> 00:05:52,130 whether it's a fair die or if it's a weighted die. -99 +100 00:05:53,450 --> 00:05:56,746 Now let me say at the top that this theorem has three different assumptions -100 +101 00:05:56,746 --> 00:06:00,130 that go into it, three things that have to be true before the theorem follows. -101 +102 00:06:00,430 --> 00:06:03,790 And I'm actually not going to tell you what they are until the very end of the video. -102 +103 00:06:04,270 --> 00:06:07,032 Instead I want you to keep your eye out and see if you can notice -103 +104 00:06:07,032 --> 00:06:09,670 and maybe predict what those three assumptions are going to be. -104 +105 00:06:10,710 --> 00:06:13,936 As a next step, to better illustrate just how general this theorem is, -105 +106 00:06:13,936 --> 00:06:17,390 I want to run a couple more simulations for you focused on the dice example. -106 +107 00:06:20,910 --> 00:06:24,223 Usually if you think of rolling a die you think of the six outcomes as -107 +108 00:06:24,223 --> 00:06:27,630 being equally probable, but the theorem actually doesn't care about that. -108 +109 00:06:27,830 --> 00:06:31,216 We could start with a weighted die, something with a non-trivial -109 +110 00:06:31,216 --> 00:06:34,550 distribution across the outcomes, and the core idea still holds. -110 +111 00:06:35,030 --> 00:06:37,661 For the simulation what I'll do is take some distribution -111 +112 00:06:37,661 --> 00:06:39,930 like this one that is skewed towards lower values. -112 +113 00:06:40,250 --> 00:06:43,872 I'm going to take 10 distinct samples from that distribution and -113 +114 00:06:43,872 --> 00:06:47,550 then I'll record the sum of that sample on the plot on the bottom. -114 +115 00:06:48,630 --> 00:06:52,610 Then I'm going to do this many many different times, always with a sum of size 10, -115 +116 00:06:52,610 --> 00:06:56,590 but keep track of where those sums ended up to give us a sense of the distribution. -116 +117 00:06:59,970 --> 00:07:02,422 And in fact let me rescale the y direction to give -117 +118 00:07:02,422 --> 00:07:04,730 us room to run an even larger number of samples. -118 +119 00:07:05,030 --> 00:07:07,891 And I'll let it go all the way up to a couple thousand, -119 +120 00:07:07,891 --> 00:07:12,490 and as it does you'll notice that the shape that starts to emerge looks like a bell curve. -120 +121 00:07:12,870 --> 00:07:16,461 Maybe if you squint your eyes you can see it skews a tiny bit to the left, -121 +122 00:07:16,461 --> 00:07:20,483 but it's neat that something so symmetric emerged from a starting point that was so -122 +123 00:07:20,483 --> 00:07:21,010 asymmetric. -123 +124 00:07:21,470 --> 00:07:24,737 To better illustrate what the central limit theorem is all about, -124 +125 00:07:24,737 --> 00:07:27,212 let me run four of these simulations in parallel, -125 +126 00:07:27,212 --> 00:07:31,221 where on the upper left I'm doing it where we're only adding two dice at a time, -126 +127 00:07:31,221 --> 00:07:34,885 on the upper right we're doing it where we're adding five dice at a time, -127 +128 00:07:34,885 --> 00:07:38,300 the lower left is the one that we just saw adding 10 dice at a time, -128 +129 00:07:38,300 --> 00:07:41,370 and then we'll do another one with a bigger sum, 15 at a time. -129 +130 00:07:42,250 --> 00:07:45,110 Notice how on the upper left when we're just adding two dice, -130 +131 00:07:45,110 --> 00:07:48,154 the resulting distribution doesn't really look like a bell curve, -131 +132 00:07:48,154 --> 00:07:52,030 it looks a lot more reminiscent of the one we started with, skewed towards the left. -132 +133 00:07:52,810 --> 00:07:55,428 But as we allow for more and more dice in each sum, -133 +134 00:07:55,428 --> 00:07:59,810 the resulting shape that comes up in these distributions looks more and more symmetric. -134 +135 00:07:59,950 --> 00:08:03,890 It has the lump in the middle and fade towards the tail's shape of a bell curve. -135 +136 00:08:07,050 --> 00:08:10,490 And let me emphasize again, you can start with any different distribution. -136 +137 00:08:10,490 --> 00:08:13,990 Here I'll run it again, but where most of the probability is tied up -137 +138 00:08:13,990 --> 00:08:17,490 in the numbers 1 and 6, with very low probability for the mid values. -138 +139 00:08:18,190 --> 00:08:22,124 Despite completely changing the distribution for an individual roll of the die, -139 +140 00:08:22,124 --> 00:08:26,550 it's still the case that a bell curve shape will emerge as we consider the different sums. -140 +141 00:08:27,270 --> 00:08:30,384 Illustrating things with a simulation like this is very fun, -141 +142 00:08:30,384 --> 00:08:33,141 and it's kind of neat to see order emerge from chaos, -142 +143 00:08:33,141 --> 00:08:35,030 but it also feels a little imprecise. -143 +144 00:08:35,390 --> 00:08:38,559 Like in this case, when I cut off the simulation at 3000 samples, -144 +145 00:08:38,559 --> 00:08:40,865 even though it kind of looks like a bell curve, -145 +146 00:08:40,865 --> 00:08:43,891 the different buckets seem pretty spiky, and you might wonder, -146 +147 00:08:43,891 --> 00:08:47,157 is it supposed to look that way, or is that just an artifact of the -147 +148 00:08:47,157 --> 00:08:48,550 randomness in the simulation? -148 +149 00:08:49,010 --> 00:08:52,151 And if it is, how many samples do we need before we can be sure that -149 +150 00:08:52,151 --> 00:08:55,110 what we're looking at is representative of the true distribution? -150 +151 00:08:59,190 --> 00:09:02,376 Instead moving forward, let's get a little more theoretical and show -151 +152 00:09:02,376 --> 00:09:05,470 the precise shape these distributions will take on in the long run. -152 +153 00:09:06,130 --> 00:09:10,367 The easiest case to make this calculation is if we have a uniform distribution, -153 +154 00:09:10,367 --> 00:09:13,970 where each possible face of the die has an equal probability, 1 6th. -154 +155 00:09:13,990 --> 00:09:18,271 For example, if you then want to know how likely different sums are for a pair of dice, -155 +156 00:09:18,271 --> 00:09:21,726 it's essentially a counting game, where you count up how many distinct -156 +157 00:09:21,726 --> 00:09:24,694 pairs take on the same sum, which in the diagram I've drawn, -157 +158 00:09:24,694 --> 00:09:28,490 you can conveniently think about by going through all the different diagonals. -158 +159 00:09:31,410 --> 00:09:34,266 Since each such pair has an equal chance of showing up, -159 +160 00:09:34,266 --> 00:09:37,530 1 in 36, all you have to do is count the sizes of these buckets. -160 +161 00:09:38,190 --> 00:09:42,428 That gives us a definitive shape for the distribution describing a sum of two dice, -161 +162 00:09:42,428 --> 00:09:45,708 and if we were to play the same game with all possible triplets, -162 +163 00:09:45,708 --> 00:09:48,130 the resulting distribution would look like this. -163 +164 00:09:48,690 --> 00:09:51,292 Now what's more challenging, but a lot more interesting, -164 +165 00:09:51,292 --> 00:09:54,990 is to ask what happens if we have a non-uniform distribution for that single die. -165 +166 00:09:55,550 --> 00:09:57,970 We actually talked all about this in the last video. -166 +167 00:09:58,450 --> 00:10:00,966 You do essentially the same thing, you go through all -167 +168 00:10:00,966 --> 00:10:03,670 the distinct pairs of dice which add up to the same value. -168 +169 00:10:03,970 --> 00:10:06,478 It's just that instead of counting those pairs, -169 +170 00:10:06,478 --> 00:10:10,868 for each pair you multiply the two probabilities of each particular face coming up, -170 +171 00:10:10,868 --> 00:10:12,750 and then you add all those together. -171 +172 00:10:13,290 --> 00:10:16,682 The computation that does this for all possible sums has a fancy name, -172 +173 00:10:16,682 --> 00:10:20,361 it's called a convolution, but it's essentially just the weighted version of -173 +174 00:10:20,361 --> 00:10:24,470 the counting game that anyone who's played with a pair of dice already finds familiar. -174 +175 00:10:25,030 --> 00:10:28,944 For our purposes in this lesson, I'll have the computer calculate all that, -175 +176 00:10:28,944 --> 00:10:33,064 simply display the results for you, and invite you to observe certain patterns, -176 +177 00:10:33,064 --> 00:10:35,330 but under the hood, this is what's going on. -177 +178 00:10:36,650 --> 00:10:39,924 So just to be crystal clear on what's being represented here, -178 +179 00:10:39,924 --> 00:10:43,779 if you imagine sampling two different values from that top distribution, -179 +180 00:10:43,779 --> 00:10:46,895 the one describing a single die, and adding them together, -180 +181 00:10:46,895 --> 00:10:50,804 then the second distribution I'm drawing represents how likely you are to -181 +182 00:10:50,804 --> 00:10:52,230 see various different sums. -182 +183 00:10:52,890 --> 00:10:57,068 Likewise, if you imagine sampling three distinct values from that top distribution, -183 +184 00:10:57,068 --> 00:11:00,500 and adding them together, the next plot represents the probabilities -184 +185 00:11:00,500 --> 00:11:02,490 for various different sums in that case. -185 +186 00:11:03,510 --> 00:11:08,002 So if I compute what the distributions for these sums look like for larger and larger -186 +187 00:11:08,002 --> 00:11:12,390 sums, well you know what I'm going to say, it looks more and more like a bell curve. -187 +188 00:11:13,350 --> 00:11:16,450 But before we get to that, I want you to make a couple more simple observations. -188 +189 00:11:17,450 --> 00:11:20,892 For example, these distributions seem to be wandering to the right, -189 +190 00:11:20,892 --> 00:11:24,790 and also they seem to be getting more spread out, and a little bit more flat. -190 +191 00:11:25,250 --> 00:11:27,911 You cannot describe the central limit theorem quantitatively -191 +192 00:11:27,911 --> 00:11:30,136 without taking into account both of those effects, -192 +193 00:11:30,136 --> 00:11:33,190 which in turn requires describing the mean and the standard deviation. -193 +194 00:11:33,950 --> 00:11:37,664 Maybe you're already familiar with those, but I want to make minimal assumptions here, -194 +195 00:11:37,664 --> 00:11:40,610 and it never hurts to review, so let's quickly go over both of those. -195 +196 00:11:43,410 --> 00:11:47,199 The mean of a distribution, often denoted with the Greek letter mu, -196 +197 00:11:47,199 --> 00:11:50,710 is a way of capturing the center of mass for that distribution. -197 +198 00:11:51,190 --> 00:11:54,431 It's calculated as the expected value of our random variable, -198 +199 00:11:54,431 --> 00:11:58,614 which is a way of saying you go through all of the different possible outcomes, -199 +200 00:11:58,614 --> 00:12:02,850 and you multiply the probability of that outcome times the value of the variable. -200 +201 00:12:03,190 --> 00:12:06,410 If higher values are more probable, that weighted sum is going to be bigger. -201 +202 00:12:06,750 --> 00:12:09,950 If lower values are more probable, that weighted sum is going to be smaller. -202 +203 00:12:10,790 --> 00:12:14,681 A little more interesting is if you want to measure how spread out this distribution is, -203 +204 00:12:14,681 --> 00:12:17,130 because there's multiple different ways you might do it. -204 +205 00:12:18,530 --> 00:12:20,290 One of them is called the variance. -205 +206 00:12:20,830 --> 00:12:25,367 The idea there is to look at the difference between each possible value and the mean, -206 +207 00:12:25,367 --> 00:12:28,270 square that difference, and ask for its expected value. -207 +208 00:12:28,730 --> 00:12:31,577 The idea is that whether your value is below or above the mean, -208 +209 00:12:31,577 --> 00:12:34,247 when you square that difference, you get a positive number, -209 +210 00:12:34,247 --> 00:12:36,650 and the larger the difference, the bigger that number. -210 +211 00:12:37,370 --> 00:12:41,165 Squaring it like this turns out to make the math much much nicer than if we did -211 +212 00:12:41,165 --> 00:12:45,150 something like an absolute value, but the downside is that it's hard to think about -212 +213 00:12:45,150 --> 00:12:48,044 this as a distance in our diagram because the units are off, -213 +214 00:12:48,044 --> 00:12:52,123 kind of like the units here are square units, whereas a distance in our diagram would -214 +215 00:12:52,123 --> 00:12:53,310 be a kind of linear unit. -215 +216 00:12:53,710 --> 00:12:57,298 So another way to measure spread is what's called the standard deviation, -216 +217 00:12:57,298 --> 00:12:59,190 which is the square root of this value. -217 -00:12:59,470 --> 00:13:03,574 -That can be interpreted much more reasonably as a distance on our diagram, - 218 -00:13:03,574 --> 00:13:06,585 -and it's commonly denoted with the Greek letter sigma, +00:12:59,470 --> 00:13:03,326 +That can be interpreted much more reasonably as a distance on our diagram, 219 -00:13:06,585 --> 00:13:09,650 -so you know m for standard deviation, but both in Greek. +00:13:03,326 --> 00:13:06,153 +and it's commonly denoted with the Greek letter sigma, 220 +00:13:06,153 --> 00:13:09,650 +so you know m for mean as for standard deviation, but both in Greek. + +221 00:13:11,870 --> 00:13:13,965 Looking back at our sequence of distributions, -221 +222 00:13:13,965 --> 00:13:16,150 let's talk about the mean and standard deviation. -222 +223 00:13:16,630 --> 00:13:19,295 If we call the mean of the initial distribution mu, -223 +224 00:13:19,295 --> 00:13:21,859 which for the one illustrated happens to be 2.24, -224 +225 00:13:21,859 --> 00:13:25,191 hopefully it won't be too surprising if I tell you that the mean -225 +226 00:13:25,191 --> 00:13:26,730 of the next one is 2 times mu. -226 +227 00:13:27,130 --> 00:13:30,631 That is, you roll a pair of dice, you want to know the expected value of the sum, -227 +228 00:13:30,631 --> 00:13:32,810 it's two times the expected value for a single die. -228 +229 00:13:33,850 --> 00:13:39,410 Similarly, the expected value for our sum of size 3 is 3 times mu, and so on and so forth. -229 +230 00:13:39,630 --> 00:13:41,708 The mean just marches steadily on to the right, -230 +231 00:13:41,708 --> 00:13:44,870 which is why our distributions seem to be drifting off in that direction. -231 +232 00:13:45,350 --> 00:13:47,559 A little more challenging, but very important, -232 +233 00:13:47,559 --> 00:13:49,910 is to describe how the standard deviation changes. -233 +234 00:13:50,490 --> 00:13:53,905 The key fact here is that if you have two different random variables, -234 +235 00:13:53,905 --> 00:13:56,881 then the variance for the sum of those variables is the same -235 +236 00:13:56,881 --> 00:13:59,370 as just adding together the original two variances. -236 +237 00:13:59,930 --> 00:14:03,630 This is one of those facts that you can just compute when you unpack all the definitions. -237 +238 00:14:03,630 --> 00:14:06,210 There are a couple nice intuitions for why it's true. -238 +239 00:14:06,630 --> 00:14:10,033 My tentative plan is to just actually make a series about probability and -239 +240 00:14:10,033 --> 00:14:13,530 talk about things like intuitions underlying variance and its cousins there. -240 +241 00:14:14,010 --> 00:14:18,196 But right now, the main thing I want you to highlight is how it's the variance that adds, -241 +242 00:14:18,196 --> 00:14:20,150 it's not the standard deviation that adds. -242 +243 00:14:20,410 --> 00:14:24,926 So, critically, if you were to take n different realizations of the same random -243 +244 00:14:24,926 --> 00:14:29,273 variable and ask what the sum looks like, the variance of sum is n times the -244 +245 00:14:29,273 --> 00:14:33,112 variance of your original variable, meaning the standard deviation, -245 +246 00:14:33,112 --> 00:14:37,685 the square root of all this, is the square root of n times the original standard -246 +247 00:14:37,685 --> 00:14:38,250 deviation. -247 -00:14:39,290 --> 00:14:42,034 -For example, back in our sequence of distributions, - 248 -00:14:42,034 --> 00:14:45,517 -if we label the standard deviation of our initial one with sigma, +00:14:39,290 --> 00:14:41,915 +For example, back in our sequence of distributions, 249 -00:14:45,517 --> 00:14:49,844 -then the next standard deviation is going to be the square root of 2 times sigma, +00:14:41,915 --> 00:14:45,248 +if we label the standard deviation of our initial one with sigma, 250 -00:14:49,844 --> 00:14:54,014 -and after that it looks like the square root of 3 times sigma, and so on This, +00:14:45,248 --> 00:14:49,388 +then the next standard deviation is going to be the square root of 2 times sigma, 251 -00:14:54,014 --> 00:14:55,650 -like I said, is very important. +00:14:49,388 --> 00:14:53,781 +and after that it looks like the square root of 3 times sigma, and so on and so forth. 252 +00:14:53,781 --> 00:14:55,650 +This, like I said, is very important. + +253 00:14:56,070 --> 00:14:59,048 It means that even though our distributions are getting spread out, -253 +254 00:14:59,048 --> 00:15:01,676 they're not spreading out all that quickly, they only do so -254 +255 00:15:01,676 --> 00:15:04,130 in proportion to the square root of the size of the sum. -255 +256 00:15:04,710 --> 00:15:08,794 As we prepare to make a more quantitative description of the central limit theorem, -256 +257 00:15:08,794 --> 00:15:12,830 the core intuition I want you to keep in your head is that we'll basically realign -257 +258 00:15:12,830 --> 00:15:15,990 all of these distributions so that their means line up together, -258 +259 00:15:15,990 --> 00:15:19,977 and then rescale them so that all of the standard deviations are just going to be -259 +260 00:15:19,977 --> 00:15:20,610 equal to one. -260 +261 00:15:21,290 --> 00:15:25,727 And when we do that, the shape that results gets closer and closer to a certain universal -261 +262 00:15:25,727 --> 00:15:29,870 shape, described with an elegant little function that we'll unpack in just a moment. -262 +263 00:15:30,470 --> 00:15:34,851 And let me say one more time, the real magic here is how we could have started with -263 +264 00:15:34,851 --> 00:15:39,284 any distribution, describing a single roll of the die, and if we play the same game, -264 +265 00:15:39,284 --> 00:15:43,144 considering what the distributions for the many different sums look like, -265 +266 00:15:43,144 --> 00:15:45,595 and we realign them so that the means line up, -266 +267 00:15:45,595 --> 00:15:48,986 and we rescale them so that the standard deviations are all one, -267 +268 00:15:48,986 --> 00:15:52,950 we still approach that same universal shape, which is kind of mind-boggling. -268 +269 00:15:54,810 --> 00:15:57,749 And now, my friends, is probably as good a time as any -269 +270 00:15:57,749 --> 00:16:00,850 to finally get into the formula for a normal distribution. -270 +271 00:16:01,490 --> 00:16:03,648 And the way I'd like to do this is to basically peel -271 +272 00:16:03,648 --> 00:16:05,930 back all the layers and build it up one piece at a time. -272 +273 00:16:06,530 --> 00:16:10,610 The function e to the x, or anything to the x, describes exponential growth, -273 +274 00:16:10,610 --> 00:16:15,008 and if you make that exponent negative, which flips around the graph horizontally, -274 +275 00:16:15,008 --> 00:16:17,870 you might think of it as describing exponential decay. -275 +276 00:16:18,510 --> 00:16:21,926 To make this decay in both directions, you could do something to make sure the -276 +277 00:16:21,926 --> 00:16:25,430 exponent is always negative and growing, like taking the negative absolute value. -277 +278 00:16:25,930 --> 00:16:29,097 That would give us this kind of awkward sharp point in the middle, -278 +279 00:16:29,097 --> 00:16:32,122 but if instead you make that exponent the negative square of x, -279 +280 00:16:32,122 --> 00:16:35,810 you get a smoother version of the same thing, which decays in both directions. -280 +281 00:16:36,330 --> 00:16:38,190 This gives us the basic bell curve shape. -281 +282 00:16:38,650 --> 00:16:41,006 Now if you throw a constant in front of that x, -282 +283 00:16:41,006 --> 00:16:44,197 and you scale that constant up and down, it lets you stretch and -283 +284 00:16:44,197 --> 00:16:48,370 squish the graph horizontally, allowing you to describe narrow and wider bell curves. -284 +285 00:16:49,010 --> 00:16:52,434 And a quick thing I'd like to point out here is that based on the -285 +286 00:16:52,434 --> 00:16:55,599 rules of exponentiation, as we tweak around that constant c, -286 +287 00:16:55,599 --> 00:16:59,750 you could also think about it as simply changing the base of the exponentiation. -287 +288 00:17:00,150 --> 00:17:03,630 And in that sense, the number e is not really all that special for our formula. -288 +289 00:17:04,050 --> 00:17:06,924 We could replace it with any other positive constant, -289 +290 00:17:06,924 --> 00:17:10,490 and you'll get the same family of curves as we tweak that constant. -290 +291 00:17:11,510 --> 00:17:13,109 Make it a 2, same family of curves. -291 +292 00:17:13,329 --> 00:17:15,069 Make it a 3, same family of curves. -292 +293 00:17:15,750 --> 00:17:19,490 The reason we use e is that it gives that constant a very readable meaning. -293 +294 00:17:20,109 --> 00:17:24,315 Or rather, if we reconfigure things a little bit so that the exponent looks -294 +295 00:17:24,315 --> 00:17:27,636 like negative 1 half times x divided by a certain constant, -295 +296 00:17:27,636 --> 00:17:31,786 which we'll suggestively call sigma squared, then once we turn this into a -296 +297 00:17:31,786 --> 00:17:36,047 probability distribution, that constant sigma will be the standard deviation -297 +298 00:17:36,047 --> 00:17:37,210 of that distribution. -298 +299 00:17:37,810 --> 00:17:38,570 And that's very nice. -299 +300 00:17:38,910 --> 00:17:42,201 But before we can interpret this as a probability distribution, -300 +301 00:17:42,201 --> 00:17:44,310 we need the area under the curve to be 1. -301 +302 00:17:44,830 --> 00:17:46,910 And the reason for that is how the curve is interpreted. -302 +303 00:17:47,370 --> 00:17:50,651 Unlike discrete distributions, when it comes to something continuous, -303 +304 00:17:50,651 --> 00:17:53,370 you don't ask about the probability of a particular point. -304 +305 00:17:53,790 --> 00:17:58,230 Instead, you ask for the probability that a value falls between two different values. -305 +306 00:17:58,750 --> 00:18:02,147 And what the curve is telling you is that that probability -306 +307 00:18:02,147 --> 00:18:05,430 equals the area under the curve between those two values. -307 +308 00:18:06,030 --> 00:18:09,430 There's a whole other video about this, they're called probability density functions. -308 +309 00:18:09,830 --> 00:18:13,747 The main point right now is that the area under the entire curve represents -309 +310 00:18:13,747 --> 00:18:17,150 the probability that something happens, that some number comes up. -310 +311 00:18:17,410 --> 00:18:20,630 That should be 1, which is why we want the area under this to be 1. -311 +312 00:18:21,050 --> 00:18:24,981 As it stands with the basic bell curve shape of e to the negative x squared, -312 +313 00:18:24,981 --> 00:18:27,790 the area is not 1, it's actually the square root of pi. -313 +314 00:18:28,410 --> 00:18:29,150 I know, right? -314 +315 00:18:29,270 --> 00:18:30,190 What is pi doing here? -315 +316 00:18:30,290 --> 00:18:31,470 What does this have to do with circles? -316 +317 00:18:32,010 --> 00:18:35,050 Like I said at the start, I'd love to talk all about that in the next video. -317 +318 00:18:35,330 --> 00:18:38,253 But if you can spare your excitement, for our purposes right now, -318 +319 00:18:38,253 --> 00:18:41,708 all it means is that we should divide this function by the square root of pi, -319 +320 00:18:41,708 --> 00:18:43,170 and it gives us the area we want. -320 +321 00:18:43,610 --> 00:18:47,261 Throwing back in the constants we had earlier, the one half and the sigma, -321 +322 00:18:47,261 --> 00:18:51,303 the effect there is to stretch out the graph by a factor of sigma times the square -322 +323 00:18:51,303 --> 00:18:51,790 root of 2. -323 +324 00:18:52,410 --> 00:18:56,480 So we also need to divide out by that in order to make sure it has an area of 1, -324 +325 00:18:56,480 --> 00:18:59,647 and combining those fractions, the factor out front looks like -325 +326 00:18:59,647 --> 00:19:02,110 1 divided by sigma times the square root of 2 pi. -326 +327 00:19:02,910 --> 00:19:05,850 This, finally, is a valid probability distribution. -327 +328 00:19:06,450 --> 00:19:10,436 As we tweak that value sigma, resulting in narrower and wider curves, -328 +329 00:19:10,436 --> 00:19:14,310 that constant in the front always guarantees that the area equals 1. -329 +330 00:19:15,910 --> 00:19:18,809 The special case where sigma equals 1 has a specific name, -330 +331 00:19:18,809 --> 00:19:23,035 we call it the standard normal distribution, which plays an especially important role -331 +332 00:19:23,035 --> 00:19:24,510 for you and me in this lesson. -332 +333 00:19:25,130 --> 00:19:29,938 And all possible normal distributions are not only parameterized with this value sigma, -333 +334 00:19:29,938 --> 00:19:33,544 but we also subtract off another constant mu from the variable x, -334 +335 00:19:33,544 --> 00:19:37,314 and this essentially just lets you slide the graph left and right so -335 +336 00:19:37,314 --> 00:19:40,210 that you can prescribe the mean of this distribution. -336 +337 00:19:40,990 --> 00:19:43,895 So in short, we have two parameters, one describing the mean, -337 +338 00:19:43,895 --> 00:19:48,065 one describing the standard deviation, and they're all tied together in this big formula -338 +339 00:19:48,065 --> 00:19:49,190 involving an e and a pi. -339 +340 00:19:49,190 --> 00:19:54,441 Now that all of that is on the table, let's look back again at the idea of starting with -340 +341 00:19:54,441 --> 00:19:59,515 some random variable and asking what the distributions for sums of that variable look -341 +342 00:19:59,515 --> 00:19:59,810 like. -342 +343 00:20:00,130 --> 00:20:03,372 As we've already gone over, when you increase the size of that sum, -343 +344 00:20:03,372 --> 00:20:06,567 the resulting distribution will shift according to a growing mean, -344 +345 00:20:06,567 --> 00:20:09,810 and it slowly spreads out according to a growing standard deviation. -345 +346 00:20:10,330 --> 00:20:14,480 And putting some actual formulas to it, if we know the mean of our underlying -346 +347 00:20:14,480 --> 00:20:18,364 random variable, we call it mu, and we also know its standard deviation, -347 +348 00:20:18,364 --> 00:20:22,568 and we call it sigma, then the mean for the sum on the bottom will be mu times -348 +349 00:20:22,568 --> 00:20:26,772 the size of the sum, and the standard deviation will be sigma times the square -349 +350 00:20:26,772 --> 00:20:27,730 root of that size. -350 +351 00:20:28,190 --> 00:20:31,892 So now, if we want to claim that this looks more and more like a bell curve, -351 +352 00:20:31,892 --> 00:20:34,969 and a bell curve is only described by two different parameters, -352 +353 00:20:34,969 --> 00:20:37,710 the mean and the standard deviation, you know what to do. -353 +354 00:20:37,930 --> 00:20:42,382 You could plug those two values into the formula, and it gives you a highly explicit, -354 +355 00:20:42,382 --> 00:20:46,990 albeit kind of complicated, formula for a curve that should closely fit our distribution. -355 +356 00:20:48,390 --> 00:20:51,456 But there's another way we can describe it that's a little more -356 +357 00:20:51,456 --> 00:20:54,810 elegant and lends itself to a very fun visual that we can build up to. -357 +358 00:20:55,270 --> 00:20:58,576 Instead of focusing on the sum of all of these random variables, -358 +359 00:20:58,576 --> 00:21:02,492 let's modify this expression a little bit, where what we'll do is we'll look -359 +360 00:21:02,492 --> 00:21:06,358 at the mean that we expect that sum to take, and we subtract it off so that -360 +361 00:21:06,358 --> 00:21:10,173 our new expression has a mean of zero, and then we're going to look at the -361 +362 00:21:10,173 --> 00:21:13,479 standard deviation we expect of our sum, and divide out by that, -362 +363 00:21:13,479 --> 00:21:17,447 which basically just rescales the units so that the standard deviation of our -363 +364 00:21:17,447 --> 00:21:18,770 expression will equal one. -364 +365 00:21:19,350 --> 00:21:21,865 This might seem like a more complicated expression, -365 +366 00:21:21,865 --> 00:21:24,090 but it actually has a highly readable meaning. -366 +367 00:21:24,450 --> 00:21:29,670 It's essentially saying how many standard deviations away from the mean is this sum? -367 +368 00:21:30,750 --> 00:21:34,977 For example, this bar here corresponds to a certain value that you might find when you -368 +369 00:21:34,977 --> 00:21:39,156 roll 10 dice and you add them all up, and its position a little above negative one is -369 +370 00:21:39,156 --> 00:21:43,432 telling you that that value is a little bit less than one standard deviation lower than -370 +371 00:21:43,432 --> 00:21:43,870 the mean. -371 +372 00:21:45,130 --> 00:21:48,897 Also, by the way, in anticipation for the animation I'm trying to build to here, -372 +373 00:21:48,897 --> 00:21:52,757 the way I'm representing things on that lower plot is that the area of each one of -373 +374 00:21:52,757 --> 00:21:56,664 these bars is telling us the probability of the corresponding value rather than the -374 +375 00:21:56,664 --> 00:21:56,990 height. -375 +376 00:21:57,230 --> 00:21:59,482 You might think of the y-axis as representing -376 +377 00:21:59,482 --> 00:22:01,930 not probability but a kind of probability density. -377 +378 00:22:02,270 --> 00:22:05,996 The reason for this is to set the stage so that it aligns with the way we -378 +379 00:22:05,996 --> 00:22:09,873 interpret continuous distributions, where the probability of falling between -379 +380 00:22:09,873 --> 00:22:13,550 a range of values is equal to an area under a curve between those values. -380 +381 00:22:13,910 --> 00:22:16,730 In particular, the area of all the bars together is going to be one. -381 +382 00:22:18,230 --> 00:22:20,950 Now, with all of that in place, let's have a little fun. -382 +383 00:22:21,330 --> 00:22:25,357 Let me start by rolling things back so that the distribution on the bottom represents -383 +384 00:22:25,357 --> 00:22:29,010 a relatively small sum, like adding together only three such random variables. -384 +385 00:22:29,450 --> 00:22:32,430 Notice what happens as I change the distribution we start with. -385 +386 00:22:32,730 --> 00:22:36,290 As it changes, the distribution on the bottom completely changes its shape. -386 +387 00:22:36,510 --> 00:22:38,770 It's very dependent on what we started with. -387 +388 00:22:40,350 --> 00:22:44,178 If we let the size of our sum get a little bit bigger, say going up to 10, -388 +389 00:22:44,178 --> 00:22:48,465 and as I change the distribution for x, it largely stays looking like a bell curve, -389 +390 00:22:48,465 --> 00:22:51,630 but I can find some distributions that get it to change shape. -390 +391 00:22:52,230 --> 00:22:55,874 For example, the really lopsided one where almost all the probability -391 +392 00:22:55,874 --> 00:22:59,310 is in the numbers 1 or 6 results in this kind of spiky bell curve. -392 +393 00:22:59,770 --> 00:23:03,510 And if you'll recall, earlier on I actually showed this in the form of a simulation. -393 +394 00:23:04,130 --> 00:23:08,129 Though if you were wondering whether that spikiness was an artifact of the randomness -394 +395 00:23:08,129 --> 00:23:11,850 or reflected the true distribution, turns out it reflects the true distribution. -395 +396 00:23:12,290 --> 00:23:16,470 In this case, 10 is not a large enough sum for the central limit theorem to kick in. -396 +397 00:23:16,470 --> 00:23:20,752 But if instead I let that sum grow and I consider adding 50 different values, -397 +398 00:23:20,752 --> 00:23:25,419 which is actually not that big, then no matter how I change the distribution for our -398 +399 00:23:25,419 --> 00:23:30,086 underlying random variable, it has essentially no effect on the shape of the plot on -399 +400 00:23:30,086 --> 00:23:30,690 the bottom. -400 +401 00:23:31,170 --> 00:23:34,835 No matter where we start, all of the information and nuance for the -401 +402 00:23:34,835 --> 00:23:38,500 distribution of x gets washed away, and we tend towards this single -402 +403 00:23:38,500 --> 00:23:42,273 universal shape described by a very elegant function for the standard -403 +404 00:23:42,273 --> 00:23:47,070 normal distribution, 1 over square root of 2 pi times e to the negative x squared over 2. -404 +405 00:23:47,810 --> 00:23:50,810 This, this right here is what the central limit theorem is all about. -405 +406 00:23:51,130 --> 00:23:55,310 Almost nothing you can do to this initial distribution changes the shape we tend towards. -406 +407 00:23:59,030 --> 00:24:02,054 Now, the more theoretically minded among you might still be -407 +408 00:24:02,054 --> 00:24:05,431 wondering what is the actual theorem, like what's the mathematical -408 +409 00:24:05,431 --> 00:24:08,910 statement that could be proved or disproved that we're claiming here. -409 +410 00:24:09,030 --> 00:24:11,670 If you want a nice formal statement, here's how it might go. -410 +411 00:24:12,130 --> 00:24:16,640 Consider this value where we're summing up n different instantiations of our variable, -411 +412 00:24:16,640 --> 00:24:20,217 but tweaked and tuned so that its mean and standard deviation are 1, -412 +413 00:24:20,217 --> 00:24:24,520 again meaning you can read it as asking how many standard deviations away from the -413 +414 00:24:24,520 --> 00:24:25,350 mean is the sum. -414 +415 00:24:25,770 --> 00:24:30,489 Then the actual rigorous no-jokes-this-time statement of the central limit theorem -415 +416 00:24:30,489 --> 00:24:35,322 is that if you consider the probability that this value falls between two given real -416 +417 00:24:35,322 --> 00:24:40,154 numbers, a and b, and you consider the limit of that probability as the size of your -417 +418 00:24:40,154 --> 00:24:44,134 sum goes to infinity, then that limit is equal to a certain integral, -418 +419 00:24:44,134 --> 00:24:49,024 which basically describes the area under a standard normal distribution between those -419 +420 00:24:49,024 --> 00:24:49,650 two values. -420 +421 00:24:51,250 --> 00:24:55,146 Again, there are three underlying assumptions that I have yet to tell you, -421 +422 00:24:55,146 --> 00:24:57,692 but other than those, in all of its gory detail, -422 +423 00:24:57,692 --> 00:25:00,030 this right here is the central limit theorem. -423 +424 00:25:04,550 --> 00:25:07,992 All of that is a bit theoretical, so it might be helpful to bring things -424 +425 00:25:07,992 --> 00:25:12,141 back down to earth and turn back to the concrete example that I mentioned at the start, -425 +426 00:25:12,141 --> 00:25:15,536 where you imagine rolling a die 100 times, and let's assume it's a fair -426 +427 00:25:15,536 --> 00:25:18,130 die for this example, and you add together the results. -427 +428 00:25:18,870 --> 00:25:22,467 The challenge for you is to find a range of values such that -428 +429 00:25:22,467 --> 00:25:25,830 you're 95% sure that the sum will fall within this range. -429 -00:25:27,130 --> 00:25:33,118 -For questions like this, there's a handy rule of thumb about normal distributions, - 430 -00:25:33,118 --> 00:25:38,023 -which is that about 68% of your values are going to fall within two +00:25:27,130 --> 00:25:31,604 +For questions like this, there's a handy rule of thumb about normal distributions, 431 -00:25:38,023 --> 00:25:43,002 -standard deviations of the mean, and a whopping 99.7% of your values +00:25:31,604 --> 00:25:35,756 +which is that about 68% of your values are going to fall within one standard 432 -00:25:43,002 --> 00:25:46,970 -will fall within three standard deviations of the mean. +00:25:35,756 --> 00:25:39,422 +deviation of the mean, 95% of your values, the thing we care about, 433 -00:25:47,450 --> 00:25:49,428 -It's a rule of thumb that's commonly memorized +00:25:39,422 --> 00:25:42,063 +fall within two standard deviations of the mean, 434 -00:25:49,428 --> 00:25:51,450 -by people who do a lot of probability and stats. +00:25:42,063 --> 00:25:45,730 +and a whopping 99.7% of your values will fall within three standard 435 -00:25:52,490 --> 00:25:55,366 -Naturally, this gives us what we need for our example, +00:25:45,730 --> 00:25:46,970 +deviations of the mean. 436 -00:25:55,366 --> 00:25:58,504 -and let me go ahead and draw out what this would look like, +00:25:47,450 --> 00:25:49,428 +It's a rule of thumb that's commonly memorized 437 -00:25:58,504 --> 00:26:01,798 -where I'll show the distribution for a fair die up at the top, +00:25:49,428 --> 00:25:51,450 +by people who do a lot of probability and stats. 438 -00:26:01,798 --> 00:26:04,884 -and the distribution for a sum of 100 such dice on bottom, +00:25:52,490 --> 00:25:55,141 +Naturally, this gives us what we need for our example, 439 -00:26:04,884 --> 00:26:07,290 -which by now looks like a normal distribution. +00:25:55,141 --> 00:25:58,033 +and let me go ahead and draw out what this would look like, 440 +00:25:58,033 --> 00:26:01,071 +where I'll show the distribution for a fair die up at the top, + +441 +00:26:01,071 --> 00:26:04,108 +and the distribution for a sum of 100 such dice on the bottom, + +442 +00:26:04,108 --> 00:26:07,290 +which by now as you know looks like a certain normal distribution. + +443 00:26:07,950 --> 00:26:12,655 Step 1 with a problem like this is to find the mean of your initial distribution, -441 +444 00:26:12,655 --> 00:26:17,532 which in this case will look like 1 6th times 1 plus 1 6th times 2 on and on and on, -442 +445 00:26:17,532 --> 00:26:18,910 and works out to be 3.5. -443 -00:26:19,410 --> 00:26:23,750 +446 +00:26:19,410 --> 00:26:23,456 We also need the standard deviation, which requires calculating the variance, -444 -00:26:23,750 --> 00:26:28,034 -which as you know involves adding all the squares of the differences between +447 +00:26:23,456 --> 00:26:27,657 +which as you know involves adding all the squares of the differences between the -445 -00:26:28,034 --> 00:26:32,430 -the values and the means, and it works out to be 2.92, the square root of 1.71. +448 +00:26:27,657 --> 00:26:30,303 +values and the means, and it works out to be 2.92, -446 +449 +00:26:30,303 --> 00:26:32,430 +square root of that comes out to be 1.71. + +450 00:26:32,950 --> 00:26:35,743 Those are the only two numbers we need, and I will invite you -447 +451 00:26:35,743 --> 00:26:38,536 again to reflect on how magical it is that those are the only -448 +452 00:26:38,536 --> 00:26:41,690 two numbers you need to completely understand the bottom distribution. -449 +453 00:26:42,430 --> 00:26:47,665 Its mean will be 100 times mu, which is 350, and its standard deviation -450 +454 00:26:47,665 --> 00:26:52,610 will be the square root of 100 times sigma, so 10 times sigma, 17.1. -451 +455 00:26:53,030 --> 00:26:57,584 Remembering our handy rule of thumb, we're looking for values two standard -452 +456 00:26:57,584 --> 00:27:01,957 deviations away from the mean, and when you subtract 2 sigma from mean, -453 +457 00:27:01,957 --> 00:27:06,330 you end up with about 316, and when you add 2 sigma you end up with 384. -454 +458 00:27:07,350 --> 00:27:08,950 There you go, that gives us the answer. -455 +459 00:27:11,470 --> 00:27:14,855 Okay, I promised to wrap things up shortly, but while we're on this example, -456 +460 00:27:14,855 --> 00:27:17,450 there's one more question that's worth your time to ponder. -457 +461 00:27:18,250 --> 00:27:21,128 Instead of just asking about the sum of 100 die rolls, -458 +462 00:27:21,128 --> 00:27:23,588 let's say I had you divide that number by 100, -459 +463 00:27:23,588 --> 00:27:28,090 which basically means all the numbers in our diagram in the bottom get divided by 100. -460 +464 00:27:28,570 --> 00:27:31,570 Take a moment to interpret what this all would be saying then. -461 +465 00:27:32,070 --> 00:27:37,171 The expression essentially tells you the empirical average for 100 different die rolls, -462 +466 00:27:37,171 --> 00:27:40,939 and that interval we found is now telling you what range you are -463 +467 00:27:40,939 --> 00:27:43,490 expecting to see for that empirical average. -464 +468 00:27:44,350 --> 00:27:47,043 In other words, you might expect it to be around 3.5, -465 +469 00:27:47,043 --> 00:27:51,033 that's the expected value for a die roll, but what's much less obvious and what -466 +470 00:27:51,033 --> 00:27:54,973 the central limit theorem lets you compute is how close to that expected value -467 +471 00:27:54,973 --> 00:27:56,570 you'll reasonably find yourself. -468 +472 00:27:57,590 --> 00:28:00,691 In particular, it's worth your time to take a moment mulling over -469 +473 00:28:00,691 --> 00:28:03,464 what the standard deviation for this empirical average is, -470 +474 00:28:03,464 --> 00:28:07,130 and what happens to it as you look at a bigger and bigger sample of die rolls. -471 +475 00:28:12,950 --> 00:28:15,225 Lastly, but probably most importantly, let's talk -472 +476 00:28:15,225 --> 00:28:17,410 about the assumptions that go into this theorem. -473 +477 00:28:18,010 --> 00:28:20,316 The first one is that all of these variables that -474 +478 00:28:20,316 --> 00:28:22,530 we're adding up are independent from each other. -475 +479 00:28:22,850 --> 00:28:26,310 The outcome of one process doesn't influence the outcome of any other process. -476 +480 00:28:27,250 --> 00:28:30,950 The second is that all of these variables are drawn from the same distribution. -477 +481 00:28:31,310 --> 00:28:34,390 Both of these have been implicitly assumed with our dice example. -478 +482 00:28:34,790 --> 00:28:38,387 We've been treating the outcome of each die roll as independent from the outcome -479 +483 00:28:38,387 --> 00:28:42,030 of all the others, and we're assuming that each die follows the same distribution. -480 +484 00:28:42,850 --> 00:28:46,183 Sometimes in the literature you'll see these two assumptions lumped -481 +485 00:28:46,183 --> 00:28:49,910 together under the initials IID for independent and identically distributed. -482 +486 00:28:50,530 --> 00:28:55,110 One situation where these assumptions are decidedly not true would be the Galton board. -483 +487 00:28:55,710 --> 00:28:56,830 I mean, think about it. -484 +488 00:28:56,970 --> 00:29:00,177 Is it the case that the way a ball bounces off of one of the pegs -485 +489 00:29:00,177 --> 00:29:03,190 is independent from how it's going to bounce off the next peg? -486 +490 00:29:03,830 --> 00:29:04,610 Absolutely not. -487 +491 00:29:04,770 --> 00:29:07,870 Depending on the last bounce, it's coming in with a completely different trajectory. -488 +492 00:29:08,210 --> 00:29:11,633 And is it the case that the distribution of possible outcomes -489 +493 00:29:11,633 --> 00:29:14,670 off of each peg are the same for each peg that it hits? -490 +494 00:29:15,190 --> 00:29:16,710 Again, almost certainly not. -491 +495 00:29:16,710 --> 00:29:20,233 Maybe it hits one peg glancing to the left, meaning the outcomes are hugely -492 +496 00:29:20,233 --> 00:29:23,710 skewed in that direction, and then hits the next one glancing to the right. -493 +497 00:29:25,730 --> 00:29:29,171 When I made all those simplifying assumptions in the opening example, -494 +498 00:29:29,171 --> 00:29:31,630 it wasn't just to make this easier to think about. -495 +499 00:29:31,970 --> 00:29:34,565 It's also that those assumptions were necessary for this -496 +500 00:29:34,565 --> 00:29:37,070 to actually be an example of the central limit theorem. -497 +501 00:29:38,130 --> 00:29:41,470 Nevertheless, it seems to be true that for the real Galton board, -498 +502 00:29:41,470 --> 00:29:45,470 despite violating both of these, a normal distribution does kind of come about? -499 +503 00:29:46,050 --> 00:29:49,994 Part of the reason might be that there are generalizations of the theorem beyond -500 +504 00:29:49,994 --> 00:29:53,890 the scope of this video that relax these assumptions, especially the second one. -501 +505 00:29:54,490 --> 00:29:58,828 But I do want to caution you against the fact that many times people seem to assume that -502 +506 00:29:58,828 --> 00:30:03,070 a variable is normally distributed, even when there's no actual justification to do so. -503 +507 00:30:04,290 --> 00:30:06,210 The third assumption is actually fairly subtle. -504 +508 00:30:06,210 --> 00:30:10,270 It's that the variance we've been computing for these variables is finite. -505 +509 00:30:10,810 --> 00:30:13,162 This was never an issue for the dice example because -506 +510 00:30:13,162 --> 00:30:14,850 there were only six possible outcomes. -507 +511 00:30:15,030 --> 00:30:18,544 But in certain situations where you have an infinite set of outcomes, -508 +512 00:30:18,544 --> 00:30:22,510 when you go to compute the variance, the sum ends up diverging off to infinity. -509 +513 00:30:23,450 --> 00:30:27,250 These can be perfectly valid probability distributions, and they do come up in practice. -510 +514 00:30:27,550 --> 00:30:30,674 But in those situations, as you consider adding many different -511 +515 00:30:30,674 --> 00:30:34,245 instantiations of that variable and letting that sum approach infinity, -512 +516 00:30:34,245 --> 00:30:37,717 even if the first two assumptions hold, it is very much a possibility -513 +517 00:30:37,717 --> 00:30:41,190 that the thing you tend towards is not actually a normal distribution. -514 +518 00:30:42,150 --> 00:30:44,187 If you've understood everything up to this point, -515 +519 00:30:44,187 --> 00:30:47,650 you now have a very strong foundation in what the central limit theorem is all about. -516 +520 00:30:48,290 --> 00:30:52,000 And next up, I'd like to explain why it is that this particular function is the -517 +521 00:30:52,000 --> 00:30:55,990 thing that we tend towards, and why it has a pi in it, what it has to do with circles. -518 +522 00:31:11,950 --> 00:31:14,170 Thank you. diff --git a/2023/clt/english/sentence_timings.json b/2023/clt/english/sentence_timings.json index d9bde9f2b..e4a624127 100644 --- a/2023/clt/english/sentence_timings.json +++ b/2023/clt/english/sentence_timings.json @@ -35,7 +35,7 @@ 76.02 ], [ - "This lesson is meant to go back to the basics, giving you the fundamentals on what the background is.", + "This lesson is meant to go back to the basics, giving you the fundamentals on what the central limit theorem is saying, what normal distributions are, and I want to assume minimal background.", 76.64, 85.26 ], @@ -440,7 +440,7 @@ 779.19 ], [ - "That can be interpreted much more reasonably as a distance on our diagram, and it's commonly denoted with the Greek letter sigma, so you know m for standard deviation, but both in Greek.", + "That can be interpreted much more reasonably as a distance on our diagram, and it's commonly denoted with the Greek letter sigma, so you know m for mean as for standard deviation, but both in Greek.", 779.47, 789.65 ], @@ -505,7 +505,7 @@ 878.25 ], [ - "For example, back in our sequence of distributions, if we label the standard deviation of our initial one with sigma, then the next standard deviation is going to be the square root of 2 times sigma, and after that it looks like the square root of 3 times sigma, and so on This, like I said, is very important.", + "For example, back in our sequence of distributions, if we label the standard deviation of our initial one with sigma, then the next standard deviation is going to be the square root of 2 times sigma, and after that it looks like the square root of 3 times sigma, and so on and so forth. This, like I said, is very important.", 879.29, 895.65 ], @@ -885,7 +885,7 @@ 1525.83 ], [ - "For questions like this, there's a handy rule of thumb about normal distributions, which is that about 68% of your values are going to fall within two standard deviations of the mean, and a whopping 99.7% of your values will fall within three standard deviations of the mean.", + "For questions like this, there's a handy rule of thumb about normal distributions, which is that about 68% of your values are going to fall within one standard deviation of the mean, 95% of your values, the thing we care about, fall within two standard deviations of the mean, and a whopping 99.7% of your values will fall within three standard deviations of the mean.", 1527.13, 1546.97 ], @@ -895,7 +895,7 @@ 1551.45 ], [ - "Naturally, this gives us what we need for our example, and let me go ahead and draw out what this would look like, where I'll show the distribution for a fair die up at the top, and the distribution for a sum of 100 such dice on bottom, which by now looks like a normal distribution.", + "Naturally, this gives us what we need for our example, and let me go ahead and draw out what this would look like, where I'll show the distribution for a fair die up at the top, and the distribution for a sum of 100 such dice on the bottom, which by now as you know looks like a certain normal distribution.", 1552.49, 1567.29 ], @@ -905,7 +905,7 @@ 1578.91 ], [ - "We also need the standard deviation, which requires calculating the variance, which as you know involves adding all the squares of the differences between the values and the means, and it works out to be 2.92, the square root of 1.71.", + "We also need the standard deviation, which requires calculating the variance, which as you know involves adding all the squares of the differences between the values and the means, and it works out to be 2.92, square root of that comes out to be 1.71.", 1579.41, 1592.43 ], diff --git a/2023/clt/english/transcript.txt b/2023/clt/english/transcript.txt index c15db6551..1a5eaa5cf 100644 --- a/2023/clt/english/transcript.txt +++ b/2023/clt/english/transcript.txt @@ -5,7 +5,7 @@ There's a very specific function to describe this distribution, it's very pretty If you were to take a large number of people who sit in a similar demographic and plot their heights, those heights tend to follow a normal distribution. If you look at a large swath of very big natural numbers, and you ask how many distinct prime factors does each one of those numbers have, the answers will very closely track with a certain normal distribution. Now our topic for today is one of the crown jewels in all of probability theory, it's one of the key facts that explains why this distribution is as common as it is, known as the central limit theorem. -This lesson is meant to go back to the basics, giving you the fundamentals on what the background is. +This lesson is meant to go back to the basics, giving you the fundamentals on what the central limit theorem is saying, what normal distributions are, and I want to assume minimal background. We're going to go decently deep into it, but after this I'd still like to go deeper and explain why the theorem is true, why the function underlying the normal distribution has the very specific form that it does, why that formula has a pi in it, and, most fun, why those last two facts are actually more related than a lot of traditional explanations would suggest. That second lesson is also meant to be the follow-on to the convolutions video that I promised, so there's a lot of interrelated topics here. But right now, back to the fundamentals, I'd like to kick things off with an overly simplified model of the Galton board. @@ -86,7 +86,7 @@ The idea there is to look at the difference between each possible value and the The idea is that whether your value is below or above the mean, when you square that difference, you get a positive number, and the larger the difference, the bigger that number. Squaring it like this turns out to make the math much much nicer than if we did something like an absolute value, but the downside is that it's hard to think about this as a distance in our diagram because the units are off, kind of like the units here are square units, whereas a distance in our diagram would be a kind of linear unit. So another way to measure spread is what's called the standard deviation, which is the square root of this value. -That can be interpreted much more reasonably as a distance on our diagram, and it's commonly denoted with the Greek letter sigma, so you know m for standard deviation, but both in Greek. +That can be interpreted much more reasonably as a distance on our diagram, and it's commonly denoted with the Greek letter sigma, so you know m for mean as for standard deviation, but both in Greek. Looking back at our sequence of distributions, let's talk about the mean and standard deviation. If we call the mean of the initial distribution mu, which for the one illustrated happens to be 2.24, hopefully it won't be too surprising if I tell you that the mean of the next one is 2 times mu. That is, you roll a pair of dice, you want to know the expected value of the sum, it's two times the expected value for a single die. @@ -99,7 +99,7 @@ There are a couple nice intuitions for why it's true. My tentative plan is to just actually make a series about probability and talk about things like intuitions underlying variance and its cousins there. But right now, the main thing I want you to highlight is how it's the variance that adds, it's not the standard deviation that adds. So, critically, if you were to take n different realizations of the same random variable and ask what the sum looks like, the variance of sum is n times the variance of your original variable, meaning the standard deviation, the square root of all this, is the square root of n times the original standard deviation. -For example, back in our sequence of distributions, if we label the standard deviation of our initial one with sigma, then the next standard deviation is going to be the square root of 2 times sigma, and after that it looks like the square root of 3 times sigma, and so on This, like I said, is very important. +For example, back in our sequence of distributions, if we label the standard deviation of our initial one with sigma, then the next standard deviation is going to be the square root of 2 times sigma, and after that it looks like the square root of 3 times sigma, and so on and so forth. This, like I said, is very important. It means that even though our distributions are getting spread out, they're not spreading out all that quickly, they only do so in proportion to the square root of the size of the sum. As we prepare to make a more quantitative description of the central limit theorem, the core intuition I want you to keep in your head is that we'll basically realign all of these distributions so that their means line up together, and then rescale them so that all of the standard deviations are just going to be equal to one. And when we do that, the shape that results gets closer and closer to a certain universal shape, described with an elegant little function that we'll unpack in just a moment. @@ -175,11 +175,11 @@ Then the actual rigorous no-jokes-this-time statement of the central limit theor Again, there are three underlying assumptions that I have yet to tell you, but other than those, in all of its gory detail, this right here is the central limit theorem. All of that is a bit theoretical, so it might be helpful to bring things back down to earth and turn back to the concrete example that I mentioned at the start, where you imagine rolling a die 100 times, and let's assume it's a fair die for this example, and you add together the results. The challenge for you is to find a range of values such that you're 95% sure that the sum will fall within this range. -For questions like this, there's a handy rule of thumb about normal distributions, which is that about 68% of your values are going to fall within two standard deviations of the mean, and a whopping 99.7% of your values will fall within three standard deviations of the mean. +For questions like this, there's a handy rule of thumb about normal distributions, which is that about 68% of your values are going to fall within one standard deviation of the mean, 95% of your values, the thing we care about, fall within two standard deviations of the mean, and a whopping 99.7% of your values will fall within three standard deviations of the mean. It's a rule of thumb that's commonly memorized by people who do a lot of probability and stats. -Naturally, this gives us what we need for our example, and let me go ahead and draw out what this would look like, where I'll show the distribution for a fair die up at the top, and the distribution for a sum of 100 such dice on bottom, which by now looks like a normal distribution. +Naturally, this gives us what we need for our example, and let me go ahead and draw out what this would look like, where I'll show the distribution for a fair die up at the top, and the distribution for a sum of 100 such dice on the bottom, which by now as you know looks like a certain normal distribution. Step 1 with a problem like this is to find the mean of your initial distribution, which in this case will look like 1 6th times 1 plus 1 6th times 2 on and on and on, and works out to be 3.5. -We also need the standard deviation, which requires calculating the variance, which as you know involves adding all the squares of the differences between the values and the means, and it works out to be 2.92, the square root of 1.71. +We also need the standard deviation, which requires calculating the variance, which as you know involves adding all the squares of the differences between the values and the means, and it works out to be 2.92, square root of that comes out to be 1.71. Those are the only two numbers we need, and I will invite you again to reflect on how magical it is that those are the only two numbers you need to completely understand the bottom distribution. Its mean will be 100 times mu, which is 350, and its standard deviation will be the square root of 100 times sigma, so 10 times sigma, 17.1. Remembering our handy rule of thumb, we're looking for values two standard deviations away from the mean, and when you subtract 2 sigma from mean, you end up with about 316, and when you add 2 sigma you end up with 384. diff --git a/2023/clt/french/sentence_translations.json b/2023/clt/french/sentence_translations.json index 06d386cd3..938728aee 100644 --- a/2023/clt/french/sentence_translations.json +++ b/2023/clt/french/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "Et ensuite, j'aimerais expliquer pourquoi cette fonction particulière est la chose vers laquelle nous tendons, et pourquoi elle contient un pi, ce qu'elle a à voir avec les cercles. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/german/sentence_translations.json b/2023/clt/german/sentence_translations.json index e507ad7f3..0a3787391 100644 --- a/2023/clt/german/sentence_translations.json +++ b/2023/clt/german/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "Und als Nächstes möchte ich erklären, warum wir zu dieser Funktion tendieren, warum sie ein Pi enthält und was sie mit Kreisen zu tun hat. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/hebrew/sentence_translations.json b/2023/clt/hebrew/sentence_translations.json index 80c19ef95..d601a612d 100644 --- a/2023/clt/hebrew/sentence_translations.json +++ b/2023/clt/hebrew/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "ובהמשך, אני רוצה להסביר מדוע הפונקציה הספציפית הזו היא הדבר שאליו אנו נוטים לכיוונו, למה יש בה פאי, ומה זה קשור למעגלים. ", "model": "google_nmt", "n_reviews": 1, diff --git a/2023/clt/hindi/sentence_translations.json b/2023/clt/hindi/sentence_translations.json index 752c2d036..b41ee1678 100644 --- a/2023/clt/hindi/sentence_translations.json +++ b/2023/clt/hindi/sentence_translations.json @@ -1589,7 +1589,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles.", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you.", "translatedText": "और आगे, मैं यह समझाना चाहूँगा कि ऐसा क्यों है कि यह विशेष कार्य वह चीज़ है जिसकी ओर हमारा रुझान है, और इसमें एक पाई क्यों है, इसका वृत्तों से क्या लेना-देना है।", "n_reviews": 0, "start": 1848.29, diff --git a/2023/clt/hungarian/sentence_translations.json b/2023/clt/hungarian/sentence_translations.json index 77f321393..a28ad6b3c 100644 --- a/2023/clt/hungarian/sentence_translations.json +++ b/2023/clt/hungarian/sentence_translations.json @@ -56,7 +56,7 @@ "end": 76.02 }, { - "input": "This lesson is meant to go back to the basics, giving you the fundamentals on what the background is.", + "input": "This lesson is meant to go back to the basics, giving you the fundamentals on what the central limit theorem is saying, what normal distributions are, and I want to assume minimal background.", "translatedText": "Ez a lecke arra szolgál, hogy visszatérjünk az alapokhoz, és megismertessük veled az alapokat, hogy mi a háttér.", "model": "DeepL", "n_reviews": 0, @@ -704,7 +704,7 @@ "end": 779.19 }, { - "input": "That can be interpreted much more reasonably as a distance on our diagram, and it's commonly denoted with the Greek letter sigma, so you know m for standard deviation, but both in Greek.", + "input": "That can be interpreted much more reasonably as a distance on our diagram, and it's commonly denoted with the Greek letter sigma, so you know m for mean as for standard deviation, but both in Greek.", "translatedText": "Ezt sokkal ésszerűbb távolságként értelmezni a diagramunkon, és ezt általában a görög szigma betűvel jelölik, így tudod, hogy m a szórás, de mindkettő görögül.", "model": "DeepL", "n_reviews": 0, @@ -808,7 +808,7 @@ "end": 878.25 }, { - "input": "For example, back in our sequence of distributions, if we label the standard deviation of our initial one with sigma, then the next standard deviation is going to be the square root of 2 times sigma, and after that it looks like the square root of 3 times sigma, and so on This, like I said, is very important.", + "input": "For example, back in our sequence of distributions, if we label the standard deviation of our initial one with sigma, then the next standard deviation is going to be the square root of 2 times sigma, and after that it looks like the square root of 3 times sigma, and so on and so forth. This, like I said, is very important.", "translatedText": "Például az eloszlások sorozatában, ha a kezdeti eloszlásunk szórását sigmával jelöljük, akkor a következő szórás a 2-szeres sigma négyzetgyöke lesz, utána pedig a 3-szoros sigma négyzetgyöke, és így tovább Ez, mint mondtam, nagyon fontos.", "model": "DeepL", "n_reviews": 0, @@ -1416,7 +1416,7 @@ "end": 1525.83 }, { - "input": "For questions like this, there's a handy rule of thumb about normal distributions, which is that about 68% of your values are going to fall within two standard deviations of the mean, and a whopping 99.7% of your values will fall within three standard deviations of the mean.", + "input": "For questions like this, there's a handy rule of thumb about normal distributions, which is that about 68% of your values are going to fall within one standard deviation of the mean, 95% of your values, the thing we care about, fall within two standard deviations of the mean, and a whopping 99.7% of your values will fall within three standard deviations of the mean.", "translatedText": "Az ilyen kérdésekre van egy praktikus ökölszabály a normál eloszlásokra vonatkozóan, amely szerint az értékek 68%-a az átlag két szórásán belülre esik, és az értékek 99,7%-a az átlag három szórásán belülre esik.", "model": "DeepL", "n_reviews": 0, @@ -1432,7 +1432,7 @@ "end": 1551.45 }, { - "input": "Naturally, this gives us what we need for our example, and let me go ahead and draw out what this would look like, where I'll show the distribution for a fair die up at the top, and the distribution for a sum of 100 such dice on bottom, which by now looks like a normal distribution.", + "input": "Naturally, this gives us what we need for our example, and let me go ahead and draw out what this would look like, where I'll show the distribution for a fair die up at the top, and the distribution for a sum of 100 such dice on the bottom, which by now as you know looks like a certain normal distribution.", "translatedText": "Természetesen ez megadja azt, amire a példánkhoz szükségünk van, és hadd rajzoljam ki, hogy ez hogyan nézne ki, ahol felül egy tisztességes kocka eloszlását mutatom, alul pedig 100 ilyen kocka összegének eloszlását, ami mostanra már normális eloszlásnak tűnik.", "model": "DeepL", "n_reviews": 0, @@ -1448,7 +1448,7 @@ "end": 1578.91 }, { - "input": "We also need the standard deviation, which requires calculating the variance, which as you know involves adding all the squares of the differences between the values and the means, and it works out to be 2.92, the square root of 1.71.", + "input": "We also need the standard deviation, which requires calculating the variance, which as you know involves adding all the squares of the differences between the values and the means, and it works out to be 2.92, square root of that comes out to be 1.71.", "translatedText": "Szükségünk van a szórásra is, amihez ki kell számolnunk a varianciát, ami, mint tudjuk, az értékek és az átlagok közötti különbségek négyzetének összeadását jelenti, és ez 2,92, ami az 1,71 négyzetgyöke.", "model": "DeepL", "n_reviews": 0, diff --git a/2023/clt/indonesian/sentence_translations.json b/2023/clt/indonesian/sentence_translations.json index 3f7377d71..0c18a18c0 100644 --- a/2023/clt/indonesian/sentence_translations.json +++ b/2023/clt/indonesian/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "Selanjutnya, saya ingin menjelaskan mengapa fungsi khusus ini cenderung kita sukai, dan mengapa ada pi di dalamnya, apa hubungannya dengan lingkaran. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/italian/sentence_translations.json b/2023/clt/italian/sentence_translations.json index da9bbada6..1ff9dcc72 100644 --- a/2023/clt/italian/sentence_translations.json +++ b/2023/clt/italian/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "E poi vorrei spiegare perché questa particolare funzione è ciò a cui tendiamo, e perché contiene un pi greco, cosa ha a che fare con i cerchi. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/japanese/sentence_translations.json b/2023/clt/japanese/sentence_translations.json index 46a06374e..2e78bd714 100644 --- a/2023/clt/japanese/sentence_translations.json +++ b/2023/clt/japanese/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "次に、この特定 の関数がなぜ私たちが好む傾向にあるのか、なぜ円周率が含まれてい るのか、円とどのような関係があるのかを説明したいと思います。", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/korean/sentence_translations.json b/2023/clt/korean/sentence_translations.json index cf0022fe9..6b945c66f 100644 --- a/2023/clt/korean/sentence_translations.json +++ b/2023/clt/korean/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "다음으로, 왜 이 특정 함수가 우리가 지향하는 것인지, 왜 파이가 포함되어 있는지, 원과 어떤 관련이 있는지 설명하고 싶습니다. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/marathi/sentence_translations.json b/2023/clt/marathi/sentence_translations.json index 1fac17c74..2a4adfecc 100644 --- a/2023/clt/marathi/sentence_translations.json +++ b/2023/clt/marathi/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "आणि पुढे, मी हे स्पष्ट करू इच्छितो की हे विशिष्ट कार्य ज्या गोष्टीकडे आपला कल आहे आणि त्यात pi का आहे, त्याचा वर्तुळांशी काय संबंध आहे. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/persian/sentence_translations.json b/2023/clt/persian/sentence_translations.json index a7192b9dc..957209b58 100644 --- a/2023/clt/persian/sentence_translations.json +++ b/2023/clt/persian/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "و در مرحله بعد، می‌خواهم توضیح دهم که چرا این تابع خاص همان چیزی است که ما به سمت آن گرایش داریم، و چرا یک عدد pi در آن وجود دارد، چه ربطی به دایره‌ها دارد. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/portuguese/sentence_translations.json b/2023/clt/portuguese/sentence_translations.json index e52910e72..acdbc7151 100644 --- a/2023/clt/portuguese/sentence_translations.json +++ b/2023/clt/portuguese/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "E a seguir, gostaria de explicar por que essa função em particular é aquilo para o qual tendemos, e por que ela contém um pi, o que ela tem a ver com círculos. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/russian/sentence_translations.json b/2023/clt/russian/sentence_translations.json index 362e40437..e91372006 100644 --- a/2023/clt/russian/sentence_translations.json +++ b/2023/clt/russian/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "И далее я хотел бы объяснить, почему именно эта функция является тем, к чему мы склонны, и почему в ней есть число «пи», какое отношение она имеет к кругам. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/spanish/sentence_translations.json b/2023/clt/spanish/sentence_translations.json index 9b3286afa..01cb6469c 100644 --- a/2023/clt/spanish/sentence_translations.json +++ b/2023/clt/spanish/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "Y a continuación, me gustaría explicar por qué esta función en particular es hacia lo que tendemos, y por qué tiene un pi, qué tiene que ver con los círculos. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/tamil/sentence_translations.json b/2023/clt/tamil/sentence_translations.json index 7e641b5c1..3d87712ee 100644 --- a/2023/clt/tamil/sentence_translations.json +++ b/2023/clt/tamil/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "அடுத்ததாக, இந்த குறிப்பிட்ட செயல்பாடு நாம் ஏன் நோக்கி செல்கிறது என்பதை விளக்க விரும்புகிறேன், ஏன் அதில் ஒரு பை உள்ளது, வட்டங்களுடன் என்ன தொடர்பு உள்ளது. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/telugu/sentence_translations.json b/2023/clt/telugu/sentence_translations.json index 3d709e9a8..c67a4f9fd 100644 --- a/2023/clt/telugu/sentence_translations.json +++ b/2023/clt/telugu/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "మరియు తదుపరి, నేను ఈ నిర్దిష్ట ఫంక్షన్ ఎందుకు మనం వైపు మొగ్గు చూపుతుంది మరియు దానిలో పై ఎందుకు ఉంది, దానికి సర్కిల్‌లతో సంబంధం ఏమిటో వివరించాలనుకుంటున్నాను. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/thai/sentence_translations.json b/2023/clt/thai/sentence_translations.json index 3a2ff977c..2e4b7a165 100644 --- a/2023/clt/thai/sentence_translations.json +++ b/2023/clt/thai/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/turkish/sentence_translations.json b/2023/clt/turkish/sentence_translations.json index a74bfb314..13eb8eda6 100644 --- a/2023/clt/turkish/sentence_translations.json +++ b/2023/clt/turkish/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "Daha sonra, neden bu özel fonksiyonun yöneldiğimiz bir şey olduğunu, neden içinde bir pi bulunduğunu, bunun çemberlerle ne ilgisi olduğunu açıklamak istiyorum. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/ukrainian/sentence_translations.json b/2023/clt/ukrainian/sentence_translations.json index eaf107fa1..796a0be19 100644 --- a/2023/clt/ukrainian/sentence_translations.json +++ b/2023/clt/ukrainian/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "І далі я хотів би пояснити, чому саме ця функція є тією річчю, до якої ми схильні, і чому в ній є пі, що це має спільного з колами. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/urdu/sentence_translations.json b/2023/clt/urdu/sentence_translations.json index 4b6eb4ec6..d88f414de 100644 --- a/2023/clt/urdu/sentence_translations.json +++ b/2023/clt/urdu/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "اور اس کے بعد، میں یہ بتانا چاہوں گا کہ ایسا کیوں ہے کہ یہ خاص فنکشن وہ چیز ہے جس کی طرف ہمارا رجحان ہے، اور اس میں ایک pi کیوں ہے، اس کا حلقوں سے کیا تعلق ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/clt/vietnamese/sentence_translations.json b/2023/clt/vietnamese/sentence_translations.json index 56a9dcdb4..83e772a28 100644 --- a/2023/clt/vietnamese/sentence_translations.json +++ b/2023/clt/vietnamese/sentence_translations.json @@ -1816,7 +1816,7 @@ "end": 1847.65 }, { - "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. ", + "input": "And next up, I'd like to explain why it is that this particular function is the thing that we tend towards, and why it has a pi in it, what it has to do with circles. Thank you. ", "translatedText": "Và tiếp theo, tôi muốn giải thích tại sao hàm đặc biệt này lại là thứ mà chúng ta hướng tới, và tại sao nó lại có số pi trong đó, nó liên quan gì đến các vòng tròn. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/arabic/sentence_translations.json b/2023/convolutions2/arabic/sentence_translations.json index a80a3929d..f6557ef6f 100644 --- a/2023/convolutions2/arabic/sentence_translations.json +++ b/2023/convolutions2/arabic/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum. ", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum. ", "translatedText": "وبالمثل، لنجعل py هي دالة التوزيع الثاني، وpx زائد y هي الدالة التي تصف توزيع المجموع. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py. ", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y. ", "translatedText": "في اللغة، ما ستقوله هو أن px plus y يساوي الالتواء بين px وpy. ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6. ", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6. ", "translatedText": "باستثناء أن الطريقة المعتادة لكتابتها هي عدم الكتابة باستخدام x وy، ولكن بدلاً من ذلك نركز فقط على أحد تلك المتغيرات، في هذه الحالة x، مع السماح لها بالنطاق فوق جميع قيمها المحتملة، وهو ما يعني هنا فقط الانتقال من 1 إلى 6 . ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/bengali/sentence_translations.json b/2023/convolutions2/bengali/sentence_translations.json index a1ad943d9..5655169c0 100644 --- a/2023/convolutions2/bengali/sentence_translations.json +++ b/2023/convolutions2/bengali/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum. ", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum. ", "translatedText": "একইভাবে, আমাদের দ্বিতীয় ডিস্ট্রিবিউশনের জন্য py ফাংশন হওয়া যাক, এবং px প্লাস y হল যোগফলের বন্টন বর্ণনাকারী ফাংশন।", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py. ", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y. ", "translatedText": "লিঙ্গোতে, আপনি যা বলবেন তা হল px প্লাস y px এবং py এর মধ্যে একটি আবর্তনের সমান।", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6. ", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6. ", "translatedText": "সাধারণ উপায় ব্যতীত এটি x এবং y দিয়ে লিখতে হবে না, বরং আমরা কেবলমাত্র সেই ভেরিয়েবলগুলির মধ্যে একটিতে ফোকাস করি, এই ক্ষেত্রে x, এটিকে এর সম্ভাব্য সমস্ত মানগুলির উপর রেঞ্জ দেওয়া, যার অর্থ এখানে 1 থেকে 6 পর্যন্ত যাওয়া।. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/chinese/sentence_translations.json b/2023/convolutions2/chinese/sentence_translations.json index a3cd398df..e6733df25 100644 --- a/2023/convolutions2/chinese/sentence_translations.json +++ b/2023/convolutions2/chinese/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "同样，让 py 为第二个分布的函数， px 加 y 为描述总和分布的函数。", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "用行话来说，你会说 px 加 y 等于 px 和 py 之间的卷积。", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "除了通常的编写方式不是用 x 和 y 来编写之外 ，而是我们只关注其中一个变量，在本例中为 x，让 它涵盖所有可能的值，这意味着从 1 到 6。", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/french/sentence_translations.json b/2023/convolutions2/french/sentence_translations.json index e4b918f4c..7731f7306 100644 --- a/2023/convolutions2/french/sentence_translations.json +++ b/2023/convolutions2/french/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "De même, soit py la fonction de notre deuxième distribution, et px plus y la fonction décrivant la distribution de la somme.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "Dans le jargon, ce que vous diriez, c'est que px plus y est égal à une convolution entre px et py.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "Sauf que la manière habituelle de l'écrire n'est pas d'écrire avec x et y, mais plutôt de nous concentrer uniquement sur l'une de ces variables, dans ce cas x, en la laissant s'étendre sur toutes ses valeurs possibles, ce qui signifie ici simplement passer de 1 à 6.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/german/sentence_translations.json b/2023/convolutions2/german/sentence_translations.json index 2bfec6151..059b25a63 100644 --- a/2023/convolutions2/german/sentence_translations.json +++ b/2023/convolutions2/german/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "In ähnlicher Weise sei py die Funktion für unsere zweite Verteilung und px plus y die Funktion, die die Verteilung für die Summe beschreibt.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "Im Fachjargon würde man sagen, dass px plus y einer Faltung zwischen px und py entspricht.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "Die übliche Schreibweise besteht jedoch nicht darin, mit x und y zu schreiben, sondern wir konzentrieren uns einfach auf eine dieser Variablen, in diesem Fall x, und lassen sie über alle möglichen Werte schwanken, was hier nur bedeutet, von 1 bis 6 zu gehen.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/hebrew/sentence_translations.json b/2023/convolutions2/hebrew/sentence_translations.json index 70d1a12e8..e77f29fc6 100644 --- a/2023/convolutions2/hebrew/sentence_translations.json +++ b/2023/convolutions2/hebrew/sentence_translations.json @@ -483,14 +483,14 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "באופן דומה, בואו נניח ל-py להיות הפונקציה עבור ההתפלגות השנייה שלנו, ו-px פלוס y להיות הפונקציה המתארת את ההתפלגות עבור הסכום.", "n_reviews": 0, "start": 544.44, "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "בז'רגון, מה שתגידו זה ש-px פלוס y שווה לקונבולולוציה בין px ל-py.", "n_reviews": 1, "start": 553.96, @@ -532,7 +532,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "חוץ מהדרך הרגילה שבה היא כתובה היא לא עם x ו-y, אלא אנחנו פשוט מתמקדים באחד מהמשתנים האלה, במקרה הזה x, נותנים לו לנוע על פני כל הערכים האפשריים שלו, שכאן זהו רק מעבר מ-1 ל-6.", "n_reviews": 1, "start": 595.82, diff --git a/2023/convolutions2/hindi/sentence_translations.json b/2023/convolutions2/hindi/sentence_translations.json index 6a691876c..6799011ec 100644 --- a/2023/convolutions2/hindi/sentence_translations.json +++ b/2023/convolutions2/hindi/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "इसी प्रकार, मान लीजिए कि py हमारे दूसरे वितरण के लिए फ़ंक्शन है, और px प्लस y योग के लिए वितरण का वर्णन करने वाला फ़ंक्शन है।", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "सामान्य भाषा में, आप क्या कहेंगे कि px प्लस y, px और py के बीच एक कनवल्शन के बराबर है।", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "इसे लिखने के सामान्य तरीके को छोड़कर, x और y के साथ लिखना नहीं है, बल्कि इसके बजाय हम केवल उन चरों में से एक पर ध्यान केंद्रित करते हैं, इस मामले में x, इसे इसके सभी संभावित मानों तक सीमित कर देता है, जिसका अर्थ यहां केवल 1 से 6 तक जाना है।", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/indonesian/sentence_translations.json b/2023/convolutions2/indonesian/sentence_translations.json index 3ba31b51c..e865ad2ed 100644 --- a/2023/convolutions2/indonesian/sentence_translations.json +++ b/2023/convolutions2/indonesian/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "Demikian pula, misalkan py menjadi fungsi untuk distribusi kedua, dan px plus y menjadi fungsi yang mendeskripsikan distribusi penjumlahan.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "Dalam istilah tersebut, apa yang Anda katakan adalah bahwa px ditambah y sama dengan konvolusi antara px dan py.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "Kecuali cara penulisannya yang biasa adalah dengan tidak menulis dengan x dan y, melainkan kita hanya fokus pada salah satu variabel tersebut, dalam hal ini x, membiarkannya berkisar pada semua kemungkinan nilainya, yang di sini berarti mulai dari 1 hingga 6.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/italian/sentence_translations.json b/2023/convolutions2/italian/sentence_translations.json index 847501d4f..27c6e8810 100644 --- a/2023/convolutions2/italian/sentence_translations.json +++ b/2023/convolutions2/italian/sentence_translations.json @@ -483,14 +483,14 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "Allo stesso modo, lasciamo che py sia la funzione per la nostra seconda distribuzione e px più y sia la funzione che descrive la distribuzione della somma.", "n_reviews": 0, "start": 544.44, "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "Nel gergo, quello che diresti è che px più y è uguale a una convoluzione tra px e py.", "n_reviews": 0, "start": 553.96, @@ -532,7 +532,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "Solo che il solito modo in cui è scritto non è scrivere con x e y, ma invece ci concentriamo solo su una di quelle variabili, in questo caso x, lasciandola variare su tutti i suoi possibili valori, che qui significa semplicemente andare da 1 a 6 .", "n_reviews": 0, "start": 595.82, diff --git a/2023/convolutions2/japanese/sentence_translations.json b/2023/convolutions2/japanese/sentence_translations.json index 5ab746e38..ddc85c732 100644 --- a/2023/convolutions2/japanese/sentence_translations.json +++ b/2023/convolutions2/japanese/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "同様に、2 番目の分布の関数を py とし、合計の 分布を記述する関数を px と y に加えます。", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "専門用語では、px + y は px と py の間の畳み込みに等しいと言えます。", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "ただし、通常の書き方では x と y を使用するのではなく、変数の 1 つ (この場合は x) に焦点を当て、取り得るすべての値の範 囲を指定します。 これは単に 1 から 6 までを意味します。 。", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/korean/sentence_translations.json b/2023/convolutions2/korean/sentence_translations.json index 3eb020c59..9ebc7fecc 100644 --- a/2023/convolutions2/korean/sentence_translations.json +++ b/2023/convolutions2/korean/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "마찬가지로 py를 두 번째 분포에 대한 함수로 설정하고 px + y를 합계에 대한 분포를 설명하는 함수로 설정하겠습니다.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "전문용어에서는 px + y가 px와 py 사이의 컨볼루션과 같다고 말할 수 있습니다.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "일반적인 작성 방법은 x와 y로 작성하는 것이 아니라 해당 변수 중 하나(이 경우 x)에만 초점을 맞춰 가능한 모든 값에 걸쳐 범위를 지정하는 것입니다. 여기서는 1에서 6까지 가는 것을 의미합니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/marathi/sentence_translations.json b/2023/convolutions2/marathi/sentence_translations.json index 65e7c6c78..3d1b257dd 100644 --- a/2023/convolutions2/marathi/sentence_translations.json +++ b/2023/convolutions2/marathi/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "त्याचप्रमाणे, आपल्या दुसऱ्या डिस्ट्रिब्युशनसाठी py हे फंक्शन बनवू आणि px प्लस y हे बेरीजचे वितरण वर्णन करणारे फंक्शन असू द्या.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "लिंगोमध्ये, तुम्ही काय म्हणाल की px अधिक y हे px आणि py यांच्यातील संयोगाच्या बरोबरीचे आहे.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "नेहमीच्या पद्धतीने लिहिल्याशिवाय ते x आणि y ने लिहायचे नाही, परंतु त्याऐवजी आम्ही फक्त त्यापैकी एका व्हेरिएबल्सवर लक्ष केंद्रित करतो, या प्रकरणात x, त्यास त्याच्या सर्व संभाव्य मूल्यांवर श्रेणी द्या, ज्याचा अर्थ येथे फक्त 1 ते 6 पर्यंत जाणे आहे.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/persian/sentence_translations.json b/2023/convolutions2/persian/sentence_translations.json index c29883d85..0cf73cd5d 100644 --- a/2023/convolutions2/persian/sentence_translations.json +++ b/2023/convolutions2/persian/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum. ", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum. ", "translatedText": "به طور مشابه، اجازه دهید py تابع توزیع دوم ما باشد و px بعلاوه y تابعی باشد که توزیع مجموع را توصیف می کند. ", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py. ", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y. ", "translatedText": "در زبان انگلیسی، آنچه شما می گویید این است که px بعلاوه y برابر است با پیچیدگی بین px و py. ", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6. ", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6. ", "translatedText": "به جز روش معمولی که نوشته می‌شود این نیست که با x و y بنویسیم، بلکه ما فقط روی یکی از آن متغیرها تمرکز می‌کنیم، در این مورد x، به آن اجازه می‌دهیم در تمام مقادیر ممکن خود در محدوده باشد، که در اینجا فقط به معنای رفتن از 1 به 6 است. . ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/portuguese/sentence_translations.json b/2023/convolutions2/portuguese/sentence_translations.json index d0a55a92b..d41f83d86 100644 --- a/2023/convolutions2/portuguese/sentence_translations.json +++ b/2023/convolutions2/portuguese/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "Da mesma forma, seja py a função da nossa segunda distribuição e px mais y a função que descreve a distribuição da soma.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "Na linguagem, o que você diria é que px mais y é igual a uma convolução entre px e py.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "Exceto que a maneira usual de escrever não é escrever com x e y, mas apenas nos concentramos em uma dessas variáveis, neste caso x, deixando-a abranger todos os seus valores possíveis, o que aqui significa apenas ir de 1 a 6.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/russian/sentence_translations.json b/2023/convolutions2/russian/sentence_translations.json index 355e894d0..89e124e2b 100644 --- a/2023/convolutions2/russian/sentence_translations.json +++ b/2023/convolutions2/russian/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "Аналогично, пусть py будет функцией нашего второго распределения, а px плюс y будет функцией, описывающей распределение суммы.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "На жаргоне вы бы сказали, что px плюс y равно свертке между px и py.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "За исключением того, что обычно это пишется не с помощью x и y, а вместо этого мы просто фокусируемся на одной из этих переменных, в данном случае x, позволяя ей варьироваться по всем возможным значениям, что здесь просто означает переход от 1 до 6.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/spanish/sentence_translations.json b/2023/convolutions2/spanish/sentence_translations.json index f74cd2d98..bb4efbf6b 100644 --- a/2023/convolutions2/spanish/sentence_translations.json +++ b/2023/convolutions2/spanish/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "De manera similar, permitamos que py sea la función de nuestra segunda distribución y que px más y sea la función que describe la distribución de la suma.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "En la jerga, lo que dirías es que px más y es igual a una convolución entre px y py.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "Excepto que la forma habitual en que se escribe no es escribir con x e y, sino que simplemente nos centramos en una de esas variables, en este caso x, dejándola abarcar todos sus valores posibles, lo que aquí solo significa ir de 1 a 6.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/tamil/sentence_translations.json b/2023/convolutions2/tamil/sentence_translations.json index fb54868e8..8d64f3630 100644 --- a/2023/convolutions2/tamil/sentence_translations.json +++ b/2023/convolutions2/tamil/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "இதேபோல், py என்பது நமது இரண்டாவது விநியோகத்திற்கான செயல்பாடாகவும், px கூட்டல் y என்பது தொகைக்கான விநியோகத்தை விவரிக்கும் செயல்பாடாகவும் இருக்கட்டும்.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "லிங்கோவில், px மற்றும் py இடையேயான வளைவுக்கு px கூட்டல் y சமம் என்று நீங்கள் கூறுவீர்கள்.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "இது எழுதப்பட்ட வழக்கமான வழியைத் தவிர, x மற்றும் y உடன் எழுதுவது அல்ல, ஆனால் அதற்கு பதிலாக நாம் அந்த மாறிகளில் ஒன்றில் கவனம் செலுத்துகிறோம், இந்த விஷயத்தில் x, அதன் சாத்தியமான மதிப்புகள் அனைத்திலும் வரம்பில் இருக்க அனுமதிக்கிறது, அதாவது இங்கே 1 முதல் 6 வரை செல்லும்.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/telugu/sentence_translations.json b/2023/convolutions2/telugu/sentence_translations.json index a1f15fb0f..f3eba0277 100644 --- a/2023/convolutions2/telugu/sentence_translations.json +++ b/2023/convolutions2/telugu/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "అదేవిధంగా, py అనేది మన రెండవ పంపిణీకి ఫంక్షన్‌గా ఉండనివ్వండి మరియు px ప్లస్ y అనేది మొత్తానికి పంపిణీని వివరించే ఫంక్షన్.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "లింగోలో, మీరు చెప్పేది ఏమిటంటే px ప్లస్ y అనేది px మరియు py మధ్య కాన్వల్యూషన్‌కు సమానం.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "ఇది వ్రాసిన సాధారణ మార్గం తప్ప, x మరియు yతో వ్రాయడం కాదు, బదులుగా మేము ఆ వేరియబుల్స్‌లో ఒకదానిపై దృష్టి పెడతాము, ఈ సందర్భంలో x, దాని సాధ్యమయ్యే అన్ని విలువల పరిధిని అనుమతిస్తుంది, అంటే ఇక్కడ 1 నుండి 6 వరకు వెళ్లడం.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/thai/sentence_translations.json b/2023/convolutions2/thai/sentence_translations.json index 6fbd660a3..966c35cd4 100644 --- a/2023/convolutions2/thai/sentence_translations.json +++ b/2023/convolutions2/thai/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum. ", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py. ", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6. ", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/turkish/sentence_translations.json b/2023/convolutions2/turkish/sentence_translations.json index 99891fbb2..51e6a753a 100644 --- a/2023/convolutions2/turkish/sentence_translations.json +++ b/2023/convolutions2/turkish/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "Benzer şekilde, ikinci dağılımımızın fonksiyonu py olsun ve toplamın dağılımını tanımlayan fonksiyon da px artı y olsun.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "Dilde px artı y'nin px ile py arasındaki bir evrişime eşit olduğunu söyleyebilirsiniz.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "Her zamanki gibi yazılma şekli dışında, x ve y ile yazmak değil, bunun yerine sadece bu değişkenlerden birine, bu durumda x'e odaklanırız, onun olası tüm değerleri arasında değişmesine izin veririz, bu da burada sadece 1'den 6'ya gitmek anlamına gelir.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/ukrainian/sentence_translations.json b/2023/convolutions2/ukrainian/sentence_translations.json index de9c1f9e1..dbc08b955 100644 --- a/2023/convolutions2/ukrainian/sentence_translations.json +++ b/2023/convolutions2/ukrainian/sentence_translations.json @@ -483,14 +483,14 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "Подібним чином, нехай py буде функцією для нашого другого розподілу, а px плюс y буде функцією, що описує розподіл для суми.", "n_reviews": 0, "start": 544.44, "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "На жаргоні ви б сказали, що px плюс y дорівнює згортці між px і py.", "n_reviews": 0, "start": 553.96, @@ -532,7 +532,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "За винятком звичайного способу запису — не писати за допомогою x і y, а замість цього ми просто зосереджуємось на одній із цих змінних, у цьому випадку x, дозволяючи їй варіюватись серед усіх можливих значень, що тут просто означає перехід від 1 до 6 .", "n_reviews": 0, "start": 595.82, diff --git a/2023/convolutions2/urdu/sentence_translations.json b/2023/convolutions2/urdu/sentence_translations.json index 763da727c..13f988a0e 100644 --- a/2023/convolutions2/urdu/sentence_translations.json +++ b/2023/convolutions2/urdu/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum. ", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum. ", "translatedText": "اسی طرح، آئیے اپنی دوسری تقسیم کے لیے py کو فنکشن بننے دیں، اور px پلس y کو رقم کی تقسیم کو بیان کرنے والا فنکشن بنائیں۔", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py. ", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y. ", "translatedText": "لنگو میں، آپ جو کہیں گے وہ یہ ہے کہ px جمع y px اور py کے درمیان کنولیشن کے برابر ہے۔", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6. ", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6. ", "translatedText": "عام طریقہ کے علاوہ اسے x اور y کے ساتھ لکھنا نہیں ہے، بلکہ اس کے بجائے ہم صرف ان متغیرات میں سے ایک پر توجہ مرکوز کرتے ہیں، اس صورت میں x، اسے اپنی تمام ممکنہ قدروں پر رینج کرنے دیتے ہیں، جس کا مطلب یہاں صرف 1 سے 6 تک جانا ہے۔. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/convolutions2/vietnamese/sentence_translations.json b/2023/convolutions2/vietnamese/sentence_translations.json index bebb131d8..7c2c33269 100644 --- a/2023/convolutions2/vietnamese/sentence_translations.json +++ b/2023/convolutions2/vietnamese/sentence_translations.json @@ -552,7 +552,7 @@ "end": 542.98 }, { - "input": "Similarly, let's let py be the function for our second distribution, and px plus y be the function describing the distribution for the sum.", + "input": "Similarly, let's let p sub y be the function for our second distribution, and p sub x plus y be the function describing the distribution for the sum.", "translatedText": "Tương tự, hãy gọi py là hàm cho phân phối thứ hai của chúng ta và px cộng y là hàm mô tả phân phối của tổng.", "model": "google_nmt", "n_reviews": 0, @@ -560,7 +560,7 @@ "end": 553.06 }, { - "input": "In the lingo, what you would say is that px plus y is equal to a convolution between px and py.", + "input": "In the lingo, what you would say is that p sub x plus y is equal to a convolution between p sub x and p sub y.", "translatedText": "Trong biệt ngữ, điều bạn sẽ nói là px cộng y bằng tích chập giữa px và py.", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 595.82 }, { - "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 to 6.", + "input": "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x, letting it range over all of its possible values, which here just means going from 1 all the way up to 6.", "translatedText": "Ngoại trừ cách thông thường nó được viết là không viết với x và y, mà thay vào đó chúng ta chỉ tập trung vào một trong các biến đó, trong trường hợp này là x, để nó nằm trên tất cả các giá trị có thể của nó, ở đây chỉ có nghĩa là đi từ 1 đến 6.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/arabic/sentence_translations.json b/2023/gaussian-integral/arabic/sentence_translations.json index 95df9e5e2..21106c4f1 100644 --- a/2023/gaussian-integral/arabic/sentence_translations.json +++ b/2023/gaussian-integral/arabic/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared. ", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared. ", "translatedText": "وطريقة التفكير في الأمر هي أن نأخذ في الاعتبار المسافة من تلك النقطة إلى نقطة الأصل، والتي سأسميها r، ثم نعوض تلك المسافة بوظيفة منحنى الجرس الأصلية، نأخذ e إلى سالب r تربيع. ", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos. ", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you. ", "translatedText": "شكرًا جزيلاً لكيفن على مشاركته هذا، وشكرًا لجميع المستفيدين، بالمناسبة، على دعم القناة وأيضًا على جميع التعليقات التي تقدمها على المسودات المبكرة لمقاطع الفيديو. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/bengali/sentence_translations.json b/2023/gaussian-integral/bengali/sentence_translations.json index 485dabc0b..3b5dad3b7 100644 --- a/2023/gaussian-integral/bengali/sentence_translations.json +++ b/2023/gaussian-integral/bengali/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared. ", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared. ", "translatedText": "এবং এটি সম্পর্কে চিন্তা করার উপায় হল সেই বিন্দু থেকে উৎপত্তির দূরত্ব বিবেচনা করা, যাকে আমি r হিসাবে লেবেল করব, এবং তারপর সেই দূরত্বটিকে আমাদের মূল বেল কার্ভ ফাংশনে প্লাগ করতে, আমরা e কে নেতিবাচক r বর্গক্ষেত্রে নিয়ে যাই।", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos. ", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/chinese/sentence_translations.json b/2023/gaussian-integral/chinese/sentence_translations.json index 7b774bffc..26d3a483a 100644 --- a/2023/gaussian-integral/chinese/sentence_translations.json +++ b/2023/gaussian-integral/chinese/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "思考它的方法是考虑从该点到原点的 距离，我将其标记为 r，然后将 该距离代入我们原始的钟形曲线函数 ，我们将 e 取负 r 平方。", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "非常感谢凯文分享这一点，顺 便感谢所有订阅者对频道的支持以及 您对视频早期草稿提供的所有反馈。", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/english/captions.srt b/2023/gaussian-integral/english/captions.srt index dba7b8943..369cc48be 100644 --- a/2023/gaussian-integral/english/captions.srt +++ b/2023/gaussian-integral/english/captions.srt @@ -539,16 +539,16 @@ function associated with that radius, which like I said earlier you can think of the negative r squared. 136 -00:07:29,380 --> 00:07:33,157 -Now the real way you want to think about this is to give that cylinder +00:07:29,380 --> 00:07:33,213 +The real way you want to think about this is to give that cylinder a little 137 -00:07:33,157 --> 00:07:36,989 -a little bit of thickness, dr, so that the volume that it represents is +00:07:33,213 --> 00:07:37,096 +bit of thickness, which we'll call dr, so that the volume that it represents 138 -00:07:36,989 --> 00:07:40,980 -approximately that area we just looked at, multiplied by this thickness dr. +00:07:37,096 --> 00:07:40,980 +is approximately that area we just looked at multiplied by this thickness dr. 139 00:07:41,600 --> 00:07:44,860 @@ -1603,102 +1603,98 @@ Nevertheless, thinking once again of our statistician's friend, this is still not entirely satisfying. 402 -00:23:06,380 --> 00:23:09,784 -Using the Herschel-Maxwell derivation, saying this property of a +00:23:06,380 --> 00:23:10,533 +Using the Herschel-Maxwell derivation, saying this property of a multi-dimensional 403 -00:23:09,784 --> 00:23:12,874 -multi-dimensional distribution is what defines a Gaussian, +00:23:10,533 --> 00:23:14,086 +distribution is what defines a Gaussian, well that presumes that we're 404 -00:23:12,874 --> 00:23:16,644 -presumes that we're already in some kind of multi-dimensional situation +00:23:14,086 --> 00:23:17,640 +already in some kind of multi-dimensional situation in the first place. 405 -00:23:16,644 --> 00:23:17,640 -in the first place. - -406 00:23:18,120 --> 00:23:21,238 Much more commonly, the way that a normal distribution -407 +406 00:23:21,238 --> 00:23:24,640 arises in practice doesn't feel spatial or geometric at all. -408 +407 00:23:24,880 --> 00:23:27,470 It stems from the central limit theorem, which is all -409 +408 00:23:27,470 --> 00:23:30,300 about adding together many different independent variables. -410 +409 00:23:30,820 --> 00:23:34,537 So to bring it all home here, what we need to do is explain why the function -411 +410 00:23:34,537 --> 00:23:38,207 that's characterized by this Herschel-Maxwell derivation should be the same -412 +411 00:23:38,207 --> 00:23:41,780 thing as the function that sits at the heart of the central limit theorem. -413 +412 00:23:42,520 --> 00:23:46,544 And at this point, those of you following along are probably going to make fun of me, -414 +413 00:23:46,544 --> 00:23:49,680 I think it makes sense to pull this last step out as its own video. -415 +414 00:23:50,260 --> 00:23:52,180 Oh, and one final footnote here. -416 +415 00:23:52,380 --> 00:23:55,962 After making a Patreon post about this particular project, one patron, -417 +416 00:23:55,962 --> 00:24:00,151 who's a mathematician named Kevin Ega, shared something completely delightful that -418 +417 00:24:00,151 --> 00:24:04,087 I had never seen before, which is that if you apply this integration trick in -419 +418 00:24:04,087 --> 00:24:08,376 higher dimensions, it lets you derive the formulas for volumes of higher dimensional -420 +419 00:24:08,376 --> 00:24:08,780 spheres. -421 +420 00:24:09,260 --> 00:24:12,126 It's a very fun exercise, I'm leaving the details up on the screen -422 +421 00:24:12,126 --> 00:24:14,780 for any viewers who are comfortable with integration by parts. -423 +422 00:24:15,260 --> 00:24:18,482 Thank you very much to Kevin for sharing that one, and thanks to all patrons, -424 +423 00:24:18,482 --> 00:24:20,507 by the way, both for the support of the channel, -425 +424 00:24:20,507 --> 00:24:23,400 and also for all the feedback you offer on the early drafts of videos. -426 +425 00:24:34,740 --> 00:24:45,100 Thank you. diff --git a/2023/gaussian-integral/english/sentence_timings.json b/2023/gaussian-integral/english/sentence_timings.json index 941db95c3..52c5b3874 100644 --- a/2023/gaussian-integral/english/sentence_timings.json +++ b/2023/gaussian-integral/english/sentence_timings.json @@ -330,7 +330,7 @@ 448.36 ], [ - "Now the real way you want to think about this is to give that cylinder a little bit of thickness, dr, so that the volume that it represents is approximately that area we just looked at, multiplied by this thickness dr.", + "The real way you want to think about this is to give that cylinder a little bit of thickness, which we'll call dr, so that the volume that it represents is approximately that area we just looked at multiplied by this thickness dr.", 449.38, 460.98 ], @@ -915,7 +915,7 @@ 1385.96 ], [ - "Using the Herschel-Maxwell derivation, saying this property of a multi-dimensional distribution is what defines a Gaussian, presumes that we're already in some kind of multi-dimensional situation in the first place.", + "Using the Herschel-Maxwell derivation, saying this property of a multi-dimensional distribution is what defines a Gaussian, well that presumes that we're already in some kind of multi-dimensional situation in the first place.", 1386.38, 1397.64 ], diff --git a/2023/gaussian-integral/english/transcript.txt b/2023/gaussian-integral/english/transcript.txt index cbda158b6..3c3d70895 100644 --- a/2023/gaussian-integral/english/transcript.txt +++ b/2023/gaussian-integral/english/transcript.txt @@ -64,7 +64,7 @@ Here, making this a little more quantitative, let's focus on just one of those c You might imagine it as something like the label on a soup can that we can unwrap into a rectangle. The circumference of the cylinder, which is the top side of that rectangle, is going to be 2 pi times the radius. And then the height of our cylinder, the other side of our rectangle, is the height of the surface at this point, which by definition is the value of our function associated with that radius, which like I said earlier you can think of as e to the negative r squared. -Now the real way you want to think about this is to give that cylinder a little bit of thickness, dr, so that the volume that it represents is approximately that area we just looked at, multiplied by this thickness dr. +The real way you want to think about this is to give that cylinder a little bit of thickness, which we'll call dr, so that the volume that it represents is approximately that area we just looked at multiplied by this thickness dr. Our task now is to integrate together, or add together, all of these different cylinders as r ranges between 0 and infinity. Or more precisely, we consider what happens as that thickness gets thinner and thinner, approaching 0, and we add together the volumes of the many many many different thin cylinders that sit underneath that curve. You might think this is just a harder version of what we were looking at earlier, three dimensions should be more complicated than two. @@ -181,7 +181,7 @@ I mean, a key problem-solving principle in math is to use the defining features And if you think back, the essence of the proof came down to using that radial symmetry on the one hand, and then also using the ability to factor the function on the other. From this standpoint, using both those facts feels less like a trick that happened to work, and more like an inevitable necessity. Nevertheless, thinking once again of our statistician's friend, this is still not entirely satisfying. -Using the Herschel-Maxwell derivation, saying this property of a multi-dimensional distribution is what defines a Gaussian, presumes that we're already in some kind of multi-dimensional situation in the first place. +Using the Herschel-Maxwell derivation, saying this property of a multi-dimensional distribution is what defines a Gaussian, well that presumes that we're already in some kind of multi-dimensional situation in the first place. Much more commonly, the way that a normal distribution arises in practice doesn't feel spatial or geometric at all. It stems from the central limit theorem, which is all about adding together many different independent variables. So to bring it all home here, what we need to do is explain why the function that's characterized by this Herschel-Maxwell derivation should be the same thing as the function that sits at the heart of the central limit theorem. diff --git a/2023/gaussian-integral/french/sentence_translations.json b/2023/gaussian-integral/french/sentence_translations.json index c157bf7ff..35e3be309 100644 --- a/2023/gaussian-integral/french/sentence_translations.json +++ b/2023/gaussian-integral/french/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "Et la façon d'y penser est de considérer la distance entre ce point et l'origine, que j'appellerai r, puis de relier cette distance à notre fonction de courbe en cloche d'origine, nous prenons e puissance moins r carré.", "model": "google_nmt", "n_reviews": 1, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "Merci beaucoup à Kevin pour ce partage, et merci à tous les contributeurs, d'ailleurs, à la fois pour le soutien de la chaîne mais aussi pour tous les retours que vous offrez sur les premières versions de vidéos.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/german/sentence_translations.json b/2023/gaussian-integral/german/sentence_translations.json index a755e87d0..fd5faa89a 100644 --- a/2023/gaussian-integral/german/sentence_translations.json +++ b/2023/gaussian-integral/german/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "Betrachten wir den Abstand r von diesem Punkt zum Ursprung. Diesen Abstand setzen wir dann in unsere ursprüngliche Glockenkurvenfunktion einzubinden, indem wir sie als e hoch minus r Quadrat schreiben.", "model": "google_nmt", "n_reviews": 1, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "Vielen Dank an Kevin hierfür, und vielen Dank übrigens an alle Patrons, sowohl für die Unterstützung des Kanals als auch für all das Feedback, das Sie zu den ersten Videoentwürfen geben.", "model": "google_nmt", "n_reviews": 1, diff --git a/2023/gaussian-integral/hebrew/sentence_translations.json b/2023/gaussian-integral/hebrew/sentence_translations.json index 2482ebdc8..daa2025c2 100644 --- a/2023/gaussian-integral/hebrew/sentence_translations.json +++ b/2023/gaussian-integral/hebrew/sentence_translations.json @@ -274,7 +274,7 @@ }, { "input": "As a quick reminder for how you might read this notation, you might imagine approximating that area with many different rectangles under the curve, where the height of each such rectangle is the value of the function above that point, in this case, e to the negative x squared for a certain input x, and the width is some little number that we're calling dx.", - "translatedText": "כתזכורת מהירה לאופן שבו אתם יכולים לקרוא את הסימון הזה, אתם עשויים לדמיין קירוב לשטח הזה ע"י מלבנים רבים ושונים מתחת לעקומה, כאשר הגובה של כל מלבן כזה הוא הערך של הפונקציה מעל לנקודה זו, במקרה זה, e במינוס x בריבוע עבור קלט מסוים x, והרוחב הוא מספר קטן שאנו קוראים לו dx.", + "translatedText": "כתזכורת מהירה לאופן שבו אתם יכולים לקרוא את הסימון הזה, אתם עשויים לדמיין קירוב לשטח הזה ע\"י מלבנים רבים ושונים מתחת לעקומה, כאשר הגובה של כל מלבן כזה הוא הערך של הפונקציה מעל לנקודה זו, במקרה זה, e במינוס x בריבוע עבור קלט מסוים x, והרוחב הוא מספר קטן שאנו קוראים לו dx.", "n_reviews": 0, "start": 237.26, "end": 253.8 diff --git a/2023/gaussian-integral/hindi/sentence_translations.json b/2023/gaussian-integral/hindi/sentence_translations.json index fe80ca952..a936f1c53 100644 --- a/2023/gaussian-integral/hindi/sentence_translations.json +++ b/2023/gaussian-integral/hindi/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "और इसके बारे में सोचने का तरीका उस बिंदु से मूल तक की दूरी पर विचार करना है, जिसे मैं आर के रूप में लेबल करूंगा, और फिर उस दूरी को हमारे मूल घंटी कर्व फ़ंक्शन में प्लग करने के लिए, हम ई को नकारात्मक आर वर्ग में ले जाते हैं।", "model": "google_nmt", "n_reviews": 1, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "इसे साझा करने के लिए केविन को बहुत-बहुत धन्यवाद, और सभी संरक्षकों को भी, चैनल के समर्थन के लिए और वीडियो के शुरुआती ड्राफ्ट पर आपके द्वारा दिए गए फीडबैक के लिए भी धन्यवाद।", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/hungarian/sentence_translations.json b/2023/gaussian-integral/hungarian/sentence_translations.json index 389af7118..1bc5d7a23 100644 --- a/2023/gaussian-integral/hungarian/sentence_translations.json +++ b/2023/gaussian-integral/hungarian/sentence_translations.json @@ -528,7 +528,7 @@ "end": 448.36 }, { - "input": "Now the real way you want to think about this is to give that cylinder a little bit of thickness, dr, so that the volume that it represents is approximately that area we just looked at, multiplied by this thickness dr.", + "input": "The real way you want to think about this is to give that cylinder a little bit of thickness, which we'll call dr, so that the volume that it represents is approximately that area we just looked at multiplied by this thickness dr.", "translatedText": "A valódi módja annak, ahogyan ezt gondolni akarjuk, hogy adjunk a hengerhez egy kis vastagságot, dr-t, hogy az általa képviselt térfogat megközelítőleg az imént vizsgált terület és a dr vastagság szorzatának feleljen meg.", "model": "DeepL", "n_reviews": 0, @@ -1464,7 +1464,7 @@ "end": 1385.96 }, { - "input": "Using the Herschel-Maxwell derivation, saying this property of a multi-dimensional distribution is what defines a Gaussian, presumes that we're already in some kind of multi-dimensional situation in the first place.", + "input": "Using the Herschel-Maxwell derivation, saying this property of a multi-dimensional distribution is what defines a Gaussian, well that presumes that we're already in some kind of multi-dimensional situation in the first place.", "translatedText": "A Herschel-Maxwell levezetést használva, azt mondván, hogy egy többdimenziós eloszlásnak ez a tulajdonsága az, ami a Gauss-t meghatározza, azt feltételezi, hogy már eleve valamilyen többdimenziós helyzetben vagyunk.", "model": "DeepL", "n_reviews": 0, diff --git a/2023/gaussian-integral/indonesian/sentence_translations.json b/2023/gaussian-integral/indonesian/sentence_translations.json index 1315797f2..285bf3ed8 100644 --- a/2023/gaussian-integral/indonesian/sentence_translations.json +++ b/2023/gaussian-integral/indonesian/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "Dan cara memikirkannya adalah dengan mempertimbangkan jarak dari titik tersebut ke titik asal, yang akan saya beri nama r, dan kemudian memasukkan jarak tersebut ke fungsi kurva lonceng awal, kita ambil e ke r kuadrat negatif.", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "Terima kasih banyak kepada Kevin karena telah membagikan video tersebut, dan terima kasih kepada semua pelanggan, atas dukungan saluran tersebut dan juga atas semua masukan yang Anda berikan pada draf awal video.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/italian/sentence_translations.json b/2023/gaussian-integral/italian/sentence_translations.json index d7e6bfac1..8ff509b4b 100644 --- a/2023/gaussian-integral/italian/sentence_translations.json +++ b/2023/gaussian-integral/italian/sentence_translations.json @@ -385,7 +385,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "E il modo di pensarci è considerare la distanza da quel punto all'origine, che chiamerò r, e poi per collegare quella distanza alla nostra funzione curva a campana originale, prendiamo e = meno r al quadrato.", "n_reviews": 0, "start": 357.46, @@ -1365,7 +1365,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "Grazie mille a Kevin per aver condiviso questo, e grazie a tutti i sostenitori, comunque, sia per il supporto del canale sia anche per tutto il feedback che offrite sulle prime bozze dei video.", "n_reviews": 0, "start": 1455.26, diff --git a/2023/gaussian-integral/japanese/sentence_translations.json b/2023/gaussian-integral/japanese/sentence_translations.json index db81350d8..4ae404a79 100644 --- a/2023/gaussian-integral/japanese/sentence_translations.json +++ b/2023/gaussian-integral/japanese/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "それを考える方法は、その点から原点までの距 離 (ここでは r とラベル付けします) を考慮し、その距離を元の釣鐘曲線関数に代入 し、e を負の r の 2 乗とします。", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "この動画を共有してくれた Kevin に感謝します。 また、チャンネルをサポートしてくれたパトロンの皆さん、そしてビデ オの初期草案に対してフィードバックをくれた皆さんにも感謝します。", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/korean/sentence_translations.json b/2023/gaussian-integral/korean/sentence_translations.json index 23f49395e..b03afaa2b 100644 --- a/2023/gaussian-integral/korean/sentence_translations.json +++ b/2023/gaussian-integral/korean/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "이에 대해 생각하는 방법은 해당 지점에서 원점까지의 거리를 고려하는 것입니다. 이를 r로 표시한 다음 해당 거리를 원래 종 곡선 함수에 연결하고 e를 음수 r 제곱으로 가져옵니다.", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "해당 동영상을 공유해 주신 Kevin에게 진심으로 감사드리며, 채널 지원과 초기 동영상 초안에 대한 모든 피드백에 대해 모든 후원자 분들께 감사드립니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/marathi/sentence_translations.json b/2023/gaussian-integral/marathi/sentence_translations.json index 2cf306e4d..c40b1f0ac 100644 --- a/2023/gaussian-integral/marathi/sentence_translations.json +++ b/2023/gaussian-integral/marathi/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "आणि त्याबद्दल विचार करण्याचा मार्ग म्हणजे त्या बिंदूपासून उत्पत्तीपर्यंतचे अंतर विचारात घेणे, ज्याला मी r असे लेबल करेन, आणि नंतर ते अंतर आपल्या मूळ बेल वक्र फंक्शनमध्ये जोडण्यासाठी, आपण e ला ऋणात्मक r वर्गाकडे नेतो.", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "ते शेअर केल्याबद्दल केविनचे खूप खूप आभार आणि चॅनेलच्या समर्थनासाठी आणि व्हिडिओंच्या सुरुवातीच्या मसुद्यांवर तुम्ही ऑफर केलेल्या सर्व अभिप्रायाबद्दल, सर्व संरक्षकांचे आभार.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/persian/sentence_translations.json b/2023/gaussian-integral/persian/sentence_translations.json index 8d7c2bc92..b58ba0421 100644 --- a/2023/gaussian-integral/persian/sentence_translations.json +++ b/2023/gaussian-integral/persian/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared. ", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared. ", "translatedText": "و راه فکر کردن در مورد آن این است که فاصله آن نقطه تا مبدأ را در نظر بگیریم، که من آن را r برچسب می زنم، و سپس برای اتصال آن فاصله به تابع منحنی زنگ اصلی، e را به r مربع منفی می بریم. ", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos. ", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you. ", "translatedText": "از کوین بسیار سپاسگزارم که آن یکی را به اشتراک گذاشت، و اتفاقاً از همه حامیان، هم برای حمایت از کانال و هم برای همه بازخوردهایی که در مورد پیش نویس های اولیه ویدیوها ارائه می دهید، تشکر می کنم. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/portuguese/sentence_translations.json b/2023/gaussian-integral/portuguese/sentence_translations.json index 0e1f67b62..42c960dec 100644 --- a/2023/gaussian-integral/portuguese/sentence_translations.json +++ b/2023/gaussian-integral/portuguese/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "E a maneira de pensar sobre isso é considerar a distância desse ponto até a origem, que rotularei como r, e então, para substituir essa distância à nossa função de curva em sino original, elevamos e elevado a menos r ao quadrado.", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "Muito obrigado ao Kevin por compartilhar isso e, a propósito, obrigado a todos os patrocinadores, tanto pelo apoio ao canal quanto por todo o feedback que vocês oferecem sobre os primeiros rascunhos dos vídeos.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/russian/sentence_translations.json b/2023/gaussian-integral/russian/sentence_translations.json index 56507e8c9..7a637b672 100644 --- a/2023/gaussian-integral/russian/sentence_translations.json +++ b/2023/gaussian-integral/russian/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "И чтобы об этом подумать, нужно рассмотреть расстояние от этой точки до начала координат, которое я обозначим как r, а затем, чтобы подставить это расстояние к нашей исходной функции колоколообразной кривой, мы возьмем e в отрицательный квадрат r.", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "Большое спасибо Кевину за то, что поделился этим, и, кстати, спасибо всем покровителям, как за поддержку канала, так и за все отзывы, которые вы даете о ранних черновиках видео.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/spanish/sentence_translations.json b/2023/gaussian-integral/spanish/sentence_translations.json index c273962e5..2d1b5b51a 100644 --- a/2023/gaussian-integral/spanish/sentence_translations.json +++ b/2023/gaussian-integral/spanish/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "Y la forma de pensarlo es considerar la distancia desde ese punto hasta el origen, que etiquetaré como r, y luego para conectar esa distancia a nuestra función de curva de campana original, llevamos e al cuadrado negativo de r.", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "Muchas gracias a Kevin por compartirlo y, por cierto, gracias a todos los patrocinadores, tanto por el apoyo al canal como por todos los comentarios que ofrecen sobre los primeros borradores de los videos.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/tamil/sentence_translations.json b/2023/gaussian-integral/tamil/sentence_translations.json index 63e24054c..3da36df0d 100644 --- a/2023/gaussian-integral/tamil/sentence_translations.json +++ b/2023/gaussian-integral/tamil/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "அதைப் பற்றி சிந்திக்க வழி என்னவென்றால், அந்த புள்ளியிலிருந்து தோற்றத்திற்கான தூரத்தைக் கருத்தில் கொண்டு, அதை நான் r என லேபிளிடுவேன், பின்னர் அந்த தூரத்தை நமது அசல் பெல் வளைவு செயல்பாட்டிற்கு செருக, நாம் e ஐ எதிர்மறையான r ஸ்கொயர்க்கு கொண்டு செல்கிறோம்.", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "அதைப் பகிர்ந்ததற்காக கெவினுக்கு மிக்க நன்றி, மேலும் சேனலின் ஆதரவிற்காகவும், வீடியோக்களின் ஆரம்ப வரைவுகள் குறித்து நீங்கள் வழங்கிய அனைத்து கருத்துக்களுக்காகவும் அனைத்து புரவலர்களுக்கும் நன்றி.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/telugu/sentence_translations.json b/2023/gaussian-integral/telugu/sentence_translations.json index d968214fb..6a61e7354 100644 --- a/2023/gaussian-integral/telugu/sentence_translations.json +++ b/2023/gaussian-integral/telugu/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "మరియు దాని గురించి ఆలోచించడానికి మార్గం ఏమిటంటే, ఆ బిందువు నుండి మూలానికి ఉన్న దూరాన్ని పరిగణనలోకి తీసుకోవడం, నేను r అని లేబుల్ చేస్తాను, ఆపై ఆ దూరాన్ని మన అసలు బెల్ కర్వ్ ఫంక్షన్‌కు ప్లగ్ చేయడానికి, మేము eని నెగటివ్ r స్క్వేర్డ్‌కి తీసుకువెళతాము.", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "దీన్ని భాగస్వామ్యం చేసినందుకు కెవిన్‌కి చాలా ధన్యవాదాలు, అలాగే, ఛానెల్‌కు మద్దతు ఇచ్చినందుకు మరియు వీడియోల ప్రారంభ చిత్తుప్రతులపై మీరు అందించిన అన్ని అభిప్రాయాలకు కూడా అన్ని పోషకులకు ధన్యవాదాలు.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/thai/sentence_translations.json b/2023/gaussian-integral/thai/sentence_translations.json index d2ead2ace..72ee103cc 100644 --- a/2023/gaussian-integral/thai/sentence_translations.json +++ b/2023/gaussian-integral/thai/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared. ", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos. ", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you. ", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/turkish/sentence_translations.json b/2023/gaussian-integral/turkish/sentence_translations.json index 18a7d7486..829eace53 100644 --- a/2023/gaussian-integral/turkish/sentence_translations.json +++ b/2023/gaussian-integral/turkish/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "Ve bunu da şöyle hayal edebiliriz: O noktadan orijine olan mesafeyi düşünelim - r olarak etiketliyorum. Daha sonra bu mesafeyi orijinal çan eğrisi fonksiyonumuza ekliyoruz: e^(-r^2) .", "model": "google_nmt", "n_reviews": 1, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "Bunu paylaştığı için Kevin'e çok teşekkür ederim. Ve tüm destekçilere hem kanala verdikleri destek hem de videoların ilk taslaklarına sunduğunuz tüm geri bildirimler için ayrıca teşekkür ederim.", "model": "google_nmt", "n_reviews": 1, diff --git a/2023/gaussian-integral/ukrainian/sentence_translations.json b/2023/gaussian-integral/ukrainian/sentence_translations.json index 414312d51..e6592783e 100644 --- a/2023/gaussian-integral/ukrainian/sentence_translations.json +++ b/2023/gaussian-integral/ukrainian/sentence_translations.json @@ -385,7 +385,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "І спосіб подумати про це — розглянути відстань від цієї точки до початку координат, яку я позначу як r, а потім, щоб підключити цю відстань до нашої вихідної дзвоноподібної кривої, ми беремо e до від’ємного r у квадраті.", "n_reviews": 0, "start": 357.46, @@ -1365,7 +1365,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "Щиро дякую Кевіну за те, що поділився цим, і, до речі, дякую всім меценатам як за підтримку каналу, так і за всі відгуки, які ви надаєте щодо перших чернеток відео.", "n_reviews": 0, "start": 1455.26, diff --git a/2023/gaussian-integral/urdu/sentence_translations.json b/2023/gaussian-integral/urdu/sentence_translations.json index ad555b88b..973cad922 100644 --- a/2023/gaussian-integral/urdu/sentence_translations.json +++ b/2023/gaussian-integral/urdu/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared. ", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared. ", "translatedText": "اور اس کے بارے میں سوچنے کا طریقہ یہ ہے کہ اس مقام سے اصل تک کے فاصلے پر غور کیا جائے، جسے میں r کا لیبل لگاؤں گا، اور پھر اس فاصلے کو اپنے اصل گھنٹی کریو فنکشن میں لگانے کے لیے، ہم e کو منفی r مربع پر لے جاتے ہیں۔", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos. ", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you. ", "translatedText": "اس کو شیئر کرنے کے لیے کیون کا بہت بہت شکریہ، اور تمام سرپرستوں کا، ویسے بھی، چینل کی حمایت کے لیے اور ویڈیوز کے ابتدائی مسودوں پر آپ کے پیش کردہ تمام تاثرات کے لیے بھی شکریہ۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/gaussian-integral/vietnamese/sentence_translations.json b/2023/gaussian-integral/vietnamese/sentence_translations.json index 94cba0a23..32b0d0e7c 100644 --- a/2023/gaussian-integral/vietnamese/sentence_translations.json +++ b/2023/gaussian-integral/vietnamese/sentence_translations.json @@ -440,7 +440,7 @@ "end": 357.12 }, { - "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, we take e to the negative r squared.", + "input": "And the way to think about it is to consider the distance from that point to the origin, which I'll label as r, and then to plug in that distance to our original bell curve function, that is, we take e to the negative r squared.", "translatedText": "Và cách nghĩ về nó là xét khoảng cách từ điểm đó đến gốc tọa độ, mà tôi sẽ gọi là r, và sau đó thay khoảng cách đó vào hàm đường cong hình chuông ban đầu của chúng ta, chúng ta lấy e theo âm r bình phương.", "model": "google_nmt", "n_reviews": 0, @@ -1560,7 +1560,7 @@ "end": 1454.78 }, { - "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel and also for all the feedback you offer on the early drafts of videos.", + "input": "Thank you very much to Kevin for sharing that one, and thanks to all patrons, by the way, both for the support of the channel, and also for all the feedback you offer on the early drafts of videos. Thank you.", "translatedText": "Nhân tiện, xin cảm ơn Kevin rất nhiều vì đã chia sẻ điều đó và nhân tiện, xin cảm ơn tất cả những người bảo trợ vì đã hỗ trợ kênh cũng như tất cả phản hồi mà bạn đưa ra về bản nháp đầu tiên của video.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/how-they-fool-ya/english/captions.srt b/2023/how-they-fool-ya/english/captions.srt index 7a9a136be..2934cf45b 100644 --- a/2023/how-they-fool-ya/english/captions.srt +++ b/2023/how-they-fool-ya/english/captions.srt @@ -87,66 +87,6 @@ if he didn't make it about love and religion and all of those great things. So welcome to the stage yet again, to accompany on keynote Matt Parker. 23 -00:01:10,100 --> 00:01:10,580 +00:01:10,100 --> 00:03:48,120 Wonderful. -24 -00:01:11,700 --> 00:01:14,828 -And I know I know Leonard Cohen, but just imagine if you're watching - -25 -00:01:14,828 --> 00:01:17,912 -Shrek and when they get to that typical moment when you don't think - -26 -00:01:17,912 --> 00:01:20,860 -love is going to hold, you've actually heard the proper original. - -27 -00:01:27,400 --> 00:01:36,797 -Well I heard there was a sequence of chords Splits the circle to one, two, - -28 -00:01:36,797 --> 00:01:43,813 -then four And points seem to cut in powers of two Yeah, - -29 -00:01:43,813 --> 00:01:54,338 -it was run like this With fourth and fifth But something's up We go out at six It's - -30 -00:01:54,338 --> 00:02:04,988 -awesome 31 I had a piece for you I'll make it for you I'll make it for you I'll make - -31 -00:02:04,988 --> 00:02:15,513 -it for you I'll make it for you I'll make it for you Yeah When your faith is strong - -32 -00:02:15,513 --> 00:02:20,400 -you still need proof What's your faith? - -33 -00:02:20,880 --> 00:02:35,323 -Death can lead to a dew Each integral up on the left is pi over two Yeah I - -34 -00:02:35,323 --> 00:02:49,574 -think that's true For the next, which is terrible like a drum We've shown - -35 -00:02:49,574 --> 00:03:04,018 -that it's off by a hair It's a subtle slip but it's true I had a piece for - -36 -00:03:04,018 --> 00:03:18,077 -you I had a piece for you I'll make it for you I'll make it for you I'll - -37 -00:03:18,077 --> 00:03:32,135 -make it for you Now take a prime and write it in Facebook It's just like - -38 -00:03:32,135 --> 00:03:48,120 -you had before This prime gives a new prime With this rule, yeah Or does it though? - diff --git a/2023/how-they-fool-ya/english/sentence_timings.json b/2023/how-they-fool-ya/english/sentence_timings.json index fdb27e0bd..9e435aea9 100644 --- a/2023/how-they-fool-ya/english/sentence_timings.json +++ b/2023/how-they-fool-ya/english/sentence_timings.json @@ -92,21 +92,6 @@ [ "Wonderful.", 70.1, - 70.58 - ], - [ - "And I know I know Leonard Cohen, but just imagine if you're watching Shrek and when they get to that typical moment when you don't think love is going to hold, you've actually heard the proper original.", - 71.7, - 80.86 - ], - [ - "Well I heard there was a sequence of chords Splits the circle to one, two, then four And points seem to cut in powers of two Yeah, it was run like this With fourth and fifth But something's up We go out at six It's awesome 31 I had a piece for you I'll make it for you I'll make it for you I'll make it for you I'll make it for you I'll make it for you Yeah When your faith is strong you still need proof What's your faith?", - 87.4, - 140.4 - ], - [ - "Death can lead to a dew Each integral up on the left is pi over two Yeah I think that's true For the next, which is terrible like a drum We've shown that it's off by a hair It's a subtle slip but it's true I had a piece for you I had a piece for you I'll make it for you I'll make it for you I'll make it for you Now take a prime and write it in Facebook It's just like you had before This prime gives a new prime With this rule, yeah Or does it though?", - 140.88, 228.12 ] ] \ No newline at end of file diff --git a/2023/moser-reboot/dutch/sentence_translations.json b/2023/moser-reboot/dutch/sentence_translations.json index 0f2c93267..6b67a9f69 100644 --- a/2023/moser-reboot/dutch/sentence_translations.json +++ b/2023/moser-reboot/dutch/sentence_translations.json @@ -184,7 +184,7 @@ "end": 171.26 }, { - "input": "In this case, two warm-up questions that come to mind are, chords are there in this diagram, and at how many points within the circle do those chords intersect each other?", + "input": "In this case, two warm-up questions that come to mind are, how many total chords are there in this diagram, and at how many points within the circle do those chords intersect each other?", "translatedText": "In dit geval zijn er twee opwarmvragen die in je opkomen: er zijn akkoorden in dit diagram, en op hoeveel punten binnen de cirkel kruisen die akkoorden elkaar?", "model": "google_nmt", "n_reviews": 0, @@ -608,7 +608,7 @@ "end": 601.36 }, { - "input": "The total number of edges after the chopping would look like 3 plus 2 times 2, or 7.", + "input": "The total number of edges after the chopping would look like three plus two times two, or seven.", "translatedText": "Het totale aantal randen na het hakken zou er uitzien als 3 plus 2 keer 2, oftewel 7.", "model": "google_nmt", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 617.34 }, { - "input": "And for the diagram we care about, where we started off with n choose 2 separate lines, and they're getting chopped up at n choose 4 points in the middle, you would end up with n choose 2 plus 2 times n choose 4 edges.", + "input": "And for the diagram we care about where we started off with n choose two separate lines and they're getting chopped up at n choose four points in the middle, you would end up with n choose two plus two times n choose four edges.", "translatedText": "En voor het diagram waar we om geven, waar we begonnen met n kies 2 afzonderlijke lijnen, en ze worden opgedeeld bij n kies 4 punten in het midden, je zou eindigen met n kies 2 plus 2 keer n kies 4 randen .", "model": "google_nmt", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 638.56 }, { - "input": "So with all of that, you have the information you need to answer the original question, pulling up our variant of Euler's formula that counts the number of regions, plugging in the expression for the number of vertices, which is n plus the n choose 4 intersection points, and you also plug in the slightly larger expression for the new number of edges, n choose 2 plus 2 times n choose 4 plus n, and the expression has a lot of nice cancellation, for example you are adding an n but also subtracting an n, and you're adding two copies of n choose 4 but subtracting another copy of n choose 4, and when all the dust settles, the answer to the question is 1 plus n choose 2 plus n choose 4.", + "input": "So with all of that you have the information you need to answer the original question. Pulling up our variant of Euler's formula that counts the number of regions we'll plug in the expression for the number of vertices which is n plus the n choose four intersection points, and you also plug in the slightly larger expression for the new number of edges n choose two plus two times n choose four plus n, and the expression has a lot of nice cancellation, for example you are adding an n but also subtracting an n and you're adding two copies of n choose four but subtracting another copy of n choose four and when all the dust settles the answer to the question is one plus n choose two plus n choose four.", "translatedText": "Dus met dat alles heb je de informatie die je nodig hebt om de oorspronkelijke vraag te beantwoorden, door onze variant van de formule van Euler tevoorschijn te halen die het aantal regio's telt, en de uitdrukking voor het aantal hoekpunten in te vullen, namelijk n plus de n kies 4 snijpunten, en je vult ook de iets grotere uitdrukking in voor het nieuwe aantal randen, n kies 2 plus 2 keer n kies 4 plus n, en de uitdrukking heeft veel mooie annuleringen, je voegt bijvoorbeeld een n toe, maar ook Als je een n aftrekt, tel je twee exemplaren van n op, kies 4, maar trek je nog een kopie van n af, kies 4, en als al het stof is neergedaald, is het antwoord op de vraag 1 plus n kies 2 plus n kies 4.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/moser-reboot/english/captions.srt b/2023/moser-reboot/english/captions.srt index 3b2b60aa7..0c4eecf2c 100644 --- a/2023/moser-reboot/english/captions.srt +++ b/2023/moser-reboot/english/captions.srt @@ -207,16 +207,16 @@ It helps you get a foothold, and sometimes those answers are helpful in the final question. 53 -00:02:51,720 --> 00:02:55,094 +00:02:51,720 --> 00:02:54,822 In this case, two warm-up questions that come to mind are, 54 -00:02:55,094 --> 00:02:59,898 -chords are there in this diagram, and at how many points within the circle do those +00:02:54,822 --> 00:02:57,398 +how many total chords are there in this diagram, 55 -00:02:59,898 --> 00:03:01,500 -chords intersect each other? +00:02:57,398 --> 00:03:01,500 +and at how many points within the circle do those chords intersect each other? 56 00:03:02,200 --> 00:03:03,940 @@ -679,76 +679,76 @@ For example, look at this simple diagram where we have three lines and two intersection points. 171 -00:10:02,020 --> 00:10:07,580 -The total number of edges after the chopping would look like 3 plus 2 times 2, or 7. +00:10:02,020 --> 00:10:07,058 +The total number of edges after the chopping would look like three plus two times two, 172 +00:10:07,058 --> 00:10:07,580 +or seven. + +173 00:10:08,060 --> 00:10:12,054 If you had four lines that intersected at three separate points, -173 +174 00:10:12,054 --> 00:10:17,340 then the total number of little lines after chopping would be 4 plus 2 times 3, or 10. -174 -00:10:17,340 --> 00:10:22,506 -And for the diagram we care about, where we started off with n choose 2 separate lines, - 175 -00:10:22,506 --> 00:10:26,440 -and they're getting chopped up at n choose 4 points in the middle, +00:10:17,340 --> 00:10:21,438 +And for the diagram we care about where we started off with n choose two 176 -00:10:26,440 --> 00:10:30,140 -you would end up with n choose 2 plus 2 times n choose 4 edges. +00:10:21,438 --> 00:10:26,210 +separate lines and they're getting chopped up at n choose four points in the middle, 177 +00:10:26,210 --> 00:10:30,140 +you would end up with n choose two plus two times n choose four edges. + +178 00:10:30,680 --> 00:10:34,571 And actually there are a few more than that, because we're including the circle, -178 +179 00:10:34,571 --> 00:10:38,560 we also need to count the n different arcs that sit on the outside of this diagram. -179 -00:10:39,340 --> 00:10:43,993 -So with all of that, you have the information you need to answer the original question, - 180 -00:10:43,993 --> 00:10:48,064 -pulling up our variant of Euler's formula that counts the number of regions, +00:10:39,340 --> 00:10:43,823 +So with all of that you have the information you need to answer the original question. 181 -00:10:48,064 --> 00:10:50,973 -plugging in the expression for the number of vertices, +00:10:43,823 --> 00:10:48,306 +Pulling up our variant of Euler's formula that counts the number of regions we'll plug 182 -00:10:50,973 --> 00:10:53,722 -which is n plus the n choose 4 intersection points, +00:10:48,306 --> 00:10:52,377 +in the expression for the number of vertices which is n plus the n choose four 183 -00:10:53,722 --> 00:10:58,005 -and you also plug in the slightly larger expression for the new number of edges, +00:10:52,377 --> 00:10:56,757 +intersection points, and you also plug in the slightly larger expression for the new 184 -00:10:58,005 --> 00:11:02,236 -n choose 2 plus 2 times n choose 4 plus n, and the expression has a lot of nice +00:10:56,757 --> 00:11:00,158 +number of edges n choose two plus two times n choose four plus n, 185 -00:11:02,236 --> 00:11:06,096 -cancellation, for example you are adding an n but also subtracting an n, +00:11:00,158 --> 00:11:02,786 +and the expression has a lot of nice cancellation, 186 -00:11:06,096 --> 00:11:10,696 -and you're adding two copies of n choose 4 but subtracting another copy of n choose 4, +00:11:02,786 --> 00:11:07,269 +for example you are adding an n but also subtracting an n and you're adding two copies 187 -00:11:10,696 --> 00:11:15,244 -and when all the dust settles, the answer to the question is 1 plus n choose 2 plus n +00:11:07,269 --> 00:11:11,649 +of n choose four but subtracting another copy of n choose four and when all the dust 188 -00:11:15,244 --> 00:11:15,720 -choose 4. +00:11:11,649 --> 00:11:15,720 +settles the answer to the question is one plus n choose two plus n choose four. 189 00:11:16,320 --> 00:11:19,380 diff --git a/2023/moser-reboot/english/sentence_timings.json b/2023/moser-reboot/english/sentence_timings.json index b2dce57be..681d748d2 100644 --- a/2023/moser-reboot/english/sentence_timings.json +++ b/2023/moser-reboot/english/sentence_timings.json @@ -115,7 +115,7 @@ 171.26 ], [ - "In this case, two warm-up questions that come to mind are, chords are there in this diagram, and at how many points within the circle do those chords intersect each other?", + "In this case, two warm-up questions that come to mind are, how many total chords are there in this diagram, and at how many points within the circle do those chords intersect each other?", 171.72, 181.5 ], @@ -380,7 +380,7 @@ 601.36 ], [ - "The total number of edges after the chopping would look like 3 plus 2 times 2, or 7.", + "The total number of edges after the chopping would look like three plus two times two, or seven.", 602.02, 607.58 ], @@ -390,7 +390,7 @@ 617.34 ], [ - "And for the diagram we care about, where we started off with n choose 2 separate lines, and they're getting chopped up at n choose 4 points in the middle, you would end up with n choose 2 plus 2 times n choose 4 edges.", + "And for the diagram we care about where we started off with n choose two separate lines and they're getting chopped up at n choose four points in the middle, you would end up with n choose two plus two times n choose four edges.", 617.34, 630.14 ], @@ -400,7 +400,7 @@ 638.56 ], [ - "So with all of that, you have the information you need to answer the original question, pulling up our variant of Euler's formula that counts the number of regions, plugging in the expression for the number of vertices, which is n plus the n choose 4 intersection points, and you also plug in the slightly larger expression for the new number of edges, n choose 2 plus 2 times n choose 4 plus n, and the expression has a lot of nice cancellation, for example you are adding an n but also subtracting an n, and you're adding two copies of n choose 4 but subtracting another copy of n choose 4, and when all the dust settles, the answer to the question is 1 plus n choose 2 plus n choose 4.", + "So with all of that you have the information you need to answer the original question. Pulling up our variant of Euler's formula that counts the number of regions we'll plug in the expression for the number of vertices which is n plus the n choose four intersection points, and you also plug in the slightly larger expression for the new number of edges n choose two plus two times n choose four plus n, and the expression has a lot of nice cancellation, for example you are adding an n but also subtracting an n and you're adding two copies of n choose four but subtracting another copy of n choose four and when all the dust settles the answer to the question is one plus n choose two plus n choose four.", 639.34, 675.72 ], diff --git a/2023/moser-reboot/english/transcript.txt b/2023/moser-reboot/english/transcript.txt index 18da3980c..ee14acbb1 100644 --- a/2023/moser-reboot/english/transcript.txt +++ b/2023/moser-reboot/english/transcript.txt @@ -21,7 +21,7 @@ If you put n points on the boundary of a circle, and you connect them with all t What function of n should we plug in? As always with math, problem solving rule number one if you're stuck is to try solving easier questions somehow related to the problem at hand. It helps you get a foothold, and sometimes those answers are helpful in the final question. -In this case, two warm-up questions that come to mind are, chords are there in this diagram, and at how many points within the circle do those chords intersect each other? +In this case, two warm-up questions that come to mind are, how many total chords are there in this diagram, and at how many points within the circle do those chords intersect each other? The first question is relatively friendly. Every one of those chords corresponds uniquely with a pair of points on the circle. So effectively what you want to do is count how many distinct pairs of points are there. @@ -74,11 +74,11 @@ But the key is to treat this as a new graph, where those intersection points are That in turn chops up all of our chords into a larger number of edges, it's much more than just n choose 2, and initially it might seem really annoying and tricky to think about exactly how much it's chopped them up, but one way you can think about it is that every intersection point takes what started as two separate lines and then turns it into four lines. So in effect, each intersection point adds two more edges. For example, look at this simple diagram where we have three lines and two intersection points. -The total number of edges after the chopping would look like 3 plus 2 times 2, or 7. +The total number of edges after the chopping would look like three plus two times two, or seven. If you had four lines that intersected at three separate points, then the total number of little lines after chopping would be 4 plus 2 times 3, or 10. -And for the diagram we care about, where we started off with n choose 2 separate lines, and they're getting chopped up at n choose 4 points in the middle, you would end up with n choose 2 plus 2 times n choose 4 edges. +And for the diagram we care about where we started off with n choose two separate lines and they're getting chopped up at n choose four points in the middle, you would end up with n choose two plus two times n choose four edges. And actually there are a few more than that, because we're including the circle, we also need to count the n different arcs that sit on the outside of this diagram. -So with all of that, you have the information you need to answer the original question, pulling up our variant of Euler's formula that counts the number of regions, plugging in the expression for the number of vertices, which is n plus the n choose 4 intersection points, and you also plug in the slightly larger expression for the new number of edges, n choose 2 plus 2 times n choose 4 plus n, and the expression has a lot of nice cancellation, for example you are adding an n but also subtracting an n, and you're adding two copies of n choose 4 but subtracting another copy of n choose 4, and when all the dust settles, the answer to the question is 1 plus n choose 2 plus n choose 4. +So with all of that you have the information you need to answer the original question. Pulling up our variant of Euler's formula that counts the number of regions we'll plug in the expression for the number of vertices which is n plus the n choose four intersection points, and you also plug in the slightly larger expression for the new number of edges n choose two plus two times n choose four plus n, and the expression has a lot of nice cancellation, for example you are adding an n but also subtracting an n and you're adding two copies of n choose four but subtracting another copy of n choose four and when all the dust settles the answer to the question is one plus n choose two plus n choose four. On the one hand, you're done, you answered the question. I give you one of these circle diagrams with n points on the boundary, and using this formula you can figure out how many regions the circle has been cut into. But of course we're not really done, because that doesn't scratch the itch. diff --git a/2023/moser-reboot/hungarian/sentence_translations.json b/2023/moser-reboot/hungarian/sentence_translations.json index 5cdd146b3..d8f104e2a 100644 --- a/2023/moser-reboot/hungarian/sentence_translations.json +++ b/2023/moser-reboot/hungarian/sentence_translations.json @@ -184,7 +184,7 @@ "end": 171.26 }, { - "input": "In this case, two warm-up questions that come to mind are, chords are there in this diagram, and at how many points within the circle do those chords intersect each other?", + "input": "In this case, two warm-up questions that come to mind are, how many total chords are there in this diagram, and at how many points within the circle do those chords intersect each other?", "translatedText": "Ebben az esetben két bemelegítő példa jut eszembe: \"Hány darab húr van ezen az ábrán?\", és \"A körön belül összesen hány pontban metszik egymást ezek a húrok?\".", "model": "DeepL", "n_reviews": 1, @@ -608,7 +608,7 @@ "end": 601.36 }, { - "input": "The total number of edges after the chopping would look like 3 plus 2 times 2, or 7.", + "input": "The total number of edges after the chopping would look like three plus two times two, or seven.", "translatedText": "Az élek száma a vágás után úgy néz ki, hogy 3 plusz 2-szer 2, azaz 7.", "model": "DeepL", "n_reviews": 0, @@ -624,7 +624,7 @@ "end": 617.34 }, { - "input": "And for the diagram we care about, where we started off with n choose 2 separate lines, and they're getting chopped up at n choose 4 points in the middle, you would end up with n choose 2 plus 2 times n choose 4 edges.", + "input": "And for the diagram we care about where we started off with n choose two separate lines and they're getting chopped up at n choose four points in the middle, you would end up with n choose two plus two times n choose four edges.", "translatedText": "És a számunkra fontos diagram esetében, ahol n válasszunk 2 különálló vonallal kezdtük, és ezek középen n válasszunk 4 ponton feldarabolódnak, a végén n válasszunk 2 plusz 2-szer n válasszunk 4 élt.", "model": "DeepL", "n_reviews": 0, @@ -640,7 +640,7 @@ "end": 638.56 }, { - "input": "So with all of that, you have the information you need to answer the original question, pulling up our variant of Euler's formula that counts the number of regions, plugging in the expression for the number of vertices, which is n plus the n choose 4 intersection points, and you also plug in the slightly larger expression for the new number of edges, n choose 2 plus 2 times n choose 4 plus n, and the expression has a lot of nice cancellation, for example you are adding an n but also subtracting an n, and you're adding two copies of n choose 4 but subtracting another copy of n choose 4, and when all the dust settles, the answer to the question is 1 plus n choose 2 plus n choose 4.", + "input": "So with all of that you have the information you need to answer the original question. Pulling up our variant of Euler's formula that counts the number of regions we'll plug in the expression for the number of vertices which is n plus the n choose four intersection points, and you also plug in the slightly larger expression for the new number of edges n choose two plus two times n choose four plus n, and the expression has a lot of nice cancellation, for example you are adding an n but also subtracting an n and you're adding two copies of n choose four but subtracting another copy of n choose four and when all the dust settles the answer to the question is one plus n choose two plus n choose four.", "translatedText": "Mindezzel tehát megvan az eredeti kérdés megválaszolásához szükséges információ, elővesszük az Euler-képletünk azon változatát, amely a régiók számát számolja, bedugjuk a csúcsok számára vonatkozó kifejezést, ami n plusz az n választ 4 metszéspontot, és bedugjuk a kicsit nagyobb kifejezést az élek új számára, n choose 2 plusz 2-szer n choose 4 plusz n, és a kifejezésben sok szép törlés van, például hozzáadsz egy n-t, de kivonsz egy n-t is, és hozzáadod az n choose 4 két példányát, de kivonod az n choose 4 egy másik példányát, és amikor minden por leülepszik, a kérdésre adott válasz 1 plusz n choose 2 plusz n choose 4 lesz.", "model": "DeepL", "n_reviews": 0, @@ -935,4 +935,4 @@ "start": 937.28, "end": 949.86 } -] +] \ No newline at end of file diff --git a/2023/moser-reboot/spanish/sentence_translations.json b/2023/moser-reboot/spanish/sentence_translations.json index 7dff19ddd..4ab678bad 100644 --- a/2023/moser-reboot/spanish/sentence_translations.json +++ b/2023/moser-reboot/spanish/sentence_translations.json @@ -895,4 +895,4 @@ "start": 937.28, "end": 949.86 } -] +] \ No newline at end of file diff --git a/2023/prism/arabic/sentence_translations.json b/2023/prism/arabic/sentence_translations.json index d9abe71a4..a85fc7ac5 100644 --- a/2023/prism/arabic/sentence_translations.json +++ b/2023/prism/arabic/sentence_translations.json @@ -752,7 +752,7 @@ "end": 871.8 }, { - "input": "nd then the initial rotation of that vector corresponds with the phase of our wave. Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "nd then the initial rotation of that vector corresponds with the phase of our wave. And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "ثم فكر بالمثل في تلك الموجة الثانية باعتبارها تصف المركبة y لمتجه دوار آخر، حيث تتوافق السعة مرة أخرى مع طول ذلك المتجه، ويخبرنا طور الموجة بالزاوية الأولية لذلك المتجه. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/bengali/sentence_translations.json b/2023/prism/bengali/sentence_translations.json index a36a716c4..8f364619a 100644 --- a/2023/prism/bengali/sentence_translations.json +++ b/2023/prism/bengali/sentence_translations.json @@ -752,7 +752,7 @@ "end": 871.8 }, { - "input": "nd then the initial rotation of that vector corresponds with the phase of our wave. Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "nd then the initial rotation of that vector corresponds with the phase of our wave. And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "এবং তারপরে একইভাবে সেই দ্বিতীয় তরঙ্গটিকে অন্য একটি ঘূর্ণায়মান ভেক্টরের y- উপাদানের বর্ণনা হিসাবে ভাবুন, যেখানে আবার প্রশস্ততা সেই ভেক্টরের দৈর্ঘ্যের সাথে মিলে যায় এবং তরঙ্গের পর্যায়টি আমাদেরকে সেই ভেক্টরের প্রাথমিক কোণ বলে।", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/chinese/sentence_translations.json b/2023/prism/chinese/sentence_translations.json index 0cc1c783e..254c18f3b 100644 --- a/2023/prism/chinese/sentence_translations.json +++ b/2023/prism/chinese/sentence_translations.json @@ -658,7 +658,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "然后类似地，将第二个波视为描述另一个旋转矢量的 y 分量，其中振幅又与该矢量的长度相对应，而波的相位告诉我们该矢量的初始角度。", "n_reviews": 0, "start": 877.69, diff --git a/2023/prism/czech/sentence_translations.json b/2023/prism/czech/sentence_translations.json index 50eeb95e8..18d40f63b 100644 --- a/2023/prism/czech/sentence_translations.json +++ b/2023/prism/czech/sentence_translations.json @@ -768,7 +768,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "Podobně si představte druhou vlnu jako popis y složky jiného rotujícího vektoru, kde amplituda odpovídá délce tohoto vektoru a fáze vlny nám udává počáteční úhel tohoto vektoru.", "model": "DeepL", "n_reviews": 0, diff --git a/2023/prism/dutch/sentence_translations.json b/2023/prism/dutch/sentence_translations.json index bdb48be89..f4d9e29bb 100644 --- a/2023/prism/dutch/sentence_translations.json +++ b/2023/prism/dutch/sentence_translations.json @@ -768,7 +768,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "Op dezelfde manier kun je de tweede golf zien als een beschrijving van de y-component van een andere roterende vector, waarbij de amplitude overeenkomt met de lengte van die vector en de fase van de golf ons de beginhoek van die vector vertelt.", "model": "DeepL", "n_reviews": 0, diff --git a/2023/prism/english/captions.srt b/2023/prism/english/captions.srt index 01c230164..0b24d6c8c 100644 --- a/2023/prism/english/captions.srt +++ b/2023/prism/english/captions.srt @@ -975,16 +975,16 @@ The length of that vector corresponds with the amplitude of our wave, and then the initial rotation of that vector corresponds with the phase of our wave. 245 -00:14:37,690 --> 00:14:42,013 -Similarly, think of the second wave as describing the y-component of another +00:14:37,690 --> 00:14:41,797 +And then similarly think of that second wave as describing the y-component of 246 -00:14:42,013 --> 00:14:46,561 -rotating vector, where the amplitude corresponds with the length of that vector, +00:14:41,797 --> 00:14:45,956 +another rotating vector, where again the amplitude corresponds with the length 247 -00:14:46,561 --> 00:14:50,380 -and the phase of the wave tells us the initial angle of that vector. +00:14:45,956 --> 00:14:50,380 +of that vector, and the phase of the wave tells us the initial angle of that vector. 248 00:14:52,780 --> 00:14:55,274 diff --git a/2023/prism/english/sentence_timings.json b/2023/prism/english/sentence_timings.json index f2d8654a5..ac5733644 100644 --- a/2023/prism/english/sentence_timings.json +++ b/2023/prism/english/sentence_timings.json @@ -480,7 +480,7 @@ 877.0 ], [ - "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", 877.69, 890.38 ], diff --git a/2023/prism/english/transcript.txt b/2023/prism/english/transcript.txt index b3497b3a2..d80733226 100644 --- a/2023/prism/english/transcript.txt +++ b/2023/prism/english/transcript.txt @@ -94,7 +94,7 @@ It has some amplitude and some phase, and if I ask you to concretely compute bot But here's a really nice way to think about it. Imagine that first wave describes the y-component of some rotating vector. The length of that vector corresponds with the amplitude of our wave, and then the initial rotation of that vector corresponds with the phase of our wave. -Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector. +And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector. Now to think about the sum of the two waves, just think about adding those two vectors tip to tail. And because they both have the same frequency as both of them rotate, their sum rotates in lockstep with them. So if you want to think about the amplitude of our resulting wave, it comes down to the length of this vector sum, and similarly the phase corresponds to the angle of that vector sum. diff --git a/2023/prism/french/sentence_translations.json b/2023/prism/french/sentence_translations.json index 107fe4ef3..6f20a8033 100644 --- a/2023/prism/french/sentence_translations.json +++ b/2023/prism/french/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "Et puis, de la même manière, considérez cette deuxième vague comme décrivant la composante y d'un autre vecteur rotatif, où encore une fois l'amplitude correspond à la longueur de ce vecteur, et la phase de l'onde nous indique l'angle initial de ce vecteur.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/german/sentence_translations.json b/2023/prism/german/sentence_translations.json index a701d6259..a00e8afd9 100644 --- a/2023/prism/german/sentence_translations.json +++ b/2023/prism/german/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "Und dann stell dir diese zweite Welle auf ähnliche Weise so vor, dass sie die y-Komponente eines anderen rotierenden Vektors beschreibt, wobei wiederum die Amplitude der Länge dieses Vektors entspricht und die Phase der Welle uns den Anfangswinkel dieses Vektors angibt.", "model": "google_nmt", "n_reviews": 1, diff --git a/2023/prism/hebrew/sentence_translations.json b/2023/prism/hebrew/sentence_translations.json index 6c25dcc9a..b21849a92 100644 --- a/2023/prism/hebrew/sentence_translations.json +++ b/2023/prism/hebrew/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "ואז חשבו באופן דומה על הגל השני הזה כמתאר את רכיב ה-y של וקטור מסתובב אחר, שבו שוב המשרעת מתכתבת עם האורך של אותו וקטור, והפאזה של הגל אומרת לנו את הזווית ההתחלתית של אותו וקטור.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/hindi/sentence_translations.json b/2023/prism/hindi/sentence_translations.json index 30c0a7925..03d1d05f6 100644 --- a/2023/prism/hindi/sentence_translations.json +++ b/2023/prism/hindi/sentence_translations.json @@ -658,7 +658,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "और फिर उसी तरह उस दूसरी लहर के बारे में सोचें जो किसी अन्य घूर्णन वेक्टर के y-घटक का दर्शाता है, जहां फिर से एंप्लीट्यूड उस वेक्टर की लंबाई से मेल खाता है, और तरंग का फेज हमें उस वेक्टर के प्रारंभिक कोण की जानकारी देता है।", "n_reviews": 1, "start": 877.69, diff --git a/2023/prism/hungarian/sentence_translations.json b/2023/prism/hungarian/sentence_translations.json index 0811c13c0..4522f532d 100644 --- a/2023/prism/hungarian/sentence_translations.json +++ b/2023/prism/hungarian/sentence_translations.json @@ -768,7 +768,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "Hasonlóképpen, gondoljunk arra, hogy a második hullám egy másik forgó vektor y komponensét írja le, ahol az amplitúdó megfelel a vektor hosszának, és a hullám fázisa megadja a vektor kezdeti szögét.", "model": "DeepL", "n_reviews": 0, diff --git a/2023/prism/indonesian/sentence_translations.json b/2023/prism/indonesian/sentence_translations.json index 158692e08..a0211f915 100644 --- a/2023/prism/indonesian/sentence_translations.json +++ b/2023/prism/indonesian/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "Dan kemudian bayangkan gelombang kedua tersebut menggambarkan komponen y dari vektor berputar lainnya, yang sekali lagi amplitudonya sesuai dengan panjang vektor tersebut, dan fase gelombang memberi tahu kita sudut awal vektor tersebut.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/italian/sentence_translations.json b/2023/prism/italian/sentence_translations.json index f0155e2e9..26b09f6d7 100644 --- a/2023/prism/italian/sentence_translations.json +++ b/2023/prism/italian/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "E poi allo stesso modo pensa a quella seconda onda come se descrivesse la componente y di un altro vettore rotante, dove ancora una volta l'ampiezza corrisponde alla lunghezza di quel vettore, e la fase dell'onda ci dice l'angolo iniziale di quel vettore.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/japanese/sentence_translations.json b/2023/prism/japanese/sentence_translations.json index d709b2772..8d4791c93 100644 --- a/2023/prism/japanese/sentence_translations.json +++ b/2023/prism/japanese/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "そして、同様に、その 2 番目の波を別の回転ベクトルの y 成分を表すものとして考えます。 ここでも振幅はそのベクトルの 長さに対応し、波の位相はそのベクトルの初期角度を示します。", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/korean/sentence_translations.json b/2023/prism/korean/sentence_translations.json index ce010e417..1298190db 100644 --- a/2023/prism/korean/sentence_translations.json +++ b/2023/prism/korean/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "그리고 유사하게 두 번째 파동을 다른 회전 벡터의 y 성분을 설명하는 것으로 생각하면 됩니다. 여기서 진폭은 해당 벡터의 길이에 해당하고 파동의 위상은 해당 벡터의 초기 각도를 알려줍니다.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/marathi/sentence_translations.json b/2023/prism/marathi/sentence_translations.json index 8834d4ad1..b40bbb740 100644 --- a/2023/prism/marathi/sentence_translations.json +++ b/2023/prism/marathi/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "आणि मग त्याचप्रमाणे दुसर्‍या फिरणार्‍या वेक्टरच्या y-घटकाचे वर्णन करणार्‍या दुसर्‍या तरंगाचा विचार करा, जिथे पुन्हा मोठेपणा त्या व्हेक्टरच्या लांबीशी सुसंगत आहे आणि तरंगाचा टप्पा आम्हाला त्या वेक्टरचा प्रारंभिक कोन सांगतो.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/persian/sentence_translations.json b/2023/prism/persian/sentence_translations.json index 01550c9ad..f6bf9d088 100644 --- a/2023/prism/persian/sentence_translations.json +++ b/2023/prism/persian/sentence_translations.json @@ -752,7 +752,7 @@ "end": 871.8 }, { - "input": "nd then the initial rotation of that vector corresponds with the phase of our wave. Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "nd then the initial rotation of that vector corresponds with the phase of our wave. And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "و سپس به طور مشابه به آن موج دوم فکر کنید که جزء y یک بردار چرخان دیگر را توصیف می کند، جایی که دامنه دوباره با طول آن بردار مطابقت دارد، و فاز موج زاویه اولیه آن بردار را به ما می گوید. ", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/polish/sentence_translations.json b/2023/prism/polish/sentence_translations.json index da91474cb..fd8dbc41e 100644 --- a/2023/prism/polish/sentence_translations.json +++ b/2023/prism/polish/sentence_translations.json @@ -672,7 +672,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "Podobnie, pomyśl o drugiej fali jako o opisie składowej y innego wirującego wektora, gdzie amplituda odpowiada długości tego wektora, a faza fali mówi nam o początkowym kącie tego wektora.", "n_reviews": 0, "start": 877.69, diff --git a/2023/prism/portuguese/sentence_translations.json b/2023/prism/portuguese/sentence_translations.json index 0442380cc..d8c0968de 100644 --- a/2023/prism/portuguese/sentence_translations.json +++ b/2023/prism/portuguese/sentence_translations.json @@ -862,7 +862,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "Da mesma forma, pense na segunda onda como descrevendo a componente y de outro vetor rotativo, onde a amplitude corresponde ao comprimento desse vetor, e a fase da onda nos diz o ângulo inicial desse vetor.", "model": "google_nmt", "from_community_srt": "E então pense da mesma forma que essa segunda onda descreve a componente y de outro vetor, onde novamente a amplitude corresponde ao comprimento desse vetor, e a fase da onda nos diz o ângulo inicial desse vetor.", diff --git a/2023/prism/russian/sentence_translations.json b/2023/prism/russian/sentence_translations.json index 54f8cfe25..4b8850ea5 100644 --- a/2023/prism/russian/sentence_translations.json +++ b/2023/prism/russian/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "А затем аналогичным образом подумайте об этой второй волне как описывающей y-компоненту другого вращающегося вектора, где амплитуда снова соответствует длине этого вектора, а фаза волны сообщает нам начальный угол этого вектора.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/spanish/sentence_translations.json b/2023/prism/spanish/sentence_translations.json index 37d67b278..d4b5d12b7 100644 --- a/2023/prism/spanish/sentence_translations.json +++ b/2023/prism/spanish/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "Y luego, de manera similar, piense en esa segunda onda como si describiera el componente y de otro vector giratorio, donde nuevamente la amplitud se corresponde con la longitud de ese vector, y la fase de la onda nos dice el ángulo inicial de ese vector.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/tamil/sentence_translations.json b/2023/prism/tamil/sentence_translations.json index d6efd7a9e..9161b8ba1 100644 --- a/2023/prism/tamil/sentence_translations.json +++ b/2023/prism/tamil/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "பின்னர் இதேபோல் அந்த இரண்டாவது அலையை மற்றொரு சுழலும் திசையனின் y-கூறு விவரிப்பதாக நினைத்துப் பாருங்கள், அங்கு மீண்டும் அலைவீச்சு அந்த திசையனின் நீளத்துடன் ஒத்துப்போகிறது, மேலும் அலையின் கட்டம் அந்த திசையனின் ஆரம்ப கோணத்தை நமக்கு சொல்கிறது.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/telugu/sentence_translations.json b/2023/prism/telugu/sentence_translations.json index cf985a9bc..2880e909f 100644 --- a/2023/prism/telugu/sentence_translations.json +++ b/2023/prism/telugu/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "ఆపై అదే విధంగా మరొక భ్రమణ వెక్టర్ యొక్క y-భాగాన్ని వివరిస్తున్నట్లు ఆ రెండవ వేవ్ గురించి ఆలోచించండి, ఇక్కడ మళ్లీ వ్యాప్తి ఆ వెక్టర్ యొక్క పొడవుకు అనుగుణంగా ఉంటుంది మరియు వేవ్ యొక్క దశ ఆ వెక్టర్ యొక్క ప్రారంభ కోణాన్ని తెలియజేస్తుంది.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/thai/sentence_translations.json b/2023/prism/thai/sentence_translations.json index 836e1de34..125c0d314 100644 --- a/2023/prism/thai/sentence_translations.json +++ b/2023/prism/thai/sentence_translations.json @@ -752,7 +752,7 @@ "end": 871.8 }, { - "input": "nd then the initial rotation of that vector corresponds with the phase of our wave. Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "nd then the initial rotation of that vector corresponds with the phase of our wave. And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/turkish/sentence_translations.json b/2023/prism/turkish/sentence_translations.json index 3c1a600c3..b4a7aac36 100644 --- a/2023/prism/turkish/sentence_translations.json +++ b/2023/prism/turkish/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "Ve sonra benzer şekilde ikinci dalganın başka bir dönen vektörün y bileşenini tanımladığını düşünün; burada genlik yine o vektörün uzunluğuna karşılık gelir ve dalganın fazı bize o vektörün başlangıç açısını söyler.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/ukrainian/sentence_translations.json b/2023/prism/ukrainian/sentence_translations.json index 0b72e363e..da9bbbfdf 100644 --- a/2023/prism/ukrainian/sentence_translations.json +++ b/2023/prism/ukrainian/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "І тоді так само подумайте про цю другу хвилю як про опис y-компоненти іншого обертового вектора, де знову ж таки амплітуда відповідає довжині цього вектора, а фаза хвилі повідомляє нам початковий кут цього вектора.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/urdu/sentence_translations.json b/2023/prism/urdu/sentence_translations.json index 096042a3f..a213b923e 100644 --- a/2023/prism/urdu/sentence_translations.json +++ b/2023/prism/urdu/sentence_translations.json @@ -752,7 +752,7 @@ "end": 871.8 }, { - "input": "nd then the initial rotation of that vector corresponds with the phase of our wave. Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "nd then the initial rotation of that vector corresponds with the phase of our wave. And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "اور پھر اسی طرح اس دوسری لہر کے بارے میں سوچیں جو کسی دوسرے گھومنے والے ویکٹر کے y-جزو کو بیان کرتی ہے، جہاں ایک بار پھر طول و عرض اس ویکٹر کی لمبائی سے مطابقت رکھتا ہے، اور لہر کا مرحلہ ہمیں اس ویکٹر کا ابتدائی زاویہ بتاتا ہے۔", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/prism/vietnamese/sentence_translations.json b/2023/prism/vietnamese/sentence_translations.json index 4034d8748..9fcc10d3d 100644 --- a/2023/prism/vietnamese/sentence_translations.json +++ b/2023/prism/vietnamese/sentence_translations.json @@ -752,7 +752,7 @@ "end": 877.0 }, { - "input": "Similarly, think of the second wave as describing the y-component of another rotating vector, where the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", + "input": "And then similarly think of that second wave as describing the y-component of another rotating vector, where again the amplitude corresponds with the length of that vector, and the phase of the wave tells us the initial angle of that vector.", "translatedText": "Và sau đó hãy nghĩ tương tự về sóng thứ hai đó khi mô tả thành phần y của một vectơ quay khác, trong đó biên độ lại tương ứng với độ dài của vectơ đó và pha của sóng cho chúng ta biết góc ban đầu của vectơ đó.", "model": "google_nmt", "n_reviews": 0, diff --git a/2023/some3-results/english/captions.srt b/2023/some3-results/english/captions.srt index ce3b5be69..ff75a8a8a 100644 --- a/2023/some3-results/english/captions.srt +++ b/2023/some3-results/english/captions.srt @@ -1403,126 +1403,6 @@ and in just 20 minutes, he hits on a lot of the really big important ideas around what exactly infinity means and how those ideas all relate to each other. 352 -00:20:03,660 --> 00:20:07,480 +00:20:03,660 --> 00:21:50,240 So that's a wrap for the ones that I wanted to mention right now and highlight to you. -353 -00:20:07,900 --> 00:20:10,410 -There are many more, of course, that were submitted and quite a - -354 -00:20:10,410 --> 00:20:12,960 -few that are worth looking at even if I didn't mention them here. - -355 -00:20:13,580 --> 00:20:16,947 -Again, I do think what makes the judging here at the end so hard is - -356 -00:20:16,947 --> 00:20:20,216 -that what makes something good is usually the sense in which it's - -357 -00:20:20,216 --> 00:20:23,435 -good for a very specific audience and trying to come up with any - -358 -00:20:23,435 --> 00:20:27,100 -notion of how to rank all of these against each other in an objective way. - -359 -00:20:27,400 --> 00:20:30,820 -I mean with different target audiences, it's just an apples to oranges comparison. - -360 -00:20:31,520 --> 00:20:34,803 -With all of that said, the ones selected as winners, - -361 -00:20:34,803 --> 00:20:40,131 -the ones collecting the golden pie creature prizes are the mathematics of string art, - -362 -00:20:40,131 --> 00:20:44,777 -minimal surfaces and the calculus of variations, rethinking the real line, - -363 -00:20:44,777 --> 00:20:49,920 -pixel art anti-aliasing, and from the non-video category how computers use numbers. - -364 -00:20:51,240 --> 00:20:55,641 -So, evidently, shortly after this point in the recording my camera battery died, - -365 -00:20:55,641 --> 00:20:57,760 -which I guess is almost perfect timing. - -366 -00:20:58,180 --> 00:21:01,331 -I do want to close out though by saying a really big thanks to James - -367 -00:21:01,331 --> 00:21:04,345 -Schloss for all the organization and helping to make this happen, - -368 -00:21:04,345 --> 00:21:08,456 -to Fred Crozettier for the web development and putting together a new peer review system, - -369 -00:21:08,456 --> 00:21:11,060 -which made everything this year just much, much smoother. - -370 -00:21:11,480 --> 00:21:14,920 -And many, many thanks to the kind folks at Jane Street for providing - -371 -00:21:14,920 --> 00:21:18,908 -the funding both for prizes associated with the winners and honorable mentions, - -372 -00:21:18,908 --> 00:21:21,900 -and also with other costs associated with running the event. - -373 -00:21:22,360 --> 00:21:25,784 -Also, at the end of August a ton of you participated in the peer review process, - -374 -00:21:25,784 --> 00:21:29,040 -and I really just want to say thank you for donating your time to doing that. - -375 -00:21:29,240 --> 00:21:31,250 -And then of course, thank you to everybody who - -376 -00:21:31,250 --> 00:21:33,260 -made some kind of math explainer for the event. - -377 -00:21:33,600 --> 00:21:37,140 -There are many really good ones that I didn't have the time to talk about in this video. - -378 -00:21:37,480 --> 00:21:40,908 -In the description you'll find a link to a playlist for all the video entries, - -379 -00:21:40,908 --> 00:21:44,120 -and then on the website we'll include a list of all the non-video entries. - -380 -00:21:44,500 --> 00:21:45,480 -So do check them out! - -381 -00:21:45,480 --> 00:21:48,769 -I can almost guarantee that if you scroll through and see what you're interested in, - -382 -00:21:48,769 --> 00:21:50,240 -you'll find some hidden gems in there. - diff --git a/2023/some3-results/english/sentence_timings.json b/2023/some3-results/english/sentence_timings.json index cc4f1af53..10dc0983a 100644 --- a/2023/some3-results/english/sentence_timings.json +++ b/2023/some3-results/english/sentence_timings.json @@ -792,71 +792,6 @@ [ "So that's a wrap for the ones that I wanted to mention right now and highlight to you.", 1203.66, - 1207.48 - ], - [ - "There are many more, of course, that were submitted and quite a few that are worth looking at even if I didn't mention them here.", - 1207.9, - 1212.96 - ], - [ - "Again, I do think what makes the judging here at the end so hard is that what makes something good is usually the sense in which it's good for a very specific audience and trying to come up with any notion of how to rank all of these against each other in an objective way.", - 1213.58, - 1227.1 - ], - [ - "I mean with different target audiences, it's just an apples to oranges comparison.", - 1227.4, - 1230.82 - ], - [ - "With all of that said, the ones selected as winners, the ones collecting the golden pie creature prizes are the mathematics of string art, minimal surfaces and the calculus of variations, rethinking the real line, pixel art anti-aliasing, and from the non-video category how computers use numbers.", - 1231.52, - 1249.92 - ], - [ - "So, evidently, shortly after this point in the recording my camera battery died, which I guess is almost perfect timing.", - 1251.24, - 1257.76 - ], - [ - "I do want to close out though by saying a really big thanks to James Schloss for all the organization and helping to make this happen, to Fred Crozettier for the web development and putting together a new peer review system, which made everything this year just much, much smoother.", - 1258.18, - 1271.06 - ], - [ - "And many, many thanks to the kind folks at Jane Street for providing the funding both for prizes associated with the winners and honorable mentions, and also with other costs associated with running the event.", - 1271.48, - 1281.9 - ], - [ - "Also, at the end of August a ton of you participated in the peer review process, and I really just want to say thank you for donating your time to doing that.", - 1282.36, - 1289.04 - ], - [ - "And then of course, thank you to everybody who made some kind of math explainer for the event.", - 1289.24, - 1293.26 - ], - [ - "There are many really good ones that I didn't have the time to talk about in this video.", - 1293.6, - 1297.14 - ], - [ - "In the description you'll find a link to a playlist for all the video entries, and then on the website we'll include a list of all the non-video entries.", - 1297.48, - 1304.12 - ], - [ - "So do check them out!", - 1304.5, - 1305.48 - ], - [ - "I can almost guarantee that if you scroll through and see what you're interested in, you'll find some hidden gems in there.", - 1305.48, 1310.24 ] ] \ No newline at end of file diff --git a/2024/shorts/cube-shadow-puzzle/french/sentence_translations.json b/2024/shorts/cube-shadow-puzzle/french/sentence_translations.json index 8f082defa..be5f948d1 100644 --- a/2024/shorts/cube-shadow-puzzle/french/sentence_translations.json +++ b/2024/shorts/cube-shadow-puzzle/french/sentence_translations.json @@ -56,7 +56,7 @@ "end": 52.4 }, { - "input": "The shadow puzzle just turns out to be a very perfect for that story.", + "input": "The shadow puzzle just turns out to be a very perfect setting for that story.", "translatedText": "L'énigme des ombres s’avère juste être un décor absolument parfait pour cette histoire.", "n_reviews": 1, "start": 52.76,